Elliptic Mixed, Transmission and Singular Crack Problems

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EMS Tracts in Mathematics 4

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EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Department of Mathematics, Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate Diffusions 2 Karl H. Hofmann and Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups 3 Ralf Meyer, Local and Analytic Cyclic Homology

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Gohar Harutyunyan B.-Wolfgang Schulze

Elliptic Mixed, Transmission and Singular Crack Problems

M

M

S E M E S

S E M E S

European Mathematical Society

Schulze_titelei

25.10.2007

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Authors: Gohar Harutyunyan Institut für Mathematik Carl von Ossietzky Universität Oldenburg 26111 Oldenburg, Germany E-mail: [email protected]

B.-Wolfgang Schulze Institut für Mathematik Universität Potsdam Am Neuen Palais 10 14469 Potsdam, Germany E-mail: [email protected]

2000 Mathematics Subject Classification: 35-02; 35J25, 35S15 Key words: Ellipticity on manifolds with singular boundaries, edge and corner pseudo-differential calculus, problems with mixed and transmission conditions, problems of Zaremba type, crack problems, meromorphic symbolic structures, relative index.

ISBN 978-3-03719-040-1 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2008 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info @ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

Preface

Elliptic mixed, transmission, or crack problems belong to the analysis on manifolds with singularities, more precisely, to the calculus of boundary value problems, where the data or the coefficients have singularities. A classical example is the Zaremba problem for the Laplace equation with mixed Dirichlet and Neumann conditions. Mixed problems in general are characterised by boundary conditions that have a jump along an interface on the boundary. At the same time the configuration may have singularities, e.g., conical points or edges, and it is an interesting task of the mathematical analysis to establish the properties of solvability. Problems of that kind are natural in many models of physics and the applied sciences. Numerous authors have contributed to this field and developed approaches of different generality. In recent years it became more and more clear that boundary value problems with singularities can be understood from the viewpoint of the analysis on manifolds with singularities and that a transparent description of solvability can be achieved by applying suitable extensions of the pseudo-differential calculus, combined with operator algebra aspects, elements of the index theory, and other areas of pure mathematics. Singularities are employed as a source of symbolic information in the form of higher generations of (operator-valued) amplitude functions, which are involved in the construction of parametrices and the characterisation of regularity and asymptotics of solutions. The present exposition is aimed at developing these ideas in a way that mixed, transmission and crack problems appear as a natural generalisation of standard boundary value problems, here as edge and corner problems. Our approach is completely general; examples and applications in mixed and crack problems will mainly concern second order elliptic differential operators with specific boundary conditions, e.g., of Zaremba type. First we treat the case of smooth interfaces in the framework of the edge calculus and then admit interfaces with conical singularities; this requires the tools of the corner calculus of boundary value problems. The task to express parametrices on the level of symbols generates these calculi in large generality, because the amplitude functions are families of boundary value problems combined with different degeneracies in the parameters. In the description of the structures it may be even more convenient to refer to the general properties. In Chapter 1 we give an introduction into mixed and transmission problems for differential operators and their symbolic structure. Chapter 2 contains general tools of the pseudo-differential calculus. This will partly be done in a ‘non-orthodox’ form, although the starting point are operators with standard interior and boundary symbols. However, the applications require a variety of new structures, such as symbols with twisted homogeneity, exit properties at infinity, meromorphy in the covariables, etc.

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In Chapter 3 we outline pseudo-differential boundary value problems with the transmission property at the boundary, including the calculus on a manifold with conical exit to infinity. In Chapter 4 we treat mixed elliptic problems in standard Sobolev spaces and with extra interface conditions. Here we employ specific reductions of orders and a kind of plus/minus calculus of boundary value problems. Chapter 5 is devoted to mixed problems in weighted edge spaces, also with extra conditions of trace and potential type at the interface. In Chapters 6 and 7 we develop the calculus of boundary value problems on manifolds with conical singularities and smooth edges, i.e., we establish the corresponding cone and edge algebras. In Chapter 8 we consider mixed and crack problems with conical singularities at the interface. We give a homotopy argument that allows us to reduce crack problems to mixed problems. In Chapter 9 we study operators on infinite cylinders with meromorphic amplitude functions for different cases of cross sections, especially, with boundary, and also with conical singularities. Moreover, we formulate relative index results. Chapter 10 contains a general discussion on motivations of the approach and on expected future developments, with open problems and new challenges. Compared with other monographs in the field of the (pseudo-differential) analysis of singular problems, e.g., [90], the present book gives for the first time a general approach to mixed elliptic problems with singularities of the interface, together with explicit tools for computing the number of extra interface conditions. The new results are mainly based on the authors’joint works [71]–[78] and [34], the latter in cooperation with Dines. This book is addressed to mathematicians and physicists interested in models with singularities, associated boundary value problems, and their solvability strategies based on pseudo-differential operators. The material is also useful for students in higher semesters and young researchers as well as for experienced specialists who are active in the fields of the analysis on manifolds with geometric singularities, applications of index theory and spectral theory, operator algebras with symbolic structures, quantisation, and asymptotic analysis. Acknowledgement. The authors thank Nicoleta Dines (University of Göttingen) and Gerd Grubb (University of Copenhagen) for valuable remarks on the manuscript. Moreover we thank Irene Zimmermann (Freiburg) for her careful editorial work, which greatly improved the exposition. Oldenburg and Potsdam, October 2007

Gohar Harutyunyan B.-Wolfgang Schulze

Contents Preface

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Introduction

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Boundary value problems with mixed and interface data 1.1 Elliptic boundary value problems . . . . . . . . . . . . . . . . 1.1.1 Differential operators with classical boundary conditions 1.1.2 The Poisson kernels in the half-space . . . . . . . . . . 1.1.3 Reduction to the boundary . . . . . . . . . . . . . . . . 1.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mixed and transmission problems . . . . . . . . . . . . . . . . 1.2.1 Mixed problems in weighted edge spaces . . . . . . . . 1.2.2 Additional conditions at the interface . . . . . . . . . . 1.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Reduction of orders and reduction to the boundary . . . 1.2.5 Mixed problems in standard Sobolev spaces . . . . . . . 1.3 Problems with several types of interfaces . . . . . . . . . . . . 1.3.1 Transmission problems with smooth interfaces . . . . . 1.3.2 Transmission problems with singular interfaces . . . . .

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Symbolic structures and associated operators 2.1 Scalar pseudo-differential calculus . . . . . . . . . . . 2.1.1 Spaces of symbols and basic operations . . . . . 2.1.2 Pseudo-differential operators and Sobolev spaces 2.1.3 Operators on manifolds . . . . . . . . . . . . . . 2.2 Calculus with operator-valued symbols . . . . . . . . . 2.2.1 Symbols and operators with twisted homogeneity 2.2.2 Abstract edge spaces . . . . . . . . . . . . . . . 2.2.3 Elements of the calculus . . . . . . . . . . . . . 2.3 Operators on manifolds with conical exit to infinity . . 2.3.1 Symbols with exit behaviour . . . . . . . . . . . 2.3.2 Operators globally in the Euclidean space . . . . 2.3.3 Operators on manifolds . . . . . . . . . . . . . . 2.3.4 Ellipticity in the scalar case . . . . . . . . . . . 2.3.5 Calculus with operator-valued symbols . . . . . 2.4 Mellin operators . . . . . . . . . . . . . . . . . . . . . 2.4.1 The Mellin transform . . . . . . . . . . . . . . . 2.4.2 Weighted Sobolev spaces . . . . . . . . . . . . . 2.4.3 Degenerate differential operators . . . . . . . . . 2.4.4 Mellin operators and quantisation . . . . . . . .

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2.4.5 A connection between edge-degenerate operators and exit calculus . . . . . . . . . . . . . . . . . . . . . . . . . 146 2.4.6 Green operators for conical singularities . . . . . . . . . . . . . 155 3

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Boundary value problems with the transmission property 3.1 Interior and boundary symbols . . . . . . . . . . . . . . . . . . . . 3.1.1 Symbols with the transmission property . . . . . . . . . . . . 3.1.2 Operators with the transmission property . . . . . . . . . . . 3.1.3 Green operators on the half-axis . . . . . . . . . . . . . . . . 3.1.4 Boundary value problems on the half-axis . . . . . . . . . . . 3.1.5 Operators on an interval . . . . . . . . . . . . . . . . . . . . 3.2 The algebra of boundary value problems . . . . . . . . . . . . . . . 3.2.1 Global smoothing operators . . . . . . . . . . . . . . . . . . 3.2.2 Green operators . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Boundary value problems . . . . . . . . . . . . . . . . . . . 3.3 Ellipticity and parametrices . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Elliptic boundary value problems . . . . . . . . . . . . . . . 3.3.2 Parametrices and inverses . . . . . . . . . . . . . . . . . . . 3.3.3 Parameter-dependent ellipticity . . . . . . . . . . . . . . . . . 3.3.4 Fredholm families and block matrix isomorphisms . . . . . . 3.4 The calculus on manifolds with conical exit to infinity . . . . . . . . 3.4.1 Motivation in terms of principal edge symbols . . . . . . . . . 3.4.2 Global operators in the half-space . . . . . . . . . . . . . . . 3.4.3 Operators on a manifold with conical exit . . . . . . . . . . . 3.4.4 A relation between edge-degenerate families and exit calculus

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Mixed problems in standard Sobolev spaces 4.1 Reductions of orders on a manifold with boundary . . . . . 4.1.1 Order reducing symbols in the half-space . . . . . . 4.1.2 Actions in Sobolev spaces . . . . . . . . . . . . . . 4.1.3 A relationship with classical Volterra symbols . . . . 4.1.4 Global reduction of orders . . . . . . . . . . . . . . 4.1.5 General operators with plus/minus-symbols . . . . . 4.2 Mixed elliptic problems . . . . . . . . . . . . . . . . . . . 4.2.1 Mixed problems for differential operators . . . . . . 4.2.2 Ellipticity with additional conditions at the interface 4.2.3 The Zaremba problem . . . . . . . . . . . . . . . . 4.2.4 Jumping oblique derivatives and other examples . . .

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Mixed problems in weighted edge spaces 5.1 Mixed problems in edge spaces . . . . . . . . . 5.1.1 Basic observations . . . . . . . . . . . . 5.1.2 Green symbols . . . . . . . . . . . . . . 5.1.3 The Zaremba problem as an edge problem

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5.2 Relations between edge and standard Sobolev spaces . . . . . . 5.2.1 Spaces on the boundary . . . . . . . . . . . . . . . . . . 5.2.2 Edge spaces in the stretched domain . . . . . . . . . . . 5.2.3 A reformulation of mixed problems from standard Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . 5.3 Elliptic interface conditions . . . . . . . . . . . . . . . . . . . 5.3.1 Mixed problems in spaces of arbitrary weights . . . . . 5.3.2 Construction of elliptic interface conditions . . . . . . . 5.3.3 Parametrices and regularity of solutions for the Zaremba problem . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Jumping oblique derivatives and other examples . . . . . 5.4 Edge calculus, specified to mixed problems . . . . . . . . . . . 5.4.1 Edge amplitude functions . . . . . . . . . . . . . . . . . 5.4.2 Edge-boundary value problems . . . . . . . . . . . . . . 5.4.3 Ellipticity and parametrices . . . . . . . . . . . . . . . 5.4.4 Asymptotics of solutions . . . . . . . . . . . . . . . . . 6

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Operators on manifolds with conical singularities and boundary 6.1 Fuchs type operators and Mellin quantisation . . . . . . . . . . . . 6.1.1 Mellin quantisation . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Kernel cut-off . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Meromorphic Fredholm families and ellipticity of conormal symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Green operators . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Mellin operators with smoothing symbols . . . . . . . . . . 6.1.6 Operators with holomorphic Mellin symbols . . . . . . . . 6.2 The cone algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Operators on a compact manifold with conical singularities and boundary . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Operators on an infinite cone with boundary . . . . . . . . . 6.3 Boundary value problems in plane domains . . . . . . . . . . . . . 6.3.1 The Dirichlet problem in a strip . . . . . . . . . . . . . . . 6.3.2 The Neumann and the Zaremba problem in a strip . . . . . . 6.3.3 The Dirichlet problem in an angle . . . . . . . . . . . . . . 6.3.4 The Neumann and the Zaremba problem in an angle . . . . 6.4 Special operators of the cone calculus . . . . . . . . . . . . . . . . 6.4.1 Reduction of orders . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Operators on a cone with arbitrary weights at infinity . . . . 6.4.3 Cone operators with parameters . . . . . . . . . . . . . . . 6.4.4 Elliptic regularity for some Schrödinger equation . . . . . .

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Operators on manifolds with edges and boundary 7.1 Differential operators on manifolds with edges . . . . . . . . . . . . . 7.1.1 Edge-degenerate differential operators . . . . . . . . . . . . . . 7.1.2 Weighted edge spaces . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Edge-boundary value problems as operators in weighted spaces 7.1.4 Operators in alternative weighted edge spaces . . . . . . . . . . 7.2 The edge algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Edge-degenerate symbols and operator conventions . . . . . . . 7.2.2 Global smoothing operators . . . . . . . . . . . . . . . . . . . 7.2.3 Green and smoothing Mellin symbols . . . . . . . . . . . . . . 7.2.4 Edge amplitude functions . . . . . . . . . . . . . . . . . . . . . 7.2.5 The edge algebra . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Ellipticity and reductions of orders . . . . . . . . . . . . . . . . 7.3 Mellin-edge representations of elliptic operators . . . . . . . . . . . . 7.3.1 Decomposition of classical Sobolev spaces . . . . . . . . . . . 7.3.2 Edge decompositions of differential operators . . . . . . . . . . 7.3.3 Global constructions . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Edge representation of boundary value problems . . . . . . . . 7.3.5 Relative index results . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Interface conditions for small weights . . . . . . . . . . . . . . 7.4 The Laplacian in a wedge, and other elliptic operators of the edge calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 The Dirichlet problem in a wedge . . . . . . . . . . . . . . . . 7.4.2 The Neumann and the Zaremba problem in a wedge . . . . . . 7.4.3 Other examples of elliptic edge operators . . . . . . . . . . . .

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Corner operators and problems with singular interfaces 8.1 Singular mixed problems and corner manifolds . . . . . . 8.1.1 The singular Zaremba problem . . . . . . . . . . . 8.1.2 Operators in edge representation . . . . . . . . . . 8.1.3 Principal symbols and edge conditions . . . . . . . 8.1.4 Corner manifolds . . . . . . . . . . . . . . . . . . 8.2 Corner operators in spaces with double weights . . . . . . 8.2.1 Transformation to a corner boundary value problem 8.2.2 Corner spaces with double weights . . . . . . . . . 8.2.3 Continuity in corner spaces . . . . . . . . . . . . . 8.2.4 Holomorphic corner symbols . . . . . . . . . . . . 8.2.5 Corner boundary value problems . . . . . . . . . . 8.2.6 Ellipticity near the corner . . . . . . . . . . . . . . 8.3 Corner edge operators . . . . . . . . . . . . . . . . . . . 8.3.1 Global corner boundary value problems . . . . . . 8.3.2 Ellipticity and parametrices . . . . . . . . . . . . 8.3.3 The singular Zaremba problem . . . . . . . . . . . 8.3.4 Remarks . . . . . . . . . . . . . . . . . . . . . .

475 475 475 477 479 481 484 484 486 491 495 497 501 503 503 508 509 512

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8.4 Cracks with singularities at the boundary . . . . . . . . . . . . 8.4.1 Crack problems as edge-corner boundary value problems 8.4.2 Operators near the smooth part of the crack boundary . . 8.4.3 Parameter-dependent crack operators on a sphere . . . . 8.4.4 The local corner-crack calculus . . . . . . . . . . . . . . 8.4.5 Singular crack problems . . . . . . . . . . . . . . . . . 8.4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.7 Further comments . . . . . . . . . . . . . . . . . . . . 8.5 Mixed problems with singular interfaces . . . . . . . . . . . . 8.5.1 Mixed problems in an infinite cylinder . . . . . . . . . . 8.5.2 Reduction to the boundary . . . . . . . . . . . . . . . . 8.5.3 Ellipticity with interface conditions . . . . . . . . . . . 9

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514 514 519 522 526 530 532 537 538 538 541 545

Operators in infinite cylinders and the relative index 9.1 Calculus with operator-valued meromorphic families . . . . . . . . . . 9.1.1 Characteristic values and factorisation of meromorphic families 9.1.2 The inhomogeneous equation . . . . . . . . . . . . . . . . . . 9.1.3 An index formula . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Boundary value problems in infinite cylinders . . . . . . . . . . . . . 9.2.1 Operators in cylindrical Sobolev spaces . . . . . . . . . . . . . 9.2.2 Characteristic values and factorisation of meromorphic families 9.2.3 The relative index . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The relative index for corner singularities . . . . . . . . . . . . . . . . 9.3.1 Parameter-dependent cone calculus . . . . . . . . . . . . . . . 9.3.2 Meromorphic families . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Characteristic values and factorisation . . . . . . . . . . . . . . 9.3.5 Operators on the infinite cylinder . . . . . . . . . . . . . . . . . 9.3.6 The relative index . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Cutting and pasting of elliptic operators . . . . . . . . . . . . . . . . . 9.4.1 The locality of the index in the smooth case . . . . . . . . . . . 9.4.2 Operators in bottleneck spaces . . . . . . . . . . . . . . . . . . 9.4.3 A general locality principle for the index . . . . . . . . . . . . .

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10 Intuitive ideas of the calculus on singular manifolds 10.1 Simple questions, unexpected answers . . . . . . . 10.1.1 What is ellipticity? . . . . . . . . . . . . . . 10.1.2 Meromorphic symbolic structures . . . . . . 10.1.3 Naive and edge definitions of Sobolev spaces 10.2 Are regular boundaries harmless? . . . . . . . . . . 10.2.1 What is a boundary value problem? . . . . . 10.2.2 Quantisation . . . . . . . . . . . . . . . . . 10.2.3 The conormal cage . . . . . . . . . . . . . . 10.3 How interesting are conical singularities? . . . . . .

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10.3.1 The iterative construction of higher singularities . . . . . . . . . 10.3.2 Operators with sleeping parameters . . . . . . . . . . . . . . . 10.3.3 Smoothing operators may contribute to the index . . . . . . . . 10.3.4 Are cylinders the genuine corners? . . . . . . . . . . . . . . . . 10.4 Is ‘degenerate’ bad? . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Operators on stretched spaces . . . . . . . . . . . . . . . . . . 10.4.2 What is ‘smoothness’ on a singular manifold? . . . . . . . . . . 10.4.3 Schwartz kernels and Green operators . . . . . . . . . . . . . . 10.4.4 Pseudo-differential aspects, solvability of equations . . . . . . . 10.4.5 Discrete, branching, and continuous asymptotics . . . . . . . . 10.5 Higher generations of calculi . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Higher generations of weighted corner spaces . . . . . . . . . . 10.5.2 Additional edge conditions in higher corner algebras . . . . . . 10.5.3 A hierarchy of topological obstructions . . . . . . . . . . . . . 10.5.4 The building of singular algebras . . . . . . . . . . . . . . . . . 10.6 Historical background and future program . . . . . . . . . . . . . . . 10.6.1 Achievements of the past development . . . . . . . . . . . . . . 10.6.2 Conification and edgification . . . . . . . . . . . . . . . . . . . 10.6.3 Similarities and differences between ellipticity and parabolicity 10.6.4 Open problems and new challenges . . . . . . . . . . . . . . . 10.6.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . .

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Bibliography

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List of Symbols

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Index

752

Introduction Boundary value problems on configurations with (geometric) singularities are an important and beautiful area of mathematical research, motivated by models of the applied sciences and engineering, for instance, of mechanics, elasticity, crack theory, and mathematical physics. Operators on manifolds with singularities are also of interest in pure mathematics, such as geometry, topology, index theory, spectral theory, and operator algebras. It is known that the parametrices of elliptic boundary value problems for differential operators in smooth domains are pseudo-differential operators (more precisely, pseudo-differential boundary value problems with the transmission property). The classical pseudo-differential techniques contain many elements of the general strategy to studying boundary value problems on singular configurations (stratified spaces), in particular, with conical, edge, corner or higher polyhedral singularities. A category of interesting models are the so-called mixed problems in which the boundary conditions to an elliptic equation have a jump along a submanifold of the boundary of codimension 1. The reduction of such problems to the boundary gives rise to pseudo-differential equations without the transmission property at the interface on the boundary. Examples are operators with symbols associated with fractional powers of the symbol of (minus) the Laplacian. For instance, the square root belongs to the Zaremba problem (i.e., with mixed Dirichlet and Neumann conditions). Boundary value problems (with or without the transmission property) have much in common with edge problems, with the boundary as the edge and the inner normal as the model cone. Also mixed or crack problems can be interpreted as edge problems; in this case the interface with the jump of the mixed conditions or the boundary of the crack play the role of an edge, and the normal planes contain the corresponding model cones. The present book is aimed at studying problems of that type, based on the general calculus of operators on a manifold with edges. First this concerns the case of smooth edges (i.e., interfaces or crack boundaries). We also treat mixed and crack problems when the interfaces or crack boundaries have conical singularities, using a corresponding corner calculus of boundary value problems. In Chapter 1 we sketch a number of mixed and transmission problems and illustrate their symbolic structure. The original problems essentially refer to differential operators with differential conditions. However, reductions to the boundary give rise to pseudo-differential operators and associated boundary or transmission problems. Let us briefly recall a few ideas from the pseudo-differential calculus. If X is a C 1 manifold and A a differential operator of order  on X , we have the homogeneous principal symbol  .A/ of A of order  which is a function in C 1 .T  X n 0/, with T  X being the cotangent bundle of X and 0 the zero section of T  X. Ellipticity of A is defined by the condition  .A/ 6D 0 on T  X n 0. The solvability of the equation Au D f and the regularity of solutions u are essentially characterised by the properties of a parametrix of A. It is therefore an interesting task to construct parametrices of elliptic differential operators. Those belong to the space

2

Introduction

of classical pseudo-differential operators on X . In other words, the problem of describing parametrices leads to pseudo-differential operators. They enlarge the class of differential operators. Classical pseudo-differential operators A of order  2 R form a space Lcl .X/ with principal symbols  .A/. The notion of ellipticity is analogous to that of differential operators. Parametrices of elliptic elements belong to L cl .X /. Another important property is that the pseudo-differential operators form an algebra. In particular, compositions are defined, and the principal symbols satisfy the relation  .AB/ D  .A/ .B/ (for convenience, we talk about an algebra, even if the algebraic operations are only possible under some natural extra assumptions; for instance, the sum of two classical operators is classical when the difference of their orders is an integer; in matrix-valued scenarios we assume that rows and columns of the involved operators fit together, etc.). All these aspects have been well known for a long time and belong to the basics of the pseudo-differential calculus. There is a large variety of generalisations, e.g., operators with other kinds of symbols than classical ones. In Chapter 2 we recall some technicalities from the standard calculus. We also outline a number of specific variants, especially, with operator-valued symbols and twisted homogeneity, the calculus on manifolds with conical exit to infinity, and operators based on the Mellin transform. Our approach of mixed, transmission or singular crack problems is mainly oriented towards the machinery of operators on manifolds with geometric singularities as is developed in [180] and [183]; see also the monographs [182] or [43]. An element of the philosophy here is that singularities of the underlying configuration or discontinuities of coefficients in the involved operators are recognised and employed as a source of extra symbolic information, apart from the ‘interior’ symbol  . In that sense the problems are described (modulo lower order terms) by a principal symbolic hierarchy with different components that take part in the definition of ellipticity and the construction of parametrices. A non-trivial example is the pseudo-differential calculus of boundary value problems with the transmission property at the boundary, cf. [15]. In Chapter 3 we outline some elements of that calculus which is a necessary tool for establishing the more singular ‘higher’ calculi connected with mixed and crack problems. The  operatorsrepresenting boundary value problems consist of 2  2 block K 1 matrices A D ACG maniT Q , where A is a pseudo-differential operator on a C fold X with boundary, with the transmission property at @X, and G is a so-called Green operator. T is a trace operator, responsible for boundary conditions; it maps distributions on int X to distributions on @X . Moreover, K is a potential operator, mapping in the opposite direction, and Q is a classical pseudo-differential operator on @X. The principal symbolic hierarchy in the case of boundary value problems consists of two components, namely,  .A/ D . .A/; @ .A//; with the interior symbol  .A/ WD  .A/ and the boundary symbol @ .A/. First simple examples may be found in Section 1.1.1; we see, in particular, how an operator A in Rn acquires a boundary symbol from any given (smooth) boundary, prescribed as an additional information, and how extra trace operators become natural in the

Introduction

3

ellipticity. Further intuitive ideas in connection with symbols which are contributed by hypersurfaces of a configuration are given in Section 10.1.1. Chapter 4 is devoted to mixed elliptic boundary value problems, realised as continuous operators in standard Sobolev spaces. An example is the Zaremba problem for a second order elliptic differential operator A in a smooth bounded domain G. The boundary Y WD @G is subdivided into smooth submanifolds Y˙ with a common boundary Z D Y \ YC . The Zaremba problem is represented by a column matrix A WD t .A T TC /, where T are the trace operators describing Dirichlet and Neumann boundary conditions on the minus and the plus side, respectively. The interpretation of A as an operator on standard Sobolev spaces H s .G/ for s > 32 rules out specific singularities of solutions that may be expected when we prescribe independent Dirichlet and Neumann data g on Y . Therefore, the image im A has infinite codimension in 1 3 the space H s2 .G/ ˚ H s 2 .int Y / ˚ H s 2 .int YC /; thus the solvability in this case is only guaranteed when we impose suitable restrictions on the data on the right-hand sides. Nevertheless, mixed problems in this formulation already give us important partial answers for the general case with arbitrary g . Depending on s we complete A by a potential operator K acting on distributions on the interface such that the row matrix A WD .A K/ is a Fredholm operator between the respective Sobolev spaces. The solutions of mixed elliptic problems with arbitrary conditions g may have singularities along the interface Z, inherited by the jumping boundary conditions. This means that the standard Sobolev spaces in G should be replaced by other spaces that admit such singularities. In Chapter 5 we show that weighted edge spaces are an adequate choice, where x as a manifold with edge Z and boundary Y . Mixed problems we interpret X WD G are identified as elements of the edge algebra of boundary value problems on such a manifold. The tools around the edge algebra are developed in Chapter 7; basic notions are already introduced in Chapter 2. By ‘edge algebra’ we understand a calculus of pseudo-differential boundary value problems with edge Z and boundary.   on a manifold ACG K Compared with the block matrices A D T Q of Chapter 3 we now have larger  ACMCG K  with A in the upper left corner, together with a block matrices A D T Q so-called Green operator G with respect to Z and a smoothing Mellin operator M, also referring to Z. The meaning of T ; K and Q with respect to the edge is similar to that of the corresponding operators T; K and Q in A with respect to the boundary. T is a trace operator, K a potential operator with respect to Z, and Q is a classical pseudo-differential operator on Z. Modulo lower order terms an operator A in the edge algebra is described by a principal symbolic hierarchy  .A/ D . .A/; @ .A/; ^ .A//;

(0.0.1)

consisting of the interior symbol  .A/ D  .A/, the boundary symbol @ .A/ D @ .A/ and the edge symbol ^ .A/. In the case of mixed problems the operators A act between spaces of the kind 1

1

1

1

W s; .X/ ˚ W s 2 ; 2 .Y / ˚ W s 2 ; 2 .YC / ˚ H s1 .Z/

4

Introduction

(in general, the components may also be vector-valued; here, for convenience, of unified smoothness on Y and YC by means of suitable reductions of orders). X is the stretched manifold associated with X and the chosen edge Z of codimension 1 on the boundary Y , and W s; .X/ are weighted edge spaces on X of smoothness s 2 R and 1 1 weight  2 R; the spaces W s 2 ; 2 .Y˙ / are of a similar meaning; the edge Z in this case is the boundary of Y˙ . The shifts of s and  by 12 make sense in connection with restriction operators to Y˙ contained in the trace operators in A on the ˙ sides of the boundary. In contrast to Chapter 4 the operators A may contain trace and potential operators with respect to Z at the same time, when the weight  is negative; such weights correspond to singularities of solutions at Z. In the theory of elliptic (with respect to (0.0.1)) operators A the additional trace and potential conditions occur for similar reasons as standard boundary and potential conditions in the context of boundary value problems. However, because of the influence of the choice of weights we may have potentials already in problems with differential operators. In differential boundary value problems in standard Sobolev spaces the potentials do not occur, however, they are generated in parametrices. Apart from the observation that mixed problems are particular edge problems, in Chapter 5 we compute the number of trace and potential operators for concrete examples, e.g., the Zaremba problem, or problems with jumping oblique derivatives. The calculations are based on the results of Chapter 4, together with reformulations of standard Sobolev spaces in terms of weighted edge spaces for  D s and combined with relative index results for the case  < s. Chapter 6 contains a survey on the calculus of (pseudo-differential) boundary value problems on a manifold with conical singularities, with the transmission property at the boundary. We establish several variants of the cone algebra of boundary value problems, including the case of an infinite cone which is necessary for the edge symbolic calculus. The operators have a principal symbolic hierarchy of the form .A/ D . .A/; @ .A/; c .A// consisting of the interior symbol  .A/ D  .A/, the boundary symbol @ .A/ and the so-called conormal symbol c .A/. The cone algebras are closed under parametrix construction of elliptic elements and under inversion (in the case of invertibility of operators in weighted Sobolev spaces). As an example we treat the Laplacian in a spindle (represented in stretched form by I ^ WD RC  I for an interval I , with two conical points corresponding to f0g and f1g) with Dirichlet, Neumann, or Zaremba conditions, and show a number of results on unique solvability in H s; .I ^ /-spaces, with weights  depending on the length of I (concerning notation, see Section 2.4.2). In the case of an infinite (open stretched) cone X ^ D RC  X , where the base X is a compact C 1 manifold with boundary @X , the cone operators A act between spaces of the kind 1 1 K s; .X ^ / ˚ K s 2 ; 2 ..@X /^ / (the components may also be vector-valued). The K s; -spaces are based on the Mellin transform in the axial variable r near r D 0, while for r ! 1 they are modelled on the standard Sobolev spaces up to infinity. The principal symbolic hierarchy in this case also contains a component that controls the behaviour of A up to infinity, called the

Introduction

5

exit symbol .E .A/; E0 .A//. The principal symbolic hierarchy for the infinite cone X ^ is a tuple  .A/ D . .A/; @ .A/; c .A/; E .A/; E0 .A//: In Chapter 7 we formulate the (pseudo-differential) calculus of boundary value problems on a manifold W with boundary V and edge Y , with the transmission property at the boundary (W and V will denote the stretched manifolds associated with W and V , respectively, and n the dimension of the base X of the model cones, locally near Y ). As a typical element of that calculus we establish the 3  3 block matrix structure of operators with the principal symbolic hierarchy (0.0.1). We study the edge algebra in 1 1 n weighted edge spaces W s; .W / ˚ W s 2 ; 2 .V / ˚ H s 2 .Y / and consider ellipticity with respect to the components of the symbols. The edge algebra is closed under parametrix construction of elliptic elements and under inverses (in the case of invertibility of operators in weighted edge spaces). Locally near the singularity the weighted edge spaces are modelled on abstract edge spaces W s .Rq ; H / for Hilbert spaces H which are equipped with the action of a strongly continuous group of isomorphisms f g2RC ,  W H ! H ,  2 RC , here for the case H D K s; .X ^ /. We show that there are several scales of such spaces in which the concept of edge operators with twisted homogeneity applies. Although in this exposition we mainly prefer spaces modelled on W s .Rq ; K s; .X ^ // which are defined in terms of group actions on the parameter spaces K s; .X ^ / independent of s;  , we also discuss alternative possibilities with parameter spaces hris K s; .X ^ / and .s;  /-dependent group actions. In a final section we show that the edge calculus also covers mixed problems in wedges of arbitrary opening angle for the Laplacian and with Dirichlet, Neumann, or Zaremba conditions, where we explicitly compute the (difference of the) number of additional trace and potential conditions on the edge, depending on the chosen weight. Chapter 8 is devoted to mixed elliptic and crack problems when the interface (the boundary of the crack) has conical singularities. We essentially content ourselves with a number of concrete cases for the Laplacian, especially, problems of Zaremba type; but after the material of Chapters 6 and 7 it becomes clear that the approach is completely general. In Chapter 9 operators with meromorphic symbols on infinite cylinders are studied. These operators are generated in the Mellin symbolic structure on manifolds with conical singularities, edges or corners. Applying the work of Gohberg and Sigal [60] we specify the structures for meromorphic pseudo-differential families when the cross section is closed and compact, and, moreover, for boundary value problems for a compact smooth cross section with boundary, and for families in the cone algebra when the cross section is a manifold with conical singularities. Chapter 10 contains a general discussion of the approach with operator algebras and symbolic structures on manifolds with singularities. We describe motivations and intentions that have played a role in the development of the past years, and we show that the theory as a whole is at a new beginning, with new challenges and open problems.

6

Introduction

The analysis of operators on configurations with singularities has a long history. It goes back to the 19th century, starting from observations in models of physics, such as potential theory, heat distribution, wave propagation, etc. For instance, the solutions of elliptic boundary value problems in domains with conical singularities at the boundary may have singularities which can be expressed in terms of asymptotic expansions in the distance variable r to the conical point. The data in such expansions (powers of r, logarithmic terms, and coefficients) are determined by the poles of the inverse of the conormal symbol of the given operator. There is a huge literature on this subject matter, and we cannot give a complete appreciation of all merits and achievements of the past. The work of Kondratyev [100] on elliptic boundary value problems for the case with conical singularities at the boundary is quoted here as a substitute for papers of many other authors. The subsequent development has extended and deepened the ideas for large classes of concrete boundary value problems, also with edges and for many other cases of non-compact domains; more references will be given below. The present book is based on pseudo-differential strategies. Their design is directly deduced from the task to express parametrices of elliptic problems on manifolds with singularities, for instance, of elliptic mixed, transmission and crack problems. As is known from ‘standard’ elliptic boundary value problems, the parametrices, together with their continuity in Sobolev spaces, give rise to the regularity of solutions and to other qualitative results. This concerns, in particular, the nature of Green functions. Applied to Au D f these produce solutions u with vanishing boundary data. Apart from a fundamental solution (or a parametrix) of the given elliptic operator A the Green function of a boundary value problem can be characterised as a pseudo-differential operator on the boundary. Its symbol is operator-valued and acts on spaces in normal direction to the boundary. Such so-called Green symbols are very close to the symbols of trace and potential operators referring to the boundary. Knowing the latter ones we can predict the structure of the Green functions, and vice versa. In other words, if (for some practical reason) we are interested in solutions of a boundary value problem with vanishing boundary data, we should be aware of the case with arbitrary data, since the Green functions contain the same ‘complexity’ of information from the boundary anyway. This may be a key for understanding the relevance of additional trace and potential data on interfaces also in the case of mixed, transmission, or crack problems. Formally they are generated from the notion of ellipticity of the (in this case edge) symbol itself as an analogue of Shapiro–Lopatinskij elliptic conditions, because ellipticity should mean that all components of the principal symbolic hierarchy are bijective. The analytic structure of trace and potential operators of the calculus in the edge case generates analogues of the Green functions, here referring to the interfaces of the singular configuration (i.e., discontinuities of boundary data in mixed problems or crack boundaries). The program to investigate parametrices and their symbolic structure is interesting also on configurations with ‘higher’ singularities. They occur, for instance, when

Introduction

7

the above-mentioned interfaces themselves have singularities, e.g., conical ones as in Chapter 8. A transparent management of corresponding operator calculi requires a well organised building of symbolic and operator levels. Let us continue illustrating several features of this approach by the aspect of Green, trace, and potential symbols. Their role does not only consist of encoding extra data on the lower-dimensional strata of a configuration with singularities. They contain the information of the corresponding symbolic and operator calculi on the lower-dimensional strata which are themselves manifolds with singularities. For instance, the set of all elliptic boundary value problems for a fixed elliptic operator on a compact C 1 manifold X with boundary Y is classified by the space of all elliptic pseudo-differential operators on Y (via reduction to the boundary). More generally, if the boundary Y has, say, conical singularities, elliptic boundary value problems for a fixed elliptic (Fuchs type) operator on X are classified by the elliptic elements of the cone algebra on Y , etc. In general, the calculus contains reductions to all lower-dimensional strata, and there are Green, trace, potential, etc., operators for all of them. The technical tool to express these elements of the calculi are operator-valued symbols with ‘twisted homogeneity’. This concept was first systematically developed in connection with edge algebras, cf. [163], [180], and then applied to several higher algebras, e.g., on manifolds with corners, cf. [183], [192]. At the same time the Green, trace, and potential symbols are a convenient tool to describe phenomena with asymptotics of solutions, especially, edge asymptotics and their singular functions. This is a voluminous program, worth to be presented in a separate exposition in more detail. The technique with asymptotics in that spirit may be found, e.g., in [182], [186], [187], [185], [188], or [90]. We comment more on that in Chapter 10. The story with additional elliptic trace and potential conditions on lower-dimensional strata has also other remarkable sides. For instance, it may be an ambitious task to determine the necessary number of such conditions (depending on the chosen weights  in the distribution spaces). Therefore, we study explicit examples in connection with mixed and crack problems (see Chapters 4, 5 and 8), but also other concrete elliptic cone and edge operators; see, for instance, Sections 6.3.3, 6.3.4, 6.4.1, or 7.3.3, 7.3.4. If W is a manifold with boundary and (say, a compact smooth) edge Y , the consideration starts from the homogeneous principal edge symbol 1

1

^ .A/.y; / W K s; .X ^ / ˚ K s 2 ; 2 ..@X /^ / 1

1

! K s;  .X ^ / ˚ K s 2 ; 2 ..@X /^ /;

(0.0.2)

.y; / 2 T  Y n0, which is for elliptic A on W nY (in the sense of Chapter 3) and under a suitable choice of the weight  a family of Fredholm operators (for simplicity, at the moment we consider spaces that only contain one component over X ^ and .@X /^ , respectively; in general, we have spaces of distributional sections in several vector bundles).

8

Introduction

The additional trace symbols, potential symbols, etc., ^ .T /.y; /, ^ .K/.y; / and ^ .Q/.y; / play the role to fill up (0.0.2) to a block matrix family   ^ .A/ ^ .K/ .y; / W K s; .X ^ / ˚ C j ! K s; .X ^ / ˚ C jC ^ .T / ^ .Q/ for suitable j˙ 2 N (where K s; .X ^ /, K s; .X ^ / are abbreviations for the spaces in (0.0.2)). Because of the (twisted) homogeneity of ^ .A/.y; / in  6D 0 it suffices to do that for .y; / 2 S  Y , the unit cosphere bundle in T  Y n 0. In order to find ^ .T /; ^ .K/, ^ .Q/ we have to know at least the index ind ^ .A/.y; / which is equal to jC  j . In other words, in order to characterise the extra conditions, we suddenly have to solve an index problem on the infinite cone X ^ with boundary (and, of course, much more, to know the dimensions j˙ themselves, together with kernels and cokernels). Let us stress at this moment that, in general, the above-mentioned spaces C j˙ have to be interpreted as the fibres of (smooth complex) vector bundles J˙ over the edge Y (they may be non-trivial, unless we do not impose corresponding assumptions on A). In analogous form, this is also the case in boundary value problems for an elliptic operator A on a compact C 1 manifold X with boundary, where the role of trace, potential, etc., boundary symbols @ .T /.x 0 ; 0 /; @ .K/.x 0 ; 0 / and @ .Q/.x 0 ; 0 / is to fill up the Fredholm family @ .A/.x 0 ; 0 / WH s .RC / !H s .RC / to a block matrix .A/ @ .K/ .x 0 ; 0 / W H s .RC / ˚ C l ! family of isomorphisms @ .A/.x 0 ; 0 / D @ .T /  .Q/ @

@

H s .RC / ˚ C lC ; .x 0 ; 0 / 2 T  .@X / n 0. Clearly, the formalism is possible in analogous form for operators A, acting between distributional sections of vector bundles. If C l˙ are the fibres of vector bundles L˙ on @X , then the Fredholm family @ .A/.x 0 ; 0 /, parametrised by the compact space S  .@X /, has an index element  indS  .@X/ @ .A/ D ŒLC  ŒL 2 @X K.@X /;

(0.0.3)

 where K.  / denotes the K-group over the space in parentheses and @X W K.@X / !   K.S .@X// is the pull back under the canonical projection @X W S .@X / ! @X. It is well known that there are elliptic operators A such that

indS  .@X/ @ .A/ 2 K.S  .@X //

(0.0.4)

is not the pull back of an element of K.@X /. This is, for instance, the case when A is a Dirac operator in even dimensions. Then the condition (0.0.3) represents a topological obstruction for the existence of additional trace and potential operators which complete A to a 2  2 block matrix Fredholm operator A between standard Sobolev spaces, with A in the upper left corner. More precisely, (0.0.3) means vanishing of the obstruction, and then A exists as desired, cf. Atiyah and Bott [8] and Boutet de Monvel [15]. We do not study here the general case, i.e., when (0.0.3) is violated, and refer to the article [190], see also [194], with global projection conditions of trace and potential type rather than usual ones.

Introduction

9

What concerns the existence of additional trace, potential, etc., conditions for A in the case of a manifold with edge there is a similar (necessary and sufficient) condition, namely, indS  Y ^ .A/ 2 Y K.Y / (0.0.5) with Y W S  Y ! Y being the canonical projection. This situation for the analogous case of a manifold without boundary and edges has been investigated in [182] and in Schulze and Seiler [198]; in the latter article this has been performed also in the sense of global projection conditions with respect to the edge Y . Throughout this book we assume that (0.0.5) is satisfied. Let us note for completeness that the phenomenon of edge operators where (0.0.5) is violated has been studied under different aspects in Nazaikinskij, Savin, Schulze and Sternin [131], [132], (see also Nazaikinskij, Schulze, Sternin, and Shatalov [138] or Schulze, Sternin, and Shatalov [199] for the case of boundary value problems in the case of differential operators). Operators on manifolds with singularities from the viewpoint of ellipticity and index lead to a large variety of beautiful and new questions of K-theoretic nature that are far from being solved in general. For instance, in connection with such problems on stratified spaces we have to expect hierarchies of topological obstructions for the extra conditions, cf. Section 10.5.3. The complexity of several phenomena makes it desirable to establish explicit examples of reasonable generality. That is why in Chapter 7 of this book we report the construction of extra conditions by reformulating elliptic operators on a smooth manifold. The results belong to a cycle of papers [36], [35], [113] of Schulze in cooperation with Dines and Liu, in which such questions are studied in different situations. The technique is also used here in Chapters 5 and 8 to explicitly computing the abovementioned index of the Fredholm family ^ .A/. It is a typical effect that ind ^ .A/ depends on the chosen weights ; the difference of indices for different weights can be characterised by relative index formulas. This is also illustrated here in Chapter 5 for the case of mixed problems. In the present book we mainly focus on the analytic part of the elliptic theory on manifolds with singularities, here in the context of mixed, transmission and crack problems; those can be regarded as examples of the edge and corner calculus of boundary value problems. The symbolic structures show that concrete cases require in a very early phase of the consideration the cone, edge and corner calculi of high generality, because the operator-valued symbols are families operating on configurations that have again singularities. As amplitude functions they have to be composed and inverted within the calculus, and this generates large operator algebras. Therefore, a part of this exposition is devoted to the general aspects of operators on manifolds with boundary and conical points and edges. In particular, (pseudo-differential) boundary value problems are a necessary element of the theory both in local form on a C 1 manifold with boundary, and from the point of view of the exit calculus, i.e., when the underlying manifold is an infinite cone with conical exit to infinity. As noted before, all these structures are motivated by the various ingredients of parametrices of elliptic operators, realised here as edge boundary value problems. A

10

Introduction

short characterisation of our calculus is that it describes in advance the structure of parametrices. This is connected with the functional analytic aspects around weighted cone and edge spaces which determine in advance different facets of the regularity of solutions. Thus the ‘technical’ parts of this exposition directly belong to the concrete problems. Moreover, we believe that the tools can be employed to other areas of the analysis on manifolds with singularities, much more than it is already the case in the papers of Krainer [104], [105], [106] on parabolic operators, of Gil, Krainer and Mendoza [55], [54] on resolvents of elliptic operators on manifolds with singularities, of Liu and Witt [114], [115] on explicit information on asymptotic types, or Dreher and Witt [40], [42], [41] on modifications and applications of edge spaces in hyperbolic equations. The ellipticity of interesting operator classes is often a starting point to develop the corresponding index theories and to derive index formulas, see Nazaikinskij, Savin, Schulze and Sternin [133], [134], [135], and Fedosov, Schulze, and Tarkhanov [48], [50], [51] and more references below. Other interesting results on the Mellin-edge calculus with meromorphic symbols and asymptotic data are contained in papers of Witt [227], [228]. Relations to models motivated by problems of elasticity and material sciences may be found in Kapanadze and Schulze [96], [92], [93]. Contributions to the functional analytic machinery of the cone and edge calculus are given in Hirschmann [79] on edge spaces, Gil, Schulze and Seiler [56], [57] on edge quantisations, Seiler [205] on a Calderón–Vaillancourt theorem in abstract edge spaces, Seiler [206], Schulze and Volpato [201] on kernel characterisations of Green operators, Kapanadze, Schulze, and Seiler [95] on edge operators with singular trace conditions, see also Liu and Witt [115], moreover, Coriasco and Schulze [28] on the edge calculus with model cones of different dimensions, Airapetyan and Witt [4], or Tarkhanov [217] on a motivated choice of edge spaces. Let us also mention the development towards calculi with higher singularities, especially, [183], [192], [75], and the joint papers of Schulze with Maniccia [118], Krainer [109], [108], and Calvo and Martin [20], [18]. Concerning the literature on mixed, crack, or other singular boundary value problems, there is a huge number of investigations devoted to specific topics or different approaches, and we cannot give a complete review here. In the past decades the analysis around manifolds with singularities, especially extensions of the index theory, attracted many mathematicians. There are now different schools and confessions, partly with a common terminology, such as corner manifolds or manifolds with boundary, although for quite different species of operators, see also the remarks in Section 10.3.4. If we now give a list of more references that have from different point of view connections with this exposition, we are aware of such ambiguities. Let us mention, in particular, Agmon [2], Agranovich and Vishik [3], Kohn and Nirenberg [98], Vishik and Eskin [221], Atiyah and Singer [11], Atiyah [7], Kasparov [97], Eskin [44], Vishik and Grushin [224], Sternin [214], [215], Kondratyev [100], Plamenevskij [150], [152], [151], Rabinovich [153], Gramsch [61], Gohberg and Sigal [60], Fedosov [46], Seeley [202], [203], Grushin [70], Boutet de Monvel [14],

Introduction

11

[15], Atiyah, Patodi and Singer [9], Maz’ja and Paneah [121], Shubin [208], Parenti [145], Cordes [25], [26], Fichera [52], Teleman [218], Cheeger [21], Melrose [124], [125], Melrose and Mendoza [127], Rempel and Schulze [159], [162], [163], [155], Luke [117], Grubb [69], [68], Kondratyev and Oleynik [101], Grisvard [65], Maz’ja and Rossmann [122], Dauge [30], Chkadua and Duduchava [22], [23], Costabel and Dauge [29], Shaw [207], Brüning and Seeley [16], Mazzeo [123], Piazza [147], Roe [165], Melrose [126], Rozenblum [166], Egorov and Schulze [43], Booss-Bavnbek and Wojciechowski [13], Mantlik [119], [120], Fedosov and Schulze [47], Nistor [142], [143], Melrose and Nistor [128], Melrose and Piazza [129], Witt [228], [227], Fedosov, Schulze, and Tarkhanov [50], [49], [51], Nazaikinskij and Sternin [140], [141], Nazaikinskij, Savin and Sternin [135], [136], [137], Grieser and Lesch [64], Krainer [106], [103], Seiler [205], [206], [204], Gil, Krainer, and Mendoza [55], [54], Ammann, Lauter, and Nistor [5]. Let us finally make a few remarks on how the present exposition is organised. Our approach to mixed elliptic boundary value problems in Chapters 1, 4 and 5 employs elements of the pseudo-differential calculus and of the formalism of boundary value problems with the transmission property at the boundary. These theories, developed in Chapters 2 and 3 before, are independently readable as introductions. In Chapter 5 on mixed problems in weighted edge spaces we make use of boundary value problems on manifolds with conical singularities and edges; they belong to the necessary tools for the construction of parametrices. In order to come to mixed problems in this exposition as early as possible we develop the details on cone and edge algebras afterwards in Chapters 6 and 7. They can be consulted if something remains to be completed in the considerations before. This material on cone and edge boundary value problems is also an introduction; at the same time it prepares the applications in Chapter 8, namely, mixed problems with singular interfaces and crack problems with singular crack boundaries. Such problems belong to the corner theory of singularity order 2 and with boundary. As such they are of some complexity. In order to illustrate the main ideas which are nevertheless simple, in Chapter 10 we try to give a feeling for the iterative way of treating problems with higher singularities and for the several sources and mathematical ingredients in connection with the building of higher singular algebras.

Chapter 1

Boundary value problems with mixed and interface data

We study elliptic boundary value problems for differential operators with smooth coefficients, furthermore, mixed elliptic problems, where the boundary conditions are discontinuous along an interface Z of codimension 1 on the boundary Y . The reduction of mixed problems to the boundary produces pseudo-differential operators on the boundary with a discontinuity at the interface. These operators have not the transmission property at Z (except for trivial cases). There are different ways to investigate the solvability. We mainly ask the Fredholm property in weighted edge spaces under additional elliptic conditions on Z. Another aspect is to first solve elliptic transmission (or boundary value) problems for the reduced operators on the boundary and then to return to the original problem. The material of this chapter plays the role of an introduction.

1.1 Elliptic boundary value problems Elliptic boundary value problems for differential operators with constant coefficients in the half-space admit explicit solutions with so-called Poisson kernels. We consider reductions of boundary value problems to the boundary and obtain pseudo-differential operators.

1.1.1 Differential operators with classical boundary conditions In this section we consider differential operators X a˛ .x/ Dx˛ AD j˛j  

x Here, as usual, Dx˛ D in a domain G  Rn with coefficients a˛ .x/ 2 C 1 .G/. ˛1 ˛n 1 @ Dx1 : : : Dxn , Dxj D i @xj , for any multi-index ˛ D .˛1 ; : : : ; ˛n / 2 N n , N D f0; 1; 2; : : : g, j˛j D ˛1 C    C ˛n . In order to motivate the basic elements of our theory of mixed and crack problems we first assume that G is the half-space RnC WD fx D .y; t / W y 2 Rn1 ; t > 0g or a bounded domain with C 1 boundary Y . The operator A is called elliptic (of order ) if the homogeneous principal symbol X a˛ .x/ ˛  .A/.x; / D j˛jD

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1 Boundary value problems with mixed and interface data

x n 0; as usual T  X denotes the cotangent bundle does not vanish for all .x; / 2 T  G 1 x as a C manifold with boundary), and 0 indicates the zero section of X (here X D G which corresponds to D 0. (In a notation of the kind  .  / the order  of the operator in parentheses will be clear from the context; if necessary we indicate it explicitly as   .  /). We identify neighbourhoods of points in Y with the half-space, where t 2 Œ0; 1/ also plays the role of a global normal variable in a corresponding collar neighbourhood Y  Œ0; 1/ of the boundary. A boundary value problem for the operator A is represented by equations Au D f in G;

T u D g on Y ;

(1.1.1)

where T D .T1 ; : : : ; TN / is a vector of boundary (or trace) operators that are of the form Tl u D rY Bl u; l D 1; : : : ; N; (1.1.2) P ˇ with differential operators Bl D jˇ j  ml bl;ˇ .x/Dx with smooth coefficients in a neighbourhood of Y and the operator rY of restriction to Y . The number N of the boundary conditions will be specified below, cf. the formula (1.1.9). The problem consists of finding solutions u of (1.1.1) with given right-hand sides f and boundary data g. They will belong to various classes of distribution spaces. First we can interpret (1.1.1) as a column matrix of operators   A x ! AD W C 1 .G/ T

x C 1 .G/ ˚ : C 1 .Y; C N /

For bounded G or the half-space when the coefficients are constant outside a compact set, the operator A extends to a continuous mapping between standard Sobolev spaces   A AD W H s .G/ ! T

H s .G/ ˚ sml  1 2 .Y / ˚N H lD1

(1.1.3)

for every s > maxfml C 12 W l D 1; : : : ; N g. An example is the Laplace operator Pn @2 A D D iD1 @x 2 with Dirichlet conditions T u D rY u. The corresponding i

operator A D

A T

W H s .G/ !

H s2 .G/ ˚ 1 H s 2 .Y /

is an isomorphism (when G is bounded)

for all s 2 R, s > 32 . Let us describe the relationship between the operator A and the trace operator T in the half-space RnC which is the local model of G near a boundary point. The covariable splits into .; /;  2 Rn1 ; 2 R. It will often be convenient to localise y in an open subset   Rn1 . In the homogeneous principal symbol  .A/.y; t; ; / we freeze the coefficients at t D 0 and apply the differentiations in t . Then the resulting family of operators @ .A/.y; / WD  .A/.y; 0; ; D t /

1.1 Elliptic boundary value problems

15

for y 2 ;  ¤ 0 is called the homogeneous principal boundary symbol of the operator A. We have @ .A/.y; / W H s .RC / ! H s .RC / (1.1.4) for all s and

x C / ! S.R x C /; @ .A/.y; / W S.R

(1.1.5)

x C / WD S.R/j x ; here S.R/ is the Schwartz space on the real axis, and H .RC / WD S.R RC s H .R/jRC with the standard Sobolev space H s .R/ on R. s

1

Remark 1.1.1. Setting . u/.t / D  2 u. t / for  2 RC , we obtain a family of x C / ! S.R x C //, and we have isomorphisms  W H s .RC / ! H s .RC / (or  W S.R @ .A/.y; / D   @ .A/.y; /1

(1.1.6)

for all  2 RC . Proposition 1.1.2. Let A be an elliptic differential operator of order . Then (1.1.4) and (1.1.5) are families of surjective Fredholm operators for all  ¤ 0, and for the x C /, both in Sobolev spaces (1.1.4) for all kernels we have ker @ .A/.y; /  S.R 1 s >   2 as well as in the sense (1.1.5). This result is well known. The tools with the corresponding arguments for Proposition 1.1.2 will be developed in the pseudo-differential calculus below. As a consequence of the homogeneity relation (1.1.6) we have the following observation. Corollary 1.1.3. Under the conditions    of Proposition 1.1.2 the dimensions of the kernels of @ .A/.y; / and @ .A/ y; jj coincide for all  ¤ 0. For the boundary operators Tl ; l D 1; : : : ; N , we set @ .Tl /.y; / WD r 0  .Bl /.y; 0; ; D t /; where r 0 is the restriction to t D 0 (and  .Bl / refers to ml D ord Bl ). This gives us families of linear operators @ .Tl /.y; / W H s .RC / ! C x C / ! C, l D 1; : : : ; N , for all y 2 ,  ¤ 0. for all s > ml C 12 or @ .Tl /.y; / W S.R We then have 1

@ .Tl /.y; / D ml C 2 @ .Tl /.y; /1

for all  2 RC .

The column vector @ .T /.y; / WD t .@ .T1 /.y; /; : : : ; @ .TN /.y; //, interpreted as a family of maps @ .T /.y; / W H s .RC / ! C N ; x C / ! C N , is called the homogeneous principal boundary symbol or @ .T /.y; / W S.R of the trace operator T .

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1 Boundary value problems with mixed and interface data

Definition 1.1.4. Let A be an elliptic differential operator of order  with smooth x Then the boundary value problem (1.1.1) is called elliptic, if in local coefficients in G. x C near the boundary the family coordinates x D .y; t / 2   R   @ .A/ .y; / @ .A/.y; / WD @ .T / defines isomorphisms H s .RC / ˚ @ .A/.y; / W H .RC / ! CN s

(1.1.7)

for some s > max.; d/  12 for d WD maxfml C 1 W l D 1; : : : ; N g and all y 2 ,  ¤ 0. In that case we also say that the boundary conditions are elliptic, or, equivalently, satisfy the Shapiro–Lopatinskij condition (with respect to the elliptic operator A). We shall see in Chapter 3 below that this property does not depend on the specific choice of s and that it is equivalent to the condition that x C / ! @ .A/.y; / W S.R

x C/ S.R ˚ CN

(1.1.8)

is an isomorphism for all y 2 ,  ¤ 0. For this reason we may refer to families of operators in the Schwartz space. It can easily be proved that the ellipticity of a boundary value problem is independent of the choice of local coordinates near the boundary. The operator family (1.1.7) (or (1.1.8)) is called the homogeneous principal boundary symbol of the boundary value problem A. Remark 1.1.5. The map (1.1.8) for an elliptic differential operator A is an isomorphism if and only if @ .T /.y; / W ker @ .A/.y; / ! C N is an isomorphism for every y 2 ,  ¤ 0. Example 1.1.6. Consider the Laplace operator in the half-space RnC . Then in the x C / ! S.R x C/ splitting of coordinates .y; t / we have @ . /./ D jj2 C @2t W S.R tjj which is surjective for  ¤ 0, and ker @ . /./ D fc e W c 2 Cg is one-dimensional. The restriction operator r 0 composed from the left with @jt , j D 0; 1, defines trace operators Tj D r 0 @jt , where T0 corresponds to the Dirichlet, T1 to the Neumann problem. Both are elliptic, as they induce isomorphisms r 0 @jt W ker @ . /./ ! C, j D 0; 1, for  ¤ 0. Remark 1.1.7. Let G  Rn be a bounded domain with C 1 boundary Y , and let A be an elliptic differential operator of order . Then the kernels of the Fredholm operators @ .A/.y; / W H s .RC / ! H s .RC /

1.1 Elliptic boundary value problems

17

form a smooth complex vector bundle JzC on T  Y n 0. We assume that JzC is the pull back under the canonical projection Y W T  Y n 0 ! Y of a vector bundle JC on Y . Then the global (on Y ) interpretation of the family of isomorphisms (1.1.7) is that 0 s 1 .RC / H A ˚ @ .A/ W Y H s .RC / ! Y @ JC is an isomorphism between the respective fibre bundles over Y ; here H s .RC / is interpreted as the trivial bundle with fibre H s .RC /. We will return to this point of view in Chapter 3 below. In concrete elliptic boundary value problems to differential operators it is often convenient to employ another equivalent description of the ellipticity of boundary conditions. Let us briefly discuss this approach for the case of operators A of even order  D 2m with constant coefficients in the half-space RnC for n  3, cf. Agmon, Douglis, and P Nirenberg [2]. ˛ Let A D j˛jD2m P a˛ Dx ; ˛a˛ 2 C, and assume that A is uniformly elliptic, i.e., that  .A/. / D j˛jD2m a˛ satisfies the following condition: there exists a ı > 0 such that ı 1 j j2m  j .A/. /j  ı j j2m for all 2 Rn . Let, furthermore, Bj D P ˇ jˇ jDmj bj;ˇ Dx ; j D 1; : : : ; m, be differential operators with constant coefficients, x n / of the boundary value problem and ask for solutions u 2 C 1 .R C Au D 0

for x 2 RnC ;

Bj uj tD0 D gj ; j D 1; : : : ; m;

(1.1.9)

with boundary data gj 2 C01 .Rn1 /. Let be a root of  .A/.; / D 0 for  ¤ 0. Then  is a root for . There are m roots k˙ ./; k D 1; : : : ; m, with positive and negative imaginary parts, respectively, for  ¤ 0. From the uniform ellipticity of A it follows that j k˙ ./j; j Im k˙ ./j1  C , for k D 1; : : : ; m, for all jj D 1, with some constant C , depending only on A and m. Consider the polynomials M ˙ .; / D

m Y

.  k˙ .//

kD1

Pm

which are of the form lD0 cl˙ ./ ml , with coefficients cl˙ ./ that are analytic in Pj  2 Rq n f0g and homogeneous of order l. Set MjC .; / D c C ./ j l , lD0 l j D 0; : : : ; m  1. Lemma 1.1.8. Let L be a rectifiable Jordan curve in the complex -plane surrounding R MC .; / k all kC ./, 1  k  m, for jj D 1. Then we have 21 i L m1j

d D ıj k M C .; / for 0  j; k  m  1. C .; / k is equal to c0C ./ m1j Ck . If Proof. The highest order term of Mm1j j > k the degree of the denominator in under the integral is  2 plus the degree of the numerator. Therefore, choosing for L a circle with a radius tending to infinity,

18

1 Boundary value problems with mixed and interface data

the integral tends to zero. Since it is independent of L, it vanishes. For j < k the polynomial C Mm1j .; / k

D

m1j X

clC ./ m1j Ckl

D

kj 1

lD0

m1j X

clC ./ ml

lD0

differs from kj 1 M C .; R/ by a polynomial Q.; / of degree < m  1. Thus the integral is equal to 21 i L MQ.; / C .; / d which vanishes again, since the degree C of M is  2 plus the degree of Q. For j D k we have an integral of the form R c0C ./ m1 C 1 d which is equal to 1 by the Residue Theorem. C 2 i L m c0 ./ C

Consider the principal symbol  .Bj /. / D

P jˇ jDmj

bj;ˇ ˇ of Bj and form

f .Bj /.; / WD  .Bj /.; / mod M C .; /

(1.1.10)

which is the remainder under division of  .Bj /.; / (as a polynomial in ) by M C .; /,  ¤ 0. Write f .Bj /.; / D

m X

dj k ./ k1 :

kD1

Definition 1.1.9. The operators Bj in the boundary value problem (1.1.9) are said to satisfy the complementing condition if d./ D det.dj k .// ¤ 0 for all real  ¤ 0. The complementing condition is equivalent to the ellipticity of the associated trace operator T , cf. Definition 1.1.4. Remark 1.1.10. Let .d j k .// be the inverse matrix to .dj k .//, and set Nk .; / WD Pm jk ./ MmC j .; /, k D 1; : : : ; m. Then we have j D1 d Z Nk .; / .Bj /.; / 1 d D ıj k ; j; k D 1; : : : ; m; (1.1.11) 2 i L M C .; / where L is a curve in the upper complex half-plane enclosing the roots of M C .; /. In fact, first  .Bj / may be replaced by f .Bj / without changing the integral, but then the relation (1.1.11) is an obvious consequence of Lemma 1.1.8.

1.1.2 The Poisson kernels in the half-space Let us set ck WD ..2 i /n1 kŠ/1 for k 2 N .0Š D 1/, ck WD .1/k .k  1/Š .2 i/.n1/ for k 2 N n f0g, and define the following function in z 2 C: 

F .z; k/ D ck z

k

z X1 log  i j k

j D1



for k > 0;

1.1 Elliptic boundary value problems

19

z for k D 0; F .z; k/ D ck z k for k < 0; i with the principal branch of the logarithm in C slit along the negative real axis. Notice  d l that dz F .z; k/ D F .z; k  l/ for each l. F .z; k/ D ck log

Theorem 1.1.11 (John’s identity). Let g.y/ 2 C01 .Rn1 / and l 2 N; n  1 C l even. Then we have Z Z n1Cl 0 0 2 g.y / F ..y  y /; l/d! dy 0 g.y/ D y Rn1 jjD1 Z Z n1Cl D y 0 2 g.y 0 / F ..y  y 0 /; l/d! dy 0 Rn1

jjD1

with d! being the area element on the unit sphere jj D 1. A proof of John’s identity may be found in the book [87]. Remark 1.1.12. John’s identity can also be written in the form g.y/ D

Z Z n1Cl 1 .y  y 0 / 0 0 l 0 2 g.y / Œ.y  y /

log d!  dy : y .2 i/n1 lŠ i Rn1 jjD1 R In fact, jjD1 Œ.y  y 0 / l d! is a homogeneous polynomial in y  y 0 of degree l and thus  n1Cl Z Z g.y 0 / y 2 Œ.y  y 0 / l d! dy 0 D 0: 

Rn1

Let us set Kj .y; t/ WD

jjD1

Z jjD1

1 2 i

Z F .y C t ; mj  n C 1/ L

Nj .; / d d! M C .; /

with the curve L as in the relation (1.1.11) and mj D ord Bj ; j D 1; : : : ; m. We denote by Kj;l .y; t/ the function defined by an analogous expression, where mj  n C 1 is replaced by mj C l. We then have n1Cl 2

y

Kj;l .y; t / D Kj .y; t /

(1.1.12)

for each l 2 N with n  1 C l even. Theorem 1.1.13. Assume A and fB1 ; : : : ; Bm g satisfy the above-mentioned assumptions. Then the function m Z X Kj .y  y 0 ; t /gj .y 0 /dy 0 u.y; t / D n1 j D1 R

for gj 2 C01 .Rn1 /; j D 1; : : : ; m, is a solution of the boundary value problem (1.1.9) in the half space RnC .

20

1 Boundary value problems with mixed and interface data

x n / for Note that for each k 2 N we find an l.k/ such that Kj;l .y; t / 2 C k .R C l  l.k/. The solution of the boundary value problem (1.1.9) can also be written m Z X n1Cl u.y; t / D Kj;l .y  y 0 ; t/ y 0 2 gj .y 0 / dy 0 (1.1.13) n1 j D1 R

for each l with n  1 C l even. In fact, it suffices to integrate by parts and to apply (1.1.12). The functions Kj .y  y 0 ; t / are called the Poisson kernels of the boundary value problem (1.1.9). The proof of Theorem 1.1.13 follows by a direct computation, using John’s identity. x n /. In fact, choosing the number l in the formula The solutions u.y; t / belong to C 1 .R C x n reaches every (1.1.13) sufficiently large, we see that the smoothness of u.y; t / in R C prescribed order. More details may be found in the paper [2] of Agmon, Douglis, and Nirenberg (part II treats the case of systems that are elliptic in the sense of Douglis and Nirenberg). Also the book of Lions and Magenes [111] is a reference to this method. Let us write the given boundary value problem (1.1.9) with m trace operators  Tj u D Bj ujRn1 , j D 1; : : : ; m, as a column matrix of operators A D TA , and consider the boundary symbol, cf. the formula (1.1.8). Theorem 1.1.14. Let A be an elliptic differential operator in RnC of order 2m with the above properties. Then the system of boundary operators fB1 ; : : : ; Bm g satisfies the complementing condition with respect to A if and only if the operator x C / ! @ .A/./ W S.R

x C/ S.R ˚ Cm

(1.1.14)

is an isomorphism for each  2 Rn1 n f0g. A proof of this result may be found in Lions and Magenes [111, Sections 4.1, 4.2]. There is also noted that the mapping is an isomorphism if and only if @ .A/./ W H

2m

L2 .RC / ˚ .RC / ! Cm

is an isomorphism for each  ¤ 0. The inverse of the boundary symbol has the form of a row matrix   @ .A/1 ./ D @ .P /./ @ .K/./ ;  ¤ 0;

(1.1.15)

x C / is an injective map that can be expressed as follows. where @ .K/./ W C m ! S.R Write D./ WD dj k ./ j;kD1;:::;m and choose a vector c 2 C m . Then @ .K/./c is equal to Z X m M C l .; / it 1 .D 1 ./c/l mC (1.1.16) e d u.; t / D 2 i L M .; / lD1

1.1 Elliptic boundary value problems

21

with the above-mentioned curve L surrounding the roots kC ./; k D 1; : : : ; m. It is x C / as a function of t and @ .A/./u.; t / then easy to verify that u.; t / belongs to S.R equals 0. Moreover, we have @ .T /./u.; t / D c. In fact, @ .Tj /./u.; t / D f .Bj /.; D t / u.; t /j tD0 Z X m M C .; / 1 D .D 1 ./c/l f .Bj /.; / ml d 2 i L M C .; / lD1 Z X m m X M C .; / 1 D d dj k ./ k1 .D 1 ./c/l ml 2 i L M C .; / lD1 kD1 Z m m X C X Mml .; / k1 1 1 D dj k ./.D ./c/l d D cj :

2 i L M C .; / lD1 kD1 P 1 ./c/k D cj . Here we used Lemma 1.1.8 and the relation m kD1 dj k ./.D

1.1.3 Reduction to the boundary   z D A (not necesWe now discuss the idea of reducing a boundary value problem A Tz sarily elliptic with respect to the elliptic differential operator A) to the boundary, using   a ‘reference’ boundary value problem A D TA for A which is assumed to be elliptic. The solution operator P to an elliptic boundary value problem   A x ! AD W C 1 .G/ T

x C 1 .G/ ˚ C 1 .Y; C m /

in a (say, bounded) domain G with C 1 boundary Y may be regarded as a row matrix of operators x C 1 .G/ x ! C 1 .G/: P D .P K/ W ˚ 1 m C .Y; C / The structure of the operators P and K will be characterised in Chapter 3 below in the analysis of boundary value problems with the transmission property. For the moment we only want to say that, as a result of that calculus, the remainders C and Cij in the expressions P A D PA C K T D 1 C C and       AP AK 1 0 C11 C12 AP D D C (1.1.17) TP TK 0 1 C21 C22 are smoothing operators with compact extensions in the Sobolev spaces in question (here 1 denotes corresponding identity operators). If an equation holds modulo such remainders we will write . In other words, we have P A  1;

AP  diag.1; 1/:

22

1 Boundary value problems with mixed and interface data

In such a case P will be called a parametrix of A (and A a parametrix of P ). The parametrix of an operator A will also be denoted by A.1/ . Example 1.1.15. Let x   C 1 .G/ A 1 x AD W C .G/ ! ˚ T C 1 .Y /

(1.1.18)

be the operator corresponding to the Dirichlet problem for the Laplace operator , T u D ujY . Then (1.1.18) is an isomorphism. Set A1 WD .P K/I then P is Green’s function of the Dirichlet problem (with a natural identification between the operator and its kernel) and K is the double layer potential. Let

  z D A W C 1 .G/ x ! A Tz

x C 1 .G/ ˚ C 1 .Y; C m /

be another boundary value problem for A. Using the relation (1.1.17) we obtain     AP AK 1 0 z AP D z  z : (1.1.19) T P Tz K T P Tz K z ! Tz K is called the reduction to the boundDefinition 1.1.16. The correspondence A z ary of the boundary value problem A (with respect to the reference operator A). Alternatively, Tz K will also be called the reduction of the boundary condition Tz to the boundary. We will see in the pseudo-differential calculus of boundary value problems that the ellipticity remains preserved under parametrix constructions and compositions. In this z is elliptic as soon framework P is an elliptic boundary value problem for P . Also AP z is elliptic. Tz K is an mm matrix of classical pseudo-differential operators on the as A boundary Y (here and in the sequel we tacitly use notions from the pseudo-differential z is equivalent to a triangular matrix with the calculus, see Chapter 2 below). Since AP identity operator in the upper left corner, we obtain the following result: z to the boundary Proposition 1.1.17. Let Tz K be the operator, obtained by reducing A (with respect to an elliptic boundary value problem A for A). Then the following properties are equivalent: z is elliptic (in the sense of Definition 1.1.4 above). (i) A (ii) Tz K is an elliptic system (in the sense of Douglis–Nirenberg) of pseudo-differential operators on the boundary. In order to explain (ii) we want to recall the notion of Douglis–Nirenberg ellipticity. First let B D .Bij /i;j D1;:::;m be a system of classical pseudo-differential operators on

1.1 Elliptic boundary value problems

23

Y , ord Bij D  2 R for all i , j . Then B is elliptic if det  .Bij /.y; / ¤ 0 on T  Y n 0. Furthermore, let Ri and Sj be scalar elliptic operators on Y of orders ri and sj , respectively, i; j D 1; : : : ; m, and set R WD diag.Ri /, S WD diag.Sj /, S .1/ WD diag.Sj.1/ /. Then a system D D .Dij /i;j D1;:::;m of pseudo-differential operators on Y is called Douglis–Nirenberg elliptic, if it can be written in the form D D RBS .1/ for some elliptic m  m system B of order . Notice that ord Dij D ri  sj C ; i; j D 1; : : : ; m. The construction of a parametrix B .1/ of B on Y (as a closed compact C 1 manifold) is a standard procedure. Then D .1/ D SB .1/ R.1/ is a parametrix of D. z is an elliptic boundary value problem for A, from Proposition 1.1.17 we Now if A have (Douglis–Nirenberg) ellipticity of Tz K on Y which yields a parametrix .Tz K/.1/ . Then, setting     1 0 1 0 .1/ ; WD R WD z ; R .Tz K/.1/ Tz P .Tz K/.1/ T P Tz K     we obtain RR.1/  10 01 , R.1/ R  10 01 . Applying the relation (1.1.19), i.e.,   z z R.1/  1 0 . This gives us a parametrix Pz D AP  R, it follows that AP 01 z namely, Pz D .P  K.Tz K/.1/ Tz P K.Tz K/.1/ /. P R.1/ of A, In other words we obtain the following result: z be elliptic boundary value problems to A, let P D Theorem 1.1.18. Let A and A z to the boundary. Then .P K/ be a parametrix of A and R D Tz K the reduction of A .1/ z .1/ z z z z z z P D .P K/ for P D P  KR is a parametrix of A. T P , K D KR Remark 1.1.19. Below we shall see that elliptic boundary value problems induce Fredholm operators between the considered Sobolev spaces (as well as in the C 1 x is compact. Then (1.1.19) yields spaces above) when G z  ind A D ind R: ind A This is the Agranovich–Dynin formula which compares the indices of two elliptic boundary value problems for an elliptic operator A in terms of an elliptic operator R on the boundary. z and A in the process of reduction to Remark 1.1.20. By interchanging the role of A the boundary, i.e., when we reduce A to the boundary with respect to the reference z instead of (1.1.19) we obtain problem A,     1 0 APz AKz APz D  ; T Pz T Kz T Pz T Kz z Then T Kz is a parametrix of Tz K in the sense z is a parametrix of A. where Pz D .Pz K/ of Douglis–Nirenberg elliptic systems on the boundary. In particular, for the respective z D . .Tz K//1 . matrices of principal symbols, we have  .T K/

24

1 Boundary value problems with mixed and interface data

Theorem 1.1.18 and Remark 1.1.19 show the relevance of the reduction of a boundary value problem to the boundary. Of course, this construction is of interest also for the case when Tz is not elliptic with respect to A or when Tz has discontinuous coefficients with jumps along a submanifold Z of Y of codimension 1. In the latter 1 1 case we assume that Y D Y [ YC  fortwo C manifolds Y˙ with a common C  is represented by a pair of elliptic boundary boundary Z D Y \ YC , and Tz D TTC conditions over YC and Y , respectively. This is the situation of the so-called elliptic mixed problems, where the reduction of Tz to the boundary leads to a pair of pseudodifferential operators R˙ on Y˙ with smooth symbols up to Z from the corresponding z in the sides, namely, R˙ D T˙ K. In this case we cannot expect a parametrix of A above simple form. The reason is that there are no parametrices in terms of the operators R˙ without specific contributions from to the interior boundary Z. An answer in terms of ellipticity and parametrices can be given when we formulate (an then solve) corresponding transmission problems for R˙ with additional conditions along Z, cf. Section 1.2.5 below. This is similar to the problem to solve elliptic boundary value problems for RC in YC (or R in Y ) with respect to the boundary Z, now in the framework of pseudo-differential boundary value problems. Example 1.1.21. Consider the Laplacian in a bounded domain G in Rn with C 1 boundary Y , and let Y D Y [ YC be a decomposition in the above sense. Then the problem x C 1 .G/ 0 1 ˚ z D @ T A W C 1 .G/ x ! C 1 .Y / A TC ˚ C 1 .YC / for T u D ujY ; TC u D @ ujYC with @ being the differentation in the (outer) normal direction is called the Zaremba problem. More precisely, the problem consists of describing the solvability of u D f;

TC u D gC ;

T u D g ;

(1.1.20)

to characterise its behaviour with respect to a parametrix construction and to obtain regularity and asymptotics of solutions near Z D Y \ YC . In the theory of pseudo-differential boundary value problems it is essential whether or not the given elliptic pseudo-differential operator has the so-called transmission property at the boundary. In Chapter 3 below we shall return to this class of pseudodifferential operators and give the precise definition. For instance, if we look at the local situation, the half-space RnC 3 .y; t /, then a classical symbol (say, with constant coefficients and of integer order ) p.; / has the transmission property when D˛ fp.  j / .; /  .1/  j p.  j / .;  /g D 0

(1.1.21)

on the set f.; / W ¤ 0;  D 0g for all j 2 N and ˛ 2 N n1 (here p.  j / .; / is the homogeneous component of order   j of the symbol p.; /).

1.1 Elliptic boundary value problems

25

The pseudo-differential calculus of boundary value problems with the transmission property (in terms of an algebra with symbolic structure) goes back to Boutet de Monvel [15], see also the monographs of Rempel and Schulze [160] or Grubb [66]. This concerns the algebra aspect and the typical Green operators; a theory for general symbols (i.e., including the case without the transmission property) was earlier developed in a series of papers by Vishik and Eskin [221], [222] (see also the monograph of Eskin [44] and the references there). In Rempel and Schulze [155] the calculus of Vishik and Eskin was later on completed to an algebra (see also the monographs [186], [188] or the paper [196] of Schulze and Seiler). More remarks on the development of the pseudo-differential calculus of boundary value problems are given in Sections 10.6.1 and 10.6.2 below. Although this theory is well understood today, there is a considerable distance in terms of the complexity of phenomena between the theories for the case with or without the transmission property. Therefore, it is a first important question, whether the operators R˙ have the transmission property with respect to Z. The answer is that this is not the case, even for the simplest mixed elliptic problems such as the Zaremba problem. A role of our discussion of the explicit form of the Poisson kernels in the half-space is to see this directly by calculating the symbols of Tz K for an elliptic boundary condition Tz , where K is the vector of operators given by the Poisson kernels. Since the homogeneous principal symbols are invariant, it suffices to x n 3 .y; t /. The calcuconsider the operators in local coordinates in the half space R C n1 lations are valid pointwise for every . Thus it is adequate to look at elliptic Ay 2 R boundary value problems A D T with constant coefficients in the sense of (1.1.9), Tj u D Bj ujRn1 ; j D 1; : : : ; m. The inverse of the operator family (1.1.14) is a row matrix (1.1.15). Recall that the second component @ .K/./ defines an injective map x C/ @ .K/./ W C m ! S.R   z D A be another operator with a column of trace for each  ¤ 0. Now let A Tz z z operators Tj u D Bj ujRn1 ; j D 1; : : : ; m. Then, according to Definition 1.1.16, we consider the composition @ .Tz /./ @ .K/./, where x C/ ! Cm @ .Tz /./ W S.R

˚

is the boundary symbol of Tz , with the vector of orders m zj C 12 j D1;:::;m . In the systematic theory of pseudo-differential boundary value problems below we shall see that @ .Tz /./ @ .K/./ D @ .Tz K/./; where @ .Tz K/./ is an m  m matrix of homogeneous functions in  ¤ 0, where the j k-th entry is of order m zj  mk ; j; k; D 1; : : : ; m, which is just the homogeneous principal symbol of Rj k for R D .Rj k /j;kD1;:::;m ; R D Tz K. From the relation (1.1.16) we obtain Z X m m X M C l .; / 1 @ .Tzj /./u./ D dzj k ./ k  1 .D 1 ./c/l mC d 2 i L M .; / lD1 kD1

26

1 Boundary value problems with mixed and interface data

with coefficients dzj k D dzj k .Tz / analogously defined as dj k D dj k .T /. By Lemma 1.1.8 P 1 z it follows that @ .Tzj /./u./ D m ./c/k . This gives us kD1 dj k ./.D @ .R/./ D

m X



dzj k ./d kl ./

kD1

j;lD1;:::;m

:

(1.1.22)

In this way we obtain the matrix of homogeneous principal symbols of the pseudodifferential operator R D Tz K on the boundary, the result of reducing Tz to the boundary by means of A. Let  usnow consider some examples. First we want to reduce the Neumann problem z AD

z for the Laplace operator to the boundary by means of the Dirichlet problem  T  A D T . The boundary symbol of A is equal to 

jj2 C @2t r0



x C / ! W S.R

x C/ S.R ; ˚ C

cf. Example 1.1.6, and for the inverse .p./ k.// we have      jj2 C @2t  1 0 p./ k./ D : 0 1 r0 k./ is characterised by the properties r 0 k./ D 1; k./ W C ! ker.jj2 C @2t / 1 injective and k./ D  2  k./ for all  2 RC ;  ¤ 0. The explicit form of k./ is c ! e tjj c, c 2 C. Applying now the Neumann boundary conditions that are nothing other than r 0 @ t , we obtain in the above notation @ .Tz K/./ D jj:

(1.1.23)

Of course, we also could apply the formula (1.1.22) directly to obtain (1.1.23). We see that the transmission property is not satisfied for the symbol (1.1.23). More precisely, if Z is an .n  2/-dimensional hyperplane in Rn1 , i.e., Rn1 D Z  R; y D .z; xn1 /;  D .; n1 /, then the transmission property requires that 1 1 .j n1 j2 C jj2 / 2 C .j  n1 j2 C j  j2 / 2 vanishes for n1 ¤ 0;  D 0, cf. (1.1.21). The formula (1.1.23) shows that the reduction of the Neumann conditions to the boundary give rise to the square root of the symbol of minus the Laplacian on the boundary. As noted above it is interesting to study boundary value problems for the corresponding operators in YC or Y (or transmission problems on Y [ YC with respect to Z). Let us now pass to further interesting examples.

1.1.4 Examples In this section we shall explicitly calculate some reductions of boundary value problems for the Laplacian or the bi-Laplacian 2 to the boundary. Because of the invariance

1.1 Elliptic boundary value problems

27

of the principal symbols we may consider the operators in the half-space RnC . Let us start from 2 . In the splitting of variables x D .y; t / and covariables D .; / we have  . 2 /./ D 2 2 jj2 C jj4 C 4 ;

@ . 2 /./ D 2jj2 @2t C jj4 C @4t :

The boundary value problem represented by    2 T1 ; ; T D AD T T2 ˇ ˇ with T1 u D uˇRn1 ; T2 u D @ t uˇRn1 is elliptic. Other examples for elliptic operators  2   z z D ; Tz D T1 A z T Tz2 are as follows:

ˇ ˇ Example 1.1.22. Tz1 u D uˇRn1 , Tz2 u D @2t uˇRn1 . ˇ ˇ Example 1.1.23. Tz1 u D @ t uˇRn1 , Tz2 u D @2t uˇRn1 . ˇ ˇ Example 1.1.24. Tz1 u D uˇRn1 , Tz2 u D uˇRn1 . Let us verify that A is elliptic. The boundary symbol of 2 x C / ! S.R x C/ @ . 2 /./ W S.R is surjective. Here and in the sequel we fix  6D 0. We have a 2-dimensional kernel ker @ . 2 /./ D fc1 e tjj C c2 t e tjj W c1 ; c2 2 Cg: Moreover, the mapping @ .T1 /./ W ker @ . 2 /./ ! C is surjective with a 1-dimensional kernel fcte tjj W c 2 Cg; i.e., x C/  S.R @ . 2 /./ x W S.RC / ! ˚ @ .T1 /./ C



is surjective, and

@ .T2 /./ W fcte tjj W c 2 Cg ! C

is an isomorphism. By Remark 1.1.5 this implies altogether that 0 1 x C/ @ . 2 /./ S.R x @ A @ .A/./ D @ .T1 /./ W S.RC / ! ˚ @ .T2 /./ C2

28

1 Boundary value problems with mixed and interface data

is an isomorphism. In an analogous manner we can check that the boundary value problems in the Examples 1.1.22, 1.1.23, and 1.1.24 are elliptic. Let us now express the potential part of   @ .A/1 ./ D @ .P /./ @ .K/./ for  6D 0. We claim that @ .K/./c D c1 .e tjj C jjt e tjj /  c2 t e tjj , where c D .c1 ; c2 / 2 C 2 . In fact, from (1.1.16) we obtain    Z X 2  C M2l .; / it 1 1 0 c1 @ .K/./c D e d c2 l .  i jj/2 0 i 2 i L lD1 Z 1 c1  2ijjc1 C i c2 it D e d 2 i L .  ijj/2 D c1 .e tjj C jjt e tjj /  c2 t e tjj : We now express the symbols on Rn1 that appear as reductions to the boundary of the boundary conditions in Examples 1.1.22, 1.1.23, 1.1.24. To this end we specify the calculations of Sections 1.1.2, 1.1.3 to the given reference problem A and the problems in the examples. If  6D 0 is fixed, for the equation  . 2 /. ; / D 0 we obtain the roots 1C ./ D

2C ./ D ijj; 1 ./ D 2 ./ D i jj. We have M C .; / D .  i jj/2 ;

c0C D 1;

M0C .; / D 1;

c1C D 2i jj;

c2C D jj2 ;

M1C .; / D  2i jj;

and for the problem A we obtain  .B1 /.; / D 1; f .B1 /.; / D 1;

 .B2 /.; / D i ; f .B2 /.; / D i :

Here and in the sequel the operators Bj are associated with Tj via (1.1.2);   concerning the meaning of f .Bj /, see (1.1.10). Hence D D .dj k .//j;kD1;2 D 10 0i . The matrix of principal symbols of the operator Tz K is as follows. For Example 1.1.22   1 0 ; @ .Tz K/./ D jj2 2jj  1 0  since f .Bz1 /.; / D 1, f .Bz2 /.; / D .2ijj C jj2 /, D D jj 2 2ijj . For Example 1.1.23 we have  0 1  @ .Tz K/./ D jj 2 2jj ;   0 i because of f .Bz1 /.; / D i , f .Bz2 /.; / D .2i jj C jj2 /, D D jj 2 2ijj . Finally for Example 1.1.24 we have  1 0  @ .Tz K/./ D 2jj 2 2jj ;  1 0  since f .Bz1 /.; / D 1; f .Bz2 /.; / D 2.ijj  jj2 /, D D 2jj 2 2ijj .

1.1 Elliptic boundary value problems

29

Example 1.1.25. The following example refers to the Laplace operator in  D RnC D fx W xn > 0; x D .x1 ; x2 ; : : : ; xn /g. Set YC D fx W xn D 0; xn  1 > 0g, Y D fx W xn D 0; xn1 < 0g,Z Dfx W xn D xn1 D 0g. z , where Tz D TzC with TzC D rYC BzC ; Tz D rY Bz , for Let Az D

Tz T

BzC D

n2 X

Bz D

˛i Dxi C ˛Dxn1 C Dxn ;

iD1

n2 X

ˇi Dxi C ˇDxn1 C ı Dxn :

iD1

rY˙ are the restrictions to YC and Y , respectively. For convenience we assume that the coefficients are constant. Symbols under reduction to the boundary can be computed pointwise; sothe method also applies for variable coefficients. Let A D

T be an elliptic boundary value problem, where T is an extension from Tz to the whole boundary, i.e., r  T D Tz . Let .P K/ be a parametrix of A. Then the operator R defined by R WD

n2 nX

o ˛i Dxi C ˛Dxn1 C  Dxn K

iD1

on W D f.x1 ; : : : ; xn1 / W xn1 > "g  Rn , " > 0, where the coefficients ˛i ; i D 1; : : : ; n  2, and ˛;  are assumed to be extended from YC to W , has the principal symbol @ .R/./ D

n2 X

n2  X

˛i i C ˛% C i jj

iD1

ˇi i C ˇ% C i ıjj

1 ;

(1.1.24)

iD1

cf. the notation (1.1.22). Here  D .; %/, and  D .1 ; : : : ; n  2 / is the covariable to z D .x1 ; : : : ; xn2 /; % the covariable to xn1 . In fact, let be the covariablePof xn I then for the Laplace operator we have M .; / D  i jj. For BzC D n2 iD1 ˛i Dxi C ˛Dxn1 C Dxn we obtain C

 .BzC /. / D

n2 X

˛i i C ˛% C 

for D .; /;

iD1

Pn2

and f .BzC /./ D iD1 ˛i i C ˛% C i jj. For B D associated with T , cf. (1.1.2), we have  .B/. / D

n2 X

ˇi i C ˇ% C ı

Pn2 iD1

ˇi Dxi C ˇDxn1 C ıDxn ,

for D .; /;

iD1

P and f .B/./ D n2 iD1 ˇi i C ˇ% C i ıjj, cf. (1.1.10). Hence, from (1.1.22) we get (1.1.24). Pn2 For example, we obtain @ .R/./ D iD1 ˛i i C ˛% C i jj when T is the Dirichlet condition on Y .

30

1 Boundary value problems with mixed and interface data

1.2 Mixed and transmission problems The investigation of mixed elliptic boundary value problems belongs to the main content of the present book. We discuss some ideas of our approach which motivate the study of necessary tools in the following chapters. In Chapter 5 we present the calculus in detail.

1.2.1 Mixed problems in weighted edge spaces By a mixed boundary value problem for an elliptic differential operator A of order  we understand the problem to find solutions u of the equations Au D f T u D g on Y ;

in G;

(1.2.1)

TC u D gC on YC :

(1.2.2) 1

Here Y D @G is given as a union Y D Y [ YC of two C manifolds with common C 1 boundary Z D Y \ YC . The operators T and TC are assumed to define elliptic boundary conditions with respect to Y and YC , respectively, of the form T˙ D .T˙;l /lD1;:::;m ; T˙;l u D rY˙ B˙;l u, with differential operators B˙;l D P ˇ 1 coefficients in a neighbourhood of Y˙ and rY˙ being jˇ jm˙;l b˙;lˇ .x/ Dx with C the operator of restriction to Y˙ . For simplicity we assume that the differential operator A is scalar and of order  D 2m. In the context of ellipticity we may also consider systems of (Douglis–Nirenberg elliptic) operators. They can formally be treated in an analogous manner by our methods. The problem (1.2.1), (1.2.2) defines an operator x C 1 .G/ A ˚ x ! C 1 .Y ; C m / A D @ T A W C 1 .G/ TC ˚ C 1 .YC ; C m / 0

1

which extends to a continuous operator H s .G/ ˚ s m sm;l  1 2 .int Y / A W H .G/ ! ˚lD1 H ˚ smC;l  1 2 .int YC / ˚m H lD1

(1.2.3)

for s > maxfm˙;l C 12 W l D 1; : : : ; mg. It is then a question whether we can expect reasonable solvability properties in those spaces. It turns out that it is more adequate to pass to weighted Sobolev spaces (or weighted spaces of C 1 functions) with a controlled jump along Z. The precise definitions will be given below. To give a first impression we choose local coordinates .z; x/ Q in a neighbourhood n U of a point Z in R such that z D .x1 ; : : : ; xn2 / varies in an open set  Rn2

31

1.2 Mixed and transmission problems

and xQ WD .xn1 ; xn / in R2 , where xn > 0 corresponds to the domain G and xn1 ? 0 to int Y˙ , such that R2 represents the normal plane of Z in Rn and xn the normal direction to Y D @G in R2 and xn1 the normal direction to Z tangent to Y . The set U can be interpreted as a wedge U D   R2 with edge  and model cone R2 , where x 2C UC WD   R

x 2C WD f.xn1 ; xn / 2 R2 W xn  0g for R

x Let us pass to the corresponding stretched wedges is just the local model of U \ G. x C  S 1/ U WD   .R

and

1 x C  SC UC WD   .R /;

1 x 2 \ S 1 . In polar respectively, with S 1 being the unit circle in R2 and SC WD R C 1 2 coordinates .r; / 2 RC  S in R n f0g the operator A takes the form

A D r 

X j Cj˛j

 @ j aj˛ .z; r/ r .rDz /˛ @r

(1.2.4)

x C ; Diff .j Cj˛j/ .S 1 //. Here Diff  .  / denotes with coefficients aj˛ .z; r/ 2 C 1 . R the space of all differential operators of order  (with smooth coefficients) on the manifold in the parentheses. Operators of the form (1.2.4) will also be called edgedegenerate. Let us now formulate so-called weighted edge spaces in which A acts as a continuous operator. First there are the weighted cone Sobolev spaces H s; .RC  S 1 /

for s;  2 R: 1

This space for s 2 N is defined to be the set of all u.r; / 2 r  2 L2 .RC  S 1 / such that  j 1 r@r D u.r; / 2 r  2 L2 .RC  S 1 / for all D 2 Diff sj .S 1 / and j D 0; : : : ; s (the space L2 .RC  S 1 / is defined in terms of the measure drd). For arbitrary s 2 R the spaces H s; .RC  S 1 / may be defined by duality and interpolation (an alternative definition will be given in Definition 2.4.5 below). In this book by a cut-off function on the half-axis we understand any function x C / which is equal to 1 in a neighbourhood of zero. We now define ! 2 C01 .R K s; .RC  S 1 / WD f!u C .1  !/v W u 2 H s; .RC  S 1 /; v 2 H s .R2 /g s .R2 n f0g/ for any cut-off function ! D !.r/. Because of H s; .RC  S 1 /  Hloc s; the specific choice of ! is unessential. We can easily endow H .RC  S 1 / and K s; .RC  S 1 / with the structure of a Hilbert space for every s;  2 R, such that 1

H 0;0 .RC  S 1 / D K 0;0 .RC  S 1 / D r  2 L2 .RC  S 1 /:

32

1 Boundary value problems with mixed and interface data

In K s; .RC  S 1 / we consider a group of isomorphisms  W K s; .RC  S 1 / ! K s; .RC  S 1 /; given by . u/.r; / WD u.r; /;

 2 RC :

Note that f g2RC is strongly continuous (which means  u 2 C.RC ; K s; .RC  S 1 // for every u 2 K s; .RC  S 1 /). Definition 1.2.1. Let E be a Hilbert space which is equipped with a strongly continuous group of isomorphisms  W E ! E;  2 RC : Then W s .Rq ; E/ (the so-called abstract edge space of smoothness s 2 R, with parameter space E) is defined to be the completion of S.Rq ; E/ (the Schwartz space of E-valued functions on Rq ) with respect to the norm Z 1=2 2s 1 2 hi kh i u./k O I Ed here hi WD .1 C jj2 /1=2 , and uO is the Fourier transform of u in Rq . We will return later on to more details on abstract edge spaces, see Sections 2.2.2, 2.4.2, and 7.1.2 below. In particular, for every open set  Rq we have ‘comp’ and ‘loc’ versions of these spaces, namely, s s .; E/ and Wloc .; E/; Wcomp

respectively. These constructions will be applied to E D K s; .RC  S 1 / or E D ˚ ˇ 1 s; ˇ K .RC  SC / WD u R .intS 1 / W u 2 K s; .RC  S 1 / . C C We also have the spaces H s; .RC / D fu 2 r  L2 .RC / W .r@r/j u 2 r  L2 .RC /; j D 0; : : : ; sg (first for s 2 N and then again defined by duality and interpolation), and K s; .RC / D f!u C .1  !/v W u 2 H s; .RC /; v 2 H s .RC /g: 1

The group action in K s; .RC / will we defined by . u/.r/ D  2 u.r/;  2 RC . Now Definition 1.2.1 gives us weighted edge spaces W s; .Rq  RC  S 1 / WD W s .Rq ; K s; .RC  S 1 //; and 1 1 / WD W s .Rq ; K s; .RC  SC // W s; .Rq  RC  SC

˚ ˇ D uˇRq R

1 C SC

Wu2W

s;



.R  RC  S / q

1

(1.2.5)

33

1.2 Mixed and transmission problems

as well as the corresponding ‘comp’ and ‘loc’ versions with respect to z in an open set  Rq (in that case we also write subscripts ‘comp.z/’ and ‘loc.z/’, respectively). Moreover, we have the spaces W s; .Rq  RC / WD W s .Rq ; K s; .RC //;

(1.2.6)

1 etc. Let us write SC D f 2 R W 0    g. Then the operators of restriction

u.z; r; / ! u.z; r; 0/ or

u.z; r; / ! u.z; r; /

induce continuous operators 1

1

1 W s; .Rq  RC  SC / ! W s 2 ; 2 .Rq  RC /

for every real s > 12 and  2 R. In addition a differential operator A of the kind (1.2.4) (whose coefficients aj˛ are independent of r for large r) induces continuous operators s; s; 1 1 A W Wcomp.z/ .  RC  SC / ! Wcomp.z/ .  RC  SC /

for all s 2 R (and the same for the spaces with ‘loc.z/’). These observations allow us to replace operators (1.2.3) connected with (1.2.1) and (1.2.2) by operators in global weighted edge spaces. Instead of G we consider the set x as a manifold with edge Z and boundary, which means that locally near points X WD G of Z we interpret X as a wedge 1

/ (1.2.7)   .SC 1

x C  S 1 /=.f0g  S 1 / Š R x 2 and edge  Rn2 . / WD .R with model cone .SC C C C x 2 n f0g makes it reasonable to consider the stretched The use of polar coordinates in R C wedge 1 x C  SC / (1.2.8)   .R

instead of (1.2.7). This construction is invariant in a natural way under changes of local coordinates on Z. In this way we can define the global stretched space X associated with X, locally modelled on (1.2.8), such that X n Z locally near Z corresponds to 1   .RC  SC / for an open set  Rn2 . For purposes below we set Xreg D X n Z;

Xsing D X n Xreg I

1 then Xsing locally corresponds to   .f0g  SC /. Also the weighted edge spaces can be globalised on X to spaces

W s; .X/; s; 1 that are locally near Xsing given by Wloc.z/ .  RC  SC /, and have the property s .G/ 'W s; .X/ D 'Hloc

34

1 Boundary value problems with mixed and interface data

for every ' 2 C01 .G/. Similarly, we have the spaces W s; .Y˙ / which are locally near s; Z characterised by Wloc.y/ .  RC / (with RC being the normal to Z tangent to the s boundary) and W s; .Y˙ / D Hloc .int Y˙ / for every 2 C01 .int Y˙ /. Now the operator A, first defined on C01 .G/, extends to continuous operators W s; .X/ ˚ A W W s; .X/ ! W r ;% .Y ; C m / ˚ W rC ;%C .YC ; C m /

(1.2.9)

for all s;  2 R, s > maxfm˙;l C 12 W l D 1; : : : ; mg, where r˙ D .r˙;l /lD1;:::;m WD .s  m˙;l  12 /lD1;:::;m ; %˙ D .%˙;l /lD1;:::;m WD .  m˙;l  12 /lD1;:::;m , L r˙;l ;%˙;l .Y˙ /: W r˙ ;%˙ .Y˙ ; C m / WD m lD1 W Remark 1.2.2. Our weighted edge spaces occurring in (1.2.9) or their local versions (1.2.5), (1.2.6), can be regarded as a special choice among continuously parametrised families with parameter g 2 R. Setting 1 1 K s;Ig .RC  SC / WD hrig K s; .RC  SC /;

K s;Ig .RC / WD hrig K s; .RC /; 1

endowed with the group actions u.r; / ! 1Cg u.r; / and v.r/ !  2 Cg v.r/, respectively, we obtain associated spaces 1 W s .Rq ; K s; Ig .RC  SC //; W s .Rq ; K s;Ig .RC //

and corresponding global versions over X and Y˙ , respectively. Denoting the latter ones by W s;Ig .X/ and W s; Ig .Y˙ /, instead of (1.2.9) we then have continuous operators W s;Ig .X/ ˚ A W W s; Ig .X/ ! W r ;% Ig .Y ; C m / ˚ rC ;%C Ig W .YC ; C m / for all s; ; g 2 R, with s sufficiently large as before. For convenience we will formulate the main part of our calculus for the case g D 0. However, we could take any other g 2 R. Spaces of a particularly natural behaviour are generated when we set g WD s  for the spaces over X and g˙;l WD r˙;l  %˙;l in the components over Y . We will return to this aspect later on in Section 7.1.2 (see also Section 10.1.3).

1.2.2 Additional conditions at the interface Solutions of the problem (1.2.1), (1.2.2) with ellipticity of the boundary conditions T˙ over Y˙ , cf. Definition 1.1.4, will be constructed in terms of parametrices of the operator

1.2 Mixed and transmission problems

35

(1.2.9). However, this operator will not be Fredholm in general. The discontinuity of the boundary conditions over Z gives rise to the interpretation of the interface Z as a kind of interior boundary on Y . Therefore, we should impose once again additional conditions over Z. The idea of this construction is as follows: We consider the operator A in the local representation (1.2.4), i.e., X

A D r 

j Cj˛j

 @ j aj˛ .z; r/ r .rDz /˛ ; @r

(1.2.10)

 D 2m. The differential operators B˙ D .B˙;l /lD1;:::;m in the conditions T˙ D rY˙ B˙ , on Y˙ can be represented in a similar form, namely, B˙;l D r m˙;l

X kCjˇ jm˙;l

 @ k b˙;l;kˇ .z; r/ r .rDz /ˇ ; @r

(1.2.11)

l D 1; : : : ; m. Now with A we associate the family of continuous operators X

^ .A/.z; / WD r 

j Cj˛j

 @ j aj˛ .z; 0/ r .r/˛ ; @r

1 1 ^ .A/.z; / W K s; .RC  SC / ! K s; .RC  SC /;

(1.2.12)

.z; / 2 T  Z n 0. The same can be done with the operators B˙;l combined with the restriction to Y˙ . We then obtain ^ .T˙;l /.z; /u.r/ D r m˙;l

X kCjˇ jm˙;l

 ˇ @ k b˙;l;kˇ .z; 0/ r .r/ˇ u.r; /ˇ D0 ; @r D

where  D 0 corresponds to the restriction to YC and  D  to the restriction to Y . This gives us continuous operators 1

1

1 ^ .T˙;l /.z; / W K s; .RC  SC / ! K sm˙;l  2 ;m˙;l  2 .R˙ /;

.z; / 2 T  Z n 0, s > maxfm˙;l C

1 2

W l D 1; : : : ; mg. Here

s K s; .R / WD fu 2 Hloc .R / W u.r/ 2 K s; .RC /g:

If we set 1 ..1/ u/.r; / WD u.r; / for  2 RC ; u 2 K s; .RC  SC /;

and 1

..0/ v/.r/ WD  2 v.r/ for  2 RC ; v 2 K s; .R˙ /;

(1.2.13)

36

1 Boundary value problems with mixed and interface data

we obtain the ‘twisted homogeneities’ ^ .A/.z; / D  .1/  .A/.z; /..1/ /1 ;

 2 RC ;

(1.2.14)

and 1

^ .T˙;l /.z; / D m˙;l C 2 .0/ ^ .T˙;l /.z; /..1/ /1 ;

 2 RC :

(1.2.15)

Let us set ^ .T˙ /.z; / WD t .^ .T˙;l /.z; //lD1;:::;m and

1 ^ .A/ ^ .A/.z; / WD @ ^ .T / A .z; /: ^ .TC / 0

We then have a family of continuous operators 1 / K s;  .RC  SC ˚ 1 1 ^ .A/.z; / W K s; .RC  SC / ! K r ;% .R ; C m / DW K s; .RC  SC / ˚ K rC ;%C .RC ; C m / (1.2.16) L r˙;l ;%˙;l for K r˙ ;%˙ .R˙ ; C m / WD m K .R /, cf. also the abbreviations in connec˙ lD1 tion with the operator (1.2.9). We call ^ .A/ the homogeneous principal edge symbol of the operator A. It may be regarded as an analogue of the homogeneous principal boundary symbol @ .A/ which was discussed in Section 1.1 before. It would be reasonable now to have an analogue of Proposition 1.1.2 for ^ .A/. Such a result holds, indeed, however with some essential modifications.

Theorem 1.2.3. Let A be elliptic with respect to the principal symbolic levels . .A/; @ .T /; @ .TC //: Then for every z 2 Z there exists a discrete countable set D.z/  R such that the operators (1.2.16) are Fredholm for all  2 R n D.z/, all  6D 0 and all s > ˚ max   12 ; m˙;1 C 12 ; : : : ; m˙;m C 12 . The dimensions of ker ^ .A/.z; / and coker ^ .A/.z; / are independent of s and , however they may depend on  . The proof of Theorem 1.2.3 employs tools that we develop later on. The idea is illustrated in the proof of Theorem 5.1.4 and the subsequent Remark 5.1.5. The case of general boundary value problems can be treated in an analogous manner. Let us set 1 1 1 x C ; C 1 .SC / WD f!u C .1  !/v W u 2 K 1; .RC  SC /; v 2 S.R //g S  .RC  SC

1.2 Mixed and transmission problems

37

and, analogously, x C /g; S  .RC / WD f!u C .1  !/v W u 2 K 1; .RC /; v 2 S.R  1  S .R / WD fu 2 C .R / W u.r/ 2 S .RC /g: L %˙;l Moreover, let S %˙ .R˙ ; C m / WD m .R˙ /. lD1 S 1 Remark 1.2.4. As we shall see below, we have ker ^ .A/.z; /  S  .RC  SC / for every  2 R n D.z/,  6D 0, and there is a finite-dimensional subspace V 1 S  .RC  SC / ˚ 1 V  S % .R ; C m / DW S  .RC  SC / ˚ S %C .RC ; C m / 1 with im ^ .A/.z; / \ V D f0g and im ^ .A/.z; / C V D K s; .RC  SC /; this holds for all s as in Theorem 1.2.3. Because of the homogeneities (1.2.14) and (1.2.15) the dimensions of ker ^ .A/ and coker ^ .A/ only depend on =jj.

Remark 1.2.5. The operators (1.2.16) belong to a pseudo-differential algebra of bound1 ary value problems on the infinite (stretched) cone RC  SC , for every fixed .z; / 2  T Z n 0. The operators in that algebra have a principal symbolic structure consisting of a hierarchy of scalar and operator-valued components  D . ; @;˙ ; c ; E ; E0 /

(1.2.17)

in the meaning of interior, boundary (on R˙ ), conormal and exit symbols, respectively. We postpone the discussion of details to Section 6.2.2 below. Let us only note that the symbols (1.2.17) determine ellipticity and the Fredholm property of the operators (1.2.16). In particular, the principal conormal symbol which is independent of  represents 1 a family of boundary value problems on SC , the base of the cone, namely, c ^ .A/.z; w/ W H

s

1 .int SC /

1 / H s .int SC ˚ ! ; Cm ˚ Cm

(1.2.18)

z 2 Z; w 2 C. We shall illustrate this by an example in Section 1.2.3 below. Under the assumptions of Theorem 1.2.3 the operators (1.2.18) form a holomorphic family of Fredholm operators for every fixed z, and the set D.z/ is determined by the real parts of those w 2 C where (1.2.18) is not a bijection. Let us now recall from Definition 1.1.4 and Proposition 1.1.2 that the role of the boundary conditions for the operator A is to fill up the family of Fredholm operators @ .A/.y; / W H s .RC / ! H s .RC /

(1.2.19)

38

1 Boundary value problems with mixed and interface data

to a family of isomorphisms  H s .RC / @ .A/ s ˚ @ .A/.y; / D .y; / W H .RC / ! : @ .T / Cm 

(1.2.20)

In a similar manner we now proceed for the Fredholm family (1.2.16) under the condition that there is a weight  2 R n D.z/ which is independent of z 2 Z. The first step of the construction is to choose an N 2 N and a smooth family of linear maps 1 / ^ .K/.z; / W C N ! S  .RC  SC

such that 1 s;   K .RC  SC / 1 ˚ ^ .A/ ^ .K//.z;  W ! K s; .RC  SC / N C

(1.2.21)

is surjective for all .z; / 2 T  Z n 0. This is always possible. It suffices to first look at .z; / in the unit cosphere bundle S  Z D f.z; / 2 T  Z W jj D 1g. In this case (1.2.16) is a surjective family of Fredholm operators parametrised by the compact topological space S  Z. The general background of this construction is the following result. Let Vect.  / denote the set of all smooth complex vector bundles on the C 1 manifold in the parentheses (on a topological space without C 1 structure Vect.  / denotes the set of continuous z , by F .H; H z / we denote the complex vector bundles). Given Hilbert spaces H; H z z/ space of all Fredholm operators H ! H in the norm topology induced by L.H; H z (the space of all linear continuous operators H ! H /. z/ a Proposition 1.2.6. Let M be a compact topological space and A W M ! F .H; H N z such !H continuous map. Then there exists an N 2 N and a linear map K W C that   H z A K W ˚ ! H (1.2.22) C N is surjective for every m 2 M . Moreover, there exists a Jz 2 Vect.M / and a continuous H

family of linear maps .T .m/ Q.m// W ˚ ! Jzm (with Jzm being the fibre of Jz over C N

m 2 M ) such that

z  H H A.m/ K W ˚ ! ˚ ; T .m/ Q.m/ C N Jzm



is a family of isomorphisms.

m 2 M;

(1.2.23)

1.2 Mixed and transmission problems

39

We shall return in Section 3.3.4 below to more information around this type of considerations. In the present case we have M D S  Z, and (1.2.21) plays the role of (1.2.22). We now impose an additional condition, namely, that (if necessary, for a sufficiently large choice of N ) the bundle Jz is the pull back of some J 2 Vect.Z/ under the canonical projection S  Z ! Z. Instead of the operator functions T and Q in (1.2.23) we write 1 s;  K .RC  SC /  ˚ ! Jz ^ .T / ^ .Q//.z;  W C N for the choice of a corresponding second row of a smooth family of isomorphisms    .A/ ^ .K/ ^ .A/.z; / WD ^ .z; /; (1.2.24) ^ .T / ^ .Q/ 1 1 / K s; .RC  SC / K s; .RC  SC ˚ ˚ ^ .A/.z; / W ! ; Jz C N

(1.2.25)

.z; / 2 S  Z. Remark 1.2.7. By virtue of Remark 1.2.4 the map ^ .K/ can (and will) be chosen in 1 such a way that it takes values in the space S  .RC SC /. Moreover, the second row in (1.2.24) will be chosen as a family of projections to ker.^ .A/.z; / ^ .K/.z; // composed with an isomorphism from that kernel to Jz . This gives us integral kernels 1 /. of ^ .T /.z; / in the space S  .RC  SC In order to interpret (1.2.24) as the homogeneous principal edge symbol of a corresponding operator 

A K A WD T Q



W s; .X/ W s; .X/ ˚ ˚ W ! s1 s1 N H .Z; J / H .Z; C /

(1.2.26)

for W s; .X/ WD W s; .X/ ˚ W r ;% .Y ; C m / ˚ W rC ;%C .YC ; C m / we extend the entries of (1.2.24) from .z; / 2 S  Z by homogeneity to T  Z n 0, by setting     ^ .A/.z; / D diagfjj ; diag jjmC;l lD1;:::;m ; diag jjm ;l lD1;:::;m ; jj g ˚ .1/  .0/   .0/ 

1 1  diag j j ; jj 2 diag j j ; jj 2 diag j j ; jj ; lD1;:::;m lD1;:::;m   

1 ˚ .1/  ^ .A/ z; ; jj : (1.2.27) diag j j jj

40

1 Boundary value problems with mixed and interface data

Here diag.˛l /lD1;:::;m means the m  m diagonal matrix with entries ˛l . Let ./ 2 C 1 .Rn2 / be an excision function (i.e., ./ D 0 for jj < c0 , ./ D 1 for jj > c1 for certain 0 < c0 < c1 ). Then ./^ .K/.z; /, ./^ .T /.z; / and ./^ .Q/.z; / are (local) operator-valued symbols, cf. Definition 2.2.3 and Example 2.2.8 (ii) below; then a global construction with a partition of unity gives us operators K, T and Q globally with respect to Z. More details of the constructions of A will be discussed below. Summing up, the operator A has a hierarchy of principal symbols  .A/ WD . .A/; @ .A/; ^ .A//;

(1.2.28)

where  .A/ WD  .A/ and @ .A/ WD @ .A/ D .@;C .A/; @; .A// are the principal symbols of the upper left corner A, and ^ .A/ is given by (1.2.27). The operator A is elliptic in the sense that the three components of (1.2.28) are elliptic, i.e.,  .A/ 6D 0 on T  X n 0; @;˙ .A/ bijective on T  Y˙ n 0; ^ .A/ bijective on T  Z n 0;

(1.2.29) (1.2.30) (1.2.31)

@;˙ .A/ WD @;˙ .A/, and the bijectivities refer to the values of the operator functions between the corresponding spaces, cf. Definition 1.1.4 and (1.2.20). Theorem 1.2.8. Let A be an edge boundary value problem for the operator A with mixed boundary operators T˙ on Y˙ and edge conditions T , K and Q on Z (of trace and potential type). Then the ellipticity of A (in the sense of (1.2.29), (1.2.30), and (1.2.31)) entails the Fredholm property of the operator (1.2.26) for every real s > max.; d/  12 for d WD maxfm˙;l C 1 W l D 1; : : : ; N g. This result will be a consequence of our calculus of mixed elliptic problems. It can be proved that the ellipticity of A is also necessary for the Fredholm property of (1.2.26). Remark 1.2.9. The number N of potential conditions on the interface Z and the bundle J (the fibre dimension of which determines the number of trace conditions on Z) depend on the chosen weight  2 R. Remark 1.2.10. The proof of Theorem 1.2.8 will be based on the construction of a parametrix A.1/ of A within a corresponding pseudo-differential calculus of edge boundary value problems, and A.1/ will belong to  .A/1 , the triple of inverses of the components of  .A/. Remark 1.2.11. The calculus of edge boundary value problems will contain a specific class of smoothing operators who are compact in the chosen scales of spaces and map to subspaces with asymptotics in the distance variable r 2 RC to Z for r ! 0. If A is elliptic and A.1/ a parametrix we then obtain A.1/ A D I  Cl , AA.1/ D I  Cr ,

1.2 Mixed and transmission problems

41

with I being the identity operators in the respective spaces and smoothing operators Cl and Cr . This will imply elliptic regularity of solutions in our spaces and subspaces with asymptotics. Note that in general the extension of the Fredholm family (1.2.16) to a family of isomorphisms (1.2.25) by adding extra entries requires non-trivial potential parts, cf. Remark 1.2.9, although at the moment we are speaking of differential mixed problems. In the similar case of boundary value problems for a differential operator A we know that the boundary symbol (1.2.19) is surjective. Then, in order to pass to an isomorphism (1.2.20) only trace operators are necessary (when a topological condition of the abovementioned kind A holds). If A D T is a boundary value problem for a differential operator A on a compact C 1 manifold X with boundary Y and T a trace operator which consists of a composition of a differential operator B near Y with the restriction to Y , then the parametrix P D .P K/ contains a potential K. In general, if A is a pseudo-differential operator on X with the transmission property at T (plus a so-called Green operator), cf. Chapter 3 below, to associate with A a Fredholm operator we need trace and potential entries at the same time. In other words, the context is to consider 2  2 block matrices  H s .X; E/ H s .X; F / A K ˚ ˚ AD W ! T Q H s .Y; G/ H s .Y; J / 

(1.2.32)

for certain E; F 2 Vect.X / and G; J 2 Vect.Y / and s 2 R large enough. The unified smoothness in the Sobolev spaces on Y is assumed without loss of generality. In general it suffices to apply a reduction of orders, cf. Theorem 1.2.19 below. In Chapter 3 below we will give a concise introduction into the main ideas on pseudo-differential boundary value problems of the form (1.2.32) with the transmission property. The calculus will refer to classical operators of integer order  2 Z. Similarly as in the special cases considered so far such operators have a homogeneous principal symbol  .A/ W X E ! X F (1.2.33) defined  .A/ WD  .A/, with the canonical projection X W T  X n 0 ! X (of the cotangent bundle of X minus the zero section) and the associated bundle pull back X (the reader who is not at once familiar with this may imagine trivial bundles X  C e and X  C f , respectively; then (1.2.33) is nothing other than an f  e matrix of C 1 functions on T  X n 0, homogeneous of order  in the covariables 6D 0). In addition, similarly as (1.2.20), we have a homogeneous principal boundary symbol which is now of the form 1 1 0 0 0 0 E ˝ H s .RC / F ˝ H s .RC / A ! Y @ A ˚ ˚ @ .A/ W Y @ (1.2.34) G J with ‘prime’ denoting the restriction to the boundary and Y W T  Y n 0 ! Y being the canonical projection.

42

1 Boundary value problems with mixed and interface data

Let us set  .A/ WD . .A/; @ .A//: Definition 1.2.12. The operator (1.2.32) is called elliptic if both (1.2.33) and (1.2.34) are isomorphisms (for sufficiently large s 2 R in (1.2.34)). If (1.2.34) is an isomorphism we also speak about the Shapiro–Lopatinskij condition for the operators T , K, Q with respect to A in the block matrix (1.2.32). This is equivalent to the Fredholm property of the operator (1.2.32) for sufficiently large s. More precise information will be given in Chapter 3 below. The set of all pseudo-differential boundary value problems (1.2.32) of order  2 Z will be denoted by B  .X /: The bundles E; F and G; J depend T on the operators and are known in each individual situation. We set B 1 .X / WD  B  .X /. z 2 B Q .X / implies AA z 2 B CQ .X / (if the Theorem 1.2.13. A 2 B  .X / and A z D  .A/ .A/, z @ .AA/ z D bundles in the middle fit together), and we have  .AA/ z @ .A/@ .A/. Theorem 1.2.14. Let A 2 B  .X / be elliptic. Then there is a parametrix P 2 B  .X/ in the sense that I  P A; I  AP 2 B 1 .X / (with I denoting the identity operators in the respective spaces), and we have  .P / D . .A//1 , @ .P / D .@ .A//1 . A proof of Theorems 1.2.13 and 1.2.14 will be given in Chapter 3 below (cf. Theorems 3.2.22 and 3.3.7 (iii), respectively). t Remark 1.2.15. We saw that boundary value problems S .A T / for a differential operator A can be embedded into a block matrix ‘algebra’  B .X / of operators (with algebraic operations when the bundle data are compatible), closed under parametrix construction of elliptic elements. A similar construction can be asked for the operators (1.2.32), cf. Remark 1.2.10. The general answer will be given in terms of a so-called edge algebra, cf. Chapter 7 below.

1.2.3 Examples We now illustrate the general results of the preceding sections by a number of concrete mixed problems. The most prominent example is the Zaremba problem, where the operator A is the Laplacian and TC the Neumann, T the Dirichlet condition. We assume here that the interface Z is smooth. The more general case of conical singularities on Z will be treated in Chapter 8 below.

1.2 Mixed and transmission problems

43

The Zaremba problem represents continuous operators W s2; 2 .X/ ˚ 1 1 A D @ T A W W s; .X/ ! W s 2 ; 2 .Y / ˚ TC 3 3 W s 2 ; 2 .YC / 0

1

(1.2.35)

for all s;  2 R, s > 32 . Instead of A we also write A. / if we realise the operator with respect to the weight  . Let us now specify the principal symbolic structure of the operator (1.2.35). First we have the homogeneous principal interior symbol  .A/. / D j j2 , 6D 0, which is the standard principal symbol of the Laplace operator. To express the pair of homogeneous boundary symbols we choose a collar neighbourhood Š Y  Œ0; 1/ of Y in X with the variables x D .y; xn /, xn normal to the boundary, and covariables D .; n /. We then have  @;˙

 H s2 .RC / s ˚ ./ W H .RC / ! ; T˙ C

 6D 0, where @ . /./ D jj2 C @2xn and ˇ @; .T /u D uˇxn D0 ;

 ˇ @;C .TC /u D @xn u ˇxn D0 :

We have @ . /./ D 2  @ . /./1 1

for  u.xn / D  2 u.xn /,  2 RC , and 1

@; .T /./ D  2 @; .T /./1 ; 3

@;C .TC /./ D  2 @;C .TC /./1 :

(1.2.36)

Note that the fact that the boundary symbols @;˙ .T˙ / do not depend on  does not contradict (1.2.36). For the homogeneous principal edge symbol we use the fact, cf. [53], that the Laplace operator in Rn can be reformulated near Z in the form D

@2 @2 C C Z C L; 2 @xn2 @xn1

where xn1 is normal to Z and tangent to Y , and Z is the Laplacian on Z (belonging to the metric on Z induced by the Euclidean metric) and L a first order differential operator.

44

1 Boundary value problems with mixed and interface data

The homogeneous principal edge symbol of the Laplace operator in polar coordi1 x 2 n f0g to Z has the nates .r; / 2 RC  SC in the .xn1 ; xn /-normal half-plane R C form   @ 2 @2 2 2 ; C  r jj ^ . /./ D r 2 r @r @ 2  6D 0, and defines a family of continuous operators 1 1 ^ . /./ W K s; .RC  SC / ! K s2; 2 .RC  SC /;

with the homogeneity ^ . /./ D 2 .1/ ^ . /./..1/ /1 ;  2 RC . The edge symbols ˇ ^ .T /u WD uˇD0 ;

^ .TC / WD r 1

@ ˇˇ u @ D

define continuous operators 1

1

3

3

1 / ! K s 2 ; 2 .R /; ^ .T /./ W K s; .RC  SC 1 ^ .TC /./ W K s; .RC  SC / ! K s 2 ; 2 .RC /

for s >

3 2

and have the homogeneities 1

^ .T /./ D  2 .0/ ^ .T /./..1/ /1 ; 3

^ .TC /./ D  2 .0/ ^ .TC /./..1/ /1 ;  2 RC , although there is no dependence on the covariables . The operators 1 / K s2; 2 .RC  SC ˚ ^ . / 1 1 1 ^ .A/./ WD @ ^ .T / A ./ W K s; .RC  SC / ! K s 2 ; 2 .R / (1.2.37) ^ .TC / ˚ 3 3 K s 2 ; 2 .RC /

0

1

belong to a pseudo-differential algebra of boundary value problems on the infinite 1 (stretched) cone RC  SC , for every  6D 0, cf. Remark 1.2.5. The conormal symbol of the principal edge symbol has the form 1 0 1 / c ^ . / H s2 .int SC s 1 ˚ ; c ^ .A/.w/ D @ c ^ .T / A .w/ W H .int SC / ! C˚C c ^ .TC / ˇ @2 @ ˇ where c ^ . /.w/ D w 2 C @ 2 and c ^ .T /u D ujD0 , c ^ .TC /u D @ u D .

1.2 Mixed and transmission problems

45

Theorem 1.2.16. The operators (1.2.37) for  6D 0 form a family of Fredholm operators for all s > 32 and  62 Z C 12 . The dimensions of kernels and cokernels are independent of s and . In order to fill up the Fredholm operators (1.2.37) to a family of isomorphisms of the kind (1.2.25) we should know the index.   Theorem 1.2.17. For every k 2 Z and  2 12  k; 32  k we have ind ^ .A/./ D k for every  6D 0: A proof of Theorems 1.2.16 and 1.2.17 will be given in Chapter 5 below (cf. Theorem 5.1.4 and Corollary 5.3.3, respectively).   Theorem 1.2.18. For every  2 12  k; 32  k , k 2 Z, the operator W s2; 2 .X/ ˚ s2;2 s; s 1 ; 1 2 2 .Y / ; A. / W W .X/ ! W .X/ WD W ˚ 3 3 W s 2 ; 2 .YC / given by (1.2.35) can be completed by additional interface conditions T . /, K. / and Q./ to an operator 

 A. / K. / A./ WD W T . / Q. /

W s; .X/ W s2; 2 .X/ ˚ ˚ ! ; H s .Z; C d./ / H s2 .Z; C e./ /

e./  d./ D k, which is Fredholm for every s > 32 . This result will be proved in Chapter 5 below, together with other theorems of that type, e.g., for problems with jumping oblique derivatives.

1.2.4 Reduction of orders and reduction to the boundary It is often convenient to unify the orders by applying order reducing operators. Since our boundary value problems and their parametrices are expected to represent elements of suitable pseudo-differential algebras, also the order reductions should belong to the algebras (as elliptic elements). Their construction is not always trivial; however, the case of pseudo-differential operators on a closed compact C 1 manifold Y is particularly simple. Let L .Y / denote the space of pseudo-differential operators of order  on Y (in this notation Y is not necessarily compact), and let L cl .Y / be the subspace of classical operators. The local amplitude functions are of the Hörmander classes S  and Scl ,

46

1 Boundary value problems with mixed and interface data

respectively. Concerning generalities, cf. Chapter 2 below. If Y is compact, every A 2 L .Y / induces continuous operators. A W H s .Y / ! H s   .Y /

(1.2.38)

for all s 2 R. A standard result is now the following theorem: Theorem 1.2.19. For every  2 R there exists an A 2 L cl .Y / such that (1.2.38) is an isomorphism for all s 2 R, and we have A1 2 L .Y /. cl We can choose A as an operator whose homogeneous principal symbol of order  is equal to jj (with  being the covariable in T  Y ). Every A 2 L cl .Y / as in Theorem 1.2.19 is called a reduction of orders (in the space of classical pseudo-differential operators on Y ). Recall that we used such operators in the above description of Douglis–Nirenberg elliptic systems. Next we consider a compact C 1 manifold X with C 1 boundary Y (for instance, x for a bounded domain G  Rn with smooth boundary). Choose a compact X DG 1 C manifold Xz with dim X D dim Xz and X  Xz . In the following we simply set Xz D 2X which is the double of X , obtained by gluing together two copies X˙ of X along the common boundary Y ; we then identify X with XC . Define the extension operators by zero ( ( u on int X , 0 on X , eC u D e v D z 0 on X n int X , v on Xz n X , where u and v are distributions on int X and Xz n X , respectively. Moreover, let r ˙ be the operators of restrictions to int X and Xz n X , respectively, acting on distributions z Then for every pseudo-differential operator Az on Xz we can form on X. r C Az eC W C01 .int X / ! C 1 .int X / as well as the other possible combinations r C Az e , r  Az eC , r  Az e , for example r C Az e W C01 .int X / ! C 1 .int XC /. z denotes the standard Sobolev space of smoothness s 2 R on Xz we have If H s .X/ the spaces H s .int X / D H s .int XC / WD r C H s .Xz /; H0s .X / WD fu 2 H s .Xz / W supp u X g; where H s .int XC / Š H s .Xz /=H0s .X / is equipped with the quotient topology. Analogously, we have the spaces H s .int X / and H0s .XC /, respectively. The operators eC ; e may be interpreted as maps e˙ W H s .int X˙ / ! D 0 .Xz / for s >  12 . For every z we thus obtain continuous maps Az 2 L .X/ s s z C W H s .int XC / ! Hloc z  W H s .int X / ! Hloc .int XC /; r  Ae .int X / r C Ae

z ˙ W H s .int X˙ / ! C 1 .int X / for s >  1 . as well as r  Ae 2

1.2 Mixed and transmission problems

47

z 2 L z Theorem 1.2.20. For every  2 R there exist elements R cl .X / which are order ˙ reductions on Xz in the sense of Theorem 1.2.19, such that the operators  z eC W H s .int XC / ! H s .int XC /; RC WD r C R C   z  R WD r R e W H s .int X / ! H s .int X /

define isomorphisms for every real s > maxf 12 ;  12 C g. A proof of Theorem 1.2.20 will be given in Chapter 4 below. We will apply this result for Y and Y˙ , instead of Xz and X˙ , respectively, but use, for convenience, the same notation for the operators themselves. We now consider the operator A, given by (1.2.3), and apply reductions of orders combined with a reduction to the boundary. To this end we choose an elliptic boundary value problem   H s .G/ A s z WD ˚ W H .G/ ! A (1.2.39) Tz m sml  1 2 ˚lD1 H .Y / z where the boundary operators Tz have no jump along the interface. Let Pz WD .Pz K/ z be a parametrix of A which we choose in Boutet de Monvel’s calculus. We now fix s z to zero, i.e., pass to and reduce the orders of the trace operators in A z0 WD A



A RTz



H s .G/ ˚ W H .G/ ! ; L2 .Y; C m / s

smj  1 2

where R D diag.R1 ; : : : ; Rm / is a diagonal matrix of operators Rj 2 Lcl in Theorem 1.2.19. Then 

z Pz0 WD Pz KR

1

s .G/  H ˚ ! H s .G/ W 2 m L .Y; C /

.Y / as

(1.2.40)

is a parametrix of Az0 . Since (1.2.39) is a Fredholm operator, also (1.2.40) is Fredholm. Thus, composing (1.2.40) with the isomorphism 

e



2 m  L .Y ; C / ˚ ! L2 .Y; C m / e W L2 .YC ; C m / C

we obtain a Fredholm operator  z 1 e L WD Pz KR

H s .G/ ˚  z 1 eC W L2 .Y ; C m / ! H s .G/: KR ˚ L2 .YC ; C m /

48

1 Boundary value problems with mixed and interface data

We then pass to the continuous operator H s .G/ H s .G/ ˚ ˚ m 2 m sm;l  1 2 .int Y / : AL W L .Y ; C / ! ˚lD1 H ˚ ˚ 1 L2 .YC ; C m / ˚m H smC;l  2 .int YC / lD1

(1.2.41)

zPz  diag.1; 1/ (where  indicates equality modulo a compact By virtue of A operator) we obtain 0 1 1 0 0 z 1 e T KR z 1 eC A : AL  @ T Pz T KR (1.2.42) 1  z z 1 eC TC Pz TC KR e TC KR The program of the consideration is to associate with A a Fredholm operator. This will be done first for (1.2.42), and then by returning to A itself. We compose (1.2.41) from the left by another reduction of orders on Y and YC , namely, H s .G/ H s .G/ ˚ ˚ sm;l  1 2 .int Y / ! L2 .Y ; C m / ; H Q WD diag.1; Q ; QC / W ˚m lD1 ˚ ˚ 2 m smC;l  1 L .Y ; Cm/ 2 C ˚lD1 H .int YC / 1

zsm;l  2 e /lD1;:::;m which are diagonal matrices of reductions for Q WD diag.r  R  z 2 Lcl .Y / are order reof orders in the sense of Theorem 1.2.20. Recall that R ˙  z e˙ W H s .int Y˙ / ! H s .int Y˙ / are ducing operators on Y such that R˙ WD r ˙ R ˙ isomorphisms for s > maxf 12 ;  12 C g. It follows that 0

1 @ C WD QAL  Q T Pz QC TC Pz

0  r C e r C Be

1 0 r  C eC A r C BeC

with the operators sm;l  1  z z 1 2/ C WD diag.R lD1;:::;m e T KR ; 1

zsmC;l  2 /lD1;:::;m eC TC KR z 1 : B WD diag.R C The 2  2 lower right corner of QAL 

r  C e B WD C  r Be

r  C eC r C BeC

 L2 .Y ; C m / L2 .Y ; C m / ˚ ˚ ! W L2 .YC ; C m / L2 .YC ; C m /

(1.2.43)

1.2 Mixed and transmission problems

49

represents the reduction of the mixed problem A to the boundary, combined with a reduction of orders, and we have B; C 2 L0cl .Y /. The operator (1.2.43) simplifies considerably if the auxiliary boundary condition Tz in (1.2.39) satisfies the condition ˇ Tz ˇint Y D T ; (1.2.44) and if we choose R D diag.Rl /lD1;:::;m in such a way that sm;l  2 z Rl D R 1

for all l D 1; : : : ; m

(1.2.45)

with the order reducing operators of Theorem 1.2.20, with Y and Y˙ in place of Xz and X˙ , respectively. By virtue of Theorem 1.2.20 the property (1.2.45) can be assumed without loss of generality. The assumption (1.2.44) on T is satisfied in all relevant concrete examples. From now on we assume that (1.2.44) and (1.2.45) are satisfied. Then we have     C   r Ce r Ce 1 0  : r C Be r C BeC 0 r C BeC

1.2.5 Mixed problems in standard Sobolev spaces Mixed problems in standard Sobolev spaces H s .G/ ˚ A s m sm;l  1 @ A 2 .int Y / ; A D T W H .G/ ! ˚lD1 H ˚ TC smC;l  1 2 .int YC / ˚m H lD1 0

1

(1.2.46)

s > maxfm˙;l C 12 W l D 1; : : : ; mg, can also be studied from the point of view of additional conditions along Z such that the extended block matrix operator is Fredholm. As before we assume A to be elliptic with T˙ satisfying the Shapiro–Lopatinskij conditions on Y˙ (up to Z from the respective sides). It is clear that when we admit independent boundary data T˙ u D g˙ in the Sobolev spaces we cannot always expect a solution u 2 H s .G/. In order to obtain a Fredholm operator from (1.2.46) we should at least impose extra potential conditions. This effect will depend on s (the number of such conditions is increasing with s). In this section we formulate some results of this type. More details will be given in Chapter 4 below. As we shall see, results on the solvability in standard Sobolev spaces are essential also for mixed elliptic problems in edge Sobolev spaces for arbitrary weights, cf. Chapter 7. Let us assume, for simplicity, that there is a trace operator TzC for the operatorˇ A on z X D 2X which satisfies the Shapiro–Lopatinskij condition, such that TC D TzC ˇint Y . C Then the operator 1

zsmC;l  2 /lD1;:::;m TzC KR z 1 2 L0cl .Y / B WD B.s/ WD diag.R C

50

1 Boundary value problems with mixed and interface data

is elliptic, cf. the notation in Theorem 1.2.20. The pseudo-differential operator B has, in general, not the transmission property at the interface Z. Nevertheless, similarly as in standard boundary value problems, for the operator r C BeC W L2 .YC ; C m / ! L2 .YC ; C m /

(1.2.47)

we can apply a corresponding calculus of boundary value problems. It allows us, under some (in this category of questions not too restrictive topological) assumptions to pass to a 2  2 block matrix operator with (1.2.47) in the upper left corner. Let 

r C BeC F

H L

 L2 .YC ; C m / L2 .YC ; C m / ˚ ˚ W ! L2 .Z; M / L2 .Z; MC /

(1.2.48)

be such an operator that represents an elliptic boundary value problem on YC with respect to the boundary Z, with a certain potential operator H , a trace operator F and a (classical) pseudo-differential operator L on Z, and suitable vector bundles M˙ 2 Vect.Z/. Although in general the operator B has not the transmission property at Z there exists a calculus of such boundary value problems. The corresponding results are developed in Chapter 4 below. The ellipticity of (1.2.48) then implies the Fredholm property of this operator. Then also the operator 0

1 @r C Be 0

0 r C BeC F

2 m L2 .Y ; C m / 1 L .Y ; C / 0 ˚ ˚ H A W L2 .YC ; C m / ! L2 .YC ; C m / L ˚ ˚ L2 .Z; M / L2 .Z; MC /

is Fredholm, and we can pass to the Fredholm operator 0

1 B Q T Pz C WD B @QC TC Pz 0

0 1 r C Be 0

H s .G/ H s .G/ ˚ ˚ 0 2 m 2 m L L .Y ; C / .Y  ; C / 0C CW ˚ ˚ ! H A L2 .Y ; C m / 2 m L .Y C C; C / L ˚ ˚ L2 .Z; MC / L2 .Z; MC / 1

0 0 r C BeC F

with C in the upper left 3  3corner and theadditional  operators H;F and L. Let us now define L WD

L 0 0 idL2 .Z;M

/

, Q WD

Q 0 0 idL2 .Z;M

C/

. We then have

a parametrix L DL ˚ idL2 .Z;M / of L (where L denotes a parametrix 1 .1/ of L), and Q is invertible. The operator Q CL is equal (modulo a compact .1/

.1/

.1/

1.2 Mixed and transmission problems

51

remainder) to a block matrix operator of the form H s .G/ ˚ m s sm;l  1   2 .int Y / H .G/ ˚lD1 H A K ˚ ˚ A WD D ! T R smC;l  1 L2 .Z; M / 2 .int YC / ˚m H lD1 ˚ L2 .Z; MC /

(1.2.49)

which is a Fredholm operator. The operator A can be interpreted as an extension of the mixed boundary value problem A to a Fredholm operator with additional potential conditions K, trace conditions T and pseudo-differential operators R on the interface Z. Let us consider, as an example, the Zaremba problem, realised as a continuous operator H s2 .G/ 0 1 ˚ s s 1 @ A 2 .int Y / ; A D T W H .G/ ! H (1.2.50) TC ˚ 3 H s 2 .int YC / s > 32 . The following result will be proved in Chapter 4 below. Theorem 1.2.21. For s > 32 , s 62 N C 12 , the Zaremba problem (1.2.50) can be  completed by j WD s  12 interface conditions K of potential type to a Fredholm operator H s2 .G/ ˚ H s .G/   s 1 ˚ AD A K W ! H 2 .int Y / : ˚ L2 .Z; C j / s 3 H 2 .int YC / Other mixed problems in standard Sobolev spaces, e.g., problems with oblique derivatives, will also be studied in Chapter 4 below. Remark 1.2.22. Similarly as (1.2.48) to a Fredholm operator with the operator (1.2.43) we can associate a Fredholm block matrix 

B F

L2 .Y ; C m / L2 .Y ; C m /  ˚ ˚ H W L2 .YC ; C m / ! L2 .YC ; C m / L ˚ ˚ L2 .Z; M / L2 .Z; MC /

which is an elliptic transmission problem on Y with extra entries F ; H and L that encode transmission conditions of trace and potential type on Z. Then a purely algebraic

52

1 Boundary value problems with mixed and interface data

construction yields an analogue of (1.2.49) for this case, a Fredholm operator between the respective spaces, with (1.2.46) in the upper left corner.

1.3 Problems with several types of interfaces We consider transmission problems with regular and singular interfaces and several modifications.

1.3.1 Transmission problems with smooth interfaces Mixed elliptic problems have relations to transmission problems on the boundary, as we have seen in connection with the operator (1.2.43), cf. also Remark 1.2.22. The diagonal entries of the operator (1.2.43) have not the transmission property at Z, and the complete calculus requires more tools than introduced so far; in Chapter 4 below we give more complete information. Transmission problems are also of interest for differential operators (and pseudo-differential operators with the transmission property at the interface Z). For transmission problems there are different possible configurations. In the simplest case we have a closed compact C 1 manifold Y , subdivided into C 1 submanifolds Y˙ with common boundary Z such that Y D Y [ YC , Z D Y \ YC . Let A˙ be elliptic differential operators on Y˙ of order  2 N with smooth coefficients up to Z from the respective sides. Assume that T˙;l ;

l D 1; : : : ; ;

are trace operators from the plus or minus side of the same nature as standard trace operators (compositions of differential operators B˙;j of some order mj with the restriction to Z; the coefficients of the operators are assumed to be smooth in a neighbourhood of Z). Then we can consider a so-called transmission problem that consists of finding solutions u˙ 2 H s .int Y˙ / of A˙ u˙ D f T u C TC uC D g

in int Y˙ ; on Z

(1.3.1) (1.3.2)

for T˙ WD .T˙;l /lD1;:::; ; g D .gl /lD1;:::; . The relations (1.3.2) are called transmission conditions. In applications it is also typical that A˙ are systems of operators; then we have to pose a corresponding larger number of transmission conditions. Let us content ourselves with the scalar case; systems may be treated in a completely analogous manner. An example is A˙ D c˙ jint Y˙

53

1.3 Problems with several types of interfaces

for different constants c˙ with being the Laplacian on Y , and T˙ D .T˙;j /j D1;2 , with TC;1 and T;1 Dirichlet conditions;

TC;2 and T;2 Neumann conditions

from the respective sides. If T˙;l is given as the composition of a differential operator B˙;l of order ml (in a neighbourhood of Z, with smooth coefficients) with the operator of restriction to Z, then the transmission problem may be interpreted as a continuous operator 0 A AD@ 0 T

H s .int Y / ˚ 0 H .int Y / H s .int YC / ; AC A W ˚ ! ˚ H s .int YC / TC  sml  1 2 .Z/ ˚lD1 H 1

s

(1.3.3)

s > maxfml C 12 W l D 1; : : : ; g. Let V  Y be a tubular neighbourhood of Z, i.e., an open submanifold of the form V Š Z  .1; C1/ where .1; C1/ denotes the global normal interval to Z (with respect to a Riemannian metric on Y near Z). We then have a diffeomorphism  W V ! VC , .z; t / WD .z; t / between the smooth manifolds V˙ WD V \ Y˙ , with boundary, and we can pass to the operator 1 0 0  .A jint V / 0 AC jint VC A : AV WD @  T TC ˇ ˇ Here  .A ˇint V / WD .  /1 .A ˇint V / (with   being the function pull back ˇ under ) and . T /u WD . B;j ˇint V /ujZ for any function u on int VC . The operator AV then represents a differential boundary value problem on VC which has a boundary symbol H s .RC / s ˚ H .RC / ˚ @ .AV /.z; / W ! H s .RC /; ˚ H s .RC / C .z; / 2 T  Z n 0, as is known from the usual context of boundary value problems. This gives rise to a so-called transmission symbol for the original operator A H s .R / ˚ H .R / ˚ tr .A/.z; / W ! H s .RC /; H s .RC / ˚ C s

(1.3.4)

.z; / 2 T  Z n 0, which is connected with @ .AV /.z; / in an analogous manner as A to AV .

54

1 Boundary value problems with mixed and interface data

Definition 1.3.1. The transmission problem A is called elliptic, if the operators A˙ are elliptic in the usual sense, and if ˚

(1.3.4) is a family of isomorphisms for any s > max   12 , m1 C 12 ; : : : ; m C 12 . ˚ Theorem 1.3.2. The operator A given by (1.3.3) is Fredholm for any s > max   12 ,

m1 C 12 ; : : : ; m C 12 if and only if it is elliptic. The proof can be reduced to a corresponding result on standard elliptic boundary value problems, see the monograph [154, Section 3.1.1.1, Theorem 7].

1.3.2 Transmission problems with singular interfaces We now turn to transmission problems for the case of interfaces with conical singularities. Details on that type of problems may be found in [92]. Here we only sketch a few results. The manifold Y is again subdivided into subsets Y˙ which are assumed to be manifolds with a common boundary that has a conical singularity v (the case of more than one conical singularities can be treated as well). The interface Z D YC \ Y is then a manifold with conical singularity v without boundary. The transmission problem on Y is formulated similarly as (1.3.1), (1.3.2); however, locally near v the involved operators A˙ and the trace operators T˙ are formulated in polar coordinates with respect to the conical point. The operators A˙ have the form A˙ D r 

 X j D0

 @ j a˙;j .r/ r ; @r

x C ; Diff j .N˙ // where N˙ is the cross section of with coefficients a˙;j .r/ 2 C 1 .R the local cone with vertex v and boundary, locally represented by Y˙ ; the manifolds N˙ are compact, C 1 , with common boundary S WD @N D @NC . Moreover, the trace operators are given as T˙ D rZnfvg B˙ with differential operators B˙ D .B˙;l /lD1;:::; , B˙;l D r ml

ml X kD0

 @ k b˙;l;k .r/ r ; @r

x C ; Diff ml k .U˙ // for collar neighbourhoods U˙ with coefficients b˙;l;k .r/ 2 C 1 .R of S, cf., similarly, the formulas (1.2.10), (1.2.11). The sets Y˙ and Z are manifolds with conical singularity in the sense of the notation of Section 2.4.2 below. Let Y˙ and Z denote the associated stretched manifolds (locally near Y˙;sing the space Y˙ is x C  S ). x C  N˙ , while Z locally near Zsing is identified with R modelled on a cylinder R With the notation of Section 2.4.2 we have weighted Sobolev spaces H s; .Y˙ /

and

H s; .Z/;

1.3 Problems with several types of interfaces

55

respectively. Our transmission problem then represents continuous operators 0

A AD@ 0 T

H s; .Y / ˚ 0 H .Y / H s; .YC / AC A W ˚ ! ; s; ˚ H .YC / TC 1 1 ˚ H sml  2 ;ml  2 .Z/ lD1 1

s;

(1.3.5)

cf. Definition 2.4.22 and the formula (2.4.22), for all s 2 R, s > maxfml C 12 W l D 1; : : : ; g,  2 R. Operators of that kind belong to the category of transmission problems with interfaces with conical singularities. They are a modification of boundary value problems on manifolds with conical singularities which we discuss in Chapter 6 below. The ellipticity of such problems is described by a principal symbolic hierarchy consisting of three components . .A/; @ .A/; c .A//, the interior, the boundary, and the conormal symbol. In the case of transmission problems, similarly as in the preceding section, there is a transmission symbol tr .A/ that is related via a local reflection argument with the boundary symbol of a corresponding (local) boundary value problem. In the present case the operator (1.3.5) has the principal symbol  .A/ D . .A/; tr .A/; c .A//:

(1.3.6)

The first component  .A/ WD . .A /;  .AC // is nothing other than the pair of homogeneous principal symbols of the operators A˙ in the usual sense. Locally near v in the splitting of variables .r; / the symbols depend on .r; ; %; #/ (with .%; #/ being the covariables to .r; /), and we can form Q .A˙ /.r; ; %; #/ WD r   .A˙ /.r; ; r 1 %; #/

(1.3.7)

for .%; #/ 6D 0, which are smooth up to r D 0. The second component of (1.3.6) is a family of operators H s .R / ˚ H .R / ˚ tr .A/.z; / W ! H s .RC / H s .RC / ˚ C s

for .z; / 2 T  .Z n fvg/ n 0. Locally near v in the splitting of variables .r;  0 / (with the covariables .%; # 0 /) we form Q tr .A/.r;  0 ; %; # 0 / WD diag.r  ; .r ml /lD1;:::; /tr .A/.r;  0 ; r 1 %; # 0 / for .%; # 0 / 6D 0, which is smooth up to r D 0.

(1.3.8)

56

1 Boundary value problems with mixed and interface data

As the third component of (1.3.6) we have the so-called principal conormal symbol H s .N / ˚ H .N / H s .NC / ˚ c .A/.w/ W ! ˚ H s .NC / sml  1 2 .S / ˚ H lD1 s

given as

(1.3.9)

0

1 0 c .A / c .AC /A .w/; c .A/.w/ D @ 0 c .T / c .TC /  P ml  P b˙;l;k .0/w k lD1;:::; . for c .A˙ /.w/ WD j D0 a˙;j .0/w j , c .T˙ /.w/ WD t rS kD0 Definition 1.3.3. The transmission problem A is called elliptic (with respect to a weight  2 R) if (i)  .A˙ / 6D 0 on T  .Y˙ n fvg/ n 0, and the functions (1.3.7) do not vanish for .%; #/ 6D 0, up to r D 0; (ii) tr .A/.z; / is an isomorphism for every .z; / 2 T  .Z n fvg/ n 0, and (1.3.8) is an isomorphism for .%; # 0 / 6D 0, up to r D 0, for any sufficiently large s; (iii) c .A/.w/ defines an isomorphism (1.3.9) for all w 2  q  and any sufficiently 2 large s: here q D dim Y , ˇ WD fw 2 C W Re w D ˇg for any real ˇ. ˚ Theorem 1.3.4. The operator (1.3.5) is Fredholm for any s > max   12 , m1 C

1 ; : : : ; m C 12 if and only if it is elliptic (with respect to the weight ). 2 This follows from the results of [174], [175], combined with the results on necessity of ellipticity under the assumption of the Fredholm property from [182, Section 2.2.1, Theorem 14], here adapted to boundary value problems. Remark 1.3.5. Mixed problems are also meaningful for the case of interfaces with edges. The corresponding calculus is elaborated in [94].

Chapter 2

Symbolic structures and associated operators

The methods in this book are based on ideas from the classical pseudo-differential calculus and its various generalisations. We first consider scalar symbols and operators. Then we discuss operators with operator-valued symbols. Other aspects concern operators on manifolds with conical exits to infinity, or with edges and conical points ‘in the finite’.

2.1 Scalar pseudo-differential calculus The role of the material of this section is to formulate basics on symbol spaces and associated pseudo-differential operators in the scalar case. More details and proofs (as far as they are not given here) may be found in textbooks, such as Hörmander [83], Kumano-go [110], or Treves [219], and also in Section 2.2 below, where we consider generalisations with operator-valued symbols.

2.1.1 Spaces of symbols and basic operations In this section we introduce notation and results on scalar amplitude functions in a corresponding pseudo-differential calculus. Let us first recall a standard notation: A Fréchet space E is a locally convex and complete metric vector space. In the following considerations many vector spaces E will be Fréchet spaces with P k pk .f g/ a metric of the form d.f; g/ D 1 , f; g 2 E. Here .pk /k2N is a kD0 2 1Cpk .f g/ countable sequence of semi-norms on E, where for every f 2 E; f 6D 0, there is a k 2 N such that pk .f / 6D 0. The sequence .pk /k2N is often given in direct connection with the definition of E. We then say that .pk /k2N defines the Fréchet space structure (or the Fréchet topology) of E. Definition 2.1.1. (i) The space S  .U  Rn / of symbols a.x; / of order  2 R on an open set U Rm is defined as the set of all a.x; / 2 C 1 .U  Rn / such that sup .x; /2KRn

h iCjˇ j jDx˛ D ˇ a.x; /j

(2.1.1)

is finite for every K b U , and arbitrary multi-indices ˛ 2 N m ; ˇ 2 N n I here h i WD 1 .1 C j j2 / 2 . If a 2 S  .U  Rn / we also write  D ord a.

58

2 Symbolic structures and associated operators

(ii) Let S ./ .U .Rn nf0g// denote the space of all elements a./ .x; / 2 C 1 .U  .R n f0g// such that a./ .x;  / D  a./ .x; / (2.1.2) n

for all  2 RC ; .x; / 2 U  .Rn n f0g/. Then Scl .U  Rn /, the space of classical symbols of order , is defined as the subspace of all a.x; / 2 S  .U  Rn / such that there are elements a.j / .x; / 2 S .j / .U  .Rn n f0g//, j 2 N, with the property rN .x; / WD a.x; /  . /

N X

a.j / .x; / 2 S .N C1/ .U  Rn /

j D0

for every N 2 N and any excision function . / (i.e., . / 2 C 1 .Rn / such that . / D 0 for j j < c0 ; . / D 1 for j j > c1 for certain 0 < c0 < c1 ). If a relation (for symbols and later on for operators) holds both for the classical and  .U  Rn /. the general case, we often write subscript ‘.cl/’, e.g., S.cl/ Remark 2.1.2. (i) The space S  .U  Rn / is a Fréchet space with the system of semi-norms (2.1.1), ˛ 2 N m , ˇ 2 N n , K b U . (ii) The (so-called homogeneous) components a.j / ; j 2 N, of a classical symbol a.x; / 2 Scl .U Rn / are uniquely determined by a.x; /, and Scl .U Rn / is a nuclear Fréchet space in the topology of the projective limit with respect to the maps Scl .U  Rn / ! S .j / .U  .Rn n f0g//;

a.x; / ! a.j / .x; /; j 2 N;

and Scl .U  Rn / ! S .N C1/ .U  Rn /;

a.x; / ! rN .x; /; N 2 N:

 .Rn / of all elements a. / with constant coefficients (i.e., inde(iii) The space S.cl/  pendent of x) is closed in S.cl/ .U  Rn /, and we have    y  S.cl/ S.cl/ .U  Rn / D C 1 .U; S.cl/ .Rn // D C 1 .U / ˝ .Rn /

y  means the completed projective tensor product between the respective spaces). (˝ T Setting S 1 .U  Rn / WD 2R S  .U  Rn /, we then have S 1 .U  Rn / D C 1 .U; S.Rn // with the Schwartz space S.Rn / D S 1 .Rn /. Given a Fréchet space F with a semi-norm system .p /2N , we denote by S  .U  Rn I F /

(2.1.3)

for U Rm open,  2 R, the space of all a.x; / 2 C 1 .U  Rn ; F / such that sup .x; /2KRn

h iCjˇ j p .Dx˛ D ˇ a.x; //

(2.1.4)

2.1 Scalar pseudo-differential calculus

59

is finite for every K b U , arbitrary multi-indices ˛ 2 N m ; ˇ 2 N n , and  2 N. The space (2.1.3) is Fréchet in the semi-norm system (2.1.4). Many natural properties and constructions for the scalar symbol spaces of Definition 2.1.1, i.e., for F D C, extend to the case of arbitrary Fréchet spaces F . Let S ./ .U  .Rq n f0g/I F / denote the space of all a./ .x; / 2 C 1 .U  .Rq n f0g/; F / such that the relation (2.1.2) holds for all  2 RC ; .x; / 2 U  .Rn n f0g/. This gives rise to the subspace Scl .U  Rn I F /

(2.1.5)

of classical F -valued symbols, defined by an obvious generalisation of the corresponding space of classical scalar symbols in Remark 2.1.2 (ii). Let us now return to scalar symbols and formulate some important properties. Theorem 2.1.3. Let aj .x; / 2 S j .U  Rn /, j 2 N, be an arbitrary sequence, j ! 1 as j ! 1. Then exists an a.x; / 2 S  .U  Rn / for  WD maxfj W P j 2 Ng such that ord.a  jND0 aj / ! 1 as N ! 1. Any such a.x; / is uniquely determined mod S 1 .U  Rn /. An explicit proof may be found, for instance, in [188, Section 1.1.2]. P If a.x; / is a symbol as in Theorem 2.1.3 we also write a.x; /  j1D0 aj .x; / and call a.x; / an asymptotic sum of the symbols aj .x; /. We can construct a.x; / as a convergent series 1   X  (2.1.6) aj .x; / a.x; / D cj j D0

in S .U  R /, with an excision function . / and constants cj > 0 tending to 1 sufficiently Pfast as j ! 1, where for every M > 0 there exists an N D N.M / 2 N such that j1DN C1 . =cj /aj .x; / converges in S M .U  Rn /. In future we will have many constructions in terms of Hilbert or Fréchet spaces. For later use we introduce the following general notation. 

n

Definition 2.1.4. (i) If E0 ; E1 are Fréchet spaces continuously embedded in a Hausdorff topological vector space V , we set E0 C E1 WD fe0 C e1 W e0 2 E0 ; e1 2 E1 g and endow this space with the quotient topology under the identification E0 C E1 D E0 ˚ E1 = for D f.e; e/ W e 2 E0 \ E1 g. Incidentally, we call E0 C E1 the non-direct sum of the spaces E0 ; E1 . (ii) Let E be a Fréchet space which is a left module over an algebra A. Then we set Œa E WD closure of fae W e 2 Eg for every a 2 A. In a similar manner we define EŒb

or Œa EŒb for a 2 A, b 2 B when E is a right or two-sided module over algebras A, B. Remark 2.1.5. If E0 ; E1 in Definition 2.1.4 are Hilbert spaces, also E0 C E1 is a Hilbert space with the scalar product of the orthogonal complement of in E0 ˚ E1 . Definition 2.1.6. Given Hilbert spaces E; E0 ; E 0 contained in a Hausdorff topological vector space V such that E \ E0 \ E 0 is dense in E; E0 and E 0 , we speak about a Hilbert space triple fE; E0 ; E 0 g;

60

2 Symbolic structures and associated operators

if the E0 -scalar product extends from .E \ E0 /  .E0 \ E 0 / to a continuous, nondegenerate sesquilinear form E  E 0 ! C, such that kek D supfj.e; e 0 /j=ke 0 kE 0 W e 0 2 E 0 ; e 0 6D 0g; ke 0 k D supfj.e; e 0 /j=kekE W e 2 E; e 6D 0g are equivalent norms in E and E 0 , respectively, such that E 0 is identified with the dual of E and vice versa. If fE; E0 ; E 0 g and fF; F0 ; F 0 g are Hilbert space triples and A W E ! F a continuous operator, then we have its adjoint A W F 0 ! E 0 defined by .Au; v/F0 D .u; A v/E0 for all u 2 E, v 2 F 0 . The following observation will play a role later on in certain kernel characterisations of operators. Proposition 2.1.7. Let fE; E0 ; E 0 g and fF; F0 ; F 0 g be Hilbert space triples, cf. Definition 2.1.6, and let S  E 0 and T  F be nuclear Fréchet subspaces. Moreover, assume that G 2 L.E; F / induces continuous operators G W E ! T , G  W F 0 ! S. y  T / \ .S ˝ y  F / where the Then G can be identified with a kernel G 2 .E 0 ˝ y intersection refers to the space of trace class operators E ˝ F . A proof is given in [86]. For future reference we want to formulate the following well-known result on y  of Fréchet spaces. projective tensor products ˝ y  F has Proposition 2.1.8. Let E and F be FréchetPspaces. Then every h 2 E ˝ a representation as a convergent sum h D j1D0 j ej ˝ fj with ej 2 E; fj 2 F , P tending to zero for j ! 1 in the respective spaces, and j 2 C, j1D0 jj j < 1. A proof may be found in [170]. If E and Ez are Hilbert spaces, we can endow the algebraic tensor product E ˝ Ez  with a scalar product by the rule .e1 ˝ eQ1 /; .e2 ˝ eQ2 / E ˝ Ez WD .e1 ; e2 /E .eQ1 ; eQ2 /Ez H z By E ˝H Ez we denote and then extended by linearity to arbitrary elements of E ˝ E. z the completion of E ˝ E with respect to the norm associated with .; /E ˝H Ez and z An example is E D L2 .Rn /, call it the Hilbert tensor product between E and E. 2 m 2 nCm z z E D L .R /; then E ˝H E D L .R /. Observe that when Ez is a Hilbert space and F a nuclear Fréchet space, written as a projective limit of Hilbert spaces F j ; j 2 N, with nuclear embeddings F j C1 ,! F j for all j , we have z y  Ez D lim F j ˝H E: (2.1.7) F ˝  If  Rn is an open set, and E a Fréchet space, by I.; E/ we denote the set of all f 2 C.; E/ such that p ı f 2 L1 ./ for every continuous semi-norm p on E. Proposition 2.1.9. For every f 2 I.; E/ there exists a unique e 2 E such that Z he 0 ; ei D he 0 ; f .x/idx for all e 0 2 E 0 : (2.1.8) 

2.1 Scalar pseudo-differential calculus

For every continuous semi-norm p on the space E we have Z p.f .x//dx: p.e/ 

61

(2.1.9)



We then write e WD

R 

f .x/dx.

Proof. The uniqueness of e is clear. To show the existence of e we first assume that f has compact support. Then e 2 E with the property (2.1.8) exists by [168, Q  1 for all eQ 2 E with p.e/ Q  1g be Theorem 3.2.7]. Let Upo WD fe 0 2 E 0 W jhe 0 ; eij the polar of p. Then we have p.e/ Q D supfjhe 0 ; eij Q W e 0 2 Upo g for all eQ 2 E. Choose a sequence .ej0 /j 2N  Upo such that limj !1 jhej0 ; eij D p.e/. Then jhej0 ; eij

Z  

jhej0 ; f .x/ijdx

Z 

p.f .x//dx 

shows that e satisfies the relation (2.1.9). For an arbitrary element f 2 I.; E/ we choose a sequence .'j /j 2N 2 C0 ./, 0  'j  1, such that 'j ! 1 pointwise on  for j ! 1. Let ej 2 E denote Rthe element from the first part of the proof belonging to 'j f . Then p.ej  ek /   j'j .x/  'k .x/jp.f .x//dx. Using Lebesgue’s theorem on dominated convergence we obtain that .ej /j 2N is a Cauchy sequence in E and hence it converges to an e 2 E. That element obviously satisfies (2.1.8) and (2.1.9).

2.1.2 Pseudo-differential operators and Sobolev spaces In this exposition we employ standard material on distribution theory, Fourier transform, Sobolev spaces, etc. Let us recall some basic notation and theorems. By D 0 ./ we denote the set of distributions (the dual of C01 ./) in an open set   Rn and S 0 .Rn / the space of temperate distributions in Rn (the dual of S.Rn /). The support supp u of an u 2 D 0 ./ is the complement of the largest open set G such that ujG D 0. Moreover, the singular support sing supp u is the complement of the largest open set G such that ujG 2 C 1 .G/. Set Z F u. / D u. / O D e ix u.x/dx which is the Fourier transform in Rn ; x WD isomorphisms F W S.Rn / ! S.Rn /;

Pn

j D1 xj j .

We use the fact that F induces

F W S 0 .Rn / ! S 0 .Rn /;

(2.1.10)

and F W L2 .Rn / ! L2 .Rn /:

(2.1.11)

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2 Symbolic structures and associated operators

The space L2 .Rn / is endowed with the scalar product .u; v/ D inverse Fourier transform is given by the formula Z 1 .F g/.x/ D e ix g. /μ ;

R Rn

u.x/v.x/dx. The

(2.1.12)

1 μ WD .2/n d . From time to time we also write F DW Fx! ; F 1 DW F !x . Let us recall Plancherel’s formula, namely,

kF ukL2 .Rn / D .2/n=2 kukL2 .Rn / for every u 2 L2 .Rn /. Definition 2.1.10. The set H s .Rn / WD fu 2 S 0 .Rn / W u. / O 2 L1loc .Rn /; h is u. / O 2 2 n n L .R /g is called the Sobolev space in R of smoothness s 2 R. If   Rn is an open set, we have the spaces s s ./ and Hloc ./ Hcomp s ./ is the subspace of all u 2 H s .Rn / such that supp u is a for every s 2 R. Here Hcomp compact subset of  (with a canonical identification with distributions in ). Moreover, s s Hloc ./ is defined to be the set of all distributions in  such that 'u 2 Hcomp ./ for 1 every ' 2 C0 ./.

Remark 2.1.11. The function h i ,  2 R, belongs to the space Scl .Rn /, and u.x/ ! .F 1 h i F u/.x/; first regarded as an operator S.Rn / ! S.Rn /, extends by continuity to an isomorphism H s .Rn / ! H s .Rn / for every s 2 R. Let  Rn be an open set, and write .x; x 0 / for the variables in the Cartesian product   . We then define “ 0 Op.a/u.x/ D e i.xx / a.x; x 0 ; /u.x 0 /dx 0 μ (2.1.13) for a.x; x 0 ; / 2 S  .    Rn /, first for u 2 C01 ./ (and later on extended to more general distribution spaces). The expression is interpreted as an oscillatory ’ i.xx 0(2.1.13) / integral, i.e., Op.a/u.x/ D lim"!0 e ." //a.x; x 0 ; /u.x 0 /dx 0 μ for any 1 n . / 2 C0 .R / such that .0/ D 1 (oscillatory integrals will be studied in more detail in Section 2.2.3 below). Then Op.a/ defines a continuous operator Op.a/ W C01 ./ ! C 1 ./:  .    Rn /g. ./ WD fOp.a/ W a.x; x 0 ; / 2 S.cl/ Definition 2.1.12. We set L .cl/ The elements of L ./ are called (classical) pseudo-differential operators of order .cl/ T 1  on . Let L ./ WD 2R L ./.

2.1 Scalar pseudo-differential calculus

63

Occasionally, if A 2 L ./ is written as A D Op.a/ for a.x; / 2 S  .x  Rn / .a.x 0 ; / 2 S  .x 0  Rn // we call a a left symbol (right symbol), also denoted by aL .x; / and

aR .x 0 ; /;

(2.1.14)

respectively. For A D Op.a/; a.x; x 0 ; / 2 S  .    Rn /, we also call a.x; x 0 ; / a double symbol. Instead of (2.1.13) for A D Op.aL / we may also write Z O : (2.1.15) Au.x/ D e ix aL .x; /u. /μ Moreover, for A D Op.aR / we have Z Z 0 e ix aR .x 0 ; /u.x 0 /dx 0 μ ; Au.x/ D e ix i.e., Fx! Au. / D

R

0

e ix aR .x 0 ; /u.x 0 /dx 0 D Fx 0 ! .aR .x 0 ; /u.x 0 //. /.

 .    Rn / there are left and right Theorem 2.1.13. For every a.x; x 0 ; / 2 S.cl/  symbols (2.1.14) belonging to S.cl/ .  Rn / such that

Op.a/ D Op.aL / mod L1 ./;

Op.a/ D Op.aR / mod L1 ./;

(2.1.16)

and X 1 ˇ D ˛ @˛x0 a.x; x 0 ; /ˇx 0 Dx ; ˛Š ˛ X 1 ˇ aR .x 0 ; /  .1/j˛j D ˛ @˛x a.x; x 0 ; /ˇxDx 0 : ˛Š ˛ aL .x; / 

A proof for a modified situation is given in connection with Theorem 2.2.52 below. This can easily be adapted to the present scalar case. For the moment let us give a simple argument, to produce, for instance, (2.1.16). By Taylor expansion of a.x; x 0 ; / at diag.  / we obtain a.x; x 0 ; / D

X 1 ˇ .x 0  x/˛ @˛x0 a.x; x 0 ; /ˇx 0 Dx C rM .x; x 0 ; / ˛Š

j˛jM

with a remainder rM .x; x 0 ; / 2 S  .    Rn /. For prescribed N and M large enough we also have jx  x 0 j2N rM .x; x 0 ; / 2 S  .    Rn /. Applying the identity Op..x 0  x/˛ b/ D Op.D ˛ b/ we thus obtain Op.a/ D

X 1 ˇ Op.D ˛ @˛x0 a.x; x 0 ; /ˇx 0 Dx / C Op.rQN / ˛Š

j˛jM

for a symbol rQN .x; x 0 ; / 2 S 2N .    Rn /. This gives us the relation (2.1.16).

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2 Symbolic structures and associated operators

1 There is an identification between C 1 .  / 3 c.x; x 0 / and where R L ./, 0 1 c.x; x / 2 C .  / corresponds to the operator C u.x/ WD  c.x; x 0 /u.x 0 /dx 0 , u 2 C01 ./. We have a linear injective map K W L ./ ! D 0 .  / from operators to their distributional (Schwartz) kernels K.A/, such that Z Au.x 0 /v.x 0 /dx 0 D hK.A/.x; x 0 /; u.x/v.x 0 /i for u; v 2 C01 ./:

Writing diag.  / D f.x; x 0 / 2    W x D x 0 g we have sing supp K.A/ diag.  /. A relatively closed subset V     is called proper if the sets f.x; x 0 / 2 V W x 2 M g;

f.x; x 0 / 2 V W x 0 2 M g

are both compact for arbitrary M; M 0 b . Definition 2.1.14. An operator A 2 L ./ is called properly supported if the set supp K.A/ is proper. Remark 2.1.15. A properly supported operator A induces continuous operators A W C01 ./ ! C01 ./ and

A W C 1 ./ ! C 1 ./:

(2.1.17)

As an easy consequence of the relation sing supp K.A/ diag.  / we see that every A 2 L ./ can be written as a sum A D A0 C C

(2.1.18)

for a properly supported operator A0 2 L ./ and an element C 2 L1 ./. In fact, if V     is any proper subset that contains diag.  / in its interior, we can choose an element .x; x 0 / 2 C 1 .  / with supp V such that 1 in an open neighbourhood of diag.  /, and then set A0 WD Op. a/, C WD Op..1  /a/ when A D Op.a/. ./V WD fA 2 L ./ W supp K.A/ V g for any such V , we then Writing L .cl/ .cl/ have ./V =L1 ./V Š L ./=L1 ./: L .cl/ .cl/ R Let A be properly supported, and apply A to u.x/ D e ix u. /μ O under the integral sign, cf. the second relation of (2.1.17). For e .x/ WD e ix we then obtain Z O ; Au.x/ D e ix fe .x/Ae .  /gu. /μ cf. the expression (2.1.15).

2.1 Scalar pseudo-differential calculus

65

Theorem 2.1.16. Let A 2 L ./ be a properly supported operator. Then we have .cl/  aL .x; / WD e .x/Ae .  / 2 S.cl/ .Rn / and A D Op.aL /. For every proper subset  V    ; diag.  /  int V , the space S.cl/ .  Rn /V WD fe Ae W A 2  L .  Rn /. ./V g is closed in S.cl/ .cl/ A proof is given, for instance, in [188, Section 1.1.3]. It follows that  ./=L1 ./ Š S.cl/ .  Rn /=S 1 .  Rn /: L .cl/  From the bijection Op.  / W S.cl/ .Rn /V ! L ./V for V as in Theorem 2.1.16 .cl/ we obtain a Fréchet topology in the space L ./ . V Then the relation .cl/

L ./ D L ./V C L1 ./ .cl/ .cl/

(2.1.19)

(which is a consequence of (2.1.18)) gives us the Fréchet topology of the non-direct sum ./, cf. Definition 2.1.4. It can easily be proved that it is independent in the space L .cl/ of the choice of V .  0 n For A 2 L cl ./ written as Op.a/ for a.x; x ; / 2 Scl .    R / we set  .A/.x; / WD a./ .x; x 0 ; /jx 0 Dx ; called the homogeneous principal symbol of A of order .

2.1.3 Operators on manifolds Let L .I C j ; C k / denote the space of all k  j matrices A with entries in L ./, .cl/ .cl/  Rn open, regarded as operators A W C01 .; C j / ! C 1 .; C k /. Moreover, let M be a C 1 manifold, and E; F (smooth complex) vector bundles on M of fibre dimension j and k, respectively. Given a chart  W D !  on M we consider trivialisations  W EjD !   C j

and

# W F jD !   C k

such that  ı pD D p ı  for the corresponding projections pD W EjD ! D, p W   C j ! , and the same for F jD . We then have pull backs  W C01 .; C j / ! C01 .D; EjD /; and operators

#  W C 1 .; C k / ! C 1 .D; F jD /

#  A. /1 W C01 .D; EjD / ! C 1 .D; F jD /:

Here C 1 .M; E/ denotes the space of all smooth sections in the vector bundle E and C01 .M; E/ is the subspace of elements with compact support. Given a locally finite open covering .D /2I of M by coordinate neighbourhoods, a subordinate partition of

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2 Symbolic structures and associated operators

unity .' /2I , and a system .  /2I of elements supp ' for all  2 I , we set .M I E; F / L .cl/ nX WD ' f# A . /1 g





2 C01 .D / such that



1 on

o j k 1 C C W A 2 L . I C ; C /; C 2 L .M I E; F /  .cl/

2I

(2.1.20) with  W EjD !  C j , # W F jD !  C k being associated with charts  W D !  that are of analogous meaning as before. Moreover, L1 .M I E; F / will denote the space of smoothing operators on M , i.e., the space of all integral operators Z hc.x; x 0 /; u.x 0 /idx 0 C u.x/ D M

y  C 1 .M; E 0 / where E 0 is the dual of E, and with kernels c.x; x 0 / 2 C 1 .M; F / ˝ h; i the pointwise bilinear pairing between vectors in E 0 and E; dx 0 refers to a fixed Riemannian metric on M . If A belongs to the space (2.1.20) we also write  DW ord A, called the order of the operator A. The definition of (2.1.20) is correct by virtue of the coordinate invariance of the space of pseudo-differential operators; a proof may be found, for instance, in [188, Section 1.1.3]. For the trivial bundles E D M  C, F D M  C we have the space L .M / .cl/ of scalar pseudo-differential operators of order  2 R on M . s s The standard spaces Hcomp .M / and Hloc .M / of Sobolev distributions on M of s s smoothness s 2 R have natural analogues as spaces Hcomp .M; E/ and Hloc .M; E/ of distributional sections in a vector bundle E on M . For compact M we simply write H s .M; E/. .M I E; F / is correct in the sense that it is Remark 2.1.17. (i) The definition of L .cl/ independent of the specific choice of data  ;  ; # , etc. .M I E; F / there is a properly supported operator A0 in (ii) For every A 2 L .cl/  L.cl/ .M I E; F / such that A D A0 mod L1 .M I E; F / (‘properly supported’ in terms of a proper support of the distributional kernel is defined in an analogous manner as above in the local situation). (iii) L .M I E; F / is a Fréchet space in a natural way. .cl/ To give an idea for (iii) assume that E and F are trivial and of fibre dimension 1 (the general case is analogous). Then we have L1 .M / Š C 1 .M  M / via a fixed Riemannian metric on M that admits to identify kernels in C 1 .M  M / with corresponding smoothing operators. For  2 R in general and an arbitrary chart  W D !  on M we set j; .A/ WD j . AjD /, A 2 L .M /, j 2 N, where .cl/

2.1 Scalar pseudo-differential calculus

67

.j /j 2N denotes a semi-norm system for the Fréchet space L ./; then j; is a .cl/  semi-norm system for the Fréchet space L.cl/ .M /. Let A 2 L .M I E; F /, and let .Apq /1pk;1qj 2 L .I C j ; C k / be a local .cl/ .cl/ representative of A on . Then the system of matrices . .Apq //1pk;1qj has an invariant meaning as a bundle morphism    .A/ W M E ! M F;

(2.1.21)

called the homogeneous principal symbol of A; here M W T  M n 0 ! M is the canonical projection of the cotangent bundle of M (minus the zero section) to M .   E ! M F Let S ./ .T  M n 0I E; F / denote the set of all morphisms p./ W M   such that p./ .x;  / D  p./ .x; / for all .x; / 2 T M n 0,  2 RC . 

j .M I E; F /, j 2 N, be an arbitrary sequence where Remark 2.1.18. Let Aj 2 L.cl/ j ! 1 for j ! 1 (and j D   j for some  2 R in the classical case). Then there is an A 2 L .M I E; F / for  D maxfj g, unique mod L1 .M I E; F /, such  PN.cl/  P that ord A  j D0 Aj ! 1 as N ! 1. We then write A  j1D0 Aj and call A an asymptotic sum of the operators Aj .

Remark 2.1.18 is a simple consequence of Theorem 2.1.3. Proposition 2.1.19. The principal symbolic map ./  W L .T  M n 0I E; F / cl .M I E; F / ! S

is surjective, and there is a linear map op W S ./ .T  M n 0I E; F / ! L cl .M I E; F / such that  ı op D id. Moreover, we have ker  map op is called an operator convention.)

(2.1.22)

D L1 .M I E; F /. (Any such cl

A choice of (2.1.22) directly follows from the existence of local operators in j k L cl .I C ; C / associated with principal symbols that correspond to local representations of a given element p./ 2 S ./ .T  M n 0I E; F /, multiplied by an excision function. A subsequent globalisation, according to the expressions in (2.1.20), then gives us (2.1.22). The statement on ker  is evident. Theorem 2.1.20. Let M be a closed, compact C 1 manifold. (i) Every A 2 L .M I E; F / for E; F 2 Vect.M / induces continuous operators A W H s .M; E/ ! H s .M; F /

(2.1.23)

for all s 2 R. (ii) A 2 L cl .M I E; F / and  .A/ D 0 implies that the operator (2.1.23) is compact.

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2 Symbolic structures and associated operators

(iii) A 2 L .M I E0 ; F /, B 2 L.cl/ .M I E; E0 / for E; E0 ; F 2 Vect.M / implies .cl/ AB 2 LC .M I E; F /. In the classical case we have .cl/  .AB/ D  .A/ .B/: (iv)

.M I E; F /, and let A denote the formal adjoint of A (defined by Let A 2 L .cl/ .Au; v/L2 .M;F / D .u; A v/L2 .M;E / for all u 2 C 1 .M; E/, v 2 C 1 .M; F /). Then we have A 2 L .M I F; E/ and, in the classical case, .cl/  .A / D  .A/ (the adjoint on the right-hand side refers to the Hermitian metrics in the bundles.)

The property (i) follows from a corresponding local result on continuity of pseudodifferential operators, see, for instance, [182, Section 1.1.4]. We will give another proof below in a slightly modified situation, cf. Theorem 2.2.20 and Theorem 2.3.43, which can easily be adapted to the present case. .M I E; F /, and as a consequence (2.1.23) is If  .A/ D 0 we have A 2 L1 cl compact, since A W H s .M; E/ ! H sC1 .M; F / is continuous and the embedding H sC1 .M; F / ! H s .M; F / compact. The statements (iii) and (iv) on compositions and formal adjoints, respectively, follow from corresponding local results; proofs are given in [188, Section 1.1.3]. For a more general situation we give self-contained proofs below in connection with Theorem 2.2.54 and Theorem 2.2.58. In the sequel we mainly concentrate on classical pseudo-differential operators. Definition 2.1.21. Let A 2 L cl .M I E; F /,  2 R, E; F 2 Vect.M /.   E ! M F is (i) The operator A is said to be elliptic (of order ), if  .A/ W M an isomorphism.

(ii) An operator P 2 L cl .M I F; E/ is called a parametrix of A, if (for the case of closed compact M ) P satisfies the following relations: Cl WD I  PA 2 L1 .M I E; E/;

Cr WD I  AP 2 L1 .M I F; F /; (2.1.24) where I denotes the corresponding identity operators (otherwise, for non-compact M we require (2.1.24) for a properly supported representative P in the coset modulo smoothing operators). Notice that when P is a parametrix of A we have  .P / D  .A/1 . Moreover, if A is elliptic, so is the formal adjoint A . We call an operator A 2 L cl .M I E; F / underdetermined (overdetermined) elliptic,   if  .A/ W M E ! M F is injective (surjective). Remark 2.1.22. (i) A 2 L cl .M I E; F / is underdetermined elliptic if and only if   A 2 Lcl .M I F; E/ is overdetermined elliptic. (ii) If A is underdetermined (overdetermined) elliptic, then AA 2 L2 cl .M I E; E/ 2  (AA 2 Lcl .M I F; F / ) is elliptic.

2.1 Scalar pseudo-differential calculus

69

This follows easily from Theorem 2.1.20 (iv). Theorem 2.1.23. Let M be a closed, compact C 1 manifold and A 2 L cl .M I E; F /,  2 R, E; F 2 Vect.M /. (i) The operator A is elliptic (of order ), if and only if A W H s .M; E/ ! H s .M; F /

(2.1.25)

is a Fredholm operator for an s D s0 2 R. (ii) If A is elliptic, (2.1.25) is a Fredholm operator for all s 2 R, and dim ker A and dim coker A are independent of s. We have V WD ker A  C 1 .M; E/, and there is a finite-dimensional subspace W  C 1 .M; F / such that W C im A D H s .M; F / and W \ im A D f0g for every s 2 R. (iii) An elliptic operator A has a parametrix P 2 L cl .M I F; E/, cf. Definition 2.1.21 (ii), and P can be chosen in such a way that the remainders in the relation (2.1.24) are projections Cl W H s .M; E/ ! V , Cr W H s .M; F / ! W to the subspaces V and W of (ii), for all s 2 R. Remark 2.1.24. Let M be a C 1 manifold, not necessarily compact. (i) Theorem 2.1.20 (i) has a generalisation to the continuity s s .M; E/ ! Hloc .M; F / A W Hcomp

for every s 2 R; moreover, Theorem 2.1.20 (iii) remains true if we assume one of the factors to be properly supported. (ii) An elliptic operator A 2 L cl .M I E; F / has a properly supported parametrix P 2 L cl .M I F; E/ in the sense that the relations (2.1.24) hold, and we have  .P / D  .A/1 . The Fredholm property of (2.1.25) follows from the fact that A has a parametrix P 1 which can˚easily P1 be obtainedj by setting P0 WD op. .A/ /, cf. Proposition 2.1.19, and P WD j D0 .I  P0 A/ P0 , where the first factor is understood as an asymptotic

sum, using .I  P0 A/j 2 Lj .M I E; E/, j 2 N, cf. Remark 2.1.18. P1.cl/ The asymptotic sum j D0 .I  P0 A/j is referred to as a formal Neumann series. The necessity of the ellipticity for the Fredholm property is proved, for instance, in [154]. As a consequence of Theorem 2.1.23 (ii) we see that ind A WD dim ker A  dim coker A, the index of A, is independent of s. Theorem 2.1.23 (iii) yields elliptic regularity of solutions u 2 H 1 .M; E/ to an elliptic equation Au D f 2 H s .M; F /, s 2 R, namely, u 2 H sC .M; E/. In fact, using a parametrix P of A as a left parametrix, we obtain PAu D Pf 2 H sC .M; E/, but Cl D I  PA implies PAu D u  Cl u where Cl u 2 H 1 .M; F /, and it follows that u D Pf C Cl u 2 H sC .M; F /.

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2 Symbolic structures and associated operators

The latter consideration gives us, in particular, V D ker A  C 1 .M; E/. Moreover, the relation ker A C im A D H s .M; F / (which is a direct decomposition, i.e., ker A \ im A D f0g) allows us to set W D ker A  C 1 .M; F /, cf. Theorem 2.1.23 (ii). The assertions of Theorem 2.1.23 (iii) have an abstract background which will be useful in other situations below. We consider operators A between Hilbert spaces that belong to scales fHis gs2R , i D 1; 2, where first A W H11 ! H21 (2.1.26) T is continuous, Hi1 WD s2R His . Concerning the scales we assume the following properties: 0

(i) There are continuous embeddings His ,! His for s 0  s that are compact for s 0 > s. (ii) The space Hi1 is dense in His for every s 2 R, i D 1; 2. (iii) If V  Hi1 , i D 1; 2, is a finite-dimensional subspace and CV W Hi0 ! V a projection, then CV induces continuous operators CV W His ! V for all s > 0. Moreover, assume that A has an order  2 R in the sense that there is a constant c > 0 such that (2.1.26) extends to continuous operators As W H1s ! H2s

(2.1.27)

for all s > c (the aspect with the constant c D c.A/ is not relevant for Theorems 2.1.20 and 2.1.23, but in boundary value problems it will play a role). If it is clear from the context which s is considered for the operator, we also write A instead of As . In addition we assume (iv) If (2.1.26) extends to a Fredholm operator (2.1.27) for all s > c, then there is a continuous operator P W H21 ! H11 that extends to a parametrix Ps W H2s ! H1s of As for every s > c, i.e., the remainders Cs;l WD I  Ps As ;

Cs;r WD I  As Ps

are compact in the respective spaces and they induce continuous operators Cs;l W H1s ! H11 , Cs;r W H2s ! H21 for all s > c. Remark 2.1.25. Under the above-mentioned conditions the dimensions of ker As and coker As of the Fredholm operator A W H1s ! H2s are independent of s > c, we have V WD ker As  H11 , and there is a finite-dimensional subspace W  H21 such that W C im As D H2s , W \ im As D f0g for every s > c. Moreover, the parametrix P can be chosen in such a way that the remainders Cl D I  PA and Cr D I  AP are projections to V and W , respectively.

2.1 Scalar pseudo-differential calculus

71

For later use we now prepare some well-known material on parameter-dependent pseudo-differential operators. First we have the Fréchet space L1 .M I E; F / of smoothing operators on M , and we set L1 .M I E; F I Rl / WD S.Rl ; L1 .M I E; F // which is the space of smoothing parameter-dependent operators (between sections of E; F 2 Vect.M //. The space L .M I E; F I Rl / .cl/

(2.1.28)

of parameter-dependent pseudo-differential operators of order  is defined in a similar manner as (2.1.20), now for C./ 2 L1 .M I E; F I Rl / and local operators A ./ 2 L . I C j ; C k I Rl /. Here L .I C j ; C k I Rl / means the space of all kj matrices .cl/ .cl/ of operators of the form Op.a/./ for arbitrary a.x; x 0 ; ; / 2 Scl .    Rn  Rl /. Remark 2.1.26. (i) For every A 2 L .M I E; F I Rl / there is a properly supported .cl/ A0 2 L .M I ; E; F I Rl / (in the sense that the support of the distributional kernel is .cl/ proper and independent of  2 Rl ) such that A  A0 2 L1 .M I E; F I Rl /; (ii) L .M I E; F I Rl / is a Fréchet space in a natural way. .cl/ Theorem 2.1.27. For every A./ 2 L .M I E; F I Rl / on a closed, compact C 1 manifold M and    we have ( for   0; .1 C jj/ kA./kL.H s .M;E /;H s .M;F //  c  .1 C jj/ for   0; with constants c D cs; > 0. For a proof see, for instance, [188, Theorem 1.2.19]. l For classical parameter-dependent operators A./ 2 L cl .M I E; F I R / we have a parameter-dependent homogeneous principal symbol (2.1.21), now with the projection M W .T  M  Rl / n 0 ! M

.0 means . ; / D 0/:

In this case,  .A/.x; ; / is defined in terms of the homogeneous principal components of local amplitude functions in . ; / ¤ 0 (at x D x 0 ). l Definition 2.1.28. An A./ 2 L cl .M I E; F I R / is called parameter-dependent elliptic (of order ) if    .A/ W M E ! M F

for M W .T  M  Rl / n 0 ! M , is an isomorphism. If necessary we write  instead of  .A/ in the parameter-dependent case.

; .A/

Theorem 2.1.29. Let M be a closed, compact C 1 manifold, and let A./ 2 l L cl .M I E; F I R / be parameter-dependent elliptic (of order ). Then

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2 Symbolic structures and associated operators

(i) A./ W H s .M; E/ ! H s .M; F /

(2.1.29)

is a family of Fredholm operators of index 0. Moreover, there is a constant C > 0 such that the operators (2.1.29) are isomorphisms for all jj  C . This holds for all s 2 R. l (ii) A./ has a parameter-dependent parametrix P ./ 2 L cl .M I F; EI R /, i.e., 1 l 1 I  P ./A./ 2 L .M I E; EI R /, I  A./P ./ 2 L .M I F; F I Rl /.

The existence of a parameter-dependent parametrix can be proved in an analogous manner as in Theorem 2.1.23. From the relations of Theorem 2.1.29 (ii) it follows easily that (2.1.29) become isomorphisms for large jj. Since A./ is elliptic in the usual sense for every fixed , Theorem 2.1.23 (i) gives us the Fredholm property of (2.1.29) for all . The continuous dependence on  of the Fredholm operators then yields ind A./ D 0 for all . Remark 2.1.30. Let M be a C 1 manifold (not necessarily compact), and consider l a parameter-dependent elliptic A./ 2 L cl .M I E; F I R /. Then A./ has a properly  l supported parametrix P ./ 2 Lcl .M I F; EI R / in the sense that the identities of Theorem 2.1.29 (ii) hold. Theorem 2.1.31. Let M be closed and compact.Then for every  2 R there exists   l s an RE ./ 2 L cl .M I E; EI R / that induces isomorphisms RE ./ W H .M; E/ ! s l H .M; E/ for all s 2 R and  2 R .  Let us give an idea on how to construct elements RE ./, for convenience, for the case of the trivial bundle of fibre dimension 1. Taking a pseudo-differential family with the local amplitude functions .1 C j j2 C jj2 C c 2 /=2 , c 2 R, we obtain a global lC1 classical pseudo-differential operator R .; c/ 2 L cl .M I R;c / with the parameterdependent homogeneous principal symbol j j2 C jj2 C c 2 . From Theorem 2.1.29 we see that R .; c/ W H s .M / ! H s .M / induces isomorphisms for j; cj  C . Thus, setting R ./ WD R .; c1 / for a real c1 such that jc1 j  C we obtain an l l element in L cl .M I R / which induces isomorphisms for all  2 R . s l ./ 2 L Remark 2.1.32. If RE cl .M I E; EI R / is an order reducing family of order s s 2 R as in Theorem 2.1.31, then kRE ./ukL2 .M;E / is a parameter-dependent family of norms on the space H s .M; E/, equivalent to each other for different  2 Rl . We shall employ such families in Section 3.3 below for constructing Sobolev spaces on an infinite cylinder M  R.

2.2 Calculus with operator-valued symbols The calculi of this book employ several extensions of scalar pseudo-differential operators (and adapted scales of Sobolev spaces) to the case of operator-valued amplitude functions. We discuss

2.2 Calculus with operator-valued symbols

73

here amplitude functions with twisted homogeneity. Another variant is based on parameterdependent operators with parameters as covariables.

2.2.1 Symbols and operators with twisted homogeneity The generalisation of standard elements of the pseudo-differential calculus from scalar to operator-valued symbols is straightforward as far as absolute values in the symbolic estimates are replaced by operator norms (referring to, say, Hilbert spaces). However such a ‘naive’ approach is not sufficient for our applications. There are many other possible extensions to the operator-valued set-up. In the present section we consider symbols with ‘twisted homogeneity’. Although the structure of this calculus with symbolic estimates, operator conventions, etc., is similar to the scalar case, there are many new surprising and beautiful examples, as we shall see below. Let H be a Hilbert space, and  WD f g2RC a strongly continuous group of isomorphisms  W H ! H (‘strongly continuous’ means that  u as a function of  2 RC belongs to C.RC ; H / for every u 2 H ) such that  % D % for all ; % 2 RC . We then say that H is endowed with a group action . It is well known, that there are constants c; M > 0 such that k kL.H /  c.max.; 1 //M (2.2.1)  ˚

 for all  2 RC . We may set M WD log sup k kL.H / W e 1    e with Euler’s constant e (M < 1 is a consequence of the Banach–Steinhaus theorem). If necessary we write M D M./ (2.2.2) for a possible choice of the constant M in (2.2.1). Remark 2.2.1. Let f g2RC be a group action in H . Then f g2RC , the family of adjoint operators, is again a group action in H . A proof may be found in [146, Corollary 1.10.6]. Combined with Peetre’s inequality .1 C j  j/s  .1 C jj/s .1 C j j/jsj for every real s and ;  2 Rq (or, in a slightly modified form, h  is  c jsj his h ijsj for some constant c > 0) the relation (2.2.1) yields 1 kh i hi kL.H /  ch  iM

(2.2.3)

1 hi kL.H /  for every ;  2 Rq , with some c > 0. In fact, from (2.2.1) we have kh i   M  hi h i M c .max h i ; hi . Moreover, Peetre’s inequality for s D M gives us hi  h i M ch  i for another c > 0. Changing the roles of and  we get the same estimate   h i M for hi .

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2 Symbolic structures and associated operators

The following Definition 2.2.3 is motivated by a certain reformulation of standard pseudo-differential operators in a Cartesian product (although most of the examples in our applications will be of a more subtle structure).  Let  Rq be an open set and choose an element p.y; ; / 2 S.cl/ .  RnCq /.

; Setting “ 0 a.y; / WD opx .p/.y; / W u.x/ ! e i.xx / p.y; ; /u.x 0 /dx 0 μ (2.2.4) we obtain a family a.y; / 2 C 1 .  Rq ; L.H s .Rn /; H s .Rn /// for every s 2 R. It is clear that then Opy .a/ D Opy .opx .p// (2.2.5) is the pseudo-differential operator in Rn   associated with the symbol p. A sim1 opx .p/.y; /hi D opx .p /.y; / for p .y; ; / WD ple calculation gives us hi p.y; hi ; /. From the symbolic estimates for p, i.e., jp.y; ; /j  ch ; i for a constant c > 0; y 2 K b , we obtain jp.y; hi ; /j  chi h i ;

(2.2.6)

using the relation hhi ; i D hih i. Now (2.2.6) yields Z 2 2 k opx .p /.y; /ukH D h i2.s/ jp.y; hi ; /u. /j O μ s .Rn / Z 2 2  chi2 h i2s ju. /j O μ D chi2 kukH s .Rn / 1 for every s. It follows that khi opx .p/.y; /hi kL.H s .Rn /;H s .Rn //  chi for every s 2 R. More generally, we obtain 1 khi fDy˛ Dˇ opx .p/.y; /ghi kL.H s .Rn /;H s .Rn //  chijˇ j

for arbitrary ˛; ˇ 2 N q , y 2 K b , with a constant c D c.˛; ˇ; K/ > 0. Similar estimates are true when p depends on x in a controlled way, e.g., in the simplest case, when p is independent of x for jxj > R for some R > 0. Remark 2.2.2. Observe the following result in connection with anisotropic reformulations of the kind given in (2.2.5) of pseudo-differential operators. For all  2 R,  the map p.y; ; / ! p.y; hi ; / defines continuous operators S.cl/ .  RnCq /!

;   S.cl/ .  Rq ; S.cl/ .Rn //; cf. also the notation (2.1.3). z be Hilbert spaces with group actions  and , Definition 2.2.3. (i) Let H and H Q z /,  2 R, of operator-valued symbols respectively. Then the space S  .U  Rq I H; H of order  2 R on an open set U Rp is defined to be the set of all a.y; / 2 z // such that C 1 .U  Rq ; L.H; H sup .y;/2KRq

1 hiCjˇ j kQ hi .Dy˛ Dˇ a.y; //hi kL.H;Hz /

(2.2.7)

75

2.2 Calculus with operator-valued symbols

is finite for every K b U and arbitrary multi-indices ˛ 2 N p ; ˇ 2 N q . As in the z /. scalar case we write  D ord a when a 2 S  .U  Rq I H; H z / denote the space of all elements a./ .y; / 2 (ii) Let S ./ .U  .Rq n f0g/I H; H z // such that C 1 .U  .Rq n f0g/; L.H; H a./ .y; / D  Q  a./ .y; /1

(2.2.8)

z / of for all  2 RC ; .y; / 2 U  .Rq n f0g/. Moreover, the space Scl .U  Rq I H; H all classical symbols of order  (with twisted homogeneity in the sense of the relations z / such that there (2.2.8)) is defined as the subspace of all a.y; / 2 S  .U  Rq I H; H .j / q z are elements a.j / .y; / 2 S .U  .R n f0g/I H; H /; j 2 N, with the property rN .y; / WD a.y; /  ./

N X

z/ a.j / .y; / 2 S .N C1/ .U  Rq I H; H

j D0

for every N 2 N and any excision function . Remark 2.2.4. The components a.j / .y; /, j 2 N, of an a.y; / 2 Scl .U  z / are uniquely determined and can be represented in the form Rq I H; H jX 1 n o a.l/ .y; /  a.j / .y; / D lim Cj Q 1 a.y; /  !1

(2.2.9)

lD0

(the sum on the right-hand side vanishes for j D 0). The convergence in (2.2.9) refers z /, uniformly in any compact subset of U  .Rq n f0g/ 3 .y; /. to the norm in L.H; H In fact, let us choose an excision function ./; then we have ./ D 1 for every .y; / varying in a compact set M  U .Rq nf0g/ and  sufficiently large. Therefore, for .y; / 2 M and large  we have for suitable constants C; Cz > 0 jX 1

n o

Cj 1 Q  a.y; /  a.l/ .y; /   a.j / .y; /

 lD0 j

n o X

Cj 1 D  Q  a.y; /  ./a.l/ .y; /  lD0

z/ L.H;H

z/ L.H;H

 C kQ 1 hi kL.Hz / Cj hi.j C1/ khi1 kL.H /  Cz Cj hi.j C1/ which tends to zero for  ! 1, uniformly in .y; / 2 M . z / is a Fréchet space with the system Remark 2.2.5. (i) The space S  .U  Rq I H; H p q of semi-norms (2.2.7), ˛ 2 N , ˇ 2 N , K b U . z / is Fréchet in the topology of the projective limit (ii) The space Scl .U  Rq I H; H with respect to the maps z / ! S .j / .U  .Rq n f0g/I H; H z /; Scl .U  Rq I H; H

a ! a.j / ; j 2 N;

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2 Symbolic structures and associated operators

and z / ! S .N C1/ .U  Rq I H; H z /; Scl .U  Rq I H; H

a ! rN ; N 2 N:

 z / of all elements a./ with constant coefficients (iii) The subspace S.cl/ .Rq I H; H z /, and we have (i.e., independent of y) is closed in S  .U  Rq I H; H .cl/

  z / D C 1 .U; S  .Rq I H; H z // D C 1 .U / ˝ z /: y  S.cl/ S.cl/ .U  Rq I H; H .Rq I H; H .cl/

T  q z / WD z Observe that the space S 1 .U  Rq I H; H 2R S .U  R I H; H / (in z //. the Fréchet topology of the projective limit) coincides with C 1 .U; S.Rq ; L.H; H We employ analogous notation in the case of a Fréchet space E that is written as a projective limit of Hilbert spaces Ek ; k 2 N, with continuous embeddings    ,! EkC1 ,! Ek ,!    ,! E0 such that E0 is endowed with a group action f g2RC that restricts to a group action on Ek for every k. In this case we have spaces of symbols  S.cl/ .U Rq I H; Ek / for the pairs of Hilbert spaces H; Ek with group action, and there   are continuous embeddings S.cl/ .U  Rq I H; EkC1 / ,! S.cl/ .U  Rq I H; Ek / for all k (this refers to natural Fréchet topologies in the respective spaces of symbols). We then set   S.cl/ .U  Rq I H; E/ WD lim S.cl/ .U  Rq I H; Ek /:  k2N

z written as E D lim Finally, in the case of Fréchet spaces E and E, E ; Ez D k2N k Ez with Hilbert spaces Ek ; Ezk and group actions f g2RC ; fQ  g2RC , relim k2N k  z r WD .U  Rq I E; E/ spectively, we fix a map r W N ! N, form the space S.cl/ lim S  .U  Rq I Er.k/ ; Ezk /, and define k2N .cl/ [   z WD z r: S.cl/ .U  Rq I E; E/ S.cl/ .U  Rq I E; E/ (2.2.10) r

The groups  and Q on the spaces are kept fixed and known in a concrete context.  z If we want to Clearly their choice affects the symbol spaces S.cl/ .U  Rq I E; E/. indicate the groups in the notation explicitly, we also write  z ;Q .U  Rq I E; E/ S.cl/

(2.2.11)

 z instead of S.cl/ .U  Rq I E; E/.

Remark 2.2.6. In the general discussion of operator-valued symbols we will often z are Hilbert spaces with group action. Most of the assume that the spaces H and H considerations will have a straightforward extension to the case of Fréchet spaces with group action. We do not always mention this explicitly; if necessary we tacitly use the corresponding generalisation.

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77

Example 2.2.7. Let p.x; y; ; / 2 S  .Rn    RnCq / be independent of x for large jxj, and define a.y; / by the expression (2.2.4). Then we obtain a.y; / 2 S  .  Rq I H s .Rn /; H s .Rn // and even Dy˛ Dˇ a.y; / 2 S jˇ j .Rq I H s .Rn /; H sCjˇ j .Rn // for all ˛; ˇ 2 N q , s 2 R. Such a smoothing effect in the image after differentiation in the covariable is typical in a similar form also for the operator-valued symbols of the edge calculus below. However, we may (and will) ignore this, since we keep in mind also the role of the interior symbol (the analogue of the present p.x; y; ; //. Note that (in the x-independent case) p.y; ; / 2 Scl .  RnCq / implies the property a.y; / 2 Scl .  Rq I H s .Rn /; H s .Rn //: If p.j / .y; ; / is the homogeneous component of p of order   j in . ; / 6D 0 then a.j / .y; / D opx .p.j / /.y; / is the (twisted) homogeneous component of a of order j in .y; / 2 .Rq nf0g/, j 2 N. Let  ! Œ be a function in C 1 .Rq / with the properties Œ > 0 for all  2 Rq , and Œ D jj for jj  C

(2.2.12)

for some C > 0. z be Hilbert spaces with group actions f g2R and Examples 2.2.8. (i) Let H and H C z // is an arbitrary fQ  g2RC , respectively. Then, if a.y; / 2 C 1 .U  Rq ; L.H; H element that is homogeneous of order  2 R in  for large jj, i.e., a.y; / D  Q  a.y; /1 z /. for all   1 and jj  c for some c > 0, then we have a.y; / 2 Scl .U  Rq I H; H z are Fréchet spaces with group action. A similar observation is true, if H or H z N N z (ii) For H D C .H D C / we usually assume  D idH .Q  D idHz /, for all  2 RC (otherwise we indicate the specific groups explicitly). In particular, for z D C we simply write H DH  S.cl/ .U  Rq / (2.2.13)  z /. The spaces (2.2.13) are the symbol spaces from the instead of S.cl/ .U  Rq I H; H standard (scalar) pseudo-differential calculus. For general

z ; fQ  g2R // .H; f g2RC / ..H C z

  z // play the role of the elements of S.cl/ .U  Rq I H; C N / and .S.cl/ .U  Rq I C N ; H trace (potential) symbols.

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2 Symbolic structures and associated operators

xC R x C //, cf. the notation (2.1.5), and set (iii) Let f .y; I t; t 0 / 2 Scl .U  Rq I S.R Z 1  g.y; /u.t / WD Œ

f .y; I t Π; t 0 Π/u.t 0 /dt 0 0

for ;  2 R; u 2 L2 .RC /. Then we have CC 1 2

g.y; / 2 Scl and also

x C // .U  Rq I L2 .RC /; S.R

CC 1 2

g  .y; / 2 Scl

x C // .U  Rq I L2 .RC /; S.R

(2.2.14)

(2.2.15)

(for the .y; /-wise adjoint g  with respect to the L2 .RC /- scalar product). In the x C / are endowed with group relations (2.2.14) and (2.2.15) the spaces L2 .RC / and S.R 1 x C / is regarded as the actions . u/.t/ D  2 u.t /;  2 RC I the Fréchet space S.R x C / D lim ht ik H k .RC / of the Hilbert spaces ht ik H k .RC /, projective limit S.R k2N endowed with the group action f g2RC for every k 2 N. Further interesting examples will be discussed below, especially, in Section 6.4.3 and Chapter 7. It may be instructive to slightly reformulate the symbolic estimates in Definition 2.2.3 (i) by using parameter-dependent norms in the Hilbert spaces with group z be Hilbert spaces with group actions  D f g2R and action. Let H and H C Q D fQ  g2RC , respectively. Let H denote the space H endowed with the norm 1 ukH kukH WD khi

(2.2.16)

z be the space H z with for any fixed norm k  kH in the space H . Analogously, let H 1 the norm kvkHz WD kQ hi vkHz . We then obtain a parameter-dependent operator norm z / 3 A by in the space L.H; H 1 kAkL.H ;Hz / WD kQ hi Ahi kL.H;Hz / :

(2.2.17)

Let us call two parameter-dependent norms k  kH and k  kH0 in the same space H equivalent, if there are constants c; C > 0 such that ckukH  kukH0  C kukH z that is for all  2 Rq , u 2 H . Then, if k  kHz0 is a parameter-dependent norm in H equivalent to k  kHz , we obtain mkAkL.H ;Hz /  kAkL.H0 ;Hz0 /  M kAkL.H ;Hz / z /, with suitable constants m; M > 0. for all  2 Rq , A 2 L.H; H Remark 2.2.9. From (2.2.17) it follows that the expressions (2.2.7) for the symbolic z / take the form estimates for a.y; / 2 S  .U  Rq I H; H sup .y;/2KRq

hiCjˇ j kDy˛ Dˇ a.y; /kL.H ;Hz / < 1

for K b U; ˛ 2 N p ; ˇ 2 N q .

(2.2.18)

2.2 Calculus with operator-valued symbols

79

1 1 Remark 2.2.10. The parameter-dependent norms kŒ ukH and khi ukH in H are equivalent. This allows us to equivalently formulate the estimates (2.2.18) by using Œ

instead of hi. z /, j 2 N, be an arbitrary seTheorem 2.2.11. Let aj .y; / 2 S j .U  Rq I H; H z/ quence, j ! 1 as j ! 1. Then there exists an a.y; / 2 S  .U  Rq I H; H for  D maxfj W j 2 Ng such that N   X aj ! 1 as N ! 1: ord a 

(2.2.19)

j D0

z /. Moreover, let E Any such a.y; / is uniquely determined mod S 1 .U  Rq I H; H and Ez be Fréchet spaces with group actions  and , Q respectively, and let aj .y; / 2 z r (cf. the notation in (2.2.10)), j 2 N, be an arbitrary sequence, S j .U  Rq I E; E/ where the function r W N ! N is independent of j . Then there is an a.y; / 2 z r such that the relation (2.2.19) holds, and a.y; / is uniquely S  .U  Rq I E; E/ T  q z r .WD z determined mod S 1 .U  Rq I E; E/ 2R S .U  R I E; E/r /. P1As in the scalar case if a.y; / is a symbol as in Theorem 2.2.11 we write a.y; /  j D0 aj .y; / and call a.y; / an asymptotic sum of the symbols aj .y; /. Similarly as (2.1.6) we can produce an asymptotic sum as a convergent series.

2.2.2 Abstract edge spaces Parallel to the spaces of operator-valued symbols of Definition 2.2.3 there is a class of natural so-called abstract edge spaces with parameter spaces H , endowed with a group action  D f g2RC . This category of spaces exists since 1988, cf. [180]. We content ourselves here with the case that H is a Hilbert space or a Fréchet space with group action, written as a projective limit of Hilbert spaces. Many assertions remain true when H is a Banach space with group action, cf. [79]. Definition 2.2.12. Let H be a Hilbert space with group action  D f g2RC . Then W s .Rq ; H /, s 2 R, denotes the completion of the space S.Rq ; H / with respect to the norm 12 Z 2s 1 2 O : (2.2.20) hi khi u./k H d m

Example 2.2.13. Let H WD H s .Rm /, s 2 R, and . u/.x/ WD  2 u.x/,  2 RC . Then we have W s .Rq ; H s .Rm // D H s .Rm  Rq /: (2.2.21) 1

Similarly, for H D H s .RC / and . u/.r/ D  2 u.r/,  2 RC , we have W s .Rq ; H s .RC // D H s .Rq  RC / ˇ (H s .Rq  RC / WD H s .Rq  R/ˇRq R ). C

(2.2.22)

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2 Symbolic structures and associated operators

Let us check (2.2.21). We have for an element u.x; y/ 2 S.Ryq ; H s .Rm x // with f .x; / WD .Fy! u/.x; /; g. ; / WD .Fx! f /. ; / Z 1 2 kuk2W s .Rq ;H s .Rm // D hi2s khi f .; /kH s .Rm / d  Z Z 2s 2s m 2 D hi h i hi jg.hi ; /j d d Z Z ˝ ˛2s D hi2s jg. ; /j2 d d hi “ 2 D h ; i2s jg. ; /j2 d d D kukH s .RmCq / : Here we employed the relation  Fx! D Fx! 1 for every  2 RC , and hi2 h

  2 j j2 D hi2 C j j2 D h ; i2 : i D hi2 1 C hi hi2

The proof of (2.2.22) is of a similar structure. The general properties of the spaces W s .Rq ; H / are investigated in [182], see also [188], or [79]. In particular, we have that W s .Rq ; H / is contained in S 0 .Rq ; H / .WD L.S.Rq /; H //. Moreover, observe that we have continuous embeddings 0

W s .Rq ; H / ,! W s .Rq ; H / for every real s 0  s. If  Rq is an open set, we have the spaces s s Wcomp .; H / and Wloc .; H /; s .; H / is defined to be the set of all u 2 D 0 .; H /.WD L.C01 ./; H // where Wcomp with compact support such that their extension by zero to Rq n belongs to W s .Rq ; H /; s furthermore Wloc .; H / denotes the space of all u 2 D 0 .; H / such that 'u 2 s Wcomp .; H / for every ' 2 C01 ./.

Remark 2.2.14. (i) The space W s .Rq ; H / depends on the group action  D f g2RC and (if necessary) we also write W s .Rq ; H / . (ii) A possible choice for the group action is  D idH for all  2 RC . In that case we employ the notation H s .Rq ; H / instead of W s .Rq ; H /idH . We then have 1 kukW s .Rq ;H / D k.F 1 hi F /ukH s .Rq ;H / ;

i.e., the operator

1 F 1 hi F W W s .Rq ; H / ! H s .Rq ; H /

is an isomorphism for every s 2 R. More generally, if  and Q are group actions in 1 H , then F 1 Q hi hi F W W s .Rq ; H / ! W s .Rq ; H /Q is an isomorphism for every s 2 R.

2.2 Calculus with operator-valued symbols

81

T s q 1 q (iii) Setting W 1 .Rq ; H / WD s2R W .R ; H / we have W .R ; H / D 1 q H .R ; H / for every . (iv) Using the notation (2.2.16) the norm (2.2.20) takes the form nZ o1=2 2 p 2s ./ku./k O ; p./ WD hi: H d Replacing kkH by an equivalent parameter-dependent norm or hi by another function p./ satisfying chi  p./  C hi for certain c; C > 0 we obtain equivalent norms in the space W s .Rq ; H /; s 2 R. For reference below we want to recall a number of general properties of abstract edge spaces. Let H0 ; H1 be Hilbert spaces, continuously embedded in a Hausdorff topological vector space V . Let f g2RC be a group action on H0 C H1 that restricts to group actions fi g2RC on Hi for i D 0; 1. Then we have W s .Rq ; H0 C H1 / D W s .Rq ; H0 / C W s .Rq ; H1 /; cf. [188, Section 1.3.2]. According to the complex interpolation method, cf. [111], we have an interpolation space ŒH0 ; H1  for every  2 Œ0; 1 . Then f g2RC induces strongly continuous group actions f g2RC on ŒH0 ; H1  for all  2 Œ0; 1 . Theorem 2.2.15 ([79]). For every s0 ; s1 2 R;  2 Œ0; 1 and s WD .1   /s0 C s1 we have ŒW s0 .Rq ; H0 /; W s1 .Rq ; H1 /  D W s .Rq ; ŒH0 ; H1  /: Let us come back to the relation in Remark 2.2.14 (iii) (which is a consequence of the estimate (2.2.1)). It follows that \ W 1 .Rq ; H / D H s .Rq ; H /; s2R

R 2 where H s .Rq ; H / WD fu 2 S 0 .Rq ; H / W hi2s ku./k O H d < 1g. From Theorem 2.2.15 we obtain \ H s .Rq ; H /: W 1 .Rq ; H / D s2N

Using the common identification L2 .Rq ; H / D H 0 .Rq ; H / which is justified by Plancherel’s theorem, we have for s 2 N H s .Rq ; H / D fu 2 L2 .Rq ; H / W Dy˛ u 2 L2 .Rq ; H / for all ˛ 2 N q ; j˛j  sg: This gives us the following result. Proposition 2.2.16. We have W 1 .Rq ; H / D fu 2 L2 .Rq ; H / W Dy˛ u 2 L2 .Rq ; H / for all ˛ 2 N q g:

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2 Symbolic structures and associated operators

Theorem 2.2.17 ([39], [188]). Let H be a Hilbert space and f g2RC a group action on H , where  is unitary for every  2 RC . Let  W Rq ! Rq be a diffeomorphism satisfying c 1  j det.d1 /.y/j  c for all y 2 Rq for some c  1. Then the pull back  W C01 .Rq ; H / ! C01 .Rq ; H / extends to an isomorphism  W W 0 .Rq ; H / ! W 0 .Rq ; H /: Remark 2.2.18. Let H be a Hilbert space and f g2RC be a group action on H . Moreover, let  W Rq ! Rq be a diffeomorphism as in Theorem 2.2.17. Then the pull back  W C01 .Rq ; H / ! C01 .Rq ; H / extends to an isomorphism  W W 1 .Rq ; H / ! W 1 .Rq ; H /: We now formulate a version of the Calderón–Vaillancourt theorem for abstract edge Sobolev spaces. We give a proof that extends a method of Hwang [84] from the scalar case to the case of operator-valued symbols, cf. Seiler [205]. Theorem 2.2.20 is much more general than really necessary for the edge pseudodifferential calculus below, but we think that it is of independent interest. For the purposes of the edge calculus more specific versions of the continuity of Op.a/ were known before, see, for instance, [182]. In the sequel, when ˛; ˛Q 2 N q are multi-indices, we write ˛  ˛Q if ‘’ holds for every component. z be Hilbert spaces and Q a group action in H z . Let Lemma 2.2.19. Let H and H M D M./ Q 2 N be a constant in the meaning of (2.2.2) for . Q Assume that a.y; / 2 z // satisfies the condition C 1 .R2q ; L.H; H L .a/ WD supfhiL kDy˛ Dˇ a.y; /kL.H;Hz / W .y; / 2 R2q ; ˛  ˛; ˇ  ˇg < 1 for a certain L 2 N, and ˛ WD .M; : : : ; M /; ˇ WD .1; : : : ; 1/; here M 2 N is a constant such that (2.2.1) holds for . Q Moreover, let ' 2 C01 .R2q / be equal to 1 near 0, and set a" .y; / WD '."y; "/a.y; / for 0 < "  1. Finally, let u 2 S.Rq ; H /; v 2 z /. Then we have S.Rq ; H z /; (i) Op.a/u 2 W 0 .Rq ; H (ii) lim"!0 .Op.a" /u; v/W 0 .Rq ;Hz / D .Op.a/u; v/W 0 .Rq ;Hz / . R Proof. Set A WD Op.a/ D e iy a.y; /u./μ. O Write .i C y/ˇ WD .i C y1 /ˇ1 : : : .i C yq /ˇq ; .i C y/ˇ WD ..i C y/ˇ /1 ; .i C Dy /ˇ WD .i C Dy1 /ˇ1 : : : .i C Dyq /ˇq for a multi-index ˇ D .ˇ1 ; : : : ; ˇq / 2 N q and Dyj D i @y@j . We then have .i C Dy /ˇ e ixy D .i C x/ˇ e ixy

(2.2.23)

2.2 Calculus with operator-valued symbols

for arbitrary x; y 2 Rq . Integration by parts gives us Z Au.y/ D .i C y/ˇ e iy .i  D /ˇ fa.y; /u./gμ: O

83

(2.2.24)

because of uO 2 S.Rq ; H / The factor .i C y/ˇ is square integrable. R iy Moreover, ˛ ˇ O and the assumptions on a we have kDy e .i  D / fa.y; /u./gμk z < 1 for H q ˛ 2 q z every ˛ 2 N , ˛  ˛. This yields .i C Dy / Au 2 L .R ; H / and Z Z 2 1 2 kAukW 0 .Rq ;Hz / D khi F .Au/./kHz d  c k.i C /˛ F .Au/./k2Hz d Z D c kF ..i C Dy /˛ Au/./k2Hz d Z D c.2/q k.i C Dy /˛ Au.y/k2Hz dy < 1 which is just the relation (i). For (ii) we set b" WD a"  a, B" WD Op.b" /, and verify lim"!0 .B" u; v/W 0 .Rq ;Hz / D 0. The relation Z .B" u; v/W s .Rq ;Hz / D



 1  1 F .B" u/./; .Q hi / Q hi v./ O z d H

z/ together with Plancherel’s formula shows that it suffices to prove B" u ! 0 in L2 .Rq ; H for " ! 0. To this end we observe that L .b" / < C for all 0 < "  1 for some C > 0 independent of " (after a straightforward computation using the Leibniz rule). Thus, with b" instead of a in (2.2.24), it follows that kB" u.y/k2z  cj.i Cy/2ˇ j 2 L1 .Rq /, H with a constant c > 0 independent of "; 0 < "  1. Applying Lebesgue’s theorem on dominated convergence it remains to show that kB" u.y/kHz tends to zero for " ! 0 for every y 2 Rq ; this is again a consequence of Lebesgue’s theorem on dominated convergence. z be Hilbert spaces with group actions  and , Theorem 2.2.20. Let H and H Q respecz // satisfies the estimate tively. Assume that a function a.y; / 2 C 1 .R2q ; L.H; H

˚ 1 ˛ ˇ fDy D a.y; /ghi L.H;Hz / W .a/ WD sup Q hi

.y; / 2 R2q ; ˛  ˛; ˇ  ˇ < 1 for ˛ WD .M C 1; : : : ; M C 1/, ˇ WD .1; : : : ; 1/, with M 2 N being a constant such that (2.2.1) holds for . Q Then Op.a/ induces a continuous operator z /; Op.a/ W W 0 .Rq ; H / ! W 0 .Rq ; H and we have k Op.a/kL.W 0 .Rq ;H /;W 0 .Rq ;Hz //  c.a/ for a constant c greater than 0 and independent of a.

84

2 Symbolic structures and associated operators

Proof. Let A WD Op.a/. We first assume that a has a compact support with respect to z / we have .y; / 2 R2q . Then for u 2 C01 .Rq ; H /, v 2 S.Rq ; H Z  1  1 .Au; v/W 0 .Rq ;Hz / D Q hi F .Au/./; Q hi v./ O z d H ZZZZ  1  1 D e i. /yi x Q hi a.y; /u.x/; Q hi v./ O z dydxμ d: H The relation (2.2.23) allows us to rewrite the latter expression, after integration by parts, as ZZZZ e i. /yi x .i C y  x/ˇ .i  D /ˇ   1 1 f.i C  /˛ Q hi .i  Dy /˛ a.y; /u.x/; Q hi v./ O z gdydxμ d: H P Using the relation .i  D/ˇ .ab/ D ˇ .1/jj Œ.i  D/ˇ a D  b (obtained by induction) it follows that .Au; v/W 0 .Rq ;Hz / X“  1   e iy ŒQ h i D D .i  Dy /˛ a.y; /h i f .y; /; g .y; / Hz dyμ  ˇ

z -valued functions with (continuous) H Z 1 f .y; / WD h i e i x .i C y  x/ˇ u.x/dx; Z 1 1 g .y; / WD .1/jj e iy .Q hi Q h i / .i  D /ˇ .i C  /˛ Q hi v./d: O This yields the estimate j.Au; v/W 0 .Rq ;Hz / j  c.a/

X

kf kL2 .R2q ;H / kg kL2 .R2q ;Hz / :

ˇ

Applying Plancherel’s formula we obtain “

 1

2  2 q ˇ 1

Q Q kg kL D .2/ .i C  /˛ Q hi v./ O Hz dd : 2 .R2q ;H z/ hi h i .i  D / Using the estimate (2.2.3) it follows that

 1  ˇ

Q Q .i C  /˛ L.Hz / hi h i .i  D /  ch  iM j.i C  /˛ j  ch 1  1 iM : : : h q  q iM j.i C  /˛ j which gives us

Z 2 kg kL 2 .R2q ;H z/  c

j.i C /2ˇ jd kvk2W 0 .Rq ;Hz / :

2.2 Calculus with operator-valued symbols

85

Let h.z/ WD .i  z/ˇ , hy .z/ WD h.z  y/, and observe that O O hOy . / D e i y h. /; h i2L h. / D F Œ.1  /L h . /:

(2.2.25)

Then, because of F .' / D .2/q 'O  O and the relations (2.2.25) we obtain f .y; /

Z

hOy .  x/u.x/dx O Z 1 1 D .2/q e i y e ixy h  xi2L h i hxi F Œ.1  /L h .  x/hxi u.x/dx: O

D

1 .2/q h i

We now fix L 2 N in such a way that 2L  M./ where M./ is the constant in (2.2.1) belonging to . Then Plancherel’s formula together with (2.2.3) gives us 2 kf kL 2 .R2q ;H / “ Z 1 1 2 D k.2/q e ixy h  xi2L h i hxi F Œ.1 /L h . x/hxi u.x/dxk O H dyd “ q 1 1 2 D .2/ kh  xi2L h i hxi F Œ.1  /L h .  x/hxi u.x/k O H dxd “ 1 2 c kF Œ.1  /L h .  x/hxi u.x/k O H d dx Z Z L 2 1 2 D c jF Œ.1  / h . /j d khxi u.x/k O H dx nZ o q D c.2/ j.1  /L h. /j2 d kuk2W 0 .Rq ;H / :

Thus we obtain altogether the estimate j.Au; v/W 0 .Rq ;Hz / j  c.a/kukW 0 .Rq ;H / kvkW 0 .Rq ;Hz /

(2.2.26)

with a constant c > 0 independent of a; u and v. We finally consider the general case. Choose any ' 2 C01 .R2q / that is equal to 1 near 0, and set a" .y; / WD '."y; "/a.y; /. Using the Leibniz rule it is easy to verify that .a" /  c.a/ with a constant c > 0 independent of ", 0 < "  1. Then from Lemma 2.2.19 (ii) and the estimate (2.2.26) it follows that j.Au; v/W 0 .Rq ;Hz / j D lim j.Op.a" /u; v/W 0 .Rq ;Hz / j "!0

 c.a/kukW 0 .Rq ;H / kvkW 0 .Rq ;Hz / : Since S.Rq ; H / is dense in the space W 0 .Rq ; H / (for any Hilbert space H with group action) we finally obtain kAukW 0 .Rq ;Hz /  c.a/kukW 0 .Rq ;H / . Remark 2.2.21. The continuity of operators of arbitrary order in edge spaces will be proved in a number of cases below, cf. Section 2.3.5.

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2 Symbolic structures and associated operators

Let us have a look at the continuity of pseudo-differential operators in a simple special case and formulate the corresponding result. z be Hilbert spaces with group actions  and , Theorem 2.2.22. Let H and H Q respec q z /. Then tively, and let a./ 2 S .R I H; H z/ Op.a/ W S.Rq ; H / ! S.Rq ; H

(2.2.27)

is continuous and extends to a continuous operator z/ Op.a/ W W s .Rq ; H / ! W s .Rq ; H

(2.2.28)

for every s 2 R. Proof. The proof of (2.2.27) is analogous to a corresponding result in the scalar case and left to reader (see Kumano-go [110]). The continuity of (2.2.28) follows from the estimates Z 2 1 1 2 k Op.a/ukW s .Rq ;Hz / D hi2.s/ kQ hi a./hi hi u./k O z d H n o 1  sup hi2 kQ hi a./hi k2L.H;Hz / kuk2W s .Rq ;H / :  2Rq

Let us consider some examples for the latter theorem. The operator r C W H s .R/ ! H s .RC / of restriction to RC can be interpreted as an element r C 2 Scl0 .Rq I H s .R/; H s .RC //. It is independent of the covariable  but satisfies the homogeneity relation r C D  r C 1 for every  2 RC . The operator RC W H s .Rq  R/ ! H s .Rq  RC / of restriction to Rq  RC represents a continuous map RC D Op.r C / W W s .Rq ; H s .R// ! W s .Rq ; H s .RC //, cf. the relations (2.2.21) and (2.2.22). For later reference another example will be given as a remark. Remark 2.2.23. The operator of restriction r 0 W S.R/ ! C, r 0 u WD u.0/, extends by continuity to an operator r 0 W H s .R/ ! C for every s > 12 , and we have r 0 2 1

Scl2 .Rq I H s .R/; C/ (with respect to the trivial group action in C). This gives us the 1 continuity of the restriction R0 WD Op.r 0 / W H s .Rq  R/ ! H s 2 .Rq /, u.y; t / ! u.y; 0/, for s > 12 , when we apply Theorem 2.2.22 together with Example 2.2.13. R In fact, from the Fourier inversion formula we have u.0/ D u. /μ O and we thus obtain Z Z O ju.0/j  ju. /jd O D h is jh is u. /jd Z Z 1 2  h i2s ju. /j O d h i2s d < 1 for s > : 2 1

This proves the first statement. Moreover, since . u/.x/ D  2 u.x/ for u 2 H s .R/ 1

1 2

we have r 0 D  2 r 0 1 ;  2 RC , which gives us r 0 2 Scl .Rq I H s .R/; C/, although r 0 is independent of the covariable.

2.2 Calculus with operator-valued symbols

87

z be an Remark 2.2.24. Let a./ be as in Theorem 2.2.22, and let a./ W H ! H q isomorphism for every  2 R . Then the operators (2.2.27), (2.2.28) are isomorphisms. Example 2.2.25. Let k  .r/ for  2 R be a strictly positive function in C 1 .RC / and k  .r/ D r  for 0 < r < c0 ;

k  .r/ D 1 for c1 < r

with constants 0 < c0 < c1 . Set H WD r n=2 k ı .r/L2 .RC  X /;

z WD r n=2 k ıC .r/L2 .RC  X / H

for a closed compact C 1 manifold X of dimension n, and define a ./u.r; x/ WD k  .rΠ/u.r; x/;

u.r; x/ 2 H:

z // and a ./ D  a ./ 1 for all  2 Then we have a ./ 2 C 1 .Rq ; L.H; H  z /, cf. RC ; jj  1, and jj  c for some c > 0. This entails a ./ 2 Scl0 .Rq I H; H Example 2.2.8 (i), and we can apply Theorem 2.2.22 and Remark 2.2.24.

2.2.3 Elements of the calculus Let us now formulate some elements of the pseudo-differential calculus in the Euclidean space. The symbols are assumed to be operator-valued, first with ‘twisted homogeneity’, later on as families of operators with another dependence on the covariables. A special case are operators with scalar symbols. In that sense the material of this section is also a supplement of the scalar calculus at the beginning of this chapter. z be Hilbert spaces with group actions f g2R Definition 2.2.26. (i) Let H and H C and fQ  g2RC , respectively. Then the space z /b I H; H S  .Ryp  RqCl ;

(2.2.29)

for  2 R, p; q; l 2 N, is defined to be the set of all a.y; ; / 2 C 1 .Rp  z // such that RqCl ; L.H; H

1 ˚ ˛ ˇ

sup h; iCjˇ j Q h;i (2.2.30) Dy D; a.y; ; / h;i L.H;Hz / .y;;/2RpCqCl

is finite for every ˛ 2 N p , ˇ 2 N qCl . z /b denote the space of all elements (ii) Let S ./ .Rp  .RqCl n f0g/I H; H 1 p qCl z //b (where the subscript ‘b’ at the a./ .y; ; / 2 C .R  .R n f0g/; L.H; H latter space indicates boundedness of all derivatives in y 2 Rp , globally in Rq , as a z //-valued function), such that C 1 .RqCl n f0g; L.H; H a./ .y; ı; ı/ D ı  Q ı a./ .y; ; /ı1

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2 Symbolic structures and associated operators

for all ı 2 RC , .y; ; / 2 Rp  .RqCl n f0g/. Moreover, z /b Scl .Rp  RqCl I H; H

(2.2.31)

denotes the subspace of all elements of (2.2.29) that are classical in .; /, cf. Definition 2.2.3 (ii). Remark 2.2.27. The space (2.2.29) is a Fréchet space with the semi-norm system (2.2.30); also (2.2.31) is Fréchet with a semi-norm system that is defined in an analogous manner as in Remark 2.1.2.  z /b of symbols a.; / in (2.2.29) or (2.2.31) Clearly, the subspaces S.cl/ .RqCl I H; H  z /, cf. the notation in Rewith constant coefficients coincide with S.cl/ .RqCl I H; H mark 2.2.5 (iii). If E is a Fréchet space with the semi-norm system .k /k2N , we define the space Cb1 .Rp ; E/ as the set of all u 2 C 1 .Rp ; E/ such that supy2Rp k .Dy˛ u.y// < 1 for all ˛ 2 N p , k 2 N. Observe that we have  z /b D Cb1 .Rp ; S  .RqCl I H; H z //: .Rp  RqCl I H; H S.cl/ .cl/

T  p qCl z /b WD z /b . We set S 1 .Rp  RqCl I H; H I H; H 2R S .R  R In the following we also write  D ord a when a belongs to the symbol space (2.2.29). z /b , j 2 N, be an arbitrary sequence, Theorem 2.2.28. Let aj 2 S j .Rp RqCl I H; H z /b for and let j ! 1 as j ! 1. Then there is an a 2 S  .Rp  RqCl I H; H PN  D maxfj W j 2 Ng such that ord.a  j D0 aj / ! 1 as N ! 1, and a is z /b . unique mod S 1 .Rp  RqCl I H; H As Pin the scalar case, the symbol a is called an asymptotic sum of the aj , written a  j 2N aj . We can obtain an asymptotic sum by an expression similar to (2.1.6). Remark 2.2.29. Denoting the space (2.2.29) for the case of trivial group actions, i.e.,  D idH , Q  D idHz for all  2 RC , by  z /b ; S.1/ .Rp  RqCl I H; H

we have continuous embeddings z

CM CM z /b ,! S  .Rp  RqCl I H; H z /b S.1/ .Rp  RqCl I H; H z z /b ,! S  .Rp  RqCl I H; H z /b , where M and M z and S CM CM .Rp  RqCl I H; H .1/ are the constants belonging to f g2RC and fQ  g2RC in the sense of (2.2.1).

Let us define the spaces of parameter-dependent pseudo-differential operators z I Rl /b L .Rq I H; H .cl/  z /b g; WD fOp.a/./ W a.y; y 0 ; ; / 2 S.cl/ .Rq  Rq  RqCl I H; H

(2.2.32)

2.2 Calculus with operator-valued symbols

“ Op.a/./u.y/ D

0

e i.yy / a.y; y 0 ; ; /u.y 0 /dy 0 μ:

89

(2.2.33)

z /b . .Rq I H; H For l D 0 we simply write L .cl/ Remark 2.2.30. We have  z /b .Rp  RqCl I H; H a.y; ; / 2 S.cl/  z /b H) a.y; ; / DW a .y; / 2 S.cl/ .Rp  Rq I H; H

for every fixed  2 Rl . Remark 2.2.31. The spaces (2.2.29), (2.2.31) and the operators (2.2.32) are contained in  z / and L .Rq I H; H z I Rl /; .Rp  RqCl I H; H (2.2.34) S.cl/ .cl/ respectively, with the symbols from Definition 2.2.3 (with U  Rq replaced by Rp  RqCl ). In Section 2.3.5 below we shall investigate subspaces with an extra control of symbols for jyj ! 1. z D C, All considerations of this section specialise to the case l D 0 or H D C, or H with trivial group action. We now investigate the operators (2.2.33) in more detail. For the scalar case we refer to [110]. Concerning the vector-valued case we employ an operator-valued generalisation of material from Kumano-go [110], and we partly follow the exposition of Seiler [204] and Krainer [103]. Let us first assume l D 0. The case l > 0 can be reduced to l D 0 by looking at subspaces of operators that are translation invariant in some directions. The aspect of classical symbols and operators will not be dominating in this discussion. Therefore, we mainly consider the general case. Definition 2.2.32. Let V be a Fréchet space, and let P .R2q ; V / defined to be the set of all a.x; / 2 C 1 .Rqx  Rq ; V / such that for every continuous semi-norm  on V there exist reals  D ./, ı D ı./ such that sup .x; /2R2q

hxiı h i .Dx˛ D ˇ a.x; // < 1

(2.2.35)

for every ˛; ˇ 2 N q . Let  WD f g2N ; ı WD fı g2N be arbitrary sequences, and fix a semi-norm system . /2N that defines the Fréchet topology of the space V . Then P ;ı .R2q ; V / denotes the subspace of all a.x; / 2 P .R2q ; V / such that sup .x; /2R2q

hxiı h i  .Dx˛ D ˇ a.x; //

is finite for every ˛; ˇ 2 N q ,  2 N.

(2.2.36)

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2 Symbolic structures and associated operators

Remark 2.2.33. The space P ;ı .R2q ; V / depends on the semi-norm system . /2N that defines the Fréchet topology of V . However, P .R2q ; V / is independent of this choice. Lemma 2.2.34. Let .aj /j 2N be a sequence in P ;ı .R2q ; V / that is bounded with respect to the semi-norms (2.2.36). Moreover, let a 2 C 1 .R2q ; V / and limj !1 aj D a in C 1 .R2q ; V /. Then we have a 2 P ;ı .R2q ; V /. Proof. By assumption there are constants c independent of .x; / 2 R2q such that  .Dx˛ D ˇ aj .x; //  c hxiı h i : Moreover, we have  .Dx˛ D ˇ a.x; // D limj !1  .Dx˛ D ˇ aj .x; //. This implies that (2.2.36) is finite for every . Corollary 2.2.35. The space P ;ı .R2q ; V / is Fréchet with the semi-norm system (2.2.36). In fact, every Cauchy sequence in P ;ı .R2q ; V / is bounded in that space and convergent in C 1 .R2q ; V /. Then by Lemma 2.2.34 it also converges in P ;ı .R2q ; V /. Proposition 2.2.36. (i) a 2 P ;ı .R2q ; V / implies Dx˛ D ˇ a 2 P ;ı .R2q ; V / for every ˛; ˇ. (ii) If T W V ! Vz is a continuous operator, then a 2 P .R2q ; V / implies T a WD ..x; / ! T .a.x; /// 2 P .R2q ; Vz /; more precisely, a ! T a defines a continuous map Q ıQ P ;ı .R2q ; V / ! P ; .R2q ; Vz / Q Q ı/. for every .; ı/ with resulting sequences of orders .; (iii) Let V be the projective limit of Fréchet spaces Vj with respect to linear maps Tj W V ! Vj j 2 I . Then a 2 P .R2q ; V / is equivalent to Tj a 2 P .R2q ; Vj / for every j 2 I . (iv) Given two Fréchet spaces V0 and V1 and a continuous bilinear map h; i W V0  V1 ! V for a Fréchet space V , then ak 2 P .R2q ; Vk /; k D 0; 1, implies ha0 ; a1 i 2 P .R2q ; V /; more precisely, .a0 ; a1 / ! ha0 ; a1 i induces continuous maps 0

0

P  ;ı .R2q ; V0 /  P 

00 ;ı 00

.R2q ; V1 / ! P ;ı .R2q I V /

for every . 0 ; ı 0 /; . 00 ; ı 00 / with some resulting sequences .; ı/. (v) If W is a closed subspace of V , then a 2 P .R2q ; V / implies Œa 2 P .R2q ; V =W /, where Œa denotes the image under the quotient map V ! V =W . These properties are immediate consequences of Definition 2.2.32. Following the lines of [110, Chapter 1] for every a.x; / 2 P .R2q ; V / we can form the oscillatory integral “ “ ix OsŒa WD e e ix ."x; " /a.x; /dxμ (2.2.37) a.x; /dxμ WD lim "!0

with any fixed  2 S.R2q /, .0; 0/ D 1. x;

2.2 Calculus with operator-valued symbols

91

Theorem 2.2.37. For every a.x; / 2 P .R2q ; V / and .x; / 2 S.R2q /, .0; 0/ D 1, the oscillatory integral (2.2.37) exists in V and is independent of the choice of . Moreover, a.x; / ! OsŒa induces a continuous map OsŒ W P ;ı .R2q ; V / ! V

(2.2.38)

for every ; ı. Proof. If the limit OsŒa 2 V exists, we can apply any linear continuous functional v 0 2 V 0 and obtain hv 0 ; OsŒa i D OsŒhv 0 ; ai , cf. Proposition 2.2.36 (ii). Since hv 0 ; ai 2 P .R2q ; C/ we can apply the result that the oscillatory integral in the scalar case is well defined. This gives us independence of the limit (2.2.37) also in the V -valued case. Let us now assume a.x; / 2 P ;ı .R2q ; V / and show that “ e ix ."x; " /a.x; /dxμ A" WD is a Cauchy sequence in V . Fix a semi-norm  2 . /2N and let  D ./, ı D ı./, cf. Definition 2.2.32. Then choosing l; m 2 N so large that 2l > ı C q;

2m >  C q;

(2.2.39)

integration by parts yields that .A"  A"0 / is dominated by “   hxi2l  .1 C /l fh i2m .1 C x /m Œ."x; " /  ."0 x; "0 / a.x; /g dxμ : ˛ Since h  i2l 2 Cb1 .Rq / and supfjDx; ."x; " /j W 0 < "  1; .x; / 2 R2q g < 1 for every ˛ 2 N 2q , the integrand can be estimated by c.a/hxi2l h i2m hxiı h i D c.a/hxiı2l h i2m uniformly in ", "0 , 0 < "; "0  1, with a constant c > 0 independent of a, and

.a/ WD supf.Dx˛ D ˇ a.x; //hxiı h i W .x; / 2 R2q ; j˛j C jˇj  2.l C m/g: From hxiı2l h i2m 2 L1 .R2q /, cf. (2.2.39), by Lebesgue’s theorem on dominated convergence we obtain .A"  A"0 / ! 0 as ";’"0 ! 0. Finally, integrating by parts, similarly as before we obtain .A" /  c.a/ hxiı2l h i2m dxμ  c.a/ with different constants c, and .A" / ! .OsŒa / which shows the existence of the limit and the continuity of (2.2.38). Definition 2.2.38. A function " .x/ W .0; 1  Rm ! C is called regularising if (i) " .x/ 2 S.Rm / for every " 2 .0; 1 ; (ii) sup.";x/2.0;1Rm jDx˛ " .x/j < 1 for every ˛ 2 N m ; ( 1 for ˛ D 0 pointwise in Rm for " ! 0. (iii) Dx˛ " .x/ ! 0 for ˛ 6D 0

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2 Symbolic structures and associated operators

If .x/ 2 S.Rm /, .0/ D 1, then " .x/ WD ."x/ is regularising in the sense of Definition 2.2.38. Remark 2.2.39. For every regularising function " .x; / and a.x; / 2 P .R2q ; V / we have “ OsŒa D lim e ix " .x; /a.x; /dxμ : (2.2.40) "!0

Proposition 2.2.40. For every a.x; / 2 P .R2q ; V / we have: ’ (i) a.x; / 2 L1 .R2q ; V / ) OsŒa D e ix a.x; /dxμ ; (ii) OsŒx ˛ a D OsŒD ˛ a ; OsŒ ˇ a D OsŒDxˇ a for every ˛; ˇ 2 N q ; (iii) OsŒa D OsŒe i.x 0 Cx0 Cx0 0 / a.x C x0 ; C 0 / for every x0 ; 0 2 Rq ; (iv) if a is independent of , then OsŒa D a.0/; (v) if T W V ! Vz is a linear, continuous operator, then T OsŒa D OsŒT a . Proof. (i) is a consequence of Lebesgue’s theorem on dominated convergence, here in the vector-valued case. The properties (ii), (iii), and (iv) are known in the scalar case V D C, cf. [110, Chapter 1, Theorems 6.7, 6.8]; then the V -valued case follows by testing on any continuous functional v 0 on V . The relation (v) is elementary. Lemma 2.2.41. Let V be a Fréchet space, I D Œ0; 1 , and f 2 C 2 .I; V /. Moreover, let  be any continuous semi-norm on V . Then there is a constant c > 0 such that sup .@ t f .t//  c sup .f .t //fsup .f .t // C sup .@2t f .t //g: I

I

I

I

Proof. The assertion holds for V D C, cf. [110, Chapter 1, Lemma 6.5]. The general case reduces to this situation, since .v/ D supfjhv 0 ; vij W v 0 2 Uo g for every v 2 V , where Uo denotes the polar of , cf. the proof of Proposition 2.1.9. Lemma 2.2.42. Let .aj /j 2N be a bounded sequence in C 1 .Rm ; V /, a 2 C 1 .Rm ; V /, and assume aj ! a in C.Rm ; V / for j ! 1. Then it follows that aj ! a in C 1 .Rm ; V / for j ! 1. Proof. Set fj .t; x/ WD aj .x C t ek /  a.x C t ek / for t 2 Œ0; 1 , x 2 Rm , ek D .0; : : : ; 1; : : : ; 0/ (with 1 as the k th component). The convergence of .aj /j 2N to a in the space C.Rm ; V / implies supI;K .fj .t; x// ! 0 for every compact set K  Rn and each continuous semi-norm . From the boundedness of .aj /j 2N in C 1 .Rm ; V / we obtain sup .fj .t; x// C sup .@2t fj .t; x// < c I;K

I;K

for a constant c > 0, independent of j . By virtue of Lemma 2.2.41 it follows that sup .@xk aj .x/  @xk a.x// D sup .@ t fj .0; x// ! 0 K

K

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93

for j ! 1. By iteration of these arguments we obtain supK .@˛x aj .x/@˛x a.x// ! 0 for arbitrary ˛ 2 N m . Proposition 2.2.43. Let .aj /j 2N be a bounded sequence in the space P ;ı .R2q ; V /, and let a.x; / 2 C 1 .R2q ; V /. Assume that one of the following conditions is satisfied: (i) aj ! a in C.R2q ; V / for j ! 1; (ii) V is nuclear, and aj ! a pointwise in R2q for j ! 1. Then we have a 2 P ;ı .R2q ; V / and OsŒa D lim OsŒaj : j !1

(2.2.41)

Proof. By virtue of Lemma 2.2.34 it suffices to show that aj ! a in C 1 .R2q ; V / for j ! 1. Under the condition (i) this follows from Lemma 2.2.42. Let us now assume that (iii) is satisfied. If V is nuclear so is C 1 .R2q ; V /. Thus, every sequence .aj /j 2N which is bounded in C 1 .R2q ; V / has a convergent subsequence. Its limit coincides with a since aj ! a converges pointwise. Thus, .aj /j 2N converges to a in C 1 .R2q ; V / and hence a 2 P ;ı .R2q ; V / as before. In order to show the relation (2.2.41) we verify that aj ! a in P C;ıC .R2q ; V / for every  > 0 (adding a real to a sequence means componentwise addition). Then it suffices to apply (2.2.38). Fix any semi-norm  ; then for every " > 0 there is an N."/ such that for j  N."/ sup .x; /2R2q

fhxiı  h i   .Dx˛ D ˇ .aj  a/.x; //g < ":

In fact, since the set faj  a W j 2 Ng is bounded in P ;ı .R2q ; V / and by virtue of the extra factor hxi h i there is a compact subset K in R2q such that sup.x; /2R2q nK f   g < " for every j 2 N; moreover, aj ! a in C 1 .R2q ; V / implies sup.x; /2K f   g < " for all j  N."/. Lemma 2.2.44. Let V ,! Vz be a continuous embedding of Fréchet spaces,  Rym open, and let f 2 C 1 .; Vz / such that Dy˛ f 2 C.; V / for every ˛ 2 N m . Then we have f 2 C 1 .; V /, and the derivatives with respect to both topologies coincide. Proof. Let us show that f 2 C 1 .; V /; then by induction it follows that f 2 C 1 .; V /. For sufficiently small h > 0 we have Z 1 @  f .y C hej /d; f .y C hej /  f .y/ D @yj 0 where ej D .0; : : : ; 1; : : : ; 0/ denotes the unit vector with 1 at the j th place. The relation holds in Vz by the fundamental theorem of calculus. Because of @yj f 2

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2 Symbolic structures and associated operators

C.; V / the integral on the right exists in V . Thus, for any continuous semi-norm  on V we have   f .y C hej /  f .y/ @  f .y/  h @yj  Z 1  @  @   f .y C hej /  f .y/ d ! 0 @yj @yj 0 for h ! 0 by dominated convergence. Thus the derivative @yj f exists in V and is continuous by assumption. 2q Proposition 2.2.45. Let a.x; y; ; / 2 P ;ı .R2q x;y  R ; ; V /. Then 2q b.x; / WD ..y; / ! a.x; y; ; // 2 P .R2q ; P ;ı .Ry; ; V //: x;

(2.2.42)

2q In particular, Proposition 2.2.36 (ii) gives us OsŒb 2 P .Ry; ; V /. Moreover, we have OsŒa D OsŒOsŒb

.

Proof. First it is easy to verify (by the fundamental theorem of calculus) that b 2 C 1 .R4q ; V / and Dx˛ D ˇ b.x; / 2 C.R2q ; P ;ı .R2q ; V // for every ˛; ˇ 2 N q . Lemma 2.2.44 then yields b.x; / 2 C 1 .R2q ; P ;ı .R2q ; V //. The relation (2.2.42) follows from the estimate  .Dx˛ D ˇ Dy D a.x; y; ; //  .a/hx; yiı h ; i C

C

 c.a/hxiı h i hyiı hi for every semi-norm  on V and some constant c > 0 from Peetre’s inequality, %C WD max.%; 0/, and a continuous semi-norm .  / on the space P ;ı .R2q R2q ; V /. The relation OsŒa D OsŒOsŒb

is known for the case V D C, cf. [169, Theorem 2.5.d]. For arbitrary v 0 2 V 0 by repeatedly applying Proposition 2.2.36 (ii) and Proposition 2.2.40 (v) we obtain hv 0 ; OsŒOsŒb

i D hv 0 ; OsŒa i and then the assertion in the vector-valued case. We now turn to a modification of oscillatory integrals for amplitude functions with an exponential growth in the x-variables (rather than a polynomial growth as in Definition 2.2.32), together with a holomorphic extension in the -variable. For our applications we are interested in the case q D 1. By A.U; E/ for an open U C and a Fréchet space E we denote the (Fréchet) space of holomorphic functions on U with values in E. Definition 2.2.46. Let V be a Fréchet space with the semi-norm system . /2N ; then P .R  C; V / is defined to be the set of all a.x; w/ 2 C 1 .R; A.C; V // such that for every  there exist reals  , ı , such that supf .Dxk D l a.x; C i%//e ı hxi h i W .x; / 2 R2 ; j%j  N;  C k C l  N g (2.2.43)

2.2 Calculus with operator-valued symbols

95

is finite for every N 2 N. The subspace of all such a 2 P .R  C; V / for given sequences  WD . /2N , ı WD .ı /2N will be denoted by P ;ı .R  C; V /. Definition 2.2.47. A function " .x; w/ W .0; 1 RC ! C is called holomorphically regularising if (i) ."; x; / ! " .x; C i%/ is regularising in the sense of Definition 2.2.38 for every % 2 R; (ii) w ! " .x; w/ is an entire function in w 2 C, and we have ! " .x; C i%/ 2 S.R / for every % 2 R, uniformly in compact intervals; (iii) for every " 2 .0; 1 there is a compact set K"  R such that " .x; w/ D 0 for x 62 K" . An example of a holomorphically regularising function is " .x; w/ D '."x/'."w/ O for any ' 2 C01 .R/ with '.0/ D '.0/ O D 1. For a.x; / 2 P .RC; V / we now form OsŒa by (2.2.40) for any holomorphically regularising function " .x; /. Theorem 2.2.48. The oscillatory integral OsŒa exists as a limit (2.2.40) for every a.x; / 2 P .R  C; V / and is independent of the choice of the holomorphically regularising function " .x; w/. It defines a continuous map OsŒ W P ;ı .R  C; V / ! V;

a ! OsŒa ;

for every ; ı. For a 2 P .RC; V /\P .RR; V / the oscillatory integral (obviously) coincides with that defined before for the case of real covariables. ’ Proof. Let a 2 P ;ı .R  C; V /, and let A" WD e ix " .x; /a.x; /dxμ . First we show the existence of lim"!0 A" . Fix a  from the semi-norm system on V , and choose a ' 2 C 1 .R/ such that '.x/ 0 for x < c0 and '.x/ 1 for x > c1 for certain c0
0

Setting .a/ WD supf .Dxk D l a.x;  iı //e ı hxi h i W .x; / 2 R2q ; k 0 C 0 l  k C lg for every k; l 2 N, it follows that  .Dxk D l fe ı x '.x/a.x;  i ı /g/  c.a/h i

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2 Symbolic structures and associated operators

for a constant c > 0 depending on k; l; ', and on the regularising function. A similar estimate holds for e ı x .1  '.x//a.x; C i ı /. Because of condition (i) of Definition 2.2.47 we can apply integration by parts analogously as in the proof of Theorem 2.2.37 (here we choose 2k 0 > 1, 2k >  C 1) to show that  .A"  A"0 ) tends to zero with "; "0 , and also the continuity of OsŒ . To show the independence of the limit of the choice of " it suffices (after testing the expressions on functional v 0 2 V 0 ) to assume V D C, i.e., a 2 P ;ı .R  C/. Both a' .x; / WD e ıx '.x/a.x;  i ı/ and a.1'/ .x; / WD e ıx .1  '.x//a.x; C i ı/ are elements of P ;0 .R  R/. It follows that OsŒa D lim A" D OsŒa' C OsŒa.1'/

(2.2.44) "!0

with oscillatory integrals in the sense of Theorem 2.2.37 which are independent of the regularising function. Proposition 2.2.49. For every a 2 P .R  C; V / we have l (i) OsŒx l a D OsŒDw a , OsŒw k a D OsŒDxk a for every k; l 2 N;

(ii) OsŒa D OsŒe i.xw0 Cx0 wCx0 w0 / a.x C x0 ; w C w0 / for every x0 2 R, w0 2 C. Proof. After testing on functionals v 0 2 V 0 it suffices to show the assertions for V D C. However, this is an immediate consequence of the relation (2.2.44). Proposition 2.2.50. Let .aj /j 2N be a bounded sequence in the space P ;ı .R  C; V / and a.x; w/ 2 C 1 .R; A.C; V //. Assume that one of the following conditions is satisfied: (i) aj ! a in C.R  C; V / for j ! 1; (ii) V is nuclear, and aj ! a pointwise in R  C for j ! 1. Then we have a 2 P ;ı .R  C; V /, and OsŒa D limj !1 OsŒaj . The proof is analogous to that of Proposition 2.2.43. Proposition 2.2.51. Let a.x; / 2 P ;ı .R2q ; V / and set b .x; / WD a.x;  / for  2 R; .x; / 2 R2q : C

Then we have that b .x; / lies in C.R; P  C1;ı .R2q ; V //, where  C is defined by .max. ; 0//2N , and the map a.x; / ! b .x; / is continuous. In particular,  ! R1 OsŒb defines an element of C.R; V /, and 0 OsŒb d exists. Analogous relations hold for holomorphic amplitude functions. Proof. Applying the inequality h i%  max.1; j j/% h i% for %  0 we obtain C

C

 .Dx˛ D ˇ b .x; //  .a/jjjˇ j max.1; j j/ hxiı h i

2.2 Calculus with operator-valued symbols

97

for .a/ WD supf .Dx˛ D ˇ a.x; //hxiı h i W .x; / 2 R2q g. This shows that C

b .x; / 2 P  C1;ı .R2q ; V / for every  2 R, and then we obtain continuity of the C map P ;ı .R2q ; V / ! P  C1;ı .R2q ; V / when we verify the continuous dependence of b .x; / on  . Using the fundamental theorem of calculus we obtain @˛x @ˇ .b .x; /  b 0 .x; // Z 1 X  ˛ ˇ Cˇ 0  0 D . 0 /jˇ j .   0 / ˇ a .x; ..1  t / 0 C t  / /dt @x @ 0

jˇ 0 jD1

  C . jˇ j  . 0 /jˇ j / @˛x @ˇ a .x;  /:

By similar arguments as before it follows that C

 .@˛x @ˇ .b .x; /  b 0 .x; ///  h.;  0 /hxiı h i

C1

with a continuous function h that tends to zero with   0 . Since ; ˛; ˇ are arbitrary we obtain the continuity of  ! b . The arguments for holomorphic amplitude functions are similar. From now on, for abbreviation, we will omit OsŒ when the interpretation of integrals as oscillatory integrals is clear from the context. z be Hilbert spaces with group actions fı gı2R and fQ ı gı2R , reLet H and H C C z /b we spectively. For a double symbol a.y; y 0 ; ; / 2 S  .Rq  Rq  RqCl I H; H set Z Z 0 Opy .a/./u.y/ WD e i.yy / a.y; y 0 ; ; /u.y 0 /dy 0 μ; (2.2.45) Rq

Rq

μ D .2/q d, u 2 S.Rq ; H /. We also write Op.  / instead of Opy .  / if the meaning is clear. For every  2 Rl we then obtain a continuous operator z /: Op.a/./ W S.Rq ; H / ! S.Rq ; H z D C with trivial group action we simply omit H and H z. In the case H D H z /b ,  2 R. Theorem 2.2.52. Let a.y; y 0 ; ; / 2 S  .Rq  Rq  RqCl I H; H z /b (i) There exist unique elements aL .y; ; /, aR .y 0 ; ; / 2 S  .Rq  RqCl I H; H (called left and right symbols, respectively) such that Op.a/./ D Op.aL /./ D Op.aR /./ z //, for all  2 Rl . as operators in L.S.Rq ; H /; S.Rq ; H

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2 Symbolic structures and associated operators

(ii) The symbols aL and aR are given by the following oscillatory integrals “ aL .y; ; / D e ix a.y; y C x;  C ; /dxμ ; “ aR .y 0 ; ; / D e ix a.y 0 C x; y 0 ;   ; /dxμ : The symbols have asymptotic expansions X 1 Dy˛0 @˛ a.y; y 0 ; ; /jy 0 Dy ; ˛Š ˛2N q X 1 aR .y 0 ; ; /  .1/j˛j Dy˛ @˛ a.y; y 0 ; /jyDy 0 : ˛Š q aL .y; ; / 

˛2N

(iii) The mappings a ! aL and a ! aR are continuous. (iv) The symbols aL and aR can be written as follows: aL .y; ; / D

X 1 D ˛0 @˛ a.y; y 0 ; ; /jy 0 Dy C rL;N .y; ; /; ˛Š y 

(2.2.46)

j˛jN

aR .y 0 ; ; / D

X 1 .1/j˛j Dy˛ @˛ a.y; y 0 ; ; /jyDy 0 C rR;N .y 0 ; ; /I ˛Š

j˛jN

(2.2.47) for every N 2 N, where the remainder terms rL;N ; rR;N 2 S .N C1/ .Rq  z /b are given as RqCl I H; H X Z 1 .1  t /N rL;N .y; ; / D .N C 1/ ˛Š 0 j˛jDN C1 (2.2.48) “ ix ˛ ˛ e .Dy 0 @ a/.y; y C x;  C t ; /dxμ dt; X

0

rR;N .y ; ; / D .N C 1/

N C1

e

ix

1

.1/

j˛jDN C1

“

Z

.Dy˛ @˛ a/.y 0

0

.1  t /N ˛Š (2.2.49)

0

C x; y ;   t ; /dxμ dt:

(v) The mappings a ! rL;N and a ! rR;N are continuous. (vi) In the case of a classical symbol a also aL , aR and rL;N , rR;N are classical, and (iii) and (v) hold in the stronger topologies of classical symbols.

2.2 Calculus with operator-valued symbols

99

Proof. Let us consider the case of left symbols, the assertions on right symbols can be proved in an analogous manner. The proof follows Kumano-go [110], so we only sketch the main steps; details for the operator-valued case are elaborated in [38], see also [204], and [103]. First the oscillatory integral expression for aL in Theorem 2.2.52 (ii), and Op.a/./ D Op.aL /./ yield the existence and uniqueness of the left symbol. Differentiation under the integral sign and inserting the group actions show that aL is a symbol of the desired class; here we employ the symbolic estimates for a and Peetre’s inequality. At the same time we obtain the continuity a ! aL . In order to see the asymptotic expansion of aL we employ the Taylor expansion of a.y; y C x;  C ; / at D 0 which gives us a.y; y C x;  C ; / X 1 D @˛ a.y; y C x; ; / ˛ ˛Š  j˛jN Z 1 X .1  t /N ˛ C .N C 1/ ˛ @ a.y; y C x;  C t ; /dt ˛Š 0 j˛jDN C1

for every N 2 N. The summands of the Taylor polynomial are amplitude functions in the variables .x; /. Moreover, Peetre’s inequality shows that the integrand in the remainder term is a continuous function in t 2 Œ0; 1 with values in the space of amplitude functions in .x; /. Integrating by parts in the oscillatory integral for a and interchanging integrals in the remainder term gives us the formulas (2.2.46) and (2.2.48). We obtain z /b ; @˛ Dy˛0 a.y; y 0 ; ; /jy 0 Dy 2 S j˛j Rq  RqCl I H; H N z /b ; rL;N .y; ; / 2 S.1/ .Rq  RqCl I H; H

which yields the desired asymptotic expansion and also the continuity of the mapping a ! rL;N . The expansions show that aL and rL;N are also classical as soon as a is classical, and the continuities a ! aL , a ! rL;N with respect to the stronger topologies of classical symbols follow from the closed graph theorem. Remark 2.2.53. Theorem 2.2.52 shows that the map  z /b ! L .Rq I H; H z I Rl /b Op W S.cl/ .Rq  RqCl I H; H .cl/

is an isomorphism. This allows us to carry over the Fréchet topology of the symbol z I Rl /b . Also space to L .Rq I H; H .cl/ z /b ! L1 .Rq I H; H z I Rl /b Op W S 1 .Rq  RqCl I H; H is an isomorphism; here z I Rl /b WD L1 .Rq I H; H

\ 2R

z I Rl /b D S.Rl ; L1 .Rq I H; H z /b / L .Rq I H; H

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2 Symbolic structures and associated operators

is independent of the group actions f g2RC and fQ  g2RC . z z , and H z be Hilbert spaces with group actions fı gı2R , Theorem 2.2.54. Let H; H C fQ ı gı2RC and fQQ ı gı2RC , respectively. Moreover, let  z z; H z /b ; .Rq  RqCl I H a.y; ; / 2 S.cl/

 z /b ; b.y; ; / 2 S.cl/ .Rq  RqCl I H; H

and A./ WD Op.a/./, B./ WD Op.b/./. Then for the composition (as operators z z I Rl /b , and there is in S.Rq ; H /) we have C./ WD A./B./ 2 LC .Rq I H; H .cl/

C z z /b , (called the Leibniz .Rq  RqCl I H; H a (unique) symbol a # b.y; ; / 2 S.cl/ product of a and b), such that C./ D Op.a # b/./. The Leibniz product can be expressed by the oscillatory integral formula “ a # b.y; ; / D e ix a.y;  C ; /b.y C x; ; /dxμ : (2.2.50)

We can write a # b.y; ; / D

X 1 @˛ a.y; ; /Dy˛ b.y; ; / C rN .y; ; / ˛Š 

(2.2.51)

j˛jN

C.N C1/ z z /b , that is, a # b  with a symbol rN .y; ; / 2 S.cl/ .Rq  RqCl I H; H P 1 ˛ ˛ ˛ ˛Š @ aDy b, and rN is of the form

rN .y; ; / D .N C 1/

X j˛jDN C1

“

Z 0

1

.1  t /N ˛Š

e ix .@˛ a/.y;  C t ; /.Dy˛ b/.y C x; ; /dxμ dt (2.2.52)

for every N 2 N. The mappings .a; b/ ! a # b and .a; b/ ! rN are bilinear continuous. Proof. The proof immediately follows again from the technique of [110], here modified to the operator-valued set-up. Remark 2.2.55. From the formula (2.2.51) and the order of the remainder term it follows that a # b D ab modulo a symbol of order  C   1: Let fH; H0 ; H 0 g be a Hilbert space triple in the sense of Definition 2.1.6, with H; H0 ; H 0 being embedded in a Hausdorff topological vector space V . Definition 2.2.56. A triple fH; H0 ; H 0 g is said to be a Hilbert space triple with group action  WD f g2RC , written fH; H0 ; H 0 I g, if

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101

(i)  is a representation of the multiplicative group RC in L.V / which restricts to (strongly continuous) group actions in H; H0 , and H 0 ; (ii)  acts on H0 as a unitary group. z; H z0 ; H z 0 I g Remark 2.2.57. Let fH; H0 ; H 0 I g and fH Q be Hilbert space triples with group action. (i) The scalar product on L2 .Rq ; H0 / induces a non-degenerate sesquilinear pairing .; /L2 .Rq ;H0 / W S.Rq ; H /  S.Rq ; H 0 / ! C: z / there is a unique operator a./ 2 L.H z 0 ; H 0 / such that (ii) For every a 2 L.H; H .ah; hQ 0 /Hz0 D .h; a./ hQ 0 /H0

z 0: for all h 2 H; hQ 0 2 H

z/ ! The mapping a ! a./ represents an antilinear isomorphism L.H; H z 0 ; H 0 /. L.H (iii) The mapping from (ii) induces an antilinear isomorphism  z /b ! S  .Rq  Rq  RqCl I H z 0 ; H 0 /b ΠW S.cl/ .Rq  Rq  RqCl I H; H .cl/

given by aΠ.y; y 0 ; ; / WD .a.y 0 ; y; ; //./ (in particular, left symbols are mapped to right symbols and vice versa). z; H z0 ; H z 0 I g Theorem 2.2.58. Let fH; H0 ; H 0 I g and fH Q be Hilbert space triples  q z I Rl /b with an a.y; ; / 2 with group action and A./ D Op.a/./ 2 L.cl/ .R I H; H z /b . Then the formal adjoint A ./, defined by S  .Rq  RqCl I H; H .cl/

.A./u; v/L2 .Rq ;Hz0 / D .u; A ./v/L2 .Rq ;H0 / z 0 /, belongs to L .Rq I H z 0 ; H 0 I Rl /b and has for all u 2 S.Rq ; H /, v 2 S.Rq ; H .cl/  z 0 ; H 0 /b the form A ./ D Op.a /./ with a symbol a .y; ; / 2 S.cl/ .Rq  RqCl I H given by the oscillatory integral formula “ e ix aŒ .y C x;  C ; /dxμ ; a .y; ; / D P

which can be written as a .y; ; / D N 2 N, with a remainder rN .y; ; / 2 X Z rN .y; ; / D .N C 1/ j˛jDN C1

“

1 ˛ ˛ Œ j˛jN ˛Š @ Dy a .y; ; / C rN .y; ; /, .N C1/ z 0 ; H 0 / of the form S.cl/ .Rq  RqCl I H

0

1

.1  t /N ˛Š

e ix .@˛ Dy˛ aŒ /.y C x;  C t ; /dxμ dt:

The mappings a ! a and a ! rN are antilinear and continuous. Proof. The results follow immediately from Theorem 2.2.52, using the fact that a./ .y 0 ; ; / is a right symbol of A ./.

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2 Symbolic structures and associated operators

2.3 Operators on manifolds with conical exit to infinity Operator-valued symbols on manifolds with singularities contain families of operators on infinite 1 cones, for instance, RC in the case of boundary value problems, or RC SC in the case of mixed problems, cf. the formulas (1.1.4) and (1.2.12), respectively. The general tool is the analysis on manifolds with conical exits to infinity. We first consider scalar operators. After that we pass to the operator-valued case. In Section 2.4.5 below we give a crucial example that connects ‘exit ellipticity’ with ‘edge-degenerate’ ellipticity. Complete proofs of the results of this sections (as far as they are not given here) may be found in [90].

2.3.1 Symbols with exit behaviour The simplest case of a manifold with conical exit to infinity is the Euclidean space Rm . Symbols and operators in this context have specific properties globally ‘up to infinity’. The general calculus has been developed by Shubin [208], Parenti [145] and Cordes [25], [26]. Several extensions and refinements have been contributed by Schrohe [172] (the manifold case with transition maps compatible with the exit symbolic estimates), Hirschmann [80] and Schulze [188] (calculus with double homogeneities), Dorschfeldt, Grieme and Schulze [38] (operator-valued symbols with twisted homogeneity), Schrohe [173], Kapanadze and Schulze [89], [90] (boundary value problems), Calvo and Schulze [20] (manifolds with edges that have conical exits to infinity), and by other authors. Definition 2.3.1. (i) The space S I .Rm  Rm / of (left) symbols a.x; / of order  2 R in and order  2 R in x is defined to be the set of all a.x; / 2 C 1 .Rm  Rm / such that sup h iCjˇ j hxiCj˛j jDx˛ D ˇ a.x; /j (2.3.1) .x; /2R2m

is finite for every pair of multi-indices ˛; ˇ 2 N m . (If a 2 S I .Rm  Rm / we also write  DW ord a which is the ordinary order in and  DW orde a the exit order in x). 0 0 (ii) The space S ; I; .R2m  R2m / of (double in variables and covariables) symbols a.x; x 0 ; ; 0 / of order .; 0 / 2 R2 in . ; 0 / and .;  0 / 2 R2 in .x; x 0 / is 2m defined to be the set of all a.x; x 0 ; ; 0 / 2 C 1 .R2m x;x 0  R ; 0 / such that 0

0

0

0

0

0

suph iCjˇ j h 0 i Cjˇ j hxiCj˛j hx 0 i Cj˛ j jDx˛ Dx˛0 D ˇ D ˇ0 a.x; x 0 ; ; 0 /j (2.3.2) taken over .x; x 0 ; ; 0 / 2 R4m is finite for every tuple of multi-indices ˛; ˛ 0 ; ˇ; ˇ 0 2 N m . (iii) In an analogous manner we define the spaces of symbols when one or several 0 variables x; x 0 ; or 0 do not occur, e.g., the space S I; .R2m  Rm / 3 a.x; x 0 ; / (of double symbols in the variables .x; x 0 /). /3 This part of the definition also contains symbols with constant coefficients S  .Rm

a. / (where we identify S  .Rm / with a corresponding subspace of S I0 .Rm  Rm /) 0I .Rm  Rm /). or S  .Rm x / 3 a.x/ (identified with a corresponding subspace of S

2.3 Operators on manifolds with conical exit to infinity 0

103

0

The space S ; I; .R2m  R2m / is Fréchet with the semi-norm system (2.3.2), 0 and the spaces S I; .R2m  Rm /, S I .Rm  Rm /, etc., can be regarded as closed subspaces. T Set S 1I1 .Rm  Rm / D ;2R S I .Rm  Rm /; this space coincides with S.Rm  Rm /. In a similar manner we define the spaces S 1I .Rm  Rm /, etc. 0 A symbol a.x; x 0 / 2 S ; .Rm  Rm / can be interpreted as an element of 0 S 0I; .R2m  Rm / which is independent of the covariable . In this sense we have the following remark. Remark 2.3.2. There exists a symbol !.r; r 0 / 2 S 0I0;0 .R2  R/ with the following property: !.r; r 0 / D 1 for jr  r 0 j < 1; !.r; r 0 / D 0 for jr  r 0 j > 2. In fact, it suffices to set   .r  r 0 /2 !.r; r 0 / WD 1 C .r  r 0 /2 for any function

x C / such that 2 C01 .R

.t / 1 for t < 12 ;

.t / 0 for t > 23 .

Theorem 2.3.3. Let aj .x; / 2 S j Ij .Rm  Rm /, j 2 N, be an arbitrary sequence, j ! 1, j ! 1 as j ! 1. Then there exists an a.x; / 2 S I .Rm  Rm / for  WD maxfj W j 2 Ng,  WD maxfj W j 2 Ng such that for every k 2 N there P is an N D N.k/ such that a.x; /  jND0 aj .x; / 2 S kIk .Rm  Rm /. Every such a.x; / (called an asymptotic sum of the symbols aj .x; /) is uniquely determined mod S 1I1 .Rm  Rm /. Similarly as (2.1.6) an asymptotic sum a.x; / can be obtained as a convergent series 1 x  X aj .x; /  ; (2.3.3) a.x; / D cj cj j D0

in the space S I .Rm  Rm / with an excision function .x; / in R2m and constants cj > 0, tending to 1 sufficiently fastP as j ! 1, where for every M > 0 there exists an N D N.M / 2 N such that j1DN C1 .x=cj ; =cj /aj .x; / converges in S M IN .Rm  Rm /. Remark 2.3.4. Theorem 2.3.3 admits many variants in connection with the symbols of Definition 2.3.1. There are also asymptotic sums of symbols in the classes 0 0 S ; I; .R2m  R2m / when all orders or only a part of them tend to 1 (in the second case the remainders have a corresponding finite order), and there are then analogues of the formula (2.3.3). Definition 2.3.5. We define  m m y  m SclI .Rm x  R / WD Scl .R / ˝ Scl .Rx /I Ix

(2.3.4)

the elements of (2.3.4) are called classical symbols (in x and ). (Concerning the nuclear Fréchet topologies of spaces of classical symbols, cf. Remark 2.1.2.)

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2 Symbolic structures and associated operators

Remark 2.3.6. (i) There is a canonical continuous embedding .Rm  Rm / ,! S I .Rm  Rm /: SclI Ix (ii) The elements a.x; / 2 SclI .Rm  Rm / have a homogeneous principal part Ix  .a/.x; / of order  in ,  .a/.x; / 2 S ./ .Rm nf0gI Scl .Rm x // (i.e., homogeneous

 m in 6D 0 of order , with values in Scl .Rx /); analogously, we have a homogeneous  m principal part e .a/.x; / 2 S ./ .Rm x n f0gI Scl .R // in x 6D 0 of order . (iii) Denoting the homogeneous principal part of  .a/.x; / in x 6D 0 of order  by e; .a/.x; / and the homogeneous principal part of e .a/.x; / in 6D 0 of order  by n f0gI S ./ .Rm  ;e .a/.x; /, we obtain  ;e .a/.x; / D e; .a/.x; / 2 S ./ .Rm x n

f0g//. The latter space coincides with the set of all p ;e .x; / 2 C 1 ..Rm n f0g/ 

m   .Rx n f0g// such that p ;e . x;  / D  p ;e .x; / for all ;  2 RC and all m .x; / 2 .Rm x n f0g/  .R n f0g/. Another equivalent definition of the space (2.3.4) may be found in [90], cf. also [80] and [188]. The definition (2.3.4) was given in [226], based on the following observations. Let Bx WD fy 2 Rm W jyj  1g, and choose a diffeomorphism  W int B ! Rm with the property .y/ D yfjyj.1  jyjg1 for 2=3 < jyj < 1. Given an a.x; / 2 C 1 .Rm  Rm / we set b.y; / WD .1  Œy / .1  Œ / a..y/; .//. Here y ! Œy is some C 1 function in Rm such that Œy D jyj for 2=3 < jyj  1 and 1  Œy 6D 0 for all jyj < 1. Then the map a.x; / ! b.y; / induces an isomorphism x .Rm  Rm / ! C 1 .Bx  B/: SclI Ix x to SclI .Rm  By carrying over the Fréchet space structure of C 1 .Bx  B/ Ix x D Rm / we obtain an identification of the projective tensor product C 1 .Bx  B/ 1 x 1 1 x x y  C .B/ with the space (2.3.4). Here we use that C .B/ itself corC .B/ ˝ responds to Sclˇ.Rm /. The ˇfunctions  .a/, ˇ e .a/ and  ;e .a/ are in a natural correˇ ˇ m1 m1 , respectively. spondence to b ˇBS ; b and b m1 x x S m1 B S S I m m For a.x; / 2 S .R  R / we set  .a/ WD . .a/; e .a/; 

;e .a//I

if necessary, we also write  .a/ WD .  .a/; e .a/;  ; ;e .a//: The pair .e .a/; 

;e .a//

will also be called the (principal) exit symbol of a.

Remark 2.3.7. (i) The conditions a 2 SclI .Rm  Rm / and  .a/ D 0 imply that Ix a 2 Scl1I1 .Rm  Rm /; Ix

2.3 Operators on manifolds with conical exit to infinity

(ii) For every triple .p ; pe ; p

;e /

of functions

 m p .x; / 2 S ./ .Rm

n f0gI Scl .Rx //;

p

;e .x; /

105

 m pe .x; / 2 S ./ .Rm x n f0gI Scl .R //;

./ 2 S ./ .Rm .Rm n f0g//

n f0gI S

with the property e .p / D  .pe / D p ;e there is an a 2 SclI .Rm  Rm / such that  .a/ D p ; e .a/ D pe ;  ;e .a/ D p ;e . In fact, it suffices to set . /p .x; /C.x/fpe .x; /. /p for an arbitrary excision function  in Rm .

;e .x; /g

DW a.x; /

I I .Rm  Rm /. .Rm  Rm / and b.x; / 2 S.cl Theorem 2.3.8. Let a.x; / 2 S.cl Ix / Ix / CIC .Rm  Rm /. For classical symbols a and b Then we have a.x; /b.x; / 2 S.cl Ix / it follows that   ; .a/ .b/ D   .a/  .b/; e .a/e .b/;  ; ;e .a/ ;e .b/ :

(Recall that subscripts ‘.cl Ix /’ are used when a relation is valid both in the classical and the general case). .Rm  Rm / with Examples 2.3.9. (i) We have a.x; / WD hxi h i 2 SclI Ix  .a/.x; / D hxi j j ;

e .a/.x; / D jxj h i ;



;e .a/.x; /

D jxj j j :

.Rm  Rm / with (ii) We have a.x; / WD hxi C h i 2 SclI Ix  .a/.x; / D j j ;

e .a/.x; / D jxj ;



;e .a/.x; /

D 0:

(iii) Let p. / 2 Scl .Rm / and  .p/. / be its homogeneous principal symbol of order ; then a.x; / WD Œx  p.Œx / 2 SclI0 .Rm  Rm / and  .a/.x; / D Œx   .p/.Œx /; e .a/.x; / D jxj p.jxj /;  ;e .a/.x; / D jxj  .p/.jxj /: Definition 2.3.10. An element a 2 S I .Rm  Rm / is called elliptic (of order .; / 2 R2 ), if there is a p 2 S I .Rm  Rm / such that ap D 1 mod S 1I1 .Rm  Rm /. .Rm  Rm / is elliptic if and only if Proposition 2.3.11. A symbol a 2 SclI Ix  .a/.x; / 6D 0 for all .x; / 2 Rm  .Rm n f0g/; e .a/.x; / 6D 0 for all .x; / 2 .Rm n f0g/  Rm ;  ;e .a/.x; / 6D 0 for all .x; / 2 .Rm n f0g/  .Rm n f0g/: Later on in Section 2.3.4 we need the following generalision of ellipticity. Let †  S m1 be an open set, and set

˚ x 2†  WD x 2 Rm W jxj > c; jxj

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2 Symbolic structures and associated operators

I  for some constant c > 0. Let S.cl .  Rm / denote the subspace of all a 2 S.cl/ .  Ix / m S R / such that for every open †0  † with compact closure †0  † and every c0 > c I .Rm  Rm / such that a0 j0 D aj0 for there is an a0 2 S.cl Ix /

˚

x 2 †0 : 0 WD x 2 Rm W jxj > c0 ; jxj I Definition 2.3.12. An element a 2 S.cl . Rm / is called elliptic (of order .; / 2 Ix /

I .Rm  R2 / in   Rm if for every †0  † as before and c0 > c there is a p0 2 S.cl Ix / ˇ 1I1 Rm / such that .ap0 /ˇ .0  Rm /. m  1 2 S 0 R

We then have a simple generalisation of Proposition 2.3.11 to the case that a is contained in SclI .  Rm / which is left to the reader.

2.3.2 Operators globally in the Euclidean space ’ 0 0 Let Op.a/u.x/ D e i.xx / a.x; x 0 ; /u.x 0 /dx 0 μ for a.x; x 0 ; / 2 S I; .R2m  Rm /, and define the space of pseudo-differential operators with exit condition at infinity of (‘usual’) order  2 R and exit order  2 R I LI .Rm / WD fOp.a/ W a.x; / 2 S.cl .Rm  Rm /g: .cl/ Ix /

Moreover, let

H sIg .Rm / WD hxig H s .Rm /

(2.3.5)

for s; g 2 R. Theorem 2.3.13. Every A 2 LI .Rm / induces continuous operators A W H sIg .Rm / ! H sIg .Rm / for every s; g; ;  2 R. By virtue of S.Rm / D lim H kIk .Rm / every A 2 LI .Rm / is also continuous k2N as an operator A W S.Rm / ! S.Rm /: Theorem 2.3.14. The map Op induces an isomorphism I Op W S.cl .Rm  Rm / ! LI .Rm / .cl/ Ix /

for every ;  2 R. Let us set L1I1 .Rm / WD 1I .Rm / and LI1 .Rm /. L

T ;2R

LI .Rm /; analogously, we define the spaces

1I1 Theorem 2.3.15. The .Rm / coincides with the space of all integral R space0 L 0 operators C u.x/ D c.x; x /u.x /dx 0 with kernels c.x; x 0 / 2 S.Rm  Rm /.

2.3 Operators on manifolds with conical exit to infinity

107

Remark 2.3.16. Let A 2 L .Rm / be an operator such that for every k 2 N there .cl/ I exists an ak .x; / 2 S.clIx / .Rm  Rm / with A D Op.ak / C Ck for some operator Ck 

IkC1

with kernel in H kIk .Rm  Rm /, and akC1 .x; /  ak .x; / 2 S.clkC1 Ix /

.Rm  Rm /

.Rm /. for certain kC1 ; kC1 that tend to 1 as k ! 1. Then we have A 2 LI .cl/

2.3.3 Operators on manifolds An example of a manifold with conical exit to infinity is the half-cylinder ."; 1/  X , 0 < " < 1, for some C 1 manifold X as the cross section (not necessarily compact), n D dim X. The conical structure is given by an atlas of charts ˚

Q Q > "; jxxj 2 †j ; (2.3.6) j W ."; 1/  Uj ! xQ 2 RnC1 W jxj Q induced by charts j .1; / W Uj ! †j ;

(2.3.7)

j 2 J (for some index set J ), with coordinate neighbourhoods Uj on X and open subsets †j  S n , such that j .r; x/ D rj .1; x/ for all .r; x/ 2 ."; 1/  Uj : The subset M1 WD Œ1; 1/  X  ."; 1/  X is a C 1 manifold with boundary @M1 Š X. The general case of a C 1 manifold M with conical exit to infinity and cross section X is as follows: M is a manifold of the form M D M0 [ M1

(2.3.8)

1

with two C submanifolds M0 and M1 with common boundary X D M0 \ M1 , where M1 D Œ1; 1/  X is as in the example before, while on M0 we do not make any special assumptions. Note that the cross section may have several connected components. Examples 2.3.17. (i) The Euclidean space M D Rd is a manifold with conical exit, where we can set M0 WD fm 2 Rd W jmj  1g; M1 WD fm 2 Rd W jmj  1g and X D S d 1 . (ii) The cylinder X WD R  X (2.3.9) for any C 1 manifold X has two conical exits r ! ˙1 (the points are denoted by .r; x/); in this case we can set M0 WD Œ1; 1  X , M1 WD ..1; 1  X / [ .Œ1; 1/  X / (disjoint union), and the cross section consists of two copies of X . Q 2 †g (iii) Let †  S n be an open set. Then the set M WD fxQ 2 RnC1 n f0g W jxxj Q Q  1g and has a conical exit jxj Q ! 1 with cross section †, where M1 WD M \ fjxj Q  1g. M0 D M \ fjxj

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2 Symbolic structures and associated operators

Remark 2.3.18. It is also interesting for several applications to admit the case when the cross section is a C 1 manifold with boundary or a manifold with singularities (e.g., conical or edge singularities). In such cases the definition of a manifold with conical exits needs a corresponding modification, cf. [20]. In the definition we do not assume that (2.3.6) is based on a maximal atlas (2.3.7) on X. However, different choices of charts (2.3.7) induce corresponding charts (2.3.6) in a unique way, and we then call the conical structures on M belonging to different such systems (2.3.6) equivalent. The relation j .r; x/ D rj .1; x/ for all r  1 allows us to define a value r.m/ 2 Œ1; 1/

(2.3.10)

for every m 2 M1 in an invariant manner, namely, by m D r.m/j .1; x/ when m 2 Œ1; 1/  Uj . Setting m WD r.m/j .1; x/ for every   1, r.m/  1, we obtain corresponding dilations on the manifold M1 . Let us call a function 2 C 1 .M / an admissible cut-off function on M , if M0 \ supp is a compact set and if .m/ D .m/ for all m 2 M1 , and   1, r.m/  1. Definition 2.3.19. Let M be a manifold with conical exit to infinity, written in the form (2.3.8). Then sIg Hloc .M / s for s; g 2 R is the subspace of all u 2 Hloc .M / such that for every chart

˚ Q j W .1; 1/  Uj ! j WD xQ 2 RnC1 W jxj Q > 1; jxxj 2 †j ; Q

j 2 J , induced by restriction of (2.3.6) to int M1 we have . u/.j1 .x// Q 2 H sIg .RnC1 / for every admissible cut-off function on M supported by int M1 . Moreover, we set \ kIk Sloc .M / WD Hloc .M /; k2N

endowed with the topology of the projective limit. For the case that both M0 and X D M0 \ M1 are compact, we omit subscripts ‘loc’ and simply write H sIg .M / and S.M /;

(2.3.11)

respectively. On M we fix a Riemannian metric g that restricts over int M1 to a metric of the form dr 2 C r 2 gX (2.3.12) with a Riemannian metric gX on X and r W int M1 ! RC being the variable defined by (2.3.10). Let d m denote the measure on M associated with g. (In (2.3.12) we observe the conical behaviour for r ! 1, in contrast to discussions around conical singularities which concern r ! 0).

2.3 Operators on manifolds with conical exit to infinity

109

We thenR denote by L1I1 .M / the space of all integral operators of the form C u.m/ D M c.m; m0 /u.m0 /d m0 with kernels c.m; m0 / in the space y  Sloc .M /: Sloc .M  M / WD Sloc .M / ˝ Definition 2.3.20. The space LI .M / for ;  2 R is defined to be the subspace of .cl/  Q all A 2 L.cl/ .M / such that A 2 L1I1 .M / for every pair of admissible cut-off functions ; Q when supp \ supp Q D ; and (in the notation of Definition 2.3.19) .j /1 . A Q /j 2 LI .RnC1 / .cl/ when ; Q are supported by a coordinate neighbourhood of the form .1; 1/  Uj for some j 2 J . .RnC1 / have Definition 2.3.20 is correct in the sense that the local operators in LI .cl/ the necessary invariance properties; this can easily be checked. Remark 2.3.21. (i) The space of operators LI .M / admits a parameter-dependent .cl/ l l variant LI .M I R / with parameters  2 R , where LI .M I Rl /  L .M I Rl /, .cl/ .cl/ .cl/ 1I1 l l 1I1 L .M I R / WD S.R ; L .M //, and local amplitude functions on the conI mCl ical ends M1 defined in terms of symbols S.cl .Rm x  R ; / with (2.3.1) < 1 .;/Ix / being replaced by sup .x; ;/2R2mCl

ˇ h ; iCjˇ j hxiCj˛j jDx˛ D ; a.x; /j < 1

for all ˛ 2 N n ; ˇ 2 N nCl . .M / to the case of operators (ii) There is also a straightforward generalisation of LI .cl/ .M I E; F / between distributional sections of bundles E; F 2 Vect.M /, denoted by LI .cl/ I l (or L.cl/ .M I E; F I R / in the corresponding parameter-dependent case). The simple details are left to the reader. Let us only note that it is natural in this case to fix a system of trivialisations of the bundles over conical sets which correspond to charts on int M1 and to assume homogeneity of the transition maps of order 1 with respect to the dilations (that we discussed in connection with (2.3.10)). Let us now fix some notation for the complete and the principal symbolic structure .M /. Writing M in the form of operators A 2 LI .cl/ z 0 [ int M1 M DM ˇ z 0 WD M0 [ .1; ı/  X for some ı > 1, the operator Aˇ z 2 L .M z 0 / has the with M .cl/ M0 standard complete symbols in local coordinates. Let ˚

Q  W .1; 1/  U !  D xQ 2 RnC1 W jxj Q > 1; jxxj 2† (2.3.13) Q

110

2 Symbolic structures and associated operators

ˇ be a chart from our atlas on int M1 ; then the operator Aˇint M in corresponding local 1 coordinates has a complete symbol Q 2 S I .RnC1  RnC1 /j  .Aj.1;1/U /.x; Q / RnC1 : .clIx /

(2.3.14)

This follows easily from our definitions. The symbols (2.3.14) are invariant modulo S 1I1 .  RnC1 / under symbol push forwards belonging to the transition maps between different such charts. In the classical case we first have the homogeneous principal symbol Q 2 S ./ .T  M n f0g/  .A/.x; Q / ˇ in Q of order  as usual. In addition there are the exit symbolic components of Aˇint M , 1 namely, Q 2 S ./ .RnC1 n f0gI S  .RnC1 //j e .A/.x; Q / (2.3.15) RnC1 cl and 

Q 2 S ./ .RnC1 n f0gI S ./ .RnC1 n f0g//j .RnC1 nf0g/ :

Q / ;e .A/.x;

(2.3.16)

Q although this comes from corresponding (Points in T  M are denoted here by .x; Q / local coordinates; we hope that this does not cause confusion.) The natural invariance properties of (2.3.15) and (2.3.16) admit the interpretation as global functions in C 1 .T  M1 /

and

C 1 .T  M1 n 0/;

(2.3.17)

Q D  e .A/.x; Q for all respectively, with the homogeneity properties e .A/. x; Q / Q /   Q Q

 1, jxj Q  1 and  ;e .A/. x; Q  / D   ;e .A/.x; Q / for all and xQ as before, and  2 RC . (In (2.3.17) we avoided writing int M1 instead of M1 because there is smoothness up to the bottom @M1 of the cylinder). Let us denote the corresponding  subspaces of (2.3.17) by Se./ .T  M1 / and S .I/ ;e .T M1 n 0/, respectively. I Summing up, every A 2 Lcl .M / has a principal symbol  .A/ WD . .A/; e .A/; 

;e .A//

(2.3.18)

belonging to  S ./ .T  M n f0g/  Se./ .T  M1 /  S .I/ ;e .T M1 n 0/:

(2.3.19)

As in the local theory there is a compatibility condition between the components. Given a triple .p ; pe ; p ;e / in the space (2.3.19) for which this compatibility condition holds there is an element A 2 LI cl .M / such that p D  .A/; pe D e .A/; p Theorem 2.3.22. Let

A 2 LI .M /; .cl/

;e

D

B 2 LI .M /: .cl/

;e .A/:

(2.3.20)

Then we have AB 2 LCIC .M / and, in the classical case,  .AB/ D  .A/ .B/ .cl/ (with componentwise multiplication).

111

2.3 Operators on manifolds with conical exit to infinity

To simplify notation we now assume that M0 and X are compact. Theorem 2.3.23. Every A 2 LI .M /, ;  2 R, induces continuous operators A W H sIg .M / ! H sIg .M /

(2.3.21)

for every s; g 2 R. As a corollary we obtain that A W S.M / ! S.M / is also continuous. Remark 2.3.24. A 2 LI cl .M / and  .A/ D 0 implies that A is compact as an operator (2.3.21) for every s; g 2 R. Remark 2.3.25. In the case that M0 or X are not necessarily compact, for every admissible cut-off function ' we have the following results: (i) A 2 LI .M / entails 'A; A' 2 LI .M /, and A' induces continuous operasIg sIg tors A' W Hloc .M / ! Hloc .M /, for all s; g 2 R and A' W Sloc .M / ! Sloc .M /. .M /, and we have  .A'B/ D (ii) Given (2.3.20) it follows that A'B 2 LCIC .cl/ .A/.'B/ in the classical case.

2.3.4 Ellipticity in the scalar case Definition 2.3.26. Let M be a manifold with conical exit and cross section X , written in the form (2.3.8). An operator A 2 LI .M / is called elliptic (of order .; / 2 in the space L .M / in the usual R2 ) if it is elliptic ˇ ˇ sense and if the local complete Q 2 S I .RnC1 RnC1 /ˇ symbol  .Aˇ.1;1/U /.x; Q / with respect to every chart RnC1  W .1; 1/  U ! , cf. (2.3.13), is elliptic in the sense of Definition 2.3.12. 2 Remark 2.3.27. For A 2 LI cl .M / the ellipticity (of order .; / 2 R ) is equivalent to the following conditions:

Q 6D 0  .A/.x; Q / Q 6D 0 e .A/.x; Q / 

Q 6D 0

Q / ;e .A/.x;

Q 2 T  M n 0; for all .x; Q / Q 2 T  M1 ; for all .x; Q / Q 2 T  M1 n 0: for all .x; Q /

Theorem 2.3.28. Let A 2 LI .M /, ;  2 R, and let M0 be compact, cf. the .cl/ decomposition (2.3.8). (i) The operator A is elliptic (of order .; / 2 R2 ) if and only if A W H sIg .M / ! H sIg .M / is a Fredholm operator for an .s0 ; g0 / 2 R2 .

(2.3.22)

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2 Symbolic structures and associated operators

(ii) If A is elliptic, (2.3.22) is a Fredholm operator for all .s; g/ 2 R2 , and dim ker A, dim coker A are independent of s; g. We have V WD ker A  S.M /, and there is a finite-dimensional subspace W  S.M / such that W C im A D H sIg .M / and W \ im A D f0g for every .s; g/ 2 R2 . .M / has a parametrix P 2 LI .M / in the (iii) An elliptic operator A 2 LI .cl/ .cl/ sense Cl WD I  PA, Cr WD I  AP 2 L1I1 .M /, and P can be chosen in such a way that the operators Cl and Cr are projections Cl W H sIg .M / ! V; Cr W H sIg .M / ! W to the spaces V and W of (ii), for all .s; g/ 2 R2 . Remark 2.3.29. Let M be a manifold with conical exit, where M0 and X D M0 \ M1 are not necessarily compact. The notion of ellipticity makes sense also in this case. Then, if A 2 LI .M / is elliptic, there is a parametrix P 2 LI .M / in the sense that 'I  'P A; 'I  A'P 2 L1I1 .M / for every choice of admissible cut-off functions ' and

such that

1 on supp '.

Remark 2.3.30. Definition 2.3.26 and Remark 2.3.27 have a straightforward extension to the parameter-dependent case in the sense of Remark 2.3.21. We then talk about parameter-dependent ellipticity. Theorem 2.3.31. Let A 2 LI .M I Rl / be parameter-dependent elliptic. Then there .cl/ is a parameter-dependent parametrix P 2 LI .M I Rl / such that the analogues .cl/ of the remainders in Remark 2.3.29 belong to L1I1 .M I Rl /. In addition, if M0 is compact, then the operators (2.3.22) for A D A./ are Fredholm of index 0, and there is a constant C > 0 such that the operators are isomorphisms for all jj  C . The proof of the existence of a parameter-dependent parametrix in the first part of Theorem 2.3.31 is straightforward. The second part is a simple consequence of that, using Theorem 2.3.28 together with the fact that A./ is elliptic in the usual sense for every fixed .

2.3.5 Calculus with operator-valued symbols We now outline a few elements of the pseudo-differential calculus with conical exits to infinity for the case of operator-valued symbols. Specialised to scalar symbols we formulate a number of additional observations and results, compared with the material of the preceding sections. For instance, in the following definition we admit the dimensions of variables and covariables to be independent. z be Hilbert spaces with group actions f g2R and Definition 2.3.32. Let H and H C z / for ;  2 R is defined fQ  g2RC , respectively. Then the space S I .Rq  Rm I H; H z // such that to be the set of all a.y; / 2 C 1 .Rq  Rm ; L.H; H



˚ 1 sup hiCjˇ j hyiCj˛j Q hi Dy˛ D ˇ a.y; / hi L.H;Hz / .y;/2RqCm

113

2.3 Operators on manifolds with conical exit to infinity

is finite for every ˛ 2 N q ; ˇ 2 N m . z / WD S.Rq  Rm ; L.H; H z //. Moreover, let We set S 1I1 .Rq  Rm I H; H z /: S I WD S I .Rq  Rm I H; H Observe that when z // W a.y; / D  Q  a.y; / 1 for SŒ WD fa.y; / 2 C 1 .Rq  Rm ; L.H; H  all   1; jj  c; y 2 Rq g and z // W a. y; / D  a.y; / for SyŒ WD fa.y; / 2 C 1 .Rq  Rm ; L.H; H all  1; jyj  c;  2 Rm g with constants c D c.a/, we have SŒ \ SyŒ  S I . Let us verify, for instance, the first symbolic estimate (i.e., without derivatives). We have     1   (2.3.23) D hi Q hi a y;  a.y; / D a y; hi hi hi hi for all y 2 Rq ; jj  c. It follows that 1 a.y; /hi kL.E;Ez / < 1: sup hi hyi kQ hi

(2.3.24)

jyjc; jj c

Analogously, a.y; / D hyi a





y ; hyi

for all jyj  c;  2 Rm gives us

1 sup hi hyi kQ hi a.y; /hi kL.E;Ez / < 1:

(2.3.25)

jyj c; jjc

 y   1 Moreover, we have a.y; / D hyi hi Q hi a hyi ; hi hi for all jyj  c; jj  c which yields 1 sup hi hyi kQ hi a.y; /hi kL.E;Ez / < 1: (2.3.26) jyj c; jj c

Then (2.3.24), (2.3.25), (2.3.26) together give rise to the desired estimate. Let SclIŒ  S I \ SyŒ denote the subspace of those a.y; / such that there  P are elements ak .y; / 2 SŒk \ SyŒ ; k 2 N, with a.y; /  N kD0 ak .y; / 2 ŒI .N C1/I S . In a similar manner we define the space Scly by interchanging the role of y and .  S I defined to be the subspace of those a.y; / such that Moreover, let SclI  P , k 2 N, with a.y; /  N there are elements ak .y; / 2 SclŒkI kD0 ak .y; / 2 

114

2 Symbolic structures and associated operators

S .N C1/I for all N 2 N. Analogously, by interchanging the role of y and  we obtain the space SclI  S I . y z // Let Scl./I defined to be the set of all p .y; / 2 C 1 ..Rq n f0g/  Rm ; L.H; H y

for every excision function ./ in Rq . Analogously, such that ./p^ .y; / 2 SclŒI y I./ z // such to be the set of all pe .y; / 2 C 1 .Rq  .Rm n f0g/; L.H; H we define Scl 

for every excision function .y/ in Rm . There are then that .y/pe .y; / 2 SclIŒ  unique linear maps ! Scl./I ; ^ W SclŒI y y

e W SclIΠ! SclI./  

(2.3.27)

such that ^ .a0 /.y; / D a0 .y; / for jj  c for some c D c.a0 / > 0, and e .b0 /.y; / D b0 .y; / for jyj  c for a c D c.b0 / > 0. z / of classical (in y and ) symbols Definition 2.3.33. The space SclI .Rq  Rm I H; H Iy z / such that there are is defined to be the set of all a.y; / 2 S I .Rq  Rm I H; H sequences ; ak .y; / 2 SclŒkI y with a.y; /  for all N 2 N.

PN kD0

bl .y; / 2 SclIŒl ; 

C1/I ak .y; / 2 Scl.N , a.y; /  y

k; l 2 N

PN lD0

C1/ bl .y; / 2 SclI.N , 

z D C this definition is equivalent to DefiniIt can be proved that for H D H tion 2.3.5. z / DW SclI we form Given an a 2 SclI .Rq  Rm I H; H Iy Iy ^ W SclI ! Scl./I ; Iy y

e W SclI ! SclI./ Iy 

obtained by composing a ! a0 , a ! b0 (with a0 ; b0 as in Definition 2.3.23) with the respective maps in (2.3.27). Moreover, we can form the principal exit symbol e ^ .a/.y; / 6D 0 (homogeneous in y 6D 0 of order ) and the principal edge symbol ^ e .a/.y; / (twisted homogeneous in  6D 0 of order ). Then we have ^;e .a/.y; / WD ^ e .a/.y; / D e ^ .a/.y; / for every a.y; / 2 SclI . We call the triple Iy  .a/ WD .^ .a/; e .a/; ^;e .a// the principal symbolic hierarchy of the symbol a. Similarly as for scalar symbols we now generalise the notation of Definition 2.3.32 as follows. By 0 z /; S I; .Rq  Rq  Rq I H; H

; ;  0 2 R,

(2.3.28)

115

2.3 Operators on manifolds with conical exit to infinity

z // such that we denote the set of all operator functions a.y; y 0 ; / 2 C 1 .R3q ; L.H; H

˚

0 0 0 sup hiCjˇ j hyiCj˛j hy 0 i Cj˛ j Q 1 D ˛ D ˛0 D ˇ a.y; y 0 ; / hi z hi

.y;y 0 ;/2R3q

y

y

L.H;H /



(2.3.29) is finite for every ˛; ˛ 0 ; ˇ 2 N q . The spaces (2.3.28) are Fréchet with the semi-norm systems (2.3.29). Analogously as in Definition 2.3.33 we have corresponding subspaces 0 z /. .Rq  Rq  Rq I H; H of classical symbols SclI; Iy;y 0 Let us give a list of simple observations in the case  0 D 0 in the sense of Definition 2.3.32; analogous statements hold in the general case. The proof of the following proposition is straightforward. Proposition 2.3.34. The symbol spaces have the following properties: (i) The differentiation induces continuous operators I z / ! S jˇ jIj˛j .Rq  Rm I H; H z/ .Rq  Rm I H; H Dy˛ Dˇ W S.cl .clIy / Iy /

for all ˛; ˇ 2 N q . (ii) The pointwise composition induces bilinear continuous maps I z /  S I S.cl .Rq  Rm I H0 ; H / .cl

Iy /

Iy

.Rq  Rm I H; H0 /

CIC z /: ! S.cl .Rq  Rm I H; H / Iy

z are the constants from the estimates (2.2.1) for  and , (iii) If M and M Q reI q m z / ,! .R  R I H; H spectively, then we have continuous embeddings S.cl / Iy

z I CM CM z /1 .Rq  Rm I H; H S.cl Iy /

with ‘1’ indicating the symbol spaces with trivial z /. In particular, we have group action (i.e., the identity, in the spaces H and H z / D S.Rq  Rm ; L.H; H z //; S 1I1 .Rq  Rm I H; H

(2.3.30)

independently of the choice of , . Q I Remark 2.3.35. Observe that fp.y; /  idH W p.y; / 2 S.cl .Rq  Rm /g is a / Iy

I subspace of S.cl .Rq  Rm I H; H / for every Hilbert space H with group action. In Iy / particular, the multiplication by p I .y; / WD hi hyi  idHz gives us isomorphisms I z / ! S CIC .Rq  Rm I H; H z /: p I W S.cl .Rq  Rm I H; H / / .cl Iy

Iy

Similar isomorphisms are obtained by corresponding multiplications from the right. 0 z / we form Given a double symbol a.y; y 0 ; / 2 S I; .Rq  Rq  Rq I H; H the associated pseudo-differential operator Op.a/, first as a mapping S.Rq ; H / ! z /, by the oscillatory integral for every fixed y S.Rq ; H “ 0 Op.a/u.y/ D e i.yy / a.y; y 0 ; /u.y 0 /dy 0 μ z /. for e iy a.y; y 0 ; /u.y 0 / in the space P .Ryq 0  Rq ; H

116

2 Symbolic structures and associated operators

Remark 2.3.36. If we reformulate Op.a/ as “ 0 Op.a/u.y/ D e iy  a.y; y 0 C y; /u.y 0 C y/dy 0 μ; then Op.a/ can be interpreted as an oscillatory integral in the space P .Ryq 0  Rq ; z //. S.Ryq ; H Let us set ˚

z / WD Op.a/ W a.y; / 2 S I .Rq  Rq I H; H z/ ; .Rq I H; H LI .cl/ .cl / Iy

z /. Later on interpreted as a space of continuous operators A W S.Rq ; H / ! S.Rq ; H in this section we show the continuity in weighted edge spaces. 0 z /, and let the Proposition 2.3.37. Let a.y; y 0 ; / 2 S I; .Rq  Rq  Rq I H; H z Cmaxj D0;:::; j sequences  D f g2N and ı D fı g2N be defined by  D M C M z are the constants associated with the j j, ı D maxj D0;:::; j 0  j j, where M and M group actions  and , Q respectively, cf. the formula (2.2.1). Then we have 0

z //; b.x; / WD ..y; / ! a.y; y C x;  C // 2 P ;ı.Rq  Rq ; S IC .Rq  Rq I H; H and the mapping a ! b is continuous. In particular, by Proposition 2.2.51 we have 0

z //; c. / WD ..x; / ! b.x;  // 2 C.Œ0; 1 ; P C1;ı.Rq  Rq ; S IC .Rq  Rq I H; H and the map a ! c is continuous. Similar relations hold for symbols in the spaces with subscript ‘clIy ’. z //. By the Proof. First note that we have a.y; y C x;  C / 2 C 1 .R4q ; L.H; H fundamental theorem of the calculus we even obtain 0 z // @˛ @ˇx b.x; / 2 C.R2q ; S IC .Rq  Rq I H; H 0 z / and Vz D for all ˛; ˇ 2 N q . Applying Lemma 2.2.44 for V D S IC .Rq Rq I H; H 1 2q 1 2q IC 0 q z z //. C .R ; L.H; H // it follows that b.x; / 2 C .R ; S .R  Rq I H; H Iı q q IC 0 q q z In order to show b.x; / 2 P .R R ; S .R R I H; H // we have to verify that qN .b/ WD supfh iN hxiıN pN .Dx˛ D ˇ b.x; // W .x; / 2 R2q g

z / of the form is finite for every semi-norm pN on the space S IC .Rq  Rq I H; H (2.3.29). The norm

1 ˚ ˛ ˇ ˛0 ˇ 0



Q z/ hi Dx D Dy D a.y; y C x;  C / hi L.H;H 0

can be estimated from above by a linear combination of terms of the form

1 ˚   0 C˛ ˇ Cˇ 0 

z c Q hC i a .y; y C x;  C / hC i L.H;Hz / h iM CM Dy Dy 0 D

2.3 Operators on manifolds with conical exit to infinity

117

z

with multi-indices ;  0 such that j C  0 j D j˛ 0 j. The factor h iM CM comes from (2.2.3). The latter expression can be estimated by 0

c'.a/hyijj hy C xi j

0 C˛j

z

0

h C ijˇ Cˇ j h iM CM

0

0

0

0

0

0

0

0

z

 c'.a/hyiC j˛ j hijˇ j hxij j k h iM CM Cjjˇ

0k

 c'.a/hyiC j˛ j hijˇ j hxiıN h iN with a constant c > 0 coming from Peetre’s inequality, and '.a/ defined by

1 ˚ 0 #

˚



0 0 sup hyij j hy 0 ij j hij#j Q hi Dy Dy 0 D a.y; y 0 ; / hi L.H;Hz / ; where the supremum is taken over all .y; y 0 ; / 2 R3q and all multi-indices ;  0 ; # 2 N q such that j C  0 C #j  N C j˛ C ˇj. It follows that qN .b/  c'.a/ which gives us the continuity of the mapping a ! b. I; Theorem 2.3.38. For every a.y; y 0 ; / 2 S.cl

0

Iy;y 0 / IC 0

z / there are .Rq  Rq  Rq I H; H

z / and aR .y 0 ; / 2 unique left and right symbols aL .y; / 2 S.clIy / .Rq  Rq I H; H 0 z /, respectively, such that Op.a/ D Op.aL / D Op.aR /. The S IC .Rq  Rq I H; H .clIy 0 /

explicit expressions and the asymptotic expansions for aL and aR are the same as in Theorem 2.2.52 (here for the case l D 0), with remainders belonging to .N C1/;C 0 .N C1/ z /, and also the other assertions of Theorem 2.2.52 .Rq Rq I H; H S.cl Iy / hold in analogous form for the present symbol spaces. Proof. The proof is similar to that of Theorem 2.2.52. The main difference is the presence of the weights ;  0 in the variables at 1, but the modifications are straightforward. Remark 2.3.39. Theorem 2.3.38 shows that the map I z / ! LI .Rq I H; H z/ .Rq  Rq I H; H Op W S.cl / .cl/ Iy

is an isomorphism for every ;  2 R. Also z / ! L1I1 .Rq I H; H z/ Op W S 1I1 .Rq  Rq I H; H is an isomorphism, cf. the formula (2.3.30). z /  S  .Rq  Rq I H; H z /b , cf. Remark 2.3.40. By virtue of S I0 .Rq  Rq I H; H I q q z Definition 2.2.26, for every a.y; / 2 S .R  R I H; H / we can write Op.a/ D hyi Op.hyi a/ with Op.  / on the right-hand side being interpreted as in (2.2.45) (as an oscillatory integral for every fixed y). In other words, many properties of the calculus with symbols in S I -spaces are a slight modification of those with symbols in the sense of Section 2.2.3.

118

2 Symbolic structures and associated operators

I z z; H z /; b.y; / 2 S I .Rq  Theorem 2.3.41. Let a.y; / 2 S.cl .Rq  Rq I H .clIy / Iy / z q Q z z z Q respectively. R I H; H / for Hilbert spaces H; H and H with group actions ; Q and , Then we have Op.a/ Op.b/ D Op.c/ CIC z z /, the Leibniz for a unique c.y; / D .a # b/.y; / 2 S.cl .Rq  Rq I H; H Iy / product between a and b, given by the corresponding formula of Theorem 2.2.54 (here for l D 0). There is an asymptotic expansion of the form (2.2.51), here with a z z / of the form (2.2.52), remainder rN .y; / 2 S C.N C1/IC .N C1/ .Rq  Rq I H; H .clIy /

and all assertions of Theorem 2.2.54 are valid in an analogous form. Proof. The proof is analogous to that of Theorem 2.2.54. z; H z0 ; H z 0 I g Theorem 2.3.42. If fH; H0 ; H 0 I g and fH Q are Hilbert space triples with I q q z / there is an a .y; / 2 group action, for every a.y; / 2 S.clIy / .R  R I H; H I z 0 ; H 0 / such that the relations of Theorem 2.2.58 hold in analogous S .Rq  Rq I H .clIy /

form.

Proof. The proof is analogous to that of Theorem 2.2.58. We now turn to the continuity of pseudo-differential operators in weighted edge spaces. If H is a Hilbert space with group action we set W sIg .Rq ; H / WD hyig W s .Rq ; H /

(2.3.31)

for s; g 2 R. z be Hilbert spaces with group action, and let a.y; / 2 Theorem 2.3.43. Let H; H I q q z S .R  R I H; H /, ;  2 R. Then Op.a/ induces a continuous operator z /; Op.a/ W W sIg .Rq ; H / ! W sIg .Rq ; H and a ! Op.a/ is continuous as an operator z / ! L.W sIg .Rq ; H /; W sIg .Rq ; H z // S I .Rq  Rq I H; H for every s; g 2 R. Proof. Let us fix s; g 2 R, and form the operators P Ig WD Op.hyig hi /

and

QIg WD Op.hi # hyig /:

They induce isometric isomorphisms P Ig W W s .Rq ; H / ! W sIg .Rq ; H /; QIg W W sIg .Rq ; H / ! W s .Rq ; H /;

2.3 Operators on manifolds with conical exit to infinity

119

z in place of H . This follows immediwhere QIg D .P Ig /1 , and the same for H ately from the definition of weighted edge spaces (2.3.31) and of Theorem 2.2.22. By z/ virtue of Theorem 2.3.41 there exists a unique element a0 .y; / 2 S 0I0 .Rq Rq I H; H z/ ! such that Op.a/ D P sIg Op.a0 /QsIg , and the mapping S I .Rq  Rq I H; H z /, defined by a ! a0 , is continuous. From Theorem 2.2.20 we S 0I0 .Rq  Rq I H; H then obtain k Op.a/kL.W sIg .Rq ;H /;W sIg .Rq ;Hz //  c.a0 / (2.3.32) with .  / as in Theorem 2.2.20, for a constant c > 0. In particular, we see the continuity of the mapping a ! Op.a/. z be Hilbert spaces with group action, and let a.y; / 2 Corollary 2.3.44. Let H; H  q z /,  Rq open,  2 R. Then Op.a/ induces continuous operators S .  R I H; H s s z/ .; H / ! Wloc .; H Op.a/ W Wcomp

for all s 2 R. z / for It suffices to show the continuity of ' Op.a/ W W s .Rq ; H / ! W s .Rq ; H 1 every ' 2 C0 ./ (functions with compact support in  are interpreted as functions in Rq by extension by zero). Then '.y/a.y; / can be regarded as a symbol in S I0 .Rq  z /, and it suffices to apply Theorem 2.3.43 together with the continuous map Rq I H; H s Wcomp .; H / ! W sI0 .Rq ; H /. z be Hilbert spaces with group action, let a.y; / 2 Theorem 2.3.45. Let H and H I q q z z be a compact S .R  R I H; H / such that  < 0,  < 0, and let a.y; / W H ! H 2q sIg q z / is operator for every .y; / 2 R . Then Op.a/ W W .R ; H / ! W sIg .Rq ; H compact for every s; g 2 R. Proof. Applying the reductions of orders and weights of the proof of Theorem 2.3.43 it suffices to assume s D g D 0. Let first a.y; / have a compact support with respect to z / is compact, cf. .y; / 2 R2q . Then the operator Op.a/ W W 0 .Rq ; H / ! W 0 .Rq ; H [188, Theorem 1.3.61]. For general a.y; / we consider a" .y; / WD ."y; "/a.y; /, " > 0, for some  2 C01 .R2q / with  1 near 0. Since a is of negative order, z / as " ! 0. Then (2.3.32) gives we then have a" ! a in S 0I0 .Rq  Rq I H; H 0 q 0 z // as " ! 0. Hence, Op.a/ is us Op.a" / ! Op.a/ in L.W .R ; H /; W .Rq ; H compact as a norm limit of compact operators. Corollary 2.3.46. Let H , H 0 be Hilbert spaces with group action and  W H 0 ,! H a compact embedding, and let the group action on H restrict to the group action on H 0 . Then for every s 0 > s, g 0 > g we have a compact embedding 0

0

W s Ig .Rq ; H 0 / ,! W sIg .Rq ; H /: 0

0

(2.3.33) 0

0

In fact, setting P WD Op.hyigg hiss /, Q WD Op.his s # hyig g /, The0 0 orem 2.3.43 gives us a continuous operator Q W W s Ig .Rq ; H 0 / ! W sIg .Rq ; H /; sIg moreover, by Theorem 2.3.45 the operator P W W .Rq ; H 0 / ! W sIg .Rq ; H / is compact. Now the embedding (2.3.33) is equal to the composition PQ.

120

2 Symbolic structures and associated operators

2.4 Mellin operators In Section 2.3 we studied a calculus of operators on X ^ D RC  X 3 .r; x/ for a C 1 manifold X , for r ! 1 from the point of view of a conical exit to infinity. We now turn to a counterpart for r ! 0. The nature of our calculus near zero is different from that near infinity; x C  X /=.f0g  X /. The typical operators zero is interpreted as the tip of the cone X D .R who reflect the conical metric near zero are of Fuchs type (in stretched coordinates .r; x/), and the Mellin transform on the half-axis is adapted to this situation. We outline here some elements of the Mellin operator calculus. This will be applied in many variants in other chapters below. Moreover, we consider edge-degenerate operators and show a connection between edgedegeneracy for r ! 0 and exit calculus for r ! 1.

2.4.1 The Mellin transform The Mellin transform

Z

1

M u.w/ WD

r w u.r/ 0

dr r

on the half-axis RC 3 r is first defined for u 2 C01 .RC /. In this case M u.w/ is an entire function in the covariable w 2 C. Later on we interpret M in an extended sense on larger (in general, vector-valued) distribution spaces. Then the variable w will vary on a weight line (2.4.1) ˇ WD fw 2 C W Re w D ˇg for some real ˇ or on another subset of the complex plane. The properties of the Mellin transform of scalar functions have natural analogues for the corresponding extensions of M . Therefore, we first consider the simplest scalar case; the generalisations will often be evident and used without further explanation. Setting 1

S u.t / WD e . 2 /t u.e t /;

t 2 R;

(2.4.2)

C 1 .RC / ! C 1 .R/:

(2.4.3)

for any fixed real  we obtain isomorphisms S W C01 .RC / ! C01 .R/; 1

For r D e t we have S1 f .r/ D r . 2 / f . log r/. By virtue of the equality R1  u.r/j2 dr D 1 jS u.t /j2 dt it follows an extension of (2.4.3) to an isomor0 jr phism (2.4.4) S W r  L2 .RC / ! L2 .R/: R 1 it% v.t /dt be the standard Fourier transform and insert v.t / D Let F W v.t/ ! 1 e 1 S u.t/ for u 2 C0 .RC /. For w D 12   C i% we then obtain R1

1  M u   C i% D .F S u/.%/: 2

(2.4.5)

121

2.4 Mellin operators

We define

ˇ M u.w/ WD M u.w/ˇ

1 

2

and call M the weighted Mellin transform with weight . Spaces of distributions that are originally given with respect to a real argument % 2 R such as L2 .R/, S.R/, S 0 .R/, etc., will often be used in an analogous meaning on lines ˇ D f.ˇ C i%/ W % 2 Rg via the bijection ˇ C i% ! %. In that way we obtain the spaces L2 .ˇ /; S.ˇ /; S 0 .ˇ /, etc. Let us set T  .RC / WD S1 S.R/;

 2 R:

(2.4.6)

Proposition 2.4.1. The weighted Mellin transform M induces isomorphisms M W r  L2 .RC / ! L2 . 1  /; 2

M W T  .RC / ! S. 1  /; 2

R and the inverse is given by the formula u.r/ D .2 i /1  1

2 

r w g.w/dw.

Proof. The continuity results immediately follow from (2.1.10), (2.1.11) and (2.4.4), (2.4.6).The inversion formula is a consequence of (2.1.12) and (2.4.5). Remark 2.4.2. u 2 T  .RC / implies r

@ u.r/ 2 T  .RC /; @r

u.r/ log r 2 T  .RC /;

and

r p u.r/ 2 T CRe p .RC /

for every p 2 C, and we have  @  d M u.w/; M  r u .w/ D wM u.w/; M ..log r/u/.w/ D @r dw MCRe p .r p u/.w/ D M u.w C p/; w 2  1 Re p : 2

Let us set  opM .f /u.r/ D

“

1 0

 r . 12 Ci%/  0 1 dr 0 f r; r ;   C i% u.r 0 / 0 μ % 0 r 2 r

(2.4.7)

for f .r; r 0 ; w/ 2 S  .RC  RC   1  /, u 2 C01 .RC /; μ % D .2/1 d%. We then 2

 .f / W C01 .RC / ! C 1 .RC /. Observe that obtain a continuous operator opM

  0 1 for p.t; t 0 ; / D f e t ; e t ;   C i ; 2 ’ i.tt 0 / cf. the formula (2.4.2); here op t .p/u.t / D e p.t; t 0 ; /u.t 0 /dt 0 μ .  .f / D S1 op t .p/S opM

xC  R x C   1 /, and let f be indeProposition 2.4.3. Assume f .r; r 0 ; w/ 2 S  .R  2

 .f / extends pendent of r for r > R and of r 0 for r 0 > R for some R > 0. Then opM    to a continuous operator opM .f / W T .RC / ! T .RC /.

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2 Symbolic structures and associated operators

Later on we shall obtain more general results on the continuity of Mellin pseudodifferential operators, cf. Section 2.4.4 below. Proposition 2.4.3 is a special case; therefore, we do not give the proof here. Another simple result is as follows.  Let A WD opM .f / with f .r; r 0 ; w/ as in Proposition 2.4.3, and define the formal adjoint A of A by .Au; v/L2 .RC / D .u; A v/L2 .RC / for all u; v 2 C01 .RC /.   Proposition 2.4.4. The formal adjoint A of A D opM .f / has the form opM .f Π/ xC  R x C   1 //. with f Π.r; r 0 ; w/ WD fx.r 0 ; r; 1  w/ x (which belongs to S  .R C 2

Later on we shall return to this kind of relations in more general form (see Theorem 2.4.60 below).

2.4.2 Weighted Sobolev spaces In Section 1.2.1 we have seen the role of weighted Sobolev spaces H s; .X ^ / and K s; .X ^ / for a compact C 1 manifold X (with or without boundary) in the special 1 cases X D S 1 or X D SC . We now give the definitions in general and formulate some useful properties. By a cut-off function (on the half-axis) we understand any real-valued !.r/ 2 x C / such that !.r/ 1 in a neighbourhood of r D 0. C01 .R Definition 2.4.5. Let X be a closed compact C 1 manifold of dimension n, and set X ^ D RC  X 3 .r; x/ regarded as an infinite stretched cone with base X. (i) The space

H s; .X ^ /

for s;  2 R

C01 .X ^ /

is defined to be the completion of with respect to the norm  12 Z 1 s 2 .2 i / kR .Im w/M u.w/kL2 .X/ dw ;  nC1 2



n D dim X , where Rs .%/ 2 Ls.cl/ .X I R% / is an order reducing family on X in the sense of Theorem 2.1.31. (ii) The space

K s; .X ^ /

for s;  2 R

(2.4.8)

s s is defined as fu 2 Hloc .X ^ / W !u 2 H s; .X ^ /; .1  !/u 2 Hcone .X ^ /g for s s ^ anyˇ cut-off function !.r/; here Hcone .X / is the subspace of all f 2 Hloc .R  ˇ X/ R X such that for every coordinate neighbourhood U on X and every C

such diffeomorphism ˇ W RC  U !  to an open conical subset   RnC1 xQ that ˇ.ır; x/ D ıˇ.r; x/ for all ı 2 RC , .r; x/ 2 RC  U , we have Q 2 H s .RnC1 / ..1  !/'u/.ˇ 1 .x// xQ for every cut-off function ! and ' 2 C01 .U /.

2.4 Mellin operators

123

ˇ s Remark 2.4.6. We have Hcone .X ^ / D H sI0 .X /ˇR X , cf. the formulas (2.3.9) and C (2.3.11). Setting ˇ sIg .X ^ / WD H sIg .X /ˇR X (2.4.9) Hcone C

we can define a generalisation of the spaces (2.4.8) to spaces with weights also for r ! 1, namely, s sIg K s;Ig .X ^ / WD fu 2 Hloc .X ^ / W !u 2 H s; .X ^ /; .1  !/u 2 Hcone .X ^ /g (2.4.10) for every s; ; g 2 R. For purposes below we set

K s; .X ^ / WD K s;Is .X ^ /: Observe that then

(2.4.11)

K s; .X ^ / D r  K s;0 .X ^ /:

Remark 2.4.7. For s D  D 0 we have K 0;0 .X ^ / D H 0;0 .X ^ / D r with L2 .RC  X / referring to the measure drdx.

(2.4.12) n 2

L2 .RC  X /

Proposition 2.4.8. (i) The spaces K s; .X ^ /, s;  2 R, are Hilbert spaces with scalar products from the identification s K s; .X ^ / D Œ! H s; .X ^ / C Œ1  ! Hcone .X ^ /;

(2.4.13)

cf. Definition 2.1.4 (ii) and Remark 2.1.5, using corresponding Hilbert space products s in the spaces H s; .X ^ / and Hcone .X ^ /. 0 0 (ii) There are continuous (dense) embeddings K s ; .X ^ / ,! K s; .X ^ / for arbitrary s 0  s,  0   . (iii) The K 0;0 .X ^ /-scalar product extends from C01 .X ^ /  C01 .X ^ / ! C to a non-degenerate sesquilinear pairing .; /K 0;0 .X ^ / W K s; .X ^ /  K s; .X ^ / ! C for all s;  2 R. Remark 2.4.9. Writing s K s; .X ^ / D Œ! H Q s; .X ^ / C Œ1  ! H Q cone .X ^ /

(2.4.14)

for another !, Q the Hilbert space norms associated with (2.4.13) and (2.4.14) are equivalent. Thus, denoting for the moment the norms in the representations (2.4.13) and (2.4.14) by k  ks; and k  k

s; , respectively, there are constants c1 ; c2 > 0 such that c1 kuks;  kuk

s;  c2 kuks;

(2.4.15)

for all u 2 K s; .X ^ /. Proposition 2.4.10. For every ı > 0 there are constants b1 .ı/; b2 .ı/ > 0 such that s .X ^ /  kukK s; .X ^ /  b2 .ı/kukH s .X ^ / b1 .ı/kukHcone cone

for all u 2 K s; .X ^ / such that u.r; x/ D 0 for r < ı.

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2 Symbolic structures and associated operators

Proof. Without loss of generality we may concentrate on elements u that are supported with respect to X in a coordinate neighbourhood U on X . An argument in terms of a partition of unity then yields the desired estimate in general. Substituting a diffeomorphism U ! V for an open set on S n it suffices to consider the case X D S n , and we can even forget about the condition on the support with respect to the angular variables. Then the desired estimate has the form b1 .ı/kukH s .RnC1 /  kukK s; ..S n /^ /  b2 .ı/kukH s .RnC1 / : Let us fix ı > 0 and apply Remark 2.4.9. Then by a suitable choice of !Q we can assume !u Q D 0 and u 2 Œ1  ! H Q s .RnC1 / when u vanishes for r < ı. Let k  kK s; ..S n /^ / denote the norm associated with (2.4.14). Then from the definition of the norm in a Q s; ..S n /^ /, E1 D Œ1  ! H Q s .RnC1 / it non-direct sum E D E0 C E1 for E0 D Œ! H follows that ˚

kukK s; ..S n /^ / D kukE D inf ku0 kE0 C ku1 kE1 W u D u0 C u1 ; ui 2 Ei ; i D 0; 1 D kukH s .RnC1 / ; since u D u0 C u1 only holds for u0 D 0; u1 D u. Moreover, we have a chain of continuous maps E1 ! E0 ˚ E1 ! E0 C E1 which shows that kukK s; ..S n /^ /  b2 .ı/kukH s .RnC1 / for a suitable constant b2 .ı/ > 0. Together with (2.4.15) we then obtain the assertion. Remark 2.4.11. The spaces fK s; .X ^ /; K 0;0 .X ^ /; K s; .X ^ /g for s;  2 R form a Hilbert space triple fE; E0 ; E 0 g in the sense of Definition 2.1.6. Proposition 2.4.12. The space H s; .X ^ / for s 2 N,  2 R, can equivalently be n described as the set of all u.r; x/ 2 r  2 C L2 .RC  X / such that  k ˛ n r@r Dx u.r; x/ 2 r  2 C L2 .RC  X / for every k 2 N, ˛ 2 N n , k C j˛j  s, where Dx˛ denotes any differential operator on X of the form v1˛1 : : : vn˛n with vector fields v1 ; : : : ; vn on X . For arbitrary real s the space H s;0 .X ^ / can be obtained by duality with respect to the scalar product of H 0;0 .X ^ / (which extends the definition to s 2 N) and then by complex interpolation. Then we can set H s; .X ^ / D r  H s;0 .X ^ /: (2.4.16) Proposition 2.4.12 can be deduced from Proposition 2.4.16 below when we employ a standard characterisation of the cylindrical Sobolev space H s .R  X / for s 2 N as the set of all elements u.t; x/ 2 L2 .R  X / such that @kt Dx˛ u.t; x/ 2 L2 .R  X / for all k C j˛j  s. We shall return to the characterisation of H s; .X ^ / for s 2 N in Section 6.4.1 below from the point of view of special elliptic operators in the cone calculus. Remark 2.4.13. The transformation I n W u.r; x/ ! r n1 u.r 1 ; x/ induces an isomorphism I n W H s; .X ^ / ! H s; .X ^ / for every s;  2 R.

125

2.4 Mellin operators

Another equivalent definition of H s; .X ^ / follows from the following cylindrical Sobolev spaces. Definition 2.4.14. We define the space H s .R  X / for s 2 R to be the completion of ˚R

12 2 . Here Rs . / 2 C01 .R  X/ with respect to the norm kRs . /F t! v. /kL 2 .X/ d s Lcl .XI R / is any order reducing family on X in the sense of Theorem 2.1.31. Remark 2.4.15. The space H s .R  X /; s s 2 R, can be characterised as the set of all u.t; x/ 2 Hloc .R  X / such that

.'u/ ı .1  ˛ 1 / 2 H s .R t  Rn / for every chart ˛ W U ! Rn on X and every ' 2 C01 .U /. Proposition 2.4.16. The first operator of (2.4.3) extends to an isomorphism S n2 W H s; .X ^ / ! H s .R  X /

(2.4.17)

for every s;  2 R. In other words, v.r; x/ ! u.t; x/ WD v.e t ; x/ induces an nC1 isomorphism H s; .X ^ / ! e . 2 / H s .R  X /. This result is a simple consequence of the Definitions 2.4.5 and 2.4.14, together with the relationship (2.4.5) between the Fourier and the Mellin transform. For purposes below we set  1 T  .X ^ / WD S n2 S.R; C 1 .X //: (2.4.18) We then have T  .X ^ / D fu 2 H 1; .X ^ / W u.r; / logk r 2 H 1; .X ^ / for all k 2 Ng: In fact, the relation (2.4.17) gives us an isomorphism S n2 W H 1; .X ^ / ! H 1 .R  X /I then, since S.R; C 1 .X // D fu 2 H 1 .R  X / W t k u.t; / 2 H 1 .R  X / for all k 2 Ng the result follows from r D e t . Remark 2.4.17. Similarly as (2.3.5) or (2.3.11) the spaces H s .R  X / can be generalised to weighted spaces htig H s .R  X / for every g 2 R. Analogously we can form hlog rig H s; .X ^ /. Then the operator S n2 induces a bijection S n2 W hlog rig H s; .X ^ / ! ht ig H s .R  X / for every s; ; g 2 R. The spaces H s; .X ^ / with power weights r  at zero (and infinity) play a crucial role in the pseudo-differential calculus on a cone. The operators are also meaningful in spaces with logarithmic weights hlog rig , though we do not deepen this aspect here.

126

2 Symbolic structures and associated operators

y s .R  X / WD f.F t!% u/.%; x/ W u.t; x/ 2 H s .R  X /g, and Remark 2.4.18. Let H s y .ˇ  X / for any fixed ˇ 2 R denote the pull back of that space under the let H map .ˇ C i%; x/ ! .%; x/. Then the weighted Mellin transform M n2 induces an isomorphism   y s  nC1  X M n2 W H s; .X ^ / ! H  2

for every s;  2 R. Theorem 2.4.19. Let s0 ; 0 ; s1 ; 1 2 R, 0    1, and set s D .1   /s0 C s1 ,  D .1   /0 C 1 . Then the complex interpolation gives us ŒK s0 ;0 .X ^ /; K s1 ;1 .X ^ /  D K s; .X ^ /; ŒH s0 ;0 .X ^ /; H s1 ;1 .X ^ /  D H s; .X ^ /: A proof may be found in [90, Theorem 2.1.16]. Definition 2.4.20. A manifold B with conical singularities is a topological space with a subspace B 0 D fv1 ; : : : ; vN g of conical points such that (i) B n B 0 is a C 1 manifold. (ii) Every v 2 B 0 has a neighbourhood V in B such that there is a homeomorphism x C  X /=.f0g  X / for a C 1 manifold  W V ! X to the cone X WD .R X D X.v/ which restricts to a diffeomorphism reg W V n fvg ! X ^ D RC  X: We say that the choice of  defines the structure of a (regular) cone on B near v 2 B 0. (iii) Another structure of a cone Q W V ! X near v is said to be equivalent to  if Q reg ı .reg /1 W RC  X ! RC  X is the restriction of a diffeomorphism R  X ! R  X to RC  X . Remark 2.4.21. A manifold B with conical singularity fvg D B 0 can equivalently be described in terms of its stretched manifold B, that is, a C 1 manifold with boundary @B Š X such that B D B=@B. In a similar manner we can proceed with more than one conical singularity. Moreover, every C 1 manifold B with boundary gives rise to a manifold B with conical singularity by squeezing down the boundary @B to a single point (which is just the meaning of the quotient space B D B=@B). From now on, for simplicity, we assume that B only has one conical singularity. Incidentally we set Breg WD B n @B; Bsing WD @B:

2.4 Mellin operators

127

Definition 2.4.22. Let B be a compact manifold with conical singularity and B its stretched manifold, X D @B. Then H s; .B/ for s;  2 R is defined to be the s .Breg / such that !.r/u.r; x/ 2 H s; .X ^ / for every cut-off subspace of all u 2 Hloc function ! supported in a collar neighbourhood Š Œ0; 1/  X 3 .r; x/ of @B. Remark 2.4.23. In Definition 2.4.20 we do not assume that X is compact. A modification of the definition allows us to include the case of a C 1 manifold X with boundary @X 6D ;, also not necessarily compact. Instead of Definition 2.4.20 (i) we require that B n B 0 is a C 1 manifold with boundary. The other conditions (ii), (iii) are as before if we talk about diffeomorphisms of C 1 manifolds with boundary. In other words we also have the notion of a manifold B with conical singularities and boundary. There is an easy process to double up a manifold B with conical singularity and boundary by gluing together two copies of B n B 0 along their common boundary and then to add B 0 again. The resulting space 2B is a manifold with conical singularities in the sense of Definition 2.4.20, and we have the corresponding stretched manifold 2B. The preimage of the original B under the quotient map 2B ! 2B, denoted by B, is the stretched manifold of B. We then set Breg WD 2Breg \ B;

Bsing WD 2Bsing \ B:

(2.4.19)

Note that both Breg and Bsing are C 1 manifolds with boundary, while B itself has corners. x C  X with 2B D R x C  2X , where X is a C 1 An example is the case B WD R manifold with boundary and 2X the double, defined by two copies X˙ of X , glued together along @X (we often identify X with XC ). From the construction it follows that 2B D B [ BC for two copies B˙ of B (with B being identified with BC /, and for the associated stretched manifolds B˙ we have 2B D B [ BC :

(2.4.20)

If B is the stretched manifold belonging to a manifold B with conical singularities and boundary we can also form .2/B WD B./ [ B.C/

(2.4.21)

by gluing together two copies B.˙/ of B along the common B.˙/;sing (and we identify B with B.C/ ). Then .2/B is a C 1 manifold with boundary @B./;reg [ @B.C/;sing [ x C  X ; in this case we have .2/B D R  X . @B.C/;reg . An example is B D R Having defined the spaces H s; ..2X /^ / and K s; ..2X /^ /, we can pass to corresponding spaces on X ^ by ˚

H s; .X ^ / WD ujint X ^ W u 2 H s; ..2X /^ / ; ˚

K s; .X ^ / WD ujint X ^ W u 2 K s; ..2X /^ / : Since K s; .X ^ / Š K s; ..2X /^ /= (where ‘’ means the equivalence relation u  v ” ujint X ^ D vjint X ^ ), we obtain the spaces K s; .int X ^ / as Hilbert spaces from

128

2 Symbolic structures and associated operators

the Hilbert space structure of K s; ..2X /^ / and the set of all elements supported by .2X/^ n .int X/^ which is a closed subspace. In a similar manner we can proceed for the space H s; .X ^ / when X is compact, C 1 , with boundary, or for H s; .B/ WD H s; .2B/jint Breg

(2.4.22)

in the case of a stretched manifold B belonging to a compact manifold B with conical singularities and boundary. Similarly as (2.4.10) we define the spaces K s;Ig .X ^ / also for the case of a compact 1 C manifold X with boundary. Remark 2.4.24. (i) Let B be the stretched manifold, associated with a compact manifold with conical singularity, with (or without) boundary. There are canonical continuous embeddings 0 0 H s ; .B/ ,! H s; .B/ for s 0  s;  0 ,  0   ; they are compact when s 0 > s,  0 >  . (ii) Let X be a compact C 1 manifold with (or without) boundary. Then we have continuous embeddings 0

K s ;

0 Ig 0

.X ^ / ,! K s;Ig .X ^ /

for s 0  s;  0  ; g 0  g; they are compact when s 0 > s,  0 >  , g 0 > g. Remark 2.4.25. (i) Let B be a compact manifold with conical singularities B 0 and boundary D WD @.B n B 0 / [ B 0 , and let B and D be the associated stretched manifolds. Then the operator of restriction C01 .Breg / ! C01 .Dreg / from Breg to Dreg extends to a continuous operator 1

1

rD W H s; .B/ ! H s 2 ; 2 .D/ for every s;  2 R, s > 12 . (ii) Let X be a compact C 1 manifold with boundary @X . Then the operator of restriction C01 .X ^ / ! C01 ..@X /^ / extends to a continuous operator 1

1

r.@X/^ W K s; .X ^ / ! K s 2 ; 2 ..@X /^ /

(2.4.23)

for every s;  2 R; s > 12 . Definition 2.4.26. A manifold W with edge is a topological space with a subspace Y , the edge, such that (i) W n Y and Y are C 1 manifolds; (ii) every y 2 Y has a neighbourhood V  W such that there is a (so-called singular) chart W V ! X   (2.4.24)

2.4 Mellin operators

129

for a C 1 manifold X D X.y/ and an open ˇset  Rq , dim Y ˇD q, such that (2.4.24) is a homeomorphism, and reg WD ˇV nY and sing WD ˇV \Y represent diffeomorphisms reg W V n Y ! X ^   and

sing W V \ Y ! ;

respectively; moreover, for every two elements  W V ! X  , Q W Vz ! z from the given system of such charts also X   ˇ ˇ z ı WD ˇV \Vz W V \ Vz ! X  †; ıQ WD Q ˇV \Vz W V \ Vz ! X  † z / z belong to the system, and (for suitable open sets † ; † z WD ıQreg ı .ıreg /1 W X ^  † ! X ^  † (2.4.25) ˇ has the property C D ˇR X† for some diffeomorphism W R  X  † ! C ˇ z furthermore, the maps ˇ form a cocycle of transition maps R  X  †; C

f0gX †

z of an X -bundle Wsing over Y . X †!X † Remark 2.4.27. A manifold W with edge Y can equivalently be described by its stretched manifold W , that is, a C 1 manifold with boundary @W , and @W is an X-bundle over the edge Y . Then W WD W = 

(2.4.26)

is a manifold with edge, where the equivalence relation  is defined by identifying points w; w 0 2 @W when they belong to the same fibre over a point y 2 Y . Moreover, every C 1 manifold W with boundary @W , where @W has the structure of an X -bundle over a C 1 manifold Y gives rise to a manifold W with edge Y by squeezing down the fibres of @W over every y 2 Y to a single point. Let us set Wreg WD W n @W ;

Wsing WD @W :

An example is W WD X  ,  R open. In this case we have Y D , x C  X/  , Wreg D X ^  , Wsing D @W D X  . In order to introduce W D .R weighted Sobolev spaces on a (stretched) manifold W with edge we first consider the case of an open stretched wedge X ^  Rq with a compact C 1 manifold X (with or without boundary). In this case the spaces will be a realisation of the abstract edge spaces, cf. Definition 2.2.3.

q

nC1

Remark 2.4.28. For u.r; x/ 2 K s; .X ^ / we set . u/.r; x/ D  2 u.r; x/ for  2 RC ; n D dim X . We then obtain a strongly continuous group of isomorphisms  W K s; .X ^ / ! K s; .X ^ / n

for every s;  2 R. On the space K 0;0 .X ^ / D r  2 L2 .RC  X / the group f g2RC is unitary, i.e., we have . f; g/K 0;0 .X ^ / D .f; 1 g/K 0;0 .X ^ / for all  2 RC , f; g 2 K 0;0 .X ^ /.

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2 Symbolic structures and associated operators

Applying Definition 2.2.12 we obtain the weighted edge spaces W s; .X ^  Rq / WD W s .Rq ; K s; .X ^ //:

(2.4.27)

Definition 2.4.29. Let W be a compact manifold with edge Y (in particular, the base X of the model cone is compact), and let W be its stretched manifold. Let us fix a system of singular charts  W V ! X  Rq in the sense of Definition 2.4.26 such that every point of Y belongs to one of the sets V and such that the transition maps (2.4.25) have the property C .r; x; y/ D .0; x; y/ for all 0  r  " for some " > 0. Then W s; .W /

for s;  2 R

s is defined to be the subspace of all u 2 Hloc .Wreg / such that for every ' 2 C01 .V \ Y / and any cut-off function ! (i.e., an element of C 1 .W / supported in a collar neighbourhood of @W and 1 near @W ) we have s; !'u ı 1 .X ^  Rq /: reg 2 W

(2.4.28)

To justify Definition 2.4.29 it is necessary to observe that s s Hcomp .X ^  Rq /  W s; .X ^  Rq /  Hloc .X ^  Rq /

(2.4.29)

for every s;  2 R. This relation is a consequence of Proposition 2.4.10, combined with the computation for the identity (2.2.21), cf. also Proposition 7.1.5 below. In addition we have to check that an isomorphism ı W X  Rq ! X  Rq of (trivial) X -bundles over Rq (which also admits a diffeomorphism ı0 W Rq ! Rq of the base) induces an isomorphism ! W s; .X ^  Rq / ! ! Q W s; .X ^  Rq / for every 2 C01 .Rq / and Q WD ı ı 1 . We return to this question in more detail in Section 7.1.2 below. 0 Definition 2.4.29 has a straightforward generalisation to the case of non-compact W (and compact base X of the local model cone). An example is W D X   for any open set  Rq . In the non-compact case we have the spaces s; Wcomp .W /

and

s; Wloc .W /;

(2.4.30)

s; s .W / is nothing other than the set of all u 2 Hloc .Wreg / such respectively, where Wloc s; s; that the conditions (2.4.28) hold, while Wcomp .W / is the subspace of all u 2 Wloc .W / with compact support in W .

Remark 2.4.30. In Definition 2.4.20 we do not assume that X is compact. A modification of the definition allows us to include the case of a C 1 manifold X with boundary @X 6D ;, also not necessarily compact. Instead of Definition 2.4.26 (i) we require that W n Y is a C 1 manifold with boundary and Y a C 1 manifold. The condition (ii) is to be replaced by an analogous one with diffeomorphisms of C 1 manifolds with boundary. In other words we also have the notion of a manifold W with edge and boundary.

2.4 Mellin operators

131

An example of a manifold with edge  and boundary is the wedge X   DW W where X is a C 1 manifold with boundary. If 2X is the double of X which is a C 1 manifold, then we can form .2X /   DW 2W which is a manifold with edge  and without boundary. The process of doubling up a manifold with edge and boundary can easily be defined in general, i.e., to W we can form 2W . Then, if 2W is the stretched manifold associated with 2W we have the projection map 2W ! 2W

(2.4.31)

as mentioned in Remark 2.4.27. We then define W , the stretched manifold to W itself, to be the preimage of W under (2.4.31). We then set Wreg WD .2W /reg \ W ;

Wsing WD .2W /sing \ W :

1

Note that Wreg is a C manifold with boundary. Moreover, Wsing as an X -bundle over Y is also a C 1 manifold with boundary @Wsing ; the latter is a @X-bundle over Y . For x C  X  , W WD X   and a C 1 manifold X with boundary we have W D R Wreg D RC  X  , Wsing D f0g  X  . If 2W is a (say, compact) stretched manifold W with edge and boundary, we set ˚ ˇ

W s; .W / WD uˇint Wreg W u 2 W s; .2W / : s; From the relation W s; .W means the equivalence relation ˇ ˇ / Š W .2W /=, where  s; ˇ ˇ u  v , u int Wreg D v int Wreg , we obtain the spaces W .W / as Hilbert spaces from a corresponding Hilbert space structure of W s; .2W /. The manifolds W with edge Y (with or without) boundary form a category (with the manifolds with conical singularities as a subcategory for which the dimension of the edge is zero). Let us first consider the case without boundary. A morphism

z W W ! W

(2.4.32)

z with edge Yz is a continuous map from a manifold W with edge Y to a manifold W z n Yz , sing W Y ! Yz , and which restricts to differentiable maps reg W W n Y ! W z there is a differentiable map  W W ! W ˇbetween the respective stretched manifolds as z sing is a homomormanifolds with C 1 boundary such that  ˇWsing DW sing W Wsing ! W z sing ! Yz z respectively; if  W Wsing ! Y , Q W W phism of bundles with fibres X and X, are the corresponding bundle projections we have sing ı  D Q ı sing ;

(2.4.33)

and (2.4.32) is the quotient map of  under the equivalence relations of the kind (2.4.26) z , respectively. for W and W z be a manifold with edge and boundary. Then a morphism  W W ! Now let W; W z is a continuous map such that there is a morphism  W 2W ! 2W z in the category of W manifolds with edges such that  D jW (with W being identified with one of the two z satisfies a relation similar to copies of W in 2W ) and the associated map  W W ! W z sing with the fibres X and Xz , which (2.4.33) for the corresponding bundles Wsing and W are now C 1 manifolds with boundary.

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2 Symbolic structures and associated operators

Remark 2.4.31. Let X be a compact C 1 manifold with boundary. Then the operator r.@X/^ defined in (2.4.23) represents an element 1

1

1

r.@X/^ 2 S 2 .Rq I K s; .X ^ /; K s 2 ; 2 ..@X /^ // for every q 2 N and s;  2 R, s > restriction

1 . 2

This entails the continuity of the operator of 1

1

1

Op.r.@X/^ / W W s .Rq ; K s; .X ^ // ! W s 2 .Rq ; K s 2 ; 2 ..@X /^ // 1

1

for such s; , or globally, of W s; .W / ! W s 2 ; 2 .V / on a compact manifold W with edge Y and boundary V D @.W n Y / [ Y , where W and V are the respective stretched manifolds. x C / the operator M' of multiplication by ' repRemark 2.4.32. For '.r/ 2 C01 .R resents an element M' 2 S 0 .Rq I K s; .X ^ /; K s; .X ^ //, and ' ! M' defines a x C / ! S 0 .Rq I K s; .X ^ /; K s; .X ^ // for every s;  2 R. continuous map C01 .R Let us return once again to Remark 2.4.28 and the subsequent definition of weighted spaces on a manifold with edge. It turns out that there is also another choice of weighted spaces in connection with boundary value problems in such spaces, see also Section 10.1.3 below. Remark 2.4.33. Consider the space K s; .X ^ / defined by (2.4.11), and set .s u/.r; x/ WD sC

nC1 2

u.r; x/

for  2 RC , u 2 K s; .X ^ /, s;  2 R. Then we have a strongly continuous group  s D fs g2RC of isomorphisms s W K s; .X ^ / ! K s; .X ^ /. Applying Definition 2.2.12 we obtain another version of weighted edge spaces, namely, W s; .X ^  Rq / WD W s .Rq ; K s; .X ^ // s : (2.4.34) Similarly as (2.4.29) it can be proved that s s Hcomp .X ^  Rq /  W s; .X ^  Rq /  Hloc .X ^  Rq /

for all s;  2 R, cf. Proposition 7.1.5 below. Moreover, there are invariance properties under transition maps belonging to the local descriptions of a manifold with edge. This allows us to form global spaces. Later on in Chapter 7 we come back to this kind of spaces.

2.4.3 Degenerate differential operators Let B be a manifold with conical singularities, for simplicity, with one conical point fvg, and let B be its stretched manifold, X Š @B, cf. Section 2.4.2. Let us consider here the case of a compact closed C 1 manifold X . Near the boundary of B we fix a xC  X. splitting of variables into .r; x/ 2 R

2.4 Mellin operators

133

 Definition 2.4.34. By Diff  deg .B/ we denote the space of all operators A in Diff .Breg / that have in a neighbourhood of @B the form

A D r 

 X j D0

 @ j aj .r/ r @r

(2.4.35)

x C ; Diff j .X // for all j . Operators in Diff  .B/ are with coefficients aj .r/ 2 C 1 .R deg also said to be of Fuchs type. Example 2.4.35. Let X be a C 1 manifold and gX a Riemannian metric on X . Then the Laplace–Beltrami operator on RC  X belonging to the metric dr 2 C r 2 gX has the form (2.4.35) for  D 2. Let X be a C 1 manifold, and let L .X I ˇ  Rq / .cl/ denote the space of all parameter-dependent pseudo-differential operators of order  on X, with parameter .Im w; / 2 R  Rq , w 2 ˇ , cf. the notation (2.1.28) and .X I ˇ /. (2.4.1). For q D 0 we simply write L .cl/ In the following discussion we assume X to be a closed compact C 1 manifold, n D dim X, and B compact. For an operator A 2 Diff  deg .B/ we form the principal symbolic hierarchy  .A/ WD . .A/; c .A// (2.4.36) with  .A/ 2 C 1 .T  Breg n 0/ being the standard homogeneous principal symbol of order . In a neighbourhood of @B in the splitting of variables .r; x/ with the covariables .%; / we set Q .A/.r; x; %; / WD r   .A/.r; x; r 1 %; /:

(2.4.37)

This function is homogeneous of order  in .%; / 6D 0 and smooth up to r D 0. Moreover, we define  X c .A/.w/ WD aj .0/w j ; (2.4.38) j D0

the principal conormal symbol of A. In (2.4.38) we usually assume w 2  nC1  2 for some fixed weight  2 R, although (2.4.38) represents a holomorphic operator function c .A/.w/ W H s .X / ! H s .X /; w 2 C, for every s 2 R. Theorem 2.4.36. Let B be a compact manifold with conical singularities and B its stretched manifold. Then an A 2 Diff  deg .B/ defines a continuous operator A W H s; .B/ ! H s; .B/ for every s;  2 R. The operator (2.4.39) is compact if  .A/ D 0 (cf. the formula (2.4.36)).

(2.4.39)

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2 Symbolic structures and associated operators

If we write the operator A in the form !A C .1  !/A for a cut-off function ! (that is equal to 1 in a collar neighbourhood of @B and 0 outside another collar neighbourhood) the main aspect of the continuity of A is that of !A, since .1  !/A is supported far from @B and continuous in standard Sobolev spaces (here we use s .1  !/H s; .B/ D .1  !/Hloc .int B/). The continuity of !A can be seen as a simpler version of Theorem 2.4.53 below. Moreover,  .A/ D 0 entails the continuity A W H s; .B/ ! H sC1;C1 .B/; then the compactness of (2.4.39) follows from the compact embedding of the latter image into H s; .B/, cf. Remark 2.4.24 (i). Let W be a manifold with edge Y (here without boundary; the case with boundary is analogous and will be considered later on), and let W be its stretched manifold. x C  X  , cf. Near Wsing D @W we fix a splitting of variables into .r; x; y/ 2 R Definition 2.4.26.  Definition 2.4.37. By Diff  deg .W / we denote the space of all A 2 Diff .Wreg / that have in the variables .r; x; y/ near the edge Y the form

A D r 

X j Cj˛j

 ˛ @ j  aj˛ .r; y/ r rDy @r

(2.4.40)

x C  ; Diff .j Cj˛j/ .X // for all j; ˛. Operators with coefficients aj˛ .r; y/ 2 C 1 .R  in Diff deg .W / will also be called edge-degenerate. P ˛ Example 2.4.38. (i) Let Az WD Q y/Dx;y be a differential operator in j˛j c˛ .x; Q nC1 q 1 nC1Cq R  R with coefficients c˛ 2 C .R /. Then,ˇ substituting polar coordinates RnC1 n f0g ! RC  S n , xQ ! .r; x/, the operator Azˇ.RnC1 nf0g/Rq takes the form (2.4.40) with coefficients as in Definition 2.4.37. (ii) Let X be a C 1 manifold and gX a Riemannian metric on X . Then the Laplace– Beltrami operator on RC  X   belonging to the metric dr 2 C r 2 gX C dy 2 has the form (2.4.40) for  D 2, more precisely, D r 2

n  Xq @ 2 @ @2 o r ;  .n  1/ r C X C r 2 lD1 @y 2 @r @r l

(2.4.41)

where X is the Laplace–Beltrami operator on X associated with gX , and n D dim X . More generally, if gX .r/ is smooth in r up to r D 0, in (2.4.41) we have to replace X by X .r/ (the Laplacian associated with gX .r/ D .gij .r//) and the factor .n  1/ in (2.4.41) by n  1 C c.r/ with c.r/ D 12 r@r log.det gij .r//. Let us introduce the principal symbolic hierarchy  .A/ WD . .A/; ^ .A//

(2.4.42)

of edge-degenerate operators A under the condition that X , the base of the local model cones, is a closed compact C 1 manifold. First we have  .A/ 2 C 1 .T  Wreg n 0/,

2.4 Mellin operators

135

the homogeneous principal symbol of order  in the standard sense. Near Wsing in the splitting of variables .r; x; y/ with covariables .%; ; / we set Q .A/.r; x; y; %; ; / WD r   .A/.r; x; y; r 1 %; ; r 1 /:

(2.4.43)

This function is homogeneous of order  in .%; ; / 6D 0 and smooth up to r D 0. Moreover, we define the homogeneous principal edge symbol of order  ^ .A/.y; / D r 

X j Cj˛j

 @ j aj˛ .0; y/ r .r/˛ : @r

This is interpreted as a family of continuous operators ^ .A/.y; / W K s; .X ^ / ! K s; .X ^ / for every s 2 R and any fixed choice of  2 R. We then have the ‘twisted’homogeneity ^ .A/.y; / D   ^ .A/.y; /1 ;

(2.4.44)

 2 RC , .y; / 2 T   n 0, cf. Remark 2.4.28. Theorem 2.4.39. Let W be a compact manifold with edge and W its stretched manifold. Then an A 2 Diff  deg .W / defines continuous operators A W W s; .W / ! W s; .W /

(2.4.45)

for every s;  2 R. The operator (2.4.45) is compact if  .A/ D 0 (cf. the formula (2.4.42)). A proof of Theorem 2.4.39 will be given below, see Theorem 7.2.24 (in the case with boundary). Let us show here the idea. If we write the operator A in the form !A C .1  !/A with a cut-off function ! (that is equal to 1 in a collar neighbourhood of @W and 0 outside another collar neighbourhood) the main aspect is the continuity of !A, since .1  !/A is supported far from @W and continuous in standard Sobolov s .int W /). After a localisation we spaces (observe that .1  !/W s; .W / D .1  !/Hloc may consider the operator in the form (2.4.40) and assume that the coefficients aj˛ are independent of r for r > R and vanish for jyj > R for some R > 0. Lemma 2.4.40. The operator function a.y; / WD r 

X j Cj˛j

 @ j aj˛ .r; y/ r .r/˛ @r

(2.4.46)

belongs to S  .Rq  Rq I K s; .X ^ /; K s;  .X ^ // for every s;  2 R. In fact, if the coefficients are independent of r, then a.y; / coincides with ^ .A/.y; /. Since this is C 1 in .y; / up to  D 0 and by virtue of the homogeneity (2.4.44) it follows that a.y; / is a symbol as asserted (even classical in this

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2 Symbolic structures and associated operators

case, cf. Example 2.2.8 (i)). In the case of variable coefficients in r we can apply Remark 2.4.32 combined with a tensor product argument, i.e., expansions of the coefficients into convergent sums, according to Proposition 2.1.8, with ej playing the role x C / and fj of r-independent coefficients, both tending to zero in of elements in C01 .R the respective spaces. Now the continuity of A between weighted edge spaces reduces to Theorem 2.3.43 (here for g D  D 0). The compactness of A under vanishing  .A/ is a consequence of Theorem 2.3.45. Remark 2.4.41. In the expression (2.4.46) we treat the variable r as an operator acting from the left; in particular, the variable r in .r/˛ is not differentiated by @r . However, it makes sense to interpret the operators also as actions from the right. If we indicate them by r 0 we obtain a0 .y; / WD

X 



j Cj˛j

@ 0 j r aj˛ .r 0 ; y/.r 0 /˛ .r 0 / : @r 0

We also may consider other combinations, e.g., a00 .y; / WD r 

X j Cj˛j

 @ j aj˛ .r; y/ r 0 .r 0 /˛ ; @r

etc.

In all those cases we obtain elements of S  .Rq  Rq I K s; .X ^ /; K s; .X ^ //. Similar notions and results make sense on a manifold B with conical singularities and boundary or on a manifold W with edge and boundary Y ; then X is a compact C 1 manifold with boundary. Let us formulate a few generalisations for the case with edge. The case with conical singularities is formally included by admitting dim Y D 0. The edge symbol is then to be replaced by the conormal symbol. Definition 2.4.42. Let W be a manifold with edge and boundary, let 2W be its double and W and 2W the associated stretched manifolds, cf. the formula (2.4.31).ˇ Then  zˇ Diff  deg .W / denotes the subspace of all A 2 Diff .int Wreg / such that A D A int Wreg for some Az 2 Diff  .2W /. For every A 2 Diff  deg .W / we have the principal interior symbol  .A/ WD ˇ zˇ   .A/ . Moreover, there is the principal boundary symbol as a family of T Wreg n0 operators @ .A/ W H s .RC / ! H s .RC / parametrised by T  .@Wreg / n 0. With  .A/ we associate Q .A/ in an analogous manner as (2.4.43). Moreover, with @ .A/ near @Wsing in the splitting of variables .r; x 0 ; y/ with the covariables .%; 0 ; / we associate Q @ .A/ by Q @ .A/.r; x 0 ; y; %; 0 ; / WD r  @ .A/.r; x 0 ; y; r 1 %; 0 ; r 1 /

2.4 Mellin operators

137

which is smooth up to r D 0. Finally, there is also the principal edge symbol ^ .A/ W K s; .X ^ / ! K s; .X ^ /; z from .2X /^ to .int X /^ . We parametrised by T  Y n 0, obtained by restricting ^ .A/ then set altogether  .A/ D . .A/; @ .A/; ^ .A//:

2.4.4 Mellin operators and quantisation An operator (2.4.35), regarded as a map A W C01 .X ^ / ! C 1 .X ^ /, can be written as a Mellin operator  A D r  opM .h/ (2.4.47)  (cf. the formula P (2.4.7) forjan arbitrary weight  2 R) with a Diff .X /-valued symbol h.r; w/ D j D0 aj .r/w . The main application of Mellin operators in this book employs symbols with values in the space of boundary value problems in a domain (or on a manifold), cf. the calculus of Chapter 3 below. In the present section we illustrate the typical structures in the case of L cl .X /-valued Mellin symbols, where X is a closed compact C 1 manifold. Analogous results are then true in boundary value problems. The operator A is called  -elliptic, if the function (2.4.37) does not vanish for all .%; / 6D 0 and all .r; x/ up to r D 0. If A is  -elliptic it is also elliptic in the standard sense, and we can construct a ^ ^ pseudo-differential parametrix P 2 L cl .X / by (locally on X ) Leibniz-inverting the amplitude functions and gluing together the associated operators by using a partition of unity. ’ i.rr 0 /% For opr .b/u.r/ WD e b.r; r 0 ; %/u.r 0 /dr 0 μ %, b.r; r 0 ; %/ 2 C 1 .RC   RC ; Lcl .XI R% //, the operator (2.4.47) can equivalently be expressed as

A D r  opr .a.r; Q r%//

(2.4.48)

x C ; L .X I R%Q // (cf. the notation (2.1.28)). C 1 .R cl

for an element a.r; Q %/ Q 2 The operator Q %/ Q is parameter-dependent elliptic of order  (cf. A is  -elliptic if and only if a.r; x C ). Definition 2.1.28 for every r 2 R The proof of the following result is left to the reader: Proposition 2.4.43. Let A be a  -elliptic differential operator of the form (2.4.35). x C ; L .X I R%Q // such that Then there exists an operator family p.r; Q %/ Q 2 C 1 .R cl P WD r  opr .p.r; Q r%//

(2.4.49)

^ 1 is a parametrix of A in the space L .X ^ /). cl .X / (i.e., AP D I; PA D I mod L  More generally, if A D r opr .a.r; Q r%// C C is given in terms of an a.r; Q %/ Q 2 x C ; L .X I R%Q // which is parameter-dependent elliptic (of order ), with paC 1 .R cl x C , and C 2 L1 .X ^ / such that A is properly supported, rameter %Q for every r 2 R there exists a parameter-dependent elliptic (of order ) family p.r; Q %/ Q such that (2.4.49) is a parametrix of A (cf. the Remarks 2.1.24 and 2.1.30).

138

2 Symbolic structures and associated operators

x C ; L .X I R%Q // are a natural Operators of the form (2.4.48) for a.r; Q %/ Q 2 C 1 .R cl analogue of Fuchs type differential operators (2.4.35) in the pseudo-differential set-up. In the differential case the equivalent formulation (2.4.47) is a trivial consequence of  opM .w/ D r@r . In other words, there is an easy Mellin quantisation a.r; Q %/ Q ! h.r; w/  .h/ for a.r; %/ WD a.r; Q r%/. For pseudo-differential operators such that opr .a/ D opM such a correspondence is not so evident; however, as we shall see, it always exists modulo smoothing remainders. To formulate the corresponding result we first give a definition of holomorphic operator-valued symbols. If E is a Fréchet space and U C an open set, by A.U; E/ we denote the space of all holomorphic functions in U with values in E, in the topology of uniform convergence on compact subsets. We 1 apply this, in particular, to E D L manifold X , cf. cl .X / for a closed, compact C Remark 2.1.17 (iii).  Definition 2.4.44. By L cl .X I C/ we denote the space of all h.w/ 2 A.C; Lcl .X //  such that h.ˇ C i%/ 2 Lcl .X I R% / for every real ˇ, uniformly in compact ˇ-intervals.

Note (as a warning) that in the latter definition the role of Re w and Im w cannot be interchanged. Below we therefore avoid notation like that and write MO .X / instead of L cl .XI C/. Moreover, in Chapter 9 we employ similar operator spaces with the real part of the complex variable as a parameter and then write C rather than C. The space L cl .XI C/ is Fréchet in a natural way. In fact, the semi-norms from the inclusion 0 L .XI C/  A.C; L cl cl .X // together with h.w/ ! supcˇ c 0 .h.ˇ C i%//, c  c ,  for every semi-norm  from the Fréchet topology of the space Lcl .X I R% / give us a Fréchet space structure in L cl .X I C/. There is an efficient way to produce elements of L cl .X I C/ by the so-called kernel cut-off procedure. Let us start from f .w/ 2 L cl .XI ˇ /

(2.4.50)

for any ˇ 2 R (the notation in (2.4.50) means f .ˇ C i%/ 2 L cl .X I R% /). The following results (up to Remark 2.4.52) will be proved in Sections 6.1.1 and 6.1.2 below. Theorem 2.4.45. For every (2.4.50) there exists an h.w/ 2 L cl .X I C/ such that ˇ hˇ D f mod L1 .X I ˇ /: ˇ

x C ; L .X I ˇ // there is an h.r; w/ 2 More generally, for every f .r; w/ 2 C 1 .R cl  x C ; L .X I C// such that C 1 .R cl ˇ 1 x 1 hˇR .X I ˇ //: x  D f mod C .RC ; L C

ˇ

The space L cl .X I C/ has many remarkable properties.

139

2.4 Mellin operators

Theorem 2.4.46. Let h.w/ 2 L cl .X I C/ and ˇ; ı 2 R. Then there are constants cj .ˇ; ı/, j 2 N, such that h.ˇ C i%/ 

X

cj .ˇ; ı/

j 2N

1 dj h.ı C i%/ j Š d%j

where c0 .ˇ; ı/ D 1 (the asymptotic sum refers to L cl .X I R% /). In particular, h.w/ 2 1 L .XI C/ and h.ı C i%/ 2 L .X I  / for a ı 2 R imply h.w/ 2 L1 .X I C/. An ı cl cl cl  x C ; L .X I C//. analogous results holds for h.r; w/ 2 C 1 .R cl Theorem 2.4.47. Let hj .w/ 2 Lj .X I C/, j 2 N, be an arbitrary sequence. Then cl P  C1/ there exists an h.w/ 2 Lcl .X I C/ such that h.w/  jND0 hj .w/ 2 L.N .X I C/ cl j 1 x for every N 2 N. An analogous result holds for hj .r; w/ 2 C .RC ; Lcl .X I C// x C ; L .X I C//. with a resulting h.r; w/ 2 C 1 .R cl Definition 2.4.48. An element h.w/ 2 L cl .X I C/ is called elliptic (of order ) if there is a ˇ 2 R such that h.ˇ C i%/ is parameter-dependent elliptic in the class L cl .X I R% /. Remark 2.4.49. The ellipticity of h.w/ is independent of the choice of ˇ. In fact, from Theorem 2.4.46 we see that the ellipticity with respect to ˇ is equivalent to the one with respect to ı for any ı 2 R. x C ; L .X I C// and  2 R we have Remark 2.4.50. For every h.r; w/ 2 C 1 .R cl  ^ opM .h/ 2 L cl .X /:

This follows from the equivalence of the phase function of local representatives of a Mellin–Fourier operator (containing the Mellin phase function .log r 0  log r/%) .log r 0  log r/% C .x  x 0 / with the ‘standard’ phase function .r  r 0 /% C .x  x 0 / of operators in the Fourier representation. Moreover,   opM .T ˇ h/r ˇ D r ˇ opM .h/

for every ˇ 2 R (T ˇ h.w/ D h.w C ˇ//. Theorem 2.4.51. For every  2 R there exists a (non-canonical) map x C ; L .X I R// ! C 1 .R x C ; L .X I C//; C 1 .R cl cl

(2.4.51)

a.r; Q %/ Q ! h.r; w/, such that for a.r; %/ WD a.r; Q r%/ we have  opr .a/ D opM .h/ mod L1 .X ^ /

(2.4.52)

for every  2 R. For a0 .r; %/ WD a.0; Q r%/; h0 .w/ WD h.0; w/ we also have  opr .a0 / D opM .h0 / mod L1 .X ^ /

for every  2 R.

(2.4.53)

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2 Symbolic structures and associated operators

(2.4.52) and (2.4.53) are interpreted as relations between (classical) pseudo-differential operators of order  on X ^ in the sense of maps C01 .X ^ / ! C 1 .X ^ /. The specific choice of the weight  will induce different extensions by continuity to maps on weighted Sobolev spaces. The analogue of (2.4.52) in the sense of a non-canonical map  1 C 1 .RC ; L cl .X I R// ! C .RC ; Lcl .X I C//

(2.4.54)

such that (2.4.52) holds for all  2 R is true as well; but the important point for us is that the smoothness of the coefficients up to r D 0 is preserved by an appropriate choice of (2.4.51). For the latter property it is essential that the operators are classical, while (2.4.54) holds in analogous form also in the non-classical case. Remark 2.4.52. The Mellin symbol h.r; w/ is uniquely determined by the operator x C ; L1 .X I C//. The map (2.4.52) induces an isomorfamily a.r; Q %/; Q mod C 1 .R phism x C ; L .X I R//=C 1 .R x C ; L1 .X I R// C 1 .R cl x C ; L .X I C//=C 1 .R x C ; L1 .X I C//: ! C 1 .R cl

x C ; L .X I  nC1 // be independent of r for Theorem 2.4.53. Let f .r; w/ 2 C 1 .R cl  large r. Then

 n opM 2 .f

2

/ induces continuous operators  n 2

opM

.f / W H s; .X ^ / ! H s; .X ^ /

for all s 2 R. Proof. If f is independent of r, the continuity is a consequence of the estimates  n 2

k opM

.f /uk2H s; .X ^ / Z 2 D .2 i /1 kRs .Im w/f .w/M u.w/kL 2 .X/ dw  nC1 2



kR

sup w2 nC1 2



1



s

.Im w/f .w/Rs .Im w/k2L.L2 .X//

(2.4.55)

Z

 .2 i /

 nC1 2

2 kRs .Im w/M u.w/kL 2 .X/ dw 

and the fact that Rs .Im w/f .w/Rs .Im w/ belongs to L0cl .X I  nC1  / which has 2 the consequence that the factor in front of the integral on the right-hand side of (2.4.55)  n is finite, cf. Theorem 2.1.27. At the same time we see that f ! opM 2 .f / induces a continuous operator s; L .X ^ /; H s; .X ^ // cl .X I  nC1  / ! L.H 2

2.4 Mellin operators

141

for every s 2 R. For f .r; w/ in general we write f D f0 C f1 for an f1 .w/ which x C ; L .X I  nC1 //. Then it remains to is independent of r and an f0 .r; w/ 2 C01 .R cl  2

 n

show that opM 2 .f0 / is continuous. There is an R > 0 such that f0 .r; w/ D 0 for r > R. Therefore, denoting by C01 .Œ0; R/0 / the Fréchet space of all ' 2 C01 .RC / y  L vanishing for r > R, we have f0 .r; w/ 2 C01 .Œ0; R/0 / ˝ /. cl .X I  nC1 2  Applying Proposition 2.1.8 we can write f0 .r; w/ as a convergent series f0 .r; w/ D

1 X

j 'j .r/fj .w/;

j D0

P j 2 C, j1D0 jj j < 1, and sequences 'j 2 C01 .Œ0; R/0 /, fj 2 L /, cl .X I  nC1 2  tending to zero in the respective spaces as j ! 1. According to the first part of  n the proof, we have opM 2 .fj / ! 0 in L.H s; .X ^ /; H s; .X ^ // as j ! 1. Moreover, the operator M'j of multiplication by 'j is continuous and satisfies the relation M'j ! 0 in L.H s; .X ^ /; H s; .X ^ // as j ! 1, for every s;  2 R. Thus we obtain a convergent series  n opM 2 .f

/D

1 X

 n 2

j M'j opM

.fj /

j D0

in the space L.H s; .X ^ /; H s; .X ^ //. Remark 2.4.54. The method of the second part of the latter proof can be adapted to many other situations and will be referred to as a tensor product argument. We now discuss some other aspects on Mellin pseudo-differential operators, connected with the material of the Sections 2.2.3 and 2.3.5. Definition 2.4.55. Let H be a Hilbert space with group action  D f g2RC , and let s;  2 R. Then V s; .RC ; H / denotes the completion of C01 .RC ; H / with respect to the norm Z kukV s; .RC ;H / D

12

1  2

1

h%i2s h%i .M u/   C i% d% : H 2

Moreover, we set V s; Ig .RC ; H / WD fhlog rig u W u 2 V s; .RC ; H /g for every s; ; g 2 R, and set T  .RC ; H / WD S1 S.R; H /, cf. also the notation of Section 2.4.1. Later on (cf. Definition 8.2.2 below) we slightly modify the weight convention in the corner spaces V s; .RC ; H / by introducing a ‘dimension effect’ from the concrete choice of H .

142

2 Symbolic structures and associated operators

Observe that the map S , cf. the formula (2.4.2), induces an isometric isomorphism S W V s; Ig .RC ; H / ! W sIg .R; H /:

(2.4.56)

This allows us to deduce the essential properties of V s;Ig .RC ; H / from those of W sIg .R; H /. z be Hilbert spaces with group actions  and , Let H and H Q respectively, and z //. Then we can form the Mellin let f .r; r 0 ; w/ 2 C 1 .RC  RC   1  ; L.H; H 2 pseudo-differential operator   0 1  .f / WD S1 op.a /S with a .t; t 0 ; / WD f e t ; e t ;   C i ; opM 2 (2.4.57) provided that a .t; t 0 ; / is an amplitude function that admits the oscillatory integral construction, e.g., when the symbolic estimates are combined with group actions, or when the covariable is a parameter in another suitable parameter-dependent calculus z /. We then have of operators in L.H; H “ 1  . 1 Ci%/   r 1 dr 0 2  opM .f /u.r/ D f r; r 0 ;   C i% u.r 0 / 0 μ % 0 r 2 r 0 “ 1   0 1 dr 1 D .r 0 / 2 Ci% f r; rr 0 ;   C i% u.rr 0 / 0 μ %; 2 r 0 u 2 T  .RC ; H /, with integrals being taken in the oscillatory integral sense. z be Hilbert spaces with group actions  and , Definition 2.4.56. Let H and H Q respectively. 0 z / for ; ;  0 ; ˇ 2 R is defined to (i) The space M S I; .RC  RC  ˇ I H; H z // such that for all be the set of all f .r; r 0 ; w/ 2 C 1 .RC  RC  ˇ ; L.H; H N 2N ˚ 0 0 sup h%il hlog rik hlog r 0 ik 

0 1 kQ h%i f.r@r /k .r 0 @r 0 /k D%l f .r; r 0 ; ˇ C i%/gh%i kL.H;Hz /

is finite, where the supremum is taken over r; r 0 2 RC , % 2 R, and kCk 0 Cl  N . The corresponding space of symbols only depending on .r; w/ 2 RC  ˇ is z /. denoted by M S I .RC  ˇ I H; H z /b is defined to be the set of all f .r; r 0 ; w/ 2 (ii) The space M S  .RC RC ˇ I H; H 1 z C .RC  RC  ˇ ; L.H; H // such that for all N 2 N ˚

0 1 sup h%il kQ h%i f.r@r /k .r 0 @r 0 /k D%l f .r; r 0 ; ˇ C i%/gh%i kL.H;Hz / is finite, where the supremum is taken over all r; r 0 2 RC ; % 2 R, and kCk 0 Cl  N . The corresponding space of symbols only depending on .r; w/ 2 RC  ˇ z /b . is denoted by M S  .RC  ˇ I H; H

143

2.4 Mellin operators

Observe that the map f ! a of the formula (2.4.57) defines isomorphisms 0 z / ! S I; 0 .R  R  RI H; H z /; M S I; .RC  RC  ˇ I H; H z /b ! S  .R  R  RI H; H z /b : M S  .RC  RC  ˇ I H; H

(2.4.58) (2.4.59)

Analogous relations hold for the corresponding spaces of symbols only depending on .r; w/ 2 RC  ˇ . Remark 2.4.57. If f .r; r 0 ; w/ belongs to one of the symbol spaces of Definition 2.4.56 for ˇ WD 12  , then we obtain continuous operators  z /: opM .f / W T  .RC ; H / ! T  .RC ; H

(2.4.60)

A similar result holds for left or right symbols in the respective symbol spaces. 0 z/ Theorem 2.4.58. (i) For every f .r; r 0 ; w/ 2 M S I; .RC  RC   1  I H; H 2 0 z / such that there exists a unique left symbol fL .r; w/ 2 M S IC .RC   1 I H; H

  opM .f / D opM .fL /, where fL .r; w/ is given by

“

1

fL .r; w/ D

s i f .r; sr; w C i / 0

2 

ds μ ; s

PN

1 k and we have fL .r; w/ D kD0 kŠ @w .r 0 @r 0 /k f .r; r 0 ; w/jr 0 Dr C rN .r; w/; N 2 N, 0 z /, given by with a remainder rN .r; w/ 2 M S .N C1/IC .N C1/ .RC   1  I H; H 2 the formula “ 1 Z 1 .1  t /N ds C1 s i .@N .r 0 @r 0 /N C1 f /.r; sr; w C i t / μ dt: rN .r; w/ D w NŠ s 0 0

The maps f ! fL and f ! rN are continuous. Analogous statements hold for right symbols, given by “ 1 ds 0 fR .r ; w/ D s i f .sr 0 ; r 0 ; w C i / μ ; s 0 PN 1 k k k 0 0 and we have fR .r 0 ; w/ D kD0 kŠ .1/ @w .r@r / f .r; r ; w/jrDr 0 C rN .r ; w/, N 2 N, with a similar remainder as for the left symbol, namely, Z 1 .1  t /N 0 rN .r ; w/ D NŠ 0 “ 1  N C1 ds C1 s i .1/N C1 .@N f /.sr 0 ; r 0 ; w C i t / μ dt; r@r w s 0 and continuity of f ! fR , f ! rN .

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2 Symbolic structures and associated operators

z z H z / and for every g.r; w/ 2 (ii) For every f .r; w/ 2 M S I .RC   1  I H; 2 z z z / (with a third Hilbert space H z with group action) there I H; H M S I .R   C

1 2 

z / such that is a unique symbol .f # g/.r; w/ 2 M S CIC .RC   1  I H; H 2

   .f / opM .g/ D opM .f # g/, and we have for the Mellin–Leibniz product opM “ 1 ds .f # g/.r; w/ D s i f .r; w C i /g.sr; w/ μ ; s 0

  P @ k 1 k and we can write .f # g/.r; w/ D N g.r; w/ C rN .r; w/, kD0 kŠ .@w f /.r; w/ r @r N 2 N, with a remainder “ Z 1  .1  t /N @ N C1  ds C1 rN .r; w/ D s i .@N f /.r; w C i t /  r g .sr; w/ μ : w NŠ @r s 0 The mappings .f; g/ ! f # g and .f; g/ ! rN are bilinear continuous. (iii) Analogous statements as (i) and (ii) hold for the symbol classes of Definition 2.4.56 (ii). Proof. Theorem 2.4.58 (i) is a direct consequence of Theorem 2.3.38 when we translate the symbols by means of (2.4.58). In a similar manner we obtain Theorem 2.4.58 (ii) from Theorem 2.3.41, and Theorem 2.4.58 (iii) from the Theorems 2.2.52 and 2.2.54, together with the bijection (2.4.59). z /; then (2.4.60) extends Theorem 2.4.59. (i) Let f .r; w/ 2 M S I .RC   1  I H; H 2 to continuous operators  z /; .f / W V s; Ig .RC ; H / ! V s;Ig .RC ; H opM  .f / induces a continuous operator and the map f ! opM

z / ! L.V s; Ig .RC ; H /; V s;Ig .RC ; H z // MS I .RC   1  I H 2

for every s; g 2 R. x C   1 I H; H z / be independent of r for large r. Then (ii) Let f .r; w/ 2 S  .R  2

 z /; opM .f / W V s; I0 .RC ; H / ! V s;I0 .RC ; H  .f / induces a continuous operator from the is continuous, and the map f ! opM z // for all s 2 R. space of these Mellin symbols to L.V s; I0 .RC ; H /; V s;I0 .RC ; H

Proof. The assertion (i) is a simple translation of Theorem 2.3.43 into the Mellin operator set-up, using the bijections (2.4.56) and (2.4.58). The continuity (ii) can be proved in an analogous manner as Theorem 2.4.53, namely, first for symbols with constant coefficients (cf., analogously, Theorem 2.2.22), and then by applying a tensor product argument (cf. Remark 2.4.54).

2.4 Mellin operators

145

In the following theorem we refer to the notation of Remark 2.2.57. z; H z0 ; H z 0 I g Q be Hilbert space triples Theorem 2.4.60. (i) Let fH; H0 ; H 0 I g and fH  z /. Then with group action and A WD opM .f / for f .r; w/ 2 M S I .RC   1  I H; H 2  the formal adjoint A defined by .Au; v/L2 .RC ;Hz0 / D .u; A v/L2 .RC ;H0 / z 0 / has the form A D op .f  / for a symbol for all u 2 C01 .RC ; H /, v 2 C01 .RC ; H M z 0 ; H 0 /, given by the oscillatory integral formula f  .r; w/ 2 M S I .RC   1 C I H 2

f  .r; w/ D

“ s i f Œ .sr; w C i /

x Moreover, for w D f Π.r; w/ WD f ./ .r; 1  w/.

1 2

ds μ ; s

C  C i we have

N  1  X @ k Œ  1 k .r; w/ C rN .r; w/ f @w r f  r; C  C i D 2 kŠ @r kD0

with a remainder  1  Z 1 .1  t /N “ 1 s i rN r; C  C i D 2 NŠ 0 0   @ N C1 Œ  ds N C1 @w r .sr; w C i t / μ dt f @r s z 0 ; H 0 /, and the mappings f ! f  belonging to M S .N C1/I.N C1/ .RC   1 C I H 2 and f ! rN are antilinear, continuous. z /b an analogous statement (ii) For a symbol f .r; w/ 2 M S  .RC   1  I H; H 2 holds. z /,  < 0;  < 0, and Proposition 2.4.61. Let f .r; w/ 2 M S I .RC   1  I H; H 2 z be a compact operator for every .r; w/ 2 RC   1 . Then let f .r; w/ W H ! H 2 

 z/ .f / W V s; Ig .RC ; H / ! V s;Ig .RC ; H opM

is a compact operator for every s; g 2 R. Proof. The proof is a consequence of Theorem 2.3.45. Similarly as Corollary 2.3.46 we obtain the following compactness result: Corollary 2.4.62. Let  W H 0 ,! H be a compact embedding between Hilbert spaces with group actions  0 and , respectively, where  0 D jH 0 . Then the embedding 0

0

V s ;Ig .RC ; H 0 / ,! V s;Ig .RC ; H / is compact when s 0 > s; g 0 > g.

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2 Symbolic structures and associated operators

Example 2.4.63. Let H s; .RC  Rn / denote the completion of C01 .RC  Rn / with respect to the norm “ kukH s; .RC Rn / D

12 ˇ ˇ2 n C 1 ˇ   C i%; ˇ d%d : h%; i ˇMF u 2 2s ˇ

We then have an isometric isomorphism S n2 W H s; .RC  Rn / ! H s .R1Cn /. If n H s .Rn / is equipped with the group action u.x/ !  2 u.x/;  2 RC , then we have n

H s; .RC  Rn / D V s;  2 I0 .RC ; H s .Rn //: Observe that the spaces H s; .X ^ / for a closed compact C 1 manifold X locally coincide with H s; .RC  Rn /, i.e., u.r; x/ 2 H s; .X ^ / is equivalent to 'u 2  H s; .RC  Rn / for any chart  W U ! X on X and arbitrary ' 2 C01 .U /. Analogously, we have n

H s; .RC  .Rn1  RC // D V s;  2 I0 .RC ; H s .RnC //:

(2.4.61)

Moreover, if X is a compact C 1 manifold with boundary, then the spaces H s; .X ^ / coincide locally near the boundary with (2.4.61).

2.4.5 A connection between edge-degenerate operators and exit calculus The spaces K s; .X ^ /, s;  2 R, form a natural scale in connection with edge-degenerate operators, as we saw in Section 2.4.3. We now deepen this information by establishing a relationship between edge-degeneracy and exit calculus in the sense of Section 2.3, in particular, between the different notions of ellipticity. We show that the ellipticity of an edge-degenerate symbol on X ^   3 .r; x; y/ ‘in the finite’, i.e., near r D 0, is perfectly coordinated with the exit ellipticity of its homogeneous principal edge symbol on X ^ 3 .r; x/ for r ! 1. To this end we return to Definition 2.4.5 (ii). Let us first observe that there is a canonical isomorphism ˇ s Hcone .X ^ / Š H s;0 .X /ˇR X ; (2.4.62) C

cf. Example 2.3.17 (ii). On the right-hand side of (2.4.62) we have identified X D R  X with a manifold with conical exits to infinity. Thus X;C WD X \ .RC  X / can be regarded as a submanifold in this category. Without loss of generality in Definition 2.4.5 (ii) we may consider coordinate neighbourhoods U  X such that there is a diffeomorphism ˇ1 W U ! B to the open unit ball in Rn 3 .xQ 1 ; : : : ; xQ n /. Set  WD f.xQ 0 ; xQ 1 ; : : : ; xQ n / 2 RnC1 W xQ 0 > 0; j.xQ 1 ; : : : ; xQ n /j < xQ 0 g;

147

2.4 Mellin operators

and define a diffeomorphism ˇ W RC  U ! 

(2.4.63)

by ˇ.r; x/ D .r; rˇ1 .x//: Then ˇ satisfies the relation ˇ.ır; x/ D ıˇ.r; x/ for all ı 2 RC , .r; x/ 2 RC  U , and we have ˇ u 2 K s; .X ^ / , !u 2 H s; .X ^ / and .1  !/'u ı ˇ 1 2 H s .RnC1 /ˇ 

for every ' 2 C01 .U / and any cut-off function !. Let us now consider an edge-degenerate differential operator  X @ j A D r  aj˛ .r; y/ r .rDy /˛ ; @r

(2.4.64)

j Cj˛j

x C  ; Diff .j Cj˛j/ .X // are indepenand assume that the coefficients aj˛ 2 C 1 .R dent of r for large r. Set  X @ j a.y; / D r  aj˛ .r; y/ r .r/˛ : (2.4.65) @r j Cj˛j

To simplify notation we identify U with B and write for the variables again x which is justified by the meaning of the diffeomorphism ˇ1 . Then the operator A.1  !/' 2 Diff  .RC  B  / takes the form X A.1  !/' D r  cj ˛ .r; x; y/r j Cj˛j Drj Dx Dy˛ : j CjjCj˛j

x C  B  / have the properties The coefficients cj ˛ .r; x; y/ 2 C 1 .R cj ˛ .r; x; y/ D 0

for 0  r < "

for some " > 0, cj ˛ .r; x; y/ is independent of r for r > R for some R > 0, and cj ˛ .r; x; y/ D 0

for x 62 K

(2.4.66)

for some compact subset K  B. Let us now interpret ˇ as a diffeomorphism ˇ W RC  B ! ;

(2.4.67)

ˇ.r; x/ D .r; rx/, and write xQ D rx, xQ 2 Rn . For the push forward ˇ under ˇ applied to the differential operators X a.y; /.1  !/' D r  cj ˛ .r; x; y/r j Cj˛j Drj Dx ˛ j CjjCj˛j

148

2 Symbolic structures and associated operators

on RC  B (pointwise for every .y; /) we then obtain X  xQ  ˇ .a.y; /.1  !/'/ D r Cj CjjCj˛j cj ˛ r; ; y Drj DxQ ˛ : (2.4.68) r j CjjCj˛j

Proposition 2.4.64. The operators (2.4.68) induce continuous maps ˇ .a.y; /.1  !/'/ W H s .RnC1 / ! H s .RnC1 / for all s 2 R and .y; / 2   Rq . Q denote one of the coProof. The differentiations Drj DxQ are continuous. Let c.r; x=r/ Q y/ for fixed y, and let l 2 N. It suffices then to note that the opefficients cj ˛ .r; x=r; erator of multiplication by r l c.r; x=r/ Q is also continuous in Sobolev spaces, because c.r; x=r/ Q behaves like an admissible cut-off function for large r, cf. Remark 2.3.25 (i), and the factor r l preserves Sobolev spaces, because it is combined with c which vanishes for small r, cf. the relation (2.4.66). As is known the manifold X can be embedded as a submanifold in RN for a sufficiently large choice of N . Without loss of generality we assume X  B N , where B N is the open unit ball in RN . Consider the map Q x/ WD .r; rx/; ˇ.r;

ˇQ W RC  B N ! RC  RN ; r 2 R, and set

ˇ ˇ WD ˇQ ˇR

C X

W RC  X ! M:

(2.4.69)

The map (2.4.69) is a global analogue of (2.4.63); therefore, we employ a similar notation. M is a manifold with conical exit to infinity. In the notation of Section 2.3.3 we can set M0 D ˇ..0; 1  X /; M1 D ˇ.Œ1; 1/  X /; and ˇ..1; 1/  X / D int M1 can be interpreted as a submanifold of X with conical exit r ! 1, cf. Example 2.3.17. Note that the relation (2.4.62) implies that u.r; x/ Q 2 H sIg .X /;

supp u M1

is equivalent to  xQ  s u r; .RC  X /; 2 r g Hcone r

u 0 for r < 1

for every s; g 2 R. Proposition 2.4.65. Let a.y; / be associated with an edge-degenerate differential operator A, cf. the expressions (2.4.65) and (2.4.64). Then we have ˇ .a.y; // 2 LI0 cl .X;C / for every .y; / 2   Rq .

(2.4.70)

2.4 Mellin operators

149

Proof. The property (2.4.70) can be checked in terms of local amplitude functions in the variables .r; x/ 2 RC B; it is then admitted to employ ˇ in the meaning (2.4.67). Since the main aspect concerns r ! 1 we may consider the localisation a.y; /.1  !/'. Q Because of (2.4.68) the complete symbol in the variables and covariables .r; x; Q %; / has the form  xQ  X r Cj Cj jCj˛j cj ˛ r; ; y %j Q  ˛ : (2.4.71) r j CjjCj˛j

Thus, for the proof it suffices to observe that  xQ  r Cj Cj jCj˛j cj ˛ r; ; y %j Q  ˛ 2 SclI0 .RnC1  RnC1 / r;xQ %; Q r for all j C j j C j˛j   and fixed .y; /, cf. Definition 2.3.5. An operator (2.4.64) is said to be elliptic in the edge-degenerate sense, if its homogeneous principal symbol of order   .A/.r; x; y; %; ; /

(2.4.72)

in the variables .r; x; y/ 2 RC  B   satisfies the condition r   .A/.r; x; y; r 1 %; ; r 1 / 6D 0

(2.4.73)

for all .%; ; / 6D 0 and all .r; x; y/ up to r D 0. Theorem 2.4.66. Let (2.4.64) be an edge-degenerate elliptic differential operator. Then for every .y; / 2   .Rq n f0g/ the operator (2.4.70) is elliptic on X;C of order .; 0/ in the sense of Definition 2.3.26, cf. also Remark 2.3.27. Proof. Because of the homogeneity of (2.4.72) in .%; ; / 6D 0 of order  the relation Q / 6D 0, Q WD r , r > 0. (2.4.73) implies  .A/.r; x; y; %; r ; / 6D 0 for all .%; ; This entails the ellipticity of ˇ .a.y; // with respect to the principal symbol in the Q for every . Alternatively we can also refer to (2.4.71) when we covariables .%; / forget about the factors .1  !/', which is admitted here because it suffices to push forward the restrictions of the differential operators to RC  U . For a similar reason we see that  X xQ  Q D e .ˇ .a.y; ///.r; x; Q %; / cj ˛ 1; ; y %j Q  ˛ 6D 0 r j CjjCj˛jD

Q 2 RnC1 (including 0),  6D 0, noting that x=r for all .%; / Q is homogeneous of order zero in the variables .r; x/. Q Finally, it follows that  X xQ  Q D  ;e .ˇ .a.y; ///.r; x; Q %; / cj 0 1; ; y %j Q  6D 0 r j CjjD

Q 2 RnC1 n f0g. for all .%; /

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2 Symbolic structures and associated operators

Let us now interpret the exit ellipticity connected with edge-degenerate ellipticity from another point of view. For simplicity we fix the variable y 2  and omit it. Set p.r%; Q r/ WD

X

aj˛ .i r%/j .r/˛

j Cj˛j

with aj˛ WD aj˛ .1; y/, cf. the formula (2.4.65). The edge-degenerate ellipticity of 1Cq (2.4.64) has the consequence that p. Q %; Q / Q 2 L / is a parameter-dependent cl .X I R%; Q Q elliptic family of (here differential) operators on X . Proposition 2.4.67. For every fixed  D 6 0 there is a constant R > 0 such that the operators r  p.r%; Q r/ W H s .X / ! H s .X / are invertible for all r  R and all % 2 R (including % D 0), s 2 R. Proof. By assumption p. Q %; Q / Q is parameter-dependent elliptic. From Theorem 2.1.29 (i) we obtain that the operators p.%; Q / W H s .X / ! H s .X / are isomorphisms for all C it follows the invertibility of j.%; /j  C and some C > 0. In particular, for R  jj p.r%; Q r/ for all r  R and % 2 R which yields the assertion. 1Cq Theorem 2.4.68. Let p. Q %; Q / Q 2 L / be a parameter-dependent family of (in cl .X I R general, pseudo-) differential operators, and let  , Q be cut-off functions. Then for every 1 2 Rq n f0g the operator

ˇ .Œr  opr ..1   .r//p.r%; Q r1 /.1  Q .r 0 ////

(2.4.74)

Q %; Q / Q is parameter-dependent elliptic, then (2.4.74) is belongs to LI0 cl .X;C /. If p. elliptic in this class for r  const for a constant > 0. 1Cq / is a finite sum of local pseudo-differential Proof. An element p. Q %; Q / Q 2 L cl .X I R operator families Opx .'a /.%; Q / Q (2.4.75)

(in local coordinates x 2 B and then pulled back to X ) with symbols a.x; ; %; Q / Q 2 1 0 / and '.x/; .x / 2 C .B/, plus a family of global smoothing Scl .B  RnC1Cq 0

;%; Q Q operators of the form Z 0 c.x; x 0 ; %; Q /u.x Q /dx 0 (2.4.76) C.%; Q /u.x/ Q D X

Q / Q 2 S.R1Cq ; C 1 .X  X //. Let us first consider (2.4.75). for a function c.x; x 0 ; %; %; Q Q From the definition of the operator push forward we have   ˇ opr Opx ..1   /'a .1  Q //.r%; r/ v.r; x// Q D .ˇ  /1 .Opr;x ..1   /'a .1  Q //.r%; r/.ˇ  v/.r; x//;

151

2.4 Mellin operators

v.r; x/ Q 2 C01 ./. The transformation x ! rx under the oscillatory integral with respect to x is simply a dilation with the factor r. Thus .ˇ  v/.r; x/ D v.r; rx/ and .ˇ  v/.r 0 ; x 0 / D v.r 0 ; r 0 x 0 /. Let us set a.r; Q r 0 ; x; x 0 ; %; Q ; / Q WD .1   .r//'.x/a.x; %; Q ; / Q .x 0 /.1  Q .r 0 //:

(2.4.77)

It follows that

  .ˇ  /1 Opr;x ..1   .r//'a .1  Q .r 0 ///.r%; r/ .ˇ  v/.r; x/ “ 0 0  1 D .ˇ / e i.rr /%Ci.xx / a.r; Q r 0 ; x; x 0 ; r%; ; r/v.r 0 ; r 0 x 0 /dr 0 dx 0 μ %μ “   x Q x Q0 xQ xQ 0 0 D e i.rr /%Ci. r  r 0 / aQ r; r 0 ; ; 0 ; r%; ; r v.r 0 ; xQ 0 /.r 0 /n dr 0 d xQ 0 μ %μ : r r (2.4.78)   x Q 1  rr 0 0 0 0 0 Let us set for the moment w WD .r; x/, Q w WD .r ; xQ /, and ˆ.w; w / WD 0 .r 0 /1 In with the unit n  n matrix In . Then we have    xQ xQ 0  % 0 0 0 .r  r /% C ;  0 D .w  w /ˆ.w; w / r r and (2.4.78) takes the form “   xQ xQ 0 0 0 t e i.ww /ˆ.w;w / .%; / aQ r; r 0 ; ; 0 ; r%; ; r v.r 0 ; xQ 0 /.r 0 /n dr 0 d xQ 0 μ %μ r r “   xQ xQ 0 0 t D e i.ww / .; / aQ r; r 0 ; ; 0 ; r C x; Q r 0 ; r v.r 0 ; xQ 0 /dr 0 d xQ 0 μ μ : r r   x Q     % rr 0 0 % which yields Here we employed the substitution D ˆ.w; w / D .r 0 /1

% D  C xrQ , μ D .r 0 /n d . In order to show that (2.4.74) belongs to LI0 Q in the latter cl .X;C / we decompose a expression as aQ D ! aQ C .1  !/a, Q where !.r; Q r 0 / is chosen as in Remark 2.3.2. Concerning ! aQ we use the following lemmas that we prove below. Lemma 2.4.69. For every fixed  6D 0 we have   xQ xQ 0 !.r; r 0 /aQ r; r 0 ; ; 0 ; r C x; Q r 0 ; r 2 S I;0 .R1Cn  R1Cn  R1Cn /: r;xQ r 0 ;xQ 0 ; r r Lemma 2.4.70. Let aQ be as in equation (2.4.77), let  6D 0 be fixed, and write b.r; r 0 ; x; x 0 ; %; ; / WD a.r; Q r 0 ; x; x 0 ; r%; ; r/. Then the operator Œr  Opr;x ..1  !/b/./ has a kernel in S.R  R; C 1 .B  B//, vanishing for r < "; r 0 < " for some " > 0 and for x, x 0 close to @B.

152

2 Symbolic structures and associated operators

Modulo pull backs from  to X the operator (2.4.74) is equal to a finite sum of expressions of the form Q Opr;xQ .! a/;

ˇ Œr  Opr;x ..1  !/a/ Q

(2.4.79)

and ˇ C.r%; r/, cf. the formula (2.4.76). By virtue of Lemmas 2.4.69 and 2.4.70 we obtain that the operators (2.4.79) give rise to elements of LI0 cl .X;C /. We now consider the contribution from the smoothing family (2.4.76). We show   that for every fixed  6D 0 the operator ˇ r op ..1   .r//C.r%; r/.1  Q .r 0 ///  r ˇ has a kernel in S.R  X  R  X /ˇŒ";1/XŒ";1/X for some " > 0. It suffices to localise the kernel with respect to x and x 0 by a suitable choice of local coordinates in the open ball B (using the fact that for every x0 ; x00 2 X there is a coordinate neighbourhood that contains these points). In other words we may consider functions '; 2 C01 .B/ and characterise   (2.4.80) ˇ r  opr ..1   .r//'.x/C.r%; r/ .x 0 /.1  Q .r 0 /// : Set c.r; Q x; r 0 ; x 0 ; r%; r/ WD r  .1   .r//'.x/c.x; x 0 ; r%; r/ .x 0 /.1  Q .r 0 // and Q cQ1 WD .1  .r  r 0 //c, Q and a function write cQ D cQ0 C cQ1 with cQ0 WD .r  r 0 /c, 1 x  2 C0 .RC / that is equal to 0 for jr  r 0 j < "0 and 1 for jr  r 0 j > "1 for certain 0 < "0 < "1 . Then the operator (2.4.80) has the form D D D0 C D1 where • 0 Di v.r; x/ Q WD .ˇ  /1 e i.rr /% cQi .r; x; r 0 ; x 0 ; r%; r/v.r 0 ; rx 0 /dx 0 dr 0 μ %; B 0 2M

i D 0; 1. Using jr  r j D%2M e i.rr after integration by parts that

0 /%

D e i.rr

0 /%

for every M 2 N it follows

Q D0 v.r; x/ •  xQ  xQ 0 0 n Dr e i.rr /% jr  r 0 j2M D%2M cQ0 r; ; r 0 ; ; r%; r v.r 0 ; xQ 0 /d xQ 0 dr 0 μ %: r r The property that cQ0 is a Schwartz function in r gives us immediately the desired characterisation of the kernel of D0 . The kernel of the operator D1 has the form Z  xQ  xQ 0 0 r n e i.rr /% cQ1 r; ; r 0 ; ; r%; r .1  .r  r 0 //μ %: r r Since cQ1 is a Schwartz function with respect to %. Q D r%/ the integral is convergent for r > 0 which is the case on the support. Moreover, the Schwartz property with respect to r yields the Schwartz property in r for every fixed  6D 0. Thus the kernel of D1 is a Schwartz function in .r; r 0 / since it vanishes for jr  r 0 j > "1 . We then obtain this property also with respect to the arguments .r; x; Q r 0 ; xQ 0 /. Let us now turn to the ellipticity of (2.4.74). The parameter-dependent ellipticity of p. Q %; Q / Q means that the homogeneous principal symbol a./ .x; ; %; Q / Q does not vanish for . ; %; Q / Q 6D 0. For  6D 0 and large r we have  xQ   .ˇ .r  opr ..1   /p.1 Q  Q //.r%; r/// D a./ ; ; %; 0 r

153

2.4 Mellin operators

for . ; %/ 6D 0, and e .ˇ .r  opr ..1   /p.1 Q  Q //.r%; r// D a./

 xQ r

 ; ; %;  ¤ 0

for all . ; %/ and large jxj. Q Finally, for . ; %/ 6D 0 and large jxj Q we have 

 opr ..1   /p.1 Q  Q //.r%; r// D a./ ;e .ˇ .r

x r

 ; ; %; 0 6D 0

Thus ˇ .r  opr ..1   /p.1 Q  Q //.r%; r// is elliptic in LI0 cl .X;C / for large r. Proof of Lemma 2.4.69. Let us first recall that the amplitude function ! aQ vanishes for r < " or r 0 < " for some " > 0 and for jr  r 0 j > 2. In addition it vanishes for x=r Q or Q r 0 ; xQ 0 ; ; / 2 xQ 0 =r 0 close to @B. Thus ! aQ can be regarded as a C 1 function in .r; x; R1Cn  R1Cn  R1Cn . On the support of ! aQ we have the estimates Q 0; cr 0  r  cr

chr 0 ; xQ 0 i  hr; xi Q  c 0 hr 0 ; xQ 0 i;

(2.4.81)

and chr; xi Q  r  chr; Q xi Q

(2.4.82)

for certain constants c; cQ > 0. Moreover, for every real  and fixed  6D 0 we have Q r 0 ; ri D r  .1 C jr C xj Q 2 C jr 0 j2 C jrj2 /=2 r  hr C x; 1 xQ r0 D .jj2 C 2 C j C j2 C j j/=2  ch; i r r r for some constant c > 0, i.e., Q  h; i hr C x; Q r 0 ; ri  cr  h; i  chr; xi

(2.4.83)

for another constant c > 0. As before c will denote different positive constants. For a.x; ; %; Q / Q in the formula (2.4.77) we have the estimates 0

ˇ .'.x/a.x; %; Q ; / Q .x 0 /j  ch%; Q ; i Q jˇ j jDx˛ Dx˛0 D%; ; Q Q

for all ˛; ˛ 0 2 N n , . ; %; Q / Q 2 RnC1Cq . This allows us to estimate xQ xQ 0 0 ˇ !.r; Q r 0 /a.r; Q r 0 ; ; 0 ; r C x; Q r 0 ; r/j jDr;˛ xQ Dr˛0 ;xQ 0 D; r r by applying Remark 2.3.2 and the estimates (2.4.81), (2.4.82). The computations are elementary; so we concentrate on a few typical points. Setting 1 WD

xQ ; r

2 WD

xQ 0 ; r0

3 WD r C x; Q

and  WD .1 ; : : : ; 5 /

4 WD r 0 ;

5 WD r

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2 Symbolic structures and associated operators



@2 @r



@ @ .a.r; Q r 0 ;  // D @r aQ .r; r 0 ;  / @r @4 3 5 D 0, @ D  , @ D . @r @r @r

we have

P5

@aQ j D1 @j

C

.r; r 0 ;  /

@j @r

where

@1 @r

D  rxQ2 ,

D We then obtain, using (2.4.83), ˇ @  ˇ ˇ ˇ aQ .r; r 0 ;  /ˇ  cr N h3 ; 4 ; 5 i  chr; xi Q N h; i ˇ @r

for every N 2 N and ˇ @aQ @1 ˇˇ ˇ .r; r 0 ;  / Q 3 ; 4 ; 5 i  C hr; xi Q 1 h; i ; ˇ ˇ  cr 2 jxjh @1 @r ˇ @aQ @3 ˇˇ ˇ .r; r 0 ;  / Q 1 h; i ; ˇ ˇ  cr 1 jjh; i1  chr; xi @3 @r ˇ @aQ @5 ˇˇ ˇ 0 .r; r ;  / Q 1 h; i ˇ ˇ  cr 1 jjh; i1  chr; xi @5 @r with different constants c > 0. Since !.r; Q r 0 / has the properties of a symbol in @ 0I0;0 S .R  R/ we obtain altogether j @r ! aj Q  chr; xi Q 1 h; i . The other derivatives can be treated in a similar manner. This yields the system of required symbolic estimates for ! a. Q Proof of Lemma 2.4.70. Since aQ vanishes for r < " for some " > 0 we simply write r  rather than Œr  . In order to compute the distributional kernel we consider the oscillatory integral r  Opr;x .1  !/b./u.r; x/ “ 0 0  Dr e i.rr /%Ci.xx / .1  !.r; r 0 //

D r 

“

 a.r; Q r 0 ; x; x 0 ; r%; ; r/u.r 0 ; x 0 /dr 0 dx 0 μ %μ e i.rr

0 /%Ci.xx 0 /r Q

.1  !.r; r 0 //

Q r/u.r 0 ; x 0 /dr 0 dx 0 μ %r n μ : Q  a.r; Q r 0 ; x; x 0 ; r%; r ; 0

0

Using the identity jr  r 0 j2N D%2N e i.rr /% D e i.rr /% for every N 2 N and integrating by parts gives us a relation of the form Z Z 1 r  Opr;x .1  !/b./u.r; x/ D gN .r; r 0 ; x; x 0 I /u.r 0 ; x 0 /dr 0 dx 0 B

for gN .r; r 0 ; x; x 0 I / D r Cn

“

 e i.rr

0

jr  r 0 j2N .1  !.r; r 0 // 0 /%Ci.xx 0 /r

r 2N .D%2N Q r 0 ; x; x 0 ; r%; r ; r/μ %μ : Q a/.r;

2.4 Mellin operators

155

The smoothness of gN with respect to x; x 0 follows from the estimates jr C2N .D%2N Q r 0 ; x; x 0 ; r%; r ; r/j  C h%; i2N Q a/.r; that are true for arbitrary N , with different constants C > 0. Furthermore, we have .1  !.r; r 0 //jr  r 0 j2N  C hriN hr 0 iN :

(2.4.84)

This gives us an estimate of the form jgN .r; r 0 ; x; x 0 I /j  C hriN hr 0 iN for arbitrary N 2 N. In a similar manner we can treat the derivatives of the kernel in .r; r 0 /. In this conclusion the .x; x 0 /-derivatives of gN only produce powers of r that are compensated by the contribution of (2.4.84) for sufficiently large N . This completes the proof.

2.4.6 Green operators for conical singularities We now turn to some specific observations on a class of smoothing operators which are of interest in connection with conical singularities. We first introduce spaces with asymptotics referring to distributions on a manifold with conical singularities and base X, n D dim X . Definition 2.4.71. (i) A set ˚

P WD .pj ; mj ; Mj / 0j N

(2.4.85)

with pj 2 C, mj 2 N, and subspaces Mj  C 1 .X / of finite dimension is said to be a discrete asymptotic type, associated with the weight data .; ‚/,  2 R, ‚ D .#; 0 , 1  # < 0, if C P WD fpj g0j N satisfies the condition C P  fw 2 C W nC1   C # < Re w < nC1   g, moreover, 2 2 N < 1 for # > 1, and Re pj ! 1 as j ! 1 for # D 1 and N D 1. (ii) We set s; .X ^ / WD lim K s;#" .X ^ / (2.4.86) K‚  ">0

for ‚ D .#; 0 and mj N X n o X EP .X ^ / WD !.r/ cj k .x/r pj logk r W cj k 2 Mj for all j; k

(2.4.87)

j D0 kD0

(with an arbitrary fixed cut-off function !) when ‚ is finite and P a discrete asymptotic type, associated with .; ‚/. Then we define s; KPs; .X ^ / WD K‚ .X ^ / C EP .X ^ /:

(2.4.88)

156

2 Symbolic structures and associated operators

In the case ‚ D .1; 0 we choose a sequence .#k /k2N , 0 > #0 >    > #k > #kC1 >    , #k ! 1 as k ! 1, and set Pk WD f.p; m; M / 2 P :   C #k g, p > nC1 2 KPs; .X ^ / WD lim KPs; .X ^ /: (2.4.89) k  k2N

Remark 2.4.72. The spaces (2.4.86), (2.4.88) and (2.4.89) are Fréchet in a natural way. The sum (2.4.88) is direct.  1; x C ; C 1 .X // (where subscript Let us set S.P .X ^ / WD Œ! K.P .X ^ /CŒ1! S.R / / ‘.P /’ means that the notation is used with or without ‘P ’) in the Fréchet topologies of non-direct sums (! is any fixed cut-off function, and the spaces are independent of the specific choice). Observe that we have

S  .X ^ / D lim hrik K k; .X ^ /; 

SP .X ^ / D lim hrik KPk; .X ^ /: 

k2N

(2.4.90)

k2N

Remark 2.4.73. The spaces (2.4.90) are Fréchet; the second one is nuclear for every discrete asymptotic type P . The spaces KPs; .X ^ / are called weighted Sobolev spaces on the infinite stretched cone X ^ with discrete asymptotics (near r D 0) of type P . If ‚ is finite, the meaning of this notation is clear from (2.4.87), (2.4.88). For ‚ D .1; 0 the relation u.r; x/ 2 KPs; .X ^ / means that for every ˇ  0 there is an N D N.ˇ/ such that u.r; x/  !.r/

mj N X X

cj k .x/r pj logk r 2 K s;ˇ .X ^ /

j D0 kD0

with coefficients cj k 2 Mj , 0  k  mj . These coefficients are uniquely determined by u. Example 2.4.74. For dim X D 0 the discrete asymptotic types have the form P WD f.pj ; mj /g0j N  C  N with the properties as in Definition 2.4.71. For P D f.j; 0/gj 2N we have x C / D S.R/j x : SP0 .RC / D S.R RC In this connection we also talk about Taylor asymptotics at 0. Definition 2.4.75. Let B be a compact manifold with conical singularities, B its stretched manifold and X Š @B. Let us fix a collar neighbourhood of @B with the variables .r; x/ 2 Œ0; 1/  X , and fix a cut-off function !.r/ supported by that neighbourhood. Given a discrete asymptotic type P we then define HPs; .B/

for s;  2 R

as the subspace of all u 2 H s; .B/ (cf. Definition 2.4.22) such that !.r/u.r; x/ 2 KPs; .X ^ /.

2.4 Mellin operators

157

Let us fix discrete asymptotic types P WD f.pj ; mj ; Mj /g0j N

associated with

.ˇ; ‚/; ‚ D .#; 0 ;

(2.4.91)

Q WD f.qj ; nj ; Nj /g0j L

associated with

.; /; D .ı; 0

(2.4.92)

and

for weights ; ˇ 2 R and (finite or infinite) weight intervals ‚; . For simplicity, in most cases we assume ‚ D . Let X be a closed compact C 1 manifold, and consider the infinite stretched cone ^ X D RC  X. Definition 2.4.76. An element G 2 L1 .X ^ / is called a Green operator in the cone algebra on X ^ with discrete asymptotics of types (2.4.91), (2.4.92), if it defines continuous operators G W K s; .X ^ / ! SPˇ .X ^ /;

 G  W K s;ˇ .X ^ / ! SQ .X ^ /

(2.4.93)

for all s 2 R. Here G  is the formal adjoint of G with respect to the scalar product of K 0;0 .X ^ /. Let CG .X ^ /P;Q denote the space of all such Green operators. Remark 2.4.77. There are several equivalent definitions of Green operators for the calculus on manifolds with conical singularities. Here we shall mainly discuss kernel characterisations of several kinds of Green operators. A consequence will be that (2.4.93) can equivalently be replaced by the conditions G W hrim K s; .X ^ / ! SPˇ .X ^ /;

 G  W hrim K s;ˇ .X ^ / ! SQ .X ^ / (2.4.94)

for all s 2 R; m 2 N. It is also admitted to require (2.4.93) only for s D 0 which then entails the conditions (2.4.94) for all s 2 R; m 2 N. Choose any strictly positive function k.r/ 2 C 1 .RC / such that k.r/ D r for 0 < r < c0 ;

k.r/ D 1 for r > c1

for certain fixed 0 < c0 < c1 . The multiplication by k % for any % 2 R gives rise to isomorphisms k % W K s; .X ^ / ! K s; C% .X ^ /;

C% ^ k % W SPˇ .X ^ / ! STˇ % P .X /

for every s;  2 R and T % P WD f.pj  %; mj ; Mj /g0j N . Remark 2.4.78. The map G ! k ˇ Gk  induces a bijection CG .X ^ /P;Q ! CG .X ^ /T ˇ P;T  Q : This allows us to pass to any other convenient weights.

158

2 Symbolic structures and associated operators

Proposition 2.4.79. Every G 2 CG .X ^ /P;Q can be written as an integral operator Z Z 1 g.r; x; r 0 ; x 0 /u.r 0 ; x 0 /r 0n dr 0 dx 0 Gu.r; x/ D X

0

with kernel g2

\˚

˚

^ y  K s; .X ^ / \ K s;ˇ .X ^ // ˝ y  S  SPˇ .X ^ / ˝ x .X / I Q

s2R

˚ xj / x WD .x is the complex conjugate of Q in an obvious sense. here Q qj ; nj ; N 0j L Proof. From the definition of the space SPˇ .X ^ / we see that there is a sequence of Hilbert spaces    ,! Ez j C1 ,! Ez j ,!    ,! Ez 0 D K 0;ˇ .X ^ / with nuclear embeddings Ez j C1 ,! Ez j for all j 2 N such that SPˇ .X ^ / D lim Ez j . The first j 2N relation of (2.4.93) shows that G W K s; .X ^ / ! Ez j is a Hilbert–Schmidt operator for every j > 0. Thus G has a kernel in Ez j ˝H K s; .X ^ /; here ˝H denotes the Hilbert space tensor product. Therefore, the kernel also belongs to y  K s; .X ^ /I lim Ez j ˝H K s; .X ^ / D SPˇ .X ^ / ˝ 

j 2N

the latter equality comes from the nuclearity of the above-mentioned embeddings, see also the formula (2.1.7) By virtue of Remark 2.4.78 it suffices to analyse the structure of Green operators for specific weights, say,  D ˇ D 0. For every G 2 CG .X ^ /P;Q in this case we obtain from Proposition 2.4.79 the relation y  K 0;0 .X ^ / \ K 0;0 .X ^ / ˝ y  S 0x .X ^ /: g 2 SP0 .X ^ / ˝ Q

(2.4.95)

It will be useful to derive more precise characterisations of the kernels of Green operators. To this end we first show the following general result. Theorem 2.4.80. For an operator G 2 L.L2 .Rn // the following conditions are equivalent: (i) G and its adjoint G  induce continuous operators G; G  W L2 .Rn / ! S.Rn /I (ii) G has an integral kernel g.x; x 0 / 2 S.Rn  Rn /, i.e., Z g.x; x 0 /u.x 0 /dx 0 : Gu.x/ D Rn

(2.4.96)

2.4 Mellin operators

159

y  L2 .Rn /g \ fL2 .Rn / ˝ y  S.Rn /g as Proof. First we have g.x; x 0 / 2 fS.Rn / ˝ a consequence of the mapping property (2.4.96). In particular, we see that G is a y  L2 .Rn / we obtain Hilbert–Schmidt operator. From g.x; x 0 / 2 S.Rn / ˝ hxik Dx˛ g.x; x 0 / 2 L2 .Rn  Rn /

for all k 2 N; ˛ 2 N n :

(2.4.97)

y  S.Rn / gives us In a similar manner g.x; x 0 / 2 L2 .Rn / ˝ hx 0 il Dxˇ0 g.x; x 0 / 2 L2 .Rn  Rn /

for all l 2 N; ˇ 2 N n :

(2.4.98)

O 0 /; h 0 ijˇ j g. ; O 0 / 2 L2 .Rn  Rn / for all ˛; ˇ 2 N n , as we see from Thus h ij˛j g. ; (2.4.97) for k D 0, (2.4.98) for l D 0, using Plancherel’s theorem, g. ; O 0 / WD 0 .Fx! Fx 0 ! 0 g/. ; /. We now use the fact that for every N 2 N there exist multiindices ˛; ˇ 2 N n such that h ; 0 iN  C.h ij˛j C h 0 ijˇ j / for some C > 0. This has the consequence that h ; 0 iN g. ; O 0 / 2 L2 .Rn  Rn / for every N 2 N. In fact, setting k  k WD k  kL2 .Rn Rn / we have kh ; 0 iN g. ; 0 /k  C kfh ij˛j C h 0 ijˇ j gg. ; 0 /k  C fkh i˛ g. ; 0 /k C kh 0 ijˇ j g. ; 0 /kg; and the right-hand side is finite by virtue of (2.4.97). Since N is arbitrary, Plancherel’s  0 2 n n 2n theorem gives us Dx;x . In particular, it 0 g.x; x / 2 L .R  R / for all  2 N 1 n n follows that g 2 C .R  R /. The relation (2.4.97) for j˛j D 0 and (2.4.98) for l D 0 yields hxik .Fx 0 ! 0 g/.x; 0 /; 0ˇ .Fx 0 ! 0 g/.x; 0 / 2 L2 .Rn  Rn / for all k 2 N, ˇ 2 N n . In an analogous manner as before we get hx; 0 iN .Fx 0 ! 0 g/.x; 0 / 2 L2 .Rn  Rn / for all N 2 N and hence hxik Dxˇ0 g.x; x 0 / 2 L2 .Rn  Rn / for all k 2 N, ˇ 2 N n . Similarly we obtain hx 0 il Dx˛ g.x; x 0 / 2 L2 .Rn  Rn /

for all l 2 N; ˛ 2 N n :

(2.4.99)

for all l 2 N; ˇ 2 N n :

(2.4.100)

for all l 2 N; ˛; ˇ 2 N n :

(2.4.101)

for all k 2 N; ˛; ˇ 2 N n :

(2.4.102)

The relations (2.4.98) are equivalent to Dxˇ0 hx 0 il g.x; x 0 / 2 L2 .Rn  Rn / Moreover, (2.4.99), (2.4.100) yield Dx˛ Dxˇ0 hx 0 il g.x; x 0 / 2 L2 .Rn  Rn / In an analogous manner we obtain Dx˛ Dxˇ0 hxik g.x; x 0 / 2 L2 .Rn  Rn /

From (2.4.101) and (2.4.102) we can conclude hx 0 il Dx˛ Dxˇ0 g.x; x 0 /; hxik Dx˛ Dxˇ0 g.x; x 0 / 2 L2 .Rn  Rn / for all k; l 2 N, ˛; ˇ 2 N n . This entails hxik hx 0 il Dx˛ Dxˇ0 g.x; x 0 / 2 L2 .Rn  Rn / for all k; l 2 N; ˛; ˇ 2 N n which is equivalent to g.x; x 0 / 2 S.Rn  Rn /.

160

2 Symbolic structures and associated operators

Corollary 2.4.81. Let X be a closed compact C 1 manifold. For an operator G 2 L.L2 .R  X// the following conditions are equivalent: (i) G and its adjoint G  induce continuous operators G; G  W L2 .R  X / ! S.R; C 1 .X //I (ii) G has an integral kernel g.t; x; t 0 ; x 0 / 2 S.R2t;t 0 ; C 1 .X  X //, i.e., Gu.t; x/ D R 0 0 0 0 0 0 RX g.t; x; t ; x /u.t ; x /dt dx . Proof. The inclusion (ii) ) (i) is evident. For (i) ) (ii) it suffices to show the analogous property for the operator 'G for arbitrary '; 2 C 1 .X / supported in a coordinate neighbourhood (those neighbourhoods may contain several connected components, and without loss of generality we assume that the charts map to an open subset of the unit ball in Rn where n D dim X ). Extending functions with respect to x or x 0 by zero to Rn , we obtain the situation of Theorem 2.4.80. A simple argument in terms of a partition of unity then yields the assertion. We now define the spaces  1; x C ; C 1 .X // S‚ .X ^ / WD !K‚ .X ^ / C .1  !/S.R

for every  2 R; ‚ D .#; 0 ; 1  # < 0, and x C ; C 1 .X // S0 .X ^ / WD !T  .X ^ / C .1  !/S.R for any cut-off function !.r/. Recall that T  .X ^ / WD fu 2 H 1; .X ^ / W u.r; / logk r 2 H 1; .X ^ / for all k 2 Ng: Corollary 2.4.82. For an operator G 2 L.K 0; .X ^ /; K 0;ˇ .X ^ // the following conditions are equivalent: (i) the operators G W K 0; .X ^ / ! S0ˇ .X/, G  W K 0;ˇ .X ^ / ! S0 .X ^ / are continuous; y  S0 .X ^ /. (ii) G has a kernel g 2 S0ˇ .X ^ / ˝ This result follows from Corollary 2.4.81 by substituting isomorphisms S.R; C 1 .X // ! S0ˇ .X ^ /;

S.R; C 1 .X // ! S0 .X ^ /

which preserve the functions for t > T , operate as in (2.4.18) for t < T for some T > 0 and transform the functions by a smooth substitution up to the multiplication by a C 1 function in t in the interval ŒT; T . Let us now consider the relation (2.4.95) for the empty asymptotic types P and Q. Then we have ˚ 0 ^

˚

0 y  K 0;0 .X ^ / \ K 0;0 .X ^ / ˝ y  S

g 2 S‚ .X / ˝ .X ^ / :

2.4 Mellin operators

This entails the relations

˚ 0 y  K 0;0 .X ^ /g \ fK 0;0 .X ^ / ˝ y  S

.X ^ / ; g 2 S00 .X ^ / ˝ ˚ 0 ^

y  K 0;0 .X ^ /g \ fK 0;0 .X ^ / ˝ y  S00 .X ^ / : g 2 S‚ .X / ˝

161

(2.4.103) (2.4.104)

Theorem 2.4.83. For every ‚ D .#; 0 ; 1  # < 0, we have

˚ 0 ^ 0 0 y  K 0;0 .X ^ /g \ fK 0;0 .X ^ / ˝ y  S‚ y  S‚ .X ^ / D S00 .X ^ / ˝ .X ^ /: S0 .X / ˝ Proof. Let us first consider the case dim X D 0. Then the assertion means

˚ 0 0 0 y  L2 .RC /g \ fL2 .RC / ˝ y  S‚ y  S‚ .RC / D S00 .RC / ˝ .RC /: S0 .RC / ˝ (2.4.105) The right-hand side of (2.4.105) is contained in the left one. Thus it suffices to show the converse inclusion. x C / for T .RC / WD T 0 .RC / and We have S00 .RC / D !T .RC / C .1  !/S.R \ 0 x C /: S‚ .RC / D !r  T .RC / C .1  !/S.R #<<0

Given any g.r; t / in the space on the left of (2.4.105) we write g D !.r/!.t/g C !.r/.1  !.t //g C .1  !.r//!.t /g C .1  !.r//.1  !.t //g: (2.4.106) Identifying the last summand on the right of (2.4.106) with its extension by 0 to r  0 y  L2 .R/g \ fL2 .R/ ˝ y and t  0 we obtain .1  !.r//.1  !.t //g 2 fS.R/ ˝ S.R/g. From the proof of Theorem 2.4.80 it follows that .1  !.r//.1  !.t //g 2 0 y  S.R/ and also .1  !.r//.1  !.t //g 2 S00 .RC / ˝ y  S‚ S.R/ ˝ .RC / because of the factors which vanish near r D 0 and t D 0. To treat the other summands we apply the isomorphism (2.4.2) for  D 0 and obtain isomorphisms S0 W L2 .RC / ! L2 .R/; T .RC / ! S.R/. Since !.r/S00 .RC / D !.r/T .RC;r / it follows that y  L2 .R/g \ fL2 .R/ ˝ y  S.R/g D S.R/ ˝ y  S.R/ S0;r !.r/.1  !.t //g 2 fS.R/ ˝ 0 y  S‚ .RC /. Set and hence !.r/.1  !.t //g 2 S00 .RC / ˝

S .Ry / WD fu 2 S.Ry / W e y u.y/ 2 S.Ry /g: For the first summand we apply the transformation S0 both with respect to r and t , \ 0 S0;r W !.r/S00 .RC / ! S.Rx /; S0;t W !.t /S‚ .RC / ! S .Ry /: #<<0

This reduces the statement to y  S .R/g D S.R/ ˝ y  S .R/: y  L2 .R/g \ fL2 .R/ ˝ fS.R/ ˝

162

2 Symbolic structures and associated operators

Set g .x; y/ D e y .S0;r S0;t !.r/!.t /g/. Then it suffices to show that when !.r/!.t/g.r; t/ belongs to the left-hand side of (2.4.105) it follows that g .x; y/ 2 S.R  R/ for all # <   0. This property for  D 0 is clear from the arguments before, since S .R/ D S.R/ for  D 0. So for # <  < 0 we have to verify hxik hyil hDx im hDy in g .x; y/ 2 L2 .R  R/ for all k; l; m; n 2 N; here hDx im WD Op.h im / for any m 2 N. Similarly as in the proof of Theorem 2.4.80 by repeatedly applying the inequality ab  a2 C b 2 for reals a; b together with Plancherel’s formula it follows for k  k WD k  kL2 .RR/ that khxik hyil hDx im hDy in g .x; y/k  khxi4k hDx im g .x; y/k C khDx im hDy i2n g .x; y/k

(2.4.107)

C khDx i2m hDy in g .x; yk C khyi4l hDy in g .x; y/k: y  S.R/. To treat the other terms we The fourth term is finite, since g 2 L2 .R/ ˝ 2 choose a p > 1 such that # < p  < p <  and define q by p1 C q1 D 1. Then we have ab  ap C b q for all a; b  0. Applying that for p D q D 2 we get khDx im hDy in g .x; y/k  khDx i2m g .x; y/k C khDy i2n g .x; y/kI as O y/ D  we know  the second summand is finite. For the first summand we pass to g. ; Fx! g . ; y/. Then khDx i2m g .x; y/k D k h i2m e y g. ; O y/k  kh i2q m g. ; O y/k C ke py g. ; O y/k D khDx i2q m g.x; y/k C ke py g.x; y/k < 1: Therefore, the second and the third term on the right-hand side of (2.4.107) are finite. Finally, khxik hDx im g .x; y/k  khxiqk hDx im g.x; y/k C khDx im e py g.x; y/k  khxiqk hDx im g.x; y/k C khDx iqk g.x; y/k C ke p

2 y

g.x; y/k < 1

shows that also first term on the right-hand side of (2.4.107) is finite. Analogous arguments are valid for the third summand of (2.4.106). We thus obtain altogether that g belongs to the space on the right of (2.4.105). Combining the proof of (2.4.105) with that of Theorem 2.4.80 we easily obtain the relation y  L2 .Rn ; L2 .RC //g \ fL2 .Rn ; L2 .RC // ˝ y  S.Rn ; S0 .RC /g fS.Rn ; S00 .RC // ˝ y  S.Rn ; S0 .RC //: D S.Rn ; S00 .RC // ˝ The assertion of Theorem 2.4.83 for dim X > 0 then follows after simple localisations with respect to charts U ! Rn on X and a subordinate partition of unity.

2.4 Mellin operators

163

Corollary 2.4.84. An operator G 2 CG .X ^ /‚; (i.e., with empty asymptotic types and referring to the weights .; ˇ/) has an integral kernel  ˇ y  S

y  S0 .X ^ /: g 2 S0ˇ .X ^ / ˝ .X ^ / \ S‚ .X ^ / ˝

(2.4.108)

Proof. We only employ the continuities  G  W K 0;ˇ .X ^ / ! S

.X ^ /

ˇ G W K 0; .X ^ / ! S‚ .X ^ /;

(2.4.109)

to obtain as in the proof of Proposition 2.4.79 the relations ˇ  y  K 0; .X ^ / \ K 0;ˇ .X ^ / ˝ y  S

g 2 S‚ .X ^ / ˝ .X ^ /:

By Remark (2.4.78) it suffices to consider the case  D ˇ D 0. Then it follows that (2.4.103) and (2.4.104) hold. Theorem 2.4.83 together with its variant with interchanged arguments then gives us (2.4.108). Theorem 2.4.85. Let p > 1, q > 1, p1 C q1 D 1, and set =p WD .ı=p; 0 , ‚=q WD .#=q; 0 . Then the continuities (2.4.109) imply ˇ  y  S =p g 2 S‚=q .X ^ / ˝ .X ^ /:

(2.4.110)

ˇ  y  S

.X ^ / ˝ .X ^ /. In particular, for ‚ D D .1; 0 it follows that g 2 S‚

Proof. First observe that an f .r; x/ 2 C 1 .X ^ / belongs to the space S0ˇ .X ^ / if and n M only if there exists a 1 < p < 1 such that r 2 ˇ logl rhriN Dr;x f .r; x/ 2 Lp .RC  M M X/ for all l; N; M 2 N and arbitrary Dr;x ; here Dr;x denotes any differential operator   P @ j on X ^ of the form jMD0 aj r @r with coefficients aj 2 Diff M j .X /. Moreover, ˇ an f .r; x/ 2 C 1 .X ^ / belongs to S‚ .X ^ /, ‚ D .#; 0 , if and only if r  f .r; x/ 2 S0ˇ .X ^ / for every 0 <  < #. Analogously, a function f .r; x; r 0 ; x 0 / 2 C 1 .X ^  X ^ / belongs to y  S0 .X ^ / S0ˇ .X ^ / ˝ (2.4.111)

if and only if n

n

0

0

0

M 2 C r 2 ˇ .logl r/hriN Dr;x r .logl r 0 /hr 0 iN DrM0 ;x 0 f .r; x; r 0 ; x 0 / 2 Lp .X ^  X ^ / 0

M for all l; l 0 ; N; N 0 ; M; M 0 2 N and arbitrary Dr;x ; DrM0 ;x 0 , for some 1 < p < 1. ˇ  y  S

y .X ^ / ˝ .X ^ / is equivalent to r  .r 0 / f 2 S0ˇ .X ^ / ˝ Moreover, f 2 S‚  ^ S0 .X / for all 0 <  < #, 0 <  < ı. By Corollary 2.4.84 for the function g we have the relation (2.4.108). After composition with corresponding weight factors without loss of generality we consider the case ˇ D  D n2 . To verify the criterion

164

2 Symbolic structures and associated operators

(2.4.111) we employ Z jr  .r 0 / gjp drdxdr 0 dx 0 Z D jr p gjj.r 0 /q gjp1 drdxdr 0 dx 0 Z 

jr p gjp drdxdr 0 dx 0

 p1 Z

j.r 0 /q gjp drdxdr 0 dx 0

(2.4.112)  q1 ;

y using q D p=.p1/. Since the functions r p g and r q g both belong to S00 .X ^ / ˝ 0 ^ S0 .X / it follows that the integral on the left-hand side of (2.4.112) is finite. An anal0 0 0 M .logl r 0 /hr 0 iN DrM0 ;x 0 g instead ogous conclusion is possible for .logl r/hriN Dr;x of g. This gives us immediately the relation (2.4.110). There are many other useful variants of kernel characterisations. Theorem 2.4.86. Let G be a Green operator in the sense of Definition 2.4.76. Then we have ^ y  S0 .X ^ /g \ fS0ˇ .X ^ / ˝ y  S  (i) g 2 fSPˇ .X ^ / ˝ x .X /g; Q

˚

(ii) if 1 < p < 1, p1 C q1 D 1, and Qq WD .w; l; N / 2 Q W Re w > nC1 C  C qı , 2 ˚

y Pp WD .z; m; M / 2 P W Re z > nC1  ˇ C p# , we have g 2 SPˇp .X ^ / ˝ 2 ˇ  ^ ^ y SQ x .X /. In particular, for # D ı D 1 it follows that g 2 SP .X / ˝ q

 ^ SQ x .X /.

This is a generalisation of (2.4.108) and Theorem 2.4.85 to the case of non-trivial asymptotic types. The technique for the proof may be found in [206], see also [196]. Another result of this category will be applied in boundary value problems with the transmission property, cf. Chapter 3 below. Theorem 2.4.87. For an operator G 2 L.L2 .RC // the following conditions are equivalent: (i) G and its adjoint G  induce continuous operators x C /I G; G  W L2 .RC / ! S.R ˇ xC  R x C / .WD S.R  R/ˇ (ii) the operator G has an integral kernel in S.R /, RC RC R1 0 0 0 0 x x i.e., Gu.r/ D 0 g.r; r /u.r /dr for a function g.r; r / 2 S.RC  RC /. Let us first show the following more general theorem. Theorem 2.4.88. For every 1 < p < 1 there is a topological isomorphism of Fréchet spaces x C /g D S.R xC  R x C /: x C/ ˝ y  Lp .RC /g \ fLp .RC / ˝ y  S.R fS.R

(2.4.113)

2.4 Mellin operators

165

Proof. First observe that the expressions n X p k .u/ WD khrik ukL p .R / C C

0j k

p kDrj ukL p .R / C

o p1

x C / which is equivalent to for k 2 N define a system of semi-norms on the space S.R the original one for describing the Fréchet topology of that space. Similarly, n X p ık .v/ WD khr; r 0 ik vkL p .R R / C C C

j˛jk

p ˛ kDr;r 0 vk p L .RC RC /

o p1

k 2 N,

;

x C /. xC  R form an equivalent system of semi-norms for the Fréchet topology of S.R Finally, the semi-norms n X p %k .v/ WD khrik vkL p .R R / C C C

0j k

p kDrj vkL p .R R / C C

o p1

˚

p1 P j p p together with %0k .v/ WD khr 0 ik vkL kD , 0 vkLp .R R / p .R R / C 0j k r C C C C k 2 N, define a semi-norm system for the Fréchet topology in the space on the leftx C/ V xC  R hand side of (2.4.113); let us denote that space by V . By virtue of S.R and since the topology of the subspace is stronger or equal to that of V , it suffices to xC R x C /. The space S.R xC R x C / is dense show the continuous embedding V ,! S.R in V , so it is enough to show inequalities of the kind ık .v/  C.%N .v/ C %0N .v//

(2.4.114)

xC  R x C / and for every k 2 N for a suitable choice of N D N.k/, with for all v 2 S.R a constant C D C.k; N / > 0, independent of v. k Applying the inequality hr; r 0 ik  2 2 1 .hrik C hr 0 ik / it follows that  p p p. k 0 k 2 1/ khrik vk p p .R R / C khr i vk khr; r 0 ik vkL p .R R /  2 L .R R / L C C C C C C   pk p p 0 k  2 2 1 khrik vkL p .R R / C khr i vkLp .R R / : C C C C Moreover, a known result from Smith [210] says that the standard norm in Sobolev spaces H kIp .RC  RC /, k 2 N, which is defined by n

p kvkL p .R R / C C C

X j˛jk

p ˛ kDr;r 0 vk p L .RC RC /

o p1

is equivalent to the norm which does not contain mixed derivatives, i.e., we have estimates X p ˛ kDr;r 0 vk p L .RC RC / j˛jk

c

 X 0j k

p kDrj vkL p .R R / C C C

X 0j p

 p kDrj0 vkL p .R R / : C C

166

2 Symbolic structures and associated operators

Combined with the former estimates it follows that ık .v/  C..%k .v//p C .%0k .v//p /1=p k

1

1

for C D max.2 2  p ; c p /. This finally yields (2.4.114) for N D k. Proof of Theorem 2.4.87. The proof of (ii) ) (i) is evident. For (i) ) (ii) we observe x C/ ˝ y  L2 .RC / that the continuity of G gives rise to a kernel function g.r; r 0 / 2 S.R  0 2 x C /, cf. Proposition 2.1.7. Thus y  S.R and that of G to g.r; r / 2 L .RC / ˝ x C/ ˝ x C /; y  L2 .RC / \ L2 .RC / ˝ y  S.R g.r; r 0 / 2 S.R and the desired characterisation follows from Theorem 2.4.88.

Chapter 3

Boundary value problems with the transmission property

Parametrices of elliptic boundary value problems for differential operators (with ellipticity in the Shapiro–Lopatinskij sense) belong to the class of pseudo-differential boundary value problems with the transmission property at the boundary. The operators that express mixed and crack problems have this property outside the singular interfaces. The calculus of these problems requires the tools from the regular case with the transmission property. We give a concise introduction into this theory. At the same time boundary value problems in this context are an interesting model for other more general theories around edge singularities.

3.1 Interior and boundary symbols Boundary value problems can be described in terms of amplitude functions with interior and boundary contributions. We first study interior symbols with the transmission property and then pass to associated operator-valued symbols. In addition we investigate Green, trace and potential symbols on the half-axis.

3.1.1 Symbols with the transmission property In this section we single out some specific classes of symbols in the sense of Definition 2.1.1. At the same time we slightly generalise the notation by setting x ˙  Rn / WD C 1 .  R x ˙ ; S  .Rn // S  .  R .cl/

.cl/

for an open set  R . Let q D n  1, and set U WD   R. Variables in the corresponding splitting are denoted by x D .y; t / and the associated covariables by D .; /. q

Definition 3.1.1. A symbol a.x; / 2 Scl .Ux  RqC1 / for  2 Z is said to have the

transmission property at t D 0 if the homogeneous components a.j / .x; / satisfy the following conditions: D tk D˛ fa.j / .y; t; ; /  .1/j a.j / .y; t; ;  /g D 0 for all y 2 ; t D 0;  D 0; 2 R n f0g, for all k 2 N, ˛ 2 N q , and for all j 2 N. Let Scl .U  RqC1 /tr denote the space of all symbols a.x; / 2 Scl .U  RqC1 / with the transmission property at t D 0. Moreover, we set ˚ ˇ

 qC1 x ˙  RqC1 /tr WD aˇ x /tr : (3.1.1) S  .  R qC1 W a 2 S .U  R cl

R˙ R

cl

168

3 Boundary value problems with the transmission property

Remark 3.1.2. The space Scl .U  RqC1 /tr is closed in Scl .U  RqC1 /; analogously, x ˙ RqC1 /tr is closed in S  . R x ˙ RqC1 /. The subspaces S  .RqC1 /tr of Scl . R cl cl symbols with the transmission property and with constant coefficients are also closed in the corresponding larger spaces; a similar observation is true of Scl .RRqC1 /tr , and x ˙  RqC1 /tr D we have Scl .  R  RqC1 /tr D C 1 .; Scl .R  RqC1 /tr /, Scl .  R  x ˙  RqC1 /tr /. C 1 .; Scl .R P Example 3.1.3. (i) Let a.x; / D j˛j a˛ .x/ ˛ , a˛ 2 C 1 .U /,  2 N. Then we have a.x; / 2 Scl .U  RqC1 /tr . (ii) Let ' 2 S.R/ such that '.0/ D 1 and supp F 1 '  R (with F being the Fourier transform on R). Then     r .; / WD ' (3.1.2) hi  i C hi for  2 Z, C > 0, belongs to Scl .RqC1 /tr , cf. Grubb [67], and Proposition 4.1.1 below. In Section 4.1.3 below we shall see that (3.1.2) is an example of a so-called minus symbol. Remark 3.1.4. The spaces of symbols with the transmission property at t D 0 remain preserved under several operations with symbols, especially, (i) asymptotic summation; (ii) pointwise products and Leibniz products; (iii) pointwise inversion and Leibniz inversion when the homogeneous principal component does not vanish on non-vanishing covectors. To illustrate the structure of the transmission property we want to have a look at the x C  R (the case R x   R is analogous; the transmission one-dimensional case, say on R property in the sense of Definition 3.1.2 is an invariant condition with respect to the reflection map t ! t ). Definition 3.1.5. Let a.t; / 2 Scl .RR/,  2 Z, and write the component a.j / .t; / of a.t; / of homogeneity   j in 6D 0 in the form a.j / .t; / D f C . /ajC .t / C   . /aj .t /g j ; j 2 N, where  ˙ is the characteristic function of R˙ 3 . The symbol a.t; / is said to have the transmission property at t D 0, written a.t; / 2 Scl .R  R/tr , if D tk ajC .t /j tD0 D D tk aj .t /j tD0

for all k; j 2 N:

(3.1.3)

˚ x ˙  R/tr WD aj x As a particular case of (3.1.1) for q D 0 we set Scl .R R˙ R W a 2

 Scl .R  R/tr .

3.1 Interior and boundary symbols

169

Remark 3.1.6. Definition 3.1.5 can also be formulated as follows. A symbol a.t; / 2 x C  R/,  2 Z, belongs to S  .R x C  R/tr if and only if the coefficients Scl .R cl P ˙ 1 aj .t/ 2 C .R/ in the expansion a.t; /  j1D0 aj˙ .t / j for ! ˙1 satisfy the condition (3.1.3). x C  RqC1 /, and set ay; .t; / WD a.y; t; ; / for Let a.y; t; ; / 2 Scl .  R x C  R/. every fixed .y; / 2   Rq ; then we have ay; .t; / 2 Scl .R Proposition 3.1.7. The following conditions are equivalent: x C  RqC1 /tr ; (i) a.y; t; ; / 2 Scl .  R x C  R/tr for every .y; / 2   Rq and all ˛ 2 N q . (ii) .D˛ a/y; .t; / 2 Sclj˛j .R Proof. For simplicity, we consider the case of symbols which are independent of .y; t /; the generalisation to the .y; t /-dependent case is easy and left to the reader. Assume that the symbol a.; / satisfies the condition (i). Then a . / belongs to the space Scl .R/tr if and only if .; /a.j / .; / has this property for every  and any excision function , for all j 2 N. Since  is equal to 1 for j; j > const for some constant > 0, it suffices to show that the symbol a.j / .; / for  6D 0 has this property. Set  WD   j for any fixed j 2 N. By assumption we have .D˛ a./ /.; /jD0; 6D0 D D˛ ..1/ a./ .;  //jD0; 6D0 ;

(3.1.4)

or, equivalently, .D˛ a./ /.0; 1/ D .1/Cj˛j .D˛ a./ /.0; 1/

(3.1.5)

for all ˛ 2 N q . Consider the expansion of a./ .; / for ! ˙1, and set % WD j j1 . Then a./ .; / D a./ .%1 %; %1 / D % a./ .%; 1/  %

X %j˛j ˛ ˛Š

˛

.D˛ a./ /.0; 1/

1 n X o X ˛ ˛ D .D a./ /.0; 1/ k ˛Š

for > 0;

kD0 j˛jDk

and a./ .; / D a./ .%1 %; %1 / D % a./ .%; 1/  %

X %j˛j ˛ ˛

D

1 n X X kD0 j˛jDk

.1/Cj˛j

˛Š

.D˛ a./ /.0; 1/

o  .D˛ a./ /.0; 1/ k ˛Š ˛

for < 0:

170

3 Boundary value problems with the transmission property

By virtue of (3.1.5) and Remark 3.1.6 we then obtain (i) ) (ii). In order to show (ii) ) (i) we observe that the assumption gives us lim  a.; / D lim  a.; /: !1

!C1

We have

lim  a.; / D lim  a./ .; / D a./ .0; 1/:

!C1

!C1

In a similar manner we obtain lim  a.; / D .1/ a./ .0; 1/:

!1

More generally, for the -derivatives we obtain the identities (3.1.5) first for  D  and then successively for all  D   j . Since this system of relations is equivalent to (3.1.4) for all ˛ 2 N q and , the proof is complete. Note that the symbols .1 ˙ i / for every  2 Z belong to Scl .R/tr and that the multiplication a. / ! .1 ˙ i / a. / defines an isomorphism Scl .R/tr ! SclC .R/tr for every ;  2 Z. Remark 3.1.8. Observe that a.y; t; ; / ! a.y; t; ; hi / defines a continuous map  x C  RqC1 / ! S  .  Rq ; S  .R x C  R//; S.cl/ .  R .cl/ .cl/

x C  RqC1 /tr is  2 Z, cf. the notation (2.1.3), (2.1.5). Then a.y; t; ; / 2 Scl .  R equivalent to the property x C  R/tr / a.y; t; ; hi / 2 Scl .  Rq ; Scl .R x  or R instead of R x C ). (similar identities are true for R To illustrate more details on the nature of the transmission property we want to have a look at the case a. / 2 Scl0 .R/tr . Recall that the transmission property in this case is characterised by the conditions ajC D aj for all j 2 N in the asymptotic expansion P a. /  j1D0 aj˙ j for ! ˙1. Remark 3.1.9. Let a. / 2 Scl0 .R/, and set L.a/ WD fa. / 2 C W 2 Rg which is a bounded set in the complex plane. Then we have a. / 2 Scl0 .R/tr if and only if L.a/ is a C 1 curve, including the end point a0C D a0 . To define operators on a half-axis RC (or R ) we set r ˙ f WD f jR˙ for f 2 L2 .R/, and ( ( u.t / for t 2 RC 0 for t 2 RC C  e u.t / WD ; e v.t / WD ; 0 for t 2 R v.t / for t 2 R for u 2 L2 .RC /; v 2 L2 .R /. Then opC .a/ WD r C op.a/eC W L2 .RC / ! L2 .RC /

(3.1.6)

171

3.1 Interior and boundary symbols

is a continuous operator for every a. / 2 Scl0 .R/. More generally, we also may consider operators r  Op.a/eC and r C Op.a/e , r  Op.a/e on L2 .RC / and L2 .R /, respectively. It will be instructive to give equivalent formulations of the transmission property of symbols as well as of the operators (3.1.6). Let us set x C /g; H0C WD f.F t! eC fC /. / W fC .t / 2 S.R x  /g H0 WD f.F t! e f /. / W f .t / 2 S.R x ˙ / WD S.R/j x and H0 WD H C C H  . Moreover, let H 0 for d 2 N; d  1, for S.R 0 0 R˙ d denote the space of all polynomials in of degree d  1, and set Hd WD H0C C H0 C Hd0 Remark 3.1.10.

for d  1:

An equivalent definition of the space Scl .R/tr Scl .R/tr D HC1 :

(3.1.7)

for  2 N is the relation (3.1.8)

x C  RqC1 /; then a consequence of R Remark 3.1.11. Let a.y; t; ; / 2 the characterisation of Remark 3.1.8 is that a has the transmission property at t D 0 if and only if b.y; ; / WD D tk a.y; t; ; hi /j tD0 for every k 2 N belongs to y  HC1 for  2 N and to Scl .  Rq / ˝ y  H0 for   1 2 N. A Scl .  Rq / ˝ x C. x  or R instead of R similar identification is true for R Scl .

The operators of restriction r ˙ W L2 .R/ ! L2 .R˙ / are complementary orthogonal projections when we identify the spaces L2 .R˙ / with the subspaces e˙ L2 .R˙ /  L2 .R/. It will also be convenient to write e˙ L2 .R˙ / D  ˙ L2 .R/ with  ˙ .t/ being the characteristic function of R˙ . Set V ˙ WD F t! . ˙ L2 .R//. We then also have an orthogonal decomposition L2 .R/ D V C ˚ V  and corresponding complementary projections …˙ W L2 .R/ ! V ˙ : (3.1.9) The operator (3.1.6) is then equivalent to …C a W V C ! V C : Remark 3.1.12. By virtue of the commutative diagram

(3.1.10) L2 .R/ # F 2

L .R/

C

! …C

!

 C L2 .R/ # F V

the

C

projections (3.1.9) are classical pseudo-differential operators of order zero on the -axis R, acting by the formula “ “ 0 ˙ i. 0  /t ˙ 0 0 .… u/. / D e  .t /u. /μ dt D e i.  /t  ˙ .t /u. 0 /μ 0 dt “ 0 D e i.  /t ..t / C .1  .t /// ˙ .t /u. 0 /μ 0 dt

172

3 Boundary value problems with the transmission property

for an excision function .t /. The variable t plays the role of a covariable in the oscillatory integral, and  ˙ .t / is the homogeneous principal symbol of …˙ , i.e., the homogeneous component of order zero of .t / ˙ .t / 2 Scl0 .R/, while the contribution with .1  .t// ˙ .t / 2 S 1 .R/ is smoothing. Remark 3.1.13. The functions .1  i /j ; .1 C i /j C1

.1 C i /j ; .1  i /j C1

j 2N

(3.1.11)

are elements of Scl1 .R/tr . At the same time they form an orthogonal base in the space L2 .R /, where V C .V  / is spanned by the first (second) sequence in (3.1.11). It will be interesting to reformulate the operators (3.1.10) in another equivalent way, namely, as operators on the unit circle S 1 D fz 2 C W jzj D 1g. To this end we set z WD

1  i 1 C i

which yields D

11z : i 1Cz

Then ! z defines a diffeomorphism  W R ! S 1 n f1g, and we have .0/ D 1; . / ! 1 for ! ˙1. From z D .1 C 2 /1 f.1  2 /  2i g we see arg . / > 0 .< 0/ for < 0 .> 0/. The pull back   under  (extended to the point f1g 2 S 1 ) induces an isomorphism   W C 1 .S 1 / ! H1 : Moreover, .T g/. / WD   ..1 C z/g.z// D

2 g.. // 1Ci

defines an isomorphism

T W L2 .S 1 / ! L2 .R/: j

j

.1i / .1Ci / j 1 In particular, we have T z j D 2 .1Ci / D 2 .1i / j C1 , T z j C1 for all j 2 N. The

operator T induces isomorphisms T W W ˙ ! V ˙ , when W C and W  are defined as the subspace of L2 .S 1 /, spanned by fz j W j 2 Ng and fz j 1 W j 2 Ng, respectively. We have an orthogonal decomposition L2 .S 1 / D W C ˚ W  with corresponding complementary projections  ˙ W L2 .S 1 / ! W ˙ . Moreover, for a D   ;  2 C 1 .S 1 /, we have …C a D T ı  C  ı T 1 : Thus,

 C W W C ! W C

is an equivalent description of (3.1.10) or (3.1.9). 0 Proposition 3.1.14. Let a. / 2 Scl .R/tr and a . / 2 H0 C HC1 , aC . / 2 H0C C 0 HC1 , cf. the characterisation of Remark 3.1.10. Then we have

opC .a / opC .a/ D opC .a a/; x C /). (say, as operators on S.R

opC .a/ opC .aC / D opC .aaC /

(3.1.12)

173

3.1 Interior and boundary symbols

Proof. We have opC .a / opC .a/u D r C op.a a/eC u C r C op.a /.1  r C / op.a/eC u: By virtue of op.a/eC u D F 1 fa. /F .eC u/. /g and F .eC u/. / 2 H0C we get a. /F .eC u/. / 2 HC1 and hence x C / C e S.R x  / C F 1 HC1 : op.a/eC u 2 eC S.R The space F 1 HC1 is spanned by all derivatives of the Dirac distribution at the origin up to order . Thus it makes sense to form .1  r C / op.a/eC u which belongs to x  / C F 1 HC1 . As such it is a distribution on R supported by R x  . However, e S.R we have x : r C op.a /v D 0 for all v 2 S 0 .R/; supp v R In a similar manner we can argue for the second relation in (3.1.12).

3.1.2 Operators with the transmission property As before, let  Rq be open, U WD   R 3 .y; t /, and set s s Hloc.y/ .  R/ WD fu 2 Hloc .U / W 'u 2 H s .RqC1 / for every '.y/ 2 C01 ./g; s Hcomp.y/ .  R/ WD fu 2 H s .RqC1 /jU W 'u D u

for some '.y/ 2 C01 ./g:

Moreover, let ˇ s s Hloc.y/ .  R˙ / WD Hloc.y/ .  R/ˇR ; ˇ ˙ s s .  R˙ / WD Hcomp.y/ .  R/ˇR : Hcomp.y/ ˙

Observe that then s s Hloc.y/ .  R/ D Wloc .; H s .R//; s s Hloc.y/ .  R˙ / D Wloc .; H s .R˙ //;

(3.1.13)

and, similarly, for the spaces with ‘comp’. Let e˙ be the extension operator of functions on   R˙ by zero to the opposite s .  R˙ / for s >  12 ). Moreover, let r ˙ denote side (applied to elements in Hloc.y/ the operator of restriction to   R˙ (applied to distributions in   R). x C RqC1 /tr and a.y; Given a.y; t;ˇ ; / 2 Scl . R Q t; ; / 2 Scl .RRqC1 /tr such that a D aQ ˇR x RqC1 we set C

OpC .a/u.x/ WD r C Op.a/e Q C u.x/;

(3.1.14)

s .RC / for s >  12 , first for u 2 C01 .RC / and later on extended to u 2 Hcomp.y/ cf. Corollary 3.1.17 below. Observe that the operator (3.1.14) is independent of the

174

3 Boundary value problems with the transmission property

choice of the extension aQ which justifies the notation. Similarly, we consider families of operators on the half-axis “ op.a/.y; Q /u.t / D

opC .a/.y; / WD r C op.a/.y; Q /eC ; e i.tt

0 /

a.y; Q t; ; /u.t 0 /dt 0 μ ; aQ 2 Scl .  R  RqC1 /tr :

(3.1.15) qC1 x Theorem 3.1.15. Let a.y; t; ; / 2  RC  R /tr , and assume that a is independent of t for t > T for some T > 0. Then opC .a/.y; / induces a family of continuous operators opC .a/.y; / W H s .RC / ! H s .RC / for every s >  12 , and we have opC .a/.y; / 2 S  .  Rq I H s .RC /; H s .RC // x C /; S.R x C //. for every s >  1 , as well as opC .a/.y; / 2 S  .  Rq I S.R Scl .

2

A proof is given in [188, Theorems 4.3.37, 4.2.38]. x C  RqC1 /tr is independent of t we have Remark 3.1.16. If a 2 Scl .  R opC .a/.y; / 2 Scl .  Rq I H s .RC /; H s .RC // x C /; S.R x C //, and the homogefor s >  12 , and opC .a/.y; / 2 Scl .  Rq I S.R neous principal component of order , the so-called principal boundary symbol, has the form @ .opC .a//.y; / WD opC .a./ /.y; /,  6D 0, with a./ .y; ; / being the homogeneous principal component of a.y; ; / of order . x C  RqC1 /tr be as in Theorem 3.1.15. Corollary 3.1.17. Let a.y; t; ; / 2 Scl .  R C Then Op .a/ induces continuous operators s s OpC .a/ W Hcomp.y/ .  RC / ! Hloc.y/ .  RC /

for all s >  12 . This is a consequence of Corollary 2.3.44 and the relations (3.1.13). Let us now pass to operators on a closed C 1 manifold M which is subdivided into C 1 submanifolds M˙ with common boundary Y , such that M D M [ MC ;

Y D M \ M C :

Let us fix a tubular neighbourhood of Y in M that we identify with Y  .1; 1/ via charts that map a neighbourhood V of a point y 2 Y to   .1; 1/,  Rq open, q C 1 WD dim M , and such that V \ M˙ is mapped to   Œ0; 1/ and   .1; 0 , respectively, and V \ Y to . It will be not essential to admit a maximal atlas on M ; therefore, for convenience, we assume that the transition maps of the latter charts near Y do not depend on the normal variable t for jt j < 12 . z Fz / of pseudo-differen.M I E; Recall from Section 2.1.3 that we have the space L .cl/ tial operators of order  on M , operating between spaces of distributional sections in (smooth complex) vector bundles Ez and Fz over M of fibre dimension n and m, respectively.

3.1 Interior and boundary symbols

175

z z Definition 3.1.18. The space L cl .M I E; F /tr of pseudo-differential operators of order  2 Z with the transmission property at Y is defined to be the subspace of all A 2 z z L cl .M I E; F / such that for every chart  W V ! U D   R near a point of Y the symbol a.x; Q / WD e f'. A/ ge with e WD e ix is an m  n matrix of elements in Scl .  R  RqC1 /tr , for arbitrary '; 2 C01 .  R/ (cf. also Theorem 2.1.16). Moreover, if X is a C 1 manifold with boundary and M WD 2X the double of X (obtained by gluing together two copies X˙ of X along the common boundary, with X being identified with XC ), z X ; F WD Fz jX , we set E WD Ej  C z C z z z L cl .X I E; F /tr WD fr Ae W A 2 Lcl .M I E; F /tr g;

(3.1.16)

where eC denotes the operator of extension by zero from int X to M and r C the restriction to int X . Example 3.1.19. For every E 2 Vect.X /,  2 Z, there exists an operator  2 L RE cl .X I E; E/tr  z  2 L z z z z eC for an elliptic operator R of the form RE D rC R cl .2X I E; E/ and E 2 z z E E z X , such that the symbols in local coordinates .y; t / 2   .1; 1/ Vect.2X/, E D Ej near Y are given by (3.1.2), tensorised by the identity map in the fibres of E, modulo lower order terms. In Section 4.1.4 we will study operators of that kind in more detail.

By  .A/ we denote the homogeneous principal symbol of A 2 L cl .X I E; F /tr of order  which is a bundle morphism  .A/ W X E ! X F; X W T  X n 0 ! X . Moreover, we have the homogeneous principal boundary symbol @ .A/ defined by @ .A/.y; / WD opC . .A/.y; 0; ; // locally in a collar neighbourhood of @X. It represents a bundle morphism @ .A/ W Y E 0 ˝ H s .RC / ! Y F 0 ˝ H s .RC / for every real s >  12 , where E 0 WD EjY , F 0 WD F jY , Y W T  Y n 0 ! Y . x C / !   F 0 ˝ S.R x C /. Alternatively, we can write @ .A/ W Y E 0 ˝ S.R Y z z Remark 3.1.20. The space L cl .M I E; F / is a Fréchet space in a natural way (cf.  z Fz /tr is a closed subspace. Moreover, the space Remark 2.1.17), and Lcl .M I E; C z C z z fAz 2 L cl .M I E; F /tr W r Ae D 0g

(3.1.17)

  z z z z is closed in L cl .M I E; F /tr , and we have Lcl .XI E; F /tr D Lcl .M I E; F /= , where =  indicates the quotient space with respect to (3.1.17). In this way the space L cl .XI E; F /tr becomes a Fréchet space in a natural way.

176

3 Boundary value problems with the transmission property

Let X be a C 1 manifold with boundary, embedded in a C 1 manifold M (of the same dimension); then we set ˇ ˇ s s s s z ˇ z ˇ H.comp/ .int X; E/ WD Hcomp .M; E/ ; H.loc/ .int X; E/ WD Hloc .M; E/ int X int X (3.1.18) ˇ for Ez 2 Vect.M /, E WD Ez ˇint X . Theorem 3.1.21. Every A 2 L cl .X I E; F /tr induces continuous operators s s A W H.comp/ .int X; E/ ! H.loc/ .int X; F /

for all s 2 R, s >  12 . Proof. Let !; !; Q !QQ 2 C 1 .X / be functions supported in a collar neighbourhood of Y , all identically 1 in a smaller collar neighbourhood of Y , such that !Q 1 on QQ Then A can be written as A D !A!Q C .1  !/A.1  !/ QQ C C supp !; ! 1 on supp !. with an operator C that has a smooth kernel up to the boundary. The continuity of s C W H.comp/ .int X; E/ ! C 1 .X; F / for s >  12 is then evident, while the continuity QQ W H s of .1  !/A.1  !/ .int X; E/ ! H s .int X; F / for all s is a consequence .comp/

.loc/

s s of Theorem 2.1.20 (i). The continuity of !A!Q W H.comp/ .int X; E/ ! H.loc/ .int X; F / 1 for s >  2 follows from Corollary 3.1.17 by a simple partition of unity argument.

3.1.3 Green operators on the half-axis Definition 3.1.22. (i) A Green operator of type 0 on the half-axis is a continuous x C / such that the adjoint G  (with respect to the L2 .RC /operator G W L2 .RC / ! S.R x C /. scalar product) defines a continuous operator G  W L2 .RC / ! S.R (ii) A Green operator of type d 2 N on the half-axis is a linear combination P dj G D jd D0 Gj dt j with arbitrary Green operators Gj of type 0. Let  d .RC / denote the space of all Green operators of type d. Theorem 3.1.23.  0 .RC / consists of the space of all operators Z 1 Gu.t / D g.t; t 0 /u.t 0 /dt 0 0

xC  R x C /.D S.R  R/j x x /. with kernel g.t; t 0 / 2 S.R RC RC This result was proved before as Theorem 2.4.87. Definition 3.1.24. (i) A trace operator is an element T 2 R 1 of 0type 00 on0 the half-axis 2 0 x C /. A trace L.L .RC /; C/ of the form T u D 0 b.t /u.t /dt for any b.t / 2 S.R

177

3.1 Interior and boundary symbols

P dj operator of type d 2 N on the half-axis is a linear combination T D jd D0 Tj dt j with arbitrary trace operators Tj of type 0. (ii) A potential operator on the half-axis is an element K 2 L.C; L2 .RC // of the x C /. form Kc.t/ D ck.t / for any k.t / 2 S.R Proposition 3.1.25. (i) Every Green operator G of type d has a unique representation of the form d1 X Kj ı  j G D G0 C j D0

 dj  0 with  j u WD dt j u .0/, potential operators Kj , and an operator G0 2  .RC /. (ii) Every T of type d has a unique representation of the form P trace operator j T D T0 C jd1 c  with constants cj 2 C, and a trace operator T0 of type 0. j D0 Proof. For (i) we first consider Gu.t / D x C /, cf. Theorem 3.1.23. From R

R1 0

xC  g.t; t 0 / dtd 0j u.t 0 /dt 0 , g.t; t 0 / 2 S.R j

  d  d j 1 j 1 j d  0 d 0 0 0 0 d g.t; t / u.t / D g.t; t / u.t / C g.t; t / u.t 0 / dt 0 dt 0j 1 dt 0 dt 0j 1 dt 0j  j 1   j 1 R1 it follows that Gu.t / D g.t; 0/ dtd 0j 1 u .0/  0 dtd 0 g.t; t 0 / dtd 0j 1 u.t 0 /dt 0 . By iterating this procedure we see that G has the asserted form. Conversely, if an operator G is given by the expression ˇ  d j 1 0 ˇ u.t / t 0 D0 dt 0j 1

Gu.t / D k.t /

(3.1.19)

x C / such that '.0/ D 1 and obtain x C / we choose any ' 2 S.R for some k 2 S.R Z Z 1 j j 1 d d j 1 0 d 0 0 0 d u.0/ D '.t / u.t /dt  '.t / u.t 0 /dt 0 : 0 0j 1 dt 0j dt dt dt 0j 1 0 Multiplying this identity by k.t / yields a reformulation of G as an operator as in Definition 3.1.22 (i). In order to show the uniqueness of the representation (3.1.19) we first observe that G0 is uniquely determined by Gu for all u 2 C01 .RC /. Then, P Pd1 j j z considering G  G0 it is enough to show that jd1 D0 Kj ı  u D j D0 Kj ı  u for x C / implies Kj D Kzj for all j . But this is trivial when we insert special all u 2 S.R elements of the form u.t / D t k !.t/, k D 0; : : : ; d  1, for a cut-off function !, which successively gives us K0 D Kz0 ; K1 D Kz1 , etc. The second part of Proposition 3.1.25 can be proved in an analogous manner.  K d x Let BG .RC I v/ with v WD .m; nI j ; jC / denote the space of all G D G T Q , where G is an n  m matrix of Green operators of type d, T an jC  m matrix of trace operators of type d, K an n  j matrix of potential operators, and Q a jC  j matrix of complex numbers.

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3 Boundary value problems with the transmission property

d x Remark 3.1.26. Operators G 2 BG .RC I v/ with v D .m; nI j ; jC / induce conm s x C ; C n / ˚ C jC for arbitrary s 2 R, tinuous maps G W H .RC ; C / ˚ C j ! S.R 1 s > d  2.

Proposition 3.1.27. The composition of operators defines a map d x e x e x BG .RC I v/  BG .RC I w/ ! BG .RC I v ı w/

for every d; e 2 N, and v D .k; nI l; jC /, w D .m; kI j ; l/ with v ıw D .m; nI j ; jC /. The elementary proof is left to the reader. Example 3.1.28. (i) Special cases of Proposition 3.1.27 are the following relations. Let K be a potential operator and T a trace operator of type d; then G WD K ı T is a Green operator of type d. Moreover, Q WD T ı K is of the type of a lower right corner. (ii) Let a. / 2 Scl .R/tr and apply r C op.a/ to cı0 , with the Dirac distribution ı0 at 0 and c 2 C. Then the map K W c ! r C op.a/.cı0 / defines a potential operator. In fact, using the representation (3.1.8) we can write a. / D aC . / C a . / C p. / with a polynomial p. / of order  (as soon as   0) and symbols a˙ . / 2 H0˙ . Now we have r C op.p/ı0 D 0, while 1 x ˙ /; op.a˙ /ı0 D F !t .a˙ . /ıO0 . // 2 S.R

see the characterisation of the spaces H0˙ in Section 3.1.1. Thus x C/ r C op.a/ı0 D r C op.aC /ı0 2 S.R x C/ (recall that r C is the restriction to the open half-axis RC , while the notation S.R means S.R/jR x C ).  K d x Theorem 3.1.29. Let G D G T Q 2 BG .RC I v/ with v D .m; mI j; j /, and assume that x C; Cm/ x C; Cm/   S.R S.R 1CG K W ! (3.1.20) ˚ ˚ T Q j j C C  1   K L is an invertible operator. Then we have 1CG for an element D 1CH T Q B R H L d x 2 B .RC I v/. B R

G

Proof. First note that (3.1.20) is invertible if and only if 

1CG T

 H s .RC ; C m / H s .RC ; C m / K ˚ ˚ W ! Q Cj Cj

179

3.1 Interior and boundary symbols

is invertible for any fixed real s > d  12 . There is a neighbourhood U of Q in the   K space of j  j matrices such that 1CG is invertible for all R 2 U . It follows that T R also     1  1CG K 1 0 1CG K DW T Q D J T R is invertible; this entails the invertibility of J . We obtain  1   1CG K 1 0 1CG D T Q J 1 D J 1 T

K R

1 :

(3.1.21)

  K 1 . This follows from the identity Thus we have to compute 1CG T R      1 C G  KR1 T 1CG K 1 0 1 KR1 1 1 D 0 1 T R R T R 0

0 1



which yields that 1 C G  KR1 T is invertible, and     1  .1 C G  KR1 T /1 0 1 KR1 1 0 1CG K : D 0 1 0 1 R1 T R1 T R (3.1.22) 1 T . By virtue of Thus it remains to characterise .1  C /1 for C WD G  KRP j Proposition 3.1.25 the operator C can be written as C D C0 C jd1 D0 Kj ı  with  j u D d tj uj tD0 , certain potential operators Kj and some Green operator C0 of type 0. Since C0 is compact as an operator in Sobolev spaces, we have ind.1  C0 / D 0. We P therefore find an operator of the form lkD1 Dk Bk , where l WD dim ker.1  C0 / and with trace operators Bk of type 0 and potential operators Dk such that 

1  C0 C

l X

 Dk Bk W H s .RC ; C m / ! H s .RC ; C m /

kD1

is invertible. The construction of Dk and Bk employs the fact that ker.1  C0 / is x C ; C m /, l D dim V , and that there is an la finite-dimensional subspace V of S.R m x dimensional subspace W of S.RC ; C / such that im.1  C0 / C W D H s .RC ; C m /. Then we choose isomorphisms t .B1 ; : : : ; Bl / W V ! C l , .D1 ; : : : ; Dl / W C l ! W of the desired structure and interpret t .B1 ; : : : ; Bl / as an extension to H s .RC ; C m / by composing it from the right with a projection H s .RC ; C m / ! V induced by the orthogonal projection L2 .RC ; C m / ! V . Let us write 

1  C D 1  C0 C

l X kD1



Dk Bk C

l X kD1

Dk Bk 

d1 X j D0

Kj ı  D 1  C1 C j

lCd X

Dk Bk

kD1

(3.1.23) P for C1 WD C0 C lkD1 Dk Bk which is Green and of type 0 and DlCj C1 WD Kj , BlCj C1 WD  j for j D 0; : : : ; d  1. We now employ the fact that there exists a Green

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3 Boundary value problems with the transmission property

operator C2 of type 0 such that .1  C1 /1 D 1  C2 . It suffices to verify the latter assertion for C1 interpreted as an operator in L.L2 .RC ; C m //. It is characterised by the property that x C; Cm/ C1 ; C1 W L2 .RC ; C m / ! S.R (3.1.24) are continuous, cf. Definition 3.1.22 (i). Thus, from the identity .1  C1 /.1  C2 / D 1  C1  C2 C C1 C2 D 1 it follows that C2 D C1 .1  C2 /. The first relation (3.1.24) together with 1C2 2 L.L2 .RC ; C m // gives us the continuity of C2 W L2 .RC ; C m / ! x C ; C m /. Moreover, .1  C2 /.1  C1 / D 1 implies .1  C  /.1  C  / D 1, and S.R 1 2 we thus obtain in an analogous manner also the continuity of C2 W L2 .RC ; C m / ! x C ; C m /. In other words, C2 is as desired. S.R In order to compute .1  C /1 we multiply (3.1.23) from the left by 1  C2 which gives us lCd X .1  C2 /.1  C / D 1 C Mk Bk (3.1.25) kD1

with potential operators Mk WD .1  C2 /Dk . This reduces the task to invert (3.1.25). Let us write the right-hand side of (3.1.25) as 1CMB W H s .RC ; C m / ! H s .RC ; C m / with M WD .M1 ; : : : ; MlCd /, B WD t .B1 ; : : : ; BlCd /. Then we can write .1 C MB/1 D 1  M.1 C BM/1 B;

(3.1.26)

lCd ! C lCd is using the fact that 1 C MB is invertible if an only if 1 C BM  W C  1 0 1 M , B WD invertible. In order to see the latter property we write M WD , B 1 0 1  1 M  F WD B and observe that MFB D diag.1 C MB; 1/, BFM D diag.1; 1 C 1 BM/. Since M and B are invertible, this shows that 1 C MB and 1 C BM are invertible at the same time, and a simple computation gives us (3.1.26). The operator L1 WD M.1 C BM/1 B is Green of type d, and (3.1.25) yields .1  C /1 D .1  L1 /.1  C2 / D 1  L2 , where L2 WD L1  L1 C2 C C2 is again Green and of type d. We now insert this in the matrix in the middle on the right-hand side of (3.1.22) and employ the result in (3.1.21). In the various compositions we repeatedly used Proposition 3.1.27.

3.1.4 Boundary value problems on the half-axis In this section we define a space of boundary value problems on the half-axis. Let us set SclI0 .R  R/tr WD S I0 .R  R/ \ Scl .R  R/tr ; ˇ x ˙  R/tr WD faˇ x cf. Definition 2.3.1 and 3.1.5, and SclI0 . R W a 2 SclI0 .R  R/tr g. R˙ R I0 x In particular, the symbols in Scl .RC  R/ have coefficients in the symbol space x C / D fcj x W c 2 S 0 .R/g with respect to the variable t . S 0 .R RC

3.1 Interior and boundary symbols

181

x C I v/ for  2 Z; d 2 N and v D .m; nI j ; jC / denote Definition 3.1.30. Let B ;d .R   d x the space of all operator matrices A D A0 00 C G for arbitrary G 2 BG .RC I v/ and C C C an n  m matrix A of operators of the form op .p/ WD r op.p/e with symbols x C  R/tr . p.t; / 2 SclI0 .R x C I v/ induces continuous operators Theorem 3.1.31. An A 2 B ;d .R A W H s .RC ; C m / ˚ C j ! H s .RC ; C n / ˚ C jC for all s 2 R, s > d  12 , and x C ; C m / ˚ C j ! S.R x C ; C n / ˚ C jC : A W S.R

(3.1.27)

Proof. The continuity of the upper left corner A is part of the technicalities for Theorem 3.1.15, while the assertion for the Green part G is contained in Remark 3.1.26. Theorem 3.1.32. The composition of operators defines a map x C I v/  B ;e .R x C I w/ ! B C;h .R x C I v ı w/ B ;d .R for h D max. C d; e/, and for v D .k; nI l; jC /; w D .m; kI j ; l/ with v ı w D .m; nI j ; jC /. Proof. It is enough to consider the operators in the sense of (3.1.27). The main aspect of the proof is that opC .a/ opC .b/ D opC .a # b/ C G x C  R/tr , b.t; / 2 S I0 .R x C  R/tr , with a Green operator for every a.t; / 2 SclI0 .R cl max.;0/ G2 . By virtue of Remark 3.1.10 we have representations (3.1.28) a.t; / D a0 .t; / C a1 .t; /; b.t; / D b0 .t; / C b1 .t; /; P P x C  R/tr and a1 .t; / D  ˛i .t / i ; b1 .t; / D  ˇj .t / j , .R a0 ; b0 2 Scl1I0 j D0 iD0 ˇ x C / .D S 0 .R/ˇ x / for all i; j . Without loss of generality we assume ˛i ; ˇj 2 S 0 .R RC ;  2 N; otherwise a1 or b1 vanish. We have X opC .a/ opC .b/ D opC .aj / opC .bk /: (3.1.29) j;kD0;1

The operators for j D k D 1 are differential operators of order  and , respectively, x C /. Thus opC .a1 / opC .b1 / is a differential operator on the with coefficients in S 0 .R x C /. Moreover, we have half-axis of order  C , with coefficients in S 0 .R opC .a0 / opC .b1 / D

 X

.opC .a0 /ˇj /D tj ; opC .a1 / opC .b0 / D opC

j D0

 X iD0

 ˛i D ti b0 :

182

3 Boundary value problems with the transmission property

x C / which shows that By virtue of Theorem 3.1.31 we have opC .a0 /ˇj 2 S 0 .R C x C /. op .a0 / op .b1 / is a differential of order  with coefficients in S 0 .R P operator 1I0 x i Furthermore, we have that iD0 ˛i .t /D t b0 .t; / 2 Scl .RC  R/tr , that is, also opC .a1 / opC .b0 / is of the desired quality. Thus it remains to consider the summand on the right-hand side of (3.1.29) for j D k D 0. This can be written in the form C

opC .a0 / opC .b0 / D r C op.a0 / op.b0 /eC C r C op.a0 /  op.b0 /eC

(3.1.30)

with the characteristic function   of the negative half-axis R . We have r C op.a0 / op.b0 /eC D r C op.aQ 0 # bQ0 /eC Q x D b0 . for symbols aQ 0 ; bQ0 2 Scl1I0 .R  R/tr such that aQ 0 jR x C D a0 , b0 jR C So we have to consider the second summand on the right of (3.1.30). Assume for the moment that both a0 and b0 are independent of t . Let " W R˙ ! R be defined by ".t/ D t. Then we can write r C op.a0 /  op.b0 /eC D .r C op.a0 /e " /." r  op.b0 /eC /

(3.1.31)

with the operators r C op.a0 /e " v.t / D

Z Z

0

C

" r op.b0 /e u.t / D

0 /

a0 . /v.t 0 /dt 0 μ

e i.tCt

0 /

a0 . /v.t 0 /dt 0 μ ;

ZR Z1 1

D  

e i.tt

ZR Z0 1 R

x C /. The functions for u; v 2 S.R Z 0 g.t; t 0 / WD e i.tCt / a0 . /μ j t;t 0 0 ;

e i.tCt

0 /

b0 . /u.t 0 /dt 0 μ

0

h.t; t 0 / WD

Z

e i.tCt

0 /

b0 . /μ j t;t 0 0

R x C R x C /. In fact, because of a0 . / 2 H C CH  we have e ir a0 . /μ 2 belong to S.R 0 0 R x C / C e S.R x  / which gives us e ir a0 . /μ jr 0 2 S.R x C / and hence the deeC S.R 0 sired property of g.t; t /. In a similar manner we can argue for h.t; t 0 /. It follows that xC  R x C /. Those (3.1.31) is the composition of integral operators with kernels in S.R 0 are Green operators in  , and the product of such operators is again of this kind. The kernel is equal to Z 1 0 xC  R x C /: g.t; t 00 /h.t 00 ; t 0 /dt 00 2 S.R (3.1.32) l.t; t / D 0

A straightforward modification of the arguments gives us the same when a0 .t; / 2 x C  R/tr , b0 .t; / 2 S 1I0 .R x C  R/tr . Let us now assume that A WD opC .a/ Scl1I0 .R cl P I0 x for a.t; / 2 Scl .RC  R/tr and G 2  e , G D je D0 Gj d tj for Gj 2  0 and

3.1 Interior and boundary symbols

183

R1 xC  R x C /. Then, decomposing the Gj u.t/ D 0 gj .t; t 0 /u.t 0 /dt 0 for gj .t; t 0 / 2 S.R C 0 symbol as in (3.1.28) it is evident that op .a1 /Gj 2  for all j , i.e., opC .a1 /G 2  e . xC  R x C / which can easily be verified using Moreover, opC .a0 /Gj has a kernel in S.R x C / ! S.R x C /, together with a tensor product argument the continuity of opC .a0 / W S.R xC  R x C / with S.R x C/ ˝ x C /. This gives us altogether AG 2 y  S.R that identifies S.R e d x C  R/tr  . Let us also check that G 2  and B WD opC .b/ for b.t; / 2 SclI0 .R Pd j Cd (say, for  2 N) entails GB 2  . In fact, writing G D j D0 Gj d t for Gj 2  0 and writing B D B0 C B1 with Bj D opC .bj /, j D 0; 1, cf. the relation (3.1.28), Pd j C then it is clear that GB1 2  Cd , while GB0 D j D1 Gj op .d t b0 / induces a x C /. The same is true of the adjoint. By virtue continuous operator L2 .RC / ! S.R of Theorem 3.1.23 this implies GB0 2  0 . The other compositions occurring in the statement of the theorem are easy as well and left to the reader. x C I v/, where v D .m; nI j ; jC /; then for the Theorem 3.1.33. Let A 2 B 0;0 .R adjoint A , defined by .Au; v/L2 .R ;C n /˚C jC D .u; A v/L2 .RC ;C m /˚C j for all C x C I v / for u 2 L2 .RC ; C m / ˚ C j ; v 2 L2 .RC ; C n / ˚ C jC , we have A 2 B 0;0 .R  v D .n; mI jC ; j /. The result is straightforward, so we omit the proof. Theorem 3.1.34. Let a. / be an m  m matrix of elements in Scl .R/tr , det a./ . / 6D 0 d x .RC I .m; mI 0; 0//. Then for 6D 0 and det a. / 6D 0 for all 2 R, and let G 2 BG the operator opC .a/ C G W H s .RC ; C m / ! H s .RC ; C m / (3.1.33) is Fredholm for every real s > max.; d/  12 , the space V WD ker opC .a/  x C ; C m / is independent of s, and there is a finite-dimensional subspace W  S.R x C ; C m / such that S.R im.opC .a/ C G/ C W D H s .RC ; C m / for every s > max.; d/  12 . The restriction x C ; C m / ! S.R x C; Cm/ opC .a/ C G W S.R

(3.1.34)

is also Fredholm, and kernel and cokernel are as before. Proof. Consider, for convenience, the case m D 1. Setting p. / D a1 . / we obtain a symbol in Scl .R/tr . The Fredholm property of (3.1.33) follows from the fact that opC .p/ is a parametrix of opC .a/ C G. In fact, using Theorem 3.1.32 it follows that opC .p/.opC .a/ C G/ D 1 C C1 C C2 ;

.opC .a/ C G/ opC .p/ D 1 C C3 C C4

for C1 WD opC .p/ opC .a/  opC .pa/ 2  max.;0/ , C2 WD opC .p/G 2  d , C3 WD opC .a/ opC .p/  opC .ap/ 2  max.;0/ , C4 WD G opC .p/ 2  max.d;0/ . In particular, this shows for the left over terms Cl WD C1 C C2 2  max.;d/ , Cr WD

184

3 Boundary value problems with the transmission property

C3 C C4 2  max.d;0/ . Applying the mapping properties of Green operators as in Remark 3.1.26 it follows that, for instance, Cl W H s .RC / ! H s .RC / is compact for every s > max.; d/  12 ; similarly, Cr is compact in Sobolev spaces of smoothness s > max.d  ; 0/  12 . Thus (3.1.33) is a Fredholm operator. At the same time, x C /. The assertion on the elliptic regularity gives us V D ker.opC .a/ C G/  S.R x C / is cokernel which can be represented by a finite-dimensional subspace W  S.R easy as well. For the formal part of the consideration we refer to Remark 2.1.25, see also [96, Lemma 1.2.94 and the subsequent remarks]. The conclusion about (3.1.34) is left to the reader. Theorem 3.1.35. Let / be an m  m matrix of elements in Scl .R/tr as in Theo Ga.  d x rem 3.1.34, let G D T K Q 2 BG .RC I v/ for v D .m; mI j ; jC /, and assume that 

 H s .RC ; C m / H s .RC ; C m / opC .a/ 0 ˚ ˚ A WD CGW ! 0 0 j C C jC

(3.1.35)

is an invertible operator for some real s > max.; d/  12 . Then (3.1.35) is invertible for all those s, and we have   C op .p/ 0 1 A D CC 0 0 C

.d/ x C I v1 / with v1 D .m; mI for some p. / 2 Scl .R/tr and an element C 2 BG .R C jC ; j /,  WD max.; 0/. In addition (3.1.35) is invertible if and only if A induces an isomorphism

x C ; C m / ˚ C jC : x C ; C m / ˚ C j ! S.R A W S.R Proof. Assume, for simplicity, m D 1; the general case is completely analogous. Applying Theorem 3.1.34 we see that opC .a/ C G W H s .RC / ! H s .RC / is a Fredholm operator for every s > max.; d/  12 , and opC .p/ for p WD a1 is a parametrix. Since (3.1.35) is invertible we have ind.opC .a/ C G/ D jC  j D  ind opC .p/. The operator opC .p/ W H s .RC / ! H s .RC / can be completed by additional entries to an isomorphism  C  op .p/ C P WD W H s .RC / ˚ C gC ! H s .RC / ˚ C g B R where ind opC .p/ D g  gC D j  jC . The operator C can easy be constructed by x C / to a complement of im opC .p/, gC WD using any injective map C W C gC ! S.R C dim coker op .p/. It is then easy to organise a trace operator B (of type 0) and a g  gC matrix Q such that P is an isomorphism. In order to invert A it is enough to do so for A ˚ idC N W H s .RC / ˚ C j CN ! H s .RC / ˚ C jC CN for some N 2 N. Similarly, from P we can pass to P ˚ idC M for any M 2 N. There is a choice of M; N such that j˙ C N D g˙ C M . Thus, without loss of generality, we assume

3.1 Interior and boundary symbols

185

j˙ D g˙ (if necessary we denote A ˚ idC N and P ˚ idC M again by A and P , respectively). This allows us to form the composition AP which is of the form   1 0 AP D C C W H s .RC / ˚ C jC ! H s .RC / ˚ C jC 0 0 h x for a C 2 BG .RC I .1; 1; jC ; jC //, h D max.d  ; 0/. The composition AP is invertible, and we can apply Theorem 3.1.29. Combined with Theorem 3.1.32 it follows that n1 0 o1 C 1 x C I .1; 1I jC ; j //: CC 2 B ;.d/ .R A DP 0 0

3.1.5 Operators on an interval The one-dimensional calculus of boundary value problems can be modified in the case of a finite interval I on the real axis. This will play a role later on in Chapter 5 and Chapter 8. Therefore, we formulate here the corresponding material. Let us set I D Œ0;  I of course,  may be replaced by any other real number, e.g., 2 as is used in Chapter 8. For simplicity, we consider the case of scalar operators in the upper left corners and j D jC D 1 for both boundary points; the generalisation to block matrices is straightforward. 1;0 Let BG .I / defined to be the space of all 3  3 block matrix operators g D .gij /i;j D1;2;3 W H s .int I / ˚ C 2 ! C 1 .I / ˚ C 2 ; s >  12 , where g11 is an integral operator with kernel in C 1 .I  I /, g1j c WD f1j ./c R for j D 2; 3, c 2 C, gi1 u WD 0 fi1 ./u./d for i D 2; 3, with arbitrary functions f1j , fi1 2 C 1 .I / for i; j D 2; 3, and an arbitrary 2  2 matrix .gij /i;j D2;3 with entries in C. The components of C 2 are associated with the end points f0g and fg of the interval I . For purposes below we also formulate a parameter-dependent variant, with parame1;0 ter  2 Rl . To this end we observe that BG .I / is a Fréchet space in a natural way (as 1;0 the direct sum of its 9 components), and we set BG .I I Rl / WD S.Rl ; B 1;0 .I //. 1;d l Moreover, for any d 2 N let BG .I I R / denote the space of all operator families g./ WD g0 ./ C

d X

gj ./ diag.@j ; 0; 0/

j D1 1;0 with arbitrary gj 2 BG .I I Rl /; here  is the variable on the interval I . Let us now consider 2  2 block matrix symbols h0 ./ of the class

x C / ˚ C/ Scl .Rl I L2 .RC / ˚ C; S.R

(3.1.36)

186

3 Boundary value problems with the transmission property 1

(with respect to group actions f g2RC of the form  .u./ ˚ c/ !  2 u./ ˚ c x C / ˚ C/ such that also h ./ (the pointwise for u./ ˚ c in L2 .RC / ˚ C or S.R 0 2 adjoint with respect to the L .RC / ˚ C scalar product) belongs to the space (3.1.36). x C , supported by Œ0; "/ for some 0 < " <  and Let !; !Q be cut-off functions on R form the operator family a./ WD !h0 ./!Q W H s .int I / ˚ C ! C 1 .I / ˚ CI

(3.1.37)

x C with f0g 2 I corresponding to the origin in here I is assumed to be embedded in R x RC . In a similar manner we can form !h ./!Q for another symbol h ./ that belongs together with h ./ to (3.1.36).We then obtain operators b./ WD  .!h ./!/ Q W H s .int I / ˚ C ! C 1 .I / ˚ C;

(3.1.38)

where  denotes the operator push forward under the map  W I ! I , defined by ./ WD  C . By definition the direct summands C in the spaces in (3.1.37) belong to f0g 2 I , those of (3.1.38) to the other end point fg 2 I . Writing (3.1.37) and (3.1.38) as a D .aij /i;j D1;2 , b D .bij /i;j D1;2 , we now form 0

a11 C b11 a21 b21

@

a12 a22 0

H s .int I / C 1 .I / b12 ˚ ˚ C C : 0 A ./ W ! ˚ ˚ b22 C C 1

(3.1.39)

More generally, we consider operator families g./ D g0 ./ C

d X

gj ./ diag.@j ; 0; 0/

(3.1.40)

j D1

for any d 2 N, where gj ./ are of the kind (3.1.39), of order   j (with  2 R from (3.1.36)). ;d Definition 3.1.36. The space BG .I I Rl / for  2 R, d 2 N is defined as the set of 1;d all operator functions of the form (3.1.40) and c./ 2 BG .I I Rl /. Moreover, let ;d ;d l BG .I I R / denote the space of upper left corners of BG .I I Rl /. ;d Remark 3.1.37. The space BG .I I Rl / has a natural Fréchet topology. So it is x C  ; B ;d .I I Rl //,  Rq open, or possible to form spaces of the kind C 1 .R G ;d A.U; BG .I I Rl //, U C open.

/ for  2 Z By # 2 R we denote the covariable to  2 I . Let Scl .I  R1Cl #; tr denote the space of all classical symbols of order  2 Z, smooth in  2 I (up to the end points), with the transmission property at @I D f0g [ fg (cf. Definition 3.1.1).

3.2 The algebra of boundary value problems

187

Recall that the transmission property (for instance, at  D 0) of a symbol a.; #; / on the homogeneous components a.j / .; #; / requires that Dk D˛ fa.j / .; #; /  .1/j a.j / .; #; /g vanishes on the set f.; #; / W  D 0; # 2 R n f0g,  D 0g for all k 2 N, ˛ 2 N l and all j 2 N. Let us set opI .a/./u./ WD r op.a/./eu./ Q (3.1.41) / , where a.; Q #; / 2 Scl .R  R1Cl / is any symbol for a.; #; / 2 Scl .I  R1Cl #; tr ˇ with a D aQ ˇI R1Cl , and e denotes the operator of extension by zero to R n .int I /, r the operator of restriction to int I , and “ 0 Q #; /u. 0 /d 0 μ #: op.a/./u./ Q D e i. /# a.; From Theorem 3.1.15 we know that (3.1.41) represents a -dependent family of continuous operators opI .a/./ W H s .int I / ! H s .int I / for every s >  12 . (Since this is independent of the choice of the extension aQ we also write opI .a/.) Definition 3.1.38. The space B ;d .I I Rl / for  2 Z, d 2 N, is defined to be the set of all operator families of the form opI .a/./ C g./ ;d for arbitrary a 2 Scl .I  R1Cl / and g 2 BG .I I Rl /. Moreover, we set #; tr ;d .I I Rl /g: B ;d .I I Rl / WD fdiag.p; 0; 0/ C g W p 2 B ;d .I I Rl /; g 2 BG

Remark 3.1.39. (i) The space B ;d .I I Rl / is Fréchet in a natural way. (ii) B ;d .I I Rl / is a special case of the class of parameter-dependent pseudodifferential boundary value problems that we introduce in Section 3.3.3 below in the case of an arbitrary C 1 manifold X .

3.2 The algebra of boundary value problems Boundary value problems for pseudo-differential operators on a C 1 manifold X with boundary with the transmission property at @X are written as 2  2 block matrix operators. The upper left corner is a sum of a pseudo-differential operator plus a Green operator. The properties of such sums are similar to those of Green functions in solution formulas for classical boundary value problems, such as Dirichlet or Neumann problems. Moreover, the trace operators represent boundary conditions, cf. the formula (1.1.3), while the potential operators are a generalisation of classical Poisson operators, cf. Section 1.1.2. The lower right corners are pseudo-differential operators on @X . The block matrices form a calculus (an ‘algebra’), with a principal symbolic structure . ; @ /, consisting of interior and boundary symbols. We establish here the basic elements of such boundary value problems.

188

3 Boundary value problems with the transmission property

3.2.1 Global smoothing operators Let X be a (not necessarily compact) C 1 manifold with boundary @X and E; F 2 Vect.X/, J˙ 2 Vect.@X /, v WD .E; F I J ; JC /. We define the space B 1;0 .X I v/ of smoothing operators of type 0 C W C01 .X; E/ ˚ C01 .@X; J / ! C 1 .X; F / ˚ C 1 .@X; JC /

(3.2.1)

as follows. For E D F WD X C and J˙ WD @X C the operator (3.2.1) is a 22 block matrix of integral operators with kernels in C 1 .X  X /; C 1 .X  @X /; C 1 .@X  X / and C 1 .@X  @X /, respectively. The identification between operators and kernels is given by integration Z Z 0 c11 .x; x/u. Q x/d Q x; Q v.x / ! c12 .x; xQ 0 /v.xQ 0 /d xQ 0 ; (3.2.2) u.x/ ! X

@X

C01 .X /,

C01 .@X /;

etc., for u 2 v 2 dx and dx 0 denote measures associated with fixed Riemannian metrics on X and @X, respectively. The generalisation to the bundle case is based on kernels that are C 1 sections in the tensor products F ˝ E  ; F ˝ J ; JC ˝ E 

and

JC ˝ J ;

(3.2.3)

respectively, smooth up to the boundary in the respective variables on X , where ‘’ indicates the dual bundles. Instead of (3.2.2) we then have Z Z 0 Q u.x/i Q E d x; Q v.x / ! hc12 .x; xQ 0 /; v.xQ 0 /iJ d xQ 0 ; u.x/ ! hc11 .x; x/; X

@X

C01 .X; E/, 1

C01 .@X; J /. 1

v2 Here h; iE ; h; iJ , denote the fibrewise etc., for u 2 pairings, moreover, C .X; E/; C .@X; J /, etc., denote the spaces of C 1 sections in the respective bundles, while subscripts ‘0’ indicate sections with compact support. 1;d .X I v/, d 2 N, we denote the space of all those operators C D C0 C Pd By B j C diag.D ; 0/ with arbitrary Cj 2 B 1;0 .X I v/ and a differential operator D j D1 j of first order on X having n ˝ idE as the principal symbol in a collar neighbourhood of @X in X, with n being the covariable to xn (where .x 0 ; xn / 2 @X  Œ0; 1/ is a corresponding splitting of variables in that collar neighbourhood). The elements of S B 1;d .X/ WD v B 1;d .X I v/ are called smoothing, of type d. Remark 3.2.1. The space B 1;d .X I v/ is Fréchet in a natural way. We then set

B 1;d .X I vI Rl / WD S.Rl ; B 1;d .X I v//:

Remark 3.2.2. Every C 2 B (3.2.1) to continuous operators

1;d

s 1

.X I v/ with v WD .E; F I J ; JC / extends from

s .int X; E/ ˚ Hcomp2 .@X; J / ! C 1 .X; F / ˚ C 1 .@X; JC / C W H.comp/

for all s > d  12 .

(3.2.4)

(3.2.5)

3.2 The algebra of boundary value problems

189

3.2.2 Green operators Green boundary symbols in the calculus of pseudo-differential boundary value problems with the transmission property are special operator-valued symbols of the kind Scl .U  Rq I H; S/, where H is a Hilbert space and S a Fréchet space, both equipped with group actions. The precise definition is as follows. Let us fix a tuple w WD .k; mI j ; jC / of dimension data, and set H WD L2 .RC ; C k / ˚ C j ; x C ; C m / ˚ C jC D f lim ht iN H N .RC ; C m /g ˚ C jC ; S WD S.R  L WD L .RC ; C / ˚ C 2

m

N 2N jC

x C ; C k / ˚ C j ; T WD S.R 1

endowed with the group action diagf ; idg2RC for . u/.t / D  2 u.t /;  2 RC . ;0 .U  Rq I w/ for w WD .k; mI j ; jC /, U Rp Definition 3.2.3. The space RG open, is defined as the space of all operator functions g.y; / W L2 .RC ; C k / ˚ C j ! L2 .RC ; C m / ˚ C jC such that 1

1

g0 .y; / WD diag.1; hi 2 /g.y; / diag.1; hi 2 /

(3.2.6)

has the properties g0 .y; / 2 Scl .U  Rq I H; S /; g0 .y; / 2 Scl .U  Rq I L; T /, for the pointwise adjoint g0 with respect to the scalar products of L2 .RC ; C m / ˚ C jC and L2 .RC ; C k / ˚ C j , respectively. ;d Moreover, RG .U  Rq I w/, d 2 N, is defined to be the space of all operator   P functions of the form g.y; / D g0 .y; / C jd D1 gj .y; / diag @jt ; 0 for arbitrary j;0 gj 2 RG .U  Rq I w/; j D 0; : : : ; d. ;d The elements of RG .U  Rq I w/ are called Green symbols of order  and type d. ;d .U  Rq I w/ we have Remark 3.2.4. For every g.y; / 2 RG 1

1

diag.1; hi 2 /g.y; / diag.1; hi 2 / x C ; C m / ˚ C jC / 2 Scl .U  Rq I H s .RC ; C k / ˚ C j ; S.R

(3.2.7)

for arbitrary s > d  12 . ;d .  Rq I w/,  Rq open, w WD Theorem 3.2.5. For every g.y; / 2 RG .k; mI j ; jC /, the associated operator G WD Opy .g/ induces continuous operators s 1

s G W Hcomp.y/ .  RC ; C k / ˚ Hcomp2 .; C j / s 1 2

s ! Hloc.y/ .  RC ; C m / ˚ Hloc

for all s 2 R; s > d  12 .

.; C jC /

190

3 Boundary value problems with the transmission property 1

1

Proof. We have diag.1; hi 2 /g.y; / diag.1; hi 2 / 2 Scl .  Rq I H s .RC ; C k / ˚ C j , H s .RC ; C m / ˚ C jC / (which is more crude than (3.2.7) but sufficient for the proof), and then s 1

s .; H s .RC ; C k // ˚ Hcomp2 .; C j // G W Wcomp s 1 2

s ! Wloc .; H s .RC ; C m // ˚ Hloc

.; C jC /

s .; H s .RC ; C k // D is continuous for s > d  12 , cf. Corollary 2.3.44. Using Wcomp s Hcomp.y/ .  RC ; C k / and, similarly, with subscripts ‘loc’, we obtain the assertion.

By definition, (3.2.6) has a homogeneous principal component g0;./ .y; / of or1 der , cf. Definition 2.2.3 and Remark 2.2.5. For g.y; / D diag.1; hi 2 /g0 .y; /  1 diag.1; hi 2 / (see the formula (3.2.6)) we set 1

1

@ .g/.y; / WD diag.1; jj 2 /g0;./ .y; / diag.1; jj 2 /; or @ .G /.y; / WD @ .g/.y; /

(3.2.8)

when G D Opy .g/. We call (3.2.8) the homogeneous principal boundary symbol of the operator G , interpreted as a family of continuous operators @ .G /.y; / W H s .RC ; C k / ˚ C j ! H s .RC ; C m / ˚ C jC ; x C ; C k / ˚ C j ! S.R x C ; C m / ˚ C jC , s > d  12 , or, alternatively, @ .G /.y; / W S.R q .y; / 2   .R n f0g/. Now let X be a C 1 manifold with boundary @X (not necessarily compact), E; F 2 Vect.X/, J˙ 2 Vect.@X /, and consider trivialisations x C  Ck ; EjV Š   R

x C  Cm; F jV Š   R

J˙ jV 0 Š   C j˙

(3.2.9) (3.2.10)

on a coordinate neighbourhood V on X near the boundary @X, such that V 0 WD V \ x C and 0 WD jV 0 W V 0 !  be corresponding charts @X 6D ;. Let  W V !   R on X and @X , respectively. Operators G D Opy .g/ are interpreted for the moment as operators x C ; C k / ˚ C01 .; C j / ! C 1 .  R x C ; C m / ˚ C 1 .; C jC /: G W C01 .  R Those can be pulled back to V with respect to the mappings (3.2.9), (3.2.10) as operators GV W C01 .V; EjV / ˚ C01 .V 0 ; J jV 0 / ! C 1 .V; F jV / ˚ C 1 .V 0 ; JC jV 0 /: Let us simply write GV D .1 / Op.g/; the pull back also refers to (3.2.9) and (3.2.10). We now fix a locally finite system fV g2I of such coordinate neighbourhoods

3.2 The algebra of boundary value problems

191

on X near @X , such that fV0 g2I is an open covering of @X . Choose functions ' ;  2 P C01 .V /,  2 I , such that L 2I '

ˇ 1 in a0 collarˇneighbourhood of @X , and  1 0 on supp ' for all , and set ' WD ' ˇV 0 ;  WD  ˇV 0 . 



;d .X I v/ BG

for v WD .E; F I J ; JC /,  2 R, d 2 N, is Definition 3.2.6. The space defined to be the set of all operators X 0 G WD diag.' ; '0 /.1  / Op.g / diag.  ;  / C C 2I ;d with arbitrary g .y; / 2 RG .  Rq I w/, w WD .k; mI j ; jC /,  2 I , and C 2 1;d B .XI v/. ;d The operators in BG .X I v/ are called Green operators of order  and type d (in the calculus of boundary value problems on the X with smooth boundary), S manifold ;d ;d and with the bundle data v. Let BG .X / WD v BG .X I v/. Moreover, let ;d BG .X I E; F /

(3.2.11)

;d and BG .X/ denote the corresponding spaces of upper left corners.

From the definition it follows that Green operators have the form of block matrices   G K G DW : (3.2.12) T Q ;d The upper left corner G 2 BG .X I E; F / will also be called a Green operator (of order  and type d) while T is called a trace operator (of order  C 12 and type d), and K a potential operator (of order   12 ). The operator Q belongs to L cl .@X I J ; JC / (cf. the notation (2.1.20)). ;d .X I v/, v D .E; F I J ; JC /. Then G is continuous as Theorem 3.2.7. Let G 2 BG 1 an operator G W C0 .X; E/˚C01 .@X; J / ! C 1 .X; F /˚C01 .@X; JC / and extends to a continuous operator s s .int X; E/ H.comp/ .int X; F / H.loc/ ˚ ˚ GW ! s 1 s 1 2 2 Hloc .@X; JC / Hcomp .@X; J /

for every s > d 

1 2

(cf. the notation (3.1.18)).

This result is a consequence of Theorem 3.2.5 and of the continuity of C , cf. the relation (3.2.5). Remark 3.2.8. The operators in (3.2.12) have the property 'G, G', 'K, T ' 2 B 1;d .X/ for every ' 2 C 1 .X /, supp ' \ @X D ;.

192

3 Boundary value problems with the transmission property

;d Remark 3.2.9. (i) Every G 2 BG .X I E; F / has a unique representation

G D G0 C

d1 X

Kj ı Tj

j D0 sj  1

;0 for an element G0 in BG .X I E; F /, potential operators Kj W Hcomp 2 .@X; E 0 / ! s H.loc/ .int X; F /, E 0 WD Ej@X , and trace operators Tj acting as Tj u D @jxn uj@X , where xn is the normal variable to @X (with respect to the given Riemannian metric on X ). (ii) Every trace operator T in the space B ;d .X I .E; F I J ; JC // has a unique representation d1 X T D D0 C Rj ı Tj j D0

.@X I E 0 ; JC /. for a trace operator D0 of type 0, operators Tj as before, and Rj 2 Lj cl This observation is a simple analogue of Proposition 3.1.25. ;d .X I v/ is Fréchet in a natural way. In order to topoloRemark 3.2.10. The space BG gise the space it suffices to do so for the entries. What concerns the case d D 0 the idea is analogous to that of topologising the space of standard pseudo-differential operators on a manifold, cf. the formula (2.1.19) and Remark 2.1.17 (iii). For arbitrary d 2 N we can employ Remark 3.2.9 to directly decompose the spaces of 11- and 21-entries and to take the topologies constructed before.

3.2.3 Boundary value problems Let X be a C 1 manifold with boundary, not necessarily compact. Fix bundle data v WD .E; F I J ; JC /, E; F 2 Vect.X /, J˙ 2 Vect.@X /. Definition 3.2.11. (i) The space B ;d .X I v/ of (pseudo-differential) boundary value problems of order  2 Z and type d 2 N (with the transmission property at @X ) is defined to be the set of all operators   A 0 AD CG (3.2.13) 0 0 ;d for arbitrary A 2SL cl .X I E; F /tr (cf. the formula (3.1.16)) and G 2 BG .X I v/. We ;d ;d set B .X/ WD v B .X I v/. (ii) We define  .A/ WD  .A/, called the (homogeneous principal) interior symbol of A of order , and    .A/ 0 C @ .G /; (3.2.14) @ .A/ WD @ 0 0

3.2 The algebra of boundary value problems

193

called the (homogeneous principal) boundary symbol of A. The pair  .A/ WD . .A/; @ .A// is called the principal symbol of the operator A. The interior symbol is a bundle morphism  .A/ W X E ! X F;

(3.2.15)

X W T  X n 0 ! X . The boundary symbol is a bundle morphism 0 0 1 0 0 1 E ˝ H s .RC / F ˝ H s .RC /  @ A !   @ A; ˚ ˚ @ .A/ W @X (3.2.16) @X J JC ˇ ˇ @X W T  .@X/ n 0 ! @X , E 0 D E ˇ@X ; F 0 D F ˇ@X , for every s > d  12 , or, alternatively, 0 0 1 0 0 1 x C/ x C/ E ˝ S.R F ˝ S.R  @ A !   @ A: @ .A/ W @X ˚ ˚ @X J JC The terminology for the boundary symbols refers to bundles with the corresponding infinite-dimensional fibres. The homogeneity of  .A/ is as usual, that means  .A/.x;  / D   .A/.x; / for all  2 RC , .x; / 2 T  X n 0. For @ .A/ we have     1 0  0  @ .A/.y; / D    (3.2.17) .A/.y; / 1 0 1=2 @ 0  2 for all  2 RC , .y; / 2 T  .@X / n 0. By B ;d .XI E; F / we denote the space of upper left corners of 2  2 block matrices in B ;d .XI v/. Occasionally we write A11 WD ulc A

when A D .Aij /i;j D1;2 2 B ;d .X I v/:

Moreover, let B ;d .X / WD

[

B ;d .X I E; F /:

E;F

Example 3.2.12. Consider a boundary value problem A of the form (1.1.3), and choose 1 1 order reducing isomorphisms Rml W H sml  2 .Y / ! H s 2 .Y / which exist in ml

Lcl

Š

.Y /, according to Theorem 2.1.31. Then we have x v/ diag.1; Rm1 ; : : : ; RmN /A 2 B ;0 .GI

194

3 Boundary value problems with the transmission property

with v WD .C; CI 0; C N /. In other words, the orders of the trace operators in A can be unified, according to the requirements of Definition 3.2.11. Clearly, the entries of A separately belong to operator classes as in Definition 3.2.11, though with individual orders. Therefore, the principal symbolic calculus makes sense without any condition on the orders, but we have to keep in mind those orders. Remark 3.2.13. The following conditions are equivalent: (i) A11 2 B ;d .X I E; F / and 'A11 C01 .int X /;

2 L1 .int X I E; F / for every ';

2

;d .X I E; F /. (ii) A11 2 BG ;d .X I E; F /  L1 .int X I E; F / which gives us In fact, we have an inclusion BG (ii) )(i). Conversely, (i) implies A11 2 L1 .int X I E; F /, but then, representing A11 in the form A11 D A C G as (3.2.13) for an A 2 L cl .X I E; F /tr and G 2 ;d 1 BG .XI E; F /, it follows that A 2 L .X I E; F /tr which yields (i) ) (ii).

Theorem 3.2.14. An A 2 B ;d .X I v/ with v D .E; F I J ; JC / induces continuous operators s 1

s 1 2

s s .int X; E/ ˚ Hcomp2 .@X; J / ! H.loc/ .int X; F / ˚ Hloc A W H.comp/

for all s > d 

1 2

.@X; JC /

(cf. the notation (3.1.18)).

Proof. Writing A in the form (3.2.13), for the operator G we may employ Theorem 3.2.7. For the operator A in the upper left corner it suffices to consider 'A for '; 2 C01 .X /, where ' runs through a partition of unity and through another system of elements of C01 .X / supported in coordinate neighbourhoods (it is enough here to take also from the functions of the partition of unity). For supp ' \ supp D ; we obtain elements in B 1;0 .X I E; F /, and then the desired continuity. For supp '; supp  int X it follows that 'A 2 L1 .int X I E; F /, and then the continuity comes from Remark 2.1.24 (i). It remains the case when supp ' and supp have a non-empty intersection with @X and are contained in a coordinate neighbourhood intersecting the boundary. In this case we have an analogue of Theorem 3.1.15, namely, opC .a/.y; / 2 S  .  Rq I H s .RC ; C k /; H s .RC ; C m // for s >  12 , referring to a symbol a.x; / of 'A in local coordinates, together with trivialisations x C  C m and   R x C  C k of E and F , respectively, over the coordinate R neighbourhood. The desired continuity is then a consequence of Corollary 2.3.44, combined with the relations (3.1.13). Remark 3.2.15. A consequence of Theorem 3.2.14 is that operators A 2 B ;d .X I v/ are also continuous as operators A W C01 .X; E/ ˚ C01 .@X; J / ! C 1 .X; F / ˚ C 1 .@X; JC /; in particular, the mappings preserve the smoothness up to the boundary.

(3.2.18)

195

3.2 The algebra of boundary value problems

We want to introduce the notion of properly supported operators in B ;d .X /. Restricting (3.2.18) to a continuous map C01 .int X; E/ ˚ C01 .@X; J / ! C 1 .int X; F / ˚ C 1 .@X; JC / we have standard Schwartz kernels of the entries that are distributional sections over int X  int X, int X  @X , etc., in the bundles (3.2.3). Write A in the form C C  z r Ae 0 AD CG 0 0 z z with Az 2 L cl .2X I E; F /tr , cf. the formula (3.1.16), with G as in Definition 3.2.6. z z z Choose a properly supported representative Az0 of Az in L cl .2X I E; F /tr such that A0 D 1 z z z A mod L .2X I E; F / (cf. Definition 2.1.14 and Remark 2.1.15). Moreover, let  Rq be an open set, and choose a function !.y; y 0 / 2 C 1 .  / with proper support in    and !.y; y 0 / 1 in an open neighbourhood of diag.  /, cf., analogously, Remark 2.1.15. Consider the operator G in Definition 3.2.6 and set X 0 diag.' ; '0 /.1 G0 WD  / Op.!g / diag.  ;  /: DI

Then we have G D G0 C D for a D 2 B 1;d .X I v/. This allows us to write the operator A in the form A D A0 C C (3.2.19) for some C 2 B 1;d .X I v/ and



r C Az0 eC A0 WD 0

 0 C G0 : 0

(3.2.20)

Let us call (3.2.20) a properly supported element of B ;d .X I v/. Summing up we proved the following result: Proposition 3.2.16. Let X be a C 1 manifold with boundary. Then every A 2 B ;d .XI v/ can be written in the form (3.2.19) for a properly supported element A0 of B ;d .XI v/. Remark 3.2.17. Let A0 2 B ;d .X I v/ be a properly supported operator. Then A0 induces continuous operators C01 .X; E/ ˚ C01 .@X; J / ! C01 .X; F / ˚ C01 .@X; JC /; C 1 .X; E/ ˚ C 1 .@X; J / ! C 1 .X; F / ˚ C 1 .@X; JC / and extends to continuous operators s 1

s 1 2

s s .int X; E/ ˚ Hcomp2 .@X; J / ! H.comp/ .int X; F / ˚ Hcomp H.comp/ s 1 2

s 1 2

s s H.loc/ .int X; E/ ˚ Hloc .@X; J / ! H.loc/ .int X; F / ˚ Hloc

for all s > d  12 .

.@X; JC /;

.@X; JC /

196

3 Boundary value problems with the transmission property

Remark 3.2.18. The space B ;d .X I v/ is Fréchet in a natural way. After Remark 3.2.10 it is enough to define the Fréchet topology of the space B ;d .X I E; F / of upper left corners which is a non-direct sum ;d B ;d .X I E; F / D L cl .X I E; F /tr C BG .X I E; F /;

(3.2.21)

cf. the formulas (3.1.16) and (3.2.11). From Remark 3.1.20, Remark 3.2.10, and Definition 2.1.4 (i) we then obtain (3.2.21) as a Fréchet space. Remark 3.2.19. In the applications below we will often speak about smooth functions in an open set on a manifold or holomorphic functions in an open set in C with values in L .M / for a C 1 manifold M (or in B ;d .X I v/ for a C 1 manifold X with bound.cl/ ary). In order to avoid too voluminous considerations with all the involved systems of semi-norms we will always keep in mind representations of our operator functions as elements in L .M / (or B ;d .X I v//, where the local amplitude functions as well as .cl/ the smoothing operators from the very beginning are C 1 or holomorphically depending functions in the respective variables. These notions of smoothness or holomorphy are quite transparent, and concern exactly the objects of interest. In other words the spaces C 1 .; L .M //; A.C; B ;d .X I v//; .cl/ etc., are well defined. In a similar manner we proceed with other spaces of operators that are defined by local amplitude functions, modulo smoothing operators. Remark 3.2.20. Let Aj 2 B j;d .X I v/, j 2 N, be an arbitrary sequence. Then P there is an A 2 B ;d .X I v/, unique modB 1;d .X I v/, such that A  jND0 Aj 2 P B .N C1/;d .XI v/ for every N 2 N. We write A  j1D0 Aj , called an asymptotic sum of the Aj , j 2 N. Remark 3.2.21. Let A 2 B ;d .X I v/, X compact, and  .A/ D 0. Then the oper1 1 ator A W H s .int X; E/ ˚ H s 2 .@X; J / ! H s .int X; F / ˚ H s 2 .@X; JC / is compact for every s > d  12 . In fact, writing A in the form (3.2.13), from  .A/ D  .A/ D 0 it follows that A 2 L1 .XI E; F /tr . We then obtain @ .A/ D 0, and hence @ .A/ D 0 entails cl 1;d @ .G / D 0. This implies G 2 BG .X I v/, and it follows that A 2 B 1;d .X I v/. Thus A can be intepreted as a continuous operator to 1

H s.1/ .int X; F / ˚ H s.1/ 2 .@X; JC /; combined with the embedding of the latter space to 1

H s .int X; F / ˚ H s 2 .@X; JC /: The compactness of this embedding then yields the compactness of A.

3.2 The algebra of boundary value problems

197

Theorem 3.2.22. Let A 2 B ;d .X I v/, v WD .E0 ; F I J0 ; JC /, and B 2 B ;e .X I w/, w WD .E; E0 I J ; J0 /, and assume that A or B is properly supported. Then AB 2 B C;h .XI v ı w/ for v ı w D .E; F I J ; JC /, h D max. C d; e/, and we have .AB/ D .A/ .B/ (with componentwise multiplication). Proof. Assume, for simplicity, that A DW A C G and B DW B C H only consist of upper left corners and that E; E0 ; F are trivial bundles of fibre dimension 1. The general case is a straightforward generalisation. Then, writing X X ' .B C H /' ' .A C G/' ; B C H D ACG D ;2I

;2I

with a partition of unity .' /2I , subordinate to an open covering of X by coordinate neighbourhoods, it suffices to characterise the compositions between every two such summands. In other words, it is enough to characterise '.A C G/ .B C H /

(3.2.22)

for suitable '; ;  2 C01 .X /. In the case supp ' \ supp \ supp  D ; it is easy to verify that (3.2.22) belongs to B 1;h .X /. To treat the remaining cases, without loss of generality we may assume that supp '; supp , and supp  belong to the same coordinate neighbourhood U . If U \ @X D ; we have a composition of pseudodifferential operators in int X which is standard. So it remains the case when U \@X 6D x C 3 .y; t /. The factor  on ;. Thus we can reduce the task to the case U D Rn1  R the right is not longer essential; so we consider (3.2.22) without  in the half space. Ignoring smoothing factors which behave well under composition (which can easily be checked) we can write '.A C G/ D Opy .opC .a/.y; / C g.y; //; C

.B C H / D Opy .op .b/.y; / C h.y; //;

(3.2.23) (3.2.24)

x n  Rn /tr , b.x; / 2 S  .R x n  Rn /tr and (scalar) Green with symbols a.x; / 2 Scl .R cl C C ;d ;e symbols g.y; / 2 RG .Rn1  Rn1 /, h.y; / 2 RG .Rn1  Rn1 /. x n and Because of the factors '; the symbols a, b have compact support in x 2 R C n1 g; h compact support in y 2 R . It follows that the product of (3.2.23) and (3.2.24) has the form Opy .p.y; // with the Leibniz product     p.y; / D opC .a/.y; / C g.y; / #y opC .b/.y; / C h.y; /

(3.2.25)

given in Theorem 2.2.54. In the present case the parameter can be omitted, and z z D H s .RC /; H z D for the involved Hilbert spaces we have H D H s .RC /; H s.C/ 1=2 H .RC /, with the group actions . u/.t / D  u.t /,  2 RC . If we want to verify that p.y; / has the form p.y; / D c.y; / C l.y; /

(3.2.26)

198

3 Boundary value problems with the transmission property

x n Rn /tr and a Green symbol l.y; / 2 RC;h .Rn1 Rn1 /, for a c.y; / 2 SclC .R C G it is first important to compute the pointwise compositions occurring under the integral of (2.2.50). This was done in the proof of Theorem 3.1.32 which, in particular, explains the type h D max. C d; e/ of the resulting Green symbol l. The computation of (2.2.50) in the case (3.2.25) and the characterisation as (3.2.26) is then elementary and left to the reader. From the asymptotic formula (2.2.51) we easily see the multiplicative behaviour of  and @ under compositions (only the terms for N D 0 contribute to the principal symbols). It may be useful to modify the orders of operators referring to the boundary as follows. Let @X be compact, and J 2 Vect.@X /,  2 R. By virtue of Theorem 2.1.31 there exists an operator RJ 2 Lcl .@X I J; J / that induces isomorphisms RJ W H s .@X; J / ! H s .@X; J / for all s 2 R. Let us set 1

1

B ;d .XI v/ WD fA WD diag.1; RJC2 /A diag.1; RJ2 / W A 2 B ;d .X I v/g 1

(3.2.27)

1

for any choice of order reducing operators RJC2 , RJ2 on @X . Moreover, let B ;d .X / WD

[

B ;d .X I v/:

v

In the case of non-compact @X we define the spaces B ;d .X I v/ in an analogous manner as (3.2.27) to be the set of all A C C with A as before and C 2 B 1;d .X I /, 1

1

by employing (properly supported) elliptic operators RJ 2 2 Lcl2 .@X I J˙ ; J˙ / that ˙ are not necessarily isomorphisms in the Sobolev scales. The space (3.2.27) itself is 1

independent of the choice of RJ 2 . ˙

The symbolic structure in B ;d .X Iv/ is defined in a similar manner as in B ;d .X Iv/. 1

1

For A0 WD diag.1; RJC2 /A diag.1; RJ2 / we simply set  .A0 / WD  .A/; 1   1 @ .A0 / WD diag 1;  .RJC2 //@ .A/ diag.1;  .RJ2 / ;   where  .RJ / W @X J ! @X J; @X W T  .@X / n 0 ! @X , is the homogeneous principal symbol of order  of the classical operator RJ on the boundary. The spaces 0 0 H.loc/ .X; E/ ˚ Hloc .@X; J / for any E 2 Vect.X /; J 2 Vect.@X / will be identified with L2loc .X; E/ ˚ L2loc .@X; J /:

Here, by L2loc we understand the set of all sections u such that 'u is square integrable over the respective space for every compactly supported smooth function '. This

3.2 The algebra of boundary value problems

199

definition refers to fixed Hermitian metrics in the involved vector bundles. In an analogous manner we can define the spaces L2comp .X; E/ ˚ L2comp .@X; J /. We then have non-degenerate sesquilinear pairings L2comp .X; E/ L2loc .X; E/ ˚ ˚  ! C: .; /L2 .X;E /˚L2 .@X;J / W loc loc L2comp .@X; J / L2loc .@X; J / For every A 2 B 0;0 .X I v/, v D .E; F I J ; JC /, we can form the formal adjoint A in the sense of .Af; g/L2

2 loc .X;F /˚Lloc .@X;JC /

D .f; A g/L2

2 loc .X;E /˚Lloc .@X;J /

for arbitrary f 2 C01 .X; E/ ˚ C01 .@X; J /; g 2 C01 .X; F / ˚ C01 .@X; JC /. If @ .A/.y; / is the boundary symbol of an operator A 2 B 0;0 .X I v/, 0 0 1 0 0 1 E ˝ L2 .RC / F ˝ L2 .RC /  @ A !   @ A; ˚ ˚ @ .A/ W @X @X J JC then we can form the adjoint 0 0 1 0 0 1 F ˝ L2 .RC / E ˝ L2 .RC /  @ A !   @ A: ˚ ˚ @ .A/ W @X @X JC J Theorem 3.2.23. A 2 B 0;0 .X I v/ implies A 2 B 0;0 .X I v / for v D .F; EI JC ; J /, and we have .A / D  .A/ D . .A/ ; @ .A/ /. The simple proof is left to the reader.  in Example 3.1.19 which Remark 3.2.24. It can be proved that the operators RE ;0 belong to B .X I E; E/ can be chosen in such a way that they induce isomorphisms  RE W H s .int X; E/ ! H s .int X; E/

(3.2.28)

 1 / 2 B ;0 .X I E; E/, cf. also Thefor all s > C  12 , cf. Section 4.1.4, where .RE orem 3.3.9 below (the type of the resulting inverse of (3.2.28) is 0, not ./C as in the general case). This allows us to reduce every A 2 B Id .X I E; F /, A W H s .int X; E/ ! H s .int X; F / for any s 2 N such that s   2 N to an operator s s A0 WD RF ARE W H 0 .int X; E/ ! H 0 .int X; F /

(3.2.29)

which belongs to B 0;0 .X I E; F /. This gives us a bijective map B ;d .X I E; F / ! B 0;0 .X I E; F /: Combined with reductions of orders on the boundary we can also define bijections B ;d .X I v/ ! B 0;0 .X I v/; respectively, for all  2 Z, d 2 N.

and B ;d .X I v/ ! B 0;0 .X I v/;

200

3 Boundary value problems with the transmission property

3.3 Ellipticity and parametrices Operators A in the calculus of boundary value problems have a principal symbolic structure .A/ D . .A/; @ .A//. The ellipticity of A is defined by the bijectivity of both components. We obtain parametrices of elliptic operators A within the calculus which belong to the inverse .A/1 D . .A/1 ; @ .A/1 /. In addition we study parameter-dependent boundary value problems and parameter-dependent ellipticity.

3.3.1 Elliptic boundary value problems Definition 3.3.1. Let A 2 B ;d .X I v/,  2 Z, d 2 N, v D .E; F I J ; JC / for E; F 2 Vect.X /, J ; JC 2 Vect.@X /. (i) The operator A is called elliptic (of order ) if both  .A/ W X E ! X F; X W T  X n 0 ! X , and 0 0 1 0 0 1 x C/ x C/ E ˝ S.R F ˝ S.R  @ A !   @ A; @ .A/ W @X ˚ ˚ @X J JC

(3.3.1)

(3.3.2)

@X W T  .@X / n 0 ! @X , are isomorphisms; (ii) an operator P 2 B ;e .X I v1 / for some e 2 N, v1 WD .F; EI JC ; J /, is called a parametrix of A, if Cl WD I  P A 2 B 1;dl .X I vl /;

Cr WD I  AP 2 B 1;dr .X I vr / (3.3.3)

for certain dl ; dr 2 N, vl WD .E; EI J ; J /, vr WD .F; F I JC ; JC /, where I denotes corresponding identity operators. (Recall that the compositions in (3.3.3) are well defined when one of the factors is properly supported.) An operator A 2 B ;d .X I v/ is called  -elliptic if (3.3.1) is an isomorphism. Remark 3.3.2. If (3.3.1) is an isomorphism, a bundle morphism (3.3.2) for A 2 B ;d .X; v/ is an isomorphism if and only if (3.2.16) is an isomorphism for any fixed s D s0 2 R, s0 > max.; d/  12 (or, equivalently, for all s > max.; d/  12 ). This is an analogue of the classical Shapiro–Lopatinskij condition, and, in fact, in future we will talk about Shapiro–Lopatinskij ellipticity as soon as such an isomorphism holds. Remark 3.3.3. If A 2 B ;d .X I E; F / is  -elliptic, i.e., (3.3.1) an isomorphism, then we have (3.3.4) E 0 Š F 0:

3.3 Ellipticity and parametrices

201

In fact, by virtue of (3.2.29) it is enough to look at the reduced operator A0 . The interior symbol  .A0 / W X E ! X F is homogeneous of order 0. In the splitting x C near the boundary with the covariables .; / the of coordinates .y; t / 2 Y  R transmission property of A0 implies  .A0 /.y; 0; 0; 1/ D  .A0 /.y; 0; 0; C1/ for all y 2 Y . Homogeneity 0 gives us  .A0 /.y; 0; 0; / D  .A0 /.y; 0; 0; / for all  2 RC but this equality smoothly extends to  D 0. Then  .A0 /.y; 0; 0; 0/ can be interpreted as an isomorphism (3.3.4). Remark 3.3.4. Let A 2 B ;d .X I E; F / be  -elliptic. Then @ .A/.y; / W Ey0 ˝ H s .RC / ! Fy0 ˝ H s .RC /

(3.3.5)

is a family of Fredholm operators for every s > max.; d/  12 , parametrised by .y; / 2 T  .@X / n 0, and ker @ .A/.y; /, coker @ .A/.y; / are independent of s. For ;d every G 2 BG .X I E; F / we have ind @ .A/.y; / D ind @ .A C G/.y; / for all  .y; / 2 T .@X / n 0. Moreover, by virtue of the homogeneity (3.2.17) it follows that ind @ .A/.y; / D ind @ .A/.y; =jj/: Thus it makes sense to interpret (3.3.5) as a family of Fredholm operators parametrised by .y; / 2 S  .@X /. Thus in the case of compact @X we have an index element indS  .@X/ @ .A/ 2 K.S  .@X //; cf. Atiyah and Bott [8], Boutet de Monvel [15], and Section 3.3.4 below. Remark 3.3.5. Let @X be compact, and let A 2 B ;d .X I v/ be elliptic in the sense of Definition 3.3.1. Then for A WD ulc A we have indS  .@X/ @ .A/ D Œ1 JC  Œ1 J ;

(3.3.6)

where 1 W S  .@X / ! @X is the canonical projection. In other words, the ellipticity of A entails the relation indS  .@X/ @ .A/ 2 1 K.@X /:

(3.3.7)

Theorem 3.3.6. Let @X be compact, and let A 2 B ;d .X I E; F / be a  -elliptic operator. Then the following conditions are equivalent: (i) There exists an elliptic element A 2 B ;d .X I v/, v WD .E; F I J ; JC /, with certain J ; JC 2 Vect.@X /, such that A D ulc A. (ii) A satisfies the relation (3.3.7). Proof. (i) ) (ii). By virtue of the ellipticity of A the boundary symbol @ .A/ represents an isomorphism (3.2.16) for any fixed s > d  12 . Let us restrict this to .y; / 2 S  .@X /. From Definition 3.3.30 below, here applied to the compact topological space S  .@X / and the Fredholm family @ .ulc A/, we obtain (3.3.6) and hence (3.3.7).

202

3 Boundary value problems with the transmission property

(ii) ) (i). Let A be  -elliptic. Then, because of Remark 3.3.4, the principal boundary symbol @ .A/ represents a family of Fredholm operators (3.3.5) that we consider for .y; / 2 S  .@X /. By specifying the constructions for Theorem 3.3.29 below we can complete @ .A/ to a (here smooth) family of isomorphisms  Ey0 ˝ H s .RC / Fy0 ˝ H s .RC / @ .A/ @ .K/ @ .A/.y; / WD .y; / W ! ˚ ˚ @ .T / @ .Q/ G;.y;/ GC;.y;/ (3.3.8) by additional entries, denoted by @ .K/; @ .T / and @ .Q/, respectively, because of their future role as boundary symbols of corresponding operators, with smooth vector bundles G ; GC 2 Vect.S  .@X // such that ŒGC  ŒG D indS  .@X/ @ .A/. By (3.3.7) the latter index element belongs to 1 K.@X /. Therefore, there is a trivial bundle C N on S  .@X / such that 

GC ˚ C N Š 1 JC ;

G ˚ C N Š 1 J

for certain J˙ 2 Vect.@X /. Passing from (3.3.8) to @ .A/ ˚ idC N we obtain a corresponding isomorphism with G˙;.y;/ replaced by J˙;y . The only remaining point is to choose @ .K/ and @ .T / as families of potential and trace operators (of type 0) on the half-axis, see the notation of Definition 3.1.24. This is possible similarly as in Theorem 3.1.34, using the characterisation of kernels and cokernels of the corresponding Fredholm operators by Schwartz functions.

3.3.2 Parametrices and inverses Theorem 3.3.7. Let X be a compact C 1 manifold with boundary and let A 2 B ;d .XI v/,  2 Z, d 2 N, v D .E; F I J ; JC /. (i) The operator A is elliptic (of order ) if and only if H s .int X; E/ H s .int X; F / ˚ ˚ AW ! s 1 s 1 2 2 H H .@X; J / .@X; JC /

(3.3.9)

is Fredholm for an s D s0 2 R, s0 > max.; d/  12 . (ii) If A is elliptic, (3.3.9) is a Fredholm operator for all s > max.; d/  dim ker A and dim coker A are independent of s.

1 2

and

(iii) An elliptic operator A 2 B ;d .X I v/ possesses a parametrix P contained in C B ;.d/ .X I v1 /, v1 D .F; EI JC ; J / (%C WD max.%; 0/ for any % 2 R) which can be chosen in such a way that the remainders in the relation (3.3.3) are

3.3 Ellipticity and parametrices

203

projections 1

Cl W H s .int X; E/ ˚ H s 2 .@X; J / ! V; 1

Cr W H s .int X; F / ˚ H s 2 .@X; JC / ! W for all s > max.; d/  12 and are of type dl D max.; d/ and dr D .d  /C , respectively, for V WD ker A  C 1 .X; E/ ˚ C 1 .@X; J / and some finitedimensional subspace W  C 1 .X; F / ˚ C 1 .@X; JC / with W C im A D 1 H s .int X; F / ˚ H s 2 .@X; JC / and W \ im A D f0g, for every s > max.; d/  12 . Proof. Let A 2 B ;d .X I v/ be elliptic. We first show the existence of a parametrix P 2 C B ;.d/ .XI v1 /. The ellipticity is defined by the invertibility of the components of .A/ D . .A/; @ .A//. The inverse  .A/1 is obviously the homogeneous principal symbol of order  belonging an operator P0 2 L cl .X I F; E/tr , cf. Remark 3.1.4. Moreover, from Theorem 3.1.35 we see that @ .A/1 can be expressed within the class C of boundary symbols belonging to an operator P0 2 B ;.d/ .X I v1 /, and we may fix P0 in such a way that ulc P0 is equal to P0 For purposes below we drop the condition that X is compact (otherwise the following precautions on properly supported representatives are unnecessary). By virtue of Proposition 3.2.16 we may assume P0 to be properly supported. Then it follows that I  P0 A 2 B 1;dl .X I vl /, cf. the notation of Definition 3.3.1 (ii). Let C denote a properly supported representative of the latterP operator. Then we can form a properly supported element D 2 B 1;dl .X I vl /, D  j1D0 C j . Thus DP0 DW P is a left parametrix of A. In a similar manner we obtain a right parametrix such that P is a right and a left parametrix. If X is compact, the rest of the assertions of (iii) can be proved by arguments analogously as for Remark 2.1.25, see also [90, Lemma 1.2.94]. The compactness of the remainders Cl and Cr shows the Fredholm property of (3.3.9). We do not prove here the necessity of the ellipticity of A for the Fredholm property and refer to [154, Section 3.1.1, Theorem 7]. Remark 3.3.8. An elliptic operator A 2 B ;d .X I v/ on a C 1 manifold X with boundC ary (not necessarily compact) has a properly supported parametrix P 2 B ;.d/ .X I v1 / with remainders as in (3.3.3) for dl D max.; d/, dr D .d  /C , and we have .P / D .A/1 (with componentwise inverses). The arguments are already contained in the proof of Theorem 3.3.7. Theorem 3.3.9 ([181]). Let X be a compact C 1 manifold with boundary, and assume that A 2 B ;d .X I v/ defines an invertible operator (3.3.9) for any s0 > max.; d/  1 . Then (3.3.9) is invertible for all s > max.; d/  12 , and we have A1 2 2 C B ;.d/ .XI v1 /. Proof. The result is essentially contained in Theorem 3.3.7 (iii); however, the explanation was very short. Therefore, we want to give more details under the extra assumption

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3 Boundary value problems with the transmission property

that A is elliptic. In our applications below the ellipticity will always be known from the context (Theorem 3.3.7 (i) tells us that the invertibility of A implies the ellipticity anyway). In our case we have ind A D 0. Let P be a parametrix of A which is also z ! H where of index zero as an operator P W H z WD H s .int X; F / ˚ H s 12 .@X; JC /; H

1

H WD H s .int X; E/ ˚ H s 2 .@X; J /;

s > max.; d/ 12 fixed. Let V and W be finite-dimensional subspaces of C 1 .X; F /˚ C 1 .@X; JC / and C 1 .X; E/ ˚ C 1 .@X; J /, respectively, such that (similarly as in Theorem 3.3.7 (ii)) V D ker P , W C im P D H . We have N WD dim V D dim W . Form a block matrix operator   P k z ˚ CN ! H ˚ CN ; (3.3.10) P WD WH t 0 z ! C N an operator which is the where k W C N ! W is an isomorphism and t W H z ! V and an isomorphism V ! C N ; the projection is composition of a projection H assumed to be induced by the orthogonal projection L2 .X; F / ˚ L2 .@X; JC / ! V . z ! H is an isomorThe operator (3.3.10) is an isomorphism. Also Pz WD P C k t W H C ;.d/ 1 phism, and we have Pz 2 B .X I v /. Thus Pz A D I  C is an isomorphism for some C 2 B 1;dl .X I vl /, and it suffices to show that .I  C /1 D I  D for C some D 2 B 1;dl .X I vl /, because then A1 D .I  D/Pz 2 B ;.d/ .X I v1 / by Theorem 3.2.21. The construction of D follows by similar algebraic manipulations as in the proof of Theorem 3.1.29, here using Remark 3.2.9 for  D 1. Remark 3.3.10. Let   A AD W H s .int X / ! T

H s .int X / ˚ s 1 2 .@X; JC / H

be an elliptic operator in B ;d .X I v/ for v D .C; CI 0; JC /. Let us write A in the z C with an elliptic differential operator Az 2 L form A D r C Ae cl .2X /, with 2X being the double of X , and eC the operator of extension by zero from int X to 2X and r C z the restriction to int X . Let Pz 2 L cl .2X / be a parametrix of A on 2X , and set C z C P WD r P e . Assume that we find a potential operator 1

K W H s 2 .@X; JC / ! H s .int X / such that u WD Kg solves the problem Au D 0;

TK D g

in the sense that AK D 0; AT D 1, modulo smoothing operators. Then P D .P  K TP K/ is a right parametrix of A.

3.3 Ellipticity and parametrices

205

3.3.3 Parameter-dependent ellipticity We now turn to a parameter-dependent version of the calculus of boundary value problems with parameters  2 Rl . Parameter-dependent smoothing operators have been introduced by the formula (3.2.4). Moreover, Definition 3.2.3 immediately extends to a space ;d RG .U  RqCl I w/ when we simply replace  2 Rq by .; / 2 RqCl . Then, as a generalisation of Definition 3.2.6 the space ;d BG .X I vI Rl / is defined to be the set of all X G ./ WD diag.' ; '0 /.1  / Opy .g /./ diag.  ;

0 /

C C ./

2I ;d .  RqCl I w/,  2 I , and C./ 2 B 1;d .X I vI Rl /. for arbitrary g .y; ; / 2 RG Moreover, there is a straightforward generalisation of (3.1.16) to the parameterdependent case which gives us the space l L cl .X I E; F I R /tr ;

(3.3.11)

based on local symbols a.x; ; / that have the transmission property at t D 0 in the sense of Definition 3.1.1, with  2 Rq being replaced by .; / 2 RqCl . Finally, we have an extension of Definition 3.2.11 to the case with parameters: Definition 3.3.11. The space B ;d .X I vI Rl / for  2 Z, d 2 N, v D .E; F I J ; JC /, is defined as the set of all operator families A./ D diag.A./; 0/ C G ./ for arbitrary ;d A./ 2 (3.3.11) and G ./ 2 BG .X I vI Rl /. By notation, for l D 0 the space ;d l ;d B .XI vI R / coincides with B .X I v/. Similarly as before we set ;d .X I Rl / WD B.G/

[

;d B.G/ .X I vI Rl /;

v

where subscript ‘.G/’ indicates Green or general operators, 1  . Moreover, let ;d ;d B.G/ .XI vI Rl / WD fulc A./ W A./ 2 B.G/ .X I vI Rl /g and ;d ;d .X I Rl / WD fulc A./ W A./ 2 B.G/ .X I Rl /g: B.G/

Remark 3.3.12. (i) The space B ;d .X I vI Rl / is Fréchet in a natural way. (ii) A./ 2 B ;d .X I vI Rl / entails A.0 / 2 B ;d .X I v/ for every 0 2 Rl . (iii) A./ 2 B ;d .X I vI Rl / entails D˛ A./ 2 B j˛j;d .X I vI Rl / for every ˛ 2 N l . (iv) Let Aj ./ 2 B j;d .X I vI Rl /, j 2 N, be an arbitrary sequence. Then there P is an A./ 2 B ;d .X I vI Rl / with A./  jND0 Aj ./ 2 B .N C1/;d .X I vI Rl / for 1;d every .X I vI Rl /. We then write A./  P1 N 2 N, and A./ is unique mod B j D0 Aj ./, called an asymptotic sum.

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3 Boundary value problems with the transmission property

The elements A./ 2 B ;d .X I vI Rl / have a parameter-dependent principal symbolic structure consisting of the interior symbol (3.2.15), in this case with the projection X W .T  X  Rl / n 0 ! X

(3.3.12)

(where 0 has the meaning of . ; / D 0, cf., analogously, Definition 2.1.28) and the boundary symbol (3.2.16), with the projection @X W .T  .@X /  Rl / n 0 ! @X

(3.3.13)

(where 0 stands for .; / D 0; this is analogous to (3.2.14) where  is replaced by .; /). If necessary, in the parameter-dependent case we also write  .A/ WD .

; .A/;

@; .A//

(3.3.14)

for the pair of parameter-dependent principal symbols. Remark 3.3.13. Let A./ 2 B ;d .X I vI Rl /; then for every fixed 0 2 Rl the symbol .A.0 // D . .A.0 //; @ .A.0 /// is independent of 0 . Remark 3.3.14. Every A./ 2 B ;d .X I vI Rl / can be written in the form A./ D A0 ./ C C./ for an A0 ./ 2 B ;d .X I vI Rl / that is properly supported (according to the construction of (3.2.20) the supports of the involved distributional kernels which may be chosen independently of  2 Rl ) and C ./ 2 B 1;d .X I vI Rl /. Theorem 3.3.15. Let A 2 B ;d .X I vI Rl / and B 2 B ;e .X I wI Rl / with bundle data v; w as in Theorem 3.2.22, and let A or B be properly supported. Then we have AB 2 B C;h .X I v ı wI Rl / with h D max. C d; e/, and  .AB/ D  .A/ .B/. The arguments for the proof are practically the same as for Theorem 3.2.22. Definition 3.3.16. An A./ 2 B ;d .X I vI Rl / is called parameter-dependent elliptic (of order ), if both (3.3.1) and (3.3.2) are isomorphisms; the projections here have the meaning of (3.3.12) and (3.3.13), respectively. Theorem 3.3.17. Let X be a compact C 1 manifold with boundary, and let A./ 2 B ;d .XI vI Rl / be parameter-dependent elliptic (of order ). Then (i)

H s .int X; F / H s .int X; E/ ˚ ˚ ! A./ W 1 1 H s 2 .@X; J / H s 2 .@X; JC /

(3.3.15)

is a family of Fredholm operators of index 0 for every real s > max.; d/ 12 . Moreover, there is a constant C > 0 such that the operators (3.3.15) are isomorphisms for all jj  C , and all s > max.; d/  12 ;

3.3 Ellipticity and parametrices

207

(ii) A./ has a parameter-dependent parametrix C

P ./ 2 B ;.d/ .X I v1 I Rl /;

v1 D .F; EI JC ; J /,

i.e., I  P ./A./ 2 B 1;dl .X I vl I Rl /; I  A./P ./ 2 B 1;dr .X I vr I Rl / (3.3.16) for vl D .E; EI J ; J /, vr D .F; F I JC ; JC /, dl D max.; d/, dr D .d  /C . Proof. If A./ is parameter-dependent elliptic, we find a parameter-dependent parametrix P ./ along the lines of the proof of Theorem 3.3.7. This is straightforward, since the parameter in the symbols is only involved as an additional covariable. The remainders in (3.3.16) are Schwartz functions in the parameters. Therefore, the operators (3.3.15) are isomorphisms for all sufficiently large jj. Since the Fredholm operators are continuous in the parameter, it follows that (3.3.15) is of index zero for every fixed . Remark 3.3.18. Let X be a compact C 1 manifold with boundary, let the operator A 2 B ;d .XI vI Rl / be parameter-dependent elliptic, and let (3.3.15) be a family of invertible operators for an s D s0 > max.; d/  12 . Then the operators are invertible C for all s > max.; d/  12 , and we have A1 2 B ;.d/ .X I v1 I Rl /. Remark 3.3.19. Let X be a C 1 manifold with boundary (not necessarily compact) and A./ 2 B ;d .X I vI Rl / parameter-dependent elliptic. Then A./ has a properly C supported parametrix P ./ 2 B ;.d/ .X I v1 I Rl / in the sense that the identities of Theorem 3.3.17 (ii) hold. Remark 3.3.20. Theorem 3.3.17 and the Remarks 3.3.18, 3.3.19 remain valid in analogous form for operator functions A.y; / 2 C 1 .U; B ;d .X I vI Rl // for an open set U Ryq , which are parameter-dependent elliptic for every y 2 U . In particular, C we then find a P .y; / 2 C 1 .U; B ;.d/ .X I v1 I Rl // which is a parameterdependent parametrix of A.y; / for every y 2 U . Similar observations are true when we replace, for instance, U by U  Œ0; R/ for a half-open interval Œ0; R/ for some R > 0.

3.3.4 Fredholm families and block matrix isomorphisms In this section we study families of Fredholm operators and block matrices of operators. Concerning more details on the K-theoretic background we refer to Atiyah [6]. z and L z be Hilbert spaces and Lemma 3.3.21. Let H; H a WD

z   H a W H ! ˚ t z L

(3.3.17)

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3 Boundary value problems with the transmission property

a linear continuous operator. Then the following properties are equivalent: (i) (3.3.17) is an isomorphism; z is surjective, and t W H ! L z restricts to an isomor(ii) the operator a W H ! H z phism ker a ! L. The elementary proof is left to the reader. z be as before and a W H ! H z a linear surjective operator. Remark 3.3.22. Let H , H z an Let p W H ! ker a be a continuous projection to the kernel and g W ker a ! L z Then (3.3.17) is an isomorphism for t WD g ı p. isomorphism to a Hilbert space L. z be Hilbert spaces and a W H ! H z a Fredholm operator. Lemma 3.3.23. Let H and H Then there is a 2  2 block matrix of continuous operators  a WD

a k t q



z H H W ˚ ! ˚ C j C jC

(3.3.18)

which is an isomorphism. If a is any isomorphism of the form (3.3.18) we have ind a D jC  j :

(3.3.19)

z be a closed subspace of finite dimension such that W C im a D H z, Proof. Let W  H j and choose a surjective linear map k W C ! W for any j  dim W . Then, setting z is surjective. Let jC WD H1 WD H ˚ C j the operator a1 WD .a k/ W H1 ! H jC dim ker a1 and choose an isomorphism g1 W ker a1 ! C . Then, as in Remark 3.3.22, for every continuous projection p1 W H1 ! ker a1 we obtain a t1 D g1 p1 W H1 ! C jC z ˚ C jC is an isomorphism. In order to show (3.3.19) we such that t .a1 t1 / W H1 ! H .1/ choose a parametrix a of the operator a. The operator c WD a.1/ a  1 is compact. Moreover, (3.3.20) c WD diag.a.1/ ; idC jC /a   a.1/ k W H ˚ C j ! H ˚ C jC and is homotopic is of the form c WD 1Cc t q through Fredholm operators to diag.idH ; 0/. A homotopy may be defined by c WD   1C c a.1/ k , 0   1. Since the index of Fredholm operators is homotopy t q invariant, this yields ind c D j  jC . On the other hand, from (3.3.20) it follows that ind c D  ind a C ind a D  ind a. We want to perform analogous constructions in the case of families of operators, continuously parametrised by the points of a topological space X. This requires some tools on vector bundles and the K-functor. By Vect.X/ we denote the set of all continuous complex vector bundles on X . If X is a C 1 manifold we use the same notation for the set of all smooth complex vector bundles on X (we hope this will not cause confusion). The equivalence relations below

209

3.3 Ellipticity and parametrices

in connection with the K-functor are known to be independent of that ambiguity when X is C 1 . In the following, for simplicity, we assume that X is compact. We introduce an equivalence relation between pairs of vector bundles by

if and only if

z Fz / .E; F /  .E;

(3.3.21)

E ˚ Fz ˚ G Š F ˚ Ez ˚ G

(3.3.22)

for some G 2 Vect.X /. An equivalent definition of (3.3.21) is E ˚ L Š Ez ˚ M;

F ˚ L Š Fz ˚ M

(3.3.23)

for certain L; M 2 Vect.X /. In fact, the condition (3.3.22) implies (3.3.23) when we set L WD Fz ˚ G and M WD F ˚ G. Conversely, (3.3.23) entails (3.3.22) for G WD L ˚ M . The equivalence class represented by .E; F / is denoted by ŒE  ŒF WD Œ.E; F / , and we also set ŒE WD Œ.E; 0/ ;

ŒF WD Œ.0; F / :

(3.3.24)

The trivial bundle on X of fibre dimension N is also denoted by C N . Observe that by adding the complementary bundle F ? of F it follows that .E; F /  .E ˚F ? ; C N /, where N is defined by F ˚ F ? D C N . This shows that every equivalence class can be written as (3.3.25) ŒE  ŒF D ŒG  ŒC N

for suitable G 2 Vect.X / and N 2 N. Definition 3.3.24. The K-group of a compact topological space X is defined as K.X / WD fŒE  ŒF W E; F 2 Vect.X /g: Remark 3.3.25. Direct sum and tensor product of vector bundles are compatible with the equivalence relation and thus induce sum and product in K.X /: .ŒE  ŒF / C .ŒG  ŒH / WD ŒE ˚ G  ŒF ˚ H ; .ŒE  ŒF /  .ŒG  ŒH / WD Œ.E ˝ G/ ˚ .F ˝ H /  Œ.E ˝ H / ˚ .F ˝ G/ : In other words, K.X / is not only an additive group but a ring. (It can be proved that these operations are commutative.) Proposition 3.3.26. Let X and Y be compact spaces and f W X ! Y a continuous map. Then the pull back of vector bundles induces a ring homomorphism f  W K.Y / ! K.X /;

ŒE  ŒF ! Œf  E  Œf  F :

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3 Boundary value problems with the transmission property

z be Hilbert spaces, and denote Let X be a compact topological space. Let H; H z / the space of Fredholm operators H ! H z , endowed with the norm by F .H; H z / (the space of all linear continuous operators). Let topology induced by L.H; H z // denote the space of continuous functions a W X ! F .H; H z /. C.X; F .H; H z // and assume that a.x/ W H ! H z is surjective Lemma 3.3.27. Let a 2 C.X; F .H; H for every x 2 X . Then the family of kernels kerX a WD fker a.x/ W x 2 X g has the structure of a vector bundle on X . Proof. Fix a point x1 2 X and let P .x1 / W H ! ker a.x1 / denote the orthogonal projection to ker a.x1 /. Then the family of operators  a.x/ W H ! P .x1 /



z H ˚ ker a.x1 /

(3.3.26)

is continuous in x, and it is an isomorphism between the spaces in (3.3.26) at x D x1 , see Lemma 3.3.21. Since the isomorphisms between Hilbert spaces form an open set in the operator norm topology, there is an open neighbourhood U.x1 / of x1 such that ˇ P .x1 /ˇker a.x/ W ker a.x/ ! ker a.x1 / (3.3.27) is a family of isomorphisms for all x 2 U.x1 /. Composing (3.3.27) with an isomorphism g W ker a.x1 / ! C j with j WD dim ker a.x1 / we obtain a trivialisation ˇ kerX aˇU.x / ! U.x1 /  C j ; .x; ker a.x// ! .x; C j /: (3.3.28) 1

The neighbourhoods U.x1 /, x1 2 X , form an open covering of X . Thus the system of maps (3.3.28) defines in kerX a the structure of a vector bundle. z // there exists a j 2 N and a linear Proposition 3.3.28. For every a 2 C.X; F .H; H j z operator k W C ! H such that H z .a.x/ k/ W ˚ ! H C j

(3.3.29)

is surjective for every x 2 X . z , j.y/ WD dim W .y/ < 1, such Proof. Let y 2 X and choose a subspace W .y/  H j.y/ z . Moreover, let k.y/ W C that W .y/ C im a.y/ D H ! W .y/ be an isomorphism. Then H z .a.x/ k.y// W ˚ ! H (3.3.30) j.y/ C is surjective for x D y. Since the surjective operators form an open set in L.H ˚ z / there is an open neighbourhood U.y/ of y such that (3.3.30) is surjective C j.y/ ; H for all x 2 U.y/.

3.3 Ellipticity and parametrices

211

This construction can be applied for every y 2 X . The sets fU.y/ W y 2 X g form an open covering of X; the compactness of X allows us to choose a finite z for subcovering fU.y1 /; : : : ; U.yN /g. Then, k WD .k.y1 /; : : : ; k.yN // W C j ! H PN z j WD lD1 j.yl / is a map that completes a.x/ W H ! H to a family (3.3.29) which is surjective for all x 2 X . z // there exist vector bundles J ; JC 2 Theorem 3.3.29. For every a 2 C.X; F .H; H Vect.X/ and a continuous family of isomorphisms 

 H z H a.x/ k.x/ a.x/ D W ˚ ! ˚ ; t .x/ q.x/ J;x JC;x

x 2 X:

(3.3.31)

Proof. Applying Proposition 3.3.28 we find a surjective family (3.3.29), and we set z // for H1 WD J WD C j . Then we have a1 .x/ WD .a.x/ k/ 2 C.X; F .H1 ; H H ˚C j , and Lemma 3.3.27 gives us a vector bundle JC WD kerX a1 . Let a2 .x/ W H1 ! ker a1 .x/ denote the orthogonal projection. Then  z H a1 .x/ W H1 ! ˚ a2 .x/ JC;x

 a.x/ WD

is a family of isomorphisms, cf. Lemma 3.3.21, which can be reinterpreted as a 2  2 block matrix family of isomorphisms (3.3.31). z // and any choice of an isomorphism (3.3.31) Definition 3.3.30. For a 2 C.X; F .H; H we set indX a WD ŒJC  ŒJ 2 K.X /; called the index element of the Fredholm family a. We shall show below that this is a correct definition, i.e., independent of the choice of the isomorphism (3.3.31). Remark 3.3.31. The considerations of this section easily generalise to the case of z of Hilbert space bundles H; H z over X , i.e., the Fredholm homomorphisms a W H ! H fibres are Hilbert spaces and a represents a family of Fredholm operators a.x/ W Hx ! zx , x 2 X . We tacitly employ this generalisation; all proofs can easily be modified, H without using Kuiper’s theorem. In particular, we also understand Definition 3.3.30 in this generalised sense. z // there is a p 2 C.X; F .H z ; H // Proposition 3.3.32. For every a 2 C.X; F .H; H such that pa D idH Cc, ap D idHz CcQ with families c 2 C.X; K.H; H //, cQ 2 z; H z // (here K.; / denotes the space of all compact operators in the Hilbert C.X; K.H spaces in brackets).

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3 Boundary value problems with the transmission property

z // can be comProof. By virtue of Theorem 3.3.29 the family a 2 C.X; F .H; H pleted to an isomorphism (3.3.31). It can easily be verified that the inverse is again z ; H //. Setting an isomorphism, where the upper left corner p belongs to C.X; F .H   a1 D ps hr we obtain pa C ht D idH , ap C ks D idHz , which gives us the assertion for c D ht and cQ D ks which are operators of finite rank and hence compact. z be a continuous family of isomorphisms and Lemma 3.3.33. Let a.x/ W H ! H J˙ 2 Vect.X/ such that   H z H a k (3.3.32) aD W ˚ ! ˚ t q J JC is an isomorphism for suitable k; t; q. Then we have J Š JC . z and a D idH ; otherwise we Proof. Without loss of generality we assume H D H compose (3.3.32) from the left by diag.a1 ; idJC /. In other words we consider the case   H H 1 k (3.3.33) W ˚ ! ˚ : t q JC J     , 0   1, which connects Let us form the -depending family  1 t 10 1t kq 10  k 1 (3.3.33) with diag.1; q  t k/ through isomorphisms. For D 1 we then obtain q  tk W J ! JC as an isomorphism. z // and Lemma 3.3.34. Let c 2 C.X; K.H; H

 1Cc k  t

q

H

z H

J

JC

W ˚ ! ˚ be a continuous

isomorphism. Then there is a G 2 Vect.X / such that J ˚ G Š JC ˚ G. z . Define c WD .1  /c, Proof. Without loss of generality we assume H D H 0   1, which can be regarded as an element c 2 C.Œ0; 1  X; K.H; H //. There is then an " > 0 such that 

1 C c b WD t

k q

 H H W ˚ ! ˚ J JC

(3.3.34)

are isomorphisms for all 0   ". Moreover, (3.3.34) is a family of Fredholm operators for all 0   1 in the interpretation of Remark 3.3.31. Let us set M WD z WD H ˚ JC . Applying the construction of Theorem 3.3.29 to obtain a H ˚ J , M block matrix family of isomorphisms we find vector bundles E; F 2 Vect.Œ0; 1  X / such that  M  z M b k (3.3.35) W ˚ ! ˚ t q E F are isomorphisms for all 0   1, for a suitable choice of t , k , q . Since b z for all 0   " by Lemma 3.3.33 we have takes values in isomorphisms M ! M

3.3 Ellipticity and parametrices

213

ˇ ˇ E ˇŒ0;"X Š F ˇŒ0;"X . It follows that E Š F Š   G for some G 2 Vect.X /, where  W Œ0; 1  X ! X is the canonical projection. Now (3.3.35) at D 1 is an isomorphism, where the upper left corner is equal to idH . From Lemma 3.3.33 we thus obtain J ˚ G Š JC ˚ G. z be Hilbert bundles on X and a W H ! H z a Fredholm Theorem 3.3.35. Let H; H homomorphism. Let a be any isomorphism (3.3.31) for suitable k; t; q and J˙ 2 Vect.X/. Then indX a D ŒJC  ŒJ

is independent of the choice of a. z are trivial Hilbert bundles, Proof. For notational convenience we assume that H; H z then H; H also denote the fibres. Given two isomorphisms 

a k a WD t q

 H z H W ˚ ! ˚ ; J JC



 H z H a h b WD W ˚ ! ˚ s r G GC

with the same upper left corner, and J˙ ; G˙ 2 Vect.X /, we can pass to the isomorphisms, 0

a k @ aQ WD t q 0 0

1 0 0 AW idG

z H H ˚ ˚ J ! JC ; ˚ ˚ GC GC

0

a 0 Qb WD @0 idJ C s 0

1 h 0A W r

z H H ˚ ˚ JC ! JC : ˚ ˚ G GC

The operator function bQ 1 aQ W H ˚J ˚GC ! H ˚JC ˚G is again an isomorphism, and the upper left corner of aQ 1 aQ has the form idH Cc for some c 2 C.X; K.H; H //. Then Lemma 3.3.34 gives us J ˚ GC ˚ L Š JC ˚ G ˚ L for some L 2 Vect.X / which implies ŒJC  ŒJ D ŒGC  ŒG in K.X /. Remark 3.3.36. Given a continuous map f W Y ! X between compact topological spaces X; Y , we have the pull backs f  W K.X / ! K.Y /;

z // ! C.Y; F .H; H z //: f  W C.X; F .H; H

z //. It can easily be proved that indY f  .a/ D f  indX a for every a 2 C.X; F .H; H z Remark 3.3.37. Let a0 ; a1 2 C.X; F .H; ˇ H //, and assume that there is an a 2 z // such that aj D aˇ for j D 0; 1. The families a0 and a1 C.Œ0; 1  X; F .H; H fj gX are then called homotopic, a0 Š a1 . We have indX a0 D indX a1 for a0 Š a1 . Let z / denote the set of all homotopy classes of elements in C.X; F .H; H z //. ŒX; F .H; H Then indX induces a map z / ! K.X /: indX W ŒX; F .H; H By a theorem of Jänich [85] this is a bijection.

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3 Boundary value problems with the transmission property

3.4 The calculus on manifolds with conical exit to infinity Boundary value problems on a manifold with edges have a principal edge symbolic structure consisting of boundary value problems on an infinite cone, with specific effects from the conical exit to infinity. We formulate here some necessary elements of the calculus on a manifold with such conical exits and boundary. The case of closed manifolds with conical exits was considered in Section 2.3 before.

3.4.1 Motivation in terms of principal edge symbols Let W be an manifold with edge Y and boundary, and A 2 Diff  deg .W /, cf. Definition 2.4.42. Recall that the operator A in the splitting of variables .r; x; y/ 2 X ^   has the form  X @ j A D r  aj˛ .r; y/ r .rDy /˛ @r j Cj˛j

x C  ; Diff .j Cj˛j/ .X // for all j; ˛. The prinwith coefficients aj˛ .r; y/ 2 C 1 .R cipal edge symbol of A is defined as X ^ .A/.y; / D r  aj˛ .0; y/.r@r/j .r/˛ ; j Cj˛j

.y; / 2 T  Y n 0, and represents a family of continuous operators ^ .A/.y; / W K s; .X ^ / ! K s; .X ^ /

(3.4.1)

for every s;  2 R. This was formulated in Section 2.4.3 as an information from the general calculus of operators on a manifold with edges (first for the case of closed compact X, but it is clear that it also holds in the case with boundary). Let us briefly summarise some aspects around the nature of operators (3.4.1). Write ^ .A/.y; / D !^ .A/.y; / C .1  !/^ .A/.y; / for a cut-off function !.r/. The continuity of !^ .A/.y; / W H s; .X ^ / ! H s; .X ^ / is easy. Moreover, s s .X ^ / ! Hcone .X ^ / .1  !/^ .A/.y; / W Hcone

(3.4.2)

is continuous, according to Proposition 2.4.64. This gives us (3.4.1), cf. Definition 2.4.5 (ii). Recall that (3.4.2) is based on information on a relationship between edgedegenerate symbols for r ! 1 and operators on a manifold with conical exit to infinity. In the calculus of edge-degenerate operators below we need similar results in the pseudo-differential case, especially, when X is a compact C 1 manifold with boundary.

3.4 The calculus on manifolds with conical exit to infinity

215

Let us apply the constructions of Section 2.4.5 concerning the exit symbols of (2.4.64) to ^ .A/.y; / for every fixed  6D 0; y 2 . We identify a coordinate neighbourhood U on int X with B D fx 2 Rn W jxj < 1g and write in the corresponding local coordinates .r; x/ X cj ˛ .x; y/r j Cj˛j Drj Dx ˛ ^ .A/.y; / D r  j CjjCj˛j

with coefficients cj ˛ 2 C 1 .B  /. In order to express symbols, for simplicity we assume cj  ˛ .x; y/ D 0 for x 62 K for some K b B. Then we have in the image under the map ˇ W RC  B !   R1Cn ; .r; x/ ! .r; rx/ D .r; x/; Q for the interior symbol Q D Q %; /  .ˇ ^ .A/.y; //.r; x;

X

cj 0

 xQ r

j CjjD

 ; y %j Q 

which is independent of . Moreover, the exit symbolic components are  xQ  X Q D Q %; / cj ˛ ; y %j Q  ˛ e .ˇ ^ .A/.y; //.r; x; r j Cj jCj˛jD

and 

Q D Q %; / ;e .ˇ ^ .A/.y; //.r; x;

X

cj 0

 xQ

j Cj jD

r

 ; y %j Q  :

For the boundary symbols we choose a neighbourhood U on X that is identified with the half-ball BC WD fx 2 Rn W jxj < 1; xn  0g. Then U \ @X is identified with B 0 D B \ fxn D 0g, and we can consider the maps ˇC W RC  BC ! C WD  \ R1Cn C with R1Cn D f.r; x/ Q 2 R1Cn W xQ n  0g and C ˇ 0 W RC  B 0 !  0 WD  \ Rn with Rn D f.r; x/ Q 2 R1Cn W xQ n D 0g. In this connection we set xQ D .xQ 0 ; xQ n / for 0 xQ D .xQ 1 ; : : : ; xQ n1 /, with the corresponding covariables Q D . Q 0 ; Qn /. In order to understand the structure of ^ .A/ up to r D 1 we also need the boundary symbols in the exit calculus of boundary value problems, more precisely, those of .ˇC / ^ .A/. They are in this case (according to notation that we explain in the following sections in a more general context)  xQ 0  X 0 @ ..ˇC / ^ .A/.y; //.r; xQ 0 ; %; Q 0 / D cj 0 (3.4.3) ; 0; y %j Q0 DxQnn r j CjjD

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3 Boundary value problems with the transmission property

with  D . 0 ; n /, for .r; xQ 0 / 2 RC  B 0 ; .%; Q 0 / 2 Rn n f0g,  xQ 0  X 0 e0 ..ˇC / ^ .A/.y; //.r; xQ 0 ; %; Q 0 / D cj ˛ ; 0; y %j Q0 Dxnn ˛ r j CjjCj˛jD

(3.4.4)

for .r; xQ 0 / 2 RC  B 0 ; .%; Q 0 / 2 Rn , and X

@;e0 ..ˇC / ^ .A/.y; //.r; xQ 0 ; %; Q0 / D

j CjjD

cj 0

 xQ 0 r

 0 ; 0; y %j Q0 Dxnn

(3.4.5)

for .r; xQ 0 / 2 RC  B 0 ; .%; Q 0 / 2 Rn n f0g. The expressions (3.4.3), (3.4.4), (3.4.5) are interpreted as families of operators 1 H s .RC / ! H s .RC / for s >  : 2 Elliptic differential operators are also equipped with boundary (or trace) operators of the kind T WD r 0 B for some B 2 Diff deg .2W /, with r 0 denoting the operator of restriction to V , cf. the notation in Section 2.4.2 (in general, such trace operators form a vector with components of different order, but for the moment, to explain notation, it suffices to consider one component); clearly we need to know B only in a neighbourhood of V . With B we can associate the family of operators 1

1

^ .T /.y; / WD r.@X/^ ^ .B/.y; / W K s; .X ^ / ! K s 2 ; 2 ..@X /^ / for s   

1 2

> 0, and we have the subordinate exit symbols

@ ..ˇC / ^ .T /.y;//.r; xQ 0 ; %; Q 0 / D rfxQn D0g @ ..ˇC / ^ .B/.y; //.r; xQ 0 ;%; Q 0 /; e0 ..ˇC / ^ .T /.y;//.r; xQ 0 ;%; 0 / D rfxQ D0g e0 ..ˇC / ^ .B/.y;//.r; xQ 0 ; %; Q 0 /; n

@;e0 ..ˇC / ^ .T /.y; //.r; xQ ;%; Q 0 / D rfxn D0g @;e0 ..ˇC / ^ .B/.y; //.r; xQ 0 ;%; Q 0 /: 0

Thus, the edge symbol ^ .A/.y; / of a boundary value problem A, represented by a column matrix A D t .A T / of operators 1

1

A W W s; .W / ! W s;  .W / ˚ W s 2 ; 2 .V /; has exit symbols which can be expressed locally in the coordinates under the maps ˇ and .ˇC ; ˇ 0 /, respectively. They form the exit symbolic hierarchy that we briefly denote by E .^ .A/.y; //; E0 .^ .A/.y; //; (3.4.6) with the components E WD .e ; 

;e /;

E0 WD .e0 ; @;e0 /:

(3.4.7)

The full principal symbolic hierarchy of ^ .A/.y; / as operators on X ^ has the components  D . ; E ; @ ; E0 /:

3.4 The calculus on manifolds with conical exit to infinity

217

3.4.2 Global operators in the half-space The considerations of the preceding section show the following. In order to understand the behaviour of ^ .A/.y; / for r ! 1 as a family of boundary value problems on the infinite stretched cone X ^ with boundary .@X /^ we may ignore for a while the relationship with edge symbols. It suffices look at boundary value problems with some additional structure at infinity. This section briefly reports elements of a corresponding calculus in the half-space. More details may be found in [90, Section 3.3]. To simplify notation we now consider Rm with the variables x D .y; t /, y 2 Rm1 , t 2 R, and covariables D .; /. Let us set SclI .Rm  Rm /tr WD Scl .Rm m /tr \ SclI .Rm  Rm /; Ix Ix cf. Definition 2.3.5 and 3.1.1. Moreover, for R > 0 we define the set TR WD fx 2 Rm W jxj  R; jtj  Rjyjg and SclI .Rm  Rm /tr; WD fa.x; / 2 SclI .Rm  Rm /tr W a.x; / D 0 Ix Ix for all .x; / 2 TR  Rm with R D R.a/ > 0g: xm  x m with respect to x we obtain the spaces S I .R By restricting symbols to R clIx C C m Q / 2 SclI .R  Rm /tr; . Given an a.x; / in the latter space we have an extension a.x; Ix m m1 R /tr; . Similarly as in Section 3.1.2 we form (3.1.14) and (3.1.15) with R instead of . x m  Rm /tr; implies Proposition 3.4.1. a.x; / 2 SclI .R C Ix (i) opC .a/.y; / 2 S I .Rm1  Rm1 I H s .RC /; H s .RC // for every s >  12 , and x C /; S.R x C //I opC .a/.y; / 2 S I .Rm1  Rm1 I S.R (ii) opC .aj tD0 /.y; / 2 SclI .Rm1  Rm1 I H s .RC /; H s .RC // for every s > Iy  12 , and x C /; S.R x C //: .Rm1  Rm1 I S.R opC .aj tD0 /.y; / 2 SclI Iy This result is a variant of Theorem 3.1.15 and Remark 3.1.16, here for the exit calculus of boundary value problems in the half-space. Definition 3.4.2. A Green symbol of order  2 R, type 0, and weight  2 R at infinity (in the sense of the exit calculus of boundary value problems in the half-space) is an element x C / ˚ C jC / g.y; / 2 SclI .Rm1  Rm1 I L2 .RC / ˚ C j ; S.R Iy

218

3 Boundary value problems with the transmission property

such that x C / ˚ C j /: g  .y; / 2 SclI .Rm1  Rm1 I L2 .RC / ˚ C jC ; S.R Iy Moreover, a Green symbol of order  2 R, type d 2 R, and weight  2 R is an operator family of the form g.y; / D g0 .y; / C

d X

gj .y; / diag.@jt ; 0/

(3.4.8)

j D1

for arbitrary Green symbols gj .y; / of order   j , type 0, and weight . Let R;dI .Rm1  Rm1 I j ; jC / denote the space of all Green symbols (3.4.8). G The notation is motivated similarly as (3.2.27), where, for simplicity, we do not shift by 1=2 the orders of operators belonging to the boundary. In the applications below we will have such shifts of orders, however, the corresponding modification is trivial and left to the reader. Applying the notation of Section 2.3.5, now with subscript ‘@’and ‘e0 ’rather than ‘^’ and ‘e’, respectively, we obtain a triple of symbols for every g.y; / 2 R;dI .Rm1  G m1 R I j ; jC /, namely,  .g/ D .@ .g/; e0 .g/; @;e0 .g//: Definition 3.4.3. The space R;dI .Rm1  Rm1 I j ; jC / for .; d; / 2 Z  N  R is defined to be the set of all families of operators a.y; / D diag.opC .a/.y; /; 0/ C g.y; / x m  Rm /tr; and g.y; / 2 R;dI .Rm1  Rm1 I j ; jC /. .R where a.y; / 2 SclI C G Ix Given an a.y; / 2 R;dI .Rm1  Rm1 I j ; jC / we set  .a/ WD . .a/; e .a/;  for  .a/ WD  .a/; e .a/ WD e .a/; 

;e .a/I @ .a/; e0 .a/; @;e0 .a//

;e .a/

WD 

;e .a/,

and

@ .a/ WD diag.@ .opC .aj tD0 //; 0/ C @ .g/; e0 .a/ WD diag.e0 .opC .aj tD0 //; 0/ C e0 .g/; @;e0 .a/ WD diag.@;e0 .opC .aj tD0 //; 0/ C @;e0 .g/: Remark 3.4.4. The components of  .a/ are uniquely determined by a.y; /, and .a/ D 0 implies a.y; / 2 R1;dI1 .Rm1  Rm1 I j ; jC /. Q

Q dIQ  Theorem 3.4.5. a.y; / 2 R;dI , b.y; / 2 R; implies – under the assumption Q  that the dimensions in the middle fit together – that .ab/.y; / 2 RCIhICQ for h D Q max.Q C d; d/, and we have  .ab/ D  .a/ .b/ (with componentwise multiplication).

219

3.4 The calculus on manifolds with conical exit to infinity

The technique of the proof is similar to that of Theorem 3.1.32. Let us now pass to the space of boundary value problems globally in the half-space. x m I j ; jC / of smoothing elements of type zero is defined The space B 1;0I1  A K.R C as the set of all A D T C where R R (i) Au.y; t/ D Rm a.y; t; y 0 ; t 0 /u.y 0 ; t 0 /dt 0 dy 0 for certain a.y; t; y 0 ; t 0 / 2 C x m /; u.y; t / 2 H sIg .Rm / D H sIg .Rm /jRm ; xm  R S.R C

C

C

C

Pj K v .y; t / for v WD .v1 ; : : : ; vj / 2 H sIg .Rm1 /, and (ii) Kv.y; t/ D R lD1 l l xm  Kl vl .y; t/ D Rn1 kl .y, t; y 0 /vl .y 0 /dy 0 for certain kl .y; t; y 0 / 2 S.R C m1 R /; R R (iii) T u.y/ D .Tr u.y//rD1;:::;jC with Tr u.y/ D Rm br .y; y 0 ; t 0 /u.y 0 ; t 0 /dt 0 dy 0 C x m /; for certain br .y; y 0 ; t 0 / 2 S.Rm1  R C

(iv) C v.y/ D

 Pj R

S.Rm1  R

lD1 m1

crl .y; y 0 /vl .y 0 /dy 0

 rD1;:::;jC

for certain crl .y; y 0 / 2

/.

x m I j ; jC / for d 2 N is the space of all operators C D Moreover, B 1;dI1 .R C Pd x m I j ; jC /, j D 1; : : : ; d. C0 C j D1 Cj diag.@jt ; 0/ for arbitrary Cj 2 B 1;0I1 .R C Let LI .Rm /` denote the subspace of all P 2 LI .Rm /, cf. Section 2.3.2, such .cl/ .cl/ that there is an R > 0 with 'P D 0 for all '; 2 C01 .Rm / with supp '; supp .Rm fx D .y; t/ 2 Rm1  R W jtj  R max.1; jyj/g. Moreover, we set LI C /` WD .cl/ I I m m z z z z m m fP jR W P 2 L .R /` g. For P D P jR , P 2 L .R /, we define C

.cl/

C

.cl/

ˇ ˇ  z ˇ xm ;  .P / D  .Pz /ˇR x m .Rm nf0g/ ; e .P / .R nf0g/Rm C

C

ˇ z ˇ

;e .P / .R x m nf0g/.Rm nf0g/ C



:

x m I j ; jC / for .; d; / 2 Z  N  R is defined Definition 3.4.6. The space B ;dI .R C to be the set of all operators A D Op.a/ C P C C

(3.4.9)

m for any a.y; / 2 R;dI .Rm1  Rm1 I j ; jC /, P D diag.P; 0/, P 2 LI cl .RC /` , 1;dI1 x m and C 2 B .RC I j ; jC /. The subset of all elements A with a.y; / 2 m1 m1 x m I j ; jC /. .R  R I j ; jC / and P D 0 will be denoted by B ;dI .R R;dI C G G

x m I j ; jC /, written in the form (3.4.9), we set Given an operator A 2 B ;dI .R C .A/ WD . .a/ C  .P /; e .a/ C e .P /;  ;e .a/ C  ;e .P /I @ .a/; e0 .a/; @;e0 .a//:

(3.4.10)

It can easily be verified that this definition is correct, i.e., independent of the choice of the representation (3.4.9).

220

3 Boundary value problems with the transmission property

x m I j ; jC / induces continuous operators Theorem 3.4.7. Every A 2 B ;dI .R C sIg sIg A W H sIg .Rm .Rm1 ; C j / ! H sIg .Rm .Rm1 ; C jC / C/ ˚ H C/ ˚ H (3.4.11) for all .s; g/ 2 R2 ; s > d  12 .

This result easily follows by combining the technique from Theorem 2.3.13 and Theorem 3.2.14. x m I j ; jC / and assume that  .A/ D 0. Then the Remark 3.4.8. Let A 2 B ;dI .R C operator (3.4.11) is compact for every .s; g/ 2 R2 , s > d  12 . The arguments for the latter observation are similar to those for Remark 3.2.21. Analogously as Theorems 3.2.22 and 3.2.23 we have the following theorem. z dQ IQ  xm x m I j0 ; jC / and B 2 B ; .RC I j ; j0 /, then Theorem 3.4.9. (i) Let A 2 B ;dI .R C C;hI;Q Q  xm Q and we have  .AB/ D AB 2 B .RC I j ; jC / for h D max.Q C d; d/, .A/.B/ (with componentwise multiplication). x m I j ; jC / implies A 2 B 0;0I0 .R x m I jC ; j / (where the adjoint (ii) A 2 B 0;0I0 .R C C 2 is based on the standard L -scalar products) and  .A / D  .A/ (with componentwise adjoints, defined in a straightforward manner).

x m I j ; jC / is called elliptic (of order .I /) if Definition 3.4.10. An A 2 B ;dI .R C x m .Rm nf0g/, e .A/.x; / for .x; / 2 .R x m nf0g/ (i)  .A/.x; / for .x; / 2 R C C x m n f0g/  .Rm n f0g/ are non-vanishing; Rm , and  ;e .A/.x; / for .x; / 2 .R C (ii) @ .A/.y; / for .y; / 2 Rm1 .Rm1 nf0g/, e0 .A/.y; / for .y; / 2 .Rm1 n f0g/  Rm1 , and @;e0 .A/.y; / for .y; / 2 .Rm1 n f0g/  .Rm1 n f0g/ are isomorphisms H s .RC / ˚ C j ! H s .RC / ˚ C jC (3.4.12) for all s > max.; d/  12 . x C / ˚ C j ! Remark 3.4.11. Instead of (3.4.12) we can alternatively require that S.R j x C / ˚ C C is an isomorphism for all those .y; /. S.R x m I j ; jC / be elliptic. Then (3.4.11) is a Fredholm Theorem 3.4.12. Let A 2 B ;dI .R C operator for every s > max.; d/  12 and g 2 R, and there is a parametrix P 2 C x m I jC ; j / in the sense that B ;.d/ I .R C

xm I  P A 2 B 1;dl I1 .R C I j ; j /;

xm I  AP 2 B 1;dr I1 .R C I jC ; jC / (3.4.13)

for dl D max.; d/, dr D .d  /C . The proof follows by combining the technique from Theorem 2.3.28 and Theorem 3.3.7.

3.4 The calculus on manifolds with conical exit to infinity

221

3.4.3 Operators on a manifold with conical exit We now consider boundary value problems on a manifold with conical exit to infinity. The half-space plays the role of a local model; the manifold case is formulated in terms of corresponding charts. An example of a manifold M with boundary and conical exit is the half-cylinder M WD ."; 1/  X; 0 < " < 1, for some C 1 manifold X with boundary, n D dim X . In order to define the conical structure we choose coordinate neighbourhoods U on X with charts 1 W U ! B 1;C W U ! BC

for U \ @X D ;; for U \ @X 6D ;;

(3.4.14) (3.4.15)

x n are the unit ball and the half-ball, respectively, cf. where B  Rn and BC  R C Section 3.4.1. This gives us a composition of maps ."; 1/  U ! ."; 1/  B !  \ .jr; xj Q > "/

(3.4.16)

defined by .r; / ! .r; 1 .  //, and .r; x/ ! .r; rx/, and, similarly, ."; 1/  U ! ."; 1/  BC ! C \ .jr; xj Q > "/:

(3.4.17)

A C 1 manifold M with boundary and conical exit to infinity and cross section X is any C 1 manifold with boundary containing ."; 1/  X as a submanifold, where the cylinder ."; 1/  X is equipped with a conical structure given by (3.4.16) and (3.4.17). A special case is X WD R  X 3 .r; /, where we have two conical exits r ! ˙1; this case can easily be subsumed under the definition before by passing to the disjoint union of two copies of X as the cross section of X . Of course, in this particular case it is easier to speak about two conical exits. There is now a straightforward generalisation of the definitions and results of the preceding section to M . Let us assume M D X , n D dim X , which is the case of interest here (more precisely, the half-cylinder RC  X D X ^ ); concerning M in general we refer to [90]. The definition of global smoothing operators of the class B 1;dI1 .X I j ; jC / is easy and left to the reader. Note that the Schwartz space on X is given by S.X / WD S.R; C 1 .X//; the boundary .@X / D @X is a closed manifold with conical exit and S.@X / WD S.R; C 1 .@X //. Let us assume that X is compact. Then we have the weighted Sobolev spaces H sIg .X /

and

H sIg .@X /;

respectively, for every s; g 2 R. Observe that when !.r/ is a cut-off function on RC x C, and 1  !.r/ interpreted as a C 1 function on R, vanishing in a neighbourhood of R then we have s .1  !/H sIg .X / D .1  !/r g Hcone .X ^ /; (3.4.18) cf. Section 2.4.2, especially, Remark 2.4.6 in the variant of a manifold with boundary. This relation can be used to define H sIg .X / as the subspace of all those

222

3 Boundary value problems with the transmission property

s u 2 Hloc ..2X/ /jint X such that (3.4.18) holds, together with a similar condition for r ! 1. For purposes below we form a set B  R1Cn 3 .r; x/ Q with B being the open unit ball in Rn , defined by

Q 2 R1Cn W jxj Q < c.r/g B WD f.r; x/ for a strictly positive function c 2 C 1 .R/ such that c.r/ D jrj with jrj > " for some 0 < " < 1. Setting ı .r; x/ Q WD .r; x/ Q for  2 RC , it follows that ı .r; x/ Q 2 B for all .r; x/ Q 2 B , r > ",   1. On X we have similar dilations, namely, .r; x/ ! .r; x/ that we also denote by ı ,  2 RC . Now if (3.4.14) is a chart on X we can form the chain of maps  W R  U ! R  B ! B with .r; / ! .r; 1 .  // as the first one and a diffeomorphism as the second one, with the property .r; x/ ! .r; rx/ DW .r; x/ Q for r > ": Then we have  ı ı D ı ı 

for all r > ",   1:

In a similar manner we can proceed with (3.4.15) and obtain a chart C W R  U ! .BC /

(3.4.19)

for a coordinate neighbourhood U  X intersecting the boundary. Let us fix such S coordinate neighbourhoods .U1 ; : : : ; UN / with @X  jND1 Uj . We then find functions 'j .r; /; j .r; / 2 C 1 .R  Uj / with compact support in the second variable such that P N j D1 'j 1 in a neighbourhood of @X , j 1 in a neighbourhood of supp 'j , and 'j .ı .r; // D 'j .r; /;

j .ı .r; //

D

j .r; /

for all   1, jrj > ", j D 1; : : : ; N . Moreover, let '0 ; 0 2 C 1 .X / be functions P with the properties '0 WD 1  jND1 'j , supp 0 \ @X D ;, and 0 1 in a neighbourhood of supp '0 . According to (3.4.19) we have charts C;j W R  Uj ! .BC /  R1Cn C ; j D 1; : : : ; N . Set 'j0 WD 'j j@X , j0 WD j j@X , j D 1; : : : ; N . We will employ operator push forwards .1 C;j / Op.aj / for symbols aj .y; / 2 ;dI R .Rn  Rn I j ; jC /, where Op.aj / is combined with the restriction to .BC / . Definition 3.4.13. The space B ;dI .X I j ; jC / for .; dI / 2 Z  N  R is defined to be the set of all operators AD

N X

diag.'j ; 'j0 /.1 C;j / Op.aj / diag.

j;

0 j/

C diag.'0 P

0 ; 0/

CC

j D1

for arbitrary aj .y; / 2 R;dI .Rn  Rn I j ; jC /, P D Pz jint X for some Pz 2 1;dI1 .X I j ; jC /. LI cl ..2X/ /, and C 2 B

3.4 The calculus on manifolds with conical exit to infinity

223

Theorem 3.4.14. Every A 2 B ;dI .X I j ; jC / induces continuous operators A W H sIg .X / ˚ H sIg .@X ; C j / ! H sIg .X / ˚ H sIg .@X ; C jC / (3.4.20) for all s; g 2 R, s > d  12 . This result follows by combining Theorem 2.3.22 and Theorem 3.4.7. Remark 3.4.15. (i) It is not essential to insist on spaces of boundary value problems of the class B ;dI .X I j ; jC /. Similarly as (3.2.27) we can pass to corresponding spaces B ;dI .X I j ; jC / by composing with reductions of orders on the boundary @X (see, analogously, the formula (3.2.27)). The operators A 2 B ;dI .X I j ; jC / (see, analogously, the formula (3.2.27)) are continuous as 1

1

A W H sIg .X / ˚ H s 2 ;g .@X ; C j / ! H sIg .X / ˚ H s 2 ;g .@X ; C jC / for all s; g 2 R, s > d  12 . (ii) In Chapter 6 below we will study operators of a similar kind on X ^ D RC  X belonging to B ;dI .X ^ I j ; jC / WD fA 2 B ;d .X ^ I j ; jC / W .1   /A.1  Q / 2 B ;dI .X I j ; jC / for any choice of cut-off functions ; Q on the half-axisg: The interpretation of .1   /A.1  Q / is that the operator is extended by zero to X n X ^ . Similarly asin the preceding section we form the principal symbolic hierarchy of  ;dI ACG K .X I j ; jC /, namely, operators A D T Q 2B  .A/ WD . .A/; @ .A/; E .A/; E0 .A// for E .A/ WD .e .A/; 

;e .A//;

E0 .A/ WD .e0 .A/; @;e0 .A//:

The components . .A/; E .A// only depend on A in the upper left corner. They are defined in an analogous manner as (2.3.18), here for X instead of M (the presence of a boundary does not cause an additional difficulty) and M1 consisting of two halfcylinders, namely, .1; 1  X and Œ1; 1/  X . The components .@ .A/; E0 .A// are a global version of those in the formula (3.4.10). Analogously as Remark 3.4.8 we have the following observation. Remark 3.4.16. The operator (3.4.20) is compact when  .A/ D 0. Remark 3.4.17. For the operator spaces of Definition 3.4.13 we have an immediate analogue of Theorem 3.4.9; we do not formulate it explicitly here.

224

3 Boundary value problems with the transmission property

Definition 3.4.18. An operator A 2 B ;dI .X I j ; jC / is called elliptic (of order .I /) if . .A/; e .A/;  ;e .A// is elliptic in the sense of Remark 2.3.27 (which is here a condition up to the boundary) and if .@ .A/; e0 .A/; @;e0 .A// is elliptic in the sense of Definition 3.4.10 (ii) (which is of invariant meaning). Theorem 3.4.19. Let A 2 B ;dI .X I j ; jC / be elliptic. Then (3.4.20) is a Fredholm operator for every s > max.; d/  12 and g 2 R, and there is a parametrix P 2 C

B ;.d/ I .X I jC ; j / which satisfies an analogue of the relation (3.4.13) with x m. X in place of R C This result is a straightforward generalisation of Theorem 3.4.12.

Remark 3.4.20. Similar results hold for boundary value problems on an arbitrary manifold with boundary and conical exit to infinity, cf. [90]. The Fredholm property holds in the case of a compact cross section X , cf. the definition at the beginning of this section, while the existence of parametrices in the respective operator classes is guaranteed when the given operator is elliptic.

3.4.4 A relation between edge-degenerate families and exit calculus As noted in the Sections 2.4.5 and 3.4.1 there is a relationship between edge-degeneracy and exit calculus. We want to formulate a result here for the case of boundary value problems. In this section X is a compact C 1 manifold with boundary. Similarly as the definition of parameter-dependence of boundary value problems, cf. Section 3.3.3, we have the parameter-dependent spaces B ;d .X I j ; jC I Rl /, including the notion of parameter-dependent ellipticity. Analogously as (2.4.68) we can define a diffeomorphism ˇ W R  X ! X Q RX for a smooth map ˇQ W RB N ! RRN with the N -dimensional unit by ˇ WD ˇj N ball B in RN , considering X as an embedded submanifold (with boundary) of B N (for N sufficiently large), such that ˇQ maps R  B N diffeomorphically to its image and Q x/ D .r; jrjx/ for r  const for a constant > 0 and x 2 B N . satisfies the relation ˇ.r; ;d / is an arbitrary parameter-dependent family, If p. Q %; Q / Q 2 B .X I j ; jC I R1Cq %; Q Q and p.r; %; / WD p.r%; Q r/, then we have opr .p/.1 / 2 B ;d .R  X I j ; jC / for 1 q every fixed  2 R n f0g. Theorem 3.4.21. (i) We have ˇ .Œr  opr .p/.// 2 B ;dI0 .X I j ; jC / for every Q %; Q / Q ! ˇ .Œr  opr .p/.//jD1 defines a continuous map 1 2 Rq n f0g, and p. 1Cq ;d B .XI j ; jC I R%; / ! B ;dI0 .X I j ; jC / for any fixed 1 2 Rq n f0g. Q Q (ii) If p. Q %; Q / Q is parameter-dependent elliptic, then ˇ .Œr  opr .p/.1 // is elliptic 1 for every  2 Rq n f0g (of order .; 0/) in the sense of Definition 3.4.18. The technicalities for the proof are very close to those of Theorem 2.4.68 and left to the reader.

Chapter 4

Mixed problems in standard Sobolev spaces

We study elliptic mixed problems as operators between standard Sobolev spaces of sufficiently large smoothness s, and we show the Fredholm property under additional interface conditions of potential type. The number of conditions will be computed as a function of s. We employ factorisations of symbols which are obtained by reducing the problems to the boundary, combined with reductions of orders. The new results of this chapter are based on the author’s joint papers [72], see Section 4.1, and [71], see Section 4.2.

4.1 Reductions of orders on a manifold with boundary The classical calculus of pseudo-differential operators on a closed, compact C 1 manifold X contains elliptic elements R of every order  2 R which induce isomorphisms R W H s .X / ! H s .X / for all s 2 R. Such order reductions also induce isomorphisms R W Lcl .X / !  Lcl .X / for all  2 R via composition of operators. In particular, every operator of order  can be transformed to an operator of order 0, and vice versa. If X is a compact C 1 manifold with boundary, the task of constructing analogous reductions of orders is not completely straightforward, since elliptic operators usually are connected with elliptic boundary conditions. The present reductions of orders (without such conditions) are based on observations on specific minus/plus symbols which have holomorphic extensions into the upper (lower) complex half-plane with respect to the covariable normal to the boundary.

4.1.1 Order reducing symbols in the half-space In this section we are interested in a particular class of standard symbols of order  that may be used for  2 Z in the calculus of pseudo-differential boundary value problems with the transmission property at the boundary, cf. Chapter 3, or Boutet de Monvel [15], Rempel and Schulze [154], Grubb [69]. We take, in particular, symbols from [67] of the following form. Set x D .y; t / with y D .y1 ; : : : ; yn1 / 2 Rn1 , t 2 R, and covariables D .; /. Choose an element ' 2 S.R/ such that '.0/ D 1 and R0 supp F 1 '  R , for instance, '. / WD c 1 1 e it .t /dt for some 2 C01 .R / R0 such that c WD 1 .t /dt 6D 0. We now set     (4.1.1) hi  i ; r .; / WD ' C hi 1

 2 R, for any constant C > 0 (recall that hi WD .1 C jj2 / 2 ). For our purposes we need the following properties:

226

4 Mixed problems in standard Sobolev spaces

Proposition 4.1.1. (i) r .; / 2 Scl .Rn /. (ii) r .; / is elliptic of order  2 R for a sufficiently large choice of C > 0 and extends with respect to to the upper complex half-plane C i;  > 0, as a holomorphic function which is C 1 for   0, such that jr .; C i/j  c.1 C jj C j j C  /

(4.1.2)

x C and a constant c > 0. for all .; ; / 2 Rn1  R  R (iii) The constant C > 0 in (ii) can be chosen in such a way that Q C jj C j j C  / jr .; C i/j  c.1

(4.1.3)

x C and a constant cQ > 0. for all .; ; / 2 Rn1  R  R Proof. (i) Let us set p. / WD h i1 r . /. By virtue of r . / 2 Scl1 .Rn /, cf. [174] or [188], we have p. / 2 Scl0 .Rn /. Moreover, the symbol p. / is elliptic of order zero, and we have p. / 6D 0 for all 2 Rn , cf. the arguments for the assertions (ii), (iii) below. To show that r . / is classical we write r . / D h i p  . /. Because of h i 2 Scl .Rn / it suffices to show that p  . / 2 Scl0 .Rn /. For every fixed 2 Rn we have by Cauchy’s theorem Z  1 p  . / D d (4.1.4) 2 i L .  p. // for any curve L in the complex plane, where   p. / does not vanish for all 2 Rn (such a curve always exists as we see from the relation (4.1.5) below). Note that  ! .  p. //1 represents a continuous map L ! Scl0 .Rn /. The formula (4.1.4) easily yields p  . / 2 Scl0 .Rn /. In fact, the integral can be written as a limit of finite P  integral sums of the form jND1 .2 i /1 j;N .j;N  p. //1 ıj;N with points j;N 2 L belonging to the i-th interval of the corresponding partition of the curve, where maxfıj;N ; j D 1; : : : ; N g ! 0 as N ! 1. We then have convergence in the Fréchet space Scl0 .Rn /. (ii), (iii) For the case  D 1 we first write  C i  Z 0 1 ' e i. Ci/.C hi/ t .t /dt D C hi 1 R0 for .t/ 2 S.R/, supp  R ; 1 .t /dt D 1. This shows that r .; / extends to a holomorphic function ˇ in C ˇi;  > 0. ˇ  c1 for all .; / 2 Rn ;   0 and some constant Moreover, we have ˇ' Ci C hi c1 > 0. This yields jr .; C i/j  c2 .1 C jj C j j C  /

(4.1.5)

for all .; / 2 Rn ;   0 and some c2 > 0. In the proof below, cf. the relation (4.1.6), we will show that r .; C i/ 6D 0 for all .; / 2 Rn ;   0. Thus

4.1 Reductions of orders on a manifold with boundary

227

log.r .; C i// is well-defined as a holomorphic function in C i for  > 0 by the branch of the logarithm that is real for positive arguments. This gives us an extension of r . / in C i;  > 0, by r .; C i/ D e  log.r .; Ci// . Now the relation (4.1.5) immediately implies the estimate (4.1.2) for   0 with a suitable constant c > 0 and (4.1.3) for   0 with a suitable cQ > 0. We now show that r .; / is elliptic for a sufficiently large C > 0. To this end it suffices to consider the case  D 1. We have     ' C hi  '.0/ ' C hi  '.0/ r .; /

D1C hi D 1 C DW 1 C ˛; hi  i hi  i C.hi  i / C hi ˇ   ˇ ˇ ' C hi '.0/ ˇ ˇ ˇ. For fixed " > 0 there exists a ı."/ > 0 such that where j˛j  ˇ ˇ C hi ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 2c1 for ˇ C hi ˇ  ı."/. Now it j˛j  C1 .j' 0 .0/j C "/ for ˇ C hi ˇ < ı."/ and j˛j  C1 ı."/ follows easily that j˛j < q for a constant q < 1 for all .; / 2 Rn , when C > 0 is sufficiently large. We thus obtain 1 C

jr .; /j  .1  q/jhi  i j  c3 h i

(4.1.6)

for some c3 > 0. This yields the estimate (4.1.2) for   0 and  D 0, and (4.1.3) for   0 and  D 0. Analogous calculations go through for C i;  > 0, where j j in the estimates is to be replaced by j j C . Remark 4.1.2. Let us set  rC .; / WD r .; /

(4.1.7)

(the complex conjugate) for every  2 R. We then have an analogue of Proposition 4.1.1 with the only exception that extensions in concern the lower complex half-plane.  . / have the transmission property at Proposition 4.1.3. For  2 Z the symbols r˙ t D 0.

Proof. First recall (cf. Remark 3.1.8 and 3.1.10) that a symbol a. / 2 Scl .Rn / of integer order  (here, with constant coefficients) has the transmission property at t D 0 if and only if y a.; hi / 2 Scl .Rn1  / ˝ HC1 a.; hi / 2

Scl .Rn1  /

y  H0 ˝

for  2 N; for   1 2 NI

concerning notation, see the formula (3.1.7). In the present case the symbol  0 O r .; hi / D hi .'. =C /  i / belongs to Scl .Rn1  /˝ .H0 ˚ HC1 / which is 0 an immediate consequence of .'. =C /  i / 2 H0 ˚ HC1 for any  2 Z.

228

4 Mixed problems in standard Sobolev spaces

4.1.2 Actions in Sobolev spaces We now turn to pseudo-differential actions between Sobolev spaces in the half-space H s .Rn˙ / WD H s .Rn /jRn , where Rn˙ WD f.y; t / 2 Rn W t 2 R˙ g. Furthermore, we set ˙ x n / D fu 2 H s .Rn / W supp u R x n g. We use the fact, that for every s 2 R there H0s .R ˙ ˙ ˙ is a continuous extension operator es W H s .Rn˙ / ! H s .Rn / such that r ˙ ı es˙ D id on the space H s .Rn˙ /I here, r ˙ f WD f jRn . ˙ The following observation is standard: x n . Then the Fourier transform Remark 4.1.4. Let u 2 S.Rn /, such that supp u R F u.; / extends with respect to to a holomorphic function in C i for  > 0 which is C 1 for   0, and for every N 2 N there is a constant cN > 0, such that .1 C jj C j j C /N jF u.; C i /j  cN :

(4.1.8)

In fact, the assertions are immediate consequences of the representation Z Z 0 F u.; C i/ D e i.yC t/ e  t u.y; t /dydt: Rn1

1

 Lemma 4.1.5. The operators Op.r˙ /;  2 R, induce continuous mappings  x n / ! H s .R xn / Op.r˙ / W H0s .R ˙ ˙ 0

for all s 2 R.  are standard Proof. First, as a consequence of Proposition 4.1.1 and Remark 4.1.2, r˙  symbols of order I then the operators Op.r˙ / W H s .Rn / ! H s .Rn / are continuous x n implies supp Op.r  /u R xn . for all s 2 R. Thus it remains to show that supp u R ˙ ˙ ˙ Let us consider, for instance, minus symbols; the plus-case is analogous and will be dropped. The arguments are the same as in Eskin’s book [44], but for completeness we x n / WD S.Rn /j x n is dense in H s .R x n /, shall recall the main steps here. Because S.R 0 R n x  /. By virtue of Proposition 4.1.1 and Remark 4.1.4 the it suffices to assume u 2 S.R function r .; Ci /F u.; Ci/R is holomorphic in  > 0 and continuous for   0. We have Op.r /u.y; t / D .2/n Rn e iyCit. Ci/ r .; Ci /F u.; Ci /dd for every   0 by Cauchy’s Theorem. Using (4.1.2) and (4.1.8) we obtain Z j Op.r /u.y; t /j  c e t .1 C jj C j j C  / jF u.; C i /jdd  ce Q t Rn

(4.1.9)

for some constants c; cQ > 0. It follows that Op.r /u.y; t / D 0 for t > 0 when we pass in (4.1.9) to the limit  ! C1.  ˙  WD r ˙ Op.r /es ;  2 R, induce isomorProposition 4.1.6. The operators R;s phisms  R;s W H s .Rn˙ / ! H s .Rn˙ / (4.1.10)

4.1 Reductions of orders on a manifold with boundary

229

for all s 2 R .they do not depend on the choice of the extension operator es˙ /, and we   ˙ have .R;s /1 D r ˙ Op.r /es .  ; the case of plus-operators is analogous and will be omitted. Proof. Let us consider R;s Let esC W H s .RnC / ! H s .Rn / be any continuous extension operator. Then the  continuity of R;s W H s .RnC / ! H s .RnC / for every s 2 R is evident. Let us show   C that R;s for any choice of es is a right inverse of R;s . In fact, for u 2 H s .RnC / we have   C R;s u D r C Op.r /esC r C Op.r /es u R;s C u C r C Op.r /v; D r C Op.r / Op.r /es

(4.1.11)

C x n /. By Lemma 4.1.5 we have where v D .esC r C  1/ Op.r /es u 2 H0s .R x n , i.e., r C Op.r /v D 0. The first summand on the right-hand supp Op.r /v R C side of (4.1.11) is equal to r C es u D u. In an analogous manner we can show   1 that R;s has a left inverse, i.e., we have calculated the inverse .R;s / as asserted. Finally, the action

r C Op.r /esC W H s .RnC / ! H s .RnC / is independent of the choice of esC , since for any other extension operator eQ sC of that kind we have r C Op.r /.esC  eQ sC /u D r C Op.r /v D 0 x n /. for v D .esC  eQ sC /u 2 H0s .R Let us define a linear map e˙ W H s .Rn˙ / ! S 0 .Rn / for s >  12 by setting ( e˙ f .x/ WD

f .x/ for t 2 R˙ ; 0 for t 2 R ;

 / in Rn to e˙ f in the x D .y; t/; f .x/ 2 H s .Rn˙ /. This allows us to apply Op.r distributional sense. In the following we use the fact that the operators e˙ W H s .Rn˙ / ! H s .Rn / (extensions by zero) are a possible choice of es˙ for all s 2 R;  12 < s < 12 .   ˙ Proposition 4.1.7. The operators R WD r ˙ Op.r /e ;  2 R, induce isomorphisms  R W H s .Rn˙ / ! H s .Rn˙ /

(4.1.12)

 1  ˙ for all s 2 R; s >  12 , and we have .R / D r ˙ Op.r /es for s     1  ˙  12 ; .R / D r ˙ Op.r /e for s   >  12 .

Proof. As noted before, by virtue of Proposition 4.1.6, it suffices to consider the case  I the plus-case is completely analogous. For u 2 H s .RnC /, s  12 . Let us discuss R

230

4 Mixed problems in standard Sobolev spaces

s     12 , we have C C u D r C Op.r /esC r C Op.r /es u r C Op.r /eC r C Op.r /es C u: C r C Op.r /.eC  esC /r C Op.r /es C Because of s  12 we have v WD r C Op.r /es u 2 H 0 .RnC / and hence eC v 2 0 n C s n 0 n C x n /. This gives H .R /, es v 2 H .R /  H .R /, i.e., .e  esC /v 2 H00 .R C  C C us r Op.r /.e  es /v D 0, and from the proof of Proposition 4.1.6 we see that C  r C Op.r /es is a right inverse of R . Moreover, for f 2 H s .RnC / we have C C r C Op.r /eC f D r C Op.r /es r C Op.r /esC f r C Op.r /es C r C Op.r /.eC  esC /f: C r C Op.r /es

x n / we have as before r C Op.r /.eC  esC /f D 0, Because of .eC  esC /f 2 H00 .R C  C  i.e., r Op.r /es is a left inverse of R . It remains to consider the case s    12 I C because for 12 > s   >  12 we may replace eC by es anyway. For u 2 H s .RnC / we have C u r C Op.r /eC r C Op.r /eC u D r C Op.r /eC r C Op.r /es C  eC /u D 0 by the same arguments as before. Moreover, because r C Op.r /.es C v WD r C Op.r /es u 2 H s .RnC /; s  12 , and we have again r C Op.r /.esC  C e /v D 0. It follows altogether that C u D u: r C Op.r /eC r C Op.r /eC u D r C Op.r /esC r C Op.r /es

Thus the operator r C Op.r /eC is a right inverse of r C Op.r /eC . It is also a left inverse, because the consideration is now symmetric, due to s  12 ; s    12 . Remark 4.1.8. We will employ below symbols in parameter-dependent form, i.e., where  2 Rn1 is replaced by .; / 2 Rn1  Rl for some l. According to Proposi  tion 4.1.6 and 4.1.7 we then have parameter-dependent operators R;s ./ and R ./ l that define isomorphisms (4.1.10) and (4.1.12), respectively, for every  2 R .

4.1.3 A relationship with classical Volterra symbols Let us first recall the following notation. If U C l is an open set and E a Fréchet space, A.U; E/ denotes the space of all holomorphic functions in U with values in E (the space A.U; E/ is endowed with the Fréchet topology that is immediate by the definition). Definition 4.1.9. Let us set H˙ WD f C i 2 C W 2 R;  2 R˙ g. We then define  S ˙ / for  2 R;  Rn1 open, to be the space of all elements .  Rn1  H S.cl/ S ˙ / with the following properties: h.y; ; C i / 2 C 1 .  Rn1  H

231

4.1 Reductions of orders on a manifold with boundary

(i) h.y; ; C i/ 2 A.H˙ ; C 1 .  Rn1  //,  x C; /, i.e., h is a classical symbol of .  Rn1  R  R (ii) h.y; ; C i/ 2 S.cl/ x ˙. order  in the covariables .; ;  / for .; / varying in Rn and  in R

The set  S.cl/ .  Rn /˙ WD fp.y; ; / WD h.y; ; C i /jD0 W  S  /g h.y; ; C i/ 2 S.cl/ .  Rn1  H

(4.1.13)

coincides with corresponding spaces of Volterra (for the case H ) and anti-Volterra (for the case HC ) symbols of anisotropy 1 and order  2 R, cf. Piriou [149], or Krainer [104]. Recall (to motivate the notation) that the inverse .i C j j2 /1 of the anisotropic homogeneous principal symbol of the heat operator  C @ t (which is of anisotropy S. 2 and order 2) is Volterra in the classical sense; in particular, it extends to Rn  H S ˙ / .S  .Rn /˙ / denote the subspace of elements of S  .  Let Scl .Rn1  H cl cl S ˙ / .S  . Rn /˙ / that are independent of y. The following theorem is Rn1  H cl valid for arbitrary  2 R.  S  /. Theorem 4.1.10. (i) We have r˙ .; C i/ 2 Scl .Rn1  H  (ii) r˙ .; C i/ is elliptic of order  with respect to the covariables .; ;  /, i.e., for the homogeneous principal symbols   C i    .r /.; ;  / D ' jj  i. C i / ; C jj   .rC /.; ;  / D  .r /.; ;  /  S  / n f0g we have  .r  /.; ;  / 6D 0. of r of order  in .; ;  / 2 .Rn1  H ˙

Proof. (i) Let us consider, for instance, the minus-case. First we verify that x C; / r .; C i/ 2 S  .Rn1  R  R  (the space on the right of the latter relation is to be interpreted as a symbol space in the x C ignoring the aspect of holomorphy). It suffices to variables .; ;  / 2 Rn1  R  R consider the case  D 1 for similar reasons as in the proof of Proposition 4.1.1 (here x C and for all , we use, in particular, that r .; C i/ 6D 0 for all .; ;  / 2 Rn  R 1 n x C / it suffices to prove cf. Proposition 4.1.1 (iii)). Because of i C  2 S .R  R that  C i  x C /: 2 Scl1 .Rn  R p.; ;  / WD hi' C hi   2 Since we have ' D Ff for a function f 2 S.R/ supported in R , we get ' Ci C x C / by Remark 4.1.4 for the case n D 1. From Proposition 2.2.1 of [196] S 1 .R  R we see that the substitution p.; ;  / ! p.; hi ; hi / induces continuous maps   x C / ! S  .Rn1 /˝ x C/ O  S.cl/ .Rn1  R  R .R  R S.cl/ .cl/

232

4 Mixed problems in standard Sobolev spaces

for all  2 R, both for classical symbols. In the present case we obviously  and general 1 n1 O x C /. Then, since the have p.; hi ; hi/ D hi' Ci 2 S .R /˝ Scl1 .R  R cl C second factor is of order 1; p.; ;  / itself is a classical symbol of first order. (ii) For the proof that r .; C i/ is elliptic of order  it suffices again to consider the case  D 1. From Proposition 4.1.1 we know that there is a constant c > 0 such x C . Together that jr .; C i/j  c.1 C jj C j j C / for all .; ;  / 2 Rn  R  with assertion (i) we conclude that r .; C i / is elliptic of order  in the sense of x C /. symbols in Scl .Rn  R Remark 4.1.11. The considerations so far have a direct generalisation to anisotropic symbols of arbitrary anisotropy m 2 N n f0g. Setting, for instance,     m r .; /m WD ' hi  i

C him we get a corresponding version of Proposition 4.1.1 when we replace jj by jjm in the  estimates (4.1.2) and (4.1.3), respectively. The analogous plus-symbols rC .; /m are then parabolic of order  and anisotropy m in the sense of the work of Krainer [105], see also Section 10.6.3 below. We now apply Definition 4.1.9 and notation (4.1.13) in the variants  S˙/ .  Rn1Cl  H S.cl/

and

 S.cl/ .  RnCl /˙ ;

i.e., where the covariable  2 Rn1 is replaced by .; / 2 Rn1Cl . In particular,  we have the .; /-dependent versions r˙ .; ; / of the symbols (4.1.1) and (4.1.7), respectively. Example 4.1.12. Let J.y/ 2 C 1 ./ ˝ Rn1 ˝ Rn1 be an .n  1/  .n  1/  matrix function on  with real-valued entries. Then p˙ .y; ; ; / WD r˙ .J.y/; ; / belongs to the space Scl .  RnCl /˙ . j Theorem 4.1.13. For every sequence pj .y; ; ; / 2 S.cl/ .  RnCl /˙ , j 2 N,  there is a sequence p.y; ; ; / 2 S.cl/ .  RnCl /˙ such that, for every N 2 N, PN p  j D0 pj 2 S .N C1/ .  RnCl /˙ , and p is unique modulo a symbol in the ˙ class of order 1.

The proof is simple and left to the reader. z be a diffeomorphism. Then the asymptotic summaExample 4.1.14. Let  W  !  tion for the symbol push forward (belonging to the push forward of associated pseudoz  Œ0; 1/, .y; t / ! ..y/; t / can be differential operators under the map   Œ0; 1/ !  carried out in Scl .RnCl /˙ . In fact, according to the standard formula in coordinate substitutions for pseudo-differential operators, the sum has the form p. Q y; Q ; Q ; /jyD.y/  Q

X ˛2N n1

1 ˛ Q ; /ˆ˛ .y; /; Q .@ p/.y;t d.y/; ˛Š 

4.1 Reductions of orders on a manifold with boundary

233

where ˆ˛ .y; / Q D Dy˛ e iı.y;z/Q jzDy for ı.y; z/ D .z/  .y/  d.y/.z  y/ are polynomials in Q of degree  j˛j=2. z be a diffeomorphism, and let g and gQ be Riemannian Remark 4.1.15. Let  W  !  z respectively, such that the associated pairings between sections metrics on  and , of cotangent bundles are invariant in the sense gy .1 ; 2 / D gQ yQ .Q 1 ; Q 2 / for yQ D .y/ Q ; / and Qj D t d.y/1 j , j D 1; 2. Consider symbols r .; ; / on  and r .; z on  with jj and jj Q belonging to g and g, Q respectively. Then, applying the symbol push forward of Example 4.1.14 to p.y; ; ; / WD r .; ; / we have z  RnCl / : Q ; / mod Scl1 . p. Q y; Q ; Q ; / D r .;

4.1.4 Global reduction of orders Let X be an oriented compact C 1 manifold with boundary Y , and let 2X denote the double of X , obtained by gluing together two copies XC ; X of X along their common boundary Y (we then identify X with XC ). Choose a collar neighbourhood V of Y in X with a global splitting of variables into .y; t / for y 2 Y , t 2 Œ0; 1/, and fix a system of charts x nC ; j D 1; : : : ; L; j W Uj ! R

j W Uj ! Rn ; j D L C 1; : : : ; N

(4.1.14)

on X with coordinate neighbourhoods Uj on X , such that Uj \Y 6D ; for j D 1; : : : ; L, and Uj \ Y D ; for j D L C 1; : : : ; N , where Uj D Uj0  Œ0; 1/, j D 1; : : : ; L, for an open covering fU10 ; : : : :UL0 g of Y by coordinate neighbourhoods. Assume for convenience that the functions y.y; Q t / and tQ.y; t / in the transition diffeomorphisms xn ! R x n , .y; t / ! .y.y; j 1 W R Q t /; tQ.y; t //, are independent of t for small t for j D C C k 1; : : : ; L. Let us fix a Riemannian metric on 2X that restricts in a tubular neighbourhood Š Y .1; 1/ of Y to a corresponding product metric with a Riemannian metric g on Y and the standard metric on .1; 1/. Absolute values of covectors  in local coordinates near Y will be taken with respect to g. We now consider local parameter-dependent symbols rQ .t; ; / WD r . ; /!.t/ h ; i.1!.t// (4.1.15) x C /, ! 1 near t D 0). on Rn , where !.t/ is a cut-off function (i.e., ! 2 C01 .R n n x Here, R is regarded as the double of RC in connection with the charts (4.1.14) for 1  j  L. Moreover, for the charts with L C 1  j  N we take symbols h ; i . Let f'1 ; : : : ; 'N g be a partition of unity on X , subordinate to fU1 ; : : : ; UN g, and let f 1 ; : : : ; N g be a system of functions j 2 C01 .Uj / that are equal to 1 on supp 'j . The charts (4.1.14) near the boundary will be chosen as restrictions of charts Qj W Uzj ! Rn for the double Uzj WD 2Uj to Uj , j D 1; : : : L. Then the sets Uz1 ; : : : UzL cover a tubular neighbourhood of Y of the form Y  .1; 1/; let Xz denote the union of X with that tubular neighbourhood. Moreover, let 'Qj 2 C01 .Uzj / be functions such that 'Qj jUj D 'j for j D 1; : : : ; L, and Qj 2 C01 .Uzj / functions that are equal to 1 on

234

4 Mixed problems in standard Sobolev spaces

supp 'Qj , and satisfy Qj jUj D j , j D 1; : : : ; L. In addition we assume the functions 'Qj and Qj to be independent of t with jt j < " for some " > 0. We now form global parameter-dependent pseudo-differential operators on Xz by z ./ WD R

L X

'Qj .Qj1 / Op.rQ /./ Qj C

j D1

N X

'j .j1 / Op.h ; i /

j:

(4.1.16)

j DLC1

l The operator family (4.1.16) (extended by zero to 2X n Xz ) then belongs to L cl .2X I R /. s C s C Set H .int X / WD r H .2X / with r being the restriction to int X , and let esC W H s .int X/ ! H s .2X / denote any continuous extension operator (i.e., r C ı esC D id on the space H s .X /). Moreover, for s >  12 we define eC to be the extension from int X to 2X by zero. The operator

z ./esC W H s .int X / ! H s .int X /; R ./ WD r C R

 2 R;

(4.1.17)

is continuous for all s 2 R (and every fixed ) and does not depend on the choice of esC . z ./eC for s >  1 . Moreover, because of Proposition 4.1.7 we have R ./ D r C R 2 Theorem 4.1.16. There exists a constant c > 0 such that (4.1.17) induces isomorphisms for all jj > c, s 2 R. Proof. Because of our assumptions on the charts and the localising functions 'Qj and Qj in (4.1.16) we may apply Remark 4.1.15; then the operators of the family R ./ have the following properties: For j D 1; : : : ; L the operators Rj ./ WD .j / R ./ in RnC have the form Rj ./u D r C Op.r /./esC u C Tj ./u on functions u 2 H s .RnC / that vanish for .y; t / 62 K  .0; "/ for some K b Rn1 and " > 0 sufficiently small, where Tj ./ 2 L1 .  RC I Rl / is a parametercl  dependent family of order   1. Moreover, Rj ./ WD .j / R ./ for arbitrary j D 1; : : : ; N acts on functions u 2 H s .RnC / for j D 1; : : : ; L and on u 2 H s .Rn / for j D LC1; : : : ; N with compact support as standard classical parameter-dependent  n l n l elliptic operators of the class L cl .RC I R / and Lcl .R I R /, respectively. We now define the system of Leibniz inverses of the local parameter-dependent symbols of the operators Rj ./ and pass to associated operators Pj ./ in RnC or Rn , according to the cases 1  j  L and L C 1  j  N , respectively. For 1  j  L we can choose Pj ./ in such a way that C Pj ./u D r C Op.r /./es u C Sj ./u

on functions u 2 H s .RnC / with support in K  Œ0; "/ for some K b Rn1 and " > 0 .  RC I Rl / . Globally, we form sufficiently small, and an element Sj ./ 2 L1 cl P N the operator family P  ./ WD j D1 'j .j1 / Pj ./ j and obtain P  ./R ./ D 1  Cl ./;

R ./P  ./ D 1  Cr ./;

(4.1.18)

4.1 Reductions of orders on a manifold with boundary

235

where Cl ./ and Cr ./ are operator families in S.Rl ; S.H s .X /; C 1 .X /// for all s 2 R. To see the invertibility of R ./ for large jj we consider, for instance, the first relation of (4.1.18). We have kCl ./kL.H s .int X/;H s .int X//  bhiN for every N 2 N, where b D b.N / > 0 is a suitable constant. We then conclude by a Neumann series argument that R ./ W H s .int X / ! H s .int X /

(4.1.19)

has a left inverse for jj  c1 for some c1 > 0. Analogously, using the second relation of (4.1.18), we also have a right inverse of (4.1.19) for jj  c2 for some c2 > 0. Thus (4.1.19) is invertible for jj  c D max.c1 ; c2 /. A simple argument in terms of elliptic regularity shows that ker R ./ and coker R ./ are independent of s. Thus, the constant c is independent of s. Remark 4.1.17. For  2 Z the operators z ./eC W H s .int X / ! H s .int X / R ./ D r C R

(4.1.20)

s > max.; 0/  12 , belong to B ;0 .X I Rl / (cf. also Remark 3.2.24). They represent reductions of orders R WD R .1 / for fixed 1 2 Rl , j1 j sufficiently large, and we have .R /1 2 B ;0 .X /. Analogous observations are true of operators acting between distributional sections in a vector bundle E. Note that reductions of orders within the calculus have been stated already in [15, Theorem 5.12], more precisely, the corresponding operator is of index 0; however, it is trivial that it can be modified by a smoothing operator to make it invertible and to see that then (using necessity of ellipticity for the Fredholm property) the inverse again belongs to the calculus (cf. also Remark 3.2.24 and Theorem 3.3.9).

4.1.5 General operators with plus/minus-symbols Let  Rn is a bounded domain with C 1 boundary Y that is written as a union Y D Y [ YC of two C 1 manifolds with common C 1 boundary Z D Y \ YC . Let Z  .1; 1/ be some fixed collar neighbourhood of Z in Y , where .Z  .1; 1// \ YC D Z  Œ0; 1/ and .Z  .1; 1// \ Y D Z  .1; 0 : Definition 4.1.18. (i) We define L1 .Y I Rl /˙ as the set of all A./ 2 L1 .Y I Rl / that induce continuous operators A./ W H0s .Y˙ / ! H0t .Y˙ / for all s; t 2 R, where H0s .Y˙ / WD fu 2 H s .Y / W supp u Y˙ g. In other words, the Schwartz kernel K.A.//.y; y 0 / of the operator A./ satisfies K.A.//.y; y 0 / 0 for y 0 2 int YC ; y 2 int Y in the C-case and for y 0 2 int Y ; y 2 int YC in the -case.

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4 Mixed problems in standard Sobolev spaces

l (ii) An operator A./ belongs to L 2 cl .Y I R /˙ iff the following holds: for all '; C 1 .Y / with disjoint supports we have 'A./ 2 L1 .Y I Rl /˙ I for all '; 2 l C01 .int Y˙ / we have 'A./ 2 L 2 C01 .Z ."; "// we have cl .Y I R /I for all '; 'A./ Opyn1 .a.yn1 ; n1 ; // mod L1 .Y I Rl /˙ , where a.yn1 ; n1 C l i; / 2 C 1 .Ryn1 ; L cl .ZI H  R //, and 0 < " 1 is sufficiently small.  Here, Lcl .ZI H  Rl / is the space of all operator families in L cl .Z/, depending on .n1 C i; / 2 H  R that are locally given as Z Z 0 .Opz .a/. Q n1 ; /u/.z/ D e i.zz / a.z; Q z 0 ; ; n1 ; /u.z 0 /dz 0 μ  Rn2



for u 2 C01 ./, where aQ 2 Scl .z  z 0  Rn1  Rl /˙ (cf. (4.1.13)) and ;n1 H WD fn1 C i 2 C W n1 2 R;  2 R g as in Definition 4.1.9. Remark 4.1.19. The (parameter-dependent) principal symbol  .A/ of A./ 2 l  L cl .Y I R /˙ that is defined as a homogeneous function on .T Y  R/ n 0 has the following property:  .A/j.T  .Z.1;1//R/n0 .z; yn1 ; ; n1 ; / extends for yn1 2 ."; "/ with respect to n1 to the half-plane H and is holomorphic in the interior there, where 0 < " 1 is sufficiently small. In the following theorem and below r ˙ are the operators of restriction to int Y˙ and e the extension operators by zero from int Y˙ to Y to the respective opposite sides. (We hope that this notation does not cause confusion; we used r ˙ , etc., before in other situations, see, for example, the beginning of Section 4.1.2.) S S  l l Theorem 4.1.20. (i) 2R L cl .Y I R /˙ form subalgebras of 2R Lcl .Y I R /. In 1 particular, the composition as operator families on C .Y / induces a bilinear map ˙

C l  l L .Y I Rl /˙ : cl .Y I R /˙  Lcl .Y I R /˙ ! Lcl l (ii) Every A./ 2 L cl .Y I R /˙ extends by continuity to a family of operators

A./ W H s .Y / ! H s .Y / for s 2 R: Moreover, it restricts to continuous operators A./ W H0s .Y˙ / ! H0s .Y˙ /. l (iii) For every s 2 R and A./ 2 L cl .Y I R /˙ the operators of the family r  A./es W H s .int Y / ! H s .int Y / are independent of the continuous extension operator es W H s .int Y / ! H s .Y /. Moreover, for B./ 2 Lcl .Y I Rl /˙ the composition  .r  A./es /.r  B./es / W H s .int Y / ! H s.C/ .int Y /

is given as r  A./ ı B./es .

237

4.1 Reductions of orders on a manifold with boundary

(iv) Let e W H s .int Y / ! D 0 .Y /; s >  12 , denote the operator of extension by l zero, and let A./ 2 L cl .Y I R /˙ . Then r  A./e W H s .int Y / ! H s .int Y / is continuous for s >  12 and is equal to r  A./es for any continuous extension operator es W H s .int Y / ! H s .Y /. In particular, for B./ 2 Lcl .Y I Rl /˙ the composition .r  A./e /.r  B./e / W H s .int Y / ! H s.C/ .int Y / is given as r  A./ ı B./e provided that s; s   >  12 .  l l Proof. We first show (i). By definition we have L cl .Y I R /˙ Lcl .Y I R /. Thus  every A./ 2 Lcl .Y I Rl /˙ is continuous as a family of operators

A./ W H s .Y / ! H s .Y /

for every s 2 R:

To prove that A./ W H0s .Y˙ / ! H0s .Y˙ / it suffices to consider the operators 'A./ with functions '; 2 C01 .Z  ."; "// (see Definition 4.1.18). Hence it remains to show that opyn1 .a.yn1 ; n1 C i; // W H0s .Z  .1; 0 / ! H0s .Z  .1; 0 / in the -case, opyn1 .a.yn1 ; n1 C i; // W H0s .Z  Œ0; 1// ! H0s .Z  Œ0; 1// l in the C-case for a.yn1 ; n1 C i; / 2 Cb1 .Ryn1 ; L cl .ZI H  R // (recall that 1 p 1 Cb .R ; E/ for a Fréchet space E denotes the subspace of all u 2 C .Rp ; E/ such that .Dy˛ u.y// is uniformly bounded in y 2 Rp for every ˛ 2 N p and every continuous semi-norm  on E). But this is a consequence of the Paley–Wiener theorem which characterises the Fourier images with respect to yn of H0s .Z  .1; 0 / and H0s .Z  Œ0; 1//, respectively, as spaces of holomorphic functions in H˙ with respect to the variable n1 , and these are respected under multiplication by the symbol a.yn1 ; n1 ; /. More details can be found, e.g., in [45]. This proves (ii). l For (i) it suffices to consider the composition of A./ 2 L cl .Y I R /˙ and B./ 2 Lcl .Y I Rl /˙ (the other properties are evident). From A./B./ 2 LC .Y I Rl / we cl C l 1 obviously obtain 'A./B./ 2 Lcl .Y I R / for '; 2 C0 .int Y˙ /. Moreover, by the already obtained mapping property (ii), we conclude that 'A./B./ 2 L1 .Y I Rl /˙ provided that '; 2 C 1 .Y / have disjoint supports. It remains to consider 'A./B./ for '; 2 C01 .Z  ."; "// for 0 < " 1 sufficiently small. Again, by (ii), we are in fact left with the composition 'A./'Q Q B./ with '; '; Q ; Q 2 C01 .Z  ."; "//, but by Definition 4.1.18 this is the sum of four terms, the only non-trivial one being of the form

opyn1 .a.yn1 ; n1 C i; // ı opyn1 .b.yn1 ; n1 C i; //

(4.1.21)

238

4 Mixed problems in standard Sobolev spaces

l 1  l for a 2 Cb1 .Ryn1 ; L cl .ZI H  R // and b 2 Cb .Ryn1 ; Lcl .ZI H  R // (all other terms belong to L1 .Y I Rl /˙ by the mapping property (ii)).The composition (4.1.21) is equal to opyn1 ..a #yn1 b/.yn1 ; n1 ; // with the Leibniz product

.a #yn1 b/.yn1 ; n1 ; / Z Z 0 0 WD e iy  a.yn1 ; n1 C 0 ; /b.yn1 C y 0 ; n1 ; /dy 0 μ0 ; R

R

and thus a #yn1 b 2 Cb1 .Ryn1 ; LC .ZI H  Rl //. This finishes the proof of (i). cl l Let us now pass to (iii). We consider A./ 2 L cl .Y I R / , the plus-case is analoC C s s .int YC / for every continuous gous. Clearly we have r A./es W H .int YC / ! H extension operator esC W H s .int YC / ! H s .Y /. Let esC ; eQ sC be two such extension operators, and consider the operator r C A./esC  r C A./QesC D r C A./.esC  eQ sC / W H s .int YC / ! H s .int YC /: We have esC  eQ sC W H s .int YC / ! H0s .Y / and (applying (ii)) A./ W H0s .Y / ! H0s .Y /, r C W H0s .Y / ! 0, i.e., r C A./esC D r C AQesC as asserted. Let B./ 2 Lcl .Y I Rl / be another minus operator, and consider the composition C C .r C A./es /.r C B./esC / D r C A./B./esC  r C A./.1  es r C /B./esC : C Note that 1  es r C 2 L.H s .Y // is a projection with range H0s .Y /, and C C consequently r A./.1  es /B./esC D 0. This proves (iii). It remains to show (iv). As eC W H t .int YC / ! H t .Y / is continuous for  12 < t < 12 we conclude that esC  eC W H s .int YC / ! H0t .Y / for s > t >  12 , t < 12 , and, consequently, r C A./.esC  eC / 0 on H s .int YC /. This shows r C A./eC W H s .int YC / ! H s .int YC / for s >  12 , and the remaining assertions in (iv) follow from (iii). l Definition 4.1.21. An operator A./ 2 L cl .Y I R /˙ is called (parameter-dependent) l ˙-elliptic provided that it is (parameter-dependent) elliptic as an element of L cl .Y I R /, and there exists some 0 < " 1 sufficiently small such that  .A/.z; yn1 ; ; n1 ; / is invertible for all jyn1 j < "; .z; ; n1 C i; / 2 .T  Z  H  Rl / n 0. l Theorem 4.1.22. For A./ 2 L cl .Y I R /˙ the following properties are equivalent:

(i) A./ is ˙-elliptic. l (ii) There exists a parametrix P ./ 2 L cl .Y I R /˙ such that both P ./A./  1 l 1 and A./P ./  1 belong to L .Y I R /˙ .

Proof. First (ii) implies (i) by the uniqueness of analytic continuation (applied to the identity  .A/ .P / D 1 D  .P / .A/, which apriori holds for real covectors n1 and extends to the half-plane H ).

4.1 Reductions of orders on a manifold with boundary

239

Assume that A./ is -elliptic (without loss of generality we just treat the -case; the C-case is completely analogous). For some suitable 0 < " < 12 we choose AC ./ 2  l l  L cl .Y I R /,  .AC / invertible on .T Y  R/ n 0; A ./ 2 Lcl .Y I R /,  .A / 1  invertible on .T YC  R/ n 0; a.yn1 ; n1 C i; / 2 Cb .Ryn1 ; L cl .ZI HC  Rl //,  .a/ invertible for jyn1 j  2", .; n1 C i; / 2 .T  Z  HC  Rl / n 0, such that for all '; 2 C01 .int Y˙ /, .supp ' [ supp / \ .Z  Œ 2" ; 2" / D ;, we have 'A./  'A˙ ./ 2 L1 .Y I Rl / . For all '; 2 C01 .Z  ."; "// we have '.A.//  ' opyn1 .a.yn1 ; n1 ; // 2 L1 .Y I Rl / . Furthermore, let l P˙ ./ 2 L 2 C01 .int Y˙ /, .supp ' [ supp / \ .Z  cl .Y I R / such that for '; " " Œ 2 ; 2 / D ;, '.A˙ ./P˙ ./  1/ , '.P˙ ./A˙ ./  1/ 2 L1 .Y I Rl / . Note that such P˙ ./ are obtained by using the standard parametrix construction l for elliptic elements in the class L cl .Y I R /.  We also choose b 2 Cb1 .Ryn1 , Lcl .ZI HC  Rl // such that for all '; 2 1 C0 .Z  ."; "// both '.opyn1 .a.yn1 ; n1 C i; // opyn1 .b.yn1 ; n1 C i; //  1/ and '.opyn1 .b.yn1 ; n1 C i; // opyn1 .a.yn1 ; n1 C i; //  1/ belong to L1 .Y I Rl / . Such a b.yn1 ; n1 ; / may be obtained according to the parametrix construction for Volterra symbols or operators (see, e.g., [107]). Locally, we first invert the principal symbol of a and ‘shift’ in direction of the half-plane HC , i.e., we consider the Volterra symbols ‘ .a/1 .yn1 ; n1 C i; /’. This gives us a ‘rough’ Volterra parametrix with remainders of order 1. The standard formal Neumann series argument can be applied to the remainders as the Volterra symbol classes are asymptotically complete. Now, having P˙ ./ and b.yn1 ; n1 ; /, we define P ./ WD 'C PC ./ C C l ' P ./  C 'O opyn1 .b/ O 2 L O is a partition of cl .Y I R / , where f'C ; ' ; 'g " unity subordinate to the covering fYC n .Z  Œ0; 2 /; Y n .Z  Œ 2" ; 0 /; Z  ."; "/g of Y , and f C ;  ; O g are supported in the same open sets of the covering as 'C ; ' ; 'O with ˙ 1 in a neighbourhood of supp '˙ , and O 1 in a neighbourhood of supp '. O  l l Then P ./ 2 L .Y I R / is the desired parametrix to A./ 2 L .Y I R / , cl cl     and the proof of Theorem 4.1.22 is complete.  Our next goal is to find ˙-elliptic elements R˙ ./ 2 L cl .Y I R /˙ which are    l 1 invertible with inverse R˙ ./ WD .R˙ .// 2 Lcl .Y I R /˙ for arbitrary  2 R.  We first show that they can be obtained from some ˙-elliptic elements RO ˙ .; ı/ 2  lC1 Lcl .Y I R;ı /˙ by fixing the value of jıj sufficiently large, and afterwards show how to construct appropriate elements RO  .; ı/. ˙

Note that the constructions so far with parameter  2 Rl could all have been performed for parameters varying in a conical set. The presence of the additional parameter causes no extra difficulties.

240

4 Mixed problems in standard Sobolev spaces

 1Cl Proposition 4.1.23. Let RO ˙ .; ı/ 2 L cl .Y I R;ı /˙ be ˙-elliptic. Then for any jı0 j >  l c 0 sufficiently large the operator RO ˙ .; ı0 / 2 L cl .Y I R /˙ is ˙-elliptic and induces isomorphisms  RO ˙ .; ı0 / W H s .Y / ! H s .Y /  l .; ı0 //1 belongs to L for all s 2 R and  2 Rl . The inverse .RO ˙ cl .Y I R /˙ .

Proof. Without loss of generality we restrict ourselves to the minus-case. lC1 O z ı/ 2 L Let R.; cl .Y I R;ı / be a parametrix of R .; ı/, according to The z ı/ D 1 C C.; ı/ W orem 4.1.22. For any fixed s0 2 R we have RO  .; ı/R.; H s0 .Y / ! H s0 .Y / with C.; ı/ 2 L1 .Y I RlC1 / . Hence, for jı0 j sufficiently ;ı  large, the operators 1 C C.; ı0 / are invertible in L.H s0 .Y // for all  2 Rl , with the inverse .1 C C.; ı0 //1 D

1 X

.1/j C.; ı0 /j

j D0

D 1  C.; ı0 / C C.; ı0 /.1 C C.; ı0 //1 C.; ı0 /: From the representation as a convergent Neumann series we obtain .1 C C.; ı0 //1 W H0s0 .Y / ! H0s0 .Y /; and the second representation then implies .1 C C.; ı0 //1 D 1 C D./

with D./ 2 L1 .Y I Rl / :

 z ı0 /.1 C D.//g D idH s0 .Y / , i.e., R.; z ı0 /.1 C D.// 2 .; ı0 /fR.; This yields RO    Lcl .Y I Rl / is a right inverse of RO  .; ı0 /. In an analogous manner we obtain that   RO  .; ı0 / has a left-inverse in Lcl .Y I Rl / for jı0 j sufficiently large. This proves the assertion. l Example 4.1.24. The operator function (4.1.16) belongs to L cl .2X I R / and is -elliptic. Applying Proposition 4.1.23 to this case with a two-dimensional parameter, then by fixing jı0 j large enough we obtain an operator family in  the inverse of which l l belongs to L cl .2XI R / for all  2 R . Then (4.1.17) is a family of isomorphisms C z 1 C with the inverse r .R .// es for all  2 Rl . In a similar manner we can proceed for the plus-case.

4.2 Mixed elliptic problems Elliptic mixed problems on a manifold with smooth boundary and jumping conditions along a smooth interface on the boundary of codimension 1 induce continuous operators between

4.2 Mixed elliptic problems

241

standard Sobolev spaces. In this interpretation the solutions of Sobolev smoothness s (sufficiently large) have boundary data which are not independent on both sides of the interface. Nevertheless, the constructions here will be a first essential step for independent conditions to be studied later on in Chapter 5 in terms of weighted edge spaces. We compute the number of additional interface conditions of potential type for a number of typical examples, in particular, for problems of Zaremba type and with jumping oblique derivatives.

4.2.1 Mixed problems for differential operators Let AD

X

a˛ .x/Dx˛

j˛j

be a differential operator in a bounded domain  Rn with C 1 boundary Y , and x for all ˛. A mixed boundary value problem for A is given as a˛ .x/ 2 C 1 ./ Au D f in ;

T˙ u D g˙ on int Y˙ ;

(4.2.1)

where Y is written as a union Y D Y [ YC of two C 1 manifolds with common C 1 boundary Z D Y \ YC . The operators T˙ are defined in the form T˙ u D r ˙ B˙ u; where B˙ D t .B1;˙ ; : : : ; BN;˙ / is a column vector of differential operators Bj;˙ of order m˙;j ; j D 1; : : : ; N , with smooth coefficients in a neighbourhood of Y˙ in Rn , and r ˙ are the operators of restriction to int Y˙ . If H s ./ and H s .int Y˙ / denote the standard Sobolev spaces of smoothness s 2 R on  and on int Y˙ , respectively, the mixed problem (4.2.1) gives rise to continuous operators 0

1

H s ./ ˚

A 1 A D @ T A W H s ./ ! ˚jND1 H sm;j  2 .int Y / TC ˚ N smC;j  1 2 .int YC / ˚j D1 H

(4.2.2)

for all real s > maxfm˙;j C 12 W j D 1; : : : ; N g. We assume that the operator A is elliptic of order  in the usual sense and that the boundary operators T˙ are elliptic on Y˙ with respect to A, i.e., satisfy the Shapiro–Lopatinskij condition up to Z from the respective ˙-sides (recall that the number N is determined by the operator A; for instance, if A is uniformly elliptic of order  D 2m and dim   3, then we have N D m, cf. Section 1.1.1).

242

4 Mixed problems in standard Sobolev spaces

A classical example is the Zaremba problem for the Laplace operator A D with Neumann conditions on int YC and Dirichlet conditions on int Y . In this case we have H s2 ./ ˚ s s 1 @ A 2 .int Y / A D T W H ./ ! H (4.2.3) ˚ TC 3 H s 2 .int YC / ˇ for all s > 32 , where T u WD ujint Y and TC u WD @ t uˇint Y (@ t is the differentiation C in the normal direction). Another example for the Laplace operator is the problem with jumping oblique derivative conditions on Y˙ , where the jump on Z is assumed to have smooth limits from the ˙-sides at Z. The Shapiro–Lopatinskij condition, say, for dim   3 is satisfied if and only if the respective vector fields are never tangent to Y and YC , respectively. To analyse our mixed elliptic problem (4.2.2) we consider another elliptic boundary value problem for the operator A   L 1 D D t A Tz W H s ./ ! H s ./ ˚ jND1 H smj  2 .Y / (4.2.4) 0

1

with smooth boundary operators Tz , satisfying the Shapiro–Lopatinskij condition with respect to A. We assume Tz D t .Tz1 ; : : : ; TzN /; Tzj u WD Bj ujY ; j D 1; : : : ; N , where Bj is a differential operator of order mj with smooth coefficients in a neighbourhood of Y ; the smoothness s > maxfmj C 12 W j D 1; : : : ; N g is kept fixed. We now employ the fact that D has a parametrix   D .1/ D Pz Kz in Boutet de Monvel’s calculus on , cf. Boutet de Monvel [15], Rempel and Schulze [154], and Theorem 3.3.7 (iii). Here, Pz is of the form Pz D rC P eC C G for a pseudox with the transmission property at Y (eC differential operator P in a neighbourhood of  denotes the operator of extension by zero from  to Rn and rC the operator of restriction to ), and G is a Green operator. Moreover, Kz is a vector of potential operators, cf. [15], [154]. Let us compose D from the left by an operator of the form diag.1; R/, where R D diag.R1 ; : : : ; RN / is a diagonal matrix of elliptic pseudo-differential operators Rj 1 of order s mj  12 on Y that induce isomorphisms Rj W H smj  2 .Y / ! L2 .Y /; j D 1; : : : ; N . We then get a Fredholm operator   t A RTz W H s ./ ! H s ./ ˚ L2 .Y; C N / z 1 /. with a parametrix .Pz KR Example 4.2.1. For the Zaremba problem (4.2.3) we take Tz u WD ujY , and the corresponding elliptic boundary value problem D is an isomorphism for all s 2 R; s > 32 . z Then D is invertible; in this case we write D 1 DW .Pz K/.

4.2 Mixed elliptic problems

Using the isomorphism .e eC / W

L2 .Y ; C N / ˚ L2 .YC ; C N /

243

! L2 .Y; C N / we can form the

Fredholm operator

 z 1 e L WD Pz KR

H s ./ ˚  z 1 eC W L2 .Y ; C N / ! H s ./: KR ˚ L2 .YC ; C N /

(4.2.5)

Composing (4.2.5) from the left with the given mixed problem (4.2.2) gives rise to an operator H s ./ H s ./ ˚ ˚ N sm;j  1 2 N 2 .int Y / : AL W L .Y ; C / ! ˚j D1 H (4.2.6) ˚ ˚ 1 L2 .YC ; C N / ˚jND1 H smC;j  2 .int YC / By virtue of DD .1/ D diag.1; 1/ modulo compact operators we obtain 0

1 AL D @ T Pz TC Pz

0 z 1 e T KR z 1 e TC KR

1 0 z 1 eC A T KR z 1 eC TC KR

(4.2.7)

modulo compact operators between the respective spaces. We want to associate with A a Fredholm operator. So we ignore compact remainders and assume, for simplicity, that (4.2.7) holds in exact form. We now employ an order reducing result from Section 4.1 that says that for every s 2 R there exists a classical elliptic pseudo-differential operator s z R 2 Lscl .Y / such that s s C z R WD r C R er W H r .int YC / ! H rs .int YC /

is an isomorphism for every r 2 R. Recall that erC may be replaced by eC for r >  12 . Analogously, we have order reducing operators on the opposite side s s  zC WD r  R er W H r .int Y / ! H rs .int Y / RC

Their explicit form will be given once again in Section 4.2.2 below. For integer s they have been employed in this form by Grubb [67]. Subscripts ‘’ at the operators is motivated by the terminology of minus- or plus-symbols relative to YC , cf. [44] or [72]; we hope this does not cause confusion. These operators allow us to reduce orders in the image of (4.2.7) to zero. In other words, we can pass to the operator A0 WD QAL

(4.2.8)

244

4 Mixed problems in standard Sobolev spaces sm

1

˙;j 2 ˙ z for Q D diag.1; Q ; QC /, Q˙ WD diag.r ˙ R e /j D1;:::;N I the image of Q  s 2 N 2 ./ ˚ L .Y ; C / ˚ L .YC ; C N /. The operator A0 is of block is the space H matrix form, namely, 1 0 1 0 0 z 1 e Q T KR z 1 eC A ; A0 D @ Q T Pz Q T KR (4.2.9) 1  z z 1 eC QC TC Pz QC TC KR e QC TC KR

and the 2  2 lower right corner 



L2 .Y ; C N / L2 .Y ; C N / ˚ ˚ W ! L2 .YC ; C N / L2 .YC ; C N / (4.2.10) represents the reduction of the mixed problem A to the boundary (up to a reduction of orders). The idea of reducing boundary value problems to the boundary is classical, see, for instance, Hörmander [82] or Seeley [203]. Explicit expressions for symbols of reduced operators on the boundary may be found, for instance, in [154, Section 4.2.2.2] (see also Section 1.1.3 and 1.1.4). z 1 e Q T KR z 1 e QC TC KR

z 1 eC Q T KR z 1 eC QC TC KR

Remark 4.2.2. There are elements B; C 2 L0cl .Y I C N ; C N / such that z 1 e D r  C e ; Q T KR

z 1 eC D r  C eC ; Q T KR

z 1 e D r C Be ; QC TC KR

z 1 eC D r C BeC : QC TC KR

z 1 is a vector of potential operators in Boutet de In fact, the operator P WD KR Monvel’s calculus. For the trace operators T˙ we easily find trace operators Tz˙ on Y such that T˙ D r ˙ Tz˙ I to define Tz˙ it suffices to go back to the definition of T˙ in terms of operators B˙ , where the coefficients may assumed to be of compact support in an open neighbourhood of Y˙ in Rn and to set Tz˙ u WD B˙ ujY . Then we have T˙ P D r ˙ Tz˙ P , where Tz˙ P are classical pseudo-differential operators on Y . The compositions with order reducing operators then allow us to write, for instance, in the C-case smC;j  2 C C z smC;j  2 z z z e /r TC P D diag.r C R /TC P; QC TC P D diag.r C R 1

1

smC;j  1 2

z and we have, in fact, B WD diag.R manner we can argue for Q T P .

/TzC P 2 L0cl .Y I C N ; C N /. In a similar

Remark 4.2.3. The principal homogeneous symbols of order zero of C and B are elliptic on Y and YC in the sense that  .C /.y; / W Y C N ! Y C N ;

 .B/.y; / W YC C N ! YC C N

are isomorphisms; Y˙ W T  Y jY˙ n 0 ! Y˙ are the corresponding canonical projections.

4.2 Mixed elliptic problems

245

In fact, because Kz is the potential symbol for a Shapiro–Lopatinskij elliptic boundary condition, the compositions with Shapiro–Lopatinskij elliptic trace conditions T˙ on the ˙-sides produce elliptic operators on the respective parts Y˙ of the boundary. Let us now analyse the elements r  C eC and r C Be . We will discuss, for instance, r  C eC I the operator r C Be is of analogous structure. From now on, for notational convenience we assume N D 1. As a consequence of Remark 4.2.2 and from the pseudo-locality we know that the operator r  C eC W L2 .YC / ! L2 .Y / is smoothing outside Z, i.e., we have r  C eC W L2 .YC / ! C 1 .int Y /. Let !; !Q be functions that are equal to 1 near Z and vanish outside a tubular neighbourhood of Z. We then have r  C eC D !r  C eC !Q C S; Q C .1  !/r  C eC !Q C !r  C eC .1  !/ Q and its where S WD .1  !/r  C eC .1  !/  adjoint S induce continuous operators S W L2 .YC / ! C 1 .int Y /, S  W L2 .Y / ! C 1 .int YC /. Q It can be described in terms of soThus the specific part consists of !r  C eC !. called smoothing Mellin and Green operators. Let U be a tubular neighbourhood of Z in Y; U Š .1; 1/  Z, in a corresponding splitting of variables into .r; z/, and let D Rn2 be an open set, corresponding to a chart on Z. In order to define so-called Green symbols on D we first specify the spaces of Definition 2.4.5 (i) for the case dim X D 0 and briefly recall the corresponding notation. Definition 4.2.4. (i) Let H s; .RC / for s;  2 R denote the completion of C01 .RC / s with respect to the norm k.1 C j Im zj2 / 2 .M u/.z/kL2 . 1 / . 2 

(ii) The space K s; .RC / for s;  2 R; s  0, is defined to be the space of all s u 2 Hloc .RC / such that for some cut-off function !.r/ !u 2 H s; .RC /; .1  !/u 2 H s .RC /.D H s .R/jRC /: Moreover, set K s;ˇ .R / WD fu.r/ W u.r/ 2 K s;ˇ .RC /g and K s;ˇ .R n f0g/ WD fu 2 L2 .R/ W ujR˙ 2 K s;ˇ .R˙ /g:

(4.2.11)

For every s;  2 R the space K s; .RC / is equipped with a scalar product and a corresponding norm such that it is a Hilbert space. Concerning other properties of the spaces K s; .RC / and H s; .RC / we refer to [182], [188] (see also Section 2.4.2). 1 Setting . u/.r/ WD  2 u.r/;  2 RC , we get a strongly continuous group of isomorphisms on the space K s; .RC / for every s;  2 R. For every " > 0 we form the space S" .RC / WD

\ k2N

Ek

k"

for Ek WD hrik r kC1 K k;0 .RC /

246

4 Mixed problems in standard Sobolev spaces

in the topology of the projective limit, and S" .R / WD fu.r/ W u.r/ 2 S" .RC /g. Moreover, let S" .R n f0g/ WD fu 2 L2 .R/ W ujR˙ 2 S" .R˙ /g: The spaces Ek , k 2 N, form a sequence of Hilbert spaces with continuous embeddings    ,! EkC1 ,! Ek ,!    ,! E0 WD L2 .RC / and group action f g2RC , such that S" .RC / D lim Ek :  k2N

A similar observation holds for the spaces S" .R / and S" .R n f0g/. As usual, for any symbol b.r; r 0 ; %/ 2 S  .R  R  R/ we have the associated pseudo-differential operator “ 0 Op.b/u.r/ D e i.rr /% b.r; r 0 ; %/u.r 0 /dr 0 μ %: In the following proposition we set, as before, y D .r; z/ and  D .%; /. Proposition 4.2.5 ([196]). Let p.y; Q / 2 Scl0 .R  D  Rn1 /; D Rn2 open, and let !.r/ and !.r/ Q be cut-off functions on RC .regarded as operators of multiplication/. Then a.z; / WD !r C Op.p/.z; Q /eC !Q belongs to S 0 .D  Rn2 I L2 .RC /; L2 .RC // and to S 0 .D  Rn2 I S" .RC /; S" .RC // for every 0 < " < 12 . The first relation can be obtained by an easy direct calculation; the tools for the second one may be found in [196, Section 2]. Definition 4.2.6. Let H˙ WD L2 .R / ˚ L2 .RC / ˚ C j˙ and F˙ ."/ WD S" .R / ˚ S" .RC / ˚ C j˙ ; " > 0, be endowed with the group action diag. ;  ; id/2RC with the above-mentioned  and the identity operator id in C j˙ . A family of operators g.z; / 2 C 1 .D  Rn2 ; L.H ; HC // is called a Green symbol of order zero if for some " D ".g/ > 0, we have g.z; / 2 Scl0 .D  Rn2 I H ; FC ."// and g  .z; / 2 Scl0 .D  Rn2 I HC ; F ."//I g  .z; / denotes the pointwise adjoint in the sense .gu; v/HC D .u; g  v/H for all u 2 H ; v 2 HC . Associated pseudo-differential operators Opz .g/ on D with respect to z will be called (local) Green operators in our transmission algebra. Note that a Green symbol g.z; / can be written as a 33 block matrix .gij /i;j D1;2;3 where g11 .z; / 2 Scl0 .D  Rn2 I L2 .R /; S" .R //, g12 .z; / 2 Scl0 .D  Rn2 I L2 .RC /; S" .R //, etc. Let  W R ! R denote the reflection diffeomorphism .r/ WD rI we also employ  in the sense  W R˙ ! R without indicating ˙-signs. We then

4.2 Mixed elliptic problems

247

have, for instance, g11 .z; /  2 Scl0 .D  Rn2 I L2 .RC /; S" .R //,   g11 .z; / 2 Scl0 .D  Rn2 I L2 .R /; S" .RC //, etc. We now turn to another class of operator-valued symbols, connected with the Mellin transform on RC , cf. the notation of Section 2.4.1, which is first defined for u 2 C01 .RC / (with the Mellin covariable w 2 C) and then extended to L2 .RC / (with w varying on fw W Re w D 12 g). Recall M induces an isomorphism M W L2 .RC / ! R thatw 2 1 1 L .RC / with M g.r/ D .2 i /  1 r g.w/dw; here ˇ WD fw 2 C W Re w D ˇg. 2

Let Scl .ˇ / for  2 R denote the space of Hörmander’s classical symbols of order  on ˇ (with constant coefficients), where Im w for w 2 ˇ is interpreted as the covariable. Then opM .h/u WD M 1 h.w/M u for h.w/ 2 Scl0 . 1 /, and u 2 L2 .RC / defines a continuous operator opM .h/ W 2

L2 .RC / ! L2 .RC /. Recall from the beginning of Section 4.1.3 that we have spaces A.U; E/ on an open set U C for any Fréchet space E. In particular, for E D C we simply write A.U /. Let M"1 denote the subspace of all h 2 A.fw W 12  " < Re w < 12 C "g/ such that hjˇ 2 S.ˇ / for every 12  " < ˇ < 12 C ", uniformly in ˇ in every compact subinterval. In our consideration we are interested, in particular, in symbols h 2 S. 1 / .D S 1 . 1 // that extend to elements in M"1 for some " > 0. 2 2 x C/ Let !.r/ and !.r/ Q be arbitrary cut-off functions on RC (i.e., elements in C01 .R that are equal to 1 in a neighbourhood of 0). Moreover, let  ! Œ denote any strictly positive function in C 1 .Rn2 / such that Œ D jj for jj > c where c > 0 is some constant. Then for Q W L2 .RC / ! L2 .RC / m.z; / WD !.rŒ / opM .h/.z/!.rŒ /

(4.2.12)

we have m.z; / 2 Scl0 .D  Rn2 I L2 .RC /; L2 .RC // 1

.D; M"1 /,

and m.z; / 2 R for every h.z/ 2 C for every "1 > 0 where "2 D min."; "1 /. Moreover, we have Scl0 .D

n2

(4.2.13)

I S"1 .RC /; S"2 .RC //

m.z; /  2 Scl0 .D  Rn2 I L2 .R /; L2 .RC //;

(4.2.14)

m.z; /  2 Scl0 .D  Rn2 I S"1 .R /; S"2 .RC //;

(4.2.15)

etc. These relations are a consequence of the  -homogeneity in  for large jjI for instance, we have m.z; / D  m.z; /1 for all   1; jj  c for some c > 0. Remark 4.2.7. Let us form g.z; / WD ' m.z; / with '.r; z/; .r; z/ contained x C  D/ and m.z; / as in (4.2.12); the functions ' and are regarded as in C01 .R operators of multiplication (as such they belong to S 0 .D  Rn1 I L2 .RC /; L2 .RC //). Then we have g.z; /; g  .z; / 2 Scl0 .D  Rn1 I L2 .RC /; L2 .RC //

248

4 Mixed problems in standard Sobolev spaces

with  denoting the pointwise adjoint. For '.r; z/ or

.r; z/ 2 C01 .RC  D/ we have

g.z; /; g  .z; / 2 S 1 .D  Rn1 I L2 .RC /; S" .RC //

(4.2.16)

for some " > 0. Analogous observations hold for m.z; /  ,   m.z; / and   m.z; /  . We now choose a covering of Y by coordinate neighbourhoods fUj gj D1;:::;N , with the property Uj0 WD Uj \ Z 6D ;, j D 1; : : : ; L, and Uj \ Z D ;, j D L C 1; : : : ; N , for some L such that the sets fUj0 gj D1;:::;L form an open covering of Z by coordinate neighbourhoods. Without loss of generality we assume that .1; 1/  Uj0 D Uj for S j D 1; : : : ; L and that .1; 1/  Z D jLD1 Uj is the above tubular neighbourhood of Z. Let f'j gj D1;:::;N be a partition of unity, subordinate to fUj gj D1;:::;N I then f'j0 gj D1;:::;L for 'j0 WD 'j jZ is a partition of unity, subordinate to fUj0 gj D1;:::;N . Choose a system of functions f j gj D1;:::;N ; j 2 C01 .Uj /, such that j 1 on supp 'j for all j D 1; : : : ; N , and set j0 WD j jZ for j D 1; : : : ; L. To introduce adequate function spaces and operators on Y n Z we first choose charts j0 W Uj0 ! Rn2 and j W Uj ! R  Rn2 for j D 1; : : : ; L, such that j .r; z/ D .r; j0 .z// for  12  r  12 . The mappings j ; 1  j  L, then induce diffeomorphisms #j WD j jUj nUj0 W Uj nUj0

!

Š

.R Rn2 /[.RC Rn2 / D Rn1 nRn2 : (4.2.17)

Let j˙ W Uj˙ n Uj0 ! R˙  Rn1 ; j D 1; : : : ; L, denote the restriction of (4.2.17) to Uj \ .int Y˙ / DW Uj˙ n Uj0 and 'j˙ and j˙ the restriction of 'j and j , respectively, to Uj˙ n Uj0 . For L C 1  j  N we employ charts of the form j W Uj ! Rn1 . Given a Hilbert space H with a strongly continuous group f g2RC of isomorphisms, we have the space W s .Rq ; H / of Definition 1.2.1, see also Section 2.2.2. s s .D; H / and Wloc .D; H / on Recall that there are also ‘comp’ and ‘loc’ versions Wcomp q open sets D R . Similar notation make sense for Fréchet spaces E with group action. z D Rq open, E and Ez (Fréchet) Proposition 4.2.8. Let a.z; / 2 S  .DRq I E;’E/; 0 spaces with group action. Then Op.a/u.z/ WD e i.zz / a.z; /u.z 0 /dz 0 μ  induces s s z for all s 2 R. .D; E/ ! Wloc .D; E/ continuous operators Op.a/ W Wcomp Proposition 4.2.8 is a simple Fréchet space analogue of Corollary 2.3.46. We will employ W s .Rq ; H / for the spaces K s;ˇ .R˙ / and K s;ˇ .R n f0g/, where in 1 (4.2.11) we assume s; ˇ  0. Clearly  W u.r/ !  2 u.r/;  2 RC , defines strongly continuous groups of isomorphisms also in the spaces K s;ˇ .R / and (4.2.11), and we can form W s;ˇ .R˙  Rn2 / WD W s .Rn2 ; K s;ˇ .R˙ //; W s;ˇ .Rn1 n Rn2 / WD W s .Rn2 ; K s;ˇ .R n f0g//:

4.2 Mixed elliptic problems

249

In a similar manner, we can define ‘comp’ and ‘loc’ spaces with respect to the variable s;ˇ s;ˇ z 2 D in an open set D Rn2 I these are denoted by Wcomp.z/ .R˙ D/, Wloc.z/ .R˙  D/, etc. For instance, we set s;ˇ s Wcomp.z/=loc.z/ .RC  D/ WD Wcomp=loc .D; K s;ˇ .RC //:

Remark 4.2.9. The operator Opz .m/ with m.z; / of the form (4.2.12) represents a s;0 s;0 continuous mapping Opz .m/ W Wcomp.z/ .RC D/ ! Wloc.z/ .RC D/ for every s  0. In fact, it suffices to apply Proposition 4.2.8 to the symbol (4.2.13) (u interpreted as an L2 .RC /-valued function on D). As a consequence of the relation (4.2.16) we have Opz .m/ 2 L1 .RC  D/ (which is the space of all smoothing operators on the respective open set). Thus we can talk about operator pull backs .jC / ' Opz .m/ ..jC /1 / of ' Opz .m/ for any x C  D/ with respect to charts C W Uj n U 0 ! RC  Rn2 to Uj n U 0 . '; 2 C01 .R j j j Definition 4.2.10. (i) Let W s;ˇ .Y n Z/ for s; ˇ  0 defined to be the set of all u 2 L2 .Y / such that .#j1 / .'j u/ 2 W s;ˇ .Rn1 n Rn2 / for j D 1; : : : ; L, and .j1 / .'j u/ 2 H s .Rn1 / for j D L C 1; : : : ; N . Moreover, we set W s;ˇ .int Y˙ / WD fr ˙ u W u 2 W s;ˇ .Y n Z/g.T T 0 (ii) Let W"s .Y nZ/ WD 0<"0 <" W s;" .Y nZ/ and S" .Y nZ/ WD s2N W"s .Y nZ/, and define by restriction to int Y˙ the spaces W"s .Y˙ n Z/ WD fr ˙ u W u 2 W"s .Y n Z/g; and S" .int Y˙ / WD fr ˙ u W u 2 S" .Y n Z/g, respectively. For the calculus of global transmission operators we also admit operators, acting between sections of (smooth, complex) vector bundles. Let Vect.  / denote the set of all smooth complex vector bundles on the space in the parentheses. Given any J 2 Vect.Z/ we fix a Hermitian metric in J , and we then have the L2 -space L2 .Z; J / of square integrable sections in J . In particular, for J D Z  C j (the trivial bundle of fibre-dimension j ) we simply have L2 .Z; C j /, the space of L2 -functions with values in C j . More generally, H s .Z; J / for J 2 Vect.Z/ denotes the Sobolev space of smoothness s 2 R of distributional sections in J . Let Y1 " .Y n ZI J ; JC / for J ; JC 2 Vect.Z/ denote the space of all operators C 2 L.L2 .Y / ˚ L2 .Z; J /; L2 .Y / ˚ L2 .Z; JC // such that, for all 0 < "0 < ", C and its adjoint C  induce continuous operators L2 .Y / S"0 .Y n Z/ ˚ ˚ CW ! 2 1 L .Z; J / C .Z; JC /

and

L2 .Y / S"0 .Y n Z/ ˚ ˚ C W ! : 2 1 L .Z; JC / C .Z; J / 

The adjoint C  is defined by .u; C  v/L2 .Y /˚L2 .Z;J / D .C u; v/L2 .Y /˚L2 .Z;JC / for all u; v in the respective L2 -spaces.

250

4 Mixed problems in standard Sobolev spaces

Alternatively, we also interpret C as a 3  3 block matrix corresponding to canonical identifications L2 .Y / D L2 .Y / ˚ L2 .YC / and S"0 .Y n Z/ D S"0 .int Y / ˚ S"0 .int YC /. Let us define the space Y0G;" .Y nZI J ; JC / to be the set of all operators H WD G CC PL for arbitrary C 2 Y1 " .Y n ZI J ; JC / and G D j D1 Gj . Here, Gj WD diag.'j ; 'jC ; 'j0 /f.j0 /1  Opz .gj /g diag.

 j ;

C j ;

0 j/

with arbitrary Green symbols gj .z; / in the sense of Definition 4.2.6, for D WD Rn2 . 0 0 n2 The operator-push forward .j0 /1 and the  refers both to the charts j W Uj ! R transition maps of bundles J and JC with the fibre dimensions j and jC , respectively. Definition 4.2.11. Let Y0" .Y n ZI J ; JC / denote the space of all operators     r A e 0 A WD diag ; 0 C diag..Mlk /l;kD1;2 ; 0/ C H ; (4.2.18) 0 r C AC e C where the summands are defined as follows: (i) A˙ 2 L0cl .Y /; H 2 Y0G;" .Y n ZI J ; JC /, P (ii) M11 WD jLD1 'j .j / Opz .m11;j /..j /1 / j , m11;j .z; / D !.rŒ / opM .h11;j /.z/!.rŒ /; Q h11;j .z/ 2 C 1 .Rn2 ; M"1 /I z P (iii) M12 WD jLD1 'j .j / Opz .m12;j   /..jC /1 / jC , m12;j .z; / D !.rŒ / opM .h12;j /.z/!.rŒ /; Q h12;j .z/ 2 C 1 .Rn2 ; M"1 /; z P (iv) M21 WD jLD1 'jC .jC / Opz .  m21;j /..j /1 / j , m21;j .z; / D !.rŒ / opM .h21;j /.z/!.rŒ /; Q h21;j .z/ 2 C 1 .Rn2 ; M"1 /; z P (v) M22 WD jLD1 'jC .jC / Opz .  m22;j   /..jC /1 / jC , m22;j .z; / D !.rŒ / opM .h22;j /.z/!.rŒ /; Q h22;j .z/ 2 C 1 .Rn2 ; M"1 /. z Notation is to be interpreted as in Remark 4.2.9 for M11 , and, analogously for the other entries, cf. also the formulas (4.2.14), (4.2.15). 0 0 Let YM CG;" .Y n ZI J ; JC / denote the space of all elements A 2 Y" .Y n ZI J ; JC / such that in the representation (4.2.18) the operators AC ; A vanish. We call the operators in Y0G;" .Y n ZI J ; JC /

0 .resp. YM CG;" .Y n ZI J ; JC //

Green (resp. smoothing Mellin plus Green) operators in the transmission algebra (of order zero) on Y with respect to the interface Z. If the fibre-dimensions of J˙ are both zero, we simply omit the bundle data in the notation. Finally, we set [ Y0 .Y n ZI J ; JC / WD Y0" .Y n ZI J ; JC /; (4.2.19) ">0 0 0 and define YM CG .Y n ZI J ; JC / and YG .Y n ZI J ; JC / in a similar manner.

4.2 Mixed elliptic problems

251

Remark 4.2.12. A crude version of the space of operators in Definition 4.2.11 has been introduced in Rempel and Schulze [160], based on the results of [155], also motivated by mixed elliptic problems, see also [158]. Theorem 4.2.13. Operators A 2 Y0" .Y n ZI J ; JC / are continuous as maps A W W"s1 .Y n Z/ ˚ H s .Z; J / ! W"s2 .Y n Z/ ˚ H s .Z; JC / for every s  0 and every "1 > 0 with "2 D min."1 ; "/. Proof. This result can be reduced to local representatives of our operators, modulo Y1 " .Y n ZI J ; JC /. In this case the asserted continuity holds by definition. In localisations over Y n Z, the operators AC and A only contribute non-smoothing terms, but those correspond to L0cl .int YC / and L0cl .int Y /, respectively, where the continuity in local Sobolev spaces is standard. Thus the specific part concerns operators in a tubular neighbourhood of Z. This allows us to apply a reflection argument, i.e., replacing r by r transforms the operators to the plus-side Œ0; 1/  Z, or, locally, to RC  Rn2 . Then A is locally nothing other than Opz .a/ with an operator-valued symbol a.z; /, cf. Definition 4.2.6, the relation (4.2.13) and Proposition 4.2.5. Then it suffices to apply Proposition 4.2.8 (cf. also Remark 4.2.9). Remark 4.2.14. The (so-called) transmission algebra (4.2.19) is a modification of the x [ R x C as the model edge algebra on Y , where Z  Y is interpreted as an edge and R cone of local wedges near Z. An analogous situation for YC (or Y ) with boundary Z has been studied in [196], and we will apply this information here. Observe that on the one hand the edge algebra of [188] with asymptotics is more specific than (4.2.19) because it contains meromorphic Mellin symbols, on the other hand the edge algebra is much larger, because it contains operators with arbitrary edgedegenerate interior symbols. From that point of view (4.2.19) is a subalgebra of the edge algebra on Y (with edge Z), although without asymptotics (except for an "-strip around the reference weight line  1 ). 2

Theorem 4.2.15. For every B 2 L0cl .Y / we have 0 r ˙ Be 2 YM CG .Y n Z/

(i.e., r  BeC is of the type of a right upper corner, r C Be of a left lower corner, etc., of the space of 2  2 block matrices consisting of smoothing Mellin plus Green operators in the transmission algebra). This result is an analogue of a characterisation of transmission operators of [196, Theorem 3.3.3]. Our operator class here is larger than that in [196], but an inspection of the proof shows that the M C G-property also holds in our case. Similar operators on the half-axis appear in Eskin [44, proof of Lemma 15.3]. In our context this corresponds to the boundary or transmission symbolic level, cf. the terminology below.

252

4 Mixed problems in standard Sobolev spaces

We want to apply this to the case of our original N  N systems, generated by the given mixed conditions. In this situation it is even more convenient, instead of corresponding trivial vector bundles Y˙  C N to admit arbitrary (smooth, complex) vector bundles E˙ ; F˙ 2 Vect.Y˙ /, where E˙ D EjY˙ ; F˙ D F jY˙ for some E; F 2 Vect.Y /. Let us set E 0 WD EjZ ; F 0 WD F jZ . We assume all bundles to be equipped with Hermitian metrics. We then have a straightforward generalisation of the spaces in Definition 4.2.10 (ii) to the case of distributional sections in our bundles, i.e., we have spaces W"s .Y n Z; E/; W"s .Y˙ n Z; E˙ /, and S" .Y n Z; E/, etc. This gives rise to an evident generalisation of the space of smoothing operators, denoted by Y1 " .Y n ZI v/ for v D .E; F I J ; JC /. More generally, using invariance of our local operator spaces (similarly as a corresponding invariance in boundary value problems without the transmission property, see [196]) we have global operator spaces 0 0 Y0G .Y n ZI v/; YM CG .Y n ZI v/ as well as Y .Y n ZI v/, based on corresponding spaces with subscript " > 0. Remark 4.2.16. An A 2 Y0" .Y n ZI v/ for v D .E; F I J ; JC / induces continuous operators A W W"s1 .Y n Z; E/ ˚ H s .Z; J / ! W"s2 .Y n Z; F / ˚ H s .Z; JC / for every "1 > 0 with "2 D min."1 ; "/, for all s  0. This is a slight generalisation of Theorem 4.2.13 to the case of actions in sections of vector bundles E and F also in the upper left corner. Let us now introduce the principal symbolic structure of elements A in Y0 .Y nZI v/. Principal symbols consist of triples  .A/ WD .  .A/;  C .A/; tr .A//; where  ˙ .A/ WD  .A˙ / are the standard homogeneous principal symbols of A˙ on the respective sides, cf. the representation (4.2.18). In local coordinates near Z, in the splitting of variables .r; z/, we have, for instance, for the plus-side,  .AC / D  .AC /.r; z; ; / which is smooth in r up to zero, and there is a so-called principal transmission symbol (which is an analogue of the former boundary symbol) tr .r C AC eC /.z; / WD r C  .AC /.0; z; Dr ; /eC W L2 .RC / ˝ E 0 ! L2 .RC / ˝ F 0 : Notice that

tr .r C AC eC /.z; / D  tr .r C AC eC /.z; /1

(4.2.20)

for all  2 RC ; .z; / 2 T  Z n 0. Analogously we have tr .r  A e /.z; / WD r   .A /.0; z; Dr ; /e W L2 .R / ˝ E 0 ! L2 .R / ˝ F 0 with  -homogeneity of order 0. Concerning M11 which is locally of the form Opz .m/ for a Mellin symbol (4.2.12) (with h11 instead of h) we set tr .M11 /.z; / WD !.rjj/ opM .h11 /.z/!.rjj/ Q

4.2 Mixed elliptic problems

253

regarded as a family of maps tr .M11 /.z; / W L2 .R / ˝ E 0 ! L2 .R / ˝ F 0 ; parametrised by .z; / 2 T  Z n 0. For M12 , locally being of the form Opz .m/  , we set Q tr .M12 /.z; / WD !.rjj/ opM .h12 /.z/  !.rjj/ which is a family of maps tr .M12 /.z; / W L2 .RC / ˝ E 0 ! L2 .R / ˝ F 0 : Analogously we form tr .M21 /.z; / and tr .M22 /.z; /. Similarly as (4.2.20) we have tr .Mlk /.z; / D  tr .Mlk /.z; /1 for all  2 RC ; .z; / 2 T  Z n 0 and l; k D 1; 2. Finally, Green operators in the transmission algebra are classical pseudo-differential operators along Z of order zero (modulo smoothing operators) with classical operator-valued symbols, cf. Definition 4.2.6. As such they have homogeneous principal symbols of order zero 0 2 1 0 2 1 L .R / ˝ E 0 L .R / ˝ F 0 B C B C ˚ ˚ B C B C  B 2  B 2 0C 0C tr .G /.z; / W Z BL .RC / ˝ E C ! Z BL .RC / ˝ F C ; @ A @ A ˚ ˚ J JC with Z W T  Z n 0 ! Z being the canonical projection. We then get altogether the homogeneous principal transmission symbol tr .A/ D diag



  tr .r  A e / 0 ; 0 C diag..tr .Mij //j D1;2 ; 0/ C tr .G /; 0 tr .r C AC eC / 0

1 0 2 1 L2 .R / ˝ E 0 L .R / ˝ F 0 B C B C ˚ ˚ B C B C  B 2  B 2 0C 0C tr .A/ W Z BL .RC / ˝ E C ! Z BL .RC / ˝ F C : @ A @ A ˚ ˚ J JC Homogeneity in this case means tr .A/.z; / D diag. ;  ; id/tr .A/.z; / diag.1 ; 1 ; id/ for all  2 RC ; .z; / 2 T  Z n 0.

(4.2.21)

254

4 Mixed problems in standard Sobolev spaces

Remark 4.2.17. It may be convenient to transform tr .A/ into the operator family @ .A_ / W D diag.1;   ; 1/@ .A/ diag.1;   ; 1/ W 0 2 1 0 2 1 L .R / ˝ E 0 L .R / ˝ F 0 B C B C ˚ ˚ B C B C  B 2  B 2 0C 0C L L .R / ˝ E .R / ˝ F Z !    ZB B C C; @ A @ A ˚ ˚ J JC where  W R ! R is the above-mentioned reflection map  W r ! r, here, interpreted as  W RC ! R or  W R ! RC . In particular, looking at r  A e in local coordix   ;  Rn2 open, the reflection produces an operator nates in the half-space R _ A from A , such that C     r C A_  e D  .r A e / : Theorem 4.2.18. A 2 Y0 .Y n ZI v1 /, v1 D .F0 ; F I J0 ; JC /, B 2 Y0 .Y n ZI v2 /, v2 D .E; F0 I J ; J0 /, implies AB 2 Y0 .Y n ZI v/; v D .E; F I J ; JC /, and we have  .AB/ D  .A/ .B/

(4.2.22)

(with component-wise multiplication). Moreover, if A or B belongs to the space with subscript M C G or G, then so is the composition AB. Theorem 4.2.18 can be proved in a similar manner as a corresponding composition result on int YC (or int Y ) of [196], though our operator class is more general than the one in [196] which is studied there including asymptotics. The reduction to the one-sided case essentially follows by a local reflection argument, similarly as in Remark 4.2.17. The operators in the transmission algebra have a subordinate symbolic structure, namely, the conormal symbols from the cone algebra on the half-axis, here, for two copies R˙ of the half-axis normal to the interface Z. By definition, A 2 Y0 .Y n ZI J ; JC / has a 2  2 matrix as upper left corner of the form A D diag.r  A e ; r C AC eC // C M C G; (4.2.23) cf. the representation (4.2.18). To express the conormal symbol M .A/.z; w/ of the operator (4.2.23), we first consider r C AC eC . Let us set g C .w/ WD .1  e 2 iw /1 ;

g  .w/ WD 1  g C .w/;

w 2 C, and let aC .r; z; %; / denote the homogeneous principal symbol (of order 0) of AC in local coordinates y D .r; z/ near Z, with covariables  D .%; /. Following Eskin [44] we now introduce the conormal symbol of the truncated operator on the plus side. Definition 4.2.19. We set c .r C AC eC /.z; w/ WD aC .0; z; C1; 0/g C .w/ C aC .0; z; 1; 0/g  .w/:

(4.2.24)

4.2 Mixed elliptic problems

255

Observe that (4.2.24) is a meromorphic function in w 2 C with simple poles at the real integers, smoothly dependent on z. Analogously, we form C _ C _  c .r C A_  e / WD a .0; z; C1; 0/g .w/ C a .0; z; 1; 0/g .w/; _ where A_  is as in Remark 4.2.17 and a .r; z; %; / its homogeneous principal symbol in local coordinates. Notice that when a .r; z; %; / is the homogeneous principal _ .r; z; %; / D a .r; z; %; /. It follows that symbol of A , then a C  C c .r C A_  e /.z; w/ D a .0; z; 1; 0/g .w/ C a .0; z; C1; 0/g .w/; C and we define c .r  A e /.z; w/ WD c .r C A_  e /.z; w/. Set

 h11 .z; w/ h12 .z; w/ I c .M /.z; w/ WD h21 .z; w/ h22 .z; w/ 

then the conormal symbol of the operator (4.2.23) is defined as c .A/.z; w/ WD c diag.r  A e ; r C AC eC /.z; w/ C c .M /.z; w/ which is a 2  2 matrix of functions in an "-strip around  1 , smoothly depending on z. 2

Proposition 4.2.20. Under the conditions of Theorem 4.2.18 we have c .AB/ D c .A/c .B/: Proof. The relation (4.2.22) implies @ .AB/ D @ .A/@ .B/ and hence (in the notation of Remark 4.2.17) @ .A_ B _ / D @ .A_ /@ .B _ /. The 2  2 upper left corners of operators @ .  /.z; w/ for every fixed z are elements in the cone algebra on RC . In multiplications these upper left corners behave like a subalgebra of the 3  3 block matrix algebra of boundary symbols, modulo Green operators coming from the entries of finite rank. The latter contributions do not affect the conormal symbols. Thus the assertion follows from the multiplicativity of principal conormal symbols in the cone algebra, see, for instance, [188, Theorem 2.4.15], or Theorem 6.2.5 below. We now turn to ellipticity, the Fredholm property, and parametrices of operators in Y .Y n ZI v/, v D .E; F I J ; JC /. 0

Definition 4.2.21. An A 2 Y0 .Y n ZI v/ with v WD .E; F I J ; JC / is called elliptic, if (i)  ˙ .A/ W Y E˙ ! Y F˙ are isomorphisms, where Y˙ W T  Y˙ n 0 ! Y˙ ˙ ˙ are the canonical projections, (ii) tr .A/ represents an isomorphism (4.2.21).

256

4 Mixed problems in standard Sobolev spaces

Condition (ii) in Definition 4.2.21 may be regarded as an analogue of the Shapiro– Lopatinskij condition for transmission operators. Let us briefly discuss the question to what extent a pair of elliptic operators A˙ on int Y˙ (that are restrictions of pseudo-differential operators on Y to the corresponding ˙-sides) admit Shapiro–Lopatinskij conditions. An analogous problem is well-investigated for standard boundary value problems for pseudo-differential operators with the transmission property at the boundary, see, for instance, Atiyah and Bott [6], Boutet de Monvel [15], Schulze [190], and Section 10.5.3 below. A situation close to the present one is the case without the transmission property, see the article [155], or Schulze and Seiler [196]. For the case of transmission operators in our algebra we consider at once the case of 2  2 block matrices A WD .Aij /i;j D1;2 2 Y0 .Y n ZI w/ for w WD .E; F I 0; 0/, i.e., A D diag.A ; AC / C M C G 0 for M C G 2 YM CG .Y n ZI w/. Consider the transmission symbol

L2 .R / ˝ E 0 L2 .R / ˝ F 0 ˚ ˚ tr .A/.z; / W ! : L2 .RC / ˝ E 0 L2 .RC / ˝ F 0

(4.2.25)

If A is the 2  2 upper left corner of an elliptic element A 2 Y0 .Y n ZI v/, the transmission symbol (4.2.25) is always a family of Fredholm operators for all .z; / 2 T  Z n 0. Remark 4.2.22. Let A 2 Y0 .Y n ZI w/ be a  -elliptic operator in the sense of Definition 4.2.21 (i). Then tr .A/.z; / is a family of Fredholm operators if and only if E0 F0 c .A/.z; w/ W ˚ ! ˚ E0 F0 is a family of isomorphisms for all z 2 Z; w 2  1 . 2

This result belongs to the information on pseudo-differential operators on the halfaxis, analogously to [44, Section 15], see also [154] or [196]. Because of  -homogeneity of tr .A/ we may restrict .z; / to the unit cosphere bundle S  Z, i.e., to jj D 1. This is a compact space, and then there is an index element indS  Z tr .A/ 2 K.S  Z/; of the Fredholm family (4.2.25), see also Section 3.3.4. Recall that K.  / denotes the K-group of the space in parentheses. Let 1 W S  Z ! Z be the restriction of Z to S  Z and 1 W K.Z/ ! K.S  Z/ induced by the pull back of bundles. The concept of ellipticity in Y0 .Y n ZI v/ is very close to that in the general edge algebra, cf. Remark 4.2.14 and the subsequent comments. Therefore, the following theorems are analogous to corresponding results in the general edge calculus; the modification of proofs is easy and left to the reader.

4.2 Mixed elliptic problems

257

Theorem 4.2.23. Let A 2 Y0 .Y n ZI w/ for w WD .E; F I 0; 0/ be an operator such that  ˙ .A/ 6D 0 on T  Y˙ n 0, and let (4.2.25) be a family of Fredholm operators for all .z; / 2 T  Z n 0. Then there is an elliptic operator A 2 Y0 .Y n ZI v/ for v WD .E; F I J ; JC / with suitable J ; JC 2 Vect.Z/, containing A as a (2  2 block matrix) upper left corner, if and only if indS  Z tr .A/ 2 1 K.Z/: Theorem 4.2.23 is an analogue of [182, Section 3.3.5, Proposition 10], see also [197, Theorem 3.2.1] for explicit constructions. Note that a similar condition is well known in classical elliptic boundary value problems, cf. Atiyah and Bott [6] (also known as the Atiyah–Bott obstruction, here for the existence of Shapiro–Lopatinskij elliptic transmission conditions). If it is violated, there is another concept of elliptic (so-called global projection) conditions, see Seeley [203], or [190], [197]. This case will not be discussed here. The following results are a special case of Theorem 7.2.38 and Remark 7.2.43 (ii) below. Theorem 4.2.24. Let A 2 Y0 .Y n ZI v/ for v WD .E; F I J ; JC / be an arbitrary transmission operator. Then the following conditions are equivalent: (i) The operator A W L2 .Y n Z; E/ ˚ L2 .Z; J / ! L2 .Y n Z; F / ˚ L2 .Z; JC / is Fredholm; (ii) A is elliptic in the sense of Definition 4.2.21. Theorem 4.2.25. Let A 2 Y0 .Y n ZI v/ for v WD .E; F I J ; JC / be elliptic. Then there is a parametrix P 2 Y 0 .Y n ZI v1 / for v1 WD .F; EI JC ; J /, i.e., I  P A 2 Y1 .Y n ZI vl /;

I  AP 2 Y1 .Y n ZI vr /

where vl D .E; EI J ; J /; vr D .F; F I JC ; JC /. Theorem 4.2.25 is a consequence of the invertibility of the components of the principal symbol of A which allows us to pass to a parametrix, first with remainders of order 1, and then by a formal Neumann series argument with remainders of order 1, cf. analogously [188, Theorem 3.1.15]. This yields the Fredholm property in Theorem 4.2.24. The necessity of the ellipticity for the Fredholm property can be obtained analogously as [37, Theorem 3.3.7].

4.2.2 Ellipticity with additional conditions at the interface We now start once again from a mixed boundary problem A as an operator (4.2.2) with x and Shapiro–Lopatinskij ellipticity of T˙ on Y˙ up ellipticity of A of order  in  to Z. In Section 4.2.1 we reduced A to an operator A0 D QAL of order zero of the form (4.2.9). Let us write A0 in the form   1 0 ; (4.2.26) A0 D T F

258

4 Mixed problems in standard Sobolev spaces



where

r  C e F D C  r Be

r  C eC r C BeC

 (4.2.27)

is equal to (4.2.10), cf. the notation of Remark 4.2.2, and T D t .Q T Pz QC TC Pz /. The operator F D F .s/ is an element of Y0 .Y n ZI w/ for w WD .C N ; C N I 0; 0/. Proposition 4.2.26. The operator F is  -elliptic in the sense that it satisfies Definition 4.2.21 (i). The next step to associate a Fredholm operator with the original mixed problem A is to add boundary and potential conditions to F . According to the criterion of Theorem 4.2.23 (with F in place of A) we assume indS  Z tr .F / 2 1 K.Z/

(4.2.28)

which is to be checked in concrete examples, cf. Sections 4.2.3, 4.2.4 below. Then Theorem 4.2.23 gives us vector bundles J ; JC 2 Vect.Z/ such that Œ1 JC ; 1 J D indS  Z tr .F / as well as a block-matrix operator F D F .s/ 2 Y0 .Y nZI v/ for v D .C N ; C N I J ; JC / that represents a Fredholm operator L2 .Y n Z; C N / L2 .Y n Z; C N / FW ! : ˚ ˚ L2 .Z; J / L2 .Z; JC /

(4.2.29)

From (4.2.26) we pass to a Fredholm operator

 A0 WD

1 Tz

H s ./ H s ./ ˚ ˚  L2 .Y ; C N / L2 .Y ; C N / 0 ˚ ˚ W ! F 2 N 2 L .YC ; C / L .YC ; C N / ˚ ˚ L2 .Z; J / L2 .Z; JC /

  for Tz WD T0 with 0 being the zero operator as a map H s ./ ! L2 .Z; JC /. From (4.2.5) we get a Fredholm operator H s ./ ˚ L2 .Y ; C N / H s ./ ˚ ˚ LW ! 2 2 N L .Z; J / L .YC ; C / ˚ L2 .Z; J /

259

4.2 Mixed elliptic problems

for L WD L ˚ idL2 .Z;J / . Moreover, we form the isomorphism H s ./ ˚

H s ./ ˚ 1 2 ˚jND1 H sm;j  2 .int Y / L .Y ; C N / ˚ ˚ Q WD Q ˚ idL2 .Z;JC / W ! ; 1 L2 .YC ; C N / ˚jND1 H smC;j  2 .int YC / ˚ ˚ 2 2 L .Z; JC / L .Z; JC / see also the notation in (4.2.8). We now set A WD Q1 A0 L.1/ which is an operator of the form H s ./ ˚  AD

A K T R



1

˚jND1 H sm;j  2 .int Y / H s ./ ˚ ˚ ! W ; 1 2 N sm C;j  2 .int Y / L .Z; J / ˚j D1 H C ˚ L2 .Z; JC /

(4.2.30)

where A is our original operator (4.2.2). Here s > maxf; dg  12 , d WD fm˙;j C 1 W j D 1; : : : ; N g, is fixed, and T D T .s/ W H s ./ ! L2 .Z; JC / represents extra trace conditions with respect to Z, moreover, K D K.s/ is a column vector of several potential operators, namely, H s ./ ˚ 1

K W L2 .Z; J / ! ˚jND1 H sm;j  2 .int Y / ; ˚ N smC;j  1 2 .int YC / ˚j D1 H and R D R.s/ is an element in L0cl .ZI J ; JC /; R W L2 .Z; J / ! L2 .Z; JC /. Clearly also J˙ depend on s. Remark 4.2.27. The relation (4.2.28) does not depend on the specific choice of the auxiliary elliptic problem D, cf. the formula (4.2.4). Theorem 4.2.28. Let A be of the form (4.2.2) with an elliptic operator A and Shapiro– Lopatinskij elliptic conditions T˙ on Y˙ . Moreover, assume that the reduced operator (4.2.27) satisfies relation (4.2.28). Then the operator A of the form (4.2.30) is a Fredholm operator for the given s > maxf; dg 12 , d WD fm˙;j C1 W j D 1; : : : ; N g, and   1 0 Q A.1/ WD L F .1/ Tz F .1/

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4 Mixed problems in standard Sobolev spaces

is a parametrix of A, where F .1/ is a parametrix of the elliptic operator (4.2.29) in the sense of Theorem 4.2.25. Proof. The Fredholm property of an operator T 2 L.H1 ; H2 / for Hilbert spaces H1 ; H2 is equivalent to the existence of an element P 2 L.H2 ; H1 / such that the operators I  P T and I  TP are compact in H1 and H2 , respectively. To verify this for our situation we fix s and take for H1 and H2 the spaces in relation (4.2.30) and consider T WD A. Let us check that P WD A.1/ is a left parametrix, i.e., that I  P T is compact in H1 . In fact, we have in this case      1 0 1 0 1 0 .1/ 1 L1 : ADL L DL A 0 F .1/ F F .1/ Tz F .1/ Tz F Since F .1/ is a parametrix of F , the operator I  F .1/ F is compact. Because L is an isomorphism between the respective spaces, we immediately obtain that A.1/ AI is a compact operator in H1 . In a similar manner we can prove that AA.1/  I is compact in H2 . Let us formulate a variant of our calculus for operators with parameters  2 Rl and argue in terms of parameter-dependent ellipticity. P 0 00 An operator A./ WD j˛j a˛ .x/.Dx˛ ˛ / for ˛ D .˛ 0 ; ˛ 00 / 2 N n  N l is said P 0 00 to be parameter-dependent elliptic, if  .A/.x; ; / WD j˛jD a˛ .x/ ˛ ˛ 6D 0 x . ; / 2 RnCl n f0g. There is also a notion of parameter-dependent for all x 2 ; ellipticity of boundary value problems. A trace condition T ./ has this property, T ./u WD rB./u for a column B./ WD t .B1 ./; : : : ; BN .// of differential operators of order mj , if locally near Y in the splitting of variables x D .y; t /, D .; /, the boundary symbol     A  .A/.y; 0; ; ; D t / @ .y; ; / WD ; .r .Bj /.y; 0; ; ; D t //j D1;:::;N T ru WD ujY , defines an isomorphism   H s .RC / A ˚ @ .y; ; / W H s .RC / ! T CN

(4.2.31)

for all .y; ; / 2 .T  Y Rl /n0; s > maxf; dg 12 , d WD fm˙;j C1 W j D 1; : : : ; N g. An example is A./ WD  2 with Dirichlet or Neumann conditions (that do not depend on ; the bijectivity (4.2.31) holds in this case for  D 2). All elements of the above considerations have a straightforward extension to the parameter-dependent elliptic case. First the mixed problems (4.2.2) may assumed to be given in parameterdependent elliptic form, as well as the auxiliary problem (4.2.4) and the order reducing operators Rj on the boundary. Here we employ the spaces Lcl .Y I Rl / of classical parameter-dependent pseudo-differential operators on Y , and we use the fact that for

261

4.2 Mixed elliptic problems

every  2 R there exists a parameter-dependent elliptic element R ./ 2 Lcl .Y I Rl / that induces isomorphisms R ./ W H s .Y / ! H s .Y / for all s 2 R;  2 Rl . Moreover, we refer to Boutet de Monvel’s calculus in a corresponding parameter-dependent variant, cf. Section 3.3.3, or [90]. In particular, if D./ is our parameter-dependent elliptic auxiliary problem (4.2.4), we find a parameter-dependent elliptic parametrix D .1/ ./ in that calculus, and D./ becomes invertible for sufficiently large jj. In addition, the order reducing operators zs ./ in Section 4.2.1 can be chosen in parameter-dependent elliptic form, cf. also R ˙ the constructions below. Then the process to produce the block matrix (4.2.10) yields z 1 on Y˙ . parameter-dependent elliptic elements Q˙ T˙ KR Finally, the transmission algebra Y0" .Y n ZI v/, v WD .E; F I J ; JC /, of Section 4.2.1 admits a parameter-dependent version Y0" .Y n ZI vI Rl /. Interior symbols  ˙ .A/.y; ; / as well as transmission symbols tr .A/.y; ; / now depend on  2 Rl as an additional covariable. The class Y1 .Y n ZI vI Rl / WD S.Rl ; Y1 .Y n ZI v// of smoothing operators refers to a canonical Fréchet topology in the spaces Y"1 .Y n ZI v/ for every " > 0. Definition 4.2.21 can easily be generalised to parameterdependent ellipticity: Instead of (i) we require isomorphisms  ˙ .A/ W Y E˙ ! ˙

Y F˙ , now for Y˙ W .T  Y˙  Rl / n 0 ! Y˙ , and (ii) is to be replaced by the ˙

condition that (4.2.21) is an isomorphism for Z W .T  Z  Rl / n 0 ! Z. In the following theorem we use the notation v; v1 ; vl ; vr as in Theorem 4.2.25.

Theorem 4.2.29. Let A./ 2 Y0 .Y nZI vI Rl / be parameter-dependent elliptic. Then there is a parametrix P ./ 2 Y0 .Y n ZI vI Rl / in the sense I  P ./A./ 2 Y1 .Y n ZI vl I Rl /; Moreover,

I  A./P ./ 2 Y1 .Y n ZI vr I Rl /:

L2 .Y n Z; F / L2 .Y n Z; E/ ˚ ˚ ! A./ W L2 .Z; J / L2 .Z; JC /

(4.2.32)

is a family of Fredholm operators, we have ind A./ D 0 for all  2 Rl , and there is a C > 0 such that the operators (4.2.32) are isomorphisms for all jj  C . The order reducing operators on Y˙ that we used in Section 4.2.1 belong to spaces of operators with a certain (isotropic) Volterra or anti-Volterra property, cf. [72, Sections 1.3 and 1.5]. For our applications we need some more material. Recall that a tubular neighbourhood of Z on the manifold Y is identified with .1; 1/  Z. We fixed corresponding charts j W Uj ! R  Rn2

for j D 1; : : : ; L

(4.2.33)

262

4 Mixed problems in standard Sobolev spaces

with Uj D .1; 1/  Uj0 ; Uj0  Z, that induce charts j0 W Uj0 ! Rn2 on Z, and interior charts j W Uj ! Rn1 for j D L C 1; : : : ; N: (4.2.34) Without loss of generality we assume that j .r; z/ D .r; j0 .z// for  23 < r < 23 , j D 1; : : : ; L, and Uj \ .. 23 ; 23 /  Z/ D ; for all j D L C 1; : : : ; N . We now introduce subspaces  L cl .Y I Z/˙  Lcl .Y /

of pseudo-differential operators that have in the strip . 23 ; 23 /  Z the Volterra-property (for the subscript C) or the anti-Volterra (for the subscript ) with respect to the Cside YC of the manifold Y . Concerning the concept of Volterra pseudo-differential operators, cf. Piriou [149], or the paper [105] of Krainer and the references there. Let us concentrate on the anti-Volterra case; the Volterra case is analogous (clearly, by interchanging the role of YC and Y Volterra turns into anti-Volterra and vice versa). Choose a partition of unity f'j gj D1;:::;N subordinate to fUj gj D1;:::;N and a system of functions f j gj D1;:::;N ; j 2 C01 .Uj /, with the following properties: (i)

j

1 on supp 'j for all j D 1; : : : ; N I

(ii) 'j0 WD 'j jUj0 for j D 1; : : : ; L is a partition of unity subordinate to fUj0 gj D1;:::;L , and 'j .r; z/ D 'j0 .z/ for  23  r  23 ; j D 1; : : : ; L;

(iii)

j .r; z/

D

2 j .0; z/ for  3

" < r <

2 3

C " for some 0 < " < 13 , j D 1; : : : ; L.

Definition 4.2.30. By L cl .Y I Z/ we denote the space of all operators N X

A WD

'j .j1 / Op.aj /

j

CG

(4.2.35)

j D1

Scl .Rn2

for any aj .z; %; / 2  Rn1 / ; j D 1; : : : ; L, and aj .y; / 2 Scl .Rn1  %;  Rn1 /, j D L C 1; : : : ; N , and G 2 L1 .Y /.  n2  In an analogous way we define L cl .Y I Z/C in terms of symbols aj in Scl .R n1 R /C , see, similarly, Definition 4.1.18.  23

Let !.r/ 2 C01 .R/ be a real-valued cut-off function on R satisfying !.r/ 1 for  r  23 , and set   aj .z; %; ; / WD '

!.r/ %  h%; ; i.1!.r// h; i  i% C h; i

for j D 1; : : : ; L and aj .; / WD h; i for j D LC1; : : : ; N ; here ' is the function involved in (4.1.1). Then the expression (4.2.35) for G D 0 gives us a parameter Q z ./ of elements in L dependent elliptic family R cl .Y I Z/ . For  WD  sufficiently large this operator induces isomorphisms H s .Y / ! H s .Y / for all s 2 R, and we   Q z z WD R ./. set R

263

4.2 Mixed elliptic problems

Remark 4.2.31. Let us explicitly observe the parameter-dependence. If we replace 0   z z in the above construction  by .; 0 / 2 RlCl we can set R .; 0 / DW R ./ for a 0 sufficiently large j j; then the corresponding operators  z rC R ./esC W H s .int YC / ! H s .int YC /

are isomorphisms for all s 2 R and all  2 Rl (esC can be replaced by eC for s >  12 ).

4.2.3 The Zaremba problem Let us consider the Zaremba problem (4.2.3) for the Laplace operator , and choose the operator (4.2.4) as in Example 4.2.1. The operator (4.2.6) then takes the form H s2 ./ H s2 ./ ˚ ˚ 2 s 1 AL W L .Y / ! H 2 .int Y / ; ˚ ˚ s 3 L2 .YC / 2 H .int YC /

(4.2.36)

s > 32 . We compose (4.2.36) from the left by the order reducing matrix Q, where s 1

z 2 e ; Q D r  R C

s 2 C z QC D r C R e : 3

Since the Dirichlet problem is an invertible operator, we have the relation (4.2.7) in exact form, and hence A0 D QAL is equal to (4.2.9) with the 2  2 lower right corner (4.2.10). Now, because of T D Tz jY we have T Kz D r  I with the identity operator I . Hence, the first row of (4.2.10) is equal to   Q r  R1 e Q r  R1 eC 1

zs 2 . Using The choice of the order reducing operator R is still free. Set R D R C s 1

s 1

z 2 /1 eC D 0, and z 2 e r  .R [72, Proposition 1.7] we obtain Q r  R1 eC D r  R C C 1

1

zs 2 /1 e D 1. The operator F (cf. (4.2.27)) is then zs 2 e r  .R Q r  R1 e D r  R C C of the form   1 0 (4.2.37) F D C  C C r Be r Be 3

1

s 2 z R zs 2 /1 2 L0cl .Y /. The homogeneous principal symbol of B of z TzC K. for B D R C order 0 is equal to

b.y; / ´  0 .B/.y; / D

s 3 2

s 2 z z .R /.y; / 1 .TzC K/.y; / 3

sC 1 2

s 1

z 2 /1 /.y; /: ..R C

264

4 Mixed problems in standard Sobolev spaces

Proposition 4.2.32. For every s … Z C

1 2

the operators

@ .r C BeC /./ D r C Op.b/./eC W L2 .RC / ! L2 .RC /: are Fredholm for all  2 Rn2 n f0g. Proof. By definition, in a tubular neighbourhood of Z in the variables .y; / D .r; z; %; / we have   %  s 32 s 3 s 3 z z 2 /.r; z; %; / D f  2 .R ;  1 .TzC K/.r; jj  i% z; %; / D jj; C jj  s 1 1    %  sC 12 z 2 R .r; z; %; / D f : jj  i% C C jj Thus, in a tubular neighbourhood of Z we have constant coefficients, i.e.,  0 .B/ is equal to b.%; / in that neighbourhood. Note that b.%; / D b.%;  0 / for jj D j 0 j. Applying the formula (4.2.24) to the present symbol we obtain 

sC 1 2

c .r C BeC /.w/ D b.C1; 0/g C .w/ C b.1; 0/g  .w/ 1 3 1 D ..1/s 2 g C .w/ C .1/sC 2 g  .w//: i We now determine s 2 R in such a way that c .r C BeC /.w/ 6D 0 for all w 2  1 , i.e., 2

s 3 2

sC 1 2

.1/ .1/ C 6D 0 1 C e 2y 1 C e 2y for every y 2 R which is equivalent to sin..s  12 //.2 C e 2y C e 2y / 6D 0, y 2 R. Thus we get M .r C BeC /.w/ 6D 0 on  1 for all s 2 R such that .s  12 / 6D k; k 2 Z, 2

i.e., s … Z C 12 . Remark 4.2.22 then gives us the assertion. Remark 4.2.33. We have indS  Z @ .r C BeC / 2 1 K.Z/. Hence, by Theorem 4.2.23 the operator (4.2.37) can be completed to an elliptic operator F 2 Y0 .Y n ZI v/, v D .C N ; C N I J ; JC / for a suitable choice of J˙ 2 Vect.Z/. Since the principal symbol of B does not depend on z 2 Z, we may set J˙ D C j˙ for suitable dimensions j˙ . Write b.%; / in the form   %    %  s 32 sC 12 1 1 b.%; / D f .jj  i%/ 2 .jj C i%/ 2 f : jj  i% jj  i% C jj C jj (4.2.38) Using notation of [44, Section 6] the function (4.2.38) has a homogeneous factorisation b.%; / D b .%; /bC .%; / with   %  s 32 1 b .%; / WD f .jj  i/ 2 ; jj  i% C jj   %  sC 12 1 bC .%; / WD .jj C i%/ 2 f : jj  i% C jj

265

4.2 Mixed elliptic problems

We have ord bC .%; / D s C 1 … Z C 12 for all s … Z C 12 . From Theorems 7.1, 7.2 and 7.3 of [44] we get following result: Proposition 4.2.34. The operators @ .r C BeC /./;  6D 0, are (i) bijective for

1 2

(ii) surjective for (iii) injective for

1 2

< s < 32 , 1 2

< s C j < 32 ; j 2 N, where dim ker @ .r C BeC /./ D j ,

< s C j < 32 ; j 2 N, where dim coker @ .r C BeC /./ D j .

Summing up, for the Zaremba problem we have proved the following result: Theorem 4.2.35. The Zaremba problem (4.2.3) can be completed by interface conditions to a Fredholm operator (4.2.30) if and only if s … Z C 12 ; s > 32 . In this case the operator A can be chosen in the form H s2 ./ ˚ H s ./   s 1 ˚ AD A K W ! H 2 .int Y / ˚ L2 .Z; C j / s 3 H 2 .int YC /  with j WD s  12 potential conditions (and no trace conditions).

(4.2.39)

For purposes below we write the operator K in the form H s2 ./ ˚ 0 2 Œs 1  0A s 1 @ 2 2 .int Y / : K D L W L .Z; C / ! H L0C ˚ 3 H s 2 .int YC / 0

1

(4.2.40)

Remark 4.2.36. Theorem 4.2.29 gives us a parametrix within our calculus, and F .1/ can be expressed as a parametrix of an elliptic operator F in the class Y0 .Y n ZI C j ; C jC /, cf. (4.2.19). The operators in the Zaremba problem, i.e., the Laplacian together with Dirichlet or Neumann conditions, may be regarded as the values of certain parameter-dependent operators with parameter  2 Rl restricted to  D 0. In fact, we can replace by 2 and also choose the auxiliary problem D as D./ with 2 and Dirichlet conditions as before. Also the involved order reducing operators can be chosen in parameterdependent elliptic form, cf. Remark 4.2.31. Finally, the construction of extra interface conditions in the sense of Theorem 4.2.23 can be carried out in this case in parameterdependent elliptic form, cf. Section 4.2.2. This gives us a parameter-dependent version A./;  2 Rl , of the operator (4.2.39), where A.0/ is the operator (4.2.39) itself. Now all involved parameter-dependent elliptic operators become isomorphisms between the

266

4 Mixed problems in standard Sobolev spaces

respective spaces for jj sufficiently large, cf. also Theorem 4.2.29 in Section 4.2.2. Thus, in particular, the operators A./ are of index 0 for all  2 Rl . This gives us the following result: Theorem 4.2.37. The operator (4.2.39) may be regarded (by an appropriate choice of K) as the value of a parameter-dependent family A./ at  D 0, such that ind A D 0: Note that parameter-dependent versions of mixed problems could possibly be used to establishing trace-expansions of the resolvents (suggested by G. Grubb in a private discussion).

4.2.4 Jumping oblique derivatives and other examples Let us now consider another example for the Laplace operator , namely, mixed conditions with jumping oblique derivatives on Y˙ . Problems of this kind have been studied already in [160], see also [156]. In this case we have H s2 ./ ˚ s s 3 @ A 2 .int Y / A D T W H ./ ! H TC ˚ 3 H s 2 .int YC /

(4.2.41)

TC D r C BC T D r  B

(4.2.42) (4.2.43)

0

1

for s > 32 , where for BC WD V C ˛Dr C D t ; for B WD W C ˇDr C ıD t :

Here, V and W are vector fields whose restrictions to Z are tangent to Z; the coefficients ˛; ˇ; ; ı are assumed to be constants such that  6D 0; ı 6D 0 (the latter condition makes sense with respect to a chosen Riemannian metric of Rn near Z; as a simple example for  we may take a sphere fx 2 Rn W jxj < Rg; R > 0, and for Z the equator fx 2 Rn W jxj D R; xn D 0g). The restrictions of V; W to Z are locally of the form V jZ D

n2 X

˛i .z/Dzi ; W jZ D

n2 X

ˇi .z/Dzi ;

iD1

iD1 1

where the coefficients   ˛i ; ˇi are C functions of z D .z1 ; : : : ; zn2 /. be an elliptic boundary value problem, where Tz is an extension Let D D

Tz z be a parametrix of D. from T to the whole boundary, i.e., r  Tz D T , and let .Pz K/

267

4.2 Mixed elliptic problems 3

3

zs 2 e˙ and R D R zs 2 . Similarly as in the case In this case we take Q˙ D r ˙ R  C of Zaremba problem we get an operator F of the form (4.2.33). The homogeneous 3 s 3 2 z z R zs 2 /1 in this case is equal to z TC K. principal symbol of the operator B D R C

  %    %  s 32 sC 32 z b.z; %; / D f  0 .TzC K/.%; : jj  i% jj  i% / f C jj C jj If v.z; / and w.z; / denote the homogeneous principal symbols of order 1 of V jZ P and W jZ , respectively, locally being of the form v.z; / D n2 ˛ iD1 i .z/i ; w.z; / D Pn2 iD1 ˇi .z/i , we have z / D .v.z; / C ˛% C i  jj/.w.z; / C ˇ% C i ıjj/1 :  0 .TzC K/.z; Applying the formula (4.2.24) we get c .r C BeC /.w/ D b.C1; 0/g C .w/ C b.1; 0/g  .w/ 3 ˛ C i  3 ˛ C i D .1/s 2 g C .w/ C .1/sC 2 g  .w/: ˇ C iı ˇ C i ı Assume that ˛; ˇ; ; ı are real numbers. By a simple calculation one can show that c .r C BeC /.w/ 6D 0 on  1 for any s 2 R satisfying the condition 2



.˛ˇ C  ı/cos s 







3 3   .ˇ  ˛ı/sin s   6D 0: 2 2

(4.2.44)

Lemma 4.2.38. Let ˛; ˇ; ; ı 2 R be arbitrary reals,  6D 0, ı 6D 0 such that ˇ D ˛ı:

(4.2.45)

Then for every s … Z the operators @ .r C BeC / W L2 .RC / ! L2 .RC / are Fredholm for all  2 Rn2 n f0g. This is an immediate consequence of the relation (4.2.44). Let us now assume that V D W D 0:

(4.2.46)

Then b.%; / has a homogeneous factorisation with   %  s 32 b .%; / WD f ; jj  i% C jj

  %  sC 32 bC .%; / WD f I jj  i% C jj

thus ord bC .%; / D s C 32 … ZC 12 for all s … Z. Analogously as Proposition 4.2.34 we get the following result: Proposition 4.2.39. The operators @ .r C BeC /./ (under the assumptions (4.2.45), (4.2.46)),  6D 0, are

268

4 Mixed problems in standard Sobolev spaces

(i) bijective for 1 < s < 2, (ii) surjective for 1 < s C j < 2; j 2 N, where dim ker @ .r C BeC /./ D j , (iii) injective for 1 < s C j < 2; j 2 N, where dim coker @ .r C BeC /./ D j . Theorem 4.2.40. The mixed problem (4.2.41) for the Laplacian with jumping oblique derivatives T˙ on Y˙ of the form (4.2.42), (4.2.43) (satisfying the relations (4.2.45), (4.2.46)) can be completed by interface conditions to a Fredholm operator (4.2.30) if and only if s … Z; s > 32 . In this case we can choose A in the form H s2 ./ ˚ H s ./   s 3 ˚ ! H 2 .int Y / AD A K W ˚ L2 .Z; C j / s 3 2 H .int YC / with j WD 0 for both cases).

3 2

(4.2.47)

< s < 2 and j WD Œs  1 for s > 2 (and no trace conditions in

Theorem 4.2.41. The operator (4.2.47) can be regarded (under the conditions of Theorem 4.2.40 and by an appropriate choice of K) as the value of a parameter-dependent family at  D 0, such that ind A D 0: (4.2.48) In particular, (4.2.48) holds for

3 2

< s < 2, where j D 0.

As a final example we consider the Laplace operator with boundary conditions TC on YC as in (4.2.42) and Dirichlet conditions T on Y . In this case QC D 1 1 s 3 2 C zs 2 , and we have z zs 2 e ; R D R e ; Q D r  R rC R C

C

z  0 .TzC K/.z; / D v.z; / C ˛% C i  jj;   %    %  s 32 sC 12 .v.z; / C ˛% C i  jj/ f ; b.z; %; / D f jj  i% jj  i% C jj C jj and c .r C BeC /.w/ 3

1

3

1

D .i /s 2 .˛ C i  /i sC 2 g C .w/ C i s 2 .˛ C i  /.i /sC 2 g  .w/ 1 3 1 D ..1/s 2 .˛ C i  /g C .w/ C .1/sC 2 .˛ C i  /g  .w//: i Let ˛;  be real numbers. The operators @ .r C BeC /./ are Fredholm if and only if s 2 R satisfies the condition         1 1 (4.2.49)  ˛cos s   C  sin s   2 C e 2y C e 2y 6D 0 2 2

269

4.2 Mixed elliptic problems

for all y 2 R. Assume that V D 0. Because of [44, Theorem 6.1] the function ˛% C i  jj has the homogeneous factorisation ˛% C i  jj D bQC .%; /bQ .%; / where 1 1 .arg .˛ C i  /  arg .˛ C i  //: ord bQ .%; / D C 2 2 To illustrate what may happen we set, for instance, ˛ D  D 1. Then 1 ord bQ .%; / D 4

and

3 ord bQC .%; / D : 4

So for b.%; / we have the homogeneous factorisation b.%; / D bC .%; /b .%; / with    sC 12 3 bC .%; / WD bQC .%; / f C%j j jj  i% , b .%; / WD .f . C%j j /jji%/s 2 bQ .%; /: The relation (4.2.49) gives us (in the case ˛ D  D 1) that @ .r C BeC /./ are Fredholm for all s 2 R, s …  14 C Z. On the other hand ord bC .%; / D s C 54 … Z C 12 for all s 2 R, s …  14 C Z. Proposition 4.2.42. The operators @ .r C BeC /./ (with conditions as in Proposition 4.2.39),  6D 0, are (i) bijective for

3 4

(ii) surjective for (iii) injective for

3 4

< s < 74 , 3 4

< s C j < 74 ; j 2 N, where dim ker @ .r C BeC /./ D j ,

< s C j < 74 ; j 2 N, where dim coker @ .r C BeC /./ D j .

Theorem 4.2.43. The mixed problem (4.2.3) for the Laplacian with oblique derivatives TC on YC as in Theorem 4.2.40 and Dirichlet conditions on Y can be completed by interface conditions to a Fredholm operator (4.2.30) if and only if s …  14 C Z, s > 32 . In this case the operator A takes the form H s2 ./ ˚ H ./   1 ˚ ! H s 2 .int Y / AD A K W ˚ L2 .Z; C j / s 3 H 2 .int YC /  with j WD s  34 (and no trace conditions). s

(4.2.50)

Remark 4.2.44. Note that Theorem 4.2.35 is not a consequence of Theorem 4.2.43. The case ˛ D 0 which corresponds to the Zaremba problem is not typical for other directions of the vector field on YC . ˛Ci Remark 4.2.45. Let ˛ 2 R, fix  D 1, and set '.˛/ WD arg ˛Ci . Then @ .r C BeC /./ is bijective for all s; s0 .˛/ < s < s1 .˛/ where

s0 .˛/ WD

1 1  '.˛/; 2 2

s1 .˛/ WD

3 1  '.˛/: 2 2

270

4 Mixed problems in standard Sobolev spaces

The function '.˛/ is continuous and satisfies '.˛/ > '.˛ 0 / for ˛ < ˛ 0 . Thus si .˛ 0 / > si .˛/ for i D 0; 1, i.e., the bijectivity interval is moving in positive direction for ˛ varying from 1 to C1. Remark 4.2.46. For the operators (4.2.50) we have ind A D 0 for an appropriate choice of K, similarly as Theorems 4.2.37 and 4.2.41.

Chapter 5

Mixed problems in weighted edge spaces

We now study mixed elliptic problems in weighted edge spaces. The advantage is that we may admit arbitrary real weights (except for a discrete set). Parametrices, Fredholm property, and elliptic regularity of solutions can be obtained in the framework of boundary value problems on a manifold with edges. In Chapter 7 below we give a systematic exposition of that calculus in general. Here we focus on some typical examples, mainly the Zaremba problem and other mixed problems for the Laplace operator. The new results of this chapter are based on the joint paper of the authors together with Dines [34].

5.1 Mixed problems in edge spaces Mixed problems, especially, the Zaremba problem, induce continuous operators in weighted edge spaces. The interface Z, i.e., the submanifold on the boundary with the jump of boundary conditions, is interpreted as an edge. We observe the principal symbolic hierarchy of such problems, namely,  D . ; @;˙ ; ^ /, and establish the structure of extra trace and potential conditions along Z in terms of Green symbols. Moreover, we study the ellipticity with respect to the symbols.

5.1.1 Basic observations By a mixed problem for a differential operator A we understand a boundary value problem on a C 1 manifold X with boundary Y of the form Au D f in int X;

T˙ u D g˙ on int Y˙ ;

(5.1.1)

cf. Chapter 1. Here Y is subdivided into C 1 submanifolds Y˙ with common boundary Z such that YC [ Y D Y and YC \ Y D Z. We assume that A is elliptic. The operators T˙ represent boundary conditions of the form r ˙ B˙ , where B˙ are differential operators in an open neighbourhood of Y˙ with smooth coefficients, r ˙ are the operators of restriction to int Y˙ . The boundary conditions on the plus and minus side are assumed to satisfy the Shapiro–Lopatinskij condition (up to Z from the respective sides). The general task is to characterise the solvability of such problems in suitable distribution spaces, to construct parametrices and to establish asymptotics of solutions near the interface Z. Let us consider as our ‘standard’ example the Zaremba problem for the Laplace operator, where T is the Dirichlet and TC the Neumann condition. Moreover, let X

272

5 Mixed problems in weighted edge spaces

be compact. Then we have a continuous operator H s2 .int X / ˚ s s 1 @ A 2 .int Y / A D T W H .int X / ! H TC ˚ 3 H s 2 .int YC / 0

1

(5.1.2)

for every real s > 32 . It is clear that when we ask solutions of the problem (5.1.1) with arbitrary boundary data in the Sobolev spaces on the boundary, we cannot expect the existence in H s .int X /. Therefore, we need another category of spaces which describe the solvability in a more adequate way. This is just the motivation for the approach in edge Sobolev spaces. Nevertheless, as we shall see, the results of the preceding chapter on solvability in standard Sobolev spaces are a useful information. In the following discussion we mainly focus on the Zaremba problem (and other mixed problems for the Laplace operator). However, the methods that we apply are completely general and can be specified to other classes of concrete examples. The manifold X together with the subspaces Y  Z will be interpreted as a manifold with boundary and edge. If we also consider the double N WD 2X of X (consisting of two copies X˙ of X , identified along the common boundary Y ), then N can be regarded as a manifold (without boundary) with edge Z of codimension 2 in N . There is a tubular neighbourhood U of Z which is a (trivial) bundle over Z with fibres being two-dimensional open disks, and there is a diffeomorphism U ! Z  R2 DW L to the normal bundle of Z in N that restricts to the identity map on Z and to fibrewise diffeomorphisms between the disks and R2 . Via polar coordinates R2 nf0g ! RC S 1 we obtain an identification between U n Z and L n Z Š Z  .RC  S 1 /, and we set x C  S 1 /. Similarly, from the manifold N with edge Z we can pass Ł WD Z  .R to the stretched manifold N by replacing U by U , which is obtained from U n Z Š Z.RC S 1 / by invariantly attaching Z.f0gS 1 /, which gives us a diffeomorphism U Š Ł. The stretched manifold N is a C 1 manifold with boundary @N Š Z  S 1 . Let us set Nreg WD N n @N; Nsing WD @N: Because of N D 2X a similar construction makes sense for X˙ separately. Identifying x 2 , ŁC WD Z  R xC  S1 X with XC we can form UC WD U \ XC and LC WD Z  R C C 1 2 1 x (with the closed half-circle SC WD RC \ S ), and we also obtain a ‘positive’ part UC of U with a corresponding bijection to ŁC . Then, replacing UC by UC from X we obtain the associated stretched manifold X with boundary. We set Xreg WD X \ Nreg ;

Xsing WD X \ Nsing :

Observe that Xreg is a (non-compact) C 1 manifold with boundary int Y [ int YC . On X we have the scale of weighted edge Sobolev spaces ˚ ˇ

W s; .X/ WD uˇint Xreg W u 2 W s; .N/ ;

5.1 Mixed problems in edge spaces

273

s;  2 R. Moreover, interpreting Y˙ as manifolds with edge Z, there are the spaces W s; .Y˙ / for every s;  2 R. Remark 5.1.1. The restriction operators r ˙ W C01 .Xreg / ! C01 .int Y˙ / extend to 1 1 continuous operators r ˙ W W s; .Xreg / ! W s 2 ; 2 .Y˙ / for every s;  2 R, s > 12 . P Remark 5.1.2. Let A D j˛j a˛ .x/Dx˛ be an arbitrary differential operator with coefficients a˛ .x/ 2 C 1 .X /, and let T˙ D r ˙ B˙ be boundary operators, where B˙ WD t .B˙;l /lD1;:::;L are vectors of differential operators in an open neighbourhood of Y˙ of order ˙;l , also with smooth coefficients. Then the corresponding mixed boundary value problem (5.1.1) induces continuous operators 0

W s; .X/ ˚

1

A 1 1 W s;l  2 ;;l  2 .Y / A D @ T A W W s; .X/ ! ˚L lD1 ˚ TC 1 L sC;l  1 2 ;C;l  2 .YC / ˚lD1 W ˚

for all s > max ˙;l C 12 and all  2 R.

(5.1.3)

In particular, the Zaremba problem represents continuous operators W s2; 2 .X/ ˚ A 1 1 s; s ; A D @ T A W W .X/ ! W 2 2 .Y / : ˚ TC 3 s 3 ; W 2 2 .YC / 0

1

(5.1.4)

The idea of solving problems of the kind (5.1.1) is to embed the associated operators (5.1.3) into a calculus of pseudo-differential boundary value problems on the manifold X with boundary and edge and to construct parametrices. This will be possible with the exception of some discrete set of weights  and under additional edge conditions (of, in general, trace and potential type) along the edge Z. Let us ignore for the moment the fact that the weighted edge spaces on int Y˙ consist of several components of different smoothness and weights. Then the general pseudo-differential edge boundary value problems, including conditions on Z, are represented by 4  4 block matrices of operators W s; .X/ ˚ 1 1 1 s  1 ;  s  ;     2 .Y / 2 2 .Y / 2 W W  ˚ ˚ A D .Aij /i;j D1;:::;4 W ! (5.1.5) sC  1 ;C  1 sC  1 ;C  1 2 2 2 2 W W .YC / .YC / ˚ ˚ H s .Z; C j / H s .Z; C jC / W s; .X/ ˚

for certain orders ; ˙ ; ˙ and dimensions j˙ .

274

5 Mixed problems in weighted edge spaces

5.1.2 Green symbols Compared with (5.1.3) the operator (5.1.5) contains additional entries which can be subsumed under the notion of Green operators in the edge operator calculus. We give a description here on the level of 4  4 block matrix Green symbols. For simplicity we now assume jC D j D 1 (the modification of the definition for the general case is obvious). We give the formulation in terms of spaces with a small ‘gain of weights’ " > 0 that may depend on the operators. Refinements including asymptotic data in the spaces are possible, too, and may be observed in connection with the general edge calculus. If M is a compact C 1 manifold (with or without boundary) we set 1

S" .M ^ / WD lim hrik K k;C".1Ck/ .M ^ /  k2N

for any " > 0. This is a Fréchet space with group action . u/.r; x/ D  m D dim M . In particular, we have the spaces S" .RC /;

S" .I ^ /;

mC1 2

u.r; x/, (5.1.6)

1 where I WD Œ0;  , identified with SC . Let us now fix a chart on Z, and let z 2 U be corresponding local coordinates, U Rq open, q WD n  2 for n D dim X . Recall that we have spaces of symbols

z Scl .U  Rq I E; E/ z endowed with group actions f g2R and fQ  g2R , for Fréchet spaces E and E, C C respectively, cf. the formula (2.2.10). According to the 4  4 block matrix structure of edge operators we have 4  4 block matrices of Green symbols g.z; / D .gij .z; //i;j D1;2;3;4 ;

(5.1.7)

where the involved orders and weights depend on i; j , motivated by the mapping properties in (5.1.5). In our case we set E1 WD K s1 ;1 .I ^ /; Ez1 WD S"ı1 .I ^ /;

E2 WD K s2 ;2 .R /; Ez2 WD S"ı2 .R /;

E3 WD K s3 ;3 .RC /; Ez3 WD S"ı3 .RC /;

E4 WD C; Ez4 WD C;

and

22

11 D 41 D 14 D 44 D ; 1 1 12 D 42 D     ; 13 D 43 D   C  ; 2 2 D    ; 32 D C   ; 23 D   C ; 33 D C  C ; 1 1 21 D 24 D  C ; 31 D 34 D C C ; 2 2

5.1 Mixed problems in edge spaces

275

moreover, 1 s2 WD s    ; 2 1 2 WD     ; 2 1 ı2 WD     ; 2

s1 WD s; 1 WD ; ı1 WD   ;

1 s3 WD s  C  ; 2 1 3 WD   C  ; 2 1 ı3 WD   C  : 2

Definition 5.1.3. A Green symbol (5.1.7) of type 0 of the edge calculus for mixed  problems is defined by gij .z; / 2 Scl ij .U  Rq I Ej ; Ezi / for some " D ".g/ > 0 and all s >  12 , and by similar conditions for the pointwise formal adjoints. A Green symbol of type d 2 N is defined by the extra property gi1 .z; / D

d X

gi1Il .z; /@l

lD0

for arbitrary symbols gi1Il .z; / of order i1 and type 0 (@ is the differentiation in  the angular variable  2 I ). We then obtain gi1 .z; / 2 Scl i1 .U  Rq I E1 ; Ezi / for all 1 s > d  2 , i D 1; : : : ; 4. Let ^ .g/.z; / D .^ .gij /.z; //i;j D1;:::;4

(5.1.8)

denote the matrix of homogeneous principal symbols of gij of order ij .

5.1.3 The Zaremba problem as an edge problem Recall that the operator A which represents the Zaremba problem induces continuous operators (5.1.4) for all s;  2 R, s > 32 . Instead of A we also write A. / if we consider different weights . We want to express the principal symbolic structure of A from the point of view of the edge calculus. The principal symbol consists of a triple  .A/ D . .A/; @ .A/; ^ .A//: Here  .A/ D j j2 is the standard homogeneous principal symbol of the Laplace operator. Moreover, @ .A/ D .@; .A/; @;C .A// is the pair of boundary symbols on the  sides of Y , and ^ .A/ is the edge symbol. In order to give the explicit expressions we choose a collar neighbourhood Š Y  Œ0; 1/ of Y in X with the variables x D .y; xn / and covariables D .; n /. The boundary symbols of T over int Y˙ have the form ˙

 @;˙

 H s2 .RC / s ˚ ./ W H .RC / ! T˙ C

276

5 Mixed problems in weighted edge spaces

0 0 for  6D 0, where @ . /./ D jj2 C @2xn and @; .T / D r , @;C .TC / D rC @xn , 0 with r˙ denoting the operators of restriction to xn D 0 for y 2 int Y˙ . To express the edge symbol we choose a tubular neighbourhood Š Z  .1; 1/ of Z in Y with the variables y D .z; xn1 / and covariables .; n1 /. We also use y , and z as local coordinates in corresponding open sets U Ryn1 and  Rn2 z respectively. In a tubular neighbourhood of the boundary Y the Laplacian in Rn can be @2 @2 C Z C L2 , where Z is the Laplacian reformulated in the form D @x 2 C @x 2 n

n1

on Z belonging to the metrics induced by the Euclidean metric in Rn , with a certain first order differential operator L2 in the .xn1 ; xn /-variables (xn is a normal variable to Y and xn1 a normal variable to Z in Y ). In order to express the principal edge symbol we introduce polar coordinates .r; / 2 RC S 1 in the .xn1 ; xn /-plane normal to Z and write the Laplacian in Rn2 R2xn1 ;xn z in the form D r 2 ..r@r /2 C @2 C r 2 Z C r 2 L2 /: Then K s2; 2 .I ^ / ˚ ^ . /./ 1 1 ^ .A/./ D @ ^ .T / A W K s; .I ^ / ! K s 2 ; 2 .R / ^ .TC / ˚ 3 3 K s 2 ; 2 .RC / 0

1

(5.1.9)

for  6D 0 is given by ^ . /./ D r 2 ..r@r /2 C @2  r 2 jj2 /; ^ .TC /u D r 1 @ ujD :

^ .T /u D ujD0 ;

(5.1.10)

The notation R˙ for the components of @I ^ , I D Œ0;  , is motivated by the identificax 2 n f0g for R x 2 D fxQ D .xn1 ; xn / 2 R2 W xn  0g, tion between I ^ D RC  I and R C C x 2 n f0g/ D R [ RC for R˙ D fxn1 2 R W xn1 ? 0g. @.R C Let us observe the homogeneities in  .A/. For the interior symbol we have  .A/. / D 2  .A/. /, 6D 0,  2 RC . Concerning the pair of boundary symbols the homogeneities are 3

@; .A/./ D 2 diag. ;  2 /@; .A/./1 ; 1

@;C .A/./ D 2 diag. ;  2 /@;C .A/./1 for all  6D 0,  2 RC . For the edge symbol we have 3

1

^ .A/./ D 2 diag.^ ;  2  ;  2  /^ .A/./.^ /1

(5.1.11)

277

5.1 Mixed problems in edge spaces

for all  6D 0,  2 RC . Here .^ u/.r; / WD u.r; / for all u 2 K s; .I ^ /,  2 RC . The edge symbol ^ .A/ consists of a family of boundary value problems on the infinite stretched cone I ^ . For every fixed  6D 0 it has a principal conormal symbol from the cone algebra of boundary value problems (which is independent of ), namely, H s2 .int I / ˚ c ^ . / C ; c ^ .A/.w/ D @ c ^ .T / A .w/ W H s .int I / ! ˚ c ^ .TC / C 0

1

(5.1.12)

where c ^ . /.w/ D w 2 C @2 , c ^ .T /u D ujD0 , c ^ .TC /u D @ ujD . Theorem 5.1.4. The operators (5.1.9) for  6D 0 form a family of Fredholm operators for all s > 32 and all  62 Z C 12 . The kernels and cokernels are independent of s and . Proof. The operators (5.1.9) which belong -wise to the cone algebra of boundary value problems on I ^ have the following principal symbolic components:  ^ .A/; @;˙ ^ .A/; c ^ .A/; E ^ .A/./; E0 ^ .A/./

(5.1.13)

from the cone algebra, cf. the expressions (3.4.6), (3.4.7). The exit symbol is responsible for the conical exit of I ^ to infinity. The Fredholm property of (5.1.9) is a consequence of the ellipticity of ^ .A/ with respect to all components of (5.1.13). It is a general property of elliptic operators in the cone algebra that kernel and cokernel are independent of s. Let us verify the ellipticity with respect to (5.1.13). x 2 n f0g 3 .xn1 ; xn / with the Dirichlet condition on We identify I ^ with R C R D fxn1 < 0g and the Neumann condition on RC D fxn1 > 0g. We have  ^ .A/. n1 ; n / D j n1 j2  j n j2 which is obviously elliptic. Moreover, 

 H s2 .RC / @;˙ ^ . / s ˚ .xn1 ; n1 / W H .RC / ! @;˙ ^ .A/.xn1 ; n1 / D @;˙ ^ .T˙ / C (5.1.14) is given by @;˙ ^ . /.xn1 ; n1 / D j n1 j2 C @2xn @; ^ .T /.xn1 ; n1 /u.xn / D u.0/ @;C ^ .TC /.xn1 ; n1 /u.xn / D @xn u.0/

on xn1 ? 0; for xn1 < 0; for xn1 > 0:

The operators (5.1.14) are isomorphisms for n1 6D 0 on xn1 < 0 and xn1 > 0, respectively, up to xn1 D 0 from the respective sides. The conormal symbol (5.1.12) will be discussed at the end the proof. The exit symbolic structure consists of two pairs E ^ and E0 ^ , belonging to the interior symbol and the boundary symbols on xn1 < 0 and xn1 > 0. In the interior we have the pair E ^ . /.; n1 ; n / D .jj2  j n1 j2  j n j2 ; j n1 j2  j n j2 /;

278

5 Mixed problems in weighted edge spaces

where the first component is the complete symbol, non-vanishing for all . n1 ; n / 2 R2 (including 0, since  6D 0), and the second component, the homogeneous principal part of the first one, is 6D 0 for . n1 ; n / 2 R2 n f0g. Moreover, on the minus and plus sides of the boundary we have the pairs       j n1 j2 C @2xn jj2  j n1 j2 C @2xn E0 ^ .; n1 / D ; ; T˙ @;˙ ^ .T˙ / @;˙ ^ .T˙ / where the first components are bijective as operator H .RC / ! s

H s2 .RC / ˚ C

for all

n1 2 R (because of  6D 0), while the second components as the  -homogeneous principal parts of the first ones are bijective for all n1 2 R n f0g. In this way we have verified the exit ellipticity of the -dependent Zaremba probx 2 n f0g; concerning the general theory of lem ^ .A/./ in the infinite half-plane R C boundary value problems on manifolds with conical exits to infinity we also refer to [90, Chapter 3]. For the ellipticity in the cone algebra it remains to check the bijectivity of the operators (5.1.12) for w 2 C, Re w 62 Z C 12 . What we know from the ellipticity of the original problem, i.e., the ellipticity of together with the Shapiro–Lopatinskij ellipticity of T is that the operators (5.1.12) form a parameter-dependent elliptic family of boundary value problems on the interval I with the parameter Im w. At the same time this family is holomorphic in w 2 C, and we know the bijectivity of (5.1.12) for large j Im wj. Generalities on holomorphic Fredholm families tell us that there is a discrete set D  C such that (5.1.12) is bijective for all w 2 C n D. In the present case we have an explicit information, namely, ˚

D D w 2 C W Re w 2 Z C 12 ; Im w D 0 : To verify this we first note that for w D 0 the problem c ^ .A/.0/u D 0 has only the trivial solution u 0. For w D a C i b 6D 0 a simple argument gives us ker.c ^ . //.w/ D fc1 e b e ia C c2 e b e ia W c1 ; c2 2 Cg. Then from the boundary conditions ujD0 D 0 and @ ujD D 0 we obtain the relations c1 C c2 D 0;

c1 e b .cos a C i sin a/  c2 e b .cos a  i sin a/ D 0:

Assuming c1 6D 0 (otherwise we have u 0) we obtain .e b C e b / cos a C i.e b  e b / sin a D 0. Since e b C e b 6D 0 for all b 2 R it follows that e b  e b D 0 and cos a D 0, i.e., b D 0 and a D k C 12 , k 2 Z. Remark 5.1.5. The non-bijectivity point of c ^ .A/ are simple. In fact, we have ker.c ^ .A//.k C 12 / D fc sin.k C 12 / W c 2 Cg, and the corresponding root functions (in the terminology of [60], see also Section 9.1.1) at the point k C 12 , k 2 Z, are c sin.w/. It is now easy to show that k C 12 , k 2 Z, is a simple zero for the holomorphic function c ^ .A/.w/ sin.w/ D t .0 0 w cos.w//.

5.2 Relations between edge and standard Sobolev spaces

279

5.2 Relations between edge and standard Sobolev spaces We establish a connection between the behaviour of mixed problems in standard Sobolev spaces as in Chapter 4 and weighted edge Sobolev spaces. This is possible via a corresponding reformulation of standard Sobolev spaces of sufficiently large smoothness. In Chapter 7 below we investigate such connections in more detail. For the reader who is more interested in mixed problems we give a brief description in more specified terms.

5.2.1 Spaces on the boundary We now construct a relation between the spaces H s .int Y˙ / and the weighted Sobolev spaces W s;s .Y˙ /, where Y˙ is regarded as a manifold with edge Z. Let us first x C in place of Y˙ . In this case we have consider the local situation with Rn2  R H s .Rn2  RC / D W s .Rn2 ; H s .RC // for all s 2 R, cf. Example 2.2.13. Let x C / WD fu 2 H s .R/ W supp u R x C g. Then for every s >  1 we have a H0s .R 2 x C / D K s;s .RC /, cf. [90, Section 2.1.2]. For s 62 N C 1 canonical isomorphism H0s .R 2 K s;s .RC /  1 s ˚ we have isomorphisms ! H .RC / for  D .s/ WD s  2 C 1. Here Œs

C

for some s 2 R denotes the integer part of s, i.e., the maximal integer  s. Let us also write s D Œs C fsg for the corresponding 0  fsg < 1. For purposes below we choose these isomorphisms parameter-dependent, with parameters .; / 2 Rn2Cl . x C and form Fix a cut-off function !.t/ on R 1

g.; /c WD Œ;  2

1 X

!.Œ;  xn1 /.Œ;  xn1 /j cj

(5.2.1)

1 j u.0/gj D0;:::;1 : @ j Š xn1

(5.2.2)

j D0

for c D .c1 ; : : : ; c / 2 C  and 1

b.; /u WD fŒ;  j  2

Then we have b.; /g.; / D idC  . Recall that in the latter expressions Πdenotes any strictly positive function in C 1 (Rq ) (here for q D n  2 C l) that is equal to jj with jj  c for some c > 0. Observe that (5.2.1) and (5.2.2) represent operator-valued symbols g.; / 2 Scl0 .Rn2Cl I C  ; H s .RC //;

b.; / 2 Scl0 .Rn2Cl I H s .RC /; C  / (5.2.3) 1

with respect to the group action u.xn1 / !  2 u.xn1 / on H s .RC / and the trivial group action on C, i.e., the identity for all  2 RC . In (5.2.2) we assume u 2 H s .RC / for s >   12 . The composition g.; /b.; / W H s .RC / ! H s .RC / is then a family of continx C /. If e W K s;s .RC / ! H s .RC / denotes the uous projections to im g.; /  C01 .R

280

5 Mixed problems in weighted edge spaces

canonical embedding, the operator s;s  K .RC / ˚ e g.; / W ! H s .RC / C



is an isomorphism for every .; / 2 Rn2Cl , s > 12 , s 62 12 C N,  D .s/. In the following global construction on a manifold M with boundary we choose a collar neighbourhood of the boundary with a fixed global normal coordinate t 2 Œ0; 1/. We employ this for M D Y˙ which is a smooth compact manifold with boundary Z. Let Uj , j D 1; : : : ; N , be coordinate neighbourhoods on M , assume that Uj \Z 6D x n1 , ; for 1  j  L and Uj \ Z D ; for L C 1  j  N , and let j W Uj ! R C n1 j D 1; : : : ; L, j W Uj ! R , j D L C 1; : : : ; N , be a system of charts. Without loss of generality the charts j for 1  j  L will be chosen in such a way that the restrictions j0 WD j jUj0 to Uj0 WD Uj \ Z, 1  j  L, form an atlas j0 W Uj0 ! Rn2 on Z. In addition the transition maps j ı 1 are assumed to be independent of t k (the normal coordinate to the boundary Z) for 0  t  12 , j; k D 1; : : : ; L. Choose P functions 'j ; j 2 C01 .Uj / for 1  j  L such that jLD1 'j D 1 in a neighbourhood of Z and j 1 on supp 'j . Then the functions 'j0 WD 'j jZ 2 C01 .Uj0 /, 1  j  L, form a partition of unity on Z subordinate to the covering fU10 ; : : : ; UL0 g of Z, and 0 1 0 0 j WD j jZ 2 C0 .Uj / are equal to 1 on supp 'j for all 1  j  L. Let us form the parameter-dependent pseudo-differential operators G./ WD

L X

'j j Opz .g/./.j01 /

0 j

W H s .Z; C  / ! H s .int M /;

j D1

B./ WD

L X

'j0 .j0 / Opz .b/./.j1 /

j

W H s .int M / ! H s .Z; C  /:

j D1

G./ is a family of potential operators, B./ a family of trace operators (of type ) in the algebra of boundary value problems on M with the transmission property at the boundary Z, and G./B./ is a Green operator in that algebra of type . Let E W W s;s .M / ! H s .int M / denote the canonical embedding. Then we have the following result: Proposition 5.2.1. There is a constant C > 0 such that the operator W s;s .M / ˚ A./ WD E G./ W ! H s .int M / H s .Z; C  / 



is an isomorphism for all jj  C , for every s > 12 , s 62 N C 12 ,  D .s/. Proof. Let us form the operator family   W s;s .M / 1  .GB/./ s ˚ B./ WD W H .int M / ! : B./ H s .Z; C  /

5.2 Relations between edge and standard Sobolev spaces

281

Then, according to the rules on operators with operator-valued symbols the composition A./B./ has locally near Z a (parameter-dependent) symbol of the form 1 C c.z; ; /, where c.z; ; / is a Green symbol of order 1. A similar relation holds for the composition B./A./ with respect to the involved operator-valued symbol as a .z; ; /-depending family of maps

K s;s .RC / ˚ C

!

K s;s .RC / ˚ . C

By Leib-

niz inverting 1 C c.z; ; / we obtain a symbol d.z; ; / of order 1 such that .1 C c.z; ; // # .1 C d.z; ; // D 1 modulo a symbol of order 1 in .; / (here # denotes the Leibniz multiplication of symbols in .z; /). On the level of operators we find an operator family P ./ such that A./P ./ D 1 modulo a family of smoothing operators which is a Schwartz function in . Thus A./ has a right inverse for large jj. In a similar manner we can proceed for B./A./, now in terms of the edge algebra on M (regarded as a manifold with edge Z). Thus A./ also has a left inverse for large jj, and hence A./ is invertible for jj  C for a sufficiently large positive constant C .

5.2.2 Edge spaces in the stretched domain In order to reinterpret the operator (5.1.2) in edge Sobolev spaces we also have to establish corresponding relations between H s .int X / and W s;s .X/, cf. the notation of Section 5.1.1. Let us first consider a manifold N without boundary, and let Z  N be of codimension 2, with a trivial normal bundle and W the associated stretched manifold. Later on we set N D 2X and pass to X and X itself. Theorem 5.2.2. For every s 2 R, s > 1, s 62 N, there exists a family of isomorphisms s;s   W .W / ˚ E K./ W ! H s .N / s  H .Z; C /

(5.2.4)

for  D .s/ D #f˛ 2 N 2 W j˛j < s  1g. (Here # indicates the number of elements in f: : : g.) Theproof of Theorem 5.2.2 will be given in Section 7.3.1.  1K./T ./ By we will denote the family of operators inverse to (5.2.4). T ./ We now illustrate the construction of analogous isomorphisms for X. As before we set I D f W 0    g where .r; / has the meaning of polar coordinates in the x 2 n f0g normal to Z. With the coordinates xQ D .xn1 ; xn / in R2 we form half plane R C the trace operators t ˛ .; /u WD Œ;  1j˛j Dx˛Q u.0/; t ˛ .; / W S.R2 / ! C, with the parameters .; / 2 RqCl , q D n  2 D dim Z. Applying t ˛ .; / to u 2 H s .R2C / for s > 12 C j˛j we obtain a symbol t ˛ .; / 2 Scl0 .RqCl I H s .R2C /; C/:

282

5 Mixed problems in weighted edge spaces

The group action in H s .R2C / is defined by . u/.x/ Q D u.x/, Q  2 RC , and on C by the identity for all  (as always when the respective space is of finite dimension). Let us x 2 / that is equal to 1 in a neighbourhood of xQ D 0. Moreover, we choose an ! 2 C01 .R C 1 form a potential operator k ˛ .; /c WD Œ;  ˛Š .Œ;  x/ Q ˛ !.Œ;  x/c, Q ˛ 2 N 2 , c 2 C; then we have k ˛ .; / 2 Scl0 .RqCl I C; H s .R2C // for arbitrary s 2 R. For every ˛ 2 N 2 we then have t ˛ .; /k ˛ .; / D idC for all .; / 2 RqCl . Let us set H0s .R2C / WD fu 2 H s .R2C / W Dx˛Q u.0/ D 0 for all j˛j < s  1g, s > 1. In a similar manner we define the space H0s .R2 /. Moreover, we have the spaces K s; ..S 1 /^ /

and

K s; .I ^ /;

respectively, with I D Œ0;  . Proposition 5.2.3. For every s  0, s 62 N, we have canonical isomorphisms K s;s .R2 n f0g/ D H0s .R2 /;

K s;s .I ^ / D H0s .R2C /:

A proof may be found in [90]. It follows that W s .Rq ; K s;s .I ^ // D W s .Rq ; H0s .R2C //; and a similar relation for R2 instead of R2C . Observe that Q 2 H s .RqC2 / W Dx˛Q u.z; 0/ D 0 for all j˛j < s  1g: W s .Rq ; H0s .R2C // D fu.z; x/ (5.2.5) Let us form the vectors of symbols t.; / WD t .t ˛ .; / W j˛j < s  1/ 2 Scl0 .RqCl I H s .R2C /; C .s/ /; k.; / WD .k ˛ .; / W j˛j < s  1/ 2 Scl0 .RqCl I C .s/ ; H s .R2C //; where .s/ is given in Theorem 5.2.2. From the construction we have t .; /k.; / D idC .s/ for all .; / 2 RqCl , while 1  k.; /t .; / W H s .R2C / ! H s .R2C / is a family of continuous projections to H0s .R2C /. We now pass to parameter-dependent continuous operators T ./ WD Opz .t /./ W W s .Rq ; H s .R2C // ! H s .Rq ; C .s/ /; K./ WD Opz .k/./ W H s .Rq ; C .s/ / ! W s .Rq ; H s .R2C //: Q 2 Rn W Observe that W s .Rq ; H s .R2C // D H s .RnC / for q D n  2, RnC D f.z; x/ xn > 0g.

5.2 Relations between edge and standard Sobolev spaces

283

Proposition 5.2.4. Let E W W s .Rq ; H0s .R2C // ! W s .Rq ; H s .R2C // be the canonical embedding. Then s 2 s q  W .R ; H0 .RC // ˚ E K./ W ! W s .Rq ; H s .R2C // s q  .s/ H .R ; C /



is an isomorphism for every  2 Rl , with the inverse t .1  K./T ./ T .//. z is a symbol which is invertible for all .; / 2 Proof. If a.; / 2 S  .RqCl I E; E/ qCl 1  z E/, then the family of pseudo-differential R such that a .; / 2 S .RqCl I E; s q s z is invertible for all  2 Rl , and we operators Op.a/./ W W .R ; E/ ! W .Rq ; E/ 1 1 have Op.a/ ./ D Op.a /./. This holds for all s 2 R. We apply this for  D 0 to the case a.; / WD .e k.; // W

H0s .R2C / ˚ C .s/

! H s .R2C /, where e W H0s .R2C / ! H s .R2C /

is the canonical embedding, and a1 .; / D t .1  k.; /t .; / t .; //. Proposition 5.2.4 admits a global variant on X; a detailed proof will be given in Section 7.3 below. Theorem 5.2.5. For every s > 1, s 62 N, there exists a family of isomorphisms 

 E K./ W

W s;s .X/ ˚ ! H s .int X / s  .s/ H .Z; C /

for all  2 Rl ,  .s/ WD #f˛ 2 N 2 W j˛j < s  1g. Remark 5.2.6. By virtue of a corresponding global version of the relation (5.2.5) for X, the space W s;s .X/ for s  0, s  1 62 N, can be identified with the subspace H0s .X / of all u 2 H s .int X / such that BujZ D 0 for every differential operator B of order < s  1. Then, if A W H s .int X / ! H sm .int X / is a differential operator of order m with smooth coefficients, A also induces (by restriction) a continuous map A0 W H0s .int X / ! H0sm .int X / for s  m. Let .Es Ks / denote the operator of Proposition 5.2.4 for any fixed s 2 R and  2 Rl , and denote by t .Ps Ts / its inverse. Then we have the identification A0 D Psm AEs , where Es W H0s .int X / ! H s .int X / is the canonical embedding and Psm W H s .int X / ! H0s .int X / a projection.

5.2.3 A reformulation of mixed problems from standard Sobolev spaces In the preceding chapter we completed the operator (5.1.2) by additional potential operators L˙ to an operator of index 0

284

5 Mixed problems in weighted edge spaces

0

Z WD @ T TC

H s2 .int X / ˚ H s .int X / 0 1 s A ˚ L W ! H 2 .int Y / 1 LC ˚ H s .Z; C Œs 2  / s 3 2 H .int YC / 1

(5.2.6)

for s > 32 , s 62 N C 12 ; recall that Œs  12 means the largest integer  s  12 . By virtue of (4.2.40) the operators L˙ have the form L˙ D L0˙ Rs with Rs W H s .Z/ ! L2 .Z/ being an order reducing operator on Z. From Proposition 5.2.1 applied to M D Y˙ we have isomorphisms 

E˙ G˙



W s;s .Y˙ / ˚ W ! H s .int Y˙ / s .s/ H .Z; C /

(5.2.7)

for s > 12 , s 62 N C 12 , .s/ WD Œs  12 C 1, where G˙ is obtained from G./ by taking  fixed and jj sufficiently large; moreover, E˙ W W s;s .Y˙ / ! H s .int Y˙ / are the respective continuous embeddings. Theorem 5.2.5 gives us isomorphisms   E K W

W s;s .X/ ˚ ! H s .int X / s  .s/ H .Z; C /

(5.2.8)

for every s > 1, s 62 N, and  .s/ as in Theorem 5.2.5 when we set K D K./ for any fixed . We also write .E˙;s G˙;s / and .Es Ks / if we want to indicate the smoothness s in (5.2.4) and (5.2.8), respectively.    s R Let us set B˙;s WD .E˙;s G˙;s /1 , P WD .Es Ks /1 . From (5.2.8) we Ts ˙;s obtain an isomorphism

 K WD

Es 0

Ks 0

W s;s .X/  ˚ H s .int X / 0 .s/ s ˚ / ! W H .Z; C 1 1 s ˚ H .Z; C Œs 2  / 1 H s .Z; C Œs 2  /

for any fixed s > 1, s 62 N. Moreover, for all s > 3, s 62 N, (5.2.8) together with

5.2 Relations between edge and standard Sobolev spaces

285

(5.2.7) yields an isomorphism

0 Ps2 B Ts2 B B 0 B L WD B 0 B B @ 0 0

0 0 R;s 1 2 B;s 1 2 0 0

1

0 0 0 0

C C C C CW C C RC;s 3 A 2 BC;s 3 2

W s2;s2 .X/ ˚ H s2 .Z; C .s2/ / ˚ H s2 .int X / 1 1 s ;s 2 2 .Y / ˚ W s 1 ˚ H 2 .int Y / ! : s 1 .s 1 / 2 2 ˚ H .Z; C / 3 ˚ H s 2 .int YC / 3 3 W s 2 ;s 2 .YC / ˚ 3 s 3 H 2 .Z; C .s 2 / /

This allows us to transform (5.2.6) to a Fredholm operator LZK for s > 3, s 62 N [ fN C 12 g between the corresponding edge Sobolev spaces plus standard Sobolev spaces on the interface Z. After an appropriate change of rows an columns in the block matrix LZK we obtain an equivalent operator W s2;s2 .X/ ˚ s 1 ;s 1 s;s 2 2 .Y / W W .X/ ˚ ˚ A.s/ W ! 3 3 H s .Z; C d / W s 2 ;s 2 .YC / ˚ ˚j3D1 H sj .Z; C dj /

(5.2.9)

for s > 3, s 62 N [ fN C 12 g and d D  .s/ C Œs  12 , s1 D s  2, s2 D s  12 , s3 D s  32 , d1 D  .s  2/, d2 D .s  12 /, d3 D .s  32 /. By construction we have ind Z D ind A.s/ D 0: Remark 5.2.7. Similarly as in Remark 5.2.6 the 3  1 upper left corner of A.s/ is nothing other than the restriction of (5.1.2) to W s;s .X/, regarded as a subspace of H s .int X/. Proposition 5.2.8. For any fixed s 2 R, s > 3, s 62 N [ fN C 12 g, the operator (5.2.9) is an element of the edge operator algebra on X (with edge Z) in the sense of Definition 5.4.2 below, belonging to the weight  D s and with the boundary orders  D 0, C D 1. Proof. The 3  1 upper left corner of the operator A.s/ belongs to the edge algebra, cf. Remark 5.1.2. The lower right corner which defines a map H s .Z; C d / !

286

5 Mixed problems in weighted edge spaces

L3

H sj .Z; C dj / is obviously a matrix of classical pseudo-differential operators on Z. The remaining entries consist of operators j D1

1

s Œs    H .Z; C 2 / 1 1 R;s 1 L R;s 1 T Ks W ! W s 2 ;s 2 .Y /; ˚ 2 2 H s .Z; C .s/ / 1

s Œs   H .Z; C 2 /  3 3 ! W s 2 ;s 2 .YC /: RC;s 3 LC RC;s 3 TC Ks W ˚ 2 2 H s .Z; C .s/ /

As is known from [196] all these operators belong to the edge calculus (here to the substructure of boundary value problems without the transmission property). The Green, potential, etc., operators are formulated in the category of S" .RC /-spaces, cf. (5.1.6). Since the compositions in question belong again to the edge calculus, we immediately obtain the desired characterisation. The composition RC;s 3 LC is a 2 potential of order 3=2; the same is true of the operator RC;s 3 TC Ks . 2

Let us now consider the principal edge symbolic structure of the operators A.s/ and its subordinate principal conormal symbol. According to the general notation the principal edge symbol consists of a family of operators K s2;s2 .I ^ / ˚ 1 s;s ^ s 1 ;s K 2 2 .R / K .I / ˚ ˚ ^ .A.s//.z; / W ! s 3 ;s 3 Cd 2 2 .RC / K ˚ C d1 Cd2 Cd3

(5.2.10)

parametrised by .z; / 2 T  Z n 0, with a corresponding scheme of Douglis–Nirenberg homogeneities. Because of the Fredholm property of (5.2.9) the operator function (5.2.10) is bijective for every .z; / 2 T  Z n 0. That means that the upper left 3  1 corner K s2;s2 .I ^ / ˚ 1 1 ^ .A.s//.z; / W K s;s .I ^ / ! K s 2 ;s 2 .R / DW K s2;s2 .I ^ / ˚ 3 ;s s 3 K 2 2 .RC / is a family of Fredholm operators for any fixed s > 3, s 62 N [ f N C 12 g. Since (5.2.10) is bijective, we have 











1 3 1 ind ^ .A.s//.z; / D d1 Cd2 Cd3 d D  .s2/C s C s  .s/ s : 2 2 2

287

5.3 Elliptic interface conditions

 Using the relations  .s/ D 12 fŒs 2 C Œs g and .s/ D s  12 C 1 it follows that 



ind ^ .A.s//.z; / D  s 

1 : 2

(5.2.11)

5.3 Elliptic interface conditions Mixed elliptic problems in weighted edge spaces are in general not Fredholm unless we do not impose a suitable number of additional interface conditions of trace and/or potential type. Their number depends on the chosen weights and is connected with the index of the principal edge symbol on a corresponding infinite cone. This corresponds to general phenomena in the calculus of operators on manifolds with edge. In general, it can be very difficult to calculate the number of interface conditions. In the present case of the Zaremba problem and other mixed problems for the Laplace operator we give explicit answers, using relative index results for the conormal symbols.

5.3.1 Mixed problems in spaces of arbitrary weights As noted before the discussion of solvability of the mixed problem (5.1.2) in standard Sobolev spaces rules out ‘most of the interesting’ solutions when we prescribe independent boundary data g˙ on Y˙ . On the other hand, because of the elliptic regularity for boundary value problems (in this case with the transmission property at int Y˙ / s 1

s 3

solutions with independently given boundary data in Hloc 2 .int Y / and Hloc 2 .int YC / s s belong to Hloc .X n Z/.WD Hloc .2X n Z/jint X / regardless of any possible jump of s .int.X n Z// solutions close to Z. The role of weighted edge spaces W s; .X/  Hloc is to reflect the standard elliptic regularity up the boundary outside Z and to admit adequate discontinuities near Z. For large s the regularity in W s;s .X/ is not really different from that in H s .int X /, as we saw before. However, the ‘realistic’ situation corresponds to weights  < s. In other words, the main task will be to pass from the W s;s .X/ case to W s; .X/ for small weights  . This is just the program of the present section. The operator A D t .A T TC / is an element in the edge algebra on X for an arbitrary weight  2 R, cf. the formula (5.1.4). For the ellipticity it is important to identify those  such that the associated principal edge symbol ^ .A. //.z; / W K s; .I ^ / ! K s2; 2 .I ^ /

(5.3.1)

represents a family of Fredholm operators, parametrised by .z; / 2 T  Z n 0; in the present case (5.3.1) is independent of z. x2 n The operators (5.3.1) belong to the cone algebra on the infinite cone I ^ Š R C f0g with corresponding mixed Dirichlet and Neumann conditions on R and RC , respectively. From the cone algebra it is known that the Fredholm property of an operator in K s; -spaces is equivalent of the bijectivity of all its principal symbols, cf. the formula (5.1.13). The components  ^ .A. //, @;˙ .A. // and E ^ .A. //,

288

5 Mixed problems in weighted edge spaces

E0 ^ .A.// are independent of the weight  . For  D s their bijectivities have been checked in the proof of Theorem 5.1.4. In other words, it remains to consider the conormal symbol and to identify those w 2 C such that the operators (5.1.12) are bijective. This has been done in the proof of Theorem 5.1.4. Theorem 5.3.1. For every s > that the operator

3 2

and  62 Z C 12 there are dimensions d. /; e. / such

W s2; 2 .X/ ˚ 1 1 A. / W W s; .X/ ! W s 2 ; 2 .Y / DW W s2; 2 .X/ ˚ 3 s 3 ; W 3 2 .YC / can be completed by extra conditions K. /, T . / and Q. / with respect to the interface Z to an elliptic operator in the edge algebra 

 A. / K. / A./ WD W T . / Q. /

W s; .X/ W s2; 2 .X/ ˚ ˚ ! : H s .Z; C d./ / H s2 .Z; C e./ /

(5.3.2)

Proof. The existence of elliptic interface conditions depends on some topological property of ^ .A.//. In this proof we interpret ^ .A. //.z; / as a family of Fredholm operators, parametrised by the points .z; / of the unit cosphere bundle S  Z induced by T  Z. The criterion is an analogue of the Atiyah–Bott condition for the existence of Shapiro–Lopatinskij elliptic boundary value problems, cf. [8]. An analogous condition for the existence of elliptic edge conditions in the edge algebra (for a closed base of the model cone) is obtained in [182], see also the papers [131] or [198]. In the present case the base I D Œ0;  of the model cone has a boundary, but the situation is similar. z / of Fredholm operators First recall that a continuous family F W X ! F .H; H z , where X is a compact topological space (for simbetween Hilbert spaces H and H plicity, arcwise connected), generates an index element K.X / in the K group of X , cf. Section 3.3.4. If dim ker F .x/ and dim coker F .x/ are constant, the families kerX F and cokerX F of kernels and cokernels, respectively, are (continuous, complex) vector bundles on X, and we have indX F D ŒkerX F  ŒcokerX F , where Œ denotes the class in K.X/, represented by the bundle in the brackets. If the dimensions are not H

z constant, is suffices to pass from F to a surjective operator family .F C / W ˚ ! H C N

z for a sufficiently large choice of N . Then for a suitable constant map C W C N ! H we can set    indX F WD kerX F C  ŒN ; (5.3.3) where N stands for the trivial bundle on X with fibre C N ((5.3.3) does not depend on the choice of N or C , cf. Theorem 3.3.35).

5.3 Elliptic interface conditions

289

In our case we have X D S  Z and F D ^ .A. //. The criterion for the existence of ‘Shapiro–Lopatinskij’ elliptic interface conditions is now indS  Z ^ .A. // 2 1 K.Z/; where 1 W

(5.3.4)



K.Z/ ! K.S Z/ is the pull back of the corresponding K groups under the canonical projection 1 W S  Z ! Z (induced by the bundle pull back). More precisely, the relation (5.3.4) is necessary and sufficient. For s D  > 3, s 62 N [ fN C 12 g the operator (5.2.9) as an element of the edge algebra (cf. Proposition 5.2.8) is Fredholm and hence elliptic (different orders of smoothness in the Sobolev spaces on Z do not affect this; we can always unify the orders by composing our operators by suitable elliptic order reducing operators on Z). Thus the property (5.3.4) holds for ^ .A.s//. For arbitrary weights  we recall that (5.3.1) is a family of elliptic operators in the cone x 2 n f0g. In this situation algebra of boundary value problems on the infinite cone R C the property (5.3.4) is independent of  as soon as (5.3.1) is Fredholm for different . The technique to prove this is similar to that in boundary value problems without the transmission property. An explicit proof for the edge situation is given in [113]. An inspection of the details shows immediately that the arguments also apply in the present situation. In other words, (5.3.4) holds for all  62 ZC 12 . This completes the proof.

5.3.2 Construction of elliptic interface conditions Our next objective is to obtain more information on the numbers d. / and e. / of extra interface conditions in the sense of Theorem 5.3.1. Theorem 5.3.2. Let s > 3, s 62 N [ fN C 12 g,  < s and  62 Z C 12 . Moreover, let n.s; / denote the number of non-bijectivity points of c ^ .A/.w/ in the strip fw 2 C W 1  s < Re w < 1  g. Then we have ind ^ .A. // D ind ^ .A.s// C n.s;  /;

(5.3.5)

and (5.3.5) is independent of s. Proof. The formula (5.3.5) can be interpreted as a relative index result on boundary value problems in an infinite cone when ˇ WD s is replaced by another weight  < ˇ. The strategy of the proof is based on index formulas for boundary value problems Bˇ and B on the infinite cone which are of Fuchs type both with respect to r D 0 and r D 1, operating in weighted Sobolev spaces with weights at zero and infinity, cf. [60], and the applications in [200], [73]. However, the operators to be considered here, denoted by Kˇ and K , respectively, (after a multiplication by suitable weight factors, cf. the notation (5.3.6) below) act in other scales of spaces. Therefore, we will compare ind B  ind Bˇ and ind K  ind Kˇ . The details are as follows. Let us set for the moment E WD H t .int I /, Ez WD

H t 2 .int I / ˚ C˚C

for any fixed t > 32 .

Moreover, set a.w/ WD c ^ .A/.w/, cf. the formula (5.1.12). Then a.w/ is a holomorphic family of Fredholm operators E ! Ez with Z C 12 as the set of points w 2 C

290

5 Mixed problems in weighted edge spaces

where a.w/ W E ! Ez is not bijective. Since the index of a continuous Fredholm function is constant, we have ind a.w/ D 0 for all w 2 C. From Section 5.1.3 it follows that dim ker a.w/ D dim coker a.w/ D 1 for all w 2 Z C 12 . These properties are independent of t . Let us fix  6D 0, t > 3, and set 1

3

K WD diag.r 2 ; r 2 ; r 2 /^ .A. //./;

(5.3.6)

K t; WD K t; .I ^ /, and z t2; WD r 2 K t2;2 .I ^ / ˚ r 12 K t 12 ; 12 .R / ˚ r 32 K t 32 ; 32 .RC /: K z t2; is a Fredholm operator, and we have Then K W K t; ! K ind ^ .A. //./ D ind K :  1

By definition we have K D opM 2 .a/ C t .r 2 jj2 0 0/ with the Mellin symbol a.w/ D t .w 2 C @2 rD0 rD0 @ /, where rD˛ denotes the restriction operator to  D ˛, ˛ D 0; .  1

Let us now set B WD opM 2 .a/ C ! t .r 2 jj2 0 0/ for some cut-off function !.r/. Moreover, form the spaces H t2;.;ı/ WD !1 H t; .I ^ / C .1  !1 /H t;ı .I ^ / and

0

Hz t2;.;ı/

0 t2;ı ^ 1 1 H t2; .I ^ / .I / H B B C C ˚ ˚ B t 1 ; B C C 1 t ;ı B B C WD !1 B H 2 .R / C C .1  !1 / B H 2 .R / C C @ @ A A ˚ ˚ 3 3 H t 2 ; .RC / H t 2 ;ı .RC /

for a weight ı   and any cut-off function !1 (the choice of !1 does not affect the spaces). Then we have a continuous operator B W H t;.;ı/ ! Hz t2;.;ı/ . The second summand in the expression for B is a compact operator in these spaces. Therefore, ellipticity and Fredholm property are determined by a.w/ alone. By assumption a.w/ has no non-bijectivity points on the weight line 1 (in the sense of weight shift corrections, according to the formula (5.4.13) below). Since the non-bijectivity points form a discrete set (here the real half-integers), we can choose ı in such a way that a.w/ is also bijective on 1ı . In a similar manner we now form the operators z t2;ˇ ; Kˇ W K t;ˇ ! K

Bˇ W H t;.ˇ;ı/ ! Hz t2;.ˇ;ı/

(5.3.7)

for another weight ˇ 2 R and ı  max.; ˇ/, such that 1ˇ does not contain nonbijectivity points of a.w/. Then the operators (5.3.7) are also Fredholm.

5.3 Elliptic interface conditions

291

We are now in a situation of Fredholm operators on manifolds as is studied in [140] (see also Section 9.4). The operators K ; B and Kˇ ; Bˇ satisfy the conditions K j0
Kˇ j0
for every R > 0 such that ! 1 on Œ0; R/. Here 0 < r < R (R < r < 1) indicates x 2 n f0g such that 0 < jxn1 ; xn j < R (R < jxn1 ; xn j < those points .xn1 ; xn / 2 R C 1). The corresponding result from [140] consists of the equality ind K  ind Kˇ D ind B  ind Bˇ :  1

ˇ 1 2

As noted before we have B D opM 2 .a/, Bˇ D opM tors. This gives us  1

(5.3.8)

.a/, modulo compact opera-

ˇ 1 2

ind B  ind Bˇ D ind opM 2 .a/  ind opM

.a/ D n.ı;  /  n.ı; ˇ/ D n.ˇ;  /: (5.3.9) The second equation is a consequence of an analogue of the results of [200] in the version of boundary value problems, see [73], as well as Section 9.2. below using the fact that the non-bijectivity points of a.w/ in the respective weight strips are all simple, cf. Remark 5.1.5. To complete the proof it suffices to combine the relations (5.3.8) and (5.3.9).   Corollary 5.3.3. For every k 2 Z and  2 12  k; 32  k we have ind ^ .A. // D k:

(5.3.10)

In other words, we have e. /  d. / D k for the dimensions e. /; d. / in Theorem 5.3.1. In fact, for a given weight  62 Z C 12 we can choose any s > 3, s 62 N [

˚ N C 12 and apply Theorem 5.3.2, combined with the relation (5.2.11). This gives us  ind ^ .A.// D  s  12 Cn.s;  /. Then (5.3.10) follows from n.s;  / D k CŒs 1,    s  12 D Œs C 1 for 0 < fsg < 12 and n.s;  / D k C Œs ,  s  12 D Œs for 1 < fsg < 1. 2

5.3.3 Parametrices and regularity of solutions for the Zaremba problem We now establish parametrices in the edge calculus and obtain regularity of solutions to our mixed problems, see also Section 7.2.6 below. Theorem 5.3.4. For every fixed  62 Z C 12 the operator A WD A. / of Theorem 5.3.1 has a parametrix P in the edge calculus, cf. Definition 5.4.2 below, i.e., we have PA D I  C, AP D I  D, where C and D are smoothing operators as in Definition 5.4.2

292

5 Mixed problems in weighted edge spaces

(iv), and I is the identity operator in the corresponding spaces. The operator P is . ; @/-regular (cf. the terminology of Section 5.4.2 below) and has the type 0 (both on the Dirichlet and the Neumann side). z is elliptic in the calculus of Section 5.4.3, Proof. By Theorem 5.3.1 the operator A i.e., all symbolic components are bijective. Thus the existence of a parametrix is a consequence of Theorem 5.4.7 below. The resulting type follows from the corresponding generalities on boundary value problems, cf. Theorem 3.3.17. In fact, the type of the parametrix of an elliptic boundary value problem of order  and type d is equal to max.d; 0/. In the present case we have  D 2 and d D 1 on the Dirichlet and d D 2 on the Neumann side. The . ; @/-regularity of the parametrix follows from the fact that A itself is . ; @/-regular and that the inversion of symbols from the edge calculus is compatible with the Leibniz inversion of smooth complete symbols, relevant for the . ; @/-regularity. Corollary 5.3.5. The operator (5.3.2) is Fredholm for every  62 Z C 12 , s > 32 , and kernel and cokernel are independent of s. Moreover, Au 2 W s2; 2 .XI Y ; YC / ˚ H s2 .Z; C e. / / and u 2 W 1; .X/ ˚ H 1 .Z; C d./ / implies u 2 W s; .X/ ˚ H s .Z; C d./ /.

5.3.4 Jumping oblique derivatives and other examples Let us now consider other examples of mixed problems for the Laplace operator, especially, conditions with jumping oblique derivatives on Y˙ . In this case we have H s2 .int X / ˚ s s 3 @ A 2 .int Y / ; A D T W H .int X / ! H ˚ TC 3 H s 2 .int YC / 0

1

(5.3.11)

s > 32 , for T WD r  B , TC WD r C BC . Here B˙ are locally of the form B D

n2 X

˛i Dzi C ˛Dxn1 C Dxn ;

BC D

iD1

n2 X

ˇi Dzi C ˇDxn1 C ıDxn

iD1

with coefficients ˛i ; ˇi , smoothly depending on z D .z1 ; : : : ; zn2 /, and constants ˛; ˇ; ; ı,  6D 0, ı 6D 0. We assume n  3; the operators T˙ then satisfy the Shapiro–Lopatinskij condition. In Section 4.2.4 we completed the operator (5.3.11) under the condition n2 X iD1

˛i .z/Dzi D

n2 X iD1

ˇi .z/Dzi D 0

5.3 Elliptic interface conditions

293

and ˇ D ˛ı by additional potential operators L to a Fredholm operator of index zero H s2 .int X / 1 0 s ˚ H .int X / 0 3 ˚ ! H s 2 .int Y / ; Z WD @ T L A W ˚ TC LC H s .Z; C l.s/ / s 3 H 2 .int YC / for s > 32 ; s 62 N, where l.s/ D 0 for 32 < s < 2 and l.s/ D Œs  1 for s > 2. The z of index zero in the edge operator can equivalently be reformulated as an operator A.s/ algebra, by applying the same technique as before for (5.2.9). In the present case this holds for s > 3; s 62 N, and we have d D  .s/ C Œs  1 ; s2 D s  32 ; d2 D .s  32 /, and di ; si ; i D 1; 3, are as in (5.2.9). Similarly as in Section 5.2.3 we can express the z principal edge symbol of the upper left 3  1 corner A.s/ of A.s/, namely, K ss;s2 .I ^ / ˚ ^ . /./ 3 3 ^ .A.s//.z; / D @ ^ .T /.z; / A W K s;s .I ^ / ! K s 2 ;s 2 .R / ˚ ^ .TC /.z; / 3 s 3 ;s K 2 2 .RC / 1

0

for ^ . /./ D r 2 ..r@r /2 C @2  r 2 jj2 / and ˇ ˛ @   @ ˇˇ  r C ; ^ .T /.z; / D r i @r i @ ˇD0 ˇ   ˇ @  ı @ ˇˇ ^ .TC /.z; / D r 1 : r  i @r i @ ˇD 1



/ is a family of Fredholm operators for any fixed s > 3, s 62 N [ ˚^ .A.s//.z;

N C 12 with index ind ^ .A.s//.z; / D Œs : The conormal symbol 1 @2 C w 2 H s2 .int I / s 1 ˚ ; c ^ .A/.w/ D @ i .@  w/jD0 A W H .int I / ! 1 C ˚ C .@ C w/jD0 i 0

I D Œ0;  , defines a family of bijective operators for all w 62 Z. In fact, first observe that w D 0 is a simple non-bijectivity point of c ^ .A/.w/. Now let w D a C i b 6D 0. Then the boundary conditions give us c1 .i  1/ D c2 .i C 1/; c1 .1  i/e b .cos a C i sin a/ C c2 .1 C i /e b .cos a  i sin a/ D 0:

294

5 Mixed problems in weighted edge spaces

Take c1 D 0. Then we obtain .e b  e b / cos a  i.e b C e b / sin a D 0, and hence b D 0; a D k; k 2 Z. We have ker c ^ .A/.k/ D fc.cos k C sin k/ W c 2 Cg. Using c ^ .A/.w/.cos w C sin w/ D t . 0 0 2i w sin w / we see that the nonbijectivity points are simple. Similarly as for the Zaremba problem we thus obtain for the edge symbol ind ^ .A. // D ind ^ .A.s// C n.s;  / for all  < s;  62 Z. More precisely, we have ind ^ .A. // D k for all  2 .k; k C 1/. This follows from n.s;  / D k C Œs . As another example we take the Laplace operator with the Dirichlet condition T on Y and the condition TP C on YC as above in this section. Applying a result of Section 4.2.4 for the case n2 iD1 ˛i .z/Dzi D 0 and ˛ D  D 1 the corresponding operator A can be completed to a Fredholm operator (5.2.6) for all s > 32 ; s 62 N C 34 ,  where in this case l.s/ D s  34 . Introducing again A.s/ as a realisation of A

˚ in weighted Sobolev spaces for s > 3; s 62 N [ N C 34 we obtain in this case  ^ .A.s//./ D  s  34 . The set of non-bijectivity points of c ^ .A/ coincides with

˚ w 2 C W w 2 Z C 14 , and the points are simple. In fact, for w D 0 the operator c ^ .A/.0/u D 0 has only the trivial solution. If w D a C i b 6D 0 the boundary conditions give us c1 C c2 D 0;

c1 .1  i /e b e ia C c2 .1 C i /e ia D 0;

(recall that ker c ^ . /.w/ D fc1 e b e ia C c2 e b e ia W c1 ; c2 2 Cg). Let us assume that c1 6D 0 (otherwise we have u D 0). Then we obtain .cos a C sin a/.1  e 2b / C i.sin a  cos a/.1 C e 2b / D 0 which implies that b D 0 and a D k C 14 ; k 2 Z. We have ker M ^ .A/.k C 14 / D fc sin.k C 14 / W c 2 Cg; k 2 Z, and using   1 c ^ .A/.w/ sin.w/ D t 0 0 .w cos.w/ C w sin.w// i we see that the non-bijectivity points are simple. Similarly as before it follows that ind ^ .A. //./ D ind ^ .A.s//./ C n.s;  / for all  < s; 1   62 Z C 14 . More precisely, ind ^ .A. //./ D k 3

 for  2 4  k; 74  k . This follows from n.s;  / D k C Œs  1, s  34 D Œs  1  for fsg < 34 and n.s;  / D k C Œs , s  34 D Œs for fsg > 34 . 

5.3 Elliptic interface conditions

295

Let us consider two other examples for the Laplace operator, namely, first with Dirichlet conditions both on Y and YC , and then with Neumann conditions on both sides. For every  2 N we have continuous operators H s2 .int X / ˚ s s 1 A @ 2 .int Y / ; A WD T; W H .int X / ! H ˚ TC; 1 H s 2 .int YC / 0

1

(5.3.12)

ˇ for all s >  C 12 where T˙; u WD @xn uˇY (the case  D 0 corresponds to Dirichlet, ˙  D 1 to Neumann conditions). Similarly as in Section 4.2.3 for the Zaremba problem, we can complete the operator (5.3.12) by additional potential operators L˙; to a Fredholm operator of index 0 0

Z WD @ T; TC;

H s2 .int X / ˚ H s .int X / 0 1 s A 2 .int Y / ˚ L; W ! H s l./ LC; ˚ H .Z; C / s 1 2 .int YC / H 1

(5.3.13)

for s >  C 12 , s 62 N, where l./ D 0 for  C 12 < s <  C 1 and l./ D Œs   for s >  C 1. In fact, using similar notation as in Section 4.2.3, @; .r C BeC /./ D r C op.b /./eC W L2 .RC / ! L2 .RC / for

s 12   %  sCC 12   %  b .%; / D f jj  i% jj  i% f C jj C jj

are Fredholm for all s 62 Z,  2 Rn2 n f0g. This follows from the fact that the corresponding conormal symbol 1

c; .r C BeC /.w/ D

1

.1/s 2 .1/sCC 2 C i2w 1e 1  e i2w

is not equal to 0 on the line  1 for all s 62 Z. Furthermore, for our example we have 2    1 bC; .%; / D f %  jj  i%/sCC 2 , and ord bC; .%; / D s C  C 12 62 Z C 12 C j j

for all s 62 Z. The analogous results of Proposition 4.2.34 give us that @; .r C BeC /./,  6D 0, are bijective for  C 12 < s <  C 1 and injective for all s >  C 1, s 62 N, with dim coker @; .r C BeC /./ D Œs   .

296

5 Mixed problems in weighted edge spaces

The operator (5.3.13) can be reformulated as an operator A .s/ of index zero in the edge algebra, namely, W s2;s2 .X/ ˚ 1 1 W s 2 ;s 2 .Y / W s;s .X/ ˚ ˚ A .s/ W ! 1 1 s d./ s ;s 2 2 .YC / / H .Z; C W ˚ ˚j3D1 H sj ./ .Z; C dj ./ /

(5.3.14)

for s > 3, s 62 N and d./ D  .s/ C Œs   , s1 ./ D s  2, s2 ./ D s3 ./ D s    12 , d1 ./ D .s  2/, d2 ./ D d3 ./ D .s    12 /. Because of the Fredholm property of (5.3.14) the operator K s2;s2 .I ^ / ˚ 1 1 K s 2 ;s 2 .R / K s;s .I ^ / ˚ ˚ ^ .A .s//.z; / W ! 1 1 d./ s ;s C 2 2 .RC / K ˚ C d1 Cd2 Cd3 is bijective for every .z; / 2 T  Z n 0, and hence the upper left 3  1 corner is of index ind ^ .A .s//./ D d1 C d2 C d3  d 1 D .Œs  2 2 C Œs  2  Œs 2  Œs / C 2.Œs    1 C 1/  Œs  

2 D Œs C 1  : We now pass to weighted edge spaces with arbitrary weights. We have continuous operators W s2; 2 .X/ ˚ 1 1 A . / W W s; .X/ ! W s 2 ; 2 .Y /; ˚ 1 s 1  2 2 .YC / W s >  C 12 ,  2 R. The corresponding principal edge symbol 0 1 ^ . / ^ .A . //./ D @ ^ .T; / A ./ ^ .TC; /

5.3 Elliptic interface conditions

297

where ^ . /./ D r 2 ..r@r /2 C @2  r 2 jj2 /, ^ .T; /u D @ ujD0 , ^ .TC; /u D @ ujD , represents a family of Fredholm operators for all  62 Z n f1g in the case of  D 0, and  62 Z for  D 1. This can be proved in the same way as the assumption of Theorem 5.1.4. We only check the bijectivity of the operators c ^ .A /.w/ for w 2 C, Re w 62 Z n f0g in the case of  D 0 and Re w 62 Z for  D 1. We have 1 @2 u C w 2 u c ^ .A /.w/u D @ @ ujD0 A : @ ujD 0

(5.3.15)

(i) Let w D 0. From c .^ .A //.0/u D 0 it follows that u 0 in the case  D 0, and u c 2 C for  D 1. (ii) Let w D a C i b 6D 0. The equation M ^ .A /.w/u D 0 is equivalent to the system ( c1 C c2 D 0 for  D 0; c1 e b e ia C c2 e b e ia D 0 and to

(

c1  c2 D 0 c1 e b e ia  c2 e b e ia D 0

for  D 1:

This shows that the set of non-zero non-bijectivity points of (5.3.15) coincides with Z n f0g both for  D 0 and  D 1. Summing up, the set of non-bijectivity points of (5.3.15) is equal to Z n f0g for  D 0 and Z for  D 1. Note that the non-bijectivity points of (5.3.15) are simple, except for the point w D 0 (in the case  D 1) which is of multiplicity 2. a) The case  D 0. Let s > 3, s 62 N,  62 Z n f1g and  < s. Then we have 8 ˆ <Œs C k  1 for  2 .k; k C 1/; k  1; n.s;  / D Œs C k for  2 .k; k C 1/; k  2; ˆ : Œs  1 for  2 .0; 2/: A corresponding analogue of Theorem 5.3.2 gives us ind ^ .A0 . // D ind ^ .A0 .s// C n.s;  / 8 ˆ
(5.3.16)

298

5 Mixed problems in weighted edge spaces

b) The case  D 1. Let s > 3, s 62 N,  62 Z and  < s. Then ( Œs C k for  2 .k; k C 1/; k  1; n.s;  / D Œs C k C 1 for  2 .k; k C 1/; k  0; and a corresponding analogue of Theorem 5.3.2 yields ( k for  2 .k; k C 1/; k  1; ind ^ .A1 . // D k C 1 for  2 .k; k C 1/; k  0:

(5.3.17)

5.4 Edge calculus, specified to mixed problems The calculus of edge-boundary value problems admits the construction of pseudo-differential parametrices of elliptic mixed problems. The general calculus will be established in Chapter 7. In this section we present a version which is adapted to specific orders and other concrete features.

5.4.1 Edge amplitude functions In Section 3.1.5 we introduced the space B ;d .I I Rl / of parameter-dependent pseudo-differential boundary value problems on the interval I D Œ0;  . This will be employed as the model algebra for establishing edge amplitude functions that refer to the stretched wedge   I ^ ,  Rq open (where z 2  are local coordinates on Z, q D n  2). The edge amplitude functions are a basic ingredient of the parametrices in Theorem 5.3.4. First we define the space B ;d .I I C  Rq / of all functions f .w; / 2 A.C; B ;d .I I Rq // such that

f .ˇ C i%; / 2 B ;d .I I R1Cq / %;

for every ˇ 2 R, uniformly in c  ˇ  c 0 for every c  c 0 . Weighted Mellin  .f /./ for amplitude functions pseudo-differential operators opM f .r; r 0 ; w; / 2 C 1 .RC  RC ; B ;d .I I C  Rq // are defined by the formula (2.4.7) for every  2 Rq . In the present case f may also  smoothly depend on z 2  which gives us opM .f /.z; /. For pseudo-differential actions on RC based on the Fourier transform we write op.p/.z; / (or also opr .p/.z; /). The amplitude functions are taken in the form p.r; z; %; / D p.r; Q z; r%; r/ Q 2 C 1 .R x C  ; B ;d .I I R1Cq //. with p.r; Q z; %; Q / Q %; Q

(5.4.1)

5.4 Edge calculus, specified to mixed problems

299

Theorem 5.4.1. For every p.r; z; %; / of the form (5.4.1) there exists a Mellin symbol Q z; w; r/ h.r; z; w; / D h.r;

(5.4.2)

Q z; w; / Q 2 C 1 .R x C  , B ;d .I I C  Rq // such that for an h.r; Q

 opr .p/.z; / D opM .h/.z; / mod C 1 .; B 1;d .I ^ I Rq //

for every  2 R. Moreover, setting Q z; r%; r/; p0 .r; z; %; / WD p.0;

Q z; w; r/ h0 .r; z; w; / WD h.0;

we have  .h0 /.z; / mod C 1 .; B 1;d .I ^ I Rq //: opr .p0 /.z; / D opM

Theorem 5.4.1 is a simple special case of Theorem 6.1.17 below. If (5.4.2) is connected with (5.4.1) via the latter theorem we will say that h is a (holomorphic) Mellin quantisation of p. We now fix arbitrary cut-off functions !j .r/, j D 0; 1; 2, such that !1 1 on supp !0 , !0 1 on supp !2 , and cut-off functions  .r/; Q .r/. Set a.z; / WD  .r/faM .z; / C a .z; /gQ .r/

(5.4.3)

with  1

aM .z; / WD r  !0 .rΠ/ opM 2 .h/.z; /!1 .rΠ/; a .z; / WD r  .1  !0 .rΠ// opr .p/.z; /.1  !2 .rΠ//; where h is a Mellin quantisation of p. Let ^ .a/.z; / WD r  f!0 .rjj/ opM .h0 /.z; /!1 .rjj/ C .1  !0 .rjj// opr .p0 /.z; /.1  !2 .rjj//g Then we have

^ .a/.z; / D   ^ .a/.z; /1 

(5.4.4)

(5.4.5)

for all  2 RC , .z; / 2   .Rq n f0g/, and  WD diag.^ ;  ;  /, cf. the notation in (5.1.11). Comparing the relations (5.1.11) and (5.4.5) we see a difference in the homogeneities which comes from the assumed unified order in the construction of (5.4.3). Moreover, consider c ^ .a/.z; w/ D h0 .0; z; w; 0/;

w 2 1 :

(5.4.6)

By definition (5.4.3) is a 33 matrix a.z; / D .aij .z; //i;j D1;2;3 , associated with an order  2 R and a weight  2 R.

300

5 Mixed problems in weighted edge spaces

For our application we need orders and weights depending on i; j . We take aij .z; / 2 S ij .  Rq I Ej ; Ezi /

(5.4.7)

with E1 WD K s1 ;1 .I ^ /; E2 WD K s2 ;2 .R /; E3 WD K s3 ;3 .RC /; Ez1 WD K t1 ;ı1 .I ^ /; Ez2 WD K t2 ;ı2 .R /; Ez3 WD K t3 ;ı3 .RC /;

E4 WD C; Ez4 WD C

for orders and weights as at the end of Section 5.1.2, together with t1 WD s  ;

1 t2 WD s    ; 2

1 t3 D s  C  : 2

The technique of proving relations of the kind (5.4.7) may be found in Chapter 7 below (see Theorem 7.2.3). In order to express the complete edge amplitude functions we need a further category of operator-valued symbols, the so-called smoothing Melling symbols. They also refer to orders ij and weights ij , i; j D 1; 2; 3. For convenience we first give the definition for ij -independent orders and weights. Let M1;d .I I ˇ / for ˇ 2 R, d 2 N, denote the space of all operator functions f .w/ 2 A.fˇ  " < Re w < ˇ C "g; B 1;d .I // for some " D ".f / > 0, such that f .Ci%/ 2 B 1;d .I I R% / for every  2 .ˇ "; ˇ C "/, uniformly in compact subintervals. Denoting for the moment by M1;d .I I ˇ /" the set of all f 2 M1;d .I I ˇ / belonging to a fixed " S > 0, we obtain a Fréchet space. 1 1;d .I I ˇ // D ">0 C 1 .; M 1;d .I I ˇ /" /. This allows us to form C .; M Setting  1

m.z; / WD r  !.rΠ/ opM 2 .f /.z/!.rΠ/ Q for an f .z; w/ 2 C 1 .; M 1;d .I I 1 // we obtain a classical operator-valued symbol. The homogeneous principal component of m.z; / of order  has the form  1

^ .m/.z; / D r  !.rjj/ opM 2 .f /.z/!.rjj/: Q

(5.4.8)

Let us set c ^ .m/.z; w/ WD f .z; w/;

w 2 1 :

(5.4.9)

We now assume  as well as the weight  to depend on i; j as at the end of Section 5.1.2. For fij 2 C 1 .; M 1;d .I I 1ij // we then obtain a matrix m.z; / WD .mij .z; //i;j D1;2;3 of elements mij .z; / 2 Scl ij .  Rq I Ej ; Ezi / 

(5.4.10)

for every s > d  12 (clearly in the spaces Ezi in the range the smoothness may be replaced by 1 in this case).

5.4 Edge calculus, specified to mixed problems

301

An edge amplitude function is defined as a 4  4 block matrix operator function of the form a.z; / WD p.z; / C m.z; / C g.z; / (5.4.11) for

 p.z; / WD

 a.z; / 0 ; 0 0



 m.z; / 0 ; 0 0

m.z; / D

with a.z; / D .aij .z; //i;j D1;2;3 as in equation (5.4.3), m.z; / D .mij .z; //i;j D1;2;3 as in (5.4.10), and g.z; / D .gij .z; //i;j D1;:::;4 as in (5.1.6). From (5.1.8), (5.4.8) and (5.4.4) we obtain a 4  4 matrix of homogeneous principal symbols ^ .a/.z; / W

L4

j D1

Ej !

L4 iD1

Ezi

(5.4.12)

with the above-mentioned spaces Ej ; Ezi , i; j D 1; : : : ; 4, and s > d  12 , with entries of order ij . The 3  3 upper left corner of the operator (5.4.12) represents a family x 2 n f0g D I ^ with the boundary of boundary value problems on the infinite cone R C components R˙ . As such they have a 3  3 matrix of subordinate conormal symbols H s .int I / H s .int I / ˚ ˚ c ^ .a/.t; w/ W ! C˚C C˚C depending on z 2 Z and the complex variable w. According to (5.4.6) and (5.4.9) the entries c ^ .aij / of c ^ .a/ are given for w on the weight lines 1ij for i; j D 1; 2; 3. It is convenient to normalise the representation by setting c ^ .aij /.z; w/ WD c ^ .aij /.z; w C ij   /;

(5.4.13)

such that all entries are defined on 1 .

5.4.2 Edge-boundary value problems Our next objective is to formulate a pseudo-differential scenario to express parametrices of mixed elliptic problems. It can be subsumed under the calculus of edge-boundary value problems of Chapter 7. Here we give a survey on elements of that calculus in a form which is adapted to our application. Definition 5.4.2. The edge algebra of boundary value problems on X is defined as the space of all 4  4 block matrices (5.1.5), i.e., A D .Aij /i;j D1;:::;4 with orders .ij /i;j D1;:::;4 as at the end of Section 5.1.2, which have (modulo smoothing operators to be described under (iv) below) the following properties. (i) A11 jint X belongs L cl .int X /.

302

5 Mixed problems in weighted edge spaces

(ii) The 2  2 submatrices .Aij /i;j D1;2 (.Aij /i;j D1;3 ) locally near int Y (int YC ) belong to Boutet de Monvel’s calculus of boundary value problems with the transmission property at int Y (int YC ) and with the respective Douglis–Nirenberg order conventions for the trace and potential parts. (iii) A is locally in coordinates .z; xn1 ; xn / 2 Rn2  R2 of the form Opz .a/ for an edge amplitude function as in (5.4.11). (iv) For convenience, in contrast to (5.1.5), we formulate the operators in spaces over Z for j D jC D 1; the general case is completely analogous. An operator C is called a regularising Green operator in the edge algebra of boundary value problems and of type 0, if C induces continuous operators W s; .X/ W 1;C" .X/ ˚ ˚ s 0 ;  1 1; 1  2 .Y / 2 C" .Y / W W ˚ ˚ CW ! s 00 ;C  1 1; 1 C 2 2 C" .YC / W .YC / W ˚ ˚ 000 H 1 .Z/ H s .Z/ for some " D ".C/ > 0, for all s; s 0 ; s 00 ; s 000 2 R, s >  12 , and if also the formal adjoint C has analogous mapping properties, now with respect to the modified weights  C ,  C   12 ,  C C  12 in the preimage and  C ",  C   12 C ",  C C  12 C " in the spaces in the image. The formal adjoint is defined via .Cu; v/ D .u; C v/ for all u; v 2 W 1;1 .X/˚W 1;1 .Y /˚W 1;1 .YC /˚H 1 .Z/, with the scalar 1 1 products of W 0;0 .X/ ˚ W 0; 2 .Y / ˚ W 0; 2 .YC / ˚ H 0 .Z/. An operator C is called smoothing and of type d 2 N if C has the form C D C0 C

d X

Cj diag.D j ; 0; 0; 0/

j D1

for arbitrary smoothing operators Cj of type 0 as before and a differential operator D on X of first order (with coefficients that are up to the boundary) which is equal to @xn in a collar neighbourhood of @X (with xn being a global coordinate in normal direction). Remark 5.4.3. Definition 5.4.2 is adapted to the case of mixed problems for second order elliptic operators A with conditions T˙ of arbitrary order; this is just the structure of the Zaremba problem for the Laplacian (where some of the entries are simply zero). There is also a more general version with block matrices Aij rather than ‘scalar’ entries and vectors of orders ˙ and ˙ , respectively, and we can also have schemes of

5.4 Edge calculus, specified to mixed problems

303

Douglis–Nirenberg orders for the operators on Z as in (5.2.9). As before some components may be zero, so there are also row and column matrix versions, cf. Remark 5.1.2. Another variant concerns operators between distributional sections of vector bundles on the various components X, Y˙ and Z of the configuration. Such generalisations are also subsumed under our notation ‘edge algebra’. The upper left corners of our edge algebra have a subtle structure. By Definition 5.4.2 (i) they induce elements of L cl .int X /, and by (ii) the operators have the transmission property at int Y˙ . They also contain Green terms near int Y˙ with the same behaviour as Green operators of some type d in Boutet de Monvel’s calculus. Near the interface Z the operators A11 have an edge-degenerate non-smoothing part, cf. the requirements (5.4.1) for the local symbols, and they also contain the other ‘smoothing’ ingredients such as Green and Mellin operators in X n Z from the edge algebra, cf. (5.4.11). In our application the original operator (the Laplace operator) has a smooth symbol across the interface Z; the symbol becomes edge-degenerate of the form (5.4.1) when we transform it to polar coordinates in the .xn1 ; xn / plane normal to Z. In other words, there is a smooth symbol behind the edge-degenerate structure, with the transmission property everywhere at Y D @X. Let us call an element A of the edge algebra regular at Y , if it has such a classical interior symbol which is smooth across Z. The requirement of regularity singles out a particularly convenient substructure. Those operators A have a standard homogeneous principal symbol of order   .A/.x; / WD  .A11 /.x; /; .x; / 2 T  X n 0. In general, near Z the operators A of Definition 5.4.2 have edgedegenerate homogeneous principal interior symbols, coming from the representation of local symbols in the upper left corners which have the form r  a.r; ; z; %; #; / Q 2 C 1 .R xC  where a.r; ; z; %; #; / D a.r; Q ; z; r%; #; r/ for an a.r; Q ; z; %; Q #; /  1Cq ; Scl .I  R#  R Q /tr /, cf. Section 3.1.5. In this case we set %; Q

 .A/.x; ; z; %; #; / D r  Q .A/.r; ; z; r%; #; r/ Q WD aQ ./ .r; ; z; %; Q (aQ ./ denotes the homogeneous prinfor Q .A/.r; ; z; %; Q #; / Q #; / Q cipal component of aQ of order  in .%; Q #; / 6D 0/. Let us also consider the operators B WD .Aij /i;j D1;2 and BC WD .Aij /i;j D1;3 of Definition 5.4.2. In differential mixed problems such as the Zaremba problem they can z in Boutet be regarded as restrictions to int Y  Œ0; 1/ of corresponding operators B de Monvel’s calculus in a collar neighbourhood Š Y  Œ0; 1/ of the boundary Y . In the pseudo-differential case we can ask a similar property, modulo the contributions of the smoothing Mellin and Green operators near Z and the smoothing operators from

304

5 Mixed problems in weighted edge spaces

Definition 5.4.2 (iv). Let us call an element A of the edge algebra . ; @/ regular, if it has this property (clearly any such operator is necessarily regular). In that case we have pairs of boundary symbols H s .RC / H s .RC / z /.y; / W ˚ ˚ ! @; .A/.y; / WD @ .B n C C m

(5.4.14)

for .y; / 2 T  .int Y / n 0, smooth up to Z. In the latter relations we assume on the operator A that the boundary value problems B contain n potential and m trace conditions with respect to int Y . In general, near Z the operators A of Definition 5.4.2 have edge-degenerate homogeneous principal boundary symbols which come from (5.4.1) together with the corresponding vectors of weights. For instance, for the upper left corner A11 we have @;C .A11 /.r; z; %; / D r  Q @;C .A11 /.r; z; r%; r/ for Q WD r C aQ ./ .0; z; %; Q W H s .RC / ! H s .RC /; Q @;C .A11 /.r; z; %; Q / Q D ; / and, similarly, for the minus sign, where aQ ./ is to be frozen at  D  and D replaced by D (because of the opposite orientation of the half-axis in this case). For the other components of the boundary symbols near Z (i.e., trace and potential symbols as well as standard pseudo-differential symbols on the boundary) we have similar expressions (with exponents in the weight factors depending on the orders of the boundary operators). Operators in the edge algebra of type d define continuous operators AW

L4

j D1

Ej !

L4 iD1

Ezi

(5.4.15)

for the spaces Ez1 WD W s; .X/;

E1 WD W s; .X/; 1

1

1 1 Ez2 WD W s  2 ;  2 .Y ; C m /;

1

1

1 1 Ez3 WD W sC  2 ;C  2 .YC ; C mC /; Ez4 WD H s .Z; C e /;

E2 WD W s  2 ;  2 .Y ; C n /; E3 WD W sC  2 ;C  2 .YC ; C nC /; E4 WD H s .Z; C d /;

s > d  12 . Here, for simplicity, the upper left corners are assumed to be scalar, while the other entries Aij of A are now block matrices, e.g., A22 is an m  n matrix, A44 an e  d matrix, etc. Clearly, as in Remark 5.1.2, the orders  and  may also assumed to be vectors, according to the components of C n and C m , respectively.

5.4 Edge calculus, specified to mixed problems

305

5.4.3 Ellipticity and parametrices Definition 5.4.4. The operator A is called elliptic, if the components of  .A/ D . .A/; @;˙ .A/; ^ .A// are bijective in the following sense: Q Q #; / (i)  .A/ WD  .A11 / does not vanish on T  Xreg n 0, and Q .A/.r; ; z; %; Q does not vanish for .%; Q #; / 6D 0 up to r D 0; (ii) The boundary symbols (5.4.14) are isomorphisms for all .y; / 2 T  .int Y˙ / n 0, and H s .RC / H s .RC / Q ˚ ˚ Q / W ! Q @;˙ .A/.r; z; %; C n˙ C m˙ Q 6D 0 up to r D 0; these conditions are required for are isomorphisms for .%; Q / 1 any s > d  2 . (iii) The edge symbol (5.4.12) is an isomorphism for every .z; / 2 T  Z n 0 and s > d  12 ; in the present notation the spaces Ej and Ezi are vector-valued and of different smoothness and weight, i.e., E1 D K s; .I ^ /, Ez1 D K s; .I ^ /, 1 1 1 1 E2 D K s  2 ;  2 .R ; C n /, Ez2 D K s  2 ;  2 .R ; C m /, 1 1 1 1 E3 D K sC  2 ;C  2 .RC ; C nC /, Ez3 D K sC  2 ;C  2 .RC ; C mC /, d z e E4 D C , E4 D C , and the homogeneities in  correspond to the scheme of Douglis–Nirenberg orders. Remark 5.4.5. Every in the edge algebra is determined by  .A/ modulo Loperator AL a compact operator j4D1 Ej ! 4iD1 Ezi , s > d  12 . This is a general property in edge algebras, cf. Chapter 7 below. Remark 5.4.6. The principal symbols  .A/ of operators A in the edge algebra form an algebra under componentwise composition. For the components belonging to boundary problems with the transmission property this is known by Chapter 3. The edge symbols are families of boundary value problems on the infinite cone I ^ with the boundary components R . Their composition behaviour is known in the edge algebra, cf. Chapter 7 below. Let A and P be operators in the edge algebra; then P is called a parametrix of A if I PA and I AP are regularising Green operators in the sense of Definition 5.4.2 (iv). Theorem 5.4.7. Every elliptic operator A has a parametrix P in the edge algebra, where .P/ D  1 .A/ with componentwise inversion. Proof. Details of the proof of the construction of parametrices in the edge calculus will be given in Chapter 7 below (see Theorem 7.2.36). Let us give here the main

306

5 Mixed problems in weighted edge spaces

idea. If A is elliptic, we can form  1 .A/ and find an element P0 in the edge calculus such that .P0 / D  1 .A/. We then obtain P0 A D I  C0 for an operator C0 in the edge calculus such that  .C0 / D 0. The asymptotic summation in the formal P Neumann series .I  C0 /1 D I  C1  j1D0 .1/j Cj0 can be carried out within the edge calculus; remainders are smoothing in the sense of Definition 5.4.2 (iv). Setting P WD .I  C1 /P0 we obtain an operator with the property that PA  I is smoothing, i.e., we get a left parametrix of A. In a similar manner we construct a right parametrix; thus P is a parametrix as desired. Corollary 5.4.8. The ellipticity of A entails the Fredholm property of (5.4.15) for every s > max.; d/  12 (where d is the type of A), and kernel and cokernel are independent of s. Furthermore, we have elliptic regularity in our weighted Sobolev spaces.

5.4.4 Asymptotics of solutions From elliptic boundary value problems in domains with conical singularities it (cf. Kondratyev [100]) that solutions have asymptotics of the form PisPknown mj c .x/r pj logk r for r ! 0 modulo flat remainders. Here r 2 RC is j k j kD0 the distance variable to the singularity. The coefficients cj k are smooth on the base of the local cone. The exponents pj 2 C and the number of logarithmic terms are determined by the non-bijectivity points of the conormal symbol of the given elliptic operator. (General information on asymptotics of that kind is given in Sections 10.1.2 and 10.4.5 below.) For the Zaremba problem these points are calculated in Section 5.1.3 and for other mixed problems in Section 5.3.4. In general we have to expect a dependence on the interface variable z (also the multiplicities may change under varying z). Such phenomena can be described in terms of continuous asymptotics (cf. [188] and the references there, and Section 10.4.5 below). Here we content ourselves with the case of constant discrete asymptotics. The structure is as follows. Let M be a compact C 1 manifold with boundary. Denote by P D f.pj ; mj /gj 2N the sequence of data which characterise asymptotics for r ! 0, with pj 2 C, Re pj ! 1 as j ! 1, mj 2 N. Then the space KPs; .M ^ / is defined to be the set of all u.r; x/ 2 K s; .M ^ / having such asymptotics with coefficients cj k 2 C 1 .M / for all 0  k  mj , j 2 N (for the given weight  this implies Re pj < mC1  ; m D dim M , for all j ). More precisely, an element 2 s; ^ u.r; x/ 2 K .M / belongs to KPs; .M ^ / if there are coefficients cj k 2 C 1 .M /, such that for every   0 there is an N D N./ such that u.r; x/ 

mj N X X

cj k .x/!.r/r pj logk r 2 K s;C .M ^ /

j D0 kD0

for a cut-off function !. Inserting the space E D KPs; .M ^ / in W s .Rq ; E/, based on the group action  u.r; x/ D  WPs; .M ^

mC1 2

u.r; x/, we obtain weighted spaces

 Rq / WD W s .Rq ; KPs; .M ^ //;

5.4 Edge calculus, specified to mixed problems

307

or spaces WPs; .W / globally on a (stretched, say compact) manifold W with edge. In this construction we assume that the transition diffeomorphisms between local wedges are independent of the axial variable r for 0  r  " for some " > 0. The dimension of M may be zero; so the same can be done for the half-space RC  Rq , or, globally, on a manifold with boundary, such as the manifolds Y˙ . Note that the singular functions of edge asymptotics in the space WPs; .RC Rq / are a generalisation of the ‘Taylor’ edge asymptotic terms of standard Sobolev distributions H s .RC  Rq /, cf. the formula (5.2.1). In the present case from the definition of W s .Rq ; E/ we see that the space WPs; .RC  Rq / is characterised as the set of all y s .Rq / such that for u.r; y/ 2 W s; .RC  Rq / such that there are elements cj k ./ 2 H every   0 there is an N D N./ such that N mj n o 1 XX 1 u.r; y/  F !z Œ 2 cj k ./!.rŒ /.rŒ /pj logk .rŒ / j D0 kD0

2W

s; C

(5.4.16)

.RC  R /; q

y s .Rq / WD fv./ O W v.z/ 2 H s .Rq /g. In a similar manner we can express the where H asymptotic terms of the edge asymptotics for a non-trivial (stretched) wedge M ^  Rq ; mC1 1 it suffices in (5.4.16) to formally replace RC by M ^ and Œ 2 by Œ 2 ; the coefficients y s .Rq //. cj k now belong to C 1 .M; H In the case of our mixed problems we have M D I D Œ0;  . The result on elliptic regularity from Corollary 5.4.8 then specifies to spaces with asymptotics as follows: Theorem 5.4.9. Set s 1 ; 1 2

2 WPs2;2 .XI Y ; YC / WD WPs2; .X/ ˚ WP 2 0 3 s 3 2 ; 2 PC

˚W

.Y /

.YC / ˚ H s2 .Z; C e./ /

(5.4.17)

for a triple P D .P0 ; P ; PC / of discrete asymptotic types (constant in z 2 Z). Then Au 2 (5.4.17) for  62 Z C 12 , s > 32 and u 2 W 1; .X/ ˚ H 1 .Z; C d./ / entails s; .X/ ˚ H s .Z; C d./ / for some resulting (constant in z 2 Z) asymptotic type u 2 WQ 0 Q0 . A similar result is true with respect to spaces without asymptotics. Proof. First we apply a parametrix P on both sides of Au D f . This yields P Au D Pf where C D I  PA is a smoothing operator. We then obtain asymptotics of solutions if P can be chosen in such a way that Pf 2 WRs;0 .X/ and C.W 1; .X//  .X/ for certain asymptotic types R0 and S0 , respectively. Parametrices of that WS1; 0 kind can be constructed in a refined version of the edge algebra, namely, with Mellin and Green symbols with asymptotics. Such a calculus in the framework of the so-called continuous asymptotics is developed in [90, Chapter 4], see also Section 10.4.5 below. This concept contains asymptotic types of the present constant discrete type as a special case. If we know that the non-bijectivity points of the conormal symbol c ^ .A/ of the given operator A are independent of z 2 C, the poles of the conormal symbol c ^ .P/

308

5 Mixed problems in weighted edge spaces

of the parametrix P are also independent of z 2 Z because c ^ .P/ is simply the inverse of c ^ .A/ (up to a translation in the complex w-plane). An inspection of the arguments of [90] shows that then the operators P and C have the desired mapping properties. In the case of the Zaremba problem we proved in Section 5.1.3 that the non-bijectivity points of the conormal symbol are just as we want. Remark 5.4.10. The mechanism to compute Q0 in terms of the given .P0 ; P ; PC / and of the poles of c ^ .P/ is similar to that in the theory of operators on manifolds with conical singularities, cf. [182] or [196]. The result is that the solutions have the asymptotic types from the data (together with possible translations to the left in the complex plane by integers) and in addition the poles (plus multiplicities) from c ^ .P/, in this case j C 12 , j 2 Z. Thus the specific extra singular functions for the Zaremba problem in the space W s; .X/ are locally near Z of the form 1 F !z

N X

1

fΠcj k .; /!.rΠ/.rΠ/j C 2 g;

j 

1 2

< 1  ;

j D0

y s .Rq // modulo remainders of flatness  C ; N D N./ (cf. the cj k .; / 2 C 1 .I; H formula (5.4.16)). Analogous relations hold for the mixed problems of Section 5.3.4.

Chapter 6

Operators on manifolds with conical singularities and boundary

In this chapter we study operators on manifolds with conical singularities in terms of the socalled cone algebra. Starting from the typical differential operators we characterise a pseudodifferential algebra that contains these operators together with the parametrices of elliptic elements. The ellipticity of an operator A is determined by a principal symbolic hierarchy  .A/ D . .A/; @ .A/; c .A// consisting of interior, boundary, and conormal components. We also prepare material on parameter-dependent cone operators and kernel cut-off results that will be necessary later on in the edge calculus.

6.1 Fuchs type operators and Mellin quantisation The typical (pseudo-)differential operators on a (stretched) manifold with conical singularities are of Fuchs type. We investigate here the basic properties, in particular, in connection with the Mellin quantisation. Operators on a manifold with conical singularities without boundary are a model situation for the case of boundary value problems which will be studied later on. In addition we consider cone operators with extra parameters; this prepares the material for operators on manifolds with edges, cf. Chapter 7 below.

6.1.1 Mellin quantisation In the Sections 2.4.1, 2.4.2 we introduced basics on the Mellin transform, manifolds with conical singularities, and weighted Sobolev spaces. In the Sections 2.4.3 and 2.4.4 we studied operators of Fuchs type and aspects of the Mellin quantisation. We now deepen this material and prepare further necessary tools both for the cone algebra as well as for the edge algebra in Chapter 7 below. Let X be a C 1 manifold and consider the open stretched cone X ^ D RC  X 3 .r; x/. Recall that differential operators of Fuchs type on X ^ have the form ADr



 X j D0

 @ j aj .r/ r @r

(6.1.1)

x C ; Diff j .X //, j D 0; : : : ; , with Diff  .X / being with coefficients aj .r/ 2 C 1 .R the space of differential operators on X of order  with smooth coefficients.

310

6 Operators on manifolds with conical singularities and boundary

P Remark 6.1.1. Let Az D j˛j aQ ˛Q .x/D Q x˛QQ be a differential operator in RnC1 3 xQ with Q coefficients aQ ˛Q .x/ Q 2 C 1 .RnC1 /. Then, by introducing polar coordinates xQ ! .r; x/, ˇ nC1 n ˇ z R n f0g ! RC  S , the operator A WD A RnC1 nf0g takes the form (6.1.1) with x C ; Diff j .S n //, j D 0; : : : ; . coefficients aj .r/ 2 C 1 .R In fact, it suffices to consider the transformation xQ ! .r; x/ in a conical set  WD fxQ 2 RnC1 W x=j Q xj Q 2 Gg for an open subset G  S n . Up to a rotation in nC1 we may assume that  is the union of all half-lines which connect the origin R of RnC1 with any point of the set fxQ 2 RnC1 W j.xQ 2 ; : : : ; xQ nC1 /j < 1; xQ 1 D 1g. Those half-lines define a diffeomorphism between G D  \ S n and the open unit ball B WD f.xQ 2 ; : : : ; xQ nC1 / 2 Rn W j.xQ 2 ; : : : ; xQ nC1 /j < 1g. Let us write .x1 ; : : : ; xn / WD .xQ 2 ; : : : ; xQ nC1 /. By ˇ.r; x/ WD .r; rx/ we obtain a diffeomorphism ˇ W RC  B !  n f0g: To calculate ˇA D

X

.ˇ  aQ ˛Q /.ˇ  Dx˛QQ /

j˛j

x C  B/. Moreover, ˇ  .@xQ / D @r we first observe that ˇ  C 1 .RnC1 /jnf0g C 1 .R 1  1 and ˇ @xQj C1 D r @xj , j D 1; : : : ; n. This gives us   ˛Q ˛Q ˛Q  ˛Q nC1  ˛Q Q ˇ  @x˛QQ11 : : : @xQnC1 D r j˛j r 1 @r 1 @x12 : : : @xnnC1 ; P and we obtain ˇ  A D r  j Cj˛j aj˛ .r; x/Dx˛ .r@r /j with coefficients aj˛ 2 x C  B/. Substituting the diffeomorphism G ! B we obtain an analogous form C 1 .R with x being interpreted as local angular variables on G. The global form (6.1.1) is then an immediate consequence when we glue together the local operators by a partition of unity on S n . P Example 6.1.2. The operator Az WD jnC1 Qj @xQj in polar coordinates takes the form r@r . D1 x In the edge calculus we are interested in edge-degenerate operators on an open stretched wedge X ^   3 .r; x; y/,  Rq open, which are of the form A D r 

X j Cj˛j

 @ j aj˛ .r; y/ r .rDy /˛ @r

x C  ; Diff j .X //. Then with coefficients aj˛ .r; y/ 2 C 1 .R a.y; / D r 

X j Cj˛j

 @ j aj˛ .r; y/ r .r/˛ @r

is a family of Fuchs type differential operators, parametrised by .y; / 2   Rq .

6.1 Fuchs type operators and Mellin quantisation

311

In local coordinates x 2 † on X , † Rn open, n D dim X , the symbols in the variables .r; x; y/ 2 RC  †   and covariables .%; ; / 2 R1CnCq have the form r  p.r; x; y; %; ; /;

(6.1.2)

where

p.r; Q x; y; %; ; / WD p.r; x; y; r 1 %; ; r 1 / (6.1.3) 1 x is a polynomial in .%; ; / of order  with coefficients in C .RC  †  /. In the following we ignore the weight factor r  for a while and consider pseudo-differential symbols of an analogous structure. We are interested in a Mellin quantisation which reformulates opr .  / as a Mellin  .  / for any weight  2 R. If p is a polynomial as in (6.1.2) we have operator opM  opr opx .p/.y; / D opM opx .h/.y; /

with h.r; x; y; w; ; / WD p.r; Q x; y; iw; ; r/, for any  2 R. In the pseudo-differential case we introduce the following spaces of symbols.  x Definition 6.1.3. (i) Let Sz.cl/ .RC  †    R1CnCq /,  2 R, denote space of all functions p.r; x; y; %; ; / WD p.r; Q x; y; r%; ; r/ (6.1.4)  x 1CnCq /. with p.r; Q x; y; %; Q ; / Q 2 S.cl/ .RC  †    R%; ; Q Q  x nCq (ii) By S.cl/ .RC  †    C  R / we denote the space of all h.r; x; y; w; ; / x C  †    RnCq / such that which are holomorphic in w 2 C with values in C 1 .R  x h.r; x; y; ˇ C i%; ; / 2 S.cl/ .RC  †    R1CnCq / %; ;

for every ˇ 2 R, uniformly in compact ˇ-intervals.  x .RC  †    C  RnCq / denote the space of all Moreover, let Sz.cl/ Q x; y; w; ; r/ h.r; x; y; w; ; / WD h.r;

(6.1.5)

 x Q x; y; w; ; / for arbitrary h.r; Q 2 S.cl/ .RC  †    C  RnCq /.

;Q

Remark 6.1.4. The use of C within Cartesian products of covariables might suggest that the complex variable w is treated as a parameter also in direction of Re w. The warning is that this is not the case; only in imaginary direction we have the interpretation of w as a parameter in the sense of a covariable. Later on, from time to time we interchange the role of real and imaginary part and then employ C instead of C. The reason for notation with such complex parameters is that in the calculus there are too many classes of objects with such parameter-dependent modifications (however, it happens that we also break this rule, cf. the formulas (6.1.51) or (6.1.80)). In the following theorem we establish a map  x  x m W Sz.cl/ .RC  †    R1CnCq / ! Sz.cl/ .RC  †    C  RnCq /;

(6.1.6)

312

6 Operators on manifolds with conical singularities and boundary

which has the meaning of a Mellin quantisation. The considerations are valid for arbitrary dimensions q; in particular, we admit q D 0. In the latter case the results concern symbols of Fuchs type p.r; x; %; / D p.r; Q x; r%; /;  x Q x; w; / 2 p.r; Q x; %; Q / 2 S.cl/ .RC †R1Cn / with Mellin symbols h.r; x; w; / D h.r;  x S.cl/ .RC  †  C  Rn /.  x Theorem 6.1.5. For every p.r; x; y; %; ; / 2 Sz.cl/ .RC  †    R1CnCq / there  x C  †    C  RnCq / such that exists an h.r; x; y; w; ; / 2 Sz .R .cl/

 opr;x .p/.y; / D opM .opx .h//.y; /

(6.1.7)

Q x; y; w; ; / Q (cf. mod C 1 .; L1 .RC  †I Rq //, for every  2 R. The symbol h.r; the formula (6.1.5)) admits the asymptotic expansion Q x; y; i%; ; / h.r; Q  p.r; Q x; y; %; ; / Q C

1 1 X X

ckj .@kCj p/.r; Q x; y; %; ; /% Q j %

kD1 j D0

(6.1.8) with certain ckj 2 R that are independent of p. Q The asymptotic summation can be  x carried out in S.cl/ .RC  †    R1CnCq /. In the classical case for the homogeneous components we have Q D hQ .l/ .r; x; y; i%; ; /

k l X X

ckj .@kCj p/ Q .lj / .r; x; y; %; ; /% Q j : (6.1.9) %

kD0 j D0

Proof. The relation (6.1.7) is to be verified for functions u 2 C01 .RC ; C 1 .†//. We have opr;x .p/.y; /u.r/ Z 1Z 1 0 D e i.rr /% opx .p/.r; y; %; /u.r 0 /dr 0 μ % 0 Z1 1 Z 1  i% 0 r 0 0 0 0 dr M.r; r /r op .p/.r; y; M.r; r /%; /u.r / μ% D x r0 r0 1 0 with the substitution % ! M.r; r 0 /% where M.r; r 0 / WD We have M.r; r 0 / 2 C 1 .RC  RC /; M > 0. For

log rlog r 0 rr 0

for r; r 0 2 RC .

g.r; r 0 ; x; y; i%; ; / WD M.r; r 0 /r 0 p.r; x; y; M.r; r 0 /%; ; / we have g 2 S  .RC  RC  †    0  RnCq / and 1

2 .opx .g//.y; / opr;x .p/.y; / D opM

313

6.1 Fuchs type operators and Mellin quantisation

for every .y; / 2   Rq . We now choose a function ' 2 C01 .RC / with '.r/ 1 near r D 1 and set r0  1 2 h.r; x; y; w; ; / WD r w opM .' g/.x; y; ; /.r 0 /w : r Then, applying the change of variables r 0 ! rt it follows that Z 1 Z 1  i%Cw  0  r r dr 0 h.r; x; y; w; ; / D g.r; r 0 ; x; y; i%; ; / 0 μ % ' 0 r r r 0 Z1 1 Z 1 dt D t i%w '.t /g.r; rt; x; y; i%; ; / μ % t 1 0 (the integrals are interpreted as oscillatory integrals). By virtue of g.r; rt; x; y; i%; ; / D tM.t; 1/p.r; Q x; y; M.t; 1/%; ; r/ it follows that

Z

1

Z

1

h.r; x; y; w; ; / D

t i%w '.t /M.t; 1/p.r; Q x; y; M.t; 1/%; ; r/dt μ %: 1

0

(6.1.10)

We now verify that Q x; y; ı C i%; ; / 2 S  .R x C  †    R1CnCq / h.r; x; y; ı C i%; ; r 1 / D h.r; %; ; uniformly in ı 2 Œc; c 0 for arbitrary c < c 0 . The symbolic estimates for hQ will be recovered from those of p; Q this will show that Q h depends continuously on p. Q The change of variables % ! % C gives us Q x; y; ı C i%; ; / h.r; Z 1Z 1 dt D t 1Ci% ı .t /p.r; Q x; y; M.t; 1/.% C /; ; / μ % t 0 Z1 1 Z 1 D .1 C i%/N t 1Ci% .t @ t /N 1

0

f

Q x; y; M.t; 1/.% ı .t /p.r;

C /; ; /g

dt μ% t

with ı .t/ WD t ı '.t /M.t; 1/. The integral on the right-hand side converges for N large enough. We have 

t

@ N f ı .t /p.r; Q x; y; M.t; 1/.% C /; ; /g @t N X   D ck .t / @k% pQ .r; x; y; M.t; 1/.% C /; ; /.% C /k kD0

314

6 Operators on manifolds with conical singularities and boundary

with certain ck 2 C01 .RC /. For the symbolic estimates of hQ we only have to investigate the derivatives of p. Q For every l 2 N; ˛ 2 N nCq ; ˇ 2 N 1CnCq , we have Il;˛;ˇ

ˇ ˚ k

ˇ ˛ ˇ WD ˇ@l D ; Dr;x;y Q x; y; M.t; 1/.% C /; ; /.% C /k ˇ .@% p/.r; 

l X ˇ j˚ k ˛ ˇ ˇ

kˇ ˇcj @ .@ D D Q x; y; M.t; 1/.% C /; ; / @lj % ; r;x;y p/.r; .% C / : j D0

Since ˇ j˚ k ˛ ˇ

ˇ ˇ@ .@ D D Q x; y; M.t; 1/.% C /; ; / ˇ % ; r;x;y p/.r; ˇ ˇ  ˛ ˇ D ˇ @j% Ck D ; Dr;x;y pQ .r; x; y; M.t; 1/.% C /; ; /.M.t; 1//j ˇ  c.t/h% C ; ; ij kj˛j k j Ckl for some c.t/ 2 C 1 .RC / and because of j@lj we .% C / j ˇ ch% C ˇ ; ; i ˇ ˇ lj˛j lj˛j lj˛j h ; ; i obtain Il;˛;ˇ  cQl;˛;ˇ .t /h% C ; ; i  cl;˛;ˇ .t /h%i by applying Peetre’s inequality. Moreover, since @kt ı .t / depends continuously on ı for every k 2 N, the symbolic estimates for hQ are uniform in c  ı  c 0 for every c  c 0 . x C †RnCq / If we calculate in an analogous manner the semi-norms of hQ in S  .R Q we see that h depends holomorphically on w. Applying standard manipulations of the pseudo-differential calculus for every N 2 N, N  1, we obtain the expansion

Q x; y; i%; ; / h.r; D

oˇ 1  0 @ k k n  r 0  r 0 @% ' g.r; r 0 ; x; y; i%; ; r 1 / ˇr 0 Dr kŠ @r r kD0 Z 1 .1  /N 1 Q C hN .r; x; y; %; ; ;  /d .N  1/Š 0

N 1 X

(6.1.11)

with hQ N .r; x; y; %; ; ;  / Z 1 Z 1  i  r 0   r dr 0 N N 0 1 .i

/ ' g .r; r ; x; y; i.% C 

/; ; r / μ : @ WD % r0 r r0 1 0 Moreover, we have @k% g.r; r 0 ; x; y; i%; ; r 1 /   D M.r; r 0 /r 0 @k% pQ .r; x; y; M.r; r 0 /r%; ; /.M.r; r 0 /r/k :

6.1 Fuchs type operators and Mellin quantisation

315

Thus the sum on the right-hand side of (6.1.11) is equal to N 1 X kD0

oˇ   1  0 @ k n k  r 0  r ' r 0 M.r; r 0 /kC1 r 0 @k% pQ .r; x; y; M.r; r 0 /r%; ; / ˇr 0 Dr : kŠ @r r

The function hQ N .r; x; y; %; ; ;  / is equal to Z 1Z 1   t  .i /N '.t /M.t; 1/N C1 @N % pQ .r; x; y; M.t; 1/.% C i /; ; /dt μ : 1

0

The latter identity shows, similarly as before, that the remainder in the expansion x C  †    R1CnCq /. The asymptotic summation for hQ (6.1.11) belongs to S N .R %; ; can be carried out in the space of symbols that are smooth up to r D 0. Finally, the relation (6.1.8) is a consequence of the expansion (6.1.11) when we PL P j j 1 0 apply the formula @L 1 CCj DL cj .@r 0 u/.v.r //@r 0 v : : : @r 0 v for j D1 r 0 .u ı v/ D ˇ 1 L; j ; cj 2 N, together with the identity @k0 M.r; r 0 /ˇ 0 D .1/k kŠ kC1 . r

r Dr

kC1 r

The relation (6.1.7) is now proved for  D 12 . For arbitrary  2 R it suffices to apply Cauchy’s integral formula, using the holomorphy of h with respect to w. From (6.1.8) we see that hQ is classical as soon as pQ is a classical symbol, and we immediately obtain the formula (6.1.9).  x Remark 6.1.6. Let the symbols p.r; Q x; y; %; ; / 2 Sz.cl/ .RC  †    R1CnCq / and  Q x; y; w; ; / 2 Sz .R x C  †    C  RnCq / be as in Theorem 6.1.5. Then by h.r; .cl/ Taylor expansion we obtain

Q x; y; ı C i%; ; / D h.r;

N 1 X kD0

ık  k Q @ h .r; x; y; i%; ; / C rN;ı .r; x; y; %; ; / kŠ w

  R1 N Q with rN;ı .r; x; y; %; ; / D .Nı1/Š 0 .1  //N 1 @N w h .r; x; y; ı C i%; ; /d.   N x 1CnCq Q Since @N w h .r; x; y; ı C i%; ; / 2 S.cl/ .RC  †    R%; ; / uniformly xC  †    in 0    1, c  ı  c 0 for arbitrary c  c 0 , we have rN;ı 2 S N .R 1CnCq R / uniformly for ı in compact intervals. This yields the following properties:  x C  †    0  RnCq / for some  <  (   2 N in the (i) If hQ 2 S.cl/ .R x C  †    C  RnCq /; classical case) then hQ 2 S  .R .cl/

1 x Q x; y; ı C i%; ; /  p.r; (ii) we have h.r; Q x; y; %; ; / 2 S.cl/ .RC  †   

R1CnCq / for all ı 2 R. %; ; Q is classical, the homogeneous principal symbol (with respect (iii) If pQ (and hence h) Q to .%; ; /) of h.r; x; y; ı C i%; ; / is independent of ı 2 R.

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6 Operators on manifolds with conical singularities and boundary

Q x; y; w; ; / Corollary 6.1.7. The correspondence p.r; Q x; y; %; Q ; / Q ! h.r; Q of the proof of Theorem 6.1.5, given by the formula o 1 n r0  2 Q x; y; w; ; / h.r; Q D r w opM ' Q x; y; M.r; r 0 /%; Q ; / Q .r 0 /w M.r; r 0 /r 0 p.r; r (6.1.12) for any fixed choice of the function ', defines a continuous map  x  x m W Sz.cl/ .RC  †    R1CnCq / ! Sz.cl/ .RC  †    C  RnCq /:

(6.1.13)

The proof gives us first the continuity in the non-classical case; then the closed graph theorem yields the continuity also in the classical case. Q induces an isomorphism Theorem 6.1.8. The map (6.1.13), pQ ! h,  x x C  †    R1CnCq / Sz.cl/ .RC  †    R1CnCq /=Sz1 .R  x x C  †    C  RnCq / ! Sz.cl/ .RC  †    C  RnCq /=Sz1 .R (6.1.14)

for every  2 R.  x Proof. Let m.p/ WD h 2 Sz.cl/ .RC  †    C  RnCq / be associated with p 2 x C  †    R1CnCq / according to Theorem 6.1.5, written as (6.1.12) with a Sz.cl/ .R fixed '. By virtue of the relation (6.1.8) the map (6.1.14) is obviously injective. To  x prove the surjectivity we start from an element h 2 Sz.cl/ .RC  †    C  RnCq /. Choose a function Q 2 C01 .R/ with the property Q 1 near 0, and set .r; r 0 / WD Q .log r 0  log r/, .t / WD Q .log t / for r; r 0 ; t 2 RC . Let

q.r; r 0 ; x; y; %; ; / WD M.r; r 0 /1 .r 0 /1 h.r; x; y; iM.r; r 0 /1 %; ; /; p.r; x; y; %; ; / WD e ir% op. .r; r 0 /q.r; r 0 ; x; y; %; ; //e ir% : Then, similarly as in the proof of Theorem 6.1.5 it follows that “ 0 p.r; x; y; %; ; / D e i.rr /. %/ .r; r 0 /q.r; r 0 ; x; y; ; ; /dr 0 μ “ dt D t i e i.1t/r% .t /h.r; x; y; i ; ; / μ ; t applying the substitutions r 0 ! rt, % ! r 1 M.t; 1/ . In this way we obtain “ Q x; y; i ; ; / dt μ (6.1.15) p.r; Q x; y; %; ; / D t i e i.1t/% .t /h.r; t  x .RC  †    R1CnCq /. According to (6.1.10) the Mellin which belongs to S.cl/

6.1 Fuchs type operators and Mellin quantisation

symbol m.p/ has the form

317

“

m.p/.r; x; y; i ; ; / D

s i.% / '.s/M.s; 1/p.r; Q x; y; M.s; 1/%; ; r/dsμ “

D

s i e i.s1/% '.s/p.r; Q x; y; %; ; r/dsμ ;

where we applied % ! M.s; 1/1 %. Inserting (6.1.15) we obtain m.p/.r; x; y; i ; ; / “ “ dt i i.s1/% i i.1t/% '.s/ t e .t /h.r; x; y; i ; ; / μ dsμ % D s e t “ D lim s i e is% '.s/ "!0 “ dt i it% Q t e .t / ." /h.r; x; y; i ; ; / μ dsμ % t Z Z Z D lim s i '.s/ e it% .t / e is% "!0 Z dt i Q t ." /h.r; x; y; i ; ; /μ μ % ds t Z Z ds D lim s i .s/ s i Q ." /h.r; x; y; i ; ; /μ "!0 s “ ds D s i.  / .s/h.r; x; y; i ; ; / μ s for .s/ WD '.s/ .s/ which has the properties .s/ 2 C01 .RC / and  1 near 1. We have h.r; x; y; i ; ; /  m.p/.r; x; y; i ; ; / “ ds D h.r; x; y; i ; ; /  s i. / .s/h.r; x; y; i ; ; / μ s Z D h.r; x; y; i ; ; /  h.r; x; y; i. C  /; ; /u.i /μ ; where u.i / is the Mellin transform of . The difference h  m.p/ belongs to x C  †    0  RnCq /. By Remark 6.1.6 (i) we even obtain Sz1 .R x C  †    C  RnCq /: .h  m.p//.r; x; y; w; ; / 2 Sz1 .R In order to verify that .h  m.p//.r; x; y; i ; ; r 1 / is of order 1 we consider the expansion Q x; y; i. C  /; ; / D h.r;

N 1 X kD0

.i /k  k Q  @w h .r; x; y; i ; ; / C rN; .r; x; y; i ; ; / kŠ

318

6 Operators on manifolds with conical singularities and boundary

  R1 .i /N Q with the remainder rN; D .N .1  /N 1 @N w h .r; x; y; i. C  /; ; /d. 1/Š 0R   The property of the Mellin transform .i /k u.i /μ D .t @ t /k  .1/ implies N 1 Z X



 Q x; y; i ; ; /: @kw hQ .r; x; y; i ; ; /.i /k u.i /μ D h.r;

kD0

Finally, as in the proof of Theorem 6.1.5 it follows that the symbol Z .h  m.p//.r; x; y; i ; ; r 1 / D rN; .r; x; y; i ; ; /u.i /μ x C  †    0  RnCq / which completes the proof. belongs to S N .R Remark 6.1.9. The formula (6.1.15) gives rise to an inverse Mellin quantisation  x  x .RC  †    C  RnCq / ! Sz.cl/ .RC  †    R1CnCq /; (6.1.16) m1 W Sz.cl/

Q pQ by (6.1.5) and (6.1.4), respectively. m W h ! p, where h, p are associated with h, 1 The interpretation of (6.1.16) is that mm and m1 m are the respective identities modulo symbols of order 1. Remark 6.1.10. The new point in Theorem 6.1.8 is the surjectivity of the map (6.1.14). This can also be obtained directly from Remark 6.1.6 (ii).  x .RC  †    C  RnCq / we form the In fact, starting from a function h 2 Sz.cl/ corresponding hQ via (6.1.5) and set

Q x; y; i%; ; / pQ0 .r; x; y; %; ; / WD h.r;  x .RC  †    R1CnCq /. For p0 associated with pQ0 via (6.1.4) which belongs to S.cl/ we obtain

m.p0 / D h C h1

1 x for an h1 2 Sz.cl/ .RC  †    C  RnCq /:

We now form hQ 1 via (6.1.5) and pass to pQ1 .r; x; y; %; ; / WD hQ 1 .r; x; y; i%; ; / which gives us m.p1 / D h1 C h2 , i.e., m.p0 C p1 / D h C h2 ;

2 x h2 2 Sz.cl/ .RC  †    C  RnCq /:

j x By iterating this process we can produce a sequence pj 2 Sz.cl/ .RC †R1CnCq / PN  N x such that m j D0 pj D h C hN with hN 2 S.cl/ .RC  †    C  RnCq / for P1 every N 2 N. Then, setting p  j D0 pj (where the asymptotic sum is carried out in  x x C †CRnCq /. S.cl/ .RC †R1CnCq /) it follows that m.p/h 2 Sz1 .R

319

6.1 Fuchs type operators and Mellin quantisation

Proposition 6.1.11. For every  2 R we have x C †    R1CnCq / \ S 1 .RC  †    R1CnCq / p 2 Szcl .R x C  †    R1CnCq /: H) p 2 Sz1 .R x C  †    R1CnCq / be associated with p via (6.1.4). Then Proof. Let pQ 2 Scl .R there is a sequence pQ.j / of homogeneous components of order   j in .%; ; / 6D 0 P which are C 1 up to r D 0 such that pQ  j1D0 pQ.j / for an arbitrary excision x C †R1CnCq /. function  in .%; ; /. The asymptotic sum is carried out in Scl .R By virtue of the assumption on p we have pQ.j / .r; x; y; %; ; / D 0 for r > 0, and because of the continuity also including r D 0, for every j 2 N. This gives us x C  †    R1CnCq /. p 2 Sz1 .R x C  †    R1CnCq / is not classical and restricts to Remark 6.1.12. If p 2 Sz .R 1 an element in S .RC  †    R1CnCq / for r > 0 it may happen that p does not x C  †    R1CnCq /. An example is belong to Sz1 .R p.r; %; ; / WD !.r%; ; r/ with a function !.%; ; / 2 C01 .R1CnCq / such that ! 1 near 0. We then have x C  †    R1CnCq / is not pjr>0 2 S 1 .RC  †    R1CnCq /, but p 2 S 0 .R of order 1 up to r D 0. z for z be a diffeomorphism and write 1   W RC  † ! RC  † Let  W † ! † the diffeomorphism .r; x/ ! .r; x/, Q xQ WD .x/. With 1   we can associate the .y; /-wise operator push forward of opr;x .p/.y; /. Proposition 6.1.13. There is a map x C  †    R1CnCq / ! Sz .R xC  † z    R1CnCq / (6.1.17) .1  / W Szcl .R cl (the so-called symbol push forward associated with 1  ) such that .1  / opr;x .p/.y; / D opr;xQ ..1  / p/.y; /

(6.1.18)

mod C 1 .; L1 .RC  †I Rq //. The map (6.1.17) induces an isomorphism x C †    R1CnCq /=Sz1 .R x C  †    R1CnCq / Szcl .R (6.1.19) xC  † xC  † z    R1CnCq /=Sz1 .R z    R1CnCq / ! Szcl .R for every  2 R. z † z Proof. As is well known there is an open neighbourhood Uz of the diagonal of † 1 z 1 1 0 0 Q 0 Q and a ‰ 2 C .U ; GL.n; R// such that . .x/ Q   .xQ //‰.x; Q xQ / D .xQ  xQ /

320

6 Operators on manifolds with conical singularities and boundary

for all .x; Q xQ 0 / 2 Uz and Q 2 Rn . Let 'Q 2 C 1 .Uz / be properly supported, and define 1 ' 2 C .†  †/ by '.x; x 0 / D '..x/; Q .x 0 //. Define “ Q .1  / p.r; x; Q y; %; ; / WD e iz '.x; Q xQ C z/j detŒ.d1 .xQ C z// (6.1.20) 1 Q ‰.x; Q xQ C z/ jp.r;  .x/; Q y; ; ‰.x; Q xQ C z/. C /; /dzμ : Then the .y; /-wise operator push forward can be expressed as .1  / opr;x .p/.y; / D opr;xQ ..1  / p/.y; / C .1  / opr;x ..1  '/p/.y; /: The second summand on the right-hand side belongs to C 1 .; L1 .RC  †I Rq //. Q / defines an element of S  .R xC † z    R1CnCq / we Since .1  / p.r; x; Q y; %; ; cl have a map (6.1.17) such that (6.1.18) holds. Since this construction also works in the converse direction we obtain the isomorphism (6.1.19). Remark 6.1.14. In an analogous manner as (6.1.20) we can define a push forward of holomorphic Mellin symbols x C  †    C  RnCq / ! Sz .R xC  † z    C  RnCq / .1  / W Szcl .R cl such that

  opx .h/.y; / D opM opx ..1  / h/.y; / .1  / opM

holds mod C 1 .; L1 .RC  †I Rq //, and there is a corresponding isomorphism x C  †    C  RnCq /=S 1 .R x C  †    C  RnCq / Szcl .R xC  † xC  † z    C  RnCq /=S 1 .R z    C  RnCq /: ! Szcl .R Moreover, the map (6.1.6) satisfies the relation m ı .1  / D .1  / ı m. Definition 6.1.15. Let X be a closed compact C 1 manifold, and j W Uj ! †j , j D 1; : : : ; N , an atlas on X , †j Rn open. A tuple of symbols x C  †j    R1CnCq /; pj 2 Szcl .R

j D 1; : : : ; N;

(6.1.21)

is called a (parameter-dependent) complete symbol on RC  X if pj D .1  .j ı i1 // pi on j .Ui \ Uj /; i; j D 1; : : : ; N , modulo elements of order 1. Analogously, a tuple of x C  †j    C  RnCq /; hj 2 Szcl .R

j D 1; : : : ; N;

is called a (parameter-dependent) complete Mellin symbol on RC  X if hj D .1  .j ı i // hi on i .Ui \ Uj /; i; j D 1; : : : ; N , modulo elements of order 1.

(6.1.22)

321

6.1 Fuchs type operators and Mellin quantisation

Let us now fix functions f'1 ; : : : ; 'N g and f 1 ; : : : ; N g in C01 .Uj / such that f'1 ; : : : ; 'N g is a partition of unity subordinate to fU1 ; : : : ; UN g and j 1 on supp 'j for all j D 1; : : : ; N . Given complete symbols (6.1.21), (6.1.22) we form operator families p.r; y; %; / and h.r; y; w; / by p.r; y; %; / WD

N X

'j .1  j1 / opx .pj /.r; y; %; /

j;

(6.1.23)

'j .1  j1 / opx .hj /.r; y; w; /

j:

(6.1.24)

j D1

h.r; y; z; / WD

N X j D1

Let

1Cq x x C  ; L .X I R1Cq //; / WD C 1 .R L cl .X I RC    R cl

and set 1Cq x z L / cl .XI RC    R ˚ 1Cq

x WD p.r; y; %; / D p.r; Q y; r%; r/ W p.r; Q y; %; Q / Q 2 L / : cl .X I RC    R%; Q Q  q q Let further L cl .X I C  R / denote the subspace of all f .w; / 2 A.C; Lcl .X I R //  1Cq such that f .ˇ C i%; / 2 Lcl .X I R%; / for every ˇ 2 R, uniformly in compact q ˇ-intervals. Using the Fréchet topology in the space L cl .X I C  R / we then define  q 1 x q x L cl .X I RC    C  R / WD C .RC  ; Lcl .X I C  R //;

and q x z L cl .XI RC    C  R / Q y; w; r/ W h.r; Q y; w; / x C    C  Rq /g: WD fh.r; y; w; / D h.r; Q 2 L .X I R cl

Q

Remark 6.1.16. There are operator families 1Cq q Q y; w; / x x p.r; Q y; %; Q / Q 2 L /; h.r; Q 2 L cl .X I RC    R%; cl .X I RC    C  RQ /; Q Q

cf. Definition 2.4.44, such that p.r; y; %; / D p.r; Q y; r%; r/;

Q y; w; r/: h.r; y; w; / D h.r;

If hj is related to pj by hj D m.pj / for all j D 1; : : : ; N , we write h D m.p/ and 1Cq q x x z z /!L obtain in this way a map m W L cl .X I RC    R cl .X I RC    C  R / such that  .h/.y; / opr .p/.y; / D opM mod C 1 .; L1 .RC  X I Rq //, for all  2 R. This is a consequence of Theorem 6.1.5.

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6 Operators on manifolds with conical singularities and boundary

We now formulate analogous results for a compact C 1 manifold X with boundary @X . In this case we start from the space B ;d .X I vI Rl / of parameter-dependent boundary value problems of order  2 Z and type d 2 N, H s .int X; F / H s .int X; E/ ˚ ˚ ! p./ W 1 1 H s 2 .@X; G / H s 2 .@X; GC /

(6.1.25)

with vector bundles E; F 2 Vect.X /, G˙ 2 Vect.@X /, v WD .E; F I G ; GC /. For notational convenience we consider the case of trivial bundles of fibre dimension 1, i.e., E D F D X  C, G˙ D .@X /  C; thus we omit v. The general case is completely analogous and left to the reader. We admit again extra variables .r; y/ 2 RC  ,  Rq open, and set  WD .%; Q / Q 2 R1Cq . Let x C    R1Cq / WD C 1 .R x C  ; B ;d .X I R1Cq //; B ;d .X I R and set z ;d .XI R x C    R1Cq / B (6.1.26) ˚

1Cq ;d xC    R WD p.r; y; %; / D p.r; Q y; r%; r/ W p.r; Q y; %; Q / Q 2 B .X I R / : %; Q Q Let further B ;d .X I C  Rq / be the subspace of all f .w; / 2 A.C; B ;d .X I Rq // such that f .ˇ C i%; / 2 B ;d .X I R1Cq %; / for every ˇ 2 R, uniformly in compact ˇ-intervals. Using the Fréchet topology in the space B ;d .X I C  Rq / we then define the spaces x C    C  Rq / WD C 1 .R x C  ; B ;d .X I C  Rq // B ;d .XI R and z ;d .XI R x C    C  Rq / B (6.1.27) ˚

;d Q Q x WD h.r; y; w; / D h.r; y; w; r/ W h.r; y; w; / Q 2 B .X I RC    C  Rq / : Q

z ;d .X I R x C    R1Cq / there exists an Theorem 6.1.17. For every p.r; y; %; / 2 B ;d q z x h.r; y; w, / 2 B .X I RC    C  R / such that  opr .p/.y; / D opM .h/.y; /

mod C 1 .; B 1;d .X ^ I Rq //, for every  2 R. Writing p.r; y; %; / D p.r; Q y; r%; r/;

Q y; w; r/ h.r; y; w; / D h.r;

Q y; w; / z ;d .X I R x C    R1Cq / and with corresponding p.r; Q y; %; Q / Q and h.r; Q in B %; Q Q

6.1 Fuchs type operators and Mellin quantisation

323

z ;d .XI R x C    C  Rq /, respectively, we have an asymptotic expansion B Q Q y; i%; / h.r; Q  p.r; Q y; %; / Q C

1 1 X X

  pQ .r; y; %; /% Q j; ckj @kCj %

kD1 j D0

cf. Remark 3.3.12 (iv). The proof of Theorem 6.1.17 is completely analogous to that of Theorem 6.1.5. The only additional aspect is that the specific properties of the amplitude functions in boundary value problems survive under the Mellin quantisation, for instance, the transmission property at the boundary. The correspondence p ! h of Theorem 6.1.17 gives rise to a (non-canonical) map z ;d .X I R z ;d .X I R x C    R1Cq / ! B x C    C  Rq /I mW B the image under the map m.p/ is called a Mellin quantisation of p. x C    C  Rq / and of the nature of Further properties of the space B ;d .X I R the map m will be formulated in the following section.

6.1.2 Kernel cut-off Let  Rn be open and a.x; x 0 ; / 2 S  .    Rn / a symbol of order  2 R. By ‘kernel cut-off’ in its simplest form we understand a procedure to pass from a symbol a.x; x 0 ; / 2 S  .    Rn / to another symbol h.x; x 0 ; / that extends to a holomorphic function h.x; x 0 ; / in the n-dimensional complex variable  D C i in such a way that a.x; x 0 ; / D h.x; x 0 ; C i/jD0 mod S 1 .    Rn /: R The method is to form the family of distributions k.a/.x; x 0 ;  / D e i a.x; x 0 ; /μ which makes sense as the inverse Fourier transform of a.x; x 0 ; / 2 S 0 .Rn /, for every fixed x; x 0 , and then to set h.x; x 0 ; / WD

Z

e i . /k.a/.x; x 0 ;  /d

(6.1.28)

for any cut-off function . / (i.e., . / 2 C01 .Rn /, . / 1 near  D 0). The ’ 0 distributional kernel of the operator Op.a/u.x/ D e i.xx / a.x; x 0 ; /u.x 0 /dx 0 μ 0 can be written as k.a/.x; x ;  /j Dxx 0 ; thus the formula (6.1.28) contains a cut-off of the kernel near diag.  / which is just a motivation of the terminology.

324

6 Operators on manifolds with conical singularities and boundary

Observe that we can also write Z nZ o Q Q Q d h.x; x 0 ; / D e i . / e i a.x; x 0 ; /μ “ Q Q d Q D e i .  / . /a.x; x 0 ; /μ “ D e i # . /a.x; x 0 ;  #/μ #d:

(6.1.29)

Remark 6.1.18. Kernel cut-off constructions are useful in many variants, in particular, for (i) operator-valued symbols with ‘twisted homogeneity’, cf. Definition 2.2.3; (ii) parameter-dependent symbols, i.e., with respect to a subtuple of covariables, while the remaining variables are treated as parameters; (iii) operator-valued amplitude functions of more general form, e.g., parameter-dependent pseudo-differential operators on a manifold, where the parameters play the role of covariables, cf. the formula (2.1.28); (iv) the Mellin instead of the Fourier transform, cf. the proof of Theorem 6.1.5. As the formula (6.1.29) shows the kernel cut-off construction only acts on covariables, i.e., x; x 0 remain untouched. For that reason we mainly discuss the case without such extra variables (in applications we tacitly employ the corresponding generalisation). Before we investigate the properties of the kernel cut-off in more detail we want to observe a few elementary properties in the case of scalar symbols a. / 2 S  .Rn /. Proposition 6.1.19. We have k. ˛ D ˇ a/. / D .1/jˇ j D˛  ˇ k.a/. / for every a. / 2 S  .Rn / and arbitrary ˛; ˇ 2 N n . Proposition 6.1.20. Let a. / 2 S  .Rn / and write D .;R/ for  D . 1 ; : : : ; q /,  D . qC1 ; : : : ; n / for any 1  q  n. Set k 0 .a/.;  / WD e i a.; /μ , and let /,  0 near 0 and  1 outside ./ be an excision function, (i.e.,  2 C 1 .Rnq  another neighbourhood of 0). Then we have . /k 0 .a/.;  / 2 S.Rn; /. Propositions 6.1.19 and 6.1.20 are immediate consequences of the definition of k.  / and k 0 .  /, respectively. Corollary 6.1.21. Let

. / 2 C01 .Rnq /,

. / 1 near  D 0, and set

H. /a. / WD F ! . /k 0 .a/.;  /: Then we have H. /a. / D a. / mod S 1 .Rn /.

6.1 Fuchs type operators and Mellin quantisation

325

We now study the kernel cut-off operator in the case of operator-valued symbols with twisted homogeneity and uniform symbolic estimates in the sense of Definition 2.2.26. For the applications it will be enough to consider the cut-off process with respect to one component of the covariables. Moreover, since only the covariables are affected, z / with constant coefficients, cf. we look at symbols a.; / 2 S  .Rq  R I H; H the notation after Remark 2.2.27 (in this case we may omit the subscript ‘b’). Let '. / 2 Cb1 .R/ and set Z Z e i # '. /a.;   #/d μ #; (6.1.30) H.'/a.; / WD R

R

cf. the formula (6.1.29). Observe that the integrand '. /a.;   #/ is a smooth funcz /-valued amplitude functions in .; #/ tion in .; / with values in L.H; H which follows from the symbolic estimates. Thus H.'/a is defined as a function z //. in C 1 .Rq  R; L.H; H Theorem 6.1.22. The operator (6.1.30), .'; a/ ! H.'/a, defines a bilinear and continuous map z / ! S  .RqC1 I H; H z /; I H; H Cb1 .R /  S  .RqC1 ; ;

(6.1.31)

and we have an asymptotic expansion H.'/a.; / 

1 X .1/k k .D '/.0/@k a.; /: kŠ

(6.1.32)

kD0

Proof. First consider z / ! C 1 .RqC1 ; S  .R  R# I H; H z /b / Cb1 .R /  S  .RqC1 I H; H ; ; defined by .'; a/ ! '. /a.;   #/ which is bilinear and continuous. Then, in order z/ to show the continuity of (6.1.31) it suffices to verify that H.'/a 2 S  .RqC1 I H; H and then to apply the closed graph theorem. ˇ ˇ H.'/a.; / D H.'/.D; a.; // for (6.1.31) it remains Using the identity D; to show 1 kQ h;i H.'/a.; /h;i kL.H;Hz /  ch; i (6.1.33) for all .; / 2 RqC1 and a constant c > 0. We regularise the oscillatory integral for H.'/a in the following manner: “ H.'/a.; / D e i # h i2 f.1  @2 /N '. /gaN .; ; #/d μ # where aN .; ; #/ D .1  @2# /fh#i2N a.;   #/g

(6.1.34)

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6 Operators on manifolds with conical singularities and boundary

for a sufficiently large N 2 N. The expression (6.1.34) is a linear combination of the terms .@j# h#i2N /.@k a/.;   #/ for 0  j; k  2. Applying the inequality Q s Q s  c jsj h ijsj h i h C i for all ; Q 2 Rm , s 2 R, with a suitable constant c > 0 (which is a version of Peetre’s z inequality) we obtain kQ h;#ih;i1 kL.Hz /  C h#iM , kh;#i1 h;i kL.H /  z belonging to the group actions fı gı2R and C h#iM , with the constants M and M C fQ ı gı2RC , respectively, cf. the formula (2.2.1). It follows that

1 ˚ j

2N

Q /.@k a/.;   #/ h;i L.H;Hz / h;i .@# h#i ˇ

ˇ  ˇ@j# h#i2N ˇ Q h;#ih;i1 L.Hz /



1  Q h;#i .@k a/.;   #/h;#i L.H;Hz / h;#i1 h;i L.H / z

 ch#iM CM C2N h; i for some c > 0. This implies an analogous estimate for the function (6.1.34). Taking z C   2N  0 we obtain the estimate (6.1.33). N large enough so that M C M Let us now show the asymptotic expansion (6.1.32). First we have the Taylor expansion N X 1 k .@ '/.0/ k C  N C1 '.N C1/ . / '. / D kŠ  kD0 R 1 C1 with '.N C1/ ./ D N1Š 0 .1  t /N .@N '/.t  /dt . The function '.N C1/ . / belongs  1 to Cb .R/. Integration by parts in (6.1.30) gives us “ N X 1 k H.'/a.; / D e i #  k a.;   #/d μ # .@ '/.0/ kŠ  kD0 “ 1 C e i #  N C1 '.N C1/ . /a.;   #/d μ # NŠ N X .1/k k C1 D a/.; /: .D '/.0/.@k a/.; / C i N C1 H.'.N C1/ /.@N  kŠ kD0

’ i# k e .@ a/.; In the latter equation we employed the identities .@k a/.; / D   #/dμ # and the expression (6.1.30) applied to '.N C1/ . From the first part of C1 the proof we know that H.'.N C1/ /.@N a/.; / is a symbol in S .N C1/ .RqC1 I  z /. This proves the asymptotic expansion (6.1.32). H; H Remark 6.1.23. Let ' 2 Cb1 .R/ and .@k '/.0/ D 0 for all 0  k  N . Then the associated kernel cut-off operator defines a continuous map z / ! S .N C1/ .RqC1 I H; H z/ H.'/ W S  .RqC1 I H; H

6.1 Fuchs type operators and Mellin quantisation

327

for every N 2 N. In particular, if ' vanishes of infinite order at 0 we obtain a map to z /. Moreover, if ' 2 Cb1 .R/ is a function such that '.0/ D 1 and S 1 .RqC1 I H; H k .@ '/.0/ D 0 for all k 2 N, k  1, it follows that z /: H.'/a D a mod S 1 .RqC1 I H; H This is an immediate consequence of Theorem 6.1.22. Another corollary of Theorem 6.1.22 is the following observation: Remark 6.1.24. The operator (6.1.30) defines a bilinear and continuous map z / ! Scl .RqC1 I H; H z /: I H; H Cb1 .R /  Scl .RqC1 ; ;

(6.1.35)

In fact, it suffices to apply the continuity of (6.1.30) together with the asymptotic expansion (6.1.32) and the definition of the topology of the classical symbol spaces, cf. Remark 2.2.27. Definition 6.1.25. Let

 z /b ; .Ryp  Rq  C I H; H S.cl/

(6.1.36)

 2 R, defined to be the set of all functions h.y; ; w/ that are holomorphic in w 2 z // such that h.y; ;  C i / 2 S  .Ryp  C with values in C 1 .Rp  Rq ; L.H; H .cl/ z /b for every 2 R, uniformly in compact -intervals. By S  .Rq  RqC1 I H; H .cl/ ; z / we denote the subspace of elements that are independent of y. C I H; H The space (6.1.36) has a natural Fréchet topology. In the following we first consider  z /. y-independent symbols; the corresponding spaces are denoted by S.cl/ .Rq C I H; H 1 Let us consider the case ' 2 C0 .R/ and write (6.1.30) as an iterated integral Z H.'/a.; / D e i '. /k 0 .a/.;  /d; R Q Q Q being interpreted as an element of S 0 .R; L.H; H z // k 0 .a/.; / D e i  a.; /μ q for every fixed  2 R . Then H.'/a.; / is the Fourier transform of a distribution with compact support. As such it has an extension to a holomorphic function in w D  C i 2 C , namely, Z HC .'/a.; w/ WD e i .Ci / '. /k 0 .a/.;  /d which can also be written as Z HC .'/a.; w/ D e i e  '. /k 0 .a/.;  /d D H.' /a.; / with ' ./ WD e  '. /. In particular, we have HC .'/a.;  C i /j D0 D H.'/a.; /:

(6.1.37)

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6 Operators on manifolds with conical singularities and boundary

Theorem 6.1.26. For every  2 R the operator HC defines a continuous bilinear mapping  z / ! S  .Rq  C I H; H z /: C01 .R/  S.cl/ .Rq  RI H; H .cl/  z /. Then, similarly .Rq  C I H; H Proof. It suffices to show that HC .'/a.; w/ 2 S.cl/ as in the proof of Theorem 6.1.22 the asserted continuity follows from the closed  z / is also a graph theorem. The property HC .'/a.; w/jIm wD 2 S.cl/ .Rq  RI H; H consequence of Theorem 6.1.22, using the relation (6.1.37). The uniform dependence on in compact intervals follows from the continuity of (6.1.31).

As noted in Remark 6.1.18 the kernel cut-off constructions make sense in many variants. In connection with the cone algebra (of boundary value problems) we want to take the Mellin transform rather than the Fourier transform with the covariable on a weight line ˇ D fw 2 C W Re w D ˇg. We therefore switch the notation from C to C. For future reference we want to explicitly formulate some relations of that kind with HC instead of HC (cf. also the constructions below). Remark 6.1.27. For every  2 R, ˇ 2 R, and U Rp open, the operator HC defines a continuous bilinear mapping  z / ! S  .U  C  Rq I H; H z /; C01 .RC /  S.cl/ .U  ˇ  Rq I H; H .cl/

z are assumed to be Fréchet spaces with group action. Moreover, if F is a where H; H Fréchet space, HC defines a continuous bilinear mapping   C01 .RC /  S.cl/ .U  ˇ  Rq I F / ! S.cl/ .U  C  Rq I F /;   where S.cl/ .U ˇ Rq I F / is analogously as (2.1.3), (2.1.5), and S.cl/ .U CRq I F /  q is defined to be the set of all a.x; w; / 2 A.C; S.cl/ .U  R I F // such that a.x; ˇ C  i%; / 2 S.cl/ .U  R1Cq %; I F / for every ˇ 2 R, uniformly in compact ˇ-intervals.

In our applications the kernel cut-off will refer to the spaces of ‘symbols’ B ;d .X I ˇ  Rq /

and B ;d .X; C  Rq /;

(6.1.38)

respectively, for a C 1 manifold X with boundary (not necessarily compact). Definition 6.1.28. The space B ;d .X I ˇ  Rq / is the set of all f .w; / 2 C 1 .ˇ  ;d Rq ; B ;d .X// such that f .ˇ C i%/ 2 B ;d .X I R1Cq .X I C  Rq / %; /. Moreover, B ;d q is defined to be the set of all h.w; / 2 A.C; B .X I R // such that h.ˇ C i%; / 2 B ;d .XI R1Cq %; / for every ˇ 2 R, uniformly in compact ˇ-intervals. Similar considerations make sense for the spaces L .X I ˇ  Rq / .cl/ 1

and L .X I C  Rq /; .cl/

(6.1.39)

respectively, when X is a closed C manifold,  2 R; cf. also Section 2.4.4 (the definition of the spaces (6.1.39) is analogous to that of (6.1.38)).

6.1 Fuchs type operators and Mellin quantisation

329

Remark 6.1.29. (i) Let .%;/ .h/ D . ;.%;/ .h/; @;.%;/ .h// be the parameter-dependent principal symbol of h.ˇ C i%; / DW hˇ .%; / for an element h.w; / 2 B ;d .XI C  Rq / (the parameters are .%; / 2 R1Cq , and ˇ 2 R is fixed, cf. also the notation (3.3.14)). Then  .%;/ .h/ is independent of the choice of ˇ. (ii) Let hj .w; / 2 B ;d .X I C  Rq /, j D 1; 2, and let .%;/ .h1;ˇ / D .%;/ .h2;ˇ / for some ˇ 2 R. Then it follows that h1 .w; /  h2 .w; / 2 B 1;d .X I C  Rq /. Moreover, h1;ˇ .%; /  h2;ˇ .%; / 2 B 1;d .X; R1Cq %; / for some ˇ 2 R implies h1 .w; /  h2 .w; / 2 B 1;d .X I C  Rq /. Let us set Cb1 .RC / WD f'.r/ 2 C 1 .RC / W sup j.r@r /k '.r/j < 1 for all k 2 Ng: r2RC

The transformation (2.4.2) for  D 12 , i.e., S 1 W f .t / ! f .e t /, t 2 R, induces an 2 isomorphism (6.1.40) S 1 W Cb1 .R/ ! Cb1 .RC /: 2

  R1 with the weight 12 Recall that the Mellin transform M 1 u .i%/ WD 0 r i% u.r/ dr r 2   1 induces an isomorphism r 2 L2 .RC / ! L2 .0 / with the inverse M 11 g .r/ D 2  R   i%  1 be a Mellin amplitude function (with cong.i%/μ%. Let a.w/ 2 S.cl/ 0 r 2

1

2 .a/u.r/ D stant coefficients), and consider the corresponding Mellin operator opM 1 R 1 R 1  r i% 0 2 a.i%/u.r 0 / dr μ %. The distributional kernel of opM .a/ has the form 1 0 r0 r0 0 0 1 k.a/.r=r /.r / where Z 1 k.a/./ D  i% a.i%/μ %:

1

2 C01 .RC / that is equal to 1 in a neighbourhood of  D 1 we

Given a function write Z

1

a.w/ D

 i% k.a/./ Z0 1

d 

d D  ./k.a/./ C  0 D h.w/ C c.w/; i%

Z  i% .1 

.//k.a/./

d 

 where h.w/ extends to a holomorphic function in S.cl/ .C/ and has the property ˇ hˇ D a mod S 1 .0 /, while c.w/ 2 S 1 .0 /. Similarly as (6.1.29) we can 0

330

6 Operators on manifolds with conical singularities and boundary

write

Z 1 d  i% ./  i %Q a.i %/μ Q %Q  1 Z0 1 Z 1 d Q D  i.%%/ ./a.i %/μ Q %Q  0 Z 11 Z 1 d D  i# ./a.i.%  #//μ # :  0 1 Z

1

h.w/ D

This is just the motivation for the kernel cut-off operator Z 1 d H.'/a.w/ WD  i% './k.a/./ ;  0 R 1 R 1 i# or H.'/a.w/ WD  0 1  './a.i.%#//μ # d for any ' 2 Cb1 .RC /. Writing  D e  we obtain Z 1Z 1 H.'/a.w/ D e i # '.e  /a.i.%  #//μ #d 1

1

which agrees with the expression (6.1.30), up to the transformed function '.e  / 2 Cb1 .R/, cf. (6.1.40). To every ' 2 C01 .RC / we have an extension of H.'/a.w/ into the complex w-plane C, w D ˇ C i%, Z 1 d HC .'/a.w/ D  w './k.a/./ D H.'ˇ /a.i%/  0 ˇ with 'ˇ ./ D  ˇ './, such that HC .'/aˇRe wD0 D H.'/a. Moreover, as mentioned before, if 2 C01 .RC / is equal to 1 in a neighbourhood of  D 0 it follows that H. /a D a mod S 1 .0 /:  .ˇ /, ˇ 2 R, we set More generally, for a symbol a 2 S.cl/

HC .'/a.w/ WD HC .'/.aˇ /.w  ˇ/

(6.1.41)

 .0 /, where HC .'/ on the right-hand side is defined for aˇ .w/ WD a.w C ˇ/ 2 S.cl/ in the former sense. The following result is now a consequence, applied to a.w; / 2 B ;d .X I ˇ Rq /.

Theorem 6.1.30. For every a.w; / 2 B ;d .X I ˇ  Rq / there is an h.w; / 2 B ;d .XI C  Rq / such that h.ˇ C i%; / D a.ˇ C i%; / mod B 1;d .X I ˇ  Rq /. In fact, similarly as (6.1.41) is suffices to consider ˇ D 0. Then we can set h.w; / D HC . /a.w; / for arbitrary

2 C01 .RC / which is equal to 1 in a neighbourhood of 1.

(6.1.42)

6.1 Fuchs type operators and Mellin quantisation

331

More generally, there is a kernel cut-off operator H.'/ W B ;d .X I ˇ  Rq / ! B ;d .X I ˇ  Rq /

(6.1.43)

for every ' 2 Cb1 .RC / and HC .'/ W B ;d .X I ˇ  Rq / ! B ;d .X I C  Rq /

(6.1.44)

for every ' 2 C01 .RC /, and we have analogous continuity results as for (6.1.31), (6.1.35) and (6.1.37), respectively. Moreover, as a consequence of (6.1.32) we have an asymptotic expansion 1 X  .1/k  k H.'/a.i%; /  D '.e  / .0/@k% a.i%; / kŠ

(6.1.45)

kD0

for the operator (6.1.43) for ˇ D 0. By virtue of HC .'/a.ˇ C i%; / D H.'ˇ /a.i%; / for 'ˇ ./ D  ˇ './, ˇ 2 R, it follows an asymptotic expansion 1 X  .1/k  k D 'ˇ .e  / .0/@k% a.i%; /: HC .'/a.ˇ C i%; /  kŠ kD0

Combined with a relation of the kind (6.1.41) we obtain the following result: Theorem 6.1.31. Let h.w; / D HC .'/a.w; / for an a.w; / 2 B ;d .X I 0  Rq /. Then for every ˇ; ı 2 R there are constants ck .ˇ; ı/, k 2 N, such that h.ˇ C i%; / 

1 X kD0

ck .ˇ; ı/

1 @k h.ı C i%; / kŠ @%k

(6.1.46)

where the asymptotic sum refers to B ;d .X I R1Cq %; /. In particular, if ' is equal to 1 in a neighbourhood of 1, we have c0 .ˇ; ı/ D 1. ˇ Remark 6.1.32. Let h.w; / 2 B ;d .X I C  Rq / and a.w; / WD h.w; /ˇ Rq for ˇ any fixed ˇ 2 R. Then, if 2 C01 .RC / is equal to 1 in a neighbourhood of 1, we have HC . /a D h mod B 1;d .X I C  Rq / (6.1.47) (with HC . / being defined in the sense of (6.1.44)). In other words, HC . / reproduces the space B ;d .X I C  Rq /, modulo smoothing elements. In fact, the relation (6.1.47) is a consequence of Theorem 6.1.30 with (6.1.42) together with Remark 6.1.29. Remark 6.1.33. Let h.w; / 2 B ;d .X I C  Rq /. Then for every ˇ; ı 2 R we have an asymptotic expansion (6.1.46) with c0 .ˇ; ı/ D 1. In fact, it suffices to apply Remark 6.1.32 together with Theorem 6.1.31.

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6 Operators on manifolds with conical singularities and boundary

Theorem 6.1.34. Let hj .w; / 2 B j;d .XI C  Rq /, j 2 N, be an arbitrary sequence. Then there exists an h.w; / 2 B ;d .X I C  Rq / such that h.w; /  PN .N C1/;d .X I C  Rq / for every N 2 N, and h.w; / is unique j D0 hj .w; / 2 B 1;d q .XI C  R /. mod B ˇ Proof. It suffices to apply Remark 3.3.12 (iv) to aj .w; / WD hj .w; /ˇ Rq for any ˇ P ˇ 2 R, form a.w; /  j1D0 aj .w; / in the class B ;d .X I ˇ  Rq /, and then to set h.w; / WD HC . /a.w; / for any 2 C01 .RC / that is equal to 1 near 1. Remark 6.1.35. The kernel cut-off operator (6.1.44) (for any ' 2 C01 .RC / which is equal to 1 near 1) induces an isomorphism B ;d .XI ˇ  Rq /=B 1;d .X I ˇ  Rq / Š B ;d .X I C  Rq /=B 1;d .X I C  Rq /: This is an easy consequence of Theorem 6.1.30 and Remarks 6.1.32 and 6.1.29.

6.1.3 Meromorphic Fredholm families and ellipticity of conormal symbols In this section we present some well-known material on meromorphic operator functions in an open set U C. We first consider a general functional analytic framework around so-called ‰-algebras and then specify the results to cases in connection with the conormal symbolic structure of operators on manifolds with conical singularities. Let B be a Banach space and L.B/ the space of linear continuous operators in B. Then L.B/ is a unital Banach algebra (in the operator norm topology .L.B/). Definition 6.1.36. A subalgebra ‰  L.B/ is said to be a ‰-algebra in L.B/ if it has the following properties: (i) ‰ is a (locally convex) Fréchet space with its topology .‰/ .L.B//; (ii) for the unit elements we have 1‰ D 1L.B/ ; (iii) for the groups ‰ 1 and L.B/1 of invertible elements in ‰ and L.B/, respectively, we have L.B/1 \ ‰ D ‰ 1 . In the case that B is a Hilbert space ‰ is called a ‰  -algebra if ‰ is a symmetric ‰-algebra. The notion of ‰- and ‰  -algebras has been introduced in [62]. Concerning the ‘abstract’ part of this section we follow the lines of [103, Appendix A] or [107, Section 1.2]. In Chapter 9 below we shall return to meromorphic operator functions from the point of view of the work of Gohberg and Sigal [60].

6.1 Fuchs type operators and Mellin quantisation

333

Remark 6.1.37. (i) The multiplication in a ‰-algebra is jointly continuous with respect to .‰/. The group ‰ 1 is open, and hence the inversion ‰ 1 ! ‰ 1 is continuous, since ‰ is a Fréchet space, cf. [225]. Also the -operation in a ‰  -algebra is continuous by the closed graph theorem. (ii) A ‰-algebra is invariant with respect to the (one-dimensional) holomorphic functional calculus. Definition 6.1.38. Let U C be an open set and D  U a discrete subset (i.e., D \ K is finite for every K b U ). A function f 2 A.U n D; ‰/ (i.e., holomorphic in U n D, with values in a ‰-algebra, ‰  L.B/ for a Banach space B) is called a finitely meromorphic Fredholm family, if has the following properties: (i) f takes values in the space F .B/ of Fredholm operators in B; (ii) every p 2 D has aPneighbourhood V .p/ U such that f can be written in 1 k the form f .z/ D kDm ck .z  p/ C h.z/, z 2 V .p/ n fpg with finitedimensional operators ck 2 L.B/ and h 2 A.V .p/; L.B//, h.z/ 2 F .B/ for every z 2 V .p/, m D m.p/. Remark 6.1.39. Clearly in Definition 6.1.38 we also admit the case D D ;, i.e., holomorphic Fredholm families. By Cauchy’s integral formulas the Laurent coefficients have the form Z f ./ 1 ck D d  for k D m; : : : ; 1; (6.1.48) 2 i j pjDı .  p/kC1 for any sufficiently small ı > 0. This entails ck 2 ‰ R and then h 2 A.V .p/ n fpg; ‰/. Thus, by Cauchy’s integral formula h.z/ D 21 i j pjDı h. / d , jz  pj < ı, we z obtain h 2 A.V .p/; ‰/. Theorem 6.1.40. Let U C be a connected open set and f 2 A.U; F .B//. Assume that there exists a point z1 2 U such that f .z1 / W B ! B is an invertible operator. Then (i) there is a discrete set D  U such that f .z/ W B ! B is invertible for all z 2 U n D; (ii) f 1 2 A.U n D; F .B//; (iii) every p 2 D has a neighbourhood V .p/ U such that f .z/1 D

1 X

ck .z  p/k C h.z/;

z 2 V .p/ n fpg;

kDm

for an m D m.p/, with finite-dimensional ck 2 L.B/, and h 2 A.V .p/; F .B//. Concerning the proof, see [107, Theorem 1.2.7].

334

6 Operators on manifolds with conical singularities and boundary

Lemma 6.1.41. Given a vector space B and subspaces E1 ; : : : ; EN of finite codimension in B. Then E1 \    \ EN is also of finite codimension in B. For a proof see [107, Lemma 1.2.8]. Lemma 6.1.42. Let B be a Banach space and U C be a connected open neighbourhood of the origin 0 2 C. Let a1 ; : : : ; am 2 L.B/ be finite-dimensional operators. Moreover, let h 2 A.U; L.B// be an element such that h.z/u D 0, z 2 U , for all u 2 K0 B, where K0 is a closed of B of finite codimension. P subspace k Set f .z/ WD 1 C h.z/ C 1 z 2 U n f0g, and assume that f .z1 / kDm ak z ; is invertible in L.B/ for some z1 2 U n f0g. Then there exists a ı > 0 such that f is invertible for all 0 < jzj < ı. Moreover, for those z we can write f .z/1 D P.z/ 1 k kDl ck z Cg.z/ with finite-dimensional operators ck 2 L.B/ and g.z/ 2 A.jzj < ı; L.B//. Furthermore, we have .1 C h/.U / F .B/ and g.jzj < ı/ F .B/. Concerning a proof, see [107, Lemma 1.2.9]. Theorem 6.1.43. Let ‰ be a ‰-algebra in L.B/ with a Banach space B. Let U C be a connected open set and f 2 A.U n D; ‰/ for a discrete set D  U a finitely meromorphic family in U . Let f .z1 / be invertible in L.B/ for some z1 2 U . Then z  U , D D, z such that f .z/ is invertible in L.B/ for all there exists a discrete set D 1 z Moreover, we have f z ‰/, and f 1 extends to a finitely z 2 U n D. 2 A.U n D; meromorphic family in the sense of Definition 6.1.38. For a proof, see [107, Theorem 1.2.6]. We now specify the results to the case of meromorphic B ;d .X /-valued families of Fredholm operators. Here X is a compact C 1 manifold with boundary @X . A sequence R WD f.pj ; mj ; Lj /gj 2Z (6.1.49) is called a (discrete) Mellin asymptotic type, if pj 2 C, j Re pj j ! 1 as jj j ! 1, mj 2 N, and Lj  B 1;d .X / is a finite-dimensional subspace of operators of finite rank. Set C R WD fpj gj 2N ; a function .w/ 2 C 1 .C/ is called a C R-excision function if .w/ D 0 in a neighbourhood of C R and .w/ D 0 for dist.w; C R/ > " for some " > 0. 1;d Definition 6.1.44. Let MR .X / denote the set of all meromorphic functions f .w/ 2 1;d .X // with poles at the points pj of multiplicity mj C1 for all j 2 Z, A.CnC R; B Pmj ckj .wpj /.kC1/ Chj .w/ in a neighbourhood and Laurent expansions f .w/ D kD0 of pj , with coefficients ckj 2 Lj , and holomorphic B 1;d .X /-valued functions hj in that neighbourhood; in addition we require .w/f .w/ 2 S.ˇ ; B 1;d .X // for every C R-excision function .w/, for every ˇ 2 R, uniformly in compact ˇ-intervals. 1;d We endow the space MR .X / with the structure of a Fréchet space which follows easily from the definition. Set ;d 1;d MR .X / WD B ;d .X I C/ C MR .X /

(6.1.50)

335

6.1 Fuchs type operators and Mellin quantisation

with the Fréchet space structure of the non-direct sum, cf. Definition 2.1.4. In the case ;d C R D ; we also write as subscript ‘O’. In that sense, since MO .X / B ;d .X I C/, we have ;d MO .X / D B ;d .X I C/: (6.1.51) The space (6.1.50) will be interpreted below as the space of conormal symbols in the cone algebra of boundary value problems, with asymptotics. Remark 6.1.45. Let 2 C01 .RC / be equal to 1 in a neighbourhood of 1 and form the associated kernel cut-off operator ;d HC . / W B ;d .X I ı / ! MO .X /;

(6.1.52)

;d cf. the formula (6.1.44) for q D 0 and the relation (6.1.51). Let f 2 MR .Xˇ / and choose any ı 2 R such that ı \ C R D ;. Then we have f  HC . /.f ˇ / 2 ı

1;d MR .X/. In particular, from a given meromorphic f we can recover the decomposition into a holomorphic summand and a meromorphic smoothing element, cf. the formula (6.1.50). j;d Theorem 6.1.46. Let hj 2 MO .X /, j 2 N, be an arbitrary sequence. Then P ;d .N C1/;d there exists an h 2 MO .X / such that h  jND0 hj 2 MO .X / for ev1;d ery N 2 N, and h is unique mod MO .X /. A similar result is true of hj 2 x C ; Mj;d .X // with an asymptotic sum in C 1 .R x C ; M;d .X //, unique mod C 1 .R O O x C ; M1;d .X //. C 1 .R O

Proof. Letˇ us consider the first case, the second one is completely analogous. Setting fj WD hj ˇ 2 B j;d .X I ı / for any fixed ı 2 R we have an asymptotic sum ı P f  j1D0 fj in B ;d .X I ı /, see Remark 3.3.12 (iv). Applying the kernel cut-off P ;d .X /. We then have .h jND0 hj /jı 2 operator (6.1.52) gives us an element h 2 MO B .N C1/;d .XI ı /. From Remark 6.1.29 (ii) (for the case q D 0) it follows that P .N C1/;d 1;d h  jND0 hj 2 MO .X / for every N . The uniqueness of h mod MO .X / is evident. ;d Theorem 6.1.47. Let f 2 MR .X /, g 2 MS;e .X / with Mellin asymptotic types R; S. Then we have for the pointwise composition fg 2 MPC;h .X / with another Mellin asymptotic type P , and h D max. C d; e/.

The proof is a straightforward consequence of Theorem 3.2.22. ;d Definition 6.1.48. An element f 2 ˇMR .X / is called elliptic if for some ı 2 R such ˇ that ı \C R D ; the restriction f  is parameter-dependent elliptic in B ;d .X I ı / ı (with parameter % D Im w; w 2 ı ).

Remark 6.1.49. The ellipticity is independent of the choice of ı.

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6 Operators on manifolds with conical singularities and boundary

Remark 6.1.50. Recall that we assumed the involved vector bundles to be trivial and of fibre dimension 1. Our considerations have an evident generalisation to the case of arbitrary bundles E; F 2 Vect.X /; G˙ 2 Vect.@X /, cf. the formula (6.1.25). Theorem 6.1.51. The parameter-dependent ellipticity of an element a.w/ in B ;d .XI ı / (with the parameter % D Im w/ entails the ellipticity of HC . /a.w/ (cf. the formula (6.1.52)) in the sense of Definition 6.1.48. ˇ Proof. Applying Theorem 6.1.30 with (6.1.42) we see that HC . /a.w/ˇ is also ı parameter-dependent elliptic, cf. Remark 6.1.49. The notation and constructions have simple analogues in the case of a closed compact C 1 manifold N , with the spaces Lcl .N IRl /, Lcl .N Iı /, MR .N /, MO .N /, etc.,  2 R, kernel cut-off operators HC . / W Lcl .N I ı / ! MO .N /, and an analogue of Theorem 6.1.51. Let us apply Theorem 6.1.51 to the operator family r  .ı C i%/ WD R .%/ for  2 Z and any fixed ı 2 R, where R .%/ 2 B ;0 .X I R/ is of the kind (4.1.18), z .%/ 2 L z .%/eC where R i.e., R .%/ D r C R cl .2X I R/ is an order reducing family ;d l on 2X. Recall that B .X I R / is the space of upper left corners of B ;d .X I R% /; similarly we use the notation B ;d .X I ı / for that space when R is replaced by ı ;d and MR;d .X/ for the space of upper left corners of MR .X /; especially, we have MO;d .X/. Let us set m .w/ WD HC . /r  .w/ 2 MO;0 .X /. In a similar manner, starting from an order reducing family R0 .%/ 2 Lcl .@X / on the boundary,  2 R, we can form r 0 .ı C i%/ WD R0 .%/ and m0 .w/ WD HC . /r 0 .w/ 2 MO .@X /. Then ;0 .X / h .w/ WD diag.m .w/; m0 .w// 2 MO

(6.1.53)

is an elliptic element in the sense of Definition 6.1.48. Recall that in the construction of r  .w/ we could start from parameter-dependent elliptic symbols with a parameter  2 Rl for arbitrary l. Setting  D .%; 1 / 2 R2 we also obtain a corresponding parameter-dependence of m .w; 1 / D HC . /r  .w; 1 /, where m .ˇ C i%; 1 / 2 B ;0 .XI R2%;1 / is parameter-dependent elliptic for every ˇ 2 R. Since m .ˇ C i%; 1 / W H s .X / ! H s .X /

(6.1.54)

induces isomorphisms for j%; 1 j > C for some sufficiently large constant C > 0; s > max.; 0/  12 , and every fixed ˇ 2 R, it follows that the operators are isomorphisms for all % 2 R, provided that j1 j > C . In addition it can easily be verified that for every c < c 0 the constant C can be chosen so large that (6.1.54) are isomorphisms for all c < ˇ < c 0 and all % 2 R (because ˇ only contributes lower order terms). In an analogous manner we can proceed with m0 .w/. This proves the following result: Proposition 6.1.52. For every  2 Z and c < c 0 there exists an elliptic element (6.1.53) 1 1 which induces isomorphisms h .w/ W H s .X / ˚ H s 2 .@X / ! H s .X / ˚ H s 2 .@X / 1 for all w 2 C in the strip c < Re w < c 0 , and s > max.; 0/  2 .

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6.1 Fuchs type operators and Mellin quantisation

Let us now consider a g 2 MP1;d .X /. Then 1 C g.w/ 2 MP0;d .X / is elliptic. Choose any sufficiently large s 2 N; s > d  12 ; then 1

1

1 C g.w/ W H s .X / ˚ H s 2 .@X / ! H s .X / ˚ H s 2 .@X / s is a family of continuous operators. Moreover, let hs 1 .w/ and h2 .w/ be families in 0 the sense of Proposition 6.1.52 for a fixed choice of c < c . Then 1

1

0 2 .@X / ! H 0 .X / ˚ H  2 .@X / f .w/ WD hs2 .w/.1 C g.w//hs 1 .w/ W H .X / ˚ H (6.1.55) is a finitely meromorphic Fredholm family in the strip U WD fw 2 C W c < Re w < c 0 g, with values in the ‰  -algebra B 0;0 .X /. The operators (6.1.55) are invertible for all Re w D ˇ; c < ˇ < c 0 , and j Im wj sufficiently large. Thus f .w/ satisfies the z U assumptions of Theorem 6.1.34, D D C R. We therefore find a discrete set D 1 z such that f .w/ is invertible for all w 2 U n D, and f .w/ extends to a finitely meromorphic family in the sense of Definition 6.1.38. Since the operator families hs2 1 and hs itself. 1 are holomorphic in U we obtain a similar property for .1 C g.w// 1 1;d z The values of .1 C g.w// for w 2 U n D belong to B .X /. Thus, from the formula (6.1.48) we obtain that the Laurent coefficients belong to B 1;d .X /. Since the strip c < Re w < c 0 is arbitrary, we proved altogether the following result.

Proposition 6.1.53. For every g 2 MP1;d .X / with P a Mellin asymptotic type and 1;d d 2 N, there exists an l 2 MQ .X / for some Mellin asymptotic type Q such that .1 C g/1 D 1 C l:

(6.1.56)

There is also a variant of Mellin symbols without a full control of asymptotics. In this connection we define, for any fixed ˇ 2 R, the space M1;d .X I ˇ /

(6.1.57)

of all fˇ .w/ 2 S.ˇ ; B 1;d .X // D B 1;d .X I ˇ // which extend to an element f .w/ 2 A.ˇ  " < Re w < ˇ C "; B 1;d .X // for an " D ".fˇ / > 0 such that f jı 2 B 1;d .X I ı / for every ˇ  " < ı < ˇ C ", uniformly in compact ı-intervals. For purposes below we also define finite sequences R D f.pj ; mj ; Lj /gj D0;:::;N with ˇ " < Re pj < ˇ C" for some " > 0 and fixed ˇ 2 R and spaces Lj of the above S kind. Setting C R D jND0 fpj g a C R-excision function is any  2 C 1 -function in fw W ˇ  " < Re w < ˇ C "g such that .w/ D 0 for dist.w; C R/ < c0 , .w/ D 1 for dist.w; C R/ > c1 for some 0 < c1 < c0 . Then 1;d MR .X I ˇ /"

denotes the space of all B 1;d .X /-valued functions f .w/ which are meromorphic in the strip fw W ˇ  " < Re w < ˇ C "g with poles at finitely many points pj in that

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6 Operators on manifolds with conical singularities and boundary

strip, of multiplicity mj C1 and Laurent coefficients at .w  pj /.kC1/ in the spaces Lj , 0  k  mj , and in such a way that .f /jı 2 B 1;d .X I ı / for any C R-excision function .w/, for every ˇ  " < ı < ˇ C ", uniformly in compact subintervals. 1;d .X I ˇ /" and 0 < "0 < " there exists an Proposition 6.1.54. For every g 2 MR 1;d l 2 MS .XI ˇ /"0 with a finite sequence S D f.sj ; nj ; Nj /g of analogous structure as R, where C S  fw W ˇ  "0 < Re w < ˇ C "0 g, such that the relation (6.1.56) holds for all ˇ  "0 < Re w < ˇ C "0 . ;d .X / there exists a p.w/ 2 Theorem 6.1.55. For every elliptic function a.w/ 2 MR C

MS;.d/ .X / for a suitable Mellin asymptotic type S such that p.w/ D a1 .w/ (the inverse with respect to composition of meromorphic operator functions).

Proof. By assumption there is a ı 2 R such that a.ı C i%/ 2 B ;d .X I ı / is parameter-dependent elliptic. Therefore, by Theorem 3.3.17 there exists a parameterC dependent parametrix a.1/ .ı C i%/ 2 B ;.d/ .X I ı /. Let us form b.w/ WD C ;.d/ HC . /a.1/ .w/ 2 MO .X /. Then it follows that b.w/a.w/ D 1  c.w/ for 1;dl an element c.w/ 2 MP .X / for some Mellin asymptotic type P . From Proposi1;dl tion 6.1.53 we find a d.w/ 2 MQ .X / for a Mellin asymptotic type Q such that C

.1  c/1 D 1  d . This gives us a1 D .1  d /a.1/ 2 MS;.d/ .X / for some Mellin asymptotic type S.

6.1.4 Green operators Let B be a compact manifold with conical singularities and boundary, cf. Remark 2.4.21. As above, we assume that there is only one conical point v (the case of finitely many conical singularities is similar and left to the reader). Recall that v has a neighbourhood V in B with a conical structure in terms of a homeomorphism  W V ! X

where X is a compact C 1 manifold with boundary, such that reg WD jV nfvg induces a diffeomorphism reg W V n fvg ! RC  X (6.1.58) between the respective C 1 manifolds with boundary. By (6.1.58) we have a local splitting of variables .r; x/ that we keep fixed. As in Section 2.4.2 we have the stretched manifold B associated with B, furthermore, the regular and the singular parts Breg

and Bsing ;

respectively, B D Breg [ Bsing . There are two kinds of doubles, namely, 2B and .2/B, where 2B is a compact manifold with conical singularity v and without boundary, while .2/B is a compact C 1 manifold with boundary. Finally D WD @.B n fvg/ [ fvg is a manifold with conical singularity v, without boundary, and we also have the associated stretched manifold D which is a C 1 manifold with boundary @D DW Dsing , and we

6.1 Fuchs type operators and Mellin quantisation

339

set Dreg WD D n Dsing . Observe that then the above  restricts to a homeomorphism 0 WD jV 0 W V 0 ! .@X / with V 0 WD V \ D such that 0reg WD jV 0 nfvg induces a diffeomorphism 0reg W V 0 n fvg ! RC  @X: (6.1.59) There is a collar neighbourhood 2V of .2B/sing in 2B such that V WD 2V \ B consists of .V n fvg/ [ X in the sense that (6.1.58) is the restriction of a corresponding map x C  X . Then V 0 WD V \ D is a collar neighbourhood of Dsing in D, and V ! R x C  @X . (6.1.59) is the restriction of a corresponding map V 0 ! R In the following, if we say nothing other, we assume n D dim X . If we describe distributions or operators on B locally near Bsing we refer to the chosen splittings of variables from (6.1.58) and (6.1.59), respectively, but do not explicitly indicate pull backs from RC  X to V . By a cut-off function ! on B we understand a function of the form ! D 'jB for a function ' 2 C 1 .2B/ with supp '  2V and ' 1 in a neighbourhood of .2B/sing . If ! is a cut-off function in that sense, ! 0 WD !jD is a cut-off function on D. Later on we fix cut-off functions QQ !; !; Q !QQ on B such that !Q 1 on supp !; ! 1 on supp !; and we then have corresponding cut-off functions ! 0 ; !Q 0 ; !QQ 0 on D. In Section 2.4.2 we introduced the scales of weighted Sobolev spaces H s; .B/

and

H s; .D/, s;  2 R;

cf. the formula (2.4.22) and Definition 2.4.22. For every compact manifold D with conical singularity and without boundary we have a non-degenerate sesquilinear pairing H s; .D/  H s; .D/ ! C;

s;  2 R;

(6.1.60)

defined via the H 0;0 .D/-scalar product. In the case of a manifold B with conical singularity and with boundary we define the spaces ˇ H s; .B/0 WD fu 2 H s; .2B/ W uˇint B;reg D 0g; cf. the notation (2.4.20). Similarly we have the spaces H s; .B /0 when we interchange the role of the plus/minus sides (recall that B is identified with BC ). From the canonical isomorphism H s; .B/ Š H s; .2B/=H s; .B /0 we obtain a Hilbert space structure in H s; .B/ (by identifying H s; .B/ with the orthogonal complement of H s; .B /0 in H s; .2B/). Instead of (6.1.60) we have a non-degenerate sesquilinear pairing H s; .B/  H s; .B/0 ! C via the H 0;0 .B/-scalar product. Definition 6.1.56. Let g WD .; ı/, ; ı 2 R, and j ; jC 2 N.

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6 Operators on manifolds with conical singularities and boundary

(i) The space C0G .B; gI j ; jC / is defined to be the set of all operators \   1 1 1 G2 L H s; .B/ ˚ H s 2 ; 2 .D/ ˚ C j ; H 1;ı .B/ ˚ H 1;ı 2 .D/ ˚ C jC s> 1 2

such that there is an " D ".G/ > 0 such that G induces continuous operators 0

1

1

G W H s; .B/˚H s ; 2 .D/˚C j ! H 1;ıC" .B/˚H 1;ı 2 C" .D/˚C jC (6.1.61) for every s; s 0 2 R, s >  12 , and in addition the formal adjoint G  with respect to the 1 sesquilinear pairings via the H 0;0 .B/ ˚ H 0; 2 .D/-scalar product induces continuous operators 0

1

1

G  W H s;ı .B/ ˚ H s ;ı 2 .D/ ˚ C jC ! H 1; C" .B/ ˚ H 1;  2 C" .D/ ˚ C j (6.1.62) for all s; s 0 2 R, s >  12 . (ii) The space CdG .B; gI j ; jC / for d 2 N is defined to be the set of all G WD G0 C

d X

Gk diag.D k ; 0; 0/

(6.1.63)

kD1

for arbitrary Gk 2 C0G .B; gI j ; jC /, k D 0; : : : ; d, where D is any first order differential operator which is the restriction from 2B to Breg of an operator with smooth coefficients (up to .2B/sing ) that is close to @Breg equal to the derivative in normal direction to @Breg . We will also write CGd .B; g/ WD CdG .B; gI 0; 0/:

(6.1.64)

CdG .B; gI j ; jC /

The elements of are called Green operators of type d in the cone algebra, belonging to the weight data g D .; ı/. Remark 6.1.57. Every G 2 CGd .B; g/ defines compact operators 1

1

1

1

G W H s; .B/ ˚ H s 2 ; 2 .D/ ! H s; .B/ ˚ H s 2 ; 2 .D/ for all s > d  12 . Remark 6.1.58. z 2 Ce .B; gI Q j ; j0 / for gQ WD .; ˇ/ G 2 CdG .B; gI j0 ; jC / for g WD .ˇ; ı/; G G z 2 Ce .B; g ı gI Q j ; jC / for g ı gQ D .; ı/. implies G G G Proposition 6.1.59. Let G 2 CdG .B; gI j; j /, g D .;  /, and assume that I  G is 1 1 invertible as an operator in H s; .B/ ˚ H s 2 ; 2 .D/ ˚ C j , s > d  12 . Then we have .I  G/1 D I  D for some D 2 CdG .B; gI j; j /.

6.1 Fuchs type operators and Mellin quantisation

341

The proof will be given in Section 6.2.2 below (in the variant for X ^ instead of B, cf. Remark 6.1.61). There are several other variants of Green operators on a manifold B with conical singularity and boundary. For instance, instead of (6.1.61) and (6.1.62) we can require the properties 0

1;ı 1 2

G W H s; .B/ ˚ H s ; 2 .D/ ˚ C j ! HP1;ı .B/ ˚ HP 0 0

1

.D/ ˚ C jC ; (6.1.65)

1;  1 2

1; G  W H s;ı .B/ ˚ H s ;ı 2 .D/ ˚ C jC ! HQ .B/ ˚ HQ0 1

.D/ ˚ C j (6.1.66)

for discrete asymptotic types P; Q on B, P 0 ; Q0 on D, associated with weight data connected with the weight in the respective spaces and some weight interval ‚, cf. the Definitions 2.4.71 and 2.4.75 and their analogues in the case with boundary. Then we obtain corresponding subspaces of C0G .B; gI j ; jC /. Applying this in (6.1.63) we get subspaces of CdG .B; gI j ; jC / of such Green operators of type d. Instead of P , P 0 , Q, Q0 we can also take so-called continuous asymptotic types, cf. [177] and Section 10.4.5 below. They will not explicitly be discussed here in detail. Another class of Green operators which plays a role in our applications refers to the infinite stretched cone X ^ where X is a compact C 1 manifold with boundary @X . Definition 6.1.60. Let g WD .; ı/, ; ı 2 R, and j ; jC 2 N. (i) The space C0G .X ^ ; gI j ; jC / is defined to be the set of all operators \ 1 1 G2 L.K s; .X ^ / ˚ K s 2 ; 2 ..@X /^ / s> 1 2 1

˚ C j ; K 1;ı .X ^ / ˚ K 1;ı 2 ..@X /^ / ˚ C jC / such that there is an " D ".G/ > 0 such that G and G  induce continuous operators 0

1

G W K s; .X ^ / ˚ K s ; 2 ..@X /^ / ˚ C j 1

! S ıC" .X ^ / ˚ S ı 2 C" ..@X /^ / ˚ C jC and 0

1

G  W K s;ı .X ^ / ˚ K s ;ı 2 ..@X /^ / ˚ C jC 1

! S C" .X ^ / ˚ S  2 C" ..@X /^ / ˚ C j ; respectively, for all s; s 0 2 R; s >  12 , where similarly as in Definition 6.1.56 (i) the 1 sesquilinear pairing refers to the K 0; 2 ..@X /^ /-scalar product in the second component. (ii) The space CdG .X ^ ; gI j ; jC / for d 2 N is defined to be the set of all operators of the form (6.1.63) for arbitrary Gk 2 C0G .X ^ ; gI j ; jC /, k D 0; : : : ; d, where D

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6 Operators on manifolds with conical singularities and boundary

is any first order differential operator which is the restriction from .2X /^ to X ^ of an operator with smooth coefficients (up to r D 0) that is close to .@X /^ equal to the derivative in normal direction to .@X /^ . We will also write CGd .X ^ ; g/ WD CdG .X ^ ; gI 0; 0/:

(6.1.67)

Remark 6.1.61. The elements of this space satisfy an analogue of Remark 6.1.57. In addition there are analogues of Remark 6.1.58 and Proposition 6.1.59 for the spaces of Definition 6.1.60. Let G 2 CGd .X; g/ and write G D .Gij /i;j D1;2 . Then G11 will also be called a Green operator of the cone algebra ‘in the proper sense’, while G21 is also called a trace operator (both of type d) and G12 a potential operator of the cone algebra on X ^ . Moreover G22 is a Green operator on .@X /^ , i.e., on the corresponding infinite cone without boundary. A similar terminology will be used for Green operators on B, cf. Definition 6.1.56. Proposition 6.1.62. (i) Every Green operator G on X ^ in the proper sense of type d can be written in the form d1 X Kj Tj (6.1.68) G D G0 C j D0

for a Green operator G0 (in the proper sense) of type 0, potential operators Kj and trace operators Tj of type j C 1, j D 0; : : : ; d  1. (ii) Every trace operator T of type d can be written as T D T0 C

d1 X

Rj ı  j

(6.1.69)

j D0

with  j u WD @xnj ujxn D0 in the splitting of variables x D .x 0 ; xn / in a collar neighbourhood Š @X  Œ0; 1/ of @X , Green operators Rj on .@X /^ , and a trace operator T0 of type 0. Conversely, every operator of the form (6.1.69) is a trace operator of type d. (iii) A similar result is true of Green and trace operators on B. Proof. To show (i) we assume for the moment that (ii) is already proved. Let G be a Green operator on X ^ of type 0 in the proper sense. There is then an " > 0 such that Z Z 1 g.r; x; t; y/u.t; y/t n dt dy (6.1.70) Gu.r; x/ D X

0

y  S C" .X ^ /. for an integral kernel g.r; x; t; y/ 2 S ıC" .X ^ / ˝ Let D be a first order differential operator as in Definition 6.1.56 (ii). We show that for every k there is a representation GD k u.r; x/ D G0 u.r; x/ C K T u.r; x/ for a Green operator G0 of type 0 in the proper sense, a potential operator K and a trace

6.1 Fuchs type operators and Mellin quantisation

343

operator T of type k. To this end we fix a collar neighbourhood V of @X with a splitting of variables x D .x 0 ; xn /, xn 2 Œ0; 1/. Without loss of generality we may assume that k with a first order differential the operator D k has the form !d k =dxnk C .1  !/Dint operator Dint on int X and a cut-off function !.xn / that vanishes for xn > 12 . Since k is of type 0 as we easily see by passing to the kernel obtained from G.1  !/Dint k g.r; x; t; y/ by applying the transposed of the operator .1  !/Dint to g with respect to y, it remains to look at dk  G ! k u .r; x/ D dyn 

Z 1Z 0

Z

1

g.r; x; t; y 0 ; yn /!.yn /

@X 0

dk u.t; y 0 ; yn /t n dt dy 0 dyn dynk

for k  1. Integration by parts gives us  dk  G ! k u .r; x/ dyn Z 1Z Z D 0

Z

@X 0 1

Z

C @X

1

0

 d k1 d g.r; x; t; y 0 ; yn /!.yn / u.t; y 0 ; yn /t n dt dy 0 dyn dyn dynk1

g.r; x; t; y 0 ; 0/

d k1 u.t; y 0 ; 0/t n dt dy 0 : dynk1

The second summand on the right-hand side has the form Rk1 ı  k1 for a Green operator Rk1 on the boundary .@X /^ . In the first summand we can integrate by parts again (when k  1  1, otherwise the proof is done). By iterating the arguments k times we finally obtain the assertion (i). In order to show (ii) we start from a trace operator T of type d. For proving (6.1.69) we can apply similar integrations by parts. It suffices to formally replace g.r; x; t; y/ in the expression (6.1.70) by a kernel 1 y  S C" .X ^ / for some " > 0 (concerning kernel gT .r; x 0 ; t; y/ 2 S ı 2 C" ..@X /^ / ˝ representations of several kind, see Section 2.4.6). Conversely, from an expression as on the right-hand side of (6.1.69) we can easily return to a representation in integral form as for the trace parts in Definition 6.1.56 (ii). The details are left to the reader. For B the arguments are of a similar structure as those for X ^ . Remark 6.1.63. Looking at the second summand on the right of (6.1.63) there are also operators of the kind GD k where Z f .t; y 0 /t n u.t; y 0 /dt dy 0 (6.1.71) Gu WD .@X/^

for certain f .t; y 0 / 2 S C" ..@X /^ /. P j The arguments in the latter proof show that GD k u D G0 uC jk1 D0 Rj ı u, where G0 u is of analogousRstructure as (6.1.71), while Rj represents an integration over .@X /^ of the kind Rj v D .@X/^ fj .t; y 0 /v.t; y 0 /t n dt dy 0 for an fj 2 S C" .X ^ /j.@X/^ .

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6 Operators on manifolds with conical singularities and boundary

6.1.5 Mellin operators with smoothing symbols Another ingredient of the cone algebra are the so-called smoothing Mellin operators. They are concentrated in a neighbourhood of Bsing ; therefore, we refer to a local splitting of variables .r; / 2 X ^ . The simplest smoothing Mellin operators have the form  n 2

M WD !r  opM

.f /!Q

(6.1.72)

with cut-off functions !.r/; !.r/ Q and a symbol f .w/ 2 M1;d .X I  nC1  /, cf. the 2 formula (6.1.57), n D dim X. An operator (6.1.72) induces continuous operators 1

1

1

M W H s; .B/ ˚ H s 2 ; 2 .D/ ! H 1; .B/ ˚ H 1; 2 .D/;

(6.1.73)

or, alternatively, 1

1

1

M W K s; .X ^ / ˚ K s 2 ; 2 ..@X /^ / ! S  .X ^ / ˚ S  2 ..@X /^ / (6.1.74) for every s;  2 R; s > d 12 (see the proof of Theorem 6.2.2 below). The interpretation of (6.1.72) as (6.1.73) or (6.1.74) depends on whether the cylinder .0; R/X is regarded as a neighbourhood on Breg close to Bsing or on X ^ close to r D 0 (recall that we often omit the pull backs of operators on B near Bsing when they are expressed in .r; / 2 X ^ close to r D 0). In order to simplify the consideration in this section we mainly talk about operators on X ^ ; most of the observations then remain true also on B (after obvious modifications). Remark 6.1.64. The operators (6.1.73) or (6.1.74) are in general not compact, cf. also Section 6.2.3 below.   1;d .X /, Remark 6.1.65. The space M1;d X I  nC1  contains the subspaces MR 2 (cf. Definition 6.1.44) for (discrete) Mellin asymptotic types R, C R \  nC1  D ;. 2 The operators (6.1.73), (6.1.74) with such symbols restrict to continuous operators between corresponding subspaces with asymptotics, cf. Definitions 2.4.71 and 2.4.75, and their analogues in the case of manifolds with boundary). Those are of interest in cone and edge algebras with a control of the asymptotics, cf. [161], [174], [175], or [90] for more details. In the present exposition we mainly focus on the more general operators with symbols in M1;d .X I  nC1  /. 2

x C / with '.0/ 6D 0 and '.0/ Proposition 6.1.66. For every '; 'Q 2 C01 .R Q 6D 0 we have  n 2

N WD 'r  opM

.f /'Q D M C G

with an operator M of the form (6.1.72) and a G 2 CGd .X ^ ; .;   //; a different choice of !; !Q only affects the Green remainder. If '.0/ D 0 or '.0/ Q D 0, then we have N 2 CGd .X ^ ; .;   //. Similar relations are true when the operators are realised on B.

6.1 Fuchs type operators and Mellin quantisation

345

Proof. Let us consider the case X ^ (the arguments for B will be similar and left to the reader). First it is easy to verify that C WD

 n 2

r C" opM

.f / Q r "Q

(6.1.75)

x C / is a Green operator whenever "  0, "Q  0 and " > 0 for arbitrary ; Q 2 C01 .R  n

 n

Q C C , by or "Q > 0. We can write 'r  opM 2 .f /'Q D !'.0/r  opM 2 .f /!Q '.0/ applying Taylor’s formula, with a remainder C of the kind (6.1.75), certain ; Q 2 x C / and " D 1 or "Q D 1. It is now evident that another choice of the cut-off C01 .R functions !; !Q only changes (6.1.72) by a Green operator. Definition 6.1.67. Let g WD .;   /; ;  2 R; d 2 N, and j ; jC 2 N. d ^ The space CM CG .X ; gI j ; jC / is defined to be the set of all operators diag.M; 0/ C G

(6.1.76)

for an arbitrary M of the form (6.1.72) and G 2 CdG .X ^ ; gI j ; jC / (cf. Definition 6.1.60). We set d ^ d ^ CM CG .X ; g/ WD CM CG .X ; gI 0; 0/

(cf., similarly, the notation (6.1.67)). In an analogous manner we define the operator d d spaces CM CG .B; gI j ; jC / and CM CG .B; g/, respectively (cf. Definitions 6.1.56, 6.1.60 and the formula (6.1.64)). The choice of weight data .;   / in contrast to .; ı/ in Section 6.1.4 with arbitrary ı is motivated by later applications. In the cone calculus we could replace everywhere    by ı. Remark 6.1.68. We have d ^ 1;d CM .X ^ / CG .X ; g/  B

and

d 1;d CM .B/: CG .B; g/  B

d ^ Definition 6.1.69. Let A 2 CM CG .; gI j ; jC / where ‘’ stands for X or B, written in the form  n Q 0/ C G A D diag.!r  opM 2 .f /!;

for some f 2 M1;d .X I  nC1  / and G 2 CdG .; gI j ; jC /. Then 2

c .A/.w/ WD f .w/;

(6.1.77)

w 2  nC1  , is called the conormal symbol of the operator A. 2

z are contained in Remark 6.1.70. Definition 6.1.69 is correct in the sense that if A, A d CM CG .; gI j ; jC /, and z c .A/.w/ D c .A/.w/

346

6 Operators on manifolds with conical singularities and boundary

z 2 Cd .; gI j ; jC /. We already saw in Proposifor all w 2  nC1  , then A  A G 2 tion 6.1.66 that changing !; !Q only affects an operator of the form (6.1.72) by a Green  n operator. Moreover, !r  opM 2 .f /!Q 2 CdG .; gI j ; jC / entails f D 0. The reason for the latter statement is essentially the same as that for the first bijectivity relation of z , combined with a Remark 2.3.39, applied to the case of trivial group actions in H and H modification to Mellin pseudo-differential operators on RC (cf. also Definition 2.4.56). The conormal symbol (6.1.77) is regarded as a family of continuous operators 1

c .A/.w/ W H s .int X / ˚ H s 2 .@X / ! H 1 .int X / ˚ H 1 .@X / for any s > d 

1 2

and for all w 2  nC1  . 2

d e Theorem 6.1.71. A 2 CM CG .; gI j0 ; jC /, B 2 CM CG .; hI j ; j0 / for g WD .  e ;   . C //, h WD .;   / implies AB 2 CM CG .; g ı hI j ; jC / for g ı h D .;   . C //, and we have

c .AB/.w/ D c .A/.w C /c .B/.w/ (‘’ stands for B or X ^ ). If A or B are in the space with subscript G, then the same is true of AB. Proof. The only non-trivial aspect of the assertion is the rule  n 2

r  !1 opM

 n 2

.f1 /!Q 1 !2 r  opM

 n 2

.f2 /!Q 2 D r .C/ ! opM

..T  f1 /f2 /!Q C G (6.1.78) for arbitrary cut-off functions !i ; !Q i ; i D 1; 2, !; !Q and Mellin symbols f1 ; f2 with some Green remainder G . By Proposition 6.1.66 the choice of cut-off functions is unessential modulo Green operators. The left-hand side of (6.1.78) can be written as  n 2

r .C/ !1 opM  n 2

for C D r .C/ !1 opM

 n 2

.T  f1 / opM

.f2 /!Q 2 C C

 n 2

.T  f1 /.!Q 1 !2  1/ opM

.f2 /!Q 2 . It is easily checked  n 2

that C is a Green operator. It remains to observe that opM  n opM 2 ..T  f1 /f2 /.

 n 2

.T  f1 / opM

.f2 / D

6.1.6 Operators with holomorphic Mellin symbols In this section we specify some elements of the Mellin pseudo-differential calculus when the symbols are holomorphic in the Mellin covariable. We first assume that the symbols are operator functions, acting as classical pseudo-differential operators on a closed compact C 1 manifold X . Analogous results will be valid in the case of a compact C 1 manifold X with boundary. Because of the applications to the edge calculus we admit from the very beginning the dependence on an additional parameter  2 Rq ; for the cone calculus itself it suffices to assume q D 0.

6.1 Fuchs type operators and Mellin quantisation

347

Definition 6.1.72. Let V be a Fréchet space with the semi-norm system . /2N . (i) P .RC  ˇ ; V / is defined to be the set of all a.r; w/ 2 C 1 .RC  ˇ ; V / such that for every  2 N there exist reals  ; ı , such that supfhlog riı h%i  ..r@r /k D%l a.r; ˇ C i%// W .r; %/ 2 RC  R; k C l  N g is finite for every N 2 N; (ii) P .RC  C; V / is defined to be the set of all a.r; w/ 2 C 1 .RC ; A.C; V // such that for every  2 N there exist reals  ; ı , such that l supfe ı hlog ri h%i  ..r@r /k Dw a.r; ˇ C i%// W .r; %/ 2 RC  R; jˇj  N; k C l  N g

is finite for every N 2 N. If  D f g2N , ı D fı g2N are sequences, then by P ;ı .RC  ˇ ; V / and P .RC  C; V / we denote the subspaces of the spaces in Definition 6.1.72 (i) and (ii), respectively, with finite supremums for the given  , ı ,  2 N. These are Fréchet spaces. The following proposition is analogous to Proposition 2.2.36. ;ı

Proposition 6.1.73. (i) P ;ı .RC  ˇ ; V / implies .r@r /k D%l 2 P ;ı .RC  ˇ ; V /. (ii) If T W V ! Vz is a continuous operator, then a 2 P .RC  ˇ ; V / implies T a WD ..r; w/ ! T a.r; w// 2 P .RC  ˇ ; Vz /; more precisely, a ! T a defines a continuous map Q ıQ P ;ı .RC  ˇ ; V / ! P ; .RC  ˇ ; Vz / Q Q ı/. for every .; ı/ with resulting sequences of orders .; (iii) Let V be the projective limit of Fréchet spaces Vj with respect to linear maps Tj W V ! Vj ; j 2 I . Then a 2 P .RC ˇ ; V / is equivalent to Tj a 2 P .RC ˇ ; V / for every j 2 I . (iv) Given two Fréchet spaces V0 and V1 and a continuous bilinear map h; i W V0  V1 ! V for a Fréchet space V , then ak 2 P .RC  ˇ ; Vk /; k D 0; 1, implies ha0 ; a1 i 2 P .RC  ˇ ; V /, more precisely, .a0 ; a1 / ! ha0 ; a1 i induces continuous maps 0

0

P  ;ı .RC  ˇ ; V0 /  P 

00 ;ı 00

.RC  ˇ ; V1 / ! P ;ı .RC  ˇ ; V /

for every . 0 ; ı 0 /, . 00 ; ı 00 /, with some resulting sequences .; ı/. (v) If W is a closed subspace of V , then a 2 P .RC  ˇ ; V / implies Œa 2 P .RC  ˇ ; V =W /, where Œa denotes the image of a under the pointwise quotient map V ! V =W (cf. also the property (ii)) . Analogous statements are true for holomorphic amplitude functions.

348

6 Operators on manifolds with conical singularities and boundary

Let T .RC  R/ denote the set of all functions '.r; t / such that '.e tQ; t / 2 S.R2tQ;t /. In a similar manner we define T .RC  ˇ / for any real ˇ. Definition 6.1.74. A function " .r; w/ W .0; 1  RC  ˇ ! C is called regularising, if (i) " 2 T .RC  ˇ / for every " 2 .0; 1 ; (ii) supfj.r@r /k @lw " .r; w/j W 0 < "  1, .r; w/ 2 RC  ˇ g < 1 for every k; l 2 N; ( 1 for k C l D 0; k l pointwise on RC  ˇ as " ! 0. (iii) .r@r / @w " .r; w/ ! 0 for k C l 6D 0 Example 6.1.75. Let .r; w/ 2 T .RC  0 / with .1; 0/ D 1, and set " .r; i%/ D .r " ; i"%/. Then " is regularising in the sense of Definition 6.1.74. Definition 6.1.76. A function " .r; w/ W .0; 1  RC  C ! C is holomorphically regularising, if (i) ."; r; i%/ ! " .r; ˇ C i%/ is regularising in the sense of Definition 6.1.74 for every ˇ 2 R; (ii) " .r; w/ is an entire function in w 2 C, and " .r; ˇ C i%/ 2 S.R% /, uniformly in compact ˇ-intervals; (iii) for every " 2 .0; 1 there is a compact set K"  RC such that " .r; w/ D 0 when r 62 K" . Example 6.1.77. Let ' 2 C01 .RC / and '.1/.D .M '/.0// D 1 with the Mellin transform M . Then " .r; w/ D '.r " /M '."w/ is holomorphically regularising in the sense of Definition 6.1.76. Theorem 6.1.78. Let h.r; w/ 2 P .RC  0 ; V /, and let " be regularising in the sense of Definition 6.1.74. Then “ 1 “ 1 ds ds s i% h.s; i%/ μ % WD lim s i% " .s; i%/h.s; i%/ μ % OsŒh D "!0 s s 0 0 exists in V and is independent of the choice of " . An analogous statement holds for h.r; w/ 2 P .RC  C; V / and a holomorphically regularising function " . In particular, both definitions of oscillatory integrals coincide on P .RC  0 ; V / \ P .RC  C; V /. Remark 6.1.79. The maps h ! OsŒh are continuous in the sense P ;ı .RC  0 ; V / ! V

and

P ;ı .RC  C; V / ! V ;

respectively, for every choice of sequences  D f g2N , ı D fı g2N .

349

6.1 Fuchs type operators and Mellin quantisation

Let T  .RC ; C 1 .X // denote the space of all functions u 2 C 1 .RC ; C 1 .X // such that 1 .S u/.t / WD e . 2 /t u.e t / 2 S.R t ; C 1 .X //: For every h.r; r 0 ; w/ 2 Cb1 .RC  RC ; L .X I  1  // we have a Mellin pseudo-diffe2 rential operator “ 1  . 1 Ci%/   0 r 2  0 1 0 dr opM .h/u.r/ D h r; r ; / μ%   C i% u.r r0 2 r0 0 which induces a continuous map  opM .h/ W T  .RC ; C 1 .X // ! T  .RC ; C 1 .X //:

(6.1.79)

As usual this is interpreted as an oscillatory integral with an amplitude function in P .RC  0 ; C 1 .X // for every fixed r > 0. The following observation is similar to Theorem 2.4.58. Remark 6.1.80. If h.r; r 0 ; w/ 2 Cb1 .RC  RC ; L .X I  1  //, then the function 2

a.s; i%/ WD ..r; w/ ! h.r; rs; w C i%// belongs to P .RC  0 ; Cb1 .RC ; L .X I  1  //. Thus the oscillatory integral 2

“

1

hL .r; w/ D

s i% h.r; rs; w C i%/ 0

ds μ% s

converges in Cb1 .RC ; L .X I  1  //, and it is a left symbol associated with h.r; r 0 ; w/ 2

  .h/ D opM .hL /. In an analogous manner we have a right symbol which satisfies opM hR .r 0 ; w/, cf. the corresponding expression in Theorem 2.4.58 (ii) with the property   .h/ D opM .hR /. opM

Proposition 6.1.81. Let h.r; r 0 ; w/ 2 Cb1 .RC  RC ; L .X I  1  //. Then (6.1.79) 2 extends to a continuous operator n

n

 .h/ W H s; C 2 .X ^ / ! H s;C 2 .X ^ / opM  for every s 2 R. Moreover, the map h ! opM .h/ induces a continuous operator n

n

Cb1 .RC  RC ; L .X I  1  // ! L.H s;C 2 .X ^ /; H s;C 2 .X ^ //: 2

Proof. A reduction of orders and conjugation with r  reduces the assertion to the 0 .h/ W L2 .X ^ / ! L2 .X ^ /. Since L2 .X ^ / D L2 .RC ; L2 .X // case  D 0 and opM the result then follows by a simple extension of the Calderón–Vaillancourt theorem (in the version of Mellin pseudo-differential operators) to operator-valued amplitude functions.

350

6 Operators on manifolds with conical singularities and boundary

In the following considerations we set q MO .X I Rq / WD L cl .X I C  R /:

(6.1.80)

Q w; / x C ; L .X I  1  Rq // and h.r; w; / WD Remark 6.1.82. Let h.r; Q 2 C 1 .R cl 2 ˇ Q w; r/. Then, if !.r/; !.r/ h.r; Q are arbitrary cut-off functions, for every ˇ;  2 R we have ˇ ˇ C .h/./r  !.r/ Q D !.r/r  opM .T  h/./!.r/; Q !.r/ opM

(6.1.81)

for .T  h/.r; w; / WD h.r; w C ; /. Remark 6.1.82 is formulated for later reference. Clearly the identity (6.1.81) has nothing to do with the presence of the covariable . Q The operators are interpreted in 1 ^ q ^ .X I R /, i.e., operating on C .X /. the sense of L cl 0 Q w; / x C ; M  .X I Rq // be a function which is Proposition 6.1.83. Let h.r; Q 2 C 1 .R O Q independent of r for large r. Then we have Q x C ; M  .X I Rq ///; a.s; w/ WD ..t; z; / ! h.st; w C z; s// 2 P .RC  C; C 1 .R O cf. Definition 6.1.72 (ii). Proof. The consideration will be performed in two steps. First we study the nonsmoothing effects in local terms; afterwards we look at the global situation for  D 1. Q x; w; ; / x C Rn CRq /. For the local situation we consider the case h.r; Q 2 Scl .R Q Since the calculations will not be affected by the variables .x; / 2 Rn  Rn we simply Q w; / x C CRq /. The function a.s; w/ ignore them and consider the case h.r; Q 2 Scl .R Q x C  C  Rq // ˝ y  A.C/ with A.C/ being the obviously belongs to C 1 .RC ; Scl .R x C  C  Rq / is Fréchet with Fréchet space of entire functions in w. The space Scl .R the system of semi-norms ˚

Q 0 C i ; /j ; Q D sup h ; ij˛0 jCl 0  j@k 0 @l 0 @˛0 h.t;  .h/ (6.1.82) t z   2 N, where the supremum is taken over all t  ; j0 j  ; . ; / 2 R1Cq , and k 0 C l 0  , together with the semi-norms, coming from the homogeneous components, cf. Remark 2.1.2 (ii). Let us concentrate on (6.1.82) (which corresponds to the nonclassical case), the consideration for the semi-norms associated with the homogeneous components is of a similar structure and left to the reader. We have to show the existence of numbers  ; ı such that for every k; l; N 2 N supf ..s@s /k @lw a.s; 1 C i /h i g.s/ı W j1 j  N; .s; / 2 RC  Rg < 1I (6.1.83) here g.s/ WD e hlog si . Writing .@ /˛ D .1 @1 /˛1 : : : .q @q /˛q we have X Q .s@s /k a.s; w/ D w C i ; s/: cm˛ Œ..t @ t /m .@ /˛ h .st; mCj˛jDk

6.1 Fuchs type operators and Mellin quantisation

351

x C  C  Rq / ! S  .R x C  C  Rq / defined by hQ ! .t @ t /m .@ /˛ hQ The map S  .R k ˛Q Q so in (6.1.83) is continuous, and .t @ t / .@ / h satisfies analogous assumptions as h; it suffices to consider the case k D 0. The estimate (6.1.83) is verified if we show that ˇ k 0 lCl 0 ˛0 ˇ 0 0 ı Q Q ˇ@ @ @ Œh.st; 0 C 1 C i C i ; s/ ˇ  .h/g.s/ h i h ; ij˛ jl (6.1.84) t z uniformly in t  , j0 j  , j1 j  N , k 0 C l 0 C j˛ 0 j  , . ; / 2 R1Cq , .s; / 2 x C  C  Rq /, only depending on l; , and N . RC  R, with a semi-norm .  / on S  .R  Using the elementary inequality h ; si  maxfs 1 ; sgjj h ; ijj , the left-hand side of (6.1.84) can be estimated by ˇ 0 0 ˇ 0 0 0 Q s k Cj˛ j ˇ.@k @lCl @˛ h/.st; 0 C 1 C i C i ; s/ˇ t

z



Q k 0 Cj˛0 j h C ; sill 0 j˛0 j  .h/s Q k 0 Cj˛0 j h ; sil 0 j˛0 j h ijj˛0 jl 0 j  c.h/s 0 0 0 0 k 0 Cj˛ 0 jCjl 0 j˛ 0 jj Q  c.h/g.s/ h ijl j˛ jj h ; il j˛ j 0 0 ı Q  c.h/g.s/ h i h ; il j˛ j :

Here we have set  D maxfjj j W 1  j  g, ı D Cmaxfjj j W 1  j  g, moreover, Q D supfj@k 0 @lCl 0 @˛0 h.t; Q  C i ; /jh ; ilCl 0 Cj˛0 j g .h/ t z  x C , j j  CN , k 0 Cl 0 Cj˛ 0 j  , and . ; / 2 R1Cq . with the supremum over all t 2 R This completes the proof for the non-smoothing part. For the smoothing part let hQ 2 x C ; M 1 .X I Rq //. Moreover, if H is a Hilbert space, by S  .R x C ˇ Rq ; H / C 1 .R O 1 x q we denote the space of all f .r; w; / 2 C .RC  ˇ  R ; H / such that supfh%; ij˛jCl k@kr @l% @˛ f .r; ˇ C i%; /kH W .%; / 2 R1Cq ; r 2 K; k C l C j˛j  N g < 1 x C . In an analogous manner we for every N 2 N and every compact set K  R q  x  x generalise the space S .RC  C  R / to S .RC  C  Rq ; H / which is a Fréchet space. We have x C ; M 1 .X I Rq // D lim S j .R x C  C  Rq ; H j .X  X // C 1 .R O  j 2N

with H j .X  X / being the Sobolev space of smoothness j on X  X . Analogously as x C C Rq ; H j .X  in the first part of the proof we have a.s; w/ 2 P .RC C; S j .R 1 x X/// for every j 2 N. This implies a.s; w/ 2 P .RC  C; C .RC ; MO1 .X I Rq // by Proposition 6.1.73 (iii) (for C instead of ˇ ).

352

6 Operators on manifolds with conical singularities and boundary

x C ; M  .X I Rq //, we write Finally, to consider the general case, i.e., hQ 2 C 1 .R O Q w; / D h.r;

N X

'j .1  j1 / opx .hQj /.r; w; /

j

C f .r; w; /

j D1

x C  Rn  C  RnCq / and f .r; w; / where hQj .r; x; w; ; / is contained in Scl .R 1 1 x q in C .RC ; MO .X I R //. Here 'j ; j ; j are as in (6.1.24). Thus, with obvious meaning of notation, a.s; w/ D

N X

'j .1  j1 / opx .aj /.s; w/

j

C b.s; w/:

j D1

By virtue of the first steps of the proof and Proposition 6.1.73, a.s; w/ is an amplitude function as desired. Q w; / x C ; M  .X I Rq // be independent of r for Proposition 6.1.84. Let h.r; Q 2 C 1 .R O Q Q w; r/. Then we have large r, and let h.r; w; / WD h.r; “ ds 0 Q hR .r ; w; / WD s i h.sr 0 ; w C i ; s/ μ (6.1.85) s x C ; M  .X I Rq //, and for hR .r 0 ; w; / WD hQ R .r 0 ; w; r 0 / we obtain 2 C 1 .R O   .h/./ D opM .hR /./ opM

(6.1.86)

for every  2 R. Proof. It follows from Proposition 6.1.83 that the oscillatory integral (6.1.85) exists in x C ; M  .X I Rq //. Let us now first assume that hQ has compact support in r 2 R x C. C 1 .R O .X I  //. By Remark 6.1.80 we obtain Then h.r; w; / 2 Cb1 .RC ; L 1 cl  hR .r 0 ; w; / D

“

2

s i h.sr 0 ; w C i ; /

ds μ s

  in the space Cb1 .RC ; L cl .X I  1  // and opM .hR /./ D opM .h/./ for every . If 2

0 we interpret both oscillatory integrals as such in L cl .X / it follows that hR .r ; w; / D  0 0 0 q hQ R .r ; w; r / in Lcl .X / for every r > 0, w 2 C, and  2 R . By virtue of the continuity this is then also true for r 0 D 0. Q w; / D .1  !.r//hQ 1 .w; / with hQ 1 .w; / 2 M  .X I Rq / We now assume h.r; O and some cut-off function !. The holomorphy allows us to write   .h/./ D opM .r N T N h/./r N : opM

For N  maxf; 0g we have hN .r; w; / WD r N T N h.r; w; / 2 Cb1 .RC ; L cl .X I  1  // 2

353

6.1 Fuchs type operators and Mellin quantisation

for each  (recall that hQ is supported away from 0). Then, similarly as in the first part of the proof, it follows that “ ds 0 0 N hN .r ; w; / WD .r / s i hN .sr 0 ; w C i ; / μ R s (converging in Cb1 .RC ; L cl .X I  1  //) satisfies 2

 opM .r N T N h/./r N

 D opM .hN R /./:

For fixed r 0 > 0, w, and  we obtain “ N 0 0 1 Q 0 ; w C N C i ; s/ ds μ hR .r ; w; .r / / D s i s N h.sr s  1 x q with convergence in L cl .X /. But this integral even converges in C .RC ; MO .X I R //   0 and is equal to hQ R .r ; w; /. This yields opM .hR /./ D opM .h/./. Finally, a general hQ can be decomposed into summands as in the first two steps of the proof.

The following two propositions can be proved in a similar manner. x C ; M j .X I Rq //, j D 1; 2, be inProposition 6.1.85. Let hQj .r; w; / Q 2 C 1 .R O dependent of r for large r, set hj .r; w; / WD hQj .r; w; r/, j D 1; 2, and define Q w; r/ where h.r; w; / WD h.r; “ ds Q h.r; w; / D s i hQ 0 .r; w C i ; /hQ 1 .sr; w; s/ μ (6.1.87) s x C ; M 0 C1 .X I Rq //). Then for every  2 R we have (which is convergent in C 1 .R O    .h0 /./ opM .h1 /./ D opM .h/./: opM

(6.1.88)

Actually, the Mellin symbol h of the composition (6.1.88) is given in a similar way as in Theorem 2.4.58. The extra information is the convergence of (6.1.87) in x C ; M 0 C1 .X I Rq //. C 1 .R O Q w; / x C ; M  .X I Rq // be independent of r for Proposition 6.1.86. Let h.r; Q 2 C 1 .R O large r, and define h .r; w; / D hQ  .r; w; r/ where “ ds hQ  .r; w; / WD s i hQ ./ .sr; n C 1  w x C i ; s/ μ (6.1.89) s x C ; M  .X I Rq //), with ‘./’ denoting the pointwise (which is convergent in C 1 .R O Q w; r/ we have formal adjoint in L .X /. Then for every  2 R and h.r; w; / WD h.r; cl  n  .h/./ D opM .h /./ opM

(the operator on the left-hand side is the formal adjoint with respect to the K pairing).

(6.1.90) 0;0

.X ^ /-

354

6 Operators on manifolds with conical singularities and boundary

For the cone and algebras below we need similar results for the space ;d MO .X I Rq / D B ;d .X I C  Rq /;

consisting of all h.w; / 2 A.C; B ;d .X I Rq // such that, for every ˇ 2 R, h.ˇ C 1 i%; / 2 B ;d .XI R1Cq %; / uniformly in compact ˇ-intervals; here X is a compact C manifold with boundary. Recall that in the formal considerations we assume the bundles on X and @X to be trivial and of fibre dimension 1. The straightforward extension to the case of arbitrary bundles is left to the reader. Q w; / x C ; M;d .X I Rq // be a function which is Proposition 6.1.87. Let h.r; Q 2 C 1 .R O Q independent of r for large r. Then we have ;d Q a.s; w/ WD ..t; z; / ! h.st; w C z; s// 2 P .RC  C; C 1 .RC ; MO .X I Rq //:

Proof. Let us consider, for simplicity, operator families of the type of upper left corners; the other entries of trace and potential type can be treated in a similar manner as the Green operators in the upper left corners, while the right lower corner corresponds to Q w; / the case of Proposition 6.1.83. The function h.r; Q is given as a sum Q w; / h.r; Q D fQ.r; w; / Q C g.r; Q w; / Q x C ; L .2X I C  Rq /tr /, cf., where fQ.r; w; / Q 2 r C Fz .r; w; /e Q 2 C 1 .R Q C , Fz .r; w; / cl Q x C ; B ;d .X I C  Rq //. The analogously, (3.3.11) and (3.1.16), and g.r; Q w; / Q 2 C 1 .R G Q assertion is proved when we show that .t; z; / ! Fz .rt; w C z; r/

(6.1.91)

x C ; L .2X I C  Rq /tr // and belongs to P .RC  C; C 1 .R cl .t; z; / ! g.rt; Q w C z; r/

(6.1.92)

x C ; B ;d .X I C C; C 1 .R G

to P .RC   Rq //. The desired property of (6.1.91) can be obtained in an analogous manner as Proposition 6.1.83, since the subspace of pseudodifferential operators on 2X with the transmission property at @X is closed in the general space. For (6.1.92) we first employ a generalisation of Remark 3.2.9 to the space ;d BG .XI C  Rq /. This reduces the assertion to the case of Green operator families of type 0 and potential operator families which can be treated separately. As noted before the potential operator families are of analogous structure as the Green ones of type zero (in fact, even simpler). Thus it remains to consider the functions (6.1.92) for d D 0. According to the scheme of the proof of Proposition 6.1.83 there is again a non-smoothing part which refers to the local representations in terms of Green symbols of the space x C  Rn1  C  Rq I L2 .RC /; S.R x C // Scl .R (6.1.93) such that the pointwise formal adjoints are of analogous structure, cf., analogously, Definition 3.2.3 and Remark 6.1.27. Although the symbols (6.1.93) are based on

6.2 The cone algebra

355

‘twisted’ symbolic estimates, (see, analogously, Definition 2.2.3) the conclusions of the first part of the proof of Proposition 6.1.83 remain valid up to small and evident modifications. Finally, the considerations corresponding to the second part of the proof are of a similar structure. In the present case instead of (6.1.86) we have x C ; B 1;0 .X I C  Rq // D lim S j .R x C  C  Rq ; H j .X / ˝H H j .X //; C 1 .R G  j 2N

here concerning the compact C 1 manifold with boundary. Q w; / x C ; M;d .X I Rq // be independent of r Proposition 6.1.88. Let h.r; Q 2 C 1 .R O Q w; r/. Then (6.1.85) belongs to the space for large r, and let h.r; w; / D h.r; x C ; M;d .X I Rq // and hR .r 0 ; w; / D hQ R .r 0 ; w; r 0 / satisfies the identity C 1 .R O (6.1.86). In a similar manner we can show the following results: Proposition 6.1.89. Let x C ; M;d .X I Rq /; hQ 1 .r; w; / Q 2 C 1 .R O

x C ; M;e .X I Rq // hQ 2 .r; w; / Q 2 C 1 .R O

be independent of r for large r, set hj .r; w; / WD hQj .r; w; r/; j D 1; 2, and deQ w; r/, where h.r; Q w; / fine h.r; w; / WD h.r; Q is given by the expression (6.1.89) C;h 1 x .X I Rq /// for h D max. C d; e/ (cf. also (which is convergent in C .RC ; MO Theorem 3.2.22). Then for every  2 R we have the relation (6.1.90). The proof employs the fact (which is an analogue of Theorem 3.3.15) that the ;d ;e .X I Rq / and MO .X I Rq /, pointwise composition of C 1 functions with values in MO C;h .X I Rq /. respectively, is a C 1 function with values in MO Q w; / 2 C 1 .R x C ; M0;0 .X I Rq // be independent of r for Proposition 6.1.90. Let h.r; O large r, and define h .r; w; / WD hQ  .r; w; r/ with hQ  .r; w; / given by the expression x C ; M0;0 .X I Rq ///, with  denoting the point(6.1.89) (which is convergent in C 1 .R O Q w; / Q wise formal adjoint in B 0;0 .X /. Then for every  2 R and h.r; w; / WD h.r; we have the formula (6.1.90). x C ; M;d .X // and '; 'Q 2 C 1 .R x C / with supp '\ Remark 6.1.91. Let h.r; w/ 2 C 1 .R 0 O  n 2

supp 'Q D ;. Then the operator 'r  opM

.h/'Q belongs to CGd .X ^ ; .;   //.

6.2 The cone algebra By cone algebra we understand a pseudo-differential calculus on a manifold B with conical singularities that is able to express parametrices of elliptic differential operators of Fuchs type. We consider here the case that B has conical singularities and boundary, i.e., the operators

356

6 Operators on manifolds with conical singularities and boundary

represent boundary value problems. The calculus is interesting on its own right. For the present exposition it prepares material for the conormal symbolic structure of corner boundary value problems in Chapter 8 as well for the edge symbolic structure of edge boundary value problems in Chapter 7. The latter aspect concerns the case of an infinite (stretched) cone X ^ for a compact C 1 manifold X with boundary. Further details of the cone calculus of boundary value problems may be found in [161], [174], [175], [89], [90].

6.2.1 Operators on a compact manifold with conical singularities and boundary In the following we consider a compact manifold B with conical singularities (for simplicity, with one conical point) and boundary D. Let B and D denote the corresponding stretched manifolds. We now define spaces of operators which constitute the so-called cone algebra. Definition 6.2.1. Let  2 Z; d 2 N; g WD .;   /. Then C ;d .B; g/ is defined to be the space of all operators of the form  n 2

A WD diag.!; ! 0 /r  opM

.h C f / diag.!; Q !Q 0 / C diag.1  !; 1  ! 0 /Areg diag.1  !; QQ 1  !QQ 0 / C G ;

with the following ingredients: x C ; M;d .X //; f .w/ 2 M1;d .X I  nC1 /; (i) h.r; w/ 2 C 1 .R O  2

(ii) Areg 2 B ;d .Breg /; (iii) G 2 CGd .B; g/. Moreover, !, !, Q !QQ are cut-off functions on B which are equal to 1 near Bsing , supported QQ prime denotes in a neighbourhood of Bsing and !Q D 1 on supp !; ! D 1 on supp !; the restriction to D. In the following, in order to avoid unnecessary technicalities, we denote the operator of multiplication by diag.!; ! 0 / simply by !; similarly we proceed with 1  ! for any cut-off function ! on B. Theorem 6.2.2. Every A 2 C ;d .B; g/ for g D .;   / induces a continuous operator 1

1

1

1

A W H s; .B/ ˚ H s 2 ; 2 .D/ ! H s; .B/ ˚ H s 2 ; 2 .D/ for every s;  2 R; s > d  12 .

(6.2.1)

6.2 The cone algebra

357

Proof. The continuity of G belongs to Definition 6.1.56 (i). The continuity of Areg in combination with the factors that localise outside Bsing is stated in Theorem 3.2.14. The Mellin operator in A has the form !  n  n !r  opM 2 .h11 C f11 /!Q !r  opM 2 .h12 C f12 /!Q : (6.2.2)  n  n !r  opM 2 .h21 C f21 /!Q !r  opM 2 .h22 C f22 /!Q The desired continuity of the right lower corner is contained in Theorem 2.4.53. x C; The symbol m11 WD h11 C f11 in the upper left corner belongs to C 1 .R ;d B .XI  nC1  //. It has the form of a finite sum of expressions, where the typi2

cal non-smoothing terms near the boundary have the form  ' 0 .1 / Opx 0 .a/.w/ 0 Q with C 1 functions ; Q in a collar neighbourhood Š @X  Œ0; 1/ of the boundary, vanishing for xn > 1  " for some " > 0 and being identically 1 in a neighbourhood of xn D 0, functions ' 0 , 0 2 C 1 .@X /, supported in a coordinate neighbourhood U  @,  W U ! Rn1 a chart, and n1 a.x 0 ; 0 ; w/ 2 S  .Rn1   nC1  I H s .RC /; H s .RC //; x 0  R 0 2

(6.2.3)

cf. Theorem 3.1.15 and Definition 3.2.3, both applied in the version with the parameter w 2  nC1  . Another non-smoothing summand of m11 has the form 2

.1   /p.w/.1  QQ / for a family p.w/ 2 L /; the meaning of ; ; Q QQ is as in the expression cl .2X I  nC1 2  (3.2.18), and a third summand is a Schwartz function in w 2  nC1  with values in 2

B 1;d .X/. n

 The continuity of opM 2 ..1   /p.1  QQ // in the desired sense is again a case of Theorem 2.4.53. The smoothing term is left to the reader as an exercise. It remains to  n consider opM 2 .a/ for the symbol (6.2.3). However this continuity is a special case of Theorem 2.4.59 (ii) when we take into account Example 2.4.63. The other entries of (6.2.2) can be treated in a similar manner.

For every A 2 C ;d .B; g/ we have a principal symbol  .A/ D . .A/; @ .A/; c .A//;

(6.2.4)

consisting of interior, boundary, and conormal symbol, defined as follows. From C ;d .B; g/  B ;d .Breg / we have . .A/; @ .A// in the sense of the principal symbolic structure of B ;d .Breg /. We now employ the Mellin operator structure near Bsing in the splitting of variables .r; x/ 2 RC  X and .r; x 0 / 2 RC  @X , respectively. Then the interior symbol (locally on X in the variables x with the covariables ) can be written as  .A/.r; x; %; / D r  Q .A/.r; x; r%; /

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6 Operators on manifolds with conical singularities and boundary

with a function Q .A/.r; x; %; Q / which is smooth up to r D 0. Similarly, the boundary symbol (locally on @X in the variables x 0 with the covariables 0 ) has the form @ .A/.r; x 0 ; %; 0 / D r  Q @ .A/.r; x 0 ; r%; 0 / Q 0 / which is smooth up to r D 0. Finally, the with an operator function Q @ .A/.r; x 0 ; %; conormal symbol is defined as c .A/.w/ D h.0; w/ C f .w/

(6.2.5)

and represents a family of continuous operators 1

1

c .A/.w/ W H s .int X / ˚ H s 2 .@X / ! H s .int X / ˚ H s 2 .@X / for all w 2  nC1  , s > max.; d/  12 . 2

Remark 6.2.3. Let A 2 C ;d .B; g/ and  .A/ D 0. Then the operator (6.2.1) is compact for every s > d  12 . d Remark 6.2.4. We have C ;d .B; g/ \ B 1;d .Breg / D CM CG .B; g/.

Theorem 6.2.5. A 2 C ;d .B; g/; B 2 C ;e .B; h/ for g WD .  ;   . C //, h WD .;   / implies AB 2 C C;h .B; g ı h/ for g ı h D .;   . C //, h D max. C d; e/, and we have  .AB/ D  .A/ .B/ with the componentwise composition, where c .AB/.w/ D c .A/.w C /c .B/.w/: If A or B belong to the subspace with subscript M C G .G/, then the same is true of the composition. Proof. For simplicity, assume that the operators A and B are of the form of an upper left corner; the compositions between the other entries in the 2  2 block matrices can be characterised in a similar manner. Write the given operators in the form  n 2

QQ C G1 ; .h1 C f1 /!Q C .1  !/Areg .1  !/

2  n 2

QQ C G2 ; .h2 C f2 /!Q C .1  !/Breg .1  !/

A D r 1 ! opM1 B D r 2 ! opM

with 1 WD , 2 WD , 1 WD   , 2 WD  , where the summands have the meaning as in Definition 6.2.1, with obvious modifications of notation. First it is easy to verify that when one of the factors is a Green operator, the same is true of the composition. Setting for abbreviation  n 2

Hj WD r j ! opMj

.hj /!; Q

 n 2

Fj WD r j ! opMj

.fj /!; Q

j D 1; 2;

we obtain QQ AB D .H1 C F1 /.H2 C F2 / C .H1 C F1 /.1  !/Breg .1  !/ QQ QQ C G C .1  !/Areg .1  !/.H 2 C F2 / C .1  !/P .1  !/

359

6.2 The cone algebra

C;h QQ for an operator G 2 CGC;h .B; gıh/ and P WD Areg .1!/.1!/B .Breg /. reg 2 B To characterise the compositions

QQ F1 .1  !/Breg .1  !/;

QQ 2 .1  !/Areg .1  !/F

(6.2.6)

we observe that the change of the cut-off functions in Fj only contributes Green remainders, cf. Proposition 6.1.66. Therefore, with an arbitrary cut-off function !0 such that !0 .1  !/ D 0 we obtain n

QQ QQ D r 1 ! op1  2 .f1 /.!Q  !0 /.1  !/Breg .1  !/ F1 .1  !/Breg .1  !/ M which is an element of CGC;h .B; g ı h/. The second operator in (6.2.6) is Green for an analogous reason. Next we consider the operators QQ H1 .1  !/Breg .1  !/;

QQ 2 : .1  !/Areg .1  !/H

(6.2.7)

Here we write with !0 as before QQ H1 .1  !/Breg .1  !/  n 2

D r 1 ! opM1

QQ .h1 /.!Q  !0 /.1  !/Breg .1  !/

1  n 2

D r 1 !1 opM

(6.2.8)

QQ .h1 /.!Q  !0 /.1  !/Breg .1  !/  n 2

C r 1 .!  !1 / opM1

QQ .h1 /.!Q  !0 /.1  !/Breg .1  !/;

where !1 is another cut-off function which we choose in such a way that !Q !0 D 0 on supp !1 . Then, according to Remark 6.1.91, the first summand on the right of (6.2.8) is a Green operator, while the second one is a composition purely within B C;h .Breg /, because both factors are supported away of Bsing . The second operator in (6.2.7) can be treated in an analogous manner. It remains to study the composition .H1 C F1 /.H2 C F2 / D H1 H2 C H1 F2 C h F1 H2 C F1 F2 . From Theorem 6.1.71 we have F1 F2 2 CM CG .B; g ı h/. Using Taylor’s formula we can write h1 .r; w/ D h1 .0; w/ C rh01 .r; w/ for a function x C ; M1 ;d .X //. This gives us h01 .r; w/ 2 C 1 .R O  n 2

H1 F2 D r 1 ! opM1

 n 2

D r  ! opM

 n 2

.h1 /!!r Q 2 opM2

 n 2

.h1 jrD0 /!!r Q  opM  n 2

C r C1 ! opM  n 2

D r  ! opM

.f2 /!Q .f2 /!Q

 n 2

.h01 /!!r Q  opM

.f2 /!Q

..T  .h1 jrD0 //f2 /!Q C D

where  n 2

D WD r  ! opM

C r C1 !

 n 2

.h1 jrD0 /.!! Q  1/r  opM

 n opM 2 .h01 /!!r Q 

.f2 /!Q

 n opM 2 .f2 /!: Q

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6 Operators on manifolds with conical singularities and boundary

h From h1 .0; w C /f2 .w/ 2 M 1;h .X I  nC1  / we get H1 F2 2 CM CG .BI g ı h/ 2 when we show that D is a Green operator. However, this is straightforward; the details are left to the reader (hint: in the boundaryless case one checks the mapping properties, of the operators (and the adjoints) using the continuity of Mellin operators which produce smoothness of order 1 in the image and improve the weights; in the case with boundary the work is a little longer, especially, with the types of operators). It remains to consider H1 H2 , but the desired property of the composition of Mellin operators with holomorphic amplitude functions is a direct consequence of Proposition 6.1.89. The arguments for the behaviour of symbols are easy as well; so we omit the details. From the computations it became clear that when one of the factors belongs to the M C G (or G) class, then also the composition.

Definition 6.2.6. An A 2 C ;d .B; g/ with g D .;   / is said to be elliptic if (i)  .A/ 6D 0 on T  .Breg / n 0, and Q .A/.r; x; %; Q / 6D 0 up to r D 0; (ii) @ .A/ is bijective for all points of T  .@Breg /n0, and Q @ .A/.r; x 0 ; %; Q 0 / is bijective up to r D 0; 1

1

(iii) c .A/.w/ W H s .int X / ˚ H s 2 .@X / ! H s .int X / ˚ H s 2 .@X / is bijective for all w 2  nC1  , for any s 2 R, s > max.; d/  12 . 2

Theorem 6.2.7. Let A 2 C ;d .B; g/ with g D .;   / be elliptic. Then A induces a Fredholm operator (6.2.1) for every s > max.; d/  12 . Moreover, A has C a parametrix P 2 C ;.d/ .B; g 1 / for g 1 D .  ; /, i.e., Gl WD I  P A and Gr WD I  AP belong to CGdl .B; g l / and CGdr .B; g r /, respectively, where dl D max.; d/, dr D .d  /C , and g l D .;  /, g r D .  ;   /. Proof. The operator A is elliptic as an element of B ;d .Breg / and has (by Remark 3.3.8) C a (properly supported) parametrix Preg 2 B ;.d/ .Breg /. We will construct a parametrix P in the sense of the theorem by modifying Pint close to Bsing , i.e., find elements C

x C ; M;.d/ .X //; l.w/ 2 M1;.d/C .X I  nC1 / k.r; w/ 2 C 1 .R O 2 ./ (6.2.9) such that  n 2 QQ P WD !r  opM .k C l/!Q C .1  !/Preg .1  !/ (6.2.10) x C; is as desired. Let us interpret the function h.r; w/ as an element of C 1 .R ;d B .XI ˇ //, ˇ 2 R. The ellipticity of A implies that h.r; w/ is parameter-dependent elliptic for every ˇ with Im w as parameter, for all 0  r < R for some R > 0. Without loss of generality in the following construction we assume that the supports of !; !; Q !QQ are contained in Œ0; R/. Let us consider for the moment an arbitrary k.r; w/ as in (6.2.9). Since the Mellin symbols are of interest only for r 2 Œ0; R  "/ for some small " > 0, we may assume

361

6.2 The cone algebra

that k.r; w/ does not depend on r for large r (for instance, vanishes there). The same assumption makes sense for h.r; w/. Let us consider the composition  n 2

r  opM

 n 2

.k/r  opM

.h/

(6.2.11)

which exists in the sense of operators H s; .X ^ / H s;  .X ^ / H s; .X ^ / ˚ ˚ ˚ ! ! : 1 1 1 1 1 s 1 ;  ^ s ; ^ s ; 2 2 ..@X / / H H 2 2 ..@X /^ / H 2 2 ..@X / / By virtue of the holomorphy of k.r; w/ in w we can commute r  trough the Mellin operator on the expense of a translation of the symbol by . By virtue of Proposition 6.1.89 it follows that (6.2.11) is equal to   n  n   n opM 2 T  k opM 2 .h/ D opM 2 .m/ ’ i  x C ; M0;dl .X //. Simiμ 2 C 1 .R for m.r; w/ D s .T k/.r; w C i /h.sr; w/ ds O s larly as Theorem 2.4.58 we have an asymptotic expansion m.r; w/ 

1  X 1 j  @ j @w .T k/.r; w/  r h.r; w/ jŠ @r

(6.2.12)

j D0

  x C ; M0;dl .X //. The term for j D 0 equals the product T  k .r; w/h.r; w/. in C 1 .R The relation (6.2.12) enables us to Leibniz invert h.r; w/ in Œ0; R/ modulo an ele1;dl ment in C 1 .Œ0; R/; MO .X //. More precisely, we can proceed as follows. We C choose an operator function p.r; w/ 2 C 1 .Œ0; R/; B ;.d/ .X I  nC1  // which 2 ˇ is a parameter-dependent parametrix of h.r; w/ˇ , cf. Remark 3.3.20. ApŒ0;R/ nC1 2



plying the kernel cut-off operator (6.1.52) for every fixed r 2 Œ0; R/ we obtain a ;.d/C g0 .r; w/ 2 C 1 .Œ0; R/; MO .X // which is a parameter-dependent parametrix of h.r; w/ for every r 2 Œ0; R/, cf. also Theorem 6.1.51. It follows that g0 .r; w/h.r; w/ D 1 C b.r; w/

(6.2.13)

1;dl C 1 .Œ0; R/; MO .X //

for some b 2 (1 denotes the identity Poperator). We now construct .T  k/.r; w/ as an asymptotic sum .T  k/.r; w/  j1D0 gj .r; w/ of eleC

j;.d/ ments gj .r; w/ 2 C 1 .Œ0; R/; MO .X //, j 2 N, which can be successively obtained by n X 1  o @ j gn .r; w/ D  h.r; w/ g0 .r; w/ (6.2.14) .@jw gl .r; w// r jŠ @r j ClDn l
for n D 1; 2 : : : . In this way we have h.r; w/ Mellin–Leibniz inverted in the interval Œ0; R/. The formula (6.2.14) gives us gn .0; w/ D 0 for all n > 0. Moreover, (6.2.13)

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6 Operators on manifolds with conical singularities and boundary

yields g0 .0; w/h.0; w/ D 1Cb.0; w/ which gives rise to g0 .0; w/fh.0; w/Cf .w/g D 1 C b1 .w/ for some b1 2 M 1;dl .X I  nC1  /. From Proposition 6.1.54 we know 2

that there is a b2 2 M1;dl .X I  nC1  / such that .1 C b2 .w//.1 C b1 .w// D 1. This 2 yields .1 C b2 .w//g0 .0; w/ D .h.0; w/ C f .w//1 (6.2.15) where C

.T  l/.w/ WD b2 .w/g0 .0; w/ D b2 .w/.T  k/.0; w/ 2 M1;.d/ .X I  nC1  /: 2

This gives us the smoothing Mellin symbol l.w/ as announced in the formula (6.2.9), and we can form the desired parametrix (6.2.10). The relation (6.2.15) is nothing other than c .P /.w C /c .A/.w/ D 1, cf. the formula (6.2.5). The Fredholm property follows from the existence of a two-sided parametrix and the fact that the left-over terms Gl and Gr are compact. Theorem 6.2.8. Let A 2 C ;d .B; g/ with g D .;   / be an elliptic operator such that the operator (6.2.1) is invertible for some s > max.; d/  12 . Then A is invertible C for all s > max.; d/  12 , and we have A1 2 C ;.d/ .B; g 1 /. Proof. The arguments are formally quite similar to those in the proof of Theorem 3.3.9. More details may be found in the proof of Theorem 6.2.23 in Section 6.2.2 below. Remark 6.2.9. (i) It can be proved that the ellipticity of an operator A 2 C ;d .B; g/ is necessary for the Fredholm property of (6.2.1). In particular, in Theorem 6.2.8 the condition of the ellipticity is superfluous. (ii) For the cone algebra we have a situation similarly as in Remark 2.1.25. If (6.2.1) is Fredholm, then there is an " > 0 such that 1

V WD ker As  H 1;C" .B/ ˚ H 1; 2 C" .D/ is a finite-dimensional subspace (here As denotes for the moment the operator (6.2.1) for fixed s > max.; d/  12 ), and there is a finite-dimensional subspace 1

W  H 1;C" .B/ ˚ H 1; 2 C" .D/

(6.2.16)

such that W \ im As D f0g

1

1

W C im As D H s; .B/ ˚ H s 2 ; 2 .D/ (6.2.17) for every s. More precisely, if W is a space as in (6.2.16) such that (6.2.17) holds for some s > max.; d/  12 , then (6.2.17) holds for all those s. Moreover, the parametrix P can be found in such a way that Gl and Gr are projections to V and W , respectively. and

In the following we set 1

1

H s; .B/ WD H s; .B/ ˚ H s 2 ; 2 .D/:

6.2 The cone algebra

363

Definition 6.2.10. Let C;d .B; gI j ; jC / for  2 Z, d 2 N, g D .;  /, j˙ 2 N, denote the space of all operator block matrices 

A K A WD T Q

 H s; .B/ H s; .B/ ˚ ˚ W ! j C C jC

(6.2.18)

for arbitrary A 2 C ;d .B; g/ and operators K; T and Q consisting of corresponding entries of an operator in CdG .B; gI j ; jC /, cf. Definition 6.1.56. We set  .A/ WD  .A/, see the formula (6.2.4). Remark 6.2.11. For the operator spaces of Definition 6.2.10 we have an analogue of Theorem 6.2.5 (in the composition we assume that rows and columns in the middle fit together). Theorem 6.2.12. Let A 2 C;d .B; gI j ; jC / be given in the form (6.2.18) for an elliptic operator A 2 C ;d .B; g/, and assume that (6.2.18) is invertible for some s > max.; d/  12 . Then (6.2.18) is invertible for all those s, and we have A1 2 C C;.d/ .B; g 1 I jC ; j /. Observe that Theorem 6.2.8 is a special case of Theorem 6.2.12 for j D jC D 0. We shall prove an analogue of Theorem 6.2.12 for X ^ instead of B, cf. Theorem 6.2.23 below. The technicalities for Theorem 6.2.12 are similar and left to the reader. Remark 6.2.13. For Theorem 6.2.12 it is again unnecessary to require the ellipticity of A when (6.2.18) is bijective.

6.2.2 Operators on an infinite cone with boundary In the following definition we employ the operator spaces of Remark 3.4.15 and (6.1.67). Definition 6.2.14. Let  2 Z, d 2 N, g D .;   /. Then C ;d .X ^ ; g/ is defined to be the space of all operators of the form  n 2

A D !r  opM

QQ C G .h C f /!Q C .1  !/Areg .1  !/

with the following ingredients: x C ; M;d .X //; f .w/ 2 M1;d .X I  nC1 /; (i) h.r; w/ 2 C 1 .R O  2

(ii) Areg 2 B ;dI0 .X ^ /; (iii) G 2 CGd .X ^ ; g/.

(6.2.19)

364

6 Operators on manifolds with conical singularities and boundary

Theorem 6.2.15. Every A 2 C ;d .X ^ ; g/ for g D .;   / induces a continuous operator A W K s; .X ^ / ! K s; .X ^ / (6.2.20) 1

1

for every s;  2 R, s > d  12 ; here K s; .X ^ / WD K s; .X ^ / ˚ K s 2 ; 2 ..@X /^ /. Proof. The continuity of G is contained in Definition 6.1.60 (i), cf. also the notation Q is a consequence of Theorem 3.4.14, to(6.1.67). The continuity of .1  !/Areg .1  !/ gether with the identity (3.4.18), see also the definition of K s; -spaces in Section 2.4.2, especially, Definition 2.4.5 (ii) and its analogue for the case of a manifold X with boundary. The continuity of the first summand on the right of (6.2.19) is contained in Theorem 6.2.2. The operators A 2 C ;d .X ^ ; g/ have a principal symbol  .A/ D . .A/; @ .A/; c .A/; E .A/; E0 .A//: The meaning of the first three components is the same as in (6.2.4) (even a little simpler, because, on the infinite stretched cone X ^ we have the splitting of variables .r; x/ everywhere). Also the functions Q .A/ and Q @ .A/ are as before. The novelty are the exit symbols E .A/ D .e .A/; 

;e .A//;

E0 .A/ D .e0 .A/; @;e0 .A//;

see Section 3.4. Remark 6.2.16. Let A 2 C ;d .X ^ ; g/ and  .A/ D 0. Then the operator (6.2.20) is compact for every s > d  12 . Theorem 6.2.17. There is an analogue of Theorem 6.2.5 for the operators on X ^ ; in the present case we have in addition the componentwise composition of the exit symbols. Proof. The operator (6.2.19) can equivalently be written in the form  n 2

A D !r  opM

.h1 C f1 /!Q C 'Areg 'Q C .1  !1 /Areg .1  !2 / C G1

for suitable functions '; 'Q 2 C01 .RC / and cut-off functions !1 , !2 , such that 1  !1 vanishes on supp !Q and 1  !2 on supp !, and for another Green operator G1 . Write the operator B to be composed with A in a similar form, namely,  n 2

B D !r  opM

.h2 C f2 /!Q C 'Breg 'Q C .1  !1 /Breg .1  !2 / C G2 :

Then the composition between the first two summands of A and B behave as in the proof of Theorem 6.2.5. If one of the factors is Green, it is easy to see that this is true for the composition. Compositions between first and third factors vanish because of the choice of !1 ; !2 , and compositions between second and third factors are as in the exit calculus of boundary value problems, cf. Remark 3.4.17.

6.2 The cone algebra

365

Definition 6.2.18. An A 2 C ;d .X ^ ; g/ with g D .;   / is called elliptic, if the conditions (i), (ii) of Definition 6.2.6 are satisfied (here, in (i) and (ii) with X ^ and .@X/^ in place of Breg and @Breg , respectively), and if the components of E .A/ and E0 .A/ satisfy the ellipticity conditions of the exit calculus of Section 3.4. Theorem 6.2.19. Let A 2 C ;d .X ^ ; g/ with g D .;   / be elliptic. Then A induces a Fredholm operator (6.2.20) for every s > max.; d/  12 . Moreover, A C has a parametrix P 2 C ;.d/ .X ^ ; g 1 / for g 1 D .  ; / in the sense that Gl WD I P A and Gr WD I AP belong to CGdl .C ^ ; g l / and CGdr .X ^ ; g r /, respectively with dl D max.; d/; dr D .d  /C , and g l D .;  /; g r D .  ;   /. Proof. Knowing a parametrix P we obtain the Fredholm property of A. The ellipticity of A contains the condition that A is elliptic in the sense of B ;d .X ^ /. By virtue of C Remark 3.3.8 there is a parametrix Pint 2 B ;.d/ .X /. Moreover, the ellipticity of A with respect to the symbolic components . .A/; E .A// and .@ .A/; E0 .A// yields a parametrix P1 in the exit calculus for r ! 1, cf. Theorem 3.4.19 (the shift of orders at the boundary is not essential). In the finite this is compatible with Pint C modulo B 1;.d/ .X /. For any cut-off functions ; ; Q QQ such that Q D 1 on supp  , Q  D 1 on supp Q , we set Preg WD  Pint Q C .1   /P1 .1  QQ / C

which belongs to B ;.d/ I0 .X ^ /. The ellipticity conditions close to r D 0 allow us to employ the parametrix construction of the proof of Theorem 6.2.7, cf. the first summand on the right-hand side of the formula (6.2.10). Thus we may altogether set  n 2

P WD !r  opM

QQ .k C l/!Q C .1  !/Preg .1  !/

which is a parametrix as desired (cf. also the formula (6.2.10)). Remark 6.2.20. For elliptic operators A 2 C ;d .X ^ ; g/ there is an analogue of Remark 6.2.9. Similarly as in Definition 6.2.10 there is the following useful generalisation of C ;d .X ^ ; g/. Definition 6.2.21. Let C;d .X ^ ; gI j ; jC / for  2 Z; d 2 N; g D .;   /, j˙ 2 N, denote the space of all operator block matrices 

A K A WD T Q

 K s; .X ^ / K s; .X ^ / ˚ ˚ ! W C j C jC

(6.2.21)

for arbitrary A 2 C ;d .X ^ ; g/ and operators K; T and Q consisting of corresponding entries of an operator in CdG .X ^ ; gI j ; jC /, cf. Definition 6.1.60.

366

6 Operators on manifolds with conical singularities and boundary

Remark 6.2.22. For the operator spaces of Definition 6.2.21 we have an immediate analogue of Remark 6.2.11. The following result will be essential below for the construction of parametrices of elliptic boundary value problems on a manifold with edges. Theorem 6.2.23. Let A 2 C;d .X ^ ; gI j ; jC / be given in the form (6.2.21) for an elliptic operator A 2 C ;d .X ^ ; g/, and assume that (6.2.21) is invertible for some s > max.; d/  12 . Then (6.2.21) is invertible for all those s, and we have C A1 2 C;.d/ .X ^ ; g 1 I jC ; j /. Remark 6.2.24. Similarly as Remark 6.2.13 in Theorem 6.2.23 it suffices to assume that (6.2.12) is invertible. In that case the upper left corner A is a Fredholm operator, and this entails the ellipticity of A 2 C ;d .X ^ ; g/. Proof of Theorem 6.2.23. First observe that when (6.2.21) is invertible the operator A W K s; .X ^ / ! K s; .X ^ / is Fredholm, and we have ind A D jC  j . The invertibility of A is equivalent to the following properties: s; ^   K .X / ˚ ! K s; .X ^ / A K W j C

is surjective, and



   T Q W ker A K ! C jC

(6.2.22)

(6.2.23)

is an isomorphism. The surjectivity of (6.2.22) is independent of s, since there is a finitedimensional subspace W  K 1; .X ^ / such that im ACW D K s; .X ^ / for every s, cf. Remark 6.2.20, and from the elliptic regularity of solutions of Au D Kv for u ˚ v 2 ker.A K/ it is easy to see that also (6.2.23) is independent of s. C According to Theorem 6.2.19 we find a parametrix P 2 C ;.d// .X ^ ; g 1 / of A. We then have ind P D j  jC . We can construct a block matrix of the form 

P P WD B

H R

 K s;  .X ^ / K s; .X ^ / ˚ ˚ W ! C mC C m

C

(6.2.24)

in C;.d / .X ^ ; g 1 I mC ; m / for suitable m˙ with m  mC D j  jC , such that (6.2.24) is an isomorphism. In fact, there is a finite-dimensional subspace W of K s; .X ^ / such that W C im P D K s; .X ^ /. Since C01 .X ^ / ˚ C01 ..@X /^ / is dense in K s; .X ^ /, by an approximation argument we can choose W as a subspace of C01 .X ^ / ˚ C01 ..@X /^ /. Then, for mC WD dim W , we can define H as the composition of any isomorphism C mC ! W with the embedding W ,! K s; .X ^ /. By elliptic regularity we have   1 ker P H  S C" .X ^ / ˚ S  2 C" ..@X /^ / ˚ C mC

6.2 The cone algebra

367

for some " > 0. This kernel is a subspace of finite dimension m . Then .B R/ can be defined as the composition ˇ ı …, where … W K s; .X ^ / ˚ C mC ! ker.P H / is a projection (for instance, induced by the scalar product of K 0;0 .X ^ / ˚ C mC ) and ˇ W ker.P H / ! C m any isomorphism. In this way we obtain that (6.2.24) belongs C to C;.d/ .X ^ ; g 1 I mC ; m /. From ind A D  ind P D m  mC we obtain either m˙ D j˙ C N or j˙ D m˙ C N for some N 2 N. Assume that, for instance, the first relation holds. Then, in order to invert (6.2.21) it suffices to invert the operator A ˚ idC N . Thus, after this modification, without loss of generality we may consider the case m˙ D j˙ . We now pass to the composition PA W K s; .X ^ / ˚ C j ! K s; .X ^ / ˚ C j which belongs to C0;dl .X ^ ; .;  /I j ; j /, cf. Remark 6.2.22, and is an isomorphism. By Theorem 6.2.19 we have P A D 1  Gl for some Gl 2 CGdl .X ^ ; .;  // (here and in the sequel identity operators in different spaces are simply denoted by 1). It follows that PA D 1  G for a G 2 CdGl .X ^ ; .;  /; j ; j /. By the corresponding analogue of Proposition 6.1.59 for operators G in the sense of Definition 6.1.60 (ii) (to be proved below) we have .1  G/1 D 1  D for a D 2 CdGl .X ^ ; .;  /I j ; j /. This gives us C A1 D .1  D/P 2 C;.d/ .X ^ ; g 1 I jC ; j / by Remark 6.2.22. Proof of Proposition 6.1.59. We show the variant for X ^ , the case B can be treated in a similar manner. Let us write   1G K 1G D T Q with G 2 CGd .X ^ ; .;  // in the 2  2 upper left corner. The j  j matrix Q is not necessarily invertible. However, since the invertible matrices form an open dense subset in the space of j  j matrices, there is an invertible R in a small neighbourhood of Q such that also   1G K (6.2.25) T R is Assume haveconstructed  Hinvertible.   that we   the inverse of (6.2.25) and denote it by L . Then also H L 1G K D 1 0 is invertible. This entails the invertibility D E T Q B S B S of E, and it follows that  1    1G K H L 1 0 D T Q B S E 1 D E 1 which belongs to CdG .X ^ ; .;  /I j; j /. Thus it remains to compute the inverse of (6.2.25). We have       1 KR1 1G K 1 0 1  G  KR1 T 0 D 0 1 T R R1 T R1 0 1 (6.2.26)

368

6 Operators on manifolds with conical singularities and boundary

which gives us  1 1G K T R  1 D R1 T

0 R1



.1  G  KR1 T /1 0

 1 KR1 : 0 1



0 1

Thus it suffices to find .1  C /1 where C WD G C KR1 T 2 CGd .X ^ ; .;  //. To this end we employ the fact that, according to Proposition 6.1.62, the operator C can P be written in the form C D C0 C jd1 K ı  j for some C0 2 CG0 .X ^ ; .;  // and j D0 t elements Kj of column matrix form Kj D .Kj Rj / where Kj are potential operators, and Rj Green operators on .@X /^ . The operator 1  C0 W K s; .X ^ / ! K s; .X ^ /, s > d  12 , is of Pindex zero, since C0 is compact. For m WD dim ker.1  C0 / there is then an operator m kD1 Dk Bk 1 1 1 s; s; ^ s 1 ; ^ s ; ^ 2 2 ..@X / / ! K with Bk W K .X / ! K 2 2 ..@X / /, D .X ^ /  k W K Pm  s; 0 ^ ^ belonging to CG .X ; .;  // such that 1  C0 C kD1 Dk Bk W K .X / ! K s; .X ^ / is invertible. In fact, it suffices to map the kernel ker.1  C0 / isomorphically to an arbitrary 1 subspace of S   2 ..@X /^ / of dimension m by the vector of operators t .B1 ; : : : ; Bm / and then to map that subspace isomorphically by operators .D1 ; : : : ; Dm / to a subspace of S  .X ^ / which is complementary to im.1  C0 /, using Remark 6.2.20. We now write m d1 m  X  X X Dk Bk C Kj  j C Dk Bk : 1  C D 1  C0 C Setting C1 WD C0 C 0; : : : ; d  1, we have

Pm kD1

kD1

j D0

kD1

Dk Bk , BmC1Cj WD  , DmC1Cj WD Kj for j D

1  C D 1  C1 C

j

mCd X

Dk Bk :

(6.2.27)

kD1

The operator C1 belongs to CG0 .X ^ ; .;  //, and 1  C1 is invertible. There is now a C2 2 CG0 .X ^ ; .;  // such that .1  C1 /1 D 1  C2 . From (6.2.27) we can pass to .1  C2 /.1  C / D 1 C

mCd X

zk Bk D

(6.2.28)

kD1

zk D .1  C2 /Dk . Then it suffices to invert (6.2.28) within our operator with D class. This is a purely algebraic task: Applying (6.2.26) for 1R D 1 and G D 0 K ; K WD 1 K , we obtain KMT D diag.1  KT ; 1/ if we set M WD 0 1 T 1  1 0 T WD T 1 . Similarly, we have T MK D diag.1; 1  T K/. This gives us T K1 diag.1  KT ; 1/T 1 K D diag.1; 1  T K/ and fdiag.1  KT ; 1/g1 D T 1 Kfdiag.1; 1  T K/g1 T K1 :

6.3 Boundary value problems in plane domains

369

This allows us to express .1  KT /1 by .1  T K/1 , together with compositions coming from the invertible matrices T and K. It follows that .1  KT /1 D 1 C K.1  T K/1 T :

(6.2.29)

z WD We now formally reinterpret T as B WD t .B1 ; : : : ; BmCd / and K as D z z .D1 ; : : : ; DmCd /. We then see that 1  KT is invertible if and only if so is 1  T K in the respective spaces. In our case .1  T K/1 is a simple matrix inversion; thus .1  KT /1 is obtained by (6.2.29) and hence belongs to our operator class. Since compositions are admitted within our structure, the proof is complete. Observe the similarity of the latter proof with that of Theorem 3.1.29.

6.3 Boundary value problems in plane domains We give a number of examples, namely, boundary value problems in infinite strips and infinite cones in the plane, for the Laplace operator under Dirichlet and Neumann conditions in various combinations.

6.3.1 The Dirichlet problem in a strip Let S˛ WD R  I˛ , I˛ WD Œ0; ˛ , ˛ > 0. We consider the Dirichlet boundary value problem for the Laplace operator in S˛ u.x/ D f .x/; u.x1 ; ˛˙ / D g˙ .x1 /;

x D .x1 ; x2 / 2 int S˛ ; x1 2 R;

(6.3.1) (6.3.2)

where ˛ WD 0, ˛C WD ˛, first for f 2 C01 .S˛ /, g˙ 2 C01 .R/. Applying the Fourier transform Fx1 ! to (6.3.1), (6.3.2) we obtain .@2x2  2 /u.; O x2 / D fO.; x2 /; u.; O ˛˙ / D gO ˙ ./:

x2 2 int I˛ ;

(6.3.3) (6.3.4)

If we find a solution of (6.3.3), (6.3.4) for every  2 R, the inverse Fourier transform gives us the solution of the original problem (6.3.1), (6.3.2). According to [88, Section 122] the parameter-dependent solution of (6.3.3), (6.3.4) is given by u.; O x2 / Z ˛ D PD .I x2 ; y/fO.; y/dy C K;D .I x2 ; 0/gO  ./ C KC;D .I x2 ; ˛/gO C ./; 0

(6.3.5)

370

6 Operators on manifolds with conical singularities and boundary

where cos i .jx2  yj  ˛/  cos i .x2 C y  ˛/ ; 2i  sin.i ˛/ K˙;D .I x2 ; ˛˙ / WD ˙@y PD .I x2 ; ˛˙ /:

PD .I x2 ; y/ WD

Indeed, first observe that PD .I ˛˙ ; y/ 0. Furthermore, K˙;D .I x2 ; ˛˙ / D ˙

sin i .x2  ˛ / sin.i ˛/

and @2x2 K˙;D .I x2 ; ˛˙ / D ˙

2 sin i .x2  ˛ / sin.i ˛/

entail K˙;D .I ˛ ; ˛˙ / D 0; K˙;D .I ˛˙ ; ˛˙ / D 1 and .@2x2 2 /K˙;D .I x2 ; ˛˙ / D 0; respectively. Finally, using Z ˛ cos i .x2 C y  ˛/ O 2 2 .@x2   / f .; y/dy D 0; 2i  sin.i ˛/ 0 Z x2 cos i .x2  y  ˛/ O .@2x2  2 / f .; y/dy 2i  sin.i ˛/ 0 cos i ˛ 1 D @x2 fO.; x2 / C fO.; x2 /; 2i  sin.i ˛/ 2 Z ˛ cos i .y  x  ˛/ 2 .@2x2  2 / fO.; y/dy 2i  sin.i ˛/ x2 1 cos i ˛ @x fO.; x2 / C fO.; x2 /; D 2i  sin.i ˛/ 2 2 Z

we obtain .@2x2

 / 2

˛

PD .I x2 ; y/fO.; y/dy D fO.; x2 /:

0

Then the solution of (6.3.1), (6.3.2) has the form Z ˛ Z 1 1 ix1 u.x/ D e PD .I x2 ; y/fO.; y/dy 2 1 0



C K;D .I x2 ; 0/gO  ./ C KC;D .I x2 ; ˛/gO C ./ d : (6.3.6) Let us verify that for s 2 N, s  2, the following estimate holds:  kukH s .int S˛ /  c kf kH s2 .int S˛ / C kg k s 1 C kgC k s 1 H

2 .R/

H

2 .R/



(6.3.7)

371

6.3 Boundary value problems in plane domains

for a constant c > 0 independent of u, where kukH s .int S˛ / WD

n X Z j Cks

and kvk

H

s 1 2

.@S˛ /

int S˛

o1 ˇ j k ˇ2 ˇ@ @ uˇ dx1 dx2 2 x1 x2

WD inffkukH s .int S˛ / W u D v on @S˛ g.

In fact, first from (6.3.5), using the Cauchy–Schwarz inequality, we have Z ˛ ju.; O x2 /j2 dx2 0  Z ˛ jfO.; x2 /j2 dx2 C jj1 .jgO  ./j2 C jgO C ./j2 / ;  c jj4 0 Z ˛ j@x2 u.; O x2 /j2 dx2 0  Z ˛ 2 2 2 2 O  c jj jf .; x2 /j dx2 C jj.jgO  ./j C jgO C ./j / : 0

On the other hand for k  2 from (6.3.3) we obtain O @kx2 u.; O x2 / D 2 @k2 O x2 / C @k2 x2 u.; x2 f .; x2 / which gives us Z ˛ j@kx2 u.; O x2 /j2 dx2 0  Z ˛ Z 4 k2 2  2 jj j@x2 u.; O x2 /j dx2 C 0

˛ 0

2 O j@k2 x2 f .; x2 /j dx2

:

Summing up, we get s X

Z jj

kD0

c

2.sk/ 0

X s2 kD0

˛

j@kx2 u.; O x2 /j2 dx2 Z

jj

2.s2k/ 0



˛

j@kx2 f .; x2 /j2 dx2

C jj

2s1

.jgO  ./j C jgO C ./j / : 2

2

(6.3.8) Note that (6.3.3), (6.3.4) is uniquely solvable for  D 0, and the latter inequality remains valid with 1 C jj instead of jj. Therefore, changing jj to 1 C jj in (6.3.8) and integrating the inequality with respect to x1 , the assertion follows from the relation jj Fx1 ! uj D jF .@jx1 u/j and Plancherel’s theorem. Because of the density of the sets C01 .S˛ / and C01 .R/ in H s .int S˛ / and H s .R/, respectively, we have the following result.

372

6 Operators on manifolds with conical singularities and boundary 1

Proposition 6.3.1. For every f 2 H s2 .int S˛ / and g˙ 2 H s 2 .R/, s 2 N, s  2, there exists a unique solution u 2 H s .int S˛ / of (6.3.1), (6.3.2) of the form (6.3.6), and the estimate (6.3.7) holds. The extension of PD with respect to  to the complex plane is a meromorphic function in C with simple poles at the points k D ik , k 2 Z n f0g. For every ˛ ;  6D k , there exists a unique solution of (6.3.3), (6.3.4). Let now ı 6D k , k 2 Z n f0g. Set ˛ uı .x/ D

1 2

Z

Z

1Ciı

e ix1 1Ciı

˛

PD .I x2 ; y/fO.; y/dy

0



(6.3.9)

C K;D .I x2 ; 0/gO  ./ C KC;D .I x2 ; ˛/gO C ./ d : We have  kuı kH s;ı .int S˛ /  cı kf kH s2;ı .int S˛ / C kg k

H

s 1 2 ;ı

.R/

C kgC k

H

 s 1 2 ;ı

.R/

(6.3.10) for the weighted Sobolev spaces H s;ı .R/, H s;ı .int S˛ /, defined by H s;ı .R/ WD e ıx1 H s .R/;

H s;ı .int S˛ / WD e ıx1 H s .int S˛ /:

An analogue of Proposition 6.3.1 is the following result. 1

Proposition 6.3.2. Let ı 6D k , k 2 Z n f0g, f 2 H s2;ı .int S˛ /, g˙ 2 H s 2 ;ı .R/, ˛ s 2 N, s  2. Then there exists a unique solution of (6.3.1), (6.3.2) of the form (6.3.9), and the estimate (6.3.10) holds. Next we establish a relation between solutions of different weighted spaces. For simplicity assume g˙ D 0, f 2 C01 .S˛ /. Let ı1 < ı2 , ıi 6D k , k 2 Znf0g; i D 1; 2. ˛ Using the Residue Theorem we have  Z ˛ X ix1 O Res e PD .I x2 ; y/f .; y/dy : (6.3.11) uı1 .x/ D uı2 .x/ C i 0

ı1
The residue of PD .I x2 ; y/ at k D

ik ˛

is equal to

lim .  k /PD .I x2 ; y/

!k

D lim

!k

D

cos i .x2 C y  ˛/  cos i .jx2  yj  ˛/ k/ 2˛ cos i˛.C 2

kjx2  yj k.x2 C y/  i kx2 ky i  cos  cos D sin sin : 2k ˛ ˛ k ˛ ˛

6.3 Boundary value problems in plane domains

373

Therefore, we can write (6.3.11) in the form X

uı1 .x/ D uı2 .x/  Z

˛ı1 =


e

kx1 ˛

int S˛

kx1

e ˛ k

sin

kx2 ˛

(6.3.12)

kx2 sin f .x/dx: ˛

Remark 6.3.3. The relation (6.3.12) yields exponential asymptotics of uı1 for x1 ! C1 and of uı2 for x1 ! 1 with singular functions of exponential type.

6.3.2 The Neumann and the Zaremba problem in a strip In this section we consider the Neumann problem u.x/ D f .x/; @x2 u.x1 ; ˛˙ / D g˙ .x1 /;

x 2 int S˛ ; x1 2 R;

(6.3.13) (6.3.14)

and the Zaremba problem u.x/ D f .x/; u.x1 ; 0/ D g .x1 /; @x2 u.x1 ; ˛/ D gC .x1 /;

x 2 int S˛ ; x1 2 R:

(6.3.15) (6.3.16)

We can proceed in a similar way as in Section 6.3.1. Here we only formulate the results. After applying the Fourier transform we get .@2x2  2 /u.; O x2 / D fO.; x2 /;

x2 2 int I˛ ;

O ˛˙ / D gO ˙ ./; @x2 u.; and

.@2x2  2 /u.; O x2 / D fO.; x2 /; u.; O 0/ D gO  ./;

(6.3.18) x2 2 int I˛ ;

@x2 u.; O ˛/ D gO C ./;

respectively. The solution of (6.3.17), (6.3.18) is given by Z ˛ PN .I x2 ; y/fO.; y/dy u.; O x2 / D 0

C K;N .I x2 ; 0/gO  ./ C KC;N .I x2 ; ˛/gO C ./ for cos i .jx2  yj  ˛/ C cos i .x2 C y  ˛/ ; 2i  sin.i ˛/ K˙;N .I x2 ; ˛˙ / WD PN .I x2 ; ˛˙ /: PN .I x2 ; y/ WD

(6.3.17)

(6.3.19) (6.3.20)

(6.3.21)

374

6 Operators on manifolds with conical singularities and boundary

The solution of (6.3.19), (6.3.20) has the form Z ˛ PZ .I x2 ; y/fO.; y/dy u.; O x2 / D

(6.3.22)

0

C K;Z .I x2 ; 0/gO  ./ C KC;Z .I x2 ; ˛/gO C ./ for sin i .jx2  yj  ˛/  sin i .x2 C y  ˛/ ; 2i  cos.i ˛/ K;Z .I x2 ; 0/ WD @y PZ .I x2 ; 0/; KC;Z .I x2 ; ˛/ WD PZ .I x2 ; ˛/: PZ .I x2 ; y/ D

In fact, the relation of (6.3.21) for (6.3.18) and (6.3.22) for (6.3.20) is a consequence of @x2 PN .I ˛˙ ; y/ D 0

for all y 2 int I˛ ; sin i .x2  ˛ / @x2 K˙;N .I x2 ; ˛˙ / D ˙ sin.i ˛/ and PZ .I 0; y/ 0; @x2 PZ .I ˛; y/ D 0 K;Z .I x2 ; 0/ D

cos i .x2  ˛/ ; cos.i ˛/

for all y 2 int I˛ ;

KC;Z .I x2 ; ˛/ D

sin.i x2 / ; i  cos.i ˛/

respectively. To show the assertions with respect to (6.3.17) and (6.3.19) we can argue in an analogous manner as in Section 6.3.1. For the solution of (6.3.13), (6.3.14) we then obtain Z ˛ Z 1 1 ix1 u.x/ D e PN .I x2 ; y/fO.; y/dy 2 1 0 (6.3.23) C K;N .I x2 ; 0/gO  ./ C KC;N .I x2 ; ˛/gO C ./ d ; and the solution of (6.3.15), (6.3.16) follows as Z ˛ Z 1 1 ix1 e PZ .I x2 ; y/fO.; y/dy u.x/ D 2 1 0



(6.3.24)

C K;Z .I x2 ; 0/gO  ./ C KC;Z .I x2 ; ˛/gO C ./ d : Remark 6.3.4. (i) The analogue of Proposition 6.3.1 for the Neumann case is not true: the Green function PN has a pole of second order at  D 0. (ii) For the Zaremba problem the assertion of Proposition 6.3.1 is true with g 2 3 s 1 H 2 .R/, gC 2 H s 2 .R/.

6.3 Boundary value problems in plane domains

375

(iii) The function PN .PZ / extends to a meromorphic function in C with poles at k D ik ; k 2 Z .k D i.2kC1/ ; k 2 Z/. Note that the poles of PN and PZ are ˛ 2˛ simple, except for the case 0 D 0 which is a second order pole of PN (cf. (i) of this remark). Proposition 6.3.5. Let ı 6D k ; k 2 Z (ı 6D .2kC1/ ; k 2 Z). Further, f 2 ˛ 2˛ 3 3 s2;ı s 2 ;ı s 1 ;ı H .S˛ /, g˙ 2 H .R/ ( g 2 H 2 .R/; gC 2 H s 2 ;ı .R/), s 2 N, s  2. Then there exists a unique solution of (6.3.13), (6.3.14) ((6.3.15), (6.3.16)) of 1 the form (6.3.23) ((6.3.24)) and the estimates (6.3.10) hold with H s 2 ;ı .R/ replaced 3 1 3 by H s 2 ;ı .R/ (with H s 2 ;ı .R/ replaced by H s 2 ;ı .R/ in the second term). Let ı1 < ı2 , ıi 6D k , k 2 Z, i D 1; 2. Then for the Neumann problem with ˛ homogeneous boundary conditions an analogue of the relation (6.3.11) holds with PD .; x2 ; y/ replaced by PN .I x2 ; y/. Similarly, for ıi 6D .2kC1/ , k 2 Z, i D 1; 2, 2˛ an analogue of (6.3.11) holds for the Zaremba problem with homogeneous boundary conditions with PD .I x2 ; y/ replaced by PZ .I x2 ; y/. ; k 6D 0, is equal to The residue of PN .I x2 ; y/ at k D ik ˛ lim .  k /PN .I x2 ; y/

!k

D  lim

cos i .jx2  yj  ˛/ C cos i .x2 C y  ˛/

k/ 2˛ cos i˛.C 2  i kjx2  yj i k.x2 C y/ kx2 ky D cos D C cos cos cos : 2k ˛ ˛ k ˛ ˛

!k



From lim @ .2 PN .I x2 ; y// D 0 and

!0

lim 2 PN .I x2 ; y/ D 

!0

1 ˛

it follows that the principal part of the Laurent expansion of PN .I x2 ; y/ at 0 D 0 is equal to  ˛1 2 . Hence, using the Taylor expansion of the exponential function at 0, the corresponding residue in (6.3.11) (with PN instead of PD ) is equal to Z  1 o n ix1   2C f .x/dx Res .1 C i x1 / ˛ ˛ int S˛ Z Z (6.3.25) ix1 i D f .x/dx C x1 f .x/dx: ˛ int S˛ ˛ int S˛ Then for ı1 ı2 > 0 we obtain

Z

kx1

X

uı1 .x/ D uı2 .x/ 

˛ı1 =


e int S˛

kx1 ˛

cos

e ˛ k

cos

kx2 f .x/dx: ˛

kx2 ˛ (6.3.26)

376

6 Operators on manifolds with conical singularities and boundary

For ı1 ı2 < 0 it follows that x1 uı1 .x/ D (6.3.26) C ˛

Z

1 f .x/dx  ˛ int S˛

Furthermore, the residue of PZ .I x2 ; y/ at k D

Z x1 f .x/dx: int S˛

i.2kC1/ ; 2˛

k 2 Z, is equal to

lim .  k /PZ .I x2 ; y/

!k

D lim

sin i .jx2  yj  ˛/  sin i .x2 C y  ˛/

k/ 2˛ sin i˛.C 2   .2k C 1/.x2 C y/ 2i .2k C 1/jx2  yj  cos D cos .2k C 1/ 2˛ 2˛ .2k C 1/x2 .2k C 1/y i sin sin : D  C k 2˛ 2˛ 2

!k

Thus for the Zaremba case we have X

uı1 .x/ D uı2 .x/ 

e

1 ı1 ˛= 1 2
.2k C 1/x2  sin 2˛

Z

e

.2kC1/x1 2˛

 2

C k

.2kC1/x1 2˛

int S˛

sin

.2k C 1/x2 f .x/dx: 2˛

6.3.3 The Dirichlet problem in an angle Let us first consider the Dirichlet boundary value problem for the Laplace operator in the angle

˚ K˛ WD re i W r > 0;  2 I˛ ; ˛ 2 .0; 2 : In polar coordinates we have r 2 ..r@r /2 C @2 /u.r; / D f .r; /;

u.r; ˛˙ / D g˙ .r/:

(6.3.27)

Remark 6.3.6. Passing to the stretched angle I˛^ WD f.r; / W r 2 RC ;  2 I˛ g with the boundary components I WD f0g  RC ; IC WD f˛g  RC , the boundary value problem (6.3.27) represents a continuous operator H s2; 2 .I˛^ / ˚ A 1 1 s; ^ s ; A WD @ T A W H .I˛ / ! H 2 2 .I / ˚ TC 1 s 1 ; H 2 2 .IC / 0

1

(6.3.28)

377

6.3 Boundary value problems in plane domains

for Au D r 2 ..r@r /2 C @2 /u, T˙ u D u.r; ˛˙ /, s > 12 ;  2 R. In this connection I˛^ and I˙ are treated as (stretched) manifolds with conical points at zero and infinity. Then (6.3.28) is an operator in the cone calculus of boundary value problems, cf. Definition 6.2.1 (however with a shift of s and  in the boundary spaces which does not affect the essential assertions). Therefore, we have the principal symbolic structure (6.2.4), where c .A/ WD .c;0 .A/; c;1 .A// now has two components, belonging to zero and infinity, respectively. By the diffeomorphism RC;r ! R t ; t D  log r, we go back to the case of the strip S˛ , where the problem (6.3.27) takes the form .@2t C @2 /v.t; / D F .t; /;

.t; / 2 S˛ ; v.t; ˛˙ / D G˙ .t /; t 2 R;

where v.t; / D u.x/, F .t; / D e 2t f .x/ for x D e t .cos ; sin / and G˙ .t / D g˙ .x/ for x D e t .cos.˛˙ /; sin.˛˙ //. Remark 6.3.7. Let X be a compact C 1 manifold (closed or with C 1 boundary). Then the diffeomorphism R t ! RC;r , r D e t , represents an isomorphism H s;ı .R  int X / ! H s; .RC  int X / where  D ı C

nC1 , 2

n D dim X . In particular, we have isomorphisms

H s;ı .S˛ / ! H s;ıC1 .I˛^ /;

1

H s;ı .R/ ! H s;ıC 2 .RC /: 1

1

C 1, Proposition 6.3.8. Let f 2 H s2;2 .I˛^ /, g˙ 2 H s 2 ; 2 .I˙ /,  6D k ˛ k 2 Z n f0g, s 2 N, s  2. Then there exists a unique solution u 2 H s; .I˛^ / of the problem (6.3.27) and the estimate   kukH s; .I˛^ /  c kf kH s2; 2 .I˛^ / C kg k s 1 ;  1 C kgC k s 1 ;  1 H

2

2 .I /

H

2

2 .IC /

(6.3.29)

holds. Remark 6.3.9. (i) Proposition 6.3.8 allows us to apply Remark 6.2.9 (i) for any s 2 N, s  2, i.e., the operator (6.3.28) is elliptic in the cone algebra. Therefore, by virtue of Theorem 6.2.7 the operator (6.3.28) is Fredholm for all s > 32 . By Remark 6.2.9 both kernel and cokernel are independent of s. (ii) The ellipticity of (6.3.28) for s > 32 entails the bijectivity of the conormal symbol at 0 and 1, e.g., 0

w C r rC

c;0 .A/.w/ D @

2

@2

H s2 .int I˛ / ˚ A W H s .int I˛ / ! C ˚ C 1

for all s > 32 , w 2 1 ; here r˙ u WD u.˛˙ /.

(6.3.30)

378

6 Operators on manifolds with conical singularities and boundary

To show the bijectivity of (6.3.30) first note that w D 0 is a bijectivity point. For w 6D 0, w D a C i b, we have ker.w 2 C @2 / D fc1 e b e ia C c2 e b e ia W c1 ; c2 2 Cg. The boundary conditions u.˛˙ / D 0 give us e b˛ e ia˛  e b˛ e ia˛ D 0, i.e., b D 0, a D k , k 2 Z n f0g. Hence, for any  as in Proposition 6.3.8, the bijectivity ˛ of (6.3.30) on 1 holds. Similarly as Remark 6.3.3 the following proposition gives us the difference of solutions for different weights in terms of singular functions of the cone asymptotic for r ! 0 and r ! 1. Proposition 6.3.10. Let f 2 H s2;1 2 .I˛^ / \ H s2;2 2 .I˛^ /, g˙ 0, 1 ; 2 6D k C 1, k 2 Z n f0g, 1 < 2 . Then ˛ X

u1 .r; / D u2 .r; / 

˛.1 1/=
Z 

I˛^

r

k ˛

sin

k

k r ˛ sin k ˛

k f .r; /rdrd: ˛

6.3.4 The Neumann and the Zaremba problem in an angle The boundary value problem for the Laplace operator with Neumann boundary conditions in the angle K˛ takes the form   (6.3.31) r 2 .r@r /2 C @2 u.r; / D f .r; /; r 1 @ u.r; ˛˙ / D g˙ .r/; and with Zaremba boundary conditions r 2 ..r@r /2 C @2 /u.r; / D f .r; /;

r 1 @ u.r; ˛C / D gC : (6.3.32) Analogously as in Remark 6.3.6, passing to the stretched angle I˛^ , we have continuous operators u.r; ˛ / D g ;

H s2; 2 .I˛^ / ˚ A 3 3 A W @ T A W H s; .I˛^ / ! H s 2 ; 2 .I / ; ˚ TC 3 3 H s 2 ; 2 .IC / 0

1

(6.3.33)

T˙ u D r 1 @ u.r; ˛˙ /, s > 32 ,  2 R, and H s2; 2 .I˛^ / ˚ A 1 1 s; ^ s ; A W @ T A W H .I˛ / / ! H 2 2 .I / ; ˚ TC 3 s 3 ; H 2 2 .IC / 0

1

(6.3.34)

6.3 Boundary value problems in plane domains

379

T u D u.r; ˛ /, TC u D r 1 @ u.r; ˛C /, s > 32 ,  2 R, with corresponding symbolic structure (6.2.4), for the Neumann and Zaremba cases, respectively. The following result is an analogue of Proposition 6.3.8. 3

3

C 1, Proposition 6.3.11. (i) Let f 2 H s2;2 .I˛^ /, g˙ 2 H s 2 ; 2 .I˙ /,  6D k ˛ k 2 Z, s 2 N, s  2. Then there exists a unique solution u 2 H s; .I˛^ / of the 1 1 problem (6.3.31), and the estimate (6.3.29) holds with H s 2 ; 2 .I˙ / replaced by 3 3 H s 2 ;  2 .I˙ /. 1 1 3 3 (ii) Let f 2 H s2;2 .I˛^ /, g 2 H s 2 ; 2 .I /, gC 2 H s 2 ; 2 .IC /,  6D .2kC1/ C 1; k 2 Z. Then there exists a unique solution u 2 H s; .I˛^ / of the 2˛ 1 1 problem (6.3.32), and the estimate (6.3.29) holds with H s 2 ; 2 .IC / replaced by 3 3 H s 2 ;  2 .IC / in the second term. Remark 6.3.12. (i) The operators (6.3.33) ((6.3.34)) define isomorphisms for all s > 32 . (ii) The ellipticity of (6.3.33) ( (6.3.34)) entails the bijectivity of the conormal symbol on 1 for all s > 32 and  6D k C 1 . 6D .2kC1/ C 1/, k 2 Z. ˛ 2˛ The arguments of (i) are similar to Remark 6.3.9 (i). Let us show (ii) for c;0 .A/.w/. If A is as in (6.3.33), we have c;0 .A/.w/ D t .w 2 C @2 r @ rC @ / for r˙ as in Remark 6.3.9 (ii). It is obvious that w D 0 is a non-bijectivity point. If w 6D 0, w D a C ib, the boundary conditions give us e b˛ e ia˛  e b˛ e ia˛ D 0 and hence b D 0, a D k . ˛ For A as in (6.3.34) the point w D 0 is a bijectivity point of the conormal symbol, and for w 6D 0, w D a C i b, the boundary conditions yield e b˛ e ia˛ C e b˛ e ia˛ D 0, or, b D 0, a D .2kC1/ . 2˛ Proposition 6.3.13. (i) Let f 2 H s2;1 2 .I˛^ / \ H s2;2 2 .I˛^ /, g˙ 0, 1 ; 2 6D k C 1, k 2 Z, 1 < 2 . Then ˛ X

u1 .r; / D u2 .r; / 

˛.1 1/=
Z 

I˛^

r

 k ˛

k

k r ˛ cos k ˛ (6.3.35)

k cos f .r; /rdrd ˛

for .1  1/.2  1/ > 0, while for .1  1/.2  1/ < 0 it follows that X

u1 .r; / D u2 .r; / 

˛.1 1/=
Z 

I˛^

r

k ˛

cos

k

k r ˛ cos k ˛

k f .r; /rdrd C ˛

380

6 Operators on manifolds with conical singularities and boundary

C (ii) Let 1 ; 2 6D g˙ 0. Then

log r ˛

Z I˛^

.2kC1/ C1, k 2˛

u1 .r; / D u2 .r; /   sin

f .r; /rdrd 

1 ˛

Z I˛^

log rf .r; /rdrd:

2 Z, 1 < 2 , f 2 H s2;1 2 .I˛^ /\H s2;2 .I˛^ /,

X

r

.2kC1/ 2˛

 1 2
.2k C 1/ 2˛

Z

I˛^

r

.2kC1/ 2˛

sin

C k

.2k C 1/ f .r; /rdrd: 2˛

6.4 Special operators of the cone calculus In this section we study a number of specific operators on manifolds with conical singularities, both for the case with and without boundary. First we establish order reducing isomorphisms in the cone calculus. Then we discuss a generalisation of the cone algebra on an infinite cone by admitting arbitrary weights at infinity. In a final section we give some useful information on parameter-dependent operators on cones.

6.4.1 Reduction of orders In this section we consider elements of special interest of the cone calculus, especially, reductions of orders. Theorem 6.4.1. Let B be a (stretched) compact manifold with boundary and conical singularity. For every  2 Z and  2 R there exists an elliptic operator A 2 C ;0 .B; g/ for g D .;   /, only consisting of an upper left corner (i.e., without additional trace and potential operators at the boundary) such that A W H s; .B/ ! H s; .B/ is an isomorphism for every real s > max.; 0/  12 , and for the inverse we have A1 2 C ;0 .B; g 1 /. Proof. Let .r; x/ denote the points on B close to Bsing , with the covariables .%; /. Near @Breg and close to Bsing we have a splitting x D .x 0 ; t / into the tangential variable x 0 2 @Breg and the normal variable t to the boundary, with the covariables . 0 ; /. Let us set, similarly as (4.1.1),    

0 r .r; %; 0 ; ; / WD ' ; i  i

: hk.r/%; C hk.r/%; 0 ; i Here  2 Rl is a parameter, k.r/ 2 C 1 .RC / a strictly positive function that is equal to 1 for 0 < r < " and 1 for r > 2" for some " > 0. As usual, hk.r/%; 0 ; i WD .1 C jk.r/%j2 C j 0 j2 C jj2 /1=2 .

6.4 Special operators of the cone calculus

381

Close to Bsing and near @Breg we now form the local symbols r .r; %; 0 ; ; /!.t/ hk.r/%; ; i.1!.t// for a cut-off function !.t/, similarly as (4.1.15). Close to Bsing and outside a neighbourhood of @Breg (where !.t/ vanishes) we form the local symbols hk.r/%; ; i . Outside a neighbourhood of Bsing (where k.r/ D 1) we denote variables and coQ respectively; near @Breg we have again a splitting xQ D .xQ 0 ; t / variables on B by xQ and , into the tangential variables xQ 0 to @Breg and the normal variable t , and the corresponding covariables . Q 0 ; /. In this region of B close to @Breg we form the local symbols Q i.1!.t// r . Q 0 ; ; /!.t/ h ; Q i . and outside a neighbourhood of @Breg (where !.t / vanishes) the local symbols h ; Taking into account the double 2Breg of Breg and modifying the construction of (4.1.16) for the present situation, we obtain an analogue of the operator family (4.1.16), namely, s s z ./ W Hcomp R .2Breg / ! Hloc .2Breg /. By composing with r C and eC we obtain analogously as (4.1.17) s z ./eC W H s R ./ D r C R .comp/ .Breg / ! H.loc/ .Breg /;

(6.4.1)

cf. the notation (3.1.18). The operator (6.4.1) belongs to B ;0 .Breg I Rl / and is elliptic. Locally, in an " neighbourhood of Bsing we can perform the same construction with respect to the variables x alone and keep r and %Q D r% as additional parameters. This gives us a family p.r; Q %; Q / 2 C 1 .Œ0; "/; B ;0 .X I R%Q  Rl // such that, when we set p.r; %; / WD p.r; Q r%; /, !R ./!Q D ! opr .p/./!Q for arbitrary cut-off functions !.r/; !.r/ Q on RC , supported in Œ0; "/. We are now in a situation similar to Theorem 6.1.17, here with  instead of Q from there. Mellin quan;0 tisation gives us an h.r; w; / 2 C 1 .Œ0; "/; MO .X I Rl // such that ! opr .p/./!Q D  ! opM .h/./!Q holds mod B 1;0 ..0; "/  X I Rl /. Observe that the minus property of the local symbols survives under the Mellin quantisation.  n QQ for cut-off funcSetting A1 ./ WD r  ! opM 2 .h/./!Q C .1  !/R ./.1  !/ Q tions !; !; Q !Q as in Definition 6.2.1 and supported in Œ0; "/, we obtain a family of elements in C ;0 .B; g/ for g D .;   /, which satisfies the conditions (i), (ii) of Definition 6.2.6. The conormal symbol c .A1 .//.w/ W H s .X / ! H s .X / is an element of B ;0 .X I  nC1   Rl /. As such it is bijective for all j Im w; j sufficiently 2 large, cf. Theorem 3.3.17 (i). In other words, there is a constant c > 0 such that also the condition (iii) of Definition 6.2.6 is satisfied for all jj > c. Fix any such  D 1 and set A1 WD A1 .1 /. Then, by virtue of Theorem 6.2.7 the operator A1 W H s; .B/ ! H s; .B/ is Fredholm. We now employ a lemma which will be proved afterwards.

382

6 Operators on manifolds with conical singularities and boundary

1;0 Lemma 6.4.2. For every m 2 N there exists an element f .w/ 2 MR .X / for some asymptotic type R (cf. Definition 6.1.44), C R \  nC1  D ;, such that 2

 n 2

1 C ! opM

.f /!Q W H s; .B/ ! H s; .B/

is a Fredholm operator of index m (the cut-off functions !; !Q are arbitrary).  n

For m WD  ind A1 we define A2 WD A1 .1 C ! opM 2 .f /!/ Q W H s; .B/ ! s; ;0 H .B/ which is an element in C .B; g/ (cf. Theorem 6.2.5) and of index zero. By Remark 6.2.9 there are subspaces V  H 1;C" .B/; W  H 1;C" .B/ of finite dimension such that V D ker A2 and W C im A2 D H s; .B/ for all s >   12 . Thus there is a Green operator G 2 CG0 .B; g/ of finite rank such that A WD A2 C G W H s; .B/ ! H s; .B/ is an isomorphism. As G we can take G D gV , where V W H 0; .B/ ! V is the orthogonal projection to V and g W V ! W an isomorphism (combined with the embedding in H s; .B/). The operator A is just as desired. By Theorem 6.2.12 C the inverse exists in C ;./ .B; g 1 /. However, since the local symbols contained in A have the minus-property at the boundary, we have in fact A1 2 C ;0 .B; g 1 /, cf., analogously, Proposition 4.1.23. Proof of Lemma 6.4.2. We employ the fact that for every  2 R and every m 2 Z there  1 exists an f1 .w/ 2 MO such that 1 C ! opM .f1 /!Q W K s; .RC / ! K s; .RC / is a Fredholm operator with index m, for all s 2 R. Details may be found in [196] or [112]. This has the consequence that operators of the desired kind also exist on X ^ instead of RC for any base X , for instance, when X is a compact C 1 manifold with boundary, see [185], or [112]. The cut-off functions !; !Q are arbitrary. It is now trivial that such operators can be defined for an arbitrary manifold B with conical singularities; they can be pulled back from X ^ to B, since the operators are the identity outside a neighbourhood of Bsing . In the rest of this section we assume X to be a closed compact C 1 manifold, n D dim X. We want to construct special elliptic elements in the cone calculus on X ^ of even order, where X ^ is interpreted as the (open) stretched manifold associated with a spindle with two conical points, namely, r D 0 and r D 1. First let us fix first order differential operators D1 ; : : : ; DN on X , generated by ˛N vector fields that span Tx X at every x 2 X . Setting D ˛ WD D1˛1 : : : DN for a multi-index ˛ D .˛1 ; : : : ; ˛N / we obtain continuous maps n

n

.r@r /j D ˛ W H s; 2 .X ^ / ! H s.j Cj˛j/; 2 .X ^ / for all s 2 R. We now take s 2 N and form the column vector of operators n

B WD t ..r@r /j D ˛ /j Cj˛js W H t; 2 .X ^ / !

L j Cj˛js

n

H ts; 2 .X ^ /;

(6.4.2)

6.4 Special operators of the cone calculus

383

n

t 2 R. The formal adjoint B  with respect to the scalar product of H 0; 2 .X ^ / D L2 .RC  X/ is a row matrix of continuous operators L n t; n 2 .X ^ / ! H ts; 2 .X ^ /; B W j Cj˛js H t 2 R, and we form the operator n

n

B  B W H 2t; 2 .X ^ / ! H 2.ts/; 2 .X ^ /; X B B D 1 C ..r@r /j D ˛ / .r@r /j D ˛ :

(6.4.3)

0<j Cj˛js

x C  X (with R x C in The operator (6.4.3) belongs to the cone calculus on B WD R the meaning   RC [ f0g [ f1g, according to the spindle interpretation), with the weight data n2 ; n2 . Its conormal symbol (at zero) has the form X c .B  B/.w/ D w j .1  w/j .D ˛ / D ˛ ; (6.4.4) j Cj˛js

see Proposition 2.4.4, c .B  B/.w/ W H t .X / ! H t2s .X /;

(6.4.5)

t 2 R. The conormal symbol at 1 is of the same shape. Observe that B  B is  -elliptic, i.e.,  .B  B/.r; x; %; / 6D 0 on T  X ^ n 0 and Q .B  B/ .r; x; %; / D r 2s  .B  B/.r; x; r 1 %; / 6D 0 for .%; / 6D 0, up to r D 0. It is easy to verify that B  B is also c -elliptic with respect to the weight n2 , i.e., the operators (6.4.5) are isomorphisms for all w 2  1 (and all t 2 R/. It suffices to 2 check the injectivity and surjectivity on C 1 .X/ (since the operators (6.4.4) are w-wise elliptic of order 2s on X , and kernels and cokernels are finite-dimensional subspaces of C 1 .X/). Now c .B  B/.w/u D 0, u 2 C 1 .X /, implies .c .B  B/.w/u; u/L2 .X/ D 0 which gives us X .w j D ˛ u; .1  w/ x j D ˛ u/L2 .X/ D 0: (6.4.6) j Cj˛js

On  1 we have w D 12 C i% D 1  w. x The summands of (6.4.6) are all non2 negative, and the one for j C j˛j D 0 coincides with .u; u/L2 .X/ . Thus (6.4.6) implies u D 0. Since (6.4.4) is formally self-adjoint with respect to the L2 .X /-scalar product, we have dim ker c .B  B/.w/ D dim coker c .B  B/.w/ which is equal to zero. In other words, we proved the bijectivity of (6.4.5) for all w 2  1 . 2

Theorem 6.4.3. The operator (6.4.3) is a Fredholm operator for every t 2 R. Proof. The Fredholm property is a consequence of the ellipticity of B  B with respect to the weight  D n2 .

384

6 Operators on manifolds with conical singularities and boundary

Theorem 6.4.4. The operator (6.4.3) is an isomorphism for every t 2 R. Proof. It is enough to check the assertion for any fixed t 2 R. An analogue of Remark 6.2.9 for the boundaryless case shows that kernel and cokernel are independent of the smoothness. Let us set, for instance, t D s and see that (6.4.3) is injective. The n relation u 2 ker B  B, u 2 H 2s; 2 .X ^ /, gives us X   .r@r /j D ˛ u; .r@r /j D ˛ u 0; n2 ^ D 0: .B  Bu; u/ 0; n2 ^ D H

.X /

H

j Cj˛js

.X /

Thus, since all summands on the right-hand side are non-negative and the sum contains .u; u/ 0; n2 ^ , it follows that u D 0. To show the surjectivity of (6.4.3) for t D s H .X / we first recall that by virtue of Theorem 6.4.3 the operator (6.4.3) is Fredholm, i.e., its range is closed and the cokernel can be represented by a finite-dimensional subspace n n of H 1; 2 .X ^ /. Therefore, it is enough to show that when f 2 H 1; 2 .X ^ / and n .B  Bu; f / 0; n2 ^ D 0 for all u 2 H 2s; 2 .X ^ / it follows that f D 0. We then have H .X / .u; B  Bf / 0; n2 ^ D 0 for all those u, especially, for u D f . Thus, similarly as in H .X / the first part of the proof we obtain f D 0. n

Corollary 6.4.5. Let u 2 H 0; 2 .X ^ /, s 2 N, and n

.r@r /j D ˛ u 2 H 0; 2 .X ^ / for all j C j˛j  s. n

Then u 2 H s; 2 .X ^ /.

L n n In fact, assuming, more generally, u 2 H t; 2 .X ^ / and Bu 2 j Cj˛js H 0; 2 .X ^ / L n n we obtain u D .B  B/1 B  Bu 2 H s; 2 .X ^ / as .B  B/1 B  W j Cj˛js H 0; 2 .X ^ / ! n H s; 2 .X ^ / defines a continuous operator.

Corollary 6.4.6. Let u 2 H 0; .X ^ /, s 2 N,  2 R, and .r@r /j D ˛ u 2 H 0; .X ^ / for all j C j˛j  s. Then u 2 H s; .X ^ / (cf. also Proposition 2.4.12). n

n

In fact, by assumption, we have r C 2 u 2 H 0; 2 .X ^ / and n

n

r C 2 .r@r /j D ˛ u 2 H 0; 2 .X ^ /;

j C j˛j  s:

This entails ˚  C n n

n n 2 .r@ /j r  2 D ˛ .r C 2 u/ 2 H 0; 2 .X ^ /; r r C n 2

j C j˛j  s:

 n 2

.r@r /j r D .r@r   C n2 /j for every j and applying Using the identity r the binomial formula we obtain Corollary 6.4.6 from Corollary 6.4.5. Remark 6.4.7. There is an immediate analogue of the constructions for Theorem 6.4.4 and the subsequent conclusions for the case of a (stretched) manifold D associated with a compact manifold with conical singularities, X D @D. It suffices to replace D1 ; : : : DN by !D1 ; : : : ; !DN , .1  !/DN C1 ; : : : ; .1  !/DM with the former D1 ; : : : ; DN and additional vector fields DN C1 ; : : : ; DM which span the full tangent space at every point of int D; ! is a cut-off function, supported in a collar neighbourhood of @D.

6.4 Special operators of the cone calculus

385

6.4.2 Operators on a cone with arbitrary weights at infinity In this chapter (with the exception of Section 6.1.4) we have formulated the cone calculus for weight data g D .;   / at r D 0, where  is the order of the operator, and  2 R a weight. This is motivated by our main application to the edge calculus; the weight factor r  determines the order of edge symbols. For the cone itself the essential results remain valid in analogous form also for arbitrary weight data g D .; ı/, ; ı 2 R, cf., for instance, [182], [188], [90]. On the infinite (stretched) cone X ^ 3 .r; x/ our calculus treats r ! 1 as a conical exit to infinity, and also here we may admit different weights, as we know from Sections 2.3 and 3.4. Our choice in connection with the spaces K s; .X ^ / D K s;I0 .X ^ / was the weight 0 in the domain and the range of the operators. The modification of the cone calculus on X ^ for arbitrary weights at infinity is also more or less trivial, but in the edge case it makes sense also to work with non-vanishing weights; then the effect will be non-trivial. For that reason in this section we establish some details on operators on X ^ for arbitrary weights at infinity. In the following discussion, for simplicity, we content ourselves with the case of a closed compact C 1 manifold X . Similarly as Definition 6.1.60 we have the space CG .X ^ ; g/ of Green operators G on X ^ for the weight data g D .;   ı/, defined by the conditions that G W K s; .X ^ / ! S ıC" .X ^ /;

G  W K s;Cı .X ^ / ! S C" .X ^ /

(6.4.7)

are continuous for all s 2 R and some " > 0. In Section 2.4.6 we proved that an y operator G with the properties (6.4.7) has a kernel g.r; x; r 0 ; x 0 / 2 S ıCQ" .X ^ / ˝  CQ " ^ .X / for every 0 < "Q < "=2 in the sense that S Z Z 1 g.r; x; r 0 ; x 0 /u.r 0 ; x 0 /.r 0 /n drdx 0 (6.4.8) Gu.r; x/ D X

0

(cf. Theorem 2.4.85 and Proposition 2.4.79). From this we obtain the continuity properties G W K s;Ig .X ^ / ! S ıCQ" .X ^ /, G  W K s;CıIg .X ^ / ! S CQ" .X ^ / for all s; g 2 R; recall that K s; Ig .X ^ / D hrig K s; .X ^ /. Remark 6.4.8. A Green operator G 2 CG .X ^ I g/, g D .;   ı/, regarded as a map 0

0

G W K s; Ig .X ^ / ! K s ;ıIg .X ^ / is compact for every s; s 0 ; g; g 0 2 R. Definition 6.4.9. The space C I .X ^ ; g/ for g D .;   ı/, ; ; ; ı 2 R, is defined to be the set of all operators of the form  n 2

A D !r ı opM

QQ C G .h C f /!Q C .1  !/Areg .1  !/

x C ; M  .X //, f .w/ 2 M 1 .X I  nC1 /, Areg 2 for arbitrary h.r; w/ 2 C 1 .R O  ^ LI cl .X /jRC X , and G 2 CG .X ; g/.

2

386

6 Operators on manifolds with conical singularities and boundary

We set C  .X ^ ; g/ WD C I0 .X ^ ; g/. Comparing this with Definition 6.2.1 (when @X ¤ ;) we see that the lower right corners of the 2  2 block matrices of such cone boundary value problems belong to C  ..@X /^ ; .  12 ;   ı  12 //. Remark 6.4.10. Every A 2 C I .X ^ ; g/ induces continuous operators A W K s; Ig .X ^ / ! K s;ıIg .X ^ /

(6.4.9)

for all s; g 2 R. The principal symbolic hierarchy of operators A 2 C I .X ^ ; g/ consists of a tuple  .A/ D . .A/; c .A/; e .A/; 

;e .A//;

where  .A/ is the standard homogeneous principal symbol of A of order , c .A/.w/ D h.0; w/ C f .w/ the conormal symbol, and e .A/;  in (2.3.14).

;e .A/

are the exit symbolic components, defined

Remark 6.4.11. Definition 6.2.18 of ellipticity has a straightforward generalisation to the case of operators A 2 C I .X ^ /. Clearly, for X without boundary, only condition (i) of Definition 6.2.6 is relevant, and the ellipticity at 1 is that of Remark 2.3.27. Theorem 6.4.12. Let A 2 C I .X ^ ; g/ with g D .;  ı/ be elliptic. Then A induces a Fredholm operator (6.4.9) for all s; g 2 R. Moreover, A has a parametrix P 2 C I .X ^ ; g 1 / for g 1 D .  ı;  /, i.e., Gl WD 1  PA and Gr WD 1  AP belong to CG .X ^ ; g l / and CG .X ^ ; g r /, respectively, for g l D .;  /, g r D .  ı;   ı/. This theorem can be proved in a similar manner as Theorem 6.2.19 (of course, the situation here is much easier, since we are talking about the case without boundary), using Theorem 2.3.27. Remark 6.4.13. We can specialise these considerations to the spaces K s; .X ^ / D K s;Is .X ^ /; cf. Remark 2.4.6 and  D 0. The operators A 2 C I0 .X ^ ; .;   // are continuous as A W K s; .X ^ / ! K s; .X ^ /; (6.4.10) and, in particular, the ellipticity of A entails the Fredholm property of (6.4.10) for all s 2 R. Theorem 6.4.14. Let X be a closed compact C 1 manifold. Then for every ; ; ı 2 R there exists an elliptic operator A 2 C I .X ^ ; g/ with g D .;  ı/, such that (6.4.9) are isomorphisms for all s; g 2 R, and we have A1 2 C I .X ^ ; g 1 /.

6.4 Special operators of the cone calculus

387

Proof. The proof of this theorem is similar to that of Theorem 6.4.1, in fact, much easier what concerns the construction in a neighbourhood of r D 0. The considerations of Section 6.2.1 can be generalised to the case of operators on a non-compact manifold with conical singularities and boundary. To illustrate some effects for simplicity we consider the case D WD X with the stretched space x C  X for a closed compact C 1 manifold X , n D dim X (for boundary value D WD R problems there are similar results). Let C  .X ^ ; g/.loc/ with g D .;   / denote the space of all operators  n 2

A D !r  opM

QQ C G; .h C f /!Q C .1  !/Areg .1  !/

(6.4.11)

where h; f; !; !; Q !QQ are as in Definition 6.2.1, specialised to the case without boundary, moreover, Areg 2 L cl .RC  X /, and G smoothing in the sense s; 1;C" G W H.comp/ .X ^ / ! H.loc/ .X ^ /;

s;C 1; C" G  W H.comp/ .X ^ / ! H.loc/ .X ^ /

for some " D ".G/ > 0. Here, s; s H.comp/=.loc/ .X ^ / WD !H s; .X ^ / C .1  !/Hcomp=loc .X ^ /;

for any cut-off function !, s;  2 R. Let CG .X ^ ; g/.loc/ denote the space of those G. The elements A 2 C  .X ^ ; g/.loc/ induce continuous operators s; s; A W H.comp/ .X ^ / ! H.loc/ .X ^ /

for all s 2 R. The operators A 2 C  .X ^ ; g/.loc/ have a principal symbolic structure  .A/ D . .A/; c .A//;

(6.4.12)

similarly as the one in the compact case, and we have a notion of ellipticity with respect to (6.4.12) which is practically as before. Let us call an A 2 C  .X ^ ; g/.loc/ properly supported (for large r) if the closures of the distributional kernels of Areg and x C  X / are proper (see the notation in (6.4.11)). To simplifying x C  X/  .R G in .R notation we also omit indicating ‘for large r’. From the definition it follows that every A 2 C  .X ^ ; g/.loc/ can be written as A D A0 C C for a properly supported A0 and a C 2 CG .X ^ ; g/.loc/ . A properly supported A 2 C  .X ^ ; g/.loc/ with g D .;   / induces continuous operators s; s; A D H.comp/ .X ^ / ! H.comp/ .X ^ /

for all s 2 R, and the same with ‘(loc)’ on both sides. Remark 6.4.15. There is an analogue of Theorem 6.2.5 when we assume A or B to be properly supported. Moreover, we have a notion of ellipticity, based on (6.4.12) in the same way as in the compact case.

388

6 Operators on manifolds with conical singularities and boundary

Theorem 6.4.16. Let A 2 C  .X ^ ; g/.loc/ , g D .;   /, be elliptic. Then A has a properly supported (for large r) parametrix B 2 C  .X ^ ; g 1 /.loc/ in the sense that Gl D I  BA and Gr D I  AB belong to CG .X ^ ; g l /.loc/ and CG .X ^ ; g r /.loc/ , respectively. For the proof one can construct a parametrix by a simple modification of the proof of Theorem 6.2.7 (specialised to the case without boundary), and then pass to a properly supported representative. There are subspaces C  .X ^ ; g as /.loc/ ; g as WD .;   ; .k; 0 /, k 2 N n f0g, of  n

C  .X ^ ; g/.loc/ in which the smoothing Mellin operator !r  opM 2 .f /!Q in Definition 6.2.1 (here in the variant of closed compact X ) is replaced by a sum of operators  n Q 0  j  k, for some k 2 N n f0g, weights j 2 R, of the form !r Cj opMj 2 .fj /!,   j  j   , meromorphic Mellin symbols fj 2 MR1 .X / (see Definition 6.1.44 j for closed X) with asymptotic types Rj , and where G 2 CG .X ^ ; g as /.loc/ , i.e., s; 1; G W H.comp/ .X ^ / ! HP;.loc/ .X ^ /;

s;C 1; G  W H.comp/ .X ^ / ! HQ;.loc/ .X ^ /

are continuous for all s. The latter condition refers to spaces with asymptotics of type P D f.pj ; mj ; Mj /g0j N as mentioned in Definition 2.4.75, i.e., s; s HP;.comp/ .X ^ / D !HPs; .X ^ / C .1  !/Hcomp .X ^ /

and, similarly, with Q and ‘(loc)’. Remark 6.4.17. It is known from [182] that Theorem 6.2.5 remains valid in analogous form for the operators in the subspaces with ‘as’, see also [43] or [188]. Theorem 6.4.18. Let A 2 C  .X ^ ; g as /.loc/ , g D .;   ; .k; 0 /, be elliptic. Then A has a properly supported (for large r) parametrix B 2 C  .X ^ ; g 1 as /.loc/ in the sense that Gl D I  BA and Gr D I  AB belong to CG .X ^ ; g l;as /.loc/ , and CG .X ^ ; g r;as /.loc/ , respectively; here g 1 as D . ; ; .k; 0 /, g l;as D .; ; .k; 0 /, g r;as D .  ;   ; .k; 0 /. We then have elliptic regularity with asymptotics as follows: Corollary 6.4.19. Let A 2 C 1 .X ^ ; g as /.loc/ be elliptic (and, for simplicity, properly supported for large r). Then s; Au 2 HQ;.loc/ .X ^ /;

1; u 2 H.loc/ .X ^ /

s: .X ^ / for some resulting asymptotic for an asymptotic type Q implies u 2 HP;.loc/ type P .

In fact, if B is a (properly supported) parametrix of A we can multiply Au D f from the left by B and obtain s; BAu D .I  Gl /u D Bf 2 HQ .X ^ / 1 ;.loc/

(6.4.13)

6.4 Special operators of the cone calculus

389

with some asymptotic type Q1 . This follows from the continuity of operators in the cone calculus in spaces with asymptotics. The remainder Gl 2 CG .X ^ ; g l;as /.loc/ 1; 1; .X ^ /  HQ .X ^ / for another asymptotic has the mapping property Gl H.loc/ 2 ;.loc/ s; type Q2 . From (6.4.13) we therefore obtain u 2 HP;.loc/ .X ^ / for some P .

Remark 6.4.20. In the case Au D 0 (and when in the above notation all smoothing Mellin symbols fj vanish) the shape of the asymptotic type P is particularly simple, namely, it follows directly from the meromorphic structure of c .A/1 .w/ for Re w < nC1  , n D dim X . More precisely, let 2 c .A/1 .w/ 2 MS .X / here in the analogous (simpler) for some S D f.sj ; dj ; Hj /gj 2Z in the sense of (6.1.49), S meaning for closed compact X , and set S WD f.s ; d ; Hj / 2 S W Re sj < nC1   g.  j j 2 S Then we obtain P D l2N f.sj  l; dj ; Lj / W .sj ; dj / 2 CN S g for some finitedimensional spaces Lj  C 1 .X / which are directly related to the finite rank operators Hj . Remark 6.4.21. As noted before analogous conclusions are valid for the case of boundary value problems on a (compact or) non-compact manifold with conical singularities and boundary.

6.4.3 Cone operators with parameters In Chapter 7 below we will employ families of operators in the cone algebra on X ^ (for the case with and without boundary) depending on variables and covariables .y; / 2   Rq ,  Rq open, as so-called edge symbols. This aspect is dominating for the whole exposition, and we already prepared corresponding material in Section 6.1.1. Operator-valued symbols taking values in the cone algebra have not necessarily the form of edge symbols; the latter ones represent specific quantisations of edgedegenerate (pseudo-differential) symbols. Here we want to discuss more general symbols which are useful for several purposes. Let us first make some preliminary remarks. z be a continuous operator between Hilbert spaces H and H z , with group Let f W H ! H actions f g2RC and fQ  g2RC , respectively. Then z; f ./ WD Q  f 1 W H ! H

 2 RC ;

(6.4.14)

is a family of continuous operators satisfying the homogeneity relation 1 D Q % f ./%1 f .%/ D Q % f %

for all ; % 2 RC , and f .1/ D f . The family (6.4.14) is not necessarily smooth in . However, the operator f may have internal properties that make f ./ smooth. In that case, using the covariable  2 Rq , we have z /; a./ ./ WD jj f .jj/ 2 S ./ .Rq n f0gI H; H

(6.4.15)

390

6 Operators on manifolds with conical singularities and boundary

cf. the notation in Definition 2.2.3 (ii), and z/ ./a./ ./ 2 Scl .Rq I H; H for any excision function  in Rq . There are many possibilities for such a situation. z D L2 .Rq / with the group action . u/.x/ D Example 6.4.22. (i) Let H D H n  2 u.x/,  2 RC , and let f WD M' W L2 .Rn / ! L2 .Rn / be the operator of multiplication by a function ' 2 S.Rn /. Then we have f ./ D M' with ' .x/ D '.x/, i.e., f ./ 2 C 1 .RC ; L.L2 .Rn //. z D H s .Rn / with group actions as in (i), and f D (ii) Let H D H s .Rn /, H opx .p/ W H s .Rn / ! H s .Rn / for a p.x; x 0 ; / 2 S  .Rn  Rn  Rn / which is independent of .x; x 0 / outside a compact set. Then a simple calculation shows f ./ D opx .p /

for p .x; x 0 ; / D p.x; x 0 ; 1 /:

Also in this case we have f ./ 2 C 1 .RC ; L.H s .Rn /; H s .Rn // for every s 2 R. These examples give an impression on a variety of possibilities to construct concrete operator-valued symbols when the spaces in consideration are a variant of Sobolev spaces and the operators given by symbols that are smooth in the variables and covariables. In this section we apply this idea to operators A in the cone algebra C I .X ^ I g/, g WD .;   ı/, ; ı 2 R, n

 QQ C G; A D !r ı opM 2 .h C f /!Q C .1  !/Areg .1  !/ x C ; M .X //; f .w/ 2 M1 .X I  nC1 /; h.r; w/ 2 C 1 .R

Areg 2

O I Lcl .X ^ /;

2



^

G 2 CG .X ; g/:

Theorem 6.4.23. The space C I .X ^ ; g/ remains preserved under conjugation with  ,  2 RC , i.e., A 2 C I .X ^ ; g/ H) A./ WD  A1 2 C I .X ^ ; g/ for all  2 RC ; n

 and we have A./ D ! ı r ı opM 2 .h C f /!Q  C .1  ! /Areg; .1  !QQ  / C G where ! .r/ WD !.r/, etc., and h .r; w/ D h.r; w/, f .w/ D f .w/, Areg; 2 ^ 1 ^ C 1 .RC ; LI cl .X //, and G 2 C .RC ; CG .X ; g//.

Proof. The assertion is nearly obvious when we take into account the definition of the operators. For instance, we have “  . nC1 Ci%/   r nC1 dr 0 2  n opM 2 .h/u.r/ D h r;   C i% u.r 0 / 0 μ % 0 r 2 r  n

 n

which shows that  opM 2 .h/1 D opM 2 .h /. The desired behaviour of Areg; is a consequence of Definition 2.3.20. For the Green operator we have a kernel reprey  S C" .X ^ / and some sentation (6.4.8) for some g.r; x; r 0 ; x 0 / 2 S ıC" .X ^ / ˝ " > 0, and the smooth dependence of G on  is then obvious.

391

6.4 Special operators of the cone calculus

Corollary 6.4.24. For every ; ı; ; ; g; m 2 R there exists an element a./ 2 Sclm .Rq I K s; Ig .X ^ /; K s;ıIg .X ^ // which takes values in isomorphisms a./ W K s;Ig .X ^ / ! K s;ıIg .X ^ / for all s 2 R. In fact, let A 2 C I .X ^ ; g/ for g D .;   ı/ be an operator as in Theorem 6.4.14. Using Theorem 6.4.23 we obtain an A./ 2 C 1 .RC ; L.K s;Ig .X ^ /; K s;ıIg .X ^ // with values in isomorphisms between the spaces. According to the relation (6.4.15) we can set a./ WD Œ m A.Œ /: Remark 6.4.25. The symbol a./ of Corollary 6.4.24 gives rise to isomorphisms Opy .a/ W W t .Rq ; K s; Ig .X ^ // ! W tm .Rq ; K s;ıIg .X ^ // for all t 2 R. In particular, in the case  D 0, m D , ı D  and g D s   we obtain for the spaces K s; .X ^ / D K s; Is .X ^ / a symbol a./ 2 Scl .Rq I K s; .X ^ /; K s; .X ^ //

(6.4.16)

and associated isomorphisms Opy .a/ W W s .Rq ; K s; .X ^ // ! W s .Rq ; K s; .X ^ //:

(6.4.17)

Up to this point the symbol space in (6.4.16) and (6.4.17) refers to the group action nC1  WD f0 g2RC defined by .0 u/.r; x/ D  2 u.r; x/. In Remark 2.4.33 we discussed the group action  s D fs g2RC , acting in K s; .X ^ / as .s u/.r; x/ D 0

s C

nC1 2

u.r; x/. It is clear, that we also have a./ 2 Scl .Rq I K s; .X ^ /; K s; .X ^ // s ; .s/. / ;

cf. the notation (2.2.11). Then Opy .a/ W W s .Rq ; K s; .X ^ // ! W s .Rq ; K s; .X ^ // is also an isomorphism, cf. the notation (2.4.34).

6.4.4 Elliptic regularity for some Schrödinger equation Consider the hydrogen Schrödinger operator with the Hamiltonian H D  

Z ; jxj Q

(6.4.18)

392

6 Operators on manifolds with conical singularities and boundary

where is the Laplacian in R3 3 xQ and Z the charge of the electron. In this section we ask the regularity of solutions ‰ 2 L2 .R3 / of the equation H ‰ D E‰

(6.4.19)

for any eigenvalue E. As we see the operator (6.4.18) is elliptic in R3 n f0g, and the potential has a singularity at xQ D 0. The equation (6.4.19) in polar coordinates .r; x/ 2 RC  S 2 Š R3xQ n f0g has the form Au D 0 where A D r 2

2 X j D0

 @ j aj .r/ r ; @r

x C ; Diff 2j .S 2 //, u.r; x/ WD ‰.x.r; Q x//, with coefficients aj .r/ 2 C 1 .R a0 .r/ D S 2 C b0 .r/I;

a1 .r/ D I;

a2 .r/ D I;

where b0 .r/ D rZ  r 2 E, with the identity operator I and the Laplace–Beltrami operator S 2 on the unit sphere S 2 . For convenience, from now on we shall write X WD S 2 . In the following consideration instead of b0 .r/ we could take any other x C ; C 1 .X // that vanishes at 0. We have A 2 C 2 .X ^ ; g as /.loc/ for every b0 2 C 1 .R g D .;   2; .k; 0 /, k 2 N n f0g (cf. the notation at the end of Section 6.4.2). The conormal symbol has the form c .A/.w/ D w 2  w C X W H s .X / ! H s2 .X /;

(6.4.20)

s 2 R. The bijectivity of the map (6.4.20) is violated at precisely those w 2 C where .w 2  w C X /' D 0 for some non-vanishing ' 2 C 1 .X /. This is equivalent to w 2  w C  D 0 for some eigenvalue q  of the operator X , and the solutions w 2 C are obtained as

w˙ D 12 ˙ 14  . There is the following well-known theorem on the spectrum of the operator X .

Theorem 6.4.26. Let X be the Laplace–Beltrami operator on the unit sphere X D S n , n  1. Then we have . X '; '/L2 .X/  0, the operator  X has a discrete spectrum, and the eigenvalues are l.l C n  1/, l 2 N D f0; 1; 2; : : : g. A proof, including the characterisation of the eigenfunctions as spherical harmonics, may be found in [220, Theorem 31.1], see also [209, Section 22.3]. Remark 6.4.27. For the Laplacian in RnC1 in polar coordinates which is of the form r 2 f.r@r /2 C .n  1/r@r C S n g we have theqconormal symbol w 2  w C S n  n1 2 and for the non-bijectivity points w˙ D n1 ˙  , where  runs over the 2 2 eigenvalues of S n . Especially for n D 2 it follows that r 1 1 w˙;l WD ˙ C l.l C 1/; 2 4

6.4 Special operators of the cone calculus

393

l 2 N, are the non-bijectivity points of (6.4.20), and the line  3 does not intersect the 2 set fw˙;l W l 2 Ng. It follows that the operator A is elliptic in C 2 .X ^ ; g as /.loc/ , g D .;   2; .k; 0 / for those  2 R such that ˚

w˙;l l2N \  3  D ; (6.4.21) 2

s; s2; 2 .X ^ / ! H.loc/ .X ^ / for all l 2 N. We view A as a continuous operator A W H.loc/ 3 2 0;0 ^ 1 2 for any s 2 R. Note, in particular, that L .RxQ / Š H .X / Š r L .RC  S 2 / is 0;0 a subspace of H.loc/ .X ^ /, and A is elliptic for the weight  D 0. The operator A WD r 2 f.r@r /2 C r@r C X C b0 g

belongs to C 2 .X ^ ; g as /.loc/ for g as D .;   2; .k; 0 / for every  2 R, see the notation at the end of Section 6.4.2. Theorem 6.4.28. For every  2 R such that fw˙;l W l 2 Ng \  3  D ; the 2 operator A is elliptic in C 2 .X ^ ; g/.loc/ and has a parametrix B 2 C 2 .X ^ ; g 1 as /.loc/ , cf. Theorem 6.4.18. Theorem 6.4.29. For every  2 R as in (6.4.21) and 1; u 2 H.loc/ .X ^ /;

s2; 2 Au 2 HQ;.loc/ .X ^ /

s; .X ^ / for every s 2 R and a with an asymptotic type Q it follows that u 2 HP;.loc/ resulting asymptotic type P .

This is the same as Theorem 6.4.29, here specialised to our operator, noting that A (as a differential operator) is properly supported (not only for large r). 1 In the present particular case our operator has the form A D r 2 fopM .h/Cb0 .r/g where b0 .r/ vanishes at r D 0, and h.w/ D w 2  w C X : Remark 6.4.30. For the regularity of solutions u 2 H 1; .X ^ / of Au D 0 we can apply Remark 6.4.20 together with the characterisation of c .A/1 .w/ D h1 .w/ in connection with Theorem 6.4.26 and Remark 6.4.27 (for n D 2).

Chapter 7

Operators on manifolds with edges and boundary

A manifold W with edge Y is locally (near points of the edge) modelled on wedges X   x C  X /=.f0g  X / and open  Rq . The base X of the model cone X is for X D .R assumed to be a C 1 manifold (with or without boundary). The typical differential operators A on W in stretched coordinates .r; x; y/ 2 RC  X   are edge-degenerate. Modulo lower order terms the operators are described by a two-component principal symbolic hierarchy  .A/ D . .A/; ^ .A// with  .A/ being the principal interior, ^ .A/ the principal edge symbol, the latter taking values in operators in weighted spaces on the model cone. In the case @X 6D ; we have a third principal symbolic component, namely, the boundary symbol @ .A/. Ellipticity is formulated in terms of the bijectivity of all components; in general, this requires extra edge conditions of trace and potential type (in the case of boundary value problems we also have boundary conditions). The present chapter develops the most essential material on a corresponding (pseudo-differential) edge algebra of operators on W which is closed under the parametrix construction of elliptic elements. We mainly focus on the case with boundary and discuss a number of examples.

7.1 Differential operators on manifolds with edges The edge algebra is motivated by the problem of constructing parametrices of elliptic edgedegenerate differential operators. The calculus starts from edge-degenerate differential boundary value problems. We discuss the principal symbolic hierarchy, the nature of weighted edge spaces, and ellipticity.

7.1.1 Edge-degenerate differential operators In this section we complete the material of Sections 2.4.2 and 2.4.3 on manifolds with edges and edge-degenerate differential operators, cf. the Definitions 2.4.26 and 2.4.37, respectively. Boundary value problems are considered on a manifold W with edge Y and boundary, cf. Remark 2.4.30. The local description near Y is given in coordinates of stretched wedges RC  X  , where X is a C 1 manifold with boundary @X , and  Rq for q D dim Y corresponds to a chart on Y . This refers to the stretched manifold x C  X   by invariantly W associated with W which is locally identified with R attaching f0g  X   to RC  X  . Globally we attach an X -bundle Wsing over Y to Wreg D W n Y with the trivialisations X  . Recall (see the explanations after Remark 2.4.30) that Wsing is a C 1 manifold with boundary where @Wsing is a @X -bundle

7.1 Differential operators on manifolds with edges

395

over Y , with the trivialisations @X  . Analogous constructions make sense for V WD @.W n Y / [ Y which is a manifold with edge Y (and without boundary). Let V denote the stretched x C  @X  . manifold associated with V , locally near @V described by R As noted in Section 2.4.2 there is a double 2W of W , locally near Wsing modelled x C  2X  . Then 2W is the stretched manifold belonging to a manifold 2W on R with edge Y and without boundary. Let for the moment X be a closed compact C 1 manifold. Consider the open stretched wedge X ^   D RC  X   3 .r; x; y/;  Rq open. Recall that an operator A 2 Diff  .X ^  / is called edge-degenerate if it has the form  X @ j A D r  aj˛ .r; y/ r .rDy /˛ (7.1.1) @r j Cj˛j

x C  ; Diff .j Cj˛j/ .X //. with coefficients aj˛ .r; y/ 2 C 1 .R P Q ˛ˇ Q y/Dx˛QQ Dyˇ be a differential operator on Remark 7.1.1. Let Az D j˛jCjˇ Q .x; Q j a Q y/ with coefficients aQ ˛ˇ Q y/ 2 C 1 .RnC1  /. Then by inRnC1   3 .x; Q .x; nC1 n f0g ! RC  S n , the operator troducing ˇ polar coordinates xQ ! .r; x/, R ˇ xC  z A WD A .RnC1 nf0g/ takes the form (7.1.1) with coefficients aj˛ .r; y/ 2 C 1 .R

; Diff .j Cj˛j/ .S n //. In this example RnC1   is regarded as a manifold W with x C S n  is the associated stretched manifold. Clearly not every edge , and W D R edge-degenerate operator A on RC  S n   is induced by such an Az with smooth coefficients aQ ˛ˇ Q . Let W be a manifold with edge Y . Then, by virtue of the invariance of the edgexC X  xC X  ! R z that restrict at r D 0 degeneracy under diffeomorphisms R z to a transition map X   ! X   of the X -bundle @W over Y , the edge-degeneracy makes sense for differential operators globally on the stretched manifold W , and we obtain in this way the space Diff  deg .W / of edge-degenerate differential operators on W . This first concerns the case without boundary. with edge ˇ If W is a manifold ˚

and  ˇ z z .W / WD A D A W A 2 Diff .2W / . Near boundary we simply set Diff  deg deg int Wreg 1 Wsing the shape of such operators A is the same as (7.1.1), now for a C manifold X with boundary. Example 7.1.2. Every C 1 manifold W with boundary can be regarded as a manifold with edge @W . In this case we have W D W . The space Diff  .W / of differential operators of order  with coefficients that are smooth up to @W is a proper subspace  z of Diff  deg .W /. In fact, if A 2 Diff .W / is written in a collar neighbourhood Š Œ0; 1/  @W 3 .r; y/ of the boundary as X Az D aQ kˇ .r; y/Drk Dyˇ kCjˇ j

396

7 Operators on manifolds with edges and boundary

with coefficients aQ kˇ .r; y/ 2 C 1 .Œ0; 1/  @W /, then, as in Remark 7.1.1, the opz int W can also be reformulated as (7.1.1) with coefficients aj˛ .r; y/ 2 erator A D Aj 1 C .Œ0; 1/  @W /. Recall from Section 2.4.3 that when W is a manifold with edge and boundary, every A 2 Diff  deg .W / has a principal symbolic structure  .A/ WD . .A/; @ .A/; ^ .A// with the homogeneous principal symbol  .A/ on Wreg , moreover, the homogeneous principal boundary symbol @ .A/, and the homogeneous principal edge symbol ^ .A/. Close to Y in the local splitting of variables .r; x; y/ 2 RC  †   with covariables .%; ; / (with † corresponding to a chart on X ) we have  .A/ as a function  .A/.r; x; y; %; ; /;

.r; x; y; %; ; / 2 T  .RC  †  / n 0I

then Q .A/.r; x; y; %; ; / WD r   .A/.r; x; y; r 1 %; ; r 1 / is smooth up to r D 0. Similarly, close to @Wsing in the variables .r; x 0 ; y/ 2 RC  †0   with the covariables .%; 0 ; / (with †0 corresponding to a chart on @X ) we can write @ .A/.r; x 0 ; y; %; 0 ; /;

.r; x 0 ; y; %; 0 ; / 2 T  .RC  †0  / n 0I

then Q @ .A/.r; x 0 ; y; %; 0 ; / WD r  @ .A/.r; x 0 ; y; r 1 %; 0 ; r 1 / is smooth up to r D 0. The homogeneous principal edge symbol ^ .A/, given by  X @ j aj˛ .0; y/ r .r/˛ ; .y; / 2 T   n 0; ^ .A/.y; / WD r  @r j Cj˛j

represents a family of continuous operators ^ .A/.y; / W K s; .X ^ / ! K s; .X ^ /;

(7.1.2)

s 2 R. The weight  2 R is regarded as an additional information which is given together with the operator A (of course, for edge-degenerate differential operators the map (7.1.2) is continuous for every  2 R). Homogeneity of the edge symbol means ^ .A/.y; / D   ^ .A/.y; /1

(7.1.3)

nC1 2

u.r; x/, n D dim X . for all  2 RC , for . u/.r; x/ D  Observe that when W is a C 1 manifold with boundary and Az 2 Diff  .W / as in Example 7.1.2, the homogeneous principal the coordinates .r; y/ P interior symbol in k ˇ z y; %; / D near the boundary  .A/.r; a Q .r; y/%  gives rise to the hokCjˇ jD kˇ mogeneous principal boundary symbol z z @ .A/.y; / D  .A/.0; y; Dr ; / W H s .RC / ! H s .RC /; z /u D .y; / 2 T   n 0. Writing Az in edge-degenerate form we have @ .A/.y; ^ .A/.y; /u for every u 2 C01 .RC /. In other words, the boundary symbol formally agrees with the edge symbol in this case, although the point of view is a little different. In the following we study boundary value problems on a manifold W with edge and boundary; then, of course, @ and ^ have nothing to do with each other.

397

7.1 Differential operators on manifolds with edges

Remark 7.1.3. The operators ^ .A/.y; / also form a family ^ .A/.y; / W K s; Ig .X ^ / ! K s;Ig .X ^ / for every s; g 2 R, homogeneous of order  in the sense (7.1.3). Especially, for g D s   and K s; .X ^ / D K s; Is .X ^ / we have ^ .A/.y; / W K s; .X ^ / ! K s; .X ^ / with the homogeneity ^ .A/.y; / D  s ^ .A/.y; /.s /1 for all  2 RC , .y; / 2 T   n 0 (cf. Remark 2.4.6 and 2.4.33).

7.1.2 Weighted edge spaces In Section 2.4.2 we introduced weighted edge spaces, first on an infinite open stretched wedge X ^  Rq 3 .r; x; y/ as W s; .X ^  Rq / D W s .Rq ; K s; .X ^ //

(7.1.4)

and then globally on a stretched manifold W with edge. We can apply (7.1.4) both for a closed compact manifold X and for a C 1 manifold X with boundary. To illustrate the nature of the spaces (7.1.4) we want to have a look at the norm, for simplicity, for the case when X is closed and compact. By definition the norm of an element u.r; x; y/ of (7.1.4) is given by Z 12 2s 1 2 kukW s; .X ^ Rq / D hi khi .Fy! u/.r; x; /kK s; .X ^ / d :   Let us write f .r; x; / WD Fy! u .r; x; /. Then, for n D dim X , using the relation  u.r; x/ D 

nC1 2

kukW s; .X ^ Rq /

u.r; x/,  2 RC , we obtain Z 12

 nC1

2 2s 1

D hi hi 2 f .hi r; x; / K s; .X ^ / d :

Let k ˇ .r/ 2 C 1 .RC / be any strictly positive function such that k ˇ .r/ D r ˇ in a neighbourhood of 0, and k ˇ .r/ D 1 for r > R for some R > 0. Then kukK s; .X ^ / D kk  ukK s;0 .X ^ / : In particular, for s D 0 we have (up to equivalence of norms) kukK 0; .X ^ / D kk  .r/u.r; x/kr n=2 L2 .RC X/ Z 12  2 n D jk .r/u.r; x/j r drdx : RC X

398

7 Operators on manifolds with edges and boundary

For arbitrary s 2 N the space K s; .X ^ / can be characterised as the set of all u.r; x/ 2 K 0; .X ^ / such that for any cut-off function !.r/ and any chart  W U ! B on X to the open unit ball B  Rn in local coordinates x 2 Rn for every ' 2 C01 .B/ we have  f!.r/'.x/.r@r /l Dxˇ u.r; x/g 2 K 0; .X ^ /;

(7.1.5)

cf. Proposition 2.4.12, and  f.1  !.r//'.x/r .lCjˇ j/ .r@r /l Dxˇ u.r; x/g 2 K 0;0 .X ^ /

(7.1.6)

for all l 2 N, ˇ 2 N n , l C jˇj  s (with  denoting the pull back under  with respect to the variable x). The characterisation of the space K s; .X ^ / is a consequence of the relations ˇ ˇ K s; .X ^ /ˇf0 0 together with the n

description of H s; .X ^ / by local derivatives belonging to r  2 L2 .RC  X / near s .X ^ / by standard Sobolev spaces at infinity, where the derivatives in zero, and Hcone (7.1.6) come from rephrasing derivatives in f.r; x/ Q 2 R1Cn W r 2 RC ; xQ D rx for x 2 Bg by derivatives in the variables .r; x/ 2 RC  B. Observe that these conditions once again confirm the continuity of an operator X al˛ .r@r /l .r/˛ W K s; .X ^ / ! K 0;s .X ^ / r s lCj˛js

with coefficients al˛ 2 Diff s.lCj˛j/ .X /;  2 Rq n f0g, s 2 N. When we interpret Dxˇ globally on X as an arbitrary composition v1ˇ1 : : : vnˇn of vector fields v1 ; : : : ; vn on X , the relations (7.1.5) and (7.1.6) can be unified to n

k  .r/hri.lCjˇ j/ .r@r /l Dxˇ w.r; x/ 2 r  2 L2 .RC  X / for all l C jˇj  s. This gives us the following characterisation. Proposition 7.1.4. For s 2 N,  2 R, a function u.r; x; y/ belongs to the space W s; .X ^  Rq / if and only if f .r; x; / D .Fy! u/.r; x; / satisfies the estimates Z Z 2s.nC1/ hi jk  .r/hri.lCjˇ j/ .r@r /l Rq

RC X

Dxˇ f .hi1 r; x; /j2 r n drdx d < 1

for all l C jˇj  s. It will be instructive to interpret W s; .X ^  Rq / as a special case of a continuously parametrised family of spaces, namely, W s .Rq ; K s; Ig .X ^ // g ;

g 2 R;

with  g WD fg g2RC acting on K s; Ig .X ^ / by .g u/.r; x/ WD gC  2 RC .

(7.1.7) nC1 2

u.r; x/,

7.1 Differential operators on manifolds with edges

399

Proposition 7.1.5. For every s; ; g 2 R we have s s .X ^  Rq /  W s .Rq ; K s; Ig .X ^ // g  Hloc .X ^  Rq /: Hcomp

(7.1.8)

Proof. After decomposing the functions by means of a partition of unity on X it is enough to check the desired property in RC  U  Rq for a cylindrical set RC  U 3 .r; x/, where U is a coordinate neighbourhood on X . By substituting a diffeomorphism of U to an open set in S n for n D dim X it suffices to show (7.1.8) in the case X D S n . Let K  RnC1 n f0g be a compact set, and let C 1 .K/0 denote the subspace of all u 2 C01 .RnC1 n f0g/ which are supported by K. We then prove that there are constants c1 ; c2 > 0 depending on K such that c1 kukH s .R1CnCq /  kukW s .Rq ;K s; Ig ..S n /^ /  c2 kukH s .R1CnCq /

(7.1.9)

for all u 2 C01 .Rq ; C 1 .K/0 /. For the space K s; Ig ..S n /^ / we have K s; Ig ..S n /^ / D !H s; ..S n /^ / C .1  !/H sIg .RnC1 / for any cut-off function !.r/ (where 1  ! means .1  !.jxj// Q when xQ denotes the variable in RnC1 /. Thus, if C01 .RnC1 /ı denotes the space of all v 2 C01 .RnC1 / such that supp v fjxj Q  ıg for some ı > 0, there are constants b1 .ı/, b2 .ı/ > 0 such that b1 .ı/kvg kH s .RnC1 /  kvkK s; Ig ..S n /^ /  b2 .ı/kvg kH s .RnC1 /

(7.1.10)

Q defined by vg .x/ Q WD jxj Q g v.x/. Q Given u 2 C01 .Rq ; C 1 .K/0 / we form for vg .x/ f .x; Q / WD .Fy! u/.x; Q /; fg .x; Q / WD jxj Q g f .x; Q /, and set Q / WD F Q .fg /.x; hg . ; Q /: x! Q For u.x; Q y/ 2 C01 .Rq ; C 1 .K/0 / we have Z kuk2W s .Rq ;K s; Ig ..S n /^ //

D

 g 1

2 hi2s hi f .x; Q / K s; Ig ..S n /^ / d:

(7.1.11)

We now employ the fact that there is a ı > 0 such that supp f .hi1 x; Q / fjxj Q  ıg for all  2 Rq . This allows us to apply (7.1.10) and to replace (modulo equivalence) the right-hand side of (7.1.11) by Z

g  g 1

2 hi2s jxj Q hi f .x; Q / H 2 .RnC1 / d Z

2  g 1 D hi2s hig hi fg .x; Q / H s .RnC1 / d D

400

7 Operators on manifolds with edges and boundary

Z

 0 1

2 hi2s hi fg .x; Q / H s .RnC1 / d Z Z 2sCnC1 Q 2s jhg .hi ; Q /j2 d d Q D hi h i

D

Z D “ D

Z hi2s

˝ Q ˛2s Q /j2 d d Q jhg . ; hi

Q i2s jhg . ; Q /j2 d d Q h ;

2 2 D kug kH s .RnC1Cq /  kukH 2 .RnC1 / ;

ug .x; Q y/ WD jxj Q g u.x; Q y/, with equivalence on functions u.x; Q y/ supported by K for all y. Now the first inclusion of (7.1.8) follows by a simple approximation argument s for arbitrary u 2 Hcomp ..RnC1 n f0g/  Rq /. For the second inclusion it is enough to show that for arbitrary ' 2 C01 ..RnC1 n f0g/  Rq / and u 2 W s .Rq ; K s;Ig ..S n /^ // we have 'u 2 H s .RnC1  Rq /. This is also an immediate consequence of (7.1.9). Similarly as Proposition 7.1.4 we have the following result. Proposition 7.1.6. For s 2 N, ; g 2 R, a function u.r; x; y/ belongs to the space W s .Rq ; K s;Ig .X ^ // g if and only if f .r; x; / D .Fy! u/.r; x; / satisfies the estimates Z Z 2.sg/.nC1/ hi k  .r/hrig.lCjˇ j/ RC X (7.1.12) ˇ ˇ ˇ.r@r /l D ˇ f .hi1 r; x; /ˇ2 r n drdx d < 1 x for all l C jˇj  s (with Dxˇ being interpreted in the same way as in Proposition 7.1.4). Note that Proposition 7.1.4 is a special case of Proposition 7.1.6, namely, for g D 0. For g WD s   we obtain a particularly convenient (‘canonical’) variant of edge spaces, cf. [217]. Theorem 7.1.7. For s 2 N,  2 R, a function u.r; x; y/ belongs to the space W s; .X ^  Rq / WD W s .Rq ; K s; .X ^ // s

(7.1.13)

if and only if “ ˇ ˇ2 r 2 hri2.s.lCjˇ jCj˛j// ˇ.r@r /l Dxˇ .rDy /˛ u.r; x; y/ˇ r n drdxdy < 1 (7.1.14) for all l 2 N, ˛ 2 N q , ˇ 2 N n with l C jˇj C j˛j  s. In particular, we have W s; .X ^  Rq / D r  W s;0 .X ^  Rq / for all  2 R, and W 0; .X ^  Rq / D r

 n 2

L2 .RC  X  Rq /.

(7.1.15)

401

7.1 Differential operators on manifolds with edges

Proof. First note that we have c1 k 2 .r/hri2.s/  r 2 hri2s  c2 k 2 .r/hri2.s/ for suitable constants c1 ; c2 > 0 and for all r 2 RC . The relation (7.1.12) for g D s  then takes the form Z Z ˇ ˇ2 hi2.nC1/ r 2 hri2.s.lCjˇ j// ˇ.r@r /l Dxˇ f .hi1 r; x; /ˇ r n drdx d < 1: (7.1.16) This is equivalent to “

ˇ ˇ2 r 2 .rhi/2.s.lCjˇ j// ˇ.r@r /l Dxˇ f .r; x; /ˇ r n drdxd < 1

(7.1.17)

which we immediately obtain by substituting rQ D hi1 r in (7.1.16). The estimates (7.1.17) hold for all l C jˇj  s, including l D jˇj D 0. From .1 C .rhi/2 /sljˇ j D .1 C r 2 C r 2 jj2 /sljˇ j sljˇ X j s  l  jˇj .1 C r 2 /s.lCjˇ jCm/ .rjj/2m D m mD0

it follows that the system of estimates (7.1.17) is equivalent to “

ˇ ˇ2 r 2 .rjj/2m hri2.s.lCjˇ jCm// ˇ.r@r /l Dxˇ f .r; x; /ˇ r n drdxd < 1

for all l C jˇj C m  s. This in turn is the same as “

ˇ ˇ2 r 2 hri2.s.lCjˇ jCj˛j// ˇ.r@r /l Dxˇ .r/˛ f .r; x; /ˇ r n drdxd < 1

(7.1.18)

for all l C jˇj C j˛j  s, which we obtain from the estimates c1

X

j.r/˛ j  .rjj/m  c2

j˛jm

X

j.r/˛ j

j˛jm

with suitable constants c1 ; c2 > 0. Using Parseval’s equality we see that (7.1.18) is equivalent to (7.1.14) for all l C jˇj C j˛j  s. Remark 7.1.8. Observe that for s 2 N we have   W s;s .X ^  Rq / D W s Rq ; K s;s .X ^ / : This follows from (7.1.13) together with K s;s .X ^ / D K s;s .X ^ / and  g D  for g D 0.

402

7 Operators on manifolds with edges and boundary

Let us investigate the norm of W s; .X ^  Rq / also for arbitrary s 2 R. By definition we have Z

2 nC1 kuk2W s; .X ^ Rq / D hi2s hi.sC 2 / f .hi1 r; x; / K s; Is .X ^ / d Z D hi2s hi2.s/ hi.nC1/ kk  .r/hris f .hi1 r; x; /k2K s;0 .X ^ / d Z  hi2.nC1/ kr  hris f .hi1 r; x; /k2K s;0 .X ^ / d: (7.1.19) According to Corollary 6.4.24 there exists a family of operators 1 a./ D hi ahi W K s;0 .X ^ / ! K 0;0 .X ^ /

(7.1.20)

which is a symbol of order 0 in  and takes values in isomorphisms between the nC1 spaces in (7.1.20); here we take . u/.r; x/ D  2 u.r; x/, such that  is unitary in K 0;0 .X ^ /. That means, we have an equivalence of norms khi aukK 0;0 .X ^ /  kukK s;0 .X ^ / , uniformly in  2 Rq . Therefore, the right-hand side of (7.1.19) can be replaced by Z hi2.nC1/ khi a.r  hris f .hi1 r; x; //k2K 0;0 .X ^ / d Z nC1 D hi2.nC1/ ka./hi 2 ..rhi/ hrhiis f .r; x; //k2K 0;0 .X ^ / d: It follows that

Z

kuk2W s; .X ^ Rq / 

ka./.r  hrhiis f .r; x; //k2K 0;0 .X ^ / d;

and we obtain the following result. Proposition 7.1.9. The relation (7.1.15) holds for all s;  2 R. Remark 7.1.10. An analogue of the relation (7.1.15) for the spaces (7.1.7) is not always true, e.g., when g D 0. Let us now turn to the invariance of edge spaces under diffeomorphisms  W Rq ! Rq ;

yQ D .y/;

with the property .y/ D y for all jyj  C for some C > 0. The argument for the invariance of abstract edge spaces is as follows. Let E and E 0 be Hilbert spaces, and assume that we already know the invariance of the space W 0 .Rq ; E 0 / under the function pull back, such that we have an isomorphism 0 W W 0 .RqyQ ; E 0 / ! W 0 .Ryq ; E 0 /:

7.1 Differential operators on manifolds with edges

403

In addition we assume that there is a symbol a./ 2 Scl .Rq I E; E 0 / which takes values in isomorphisms E ! E 0 , such that a1 ./ 2 Scl .Rq I E 0 ; E/. Then the associated pseudo-differential operator A D Op.a/ W W  .Ryq ; E/ ! W 0 .Ryq ; E 0 / is an isomorphism, and A1 D Op.a1 /. The operator push forward Az WD  A D . /1 A is then a pseudo-differential operator, AzW W  .RqyQ ; E/ ! W 0 .RqyQ ; E 0 /, and we express  W W  .RqyQ ; E/ ! W  .Ryq ; E/ z This shows the invariance of abstract edge spaces W  .Rq ; E/. as  D A1 0 A. For the spaces W t .Rq ; K s; Ig .X ^ // g DW W t .Rq ; K s;Ig .X ^ // this gives us the following invariance property. Theorem 7.1.11. Let  W Rq ! Rq be a diffeomorphism which is equal to the identity map for jyj > C for some C > 0. Then the function pull back  W S.Rq ; E/ ! S.Rq ; E/ with E WD K s;Ig .X ^ / extends to an isomorphism  W W t .Rq ; E/ ! W t .Rq ; E/ z is a diffeomorphism between open for every t; s; ; g 2 R. Moreover, if  W  !  1 z q  z z E/ ! sets ;  R , the pull backs  W C0 .; E/ ! C01 .; E/ and  W C 1 .; 1  t t z E/ ! Wcomp .; E/ and C .; E/ extend to isomorphisms  W Wcomp .; t z t   W Wloc .; E/ ! Wloc .; E/, respectively, for all t; s; ; g 2 R. Proof. It suffices to apply the above general construction to symbols a./, constructed in Corollary 6.4.24, here for the case E D K s;Ig .X ^ /, E 0 D K 0;0 .X ^ /; then W 0 .Rq ; E 0 / D r n=2 L2 .RC  X  Rq /, which is invariant. The assertion on invariance of ‘comp’/‘loc’ edge spaces is a simple corollary. Remark 7.1.12. Theorem 7.1.11 for t D s together with Proposition 7.1.5 for g D 0 allows us to define global spaces s; s Wloc .W /  Hloc .Wreg /

(7.1.21)

on a (not necessarily compact) stretched manifold W with edge which are locally near Wsing modelled on W s .Rq ; K s; .X ^ //. We impose here (for simplicity) the condition that @W is a trivial X -bundle over the edge Y and that the cocycle of transition maps which describes W globally near Wsing consists of diffeomorphisms as used in Theorem 7.1.11 (i.e., .r; x; y/ ! .r; Q x; Q y/ Q is independent of r, x for small r). In a similar manner we can define the global spaces s; s Wloc .W /  Hloc .Wreg /;

(7.1.22)

404

7 Operators on manifolds with edges and boundary

locally near Wsing modelled on W s .Rq ; K s; .X ^ // s . The subscript ‘loc’on the lefthand sides of (7.1.21) and (7.1.22) indicates that the distributions are defined up to Wsing in the sense of weighted edge spaces W s .Rq ; K s; .X ^ // and W s .Rq ; K s; .X ^ // s , respectively. For compact W we simply write W s; .W /

and

W s; .W /;

(7.1.23)

respectively. If the manifold with edge has a boundary, then (7.1.21), (7.1.22) or (7.1.23) are defined by restriction of corresponding spaces on the double 2W to Wreg .

7.1.3 Edge-boundary value problems as operators in weighted spaces We now turn to boundary value problems for differential operators on a manifold W with edge Y and boundary. As before, by W we denote the associated stretched manifold. The boundary V D @.W n Y / [ Y is a manifold with edge Y , and we have the corresponding stretched manifold V .  Recall that Diff  deg .W / denotes the space of all elements of Diff .Wreg / that have near @W D Wsing in the splitting of variables .r; x; y/ 2 X ^   the form  X @ j aj˛ .r; y/ r .rDy /˛ A D r  @r j Cj˛j

x C  ; Diff .j Cj˛j/ .X //. The manifold X is with coefficients aj˛ .r; y/ 2 C 1 .R 1 C , with boundary @X . A boundary value problem for the operator A in the edge calculus has the form Au D f in int Wreg ;

T u D g on Vreg :

(7.1.24)

Here T D t .T1 ; : : : ; TN / is a vector of trace operators Tj u WD rVreg Bj u;

j D 1; : : : ; N;



j .W / and the operator rVreg of restriction to Vreg . The number N is with Bj 2 Diff deg determined by A (for instance, N D =2 when  is even and 1 C n C q  3). It suffices that the operators Bj are only given in a neighbourhood of V . Assuming here, for simplicity, that W is compact, we obtain continuous operators

  A A WD W W s; .W / ! T

W s; .W / ˚ 1 N sj  1 2 ;j  2 .V / ˚j D1 W

for every s;  2 R; s > maxfj C 12 W j D 1; : : : ; N g (otherwise we have operators between the corresponding ‘comp’ or ‘loc’ spaces). The principal symbol of A consists of a triple  .A/ D . .A/; @ .A/; ^ .A//: (7.1.25)

7.1 Differential operators on manifolds with edges

405

The meaning of the interior symbol  .A/ WD  .A/ was explained in Section 7.1.1. Moreover, we define   @ .A/ WD t @ .A/ @ .T / as the boundary symbol of A; here A is regarded as an element of B ;0 .Wreg /. The boundary symbol is a family of operators @ .A/ W H s .RC / ! H s .RC / ˚ C N ;

(7.1.26)

s > maxfj C 12 W j D 1; : : : ; N g, parametrised by the points of T  Vreg n 0. Together with  .A/ we consider Q .A/.r; x; y; %; ; / D r   .A/.r; x; y; r 1 %; ; r 1 / near Wsing , smooth up to r D 0, and together with @ .A/ we consider   Q @ .A/ D t Q @ .A/ Q @ .T / near Vsing , where Q @ .A/.r; x 0 ; y; %; 0 ; / is as in Section 7.1.1 and Q @ .T / is equal to .Q @ .Tj //j D1;:::;N with

t

Q @ .Tj /.r; x 0 ; y; %; 0 ; / D r j @ .Tj /.r; x 0 ; y; r 1 %; 0 ; r 1 / where @ .Tj /.r; x 0 ; y; %; 0 ; /u.xn / D r j  .Bj /.r; x 0 ; 0; y; %; 0 ; Dxn ; /u.xn /jxn D0 ; j D 1; : : : ; N . Also the components of Q @ .A/ are smooth up to r D 0. Finally, we have the homogeneous principal edge symbol   ^ .A/ D t ^ .A/ .^ .Tj //j D1;:::;N : The first component of ^ .A/.y; / was explained in Section 7.1.1, while ˇ ^ .Tj /.y; /u.r; x 0 / WD ^ .Bj /.y; /u.r; x/ˇ.@X/^ ; j D 1; : : : ; N , .y; / 2 T  Y n 0. Remark 7.1.13. The edge symbol induces a family of continuous operators ^ .A/.y; / W K s; .X ^ / ! K s;  .X ^ / ˚

LN

1

1

K sj  2 ;j  2 ..@X /^ /; (7.1.27) s > maxfj C 12 W j D 1; : : : ; N g. The operators ^ .A/.y; / belong to the cone algebra C ;d .X ^ ; g/; g D .;   /, of boundary value problems on the infinite stretched cone X ^ , for every fixed .y; / 2 T  Y n 0, for d WD maxfj W j D 1; : : : ; N g C 1. As such they have the principal symbolic structure of that algebra, namely, . ^ .A/; @ ^ .A/; c ^ .A/; E ^ .A/; E0 ^ .A//: j D1

406

7 Operators on manifolds with edges and boundary

In particular, the principal conormal symbol c ^ .A/ represents a family of continuous operators c ^ .A/.y; w/ W H s .int X / ! H s .int X / ˚ for .y; w/ 2 Y  C; s > maxfj C

1 2

LN

j D1

1

H sj  2 .@X /

(7.1.28)

W j D 1; : : : ; N g.

Definition 7.1.14. The operator A D t .A T / is called . ; @ /-elliptic if (i)  .A/ 6D 0 on T  Wreg n 0 and Q .A/ 6D 0 up to r D 0. (ii) @ .A/ is a bijective family of operators (7.1.26) for all parameter points of T  Vreg n 0, and Q @ .A/ is bijective up to r D 0. This concerns any s > maxf; dg  12 ; d WD fj C 1 W j D 1; : : : ; N g. Theorem 7.1.15. Let A D t .A T / be . ; @ /-elliptic. Then for every y 2 Y there exists a discrete set D.y/  C (where D.y/ \ fw 2 C W c  Re w  c 0 g is finite for every c  c 0 ) such that the operators (7.1.27) are Fredholm for all s 2 R, s >   12 , s > maxfj C 12 W j D 1; : : : ; N g and  2 R such that  nC1  \ D.y/ D ; 2 (n D dim X). Proof. Let y 2 Y be fixed. From the . ; @ /-ellipticity of A it follows that c ^ .A/.y; w/ is a parameter-dependent elliptic family of boundary value problems on X with the parameter Im w on every line ˇ , ˇ 2 R. Thus (7.1.28) is a family of isomorphisms for all j Im wj sufficiently large. In addition c ^ .A/.y; w/ is a holomorphic family of Fredholm operators. Thus, by virtue of Theorem 6.1.40, there is a discrete set D.y/ with the desired properties. If we fix  2 R in such a way that  nC1  \ D.y/ D ;, then ^ .A/.y; / is elliptic in C ;d .X ^ ; .;   // 2 with respect to c , cf. Definition 6.2.18. The . ; @ /-ellipticity of A also entails the . ; @ /-ellipticity of ^ .A/.y; / in C ;d .X ^ ; .;   //. The ellipticity of ^ .A/.y; / for  6D 0 with respect to E and E0 is a consequence of Theorem 3.4.21. Thus the Fredholm property of (7.1.27) follows from Theorem 6.2.19. Explicit examples for the discrete sets D.y/ are given in Chapter 5 (see, for instance, Corollary 5.3.3) and in Section 7.4 below.

7.1.4 Operators in alternative weighted edge spaces In Section 7.1.2 we discussed an alternative scale of weighted edge spaces, namely W s; .W /, locally near Wsing modelled on W s .Rq ; K s; .X ^ //. Analogously, such spaces exist on V , and, similarly as for the W s; -spaces, one can show that the operator of restriction rVreg induces continuous operators 1

1

rVreg W W s; .W / ! W s 2 ; 2 .V /

7.1 Differential operators on manifolds with edges

407

for all s > 12 ,  2 R (we consider here compact W ; the non-compact case is analogous). Now an edge-degenerate boundary value problem (7.1.24) generates continuous operators   A AD W W s; .W / ! T

W s; .W / ˚ 1 N sj  1 2 ;j  2 .V / ˚j D1 W

(7.1.29)

for every s;  2 R, s > maxfj C 12 W j D 1; : : : ; N g. The principal symbolic structure of A is nearly the same as in the preceding section, given by a tuple (7.1.25). The interior and the boundary components are exactly as before, while the edge component is now a family of operators L 1 1 ^ .A/.y; / W K s; .X ^ / ! K s; .X ^ / ˚ jND1 K sj  2 ;j  2 ..@X /^ /; (7.1.30) s > maxfj C 12 W j D 1; : : : ; N g. The homogeneity can be formulated as before, now with respect to the group actions s ; for instance, we have ^ .A/.y; / D  s ^ .A/.y; /.s /1 for all .y; / 2 T  Y n 0,  2 RC , and similar relations for the components of ^ .T /.y; /. Theorem 7.1.16. Theorem 7.1.15 is valid in analogous form for the operator family (7.1.30) with the same discrete sets D.y/  C, y 2 Y , as for (7.1.27). Proof. The only difference between the situation for (7.1.30) and (7.1.27) consists in the realisation of the operators in spaces with different weights for r ! 1. From the calculus of elliptic operators on manifolds with conical exit to 1 we know that the Fredholm property, including dimensions of kernel and cokernel, are independent of those weights. Moreover, the conormal symbols are the same in both cases. Explicit examples for the discrete sets D.y/ are given in Chapter 5 (see, for instance, Corollary 5.3.3) and in Section 7.4 below. Remark 7.1.17. The local amplitude functions X a.y; / D r  aj˛ .r; y/.r@r /j .r/˛ j Cj˛j

(under the condition that the coefficients are independent of r for large r) represent symbols a.y; / 2 S  .  Rq I K s; .X ^ /; K s; .X ^ //; with respect to . u/.r; x/ D 

nC1 2

u.r; x/,  2 RC , and also

a.y; / 2 S  .  Rq I K s; .X ^ /; K s; .X ^ // s ; s

nC1

(7.1.31)

with respect to .s u/.r; x/ D sC 2 u.r; x/,  2 RC . A similar observation is true of the local amplitude functions for the trace operators Tj .

408

7 Operators on manifolds with edges and boundary

Instead of (7.1.31) we could realise the operators also in K s;Ig .X ^ / for arbitrary g 2 R, with the same result. For simplicity, we content ourselves with the case g D s  . The order  in the operator-valued symbols is caused by the weight factor r  in front of the operators. Therefore, it seems that, for solving a boundary value problem (7.1.29) in the sense of a parametrix construction, these weight factors are very essential. It turns out that for the spaces in (7.1.29) this effect is not so dominating. As we saw in Section 7.1.2 the weights in this set-up are multiplicative. Therefore, if we want to solve a problem of the kind A0 u0 D f 0 in int Wreg ;

T 0 u0 D g 0 on Vreg ;

(7.1.32)

where, locally, A0 is given as A0 D r 

0

X j Cj˛j

 @ j aj˛ .r; y/  r .rDy /˛ ; @r 0

and, similarly, T 0 with weight factors r j quite independent of the orders j , we could easily pass to a problem of the form (7.1.24) by composing the operators in (7.1.32) by suitable weight factors, solve the problem as before, and then return to the original problem by removing the auxiliary weight factors. In other words, (7.1.32) is solvable in the sense of a problem 0

A0 W W s; .W / !

W s; .W / ˚ 1

0

1

:

˚jND1 W sj  2 ;j  2 .V /

Clearly, this is not the complete information, since the solvability, in general, requires extra edge conditions, and the parametrix construction is to be carried out in a corresponding edge algebra, but the basic facts remain untouched. Such a simple weight shift manipulation (which we know in analogous form from the cone calculus) is not possible in the case of W s; -spaces.

7.2 The edge algebra We develop the ‘standard’ elements of the edge algebra of pseudo-differential boundary value problems on a manifold W with smooth edge Y and boundary. The main aspects are edgeamplitude functions, the principal symbolic hierarchy  D . ; @ ; ^ /, composition results, and ellipticity and parametrices. Since W n Y is a C 1 manifold with boundary, the edge algebra is a subcalculus of boundary value problems with the transmission property at the boundary (up to entries consisting of trace and potential operators on the edge). The specific feature is the edge-degenerate behaviour of various kinds of amplitude functions in stretched coordinates.

7.2 The edge algebra

409

The ellipticity gives rise to extra edge conditions which are an analogue of boundary conditions, here referring to the edge. Together with the boundary conditions on the smooth part of the boundary the operators form 3  3 block matrices, and they act in weighted edge spaces plus standard Sobolev spaces on the edge. Elliptic operators have parametrices in the calculus. If W is compact, the ellipticity entails the Fredholm property.

7.2.1 Edge-degenerate symbols and operator conventions The task to express parametrices of elliptic edge-degenerate boundary value problems (as in Section 7.1.3) gives rise to corresponding constructions on the level of pseudodifferential operators. The result will consist of the edge algebra of boundary value problems. This is, apart from edge conditions (which also may arise as additional data in the differential case), a subset of B ;d .Wreg / with a specific edge-degenerate behaviour near Wsing . For convenience, we assume here that the orders of the operators which refer to the boundary are ‘normalised’, i.e., equal to the orders of the upper left corners, up to a shift by 1=2. For the general considerations this can be done without loss of generality, since there are order reducing isomorphisms in the edge operator calculus on the boundary, cf. [112] and Section 7.2.6 below. If we want to avoid reductions of orders but treat the original problems we can simply form block matrices with entries of different order, similarly as in Douglis–Nirenberg elliptic systems. For the symbol and operator spaces themselves we may refer to the present definitions which are valid for each entry, separately for any individual order. Starting point of the constructions for the edge algebra are the families of boundary z ;d .X I R z ;d .X I R x C    R1Cq / and B x C    C  Rq /, defined value problems B by (6.1.26) and (6.1.27), respectively. z ;d , etc., in the meaning of Throughout the general exposition we employ B ;d , B 2  2 block matrices of continuous operators 1

1

H s .int X / ˚ H s 2 .@X / ! H s .int X / ˚ H s 2 .@X /; s > d 12 (when they are given for a compact C 1 manifold X with boundary, otherwise, on a non-compact manifold, in ‘comp’ and ‘loc’ spaces). In other words, we ignore for a while the aspect of systems or distributional sections in vector bundles. Below in the context of ellipticity we will employ the operators in the corresponding more general sense and tacitly use straightforward extensions of the considerations for the case of non-trivial bundles of arbitrary fibre dimension. We employ Theorem 6.1.17 as an aspect of the Mellin operator convention of the edge calculus. Let us formulate a similar theorem with more information. Recall that z ;d .X I R x C    R1Cq / has the form every p.r; y; %; / 2 B p.r; y; %; / D p.r; Q y; r%; r/

(7.2.1)

x C    R1Cq /. Moreover, every h.r; y; w; / 2 for a p.r; Q y; %; Q / Q 2 B ;d .X I R %; Q Q

410

7 Operators on manifolds with edges and boundary

z ;d .XI R x C    C  Rq / has the form B Q y; w; r/ h.r; y; w; / D h.r;

(7.2.2)

Q y; w; / x C    C  Rq /. for an h.r; Q 2 B ;d .X I R Q z ;d .X I R x C    R1Cq /, and let ' 2 C 1 .RC / Theorem 7.2.1. Let p.r; y; %; / 2 B 0 be a function such that ' 1 in a neighbourhood of the point r D 1. Then there exists z ;d .X I R x C    C  Rq / such that an h.r; y; w; / 2 B ˇ .h/.y; / D opr .q/.y; / 2 C 1 .; B 1;d .X ^ I Rq // (7.2.3) opr .p/.y; /  opM

for every ˇ 2 R, for q.r; r 0 ; y; %; / WD .1  '.r 0 =r//p.r; y; %; /. Moreover, setting Q y; w; r/ h0 .r; y; w; / WD h.0;

p0 .r; y; %; / WD p.0; Q y; r%; r/;

(7.2.4)

we have ˇ opr .p0 /.y; /  opM .h0 /.y; / 2 C 1 .; B 1;d .X ^ I Rq //;

(7.2.5)

and the difference (7.2.5) is of a similar structure as (7.2.3). Theorem 7.2.1 is a refinement of Theorem 6.1.17. The shape of the remainder in the analogous case without boundary is characterised in this form in [57], the case with boundary is studied in [91]. The relation for the symbols with subscripts ‘0’ is a consequence, since the Mellin operator convention only acts on the covariables. Let us fix cut-off functions  .r/; Q .r/ and elements '.r/; '.r/ Q 2 C01 .RC /, and form a.y; / WD  r  faM .y; / C a .y; /gQ C aint .y; / (7.2.6) where aint .y; / WD 'opr .pint /.y; /', Q pint 2 B ;d .X I RC    R1Cq ; /, and  n 2

aM .y; / WD !.rΠ/ opM

.h/.y; /!.rΠ/; Q

QQ a .y; / WD .1  !.rΠ// opr .p/.y; /.1  !.rΠ//; QQ with cut-off functions !.r/; !.r/; Q !.r/ such that !Q 1 on supp !, ! 1 on supp !, QQ and operator families (7.2.1), (7.2.2) that are related to each other via Theorem 7.2.1. Moreover, set  n 2

^ .a/.y; / WD r  f!.rjj/ opM

.h0 /.y; /!.rjj/ Q

QQ C .1  !.rjj// opr .p0 /.y; /.1  !.rjj//g for .y; / 2 T   n 0.

(7.2.7)

411

7.2 The edge algebra

Remark 7.2.2. The way to pass from r  p.r; y; %; / to (7.2.6) can be interpreted as a part of an operator convention. In fact, first we can observe that when  .r/; Q .r/; QQ .r/ are cut-off functions such that Q 1 on supp ;  1 on supp QQ we have r  opr .p/.y; / D  r  opr .p/.y; /Q C .1   /r  opr .p/.y; /.1  QQ / mod C 1 .; B 1;d .X ^ I Rq //. The summand with the factors .1   / and .1  QQ / is localised outside a neighbourhood of the edge. Thus the specific summand is the one with  and . Q From the pseudo-locality and Theorem 7.2.1 it is easy to see that  r  fopr .p/.y; /  aM .y; /  a .y; /gQ 2 C 1 .; B 1;d .X ^ I Rq //: Theorem 7.2.3. The operators (7.2.6) form a family of continuous operators a.y; / 2 C 1 .  Rq ; L.K s; .X ^ /; K s; .X ^ ///

(7.2.8)

1

1

for every s > d  12 , where K s; .X ^ / WD K s; .X ^ / ˚ K s 2 ; 2 ..@X /^ /; more precisely, we have a.y; / 2 S  .  Rq I K s; .X ^ /; K s; .X ^ //;

(7.2.9)

for s > d  12 . The relation (7.2.9) refers to the group action  WD f g2RC ,  W u.r; x/ ˚ v.r; x 0 / ! 

nC1 2

u.r; x/ ˚ 

nC1 2

v.r; x 0 /;

 2 RC , both in K s; .X ^ / and K s; .X ^ /. The family of operators (7.2.7) is homogeneous in the sense ^ .a/.y; / D   ^ .a/.y; /1  ;

(7.2.10) 1

 2 RC . In addition, setting S C" .X ^ / WD S C" .X ^ / ˚ S C" 2 ..@X /^ / for any " > 0 we have a.y; / 2 S  .  Rq I S C" .X ^ /; S C" .X ^ // for every " > 0. Proof. For simplicity we assume aint .y; / D 0. Setting H s; .X ^ / WD H s; .X ^ / ˚ 1 1 H s 2 ;  2 ..@X /^ / we first observe that  r  aM .y; /Q W H s; .X ^ / ! H s; .X ^ / is continuous for every s > d  12 and C 1 in .y; / 2   Rq . The continuity is analogous to Theorem 6.2.15, and the C 1 -dependence on .y; / is evident. Also  r  a .y; /Q W K s; .X ^ / ! K s; .X ^ / is continuous for every s > d  12 and C 1 in .y; / 2   Rq , because the factors QQ .r/.1  !.rŒ // and .1  !.rŒ // Q .r/ are in C01 .RC /. This reduces the assertion ;d to the continuity of operators in B .X ^ / in standard Sobolev spaces on X ^ . Thus, if ./ is an excision function, it follows that .1  .//a.y; / 2 S 1 .  Rq I K s; .X ^ /; K s; .X ^ //. Therefore, in order to show the relation (7.2.8) it is

412

7 Operators on manifolds with edges and boundary

enough to verify it for ./a.y; /. By virtue of the observation in Remark 2.4.32 it suffices to show that ./r  aM .y; /;

./r  a .y; /

(7.2.11)

belong to the symbol space (7.2.9). We now employ the fact that x C    C  Rq / D C 1 .R x C/ ˝ y  B ;d .X I   C  Rq /: B ;d .XI R P Q y; w; / Q with null By Proposition 2.1.8 we can write h.r; Q D j1D0 j 'j .r/hQj .y; w; / P x C /; hQj 2 B ;d .X I C Rq /, and j 2 C with 1 jj j < sequences 'j 2 C 1 .R j D0 1. Since the Mellin operator in aM .y; / is multiplied by !.rŒ /, without loss of x C / W u.r/ D 0 generality we may consider the case 'j 2 C 1 .Œ0; R/0 / .WD fu 2 C 1 .R Q for r  R for some R > 0g/. Setting hj .r; y; w; / WD hj .y; w; r/ we thus obtain ./r



aM .y; / D

1 X

j M'j bj .y; /

(7.2.12)

j D0  n

for bj .y; / WD ./r  !.rŒ / opM 2 .hj /.y; /!.rŒ /. Q The operators bj .y; / are continuous in our weighted spaces and C 1 in .y; / 2   Rq . Moreover, we have bj .y; / D   bj .y; /1  for all   1, jj  c for some c > 0. This gives us bj .y; / 2 Scl .Rq I K s; .X ^ /, K s; .X ^ //, cf. Example 2.2.8 (i). In addition it is easy to verify that hQj ! 0 entails bj ! 0 in the space of symbols. Together with the second part of Remark 2.4.32 it follows that (7.2.12) is convergent as a series of symbols which yields that ./r  aM .y; / is a symbol as desired. The x C    R1Cq /. Since second term in (7.2.11) is based on p.r; Q y; %; Q / Q 2 B ;d .X I R %; Q Q we study operator functions (7.2.6) with cut-off factors ; Q , it is enough to assume // p.r; Q y; %; Q / Q 2 C 1 .Œ0; R/0 ; B ;d .X I   R1Cq %; Q Q y  B ;d .X I   R1Cq D C 1 .Œ0; R/0 / ˝ / %; Q Q for some R > 0. Applying again Proposition 2.1.8 we obtain a representation p.r; Q y; %; Q / Q D

1 X

j 'j .r/pQj .y; %; Q / Q

j D0

with null sequences 'j 2 C 1 .Œ0; R/0 /; pQj 2 B ;d .X I   R1Cq /, and j 2 C, %; Q Q P1 j j < 1. j D0 j It follows that 1 X j M'j cj .y; / (7.2.13) ./r  a .y; / D j D0

7.2 The edge algebra

413

QQ for cj .y; / WD ./r  .1  !.rΠ// opr .pj /.y; /.1  !.rΠ//, pj .r; y; %; / WD pQj .y; r%; r/. From Theorem 3.4.21 we easily see that the operators cj .y; / are continuous in the weighted spaces and C 1 in .y; / 2   Rq , cf. Definition 2.4.5 (ii), the relation (3.4.18) and Theorem 3.4.14. for all   1, jj  c for some Moreover, we have cj .y; / D   cj .y; /1  c > 0. This yields cj .y; / 2 Scl .  Rq I K s; .X ^ /; K s; .X ^ //. In addition from Theorem 3.4.21 (i) it follows that pQj ! 0 implies cj ! 0 in the space of symbols. Again by the second part of Remark 2.4.32 we obtain the convergence of (7.2.13) as a series of symbols. Thus the second operator function in (7.2.11) belongs to our symbol space which completes the proof of (7.2.9). The proof of the homogeneity relation (7.2.10) is straightforward. Corollary 7.2.4. Let a.y; / be as in (7.2.6); then Op.a/ D Opy .a/ induces continuous operators s 1

1

1

s Op.a/ W Wcomp .; K s; .X ^ // ˚ Wcomp2 .; K s 2 ; 2 ..@X /^ // s 1 2

s ! Wloc .; K s;  .X ^ // ˚ Wloc

1

1

.; K s 2 ; 2 ..@X /^ //

for all s > d  12 . This is a consequence of Theorem 7.2.3 together with Corollary 2.3.44.

7.2.2 Global smoothing operators Let W be a manifold with boundary and edge Y . In this section we study global smoothing operators in the edge calculus, associated with weight data g D .; ˇ/ 2 R2 . Let W be the stretched manifold associated with W , and let 2W be the double of W which is a compact manifold with C 1 boundary. 2W is the stretched manifold of the manifold 2W with edge Y (and without boundary). Recall that 2W is obtained by gluing together two copies W˙ of W along @.W n Y /, where we identify W with WC ; similarly, 2W is defined in terms of copies W˙ of W with W being identified with WC . Recall that when X is the base of the local model cones of W (which is a compact C 1 manifold with boundary @X ) then 2X is the base of the local model cones of 2W . Moreover, Wsing is an X -bundle over Y and .2W /sing a 2X -bundle over Y . For simplicity, we always assume that these bundles are trivial. We fix a collar neighbourhood of .2W /sing in 2W of the form 2N WD Œ0; 1/  .2X /  Y and a diffeomorphism x C  .2X /  Y Œ0; 1/  .2X /  Y ! R

(7.2.14)

which is the identity in the first variable in a neighbourhood of zero and the identity in the other variables.

414

7 Operators on manifolds with edges and boundary

By restriction to W \ 2N DW N we obtain a corresponding map x C  X  Y: N D Œ0; 1/  X  Y ! R

(7.2.15)

Let h denote any strictly positive C 1 function on .2W /reg which is near .2W /sing the pull back under (7.2.14) of a function k.r/ that is strictly positive, C 1 on RC , and k.r/ D r for r < c0 , k.r/ D 1 for r > c1 for certain 0 < c0 < c1 . Let us fix a Riemannian metric on 2W that is equal to the product metric of Œ0; 1=2/  .2X /  Y with the Lebesgue measure on Œ0; 1=2/ and measures from Riemannian metrics on 2X and Y . We then have the spaces L2comp .2W / and L2loc .2W / which are the corresponding L2 spaces on 2W of elements with compact support and of those which belong to L2comp .2W / after multiplication by a compactly supported C 1 function. If 2W is compact we simply have L2 .2W /. s; s; On 2W we have the global weighted edge spaces Wcomp .2W / and Wloc .2W /, s; respectively, for every s;  2 R. The space Wcomp .2W / is defined to be the subset of s all u 2 Hloc ..2W /reg / such that u D u for a certain 2 C01 .2W / and such that .!'u/ ı # 1 2 W s .Rq ; K s; ..2X /^ // for every chart U ! Rq on Y , ' 2 C01 .U / and any cut-off function ! (i.e., ! 2 C01 .Œ0; 1//, ! 1 near 0), where # W .0; 1/  2X  U ! .2X /^  Rq as in (7.2.14) for the first two variables and to the chosen chart on Y for the third variable. n 0;0 Observe that Wloc .2W / D h 2 L2loc .2W /, and, similarly, with ‘comp’. We then s; s; have a non-degenerate sesquilinear pairing Wcomp .2W /Wloc .2W / ! C for every s;  2 R, via the .; /W 0;0 .2W / -pairing which makes sense when one of the factors has loc compact support. Let us first define smoothing operators C D .Cij /i;j D1;2 2 Y1 .2W ; g/ in the edge calculus on 2W , associated with the weight data g WD .; ˇ/ 2 R2 . Let us look at the entries separately. An operator C11 belongs to Y1 .2W ; g/, if there  (with respect to the is an " D ".C11 / > 0 such that C11 and its formal adjoint C11 0;0 Wloc .2W /-pairing) induce continuous operators 1;ˇ C" s; C11 W Wcomp .2W / ! Wloc .2W /;

1; C"  s;ˇ C11 W Wcomp .2W / ! Wloc .2W /

for all s 2 R. Similarly, C12 and C21 in Y1 .2W ; g/ are defined by the conditions 1;ˇ C" s C12 W Hcomp .Y / ! Wloc .2W /;

 s;ˇ 1 C12 W Wcomp .2W / ! Hloc .Y /;

s; 1 .2W / ! Hloc .Y /; C21 W Wcomp

1; C"  s C21 W Hcomp .Y / ! Wloc .2W /;

for all s 2 R, with " > 0 depending on the corresponding operator. The formal adjoints   and C21 are defined, respectively, by C12  v/L2 .C12 u; v/W 0;0 .2W / D .u; C12 loc

loc .Y /

and .u; C21 v/L2

loc .Y /

 D .C21 u; v/W 0;0 .2W / ; loc

7.2 The edge algebra

415

for arbitrary u 2 C01 .Y /, v 2 C01 ..2W /reg /. Finally, C22 2 Y1 .2W ; g/ simply means C22 2 L1 .Y /. In this definition it is, of course, not essential that the stretched manifold with edge is the double of some other stretched manifold with edge and boundary. Thus, for V WD @.W n Y / [ Y and the associated stretched manifold s; s; V we also have the spaces Wcomp .V / and Wloc .V /, s;  2 R, with corresponding s; s; sesquilinear pairings between Wcomp .V / and Wloc .V /, and the space Y1 .V ; g/ of smoothing operators. Remark 7.2.5. If Y1 .2W ; g/" denotes for the moment the set of all those C 2 Y1 .2W ; g/ that have the above-mentioned mapping properties for fixed " > 0, we obtain a Fréchet space. Thus it makes sense to form S.Rl ; Y1 .2W ; g/" / DW Y1 .2W ; gI Rl /" , or, more generally, C 1 .U; Y1 .2W ; gI Rl /" / for any open set U Rp , and we set [ Y1 .2W ; gI Rl / WD Y1 .2W ; gI Rl /" : ">0

In a similar manner we obtain Y1 .V ; gI Rl /, as well as the corresponding spaces of C 1 functions on U with values in Y1 .2W ; gI Rl / and Y1 .V ; gI Rl /, respectively. Remark 7.2.6. The operators D D .Dij /i;j D1;2 2 Y1 .V ; g/ have integral kernels K.D11 / 2 C 1 .Vreg  Vreg /; K.D21 / 2 C 1 .Y  Vreg /;

K.D12 / 2 C 1 .Vreg  Y /; K.D22 / 2 C 1 .Y  Y /:

They have a specific behaviour in the variables on Vreg close to Vsing which depends on the chosen weights and on " > 0; this will not be discussed here in detail. Similarly, the entries of C D .Cij /i;j D1;2 2 Y1 .2W ; g/ have smooth integral kernels. In the parameter-dependent case the kernel functions depend on the parameter  2 Rl as Schwartz functions. Our next objective is to define smoothing operators C 2 Y1;d .W ; g/;

g D .; ˇ/; d 2 N;

(7.2.16)

for a stretched manifold W , associated with a manifold W with edge Y and boundary. The number d plays the role of a type. Operators (7.2.16) will induce continuous operators s 0  1 ; 1 2

s; C W Wcomp .W / ˚ Wcomp2

!

1;ˇ C" Wloc .W /

s 00  nC1 2

.V / ˚ Hcomp 1;ˇ C" 1 2

˚ Wloc

.Y /

.V / ˚

1 Hloc .Y /

(7.2.17)

for arbitrary s; s 0 ; s 00 2 R, s > d  12 and an " D ".C/ > 0. For convenience, the entries of C D .Cij /i;j D1;2;3 are again assumed to be scalar; the generalisation to the case of distributional sections of vector bundles is straightforward and left to the reader.

416

7 Operators on manifolds with edges and boundary

By definition we have s; s; .W / D r C Wloc .2W /; Wloc

where r C is the operator of restriction of distributions on .2W /reg to int Wreg . The s; s; .W /0 of all u 2 Wloc .2W / that vanish on int.W /reg is closed in subspace Wloc s; s; s; s; Wloc .2W /, and we have Wloc .W / D Wloc .2W /=Wloc .W /0 . Similar relations are true of the ‘comp’ spaces. Let eC denote the operator of extension by zero from int Wreg to .2W /reg . s; s; Remark 7.2.7. The operator eC W Wcomp .W / ! Wcomp .2W / is continuous for s;  2 1 R, jsj < 2 ; similar observations are true of the corresponding ‘loc’ spaces.

To prepare Definition 7.2.8 let Y1 .2W ; V ; g/ denote the space of 3  3 matrices D D .Dij /i;j D1;2;3 such that .Dij /i;j D1;3 2 Y1 .2W ; .; ˇ//, .Dij /i;j D2;3 2 Y1 .V ; .  12 ; ˇ  12 //, while D12 and D21 induce continuous operators s; 1

1;ˇ C" D12 W Wcomp 2 .V / ! Wloc .2W /; 1;ˇ  1 2 C"

s; D21 W Wcomp .W / ! Wloc

.V /;

1

 s;ˇ D12 W Wcomp .2W / ! W 1;  2 C" .V /; s;ˇ  1 2

 D21 W Wcomp

1; C" .V / ! Wloc .2W /;

for all s 2 R and some " D ".D/ > 0; the formal adjoints in the components over V 1 refer to the W 0; 2 .V /-scalar product. Definition 7.2.8. (i) Let Y1;0 .W ; g/ defined to be the set of all C D .Cij /i;j D1;2;3 such that for some D D .Dij /i;j D1;2;3 2 Y1 .2W ; V ; g/ we have C11 WD r C D11 eC ;

C12 WD r C D12 ; C13 WD r C D13 ;

Cj1 WD Dj1 eC ;

C2j WD D2j ;

C3j WD D3j ;

j D 2; 3I

(ii) for dP2 N n f0g we denote by Y1;d .W ; g/ the set of all operators of the form C D C0 C jd D1 Cj diag.D j ; 0; 0/ with arbitrary Cj 2 Y1;0 .W ; g/, j D 0; : : : ; d. Here D is any first order differential operator that is equal to @=@t in a neighbourhood x C  .@X/  Y in R x C  X  Y , cf. the formula (7.2.15), where t denotes the normal of R variable to @X in X in a collar neighbourhood of @X. Remark 7.2.9. The continuity (7.2.17) is a direct consequence of Definition 7.2.8.

7.2.3 Green and smoothing Mellin symbols The edge calculus will contain another category of operators which are smooth on int Wreg but not ‘uniformly’ up to Wsing , and, similarly, on int Vreg . These operators have a non-trivial (operator-valued) symbolic structure along the edge Y and the values of the symbols will consist of specific smoothing operators on the infinite stretched

417

7.2 The edge algebra

cones .int X/^ and .@X /^ , respectively. We know them from the cone calculus as Green and smoothing Mellin operators. In order to formulate Green symbols it will be convenient to refer for the moment to model cones N ^ and N 0^ for closed compact manifolds N and N 0 of dimensions n and n  1, respectively. Definition 7.2.10. Let  Rp be an open set. A Green symbol of order  2 R with respect to the pair of cones N ^ and N 0^ and weights ; ˇ 2 R is an operator function g.y; / 2 C 1 .  Rq ;

T

s;s 0 2R

0

1

L.K s; .N ^ / ˚ K s ; 2 .N 0^ / 1

˚ C; K 1;ˇ .N ^ / ˚ K 1;ˇ  2 .N 0^ / ˚ C// 1

such that g0 .y; / WD diag.1; hi 2 ; hi the relations g0 .y; / 2 Scl .  Rq I E; F /;

nC1 2

1

/g.y; / diag.1; hi 2 ; hi

nC1 2

g0 .y; / 2 Scl .  Rq I E  ; F  /;

/ satisfies

(7.2.18)

for 0

1

E WD K s; .N ^ / ˚ K s ; 2 .N 0^ / ˚ C; 1

F WD S ˇ C" .N ^ / ˚ S ˇ  2 C" .N 0^ / ˚ C; 0

(7.2.19)

1

E  WD K s;ˇ .N ^ / ˚ K s ;ˇ  2 .N 0^ / ˚ C; 1

F 0 WD S C" .N ^ / ˚ S  2 C" .N 0^ / ˚ C

(7.2.20)

for all s; s 0 2 R and some " D ".g/ > 0, where the spaces of symbols refer to the nC1 n group actions  W u.r; x/ ˚ u0 .r; x 0 / ˚ c !  2 u.r; x/ ˚  2 u0 .r; x 0 / ˚ c in the respective spaces. A Green symbol g.y; / with respect to the pair .N ^ ; N 0^ / is said to be flat (of infinite order) if it satisfies the relation (7.2.18) for the spaces (7.2.19) and (7.2.20) for all " > 0. Remark 7.2.11. An alternative description of Green symbols g.y; / of order  with respect to the pair of cones N ^ , N 0^ is given by the conditions g.y; / 2 Scl .  Rq I E; F /; ;

g .y; / 2 Scl .  Rq I E  ; F  / ;

for all s; s 0 2 R, " D ".g/ > 0, where  D f g2RC and  D f g2RC are defined by  W u.r; x/ ˚ v.r; x 0 / ˚ c !  cf. also the notation (2.2.11).

nC1 2

fu.r; x/ ˚ v.r; x 0 / ˚ cg;

 WD 1 ;

418

7 Operators on manifolds with edges and boundary

We apply this to the case N D 2X and N 0 D @X and consider operator functions g.y; / WD diag.r C ; 1; 1/g.y; Q / diag.eC ; 1; 1/

(7.2.21)

for Green symbols g.y; Q / with respect to the cones .2X /^ and .@X /^ , where r C is the restriction of distributions on .2X /^ to .int X /^ , and eC the operator of extension by zero from .int X /^ to .2X /^ . Then we have 0

1

g.y; / 2 Scl .  Rq I K s; .X ^ / ˚ K s ; 2 ..@X /^ / 1

˚ C; S ˇ C" .X ^ / ˚ S ˇ  2 C" ..@X /^ / ˚ C/; for every s; s 0 2 R, s >  12 . ;0 Definition 7.2.12. (i) By RG .  Rq ; g/ for g D .; ˇ/ we denote the space of all operator functions (7.2.21) for arbitrary Green symbols g.y; Q / with respect to the ^ ^ cones .2X/ , .@X / ; ;d .  Rq ; g/ for d 2 N n f0g we denote the space of all g.y; / WD (ii) by RG Pd ;0 g0 .y; / C j D1 gj .y; / diag.D j ; 0; 0/ for arbitrary gj .y; / 2 RG .  Rq ; g/, j D 0; : : : ; d; the operator D j is the same as that in Definition 7.2.8; (iii) an element g.y; / 2 R;d .  Rq ; g/ is called flat, if the elements gQ j .y; / belonging to gj .y; /; j D 0; : : : ; d, via the relations (7.2.21) are flat in the sense of ;d Definition 7.2.10. Let RO .  Rq ; g/ denote the space of all flat Green symbols. ;d The space RO .  Rq ; g/ does not depend on the weights g D .; ˇ/; therefore we also write ;d ;d .  Rq / or, simply, RO RO

for this space. Observe that the entries of g.y; / D .gij .y; //i;j D1;2;3 are classical operatorvalued symbols between the respective spaces, with orders ij D ord gij , given by the block matrix 0 1    12   .n C 1/=2 @ (7.2.22)  C 12    n=2 A :  C .n C 1/=2  C n=2  Let ^ .g/.y; / WD .^ .gij /.y; //i;j D1;2;3

(7.2.23)

denote the matrix of homogeneous principal components of orders ij . Q y; w; r/, h.r; Q y; w; / Example 7.2.13. Let h.r; y; w; / D h.r; Q be elements of x C  ; M;d .X //, let '0 .r/; '1 .r/ 2 C 1 .R x C /, and suppose that '1 0 C 1 .R 0 O on supp '0 (e.g., '1 may be a cut-off function, '0 2 C01 .RC /). Then it follows that  n 2

'1 .rΠ/r  opM

.h/.y; /'0 .rΠ/ DW g.y; / is a flat Green symbol of order .

419

7.2 The edge algebra

Another ingredient of the symbolic algebra of edge boundary value problems are the so-called smoothing Mellin symbols. These are operator functions taking values in B 1;d .X/ as follows. Recall, cf. Section 6.1.3, that M 1;d .X I ˇ / is defined as 1 the set of all operator functions f .w/ W H s .X / ˚ H s 2 .@X / ! C 1 .X / ˚ C 1 .@X / with values in B 1;d .X /, holomorphic in a strip fw W ˇ  " < Re w < ˇ C "g for some " D ".f / > 0 and f .ı C i%/ 2 B 1;d .X I ı / for every ˇ  " < ı < ˇ C ", uniformly in compact subintervals. We then have [ M1;d .X I ˇ / D M1;d .X I ˇ /" ; ">0

where subscript ‘"’ denotes the space of all f which satisfy the condition for a given " > 0. The space M1;d .X I ˇ /" is Fréchet, and we set [ C 1 .; M 1;d .X I ˇ // WD C 1 .; M 1;d .X I ˇ /" / ">0

for any open set  R . q

Definition 7.2.14. A smoothing Mellin symbol of the calculus of boundary value problems with weight data g WD .; ˇ/ 2 R2 is an operator function of the form  n 2

m.y; / WD r ˇ  !.rŒ / opM 1

for an f .y; w/ 2 C .; M

1;d

.f /.y/!.rΠ/ Q

.X I  nC1  // and arbitrary cut-off functions !; !. Q 2

1

1

Remark 7.2.15. Observe that m0 .y; / WD diag.1; hi 2 /m.y; / diag.1; hi 2 / for m.y; / as in Definition 7.2.14 satisfies the relation 0

m0 .y; / 2 Sclˇ C .  Rq I K s; .X ^ / ˚ K s ; 2 ..@X /^ /; 1

1

S ˇ .X ^ / ˚ S ˇ  2 ..@X /^ // for all s; s 0 2 R, s > d  12 . For m.y; / itself we have 0

m.y; / 2 Sclˇ C .  Rq I K s; .X ^ / ˚ K s ; 2 ..@X /^ /; 1

1

S ˇ .X ^ / ˚ S ˇ  2 ..@X /^ //; nC1

for the same s, s 0 , and  W u.r; x/ ˚ u0 .r; x 0 / !  2 fu.r; x/ ˚ u0 .r; x 0 /g,  WD f g2RC . Analogous relations are true for the pair of spaces 1

1

S  C" .X ^ / ˚ S C" 2 ..@X /^ /; S ˇ CQ" .X ^ / ˚ S ˇ CQ" 2 ..@X /^ / for every " > 0 with another "Q > 0, depending on " and the symbol m. The operator functions m.y; / D .mij .y; //i;j D1;2 are classical operator-valued symbols of the same orders as in the 2  2 upper left corner of (7.2.22), and we have a matrix of corresponding homogeneous principal components ^ .m/.y; / WD .^ .mij /.y; //i;j D1;2 of order ij , i; j D 1; 2.

(7.2.24)

420

7 Operators on manifolds with edges and boundary

7.2.4 Edge amplitude functions In this section we consider so-called edge amplitude functions which are symbols of operators in the edge algebra. Definition 7.2.16. By R;d .  Rq ; g/ for  2 Z, d 2 N, g D .;   /, we denote the space of all families of operators a.y; / WD diag.a.y; / C m.y; /; 0/ C g.y; / for arbitrary a.y; / of the form (7.2.6),  n 2

m.y; / D r  !.rΠ/ opM

.f /.y/!.rΠ/; Q

(7.2.25)

for f .y; w/ 2 C 1 .; M 1;d .X I  nC1  // and cut-off functions !; !, Q cf. Defini2

;d tion 7.2.14, and g.y; / 2 RG .  Rq ; g/, cf. Definition 7.2.12. Moreover, let ;d q RM CG .  R ; g/ denote the subspace of all elements of R;d .  Rq ; g/ such that a.y; / vanishes.

Remark 7.2.17. Theorem 7.2.3, Definition 7.2.10 and Remark 7.2.15 give us z ; R;d .  Rq ; g/ S  .  Rq I E; E/ with E WD K s; .X ^ / ˚ C; Ez WD K s;  .X ^ / ˚ C, (cf. the notation in Remark 7.2.11) for every s > d  12 . Alternatively, one may interpret the entries of a.y; / D .aij .y; //i;j D1;2;3 as elements aij .y; / 2 S ij .  Rq I Ej ; Ezi /

(7.2.26)

with orders ij as in the block matrix (7.2.22), where 1

1

E1 WD K s; .X ^ /;

E2 WD K s 2 ; 2 ..@X /^ /;

Ez1 WD K s; .X ^ /;

Ez2 WD K

1 s 1 2 ; 2

E3 WD C;

..@X /^ /; Ez3 WD C;

nC1 with the group actions u.r; x/ !  2 u.r; x/ in E1 ; Ez1 , moreover, v.r; x 0 / ! n  2 v.r; x 0 / in E2 ; Ez2 , and the identity map in C. The relation (7.2.26) also holds for the spaces 1

E1 WD S C" .X ^ /;

E2 WD S C" 2 ..@X /^ /;

Ez1 WD S CQ" .X ^ /;

Ez2 WD S

CQ " 1 2

E3 WD C;

..@X /^ /; Ez3 WD C

for every " > 0 with another "Q > 0, depending on " and the symbol a. For a.y; / 2 R;d .  Rq ; g/ we set ^ .a/.y; / WD diag.^ .a/.y; / C ^ .m/.y; /; 0/ C ^ .g/.y; /;

421

7.2 The edge algebra

cf. (7.2.7), (7.2.24) and (7.2.23). We then have ^ .a/.y; / D   ^ .a/.y; /1 

(7.2.27)

for all .y; / 2   .Rq n f0g/;  2 RC . Observe that for a.y; / D .aij .y; //i;j D1;2;3 2 R;d .  Rq ; g/ we have .aij .y; //i;j D1;2 2 C 1 .; B ;d .X ^ I Rq //: This gives rise to parameter-dependent (with parameter  2 Rq and smoothly dependent on y 2 ) homogeneous principal interior and boundary symbols, namely,  .a/.r; x; y; %; ; / WD  .a11 /.r; x; y; %; ; /; .%; ; / 6D 0, and @ .a//.r; x 0 ; y; %; 0 ; / WD @ ..aij /i;j D1;2 /.r; x 0 ; y; %; 0 ; /; .%; 0 ; / 6D 0, respectively. As usual, we have  .a/ 2 C 1 .T  .  X ^ / n 0/ with standard homogeneity of order  in the covariables .%; ; /, while @ .a/ is a family of operators @ .a/.r; x 0 ; y; %; 0 ; / W H s .RC / ˚ C ! H s .RC / ˚ C, s > d  12 , smoothly parametrised by .r; x 0 ; y; %; 0 ; / 2 T  ..@X ^ //n0, with the homogeneity @ .a/.r; x 0 ; y; %;  0 ; / 1

(7.2.28)

1

D  diag. ;  2 /@ .a/.r; x 0 ; y; %; 0 ; / diag. ;  2 /1 ;

. u/.xn / D 1=2 u.xn /;  2 RC . From the edge-degeneracy of the involved amplitude functions we have the relations  .a/.r; x; y; %; ; / D r  Q .a/.r; x; y; r%; ; r/; @ .a/.r; x 0 ; y; %; 0 ; / D r  Q @ .a/.r; x 0 ; y; r%; 0 ; r/; with homogeneous functions Q .a/.r; x; y; %; Q ; / Q and Q @ .a/.r; x 0 ; y; %; Q 0 ; / Q of analogous nature as  .a/ and @ .a/, respectively, but smooth in r up to 0. Proposition 7.2.18. Let a.y; / 2 R;d .  Rq ; g/,  Rq open, g D .;   /. Then Op.a/ induces continuous operators s 1

1

s nC1

1

s Op.a/ W Wcomp .; K s; .X ^ // ˚ Wcomp2 .; K s 2 ; 2 ..@X /^ // ˚ Hcomp 2 ./ s 1 2

s ! Wloc .; K s; .X ^ // ˚ Wloc

s nC1 2

˚ Hloc

1

1

.; K s 2 ; 2 ..@X /^ // ./

for all s > d  12 . Proof. By virtue of the relations (7.2.26) it suffices to apply Corollary 2.3.44.

422

7 Operators on manifolds with edges and boundary

Theorem 7.2.19. a.y; / 2 R;d .Rq ; g/, b.y; / 2 R;e .Rq ; l / for ;  2 Z, d; e 2 N, g WD . ;  .C//, l WD .;  /, implies .ab/.y; / 2 RC;h . Rq , g ı l / for h D max. C d; e/, g ı l D .;   . C //, and we have  .ab/ D  .a/ .b/;

@ .ab/ D @ .a/@ .b/;

^ .ab/ D ^ .a/^ .b/:

Moreover, if a or b belongs to the subspace with subscript ‘G’ (or ‘M C G’), then the same is true of ab. The details of the proof of Theorem 7.2.19 are voluminous; some elements are nevertheless straightforward. For simplicity we assume that the functions '; '; Q ::: 2 C01 .RC / involved in a, b (see the formula (7.2.6)) vanish; the general case is simple and left to the reader. For the proof we first show a number of lemmas. Lemma 7.2.20. Let Q a.y; / WD  r  faM .y; / C a .y; /g;  b.y; / WD  r fbM .y; / C b .y; /gQ for ;  2 Z, where a.y; / is as in the formula (7.2.6) and b.y; / of analogous structure, cf. the notation in the proof. Then there is a flat (of infinite order, see Definition 7.2.12) Green symbol g.y; / of order  C  such that .ab/.y; / D  r .C/ fcM .y; / C c .y; /gQ C g.y; /; where cM .y; / and c .y; / are again of analogous structure as aM .y; / and a .y; /, respectively, now of order  C . We have  .ab/ D  .a/ .b/;

@ .ab/ D @ .a/@ .b/;

^ .ab/ D ^ .a/^ .b/: (7.2.29)

Proof. The arguments of the proof will not depend in an essential way on the presence of the variables y. Therefore, for convenience, we content ourselves with the y-independent case. Let us write a./ D ! a0 ./!Q  C .1  ! /a1 ./.1  !QQ  / for ! .r/ WD !.rΠ/, etc., and  n 2

a0 ./ WD  r  opM

.h/./; Q

a1 ./ D  r  opr .p/./: Q

Similarly, we set b./ D ! b0 ./!Q  C .1  ! /b1 ./.1  !QQ  / for

 n 2

b0 ./ D  r  opM

.f /./; Q b1 ./ D  r  opr .d /./: Q

423

7.2 The edge algebra

We will often omit  in the following computations. Then we obtain ab D S C R1 C R2 for QQ S WD !a0 !b0 !Q C .1  !/a1 .1  !/b1 .1  !/; QQ 0 !: R1 WD !a0 .!Q  !/b1 .1  !/; QQ R2 WD .1  !/a1 .!  !/b Q Here and in the sequel we systematically apply the properties !Q 1 on supp !, ! 1 QQ which implies ! !Q D !, etc. We then obtain on supp !, QQ 0 .!Q  !/b1 .1  !/ QQ C .!  !/a QQ 0 .!Q  !/b1 .1  !/ QQ D G1 C R z1 R1 D !a for QQ 0 .!Q  !/b1 .1  !/ QQ C .!  !/a QQ 0 .!Q  !/b1 .1  !/; Q G1 WD !a Q Q z R1 WD .!  !/a Q 0 .!Q  !/b1 .!Q  !/: Q

(7.2.30)

The operator family G1 is smoothing, i.e., -wise in B 1;h .X ^ /; this follows from QQ !Q  !/ D 0 and .!  !/.1 QQ the pseudo-locality, using the relations !.  !/ Q D !  !QQ  Q ! !Q C !Q !Q D 0. For similar reasons also the remainders Gk below will be smoothing; more precisely, they will be identified as flat Green symbols in . Moreover, we have QQ 0 !Q C .!Q  !/a1 .!  !/b QQ 0 !Q D G2 C R z2 Q 1 .!  !/b R2 D .1  !/a for QQ QQ 0 !Q C .!Q  !/a1 .!  !/b QQ 0 !; Q 1 .!  !/b G2 WD .1  !/a z2 WD .!Q  !/a1 .!  !/b R QQ 0 .!Q  !/: QQ For S we have S D P C T1 C T2 where QQ 1 .1  !/; QQ Q 0 !Q C .1  !/a1 .1  !/b P WD !a0 !b QQ T1 WD !a0 .!  !/b Q 0 !; Q T2 WD .1  !/a1 .!QQ  !/b1 .1  !/: Reformulating these expressions gives us QQ 0 .!  !/b QQ 0 .!  !/b Q 0 !Q C !a Q 0 !Q D G3 C Tz1 T1 D .!  !/a for QQ 0 .!  !/b QQ 0 .!  !/b Q 0 !QQ C !a Q 0 !; Q G3 WD .!  !/a QQ 0 .!  !/b QQ Tz1 WD .!  !/a Q 0 .!Q  !/:

424

7 Operators on manifolds with edges and boundary

Moreover, QQ C .!Q  !/a1 .!QQ  !/b1 .1  !/ QQ D G4 C Tz4 Q 1 .!QQ  !/b1 .1  !/ T2 D .1  !/a for QQ C .!Q  !/a1 .!QQ  !/b1 .1  !/; Q 1 .!QQ  !/b1 .1  !/ Q G4 WD .1  !/a Q Q z T2 WD .!Q  !/a1 .!Q  !/b1 .!Q  !/: Q It follows altogether ab D P C T1 C T2 C R1 C R2 P z1 C R z2 C 4 Gk and where T1 C T2 C R1 C R2 D Tz1 C Tz2 C R kD1

(7.2.31)

z1 C R z2 Tz1 C Tz2 C R QQ 0 .!  !/b QQ C .!Q  !/a1 .!QQ  !/b1 .!Q  !/ QQ D .!  !/a Q 0 .!Q  !/ QQ 0 .!Q  !/b1 .!Q  !/ QQ C .!Q  !/a1 .!  !/b QQ 0 .!Q  !/ QQ C .!  !/a D .!  !/a QQ 0 .!  !/b Q 0 .!Q  !/ QQ C .!Q  !/a0 .!QQ  !/b0 .!Q  !/ QQ QQ 0 .!Q  !/b0 .!Q  !/ QQ C .!Q  !/a0 .!  !/b QQ 0 .!Q  !/ QQ C .!  !/a C G5 for G5 WD L1 C L2 C L3 where QQ  .!Q  !/a0 .!QQ  !/b0 .!Q  !/; QQ L1 WD .!Q  !/a1 .!QQ  !/b1 .!Q  !/ L2 WD .!  !/a QQ 0 .!Q  !/b1 .!Q  !/ QQ  .!  !/a QQ 0 .!Q  !/b0 .!Q  !/; QQ QQ 0 .!Q  !/ QQ  .!Q  !/a0 .!  !/b QQ 0 .!Q  !/: QQ L3 WD .!Q  !/a1 .!  !/b Thus T1 C T2 C R1 C R2 D

5 X

Gk :

kD1

To analyse (7.2.31) we write QQ C G6 C G7 P D !a0 b0 !Q C .1  !/c1 .1  !/ for Q G6 WD !a0 .!Q  1/b0 !; QQ 1  c1 g.1  !/: QQ G7 WD .1  !/fa1 .1  !/b Here c1 is defined by

c1 ./ WD  r .C/ opr .v/./Q

(7.2.32)

with an operator function v.r; %; / D v.r; Q r%; r/, where x C  R1Cq / v.r; Q %; Q / Q 2 B C;h .X I R %; Q Q

(7.2.33)

7.2 The edge algebra

425

is defined by the Leibniz product with respect to r, namely, Q r%; r/: Q r%; r/ D r  p.r; Q r%; r/ #r Q .r/ .r/r  d.r; r .C/ v.r; Recall that the involved factors come from p.r; %; / D p.r; Q r%; r/;

Q r%; r/ d.r; %; / D d.r;

Q %; x C R1Cq /, d.r; x C R1Cq /. This Q / Q 2 B ;e .X I R with given p.r; Q %; Q / Q 2 B ;d .X I R %; Q Q %; Q Q allows us to compute (7.2.33); in particular, v.r; Q %; Q / Q can be chosen to be smooth up to r D 0. Moreover, we have  n 2

.a0 b0 /./ D  r  opM D

 n 2

.h/./Q  r  opM

.f /./Q

 n  n  r .C/ opM 2 .T  h/./ opM 2 .Q f

/./: Q

In the latter relation we employed Remark 6.1.82. By virtue of the analogue of Proposition 6.1.81 for the case of a compact C 1 manifold X with boundary we therefore obtain  n .a0 b0 /./ D  r .C/ opM 2 .l/./Q (7.2.34) Q w; r/, l.r; Q w; / x C ; MC;h .X I Rq //. for a Mellin symbol l.r; w; / D l.r; Q 2 C 1 .R O Q Summing up it follows that .ab/./ D ! c0 ./!Q  C .1  ! /c1 ./.1  !QQ  / C g./; where c0 ./ D .a0 b0 /./ is of the form (7.2.34), moreover, c1 ./ is given by (7.2.32), and 7 X Gk ./: g./ WD kD1

The next point is to verify that g./ is a flat Green symbol in the sense of Definition 7.2.10. This can be done for every Gk ./, k D 1; : : : ; 7. Let us consider for instance, G1 ./. By the formula (7.2.30) we have two summands, and we may consider them separately. The first summand has the form G D CD QQ and any ' 2 C01 .RC / that is equal to 1 for C WD !a QQ 0 .!Q  !/ and D WD 'b1 .1  !/ on supp.!Q  !/. The function ' is interpreted, similarly as the cut-off functions, as an -dependent factor, i.e., ' D ' for ' .r/ WD '.rΠ/. The operator family C is flat and Green by Example 7.2.13. It is then an easy task to verify that the composition with D gives us again a flat Green symbol. The second summand of G1 has the form QQ 0 .!Q  !/b1 .1  !/: Q G WD .!  !/a

426

7 Operators on manifolds with edges and boundary

Choose another cut-off function !0 that is equal to 1 on supp ! and such that !Q is equal to 1 on supp !0 (this is always possible). Then G can be written in the form G DH CL QQ 0 !0 .!Q  !/b1 .1  !/. Q L WD .!  !/a Q for H WD .!  !/a QQ 0 .1  !0 /.!Q  !/b1 .1  !/, We then have H D CD for QQ 0 .1  !0 /'; C WD .!  !/a

D WD .!Q  !/b1 .1  !/ Q

for any ' 2 C01 .RC / such that ' 1 on supp.!Q  !/. (In this proof, C and D occur in different meaning which will be clear the context). Since 1  !0 vanishes QQ the factor C is smoothing and can be treated in a similar manner as on supp.!  !/, the corresponding operator C occurring in the first summand. Similarly as above we obtain that H is a flat Green symbol. Moreover, taking some ' 2 C01 .RC / that is equal to 1 on supp.!Q  !/ we can write L D DC for QQ 0 .!Q  !/; D WD .!  !/a

C WD !0 'b1 .1  !/: Q

Since 1  !Q vanishes on supp !0 , the family C is smoothing and a flat Green symbol. This implies the same for L. The arguments for Gk , k D 2; : : : ; 8, are similar and left to the reader. The symbolic relations (7.2.29) concerning  and @ are a direct consequence of the behaviour of interior and boundary symbols under composition of parameter-dependent boundary value problems. The composition rule of edge symbols ^ follows from the above computations for the composition of corresponding operator functions, where Œ is replaced by jj and all the first r-variables are frozen at zero (cf. the formula (7.2.4)). ;d ;e q q Lemma 7.2.21. Let a 2 RM CG .  R ; g/, b 2 RM CG .  R ; l / for ;  2 R, C;h d; e 2 N, and g; l as in Theorem 7.2.19. Then we have ab 2 RM CG .  Rq ; g ı l / for h D max. C d; e/ and ^ .ab/ D ^ .a/^ .b/. If a or b belongs to the subspace with subscript ‘G’, then the same is true of the composition.

Proof. If both factors belong to the space with subscript ‘G’, then ab is a composition between Green symbols and the result is again of Green type; this is elementary. The assertion in general is a consequence of the fact that the pointwise composition of the corresponding classical operator-valued symbols is again a classical operator-valued symbol, and the values of the symbols are to be composed as in the cone algebra on X ^ as in Chapter 6. Thus the latter compositions are smoothing Mellin plus Green and Green if so is one of the factors. The symbolic rule is the same as for the principal symbols of composition of classical operator-valued symbols. Lemma 7.2.22. Let a; b be as in Theorem 7.2.19, and assume one of the factors to be of Green (or smoothing Mellin plus Green) type. Then the same is true of the composition, and we have ^ .ab/ D ^ .a/^ .b/.

7.2 The edge algebra

427

Proof. Let us assume, for instance, that b is of Green (or Mellin plus Green) type (the case when a is of that kind can be treated in a similar manner). Then, after Lemma 7.2.21 it is enough to assume that a D diag.a; 0/ for a symbol a.y; / as in Lemma 7.2.20. It is obviously enough to look at a.y; /b.y; / where b.y; / is the upper left 22 corner of b.y; /. Moreover, without loss of generality, in the present discussion we may omit the factors .r/ and Q .r/ at the symbol, since Q b is of the same nature as b itself, and after having proved that r  aM .y; /b.y; / and r  a .y; /b.y; / is of Green (or smoothing Mellin plus Green) type we have the same for the composition from the left with  . In other words, it is sufficient to look at r  aM .y; / and r  a .y; /b.y; /. Let us discuss the second term, the first one is of analogous nature. We have QQ a .y; / D .1  !.rΠ// opr .p/.y; /.1  !.rΠ// for p.r; y; %; / D p.r; Q y; r%; r/, cf. the formula (7.2.1), p.r; Q y; %; Q / Q 2 B ;d .X I PN 1Cq x RC    R%; /. The Taylor formula gives us p.r; Q y; %; Q / Q D j D0 r j pQj .y; %; Q / Q C Q Q r N C1 pQ.N C1/ .r; y; %; Q / Q with functions pQj .y; %; Q / Q 2 B ;d .X I   R1Cq / and %; Q Q 1Cq ;d x pQ.N C1/ .r; y; %; Q / Q 2 B .X I RC    R%; /. This yields a corresponding repQ Q resentation of p itself, namely, p.r; y; %; / D

N X

r j pj .y; %; / C r N C1 p.N C1/ .r; y; %; /;

j D0

with obvious meaning of notation on the right-hand side. It follows that r



a .y; /b.y; / D

N X

cj .y; / C c.N C1/ .y; /

j D0

QQ for cj .y; / D r  .1!.rΠ//r j opr .pj /.y; /.1 !.rΠ//b.y; / and c.N C1/ .y; / is defined in an analogous manner in terms of opr .p.N C1/ /. The values of b.y; / are Green (or smoothing Mellin plus Green) in the cone algebra on X ^ . Therefore, also cj .y; / takes values in the same kind of operators. Moreover, cj .y; / is a classical symbol of order C j . Computing the homogeneous components we obtain Green (or smoothing Mellin plus Green) operator-valued functions. This shows altogether that every cj .y; / is of the desired type, namely, smoothing Mellin plus Green if so is b.y; /, and always Green for j > 0. Concerning c.N C1/ .y; / we apply a tensor product argument and write pQ.N C1/ .r; y; %; Q / Q as a convergent sum Q / Q D pQ.N C1/ .r; y; %;

1 X

k 'k pQ.N C1/;k .y; %; Q / Q

kD0

P 1 with 1 kD0 jk j < 1, 'k .r/ 2 C0 .Œ0; R/0 / for some R > 0 (using that, without loss of generality, the function pQ.N C1/ may assumed to be vanishing for r > R),

428

7 Operators on manifolds with edges and boundary

with null-sequences 'k , pQ.N C1/;k in the spaces C01 .Œ0; R/0 / and B ;d .X I   R1Cq /, respectively. Now for p.N C1/;k .r; y; %; / WD pQ.N C1/;k .y; r%; r/, we have c.N C1/ .y; / D

1 X

k 'k .r/d.N C1/;k .y; /;

(7.2.35)

kD0

d.N C1/;k .y; / D r CN C1 .1  !.rΠ// opr .p.N C1/;k /.y; /.1  !.rΠ//b.y; QQ /. After the discussion before, the latter symbols are Green, and it can easily be proved that they tend to zero in the space of Green symbols as k ! 1. The same is true of 'k d.N C1/;k .y; /, which means that (7.2.35) converges in the space of Green symbols. Thus the composition r  a .y; /b.y; / is completely characterised. Proof of Theorem 7.2.19. Writing the symbols a.y; / and b.y; / as in Definition 7.2.16 we see that all occurring summands in the composition are treated by the Lemmas 7.2.21 and 7.2.22.

7.2.5 The edge algebra We now establish the algebra of boundary value problems on a manifold with edges. The main ingredients close to the edge in the local splitting of variables .r; x; y/ 2 X ^   have the form Op.a/ for some a.y; / 2 R;d .  Rq ; g/, cf. Proposition 7.2.18. For the complete definition we need a few preparations. Let W be a manifold with edge Y and boundary, not necessarily compact, let W be the associated stretched manifold, and 2W its double. By definition, a cut-off functions on W is any function of the form  D !jW for an ! 2 C 1 .2W / supported in a collar neighbourhood of .2W /sing , and equal to 1 in a neighbourhood of .2W /sing . Let .U00 /2I be a locally finite open covering of Y by coordinate neighbourhoods, with charts 00 W U00 ! Rq . Moreover, let .U /2I be a system of wedge neighbourhoods on W such that U00 D U \ Y , with singular charts U ! X  Rq that induce (by restriction) the charts 00 and restrict to diffeomorphisms   W U  n Y ! X ^  Rq ;

 2 I:

The sets U0 WD @.U n Y / [ U00 are wedge neighbourhoods on V D @.W n Y / [ Y with singular charts U0 ! .@X /  Rq (obtained by restriction of the singular charts U ! X  Rq ) which restrict to diffeomorphisms 0 W U0 n Y ! .@X /^  Rq ;

 2 I:

Let U and U0 be the stretched manifolds associated with U and U0 , respectively, and let 2U and 2U0 be the corresponding doubles. 1 P Let us fix a system of functions .' /2I belonging to C0 .2U /jU , such that 2I ' 1 in a neighbourhood of Wsing , and let .  /2I be another system of functions in C01 .2U /jU , such that  1 on supp ' for all . Set '0 WD ' jU0 ,

7.2 The edge algebra 0 

429

 2 I . Moreover, let .'00 /2I be a partition of unity subordinated to and . 00 /2I be another system of functions 00 2 C01 .U00 / such that 00 1 on supp '00 for all . Q #QQ be cut-off functions on W such that #Q 1 on supp #, # 1 on Finally, let #; #; QQ are cut-off functions on V . We now QQ Then # 0 WD #j , #Q 0 WD #j Q V , #QQ 0 WD #j supp #. V V give the definition of the so-called edge algebra. WD

.U00 /2I

 jU0 ;

Definition 7.2.23. Let  2 Z; d 2 N, and g D .;   /. Then Y;d .W ; g/ denotes the space of all operators of the form A D Aedge C Aint C C;

(7.2.36)

where

P 0 00 1 0 0 00 Q (i) Aedge WD 2I diag.. ;  ;  / / diag.#' ; # ' ; ' / Opy .a / diag.#  ; 0 0 00 ;d q q Q #  ;  / for arbitrary a .y; / 2 R .R  R ; g/,  2 I ;   QQ 1  #QQ 0 ; 0/ for an arbitrary (ii) Aint WD diag.1  #; 1  # 0 ; 0/ A0int 00 diag.1  #; Aint 2 B ;d .W n Y /;

(iii) C 2 Y1;d .W ; g/. ;d ;d ;d Let YM .W ; g/ such CG .W ; g/ (YG .W ; g/) denote the subspace of all A 2 Y ;d ;d q q q q that Aint D 0 and a .y; / 2 RM CG .R  R ; g/ .RG .R  R ; g// for all  2 I , see Definition 7.2.12 and 7.2.16, respectively.

The operators A 2 Y;d .W ; g/ are 3  3 block matrices. Writing A D .Aij /i;j D1;2;3 ;

A D .Aij /i;j D1;2

(7.2.37)

we have A 2 B ;d .W n Y /, and we set  .A/ WD  .A/;

@ .A/ WD @ .A/;

which are the homogeneous principal interior and boundary symbols, respectively. We have  .A/ 2 C 1 .T  .W nY /n0/ with (standard) homogeneity of order  in the fibre variables of T  .W n Y / n 0, and @ .A/ is a family of operators @ .A/ W H s .RC / ! H s .RC /, s > d  12 , smoothly parametrised by the points of T  .V n Y / n 0 and homogeneous of order  in the fibre variables of T  .V n Y / n 0 in the same sense as (7.2.28). Moreover, we have Q ; /; Q Q .A/.r; x; y; %;

Q @ .A/.r; x 0 ; y; %; Q 0 ; / Q

Q ; ) Q and Q @ .a/.r; x 0 ; y; %; Q 0 ; / Q in Secof analogous meaning as Q .a/.r; x; y; %; tion 7.2.4, smooth in r up to 0, and satisfying the identities  .A/.r; x; y; %; ; / D r  Q .A/.r; x; y; r%; ; r/; @ .A/.r; x 0 ; y; %; 0 ; / D r  Q @ .A/.r; x 0 ; y; r%; 0 ; r/:

430

7 Operators on manifolds with edges and boundary

Finally, close to the edge Y we have the homogeneous principal edge symbol ^ .A/ which follows from ^ .a / in terms of the local amplitude functions a .y; /. The homogeneous principal edge symbol is a family of operators ^ .A/.y; / W K s; .X ^ / ˚ C ! K s; .X ^ / ˚ C 1

(7.2.38) 1

for s > d  12 . Recall that K s; .X ^ / WD K s; .X ^ / ˚ K s 2 ; 2 ..@X /^ /. The homogeneity of (7.2.38) is as in (7.2.27). Summing up, the operators A 2 Y;d .W ; g/ have the principal symbolic hierarchy  .A/ D . .A/; @ .A/; ^ .A//: Observe that ^ .A/.y; / 2 C;d .X ^ ; g/ for every fixed .y; / 2 T  Y n 0. Thus we have the subordinate principal symbolic structure from the cone calculus, see Definition 6.2.10 and the formula (6.2.18). In particular, we have the (principal) conormal symbol as a family of operators 1

1

c ^ .A/.y; w/ W H s .int X / ˚ H s 2 .@X / ! H s .int X / ˚ H s 2 .@X / for every s > d  12 . Theorem 7.2.24. Every A 2 Y;d .W ; g/ with g D .;   / induces continuous s nC1

s nC1 2

2 operators A W W s; .Y / ! W s; .W / ˚ Hloc comp .W / ˚ Hcomp loc

.Y / for

1 s 1 2 ; 2

s; .V /, and the same with all s > d  12 ; here W s; comp .W / WD Wcomp .W / ˚ Wcomp ‘loc’. In the case of compact W we may omit the ‘comp’ or ‘loc’ subscripts and obtain continuous operators

A W W s; .W / ˚ H s

nC1 2

.Y / ! W s; .W / ˚ H s

nC1 2

.Y /

(7.2.39)

for all s > d  12 . Proof. Using Definition 7.2.23 the continuity of Aedge follows from Proposition 7.2.18, the one of Aint from the standard continuity of boundary value problems with the transmission property in Sobolov spaces, and the continuity of C from Definition 7.2.8 and the properties of the various entries. Remark 7.2.25. Let A 2 Y;d .W ; g/, g D .;   /, W compact, and  .A/ D 0. Then the operators (7.2.39) are compact for all s > d  12 . Remark 7.2.26. Let A 2 Y;d .W ; g/, and let A 2 B 1;d .W n Y / (in the notation ;d (7.2.37)). Then we have A 2 YM CG .W ; g/. Remark 7.2.27. A consequence of Theorem 7.2.24 is that A W C01 .W n Y / ˚ C01 .V n Y / ˚ C01 .Y / ! C 1 .W n Y / ˚ C 1 .V n Y / ˚ C 1 .Y / is a continuous operator. Thus the entries of A D .Aij /i;j D1;2;3 have distributional kernels K.A11 / 2 D 0 .int.W nY /int.W nY //, K.A12 / 2 D 0 .int.W nY /.V nY //, K.A13 / 2 D 0 .int.W n Y /  Y /, etc.

7.2 The edge algebra

431

We want to introduce properly supported operators in Y;d .W ; g/. To this end we observe that when !.y; y 0 / 2 C 1 .Rq  Rq / is a function with proper support in Rq  Rq such that !.y; y 0 / D 1 in an open neighbourhood of diag.Rn  Rn / (cf. the notation after Remark 3.2.15) the operator X diag.. ; 0 ; 00 /1 / diag.#' ; # 0 '0 ; '00 /Opy .!a /diag.#Q  ; #Q 0 0 ; 00 / Aedge;0 WD 2I

(in the notation of Definition 7.2.23 (i)) belongs to Y;d .W ; g/ and has the property Aedge D Aedge;0 C G

(7.2.40)

for some G 2 Y1;d .W ; g/. Moreover, writing Aint (in the notation of Definition 7.2.23 (ii)) in the form Aint D Aint;0 C G for a G 2 B 1;d .W n Y / and a properly supported element Aint;0 2 B ;d .W nY / (which exists according to Proposition 3.2.16) we obtain a decomposition of Aint as Aint D Aint;0 C G for another G 2 Y1;d .W ; g/ when we set  Aint;0 0 Aint;0 WD diag.1  #; 1  # ; 0/ 0

(7.2.41)

 0 QQ 1  #QQ 0 ; 0/: diag.1  #; 0

Let us call A0 WD Aedge;0 C Aint;0 a properly supported element of Y;d .W ; g/. According to (7.2.40), (7.2.41) it follows that A D A0 C C for a C 2 Y1;d .W ; g/: (7.2.42) Summing up we obtain the following result: Proposition 7.2.28. Let W be a manifold with edge Y and boundary. Then every A 2 Y;d .W ; g/ can be written in the form (7.2.42) for a properly supported element A0 2 Y;d .W ; g/. Remark 7.2.29. Let A0 2 Y;d .W ; g/ be properly supported; then A0 induces continuous operators C01 .W n Y / ˚ C01 .V n Y / ˚ C01 .Y / ! C01 .W n Y / ˚ C01 .V n Y / ˚ C01 .Y /; C 1 .W n Y / ˚ C 1 .V n Y / ˚ C 1 .Y / ! C 1 .W n Y / ˚ C 1 .V n Y / ˚ C 1 .Y / and extends to continuous operators s nC1

s nC1 2

2 W s; .Y / ! W s; .W / ˚ Hcomp comp .W / ˚ Hcomp comp

W s; loc .W / ˚ for all s > d  12 .

s nC1 Hloc 2 .Y /

! W s; .W / ˚ loc

.Y /;

s nC1 2 Hloc .Y /

432

7 Operators on manifolds with edges and boundary

Theorem 7.2.30. Let A 2 Y;d .W ; g/, B 2 Y;e .W ; l / for ;  2 Z, d; e 2 N, g WD .  ;   . C //, l WD .;   /, and let A or B be properly supported. Then we have AB 2 YC;h .W ; g ı l / for h D max. C d; e/, g ı l D .;   . C //, and .AB/ D  .A/ .B/ with componentwise multiplication on the right-hand side. Moreover, if A or B belongs to the subspace with subscript ‘G’ or ‘M C G’, then the same is true of AB. Proof. First note that A 2 Y1;d .W ; g/ or B 2 Y1;e .W ; l / implies AB 2 Y1;h .W ; g ı l /. The details are voluminous but altogether elementary and left to the reader. Similarly as (7.2.36) we employ the notation Bedge and Bint for the corresponding summands of B. The composition between Aint and Bint is entirely an operation between boundary value problems with the transmission property at the (smooth part) of the boundary and the operators are supported outside Wsing . Therefore, the composition Aint Bint is of the desired structure. The compositions Aint Bedge and Aedge Bint can be reduced to the composition between ‘int’ ingredients, modulo operators in Y1;h .W ; g ı l /. The reason is that when Aedge or Bedge is multiplied from one side by a C 1 function that vanishes close to Wsing , this composition can be rewritten as an operator which is supported away from Wsing , plus a smoothing operator; this is a simple effect of pseudo-locality. The composition Aedge Bedge can be treated in a similar manner as the composition between pseudo-differential operators, here with operator-valued symbols. The local amplitude functions in the present case are as in Theorem 7.2.19. The pointwise composition between them is again of the desired structure, cf. Theorem 7.2.19, and this implies a similar property between the Leibniz products. Thus Aedge Bedge is locally near Wsing a pseudo-differential operator with a symbol in RC;h .  Rq I g ı l /, modulo a smoothing operator of the calculus. At the same time from Theorem 7.2.19 we see that if one of the factors is Green or smoothing Mellin plus Green, the same is true of the composition. The composition behaviour .A/.B/ D  .AB/ follows from Theorem 7.2.19 together with the behaviour of symbols (interior/boundary) in the composition of boundary value problems. Corollary 7.2.31. Under the conditions of Theorem 7.2.30 we have c ^ .AB/.y; w/ D c ^ .A/.y; w C /c ^ .B/.y; w/: The calculus of Y;d .W ; g/ has a natural extension to systems, or, more generally, to operators between distributional sections of vector bundles. For the ellipticity this is a natural context, even if we start from ‘scalar’ upper left corners. Let us briefly give the corresponding formulations. Recall that Vect.  / denotes the set of all smooth complex vector bundles on the manifold in the brackets. In all such bundles in consideration we fix Hermitian metrics. There is a straightforward generalisation of the s; s; s; s; spaces Wcomp .W /, Wloc .W / to spaces Wcomp .W ; E/ and Wloc .W ; E/ of distributional sections in E 2 Vect.W /. Similarly as pseudo-differential operators on a C 1 manifold we have an analogue of Definition 7.2.23 in the case of bundles v WD ..E; E 0 ; J /; .F; F 0 ; JC //;

(7.2.43)

7.2 The edge algebra

433

for E; F 2 Vect.W / (smoothly extendible to bundles on 2W ), E 0 ; F 0 2 Vect.V /, J ; JC 2 Vect.Y /. The generalisation of Y;d .W ; gI v/ consists of a space Y;d .W ; gI v/ of continuous operators s nC1

s nC1 2

2 A W W s; .Y; J / ! W s; .W ; F / ˚ Hloc comp .W ; E / ˚ Hcomp loc

.Y; JC /

s 1 ; 1

2 2 s; s > d  12 , for W s; .V ; E 0 /, and, simicomp .W ; E / WD Wcomp .W ; E/ ˚ Wcomp 0 0 larly, with ‘loc’, or with F WD .F; F / instead of E WD .E; E /. The definition of Y;d .W ; gI v/ is nearly the same as for trivial bundles of fibre dimension 1. The main difference is that the local amplitude functions a .y; / now refer to the involved bundles and that the pull backs of operators to the manifold are defined in combination with the transition maps of the bundles referring to different localisations. Another point is that the bundles E; E 0 and F; F 0 are not entirely trivialised in a .y; / but induce bundles on the infinite stretched cones X ^ and .@X /^ , respectively. For brevity we denote them by E^ ; E^0 and F^ ; F^0 , respectively. The components of  .A/ for A 2 Y;d .W ; gI v/ are now bundle morphisms

  E ! W F;  .A/ W W

W W T  Wreg n 0 ! Wreg , 0 1 0 1 E@ ˝ H s .RC / F@ ˝ H s .RC / A ! V @ A; ˚ ˚ @ .A/ W V @ 0 E F0 ˇ ˇ V W T  Vreg n 0 ! Vreg , E@ WD E ˇVreg , F@ WD F ˇVreg , and 0

1

0

(7.2.44)

(7.2.45)

1

E^ ˝ K s; .X ^ / F^ ˝ K s;  .X ^ / ˚ ˚ B C B C B 0 C C 1 1 1 1 B ^ .A/ W Y BE^ ˝ K s 2 ;  2 ..@X /^ /C ! Y BF^0 ˝ K s 2 ;  2 ..@X /^ /C ; @ A @ A ˚ ˚ J JC

(7.2.46) Y W T  Y n 0 ! Y . Clearly we also have the variants of the symbols Q and Q @ close to the edge, in the variables .r; x; y; %; Q ; / Q and .r; x 0 ; y; %; Q 0 ; /, Q respectively, smooth in r up to 0. Remark 7.2.32. The results and observations on edge boundary value problems have obvious generalisations to the case of operators between distributional sections of vector bundles. Another generalisation of the calculus of Y;d .W ; g/ concerns the case of parameter-dependent operators. Let Y;d .W ; gI Rl / denote the space of all families of operators of the form A./ D Aedge ./ C Aint ./ C C./;

434

7 Operators on manifolds with edges and boundary

 2 Rl , where Aedge ./ is defined in an analogous manner as Aedge in Definition 7.2.23 (i) with the only difference that the local amplitude functions are elements a .y; ; / 2 R;d .Rq  RqCl ; g/,  2 I ; moreover, Aint ./ is defined by inserting ; arbitrary Aint ./ 2 B ;d .W n Y I Rl / in Definition 7.2.23 (ii); finally, C./ 2 S.Rl ; Y1;d .W ; g// DW Y1;d .W ; gI Rl /: Clearly we have also the parameter-dependent versions Y;d .W ; gI vI Rl /

(7.2.47)

in the case of tuples (7.2.43) of vector bundles. The symbols of elements A./ in (7.2.47) contain  2 Rl as an additional covariable, with the same degeneracy as in  (i.e., combined with the factor r 2 RC ). Observe that when A./ belongs to (7.2.47) we have A.1 / 2 Y;d .W ; gI v/

(7.2.48)

for every fixed 1 2 Rl . Remark 7.2.33. The results and observations on Y;d .W ; gI v/ have obvious generalisations to Y;d .W ; gI vI Rl /.

7.2.6 Ellipticity and reductions of orders In this section we study ellipticity, parametrices, and other useful properties of operators in the edge calculus. For notational convenience we mainly consider the case of trivial bundles of fibre dimension 1; analogous results will be true of operators between spaces of distributional sections in vector bundles (7.2.43). Definition 7.2.34. An element A 2 Y;d .W I g/, g D .;   /, is called elliptic, if (i)  .A/ 6D 0 on T  .W n Y / n 0 and Q .A/.r; x; y; %; ; / 6D 0 for .%; ; / 6D 0, up to r D 0, in local splittings of variables near the edge; (ii) the operators @ .A/ W H s .RC / ˚ C ! H s .RC / ˚ C are bijective for all points of T  .V n Y / n 0, s > d  12 , and Q @ .A/.r; x 0 ; y; %; 0 ; / are bijective for all .%; 0 ; / 6D 0, up to r D 0; (iii) the operators ^ .A/ W K s; .X ^ / ˚ C ! K s; .X ^ / ˚ C

(7.2.49)

are bijective for all points of T  Y n 0, s > d  12 . If an A 2 Y;d .W ; g/ only satisfies (i) (or (i) and (ii)) of the latter definition we also speak about  - (or . ; @ /-) ellipticity.

7.2 The edge algebra

435

Remark 7.2.35. The definition of ellipticity of operators A 2 Y;d .W ; gI v/ for (7.2.43) in general consists of the requirement that (7.2.44), (7.2.45) and (7.2.46) are isomorphisms, s > max.; d/ >  12 , and that Q .A/ and Q @ .A/ are corresponding isomorphisms up to r D 0. Theorem 7.2.36. Let A 2 Y;d .W ; g/ with g D .;   / be elliptic. Then there is a (properly supported) parametrix C

P 2 Y;.d/ .W ; g 1 / for g 1 D .  ; /; i.e., Gl WD I  PA and Gr WD I  AP belong to Y1;dl .W ; g l / and Y1;dr .W ; g r /, respectively, where dl D max.; d/, dr D .d  /C , and g l D .;  /, g r D .  ;   /. We have  .P/ D  .A/1 with componentwise inverses. Proof. Writing A D .Aij /i;j D1;2;3 we have Aint WD .Aij /i;j D1;2 2 B ;d .Wreg /, and Aint is elliptic with respect to  and @ . Therefore, Aint has a properly supported C parametrix Pint 2 B ;.d/ .Wreg /. According to Definition 7.2.23 we will obtain the parametrix P in the form P D Pedge C Pint where Pint WD diag.1  #; 1  # 0 ; 0/ Pedge WD

X 2I

P

int 0 0 0



(7.2.50)

QQ 1  #QQ 0 ; 0/ and diag.1  #;

diag.. ; 0 ; 00 /1 / diag.#' ; # 0 '0 ; '00 / Opy .p / diag.#Q

Q0 ; #

0 ;

00  /

C

(7.2.51)

for suitable p .y; / 2 R;.d/ .Rq  Rq ; g 1 /,  2 I . Therefore, it is enough to construct the amplitude functions p .y; /. We fix  and then omit it. In other words, the starting point is an a.y; / 2 R;d .Rq  Rq ; g/ and we find p.y; /. As in Definition 7.2.16 we write a.y; / D diag.a.y; / C m.y; /; 0/ C g.y; / with a.y; / of the form (7.2.6), a Mellin edge amplitude function (7.2.25) and a Green symbol g.y; /, of order  and type d, see Definition 7.2.12. Without loss of generality we assume that the cut-off functions  and Q are chosen in such a way that (in the splitting Q of variables .r; x; y/ near Wsing )  1 on supp #, Q 1 on supp #. The first essential aspect for the construction of p.y; / is the observation that the ellipticity of A with respect to  and @ implies that the operator function x C  Rq  R1Cq / which is involved in (7.2.6) is parameterp.r; Q y; %; Q / Q 2 B ;d .X I R %; Q Q x C Rq . This allows us dependent elliptic with the parameters .%; Q /, Q for every .r; y/ 2 R .1/ ;.d/C x C Rq R1Cq / to construct an operator function pQ .r; y; %; Q / Q 2B .X I R %; Q Q such that C

Opr;y .r  pQ .1/ .r; y; r%; r// 2 B ;.d/ .RC  X  Rq /

436

7 Operators on manifolds with edges and boundary

is a parametrix of Opr;y .r  p.r; Q y; r%; r//. The construction of pQ .1/ follows from  Leibniz-inverting r p.r; Q y; r%; r/ with respect to the variables .r; y/, such that fr  pQ .1/ .r; y; r%; r/g #r;y fr  p.r; Q y; r%; r/g X 1 ˛  .r  p.r; Q y; r%; r// @˛%; .r  pQ .1/ .r; y; r%; r//Dr;y ˛Š ˛ is equal to the identity, modulo an operator family in B 1;dl .X I RC  Rq  R1Cq %; /. z ;.d/C .X I R x C  Rq  R1Cq / Now p .1/ .r; y; %; / WD pQ .1/ .r; y; r%; r/ 2 B z ;.d/C .X I R x C  Rq  C  Rq / gives rise to a Mellin symbol h.1/ .r; y; w; / 2 B via Theorem 7.2.1 which allows us to form an operator function .1/ .y; / C a.1/ .y; /gQ a.1/ .y; / WD  r  faM

(7.2.52)

with  n 2

.1/ aM .y; / WD !.rΠ/ opM

.h.1/ /.y; /!.rΠ/; Q

QQ a.1/ .y; / WD .1  !.rΠ// opr .p .1/ /.y; /.1  !.rΠ//; analogously as (7.2.6). If we take (7.2.52) as p.y; / DW p .y; / (where the third row and column is zero) and carry out the construction for every  2 I we can form the operator (7.2.50) in terms of (7.2.51) and Pint that was constructed before. This C gives us an element in Y;.d/ .W ; g 1 /. By Proposition 7.2.28 we can replace P by a properly supported representative, for simplicity, again denoted by P. Then we can form the operator I  PA. By construction this satisfies the assumption of Remark 7.2.26, and it follows that 0;dl I  PA 2 YM CG .W ; g l /:

(7.2.53)

This shows that P is not yet the final parametrix. In order to improve P we first replace in the above construction a.1/ .y; / by  n 2

a.1/ .y; / C r  !.rΠ/ opM

.f .1/ /.y/!.rΠ/ Q

(7.2.54)

C

for a suitable f .1/ .y; w/ 2 C 1 .Rq ; M1;.d/ .X I  nC1 ./ // (for conve2 nience, we talk about C 1 functions in y 2 Rq , although in (7.2.51) every summand contains a localisation on a compact set with respect to y). The principal conormal symbol of P then has the form c ^ .P/.y; w/ D h.1/ .0; y; w; 0/ C f .1/ .y; w/:

(7.2.55)

The principal conormal symbol of the given operator A is equal to c ^ .A/.y; w/ D h.0; y; w; 0/ C f .y; w/

(7.2.56)

7.2 The edge algebra

437

with h being given by (7.2.2) and f .y; w/ as in the formula (7.2.25). Applying Corollary 7.2.31 we have c ^ .PA/.y; w/ D .h.1/ .0; y; w C ; 0/ C f .1/ .y; w C //.h.0; y; w; 0/ C f .y; w//: From h.1/ .0; y; w C ; 0/h.0; y; w; 0/ D id Ck.y; w/ and h.1/ .0; y; w C ; 0/f .y; w/ D l.y; w/ for some k.y; w/; l.y; w/ 2 C 1 .Rq ; M1;dl .X I  nC1  // it follows that 2

h.1/ .0; y; w C ; 0/c ^ .A/.y; w/ D id Cm.y; w/ for m.y; w/ D .k C l/.y; w/ 2 C 1 .Rq ; M1;dl .X I  nC1  //. 2 is then Let us now fix for the moment the point y ˇD y0 . There  ˇ an operator ˇ < " for some family g.y0 ; w/ which is meromorphic in a strip ˇRe w  nC1   2 " > 0 with finitely many poles p0 ; : : : ; pN of finite multiplicity and finite rank Laurent coefficients at the negative powers of .pj  w/ belonging to B 1;dl .X / such that .id Cg.y0 ; w//.id Cm.y0 ; w// D id for all w in a little smaller strip, cf. Proposition 6.1.54. That means .id Cg.y0 ; w//h.1/ .0; y0 ; w C ; 0/ D h.1/ .0; y0 ; w C ; 0/ C g.y0 ; w/h.1/ .0; y0 ; w C ; 0/ is the inverse of c ^ .A/.y0 ; w/, i.e., we find f .1/ .y0 ; w C / D g.y0 ; w/h.1/ .0; y0 ; w C ; 0/: C

We have f .1/ .y0 ; w/ 2 M 1;.d/ .X I  nC1 ./ / because c ^ .A/.y0 ; w/ is 2

invertible for all w 2 fw 2 C W nC1    " < Re w < nC1   C "g for some " > 0; 2 2 thus the inverse is also holomorphic in such a strip. Varying y0 we can choose " > 0 independent of y0 in any compact subset of Rq . Since, as noted before, the expression (7.2.51) contains local amplitude functions localised over compact sets with respect to y, for convenience we admit y0 to vary over Rq . The smoothness of f .1/ .y; w/ in y is clear; so we have constructed the conormal symbol (7.2.55). Observe that when we construct P by using the local amplitude functions (7.2.54) rather than a.1/ .y; / we obtain the relation l I  PA 2 Y0;d G .W ; g l /

(7.2.57)

instead of (7.2.53). Another step for the construction of a parametrix of A is to invert the principal edge symbol ^ .A/.y; /. From (7.2.57) we see that when we set

438

7 Operators on manifolds with edges and boundary

A WD .Aij /i;j D1;2 and P WD .Pij /i;j D1;2 (while A D .Aij /i;j D1;2;3 and P D .Pij /i;j D1;2;3 ) the operators ^ .P /.y; / W K s; .X ^ / ! K s; .X ^ / are .y; /-wise a parametrix of ^ .A/.y; / W K s; .X ^ / ! K s; .X ^ / in the sense that ^ .P /.y; /^ .A/.y; / and ^ .A/.y; /^ .P /.y; / are identity operators modulo Green operator families on the infinite cone, cf. Definition 6.1.46. Since the Green operators are compact, it follows that ind ^ .A/.y; / D  ind ^ .P /.y; /

(7.2.58)

for all .y; / 2 T  Y n 0. Recall that for notational convenience we talk about operators with exactly one trace and one potential condition with respect to the edge Y , although all constructions have a straightforward extension to the general case where the principal symbols have the form (7.2.44), (7.2.45) and (7.2.46), cf. the remarks below. If A is elliptic with one trace and potential condition, ^ .A/ is as in Definition 7.2.34 (ii), i.e., ind ^ .A/.y; / D 0 for every .y; / 2 T  Y n 0 (because dim ker ^ .A/.y; / D dim coker ^ .A/ D 1). Thus, by virtue of (7.2.58) we have also ind ^ .P /.y; / D 0. We now employ the fact that for every .y0 ; 0 / 2 S  Y (represented under a chart as a point in Rq  S q1 ) there is a neighbourhood of .y0 ; 0 / in Rq  S q1 with compact closure of the form K  S q1 , K b Rq , and a C 1 family of isomorphisms 

 K s; .X ^ / K s; .X ^ / ^ .P / ^ .K/ ˚ ˚ b.y; / D .y; / W ! ; (7.2.59) ^ .T / ^ .Q/ CN CN .y; / 2 K S q1 . Let us briefly speak about isomorphisms in this context. The formal construction of ^ .K/; ^ .T /, ^ .Q/ is analogous to the proof of Theorem 6.2.23. However, here we can choose these families in a more specific way, namely, smooth in .y; / and with ranges of compact support in r 2 RC (since those functions are dense in the respective weighted spaces, cf. a similar conclusion in the proof of Theorem 6.2.23). In order to invert (7.2.49) it suffices to do so for ^ .A/.y; / ˚ idC N 1 DW a.y; / and then pass to the upper left corner of the resulting inverse. Since (7.2.56) and (7.2.59) are isomorphisms, also the composition b.y; /a.y; / W K s; .X ^ / ˚ C N ! K s; .X ^ / ˚ C N is an isomorphism. As noted before, we have ^ .P /.y; /^ .A/.y; / D 1 C ^ .G /.y; / for a family ^ .G /.y; / of Green operators in C dl .X ^ ; .;  //, cf. the notation (6.1.64). Thus, since the other contributions in the entries of ba are also of Green type, it follows that b.y; /a.y; / D 1  g.y; /

(7.2.60)

for a g.y; / 2 Cdl .X ^ ; .;  /I N; N /, cf. Definition 6.1.60 (ii), and (7.2.60) is invertible. This allows us to apply Proposition 6.1.59 in the variant for X ^ instead of B, cf

7.2 The edge algebra

439

Remark 6.1.61 and the corresponding proof in Section 6.2.2. In other words, we find a family of Green operators d.y; / 2 Cdl .X ^ ; .;  /I N; N / such that .1g.y; //1 D 1d.y; /. Multiplying (7.2.60) from the left by 1d.y; / it follows that a1 .y; / D C .1  d.y; //b.y; / which belongs .y; /-wise to C;.d/ .X ^ ; .  ; /I N; N / and is C 1 in .y; / 2 S  Y . Here we employed several times that compositions exist within the operator spaces in question. In sum we obtain that ^1 .A/.y; / is constructed when we extend the inverse from S  Y by twisted homogeneity in  to T  Y n 0. Moreover, we know that diag.^ .P /.y; /; 0/  ^1 .A/.y; / DW C

l.0/ .y; / D ^ .L/.y; / for a certain L 2 Y0;.d/ .W ; g 1 /. We can obtain L G by forming an operator by using l.y; / WD ./l.0/ .y; / with an excision function ./ as local amplitude functions in an expression of the kind (7.2.51) with l.y; / instead of p . According to Proposition 7.2.28 both P and L will be chosen to be properly supported. Now, setting P0 WD diag.P ; 0/  L we obtain an operator in C Y;.d/ .W ; g 1 / such that  .P0 / is the inverse of (7.2.49). P The final step of constructing a parametrix P is to form an asymptotic sum P WD j1D0 .I  P0 A/j P0 , when we insert for A a properly supported representative of the original operator. Corollary 7.2.37. Let A 2 Y;d .W ; g/, g D .;   /, be elliptic. Then s nC1 2

 .W / ˚ Hloc Au D f 2 W s; loc

for s > max.; d/  implies

1 2

.Y /

1 and u 2 W r; comp .W / ˚ Hcomp .Y / for any r > max.; d/ 

1 2

s u 2 W s; comp .W / ˚ Hcomp .Y /:

In fact, from Theorem 7.2.36 we have a properly supported parametrix P of A. From the continuity of s nC1 2

.W / ˚ Hloc P W W s; loc

s nC1 2

.Y / ! W s; loc .W / ˚ Hloc

.Y /

1; 1 r 0 and Gl W W r; comp .W / ˚ Hcomp .Y / ! W loc .W / ˚ Hloc .Y / for every r; r 2 R, r > max.; d/  12 , we obtain PAu D .I  Gl /u D Pf , i.e., 0

˚

˚ s nC1 1 2 .Y / C W 1; u 2 W s; comp .W / ˚ Hcomp .Y / loc .W / ˚ Hloc s nC1

2 .Y /: D W s; comp .W / ˚ Hcomp

Theorem 7.2.38. If W is compact and A 2 Y;d .W ; g/ elliptic, then A W W s; .W / ˚ H s

nC1 2

.Y / ! W s; .W / ˚ H s

nC1 2

.Y /

(7.2.61)

is a Fredholm operator for every s > max.; d/  12 . Proof. By virtue of Theorem 7.2.36 the operator A has a two-sided parametrix P, and the remainders Gl and Gr are compact between the respective spaces.Therefore, A is a Fredholm operator (7.2.61) for s > max.; d/  12 .

440

7 Operators on manifolds with edges and boundary

Remark 7.2.39. Let W be compact and A 2 Y;d .W ; g/ be elliptic. Then, if A is realised as a Fredholm operator (7.2.61), we have V WD ker A  W 1; .W / ˚ H 1 .Y /, and there is a finite-dimensional subspace W  W 1; .W / ˚ H 1 .Y / with W \ im A D f0g and im A C W D W s; .W / ˚ H s

nC1 2

.Y /:

This holds for all s > max.; d/  12 , and the spaces V and W are independent of s. In addition the parametrix P can be chosen in such a way that the remainders Gl and Gr are projections Gl W W s; .W / ˚ H s

nC1 2

.Y / ! V;

Gr W W s; .W / ˚ H s

nC1 2

.Y / ! W

to spaces V , W of the above-mentioned kind, for all s > max.; d/  12 . Remark 7.2.40. Definition 7.2.34 has an immediate generalisation to the space Y;d .W ; g; vI Rl / of parameter-dependent edge boundary value problems with parameter  2 Rl , l 2 N (including l D 0), and tuples of vector bundles (7.2.43). Instead of Definition 7.2.34 (i), (ii), and (iii) we ask corresponding isomorphisms (7.2.44), (7.2.45), and (7.2.46), respectively, where in the parameter-dependent case the projections are replaced by W W .T  Wreg  Rl / n 0 ! Wreg , V W T  .Vreg  Rl / n 0 ! Vreg , and Y W .T  Y Rl /n0 ! Y ; the 0’s are used in the meaning of vanishing of covectors together with ; in addition the corresponding bijectivities of the symbols Q and Q @ up to r D 0 are required. The ellipticity for l > 0 will also be called parameter-dependent ellipticity. Remark 7.2.41. Theorem 7.2.36 has a generalisation to the parameter-dependent case as follows: A parameter-dependent elliptic element A 2 Y;d .W ; gI vI Rl /, v WD ..E; E 0 ; J /; .F; F 0 ; JC //, possesses a parameter-dependent parametrix P 2 C Y;.d/ .W ; g 1 I v1 I Rl /, v1 WD ..F; F 0 ; JC /, .E; E 0 ; J // belonging to the inverted principal symbol, and the analogues of the remainders Gl and Gr belong to Y1;dl .W ; g l I vl I Rl / and Y1;dr .W ; g r I vr I Rl /, respectively, with dl , dr , g l , g r being defined as in Theorem 7.2.36, and vl WD ..E; E 0 ; J /; .E; E 0 ; J //, vr WD ..F; F 0 ; JC /; .F; F 0 ; JC //. Theorem 7.2.42. Let W be compact, and let A 2 Y;d .W ; gI vI Rl / be parameterdependent elliptic. Then A./ W W s; .W ; E / ˚ H s

nC1 2

.Y; J /

! W s;  .W ; F / ˚ H s

nC1 2

(7.2.62) .Y; JC /;

s > max.; d/  1=2, is a family of Fredholm operators of index zero, and there is a C > 0 such that the operators (7.2.62) are invertible for all  2 Rl ; jj  C . Proof. A slight modification of the constructions of the proof of Theorem 7.2.36 C gives us a parameter-dependent parametrix P 2 Y;.d/ .W ; g 1 I v1 I Rl /

7.2 The edge algebra

441

such that the analogues of the remainders Gl and Gr of Theorem 7.2.36 belong to Y1;dl .W ; g l I vl I Rl / and Y1;dr .W ; g r I vr I Rl /, respectively, with dl ; g l and dr ; g r as in Theorem 7.2.36, and v1 ; vl ; vr as in Remark 7.2.41. Since the parameterdependent smoothing operators are Schwartz functions with values in the space of smoothing operators without parameters, they are also Schwartz functions with values in continuous operators between the respective weighted spaces. Therefore, the operators Gl and Gr (in the parameter-dependent meaning) are invertible between those spaces for all sufficiently large jj. This entails the invertibility of A for those . Remark 7.2.43. (i) Let A 2 Y;d .W ; gI vI Rl /, and set A1 WD A.1 / for a sufficiently large 1 2 Rl such that A1 is invertible; then C

;.d/ P1 WD A1 .W ; g 1 I v1 /: 1 2Y

(ii) It can be proved that the invertibility of an A 2 Y;d .W ; gI v/ as an operator C (7.2.62) entails the ellipticity of A and A1 2 Y;.d/ .W ; g 1 I v1 /. If an operator A of the form of an upper left corner in the space Y;d .W;gI.E; F // is  -elliptic, i.e., (7.2.44) is an isomorphism together with its Q -variant up to r D 0, the question is whether there is a . ; @ /-elliptic element A D .Aij /i;j D1;2 2 Y;d .W ; gI ..E; E 0 /, .F; F 0 // for suitable E 0 ; F 0 2 Vect.V /, such that A D A11 , and, furthermore, if this the case, whether there is an elliptic element A D .Aij /i;j D1;2;3 2 Y;d .W ; gI v/ for suitable J˙ 2 Vect.Y / so that A D .Aij /i;j D1;2 . The answer is that there are two topological obstructions for that. The first one concerns a generalisation of the Atiyah–Bott obstruction in boundary value problems, cf. [8], [15], see also [190], [194], which is a condition on @ .A/. The second one is a condition on ^ .A/, cf. [182] and [198] in the case of manifolds with edges (without boundary), see also [131], [132]. We do not go into details here, but refer to the general discussion of Section 10.5.3 below. Another interesting question is whether for every  2 Z there exists a  elliptic element A 2 Y;d .W ; gI .E; E//, E 2 Vect.W /, which is at the same time . ; @ ; ^ /-elliptic, without extra boundary conditions on Vreg and edge conditions on Y . Let us formulate such a theorem. Theorem 7.2.44. (i) Let W be a compact manifold with boundary and edge Y . Then for every  2 Z;  2 R, and E 2 Vect.W / there exists an elliptic element A 2 Y;0 .W ; gI .E; E//, g D .;   /, such that A W W s; .W ; E/ ! W s; .W ; E/ is an isomorphism for every s 2 R, s >   12 , and we have A1 2 Y;0 .W ; g 1 I .E; E//. (ii) Let V be a compact manifold (without boundary) and edge Y . Then for every ;  2 R and E 2 Vect.V / there exists an elliptic element B 2 Y .V ; hI .E 0 ; E 0 //, h WD .;   /, such that B W W s; .V ; E 0 / ! W s; .V ; E 0 / is an isomorphism for every s 2 R, and we have B 1 2 Y .V ; h1 I .E 0 ; E 0 //. A proof of Theorem 7.2.44 (i) is given in [112]. The ideas are based on a parameterdependent version of the order reducing symbols (4.1.1), combined with other insight on smoothing Mellin edge symbols that admit manipulations of Fredholm indices

442

7 Operators on manifolds with edges and boundary

of edge symbols. Theorem 7.2.44 (ii) is of simpler structure, but it also employs specific smoothing Mellin edge symbols. In both cases the construction first produces parameter-dependent elliptic elements A./ 2 Y;0 .W ; gI .E; E/I Rl /

and

B./ 2 Y .V ; hI .E 0 ; E 0 /I Rl /;

respectively, without extra conditions, and the parameter  2 Rl is then fixed and sufficiently large, cf. also Remark 7.2.43. Operators as in Theorem 7.2.44 are called reductions of orders in the edge calculus.

7.3 Mellin-edge representations of elliptic operators We construct a class of elliptic operators in the edge algebra on a C 1 manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to the principal symbolic structure of the edge algebra consisting of the standard principal symbol and the operator-valued principal edge symbol. For every differential operator A on M and any s 2 R (sufficiently large and with the exception of some discrete values) we construct an associated operator A.s/ in the edge calculus. We show that the ellipticity of A entails the ellipticity of A.s/ as an edge operator. Parametrices P of A then correspond to parametrices P .s/ of A.s/ in the edge calculus. P .s/ may be regarded as a Mellin quantisation of P . A reference of the results of this section are the papers [33], [35], [36], [113].

7.3.1 Decomposition of classical Sobolev spaces Sobolev spaces H s .Rm /, s 2 R, can be written as edge spaces with respect to any coordinate hypersurface of Rm . Let us write Rm D Rd  Rq ; i.e., m D d Cq, for d  1 and q  1. Recall that we have the following representation, cf. Example 2.2.13. Proposition 7.3.1. Let H s .Rd / be endowed with the group action . u/.x/ Q WD d Q  2 RC . Then we have  2 u.x/, H s .Rd Cq / D W s .Rq ; H s .Rd //

(7.3.1)

for every s 2 R. We want to decompose the spaces H s .Rm / by using trace and potential symbols of the following kind. Let u 2 H s .RdxQ /, ˛ 2 N d ; s > d2 C j˛j, and set   d t ˛ ./u WD Œ  2 j˛j Dx˛Q u .0/:

443

7.3 Mellin-edge representations of elliptic operators

Then we have t ˛ ./ 2 Scl0 .Rq I H s .Rd /; C/: Q 1 in a neighbourhood of xQ D 0. Then Moreover, fix an !.x/ Q 2 C01 .Rd /; !.x/ d

k ˛ ./c WD Œ 2

1 Q .Œ x/ Q ˛ !.Œ x/c ˛Š

(7.3.2)

for c 2 C defines a symbol k ˛ ./ 2 Scl0 .Rq I C; H s .Rd //   for arbitrary ˛ 2 N d ; s 2 R. Incidentally, we set  ˛ u WD Dx˛Q u .0/. Remark 7.3.2. We have 1 Q .jjx/ Q ˛ !.jjx/: ˛Š     Definition 7.3.3. Set t ./ WD t t ˛ ./ j˛j<s d ; k./ WD k ˛ ./ j˛j<s d for fixed s > d

^ .t ˛ /./ D jj 2 j˛j  ˛

and

d

^ .k ˛ /./ D jj 2

2

d 2

2

C j˛j.

Remark 7.3.4. We have t./ 2 Scl0 .Rq I H s .Rd /; C .s;d / /

and

k./ 2 Scl0 .Rq I C .s;d / ; H s .Rd //

for .s; d / WD #f˛ 2 N d W j˛j < s  d2 g, and t ./k./ D idC .s;d / for every  2 Rq . Moreover, k./t ./ 2 Scl0 .Rq I H s .Rd /; H s .Rd // is a family of projections in the space H s .Rq /. ˚

Definition 7.3.5. We set H0s .Rd / WD u 2 H s .Rd / W Dx˛Q u.0/ D 0 for all j˛j < s d2 for s > d2 , s  d2 62 N. Remark 7.3.6. The space H0s .Rd / is closed in H s .Rd /, and we have H0s .Rd / D K s;s ..S d 1 /^ / for s > d2 , s  rem 2.1.32].

d 2

(7.3.3)

62 N. A proof of the relation (7.3.3) may be found in [90, Theod

The group action . u/.x/ Q D  2 u.x/; Q  2 RC , on the space H s .Rd / induces a s d group action on H0 .R /. Remark 7.3.7. For s >

d ;s 2



d 2

62 N we have

W s .Rq ; H0s .Rd // ˚

D u.x; Q y/ 2 H s .Rd Cq / W Dx˛Q u.0; y/ D 0 for all j˛j < s  d2 :

444

7 Operators on manifolds with edges and boundary

The symbols of Remark 7.3.4 can be subsumed under the following consideraz be Hilbert spaces with group actions f g2R and fQ  g2R , tion. Let H and H C C respectively. Let z /; t ./ 2 Scl0 .Rq I H; H

z; H/ k./ 2 Scl0 .Rq I H

be symbols such that t ./k./ D idHz

for all  2 Rq :

(7.3.4)

Then k./t./ 2 Scl0 .Rq I H; H /, k./t ./ W H ! im k./, takes values in continuous projections, and the symbol 1  k./t ./ 2 Scl0 .Rq I H; H / represents the family of complementary projections 1  k./t ./ W H ! ker t ./. In particular, we have ker t ./ ˚ im k./ D H

for all  2 Rq :

Similar relations hold on the level of associated operators z /; T WD Op.t / W W s .Rq ; H / ! W s .Rq ; H z / ! W s .Rq ; H / K WD Op.k/ W W s .Rq ; H for any fixed s 2 R. The relation (7.3.4) implies TK D idW s .Rq ;Hz / : Moreover, K T W W s .Rq ; H / ! im K, 1  K T W W s .Rq ; H / ! ker T are complementary projections, and we have W s .Rq ; H / D ker T ˚ im K: Assume that the space H0 WD ker t ./ is independent of  and preserved under the action of f g2RC . Then we have ker T D W s .Rq ; H0 /: It follows that W s .Rq ; H / D W s .Rq ; H0 / ˚ im K:

(7.3.5)

We can apply these relations to the symbols t ./ and k./ from Remark 7.3.4 and H WD H s .Rd /;

z WD C .s;d / ; H

H0 WD H0s .Rd /:

Then we obtain W s .Rq ; H / D H s .Rd Cq / and T D Op.t / W W s .Rq ; H s .Rd // ! H s .Rq ; C .s;d / /; K D Op.k/ W H s .Rq ; C .s;d / / ! W s .Rq ; H s .Rd //:

445

7.3 Mellin-edge representations of elliptic operators

Let us set

V s .Rd Cq / WD im K:

From (7.3.5) we then obtain a direct decomposition H s .Rd Cq / D W s .Rq ; H s .Rd // D W s .Rq ; H0s .Rd // ˚ V s .Rd Cq / for every s >

d ;s 2



d 2

(7.3.6)

62 N.

Proposition 7.3.8. Let E W W s .Rq ; H0s .Rd // ! H s .Rd Cq / be the canonical embedding, and let s > d2 ; s  d2 62 N. Then s s q d   W .R ; H0 .R // ˚ E K W ! H s .Rd Cq / s q .s;d / H .R ; C /

is an isomorphism, and we have .E K/1 D t .1  K T T /. Proof. The isomorphism is an immediate consequence of the relation (7.3.5). For P 

we have T operators and

P WD 1  K T W H s .Rd Cq / ! W s .Rq ; H0s .Rd //     PK .E K/ D PE D 10 01 with 1 denoting the respective identity TE TK     P E K D EP C K T D 1: T

Remark 7.3.9. Let  2 C01 .Rd / be a function such that  1 on supp !.Πx/ Q for all  2 Rq , where ! is the cut-off function in (7.3.2) (observe that there is a compact Q K for all  2 Rq ). Then we have set K  Rq such that supp !.Πx/ u D u

for all u 2 V s .Rd Cq /:

This is a consequence of f 2 im k./ ) f D f for every  2 Rq . Let us now formulate analogous relations for Sobolev spaces in the half-space d q Rm C D RC  R ;

RdC D fxQ D .x; Q : : : ; xQ d / 2 Rd W xd > 0g. Analogously as (7.3.1) we have the edge representation H s .RdC  Rq / D W s .Rq ; H s .RdC //: Moreover, the operators t ˛ ./ W H s .RdC / ! C for s > d

k ˛ ./c WD Œ 2

1 Q .Œ x/ Q ˛ !C .Œ x/c; ˛Š

d 2

C j˛j and

ˇ c 2 C; !C WD ! ˇRd ; C

generate symbols t ./ 2 and k ./ 2 respectively. For simplicity, here and in the remaining part of this section we employ the notations t ˛ ./; k ˛ ./, etc., also in connection with the half-space case; we hope this does not cause confusion. ˛

Scl0 .Rq I H s .RdC /; C/

˛

Scl0 .Rq I C; H s .RdC //,

446

7 Operators on manifolds with edges and boundary

Remark 7.3.10. We have t ./ WD t .t ˛ .//j˛j<s d 2 Scl0 .Rq I H s .RdC /; C .s;d / / 2

for s >

d , 2

and k./ WD .k ˛ .//j˛j<s d 2 Scl0 .Rq I C .s;d / ; H s .RdC //; 2

˚ .s; d / D # ˛ 2 N d W j˛j < s  d2 . Moreover, t ./k./ D idC .s;d / for every  2 Rq , and k./t ./ 2 Scl0 .Rq I H s .RdC /; H s .RdC // is a family of projections in the space H s .RdC /. Remark 7.3.11. Setting ˇ H0s .RdC / WD H0s .Rd /ˇRd

and

C

ˇ d 1 ^ K s;s ..SC / / WD K s;s ..S d 1 /^ /ˇint.S d 1 /^ C

d 1 x d , we have H s .Rd / D K s;s ..S d 1 /^ / for every s > for SC WD S d 1 \ R C C C 0 d s  2 62 N.

d , 2

Observe that ˚

H0s .RdC / D u 2 H s .RdC / W Dx˛Q u.0/ D 0 for all j˛j < s  d2 and ˚

W s .Rq ; H0s .RdC // D u.x; Q y/ 2 H s .RdCCq / W Dx˛Q u.0; y/ D 0 for all j˛j < s  d2 : The operators T WD Op.t / W W s .Rq ; H s .RdC // ! H s .Rq ; C .s;d / /; K WD Op.k/ W H s .Rq ; C .s;d / / ! W s .Rq ; H s .RdC // are continuous. Setting V s .RdCCq / WD im K we then obtain a direct decomposition H s .RdCCq / D W s .Rq ; H0s .RqC // ˚ V s .RdCCq / for s >

d , 2

s

d 2

62 N.

Proposition 7.3.12. Let E W W s .Rq ; H0s .RdC // ! H s .RdCCq / be the canonical embedding, and let s >

d , 2

s

d 2

62 N. Then .E K/ W

W s .Rq ; H0s .Rd C // ˚ H s .Rq ; C .s;d / /

is an isomorphism, and we have .E K/1 D t .1  K T T /.

The proof is practically the same as that of Proposition 7.3.8.

! H s .RdCCq /

7.3 Mellin-edge representations of elliptic operators

447

Remark 7.3.13. The operators K and T in Proposition 7.3.8 or 7.3.12 can also be chosen in parameter-dependent form K./ WD Op.k/./, T ./ WD Op.t /./ with the parameter  2 Rl , when we set t.; / WD t .t ˛ .; //j˛j<s d ; 2

d 2

 j˛j

k.; / WD .k ˛ .; //j˛j<s d ; 2



d 2

1 t ˛ .; / WD Œ; 

Dx˛Q u .0/, k ˛ .; /c WD Œ;  ˛Š .Œ;  x/ Q ˛ !.C/ .Œ;  x/c, Q d d c 2 C, where the subscript ‘.C/’ means ! for R and !C for RC . Then Proposition 7.3.8 and 7.3.12 are valid for every fixed  2 Rl .

7.3.2 Edge decompositions of differential operators We now consider a differential operator X AD a˛ˇ .x; Q y/Dx˛Q Dyˇ j˛jCjˇ j

with coefficients a˛ˇ 2 C 1 .Rd Cq / for all ˛ 2 N d ; ˇ 2 N q . For simplicity, we assume that the coefficients are independent of .x; Q y/ for jx; Q yj > C for some C > 0. Then A induces continuous operators A W H s .Rd Cq / ! H s .Rd Cq / for all s 2 R. Applying Proposition 7.3.8 for s   >

d , 2

s

d 2

(7.3.7)

62 N, we have isomorphisms

s s q d  W .R ; H0 .R // ˚ Es Ks W ! H s .Rd Cq / s q .s;d / H .R ; C /



and 

Ps Ts

 WH

s

d Cq

.R

W s .Rq ; H0s .Rd // ˚ / ! : s q .s;d / H .R ; C /

Together with (7.3.7) we obtain a continuous operator    Ps Ps AEs A.Es Ks / D A.s/ WD Ts Ts AEs

 Ps AKs ; Ts AKs (7.3.8)

W .R

A.s/ W

; H0s .Rd //

.R W ˚ ˚ ! : s q .s;d / s q .s;d / H .R ; C H / .R ; C / s

q

; H0s .Rd //

s

q

(7.3.9)

448

7 Operators on manifolds with edges and boundary

Remark 7.3.14. The operator A restricts to a continuous map A W W s .Rq ; H0s .Rd // ! W s .Rq ; H0s .Rd //; ˇ and we have Ps AEs D AˇW s .Rq ;H s .Rd // and Ts AEs D 0. It follows that 0   A0 .s/ Ps AKs (7.3.10) A.s/ D 0 Ts AKs ˇ for A0 .s/ WD AˇW s .Rq ;H s .Rd // . 0

Let us form the operator family X a./ .y; / WD a˛ˇ .0; y/Dx˛Q ˇ W H s .Rd / ! H s .Rd /

(7.3.11)

j˛jCjˇ jD

for .y; / 2 T  Rq n0. Observe that a./ .y; / D   a./ .y; /1 for all  2 RC . Proposition 7.3.15. Let A be an elliptic operator in the standard sense, i.e., it holds P Q / 2 T  Rd Cq n 0. Then (7.3.11) that j˛jCjˇ jD a˛ˇ .x; Q y/ Q ˛ ˇ 6D 0 for all .x; Q y; ;  q is a family of isomorphisms for all .y; / 2 T R n 0 and s 2 R. Q WD P Q˛ ˇ Proof. Fix .y; / 2 T  Rq n 0 and set p.y;/ . / j˛jCjˇ jD a˛ˇ .0; y/  . Then Q 6D 0 for all Q 2 Rd and p 1 . / Q 2 S  .Rd /. We have a./ .y; / D p.y;/ . / cl

.y;/

Q and the inverse is obviously equal to a1 .y; / D Op .p 1 . //. Q OpxQ .p.y;/ . //, xQ ./ .y;/ From the preceding section we have symbols ts ./ 2 Scl0 .Rq I H s .Rd /; C .s;d / /;

ks ./ 2 Scl0 .Rq I C .s;d / ; H s .Rd //;

cf. Remark 7.3.4 (we indicate the smoothness as subscript when the symbols refer to different s). We have ^ .ts /./^ .ks /./ D idC .s;d / which yields families of projections ^ .ks /./^ .ts /./ D ^ .ks ts /./ W H s .Rd / ! V s .Rd I / for V s .Rd I / WD im ^ .ks /./ and ^ .ps /./ W H s .Rd / ! H0s .Rd / for ps ./ WD 1  ks ./ts ./. An analogue of Proposition 7.3.8 is then the following result: Proposition 7.3.16. Let es W H0s .Rd / ! H s .Rd / be the canonical embedding, and let H0s .Rd / d d ˚ s > 2 , s  2 62 N. Then .es ^ .ks /.// W ! H s .Rd / is an isomorphism .s;d / C for every  6D 0, and we have   1  ^ .ps /./ : D es ^ .ks /./ ^ .ts /./

7.3 Mellin-edge representations of elliptic operators

Theorem 7.3.17. Let A be elliptic in the  standard sense, and let s  > Then ^ .A.s//.y; / WD isomorphisms

^ .ps /./ ^ .ts /./

d , s  d2 2

449 62 N.

a./ .y; /.es ^ .ks /.// defines a family of

H0s .Rd / H0s .Rd / ˚ ˚ ^ .A.s//.y; / W ! C .s;d / C .s;d /

for .y; / 2 T  Rq n 0. Proof. It suffices to combine Proposition 7.3.16 with Proposition 7.3.15. We now interpret the space Rd Cq as a manifold W with edge Rq and model cone x C S d 1 Rq 3 .r; x; y/. R D .S d 1 / . The stretched manifold is equal to W D R d By introducing polar coordinates xQ D .r; x/ in R n f0g we obtain the operator A on .Rd n f0g/  Rq in edge-degenerate form  X @ j A D r  aj˛ .r; y/ r .rDy /˛ @r d

j Cj˛j

x C ; Diff .j Cj˛j/ .S d 1 //. In the following we with coefficients aj˛ .r; y/ 2 C 1 .R d d assume s   > 2 , s  2 62 N. By virtue of (7.3.3) and the identifications W s .Rq ; K s;s ..S d 1 /^ // D W s .Rq ; H0s .Rd // we obtain an identification of A0 .s/ W W s .Rq ; H0s .Rd // ! W s .Rq ; H0s .Rd // with the operator A0 .s/ W W s .Rq ; K s;s ..S d 1 /^ // ! W s .Rq ; K s;s ..S d 1 /^ //: Moreover, the operators Ks W H s .Rq ; C .s;d / / ! H s .Rd Cq /; Ts W H s .Rd Cq / ! H s .Rq ; C .s;d / /; Ps W H s .Rd Cq / ! W s .Rq ; H0s .Rd // together with their ‘principal edge symbols’ ^ .ks /./ W C .s;d / ! H s .Rd /; ^ .ts /./ W H s .Rd / ! C .s;d / ; ^ .ps /./ W H s .Rd / ! H0s .Rd / are reformulated in combination with (7.3.8) as A0 .s/ D Ps AEs W W s .Rq ; K s;s ..S d 1 /^ / ! W s .Rq ; K s;s ..S d 1 /^ //; Ps AKs W H s .Rq ; C .s;d / / ! W s .Rq ; K s;s ..S d 1 /^ //; Ts AKs W H s .Rq ; C .s;d / / ! H s .Rq ; C .s;d / /;

450

7 Operators on manifolds with edges and boundary

and X

^ .A0 .s/.y; / D r 

j Cj˛j

 @ j .r/˛ W aj˛ .0; y/ r @r K s;s ..S d 1 /^ / ! K s;s ..S d 1 /^ /;

^ .Ps AKs /.y; / D ^ .ps /./a./ .y; /^ .ks /./ W C .s;d / ! K s;s ..S d 1 /^ /; ^ .Ts AKs /.y; / D ^ .ts /./a./ .y; /^ .ks /./ W C .s;d / ! C .s;d / ; respectively. Theorem 7.3.18. Let A be an elliptic differential operator of order  in Rd Cq 3 .x; Q y/, and let s   > d2 , s  d2 62 N. Then the associated operator (7.3.10) can be identified with an operator   A0 .s/ K0 .s/ W A.s/ D 0 Q0 .s/ W s .Rq ; K s;s ..S d 1 /^ // W s .Rq ; K s;s ..S d 1 /^ // ˚ ˚ ! C .s;d / C .s;d / x C  S d 1  Rq and is elliptic in that which belongs to the edge algebra on W D R algebra, i.e., with respect to the principal symbolic hierarchy  .  / D . .  /; ^ .  //. Proof. We set K0 .s/ WD Ps AKs , Q0 WD Ts AKs . Then the ellipticity of A gives the  -ellipticity in the sense of  .A0 .s//.r; x; y; %; ; / 6D 0 on T  Wreg n 0, Wreg D RC  S d 1  Rq , together with Q .A0 .s//.r; x; y; %; ; / 6D 0

for .%; ; / 6D 0; up to r D 0.

In addition Theorem 7.3.17 gives us the bijectivity of K s;s ..S d 1 /^ / K s;s ..S d 1 /^ / ˚ ˚ ^ .A.s//.y; / W ! .s;d / .s;d / C C for all .y; / 2 T  Rq n 0. Corollary 7.3.19. Let s   >

d , 2

s

d 2

62 N. Then

^ .A0 .s//.y; / W K s;s ..S d 1 /^ / ! K s;s ..S d 1 /^ /; .y; / 2 T  Rq n 0, is a family of Fredholm operators, and we have ind ^ .A0 .s//.y; / D .s  ; d /  .s; d /:

7.3 Mellin-edge representations of elliptic operators

451

The subordinate conormal symbol c ^ .A0 .s//.y; w/ D

 X

aj 0 .0; y/w j W H sQ .S d 1 / ! H sQ .S d 1 /

j D0

is a family of isomorphisms parametrised by w 2  d s and y 2 Rq . 2

7.3.3 Global constructions Let M be a closed compact C 1 manifold, and let Y  M be a closed compact C 1 submanifold of codimension d > 1. We first assume that Y has a trivial normal bundle in M (this assumption is not really essential and will be dropped later on). We interpret M as a manifold with edge Y . In a collar neighbourhood V of Y we fix a splitting of variables .x; Q y/ 2 Rd  Rq (dim M D d C q), and by .r; x/ 2 RC  S d 1 we denote polar coordinates in Rd n f0g. With M we can associate the stretched manifold M x C  S d 1  Rq ) such that Mreg D M n Y (locally near @M D Msing modelled on R and Msing D S d 1  Y . On M we have the standard Sobolev spaces H s .M /, and we form subspaces ˚

H0s .M / WD u 2 H s .M / W Dx˛Q u.0; y/ D 0 locally near Y for all j˛j < s  d2 ; for s >

d , 2

s

d 2

62 N. There is then a canonical isomorphism H0s .M / Š W s;s .M/

(7.3.12)

for every such s. Theorem 7.3.20. For every fixed s >   E K./ W

d , 2

s  d2 62 N, there is a family of isomorphisms

W s;s .M/ ˚ ! H s .M / s .s;d / / H .Y; C

(7.3.13)

for every  2 Rl , jj sufficiently large, where (7.3.13) localises near Y to the operators  ./ localise .E Op.k/.// as in Remark 7.3.9, and the entries of .E K.//1 DW P T ./ to T ./ D Op.t /./ and P ./ D 1  K./T ./, respectively, modulo lower order terms. The proof of Theorem 7.3.20 will be given below which also explains the nature of lower order terms. Let W s;s .M/   ˚ Es Ks W ! H s .M /; s .s;d / H .Y; C /   W s;s .M/ Ps s ˚ .M / ! WH Ts s .Y; C .s;d / / H

452

7 Operators on manifolds with edges and boundary

denote the isomorphisms in the sense of Theorem 7.3.20 for a fixed sufficiently large jj. Given an elliptic differential operator A W H s .M / ! H s .M / we form the composition      Ps Ps AEs A Es Ks D A.s/ WD Ts Ts AEs

 Ps AKs ; Ts AKs

W s;s .M/ W s;s .M/ ˚ ˚ ! : A.s/ W H s .Y; C .s;d / / H s .Y; C .s;d / / ˇ Note that we have AEs D AˇW s;s .M/ (where we employ the identification (7.3.12)) and AEs W W s;s .M/ ! W s;s .M/ which implies Ts AEs D 0, since Ts vanishes on W s;s .M/. Moreover, s;s since .M/ it also follows that A.s/ WD Ps AEs D ˇ Ps f D f for f 2 W ˇ A W s;s .M/ . We thus obtain   A.s/ Ps AKs : (7.3.14) A.s/ D 0 Ts AKs Theorem 7.3.21. Let A be an elliptic differential operator of order  on M , and form the operator (7.3.14) for s   > d2 , s  d2 62 N. Then A.s/ is elliptic in the edge algebra on M with respect to the symbols  .A/ D . .A/; ^ .A//, and we have ind A D ind A.s/: Remark 7.3.22. As a corollary of Theorem 7.3.21 we obtain, when we replace s by  , an elliptic operator A. / in the edge algebra which is continuous as W s; .M/ W s; .M/ ˚ ˚ ! A. / W H s .Y; C .;d / / H s .Y; C .;d / / for every s 2 R and fixed  2 R,    > every s 2 R.

d , 2



d 2

62 N, and ind A D ind A. / for

In the following considerations we repeatedly employ the fact that the operator of multiplication by a '.x; Q y/ 2 C01 .Rd Cq / represents an operator-valued symbol 0 qCl ' 2 S .R I E; E/ for E D H s .Rd /; S.Rd /, or H0s .Rd /. It does not depend on the covariables but it is not a classical symbol in the sense of Definition 2.2.3 (ii). In the following we also use the fact (proved in Section 2.2.3) that  z b ! L .Rq I E; EI z Rl /b ; Op.  / W S.cl/ .Rq  RqCl I E; E/ .cl/

a.y; ; / ! Op.a/./, induces a bijection, including  D 1.

(7.3.15)

7.3 Mellin-edge representations of elliptic operators

453

Remark 7.3.23. For every '.x; Q y/ 2 C01 .Rd Cq /, ˇ 2 C01 .Rq /, supp ' \fxQ D 0g D ;, we have ' Op.k/./ˇ D Op.c/./ for a c.y; ; / 2 S 1 .Rq  RqCl I C .s;d / ; S.Rd //b . In fact, we can write ' Op.k/./ˇ D 'jxj Q 2M Œ;  2M Op.jŒ;  xj Q 2M k/./ˇ: By virtue of 'jxj Q 2M 2 C01 .Rd Cq / for every M 2 N the operator of multiplication by that function belongs to S 1 .Rq  RqCl I S.Rd /; S.Rd //b . We have jŒ;  xj Q 2M k.; / 2 Scl0 .RqCl I S.Rd /; S.Rd //, i.e., Q 2M k.; // 2 Scl2M .RqCl I C .s;d / ; S.Rd //b : k2M .; / WD Œ;  2M .jŒ;  xj Then ' Op.k/./ˇ D Op.r/./ for a symbol r.y; ; / 2 S 2M .RqCl I C .s;d / , S.Rd //b . Since this holds for all M 2 N, the relation (7.3.15) for  D 1 gives us the assertion. Proof of Theorem 7.3.20. On M we fix a Riemannian metric such that a tubular neighbourhood of Y is equipped with the product metric of B  Y , where B is the unit ball in Rd . We fix a covering of M by coordinate neighbourhoods U1 ; : : : ; UL , ULC1 ; : : : ; UN such that Uj \ Y 6D ; for 1  j  L, Uj \ Y D ; for L C 1  j  N . Without loss of generality we assume that for every two neighbourhoods Uj ; Uk of our system also Uj [ Uk is a coordinate neighbourhood. If j W Uj ! j .Uj / Rd Cq is a chart and Uj0 WD Uj \ Y 6D ; we assume that j0 WD j jUj0 W Uj0 ! j0 .Uj0 / Rq is a chart on Y . Suppose that f'1 ; : : : ; 'N g is a partition of unity subordinate to fU1 ; : : : ; UN g, 1 and let f 1 ; : : : ; N g be a system of functions j 2 ˇ C0 .Uj / such that j 1 ˇ on supp 'j for all j and set 'j0 WD 'j ˇU 0 , j0 WD j ˇU 0 . We then have functions j

j

'Qj ; Qj 2 C01 .j .Uj // and 'Qj0 ; Qj0 2 C01 .j0 .Uj0 // such that 'j D j 'Qj , j D j Qj , etc. Without loss of generality we assume that the functions 'j and j for 1  j  L have the form 'j D  'j0 , j D j0 for functions ; 2 C01 .Rd / that are equal to 1 near the origin and such that 1 on supp  . We construct the operators .E K.// in the form 

L X  E K./ WD .'j E

j

'j K./

j D1

0 j/

C

N X

.'j 0/;

j DLC1

where E W W s;s .M/ ! H s .M / is the canonical embedding, and 'j K./ j0 D j 'Qj Op.k/./ Qj0 ..j0 /1 / . The inverse of .E K.// will be approximated by  L  X P0 ./ WD T0 ./



kD1

k P ./'k 0 T ./'k k



  N X 'k C 0 kDLC1

454

7 Operators on manifolds with edges and boundary

for large jj, where  D k Q k Op.p/./'Qk .1 k / ; 0 0  Q0  Qk .1 k T ./'k D .k / k Op.t /./' k / ;

k P ./'k

for the above symbols t .; /, k.; /, and p.; / D 1  k.; /t .; /, cf. Remark 7.3.13.     P P Pk ./ 0 ./ We have .E K.// D jND1 .Ej Cj .//, P D N kD1 Tk ./ , when we T0 ./ set ( 'j E j for 1  j  L; Ej WD 'j for L C 1  j  N ; ( 'j K./ j0 for 1  j  L; Cj ./ WD 0 for L C 1  j  N ; and ( Pk ./ WD ( Tk ./ WD

k P ./'k

'k 0 T ./'k k

0

for 1  k  L; for L C 1  k  N ; for 1  k  L; for L C 1  k  N :

We then consider   N  X P0 ./ Pk ./Ej .E K.// D T0 ./ Tk ./Ej j;kD1

Pk ./Cj ./ Tk ./Cj ./

 (7.3.16)

and  .E K.//

 N X P0 ./ D fEj Pk ./ C Cj ./Tk ./g: T0 ./

(7.3.17)

j;kD1

Let us first characterise the entries of (7.3.16). For 1  j; k  L we have Pk ./Ej D Tk ./Ej D

Pk ./Cj ./ D

k P ./'k 'j E j ; 0 k T ./'k 'j E j ;

Tk ./Cj ./ D

0 k P ./'k 'j K./ j ; 0 0 k T ./'k 'j K./ j :

Let us express these operators in local coordinates with respect to charts that are suitable both for j and k. For abbreviation we write k ; 'k ; : : : instead of Q k ; 'Qk , etc., but insert P ./ D Op.p/./, etc. For u.y/ 2 W s .Rq ; H0s .Rd // we have Pk ./Ej u.y/ D f'k 'j 

k

Op.k t /./'k 'j E

j gu.y/;

(7.3.18)

455

7.3 Mellin-edge representations of elliptic operators

Op.k t /./'k 'j E

k

j u.y/

D

0 k

Op.k t /./'k0  'j0 

0 j u.y/:

Since 'k0 'j0  j0 u.y/ takes values in ker Op.t /./ we see that the second summand of (7.3.18) vanishes, i.e., we have Pk ./Ej D 'k 'j for 1  j; k  L. For v.y/ 2 H s .Rq ; C .s;d / / we obtain for ˇ.y/ WD 'k0 .y/'j0 .y/ Pk ./Cj ./v.y/ D

k

Op.p/./'k 'j Op.k/./

D

k

Op.p/./ˇ.y/ Op.k/./

0 j v.y/ 0 j v.y/;

where we used the fact that (by an appropriate choice of the cut-off function ! involved in k./)  2 k./ D k./. Applying Remark 2.2.55 we obtain Op.p/./ˇ.z/ D Op.p # ˇ/./ D ˇ Op.p/./ C Op.r/./ for a symbol r.y; ; / of order 1, and it follows that Pk ./Cj ./ D

k

Op.pk/./

0 j

C

k

Op.rk/./

0 j

D

k

Op.rk/./

0 j

0 k Op.rk/./ j D Op.rkj /./ DW qCl Scl1 .Rq  R; I C .s;d / ; H0s .Rd / \ S.Rd //b . In

as pk D 0. By Remark 2.2.55 we can write Rkj ./ for a symbol rkj .y; ; / 2 other words, we obtain

Pk ./Cj ./ D Rkj ./

for 1  j; k  L:

Furthermore, Tk ./Ej u.y/ D since 'k 'j E we obtain

0 k

Op.t /./'k 'j E

j u.y/

j u.y/ takes values in ker Op.t /./.

Tk ./Cj ./v.y/ D

0 k

Op.t /./'k 'j Op.k/./

D 0 for 1  j; k  L;

Moreover, again by Remark 2.2.55, 0 0 0 j v.y/ D 'k 'j v.y/

0 C Rkj ./v.y/

0 0 0 ./ D k0 Op.rkj /./ j0 , with a symbol rkj .y; ; / 2 Scl1 .Rq  RqCl I for Rkj C .s;d / ; C .s;d / /b . For L C 1  j  N , 1  k  L we have

Pk Ej D

k P ./'k 'j

D 'k 'j ;

Pk ./Cj ./ D Tk ./Cj ./ D 0

and Tk ./Ej D k0 T ./'k 'j D 0, since supp 'j \ Y D ;, i.e., 'k 'j 2 ker T ./. Moreover, for 1  j  L, L C 1  k  N , we have Pk ./Ej D 'k 'j E

j

D 'k 'j ;

Tk ./Ej D Tk ./Cj ./ D 0;

and Pk ./Cj ./ D 'k 'j K./ j0 WD Ckj ./ is of order 1 in  2 Rl , cf. Remark 7.3.23. Finally, for L C 1  j , k  N it follows that Pk ./Ej D 'k 'j ;

Pk ./Cj ./ D Tk ./Ej D Tk ./Cj ./ D 0:

456

7 Operators on manifolds with edges and boundary

For the expression (7.3.17) we first assume 1  j , k  L. Then Ej Pk ./ C Cj ./Tk ./ D 'j E D 'j 'k  'j zj k ./ D 'j for R Remark 2.2.55 that

k

k P ./'k

j

k K./T ./'k

C 'j K./

C 'j K./

0 j

Op.k t /./'k C'j Op.k/./ Op.

'j Op.k/./ Op. D 'j D 'j

0 j 0 k

0 k

0 j

0 j

0 k T ./'k

0 k T ./'k 0 j

zj k ./ D 'j 'k C R

0 t /./'k . k

We obtain from

0 k t /./'k

Op.k t /./'k C 'j Op.rQj k /./'k

Op.k t /./'k C 'j Op.rQj k /./'k

for a symbol rQj k .y; ; / 2 Scl1 .Rq  RqCl I H s .Rd /; S.Rd //b . In the latter relation we employed j D j0 ; 'j j D 'j and k./ D k./. For L C 1  j  N , 1  k  L we have Ej Pk ./ C Cj ./Tk ./ D 'j D 'j

k P ./'k k .1

 Op.k t /.//'k D 'j 'k C Czj k ./;

Czj k ./ D 'j k Op.k t /./'k D Op.cj k /./ for a symbol cQj k .y; ; / 2 S 1 .Rq  RqCl I H s .Rd /; S.Rd //b . The latter relation can be obtained by similar arguments as in Remark 7.3.23. For 1  j  L, L C 1  k  N we have Ej Pk ./ C Cj ./Tk ./ D 'j E

j 'k

D 'j 'k :

Finally, in the case L C 1  j , k  N we have Ej Pk ./ C Cj ./Tk ./ D 'j 'k : Summing up we have obtained the following relations:     'j 'k Rkj ./ C Ckj ./ Pk ./Ej Pk ./Cj ./ D 0 ./ 0 'k0 'j0 C Rkj Tk ./Ej Tk ./Cj ./ and zj k ./ C Czj k ./; Ej Pk ./ C Cj ./Tk ./ D 1 C R zj k ./ are of order 1, concentrated near Y , and Ckj ./, where Rj k ./; Rj0 k ./; R Czkj ./ are of Schwartz behaviour in . Taking the sums over j; k D 1; : : : ; N it follows that (7.3.16) is equal to       1 0 0 R./ C C./ P0 ./ .E K.// D C D 1 C R./ C C./ T0 ./ 0 1 0 R0 ./

7.3 Mellin-edge representations of elliptic operators

457

 0 R./ , and (7.3.17) is equal to 0 R0 ./

 for R./ WD

 .E K.//

 P0 ./ z D 1 C R./ C Cz ./ T0 ./

(7.3.19)

z with remainders R./; R0 ./; R./ of order 1 in  2 Rl in the sense of the pseudodifferential calculus along Y with corresponding operator-valued symbols, while C./ and Cz ./ are Schwartz functions in  with values in the respective operators. The operator R./ is locally on Y in coordinates y 2 Rq a parameter-dependent pseudo-differential operator with symbol in 0 1 H0s .Rd / H0s .Rd / \ S.Rd / A: ˚ S 1 @Rq  RqCl I ˚ ; .s;d / .s;d / C C Applying asymptotic summation  of symbols   and associated operators   allows us to  P1 P0 ./ P1 ./ P1 ./ j j form j D0 .1/ R ./ T0 ./ DW T1 ./ . This gives us T1 ./ .E K.// D 1 C D./ for an operator function D./ which is Schwartz in  2 Rl . Since 1 C C./ is invertible for jj large enough we conclude that .E K.// is invertible for those , and we obtain a left inverse of .E K.//, namely,     P ./ P1 ./ D .1 C D.//1 : T ./ T1 ./ In a similar manner, using the relation (7.3.19) we obtain a right inverse which completes the proof of Theorem 7.3.20.

7.3.4 Edge representation of boundary value problems Let M be a compact C 1 manifold with boundary @M , m D dim M , and let Y  @M be a compact C 1 manifold of codimension d > 1 in M . For simplicity, we assume that the normal bundle of Y in @M is trivial. We interpret M as a manifold W with edge Y and boundary, and @M as a manifold with edge Y (and without boundary). Locally near Y the manifold @M is modelled on Rd 1  Rq for q D m  d , and M x C /  Rq . Let W and V denote the stretched manifolds associated itself by .Rd 1  R with M and @M , respectively, with edge Y . Theorem 7.3.24. For every fixed s  0, s  

 E K./ W

d 2

62 N, there is a family of isomorphisms

W s;s .W / ˚ ! H s .int M / s .s;d / H .Y; C /

(7.3.20)

458

7 Operators on manifolds with edges and boundary

for every  2 Rl , jj sufficiently large, which localise near Y to the operators Op.k/./ ./ as in Remark 7.3.13, and the entries of .E K.//1 DW P localise to T ./ D T ./ Op.t/./ and P ./ D 1  K./T ./, respectively, with t .; / as in Remark 7.3.13, modulo lower order terms. The proof of Theorem 7.3.24 is similar to that of Theorem 7.3.20 and will be dropped here. Let A W H s .int M / ! H s2m .int M / be an elliptic differential operator of order 2m, and consider an elliptic boundary value problem   A AD W H s .int M / ! T

H s2m .int M / ˚ m sj  1 2 .@M / ˚j D1 H

(7.3.21)

associated with A. By notation, T D t .T1 ; : : : ; Tm / is a column matrix of operators of the form Tj u WD Bj uj@M for differential operators Bj of order j that are smooth in a neighbourhood of @M (on, say, the double of M ). Applying Theorem 7.3.20 we obtain a family of isomorphisms that we now indicate with ‘primes’, namely, 1

1

˚jmD1 W sj  2 ;sj  2 .V / 1 ˚ ! ˚jmD1 H sj  2 .@M / 1 1 ˚jmD1 H sj  2 .Y; C .sj  2 ;d 1/ / (7.3.22) 1 d 1 for every s  j  12 > d 1 , s     2 6 N. j 2 2 2 Combining (7.3.21) with the isomorphisms (7.3.20) and (7.3.22) we obtain a Fredholm operator  0 0  E K ./ W

W s;s .W / W s2m;s2m .W / ˚ ˚ A.s/ W ! H s .Y; C .s;d / / H s2m .Y; C .s2m;d / /

(7.3.23)

where W

H

s2m

s2m;s2m

.Y; C

W s2m;s2m .W / ˚ .W / WD ; m sj  1 ;sj  1 2 2 ˚j D1 W .V /

.s2m;d /

H s2m .Y; C .s2m;d / / ˚ / WD : 1 1 ˚jmD1 H sj  2 .Y; C .sj  2 ;d 1/ /

This reformulation of (7.3.21) is valid for s  2m > d2 , s  2m  s  j  12 > d 1 , s  j  12  d 1 62 N, j D 1; : : : ; m. 2 2

d 2

62 N and

7.3 Mellin-edge representations of elliptic operators

459

The operator (7.3.23) also induces continuous operators W t; .W / W t2m; 2m .W / ˚ ˚ A. / W ! H t .Y; C .;d / / H t2m .Y; C . 2m;d / /

(7.3.24)

for every t 2 R, t > j C 12 , j D 1; : : : ; m, and  2 R with 1 d 1  j   62 N: 2 2 (7.3.25) The entries of the block matrix A. / belong to the edge algebra of boundary value problems of the respective orders.  2m >

d ; 2

 2m

d 62 N; 2

 j 

d 1 1 > ; 2 2

Theorem 7.3.25. The operator (7.3.24) is elliptic in the edge algebra for all weights (7.3.25), and we have ind A D ind A. /: In fact, the operator (7.3.24) is the result of reformulating the Fredholm operator (7.3.21) as (7.3.23) and then using the fact that the smoothness index in the spaces can be chosen independently of the weights. The Fredholm property in the edge algebra is equivalent to the ellipticity. In particular, we have a family of isomorphisms K t; .X ^ / K t2m; 2m .X ^ / ˚ ˚ ^ .A. //.y; / W ! C .;d / C . 2m;d / for

(7.3.26)

d 1 X WD SC ;

@X D S d 2 ; L 1 1 and K t2m;2m .X ^ / D K t2m;2m .X ^ / jmD1 K tj  2 ;j  2 ..@X /^ /, .y; / 2 T  Y n 0, t  2m >  12 , t > j C 12 , j D 1; : : : ; m, and (7.3.25), m   X 1  s  j  ; d  1 : .s  2m; d / D .s  2m; d / C 2 j D1

Similarly as in the boundaryless case the upper left 21 corner A.s/ of the operator (7.3.23) is nothing other than the restriction of the operator A from H s .int M / to the space W s;s .W /. Replacing the notation of the weight s by  and of s in the meaning of smoothness by t, we can again write A. / as an operator A. / W W t; .W / ! W t2m; 2m .W /

(7.3.27)

for the above-mentioned t and  . Corollary 7.3.26. For the index of the family of Fredholm operators ^ .A. /.y; / W K t; .X ^ / ! K t2m; 2m .X ^ / (7.3.28) Pm we have ind ^ .A. /.y; / D .  2m; d / C j D1 .  j  12 ; d  1/  .; d /.

460

7 Operators on manifolds with edges and boundary

Another consequence is that (7.3.29) c ^ .A. //.y; w/ W H t .int X / ! H t2m .int X / L 1 for H t2m .int X / WD H t2m .int X / ˚ jmD1 H tj  2 .@X / is a family of isomorphisms for all y 2 Y and w 2  d  , t  2m >  12 , t > j C 12 , j D 1; : : : ; m. 2

Remark 7.3.27. Let A D .A T / be an elliptic boundary value problem on M , and  2 R be weight such that (7.3.28) is a family of Fredholm operator for any t > 2m 12 , t > j C 12 , j D 1; : : : m, and all y 2 Y . Then we have t

indS  Y ^ .A. // 2 1 K.Y /I

(7.3.30)

1 W S  Y ! Y is the canonical projection of the unit cosphere bundle in T  Y to Y and K.  / the K-group on Y . In fact, if  satisfies the conditions (7.3.25) there is an elliptic edge problem A. / with A./ in the upper left corner. In particular, (7.3.26) is a family of isomorphisms, and we have ind ^ .A. // in Corollary 7.3.26 corresponds to the relation indS  Y ^ .A. // D ŒC .s2m;d /  ŒC .;d / ; i.e., (7.3.30) is satisfied. On the other hand we know that (7.3.30) is independent of the weight, cf. [113].

7.3.5 Relative index results Given an elliptic differential boundary value problem A D t .A T / as in the preceding section, realised as a continuous operator (7.3.27) in the edge calculus for any t  2m > 12 , t > j C 12 , j D 1; : : : ; m, and arbitrary  2 R, we can complete A. / to a matrix A. / which is Fredholm with the corresponding additional entries, and ask how to compute the number of the extra trace and potential entries. An answer for  as in (7.3.25) is given by the constructions for Theorem 7.3.25. The case of arbitrary  2 R (up to a possible set of exceptional values) can be treated in terms of the relative index behaviour of the principal conormal symbol. We saw such a construction already in Section 5.3.2 before. Now we want to discuss the general case. For this consideration it suffices to assume the following. A D t .A T / is a general edge boundary value problem on a manifold W with edge Y and boundary V . Let W and V denote the stretched manifolds associated with W and V , respectively, and let A and T be given in the local splitting of variables near Y as X A D r 2m ak˛ .r; y/.r@r /k .rDy /˛ kCj˛j2m

x C  ; Diff 2m.kCj˛j/ .X //, and with coefficients ak˛ .r; y/ 2 C 1 .R T D t .T1 ; : : : ; Tm /

461

7.3 Mellin-edge representations of elliptic operators

for

X

Tj D r.@X/^ r j

bj;kˇ .r; y/.r@r /k .rDy /ˇ ;

kCjˇ jj

x C  , Diff j .kCjˇ j/ .@X //, and r.@X/^ as the restriction. bkˇ .r; y/ 2 C 1 .R d 1 and @X D S d 2 It is not necessary here to assume that X is the half-sphere SC its boundary, but we now may admit X (the base of the local model cone) to be an arbitrary compact C 1 manifold with boundary @X. Let us write A. / rather than A when the operator is regarded as a continuous mapping in spaces of weight  . Then we have 1 0 P r 2m kCj˛j2m ak˛ .0; y/.r@r /k .r/˛ A; ^ .A.//.y; / D @ P  k ˇ t j .r.@X/^ r kCjˇ jj bj;kˇ .0; y/.r@r / .r/ /j D1;:::;m (7.3.31) ^ .A. //.y; / W K t; .X ^ / ! K t2m; 2m .X ^ / and

0

P2m kD0

c ^ .A. //.y; w/ D @ t

.r@X

1

ak0 .0; y/w k

A;

Pj

(7.3.32)

b .0; y/w k /j D1;:::;m kD0 j;k0

c ^ .A. //.y; w/ W H t .int X / ! H t2m .int X /: Recall that (7.3.31) is a Fredholm operator for .y; / 2 T  Y n 0, t sufficiently large (as above) if and only if the operator family (7.3.32) is a family of isomorphisms for all y 2 Y , w 2  d  . Let us fix t and .y; /, and form the operator 2

K WD diag.r 2m ; .r j /j D1;:::;m /^ .A. /.y; /; z t2m; K W K t; ! K for K

t;

WD K

t;

(7.3.33)

^

.X /, and

z t2m; WD r 2m K t2m;2m .X ^ / ˚ K

Lm

j D1

1

1

r j K tj  2 ;j  2 ..@X /^ /:

Setting a.y; w/ WD c ^ .A. //.y; w/; 0 P B a0 .y; w/ WD B @ t

r@X

P

kCj˛j2m j˛j>0 kCjˇ jj jˇ j>0

ak˛ .0; w/.r@r /k .r; /˛ k

ˇ

bj;kˇ .0; w/.r@r / .r; /

 j D1;:::;m

1 C C A

462

7 Operators on manifolds with edges and boundary  d 1

 d 1

we have K D opM 2 .a/ C opM 2 .a0 /. Let us do the same for another weight ˇ, namely, z Kˇ W K t;ˇ ! K

t2m;ˇ

(7.3.34)

(the subordinate principal conormal symbol M ^ .A.ˇ//.y; w/ is then bijective for all y 2 Y , w 2  d ˇ . Choose a cut-off function !.r/, and form the operator 2

 d 1 2

B WD opM

 d 1 2

.a/ C ! opM

.a0 /;

t2m;.;ı/ B W H t;.;ı/ ! Hz

(7.3.35)

for H t;.;ı/ WD !1 H t;ı .X ^ / C .1  !1 /H t;ı .X ^ /, and

˚ L 1 1 Hz t2m;.;ı/ WD !1 H t2m; .X ^ / ˚ jmD1 H tj  2 ; 2 ..@X /^ / ˚

L 1 1 C .1  !1 / H t2m;ı .X ^ / ˚ jmD1 H tj  2 ;ı 2 ..@X /^ / for a cut-off function !1 and weights  < ı. In a similar manner we consider the operator Bˇ W H t;.ˇ;ı/ ! Hz

t2m;.ˇ;ı/

(7.3.36)

for another weight ˇ < ı. Observe that for every R > 0 such that ! D 1 on Œ0; R/ we have K j0
(7.3.37) (7.3.38)

where (7.3.38) is valid modulo compact operators in the respective spaces. In the following assertion without loss of generality we assume  < ˇ. Theorem 7.3.28. Let  < ˇ < ı be arbitrary weights such that for fixed y 2 Y the operators a.y; w/ W H t .int X / ! H t2m .int X / are isomorphisms for all w 2  d  [  d ˇ [  d ı (t sufficiently large). Then 2 2 2 (7.3.33), (7.3.34), (7.3.35) and (7.3.36) are Fredholm operators, and we have ind K  ind Kˇ D ind B  ind Bˇ D n.ˇ;  /;

(7.3.39)

where n.ˇ; / is the sum of all null-multiplicities of the non-bijectivity points of a.y; w/ in the strip fw 2 C W d2  ˇ < Re w < d2   g (in the sense of Gohberg and Sigal [60], cf. also [73], and Section 9.2 below).

7.3 Mellin-edge representations of elliptic operators

463

Proof. The Fredholm property of K  and K ˇ is satisfied when ^ .A. //.y; / and ^ .A.ˇ//.y; / are Fredholm. From the structure of these operators we know that the ellipticity conditions for r > 0 (up to r D 1 in the sense of exit ellipticity) are satisfied. The Fredholm property is then equivalent to the bijectivity of c ^ .A. //.y; w/ and c ^ .A.ˇ//.y; w/ for w 2  d  and w 2  d ˇ , respectively. Concerning the 2 2 operators B and Bˇ we are in the situation of Section 9.2, up to a transformation of the infinite (stretched) cone RC  X to the infinite cylinder R  X . From there we know that when a.y; w/ has no non-bijectivity points on  d  and 2  d ı for  < ı, the operator 2

 d 1 2

opM

t2m;.;ı/ .a/ W H t;.;ı/ ! Hz  d 1

 d 1

is Fredholm, and we have ind opM 2 .a/ D n.ı;  /. Since B D opM 2 .a/ modulo a compact operator, it follows that ind B  D n.ı;  /. Analogous arguments yield the Fredholm property of Bˇ with ind Bˇ D n.ı; ˇ/. This gives us ind B  ind Bˇ D n.ı;  /  n.ı; ˇ/ D n.ˇ;  /:

(7.3.40)

Because of the compatibility conditions (7.3.37), (7.3.38) between the operators K ; Kˇ and B ; Bˇ over corresponding subregions of RC  X and compatibilities of the respective spaces, we can apply a relative index result of [140], cf. also Section 9.4 below. It says that (7.3.41) ind K  ind Kˇ D ind B  ind Bˇ : Combining (7.3.40) and (7.3.41) yields the relation (7.3.39).

7.3.6 Interface conditions for small weights Let A D t .A T / be an elliptic boundary value problem on M . As we know, A induces a continuous operator (7.3.28) for every t 2 R, t > j C 12 , j D 1; : : : ; m, and every  2 R. As we saw in Section 7.3.4 we can complete A. / to a block matrix operator A./ of the form (7.3.24) for all  as in (7.3.25) which is a Fredholm operator for t  2m >  12 . According to Remark 7.3.27 the condition (7.3.30) is satisfied whenever (7.3.29) is bijective for all w 2  d  and all y 2 Y . Combined with 2 Theorem 7.3.28 we obtain the following result: Theorem 7.3.29. Let A D t .A T / be an elliptic boundary value problem on M , and assume that we have chosen a weight  2 R such that (7.3.29) is bijective for all w 2  d  and all y 2 Y . Moreover, fix any weight ˇ > , with 2

ˇ  2m >

1 d 1 1 d 1 d d ; ˇ  2m  62 N; ˇ  j  > ; ˇ  j   62 N: 2 2 2 2 2 2

464

7 Operators on manifolds with edges and boundary

Then there are elements J˙ 2 Vect.Y /, the fibre dimensions j˙ of which satisfy the relations jC  j D n.ˇ;  / C .ˇ  2m; d / C

m   X 1  ˇ  j  ; d  1  .ˇ; d /; (7.3.42) 2

j D1

such that A./ can be completed to an elliptic operator A./ W W t; .W / ˚ H s .Y; J / ! W t2m; 2m .W / ˚ H t2m .Y; JC / in the edge algebra. Proof. From the proof of Theorem 7.3.28 we know that ind ^ .A.ˇ//.y; / D ind Kˇ , ind ^ .A.//.y; / D ind K for every fixed y 2 Y and  6D 0. From Remark 7.3.27 we know that the condition for the existence of extra trace and potential conditions on Y is satisfied. This allows us the construction of vector bundles J˙ 2 Vect.Y / and a family of block matrix isomorphisms ^ .A. //.y; / W K t; .X ^ / ˚ J;y ! K t2m; 2m .X ^ / ˚ JC;y first for .y; / 2 S  Y , and then extended to T  Y n 0 by twisted homogeneity. It is clear that then ind ^ .A. //.y; / D jC  j for every .y; /. Thus (7.3.42) is a consequence of (7.3.39).

7.4 The Laplacian in a wedge, and other elliptic operators of the edge calculus The considerations of Section 6.3 yield useful information for the edge symbolic calculus of concrete boundary value problems on a manifold with edges and boundary, locally modelled on a stretched wedge I˛^  Rq , I˛ D Œ0; ˛ , 0 < ˛  2. The role of the present section is to observe the Fredholm property of edge symbols for the Laplacian in such a wedge, embedded in R2Cq , with Dirichlet, Neumann, and Zaremba conditions. The material of this section plays the role of further examples of the edge calculus. More details may be found in [78]. Moreover, similarly as in Section 6.4.1 we construct other special elliptic operators, here of the edge calculus.

7.4.1 The Dirichlet problem in a wedge Consider the Laplace operator D

q 2 X X @2 @2 C @xj2 @yl2 j D1 lD1

(7.4.1)

465

7.4 The Laplacian in a wedge, and other elliptic operators of the edge calculus

in a wedge fre i W r 2 RC ,  2 I˛ g  Rq , I˛ D Œ0; ˛ , 0 < ˛  2, where .r; / 2 RC  S 1 are polar coordinates in R2x1 ;x2 n f0g. Write (7.4.1) as ˚

D r 2 .r@r /2 C @2 C r 2 Rq with Rq D

Pq

@2 , lD1 @y 2 l

and consider the Dirichlet problem

u D f in int I˛^  Rq ;

T˙ u D g˙ on I˙  Rq ;

(7.4.2)

where I˙ D f˛˙ g  RC for ˛ D f0g, ˛C D f˛g, T˙ u WD ujI˙ . The problem fits to the formalism of Section 7.1.3 for W D I˛  Rq , W D x RC  I˛  Rq , where here @Wreg consists of two components RC  f˛˙ g  Rq (also in the case ˛C D 2 which is a crack situation, see also Section 8.4). Similarly as in Section 5.1.3 we identify RC  f˛˙ g with R˙ . The boundary value problem (7.4.2) represents a continuous operator   AD WD t T TC W W s .Rq ; K s; .I˛^ // ! W s2 .Rq ; K s2; 2 .I˛^ // 1

1

1

1

1

1

˚W s 2 .Rq ; K s 2 ; 2 .R // ˚ W s 2 .Rq ; K s 2 ; 2 .RC //; s > 12 ,  2 R and belongs to the edge algebra of boundary value problems. The principal symbol of A consists of the triple (7.1.25) where   @ .AD / D t @ . / @ .T / for T WD t .T TC /. Since T˙ both represent Dirichlet trace operators for the Laplacian, the operator AD is elliptic in the sense of Definition 7.1.14. The homogeneous principal edge symbol ^ .A/ has the form   ^ .AD /./ D t ^ . /./ ^ .T /./ ^ .TC /./ ; 1

1

1

1

^ .AD /./ W K s; .I˛^ / ! K s2;2 .I˛^ / ˚ K s 2 ; 2 .R / ˚ K s 2 ; 2 .RC /; (7.4.3) (cf., similarly, the formula (5.1.9)),  6D 0, where ^ . /./ D r 2 ..r@r /2 C @2  r 2 jj2 /;

^ .T˙ /./u D ujD˛˙ :

We see that ^ .A/ is independent of y and ^ .T˙ / also independent of . For the ellipticity of A in the edge calculus it is a first essential information that (7.4.3) is a family of Fredholm operators. Theorem 7.4.1. The operators (7.4.3),  6D 0, are Fredholm for arbitrary s > all  2 R for  6D k C 1, k 2 Z n f0g. ˛

3 2

and

Proof. The arguments are similar to those for Theorem 5.1.4. We have a principal symbolic tuple of the kind (5.1.13), and the ellipticity of ^ .AD /./ with respect to  ^ .AD /./;

@ ^ .AD /./;

E ^ .AD /./;

E0 ^ .AD /./

466

7 Operators on manifolds with edges and boundary

for  6D 0 is satisfied. The only specific point is the principal conormal symbol c .AD /.w/ which coincides with c;0 .AD /.w/ of Remark 6.3.9 (ii). By virtue of Proposition 6.3.8 the bijectivity of (6.3.30) holds for all w 2 1 for  6D k C 1, ˛ k 2 Z n f0g. Remark 7.4.2. Let us recall from Section 5.3.4 that in the case ˛ D  and  62 Z n f1g 8 ˆ for  2 .k C 1; k C 2/; k  1; <k ind ^ .AD /./ D .k C 1/ for  2 .k C 1; k C 2/; k  2; ˆ : 0 for  2 .0; 2/: The non-bijectivity points of c ^ .AD /.w/ consist of the set Z n f0g, and they are all simple. Theorem 7.4.3. For every 0 < ˛  2 and  6D

k ˛

C 1, k 2 Z n f0g, we have

8   ˆ C 1; .kC1/ C 1 ; k  1; for  2 k <k ˛ ˛  k  ind ^ .AD /./ D .k C 1/ for  2 ˛ C 1; .kC1/ C 1 ; k  2; ˛ ˆ   : 0 for  2  ˛ C 1; ˛ C 1 : Proof. Applying Remark 7.4.2 we obtain the result in the case ˛ D  from (5.3.16). In the general case we look at the conormal symbol H s2 .int I˛ / ˚ c ^ .AD /.w/ W H .int I˛ / ! C2 s

and find k , k 2 Z n f0g, as the non-bijectivity points; they are all simple. Then a ˛ homotopy argument gives us the index expression in general. Having established the Fredholm property of (7.4.3) we are now in the position to fill up the operators (7.4.3) to a block matrix of isomorphisms 

 K s; .I˛^ / K s2; 2 .I˛^ / ^ .AD /./ ^ .KD /./ ˚ ^ .AD /./ WD ! ; W ˚ ^ .TD /./ ^ .QD /./ C j C jC  6D 0, where K s2;2 .I˛^ / means the space on the right-hand side of (7.4.3) and jC  j D ind ^ .AD /./:   D KD We then find an operator AD D A 2 Y;d .W ; g/ for g D .;   2/ which TD QD is elliptic in the sense of Definition 7.2.34, and we can apply the elements of the edge calculus, especially Theorem 7.2.36.

467

7.4 The Laplacian in a wedge, and other elliptic operators of the edge calculus

7.4.2 The Neumann and the Zaremba problem in a wedge We now consider boundary value problems of the form (7.4.2) for other choices of trace operators T˙ , namely, Neumann conditions on both boundary components, i.e., ˇ ˇ T u WD r 1 @ uˇI ; TC u WD r 1 @ uˇI C

and Zaremba conditions ˇ T u WD uˇI ;

ˇ TC u WD r 1 @ uˇI : C

We then obtain continuous operators   AN WD t T TC W W s .Rq ; K s; .I˛^ // 3

3

3

3

3

3

! W s2 .Rq ; K s2;2 .I˛^ // ˚ W s 2 .Rq ; K s 2 ; 2 .R // ˚ W s 2 .Rq ; K s 2 ; 2 .RC // and

  AZ WD t T TC W W s .Rq ; K s; .I˛^ // 1

1

1

3

3

3

! W s2 .Rq ; K s2;2 .I˛^ // ˚ W s 2 .Rq ; K s 2 ; 2 .R // ˚ W s 2 .Rq ; K s 2 ; 2 .RC //; respectively, s > 32 ,  2 R. Similarly as in the preceding section the essential specific information, compared with the general formalities in boundary value problems on a manifold with edges is the index of the edge symbols 3

3

3

3

1

1

3

3

^ .AN /./ W K s; .I˛^ / ! K s2;2 .I˛^ / ˚ K s 2 ; 2 .R / ˚ K s 2 ; 2 .RC / (7.4.4) and ^ .AZ /./ W K s; .I˛^ / ! K s2;2 .I˛^ / ˚ K s 2 ; 2 .R / ˚ K s 2 ; 2 .RC /; (7.4.5) respectively, s > 32 ,  2 R,  6D 0. It is again clear that ^ .AN /./ and ^ .AZ /./ are elliptic with respect to  ^ .  /./; @ ^ .  /./; E ^ .  /./; E0 ^ .  /./

(7.4.6)

where ‘’ stands for AN or AZ . For the Fredholm property it remains to consider the respective principal conormal symbols. From Proposition 6.3.11 we obtain that 1 0 2 w C @2 H s2 .int I˛ / s A @ ˚ W H .int I˛ / ! c ^ .AN /.w/ r @ C2 rC @

468

7 Operators on manifolds with edges and boundary

is bijective for all w 2 1 ,  6D k C 1, k 2 Z, (the non-bijectivity points are ˛ and they are all simple, except for k D 0). Moreover 1 0 2 w C @2 H s2 .int I˛ / s ˚ c ^ .AZ /.w/ D @ r A W H .int I˛ / ! 2 C rC @ is bijective for all w 2 1 ,  6D .2kC1/ , they are all simple). 2˛

.2kC1/ 2˛

k , ˛

C 1, k 2 Z, (the non-bijectivity points are

Theorem 7.4.4. (i) The operators (7.4.4),  6D 0, are Fredholm for arbitrary s > and all  2 R for  6D k C 1, k 2 Z, and we have ˛ (   C 1; .kC1/ C 1 ; k  1; k for  2 k ˛ ˛   ind ^ .AN /./ D .k C 1/ for  2 k C 1; .kC1/ C 1 ; k  2: ˛ 2 (ii) The operators (7.4.5),  6D 0, are Fredholm for arbitrary s > for  6D .2kC1/ C 1, k 2 Z, and we have 2˛ 

ind ^ .AZ /./ D .k C 1/ for  2

3 2

3 2

and all  2 R 

.2k C 3/ .2k C 1/ C 1; C1 : 2a 2˛

Proof. The operators AN and AZ are elliptic with respect to the symbols (7.4.6), and the conormal symbols are bijective on 1 for the indicated values of  . From the formula (5.3.17) and Corollary 5.3.3 (which corresponds to the case ˛ D ) we obtain the index expressions for general 0 < ˛  2 by a homotopy argument. Similarly as at the end of the preceding section we can construct block matrices of isomorphisms ^ .AN /./ and ^ .AZ /./, where the difference of the numbers of the trace and potential entries, depending on , is given by ind ^ .AN /./ and ind ^ .AZ /./, respectively. Again we can proceed as in the general edge calculus and apply, in particular, Theorem 7.2.36. More details on mixed problems in the case 0 < ˛  2 may be found in [78]. The case ˛ D 2 is separately treated once again in Section 8.4 below.

7.4.3 Other examples of elliptic edge operators Let W be the stretched manifold associated with a compact manifold W with edge, and, in this section, without boundary. From the definition we have a finite system of singular charts close to @W x C  X  Rq ; W V ! R

(7.4.7)

where V denotes a neighbourhood in W intersecting @W (observe that (7.4.7) is the stretched variant of (2.4.24)). Moreover, let the transition maps to different (2.4.24) of

7.4 The Laplacian in a wedge, and other elliptic operators of the edge calculus

469

our system be independent of r for small r. Furthermore, choose a finite number of coordinate neighbourhoods U on Wreg D W n @W such that the sets V and U form a covering of W , and fix charts  W U ! R1CnCq :

(7.4.8)

Moreover, choose a partition of unity subordinate to the covering of W by the sets V and U . Let !V and 'U denote the corresponding functions and set ! WD  !V , x C  X  Rq / in such a way that they ' WD  'U . We choose the functions ! 2 C01 .R x C  Rq . Observe that the sets V \ Y form an open covering only depend on .r; y/ 2 R of Y with charts V \ Y ! Rq induced by (7.4.7) and that the functions !V jV \Y form a subordinate partition of unity on Y ; in local coordinates the corresponding functions are given by !.0; y/. Let us choose operators Dj 2 Diff 1 .X /, j D 1; : : : ; N (for a suitable N ), such that Dj are given by vector fields on X which span the tangent space of X at every point. Similarly, let Hj 2 Diff 1 .Wreg /, j D 1; : : : ; M (for a suitable M ), be operators given by vector fields on Wreg which span the tangent space of Wreg at every point. We fix an s 2 N and form the differential operators depending on a parameter  2 Rl  1    !.r; y/r jˇ j .r@r /j D ˛ Dyˇ  r s (7.4.9) for every V with the associated  and !, for arbitrary j 2 N, ˛ 2 N N , ˇ 2 N q , ˛N , and  2 N l , such that j C j˛j C jˇj C jj  s, D ˛ WD D1˛1 : : : DN . 1 / '. H ı /

(7.4.10)

for every U with the associated  and ', for arbitrary ı 2 N M ,  2 N l , such that ıM . jıj C jj  s, H ı WD H1ı1 : : : HM In this way we obtain a column vector of parameter-dependent differential operators   t (7.4.11) Bk ./ kD0;:::;K for some K, determined by all possible combinations of the involved multi-indices and the number of sets V , U of the open covering of W . Remark 7.4.5. In (7.4.9) we interpret the r-variables as operators of multiplication from the left, with the exception of the factor r s . More precisely, (apart from .1 / / the operator has the form u.r; / ! !.r; y/r jˇ j .r@r /j D ˛ Dyˇ  .r s u.r; //: In formal adjoints of such operators r s becomes an operator from the left, while the other variables are acting from the right. Observe that (7.4.11) induce continuous operators L n ts; n 2 C.s/ .W / B./ W W t; 2 C .W / ! K kD0 W

470

7 Operators on manifolds with edges and boundary

for every t;  2 R,  2 Rl ; n D dim X . Similarly, formal adjoints with respect to the n scalar product of L2 .W / D W 0; 2 .W / induce continuous operators LK n t; n 2 Cı .W / ! W ts; 2 C.ıs/ .W / B  ./ W kD0 W for every t; ı 2 R,  2 Rl . Let us now interpret B  ./B./ D

K X

Bk ./Bk ./

(7.4.12)

kD0

as a family of continuous operators n

n

B  ./B./ W W 2t; 2 Cs .W / ! W 2.ts/; 2 s .W /:

(7.4.13)

l We have B  ./B./ 2 Diff 2s deg .W / for every  2 R . We are interested in the question of ellipticity of (7.4.13) with respect to the principal symbolic structure  D . ; ^ / of the edge calculus, mainly for single operators, say, when  D 0. For analysing this case we will employ the fact that

B  B WD B  .0/B.0/ is embedded in a parameter-dependent family of operators. The operators B  ./B./ are parameter-dependent elliptic of order 2s on Wreg , i.e., the parameter-dependent principal symbol  .B  B/.; / with  2 Rl as an additional covariable does not vanish on T  Wreg  Rl n 0. Moreover, in the local splitting of variables .r; x; y/ close to @W we can write  .B  B/.; / D r 2s Q .B  B/.r; x; y; r%; ; r; / with a function Q .B  B/.r; x; y; %; Q ; ; Q / which is homogeneous of order 2s in .%; Q ; ; Q / 6D 0 and non-vanishing up to r D 0. In particular, the operators B  ./B./ are  -elliptic in the sense of the edgedegeneracy, for every fixed  2 Rl . Let us now turn to the principal edge symbol ^ .B  B/ as a family of continuous operators n

n

^ .B  B/.y; ; / W K t; 2 Cs .X ^ / ! K t2s; 2 s .X ^ /;

(7.4.14)

t 2 R. We can write (7.4.14) as a composition of L n n (7.4.15) ^ .B/.y; ; / W K t; 2 Cs .X ^ / ! j Cj˛jCjˇ jCjjs;! K ts; 2 .X ^ /; L n n  ts; 2 ^ .B /.y; ; / W .X ^ / ! K t2s; 2 s .X ^ /; (7.4.16) j Cj˛jCjˇ jCjjs;! K where ^ .B/.y; ; / is a vector of operator functions of the form !.0; y/r jˇ j .r@r /j D ˛ ˇ  r s ;

471

7.4 The Laplacian in a wedge, and other elliptic operators of the edge calculus

parametrised by .j; ˛; ˇ; / with j C j˛j C jˇj C jj  s, and !.0; y/ belonging to V (the latter correspondence is just the meaning of ! in the direct sums in (7.4.15) and (7.4.16)). The subordinate conormal symbols are vectors of operator functions on X , namely, c ^ .B/.y; w; / D t .!.0; y/.w C s/j D ˛  /j Cj˛jCjjs;! ; c ^ .B  /.y; w; / D .!.0; y/.1  .w C s//j .D ˛ /  /j Cj˛jCjjs;! : They represent families of continuous maps L c ^ .B/.y; w; / W H t .X / ! j Cj˛jCjjs;! H ts .X /; L ts c ^ .B  /.y; w; / W .X / ! H t2s .X /; j Cj˛jCjjs;! H t 2 R. The operators c ^ .B  B/.y; w; / W H t .X / ! H t2s .X /

(7.4.17)

are parameter-dependent elliptic on X , for every fixed y 2 Y , with the parameters .Im w; / 2 R1Cl , w 2 ˇ for every real ˇ. In particular, we consider (7.4.17) on the weight line  1 s , where 2



c ^ .B  B/ y; D

X



1  s C i%;  2  j  j X 1 2 ˛   1 ! .0; y/  i% .D /  C i% D ˛  2 2

(7.4.18)

! j Cj˛jCjjs

(similarly as before the sum over ! means summation over all contributions from V \Y for V varying over the above-mentioned system of neighbourhoods on W . Proposition 7.4.6. The operators (7.4.17) form a family of isomorphisms for all y 2 Y , w 2  1 s ,  2 Rl , and all t 2 R. 2

This is evident for algebraic reasons, taking into account the form (7.4.18). Theorem 7.4.7. The operators (7.4.14) form a family of isomorphisms for all .y; / 2 T  Y n 0,  2 Rl , and for all t 2 R. Proof. We first show that (7.4.14) is a family of Fredholm operators. To this end we have to verify that these operators are elliptic in the cone calculus on the infinite (stretched) cone X ^ with r ! 1 being treated as a conical exit to infinity. The principal symbolic hierarchy of the cone calculus on X ^ contains the conormal, interior and exit components. By Proposition 7.4.6 the conormal symbol is elliptic on the weight line  1 s . Moreover, the interior symbol  ^ .B  B/.r; x; %; / does not vanish for 2

.%; / 6D 0, and its reduced version r 2s  ^ .B  B/.r; x; r 1 %; / does not vanish for .%; / 6D 0, up to r D 0. The exit symbol E ^ .B  B/ can be expressed in variables

472

7 Operators on manifolds with edges and boundary

Q The exit ellipticity, as xQ varying in a conical set of R1Cn , with the covariables . we know from Section 2.3.4, contains two conditions, namely, non-vanishing of the homogeneous component in xQ of order 0 for jxj Q ! 1 and all Q 2 R1Cn , and, moreover, non-vanishing of the homogeneous component of the latter function in Q of order 2s for Q 6D 0. This property is satisfied for  6D 0, cf. Section 2.4.5. Thus we obtain altogether the Fredholm property of (7.4.14). Next we observe that the parameter  2 Rl behaves as an extra covariable in the exit calculus, and our operators are parameter-dependent elliptic with parameter . In particular, the above-mentioned homogeneous component in xQ of order zero for Q / 2 R1CnCl , and its principal homogeneous jxj Q ! 1 does not vanish for all . ; Q Q / 6D 0. Of course,  is also an part in . ; / of order 2s does not vanish for . ; extra covariable in the pseudo-differential calculus on the open manifold X ^ , and our operators are parameter-dependent elliptic also in that sense. Finally the conormal symbol is a bijective family of parameter-dependent elliptic operators on X . This has the consequence that we can construct a parametrix of (7.4.14) in the cone algebra on X ^ with exit property at 1, where the left-over terms are Schwartz functions in  2 Rl with values in Green operators on X ^ . In other words we conclude that the operators (7.4.14) are isomorphisms for all jj  C for some positive C . Thus, since they are Fredholm for all  2 Rl we obtain ind ^ .B  B/.y; ; / D 0 for all  2 Rl , .y; / 2 T  Y n 0. To complete the proof of Theorem 7.4.7 it is enough to show that (7.4.14) has a trivial kernel for all  2 Rl , .y; / 2 T  Y n 0. n Assuming u 2 ker ^ .B  B/.y; ; / we know u 2 K 1; 2 Cs .X ^ /; this is contained n in K 0; 2 .X ^ /. Thus we can form 0 D .^ .B  B/.y; ; /u; u/ D .^ .B/.y; ; /u; ^ .B/.y; ; /u/

˚K

0; n 2

.X ^ /

where the direct sum in the subscript on the right-hand side is taken over ! and the tuples .j; ˛; ˇ; / with j C j˛j C jˇj C jj  s. In other words, all terms ..!.0; y/r jˇ j .r@r /j D ˛ ˇ  r s u; !.0; y/r jˇ j .r@r /j D ˛ ˇ  r s u/

K

0; n 2

.X ^ /

vanish. This is the case, in particular, for j C j˛j C jˇj C jj D 0. It follows that u 0. Corollary 7.4.8. The operators B  ./B./ are elliptic of order 2s in the edge calculus on W , for every  2 Rl . As such they induce Fredholm operators (7.4.13) for all t 2 R,  2 Rl . We now modify the construction of operators B  ./B./ in order to obtain isomorphisms (7.4.13) in the edge algebra. To this end, instead of (7.4.9) and (7.4.10) we start from families of operators .1 / !.r; y/r jˇ j .r@r /j D ˛ .Dy ; #/ˇ  r s

(7.4.19)

473

7.4 The Laplacian in a wedge, and other elliptic operators of the edge calculus

for every V with the associated  and !, where # D .#1 ; : : : ; #p /, is an additional ˇ ˇ ˇ parameter, p  1, ˇ 2 N qCp , .Dy ; #/ˇ WD Dyˇ11 : : : Dyqq #1 qC1 : : : #p qCp , and j 2 N, ˛ 2 N N ,  2 N l , such that j C j˛j C jˇj C jj  s, and . 1 / ' H ı .; #/

(7.4.20)

for every U with the associated  and ', for arbitrary ı 2 N M ,  2 N lCp , such that    jıj C jj  s, .; #/ WD 11 : : : ll #1lC1 : : : #plCp . We then obtain an analogue of (7.4.11), namely, a column vector of .; #/-depending operators   t Bk .; #/ kD0;:::;K (of course, with another K). Similarly as (7.4.12), (7.4.13) we obtain operators B  .; #/B.; #/ 2 Diff 2s deg .W /; n

n

B  .; #/B.; #/ W W 2t; 2 Cs .W / ! W 2.ts/; 2 s .W /;

(7.4.21)

and we have B  .; 0/B.; 0/ D B  ./B./ with the right-hand side in the sense of (7.4.12). Remark 7.4.9. The operators B  .; #/B.; #/ are elliptic of order 2s in the edge calculus on W . As such they induce Fredholm operators (7.4.21) for all t 2 R and .; #/ 2 RlCp . In fact, the arguments for Corollary 7.4.8 can easily be modified in the case of .; #/-depending families. Theorem 7.4.10. The operators (7.4.21) are isomorphisms for all t 2 R, .; #/ 2 RlCp . Proof. The operators (7.4.21) form a parameter-dependent elliptic family of the edge calculus. In particular, the analogue of (7.4.14) n

n

^ .B  B/.y; ; ; #/ W K t; 2 Cs .X ^ / ! K t2s; 2 s .X ^ /

(7.4.22)

is a family of isomorphisms for all .y; ; #/ 2 T  Y Rp n0 (where 0 means .; #/ D 0) and all  2 Rl . This allows us the construction of a parameter-dependent parametrix of (7.4.21) with parameter # 2 Rp , for every fixed  2 Rl , where the left-over terms are Schwartz functions in # 2 Rp with values in the space of smoothing operators of the edge calculus on W . This shows that the operators (7.4.21) become invertible when j#j is sufficiently large. Thus the Fredholm operators (7.4.21) are of index 0 for all # 2 Rp and  2 Rl . To show that the operators are isomorphisms it is enough to check that the kernel is trivial for all ; #. This can be done in a similar manner as in

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7 Operators on manifolds with edges and boundary

the last part of the proof of Theorem 7.4.7. In fact, u 2 ker.B  .; #/B.; #/ shows n n that u 2 W 1; 2 Cs .W /; this is contained in W 0; 2 .W /. We have 0 D .B  .; #/B.; #/u; u/L2 .W / D

K X

.Bk .; #/u; Bk .; #/u/L2 .W /

kD0

which gives us .Bk .; #/u; Bk .; #/u/L2 .W / D 0 for every k. Going back to the original meaning of the operators Bk .; #/, namely, (7.4.19) or (7.4.20), we see that, when we insert, for instance,  D 0, # D 0, j C j˛j C jˇj C jj D 0 or jıj C jj D 0, the functions !V r s u and 'U u have a vanishing L2 .W /-norm for all V and U , and this gives us u 0.

Chapter 8

Corner operators and problems with singular interfaces

In Chapter 4 and 5 we have studied mixed elliptic boundary value problems in the case of a smooth interface, i.e., the submanifold of the boundary with the jump of the conditions was assumed to be smooth. In the present chapter we treat the case of interfaces with conical singularities. Here we mainly concentrate on the case of the Zaremba problem. The approach from the pseudodifferential calculus of corner operators shows that other mixed problems can be treated in a similar manner. A situation of different geometry is realised in crack problems, cf. [90]. We apply the corner operator calculus also to the case of cracks that have a boundary with conical singularities. The results of this chapter are based on the author’s joint papers [76], [77], see Sections 8.1–8.3 and 8.5, and [193], see Section 8.4.

8.1 Singular mixed problems and corner manifolds By a manifold with corners (of singularity order 2) we understand a topological space with finitely many corner points, locally modelled on a cone with base that has itself conical singularities. In the present case we admit the base to be a manifold with conical singularities and boundary. Outside the corner points the manifold is assumed to have edges and boundary. Corner operators are defined in terms of the Mellin transform in the corner axis variable t 2 RC with amplitude functions taking values in cone boundary value problems on the base. A typical novelty are corner Sobolev spaces with double weights, responsible for t ! 0 and r ! 0 (with r 2 RC being the cone half-axis variable on the base).

8.1.1 The singular Zaremba problem The ideas for studying singular mixed problems will be illustrated by the Zaremba problem for a second order elliptic differential operator A in a smooth boundary domain G  R3 when the interface, i.e., the subset Z of Y WD @G with the jump of the boundary x The problem is represented by conditions, has conical singularities. Set X WD G. equations Au D f in int X; T˙ u D g˙ on int Y˙ ; (8.1.1) where Y D Y [ YC , with closed subsets Y˙  Y , and a curve Z WD Y \ YC with finitely many conical singularities. For convenience we consider the case of one conical point v 2 Z (there are no additional difficulties with finitely many such

476

8 Corner operators and problems with singular interfaces

points). The assumption that X is of dimension 3 is also not essential (recall that in Chapter 5 we treated mixed problems without any restriction on the dimension), but the technicalities in the three-dimensional case are a little easier. X may be a C 1 manifold with boundary, not necessarily the closure of a smooth domain. Our approach admits also the case when X n fvg is a manifold with edge and boundary and with a specific corner structure near v, cf. Remark 8.3.19 below. The operators T˙ which represent the boundary conditions are assumed to be of the form T˙ WD r˙ B˙ with differential operators B˙ with smooth coefficients, given in an open neighbourhood of Y˙ ; r˙ denote the operators of restriction to int Y˙ . We assume that T˙ satisfy the Shapiro–Lopatinskij condition on int Y˙ with respect to A, uniformly up to Z from the respective sides. The classical Zaremba problem corresponds to the case A D (the Laplacian) with Dirichlet and Neumann conditions on int Y and int YC , respectively, cf. the notation in Chapter 5. Similarly as (5.1.2) the operator   A WD t A T TC (8.1.2) (in the Zaremba case) induces continuous maps 1

3

A W H s .int X / ! H s2 .int X / ˚ H s 2 .int Y / ˚ H s 2 .int YC / ˇ for s > 32 ; H s .int Y˙ / D H s .Y /ˇint Y . It is again clear that when we ask solutions u ˙ of (8.1.1) in standard Sobolev spaces we rule out most of the interesting solutions that may have singularities close to Z and, especially, close to v. The strategy will be to realise A as a continuous operator A W V s;.;ı/ .X/ ! V s2;.2;ı2/ .X/

(8.1.3)

between corner Sobolev spaces with double weights .; ı/ 2 R2 , s > 32 ; the space on the right-hand side of (8.1.3) is a direct sum 1

1

1

3

3

3

V s2;. 2;ı2/ .X/ ˚ V s 2 ;. 2 ;ı 2 / .Y / ˚ V s 2 ;. 2 ;ı 2 / .YC /:

(8.1.4)

The definition of these spaces will be given in Section 8.2.2 below; concerning the stretched spaces X and Y˙ associated with X and Y˙ , respectively, cf. Sections 8.1.4 and 8.2.2. Similarly as (5.1.5) we will find block matrices 

A K AD T Q



V s;.;ı/ .X/ V s2;.2;ı2/ .X/ ˚ W ! ˚ H s1;ı1 .Z; C j / H s3;ı3 .Z; C jC /

(8.1.5)

for suitable dimensions j˙ , depending on the weights  , such that (8.1.2) is a Fredholm operator and has a parametrix in a so-called corner calculus of boundary value problems. Here  2 R is arbitrary, with the exception of a certain discrete set, and then ı 2 R is also arbitrary, up to a discrete set which depends on  and the chosen operators T ; K; Q.

8.1 Singular mixed problems and corner manifolds

477

8.1.2 Operators in edge representation We now consider our operators in local coordinates close to Z n fvg and summarise a few results of Section 5.1.3, adapted to the present situation. After that, in Section 8.2 below, we study the operators close to v. Let U  X be a coordinate neighbourhood with U \ Z 6D ;, v 62 U , and let x 3C D fx D .x1 ; x2 ; x3 / 2 R3 W x3  0g W U ! R be a chart which restricts to charts 0 W U 0 ! R2 D fx 2 R3 W x3 D 0g;

00 W U 00 ! R D fx 2 R3 W x2 D x3 D 0g

for U 0 WD U \ Y , U 00 WD U \ Z. In particular, x1 2 R plays the role of a local coordinate on Z which we also call z. Let us write the operator A in local coordinates x 3 in the form x2R C X AD a˛ .x/Dx˛ ; (8.1.6) j˛j2

x 3 /. Inserting polar coordinates .r; / in R x 2 n f0g for R x 2 D f.x2 ; x3 / 2 a˛ 2 C 1 .R C C C 2 R W x3  0g gives us X A D r 2 aj k .r; z/.r@r /j .rDz /k (8.1.7) j Ck2

x C  R; Diff 2.j Ck/ .S 1 //. Here S 1 WD f 2 S 1 W 0  with coefficients aj k 2 C 1 .R C C   g. In a similar manner we reformulate the boundary operators T˙ which have the form T˙ W u ! r˙ B˙ u ˇ for r˙ f WD f .x1 ; x2 ; 0/ˇx

2 ?0

, and

B˙ u D

X

b˙;˛ .x/Dx˛

(8.1.8)

j˛j˙

(in the Zaremba case for C D 1;  D 0) with smooth coefficients b˙;˛ in a neighx 2 n f0g we then bourhood of f.x1 ; x2 ; 0/ W x2 ? 0g in R3 . In polar coordinates in R C obtain X T˙ D r˙ r ˙ b˙;j k .r; z/.r@r /j .rDz /k (8.1.9) j Ck˙

x C  R; Diff ˙ .j Ck/ .S 1 // where r˙ u.r; ; z/ WD with coefficients b˙;j k 2 C 1 .R C u.r; ˙ ; z/ for C D 0,  D . The choice of our formulations is motivated by the fact that the constructions also make sense for (elliptic) differential operators A of arbitrary order, together with mixed elliptic boundary conditions. With (8.1.7) and

478

8 Corner operators and problems with singular interfaces

(8.1.9) we associate continuous operators in weighted edge spaces, defined by the operator-valued amplitude function 1 0 P r 2 j Ck2 aj k .r; z/.r@r /2 .r/k P C B (8.1.10) a.z; / WD @ r r  j Ck b;j k .r; z/.r@r /j .r/k A ; P rC r C j CkC bC;j k .r; z/.r@r /j .r/k where r˙ denotes the operator of restriction v.r; / ! v.r; ˙ / with .r; ˙ / for C D 0 and ' D  being identified with RC and R , respectively. Then the local representative of (8.1.2) is equal to Opz .a/ D A: (8.1.11) Since the global operators are then defined by local terms, combined with pull backs to the original configuration and multiplied by the functions of a partition of unity, cf. the notation in Section 8.2.2 below, without loss of generality we assume that the coefficients aj k .r; z/ in (8.1.7) and b˙;j k .r; z/ in (8.1.9) are independent of r for 1 ^ / / and large r. We employ the spaces K s; ..SC 1

1

3

3

1 ^ 1 ^ K s2;2 ..SC / / WD K s2; 2 ..SC / / ˚ K s 2 ; 2 .R / ˚ K s 2 ; 2 .RC /

endowed with the group actions 1 ^ 1 ^ .1/ W K s; ..SC / / ! K s; ..SC / /;

u.r; / ! u.r; /;

and 1 ^ 1 ^  W K s2; 2 ..SC / / ! K s2; 2 ..SC / /; 3

1

u.r; / ˚ v .r/ ˚ vC .r/ ! .1/ u.r; / ˚  2 .0/ v .r/ ˚  2 .0/ vC .r/; 1

 2 RC ; recall that .0/ v.r/ D  2 v.r/. We then have the following property: Proposition 8.1.1. The operators (8.1.10) define a family of continuous operators 1 ^ 1 ^ a.z; / W K s; ..SC / / ! K s2; 2 ..SC / /

for every s;  2 R, s > 32 , and we have 1 ^ 1 ^ / /; K s2; 2 ..SC / //; a.z; / 2 S 2 .Rz  R I K s; ..SC

with the group actions  WD f.1/ g2RC and  WD f g2RC (cf. the notation (2.2.11)). Proof. The arguments are analogous to those of the proof for Theorem 7.2.3. Proposition 8.1.2. The operator A D t .A T TC /, given by (8.1.7) and (8.1.9), induces continuous operators 1 ^ 1 ^ A W W s .R; K s; ..SC / // ! W s2 .R; K s2; 2 ..SC / //

(8.1.12)

8.1 Singular mixed problems and corner manifolds

479

for every s;  2 R, s > 32 , where the space on the right-hand side is defined by 1

1

1

3

3

3

1 ^ W s2 .R; K s2; 2 ..SC / // ˚ W s 2 .R; K s 2 ; 2 .R //

˚ W s 2 .R; K s 2 ; 2 .RC //: 1 ^ 1 ^ Proof. We have W s2 .R; K s2; 2 ..SC / // D W s2 .R; K s2; 2 ..SC / //, cf. the notation in Remark 2.2.14. Then it suffices to apply Proposition 8.1.1 and the continuity of edge operators (cf. Corollary 7.2.4) to the operator (8.1.11).

8.1.3 Principal symbols and edge conditions We now recall the principal symbolic structure of edge boundary value problems, specified to the operator (8.1.12). From the general calculus of Chapter 7 we know that such operators have a principal symbolic hierarchy  .A/ D . .A/; @ .A/; ^ .A//: The interior symbol  .A/ WD  .A/ than the standard homogeneous P is nothing other ˛ principal symbol  .A/.x; / D a .x/ of the operator (8.1.6), .x; / 2 j˛jD2 ˛  x3 T RC n 0. The boundary symbol, according to the notation of Chapter 3, has two components @ .A/ D .@; .A/; @;C .A//, namely,   @ .A/.x1 ; x2 ; 1 ; 2 / @;˙ .A/.x1 ; x2 ; 1 ; 2 / WD ; @ .T˙ /.x1 ; x2 ; 1 ; 2 / where @ .A/.x1 ; x2 ; 1 ; 2 / WD  .A/.x1 ; x2 ; 0; 1 ; 2 ; Dx3 /; @ .T˙ /.x1 ; x2 ; 1 ; 2 / WD r˙  .B˙ /.x1 ; x2 ; 0; 1 ; 2 ; Dx3 / for x2 ? 0; . 1 ; 2 / 6D 0. For the Dirichlet condition on the minus side we simply have @ .T /.x1 ; x2 ; 1 ; 2 / W u ! ujx3 D0;x2 <0 . The boundary symbols @;˙ .A/ represent operator families H s2 .RC / ˚ @;˙ .A/.x1 ; x2 ; 1 ; 2 / W H .RC / ! C s

for s > 32 , with the ‘standard’  -homogeneity in . 1 ; 2 / 6D 0. The principal edge symbol ^ .A/ has the form 0 1 ^ .A/ ^ .A/.z; / WD @ ^ .T / A .z; / ^ .TC /

480

8 Corner operators and problems with singular interfaces

for .z; / 2 T  R n 0, where ^ .A/.z; / WD r 2

X j Ck2

^ .T˙ /.z; / WD r˙ r ˙

aj k .0; z/.r@r /j .r/k ; X

b˙;j k .0; z/.r@r /j .r/k :

(8.1.13) (8.1.14)

j Ck˙

The edge symbol ^ .A/ represents a family of continuous operators 1 ^ 1 ^ ^ .A/.z; / W K s; ..SC / / ! K s2; 2 ..SC / /

(8.1.15)

for all s;  2 R, s > 32 , and has the homogeneity ^ .A/.z; / D 2  ^ .A/.z; /..1/ /1

(8.1.16)

for all  2 RC , with  as before. The realisation (8.1.15) of ^ .A/.z; / in the spaces with weight  2 R will now be denoted by ^ .A. //.z; /. Let us recall from Corollary 5.3.3 the following result.   Theorem 8.1.3. For every k 2 Z and  2 12  k; 32  k the operators (8.1.15) are Fredholm, and we have ind ^ .A. //.z; / D k for every s 2 R; s > 32 , and .z; / 2 T  R n 0. Remark 8.1.4. For every .z; / 2 T  Rn0 and  62 ZC 12 the operator (8.1.15) belongs to the cone algebra of boundary value problems (with the transmission property) on 1 ^ the infinite (stretched) cone .SC / and is elliptic with respect to the principal symbolic structure, i.e., . ; @ ; c /, the interior, the boundary and the conormal symbol in the cone algebra. As in Chapter 5 we are now able to fill up the Fredholm family (8.1.15) to a family of isomorphisms by additional entries of trace and potential type with respect to the interface. Their number is determined by the dimension of kernels and cokernels. By virtue of (8.1.16) we have    ind ^ .A. // z; D ind ^ .A. //.z; / jj

(8.1.17)

for all .z; / 2 T  R n 0. Since z is a one-dimensional variable, the cosphere bundle S  R consists of two copies of R, characterised by .z; 1/ and .z; C1/; z 2 R. As we already know, the indices (8.1.17) on the  plus and the minus side coincide. Now we can choose dimensions j˙ . / for  2 12  k; 32  k such that jC . /  j . / D k and we find isomorphisms   ^ .A. // ^ .K/ .z; /; ^ .a/.z; / WD ^ .T / ^ .Q/ (8.1.18) s2; 2 s; 1 ^ j 1 ^ jC ^ .a/.z; / W K ..SC / / ˚ C ! K ..SC / / ˚ C ;

481

8.1 Singular mixed problems and corner manifolds

first for jj D 1, and then extended by homogeneity to all  6D 0, by the rule ^ .a/.z; / D 2 Q  ^ .a/.z; /1  ;

 2 RC ;

(8.1.19)

where  WD diag..1/ ;  idC j /, Q  WD diag. ;  idC jC /. We assume that the upper left corner is independent of z for large jzj, and we can (and will) choose the other entries of (8.1.18) also independent of z for large jzj. Setting   a k a.z; / WD .z; / (8.1.20) t q with t.z; / WD ./^ .T /.z; /, k.z; / WD ./^ .K//.z; /, and q.z; / WD ./^ .Q/.z; / and any fixed excision function ./ we obtain an element 1 ^ 1 ^ a.z; / 2 S 2 .R  RI K s; ..SC / / ˚ C j ; K s2; 2 ..SC / / ˚ C jC /;Q

for  WD f g2RC , Q WD fQ  g2RC . This gives rise to continuous operators 1 ^ 1 ^ Op.a/ W W s .R; K s; ..SC / / ˚ C j / ! W s2 .R; K s2; 2 ..SC / / ˚ C jC /Q (8.1.21) for all s;  2 R, s > 32 .

8.1.4 Corner manifolds Our next objective is to show that a mixed boundary value problem with singular interface can be interpreted as a problem on a manifold with corner and boundary. We will first discuss corner manifolds in terms of the present special case; more information may be found in Section 8.3.4 below, cf. also [18]. The situation in the case of mixed x we associate a chain of subspaces problems is as follows. With X D G X  X 0  X 00  X 000

(8.1.22)

where X 0 WD @X , X 00 WD Z, X 000 WD fvg, and X n X 0 , X 0 n X 00 D int Y [ int YC , X 00 n X 000 D Z n fvg DW Zreg , and X 000 are C 1 manifolds of decreasing dimensions. Moreover, X is equal to the disjoint union X D .X n X 0 / [ .X 0 n X 00 / [ .X 00 n X 000 / [ X 000 : For local descriptions we observe that X n @X is a C 1 manifold; X n Z is a C 1 manifold with boundary @X n Z; X n fvg is a manifold with edge Z n fvg and boundary @X n fvg; and X is a manifold with corner v and boundary @X :

(8.1.23)

482

8 Corner operators and problems with singular interfaces

The latter relation is the new aspect compared with Chapter 5. As the base of the corner point we have 2 x 3C K WD SC D S2 \ R 2 which is regarded as a manifold with boundary K 0 WD S 1 D @SC and two conical 00 1 singularities K D fv1 g [ fv2 g  S , where v1 ; v2 are the points which subdivide the boundary into sets with the Dirichlet or Neumann conditions (in the Zaremba case). In other words, similarly as (8.1.22), (8.1.23) we have a chain of subspaces K  K 0  K 00 and a disjoint union of C 1 submanifolds

K D .K n K 0 / [ .K 0 n K 00 / [ K 00 : 2 2 as a manifold with conical points v1 ; v2 2 @SC In order to stress the interpretation of SC 2 2 and boundary we also write SC rather than SC . x C  K/=.f0g  K/ with K The space X is locally near v modelled on K D .R 2 being identified with SC . More precisely, we fix a neighbourhood U0 of v in X and a ‘singular corner chart’ 2

 W U0 ! .SC / (8.1.24)

which restricts to an isomorphism 2 0 W U0 n fvg ! RC  SC

(8.1.25)

2

/ is any in the category of C 1 manifolds with edge and boundary. If Q W U0 ! .SC 1 other homeomorphism of that kind then Q 0 ı 0 required to be the restriction of an 2 2 isomorphism R  SC ! R  SC in the category of manifolds with edge and boundary 2 to RC  SC . We then call  and Q equivalent. The map (8.1.25) is compatible with ‘singular edge charts’ from neighbourhoods U1 of points z 2 Z n fvg, U1 \ fvg D ;,

restricting to

1

/ R U1 ! .SC

(8.1.26)

1 ^ U1 n Z ! .SC /  R:

(8.1.27)

(8.1.26) is an isomorphism in the category of manifolds with edge and boundary. The 2 identification (8.1.25) allows us to invariantly attach f0g  SC to X n fvg. This gives us the stretched manifold X associated with X : 2 In this connection the bottom SC of the cylinder that completes X n fvg to X will also be called Xsing and we then set Xreg WD X n Xsing which is the same as X n fvg. Thus there is a continuous map W X ! X (8.1.28)

which projects Xsing to v and restricts to the identity map Xreg ! X n fvg. Moreover, W WD X n fvg as a manifold with edge and boundary gives rise to the stretched manifold W associated with W :

8.1 Singular mixed problems and corner manifolds

483

The latter notion was explained in Section 2.4.3 before. The base of the model cones 1 of local wedges is the half-circle SC , and (8.1.27) gives us a splitting of variables 1 ^ .r; ; z/ 2 .SC /  R:

The variable r 2 RC will be referred to as the cone axis variable. By (8.1.25) we obtain a splitting of variables 2 .t; / 2 RC  SC with t 2 RC being the corner axis variable near the corner point v. Globally on W we have the weighted edge Sobolev spaces ˚ ˇ

s; s; Wcomp.loc/ .W / WD uˇint Wreg W u 2 Wcomp.loc/ .2W / :

(8.1.29)

Moreover, we interpret the sets Y˙;reg WD Y˙ n fvg as manifolds with edge Zreg D Z n fvg. Then we have the global weighted edge spaces s; s Wcomp.loc/ .Y˙;reg / that are far from Zreg modelled on Hcomp.loc/ .int Y˙ / and locally close to Zreg by finite (locally finite) sums of pull backs from RC  R of elements of W s .R; K s; .R˙ // (with bounded supports in r; z). Far from v the mixed problem (8.1.1) will be interpreted as a continuous operator 2 s; A W Wcomp .W / ! W s2; .W / comp

where s 1 ; 1 2

s2; 2 W s2;2 .W / WD Wcomp .W / ˚ Wcomp2 comp

s 3 ; 3 2

.Y;reg / ˚ Wcomp2

.YC;reg /: (8.1.30) A similar continuity together with the trace and potential operators along Zreg follows from (8.1.21). In order to fix notation we choose (8.1.24) (and then the associated map (8.1.25)) in the following manner. First there is a diffeomorphism x 3C Q W U0 ! R

(8.1.31)

in the sense of C 1 manifolds with boundary such that .v/ Q D 0, and Q 0 WD j Q U00 W U00 ! R2 x 3 . The restriction where U00 WD U0 \ @X is a diffeomorphism to R2 D @R C z1 [ L z2 Q 00 WD Q 0 jU000 W U000 ! L where U000 WD U00 \ Z is a map which restricts to a diffeomorphism Q 00 jU000 nfvg W U000 n z 2 / n f0g; the image consists of two smooth curves such that L z1 [ L z2 z1 [ L fvg ! .L is a manifold (of dimension 1) with conical singularity 0.

484

8 Corner operators and problems with singular interfaces

Choosing the diameter of U0 sufficiently small we find a diffeomorphism  W R2 ! R2

with .0/ D 0

z k / DW Lk is a half-line starting from the origin for k D 1; 2 and L1 \L2 D such that .L f0g and such that  ı Q 00 W U000 ! L1 [ L2 is an isomorphism between the manifolds U000 and L1 [ L2 with conical singularities. Then, if we replace (8.1.31) by x 3C  W U0 ! R

(8.1.32)

with  WD . ˚ idRx3 / ı Q we obtain another homeomorphism which is equivalent to Q in the above-mentioned sense, where 00 WD jU0 \Z W U000 ! L1 [ L2 coincides with  ı Q 00 . Instead of (8.1.32) we now write again (8.1.24), where the 2 conical points v1 ; v2 on S 1 D @SC are given by vk D S 1 \ Lk

for k D 1; 2:

(8.1.33)

8.2 Corner operators in spaces with double weights Our singular mixed problem will be interpreted as a boundary value problem on a manifold with corner and boundary. In (stretched) local coordinates the operators are corner degenerate in a specific way. We introduce corner Sobolev spaces with double weights and an algebra of corner boundary value problems. This will be a combination of the edge algebra of boundary value problems and the cone algebra in corner axis direction. In the calculus we admit the base of the corner to be an arbitrary manifold with conical singularities and boundary.

8.2.1 Transformation to a corner boundary value problem We now consider the mixed elliptic problem (8.1.1) in a neighbourhood of the conical point v 2 Z. Local stretched coordinates will be chosen as in Section 8.1.4. Writing the operators A and B˙ first in the form (8.1.6) and (8.1.8), respectively, polar coordinates x 3 n f0g with the axial variable t 2 RC give us in R C ADt

2

2 X j D0

aj .t /.t @ t /

j

and

B˙ D t

˙

˙ X

b˙;j .t /.t @ t /j

(8.2.1)

j D0

x C ; Diff 2j .S 2 // and b˙;j 2 C 1 .R x C ; Diff ˙ j .S 2 //, with coefficients aj 2 C 1 .R C C respectively. We now interpret the problem (8.1.1) as a corner problem, formulated in the Mellin operator calculus along the corner half-axis t 2 RC and the covariable w 2 C. In

8.2 Corner operators in spaces with double weights

485

other words, we form a .t; w/-depending family of mixed problems on the corner base 2 SC which is a manifold with boundary S 1 D f.x1 ; x2 / 2 R2 W x12 C x22 D 1g. The boundary is subdivided into intervals IC WD f# 2 S 1 W 0  #  ˛g;

I WD f# 2 S 1 W ˛  #  2g

for a certain 0 < ˛C < 2 (without loss of generality we take 0 and 2 as the respective end points). According to (8.2.1) we form h.t; w/ WD t

2 X

aj .t /w j ; r

j D0

 X

b;j .t /w j ; rC

j D0

C X

 bC;j .t /w j ;

(8.2.2)

j D0

where r˙ denote the operators of restriction to int I˙ . Then our mixed problem takes the form   ı1 Au D diag.t 2 ; t  ; t C / opM .h/u D f g gC : The weight ı 2 R is arbitrary for the moment; it will be specified below. Observe that (8.2.2) represents a family of mixed boundary value problems on the 2 half-sphere SC with respect to the subdivision of the boundary S 1 D I [ IC . Formally, we now proceed in a similar manner as in Section 8.1.2. Let v denote x 2 3 .x2 ; x3 / on one of the points v1 or v2 , cf. (8.1.33), and choose a chart U ! R C 2 x 2 and SC for a neighbourhood U 3 v such that v is transformed to the origin of R C 2 2 x n f0g we can identify U \ @SC to x3 D 0. If .r; / are polar coordinates in R C 2 U n fvg with f.r; / W r 2 RC , 0    g and .U n fvg/ \ @SC with R n f0g, where R D f.r; / W r 2 RC g  R n f0g corresponds to the Dirichlet side and RC D f.r; 0/ W r 2 RC g  R n f0g to the Neumann side, or conversely. In order to unify some descriptions for the operators A and B˙ we replace for the moment x C ; Diff j .S 2 //, the order 2 by any  2 N. Then, with coefficients aj .t / 2 C 1 .R C A takes the form  X  ADt aj .t /.t @ t /j (8.2.3) j D0 1 Writing the operators locally in U  in polar coordinates .r; / 2 RC  SC we can Pj Cj k c .r; t /.r@ / in (8.2.3) for small r 2 R and then insert aj .t/ D r r C kD0 j k obtain X A D t  r  aj k .r; t /.r@r /k .rt @ t /j (8.2.4) 2 SC

j Ck

x C R x C ; Diff .j Ck/ .S 1 //. A differential operator with coefficients aj k .r; t / 2 C .R C of the form (8.2.4) is called corner-degenerate. The operators B˙ which are involved in the boundary conditions can also be expressed in corner-degenerate form. x 3 in corner-degenerate form we have Example 8.2.1. For the Laplace operator in R 1

C

Dt

2 2

r

f.rt @ t / C .r@r /2 C r 2 t @ t C @2 g; 2

cf. also the choice of local coordinates in connection with (8.2.10) below.

486

8 Corner operators and problems with singular interfaces

8.2.2 Corner spaces with double weights In order to define corner spaces with double weights belonging to t ! 0 and r ! 0, respectively, we first introduce an analogue of the abstract edge spaces, cf. Definition 2.2.12. Definition 8.2.2. Let H be a Hilbert space with group action f g2RC . In addition, together with H we assume to be given an m 2 N, also referred to as a dimension number (in concrete cases associated with the dimension of a corner base). Then the space V s;ı .RC ; H / of smoothness s 2 R and weight ı 2 R is defined to be the completion of C01 .RC ; H / ˚

1 R 1 2 with respect to the norm 21 i  mC1 hwi2s khwi .M t!w u/.w/kH dw 2 . If we want 2

ı

to indicate the number m we also write V s;ı .RC ; H /.m/ . Observe that for m D 0 and H D C with the trivial group action we have V s;ı .RC ; C/.0/ D H s;ı .RC /: For different choices m and m0 we have the relation V s;ı .RC ; H /.m/ D V s;ıC

m0 m 2

.RC ; H /.m0 / :

(8.2.5)

This is a consequence of the property V s;ıCˇ .RC ; H / D t ˇ V s;ı .RC ; H / for every s; ı; ˇ 2 R. In our concrete situation for H we insert the spaces 1 ^ H D K s; ..SC / / with m D 2

(8.2.6)

and the group action .1/ W u.r; / ! u.r; /,  2 RC , and H D K s; .R˙ /

with m D 1

(8.2.7)

1

and .0/ W v.r/ !  2 v.r/,  2 RC . We then obtain the spaces 1 ^ 1 ^ / // DW V s;.;ı/ .RC  .SC / / V s;ı .RC ; K s; ..SC

and V s;ı .RC ; K s; .R˙ // DW V s;.;ı/ .RC  R˙ /; respectively.

(8.2.8)

487

8.2 Corner operators in spaces with double weights

Remark 8.2.3. The V s;ı -spaces contain elements both of the H s;ı - as well as of the W s -spaces, cf. Definitions 2.4.5 and 2.2.12, respectively. For instance, we have 1 ^ s 1 ^ / //  Wloc .RC ; K s; ..SC / //; V s;ı .RC ; K s; ..SC

cf. the notation (2.2.20). Moreover, we have 1 ^ / //  H s;ı .RC  M / '.r; /V s;ı .RC ; K s; .SC

for any 2-dimensional closed compact C 1 manifold M and for every '.r; / 2 1 / the support of which with respect to  is contained in a coordiC01 .RC  int SC nate neighbourhood on M . s More generally, we have V s;ı .RC ; H /  Wloc .RC ; H /. These compatibility properties allow us to form global weighted corner spaces. On IC we choose a partition of unity consisting of functions f!1 ; !2 g such that !1 C !2 D 1, !1 1 near 0, !2 1 near ˛, and set

V s;.;ı/ .RC  IC / WD f!1 g1 C !2 g2 W g1 2 V s;ı .RC ; K s; .RC //; g2 2 V s;ı .RC ; K s; .R //g (8.2.9) x C .R x  /. when we identify a neighbourhood of 0 .˛/ in IC with R 2 s;.;ı/ We now define corner spaces V .RC  SC / as follows. First recall that there 2 ^ 2 are the spaces H s;ı ..SC / / with SC as a ‘usual’ compact C 1 manifold with boundary, x C be the half-line through the origin in R x3 cf. the formula (2.1.14). Moreover, let R C 2 1 3 1 x and the point vi 2 S D @SC , and let SC  RC be the unit half-circle with center vi and orthogonal to RC (i.e., if Hi  R3 is the two-dimensional hyperplane orthogonal 1 x 3 with the unit D S1 \ R to RC and vi 2 Hi taken as the origin of Hi Š R2 , then SC C 1 circle S in Hi ). 1 ^ We then denote by V s;.;ı/ .RC  .SC / /i the space (8.2.8), where i indicates the 2 by neighbourhoods reference point vi , i D 1; 2. Now we choose a covering of SC 2 Ui , i D 1; 2; 3 on SC , where U3 is disjoint to fv1 g [ fv2 g and vi 2 Ui , i D 1; 2, U1 \ U2 D ;. In order to fix Ui we first consider the closed unit half-disk Bi with center vi and 1 1 form Di WD Bi n .int SC / with the above-mentioned half-circle SC centered at vi . We then set Ui D ı" Di , where ı" denotes the dilation to a disc of some radius " > 0 and 1 2 . This gives us an  the projection to SC orthogonal to the plane Hi that contains SC 2 open covering fU1 ; U2 ; U3 g of SC . Let f'1 ; '2 ; '3 g be a subordinate partition of unity. We then define 2 V s;.;ı/ .RC  SC / 3 nX o 2 ^ 1 ^ WD 'i ui W u3 2 H s;ı ..SC / /; ui 2 V s;.;ı/ .RC  .SC / /i ; i D 1; 2 : iD1

(8.2.10)

488

8 Corner operators and problems with singular interfaces

On the stretched manifold W associated with W D X n fvg as a manifold with s; edges we have the weighted edge spaces Wloc .W /. In the following definition we s; identify Wloc .W / with a corresponding space of distributions on X n Z. Moreover, Y˙ are manifolds with corner v and edge Zreg , locally near v modelled on .I˙ / . Let Y˙ denote the associated stretched manifolds, locally modelled on x C  I˙ . Here Y˙;reg WD Y˙ n fvg is a manifold with smooth boundary Zreg , regarded R as an edge; according to general notation the singular part of Y˙ is equal to I˙ . Definition 8.2.4. Let s; ; ı 2 R; then we define the spaces (i) ˚  ˇ s; V s;.;ı/ .X/ WD u 2 Wloc .W / W !u 2 ˇU



0 nfvg



2 V s;.;ı/ .RC  SC /

for any fixed ! 2 C01 .U0 / which is equal to 1 near the point v, cf. the formula (8.1.25); (ii) ˚  ˇ s; V s;.;ı/ .Y˙ / WD u 2 Wloc .Y˙;reg / W ! 0 u 2 ˙ ˇU



˙ nfvg



V s;.;ı/ .RC  I˙ /

for any fixed ! 0 2 C01 .U 0 /, U 0 WD U0 \ @X , which is equal to 1 near the point v; (iii) H s;ı .Z/ is the weighted Sobolev space on Z as a one-dimensional manifold with conical singularity v, cf. Definition 2.4.5. Let fU0 ; U1 ; : : : ; UK ; UKC1 ; : : : ; UL ; ULC1 ; : : : ; UN g

(8.2.11)

be an open covering of X by coordinate neighbourhoods, such that U0 is a neighbourhood of the corner point v, and v 62 Uj ; j > 0; Uj \ Z 6D ;; 1  j  K; Uj \ Z D ;, Uj \ @X 6D ;, K C 1  j  L; Uj \ @X D ;, L C 1  j  N . Fix a partition of unity f'j g0j N subordinate to (8.2.11) and a system f j g0j N of functions j 2 C01 .Uj / such that j 1 on supp 'j for all j . Set Uj0 WD Uj \ @X; 0  j  L; Uj00 WD Uj \ Z; 0  j  K; and 'j0 WD 'j jUj0 ;

0 j

WD

j jUj0 ; 0

 j  L;

'j00 WD 'j jUj00 ;

00 j

WD

j jUj00 ; 0

 j  K;

'j0 D j0 D 0; L C 1  j  N; 'j00 D j00 D 0; K C 1  j  N . x 3 ; 0  j  L, j W Uj ! R3 ; L C 1  j  Moreover, we fix charts j W Uj ! R C N in the following way. For L C 1  j  N we take diffeomorphisms as usual, while x 3 (as charts on a C 1 manifold with j for K C 1  j  L are diffeomorphisms to R C 0 0 2 boundary such that j WD j jUj0 W Uj ! Rx1 ;x2 ). Concerning 1  j  K we assume

8.2 Corner operators in spaces with double weights

489

that j is an isomorphism in the category of manifolds with edge and boundary (such that j0 WD j jUj0 W Uj0 ! R2x1 ;x2 , j00 WD j jUj00 W Uj00 ! Rx1 ). Finally, x 3C 0 W U0 ! R is assumed to be an isomorphism in the category of manifolds with corner and boundary. In order to avoid general definitions we generate that map as explained before in x 3 nf0g is an isomorphism Section 8.1.4. Then 0 .v/ D 0 and 0 jU0 nfvg W U0 nfvg ! R C in the category of manifolds with edge and boundary. Moreover, we then have ˇ 00 WD 0 ˇU 0 W U00 ! R2 ; 000 WD 0 jU000 W U000 ! L1 [ L2 ; 0

where 00 is an isomorphism in the category of manifolds with corner (‘without boundary’), and 000 is an isomorphism between the respective manifolds with conical singularities. Then 00 restricts to an isomorphism 00 jU00 nfvg W U00 n fvg ! R2 n f0g between manifolds with edges (‘without boundary’) and ˇ ^ 00;˙ WD 00 ˇ.U 0 nfvg/\Y .U00 n fvg/ \ Y˙ ! RC  I˙ D I˙ ˙

0

to diffeomorphisms between C 1 manifolds with boundary. Finally, the mappings ˇ 000 ˇU 00 nfvg W U000 n fvg ! .L1 [ L2 / n f0g; 0 ˇ (8.2.12) 000;k WD 000 ˇ.U 00 nfvg/\L W .U000 n fvg/ \ Zk ! Lk n f0g 0

k

ˇ for Zk WD .000 ˇU 00 nfvg /1 .Lk n f0g/, k D 1; 2, are diffeomorphisms. 0 The following theorem is of a similar structure as the assertion in Remark 2.4.31. ˇ Theorem 8.2.5. The restrictions r˙ W u ! uˇint Y induce continuous operators ˙

1

1

1

r˙ W V s;.;ı/ .X/ ! V s 2 ;. 2 ;ı 2 / .Y˙ / for all s; ; ı 2 R, s > 12 . P Proof. Every u 2 V s;.;ı/ .X/ can be written as u D jND0 'j u. Let us consider, for instance, rC (the minus-case is analogous). We then have rC u D rC .'0 u/ C PN PK j D1 rC .'j u/ C j DKC1 rC .'j u/. From the properties of the restriction operator between standard Sobolev spaces we see that the operators rC 'j for K C 1  j  L have the desired continuity property with a shift of smoothness s by 12 . Analogously, as is known of edge spaces, the operators rC 'j for 1  j  K are continuous between those spaces, with a shift of smoothness s and weight  by 12 . Recall that the latter

490

8 Corner operators and problems with singular interfaces

continuity comes from the fact that the restriction to the boundary between edge spaces is induced by 1

1

1 ^ / / ! K s 2 ; 2 .RC / b W K s; ..SC 1

1

(8.2.13)

1

1 ^ / /; K s 2 ; 2 .RC //. Thus the restricfor s > 12 , and we have b 2 Scl2 .R I K s; ..SC tion operator rC is equal to Op.b/, taken along R t (t is the local variable of Z).This gives us 1

1

1

1 ^ / // ! W s 2 .R; K s 2 ; 2 .RC //: rC Op.b/ W W s .R; K s; ..SC

(8.2.14)

The conclusion for j D 0 is analogous when we apply the formula (8.2.10) together with (8.2.8) and Definition 8.2.2. The change is that instead of (8.2.14) we 1 1 1 1 1 ^ take rC W V s;ı .RC ; K s; ..SC / // ! V s 2 ;ı 2 .RC ; K s 2 ; 2 .RC //. In this case the continuity of rC is a consequence of Theorem 8.2.6 below when we first interpret 1 1 1 1 ^ (8.2.13) as an element of S 2 . mC1 ı I K s; ..SC / /; K s 2 ; 2 .RC // (independent 2 of the covariable w) for m D 2, which yields ı m 2

rC opM

1

1

1

1 ^ .b/ W V s;ı .RC ; K s; ..SC / //.m/ ! V s 2 ;ı .RC ; K s 2 ; 2 .RC //.m/ 1

1

1

1

D V s 2 ;ı 2 .RC ; K s 2 ; 2 .RC //.m0 / for m0 D 1, cf. the formulas (8.2.5), (8.2.6), (8.2.7). Finally, in order to show the 1 1 1 2 / ! V s 2 ;. 2 ;ı 2 / .RC  RC /, cf. the formula continuity of V s;.;ı/ .RC  SC 2 ^ (8.2.10), we employ the continuity of the operator of restriction H s;ı ..SC / / ! 1 1 s 2 ;ı 2 1 ^ H ..S / /. The mixed boundary value problem (8.1.1) which is represented as the column matrix (8.1.2) can be decomposed in the form AD

N X

0 0 diag.'j ; ';j ; 'C;j /A

j

D

j D0

N X  t

'j A

j

0 ';j T

j

0 'C;j TC

 j

j D0

ˇ 0 where ';j WD 'j0 ˇY . As we know from the considerations of Chapter 5 the operator 

N X  t

'j A

j

0 ';j T

j

0 'C;j TC

 j

2 s; .W / ! W s2; .W / W Wcomp comp

j D1

is continuous, cf. the formulas (8.1.29), (8.1.30). The continuity close to the point v in weighted corner spaces will be obtained in the following section.

8.2 Corner operators in spaces with double weights

491

8.2.3 Continuity in corner spaces z / for Hilbert spaces H and H z Consider the space of symbols S  .RC  RC  ˇ I H; H  x x C  ˇ I H; H z / denotes the subspace with group action, ˇ 2 R. As usual, S .RC  R of symbols that are smooth in t; t 0 up to 0. Moreover, “  . 1 ıCi /   0 t 1 2 ı 0 1 0 dt opM .f /u.t / D  ı C i

u.t f t; t ; / μ ; 2 i t0 2 t0 z /, defines a continuous operator for a symbol f .t; t 0 ; w/ 2 S  .RC  RC   1 ı I H; H 2 z /. C 1 .RC ; H / ! C 1 .RC ; H 0

xC  R x C   mC1 I H; H z / with a ‘dimension Theorem 8.2.6. Let f .t; t 0 ; w/ 2 S  .R 2 ı z . Assume that f .t; t 0 ; w/ does not depend number’ m which is the same for H and H ı m 2

on t for jt j > c and t 0 for jt 0 j > c for some c > 0. Then opM continuous operator ı m 2

opM

.f / extends to a

z/ .f / W V s;ı .RC ; H / ! V s;ı .RC ; H

for every s 2 R. Proof. If the Mellin symbol f is independent of t; t 0 , then the asserted continuity is an easy consequence of Definition 8.2.2. The operator norm tends to zero when f tends to zero in the space of symbols. In the general case we can write f D f0 C f1 for a .t; t 0 /-independent symbol f0 and a symbol f1 with compact support in .t; t 0 /. In that case we can apply a tensor product argument combined with the observation that the x C / defines a continuous operator in operator of multiplication by some '.t / 2 C01 .R x C /, cf. also the technique V s;ı .RC ; E/ and its norm tends to zero for ' ! 0 in C01 .R in [37, Section 1.2.2]. 1 ^ z D As noted before, in our case we have, for instance, H D K s; ..SC / / and H 1 ^ ..SC / /, with the corresponding dimension number m D 2. K In order to construct concrete symbols which also play a role in parametrices of our corner boundary value problems we consider an element s;

xC  R x C ; B ;d .N I R2 // Q t; %; Q / Q 2 C 1 .R p.r; Q Q %;

(8.2.15)

for some compact C 1 manifold N with boundary, n D dim N . Later on we employ the parameters in the meaning %Q WD r%;

Q WD r Q

and

Q D t :

For convenience, we assume that pQ is independent of t for large t . Let Q t; r%; r /; p.r; t; %; / Q WD p.r; Q

and

Q t; w; r Q / h.r; t; w; Q / WD h.r;

(8.2.16)

492

8 Corner operators and problems with singular interfaces

Q t; w; / xC  R x C ; B ;d .N I C  RQ //, that we choose in such a way Q 2 C 1 .R for h.r; that Q D opˇM .h/.t; / Q mod C 1 .RC ; B 1;d .N ^ I R Q // opr .p/.t; / for every ˇ 2 R. Here opˇM .  / indicates the weighted Mellin action in r 2 RC (while ˇ opM .  / in this context denotes the Mellin action in the corner axis variable t 2 RC ). Moreover, setting Q t; w; r Q / h0 .r; t; w; Q / WD h.0;

Q t; r%; r /; p0 .r; t; %; / Q WD p.0; Q

Q D opˇM .h0 /.t; / Q mod C 1 .RC ; B 1;d .N ^ I R Q //. Choose we have opr .p0 /.t; / cut-off functions QQ !.r/; !.r/; Q !.r/ and  .r/; Q .r/; QQ According to the quantisations and assume that !Q 1 on supp !, ! 1 on supp !. of the edge calculus we form  n

Q opM 2 .h/.t; / Q !.rΠQ Q / a.t; Q / WD .r/r  f!.rΠ/ (8.2.17) QQ Q //gQ .r/ C aint .t; Q /; C .1  !.rΠQ // opr .p/.t; Q /.1  !.rΠwhere aint .t; Q / is of analogous form as the corresponding summand in (7.2.6). Moreover, we set  n

Q WD r  f!.rj j/ Q opM 2 .h0 /.t; Q /!.rj Q Q j/ ^ .a/.t; / QQ j//g: C .1  !.rj j// Q opr .p 0 /.t; Q /.1  !.rj Q 1

1

(8.2.18) 1

Let K s; .N ^ / WD K s; .N ^ / ˚ K s 2 ; 2 ..@N /^ / with  WD diag..n/ ;  2 .n1/ /. x C  RI K s; .N ^ /; K s; .N ^ //; Proposition 8.2.7. We have a.t; / Q 2 S  .R for every s;  2 R. Proposition 8.2.7 can be proved in an analogous manner as Theorem 7.2.3. Another important ingredient of the corner symbolic structure are the smoothing Mellin symbols and Green symbols. The smoothing Mellin symbols are of the form  n

m.t; / Q WD r  !.rΠ/ Q opM 2 .f /.t /!.rΠQ Q /

(8.2.19)

x C ; B 1;d .N I  nC1 /" / for some " > 0 and cut-off for an element f .t; w/ 2 C 1 .R 2  functions !; !. Q For E WD K s; .N ^ /; we have

Ez WD K s; .N ^ /

z ; x C  RI E; E/ m.t; / Q 2 Scl .R

(8.2.20)

8.2 Corner operators in spaces with double weights

493

for all s > d  12 . We set  n

Q WD r  !.rj j/ Q opM 2 .f /.t /!.rj Q Q j/: ^ .m/.t; /

(8.2.21)

Let us now pass to Green symbols, where we employ the spaces E WD K s; .N ^ / ˚ C j ; S" WD S" .N ^ / ˚

 1 2 S" ..@N /^ /

(8.2.22) ˚ C jC :

(8.2.23)

A Green symbol of order  2 R and type 0, and with the weight data g cone WD .;  / is defined as an operator family x C  R; L.E; S" //; g.t; / Q 2 C 1 .R

(8.2.24)

" D ".g/ > 0, s >  12 , such that x C  RI E; S" /;Q g.t; / Q 2 Scl .R nC1

nC1

where  WD  ˚  2 idC j , Q WD  ˚  2 idC jC ; a similar property is required for g .t; Q /, the pointwise formal adjoint. A family (8.2.24) is calledP a Green symbol of order  2 R and type d 2 N if it has the form g.t; Q / D g0 .t; / Q C jd D1 gj .t; / Q diag.D j ; 0; 0/ for a first order differential operator D on N which is defined in a neighbourhood of @N by the normal vector field to @N , and Green symbols gj .t; / Q of order  and type 0 for j D 0; : : : ; d. ;d x Let RG .RC  R; g cone ; j ; jC /" denote the space of all such Green symbols. 1;d x In particular, we have the space RG .RC  R; g cone ; j ; jC /" . For a g.t; Q / 2 ;d x RG .RC  R; g cone ; j ; jC /" we denote by ^ .g/.t; Q / the matrix of homogeneous principal symbols. x C  R; g cone ; j ; jC /" ,  2 Z, d 2 N, we denote the Definition 8.2.8. By R;d .R space of all families of operators a.t; / Q WD p.t; / Q C m.t; / Q C g.t; Q / W E ! Ez with E WD K s; .N ^ / ˚ C j as before, cf. (8.2.22), and Ez WD K s; .N ^ / ˚ C jC ; (8.2.25)    / 0  / 0 s > d  12 , such that p.t; / , m.t; Q / D m.t;Q , with the 2  2 upper Q D a.t;Q 0 0 0 0 left corners (8.2.17) and (8.2.19), respectively, and an element ;d x g.t; / Q 2 RG .RC  R; g cone I j ; jC /"

(with " > 0 being involved in the corresponding spaces).

494

8 Corner operators and problems with singular interfaces

;d x Let R.G/ .RC  R; g cone ; j ; jC / WD

S ">0

;d x R.G/ .RC  R; g cone ; j ; jC /" (sub-

;d script ‘.G/’ indicates Green or general operator families). Moreover, R.G/ .R; g cone ; j ; jC / denotes the corresponding elements which are independent of t . The operator families a.t; / Q have principal symbols  .a/ WD . .a/; @ .a/; ^ .a//, consisting of the interior symbol  .a/, the boundary symbol @ .a/, and the edge symbol ^ .a/. The interior symbol is determined by the upper left corner a.t; Q / of p.t; Q / and defined as ˇ Q Q ; Q /ˇ%Dr%; t; x; %; ; (8.2.26)  .a/.r; t; x; %; ; / D r   .p/.r; Q Q Drt

Q is the parameter-dependent homogeneous principal symbol of (8.2.15) where  .p/ Q , Q in local coordinates x on N with the covariables . of order  with the parameters %; The boundary symbol is also determined by p.t; Q /, namely, as ˇ Q Q ; Q 0 /ˇ%Dr%; t; x 0 ; %; (8.2.27) @ .a/.r; t; x 0 ; %; ; 0 / D r  @ .p/.r; Q Q Drt with the parameter-dependent homogeneous principal boundary symbol of (8.2.15) of Q , Q in local coordinates x 0 on @N with the covariables 0 . order , with the parameters %; We set     Q 0 ^ .m/.t; Q / 0 ^ .a/.t; / ^ .a/.t; Q / WD C C ^ .g/.t; Q /; (8.2.28) 0 0 0 0 x C  .R n f0g/, cf. (8.2.18), (8.2.21). .t; Q / 2 R x C  R Q ; g cone I j ; jC / be independent of t for Theorem 8.2.9. Let a.t; / Q 2 R;d .R large t , and form f.t; w/ WD a.t; / Q for Q D Im w; w 2  nC2 ı . Then 2

ı nC1 2

opM

z .nC1/ .f/ W V s;ı .RC ; E/.nC1/ ! V s;ı .RC ; E/

(8.2.29)

is continuous for every s 2 R, s > d  12 .

 z x C   nC2 I E; E/. Proof. Similarly as in Proposition 8.2.7 we have f.t; w/ 2 S  R 2  Then the continuity of the operator (8.2.29) is a consequence of Theorem 2.4.59 (ii). Remark 8.2.10. According to the identity (8.2.5) the space V s;ı .RC ; E/.nC1/ is defined by 1

1

1

V s;ı .RC ; K s; .N ^ //.nC1/ ˚ V s;ı 2 .RC ; K s 2 ; 2 ..@N /^ //.n/ ˚ V s;ı

nC1 2

.RC ; C j /.0/ :

Theorem 8.2.9 can be modified to a continuity between spaces which are obtained by , respectively. replacing s in the second and the third component by s  12 and s  nC1 2 This will be the smoothness convention in the spaces in our applications below, cf. also the formula (8.1.5) where n D 1.

495

8.2 Corner operators in spaces with double weights

8.2.4 Holomorphic corner symbols We now turn to a category of Mellin amplitude functions in corner axis direction t 2 RC that are holomorphic in the complex Mellin covariable w. Definition 8.2.11. By R;d .C; g cone I j ; jC / for  2 Z, d 2 N, we denote the space z with the spaces (8.2.22), (8.2.25), s > d  1 , such that of all h.w/ 2 A.C; L.E; E// 2 h.ı C i Q / 2 R;d .R Q ; g cone I j ; jC / for every ı 2 R, uniformly in compact intervals. ;d In a similar manner we define the space RG .C; g cone I j ; jC /. Remark 8.2.12. Let R1;d .C; g cone I j ; jC /" be defined as the set of all those c.w/ in A.C; L.E; S" // with the above-mentioned spaces (8.2.22) and (8.2.23) such that c.ı C i Q / 2 S.R Q ; L.E; S" // holds for every ı 2 R, uniformly in compact intervals, and a similar condition holds for c .w/. The space R1;d .C; g cone I j ; jC /" is Fréchet, and we set x C  C; g cone I j ; jC /" WD C 1 .R x C ; R1;d .C; g cone I j ; jC /" /; R1;d .R [ x C  C; g cone I j ; jC / WD x C  C; g cone I j ; jC /" : R1;d .R R1;d .R ">0

We have R1;d .C; T g cone I j ; jC /  R;d .C; g cone I j ; jC /, but the left-hand side does not coincide with 2Z R;d .C; g cone I j ; jC /, since g cone D .;   / is the same on both sides of the inclusion. Remark 8.2.13. Definition 8.2.11 can also be specified for a fixed " > 0 as in Definition 8.2.8 before; then the spaces in Definition 8.2.11 are the union of all x C  C; g cone I j ; jC / WD those spaces over " > 0. In addition we can form R;d .R x C ; R;d .C; g cone I j ; jC //. C 1 .R By definition the operator families a.t; / Q of Definition 8.2.8 belong to z ;Q x C  R Q I E; E/ S  .R

(8.2.30)

Q mentioned before. with the spaces (8.2.22), (8.2.25) and the group actions ; , Thus, if we form op t .a/ for such symbols, we obtain continuous operators s s z Q op t .a/ W Wcomp .RC ; E/ ! Wloc .RC ; E/

z They for every s 2 R, s > d  12 (recall that s is also involved in the spaces E and E). belong to the space Y;d .RC  N ; g cone I j ; jC / of edge boundary value problems of order  and type d; here RC  N is regarded as a manifold with edge RC and boundary RC .@N / (in Chapter 7 we mainly preferred writing the stretched manifold x C  N / rather than RC  N ). RC  .R In Chapter 7 we also introduced the space Y1;d .RC  N ; g cone I j I jC /

(8.2.31)

496

8 Corner operators and problems with singular interfaces

of smoothing operators. Symbols admit kernel cut-off operations that produce holomorphic families. Kernel cut-off in corner-degenerate situations in the case without boundary is studied in [118]. In the version of boundary value problems the details may be found in [144] and [31]. The process in general is as follows. Given a symbol a.t; / Q in the space (8.2.30) with Q being interpreted as the covariable on 0  C we form Z k.a/.t; b/ WD b i Q a.t; Q /μ : Q R

Then, for any

2

C01 .RC /

such that

h.t; w/ WD HC . /a.t; w/ WD

Z

.b/ D 1 in a neighbourhood of b D 1 we set 1

b w1 .b/k.a/.t; b/db:

(8.2.32)

0

z and x C ; L.E; E// We then obtain h.t; w/ 2 A.C; C 1 .R z ;Q x C  R Q I E; E/ h.t; ı C i / Q 2 S  .R for every ı 2 R, uniformly in compact intervals. x C  C; g cone I j ; jC / D Remark 8.2.14. For  2 Z, d 2 N, we have R;d .R xC  fh.t; w/ C c.t; w/ W h.t; w/ D HC . /a.t; w/ for arbitrary a.t; Q / 2 R;d .R 1;d x R Q ; g cone I j ; jC / and c.t; w/ 2 R .RC  C; g cone I j ; jC /g. x C  R Q ; g cone I j ; jC / and every Theorem 8.2.15. (i) For every a.t; Q / Q 2 R;d .R x C  C; g cone I j ; jC / such that for a.t; / WD ˇ 2 R there exists an h.t; w/ 2 R;d .R a.t; Q t / we have ˇ  nC1 2

op t .a/ D opM

.h/

modulo a smoothing operator defined in (8.2.31). (ii) We have Q  / D ^ .h/.t; ı C i / ^ .a/.t; for aQ and h as in (i), for every ı 2 R. The proof of Theorem 8.2.15 is based on the kernel cut-off technique combined with a Mellin operator convention as is developed in the present variant in [75], see also [31] and [144] for the case of boundary value problems. x C R Q ; g cone I j ; jC / is called elliptic Definition 8.2.16. An element a.t; Q / Q 2 R;d .R if  .a/ Q is elliptic, i.e.,  .a/ is non-vanishing on T  .RC  RC  N / n 0 and (in the Q Q ; Q / 6D 0 for .%; Q ; Q / 6D 0 up to r D t D 0, and, notation (8.2.26))  .p/. ; t; x; %; moreover, if @ .a/ is a family of isomorphisms in the sense @ .a/.r; t; x 0 ; %; ; 0 / W H s .RC / ˚ C j ! H s .RC / ˚ C jC ; parametrised by .r; t; x 0 ; %; ; 0 / 2 T  .RC  RC  @N / n 0, and (in the notation Q Q ; Q 0 / are isomorphisms for .%; Q ; Q 0 / 6D 0 up to r D t D 0, (8.2.27)) @ .p/.r; t; x 0 ; %; x C  .R n f0g/. and if ^ .a/.t; Q

Q / W E ! Ez is a family of isomorphisms for all .t; Q / 2 R

8.2 Corner operators in spaces with double weights

497

x C  R; g cone I j ; jC / be elliptic; then also Remark 8.2.17. Let a.t; Q / Q 2 R;d .R ;d x h.t; w/ 2 R .RC  C; g cone I j ; jC /, associated with aQ via Theorem 8.2.15, is x C  R Q ; g cone I j ; jC / for elliptic in the sense that h.t; ı C i / Q is elliptic in R;d .R some ı 2 R (this is then true for all ı 2 R).

8.2.5 Corner boundary value problems Let D be a manifold with conical singularity v and boundary; in particular, D n fvg is x C N /=.f0gN / then a C 1 manifold with boundary. If D is locally near v written as .R 1 for a compact C manifold N with smooth boundary, n D dim N , we can pass to the double 2D DW B which is a manifold with conical singularity v without boundary, x C  2N /=.f0g  2N /. We then have the stretched locally near v identified with .R manifold B with Breg WD B n @B, Bsing WD @B. In this case B can be written as the double 2D of the space D which is the stretched manifold of D with subsets Dreg WD D \ Breg ;

Dsing WD D \ Bsing ;

cf. the notation (2.4.19). In particular, Dreg is a C 1 manifold with boundary isomorphic to @.D n fvg/, and Dsing is diffeomorphic to N , the base of the local cone near v. Observe that a stretched manifold D belonging to a manifold D with conical singularities and boundary can also be doubled up in another way, namely by gluing together z with boundary, by identifying D;sing two copies D and DC WD D to a C 1 manifold D and DC;sing (this operation has nothing to do with the former double 2D). We then have the spaces V s;ı .RC ; K s; .N ^ //

and

z H s;ı .RC  D/

for every s; ; ı 2 R. Let us fix a cut-off function ! 2 C 1 .B/ which is equal to 1 in a collar neighbourhood of Bsing and write ! also for the restriction to D. The function z vanishing in a neighbourhood of D . 1  ! will also be interpreted as a function on D, Let us set z V s;.;ı/ .D ^ / WD f!v C .1  !/u W v 2 V s;ı .RC ; K s; .N ^ //; u 2 H s;ı .RC  D/g: Similarly we can form the spaces V s;.;ı/ ..@D/^ /, based on V s;ı .RC ; K s; ..@N /^ // z respectively. Here @D is the stretched manifold belonging to and H s;ı .RC  @D/, @.D n fvg/ [ fvg which is a manifold with conical singularity and without boundary. We then form 1

1

1

V s;.;ı/ .D ^ / WD V s;.;ı/ .D ^ / ˚ V s 2 ;. 2 ;ı 2 / ..@D/^ /:

(8.2.33)

.s;s 0 /;.;ı/

.D ^ / which is similar For purposes below we also introduce the notation V 0 to (8.2.33) but with s rather than s in the second component. This will be applied below to the case 2 D D SC

and @D D S1 ;

(8.2.34)

498

8 Corner operators and problems with singular interfaces

cf. Section 8.1.4. Let us stress once again that from the geometric point of view the 2 conical points v1 ; v2 2 @SC are only fictitious. That is why in the definition of the 2 1 notation of SC and S we did not really form the stretched manifolds although the relation (8.2.34) suggests so (we hope this will not cause confusion). In the following definition we need operators G that are continuous as 0

1

1

G W H s; .D/ ˚ H s ; 2 .@D/ ˚ C j ! H 1;C" .D/ ˚ H 1; 2 C" .@D/ ˚ C jC for some " D ".G/ > 0 and all s; s 0 2 R, s >  12 , such that the formal adjoint has analogous mapping properties. Let C1;d .D; g cone PId j ; jC /" for gjcone D .;   /, d 2 N, denote the space of all operators G0 C j D1 Gj diag.D ; 0; 0/ where Gj , 0  j  d, are as described before, and D is a first order differential operator on D which is near @Dreg [ @Dsing equal to @ , where  is the normal coordinate to the boundary @N . The space C1;d .D; g cone I j ; jC /" is Fréchet in a natural way. Definition 8.2.18. (i) By M1;d .D; g cone I j ; jC I ˇ /" we denote the set of all f .w/ 2 A.ˇ  " < Re w < ˇ C "; C1;d .D; g cone I j ; jC /" / such that f .ı C i / 2 S.R ; C1;d .D; g cone I j ; jC /" / for every ı 2 .ˇ  "; ˇ C "/, uniformly in compact subintervals. Let [ M1;d .D; g cone I j ; jC I ˇ /" : M1;d .D; g cone I j ; jC I ˇ / WD ">0

T 1;d .D; g cone I j ; jC / WD ">0 M1;d .D; g cone I j ; jC I ˇ /" Moreover, we set MO (which is, of course, independent of the choice of ˇ). (ii) The space ;d MO .D; g cone I j ; jC /;

 2 Z; d 2 N;

is defined to be the set of all operator functions h.w/ C f .w/ where h.w/ D sing h.w/Qsing C reg diag.hreg .w/; 0/Qreg

(8.2.35)

z C/, cf. Definifor arbitrary h.w/ 2 R;d .C; g cone I j ; jC /, hreg .w/ 2 B ;d .DI 1;d tion 6.1.28 and f .w/ 2 MO .D; g cone I j ; jC / with a partition of unity .sing ; reg / on D, where sing 0 outside a neighbourhood of Dsing and sing 1 near Dsing ; moreover, Qsing ; Qreg are C 1 functions on D, and Qsing also supported in a neighbourhood of Dsing , supp Qreg \ Dsing D ;, Qsing 1 on supp sing , Qreg 1 on supp reg . In the following by ˆ, ‰, etc., we denote diagonal matrices ˆ WD diag.'; '; '/; for arbitrary ';

‰ WD diag. ; ; /

x C /, acting as operators of multiplication. 2 C01 .R

(8.2.36)

499

8.2 Corner operators in spaces with double weights

By C1;0 .D ^ ; gI j ; jC / for g WD .;   I ı; ı  / we denote the space of all G that induce continuous operators 0

V 1;. C";ıC"/ .D ^ / V .s;s /;.;ı/ .D ^ / ˚ ˚ ˆG‰ W ! 1;ı nC1 C" s 00 ;ı nC1 j 2 2 .RC ; C jC / H .RC ; C / H for some " D ".G/ > 0 and every s; s 0 ; s 00 2 R, s >  12 , and for arbitrary ˆ; ‰ of the kind (8.2.36); an analogous condition is required for the formal adjoint G  , with the pair of weights  C , ı C  in the preimage and  C ", ı C " in the 1;d image. Moreover, .D ^ ; gI j ; jC / for d 2 N is defined to be the space of all Pd C G D G0 C j D1 Gj diag.D j ; 0; 0/ for arbitrary Gj 2 C1;0 .D ^ ; gI j ; jC / and a first order differential operator D that is locally near @N equal to @xn , where xn is the normal coordinate to the boundary. Definition 8.2.19. The space of local corner operators C;d .D ^ ; gI j ; jC /;

g D .;   I ı; ı  /;

 2 Z; d 2 N;

is defined as the set of all operators A D Lcorner C G with the following ingredients: (i) ı nC1 2

Lcorner D t  opM

.h C f /;

(8.2.37)

x C ; M;d .D; g cone I j ; jC //; h.t; w/ 2 C 1 .R O

f .w/ 2 M1;d .D; g cone I j ; jC I  nC2 ı /I 2

(ii) G 2 C1;d .D ^ ; gI j ; jC /. Remark 8.2.20. If we change the cut-off functions sing ; Qsing and reg ; Qreg in the formula (8.2.35) we obtain in (8.2.37) a remainder in C1;d .D ^ ; gI j ; jC /. x C ; M1;d .D; g cone I j ; jC I  nC2 // vanishes at t D 0, then If f .t; w/ 2 C 1 .R ı 2

ı nC1 2

t  opM

.f / is also smoothing in that sense.

x C / and A 2 C;d .D ^ ; gI j ; jC / we have Remark 8.2.21. For every '; 2 C01 .R ;d ^ ˆA; A‰ 2 C .D ; gI j ; jC /, cf. the formula (8.2.36). Theorem 8.2.22. For every A 2 C;d .D ^ ; gI j ; jC /, g D .;   I ı; ı  /, we have continuous operators V s;.;ı/ .D ^ / V s;. ;ı/ .D ^ / ˚ ˚ ˆA‰ W ! nC1 nC1 nC1 s nC1 ;ı s ;ı j  2 2 .RC ; C 2 2 .RC ; C jC / / H H for every s 2 R, s > d  12 , and arbitrary ˆ; ‰ as in (8.2.36).

500

8 Corner operators and problems with singular interfaces

Proof. The result is an immediate consequence of Theorem 8.2.9 and of the mapping properties of the smoothing operators in Definition 8.2.19 (ii). Remark 8.2.23. If the operator A in Theorem 8.2.22 satisfies a suitable condition for large t, e.g., that h.t; w/ is independent of t for large t and the smoothing summand vanishes, then A is continuous between the spaces without the factors ˆ and ‰. The operators A 2 C;d .D ^ ; gI j ; jC / have a principal symbolic hierarchy  .A/ D . .A/; @ .A/; ^ .A/; c .A// which is defined as follows. Writing A D .Aij /i;j D1;3;2

ˇ we have A11 ˇ

L cl .RC

(8.2.38) RC .int Dreg /

2

 .int Dreg // with the homogeneous principal symbol  .A/ WD  .A11 / of order . Locally near t D 0 and Dsing it has the form  .A/.r; t; x; %; ; / D t  r  Q .A/.r; t; x; r%; rt ; / Q ; Q / homogeneous of order  in .%; Q ; Q / 6D 0 and for a function Q .A/.r; t; x; %; smooth up to t D r D 0. ^ Moreover, we have .Aij /i;j D1;2 2 B ;d .Dreg /, and there is then a principal bound^ / n 0. Locally near ary symbol @ .A/ WD @ ..Aij /i;j D1;2 / parametrised by T  .@Dreg ^ t D 0 and @Dsing the boundary symbol is a family of maps @ .A/.r; t; x 0 ; %; ; 0 / D t  r  Q @ .A/.r; t; x 0 ; r%; rt ; 0 / Q ; Q 0 / which is  -homogeneous of order  in for a function Q @ .A/.r; t; x 0 ; %; 0 Q ; Q / 6D 0 and smooth up to r D t D 0. .%; From Definition 8.2.19 it follows that A 2 Y;d .D ^ ; g cone I j ; jC / with D ^ D RC  D being regarded as a (stretched) manifold with edge RC 3 t , cf. Section 7.2.5. From the edge calculus of boundary value problems we have a homogeneous principal edge symbol of order , namely, ^ .A/.t; / W K s; .N ^ / ˚ C j ! K s; .N ^ / ˚ C jC

(8.2.39)

for N D Dsing , .t; / 2 T  RC n 0. In the present case there is an analogous operator function Q ^ .A/.t; / Q in .t; /, Q Q 6D 0, smooth in t up to t D 0, such that ^ .A/.t; / D t  Q ^ .A/.t; t /:  ı C i Q / we have ^ .b/.t; Q / D Observe that when we form b.t;  / Q WD h.t; nC2 2 Q ^ .A/.t; Q /. Finally, in order to define the so-called corner conormal symbol we set H s;ı .D/ WD 1 1 H s;ı .D/ ˚ H s 2 ;ı 2 .@D/; then c .A/.w/ W H s; .D/ ˚ C j ! H s; .D/ ˚ C jC ;

(8.2.40)

w 2  nC2 ı , is defined by 2

c .A/.w/ WD h.0; w/ C f .w/; cf. the formula (8.2.37).

(8.2.41)

8.2 Corner operators in spaces with double weights

501

Theorem 8.2.24. Let A 2 C;d .D ^ ; aI j0 ; jC /, B 2 C;e .D ^ ; bI j ; j0 / and ˆ, ‰ be as in (8.2.36), where ;  2 Z, d; e 2 N, and a WD .  ;   . C /I ı  ; ı  . C //, b WD .;   I ı; ı  /. Then we have AˆB‰ 2 CC;h .D ^ ; a ı bI j ; jC / with h D max. C d; e/, a ı b D .;   . C /I ı; ı  . C //, and  .AˆB‰/ D  .Aˆ/ .B‰/ with componentwise multiplication, and the rule for the conormal symbols c .AˆB‰/.w/ D c .Aˆ/.w C /c .B‰/.w/: Theorem 8.2.24 states that the operator spaces of Definition 8.2.19 form (up to the localising factors ˆ and ‰) an algebra of boundary value problems on the stretched corner D ^ . An analogous calculus in the case of corners without boundary is given in [183]. The technique of proving Theorem 8.2.24 is quite similar to that in the edge algebra of boundary value problems, cf. Theorem 7.2.30, combined with ideas of the cone algebra, cf. Theorem 6.2.5.

8.2.6 Ellipticity near the corner Definition 8.2.25. An operator A as in Definition 8.2.19 is said to be elliptic with respect to the symbols (8.2.38) if Q ; Q / 6D 0 for (i)  .A/ 6D 0 on T  .RC  .int Dreg // n 0 and Q .A/.r; t; x; %; Q ; Q / 6D 0, up to r D t D 0; .%; ^ Q ; Q 0 / is (ii) @ .A/ is bijective for all points of T  .@Dreg / n 0, and Q @ .A/.r; t; x 0 ; %; 0 Q ; Q / 6D 0, up to r D t D 0; bijective for .%;

(iii) ^ .A/.t; / defines isomorphisms (8.2.39) for all .t; / 2 T  RC n 0, and Q ^ .A/.t; / Q defines analogous isomorphisms for Q 6D 0, up to t D 0; (iv) c .A/.w/ is a family of isomorphisms (8.2.40) for all w 2  nC2 ı . 2

Remark 8.2.26. The conditions (ii)–(iv) of the latter definition are required for any s 2 R, s > max.; d/  12 ; they are then independent of s. Theorem 8.2.27. An elliptic operator A 2 C;d .D ^ ; gI j ; jC / has a parametrix C P 2 C;.d/ .D ^ ; g 1 I jC ; j /, g 1 D .  ; I ı  ; ı/,  C WD max.; 0/, in the following sense: z z ˆ z D ˆI ˆPˆA

z z ˆ z D ˆI and ˆAˆP

502

8 Corner operators and problems with singular interfaces

modulo C1;dl .D ^ ; g l I j ; j / and C1;dr .D ^ ; g r I jC ; jC /, respectively; here g l D .; I ı; ı/, dl D max.; d/, g r D .  ;   I ı  ; ı  /, dr D .d  /C , and z QQ '; QQ '/ QQ with '; '; z D diag.'; z D diag.'; arbitrary ˆ D diag.'; '; '/, ˆ Q '; Q '/, Q ˆ Q 'QQ 2 1 C0 .RC / such that 'Q 1 on supp '; 'QQ 1 on supp '. Q Proof. By virtue of Definition 8.2.19 (i) it suffices to assume that A has the form (8.2.37). The ellipticity of Lcorner gives us the existence of elements 1 x ;dr z .DI  nC2 ı // h.1/ reg .t; w/ 2 C .RC ; B 2

and

x C  C; g 1 h.1/ .t; w/ 2 R;dr .R cone I jC ; j /

Q such that for hŒ1 .t; w/ WD sing h.1/ .t; w/Qsing Creg diag.h.1/ reg .t; w/; 0/reg we have ˇ hŒ1 .t; w  /h.t; w/ˇ

nC2 2 ı

D 1 C f0 .t; w/

(8.2.42)

x C ; M1;dl .D; g cone;l I j ; j I  nC2 //, g cone;l WD .;  /. for some f0 .t; w/ 2 C 1 .R 2 ı Note that, because of the holomorphy of the involved Mellin operator functions in w 2 C this relation holds for all ı 2 R, although we only employ it for the prescribed fixed corner weight ı. For the construction of the parametrix we also have to invert the principal conormal symbol (8.2.41). From (8.2.42) we obtain hŒ1 .0; w  /.h.0; w/ C f .w// D 1 C f1 .w/ for some f1 .w/ 2 M1;dl .D; g cone;l I j ; j I  nC2 ı /. We now use the fact that there 2

exists an f2 .w  / 2 M1;dl .D; g cone;l I j ; j I  nC2 ı / such that .1 C f2 .w  2 //.1 C f1 .w// D 1. This yields .1 C f2 .w  //hŒ1 .0; w  /.h.0; w/ C f .w// D 1: From f2 .w/hŒ1 .0; w/ DW g.w/ 2 M1;dr .D; g 1 cone I jC ; j I  nC2 ı / we 2

have .hŒ1 .0; w/Cg.w//.h.0; w/Cf .w// D 1. It follows that c .A/1 .w/ D hŒ1 .0; w  / C g.w  / for w 2  nC2 ı . We now employ the relation (8.2.42) 2 which entails x C ; M1;dl .D; g cone;l I j ; j //: hŒ1 .t; w  / # h.t; w/  1 DW h1 .t; w/ 2 C 1 .R O (8.2.43) Here # denotes the Mellin–Leibniz product between Mellin symbols. The relation (8.2.43) allows us to construct an h2 .t; w/ in the same space as h1 .t; w/ with the property x C ; M1;dl .D; g cone;l I j ; j //: .1 C h2 .t; w// # .1 C h1 .t; w// D 1 mod C 1 .R O

8.3 Corner edge operators

503

It follows modulo a remainder of the same kind that h.1/ .t; w  / # h.t; w/ D 1 where h.1/ .t; w  / WD .1 C h2 .t; w  // # hŒ1 .t; w  / belongs to the space C ı nC1 2  x C , M ;.d/ .D; g 1 C 1 .R .h.1/ C g/ is a cone I jC ; j //. Now P WD t opM O parametrix as desired.

8.3 Corner edge operators We specify the corner boundary value problems of Section 8.2 to the situation of the singular Zaremba problem in three dimensions. We could treat singular mixed problems in full generality, especially, without any restriction on the dimensions. But the advantage of the Zaremba problem is that we have an explicit example, including information on the extra interface conditions. In the first subsections we adopt the smoothness and weight shift conventions from Section 8.2 in the operators and spaces referring to the boundary. After that, for the case of the Zaremba problem, we take those shifts which correspond to the given orders in the boundary conditions.

8.3.1 Global corner boundary value problems We now return to the stretched configuration X in connection with the mixed problem on X with the interface Z. The most specific part comes from the (stretched) neigh2 ^ bourhood .SC / of the corner point v 2 Z. The necessary material on corner boundary value problems for a corner of the kind D ^ is prepared in Section 8.2.6. This will be ap2 2 ^ plied to D WD SC . From Definition 8.2.19 we have the space C;d ..SC / ; gI j; jC / K of corner boundary value problems, represented by block matrices A D A T Q that are (after a localisation by cut-off factors as in Theorem 8.2.22) continuous in the sense 2 ^ 2 ^ / / / / V s;. ;ı/ ..SC V s;.;ı/ ..SC ! AW ˚ ˚ H s1;ı1 .RC ; C j / H s1;ı1 .RC ; C jC /

where 1

1

1

1

1

1

2 ^ 2 ^ V s;.;ı/ ..SC / / WD V s;.;ı/ ..SC / / ˚ V s 2 ;. 2 ;ı 2 / .I^ / ^ ˚ V s 2 ;. 2 ;ı 2 / .IC /: 0

2 ^ In the following notation we employ spaces V .s;s /;.;ı/ ..SC / / of a similar kind s;.;ı/ 1 2 ^ ..SC / / with the only difference that s  2 is replaced by s 0 2 R. as V

504

8 Corner operators and problems with singular interfaces

By C1;0 .X; gI j ; jC / for g D .;   I ı; ı  / we denote the space of operators G that are continuous as 0

G W V .s;s /;.;ı/ .X/ ˚ H s

00 ;ı1

.Z; C j /

! V 1;.C";ıC"/ .X/ ˚ H 1;ı1C" .Z; C jC /

(8.3.1)

for some " D ".G/ > 0, for every s; s 0 ; s 00 2 R, s >  12 ; an analogous condition is required for the formal adjoint G  , with the pair of weights  C ; ı C  in the preimage and  C "; ı C " in the image. Moreover, C1;d .X; gI j ; jC / for d 2 N is defined to be the space of all operators G D G0 C

d X

Gj diag.D j ; 0; 0; 0/

(8.3.2)

j D1

for arbitrary Gj 2 C1;0 .X; gI j ; jC / and a first order differential operator D that is locally near @X of the form @xn with xn being the normal to the boundary. The following definition will refer to localisations of operators on X near different singular strata. Let us set (with the notation (8.2.11)) U0 WD U0 ;

U1 WD

K [

Uj ;

N [

U2 WD

j D1

Uj

j DKC1

which form an open covering fU0 ; U1 ; U2 g of X , and let f'0 ; '1 ; '2 g be a subordinate partition of unity. Moreover, let f 0 ; 1 ; 2 g be C01 functions in the respective neighbourhoods such that j 1 on supp 'j ; j D 1; 2; 3. The functions 'j ; j may also be regarded as functions on X when we identify them with the pull backs under the map (8.1.28). Moreover, we form 4  4 diagonal matrices ˆcorner ; ‰corner ;

and ˆedge ; ‰edge

defined by ˆcorner WD diag.'0 ; '0 jint Y ; '0 jint YC ; '0 jZ / and, similarly, ‰corner in terms of 0 , furthermore, ˆedge WD diag.'1 ; '1 jint Y ; '1 jint YC ; '1 jZ / and, similarly, ‰edge in terms of 1 . Finally, we form ˆreg WD diag.'2 ; '2 jint Y ; '2 jint YC ; 0/ and, similarly, ‰reg in terms of 2 . Definition 8.3.1. Let X be the (stretched) corner configuration as described in Section 8.1.4. Moreover, let  2 Z, d 2 N, and let g D .;   I ı; ı  / be weight data with a cone weight  2 R and a corner weight ı 2 R (associated with the local axial variables r 2 RC and t 2 RC , respectively). Then C;d .X; gI j ; jC /

(8.3.3)

is defined to be the set of all operators A D Acorner C Aedge C Areg C G with the following summands:

(8.3.4)

8.3 Corner edge operators

505

2 ^ (i) Acorner WD ˆcorner Lcorner ‰corner where Lcorner 2 C;d ..SC / ; gI j ; jC /, cf. Def2 inition 8.2.19 for the case D D SC ;

(ii) Aedge WD ˆedge Ledge ‰edge where Ledge 2 Y;d .U ; g cone I j ; jC /, cf. Definition 7.2.23 in the generalisation of the notation in connection with (7.2.43) for the trivial one-dimensional bundles E; E 0 ; F; F 0 , and J˙ D C j˙ . (iii) Areg WD ˆreg Lreg ‰reg for Lreg 2 B ;d .X n Z/; (iv) G 2 C1;d .X; gI j ; jC /. Let V s;.;ı/ .X/ be defined as 1

1

1

1

1

1

V s;.;ı/ .X/ ˚ V s 2 ;. 2 ;ı 2 / .Y / ˚ V s 2 ;. 2 ;ı 2 / .YC /; cf. Definition 8.2.4. Theorem 8.3.2. Every A 2 C;d .X; gI j ; jC / induces continuous operators A W V s;.;ı/ .X/ ˚ H s1;ı1 .Z; C j / ! V s;.;ı/ .X/ ˚ H s1;ı1 .Z; C jC /;

(8.3.5)

s > d  12 . Proof. In order to prove (8.3.5) it suffices to consider the summands in (8.3.4) separately. The desired continuity of Acorner was given in Theorem 8.2.22. The continuity of Aedge is the same as in the edge calculus, cf. Section 7.2.5. The operator Areg corresponds to a standard pseudo-differential boundary value problem with the transmission property at the boundary; thus it is continuous in standard Sobolev spaces, cf. Section 3.2.3. The smoothing term G is continuous by the properties (8.3.1) and (8.3.2). The second property is easy as well when we employ the holomorphy of the involved Mellin symbols in an "-strip around the weight line and for G once again (8.3.1), (8.3.2). We now turn to the principal symbolic structure of operators A 2 C;d .X; gI j ; jC /; namely,  .A/ D . .A/; @ .A/; ^ .A/; c .A//;

(8.3.6)

where the components are of analogous meaning as in (8.2.38). Writing A D .Aij /i;j D1;2;3;4 from Definition 8.3.1 it follows that A11 2 L cl .int X /. Thus we have the homogeneous principal symbol of order   .A/ WD  .A11 / as a function on T  .int X / n 0 in the standard sense.

506

8 Corner operators and problems with singular interfaces

1 Close to Zreg D Z n fvg in local coordinates .r; ; z/ 2 RC  SC  R with the covariables .%; #; / we have

 .A/.r; ; z; %; #; / D r  Q .A/.r; ; z; r%; #; r/

(8.3.7)

Q .%; Q 6D 0, which is smooth up to r D 0. Q #; /, Q #; / for a function Q .A/.r; ; z; %; Moreover, close to v near the branch Zk  Z, k D 1; 2, cf. the formula (8.2.12), 1  RC with the covariables .%; #; / we have in local coordinates .r; ; t / 2 RC  SC  .A/.r; ; t; %; #; / D t  r  Q .A/.r; ; t; r%; #; rt /

(8.3.8)

Q #; /, Q .%; Q #; / Q 6D 0, which is smooth up to r D t D 0. for a function Q .A/.r; ; t; %; The descriptions (8.3.7) and (8.3.8) are compatible close to v when we substitute  by t and replace the dependence on the variable z by t , including the weight factor in front of the function. For the second component of (8.3.6) which is the boundary symbol we employ that .Aij /i;j D1;2;3 belongs to B ;d .X n Z/. The boundary of X n Z consists of int Y˙ ; so there are two components, namely, @ .A/ WD .@; .A/; @;C .A//

(8.3.9)

with @; .A/ WD @ ..Aij /i;j D1;2 /;

@;C .A/ WD @;C ..Aij /i;j D1;3 /:

(8.3.10)

The latter boundary symbols are those of the boundary value problems .Aij /i;j D1;2 on int Y and .Aij /i;j D1;3 on int YC in the standard sense, i.e., operator families parametrised by T  .int Y / n 0 and homogeneous of order . 1 Close to Zreg in the variables .r; ; z/ 2 RC  SC  R we have @; .A/.r; z; %; / D r  Q @; .A/.r; z; r%; r/

(8.3.11)

Q .%; Q 6D 0, which are smooth up to r D 0. Q /, Q / for operator functions Q @; .A/.r; z; %; Moreover, close to v near the branch Zk  Z, k D 1; 2, in the variables .r; ; t / 2 1  RC we have RC  SC @; .A/.r; t; %; / D t  r  Q @; .A/.r; t; r%; rt /

(8.3.12)

Q /, Q .%; Q / Q 6D 0, which are smooth up to r D for operator functions Q @; .A/.r; t; %; t D 0. The third component of (8.3.6) is the edge symbol, where the operator A on X nfvg is 1 interpreted as an edge boundary value problem. In the variables .r; ; z/ 2 RC SC R we have 1 ^ 1 ^ ^ .A/.z; / W K s; ..SC / / ˚ C j ! K s; ..SC / / ˚ C jC ;

(8.3.13)

507

8.3 Corner edge operators 1

1

1 ^ 1 ^ / / WD K s; ..SC / / ˚ K s 2 ; 2 .R / ˚ .z; / 2 T  .Zreg / n 0, where K s; ..SC 1 1 K s 2 ;  2 .RC /. Moreover, close to v near the branch Zk  Z, k D 1; 2, in the 1 variables .r; ; t / 2 RC  SC  RC we have

^ .A/.t; / D t  Q ^ .A/.t; t / Q Q 6D 0, smooth up to t D 0. for an operator function Q ^ .A/.t; /, Finally, the principal conormal symbol of A is a family of maps 2 2 / ˚ C j ! H s; .SC / ˚ C jC c .A/.w/ W H s; .SC 1

1

1

1

2 2 with H s; .SC / WD H s; .SC / ˚ H s 2 ; 2 .I / ˚ H s 2 ; 2 .IC /, w 2  3 ı . 2

Remark 8.3.3. Let A 2 C .X; gI j ; jC / and  .A/ D 0. Then the operator (8.3.5) is compact for every s > d  12 . ;d

Theorem 8.3.4. For A 2 C;d .X; aI j0 ; jC /, B 2 C;e .X; bI j ; j0 / for ;  2 Z, d; e 2 N, a WD .  ;   . C /; ı  ; ı  . C //, b WD .;   I ı; ı  /, we have AB 2 CC;h .X; a ı bI j ; jC / for h D max. C d; e/, a ı b D .;   . C /I ı; ı  . C //, where  .AB/ D  .A/ .B/ with componentwise multiplication, and the rule c .AB/.w/ D c .A/.w C /c .B/.w/ for the conormal symbols. The proof of Theorem 8.3.4 is a combination of Theorem 8.2.24, specified to 2 , with the corresponding known composition behaviour in the edge algebra of D D SC boundary value problems outside fvg. Remark 8.3.5. Our applications will refer to a slightly modified definition of the operator space C;d .X; gI j ; jC /. In Definition 8.3.1 and the subsequent considerations we tacitly assumed the number of trace and potential conditions on Y˙ nZ to be one and their orders  C 12 and   12 , respectively. In our application to the Zaremba problem we have one trace (no potential) condition and orders 12 and 32 , respectively, and in parametrices one potential (no trace) condition. From now on we assume those numbers to be arbitrary, more precisely, lC trace and l potential conditions, for simplicity, both on Y and YC . Nevertheless, we use the same notation as in Definition 8.3.1, i.e., we assume that the numbers l˙ are from that context. There is then a straightforward generalisation of Theorem 8.3.4 to the case of such operators when the rows and columns in the middle fit together. For future reference we want to indicate the operator families of @ .A/, ^ .A/, c .A/ for arbitrary dimensions l˙ ; j˙ 2 N (including 0), namely, @;˙ .A/ W H s .RC / ˚ C l ! H s .RC / ˚ C lC ; 1

1

1

(8.3.14)

1

1 ^ ^ .A/ W K s; ..SC / / ˚ K s 2 ; 2 .R ; C l / ˚ K s 2 ; 2 .RC ; C l / ˚ C j 1

1

2 ^ ! K s; ..SC / / ˚ K s 2 ; 2 .R ; C lC /

˚K

1 s 1 2 ; 2

.RC ; C lC / ˚ C jC ;

(8.3.15)

508

8 Corner operators and problems with singular interfaces 1

1

1

1

2 c .A/ W H s; .SC / ˚ H s 2 ; 2 .I ; C l / ˚ H s 2 ; 2 .IC ; C l / ˚ C j 1

1

2 ! H s; .SC / ˚ H s 2 ; 2 .I ; C lC /

˚H

1 s 1 2 ; 2

(8.3.16)

.IC ; C lC / ˚ C jC :

8.3.2 Ellipticity and parametrices Recall that in this section we employ the definition of the class C;d .X; gI j ; jC / in the sense of Remark 8.3.5, i.e., with lC trace and l potential conditions both on Y and YC . Definition 8.3.6. An operator A 2 C;d .X; gI j ; jC / is called elliptic, if Q (i)  .A/ 6D 0 on T  .int X / n 0, and near Zreg the function Q .A/.r; ; z; %; Q #; / Q does not vanish for .%; Q #; / 6D 0, up to r D 0, and near v and Zk , k D Q #; / Q does not vanish for .%; Q #; / Q 6D 0 up to 1; 2, the function Q .A/.r; ; t; %; r D t D 0; (ii) @;˙ .A/ defines isomorphisms (8.3.14) for all points of T  .int Y˙ / n 0, and Q are isomorphisms for .%; Q 6D 0, up near Zreg the mappings Q @;˙ .A/.r; z; %; Q / Q / Q / Q are to r D 0, and near v and Zk , k D 1; 2, the mappings Q @;˙ .A/.r; t; %; Q / Q 6D 0, up to r D t D 0; isomorphisms for .%; (iii) ^ .A/ defines isomorphisms (8.3.15) for all .z; / 2 T  .Zreg / n 0, and near v the mappings Q ^ .A/.t; / Q are isomorphisms for Q 6D 0, up to t D 0; (iv) c .A/ defines isomorphisms (8.3.16) for all w 2  3 ı . 2

Remark 8.3.7. The conditions (ii), (iii), (iv) are required for all s > max.; d/  12 , and they are independent of the choice of s. Remark 8.3.8. The operator spaces C;d .X; gI j ; jC / contain subspaces with more regularity with respect to  and @ . Let C;d .X; gI j ; jC /reg denote the class of all A such that (i)reg A11 is regular in the sense that there is a ‘smooth’ element A11 2 B ;d .X / such that A11 jXnZ  A11 2 B 1;d .X n Z/; (ii)reg .Aij /i;j D1;2 is regular in the sense that there are ‘smooth’ elements A˙ 2 B ;d .X/ such that .A˙  .Aij /i;j D1;2 /jXnY 2 B 1;d .X n Y /. If A˙ are elliptic as elements in that smooth calculus of boundary value problems with respect to  and @ , then the conditions (i), (ii) of Definition 8.3.6 with respect to Q ; Q and Q @;˙ ; Q @;˙ are automatically satisfied.

509

8.3 Corner edge operators

Theorem 8.3.9. Let A 2 C;d .X; gI j ; jC /, g D .;  I ı; ı/, be elliptic. Then C there exists a parametrix P 2 C;.d/ .X; g 1 I jC ; j /, g 1 WD .  ; I ı  ; ı/, i.e., we have I  PA 2 C1;dl .X; g l I j ; j /;

I  AP 2 C1;dr .X; g r I jC ; jC /

(8.3.17)

with the same meaning of g l ; dl , etc., as in Theorem 8.2.27. Proof. We construct the parametrix in the form P D Pcorner C Pedge C Preg where the summands have a similar meaning as in Definition 8.3.1. The ellipticity of A implies that the operators Ledge D Y;d .U ; g cone I j ; jC / and Lreg 2 B ;d .X n Z/ are elliptic in the respective classes. Therefore, we have corresponding parametrices C

Medge 2 Y;.d/ .U ; g 1 cone I jC ; j /

and

C

Mreg 2 B ;.d/ .X n Z/;

respectively. Concerning Medge we refer to Section 7.2.6. The construction of Mreg is standard, cf. Section 3.3.2. We then set Pedge WD ‰edge Medge ˆedge and Preg WD ‰reg Mreg ˆreg . Thus the main step is the construction of a parametrix Mcorner of Lcorner , cf. the notation of Definition 8.2.19. However, this is done in Theorem 8.2.27 for the 2 case D D SC and with a slight modification; here, because of the two components Y˙ z z ˆ. z We of @.X n Z/, we have 4  4 matrices with a corresponding meaning of ˆ; ˆ; then set Pcorner WD ‰corner Mcorner ˆcorner . Corollary 8.3.10. Let A be as in Theorem 8.3.9. Then the associated operator (8.3.5) is Fredholm for every s > max.; d/  12 . In fact, the parametrix P of A has a principal symbolic hierarchy which is componentwise inverse to  .A/. Then, because of Theorem 8.3.4 the remainders in (8.3.17) have vanishing symbols and hence are compact operators, cf. Remark 8.3.3. This entails the Fredholm property of A.

8.3.3 The singular Zaremba problem Our next objective is to apply the results of the preceding section to the Zaremba problem for a second order elliptic differential operator A with Dirichlet conditions T on int Y and Neumann conditions TC on int YC that are elliptic with respect to A. Recall that the column matrix A D t .A T TC / induces continuous operators (8.1.3) between the corresponding weighted corner spaces. From the orders of the involved operators we see that, compared with Remark 8.3.5, the Definition 8.3.1 has to be generalised once again. For simplicity, we do not change the notation (8.3.3) but interpret the space C;d .X; gI j ; jC / as a space of corner boundary value problems A D .Aij /i;j D1;:::;4 , where .Aij /i;j D1;2;3 belongs to the space B ;d .X n Z/ in the sense that the upper left corner A11 is of order  but that we admit lC trace and l potential conditions on Y˙ , l˙ 2 N (including zero), and the components may have

510

8 Corner operators and problems with singular interfaces

arbitrary orders that are assumed to be known in connection with A. This should not cause confusion, since we consider very special operators around the Zaremba problem. In other words, in that sense our operator A belongs to C2;2 .X; g/. Remark 8.3.11. The operator A satisfies the ellipticity conditions (i), (ii) of Definition 8.3.6. In fact, the condition (i) is the ellipticity of A itself which remains untouched when we introduce polar coordinates and pass to the symbols Q and Q . The condition (ii) is the Shapiro–Lopatinskij ellipticity of the Dirichlet and Neumann problem on the respective sides of the boundary of X n Z, and the corresponding isomorphisms also remain satisfied when we pass to Q @; and Q @; .   Theorem 8.3.12. For every  2 12 k; 32 k , k 2 Z, we find dimensions j˙ D j˙ . /  K with jC  j D k, and elements T ; K and Q such that A WD A T Q belongs to the corner operator space C2;2 .X; gI j ; jC / for g D .;   2I ı; ı  2/ for every ı 2 R, with orders corresponding to the mapping properties (8.1.4), such that (i) A is elliptic with respect to ^ and Q ^ in the sense of Definition 8.3.6 (iii), here referring to isomorphisms 1 ^ 1 ^ K s; ..SC / / ˚ C j ! K s2; 2 ..SC / / ˚ C jC

for 1 ^ / / K s2;2 ..SC 1

1

3

3

1 ^ D K s2; 2 ..SC / / ˚ K s 2 ; 2 .R / ˚ K s 2 ; 2 .RC /I

(ii) there is a discrete set D of reals such that A is elliptic with respect to c in the sense of Definition 8.3.6 (iv), here referring to isomorphisms 2 2 / ˚ C j ! H s2; 2 .SC / ˚ C jC c .A/.w/ W H s; .SC 1

1

(8.3.18) 3

3

2 2 for H s2;2 .SC / D H s2;2 .SC / ˚ H s 2 ; 2 .I / ˚ H s 2 ; 2 .IC / and all w 2  3 ı and ı 2 R such that D \  3 ı D ;. 2

2

 Proof. Theorem 8.1.3 established the Fredholm property of (8.1.15) for all  2 12   k; 32  k , k 2 Z. This enabled us, similarly as in Section 5.3, to fill up ^ .A/.z; / to a family of isomorphisms (8.1.18), and to construct local amplitude functions t.z; /; k.z; /, and q.z; /. Close to v on every branch of Z emanated from v we can carry out such a construction by starting with the Fredholm family Q ^ .A/.t; Q / up to t D 0, and, analogously as (8.1.20), we obtain in this way an amplitude function x C  R Q ; g cone I j ; jC / for g cone D .;   2/ (the latter space is also a.t; Q Q / 2 R2;2 .R understood in the generalised sense, cf. Remark 8.3.5 and the comment at the beginning

511

8.3 Corner edge operators

of this section). By virtue of Theorem 8.2.15 from a.t; / WD a.t; Q t / we can pass to x C  C; g cone I j ; jC / such that op t .a/ D opı1 .h/, modulo a an h.t; w/ 2 R2;2 .R M smoothing operator. In the present case, in order to construct h, we may concentrate on the entries aQ ij for i; j 6D 1, because aQ 11 which comes from A has a holomorphic Mellin symbol h11 anyway which we can take. Now we have all ingredients to construct a corresponding operator A as desired. In particular, the latter step gives us Acorner , cf. Definition 8.3.1, while Aedge and Areg are known from Section 7.2.5. The ellipticity of A with respect to ^ and Q ^ is a consequence of the construction of the amplitude functions a.z; /, and a.t; Q / Q and the ‘ ’-homogeneous principal edge symbolic components are isomorphisms, the second one up to t D 0. Thus A has the asserted property (i). For (ii) we observe that (8.3.18) is a holomorphic family of operators which is on every weight line ˇ , ˇ 2 R, a parameter-dependent elliptic 2 family of boundary value problems on the manifold SC with conical singularities on the boundary; the notion ‘boundary value problem’is used here in the extended sense of additional entries of finite rank in the block matrices which correspond to the spaces C j˙ . Parameter-dependent ellipticity means that all symbolic components . ; @ ; ^ / are families of bijections, cf. Section 7.2.6. This has the consequence, that the operators (8.3.18) are bijective for large j Im wj. Moreover, they form a holomorphic family of Fredholm operators. This allows us to apply Theorem 6.1.43 which shows that there is a discrete set D  C such that (8.3.18) are isomorphisms for all w 2 C n D. In the present case the intersection D \ fc  Re w  c 0 g is finite for every c  c 0 , and we can set D WD fˇ 2 R W D \ ˇ 6D ;g. This completes the proof.   Theorem 8.3.13. For every  2 12  k; 32  k , k 2 Z, the operator A D t .A T TC / which represents the singular Zaremba problem A W V s;.;ı/ .X/ ! V s2;.2;ı2/ .X/ with 1

1

1

V s2;.2;ı2/ .X/ WD V s2;.2;ı2/ .X/ ˚ V s 2 ;. 2 ;ı 2 / .Y / 3

3

3

˚ V s 2 ;. 2 ;ı 2 / .YC / can be completed by additional interface conditions T ; K and Q to an element A 2 C2;2 .X; gI j ; jC /;

g D .;   2I ı; ı  2/;

where jC  j D k, such that 

A K AD T Q



V s2;.2;ı2/ .X/ V s;.;ı/ .X/ ˚ ˚ ! W s 3 ;ı 3 2 2 .Z; C jC / H s1;ı1 .Z; C j / H

(8.3.19)

is a Fredholm operator for all ı 2 R n D for a certain discrete set D, for all s > 32 . Moreover, A has a parametrix P in the corner calculus of boundary value problems on X in the sense of Theorem 8.3.9 (with the corresponding modification of orders).

512

8 Corner operators and problems with singular interfaces

Proof. For A we take the operator of Theorem 8.3.12. By virtue of Remark 8.3.11 together with Theorem 8.3.12 the operator A is elliptic in the sense of Definition 8.3.6. Thus, according to Theorem 8.3.9 there is a parametrix P in the corner calculus. Then Corollary 8.3.10 gives us the Fredholm property of A. Corollary 8.3.14. Let A be as in Theorem 8.3.13, let Au D f with f belonging to the space on the right of (8.3.19), s > 32 , and u 2 V 1;.;ı/ .X// ˚ H 1;ı1 .Z; C j /. Then we have u 2 V s;.;ı/ .X/ ˚ H s1;ı1 .Z; C j /. In fact, it suffices to compose the equation Au D f from the left with a parametrix P which gives us u D .I  PA/u C Pf . We then obtain the desired property of u when we apply Theorem 8.3.2 to P and the mapping properties of the smoothing remainder I  PA. Remark 8.3.15. The method of treating mixed elliptic problems in the sense of Theorem 8.3.13 and Corollary 8.3.14 can be applied to other mixed conditions rather than Dirichlet/Neumann conditions, and to other elliptic operators (also systems) instead of the Laplacian. The problem in concrete cases is to determine the number of extra trace and potential conditions j˙ and the admitted corner weights ı 2 R. This is a separate task for every individual operator. In Section 5.3.4 we obtained the dimensions j˙ for several other mixed problems for the Laplace operator. Also the dimension of the underlying configuration for mixed problems and the smoothness of the domain G (cf. the notation at the beginning of Section 8.1.1) are not too essential for the methods in general. For instance, Z may be a manifold of arbitrary dimension, with conical singularities. The existence of extra edge-elliptic interface conditions is guaranteed when a topological criterion of Atiyah–Bott type is satisfied (this is often automatically the case).

8.3.4 Remarks In Section 8.1.4 we considered corner manifolds with boundary. We now return to this discussion and complete the terminology (see also Sections 10.3.1 and 10.4.1 below). A manifold X with corners and boundary is a topological space with a finite subset of corner points X 000  X such that W WD X n X 000 is a manifold with edge and boundary, cf. Definition 2.4.26 and Remark 2.4.30, and every v 2 X 000 has a neighbourhood U0  X such that there is a homeomorphism  W U0 ! V

(8.3.20)

for a manifold V with edge and boundary which restricts to an isomorphism 0 W U0 n fvg ! RC  V in the category of manifolds with edge and boundary, cf. the terminology at the end of Section 2.4.2.

8.3 Corner edge operators

513

More material on manifolds with higher corners may be found in [18]. In our application, cf. Section 8.4.1, we consider the simpler case of a manifold V with conical singularities and boundary. Let Q W U0 ! V be another homeomorphism of the kind (8.3.20) such that the restriction Q 0 W U0 n fvg ! RC  V is an isomorphism in the category of manifolds with edge and boundary. Then  and Q are said to be equivalent, if the transition map Q 0 ı 1 0 W RC  V ! RC  V is the restriction of an isomorphism R  V ! R  V in the category of manifolds with edge and boundary to RC  V . The neighbourhood U0 itself is also a manifold with corners and boundary. Remark 8.3.16. If  and Q are equivalent, the homeomorphism

WD 1 ı Q W U0 ! U0

(8.3.21)

is a special (typical) case of an isomorphism in the category of manifolds with corners and boundary (we do not give the definition in full generality; this needs more notation). The boundary of X may be empty; in that case we speak about a manifold with corners and without boundary. x 3 be the upper half-plane fx D .x1 ; x2 ; x3 / 2 R3 W Example 8.3.17. Let U0 WD R C 2 x 3 , fix points v1 ; v2 on S 1 D @S 2 . Then S 2 WD S 2 \ R x3  0g, consider SC C C C can be regarded as a manifold with conical singularities v1 ; v2 on the boundary, and 2 2 we then write SC for SC in this interpretation. Via polar coordinates x ! .t; /, 2 U0 n f0g ! RC  SC we have a natural homeomorphism 2

 W U0 ! .SC / ;

cf. the formula (8.1.24). Let W U0 ! U0 be a diffeomorphism in the category of manifolds with C 1 boundary. Then can also be interpreted as an isomorphism in the category of manifolds with corners and boundary which is of the form (8.3.21) when we set Q WD  ı . If X is a manifold with corners X 000 and boundary, then the boundary Y n X 000 of X nX 000 is a manifold with edge and without boundary, and we call Y WD .Y nX 000 /[X 000 the boundary of X which is a manifold with corners and without boundary. Let us consider once again the special case U0 where V is a manifold with conical singularities and boundary. Then V 0  V , the boundary of V including its conical singularities gives rise to a corresponding subset U00 WD 1 ..V 0 / /  U0 . Moreover, if V 00 denotes the conical singularities of V 0 we obtain a subset U000 WD 1 ..V 00 / / which is a manifold with conical singularities. Remark 8.3.18. The isomorphism (8.3.21) induces (by restriction) isomorphisms

0 W U00 ! U00 ;

00 W U000 ! U000 ;

where 0 refers to manifolds with corners and without boundary and 00 to manifolds with conical singularities.

514

8 Corner operators and problems with singular interfaces

Remark 8.3.19. The notation ‘corner manifolds’ in Section 8.1.4 is employed because of the discontinuities of the coefficients in the boundary conditions of the singular mixed problems. The local transformations of the operators and boundary conditions into corner degenerate form, cf. (8.2.4), lead to a more general operator class than necessary for the original problem, namely, that of Definition 8.2.19 or Definition 8.3.1. x as well as A is smooth and also The situation in the original problem is that X D G B˙ are smooth in a neighbourhood of Y˙ . In particular, U0 in the formula (8.1.24) can also be interpreted as a neighbourhood of v 2 @X on the manifold X with smooth boundary. However, (8.1.24) is assumed to be an isomorphism of U0 as a manifold with corners and boundary, cf. Remark 8.3.16. Thus the results of the Sections 8.3.2 and 8.3.3 admit more general assumptions on the singular mixed problem (8.1.1), namely, x is a manifold with corner point v, especially, from the very beginning, that X D G that X n fvg is a manifold with edge (both with boundary) and, in addition, that the operators are even corner degenerate in the corresponding stretched coordinates.

8.4 Cracks with singularities at the boundary A crack in a medium is modelled on an oriented hypersurface Y of codimension 1 in a (bounded, smooth) domain G. We assume that Y is a manifold with conical singularities at the boundary Z WD @Y , i.e., Y n Z is C 1 , and Z is a manifold with conical singularities. By a crack problem we understand an elliptic boundary value problem Au D f in G n Y ,

T˙ u D g˙ on int Y˙ ,

T u D g on @G;

where A is an elliptic operator, and T˙ , T represent elliptic boundary conditions on int Y˙ and @G, respectively; here int Y˙ denote the ˙ sides of int Y . If Z is smooth, we have an edge x C , where boundary value problem, with edge Z. In this case the model cone is equal to R2 n R 2 x R is the two-dimensional normal plane to Z and RC its intersection with Y (in suitable local coordinates), cf. [90]. If Z has conical singularities, we can apply a variant of the corner calculus of boundary value problems. The corner axis t 2 RC measures the distance to a conical singularity v, and the base of the corner is equal to S 2 n I with S 2 being centered at v, and I  S 2 is an interval which corresponds to the intersection of S 2 with the crack.

8.4.1 Crack problems as edge-corner boundary value problems Let X be a compact C 1 manifold with boundary and Y  int X a compact hypersurface of codimension 1 with boundary Z that has conical singularities. In ‘reality’ X is the closure of a smooth bounded domain G in Euclidean space, e.g., G WD fx 2 R3 W jxj < 1g and Y WD fx 2 R3 : jx1 j C jx2 j  12 , x3 D 0g. In general, we assume int Y to be an oriented C 1 manifold of dimension dim X  1; so we have ˙-sides int Y˙ . A crack problem is represented by a boundary value problem Au D f in .int X / n Y ,

T˙ u D g˙ on int Y˙ , T u D g on @X ,

(8.4.1)

8.4 Cracks with singularities at the boundary

515

for an elliptic differential operator A (for instance, Lamé’s system in a three-dimensional medium G D int X) and operators T˙ and T of the form T˙ u WD r˙ B˙ u;

T u WD r@X Bu;

(8.4.2)

for differential operators T and T˙ given in open neighbourhoods of Y˙ D int Y ˙ and @X , respectively, with smooth coefficients, and the operators of restriction r˙ and r@X to int Y˙ and @X , respectively. Our approach is valid in full generality, i.e., for elliptic systems A and arbitrary boundary operators T˙ that satisfy the Shapiro–Lopatinskij condition on int Y˙ (uniformly up to Z from the respective sides) and arbitrary Shapiro–Lopatinskij elliptic T at @X . Nevertheless, in order to keep the exposition transparent, we illustrate the situation in the case dim X D 3 and a second order differential operator A and Dirichlet conditions on int Y and @X , Neumann conditions on int YC . In that case (8.4.1) represents a continuous operator H s .int X / ˚ 0 1 A s 1 2 H .int Y / B T C C W H s .int X / ! ˚ A WD B @TC A 3 H s 2 .int YC / T ˚ 1 H s 2 .@X /

(8.4.3)

for every s > 32 . Here H s .int X / is the standard Sobolev space on int X of smoothness s 2 R (more precisely, H s .int X / WD H s .2X /jint X , where 2X denotes the double s z of X), and H s .int Y / WD Hloc .Y /jint Y for any C 1 manifold Yz  int X of the same dimension as Y which contains Y as a compact subset. Similarly as for mixed problems, the spaces in (8.4.3) are too restrictive for an adequate discussion of solvability. Therefore, we realise the operators on weighted corner spaces. In particular, instead of (8.4.3) we consider A as a continuous operator 1

A W V s;.;ı/ .X/ ! V s2;.2;ı2/ .X/ ˚ H s 2 .@X /

(8.4.4)

for s > 32 , V s2;. 2;ı2/ .X/ 1

1

1

3

3

3

WD V s2;.2;ı2/ .X/ ˚ V s 2 ;. 2 ;ı 2 / .Y / ˚ V s 2 ;. 2 ;ı 2 / .YC /; (8.4.5) cf. the notation in Section 8.4.4 below. The program of the consideration is to associate Fredholm operators with A. This will restrict the choice of weights  , i.e., we will admit  2 R with the exception of some discrete set C . Then for every real ı 2 R n D

516

8 Corner operators and problems with singular interfaces

for some other discrete set D (which depends on  ) we complete the operator (8.4.4) to a block matrix operator V s2;.2;ı2/ .X/   ˚ V .X/ A K s 1 ˚ AD W ! 2 .@X / H T Q H s1;ı1 .Z; C j / ˚ H s3;ı3 .Z; C jC / s;.;ı/

(8.4.6)

which is Fredholm and belongs to a corner algebra of boundary value problems with additional edge conditions on @S and in such a way that A has a parametrix in that calculus. The operators in this so-called crack algebra are determined by a principal symbolic hierarchy  .A/ D . .A/; @ .A/; ^ .A/; c .A// (8.4.7) modulo lower order terms that are compact operators in weighted corner spaces. Having in mind the calculus of mixed problems with conical singularities at the interface, the main effort for crack problems with conical singularities at the crack boundary is to translate the approach to the new situation. For simplicity, we assume that Z has only one conical point v. Moreover, the main aspect is to study crack problems locally near the crack Y ; the boundary @X with its elliptic conditions can be treated separately, and the behaviour of operators close to @X is essentially that of Chapter 3, i.e., of boundary value problems with smooth boundary and the transmission property. An exception are the global smoothing operators that may contain an interaction of influences near Y and @X ; we will come back to that point at the end of the consideration, cf. Section 8.4.7. Up to that modification, for the main part of the exposition we assume that X is a closed compact C 1 manifold; Y  int X is a crack with Z D @Y ; and the boundary Z of the crack has a conical singularity v:

(8.4.8)

The other assumptions, i.e., Y compact, int Y oriented, also remain in force. In addition, for the corner part of the theory, we assume dim X D 3 (although, as we mentioned before, this is not really essential). From the point of view of local descriptions we consider suitable open coordinate neighbourhoods U  X with charts  W U ! R3 :

(8.4.9)

We distinguish the cases U U U U

\ Y D ;; \ Y 6D ;; U \ Z D ;; \ Z 6D ;; U \ fvg D ;; \ fvg 6D ;:

(8.4.10) (8.4.11) (8.4.12) (8.4.13)

8.4 Cracks with singularities at the boundary

517

The case (8.4.10) concerns the interior part of the configuration, and (8.4.9) may be taken as any diffeomorphism. For (8.4.11) we choose (8.4.9) as a diffeomorphism such that 0 WD jU \Y W U \ Y ! R3 is a chart on int Y with R2 D fx D .x1 ; x2 ; x3 / 2 R3 W x3 D 0g, such that  restricts to charts x3 ˙ W U˙ ! R ˙ x 3 /, R x 3 WD fx 2 R3 W x3 ? 0g. for U˙ WD 1 .R ˙ ˙ In the case (8.4.12) we take the neighbourhood U in such a way that U can be interpreted as a manifold with edge U \ Z and boundary U \ Y˙ ; then (8.4.9) is assumed to be an isomorphism in the category of manifolds with edge and boundary such that x 2C 0 WD jU \Y W U \ Y ! R x 2 WD fx 2 R3 W x3 D 0; x2 = 0g in the category of C 1 is a diffeomorphism to R C manifolds with boundary; then 00 WD jU \Z W U \ Z ! R D fx 2 R3 W x2 D x3 D 0g is a chart on Z n fvg. In other words, U is locally identified with R3 such that x 2 and the crack boundary to R. Since we the crack corresponds to the half-plane R C distinguish between the ˙ sides of the crack, locally described by x3 & 0 and x3 % 0, x 2 that are identified over R D @R x2 . we imagine in R3 two copies of half-planes R C C 2 2 Introducing polar coordinates in R n f0g for the .x2 ; x3 / plane R (normal to the crack boundary R), .x2 ; x3 / ! .r; / 2 RC  Œ0; 2 (where we distinguish the end points of the interval Œ0; 2 ), we therefore identify the (local) crack configuration R3 with x C  Œ0; 2 /=.f0g  Œ0; 2 /g, or, in stretched form, with R  f.R R  RC  Œ0; 2 3 .x1 ; r; /;

(8.4.14)

with  D 0 for the plus side,  D 2 for the minus side of the crack. In the case (8.4.13) we interpret U as a manifold with corner v and boundary and (8.4.9) as an isomorphism in the corresponding category, such that the local model R3 is identified with a cone x C /=.S 2  f0g/; .S 2  R or, in stretched form, with S 2  RC 3 .; t /; in such a way that

(8.4.15)

x 2C;C 0 WD jU \Y W U \ Y ! R

x 2 D fx 2 R3 W x3 D 0; x1  0; x2  0g is an isomorphism in the category for R C;C of manifolds with conical singularities and boundary. As a consequence x C;x [ R x C;x 00 WD jU \Z W U \ Z ! R 1 2

518

8 Corner operators and problems with singular interfaces

is an isomorphism between the (one-dimensional) manifolds with conical singularities v and 0, respectively, where 000 WD jU \fvg W v ! 0: This leads to a more precise description of the local stretched model (8.4.15) of the singular crack configuration, namely, as S2  RC 3 .; t /;

(8.4.16)

 W U n fvg ! S2  RC ;

(8.4.17)

and a corresponding chart

x 2 . Moreover, where S2 means the sphere with a one-dimensional crack I WD S 2 \ R C;C we have the local stretched model of Y n fvg as I  RC 3 .; t /:

(8.4.18)

S2 will be treated as a manifold with boundary I WD I [ IC =  and conical singularities on the boundary; ‘’ means the identification of the end points vj on I with those on IC for j D 1; 2. Let us set H1 WD fx 2 R3 W x1 D 1g;

H2 WD fx 2 R3 W x2 D 1g;

and let j W R3 ! Hj denote the orthogonal projection to the plane Hj , j D 1; 2. Then there is a neighbourhood V1 on S 2 of the point v1 such that 

1 W V1 ! x 2 H1 W

x22

C

x32

1 < 3



is a diffeomorphism, where V1 \ I is mapped to a set J  H1 which consists of two copies of fx 2 R3 W x1 D 1; 0  x2 < 12 ; x3 D 0g identified at x2 D 0. It can easily be verified that there is another neighbourhood U1  V1 on S 2 of v1 and a diffeomorphism 1 W U1 ! H1 such that 1 jV1 D 1 , where U1 \ I is mapped to two copies of fx 2 R3 W x1 D 1, 0  x2 < 1; x3 D 0g, identified at x2 D 0. In a similar manner we can proceed with neighbourhoods of V2  U2 of v2 2 S 2 . Let U1 and U2 denote the sets U1 and U2 , respectively, in which we keep in mind the slits Uj \ I, j D 1; 2. We now identify the plane Hj with R2 of the variables y D .y1 ; y2 / in which the xC [ R x C =  for R x C D fy 2 R2 W slit (the x2 or x1 half-axis) is identified with R

8.4 Cracks with singularities at the boundary

519

y1  0; y2 D 0g. Composing j W Uj ! Hj with Hj Š R2 and introducing polar coordinates R2 n f0g ! RC  S 1 3 .r; /; (8.4.19) where we distinguish between  D 0 and  D 2, we obtain bijections Uj n fvj g ! RC  Œ0; 2 :

(8.4.20)

Summing up, together with (8.4.15) on S2 RC 3 .; t / near t D 0 (which corresponds to z 2 Z) we obtain two (stretched) charts .Uj n fvj g/  RC ! RC  Œ0; 2  RC 3 .r; ; t /;

(8.4.21)

j D 1; 2. Let X  R3 be a closed compact C 1 manifold and Y  X a compact hypersurface of codimension 1 satisfying the assumptions made at the beginning. In particular, Z D @Y is a (one-dimensional) manifold with conical singularity v. We then set W WD X n fvg; (8.4.22) regarded as a manifold with edge Zreg D Z n fvg and boundary which consists of two components Y˙;reg WD Y˙ n fvg, identified along Zreg . By W we denote the associated stretched manifold with (stretched) local coordinates (8.4.14) near Zreg . Moreover, by X we denote the stretched corner manifold associated with X and the subsurface Y which is obtained from X n fvg by invariantly attaching S2  f0g to the local cylinder (8.4.16) that describes X n fvg locally near v. Similarly, by Y we denote the stretched corner manifold associated with Y which is obtained from Y n fvg by invariantly attaching I  f0g to the local cylinder (8.4.18). From the two copies Y˙ of Y we then have the stretched manifolds Y˙ . For reference below we also introduce the double 2S2 of S2 (belonging to the double of the original configuration) which is a manifold with conical singularities v1 ; v2 and without boundary, cf. also the general notation in connection with the formula (2.4.19). z 2 , cf. the formula (2.4.21), which is a Moreover, we have another double, namely, S 1 compact C manifold with boundary.

8.4.2 Operators near the smooth part of the crack boundary The crack problem Au D f on X n Y ,

T˙ u D g˙ on int Y˙

will be analysed first in local coordinates far from z (recall that X is closed and compact; the case with non-trivial boundary @X , cf. (8.4.1), only causes small modifications of the consideration, cf. Section 8.4.7 below). Let us establish the principal symbolic structure .A/ D . .A/; @ .A/; ^ .A/) of   A D t A T TC

520

8 Corner operators and problems with singular interfaces

in X n fvg. First we have the standard homogeneous principal symbol  .A/ of the operator A as a function on C 1 .T  X n0/ (by assumption, in our model the coefficients of A are smooth across Z which is not essential for the corner calculus). The second symbolic level is the pair of boundary symbols @ .A/ D .@; .A/; @;C .A// belonging to the boundary value problems t .A T / and t .A TC / near the  sides int Y . These are operators in the standard calculus of Chapter 3, and we set   @;˙ .A/ D @ t .A T˙ / which are operator functions parametrised by T  .int Y˙ / n 0, smooth up to Z n fvg on the respective sides, cf., analogously, (8.3.14). Next we express the operators A and T˙ in (stretched) local coordinates (8.4.14) near points of Z n fvg. Denoting by z 2 R the local edge coordinate (instead of x1 ) the operators A and T˙ take the form X A D r 2 aj k .r; z/.r@r /j .rDz /k j Ck2

x C  R; Diff 2.j Ck/ .Œ0; 2 // and with coefficients aj k 2 C 1 .R X T˙ D r˙ r ˙ b˙;j k .r; z/.r@r /j .rDz /k ; j Ck˙

x C  R; Diff ˙ .j Ck/ .Œ0; 2 //, where r˙ u.r; ; z/ WD u.r; ˙ ; z/ for b˙;j k 2 C 1 .R C D 0,  D 2. This is similar to (8.1.6) and (8.1.8), respectively. We assume again  D 0, C D 1; our formulations are chosen in such a way that they easily generalise to other orders. Treating the interval Œ0; 2 as a manifold with boundary f0g [ f2g we have the spaces K s; .Œ0; 2 ^ / and 1

1

3

3

K s2;2 .Œ0; 2 ^ / WD K s2; 2 .Œ0; 2 ^ / ˚ K s 2 ; 2 .R / ˚ K s 2 ; 2 .RC / (8.4.23) when we identify the boundary components RC  f0g and RC  f2g of Œ0; 2 ^ D RC  Œ0; 2 with R and RC , respectively. The principal edge symbol ^ .A/.z; / for .z; / 2 T  R n 0 will be defined by expressions analogous to (8.1.13), (8.1.14), now with r˙ W v.r; / ! v.r; ˙ / for C D 0,  D 2. Then ^ .A/ represents a family of continuous operators ^ .A/.z; / W K s; .Œ0; 2 ^ / ! K s2; 2 .Œ0; 2 ^ /

(8.4.24)

for every s;  2 R, s > 32 , with a corresponding analogue of the homogeneity (8.1.16). The analogue a.z; / of the amplitude function (8.1.10) has the property a.z; / 2 S 2 .Rz  R I K s; .Œ0; 2 ^ /; K s2; 2 .Œ0; 2 ^ //; : This gives us the operator A itself as A D Opz .a/;

(8.4.25)

8.4 Cracks with singularities at the boundary

521

cf. also Proposition 8.1.1 and the formula (8.1.11). On the stretched manifold W associated with (8.4.22) we have the weighted edge spaces s; Wcomp.loc/ .W / locally near Wsing modelled on W s .Rz ; K s; .Œ0; 2 ^ //. Moreover, we have s; .Y˙;reg /; Wcomp.loc/

locally near Zreg modelled on W s .Rz ; K s; .R˙ //. The spaces are based on the usual group actions on K s; .Œ0; 2 ^ / and K s; .R˙ /, i.e., .1/ u.r; / D u.r; / and 1 .0/ v.r/ D  2 v.r/,  2 RC , respectively. Globally on W the operator (8.4.1) is continuous in the sense 2 s; .W / ! W s2; .W / A W Wcomp comp

(8.4.26)

for every s > 32 ;  2 R; here s 1 ; 1 2

s2; 2 W s2;2 .W / WD Wcomp .W / ˚ Wcomp2 comp

3

3

.Y;reg / ˚ W s 2 ; 2 .YC;reg /:

Similarly as in the general calculus of edge boundary value problems we can ask to what extent the operator (8.4.26) can be completed to a . ; @ ; ^ /-elliptic block matrix 

A K AD T Q



2 s; .W / Wcomp W s2; .W / comp ˚ ! W ˚ s1 s3 Hcomp .Zreg ; C j / Hloc .Zreg ; C jC /

(8.4.27)

with suitable dimensions j˙ , such that (8.4.27) admits a parametrix in the calculus. In this connection we always assume that A is elliptic and that the boundary operators T˙ satisfy the Shapiro–Lopatinskij condition on int Y˙ , uniformly up to Zreg from the respective sides. For convenience, we consider a specific example, namely, Dirichlet conditions on the minus and Neumann conditions on the plus side. From the generalities of the edge calculus of Chapter 7 we have the following result: Proposition 8.4.1. For every z 2 Zreg there exists a discrete set D.z/  R such that (8.4.24) is a family of Fredholm operators for all z 2 Zreg ,  6D 0, s > 32 , and all  2 R n D.z/. From the homogeneity of ^ .A/.z; / in  6D 0 we obtain an analogue of the relation (8.1.13) when we denote the operator (8.4.26) by A. /. Under the condition that there is a choice of a fixed weight  2 R n D.z/ for all z 2 Zreg and that ind.A.//.z; / coincide for  D 1 and  D C1 we find an analogue of the block matrix (8.1.18) which defines a family of isomorphisms ^ .A/.z; / W K s; .Œ0; 2 ^ / ˚ C j ! K s2; 2 .Œ0; 2 ^ / ˚ C jC first for jj D 1, and then extended by homogeneity to all  6D 0 by the rule (8.1.19).

522

8 Corner operators and problems with singular interfaces

It remains the problem to determine concrete weights  and suitable dimensions j˙ in this process. For the case of the Laplace operator A D and Dirichlet/Neumann conditions on the  sides we will give an answer in Section 8.4.6 below, cf. also the paper [193]. This will employ the known result for the Zaremba problem given in Theorem 8.1.3. Let us already announce what we obtain: Theorem 8.4.2. For A D and T representing Dirichlet/Neumann   conditions on the -sides of the crack Y , for every k 2 Z and  2 12 32  k; 52  k , the operators (8.4.24) for A D A. / are Fredholm, and we have ind ^ .A. //.z; / D k for every s 2 R, s >

3 , 2

and .z; / 2 T  Zreg n 0.

Corollary 8.4.3. For every  as in Theorem 8.4.2 the operator A D A. / (i.e., A realised as (8.4.26)) can be completed to a . ; @ ; ^ /-elliptic operator (8.4.27) in the edge calculus of boundary value problems with jC trace and j potential conditions on the crack boundary Zreg where k D jC  j . Based on other examples in terms of mixed problems for the Laplace operator we can also derive corresponding examples in the situation of a crack (with smooth crack boundary). The pseudo-differential analysis which produces parametrices of . ; @ ; ^ /elliptic operators on W is practically the same as that in Chapter 7 or that in the considerations on mixed problems. For the general pseudo-differential formalism it is always convenient to first consider a ‘normalised’ situation in terms of the same orders of trace and potential operators on the ˙-sides and the same number of trace and potential conditions with respect to the smooth part of the boundary, here int Y˙ , where the number is taken to be 1. The applications then require admitting lC trace and l potential conditions and vectors of different orders, cf. Remark 8.4.17 below. A similar method will be applied here.

8.4.3 Parameter-dependent crack operators on a sphere Let us fix a neighbourhood of the corner point v 2 Z in X and introduce coordinates (8.4.16). The slit sphere is a manifold with conical singularities v1 ; v2 and boundary. So we have the parameter-dependent calculus of the class C ;d .S2 ; g cone I Rl /, g cone WD .;   /, of cone boundary value problems on S2 , with the transmission property at the smooth part of the boundary and with parameters  2 Rl . It consists of -dependent families of continuous operators f ./ W H s; .S2 / ! H s; .S2 /;

(8.4.28)

s 2 R, s > d  12 , when we set 1

1

1

1

H s; .S2 / D H s; .S2 / ˚ H s 2 ; 2 .I / ˚ H s 2 ; 2 .IC /:

(8.4.29)

523

8.4 Cracks with singularities at the boundary

(Recall that, for convenience, in the general considerations we take the same orders of operators with respect to the ˙-sides.) We will also employ a small modification 0 of these spaces, namely, H .s;s /; .S2 /, defined analogously as (8.4.29) but with s 0 rather than s in the second and third component. As emphasized before, in general considerations we take the same orders on the ˙ parts of the boundary and with one trace and potential operator with respect to the components of the boundary. For the generalisation to vectors of such operators of different order (which are known in each concrete case) we then employ the same notation. The definition of C ;d .S2 ; g cone I Rl / for  2 Z, d 2 N was given in Chapter 6 in general form. Here we need a small modification. In order to keep the consideration self-contained we recall some definitions in the present situation. In addition, instead of (8.4.28) we need block matrices k./ W H s; .S2 / ˚ C j ! H s; .S2 / ˚ C jC ;

(8.4.30)

k./ D .kij .//i;j D1;2;3;4 . More generally, we will have an additional dependence on the variable t 2 RC which appears in (8.4.16) as the local corner axis variable. In the corner calculus we only need the case l D 1. Let us set  WD Q . First, by Definition 8.2.8, applied to the case N D Œ0; 2 , we have the space x C  R Q ; g cone I j ; jC /; R;d .R

(8.4.31)

 2 Z, d 2 N, of operator families a.t; / Q WD p.t; / Q C m.t; / Q C g.t; Q Q / W E ! Ez

(8.4.32)

with E WD K s; .Œ0; 2 ^ / ˚ C j ; Ez WD K s; .Œ0; 2 ^ / ˚ C jC , where 1

1

1

1

K s; .Œ0; 2 ^ / WD K s; .Œ0; 2 ^ / ˚ K s 2 ; 2 .R / ˚ K s 2 ; 2 .RC /; 1

1

K s; .Œ0; 2 / WD K s;  .Œ0; 2 /^ / ˚ K s 2 ; 2 .R / 1

1

˚ K s 2 ; 2 .RC /; cf., similarly, the formula (8.4.23). Recall that the ‘non-smoothing’ part p.t; Q / of the definition of (8.4.32) contains operator functions of the kind (8.2.15), where the parameters are employed in the meaning (8.2.16) with the covariables .%; / dual to .r; t/ in the formula (8.4.21). The main part of the general calculus of crack problems can be carried out by exactly the same scheme as for mixed boundary value problems. So we can simplify the process a little or slightly change the point of view, compared with the Sections 8.1–8.3. By R;d .R Q ; g cone I j ; jC / we denote the subspace of all t -independent elements of (8.4.31). On S2 with the crack I and the end points v1 ; v1 we choose an open covering by sets fU1 ; U2 ; U3 g (8.4.33)

524

8 Corner operators and problems with singular interfaces

where Uj , j D 1; 2, are neighbourhoods of vj , U1 \U2 D ; and U3 \.fv1 g[fv2 g/ D ;. Let f1;sing ; 2;sing ; reg g be functions of analogous meaning as those in Definition 8.2.18 (ii). Moreover, fix functions Qj;sing 2 C01 .Uj /, j D 1; 2, and Qreg 2 C01 .U3 / that are equal to 1 on the supports of j;sing and reg , respectively. The neighbourhoods U1 ; U2 will be chosen in such a way that they admit local stretched coordinates of the kind (8.4.21). More precisely, the coordinates (8.4.21) refer to the open sets Uj n fvj g endowed with the double slit coming from I . Let Uj n fvj g denote this subset of S2 when Uj n fvj g is interpreted in that way. Operator families f. / Q 2 R;d .R Q ; g cone I j ; jC / can be pulled back to operators on Uj n fvj g under the corresponding singular charts Uj n fvj g ! RC  Œ0; 2 , cf. the formula (8.4.20); for abbreviation they will be Q again. Setting denoted by fj . / Q Q1;sing C 2;sing f2 . Q /Q2;sing (8.4.34) f. / Q WD 1;sing f1 . / S we obtain an operator family concentrated on j D1;2 .Uj nfvj g/. Furthermore, observe z 2 , cf. the notation that the functions reg and Qreg can also be interpreted as functions on S z 2 I R Q / we can form an at the end of Section 8.4.1. Given an element r. Q / 2 B ;d .S operator function r. / Q WD diag.reg r. Q /Qreg ; 0/: (8.4.35) Another ingredient of parameter-depending families on S2 are the smoothing elements. The space C1;0 .S2 ; g cone I j ; jC / is defined as the set of all continuous operators 0

G W H .s;s /; .S2 / ˚ C j ! H 1;C" .S2 / ˚ C jC for all s; s 0 2 R; s >  12 and for some " D ".G/ > 0 such that the formal adjoint has an analogous mapping property with the weight  C  in the preimage and  C " inPthe image. Moreover, C1;d .S2 ; g cone I j ; jC / is defined to be the set of all G0 C jd D1 Gj diag.D j ; 0; 0/, where Gj , 0  j  d, are described as before, and D is a first order differential operator on S2 which is equal to @ near @Sreg , where  is the normal variable to the boundary. We then set C1;d .S2 ; g cone I j ; jC I R Q / WD S.R Q ; C1;d .S2 ; g cone I j ; jC //: Definition 8.4.4. The space C;d .S2 ; g cone I j ; jC I R Q / for  2 Z, d 2 N, g cone D .;   / is defined as the set of all operator families k. Q / WD f. Q / C r. Q / C c. Q / for arbitrary (8.4.34), (8.4.35), and c. / Q 2 C1;d .S2 ; g cone I j ; jC I R Q /. Remark 8.4.5. The elements k. / Q 2 C;d .S2 ; g cone I j ; jC I R Q / define families of continuous operators (8.4.30) (for  D ) Q for all s 2 R, s > d  12 .

525

8.4 Cracks with singularities at the boundary

Remark 8.4.6. There is also a variant of Definition 8.4.4 without parameter, denoted by 1;d C;d .S2 ; g cone I j ; jC /. Modulo .S2 ; g cone I j ; jC / this is nothing other than   C A 0 the space of all operators 0 0 for A 2 C ;d .S2 ; g cone / with S2 being treated as a manifold with conical singularities at the boundary (which are just the end points of the slit I ), and C ;d .S2 ; g cone / as the standard cone algebra of boundary value problems on S2 . The families k. / Q have a principal symbolic structure, namely,  .k/ D . .k/; @ .k/; ^ .k// with  ; @ , and ^ denoting the interior, the boundary, and the edge component. The definition is as follows: Writing k. / Q D .kij . Q //i;j D1;:::;4 we have 2 k11 . Q / 2 L cl .int Sreg I R Q /;

and

2 .kij . Q //i;j D1;2;3 2 B ;d .Sreg I R Q /:

 .k/ is nothing other than the parameter-dependent homogeneous principal sym2 bol of k 11 . Q / 2 L cl .int Sreg I R Q /. Moreover, close to each of the end points v of the slit on S2 in the splitting of variables .r; / 2 RC  Œ0; 2 and covariables .%; #/, cf. also (8.4.21), we have a representation Q D r  Q .k/.r; ; r%; #; r Q /  .k/.r; ; %; #; / Q #; / Q which is homogeneous in .%; Q #; / Q 6D 0 of order  for a function Q .k/.r; ; %; and smooth up to r D 0. 2 Moreover, the property .kij . // Q i;j D1;2;3 2 B ;d .Sreg I R Q / gives rise to the parameter-dependent homogeneous principal boundary symbol @ .k/ D .@; .k/; @;C .k// with minus and plus components, belonging to the corresponding sides of the slit int I , cf. (8.3.14). Close to the end points v of the slit we have a representation Q D r  Q @; .k/.r; r%; r Q / @; .k/.r; %; / Q /, Q homogeneous in .%; Q / Q 6D 0 and smooth up to for operator functions Q @; .k/.r; %; r D 0. It remains the homogeneous principal edge symbol which is also a pair of operator functions belonging to the end points of the slit, where Q is treated as an edge covariable ^ .k/ W K s; .Œ0; 2 ^ / ˚ C j ! K s; .Œ0; 2 ^ / ˚ C jC ;

(8.4.36)

cf. also the formula (8.4.30). The ingredients of ^ .k/. Q / are defined by the same scheme as in (8.2.28). Definition 8.4.7. An operator family k. / Q 2 C;d .S2 ; g cone I j ; jC I R Q / is called parameter-dependent elliptic if 2 Q #; / Q 6D 0 for .%; Q #; / Q 6D (i)  .k/ 6D 0 on .T  .int Sreg /  R/ n 0 and Q .k/.r; ; %; 0, up r D 0;

526

8 Corner operators and problems with singular interfaces

Q / Q is (ii) @; .k/ is bijective for all points of .T  .int I /  R/ n 0, and Q @; .k/.r; %; Q / Q 6D 0, up to r D 0; bijective for .%; (iii) ^ .k/. Q / is a family of isomorphisms (8.4.36). Theorem 8.4.8. An elliptic k. / Q 2 C;d .S2 ; g cone I j ; jC I R Q / with g cone D .;   / C has a (parameter-dependent) parametrix p. / Q 2 C;.d/ .S2 ; g 1 cone I jC ; j I R Q / with g 1 cone D .  ;  / in the sense that p. /k. Q / Q D i and

k. Q /p. Q / D i

modulo C1;dl .S2 ; g l;cone I j ; j I R Q / and C1;dl .S2 , g r;cone I jC ; jC I R Q /, respectively, with g l;cone D .;  /, g r;cone D .  ;   / and dl D max.; d/, dr D .d  /C . The proof is essentially the same as the part of the proof of Theorem 7.2.36 which concerns the construction of P.y; /, separately for the summands of (8.4.34). The part containing (8.4.35) corresponds to the construction of a parameter-dependent parametrix in the sense of Theorem 3.3.17. By M 1;d .S2 ; g cone I j ; jC I ˇ / for some ˇ 2 R we denote the space of all c.w/ 2 A.ˇ  " < Re w < ˇ C "; C1;d .S2 ; g cone I j ; jC // for some " D ".c/ > 0 such that c.ı C i / Q 2 C1;d .S2 I g cone I j ; jC I R Q / for every ı 2 .ˇ  "; ˇ C "/, uniformly in compact subintervals. Moreover, let ;d .S2 ; g cone I j ; jC / MO

(8.4.37)

denote the space of all h.w/ 2 A.C; C;d .S2 ; g cone I j ; jC // such that h.ı C i / Q 2 C;d .S2 ; g cone I j ; jC I R Q / for every ı 2 R, uniformly in compact intervals. Observe that the space (8.4.37) is equal to the space of all l.w/ C c.w/ for arbitrary 1;d .S2 ; g cone I j ; jC / c.w/ 2 MO

a. / Q 2C

;d

and

l.w/ D HC . /.a/.w/;

.S ; g cone I j ; jC I R Q / 2

where .b/ is any function in C01 .RC / that is equal to 1 in a neighbourhood of b D 1, cf. the kernel cut-off expression (8.2.32).

8.4.4 The local corner-crack calculus In Section 8.4.1 we described a crack with conical singularity locally by the stretched cone S2  RC 3 .; t /, cf. the formula (8.4.16). Let us now define the weighted corner spaces for the present situation.

527

8.4 Cracks with singularities at the boundary

In local coordinates Œ0; 2 ^ 3 .r; / near the point vj 2 S2 , cf. the formula (8.4.20), we have K s; .Œ0; 2 ^ / and associated corner spaces V s;.;ı/ .Œ0; 2 ^ RC /j obtained as the pull back of V s;ı .RC ; K s; .Œ0; 2 ^ // to Uj  RC , j D 1; 2. We then set V s;.;ı/ .S2  RC / 3 nX o D 'j uj W u3 2 H s;ı ..S 2 /^ /; uj 2 V s;.;ı/ .Œ0; 2 ^  RC /j ; j D 1; 2 ; j D1

where f'1 ; '2 ; '3 g denotes a partition of unity on S 2 subordinate to an open covering (8.4.33). Moreover, for W as at the end of Section 8.4.1 we have the weighted edge s; spaces Wloc .W /. We then define s; V s;.;ı/ .X/ WD fu 2 Wloc .W / W !u 2  V s;.;ı/ .S2  RC /g

with  being given by the formula (8.4.17), for any ! 2 C01 .U / which is equal to 1 in a neighbourhood of the point v. The plus and minus parts Y˙ of the crack, cf. the notation in (8.4.1), have a conical point v on the boundary. In the present case we interpret Y˙ as manifolds with corner v, where the base is I˙ with end points treated as conical singularities. We then have the s; weighted edge spaces Wcomp;loc .Y˙ n fvg/; s;  2 R, and corner spaces V s;.;ı/ .Y˙ /  s; Wloc .Y˙ n fvg/ with double weights, locally near v defined as (8.2.9). The spaces (8.2.9) will also be employed in the meaning ^ / V s;.;ı/ .I˙

from the two sides I˙ of the slit I on S 2 , cf. the notation of Section 8.4.1. The following material is analogous to that of Section 8.2.5, and we do not repeat all details. By ˆ; ‰, etc., we denote functions of the form ˆ WD diag.'; '; '; '/; ‰ WD diag. ; ; ; / for x C /, interpreted as operators of multiplication on direct sums arbitrary '; 2 C01 .R of 4 spaces, referring to RC 3 t. Let us set 1

1

1

1

1

1

^ V s;.;ı/ ..S2 /^ / WD V s;.;ı/ ..S2 /^ / ˚ V s 2 ;. 2 ;ı 2 / .I^ / ˚ V s 2 ;. 2 ;ı 2 / .IC /: (8.4.38) 0 Moreover, let V .s;s /;.;ı/ ..S2 /^ / denote the space which is analogous to (8.4.38) but with s 0 instead of s in the second and third component. By C1;0 ..S2 /^ ; gI j ; jC /, g WD .;   I ı; ı  /, we denote the space of all G that induce continuous operators 0

V .s;s /;.;ı/ ..S2 /^ / V 1;.C";ıC"/ ..S2 /^ / ˆG‰ W ! ˚ ˚ s 00 ;ı1 j H 1;ı1C" .RC ; C jC / H .RC ; C / for some " D ".G/ > 0, for every s; s 0 ; s 00 2 R, s >  12 ; an analogous condition is required for the formal adjoint G  with respect to the pair of weights  C ; ı C 

528

8 Corner operators and problems with singular interfaces

1;d in the preimage and  C "; ı C " in the image. Moreover, ..S2 /^ ; gI j ; jC / PdC for d 2 N is defined to be the space of all G D G0 C j D0 Gj diag.D j ; 0; 0/ for arbitrary Gj 2 C1;0 ..S2 /^ ; gI j ; jC / and a first order differential operator D that is locally near the slit I equal to @ where  is the normal coordinate to I .

Definition 8.4.9. The space C;d ..S2 /^ ; gI j ; jC /; g WD .;  I ı; ı /,  2 Z, d 2 N, is defined as the set of all operators ı1 A WD t  opM .h C f / C G

(8.4.39)

with x C ; M;d .S2 ; g cone I j ; jC //; h.t; w/ 2 C 1 .R O f .w/ 2 M1;d .S2 ; g cone I j ; jC I  3 ı /; 2

G 2 C1;d ..S2 /^ ; gI j ; jC /: x C / and A 2 C;d ..S2 /^ ; gI j ; jC / we have Remark 8.4.10. For every '; 2 C01 .R ;d 2 ^ ˆA; A‰ 2 C ..S / ; gI j ; jC /. Theorem 8.4.11. For every A 2 C;d ..S2 /^ ; gI j ; jC /, g D .;   I ı; ı  / we have continuous operators V s;. ;ı/ ..S2 /^ / V s;.;ı/ ..S2 /^ / ! ˆA‰ W ˚ ˚ H s1;ı1 .RC ; C j / H s1;ı1 .RC ; C jC / for every s 2 R, s > d  12 , and arbitrary ˆ; ‰. The proof can easily be reduced to Theorem 2.4.59 (ii). Let us now establish the principal symbolic structure  .A/ D . .A/; @ .A/; ^ .A/; c .A//

(8.4.40)

of operators A 2 C;d ..S2 /^ ; gI j ; jC /. The components of (8.4.40) are of analogous meaning as those in Section 8.2.5. Writing A D .Aij /i;j D1;:::;4 we have A11 2 2 L cl ..S n I /  RC /. This gives us the homogeneous principal symbol  .A/ WD  .A11 / of order . Moreover, near the end points v1 ; v2 of the slit in the variables .r; ; t/ 2 RC  Œ0; 2  RC and covariables .%; #; /, cf. the formula (8.4.21), we can write  .A/ in the form  .A/.r; ; t; %; #; / D t  r  Q .A/.r; ; t; r%; #; rt /

(8.4.41)

Q #; / Q homogeneous of order  in .%; Q #; / Q 6D 0 and for a function Q .A/.r; ; t; %; smooth up to r D t D 0. Moreover, .Aij /i;j D1;2 and .Aij /i;j D1;3 are locally near int I boundary value problems of the class B ;d with boundary symbols @; .A/ WD @ ..Aij /i;j D1;2 /;

@;C .A/ WD @ ..Aij /i;j D1;3 /:

8.4 Cracks with singularities at the boundary

529

Locally near the end points v1 ; v2 they can be written in the form @; .A/.r; t; %; / D t  r  Q @; .A/.r; t; r%; rt / Q / Q in .%; Q / Q 6D 0, which are for corresponding operator functions Q @; .A/.r; t; %; smooth up to r D t D 0. From Definition 8.4.9 it follows that A 2 Y;d .W /, and hence we have the corresponding homogeneous principal edge symbol ^ .A/.t; / W K s; .Œ0; 2 ^ / ˚ C j ! K s; .Œ0; 2 ^ / ˚ C jC

(8.4.42)

cf. the formula (8.4.23), .t; / 2 T  RC n 0. There is then a representation ^ .A/.t; / D t  Q ^ .A/.t; t /

(8.4.43)

for an operator function Q ^ .A/.t; / Q of analogous structure, Q 6D 0, smooth up to t D 0. Finally, A has a corner conormal symbol c .A/.w/ W H s; .S2 / ˚ C j ! H s; .S2 / ˚ C jC ; w 2  3 ı , defined by 2

(8.4.44)

c .A/.w/ D h.0; w/ C f .w/;

cf. the formulas (8.4.29) and (8.4.39). Remark 8.4.12. There is an analogue of Theorem 8.2.24 on compositions of operators. Definition 8.4.13. An A 2 C;d ..S2 /^ , gI j ; jC / is said to be elliptic, if its symbol satisfies the following conditions: Q #; / Q 6D 0 for .%; Q #; / Q 6D (i)  .A/ 6D 0 on T  .S 2 nI /RC / and Q .A/.r; ; t; %; Q #; / Q 6D 0; 0, up to .%; (ii) @; .A/ are bijective for all points of T  .int I˙  RC / n 0, and so are Q / Q for .%; Q / Q 6D 0, up to r D t D 0; Q @; .A/.r; t; %; (iii) ^ .A/ is bijective for all points of T  RC n 0, and so is Q ^ .A/.t; Q / for Q 6D 0, up to t D 0; (iv) c .A/.w/ defines a family of isomorphisms (8.4.44) for all w 2  3 ı . 2

Theorem 8.4.14. An elliptic operator A 2 C;d ..S2 /^ ; gI j ; jC / has a parametrix C P 2 C;.d/ ..S2 /^ ; g 1 I jC ; j / in the following sense: z z ˆ z D ˆI ˆPˆA

z z ˆ z D ˆI and ˆAˆP

modulo C1;dl ..S2 /^ ; g l I j ; j / and C1;dr ..S2 /^ ; g r I jC ; jC /, respectively, with analogous meaning of g l , g r , dl , dr , etc., as in Theorem 8.2.27. The result is an analogue of Theorem 8.2.27.

530

8 Corner operators and problems with singular interfaces

8.4.5 Singular crack problems We now turn to global singular problems on the stretched manifold X, cf. the notation at  Athe end of Section 8.4.1. Similarly as in Section 8.3.1 we consider operators A D K which are continuous in the sense T Q V s;.;ı/ .X/ V s;. ;ı/ .X/ AW ! ˚ ˚ s1;ı1 j H H s1;ı1 .Z; C jC / .Z; C / 1

1

1

(8.4.45)

1

1

1

where V s;.;ı/ .X/ WD V s;.;ı/ .X/ ˚ V s 2 ;. 2 ;ı 2 / .Y / ˚ V s 2 ;. 2 ;ı 2 / .YC /. The first step of the definition are again the smoothing operators, of the class C1;d .X; g; j ; jC /. The properties are formally the same as for the corresponding operators in Section 8.3.1, cf. the formulas (8.3.1) and (8.3.2) and operators D j in the meaning of those of the preceding section. The space X can be covered by open neighbourhoods fU0 ; U1 ; U2 g where U0 is a neighbourhood of the point v 2 Z, U1 a neighbourhood such that z 62 U1 but U0 [ U1 is a neighbourhood of Z, and U2 WD X n Z. More precisely, U0 corresponds to U in (8.4.13) and U1 is a union of neighbourhoods U of the kind (8.4.12). Let f'0 ; '1 ; '2 g; f 0 ; 1 ; 2 g be of analogous meaning as the corresponding functions 'j ; j 2 C01 .Uj / in Section 8.3.1. They will also be regarded as functions on X. Moreover, we form 4  4 diagonal matrices ˆcorner;edge;reg and ‰corner;edge;reg of formally the same shape as in Section 8.3.1. Definition 8.4.15. Let X be the stretched singular crack configuration as described in Section 8.4.1. Then C;d .X; gI j ; jC / with g D .;   I ı; ı  / and  2 Z, d 2 N, is defined to be the set of all operators A D Acorner C Aedge C Areg C G

(8.4.46)

with the following summands: Acorner WD ˆcorner Lcorner ‰corner ; Areg WD ˆreg Lreg ‰reg ;

Aedge WD ˆedge Ledge ‰edge ;

G 2 C1;d .X; gI j ; jC /;

where Lcorner 2 C;d ..S2 /^ ; gI j ; jC /, cf. Definition 8.4.9, interpreted as an operator on U0 , moreover, Ledge 2 Y;d .U1 ; g cone I j ; jC //, and Lreg 2 B ;d .X n Z/. Remark 8.4.16. Each A 2 C;d .X; gI j ; jC / induces continuous operators (8.4.45) for every s 2 R, s > d  12 . This is a direct consequence of the definition of the spaces in (8.4.45) and of the continuity results for the summands in (8.4.46) in the corresponding localised form.

8.4 Cracks with singularities at the boundary

531

The principal symbols of operators A 2 C;d .X; gI j ; jC / consist of tuples (8.4.40) with the following components: interior symbol  .A/; boundary symbol @ .A/; edge symbol ^ .A/; and conormal symbol c .A/: The definitions and properties follow from the representation (8.4.46). According to (8.4.45) we have a block matrix structure A D .Aij /i;j D1;:::;4 :  Here A11 2 L cl .X nY /, and we set  .A/ WD  .A11 / as a function on T .X nY /n0. Locally near Zreg D Z n fvg in the stretched variables .r; ; z/ 2 RC  Œ0; 2  R we can express  .A/ in the form (8.3.7); moreover, locally near v in the stretched variables .r; ; t / 2 RC  Œ0; 2  RC we have a representation of the form (8.4.41). The involved functions

Q Q .A/.r; ; z; %; Q #; /

and

Q #; / Q Q .A/.r; ; t; %;

are smooth up to r D 0 and r D t D 0, respectively. The boundary symbol @ .A/ admits a similar splitting as (8.3.9) for (8.3.10), where the ˙-components refer to both sides of the crack int Y˙ , and the boundary symbols are taken in the sense of the B ;d calculus. Close to Zreg in stretched variables .r; ; z/ 2 RC  Œ0; 2  R we have a representation of the form (8.3.11); moreover, locally near v in stretched variables .r; ; t / 2 RC  Œ0; 2  RC we have an expression of the form (8.3.12). The involved operator functions Q Q @;˙ .A/.r; z; %; Q /

and

Q / Q Q @;˙ .A/.r; t; %;

are smooth up to r D 0 and r D t D 0, respectively. The edge symbol ^ .A/ as an operator function on T  Zreg n 0 appears in a similar manner as (8.3.13), here as a family of maps (8.4.42). Close to v we have a representation of the kind (8.4.43) with an operator function Q ^ .A/.t; Q / of analogous structure which is smooth up to t D 0. The corner conormal symbol c .A/.w/ is determined by Acorner in the representation (8.4.46), more precisely, by Lcorner , and is given by (8.4.44) for w 2  3 ı , with the 2 spaces (8.4.29). Remark 8.4.17. The calculus of crack operators in the sense of Definition 8.4.15 has a generalisation analogously as Remark 8.3.5 that we employ below, including a further generalisation concerning the involved orders of the operators. For reference below we formulate the analogues operator families @ .A/, ^ .A/, c .A/ for arbitrary dimensions l˙ ; j˙ 2 N (including 0), namely, @;˙ .A/ W H 3 .RC / ˚ C l ! H s .RC / ˚ C lC ;

(8.4.47)

532

8 Corner operators and problems with singular interfaces 1

1

1

1

^ .A/ W K s; .Œ0; 2 ^ / ˚ K s 2 ; 2 .R ; C l / ˚ K s 2 ; 2 .RC ; C l / ˚ C j 1

1

! K s; .Œ0; 2 ^ / ˚ K s 2 ; 2 .R ; C lC / ˚K

1 s 1 2 ; 2

1

(8.4.48)

.RC ; C lC / ˚ C jC ; 1

1

1

c .A/ W H s; .S2 / ˚ H s 2 ; 2 .I ; C l / ˚ H s 2 ; 2 .IC ; C l / ˚ C j 1

1

! H s; .S2 / ˚ H s 2 ; 2 .I ; C lC / ˚H

1 s 1 2 ; 2

(8.4.49)

.IC ; C lC / ˚ C jC :

For A 2 C;d .X; gI j ; jC / (in the generalised meaning of notation, according to Remark 8.4.17) in (8.4.47), (8.4.48), (8.4.49) we assume s > max.; d/  12 . Remark 8.4.18. Let A 2 C;d .X; gI j ; jC / and  .A/ D 0. Then the operator (8.4.45) is compact for every s > d  12 . Remark 8.4.19. For the operators of Definition 8.4.15 we have a direct analogue of Theorem 8.3.4. Definition 8.4.20. An A 2 C;d .X; gI j ; jC / is called elliptic, if the corresponding analogues of Definition 8.3.6 are satisfied. For  .A/ the condition is formally the same, while for @ .A/; ^ .A/ and c .A/ the bijectivities refer to the spaces in (8.4.47), (8.4.48) and (8.4.49), respectively. Theorem 8.4.21. An elliptic element A 2 C;d .X; gI j ; jC /, g D .;  I ı; ı/, C has a parametrix P 2 C;.d/ .X; g 1 I jC ; j /, g 1 WD .  ; I ı  ; ı/ in the sense of an analogue of the relation (8.3.17), cf. also the notation in Theorem 8.2.27. The result can be proved by combining Theorem 8.4.14 and Theorem 7.2.36. Corollary 8.4.22. Let A be as in Theorem 8.4.21. Then the associated operator (8.4.45) is Fredholm for every s > max.; d/  12 (according to the number of components in A the spaces in (8.4.45) are interpreted in the corresponding modified sense).

8.4.6 Examples In the preceding section we studied the general structure of pseudo-differential crack problems with conical singularities. We now specify the considerations in such a way that the operators (8.4.4), (8.4.6) (first, for simplicity, for the case @X D ;) belong to the calculus, especially, when the boundary operators B˙ in (8.4.2) have the orders 1 and 0, respectively. The generalision of Definition 8.4.15 to the case of an arbitrary number l˙ 2 N of components of trace and potential operators on int Y˙ was explained in Remark 8.4.17 (clearly we always admit l˙ or j˙ to be zero).

8.4 Cracks with singularities at the boundary

533

In order to cover operators of the kind (8.4.4) or (8.4.3) we interpret the definition of the space C;d .X; gI j ; jC / (8.4.50) once again in a wider sense, namely, that the components of the trace and   potential conditions on int Y˙ may have arbitrary orders. More precisely, if A D

A K KC T Q QCC TC QC QCC

is the upper left corner of A, the components of T , K , etc., may have different orders. In every concrete example, such as (8.4.3), they are known from the context, and also the resulting orders in parametrices are known. In compositions of operators we always assume that the numbers of entries of the involved block matrices fit together in the middle. For the calculus itself we do not need any generalisation of proofs, since the single entries belong to the former calculus with unified orders. Remark 8.4.23. Definition 8.4.20 of ellipticity, Theorem 8.4.21 on the existence of a parametrix in the calculus, and Corollary 8.4.22 on the Fredholm property in weighted Sobolev spaces have a straightforward extension to the case of operators in (8.4.50). As an example we consider the case that in a neighbourhood of Y our manifold X is identified with an open set of R3 in the standard Euclidean metric and that the second order elliptic operator A is equal to the Laplace operator in this neighbourhood. For T we take T u D r u; TC u D rC @ u; where @ is the derivative to int YC in normal direction. Then   A D t A T TC is continuous as an operator A W V s;.;ı/ .X/ ! V s2;.2;ı2/ .X/ for every s 2 R, s > 32 , and ; ı 2 R (cf. the formula (8.4.5)). We complete A to an elliptic operator 

A K AD T Q



V s;.;ı/ .X/ V s2;.2;ı2/ .X/ ˚ ! W ˚ s1;ı1 j s3;ı3 H .Z; C / H .Z; C jC /

(8.4.51)

within our calculus.   Theorem 8.4.24. For every  2 12 32  k; 52  k , k 2 Z, there exists a choice of T ; K and Q, for suitable dimensions j˙ 2 N with jC  j D k, such that the operator A defined by (8.4.51) belongs to C2;2 .X; gI j ; jC / for g D .;   2I ı; ı  2/ and is elliptic for every ı 62 D for a certain discrete set D  R, depending of ; T ; K; Q.

534

8 Corner operators and problems with singular interfaces

Proof. The first step of constructing the operators T ; K and Q is to fill up the edge symbol ^ .A/.z; / W K s; .Œ0; 2 ^ / ! K s2;2 .Œ0; 2 ^ / for 1

1

3

3

K s2;2 .Œ0; 2 ^ / D K s2;2 .Œ0; 2 ^ / ˚ K s 2 ; 2 .R / ˚ K s 2 ; 2 .RC / to a family of isomorphisms K s; .Œ0; 2 ^ / K s2; 2 .Œ0; 2 ^ / ˚ ! ; ^ .A/.z; / W ˚ C j C jC

(8.4.52)

.z; / 2 T  Zreg n 0. This process is standard when we know the admitted  and associated dimensions j˙ , obtained from jC  j D ind ^ .A/.z; /. The index will be derived in Theorem 8.4.2 below. At the same time, by a homotopy argument and the reduction of the crack edge symbol (8.4.52) to the edge symbol of the Zaremba problem we will see that the topological obstruction for the existence of extra conditions also vanishes in the crack case. In other words, over Zreg we can construct a family of isomorphisms  K s; .Œ0; 2 ^ / K s2; 2 .Œ0; 2 ^ / ^ .A/ ^ .K/ ˚ ^ .A/.z; / D .z; / W ! ; ˚ ^ .T / ^ .Q/ C j C jC (8.4.53) .z; / 2 T  Zreg n 0. The nature of the symbols ^ .T /; ^ .K/ and ^ .Q/ is analogous to that of the edge calculus, cf. Chapter 7. The construction can be performed in that sense up to the end points of the branches Zk close to v, k D 1; 2, such that we obtain a structure of the form 

^ .A/.t; / D t 2 Q ^ .A/.t; t /; Q is a family of isomorphisms for Q 6D 0, smooth up to t D 0. where Q ^ .A/.t; / We are now in the position to associate with the principal edge symbols ^ .T /, ^ .K/ and ^ .Q/ corresponding operators T ; K, and Q, respectively, in the space C;d .X; gI j ; jC /. Similarly as in the edge calculus the method is to first multiply the symbols by an excision function ./ to obtain local operator-valued symbols with respect to .z; /. Then we can construct associated operators locally near Zreg , based on the Fourier transform, combined with localising functions from a partition of unity, etc. The most specific part is contributed by a neighbourhood of z. In this case we form .t /Q ^ .T /.t; t /, etc., pass by kernel cut-off to holomorphic amplitude functions in the Mellin covariable w and then form the associated weighted Mellin pseudo-differential operators, according to the first summand on the right of (8.4.46). The obtained operator (8.4.51) is elliptic with respect to  .A/ and ^ .A/ which is a consequence of the construction. The conormal symbol c .A/ is a holomorphic family of Fredholm operators and bijective for large Im w. This gives us the bijectivity for all w 2 C up to a discrete set, and D is then determined by those ı 2 R such that  3 ı 2 intersects that set.

8.4 Cracks with singularities at the boundary

535

The edge symbol (8.4.52) is very similar to (5.1.10), the components of the edge symbol (5.1.9) of the Zaremba problem. The interface Z in this case is locally represented by R 3 z, and .r; / 2 RC  Œ0;  D I ^ are polar coordinates in the normal x 2 D fxQ D .x2 ; x3 / 2 R2 W x3  0g). In order half-plane (for n D 3 identified with R C to unify (8.4.53) and (5.1.9) we set I WD Œ0; .1 C / for 0    1. Then, setting 1 0 ^ . /./ A rf0g ^ .A /./ WD @ (8.4.54) rf.1C/g @ with rf˛g denoting the operator of restriction of a function in the interval I to  D ˛, and ^ . /./ operating on functions in K s; .I^ / for some -depending weight  , the operator families ^ .A /.z; / W K s; .I^ / ! K s2; 2 .I^ /

(8.4.55)

coincide for  D 0 with ^ .A0 /.z; / for the Zaremba problem and for  D 1 with ^ .A1 /.z; / for the crack problem. As we know from the calculus of boundary value problems on the stretched cone I^ , the operators (8.4.54) for a weight  2 R are Fredholm when the subordinate conormal symbol H s2 .int I / ˚ c ^ . / C c ^ .A /.w/ D @ c ^ .T / A .w/ W H s .int I / ! ˚ c ^ .TC / C 0

1

(8.4.56)

for c ^ . /.w/ D @2 C w 2 , c ^ .T / D rf0g , c ^ .TC / D rf.1C/g @ is a family of isomorphisms for all w 2 1 . In Section 5.1.3 we found that the set of non-bijectivity points of (8.4.56) for  D 0 is equal to ˚

w 2 C W Re w 2 Z  12 ; Im w D 0 ; (8.4.57) and that the points are all simple.   Recall from Corollary 5.3.3 that for every k 2 Z, 0 2 12  k; 32  k , we have ind ^ .A0 / D k:

(8.4.58)

Theorem 8.4.25. The set of non-bijectivity points of c ^ .A /, 0    1, is equal to ˚  

1 w 2 C W Re w D 1C k  12 ; k 2 Z; Im w D 0 ; (8.4.59)  1 1  1 3  and the points are simple. For every  2 1C 2 C   k ; 1C 2 C   k/ , k 2 Z, we have ind ^ .A / D k: (8.4.60)

536

8 Corner operators and problems with singular interfaces

Proof. Writing w D a C i b we see that for w 6D 0 the kernel of c ^ . /.w/ consists of all functions u./ D c1 e b e ia C c2 e b e ia for c1 ; c2 2 C. The condition u.0/ D 0 for such a function implies c WD c1 D c2 . Moreover, we have u0 ./ D c.b C i a/.e b e ia C e b e ia /: Assuming c 6D 0 (otherwise we have u 0) from u0 . / D 0,  WD .1 C /, we obtain the condition e b fcos.a / C i sin.a /g C e b fcos.a / C i sin.a /g D .e b C e b / cos.a / C i.e b  e b / sin.a / D 0: Since e b C e b never vanishes we obtain cos.a / D 0; then  sin.a   / 6D 0 yields 1 b b e D 0 and then b D 0. This gives us a D k  2 , k 2 Z, i.e., e   a D .1 C /1 k  12 , which proves (8.4.59). The property of the non-bijectivity points to be simple follows in an analogous manner as Remark 5.1.5. We now obtain that (8.4.55) is a family of Fredholm operators for all  2 R such that 1 does not  the set  (8.4.59). In order to show (8.4.60) we observe that  intersect 1 1 1 1 the relation  2 1  1C k C 2 ; 1  1C k  2 is equivalent to 

1   2







1 1 1 1 k ; kC 1C 2 1C 2



:

(8.4.61)

It entails the Fredholm property of (8.4.55), since the weight line 1 is then off the set (8.4.59). The relation (8.4.61) is equivalent to 









1 1 3 1 Ck ; Ck : (8.4.62) 1C 2 1C 2   For  D 0 this corresponds to 0 2 12  k; 32  k from the Zaremba problem. In that case we have the relation (8.4.58). For every 0  1  1 we find a family of weights  which smoothly depends on  2 Œ0; 1 such that (8.4.62) holds. The associated operators (8.4.55) are Fredholm for all  2 Œ0; 1 . Consider the family of isomorphisms  2

K s; .I^ / ! K s;0 .I0^ / defined by u.r; / ! fr 0  !.r/ C .1  !.r//gu.r; .1 C /1 /,  2 Œ0; 1

for any cut-off function !.r/, and define analogous isomorphisms on the spaces K s2;2 .I^ /. Then the associated substitution transforms ^ .A / into a family of Fredholm operators a W K s;0 .I0^ / ! K s2;0 2 .I0^ /, continuous in  2 Œ0; 1 , and we have a0 D ^ .A0 /. By virtue of ind a0 D ind a D k for all  2 Œ0; 1

and ind a D ind ^ .A / we finally obtain (8.4.60).

8.4 Cracks with singularities at the boundary

537

8.4.7 Further comments A crack boundary value problem in the sense of (8.4.1) also contains additional boundary conditions on @X . This case can be treated in a similar manner. Formally, we only have to slightly generalise Definition 8.4.15 by adding the effects from the boundary @X . Recall that now X is a (compact) C 1 manifold with boundary. We then have the operator space B ;d .X /, cf. Chapter 3. Moreover there is a simple generalisation of the corner Sobolev spaces V s;.;ı/ .X/. In the case of @X 6D 0 we fix a collar neighbourhood V Š @X  Œ0; 1/ of the boundary, functions '; 2 C 1 .X / supported in V where ' 1 in a neighbourhood of @X; 1 on supp '. Since by assumption the crack does not intersect @X we can talk about spaces of the kind V s;.;ı/ on the double 2X of X, denoted for the moment by V s;.;ı/ .2X/ which makes sense when we assume the crack to be located in the plus copy of X in the double, while the minus copy is assumed to be free of any crack (i.e., smooth). Then we have V s;.;ı/ .2X/jint V D H s .2X /jint V ; and we now define V s;.;ı/ .X/ WD f'u C .1  '/v W u 2 H s .int X /; v 2 V s;.;ı/ .2X/g: Instead of (8.4.45) we talk about operators V s;.;ı/ .X/ V s;. ;ı/ .X/ ˚ ˚ s1;ı1 j s;ı1 H H .Z; C / .Z; C jC / AW ! ˚ ˚ 1 1 H s 2 .@X / H s 2 .@X / (in the simplified set-up with one trace and potential operator with respect to @X , otherwise, in the elliptic theory, with an arbitrary number of components). The generalisation of Definition 8.4.13 is that we replace (8.4.46) by operators of analogous structure, where in the condition Lreg 2 B ;d .X n Z/ the B ;d set-up now refers both to the boundary components int Y˙ and @X ; moreover, the smoothing operators G now refer to global mapping properties in spaces with an extra component on @X . The definition is straightforward after the requirements for operators in C1;d .X; gI j ; jC / in Section 8.4.5. We thus obtain altogether an operator space C;d .X; gI j ; jC / with a principal symbolic hierarchy  .A/ D . .A/; @ .A/; ^ .A/; c .A/; @X .A//; where the first 4 components are as in Section 8.4.5, and @X .A/ is the boundary symbol referring to @X . The composition properties of operators are similar as before. Moreover, we have a straightforward extension of the notion of ellipticity, as well as of Theorem 8.4.21 and Corollary 8.4.22.

538

8 Corner operators and problems with singular interfaces

Remark 8.4.26. The results of Section 8.4.6 easily extend to the case of a (compact) C 1 manifold X with boundary. Remark 8.4.27. The results of Section 8.4.5 have a straightforward extension to the case of systems of operators in the upper left corner, including the case of @X 6D ;.

8.5 Mixed problems with singular interfaces We return once again to the technique of Section 4.2 to study mixed elliptic problems, here from the point of view of singular interfaces. The configuration in Section 4.2 is interpreted as the base of a cone. In ‘direction’ of the cone base we take standard Sobolev spaces and in the axial direction weighted spaces from the cone calculus. In the present section we mainly consider the conormal symbolic structure. Therefore, it is convenient to transform the cone to an infinite cylinder, where we imitate the approach of Section 4.2.

8.5.1 Mixed problems in an infinite cylinder In contrast to the notation in the introduction we now slightly change the context and consider mixed elliptic problems in an infinite cylinder. x be the closure of a smooth bounded domain in Rm and let M WD 2N Let N D  denote the double (obtained by gluing together two copies N˙ of N along the common boundary to a closed compact C 1 manifold; we then identify N with NC ). Let H s .R M / denote the cylindrical Sobolev space on RM of smoothness s 2 R. Let us briefly l recall the definition. The space L cl .M I R / of parameter-dependent (with parameter l  2 R ) pseudo-differential operators on M of order  contains an element R ./ which is parameter-dependent elliptic and induces isomorphisms R ./ W H s .M / ! H s .M / for all s 2 R;  2 Rl . Then H s .R  M / is defined as the closure of ˚R

12 2 where C01 .R; C 1 .M // with respect to the norm R jjRs . /.F u/. /jjL 2 .M / d Rs . / 2 Lscl .M I R / is an order reducing element as mentioned before, and F is the Fourier transform in t 2 R, cf. Definition 2.4.14. The space L2 .M / refers to a fixed Riemannian metric on M . Moreover, let H s .R  int N / WD fujRint N W u 2 H s .R  2N /g: In order to investigate conormal symbols in a corner situation we now study mixed elliptic problems in an infinite cylinder. Let X be a compact C 1 manifold with boundary Y D @X , and assume Z to be a C 1 submanifold of Y of codimension 1, Y˙ compact C 1 manifolds with common boundary Z, Y [ YC D Y , Y \ YC D Z. According to the above definition we have the cylindrical spaces H s .R  int X/; H s .R  int Y˙ /; H s .R  Z/. Let us set H s;ı .R  int X / WD e tı H s .R  int X /; H s;ı .R  int Y˙ / WD e tı H s .R  int Y˙ /:

8.5 Mixed problems with singular interfaces

Let

X

AD

539

˛ a˛ .t; x/D t;x

j˛j2

be an elliptic differential operator of second order with coefficients a˛ 2 C 1 .R  X /. Moreover, let T˙ WD r˙ B˙ be boundary operators with r˙ being the restriction to R  int Y˙ and X ˇ B˙ D bˇ;˙ .t; x/D t;x jˇ j˙

differential operators with coefficients bˇ;˙ 2 C 1 .R  U˙ /, where U˙ are open neighbourhoods of Y˙ . We assume that the boundary operators T˙ are elliptic on Y˙ with respect to A, i.e., satisfy the Shapiro–Lopatinskij condition uniformly up to Z from the respective ˙-sides. Under suitable assumptions on the coefficients for jtj ! 1 we then obtain continuous operators H s2;ı2 .R  int X / ˚ A 1 A D @ T A W H s;ı .R  int X / ! H s  2 ;ı .R  int Y / TC ˚ sC  1 ;ı C .R  int Y / 2 H C 0

1

(8.5.1)

˚ for every fixed choice of ı and for all s 2 R; s > max ˙ C 12 . Let us compare (8.5.1) with a mixed boundary value problem on the infinite stretched cone .int X /^ D RC  int X 3 .r; x/ that we obtain by substituting the diffeomorphism R ! RC ; t ! e t D r. To this end we introduce the space H s; .M ^ /; s;  2 R; M ^ WD RC  M 3 .r; x/; for a closed compact C 1 manifold M of dimension d as the completion of the space C01 .RC ; C 1 .M // with respect to the norm 

1 2 i

12

Z kR  d C1 2

s

2 .Im w/Mr!w .w/kL 2 .M / d w

;



where Mr!w is the Mellin transform on u.r/ 2 C01 .RC ; C 1 .M // (which is holomorphic in w), ˇ WD fw 2 C W Re w D ˇg, and Rs . / 2 Lscl .M I R / is an order reducing family of order s. If X is a compact C 1 manifold with C 1 boundary @X we define H s; ..int X /^ / WD fuj.int X/^ W u 2 H s; ..2X /^ /g; where 2X is the double of X, obtained by gluing together two copies X˙ along the common boundary @X , with XC being identified with X . Then v.t; x/ ! u.r; x/, defined by u.e t ; x/ D v.t; x/, induces an isomorphism H s;ı .R  X / ! H s; .X ^ / for  D ı C d C1 ; d D dim X . 2

540

8 Corner operators and problems with singular interfaces

The operator A then takes the form A WD r 2

2 X

aj .r/.r@r /j ;

j D0

i.e., is a Fuchs type differential operator on the infinite stretched cone .int X /^ D x C  Diff 2j .X //. Moreover, the boundary RC  int X , with coefficients aj 2 C 1 .R operators are transformed into T ˙ WD r˙ B ˙

for B ˙ WD r ˙

˙ X

bk;˙ .r/.r@r /k

kD0

x C ; Diff ˙ k .U˙ // for U˙ as above. Assuming the with coefficients bk;˙ 2 C 1 .R coefficients aj and bk;˙ to be independent of r for large r, we obtain continuous operators H s2; 2 ..int X /^ / ˚ A s; ^ s  1 ;  1 @ A 2 2 ..int Y /^ / A WD T  W H ..int X / / ! H ˚ TC 1 1 H sC  2 ;C  2 ..int YC /^ / 0

1

(8.5.2)

˚

for all s 2 R; s > max ˙ C 12 . Here and in future ı is fixed and chosen below in a suitable way. The operator (8.5.2) represents a mixed boundary value problem in a cone .int X /^ with a subdivision of @X ^ into Y˙^ , where now the interface Z ^ (written in stretched form) has conical singularities (in this description at r D 0). According to the calculus of operators on a manifold with conical points the operator (8.5.2) has a conormal symbol, defined as the operator family H s2 .int X / ˚ c .A/ s s  1 @ A 2 .int Y / H .T /  c .A/.w/ WD .w/ W H .int X / ! c  c .T C / ˚ 1 H sC  2 .int YC / 0

1

holomorphic in w 2 C , where c .A/.w/ D

2 X j D0

aj .0/w ; j

c .T ˙ /.w/ D r˙

˙ X kD0

bk;˙ .0/wk :

8.5 Mixed problems with singular interfaces

541

8.5.2 Reduction to the boundary Let us consider another boundary operator T D rB

for B D r



 X

bk .r/.r@r /k

kD0

x C  Diff k .U // for a neighbourhood U of Y . Let us with coefficients bk 2 C 1 .R assume, as above, that aj ; bk are independent of r for large r. Then the operator  P2  H s2; ..int X /^ / aj .r/.r@r /j s; ^ j D0 ˚ D WD P W H ..int X / / ! r kD0 bk .r/.r@r /k s 1 2 ; .Y ^ / H (8.5.3) for all ; s 2 R; s >  C 12 , represents a boundary value problem on X ^ . Assume that T satisfies the Shapiro–Lopatinskij condition with respect to A (in the Fuchs type sense, cf. [100]). Then the conormal symbol   P2 a .0/wj c .D/.w/ WD PjD0 j (8.5.4) r kD0 bk .0/wk is a holomorphic (operator-valued) function in w 2 C and defines a parameterdependent elliptic family of boundary value problems on X with the parameter D Im w. There is then a countable set D  C with finite intersection D \ fw 2 C W c  Re w  c 0 g for every c  c 0 such that the operators (8.5.4) define isomorphisms H s2 .int X / ˚ c .D/.w/ W H s .int X / ! s 1 2 .Y / H for all w 2 C n D and all sufficiently large s 2 R. Since the main purpose of our investigation is to determine admissible corner weights of mixed problems, from now on for simplicity we assume the coefficients aj and bk to be independent of r. We then have  d D D opM 2 .c .D//; and D defines isomorphisms (8.5.3) for all  2 R such that  d C1  \ D D ; and all 2 sufficiently large s 2 R. Here, ˇ WD fw 2 C W w D ˇ C i ; 2 Rg. We have Z 1  w Z r 1 dr 0  d c .D/.w/u.r 0 / 0 d w opM 2 .c .D// D 0 2 i  d C1 r r 0 2 

Z Z   1  d C1 C 1 1  r i d C1 dr 0 0 d C1 2  u.r 0 / D r 2   C i

 .D/ .r / d : c 2 r0 2 r0 1 0

542

8 Corner operators and problems with singular interfaces

As noted before the transformation u.r; x/ ! u.e t ; x/ induces an isomorphism d C1 H s; 2 .X ^ / ! H s .R  int X / for all s 2 R. Hence it follows an operator D WD op .d/ D F ı

1

d.w/F W H

s;ı

H s2;ı .R  int X / ˚ .R  int X / ! ; (8.5.5) s 1 ;ı 2 H .R  Y /

where d.w/ WD t .e.w/ t.w// D c .D/.w/; w D iw, that is an isomorphism for all ı 2 R such that Iı \ D D ;. Here Iˇ WD fw 2 C W Im w D ˇg; ˇ 2 R, and D D fw 2 C W iw 2 Dg. l If M is a compact C 1 manifold by L cl .M I R / we denote the space of all classical pseudo-differential operators of order  2 R on M depending on a parameter  2 R. Moreover, if F is a Fréchet space and U C an open set by A.U; F / we denote the space of all holomorphic functions in U with values in F . We now employ the fact that for every constants c  c 0 there exists a holomorphic s 1

s 1 2

2 operator function r.w/ 2 A.C; Lcl .Y // such that r. C iˇ/ 2 Lcl for every ˇ 2 R, uniformly in compact ˇ-intervals, and

.Y I R /

1

r. C iˇ/ W H s 2 .Y / ! L2 .Y / is a family of isomorphisms for all 2 R and all c  ˇ  c 0 (s 2 R is now fixed). Then, in particular, we obtain an isomorphism 1

opı .r/ W H s 2 ;ı .R  Y / ! H 0;ı .R  Y /; ı 2 R. We will choose r.w/ as follows. Let ˛ 2 R (which plays the role of s    12 /, and fix a collar neighbourhood Š Œ1; 1  Z of the interface Z  Y . Choose local coordinates .n; z/ 2 Œ1; 1  U for an open set U  Rd 2 with covariables .; / 2 Rd 1 , and form a symbol of the following kind: ˛!.n/     ˛ p h; i  i  .n; ; ; / WD f h; ; i˛.1!.n// : (8.5.6) C h; i Here ! 2 C01 .1; 1/ is a real-valued function, 0  !  1, that is equal to 1 in a neighbourhood of the origin,  2 Rl , and f ./ 2 S.R/ is a function such that f .0/ D 1 and supp F 1 f  R (with the Fourier transform on the n-axis). We 1Cl ˛ ˛ then have p .n; ; ; / 2 Scl˛ .R  Rd; ; /, and p is elliptic with respect to the covariable .; ; / when C > 0 is chosen sufficiently large. On Y we now define ˛ a parameter-dependent elliptic operator p˛ ./ 2 L˛cl .Y I Rl / taking p .n; ; ; / as local amplitude functions in the collar neighbourhood of Z and h; i˛ outside that neighbourhood, with  being the covariable on Y . The precise (standard) construction in terms of an open covering of Y by charts, a subordinate partition unity, etc., may be ˛ found in [71]. In a similar manner, starting from pC .n; ; ; /, defined as the complex ˛ conjugate of (8.5.6), we obtain a family pC ./ 2 L˛cl .Y I Rl /.

8.5 Mixed problems with singular interfaces

543

By virtue of the specific properties of the symbol (8.5.6) in a neighbourhood of Z we have the following results. s s Let eC W H s .int YC / ! H s .Y / denote a continuous operator such that rC eC D 1 s min .s;0/ .Y /, the idH s .int YC / . Moreover, for s >  2 we consider eC W H .int YC / ! H operator of extension by 0 to the opposite side of Y . Theorem 8.5.1. There is a constant M > 0 such that the operators p˛ ./ W H s .Y / ! H s˛ .Y / and s rC p˛ ./eC W H s .int YC / ! H s˛ .int YC /

are isomorphisms for all  2 Rl ; jj  M . For s >  12 also rC p˛ ./eC W H s .int YC / ! H s˛ .int YC / is a family of isomorphisms for all  2 Rl , jj  M . We then have .rC p˛ ./eC /1 D rC .p˛ /1 ./eC . An analogous result holds for pC ./ when we interchange C and  signs. Let L˛cl .Y I C  Rl / denote the space of all h.w; / 2 A.C; L˛cl .Y I Rl // such that / h. C iˇ; / 2 L˛cl .Y I R1Cl ; for all ˇ 2 R, uniformly in compact ˇ-intervals. Let us now replace the parameter  by . ; / 2 R1Cl and consider the corresponding families p˛˙ . ; /. Choose a .b/ 2 C01 .R/ such that .b/ 1 in a neighbourhood of b D 0. Then, setting Z Z ˛ iwb i b ˛ r˙ .w; / WD e .b/e p˙ . ; /μ db (8.5.7) R

R

we obtain an operator function in L˛cl .Y I C  Rl /. Theorem 8.5.2. For every constants c  c 0 there exists an M > 0 such that r˛˙ .w; / W H s .Y / ! H s˛ .Y / and s rC r˛ .w; /eC W H s .int YC / ! H s˛ .int YC /; ˛ s r rC .w; /e W H s .int Y / ! H s˛ .int Y /

are isomorphisms for all c  Im w  c 0 and all  2 Rl ; jj  M . For s >  12 also rC r˛ .w; /eC W H s .int YC / ! H s˛ .int YC /; r r˛C .w; /e W H s .int Y / ! H s˛ .int Y / are families of isomorphisms for those w and .

544

8 Corner operators and problems with singular interfaces

The proof follows directly from the results of Section 4.1. s In the following we also use the notation e˙ and e˙ for the corresponding extension operators H s;ı .R  .int Y˙ // ! H s;ı .R  Y /; s 2 R, and H s;ı .R  .int Y˙ // ! H min .s;0/;ı .R  Y /; s 2 R; s >  12 , respectively, for any ı 2 R. Corollary 8.5.3. Let r .w; / denote the operator function of Theorem 8.5.2. Then opı .r˛ /./ W H s;ı .R  Y / ! H s˛;ı .R  Y /; s W H s;ı .R  int YC / ! H s˛;ı .R  int YC / rC opı .r˛ /./eC

for ı; s 2 R and rC opı .r˛ /./eC W H s;ı .R  int YC / ! H s˛;ı .R  int YC / for ı; s 2 R; s >  12 , are isomorphisms for all jj  M for a suitable M > 0. Analogous relations hold when we interchange C and  signs. We now fix a 1 2 Rl ; j1 j > M , and set r˛˙ .w/ WD r˛˙ .w; 1 /. It is known that there is a meromorphic inverse .r˛˙ /1 .w/, and we then have opı ..r˛˙ /1 / D s s .opı .r˛˙ //1 . Similarly, the operators rC opı .r˛ /eC , rC opı .r˛ /eC and r opı .r˛C /e , ı ˛ r op .rC /e can be inverted. From the operator (8.5.5) we now pass to a reduction of orders to 0 on the boundary. As above we write d.w/ WD t .e.w/ t.w// and form  ı

diag.1; op

.r˛C //opı .d/

D op

ı

e r˛C t

 WH

s;ı

H s2;ı .R  int X / ˚ .R  int X / ! L2;ı .R  Y /

where ˛ D s  12 . The order reduction with the C operator is taken for convenience; we could take any other order reduction as well. With the restriction operators r˙ to R  int Y˙ and the extensions e˙ by zero to R  int Y we have an isomorphism 

e eC



L2;ı .R  Y / ˚ W ! L2;ı .R  Y / L2;ı .R  YC /

with the inverse t .r rC /. Similarly as in the calculus of pseudo-differential boundary value problems the operator function d.w/ has a meromorphic inverse d1 .w/ DW .g.w/ k.w//. Remark 8.5.4. It is known that the Laurent coefficients of d1 are smoothing operators of finite rank, more precisely, smoothing in the calculus of boundary value problems on X with the transmission property at Y . Let us now form the operator L WD .opı .g/ opı .k.r˛C /1 /e opı .k.r˛C /1 /eC / H s2;ı .R  int X / ˚ L W L2;ı .R  Y / ! H s;ı .R  int X / ˚ L2;ı .R  YC /

8.5 Mixed problems with singular interfaces

545

which is an isomorphism (recall that ˛ D s    12 ). Multiplying L from the left by A, cf. the formula (8.5.1), we obtain the operator H s2;ı .R  int X / H s2;ı .R  int X / ˚ ˚ 1 AL W ! H s  2 ;ı .R  int Y / : L2;ı .R  Y / ˚ ˚ 1 2;ı s  ;ı C L .R  YC / 2 .R  int YC / H

(8.5.8)

By virtue of DD 1 D diag.1; 1/ we obtain the operator AL in the form 1 0 1 0 0 AL D @ T G T KR1 e T KR1 eC A TC G TC KR1 e TC KR1 eC where we employ the abbreviation G WD opı .g/, K WD opı .k/, R WD opı .r˛C /, ˛ D s    12 . We also want to reduce the spaces on R  int Y on the right of (8.5.8) to zero. To ˛ this end we take the elements r˙ .w/; ˛ D s    12 . Set R WD r opı .r˛C /e ; RC WD rC opı .r˛C /eC for s > max fC ;  g. Setting R WD diag.1; R ; RC /, and multiplying (8.5.8) from the left by R we get an operator 1 0 1 0 0 A0 WD RAL D @ R T G R T KR1 e R T KR1 eC A RC TC G RC TC KR1 e RC TC KR1 eC with the 2  2 lower right corner 



L2;ı .R  Y / L2;ı .R  Y / ˚ ˚ W ! : L2;ı .R  YC / L2;ı .R  YC / (8.5.9) The latter operator represents the reduction of our mixed problem to the boundary, combined with suitable reductions of orders. R T KR1 e RC TC KR1 e

R T KR1 eC RC TC KR1 eC

8.5.3 Ellipticity with interface conditions We assume that the boundary condition T is the restriction of T to int Y , that means,  D  , or ˛ D ˛ . In that case, since the order reducing operators R and R are connected by the relation R D r Re , we obtain R T KR1 e D idL2;ı .RY /

(8.5.10)

546 and

8 Corner operators and problems with singular interfaces

R T KR1 eC D 0:

(8.5.11)

In fact, from T D rB for a differential operator B in a neighbourhood of Y it follows that T D r rB which implies that rBK D 1 and R T KR1 e D r Re r R1 e , i.e., we obtain (8.5.10). Also, (8.5.11) is equal to r Re r R1 eC which vanishes because of r R1 eC D r opı .r˛ C /eC D 0. Thus the operator (8.5.9) is a triangular matrix with the lower right corner F WD RC TC KR1 eC W L2;ı .R  YC / ! L2;ı .R  YC /:

(8.5.12)

The operator (8.5.12) can be written in the form F D opı .f/ for a meromorphic operator family 2 2  f.w/ D rC r˛C .w/eC tC .w/k.w/r˛ C .w/eC W L .YC / ! L .YC /:

The operators f.w/ are parameter-dependent elliptic of order zero, with the parameter Re w D 2 R. The homogeneous principal boundary symbol @ .f/.z; ; / is a family of continuous operators @ .f/.z; ; / W L2 .RC / ! L2 .RC /;

(8.5.13)

independent of ı, and @ .f/.z;  ; / D @ .f/.z; ; / for each  2 RC ; . ; / 6D 0. By construction the operator family f .w/ depends on s 2 R. We now assume s 2 R to be chosen in such a way that (8.5.13) is a family of Fredholm operators for all . ; / 6D 0. The criterion for that is that the subordinate conormal symbol has no zeros on the line  1 . In the case of the Fredholm property we have a K-theoretic 2 index element indS  Z @ .f/ 2 K.S  Z/; here S  Z is defined as the compact space f.z; ; / 2 R  T  Z W j ; j D 1g with the canonical projection 1 W S  Z ! Z. Another condition to be imposed is indS  Z @ .f/ 2 1 K.Z/: There is a block matrix family of isomorphisms 0 2 1 0 2 1   L .RC / L .RC / @ .f/.z; ; / @ .k/.z; ; / W 1 @ ˚ A ! 1 @ ˚ A @ .t/.z; ; / @ .q/.z; ; / J JC for suitable J˙ 2 Vect.Z/ between the corresponding pull backs with respect to 1 . We now choose a system of charts j W Uj ! Rd 2 on Z for an open covering .Uj /j D1;:::;N of Z. Let .'j /j D1;:::;N be a subordinate partition of unity and . j /j D1;:::;N a system of functions j 2 C01 .Uj / such that j 1 on supp 'j for all j . Moreover, let ; Q 2 C 1 .YC / be supported in a collar neighbourhood of Z; Q 1 in a neighbourhood of supp  , and  1 in a neighbourhood of Z. We then define the operator family     N  X .j  id/1 Q j 'j .j  id/ 0 0 Opz .gj /. / (8.5.14) 0 'j j 0 .j /1 j j D1

8.5 Mixed problems with singular interfaces



0 @ .k/ @ .t/ @ .q/

547



.z; ; / in local coordinates with x C on Z respect to the charts j W Uj ! Rd 2 and j  id W Uj  Œ0; 1/ ! Rd 2  R and in a collar neighbourhood of Z with the normal variable in Œ0; 1/. Now (8.5.14) is a block matrix family of operators where gj .z; ; / is given by . ; /



0 g. / WD g21 Moreover,

 L2 .YC / L2 .YC / g12 ˚ ˚ . / W ! : g22 L2 .Z; J / L2 .Z; JC / 

 f. C i ı/ g12 . / g22 . / g21 . /

(8.5.15)

is a family of Fredholm operators which defines isomorphisms for j j > C for some constant C > 0. Similarly as (8.5.7) we now pass to a holomorphic operator function Z nZ o iwb g.w/ WD .b/e i b g. /μ db e R

C01 .R/ that is equal to 1 near the origin (clearly, the integrals may be carried

for a 2 out for the entries separately). We then have g.w/ D .gij .w//i;j D1;2 with g11 .w/ D 0. This gives us a family of operators 

 L2 .YC / L2 .YC / f.w/ g12 .w/ ˚ ˚ ! W g21 .w/ g22 .w/ L2 .Z; J / L2 .Z; JC /

(8.5.16)

which is meromorphic in w 2 C. Proposition 8.5.5. There is a discrete set M  R such that (8.5.16) is a family of isomorphisms for all w D C i ı; 2 R; ı 2 R n M . Proof. The family (8.5.16) is parameter-dependent elliptic in the class of boundary value problems on YC (of order zero and without the transmission property at Z), cf. [44], [155], with parameter D Re w. The meromorphy is clear by construction; gij .w/ are even holomorphic for all i; j . Let us assume for the moment that also f.w/ is holomorphic in the complex plane. The operators (8.5.15) are parameter-dependent elliptic, and the principal parameter-dependent interior and boundary symbols are independent of ı. The same is true of (8.5.16), i.e., (8.5.16) is Fredholm for every w 2 C and a holomorphic operator function. Moreover, there is a constant c > 0 such that (8.5.15) are isomorphisms for all j j > c. Thus our operator function satisfies a wellknown condition on holomorphic Fredholm families which are isomorphic for at least one value of the complex parameter. This gives us the invertibility for all w with Im w outside some discrete set. In the case that f.w/ is meromorphic we can argue in a similar manner when we take into account that the Laurent coefficients are smoothing and of finite rank, cf. Remark 8.5.4.

548

8 Corner operators and problems with singular interfaces

We have (8.5.16) as a meromorphic operator function which is invertible for all w 2 C n N for some discrete set N  C such that N \ fc  Im w  c 0 g is finite for every c  c 0 . Our next objective is to pass from the symbol a.w/ W H s .int X / ! z s2 .int X/ for H z s2 .int X / WD H s2 .int X / ˚ H s  12 .int Y / ˚ H sC  12 .int YC / H to an operator function a.w/ Q by adding extra entries of trace and potential type such z s2 .int X / ˚ L2 .Z; JC / are meromorphic that a.w/ Q W H s .int X / ˚ L2 .Z; J / ! H and invertible in such a sense. To this end we form the block matrix operator family L2 .Y / L2 .Y / ˚ ˚ 1 0 0 @m.w/ f.w/ g12 .w/A W L2 .YC / ! L2 .YC / 0 g21 .w/ g22 .w/ ˚ ˚ L2 .Z; J / L2 .Z; JC / 1

0

(8.5.17)

 for the meromorphic operator function m.w/ WD rC r˛C .w/eC tC .w/k.w/r˛ C .w/e which has the property opı .m/ D RC TC KR1 e :

˛

Moreover, for n˙ .w/ WD r˙ r˙ .w/e˙ t˙ .w/g.w/ we have opı .n˙ / D R˙ T˙ G. Setting 0

1

1 0 0 0 B n .w/ 1 0 0 C C aQ 0 .w/ D B @nC .w/ m.w/ f.w/ g12 .w/A W 0 0 g21 .w/ g22 .w/

H s2 .int X / H s2 .int X / ˚ ˚ L2 .Y / L2 .Y / ˚ ˚ ! L2 .YC / L2 .YC / ˚ ˚ L2 .Z; J / L2 .Z; JC /

z0 D opı .aQ 0 / that has A0 as the upper left corner. gives us an operator A Setting l.w/ WD diag.g.w/; k.w/.r˛C .w//1 e ; k.w/.r˛ .w//1 eC / and r.w/ WD diag.1; r r˛C .w/e ; rC r˛C .w/eC / we have L D opı .l/ and R D opı .r/. Moreover, let Q l.w/ WD diag.l.w/; idL2 .Z;J / / and rQ .w/ D diag.r.w/; idL2 .Z;JC / /: We then obtain an operator function H s2 .int X / ˚ s s  1 2 .int Y / H H .int X / 1 1 Q ˚ ˚ a.w/ Q WD rQ .w/aQ 0 .w/l .w/ W ! : (8.5.18) 1 L2 .Z; J / H sC  2 .int YC / ˚ L2 .Z; JC /

8.5 Mixed problems with singular interfaces

549

  kZ .w/ where a.w/ is the symbol of the Remark 8.5.6. We have a.w/ Q D ta.w/ .w/ q .w/ Z Z original mixed problem (8.5.1). The other entries play the role of trace, potential, etc., symbols with respect to the interface Z. Theorem 8.5.7. There is a discrete set N  C; N \ fc  Im w  c 0 g finite for every c  c 0 , such that a.w/ Q is a family of isomorphisms for all w 2 C n N . Proof. From Proposition 8.5.5 we have an operator function of the asserted kind. Thus (8.5.17) as well as aQ 0 .w/ also have this property. Finally, a.w/ Q itself is as desired, since the factors at aQ 0 .w/ on the left-hand side of (8.5.18) preserve this structure. Corollary 8.5.8. The operator H s2 .R  int X / ˚ 1 H s  2 .R  int Y / H s .R  int X / z WD opı .a/ ˚ ˚ A Q W ! 1 2 s  C L .R  Z; J / 2 .R  int YC / H ˚ L2 .R  Z; JC / is an isomorphism for all ı 2 R such that Iı \ N D ;. Let us consider the Zaremba problem H s2;ı2 .R  int X / ˚ 1 A D @ T A W H s;ı .R  int X / ! H s 2 ;ı .R  int Y / TC ˚ s 3 ;ı1 H 2 .R  int YC / 0

1

(8.5.19)

on a cylinder RX for the Laplace operator with Dirichlet and Neumann conditions on Y and YC , respectively, where X WD fx D .x1 ; x2 / 2 R2 W jxj  1g and R2 identified with the complex plane, Y WD fx D e i W 0    ˛g; YC WD fx D e i W ˛    2g for some 0 < ˛ < 2. We have v D e 2t f@2t v  @ t v C X vg;

(8.5.20)

T v D v.t; e /j0˛ ; TC v D  i

1

i

@ v.t; e /j˛2 ;

v.t; x/ 2 H s;ı .R  int X /, where  is the exterior normal direction to Y . We have continuous operators (8.5.19) for every fixed ı and all s 2 R; s > 32 . Substituting the diffeomorphism H s;ı .R  X / ! H s; .X ^ /;

R ! RC ;

e t ! r;

550

8 Corner operators and problems with singular interfaces

 D ı C 32 , the operators in (8.5.20) take the form

u D r 2 f.r@r /2 u C r@r u C X ug; T  u D u.r; e i /j0˛ ; T C u D 1 @ u.r; e i /j˛2 ; and we get the continuous operators H s2; 2 ..int X /^ / ˚

s; ^ s 1 ; @ A 2 ..int Y /^ / ; A D T  W H ..int X / / ! H TC ˚ s 3 ; 2 H ..int YC /^ / 0

1

v.t; x/ D u.e t ; x/, for every fixed  and for all s 2 R; s > conormal symbols have the form

3 . 2

The corresponding

c . /.w/u D w2 u  wu C X u; c .T  /u D u.e i' /j0'˛ ; c .T C /u D 1 @ uj˛'2 ; u 2 H s .X/. Let us take as another boundary operator ru WD T u D u.r; e i / which represents the Dirichlet condition on Y . Then we have 

.r@r /2 C r@r C X DD r for all s;  2 R; s > 12 , and

 WH

s;

H s2; 2 ..int X /^ / ˚ ..int X / / ! 1 1 H s 2 ; 2 .Y ^ / ^

 w2 u  wu C X u c .D/.w/u D ; ru 

1 u 2 H s .X/. We have D D opM .c .D// and D D opı .d/ for d.w/ D t .e.w/ t.w//, 2 where e.w/ D w iwC X ; t.w/ D r. According to [102, Section 11.1] the symbol d.w/ defines isomorphisms

H s2 .int X / ˚ H .int X / ! 1 H s 2 .Y / s

for all w D C i ı; ı 2 Œ1; 0 . Let us fix such a ı. In this case we have ˛ D ˛ D s  12 ; ˛C D s  32 . As order reducing operators we take s 32     3 2 .w/ D f rs h i  i  ;  C h i

sC 1 2

rC

    sC 12 h i  i  .w/ D f ; C h i

551

8.5 Mixed problems with singular interfaces

w D C iı. The corresponding family (8.5.13) is a family of Fredholm operators for all . ; / 6D 0 if s … Z C 12 , cf. Proposition 4.2.32. In our example we have @ .f /. / D r C op.b/. /eC

(8.5.21)

 s 3    sC 12   . for b.; / D f Cj j j j  i  2 j; j f Cj j j j  i  According to Proposition 4.2.34 the operator (8.5.21) is (i) bijective for

1 2

(ii) surjective for (iii) injective for

1 2

< s < 32 , 1 2

< s C j < 32 ; j 2 N, where dim ker @ .f /. / D j ,

< s C j < 32 ; j 2 N, where dim coker @ .f /. / D j . L2 .RC /

˚ Then there is a family of isomorphisms .@ .f/. / @ .k/. // W ! L2 .RC / Z  C j  for j WD s  12 .

Remark 8.5.9. In problems of Zaremba type, given as meromorphic families of conormal symbols there is also a parameter-dependent variant, i.e., where w is replaced by .w; / and .Re w; / 2 R1Cl is the parameter. The construction of extra conditions (here of potential type) is possible also in -dependent form. Then for every weight ı there is a  such that a.w; Q /, the parameter-dependent version of a.w/, Q is a family of isomorphisms (8.5.18) for all Re w D ı, provided that jj is chosen sufficiently large.

Chapter 9

Operators in infinite cylinders and the relative index

The structure of operator-valued meromorphic Mellin symbols has an influence to the relative index of associated operators in weighted Sobolev spaces. In this chapter we develop elements of the theory of meromorphic Fredholm functions from the work of Gohberg and Sigal [60], specialised to the case of functions with values in boundary value problems or in the cone calculus. In addition we discuss the concept of cutting and pasting elliptic operators, following the lines of Nazaikinskij and Sternin [140], see Section 9.4. Other material of this chapter is based on the author’s joint papers [73], see Section 9.2, and [75], see Section 9.3.

9.1 Calculus with operator-valued meromorphic families We investigate meromorphic operator functions in the complex plane from the point of view of so-called partial null and polar indices.

9.1.1 Characteristic values and factorisation of meromorphic families In this chapter the complex plane is denoted by C (in contrast to C before), since we now take the parameter-dependence of operators parallel to the real axis (rather than the imaginary axis). Let X be a compact closed C 1 manifold and L cl .X I E; F / for E; F 2 Vect.X / the space of all classical pseudo-differential operators of order  on X . Denote by  M O .XI v/, v WD .E; F /, the space of all operator families f .w/ 2 A.C ; Lcl .X I E; F // such that f . C i ı/ 2 L cl .XI E; F I R /; w D  C iı, for every ı 2 R, uniformly in compact ı-intervals. Let R D f.pj ; mj ; Lj /gj 2Z

(9.1.1)

be a sequence for pj 2 C ; j Im pj j ! 1 as jj j ! 1, mj 2 N, and Lj  L1 .X / finite-dimensional subspaces of operators of finite rank. Set C R WD fpj gj 2Z . Theorem 9.1.1. For every g./ 2 L cl .X I E; F I R / there exists an operator family f .w/ 2 M  .XI v/ such that f ./  g./ 2 L1 .X I E; F I R /. O Proof. The result follows by a kernel cut-off argument, similarly as Theorem 6.1.30.

9.1 Calculus with operator-valued meromorphic families

553

Remark 9.1.2. If g./ is parameter-dependent elliptic, then f , associated with g via Theorem 9.1.1, is parameter-dependent elliptic (as a parameter-dependent pseudo-differential operator on X of order ) for every ı 2 R (the principal symbol  .f . C i ı// is independent of ı 2 R). Then we say that f is elliptic. Definition 9.1.3. The space M 1 R .X I v/, v D .E; F /, with an asymptotic type R of the form (9.1.1), is defined to be the set of all f .w/ 2 A.C n C R; L1 .X I E; F // such that f is meromorphic with poles at the points pj of multiplicity mj C 1 and Laurent coefficients at .w  pj /.kC1/ belonging to Lj for all 0  k  mj ; j 2 Z, such that . C i ı/f . C i ı/ 2 L1 .X I E; F I R / for every ı 2 R, uniformly in compact ı-intervals, for any C R-excision function  (i.e.,  2 C 1 .C /,  0 for dist.w; C R/ < c 0 ,  1 for dist.w; C R/ > c 00 for certain 0 < c 0 < c 00 ). The space M 1 R .X I v/ is Fréchet in a natural way. We define  1 M R .X I v/ WD M O .X I v/ C M R .X I ; v/

(9.1.2)

in the sense of a non-direct sum of Fréchet spaces.  Theorem 9.1.4. f .w/ 2 M  R .X I v/ with v D .J; F / and g.w/ 2 M S .X I w/ with w D .E; J / implies f .w/g.w/ 2 M PC .X I v ı w/

for v ı w D .E; F / and some resulting asymptotic type P . Proof. Cf. [182, Section 2.1, Proposition 6]. Remark 9.1.5. Let us write f .w/ 2 M  R .X I v/ in the form f .w/ D fo .w/ C fr .w/

(9.1.3)

1 for fo .w/ 2 M  O .X I v/ and fr .w/ 2 M R .X I v/. Then we say that f is elliptic if so is fo . This definition is correct, i.e., independent of the particular choice of the decomposition (9.1.3).

Let us set

S.c 0 ;c 00 / WD fw 2 C W c 0 < Im w < c 00 g:

(9.1.4)

Theorem 9.1.6. Let f .w/ 2 M  R .X I v/; v D .E; F /, be elliptic. Then there exists a discrete set D  C with finite intersections D \ S.c 0 ;c 00 / for every c 0  c 00 such that f .w/ W H s .X; E/ ! H s .X; F / 1 1 is invertible for all w 62 D, with the inverse f 1 .w/ 2 M  D .F; E/, S .X I v /, v for a resulting asymptotic type S.

554

9 Operators in infinite cylinders and the relative index

Proof. Write f in the form (9.1.3). The ellipticity of fo yields that there is an 1 1 fo1 .w/ 2 M  Q .X I v / for an asymptotic type Q such that fo .w/fo .w/ D fo .w/fo1 .w/ D 1 for all w 2 C (cf. [188, Section 1.2.4]). By Theorem 9.1.4 we have fo1 .w/fr .w/ 2 M P1 .X I w/; w D .E; E/, for some resulting asymptotic type P . Hence (cf. [174, Lemma 4.3.13]) there exists a discrete set D  C such that 1 C fo1 .w/fr .w/ is invertible for all w 62 D, and its inverse is of the form 1 C g.w/ with g.w/ 2 M 1 T .X I w/ for a certain asymptotic type T . Then it is easy to see that f 1 .w/ D .1 C g.w//fo1 .w/. The operator f 1 .w/ is called the resolvent of f .w/ and is again elliptic. Definition 9.1.7. A point w0 2 C is called a characteristic value of f .w/ if there exists a holomorphic function u.w/ in a neighbourhood of w0 with values in H s .X; E/ such that u.w0 / 6D 0, f .w/u.w/ is holomorphic near w0 and f .w/u.w/jwDw0 D 0. The function u is called a root function of f .w/ at w0 . The order of w0 as a zero of f .w/u.w/ is called the multiplicity of u.w/, and u0 WD u.w0 / an eigenvector of f .w/ at w0 . The eigenvectors of f .w/ at w0 (together with the zero element) form a vector space. This space is called the kernel of f .w/ at w0 and denoted by ker f .w0 /. The supremum of the multiplicities of all root functions u.w/ such that u.w0 / D u0 is called the rank of the eigenvector u0 . Remark 9.1.8. Definition 9.1.7 and the subsequent notation make sense in analogous z / of form for meromorphic operator-valued functions with values in the space L.H; H z linear operators between Hilbert spaces H and H . Proposition 9.1.9. Let f .w/ 2 M  R .X I v/; v D .E; F /, be elliptic (in the sense of Remark 9.1.5), and let w0 be a characteristic value of f .w/. Then (i) ker f .w0 / is a finite-dimensional subspace of C 1 .X; E/; (ii) the rank of each eigenvector of f .w/ at w0 is finite. Proof. (i) By assumption we have f .w/ D

1 X

fj .w  w0 /j C h.w/

(9.1.5)

j Dm

in a neighbourhood of w0 with coefficients fj 2 L1 .X I E; F /, j D m; : : : ; 1, of finite rank, and h.w/ is a holomorphic function near w0 with values in L cl .X I E; F /. First observe that h.w0 / is an elliptic pseudo-differential operator. In fact, writing f in the form (9.1.3) and comparing that with (9.1.5), we see that the homogeneous principal symbols of fo and h coincide at the point w0 . Since fo .w0 / is elliptic, so is h.w0 /. Now a function u.w/ is a root function of f .w/ at w0 if and only if u.w0 / 6D 0 and h.w0 /u.w0 / D 

m X 1 fk u.k/ .w0 /; kŠ

kD1

mC X kD0

1 fk u.k/ .w0 / D 0 kŠ

(9.1.6)

9.1 Calculus with operator-valued meromorphic families

555

for all  D m; : : : ; 1. The first equation of (9.1.6) shows that h.w0 /u.w0 / belongs to a finite-dimensional subspace of C 1 .X; F /, and the ellipticity of h.w0 / implies that u.w0 / belongs to a finite-dimensional subspace of C 1 .X; E/ which is determined by the operators fm ; : : : ; f1 and h.w0 /. (ii) Let u.w/ be a root function of f .w/ at w0 . Then g.w/ WD f .w/u.w/ is holomorphic near w0 , and we have g.w0 / D 0. Choose a neighbourhood U of w0 such that f .w/ is invertible for all w 2 U n fw0 g (cf. Theorem 9.1.6). Then we have u.w/ D f 1 .w/g.w/ for w 2 U n fw0 g: Because of u.w0 / 6D 0 the order of w0 as a zero of g.w/ does not exceed the order of w0 as a pole of f 1 .w/ which is finite. Let us set N WD dim ker f .w0 /. By a canonical system of eigenvectors of f .w/ at .N / w0 we understand a system of eigenvectors u.1/ such that 0 ; : : : ; u0 r1 WD rank u.1/ 0 D maxfrank u0 W u0 2 ker f .w0 /g; :: : .1/ .i1/ ri WD rank u.i/ gg; 0 D maxfrank u0 W u0 2 ker f .w0 /  spanfu0 ; : : : ; u0

i D 2; : : : ; N . Here spanf  g denotes the linear span of the vectors in the braces. Note that, in general, a canonical system of eigenvectors is not uniquely determined, but the sequence of numbers fr1 ; : : : ; rN g is uniquely determined by the function f .w/. Definition 9.1.10. The numbers fr1 ; : : : ; rN g, N D dim ker.f .w0 //, are called the partial null-multiplicities and n.f .w0 // WD

N X

ri

iD1

the null-multiplicity of the characteristic value w0 of f .w/. If f .w/ has no root function at w0 we set n.f .w0 // D 0. Moreover, let M D dim ker f 1 .w0 / and fp1 ; : : : ; pM g be the partial null-multiplicities of the characteristic value w0 of f 1 .w/. We then call fp1 ; : : : ; pM g the partial polar-multiplicities and p.f .w0 // WD

M X

pj

j D1

the polar-multiplicity of the singular value w0 of f .w/ (characteristic values of f .w/ or f 1 .w/ are called the singular values of f .w/), and m.f .w0 // WD n.f .w0 //  p.f .w0 // is called the multiplicity of the singular value w0 . If f .w/ is holomorphic in a neighbourhood of a point w0 , and if f .w0 / is invertible, then w0 is said to be a regular point of f .w/.

556

9 Operators in infinite cylinders and the relative index

Remark 9.1.11. According to Theorem 9.1.6 for every elliptic f .w/ 2 M  R .X I v/, v D .E; F /, the set D of singular values of f .w/ is countable, and D \ S.c 0 ;c 00 / is finite for every c 0  c 00 . Remark 9.1.12. Let w0 be a characteristic value of f .w/ 2 M  R .X I v/, and let 2 1 b1 .w/ 2 A.U; L .X I E; E//, b .w/ 2 A.U; L .X I F; F // be invertible for all 2 cl cl w in a neighbourhood U of w0 . Then w0 is a characteristic value of g.w/ WD b2 .w/f .w/b1 .w/ and the partial null- and polar-multiplicities of w0 for f .w/ and g.w/ coincide. Indeed, the multiplicity of any root function u.w/ of f .w/ at w0 is equal to the multiplicity of the root function b11 .w/u.w/ of g.w/ at w0 . In particular, the kernels of f .w/ and g.w/ at w0 are isomorphic. In an analogous way we show that the partial polar-multiplicities of w0 for f .w/ and g.w/ coincide. Theorem 9.1.13. Let f .w/ 2 M  R .X I v/ be elliptic and w0 a singular value of f .w/. Then there is a neighbourhood U of w0 where f has a representation L o n X f .w/ D b2 .w/ 0 C l .w  w0 /ml b1 .w/

(9.1.7)

lD1

A.U; L cl .X I E; E//,

b2 .w/ 2 A.U; L0cl .X I F; F //, taking values in with b1 .w/ 2 invertible operators, and mutually orthogonal projections l for l D 0; : : : ; L, 0 2 L0cl .XI E; F /, and l 2 L1 .X I E; F / being of rank 1 for l D 1; : : : ; L, and integers m1  m2      mL . Proof. Write f .w/ in the form (9.1.5) in a neighbourhood U of the point w0 . By the above considerations, h.w0 / is an elliptic pseudo-differential operator of order  on X . Since f .w/ is invertible in a punctured neighbourhood of w0 (we may take U n fw0 g), and the homogeneous principal symbols of h.w/ and f .w/ coincide for all w 2 U nfw0 g, it follows that the index of the operator h.w0 / is equal to 0. Hence, there is a smoothing operator s0 of finite rank on X , such that the operator e0 WD s0 C h.w0 / is invertible. By continuity, the operator e.w/ WD s0 C h.w/ is invertible in some neighbourhood (we may assume that in U ) of w0 . Then we have f .w/ D g.w/ C e.w/ D e.w/.1 C e 1 .w/g.w// for all w 2 U; P j where g.w/ WD j1 Dm fj .w  w0 /  s0 . The operator-valued function s.w/ WD e 1 .w/g.w/ is holomorphic in U n fw0 g whose values are smoothing operators of finite rank on X. It has a representation s.w/ D

1 X

sj .w  w0 /j C t .w/;

j Dm

where sm ; : : : ; s1 are smoothing operators on X of finite rank and t .w/ is a holomorphic function in U with values in smoothing operators of finite rank.

9.1 Calculus with operator-valued meromorphic families

557

T1  T0  0 Let N WD j Dm ker fj \ j Dm ker sj in D .X; E/. Since the operators fj ; j D m; : : : ; 1, and sj ; j D m; : : : ; 0, are of finite rank, the subspace N of D 0 .X; E/ is of finite codimension. Moreover, there exists a direct complement D 0 .X; E/  N of N in D 0 .X; E/ which is invariant with respect to each of the operators sm ; : : : ; s1 . Finally, since all the operators fj ; j D m; : : : ; 1, and sj ; j D m; : : : ; 0, are smoothing, it follows that D 0 .X; E/  N  C 1 .X; E/. Let  be the projection from D 0 .X; E/ onto D 0 .X; E/  N parallel to N . Clearly  is a smoothing operator. Set 0 WD 1  . Using the relation 0 s.w/ D 0 t .w/, we deduce that 1 C s.w/ D .1 C s.w//.1 C 0 t .w//: For d.w/ WD 1 C s.w/, w 2 U , the operator d.w/ can be regarded as acting in the finite-dimensional space D 0 .X; E/. This operator can be represented in the form Q d.w/ D d1 .w/d.w/d 2 .w/; where d1 .w/; d2 .w/ are holomorphic in U with values in the invertible linear operators Q in D 0 .X; E/, and d.w/ is of the form Q d.w/ D

L X

Q l .w  w0 /ml ;

lD1

cf. [60, §3]. Here, m1  m2      mL are integer numbers, Q 1 ; : : : ; Q L are pairwise orthogonal projections acting in the space D 0 .X; E/, and L is the rank of P the projection L Q l in D 0 .X; E/. Setting l WD Q l , it is easy to see that lD1  L   X l .w  w0 /ml .0 C d2 .w//: d.w/ D .0 C d1 .w// 0 C lD1

With the notation b1 .w/ WD .0 C d2 .w//.1 C 0 t .w//; b2 .w/ WD e.w/.0 C d1 .w// we then obtain (9.1.7). Corollary 9.1.14. Let f .w/; b1 .w/; b2 .w/ be as in Theorem 9.1.13. Then there is a neighbourhood U of w0 such that f

1

.w/ D

n

b11 .w/

0 C

L X

o l .w  w0 /ml b21 .w/

lD1

for all w 2 U n fw0 g. Moreover, if the numbers fml g1lL satisfy the conditions m1      mM < 0, mM C1 D    D mM CJ D 0, 0 < mM CJ C1      mM CJ CN , L D M C J C N , the partial null-multiplicities of the singular value w0 of f .w/ are equal to ri D mM CJ Ci , i D 1; : : : ; N , and the partial polar-multiplicities of the singular value w0 of f .w/ are equal to pj D mj , j D 1; : : : ; M .

558

9 Operators in infinite cylinders and the relative index

t For f .w/ 2 M  R .X I v/; v D .E; F /, let f .w/ denote the transposed pseudo0 differential operator of f .w/ for any w 2 C . It is clear that tf .w/ 2 M  S .X I v / for 0 0 0 0 0 v D .F ; E / with the dual bundles F and E of F and E, respectively, and some resulting S. Moreover, tf .w/ is elliptic if this is true for f .w/.

Corollary 9.1.15. Let f .w/ be elliptic. Then f .w/ and tf .w/ have the same singular values with the same partial null- and polar-multiplicities. This implies m. tf .w0 // D m.f .w0 // for any singular value w0 of f .w/. Let f .w/ 2 M  R .X I v/ be elliptic. Assume w0 is a characteristic value of f .w/ and u.w/ a root function of f .w/ at w0 . Denote by r the multiplicity of u.w/. The vector-valued functions 1  @ k u.w0 /; k D 1; : : : ; r  1; kŠ @w are said to be associated vectors for the eigenvector u0 D u.w0 /. Proposition 9.1.16. For each characteristic value w0 of f .w/ the associated vectors of f .w/ at w0 belong to a finite-dimensional subspace of C 1 .X; E/. PC1 P j k Proof. Let us write f .w/ D jC1 Dm fj .w  w0 / , u.w/ D kD0 uk .w  w0 / in a neighbourhood of w0 . A root function u.w/ of f .w/ at w0 is of multiplicity r if and only if mC X fk uk D 0 kD0

for all  D m; : : : ; 1, and f0 u0 C

m X kD1

fk uk D 0;

f0 u C

1 X kD0

fk uk C

mC X

fk uk D 0

kDC1

for all  D 1; : : : ; r  1. Now by induction (with respect to k), as in proof of Proposition 9.1.9 (i), we obtain that uk , k D 0; : : : ; r  1, belongs to a finite-dimensional subspace of C 1 .X; E/ which is determined by fm ; : : : ; f0 . .N / Let u.1/ 0 ; : : : ; u0 , N D dim ker f .w0 /, be a canonical system of eigenvectors of f .w/ at w0 , and let ri , i D 1; : : : ; N , denote the rank of u.i/ 0 . Moreover, let .i/ .i/ u1 ; : : : ; uri 1 , i D 1; : : : ; N , be associated vectors for the eigenvector u.i/ 0 . Then the system .i/ .i/ .u.i/ 0 ; u1 ; : : : ; uri 1 /iD1;:::;N

is called a canonical system of eigenvectors and associated vectors of f .w/ at w0 .

9.1 Calculus with operator-valued meromorphic families

559

Proposition 9.1.17. For each characteristic value w0 of f .w/ there are canonical .i/ .i/ .i/ .i/ .i/ systems .u.i/ 0 ; u1 ; : : : uri 1 /iD1;:::;N and .v0 ; v1 ; : : : ; vri 1 /iD1;:::;N of eigenvectors and associated vectors of f .w/ and t f .w/ at w0 , respectively, such that p:p:f 1 .w/ D

1 nrX N X i Cj X iD1 j Dri

o .w  w0 /j hvk.i/ ; iu.i/ ri Cj k

kD0

in a neighbourhood of w0 . .Here p.p. means the principal part of the Laurent expansion./ Proof. The meromorphic operator functions in our context may be regarded as a special case of the ones in the paper [60]. Hence, we can directly apply [60, Theorem 7.1] in the present situation.

9.1.2 The inhomogeneous equation  Let X be a closed compact C 1 manifold and RE ./ 2 L cl .X I E; EI R / be a parameter-dependent elliptic family of classical pseudo-differential operators on X  of order  which induce isomorphisms RE ./ W H s .X; E/ ! H s .X; E/ for all s;  2 R.

Definition 9.1.18. The Sobolev space H s .R  X; E/ of distributional sections on the infinite cylinder R  X of Sobolev smoothness s 2 R is defined as the completion of ˚R

12 s 2 C01 .R  X; E/ with respect to the norm R kRE ./F u./kL , with the 2 .X;E / d Fourier transform F D F t! on the real t-axis. (For convenience, E 2 Vect.X / is identified with the pull back of that bundle to R  X under the canonical projection R  X ! X, .t; x/ ! x). It can easily be verified that the space H s .R  X; E/ is independent of the choice of the order reducing family. For every ı 2 R we set H s;ı .R  X; E/ WD e ıt H s .R  X; E/: Remark 9.1.19. Recall that the (scalar) cylindrical Sobolev spaces have been introduced in Definition 2.4.14. The transformation 1

S0 u.t; / WD e  2 t u.e t ; / n

for r D e t induces isomorphisms S0 W H s; 2 .X ^ ; E/ ! H s .R  X; E/ for every n s 2 R, or, more generally, S0 W H s;ıC 2 .X ^ ; E/ ! H s;ı .RX; E/ for every s; ı 2 R. We now introduce spaces with double weights ı WD .ı ; ıC /; ı˙ 2 R, by H s;ı .R  X; E/ WD Œ H s;ı .R  X; E/ C Œ1   H s;ıC .R  X; E/

560

9 Operators in infinite cylinders and the relative index

in the sense of a non-direct sum, cf. Definition 2.1.4, for any cut-off function  .t / for the point t D 1 on the real axis, i.e.,  .t / D 1 for t < c0 and  .t / D 0 for t > c1 for certain c0 < c1 . A norm on the space H s;ı .R  X; E/ is defined by  2 2 kukH s;ı .RX;E / WD k ukH s;ı .RX;E / C k.1   /uk

 12

H s;ıC .RX;E /

:

(9.1.8)

Lemma 9.1.20. Let ı D .ı ; ıC / satisfy ı  ıC . Then a function u belongs to H s;ı .R  X; E/ if and only if u 2 H s;ı .R  X; E/ for each ı 2 Œı ; ıC . Proof. For ı 2 Œı ; ıC we have   2 2 2 kukH s;ı .RX;E /  2 k ukH s;ı .RX;E / C k.1   /ukH s;ı .RX;E / 2  ckukH s;ı .RX;E / ;

since 2 2 k ukH s;ı .RX;E /  c1 k ukH s;ı .RX;E /

and 2 2 k.1   /ukH s;ı .RX;E /  c2 k.1   /uk

H s;ıC .RX;E /

:

Here the constants c1 ; c2 and hence c are independent of u. Conversely, let u 2 H s;ı .R  X; E/ for any ı 2 Œı ; ıC . The relation (9.1.8) gives us

 2 2 2 kukH s;ı .RX;E /  c kukH s;ı .RX;E / C kuk



H s;ıC .RX;E /

(9.1.9)

for a constant c, depending only on . The estimate (9.1.9) gives more, namely, when u belongs both to H s;ı .R  X; E/ and H s;ıC .R  X; E/ it follows that u 2 H s;ı .R  X; E/. Let f .w/ 2 M  R .X I v/; v D .E; F /, be elliptic, and assume that f has no poles on the line Iı WD fw 2 C W w D  C i ı;  2 Rg for any fixed ı 2 R. Consider the operator opı .f /u.t / WD

1 2

Z e itw f .w/F u.w/dw

(9.1.10)

Iı

first for u.t/ 2 C01 .R  X; E/. Then F u.w/ is an entire function in w, and we have F .opı .f /u/.w/ D f .w/F u.w/ for all w 2 Iı . Proposition 9.1.21. The operator opı .f /, defined as in (9.1.10), induces a continuous map opı .f / W H s;ı .R  X; E/ ! H s;ı .R  X; F / (9.1.11) for all s 2 R.

9.1 Calculus with operator-valued meromorphic families

561

Proof. Fix a family of order reductions Rs ./. Then we have Z s 2 ı 2 k op .f /ukH s;ı .RX;F / D kRF ./f . C i ı/F u. C i ı/kL 2 .X;F / d  sup 2R

s kRF ./f .

C

R s 2 i ı/RE ./k2L.L2 .X;E /;L2 .X;F // kukH s;ı .RX;E /

for all u 2 C01 .RX; E/ which gives us the continuity of f .w/ between the respective spaces. Remark 9.1.22. We have the commutative diagram opı .f /

H s;ı .R  X; E/ ! H s;ı .R  X; F / x x ?S ?S ? 0 ? 0 n

ı .h/ opM

n

H s;ıC 2 .X ^ ; E/ ! H s;ıC 2 .X ^ ; F / for f .w/ 2 M  O .X I v/, w D  C i ı, and h.z/ D f .w/ for z D iw. Proposition 9.1.23. Let f .w/ be elliptic and without singular values on Iı . Then (9.1.11) extends to an isomorphism. Moreover, the inverse is given by the formula Z 1 ı 1 .op .f // v.t / D e itw f 1 .w/F v.w/dw: 2 Iı Proof. This proposition is a consequence of Theorem 9.1.6 and Proposition 9.1.21. Remark 9.1.24. For any u 2 H s;ı .R  X; E/ the Fourier transform F u.w/ is holomorphic in the strip S.ı ;ıC / . Indeed, this is an immediate consequence of the representation of u, i.e., u contains the factors e ı t and e ıC t in a neighbourhood of 1 and C1, respectively. Note that the integral on the right side of (9.1.10) depends on the choice of ı in the interval Œı ; ıC which is a problem for the definition of an operator. Therefore, we admit in the domain of opı .f / only those u 2 H s;ı .R  X; E/ for which (9.1.10) is independent of ı 2 Œı ; ıC . Proposition 9.1.25. Let f .w/ 2 M  R .X I v/; v D .E; F /, be elliptic and without poles on the lines Iı and IıC , where ı  ıC . Then for each u 2 H s;ı .R  X; E/ we have opı .f /u.t /  opıC .f /u.t / D 2 i

X p2S.ı ;ıC /

resp e itw f .w/F u.w/:

562

9 Operators in infinite cylinders and the relative index

Proof. According to (9.1.2) the operator-valued function f .w/ has a representation f .w/ D fo .w/ C fr .w/ 1 for certain fo .w/ 2 M  O .X I v/ and fr .w/ 2 M R .X I v/ such that fr .w/ has no poles on the lines Iı and IıC . We show that

opı .fr /u.t /  opıC .fr /u.t / D 2 i

X

resp e itw fr .w/F u.w/

(9.1.12)

p2S.ı ;ıC /

and opı .fo /u.t / D opıC .fo /u.t /:

(9.1.13)

The relation (9.1.12) is a consequence of Cauchy’s integral formula and the Residue Theorem. Concerning (9.1.13) the relation is true for any u 2 C01 .R  X; E/ by Cauchy’s theorem. Since C01 .R  X; E/ is dense in H s;ı .R  X; E/ for every ı 2 R, by Proposition 9.1.21 and Lemma 9.1.20, the relation (9.1.13) is true for any u 2 H s;ı .R  X; E/. Fix s 2 R and weight data ı D .ı ; ıC / satisfying ı  ıC . Let us set D.f / WD fu 2 H s;ı .R  X; E/ W resp e itw f .w/F u.w/ D 0 for all p 2 S.ı ;ıC / g: Then opı .f /jD.f / is independent of the choice of ı (also of s). Lemma 9.1.26. Let f .w/ be elliptic. Then D.f / is a closed subspace of H s;ı .R  X; E/ of finite codimension codim D.f / D

X

p.f .p//

p2S.ı ;ıC /

(which is independent of ı and s). Proof. Let p be a pole of f .w/ in the strip S.ı ;ıC / , then p is a characteristic value of the inverse f 1 .w/. Because of Proposition 9.1.16, there are canonical systems .i/ .i/ .u.i/ 0 ; u1 ; : : : ; upi 1 /iD1;:::;M

and .v0.i/ ; v1.i/ ; : : : ; vp.i/ /iD1;:::;M i 1

of eigenfunctions and associated functions of f 1 .w/ and t .f 1 .w// at p, respectively, such that p:p: f .w/ D p:p: .f 1 /1 .w/ D

1 n pX M X i Cj X iD1 j Dpi

kD0

o hvk.i/ ; iup.i/i Cj k .w  p/j

9.1 Calculus with operator-valued meromorphic families

563

in a neighbourhood of p. This gives us resp e itw f .w/F u.w/ D resp .p:p: f .w//e itw F u.w/ D

 @ j 1 1 .j  1/Š @w kD0 Z n oˇ 0 ˇ itw e e it w hvk.i/ ; u.t 0 /idt 0 ˇ

1 pX M X i Cj X iD1 j Dpi

D e itp

M X

R pi Cj 1 X X

wDp

up.i/i Cj k

iD1 j Dpi kD0

Z o 1 0 e it p .i t  i t 0 /j 1 hvk.i/ ; u.t 0 /idt 0 up.i/i Cj k .j  1/Š R Z pi 1 n pi ˛ M o XX X 1 0 itw 0 0 u.i/ De e it p .i t  i t 0 /ˇ 1 hvp.i/ ; u.t /idt ˛ ; i ˛ˇ .ˇ  1/Š R ˛D0 n

iD1

ˇ D1

where ˛ WD pi C j  k; ˇ WD j . Hence, we see that u 2 D.f / if and only if u P satisfies a system of M iD1 pi D p.f .p// linearly independent conditions. Now starting from a parameter-dependent element f .w/ 2 M  R .X I v/, v D .E; F /, for u 2 D.f / we define opı .f /u by formula (9.1.10) with any ı in the interval Œı ; ıC , such that the line Iı is free from the poles of f .w/. Proposition 9.1.27. Let f .w/ be elliptic and without poles on the lines Iı and IıC . Then opı .f / induces a continuous operator opı .f / W D.f / ! H s;ı .R  X; F /:

(9.1.14)

Proof. Because of Proposition 9.1.25 for u 2 D.f / we have Z Z 1 1 ı itw op .f /u.t/ D e f .w/F u.w/dw D e itw f .w/F u.w/dw: 2 Iı 2 Iı C

Proposition 9.1.21 yields opı .f /u 2 H s;ı˙ .R  X; E/, and the proof follows from Lemma 9.1.20. The following proposition describes the set of all elements v 2 H s;ı .R  X; F / for which the inhomogeneous equation opı .f /u D v has a solution in D.f /. Proposition 9.1.28. Let f .w/ be elliptic and without singular values on the lines Iı and IıC . Then, for v 2 H s;ı .R  X; F / there exists a solution u 2 D.f / of the equation opı .f /u D v if and only if resp e itw f 1 .w/F v.w/ D 0 for all p 2 S.ı ;ıC / :

(9.1.15)

564

9 Operators in infinite cylinders and the relative index

We define R.f / WD fv 2 H s;ı .R  X; F / W the relation .9.1.15/ holdsg: Proof. Let v D opı .f /u for some u 2 D.f /. Then F v.w/ D f .w/F u.w/ or f 1 .w/F v.w/ D F u.w/. Since F u.w/ is holomorphic in the strip S.ı ;ıC / , we obtain v 2 R.f /. Conversely, let v 2 R.f /. Then, because of Proposition 9.1.25 the integral Z 1 u.t / D e itw f 1 .w/F v.w/dw 2 Iı is independent of ı 2 Œı ; ıC , if f 1 .w/ has no poles on Iı . In particular, taking ı D ı˙ we conclude from Lemma 9.1.20 that u 2 H s;ı .R  X; E/. Since F v.w/ is holomorphic in the strip S.ı ;ıC / , the relation f .w/F u.w/ D F v.w/ implies that u 2 D.f /. Finally, a simple calculation shows that opı .f /u D v. Corollary 9.1.29. The operator (9.1.14) is injective, and R.f P / is a closed subspace of H s;ı .R  X; F / of finite codimension codim R.f / D p2S.ı ;ı / n.f .p//. 

C

In fact, this is a consequence of Proposition 9.1.28 and Lemma 9.1.26. Let us now define the operator A.f / W H s;ı .R  X; E/ ! H s;ı .R  X; F /

(9.1.16)

by setting A.f /u D opı .f /u

for u 2 D.f /;

A.f /u D 0

for u 2 H s;ı .R  X; E/  D.f /:

(9.1.17)

Summing up, Lemma 9.1.26 and Corollary 9.1.29 give us the following theorem. Theorem 9.1.30. Let f .w/ 2 M  R .X I v/ be elliptic and without singular values on the lines Iı and IıC ; ı  ıC . Then the operator (9.1.16) defined by (9.1.17) is Fredholm, and we have X ind A D fp.f .p//  n.f .p//g: p2S.ı ;ıC /

Next we assume ı > ıC . To define an operator (9.1.16) for a parameter-dependent element f .w/ 2 M  R .X I v/ in this case we pass to the formal transposed operator with respect to the bilinear pairings H 0;ı .R  X; E/  H 0;ı .R  X; E 0 / ! C induced by hu; vi WD .u; vx/ with the H 0;.0;0/ .R  X; E/-scalar product .; /; here E 0 is the dual bundle of E.

9.1 Calculus with operator-valued meromorphic families

565

Proposition 9.1.31. Let f .w/ have no poles on a line Iı . Then the transposed operator of the operator (9.1.11) is given by Z 1 opı .f 0 /v.t / WD e itw t f .w/F v.w/dw; t 2 R; (9.1.18) 2 Iı for any v 2 H sC;ı .R  X; F 0 /. Proof. It suffices to verify the relation (9.1.18) for elements v 2 C01 .R  X; F 0 /. For u 2 C01 .R  X; E/ we have Z D Z E 1 v.t /; hv; opı .f /ui D e itw f .w/F u.w/dw dt 2 Iı R Z Z D E 1 0 e it w t f .w/F v.w/dw; u.t 0 / dt 0 D R 2 Iı Z D Z E 1 0 D e it w t f .w/F v.w/dw; u.t 0 / dt 0 D hopı .f 0 /v; ui R 2 Iı which shows the assertion. We see that the formulas (9.1.10) and (9.1.18) correspond to each other when f .w/ and ı are replaced by f 0 .w/ and ı, respectively. We now are in the situation with weights ı D .ı ; ıC / as before. Hence there is an associated operator denoted by t A, continuous in the sense t

A W H sC;ı .R  X; F 0 / ! H s;ı .R  X; E 0 /:

Analogously we define ˚ D.f 0 / WD v 2 H sC;ı .R  X; F 0 / W resp e itw f 0 .w/F v.w/ D 0 for all p 2 S.ı ;ıC /



which is a closed subspace of H sC;ı .R  X; F 0 / of finite codimension X X codim D.f 0 / D p.f 0 .p/// D p.f .p//: p2S.ı ;ıC /

p2S.ıC ;ı /

Corollary 9.1.29 shows that when f .w/ has no singular values on the lines Iı˙ the analogue of the operator (9.1.14) D.f 0 / ! H s;ı .R  X; E 0 / is injective, and R.f 0 / is a closed subspace of H s;ı .RX; E 0 / of finite codimension X X codim R.f 0 / D p.f 0 .p// D p.f .p//: p2S.ı ;ıC /

p2S.ıC ;ı /

To define the analogous operator of (9.1.16) in the case ı > ıC we need the following result.

566

9 Operators in infinite cylinders and the relative index

Lemma 9.1.32. For each u 2 H s;ı .RX; E/ there is a unique g 2 H s;ı .RX; F / such that hv; gi D h t Av; ui

for v 2 D.f 0 /;

hv; gi D 0

for v 2 H sC;ı .R  X; F 0 /  D.f 0 /:

(9.1.19)

Proof. Let  be the projection operator of H sC;ı .R  X; F 0 / to D.f 0 /. Then for u 2 H s;ı .R  X; E/ we define hv; gi D h t Av; ui for v 2 H sC;ı .R  X; F 0 /. Obviously, g is a continuous linear functional on H sC;ı .R  X; F 0 / (therefore, it can be identified with an element of H s;ı .R  X; F /) and satisfies (9.1.19). Furthermore, if g1 ; g2 2 H s;ı .R  X; F / satisfy the relations (9.1.19), we have hv; g1  g2 i D hv; g1  g2 i C h.1  /v; g1  g2 i D 0 for all v 2 H sC;ı .R  X; F 0 /, i.e., g1 D g2 . For u 2 H s;ı .R  X; E/ we set Au WD g

(9.1.20)

for g associated with u via Lemma 9.1.32. This defines a linear continuous operator A W H s;ı .R  X; E/ ! H s;ı .R  X; F /:

(9.1.21)

Theorem 9.1.33. Let f .w/ be an elliptic operator without singular values on the lines Iı and IıC . The operator (9.1.21) defined P as in (9.1.20) is a Fredholm operator. More precisely, we have dim ker A D p2S.ı ;ı / n.f .p//, dim coker A D C  P p2S.ı ;ı / p.f .p//, i.e., C



ind A D

X

fn.f .p//  p.f .p//g:

p2S.ıC ;ı /

Proof. The relation (9.1.19) shows that u 2 ker A is equivalent to h t Av; ui D 0 for all v 2 H sC;ı .R  X; F 0 /, and Au D g has a solution for g 2 H s;ı .R  X; F / if and only if hv; gi D 0 for all v 2 H sC;ı .R  X; F 0 /  D.f 0 /. Hence, codim D.f / D codim R.f 0 / and codim R.f / D codim D.f 0 /.

9.1.3 An index formula Let f .w/ 2 M  R .X I v/; v D .E; F /, be elliptic and let w0 be a singular value of f .w/. The principal part (denoted by p.p.) of the Laurent expansion (9.1.5) of f .w/ is a smooth operator of finite rank on X for all w in a punctured neighbourhood of w0 . Hence the trace of p:p:f 0 .w/f 1 .w/ is well defined.

9.2 Boundary value problems in infinite cylinders

Lemma 9.1.34. Let f .w/ and w0 be as above. Then tr p:p:f 0 .w/f 1 .w/ D

567

m.f .w0 // . ww0

.w0 // tr p:p:.b10 .w/b11 .w/C Proof. Indeed, (9.1.7) gives tr p:p:f 0 .w/f 1 .w/ D m.f ww0 0 1 b2 .w/b2 .w// in a neighbourhood of w0 . Since b1 .w/; b2 .w/ are holomorphic and invertible near w0 , we have p:p: .b10 .w/b11 .w/ C b20 .w/b21 .w// D 0 which gives us the assertion.

The following theorem yields an explicit formula for the index. Theorem 9.1.35. Let f 2 M  R .X I v/ be elliptic and without singular values on the lines Iı and IıC . Then Z Z   1 1 f 1 .w/f 0 .w/dw  f 1 .w/f 0 .w/dw : (9.1.22) ind A D tr 2 i Iı 2 i Iı C

Proof. Let ı  ıC and QT be a rectangle with vertices ˙T C i ı˙ , that contains all singular values of a.w/ in the strip Im w 2 .ı ; ıC /. Using Corollary 9.1.34 and the Residue Theorem, we obtain Z   X 1 1 0 f .w/f .w/dw D m.f .p// D ind A: (9.1.23) tr 2 i QT p2S.ı ;ıC /

Now, for T ! 1 on the left of (9.1.23), we obtain the assertion. Analogously, we can argue for the case ıC < ı .

9.2 Boundary value problems in infinite cylinders In our applications we need a variant of the material of Section 9.3 in the case of an infinite cylinder when the cross section is a compact C 1 manifold with boundary. This is the program of the following consideration. The ideas are analogous as before. In the present situation we employ parameter-dependent pseudo-differential boundary value problems with the transmission property at the boundary.

9.2.1 Operators in cylindrical Sobolev spaces Let X be a compact C 1 manifold with boundary @X . Set H s .X I E; J / WD H s .int X; E/ ˚ H s .@X; J / for E 2 Vect.X /; J 2 Vect.@X /. Let B ;d .X I Rl /, d 2 N, denote the space of all parameter-dependent pseudodifferential boundary value problems; those represent continuous operators A./ W H s .X I E; J / ! H s .X I F; G/

568

9 Operators in infinite cylinders and the relative index

for E; F 2 Vect.X /, J; G 2 Vect.@X /, s > max.; d/  12 . More precisely, B ;d .XI Rl / 3 A./ is defined by the condition 1

1

diag.1; RG2 .//A./ diag.1; RJ 2 .// 2 B ;d .X I Rl / with B ;d .XI Rl / from Chapter 3 and order reducing families 1

1

RG2 ./ 2 Lcl2 .@X I G; GI Rl /

and

1

1

RJ 2 ./ 2 Lcl 2 .@X I J; J I Rl /

as mentioned at the beginning of Section 9.1.2. The elements of the calculus of operators in B ;d .X I Rl / or B ;d .X I Rl / are practically the same, and we use the results on B ;d .X I Rl / in the corresponding modified form for B ;d .X I Rl /. As we know from the calculus of boundary value problems on a manifold with conical singularities, parameter-dependent operators are the raw material of operator-valued symbols. Here we employ a similar scheme and ;d denote by M;d .X // such O .X / the space of all operator families h.w/ 2 A.C ; B ;d that h. C iı/ 2 B .X I R / for every ı 2 R, uniformly in compact ı-intervals. (In contrast to the notation before we now denote families of boundary value problems by small letters because of their role as operator-valued symbols; the bundles on X and @X are assumed to be known in every concrete case.) Let R D f.pj ; mj ; Gj /gj 2Z ; (9.2.1) be a sequence, for pj 2 C ; mj 2 N, j Im pj j ! 1 as jj j ! 1, and finitedimensional subspaces Gj  B 1;d .X / of operators of finite rank. Set C R WD fpj gj 2Z . .X / defined to be the space of all f .w/ 2 A.C n C R, Definition 9.2.1. Let M1;d R 1;d B .X// such that f is meromorphic with poles at the points pj of multiplicity mj C 1 and Laurent coefficients at .w  pj /.kC1/ belonging to Gj for all 0  k  mj ; j 2 Z, and such that . C i ı/f . C i ı/ 2 B 1;d .X I R / for every ı 2 R, uniformly in compact ı-intervals for any C R-excision function .w/. If we want to indicate the tuple v WD .E; F I J; G/ of the involved vector bundles we also write M1;d .X I v/. This space is Fréchet in a natural way, and we set R ;d 1;d M;d .X I v/; R .X I v/ WD M O .X I v/ C M R

(9.2.2)

equipped with the topology of the non-direct sum. Clearly the operator families in (9.2.2) are meromorphic with corresponding poles at the points of C R. The operator functions of the class (9.2.2) are a slight modification of the operatorvalued Mellin symbols of Chapter 6. In particular, we have the algebra property which admits composing two such functions (when the bundle data in the middle fit together). Moreover, we have a notion of ellipticity which is nothing other than parameterdependent ellipticity on a line Iı D f C i ı W  2 Rg for a ı 2 R with parameter

9.2 Boundary value problems in infinite cylinders

569

Re w, such that C R \ Iı D ;. This condition is independent of the choice of ı. For purposes below we formulate some properties in the present version. Incidentally, B ;d .X I vI Iı / denotes the space of all operator families f .w/ depending on the parameter w 2 Iı and belonging to B ;d .X I vI R / for  D Re w. ;e Theorem 9.2.2. f .w/ 2 M;d R .X Iv/ with v D .E0 ; F IJ0 ; G/ and g.w/ 2 M S .X Iw/ with w D .E; E0 I J; J0 / implies

f .w/g.w/ 2 MPC;h .X I v ı w/ for h D max. C d; e/, v ı w D .E; F I J; G/ and some resulting asymptotic type P . The proof is a simple consequence of Theorem 3.3.15 combined with arguments of complex analysis which show that the product of meromorphic functions is again meromorphic, see also Theorem 6.1.47. Proposition 9.2.3. (i) Let f 2 M;d R .X I v/, v D .E; F I J; G/, be elliptic. Then there C

is an element f 1 2 M ;.d/ .X I v1 /, v1 D .F; EI G; J /, for an asymptotic S 1 1 and f f are both the identity operators in the respective type S such that ff spaces for all w 2 C . (ii) Let f1 2 M 1;d .X I b/ belong to the bundle data b D .E; EI J; J /. Then R there exists an f2 2 M1;d .X I b/ for some asymptotic type S such that S .1 C f1 .w//1 D 1 C f2 .w/ (here 1 stands for the corresponding identity operator).

The assertion (ii) is essentially the same as Proposition 6.1.53 and (i) as Theorem 6.1.55. Remark 9.2.4. Let f 2 M;d R .X I v/ be elliptic. Then there is a discrete set D  R (i.e., countable, and intersecting each compact subinterval in a finite set) such that the operators f .w/ W H s .X I E; J / ! H s .X I F; G/ (9.2.3) are bijections for all w 2 Iı , ı 62 D, and all s 2 R, s > max.; d/  12 . Remark 9.2.5. If we write f 2 M;d R .X I v/ in the form f D fo C fr 1;d with fo 2 M;d .X I v/, then the ellipticity of f is equivalent to O .X I v/; fr 2 M R that of fo .

570

9 Operators in infinite cylinders and the relative index

Theorem 9.2.6. For every f .w/ 2 B ;d .XˇI vI Iı /, ı 2 R, there exists a holomor1;d ˇ phic family h.w/ 2 M;d .X I vI Iı /. If f .w/ is O .X I v/ such that h Iı  f 2 B parameter-dependent elliptic, also h.w/ is elliptic. This result is a simple modification of Theorem 6.1.30 (for the case q D 0). Theorem 9.2.7. For every  2 Z and b WD .E; EI J; J /; E 2 Vect.X /; J 2 Vect.@X/, there exists an elliptic element r.w/ 2 M ;0 O .X I b/ which induces isomorphisms r.w/ W H s .X I E; J / ! H s .X I E; J / for all s >   12 , in a strip of arbitrary fixed prescribed width and such that r 1 .w/ 2 M;0 .XI b/ for some R as in (9.2.1). R Theorem 9.2.7 is true in analogous form for arbitrary s 2 R; in this case the operator in the upper left corner which contains an extension eC by zero to a neighbouring manifold Xz of int X is to be replaced by esC , an extension of u 2 H s .int X; E/ to a Sobolev distribution of smoothness s on Xz . We easily find r.w/ in the form r.w/ D r1 .w/ ˚ r2 .w/ where r2 .w/ refers to @X and the bundle J and r1 .w/ to X and E, based on a family of minus-operators in the sense of (4.1.17), modified to operators between distributional sections in E. According to (4.1.17) and Theorem 4.1.16 we first have a parameterdependent elliptic family of isomorphisms which can be turned into a holomorphic family of the asserted quality by applying kernel cut-off with respect to the parameter. In a similar manner we can proceed with the second component. Corollary 9.2.8. Given an elliptic operator f 2 M;d R .X I v/ as in Remark 9.2.4, realised as a family of maps (9.2.3), we can choose order reducing isomorphisms r.w/ W H 0 .XI E; J / ! H s .X I E; J /;

r.w/ Q W H s .X I F; G/ ! H 0 .X I F; G/;

w 2 C , such that f0 .w/ WD r.w/f Q .w/r.w/ 2 M0;0 R .X I v/ is elliptic. If R is given by z zj /g for certain finite(9.2.1), the new asymptotic type has the form R D f.pj ; mj ; G 1;0 zj 2 B dimensional subspaces of operators G .X I v/ of finite rank. In the sequel, for convenience we content ourselves with the case  D d D 0. Let f .w/ 2 M0;0 R .X I v/ for C R \ Iı D ;; ı 2 R. Then we can form the operator Z ı 1 op .f /u.t / WD .2/ e itw f .w/F u.w/dw (9.2.4) Iı

for u.t/ 2 C01 .R  X; E/ ˚ C01 .R  @X; J / (in this case F u.w/ is an entire function in w).

9.2 Boundary value problems in infinite cylinders

571

Let us also set H s;ı .R  X I E; J / WD H s;ı .R  int X; E/ ˚ H s;ı .R  @X; J / for E 2 Vect.X/; J 2 Vect.@X /, where z Rint X H s;ı .R  int X; E/ WD H s;ı .R  2X; E/j

(9.2.5)

z RX . Here 2X is the double of X (cf. Section 9.1.2) for Ez 2 Vect.R  2X /, E WD Ej (obtained by gluing together two copies X˙ along the common boundary @X , where we identify XC with X ), and (9.2.5) is equipped with the norm from an isomorphism s;ı s;ı z z z to H s;ı .R  2X; E/=H 0 .R  X ; E/, where H0 .R  X ; E/ means the (closed) s;ı z that vanish on R  int X . subspace of all u 2 H .R  2X; E/ Proposition 9.2.9. Let f .!/ 2 M 0;0 R .X I v/ and C R \ Iı D ;, ı 2 R. Then (i) the operator opı .f / W C01 .RX; E/˚C01 .R@X;J / ! C 1 .RX; F /˚C 1 .R@X;G/ extends to a continuous operator opı .f / W H s;ı .R  X I E; J / ! H s;ı .R  X I F; G/

(9.2.6)

for every real s >  12 ; (ii) if f is as in Remark 9.2.4 (for  D d D 0) the operator (9.2.6) is bijective for all ı 62 D and s >  12 . Proof. The assertions can be proved in an analogous manner as Propositions 9.1.21 and 9.1.23.

9.2.2 Characteristic values and factorisation of meromorphic families In the following consideration we consider, for simplicity, operator functions of the spaces B 0;0 R .XI C / DW M R .X /. Recall that the bundle data v D .E; F I J; G/ are assumed to be known in every concrete case; if necessary, we write M R .X I v/. The operator functions f .w/ 2 MR .X I v/ for v D .E; F I J; G/ are families of continuous operators f .w/ W H s .X I E; J / ! H s .X I F; G/, w 62 C R. Analogously as in Definition 9.1.7 a point w0 2 C is called a characteristic value of f .w/ 2 MR .X I v/, if there exists an open neighbourhood U of w0 and an element u 2 A.U; H s .XI E; J // with the properties f .w/u.w/ 2 A.U; H s .X I F; G// and ˇ u.w0 / 6D 0; f .w/u.w/ˇwDw D 0: 0

Also other notions in this context will be used analogously as in Section 9.1.2. For instance, the characteristic values of f .w/ or f 1 .w/ are called singular values of f .w/. Moreover, if f .w/ is holomorphic in a neighbourhood of a point w0 , and if f .w0 / is invertible, then w0 is said to be a regular point of f .w/.

572

9 Operators in infinite cylinders and the relative index

Proposition 9.2.10. Let f .w/ be elliptic, and let w0 be a characteristic value of f .w/. Then (i) ker f .w0 /  H 1 .X I E; J / is of finite dimension; (ii) the rank of each eigenvector of f .w/ at w0 is finite. Proof. The proof is analogous to that of Proposition 9.1.9. Remark 9.2.11. For every elliptic f 2 M R .X I v/ the set D of singular values of f is countable, and D \ S.c 0 ;c 00 / is finite for every c 0  c 00 . The explanation of the latter observation is similar to that of Remark 9.1.11. Remark 9.2.12. Suppose that w0 is a characteristic value of f .w/ 2 MR .X I v/ with v D .E; F I J; G/. Let further b1 .w/ 2 A.U; B 0;0 .XI E; EI J; J //; b2 .w/ 2 A.U; B 0;0 .XI F; F I G; G// be invertible for all w in a neighbourhood U of w0 . Then w0 is also a characteristic value of g.w/ WD b2 .w/f .w/b1 .w/, and the partial null- and polar-multiplicities of f .w/ and g.w/ at w0 coincide. The arguments are analogous to those for Remark 9.1.12. Theorem 9.2.13. Let f .w/ 2 MR .X I v/, v D .E; F I J; G/, be elliptic, and let w0 2 C be a singular value of f . Then there are integers m1  m2      mL and mutually orthogonal projections l , l D 0; : : : ; L, such that l 2 B 1;0 .X I E; EI J; J / are P of rank 1 for l D 1; : : : ; L, 0 2 B 0;0 .X I E; EI J; J /, and 0 C L lD1 l D 1, such that in a neighbourhood U of w0 the function f .w/ has the representation L o n X f .w/ D b2 .w/ 0 C .w  w0 /ml l b1 .w/ lD1

with operator functions b1 .w/; b2 .w/ as in Remark 9.2.12. The proof may be given along the lines of Theorem 9.1.13. Corollary 9.2.14. Let f .w/; b1 .w/; b2 .w/ be as in Theorem 9.2.13. Then there is a neighbourhood U of w0 such that L o n X f 1 .w/ D b11 .w/ 0 C .w  w0 /ml l b21 .w/ lD1

for all w 2 U n fw0 g. If the numbers fml g1lL satisfy the conditions m1      mM < 0, mM C1 D    D mM CJ D 0, mM CJ C1      mM CJ CN , L D M CJ CN the partial null-multiplicities of f .w/ at w0 are equal to ri D mM CJ Ci , i D 1; : : : ; N and the partial polar-multiplicities of f .w/ at w0 are equal to pj D mj , j D 1; : : : ; M .

9.2 Boundary value problems in infinite cylinders

573

9.2.3 The relative index We now introduce spaces with double weights ı WD .ı ; ıC / 2 R2 as H s;ı .R  X I E; J / WD Œ H s;ı .R  X I E; J / C Œ1   H s;ıC .R  X I E; J / in the sense of a non-direct sum for any  .t / 2 C 1 .R/ such that  .t / D 1 for t < c 0 and .t/ D 0 for t > c 00 for certain c 0 < c 00 . A norm on the space H s;ı .R  X I E; J / is defined by ˚ 2 2 kukH s;ı .RXIE;J / D k ukH s;ı .RXIE;J / Ck.1 /uk

H s;ıC .RXIE;J /

12

: (9.2.7)

The choice of  only affects (9.2.7) up to equivalence. The following assertion is an obvious analogue of Lemma 9.1.20. Lemma 9.2.15. For every ı  ıC we have u 2 H s;ı .R  X I E; J / if and only if u 2 H s;ı .R  X I E; J / for every ı 2 Œı ; ıC . Remark 9.2.16. From u 2 H s;ı .R  X I E; J / it follows that .F u/.w/ is holomorphic in the strip S.ı ;ıC / . This allows us to apply (9.2.4) for every ı  ı  ıC , Iı \ C R D ;. As for Remark 9.1.24 this follows from the definition of weighted Sobolev spaces, i.e., the elements u contain the factors e ı t and e ıC t in a neighbourhood of 1 and C1, respectively. Proposition 9.2.17. Let f .w/ 2 M R .X I v/, and let c 0 < ı  ıC < c 0 , C R \ .Iı [ IıC / D ;. Then for every u 2 H s;ı .R  X I E; J / we have X opı .f /u.t /  opıC .f /u.t / D 2 i resp e itw f .w/F u.w/: p2S.ı ;ıC /

This result is completely analogous to Proposition 9.1.25. Our next objective is to associate with any f 2 MR .X I v/ a continuous operator A W H s;ı .R  X I E; J / ! H s;ı .R  X I F; G/

(9.2.8)

in the case ı  ıC . Assuming first s D 0 the conclusions then hold for arbitrary reals s >  12 . Let us set D.f / WD fu 2 H s;ı .R  X I ; E; J / W resp e itw f .w/F u.w/ D 0 for all p 2 S.ı ;ıC / g: ˇ Thus opı .f /ˇD.f / is independent of the choice of ı 2 Œı ; ıC (and also of s). Remark 9.2.18. For the case of boundary value problems we have immediate analogues of Proposition 9.1.16 and 9.1.17.

574

9 Operators in infinite cylinders and the relative index

Moreover, as Lemma 9.1.26 we have the following lemma. s;ı Lemma 9.2.19. Let f .w/ be elliptic. Then D.f P/ is a closed subspace of H .R  XI E; J / of finite codimension codim D.f / D p2S.ı ;ı / p.f .p// (which is inde C pendent of ı and s).

Proposition 9.2.20. (i) Let f .w/ 2 MR .X I v/ be elliptic, and let Iı˙ \ C R D ;. Then opı .f / induces a continuous operator opı .f / W D.f / ! H s;ı .R  X I F; G/

(9.2.9)

for every s >  12 . (ii) Let f .w/ have no singular values on the lines Iı˙ . Then for every v 2 H s;ı .R XI F; G/ there exists a solution u 2 D.f / of the equation opı .f /u D v if and only if resp e itw f 1 .w/F v.w/ D 0 for all p 2 S.ı ;ıC / :

(9.2.10)

We define R.f / WD fv 2 H s;ı .R  X I F; G/ W the relation (9.2.10) holdsg. Proof. (i) Because of Proposition 9.2.17 for u 2 D.f / we have Z ı 1 e itw f .w/F u.w/dw op .f /u.t / D .2/ Iı

D .2/1

Z

e itw f .w/F u.w/dw: IıC

Proposition 9.2.9 (i) yields opı .f /u 2 H s;ı˙ .R  X I F; G/, and the result follows from Lemma 9.2.15. (ii) Let v D opı .f /u for some u 2 D.f /. Then we have F v.w/ D f .w/F u.w/ or f 1 .w/F v.w/ D F u.w/. Since F u.w/ is a holomorphic function in the strip S.ı ;ıC / , we obtain v 2 R.f /. Conversely, let v 2 R.f /. Then it is easy to show that the function u defined by the integral Z 1 u.t / D e itw f 1 .w/F v.w/dw 2 Iı (which is independent of ı 2 Œı ; ıC , cf. Proposition 9.2.17) belongs to D.f / and opı .f /u D v. Corollary 9.2.21. The operator (9.2.9) is injective, and R.f P / is a closed subspace of H s;ı .R  XI F; G/ of finite codimension codim R.f / D p2S.ı ;ı / n.f .p//. 

C

This is a consequence of Proposition 9.2.20 (i) and Lemma 9.2.19. We now define the operator A.f /u D opı .f /u

for u 2 D.f /;

A.f /u D 0

for u 2 H s;ı .R  X I E; J /  D.f /:

(9.2.11)

9.2 Boundary value problems in infinite cylinders

575

The latter orthogonal complement refers to the scalar product of H 0;ı .RX I E; J /. Summing up, Lemma 9.2.19 and Corollary 9.2.21 give us the following theorem. Theorem 9.2.22. Let f .w/ be elliptic and without singular values on the lines Iı˙ , ı  ıC . Then the operator (9.2.8) defined by (9.2.11) is Fredholm, and we have X fp.f .p//  n.f .p//g: ind A D p2S.ı ;ıC /

Next we assume ı > ıC . With elements f .w/ 2 B 0;0 R .X I vI C / we associate operators (9.2.8). To this end we pass to the formal transposed operators with respect to the bilinear pairings H 0;ı .R  X I E; J /  H 0;ı .R  X I E 0 ; J 0 / ! C induced by hu; vi WD .u; vx/ with the H 0;.0;0/ .RX I E; J /-scalar product .; /; here E 0 and J 0 are the dual bundles of E and J , respectively. The formal transposed of opı .f / is equal to opı .f 0 / for f 0 .w/ WD t f .w/, where the latter notation ‘t ’ means the pointwise 0 0 transposed of f which gives us an element in B 0;0 R0 .X I v I C / for some resulting R , 0 0 0 0 0 t and v WD .F ; E I G ; J /. Analogously as in Corollary 9.1.15 for f .w/ we are now in the situation with weights ı < ıC as before. Hence there is a corresponding operator denoted by t A, continuous in the sense t

A W H s;ı .R  X I F 0 ; G 0 / ! H s;ı .R  X I E 0 ; H 0 /;

for s >  12 (first for s D 0, but then for all s >  12 , since the operators belong to Boutet de Monvel’s calculus). Analogously we define D.f 0 / ˚

WD v 2 H s;ı .R  X I F 0 ; G 0 / W resp e itw f 0 .w/F v.w/ D 0 for p 2 S.ı ;ıC / which is a closed subspace of H s;ı .R  X I F 0 ; G 0 / of finite codimension X X p.f 0 .w// D p.f .w//: codim D.f 0 / D p2S.ı ;ıC /

p2S.ıC ;ı /

Corollary 9.2.21 shows that when f .w/ has no non-bijectivity points on the lines Iı˙ the analogue opı .f 0 / W D.f 0 / ! H s;ı .R  X I E 0 ; J 0 / of the operator (9.2.9) is injective, and R.f 0 / is a closed subspace of H s;ı .RX I E 0 ; J 0 / of finite codimension X X n.f 0 .p// D n.f .p//: codim R.f 0 / D p2S.ı ;ıC /

p2S.ıC ;ı /

Using the fact (analogously as in Lemma 9.1.32) that for every u 2 H s;ı .R  X I E; J / there is a unique g 2 H s;ı .R  X I F; G/ such that hv; gi D ht Av; ui for v 2 D.f 0 /; hv; gi D 0

for v 2 H s;ı .R  X I F 0 ; G 0 /  D.f 0 /;

(9.2.12)

576

9 Operators in infinite cylinders and the relative index

we can define a continuous operator (9.2.8) in the case ı > ıC by Au WD g for g associated with u via (9.2.12). Theorem 9.2.23. Let f .w/ 2 B 0;0 R .X I vI C / be elliptic and have no non-bijectivity points on the lines I , ı > ı . operator (9.2.8), where C Then A defines a Fredholm ı˙  P P dim ker A D p2S.ı ;ı / n.f .p//; dim coker A D p2S.ı ;ı / p.f .p//, i.e., C



ind A D

C

X



fn.f .p//  p.f .p//g:

p2S.ıC ;ı /

The proof is analogous to that of Theorem 9.1.33. Remark 9.2.24. For boundary value problems we have an immediate analogue of the content of Section 9.1.3. This follows from the fact that the constructions are only based on the general properties of meromorphic operator functions.

9.3 The relative index for corner singularities We study elliptic operators on an infinite cylinder, where the cross section has conical singularities. The results will be of analogous structure as those in the case of boundary value problems before. A tricky point is the nature of holomorphic families of cone operator-valued families and the way to construct them by a kernel cut-off procedure. This is a link to Mellin symbols for higher corner singularities.

9.3.1 Parameter-dependent cone calculus Let B be a compact manifold with conical singularity v (for convenience, we content ourselves with one conical point; the case of finitely many conical points is analogous and left to the reader). Let B be the associated stretched manifold. Recall that B is a compact C 1 manifold with boundary @B Š X for a closed compact C 1 manifold x C  X with the splitting of X. We often identify a collar neighbourhood of @B with R x C/ variables .r; x/. A cut-off function on the half-axis is any (real-valued) ! 2 C 1 .R such that !.r/ 1 in a neighbourhood of r D 0. In the following considerations we focus on the case of operators between scalar weighted Sobolev spaces; the generalisation to distributional sections in vector bundles is straightforward and left to the reader. Let us briefly recall some notation from the edge symbolic calculus. On B we have the scale of weighted spaces H s; .B/ of smoothness s 2 R and weight  2 R. Moreover, on the infinite open stretched cone X ^ D RC  X we

9.3 The relative index for corner singularities

577

have the weighted spaces K s; .X ^ /. Moreover, we have the subspaces HPs; .B/ and KPs; .X ^ /, respectively, with discrete asymptotics of type P D f.pj ; mj ; Lj /gj D0;:::;N ; associated with the weight data .; ‚/, ‚ D .#; 0 , 1  # < 0, with N < 1 for finite #. Recall that (cf. the formula (2.4.90)) SP .X ^ / D lim hriN KPN; .X ^ /:  N 2N

Parameter-dependent operators on the cylinder Rq  B will be defined in terms of 2  2 block matrices H s; .B/ ˚ C j ! H s; .B/ ˚ C jC depending on the parameter  2 Rq . In a collar neighbourhood of @B the operator families will have the form of edge symbols with the covariable  and with constant coefficients. An ingredient are the Green symbols with discrete asymptotics.  The space RG .Rq I gI j ; jC /P;Q for g D .;   / and asymptotic types P and Q associated with weight data .  ; ‚/ and .; ‚/, respectively, is defined as the set of all operator-valued symbols \  z g./ 2 S .Rq I E s ; S/ cl

s2R

with E s WD K s; .X ^ / ˚ C j ; Sz WD SP .X ^ / ˚ C jC , such that the -wise formal adjoint with respect to the K 0;0 .X ^ / ˚ C j˙ scalar products has the property g  ./ 2 T   q zs s; C zs .X ^ / ˚ C jC ; S WD SQ .X ^ / ˚ C j . s2R Scl .R I E ; S/, E WD K  By RG .Rq I gI j ; jC / with g D .;   / we denote the union of all spaces  q RG .R I gI j ; jC /P;Q over P; Q. From now on we assume the weight interval ‚ to be finite, ‚ D ..k C 1/; 0 for some k 2 N: As we know there is another category of operator-valued symbols in the edge calculus, namely, the Mellin operator families  n 2

aM ./ D r  !.rΠ/ opM

.h/./!.rΠ/ Q

with cut-off functions !; !, Q and Q z; r/; h.r; z; / D h.r;

(9.3.1)

Q z; / x C ; L .X I C  Rq //; n D dim X . h.r; Q 2 C 1 .R cl Q x C ; L .X I R1Cq // is an arbitrary Recall the fact that when p.r; Q %; Q / Q 2 C 1 .R cl %; Q Q Q z; / x C ; L .X I C  Rq // parameter-dependent family on X there is an h.r; Q 2 C 1 .R cl Q such that for p.r; %; / WD p.r; Q r%; r/

578

9 Operators in infinite cylinders and the relative index

and (9.3.1) we have

 n 2

opM mod L

1

.X

^

.h/./ D opr .p/./

(9.3.2)

I Rq /,

for all  2 R. Here “ 0 opr .p/./v.r/ WD e i.rr /% p.r; %; /v.r 0 /dr 0 μ %:

Q z; r/, we have Q r%; r/; h0 .r; z; / WD h.0; In addition, for p0 .r; %; / WD p.0;  n 2

opM

.h0 /./ D opr .p0 /./

mod L1 .X ^ I Rq /. Given p and h, connected via the relation (9.3.2), we now form pc ./ WD  .r/faM ./ C a ./gQ .r/

(9.3.3)

where

QQ a ./ WD r  .1  !.rΠ// opr .p/./.1  !.rΠ// with cut-off functions !; !; Q !QQ and ; Q such that !Q 1 on supp !;

QQ ! 1 on supp !;

Q 1 on supp :

Finally, we have the smoothing Mellin operator families with asymptotics, namely, m./ WD r  !.rΠ/

k X j D0

rj

X



opMj˛

n 2

.mj˛ /˛ !.rŒ / Q

(9.3.4)

j˛jj

1 with Mellin symbols mj˛ .z/ 2 MR .X / for certain discrete Mellin asymptotic types j˛ Rj˛ , C Rj˛ \  nC1  D ;, and weights j˛ satisfying   j  j˛   for all j; ˛. j˛ 2 Then R .Rq ; gI j ; jC / for g D .;   / denotes the space of all operator block matrix families   a./ WD diag pc ./ C m./; 0 C g./  for arbitrary families (9.3.3) and (9.3.4) and Green symbols g./ 2 RG .Rq ; g; j ; jC /. Let CG .B; gI j ; jC /P;Q with g D .;   / denote the space of all operators \ G2 L.H s; .B/ ˚ C j ; H 1; .B/ ˚ C jC / s2R

such that G 2 L.H s; .B/ ˚ C j ; HP1; .B/ ˚ C jC /; 1; G  2 L.H s; C .B/ ˚ C jC ; HQ .B/ ˚ C j /

for all s 2 R and discrete asymptotic types P and Q, associated with the weight data .  ; ‚/ and .; ‚/, respectively, ‚ D ..k C 1/; 0 for some k 2 N. The formal adjoint G  refers to the scalar products of H 0;0 .B/ ˚ C j˙ . The space CG .B; gI j ; jC /P;Q is Fréchet in a natural way. Let CG .B; gI j ; jC / denote the union of all these spaces over P; Q.

579

9.3 The relative index for corner singularities

Definition 9.3.1. Let us fix a finite weight interval ‚ D ..k C 1/; 0 , and let C .B; gI j ; jC I Rq /;

g D .;   /;

denote the space of all 2  2 block operator families   a./ WD diag pc ./ C pint ./ C m./; 0 C g./

(9.3.5)

for arbitrary operator families of the form (9.3.3) and (9.3.4), moreover, pint ./ WD q QQ such .1  /f ./.1  QQ / for any f ./ 2 L cl .int BI R / and another cut-off function  that  1 on supp QQ , and an operator function g./ D diag.; 1/l./ diag.; Q 1/ C c./  for arbitrary l./ 2 RG .Rq ; gI j ; jC /, and c./ 2 S.Rq ; CG .B; gI j ; jC //; g D .;   /. In the case ‚ D .1; 0 we define C .B; gI j ; jC I Rq / as the intersection of the corresponding operator spaces belonging to weight intervals ..k C 1/; 0 , k 2 N.

Remark 9.3.2. The operator family pc ./ in Definition 9.3.1 can be replaced by n

pc ./ WD  .r/r  op 2 .h/./Q .r/

(9.3.6)

 .Rq ; gI 0; 0/; g D .;   /, cf. [57] (an analogous result modulo an element of RG for the case of boundary value problems is given in [89] or [90]).

Observe that every a./ 2 C .B; gI j ; jC I Rq /; g D .;   /, induces families of continuous operators a./ W H s; .B/ ˚ C j ! H s; .B/ ˚ C jC

(9.3.7)

for every s 2 R as well as a./ W HSs; .B/ ˚ C j ! HTs; .B/ ˚ C jC ; s 2 R, for every discrete asymptotic type S , associated with the weight data .; ‚/, with some resulting discrete asymptotic type T , associated with .  ; ‚/. The elements of C .B; gI j ; jC I Rq / have a parameter-dependent principal symbolic hierarchy consisting of a tuple  .a/ D . .a/; ^ .a// which is defined as follows. q Writing a./ D .aij .//i;j D1;2 , we have a11 ./ 2 L cl .int BI R /. Thus there exists q the parameter-dependent (with parameter  2 R ) homogeneous principal symbol Q / 2 C 1 .T  .int B/  Rq n 0/: Q ;  .a11 /.x;

580

9 Operators in infinite cylinders and the relative index

Because of the specific edge-degenerate structure near @B in the splitting of variables xQ D .r; x/, Q D .%; / the function Q .a11 /.r; x; %; ; / WD r   .a11 /.r; x; r 1 %; ; r 1 / is C 1 up to r D 0. Let us set  .a/ WD  .a11 /;

Q .a/ WD Q .a11 /:

(9.3.8)

 .Rq ; gI j ; jC /; g D From Definition 9.3.1 we have a Green family l./ 2 RG .;   /, with its homogeneous principal symbol of order 

^ .l/./ W K s; .X ^ / ˚ C j ! K s; .X ^ / ˚ C jC (of course, the image of the upper left corner is contained in K 1; .X ^ /). In addition we set  n 2

^ .pc /./ WD r  f!.rjj/ opM

.h0 /./!.rjj/ Q

QQ C .1  !.rjj// opr .p0 /./.1  !.rjj//g; ^ .m/./ WD r  !.rjj/

k X

rj

j D0

X



opMj˛

n 2

.mj˛ /˛ !.rjj/: Q

j˛jDj

We then define   ^ .a/./ WD diag ^ .pc /./ C ^ .m/./; 0 C ^ .l/./ which is a family of continuous operators ^ .a/./ W K s; .X ^ / ˚ C j ! K s; .X ^ / ˚ C jC ;

(9.3.9)

 2 Rq n f0g. For a./ 2 C .B; gI j ; jC / we set  .a/ D . .a/; ^ .a//:

(9.3.10)

Theorem 9.3.3. a./ 2 C .B; gI j0 ; jC I Rq /, g D .  ;   . C //, b./ 2 C .B; hI j ; j0 ; Rq /; h D .;   /, implies a./b./ 2 CC .B; g ı hI j ; jC I Rq / where g ı h D .;   . C //, and we have  .ab/ D  .a/ .b/ with componentwise multiplication. The proof of Theorem 9.3.3 is very close to that of Theorem 7.2.19. The evident modifications are left to the reader. Definition 9.3.4. An operator family a./ 2 C .B; gI j ; jC I Rq /, g D .;   /, is said to be elliptic if Q / 6D 0 for all .x; Q / 2 T  .int B/  Rq n 0, and, for .%; ; / 6D 0, (i)  .a/.x; Q ; Q ; Q .a/.r; x; %; ; / 6D 0, up to r D 0;

9.3 The relative index for corner singularities

581

(ii) ^ .a/./ defines a family of isomorphisms (9.3.9) for any fixed s 2 R. Theorem 9.3.5. Let a./ 2 C .B; gI j ; jC I Rq /, g D .;   /, be elliptic. Then there is a parametrix p./ 2 C .B; g 1 I jC ; j I Rq /, g 1 WD .  ; /, in the sense p./a./ D 1  c./; a./p./ D 1  d./ with remainders c./ 2 S.Rq ; CG .B; g l I j ; j //, g l D .;  /, d./ 2 S.Rq ; CG .B; g r I jC ; jC //, g r D .  ;   /. In addition (9.3.7) is a Fredholm operator for every  2 Rq , s 2 R, and there is a constant C > 0 such that the operators (9.3.7) are isomorphisms for all jj  C and all s 2 R. The proof of Theorem 9.3.5 essentially corresponds to an aspect of the proof of Theorem 7.2.36, namely, concerning the computation of P.y; /. The simple details will be omitted. Remark 9.3.6. Let a./ 2 C .B; gI j ; jC I Rq /, g D .;   /, be elliptic and assume that (9.3.7) is a family of isomorphisms for all  2 Rq and some s 2 R. Then a./ defines a family of isomorphisms for all s 2 R, and we have a1 ./ 2 C .B; g 1 I jC ; j I Rq /; g 1 D .  ;  /.

9.3.2 Meromorphic families We shall now introduce holomorphic families of operators in the cone algebra C .B; gI j ; jC /; g D .;   /. Let us first recall the notion of operator families on a C 1 manifold X that are holomorphic in a parameter. Let X be a C 1 manifold (not necessarily compact), and let L cl .X I C / denote the   space of all b.w/ 2 A.C ; Lcl .X // such that b. C i/ 2 Lcl .X I R / for every  2 R, uniformly in compact -intervals. Recall that the kernel cut-off construction allows us to produceR elements of i  L .XI C /. In fact, given any f ./ 2 L f ./μ. cl cl .X I R / we first set k.f /. / WD e 1 Moreover, we choose some '. / 2 C0 .R/. Then Z H.'/f . C i/ WD e i .Ci / '. /k.f /. /d (9.3.11) belongs L cl .XI C /. Remark 9.3.7. If 2 C01 .R/ is a cut-off function with respect to D 0, i.e., in a neighbourhood of the origin, we have

1

H. /f . C i/j D0 D f ./ mod L1 .X I R/: In a similar manner we can apply the kernel cut-off operator H. / to an operator function a./ 2 C .B; gI j ; jC I R /; g D .;   /, cf. Definition 9.3.1, and form  H. /a. C i/ D diag H. /pc . C i/ C H. /pint . C i/  (9.3.12) C H. /m. C i/; 0 C H. /g. C i/;

582

9 Operators in infinite cylinders and the relative index

cf. the formula (9.3.5) and Remark 9.3.2. Let us first consider H. /p c . C i/, cf. the  n 2

formula (9.3.6). Because of H. /pc .Ci/ D  .r/r  opM it suffices to analyse H. /h. C i/. First we have

.H. /h/.Ci/Q .r/

Q z; r/ h.r; z; / D h.r;

(9.3.13)

Q z; / x C ; L .X I C  RQ //. Let us set for an operator function h.r; Q 2 C 1 .R cl Z Z Q z; r/μ; kQ .h/.r; Q z; Q / WD e i Q Q h.r; z; /μ Q : Q k .h/.r; z; / WD e i  h.r; R Q z; /μ Q Q Q D r 1 kQ .h/.r; z; r /. Then, if This gives us k .h/.r; z; / D r 1 e i r Q h.r; . / is any cut-off function, it follows that Z H. /h.r; z;  C i/ D e i .Ci / . /k .h/.r; z; /d Z

Q D r 1 e i .Ci / . /kQ .h/.r; z; /d r Z Q D e ir Q .Ci / .r Q /kQ .h/.r; z; Q /d Q D .H. for

r . Q /

WD

Q

r /h/.r; z; r.

C i//

.r / Q which is an r-dependent cut-off function.

x Definition 9.3.8. Let L cl .X I RC  C  C / denote the space of all operator functions Q x C ; L .X I Cz // such that f .r; z; w/ D f .r; z; rw/. Here fQ.r; z; w/ Q 2 A.C wQ ; C 1 .R cl

and

fQ.r; z; rw/ 2 A.C w ; C 1 .RC ; L cl .X I Cz //

(9.3.14)

x C ; L .X I Cz  RQ // fQ.r; z; Q C i r/ 2 C 1 .R cl

(9.3.15)

for every  2 R, uniformly in c    c 0 for every c  c 0 . Proposition 9.3.9. Let 2 C01 .R/ be a cut-off function. Then, if h.r; z; / is defined Q z; / x C ; L .X I C  RQ //, we have by (9.3.13) for an h.r; Q 2 C 1 .R cl x H. /h.r; z; w/ 2 L cl .X I RC  C  C / and

Q z; 0/ D M .H. /h/.z/: M .h/.z/ D h.0;

(9.3.16)

Proof. The property (9.3.14) for f .r; z; w/ WD H. /h.r; z; w/ is an immediate consequence of the kernel cut-off procedure which generates holomorphic functions in w 2 C. For (9.3.15) we first consider .H.

Q

Q r1 /h/.r; z; 

C i r2 /;

(9.3.17)

9.3 The relative index for corner singularities

583

where r1 in the first argument indicates dependence on the half-axis variable coming from r1 , while r2 comes from the factor at  (the other r-variable does not affect the conclusions). To verify the smoothness of (9.3.15) in r up to r D 0 it suffices to show xC  R x C . The smoothness in r2 2 R x C is evident that (9.3.17) is C 1 in .r1 ; r2 / 2 R for every  2 R, uniformly in finite intervals. The crucial point is the smoothness in r1 which is a parameter in the family of cut-off functions .r1 Q / that tends to 1 for r1 ! 0. Here we can apply a property on the kernel cut-off that shows that the desired smooth dependence in the parameter up to zero concerns the topology of symbols. At the same time we obtain the relation (9.3.16). Concerning the summand H. /pint . C i/ for pint ./ as in (9.3.5) we have H. /pint . C i/ D .1   /H. /f . C i/.1  QQ /; where H. /f . C i/ is obtained from the formula (9.3.11) with and int B instead of ' and X, respectively. Let us now turn to the Green summand on the right of (9.3.5). According to Definition 9.3.1 we set H. /g. C i/ D diag.; 1/H. /l. C i/ diag.; Q 1/ C g1 . C i/: (9.3.18) Here g1 . C i/ denotes any element of A.C; CG .B; gI j ; jC //; g D .;   /, such that g1 . C i/ 2 S.R ; CG .B; gI j ; jC // (9.3.19) for every  2 R, uniformly in compact -intervals. Moreover, we set, analogously as (9.3.11) Z H. /l. C i/ D

e i .Ci / . /k.l/. /d :

Let M G;O .B; gI v/, g D .;   /; v WD .j ; jC /, denote the space of all operator  functions of the kind (9.3.18) for arbitrary l./ 2 RG .R; gI j ; jC /. Observe that  H. /l. C i/ 2 RG .R ; gI j ; jC / for every  2 R, uniformly in compact -intervals. To complete (9.3.12) it remains to characterise H. /m. C i/. By definition the operators m./ are given by the expression (9.3.4). Formally we proceed as before, namely, we set Z H. /m. C i/ D e i .Ci / . /k.m/. /d : (9.3.20) R with k.m/. / WD e i  m./μ. In order to characterise the structure of (9.3.20) we first observe that m./ 2 Scl .RI K s; .X ^ /; K 1; .X ^ // for every s 2 R. The space in the image may even be replaced by S  .X ^ /; recall that x C ; C 1 .X //g: S  .X ^ / D f!u C .1  !/v W u 2 K 1; .X ^ /; v 2 S.R

584

9 Operators in infinite cylinders and the relative index

That means the kernel cut-off expression (9.3.20) can be seen from the point of view of operator-valued symbols (similarly as in the case of Green symbols before). From that interpretation we know that for every fixed  2 R H. /m. C i/ 2 Scl .RI K s; .X ^ /; K 1; .X ^ // and H. /m. C i/ 

1 X j D0

cj

 dj  H. /m . C i0 / dj

(9.3.21)

for every 0 ;  2 R, with constants cj depending on 0 ; , and c0 D 1. In addition we know that H. /m./ D m./ mod S 1 .RI K s; .X ^ /; K 1; .X ^ //: This gives us together with the relation (9.3.21) H. /m. C i/  m./ C

1 X j D1

cj

dj H. /m./: dj

Looking at (9.3.4) we see that for every fixed  2 R the family H. /m. C i/ is again an expression of the kind (9.3.4), modulo a Green symbol as considered before, and that M .H. /m. C i//.z/ D M .m/.z/ D m00 .z/: Definition 9.3.10. The space M O .B; gI v/;

g D .;   /; v D .j ; jC /;

is defined to be the set of all operator functions of the form   a.w/ D diag aM .w/ C aint .w/ C m.w/; 0 C g.w/ where  n 2

(i) aM .w/ D  .r/r  opM Definition 9.3.8;

x .f /.w/Q .r/, f .r; z; w/ 2 L cl .X I RC  C  C /, cf.

(ii) aint .w/ D .1   .r//b.w/.1  QQ .r// for a element b.w/ 2 L cl .int B; C /, with Q cut-off functions ; ; Q Q supported near @B such that Q 1 on supp  ,  1 on QQ supp ; (iii) m.w/ is of the form m.w/ D H. /m0 .w/ of analogous structure as (9.3.20) for some m0 of the form (9.3.4), w D  C i;

9.3 The relative index for corner singularities

585

(iv) g.w/ is of the form g.w/ WD diag.; 1/H. /l0 .w/ diag.; Q 1/ C g1 .w/  .R; gI j ; jC /; g D .;   /, and an element (9.3.19), for an l0 ./ 2 RG w D  C i.

For Iı D fw 2 C W Im w D ıg and g D .;   / let C .B; gI j ; jC I Iı / denote the space of all operator functions a W Iı ! C .B; gI j ; jC / such that a. C iı/ 2 C .B; gI j ; jC I R /, cf. Definition 9.3.1. Let R D f.pj ; mj ; Nj /gj 2Z (9.3.22) be a sequence, with points pj 2 C , j Im pj j ! 1 as jj j ! 1, mj 2 N, and finitedimensional subspaces Nj  CG .B; gI j ; jC / of operators of finite rank. The weight interval ‚ contained in the operators of Nj is fixed, ‚ WD ..l C1/; 0 , l 2 N [fC1g. Set C R WD fpj gj 2Z . Sequences (9.3.22) will also be called corner asymptotic types. The constructions that contributed to Definition 9.3.10 can be interpreted as the following result. Theorem 9.3.11. For everyˇ f 2 C .B; gI j ; jC I Iı / there exists an h 2 M O .B; gI v/, v D .j ; jC /, such that .hˇI  f /.w/ 2 S.Iı ; C .B; gI j ; j // and, in particular,  C G ı ˇ (9.3.23)  .f / D  .hˇI /; ı

cf. the notation (9.3.10).

ˇ ˇ ˇ / D  .hˇ / for every Remark 9.3.12. For every h 2 M .B; gI v/ we have  .h O Iı IıQ ı; ıQ 2 R. Theorem 9.3.13. Let f .w/ 2 M O .B; gI v/ for g D . ;  .C//, v D .j0 ; jC /, and g.w/ 2 MO .B; hI w/ for h D .;   /; w D .j ; j0 /. Then the composition of operators defines a map C  M .B; g ı hI v ı w/ O .B; gI v/  MO .B; hI w/ ! MO

and .ab/ D  .a/ .b/ with componentwise composition. Definition 9.3.14. Let M1 G;R .B; gI v/, g D .;   /, v D .j ; jC /, defined to be the space of all operator families f .w/ 2 A.C n C R; CG .B; gI j ; jC // such that (i) for any C R-excision function  we have .f /. C i ı/ 2 S.R ; CG .B; gI j ; jC // for every ı 2 R, uniformly in compact ı-intervals;

586

9 Operators in infinite cylinders and the relative index

(ii) f .w/ is meromorphic with poles at the points pj of multiplicity mj C 1 and Laurent coefficients at .w  pj /.kC1/ belonging to Nj for all 0  k  mj , j 2 Z. Let us set  1 M R .B; gI v/ WD MO .B; gI v/ C MG;R .B; gI v/

(9.3.24)

in the sense of a non-direct sum.  Theorem 9.3.15. Let f .w/ 2 M R .B; gI v/ and g.w/ 2 MS .B; hI w/ for g; v; h; w as in Theorem 9.3.13. Then we have

f .w/g.w/ 2 MPC .B; g ı hI v ı w/ with a resulting asymptotic type determined by f and g. Let f .w/ 2 M R .B; gI v/. Then (9.3.24) gives us a decomposition f .w/ D fo .w/ C fr .w/ where fo .w/ 2

M O .B; gI v/

and fr .w/ 2

(9.3.25)

M1 G;R .B; gI v/.

Remark 9.3.16. For every g.w/ 2 M1 G;P .B; gI v/ for g D .;  /, v D .j; j / and a corner asymptotic type P there exists a g.w/ Q 2 M1 G;Q .B; gI v/ for another corner asymptotic type Q such that .1  g/1 D 1  g: Q In fact, this can be obtained in an analogous manner as [174, Lemma 4.3.13]. Definition 9.3.17. An element f 2 M O .B; gI v/; g D .;   /;

v D .j ; jC /;

(9.3.26) ˇ is called elliptic if for some ı 2 R with C R \ Iı D ; the operator family f ˇI 2 ı C .B; gI j ; jC I Iı / is elliptic with parameter  D Re w for w 2 Iı , cf. Definition 9.3.4. Remark 9.3.18. The ellipticity in the sense of Definition 9.3.17 is independent of the Q for specific choice of the weight ı. In fact, we have  .f . C i ı// D  .f . C i ı// Q every ı; ı 2 R, cf. formula (9.3.23). Theorem 9.3.19. Let (9.3.26) be elliptic. Then there is a discrete set of points D  C with finite intersection D \ fw W c 0  Im w  c 00 g for every c 0  c 00 such that the operators f .w/ W H s; .B/ ˚ C j ! H s; .B/ ˚ C jC are invertible for all w 2 C n D. The inverse extends to an element 1 1 f 1 2 M Iv / S .B; g

for g 1 D .  ;  /; v1 D .jC ; j / and some corner asymptotic type S , and f 1 is the inverse of f in the sense of Theorem 9.3.13.

9.3 The relative index for corner singularities

587

ˇ Proof. By virtue of the ellipticity of (9.3.26) there is a ı 2 R such that fı WD f ˇI ı is elliptic in the sense of Definition 9.3.4. Then Theorem 9.3.5 gives us a parametrix pı .w/ 2 C .B; g 1 I jC ; j I Iı /. Theorem 9.3.11 applied toˇpı yields an element 1 1 I v / which is also elliptic such that hı WD hˇI satisfies h 2 M O .B; g ı

.hı  pı /.w/ 2 S.Iı ; CG .B; gI j ; jC //: It follows that 1  .hı fı /.w/ 2 S.Iı ; CG .B; gI j ; j // and hence g.w/ WD 1  .hf /.w/ 2 M1 G;P .B; g l I vl / for g l D .; /, vl D .j ; jC / and some corner asymptotic type P . Now, according .B; g l I vl / such that 1  g.w/ Q D .1  to Remark 9.3.16, there is a g.w/ Q 2 M1 G;Pz 1 1 g.w//1 . This gives us f 1 .w/ WD .1  g.w//h.w/ Q 2 M I v / for some S .B; g corner asymptotic type S. The existence of the discrete set D is a consequence of the fact that h.w/ is a holomorphic family of Fredholm operators in L.H s; .B/ ˚ C jC ; H s; .B/ ˚ C j / which are invertible for large j Re wj. Hence the set D0 of points w 2 S.c 0 ;c 00 / where the invertibility of h.w/ is violated is discrete and intersects every strip c 0  Im w  c 00 for c 0  c 00 in a finite set. A similar observation is true of the set of non-invertibility points of the operator function 1 C g.w/ Q which can be obtained analogously as [174, Lemma 4.3.13].

Definition 9.3.20. An element f 2 M R .B; gI v/, where g D .;   /, v D .j ; jC /, is said to be elliptic if so is fo for any decomposition of f of the form (9.3.25). The definition is correct, i.e., independent of the particular choice of the splitting (9.3.25) of f . Theorem 9.3.21. Let f .w/ 2 M R .B; gI v/, g D .;  /, v D .j ; jC /, be elliptic. Then there is a countable subset D  C , with finite intersection D \fw W c 0  Im w  c 00 g for every c 0  c 00 , such that f .w/ W H s; .B/ ˚ C j ! H s; .B/ ˚ C jC

(9.3.27)

is invertible for all w 2 C n D. Moreover, there is an inverse (in the sense of the 1 1 composition of Theorem 9.3.15) belonging to M I v /, g 1 D .  ; /, S .B; g v1 D .jC ; j /, for a resulting asymptotic type S .

9.3.3 Examples Examples 9.3.22. (i) Set n

h.w/ WD !.r/ r



X j Ck

j

k

aj k .r/.r@r / .rw/

o

C .1  !.r//

 X

ˇ bk ˇint B w k ;

kD0

(9.3.28)

588

9 Operators in infinite cylinders and the relative index

x C ; Diff .j Ck/ .X // and bk 2 Diff k .2B/. We !.r/ a cut-off function, aj k 2 C 1 .R  then obtain an element of MO .B; gI v/, g D .;   /, v D .0; 0/. (ii) (9.3.28) represents a holomorphic family h.w/ W H s; .B/ ! H s; .B/, s 2 R. For instance, we can assume  D 2 and take h11 ./ D B  jj2 ;

(9.3.29)

where B is the Laplace–Beltrami operator on B associated with a Riemanian metric on B that has locally near @B the form dr 2 C r 2 gX for a Riemanian metric gX on X , j  P i.e., B D r 2 j Ck2 aj k .r/ r@r near r D 0 for suitable coefficients aj k .r/ 2 x C ; Diff 2.j Ck/ .X //. C 1 .R Example 9.3.23. In order to construct an elliptic element h.w/ 2 M O .B; gI v/ we start from a family of differential operators as in Example 9.3.22. Since elliptic operator families consist of 2  2 block matrices we now denote (9.3.28) by h11 .w/. Let us start from a family of that kind such that h11 ./ is parameter-dependent elliptic with the parameter  2 R. In addition we choose a weight  2 R such that X j  aj k .0/ r@r .r/k ^ .h11 /./ D r  j Ck

induces a family of Fredholm operators ^ .h11 /./ W K s; .X ^ / ! K s; .X ^ / for one (and then every) s 2 R and all  6D 0. This is always possible; it suffices to look at the non-bijectivity points of the principal conormal symbol c ^ .h11 /.z/ W H s .X/ ! H s .X /, z 2 C, and to take  in such a way that  nC1  does not contain 2 such a point. Moreover, we assume that ind ^ .h11 /.1/ D ind ^ .h11 /.C1/ which is a very mild topological condition on the index, satisfied, for instance, in the case (9.3.29). This condition does not depend on  . In concrete cases, e.g., for the operator (9.3.29), possible choices of admitted weights  can easily be calculated, cf. [182, Section 2.2.3]. As is well known there exist numbers j˙ and a family of isomorphisms ^ .h/./ W K s; .X ^ / ˚ C j ! K s; .X ^ / ˚ C jC .hij /i;j D1;2 , ^ .h/./

(9.3.30)

 diag. ; 1/^ .h/./ diag.1 ; 1/.

for  6D 0, h D D Let us form the symbols fij ./ WD ./^ .hij /./ for i; j D 1; 2; i C j > 2, with an excision function ./. The entries ^ .hij /./ for i C j > 2 can be chosen in such a way that  f12 ./ 2 Scl .RI C j ; S‚ .X ^ //;

  f21 ./ 2 Scl .RI C jC ; S‚ .X ^ //:

Applying kernel cut-off with respect to the covariable , similarly as at the beginning of Section 9.3.2, we obtain holomorphic symbols mij .m/ such that  m12 ./  f12 ./ 2 S 1 .RI C j ; S‚ .X ^ //;   m21 ./  f21 ./ 2 S 1 .RI C jC ; S‚ .X ^ //;

9.3 The relative index for corner singularities

589

and m22 ./  f22 ./ 2 S 1 .RI C j ; C jC /. Now let h11 be as before, and consider the holomorphic operator families h12 .w/ WD !m12 .w/ W C j ! H 1; .B/; h21 .w/ WD m21 .w/! W H s; .B/ ! C jC ; and h22 .w/ WD m22 .w/ W C j ! C jC , for any fixed cut-off function !.r/. Then h.w/ WD .hij .w//i;j D1;2 2 M O .B; gI v/, g D .;   /, v D .j ; jC /, is elliptic in the sense of Definition 9.3.20. Example 9.3.24. Let us consider the unit interval B WD I 3 r as a manifold with conical singularities r D 0 and r D 1. We take the operator family ˚

˚

h11 .w/ WD !0 .r/r 2 .r@r /2  r@r  r 2 w 2 C !1 .t /t 2 .t @ t /2  t @ t  t 2 w 2 for t WD 1  r, 0 < r < 1, and w D  C i. Here !0 .r/ is a cut-off function with respect to r D 0, i.e., !0 .r/ D 1 near r D 0 such that !0 .r/ D 0 for r > 12 , and !1 .t/ WD 1  !.1  t /. We then have a holomorphic family of operators h11 .w/ W H s; .I / ! H s2; 2 .I / for arbitrary s;  2 R. We want to find dimensions j˙ such that the principal edge symbol (9.3.30) is a family of isomorphisms for  6D 0. Let us first consider the case s D  D 2 (clearly the choice of s is not essential, in contrast to ).Then we have h11 .w/ W H 2;2 .I / ! H 0;0 .I / D L2 .I /: Since the operators for r D 0 and r D 1 are of a similar nature, we consider the case r D 0. The edge symbol of h11 .w/ at r D 0 is equal to ^;0 .h11 /./ D

@2  2 W K 2;2 .RC / ! L2 .RC /; @r 2

(9.3.31)

 6D 0, which is injective with dim coker equal to 1. Since the kernel and cokernel of (9.3.31) are independent of s, we have the family of injective operators ^;0 .h11 /./ W K s;2 .RC / ! K s2;0 .RC / for all s 2 R. Moreover, c .^;0 /.h11 /.z/ D z 2 C z shows that ^;0 .h11 /./ W K s; .RC / ! K s2; 2 .RC /

(9.3.32)

is a family of Fredholm operators for all  2 R,  6D 12 ; 32 . It can easily be proved that 8 ˆ for  < 12 ; <1 ind ^;0 .h11 /./ D 0 for 12 <  < 32 ; ˆ : 1 for  > 32 :

590

9 Operators in infinite cylinders and the relative index

More precisely, for  < 12 we have surjectivity of (9.3.32) and a kernel of dimension 1; for  > 32 we obtain injectivity and a cokernel of dimension 1, while 12 <  < 32 corresponds to isomorphisms of (9.3.32) for all s 2 R. As noted before, similar considerations are valid with respect to r D 1. Then the constructions for Example 9.3.23 contain contributions from both end points. Identifying I with B we thus arrive at elliptic elements (9.3.27) for  D 2 and j D 0, jC D 2 for  < 12 , and j D 2, jC D 0 for  > 32 , while j D jC D 0 for 12 <  < 32 .

9.3.4 Characteristic values and factorisation Let f .w/ 2 M R .B; gI v/, g D .;   /, v D .j ; jC /, be elliptic. For a fixed w 2 C away from the set of poles, f .w/ defines an operator H s; .B/ ˚ C j ! H s; .B/ ˚ C jC . The particular choice of s is not import