Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends

Operator Theory: Advances and Applications Volume 219 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Black...

Author: B. Malcolm Brown | Jan Lang | Ian G. Wood

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Operator Theory: Advances and Applications Volume 219 Founded in 1979 by Israel Gohberg

Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)

B. Malcolm Brown Jan Lang Ian G. Wood Editors

Spectral Theory, Function Spaces and Inequalities New Techniques and Recent Trends

Editors B. Malcolm Brown School of Computer Science and Informatics Cardiff University Cardiff, CF24 3XF UK

Jan Lang Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, OH 43210 USA

Ian G. Wood School of Mathematics, Statistics and Actuarial Science University of Kent Cornwallis Building Canterbury, Kent CT2 7NF UK

ISBN 978-3-0348-0262-8 e-ISBN 978-3-0348-0263-5 DOI 10.1007/978-3-0348-0263-5 Springer Basel Dordrecht Heidelberg London New York Library of Congress Control Number: 2011941087 Mathematics Subject Classification (2010): Primary: 11R09, 26A12, 26D10, 34A40, 34B24, 35J60, 35Q40, 35P99, 42B20, 46E30, 46E35, 47B25, 47F05, 47G30; Secondary: 26D07, 26D15, 34L05, 34L40, 35J70, 35P15, 35S05, 47B10, 47B35, 81Q35  Springer Basel AG 2012 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. Printed on acid-free paper

Springer Basel AG is part of Springer Science + Business Media (www.birkhauser-science.com)

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii B.M. Brown, J. Lang and I.G. Wood David Edmunds’ Mathematical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix B.M. Brown, J. Lang and I.G. Wood Desmond Evans’ Mathematical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi B.M. Brown and M.S.P. Eastham Generalised Meissner Equations with an Eigenvalue-inducing Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 B.M. Brown and K.M. Schmidt On the HELP Inequality for Hill Operators on Trees . . . . . . . . . . . . . . . 21 F. Cobos, L.M. Fern´ andez-Cabrera and A. Mart´ınez Measure of Non-compactness of Operators Interpolated by Limiting Real Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 R.L. Frank and E.H. Lieb A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality . . . . . . . . . . . . . . . . . . . . . . . 55 D.D. Haroske Dichotomy in Muckenhoupt Weighted Function Space: A Fractal Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 I. Knowles and M.A. LaRussa Lavrentiev’s Theorem and Error Estimation in Elliptic Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 V. Kokilashvili and A. Meskhi Two-weighted Norm Inequalities for the Double Hardy Transforms and Strong Fractional Maximal Functions in Variable Exponent Lebesgue Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 J. Lang and O. M´endez Modular Eigenvalues of the Dirichlet p(·)-Laplacian and Their Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 R.T. Lewis Spectral Properties of Some Degenerate Elliptic Diﬀerential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

vi

Contents

B. Opic Continuous and Compact Embeddings of Bessel-Potential-Type Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Y. Sait¯ o and T. Umeda A Sequence of Zero Modes of Weyl–Dirac Operators and an Associated Sequence of Solvable Polynomials . . . . . . . . . . . . . . . . . . . . . 197 A.V. Sobolev A Szeg˝o Limit Theorem for Operators with Discontinuous Symbols in Higher Dimensions: Widom’s Conjecture . . . . . . . . . . . . . . 211 V.D. Stepanov On a Supremum Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 H. Triebel Entropy Numbers of Quadratic Forms and Their Applications to Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Preface This is a a collection of contributed papers by David and Des’s friends and colleagues and is issued to mark their respective 80th and 70th birthday. For the past forty years they have made fundamental contributons in the area of diﬀerential equations, operator theory and function space theory, and it is ﬁtting that these contributions reﬂect that. Our thanks must also go to Dr Thomas Hempﬂing, Executive Editor, Mathematics Birkh¨ auser and Ms Sylvia Lotrovsky for the help and assistance given to us during the preparation of this volume. Finally we thank all the authors who contributed papers to this special edition to mark David and Des’s birthdays. B.M. Brown J. Lang I.G. Wood

Operator Theory: Advances and Applications, Vol. 219, ix-x c 2012 Springer Basel AG

David Edmunds’ Mathematical Work B.M. Brown, J. Lang and I.G. Wood

David Edmunds has inﬂuenced and made major contributions to numerous branches of mathematics. These include spectral theory, functional analysis, approximation theory, the theory of function spaces, operator theory, ordinary and partial diﬀerential equations. The breadth of his impact is demonstrated by his publication record, which consists of 5 books and more than 190 research papers, and by his winning the LMS P´ olya prize in 1996 and the Bolzano Medal of the Czech Academy of Sciences, in 1998. He was awarded the Ph.D. degree by the University of Wales in 1955, having been supervised by R.M. Morris. After some years working for EMI Electronics on guided missiles, he held positions of Lecturer and then Senior Lecturer at the University of Wales, Cardiﬀ, leaving in 1966 to take up a Readership at the University of Sussex. He was awarded a Personal Chair there in 1970 and is still aﬃliated to Sussex as well as additionally being appointed Honorary Professor, School of Mathematics, Cardiﬀ University, 2004. In his early works he focused on problems of ﬂuid dynamics, including moving aerofoils, magneto-hydrodynamics and the nature of solutions of the Navier-Stokes equations (studying questions of stability, backward uniqueness, asymptotic behaviour and removable singularities). Then his interests shifted towards the study of more general non-linear problems, elliptic equations and inequalities, and functional analysis. His ﬁrst joint paper with W.D. Evans appeared in 1973. In this, by deriving new weighted embeddings on unbounded domains in Lp spaces, results were obtained for the Dirichlet problem concerning elliptic equations. This work began their long and fruitful collaboration and established their common interest in the properties of function spaces, embedding theorems, integral operators and spectral theory. These matters were also the main topics of their two joint books, the well-known ‘Spectral theory and diﬀerential operators’ (OUP) and the more recent ‘Hardy operators, function spaces and embeddings’ (Springer). Edmunds’ work on the properties of Besov and Lizorkin spaces has often been motivated by his interest in the nature of eigenvalues and eigenvectors of operators acting on non-Hilbert spaces. Many of his papers on this topic are

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concerned with the qualitative and quantitative properties of embeddings of function spaces, such as the behaviour of their entropy and s-numbers. These results, including those obtained with Hans Triebel, formed the basis of their joint book, ‘Function spaces, entropy numbers, diﬀerential operators’ (CUP). Interest in functional analysis led him to study interpolation theory. His recent results with Yuri Netrusov settled a long-standing conjecture concerning the behaviour of entropy numbers under real interpolation. In the theory of integral operators he has concentrated on maps of Hardy or Volterra type, acting on function spaces with and without weights; many of his results in this area are presented in his book ‘Bounded and compact integral operators’, with V. Kokilashvili and A. Meskhi. His very recent book ‘Eigenvalues, embeddings and generalised trigonometric functions’, with J. Lang, has as its basis their work on the properties of s-numbers of Hardy-type operators, which involves the study of eigenfunctions of integral and diﬀerential operators together with certain generalisations of the trigonometric functions that are of importance in non-linear analysis. Another of his interests is the currently popular theory of Lp(x) spaces, the so-called variable exponent spaces, fundamental work on which was carried out by J. R´akosn´ık, and together with whom results were obtained that have proved stimulating for many analysts. His standing attracted a number of mathematicians, such as D. Vassiliev and A. Sobolev, who started their professional careers in the UK at Sussex. He was also active in making contacts with colleagues from other countries. He has supervised 18 Ph.D. students, including J.M. Ball (Oxford), J.R.L. Webb (Glasgow) and V. Mustonen (Oulu); according to the Mathematics Genealogy project he has 73 descendants. David Edmunds’ contribution to mathematics is not only long, but wide and deep. His superb professional work and his warm personality have deeply inﬂuenced a large part of the mathematical community. B.M. Brown School of Computer Science, Cardiﬀ University, Cardiﬀ CF24 3XF UK e-mail: [email protected] J. Lang Department of Mathematics, The Ohio State University, 231 West 18th Avenue Columbus, OH 43210-1174 USA e-mail: [email protected] I.G. Wood School of Mathematics, Statistics and Actuarial Science, Cornwallis Building University of Kent, Canterbury, Kent CT2 7NF UK e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, xi-xii c 2012 Springer Basel AG

Desmond Evans’ Mathematical Work B. M. Brown, J. Lang and I. G. Wood

Desmond Evans has made major inﬂuential contributions to numerous branches of mathematical analysis which include the spectral theory of both ordinary and partial diﬀerential equations, mathematical physics and functional analysis. The breadth and depth of his achievements are recorded in his 142 published papers and 3 books. Following a B.Sc (Wales) from University College, Swansea in 1961 he went up to Jesus College, Oxford to work for a D.Phil. under the guidance of E. C. Titchmarsh, one of the leading analysts of the day. After the sudden death of Titchmarsh in 1963, Des completed his studies under the direction of another leading analyst, J.B.McLeod. The degree was awarded in 1965. In 1964 he was appointed to a lectureship in Pure Mathematics at the then University College of South Wales and Monmouthshire, which later became Cardiﬀ University, and has remained at Cardiﬀ all his working life, progressing through the grades of senior lecturer and reader before being awarded a Personal Chair by the University of Wales in 1977. During this time at Cardiﬀ he has supervised 13 Ph.D. students. Following some early work on the Dirac system he worked on the limitpoint, limit-circle classiﬁcation problem for ordinary diﬀerential equations, inequalities related to diﬀerential and diﬀerence equations (in particular the HELP inequality and its later variants), and on spectral problems associated with non-selfadjoint diﬀerential systems. His work on partial diﬀerential equations has often been motivated by physical questions, a signiﬁcant portion having been concerned with problems arising in the study of non-relativistic quantum mechanics. This research contains work on the spectrum of relativistic one-electron atoms and on the zero modes of Pauli and Weyl-Dirac operators. His many papers in this and related areas cover an impressive range of topics. These include the spectral analysis of N-body operators for atoms and molecules; quantum graphs; Hardy and Rellich inequalities with magnetic potentials; Schr¨odinger operators and biharmonic operators with magnetic ﬁelds. He has been active also in functional analysis and operator theory, especially in areas concerning the properties of Hardy-type operators acting

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between function spaces, estimates and asymptotic results for their approximation numbers and related inequalities. Much of this work was motivated by his study of the properties of embedding maps between Sobolev spaces deﬁned on irregular (including fractal) domains, in which he and Desmond Harris introduced the notion of a generalised ridged domain and developed techniques for reducing problems to analogous ones on associated trees. His two research monographs with David Edmunds, “Spectral Theory and Diﬀerential Equations” and “Hardy Operators, Function Spaces and Embeddings” have become standard texts in his main areas of activity. Recently, with Alex Balinsky, he has published “Spectral analysis of relativistic operators”, which includes, in particular, an account of their numerous contributions to problems concerning the stability of matter. As well as his mathematical contributions, Des has been active in various administrative roles both within and outside Cardiﬀ University. In particular, he has been Head of School at Cardiﬀ on several occasions, and served the London Mathematical Society over many years as Editorial Advisor, Editor of the Proceedings and Council member. Recently he has played an important part in establishing the Wales Institute of Mathematical and Computational Sciences. In recognition of these achievements he was elected a fellow of the Learned Society of Wales in 2011. (This is an expanded version of the Laudatum in JCAM V208 1 November 2007) B. M. Brown School of Computer Science Cardiﬀ University, Cardiﬀ, CF24 3XF, U.K. e-mail: [email protected] J. Lang Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210-1174 USA e-mail: [email protected] I. G. Wood School of Mathematics, Statistics and Actuarial Science, Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, UK e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 1–20 c 2012 Springer Basel AG

Generalised Meissner Equations with an Eigenvalue-inducing Interface B.M. Brown and M.S.P. Eastham To David and Des

Abstract. An interface situation is considered, where a periodic diﬀerential equation is given on one side x > 0 of the interface and a general Sturm–Liouville equation is given on a ﬁnite interval (−X, 0) on the other side of the interface. A boundary condition is imposed at −X. The emphasis is on a periodic discontinuous weight function, which has the eﬀect of widening the spectral gaps (instability intervals). It is shown that the interface can induce eigenvalues in all the gaps beyond some point. The dependence on X of the number of eigenvalues in each gap is noted. The general theory is supported by step-function examples. Mathematics Subject Classiﬁcation (2010). Primary 34B24; Secondary 34L05. Keywords. Periodic, interface, Meissner.

1. Introduction A basic property of the periodic diﬀerential equation y (x) + {λw(x) − q(x)}y(x) = 0

(1.1)

on an unbounded x-interval I is the existence of stability and instability λintervals on the real λ-axis. Here the weight function w(x) and the potential function q(x) are real-valued and Lloc (I) with w(x) > 0, and they have a common period a; I can be either (−∞, ∞) or a semi-inﬁnite interval, let us say [0, ∞). The stability intervals are speciﬁed in terms of the eigenvalues λn and μn (n ≥ 0) of (1.1) arising respectively from the periodic boundary conditions y(0) = y(a), y (0) = y (a) (1.2) and the semi-periodic boundary conditions y(0) = −y(a), y (0) = −y (a).

(1.3)

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B.M. Brown and M.S.P. Eastham

These eigenvalues have the ordering λ0 < μ0 ≤ μ1 < λ1 ≤ λ2 < μ2 ≤ μ3 < λ3 · · · .

(1.4)

Then the stability intervals are the open intervals (λ2m , μ2m ),

(μ2m+1 , λ2m+1 ) (m ≥ 0)

(1.5)

which, together possibly with their end-points, comprise the values of λ for which all solutions of (1.1) are bounded in I. In terms of the spectral theory of (1.1) in the Hilbert space L2 (I), the intervals (1.5) exhibit the band structure of the associated spectrum in the sense that the essential spectrum is formed by the closures of (1.5). We refer to the standard books [4, chapters 8–9], [6], [17, chapter 13] and [22, chapter 21] for all this basic theory of (1.1). We also refer to [8] for (1.4) in a wider context of more general boundary conditions. Moving on to the particular topic considered in this paper, we note ﬁrst that, when I is (−∞, ∞), (1.1) has no eigenvalues in the spectral gaps between the intervals (1.5) but, when I is [0, ∞) and the usual type of boundary condition is imposed at x = 0, any given λ in a gap can be an eigenvalue arising from an associated boundary condition [4, p. 257], [9, Theorem 2.5.3]. If however the periodicity of (1.1) is compromised by some perturbation, there is the possibility that eigenvalues can appear in the gaps as a result of the perturbation, that is, new spectral points may arise. Historically, the ﬁrst type of perturbation to be considered in this context is the addition to q(x) of a non-periodic function p(x) which is small in some sense when |x| is large. On the one hand, we have the result that if p(x) = O(|x|)−2 ) (|x| → ∞), then only a ﬁnite number of eigenvalues can appear in suﬃciently distant spectral gaps [19, Corollary 3] (see also [18]). On the other hand, if a ﬁxed spectral gap is considered and p(x) contains a coupling parameter c, the number of eigenvalues in the gap can become large for large c, and the asymptotic distribution is investigated in [3], [20], [21] (see also [1]). Another type of perturbation is based on the idea of introducing an interface where (1.1) holds on one side of the interface (say for x > 0) and a diﬀerent periodic equation holds on the other side x < 0. In [13] a dislocation situation is considered in which w(x) = 1 and the potential for x < 0 is q(x + t) where t ∈ R is the dislocation parameter. It is shown that a spectral point (an eigenvalue or a resonance) λ(t) is produced in each spectral gap, and its behaviour in terms of t is discussed [13, pp. 474, 480]. A biperiodic situation is introduced in [14] where q has a diﬀerent period for x < 0, and here it is shown that up to two spectral points appear in each spectral gap. Recently, similar interface problems have been considered in [5] again with w(x) = 1 and using the method of C 1 gluing across the interface. In [5, section 3] explicit conditions on q are derived which guarantee the appearance of up to two eigenvalues in the ﬁrst two spectral gaps (−∞, λ0 ) and (μ0 , μ1 ). In this paper we consider a diﬀerent interface situation where a new feature is that arbitrarily many eigenvalues can occur in the spectral gaps (μ2m , μ2m+1 ) and (λ2m+1 , λ2m+2 ) (m ≥ 0). Our main focus is on the weight

Generalised Meissner Equations

3

function w(x) and, in (1.1), we take w(x) to be non-constant for x > 0 and, in particular, w(x) has discontinuities. This has the eﬀect of widening the gaps [6, section 4.5], [7], [16]. On the other side (x < 0) of the interface the diﬀerential equation is y (x) + {λ − q1 (x)}y(x) = 0

(1.6)

on a ﬁnite interval [−X, 0) with an arbitrary q1 (x) and a non-trivial boundary condition c1 y(−X) + c2 y (−X) = 0. (1.7) Our spectral setting is therefore on the x-interval [−X, ∞) with (1.1) for x > 0, (1.6) for x < 0 and the boundary condition (1.7). A simple relative compactness argument shows that the essential spectrum of our interface problem retains the band structure noted above. In section 2, we recall the basic Floquet theory from (for example) [6] which we require, and formulate the eigenvalue equation arising from (1.7). Then in section 3 we formulate and prove a general theorem (Theorem 3.1) on the existence of interfaceinduced eigenvalues in suﬃciently distant spectral gaps, subject to a condition concerning the length of the gaps. The dependence of the number of these eigenvalues on the value of X is noted. In sections 4 and 5, we consider the case where w(x) has two discontinuities in its period, a two-valued stepfunction being an example, and we lead up to situations where the length condition in Theorem 3.1 is satisﬁed. We mention here that the eigenvalues λn and μn for step-function examples have been discussed in [6, section 2.2], [10, section 50], [12], and one contribution of our paper in sections 4 and 5 is to develop properties of these eigenvalues which are not conﬁned to the stepfunction case. In section 6, we discuss brieﬂy the case where w(x) has just one discontinuity in its period, that is, w(0) = w(a). Finally in section 7 we discuss in more detail some step-function examples where induced eigenvalues appear in all the spectral gaps (except (−∞, λ0 )), not just the distant ones. We conclude this introduction by mentioning that the adjective Meissner is applied to any periodic equation (1.1) in which w and q are step functions (cf. [2], [10], [12]). This follows the original equation of this kind formulated by Meissner [15] (concerning locomotive coupling rods). In our paper we are not conﬁned to step-functions, but the discontinuities in w(x) are essential.

2. Formulation of the eigenvalue problem We begin with the solutions φ1 (x) and φ2 (x) of (1.1) which have the initial values 1, 0 and 0, 1 (2.1) respectively at x = 0, with the dependence on λ not indicated until necessary. Since we are dealing with λ in a spectral gap, the basic theory of [6, chapters 1 and 2] shows that there are also solutions ψk (x) (k = 1, 2) of (1.1) such that ψk (x + a) = ρk ψk (x) (x ≥ 0) (2.2)

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where the ρk are the two distinct and real solutions of the quadratic ρ2 − Dρ + 1 = 0 with D = φ1 (a) + φ2 (a).

(2.3)

Further [6, section 1.1], ψk (x) can be written as either ψk (x) = φ2 (a)φ1 (x) − {φ1 (a) − ρk }φ2 (x).

(2.4)

or, in case (2.4) is a trivial linear combination of φ1 (x) and φ2 (x), ψk (x) = {φ2 (a) − ρk }φ1 (x) − φ1 (a)φ2 (x).

(2.5)

We shall generally keep to (2.4) and comment on the change to (2.5) as necessary. Since ρ1 ρ2 = 1, we take it that |ρ1 | < 1 and |ρ2 | > 1.

(2.6)

We are looking for a solution of (1.1) which is L2 (0, ∞), and it follows from (2.2) and (2.6) that this solution must be y(x) = ψ1 (x)

(x ≥ 0)

(2.7)

to within a constant multiple. We have now to continue this solution into x < 0 and substitute the result into the boundary condition (1.7) to complete the formulation of the eigenvalue problem. We continue to denote by φ1 (x) and φ2 (x) the solutions now of (1.6) in [−X, 0) but still satisfying (2.1). We also note that, by (2.1) and (2.4), ψ1 (x) has the initial values φ2 (a), −{φ1 (a) − ρ1 } at x = 0. Hence, as a linear combination of φ1 (x) and φ2 (x), (2.7) is continued into x < 0 as y(x) = φ2 (a)φ1 (x) − {φ1 (a) − ρ1 }φ2 (x) (x < 0).

(2.8)

Then (1.7) gives the equation to determine the eigenvalues in the spectral gaps as φ2 (a){c1 φ1 (−X) + c2 φ1 (−X)} − {φ1 (a) − ρ1 }{c1 φ2 (−X) + c2 φ2 (−X)} = 0.

(2.9)

We shall examine (2.9) ﬁrstly for suﬃciently distant gaps and then, in an example, for all gaps. To prepare for the former in the next section, we require a slightly more precise version of a familiar result for the SturmLiouville equation y (x) + {λ − Q(x)}y(x) = 0

(0 ≤ x ≤ A) √ with any Q in Lloc [0, A]. We consider λ > 0 and write ν = λ. Lemma 2.1. Let y(x) satisfy (2.10) and let A ν≥ |Q(t)|dt. 0

(2.10)

(2.11)

Generalised Meissner Equations

5

Then 1 1 E(x)} cos νx + {y (0) + F (x)} sin νx, ν ν 1 y (x) = −ν{y(0) + E(x)} sin νx + {y (0) + F (x)} cos νx, ν y(x) = {y(0) +

where |E(x)|, |F (x)| ≤ (|y(0)| +

1 |y (0)|)(e − 1) ν

(2.12) (2.13)

A

0

|Q(t)|dt.

(2.14)

Proof. The integral formulation of (2.10) is 1 1 x y(x) = y(0) cos νx + y (0) sin νx + sin{ν(x − t)}Q(t)y(t)dt. (2.15) ν ν 0 Hence 1 1 x |y(x)| ≤ |y(0)| + |y (0)| + |Q(t)||y(t)|dt, ν ν 0 and this Gronwall inequality gives x 1 1 |y(x)| ≤ (|y(0)| + |y (0)|) exp |Q(t)|dt . (2.16) ν ν 0 We now write (2.15) in the form (2.12) with x E(x) = − (sin νt)Q(t)y(t)dt 0

and similarly for F (x) with cos νt in place of − sin νt. Then, by (2.16), x 1 t 1 |E(x)|, |F (x)| ≤ (|y(0)| + |y (0)|) |Q(t)| exp |Q(u)|du dt ν ν 0 0 x 1 1 = (|y(0)| + |y (0)|)ν{exp |Q(t)|dt − 1} ν ν 0 yielding (2.14) when (2.11) holds. This proves (2.12), and (2.13) follows similarly from diﬀerentiation of (2.15). We note that the lemma also holds for A < 0 if the integration range in (2.11) and (2.14) is replaced by (A, 0).

3. Dirichlet and Neumann boundary conditions The Dirichlet condition is the case c2 = 0 of (1.7), and then (2.9) becomes φ2 (a, λ)φ1 (−X, λ) − {φ1 (a, λ) − ρ1 (λ)}φ2 (−X, λ) = 0,

(3.1)

where we are now indicating the dependence on λ. The Neumann condition is the case c1 = 0 of (1.7) and, as usual, it is typical of the situation when c2 = 0. When c1 = 0, (2.9) becomes φ2 (a, λ)φ1 (−X, λ) − {φ1 (a, λ) − ρ1 (λ)}φ2 (−X, λ) = 0.

(3.2)

Before proceeding further with (3.1) and (3.2), it is convenient at this point to refer to the familiar Dirichlet and Neumann problems for (1.1) over

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B.M. Brown and M.S.P. Eastham

the basic periodicity interval (0, a) [6, section 3.1]. In any instability interval, each of these problems has a unique eigenvalue ΛD and ΛN for which φ1 (a, ΛN ) = 0

φ2 (a, ΛD ) = 0,

[6, Theorem 3.1]. As we indicated in section 2, there is therefore the possibility that (2.4) is a trivial linear combination when λ = ΛD . But both cannot occur together should ΛD = ΛN . In case (2.4) is trivial, (2.9), (3.1) and (3.2) can be expressed instead interms of (2.5). Then (3.2) for example becomes {φ2 (a, λ) − ρ1 (λ)}φ1 (−X, λ) − φ1 (a, λ)φ2 (−X, λ) = 0.

(3.3)

In what follows, we avoid this slight complication by simply excluding the value ΛD from our considerations. Thus we work with (3.2) (λ = ΛD ) rather than (3.3). We can now state and prove a theorem which, in general terms, is the main result of the paper. It contains a general condition (3.4) which will be analysed in subsequent sections. The theorem concerns λ-solutions of (3.2) lying in a spectral gap (λ , λ ) of (1.1), and we recall that these solutions are the eigenvalues induced in the gap by the interface represented by (1.6) and (1.7) with a√variable X being √ allowed for. A similar result holds for (3.1). We write ν = λ and ν = λ . Theorem 3.1. Suppose that there exist a ﬁxed number K (K > 0) such that ν − ν ≥ K

(3.4)

for a sequence of spectral gaps receding to inﬁnity. Then there is a number ν0 (X) such that (3.2) has λ-solutions in (λ , λ ) when ν ≥ ν0 (X), and the number of such solutions exceeds 3KX/4π − 4.

(3.5)

Further, if q1 is deﬁned in (−∞, 0) and satisﬁes q1 ∈ L(−∞, 0),

(3.6)

then ν0 (X) can be taken to be independent of X. Proof. We begin by applying Lemma 2.1 to (1.6) on [−X, 0). By (2.11) we are considering values of ν such that 0 ν≥ |q1 (t)|dt. (3.7) −X

Then, recalling the initial values (2.1), we obtain from (2.13) and (2.14) φ1 (−X, λ) = ν(1 +

1 E1 ) sin νX + F1 cos νX ν

φ2 (−X, λ) = E2 sin νX + (1 + F2 ) cos νX, where E1 = E1 (−X) etc, and |E1 |, |F1 | ≤ (e − 1)

0 −X

|q1 (t)|dt := I(X),

(3.8)

Generalised Meissner Equations

7

say, and |E2 |, |F2 | ≤

1 I(X). ν

(3.9)

On substituting into (3.2), we obtain (1 + ν1 E1 ) sin νX + ν1 F1 cos νX 1 = H(ν), E2 sin νX + (1 + F2 ) cos νX ν

(3.10)

H(ν) = {φ1 (a, λ) − ρ1 (λ)}/φ2 (a, λ) (λ = ΛD ).

(3.11)

where Let us now denote the left-hand side of (3.10) by T (ν) (ν < ν < ν ). We wish to show that T (ν) behaves suﬃciently like tan νX, the relevant property of the latter being that it increases from −∞ to +∞ in any ν-interval 1 1 ((k − )π/X, (k + )π/X) 2 2

(3.12)

with k an integer. Then, in any such interval which lies within (ν , ν ), the graph of tan νX crosses that of ν1 H(ν), producing a solution of the equation tan νX = ν1 H(ν) (excepting possibly only an interval (3.12) which contains the point ΛD ). Since, however, we have T (ν) rather than tan νX, we begin by considering instead of (3.12) intervals I(k) of the form 1 1 ({(k − )π + η}/X, {(k + )π − η}/X), 2 2

(3.13)

sin η = I(X)/{ν − I(X)}.

(3.14)

where Also, further to (3.7), we assume that ν > 3I(X),

(3.15)

where I(X) is as in (3.8). Then (3.14) and (3.15) imply that 0 < η < π/6.

(3.16)

Denoting the denominator in T (ν) by C(ν), we show ﬁrst that C(ν) > 0 or C(ν) < 0 in I(k) according to k being even or odd. Taking k even (odd is similar), it follows from (3.9) that, in I(k), C(ν) := E2 sin νX + (1 + F2 ) cos νX > −

1 1 I(X) + {1 − I(X)} sin η = 0 ν ν

by (3.14), as required. Next, considering also I(k + 1) in (3.13), it follows from what we have just proved that, in the interval 1 1 L(k) = [{(k + )π − η}/X, {(k + )π + η}/X], 2 2

(3.17)

8

B.M. Brown and M.S.P. Eastham

C(ν) has a least zero νl (k) and a greatest zero νg (k). Also, in L(k), the numerator S(ν) in (3.10) is positive (k even) or negative (k odd). This is easily seen because (if k is even for example) 1 1 E1 ) sin νX + F1 cos νX ν ν 1 1 ≥ {1 − I(X)} cos η − I(X) sin η ν ν 1 > (2 cos η − sin η) > 0. 3

S(ν) := (1 +

Altogether, then, we have shown that T (ν) → +∞ as

ν → νl (k) − 0

and T (ν) → −∞ as

ν → νg (k − 1) + 0

with T (ν) continuous in (νg (k − 1), νl (k)). Further, both νg (k − 1) and νl (k) lie in the interval L(k − 1) ∪ I(k) ∪ L(k),

(3.18)

which is an interval of total length (π + 2η)/X. It follows that (3.10) has at least one solution ν in each interval (3.18), with the possible exception of at most two intervals which contain ΛD . The number of complete intervals (3.18) which lie in [ν , ν ] is at least KX/(π + 2η) − 2, by (3.4). Then, discounting at most two intervals which contain ΛD and using (3.16), we arrive ﬁnally at (3.5) with 0 ν0 (X) = 3(e − 1) |q1 (t)|dt (3.19) −X

by (3.8) and (3.15). Also, the concluding statement of the theorem, where (3.6) holds, is now clear There is one further observation to be made concerning the condition (3.6). If we consider X → ∞ in (1.7), our eigenvalue problem approximates to the problem with two singular end-points where the diﬀerential equation is (1.6) in (−∞, 0) and (1.1) in (0, ∞).When (3.6) holds, this latter problem has [λ0 , ∞) as an interval of continuous spectrum [23, sections 3.1, 3.8, 5.6 and 5.14]. Our Theorem 3.1 is therefore in accord with this property in that, at least for ν > ν0 (∞) (see(3.19)), (3.13), (3.17) and (3.18) show that the interface eigenvalues become everywhere dense in (ν , ν ) as X → ∞, thus ﬁlling up the spectral gaps of (1.1). Finally, we note that Theorem 3.1 continues to apply when (3.4) holds, not for a ﬁxed K, but for a sequence Kn → 0 as n → ∞. Then for (3.5) to guarantee at least one eigenvalue, we require Kn ≥ 20π/3X. Thus only a ﬁnite number of the Kn is allowed for a ﬁxed X.

Generalised Meissner Equations

9

4. Two discontinuities In this section and the next we suppose that the weight function w(x) in (1.1) is given by w1 (x) (0 ≤ x < a1 ) w(x) = (4.1) w2 (x) (a1 ≤ x < a), where w1 and w2 have continuous second derivatives in [0, a1 ] and [a1 , a] respectively, but w1 (a1 − 0) = w2 (a1 ) and w2 (a − 0) = w1 (0). (To be brief, we omit “−0” in the sequel.) We write σ1 = (w2 /w1 )1/4 (a1 ),

σ2 = (w1 /w2 )1/4 (a).

(4.2)

At this point we note that, in the step-function case where w1 and w2 are diﬀerent constants, σ1 and σ2 are connected by the relation σ1 σ2 = 1. We shall refer later to this relation, but our analysis is not dependent on it. In order to examine the spectral gaps in the case (4.1), and in particular to verify (3.4), we need to determine the eigenvalues λn and μn associated with (1.2) and (1.3). These eigenvalues are the solutions of the equations D(λ) = ±2 with D as in (2.3) [6, chapter√2]. In the following lemma, we obtain the form of D(λ), at least for ν(= λ) large enough. In the lemma and its proof we use the general notation M (ν) to denote any expression satisfying an inequality |M (ν)| ≤ C (4.3) where C is independent of ν and is expressible explicitly in terms of w1 , w2 and q. Lemma 4.1. With the notation (4.1)–(4.3), 1 φ1 (a, λ) + φ2 (a, λ) = (σ1 σ2 + ) cos νA1 cos νA2 σ1 σ2 σ1 σ2 1 −( + ) sin νA1 sin νA2 + M (ν), σ2 σ1 ν where a a 1

A1 =

0

and ν≥

a

0

1/2

w1 (x)dx,

1/2

A2 =

(4.4)

w2 (x)dx

(4.5)

|(qw−1/2 − w−1/4 (w−1/4 ) )(x)|dx.

(4.6)

a1

Proof. We make the Liouville transformation of (1.1) in each of the two intervals indicated in (4.1): x 1/2 −1/4 y = w1 z1 , t1 = 0 w1 (u)du (0 ≤ x < a1 ) x 1/2 (4.7) −1/4 y = w2 z2 , t2 = a1 w2 (u)du (a1 ≤ x < a) This gives where

d2 zj /dt2j + {λ − Qj (tj )}zj = 0 −3/4

Qj (tj ) = (q/wj − wj

−1/4

(wj

) )(x)

(4.8) (4.9)

10

B.M. Brown and M.S.P. Eastham

and 0 ≤ tj < Aj (j = 1, 2) [6, section 4.1]. The solutions φ1 and φ2 of (1.1) are transformed into solutions Φ1,j and Φ2,j of (4.8), and we shall obtain (4.4) by applying Lemma 2.1 to (4.8) We ﬁrst note the connection between the values of z1 and z2 and their derivatives which follows from the continuity of y and y at x = a1 . Thus, from (4.7), we have 1/4

z2 (0) = [w2 y](a1 ) = σ1 z1 (A1 ) dz2 1 dz1 (0) = W (a1 )z1 (A1 ) + (A1 ) dt2 σ1 dt1 where σ1 is as in (4.2) and −1/2

W = w2

1/4

−1/4

(w2 w1

).

(4.10) (4.11)

(4.12)

Let us now consider φ1 and its transforms Φ1,1 and Φ1,2 . By (2.1) and (4.7), Φ1,1 (t1 , λ) has the initial values 1/4

−1/2

w1 (0),

[w1

1/4

(w1 ) ](0)

at t1 = 0. Then Lemma 2.1 gives 1 −1/2 1/4 1 1/4 Φ1,1 (A1 , λ) = w1 (0) cos νA1 + [w1 (w1 ) ](0) sin νA1 + E1 (ν) (4.13) ν ν dΦ1,1 1/4 −1/2 1/4 (A1 , λ) = −νw1 (0) sin νA1 + [w1 (w1 ) ](0) cos νA1 + F1 (ν) dt1 (4.14) where A1 1 −1/2 1/4 1/4 |E1 (ν)|, |F1 (ν)| ≤ {w1 (0) + |[w1 (w1 ) ](0)|}(e − 1) |Q1 (t1 )|dt1 ν 0 (4.15) and ν≥

A1

0

|Q1 (t1 )|dt1

(4.16)

by (2.11). Using | sin νA1 | ≤ 1 and | cos νA1 | ≤ 1, we can write (4.13) and (4.14) as 1 1/4 Φ1,1 (A1 , λ) = w1 (0) cos νA1 + M (ν) ν dΦ1,1 1/4 (A1 , λ) = −νw1 (0) sin νA1 + M (ν) dt1 where M (ν) has the form indicated in (4.3). It then follows from (4.10) and (4.11) that the initial values of Φ1,2 (t2 ) at t2 = 0 are 1 1/4 σ1 w1 (0) cos νA1 + M (ν) ν and ν 1/4 − w1 (0) sin νA1 + M (ν). (4.17) σ1

Generalised Meissner Equations

11

Turning to Lemma 2.1 again, applied now to Φ1,2 (t2 , λ) over the interval 0 ≤ t2 ≤ A2 , and using the initial values (4.17), we have 1 1 sin νA1 sin νA2 } + M (ν) σ1 ν (4.18) dΦ1,2 1 1/4 (A2 , λ) = −w1 (0)ν{σ1 cos νA1 sin νA2 + sin νA1 cos νA2 } + M (ν) dt2 σ1 (4.19) where, by (2.11), A2 ν≥ |Q2 (t2 )|dt2 . (4.20) 1/4

Φ1,2 (A2 , λ) = w1 (0){σ1 cos νA1 cos νA2 −

0

Next we consider similarly φ2 and its transforms Φ2,1 and Φ2,2 . By (2.1) and (4.7), Φ2,1 (t1 , λ) has the initial values 0,

−1/4

w1

(0)

at t1 = 0. Then Lemma 2.1 gives 1 −1/4 1 w (0) sin νA1 + 2 M (ν) ν 1 ν dΦ2,1 1 −1/4 (A1 , λ) = w1 (0) cos νA1 + M (ν), dt1 ν Φ2,1 (A1 , λ) =

where (4.16) holds. It then follows from (4.10) and (4.11) that the initial values of Φ2,2 (t2 , λ) at t2 = 0 are σ1 −1/4 1 w (0) sin νA1 + 2 M (ν) ν 1 ν and 1 −1/4 1 w1 (0) cos νA1 + M (ν). σ1 ν

(4.21)

Turning to Lemma 2.1 once again, applied now to Φ2,2 (t2 , λ) over the interval 0 ≤ t ≤ A2 , and using the initial values (4.21), we have 1 −1/4 1 1 w (0){σ1 sin νA1 cos νA2 + cos νA1 sin νA2 } + 2 M (ν) ν 1 σ1 ν (4.22) dΦ2,2 1 1 −1/4 (A2 , λ) = w1 (0){−σ1 sin νA1 sin νA2 + cos νA1 cos νA2 }+ M (ν) dt2 σ1 ν (4.23) where (4.20) holds. To complete the proof of (4.4), we use (4.7), (4.18), (4.22) and (4.23) to revert to the values of φ1 and φ2 at x = a. Thus, with σ2 as in (4.2), Φ2,2 (A2 , λ) =

−1/4

φ1 (a, λ) = w2

(a)Φ1,2 (A2 , λ)

= σ1 σ2 cos νA1 cos νA2 −

σ2 1 sin νA1 sin νA2 + M (ν) σ1 ν

(4.24)

12

B.M. Brown and M.S.P. Eastham

and dΦ2,2 (A2 , λ) dt2 1 σ1 1 = cos νA1 cos νA2 − sin νA1 sin νA2 + M (ν). σ1 σ2 σ2 ν −1/4

φ2 (a, λ) = (w2

1/4

) (a)Φ2,2 (A2 , λ) + w2 (a)

(4.25)

Now (4.4) follows since (4.6) accommodates both (4.16) and (4.20), and the lemma is proved. Let us now write (4.4) as D(λ) = where

1 1 1 1 1 1 1 (σ1 + )(σ2 + ) cos νI + (σ1 − )(σ2 − ) cos νJ + M (ν), 2 σ1 σ2 2 σ1 σ2 ν (4.26) a I = A1 + A2 = w1/2 (x)dx, J = |A1 − A2 |. (4.27) 0

Then the equations D(λ) = ±2 for the periodic and semi-periodic (respectively) eigenvalues become cos νI = f± (ν) +

1 M (ν), ν

(4.28)

where f± (ν) = (σ1 +

1 −1 1 1 1 ) (σ2 + )−1 {−(σ1 − )(σ2 − ) cos νJ ± 4}. (4.29) σ1 σ2 σ1 σ2

In the next section, we discuss the solutions of (4.28), and we are interested in identifying situations where the main condition (3.4) in Theorem 3.1 is satisﬁed, and the ν1 term in (4.28) does not materially aﬀect the analysis for large ν. We note in passing that a similar type of equation to (4.28) and (4.29) (with σ1 σ2 = 1) appears in [11, (1.5)] for a diﬀerent step-function example, and the diﬃculty of analysing the equation is commented upon. Nevertheless, even without the restriction to the step-function relation σ1 σ2 = 1, we shall extract suﬃcient information for our purposes from (4.28) and (4.29).

5. Lower bounds concerning the spectral gaps We ﬁrst note that the solutions of (4.28)–(4.29) are known from other sources, but not

suﬃciently

accurately for our purposes. Thus [7, Theorem 1], [16], for ν = λ2m+1 or λ2m+2 , we have |ν − 2(m + 1)πI −1 | ≤ I −1 (ω1 + ω2 ) + o(1) (m → ∞), where

(5.1)

1 1 1 ωj = tan−1 ( |σj − |) (0 < ωj < π) (5.2) 2 σj 2 √ √ and I is as in (4.27). For ν = μ2m or μ2m+1 , we replace 2(m + 1) by 2m + 1 in (5.1). However, (5.1) only requires w1 and w2 to be diﬀerentiable

Generalised Meissner Equations

13

once. Under our assumptions of twice diﬀerentiability, we can go further in the following proposition which identiﬁes a situation where (3.4) is satisﬁed. Proposition 5.1. Suppose that σ1 = σ2 and that σ1 σ2 = 1. Then

λ − λ2m+1 } ≥ 2α/I + O(m−1 ) √ √ 2m+2 μ2m+1 − μ2m

(5.3)

as m → ∞, where cos α = (σ1 +

1 −1 1 1 1 ) (σ2 + )−1 {|(σ1 − )(σ2 − )| + 4} σ1 σ2 σ1 σ2

(5.4)

and 0 < α < π/2. Proof. We ﬁrst check that α is well deﬁned by (5.4). Thus we need to check that 1 1 1 1 |(σ1 − )(σ2 − )| + 4 < (σ1 + )(σ2 + ). (5.5) σ1 σ2 σ1 σ2 There are two cases to consider: (i) σ1 > 1 and σ2 < 1 (or vice versa) (ii) σ1 > 1 and σ2 > 1 (or σ1 < 1 and σ2 < 1). In case (i), (5.5) simpliﬁes to σ1 σ2 +1/σ1 σ2 > 2, which is true when σ1 σ2 = 1. In case (ii), (5.5) simpliﬁes to σ1 /σ2 + σ2 /σ1 > 2, which is true when σ1 = σ2 . To obtain the upper inequality in (5.3) we ﬁrst note from (5.1) and (5.2)

that λ2m+1 and λ2m+2 lie in the open interval ((2m+1)π/I, (2m+3)π/I). We therefore consider ν to lie in this interval and let ν1 and ν2 be the solutions of 1 cos νI = cos α + M (ν), ν where M (ν) is as in (4.28). Thus ν1 , ν2 = 2(m + 1)π/I ± α/I + O(m−1 ).

(5.6)

In (ν1 , ν2 ) we have 1 1 M (ν) ≥ f+ (ν) + M (ν) ν ν

by (4.29) and (5.4), and hence D(λ) > 2 in (ν1 , ν2 ). Hence λ2m+1 ≤ ν1 and

√ √ λ2m+2 ≥ ν2 , and (5.3) follows from (5.6). The proof for μ2m+1 − μ2m is similar , completing the proof of the proposition. cos νI > cos α +

There are two, more specialised situations where (5.3) can be improved to an asymptotic formula. The cases are described by I and J in (4.27), and we present them in the following subsections.

14

B.M. Brown and M.S.P. Eastham

5.1. The case I = 2J (= 0) Proposition 5.2. Let I = 2J. Then there are numbers ψ± and ω± with ψ− ∈ (π/2, π], ω− ∈ [0, π/2), ψ+ ∈ (π/2, π), ω+ ∈ (0, π/2) such that, as m → ∞,

λ4m−1

λ4m

λ4m+1

λ4m+2 √ μ4m √ μ4m+1 √ μ4m+2 √ μ4m+3

= 2(2mπ − ω− )/I + O(m−1 ) = 2(2mπ + ω− )/I + O(m−1 ) = 2(2mπ + ψ− )/I + O(m−1 ) = 2{2(m + 1)π − ψ− }/I + O(m−1 ) = 2(2mπ + ω+ )/I + O(m−1 ) = 2(2mπ + ψ+ )/I + O(m−1 ) = 2{2(m + 1)π − ψ+ }/I + O(m−1 ) = 2{2(m + 1)π − ω+ }/I + O(m−1 ).

Further, ψ− < π if σ1 = σ2 and ω− > 0 if σ1 σ2 = 1. Proof. Deﬁning θ = νJ and using I = 2J, we write (4.28)-(4.29) as 1 1 1 1 1 (σ1 + )(σ2 + )(2 cos2 θ−1)+(σ1 − )(σ2 − ) cos θ∓4 = M (ν). (5.7) σ1 σ2 σ1 σ2 ν The left-hand side here is a quadratic p− (c) (or p+ (c)) in cos θ (c = cos θ). It is easy to check that p± (0) < 0 and √ 1 p± (1) = 2( σ1 σ2 ± √ )2 ≥ 0 σ1 σ2

p± (−1) = 2( σ1 /σ2 ± σ2 /σ1 )2 ≥ 0, where the inequalities are strict for p+ while, for p− , the ﬁrst inequality is strict if σ1 σ2 = 1 and the second if σ1 = σ2 . Thus each p± (c) has two distinct zeros c± and d± with −1 < c+ < 0,

0 < d+ < 1

−1 ≤ c− < 0, 0 < d− ≤ 1 and c− > −1 if σ1 = σ2 and d− < 1 if σ1 σ2 = 1.

Now deﬁne ψ± and ω± by cos ψ± = c± cos ω± = d±

(π/2 < ψ+ < π, π/2 < ψ− ≤ π) (0 < ω+ < π/2 0 ≤ ω− < π/2)

Then, for any integer m > 0, consider θ in the ranges 2mπ − ω− < θ < 2mπ + ω− 2mπ + ψ− < θ < 2(m + 1)π − ψ−

.

(5.8)

2mπ + ω+ < θ < 2mπ + ψ+ 2(m + 1)π − ψ+ < θ < 2(m + 1)π − ω+

(5.9) .

(5.10)

Generalised Meissner Equations

15

The ranges (5.9) are maximal ones in which p− (c) > 0, and (5.10) are maximal ones in which p+ (c) < 0. By (4.26) and (5.7), these ranges (5.9) and (5.10) correspond to the spectral gaps in which D(λ) > 2 and D(λ) < −2 respectively, with an error O(m−1 ) when m is large. The spectral gaps are the intervals (λ2n+1 , λ2n+2 ) and (μ2n , μ2n+1 ) and, since θ = νJ = 12 νI, the values stated in the proposition follow from (5.9) and (5.10),√where the choices n = 2m − 1, n = 2m, n = 2m + 1 tally with (5.1) (for λ) and with the √ corresponding formula for μ. By (4.27), the criterion I = 2J means that A1 = 3A2 (or A2 = 3A1 ), and a step-function of this type was examined in [6, section 2.2] (see also [10, section 50]). Proposition 5.2 generalises these ﬁndings in [6]. It is also possible to consider in the same way (N − 1)A1 = (N + 1)A2 , N being a positive integer, and the criterion is then I = N J. As a corollary, we state the conclusions of Proposition 5.2 in a form which shows that (3.4) is satisﬁed. Corollary 5.3. Let I = 2J. Then

4ω− /I + O(m−1 ) (n = 2m − 1) λ2n+2 − λ2n+1 = 4(π − ψ− )/I + O(m−1 ) (n = 2m), √ √ μ2n+1 − μ2n = 2(ψ+ − ω+ )/I + O(m−1 ), where ω− > 0 if σ1 σ2 = 1 and π − ψ− > 0 if σ1 = σ2 5.2. The case J = 0 When J = 0 (i.e., A1 = A2 ), (4.28) simpliﬁes to 1 −1 1 −1 1 1 1 ) (σ2 + ) {−(σ1 − )(σ2 − ) ± 4} + M (ν). σ1 σ2 σ1 σ2 ν In the same way as in the proof of Proposition 5.2 we write this equation as cos νI = (σ1 +

cos νI = cos α± +

1 M (ν) ν

(5.11)

where (as is easily veriﬁed) 0 ≤ α+ < π/2,

π/2 < α− ≤ π.

Further, α+ > 0 if σ1 σ2 = 1 and α− < π if σ1 = σ2 . Then (5.11) gives

λ2m+2 , λ2m+1 = 2(m + 1)π/I ± α+ /I + O(m−1 ) √ √ μ2m+1 , μ2m = (2m + 1)π/I ± (π − α− )/I + O(m−1 ). Thus, corresponding to Corollary 5.3, we have the simpler formulae

λ2m+2 − λ2m+1 = 2α+ /I + O(m−1 ) √ √ μ2m+1 − μ2m = 2(π − α− )/I + O(m−1 ), giving another situation where (3.4) is satisﬁed.

(5.12)

16

B.M. Brown and M.S.P. Eastham

6. One discontinuity Here we mention brieﬂy the situation where w(x) has just one discontinuity, which we can take as w(0) = w(a), and w(x) is otherwise twice diﬀerentiable as before. The periodic and semi-periodic eigenvalues have been determined asymptotically in [7, section 3] (see also [16]) using the Liouville transformation as in (4.7). In place of (4.28), we have the simpler equation cos νI = ±2σ/(σ 2 + 1) +

1 M (ν), ν

(6.1)

where σ = {w(0)/w(a)}1/4 . Formally, this tallies with (4.28)–(4.29) if the discontinuity at a1 is made to disappear by taking σ1 = 1. If we now deﬁne α by cos α = 2σ/(σ 2 + 1) (0 < α < π/2), we have the same situation in (6.1) as in (5.11) but now with α+ = α and α− = π − α. Then, referring again to (3.4), (5.12) shows that ν − ν = 2α/I + O(m−1 ) for all spectral gaps as m → ∞. Although Theorem 3.1 is applicable, there is no simple illustrative example when w(x) has just the one discontinuity. A step-function is excluded, and the next obvious example is w(x) = x + 1 and q(x) = 0 for which (1.1) has the Bessel function solutions (x + 1)1/2 J 13 { 23 λ1/2 (x + 1)3/2 } and (x + 1)1/2 Y 13 { 32 λ1/2 (x + 1)3/2 } [23, section 4.13].

7. Step-function examples In this section we take q = 0, q1 = 0 and w to be a step-function. Then (1.1) and (1.6) can be solved explicitly in terms of trigonometric or hyperbolic functions. The E and F terms in (3.8)-(3.10), and the M (ν) terms in section 4, are now all zero, and there is no need for a restriction on ν such as (3.15). Since w is a step function, (4.2) gives σ1 = 1/σ2

(= σ, say),

(7.1)

and the formulae (4.24)-(4.26) becomes D(λ) =

1 1 1 1 (σ + )2 cos νI − (σ − )2 cos νJ 2 σ 2 σ

1 1 1 σ{(σ + ) cos νI + (σ − ) cos νJ} 2 σ σ 1 −1 1 1 φ2 (a, λ) = σ {(σ + ) cos νI − (σ − ) cos νJ}. 2 σ σ We also require the similar formula for φ2 (a, λ) which, by (4.22) and is now 1 1 1 φ2 (a, λ) = ν −1 {w1 (0)w2 (a)}−1/4 {(σ + ) sin νI + (σ − ) sin νJ}. 2 σ σ φ1 (a, λ) =

(7.2) (7.3) (7.4) (4.7), (7.5)

Generalised Meissner Equations

17

In section 1, we have indicated that there are no induced eigenvalues in the ﬁrst spectral gap, which is now (−∞, 0), and we deal with this point ﬁrst. Then we move on to the other spectral gaps. 7.1. The ﬁrst gap (−∞, 0) Proposition 7.1. Let q = q1 = 0 and let w be a step-function. Then (3.2) has no solution in (−∞, 0). Proof. In √ the ﬁrst gap, D(λ) > 2 [6, section 2.3] and hence (2.6) gives ρ1 = 1 {D − D 2 − 4}. Then, by (2.3), (3.10) becomes 2

tan νX = {φ1 (a, λ) − φ2 (a, λ) + D2 (λ) − 4}/{2νφ2 (a, λ)}. (7.6) √ Since we are considering λ < 0, we write ν = λ = iμ with real μ > 0. By (7.2)-(7.4), the numerator in (7.6) is real, and we claim now that it is positive, that is, D2 (λ) − 4 > {φ1 (a, λ) − φ2 (a, λ)}2 . By (2.3), this reduces to φ1 (a, λ)φ2 (a, λ)} > 1 which, by (7.3) and (7.4), in turn becomes 1 1 (σ + )2 cosh2 μI − (σ − )2 cosh2 μJ > 4, σ σ i.e., 1 (σ 2 + 2 )(cosh2 μI − cosh2 μJ) + 2(sinh2 μI + sinh2 μJ) > 0. σ This inequality is clearly true since I > J by (4.27) . Thus, by (7.5), (7.6) can be written 1 1 tanh μX = −{positive}{w1 (0)w2 (a)}1/4 /{(σ + ) sinh μI + (σ − ) sinh μJ}. σ σ Since the denominator here is also positive, there are no solutions for μ > 0 as required. 7.2. Induced eigenvalues in all other gaps We now show that it is possible for eigenvalues to appear in all the spectral gaps (except (−∞, 0)) if X is chosen suitably. The example which we highlight here is where 1 a = 1, a1 = , w1 (x) = 9, w2 (x) = 1 2 and, consequently, σ(= σ1 ) = √13 [6, pp. 25-26]. Also, (4.27) gives I = 2 and J = 1. These values are substituted into (7.2)–(7.5). Then, for λ > 0 and ν real, (3.10) becomes tan νX = H± (ν) (7.7) where, as in (7.6), H± (ν) =

1 {4(cos 2ν − cos ν) ± 3 D2 (λ) − 4}/(2 sin 2ν − sin ν) 2

(7.8)

18

B.M. Brown and M.S.P. Eastham

and 2 (4 cos 2ν − cos ν). 3 Here H+ refers to a λ-gap in which D(λ) > 2, and H− to a μ-gap in which D(λ) < −2. The periodic eigenvalues are given by

λ4m−1 = λ4m = 2mπ,

λ4m+1 = 2mπ + α, λ4m+2 = 2(m + 1)π − α, (7.9) D(λ) =

and the semi-periodic eigenvalues by √ √ μ4m = 2mπ + β, μ4m+1 = 2mπ + γ, √ √ μ4m+2 = 2(m + 1)π − γ, μ4m+3 = 2(m + 1)π − β

(7.10)

where β = cos−1 {(1 +

√

α = cos−1 (−7/8)

(π/2 < α < π), √ 33)/16}, γ = cos−1 {(1 − 33)/16}

(0 < β < γ < π)

[6, p. 26]. These values are in line with the case ω− = 0, ψ− < π of Proposition 5.2. We begin our consideration of (7.7) by examining the ﬁrst λ-gap (λ1 , λ2 ). The corresponding ν-interval as given by (7.9) is (α, 2π − α) and, in this interval, H+ (ν) has a vertical asymptote at ν = π (corresponding to the Dirichlet eigenvalue ΛD = π 2 ). The graph of H+ (ν) can be exhibited using Mathematica. Consequently, √ we can say that, as ν increases from α to π, H+ (ν) decreases from − 13 15 to −∞ and, as ν increases from π to 2π − α, √ H+ (ν) decreases from +∞ to 13 15. The solutions of (7.7) in (α, 2π − α) can be obtained computationally for a mesh of values of X. This interval is (2.64, 3.65) to 2d.p. and there are no solutions for X = 1 and X = 2. For other integer values of X there are solutions 2.73 and 3.55 (X = 3), 2.81 and 3.47 (X = 4), 2.86 and 3.42 (X = 5)), and there are four solutions when X = 10. The number of solutions generally increases with X in line with the method of proving Theorem 3.1. The situation for a general λ-gap (λ4m+1 , λ4m+2 ) is similar, and it is particularly simple when X is an integer because then, by (7.9), the problem in terms of ν is just translated by 2mπ, and (7.7) is invariant under this translation. The situation is also similar for the μ-gaps, and we only mention the ﬁrst one (μ0 , μ1 ). The corresponding ν-interval as given by (7.10) is (β, γ), which is (1.14, 1.87). In this interval, H− (ν) has a vertical asymptote at ν = cos−1 (1/4) = 1.32 (corresponding to a Dirichlet eigenvalue). As with H+ (ν), H− (ν) decreases from −3.43 to −∞ and from +∞ to 0.51 in the two parts of (β, γ). The ν-solutions of (7.7) are 1.68 (X = 1), none (X = 2), 1.52 (X = 3), 1.18 and 1.81 (X = 4), 1.53 (X = 5), 1.41 and 1.69 (X = 10). Again, these values can be translated by 2mπ for other μ-gaps.

Generalised Meissner Equations

19

7.3. Repeated eigenvalues The step-function example in section 7.2 is described in [10, section 50] as exhibiting a remarkable distribution of single and double eigenvalues for periodic boundary conditions (1.2) (i.e., inﬁnitely many of both simple and double periodic eigenvalues). Here, as an aside to the main topic of the paper, we take the opportunity to develop this observation from [10] by considering I = N J in (7.2), where N is an integer. Thus (7.2) is now 1 1 1 D(λ) = (σ + )2 cos(νN J) − (σ − )2 cos νJ 2 σ σ and we consider the following choices for ν with m as a positive integer. 1. ν = 2mπ/J (= 2mN π/I). Here D(λ) = 2 and D (λ) = 0. Hence (2N mπ/I)2 are double periodic eigenvalues, as in section 7.2 where N = 2. 2. ν = (2m + 1)π/J (= (2m + 1)N π/I), N odd. This time D(λ) = −2 and D (λ) = 0. Hence {(2m + 1)N π/I}2 are double semi-periodic eigenvalues. Thus we can add to the observation in [10] by noting that, when N is odd, there are inﬁnitely many double eigenvalue in both the periodic and semiperiodic problems. For further information on double eigenvalues, we refer to [12].

References [1] L. Aceto, P. Ghelardoni, and M. Marletta. Numerical computation of eigenvalues in spectral gaps of Sturm-Liouville operators. J. Comput. Appl. Math., 189(1-2):453–470, 2006. [2] J. M. Almira and P. J. Torres. Invariance of the stability of Meissner’s equation under a permutation of its intervals. Ann. Mat. Pura Appl. (4), 180(2):245–253, 2001. [3] B. M. Brown, M. S. P. Eastham, A. M. Hinz, and K. M. Schmidt. Distribution of eigenvalues in gaps of the essential spectrum of Sturm-Liouville operators—a numerical approach. J. Comput. Anal. Appl., 6(1):85–95, 2004. [4] E. A. Coddington and N. Levinson. Theory of ordinary diﬀerential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [5] T. Dohnal , M. Plum, and W. Reichel. Localized modes of the linear periodic Schr¨ odinger operator with a nonlocal perturbation. SIAM J. Math. Anal., 41(5):1967-1993, 2009. [6] M. S. P. Eastham. The spectral theory of periodic diﬀerential equations. Scottish Academic Press, Edinburgh, 1973. [7] M. S. P. Eastham. Results and problems in the spectral theory of periodic differential equations. In Spectral theory and diﬀerential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad J¨ orgens), pages 126–135. Lecture Notes in Math., Vol. 448. Springer, Berlin, 1975. [8] M. S. P. Eastham, Q. Kong, H. Wu, and A. Zettl. Inequalities among eigenvalues of Sturm-Liouville problems. J. Inequal. Appl., 3(1):25–43, 1999.

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[9] M. S. P. Eastham and H. Kalf. Schr¨ odinger-type operators with continuous spectra, volume 65 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass., 1982. [10] W. N. Everitt. A catalogue of Sturm-Liouville diﬀerential equations. In SturmLiouville theory, pages 271–331. Birkh¨ auser, Basel, 2005. [11] S. Gan and M. Zhang. Resonance pockets of Hill’s equations with two-step potentials. SIAM J. Math. Anal., 32(3):651–664 (electronic), 2000. [12] H. Hochstadt. A special Hill’s equation with discontinuous coeﬃcients. Amer. Math. Monthly, 70:18–26, 1963. [13] E. Korotyaev. Lattice dislocations in a 1-dimensional model. Comm. Math. Phys., 213(2):471–489, 2000. [14] E. Korotyaev. Schr¨ odinger operator with a junction of two 1-dimensional periodic potentials. Asymptot. Anal., 45(1-2):73–97, 2005. ¨ [15] E. Meissner. Uber Sch¨ uttelswingungen in Systemen mit periodisch ver¨ anderlicher Elastizit¨ at. Schweizer. Bauzeitung, 72:95–98, 1918. [16] A. A. Ntinos. Lengths of instability intervals of second order periodic diﬀerential equations. Quart. J. Math. Oxford (2), 27(107):387–394, 1976. [17] M. Reed and B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York, 1978. [18] F. S. Rofe-Beketov. A ﬁniteness test for the number of discrete levels which can be introduced into the gaps of the continuous spectrum by perturbations of a periodic potential. Soviet Math., 5 689-692, 1964. [19] F. S. Rofe-Beketov. Kneser constants and eﬀective masses for band potentials. Soviet Physics, 29:391–393, 1984. [20] K. M. Schmidt. Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators. Comm. Math. Phys., 211(2):465– 485, 2000. [21] A. V. Sobolev. Weyl asymptotics for the discrete spectrum of the perturbed Hill operator. In Estimates and asymptotics for discrete spectra of integral and diﬀerential equations (Leningrad, 1989–90), volume 7 of Adv. Soviet Math., pages 159–178. Amer. Math. Soc., Providence, RI, 1991. [22] E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Part 2. Clarendon Press, Oxford, 1958. [23] E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Part 1. Second Edition. Clarendon Press, Oxford, 1962. B.M. Brown School of Computer Science Cardiﬀ University, Cardiﬀ CF24 3XF UK e-mail: [email protected] M.S.P. Eastham School of Computer Science Cardiﬀ University, Cardiﬀ CF24 3XF UK

Operator Theory: Advances and Applications, Vol. 219, 21–36 c 2012 Springer Basel AG

On the HELP Inequality for Hill Operators on Trees B.M. Brown and K.M. Schmidt To David and Des

Abstract. The validity of a generalised HELP inequality for a Schr¨ odinger operator with periodic potential on a rooted homogeneous tree is related to the quasi-stability or quasi-instability of the associated differential equation. A numerical approach to the determination of the optimal constant in the HELP inequality is presented. Moreover, we give an example to illustrate that the generalised Weyl–Titchmarsh m function for the tree operator fails to capture all of its spectral properties. Mathematics Subject Classiﬁcation (2010). Primary 34A40; Secondary 34B24, 34L40, 81Q35. Keywords. Diﬀerential inequality, regular quantum trees, Hill operator, Weyl–Titchmarsh function.

1. Introduction The classical HELP inequality for a Sturm–Liouville operator τf =

1 (−(pf ) + qf ) w

with locally integrable real-valued coeﬃcients w, p1 , q; w, p > 0 on an interval (a, b) is 2 b b b 2 2 (p |f | + q |f | ) ≤K |f |2 w |τ f |2 w a

a

a

[6]. Typically one considers a situation where a is a regular, b a singular end-point in the limit-point case; the crucial point about the inequality is that, when valid, it holds for all functions f for which the derivatives exist in a weak sense and the right-hand side is ﬁnite, irrespective of the boundary values at a.

22

B.M. Brown and K.M. Schmidt

HELP is a generalisation of an inequality of Hardy and Littlewood [8] which covers the case w = p = 1, q = 0 and is valid with the optimal constant K = 4. The validity of the more general HELP inequality with some constant K can be shown to be equivalent to a certain property of the Weyl–Titchmarsh m function for the Sturm–Liouville operator in a cone-like neighbourhood of the imaginary axis [7]. The recent work [3] has shown that a similar criterion using a generalised m function can be obtained for a HELP inequality on trees of inﬁnite length; in fact this is a particular instance of the abstract HELP inequality established by [1] and [10]. In the present study we focus on a situation which is analogous to Hill’s equation (the periodic Sturm–Liouville equation) on a half-line; we consider a tree composed of inﬁnitely many identical intervals, each carrying the same potential q, and such that at each end-point with a single exception (the tree root) a ﬁxed number of intervals are joined together. The spectrum of the graph Laplacian on such trees has been studied by [11]; it consists of bands of purely absolutely continuous spectrum with an additional eigenvalue in each gap and thus is analogous to that of a Hill operator on a half-line, except that the eigenvalues of the tree Laplacian have inﬁnite multiplicity. [11] then went on to consider the eﬀect of adding a decaying potential which is symmetric in the sense of only depending on the distance from the tree root. The generalised Hill operator on a perfectly homogeneous, rootless tree has been analysed in detail by [5] under the assumption that the potential, equal on each tree edge, is an even function on the interval. Our paper is organised as follows. In Section 2 we show, based on its strong limit-point property, that the maximal tree-Hill operator has deﬁciency indices (1, 1) and that, as a consequence, the Weyl solutions for nonreal spectral parameter are symmetric on the tree branches and their evolution is governed by a period-transfer matrix. This leads to an analogue to Floquet theory which we pursue in Section 3. In Section 4 we state the HELP inequality and study the associated generalised m function to relate the validity of the inequality to the quasi-stability or quasi-instability, in the sense of the generalised Floquet theory, of the tree-Hill equation at spectral parameter 0. Section 5 reports on a numerical approach to calculate the optimal HELP constant and shows results in the example of a shifted piecewise linear (sawtooth-type) potential in dependence on the constant oﬀset. Finally, an appendix illustrates the observation that, due to the discrepancy between the ﬁnite deﬁciency indices of the operator and the a priori unboundedness of the multiplicity of its eigenvalues, the generalised m function does not carry complete spectral information, unlike the classical Sturm–Liouville case.

2. Hill operators on trees Consider a regular tree Γ with constant branching number b ∈ N+1 and ﬁxed edge length l. Thus, b copies of the interval [0, l] (1st generation edges) are attached to the right-hand end-point of the interval [0, l] (the 0th generation),

Hill Operators on Trees

23

b copies (2nd generation edges) are attached to each of the b right-hand endpoints of 1st generation edges, and so on ad inﬁnitum. The maximal domain for the tree Laplacian on Γ in the Hilbert space L2 (Γ) is D := {f : Γ → C | f, f a.c. on edges, f continuous, Kirchhoﬀ conditions, f, f ∈ L2 (Γ)} (Kirchhoﬀ conditions meaning, as usual, that at each junction the outgoing derivatives of f add up to 0). As shown in [3] Thm 4.1, the tree Laplacian on Γ, generally deﬁned as −f for functions f satisfying the regularity, continuity and Kirchhoﬀ conditions of D, has the Strong Limit-Point Property that for all f, g ∈ D,

lim f (x)g (x) = 0. r→∞

|x|=r

(here | · | denotes the metric distance from the tree root, 0). In fact, this property holds for general trees of inﬁnite length, i.e., those for which any forward path can be extended indeﬁnitely. In the following, we consider the tree-Hill operator on Γ, i.e., the operator −f + qf

(f ∈ D),

where q : Γ → R is an l-periodic, bounded function of the distance from the root only: in other words, each edge carries a (directed) copy of the same potential function. Thus the tree-Hill operator is a generalisation of the classical Hill, or one-dimensional Schr¨odinger, operator on a half-line. Let θ φ Φ(·, λ) = (·, λ) θ φ be the canonical fundamental system of the one-dimensional Schr¨odinger equation −u + qu = λu on the interval [0, l], λ ∈ C; then Φ(x, ·) is an entire function for each x ∈ [0, l]. For non-real spectral parameter, the strong limit-point property of the tree Laplacian yields the following characterisation of square-integrable solutions of the eigenvalue equation for the tree Laplacian. Theorem 2.1. Let λ ∈ C \ R and ψ ∈ D a solution of −f + qf = λf

(1)

on Γ. Then a) ψ can be taken to be non-trivial, and any other solution of (1) in D is linearly dependent on ψ; ˜ b) there is a function ψ˜ : [0, ∞) → C such that ψ(x) = ψ(|x|) (x ∈ Γ);

24 c)

B.M. Brown and K.M. Schmidt

˜ ˜ ψ(nl+) ψ((n − 1)l+) = A(λ) ˜ ψ˜ (nl+) ψ ((n − 1)l+) for all n ∈ N, with the transfer matrix θ(l, λ) φ(l, λ) A(λ) = 1 . 1 b θ (l, λ) b φ (l, λ)

Remark 2.2. Part a) shows that the minimal tree-Hill operator, deﬁned on the subspace of D of functions of compact support in the interior of Γ, has deﬁciency indices (1, 1). Self-adjoint realisations are obtained by restricting the maximal operator, deﬁned on D, by means of a boundary condition, e.g. of Dirichlet or Neumann type, at the tree root. Part b) shows that the generalised Weyl solutions, i.e., solutions in D for non-real spectral parameter, are all what we shall call symmetric functions in the following; these are functions which are symmetric under arbitrary permutations of the tree branches which leave the tree structure intact. We shall call Dsym := {f ∈ D | f symmetric} the symmetric subspace. Proof. a) Assume we have two linearly independent solutions ψ, ξ ∈ D of (1); let us assume for the moment that ψ(0), ξ(0) = 0 (the following argument ψ(0) will also show that this must always be the case). Then α := − ∈ C and ξ(0) f := ψ + αξ ∈ D is also a solution of (1), with f (0) = 0. Integrating by parts, we ﬁnd

2 |f | = − f f + f (x) f (x) − f (0) f (0), Γr

Γr

|x|=r

where Γr := {x ∈ Γ | |x| ≤ r}; so passing to the limit and using the strong limit-point property, the boundary value at the root and the eigenvalue equation, Γ

|f |2 =

Γ

(λ − q) |f |2 .

Taking the imaginary parts on either side gives 0 = (Im λ)

Γ

|f |2 . In particu-

lar, if λ ∈ / R, then ψ and ξ are linearly dependent. Clearly, if either ψ(0) = 0 or ξ(0) = 0, the same argument gives ψ = 0 or ξ = 0, respectively. The existence of a non-trivial, symmetric solution of (1) can be inferred from the fact that, on the symmetric subspace, the tree-Hill operator is equivalent to a Sturm–Liouville operator with a singular right-hand end-point in the limit-point case; see [3] Section 3 for details. b) By part a), the function arising from ψ by any rearrangement of tree branches which leaves the distance to the root invariant must be a solution of (1) linearly dependent on ψ; as it coincides with ψ on the ﬁrst edge, the two must be identical. Hence ψ itself is a symmetric function. c) In view of the symmetry shown in b), the continuity and Kirchhoﬀ condition at the junctions joining the (n − 1)-st to the n-th generation edges

Hill Operators on Trees imply

25

˜ ˜ ψ(nl+) ψ(nl−) = 1 ˜ ψ˜ (nl+) ψ (nl−) b

for all n ∈ N. Also

˜ ˜ ψ(nl−) ψ((n − 1)l+) = Φ(l, λ) ψ˜ (nl−) ψ˜ ((n − 1)l+)

by solving the diﬀerential equation along the edges, so we get the transfer 1 0 relation with A(λ) = Φ(l, λ). 0 1b

3. Quasi-Floquet theory The transfer matrix A(λ) plays a similar role to the standard monodromy matrix in the Floquet theory of Hill’s equation. There is, however, the essential diﬀerence that it has determinant 1b instead of 1. From Theorem 2.1 c) it is clear that if ψ ∈ D is a non-trivial solution of (1) for λ ∈ C \ R, then ψ(0) will be an eigenvector of A(λ). The corresponding eigenvalue will ψ (0) have modulus < √1b for the following reason. Generally, if for any λ ∈ C, μ is an eigenvalue of A(λ) and u : Γ → C is the corresponding unique symmetric solution of (1) whose start phase vector is an eigenvector, u u A(λ) (0) = μ (0), u u then u will be equal on all tree edges, except for a factor μn on the n-th generation edges (of which there are bn copies). Hence the square integral of u is ∞ l ∞ l

√ 2 n 2n 2 2n |u| = b |μ| |u| = ( b|μ|) |u|2 , Γ

n=0

0

n=0

0

which is ﬁnite if and only if |μ| < √1b . Let us now consider a real spectral parameter. As uniqueness of solution of initial-value problems holds only on the symmetric subspace, the quasiFloquet theory will not give a complete overview of the solutions of (1), but only of symmetric solutions; however, we shall be interested in the situation of real λ as limiting values of λ in the complex upper half-plane where all solutions in D are symmetric. The two eigenvalues of A(λ) satisfy μ1 μ2 = 1b , μ1 + μ2 = D(λ), where D(λ) := traceA(λ) = θ(l, λ)+ 1b φ (l, λ) is a quasi-discriminant ; if λ ∈ R, then D(λ) ∈ R. Solving the characteristic equation, we ﬁnd D(λ) ± D(λ)2 − 4b μ= . 2

26

B.M. Brown and K.M. Schmidt

Thus the eigenvalues are real and of equal sign if D(λ)2 > 4/b (and then both eigenvalues have equal sign), complex conjugates if D(λ)2 < 4/b, and in the limiting case they are ± √1b , with the same sign as D(λ).

In the case of real eigenvalues, the one closer to 0 has modulus |μ| < √1b . Hence the corresponding symmetric (quasi-Floquet) solution u whose start phase vector is an eigenvector for this eigenvalue, will be square-integrable; in the exceptional case where it also satisﬁes the boundary condition imposed at the tree root, λ will be an eigenvalue of the corresponding self-adjoint tree-Hill operator with symmetric eigenfunction. If the eigenvalues of A(λ) are non-real complex conjugates, they will have modulus |μ| = √1b , so the corresponding quasi-Floquet solutions are not square-integrable. Remark 3.1. We remark that the intervals on the real λ-axis where D(λ)2 > 4/b and where D(λ)2 < 4/b formally correspond to the instability and stability intervals of Hill’s equation, respectively; the terminology refers to the fact that for the periodic Sturm–Liouville equation the trivial solution u = 0 is respectively unstable or stable in the two situations. By analogy, we shall refer to them as quasi-instability and quasi-stability intervals; but note that the trivial solution of the equation on the tree may still be asymptotically stable for λ inside a quasi-instability interval: this will happen when √1b < |μ| < 1 for the eigenvalue of larger modulus. For the purpose of locating the quasi-stability and quasi-instability intervals, we observe that the transition points, i.e., the values of λ ∈ R where the quasi-discriminant is D(λ) = ± √2b and μ = ± √1b correspondingly, are the eigenvalues of an associated quasi-(anti-)periodic boundary value problem on the interval [0, l]. To see this, assume μ is an eigenvalue of A(λ) — for real λ — and v a corresponding eigenvector. Then the solution u of the diﬀerential equation −u + qu = λu u u on [0, l] with initial data (0) = v is given as = Φ(·, λ) v, and hence u u u 1 0 1 0 u(0) (l) = Φ(l, λ) v = A(λ) v = μ v=μ . u 0 b 0 b bu (0) At end-points of the quasi-stability intervals, μ = ± √1b , so we ﬁnd that, depending on the sign, u will be a solution of the quasi-periodic boundary value problem √ 1 u(l) = √ u(0), u (l) = b u (0), b or of the quasi-anti-periodic boundary value problem √ 1 u(l) = − √ u(0) u (l) = − b u (0). b

Hill Operators on Trees

27

4. Weyl–Titchmarsh m function and HELP inequality For spectral parameter λ in the complex upper half-plane, we can construct a Weyl–Titchmarsh m function in analogy to the standard Sturm–Liouville theory. Indeed, by Theorem 2.1 there is exactly one non-trivial solution ψ ∈ D (up to multiplication with a constant) and it is symmetric under branch permutations of the tree. The solution ψ cannot have ψ (0) = 0, as λ would then be a non-real eigenvalue of the self-adjoint realisation of our operator with Neumann boundary condition at the root; so by multiplication with a suitable constant we can assume that ψ (0) = 1 and hence represent ψ on [0, l] in terms of the canonical fundamental system, ψ = φ − mθ. The coeﬃcient m gives the Weyl–Titchmarsh function; clearly ψ(0) = −m. In fact, m is the standard Weyl–Titchmarsh function for the Sturm–Liouville operator equivalent to the tree-Hill operator restricted to the symmetric subspace. In particular, m has the property that Im m(λ) > 0 (Im λ > 0). As observed of the previous section, the initial phase vector atthe beginning ψ(0) −m = is an eigenvector of the transfer matrix A(λ) with eigenψ (0) 1 value μ(λ) of modulus |μ(λ)| < √1b , which gives the following characterisation of the m as a function of the spectral parameter. (Note that μ(λ) denotes the eigenvalue of A(λ) of smaller modulus.) Lemma 4.1. For λ ∈ C \ R, m(λ) =

φ(l, λ) φ (l, λ) − bμ(λ) = . θ(l, λ) − μ(λ) θ (l, λ)

In particular, although m is the generalised Weyl–Titchmarsh function for the tree-Hill operator, it can be calculated from the solutions on the single interval [0, l] only. We remark that the relationship of this m function to the spectral properties of the tree-Laplace operator turns out to be more tenuous than in the case of Sturm–Liouville operators, where knowing the m function is equivalent to knowing the operator’s spectral function. From the point of view of Sturm–Liouville theory, it may already appear paradoxical that the minimal tree Laplacian has ﬁnite deﬁciency indices, as shown in Theorem 2.1, while it is well known [11] that the self-adjoint realisation of the Laplacian on Γ with Dirichlet boundary condition has eigenvalues of inﬁnite multiplicity. Here it is important to observe that the statement of Theorem 2.1 a) only holds for non-real λ, but will be false for real λ in general, as is obvious from the proof. In particular, eigenfunctions for real λ need not be symmetric functions under branch permutations — each of the eigenvalues given in [11] has one symmetric eigenfunction, whereas the inﬁnitely many additional eigenfunctions are not symmetric.

28

B.M. Brown and K.M. Schmidt

This observation may indicate some severe limitations to the scope of Titchmarsh-Weyl m function theory when applied to trees (or, more generally, graphs). The deﬁnition of the m function as characteristic coeﬃcient in the representation of the distinguished square-integrable solution for non-real λ only captures the behaviour of the operator on the symmetric subspace and, in particular, will only ﬂag a simple real eigenvalue where the tree Laplacian has in fact an eigenvalue of inﬁnite multiplicity. The underlying reason for this eﬀect is the lack of uniqueness of the solution of initial-value problems on the tree. Indeed, there are non-trivial solutions of (1) which vanish identically near the root. This peculiarity may seem inconsequential in the present case of a fully regular tree with constant branching number and edge lengths, where the spectrum of the Laplacian just consists of inﬁnitely many copies of the spectrum on the symmetric subspace; but for more general trees one may expect to lose spectral information in the m function. We illustrate this eﬀect with a simple example in the Appendix below. A similar spectral incompleteness of the generalised m function has been observed for operators of a diﬀerent type by [2]. The properties of the function m play a decisive role for the validity of a HELP inequality for the tree-Hill operator, as shown in [3]. Theorem 4.2. a) The following statements are equivalent. (i) (HELP Inequality) There is a constant K > 0 such that 2 2 2 2 (|f | + q|f | ) ≤ K |f | | − f + qf |2 (f ∈ D). Γ

Γ

Γ

(ii) There exist ϑ+ , ϑ− ∈ [0, π/2) such that Im(−λ2 m(λ)) ≥ 0

(λ ∈ C \ {0}, arg λ ∈ [ϑ+ , π − ϑ− ]).

(2)

ˆ −2 , The optimal constant for the HELP inequality is given by K = (cos ϑ) ˆ where ϑ := min{ϑ ∈ [0, π/2) | (2) holds for all λ ∈ C \ {0}, arg λ ∈ [ϑ, π − ϑ]}. b) (2) is satisﬁed if and only if it holds locally both at 0 and at ∞. If m(λ) ∼ cλα (λ → ∞ in an open sector containing the imaginary axis) with c = 0, α ∈ [−1, 1] \ {0}, then (2) holds at inﬁnity. If the non-tangential limit m0 := limλ↓0 m(λ) exists, then (2) is satisﬁed at 0 if m0 ∈ C \ R and (2) is not satisﬁed at 0 if m0 ∈ R \ {0}. 1 Moreover, if either m is analytic at 0 and m(0) = 0 or if m is 1 analytic at 0 and m (0) = 0, then (2) is satisﬁed at 0. For the proof of these statements, we refer to the cited references, with the exception of the last sentence in the theorem, for which we give a proof now. If m is analytic at 0 and m(0) = 0, then m(λ) = aλ + o(|λ|) with a > 0 (since Im m(λ) > 0 in the complex upper half-plane). Hence we ﬁnd for all λ = reiϑ with suﬃciently small r > 0 and ϑ ∈ ( π3 , 2π 3 ) that Im(−λ2 m(λ)) = −ar3 Im e3iϑ + o(r3 ) ≥ 0.

Hill Operators on Trees

29

1 1 1 Similarly, if m is analytic at 0 and m (0) = 0, then m (λ) = aλ + o(λ) with iϑ a < 0, and therefore for all λ = re with suﬃciently small r > 0 and ϑ ∈ (0, π)

Im(−λ2 m(λ)) = |m(λ)|2 Im(−λ2 (aλ + o(|λ|)) = |m(λ)|2 (−ar3 Im eiϑ + o(r3 )) ≥ 0. Part b) of the preceding theorem shows that, in order to decide whether a HELP inequality holds, we only need to study the limiting behaviour of the m function at ∞ and at 0. In the following we shall prove that for the tree-Hill operator, the condition at ∞ is always satisﬁed; the behaviour of m at 0 depends on whether we have quasi-stability or quasi-instability at λ = 0. The behaviour of m at ∞ We have the following observation, valid under general hypotheses. Lemma 4.3. Let q : [0, l] → ∞ be integrable and Z := {z ∈ C \ {0} | arg z ∈ / (−α, α) mod 2π}, with any α > 0. Then m(λ) ∼ √iλ (λ → ∞, λ ∈ Z). Proof. We begin with obtaining the asymptotics of φ(l, λ) and θ(l, λ) by a variation of constants estimate (cf., e.g., [4]). Let λ ∈ Z. Then the free Schr¨ odinger equation on [0, l], −y = λy, has the canonical fundamental system √ √cos( λx) √ − λ sin( λx)

Φ0 (x) =

√ √ sin( λx)/ λ √ cos( λx)

(x ∈ [0, l]),

√ using the convention Im λ > 0; so solving the initial-value problem −u + q(x)u = λu with given u(0), u (0) by variation of constants, we ﬁnd

u(x) u (x)

l u0 (x) 0 −1 = + Φ0 (x)Φ0 (t) u0 (x) q(t) 0

0 0

u(t) u (t)

dt

(x ∈ [0, l])

and hence u(x) − u0 (x) =

0

x

√ sin( λ(x − t)) √ q(t) u(t) dt λ

(x ∈ [0, l]),

where u0 is the solution of −y = λy with the same initial data as u. Thus, writing √ √ sin( λt) √ u(t) = (u(t) − u0 (t)) + cos( λt) u(0) + u (0), λ

30

B.M. Brown and K.M. Schmidt

we can estimate √

|ei

λx

| |u(x) − u0 (x)| √ x 1 |e2i λ(x−t) − 1| i√λt ≤ √ |q(t)| |e | |u(t) − u0 (t)| 2 |i λ| 0 √

≤

x 0

√

|e2i λt − 1| √ + |u(0)| + |u (0)| dt 2 2|i λ| x √ |q(t)| i λt |q(t)| |u (0)| √ |e √ dt (|u(0)| + √ ). | |u(t) − u0 (t)| dt + | λ| | λ| 0 | λ| |e2i

λt

+ 1|

This is an integral inequality of Gronwall type, x x f (x) ≤ c g+ fg 0

0

with g ≥ 0; it follows that x x x x f (x) e− 0 g ≤ c g+ f g e− 0 g 0 0 x y y y = c + (f (y) − c g− f g) g(y) e− 0 g dy 0 0 0 x y x ≤c f (y) e− 0 g dy = c(1 − e− 0 g ) 0

(for the second line, we rewrite the previous expression as the integral of its derivative). Thus we obtain l √ |u (0)| √1 |q| |u(l) − u0 (l)| ≤ (|u(0)| + √ ) |e−i λl | (e λ 0 − 1). | λ|

In particular, √

√ e−i θ0 (l, λ) = cos( λl) ∼ 2

λl

√ (Im λ → ∞)

and |θ(l, λ) − θ0 (l, λ)| ≤ |e so θ(l, λ) ∼

and

√ −i λl

| (e

√1 | λ|

l 0

|q|

√ (Im λ → ∞); similarly, √ √ sin( λl) i e−i λl √ √ φ0 (l, λ) = ∼ λ 2 λ √

e−i 2

√ −i λl

− 1) ∼ |e

1 | √ | λ|

l

0

|q|,

λl

√

(Im

√

λ → ∞)

√ |e−i λl | |√1λ| 0l |q| 1 |φ(l, λ) − φ0 (l, λ)| ≤ √ (e − 1) ∼ |e−i λl | |λ| | λ| √ √ −i λl so φ(l, λ) ∼ i e2√λ (Im λ → ∞).

l 0

|q|,

Hill Operators on Trees

31 √

The assertion now follows, bearing in mind that |μ(λ)| ≤ √1b = o(|e−i √ and that Im λ → ∞ whenever λ → ∞ in Z.

λl

|)

The behaviour of m at 0 The type of limiting value of m at 0 essentially depends on whether the tree-Hill equation with spectral parameter 0 has quasi-stability or quasiinstability. Theorem 4.4. a) If λ = 0 is a point of quasi-stability, then limλ↓0 m(λ) ∈ C \ R. Consequently, a HELP inequality holds. b) If λ = 0 is a point of quasi-instability, then either • limλ↓0 m(λ) ∈ R \ {0} and there is no valid HELP inequality, or 1 • m(0) = 0, 0 is a Dirichlet eigenvalue, or m (0) = 0, 0 is a Neumann eigenvalue, and in either case, a HELP inequality holds. Proof. First we observe that if φ(l, 0) = 0 or θ (l, 0) = 0, then 0 is not a point of quasi-stability. Indeed, then either the top-right or the bottom-left entries 1 of A(0) vanish, so in view of the determinant of A(0), φ (l, 0) = θ(l,0) , and the quasi-discriminant satisﬁes 2 4 1 2 1 √ 1 D(0)2 = θ(l, 0) + = bθ(l, 0) + √ ≥ . θ(l, 0)b b b bθ(l, 0) a) If 0 is a point of quasi-stability, then observing that all entries of A are entire functions of λ and hence approach a ﬁnite real limit as λ → 0, while μ(λ) ∈ C \ R in the limit, the formulae given in Lemma 4.1 show that m has a non-real limit, and the assertion follows. 1 b) If 0 is a point of quasi-instability and m(0) = 0 = m (0), then m will have a ﬁnite limit as the ratio of non-vanishing analytic functions given in Lemma 4.1 (note μ is analytic except at the quasi-(anti-)symmetric eigenvalues; cf. [9] p. 64); moreover, as all entries of A and μ are real in the limit, so will m be, and by Theorem 4.2 b) there will be no valid HELP inequality. If m(0) = 0, then the formulae of Lemma 4.1 show that φ(l, 0) = 0 and φ (l, 0) = bμ(0), so the transfer matrix must have the form 1 0 μ(0)b A(0) = θ (l,0) . μ(0) b 0 This matrix clearly has eigenvector with eigenvalue μ(0) of modulus 1 |μ(0)| < √1b ; in other words, the extension of φ as a symmetric function on the tree is a Dirichlet eigenfunction. As the two eigenvalues of A(0) are distinct, θ(l, 0) − μ(0) = 0, so m is analytic at 0. Theorem 4.2 b) shows that a HELP inequality holds. 1 The situation m (0) = 0 is analogous; in this case, μ(0) φ(l, 0) A(0) = , 1 0 μ(0)b

32

B.M. Brown and K.M. Schmidt

Now the extension of θ as a symmetric function on the tree is a Neumann 1 eigenfunction, φ (l, 0) = bμ(0), and m is analytic at 0. Again by Theorem 4.2 b), we conclude that a HELP inequality holds. Remark 4.5. If 0 is a point of transition between quasi-stability and quasiinstability, i.e., a quasi-(anti-)periodic eigenvalue, then the analysis is very much like part b) of the preceding theorem, with the diﬀerence that in the 1 exceptional cases m(0) = 0 or m (0) = 0, the two eigenvalues of A(0) are equal and φ or θ, resp., will not be Dirichlet or Neumann eigenfunctions. It would seem that in this exceptional situation the local validity of (2) will need to be veriﬁed separately for the particular operator under study.

5. Calculating the optimal constant Theorem 4.4 only provides a criterion for the existence or otherwise of a HELP inequality with a ﬁnite constant; however, the actual determination of that constant will require more detailed knowledge of the m function, beyond the bare asymptotics at 0 and at ∞. To illustrate the process, we studied the tree-Hill operator with edge length l = 1, branching number b = 3 and the potential qτ (x) := x + τ (x ∈ [0, 1]) on each edge. The oﬀset τ is introduced with a view to investigating the eﬀect of closeness of λ = 0 to an end-point of the region of quasi-stability. More precisely, the ﬁrst two quasistability intervals for potential q0 are approximately [0.68772, 7.15072] and [14.0335, 33.5907]; we therefore considered the cases τ = −3, τ = −0.7 and τ = −1 in order to study situations where 0 is near the middle, close to the end, or at an intermediate position in the quasi-stability interval, respectively. For the calculation, we use (2) directly on semicircular arcs in the complex upper half-plane, i.e., for λ = Reiϑ , ϑ ∈ (0, π); r > 0. The Weyl– Titchmarsh function m is computed, using Lemma 4.1, with Mathematica software from numerical solutions of −y + qτ y = λy on [0, 1]. For ﬁxed r > 0, 1 Im(λ2 m(λ)) = Im(e2iϑ m(reiϑ )) |λ|2 then is a function taking a non-negative value at ϑ = 0, a negative value at ϑ = π2 and a non-negative value at ϑ = π (see Fig. 1). To determine the maximal symmetric interval of negativity around ϑ = π2 for this function, we computed its nearest zero ϑ0 (r) by the bisection method, taking ϑ0 (r) ∈ [0, π2 ) w.l.o.g. (otherwise subtract from π). The supremum ϑˆ of ϑ0 (r), taken over all r > 0, will then determine the optimal constant ˆ −2 . K = (cos ϑ) The numerical results show three diﬀerent situations.

Hill Operators on Trees

33

1.0

0.5

0.5

1.0

1.5

2.0

2.5

3.0

0.5

Figure 1. Fig 1. Plot of Im(e2iϑ m(eiϑ )). In the ﬁrst case, we have τ = −3, and the proﬁle curve of the zero is given in Fig. 2 a). The curve approaches its supremum asymptotically as r → ∞, and ϑˆ = π3 gives the optimal constant K = 4. Since the asymptotic value at inﬁnity will always be the same, as shown in Lemma 4.3, K = 4 is clearly the minimum value for an optimal HELP constant. In the second case, where τ = −0.7, the behaviour of the curve near ∞ is the same as before, but the supremum is now achieved near 0, giving a HELP constant of about 60, cf. Fig. 2 b). This behaviour seems to be typical of a situation where 0 is close to the end of an interval of quasi-stability; it is plausible that the HELP constant should tend to inﬁnity as 0 approaches a region of quasi-instability, in the interior of which the inequality was shown to fail. Finally, the intermediate choice τ = −1 gives rise to a situation where the optimal constant is determined by a local maximum, cf. Fig. 2c), illustrating the point that knowledge of the asymptotics is not suﬃcient in general.

6. Appendix: m function and spectra We give a simple example to show that the m function for the Schr¨odinger operator on a tree does not provide full information on its spectral properties. Consider the tree-Hill operator with potential q = 0, i.e., the tree Laplacian on our regular tree Γ, with a Dirichlet boundary condition at 0. As shown in [11], the spectrum of this operator consists of absolutely continuous bands coinciding with the closure of the quasi-stability intervals and of eigenvalues of inﬁnite multiplicity, situated in quasi-instability intervals. The high multiplicity arises from the fact that any Dirichlet eigenfunction can be shifted down, multiplied with the b-th roots of unity, into the ﬁrst-generation

34

B.M. Brown and K.M. Schmidt

1.0

0.9

0.8

0.7

10

8

6

4

2

2

4

2

4

2

4

1.4

1.3

1.2

1.1

1.0

10

8

6

4

2

1.05

1.00

0.95

10

8

6

4

2

Figure 2. Fig. 2. Zero ϑ0 as a function of log r for τ = −3 (a), τ = −0.7 (b), and τ = −1 (c).

subtrees and extended by 0 on the 0-th generation edge, to obtain another Dirichlet solution in D. Thus, starting from a symmetric eigenfunction, an

Hill Operators on Trees

35

unlimited number of linearly independent eigenfunctions can be created by repeated application of this mechanism. As apparent from the considerations in the proof of Theorem 4.4 b) above, the Dirichlet eigenvalue will, by virtue of its symmetric eigenfunction, be accompanied with a zero of the m function. While the m function thus ﬂags up all eigenvalues for the operator with full tree-Hill symmetry, although not their multiplicities, the disparity becomes more obvious when this symmetry of periodic type is broken. Let us therefore consider a modiﬁed operator which diﬀers from the above by the addition of a potential q˜ of compact support in (0, l) on the 0th generation edge only. The essential spectrum of this perturbed operator will be the same as before, and the inﬁnitely many non-symmetric Dirichlet eigenfunctions constructed above will still be Dirichlet eigenfunctions, as they vanish on the 0-th generation edge and hence know nothing about the potential. However, the Dirichlet eigenvalues on the symmetric subspace will feel the perturbation and hence change in general, as will the m function. Regarding the latter, we can calculate it for the perturbed operator as follows. Let m be the original m function, calculated as previously for the tree-Hill operator from the canonical fundamental system Φ, which in our case is equal to the Φ0 of the proof of Lemma 4.3 above. Now let ˜ ˜ ˜= θ φ Φ θ˜ φ˜ be the canonical fundamental system on [0, l] of −u + q˜ u = λu. For λ ∈ C \ R, the distinguished solution ψ ∈ D will be the same, except on the 0-th generation edge, for the unperturbed and perturbed operators, so ˜ we ﬁnd ψ(l−) ˜ denoting the solution in the perturbed case with ψ, = ψ(l−), ψ˜ (l−) = ψ (l−) and therefore ψ˜ ψ˜ ψ −1 −1 ˜ ˜ (0) = Φ (l) ˜ (l) = Φ (l)Φ(l) (0) ψ ψ˜ ψ ˜ (θ φ˜ − φ θ˜ )(l) ψ(0) + (φ θ˜ − θ φ)(l) ψ (0) = . ˜ (θ φ˜ − φ θ˜ )(l) ψ(0) + (φ θ˜ − θ φ)(l) ψ (0). For the perturbed m function, we thus obtain m ˜ =−

˜ ˜ ψ(0) (θ φ˜ − φ θ˜ )(l) m − (φ θ˜ − θ φ)(l) = . ˜ ˜ ˜ ˜ ˜ ψ (0) (φ θ − θ φ)(l) − (θ φ − φ θ )(l) m

This function m ˜ has a zero at the position of a simple Dirichlet eigenvalue for the perturbed operator with symmetric eigenfunction. However, at the position of the Dirichlet eigenvalue of inﬁnite multiplicity, m has a zero and hence (cf. Lemma 4.1) φ(l) = 0 and, by linear independence, θ(l) = 0. Thus we obtain ˜ θ(l) φ(l) m ˜ = , ˜ − θ (l) φ(l) ˜ φ (l) θ(l)

36

B.M. Brown and K.M. Schmidt

˜ = 0. This will not be the case for most perwhich will be zero only if φ(l) turbations q˜, and hence m ˜ cannot be used to detect the eigenvalue of inﬁnite multiplicity inherited from the unperturbed problem.

References [1] Bennewitz C: A general version of the Hardy-Littlewood-P´ olya-Everitt (HELP) inequality. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984) 9–20 [2] Brown BM, Hinchcliﬀe J, Marletta M, Naboko S, Wood IG: The abstract Titchmarsh-Weyl M -function for adjoint operator pairs and its relation to the spectrum. Integr. Equ. Oper. Theory 63 (2009) 297–320 [3] Brown BM, Langer M, Schmidt KM: The HELP inequality on trees. Proceedings of Symposia in Pure Mathematics 77 (2008) 337–354 [4] Brown BM, Peacock RA, Weikard R: A local Borg-Marchenko theorem for complex potentials. J. Comput. Appl. Math. 148 (2002) 115-131 [5] Carlson R: Hill’s equation for a homogeneous tree. Electron. J. Diﬀerential Equations (1997) No. 23, 30 pp [6] Everitt WN: On an extension to an integro-diﬀerential inequality of Hardy, Littlewood and P´ olya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1972) 295–333 [7] Evans WD, Everitt WN: A return to the Hardy-Littlewood integral inequality. Proc. Roy. Soc. London Ser. A 380 (1982) 447–486 [8] Hardy GH, Littlewood JE: Some integral inequalities connected with the calculus of variations. Quart. J. Math. Oxford 3 (1932) 241–252 [9] Kato T: Perturbation Theory for Linear Operators. Springer, Berlin 1980 [10] Langer M: A general HELP inequality connected with symmetric operators. Proc. Roy. Soc. London Ser. A 462 (2006) 587–606 [11] Sobolev AV, Solomyak MS: Schr¨ odinger operators on homogeneous metric trees: spectrum in gaps. Rev. Math. Phys. 14 (2002) 421–467 B.M. Brown School of Computer Science Cardiﬀ University, Cardiﬀ CF24 3XF UK e-mail: [email protected] K.M. Schmidt School of Mathematics Cardiﬀ University, Cardiﬀ CF24 4AG UK e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 37–54 c 2012 Springer Basel AG

Measure of Non-compactness of Operators Interpolated by Limiting Real Methods Fernando Cobos, Luz M. Fern´andez-Cabrera and Ant´on Mart´ınez Dedicated to Professor David E. Edmunds and Professor W. Desmond Evans on the occasion of their 80th and 70th birthday, respectively.

Abstract. We establish formulae for the measure of non-compactness of operators interpolated by limiting methods that come up by the choice θ = 0 and θ = 1 in the deﬁnition of the real method. Mathematics Subject Classiﬁcation (2010). 46B70, 47B06. Keywords. Limiting interpolation methods, real interpolation, measure of non-compactness, compact operators.

1. Introduction Interpolation theory is a branch of functional analysis with many important applications in partial diﬀerential equations, approximation theory, harmonic analysis, operator theory and other areas of mathematics (see, for example, the monographs by Butzer and Berens [4], Bergh and L¨ofstr¨ om [3], Triebel [24], Bennett and Sharpley [2], Connes [13] or Amrein, Boutet de Monvel and Georgescu [1]). Two main interpolation methods have been developed: The real method (A0 , A1 )θ,q , connected with the Marcinkiewicz interpolation theorem, and the complex method (A0 , A1 )[θ] , associated with the RieszThorin theorem. Here 0 < θ < 1 and 1 ≤ q ≤ ∞. The real method can be described using the Peetre’s K- and J-functionals. If we restrict to ordered couples of Banach spaces, i.e., A0 → A1 , and we make a natural change in the usual deﬁnition, then it was shown by Gomez and Milman [20] that the real method in the K-form is meaningful when θ = 1 and produces very interesting spaces (see also [21]). The extreme Jspaces with θ = 0 have been studied more recently by K¨ uhn, Ullrich and two of the present authors in [7] (see also [9] and [11]). The authors have been supported in part by the Spanish Ministerio de Educaci´ on y Ciencia (MTM2010-15814).

38

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez

The limiting spaces with θ = 1 are very close to A1 and, if θ = 0, they are very near to A0 . This fact produces a number of important changes in their theory in comparison with the real method. For example, consider the behaviour of compact operators. Cwikel [14] and Cobos, K¨ uhn and Schonbek [12] have shown that if any restriction of the operator is compact, then the interpolated operator by the real method is also compact (see [6] for the background of this result). In contrast to this, compactness of T : A0 −→ B0 is not enough to imply that the interpolated operator by the limiting Kmethod is compact (see [7]). The limiting J-method has a similar negative behaviour but now for the restriction T : A1 −→ B1 . However, if T : Ai −→ Bi is compact, then the interpolated operator by the extreme method with θ = i is compact as well. This result has been proved in [7] as well as some estimates for entropy numbers in degenerated cases. In this paper we continue their research by establishing formulae for the measure of non-compactness of operators interpolated by the limiting methods. In the case of the real method, this problem was settled by Fern´ andezMart´ınez and two of the present authors in [10]. They proved that β(T : (A0 , A1 )θ,q −→ (B0 , B1 )θ,q )

(1.1)

≤ C β(T : A0 −→ B0 )1−θ β(T : A1 −→ B1 )θ . No similar result is known for the complex method. Inequality (1.1) led to the problem if a similar formula holds for entropy numbers. This question has been recently solved in the negative by Edmunds and Netrusov [16]. An extension of (1.1) to the so-called abstract real method was given by Szwedek [23] and the present authors [8]. The techniques we use here are based on the tools developed in [10] and [8]. The organization of the paper is simple. In Section 2 we recall some basic results on the measure of non-compactness and we introduce limiting interpolation methods. Section 3 contains the formula for the limiting Jmethod and, in Section 4, we consider the case of the limiting K-method.

2. Preliminaries Let A, B be Banach spaces and let T ∈ L(A, B) be a bounded linear operator acting from A into B. The (ball) measure of non-compactness β(T ) = β(TA,B ) = β(T : A −→ B) of T is deﬁned to be the inﬁmum of the set of numbers σ > 0 for which there is a ﬁnite subset {b1 , . . . , bn } ⊆ B such that T (UA ) ⊆

n

{bj + σUB },

j=1

where UA (respectively, UB ) stands for the closed unit ball in A (respectively, B).

Measure of Non-compactness of Operators Interpolated

39

It is easy to check that β(T ) = 0 if and only if T is compact. Note that β(T ) is the limit of the sequence of entropy numbers of the operator T (see [17]). Clearly, β(TA,B ) ≤ T A,B . Other properties of β can be found in [15, 5, 22]. Let K(A, B) be the set of compact linear operators from A into B. One can show that β(TA,B ) = inf{σ > 0 : there is a Banach space E and V ∈ K(E, B) such that T (UA ) ⊆ σUB + V (UE )}. (2.1) Non-compactness of T ∈ L(A, B) can be also measured by the seminorm γ(T ) = γ(TA,B ) deﬁned to be the inﬁmum of the set of numbers σ > 0 for which there is a Banach space Z and a compact operator V ∈ K(A, Z) such that

T x B ≤ σ x A + V x Z for all x ∈ A. The seminorm γ(T ) coincides with the inﬁmum of all η > 0 such that there is a subspace M of A with ﬁnite codimension, such that

T x B ≤ η x A

for all x ∈ M.

Under this form, the seminorm γ(T ) is studied in [22]. It is shown there that β and γ are equivalent: 1 β(T ) ≤ γ(T ) ≤ 2 β(T ) for any T (2.2) 2 (see [22, Thm. 14.36]). Let A¯ = (A0 , A1 ) be a couple of Banach spaces with A0 → A1 . Here the symbol → means continuous inclusion. For t > 0, the Peetre’s K- and J-functionals are deﬁned by K(t, a) = inf{ a0 A0 + t a1 A1 : a = a0 + a1 , ai ∈ Ai } ,

a ∈ A1 ,

and J(t, a) = max{ a A0 , t a A1 } , a ∈ A0 . Let 1 ≤ q ≤ ∞. The limiting J-space A¯0,q;J = (A0 , A1 )0,q;J , realized in discrete way, is formed by all those a ∈ A1 which can be represented as a=

∞

un

(convergence in A1 ), with (un ) ⊆ A0

(2.3)

n=1

and

∞

q1 J(2n , un )q

<∞

(2.4)

n=1

(the sum must be replaced by the supremum if q = ∞). The norm in A¯0,q;J is ⎧ 1q ⎫ ∞ ⎨ ⎬

a 0,q;J = inf J(2n , un )q ⎩ ⎭ n=1

40

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez

where the inﬁmum is taken over all sequences (un ) satisfying (2.3) and (2.4). Spaces (A0 , A1 )0,q;J have been introduced in [7]. They correspond to the extreme choice θ = 0 in the deﬁnition of the real interpolation method. These spaces are very close to A0 and (A0 , A1 )0,1;J = A0 . For 1 < q ≤ ∞, they can be also described by using the K-functional. Namely, A¯0,q;J coincides with A¯log,q;K = (A0 , A1 )log,q;K ⎧ ⎫ ∞ 1 ⎨ ⎬

K(2n , a) q q = a ∈ A1 : a log,q;K = <∞ ⎩ ⎭ n n=1

and the norms · 0,q;J and · log,q;K are equivalent (see [7, Thm. 4.2]). Note that in the deﬁnition of A¯log,q;K we have replaced the usual power weight of the real method (2n )−θ , by the logarithmic weight (log2 2n )−1 = n−1 . This is the reason for the notation we use. We refer to [18], [19], [11] and the references given there for properties of K-interpolation methods with logarithmic weights. Working on a ﬁnite measure space (Ω, μ) and with 1 < q ≤ ∞, we have (L∞ , L1 )0,q;J = L∞,q (log L)−1 ⎧ ⎫ q1 ⎨ ⎬ μ(Ω) dt q = f : f L∞,q (log L)−1 = (1 + | log t|)−1 f ∗∗ (t) <∞ . ⎩ ⎭ t 0 t Here f ∗∗ (t) = (1/t) 0 f ∗ (s)ds and f ∗ is the non-increasing rearrangement of the measurable function f deﬁned by f ∗ (t) = inf {s > 0 : μ({x ∈ Ω : |f (x)| > s}) ≤ t} . We are also interested in the corresponding spaces to the limit choice θ = 1 in the deﬁnition of the real method (see [20]). Let A0 , A1 and q be as above. The space A¯1,q;K = (A0 , A1 )1,q;K consists of all a ∈ A1 which have a ﬁnite norm ∞ 1q

q

a 1,q;K = 2−n K(2n , a) . n=1

Spaces A¯1,q;K are very near to A1 , with (A0 , A1 )1,∞;K = A1 . They are related with limiting J-spaces by duality: If A0 is dense in A1 , 1 < q < ∞ and 1/q + 1/q = 1, then (A0 , A1 )0,q;J = (A1 , A0 )1,q ;K (see [7, Thm. 8.1]). Let ρ(t) = t−1 (1 + log t) and deﬁne A¯ρ,q;J = (A0 , A1 )ρ,q;J as the collection of all those a ∈ A1 for which there is a sequence (un ) ⊆ A0 such that ∞

a= un (convergence in A1 ) n=1

and

∞

n=1

1q n

n

(ρ(2 )J(2 , un))

q

< ∞.

Measure of Non-compactness of Operators Interpolated We put

a ρ,q;J = inf

⎧ ∞ ⎨ ⎩

1q (ρ(2n )J(2n , un ))

q

n=1

: a=

∞

41

⎫ ⎬ un

n=1

⎭

.

For 1 ≤ q < ∞, it turns out that (A0 , A1 )1,q;K = (A0 , A1 )ρ,q;J with equivalence of norms (see [7, Thm. 7.6]). ¯ = (B0 , B1 ) be another pair of Banach spaces with B0 → B1 . We Let B ¯ B) ¯ to mean that T ∈ L(A1 , B1 ) and the restriction of T to A0 write T ∈ L(A, deﬁnes a bounded linear operator from A0 into B0 . We put Mj = T Aj ,Bj , j = 0, 1. It is easy to check that the restrictions T : (A0 , A1 )0,q;J −→ (B0 , B1 )0,q;J

and T : (A0 , A1 )1,q;K −→ (B0 , B1 )1,q;K

are also bounded. As for the norms, it is shown in [7] that C M0 if M1 ≤ M0 ,

T A¯0,q;J ,B¯0,q;J ≤ C M0 (1 + log(M1 /M0 )) if M0 ≤ M1 , and

T A¯1,q;K ,B¯1,q;K ≤

C M1 C M1 (1 + log(M0 /M1 ))

if M0 ≤ M1 , if M1 ≤ M0 .

(2.5)

(2.6)

Here C is a constant independent of T . To describe the behaviour of the measure of non-compactness under the limiting K- and J-methods, we shall work with certain vector-valued sequence spaces and families of projections on them. Given any sequence (Wn ) of Banach spaces and any sequence (λn ) of non-negative numbers, we write q (λn Wn ) for the vector-valued q space q (λn Wn )

∞ 1/q

q = w = (wn ) : wn ∈Wn and w q (λn Wn ) = (λn wn Wn ) <∞ . n=1

When λn = 1 for all n ∈ N, we write simply q (Wn ). Given two Banach spaces V , W , we denote by (V ⊕ W ) 1 the direct sum of V and W , normed by (x, y) = x V + y W . The space (V ⊕ W ) ∞ is deﬁned similarly.

3. Limiting J-method and measure of non-compactness Let (Wn ) be a sequence of Banach spaces. It was shown in [7, Lemma 6.3] that for 1 ≤ q ≤ ∞ we have q (Wn ) → 1 (Wn ), 1 (2−n Wn ) 0,q;J . (3.1) Next we establish another formula of this kind.

42

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez

Lemma 3.1. Let (Wn ) be a sequence of Banach spaces and let 1 < q ≤ ∞. Then ∞ (Wn ), ∞ (2−n Wn ) 0,q;J → q (n−1 Wn ). Proof. We consider in (∞ (Wn ), ∞ (2−n Wn ))0,q;J the equivalent norm

· log,q;K . Take any x = (xn ) ∈ (∞ (Wn ), ∞ (2−n Wn ))0,q;J and let x = y +z with y = (yn ) ∈ ∞ (Wn ) and z = (zn ) ∈ ∞ (2−n Wn ). For any k ∈ N, we obtain

xk Wk ≤ yk Wk + zk Wk ≤ y ∞(Wn ) + 2k z ∞ (2−n Wn ) . So, xk Wk ≤ K(2k , x). Therefore, ∞ q 1q

1

x q (n−1 Wn ) =

xn Wn n n=1 ∞ q 1q

1 n ≤ K(2 , x) = x log,q;K . n n=1

We show now the behaviour of the measure of non-compactness under the limiting J-method. Given any real number x we put x+ = max{x , 0}. ¯ = (B0 , B1 ) be pairs of Banach spaces with Theorem 3.2. Let A¯ = (A0 , A1 ), B ¯ B). ¯ Then we have: A0 → A1 , B0 → B1 , let 1 ≤ q ≤ ∞ and T ∈ L(A, (a) β(TA¯0,q;J ,B¯0,q;J ) = 0 if β(TA0 ,B0 ) = 0, (b) β(TA¯0,q;J ,B¯0,q;J ) ≤ Cβ(TA0 ,B0 ) if β(TA1 ,B1 ) = 0, β(TA1 ,B1 ) (c) β(TA¯0,q;J ,B¯0,q;J ) ≤ Cβ(TA0 ,B0 ) 1 + log β(TA ,B ) if β(TAi ,Bi ) > 0 0

0

+

for i = 0, 1. Here C is a constant independent of T . Proof. Since (A0 , A1 )0,1;J = A0 and (B0 , B1 )0,1;J = B0 the result is clear for q = 1. Suppose then 1 < q ≤ ∞. In what follows, we let Gn to be A0 endowed with the norm J(2n , ·), and we put Fn for B1 normed by K(2n , ·). The space A¯0,q;J is the quotient space of q (Gn ) given by the surjective map π : q (Gn ) −→ A¯0,q;J deﬁned by ∞

π(un ) = un (in A1 ). Note also that the maps n=1

π : 1 (Gn ) −→ A0

and

π : 1 (2−n Gn ) −→ A1

are bounded with norm ≤ 1. Consider now (B0 , B1 )0,q;J realized as a K-space and let jb = {b, b, . . . }. ¯log,q;K −→ q (n−1 Fn ) is a metric injection. Moreover, the maps The map j : B j : B0 −→ ∞ (Fn ) are bounded with norm ≤ 1.

and

j : B1 −→ ∞ (2−n Fn )

Measure of Non-compactness of Operators Interpolated

43

The following diagram of bounded operators holds: π

T

j

1 (Gn ) −−−−→ A0 −−−−→ B0 −−−−→ ∞ (Fn ) j

1 (2−n Gn ) −−−−→ A1 −−−−→ B1 −−−−→ ∞ (2−n Fn ) π

T

——————————————————————————– j π T ¯log,q;K −−− q (Gn ) −−−−→ A¯0,q;J −−−−→ B −→ q (n−1 Fn )) ¯log,q;K , we are going Instead of working directly with T : A¯0,q;J −→ B to estimate the measure of non-compactness of jT π : q (Gn ) −→ q (n−1 Fn ) . The reason is that on the vector-valued sequence spaces, we can take advantage of the following families of projections. Given any m ∈ N, we put Pm (un ) = (u1 , . . . , um , 0, 0, . . . ), Qm (un ) = (0, . . . , 0, um+1 , um+2 , . . . ). It is clear that the identity operator I can be written as I = Pm + Qm . Moreover

Pm 1 (Gn ), 1 (Gn ) = Pm 1 (2−n Gn ), 1 (2−n Gn ) = Pm 1 (Gn ), 1 (2−n Gn ) = 1 and

Pm 1 (2−n Gn ), 1 (Gn ) = 2m . Operators Qm satisfy

Qm 1 (Gn ), 1 (Gn ) = Qm 1 (2−n Gn ), 1 (2−n Gn ) = 1 and

Qm 1 (Gn ), 1 (2−n Gn ) = 2−(m+1) . We denote by Rm , Sm the corresponding families of projections acting on the pair (∞ (Fn ) , ∞ (2−n Fn )) . They satisfy analogous norm estimates. We proceed to estimate the measure of non-compactness. Since π is a metric surjection and j is a metric injection, we have for each m ∈ N ¯ log,q;K β TA¯0,q;J ,B¯log,q;K = β T π : q (Gn ) −→ B ¯log,q;K + 2β jT πQ2m : q (Gn ) −→ q (n−1 Fn ) ≤ β T πP2m : q (Gn ) −→ B ¯log,q;K ≤ β T πP2m : q (Gn ) −→ B + 2β Rm jT πQ2m : q (Gn ) −→ q (n−1 Fn ) + 2β Sm jT πQ2m : q (Gn ) −→ q (n−1 Fn ) . Take σi > β(TAi ,Bi ) for i = 0, 1. Our plan is to show ﬁrst that σ 1 ¯log,q;K ≤ C1 σ0 1 + log β T πP2m : q (Gn ) −→ B , for all m ∈ N, σ0 + (3.2) then β Rm jT πQ2m : q (Gn ) −→ q (n−1 Fn ) −→ 0 as m → ∞, (3.3)

44

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez

and ﬁnally that there is N ∈ N such that for any m ≥ N we have σ1 −1 β Sm jT πQ2m : q (Gn ) −→ q (n Fn ) ≤ C2 σ0 1 + log . (3.4) σ0 + Assuming these three facts, the choice of m ∈ N big enough yields that σ1 β(TA¯0,q;J ,B¯log,q;K ) ≤ Cσ0 1 + log . σ0 + Consequently, if β(TA0 ,B0 ) = 0 , letting σ0 → 0, we obtain case (a) of the theorem. If β(TA1 ,B1 ) = 0 , letting σ1 → 0, we derive case (b). Finally, if β(TAi ,Bi ) > 0 for i = 0, 1, take ε > 0 and write σi = (1 + ε)β(TAi ,Bi ). It follows that β(TA1 ,B1 ) β(TA¯0,q;J ,B¯log,q;K ) ≤ C(1 + ε)β(TA0 ,B0 ) 1 + log . β(TA0 ,B0 ) + This gives case (c) by letting ε → 0. We proceed to prove (3.2). By the equivalence (2.2) between the measures β and γ, we can ﬁnd Banach spaces Zi and compact linear operators Vi ∈ K(Ai , Zi ), i = 0, 1, such that

T x Bi ≤ 2σi x Ai + Vi x Zi ,

x ∈ Ai ,

i = 0, 1.

(3.5)

For any n ∈ N, put Wn = (Z0 ⊕ Z1 ) 1 . Let λ > 0 and r ∈ N ∪ {0} be ﬁxed numbers to be speciﬁed later and consider the operators V, Lλ,r : q (Gn ) −→ q (Wn ) deﬁned by V (un ) = (vn ) and Lλ,r (un ) = (wn ) where (V0 un , 2n V1 un ) if 1 ≤ n ≤ 2m, vn = (0 , 0) otherwise, and

⎧ r+1 ⎪ ⎪ (λV u , 0) if n = 1, 0 j ⎨ j=1 n wn = (V0 ur+n , 2 V1 ur+n ) if 2 ≤ n ≤ 2m − r, ⎪ ⎪ ⎩ (0 , 0) otherwise.

It is easy to check that V and Lλ,r are linear and compact. Now we distinguish two cases. If β(TA1 ,B1 ) < β(TA0 ,B0 ), then we can suppose that 1 < σ0 /σ1 . For u = (un ) ∈ q (Gn ), we have

un Ai ≤ 2−in J(2n , un ) ≤ 2−in (σ0 /σ1 )i J(2n , un ),

n ∈ N,

Using (3.5), we derive

T un Bi ≤ 2σi 2−in (σ0 /σ1 )i J(2n , un) + Vi un Zi .

i = 0, 1.

Measure of Non-compactness of Operators Interpolated

45

It follows that

T πP2m u 0,q;J ≤ ≤

2m

2m

1/q n

q

J(2 , T un )

n=1

1/q q

(max{2σ0 J(2 , un ) + V0 un Z0 , 2σ0 J(2 , un ) + 2 V1 un Z1 }) n

n

n=1

≤ 2σ0

∞

n

1/q n

J(2 , un )

q

+ V u q (Wn ) .

n=1

Since V is compact, this yields that

β (T πP2m ) ≤ C1 σ0 = C1 σ0

σ1 1 + log . σ0 +

Suppose now β(TA0 ,B0 ) ≤ β(TA1 ,B1 ). This time we may assume that 1 ≤ σ1 /σ0 . Let r = [log2 (σ1 /σ0 )] where the logarithm is taken in the base 2 and [·] is the greatest integer function. Clearly r ≥ 0 and 2r ≤ σ1 /σ0 < 2r+1 . Given any u = (un ) ∈ q (Gn ), let v = (vn ) be the sequence deﬁned by v1 =

r+1

uj , v2 = ur+2 , . . . , v2m−r = u2m and vn = 0 for n > 2m − r.

j=1

Then

v1 A0 ≤

r+1

uj A0 ≤ (r + 1) max

1≤j≤r+1

j=1

Using (3.5), we get

T v1 B0 ≤ 2σ0 (r + 1) max

1≤j≤r+1

2 T v1 B1 ≤ 2C3 T v1 B0 ≤ 4C3 σ0 (r + 1) max

1≤j≤r+1

J(2, T v1 ) ≤ 4(1 + C3 )σ0 (r + 1) max

1≤j≤r+1

J(2j , uj ) .

J(2j , uj ) + V0 v1 Z0 .

Moreover, since B0 → B1 , we obtain

So,

J(2j , uj ) + 2C3 V0 v1 Z0 .

J(2j , uj ) + (1 + 2C3 ) V0 v1 Z0 .

For 2 ≤ n ≤ 2m − r, using again (3.5), we obtain with i = 0, 1 2in T vn Bi ≤ 2in 2σi 2i(−r−n) J(2r+n , ur+n ) + 2in Vi ur+n Zi σ i 0 ≤ 2σi 2 J(2r+n, ur+n ) + 2in Vi ur+n Zi σ1 ≤ 4σ0 J(2r+n , ur+n ) + 2in Vi ur+n Zi . Whence, J(2n , T vn) ≤ 4σ0 J(2r+n , ur+n ) + max{ V0 ur+n Z0 , 2n V1 ur+n Z1 }.

46

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez Let λ = 1 + 2C3 and consider the compact operator Lλ,r . Since

T πP2m u 0,q;J = T πP2m v 0,q;J ≤

2m−r

1/q n

q

J(2 , T vn )

n=1

≤ 4(1 + C3 )σ0 (r + 1) (un ) q (Gn ) + Lλ,r (un ) q (Wn ) and r ≤ C4 log(σ1 /σ0 ), it follows that β (T πP2m ) ≤ C1 σ0

σ1 1 + log . σ0 +

This completes the proof of (3.2). To establish (3.3), we ﬁrst use (3.1), Lemma 3.1 and the norm estimate (2.5) to obtain β Rm jT πQ2m ≤ Rm jT πQ2m q (Gn ), q (n−1 Fn ) ≤ C5 Rm jT πQ2m ( 1 (Gn ), 1 (2−n Gn ))0,q,J ,( ∞ (Fn ), ∞ (2−n Fn ))0,q,J ≤ C6 Rm jT πQ2m 1 (Gn ), ∞ (Fn )

Rm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) × 1 + log .

Rm jT πQ2m 1 (Gn ), ∞ (Fn ) + Using the factorization Q2m

jT π

1 (Gn ) −−−−−−→ 1 (2−n Gn ) −−−−−−→ ∞ (2−n Fn ) −−−−m−−→ ∞ (Fn ) R

and the estimates for the norms of the projections, we derive

Rm jT πQ2m 1 (Gn ), ∞ (Fn ) ≤ 2−(2m+1) T A1 ,B1 2m = 2−(m+1) T A1,B1 . On the other hand, the factorization Q2m

jT π

1 (2−n Gn ) −−−−−−→ 1 (2−n Gn ) −−−−−−→ ∞ (2−n Fn ) −−−−m−−→ ∞ (2−n Fn ) R

yields that

Rm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) ≤ T A1,B1 for all m ∈ N. Hence, lim Rm jT πQ2m 1 (Gn ), ∞ (Fn )

Rm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) × 1 + log =0

Rm jT πQ2m 1 (Gn ), ∞ (Fn ) +

m→∞

and (3.3) follows.

Measure of Non-compactness of Operators Interpolated

47

Finally, to prove (3.4), we use again (3.1), Lemma 3.1 and (2.4) to get β Sm jT πQ2m ≤ Sm jT πQ2m q (Gn ), q (n−1 Fn ) ≤ C5 Sm jT πQ2m ( 1 (Gn ), 1 (2−n Gn ))0,q,J ,( ∞ (Fn ), ∞ (2−n Fn ))0,q,J ≤ C6 Sm jT πQ2m 1 (Gn ), ∞ (Fn )

Sm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) × 1 + log .

Sm jT πQ2m 1 (Gn ), ∞ (Fn ) + Clearly,

Sm jT πQ2m 1 (Gn ), ∞ (Fn ) ≤ jT πQ2m 1 (Gn ), ∞ (Fn ) . Moreover,

jT πQ2 1 (Gn ), ∞ (Fn ) ≥ jT πQ4 1 (Gn ), ∞ (Fn ) ≥ · · · ≥ 0. Whence, there is λ ≥ 0 such that

jT πQ2m 1 (Gn ), ∞ (Fn ) −→ λ as m → ∞. We are going to estimate λ from above by using σ0 . For this aim, let wm ∈ U 1 (Gn ) such that

jT πQ2mwm ∞ (Fn ) −→ λ as m → ∞, and ﬁnd vectors {b1 , . . . , bs } ⊆ B0 such that s T π U 1 (Gn ) ⊆ {bk + σ0 UB0 } . k=1

Since the number of bk is ﬁnite, there is some k, say k = 1, and a subsequence (m ) of N such that

T πQ2m wm − b1 B0 ≤ σ0 for all m .

Using that Q2m 1 (Gn ), 1 (2−n Gn ) = 2−(2m +1) , we derive for any n ∈ N K(2n , b1 ) ≤ b1 − T πQ2m wm B0 + 2n T πQ2m wm B1 ≤ σ0 + 2n T A1,B1 Q2m wm 1 (2−n Gn )

≤ σ0 + 2n−(2m +1) T A1 ,B1 −→ σ0

as m → ∞.

Hence,

jb1 ∞ (Fn ) = sup {K(2n , b1 )} ≤ 2σ0 . n∈N

It follows that λ = lim

jT πQ2m wm ∞ (Fn ) m →∞ ≤ sup jT πQ2m wm − jb1 ∞ (Fn ) + jb1 ∞(Fn ) m

≤ sup { T πQ2m wm − b1 B0 + 2σ0 } ≤ 3σ0 . m

This yields that there is N1 ∈ N such that for all m ≥ N1 we have

jT πQ2m 1 (Gn ), ∞ (Fn ) ≤ 4σ0 .

48

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez

Therefore

Sm jT πQ2m 1 (Gn ), ∞ (Fn ) ≤ 4σ0 for all m ≥ N1 . Now we turn our attention to Sm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) . We can ﬁnd a ﬁnite set {d1 , . . . , dr } ⊆ U 1 (2−n Gn ) , formed by sequences having only a ﬁnite number of non-zero coordinates, such that r jT π U 1 (2−n Gn ) ⊆ jT πdk + 2σ1 U ∞ (2−n Fn ) . k=1

Since Sm ∞ (Fn ), ∞ (2−n Fn ) = 2−(m+1) and {jT πdk : 1 ≤ k ≤ r} ⊆ ∞ (Fn ), there is N2 ∈ N such that

Sm jT πdk ∞ (2−n Fn ) ≤ σ1 for 1 ≤ k ≤ r and m ≥ N2 . We claim that

Sm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) ≤ 3σ1 for all m ≥ N2 . Indeed, take any u ∈ U 1 (2−n Gn ) . Since Q2m u ∈ U 1 (2−n Gn ) , we can ﬁnd 1 ≤ k ≤ r such that jT πQ2m u − jT πdk ∞ (2−n Fn ) ≤ 2σ1 . Therefore, for m ≥ N2 , we derive

Sm jT πQ2m v ∞ (2−n Fn ) ≤ Sm jT πQ2m v − Sm jT πdk ∞ (2−n Fn ) + Sm jT πdk ∞ (2−n Fn ) ≤ jT πQ2m v − jT πdk ∞ (2−n Fn ) + σ1 ≤ 3σ1 . Consequently, for m ≥ max{N1 , N2 } we conclude that β Sm jT πQ2m ≤ C6 Sm jT πQ2m 1 (Gn ), ∞ (Fn )

Sm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) × 1 + log

Sm jT πQ2m 1 (Gn ), ∞ (Fn ) + σ1 ≤ C7 σ0 1 + log . σ0 + This establishes (3.4) and completes the proof.

As a direct consequence of Theorem 3.2, we recover the compactness result for the limiting J-method proved in [7]. ¯ = (B0 , B1 ) be pairs of Banach spaces with Corollary 3.3. Let A¯ = (A0 , A1 ), B ¯ B). ¯ If T : A0 −→ B0 A0 → A1 , B0 → B1 , let 1 ≤ q ≤ ∞ and let T ∈ L(A, is compact, then T : (A0 , A1 )0,q;J −→ (B0 , B1 )0,q;J is also compact.

Measure of Non-compactness of Operators Interpolated

49

4. Limiting K-method and measure of non-compactness To deal with the limiting K-method, we shall need the following interpolation formula for vector-valued sequence spaces established in [7, Lemma 7.12]: ∞ (Wn ), ∞ (2−n Wn ) 1,q;K → q (2−nWn ). (4.1) We shall also require the following result. Lemma 4.1. Let (Wn ) be a sequence of Banach spaces and let 1 ≤ q < ∞. Then 1 + n q Wn → 1 (Wn ), 1 (2−n Wn ) 1,q;K . n 2 Proof. We view (1 (Wn ), 1 (2−n Wn ))1,q;K as a J-space, endowed with the equivalent norm · ρ,q;J (see Section 2). Take any w = (wn ) ∈ q 1+n 2n Wn . m m Let um = (um n )n∈N with un = 0 if n = m and un = wm if n = m. We have J(2m , um ) = max um 1 (Gn ) , 2m um 1 (2−n Gn ) = max wm Gm , 2m 2−m wm Gm = wm Gm . Whence,

w ρ,q;J ≤

∞

1+n

2n

n=1

=

∞

1+n

2n

n=1

q 1q J(2 , u ) n

n

q 1q

wn Gn

= w 1+n

q

2n

Wn

.

For the limiting K-method the result is as follows ¯ = (B0 , B1 ) be pairs of Banach spaces with Theorem 4.2. Let A¯ = (A0 , A1 ), B ¯ B). ¯ Then we have: A0 → A1 , B0 → B1 , let 1 ≤ q ≤ ∞ and T ∈ L(A, (a) β(TA¯1,q;K ,B¯1,q;K ) = 0 if β(TA1 ,B1 ) = 0, (b) β(TA¯1,q;K ,B¯1,q;K ) ≤ Cβ(TA1 ,B1 ) if β(TA0 ,B0 ) = 0, β(TA0 ,B0 ) (c) β(TA¯1,q;K ,B¯1,q;K ) ≤ Cβ(TA1 ,B1 ) 1+ log β(TA ,B ) if β(TAi ,Bi ) > 0 1

1

+

for i = 0, 1. Here C is a constant independent of T . Proof. This time the trivial case corresponds to q = ∞ because (A0 , A1 )1,∞;K = A1 and (B0 , B1 )1,∞;K = B1 . Assume then 1 ≤ q < ∞. We shall use the notation introduced in the proof of Theorem 3.2 for vector-valued sequence spaces, maps and projections. The relevant diagram says now π

T

j

1 (Gn ) −−−−→A0 −−−−→ B0 −−−−→ ∞ (Fn ) j

1 (2−n Gn ) −−−−→A1 −−−−→ B1 −−−−→ ∞ (2−n Fn ) π

q

1 + n 2n

T

j π T ¯1,q;K −−− Gn −−−−→ A¯ρ,q;J =A¯1,q;K −−−−→ B −→ q (2−n Fn )

50

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez

For each m ∈ N, we have β TA¯1,q;K ,B¯1,q;K ≤ 2β jT : A¯1,q;K −→ q (2−n Fn ) ≤ 2β R2m jT : A¯1,q;K −→ q (2−n Fn ) 1 + n + C1 β S2m jT π : q Gn −→ q (2−n Fn ) n 2 ≤ 2β R2m jT : A¯1,q;K −→ q (2−n Fn ) 1 + n −n + C1 β S2m jT πPm : q G −→ (2 F ) n q n 2n 1 + n −n + C1 β S2m jT πQm : q G −→ (2 F ) . n q n 2n Take σi > β(TAi ,Bi ) for i = 0, 1. Proceeding as in the proof of (3.3) but using now (4.1), Lemma 4.1, (2.6) and the factorization

jT π

2m 1 (2−n Gn ) −−−−m−−→ 1 (Gn ) −−−−−−→ ∞ (Fn ) −−−− −−→ ∞ (2−n Fn )

P

S

we derive 1 + n −n β S2m jT πPm : q G −→ (2 F ) −→ 0 as m → ∞. (4.2) n q n 2n Moreover, using a similar argument as in the proof of (3.4), we obtain that there is N ∈ N such that for any m ≥ N 1 + n σ0 −n β S2m jT πQm : q Gn −→ q (2 Fn ) ≤ C2 σ1 1 + log . 2n σ1 + (4.3) For the remaining term, we claim that σ0 −n ¯ β R2m jT : A1,q;K −→ q (2 Fn ) ≤ C3 σ1 1 + log . (4.4) σ1 + Indeed, by (2.1), there are Banach spaces Ei , i = 0, 1, and compact linear operators Vi : Ei −→ Bi such that T (UAi ) ⊆ σi UBi + Vi (UEi ),

i = 0, 1.

(4.5)

For each n ∈ N, let En = (E0 ⊕ E1 ) ∞ . Let r ∈ N ∪ {0} be a ﬁxed number to be determined later, and deﬁne operators V, Lr : q (2−n En ) −→ q (2−n Fn ) by V (x0,n , x1,n ) = (vn ), Lr (x0,n , x1,n ) = (wn ) where 2 (V0 x0,n + 2−n V1 x1,n ) if 1 ≤ n ≤ 2m, vn = 0 otherwise, and

4(1 + r) V0 x0,n + (2n σ1 /σ0 )−1 V1 x1,n if 1 ≤ n ≤ 2m, wn = 0 otherwise.

It is not hard to check that V, Lr ∈ K(q (2−n En ), q (2−n Fn )). Suppose ﬁrst β(TA0 ,B0 ) ≤ β(TA1 ,B1 ). Then we may assume that 0 < σ0 ≤ σ1 . Take any 0 = a ∈ UA¯1,q;K and put dn = dn (a) = 2K(2n, a).

Measure of Non-compactness of Operators Interpolated

51

Since K(2n , a) < dn , we can decompose a = a0,n + a1,n with ai,n ∈ Ai , n −1

d−1 n a0,n A0 ≤ 1 and 2 dn a1,n A1 ≤ 1. Using (4.5), we can ﬁnd xi,n ∈ UEi such that

T ai,n − Vi (2−in dn xi,n ) Bi ≤ 2−in dn σi ≤ 2−in dn σ1 .

(4.6)

Write x = ((dn /2)(x0,n , x1,n )), which belongs to the closed unit ball of q (2−n Wn ). Let V x = (vn ). We have vn = dn (V0 x0,n + 2−n V1 x1,n ),

1 ≤ n ≤ 2m.

Using (4.6), we get K(2n , T a − vn ) ≤ 2dn σ1 ,

1 ≤ n ≤ 2m.

Therefore

R2m jT a − V x q (2−n Fn ) ≤ 2σ1 (dn ) q (2−n Fn ) ≤ 4σ1 . This yields that

β R2m jT ≤ 4σ1 = 4σ1

σ0 1 + log σ1

. +

Suppose now β(TA1 ,B1 ) < β(TA0 ,B0 ). We may assume this time that 0 < σ1 < σ0 . Let r = [log2 (σ0 /σ1 )]. Then r ≥ 0 and 2r ≤ σ0 /σ1 < 2r+1 . Given any 0 = a ∈ UA¯1,q;K , let again dn = dn (a) = 2K(2n , a), n ∈ N. If r = 0, we put dk = d1 for 1 − r ≤ k ≤ 0. We have K(2n (σ1 /σ0 ), a) ≤ K(2n−r , a) < dn−r , n ∈ N. Hence, we can decompose a = a0,n + a1,n with ai,n ∈ Ai , d−1 n−r a0,n A0 ≤ 1 and (2n σ1 /σ0 )d−1 a

≤ 1. By (4.5), there exist x ∈ UEi such that 1,n A i,n n−r 1

T ai,n − Vi ((2n σ1 /σ0 )−i dn−r xi,n ) Bi ≤ (2n σ1 /σ0 )−i dn−r σi = 2−in σ0 dn−r . (4.7) Let x = 4−1 (1 + r)−1 (dn−r x0,n , dn−r x1,n ). Then x ∈ U q (2−n Wn ) because 1/q r ∞

1

x q (2−n Wn ) = (2−j d1 )q + (2−j dj−r )q 4(1 + r) j=1 j=r+1 r ∞ 1/q 1/q 1 −j q −r −j q ≤ (2 d1 ) +2 (2 dj ) 4(1 + r) j=1 j=1 ≤

1 (4r + 2−r 2) ≤ 1. 4(1 + r)

Put Lr x = (wn ). So, wn = V0 (dn−r x0,n ) + (2n σ1 /σ0 )−1 V1 (dn−r x1,n ),

1 ≤ n ≤ 2m.

Using (4.7) we get K(2n , T a − wn ) ≤ 2σ0 dn−r ,

1 ≤ n ≤ 2m.

52

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez

Consequently,

R2m jT a − Lr x q (2−n Fn ) ≤ 2σ0 2−r

2m

(2−(j−r) dj−r )q

1/q

j=1

σ0 σ0 ≤ 8σ0 2−r (1 + r) ≤ 16σ1 1 + log2 ≤ C4 σ1 1 + log . σ1 σ1 This establishes (4.4). From (4.2), (4.3) and (4.4), we conclude the proof of the theorem by letting σi −→ β(TAi ,Bi ) , i = 0, 1. As a direct consequence of Theorem 4.2 we can derive the compactness result for the limiting K-method given in [7, Thm. 7.14]. ¯ = (B0 , B1 ) be pairs of Banach spaces with Corollary 4.3. Let A¯ = (A0 , A1 ), B ¯ B). ¯ If T : A1 −→ B1 A0 → A1 , B0 → B1 , let 1 ≤ q ≤ ∞ and let T ∈ L(A, is compact, then T : (A0 , A1 )1,q;K −→ (B0 , B1 )1,q;K is also compact.

References [1] W.O. Amrein, A. Boutet de Monvel and V. Georgescu, “C0 -Groups, Commutator Methods and Spectral Theory of N -Body Hamiltonians”, Progress in Math. 135, Birkh¨ auser, Basel, 1996. [2] C. Bennett and R. Sharpley, “Interpolation of Operators”, Academic Press, Boston, 1988. [3] J. Bergh and J. L¨ ofstr¨ om, “Interpolation Spaces. An Introduction”, Springer, Berlin 1976. [4] P.L. Butzer and H. Berens, “Semi-Groups of Operators and Approximation”, Springer, New York, 1967. [5] B. Carl and I. Stephani, “Entropy, Compactness and the Approximation of Operators”, Cambridge Univ. Press, Cambridge 1990. [6] F. Cobos, Interpolation Theory and Compactness, in “Function Spaces, Inequalities and Interpolation (Paseky 2009), pages 31-75, Ed. J. Lukeˇ s and L. Pick, Matfyzpress, Prague, 2009. [7] F. Cobos, L.M. Fern´ andez-Cabrera, T. K¨ uhn and T. Ullrich, On an extreme class of real interpolation spaces, J. Funct. Anal. 256 (2009) 2321-2366. [8] F. Cobos, L.M. Fern´ andez-Cabrera and A. Mart´ınez, Abstract K and J spaces and measure of non-compactness, Math. Nachr. 280 (2007) 1698-1708. [9] F. Cobos, L.M. Fern´ andez-Cabrera and M. Mastylo, Abstract limit J-spaces, J. London Math. Soc. (2) 82 (2010), 501–525. [10] F. Cobos, P. Fern´ andez-Mart´ınez and A. Mart´ınez, Interpolation of the measure of non-compactness by the real method, Studia Math. 135 (1999) 25-38. [11] F. Cobos and T. K¨ uhn, Equivalence of K- and J-methods for limiting real interpolation spaces, J. Funct. Anal. (to appear).

Measure of Non-compactness of Operators Interpolated

53

[12] F. Cobos, T. K¨ uhn and T. Schonbek, One-sided compactness results for Aronszajn-Gagliardo functors, J. Funct. Anal. 106 (1992) 274-313. [13] A. Connes, “Noncommutative Geometry”, Academic Press, San Diego, 1994. [14] M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992) 333-343. [15] D. E. Edmunds and W. D. Evans, “Spectral Theory and Diﬀerential Operators”, Clarendon Press, Oxford 1987. [16] D. E. Edmunds and Yu. Netrusov, Entropy numbers and interpolation, Math. Ann., DOI:10.1007/s00208-010-0624-1. [17] D. E. Edmunds and H. Triebel, “Function Spaces, Entropy Numbers, Diﬀerential Operators”, Cambridge Univ. Press, Cambridge 1996. [18] W. D. Evans, B. Opic and L. Pick, Real Interpolation with logarithmic functors, J. of Inequal. & Appl. 7 (2002) 187-269. [19] A. Gogatishvili, B. Opic and W. Trebels, Limiting reiteration for real interpolation with slowly varying functions, Math. Nachr. 278 (2005) 86-107. [20] M.E. Gomez and M. Milman, Extrapolation spaces and almost-everywhere convergence of singular integrals, J. London Math. Soc. 34 (1986) 305-316. [21] M. Milman, “Extrapolation and Optimal Decompositions”, Springer, Lect. Notes in Math. 1580, Berlin, 1994. [22] M. Schechter, “Principles of Functional Analysis”, Amer. Math. Soc., Providence 2002. [23] R. Szwedek, Measure of non-compactness of operators interpolated by real method, Studia Math. 175 (2006) 157-174. [24] H. Triebel, “Interpolation Theory, Function Spaces, Diﬀerential Operators”, North-Holland, Amsterdam 1978.

Fernando Cobos Departamento de An´ alisis Matem´ atico Facultad de Matem´ aticas Universidad Complutense de Madrid Plaza de Ciencias 3 28040 Madrid Spain e-mail: [email protected] Luz M. Fern´ andez-Cabrera Secci´ on Departamental de Matem´ atica Aplicada Escuela de Estad´ıstica Universidad Complutense de Madrid 28040 Madrid Spain e-mail: luz [email protected]

54

F. Cobos, L.M. Fern´andez-Cabrera and A. Mart´ınez

Ant´ on Mart´ınez Departamento de Matem´ atica Aplicada E.T.S. Ingenieros Industriales Universidad de Vigo 36200 Vigo Spain e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 55–67 c 2012 by the authors

A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality Rupert L. Frank and Elliott H. Lieb Dedicated to D. E. Edmunds and W. D. Evans

Abstract. We show that the sharp constant in the Hardy–Littlewood– Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the ﬁrst one to do so for the whole parameter range. Mathematics Subject Classiﬁcation (2010). Primary 39B62; Secondary 26A33, 26D10, 46E35. Keywords. Sharp constants, Sobolev inequality, Hardy–Littlewood–Sobolev inequality.

1. Introduction In a recent paper [11] we showed how to compute the sharp constants for the analogue of the Hardy–Littlewood–Sobolev (HLS) inequality on the Heisenberg group. Unlike the situation for the usual HLS inequality on RN , there is no known useful symmetric decreasing rearrangement technique for the Heisenberg group analogue. A radically new approach had to be developed and that approach can, of course, be used for the original HLS problem as well, thereby providing a genuinely rearrangement-free proof of HLS on RN . That will be given here. The HLS inequality (more precisely, the diagonal case) on RN is 1−λ/N f (x) g(y) Γ(N ) λ/2 Γ((N − λ)/2) dx dy ≤ π

f p g p λ Γ(N − λ/2) Γ(N/2) RN ×RN |x − y| (1.1) where 0 < λ < N and p = 2N/(2N − λ). The constant in (1.1) is sharp and inequality (1.1) is strict unless f and g are proportional to a common This paper may be reproduced, in its entirety, for non-commercial purposes. Support by U.S. NSF grant PHY 0965859 (E.H.L.) is acknowledged.

56

R.L. Frank and E.H. Lieb

translate or dilate of −(2N −λ)/2 H(x) = 1 + |x|2 .

(1.2)

An equivalent formulation of (1.1), which has been noted before in the special cases s = 1 and s = 1/2 [19, Thms. 8.3 and 8.4], is the sharp fractional Sobolev inequality (−Δ)s/2 u

2

≥

22s π Γ((N + 2s)/2) Γ(s) Γ((N − 2s)/2)

Γ(N/2) Γ(N )

2s/N

u 2q

(1.3)

for 0 < s < N/2 and q = 2N/(N − 2s). This follows from (1.1) by a duality argument (see [19, Thm. 8.3]), using the fact that the Green’s function of (−Δ)s is 2−2s π −N/2 Γ((N − 2s)/2)/(Γ(s)|x|N −2s ) for 0 < s < N/2 [19, Thm. 5.9]. In particular, for s = 1, (1.3) is the familiar Sobolev inequality 2/N N (N − 2) 2π (N +1)/2 2 |∇u| dx ≥

u 2q (1.4) 4 Γ((N + 1)/2) RN for N ≥ 3 and q = 2N/(N − 2) in the sharp form of [22, 2, 24]. To recall, brieﬂy, the previous proofs of (1.1) we ﬁrst mention the papers [13, 14, 23], where the inequality was initially derived, but with a nonsharp constant. The sharp version was found in [18] by noting the conformal invariance of the problem and relating it, via stereographic projection, to a conformally equivalent, but more tractable problem on the sphere SN . Riesz’s rearrangement inequality (see [19, Thm. 3.7]) was used in the proof of the existence of a maximizer, and its strong version ([17], see also [19, Thm. 3.9]) was used to prove that the constant function is a maximizer – in the spherical version. There are other, by now standard, ways to prove the existence of a maximizer; that is not the issue. The main point is to prove that (1.2) is a maximizer and that it is, essentially, unique. Then Carlen and Loss [6] cleverly utilized the translational symmetry in RN in competition with the rotational symmetry on the sphere, together with the strong Riesz inequality, to conclude the same thing. Another proof, but only for N − 2 ≤ λ < N , was recently given in [10]. This was done by proving a form of reﬂection positivity for inversions in spheres in RN , and generalizing a theorem of Li and Zhu [16]. This is the ﬁrst rearrangement-free proof of HLS, but it is not valid for 0 < λ < N − 2. An elegant, rearrangement-free proof, this time only for λ = N − 2, is in [5]. In this note we show how the new method developed in [11] can be adapted to the HLS problem to yield a proof for all 0 < λ < N . We also apply the method directly to a proof of (1.4) in Section 3.

2. Main result We shall prove

Sharp HLS Inequality

57

Theorem 2.1. Let 0 < λ < N and p := 2N/(2N − λ). Then (1.1) holds for any f, g ∈ Lp (RN ). Equality holds if and only if f (x) = c H(δ(x − a)) ,

g(y) = c H(δ(x − a))

for some c, c ∈ C, δ > 0 and a ∈ RN (unless f ≡ 0 or g ≡ 0). Here H is the function in (1.2). In other words, we prove that the function H in (1.2) is the unique optimizer in inequality (1.1) up to translations, dilations and multiplication by a constant. The stereographic projection (see Appendix A) deﬁnes a bijection between RN and the punctured sphere SN \ {(0, . . . , 0, −1)}. We consider the sphere SN as a subset of RN +1 with coordinates (ω1 , . . . , ωN +1 ) satisfying N +1 2 j=1 ωj = 1, and (non-normalized) measure denoted by dω. Via stereographic projection Theorem 2.1 is equivalent to Theorem 2.2. Let 0 < λ < N and p := 2N/(2N − λ). Then for any f, g ∈ Lp (SN ) 1−λ/N f (ω) g(η) Γ(N ) λ/2 Γ((N − λ)/2) dω dη ≤ π

f p g p SN ×SN |ω − η|λ Γ(N − λ/2) Γ(N/2) (2.1) with equality if and only if f (ω) =

c , (1 − ξ · ω)(2N −λ)/2

g(ω) =

c , (1 − ξ · ω)(2N−λ)/2

(2.2)

for some c, c ∈ C and some ξ ∈ RN +1 with |ξ| < 1 (unless f ≡ 0 or g ≡ 0). In particular, with ξ = 0, f = g ≡ 1 are optimizers. We conclude this section by recalling that (2.1) can be diﬀerentiated at the endpoints λ = 0 and λ = N , where the inequality turns into an equality. In this way one obtains the logarithmic HLS inequality [7, 4] and a conformally invariant logarithmic Sobolev inequality [3].

3. The sharp Sobolev inequality on the sphere In this section we derive the classical Sobolev inequality (1.4). This case is simpler than the general λ case of the HLS inequality, but it already contains the main elements of our strategy. It is easiest for us to work in the formulation on the sphere SN . N +1 2 We consider SN as a subset of RN+1 , i.e., {(ω1 , . . . , ωN+1 ) : j=1 ωj = 1}. We recall that the conformal Laplacian on SN is deﬁned by L := −Δ +

N (N − 2) , 4

58

R.L. Frank and E.H. Lieb

where Δ is the Laplace-Beltrami operator on SN , and we denote the associated quadratic form by N (N − 2) 2 E[u] := |∇u|2 + |u| dω . 4 SN The sharp Sobolev inequality on SN is Theorem 3.1. For all u ∈ H 1 (SN ) one has 2/N (N −2)/N N (N − 2) 2π (N+1)/2 E[u] ≥ |u|2N/(N −2) dω , 4 Γ((N + 1)/2) SN (3.1) with equality if and only if u(ω) = c (1 − ξ · ω)−(N −2)/2 for some c ∈ C and some ξ ∈ R

N +1

(3.2)

with |ξ| < 1.

See Appendix A for the equivalence of the RN -version (1.4) and the S -version (3.1) of the Sobolev inequality. In the proof of Theorem 3.1 we shall make use of the following elementary formula. N

Lemma 3.2. For all u ∈ H 1 (SN ) one has N +1

E[ωj u] = E[u] + N

j=1

SN

|u|2 dω .

(3.3)

Proof. We begin by noting that for any smooth, real-valued function ϕ on SN one has |∇(ϕu)|2 = ϕ2 |∇u|2 + |u|2 |∇ϕ|2 + ϕ∇ϕ · ∇(|u|2 ) . Hence an integration by parts leads to 2 |∇(ϕu)|2 dω = ϕ |∇u|2 − ϕ(Δϕ)|u|2 dω . SN

SN

We apply this identity to ϕ(ω) = ωj . Using the fact that −Δωj = N ωj , we ﬁnd

SN

|∇(ωj u)|2 dω =

SN

ωj2 |∇u|2 + N |u|2 dω .

Summing over j yields (3.3) and completes the proof.

We are now ready to give a short Proof of Theorem 3.1. It is well-known that there is an optimizer U for inequality (3.1). (Using the stereographic projection, one can deduce this for instance from the existence of an optimizer on RN ; see [18].) As a preliminary remark we note that any optimizer is a complex multiple of a non-negative function. Indeed, if u = a + ib with a and b

Sharp HLS Inequality

59

real functions, then E[u] = E[a] + E[b]. We also note that the right side of (3.1) is a2 + b2 q/2 with q = 2N/(N − 2) > 2. By the triangle inequality, a2 + b2 q/2 ≤ a2 q/2 + b2 q/2 . This inequality is strict unless a ≡ 0 or b2 = λ2 a2 for some λ ≥ 0. Therefore, if U = A + iB is an optimizer for (3.1), then either one of A and B is identically equal to zero or else both A and B are optimizers and |B| = λ|A| for some λ > 0. For any real u ∈ H 1 (SN ) its positive and negative parts u± belong to H 1 (SN ) and satisfy ∂u± /∂ωk = ±χ{±u>0} ∂u/∂ωk in the sense of distributions. (This can be proved similarly to [19, Thm. 6.17].) Thus E[u] = E[u+ ] + E[u− ] for real u. Moreover, u 2q ≤ u+ 2q + u− 2q for real u with strict inequality unless u has a deﬁnite sign. Therefore, if U = A + iB is an optimizer for (3.1), then both A and B have a deﬁnite sign. We conclude that any optimizer is a complex multiple of a non-negative function. Hence we may assume that U ≥ 0. It is important for us to know that we may conﬁne our search for optimizers to functions u satisfying the ‘center of mass condition’ ωj |u(ω)|q dω = 0 , j = 1, . . . , N + 1 . (3.4) SN

It is well known, and used in many papers on this subject (e.g., [15, 21, 8]), that (3.4) can be assumed, and we give a proof of this fact in Appendix B. It uses three facts: one is that inequality (3.1) is invariant under O(N + 1) rotations of SN . The second is that the stereographic projection, that maps RN to SN , leaves the optimization problem invariant. The third is that the RN -version, (1.4), of inequality (3.1) is invariant under dilations F (x) → δ (N −2)/2 F (δx). Our claim in the appendix is that by a suitable choice of δ and a rotation we can achieve (3.4). Therefore we may assume that the optimizer U satisﬁes (3.4). Imposing this constraint does not change the positivity of U . We shall prove that the only optimizer with this property is the constant function (which leads to the stated expression for the sharp constant). It follows, then, that the only optimizers without condition (3.4) are those functions for which the dilation and rotation, just mentioned, yields a constant. In Appendix B we identify those functions as the functions stated in (3.2). The second variation of the quotient E[u]/ u 2q around u = U shows that q E[v] U dω − (q − 1)E[U ] U q−2 |v|2 dω ≥ 0 (3.5)

SN

SN

for all v with U q−1 v dω = 0. Because U satisﬁes condition (3.4) we may choose v(ω) = ωj U (ω) in (3.5) and sum over j. We ﬁnd N +1

j=1

E[ωj U ] ≥ (q − 1) E[U ] .

(3.6)

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R.L. Frank and E.H. Lieb

On the other hand, Lemma 3.2 with u = U implies N +1

E[ωj U ] = E[U ] + N U 2 dω , SN

j=1

which, together with (3.6), yields N U 2 dω ≥ (q − 2) E[U ] . SN

Recalling that q − 2 =

4 , N −2

we see that this is the same as |∇U |2 dω ≤ 0 . SN

We conclude that U is the constant function, as we intended to prove.

4. The sharp HLS inequality on the sphere Our goal in this section is to compute the sharp constant in inequality (2.1) on the sphere SN . We outline our argument in Subsection 4.1 and reduce everything to the proof of a linear inequality. After some preparations in Subsection 4.2 we shall prove this inequality in Subsection 4.3. 4.1. Strategy of the proof Step 1. The optimization problem corresponding to (2.1) admits an optimizing pair with f = g. The fact that one only needs to consider f = g follows from the positive deﬁniteness of the kernel |x − y|−λ . The existence of an optimizer has been proved in [18] for the inequality (1.1) on RN and follows, as explained in Appendix A, via stereographic projection for the inequality on the sphere; for a rearrangement-free proof, see [20] and also the arguments in [11], which easily carry over to the RN case. We claim that any optimizer for problem (2.1) with f = g is a complex multiple of a non-negative function. Indeed, if we denote the left side of (1.1) with g = f by I[f ] and if f = a + ib for real functions a and b, then I[f ] =

I[a] + I[b].

Moreover, for any numbers α, β, γ, δ ∈ R one has 2 2 αγ + βδ ≤ α + β γ 2 + δ 2 with strict inequality unless αγ + βδ ≥ 0 −λ and αδ = βγ. Since is strictly positive, we infer that √ the kernel |x − y| 2 2 I[a] + I[b] ≤ I[ a + b ] for any real functions a, b with strict inequality unless a(x)a(y) + b(x)b(y) ≥ 0 and a(x)b(y) = a(y)b(x) for almost every x, y ∈ RN . From this one easily concludes that any optimizer is a complex multiple of a non-negative function. We denote a non-negative optimizer for problem (2.1) by h := f = g. Since h satisﬁes the Euler-Lagrange equation h(η) dη = c hp−1 (ω) , λ |ω − η| N S we see that h is strictly positive.

Sharp HLS Inequality

61

Step 2. As in the proof of Theorem 3.1, we may assume that the center of mass of hp vanishes, that is, ωj h(ω)p dω = 0 for j = 1, . . . , N + 1 . (4.1) SN

We shall prove that the only non-negative optimizer satisfying (4.1) is the constant function. Then, for exactly the same reason as in the proof of Theorem 3.1, the only optimizers without condition (4.1) are the ones stated in (2.2). We also note that, once we know that a constant is the optimizer, the expression for the sharp constant follows by a computation (see the l = 0 case of Corollary 4.3 below). Step 3. The second variation around the optimizer h shows that f (ω) f (η) h(ω) h(η) p dω dη h dω − (p − 1) dω dη hp−2 |f |2 dω ≤ 0 |ω − η|λ |ω − η|λ (4.2) for any f satisfying hp−1 f dω = 0. Note that the term hp−2 causes no problems (despite the fact that p < 2) since h is strictly positive. Because of (4.1) the functions f (ω) = ωj h(ω) satisfy the constraint p−1 h f dω = 0. Inserting them in (4.2) and summing over j we ﬁnd h(ω) ω · η h(η) h(ω) h(η) dω dη − (p − 1) dω dη ≤ 0 . (4.3) λ |ω − η| |ω − η|λ Step 4. This is the crucial step! The proof of Theorem 2.2 is completed by showing that for any (not necessarily maximizing) h the inequality opposite to (4.3) holds and is indeed strict unless the function is constant. This is the statement of the following theorem with α = λ/2, noting that p − 1 = α/(N − α). Proposition 4.1. Let 0 < α < N/2. For any f on SN one has f (ω) ω · η f (η) α f (ω) f (η) dω dη ≥ dω dη |ω − η|2α N −α |ω − η|2α

(4.4)

with equality iﬀ f is constant. This proposition will be proved in Subsection 4.3. 4.2. The Funk–Hecke theorem We decompose L2 (SN ) into its O(N + 1)-irreducible components, ! L2 (SN ) = Hl .

(4.5)

l≥0

The space Hl is the space of restrictions to SN of harmonic polynomials on RN +1 which are homogeneous of degree l. It is well known that integral operators on SN whose kernels have the form K(ω · η) are diagonal with respect to this decomposition and their eigenvalues can be computed explicitly. A proof of the following Funk–Hecke

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formula can be found, e.g., in [9, Sec. 11.4]. It involves the Gegenbauer poly(λ) nomials Cl , see [1, Chapter 22]. Proposition 4.2. Let K ∈ L1 ((−1, 1), (1 − t2 )(N −2)/2 dt). Then the operator on SN with kernel K(ω · η) is diagonal with respect to decomposition (4.5), and on the space Hl its unique eigenvalue is given by 1 (N −1)/2 κN,l K(t)Cl (t)(1 − t2 )(N −2)/2 dt , (4.6) −1

where κN,l

⎧ ⎪ ⎨2 = l ⎪ ⎩(4π)(N −1)/2

l! Γ((N −1)/2) (l+N −2)!

if N = 1 , l = 0 , if N = 1 , l ≥ 1 , if N ≥ 2 .

This proposition allows us to compute the eigenvalues of the family of operators appearing in Proposition 4.1. Corollary 4.3. Let −1 < α < N/2. The eigenvalue of the operator with kernel (1 − ω · η)−α on the subspace Hl is El = κN 2−α (−1)l where κN

Γ(1 − α) Γ(N/2 − α) , Γ(−l + 1 − α) Γ(l + N − α)

2π 1/2 = 22(N −1) π (N −1)/2

Γ((N−1)/2) Γ(N/2) (N −2)!

(4.7)

if N = 1 , if N ≥ 2 .

When α is a non-negative integer, formula (4.7) is to be understood by taking limits with ﬁxed l. This result appears already (without proof) in [4]. Proof. By Proposition 4.2 we have to evaluate the integral (4.6) for the choice K(t) = (1 − t)−α . Our assertion follows from the β = (N − 2)/2 − α case of the formula 1 (N−1)/2 (1 + t)(N −2)/2 (1 − t)β Cl (t) dt (4.8) −1

= (−1)l

2N/2+β Γ(1 + β) Γ(N/2) Γ(l + N − 1) Γ(−N/2 + 2 + β) . l! Γ(N − 1) Γ(−l − N/2 + 2 + β) Γ(l + N/2 + 1 + β)

This formula, which is valid for β > −1, follows from [12, (7.311.3)] together (λ) (λ) with the fact that Cl (−t) = (−1)l Cl (t). As it stands, (4.8) is only valid for N ≥ 2. For N = 1 and l = 0, the (divergent) factors Γ(l + N − 1) and Γ(N − 1) need to be omitted, and for N = 1 and l ≥ 1, the divergent factor Γ(N − 1) in the denominator needs to be replaced by 12 .

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4.3. Proof of Proposition 4.1 Using the fact that |ω − η|2 = 2(1 − ω · η), we see that the assertion is equivalent to f (ω) f (η) N − 2α f (ω) f (η) dω dη ≤ dω dη . (1 − ω · η)α−1 N −α (1 − ω · η)α Both quadratic forms are diagonal with respect to decomposition (4.5) and their eigenvalues on the subspace Hl are given by Corollary 4.3. For simplicity, we ﬁrst assume that α = 1. The eigenvalue of the right side is (N −2α)El /(N − ˜l , which is α), with El given by (4.7), and the eigenvalue of the left side is E El with α replaced by α − 1. Noting that ˜l = El E

(α − 1)(N − 2α) (l − 1 + α)(l + N − α)

and that El > 0 and α < N/2, we see that the conclusion of the theorem is equivalent to the inequality α−1 1 ≤ (l − 1 + α)(l + N − α) N −α for all l ≥ 0. This inequality is elementary to prove, distinguishing the cases α > 1 and α < 1. Finally, the case α = 1 is proved by letting α → 1 for ﬁxed l. Strictness of inequality (4.4) for non-constant f follows from the fact that the above inequalities are strict unless l = 0. This completes the proof of Proposition 4.1.

Appendix A. Equivalence of Theorems 2.1 and 2.2 In this appendix we consider the stereographic projection S : RN → SN and its inverse S −1 : SN → RN given by 2x 1 − |x|2 ω1 ωN −1 S(x) = , , S (ω) = , . . . , . 1 + |x|2 1 + |x|2 1 + ωN +1 1 + ωN +1 The Jacobian of this transformation (see, e.g., [19, Thm. 4.4]) is N 2 JS (x) = , 1 + |x|2 which implies that

ϕ(ω) dω = SN

RN

ϕ(S(x))JS (x) dx

(A.1)

for any integrable function ϕ on SN . We now explain the equivalence of (1.1) and (2.1) for each ﬁxed pair of parameters λ and p with p = 2N/(2N − λ). There is a one-to-one correspondence between functions f on SN and functions F on RN given by F (x) = |JS (x)|1/p f (S(x)) .

(A.2)

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R.L. Frank and E.H. Lieb

It follows immediately from (A.1) that f ∈ Lp (SN ) if and only if F ∈ Lp (RN ), and in this case f p = F p . Moreover, we note the fact that 2 2 2 2 |ω − η| = |x − y| 1 + |x|2 1 + |y|2 for ω = S(x) and η = S(y), where |ω − η| is the chordal distance between ω and η, i.e., the Euclidean distance in RN +1 . With the help of this relation one easily veriﬁes that F (x) F (y) f (ω) f (η) dx dy = dω dη . λ |x − y| |ω − η|λ N N N N R ×R S ×S This shows that the sharp constants in (1.1) and (2.1) coincide and that there is a one-to-one correspondence between optimizers. In particular, the function f ≡ 1 on SN corresponds to the function |JS (x)|1/p = 2N/p H(x) on RN with H given in (1.2). Similarly, when p = 2N/(N − 2), and F and f are related via (A.2), then N (N − 2) 2 2 2 |∇F | dx = |∇f | + |f | dω , (A.3) 4 RN SN as can be checked by a direct computation.

Appendix B. The center of mass condition Here, we prove that by a suitable inequality preserving transformation of SN we may assume the center of mass conditions given in (3.4) and (4.1). We shall deﬁne a family of maps γδ,ξ : SN → SN depending on two parameters δ > 0 and ξ ∈ SN . To do so, we denote dilation on RN by Dδ , that is, Dδ (x) = δx. Moreover, for any ξ ∈ SN we choose an orthogonal (N + 1) × (N + 1) matrix O such that Oξ = (0, . . . , 0, 1) and we put γδ,ξ (ω) := OT S Dδ S −1 (Oω) for all ω ∈ SN \ {−ξ} and γδ,ξ (−ξ) := −ξ. This transformation depends only on ξ (and δ) and not on the particular choice of O. Indeed, a straightforward computation shows that 2δ γδ,ξ (ω) = (ω − (ω · ξ) ξ) (1 + ω · ξ) + δ 2 (1 − ω · ξ) (1 + ω · ξ) − δ 2 (1 − ω · ξ) + ξ. (1 + ω · ξ) + δ 2 (1 − ω · ξ) Lemma B.1. Let f ∈ L1 (SN ) with SN f (ω) dω = 0. Then there is a transformation γδ,ξ of SN such that γδ,ξ (ω)f (ω) dω = 0 . SN

Sharp HLS Inequality

65

Proof. We may assume that f ∈ L1 (SN ) is normalized by SN f (ω) dω = 1. We shall show that the RN+1 -valued function F (rξ) := γ1−r,ξ (ω)f (ω) dω , 0 < r < 1 , ξ ∈ SN , SN

has a zero. First, note that because of γ1,ξ (ω) = ω for all ξ and all ω, the limit of F (rξ) as r → 0 is independent of ξ. In other words, F is a continuous function on the open unit ball of RN +1 . In order to understand its boundary behavior, one easily checks that for any ω = −ξ one has limδ→0 γδ,ξ (ω) = ξ, and that this convergence is uniform on {(ω, ξ) ∈ SN × SN : 1 + ω · ξ ≥ ε} for any ε > 0. This implies that lim F (rξ) = ξ

r→1

uniformly in ξ .

Hence F is a continuous function on the closed unit ball, which is the identity on the boundary. The assertion is now a consequence of Brouwer’s ﬁxed point theorem. In the proof of Theorem 3.1 we use Lemma B.1 with f = |u|q . Then the new function u ˜(ω) = |Jγ −1 (ω)|1/q u(γ −1 (ω)), with γ = γδ,ξ of Lemma B.1, satisﬁes the center of mass condition (3.4). Moreover, since rotations of the sphere, stereographic projection S and the dilations Dδ leave the inequality invariant, u can be replaced by u ˜ in (3.1) without changing the values of each side. In particular, if U is an optimizer, our proof in Section 3 shows that the ˜ is a constant, which means that the original U is a constant corresponding U 1/q times |Jγ | . It is now a matter of computation, using the explicit form of γδ,ξ , to verify that all such functions have the form of (3.2). Conversely, let us verify that all the functions given in (3.2) are optimizers. By the rotation invariance of inequality (3.1), we can restrict our attention to the case ξ = (0, . . . , 0, r) with 0 < r < 1. These functions correspond via stereographic projection, (A.2), to dilations of a constant times the function H in (1.2). Because of the dilation invariance of inequality (1.4) and because of the fact that we already know that H, which corresponds to the constant on the sphere, is an optimizer, we conclude that any function of the form (3.2) is an optimizer. We have discussed the derivative (Sobolev) version of the λ = N − 2 case of (2.1). Exactly the same considerations show the invariance of the fractional integral for all 0 < λ < N . Acknowledgement We thank Richard Bamler for valuable help with Appendix B.

References [1] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 edition. Dover Publications, New York, 1992.

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[2] Th. Aubin, Probl` emes isoperim´etriques et espaces de Sobolev. J. Diﬀer. Geometry 11 (1976), 573–598. [3] W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn . Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 11, 4816–4819. [4] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. of Math. (2) 138 (1993), no. 1, 213–242. [5] E. A. Carlen, J. A. Carrillo, M. Loss, Hardy–Littlewood–Sobolev inequalities via fast diﬀusion ﬂows, Proc. Nat. Acad. USA 107 (2010), no. 46, 19696–19701. [6] E. A. Carlen, M. Loss, Extremals of functionals with competing symmetries. J. Funct. Anal. 88 (1990), no. 2, 437–456. [7] E. A. Carlen, M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on Sn . Geom. Funct. Anal. 2 (1992), no. 1, 90–104. [8] S-Y. A. Chang, P. C. Yang, Prescribing Gaussian curvature on S 2 . Acta Math. 159 (1987), no. 3–4, 215–259. [9] A. Erd´elyi, M. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions. Vol. II. Reprint of the 1953 original. Robert E. Krieger Publishing Co., Melbourne, FL, 1981. [10] R. L. Frank, E. H. Lieb, Inversion positivity and the sharp Hardy–LittlewoodSobolev inequality. Calc. Var. PDE 39 (2010), no. 1–2, 85–99. [11] R. L. Frank, E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group. Preprint (2010), arXiv:1009.1410. [12] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products. Seventh edition. Elsevier/Academic Press, Amsterdam, 2007. [13] G. H. Hardy, J. E. Littlewood, Some properties of fractional integrals (1). Math. Z. 27 (1928), 565–606. [14] G. H. Hardy, J. E. Littlewood, On certain inequalities connected with the calculus of variations. J. London Math. Soc. 5 (1930), 34–39. [15] J. Hersch, Quatre propri´et´es isop´ erim´etriques de membranes sph´ eriques homog`enes. C. R. Acad. Sci. Paris Sr. A-B 270 (1970), A1645–A1648. [16] Y. Y. Li, M. Zhu, Uniqueness theorems through the method of moving spheres. Duke Math. J. 80 (1995), no. 2, 383–417. [17] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies in Appl. Math. 57 (1977), no. 2, 93–105. [18] E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. of Math. (2) 118 (1983), no. 2, 349–374. [19] E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics 14, Amer. Math. Soc., Providence, RI, 2001. [20] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. [21] E. Onofri, On the positivity of the eﬀective action in a theory of random surfaces. Comm. Math. Phys. 86 (1982), no. 3, 321–326. [22] G. Rosen, Minimum value for c in the Sobolev inequality ϕ3 ≤ c∇ϕ3 . SIAM J. Appl. Math. 21 (1971), 30–32. [23] S. L. Sobolev, On a theorem of functional analysis. Mat. Sb. (N.S.) 4 (1938), 471–479; English transl. in Amer. Math. Soc. Transl. Ser. 2 34 (1963), 39–68.

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[24] G. Talenti, Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372. Rupert L. Frank Department of Mathematics Princeton University Princeton NJ 08544 USA e-mail: [email protected] Elliott H. Lieb Department of Mathematics Princeton University Princeton NJ 08544 USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 69–89 c 2012 Springer Basel AG

Dichotomy in Muckenhoupt Weighted Function Space: A Fractal Example Dorothee D. Haroske Dedicated to Professor David E. Edmunds and Professor W. Desmond Evans on the occasion of their 80th and 70th birthday, respectively.

Abstract. We study dichotomy questions for weighted function spaces of Besov and Triebel–Lizorkin type where the weight is related to the distance of some point to a d-set Γ, 0 < d < n, and the trace is taken on Γ. We can prove that – depending on the function space, the weight and the set Γ – there occurs an alternative: either the trace on Γ exists, or smooth functions compactly supported outside Γ are dense in the space. Such a phenomenon is called dichotomy. The paper is based on trace results in [33] and the atomic decomposition in [18] and perfectly ﬁts together with corresponding unweighted results in [51]. Mathematics Subject Classiﬁcation (2010). 46E35. Keywords. Muckenhoupt weights, function spaces, traces, density, d-sets, atoms.

1. Introduction The main goal of this paper is to demonstrate the strength of the recently developed atomic approach in weighted function spaces in view of trace results. In particular, we concentrate on spaces of Besov and Triebel–Lizorkin type, thus including Sobolev spaces, where the weight function belongs to some Muckenhoupt class. Such questions are of particular interest in view of boundary value problems of elliptic operators, where some singular behaviour near the boundary (characterised by the appropriate Muckenhoupt class) is admitted. Usually one starts with assertions about traces on hyperplanes and tries to transfer these results to bounded domains with suﬃciently smooth boundary afterwards. Further studies may concern compactness or regularity results, leading to the investigation of spectral properties. However, the problem is not so simple and little is known so far. First partial results can be found in [45] and [30] for domains Ω with smooth boundaries ∂Ω and

70

D.D. Haroske γ

Muckenhoupt weights of type w(x) = (dist(x, ∂Ω)) , γ > −1. This was further extended to fractal d-sets Γ in [34], using the atomic approach [18] and based on ideas for the unweighted case in [48]. Recently we noticed renewed interest in trace questions in weighted spaces which led to the papers [1, 19] dealing with some modiﬁcation of the most prominent Muckenhoupt weight function, w(x) = |x| . In this paper we stick to the situation where the trace is taken on a compact d-set Γ ⊂ Rn , 0 < d < n. We rely on results obtained in [33, 34] and combine it with the recently studied dichotomy question explained in more detail below. Roughly speaking, one links two phenomena: the existence of a trace on Γ (by completion of pointwise traces) on the one hand, and the density of smooth functions compactly supported outside Γ, denoted by D(Rn \ Γ), on the other hand. Though it is rather obvious that the density of D(Rn \ Γ) in some space prevents the existence of a properly deﬁned trace, it is not clear (and also not true in general) that there is some close connection vice versa. However, in our setting there appears an alternative that either we have an aﬃrmative answer to the density question or traces exist. The criterion which case occurs naturally depends on the function spaces, the weight and the set Γ. More precisely, for a d-set Γ, 0 < d < n, we deal with the weight dist (x, Γ)κ , if dist (x, Γ) ≤ 1, wκ,Γ (x) = 1, if dist (x, Γ) ≥ 1, when κ > −(n − d). Then wκ,Γ is a Muckenhoupt weight and ﬁts in our s scheme for weighted spaces of Besov type Bp,q (Rn , w) or Triebel–Lizorkin s n type Fp,q (R , w). As mentioned above, corresponding trace results were obtained in [33], complemented by some ﬁrst (suﬃcient) conditions when D(Rn \ s s Γ) is dense in Bp,q (Rn , wκ,Γ ) or Fp,q (Rn , wκ,Γ ). We can strengthen this now as follows: in case of Besov spaces we obtain that s > n−d+κ , 0 < q < ∞, s n p trΓ Bp,q (R , wκ,Γ ) exists for s = n−d+κ , 0 < q ≤ min(p, 1), p and D(R \ Γ) is dense in n

s Bp,q (Rn , w)

s= for s<

n−d+κ , p n−d+κ , p

min(p, 1) < q < ∞, 0 < q < ∞.

There are parallel F -results in Theorem 3.9 below. Note that we always deal s s with p < ∞, q < ∞ due to the density of D(Rn ) in Bp,q (Rn , w) or Fp,q (Rn , w), respectively. The paper is organised as follows. First we collect some fundamentals about Muckenhoupt weights, weighted function spaces, their atomic decomposition, and corresponding continuous embeddings. In Section 3 we turn to trace questions and dichotomy questions with our main result Theorem 3.9. In the end we add some further comments and discussion.

Dichotomy in Muckenhoupt Weighted Function Space

71

2. Weighted function spaces We ﬁx some notation. By N we mean the set of natural numbers, by N0 the set N ∪ {0}, and by Zn the set of all lattice points in Rn having integer components. The positive part of a real function f is denoted by f+ (x) = max(f (x), 0), the integer part of a ∈ R by a = max{k ∈ Z : k ≤ a}. If 1 < u ≤ ∞, the number u is given by u1 = 1 − u1 , in case of 0 < u ≤ 1 we put u = ∞. The set of multi-indices α = (α1 , . . . , αn ), αi ∈ N0 , i = 1, . . . , n, is denoted by Nn0 , with |α| = α1 + · · · + αn , as usual. Moreover, we put Dα =

1 ∂xα 1

∂ |α| , n . . . ∂xα n

and ξ α = ξ1α1 · · · ξnαn ,

α ∈ Nn0 ,

ξ ∈ Rn .

For two positive real sequences {αk }k∈N and {βk }k∈N we mean by αk ∼ βk that there exist constants c1 , c2 > 0 such that c1 αk ≤ βk ≤ c2 αk for all k ∈ N; similarly for positive functions. Given two (quasi-) Banach spaces X and Y , we write X → Y if X ⊂ Y and the natural embedding of X in Y is continuous. Let for m ∈ Zn and ν ∈ N0 , Qν,m denote the n-dimensional cube with sides parallel to the axes of coordinates, centered at 2−ν m and with side length 2−ν . For x ∈ Rn and r > 0, let B(x, r) denote the open ball B(x, r) = {y ∈ Rn : |y − x| < r}. For convenience, let both dx and |·| stand for the (n-dimensional) Lebesgue measure in the sequel. All unimportant positive constants will be denoted by c, occasionally with subscripts. As we shall always deal with function spaces on Rn , we may often omit the ‘Rn ’ from their notation for convenience. 2.1. Muckenhoupt weights We brieﬂy recall some fundamentals on Muckenhoupt classes Ap . By a weight n w we shall always mean a locally integrable function w ∈ Lloc 1 (R ), positive a.e. in the sequel. Let M stand for the Hardy-Littlewood maximal operator given by 1 M f (x) = sup |f (y)| dy, x ∈ Rn , (2.1) B(x,r)∈B |B(x, r)| B(x,r) where B is the collection of all open balls B(x, r) centred at x ∈ Rn , r > 0. Deﬁnition 2.1. Let w be a weight on Rn . (i) Then w belongs to the Muckenhoupt class Ap , 1 < p < ∞, if there exists a constant 0 < A < ∞ such that for all balls B the following inequality holds, 1/p 1/p 1 1 −p /p w(x) dx · w(x) dx ≤ A. (2.2) |B| B |B| B (ii) Then w belongs to the Muckenhoupt class A1 if there exists a constant 0 < A < ∞ such that the inequality M w(x) ≤ Aw(x) holds for almost all x ∈ R . n

(2.3)

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D.D. Haroske

(iii) The Muckenhoupt class A∞ is given by A∞ = Ap .

(2.4)

p>1

Since the pioneering work of Muckenhoupt [27–29], these classes of weight functions have been studied in great detail, we refer, in particular, to the monographs [16], [42, Ch. V], [43], and [44, Ch. IX] for a complete account on the theory of Muckenhoupt weights. As usual, we use the abbreviation w(Ω) = w(x) dx, (2.5) Ω

where Ω ⊂ Rn is some bounded, measurable set. Then a weight w on Rn belongs to Ap , 1 ≤ p < ∞, if, and only if, 1/p 1 c f (y) dy ≤ f p (x)w(x) dx |B| B w(B) B holds for all non-negative f and all balls B. In particular, with E ⊂ B and f = χE , this implies that 1/r |E| w(E) ≤c , E ⊂ B, w ∈ Ar , r ≥ 1. (2.6) |B| w(B) Another property of Muckenhoupt weights that will be used in the sequel is that w ∈ Ap , p > 1, implies the existence of some number r < p such that w ∈ Ar . This is closely connected with the so-called ‘reverse H¨older inequality’, see [42, Ch. V, Prop. 3, Cor.]. In our case this fact will re-emerge in the number rw = inf{r ≥ 1 : w ∈ Ar }, w ∈ A∞ , (2.7) that plays an essential rˆ ole later on. Obviously, 1 ≤ rw < ∞, and w ∈ Arw if, and only if, rw = 1. Example 2.2. We restrict ourselves to a ‘fractal’ example studied in [18], and refer for further examples to [12, 20, 21]. Let Γ ⊂ Rn be a d-set, 0 < d < n, in the sense of [48, Def. 3.1], see also [23, 24] (which is diﬀerent from [11]), i.e., there exists a Borel measure μ in Rn such that supp μ = Γ compact, and there are constants c1 , c2 > 0 such that for arbitrary γ ∈ Γ and all 0 < r < 1 holds c1 rd ≤ μ(B(γ, r) ∩ Γ) ≤ c2 rd . Recall that some self-similar fractals are outstanding examples of d-sets; for instance, the usual (middle-third) Cantor set in R1 is a d-set for d = ln 2/ ln 3, and the Koch curve in R2 is a d-set for d = ln 4/ ln 3. We proved in [18] that the weight wκ,Γ , given by dist (x, Γ)κ , if dist (x, Γ) ≤ 1, wκ,Γ (x) = (2.8) 1, if dist (x, Γ) ≥ 1, satisﬁes wκ,Γ ∈ Ap

if, and only if,

− (n − d) < κ < (n − d)(p − 1),

1 < p < ∞,

Dichotomy in Muckenhoupt Weighted Function Space and wκ,Γ ∈ A1 if −(n − d) < κ ≤ 0. Consequently, rwκ,Γ = 1 +

73

max(κ,0) . n−d

Remark 2.3. For a reﬁned study of the singularity behaviour of Muckenhoupt A∞ weights we introduced in [21] the notion of the set of singularities Ssing (w) for w ∈ A∞ by " w(Qν,m ) n Ssing (w) = x0 ∈ R : inf =0 Qν,m x0 |Qν,m | w(Q ) ν,m ∪ x0 ∈ Rn : sup =∞ . (2.9) Qν,m x0 |Qν,m | This is a special case of Ssing (w1 , w2 ) deﬁned in [21] with w2 ≡ 1, w1 ≡ w. In case of the weight wκ,Γ introduced in (2.8) where Γ is a d-set in Rn with 0 < d < n and κ > −(n − d), one can prove that Γ, if κ = 0, Ssing (wκ,Γ ) = ∅, if κ = 0, based on the estimate wκ,Γ (Qν,m ) ∼ |Qν,m |

if 2Qν,m ∩ Γ = ∅,

1, 2

−νκ

(2.10)

, otherwise,

see [18]. Note that we have |Ssing (wκ,Γ )| = 0 which reﬂects the general fact |Ssing (w)| = 0 for all w ∈ A∞ , cf. [21]. s s 2.2. Function spaces of type Bp,q (Rn , w) and Fp,q (Rn , w) with w ∈ A∞ Let w ∈ A∞ be a Muckenhoupt weight and 0 < p < ∞. Then the weighted Lebesgue space Lp (Rn , w) contains all measurable functions such that 1/p

f |Lp (Rn , w) = |f (x)|p w(x) dx (2.11) Rn

is ﬁnite. For p = ∞ one obtains the classical (unweighted) Lebesgue space, L∞ (Rn , w) = L∞ (Rn ),

w ∈ A∞ ;

(2.12)

we thus mainly restrict ourselves to p < ∞ in what follows. The Schwartz space S(Rn ) and its dual S (Rn ) of all complex-valued tempered distributions have their usual meaning here. Let ϕ0 = ϕ ∈ S(Rn ) be such that supp ϕ ⊂ {y ∈ Rn : |y| < 2}

and ϕ(x) = 1 if |x| ≤ 1 ,

−j

and for each j ∈ N let ϕj (x) = ϕ(2 x) − ϕ(2−j+1 x). Then {ϕj }∞ j=0 forms a smooth dyadic resolution of unity. Given any f ∈ S (Rn ), we denote by Ff and F −1 f its Fourier transform and its inverse Fourier transform, respectively. Let f ∈ S (Rn ), then the Paley–Wiener–Schwartz theorem implies that F −1 (ϕj F f ) is an entire analytic function on Rn . Deﬁnition 2.4. Let w ∈ A∞ , 0 < q ≤ ∞, 0 < p < ∞, s ∈ R and {ϕj }j∈N0 a smooth dyadic resolution of unity.

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D.D. Haroske

s (i) The weighted Besov space Bp,q (Rn , w) is the set of all distributions f ∈ n S (R ) such that s f |Bp,q (Rn , w) = 2js F −1 (ϕj F f )|Lp (Rn , w) j∈N |q (2.13) 0

is ﬁnite. s (ii) The weighted Triebel–Lizorkin space Fp,q (Rn , w) n butions f ∈ S (R ) such that js −1 s f |Fp,q (Rn , w) = 2 |F (ϕj F f )(·)| j∈N0 |q

is the set of all distri Lp (Rn , w)

(2.14)

is ﬁnite. s s Remark 2.5. The spaces Bp,q (Rn , w) and Fp,q (Rn , w) are independent of the particular choice of the smooth dyadic resolution of unity {ϕj }j appearing in their deﬁnitions. They are quasi-Banach spaces (Banach spaces for p, q ≥ 1), s and S(Rn ) → Bp,q (Rn , w) → S (Rn ), where the ﬁrst embedding is dense if q < ∞, similarly for the F -case; cf. [4]. Moreover, for w0 ≡ 1 ∈ A∞ these are the usual (unweighted) Besov and Triebel–Lizorkin spaces; we refer, in particular, to the series of monographs [46–50] for a comprehensive treatment of the unweighted spaces. The above spaces with weights of type w ∈ A∞ have been studied systematically in [4,5], with subsequent papers [6,7]. It turned out that many of the results from the unweighted situation have weighted counterparts: e.g., we 0 have Fp,2 (Rn , w) = hp (Rn , w), 0 < p < ∞, where the latter are Hardy spaces, 0 see [4, Thm. 1.4], and, in particular, hp (Rn , w) = Lp (Rn , w) = Fp,2 (Rn , w), 1 < p < ∞, w ∈ Ap , see [43, Ch. VI, Thm. 1]. Concerning (classical) Sobolev spaces Wpk (Rn , w) (built upon Lp (Rn , w) in the usual way) it holds that k Wpk (Rn , w) = Fp,2 (Rn , w),

k ∈ N0 ,

1 < p < ∞,

w ∈ Ap ,

(2.15)

cf. [4, Thm. 2.8]. Further details can be found in [2–5,15,16,36,37]. In [38] the above class of weights was extended in order to incorporate locally regular weights, too, creating in that way the class A oc p . We partly rely on our approaches in [18, 20–22]. We brieﬂy recall the deﬁnition of atoms. Deﬁnition 2.6. Let K ∈ N0 and b > 1. (i) The complex-valued function a ∈ C K (Rn ) is said to be an 1K -atom if supp a ⊂ bQ0,m for some m ∈ Zn , and |Dα a(x)| ≤ 1 for |α| ≤ K, x ∈ Rn . (ii) Let s ∈ R, 0 < p ≤ ∞, and L + 1 ∈ N0 . The complex-valued function a ∈ C K (Rn ) is said to be an (s, p)K,L -atom if for some ν ∈ N0 , supp a ⊂ bQν,m for some m ∈ Zn , |Dα a(x)| ≤ 2−ν(s− p )+|α|ν for |α| ≤ K, x ∈ Rn , n

Rn

xβ a(x) dx = 0 for |β| ≤ L.

Dichotomy in Muckenhoupt Weighted Function Space

75

We shall denote an atom a(x) supported in some Qν,m by aν,m in the sequel. Choosing L = −1 in (ii) means that no moment conditions are required. (p) For 0 < p < ∞, ν ∈ N0 , m ∈ Zn , let χνm denote the p-normalised characteristic function of Qνm , that is νn νn 2 p , for x ∈ Qνm , p χ χ(p) (2.16) νm (x) = νm (x) = 2 0, for x ∈ / Qνm , (p)

such that χνm |Lp (Rn ) = 1. Now we can introduce suitable sequence spaces bpq (w) and fpq (w) for 0 < p < ∞, 0 < q ≤ ∞, w ∈ A∞ , by bpq (w) =

λ = {λν,m }ν,m : λν,m ∈ C,

λ|bpq (w) =

n λνm χ(p) νm |Lp (R , w)

m∈Zn

and

ν∈N0

|q < ∞

(2.17)

fpq (w) =

λ = {λν,m }ν,m : λν,m ∈ C,

∞ q 1/q (p) n λ |fpq (w) = (2.18) λνm χνm (·) Lp (R , w) < ∞ ν=0 m∈Zn

(with the usual modiﬁcation if q = ∞). For convenience we adopt the usual notation 1 1 σp = n −1 , σp,q = n −1 , (2.19) p min(p, q) + + where 0 < p, q ≤ ∞. Then the atomic decomposition result used below reads as follows. Theorem 2.7. Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and w ∈ A∞ be a weight with rw given by (2.7). (i) Let K, L + 1 ∈ N0 with K ≥ (1 + s)+

and

L ≥ max −1, σp/rw − s .

(2.20)

Then f ∈ S(R ) belongs to if, and only if, it can be written as a series ∞

f= λνm aν,m (x), converging in S (Rn ), (2.21) n

s Bp,q (Rn , w)

ν=0 m∈Zn

where aν,m (x) are 1K -atoms (ν = 0) or (s, p)K,L -atoms (ν ∈ N) and λ ∈ bpq (w). Furthermore, inf λ|bpq (w)

(2.22)

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D.D. Haroske

s is an equivalent quasi-norm in Bp,q (Rn , w), where the inﬁmum ranges over all admissible representations (2.21). (ii) Let K, L + 1 ∈ N0 with K ≥ (1 + s)+ and L ≥ max −1, σp/rw ,q − s . (2.23) s Then f ∈ S(Rn ) belongs to Fp,q (Rn , w) if, and only if, it can be written as a series (2.21) where aν,m (x) are 1K -atoms (ν = 0) or (s, p)K,L atoms (ν ∈ N) and λ ∈ fpq (w). Furthermore,

inf λ|fpq (w)

(2.24)

s is an equivalent quasi-norm in Fp,q (Rn , w), where the inﬁmum ranges over all admissible representations (2.21).

Remark 2.8. The above result coincides with [18, Thm. 3.10], cf. also [2, Theorem 5.10]. Notational agreement. We adopt the nowadays usual custom to write Asp,q s s instead of Bp,q or Fp,q , respectively, when both scales of spaces are meant simultaneously in some context. 2.3. Continuous Embeddings We collect some embedding results for weighted spaces that will be used later. Recall that we deal with function spaces on Rn exclusively, and will thus omit the ‘Rn ’ from their notation. Proposition 2.9. Let w1 and w2 be two A∞ weights and let −∞ < s2 ≤ s1 < ∞, 0 < p1 , p2 ≤ ∞, 0 < q1 , q2 ≤ ∞. We put 1 1 1 1 1 1 = − and = − . (2.25) p∗ p2 p1 + q∗ q2 q1 + Then id : Bps11 ,q1 (w1 ) → Bps22 ,q2 (w2 ) is continuous if, and only if, " w2 (Qν,m )1/p2 −ν(s1 −s2 ) 2 |p∗ w1 (Qν,m )1/p1 m∈Zn

(2.26)

∈ q ∗ .

(2.27)

ν∈N0

Remark 2.10. For the proof and further details, also concerning questions of compactness, we refer to [20]. In view of (2.12) it is clear that we obtain unweighted Besov spaces if p1 = p2 = ∞. Then by (2.5), w1 (Qν,m ) = w2 (Qν,m ) = 2−νn for all ν ∈ N0 and m ∈ Zn , such that (2.27) leads to p∗ = ∞, i.e., p1 ≤ p2 , and n n δ∗ = s1 − − s2 + > 0, (2.28) p1 p2 with the extension to δ∗ = 0 if q1 ≤ q2 , i.e., q ∗ = ∞. In [20,21] we concentrated on the interplay between smoothness parameters and properties of the weight. We exemplify it for our special weight wκ,Γ .

Dichotomy in Muckenhoupt Weighted Function Space

77

Corollary 2.11. Let Γ ⊂ Rn be a d-set, 0 < d < n, and wκ,Γ be given by (2.8) with κ > −(n − d). Let the parameters satisfy −∞ < s2 ≤ s1 < ∞,

0 < p1 < ∞,

0 < p2 ≤ ∞,

0 < q1 , q2 ≤ ∞. (2.29)

The embedding idκ,Γ : Bps11 ,q1 (wκ,Γ ) → Bps22 ,q2 is continuous if, and only if, p1 ≤ p2

and

2

−ν(δ∗ −

max(κ,0) ) p1

ν∈N0

(2.30)

∈ q ∗ .

(2.31)

In [20, 21] we also considered situations where both source and target space are weighted with the same w ∈ A∞ . Here we shall only need the following basic observation. Proposition 2.12. Let 0 < q ≤ ∞, 0 < p < ∞, s ∈ R and w ∈ A∞ . (i) Let −∞ < s1 ≤ s0 < ∞ and 0 < q0 ≤ q1 ≤ ∞. Then 0 1 Asp,q (w) → Asp,q (w)

and

Asp,q0 (w) → Asp,q1 (w).

(2.32)

(ii) We have s s s Bp,min(p,q) (w) → Fp,q (w) → Bp,max(p,q) (w).

(2.33)

(iii) Assume that there are numbers c > 0, d > 0 such that for all cubes, w (Qν,m ) ≥ c2−νd ,

ν ∈ N0 ,

m ∈ Zn .

(2.34)

Let 0 < p0 < p < p1 < ∞, −∞ < s1 < s < s0 < ∞ satisfy s0 −

d d d = s − = s1 − . p0 p p1

(2.35)

Then Bps00 ,q (w) → Bps11 ,q (w)

and

s Bps00 ,p (w) → Fp,q (w) → Bps11 ,p (w). (2.36)

Remark 2.13. These embeddings are natural extensions from the unweighted case w ≡ 1, see [46, Prop. 2.3.2/2, Thm. 2.7.1] and [41, Thm. 3.2.1]. The above result essentially coincides with [4, Thm. 2.6] and can be found in [20, Prop. 1.8]. Example 2.14. Assume that inf m∈Zn w(Q0,m ) ≥ c > 0, then (2.6) implies d ≥ nrw in (2.34). In particular, for our model weight wκ,Γ the embeddings (2.36) can be exempliﬁed as follows, recall also (2.10). Let 0 < p0 −(n − d). Assume that s0 −

max(κ, 0) + n max(κ, 0) + n max(κ, 0) + n =s− = s1 − . p0 p p1

Then (2.36) holds for w = wκ,Γ .

(2.37)

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D.D. Haroske

3. Dichotomy 3.1. Traces on hyperplanes and d-sets Let Γ ⊂ Rn , 0 σp , w ∈ A∞ . Suppose there exists some c > 0 such that for all ϕ ∈ D(Rn ), ϕ|Γ |Lp (Γ) ≤ c ϕ|Asp,q (Rn , w) ,

(3.1)

where the trace on Γ is taken pointwise. By the density of D(Rn ) in Asp,q (Rn , w) for p, q < ∞ and the completeness of Lp (Γ) one can thus deﬁne for f ∈ Asp,q (Rn , w) its trace trΓ f = f |Γ on Γ in the standard way. Remark 3.1. Since our main goal in this paper are density questions in contrast to the existence of traces, we shall not further discuss possible extensions of the above approach to q = ∞. But something can be done as far as traces are concerned; we refer to [48, Cor. 18.12], [50, Prop. 1.172]. We recall what is known in the unweighted situation for hyperplanes and d-sets. For m ∈ N, 1 ≤ m ≤ n − 1, we adopt the usual convention to identify an m-dimensional hyperplane in Rn with Rm . Proposition 3.2. Let 0 

n−m p

A = B, A = F,

+ m( p1 − 1)+ ,

s s− n−m trRm Bp,q (Rn ) = Bp,q p (Rm ), s s− n−m s− n−m trRm Fp,q (Rn ) = Fp,p p (Rm ) = Bp,p p (Rm ). (ii) Let Γ be a compact d-set, 0 < d < n. Then n−d 0 < q ≤ min(p, 1), p n trΓ Ap,q (R ) = Lp (Γ) if 0 

A = B, A = F,

n−d , p

s s− n−d trΓ Bp,q (Rn ) = Bp,q p (Γ). Remark 3.3. The above trace results have a long history, we refer to [8, 17, 31, 32] for the Besov cases p ≥ 1, and [13], whereas their F -counterparts can be found in [14,47]. The situation (ii) was essentially solved in [48] with some later additions, see also [50].

Dichotomy in Muckenhoupt Weighted Function Space

79

In case of Muckenhoupt weighted spaces of the above type and traces on hyperplanes one can ﬁnd ﬁrst trace results in [45, Thm. 3.6.4/2], and recently for spaces of type Asp,q (wα,β ) with |x|α , if |x| ≤ 1 , wα,β (x) = with α > −n, β > −n, |x|β , if |x| > 1 , in [1, 19], where the latter is also based on the atomic approach. We study traces of weighted spaces on d-sets and rely on the following related results in [33, 34]. Proposition 3.4. Let 0 −(n − d). Then κ+n−d 0 < q ≤ min(p, 1), A = B, p trΓ Ap,q (Rn , wκ,Γ ) = Lp (Γ) if (3.2) 0 

κ+n−d , p

s s− κ s− κ+n−d trΓ Bp,q (Rn , wκ,Γ ) = trΓ Bp,q p (Rn ) = Bp,q p (Γ), s s− κ s− κ+n−d trΓ Fp,q (Rn , wκ,Γ ) = trΓ Bp,p p (Rn ) = Bp,p p (Γ).

(3.3) (3.4)

3.2. Dichotomy If Γ is a compact d-set in Rn or the hyperplane Rm in Rn , then we abbreviate now DΓ = D(Rn \ Γ). (3.5) Recall that D(Rn ) is dense in all spaces Asp,q (Rn , w) with p < ∞, q < ∞, independent of s ∈ R and w ∈ A∞ . So removing from Rn only ‘small enough’ Γ one can ask whether (still) DΓ is dense in Asp,q (Rn , w).

(3.6)

Conversely, we have the aﬃrmative trace results mentioned above, but one can also ask for what (‘thick enough’) Γ there exists a trace of Asp,q (Rn , w) in Lp (Γ)

(3.7)

(for suﬃciently high smoothness). Though these questions may arise independently, it is at least clear that whenever DΓ is dense in Asp,q (Rn , w), then there cannot exists a trace according to (3.1); see [51] and [52, Chapter 6.4] for the corresponding argument in the unweighted case. However, it is not always clear that one really has an alternative in the sense that either there is a trace or DΓ is dense in Asp,q (Rn , w). More precisely, there is described a situation in [51] in the unweighted context where a gap remains: traces can only exist for spaces Asp,q with smoothness s ≥ s0 , whereas density requires s ≤ s1 and s1 < s0 . However, in the setting described here, we are in the lucky situation that we have an alternative between (3.6) and (3.7); following Triebel in the unweighted setting we call this phenomenon dichotomy.

80

D.D. Haroske We introduce the following notation. Let 0 < p < ∞, w ∈ A∞ . Then Ap (Rn , w) = {Asp,q (Rn , w) : 0 < q < ∞, s ∈ R}.

(3.8)

In the spirit of our notational agreement above we shall also use Bp (Rn , w) and Fp (Rn , w) occasionally. Moreover, when w ≡ 1, we shall write Ap (Rn ) = Ap (Rn , 1) for the unweighted situation. Let trΓ : Asp,q (Rn , w) → Lp (Γ)

(3.9)

be the trace operator deﬁned by completion from the pointwise trace according to (3.1). Deﬁnition 3.5. Let n ∈ N, Γ ⊂ Rn , 0 sΓ , 0 < q < ∞, trΓ exists for s = sΓ , 0 < q ≤ qΓ ,

and DΓ is dense in

Furthermore, D Ap (Rn , w), Lp (Γ) = (sΓ , 0) means that

trΓ DΓ

trΓ DΓ

(3.11)

s = sΓ , qΓ < q < ∞, for s < sΓ , 0 < q < ∞. (3.12)

(3.13)

exists for s > sΓ , 0 < q < ∞, is dense in Asp,q (Rn , w) for s ≤ sΓ , 0 < q < ∞, (3.14)

and D Ap (Rn , w), Lp (Γ) = (sΓ , ∞) means that

Asp,q (Rn , w)

(3.10)

(3.15)

exists for s ≥ sΓ , 0 < q < ∞, is dense in Asp,q (Rn , w) for s < sΓ , 0 < q < ∞. (3.16)

Dichotomy in Muckenhoupt Weighted Function Space

81

Remark 3.6. Recall that D(Rn ) is dense in Asp,q (Rn , w) with p < ∞, q < ∞. In view of the continuous embeddings in Proposition 2.12, in particular (2.32), this deﬁnition makes sense; as discussed in [52, Sect. 6.4.3] already, it might be more reasonable in general, to exclude the limiting case q = qΓ in (3.11) or shift it to (3.12). But as will turn out below, in our context the breaking point q = qΓ is always on the trace side. For convenience, we ﬁrst collect what is known in the unweighted situation, w ≡ 1, for hyperplanes Γ = Rm or d-sets Γ, 0 < d < n. Proposition 3.7. Let 0 1, p ≤ 1.

(ii) Let Γ be a compact d-set, 0 < d < n. Then n−d n D(Bp (R ), Lp (Γ)) = , min(p, 1) p and

⎧ ⎨ n−d , 0 , p D(Fp (Rn ), Lp (Γ)) = ⎩ n−d , ∞ , p

p > 1, p ≤ 1.

Remark 3.8. The result (i) is proved in this explicit form in [51], see also [52, Cor. 6.69], for a diﬀerent approach see also [39,40] and Remark 3.3 for further literature. The second part (ii) can be found in [51] and [52, Thm. 6.68]. Now we state our main result in this paper. Theorem 3.9. Let Γ be a compact d-set, 0 < d < n, 0 −(n − d). Then n−d+κ D(Bp (Rn , wκ,Γ ), Lp (Γ)) = , min(p, 1) (3.17) p and ⎧ ⎨ n−d+κ ,0 , p D(Fp (Rn , wκ,Γ ), Lp (Γ)) = ⎩ n−d+κ , ∞ , p

p > 1, p ≤ 1.

(3.18)

82

D.D. Haroske

Proof. Step 1. In view of Proposition 3.4, in particular the transfer formulas between trace spaces in the weighted and unweighted setting (3.3), (3.4), together with the (unweighted) dichotomy result Proposition 3.7 it is clear that all the corresponding trace assertions in Deﬁnition 3.5, that is, (3.11) and the upper lines in (3.14), (3.16), are already covered for our situation. Furthermore, in view of the embeddings (2.32) and the density of D(Rn ) in all spaces Asp,q (Rn , wκ,Γ ) it is only left to prove that DΓ

n−d+κ

is dense in Bp,q p

(Rn , wκ,Γ ) if 0 min(1, p),

(3.19)

(Rn , wκ,Γ ) if 1 < p < ∞, 0 < q < ∞.

(3.20)

and DΓ

n−d+κ

is dense in Fp,q p

This will immediately imply the density of DΓ in Asp,q (Rn , wκ,Γ ) with s < sΓ = n−d+κ and thus already ﬁnish the F -case with p ≤ 1. p The plan is as follows. We adapt ideas presented in [51] for the unweighted case, see also [52, Thm. 6.68]. Roughly speaking, the clue is to n−d+κ

construct for given f ∈ Bp,q p ﬁes n−d+κ

f − fJ |Bp,q p

(Rn , wκ,Γ ) a sequence {fJ }J∈N which satis-

(Rn , wκ,Γ ) −−−−→ 0 J→∞

and fJ ∈ DΓ .

These approximating functions fJ are based on special atomic decompositions which are appropriately adapted to the diﬀerent q-cases in (3.19): q > 1 (studied in Steps 2 and 3) and q > p (postponed to Step 5). This will cover the B-case. For simplicity we shall describe the easiest case (that is, when no moment conditions in (2.20) are required) in some detail; the extension to all parameters needs some further tricky technical reﬁnement, but is essentially covered by parallel arguments in the unweighted situation. Here one essentially beneﬁts from the so-called porosity of d-sets, cf. [24, p.156] and [50, Sects. 9.16–9.19], to circumvent this diﬃculty. ∞ Step 2. We begin with a preparation and construct a sequence ϕJ J=1 ⊂ D(Rn ) with ϕJ (x) = 1

in an open neighbourhood of Γ

(3.21)

(depending on J) and n−d+κ

ϕJ −−−−→ 0 in Bp,q p J→∞

(Rn , wκ,Γ ),

p ≥ rwκ,Γ ,

q > 1.

(3.22)

We proceed as in [51] and cover for given j ∈ N a neighbourhood of Γ with balls Bj,m centred at Γ and of radius 2−j , where m = 1, . . . , Mj with Mj ∼ Mj 2jd . Accordingly we choose a resolution of unity {ϕj,m }m=1 , with ϕj,m ∈

Dichotomy in Muckenhoupt Weighted Function Space

83

D(Bj,m ), ϕj,m ≥ 0, |Dα ϕj,m (x)| ≤ cα 2j|α| , α ∈ Nn0 , and Mj

ϕj,m (x) = 1 near Γ.

m=1

J0 1 J0 +1 1 Assume J ≥ 2 and determine J0 ∈ N such that j=J < 1 ≤ j=J . j j Thus we obtain that J0 1

, j = J, . . . , J0 − 1, rj = 1 with rj = j J0 1 1 − j=J j , j = J0 . j=J Then J

ϕ (x) =

J0

rj 2

−j d−κ p

Mj

2

j(d−κ) p

x ∈ Rn ,

ϕj,m (x),

(3.23)

m=1

j=J

n−d+κ

is an atomic decomposition in Bp,q p (Rn , wκ,Γ ) according to Theorem 2.7(i) and Deﬁnition 2.6 for K > 1 + n−d+κ and L = −1 (no moment conditions p needed). Thus Theorem 2.7(i) together with (2.17) and (2.10) implies that n−d+κ

ϕJ |Bp,q p

(Rn , wκ,Γ )

q

≤ c1

J0

rjq 2−j

d−κ p

q

Mj

≤ c2 ≤ c3

j=J ∞

q/p

m=1

j=J J0

2jn wκ,Γ (Bj,m )

rjq 2−j

d−κ p

q −j κq p

2

q

Mjp

j −q ∼ J 1−q .

(3.24)

j=J

This gives (3.22). Step 3. We show (3.19) for p > rwκ,Γ ≥ 1 and q > min(p, 1) = 1. Note that by density arguments it is suﬃcient to approximate f ∈ D(Rn ) in n−d+κ

Bp,q p (Rn , wκ,Γ ) by functions fJ ∈ DΓ . Let ϕJ be the functions according to (3.21), (3.22), put f J = ϕJ f such that f can be decomposed into f = f J + fJ

with

f J = ϕJ f

and

fJ = (1 − ϕJ )f ∈ DΓ .

By an appropriately adapted weighted counterpart of the pointwise multiplier theorem in [47, Sect. 4.2.2] one has for suﬃciently large smoothness that there is some c > 0, such that for all f ∈ D(Rn ) and all ϕJ , n−d+κ

f J |Bp,q p

n−d+κ

(Rn , wκ,Γ ) ≤ c f |C (Rn ) · ϕJ |Bp,q p

(Rn , wκ,Γ ) −−−−→ 0 J→∞

according to (3.24). This completes the argument for (3.19) when p > rwκ,Γ , q > 1. Step 4. We prove (3.20) and use the fact that d-sets are porous for d < n, we refer to [24, p.156] and [50, Sects. 9.16-9.19] for further details. Let ﬁrst q ≥ 1. Then by the same reasoning as above we consider the atomic

84

D.D. Haroske n−d+κ

decomposition (3.23) in Fp,q p (Rn , wκ,Γ ) with p > rwκ,Γ , q ≥ 1 (where we may choose L = −1 in Theorem 2.7(ii), i.e., no moment conditions needed) leading to n−d+κ

lim ϕJ |Fp,q p

J→∞

n−d+κ

(Rn , wκ,Γ ) = lim ϕJ |Bp,p p J→∞

(Rn , wκ,Γ ) = 0.

For arguments of this type we refer to [49, pp. 142/143]. In view of (2.23) the situation is more complicated when 0 < q < 1 or 1 < p ≤ rwκ,Γ , since one needs moment conditions L ≥ 0 in that case. So it may happen that (3.23) n−d+κ

is no longer an atomic decomposition in Fp,q p (Rn , wκ,Γ ) and we cannot apply Theorem 2.7(ii) as above. But the porosity of (the d-set) Γ ensures that one can complement ϕj,m outside of Γ in an appropriate way in order to obtain the needed moment conditions. For details and further discussion one may consult [48, p. 143] based on [53]. This completes the proof of (3.20) and thus of (3.18). Moreover, by the same reasoning we can also cover the B-case in (3.19) when 1 min(p, 1) = 1, that is, when moment conditions are needed in (2.20), which we excluded in Step 3 for simplicity. Step 5. It remains to verify (3.19) in case of 0 min(p, 1) = p. (In fact, the proof below will cover all situations when q > p.) For this reason we have to reﬁne our covering and decomposition argument from Step 2. Let L ∈ N and assume, for convenience, that μ(Γ) = 1. We use the same decomposition of Γ by other d-sets Γl , l = L, . . . , L0 , as described in [51], resulting in Γ=

L0

Γl ,

μ(Γl ) ∼ l−1 ,

l=L

L0

μ(Γl ) ∼ μ(Γ) = 1,

l=L

where L0 ∈ N with L0 > L is chosen appropriately. Moreover, according to [52, p. 224] this can be done in such a way that there are non-negative functions ψl ∈ D(Rn ) with L0

ψl (γ) = 1 if γ ∈ Γ,

Γl ⊂ supp ψl ⊂ {y ∈ Rn : dist (y, Γl ) < εl }

l=L

for some εl > 0. For given l ∈ N between L and L0 and appropriately chosen j(l) ∈ N we mimic the construction from Step 2 and arrive at

Mj(l)

ϕj(l),m (x) = 1 near Γ,

0 ≤ ϕj(l),m ∈ D Bj(l),m ,

m=1

with Mj(l) ∼ 2j(l)d . We assume that j(L) < · · · < j(l) < j(l + 1) < · · · < j(L0 ), and in analogy to (3.23), L

ϕ (x) =

L0

l=L

− j(l)(d−κ) p

ψl (x) 2

Mj(l)

m=1

2

j(l)(d−κ) p

ϕj(l),m (x),

x ∈ Rn . (3.25)

Dichotomy in Muckenhoupt Weighted Function Space

85

If p > pκ = max(κ,0) + d−κ , i.e., when s = n−d+κ > σp/rwκ,Γ , we do not n−d n p need moment conditions in (2.20) and can apply Theorem 2.7(i): for large j(l) (3.25) represents an atomic decomposition, ϕL (x) =

L0

2−

j(l)(d−κ) p

Nj(l)

x ∈ Rn ,

aj(l),m (x),

m=1

l=L −1

with Nj(l) ∼ μ(Γl ) 2 ∼ l 2 , counting only non-vanishing terms, where the equivalence constants are independent of L. Clearly, ϕL (x) = 1 near Γ, and by Theorem 2.7(i) for q > p, j(l)d

n−d+κ p

ϕ |Bp,q L

q

n

(R , wκ,Γ )

≤ c1

j(l)d

L0

2

−j(l) d−κ p q

≤ c2

q/p 2jn wκ,Γ (Bj(l),m )

m=1

l=L L0

j(l) N

2−j(l)

d−κ p q

κ

q

p 2−j(l) p q Nj(l)

l=L

≤ c3

∞

q

l −q/p ∼ L1− p .

l=L

This replaces (3.24) in this case and the counterpart of Step 3 concludes the argument. In case of p ≤ pκ we need moment conditions in (2.20), but the same type of modiﬁcation as mentioned in Step 4 works here as well, where we beneﬁt from the porosity of the d-set Γ. This completes the proof. Remark 3.10. The above theorem extends partial results in [34] related to the case 1 < p < ∞, cf. [34, Corollars 4.8, 4.13]. Moreover, in view of (2.31) we know that s+

Bp,q

max(κ,0) p

s (Rn , wκ,Γ ) → Bp,q (Rn ),

that is, for κ ≥ 0, s+ κ

s Bp,q p (Rn , wκ,Γ ) → Bp,q (Rn ).

Though these spaces are only continuously embedded (but diﬀer from each other), their dichotomy parameters just scale as expected: by Proposition 3.7(ii) and Theorem 3.9 we have that D(Bp (Rn ), Lp (Γ)) = sΓ , qΓ and κ D(Bp (Rn , wκ,Γ ), Lp (Γ)) = sΓ + , qΓ . p A similar phenomenon occurs in view of the continuously embedded (but diﬀerent) spaces from Besov and Triebel–Lizorkin type, see Proposition 2.12 together with Example 2.14 in contrast to Theorem 3.9, in particular (3.17) and (3.18). The stronger assumption in (2.31) and Example 2.14 for negative κ > −(n − d) is due to the embedding of the whole Rn ‘far away’ which

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cannot be better than in the unweighted situation. However, since our trace questions are of purely local nature in a close neighbourhood of the compact d-set Γ, it is less astonishing that we have no extra assumption. Just to the contrary we have the meanwhile well-known phenomenon, that for κ < 0 (that is, where the weight tends to inﬁnity when approaching Γ) we have an increased smoothness in the sense of (3.3), (3.4): the trace of the weighted space results in the same space on Γ like the trace of an even smoother unweighted space, such that the special weight wκ,Γ can compensate some lack of smoothness in this situation. Remark 3.11. The concept of d-sets Γ can be generalised in a natural way to (d, Ψ)-sets studied in [10,26], where Ψ is a so-called ‘admissible’ function or a slowly varying function; we also refer to [24, 35] for more general background material. Then it turns out that appropriate spaces in the context of trace or dichotomy questions as discussed above are spaces of generalised smoothness (s,Ψ) Ap,q (Rn ) studied in [25, 26] in great detail. Based on these observations in the unweighted setting there are some ﬁrst consequences in the weighted case in [34, Chapter 5], where the idea is to modify the weight function wκ,Γ appropriately such as to obtain the counterpart of (3.3), (3.4) in this context: Ψ if vκ,Γ is this modiﬁed weight function corresponding to the admissible Ψ function Ψ and the (d, Ψ)-set ΓΨ , then s (s− κ ,Ψ1/p ) Ψ trΓΨ Bp,q (Rn , vκ,Γ ) = trΓΨ Bp,q p (Rn ) Ψ and one can use unweighted trace results from [25,26]. It would be interesting to determine the inﬂuence of the function Ψ one the dichotomy results. Some ﬁrst approach in the unweighted setting, but dealing with the even more general h-sets Γ (and thus requiring more general spaces, too) is suggested in [9]; one can ﬁnd further discussions, curiosities and applications in [52, Chapter 6.4]. Remark 3.12. Instead of playing with the set Γ where the trace is taken, one can likewise inquire into the inﬂuence of the weight function w ∈ A∞ in a more qualitative way. Certainly one can expect that the singular set Ssing (w) introduced in (2.9) should have some essential inﬂuence on trace questions, and possibly also on dichotomy matters. But nothing seems to be done yet in this direction, though in view of some application to sampling numbers as explained in [52, Chapter 6.4] this could also be of wider interest.

References [1] H. Abels, M. Krbec, and K. Schumacher. On the trace space of a Sobolev space with a radial weight. J. Funct. Spaces Appl., 6(3):259–276, 2008. [2] M. Bownik. Atomic and molecular decompositions of anisotropic Besov spaces. Math. Z., 250(3):539–571, 2005. [3] M. Bownik and K.P. Ho. Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces. Trans. Amer. Math. Soc., 358(4):1469–1510, 2006.

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[4] H.-Q. Bui. Weighted Besov and Triebel spaces: Interpolation by the real method. Hiroshima Math. J., 12(3):581–605, 1982. [5] H.-Q. Bui. Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures. J. Funct. Anal., 55(1):39–62, 1984. [6] H.-Q. Bui, M. Paluszy´ nski, and M.H. Taibleson. A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Studia Math., 119(3):219–246, 1996. [7] H.-Q. Bui, M. Paluszy´ nski, and M.H. Taibleson. Characterization of the BesovLipschitz and Triebel-Lizorkin spaces. The case q < 1. J. Fourier Anal. Appl., 3 (Spec. Iss.):837–846, 1997. [8] V.I. Burenkov and M.L. Gol’dman. Extension of functions from Lp . Trudy Mat. Inst. Steklov., 150:31–51, 1979. Studies in the theory of diﬀerentiable functions of several variables and its applications, VII. [9] A.M. Caetano and D.D. Haroske. Traces for Besov spaces on fractal h-sets and dichotomy results. Preprint. [10] D.E. Edmunds and H. Triebel. Spectral theory for isotropic fractal drums. C. R. Acad. Sci. Paris, 326(11):1269–1274, 1998. [11] K.J. Falconer. The geometry of fractal sets. Cambridge Univ. Press, Cambridge, 1985. [12] R. Farwig and H. Sohr. Weighted Lq -theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan, 49(2):251–288, 1997. [13] M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J., 34(4):777–799, 1985. [14] M. Frazier and B. Jawerth. A discrete transform and decomposition of distribution spaces. J. Funct. Anal., 93(1):34–170, 1990. [15] M. Frazier and S. Roudenko. Matrix-weighted Besov spaces and conditions of Ap type for 0 < p ≤ 1. Indiana Univ. Math. J., 53(5):1225–1254, 2004. [16] J. Garc´ıa-Cuerva and J. L. Rubio de Francia. Weighted norm inequalities and related topics, volume 116 of North-Holland Mathematics Studies. NorthHolland, Amsterdam, 1985. [17] M.L. Gol’dman. Extension of functions in Lp (Rm ) to a space of higher dimension. Mat. Zametki, 25(4):513–520, 1979. [18] D.D. Haroske and I. Piotrowska. Atomic decompositions of function spaces with Muckenhoupt weights, and some relation to fractal analysis. Math. Nachr., 281(10):1476–1494, 2008. [19] D.D. Haroske and H.-J. Schmeißer. On trace spaces of function spaces with a radial weight: the atomic approach. Complex Var. Elliptic Equ., 55(8-10):875– 896, 2010. [20] D.D. Haroske and L. Skrzypczak. Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, I. Rev. Mat. Complut., 21(1):135–177, 2008. [21] D.D. Haroske and L. Skrzypczak. Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, II. General weights. Annales Academiæ Scientiarum Fennicæ, 36(1):111–138, 2011.

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[22] D.D. Haroske and L. Skrzypczak. Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases. J. Funct. Spaces Appl., 9(2):129–178, 2011. [23] A. Jonsson and H. Wallin. Function spaces on subsets of Rn . Math. Rep. Ser. 2, No.1, xiv + 221 p., 1984. [24] P. Mattila. Geometry of sets and measures in euclidean spaces. Cambridge Univ. Press, Cambridge, 1995. [25] S.D. Moura. Function spaces of generalised smoothness. Dissertationes Math., 398:88 pp., 2001. [26] S.D. Moura. Function spaces of generalised smoothness, entropy numbers, applications. PhD thesis, Universidade de Coimbra, Portugal, 2002. [27] B. Muckenhoupt. Hardy’s inequality with weights. Studia Math., 44:31–38, 1972. [28] B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165:207–226, 1972. [29] B. Muckenhoupt. The equivalence of two conditions for weight functions. Studia Math., 49:101–106, 1973/74. [30] S.M. Nikol’ski˘ı. Approximation of functions of several variables and embedding theorems. Nauka, Moscow, second, revised and supplemented edition, 1977. Russian; First edition 1969. English Transl.: Grundlehren der Mathematischen Wissenschaften, vol. 205, Springer, New York-Heidelberg, 1975. [31] J. Peetre. The trace of Besov space – a limiting case. Technical report, University Lund, Sweden, 1975. [32] J. Peetre. A counterexample connected with Gagliardo’s trace theorem. Comment. Math. Special Issue, 2:277–282, 1979. Special issue dedicated to Wladyslaw Orlicz on the occasion of his seventy-ﬁfth birthday. [33] I. Piotrowska. Traces on fractals of function spaces with Muckenhoupt weights. Funct. Approx. Comment. Math., 36:95–117, 2006. [34] I. Piotrowska. Weighted function spaces and traces on fractals. PhD thesis, Friedrich-Schiller-Universit¨ at Jena, Germany, 2006. [35] C.A. Rogers. Hausdorﬀ measures. Cambridge Univ. Press, London, 1970. [36] S. Roudenko. Matrix-weighted Besov spaces. Trans. Amer. Math. Soc., 355:273–314, 2002. [37] S. Roudenko. Duality of matrix-weighted Besov spaces. Studia Math., 160(2):129–156, 2004. [38] V.S. Rychkov. Littlewood-Paley theory and function spaces with Aloc p weights. Math. Nachr., 224:145–180, 2001. [39] C. Schneider. Besov spaces with positive smoothness. PhD thesis, Universit¨ at Leipzig, 2009. [40] C. Schneider. Trace operators in Besov and Triebel-Lizorkin spaces. Z. Anal. Anwendungen, 29(3):275–302, 2010. [41] W. Sickel and H. Triebel. H¨ older inequalities and sharp embeddings in function s s spaces of Bp,q and Fp,q type. Z. Anal. Anwendungen, 14:105–140, 1995. [42] E.M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, 1993.

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[43] J.-O. Str¨ omberg and A. Torchinsky. Weighted Hardy spaces, volume 1381 of Lecture Notes in Mathematics. Springer, Berlin, 1989. [44] A. Torchinsky. Real-variable methods in harmonic analysis, volume 123 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1986. [45] H. Triebel. Interpolation theory, function spaces, diﬀerential operators. NorthHolland, Amsterdam, 1978. [46] H. Triebel. Theory of function spaces. Birkh¨ auser, Basel, 1983. [47] H. Triebel. Theory of function spaces II. Birkh¨ auser, Basel, 1992. [48] H. Triebel. Fractals and spectra. Birkh¨ auser, Basel, 1997. [49] H. Triebel. The structure of functions. Birkh¨ auser, Basel, 2001. [50] H. Triebel. Theory of function spaces III. Birkh¨ auser, Basel, 2006. [51] H. Triebel. The dichotomy between traces on d-sets Γ in Rn and the density of D(Rn \ Γ) in function spaces. Acta Math. Sinica, 24(4):539–554, 2008. [52] H. Triebel. Function Spaces and Wavelets on domains. EMS Tracts in Mathematics (ETM). European Mathematical Society (EMS), Z¨ urich, 2008. [53] H. Triebel and H. Winkelvoß. Intrinsic atomic characterizations of function spaces on domains. Math. Z., 221(4):647–673, 1996. Dorothee D. Haroske Mathematical Institute Friedrich-Schiller-University Jena D-07737 Jena Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 91–103 c 2012 Springer Basel AG

Lavrentiev’s Theorem and Error Estimation in Elliptic Inverse Problems Ian Knowles and Mary A. LaRussa Abstract. Lavrentiev’s theorem provides bounds for analytic functions known to be small at a ﬁnite number of points in a bounded region. An analogous result is established for solutions of elliptic equations on bounded regions in R2 and applied to estimating non-uniqueness error in elliptic inverse problems. Mathematics Subject Classiﬁcation (2010). 35R30, 35J25, 86A22. Keywords. Quasiconformal mapping, Beltrami equation, Stoilow factorization, Beurling and Cauchy transforms, elliptic error estimates.

1. Introduction It has long been known in the complex analysis world that surfaces deﬁned by analytic functions have a certain “rigidity” in that at points where the derivative is non-zero (so the analytic function becomes a conformal mapping), the surface is locally close to its linearization in a very conﬁning manner (see [1, §2.10]). It is also known that an analytic function that is zero on a set with a limit point (which can be a rather small set in the plane) must be identically zero. Of course, just knowing that an analytic function is zero on a ﬁnite set does not make it zero everywhere, but when one adds these rigidity properties, it is natural to speculate that if such a function is zero (or small) at a few points, then it should not be able to “wobble” much in between, with the wobble becoming less as the number of such points increases. The remarkable theorem of Lavrentiev (see Theorem 2.3 below) indicates, in a precise and quantitative fashion, that analytic functions do indeed behave in this way. Now, if f is analytic and f = σ + iω, it is known that the real functions σ and ω are solutions of the Laplace equation. It follows from the Lavrentiev theorem that these conjugate harmonic functions inherit the same rigidity properties. In this paper, using the Lavrentiev theorem and ideas from the

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theory of quasi-conformal mappings in the plane, our primary goal is to show that solutions u of the elliptic equation ∇ · p(x)∇u = 0,

x ∈ Ω ⊂ R2 .

(1.1)

have similar behaviour. In the ﬁnal sections we apply this theory to the estimation of the “non-uniqueness” error, an important component of the modelling error in the inverse groundwater problem.

2. Some complex function theory We assume throughout that Ω ⊂ R2 is a bounded domain with a connected complement, and that p : Ω → (0, ∞) is measurable and bounded away from zero and inﬁnity. Given the famous connection between analytic functions and harmonic functions aﬀorded by the Cauchy–Riemann equations, it is perhaps not surprising that the solutions of the more general elliptic equation (1.1) belong to a similar framework. In particular we have [2, Lemma 2.1] Theorem 2.1. Assume that the function u lies in the Sobolev space H1 (Ω), is real-valued, and satisﬁes (1.1). Then there exists a function v ∈ H1 (Ω), called the p-harmonic conjugate of u, unique up to an additive constant, such that ∂x v = −p∂y u, ∂y v = p∂x u, (2.1) and 1 ∇ · ∇v = 0. (2.2) p Also f = u + iv satisﬁes the R-linear Beltrami equation ∂f = μ∂f , where ∂ =

1 (∂ 2 x

− i∂y ), ∂ =

1 (∂ 2 x

(2.3)

+ i∂y ), and μ = (1 − p)/(1 + p).

H1loc (Ω)

A function f ∈ that satisﬁes (2.3) is called a quasi-regular mapping, and if it is also a homeomorphism it is called quasiconformal. From the Stoilow factorization [1, Theorem 5.5.1] we know that H1loc (Ω) solutions f of (2.3) may be written in the form f = ψ ◦ h where ψ is C-analytic and h is a quasiconformal homeomorphism. Let E be a set of points of the real axis in C, and let Φn = {Δk : 1 ≤ k ≤ n} be a collection of n intervals whose union contains the set E, and denote by k the length of Δk . Deﬁne the Lavrentiev set function, μn by n

μn (E) = inf k . Φn

k=1

Let D = {z ∈ C : |z| < 1},

D1/4 = {z ∈ C : |z| < 1/4}

and consider the ﬁnite set of points (i)

(i)

A = {x(i) = (x1 , x2 ) : 1 ≤ i ≤ m} ⊂ D1/4 ,

Lavrentiev’s Theorem and Error Estimation

93

where we assume that (1)

(2)

(m)

x1 < x1 < · · · < x1 . For later use we have, more or less directly from the deﬁnition of the Lavrentiev set function, Theorem 2.2. Let A˜ denote the projection of the ﬁnite set A onto the real axis in C. Let m denote the number of points in A. ˜ = 0 for all k ≥ 1. (i) If m = 1, then μk (A) (ii) If m = 2, then ˜ = x(2) − x(1) ; μ1 (A) 1 1 ˜ = 0 for k ≥ 2. μk (A) (iii) If m = 3, then ˜ = x(3) − x(1) ; μ1 (A) 1 1 (2)

(1)

(3)

(2)

˜ = min{x − x , x − x }; μ2 (A) 1 1 1 1 ˜ μk (A) = 0 for k ≥ 3. (iv) If m = 4, then (4)

(1)

˜ =x −x ; μ1 (A) 1 1 ˜ = min{x(4) − x(2) , x(3) − x(1) , (x(4) − x(3) ) + (x(2) − x(1) )}; μ2 (A) 1 1 1 1 1 1 1 1 (j+1)

(j)

˜ = min{x μ3 (A) − x1 : 1 ≤ j ≤ 3}, 1 ˜ = 0, for k ≥ 4. μk (A) Note in particular that these measure values are either zero or can be made as small as we like, depending on the placement of the points x(i) , (i) 1 ≤ i ≤ m, in the region Ω. In particular, if the projected coordinates x1 , 1 ≤ i ≤ m, are all close together on the real axis, these measures would all be small, even with the wells being well spaced in the y-direction. We comment on this later in §5. Of central interest to us here is the following theorem of Lavrentiev [10, p. 58, Theorem 1]: Theorem 2.3. Suppose that the set A ⊂ D1/4 and let A˜ be the projection of ˜ A onto the real axis in C. Assume that n ∈ N and , where 0 < ≤ μ1 (A)/4, are related by the inequalities n+1 n ˜ ˜ 1 μn+1 (A) 1 μn (A) << . (2.4) 2 2 2 2 Assume also that the function g is analytic on D and that |g(z)| ≤ 1 for all z ∈ D. If |g(z)| ≤ , z ∈ A,

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then, for each z ∈ D1/4 , one of the two inequalities |g(z)| ≤ 4/25 ,

|g(z)| ≤ (6/7)n

holds. This theorem tells us, inter alia, that if an analytic function is bounded on an open region, and small on a ﬁnite subset A of a strictly interior region, then it is small throughout the interior region, in a way that depends explicitly on the number of points in A. It is a precise complement to the theorem that any analytic function that is zero on a set with a limit point, is zero everywhere in the region of analyticity.

3. Bounds on solutions of elliptic equations In the sequel we assume for simplicity that for some η > 0 Ω = {x ∈ R2 : |x| < 1 + η}, and that the coeﬃcient function p is bounded and measurable on Ω, and satisﬁes the ellipticity condition 1 ≥ p(x) ≥ ν > 0, ν for some constant ν, and all x ∈ Ω. We now establish

(3.1)

Theorem 3.1. Let A be a ﬁnite subset of D1/4 and let u be an H1 (Ω) solution of (1.1) satisfying the conditions |u(x)| ≤ 1, for all x ∈ Ω, and |u(x)| ≤ for ˜ some for which 0 ≤ ≤ μ1 (A)/4, and all x ∈ A. Let n be deﬁned by the inequality (2.4). Then, for all x in the set D1/4 , one of the inequalities |u(x)| ≤ 4/25 ,

|u(x)| ≤ (6/7)n

(3.2)

holds. Proof. Let v be a p-harmonic conjugate of u. From standard elliptic regularity results (for example [9, Chapter 3, §14]) we know that v, which satisﬁes (2.2), is continuous on the interior closed unit ball D ⊂ Ω. So |v(x)| < M1 for some constant M1 > , and all x ∈ D. Let α > 0 be chosen arbitrarily, and consider the complex function fα =

1 (u + iαv). 1 + αM1

One can see after multiplying both equations in (2.1) by α, that the function fα lies in H1loc (D) and satisﬁes the Beltrami equation (2.3) with μ = μα = (1 − αp)/(1 + αp). So, by the Stoilow factorization theorem [1, Theorem 5.5.1] (taking Ω = D in that theorem) fα may be written in the form fα = ψα ◦ hα

Lavrentiev’s Theorem and Error Estimation

95

where hα : D → hα (D) is a quasiconformal homeomorphism and ψα is Canalytic on hα (D). Further, as |u(x)| + α|v(x)| < 1, 1 + αM1 for all y ∈ hα (D), we have ψα : hα (D) → D here. Fix α. For all y ∈ Bα = hα (A) ∩ D1/4 , we have |ψα (y)| = |ψα (hα (x))| = |fα (x)| ≤

+ αM1 . 1 + αM1 one of the inequalities

|ψα (y)| ≤ α = So, by Theorem 2.3, for each y ∈ D1/4 |ψα (y)| ≤ 4/25 , α

|ψα (y)| ≤ (6/7)m

holds, where m = m(α) is deﬁned by m+1 m ˜α ) ˜α ) 1 μm+1 (B 1 μm (B < α ≤ . 2 2 2 2

(3.3)

In particular, as 1 |u(x)| ≤ |fα (x)| = |ψα (hα (x))|, 1 + αM1 it follows that, for all x for which hα (x) ∈ D1/4 , one of the inequalities 1 |u(x)| ≤ 4/25 , α 1 + αM1

1 |u(x)| ≤ (6/7)m 1 + αM

(3.4)

holds. We show below that as α → 0, hα → h pointwise uniformly, where the limit function h : D → D is the projection onto the real axis in D. Let us assume this for the moment. So, if x ∈ D1/4 , then, as D1/4 is open and hα → h pointwise, for all α small enough hα (x) ∈ D1/4 . Hence, for all x ∈ D1/4 u(x) satisﬁes (3.4), where m = m(α) and α satisfy (3.3), for all α small enough. ˜ α = h(hα (A)) for all α small enough. Deﬁne Also, as A is ﬁnite, B n ˜ 1 μn (A) qn = . 2 2 We also have that, as α → 0, n ˜α ) 1 μn (B → qn , 2 2

1 2

˜α ) μn+1 (B 2

n+1 → qn+1 .

(3.5)

This follows from the pointwise convergence of the functions hα and the ﬁniteness of A, via Theorem 2.2. If we set + αM1 α = g(α) = , 1 + αM1 then M1 (1 − ) g (α) = > 0. (1 + αM1 )2

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So we see that α decreases to as α decreases to 0. Choose α0 so that for 0 < α < α0 we have 1 < α < ( + qn ). 2 In addition, by making α0 smaller if necessary, we have from (3.5) that n 1 μn (B˜α ) 1 | − qn | < (qn − ) 2 2 2 and 1 | 2

μn+1 (B˜α ) 2

n+1 − qn+1 | <

1 ( − qn+1 ), 2

for all 0 < α < α0 . This means that n+1 n 1 μn+1 (B˜α ) 1 μn (B˜α ) < α < . 2 2 2 2 so that in (3.3) and (3.4), for 0 < α < α0 we can set m(α) = n. The desired result follows by letting α → 0 in (3.4). So it remains to prove that, as α → 0, hα → h pointwise uniformly, where the limit function h : D → D is the projection onto the real axis in D. Observe from the proof of the Stoilow factorization theorem that the map hα can be any homeomorphic solution of the Beltrami equation ∂hα ∂hα = μα χD ∂z ∂z on C, where

1 − αp 2αp =1− , (3.6) 1 + αp 1 + αp and χD denotes the characteristic function for the set D. We choose hα to be one-half of the principal solution of the Beltrami equation, so that [1, Theorem 5.1.2] we may write hα (z) = (z + ωα (z))/2, where the function ωα satisﬁes the inhomogeneous Beltrami equation μα =

∂ωα ∂ωα = μα χ D + μα χD . (3.7) ∂z ∂z Notice that, as α → 0, the Beltrami coeﬃcient μα → 1, so the Beltrami equation becomes degenerate. Clearly, some care is required in taking such a limit. Let hα be one-half the principal solution of the Beltrami equation ∂hα ∂h = μα χD α ∂z ∂z on C, where μα =

1 − αν 2αν =1− , 1 + αν 1 + αν

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and ν is deﬁned by (3.1). Then hα (z) = (z + ωα (z))/2 where the function ωα satisﬁes the inhomogeneous Beltrami equation ∂ωα ∂ω = μα χD α + μα χD . (3.8) ∂z ∂z Using the Neumann series method of solving the basic inhomogeneous Beltrami equation [1, Theorem 5.1.1] we have that ωα = C (I − μα χD S)−1 (μα χD ) . (3.9) Here [1, Chapter 4] C denotes the Cauchy transform, S denotes the Beurling transform, and (I − μα χD S)−1 φ = φ + μα χD Sφ + (μα χD S)2 φ + · · · ,

(3.10)

where the right side is a convergent series, given that S L2 = 1 [1, Theorem 4.5.3], and |μα | < 1 for α > 0. Now, note from [1, (4.24)] that 1 χ , z 2 C\D

(3.11)

z, |z| ≤ 1, 1 z , |z| > 1.

(3.12)

S χD = − and from [1, (4.38)]

C χD =

So as μα is a constant, 2

μ α χ χ = 0. z 2 D C\D So, from (3.9), (3.10), (3.12) and (3.13), z, |z| ≤ 1, ωα = C(μα χD ) = μα 1 , |z| > 1, z μα χD S(μα χD ) = −

(3.13)

(3.14)

and consequently, as α → 0, we have μα → 1 and hα → h0 where z z/2, |z| ≤ 1, h0 (z) = + 1/2z, |z| > 1. 2 Notice that h = h0 |D : D → D is the projection map of D onto the real axis in D. Next, subtracting equations (3.7) and (3.8) we get ∂(ωα − ωα ) ∂(ωα − ωα ) ∂ωα = μα χD + .(μα −μα )χD +(μα −μα )χD . (3.15) ∂z ∂z ∂z Observe that from (3.14) we have ∂ωα 0, |z| ≤ 1, = μα , 1 − , |z| > 1, ∂z z2 so that the second term on the right of (3.15) vanishes identically. Consequently ωα − ωα = C (I − μα χD S)−1 ((μα − μα )χD ) . (3.16)

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Now by (3.11) μα χD S((μα − μα )χD ) = μα χD S(μα χD ). From (3.1) and (3.6) we have 2α 2αν μα = 1 − ≤ μα ≤ 1 − = μα , α+ν 1 + αν so that μα χD μα χ D μα χD dτ ≤ dτ ≤ dτ. 2 2 2 |z−τ |> (z − τ ) |z−τ |> (z − τ ) |z−τ |> (z − τ ) If we multiply by −1/π and let → 0, it follows from (3.11) that 1 1 −μα 2 χC\D ≥ S(μα χD ) ≥ −μα 2 χC\D , z z so that μα χD S(μα χD ) = 0. This means that, by (3.16) ωα − ωα = C (μα − μα )χD ν−p χ = 2αC , (1 + αp)(1 + αν) D which approaches zero pointwise uniformly as α → 0 because the Cauchy transform of a function in L3 (C) lies in the H¨ older space C 1/3 (C) by [1, Theorem 4.3.13]. Finally, we have hα − h0 = (hα − hα ) + (hα − h0 ) = (ωα − ωα )/2 + (hα − h0 ) → 0 pointwise uniformly as α → 0, which completes the proof.

4. Error estimation in the inverse groundwater problem Underground aquifer systems (as well as oil reservoirs) are often modeled by the diﬀusion equation: ∂w q(x) = ∇ · [p(x)∇w(x, t)] + R(x, t), (4.1) ∂t for time 0 ≤ t ≤ T , and for x = (x1 , x2 ) in a two-dimensional aquifer region Ω, which we assume to be equal to D for notational convenience. Here, w represents the piezometric head, p the aquifer transmissivity, R the recharge, and q ≥ 0 the storativity of the aquifer (see, for example, [3]). It is well known among hydrologists that the inability to obtain reliable values for the coeﬃcients in (1.1) from measured values of the water levels w taken over time at a collection of wells in the aquifer, together with reasonable estimates of the associated inverse recovery error, is a serious impediment to the conﬁdent use of such models. While new methods for the simultaneous recovery of p, q, and R (including its time dependence) from a known w have recently been developed [6, 7, 11], the nagging and fundamentally important problem of properly quantifying the associated error remains. In discussing the recovery error for the inverse groundwater problem we assume in advance that one is using a recovery algorithm that is provably

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well-posed (see for example [5]) so that in theory at least, the associated recovery error may be properly quantiﬁed. With this in mind, the recovery error that we seek to quantify centers on the error in the head data function w(x, t) that we use, given that the theorems guaranteeing unique recovery of the coeﬃcient functions in (1.1) [4] require that we know the head function w(x, t) at all points x ∈ Ω and for all 0 ≤ t ≤ T . We restrict attention here to a choice of w(x, t) obtained by linear interpolation in space and time from the ﬁnite set of well data values. The error in our interpolated head function w now has basically three components. First, the head data may have measurement error. Second, we observe that in practice one only knows w at discrete times and at a ﬁnite (often small) number, m, of wells located at points {x(i) : 1 ≤ i ≤ m} in Ω. The solution w is certainly not speciﬁed uniquely by a knowledge of its values at a ﬁnite number of interior points in Ω. In general one needs to know something like the solution values at all points of the boundary at a particular time. So we have a “non-uniqueness” error to contend with, and in particular, while our solution is tied-down nicely at the well points {x(i) : 1 ≤ i ≤ m} we need to know how much “non-uniqueness wobble” in w is possible in between the well points. Finally, as aquifers change only slowly over time, interpolation in t is generally not a source of signiﬁcant error. However, as a practical matter, and given that one typically does not have very many measurement wells available in a given aquifer, the interpolation of w values in x at ﬁxed t is a third, and possibly large, source of error in the inverse recovery process. We outline how to quantify the ﬁrst and second type of error below, and a method for estimating the third type of error may be found in [8]. Speciﬁcally, assume that we are given measured values of the solution w of (4.1) at the well points {x(i) : 1 ≤ i ≤ m} at times t satisfying 0 ≤ t ≤ 1. We assume that the term q(x) ∂w in (4.1) contributes little to this discussion, ∂t because the storativity values q(x) are typically quite tiny relative to the conductivity term, and in any event, aquifers tend to change only slowly in time, so the factor ∂w/∂t is also rather small. There are inﬁnitely many solutions of (4.1) equal to the measured values w at the points {x(i) : 1 ≤ i ≤ m}. Let w1 and w2 be two such solutions, and set w∗ = w1 − w2 . Then for any given time t, w∗ (x(i) , t) = 0

(4.2)

for all 1 ≤ i ≤ m. Also as the two solutions wi share the right-hand side function R in (4.1), and we are ignoring the term q(x) ∂w ∂t , for ﬁxed t the function w∗ is a solution of our elliptic equation (1.1). We are interested in how large |w∗ | can be in a connected neighbourhood N ⊂ Ω of the set of well points {x(i) : 1 ≤ i ≤ m} as w1 and w2 range over all possible solution choices. With no further restrictions, bounds on |w∗ | are problematical. However, for any given aquifer, in practice there are physical limits on how much the heads can vary over the entire aquifer over time. So, guided by these physical

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considerations, we make the assumption that for any choice of the solutions w1 and w2 the error w∗ is bounded, |w∗ (x, t)| < M,

for all x ∈ Ω and 0 ≤ t ≤ 1,

(4.3)

for some known constant M . In [8], a method for estimating M from groundwater well data is given; we return to this later. It is convenient to consider the normalized error function u = w∗ /M , so we actually assume that |u(x, t)| < 1,

for all x ∈ Ω,

(4.4)

at all points x in the aquifer region Ω, and at all times 0 ≤ t ≤ 1. As we consider the time t to be ﬁxed in this discussion, we henceforth omit reference to this variable. If there are measurement errors in the head data, the equality (4.2) may be modiﬁed to |u(x(i) )| ≤ (4.5) for some 0 ≤ < 1 and all 1 ≤ i ≤ m. We then have, from Theorem 3.1, an explicit bound on the scaled head error, u, in terms of m and the scaled measurement error , over the quarter disk N = D1/4 . This allows us to estimate the measurement and “non-uniqueness” errors in a groundwater model as we see next.

5. Estimating “non-uniqueness” head error We are interested in situations in which the “non-uniqueness” head error, |u|, is small on D1/4 in situations when m, the number of wells, is relatively small as well. Using Theorem 2.2 one sees immediately that one should not consider m = 1, as there is no estimate in this case. For each m > 1 we # wells m n 2 1 3 2 4 3 5 4 6 5 7 6 8 7 11 10

max scaled absolute head error 0.38 0.86 0.15 0.73 0.06 0.63 0.02 0.54 0.008 0.46 0.003 0.40 0.001 0.34 0.00007 0.21

˜ = 0 henceforth. For certain exceptional assume for simplicity that μm−1 (A) arrangements of the wells one would need an appropriate modiﬁcation of the following discussion. By way of example, if all the wells were on a line parallel to the vertical axis, then by Theorem 2.2 (and analoguous formulae ˜ = 0 for all k and the method fails. Of course, in this case for m > 4), μk (A) one can rotate the plane by π/2 and successfully re-apply the method.

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For m = 2 one is restricted to choosing n = m − 1 = 1 in applying (3.2); here, for small enough, the second estimate in (3.2) dominates, and we see that |u| ≤ 6/7 ≈ 0.86. For m = 3, from Theorem 2.2 the largest feasible n is n = 2, and we see that |u| < (6/7)2 ≈ 0.73 provided that < 0.14. In general, for small enough, one chooses n = m − 1, and the ﬁrst few cases for the largest compatible , i.e. the that solves 6 n 4/25 = , (5.1) 7 and the resultant maximum scaled head errors |u| = (6/7)n on D1/4 are given in the table above. These are hard estimates for the worst case scenario for the “non-uniqueness” head error. If, , the absolute head measurement error at the wells, is larger than that listed in the “max ” column, one needs to substitute the value 4/25 instead of (6/7)n in the last column. In particular, if one ﬁxes the number of wells, m, and checks the variation of error with a somewhat diﬀerent calculation ensues. We note that the speciﬁc numbers, such as 6/7 and 4/25, appearing in Theorem 3.1 (and, consequently, (5.1)) are unlikely to be optimal for this problem. Recall that u here represents the scaled absolute “non-uniqueness” head error. If the actual absolute error w∗ = w1 − w2 satisiﬁes |w∗ | ≤ M , then u = w∗ /M . From the table one can see, for example, that with four wells

Figure 1. Positions of 4 wells. (m = 4), each with an absolute head measurement error of 0.06M , one should expect a worst case absolute head non-uniqueness error of 0.62M . In general the aquifer “wobble”, M , which is an estimate for the maximal head variation in a given aquifer region over time, would be known to a ﬁeld hydrologist with local knowledge of a particular groundwater system. In particular, data from the Port Willunga aquifer in south-eastern South Australia over the 13 year

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period January 1995 to December 2007 using the 4 wells WLG014, WLG051, WLG055, and WLG060 pictured in Figure 1, and assuming that the circle shown is scaled and translated to D1/4 , we have that M ≈ 3.4 meters. For the shorter one-year time period January 1998 to December 1998, M ≈ 2.2 meters. This means, for example, in the circular region around the four wells shown in Figure 1, during the year 1998 when the heads varied around 2.2 meters, (and provided the absolute measurement error for each of these head measurements is no more than 0.13 meters), we have a maximum head nonuniqueness error of 1.4 meters in a typical head measurement for this area of 40 to 80 meters. In practice, one might expect that using, for example, linear interpolation to create the “measured” head surface from the discrete well data, the actual interpolation error would be somewhat less than the ﬁgure obtained by the foregoing method. Acknowledgement The authors are grateful to the referee for a careful reading of the paper, and for catching a number of errors in the initial draft.

References [1] Kari Astala, Tadeusz Iwaniec, and Gaven Martin. Elliptic partial diﬀerential equations and quasiconformal mappings in the plane, volume 48 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2009. [2] Kari Astala and Lassi P¨ aiv¨ arinta. Calder´ on’s inverse conductivity problem in the plane. Ann. of Math. (2), 163(1):265–299, 2006. [3] J. Bear. Dynamics of Fluids in Porous Media. American Elsevier, New York, 1972. [4] Ian Knowles. Uniqueness for an elliptic inverse problem. SIAM J. Appl. Math., 59(4):1356–1370, 1999. [5] Ian Knowles and Mary A. LaRussa. Conditional well-posedness for an elliptic inverse problem. preprint. Available online at http://www.math.uab.edu/knowles/pubs.html. [6] Ian Knowles, Tuan A. Le, and Aimin Yan. On the recovery of multiple ﬂow parameters from transient head data. J. Comp. Appl. Math., 169:1–15, 2004. [7] Ian Knowles, Michael Teubner, Aimin Yan, Paul Rasser, and Jong Wook Lee. Inverse groundwater modelling in the Willunga Basin, South Australia. Hydrogeology Journal, 15:1107–1118, 2007. [8] Mary A. La Russa. Conditional well-posedness and error estimation in the groundwater inverse problem. PhD thesis, University of Alabama at Birmingham, 2010. [9] Olga A. Ladyzhenskaya and Nina N. Uraltseva. Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York, 1968. [10] M. M. Lavrentiev, V. G. Romanov, and S. P. Shishatski˘ı. Ill-posed problems of mathematical physics and analysis, volume 64 of Translations of Mathematical

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Monographs. American Mathematical Society, Providence, RI, 1986. Translated from the Russian by J. R. Schulenberger, Translation edited by Lev J. Leifman. [11] Aimin Yan. An Inverse Groundwater Model. PhD thesis, University of Alabama at Birmingham, 2004. Ian Knowles and Mary A. LaRussa Department of Mathematics University of Alabama at Birmingham Birmingham, Alabama, 35294 USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 105–124 c 2012 Springer Basel AG

Two-weighted Norm Inequalities for the Double Hardy Transforms and Strong Fractional Maximal Functions in Variable Exponent Lebesgue Space Vakhtang Kokilashvili and Alexander Meskhi Dedicated to Professor D. E. Edmunds and Professor W. D. Evans to mark their 80th and 70th birthday, respectively.

Abstract. Two-weight norm estimates for double Hardy transforms and variable order strong fractional maximal functions are established in variable exponent Lebesgue spaces. Derived conditions are simultaneously necessary and suﬃcient in the case when the exponent function of the right–hand side space is constant. In the main statements the weak logarithmic condition for exponents of the spaces is not assumed. Mathematics Subject Classiﬁcation (2010). 42B20, 46E30. Keywords. Variable exponent Lebesgue spaces, double Hardy transforms, strong fractional maximal operators, weights, two-weight inequality.

1. Introduction The paper deals with two-weight criteria for double Hardy transforms and strong fractional maximal functions in the framework of variable exponent Lebesgue spaces. Recently T. Kopaliani [14] showed that the Hardy–Littlewood strong maximal operator is bounded in Lp(·) space if and only if p is constant. The similar result occurs for fractional maximal functions. However, we discovered that situation with strong fractional maximal function of variable order and multiple Hardy transform is completely diﬀerent. One of the novelties of this research is that two-weight problem for the double Hardy transform is studied for the ﬁrst time in Lp(·) spaces. We treat the similar task for fractional maximal function of variable order and solve the trace problem. It should be emphasized that two-weight estimates are derived without the requirement of the weak logarithmic condition for exponents of spaces.

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For the weighted results regarding the classical Hardy transform in Lp(·) spaces with power weights we refer to [12] and [4]. The papers [5] and [13] deal with the Hardy inequality with weights of general type. We notice that in [5] the weak logarithmic condition is not assumed for the exponents of spaces. The boundedness of the fractional maximal operator Mα from Lp(·) (Rn ) to Lq(·) (Rn ) was proved in [1], where p and q are related by the Sobolev condition and p satisﬁes the log-H¨ older continuity condition. In [10] the authors of this paper derived criteria governing the boundedness of variable parameter q(·) fractional maximal operator Mα(·) from Lpw (Rn ) to Lv (Rn ) (see also [9] for two-weight criteria of Sawyer type). Let us recall some well-known results regarding the Hardy inequality in the classical Lebesgue spaces (see e.g., [16], [17]). The celebrated classical Hardy inequality states: Theorem A. Let p be constant satisfying the condition 1 < p < ∞ and let f be a measurable, non-negative function on (0, ∞). Then ∞ x p p1 ∞ p1 1 p p f (y)dy dx ≤ f (x)dx . x 0 p−1 0 0 Two-weighted boundedness criteria for the Hardy transform x (H1 f )(x) =

f (y)dy, 0

reads as follows: Theorem B. Let p and q be constants satisfying the condition 1 < p ≤ q < ∞. Suppose that u and v are weight functions on R+ . Then each of the following conditions are necessary and suﬃcient for the inequality ∞

H1 f

q

q1 ∞ p1 p (x)v(x)dx ≤C f (x)w(x)dx

0

(1.1)

0

to hold for all non-negative and measurable functions f on R+ : a) The Muckenhoupt condition, ⎛∞ ⎞ q1 ⎛ x ⎞ 1 p 1−p ⎠ ⎝ ⎠ ⎝ AM := sup v(t)dt w(t) dt < ∞. x>0

x

0

Moreover, the best constant C in (1.1) can be estimated as follows: 1 1 q q p p AM ≤ C ≤ 1 + 1+ AM . p q

Two-weighted Norm Inequalities

107

b) The condition of L.-E. Persson and V. D. Stepanov, ⎛ x ⎞ q1 x − p1 ⎝ q ⎠ AP S := sup W (x) v(t)W (t) dt < ∞, W (x) := w(t)1−p dt. x>0

0

0

Moreover, the best constant C in (1.1) satisﬁes the following estimates: AP S ≤ C ≤ p AP S . In 1984 E. Sawyer [20] found a characterization of the two-weight inequality in terms of three independent conditions for the double Hardy transform x y (H2 f )(x, y) = f (t, τ )dtdτ. 0

0

The following statements gives two-weight criteria written by one condition when the weight on the right-hand side is a product of two univariate weights (see [18], [11], Ch. 1): Theorem C. Let p and q be constants such that 1 < p ≤ q < ∞ and let w(x, y) = w1 (x)w2 (y). Then the operator H2 is bounded from Lpw to Lqv (1 0

y1 y2

0

0

is fulﬁlled. It should be emphasized that from the results regarding the two-weight problem derived in this paper, as a corollary, we deduce trace inequality criteria for the double Hardy transform when the exponent of the initial Lebesgue space is a constant. Another remarkable corollary is that there exists a variable exponent p(x) for which the double average operator is bounded in Lp(·) . In the paper [8] the authors established trace inequality criteria for the strong fractional maximal operator 1 (Mα,β f )(x, y) := sup |f (t, τ )|dtdτ, 0 < α, β < 1, 1−α |J|1−β I×J(x,y) |I| I×J

in constant exponent Lebesgue spaces. In particular, the next statement holds (see [8], [11], Ch. 4): Theorem D. Let p, q, α and β be constants satisfying the conditions 1 < p < q < ∞, 0 < α, β < 1/p. Then the following statements are equivalent: (i) Mα,β is bounded from Lp (R2 ) to Lqv (R2 );

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(ii)

v (x, y)dxdy |I|q(α−1/p) |J|q(β−1/p) < ∞, q

sup I,J I×J

where I and J are arbitrary bounded intervals in R. Exploring the two-weight problem for the strong fractional maximal function of variable order we prove an analog of Theorem D in Lp(·) spaces when the exponent of the initial Lebesgue space is constant. Let p be a non-negative measurable function on Rn . Suppose that E is a measurable subset of Rn . In the sequel we will use the following notation: p− (E) := inf p; p+ (E) := sup p; E

E

p− := p− (Rn ); p+ := p+ (Rn ).

Let Ω be an open set in Rn . Suppose that P (Ω) is the class of all measurable functions p : Ω → R satisfying the condition 1 < p− (Ω) ≤ p(t) ≤ p+ (Ω) < ∞, t ∈ Ω. By L such that

p(·)

(Ω) we denote the space of measurable functions f : Ω → R

f Lp(·) (Ω)

⎧ ⎨

⎫ ⎬ f (x) p(x) := inf λ > 0 : dx ≤ 1 < ∞. ⎩ ⎭ λ Ω

p(·) Lw (Ω)

In the sequel by we denote the weighted variable exponent Lebesgue space deﬁned by the norm

f Lp(·) (Ω) := f w Lp(·) (Ω) . w

Sometimes we use the symbol Lp(·,·)(R2 ) (or Lp(x,y)(R2 )) when p is deﬁned on R2 . It is known (see e.g., [15], [19], [12]) that Lp(·) (Ω) is a Banach space. For other essential properties of Lp(·) spaces we refer, e.g., to [22], [15], [19], [6]. Finally we point out that constants (often diﬀerent constants in the same series of inequalities) will generally be denoted by c or C. Throughout the paper by the symbol p (x) we denote the function p(x)/(p(x) − 1). Under rectangle we mean a rectangle with sides parallel to the coordinate axes.

2. Strong fractional maximal functions in Lp(·) spaces. Unweighted case Let S Mα f (x, y) = sup |R|α−1 |f (t, τ )|dtdτ, R(x,y)

(x, y) ∈ R2 , 0 < α < 1,

R

be the fractional maximal function, where the supremum is taken over all rectangles R ⊂ R2 containing (x, y).

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109

Theorem 2.1. Let p be a measurable function deﬁned on R2 satisfying the condition p ∈ P (R2 ). Suppose that α is a constant for which the condition p(x) 0 < α < p1− is satisﬁed. We set q(x) = 1−α·p(x) . Then MαS is bounded from Lp(·) (R2 ) to Lq(·) (R2 ) if and only if p ≡ const.

Proof. Suﬃciency can be obtained easily by using the Lp (R) → Lq (R) boundedness twice for the one-dimensional fractional maximal operator 1 (Mα f )(x) = sup 1−α |f (t)|dt, 0 < α < 1. Ix |I| I⊂R

I

Necessity. We follow T. Kopaliani [14] which proved the theorem for α = 0. First we observe that if MαS is bounded from Lp(·) (R2 ) to Lq(·) (R2 ), then 1 sup AR := sup

χR Lq(·) χR Lp (·) < ∞, 1−α R R |R| where the supremum is taken over all rectangles R in R2 . Indeed, let f Lp(·) (R2 ) ≤ 1. Then for every rectangle R we have c ≥ MαS f Lq(·) (R2 ) ≥ MαS f Lq(·) (R) ≥ χR Lq(·) |R|α−1

|f (t, τ )|dtdτ. R

Taking now the supremum with respect to f , f Lp(·) ≤ 1, we ﬁnd that |R|α−1 χR Lq(·) χR Lp (·) ≤ c for all R ⊂ R2 . Further, suppose the contrary: p is not constant, i.e. inf2 p(t) < sup p(t). R

R2

By using Luzin’s theorem we can conclude that there is a family of pairwise disjoint sets {Fi } satisfying the conditions: (i) |R2 \ ∪j Fj | = 0; (ii) functions p : Fi → R are continuous; (iii) for every ﬁxed i, all points of Fi are points of density with respect to the basis consisting of all open rectangles in R2 . Indeed, let us represent R2 as follows R2 = ∪j Qj , where {Qj } is a family of pairwise disjoint half-open unit squares. Let us ﬁxed j. Suppose that εk is a sequence converging to 0. By using Luzin’s theorem step by step, there is a family of pairwise disjoint sets Fkj in Qj such that |Qj \ (∪k Fkj )| = 0 and p is continuous on Fkj . Removing now sets of measure zero from Fkj we can assume that all points of Fki are points of density with respect to open rectangles. Further, we can ﬁnd a pair of points of the type ((x0 , y1 ), (x0 , y2 )) or ((x1 , y0 ), (x2 , y0 )) in ∪Fi such that p(x0 , y1 ) = p(x0 , y2 ) or p(x1 , y0 ) = p(x2 , y0 ). Without loss of generality, assume that this pair is ((x0 , y1 ), (x0 , y2 )) such that (x0 , y1 ) ∈ F1 , (x0 , y2 ) ∈ F2 and y1 < y2 .

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Let 0 < ε < 1 be a ﬁxed number. Then there is a number δ > 0 such that for any rectangles Q1 (x0 , y1 ) and Q2 (x0 , y2 ) with diameters less than δ, the following inequalities hold: |Q1 ∩ F1 | > (1 − ε)|Q1 |, |Q2 ∩ F2 | > (1 − ε)|Q2 |,

(2.1)

p1 = sup p(x, y) < c1 < c2 < inf p(x, y) = p2 ,

(2.2)

Q2 ∩F2

Q1 ∩F1

where c1 and c2 are some positive constants. Let Q1,τ and Q2,τ be rectangles with properties (2.1) and (2.2). Suppose that Q1,τ := (x0 − τ, x0 + τ ) × (a, b) and Q2,τ := (x0 − τ, x0 + τ ) × (c, d), where a < b < c < d. Observe now that the following embeddings hold:

Lq(·) (Q2,τ ) → Lq2 (Q2,τ ), Lp (·) (Q1,τ ) → L(p1 ) (Q1,τ ), where q2 = inf q = Q2 ∩F2

p2 1−αp2 ,

(pQ1 ) =

p1 p1 −1 .

(2.3)

Recall that (see, e.g., Theorem

2.8 in [15]) the norm of embedding operators in (2.3) does not exceed 2τ (d − c) + 1 and 2τ (b − a) + 1 respectively. Further, by using (2.1) and (2.3) we have for the rectangle Qτ := (x0 − τ, x0 + τ ) × (a, d), 1

χQτ Lq(·) χQτ Lp (·) |Qτ |1−α 1 ≥

χQ2,τ ∩F2 Lq(·) χQ1,τ ∩F1 Lp (·) [2τ (d − a)]1−α 1 C 1− 1 ≥ [2τ (d − c)] q2 [2τ (b − a)] p1 1−α [2τ (d − a)]

sup AR ≥ R

= Cτ

α−1+ q1 +1− p1 2

1

= Cτ

α− p1 − q1 1

2

.

The last expression tends to 0 as τ → 0 because α− p11 + q12 = α− p11 + p12 −α < 0 and the constant C does not depend on τ and ε for small τ and ε (recall also that a, b, c and d are ﬁxed). This contradicts the condition supR AR < ∞.

p(·)

3. Double Hardy transform in Lw spaces Let

x y (H2 f )(x, y) =

f (t, τ )dtdτ, 0

(x, y) ∈ R2+ ,

0

R2+

where := [0, ∞) × [0, ∞). First we prove the following lemma: Lemma 3.1. Let p be a constant satisfying the condition 1 < p < ∞. Suppose that 0 < b ≤ ∞. Let ρ be an almost everywhere positive function on [0, b).

Two-weighted Norm Inequalities

111

Then there is a positive constant c such that for all f ∈ Lpρ ([0, b)), f ≥ 0, the inequality b 0

x

1 λ([0, x])

p b f (t)dt λ(x)dx ≤ C (f (x)ρ(x))p dx

0

0

holds, where λ(x) = ρ−p (x) and λ([0, x]) :=

x

λ(t)dt.

0

Proof. It is enough to show that (see e.g., [17], Ch. 1) the condition b t p−1 sup λ([0, x])−p λ(x)dx λ(x)dx <∞ 0
0

t

is satisﬁed. To check that this condition holds observe that b

−p

λ([0, x])

b x λ(x)dx =

t

λ(x)dx

0

t

=

−p λ(τ )dτ

1 1−p

1 = p−1 1 ≤ p−1

1−p b x d λ(τ )dτ t

' t

0

1−p −

λ(τ )dτ 0 t

1−p (

b λ(τ )dτ 0

1−p λ(τ )dτ

.

0

To formulate the next theorem we introduce the notation ∞ 0 Jab := [a, ∞) × [b, ∞), Jab := [0, a) × [0, b).

Theorem 3.1. Let 1 < p ≤ q− ≤ q+ < ∞, where p is constant and the exponent function q is deﬁned on R2 . Suppose that v and w are weights on R2+ with w(x, y) = w1 (x)w2 (y) for some univariate weights w1 and w2 . Then q(·) H2 is bounded from Lpw (R2+ ) to Lv (R2+ ) if and only if −1 ∞ ) q(·) 0 p B := sup v(χJab χJab L (R2+ ) w L (R2 ) < ∞. +

a,b>0

Proof. Necessity follows by the standard way taking the test function

f (x, y) = w−p (x, y)χ[0,a]×[0,b] (x, y), in the two-weight inequality.

a, b > 0,

112

V. Kokilashvili and A. Meskhi

Suﬃciency. Suppose that f ≥ 0 and f Lpw (R2 ) ≤ 1. Let {xk } and {yj } be sequences of positive numbers chosen so that xk

yj

w1−p

k

=2 ,

0

w2−p = 2j .

(3.1)

0

∞ w1−p = w2−p = ∞. Then [0, ∞) = 0 0 ) ) ) [xk , xk+1 ) = [yj , yj+1 ). Hence, R2+ = Ek × Fj , where Without loss of generality assume that

∞

j

k

k,j

Ek := [xk , xk+1 ), Fj := [yj , yj+1 ). It is easy to see that (3.1) implies −p k w1 = 2 , w2−p = 2j . Ek

(3.2)

Fj

Let us choose r so that p ≤ r ≤ q− . Then

v(H2 f ) rLq(·) (R2 ) = [v(H2 f )]r Lq(·)/r (R2+ ) + ≤c sup (v(x, y))r (H2 f (x, y))r h(x, y)dxdy. h

R2

L(q(·)/r) ≤1

Let

σ1 (E) :=

w1−p , σ2 (E) :=

E

w2−p

E

for a measurable set E ⊂ R. Observe that (3.1), (3.2) and the H¨ older’s inequality in variable exponent Lebesgue spaces (see e.g., [15]) yield the following chain of inequalities: (v(x, y))r (H2 f )r (x, y)h(x, y)dxdy R2+

⎡

⎢ ≤ ⎣ k,j

⎤⎡ ⎥ v r (x, y)h(x, y)dxdy ⎦ ⎣

xk+1 yj+1

f⎦

0

Ek ×Fj

⎤r

0

⎡ xk+1 yj+1 ⎤r

r ⎣ ≤c

v Lq(·)/r (Ek ×Fj ) h L(q(·)/r) (R2 ) f⎦ +

k,j

0

⎡ xk+1 yj+1 ⎤r

≤c

v rLq(·) (Ek ×Fj ) ⎣ f⎦ k,j

≤ cB r

0

k,j

w1−1 −r Lp ([0,x

k ))

0

w2−1 −r Lp ([0,y

j ))

0

⎡ xk+1 yj+1 ⎤r ⎣ f⎦ 0

0

Two-weighted Norm Inequalities ⎡ ⎤ pr ⎡ xk+2 yj+2

⎢ ⎥ = cB r (w1 (x)w2 (y))−p dxdy ⎦ · ⎣ ⎣ k,j

⎢ ⎣ k,j

1 σ1 (Ek )σ2 (Fj )

xk+1 yj+1

⎡ ≤ cB r

113

⎡

xk+2 yj+2

[w1 (x)w2 (y)]−p ⎣

1 σ1 ([0, x])σ2 ([0, y])

xk+1 yj+1

⎡ ⎡ ⎢ ≤ cB r ⎣ [w1 (x)w2 (y)]−p ⎣ R2+

1 σ1 ([0, x])σ2 ([0, y])

x y 0

x y 0

xk+1 yj+1

⎤r

f⎦

0

0

⎤p

⎤r/p

⎥ f ⎦ dxdy ⎦

0

⎤p

⎤r/p

⎥ f ⎦ dxdy ⎦

=: S.

0

By using Lemma 3.1 twice we conclude that ⎡ ⎤ rp ⎢ ⎥ S ≤ cB r ⎣ [f (x, y)]p (w(x, y))p dxdy ⎦ ≤ c.

R2+

Corollary 3.1. (Trace inequality). Let 1 0

Deﬁnition 3.1. Let Ω be an open set in Rn . We say that the exponent function p(·) ∈ P(Ω) if there is a constant 0 < δ < 1 such that p(x)p− δ p(x)−p− dx < +∞. Ω

This condition on p comes from the continuous embeddings of Lp(·) spaces investigated in [3]. Deﬁnition 3.2. We say that p(·) ∈ P∞ (Ω) (see [2]) if c |p(x) − p(y)| ≤ ln(e + |x|) for all x, y ∈ Ω with |y| ≥ |x|. Corollary 3.2. Let 1 < p− ≤ q− ≤ q+ < ∞ with p+ < ∞. Let v and w be a. e. positive functions on R2 with w(x, y) = w1 (x)w2 (y). Suppose that p ∈ P(R2+ ). If ∞ sup vχJab

a,b>0

Lq(·) (R2+ )

0 w−1 χJab

L(p− ) (R2+ )

< ∞,

(3.3)

then H2 is bounded from Lw (R2+ ) to Lv (R2+ ). p(·)

q(·)

Proof. Recall that (see [3]) if p ∈ P(R2+ ), then Lp(·) (R2+ ) → Lp− (R2+ ). Now Theorem 3.1 completes the proof.

114

V. Kokilashvili and A. Meskhi

Corollary 3.3. Let 1 < p− ≤ q− ≤ q+ < ∞ with p+ < ∞. Suppose that the limit p(∞) := lim p(x) exists and equals to p− . Assume that p ∈ P∞ (R2+ ). x→∞

Suppose also that the weight function w satisﬁes the condition w(x, y) = p(·) q(·) w2 (x)w2 (y). If (3.3) holds, then H2 is bounded from Lw (R2 ) to Lv (R2 ). Let us now discuss the operator H2 on a bounded rectangle J := [0, a0 ]× [0, b0 ]. It is convenient to use the notation: 1 Jab := [a, a0 ] × [b, b0 ]. 0 Recall that by the symbol Jab we denote a rectangle [0, a) × [0, b). The arguments used in the proof of Theorem 3.1 enable us to formulate the next statement:

Theorem 3.2. Let 1 < p− (J) ≤ q− (J) ≤ q+ (J) < ∞ with p+(J) < ∞. Suppose that v and w are a. e. positive functions on J with w(x, y) = w1 (x)w2 (y) for some univariate weights w1 and w2 . If −1 1 q(·) 0 (p (J)) sup vχJab χJab < ∞, L (R2+ ) w L − (R2 ) +

0
p(·)

q(·)

then H2 is bounded from Lw (J) to Lv (J). Corollary 3.4. There is non-constant exponent p on [0, 2]2 such that the double average operator x y 1 (Af )(x, y) = f (t, τ )dtdτ xy 0

0

is bounded in Lp(·,·) ([0, 2]2 ). Proof. Let p be deﬁned as follows: 3, (x, y) ∈ [1, 2]2 ; p(x, y) = 2, (x, y) ∈ [0, 2]2 \[1, 2]2 . It is clear that p(0, 0) = p− = 2 and that 1

sup (xy)−1 χ[a,2]×[b,2] (x, y) Lp(·,·) (R2+ ) (ab) p (0,0) < ∞.

0
Theorem 3.2 completes the proof.

4. Two-weight estimates for strong fractional maximal functions Let D(Rn ) (or simply D) be the dyadic lattice in Rn . A weight function ρ is said to satisfy the dyadic reverse doubling condition (ρ ∈ RD(d) (R)) if for any two dyadic intervals I and I with I ⊂ I , |I| = |I2 | the inequality ρ(I ) ≤ bρ(I)

(4.1)

Two-weighted Norm Inequalities

115

holds with some constant b > 1. If (4.1) holds for arbitrary intervals I and I having a common end-point such that I ⊂ I and |I| = |I2 | , then we say that ρ satisﬁes the reverse doubling condition (ρ ∈ RD(R)). It is easy to check (see also [23]) that if ρ satisﬁes the doubling condition on R, then ρ ∈ RD(R) (hence, ρ ∈ RD(d) (R)). In order to establish two-weight estimates for strong fractional maximal function of variable order we need the following variant of the CarlesonH¨ ormander’s embedding theorem regarding the dyadic intervals. Theorem E ([24], [21], Lemma 3.10). Let p and q be constants satisfying the condition 1 < p < q < ∞ and let ρ be a weight function on R such that ρ1−p satisﬁes the dyadic reverse doubling condition. Let {cI } be a sequence of non-negative numbers corresponding to dyadic intervals I in R. Then the following two statements are equivalent: (i) There is a positive constant C such that ⎛ ⎞q ⎛ ⎞q/p

1 cI ⎝ g(x)dx⎠ ≤ C ⎝ g(x)p ρ(x)dx⎠ |I| I∈D(R)

R

I

for all non-negative g ∈ (ii) There is a positive constant C1 such that ⎛ ⎞−q/p cI ≤ C1 |I|q ⎝ ρ(x)1−p dx⎠ Lpρ (R);

I

for all I ∈ D. This result yields the following corollary. Corollary A. Let p and q be constants satisfying the condition 1 < p < q < ∞ and let ρ be a weight function on R such that ρ1−p satisﬁes the dyadic reverse doubling condition. Then the Carleson-H¨ ormander inequality ⎛ ⎞−q/p ⎛ ⎞q ⎛ ⎞q/p

⎝ ρ1−p (x)dx⎠ ⎝ f (x)dx⎠ ≤ c ⎝ f p (x)ρ(x)dx⎠ I∈D(R)

I

holds for all non-negative f ∈

R

I

Lpρ (R).

Let

S Mα(x),β(y) f (x, y) = sup |I|α(x)−1 |J|β(y)−1 |f (t, τ )|dtdτ, Ix Jy

(x, y) ∈ R2 ,

I×J

be the fractional maximal operator with variable parameters α and β, where α and β are measurable functions on R satisfying the conditions 0 < α− ≤ α+ < 1, 0 < β− ≤ β+ < 1, and the supremum is taken over all intervals I, J containing x,y respectively.

116

V. Kokilashvili and A. Meskhi In the following statements the notation p− := p− (R2 ), p+ := p+ (R2 ),

q− := q− (R2 ), q+ := q+ (R2 );

α− := α− (R), α+ := α+ (R), β− := β− (R), β+ := β+ (R) will be used. S Together with the operator Mα(·),β(·) we are interested in the dyadic strong fractional maximal operator S,(d) Mα(x),β(y) f (x, y) = sup |I|α(x)−1 |J|β(y)−1 |f (t, τ )|dtdτ, (4.2) Ix Jy I,J∈D(R)

I×J

(x, y) ∈ R2 . Feﬀerman–Stein type and trace inequalities. Formulation of results We start by the Feﬀerman–Stein type inequality. The original inequality for fractional maximal operator deﬁned on cubes in Lp spaces with constant p was derived by E.T. Sawyer. Theorem 4.1. Let 1 < p− ≤ p+ < q− ≤ q+ < ∞ and let p1− − q1+ < α− ≤ α+ < p1− , p1− − q1+ < β− ≤ β+ < p1− . Then there is a positive constant c such that S 0α(x),β(y)v) Lp(·,·) (R2 ) ,

(Mα(x),β(y) f )v Lq(·,·) (R2 ) ≤ c f (M where 0α(x),β(y) v)(x, y) := max{(M 0(1) 0(2) (M v)(x, y)(M v)(x, y)}, α(x),β(y) α(x),β(x) − p1

0 (M α(x),β(y) v)(x, y) := sup |I × J| (1)

−

Ix Jy

− p1

0 (M α(x),β(y)v)(x, y) := sup |I × J| (2)

+

v(x, y)|I|α(x) |J|β(y) Lq(x,y) (I×J) ,

v(x, y)|I|α(x) |J|β(y) Lq(x,y) (I×J) .

Ix Jy

Corollary 4.1. Let p be constant and let 1 < p < q− ≤ q+ < ∞. Suppose that 1 1 1 1 1 1 p − q+ < α− ≤ α+ < p and p − q− < β− ≤ β+ < p . Then the following inequality holds: S 0α(x),β(y)v) Lp (R2 ) .

(Mα(x),β(y) f )v Lq(·,·) (R2 ) ≤ c f (M

Corollary 4.2. (Trace inequality). Let 1 < p− ≤ p+ < q− ≤ q+ < ∞, p1− − 1 1 1 1 1 q+ < α− ≤ α+ < p− and p− − q+ < β− ≤ β+ < p− . Suppose that for the weight function v the condition sup |I|α(x) |J|β(y) v(x, y) Lq(·,·) (I×J) |I × J| I,J⊂R

holds, where pI×J

p− , = p+ ,

if |I||J| ≤ 1 . if |I||J| > 1

1 I×J

−p

< ∞,

Two-weighted Norm Inequalities S Then Mα(·),β(·) is bounded from Lp(·,·)(R2 ) to Lv

q(·,·)

117

(R2 ).

Theorem 4.2. (Criteria for the trace inequality). Let 1 < p < q− ≤ q+ < ∞. Suppose that p1 − q1+ < α− ≤ α+ < 1p and 1p − q1+ < β− ≤ β+ < 1p . Then S Mα(·),β(·) is bounded from Lp (R2 ) to Lv

q(·,·)

(R2 ) if and only if 1

sup |I|α(x) |J|β(y) v(x, y) Lq(·,·) (I×J) |I × J|− p < ∞.

I,J ⊂R

Theorem 4.3. Let p be constant and let 1 < p < q− ≤ q+ < ∞. Suppose that 0 < α− ≤ α+ < 1 and 0 < β− ≤ β+ < 1. Let v and w be weight functions in S R2 and let w be of product type, i.e. w(x, y) = w1 (x)w2 (y). Then Ma(·),β(·) is bounded from Lpw (R2 ) to Lv

q(·,·)

(R2 ) if and only if

sup (|I||J|)−1 v(x, y)|I|α(x) |J|β(y) Lq(·,·) (I×J) w−1 Lp (·,·) (I×J) < +∞,

I,J⊂R2

provided that w1−p , w2−p ∈ RD(R). Corollary 4.3. Let p, q, α, β satisfy the conditions 1 < p− < q− ≤ q+ < ∞, p+ < ∞, 0 < α− ≤ α+ < 1 and 0 < β− ≤ β+ < 1. Suppose that p ∈ P(R2 ). Assume also that v and w are weight functions on R2 and that w(x, y) = −(p ) −(p ) w1 (x)w2 (y) with w1 − , w2 − ∈ RD(R). If the condition sup (|I||J|)−1 v(x, y)|I|α(x) |J|β(y) Lq(·,·) (I×J) w−1 L(p− ) (I×J) < +∞

I,J⊂R2

(4.3) p(·,·)

S holds, then Mα(·),β(·) is bounded from Lw

(R2 ) to Lv

q(·,·)

(R2 ).

Corollary 4.4. Let 1 < p− < q− ≤ q+ < ∞, p+ < ∞, 0 < α− ≤ α+ < 1 and let 0 < β− ≤ β+ < 1. Suppose that p(∞) := lim p(x) exists and is equal x→∞

to p− . Let p ∈ P∞ (R2 ). Assume that v and w are weights on R2 and that −(p ) −(p ) w(x, y) = w1 (x)w2 (y) with w1 − , w2 − ∈ RD(R). Then condition (4.3) p(·,·) q(·,·) S guarantees the boundedness of Mα(·),β(·) from Lw (R2 ) to Lv (R2 ). Proofs of the results Proof of Theorem 4.1. Recall that the dyadic strong fractional maximal opS,(d) erator Mα(·),β(·) is deﬁned by (4.2). Without loss of generality we can assume that f ≥ 0 and f is bounded with compact support. It is obvious that for (x, y) ∈ R2 , there are dyadic intervals I x, J y such that 2 S,(d) |f (t, τ )|dtdτ > (Mα(x),β(y) f )(x, y). (4.4) |I|1−α(x) |J|1−β(y) I×J Let us introduce the set FI,J = {(x, y) ∈ R2 : x ∈ I, y ∈ J and (4.4) holds for I and J}.

118

V. Kokilashvili and A. Meskhi

Observe that R2 = ∪I,J∈D(R) FI,J and FI,J ⊂ I × J (It may happen that FI1 ,J1 ∩ FI2 ,J2 = ∅ for some diﬀerent couples of dyadic intervals (I1 , J1 ), (I2 , J2 ).) Let us take a number r so that p+ < r < q− . Then we have 1 2r S,(d) r S,(d) v Mα(x),β(y)f = v M f α(·),β(·) Lq(·,·) (R2 ) Lq(·,·)/r (R2 ) 1 2r S,(d) ≤c sup h v Mα(·),β(·)f . h

≤1 L(q(·,·)/r) (R2 )

R2

0 Lp(·,·) (R2 ) ≤ 1. Further, using the arguments obSuppose that f Mv served above, we ﬁnd that for h such that h L(q(·,·)/r) (R2 ) ≤ 1,

R2

r S,(d) h v Mα(·),β(·) f ≤

≤c

FI,J

I,J ∈D(R)

r S,(d) h v Mα(·),β(·) f

v r (x, y)(|I|α(x) |J|β(y) )r h(x, y)dxdy I×J

I,J ∈D(R)

r 1 |f (t, τ )|dtdτ |I||J| I×J (v(·, ·)|I|α(·) |J|β(·) )r q(·,·)

× ≤c

(I×J )

r

L

I,J ∈D(R)

h q(·,·) L

r

1 |f (t, τ )|dtdτ |I||J| I×J (v(·, ·)|I|α(·) |J|β(·) )r

×

r

(I×J)

r 1 |f (t, τ )|dtdτ (I×J ) |I||J| I×J I,J ∈D(R) r 1 α(·) β(·) r ≤c (v(·, ·)|I| |J| ) q(·,·) |f1 (t, τ )|dtdτ L r (I×J) |I||J| I×J I,J ∈D(R) r 1 + (v(·, ·)|I|α(·) |J|β(·) )r q(·,·) |f2 (t, τ )|dtdτ L r (I×J ) |I||J| I×J ≤c

q(·,·) L r

I,J ∈D(R)

=: c[S1 + S2 ],

where f1 = f χ{f M α(·),β(·) v≥1} , f2 = f − f1 . Now we estimate S1 and S2 separately. By using Corollary A (with ρ ≡ 1) and Minkowski’s inequality we have that S1 =

(|I||J|)

−

r (p− )

|f1 |(|I||J|) I×J

I,J ∈D(R)

≤

|I|

− r (p− )

|I|

I∈D(R)

−

−

|I|

r (p− )

I

I

r (p− )

J

R

I∈D(R)

≤c

|J|

J ∈D(R)

I∈D(R)

≤c

− r (p− )

R

I

− p1

−

v(·, ·)|I|α(·) |J|β(·) Lq(·,·) (I×J)

r (1) |f1 | M v α(·),β(·)

p− r p− (1) |f1 |[Mα(·),β(·) v] p− (1) |f1 |p− [M α(·),β(·) v]

1 p−

r

r

Two-weighted Norm Inequalities ≤c ≤c

R2

R2

(1) p |f1 |p− M v − α(·),β(·)

119

r p−

α(x),β(y)v)(x, y) f (x, y)(M

p(x,y)

dxdy

r p−

≤ c.

By the similar arguments we can see that pr + (2) p(x,y) 0 S2 ≤ c [f (x, y)(M dxdy ≤ c. α(x),β(y) v)(x, y)] R2

Thus, we established the desired estimate for the dyadic fractional maximal function. S,(d) S Now we pass from Mα(·),β(·) to Mα(·),β(·) . The following inequality for constant α and β was proved in [8] but it is true also for variable α and β: Cα,β S,(2k ) Ma(x),β(y)f (x, y) ≤ St,τ (x, y)dtdτ, (4.5) |R(0, 2k+2 )|2 R(0,2k+2 )2 where S,(2k ) Mα(x),β(y)f (x, y) :=

St,τ (x, y) :=

sup I−tx J−τ y I,J ∈D(R)

sup

|I|α(x)−1 |J|β(y)−1

I×J(x,y) |I|,|J|≤2k

|f (t, τ )|dtdτ, I×J

|I|α(x)−1 |J|β(y)−1

(I−t)×(J−τ )

|f (t, τ )|dtdτ,

R(0, r) := {t : −r ≤ t ≤ r}. Indeed, let j ∈ Z and let I be an interval such that 2j−1 < |I| ≤ 2j . Let j ≤ k, k ∈ Z. Suppose that E is the set of those t ∈ R(0, 2k+2 ) for which there is some I1 ∈ D(R) − t with |I1 | = 2j+1 and such that I ⊂ I1 , where D(R) − t := {I − t : I ∈ D(R)}. Then (see, e.g., [7], p. 431) |E| ≥ 2k+2 . By similar arguments, for another interval J ⊂ R, there is i ∈ Z such that 2i−1 < |J| ≤ 2i . Then for i ≤ k, k ∈ Z, we have that the set F of those t ∈ R(0, 2k+2 ) for which there is J1 ∈ D(R) − t such that |J1 | = 2i+1 and J ⊂ J1 has measure greater than or equal to 2k+2 . To prove (4.5) observe that for (x, y) ∈ R2 , there are intervals Q1 and Q2 such that Q1 x, Q2 y, |Q1 |, |Q2 | ≤ 2k and 2 S,(2k ) |f (t, τ )|dtdτ > (Mα(x),β(y)f )(x, y). |Q1 |1−α(x) |Q2 |1−β(y) Q1 ×Q2 Let j and i be integers such that 2j−1 ≤ |Q1 | ≤ 2j ,

2i−1 ≤ |Q2 | ≤ 2i .

It is obvious that j, i ≤ k. Let us deﬁne the following sets: E1 :={t ∈ R(0, 2k+2 ) : ∃I ∈ D(R) − t, |I| = 2j+1 , Q1 ⊂ I} F1 :={t ∈ R(0, 2k+2 ) : ∃J ∈ D(R) − t, |J| = 2i+1 , Q2 ⊂ J}.

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Then applying the arguments observed above we ﬁnd that for x ∈ Q1 ⊂ I, y ∈ Q2 ⊂ J, 1 1 S,(2k ) (Mα(x),β(y)f )(x, y) ≤ |f (t, τ )|dtdτ 2 |Q1 |1−α(x) |Q2 |1−β(y) Q1 ×Q2 cα,β ≤ 1−α(x) 1−β(y) |f (t, τ )|dtdτ ≤ cα,β St,τ (x, y) |I| |J| I×J because I ∈ D(R)−t, J ∈ D(R)−t, I x, J y. Since |E1 |, |F1 | ≥ |R(0,22 we have that c S,(2k ) (Mα(x),β(y) f )(x, y) ≤ St,τ (x, y)dtdτ |E1 × F1 | E1 ×F1 c ≤ St,τ (x, y)dtdτ. |R(0, 2k+2 )|2 R(0,2k+2 )2

k+2

)|

,

Inequality (4.5) is proved. Further, it is easy to see that (q) Dt,τ := (St,τ (x, y))q(x,y) v(x, y)q(x,y) dxdy R2 q(x−t,y−τ ) S,(d) = Mα(x−t),β(y−τ )f (· − t, · − τ ) R2

× v q(x−t,y−τ ) (x − t, y − τ )dxdy. Observe now that 1 1 − < (α(· − t, · − τ ))− (p(· − t, · − τ ))− (q(· − t, · − τ ))+ 1 ≤ (α(· − t, · − τ ))+ < . (p(· − t, · − τ ))− S,(d)

Taking now into account the two-weight result for Mα(·),β(·) proved above we (q)

ﬁnd that Dt,τ ≤ c because 0α(x−t),β(y−τ ) |f (x − t, t − τ )|p(x−t,y−τ ) M R2

p(x−t,y−τ ) × v(· − t, · − τ ) (x, y)dxdy 0α(x),β(y)v)p(x,y) (x, y)dxdy ≤ 1. = |f (x, y)|p(x,y) (M R2

This means that 0α(·),β(·)v) Lp(·,·) (R2 ) .

St,τ (·, ·)v(·, ·) Lq(·,·) (R2 ) ≤ c f (M This inequality together with (4.5) gives for h such that h Lq (·,·) (R2 ) ≤ 1, S,(2k ) (Mα(x),β(y)f )(x, y)v(x, y)h(x, y)dxdy 2 R 4 3 ≤ c|R(0, 2k+2 )|−2 St,τ (x, y)dtdτ v(x, y)h(x, y)dxdy R2

R(0,2k+2 )

Two-weighted Norm Inequalities = c|R(0, 2

k+2

)|

−2

121

St,τ (x, y)v(x, y)h(x, y)dxdy dtdτ 0α(·),β(·)v(·, ·)) Lp(·,·) (R2 ) |R(0, 2k+2 )|−2 ≤ c f (M dtdτ R(0,2k+2 )2

R2

R(0,2k+2 )2

0α(·),β(·)v(·, ·)) p(·,·) 2 . = c f (M L (R ) Passing now by k to the inﬁnity and taking the supremum with respect to h in the last inequality, we get the desired result. Proof of Corollary 4.2. This proposition will be proved if we show that 0α(x),β(y)v)(x, y) ≤ c in Theorem 4.1. Indeed, if the condition (M − 1 A := sup |I|α(x) |J|β(y)v(x, y) Lq(·,·) (I×J) |I||J| pI×J < ∞ I,J⊂R

is satisﬁed, then − 1

|I|α(x)|J|β(y) v(x, y) Lq(·,·) (I×J) |I||J| p+ ≤ A < ∞ and

− 1

|I|α(x) |J|β(y)v(x, y) Lq(·,·) (I×J) |I||J| p− ≤ A < ∞.

S,(d) Mα(·),β(·)

Proof of Theorem 4.3. Let us recall that is the dyadic strong fractional maximal operator deﬁned by (4.2). Suﬃciency. We use the notation of the proof of Theorem 4.1. First we construct the sets FI×J . Take r so that p < r < q− and observe that S,(d) S,(d)

v(Mα(·),β(·)f ) Lq(·,·) (R2 ) ≤ c sup h[vMα(·),β(·)f ]r . h

≤1 L(q(·,·)/r) (R2 )

R2

Let f Lpw (R2 ) ≤ 1. Then for h such that h L(q(·,·)/r) (R2 ) ≤ 1, we ﬁnd that

S := R2

≤c

S,(d) h[vMα(·),β(·)f ]r

I,J∈D(R)

≤

I,J∈D(R)

S,(d)

FI,J

h[vMα(·),β(·)f ]r

vr (x, y)(|I||α(x) |J|β(y) )r h(x, y)dxdy I×J

r 1 × |f (t, τ )|dtdτ |I||J| I×J

≤c

(v(x, y)|I|α(x) |J|β(y) )r Lq(x,y)/r (I×J) h L(q(·,·)/r) (I×J) I,J∈D(R)

r 1 × |f (t, τ )|dtdτ |I||J| I×J

=c

v(x, y)|I|α(x) |J|β(y) rLq(x,y) (I×J)

I,J∈D(R)

1 |I||J|

|f (t, τ )|dtdτ I×J

r

122 ≤c

V. Kokilashvili and A. Meskhi

I,J∈D(R)

w1−p

− r p

I

w2−p

− r p

J

r |f (t, τ )|dtdτ

.

I×J

Applying Corollary A (with ρ ≡ 1) we derive the following estimates: r p pr

−p − p S≤c w2 w1 (t) |f (t, τ )|dτ dt J∈D(R)

≤c

J∈D(R)

R

J

w2−p

p

J

J

≤c

R

w1p (t)|f (t, τ )|p dt

p1

r dτ

r/p |f (t, τ )| w (t, τ )dtdτ p

R2

J

− r

p

≤ c.

Thus, we established the desired inequality for the dyadic fractional maximal function. S Now we can pass to the fractional maximal function Mα(·),β(·) in the same manner as in the proof of Theorem 4.1. Details are omitted. Necessity follows easily by taking appropriate test functions in the twoweight inequality. We omit the details. Proof of Corollary 4.3. The proof is a direct consequence of Theorem 4.3 and the fact that the condition p ∈ P(R2 ) implies the inequality (see, e.g., [3])

f w Lp− (R2 ) ≤ c f w Lp(·) (R2 ) .

Corollary 4.4 follows from Corollary 4.3 and the following fact: p ∈ P∞ (R2 ) =⇒ p ∈ P(R2 ) provided that p(∞) exists and p− = p(∞). Acknowledgements The authors were partially supported by the Georgian National Science Foundation Grant (project number: No. GNSF/ST09 23 3-100). The authors express their gratitude to the referee for valuable remarks and suggestions. They are also thankful to Prof. L. Ephremidze for helpful discussions regarding Theorem 2.1.

References [1] C. Capone, D. Cruz-Uribe SFO and A. Fiorenza, The fractional maximal operator on variable Lp spaces, Revista Mat. Iberoamericana 3(23) (2007), 747–770. [2] D. Cruz-Uribe, SFO, A. Fiorenza and C.J. Neugebauer, The maximal function on variable Lp spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223–238, and 29 (2004), 247–249. [3] L. Diening, Maximal function on generalized Lebesgue spaces Lp(·) . Math. Inequal. Appl. 7(2) (2004), 245–253. [4] L. Diening and S. Samko, Hardy inequality in variable exponent Lebesgue spaces. Frac. Calc. Appl. Anal. 10 (2007), 1–18.

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[5] D. E. Edmunds, V. Kokilashvili, and A. Meskhi, On the boundedness and compactness of the weighted Hardy operators in Lp(x) spaces. Georgian Math. J. 12 (2005), 27–44. [6] D. E. Edmunds, J. Lang and A. Nekivnda, On Lp(x) norms. Proc. R. Soc. London, A 455 (1999), 219–225. [7] J. Garc´ıa-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116, Mathematical Notes, 104, North-Holland Publishing Co., Amsterdam, 1985. [8] V. Kokilashvili and A. Meskhi, Two-weight estimates for strong fractional maximal functions and potentials with multiple kernels. J. Korean Math. Soc. 46 (2009), No. 3, 523-550 [9] V. Kokilashvili and A. Meskhi, Two-weight iequalities for fractional maximal functions and singular integrals in Lp(·) spaces. J. Math. Sci. 173(2011), No. 6, 658–673. [10] V. Kokilashvili and A. Meskhi, Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent. Integr. Trans. Spec. Func. 18 (2007), 609–628. [11] V. Kokilashvili, A. Meskhi and L. E. Persson, Weighted norm inequalities for integral transforms with product kernels. Nova Science Publishers, New York, 2009 [12] V. Kokilashvili and S. Samko, Maximal and fractional operators in weighted Lp(x) spaces. Rev. Mat. Iberoamericana 20 (2004), 493–515. [13] T. S. Kopaliani, On some structural properties of Banach function spaces and boundedness of certain integral operators. Czechoslovak Math. J. 54(129) (2004), 791-805. [14] T. S. Kopaliani, Littlewood–Paley theorems on spaces Lp(t) (Rn ). Ukrainian Mathematical Journal 60, No. 12, 2008. [15] O. Kov´ aˇcik and J. R´ akosn´ık, On spaces Lp(x) and W k,p(x) . Czechoslovak Math. J. 41(4) (1991), 592–618. [16] A. Kufner and L.-E. Persson, Weighted inequalities of Hardy type. World Scientiﬁc Publishing Co, Singapore, New Jersey, London, Hong Kong, 2003. [17] V. G. Maz’ya, Sobolev spaces. Springer, Berlin, 1985. [18] A. Meskhi, A note on two-weight inequalities for multiple Hardy-type operators. J. Funct. Spaces Appl. 3 (2005), 223–237. [19] S. Samko, Convolution type operators in Lp(x) . Integral Transf. Spec. Funct. 7 (1998), No. 1–2, 123–144. [20] E. Sawyer, Weighted inequalities for two-dimensional Hardy operator. Studia Math. 82 (1985), No. 1, 1–16. [21] E. Sawyer and R. L. Wheeden, Carleson conditions for the Poisson integral. Indiana Univ. Math. J. 40 (1991), No. 2, 639–676. [22] I. I. Sharapudinov, On a topology of the space Lp(t) ([0, 1]). Mat. Zametki 26 (1979), 613–632. [23] J. O. Str¨ omberg and A. Torchinsky, Weighted Hardy spaces. Lecture Notes in Math., 1381, Springer Verlag, Berlin, 1989. [24] K. Tachizava, On weighted dyadic Carleson’s inequalities. J. Inequal. Appl. 6 (2001), No. 4, 415–433.

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Vakhtang Kokilashvili A. Razmadze Mathematical Institute I. Javakhishvili Tbilisi State University 2. University Str. 0186 Tbilisi Georgia Second Address: Faculty of Exact and Natural Sciences I. Javakhishvili Tbilisi State University 2. University Str. Tbilisi, 0143 Georgia e-mail: [email protected] Alexander Meskhi A. Razmadze Mathematical Institute I. Javakhishvili Tbilisi State University 2. University Str. 0186 Tbilisi Georgia Second Address: Department of Mathematics Faculty of Informatics and Control Systems Georgian Technical University 77, Kostava St. Tbilisi Georgia e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 125–137 c 2012 Springer Basel AG

Modular Eigenvalues of the Dirichlet p(·)-Laplacian and Their Stability Jan Lang and Osvaldo M´endez This paper is dedicated to David Edmunds on his 80th birthday and to Desmond Evans on his 70th birthday. Abstract. The concept of the modular ﬁrst Dirichlet eigenvalue for the p(·)-Laplacian is introduced as a generalization of the constant case. An important property of the corresponding eigenfunctions is obtained. We prove a qualitative stability result for such eigenvalues in terms of the magnitude of the perturbation of the variable modular exponent p(·). Mathematics Subject Classiﬁcation (2010). Primary 35J60; Secondary 15A18. Keywords. Variable exponent p-Laplacian, Sobolev embedding, stability of eigenvalues, modular spaces.

1. Introduction Etwas allgemein machen heisst, es denken. G.W.F. Hegel [9]. The eigenvalues of Laplace-type operators depend mainly on the domain, the structure of the particular operator and the underlying function space. An enormous amount of work has been devoted to the analysis of stability of the eigenvalues with respect to perturbations of the domain and the inner structure of the operator under consideration ([1], [10], [12] and the references therein). On the other hand, the eﬀect on the eigenvalues of perturbations of the underlying function space does not seem to have been studied or to be well understood. In the present article we aim at opening this direction of investigation. We aim at studying the eigenvalue problem for the generalized p(·)-Laplacian −Δp(.) u = λ|u|p(·)−2 u, (1.1) where Δp(.) u := div(|∇u|p(·)−2 ∇u).

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To situate our line of investigation in perspective, we organize our exposition by including a brief heuristic digression in Section 2. In Section 3 we present the variable exponent function spaces Lp(·) , W 1,p(·) and the necessary mathematical framework to introduce our problem. In Section 4 the concept of modular eigenvalues and corresponding eigenfunctions as extremal functions are introduced. In Section 5 the stability of the eigenvalues with respect to perturbations of the exponent is considered.

2. Preliminaries Eigenvalue problems for the classical Laplacian in L2 , namely −Δu = λu

(2.1)

are amenable to the utilization of variational techniques due to the linearity of the operator and the Hilbert space structure of the underlying space (L2 ). For 1 < p < ∞, the analysis of Lp -eigenvalue problem −Δp (u) = λ|u|p−2 u,

(2.2)

where Δp (u) := div(|∇u|p−2 ∇u), which obviously corresponds to a natural generalization of the left-hand side of (2.1), requires far more sensitive, non-linear analytic tools. We remind the reader that the ﬁrst eigenvalue λ1 of the above problem is also the minimizer of the Rayleigh quotient, namely λ1 = inf

∇u=0

∇u pp ;

u pp

1 p

thus 1/(λ1 ) is the best constant in Poincar´e’s inequality in W01,p (Ω)

u Lp(Ω) ≤ C ∇u Lp (Ω) . Equality (2.2) is the Euler equation for the above minimizing problem obtained via Gateˆaux diﬀerentiation (see [2]). We refer the interested reader to [4], [14] and [15] for a survey on this and related problems. It is certainly not surprising that due to the non-linearity of the operator Δp for p = 2, a number of important questions remain unanswered for the problem (2.2) in this case. In particular, if Ω = [0, 1]n the linear structure of the 2-Laplacian is suitable for the use of the method of separation of variables which allows a full characterization of the eigenvalues. No such characterization is known if p = 2 for n > 1. We refer the interested reader to [4] for a complete description of the eigenfunctions when n = 1. Since the introduction in 1991 of the Lp(·) spaces with variable exponent p(·) (see [11]), it seems natural to consider the eigenvalue problem (2.2) as a particular case of a family of eigenvalue problems associated to a generalized p(·)- Laplacian deﬁned on variable-exponent spaces. It is via this interpretation that (1.1) is introduced. The mere replacement of the constant p with a suitable function in (1.1) is, in some subtle sense, na¨ıve, since it does not

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127

lead to the most natural generalization of the eigenvalue problem for the p-Laplacian, as we shall discuss in detail in a forthcoming paper.

3. Modulars and generalized Lebesgue spaces Throughout this paper we will consider a bounded, Lipschitz domain Ω ⊆ Rn . For a measurable function p : Ω → Rn , we set p∗ = essinfΩ p(x) , p∗ = esssupΩ p(x) and consider the family of admissible exponent functions P(Ω) = {p : Ω →, R , p Borel-measurable, 1 < p∗ ≤ p∗ < ∞} .

(3.1)

To ensure the validity of the Sobolev embedding theorems needed in the sequel we introduce the class P log (Ω). Deﬁnition 3.1. A function v ∈ P(Ω) is said to be globally H¨ older continuous, or v ∈ P log (Ω) if there exist v∞ ∈ R and positive constants α, β such that for any x, y ∈ Ω, it holds that α |v(x) − v(y)| ≤ log (e + |x − y|−1 ) and |v(x) − v∞ | ≤

β . log (e + |x|)

In what follows, the class of exponents p(·) will be restricted to P log (Ω). In particular, this condition on the variable exponent is suﬃcient to guarantee the compactness of the Sobolev embedding (see [3], Chapter 3) 1,p(·)

E : W0

(Ω) → Lp(·) (Ω).

On the set Lp(·) of all real-valued, Borel measurable functions on Ω for which ρp(·) (f ) := |f (x)|p(x) dx < ∞, (3.2) Ω

the function ρp(·) deﬁnes a convex monotone modular and u

u Lp(·) (Ω) = inf λ > 0 : ρp(·) ≤1 λ

(3.3)

deﬁnes a norm endowed with which Lp(·) (Ω) turns into a reﬂexive, uniformly convex Banach space. We note that the latter coincides with the usual Lebesgue Lp (Ω) norm when p is constant; accordingly the scale Lp(·) (Ω) for p(·) as in (3.1) will be referred to as the variable-exponent Lebesgue scale in Ω. The variable-exponent Sobolev scale in Ω can be deﬁned analogously as W 1,p(·) (Ω) = u ∈ Lp(·)(Ω) : |∇u| ∈ Lp(·) (Ω) , endowed with the norm

u W 1,p(·) (Ω) = u Lp(·) (Ω) + ∇u W 1,p(·) (Ω) .

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In the sequel the reference to the domain will be dropped. The closure of 1,p(·) C0∞ (Ω) in W 1,p(·) (Ω) will be denoted by W0 (Ω) and will be furnished with the norm

u W 1,p(·) (Ω) = ∇u Lp(·) (Ω) . 0

The reader is referred to [3] and [11] for an exhaustive treatment of the spaces of variable-exponent Lebesgue-Sobolev spaces; the results therein will be used extensively, frequently without explicit notice, throughout this work.

4. Eigenvalue Problems for the p(·)-Laplacian In analogy with the classical (p=constant) case, the p(·)-Laplacian operator Δp(·) is deﬁned by ∗ 1,p(·) 1,p(·) Δp(·) : W0 (Ω) → W0 (Ω) (4.1) Δp(·) u, v = − |∇u|p(x)−2 ∇u∇v. Ω

In particular, it is well known that Δp(·) is the (Gateˆ aux) derivative of the functional 1 1,p(·) J : W0 (Ω) → R , J(u) = |∇u|p(x) . p(x) Ω For future reference we introduce 1 I(u) = |u|p(x) , Ω p(x) note that its (Gateˆaux) derivative is the functional I (u), v = |u|p(x)−2 uv. Ω

A real number λ is said to be an eigenvalue for the p(·)-Laplacian iﬀ there 1,p(·) exists 0 = u ∈ W0 (Ω) such that −Δp(·) u = λ|u|p(x)−2 u, from which it is immediate that

|∇u|p(x) λ = Ω . |u|p(x) Ω

We highlight the fact that if u ˜ is a critical point of |∇u|p(x) /p(x)dx J(u) = Ω I(u) |u|p(x)/p(x)dx Ω

(4.2)

(4.3)

then u ˜ is a solution of the above eigenvalue problem. It follows easily that for constant p, the “global” inﬁmum of the ratios (4.2) for u = 0 (which is the reciprocal of the best constant in Poincare’s inequality, hence, positive) coincides with the inﬁmum of (4.2) on any closed ball ρp(·) (∇u) ≤ r; due to the lack of homogeneity of the modular, this is no longer the case for a variable exponent. As a matter of fact, the following example borrowed from

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129

[3], shows that the inﬁmum of the Rayleigh quotient (4.2) in general cannot be expected to be positive on any open set containing the origin: Fix > 0, let p = 3 on (−2, −1) ∪ (1, 2), p = 2 on (− 12 , 12 ) and set u = on (−1, 1) with |u | = on (−2, −1) ∪ (1, 2). Then 2 |u (x)|p(x) dx 23 −2 ≤ 2 = 2. 2 |u(x)|p(x) dx −2

In fact, the global inﬁmum of (4.2) and of (4.3) vanishes for any continuous function p(·) with a strict local minimum or maximum in Ω (see [6]). In particular, Poincare’s inequality does not hold in modular form. An elementary calculation shows that the natural generalization of the inﬁmum of the Rayleigh quotient (4.2) in the modular setting is given by r λp(·),r := inf (4.4) ρp(·) (∇u)≤r |u(x)|p(x) Ω for each r > 0; as for the quotient (4.3) of normalized modulars, the natural deﬁnition is r ˜ p(·),r := λ inf (4.5) ρp(·) (∇u/p(.))≤r |u(x)|p(x) /p(x) Ω for each r > 0; obviously (4.4) and (4.5) coincide when p(·) is a constant and in that case, the inﬁmum is independent of the choice of r. We refer the reader to [6] for the connection of this modular eigenvalues and the variational eigenvalues obtained by the application of the Ljusternik– Schnirelmann theory. As we will show in Theorem 4.2, the inﬁmum (4.4) is strictly positive on modular balls and is attained in the modular sphere ρp(·) (∇u) = r. In what follows we study the stability of these generalized eigenvalues (4.4) and (4.5) with respect to the variable modular exponent p. A technical Lemma will shed light on the proof of the next Theorem. Lemma 4.1. For any w ∈ Lp(·) (Ω) one has the following inequalities: 1 " 1 " 1 1 p∗ p∗ p∗ p∗ min ρp(·) (w), ρp(·) (w) ≤ w p(·) ≤ max ρp(·) (w), ρp(·) (w) .

(4.6)

Proof. A closer look at Deﬁnition (3.3) reveals that for w = 0 as above and any suﬃciently small μ > 0 w ρp(·) > 1, (4.7)

w p(·) − μ which immediately yields ρp(·)

w

w p(·)

≥

w p(·) − μ

w p(·)

p∗ ,

on the other hand, if w p(·) ≤ α < w p(·) + μ with ρp(·) p∗ w α ρp(·) ≤ ≤ 1.

w p(·)

w p(·)

(4.8) w α

≤ 1, one has (4.9)

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J. Lang and O. M´endez

From (4.8) and (4.9),

ρp(·)

w

w p(·)

= 1.

(4.10)

Equality (4.10) is well known; we have included a short proof for the sake of clarity in the ﬂow of ideas. If w p(·) ≤ 1, equality (4.10) yields p∗ p∗ 1 w 1 ρp(·) (w) ≤ ρp(·) =1≤ ρp(·)(w);

w p(·)

w p(·)

w p(·) (4.11) whereas for w p(·) > 1 p∗ p∗ 1 1 ρp(·) (w) ≤ 1 ≤ ρp(·) (w). (4.12)

w p(·)

w p(·) As is apparent from Deﬁnition (3.3) and equality (4.10), ρp(·) ≤ 1 ⇐⇒

w p(·) ≤ 1, hence ρp(·) > 1 ⇐⇒ w p(·) > 1. These observations coupled with inequalities (4.11) and (4.12) readily yield (4.6). 1,p(·)

Theorem 4.2. For each r > 0, there exists up ∈ W0 (Ω) with ρp(·) (∇up ) = r such that −1 −1 1 p(x) p(x) |up | = inf |u| = λp(·),r . (4.13) ρ (∇u)≤r r p(·) Ω Ω |∇vp |p(x) Analogously, there exists vp with Ω p(x) = r for which Ω

|vp |p(x) p(x)

−1

=

inf |∇u|p(x) p(x) Ω

≤r

Ω

|u|p(x) p(x)

−1 =

1˜ λp(·),r . r

(4.14)

Proof. For the proof of (4.13) we refer to inequalities (4.6) to obtain, for any 1,p(·) u ∈ W0 (Ω), 1

1

ρp(·) (∇u) ≤ r ⇒ ∇u p(·) ≤ max{r p∗ , r p∗ } = M. The boundedness of Sobolev’s embedding 1,p(·)

E : W0

(Ω) → Lp(·) (Ω)

(4.15)

and (4.6) imply that for ρp(·) (∇u) ≤ r it holds that p∗ p∗ ρp(·) (u) ≤ E

∇u p(·) ≤ E p∗ max{r, r p∗ }

(4.16)

if ρp(·) (u) ≤ 1 and its counterpart for ρp(·) (u) ≤ 1, p∗ p∗ ∗ ρp(·) (u) ≤ E

∇u p(·) ≤ E p max{r, r p∗ }.

(4.17)

Hence, for r > 0 and u ∈ Lp(·) (Ω) with ρp(·) (∇u) ≤ r, one has the estimate ∗

∗

p∗

p∗

ρp(·) (u) ≤ max{ E p r, E p r p∗ , E p∗ r p∗ , E p∗ r} = M1 .

(4.18)

Modular Eigenvalues

131

An immediate consequence of equality (4.18) is the strict positivity of the eigenvalues deﬁned in (4.4). Indeed, since ρp(·) (1) = |Ω|, it is readily concluded that for ﬁxed r > 0, 0<

sup ρp(·) (∇u)≤r

Observing that supρp(·) (∇u)≤r

ρp(·) (u) M1 ≤ < ∞. r r

ρp(·) (u) r

−1

≤

r ρp(·) (u)

(4.19)

for all u in the modular

ball ρp(·) (∇u) ≤ r, one readily concludes −1 ρp(·) (u) sup ≤ λp(·),r . r ρp(·) (∇u)≤r

(4.20)

On the other hand, since λp(·),r has been shown to be positive, one has, for each r > 0 and ρp(·) (∇u) ≤ r ρp(·) (u) 1 ≤ . r λp(·),r

(4.21)

In the light of estimates (4.20, 4.21) it is concluded that −1 −1 |u|p(x) r Ω λp(·),r = inf = sup < ∞. r ρp(·) (∇u)≤r |u|p(x) ρp(·) (∇u)≤r Ω 1,p(·)

By virtue of the reﬂexivity of W0 (Ω) and the compactness of E it follows 1,p(·) that any maximizing sequence (un )n ⊆ W0 (Ω) with ρp(·) (∇un ) ≤ r has a convergent subsequence (still denoted by (un )n ) strongly convergent in ∗ 1,p(·) Lp(·) (Ω) to up ∈ W0 (Ω), satisfying ρp(·) (∇up ) ≤ max{M p , M p∗ }. The equality ρp(·) (un ) = ρp(·) tt−1 (un − up ) + (1 − t)(1 − t)−1 up , (4.22) valid for arbitrary t ∈ (0, 1), together with he convexity of the modular ρp(·) yield ∗ ∗ |ρp(·) (un ) − ρp(·) (up )| ≤ t1−p ρp(·) (un − up ) + (1 − t)1−p − 1 ρp(·) (up ) . (4.23) Since un → up in Lp(·) (Ω) as n → ∞, by virtue of the inequalities (4.6) the ﬁrst term in the right-hand side of (4.23) tends to 0 as n → ∞, whence the arbitrariness of t yields ρp(·) (up ) = sup |u|p(x) . (4.24) ρp(·) (∇u)≤r

Ω

Analogously, ∗

|ρp(·) (∇up ) − ρp(·) (∇un )| ≤ t1−p ρp(·) (∇up − ∇un ) ∗ + (1 − t)1−p − 1 ρp(·) (∇un ),

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from which it follows that ρp(·) (∇up(·) ) ≤ r. In fact the maximizing function up satisﬁes the condition ρp(·) (∇up ) = r, (4.25) which is reminiscent to strong maximum principle in the sense that the maximum of the functional ρp(·) on the modular ball ρp(·) (∇u) ≤ r is attained on the modular sphere ρp(·) (∇u) = r. This last assertion follows by observing that for any t > 1 it holds that ρp(·) (tup ) ≥ tp∗ ρp(·) (up ) > sup |u|p(x) , (4.26) ρp(·) (∇u)≤r

hence, the inequality

Ω

ρp(·) (∇up ) =

would imply (with t =

r ρp(·) (∇up )

Ω

p1∗

|∇up |p(x) dx < r

) ∗

ρp(·) (∇tup ) ≤ tp ρp(·) (∇up ) ≤ r which together with (4.26) contradicts (4.24); in all: ρp(·) (∇up ) = r as claimed. The proof of (4.14) is similar and will be omitted. The function up obtained in the above proof will be referred to as “the” extremal function in the modular ball of radius r (of course no uniqueness claim is implied by the use of this terminology) and will be denoted by up (r). A remark at this point is in order to highlight the contrast with the classical case. Elementary arguments show that when p is constant and u1 is any extremal function on the modular ball of radius, 1, then ur = ru1

(4.27)

is an extremal function on the modular ball of radius r. No such relation is expected among extremal functions for non-constant p. In fact, for a large class of variable exponents p(·) no multiple of an extremal function on the modular ball of radius r can be extremal for a modular ball of radius s if r = s. 1,p(·)

Lemma 4.3. Let p ∈ P(Ω) be weakly diﬀerentiable and u ∈ W0 (Ω) be an extremal function for the modular ρp(·) in the modular ball of radius r, ρp(·) (∇u) ≤ r, i.e., corresponding to the eigenvalue (4.4). Then no scalar multiple of u, αu, α = 1 can be extremal unless p is constant on Ω. 1,p(·) Similarly, if u ∈ W0 (Ω) is extremal corresponding to the eigenvalues (4.5), i.e., |u|p(x) |w|p(x) = sup , |∇w|p(x) Ω p(x) Ω p(x) Ω

p(x)

≤r

then no scalar multiple of u is extremal in the above sense.

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133

Proof. Both proofs are almost identical given the conditions on p, so we focus on the second claim. As is known from the standard theory of Lagrange multipliers (c.f. [16]), any extremal function w necessarily satisﬁes the equation −Δp(·) (w) = γ|w|p(·)−2 w

(4.28)

for some real number γ (the Lagrange multiplier). We claim that if u and αu were extremal then both u and αu would be solutions of (4.28) with the same γ. This is immediately seen from the weak form of (4.28) if p is constant on some open set O ⊂ Ω. Indeed, assuming that u and αu are extremal and setting γ and γα to be the Lagrange multipliers corresponding to u and αu respectively, using (4.28) for u and αu and choosing a test function v supported in O it is readily veriﬁed that |α|p(·)−1 |∇u|p(·)−2 ∇u∇v dx |∇u|p(·)−2 ∇u∇v dx Ω Ω γα = = = γ. |α|p(·)−1 |u|p(·)−2 uv dx |u|p(·)−2 uv dx Ω Ω For a continuous exponent p(·) the argument above can be adapted by observing that for each μ > 0 there exists an open subset Oμ of Ω on which the oscillation of p is less than μ and integrating against a test function supported in Oμ . Assuming that both u and αu are extremal and noticing that |α|p(x) v is an admissible test function for any test function v, on concludes from (4.28) for u and the above observation that |∇u|p(x)−2 ∇u|α|p(x) ∇v + |∇u|p(x)−2 ∇u ln |α||α|p(x) v∇p (4.29) Ω Ω =γ |u|p(x)−2 |α|p(x) v. (4.30) Ω

Using now (4.28) for αu, integrating versus the test function αv, the expression (4.30) is equal to the ﬁrst term in (4.29), which yields |∇u|p(x)−2 ∇u ln |α||α|p(x) v∇p = 0, Ω

which shows that the multiplicative property of extremal functions alluded to at the beginning of the paragraph is only valid if either α = 1 or when p is constant. We underline the following consequence of Lemma 4.3 as a salient feature of the variable exponent case, which stands out sharply in contrast to the case of a constant exponent: 1,p(.)

Theorem 4.4. If u ∈ W0 (Ω) is a solution to the eigenvalue problem (1.1), and p is not constant in Ω, then αu is not a solution of (1.1) for any α = 1. In particular, the solution set of (1.1) is not a vector space.

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5. Stability of the eigenvalues with respect to perturbations of p(·) The central idea of this section is an estimate for the variation of the eigenvalues deﬁned in Section 4 under perturbations of the exponent p(·). For the sake of completeness we include here a variant of Lemma 4.1 from [5]: Lemma 5.1. For p, q in P with p < q < p+ a.e. in Ω and a Borel-measurable function f : Ω → R, one has the inequality p(x) − |f (x)| dx ≤ |Ω| + |f (x)|q(x) dx. (5.1) Ω

Ω

Proof. As in [5] we write p(x) p(x) |f (x)| dx = |f (x)| dx + |f (x)|p(x) dx Ω {|f (x)|<} {<|f (x)|<1} + |f (x)|p(x) dx, {|f (x)|>1}

which leads to the bound |f (x)|p(x) dx ≤ |Ω| + |f (x)|q(x) |f (x)|p(x)−q(x) dx Ω {<|f (x)|≤1} + |f (x)|q(x) dx (5.2) {|f (x)|>1}

from which the claim is easily derived.

Lemma 5.2. For p ∈ P, r > 0 and > 0 the following inequalities hold: p∗ r+ r+ λp(·),r+ ≤ λp(·),r ≤ λp(·),r+ . (5.3) r r Proof. Set Sp (r) =

sup ρp(·) (∇u)≤r

ρp(·) (u).

Claim (5.3) is a direct consequence of Sp (r) ≤ Sp (r + ) ≤

r+ r

p∗ Sp (r).

(5.4)

To see this, observe that for s > 0, λp(·),s =

inf

ρp(·) (∇u)≤s

s . ρp(·) (u)

(5.5)

The ﬁrst inequality in (5.4) is obvious. The second one follows by observing that for ρp(·) (∇u) ≤ r + one has r r ρp(·) ∇u ≤ ρp(·) (∇u) ≤ r; r+ r+

Modular Eigenvalues consequently,

r (r + )

135

p∗

In all,

ρp(·) (u) ≤ Sp (r).

Sp (r + ) ≤

r+ r

p∗ Sp (r).

(5.6)

Lemma 5.3. For a bounded domain Ω ⊂ Rn , a function u ∈ L∞ (Ω) and p(·), q(·) as in Lemma 5.1, one has the inequality: |u(x)|q(x) dx ≤ u L∞ (Ω) + 1 |u(x)|p(x) dx. Ω

Ω

Proof. Setting Ωn = {x ∈ Ω : n − 1 ≤ u < n} one has [uL∞ +1]

q(x) |u(x)| dx = |u(x)|q(x) dx Ω

n=1

Ωn

u(x) p(x) q(x)−p(x) p(x) = n n dx n Ωn n=1 [uL∞ +1]

u(x) p(x) p(x) ≤ nn n Ω n n=1 ≤ [ u L∞ + 1] |u|p(x) dx. [uL∞ +1]

Ω

Theorem 5.4. The ﬁrst eigenvalue (4.4) of the modular p(·)-Laplacian is exponent-stable: More speciﬁcally, given a bounded domain Ω ⊂ Rn ,a ﬁxed > 0, a positive number s, and admissible exponents p(·), q(·) ∈ P log (Ω) satisfying the inequalities n + δ 0 then for s = |Ω| + − s there exists a positive constant M = M (Ω, q(·), s) such that the following inequality holds: 1 1 λp(·),s ≤ M λq(·),s . (5.7) s s Proof. In the course of the proof we retain the terminology introduced in Lemma 5.3. By virtue of Theorem 3.6 in [11], the Sobolev embedding E : W 1,p(·) (Ω) → L∞ (Ω) is continuous. If uq is a maximal function for ρq on the ball ρq (∇u) ≤ s, then ρp(·) (∇uq ) ≤ |Ω| + − s := s ,

(5.8)

as it follows from (5.1), whence ρp(·) (uq ) ≤ Sp (s ).

(5.9)

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J. Lang and O. M´endez

Let C be the norm of the Sobolev embedding E : W 1,q(·) (Ω) → L∞ (Ω),

(5.10)

and denote the best constant in the q(·)-Poincare’s inequality by C , i.e. C =

u q(·) ;

∇u q(·) 1,q(·) 0=u∈W (Ω) sup 0

Lemma 5.3 and the estimate (5.9) yield the inequality 1 1 Sq (s) ≤ uq L∞ (Ω) + 1 Sp (s ) ≤ (1 + C(C + 1) max{s q∗ , s q∗ }) Sp (s ), from which (5.7) can be easily derived.

On an analogous note, we state the corresponding version of Theorem 5.4 for the eigenvalues with respect to the normalized modular (4.5). Its proof closely follows the arguments of that of the previous theorem. Theorem 5.5. The ﬁrst eigenvalue (4.5) of the normalized modular p(·)Laplacian is exponent-stable: More speciﬁcally, given a bounded domain Ω ⊂ Rn ,a ﬁxed > 0, a positive number s, and admissible exponents p(·), q(·) ∈ P log (Ω) satisfying the inequalities n + δ 0 then for |Ω| q ∗ − s = + s p∗ p∗ there exists a positive constant M = M (Ω, q(·), s) such that the following inequality holds: 1˜ 1˜ λp(·),s ≤ M λ (5.11) q(·),s . s s

References [1] W. Arendt, S. Monniaux, Domain Perturbation for the ﬁrst Eigenvalue of the Dirichlet Schr¨ odinger Operator, Op. Th. 78 (1995), 11–19. [2] C. Bennewitz Approximation numbers=singular values, J. of Comp. and Appl. Math. 208 (1) (2007), 102–110. [3] L. Diening, P. Harjulehto, P. H¨ ast¨ o, M. Rusicka, Lebesgue and Sobolev spaces with variable exponent, Lecture notes in Mathematics, vol. 2017, Springer. [4] D.E. Edmunds, J. Lang, Eigenvalues, embeddings and generalised trigonometric functions, Lecture notes in Mathematics, vol. 2016, Springer. [5] D.E. Edmunds, J. Lang, A. Nekvinda, Some s-numbers of an integral operator of Hardy type in Lp(·) -spaces, J. Funct. Anal. 257 (2009), 219–242. [6] X. Fan, Q. Zhang, D. Zhao Eigenvalues of p(x)-Laplacian Dirichlet Problem, J. Math. Anal. Appl. 302 (2005), 306–317. [7] J. Fleckinger-Pelle, M.L. Lapidus, Vers une resolution de la conjecture de WeylBerry pour les valeurs propres du laplacien, C.R. Acad. Sci. Paris, Ser. I 306 (1988), 171–175. [8] J. Fleckinger-Pelle, D. Vasil’ev, An example of two-term asymptotics for the counting function of a fractal drum, Preprint.

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[9] G.W.F. Hegel, Grundlinien der Philosophie des Rechts, oder Naturrecht und Staatswissenschaft im Grundrisse, Berlin 1833. [10] C. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS-RCSM, 83, Am. Math. Soc., Providence, RI, 1994. [11] O. Kovacic, J. Raskosnik, Z. Rakosnik, On spaces Lp(x) and W k,p(x) , Czechoslovak Mathematical Journal 41 116 (1991), 592–618. [12] P. Lamberti, A diﬀerentiability result for the ﬁrst eigenvalue of the p-Laplacian upon domain perturbations, Nonlin. Anal. and Appl. to V. Lakshmikantham, Vol. 1, 2, Kluwer Acad. Publ., Dodrecht 2003, 741–754. [13] M.L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Am. Math. Soc. 325 (2) (1991), 465-529. [14] P. Lindquist, On the equation div(|∇u|p−2 ∇u) + λ|u|p−2 u = 0, Proc. Am. Math. Soc. 109 (1) (1990), 157–164. [15] P. Lindquist, On non-Linear Rayleigh Quotients, Pot. Anal. 2 (1993), 199–218. [16] E. Zeidler, Applied functional analysis: main principles and their applications, Appl. Math. Sci., 109, Springer, New York 1995. Jan Lang Ohio State University Department of Math. 100 Math Tower 231 West 18th Avenue Columbus, OH, 43210 USA e-mail: [email protected] Osvaldo M´endez University of Texas at El Paso Department of Math. Sciences 500 W University Ave. El Paso, TX, 79968 USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 139–156 c 2012 Springer Basel AG

Spectral Properties of Some Degenerate Elliptic Diﬀerential Operators Roger T. Lewis This paper is dedicated to David Edmunds on his 80th birthday and to Desmond Evans on his 70th birthday.

Abstract. In this paper we extend classical criteria for determining lower bounds for the least point of the essential spectrum of second-order elliptic diﬀerential operators on domains Ω ⊂ Rn allowing for degeneracy of the coeﬃcients on the boundary. We assume that we are given a sesquilinear form and investigate the degree of degeneracy of the coeﬃcients near ∂Ω that can be tolerated and still maintain a closable sesquilinear form to which the First Representation Theorem can be applied. Then, we establish criteria characterizing the least point of the essential spectrum of the associated diﬀerential operator in these degenerate cases. Applications are given for convex and non-convex Ω using Hardy inequalities, which recently have been proven in terms of the distance to the boundary, showing the spectra to be purely discrete. Mathematics Subject Classiﬁcation (2010). Primary 47F05; 47B25; Secondary 35P15, 35J70. Keywords. Essential spectrum, discrete spectrum, Hardy inequality, elliptic operators, distance function.

The classical criterion for the least point of the essential spectrum was given by Persson [22] for a Schr¨ odinger operator −Δ + q(x),

x ∈ Ω,

with the only singularity being at inﬁnity, assuming Dirichlet boundary conditions on Ω and assuming q to be bounded below at inﬁnity. For q bounded below at inﬁnity and near ∂Ω, Edmunds and Evans [10] extended this result to include singularities on the boundary ∂Ω showing that “if q ∈ L2loc (Ω) and the negative part of q behaves itself locally, then the essential spectrum” of the Friedrichs extension of the operator “is only inﬂuenced by the behaviour of q at ∂Ω and at inﬁnity in the respective cases”. Conditions (1.5) and (1.6) below give a mathematical description of the requirement that “q behaves itself locally”. Related techniques were used in [20] to establish conditions

140

R.T. Lewis

for a purely discrete spectra of second order elliptic diﬀerential operators in weighted L2 spaces including mixed boundary conditions. While still assuming that q is bounded below near singularities on ∂Ω or at ∞, Evans and Lewis [13] used techniques developed in [10] to study even-order elliptic diﬀerential operators in weighted L2w (Ω) spaces with emphasis upon the criteria for the ﬁniteness or inﬁniteness of the eigenvalues below the essential spectrum. We refer to that paper for many other references to related work. Edmunds and Evans [12] study the Neumann operator generated by the degenerate elliptic operator − div(d(x)2μ ∇ ) + d(x)−2θ ,

μ, θ ≥ 0,

on a proper open subset Ω ⊂ R where d(x) := dist(x, ∂Ω). They present upper and lower estimates for the eigenvalue counting function as well as examining the embedding properties for associated spaces. In this paper we study second-order elliptic sesquilinear forms that give rise to diﬀerential operators whose coeﬃcients may “blow-up” near parts of ∂Ω including some cases in which the potential diverges to negative inﬁnity near the boundary. Applications are given when the coeﬃcients are approximated by the distance function d(x) near ∂Ω. We follow and abbreviate the structure established in [13], but without the introduction of weights or higher-order cases. Those extensions should be clear from [13] and the presentation in this paper. n

1. Introduction Let Ω ⊂ Rn be open and connected. Throughout this paper u := u L2(Ω) . If Ω is unbounded, then ∞ is considered to be on the boundary of Ω in the sense of a one-point compactiﬁcation of Rn . The ﬁnite points of the boundary are denoted by ∂Ω. Outside some set S, which contains the singular part of ∂Ω, we assume that ∂Ω has a normal in order that certain boundary conditions are met. If Ω is unbounded then {∞} ⊆ S, but the emphasis here is upon the part of S on ∂Ω. The ﬁnite part of the singular set S \ {∞} is assumed to be a closed subset of ∂Ω. Let the singular and regular parts of the boundary be deﬁned by ΓS := NS ∩ ∂Ω

and ΓR := ∂Ω \ ΓS

where NS is an open neighborhood of S \ {∞} and N∞ := {x : |x| > K} for some large K. We may assume that NS ∩ N∞ = ∅ for unbounded domains Ω. For an Hermitian matrix A(x) = (aij (x)), real-valued q(x), x ∈ Ω, and σ(s), s ∈ ΓR , and a function c(s) that assumes either the value 1 or 0 for s ∈ ΓR , we are interested in diﬀerential operators of the form T : D(T ) → L2 (Ω) with ⎡ ⎤ n

∂ ∂ T u = ⎣− aij (x) + q(x)⎦ , x ∈ Ω, ∂x ∂x j i i,j=1

Spectral Properties

141

for D(T ) := {u : u = ϕ Ω , ϕ ∈ C0∞ (Rn \ ΓS ), T u ∈ L2 (Ω), ∂ϕ(s) + σ(s)ϕ(s) = 0, s ∈ ΓR } ∂ηA where ∂ϕ/∂ηA :=< Aη, ∇ϕ > and η is the unit outward normal on ΓR . The coeﬃcients c(s) and σ(s) are assumed not to be simultaneously zero allowing for mixed boundary conditions on ΓR . The case ΓS = ∂Ω, which requires Dirichlet boundary conditions, is included. In the case of suﬃciently smooth coeﬃcients for a symmetric operator T that is bounded from below, the sesquilinear form and c(s)

t[u, v] := (T u, v),

D(t) := D(T ),

(1.1)

is closable, Kato [19], Theorem VI.1.27, p. 318. In the absence of smooth coeﬃcients, the problem can be interpreted in a weak or variational sense initially involving only a sesquilinear form. In that case consider the form t[u, v] := [< A(x)∇u, ∇v > +quv] dx + σ(s)u(s)v(s)ds (1.2) Ω

ΓR

with domain

D(t) := {u : u = ϕ Ω , ϕ ∈ C0∞ (Rn \ ΓS )}. The value of c(s) is implicit in (1.2). At points where σ(s) = 0 Neumann conditions are implied so that c(s) = 1, and at points where σ(s) = 0 there are either Dirichlet or mixed conditions. For example, see R.E. Showalter [24], Chapter III, Theorem 3A and Example 4.1. We will give conditions which guarantee that the form is bounded below and closable. In that case the First Representation Theorem (Kato [19], §VI, Theorem 2.1) guarantees a unique self-adjoint operator T˜ associated with the closure ˜t of t for which D(T˜ ) ⊂ D(˜t). For forms deﬁned by (1.1), T˜ is the Friedrichs extension of T . Once we have established that t is bounded below and closable, we will assume that t[u] ≥ u 2 , which can be accomplished by the addition of a positive constant to T˜ merely translating σe (T˜ ). In this case, 1 according to the Second Representation Theorem [19], Theorem VI-2.23, T˜ 2 1 exists, D(T˜ 2 ) = D(˜t), and ˜t[u, v] = (T˜ 12 u, T˜ 12 v) := (u, v)˜t . 1

(1.3) 1,2

In this paper, we will use the Sobolev space H (G) = W (G) for an open set G ⊂ Rn , see Lieb and Loss [21], chapter 7. Let Ωk , k = 1, 2, . . . , be bounded domains in Rn which satisfy (i) Ωk Ωk+1 ; (ii) Ω \ S = ∪∞ k=1 (Ω ∩ Ωk ); (iii) there is a k0 ∈ N such that Ω \ Ωk ⊂ Ω ∩ (NS ∪ N∞ ) for all k ≥ k0 ; and (iv) the embedding H 1 (Ωk ) → L2 (Ωk ) is compact for each k ∈ N.

(1.4)

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R.T. Lewis

(Recall the notation Ωk Ωk+1 indicates that Ωk is compactly contained in Ωk+1 , i.e. Ωk is compact and Ωk ⊂ Ωk+1 .) This family of domains {Ωk }∞ k=1 is an S-admissible family of domains in Ω as deﬁned in Edmunds and Evans [11], p. 278. Note that (iv) holds provided the Rellich embedding theorem applies, e.g., if ∂(Ω ∩ Ωk ) has the segment property, Agmon [1], Theorem 3.8. In most applications considerable ﬂexibility in constructing each Ωk will be available. Denote the maximum and minimum eigenvalue of A(x) by νA (x) and μA (x) respectively. The notation f− (x) := − min{f (x), 0} and f+ (x) := f (x) + f− (x) will be used. Assume Hypothesis (H): For each k, assume that (a) ∂(Ω ∩ Ωk ) is C 1 ; ∞ (b) μA (x) > 0 a.e. on Ω and μ−1 A ∈ L (Ω ∩ Ωk ); α (c) q ∈ L (Ω ∩ Ωk ) with = n2 , n > 2, α > 1, n = 2;

(1.5)

(d) σ− (s) = 0 for s ∈ ΓR \ Ωk0 ; and (e) σ ∈ Lβ (ΓR ) with = n − 1, n > 2, β > 1, n = 2. The next lemma is a special case of Lemma 1 of [13]. We refer to that paper for the complete proof. It indicates the degree of unbounded behavior of q− that is allowed locally. Lemma 1. If (H) holds, then for > 0 and each k ∈ N there is a K(, k) > 0 such that 2 q− |u| dx + |σ(s)||u(s)|2 ds Ω∩Ωk ΓR ∩Ωk ≤ < A∇u, ∇u > dx + K(, k) |u|2 dx (1.6) Ω∩Ωk

Ω∩Ωk

for all u ∈ D(t). Proof. The proof follows from the Monotone Convergence Theorem, the H¨ older Inequality, and the Sobolev Inequality.

2. The main results When we know of the existence of T˜ we let e = e (T˜ ) denote the least point of its essential spectrum. The following proposition compares with Corollary 7D, Chapter III, of R.E. Showalter [24]. Proposition 1. Assume hypothesis (H), that νA (x) ∈ L∞ (Ω ∩ Ωk ),

k ∈ N,

Spectral Properties

143

and that for all k suﬃciently large t[u] + αk u 2L2(Ω∩Ωk ) ≥ ck u 2H 1 (Ω∩Ωk ) ,

u ∈ D(t),

(2.1)

for positive constants αk and ck . If t is bounded below and closable, then e := inf{λ : λ ∈ σe (T˜ )} = lim

inf {t[u] : u ∈ D(t), supp u ⊂ Ω \ Ωk } .

(2.2)

k→∞ u=1

Proof. It will suﬃce to show that the following holds (see p. 476 of [11]): (A) For each k ∈ N large enough and φ ∈ C0∞ (Rn \ ΓS ) such that 1, x ∈ Ωk , φ(x) = 0, x ∈ / Ωk+1 ,

(2.3)

with 0 ≤ φ ≤ 1, we have (i) φv ∈ D(t) for every v ∈ D(t) and (ii) if v ∈ D(t) with v ˜t = 1 and v 0 in the Hilbert space H(˜t) := (D(˜t); · ˜t ), then

(1 − φ)v ˜2t ≤ 1 + o(1) as → ∞. Part (A)(i) is immediate. Since t is bounded below, without loss of generality, we may assume that t ≥ 1 on D(t) as discussed above. Therefore (1.3) holds. For all u ∈ D(t) and any φ satisfying (2.3) 2

(1 − φ)u ˜t − σ|(1 − φ)u|2 ds ΓR = < A∇(1 − φ)u, ∇(1 − φ)u > +q|(1 − φ)u|2 dx Ω\Ωk = (1 − φ)2 < A∇u, ∇u > +q|u|2 − (2 − φ)φ q|u|2 dx Ω\Ωk ×2 Re < A1/2 (1 − φ)∇u, A1/2 u∇(1 − φ) > (Ω∩Ωk+1 )\Ωk

+ ≤

(Ω∩Ωk+1 )\Ωk

Ω\Ωk

< A∇φ, ∇φ > |u|2 dx

(1 − φ)2 < A∇u, ∇u > +q|u|2 + (2 − φ)φ q− |u|2 dx

−2 + ≤

Ω

(Ω∩Ωk+1 )\Ωk

(Ω∩Ωk+1 )\Ωk

5

(1 − φ)Re < A1/2 ∇u, A1/2 ∇φ > u

< A∇φ, ∇φ > |u|2 dx

6 < A∇u, ∇u > +q|u|2 dx +

Ω∩Ωk+1

q− |u|2 dx

144

R.T. Lewis +δ (Ω∩Ωk+1 )\Ωk

+ (1 + δ −1 )

< A∇u, ∇u > dx

(Ω∩Ωk+1 )\Ωk

for δ > 0. Similarly, σ(1 − φ)2 |u|2 ds = ΓR

ΓR

< A∇φ, ∇φ > |u|2 dx

σ|u|2 ds −

ΓR ∩Ωk+1

φ(2 − φ)σ|u|2 ds.

∞

Since νA ∈ L (Ω ∩ Ωk ) and (1.6) holds for each k, it then follows that

(1 − φ)u ˜2t ≤ u ˜2t + δ < A∇u, ∇u > dx + C(δ , k) |u|2 dx, Ω∩Ωk+1

Ω∩Ωk+1

for an arbitrarily small δ and C(δ , k) > 0. According to the coercivity requirement (2.1) and the fact that νA ∈ L∞ (Ω ∩ Ωk ) for each k ∈ N αk

(1 − φ)u ˜t ≤ (1 + ) u ˜2t + (C(δ , k) + ) u 2L2 (Ω∩Ωk+1 ) ck+1 ck+1 for an arbitrarily small . As in (A)(ii) suppose that {v } ⊂ D(t) satisﬁes v t˜ = 1 and v 0 in H(˜t). We have that

(1 − φ)v 2t˜ ≤ 1 + + C (δ , k) v 2L2 (Ω∩Ωk+1 ) . By (2.1) and the fact that t ≥ 1, it follows that the embedding H(˜t) → H 1 (Ω∩Ωk+1 ) is continuous. Since H 1 (Ω∩Ωk+1 ) → L2 (Ω∩Ωk+1 ) is compact, then v 2L2 (Ω∩Ωk+1 ) = o(1) as → ∞. Hence,

(1 − φ)v 2t˜ ≤ 1 + o(1) since can be chosen arbitrarily small. That completes the proof.

In unbounded domains Ω we will assume that q is bounded below at inﬁnity as in (2.5) below. When we know a priori that t is bounded below, we may assume without loss of generality that for k suﬃciently large q(x) > 0 for x ∈ (Ω \ Ωk ) ∩ N∞ as well as t[u] ≥ u 2 , mentioned above, since the addition of a constant only translates the spectrum. In contrast to [13], [10], and the classical criterion of Persson [22], we are not requiring that the potential q be bounded below in a neighborhood NS of the ﬁnite singularities. The next theorem shows that in the case of a coeﬃcient degenerate on S ∩ ∂Ω, the existence of a Hardy-type inequality in a neighborhood of the singularities may be suﬃcient to ensure that the form is closable and bounded below, i.e., inequality (2.4) replaces the requirement that q be bounded below on ∂Ω. Theorem 1. Assume (H) holds and that for some γ ∈ (0, 1) and k0 given in (1.4) [(1 − γ) < A∇u, ∇u > −q− |u|2 ]dx ≥ 0, u ∈ D(t), (2.4) (Ω\Ωk )∩NS

Spectral Properties

145

for all k ≥ k0 and lim

ess sup

k→∞x∈(Ω\Ωk )∩N∞

q− (x) = C∞ < ∞

(2.5)

when Ω is unbounded. Then t is bounded below and closable and (2.1) holds. Furthermore, if νA ∈ L∞ (Ω ∩ Ωk ), k ∈ N, then e (T˜) is given by (2.2). Proof. We give the proof in the case that Ω is unbounded. The proof for Ω bounded requires only slight modiﬁcation. Let t1 [u] := [< A∇u, ∇u > +q+ |u|2 ]dx + σ+ (s)|u(s)|2 ds, Ω ΓR t1 [u] := − q− |u|2 dx − σ− (s)|u(s)|2 ds Ω

ΓR

D(t1 )

with D(t) = = D(t1 ) and t = t1 + t1 . We ﬁrst show that t1 is t1 -bounded with t1 -bound less than 1. Then, in order to conclude that t is closable it will suﬃce to show that t1 is closable – see Kato [19], Theorem 1.33, p. 320. Let k ≥ k0 in (1.6) recalling that σ− (s) = 0 for s ∈ ΓR \ Ωk0 according to (H ). Without loss of generality, we may assume that for δ > 0, q− (x) < C∞ + δ,

x ∈ (Ω \ Ωk0 ) ∩ N∞ .

Then it follows from (2.4) and (1.6) that for all u ∈ D(t), ≤ (1 − γ), and α(, k) ≥ max{K(, k), C∞ + δ} + 1, |t1 [u]| ≤ (1 − γ) < A(x)∇u, ∇u > dx (Ω\Ωk )∩NS + q− |u|2 dx + q− |u|2 dx + σ− |u|2 ds (Ω\Ωk )∩N∞ Ω∩Ωk ΓR (2.6) ≤ (1 − γ)t1 [u] + (C∞ + δ) |u|2 dx (Ω\Ωk )∩N∞ + K(, k) |u|2 dx Ω∩Ωk

≤ (1 − γ)t1 [u] + α(, k) u 2L2 (Ω) . Therefore, t1 has t1 -bound less than 1. Note that (2.6) implies the inequality t[u] + α(, k) u 2L2 (Ω) ≥ γt1 [u] ≥ 0.

(2.7)

Therefore, t is bounded below. To show that t1 is closable in L2 (Ω), choose {ϕn } ⊂ D(t) such that t1 [ϕn − ϕm ] → 0,

ϕn → 0

as m, n → ∞,

(2.8)

146

R.T. Lewis

i.e., {ϕn } is t1 -convergent to 0. Then, we must show that t1 [ϕn ] → 0 as n → ∞. First, note that (2.8) implies that < A(x)∇(ϕn − ϕm ), ∇(ϕn − ϕm ) > dx → 0, m, n → ∞. Ω

It follows as in (3.13) of [13] that < A(x)∇ϕn , ∇ϕn > dx → 0, Ω

Since t1 [u] + α(, k0 ) u 2 ≥

Ω

n → ∞.

(2.9)

q+ |u|2 dx + u 2 ,

then {ϕn } must be a Cauchy sequence in L2q+ +1 (Ω). Since this space is complete, we must have that ϕn → ψ for some ψ ∈ L2q+ +1 (Ω). But, ϕn → 0 in L2 (Ω) implies that ψ ≡ 0. Consequently, we have shown that [< A∇ϕn , ∇ϕn > +q+ |ϕn |2 ]dx → 0. Ω

We need to show that

ΓR

σ+ (s)|ϕn (s)|2 ds → 0

in order to complete the proof. An analysis similar to (3.17) in [13] applies here as well since t1 [(ϕn − ϕm )] ≥ σ+ (s)|(ϕn − ϕm )|2 ds ≥ 0. ΓR

L2σ+ +1 (ΓR )

Hence, {ϕn } is Cauchy in and converges to a v ∈ L2σ+ +1 (ΓR ). By (1.6) and (2.9) we conclude that v = 0 on ΓR ∩ Ωk for each k. Since ΓR = ∪k (ΓR ∩ Ωk ), then v = 0 on ΓR which is what we wanted to show. Therefore, t is bounded below and closable. As discussed above, it will suﬃce for the remainder of the proof to assume that t ≥ 1 and q > 0 in (Ω \ Ωk ) ∩ N∞ for k large. Since q− (x) = 0 for k large and x ∈ (Ω \ Ωk ) ∩ N∞ , it follows from (2.6) that |t1 [u]| ≤ (1 − γ)t1 [u] + K(, k) |u|2 dx, u ∈ D(t), Ω∩Ωk

which implies that t[u] +

K(, k) u 2L2(Ω∩Ωk )

≥ γt1 [u] ≥ γ

Ω∩Ωk

< A∇u, ∇u > dx, u ∈ D(t).

∞ Since μ−1 A ∈ L (Ω ∩ Ωk ), then (2.1) holds. If we know that

νA ∈ L∞ (Ω ∩ Ωk ),

k ∈ N,

then it follows from Proposition 1 that (2.2) holds.

Spectral Properties

147

Note that if q is bounded below by B < 0 on (Ω \ Ωk0 ) ∩ NS as assumed in earlier work,e.g., [10], [11], and [13], then we may apply Theorem 1 to the form t[u] + |B| Ω χ(Ω\Ωk )∩NS |u|2 dx. 0

It may be advantageous to need only show that the inequality in (2.4) holds for u ∈ H01 ((Ω \ Ωk0 ) ∩ NS ). The next Theorem shows that is allowed. However, we will see in the applications below that in some cases it is best to use (2.4) directly avoiding certain convexity requirements. Theorem 2. Assume hypothesis (H), that νA ∈ L∞ (Ω ∩ Ωk ),

k ∈ N,

and (2.5) for Ω not bounded. If for all ϕ ∈ H01 ((Ω \ Ωk0 ) ∩ NS ) [(1 − γ) < A(x)∇ϕ, ∇ϕ > −q− (x)|ϕ|2 ]dx ≥ 0

(2.10)

(Ω\Ωk0 )∩NS

for some γ ∈ (0, 1), then (2.2) holds. Proof. Recall that NS is an open neighborhood of the ﬁnite singularities, S \ {∞}, with Ω \ Ωk ⊂ Ω ∩ (NS ∪ N∞ ) for k ≥ k0 . We employ a simple IMS localization formula – see [8], p. 28. Choose k2 > k1 ≥ k0 . There exists φ1 ∈ C ∞ (Rn ) for which 1, x ∈ (Ω \ Ωk2 ) ∩ NS , φ1 (x) = 0, x ∈ Ω ∩ Ωk1 , (with the support of φ extending into Rn \ Ω as needed) and φ2 such that • • • •

φj (x) ∈ [0, 1] for j = 1, 2, and all x ∈ Rn ; φ21 (x) + φ22 (x) ≡ 1 for all x ∈ Rn ; φj ∈ C ∞ (Rn ); and supx∈Rn [|∇φ1 (x)|2 + |∇φ2 (x)|2 ] < ∞.

Recall the pointwise identity that gives rise to the IMS localization formula: for u ∈ D(t) and j = 1, 2, < A∇(φj u), ∇(φj u) > = φ2j < A∇u, ∇u > + < A∇φj , ∇φj > |u|2 + e < A∇φ2j , u∇u > . Summing over j = 1, 2, and integrating yields the identity t[u] =

2

j=1

Ω

[< A(x)∇(φj u), ∇(φj u) > +q|φj u|2

− < A∇φj , ∇φj > |u|2 ]dx + since φ2 (s) = 1 on ΓR . Then φ1 u ∈

C0∞ (Ω

ΓR

\ Ωk1 ).

σ(s)|u(s)|2 ds

(2.11)

148

R.T. Lewis It follows from the pointwise identity (2.11) that

(Ω\Ωk1 )∩NS

=

2

(Ω\Ωk1 )∩NS

j=1

≥

< A∇u, ∇u > dx

(Ω\Ωk1 )∩NS

[< A∇(φj u), ∇(φj u) > − < A∇φj , ∇φj > |u|2 ]dx

< A(x)∇(φ1 u), ∇(φ1 u) > dx − Ck2

for Ck2 :=

2

sup

x∈(Ωk2 \Ωk1 )∩NS j=1

(Ωk2 \Ωk1 )∩NS

< A∇φj , ∇φj > < ∞

since νA ∈ L∞ (Ω ∩ Ωk ), k ∈ N. Since (2.10) holds for γ ∈ (0, 1), < A∇u, ∇u > dx + Ck2 (Ω\Ωk1 )∩NS

≥ (1 − γ)−1

(Ωk2 \Ωk1 )∩NS

(Ω\Ωk1 )∩NS

= (1 − γ)−1 [

(Ω\Ωk1 )∩NS

|u|2 dx

|u|2 dx

q− |φ1 u|2 dx q− |u|2 dx −

(Ωk2 \Ωk1 )∩NS

q− |φ2 u|2 dx].

As in Lemma 1 we have that for any > 0 there is a positive constant K(, k2 ) such that q− |φ2 u|2 dx] (Ωk2 \Ωk1 )∩NS

≤

(Ωk2 \Ωk1 )∩NS

< A∇u, ∇u > dx + K(, k2 )

(Ωk2 \Ωk1 )∩NS

(see (2.9) of [13]) which implies that (1 + ) < A∇u, ∇u > dx + C(, k2 ) (Ω\Ωk1 )∩NS

≥ (1 − γ)−1

(Ω\Ωk1 )∩NS

(Ωk2 \Ωk1 )∩NS

|u|2 dx

|u|2 dx

q− |u|2 dx

for C(, k2 ) := Ck2 + K(, k2 ). Then, for chosen suﬃciently small (1 + ) × (1 − γ) ∈ (0, 1). Since k1 is an arbitrary integer greater than or equal to k0 , the hypothesis of Theorem 1 holds for h[u] := t[u] − (1 − γ)C(, k2 ) χ(Ωk \Ωk )∩NS |u|2 dx, u ∈ D(t), Ω

2

1

implying that h[u] is bounded below and closable and, as shown in the proof of Theorem 1, that (2.1) holds for h. But, this implies that t is bounded below and closable (cf. (2.8)) and (2.1) holds as well for t. The conclusion follows from Proposition 1.

Spectral Properties

149

With appropriate conditions required of the coeﬃcients, inequality (2.10) is associated with the existence of a nonnegative solution of the Dirichlet problem for −(1 − γ) div(A(x)∇u) − q− (x)u = 0 on (Ω \ Ωk0 ) ∩ NS , the absence of nodal domains, and the ﬁniteness of the negative spectrum ([2], [22], [23]). Corollary 1. Assume the hypothesis of Theorem 2 and for k ≥ k0 deﬁne LS [u; k] := [< γA(x)∇u, ∇u > +q+ (x)|u|2 ]dx, u ∈ D(t). (Ω\Ωk )∩NS

Then, for Ω bounded e ≥ limk→∞ inf LS [u; k] u=1

with the inﬁmum taken over all u ∈ D(t) with supp u ⊂ (Ω \ Ωk ) ∩ NS . If Ω is unbounded and (2.5) holds, then e ≥

limk→∞ inf LS [u; k] − C∞ . u=1

Proof. We give the proof for the case in which Ω is unbounded. The adaptation for Ω bounded is straightforward. According to Theorem 2, for k ≥ k0 and ϕ := u/ u for u ∈ D(t) with supp u ⊂ Ω \ Ωk , t[ϕ] ≥ γ < A(x)∇ϕ, ∇ϕ > dx + q+ (x)|ϕ|2 dx (Ω\Ωk )∩NS (Ω\Ωk )∩NS + [< A(x)∇ϕ, ∇ϕ > +q(x)|ϕ|2 ]dx (Ω\Ωk )∩N∞ ≥ [γ < A(x)∇ϕ, ∇ϕ > +q+(x)|ϕ|2 ]dx (Ω\Ωk )∩NS − (C∞ + δ) |ϕ|2 dx (Ω\Ωk )∩N∞

for some small δ > 0. Therefore, e = lim inf t[ϕ] k→∞ ϕ

≥ lim inf LS [ϕ; k] − C∞ . k→∞ ϕ

3. Applications using Hardy inequalities in d(x) In this section we explore applications of Theorems 1 and 2 with some of the more recent results for Hardy inequalities given in terms of the distance to the boundary of the domain, i.e., d(x) := dist(x, ∂Ω). Weighted Hardy inequalities in L2 (G), which best suit our purposes, are of the following form: for an open connected set G ⊂ Rn and u ∈ H01 (G) |u(x)|2 β 2 d(x) |∇u(x)| dx ≥ κ(β) dx + λ(G) d(x)α |u(x)|2 dx (3.1) 2−β G G d(x) G

150

R.T. Lewis

with β < 1 and α > (β − 2). Here, κ(β) is assumed to be positive for each β < 1 and λ(G) ≥ 0 depends upon certain geometric properties of G, e.g., the diameter of G, the volume of G, etc. Several results of this type are discussed below. Corollary 2. Assume hypothesis (H), νA ∈ L∞ (Ω ∩ Ωk ) for all k, and that for some β < 1 μA (x) ≥ d(x)β ,

x ∈ (Ω \ Ωk0 ) ∩ NS .

(3.2)

For Ω unbounded assume that q− is bounded below at inﬁnity as in (2.5). Finally, assume that (3.1) holds for some β < 1 and for G = (Ω \ Ωk0 ) ∩ NS . If for some γ ∈ (0, 1) q− (x) ≤ (1 − γ)[

κ(β) + λ(G)d(x)α ], d(x)2−β

x ∈ G,

(3.3)

then t is bounded below and closable and the spectrum of T˜ is purely discrete. Proof. The fact that t is bounded below and closable follows from Theorem 1. By (3.1) and (3.3) the hypothesis of Theorem 2 holds. We may apply Corollary 1. For k > k0 |u|2 β 2 LS [u; k] ≥ γd(x) |∇u| dx ≥ γκ(β) dx 2−β (Ω\Ωk )∩NS (Ω\Ωk )∩NS d(x) according to (3.2) followed by (3.1). According to property (ii) of the S1 admissible family of domains {Ωk }∞ k=1 we may assume that d(x) < k for x ∈ (Ω \ Ωk ) ∩ NS and k ≥ k0 . Since the inﬁmum in Corollary 1 is taken over all u ∈ D(t) with support in (Ω \ Ωk ) ∩ NS , then for k ≥ k0 inf LS [u; k] ≥ γκ(G)k 2−β ,

u=1

β < 1.

Letting k → ∞, we conclude that e = ∞ implying that the spectrum is purely discrete. Corollary 2 indicates that if a Hardy inequality (3.1) holds, the form t can be bounded below and closable even though all coeﬃcients are degenerate at parts of the boundary ∂Ω. We review some of the earlier results in which (3.1) holds. For α = β = 0, (3.1) reduces to 1 |u(x)|2 |∇u(x)|2 dx ≥ dx + λ(Ω) |u(x)|2 dx. (3.4) 4 Ω d(x)2 Ω Ω Recent results for this inequality were motivated by work of Brezis and Mar1 cus in [7] who showed that for Ω convex with ∂Ω ∈ C 2 , λ(Ω) ≥ 4D(Ω) 2 with D(Ω) denoting the usual diameter of Ω. For the “interior diameter” deﬁned by Dint (Ω) := 2 supx∈Ω d(x), Filippas, Maz’ya, and Tertikas [15] showed that for Ω convex, λ(Ω) ≥ Dint3(Ω)2 . Subsequently, Avkhadiev and Wirths [5] have 4λ0 shown that λ(Ω) ≥ Dint (Ω)2 where λ0 ≥ 0.94. Using methods of Davies [9], M. Hoﬀmann-Ostenhof, T. Hoﬀmann-Ostenhof, and Laptev [18] answered

Spectral Properties

151

a question posed by Brezis and Marcus showing that for convex domains K(n) 1−2/n n−1 2/n λ(Ω) ≥ 4|Ω| |S | , in which |Ω| denotes the volume of 2/n , K(n) := n 3K(n) Ω. Using similar methods, Evans and Lewis [14] showed that λ(Ω) ≥ 2|Ω| 2/n . Since a ball of diameter Dint (Ω) must be contained in Ω, it follows that for n = 2, 3, the results for λ(Ω) in the paper of Filippas, Maz’ya, and Tertikas [15] are comparable to those in terms of the volume improving the inequality in the paper of M. Hoﬀmann-Ostenhof, T. Hoﬀmann-Ostenhof, and Laptev [18]. Also, there is some advantage in the fact that the inequalities of [5], [7] and [15] do not require |Ω| to be ﬁnite, e.g., Ω = ω × R with ω ⊂ Rn−1 convex. In that case |Ω| = ∞, but Dint (Ω) < ∞ if Dint (ω) < ∞. While applying some of these inequalities in Corollary 2, convexity may be required, but that requirement is diminished by the fact it is needed only on (Ω \ Ωk0 ) ∩ NS and not necessarily on Ω. In addition, a certain degree of ﬂexibility is available in constructing the family {Ωk }∞ k=1 in NS . Nevertheless, we will also be interested in inequalities not requiring convexity. In a domain Ω ⊂ Rn the distance function d(x) is uniformly Lipschitz continuous (cf. Gilbarg and Trudinger [17], §14.6) and consequently, diﬀerentiable almost everywhere according to Rademacher’s theorem. Moreover, if Ω is bounded and ∂Ω ∈ C k , k ≥ 2, then for some δ > 0 suﬃciently small, d ∈ C k (Ωδ ) in which Ωδ := {x ∈ Ω : d(x) < δ} – Lemma 14.6 of [17]. If Ω is convex, then the distance function is superharmonic, i.e., −Δd(x) is a nonnegative measure. (See Lemma 3 of [4] for a short proof). For dimension n = 2, −Δd ≥ 0 implies that Ω is convex, but not for n > 2. Armitage and Kuran [3] give an example of a torus in dimension greater than 2, which is (obviously) not convex, but −Δd(x) ≥ 0. In order to accommodate weights, we give a small extension of Theorem 3.1 of Filippas, Maz’ya, and Tertikas [15] requiring only a modiﬁcation of their change of variable. Rather than assuming convexity of Ω it suﬃces (here and in the proof of Theorem 3.1 of [15]) to assume the weaker condition that −Δd(x) ≥ 0 in Ω.

Theorem 3. If −Δd(x) ≥ 0 in a domain Ω, then for all u ∈ H01 (Ω), β < 1, and α > β − 2 (1 − β)2 |u|2 β−(α+2) β 2 d(x) |∇u| dx − dx ≥ Cα,β Dint d(x)α |u|2 dx 2−β 4 d(x) Ω Ω Ω for a constant Cα,β := 2α−β ·

(α + 2 − β)2 , (1 − β)(2α + 3 − β),

α ∈ (β − 2, −1) . α ∈ [−1, ∞)

Proof. It will suﬃce to show the inequality for real-valued u ∈ C0∞ (Ω). Let 1−β u = d 2 v. Since |∇d|2 = 1, it follows that (1 − β)2 1−β β 2 β−2 2 2 d |∇u| dx− d u dx = (−Δd)v dx+ d|∇v|2 dx. 4 2 Ω Ω Ω Ω (3.5)

152

R.T. Lewis

After noting the identity

dα u2 dx = Ω

dα+1−β v 2 dx

Ω

we estimate the integral on the right-hand side for α > β − 2 following a path similar to that of [15] to arrive at their inequality (3.4) and see that for this case 1 α+2−β (α + 2 − β − δ) dα+1−β v 2 dx ≤ Rint d|∇v|2 dx + (−Δd)v 2 dx . δ Ω Ω Ω Here Rint := follows.

1 D (Ω). 2 int

Choose δ ≤ min{ 1−β , α+2−β } and the conclusion 2 2

If we know that G in Corollary 2 is convex, then −Δd(x) is a positive measure and we may apply Theorem 3. Corollary 3. Assume the hypothesis of Corollary 2. If for γ ∈ (0, 1) and α>β−2 q− (x) ≤ (1 − γ)[

(1 − β)2 + λ(G)d(x)α ], 4d(x)2−β

x ∈ G,

(3.6)

α−(β−2)

for G = (Ω \ Ωk0 ) ∩ NS convex and λ(G) = Cα,β /Dint (G), then t is bounded below and closable and the spectrum of T˜ is purely discrete. Proof. The proof follows from Corollary 2 and Theorem 3.

In [16] Filippas, Maz’ya, and Tertikas prove a Hardy–Sobolev inequality in a tubular domain Ωδ := {x ∈ Ω : d(x) < δ} for some δ > 0. Here, we adapt some of those ideas to use as an application of Corollary 1. The next lemma allows application for the case in which −Δd(x) ≥ 0 in the whole of a nonconvex Ω, but G in Corollary 2 is not convex and d is not superharmonic in G. The prototype for Ω in this case is the torus studied by Armitage and Kuran [3]. It is important to note that in the next Lemma, d(x) = d(x; Ω), the distance from x to ∂Ω as before, as opposed to the distance from x to ∂Ωδ , d(x; Ωδ ). We will use this additional notation in some cases below to avoid confusion. Lemma 2. Assume that Ω is a bounded domain with a C 2 boundary and −Δd ≥ 0 in Ωδ for all δ > 0 suﬃciently small. Let β < 1 and α > (β − 3)/2. If 0 < δ ≤ 1−β , then for all u ∈ C0∞ (Ω) 2 (1 − β)2 dβ |∇u|2 dx − dβ−2 |u|2 dx ≥ C(α, β)δ dα |u|2 dx 4 Ωδ Ωδ Ωδ for a positive constant C(α, β) :=

2α−β+1 (2α − β + 3) . (1 − β)α−β+2

(3.7)

Spectral Properties

153

Proof. Since Ω is bounded and ∂Ω ∈ C 2 , then d ∈ C 2 (Ωδ∗ ∩ Ω) for some δ∗ (Lemma 14.16 of [17]). We may assume without loss of generality that δ∗ = δ ∈ (0, 1−β ). It will suﬃce to prove the inequality for functions u ∈ C0∞ (Ω) 2 that are real-valued and non-negative (Lieb and Loss [21], pp. 176–177). For 1−β u ∈ C0∞ (Ω) and u = d 2 v it follows from integrating by parts that (1 − β)2 1−β dβ |∇u|2 dx = d−1 v 2 dx + (−Δd)v 2 dx 4 2 Ωδ Ωδ Ωδ 1−β + (∇d · ν)v2 ds + d|∇v|2 dx 2 ∂Ωcδ Ωδ since |∇d| = 1 where ν is the unit outward normal from Ωδ on ∂Ωδ ∩Ω = ∂Ωcδ . Since ∇d · ν = 1 on ∂Ωcδ we have that (1 − β)2 β 2 d |∇u| dx − dβ−2 u2 dx 4 Ωδ Ω δ (3.8) 1−β 1−β 2 = (−Δd)v dx + d|∇v|2 dx + v2 ds. 2 2 Ωδ Ωδ ∂Ωcδ In order to estimate substitution u = d

1−β 2

Ωδ

dα u2 dx for α >

β−3 2

> β − 2, we make the

v again and use the identity

div(dα−β+2 ∇d) = (α − β + 2)dα−β+1 + dα−β+2 Δd in Ωδ . Multiply by v 2 and integrate by parts to see that (α − β + 2) dα−β+1 v 2 dx Ωδ α−β+2 α−β+2 2 = −2 d v∇d · ∇v dx + δ v ds + dα−β+2 (−Δd)v 2 dx Ωδ

Ωδ

∂Ωcδ

=: I1 + I2 + I3 .

(3.9)

Applying the Cauchy-Schwarz inequality to I1 we have that α−β+1 2 α−β+1 I1 ≤ δ d v dx + δ d|∇v|2 dx Ωδ

since d(x) ∈ (0, δ) in Ωδ . It follows from (3.9) and (3.10) that [(α − β + 2) − δ] dα−β+1 v 2 dx ≤ δ α−β+1 d|∇v|2 dx Ωδ Ωδ α−β+2 +δ (−Δd)v 2 dx Ω δ α−β+2 +δ v2 ds ∂Ωcδ

since −Δd(x) ≥ 0 in Ωδ .

(3.10)

Ωδ

(3.11)

154

R.T. Lewis Since δ ≤

1−β 2

in (3.11) we will have that α−β+2−δ dα−β+1 v 2 dx δ α−β+1 Ωδ 1−β 1−β ≤ (−Δd)v 2 dx + d|∇v|2 dx + v 2 ds c 2 2 Ωδ Ωδ ∂Ωδ 2 (1 − β) = dβ |∇u|2 dx − dβ−2 u2 dx 4 Ωδ Ωδ

according to (3.8). Finally, use the fact that C(α, β)δ ≤

α−β+2−δ δ α−β+1

to complete the proof.

Next, we present a corollary to Lemma 2 in which we can use Theorem 1 directly avoiding a convexity assumption for Ω \ Ωk0 . Then, we follow with an application on a torus in R3 in which −Δd(x) ≥ 0. Corollary 4. Let Ω be a bounded domain with a C 2 -boundary and let h[u, v] be given by (1.2) with σ ≡ 0 and D(h) = C0∞ (Ω). Set S = ∂Ω and deﬁne the S-admissible family of domains by 1 Ωk := Ω \ Ωδk , δk = k for k ∈ N. Assume (H)(a),(b),and (c); for some β < 1 μA (x) ≥ d(x)β ,

x ∈ Ωδk ;

∞

and νA ∈ L (Ω ∩ Ωk ) for k suﬃciently large. Suppose for γ ∈ (0, 1) and α satisfying 2α − β + 3 > 0 that q− (x) ≤ (1 − γ)[

(1 − β)2 + δC(α, β) d(x)α ], 4d(x)2−β

x ∈ Ωδk

and −Δd(x) ≥ 0 in Ωδk for k suﬃciently large and C(α, β) deﬁned in (3.7). Then h is bounded below and closable. The self-adjoint operator associated with h has a purely discrete spectrum. Proof. It follows from Theorem 1 and Lemma 2 that h is bounded below and closable as well as the fact that (2.2) holds. For k ≥ k0 , u ∈ D(t) with supp u ⊂ Ω \ Ωk and ϕ := u/ u t[ϕ] ≥ γ < A(x)∇ϕ, ∇ϕ > dx Ωδk

≥γ Since d(x) ≤

1 k

[ Ωδk

(1 − β)2 + δC(α, β) d(x)α ]|ϕ|2 dx. 4d(x)2−β

for x ∈ Ωδk , e = lim inf t[ϕ] = ∞ k→∞ ϕ

Spectral Properties

155

implying that the spectrum of the operator associated with the closure of h is discrete. Example 1. Let Ω ⊂ R3 be the torus obtained by rotating the disc ω = {(0, y, z) : (y − c)2 + z 2 < R}, c > 2R, about the z-axis. Armitage and Kuran [3] have shown that the distance function dΩ on the whole of Ω is superharmonic, i.e., −ΔdΩ (x) ≥ 0 in Ω although Ω is not convex. Assuming the hypothesis of Corollary 4, the operator associated with the Dirichlet form h on the torus Ω has a purely discrete spectrum. Of course, the Example 1 can be extended to the image of any unitary transformation of the torus described there since the spectrum is preserved under such transformations. Note that the distance function dΩδ (x) in Ωδ for small δ > 0 dΩ (x), x ∈ Ωδ/2 , dΩδ (x) = δ − dΩ (x), x ∈ Ωδ \ Ωδ/2 is not superharmonic. Corollary 3 does not apply to the torus of Example 1 since Ωδ for δ > 0 is not convex and dΩδ is not superharmonic. Finally, we refer the reader to recent results in [6] where Hardy inequalities are given which exploit the interesting connection between Δd(x) and the principal curvatures at the near point y ∈ ∂Ω of x. These new Hardy inequalities allow for applications of the results here to far more general nonconvex domains such as the torus discussed above. Using a representation of Δd in terms of principal curvatures, a new proof is given of Armitage and Kuran’s result discussed in Example 1.

References [1] S. Agmon. Lectures on Elliptic Boundary Value Problems, Reprinted in 2010 by AMS Chelsea Publishing, AMS, Providence, RI, 1965. [2] W. Allegretto. On the equivalence of two types of oscillation for elliptic operators. Paciﬁc J. Math., 55(2), 319–328, 1974. ¨ Kuran. The convexity of a domain and the superhar[3] D.H. Armitage and U. monicity of the signed distance function. Proc. Amer. Math. Soc., 93(4), 598– 600, April 1985. [4] F.G. Avkhadiev and A. Laptev. Hardy inequalities for non-convex domains. The Erwin Schr¨ odinger International Institute for Mathematical Physics. Preprint ESI 2185, October 2009. [5] F.G. Avkhadiev and K.J. Wirths. Uniﬁed Poincar´e and Hardy inequalities with sharp constants for convex domains. Z. Angew. Math. Mech. 87, 632–642, 2007. [6] A. Balinsky, W.D. Evans, and R.T. Lewis. Hardy’s inequality and curvature. Math. Phys. Arc. 11-81, 23 May 2011, and arXiv:1101.2331. [7] H. Brezis and M. Marcus. Hardy’s inequalities revisited. Ann. Scuola Norm. Pisa 25, 217–237, 1997. [8] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon. Schr¨ odinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987.

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[9] E.B. Davies. Spectral Theory and Diﬀerential Operators, Cambridge Studies in Advanced Mathematics, Vol. 42, Cambridge University Press, Cambridge, 1995. [10] D.E. Edmunds and W.D. Evans. Spectral theory and embeddings of Sobolev spaces. Ouart. J. Math., Oxford Ser. (2), 30(120), 431–453, December 1979. [11] D.E. Edmunds and W.D. Evans. Spectral Theory and Diﬀerential Operators, Oxford University Press, Oxford OX2 6DP, 1987. [12] D.E. Edmunds and W.D. Evans. Spectral problems on arbitrary open subsets of Rn involving the distance to the boundary. J. Computational and Applied Math., 194, 36–53, 2006. [13] W.D. Evans and R.T. Lewis. Eigenvalues below the essential spectra of singular elliptic operators. Trans. Amer. Math. Soc., 297(1), 197–222, September 1986. [14] W.D. Evans and R.T. Lewis. Hardy and Rellich inequalities with remainders. J. Math. Inequal., 1, 473–490, 2007. [15] S. Filippas, V. Maz’ya, and A. Tertikas. On a question of Brezis and Marcus. Calc. Var., 25(4), 491–501, 2006. [16] S. Filippas, V. Maz’ya, and A. Tertikas. Critical Hardy-Sobolev inequalities. J. Math. Pures Appl., 87, 37–56, 2007. [17] D. Gilbarg and N.S. Trudinger. Elliptic Partial Diﬀerential Equation of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, Heidelberg, New York, 2001. [18] M. Hoﬀmann-Ostenhof, T. Hoﬀman-Ostenhof, and A. Laptev. A geometrical version of Hardy’s inequality. J. Funct. Anal. 189, 539–548, 2002. [19] T. Kato. Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966. [20] R.T. Lewis. Singular elliptic operators of second order with purely discrete spectra. Trans. Amer. Math. Soc., 271, 653-666, 1982. [21] E.H. Lieb and M. Loss. Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edition, American Mathematical Society, Providence, R.I., 2001. [22] A. Persson. Bounds for the discrete part of the spectrum of a semi-bounded Schr¨ odinger operator. Math. Scand., 8, 143–153, 1960. [23] J. Piepenbrink. Nonoscillatory elliptic equations. J. Diﬀerential Equations., 15, 541–550, 1974. [24] Ralph E. Showalter. Hilbert Space Methods for Partial Diﬀerential Equations, Monographs and Studies in Mathematics, Vol. 1. Pitman, London, San Francisco, Melbourne, 1977. [25] B. Simon. Schr¨ odinger semigroups. Bulletin A.M.S., 7(3), 447–526, November 1982. Roger T. Lewis Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 157–196 c 2012 Springer Basel AG

Continuous and Compact Embeddings of Bessel-Potential-Type Spaces Bohum´ır Opic Abstract. We survey, comment and complement our results on embeddings of Bessel potential spaces modelled upon Lorentz–Karamata spaces. Target spaces in our embeddings are either Lorentz–Karamata spaces or generalized H¨ older spaces. In particular, we establish necessary and suﬃcient conditions both for continuous and compact embeddings of spaces in question. Mathematics Subject Classiﬁcation (2010). 26A12, 46E30, 46E35, 26A15, 26A16, 47B38, 26D10, 26D15. Keywords. Slowly varying functions, Lorentz–Karamata spaces, Bessel potentials, rearrangement-invariant Banach function spaces, (fractional) Sobolev-type spaces, H¨ older-type spaces, continuous embeddings, compact embeddings.

1. Introduction Classical Bessel potential spaces H σ,p (Rn ) = H σ Lp (Rn ), introduced in [3] and [9], have played a signiﬁcant role in mathematical analysis and in applications for many years (cf. [55], [58], [1], etc.). These spaces are modelled upon the scale of Lebesgue spaces Lp (Rn ) and they coincide with the Sobolev spaces W k,p (Rn ) = W k Lp (Rn ) when σ = k ∈ N and p ∈ (1, +∞). However, it has gradually become clear that to handle some situations (especially limiting ones) a more reﬁned tuning is desirable. For this purpose, one needs to replace the Lebesgue scale of spaces by a scale of spaces which can be more ﬁnely tuned. For example, to obtain estimates of degenerate elliptic diﬀerential operators with coeﬃcients having singular behaviour, Edmunds and Triebel (cf. [24], [25]) replaced the Lp spaces by the spaces Lp (log L)q of Zygmund type. The same replacement enables to obtain interesting results The research was supported by grant no. 201/08/0383 of the Grant Agency of the Czech Republic and by the grant MSM 0021620839 of the Czech Ministry of Education.

158

B. Opic

concerning smoothness properties of orientation-preserving maps (see [44] for references and an account of work in this direction). In a series of papers [17]–[20] a systematic research of embeddings of Bessel potential spaces with order of smoothness σ ≥ 1 modelled upon generalized Lorentz–Zygmund (GLZ) spaces was carried out. The authors of those papers established embeddings of such spaces either into GLZ-spaces or into H¨ older-type spaces C 0,λ(·) (Ω) and showed that their results are sharp (within the given scale of target spaces) and fail to be compact. They also clariﬁed the role of the logarithmic terms involved in the quasi-norms of the spaces mentioned. This role proved to be important especially in limiting cases. In particular, they obtained reﬁnements of the Sobolev embedding theorems, Trudinger’s limiting embedding as well as embeddings of Sobolev spaces into λ(·)-H¨ older continuous functions including the result of Br´ezis and Wainger about almost Lipschitz continuity of elements of the (fractional) Sobolev space H 1+n/p,p (Rn ) (cf. [8]). For a survey of these results we refer to [50]. Although GLZ-spaces form an important scale of spaces containing, for example, Zygmund classes Lp (log L)α , Orlicz spaces of multiple exponential type, Lorentz spaces Lp,q , Lebesgue spaces Lp , etc., GLZ-spaces are a particular case of more general spaces, namely the Lorentz–Karamata (LK) spaces. The embeddings mentioned above were extended in [45], [47]–[48] to the case when Bessel potential spaces are modelled upon LK-spaces. Since Neves considered more general targets (besides LK-spaces and H¨older-type spaces also generalized H¨ older spaces), in several cases he obtained improvements of embeddings from [17]–[20]. The sharpness and the non-compactness of these embeddings were proved in [28] and [29]. (An account of the principal embedding results involving Bessel potential spaces modelled upon LK-spaces is also given in [16].) We should mention that in [28] and [29] we followed [36] and used a more general deﬁnition of LK-spaces; in contrast to [47]–[48], [23] and [16], we did not require a symmetry (with respect to the point 1) of slowly varying functions given on (0, +∞). In [28], where target spaces of our embeddings are LK-spaces, main ingredients of proofs of embeddings are the O’Neil inequality for convolutions and convenient Hardy-type inequalities. The sharpness and non-compactness of embeddings in question are proved by means of suitable sequences of test functions. The method is based on results and techniques of [18] and [20]. Note that in [29] one of main steps in the proof of continuous embeddings of Bessel potential spaces with order of smoothness σ ≥ 1 into generalized H¨ older spaces consists in the application of a result due to R.A. DeVore and R.C. Sharpley (cf. [15, Lemma 2]). This result states that a function u, such that the norm of its distributional gradient |∇u| belongs locally to the Lorentz space Ln,1 (Rn ), can be redeﬁned on a set of measure zero so that the modulus of smoothness ω(u, ·) of u satisﬁes the inequality ω(u, t)

tn 0

1

s n −1 |∇u|∗ (s) ds

for all t ∈ (0, 1)

(1.1)

Continuous and Compact Embeddings

159

(here |∇u|∗ denotes the non-increasing rearrangement of |∇u|). As in [28], appropriate sequences of test functions are used to prove the sharpness and non-compactness of embeddings in question. The approach is based on results and techniques of [20]. In [21] and [22] the authors analysed the situation when the order of smoothness is less than 1. In such a case one cannot use the method in which inequality (1.1) and a lifting argument (based on [19, Lemma 4.1] or [29, Lemma 4.5], which extend the Calder´ on result [9, Theorem 7]) are applied to reduce the superlimiting case to the sublimiting one, and a new approach was used. The authors of those papers established embeddings of such spaces into H¨ older-type spaces C 0,λ(·) (Ω) and showed that their results concerning non-limiting cases are sharp (within the given scale of target spaces) and fail to be compact. Results of [22] were extended in [31], where Bessel potential spaces were modelled upon LK-spaces. In [21] also another question is treated. Namely, it is shown how to get compact embeddings of Bessel-potential-type spaces into Lorentz-type spaces from sharp continuous ones in sublimiting and limiting cases. While in the situation of classical Sobolev embeddings this can be achieved by restricting the parameter on the power-type level, in our general situation the same eﬀect is caused by an appropriate modiﬁcation of slowly varying function involved in the target space. In [32] we established necessary and suﬃcient conditions for embeddings of Bessel potential spaces H σ X(Rn ) with order of smoothness less than one, modelled upon rearrangement invariant Banach function spaces X(Rn ), into λ(·) generalized H¨older spaces Λ∞,r (Ω), 0 < r ≤ +∞. (We refer to Section 2 λ(·) for precise deﬁnitions. Note also that the space Λ∞,∞ (Ω) coincides with the 0,λ(·) space C (Ω) mentioned above provided that the function λ is continuous and non-decreasing on the interval (0, 1) and satisﬁes lim λ(t) = 0.) For this t→0+

purpose, we derived a convenient replacement of (1.1). Namely, if σ ∈ (0, 1), X = X(Rn ) is a rearrangement invariant Banach function space and the Bessel potential kernel gσ belongs to the associate space of X, then we proved that tn σ ω(f ∗ gσ , t) s n −1 f ∗ (s) ds for all t ∈ (0, 1) and every f ∈ X, (1.2) 0

∗

where f denotes the non-increasing rearrangement of f . Moreover, estimate (1.2) is sharp in the sense that tn σ ω(f ∗ gσ , t) s n −1 f ∗ (s) ds for all t ∈ (0, 1) and every f ∈ X, (1.3) 0

where f (x) = f ∗ (βn |x|n )χ{y∈Rn : y1 >0}∩B(0,1) (x),

x = (x1 , · · · , xn ) ∈ Rn ,

and βn is the volume of the unit ball in Rn . Inequalities (1.2) and (1.3) enabled us to show that the continuous embedding of the Bessel potential

160

B. Opic μ(·)

space H σ X(Rn ) into the generalized H¨older space Λ∞,r (Rn ) is equivalent to the condition that gσ belongs to the associate space of X

(1.4)

and to the boundedness of the Hardy-type operator H : X −→ Lr ((0, 1); t−1/r (μ(t))−1 ), where the operator H is deﬁned by (Hf )(t) :=

tn

s n −1 f ∗ (s) ds, σ

(1.5)

(1.6)

0

X denotes the representation space of X and Lr ((0, 1); t−1/r (μ(t))−1 ) is the weighted Lebesgue space over the interval (0, 1). Furthermore, we characterized compact subsets of generalized H¨older μ(·) spaces Λ∞,r (Ω), 0 < r < +∞, with a bounded domain Ω in Rn and then we derived necessary and suﬃcient conditions for compact embeddings of μ(·) Bessel potential spaces H σ X(Rn ) into generalized H¨older spaces Λ∞,r (Ω), 0 < r < +∞. To this end, we made use of local versions of inequalities (1.2) and (1.3) to show that the compactness of the embedding in question is equivalent to (1.4) and to the compactness of the Hardy-type operator (1.5). (Note that if r = +∞, then our conditions are suﬃcient; under some additional assumptions, they are also necessary.) In [33]–[35] results of [32] were extended to the case when σ ∈ (0, n). λ(·) 1,λ(·) To this end, ﬁrst we replaced generalized H¨ older spaces Λ∞,r (Ω) = Λ∞,r (Ω) (deﬁned by means of the ﬁrst order modulus of smoothness) with generalized k,λ(·) H¨ older spaces Λ∞,r (Ω) (deﬁned by means of the k-th order modulus of k,λ(·) smoothness, k ∈ N). Note that the spaces Λ∞,r (Ω) involve as particular cases both the classical H¨older and classical Zygmund spaces. Second, we proved analogues of inequalities (1.2) and (1.3) in the case that the role of the ﬁrst-order modulus of smoothness is played with the k-th order modulus of smoothness, k ∈ N. To prove such an analogue of inequality (1.3), one needs some knowledge about the behaviour of derivatives of the Bessel potential kernel. However, in the existing literature there were available only estimates of these derivatives from above, which was not suﬃcient for our purpose. Therefore, in [34] we calculated the derivatives ∂ j gσ /∂xj1 , σ ∈ (0, n), j ∈ N, and showed that ∂ j gσ sgn j (x) = (−1)j , σ ∈ (0, n), j ∈ N, (1.7) ∂x1 for all x in a small circular half-cone which has its vertex at the origin and its axis coincides with the positive part of x1 -axis. Note that (1.7) played the essential role in the proof of the mentioned analogue of inequality (1.3). Finally, in [33] and [35] we applied our results to the case when X(Rn ) is the Lorentz–Karamata space Lp,q;b (Rn ). Applications cover both the superlimiting case when p > n/σ and the limiting case when p = n/σ. Our results extend and improve those of [21], [22] and [31]. For instance, taking the slowly

Continuous and Compact Embeddings

161

varying function (involved in the deﬁnition of the Lorentz–Karamata space) of logarithmic type and using Example 5.19 (cf. also [21, pp. 230, 231]), we obtain an interesting result which has no analogue in the classical theory of embeddings of Sobolev–Orlicz spaces. Namely, the Sobolev–Orlicz space n W k L k (log L)α (Rn ), k ∈ N and k < n, (the Sobolev space modelled upon n the Orlicz space L k (log L)α (Rn ) ≡ LΦ (Rn ), where the Young function satisﬁes Φ(t) = [t(1 + | log t|)α ]n/k , t > 0) is continuously embedded into the λ(·)-H¨ older class C 0,λ(·) (Rn ) with λ(t) = (1 + | log t|)1−k/n−α ,

t > 0,

(1.8)

that is, W k L k (log L)α (Rn ) → C 0,λ(·) (Rn ) n

(1.9)

provided that α > 1−k/n (the function λ(t) tends to 0 as t → 0+ more slowly than any function tε with ε > 0). This complements [19, Corollary 4.6] and illustrates the important role of the logarithmic term (log L)α involved in n the Sobolev–Orlicz space W k L k (log L)α (Rn ). (By the classical result, the n k, n n Sobolev space W k (R ) ≡ W k L k (Rn ), k ∈ N and k < n, is not even continuously embedded into the space L∞(Ω) for any subdomain Ω ⊂ Rn .) If k = 1 and Rn is replaced by a bounded domain Ω ⊂ Rn , then such a result also follows from [11, Theorem 3.15]. (Note that [11, Theorem 3.15] is stronger than [2, Theorem 8.40].) Embedding (1.9) with λ from (1.8) should be also compared with the following corollary of [19, Theorem 4.11] (which extends the result of [8] about “almost Lipschitz continuity”): W k+1 L k (log L)α (Rn ) → C 0,λ(·) (Rn ), n

(1.10)

with λ(t) = t (1 + | log t|)1−k/n−α ,

t > 0,

if k ∈ N, k < n, and α < 1 − k/n. Although embeddings (1.9) and (1.10) are sharp (within the given scale of target spaces), they are consequences of more precise embeddings (5.37) and (5.26), respectively, mentioned below. In the case that k < n − 1, there is even an improvement of embedding (5.26), namely embedding (5.23) in2,λ(·) volving the space Λ∞,r (Ω) deﬁned in terms of the second order modulus of smoothness. Here we present a survey of some results proved in [28], [51], [33], [34] and [35]. We also comment and complement these results. Now we are able to look at them from a uniform point of view. The survey is organized as follows. Section 2 contains notation, definitions and basic properties. In Section 3 we characterize continuous and compact embeddings of Bessel potential spaces, modelled upon Lorentz– Karamata spaces, target spaces of our embeddings are again Lorentz–Karamata spaces. Section 4 involves auxiliary results and key estimates which we need in subsequent sections, where we study embeddings of Bessel potential spaces, modelled upon rearrangement invariant Banach function spaces

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X(Rn ), into generalized H¨older spaces involving the k-modulus of smoothness. In Section 5 we characterize continuous embeddings of spaces in question, while in Section 6 we present characterization of compact embeddings. Finally, in Section 7 we mention a few related papers and a few open problems.

2. Notation, deﬁnitions and basic properties As usual, Rn denotes the Euclidean n-dimensional space. Throughout the paper μn is the n-dimensional Lebesgue measure in Rn and Ω is a μn -measurable subset of Rn . We denote by χΩ the characteristic function of Ω and put |Ω|n = μn (Ω). The family of all extended scalar-valued (real or complex) μn -measurable functions on Ω is denoted by M(Ω). The non-increasing rearrangement of f ∈ M(Ω) is the function f ∗ deﬁned by f ∗ (t) := inf {λ ≥ 0 : |{x ∈ Ω : |f (x)| > λ}|n ≤ t}

for all t ≥ 0. t By f ∗∗ we denote the maximal function of f ∗ given by f ∗∗ (t) := t−1 0 f ∗ (τ ) dτ, t > 0. Given a rearrangement-invariant Banach function space (r.i. BFS) X, the associate space is denoted by X . For general facts about rearrangementinvariant Banach function spaces we refer to [6]. Let X and Y be two (quasi-)Banach spaces. We say that X coincides with Y (and write X = Y ) if X and Y are equal in the algebraic and topological sense (their (quasi-)norms are equivalent). The symbol X → Y or X →→ Y means that X ⊂ Y and the natural embedding of X in Y is continuous or compact, respectively. For two non-negative expressions (i.e., functions or functionals) A, B, the symbol A B (or A B) means that A ≤ c B (or c A ≥ B), where c is a positive constant independent of appropriate quantities involved in A and B. If A B and A B, we write A ≈ B and say that A and B are equivalent. Throughout the paper we use the abbreviation LHS(∗) (RHS(∗)) for the left-(right-)hand side of the relation (∗). We adopt the convention that a/+∞ = 0 and a/0 = +∞ for all a > 0. If p ∈ (0, +∞], the conjugate number p is given by 1/p + 1/p = 1. Note that p is negative if p ∈ (0, 1). The symbol . p;(c,d), p ∈ (0, +∞], stands for the usual Lp -(quasi-)norm on the interval (c, d) ⊆ Rn . For ρ ∈ (0, +∞) and x ∈ Rn , B(x, ρ) = Bn (x, ρ) denotes the open ball in Rn of radius ρ and centre x. By βn we mean the volume of the unit ball in Rn . Following [36], we say that a positive and Lebesgue-measurable function b is slowly varying on (0, +∞), and write b ∈ SV (0, +∞), if, for each > 0, t b(t) is equivalent to a non-decreasing function on (0, +∞) and t− b(t) is equivalent to a non-increasing function on (0, +∞). The family of all slowly varying functions includes not only powers of iterated logarithms and the broken logarithmic functions from [26], but also such functions as

Continuous and Compact Embeddings

163

a

t → exp (|log t| ) , a ∈ (0, 1). (The last mentioned function has the interesting property that it tends to inﬁnity more quickly than any positive power of the logarithmic function.) Some basic properties of slowly varying functions are mentioned in the following lemma. Lemma 2.1 ([33, Lemma 3.1]). Let b, b1 and b2 belong to SV (0, +∞). (i) Then b1 b2 ∈ SV (0, +∞) and br ∈ SV (0, +∞) for each r ∈ R. (ii) Given positive numbers ε and κ, there are positive constants cε and Cε such that cε min{κ−ε , κε }b(t) ≤ b(κt) ≤ Cε max{κ−ε , κε }b(t) for all t > 0. (iii) If α > 0 and q ∈ (0, +∞], then for all t > 0, ≈ tα b(t)

and

lim tα b(t) = 0

and

τ α−1/q b(τ ) q;(0,t)

τ −α−1/q b(τ ) q;(t,∞)

≈ t−α b(t).

(iv) If α > 0, then t→0+

lim t−α b(t) = +∞.

t→0+

1 t ˜ (v) If 0 b(s) ds < +∞ and ˜b(t) := 0 b(s) ds s s , t ∈ (0, 1), then b ∈ SV (0, 1). 1 (vi) If 0 b(s) ds < +∞, then s t b(s) ds s lim 0 = +∞. (2.1) t→0+ b(t) We can see from Lemma 2.1 (iii) that any b ∈ SV (0, +∞) is equivalent to a 7b ∈ SV (0, +∞) which is continuous on (0, +∞). Consequently, without loss of generality, we can assume that all slowly varying functions in question are continuous on (0, +∞). (2.2) More properties and examples of slowly varying functions can be found in [59, Chapter V, p. 186], [7], [23], [43], [47] and [36]. Let p, q ∈ (0, +∞], b ∈ SV (0, +∞) and let Ω be a measurable subset of Rn . The Lorentz–Karamata (LK) space Lp,q;b (Ω) is deﬁned to be the set of all functions f ∈ M(Ω) such that

f p,q;b;Ω := t1/p−1/q b(t) f ∗ (t) q;(0,+∞) < +∞.

(2.3)

If Ω = Rn , we simply write · p,q;b instead of · p,q;b;Rn . When 0 < p < +∞, the Lorentz–Karamata space Lp,q;b (Ω) contains the characteristic function of every measurable subset of Ω with ﬁnite measure and hence, by linearity, every μn -simple function. When p = +∞, the Lorentz–Karamata space Lp,q;b (Ω) is diﬀerent from the trivial space if and only if t1/p−1/q b(t) q;(0,1) < +∞. The Lorentz–Karamata spaces Lp,q;b (introduced in [23] in the case when the function b is symmetrical with respect to the point 1) form an important scale of spaces. They are particular cases (if q ∈ [1, ∞)) of the classical

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B. Opic

Lorentz spaces Λq (w) introduced by Lorentz (cf. [42], [13]). On the other hand, if m ∈ N, α = (α1 , . . . , αm ) ∈ Rm and m 8 i b(t) = α (t) := α (2.4) i (t) for all t > 0, i=1

where, for t > 0, 1 (t) := 1 + |log t| , i (t) := 1 (i−1 (t)) if i > 1, then the LK-space Lp,q;b (Ω) is the generalized Lorentz–Zygmund space Lp,q,α (Ω) introduced in [19] and endowed with the (quasi-)norm f p,q;α;Ω , which in turn becomes the Lorentz–Zygmund space Lp,q (log L)α1 (Ω) of Bennett and Rudnick (see [5], [6]) when m = 1. If α = (0, . . . , 0), we obtain the Lorentz space Lp,q (Ω) endowed with the (quasi-)norm . p,q;Ω , which is just the Lebesgue space Lp (Ω) equipped with the (quasi-)norm . p;Ω when p = q; if p = q and m = 1, we obtain the Zygmund space Lp (log L)α1 (Ω) endowed with the (quasi-)norm . p;α1 ;Ω . If b(t) = 1 (t)α 2 (t)β , t > 0, α, β ∈ R, then we also write Lp,q (log L)α (log log L)β (Ω) instead of Lp,q;b (Ω). By a Young function Φ we mean a continuous, non-negative, strictly increasing and convex function on [0, +∞) such that lim Φ(t)/t = lim t/Φ(t) = 0. t→+∞

t→0+

The symbol LΦ (Ω) is used to denote the corresponding Orlicz space equipped with the Luxemburg norm · Φ . Orlicz spaces and Lorentz–Karamata spaces are diﬀerent scales of spaces with a non-trivial intersection. For example, assume that b is given by (2.4). If 1 0, or α1 = 0 and α2 > 0, . . . , or α1 = . . . = αm−1 = 0 and αm > 0, then (2.5) with Φ from (2.6) remains true. Moreover, if μn (Ω) < ∞, αm = −a where a > 0, and, if m > 1, αi = 0 for i = 1, . . . , m − 1, then (cf. [52]) L∞,∞;b (Ω) = EXPm L1/a (Ω) := LΦ (Ω)

(2.7)

(if m = 1, we omit the index m in (2.7)), where Φ(t) = (exp ◦ exp ◦ . . . ◦ exp)(t1/a ) for all large t. 9 :; < m times

Note also that if b is given by (2.4) and p = q, then the space Lp,q;b (Ω) does not coincide with an Orlicz space (cf. [52, p. 444]). The Bessel kernel gσ , σ > 0, is deﬁned as that function on Rn whose −n/2 Fourier transform is g= (1+|ξ|2 )−σ/2 , ξ ∈ Rn , where the Fourier σ (ξ) = (2π) ˆ transform f of a function f is given by fˆ(ξ) = (2π)−n/2 Rn e−iξ·x f (x) dx.

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165

Let us summarize the basic properties of the Bessel kernel gσ : gσ is a positive, integrable function which is analytic except at the origin; gσ is σ−n −c2 |x|

radially decreasing;

(2.8) (2.9)

gσ (x) ≤ c1 |x| e for 0 < σ < n and all x ∈ R \{0}; gσ (x) ≈ |x|σ−n as |x| → 0 if 0 < σ < n; ∂ σ−n−1 g (x) for 0 < σ ≤ n + 1, j ∈ {1, . . . , n} ∂xj σ ≤ c|x| and all x ∈Rn \{0}; n

(2.10) (2.11) (2.12)

1/n

gσ∗ (t) t(σ−n)/n e−ct for 0 < σ < n and all t > 0 (2.13) (c, c1 and c2 are positive constants). Property (2.9) follows from formula (26) in [55, Chapter V]. For the proof of (2.8), (2.10)–(2.12) see [4], for (2.13) see [17]. Let σ > 0 and let X = X(Rn ) = X(Rn , μn ) be a r.i. Banach function space endowed with the norm · X . The Bessel potential space H σ X(Rn ) is deﬁned by H σ X(Rn ) := {u : u = f ∗ gσ , f ∈ X(Rn )} (2.14) and is equipped with the norm

u H σ X := f X .

(2.15)

Note that, given f ∈ X, the convolution u = f ∗ gσ is well deﬁned and ﬁnite μn -a.e. on Rn since the measure space (Rn , μn ) is resonant and so (cf. [6, Theorem II.6.6]) X → L1 (Rn ) + L∞ (Rn ). When σ = 0, we put H σ X(Rn ) := X(Rn ). If p ∈ (1, +∞], q ∈ [1, +∞] and b ∈ SV (0, +∞), then the space Lp,q;b (Rn ) coincides with a r.i. Banach function space X(Rn ) (the (quasi-) norm (2.3) is equivalent to the norm t1/p−1/q b(t) f ∗∗ (t) q;(0,+∞) ). Consequently, if σ > 0, p ∈ (1, +∞], q ∈ [1, +∞] and b ∈ SV (0, +∞), then H σ Lp,q;b (Rn ) := H σ X(Rn ) is the usual Bessel potential space modelled upon the Lorentz–Karamata space Lp,q;b (Rn ), which is equipped with the (quasi-) norm

u σ;p,q;b := f p,q;b . (2.16) α m When m ∈ N, α = (α , . . . , α ) ∈ R and b = , we obtain the loga1

m

rithmic Bessel potential space H σ Lp,q;α (Rn ), endowed with the (quasi-)norm

u σ;p,q;b and considered in [19]. Note that if α = (0, . . . , 0), H σ Lp,p;α (Rn ) is simply the (fractional) Sobolev space H σ,p (Rn ) of order σ. Given k ∈ N and a Banach space Y = Y (Rn ) of functions on Rn , we denote by W k Y (Rn ) the corresponding Sobolev space, that is, the space of all f on Rn whose distributional derivatives Dα f , |α| ≤ k, belong to Y (Rn ). This space is equipped with the norm

f W k Y =

D α f Y . |α|≤k

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Theorem 2.2 (Lifting argument). Let k ∈ N, σ ∈ [k, +∞) and let X = X(Rn ) = X(Rn , μn ) be a r.i. BFS. Suppose that the Boyd indices of X belong to the interval (0, 1) and that the space X has absolutely continuous norm. Then H σ X(Rn ) = W k H σ−k X(Rn ). Proof. The proof can be carried out as in [9, Theorem 7] and [55, Chap. V, Lemma 3]. When k ∈ N, p ∈ (1, +∞), q ∈ [1, +∞) and b ∈ SV (0, +∞), then, by Theorem 2.2, H k Lp,q;b (Rn ) = W k Lp,q;b (Rn ). Let Ω be a domain in Rn . We denote by B(Ω) the set of all scalar-valued functions (real or complex) which are bounded on Ω and we equip this set with the norm

f B(Ω) := sup{|f (x)| : x ∈ Ω}. The subspace of B(Ω) of all continuous functions on Ω is denoted by CB (Ω) and it is equipped with the B(Ω)-norm. By C(Ω) we mean the subspace of CB (Ω) of all uniformly continuous functions on Ω. Let Ω be a domain in Rn and let k ∈ N. For each h ∈ Rn , put Ω(kh) := {x ∈ Ω : x + th ∈ Ω, 0 ≤ t ≤ k}. The ﬁrst diﬀerence operator Δh is deﬁned on scalar functions f ∈ B(Ω) by Δh f (x) = f (x + h) − f (x) for all x ∈ Ω(h), and higher order diﬀerences are deﬁned inductively by k Δk+1 h f (x) = Δh (Δh f )(x),

x ∈ Ω((k + 1)h).

The k-modulus of smoothness of a function f in CB (Ω) is given by ωk,Ω (f, t) := sup Δkh f |B(Ω(kh)) |h|≤t

for all t ≥ 0.

If k = 1, we write ωΩ (f, t) instead of ω1,Ω (f, t). It is clear that the k-modulus of smoothness depends on a given domain Ω. In what follows we shall sometimes omit the subscript Ω at the k-modulus of smoothness since it will be always clear from the context which k-modulus of smoothness we have in mind. Let k ∈ N, r ∈ (0, +∞] and let Lkr be the class of all continuous functions λ : (0, 1) → (0, +∞) such that t−1/r

1 λ(t)

r;(0,1)

tk λ(t)

r;(0,1)

= +∞

(2.17)

< +∞.

(2.18)

and t−1/r

When r = +∞, we simply write Lk instead of Lkr .

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167

Let k ∈ N, r ∈ (0, +∞], λ ∈ Lkr and let Ω be a domain in Rn . The k;λ(·) generalized H¨ older space Λ∞,r (Ω) consists of all functions f ∈ CB (Ω) for which the quasi-norm −1/r

f |Λk,λ(·) ∞,r (Ω) := f |B(Ω) + t

ωk (f, t) λ(t)

(2.19) r;(0,1)

k,λ(·)

is ﬁnite. Standard arguments show that the space Λ∞,r (Ω) is complete (cf. k,λ(·) [45, Theorem 3.1.4]). Note that the scale Λ∞,r (Ω) contains both the classical H¨ older and classical Zygmund spaces. Conditions (2.17) and (2.18) are natural. In fact, if (2.17) does not hold, k,λ(·) then the space Λ∞,r (Ω) coincides with CB (Ω). If (2.18) does not hold, then t−1/r

ωk (f, t) λ(t)

r;(0,1)

is ﬁnite if and only if f is a polynomial of degree less or equal to k − 1. If r = +∞, we can assume without loss of generality in the deﬁnition k,λ(·) of Λ∞,r (Ω) that all the elements λ of Lkr are continuous non-decreasing functions on the interval (0, 1) satisfying lim λ(t) = 0. Indeed, if λ : (0, 1) → t→0+

1 1 (0, +∞) is deﬁned by = sup , t ∈ (0, 1), then λ is non-decreasing λ(t) t≤s<1 λ(s) and (2.17) implies that limt→0+ λ(t) = 0. Since λ is a continuous function, the function λ is continuous on (0, 1) as well. Now, by exchanging suprema, we obtain ωk (f, t) 1 = sup sup ωk (f, t) λ(s) 0
ωk (f, s) , 0<s<1 λ(s)

= sup k,λ(·)

f ∈ CB (Rn ).

k,λ(·)

Hence, Λ∞,∞ (Ω) = Λ∞,∞ (Ω). k,λ(·)

Note also that (cf. [46, Proposition 3.5]) the latter space Λ∞,∞ (Ω) coincides with the space C 0,λ(·) (Ω) deﬁned by

f |C 0,λ(·) (Ω) := sup |f (x)| + x∈Ω

sup x,y∈Ω

0<|x−y|≤1 1,λ(·)

|f (x) − f (y)| < +∞. λ(|x − y|)

If λ(t) = t, t ∈ (0, 1), then Λ∞,∞ (Ω) coincides with the space Lip(Ω) 1,λ(·) of the Lipschitz functions. If λ(t) ≡ tα , α ∈ (0, 1), then the space Λ∞,r (Ω) coincides with the space C 0,α,r (Ω) considered in [2, p. 232]. The next lemma shows that if we deﬁne the generalized H¨older space k,λ(·) Λ∞,r (Ω) as a subspace of C(Ω) rather than a subspace of CB (Ω), then both

168

B. Opic

deﬁnitions coincide provided that Ω is a domain in Rn with minimally smooth boundary (see [6, p. 430] or [55, p. 189]). Lemma 2.3 ([35, Lemma 2.1]). Let Ω be a domain in Rn with minimally smooth boundary. Let k ∈ N, r ∈ (0, +∞] and let λ ∈ Lkr . Then Λk,λ(·) ∞,r (Ω) → C(Ω). Now, we recall some more properties of moduli of smoothness. For each ﬁxed f in CB (Ω), ωk (f, ·) is a non-negative non-decreasing function on [0, +∞). Putting ω 7k (f, t) := ωk (f, t)/tk for all t > 0, one can prove that ω 7k (f, ·) is equivalent to a non-increasing function on (0, +∞). If f ∈ CB (Ω), then ωk (f, t) ≤ 2k f |B(Ω) , t > 0. (2.20) We refer to [6, pp. 331–333, 431], [13, pp. 40–50], [14] and [39] for more details about k-modulus of smoothness. The next lemma is a straightforward extension of [34, Lemma 4.2]. Lemma 2.4 ([35, Lemma 3.4]). Let Ω be a domain in Rn with minimally smooth boundary. Let k ∈ N and let S be a bounded subset of CB (Ω) such that lim sup ωk (u, t) = 0. Then lim sup ω(u, t) = 0. In particular, if u ∈ CB (Ω) t→0+ u∈S

t→0+ u∈S

satisﬁes lim ωk (u, t) = 0, then lim ω(u, t) = 0, which means that u ∈ C(Ω). t→0+

t→0+

The following lemma is related to [19, Lemma 4.5]. Lemma 2.5 ([32, Lemma 3.3]). Let X = X(Rn ) be a r.i. BFS and let Ω be a domain in Rn . Suppose that σ > 0 and let gσ be the Bessel kernel. Then H σ X(Rn ) → B(Ω) if and only if

gσ X < +∞. We shall investigate embeddings of the form H σ X(Rn ) → Y (Ω),

(2.21)

n

where σ > 0, X = X(R ) is a r.i. Banach function space, Ω is a domain in Rn and Y (Ω) is a convenient Banach space of functions deﬁned on Ω. Note that embedding (2.21) means that the mapping u → u|Ω from H σ X(Rn ) into Y (Ω) is continuous. Note also that in the whole paper we use the symbol u both for the function u and its restriction to Ω.

3. Embeddings into Lorentz–Karamata spaces In this section we present embeddings of Bessel-potential-type spaces into Lorentz–Karamata spaces, which extend and slightly improve those of [18], [20] and complement those of [47]. Our main results state that such embeddings are sharp and fail to be compact. We also show how to modify parameters of target spaces to arrive to compact embeddings.

Continuous and Compact Embeddings

169

The ﬁrst theorem concerns the sublimiting case when σ ∈ (0, n) and 1 < p < n/σ. Part (i) of this theorem extends [20, Theorem 3.1] and [47, Theorem 5.1] and corresponds to the Sobolev-type embedding. The second and the third parts concern the (local) sharpness of the embedding from part (i) while the fourth part states that the embedding from part (i) is not compact. In part (v) it is shown that this embedding becomes compact if the parameters of the target space are restricted in a proper way. Theorem 3.1 (cf. [28, Theorem 3.1]). Let σ ∈ (0, n), 1 < p < n/σ, q ∈ [1, +∞], r ∈ [q, +∞], 1/pσ = 1/p − σ/n and let b ∈ SV (0, +∞). Let Ω ⊂ Rn be a non-empty domain. (i) Then H σ Lp,q;b (Rn ) → Lpσ ,r;b (Rn ). (3.1) (ii) Let P ∈ [pσ , +∞] and let ˜b ∈ SV (0, +∞). Suppose that either P > pσ or P = pσ and

lim

t→0+

(3.2) ˜b(t) = +∞. b(t)

(3.3)

Then the embedding H σ Lp,q;b (Rn ) → LP,r;˜b (Ω)

(3.4)

does not hold. (iii) Let q ∈ (0, q). Then the embedding H σ Lp,q;b (Rn ) → Lpσ ,q;b (Ω) fails. (iv) The embedding H σ Lp,q;b (Rn ) → Lpσ ,r;b (Ω)

(3.5)

is not compact. (v) Let Ω be bounded and let ˜b ∈ SV (0, +∞). Suppose that either P ∈ (0, pσ ) or

(3.6) ˜b(t) = 0. b(t)

(3.7)

H σ Lp,q;b (Rn ) →→ LP,r;˜b (Ω).

(3.8)

P = pσ and

lim

t→0+

Then Remark 3.2 ([28, Remarks 3.2–3.6]). (i) By Theorem 3.1 (ii), all embeddings (3.1) are sharp with respect to the ﬁrst and third parameters of the target space. (This is why we consider all embeddings (3.1) in Theorem 3.1 and not only optimal embedding (3.10) mentioned below.) (ii) As Lpσ ,r;b (Rn ) → Lpσ ,s;b (Rn ) if 0 < r < s < +∞,

(3.9)

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among embeddings (3.1) the embedding H σ Lp,q;b (Rn ) → Lpσ ,q;b (Ω),

(3.10)

with Ω = Rn , is optimal. (Note that embedding (3.9) can be proved analogously as the classical embedding Lpσ ,r (Rn ) → Lpσ ,s (Rn ) if 0 < r < s < +∞.) (iii) Theorem 3.1 (iii) shows that embedding (3.10) is also sharp with respect to the second parameter. (iv) By Theorem 3.1 (i), embedding (3.10) is continuous and, by Theorem 3.1 (iv), this embedding is not compact. Moreover, Theorem 3.1 (iv) also shows that we cannot arrive to a compact embedding if we replace the target space Lpσ ,q;b (Ω) in (3.10) by a larger space Lpσ ,r;b (Ω) with r > q. (v) Put X := H σ Lp,q;b (Rn ). By Theorem 3.1 (i), sup t1/pσ b(t)f ∗ (t) f X

for all

t>0

f ∈ X,

and, by Theorem 3.1 (ii), the inequality sup t1/pσ ˜b(t)f ∗ (t) f X t>0

does not hold for all f ∈ X if ˜b ∈ SV (0, +∞) satisﬁes lim

t→0+

˜b(t) = +∞. b(t)

If we use an analogue of terminology from [38] or [57], this means that the function [t1/pσ b(t)]−1 , t > 0, is the growth envelope function of the space H σ Lp,q;b (Rn ). Using also Theorem 3.1 (iii), we can see that the couple ([t1/pσ b(t)]−1 , q) is the growth envelope of the space H σ Lp,q;b (Rn ). Remark 3.3. One can easily verify that the assumptions of part (ii) of Theorem 3.1 are equivalent to: Let P ∈ (0, +∞], ˜b ∈ SV (0, +∞) and let lim

t→0+

t1/P ˜b(t) = +∞. t1/pσ b(t)

(3.11)

Similarly, the assumptions of part (v) Theorem 3.1 can be rewritten as: Let Ω be bounded, P ∈ (0, +∞], ˜b ∈ SV (0, +∞) and let lim

t→0+

t1/P ˜b(t) = 0. t1/pσ b(t)

(3.12)

Note also that, by Lemma 2.1 (iii), t1/ρ b(t) ≈ τ 1/ρ−1/r b(τ ) r;(0, t) for all t ∈ (0, +∞) when 0 < ρ < +∞, 0 < r ≤ +∞ and b ∈ SV (0, +∞). Thus, the function t → t1/ρ b(t) is (equivalent) to the fundamental function of the Lorentz–Karamata space Lρ,r;b (we refer to [6] for this notion).

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171

Proof of Theorem 3.1. Since pσ < +∞, parts (i)–(iv) coincide with parts (i)– (iv) of [28, Theorem 3.1]. Thus, we refer to [28] for proofs of these statements. The proof of part (v) of Theorem 3.1 is similar to that of [21, Theorem 3.1]. Corollary 3.4. Let all the assumptions of Theorem 3.1 be satisﬁed. Let P ∈ (0, +∞] and ˜b ∈ SV (0, +∞). (i) Let |Ω|n < +∞. Then H σ Lp,q;b (Rn ) → LP,r;˜b (Ω)

(3.13)

if and only if lim

t→0+

t1/P ˜b(t) < +∞. t1/pσ b(t)

(3.14)

(ii) Let Ω be bounded. Then H σ Lp,q;b (Rn ) →→ LP,r;˜b (Ω)

(3.15)

t1/P ˜b(t) = 0. t→0+ t1/pσ b(t)

(3.16)

if and only if lim

Proof. (i) Since singularities of slowly varying functions b and ˜b at the interval [0, |Ω|n ] are only those at 0, part (i) of the corollary is a consequence of parts (i) and (ii) of Theorem 3.1 (see also Remark 3.3). (ii) The implication (3.16) ⇒ (3.15) holds by Theorem 3.1 (v) (see also Remark 3.3). The converse implication follows by contradiction from estimates [28, (5.4)], [28, (5.6)] and the second displayed estimate in Step 5 of the proof of [28, Theorem 3.1] (cf. also [28, (5.10)]). The next theorem represents an analogue of Theorem 3.1 and concerns the limiting case when p = n/σ. Part (i) of this theorem extends [18, Theorem 3.1 and Theorem 3.2]. Theorem 3.5 (cf. [28, Theorem 3.7]). Let σ ∈ (0, n), q ∈ (1, +∞], r ∈ [q, +∞] and let b ∈ SV (0, +∞) be such that

t−1/q [b(t)]−1 q ;(0,1) = +∞.

(3.17)

Suppose that Ω ⊂ Rn is a non-empty domain with |Ω|n < +∞ and that bqr ∈ SV (0, +∞) satisﬁes bqr (t) := [b(t)]

−q /r

2

τ

−1

[b(τ )]

−q

−1/q −1/r dτ

for all t ∈ (0, 1]. (3.18)

t

(i) Then H σ Ln/σ,q;b (Rn ) → L∞,r;bqr (Ω).

(3.19)

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(ii) If a function ˜b ∈ SV (0, +∞) is such that lim

t→0+

τ −1/r ˜b(τ ) r;(0,t) = +∞,

τ −1/r bqr (τ ) r;(0,t)

(3.20)

then the embedding H σ Ln/σ,q;b (Rn ) → L∞,r;˜b (Ω)

(3.21)

does not hold. (iii) Let q ∈ (0, q). Then the embedding H σ Ln/σ,q;b (Rn ) → L∞,q;bqq (Ω) fails, where bqq is again deﬁned by (3.18) with r replaced by q. (iv) The embedding H σ Ln/σ,q;b (Rn ) → L∞,r;bqr (Ω)

(3.22)

is not compact. (v) Let Ω be bounded and let ˜b ∈ SV (0, +∞) be such that

τ −1/r ˜b(τ ) r;(0,t) = 0. t→0+ τ −1/r bqr (τ ) r;(0,t)

(3.23)

H σ Ln/σ,q;b (Rn ) →→ L∞,r;˜b (Ω).

(3.24)

lim

Then Proof. Parts (i)–(iv) coincide with parts (i)–(iv) of [28, Theorem 3.7]. Thus, we refer to [28] for proofs of these statements. The proof of part (v) of Theorem 3.5 makes use of the same main ideas as that of [21, Theorem 3.4]. However, to estimate analogues of quantities J1 , J2 and J3 (J1 , J2 and J3 are introduced in the proof of [21, Theorem 3.4]), one has to employ the estimate 2 −1/q −1/r −1 −q

τ bqr (τ ) r;(0,t) ≈ τ [b(τ )] dτ = bq∞ (t) for all t ∈ (0, 1) (3.25) t

(which follows from (3.18) and (3.17)) and some further properties of slowly varying functions. Therefore, it is much shorter to proceed as follows: Put Y := L∞,r,˜b (Ω), X = Lp,q,b (Rn ) and K := {u ∈ H σ X : ||u||H σ X ≤ 1}. As in the proof of [21, Theorem 3.4], it is suﬃcient to verify that, given ε > 0, there is δ > 0 such that

u χM Y ≤ ε for all u ∈ K and every M ⊂ Ω with |M |n < δ. Let ε > 0. Assumption (3.23) implies that there is δ > 0 such that B1 (δ) := sup

0
τ −1/r ˜b(τ )) r;(0,R) < ε.

τ −1/r bqr (τ ) r;(0,R)

Assume that u ∈ K and let M ⊂ Ω satisfy |M |n < δ. Then

u χM Y ≤ t−1/r ˜b(t) u∗ (t) r;(0,δ) .

(3.26)

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173

Moreover, by theorem on embeddings of classical Lorentz spaces (cf., e.g., [10, Theorem 3.1]) and by (3.26),

t−1/r˜b(t) u∗ (t) r;(0,δ) ≤ B1 (δ) t−1/r bqr (t) u∗ (t) r;(0,δ) < ε u ∞,r;bqr . Finally, by Theorem 3.5 (i),

u ∞,r;bqr u H σ X ≤ 1. Combining estimates mentioned above, we arrive at u χM Y ε for all u ∈ K and every M ⊂ Ω with |M |n < δ. Corollary 3.6. Let all the assumptions of Theorem 3.5 be satisﬁed. Let ˜b ∈ SV (0, +∞). (i) Let |Ω|n < +∞. Then H σ Ln/σ,q;b (Rn ) → L∞,r;˜b (Ω)

(3.27)

if and only if lim

t→0+

τ −1/r ˜b(τ ) r;(0,t) < +∞.

τ −1/r bqr (τ ) r;(0,t)

(3.28)

(ii) Let Ω be bounded. Then H σ Ln/σ,q;b (Rn ) →→ L∞,r;˜b (Ω)

(3.29)

if and only if lim

t→0+

τ −1/r ˜b(τ ) r;(0,t) = 0.

τ −1/r bqr (τ ) r;(0,t)

(3.30)

Proof. (i) Singularities of slowly varying functions b and ˜b at the interval [0, |Ω|n ] are only those at 0. Thus, part (i) of the corollary is a consequence of parts (i) and (ii) of Theorem 3.5 and [10, Theorem 3.1]. Indeed, Theorem 3.5 (i), (3.28) and [10, Theorem 3.1] imply that H σ Ln/σ,q;b (Rn ) → L∞,r;bqr (Ω) → L∞,r;˜b (Ω). On the other hand, if (3.28) is not satisﬁed, then, by Theorem 3.5 (ii), embedding (3.27) fails. (ii) The implication (3.30) ⇒ (3.29) holds by Theorem 3.5 (v). The converse implication follows by contradiction from estimates [28, (5.16)], [28, (5.22)] and (3.25) (cf. also [28, (5.23)]). Remark 3.7 (cf. [28, Remarks 3.8–3.18]). (i) Theorem 3.5 (i) holds without assumption (3.17). However, if q ∈ (1, +∞) and t−1/q [b(t)]−1 q ;(0,1) < +∞, then H σ Ln/σ,q;b (Rn ) → CB (Rn ), c.f. [48, Proposition 5.6]. (ii) Assume that all the assumptions of Theorem 3.5 are satisﬁed. If r ∈ [q, +∞], then the embedding H σ Ln/σ,q;b (Rn ) → L∞,r;˜b (Ω)

(3.31)

174

B. Opic

with ˜b = bqr is sharp with respect to the parameter ˜b ∈ SV (0, +∞), that is, the target space L∞,r;˜b (Ω) in (3.31) and the space L∞,r;bqr (Ω) (i.e., the target space in (3.31) with ˜b = bqr ) satisfy L∞,r;bqr (Ω) → L∞,r;˜b (Ω). Indeed, the last embedding is equivalent to

t−1/r ˜b(t) f ∗ (t) r;(0,|Ω|n ) t−1/r bqr (t) f ∗ (t) r;(0,|Ω|n) for all f ∈ L∞,r;bqr (Ω). This inequality holds (cf., e.g., [10, Theorem 3.1]) if sup x∈(0,|Ω|n ))

t−1/r˜b(t) r;(0,|Ω|n ) / t−1/r bqr (t) r;(0,|Ω|n) < +∞,

which is equivalent to (3.28). The result follows from Corollary 3.6 (i) since (3.28) is satisﬁed when (3.31) holds. (iii) The target spaces in (3.19) form a scale {L∞,r;bqr (Ω)}+∞ r=q whose endpoint spaces with r = +∞ and r = q are of particular interest. The former endpoint space L∞,∞;bq∞ (Ω) corresponds to the target space in Trudinger’stype embedding while the latter endpoint space L∞,q;bqq (Ω) corresponds to the target space in the Br´ezis-Wainger-type embedding. Since the spaces {L∞,r;bqr (Ω)}+∞ r=q satisfy L∞,r;bqr (Ω) → L∞,s;bqs (Ω)

if q ≤ r ≤ s ≤ +∞,

(3.32)

the embedding H σ Ln/σ,q;b (Rn ) → L∞,q;bqq (Ω)

(3.33)

is optimal. (Making use of estimate (3.25), one can prove (3.32) analogously as the embedding Lp,r (Ω) → Lp,s (Ω), 0 < p < +∞, 0 < r ≤ s ≤ +∞. Another way how to verify (3.32) is to apply, e.g., [10, Theorem 3.1].) (iv) Theorem 3.5 (iii) shows that embedding (3.33) is also sharp with respect to the second parameter. (v) By Theorem 3.5 (i), embedding (3.33) is continuous and, by Theorem 3.5 (iv), this embedding is not compact. Moreover, Theorem 3.5 (iv) also shows that we cannot arrive to a compact embedding if we replace the target space L∞,q;bqq (Ω) in (3.33) by a larger space L∞,r;bqr (Ω) with r > q. (vi) Put X := H σ Ln/σ,q;b (Rn ). By Theorem 3.5 (i), sup bq∞ (t)f ∗ (t) f X t>0

for all f ∈ X,

and, by Theorem 3.5 (ii) (cf. also part (vii) of this remark), the inequality sup ˜b(t)f ∗ (t) f X t>0

does not hold for all f ∈ X if ˜b ∈ SV (0, +∞) satisﬁes lim

t→0+

˜b(t) = +∞. bq∞ (t)

Continuous and Compact Embeddings This means that the function [bq∞ (t)]−1 =

175

1/q 2 −1 −q τ [b(τ )] dτ , t ∈ (0, 1], t σ n H Ln/σ,q;b (R ). Using also The-

is the growth envelope function of the space orem 3.5 (iii), we can see that the pair 1/q 2 τ −1 [b(τ )]−q dτ ,q t

is the growth envelope of the space H σ Ln/σ,q;b (Rn ). (vii) Let r ∈ [q, +∞]. By (3.25),

τ −1/r ˜b(τ ) r;(0,t)

τ −1/r ˜b(τ ) r;(0,t) lim ≈ lim . −1/r t→0+ τ bq∞ (t) bqr (τ ) r;(0,t) t→0+

(3.34)

Together with the estimate

τ −1/r ˜b(τ ) r;(0,t) = τ 1−1/r [τ −1˜b(τ )] r;(0,t) t−1˜b(t) τ 1−1/r r;(0,t) ≈ ˜b(t) for all t ∈ (0, 1), this shows that condition (3.20) holds if ˜b(t) lim = +∞. t→0+ bq∞ (t)

(3.35)

(3.36)

(viii) Let r = +∞ and let ˜b be equivalent to a non-decreasing function on some interval (0, δ), δ ∈ (0, 1). Then, for all t ∈ (0, δ),

τ −1/r ˜b(τ ) r;(0,t) = ˜b(τ ) ∞;(0,t) ≈ ˜b(t). Applying this estimate in (3.34), we can see that (3.20) is now equivalent to (3.36). (ix) Let r ∈ [q, +∞) and let

τ −1/r ˜b(τ ) r;(0,T ) < +∞ for some T ∈ (0, 1). (3.37) Then τ −1/r ˜b(τ ) r;(0,t) → 0 as t → 0+ . Since also, by (3.25) and (3.17),

τ −1/r bqr (τ ) r;(0,t) → 0 as t → 0+, a convenient version of L’Hospital’s rule implies that ˜b(t)

τ −1/r ˜b(τ ) r;(0,t) lim = lim (3.38) −1/r t→0+ τ bqr (τ ) r;(0,t) t→0+ bqr (t) provided that the last limit exists. Remark 3.8. It follows from Remark 3.3 that conditions (3.2), (3.3) are of the same form as condition (3.20). The same concerns conditions (3.6), (3.7) and condition (3.23). Examples 3.9 (cf. [30, Examples 3.1]). Let σ ∈ (0, n), p = n/σ, q ∈ (1, +∞] and let r ∈ [q, +∞]. Suppose that Ω ⊂ Rn is a non-empty domain with |Ω|n < +∞. 1. Let α, β ∈ R and let b ∈ SV (0, +∞) be deﬁned by b(t) = 1 (t)α 2 (t)β ,

t > 0.

176

B. Opic

(i) If α < 1/q and β ∈ R, then H σ Lp,q (log L)α (log log L)β (Rn ) → L∞,q;bqq (Ω) → L∞,r;bqr (Ω) → L∞,∞;bq∞ (Ω) (3.39) with

bqq (t) ≈ 1 (t)α−1 2 (t)β , t ∈ (0, 1]; bqr (t) ≈ 1 (t)α−1/q −1/r 2 (t)β , t ∈ (0, 1]; bq∞ (t) ≈ 1 (t)α−1/q 2 (t)β , t ∈ (0, 1].

Note (cf. [27, Lemma 2.2]) that L∞,∞;bq∞ (Ω) coincides with the Orlicz space LΦ (Ω), where Φ(t) = exp(t1/γ (log t)β/γ )

for all large t,

γ = 1/q − α.

(ii) If α = 1/q and β < 1/q , then H σ Lp,q (log L)α (log log L)β (Rn ) → L∞,q;bqq (Ω) → L∞,r;bqr (Ω) → L∞,∞;bq∞ (Ω) (3.40) with

bqq (t) ≈ 1 (t)−1/q 2 (t)β−1 , t ∈ (0, 1]; bqr (t) ≈ 1 (t)−1/r 2 (t)β−1/q −1/r , t ∈ (0, 1]; bq∞ (t) ≈ 2 (t)β−1/q , t ∈ (0, 1].

Recall that (see (2.7)) L∞,∞;bq∞ (Ω) = EXP2 L1/γ (Ω), where γ = 1/q − β. (iii) If α = 1/q and β = 1/q , then H σ Lp,q (log L)α (log log L)β (Rn ) → L∞,q;bqq (Ω) → L∞,r;bqr (Ω) → L∞,∞;bq∞ (Ω) (3.41) with

bqq (t) ≈ 1 (t)−1/q 2 (t)−1/q 3 (t)−1 , t ∈ (0, 1]; bqr (t) ≈ 1 (t)−1/r 2 (t)−1/r 3 (t)−1/q −1/r , t ∈ (0, 1]; bq∞ (t) ≈ 3 (t)−1/q , t ∈ (0, 1].

Recall that (see (2.7)) L∞,∞;bq∞ (Ω) = EXP3 Lq (Ω). (iv) If either α > 1/q or α = 1/q and β > 1/q , then H σ Lp,q (log L)α (log log L)β (Rn ) → CB (Rn ). 2. Let α ∈ (0, 1), β ∈ R and let b ∈ SV (0, +∞) be deﬁned by

b(t) = 1 (t)−(α−1)/q exp(β 1 (t)α ),

t > 0.

(i) If β < 0, then H σ Lp,q;b (Rn ) → L∞,q;bqq (Ω) → L∞,r;bqr (Ω) → L∞,∞;bq∞ (Ω) with

bqq (t) ≈ 1 (t)(α−1)/q exp(β 1 (t)α ), bqr (t) ≈ 1 (t)(α−1)/r exp(β 1 (t)α ), bq∞ (t) ≈ exp(β 1 (t)α ), t ∈ (0, 1].

t ∈ (0, 1]; t ∈ (0, 1];

(3.42)

Continuous and Compact Embeddings

177

(ii) If β > 0, then H σ Lp,q;b (Rn ) → CB (Rn ).

4. Auxiliary results and key estimates In this section we summarize some results, which we shall need in subsequent sections. Theorem 4.4 mentioned below gives sharp estimates for the k-modulus of smoothness of the convolution of a function f from a r.i. BFS X(Rn ) with the Bessel potential kernel gσ , 0 < σ < n, with k ≥ [σ] + 1. Such estimates are essential in what follows. The following results concern embeddings of generalized H¨older spaces. Theorem 4.1 ([33, Theorem 3.6]). Let k1 , k2 ∈ N, k1 ≤ k2 , λ ∈ Lkq 1 and μ ∈ Lkr 2 . (i) If 0 < q ≤ r ≤ +∞ and sup

t

− r1

>

−1

(μ(t))

1

t− q (λ(t))−1

r;(x,1)

x∈(0,1/2)

< +∞,

(4.1)

q;(x,1)

then 1 ,λ(·) 2 ,μ(·) Λk∞,q (Rn ) → Λk∞,r (Rn ).

(ii) If 0 < r < q < +∞ and 1/2 1 1 t− r (μ(t))−1 t− q (λ(t))−1

− qr

r;(x,1)

0

where

1 u

:=

1 r

(4.2)

u (λ(x))

q;(x,1)

− 1q , then (4.2) holds.

−q

dx < +∞, x (4.3)

Theorem 4.2 ([33, Theorem 3.7]). Let k1 , k2 ∈ N, k1 < k2 , 1 ≤ q ≤ r ≤ +∞, λ ∈ Lkq 2 and μ ∈ Lkr 1 . Assume that sup

t−1/r+k1 (μ(t))−1

t−1/q −k1 λ(t) r;(0,x)

x∈(0,1)

q ;(x,1)

< +∞.

(4.4)

Then 2 ,λ(·) 1 ,μ(·) Λk∞,q (Rn ) → Λk∞,r (Rn ).

Corollary 4.3 ([33, Corollary 3.8]). Let k1 , k2 ∈ N, k1 < k2 , q ∈ [1, +∞] and let λ ∈ Lkq 1 (this implies that λ ∈ Lkq 2 as well ). Suppose that sup x∈(0,1)

t−1/q+k1 (λ(t))−1

t−1/q −k1 λ(t) q;(0,x)

q ;(x,1)

< +∞.

(4.5)

Then 2 ,λ(·) 1 ,λ(·) Λk∞,q (Rn ) = Λk∞,q (Rn ).

We continue with key estimates for the k-modulus of smoothness of the convolution of a function f from a r.i. BFS X(Rn ) with the Bessel potential kernel gσ , 0 < σ < n, with k ≥ [σ] + 1.

178

B. Opic

Theorem 4.4 ([34, Theorem 3.1]). Let σ ∈ (0, n) and let X = X(Rn ) be a r.i. BFS such that gσ X < +∞. Then f ∗ gσ ∈ C(Rn ) for all f ∈ X and tn σ ωk (f ∗ gσ , t) s n −1 f ∗ (s) ds for all t ∈ (0, 1) and every f ∈ X, (4.6) 0

where k ≥ [σ] + 1. Moreover, estimate (4.6) is sharp in the sense that given k ∈ N, there are (small enough) δ ∈ (0, 1) and (big enough) α > 0 such that tn σ ωk (f ∗ gσ , t) s n −1 f ∗ (s) ds for all t ∈ (0, 1) and every f ∈ X, (4.7) 0

where f (x) := f ∗ (βn |x|n )χCα (0,δ) (x),

x = (x1 , . . . , xn ) ∈ Rn , (4.8) n n 2 2 Cα (0, δ) := Cα ∩ B(0, δ) with Cα := {y ∈ R : y1 > 0, y1 > α i=2 yi }. Remark 4.5. A particular case of Theorem 4.4 with σ ∈ (0, 1) and k = 1 was proved in [32] (cf. [32, Theorem 1]). Remark 4.6 ([34, Remark 3.8]). We shall investigate the compactness of the embedding H σ X(Rn ) → Λk,μ(·) (4.9) ∞,r (Ω), where Ω will be a bounded domain in Rn . Note that, by (4.9), the restriction k,μ(·) to Ω of a function u ∈ H σ X(Rn ) belongs to the space Λ∞,r (Ω).∗ ) Note also n that u = f ∗ gσ for some f ∈ X(R ). Under the assumptions of Theorem 4.4, k,μ(·) u ∈ C(Rn ), which implies that u ∈ C(Ω). To calculate u|Λ∞,r (Ω) , we need the k-modulus of smoothness ωk (u, t) = ωk,Ω (u, t), t ≥ 0, of the function u. Recall also that the k-modulus of smoothness ω(f ∗ gσ , ·) involved in Theorem 4.4 is the k-modulus of smoothness with respect to the whole Rn , that is, ωk,Rn (f ∗ gσ , ·). To characterize the compactness of embedding (4.9), we shall need analogues of estimates (4.6) and (4.7) with ωk replaced by ωk,Ω . Since ωk,Ω (f ∗ gσ , t) ≤ ωk,Rn (f ∗ gσ , t),

t ≥ 0,

estimate (4.6) implies that tn σ ωk,Ω (f ∗gσ , t) s n −1 f ∗ (s) ds for all t ∈ (0, 1) and every f ∈ X, (4.10) 0

where k ≥ [σ] + 1. To get an analogue of (4.7), take x0 = (x01 , . . . , x0n ) ∈ Ω and δ1 ∈ (0, 1] so that B(x0 , δ1 ) ⊂ Ω. Then, for given k ∈ N, tn σ ωk,Ω (f ∗gσ , t) s n −1 f ∗ (s) ds for all t ∈ (0, 1) and every f ∈ X, (4.11) 0

where f (x) := f ∗ (βn |x − x0 |n )χCα (x0 ,δ2 ) (x), ∗)

x = (x1 , · · · , xn ) ∈ Rn ,

(4.12)

In the whole paper we use the symbol u both for the function u and its restriction to Ω.

Continuous and Compact Embeddings

179

Cα (x0 , δ2 ) := (x0 + Cα ) ∩ B(x0 , δ2 ) with Cα := {y ∈ Rn : y1 > 0, y12 > n α i=2 yi2 }, δ2 := min{δ, δ1 } and δ ∈ (0, 1) is given by Theorem 4.4. Indeed, to prove it, take t ∈ (0, δk2 ) and put t = (−t, 0, · · · , 0) ∈ Rn and u = f ∗ gσ . Then, instead of [34, (4.12)], we now have k k ωk,Ω (u, t) ≥ |(Δkt u)(x0 )| = (−1)k−i u(x0 + it) i i=0 k

k ∗ n k−i = f (βn |y − x0 | ) (−1) gσ (x0 + it − y) dy Cα (x0 ,δ2 ) i i=0 k

k = f ∗ (βn |y|n ) (−1)k−i gσ (y − it) dy , (4.13) Cα (0,δ2 ) i i=0

with Cα (0, δ2 ) = −x0 + Cα (x0 , δ2 ), and the same arguments as those used in part (ii) of the proof of Theorem 4.4 yield (4.11). In what follows we shall omit again the subscript Ω at k-modulus of smoothness (since it will be always clear from the context which k-modulus of smoothness we have in mind).

5. Continuous embeddings into generalized H¨ older spaces The aim of this section is to make use of results of [34], that is, estimate (4.6) and its reverse form (4.7), to characterize embeddings of the Bessel potential space H σ X(Rn ) with the order of smoothness σ ∈ (0, n), modelled upon k,μ(·) r.i. BFS X(Rn ), into generalized H¨older spaces Λ∞,r (Rn ), 0 < r ≤ +∞, involving the k-modulus of smoothness. Recall that the scale of generalized H¨ older spaces contains as particular cases the classical H¨older and classical Zygmund spaces. Our main results are the next theorem and its corollary, where we reduce the given problem to inequality (5.2) below, which involves a Hardy-type operator. Theorem 5.1 ([33, Theorem 1.1]). Let σ ∈ (0, n) and let X = X(Rn ) = X(Rn , μn ) be a r.i. BFS such that gσ X < +∞. Put k := [σ] + 1, assume that r ∈ (0, +∞] and μ ∈ Lkr . Then n H σ X(Rn ) → Λk,μ(·) ∞,r (R )

if and only if 1

t− r (μ(t))−1

0

tn

τ n −1 f ∗ (τ ) dτ σ

f X for all f ∈ X.

(5.1)

(5.2)

r;(0,1)

Remark 5.2. The implication (5.1) =⇒ (5.2) in Theorem 5.1 remains true if k ∈ N (cf. Theorem 4.4 and the proof of the necessity part of Theorem 5.1). Corollary 5.3 ([33, Corollary 1.2]). Let σ ∈ (0, n) and let X = X(Rn ) be a r.i. BFS. Put k := [σ] + 1, assume that r ∈ (0, +∞] and μ ∈ Lkr . Then embedding (5.1) holds if and only if gσ X < +∞ and (5.2) is satisﬁed.

180

B. Opic

We apply our results to the case when X(Rn ) is the Lorentz–Karamata space Lp,q;b (Rn ). The corresponding continuous embeddings are characterized in Theorems 5.4 and 5.14 below. The former theorem concerns the superlimiting case p > n/σ while the latter theorem is devoted to the limiting case when p = n/σ. Theorem 5.4 ([33, Theorem 1.3]). Let σ ∈ (0, n), p ∈ ( nσ , +∞), q ∈ [1, +∞], b ∈ SV (0, +∞), r ∈ (0, +∞], k := [σ] + 1 and μ ∈ Lkr . Let λ : (0, 1) → (0, +∞) be deﬁned by λ(x) := xσ− p (b(xn ))−1 n

for all

x ∈ (0, 1).

(5.3)

(Note that λ ∈ Lkr for any r ∈ (0, +∞]; recall that b is continuous (cf. (2.2)).) (i) If 1 ≤ q ≤ r ≤ +∞, then n H σ Lp,q;b (Rn ) → Λk,μ(·) ∞,r (R )

(5.4)

if and only if 1

t− r (μ(t))−1

lim

x→0+

λ(x) < +∞.

(5.5)

r;(x,1)

(ii) If 0 < r < q ≤ +∞ and q > 1, then n H σ Lp,q;b (Rn ) → Λk,μ(·) ∞,r (R )

if and only if 1

1

t− r (μ(t))−1 r;(x,1)

0

where

1 u

:=

1 r

λ(x)

u dx < +∞, x

(5.6)

(5.7)

− 1q .

Remark 5.5. Continuous embeddings of spaces H σ Lp,q;b (Rn ) with σ ∈ (0, 1) 1,μ(·) into generalized H¨ older spaces Λ∞,r (Rn ) in the superlimiting case (that is, when p > n/σ) were characterized in [32, Theorem 3]. Theorem 5.4 extends 1,μ(·) this result to the case when σ ∈ (0, n) and when the space Λ∞,r (Rn ) is [σ]+1,μ(·) replaced by Λ∞,r (Rn ). Here, the formulation is slightly diﬀerent because in [32] the deﬁnition of the class L1r was more restrictive (in particular, the function μ ∈ L1r was equivalent to an increasing function on the interval (0, 1]). Remark 5.6. As in Remark 5.2, we see that in Theorem 5.4 the implications (5.4) =⇒ (5.5) and (5.6) =⇒ (5.7) remain true if k ∈ N. Remark 5.7. Assume that all the assumptions of Theorem 5.4 are satisﬁed. (i) When r ∈ [1, +∞], then, by Corollary 4.3, [σ− n ]+1,λ(·)

p n Λk,λ(·) ∞,r (R ) = Λ∞,r

(Rn ).

(ii) Let m ∈ N. If r ∈ [q, +∞] and the embedding n H σ Lp,q;b (Rn ) → Λm,μ(·) ∞,r (R )

(5.8)

Continuous and Compact Embeddings

181

holds, then (5.8) with μ = λ is sharp with respect to the parameter μ, that m,μ(·) m,λ(·) is, the target space Λ∞,r (Rn ) in (5.8) and the space Λ∞,r (Rn ) (i.e., the target space in (5.8) with μ = λ) satisfy m,μ(·) n n Λm,λ(·) ∞,r (R ) → Λ∞,r (R ).

(5.9)

Indeed, if (5.8) holds, then (5.5) follows from Remark 5.6. Moreover, (5.5) is equivalent to > 1 1 sup t− r (μ(t))−1 t− r (λ(t))−1 < +∞. r;(x,1)

x∈(0,1/2)

r;(x,1)

Thus, we obtain (5.9) from Theorem 4.1 (i). (iii) Let m ∈ N. Among embeddings (5.8) that one with μ = λ and m,μ(·) r = q is optimal, that is, the target space Λ∞,r (Rn ) in (5.8) and the space m,λ(·) Λ∞,q (Rn ) (i.e., the target space in (5.8) with μ = λ and r = q) satisfy m,μ(·) n n Λm,λ(·) ∞,q (R ) → Λ∞,r (R ).

(5.10)

Indeed, if r ∈ [q, +∞], this follows as in part (ii). To verify it when 0 < r < q, note that, by Theorem 4.1 (ii), (5.10) is satisﬁed if 1/2 u dx 1 t− r (μ(t))−1 λ(x) < +∞. x r;(x,1) 0 The last condition is equivalent to (5.7). The result follows from Remark 5.6 since (5.7) is satisﬁed when (5.8) holds. (iv) If k ∈ N, the embedding n H σ Lp,q;b (Rn ) → Λk,μ(·) ∞,r (R )

(5.11)

does not hold with μ = λ and r ∈ (0, q). Indeed, by Remark 5.6, embedding 1

(5.11) implies (5.7). If μ = λ, then (5.7) is violated since t− r (λ(t))−1 r;(x,1)

≈ (λ(x))−1 for all x ∈ (0, 1/2) (cf. Lemma 2.1 (iii)).

Corollary 5.8 ([33, Corollary 1.4]). Assume that all the assumptions of The[σ− n p ]+1

orem 5.4 are satisﬁed and μ ∈ Lr

. If 1 ≤ q ≤ r ≤ +∞, then

[σ− n ]+1,μ(·)

H σ Lp,q;b (Rn ) → Λ∞,r p

(Rn )

(5.12)

if and only if (5.5) is satisﬁed. In particular, (5.12) holds with μ = λ. Consequently, if σ = np + 1 and r = q, then n H σ Lp,q;b (Rn ) → Λ2,λ(·) ∞,q (R ).

(5.13)

The next result improves [29, Theorem 3.2 (i)] provided that σ = 1+np < n. Corollary 5.9 ([33, Corollary 1.5]). Let σ ∈ (1, n), p = r ∈ [q, +∞] and let b ∈ SV (0, +∞) be such that

t−1/q [b(t)]−1 q ;(0,1) = +∞.

n σ−1 ,

q ∈ (1, +∞], (5.14)

182

B. Opic

Deﬁne λ by (5.3) and λqr by n

λqr (t) := t [b(t )]

q /r

2

τ

−1

[b(τ )]

−q

1/q +1/r dτ

,

t ∈ (0, 1).

tn

(Note that λqr ∈ L1r ; recall that b is continuous (cf. (2.2)).) Then 1,λqr (·) n H σ Lp,q;b (Rn ) → Λ2,λ(·) (Rn ). ∞,q (R ) → Λ∞,r

(5.15)

Remark 5.10. (i) Note that in [33, Corollary 1.5] we have assumed that q ∈ (1, +∞). However, this corollary holds even if q ∈ (1, +∞]. Note also that the second embedding in (5.15) follows from Theorem 4.2. 1,λ (.) (ii) Since the scale of spaces {Λ∞,rqr (Rn )}+∞ r=q from (5.15) satisﬁes qr (.) qs (.) Λ1,λ (Rn ) → Λ1,λ (Rn ) if q ≤ r ≤ s ≤ +∞, ∞,r ∞,s

(5.16)

embedding (5.15) can be rewritten as 1,λqq (·) qr (·) n H σ Lp,q;b (Rn ) → Λ2,λ(·) (Rn ) → Λ1,λ (Rn ) ∞,q (R ) → Λ∞,q ∞,r q∞ (·) → Λ1,λ (Rn ). ∞,∞

(5.17)

(Making use of the fact that

t−1/r+1 (λgr (t))−1 r ,(0,x) ≈ t−1/q (b(t))−1 −1 q ,(xn ,1) , one can prove (5.16) analogously as the embedding Lp,r (Ω) → Lp,s (Ω), 0 < p < +∞, 0 < r ≤ s ≤ +∞, provided that one replaces the role of f ∗ (t) by the role of ω 7 (f, t).) (iii) Note that the embeddings qq (·) qr (·) q∞ (·) H σ Lp,q;b (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,q ∞,r ∞,∞

(5.18)

n involved in (5.17) remain true even if σ ∈ (1, n + 1) (and p = σ−1 ) provided that q ∈ (1, +∞). Indeed, when σ ∈ (1, n + 1) and q ∈ (1, +∞), the ﬁrst embedding in (5.18) can be proved using the lifting argument (i.e., Theorem 2.2), Theorem 3.5 (i), inequality (1.1) and a convenient Hardy-type inequality (cf. the proof of [29, Theorem 3.1 (i)] or the proof of [49, Theorem 5.7]). The remaining embeddings in (5.18) follow again from part (ii) of this remark.

Using Corollary 5.9 and Remark 5.10 (ii), we obtain the following example. Examples 5.11 (cf. [30, Examples 3.2] or [51, Examples 4.9]). Let σ ∈ (1, n), p = n/(σ − 1), q ∈ (1, +∞] and let r ∈ [q, +∞]. 1. Let α, β ∈ R and let b ∈ SV (0, +∞) be deﬁned by b(t) := 1 (t)α 2 (t)β

for all

t ∈ (0 + ∞).

λ(t) := tσ− p (b(tn ))−1

for all

t ∈ (0, 1).

Put n

Continuous and Compact Embeddings

183

(i) If α < 1/q and β ∈ R, then n H σ Lp,q (log L)α (log log L)β (Rn ) → Λ2,λ(·) ∞,q (R ) qq (·) qr (·) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,q ∞,r q∞ (·) → Λ1,λ (Rn ), ∞,∞

with

(5.19)

λqq (t) ≈ t 1 (t)1−α 2 (t)−β , t ∈ (0, 1]; λqr (t) ≈ t 1 (t)1/r+1/q −α 2 (t)−β , t ∈ (0, 1]; λq∞ (t) ≈ t 1 (t)1/q −α 2 (t)−β , t ∈ (0, 1].

(ii) If α = 1/q and β < 1/q , then n H σ Lp,q (log L)α (log log L)β (Rn ) → Λ2,λ(·) ∞,q (R ) qq (·) qr (·) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,q ∞,r q∞ (·) → Λ1,λ (Rn ), ∞,∞

with

(5.20)

λqq (t) ≈ t 1 (t)1/q 2 (t)1−β , t ∈ (0, 1]; λqr (t) ≈ t 1 (t)1/r 2 (t)1/r+1/q −β , t ∈ (0, 1]; λq∞ (t) ≈ t 2 (t)1/q −β , t ∈ (0, 1].

(iii) If α = 1/q and β = 1/q , then n H σ Lp,q (log L)α (log log L)β (Rn ) → Λ2,λ(·) ∞,q (R ) qq (·) qr (·) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,q ∞,r q∞ (·) → Λ1,λ (Rn ), ∞,∞

with

(5.21)

λqq (t) ≈ t 1 (t)1/q 2 (t)1/q 3 (t), t ∈ (0, 1]; λqr (t) ≈ t 1 (t)1/r 2 (t)1/r 3 (t)1/r+1/q , t ∈ (0, 1]; λq∞ (t) ≈ t 3 (t)1/q , t ∈ (0, 1].

(iv) If either α > 1/q or α = 1/q and β > 1/q , then n n H σ Lp,q (log L)α (log log L)β (Rn ) → Λ2,λ(·) ∞,q (R ) → Lip(R ).

2. Let α ∈ (0, 1), β ∈ R and let b ∈ SV (0, +∞) be deﬁned by

b(t) = 1 (t)−(α−1)/q exp(β 1 (t)α ),

t > 0.

(i) If β < 0, then 1,λqq (·) qr (·) n H σ Lp,q;b (Rn ) → Λ2,λ(·) (Rn ) → Λ1,λ (Rn ) ∞,q (R ) → Λ∞,q ∞,r q∞ (·) → Λ1,λ (Rn ), ∞,∞

with

λqq (t) ≈ t 1 (tn )−(α−1)/q exp(−β 1 (tn )α ), λqr (t) ≈ t 1 (tn )−(α−1)/r exp(−β 1 (tn )α ), λq∞ (t) ≈ t exp(−β 1 (tn )α ), t ∈ (0, 1].

t ∈ (0, 1]; t ∈ (0, 1];

184

B. Opic

(ii) If β > 0, then n n H σ Lp,q;b (Rn ) → Λ2,λ(·) ∞,q (R ) → Lip(R ).

In particular, choosing β = 0 and q = p in part 1(i) of Example 5.11, we get the following example. n Example 5.12. If n ∈ N, n > 1, n−1 < p < +∞, b(t) = α (t), t ∈ (0, +∞), 1 and α < p , we obtain from (5.19) that 1,λpp (·) n H 1+ p Lp (log L)α (Rn ) → Λ2,λ(·) (Rn ) ∞,p (R ) → Λ∞,p n

pr (·) p∞ (·) → Λ1,λ (Rn ) → Λ1,λ (Rn ), ∞,r ∞,∞

(5.22)

where λ(x) := x −α (x) and

for all x ∈ (0, 1),

λpr (x) ≈ x ((x))1/p +1/r−α ,

Moreover, if k :=

n p

x ∈ (0, 1), r ∈ [p, +∞].

∈ N, we obtain 2,λ(·)

pp (·) W k+1 Ln/k (log L)α (Rn ) → Λ∞,n/k (Rn ) → Λ1,λ (Rn ) ∞,p pr (·) p∞ (·) → Λ1,λ (Rn ) → Λ1,λ (Rn ), ∞,r ∞,∞

(5.23)

with λ as before and λpr (x) ≈ x ((x))

1−k/n+1/r−α

,

x ∈ (0, 1), r ∈ [n/k, +∞].

If α = 0, this example shows that the Br´ezis–Wainger embedding of the Sobolev space W k+1,n/k (Rn ) := W k+1 Ln/k (Rn ), k ∈ N and k < n − 1, into the space of “almost” Lipschitz functions (cf. [8, Corollary 5]) is a consequence 2,Id(·) of a better embedding whose target is the Zygmund-type space Λ∞,n/k (Rn ) (Id(·) stands for the identity map). Note also that the space Ln/k (log L)α (Rn ) mentioned in (5.23) is the Orlicz space LΦ (Rn ), whose Young function Φ satisﬁes (cf. [52]) Φ(t) ≈ [tα (t)]n/k for all t ∈ (0, +∞).

(5.24)

Thus, the ﬁrst embedding in (5.23) states that 2,λ(·)

W k+1 LΦ (Rn ) → Λ∞,n/k (Rn ), where the Young function Φ is given by (5.24). Such an embedding is optimal and it does not follow from known results on embeddings of Sobolev–Orlicz spaces (see, for example, [2, Chap. 8], [41, Chap. 7] or [11, Theorem 3.15], where embeddings of Sobolev–Orlicz spaces with targets involving only the ﬁrst order of modulus of smoothness were investigated). Remark 5.13. By Remark 5.10 (iii), the embeddings pp (·) pr (·) H 1+ p Lp (log L)α (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,p ∞,r n

p∞ (·) → Λ1,λ (Rn ) ∞,∞

(5.25)

Continuous and Compact Embeddings

185

and pp (·) pr (·) W k+1 Ln/k (log L)α (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,p ∞,r p∞ (·) → Λ1,λ (Rn ), ∞,∞

(5.26)

involved in (5.22) and (5.23), respectively, of Example 5.12 remain true even if σ ∈ (1, n + 1) and 1 < p < +∞ provided that q ∈ (1, +∞). (A similar remark can be made about Example 5.11.) The following assertions represents a limiting case of Theorem 5.4. Theorem 5.14 ([33, Theorem 1.6]). Let σ ∈ (0, n), p = nσ , q ∈ (1, +∞], r ∈ (0, +∞], k := [σ] + 1, μ ∈ Lkr and let b ∈ SV (0, +∞) be such that −1

t q (b(t))−1 q ;(0,1) < +∞. Let λqr be deﬁned by n 1 + r1 q x dt q /r n −q λqr (x) := b (x ) b (t) , x ∈ (0, 1). (5.27) t 0 (Note that λqr ∈ Lkr ; recall that b is continuous (cf. (2.2)).) (i) If 1 < q ≤ r ≤ +∞, then n H σ Lp,q;b (Rn ) → Λk,μ(·) ∞,r (R )

(5.28)

if and only if 1

lim

x→0+

t− r (μ(t))−1 r;(x,1) 1

t− r (λqr (t))−1 r;(x,1)

< +∞.

(5.29)

(ii) If 0 < r < q ≤ +∞ and q > 1, then n H σ Lp,q;b (Rn ) → Λk,μ(·) ∞,r (R )

(5.30)

if and only if u n −1 1/2 x

t−1/r (μ(t))−1 r;(x,1) dt dx −q b (t) b−q (xn ) < +∞, −1/r (λ (t))−1 t x

t qr r;(x,1) 0 0 (5.31) where u1 := 1r − 1q .

Remark 5.15. As in Remark 5.2, we see that in Theorem 5.14 the implications (5.28) =⇒ (5.29) and (5.30) =⇒ (5.31) remain true if k ∈ N. Remark 5.16. Assume that all the assumptions of Theorem 5.14 are satisﬁed. (i) When r ∈ [1, +∞], then, by Corollary 4.3, qr (·) qr (·) Λk,λ (Rn ) = Λ1,λ (Rn ). ∞,r ∞,r

(ii) Let m ∈ N. If r ∈ [q, +∞] and the embedding n H σ Lp,q;b (Rn ) → Λm,μ(·) ∞,r (R )

(5.32)

holds, then (5.32) with μ = λqr is sharp with respect to the parameter μ, that m,λ (·) m,μ(·) is, the target space Λ∞,r (Rn ) in (5.32) and the space Λ∞,rqr (Rn ) (i.e.,

186

B. Opic m,λ

(·)

m,μ(·)

the target space in (5.32) with μ = λqr ) satisfy Λ∞,rqr (Rn ) → Λ∞,r (Rn ). Indeed, the last embedding hods if > 1 − r1 −1 sup t (μ(t)) t− r (λqr (t))−1 < +∞ x∈(0,1/2)

r;(x,1)

r;(x,1)

(cf. Theorem 4.1 (i)), which is equivalent to (5.29). The result follows from Remark 5.15 since (5.29) is satisﬁed when (5.32) holds. (iii) Let m ∈ N. Among embeddings (5.32) that one with μ = λqq and m,μ(·) r = q is optimal, that is, the target space Λ∞,r (Rn ) in (5.32) and the space m,λ (·) Λ∞,qqq (Rn ) (i.e., the target space in (5.32) with μ = λqq and r = q) satisfy qq (·) n Λm,λ (Rn ) → Λm,μ(·) ∞,q ∞,r (R ).

(5.33)

Indeed, if r ∈ [q, +∞], this follows as in part (ii). To verify it when 0 < r < q, note that, by Theorem 4.1 (ii), (5.33) is satisﬁed if u 1/2 − qr dx − 1q − 1r −1 −1 t (μ(t)) t (λqq (t)) (λqq (x))−q < +∞, x r;(x,1) q;(x,1) 0 which is equivalent to (5.31). The result follows from Remark 5.15 since (5.31) is satisﬁed when (5.32) holds. (iv) If m ∈ N, the embedding n H σ Lp,q;b (Rn ) → Λm,μ(·) ∞,r (R )

does not hold if μ = λqr and r ∈ (0, q) (this follows from Remark 5.15). Corollary 5.17 ([33, Corollary 1.7]). Assume that all the assumptions of Theorem 5.14 are satisﬁed and μ ∈ L1r . If 1 < q ≤ r ≤ +∞, then n H σ Lp,q;b (Rn ) → Λ1,μ(·) ∞,r (R )

(5.34)

if and only if (5.29) is satisﬁed. In particular, (5.34) holds with μ = λqr . Using Corollary 5.17 and Remark 5.16, we obtain the next example, which extends some embeddings of [51, Example 5.7] to the case σ ∈ (0, n). Examples 5.18. Let σ ∈ (0, n), p = n/σ, q ∈ (1, +∞] and let r ∈ [q, +∞]. 1. Let α, β ∈ R and let b ∈ SV (0, +∞) be deﬁned by b(t) = 1 (t)α 2 (t)β ,

t > 0.

(i) If α > 1/q and β ∈ R, then qq (·) qr (·) H σ Lp,q (log L)α (log log L)β (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,q ∞,r q∞ (·) → Λ1,λ (Rn ), ∞,∞

with

λqq (t) ≈ 1 (t)1−α 2 (t)−β , t ∈ (0, 1]; λqr (t) ≈ 1 (t)1/r+1/q −α 2 (t)−β , t ∈ (0, 1]; λq∞ (t) ≈ 1 (t)1/q −α 2 (t)−β , t ∈ (0, 1].

(5.35)

Continuous and Compact Embeddings

187

(ii) If α = 1/q and β > 1/q , then qq (·) qr (·) H σ Lp,q (log L)α (log log L)β (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,q ∞,r q∞ (·) → Λ1,λ (Rn ), ∞,∞

with λqq (t) ≈ 1 (t)1/q 2 (t)1−β , t ∈ (0, 1]; λqr (t) ≈ 1 (t)1/r 2 (t)1/r+1/q −β , t ∈ (0, 1]; λq∞ (t) ≈ 2 (t)1/q −β , t ∈ (0, 1]. (iii) If either α < 1/q or α = 1/q and β < 1/q or α = 1/q and β = 1/q , then we have (3.39) or (3.40) or (3.41), respectively. 2. Let α ∈ (0, 1), β ∈ R and let b ∈ SV (0, +∞) be deﬁned by

b(t) = 1 (t)−(α−1)/q exp(β 1 (t)α ),

t > 0.

(i) If β > 0, then qq (·) qr (·) q∞ (·) H σ Lp,q;b (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ), ∞,q ∞,r ∞,∞

with λqq (t) ≈ 1 (tn )−(α−1)/q exp(−β 1 (tn )α ), λqr (t) ≈ 1 (tn )−(α−1)/r exp(−β 1 (tn )α ), λq∞ (t) ≈ exp(−β 1 (tn )α ), t ∈ (0, 1].

t ∈ (0, 1]; t ∈ (0, 1];

(ii) If β < 0, then (3.42) holds. In particular, choosing β = 0 and q = p in part 1(i) of Example 5.18, we get the following example. Example 5.19. Let n ∈ N, σ ∈ (0, n), p = b(t) = α (t), t ∈ (0, +∞), and α > q1 . Then

n σ

and let r ∈ [q, +∞]. Let

pp (·) pr (·) H p Lp (log L)α (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,p ∞,r n

p∞ (·) → Λ1,λ (Rn ), ∞,∞

(5.36)

with λpr (x) := ((x)) Moreover, if k :=

n p

1/p +1/r−α

,

x ∈ (0, 1), r ∈ [p, +∞].

∈ N, then (5.36) gives

pp (·) pr (·) W k Ln/k (log L)α (Rn ) → Λ1,λ (Rn ) → Λ1,λ (Rn ) ∞,p ∞,r p∞ (·) → Λ1,λ (Rn ), ∞,∞

(5.37)

with λpr (x) := ((x))

1−k/n+1/r−α

,

x ∈ (0, 1), r ∈ [n/k, +∞],

which is another interesting result on sharp embeddings of Sobolev–Orlicz spaces W k Ln/k (log L)α (Rn ) into generalized H¨older spaces.

188

B. Opic

Remark 5.20. Assume that all the assumptions of Theorem 5.14 are satisﬁed. If 1 < q ≤ r ≤ +∞, then, by Corollary 5.17, qr (·) H σ Lp,q;b (Rn ) → Λ1,λ (Rn ). ∞,r

(5.38)

Note that embedding (5.38) continue to hold even if σ ∈ (1, n + 1) provided that q ∈ (1, +∞). This can be proved using the lifting argument, Theorem 3.1 (i), inequality (1.1) and a convenient Hardy-type inequality.

6. Compact embeddings into generalized H¨ older spaces The aim of this section is to characterize compact embeddings of the Bessel potential space H σ X(Rn ) with the order of smoothness σ ∈ (0, n), modelled k,μ(·) upon r.i. BFS X(Rn ), into generalized H¨older spaces Λ∞,r (Ω), where 0 < n r ≤ +∞ and Ω is a bounded domain in R . To this end, it is essential k,μ(·) to characterize totally bounded subsets of the space Λ∞,r (Ω), Ω having minimally smooth boundary. The result is given in the following theorem. Theorem 6.1 ([35, Theorem 1.3]). Let k ∈ N, r ∈ (0, +∞), μ ∈ Lkr and let Ω be a bounded domain in Rn with minimally smooth boundary. Then k,μ(·) k,μ(·) S ⊂ Λ∞,r (Ω) is totally bounded if and only if S is bounded in Λ∞,r (Ω) and 1 sup t− r (μ(t))−1 ωk (u, t) r;(0,ξ) → 0 as ξ → 0+ . (6.1) u∈S

Remark 6.2. (i) In Theorem 6.1 the implication S ⊂ Λk,μ(·) ∞,r (Ω) is bounded and (6.1) holds =⇒ S is totally bounded in Λk,μ(·) ∞,r (Ω)

(6.2)

remains true even if r = +∞. (This can be seen from the proof of Theorem 6.1.) (ii) If r = +∞ in Theorem 6.1, then the reverse implication to (6.2) holds k,μ(·),0 k,μ(·),0 provided that we assume S ⊂ Λ∞,∞ (Ω). Here Λ∞,∞ (Ω) is a subspace of k,μ(·) Λ∞,∞ (Ω) consisting of those functions u which satisfy lim (μ(t))−1 ωk (u, t) ∞;(0,δ) = 0.

δ→0+

(This follows from the necessity part of the proof of Theorem 6.1.) (iii) Summarizing what we have said, we arrive at the following result. Let μ ∈ Lk and let Ω be a bounded domain in Rn with minimally smooth k,μ(·),0 k,μ(·) boundary. Then S ⊂ Λ∞,∞ (Ω) is totally bounded in Λ∞,∞ (Ω) if and only k,μ(·) if S is bounded in Λ∞,∞ (Ω) and sup (μ(t))−1 ωk (u, t) ∞;(0,ξ) → 0 as ξ → 0+ .

u∈S

Our main result (which is an analogue of Theorem 5.1) reads as follows.

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189

Theorem 6.3 ([35, Theorem 1.4]). Let σ ∈ (0, n) and let X = X(Rn ) = X(Rn , μn ) be a r.i. BFS such that gσ X < ∞. Put k := [σ] + 1, assume that r ∈ (0, +∞), μ ∈ Lkr and that Ω is a bounded domain in Rn . Then †

H σ X(Rn ) →→ Λk,μ(·) ∞,r (Ω) if and only if 1

t− r (μ(t))−1

sup f X ≤1

tn

τ n −1 f ∗ (τ ) dτ σ

0

)

(6.3)

→ 0 as ξ → 0+ .

(6.4)

r;(0,ξ)

Remark 6.4. The implication (6.3) =⇒ (6.4) in Theorem 6.3 remains true if k ∈ N (cf. Theorem 4.4, Remark 4.6 and the necessity part in the proof of Theorem 6.3). Remark 6.5. (i) In Theorem 6.3 the implication (6.4) =⇒ (6.3) remains true even if r = +∞. (This can be seen from Remark 6.2 (i) and the proof of Theorem 6.3.) (ii) We see from Remark 6.2 (iii) that if we assume additionally in Theorem 6.3 that r = +∞ and the space X(Rn ) and μ ∈ Lk are such that H σ X(Rn ) → Λk,μ(·),0 ∞,∞ (Ω),

(6.5)

then (6.3) is equivalent to (6.4). (iii) For example, (6.5) is satisﬁed provided that the Schwartz space S (Rn ) is dense in H σ X(Rn ), σ

n

H X(R ) →

(6.6)

Λk,μ(·) ∞,∞ (Ω),

(6.7)

k

lim t /μ(t) = 0.

(6.8)

t→0+

Indeed, given u ∈ H σ X(Rn ) and ε > 0, there is v ∈ S (Rn ) such that

u − v H σ X < ε. Moreover, ωk (v, t) ≤ ctk for all t ∈ (0, 1), where c = c(v) is a positive constant. Thus, using also (6.7), we obtain

(μ(t))−1 ωk (u, t) ∞;(0,δ) ≤ (μ(t))−1 ωk (u − v, t) ∞;(0,δ) + (μ(t))−1 ωk (v, t) ∞;(0,δ) u − v H σ X + c tk /μ(t) ∞;(0,δ) ≤ ε + c tk /μ(t) ∞;(0,δ)

for all δ ∈ (0, 1).

Together with (6.8), this implies (6.5). For instance, (6.6) holds if the Schwartz space S (Rn ) is dense in X(Rn).

(6.9)

Indeed, this is a consequence of (2.14), (2.15), the fact that the mapping h → gσ ∗ h maps S (Rn ) on S (Rn ), and (6.9). In particular, (6.9) is satisﬁed provided that the r.i. BFS X(Rn ) has absolutely continuous norm (cf. [19, Remark 3.13]). †)

k,μ(·)

Recall that this means that the mapping u → u|Ω from H σ X(Rn ) into Λ∞,r (Ω) is compact.

190

B. Opic

Corollary 6.6 ([35, Corollary 1.5]). Let σ ∈ (0, n) and let X = X(Rn ) be a r.i. BFS. Put k := [σ] + 1, assume that r ∈ (0, +∞), μ ∈ Lkr and that Ω is a bounded domain in Rn . Then (6.3) holds if and only if gσ X < +∞ and (6.4) is satisﬁed. As in [33], we apply our main result (Theorem 6.3) to the case when X(Rn ) is a Lorentz–Karamata space Lp,q;b (Rn ). The corresponding compact embeddings are characterized in Theorems 6.7 and 6.11 (and Corollaries 6.10 and 6.13) below. The former theorem concerns the superlimiting case p > n/σ while the latter one is devoted to the limiting case p = n/σ. Theorem 6.7 ([35, Theorem 1.6]). Let σ ∈ (0, n), p ∈ ( nσ , +∞), q ∈ [1, +∞], b ∈ SV (0, +∞), r ∈ (0, +∞), k := [σ] + 1 and μ ∈ Lkr . Assume that Ω is a bounded domain in Rn . Let λ : (0, 1) → (0, +∞) be deﬁned by λ(x) := xσ− p (b(xn ))−1 for all x ∈ (0, 1). n

(6.10)

(Note that λ ∈ Lkr for any r ∈ (0, +∞]; recall that b is continuous (cf. (2.2)).) (i) If 1 ≤ q ≤ r < +∞, then H σ Lp,q;b (Rn ) →→ Λk,μ(·) ∞,r (Ω)

(6.11)

if and only if 1

lim

x→0+

t− r (μ(t))−1

λ(x) = 0.

(6.12)

r;(x,1)

(ii) If 0 < r < q ≤ +∞ and q > 1, then H σ Lp,q;b (Rn ) →→ Λk,μ(·) ∞,r (Ω) if and only if 1 where

1 u

:=

0 1 r

1

t− r (μ(t))−1

λ(x) r;(x,1)

u dx < +∞, x

(6.13)

(6.14)

− 1q .

Remark 6.8. Compact embeddings of spaces H σ Lp,q;b (Rn ) with σ ∈ (0, 1) 1,μ(·) into generalized H¨ older spaces Λ∞,r (Ω) in the superlimiting case (that is, when p > n/σ) were characterized in [32, Theorem 7]. Theorem 6.7 extends 1,μ(·) this result to the case when σ ∈ (0, n) and when Λ∞,r (Ω) is replaced by [σ]+1,μ(·) Λ∞,r (Ω). Its formulation is slightly diﬀerent because in [32] the deﬁnition of the class L1r was more restrictive (in particular, the function μ ∈ L1r was equivalent to an increasing function on the interval (0, 1]). Remark 6.9. (i) As in Remark 6.4, we see that in Theorem 6.7 the implications (6.11) =⇒ (6.12) and (6.13) =⇒ (6.14) remain true if k ∈ N. (ii) Recall that, by Remark 5.7, [σ− n ]+1,λ(·)

p n Λk,λ(·) ∞,r (R ) = Λ∞,r

(Rn ),

when σ ∈ (0, n), p ∈ ( nσ , +∞), q ∈ [1, +∞], b ∈ SV (0, +∞), r ∈ [1, +∞], k = [σ] + 1 and λ is given by (6.10).

Continuous and Compact Embeddings

191

Corollary 6.10 ([35, Corollary 1.7]). Assume that all the assumptions of [σ− n p ]+1

Theorem 6.7 are satisﬁed. Let μ ∈ Lr

H σ Lp,q;b (Rn ) →→

. If 1 ≤ q ≤ r < +∞, then

[σ− n ]+1,μ(·) Λ∞,r p (Ω)

(6.15)

if and only if (6.12) is satisﬁed. Theorem 6.11 ([35, Theorem 1.8]). Let σ ∈ (0, n), p = nσ , q ∈ (1, +∞], r ∈ (0, +∞), k := [σ] + 1, μ ∈ Lkr and let b ∈ SV (0, +∞) be such that −1

t q (b(t))−1 q ;(0,1) < +∞. Assume that Ω is a bounded domain in Rn . Let λqr be deﬁned by n 1 + r1 q x dt λqr (x) := bq /r (xn ) b−q (t) , x ∈ (0, 1). (6.16) t 0 (Note that λqr ∈ Lkr ; recall that b is continuous (cf. (2.2)).) (i) If 1 < q ≤ r < +∞, then H σ Lp,q;b (Rn ) →→ Λk,μ(·) ∞,r (Ω)

(6.17)

if and only if 1

lim

x→0+

t− r (μ(t))−1 r;(x,1) 1

t− r (λqr (t))−1 r;(x,1)

= 0.

(6.18)

(ii) If 0 < r < q ≤ +∞ and q > 1, then H σ Lp,q;b (Rn ) →→ Λk,μ(·) ∞,r (Ω)

(6.19)

if and only if u n −1 1/2 −1/r x

t (μ(t))−1 r;(x,1) dx −1 −q t b (t) dt b−q (xn ) < +∞, −1/r −1 x

t (λ (t))

qr r;(x,1) 0 0 (6.20) where u1 := 1r − 1q . Remark 6.12. (i) As in Remark 6.4, we see that in Theorem 6.11 the implications (6.17) =⇒ (6.18) and (6.19) =⇒ (6.20) remain true if k ∈ N. (ii) Recall that, by Remark 5.16 (i), qr (·) qr (·) Λk,λ (Rn ) = Λ1,λ (Rn ) ∞,r ∞,r

when σ ∈ (0, n), p = is such that t

− q1

q ∈ (1, +∞], r ∈ [1, +∞], k = [σ] + 1, b ∈ SV (0, +∞)

n , σ

−1

(b(t))

q ;(0,1) < +∞ and λqr is given by (6.16).

Corollary 6.13 ([35, Corollary 1.9]). Assume that all the assumptions of Theorem 6.11 are satisﬁed. Let μ ∈ L1r . If 1 ≤ q ≤ r < +∞, then H σ Lp,q;b (Rn ) →→ Λ1,μ(·) ∞,r (Ω) if and only if (6.18) is satisﬁed.

(6.21)

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B. Opic

Remark 6.14. (i) In Theorem 6.7 (i) the implication (6.12)

=⇒

H σ Lp,q;b (Rn ) →→ Λk,μ(·) ∞,r (Ω)

remains true even if we extend the range of q and r to 1 ≤ q ≤ r ≤ +∞. (Indeed, this can be seen from the proof of Theorem 6.7 (i), where we use Theorem 6.3 and Remark 6.5 (i) instead of Theorem 6.3.) Theorem 6.7 (i) continues to hold if we assume that 1 ≤ q ≤ r ≤ +∞, q < +∞, and (6.8) is satisﬁed. (This follows from Remarks 6.5 (ii), (iii). Note that the condition q < +∞ implies that the space Lp,q;b (Rn ) has absolutely continuous norm - cf. [48, Lemma 3.2].) (ii) Similarly, in Theorem 6.11 (i) the implication (6.18)

=⇒

H σ Lp,q;b (Rn ) →→ Λk,μ(·) ∞,r (Ω)

remains true if we extend the range of q and r to 1 < q ≤ r ≤ +∞. Theorem 6.11 (i) continues to hold if we assume that 1 < q ≤ r ≤ +∞, q < +∞, and (6.8) is satisﬁed. (This follows from Remarks 6.5 (ii), (iii).) Example 6.15. Let Ω be a bounded domain in Rn . Then Theorem 6.7 and Corollary 6.10 yield the compactness result corresponding to the ﬁrst embedding in (5.23): If k ∈ N, k < n − 1 and α < 1 − k/n, then 2,μ(·)

W k+1 Ln/k (log L)α (Rn ) →→ Λ∞,n/k (Ω), where μ(t) := t(t)−β , t > 0, β ∈ R, holds if and only if β < α. Similarly, Theorem 6.11 and Corollary 6.13 provide the compactness result corresponding to the ﬁrst embedding in (5.37) with p = nk : If k ∈ N, k < n and α > 1 − k/n, then 1,μ(·) W k Ln/k (log L)α (Rn ) →→ Λ∞,n/k (Ω), where μ(t) = (t)1−β , t ∈ (0, 1), β ∈ R, holds if and only if β ∈ (1 − k/n, α).

7. Concluding remarks In Section 3 we characterized continuous and compact embeddings of Bessel potential spaces, modelled upon Lorentz–Karamata spaces, target spaces of our embeddings were again Lorentz–Karamata spaces. The resulting necessary and suﬃcient conditions were very simple – cf. Corollaries 3.4 (i) and 3.6 (i). In the case of continuous embeddings we have also found the optimal targets (cf. (3.10) and (3.33)). As regards continuous embeddings, note that more general results can be found in [37]. Namely, in [37] necessary and suﬃcient conditions for the validity of continuous embeddings H G X(Rn) → Y (Rn )

(7.1)

were established and the optimal targets of such embeddings were described. Here G stands for the kernel of a generalized Bessel or generalized Riesz potential and X(Rn ), Y (Rn ) are rearrangement invariant Banach function spaces. However, necessary and suﬃcient conditions for the validity of (7.1)

Continuous and Compact Embeddings

193

are much more involved than those mentioned in Section 3. These criteria state that (7.1) is equivalent to the fact that X(Rn ) ∩ L∞ (Rn ) → Y (Rn ) and that a certain Hardy-type operator (with a kernel) is bounded on a convenient cone of non-negative and non-increasing functions. By Theorem 2.2, H k X(Rn ) = W k X(Rn ) provided that k ∈ N, the rearrangement invariant Banach function space X = X(Rn ) has absolutely continuous norm and the Boyd indices of X belong to the interval (0, 1). Thus, it makes sense to mention that continuous and compact embeddings of the form W k X(Rn ) → Y (Rn ) and W k X(Rn ) →→ Y (Rn ), where Y (Rn ) is a r.i. BFS, were also studied, e.g., in [23], [53], [40], [54], [12], etc. On the other hand, as regards embeddings of the Sobolev-type spaces into generalized H¨ older spaces involving the k-modulus of smoothness, k > 1, k ∈ N, the existing literature is not rich. We mention here only the recent paper [56], where the reader can ﬁnd references to particular cases of embeddings in question. Although many problems concerning embeddings of the form H σ X(Rn ) → Y (Rn ),

σ > 0,

(7.2)

or W k X(Rn ) → Y (Rn ), k ∈ N, (7.3) n n (where X(R ) is a r.i. BFS and Y (R ) is either a r.i. BFS or a generalized H¨ older space) have been solved, still there are some open problems related to this area. We are going to mention a few of them. For example, in the proof of embedding (7.2) with a generalized H¨ older space Y (Rn ) the key role was played by inequality (4.6) and its reverse form (4.7). One can expect that in the case of embedding (7.3) with a generalized H¨ older space Y (Rn ) involving the k-modulus of smoothness, k > 1, k ∈ N, a similar role will be played by a convenient replacement of inequality (1.1) (involving the k-modulus of smoothness) and its reverse form. However, such an inequality and its reverse form are not available in the existing literature. Another problem is to prove variants of inequalities mentioned in the previous paragraph which involve the moduli of smoothness ωk (f ∗ gσ , ·)p , ωk (f, ·)p with p ∈ [1, ∞] rather than ωk (f ∗gσ , ·), ωk (f, ·) corresponding to the particular case when p = ∞. Note that such estimates would be convenient tools to characterize embeddings of Bessel-potential-type and Sobolev-type spaces into Besov-type spaces. In Section 4 we found some suﬃcient conditions for the continuous emk;λ(·) bedding between generalized H¨ older spaces Λ∞,r (Ω). It would be desirable to have a complete characterization of all continuous embeddings between spaces in question.

References [1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der matematischen Wissenschaften, vol. 314, Springer-Verlag, Berlin, 1996.

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[2] R. A. Adams and J. J. Fournier, Sobolev spaces, Pure and Applied Mathematics, vol. 140, Academic Press, Amsterdam, 2003. [3] N. Aronszajn and K. Smith, Theory of Bessel potentials, Part I, Ann. Inst. Fourier 11 (1961), 395–475. [4] N. Aronszajn, F. Mulla, and P. Szeptycki, On spaces of potentials connected with Lp classes, Part I, Ann. Inst. Fourier 13 (1963), no. 2, 211–306. [5] C. Bennett and K. Rudnick, On Lorentz–Zygmund spaces, Dissertationes Math. (Rozprawy Mat.) 175 (1980), 1–72. [6] C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, New York, 1988. [7] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Cambridge University Press, Cambridge, 1987. [8] H. Br´ezis and S. Wainger, A note on limiting cases of Sobolev embeddings, Comm. Partial Diﬀerential Equations 5 (1980), 773–789. [9] A. P. Calder´ on, Lebesgue spaces of diﬀerentiable functions and distributions, in: Partial Diﬀerential Equations, Proc. Sympos. Pure Math. 4, Amer. Math. Soc., Providence, RI (1961), 33–49. [10] M. J. Carro, L. Pick, J. Soria and V. D. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl. 4 (2001), 397–428. [11] A. Cianchi, Some results in theory of Orlicz spaces and applications to variational problems, in: Nonlinear Analysis, Function Spaces and Applications, Vol. 6, (M. Krbec and A. Kufner, eds.), 50–92, Math. Inst. Acad. Sci. Czech Republic, Prague, 1999. [12] G. Curbera, Compactness properties of Sobolev imbeddings for rearrangement invariant norms, Trans. Amer. Math. Soc. 359 (2007), 1471–1484. [13] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der mathematischen Wissenschaften–A series of Comprehensive Studies in Mathematics, vol. 303, Springer-Verlag, Berlin, 1993. [14] R. A. DeVore, S. D. Riemenschneider, and R. C. Sharpley, Weak Interpolation in Banach Spaces, J. Funct. Anal. 33 (1979), 58–94. [15] R. A. DeVore and R. C. Sharpley, On the diﬀerentiability of functions in Rn , Proc. Amer. Math. Soc. 91 (1984), 326–328. [16] D. E. Edmunds and W. D. Evans, Hardy operators, Function Spaces and Embeddings, Springer-Verlag, Berlin, 2004. [17] D. E. Edmunds, P. Gurka, and B. Opic, Double exponential integrability, Bessel potentials and embedding theorems, Studia Math. 115 (1995), 151–181. [18] , Sharpness of embeddings in logarithmic Bessel potential spaces, Proc. Roy. Soc. Edinburgh, 126A (1996), 995–1009 . [19] , On embeddings of logarithmic Bessel potential spaces, J. Funct. Anal. 146 (1997), no. 1, 116–150. [20] , Optimality of embeddings of logarithmic Bessel potential spaces, Quart. J. Math. 51 (2000), 185–209. [21] , Compact and continuous embeddings of logarithmic Bessel potential spaces, Studia Math. 168 (2005), no. 3, 229–250. [22] , Non-compact and sharp embeddings of logarithmic Bessel potential spaces into H¨ older-type spaces, Z. Anal. Anwendungen 25 (2006), 73–80.

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[23] D. E. Edmunds, R. Kerman, and L. Pick, Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms, J. Funct. Anal. 170 (2000), 307–355. [24] D. E. Edmunds and H. Triebel, Logarithmic Sobolev spaces and their applications to spectral theory, Proc. London. Math. Soc. 71 (1995), 333–371. [25]

, Function spaces, entropy numbers, diﬀerential operators, Cambridge Univ. Press, Cambridge, 1996.

[26] W. D. Evans and B. Opic, Real interpolation with logarithmic functors and reiteration, Canad. J. Math. 52 (2000), 920–960. [27] W. D. Evans, B. Opic and L. Pick, Interpolation of operators on scales of generalized Lorentz–Zygmund spaces, Math. Nachr. 182 (1996), 127–181. [28] A. Gogatishvili, J. S. Neves, and B. Opic, Optimality of embeddings of Besselpotential-type spaces into Lorentz–Karamata spaces, Proc. Roy. Soc. Edinburgh 134A (2004), 1127–1147. [29]

, Optimality of embeddings of Bessel-potential-type spaces into generalized H¨ older spaces, Publ. Mat. 49 (2005), 297–327.

[30]

, Optimality of embeddings of Bessel-potential-type spaces, in: Function Spaces, Diﬀerential Operators and Nonlinear Analysis, Proc., Milovy 2004 (P. Dr´ abek and J. R´ akosn´ık, eds.), 97–112, Math. Inst. Acad. Sci. Czech Republic, Prague, 2005.

[31]

, Sharpness and non-compactness of embeddings of Bessel-potential-type spaces, Math. Nachr. 280 (2007), no. 9-10, 1083-1093.

[32]

, Optimal embeddings and compact embeddings of Bessel-potential-type spaces, Math. Z. 262 (2009), no. 3, 645-682.

[33]

, Optimal embeddings of Bessel-potential-type spaces into generalized H¨ older spaces involving k-modulus of smoothness, Potential Anal. 32 (2010), 201-228.

[34]

, Sharp estimates of the k-modulus of smoothness of Bessel potentials, J. London Math. Soc. 2 (81) (2010), 608-624.

[35]

, Compact embeddings of Bessel-potential-type spaces into generalized H¨ older spaces involving k-modulus of smoothness, Z. Anal. Anwendungen 30 (2011), 1–27.

[36] A. Gogatishvili, B. Opic, and W. Trebels, Limiting reiteration for real interpolation with slowly varying functions, Math. Nachr. 278 (2005), 86–107. [37] M. L. Goldman, Optimal embeddings of generalized Bessel and Riesz potentials, Proc. Steklov Inst. 269 (2010), 85–105. [38] D. D. Haroske, Envelopes in function spaces – a ﬁrst approach, Informatik Math/Inf/16/01, Friedrich-Schiller-Universit¨ at Jena, 2001. [39] H. Johnen and K. Scherer, On the equivalence of the K-functional and moduli of continuity and some applications, in “Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976)”, pp. 119140, Lecture Notes in Math., vol. 571, Springer, Berlim, 1977. [40] R. Kerman and L. Pick, Compactness of Sobolev imbeddings involving rearrangement-invariant quasinorms, Studia Math. 186 (2008), 127–160. [41] A. Kufner, O. John, and S. Fuˇc´ık, Function spaces, Academia, Prague, 1977. [42] G. G. Lorentz, On the theory of spaces Λ, Paciﬁc J. Math. 1 (1951), 411–429.

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[43] V. Mari´c, Regular variation and diﬀerential equations, Lecture Notes in Mathematics, vol. 1726, Springer-Verlag, Berlin, 2000. [44] M. Milman, Extrapolation and Optimal Decompositions with Applications to Analysis, Lecture Notes in Mathematics, vol. 1580, Springer-Verlag, Berlin, 1994. [45] J. S. Neves, Fractional Sobolev-type spaces and embeddings, Ph. D. Thesis, University of Sussex, 2001. [46] , Extrapolation results on general Besov-H¨ older-Lipschitz spaces, Math. Nachr. 230 (2001), 117–141. [47] , Lorentz–Karamata spaces, Bessel and Riesz potentials and embeddings, Dissertationes Math. (Rozprawy Mat.) 405 (2002), 46 pp. [48] , Spaces of Bessel-potential type and embeddings: the super-limiting case, Math. Nachr. 265 (2004), 68–86. [49] , A survey on embeddings of Bessel-potential-type spaces, in: “The J.A. Sampaio Martins Anniversary Volume” (A.J.G. Bento, A.M. Caetano, S.D. Moura, and J.S. Neves, eds.), Textos de Matem´ atica (S´erie B) 34, Departamento de Matem´ atica, Universidade de Coimbra, 2004, pp. 75-90. [50] B. Opic Embeddings of Bessel potential and Sobolev type spaces, Colloquium del Departamento de An´ alisis Matem´ atico, Secci´ on 1, no. 48, Universidad Complutense de Madrid, CURSO 1999–2000, 100-118. [51] B. Opic, Embeddings of Bessel-potential-type spaces, Preprint no. 169/2007. Institute of Mathematics, AS CR, Prague, 2007. [52] B. Opic and L. Pick, On generalized Lorentz–Zygmund spaces, Math. Inequal. Appl. 2 (1999), 391–467. [53] L. Pick, Optimality of function spaces in Sobolev embeddings, in: Sobolev spaces in mathematics. I, 249–280, Int. Math. Ser. (N.Y.), Springer, New York, 2009. [54] E. Pustylnik, On compactness of Sobolev embeddings in rearrangementinvariant spaces, Forum Math. 18 (2006), 839-852. [55] E. M. Stein, Singular integrals and diﬀerentiability properties of functions, Princeton University Press, Princeton, New Jersey, 1970. [56] S. Tikhonov, Weak type inequalities for moduli of smoothness: the case of limit value parameters, J. Fourier Anal. Appl. 16 (2010), 590–608. [57] H. Triebel, The structure of functions, Monographs in Mathematics, vol. 97, Birkh¨ auser Verlag, Basel, 2001. [58] W. Ziemer, Weakly Diﬀerentiable Functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, Berlin, 1989. [59] A. Zygmund, Trigonometric series, vol. I, Cambridge University Press, Cambridge, 1957. Bohum´ır Opic Charles University, Faculty of Mathematics and Physics Department of Mathematical Analysis, Sokolovsk´ a 83, 186 75 Prague 8 Czech Republic e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 197–209 c 2012 Springer Basel AG

A Sequence of Zero Modes of Weyl–Dirac Operators and an Associated Sequence of Solvable Polynomials Yoshimi Sait¯o and Tomio Umeda Dedicated to Professor Edmunds and Professor Evans on the occasion of their eightieth and seventieth birthdays

Abstract. It is shown that a series of solvable polynomials is attached to the series of zero modes constructed by Adam, Muratori and Nash [1]. Mathematics Subject Classiﬁcation (2010). 35Q40, 35P99, 11R09. Keywords. Weyl–Dirac operators, magnetic potentials, zero modes, solvable polynomials.

1. Introduction The aim of this note is to point out an interesting and unpredictable connection between zero modes and solvable polynomials. We shall precisely explain our aim. To this end, we ﬁrst introduce a Weyl–Dirac operator HA = σ · (D − A) =

3

σk (Dk − Ak (x)),

(1.1)

k=1

where σ = (σ1 , σ2 , σ3 ) is the triple of 2 × 2 Pauli matrices 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = , 1 0 i 0 0 −1 A(x) = (A1 (x), A2 (x), A3 (x)) is a vector potential, and ∂ ∂ ∂ D = −i∇ = − i , −i , −i . ∂x1 ∂x2 ∂x3 Supported by Grant-in-Aid for Scientiﬁc Research (C) No. 21540193, Japan Society for the Promotion of Science.

198

Y. Sait¯o and T. Umeda

If each component of the vector potential A is a bounded measureable function, then the operator σ · A is a bounded self-adjoint operator in the Hilbert space L2 := [L2 (R3 )]2 . Hence it is straightforward that the Weyl–Dirac operator HA deﬁnes the unique self-adjoint realization in L2 and its domain is given as Dom(HA ) = H1 := [H 1 (R3 )]2 whenever Aj ∈ L∞ (R3 ). Here H 1 (R3 ) denotes the Sobolev space of order 1. Deﬁnition 1.1. If ψ ∈ Ker(HA ), then ψ is called a zero mode of HA . In other words, ψ is said to be a zero mode if and only if ψ ∈ Dom(HA ) and HA ψ = 0. We should remark that the Weyl–Dirac operator is intimately related with the Pauli operator PA =

3

(Dj − Aj )2 − σ · B,

j=1

where B denotes the magnetic ﬁeld given by B = ∇ × A. This is because PA = {σ · (D − A)}2 = HA2 in a formal sense. Roughly speaking, we can say that ψ is a zero mode of the Weyl–Dirac operator HA if and only if it is a zero mode of the Pauli operator PA . It is now well understood that the existence of magnetic ﬁelds which give rise to zero modes of the Weyl–Dirac operators has signiﬁcant implications in mathematics and physics (see [1]–[7], [9]–[17]). However, Balinsky and Evans [4]–[6] and Elton [10] showed that the set of vector potentials which yield zero modes is scarce in a certain sense. Vector potentials which give rise to zero modes do exist. The ﬁrst examples of such vector potentials were given by Loss and Yau [13]. Later Adam, Muratori and Nash [1]–[3] and Elton [9] constructed further examples of zero modes, using and developing the ideas from [13]. The works [11] by Erd¨ os and Solovej generalize all these examples. The basic idea of Loss and Yau [13] is to ﬁnd a solution of the Loss–Yau equation (σ · D)ψ(x) = h(x)ψ(x), (1.2) where h is a given (real-valued) function, and then to deﬁne a vector potential A so that ψ satisﬁes the equation σ · (D − A)ψ = 0. A precise statement of their idea is the following. Proposition 1.2. (Loss–Yau [13]). Let ψ ∈ H1 be a solution to the Loss–Yau equation (1.2) with a real valued bounded function h. Then ψ is a zero mode of the Weyl–Dirac operator HA with the vector potential deﬁned by h(x) A(x) = ψ(x) · σ1 ψ(x), ψ · σ2 ψ(x), ψ · σ3 ψ(x) , (1.3) |ψ(x)|2 where, for a = t (a1 , a2 ), b = t (b1 , b2 ) ∈ C2 , a · b denotes the inner product: a · b = a1 b1 + a2 b2 .

Dirac Operators

199

In [13], choosing ψ(x) = x−3 (I2 + iσ · x)φ0

(x =

1 + |x|2 ),

(1.4)

where I2 is a 2 × 2 unit matrix and φ0 ∈ C2 a unit vector, they showed that ψ deﬁned by (1.4) satisﬁes the Loss–Yau equation (1.2) with h(x) =

3 . x2

(1.5)

It follows from (1.3) and (1.4) that A(x) = 3x−4 (1 − |x|2 )w0 + 2(w0 · x)x + 2w0 × x ,

(1.6)

w0 = φ0 · (σφ0 ) := φ0 · (σ1 φ0 ), φ0 · (σ2 φ0 ), φ0 · σ3 φ0 ) ,

(1.7)

where

and w0 · x and w0 × x denote the inner product and the exterior product respectively. Proposition 1.2 implies that ψ deﬁned by (1.4) is a zero mode of the Weyl–Dirac operator HA with the vector potential (1.6). Adam, Muratori and Nash [1] exploited ∞ the idea of Proposition 1.2, and successfully constructed a series ψ (m) m=1 , each of which satisﬁes the Loss–Yau equation (1.2) with h(m) (x) =

2m + 3 x2

(m = 1, 2, . . .).

(1.8)

It is obvious that each ψ (m) is a zero mode of the Weyl–Dirac operator HA(m) with the vector potential A(m) (x) =

h(m) (x) (m) ψ (x)·σ1 ψ (m) (x), ψ (m) ·σ2 ψ (m) (x), ψ (m) ·σ3 ψ (m) (x) . (m) 2 |ψ (x)|

The goal of this note is to show that a polynomial Pm (t) of degree m + 1 is associated with each zero mode ψ (m) in such a way that the polynomial equation Pm (t) = 0 is solvable and all of the roots of this equation determine a set of zero modes, one of which is designated as ψ (m) . Obviously, as m gets larger, it will become more diﬃcult to solve the equation Pm (t) = 0. It is wellknown [18] that “there is no formula for the roots of a ﬁfth (or higher) degree polynomial equation in terms of the coeﬃcients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc).” Here are the ﬁrst

200

Y. Sait¯o and T. Umeda

six equations of Pm (t) = 0: P1 (t) = 0 ⇐⇒ 9t2 − 34t + 25 = 0, P2 (t) = 0 ⇐⇒ 81t3 − 747t2 + 1891t − 1225 = 0, P3 (t) = 0 ⇐⇒ 81t4 − 1476t3 + 8614t2 − 18244t + 11025 = 0, P4 (t) = 0 ⇐⇒ 729t5 − 23085t4 + 256122t3 − 1206490t2 + 2306749t − 1334025 = 0, 6

5

P5 (t) = 0 ⇐⇒ 6561t − 330966t + 6206463t4 − 54143028t3 + 224657551t2 − 401846806t + 225450225 = 0, P6 (t) = 0 ⇐⇒ 6561t7 − 494991t6 + 14480613t5 − 209304603t4 + 1578233251t3 − 6018285581t2 + 10271620375t − 5636255625 = 0. It is incredable to see that all these polynomial equations are solvable. Actually, by computer-aided calculation we see that 25 P1 (t) = 0 ⇐⇒ t = 1, 9 25 49 P2 (t) = 0 ⇐⇒ t = 1, , , 9 9 25 49 P3 (t) = 0 ⇐⇒ t = 1, , , 9, 9 9 25 49 121 P4 (t) = 0 ⇐⇒ t = 1, , , 9, , 9 9 9 25 49 121 169 P5 (t) = 0 ⇐⇒ t = 1, , , 9, , , 9 9 9 9 25 49 121 169 P6 (t) = 0 ⇐⇒ t = 1, , , 9, , , 25. 9 9 9 9 Based on this observation, it is natural to predict that the roots of the equation Pm (t) = 0 must be given by 5 2 7 2 2m + 3 2 1, , , ..., (1.9) 3 3 3 for every m ∈ N. This prediction will be proven to be true in §4, though we should like to mention that these polynomials Pm (t) will be only implicitly deﬁned in a rather messy manner; see the formula (Lm ) in Proposition 2.1 as well as Proposition 2.2 in §2. In relation with this, we emphasize that the bigger m gets, the more complicated Pm (t) becomes, as can be seen from P1 (t), . . ., P6 (t) above. We should like to call Pm (t) in their monic forms the Adam–Muratori– Nash polynomials. We feel solvability of Pm (t) seems an interesting subject from the view point of Galois theory (see Edwards [8]), though it is well beyond the scope of the present note.

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2. Recurrence formulae In this section we follow the line of the arguments demonstrated in AdamMuratori-Nash [1]. For this reason, we shall use the same notation as in [1] to indicate I2 and iσ · x in the rest of this note ; namely 1 = I2 , X = iσ · x.

(2.1)

Their construction of the zero modes is based on the following ansatz: m m 1 2 ψ (m) (x) = x−(3+2m) an |x|2n 1 + bn |x|2n X φ0 , (2.2) n=0

n=0

where x = 1 + |x|2 , and φ0 = t (1, 0). We are going to study the case where h(x) in (1.2) is x−2 multiplied by a constant α. By a simple but tedious computation we have Proposition 2.1. Let ψ (m) (x) be as above with a0 = 1. (i) Then, for each m = 1, 2, 3, . . . , we have (σ · D)ψ (m) = x−(5+2m)

m 1

(2m + 3)bn |x|2n

n=0

+ x−(5+2m)

2 − (2m − 2n)bn |x|2(n+1) 1φ0 1 m−1

{(3 + 2m)an − 2(n + 1)an+1 }|x|2n

n=0

+ (3 + 2m)am |x|

2m

−

m−1

2 2(n + 1)an+1 |x|2(n+1) Xφ0 . (2.3)

n=1

(ii) The equation (σ · D)ψ (m) (x) = where α is a constant, ⎧ ⎪ (2j − 1) : ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (Lm ) (2k) : ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩(2m + 1) :

α (m) ψ (x), x2

(2.4)

is equivalent to the system 2jaj − (2m + 5 − 2j)aj−1 = −3b0bj−1 (1 ≤ j ≤ m, a0 = 1), (2k + 3)bk − (2m + 2 − 2k)bk−1 = 3b0 ak (1 ≤ k ≤ m), am = b0 bm

of (2m + 1) equations for the (2m + 1) unknowns aj (1 ≤ j ≤ m) and bk (0 ≤ k ≤ m), where the constant α turns out to be 3b0 . m It is easy to see that {an }m n=1 and {bn }n=1 can be expressed by b0 and the last equation am = b0 bm becomes a polynomial equation for unknown b0 .

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Proposition 2.2. The last equation (2m + 1) of (Lm ) : am = b0 bm takes the form Pm (b20 ) = 0, where Pm (t) is a polynomial of degree m + 1. Proof. Since a0 = 1, the ﬁrst equation (1) of (Lm ) is given as 2a1 −(2m+3) = −3b20 . Hence a1 = p1 (b20 ), where p1 (t) = 2−1 {(2m + 3) − 3t}. From the second equation (2) of (Lm ) we see that b1 takes the form b1 = b0 q1 (b20 ), where q1 (t) = (10)−1 (10m + 9 − 9t). Then, by induction, one can show that aj , 1 ≤ j ≤ m, and bk , 1 ≤ k ≤ m, are expressed as aj = pj (b20 ) and bk = b0 qk (b20 ) with polynomial pj (t) of degree j and polynomial qk (t) of degree k. Thus the equation b0 bm − am = 0 becomes a polynomial equation Pm (b20 ) = 0, where Pm (t) is a polynomial of t of degree m + 1. Remark 2.3. The case m = 1 is discussed in [1]. In this case 5 5 ψ (1) (x) = x−5 (1 − |x|2 )1 + ( − |x|2 )X φ0 . 3 3

(2.5)

Proposition 2.4. Let m be a ﬁxed non-negative integer and let an and bn be the coeﬃcients in (2.2) with a0 = 1. Then we have ⎛ ⎞ ⎛ ⎞ aj a1 ⎝ ⎠ = Kj Kj−1 · · · K2 ⎝ ⎠ (j = 2, 3, . . . , m), (2.6) bj b1 where

⎛

2m + 5 − 2p ⎜ 2p Kp = ⎜ ⎝ 3(2m + 5 − 2p)b0 2p(2p + 3)

⎞ 3b0 ⎟ 2p ⎟ 2⎠ 2p(2m + 2 − 2p) − 9b0 2p(2p + 3) −

(2.7)

for p = 2, 3, . . . , m, and ⎛ ⎞ ⎛ 2m + 3 − 3b2 ⎞ 0 a1 ⎟ 2 ⎝ ⎠=⎜ ⎝ ⎠. b 0 2 b1 (10m + 9 − 9b0 ) 10

(2.8)

Proof. We divide the proof into three steps. (I) Let 2 ≤ j ≤ m. It follows from the equation (2k) in the system (Lm ) with k replaced by j that (2j + 3)bj − (2m + 2 − 2j)bj−1 = 3b0 aj .

(2.9)

From the equation (2j − 1) in (Lm ) we have 2jaj − (2m + 5 − 2j)aj−1 = −3b0bj−1 , or aj =

(2m + 5 − 2j)aj−1 − 3b0 bj−1 . 2j

(2.10)

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203

Then we obtain from (2.9) and (2.10) (2j + 3)bj − (2m + 2 − 2j)bj−1 5 6 3b0 (2m + 5 − 2j)aj−1 − 3b0 bj−1 ) = , 2j or (2j + 3)bj =

3b0 (2m + 5 − 2j) 2j(2m + 2 − 2j) − 9b20 aj−1 + bj−1 . 2j 2j

(2.11)

(II) From the equation (2j − 1) in (Lm ) with j = 1 we see that, by noting that a0 = 1, 2a1 − (2m + 3) = −3b20 , and hence a1 =

2m + 3 − 3b20 . 2

(2.12)

We have from the equation (2k) in (Lm ) with k = 1 5b1 − (2m + 2 − 2)b0 = 3b0 a1 , which is combined with (2.12) to yield 3b0 2m + 3 − 3b20 5b1 − 2mb0 = , 2 or

b0 10m + 9 − 9b20 b1 = . 10 (III) It follows from (2.10) and (2.11) that ⎛ ⎞ ⎛ ⎞ aj aj−1 ⎝ ⎠ = Kj ⎝ ⎠ (j = 2, 3, . . . , m) bj bj−1

(2.13)

(2.14)

with Kj given by (2.7) with p replaced by j. By using (2.14) repeatedly we can obtain (2.6). As for a1 and b1 , (2.8) is justiﬁed by (2.12) and (2.13). Remark 2.5. Proposition 2.4 was used to construct a Maple program to ﬁnd polynomial Pm (t) as well as to solve the polynomial equation Pm (t) = 0. We have been able to handle the equations with the Maple program up to the case m = 26. The ﬁrst six equations were listed up at the end of §1.

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3. Monotonicity of the sequence {Rm }∞ m=1 We begin by Deﬁnition 3.1. For each m ∈ N, Rm is deﬁned to be the set of all the roots of the polynomial equation Pm (t) = 0, namely, Rm := t ∈ C Pm (t) = 0 . (3.1) Proposition 3.2. Let Rm be as above. Then we have R1 ⊂ R2 ⊂ · · · ⊂ Rm ⊂ · · · ,

(3.2)

i.e., the sequence {Rm }∞ m=1 is increasing with m. Proof. For m = 1, 2, 3, . . . , let ψ (m) (x) be given by (2.2) and suppose that the coeﬃcients of ψ (m) (x) satisfy the system (Lm ), i.e., ψ (m) (x) is a solution to the equation (2.4) and hence a zero mode of the Weyl–Dirac operator HA(m) . Thus b20 ∈ Rm . We now rewrite ψ (m) (x) as m m 1 2 ψ (m) (x) = x−(3+2(m+1)) x2 an |x|2n 1 + x2 bn |x|2n X φ0 n=0

n=0

=: ψ7(m+1) (x).

(3.3)

By using the deﬁnition x2 = 1 + |x|2 , we obtain ψ7(m+1) (x) = x−(3+2(m+1))

1 m+1

m+1 2

7bn |x|2n X φ0 , (3.4) 7 an |x| 1+

n=0

where

⎧ 7 a0 = a0 = 1, ⎪ ⎪ ⎪ ⎪ ⎪ 7 an = an−1 + an ⎪ ⎪ ⎪ ⎨7 am+1 = am , 7 ⎪ b 0 = b0 , ⎪ ⎪ ⎪ ⎪ 7bn = bn−1 + bn ⎪ ⎪ ⎪ ⎩7 bm+1 = bm .

2n

n=0

(1 ≤ n ≤ m), (3.5) (1 ≤ n ≤ m),

Therefore, noting that ψ (m) (x) satisﬁes the equation (σ · D)ψ (m) (x) = 3b0 x−2 ψ (m) (x) and that 7b0 = b0 by (3.5), we see that (σ · D)ψ7(m+1) (x) = (σ · D)ψ (m) (x) = 3b0 x−2 ψ (m) (x) = 37b0 x−2 ψ7(m+1) (x).

(3.6)

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205

Thus, since ψ7(m+1) (x) is a zero mode, we see that the coeﬃcients 7 an and 7bn satisfy the system (Lm+1 ) which is (Lm ) in Proposition 2.1(ii) with m replaced by m + 1. Thus we have the system of (2m + 3) equations for ⎧ ⎪ (2j − 1) 2j7 aj − (2m + 7 − 2j)7 aj−1 = −37b07bj−1 ⎪ ⎪ ⎪ ⎪ ⎪ (1 ≤ j ≤ m + 1, 7 a0 = 1), ⎨ 7 7 7 (2k) (2k + 3)bk − (2m + 4 − 2k)bk−1 = 3b07 ak ⎪ ⎪ ⎪ (1 ≤ k ≤ m + 1), ⎪ ⎪ ⎪ ⎩(2m + 3) 7 am+1 = 7b07bm+1 . Therefore b0 = 7b0 satisﬁes the polynomial equation Pm+1 (b20 ) = 0.

Remark 3.3. Proposition 3.2 above does not give us enough information to determine the set Rm though it signiﬁcantly clariﬁes the situation. In fact, we know that 5 2 7 2 2m + 3 2 Rm = 1, , , ..., (3.7) 3 3 3 for m = 1, . . ., 6. We also know that the polynomial Pm (t) is of degree m + 1. Therefore Proposition 3.2, together with these two facts, tells us that we can prove (3.7) by induction on m. In fact, assuming that (3.7) with m replaced by m − 1 is true, we only have to show that 2m + 3 2 ∈ Rm (3.8) 3 for every m ≥ 2 (actually m ≥ 7).

4. Construction of zero modes We are going to prove that (3.7) is true for every m ∈ N, and describe how to construct the sequence {ψ (m) }∞ m=1 of zero modes in terms of a root of the polynomial equation Pm (b20 ) = 0. Recall that in Proposition 2.2 we saw that aj = pj (b20 ) and bk = b0 qk (b20 ) with the polynomials pj (t) and qk (t) of degrees j and k respectively, where 1 ≤ j, k ≤ m. Let ⎧ ⎨cj := pj (0) (4.1) ⎩dk := lim qk (t) . t→∞ tk In other words, cj denotes the constant coeﬃcient of the polynomial pj (t) and dk denotes the coeﬃcient of tk in the polynomial qk (t), of which degree is k. Lemma 4.1. Let m be a ﬁxed non-negative integer and let an and bn be the coeﬃcients in (2.2) with a0 = 1. Then we have cm =

5 · 7 · 9 · · · (2m + 3) 2m (m!)

(4.2)

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Y. Sait¯o and T. Umeda

and

32m . 5 · 7 · 9 · · · (2m + 3)2m (m!)

dm = (−1)m

(4.3)

Proof. We divide the proof into two steps. (I) From the equation (2j − 1) of (Lm ) we have 2jaj − (2m + 5 − 2j)aj−1 = −3b0bj−1 , where the right hand side has no constant coeﬃcient as a polynomial of b0 . Hence we have 2jcj − (2m + 5 − 2j)cj−1 = 0. Therefore we obtain recursive relations 2m + 5 − 2j cj = cj−1 (j = 2, . . . , m), (4.4) 2j which implies that 2m + 5 − 2j 2m + 5 − 2(j − 1) · · · 2m + 1 2m + 3 cj = , (4.5) 2j (j!) where we should note that c1 = (2m+ 3)/2. We obtain (4.2) by setting j = m in (4.5). (II) Let 7 cj be the coeﬃcient of tj of pj (t). Then it follows from the equation (2j − 1) of (Lm ) that 2j7 cj = −3dj−1 or

3 dj−1 . (4.6) 2j On the other hand, from the equation (2k) of (Lm ) with k replaced by j we see that (2j + 3)dj = 37 cj . Thus we have 3 dj = 7 cj . (4.7) 2j + 3 It follows from (4.6) and (4.7) that cj = − 7

3 3 32 dj−1 = − dj−1 2j + 3 2j 2j(2j + 3) for j = 2, . . . , m, which implies that dj = −

dj = (−1)j

32j , 5 · 7 · 9 · · · (2j + 3)2j (j!)

(4.8)

where we should note that d1 = (−9)/(5 · 2). We thus obtain (4.3) by setting j = m in (4.8). Theorem 4.2. For each m ∈ N, we have 5 2 7 2 2m + 3 2 Rm = 1, , , ..., . 3 3 3

(4.9)

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207

Proof. We prove the theorem by induction on m. As was pointed out in Remark 3.3, we only have to show that 2m + 3 2 ∈ Rm (4.10) 3 for every m ≥ 2, assuming that (4.9) with m replaced by m − 1 is true. Let us recall that b0 bm − am = b20 qm (b20 ) − pm (b20 ) = Pm (b20 ),

(4.11)

where pm (t) and qm (t) are polynomials of degree m. It follows from (4.11) that dm is equal to the coeﬃcient of tm+1 of the polynomial Pm (t), of which degree is m + 1. Also it follows from (4.11) that the constant coeﬃcient of Pm (t) is given by −cm . By hypothesis of the induction, we have 5 2 7 2 2m + 1 2 Rm−1 = 1, , , ..., . (4.12) 3 3 3 Since Rm−1 ⊂ Rm by Proposition 3.2, we see that 5 2 7 2 2m + 1 2 1, , , ..., 3 3 3 are the roots of Pm (t). For simplicity, we put 2j + 1 2 λj = (j = 1, 2, . . . , m). (4.13) 3 Since there exists one more root λ ∈ Rm of Pm (t), we ﬁnd that Pm (t) = dm (t − λ1 )(t − λ2 ) · · · (t − λm )(t − λ).

(4.14)

Noting that Pm (0) = −cm , we get dm (−1)m+1 λ1 λ2 · · · λm λ = −cm . Hence, by using (4.2), (4.3) and (4.13), we obtain (−1)

m+1

m 18 2j + 1 2 2 j=1

3

m+1 8 2j + 1 2 cm m+1 λ=− = (−1) . dm 3 j=1

Therefore we can conclude that λ=

2m + 3 2 3

,

which implies (4.10).

For each m ∈ N, the polynomial (4.11) has 2m + 2 roots: 5 2m + 3 b0 = ±1, ± , . . . , ± . 3 3 If we choose the root b0 = +(2j + 1)/3 for a ﬁxed j with 1 ≤ j ≤ m + 1, then we can deﬁne a1 , . . ., am , b1 , . . ., bm by Proposition 2.4. With these (m) a1 , . . ., am , b1 , . . ., bm obtained, we construct ψj,+ (x) by (2.2). It follows

208

Y. Sait¯o and T. Umeda (m)

from Propositions 2.1 and 1.2 that ψj,+ is a zero mode of the magnetic Dirac (m)

(m)

operator HA(m) := σ · (D − Aj,+ ), where Aj,+ is deﬁned by (1.3) with j,+

h(x) =

2j + 1 , x2

(m)

ψ(x) = ψj,+ (x).

The sequence {ψ (m) }∞ m=1 constructed in Adam, Muratori and Nash [1] is now obtained by putting (m)

ψ (m) (x) := ψm+1,+ (x). We make a few of concluding remarks. (i) For each m ∈ N, set (m) Ψm = ψj,+ (x) j = 1, 2, . . . , m + 1}.

(4.15)

Adam, Muratori and Nash [1] pointed out that Ψ1 Ψ2 · · · Ψm · · · . (ii) In a similar manner, choosing the root b0 = −(2j + 1)/3, we can construct a diﬀerent sequence of zero modes from {ψ (m) }∞ m=1 deﬁned above. Acknowledgments. T.U. would like to thank Ryuichi Ashino for his help with our Maple program, and thank Takeshi Usa for his valuable comment on the solvability of the Adam-Muratori-Nash polynomials that Galois theory seems to play an important role behind the scene.

References [1] C. Adam, B. Muratori and C. Nash, Zero modes of the Dirac operator in three dimensions, Phys. Rev. D 60 (1999), 125001-1 – 125001-8. [2] C. Adam, B. Muratori and C. Nash, Degeneracy of zero modes of the Dirac operator in three dimensions, Phys. Lett. B 485 (2000), 314–318 [3] C. Adam, B. Muratori and C. Nash, Multiple zero modes of the Dirac operator in three dimensions, Phys. Rev. D 62 (2000), 085026-1 – 085026-9. [4] A.A. Balinsky and W.D. Evans, On the zero modes of Pauli operators, J. Funct. Analysis, 179 (2001), 120–135. [5] A.A. Balinsky and W.D. Evans, On the zero modes of Weyl–Dirac operators and their multiplicity, Bull. London Math. Soc., 34 (2002), 236–242. [6] A.A. Balinsky and W.D. Evans, Zero modes of Pauli and Weyl–Dirac operators, Advances in diﬀerential equations and mathematical physics (Birmingham, AL, 2002), 1–9, Contemp. Math., 327, Amer. Math. Soc., Providence, Rhode Island, 2003. [7] A.A. Balinsky, W.D. Evans and T. Umeda, The Dirac-Hardy and Dirac-Sobolev inequalities in L1 , Publ. Res. Inst. Math. Sci. Kyoto Univ. 47 (2011), 791–801. [8] H.M. Edwards, The construction of solvable polynomials, Bull. Amer. Math. Soc. 46 (2009), 397 – 412. [9] D.M. Elton, New examples of zero modes, J. Phys. A: Math. Gen. 33 (2000), 7297–7303.

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[10] D.M. Elton, The local structure of zero mode producing magnetic potentials, Commun. Math. Phys. 229 (2002), 121–139. [11] L. Erd¨ os and J.P. Solovej, The kernel of Dirac operators on S3 and R3 , Rev. Math. Phys. 13 (2001), 1247–1280. [12] J. Fr¨ ohlich, E.H. Lieb and M. Loss, Stability of Coulomb systems with magnetic ﬁelds. I. The one-electron Atom, Commun. Math. Phys. 104 (1986), 251–270. [13] M. Loss and H.T. Yau, Stability of Coulomb systems with magnetic ﬁelds. III. Zero energy bound states of the Pauli operators, Commun. Math. Phys. 104 (1986), 283–290. [14] P. Pickl, Generalized eigenfunctions for critical potentials with small perturbations, J. Math. Phys. 48 (2007), 123505-1 – 123505-31. [15] P. Pickl and D. D¨ urr, On adiabatic pair creation, Commun. Math. Phys. 282 (2008), 161–198. [16] P. Pickl and D. D¨ urr, Adiabatic pair creation in heavy ion and laser ﬁelds, Europhys. Lett. 81 (2008), 40001–40007. [17] Y. Sait¯ o and T. Umeda, Eigenfunctions at the threshold energies of magnetic Dirac operators, Rev. Math. Phys. 23 (2011), 155–178. [18] Wikipedia, Galois Theory, http://en.wikipedia.org/wiki/Galois theory. Yoshimi Sait¯ o Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294 USA e-mail: [email protected] Tomio Umeda Department of Mathematical Sciences University of Hyogo Himeji 671-2201 Japan e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 211–231 c 2012 Springer Basel AG

A Szeg˝o Limit Theorem for Operators with Discontinuous Symbols in Higher Dimensions: Widom’s Conjecture Alexander V. Sobolev To David and Des

Abstract. Relying on the known two-term asymptotic formula for the trace of the function f (A) of a truncated Wiener–Hopf-type operator A in dimension 1, in 1982 H. Widom conjectured a multi-dimensional generalisation of that formula for a pseudo-diﬀerential operator A with a symbol a(x, ξ) having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs on a hyperplane. The present paper outlines a proof of Widom’s Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces. Mathematics Subject Classiﬁcation (2010). Primary 47G30; Secondary 35S05, 47B10, 47B35. Keywords. Pseudo-diﬀerential operators with discontinuous symbols, quasi-classical asymptotics, Szeg˝ o formula.

1. Introduction, the main result For a symbol a = a(x, ξ), and any function u from the Schwartz class on Rd we deﬁne the pseudo-diﬀerential operator Opα (a) in L2 (Rd ) in the standard way: d α (Opα a)u(x) = eiα(x−y)ξ a(x, ξ)u(y)dξdy. (1.1) 2π For our purposes it suﬃces to assume that a ∈ Cl0 (Rd × Rd ), with some l ≥ d + 1, so that Opα (a) is not only bounded, but is trace class for all α > 0. Let Λ, Ω be two domains in Rd , and let χΛ (x), χΩ (ξ) be their characteristic functions. We always use the notation PΩ,α = Opα (χΩ ).

212

A.V. Sobolev

We study the operator Tα (a) = Tα (a; Λ, Ω) = χΛ PΩ,α Opα (a)PΩ,α χΛ ,

(1.2)

and its symmetrised version: Sα (a) = Sα (a; Λ, Ω) = χΛ PΩ,α Re Opα (a) PΩ,α χΛ . We are interested in the asymptotics of the trace tr g(Tα ) as α → ∞ with a smooth function g such that g(0) = 0. In 1982 H. Widom in [22] conjectured the asymptotic formula tr g Tα (a) = αd W0 g(a); Λ, Ω + αd−1 log α W1 A(g; a); ∂Λ, ∂Ω + o(αd−1 log α), α → ∞, (1.3) with the following coeﬃcients. For any symbol b = b(x, ξ), any domains Λ, Ω and any C1 -surfaces S, P , let 1 W0 (b) = W0 (b; Λ, Ω) = b(x, ξ)dξdx, (1.4) (2π)d Λ Ω 1 W1 (b) = W1 (b; S, P ) = b(x, ξ)|nS (x) · nP (ξ)|dSξ dSx , (1.5) (2π)d−1 S P where nS (x) and nP (ξ) denote the exterior unit normals to S and P at the points x and ξ respectively, and 1 1 g(bt) − tg(b) A(g; b) = dt, A(g) := A(g; 1). (1.6) (2π)2 0 t(1 − t) Our main objective is to prove the formula (1.3) for a large class of functions g and bounded domains Λ, Ω. The interest in the pseudo-diﬀerential operators with discontinuous symbols goes back to the classical Szeg˝o formula for the determinant of a Toeplitz matrix, see [19] and [7]. There exists a vast body of literature devoted to various non-trivial generalisations of the Szeg˝ o formula in dimension d = 1, and it is not our intention to review them here. Instead, we refer to the monographs by A. B¨ottcher–B. Silbermann [1], and by N.K. Nikolski [14] for the background reading, T. Ehrhardt’s paper [3] for a review of the pre-2001 results, and the recent paper by P. Deift, A. Its, I. Krasovsky [2], for the latest results and references. A multidimensional generalisation of the continuous variant of the Szeg˝ o formula was obtained by I.J. Linnik [13] and H. Widom [20], [21]. In fact, paper [21] addressed a more general problem: instead of the determinant, suitable analytic functions of the operator were considered, and instead of the scalar symbol matrix-valued symbols were allowed: for Ω = Rd and a(x, ξ) = a(ξ) it was shown that tr g(Tα (a)) = αd V0 + αd−1 V1 + o(αd−1 ), α → ∞, with some explicitly computable coeﬃcients V0 , V1 , such that V0 = W0 (g(a)) for the scalar case. Under some mild extra smoothness assumptions on the

Quasi-classical Asymptotics

213

boundary ∂Λ, R. Roccaforte (see [15]) found the term of order αd−2 in the above asymptotics of tr g(Tα (a)). The situation changes if we assume that Λ = Rd and Ω = Rd , i.e., that the symbol has jump discontinuities in both variables, x and ξ. As conjectured by H. Widom, in this case the second term should be of order αd−1 log α, see formula (1.3). For d = 1 this formula was proved by H. Landau–H. Widom [10] and H. Widom [22]. For higher dimensions, the asymptotics (1.3) were proved in [23] under the assumptions that one of the domains is a half-space, and that g is analytic in a disk of a suﬃciently large radius. After this paper there have been just a few publications with partial results. Using an abstract version of the Szeg˝ o formula with a remainder estimate, found by A. Laptev and Yu. Safarov (see [11], [12]), D. Gioev (see [4, 5]) established a sharp bound tr g(Tα (a)) − αd W0 (g(a)) = O(αd−1 log α). (1.7) In [6] D. Gioev and I. Klich observed a connection between the formula (1.3) and the behaviour of the entanglement entropy for free Fermions in the ground state. As explained in [6], the studied entropy is obtained as tr h Tα (1; Λ, Ω) with some bounded domains Λ, Ω, and the function h(t) = −t log t − (1 − t) log(1 − t), t ∈ (0, 1).

(1.8)

Since h(0) = h(1) = 0, the leading term, i.e., W0 (h(1)), vanishes, and the conjecture (1.3) gives αd−1 log α-asymptotics of the trace, which coincide with the expected quasi-classical behaviour of the entropy. However, the formula (1.3) is not justiﬁed for non-smooth functions, and in particular for the function (1.8). Instead, in the recent paper [8] R. Helling, H. Leschke and W. Spitzer proved (1.3) for a quadratic g. With g(t) = t − t2 this gives the asymptotics of the particle number variance, which provides a lower bound of correct order for the entanglement entropy. The operators of the form (1.2) also play a role in signal processing. Although the main object there is band-limited functions of one variable, in [16] D. Slepian considered some multi-dimensional generalisations. In particular, he derived asymptotic formulas for the eigenvalues and eigenfunctions of Tα (1) for the special case when both Λ and Ω are balls in Rd . Some of those results are used in [17]. These results, however, do not allow to study the trace tr g(Tα (1)). The next two theorems represent the main result of the present paper: Theorem 1.1. Let Λ, Ω ⊂ Rd , d ≥ 2 be bounded domains in Rd such that Λ is C1 and Ω is C3 . Let a = a(x, ξ) be a C2(d+2) (Rd × Rd )-function with a compact support. Let g be a function analytic in the disk of radius R > 0, such that g(0) = 0. If the constant R = R(a) is suﬃciently large, then tr g(Tα (a)) = αd W0 (g(a); Λ, Ω) + αd−1 log α W1 (A(g; a); ∂Λ, ∂Ω) + o(αd−1 log α), as α → ∞.

(1.9)

214

A.V. Sobolev For the self-adjoint operator Sα (a) we have a wider choice of functions

g: Theorem 1.2. Let Λ, Ω ⊂ Rd , d ≥ 2 be bounded domains in Rd such that Λ is C1 and Ω is C3 . Let a = a(x, ξ) be a C2(d+2) (Rd × Rd )-function with a compact support. Then for any function g ∈ C∞ (R), such that g(0) = 0, one has tr g(Sα (a)) = αd W0 (g(Re a); Λ, Ω) + αd−1 log α W1 (A(g; Re a); ∂Λ, ∂Ω) + o(αd−1 log α), (1.10) as α → ∞. A full proof of Theorems 1.1, 1.2 is given in [18]. The aim of the present paper is to give a detailed description of the main steps of the proof with emphasis on its pivotal points. We begin with some remarks. It would be natural to expect that the variables x and ξ in the operator Tα (a) have “equal rights”. Indeed, it was shown in [23], p. 173, by an elementary calculation, that the roles of x, ξ are interchangeable. On the other hand, the conditions on Λ and Ω in the main theorems above, are clearly asymmetric. At present it is not clear how to rectify this drawback. Theorem 1.2 can be used to study the asymptotics of the eigenvalue counting function of the operator Sα (a). Denote by n(λ1 , λ2 ; α) with λ1 λ2 > 0, λ1 < λ2 the number of eigenvalues of the operator S(a) which are greater than λ1 and less than λ2 . In other words, n(λ1 , λ2 ; α) = tr χI (Sα (a)), I = (λ1 , λ2 ). Since the interval I does not contain the point 0, this quantity is ﬁnite. Theorem 1.2 can be used to ﬁnd the leading term of the asymptotics of the counting function n(λ1 , λ2 ; α), by approximating the characteristic function χI with smooth functions g. Suppose for instance that a− < a(x, ξ) < a+ , x ∈ Λ, ξ ∈ Ω, with some positive constants a− , a+ , and that [a− , a+ ] ⊂ I. Then it follows from Theorem 1.2 that d α n(λ1 , λ2 ; α) = |Λ| |Ω| + αd−1 log α W1 A(χI ; a) + o(αd−1 log α). 2π (1.11) A straightforward calculation shows that 1 a(x, ξ) A χI ; a(x, ξ) = log −1 . (2π)2 λ1 Another interesting case is when [λ1 , λ2 ] ⊂ (0, a− ). Using (1.11) one gets n(λ1 , λ2 ; α) = αd−1 log α W1 A(χI ; a) + o(αd−1 log α), α → ∞. with

A χI ; a(x, ξ) =

1 λ2 (a(x, ξ) − λ1 ) log . (2π)2 λ1 (a(x, ξ) − λ2 ) If one of the points λ1 , λ2 is in the interval [a− , a+ ], then (1.10) gives for nα (λ1 , λ2 ) only the αd -term of the asymptotics with an o(αd )-error bound.

Quasi-classical Asymptotics

215

The proof of Theorems 1.1 and 1.2 splits in two unequal parts. The crucial and the most diﬃcult part is to justify the asymptotics for a polynomial function g: Theorem 1.3. Let Λ, Ω ⊂ Rd , d ≥ 2 be bounded domains in Rd such that Λ is C1 and Ω is C3 . Let a = a(x, ξ) be a C2(d+2) (Rd × Rd )-function with a compact support in both variables. Then for gp (t) = tp , p = 1, 2, . . . , tr gp (Tα (a)) = αd W0 (gp (a); Λ, Ω) + αd−1 log α W1 (A(gp ; a); ∂Λ, ∂Ω) + o(αd−1 log α), (1.12) as α → ∞. If Tα (a) is replaced with Sα (a), then the same formula holds with the symbol a replaced by Re a on the right-hand side. Once this theorem is proved, the asymptotics (1.9) are obtained by approximating the analytic function g by polynomials. The formula (1.10) is also derived from (1.12) using a bound of the form (1.7) with an explicit dependence of the constant on the function g. This part of the argument is technically simpler than the proof of Theorem 1.3, and we do not go into details here, but instead, we concentrate on Theorem 1.3 only. The author is happy to dedicate this paper to David Edmunds and Des Evans.

2. Proof of the Widom’s Conjecture for polynomials 2.1. Trace class estimates For the sake of presentation we make the simplifying assumption that both Λ and Ω are “graph-type” domains in the following sense. Condition 2.1. Let Φ ∈ C1 (Rd−1 ), Ψ ∈ C3 (Rd−1 ) be some real-valued functions such that Φ(0) = Ψ(0) = 0, the functions ∇Φ, ∇Ψ, ∇2 Ψ, ∇3 Ψ are uniformly bounded and ∇Φ is uniformly continuous on Rd−1 . The domains Λ and Ω are deﬁned as follows: ˆ = (x1 , x2 , . . . , xd−1 ); Λ = {x ∈ Rd : xd > Φ(ˆ x)}, x and for some index l = 1, 2, . . . , d, ◦

◦

Ω = {ξ ∈ Rd : ξl > Ψ(ξ)}, ξ = (ξ1 , . . . , ξl−1 , ξl+1 , . . . , ξd ). The above assumption does not restrict generality, since one can always satisfy the above condition locally, by choosing appropriate coordinates. Also for simplicity we assume that all symbols are C∞ 0 -functions (instead of the ﬁnite diﬀerentiability assumed in Theorems 1.1-1.3). Throughout the proof it is crucial to keep track of the dependence of the obtained estimates on the scaling properties of the symbols. From now on for any symbol b ∈ d d C∞ 0 (R × R ) we always assume that s l ∇ ∇ b(x, ξ) ≤ Cs,l −s ρ−l , (2.1) x ξ

216

A.V. Sobolev

with some ρ > 0, > 0, and that the support of b is contained in the product set B(0, ) × B(0, ρ), where B(z, r) denotes the open ball of radius r > 0, centered at z ∈ Rd . Henceforth by C or c (with or without indices) we denote various positive constants whose precise values are of no importance. The majority of the subsequent estimates are derived under the assumption that either the parameter α or one of the products αρ or αρ is separated from zero, i.e., α ≥ c or αρ ≥ c, or αρ ≥ c with some constant c. In this case the constants in the obtained estimates are uniform in α, , ρ, but allowed to depend on the bound c. The constants in all the estimates are also uniform in symbols satisfying (2.1) and functions Φ, Ψ, satisfying the bounds

∇Φ L∞ ≤ M, ∇Ψ L∞ ≤ M

(2.2)

with some constant M > 0. The notation S1 stands for the trace class. Lemma 2.2. Suppose that αρ ≥ c. Then

and

Opα (b) S1 ≤ C(αρ)d ,

(2.3)

[Opα (b), χΛ ] S1 + [Opα (b), PΩ,α ] S1 ≤ C(αρ)d−1 ,

(2.4)

tr gp (T (a; Λ, Ω)) − tr Opα (ap )gp (T (1; Λ, Ω)) ≤ Cp (αρ)d−1 ,

(2.5)

p

where gp (t) = t , p = 1, 2, . . . . The estimate (2.3) is standard. The bound (2.4) is obtained by approximating the characteristic functions χΛ , χΩ by smooth cut-oﬀ functions, and putting together a number of relatively standard trace class estimates. The estimate (2.5) is proved by using (2.4) repeatedly. In view of the estimate (2.5), in order to prove (1.12) we only need to study the trace Tα (b; Λ, Ω; g) := tr Opα (b)g(T (1; Λ, Ω)) (2.6) with a compactly supported symbol b, satisfying (2.1). If necessary, for brevity we sometimes omit some of the variables from this notation: for example, we may simply write Tα (b) or Tα (b; g). 2.2. Standard partition of unity Our study of the trace (2.6) uses the standard partition of unity which is described in [9], Ch.1. Let us state the required result in the form convenient for our purposes: Proposition 2.3. Let ∈ C1 (Rd ) be a function such that |(x) − (y)| ≤ |x − y|,

(2.7)

for all x, y ∈ R with some ∈ [0, 1). Then there exists a set xj ∈ Rd , j ∈ N such that the balls B(xj , (xj )) form a covering of Rd with the ﬁnite intersection property, i.e., each ball intersects no more than N = N () < ∞ d

Quasi-classical Asymptotics

217

d other balls. Furthermore, there exist non-negative functions ψj ∈ C∞ 0 (R ), j ∈ N, supported in B(xj , (xj )) such that

ψj (x) = 1, j

and

|∇m ψj (x)| ≤ Cm (x)−m ,

for all m uniformly in j. We let j := (xj ). For our purposes the convenient choice of (x) for all x ∈ Rd is ⎧ 1 ⎪ xd − Φ(ˆ x) , xd > Φ(ˆ x) + α−1 , ⎨ 32M (x) = 1 ⎪ ⎩ , xd ≤ Φ(ˆ x) + α−1 . 32αM

(2.8)

Here M > 0 is the constant from condition (2.2) and M 2 = 1 + M 2 . Since |∇| ≤ 1/32 a.e., the condition (2.7) is satisﬁed with = 1/32. Studying the traces Tα (ψj b; Λ, Ω; gp ) enables one to derive relatively easily a one-term asymptotic formula for Tα (b) with a correct remainder estimate. To show this, we state ﬁrst the asymptotic formula for Tα (ψj b; Λ, Ω; gp ) for the case when the support of ψj is strictly inside Λ: Lemma 2.4. Let b be a symbol satisfying (2.1) with = ρ = 1, and let ψj be a function from Proposition 2.3 such that ψj χΛ = ψj . Then | tr ψj Opα (b) gp (Tα (1)) − αd W0 (ψj b; Λ, Ω)| ≤ C(αj )d−1 . (2.9) uniformly in the index j. The observation central to the proof of this lemma is that up to an error of order O(αj )d−1 the domain Λ can be replaced by Rd . Since gp (Tα (1; Rd , Ω)) = PΩ,α , the computation of the trace becomes straightforward. The above lemma almost immediately leads to the one-term asymptotics of Tα (b; gp ) with the correct remainder estimate: Theorem 2.5. Let b satisfy (2.1) with = ρ = 1. Then | tr Opα (b) gp (Tα (1)) − αd W0 (b; Λ, Ω)| ≤ Cαd−1 log α,

(2.10)

for all α ≥ 2, with the number W0 (b; Λ, Ω) deﬁned in (1.4). Proof. Due to (2.4), replacing Opα (b)gp (Tα ) by χΛ Opα (b)gp (Tα ) produces only an error of order O(αd−1 ). Let J be the set of indices j such that ψj χΛ = 0. Thus

Tα (b) = Tα (ψj b) + O(αd−1 ). j∈J

If the support of ψj is strictly inside Λ, then we already have the asymptotics (2.9). If the support of ψj intersects with the boundary of Λ, then by deﬁnition (2.8) j = (32M α)−1 , and by (2.3), |Tα (ψj b)| ≤ ψj Opα (b) S1 ≤ C.

218

A.V. Sobolev

Since αd W0 (ψj b; Λ, Ω) ≤ C, the above estimate can be rewritten in the same asymptotic form (2.9). Thus

Tα (b; gp ) − αd W0 (ψj b) ≤ αd−1 d−1 . (2.11) j j∈J

j∈J

Remembering the ﬁnite intersection property (see Proposition 2.3), we can estimate the sum on the right-hand side by the integral −1 (x) dx ≤ C + C (x)−1 dx Λ∩B(0,1)

≤ C + C

xd >Φ(ˆ x)+α−1 ,|ˆ x|≤1 1

α−1

1 dt ≤ C(1 + log α). t

Together with (2.11) this leads to the required estimate (2.10).

We can make two important conclusions from Theorem 2.5. First, as (2.10) shows, the leading asymptotic term for Tα (b; gp ) is the same for all values of p = 1, 2, . . . . Thus, to ﬁnd the second asymptotic term for Tα (b; gp ) it suﬃces to ﬁnd the ﬁrst(leading) asymptotic term for Tα (b; g) with g(t) = tp − t.

(2.12)

Second, it is clear from the proof that the contribution of order αd−1 log α to the remainder estimate (2.10) comes from the neighbourhood of the boundary ∂Λ. In order to isolate the relevant region and subsequently calculate the asymptotics, we split Λ in the “boundary layer” and “inner region”. To this end we need one more partition of unity. 2.3. Boundary layer Here we introduce a new covering of Λ and a subordinate partition of unity. Apart from the large parameter α this construction depends on two new auxiliary parameters: r > 0, A > 0 which will be also assumed large. Let v1 , v2 ∈ C∞ (R) be two non-negative functions such that v1 (t) + v2 (t) = 1 for all t ∈ R and 0, t ≤ 1, v1 (t) = 1, t ≥ 2. Deﬁne a partition of unity subordinate to the covering of the half-axis by the intervals Δ−1 = (−2, 3), Δk = (2rk , 3rk+1 ), k = 0, 1, . . . , where r > 2. Denote t − rk t − rk+1 (k) (k) v1 (t) = v1 , v2 (t) = v2 , k = 0, 1, . . . , rk rk+1 and deﬁne ζ−1 (t) = v2 (t − 1)v1 (t + 2),

(k)

(k)

ζk (t) = v1 (t)v2 (t), k = 0, 1, 2, . . . .

Quasi-classical Asymptotics

219

It is clear that ζk (t) = 0 if t ∈ / Δk , k = −1, 0, . . . . It follows from the deﬁnition of v1 , v2 that for r > 2 and any K ≥ 0, K

(K)

ζk (t) = v2

k=−1

(t),

∞

ζk (t) = 1, t ≥ 0.

k=−1

Deﬁne two cut-oﬀ functions on Rd : (K) (K) q ↓ (x) = v2 α(xd − Φ(ˆ x)) , q ↑ (x) = v1 α(xd − Φ(ˆ x)) . To ﬁnd the asymptotics of Tα (b) we study the traces Tα (q ↓ b) and Tα (q ↑ b) for the following value of the parameter K: 3 4 log α − A K = K(α; r, A) = , (2.13) log r with some number A > 0. Here [. . . ] denotes the integer part. This value of K is chosen thus to ensure that q ↓ is supported in a thin “boundary layer” whose width does not depend on α, whereas q ↑ is supported “well inside” the domain Λ. More precisely, ⎧ ⎨q ↓ (x) = 0, xd − Φ(ˆ x) ≥ 3re−A , ⎩q ↑ (x) = 0, x − Φ(ˆ x) ≤ 2r−1 e−A . d As we show next, the asymptotics of the trace Tα (q ↓ b) (see (2.6)) “feel” the boundary ∂Λ, whereas Tα (q ↑ b) can be handled as if Λ were the entire space Rd . More precisely, the latter trace can be studied using the same ideas which were employed in the proof of Theorem 2.5. Theorem 2.6. Let b satisfy (2.1) with = ρ = 1, and let g be as in (2.12). Then for any A > 0 and r > 0, lim sup α→∞

1 Tα (q ↑ b; Λ, Ω; g) = 0. αd−1 log α

(2.14)

Proof. We use the partition of unity from Proposition 2.3 associated with the slowly-varying function deﬁned in (2.8). Let xj be the sequence constructed in Proposition 2.3, and let j = (xj ), Bj = B(xj , j ). By deﬁnition (2.8) we have (x) ≥ cr−1 e−A for all x ∈ supp q ↑ . Denote by J the set of indices j such that ψj q ↑ χB(0,1) ≡ 0. It follows from Lemma 2.4 that |Tα (q ↑ b; g)| ≤

j∈J

|Tα (ψj q ↑ b; g)| ≤ Cαd−1

d−1 . j

j∈J

As in the proof of Theorem 2.5, using the ﬁnite intersection property, we can estimate:

220

A.V. Sobolev

d−1 j

≤C

j∈J

j∈J

−1

(x)

dx ≤ C

Bj

cr−1 e−A <xd −Φ(ˆ x)

t−1 dtdˆ x ≤ C (A + log r).

≤C |ˆ x|≤1

(x)−1 dx

cr−1 e−A
Therefore |Tα (q ↑ b; g)| ≤ C(A + log r)αd−1 .

Thus the limit in (2.14) equals zero, as claimed.

The trace Tα (q ↓ b; g) requires a more careful analysis. In particular, we ˆ . Cover Rd−1 by cubes of the need a further partition of unity in variable x d−1 form Qm = Q0 + m, m ∈ Z , where Q0 = (−1, 1)d−1 . Let σm = σm (ˆ x) be a partition of unity, associated with this covering, such that σm (ˆ x) = σ0 (ˆ x − m), which guarantees that |∇jxˆ σm (ˆ x)| ≤ Cj , for all j uniformly in m ∈ Zd−1 . For each k = 0, 1, . . . we use the partition of unity σk,m (ˆ x) = σm (αˆ xr−k−1 ), m ∈ Zd−1 . Now deﬁne for all x ∈ Rd ⎧ ⎨q−1 (x) = ζ−1 α xd − Φ(ˆ x) , ⎩q (x) = q (ˆ x) σk,m (ˆ x), k = 0, 1, . . . , m ∈ Zd−1 , k,m k,m x, xd ) = ζk α xd − Φ(ˆ (2.15) so that q ↓ (x) = q−1 (x) +

K

qk,m (x).

(2.16)

k=0 m∈Zd−1

Note that ˆ k,m , xd ), x ˆ k,m = α−1 rk+1 m. qk,m (ˆ x, xd ) = qk,0 (ˆ x−x

(2.17)

For the sake of brevity we do not write out explicitly the dependence of qk,m and σk,m on the parameters α, r.

Quasi-classical Asymptotics

221

2.4. Two crucial theorems In order to study the trace Tα (q ↓ b) we ﬁnd the asymptotics of Tα (qk,m b) for each k = 0, 1, . . . and m ∈ Zd−1 . To describe the asymptotic results it is convenient to introduce the notion of a W-sequence: Deﬁnition 2.7. Let wk = wk (r, A), k ≥ 0, be a sequence of non-negative numbers, depending on the parameters r, A > 0, and let K = K(α; r, A) be as deﬁned in (2.13). We say that wk is a W-sequence if lim lim sup lim sup

r→∞ A→∞

α→∞

1 log α

K(α;r,A)

wk (r, A) = 0.

(2.18)

k=0

If a W-sequence depends on some other ﬁxed parameter, for instance M from (2.2), then it is not reﬂected in the notation and we still write simply wk (r, A). The next two theorems are central for the whole method. The ﬁrst one (Theorem 2.8) ensures that for each trace Tα (qk,m b) the domain Λ can be replaced with a suitably chosen half-space. The second one (Theorem 2.9) establishes the asymptotics of Tα (qk,m b) for this half-space. ˆ k,m deﬁned in (2.17) we deﬁne the approximating halfFor each point x space by ˆ k,m )}. Πk,m = {x : xd > Φ(ˆ xk,m ) + ∇Φ(ˆ xk,m ) · (ˆ x−x

(2.19)

In words, Πk,m is the half-space above the tangent hyperplane to the domain Λ at the point (ˆ xk,m , Φ(ˆ xk,m )). Note that Πk,m depends only on m ∈ Zd−1 and on k = 0, 1, . . . . Theorem 2.8. Let b satisfy (2.1) with = ρ = 1. Then for any k = 0, 1, . . . , K = K(α; r, A),

qk,m Opα (b)) gp (Tα (1; Λ)) − gp (Tα (1; Πk,m )) S1 ≤ r(k+1)(d−1) wk (r, A), with some W-sequence wk (r, A) independent of the point m ∈ Zd−1 . Moreover, wk (r, A) does not depend on the functions Φ, Ψ, but may depend on the parameter M . Theorem 2.9. Let b satisfy (2.1) with = ρ = 1, and let g be as in (2.12). Then for any k = 0, 1, . . . , K = K(α; r, A), Tα (qk,m b; Πk,m , Ω; g) − αd−1 log r A(g)W1 (σk,m b; ∂Λ, ∂Ω) ≤ r(k+1)(d−1) wk (r, A), with some W-sequence wk (r, A) independent of the point m ∈ Zd−1 , and numbers W1 , A which are deﬁned in (1.5) and (1.6) respectively. We provide a sketch of the proof for both theorems later, but ﬁrst we concentrate on the derivation of formula (1.12) from them.

222

A.V. Sobolev

2.5. Proof of Theorem 1.3 Theorems 2.8 and 2.9 immediately lead to the following conclusion. Corollary 2.10. Under the conditions of Theorem 2.9, Tα (qk,m b; Λ, Ω; g) − αd−1 log r A(g) W1 (σk,m b; ∂Λ, ∂Ω)

(2.20)

≤ r(k+1)(d−1) wk (r, A), with some W-sequence wk (r, A), independent of the point m ∈ Zd−1 . Now we are in a position to ﬁnd the asymptotics in the boundary layer: Theorem 2.11. Under the conditions of Theorem 2.9, lim lim sup lim sup

r→∞ A→∞

α→∞

1 Tα (q ↓ b; Λ, Ω; g) = A(g)W1 (b; ∂Λ, ∂Ω). (2.21) log α

αd−1

Proof. Represent q ↓ (x) in accordance with (2.16): K

q (x) = ζ−1 α(xd − Φ(ˆ x)) + qk,m (x), ↓

(2.22)

k=0 m∈Zd−1

and calculate the contribution of each term to the sought trace. Since the function ζ−1 α( . . . ) is supported on a layer of width α−1 near the boundary ∂Λ, using (2.3) one can show that

ζ−1 Opα (b)g(T (1; Λ)) S1 ≤ Cαd−1 .

(2.23)

Consider now the sum over m and k on the right-hand side of (2.22). For each trace Tα (qk,m b; Λ, Ω; g) we use Corollary 2.10, and then sum up the inequalities (2.20) over m ∈ Zd−1 and k = 0, 1 . . . , K. Let us handle the asymptotic coeﬃcient ﬁrst: Y (α, r, A) :=

K

αd−1 log r A(g)W1 (σk,m b; ∂Λ, ∂Ω)

k=0 m∈Zd−1

= αd−1 log r A(g)W1 (b; ∂Λ, ∂Ω) where we have used that fact that K

k=0

1=

m

K

1,

k=0

σk,m = 1 for all k = 0, 1, . . . . Since

log α − A + O(1), log r

we have 1 Y (α, r, A) = A(g)W1 (b; ∂Λ, ∂Ω), log α for any A ∈ R and r > 0. lim

α→∞

αd−1

Quasi-classical Asymptotics

223

Let us consider the remainder. To estimate the sum of the right-hand sides of (2.20) over diﬀerent values of k and m, we observe that the summation over m for each value of k, is restricted to |m| ≤ Cαr−(k+1) , since the support of the symbol b in the x-variable is contained in the unit ball. Thus Z(α, r, A) :=

K

wk (r, A)r(k+1)(d−1) ≤ Cαd−1

k=0 |m|≤Cαr −(k+1)

K

wk (r, A).

k=0

By deﬁnition of the W-sequence (see (2.18)), lim lim sup lim sup

r→∞ A→∞

α→∞

1 Z(α, r, A) = 0. αd−1 log α

Together with (2.23) this leads to (2.21).

Let us put Theorems 2.11 and 2.6 together. Corollary 2.12. Under the conditions of Theorem 2.9, lim

α→∞

1 tr Oplα (b)g(Tα (1; Λ, Ω)) = A(g)W1 (b; ∂Λ, ∂Ω). αd−1 log α

(2.24)

Proof. To avoid cumbersome formulae, throughout the proof we write G1 ≈ G2 for any two trace-class operators depending on α, such that 1 lim d−1

G1 − G2 S1 = 0. α→∞ α log α For brevity write Tα := Tα (1; Λ, Ω). By (2.4), Opα (b)g(Tα ) ≈ χΛ Opα (b)g(Tα ). Rewrite the symbol χΛ b in the form χΛ b = q ↓ χΛ b + q ↑ χΛ b. Again by (2.4), χΛ q ↓ Opα (b)g(Tα ) ≈ q ↓ Opα (b)g(Tα ), and q ↑ χΛ = q ↑ by deﬁnition of q ↑ . Thus Opα (b)g(Tα ) ≈ q ↓ Opα (b)g(Tα ) + q ↑ Opα (b)g(Tα ). Now apply (2.21) and (2.14). Since the left-hand side of (2.24) does not depend on A or r, the claimed result follows. Proof of Theorem 1.3. As in the proof of Corollary 2.12 we use the notation G1 ≈ G2 . Due to (2.5), gp (Tα (a)) ≈ Opα (ap )gp (Tα (1)). Furthermore, by (2.5) again, with the notation g(t) = tp − t, we have: Opα (ap )gp (Tα (1)) = Opα (ap )g(Tα (1)) + Opα (ap )Tα (1) ≈ Opα (ap )g(Tα (1)) + Tα (ap ).

(2.25)

224

A.V. Sobolev

By Corollary 2.12, 1 lim tr Opα (ap )g(Tα (1)) = W1 (A(gp ; a); ∂Λ, ∂Ω), α→∞ αd−1 log α where we have taken into account that A(gp − g1 ; a) = A(gp ; a). By (2.4), Tα (ap ) ≈ χΛ Opα (ap )PΩ,α χΛ . The trace of the operator on the right-hand side equals αd W0 (gp (a); Λ, Ω). Thus 1 lim tr Opα (ap )gp (Tα (1)) − αd W0 (gp (a); Λ, Ω) α→∞ αd−1 log α − αd−1 log α W1 (A(gp ; a); ∂Λ, ∂Ω) = 0. The reference to (2.25) completes the proof. The formula (1.12) for the operator Sα (a) follows by noticing that Re Opα (a) ≈ Opα (Re a).

3. Proof of Theorem 2.8 The partition of unity constructed in Proposition 2.3 is central for the proof of Theorem 2.8. Let J = Jk,m , k = 0, 1, . . . , K, m ∈ Zd−1 , be the indices j such that ψj qk,m ≡ 0. By deﬁnition (2.8) this means in particular that ψj is supported strictly inside Λ. Next we replace Lemma 2.4 by a more precise statement: we show that for the functions ψj , j ∈ Jk,m one can replace Λ with Πk,m (see deﬁnition (2.19)) in the trace Tα (ψj b). We do this in two steps. Lemma 3.1. Let b satisfy (2.1) with = ρ = 1. Suppose that ψj is supported strictly inside Λ. Then

ψj Opα (b)gp (Tα (1; Λ))(1 − χB(xj ,r j ) ) S1 ≤ C(αj )d−1 r−κ

(3.1)

with some κ > 0 and all suﬃciently large r. If the parameter r ≥ 1 were ﬁxed, then (3.1) would follow from (2.4). The proof for arbitrary r is more complicated. Let us introduce the continuity modulus for the function ∇Φ. Since ∇Φ is uniformly continuous, ε(s) :=

sup |∇Φ(ˆ x) − ∇Φ(ˆ z)| → 0 as s ↓ 0.

|ˆ x−ˆ z|<s

Lemma 3.2. Let b satisfy (2.1) with = ρ = 1. Suppose that j ∈ Jk,m . Then

ψj Opα (b) gp (Tα (1; Λ)) − gp (Tα (1; Πk,m )) S1 ≤ C(αj )d−1 R(r, j ), (3.2) with R(r, j ) = r−κ + r2(d−1)+κ (ε(rj ))κ , for all suﬃciently large r and small rε(tj ).

(3.3)

Quasi-classical Asymptotics

225

To prove the above estimate we replace the operator on the left-hand side of (3.2) with ψj Opα (b) gp (Tα (1; Λ)) − gp (Tα (1; Πk,m )) χB(xj ,r j ) , using Lemma 3.1. Now notice that the set (Λ"Πk,m )∩B(xj , rj ) is contained in the layer between two hyperplanes, of width rj ε(rj ). Here " denotes the symmetric diﬀerence. This observation eventually produces the second term in the deﬁnition (3.3). 3.1. Proof of Theorem 2.8 As before, let J = Jk,m be the set of indices j such that qk,m ψj ≡ 0, so that

qk,m ψj ≡ qk,m . j∈J

Thus by (3.2),

qk,m Opα (b) gp (Tα (1; Λ)) − gp (Tα (1; Πk,m )) S1

≤

ψj Opα (b) gp (Tα (1; Λ)) − gp (Tα (1; Πk,m )) S1 j∈J

≤ Cαd−1

d−1 R(r, j ). j

(3.4)

j∈J

Recall that k ≤ K(α; r, A). This means that j ≤ Cα−1 rk+1 ≤ C re−A for all j ∈ J. Thus the right-hand side of (3.4) does not exceed

Cαd−1 R(r, C re−A ) d−1 . j j∈J

We estimate the sum using the same idea as in the proof of Theorem 2.5. Namely, in view of the ﬁnite intersection property (see Proposition 2.3), we can estimate the sum by the integral:

d−1 j ≤ C (x)−1 dx j∈J

supp qk,m

≤ Cα

1−d (k+1)(d−1)

Cα−1 r k+1

r

cα−1 rk

t−1 dt ≤ Cα1−d r(k+1)(d−1) (1 + log r).

Consequently the right-hand side of (3.4) is bounded by Cr(k+1)(d−1) log r R(r, C re−A ). It remains to show that the constant sequence wk (r, A) := log r R(r, C re−A ) is a W-sequence. By deﬁnition (2.13), 1 log α

K(α;r,A)

k=0

wk (r, A) ≤ R(r, C re−A ).

226

A.V. Sobolev

By deﬁnition (3.3),

κ R(r, C re−A ) = r−κ + r2(d−1)+κ ε(C r2 e−A ) .

The second term tends to zero as A → ∞, since ε(s) → 0 as s → 0. The ﬁrst term in (3.3) tends to zero as r → ∞. Thus wk is indeed a W-sequence, as required. This completes the proof of Theorem 2.8.

4. Proof of Theorem 2.9 4.1. Reduction to the problem in one dimension Applying an appropriate aﬃne transformation, we may assume Φ(ˆ xk,m ) = 0 and ∇Φ(ˆ xk,m ) = 0. This ensures that Πk,m = {x : xd > 0}. Let ˆ = {t ∈ R : (ξ, ˆ t) ∈ Ω}. Ω(ξ) (4.1) A straightforward calculation shows that gp Tα (1; Πk,m , Ω) is a PDO in L2 (Rd−1 , H), H = L2 (R), with the operator-valued symbol ˆ . gp Tα (1; R+ , Ω(ξ)) Therefore the operator Xα = qk,m Opα (b)g Tα (1; Πk,m , Ω) , g(t) = tp − t can be viewed as a PDO with the operator-valued symbol ˆ = qk,m (ˆ ˆ · ) g Tα (1; R+, Ω(ξ)) ˆ , Xα (ˆ x, ξ) x, · ) Op b(ˆ x, · ; ξ, (4.2) α

i.e.,

(Xα u)(x) =

α 2π

d−1

ˆ ˆ y, · ) dˆ ˆ eiαξ·(ˆx−ˆy) Xα (ˆ x, ξ)u(ˆ ydξ,

Rd−1 Rd−1

for any u from the Schwartz class on Rd . In a standard manner one shows that d−1 α ˆ xdξ. ˆ tr Xα = tr Xα (ˆ x, ξ)dˆ (4.3) 2π Rd−1 Rd−1

The asymptotics of the trace under the integral are handled using the asymptotic result for the operators Tα in the case of one dimension. The required information is collected in the next subsection. 4.2. Asymptotics for the problem in one dimension We begin with the model operator T1 := Tα (1; R+, R+ ) = T1 (1; R+ , R+ ). As in [22], using the Mellin transform M : L2 (R+ ) → L2 (R): ∞ 1 1 u ˜(s) = √ x− 2 +is u(x)dx, 2π 0 one can easily show that the operator T1 is unitarily equivalent to the multiplication by the function 1 1 + e2πz

Quasi-classical Asymptotics

227

in L2 (R). Thus g(T1 ) is also multiplication by a function. If g satisﬁes the conditions g ∈ C1 (R), g(0) = g(1) = 0, (4.4) then this function is integrable, and hence one can write the kernel of g(T1 ): is 1 1 y K(x, y) = (xy)− 2 g (1 + e2πs )−1 ds. 2π R x We are computing the asymptotics of the trace Tα (ψb; R+ , R+ ; g) with a suitable symbol b, and a non-negative function ψ ∈ C∞ 0 (R) such that ψ(x) ≤ 1 for all x ∈ R and 0, x ∈ / (1, 4r), ψ(x) = (4.5) 1, x ∈ (4, r), with a parameter r > 4. Theorem 4.1. Suppose that the symbol b satisﬁes (2.1) with some > 0, ρ > 0, and that g is as in (4.4). Then under the assumption αρ ≥ c, for any δ > 0 and > 0, we have |Tα (ψb; R+ , R+ ; g) − A(g)b(0, 0) log r| ≤ C[1 + r−1 + (αρ)−δ log r], (4.6) with a constant C = C(δ), uniformly in r ≥ 5, the symbol b satisfying (2.1), and the function ψ satisfying (4.5). The proof is based on the asymptotic analysis of the formula α Tα (ψb; R+ , R+ ; g) = ψ(z)b(z, ξ)eiαξ(z−x) K(x, z)dξdzdx, 2π R+ R+ R and on the fact that 1 K(1, 1) = 2π

R

g (1 + e2πs )−1 ds = A(g),

where A(g) is deﬁned in (1.6). This lemma is a modiﬁed version of a similar result from [22] which gives the asymptotics of the trace Tα (b; R+ , R+ ; g) as α → ∞ (i.e., when ψ(x) = 1), and the parameters = ρ = 1. Note that it is important for our approach that the formula (4.6) contains explicit dependence on all the parameters , r, ρ, α. When we apply (4.6) to study the operator (4.2) we choose these parameters in such a way that the right-hand side of (4.6) is uniformly bounded. In order to handle the operator Xα (see (4.2)) we need to allow a more general set Ω instead of R+ . Now let Ω be a subset of real line such that A A Ω (−2, 2) = Ij (−2, 2), j

where {Ij }, j = 1, 2, . . . is a ﬁnite collection open intervals with distinct endpoints, i.e., I¯j ∩ I¯s = ∅ for j = s. Now we write a formula analogous to (4.6) for the trace Tα (ψb; R+ , Ω; g). This formula depends on the spacing between the endpoints of the intervals Ij . More precisely, let X be the set

228

A.V. Sobolev

of their endpoints which lie inside (−2, 2). If X = ∅, then for any δ ≥ 0 we deﬁne mδ (X) = 4−δ , X = ∅. If #X = N ≥ 1, then we label the points ξ ∈ X in increasing order: ξ1 , ξ2 , . . . , ξN , and deﬁne ⎧ N ⎨ρj = dist{ξj , X \ {ξj }}, N ≥ 2,

mδ (X) = ρ−δ , j ⎩ρ = 4, N = 1. j=1 1 Theorem 4.2. Suppose that the symbol b satisﬁes (2.1) with some > 0 and ρ = 1. Let g(t) = tp −t with some p = 1, 2, . . . . Assume that α ≥ c, ≥ r ≥ 5. Then

Tα (ψb; R+ , Ω; g)− A(g) b(0, ξ) log r ξ∈X

5 6 ≤ Cδ mδ (X) 1 + α1−δ r + α−δ log r ,

(4.7)

for any δ ≥ 1, where the sum on the left-hand side equals zero if X = ∅. Theorem 4.2 is derived from Theorem 4.1 by using a suitable partition of unity localizing to neighbourhoods of the endpoints of the intervals Ij . In order to apply this theorem to the trace (4.3) we write down a rescaled version of the formula (4.7). Theorem 4.3. Suppose that b satisﬁes (2.1) with = ρ = 1. Let g be as in Theorem 4.2, and let ˜ = ψ(tαr−k ), ψ(t) (4.8) with some k = 0, 1, . . . . Assume that α ≥ rk+1 , r ≥ 5. Then

5 6 ˜ R+ , Ω; g) − A(g) Tα (ψb; b(0, ξ) log r ≤ Cδ mδ (X) 1 + rk(1−δ)+1 , (4.9) ξ∈X

for any δ ≥ 1, where the sum on the left-hand side equals zero if X = ∅. Proof. The rescaling x = t, = αr−k , reduces the operator ˜ Opα (ψb)g(T α (1; R+ , Ω)) to the unitarily equivalent operator Opβ (ψ˜b)g(Tβ (1; R+ , Ω)), ˜b(x, ξ) = b(x−1 , ξ), β = rk . The symbol ˜b satisﬁes (2.1) with the parameter just deﬁned. Since αr−k ≥ r and r ≥ 5 we have ≥ r and β ≥ 1. Now applying (4.7) we get

5 6 Tβ (ψ˜b; R+ , Ω; g) − A(g) b(0, ξ) log r ≤ Cδ mδ (X) 1 + β 1−δ r + β −δ log r . ξ∈X

The sum in the square brackets on the right-hand side is bounded by 1 + rk(1−δ)+1 , whence (4.9).

Quasi-classical Asymptotics

229

4.3. Proof of Theorem 2.9, continued ˆ The set (4.1) is a union of (countably many) open intervals. Denote by X(ξ) the set of their endpoints lying inside the interval (−2, 2). Before applying Theorem 4.3 to the operator (4.2) we state the following property of the set ˆ X(ξ). ˆ have Lemma 4.4. For almost all ξˆ the open intervals-constituents of Ω(ξ) ˆ distinct endpoints, and the set X(ξ) is ﬁnite. Moreover, for any δ ∈ (0, 2) the ˆ satisﬁes the bound function mδ (X(ξ)) ˆ ξˆ ≤ C(1 + ∇2 Ψ L∞ + ∇3 Ψ L∞ ). mδ (X(ξ))d (4.10) ˆ |ξ|≤4

The constant C depends only on δ and dimension d. For δ ≤ 1 the term with ∇3 Ψ can be removed from the right-hand side. Such an estimate was proved in [23] for δ = 1. To make use of Theorem 4.3 it is imperative for us to have δ > 1, which necessitates the assumption Ψ ∈ C3 . In fact, this is the only reason why we need the assumption that Ω is a C3 -domain. Let us return to the study of the operator (4.2). Recall the assumptions Φ(ˆ xk,m ) = 0, ∇Φ(ˆ xk,m ) = 0. Under these assumptions the function ζk (α( · − Φ(ˆ x))) which enters the deﬁnition (2.15), has the form (4.8) with the function ψ described in (4.5). As k ≤ K(α; r, A) (see (2.13)), we also have rk+1 ≤ re−A α. Consequently for suﬃciently large A the bound α ≥ rk+1 holds. Thus, apˆ we get plying Theorem 4.3 to the operator Xα (ˆ x, ξ),

◦ ◦ ˆ tr Xα (ˆ ˆ x , ξ)− log r A(g)σ (ˆ x ) b x , 0; ξ, Ψ( ξ) k,m ˆ ξd ∈X(ξ)

5 6 ˆ 1 + r(1−δ)k+1 , ≤ Cδ mδ (X(ξ)) for any δ ≥ 1. Now, integrating this formula in accordance with (4.3) we get tr Xα − αd−1 log r A(g) W1 (σk,m b; ∂Πk,m , ∂Ω) (4.11) ≤ Cαd−1 1 + r(1−δ)k+1 dˆ x ≤ Cr(k+1)(d−1) 1 + r(1−δ)k+1 . |ˆ x|≤Cα−1 rk+1

Here we have used (4.10). Under the condition δ > 1 the sequence in the brackets on the right-hand side is a W-sequence (see Deﬁnition 2.7), since 1 1 1 r 1 + r(1−δ)k+1 ≤ + , log α log r log α 1 − r1−δ K

k=0

and hence the triple limit limr→∞ lim supA→∞ lim supα→∞ of the left-hand side of (4.11) equals zero. To complete the proof of Theorem 2.9 it remains

230

A.V. Sobolev

to notice that replacing Πk,m with ∂Λ in the integral W1 in (4.11) produces an error of the size r(k+1)(d−1) wk (r, A) with some W-sequence wk (r, A).

References [1] A. B¨ ottcher, B. Silbermann, Analysis of Toeplitz operators, Second edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. [2] P. Deift, A. Its, I. Krasovsky, Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. Math. 174 (2011), no. 2, 1243–1299. [3] T. Ehrhardt, A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities, Recent advances in operator theory (Groningen, 1998), 217–241, Oper. Theory Adv. Appl., 124, Birkh¨ auser, Basel, 2001. [4] D. Gioev, Generalizations of Szeg˝ o Limit Theorem: Higher Order Terms and Discontinuous Symbols, PhD Thesis, Dept. of Mathematics, Royal Inst. of Technology (KTH), Stockholm, 2001. [5] D. Gioev, Szeg˝ o Limit Theorem for operators with discontinuous symbols and applications to entanglement entropy, (2006) IMRN, article ID 95181, 23 pages. [6] D. Gioev, I. Klich, Entanglement Entropy of fermions in any dimension and the Widom Conjecture, Phys. Rev. Lett. 96 (2006), no. 10, 100503, 4pp. [7] U. Grenander, G. Szeg˝ o, Toeplitz forms and their applications, Berkeley- Los Angeles, U of California press, 1958. [8] R.C. Helling, H. Leschke, W.L. Spitzer, A special case of a conjecture by Widom with implications to fermionic entanglement entropy, Int. Math. Res. Notices 2011 (2011), 1451–1482. [9] L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators, I, Grundlehren Math. Wiss. 256, Springer-Verlag, Berlin, 1983. [10] H. Landau, H. Widom, Eigenvalue distribution of time and frequency limiting, J. Math. Analysis Appl. 77 (1980), 469–481. [11] A. Laptev, Yu. Safarov, A generalization of the Berezin-Lieb inequality, Contemporary Mathematical Physics, 69–79, Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, 1996. [12] A. Laptev and Yu. Safarov, Szeg˝ o type limit theorems, J. Funct. Anal., 138 (1996), 544-559. [13] I. J. Linnik, A multidimensional analog of a limit theorem of G. Szeg˝ o, Mathematics USSR-Izvestiya 9(1975), no. 6, 1323–1332. [14] N.K.Nikolski, Operators, functions, and systems: an easy reading. Vol.1. Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, 92, American Mathematical Society, Providence, RI, 2002. [15] R. Roccaforte, Asymptotic expansions of traces for certain convolution operators, Trans. Amer. Math. Soc. 285 (1984), no. 2, 581–602. [16] D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J. 43 (1964), 3009–3057.

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[17] D Slepian, Analytic solution of two apodization problems, J. Optical Soc. of America, 55 (1965), no 9, 1110–1115. [18] A. V. Sobolev, Quasi-classical asymptotics for the pseudo-diﬀerential operators with discontinuous symbols: Widom’s hypothesis, arXiv:1004.2576v1 [math.SP] 2010. [19] G. Szeg˝ o, On certain Hermitian forms associated with the Fourier series of a positive function, Festskrift Marcel Riesz, Lund, 1952, pp. 228–238. [20] H. Widom, A theorem on translation kernels in n dimensions, Trans. AMS 94 (1960), no. 1, 170–180. [21] H. Widom, Szeg˝ o’s limit theorem, the higher-dimensional matrix case, J. Funct. Anal. 39 (1980), 182–198. [22] H. Widom, On a class of integral operators with discontinuous symbol, Toeplitz centennial (Tel Aviv, 1981), pp. 477–500, Operator Theory: Adv. Appl., 4, Birkh¨ auser, Basel-Boston, Mass., 1982. [23] H. Widom, On a class of integral operators on a half-space with discontinuous symbol, J. Funct. Anal. 88 (1990), no. 1, 166–193. Alexander V. Sobolev Department of Mathematics University College London Gower Street, London UK e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 233–242 c 2012 Springer Basel AG

On a Supremum Operator Vladimir D. Stepanov Abstract. For a supremum operator Rϕ(t) := esssup Φ(y, t)ϕ(y) on the y∈[t,∞)

semi-axis with a measurable non-negative function Φ(x, y) the weighted Lp − Lq boundedness on the cone of non-increasing functions is characterized. Mathematics Subject Classiﬁcation (2010). Primary 26D10; Secondary 26D15, 26D07. Keywords. Integral inequalities, weights, Hardy operator, monotone functions, measures.

1. Introduction Let R+ := [0, ∞). Denote M+ the set of all non-negative measurable functions on R+ and M↓ ⊂ M+ the subset of all non-increasing functions. For a measurable non-negative function Φ(x, y), on {(x, y) : x ≥ y ≥ 0} we deﬁne the supremum operator Rϕ(t) := esssup Φ(y, t)ϕ(y), ϕ ∈ M↓ .

(1.1)

y∈[t,∞)

The paper is devoted to the necessary and suﬃcient conditions for the inequality ∞ 1/q ∞ 1/p q [Rϕ(t)] w(t)dt ≤C ϕp (t)v(t)dt , ϕ ∈ M↓ (1.2) 0

0

with non-negative locally integrable on R+ weight functions v and w and a constant C ≥ 0, independent on ϕ, which we suppose to be the least possible. This problem was studied in the paper by A. Gogatishvili, B. Opic and L. Pick ([1], Theorem 3.2) in a more simple case, when Φ(x, y) is independent on y and contnious with respect to x. In our work we use the technique of the paper [1]. With diﬀerent supremum operators some similar problems were The work of the author was partially supported by the Russian Fund for Basic Research (Projects 09-01-00093, 09-01-00586 and 10-01-91331).

234

V.D. Stepanov

studied in [2]–[6]. This area is currently developing intensively and ﬁnds many interesting applications. In Section 2 we give some preliminaries. In particular, we extend our result ([7], Theorem 4.1). In Sections 3 and 4 the cases 0 < p ≤ q < ∞ and 0 < q < p < ∞ of (1.2) are considered, respectively. We use signs := and =: for determining new quantities and Z for the set of all integers. For positive functionals F and G we write F # G, if F ≤ cG with some positive constant c, which depends only on irrelavant parameters. F ≈ G means F # G # F or F = cG. χE denotes the characteristic function 0 (indicator) of a set E. Uncertainties of the form 0 · ∞, ∞ ∞ and 0 are taken to be zero.

2. Preliminaries We need the following simple case of ([8], Lemma 1.2). Lemma 2.1. Let f ∈ M↓ . Then there exist the sequence of non-negative ﬁnitely supported integrable functions {gn } ⊂ M+ such, that the functions ∞ fn (x) := x gn (s)ds are increasing with respect to n for any x > 0 and ∞ f (x) = lim x gn (y)dy for almost all x > 0. n→∞

Let α > 0 and let Φ(y, t) ≥ 0 be a measurable function with respect to both variables on the set {(y, t) ∈ R2 : y ≥ t ≥ 0}. On the cone M↓ we deﬁne the operator ∞ 1/α α α Rα ϕ(t) := Φ (y, t)ϕ (y)dy , ϕ ∈ M↓ . (2.1) t

For p, q ∈ (0, ∞) and weight functions v and w we deﬁne ∞ 1/q [Rα ϕ]q w 0 Jα := sup ∞ 1/p ϕ∈M↓ ϕp v 0

(2.2)

In the next assertion we extend ([7], Theorem 4.1). t Lemma 2.2. Let 0 0

s

Proof. Let t > 0 and ϕt (s) := χ[0,t] (s). Then t 1/α Rα ϕt (s) = χ[0,t] (s) Φα (y, s)dy s

and

t Jα ≥

α

Φ (y, s)dy 0

s

1/q

q/α

t

w(s)ds

V −1/p (t).

On a Supremum Operator

235

Hence, Jα ≥ Aα . Conversely, as ϕα ∈ M ↓ if and only if ϕ ∈ M ↓ we can change ϕα by ϕ in Jα . Then ∞ ∞ α q/α Φ (y, t)ϕ(y)dy w(t)dt q 0 t Jα = sup ∞ q/p p/α ϕ∈M↓ ϕ v 0 ↓ ∞ Let Aα < ∞ and suppose ﬁrst for ϕ ∈ M the representation ϕ(y) = h(s)ds holds with a non-negative ﬁnitely supported integrable function y h. Then α/p p α/p ∞ ∞ αp −1 ϕ(y) = h h(s)ds . (2.3) α y s

Using this representation and Minkowskii’s inequality we ﬁnd ∞ Φα (y, t)ϕ(y)dy t

=

≤

p α/p α

∞

Φα (y, t) y

∞

αp −1

∞

t

α/p

αp −1

∞

h(s)ds

dy

s

h

α

h

t

p α/p

∞

α

h(s)

Φ (y, t)dy

s

α/p

p/α

s

ds

.

t

Now, again applying Minkowskii’s inequality, we obtain q/α ∞ ∞ Φα (y, t)ϕ(y)dy w(t)dt 0

≤ ≤

t

p q/p α

×⎝

≤ Aqα = Aqα

∞

∞

αp −1

∞

h

0

p q/p α ⎛

∞

t

0

∞

α

0 ∞ ∞

∞

p

ϕα v

s s

q/α

ds

w(t)dt

Φα (y, t)dy

p/q w(t)dt

⎞q/p ds⎠

t

q/p

αp −1 h

0

Φα (y, t)dy t

αp −1 h h(s)

q/p

p/α

s

h(s)

s

s

p q/p

h(s)V (s)ds

s

q/p .

0

The proof of the upper bound Jα ≤ Aα now follows by Lemma 2.1 and by the Theorem on Monotone Convergence.

236

V.D. Stepanov

3. The case p ≤ q Let Φ(x, y) ≥ 0 be a measurable function with respect to both variables on the set {(x, y) ∈ R2 : x ≥ y ≥ 0}. Put Φ∞ (x, y) := esssupΦ(s, y).

(3.1)

s∈[y,x]

Proposition 3.1. If ϕ ∈ M↓ , then esssup Φ(y, t)ϕ(y) = sup ϕ(s)Φ∞ (s, t). y∈[t,∞)

s∈[t,∞)

Proof. It follows from the properties of the essential supremum that esssup Φ(y, t)ϕ(y) = sup esssupΦ(y, t)ϕ(y) s≥t y∈[t,s]

y∈[t,∞)

≥ sup ϕ(s)esssupΦ(y, t) = sup ϕ(s)Φ∞ (s, t). s≥t

y∈[t,s]

(3.2)

s∈[t,∞)

To prove the inverse we use that Φ(y, t) ≤ Φ∞ (y, t) for almost all y ∈ [t, ∞). Then esssup Φ(y, t)ϕ(y) ≤ esssup Φ∞ (y, t)ϕ(y) = sup Φ∞ (y, t)ϕ(y). y∈[t,∞)

y∈[t,∞)

y∈[t,∞)

Now, on the cone M↓ we consider the operator Rϕ(t) := esssup Φ(y, t)ϕ(y), ϕ ∈ M ↓,

(3.3)

y∈[t,∞)

which might be interpreted as the extremal for the set of operators (2.1) when α → ∞. It follows from Proposition 3.1 that Rϕ(t) = sup Φ∞ (s, t)ϕ(s).

(3.4)

s≥t

Therefore, without a loss of generality, the function Φ(y, t) in the deﬁnition (3.3) we may and shall assume non-decreasing with respect to y for y ≥ t. Analogously with (2.2) we set ∞ q 1/q [Rϕ] w 0 J := sup (3.5) ∞ 1/p . ϕ∈M↓ ϕp v 0 Theorem 3.2. Let 0 0

(3.6)

0

or, equivalently, '

(q

t

A := sup t>0

esssupΦ(y, s) 0

y∈[t,s]

1/q w(s)ds

V −1/p (t).

(3.7)

On a Supremum Operator

237

Proof. Let t > 0 and ϕt (x) := χ[0,t] (x). Then Rϕt (x) = sup Φ∞ (y, x)χ[0,t] (y) = χ[0,t] (x)Φ∞ (t, x). y≥x

We have

J≥

t

0

1/q Φ∞ (t, s)w(s)ds V −1/p (t), t > 0.

Hence, J ≥ A. As ϕq ∈ M ↓ if and only if ϕ ∈ M ↓ we can change ϕq by ϕ on the right hand side of (3.5). Then, using (3.4), we ﬁnd 6 ∞5 supy≥t Φq∞ (y, t)ϕ(y)dy w(t)dt q 0 J = sup . ∞ p/q v q/p ϕ∈M↓ ϕ 0 Applying representation (2.3) α = q, we obtain q/p p α/p ∞ ∞ q −1 p q q sup Φ∞ (y, t)ϕ(y) = sup Φ∞ (y, t) h h(s)ds q y≥t y≥t y s q/p p q/p ∞ ∞ q −1 p ≤ sup h Φq∞ (s, t)h(s)ds . q y≥t y s Now, applying Minkowskii’s inequality, we get 4 ∞3 q sup Φ∞ (y, t)ϕ(y) w(t)dt 0

y≥t

q/p q/p ∞ ∞ ∞ pq −1 p ≤ h Φp∞ (s, t)h(s)ds q 0 t s q/p q/p ∞ ∞ ∞ pq −1 p q ≤A h h(s)V (s)ds q 0 t s ∞ q/p = Aq ϕp/q v . 0

The remaining part of the proof of the upper bound J ≤ A follows by applying Lemma 2.1 and the Monotone Convergence Theorem . Remark 3.3. In the case Φ(y, t) = esssup u(s), s∈[t,y]

where u(s) ≥ 0 is a measurable function, we obtain Rϕ(t) = esssup u(s)ϕ(s), s∈[t,∞]

and Theorem 3.2 extends ([1], Theorem 3.2 (i)), where the function u(s) was supposed to be continuous.

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V.D. Stepanov

4. The case q < p Deﬁnition 4.1. A measurable function Φ(x, y) ≥ 0 on {(x, y) : x ≥ y ≥ 0}, we name Oinarov kernel, Φ(x, y) ∈ O, if there exist a constant D ≥ 1, independent of x, y and z such that D−1 (Φ(x, z) + Φ(z, y)) ≤ Φ(x, y) ≤ D (Φ(x, z) + Φ(z, y))

(4.1)

for all x ≥ z ≥ y ≥ 0. Proposition 4.2. Let Φ(x, y) ∈ O. Then Φ∞ (x, y) ∈ O. Proof. For all x ≥ y ≥ 0 from (3.1) and (3.2) we have Φ∞ (x, y) = esssupΦ(s, y) = sup esssupΦ(t, y) = sup Φ∞ (s, y). s∈[y,x]

s∈[y,x] t∈[y,s]

(4.2)

s∈[y,x]

Then Φ∞ (x, y) is non-decreasing with respect to x for x ∈ [y, ∞). Similarly, using (4.1), we ﬁnd Φ∞ (x, y) = esssupΦ(s, y) = sup esssupΦ(t, y) ≥ D−1 sup Φ(s, y) s∈[y,x]

≥D

−1

s∈[y,x] t∈[s,x]

s∈[y,x]

Φ(x, y).

Therefore, Φ(x, y) ≤ DΦ∞ (x, y)

(4.3)

for all x ≥ y ≥ 0. Let x ≥ z ≥ y ≥ 0. Then it follows from (4.2), that Φ∞ (x, y) ≥ Φ∞ (z, y).

(4.4)

Moreover, again using (3.2) and (4.1), we ﬁnd Φ∞ (x, y) = sup esssupΦ(t, y) ≥ sup esssupΦ(t, y) s∈[y,x] t∈[s,x]

≥D

−1

s∈[z,x] t∈[s,x]

sup esssupΦ(t, s) = D

−1

s∈[z,x] t∈[s,x]

sup Φ∞ (x, s) ≥ D−1 Φ∞ (x, z).

(4.5)

s∈[z,x]

From this and (4.4) the left hand side of (4.1) follows for Φ∞ (x, y). Let x ≥ y ≥ 0 and z ∈ [y, x], s ∈ [y, x]. Then it follows from (4.1) that Φ(s, y) = χ[y,z] (s)Φ(s, y) + χ[z,x] (s)Φ(s, y) ≤ D χ[y,z] (s)Φ(z, y) + χ[z,x] (s) (Φ(s, z) + Φ(z, y)) = D Φ(z, y) + χ[z,x] (s)Φ(s, z) . From this and (4.3)

Φ∞ (x, y) ≤ D Φ(z, y) + esssupΦ(s, z)

≤ D2 (Φ∞ (z, y) + Φ∞ (x, z)) .

s∈[z,x]

Hence, Φ∞ (x, y) ∈ O with a constant D2 in (4.1).

Remark 4.3. If 0 < q < ∞ and Φ(x, y) ∈ O, that is (4.1) holds, then Φq (x, y) ∈ O, so that (4.1) holds with some constant Dq , dependent only on D and q.

On a Supremum Operator

239

Theorem 4.4. Let 0 < q < p < ∞, 1r := 1q − p1 and Φ(x, y) ∈ O. Suppose Φ(x, y) be continuous with respect to x for x ∈ [y, ∞) for all y ≥ 0 and t assume that the weight functions v and w such that 0 < V (t) := 0 v < t ∞, 0 < W (t) := 0 w < ∞ and V (∞) = W (∞) = ∞. If the supremum operator R and the functional J are deﬁned by (3.3) and (3.5), respectively, and r1 rq

xk+1 − rp q B := sup Φ∞ (xk+1 , t)w(t)dt [V (xk+1 )] , (4.6) {xk } k

xk

where the sup is taken over all increasing sequences {xk } ⊂ R+ , then J ≈ B.

(4.7)

Proof. We start with the proof of the upper bound J # B. To this end we note, that because of (3.4) and (4.5) we have Rϕ(s) ≤ DRϕ(t) for all s ≥ t ≥ 0, if ϕ ∈ M↓ . Let a > 1 be a number, which we choose later and let {xk }, {yk } ⊂ R+ be such increasing sequences, that W (xk ) = V (yk ) = ak , k ∈ Z. We have ∞ 0

(Rϕ)q w ≤ Dq

= D q (a − 1)

[Rϕ(xk )]

q

(4.8)

xk+1

w xk

k

q

ak sup [Φ∞ (s, xk )ϕ(s)] =: Dq (a − 1) s≥xk

k

ak Ik .

k

Since Φ(x, y) ∈ O, then Φ∞ (x, y) ∈ O by Proposition 4.2 and by Remark 4.3 we have Φq∞ (x, y) ∈ O. Let Dq ≥ 1 be a constant, for which (4.1) holds for Φq∞ (x, y). Then, applying (4.1) with Φq∞ (x, y) and Dq , we obtain Ik ≤ ≤

sup xk ≤s≤xk+1

sup xk ≤s≤xk+1

Φq∞ (s, xk )ϕ(s) + sup Φq∞ (s, xk )ϕ(s) s≥xk+1

Φq∞ (s, xk )ϕ(s)

'

+ Dq

(

Φq∞ (xk+1 , xk )ϕ(xk+1 )

≤ (1 + Dq )

sup xk ≤s≤xk+1

+ sup s≥xk+1

Φq∞ (s, xk )ϕ(s)

Φq∞ (s, xk+1 )ϕ(s)

+ Dq sup Φq∞ (s, xk+1 )ϕ(s) s≥xk+1

=: (1 + Dq )Lk + Dq Ik+1 . We ﬁnd from this

I := ak Ik ≤ (1 + Dq ) ak L k + D q ak Ik+1 k

= (1 + Dq )

k

k

k

Dq k Dq a k Lk + a Ik = (1 + Dq ) a k Lk + I. a a k

k

240

V.D. Stepanov

Now we choose a > 1 such, that a > 2Dq . Then I ≤ 2(1 + Dq ) Consequently, ' (q ∞

q k (Rϕ) w # a sup Φ∞ (s, xk )ϕ(s) . 0

k

a k Lk .

xk ≤s≤xk+1

k

It follows from the continuity of Φ(x, y) that Φ∞ (x, y) is continuous. Moreover, taking into account Proposition 3.1 without a loss of generality we may and shall assume Φq∞ (s, xk )ϕ(s) to be continuous on [xk , xk+1 ]. while the upper bound is proving. Then there exist a point zk ∈ [xk , xk+1 ] such, that sup xk ≤s≤xk+1

Φ∞ (s, xk )ϕ(s) ≤ a1/q Φ∞ (zk , xk )ϕ(zk ).

Now using (4.1) and zk−2 ≤ xk−1 < xk ≤ zk , it follows, that ∞

xk q (Rϕ) w # w(t)dt Φq∞ (zk , xk )ϕq (zk ) 0

xk−1

k

#

k

≤

zk

w(t)Φq∞ (zk , t)dt

w(t)Φq∞ (z2k , t)dt

ϕq (z2k )

z2k+1

z2k−1

k

ϕq (zk )

z2k

ϕq (zk )

w(t)Φq∞ (zk , t)dt

z2k−2

k

+

xk−1

zk−2

k

=

xk

w(t)Φq∞ (z2k+1 , t)dt ϕq (z2k+1 )

=: Seven + Sodd . Further we estimate only Seven , the arguments for Sodd are similar. Put Yk := {l ∈ Z : yl ∈ [z2k−2 , z2k ]}, Y := {k ∈ Z : Yk = ∅}, where yl are taken from the deﬁnition (4.8). Denote θk := min{yl : l ∈ Yk }, k ∈ Y ; Θ := {θk }k∈Y ⊂ {yl }l∈Z and renumerate Θ so, that Θ =: {yn }n∈Z and yn < yn+1 . It is shown in the proof of ([1], Theorem 3.2 (ii)) that if n, k ∈ Z such, that yn < z2k ≤ yn+1 , then q/p yn+1 −q/p yn+1 q p [ϕ(z2k )] # v ϕ v . (4.9) 0

yn−1

Denote An := {k ∈ Z : yn < z2k ≤ yn+1 }, n ∈ Z. Then Seven =

n

An

z2k

z2k−2

Φq∞ (z2k , t)w(t)dt ϕq (z2k ).

On a Supremum Operator

241

It frollows from the properties of Φ∞ (x, y) ∈ O that Φ∞ (z2k , t) #

sup t≤s≤yn+1

Φ∞ (s, t) # Φ∞ (yn+1 , t).

(4.10)

Applying (4.9), (4.10) and H¨ older’s inequality, we ﬁnd q/p

yn+1 −q/p yn+1 p Seven # v ϕ v n

×

yn+1 yn−1

n

⎛

≤⎝

×

Φq∞ (yn+1 , t)w(t)dt

Φq∞ (yn+1 , t)w(t)dt

yn+1 yn−1

n

yn−1

z2k−2

An

≤

0 z2k

# Bq

q/p

yn+1

v

p

ϕ v

0

yn−1

r/q Φq∞ (yn+1 , t)w(t)dt

−r/p

yn+1

v

⎞q/r ⎠

0

q/p

yn+1

p

ϕ v yn−1

n

−q/p

yn+1

∞

q/p

ϕp v

.

0

From this and analogous bound for Sodd the upper bound J # B follows. Let {xk } ⊂ R+ be an arbitrary increasing sequence, N - any positive integer. The proof of the lower bound J $ B is proceeding with the help of the test function N 1/p N 1/p N

ϕN (t) := χ(0,x−N ) (t) αi + χ[xk ,xk+1 ] (t) αi , i=−N

where

xi+1

αi := xi

k=−N

r/q Φq∞ (xi+1 , t)w(t)dt

i=k

V −r/q (xi+1 ).

We have 0

∞

q

[RϕN ] w ≥

N

k=−N

≥

N

k=−N

=

sup t≤s≤xk+1

xk xk+1

sup t≤s≤xk+1

xk

N

k=−N

xk+1

xk+1 xk

Φq∞ (s, t)w(t)dt

N

q/p αi

i=k q/p

Φq∞ (xk+1 , t)w(t)dtαk r/q

Φq∞ (xk+1 , t)w(t)dt

r V −r/p (xk+1 ) =: BN .

242

V.D. Stepanov

On the other hand ∞ N

p ϕN v = αk 0

k=−N

=

N

=

N

i=−N

v + 0

αi

i=−N

v +

xi+1

v 0

N N

k=−N i=k

x−N

0

αi

x−N

N

i=−N

αi

xk+1

αi

v xk

i

k=−N

xk+1

v

xk

r = BN .

Consequently, J $ BN and the lower bound J $ B follows.

References [1] Gogatishvili A., Opic B. and Pick L. Weighted inequalities for Hardy-type operators involving suprema. Collect. Math., 57 (2006), 227–255. [2] Gogatishvili A. and Pick L. A reduction theorem for supremum operators. J. Comp. Appl. Math., 208 (2007), 270–279. [3] Cwikel M. and Pustylnik E. Weak type interpolation near ”endpoint” spaces. J. Funct. Anal., 171 (2000), 235–277. [4] Evans W.D. and Opic B. Real interpolation with logarithmic functions and reiteration. Canad. J. Math., 52 (2000), 920–960. [5] Pick L. Optimal Sobolev Embeddings. Rudolph-Lipshitz-Vorlesungsreihe no. 43, Rheinische Friedrich-Wilhelms-Universit¨ at Bonn, 2002. [6] Prokhorov D. V. Inequalities for Riemann-Liouville operator involving suprema. Collect. Math., 61 (2010), 263–276. [7] Stepanov V.D. Integral operators on the cone of monotone functions. J. London Math. Soc. 48 (1993), 465–487. [8] Sinnamon G. Transferring monotonicity in weighted norm inequalities. Collect. Math. 54 (2003), 181–216. Vladimir D. Stepanov Department of Mathematical Analysis and Function Theory Peoples Friendship University 117198 Moscow Russia e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 243–262 c 2012 Springer Basel AG

Entropy Numbers of Quadratic Forms and Their Applications to Spectral Theory Hans Triebel Dedicated to my dear friends David Edmunds and Des Evans

Abstract. The paper deals with positive deﬁnite quadratic forms n ∂f (x) ∂g(x) Eb (f, g) = b(x) dx ∂xj ∂xj Ω j=1 in L2 (Ω) in limiting situations. We estimate the entropy numbers of related compact embeddings and apply the outcome to say something about the distribution of eigenvalues of the generated degenerate positive deﬁnite self-adjoint elliptic operators n ∂ ∂f Ab f = − b(x) ∂xj ∂xj j=1 with pure point spectrum. Mathematics Subject Classiﬁcation (2010). 46E35, 41A46, 35P15. Keywords. Quadratic forms, degenerate elliptic operators, entropy numbers, distribution of eigenvalues.

1. Introduction This paper may be considered as the continuation of [14], [15, Section 3.4], and also of [36, 37]. Let Ω be a bounded domain in Rn , n ≥ 2, and let −1 b ∈ Lloc ∈ Ln/2 (Ω). Then 1 (Ω) be a real function with b(x) > 0 a.e. and b the closure Eb (Ω) of the closable quadratic form n

∂f ∂¯ g Eb (f, g) = b(x) (x) (x) dx, f, g ∈ D(Ω), ∂xj ∂xj Ω j=1 is continuously embedded in L2 (Ω) (positive deﬁnite quadratic form). This is a limiting situation. If b−1 belongs to the Zygmund spaces Ln/2 (log L)d (Ω),

244

H. Triebel

n ≥ 3, with d > 0, then id :

Eb (Ω) → L2 (Ω)

is compact.

For d > 4/n and a suitable constant c > 0 one has for the corresponding entropy numbers ek (id), ek (id) ≤ c b−1 |Ln/2 (log L)d (Ω) 1/2 k −1/n ,

k ∈ N.

For 0 < d ≤ 4/n, ε > 0, and suitable constants cε > 0 one has ek (id) ≤ cε b−1 |Ln/2 (log L)d (Ω) 1/2 k − 4 +ε , d

k ∈ N.

This will be complemented by corresponding assertions for bounded domains Ω in the plane R2 , hence for n = 2. We rely on extrapolation techniques which go back (in this context) to [33, 14, 15]. Section 2 deals with the indicated assertions about entropy numbers. Our main results here are the Theorems 2.3, 2.4. In Section 3 we apply these results to spectral assertions for generated positive deﬁnite self-adjoint elliptic operators in L2 (Ω) with pure point spectrum of type Ab f = −

n

∂ ∂f b(x) ∂x ∂x j j j=1

with Theorem 3.1 as our related main result. This will be complemented in Section 4 by some discussions, examples and further references.

2. Entropy numbers of quadratic forms 2.1. Preliminaries Let Ω be a bounded domain in the Euclidean n-space Rn , n ≥ 2. Domain means open set. We use standard notation. In particular, Lp (Ω) with 1 ≤ p ≤ ∞ are the usual complex Lebesgue spaces with respect to the Lebesgue measure indicated by dx. The space of all locally Lebesgueintegrable functions in Ω is denoted by Lloc 1 (Ω). Let D(Ω) be the collection of all complex-valued C ∞ functions on Ω with compact support and let D (Ω) be the related dual space of all distributions. We are interested in symmetric quadratic forms n

∂f ∂¯ g Eb (f, g) = b(x) (x) (x) dx, f, g ∈ D(Ω), (2.1) ∂x ∂x j j Ω j=1 in L2 (Ω) with b ∈ Lloc 1 (Ω) and b(x) > 0 a.e. (almost everywhere) in Ω. We recall some notation in an abstract setting. Let H be a separable complex Hilbert space with scalar product (u, v)H 1/2 and norm u |H = (u, u)H . Let D be a dense linear subset of H and E : D × D → C (complex numbers) be a bilinear symmetric map, hence E(λ1 u1 + λ2 u2 , v) = λ1 E(u1 , v) + λ2 E(u2 , v),

E(u, v) = E(v, u),

Entropy Numbers of Quadratic Forms

245

where u1 , u2 , u, v ∈ D and λ1 , λ2 ∈ C. A positive deﬁnite quadratic form is a densely deﬁned bilinear symmetric map such that E(u, u) ≥ c u |H 2

for some c > 0 and all u ∈ D.

(2.2)

Recall that a positive deﬁnite quadratic form E is called closable on H when E(uk , uk ) → 0 for k → ∞ for all sequences {uk } ⊂ D with uk → 0 in H and for any ε > 0, E uk − ul , uk − ul ≤ ε if k ≥ l ≥ l(ε) (Cauchy E-sequence). Then the abstract completion of D with respect to the E-norm (the space of all E-Cauchy sequences) can be identiﬁed in a one-toone way with a related linear subset in H, the domain of deﬁnition dom E of the closure of E. One has (2.2) now with u ∈ dom E. Let dom A = u ∈ dom E : E(u, v) = (u , v)H for some u ∈ H and all v ∈ dom E . (2.3) Then u is uniquely determined and Au = u generates a positive deﬁnite self-adjoint operator in H. As usual, dom E = dom A1/2 is called the energy space, and A1/2 dom E = H, A−1/2 H = dom E, (2.4) are isomorphic maps. Details about closable and closed (positive deﬁnite) forms may be found in [28]. The construction (2.3) coincides essentially with Friedrichs’ extension of positive deﬁnite symmetric operators in complex Hilbert spaces. We refer for details to [10, Section IV,2, pp. 172–180] and [32, Sections 4.1.9, 4.4.3, pp. 213–215, 253]. A description may also be found in [34, pp. 190/191] with a reference to [9, Section 4.4, pp. 81–84] for a short direct proof. We return to the quadratic form Eb (f, g) in a bounded domain Ω in Rn , n ≥ 2, according to (2.1). Let Lp (Ω) with 1 ≤ p ≤ ∞ be the usual complex ◦ ∂f n Lebesgue spaces and let ∇f (x) = ∂x . Then W 1p (Ω) with 1 ≤ p < ∞ j=1 j is the completion of D(Ω) in the norm n

◦

f |W 1p (Ω) = |∇f | Lp (Ω) ∼ j=1

∂f |Lp (Ω) . ∂xj

(2.5)

Recall that for some c > 0, ◦

f |L2 (Ω) ≤ c f |W 12n (Ω) , n+2

◦

f ∈ W 12n (Ω).

(2.6)

n+2

If n ≥ 3, then (2.6) is the well-known Sobolev embedding. In case of n = 2 we refer for a short elegant proof of

f |L2 (Ω) ≤ c

∂f ∂f |L1 (Ω) + c |L1 (Ω) , ∂x1 ∂x2

to [39, Theorem 2.4.1, p. 56].

f ∈ D(Ω),

246

H. Triebel

If Eb (f, g) is closable, then the domain of its closure dom Eb will be denoted by Eb (Ω). Proposition 2.1. Let Ω be a bounded domain in Rn , n ≥ 2. Let b ∈ Lloc 1 (Ω), b(x) > 0 a.e. in Ω and b−1 ∈ Ln/2 (Ω). Then Eb (f, g) according to (2.1) is a closable positive deﬁnite quadratic form in L2 (Ω), and 1/2

f |L2 (Ω) ≤ c b−1 |Ln/2 (Ω) 1/2 b(x) |∇f (x)|2 dx , (2.7) Ω

for some c > 0 and all f ∈ Eb (Ω). Proof. Step 1. Let f ∈ D(Ω). Then (2.7) follows from (2.6) and H¨older’s inequality, n+2 n n 2n

f |L2 (Ω) ≤ c b− n+2 (x) b(x) |∇f (x)|2 n+2 dx Ω (2.8) 1/n 2 1/2 −n/2 ≤c b (x) dx b(x) ∇f (x) dx . Ω

Ω

Step 2. We prove that Eb is closable. Let {fk }∞ k=1 ⊂ D(Ω) be a Cauchy sequence in the Eb -norm. Let in addition fk → 0 in L2 (Ω). In particular ∂f ∂ϕ k , ϕ = − fk , → 0 if k → ∞, ϕ ∈ D(Ω), ∂xl ∂xl where l = 1, . . . , n. By (2.8) it follows that {fk }∞ k=1 is also a Cauchy sequence ◦

in W 12n (Ω). Then n+2

∂f k (x) → 0 if k → ∞ in L 2n (Ω), n+2 ∂xl

(2.9)

l = 1, . . . , n. It remains to prove that Eb (fk , fk ) → 0

if k → ∞.

(2.10)

By Eb (fk , fk ) = Eb (fk − fm , fk − fm ) + Eb (fk − fm , fm ) + Eb (fm , fk ) ≤ Eb (fk − fm , fk − fm ) + Eb (fk − fm , fk − fm )1/2 Eb (fm , fm )1/2 + Eb (fm , fk ) (2.11) one can reduce (2.10) to Eb (fk , ϕ) → 0 if k → ∞ for any ϕ ∈ D(Ω).

(2.12)

BK = {x ∈ Ω : b(x) < K}.

(2.13)

Let Then it follows from (2.9) that n

∂fk ∂ ϕ ¯ b(x) · dx → 0 if k → ∞. ∂xl ∂xl BK l=1

(2.14)

Entropy Numbers of Quadratic Forms

247

Let Eb (f, g)K be given by (2.1) where the integration over Ω is replaced by the integration over Ω \ BK . One has by the triangle inequality that 1/2

1/2

Eb (fk , fk )K ≤ Eb (fk − fm , fk − fm )1/2 + Eb (fm , fm )K . Now (2.10) follows from (2.11)–(2.15) by standard arguments.

(2.15)

We need a second preparation. Let again Ω be a bounded domain (= open set) in Rn , n ≥ 2. Let f be a complex-valued a.e. ﬁnite Lebesgue measurable function on Ω. The distribution function μf and the non-increasing rearrangement f ∗ of f have the usual meaning, μf (λ) = {x ∈ Ω : |f (x)| > λ}, λ ≥ 0, and

f ∗ (t) = inf λ : μf (λ) ≤ t , t ≥ 0. Let 0 < p < ∞ and a ∈ R. Then the Zygmund space Lp (log L)a (Ω) consists of all complex-valued a.e. ﬁnite Lebesgue measurable functions on Ω for which |Ω| ap 1/p

f |Lp (log L)a (Ω) = f ∗ (t)p 1 + | log t| dt < ∞. (2.16) 0

Detailed information about rearrangement of functions and Zygmund spaces may be found in the standard references [2, 11]. We refer in particular to [11, Sections 3.2–3.4] and [2, Chapter 2, Section 4.6]. If a = 0, then Lp (log L)0 (Ω) = Lp (Ω) quasi-normed as usual by 1/p

f |Lp (Ω) = |f (x)|p dx . Ω

We need the following extrapolation characterisation of Lp (log L)a (Ω) with a = 0. Let 0 0, j ∈ N, j ≥ j0 (p). (2.17) j p(j) p n·2 p[j] p n · 2j Here N is the collection of all natural numbers. (i) Let a < 0. Then Lp (log L)a (Ω) is the collection of all Lebesgue measurable a.e. ﬁnite functions f on Ω such that ∞ 1/p 2jap f |Lp(j) (Ω) p < ∞, (2.18) j=j0

equivalent quasi-norm. (ii) Let a > 0. Then Lp (log L)a (Ω) is the collection of all Lebesgue measurable a.e. ﬁnite functions f on Ω which can be represented as ∞

f= fj , fj ∈ Lp[j] (Ω), (2.19) j=j0

such that

∞ j=j0

2jap fj |Lp[j] (Ω) p

1/p

< ∞.

(2.20)

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Furthermore, the inﬁmum of all expressions (2.20) taken over all representations (2.19), (2.20) is an equivalent quasi-norm on Lp (log L)a (Ω). This is one of the main results in [8, Corollary 3.1, p. 74]. Assertions of this type go back to [14] and [15, Section 2.6] with p ≥ 1 in case of a > 0. As for further information, generalisations and abstract versions we refer to [16, 8]. Let 0 < ε < p < ∞ and −∞ < a2 < a1 < ∞. Then Lp+ε (Ω) → Lp (log L)a1 (Ω) → Lp (log L)a2 (Ω) → Lp−ε (Ω)

(2.21)

and Lp (log L)ε (Ω) → Lp (Ω) → Lp (log L)−ε (Ω).

(2.22)

Here → means continuous embedding. Hence the Zygmund spaces Lp (log L)a (Ω) are reﬁnements of the Lebesgue spaces Lp (Ω). For our purpose it is reasonable to introduce the auxiliary modiﬁcations Gp,a (Ω) of Lp (log L)a (Ω). If a < 0, then one replaces (2.18) in part (i) by

f |Gp,a (Ω) = sup 2ja f |Lp(j) (Ω) . j≥j0

If a > 0, then Gp,a (Ω) is the collection of all f which can be represented by (2.19) with sup 2ja fj |Lp[j] (Ω) < ∞

(2.23)

j≥j0

quasi-normed by f |Gp,a (Ω) which is the inﬁmum over (2.23) for f represented by (2.19). We need the following simple observation. Proposition 2.2. Let Ω be a bounded domain in Rn , n ≥ 2. Let 0 < p, r < ∞ and 1t = 1p + 1r . Let a = 0 and ε > 0. Then Lp (log L)a (Ω) → Gp,a (Ω) → Lp (log L)a−ε (Ω)

(2.24)

gf |Gt,a (Ω) ≤ g |Lp (log L)a (Ω) · f |Lr (Ω) ,

(2.25)

and where g ∈ Lp (log L)a(Ω) and f ∈ Lr (Ω). Proof. The embedding (2.24) is obvious, whereas (2.25) follows from H¨ older’s inequality for Lebesgue spaces. Finally we recall the deﬁnition of the (fractional) Sobolev spaces Hps with 0 < p < ∞ and s ∈ R on Rn and on domains. We use standard notation. In particular, the Schwartz space S(Rn ), the space S (Rn ) of tempered distributions, the Fourier transform fB and its inverse f ∨ of S (Rn ) have the usual meaning. Let ϕ0 ∈ S(Rn ) with ϕ0 (x) = 1 if |x| ≤ 1 and let

and ϕ0 (y) = 0 if |y| ≥ 3/2,

ϕk (x) = ϕ0 2−k x − ϕ0 2−k+1 x ,

x ∈ Rn ,

k ∈ N.

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249

n Then ϕ = {ϕj }∞ j=0 is a dyadic resolution of unity in R . Let 0 < p < ∞ and s n s ∈ R. Then the (fractional) Sobolev space Hp (R ) is the collection of all f ∈ S (Rn ) such that ∞ 2 1/2

f |Hps (Rn ) ϕ = 22js (ϕj fB)∨ (·) |Lp (Rn ) (2.26) j=0

is ﬁnite (equivalent quasi-norms for diﬀerent choices of ϕ). Of interest for us are the classical Sobolev spaces Wp1 (Rn ) with 1 ≤ p < ∞, normed by

f

|Wp1 (Rn )

n

∂f = f |Lp (R ) + |Lp (Rn ) . ∂xl n

l=1

Wp1 (Rn )

Hp1 (Rn ).

If 1 0, c2 > 0 such that for all p with 1 < p < ∞ and all f ∈ Lp (Rn ), c1 (p + p )−1 f |Hp0 (Rn ) ≤ f |Lp (Rn ) ≤ c2 (p + p ) f |Hp0 (Rn ) ,

(2.27)

1 + p1 p n

where = 1. This is the well-known Littlewood–Paley characterisation of Lp (R ) with 1 < p < ∞. It is essentially covered by [17, p. 339]. There is a similar inequality with Hp1 (Rn ) in place of Hp0 (Rn ) and Wp1 (Rn ) in place of Lp (Rn ). 2.2. Quadratic forms in higher dimensions Higher dimensions means n ≥ 3. But ﬁrst we assume n ≥ 2 and indicate where the cases n ≥ 3 and n = 2 diﬀer. Let Ω be a bounded domain in Rn , n ≥ 2, and let n

∂f ∂¯ g Eb (f, g) = b(x) (x) (x) dx, f, g ∈ D(Ω), (2.28) ∂xj ∂xj Ω j=1 be the above quadratic form where b ∈ Lloc 1 (Ω), b(x) > 0 a.e. in Ω and b−1 ∈ Ln/2 (Ω). By Proposition 2.1 this quadratic form Eb (f, g) is closable in L2 (Ω) and positive deﬁnite and one has (2.7) on the closure Eb (Ω). In particular the embedding id :

Eb (Ω) → L2 (Ω)

(2.29)

is continuous. But we do not know whether it is always compact. The situation improves if one strengthens b−1 ∈ Ln/2 (Ω) by b−1 ∈ Ln/2 (log L)d (Ω) with d > 0, (2.22). We measure compactness in terms of entropy numbers and recall brieﬂy the standard deﬁnition of entropy numbers in an abstract setting. Let A and B be two (complex) quasi-Banach spaces with the respective unit balls UA and UB . Let T ∈ L(A, B) be a linear and bounded operator from A into B. Then for k ∈ N the k-th entropy number ek (T ) of T is deﬁned as the inﬁmum of all ε > 0 such that T (UA ) ⊂

k−1 2

j=1

bj + ε U B

for some b1 , . . . , b2k−1 ∈ B.

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H. Triebel

We assume that the reader is familiar with basic assertions for entropy numbers of compact embeddings between function spaces. In [15] and also in [24, 36] one ﬁnds historical comments. A few more recent and more speciﬁc papers will be mentioned later on. Theorem 2.3. Let Ω be a bounded domain in Rn , n ≥ 3. Let b be a real Lebesgue-measurable function on Ω with b ∈ Lloc 1 (Ω),

b(x) > 0 a.e. in Ω,

b−1 ∈ Ln/2 (log L)d (Ω)

with d > 0. Then the embedding id in (2.29) is compact. If d > 4/n, then ek (id) ≤ c b−1 |Ln/2 (log L)d (Ω) 1/2 k −1/n ,

k ∈ N,

(2.30)

for some c > 0 which is independent of b and k. If 0 < d ≤ 4/n, then for any ε > 0 there is a constant cε > 0 such that for all b and k, ek (id) ≤ cε b−1 |Ln/2 (log L)d (Ω) 1/2 k − 4 +ε , d

k ∈ N.

(2.31)

2n Proof. Step 1. First we assume that n ≥ 2. Let t = n+2 be the Sobolev index according to (2.6). Let Gt,a (Ω) be the spaces introduced in (2.23) with

a>0

and

1 1 1 = − > 0, t[j] t n · 2j

j ∈ N, j ≥ j0 (t),

according to (2.17). With f = b−1/2 b1/2 f ∈ D(Ω) and from (2.25) that

1 t

=

1 2

+

f |Gt,a (Ω) ≤ b−1/2 |Ln (log L)a (Ω) · b1/2 f |L2 (Ω) .

(2.32) 1 n

it follows (2.33)

If n ≥ 3, then t > 1. But if n = 2, then we have t = 1. This requires some extra care. In order to cover both cases, n ≥ 3 and n = 2, we introduce the Littlewood–Paley counterpart of Gt,a (Ω) with a > 0 according to (2.23). We use the norms in (2.26) (without the subscript ϕ). Let t be as above and a, 0 t[j] as in (2.32). Then Ht,a (Ω) is the collection of all f ∈ L1 (Ω) which can be represented as ∞

f= fj , fj ∈ Lt[j] (Ω), (2.34) j=j0

such that 0 sup 2ja fj |Ht[j] (Rn ) < ∞

(2.35)

j≥j0

with fj extended by zero outside of Ω. Furthermore, 0 0

f |Ht,a (Ω) = inf sup 2ja fj |Ht[j] (Rn )

(2.36)

j≥j0

where the inﬁmum is taken over all representations (2.34) (since t[j] > 1 one 0 has Lt[j] (Rn ) = Ht[j] (Rn ) with the Littlewood–Paley assertions in (2.27)). 0 We assume now n ≥ 3. Then t > 1 and the step from Gt,a to Ht,a would not 0 be necessary. But we stick at formulations in terms of Ht,a -spaces in order

Entropy Numbers of Quadratic Forms

251

to prepare our later considerations of n = 2. Let f ∈ D(Ω). Then it follows from (2.33) that n 1/2

∂f 0 −1/2 Ht,a (Ω) ≤ c b |Ln (log L)a (Ω) b(x) |∇f (x)|2 dx . ∂xl Ω l=1 (2.37) In [15, Section 2.6.3] we introduced logarithmic Sobolev spaces Hps (log H)a (B) in smooth domains B, say a ball B with Ω ⊂ B. In Theorem 2.6.3(iv), p. 79, based on (29), (30), p. 80, in [15] we justiﬁed that optimal decompositions of f ∈ Hps (log H)a (B) with 1 0, can be reduced to corresponding optimal decompositions of Dα f ∈ Lp (log L)a (B), 1 0 ≤ |α| ≤ s. This applies also to obviously deﬁned spaces Hp,a (B) based 0 on Gp,a (B). If one replaces Gp,a (B) by Hp,a (B), then one can extend these assertions also to p = 1, which will be needed later on in case of n = 2. We return to n ≥ 3. Let 1/2

b−1/2 |Ln (log L)a (Ω) b(x) |∇f (x)|2 dx ≤1 Ω

in (2.37). Then it follows from the above considerations that f can be decomposed by ∞

f=

fj ,

1 fj ∈ Ht[j] (B),

1

fj |Ht[j] (B) ≤ c 2−ja

(2.38)

j=j0

where c > 0 is independent of j. Step 2. We apply [15, Proposition 1, p. 139] to the embedding idj : where p1j = such that

1 Ht[j] (B) → L2 (Ω) 1 2

with

1 n+2 1 1 1 = − = + j t[j] 2n n2 pj n

− n21 j . One obtains that for any ε > 0 there is a constant cε > 0 2

ek (idj ) ≤ cε 2j(ε+ n ) k −1/n ,

k ∈ N.

(2.39)

Here cε is independent of j and k. Hence the image of a ball of radius c 2−ja 2 −1/n 1 in Ht[j] (B) in L2 (Ω) can be covered by 2kj balls of radius c 2j(ε+ n −a) kj centred at glj ∈ L2 (Ω) where l = 1, . . . , 2kj . Let J > j0 . Then f in (2.38) can be approximated by f−

J

j=j0

gljj =

J

fj − gljj + f J ,

where f J =

j=j0

fj .

j>J

Choosing gljj optimally one obtains that f−

J

j=j0

gljj |L2 (Ω) ≤ cε

J

j=j0

2

−1/n

2j(ε+ n −a) kj

+ c 2−Ja .

(2.40)

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H. Triebel

Step 3. Let a > 2/n and 0 < δ < a. We choose kj > 0 such that −1/n

2

2j(ε+ n −a) kj

= 2−Ja+(J−j)δ ,

j = j0 , . . . , J.

(2.41)

To avoid clumsy formulations we identify kj with its nearest natural number, hence assuming kj ∈ N. Choosing ε > 0, δ > 0 small one obtains that k=

J

J

2

kj = 2Jn( n +ε)

j=j0

2

2(J−j)n(a− n −ε−δ) ∼ 2Jn(a−δ) .

(2.42)

j=j0

By (2.40)–(2.42) we have f−

J

gljj |L2 (Ω) ≤ c 2−J(a−δ) ∼ c k −1/n

(2.43)

j=j0

J for one of the the ∼ 2k elements of j=j0 gljj . Then one obtains that ek id : Eb (Ω) → L2 (Ω) ≤ c b−1/2 |Ln (log L)a (Ω) k −1/n , k ∈ N, (2.44) 5 62 where a > 2/n. Recall that (b−1 )∗ (t) = (b−1/2 )∗ (t), [2, p. 41, (1.20)], hence by (2.16)

b−1/2 |Ln (log L)a (Ω) ∼ b−1 |Ln/2 (log L)2a (Ω) 1/2 .

(2.45)

Then (2.30) follows from (2.44) with d = 2a > 4/n. Step 4. Let 0 < a ≤ 2/n. Instead of (2.41) we choose 1/n

kj

2

= 2j(ε+ n −a) 2Ja 2jκ ,

j = j0 , . . . , J,

(2.46)

where κ > 0. Then the counterpart of (2.42) is given by k=

J

kj = 2nJa

j=j0

J

2j(nε+2−na+nκ) ≤ cσ 22J(1+σ)

(2.47)

j=j0

where σ > 0 can be chosen arbitrarily small. We insert (2.46) in (2.40). Using (2.47) one obtains that

f −

J

gljj |L2 (Ω) ≤ c 2−Ja ≤ c k − 2(1+σ) . a

(2.48)

j=j0

This is the counterpart of (2.43). The rest is now the same as in Step 3. This proves (2.31). 2.3. Quadratic forms in two dimensions Let Ω be a bounded domain (= open set) in the plane R2 and let 2

∂f ∂¯ g Eb (f, g) = b(x) dx, f, g ∈ D(Ω), ∂x ∂x j j Ω j=1

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253

be the two-dimensional version of (2.28) with b ∈ Lloc 1 (Ω), b(x) > 0 a.e. in Ω and b−1 ∈ L1 (Ω). By Proposition 2.1 the quadratic form Eb (f, g) is closable and positive deﬁnite. One has 1/2 −1 1/2

f |L2 (Ω) ≤ c b |L1 (Ω) b(x) |∇f (x)|2 dx Ω

on its closure Eb (Ω). In particular, id :

Eb (Ω) → L2 (Ω)

(2.49)

is continuous. Again we strengthen b−1 ∈ L1 (Ω) by b−1 ∈ L1 (log L)d (Ω) with d > 0, (2.22), and ask for compactness. Let ek (id) be the corresponding entropy numbers. Then one has the following counterpart of Theorem 2.3. Theorem 2.4. Let Ω be a bounded domain in the plane R2 . Let b be a real Lebesgue-measurable function on Ω with b ∈ Lloc 1 (Ω),

b(x) > 0 a.e. in Ω,

b−1 ∈ L1 (log L)d (Ω)

where d > 2. Then the embedding (2.49) is compact. If d > 4, then ek (id) ≤ c b−1 |L1 (log L)d (Ω) 1/2 k −1/2 ,

k ∈ N,

for some c > 0 which is independent of b and k. If 2 < d ≤ 4, then for any ε > 0 there is a constant cε > 0 such that for all b and k, 1

ek (id) ≤ cε b−1 |L1 (log L)d (Ω) 1/2 k − 4 + 2 +ε , d

k ∈ N.

Proof. We follow the proof of Theorem 2.3 and indicate the necessary modiﬁcations. Now one has t = 1 and the counterparts a > 0,

1 1 = 1 − j+1 , t[j] 2

j ∈ N,

and

f |G1,a (Ω) ≤ b−1/2 |L2 (log L)a (Ω) · b1/2 f |L2 (Ω) 0 of (2.32), (2.33). Let again H1,a (Ω) be the Littlewood–Paley version of G1,a (Ω) according to (2.34)–(2.36). In contrast to n ≥ 3 one must now apply in addition (2.27) with p = t[j]. This produces an extra factor 2j and the counterpart of (2.37) is now given by 2 1/2

∂f 0 −1/2 |H1,a (Ω) ≤ c b |L2 (log L)a+1 (Ω) b(x) |∇f (x)|2 dx . ∂xl Ω l=1

Otherwise one can follow the arguments from the proof of Theorem 2.3 with a > 1 in Step 3 and 0 < a ≤ 1 in Step 4. The counterpart of (2.45) is now given by b−1/2 |L2 (log L)a+1 (Ω) ∼ b−1 |L1 (log L)d (Ω) ,

d = 2 + 2a.

Hence d > 4 if a > 1 and 2 < d ≤ 4 if 0 < a ≤ 1. The rest is the same as in the proof of Theorem 2.3 based on (2.44), (2.48).

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H. Triebel

3. Spectral theory 3.1. Preliminaries We complement the abstract preliminaries from Section 2.1. Let again H be a separable complex Hilbert space and let E be a positive deﬁnite closed quadratic form with E(u, u) ≥ c u |H 2

for some c > 0 and all u ∈ dom E.

Let A be the related positive deﬁnite self-adjoint operator with dom A according to (2.3). One has the isomorphic maps (2.4). Recall that a positive deﬁnite self-adjoint operator A is said to be an operator with pure point spectrum if its spectrum consists solely of eigenvalues {λk }∞ k=1 of ﬁnite (geometric) multiplicity, repeated according to their geometric multiplicity, and ordered by 0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ · · · → ∞ if k → ∞. (3.1) The necessary information may be found in, for example, in [32, Section 4.5]. In particular, A is an operator with pure point spectrum if, and only if, id :

dom E → H

is compact. Furthermore, −1/2

λk

≤ c ek (id),

k ∈ N,

(3.2)

which is essentially an easy consequence of Carl’s inequality. This observation underlies the related spectral assertions in [14, 15]. We refer to [24, Chapter 6]. There one ﬁnds also the following assertions about approximation numbers. Recall that the k-th approximation number ak (T ) of a compact linear operator T ∈ L(B1 , B2 ) between the complex quasi-Banach spaces B1 and B2 is given by ak (T ) = inf T − S : S ∈ L(B1 , B2 ), rank S < k , k ∈ N, (3.3) where rank S = dim(image S). With B1 = dom E, B2 = H and the above self-adjoint positive deﬁnite operator A with pure point spectrum and the eigenvalues λk ordered by (3.1) one has −1/2 λk = ak A−1/2 : H → H ∼ ak id : dom E → H , k ∈ N. (3.4) This equality between eigenvalues and approximation numbers is well known since a long time. Proofs may be found in [10, p. 91] and [24, Theorem 6.21, p. 195]. 3.2. Distributions of eigenvalues Let Ω be a bounded domain in Rn with n ≥ 2 and let Eb (f, g) be the closed quadratic form on dom Eb = Eb (Ω) according to Proposition 2.1. Let Ab be the generated positive deﬁnite self-adjoint operator in L2 (Ω). If ϕ ∈ D(Ω) ∩ dom Ab , then Ab ϕ ∈ D (Ω) can be written as n

∂ ∂ϕ Ab ϕ = − b(x) ∈ L2 (Ω). ∂xj ∂xj j=1

Entropy Numbers of Quadratic Forms

255

This follows from the deﬁnition of Ab according to (2.3). If id :

Eb (Ω) → L2 (Ω)

is compact, then Ab is an operator with pure point spectrum and its eigenvalues λk (Ab ) can be ordered as in (3.1) by 0 < λ1 (Ab ) ≤ λ2 (Ab ) ≤ · · · ≤ λk (Ab ) ≤ · · · → ∞

if k → ∞.

Theorem 3.1. (i) Let Ω be a bounded domain in Rn with n ≥ 3. Let b ∈ Lloc 1 (Ω),

b(x) > 0 a.e. in Ω,

b−1 ∈ Ln/2 (log L)d (Ω)

(3.5)

with d > 0. Then Ab is a positive deﬁnite self-adjoint operator in L2 (Ω) with pure point spectrum {λk (Ab )}∞ k=1 . If, in addition, d > 4/n, then λk (Ab ) ≥ c b−1 |Ln/2 (log L)d (Ω) −1 k 2/n ,

k ∈ N,

(3.6)

where c > 0 is independent of b and k. If, in addition, 0 < d ≤ 4/n, then for any ε > 0 there is a constant cε > 0 such that for all b and k, λk (Ab ) ≥ cε b−1 |Ln/2 (log L)d (Ω) −1 k 2 −ε , d

k ∈ N.

(3.7)

b−1 ∈ L1 (log L)d (Ω)

(3.8)

(ii) Let Ω be a bounded domain in the plane R2 . Let b ∈ Lloc 1 (Ω),

b(x) > 0 a.e. in Ω,

with d > 2. Then Ab is a positive deﬁnite self-adjoint operator in L2 (Ω) with pure point spectrum {λk (Ab )}∞ k=1 . If, in addition, d > 4, then λk (Ab ) ≥ c b−1 |L1 (log L)d (Ω) −1 k,

k ∈ N,

(3.9)

where c > 0 is independent of b and k. If, in addition, 2 < d ≤ 4, then for any ε > 0 there is a constant cε > 0 such that for all b and k, λk (Ab ) ≥ cε b−1 |L1 (log L)d (Ω) −1 k 2 −1−ε , d

k ∈ N.

(3.10)

(iii) Let Ω be a bounded domain in Rn with n ≥ 2. Let b be as in part (i) if n ≥ 3 and as in part (ii) if n = 2. Then there is a constant c > 0 such that 2 λk (Ab ) ≤ c inf |B|−1− n b(x) dx · k 2/n , k∈N (3.11) B

for all admitted b and k, where the inﬁmum is taken over all balls B with B ⊂ Ω. Proof. Step 1. Part (i) follows from Theorem 2.3 and (3.2). Similarly one obtains part (ii) from Theorem 2.4. Step 2. We prove part (iii) and begin with a preparation. Let Br = x ∈ Rn : |x| < r , r > 0, ◦

and let C 1 (Br ) be the completion of D(Br ) in the norm ◦

f |C 1 (Br ) =

n

∂f (x) . sup ∂x x∈B j r j=1

256

H. Triebel

Let ak (idr ), k ∈ N, be the approximation numbers of the compact embedding idr :

◦

C 1 (Br ) → L2 (Br ),

r > 0.

(3.12)

We wish to show that ak (idr ) = r1+ 2 ak (id1 ) ∼ r1+ 2 k −1/n , n

n

k ∈ N.

(3.13)

The equivalence in (3.13) is covered by [15, Theorem, Section 3.3.4, p. 119]. The ﬁrst equality in (3.13) follows from the isometries ◦

◦

f (r·) |C 1 (B1 ) = r f |C 1 (Br ) ,

g(r−1 ·) |L2 (Br ) = rn/2 g |L2 (B1 ) , and the well-known properties of approximation numbers. If B is a ball in Rn with B ⊂ Ω, then n 1/2

◦ ∂f (x) 2 1/2 b(x) dx ≤c b(x) dx

f |C 1 (B) , f ∈ D(B). ∂x j Ω B j=1

One has for the corresponding approximation numbers 1/2 ◦ ak id : C 1 (B) → L2 (Ω) ≤ c b(x) dx ak id : Eb (Ω) → L2 (Ω) , Ω

and by (3.13) that −1/2 1 1 ak id : Eb (Ω) → L2 (Ω) ≥ c |B| 2 + n b(x) dx k −1/n . Ω

Using (3.4) one obtains (3.11). Remark 3.2. If d > 4/n in case of n ≥ 3 and d > 4 in case of n = 2, then λk (Ab ) ∼ k 2/n ,

k ∈ N,

as in the classical case. This applies to any admitted b how rough it might be. One may think about ∞

cj b(x) = , x ∈ Ω, |x − xj | j=1 where {xj } ⊂ Ω collects all points with rational components and cj > 0 appropriately chosen. The breaking points d = 4/n if n ≥ 3 in part (i) and d = 4 if n = 2 in part (ii) can surely be improved. There are good reasons to believe that the alternative decomposition techniques used in the remarkable paper [13] by D.E. Edmunds and Yu. Netrusov can also be employed in the above context supporting the following expectation. Conjecture 3.3. (i) Let n ≥ 3. Then (3.6) remains valid for all d > 2/n. If 0 < d < 2/n, then (3.7) can be improved by λk (Ab ) ≥ c b−1 |Ln/2 (log L)d (Ω)

−1

for some c > 0 which is independent of b and k.

kd ,

k ∈ N,

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(ii) Let n = 2. Then (3.9) remains valid for all d > 3. If 2 < d < 3, then (3.10) can be improved by λk (Ab ) ≥ c b−1 |L1 (log L)d (Ω)

−1

k d−2 ,

k ∈ N,

for some c > 0 which is independent of b and k.

4. Complements 4.1. An example One may ask to which extent the conditions for b in Theorems 2.3, 2.4, 3.1 and in Conjecture 3.3 are natural. But it seems to be a sophisticated interplay between local and global smoothness and singularity properties of b and b−1 . We illuminate the situation by a simple example. Let Ω = B = {x ∈ R2 : |x| < 1} be the unit circle in the plane and let λ b(x) = |x|2 1 + log |x| , x ∈ B, λ ≥ 0. (4.1) Let Eb (f, g) =

b(x) B

2

∂f ∂¯ g (x) (x) dx, ∂xj ∂xj j=1

f, g ∈ D(B),

(4.2)

be as in (2.1). Proposition 4.1. Eb (f, g) according to (4.1), (4.2) is a closable positive deﬁnite quadratic form in L2 (B) and 1/2 λ

f |L2 (B) ≤ |x|2 1 + log |x| |∇f (x)|2 dx (4.3) B

on its closure Eb (B). The embedding id :

Eb (B) → L2 (B)

(4.4)

is compact if, and only if, λ > 0. Proof. Step 1. Let f ∈ D(B) be real. Then (4.3) with λ = 0 (and hence also with λ ≥ 0) follows from 0≤

2 xj B j=1

= B

=

B

|x|

f (x) + |x|

f 2 (x) +

2

j=1

xj

∂f 2 dx ∂xj

∂f 2 (x) + |x|2 |∇f (x)|2 dx ∂xj

2 |x| |∇f (x)|2 − f 2 (x) dx.

(4.5)

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We used integration by parts. We prove that Eb (f, g) is closable. Let ∞ {fk }∞ k=1 ⊂ D(B), with fk → 0 in L2 (B) and let {fk }k=1 be a Cauchy sequence in the Eb -norm. Let ϕ ∈ D(B). Then √ ∂f ∂f √ k k b ,ϕ = b ϕ¯ dx ∂xj ∂x 1 B ∂ √b √ ∂ϕ ¯ =− fk ϕ¯ + b dx → 0 if k → ∞. ∂xj ∂xj B √ k Then b ∂f → 0 in L2 (B). This proves that Eb (f, g) is closable and by (4.3) ∂xj positive deﬁnite on its closure Eb (B). Step 2. Let ϕj (x) = 2j/2 ϕ(2j x) with ϕ ∈ D(B). One can choose ϕ such that the functions ϕj have disjoint supports, 2

ϕj |L2 (R) = 1 and |x|2 ∇ϕj (x) dx ∼ 1. B

This shows that the embedding id in (4.4) with λ = 0 is not compact. Let λ > 0 and let ψ ∈ D(B)

with

ψ(x) = 1 if

|x| ≤ 1/2.

Let id = idj + idj with idj = ψ(2j ·) id, j ∈ N. Then idj : f → ( 1 − ψ(2j ·) f is compact and idj ≤ c j −λ/2 → 0 if j → ∞ as will be justiﬁed below. Hence id is compact. Remark 4.2. First we prove that idj ≤ c j −λ/2 . Let g ∈ C ∞ (B) with supp g ⊂ {y : |y| ≤ δ}

where δ < 1.

Then it follows from (4.3) that 2

2 ∂ log |x|λ |g(x)|2 dx ≤ |x|2 | log |x| |λ/2 g(x) dx ∂xj B B j=1 λ 2 log |x| λ−2 |g(x)|2 dx. ≤c |x|2 log |x| ∇g(x) dx + c B

B

If δ > 0 is small, then one obtains λ 2 log |x|λ |g(x)|2 dx ≤ c |x|2 log |x| ∇g(x) dx. B

B

Inserting g = id f one obtains id ≤ c j −λ/2 . We add a few further discussions. With b as in (4.1) one has b−1 ∈ L1 (B) if, and only if, λ > 1. Then one can rely on Proposition 2.1 with n = 2. One can calculate for which λ > 1 Theorems 2.4 and 3.1(ii) can be applied. But it seems to be more eﬀective to deal directly with problems of this type for all λ > 0. This will be done in [38] in a more general context. We add a comment about the inequality (4.5) and (4.3) with λ = 0. The proof is very easy but the outcome is nevertheless remarkable and (of course) well known. see, e.g., [29, p. 97]. Let κ(t) with j

j

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0 < t ≤ 1 be a positive monotonically decreasing function. Then there is a constant c > 0 with κ(|x|) |f (x)|2 dx ≤ c |∇f (x)|2 dx for all f ∈ D(B) 2 |x|2 B 1 + | log |x| | B if, and only if, κ is bounded. This is a special case of [35, Theorem 16.2(i), p. 237]. It shows that the weight |x|2 in (4.3) with λ = 0 can not be shifted from the right-hand side to the left-hand side as it is quite often the case in non-limiting situations (as a rule of thumb). 4.2. Extrapolations, decompositions, weights We add a few (partly historical) comments. Remark 4.3. Let again Ω be a bounded domain Rn , n ≥ 2. If a < 0, then the deﬁnition of Lp (log L)a (Ω) can be extended to p = ∞, where (2.16) must be modiﬁed by

f |L∞ (log L)a (Ω) =

sup (1 + | log t|)a f ∗ (t) < ∞.

0
These spaces coincide with the Orlicz spaces Lexp,−a (Ω). They can be described in terms of the extrapolation (2.18) with p = ∞ (obviously modiﬁed) and p(j) = n 2j . This characterisation had been used in [33] to study entropy numbers (and also approximation numbers) of the compact limiting embeddings n/p id : Bpp (Ω) → L∞ (log L)a (Ω),

id : Hpn/p (Ω) → L∞ (log L)a (Ω), (4.6) 1 < p < ∞. We refer also to [15, Section 3.4]. This approach estimating entropy numbers of limiting embeddings of the above type based on extrapolation assertions for Lp (log L)a (Ω) has been used afterwards by several authors improving and generalising substantially the original results. We refer in particular to [1, 25, 26, 27] and most recently [12] to mention only a few papers which are directly related to embeddings of type (4.6) based on extrapolation arguments. This technique has also been used in [14] and related parts of [15] to study entropy numbers of limiting embeddings into target spaces of type Lq (log L)a (Ω) with 1 < q < ∞, called interior estimates. As said the present paper is a continuation of these considerations. The corresponding results in [14, 15] have been improved essentially in [13] based on other methods which will be commented in Remark 4.5 below. Extrapolation assertions for Zygmund spaces Lp (log L)a (Ω) as needed in [14, 15] and also in the present paper are covered by these sources. More general interpolation and extrapolation assertions for Lorentz-Zygmund spaces Lpq (log L)a (Ω) and related logarithmic Sobolev spaces can be found in [8]. This may be of some use for later research. Remark 4.4. We expressed the admitted singularity behaviour of b−1 in Proposition 2.1 and Theorems 2.3, 2.4, 3.1 in terms of Lebesgue spaces Lr (Ω) and Zygmund spaces Lr (log L)a (Ω) using a modest version of a related H¨ older

260

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inequality according to Proposition 2.2. Proposition 4.1 shows that there are interesting weights b which do not ﬁt in this scheme. Looking for other reasonable classes of weights one may think about the Muckenhoupt classes. It is also known that there are substantial connections to Zygmund spaces Lr (log L)a (Ω) and Lorentz-Zygmund spaces Lpq (log L)a (Ω), including related H¨ older inequalities. We refer to [18, Lemma 3.33, p. 61], [19] and the series [21, 22, 23], especially part III, and [20]. In other words, it seems to be reasonable to express the needed properties of b−1 in terms of Muckenhoupt classes, to check what is covered by the above approach and to have a closer look at cases which are not included so far. Remark 4.5. As mentioned in Remark 4.3 there is a very eﬀective alternative approach to problems of the above type based on decomposition techniques and piecewise polynomial approximations in function spaces. This goes back to the seminal paper [3] in 1967 by M.S. Birman and M.Z. Solomyak. These techniques have been used to study afterwards the distribution of eigenvalues for weighted non-smooth elliptic operators, [4, 5, 6] and to the so-called negative spectrum resulting in assertions of Rozenblum–Lieb–Cwikel type, [7]. The state of the art but also outlines of the underlying decomposition techniques may be found in the two recent surveys [30, 31]. This approach has been used in [13] to improve substantially some so-called interior estimates of entropy numbers in limiting situations, typically with target spaces of type Lp (log L)a (Q), 1 < p < ∞, where Q is a cube in Rn . We refer to [13, Theorems on pp. 84,90]. This type of arguments may also work in the above context. It may well be possible that one can combine extrapolation and decomposition techniques. We describe an example although we did not check in detail all technical adaptations. Using [13, Theorem 1, p. 84] there is a good chance to improve (2.39) by 1

ek (idj ) ≤ cε 2j(ε+ n ) k −1/n ,

k ∈ N,

hence to replace 2/n by 1/n (maybe even with ε = 0). But this has the consequence that the breaking point d = 4/n with n ≥ 3 in Theorem 2.3 can be improved by the more natural d = 2/n. Then one has (2.30) for d > 2/n d d and one can improve k − 4 +ε in (2.31) by k − 2 +ε if 0 < d ≤ 2/n. If n = 2, then the situation might be a little bit diﬀerent. Let again n ≥ 3. Then one obtains corresponding improvements in Theorem 3.1(i). The breaking point is shifted from 4/n to 2/n and one has (3.6) for all d with d > 2/n. If 0 < d ≤ 2/n, d then one can replace k 2 −ε in (3.7) by k d−ε . This supports Conjecture 3.3. But as said details remain to be checked.

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