PSEUDO-DIFFERENTIAL OPERATORS ON MANIFOLDS WITH SINGULARITIES
STUDIES IN MATHEMATICS AND ITS APPLICATIONS
VOLUME 24 Editors: J.L. LIONS, Paris G. PAPANICOLAOU, New York H. FUJITA, Tokyo H.B. KELLER, Pasadena
NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO
PSEUDO-DIFFERENTIAL OPERATORS ON MANIFOLDS WITH SINGULARITIES
B.-W. SCHULZE Karl- Weierstrass-Institut fur. Mathematik Berlin, Germany
1991
NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO
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Contents
Introduction
. . . . . . . . . . . . . . . . . . . . .
1
1.
The Conormal Asymptotics on R+
. . . . . . . . . . .
17
1.1. 1.1.1. 1.1.2. 1.1.3. 1.1.4. 1.1.5.
The Mellin Transform . . . . . . . . . . . . . . . Basic Properties . . . . . . . . . . . . . . . . . The Spaces LWY(R+),Wy(R+) . . . . . . . . . . . . Examples and Special Formulas . . . . . . . . . . . . Holomorphic Functions in a Strip . . . . . . . . . . . Analytic Functionals . . . . . . . . . . . . . . .
17 17 23 30 31 36
1.2. 1.2.1. 1.2.2. 1.2.3. 1.2.4. 1.2.5.
Spaces with Conormal Asymptotics . . The Spaces with Discrete Asymptotics . The Spaces with Continuous Asymptotics Mellin Symbols and Actions . . . . Mellin Pseudo-Differential Operators . Green and Flat Operators on R+ . . .
42 42 51 60 72 88
1.3. 1.3.1. 1.3.2. 1.3.3. 1.3.4.
The Mellin Expansion of Operators . . . . . . . . . . . 97 Operators of Lower Conormal Order . . . . . . . . . . 97 The Mellin Expansions . . . . . . . . . . . . . . . 115 . . . . . . . . 119 Operators with Mellin Expansions at Zero Examples and Remarks (The Mellin Calculus of Standard lyDO-s on R+) . . . . . . . . . . . . . . 125
2.
Operators on Manifolds with Conical Singularities . . . . . . . . .
2.0.
Preliminary Remarks
2.1. 2.1.1.
SpaceswithConormalAsymptoticsfortheCone . . . . . . 137 The Spaces with Continuous and Discrete Asymptotics . . . . . . . . . . . . . . . . . . 137 Operator-Valued Mellin Symbols . . . . . . . . . . . 153 Mellin-Fourier Pseudo-Differentialoperators1 . . . . . . . 161 Mellin-Fourier Pseudo-Differential Operators I1 . . . . . . 171 Green and Flat Cone Operators . . . . . . . . . . . . 184 Appendix I (Mellin Sobolev Spaces and Reductions of Orders) . . . . . . . . . . . . . . . . . . . 189 Appendix 11 (Polar Coordinates in 1pD0-s) . . . . . . . . 195
2.1.2. 2.1.3. 2.1.4. 2.1.5. 2.1.6. 2.1.7.
-
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. . . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
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. . . . . . . . . 135
. . . . . . . . . . . . . . . 135
VI
Contents
2.2. 2.2.1. 2.2.2. 2.2.3. 2.2.4. 2.2.5.
The Mellin Expansions for the Cone . . . . . . . . . . 202 Operators with Finite Mellin Expansions . . . . . . . . . 202 Operators with Asymptotic Mellin Expansions . . . . . . . 214 Operators on Manifolds with Conical Singularities . . . . . . 216 . . . . . . . . . 228 Operators on R, X X in the XSYSpaces Appendix (Meromorphic OperatorFunctions) . . . . . . . 236
2.3. 2.3.1. 2.3.2. 2.3.3.
The Parameter-Dependent Cone Calculus . . . . . . . . The Cone Algebra with Parameters . . . . . . . . . . . C" Dependence of the Point-Wise Discrete Class . . . . . . Branching of Asymptotics Near an Edge . . . . . . . . .
3.
Operators on Manifolds with Edges
3.1. 3.1.1. 3.1.2.
Preliminary Constructions . . . . . . . . . . . . . . 268 Motivation of the Approach . . . . . . . . . . . . . 268 Abstract Sobolev Spaces with Respect to an Edge . . . . . . 272
3.2.
Pseudo-Differential Operators with Operator-Valued . . . . . . . . . . . . . . Symbols Amplitude Functions and Continuity of vDO-s . . The Standard Elements of the Calculus . . . . The Scale Axiom . . . . . . . . . . . . Amplitude Functions with Reductions of Orders . Edge Spaces with Asymptotics . . . . . . .
3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.3. 3.3.1. 3.3.2. 3.3.3. 3.3.4. 3.3.5. 3.3.6.
242 242 250 258
. . . . . . . . . . 266
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
280 281 287 299 303 307
Pseudo-Differential Operators on Manifolds with Edges . . . . . . . . . . . . . . . . . . . 325 Pseudo-Differential Operators ofWedgeTypeI . . . . . . . 325 Pseudo-Differential Operators of Wedge Type I1 . . . . . . 337 Subalgebras with Asymptotics . . . . . . . . . . . . 347 Edge Boundary Value Problems . . . . . . . . . . . . 358 Ellipticity, Fredholm Property, Asymptotics of Solutions . . . . . . . . . . . . . . . . . . 312 Examples and Remarks (1pD0-s without . . . . . . . . . . . . . 384 the Transmission Property)
References . . . . . . . . . . . . . . . . . . . . . .
395
Index . . . . . . . . . . . . . . . . . . . . . . . .
402
. . . . . . . . . . . . . . . . . . . . .
407
Symbol Index
Introduction
The analysis of partial differential equations employs as a basic information the type. The equations are often the models of physical or geometric situations that are classified to some extent by the types. It is useful to look at the symbols of associated operators (after establishing a relation between equations and operators in distribution spaces). The past few decades have seen a fruitful development of an analysis with symbols (the micro-local analysis). Essential information and constructions were expressed on symbolic level. The properties of solvability became transparent for large classes of equations. In particular the concept of ellipticity on compact C” manifolds (with and without boundary) is well-understood and can easily be illustrated today in elementary courses for students. Many applications of physics and applied sciences as well as of the structure mathematics lead to “elliptic” problems with singularities in the sense of nonsmoothness of boundaries, singularities of the metric of the underlying space, noncompactness of the domain, singularities or discontinuities of the coefficients of the operators or the boundary conditions, and combinations of these cases. In such situations the usual ellipticity degenerates in a certain way. A deeper analysis then often leads to serious difficulties, incidentally quite unexpected at the first glance. The applications give rise also to parabolic and hyperbolic equations, boundary and initial value problems, and non-linear equations, with a singular or degenerate behaviour. A classification of problems (even around elliptic operators) is difficult. On the other hand concrete investigations of rather different questions have shown common properties, for instance, the conormal asymptotics of solutions. So, one may hope that there exist common sources and supporting principles that allow a general approach with sufficiently aesthetic mathematical structures. After the experience with the “usual” pseudo-differential operators one may expect that it is adequate to introduce (i) algebras of operators (if possible *-algebras) filtered by the orders, (ii) adequate scales of distribution spaces, (iii) symbolic structures with principal and complete symbols and a composition rule on symbolic level, (iv) ideals of “smoothing operators”, where an operator is determined by the complete symbol, modulo a smoothing one, (v) concepts of ellipticiw, parametrix constructions in the algebras and the elliptic regularity in the considered distribution classes and the index theory. The classical calculus of pseudo-differential operators (I~DO-s)and the theory of el[S27], [Pl],[A8]) including the liptic operators contains all these elements (cf. [H4], Atiyah-Singer index theorem. Elliptic boundary value problems also admit a theory in this sense in the context of Boutet de Monvel’s algebra (cf. [B9],[R8],[GlO]). Subject of this monograph is a theory for general classes of singular problems in terms of operator algebras with symbolic structures that contain the classical the-
2
Introduction
ories as special cases. For illustrating the range of our considerations we shall first sketch some problems in more detail. First there are operators on manifolds with singularities, in particular (i)l
Cq-1
conical singularities
\
2
w
e
R+xX
Loco lly
Fig. 1
(one considers a stretched object C and there is a canonical map C +M),
(ii)l edges
(iii)l corners
\
locally
Fig.3
and further iterations of forming edges and cones ("higher singularities"), (iv)l boundary value problems for domains and manifolds, respectively, where the boundaries have such singularities, (v)] topological spaces such as skeletons of the various dimensions of polyhedra, for instance, the system of one-dimensional edges of a cube in R'. Such spaces are C" manifolds outside an exceptional set. Close to this set the spaces have branchings in a well-defined way.
3
Introduction
A second class can be described by the following key words: mixed, transmission-, crack- (screen-) problems, boundary value problems for pseudo-differential operators without (or with) the transmission property. More precisely we have (i)2 transmission and mixed problems (for example the Zaremba problem or problems with jumping oblique derivatives),
Fig. 4
a domain (ii)2 crack problems: Q is a domain of the form d\S, S being a C" manifold with boundary Y of codimension 1, d a domain with smooth boundary, Au=f
inR,
TuI, = S , being
two
-
sided boundary
conditions on S
Fig. 5
(iii)z boundary value problems for pseudo-differential operators (with or without the transmission property with respect to the boundary Y);
locally near Y the manifold can be identified with the half space R:=[x: x,, > 0) and Y = {x, = 0), and the operator (for example in the zero order case) has the form r+ Op(a)e+ : Lz(R:)
+ Lz(R:),
where a is a WDO-symbol, Op(.) the associated operator, based on the Fourier transform in R",e + : L2(R:) +Lz(R") the extension by zero, r+ the operator of restriction to R:, (iv)z combinations of such problems, for example when Y = aS in (i& has singularities in the sense of the fust class or when in (i)l the base of the cone has a boundary and the boundary problems are understood in the sense of (iii)*. Problems of this sort have been studied in the literature by many authors, often under very special assumptions. The vast variety of particular results shows that it is
4
Introduction
necessary to look for the general properties. A theory should also deal with information that is useful for numerical computations in concrete models (such as the aymtotics of solutions near the singular sets). Let us also mention the following aspect. In the “structure mathematics” (representation theory of Lie groups, differential geometry, topology, . ..) the problem of an analysis on “manifolds with singularities“ becomes more and more interesting. In recent years there appeared many investigations on the index theory of elliptic operators on such spaces for obtaining adequate analogues of the classical Atiyah-Singer index theorem. This is a parallel of a similar (already classical) development for the standard elliptic operators. Remember that the calculus of WDO-shas provided the analytical tools for solving long-standing practical and theoretical problems for elliptic differential equations in a very general framework. After these achievements it was reasonable to look for an index theorem in terms of a stable homotopy classification. This was solved by constructions of K-theory and differential geometry and led to the famous development around the index theory and the global analysis, cf. [Pl],where in particular the analysis was used and interpreted in geometric terms and conversely. For the problems with singularities one may expect a similar synthesis of questions and ideas from different areas of mathematics. The classical calculus has shown that many problems remain obscure without the analytic theory in sufficient generality, in particular on the symbolic structure in algebraic operations and parametrix constructions or the natural distribution spaces. The latter aspect is a program on its own and subject of numerous papers in the literature. It has its own beauty with new deep phenomena and rich structures compared with the standard situation. Even the “adequate” notion of ellipticity itself leads to interesting considerations. Now let us discuss some elements of the approach in our book in more detail. A case of crucial importance is that of conical singularities. Then we can locally generate wedges by multiplying a cone by Rq, q E IN, and also comers by considering cones with bases that have conical singularities. A “manifold” M with conical singularities is intuitively defined as a compact topological space with exceptional points v l , ..., v N ,where M\ {vl, ... ,vN} is a C” manifold, and M may be identified locally near any vertex vj with R, xX,/[O} x X,, for some closed compact C” manifold X, (the base of the cone near vj), where we also keep in mind the local R, actions 1 ( t , x ) = (At, x ) , (t, x) E R, x X j , 1E R,. Since we do not necessarily assume that Xj has only one connected component, we might pass to another space M by identifying all vj with a single point v, where X = XI u . . . u X, is the new base of the cone close to v. It is custom anyway to “blow up” the singularity, i.e. to consider the stretched manifold C = M o which is a compact C” manifold with boundary X where we have a tubular neighbourhood V z [0,1) x X with the corresponding local R, action. M is obtained from M oby contracting X, to vj. In an analogous manner we can talk of manifolds with conical singularities with boundary. In the introduction we mainly restrict ourselves to the case when X is closed. Boundary value problems on manifolds with conical singularities correspond to ax 0, and the basic ideas are analogous. The investigation of differential operators on M leads to operators over R, XX of
*
5
Introduction
where a j ( t ) E Cm(a+,Diffr-J(X)), Here Diff”(X) denotes the class of all differential operators on X of order v with C” coefficients in local coordinates. Diff’(X) has a natural FrCchet structure. The operators (1) can invariantly be defined in terms of polynomials of vector fields on R, XX which are tangent to the boundary. Operators containing the derivatives with respect to a variable ? in the combination tala? are called totally characteristic (or also of Fuchs type). We get such Cperators, for instance, by inserting polar coordinates into a differential operator A in R” 3 2. It 21 (up to a weight factor t P , p = ord A ) , and the is then totally characteristic in t = 1 role of the above X plays the sphere S” = {I21= I}. This transform already suggests the following specific point of view: For studying effects in terms of the distance t to a singularity (here the vertex of a cone) we should have a description by means of totally characteristic actions. The precise theory shows that in many cases yDO-s can actually be written in this form. So the totally characteristic aspect is one of the first essential observations for the general calculus. Singularities can often be thought as geometric ones, formulated by Riemannian metrics. For instance, the Laplace-Beltrami operator on R+XXwhich belongs to the metric d t 2 + t 2 g , g being a Riemannian metric on X, has the form +
(
r 2(I:)’
+ (dimX-
Ax being the Laplace Beltrami operator on X for
g. (2) is just of the form we are talking about (the weight factors are not essential for the moment). The operators (1) lead in a natural way to the “totally characteristic” Sobolev spaces X s ( C ) ,C = M o being the stretched manifold to M. For s E N we set
X‘“(C)= { u E L’(C): L1’ . . . . Lsu E L q c ) for arbitrary vector fields Lj that are tangent to aC}. For s E IR we then obtain a definition by duality and interpolation. In particular we have a canonical embedding P ( C ) 4 Hs,,,(int C), where Hs is the standard Sobolev space. It is reasonable also to consider spaces with weights %S,Y(C) := gY%S(C),
3,
yE
w.
Here gY is a strictly positive C” function on int C which equals t Y close to aC. The vector fields tangent to aC suggest the definition of the “compressed cotangent bundle” TgC which is the usual one over int C where close to aC the C” sections are generated by n
t-’P0(t, X ) dt +
C Pj(C X ) dxj,
j = 1
p j E C” up to t = 0. These sections can invariantly be paired with vector fields tan[M9]. The differential operators on C of order p which gent to aC, cf. also MELROSE are close to aC of the form (1) induce continuous operators
A:
%SY(C)
+-%s-r,Y(C)
(3)
for all s, y E R. A basic question is now to characterize those A for which (3) is a Fredholm operator. This is just the elliptici& defined in terms of the symbolic stmc-
6
Introduction
ture. First we have the homogeneous principal symbol of A which is close to aC of the form
where $--’(aj) denotes the homogeneous principal symbol of a’. For Z= tzwe get a smooth function in t, x, Z, 5 up to t = 0. We set It is then easily seen that these local expressions admit an invariant interpretation as a C” function @ ; ( A )on TZC (homogeneous of order p ) . It is called the homogeneous principal symbol of A (in the compressed sense). Moreover we have the operator-valued “Mellin symbol” of the “conormal order” zero, namely
which is holomorphic in @ and Diff p(X)-valued. Thus &(A) ( z ) : H”X) +H”-”(X)
(4)
is continuous for every z E @. Now A is called elliptic with respect to the weight y if (i) a;(A)+O on T$C\O, (ii) (4) is an isomorphism for all z with Re z = 1/2 - y. A theorem is then that the ellipticity of A is equivalent to the Fredholm property of (3). It is a supporting principle in a well-organized operator calculus with symbolic structures that the Fredholm property in the adequate Sobolev spaces is equivalent to the bijectivity of the leading symbols. Here we have a pair of leading symbols (cr$(A), aL(A)),one of them being operator-valued. This is now another important observation for the general calculus, namely that we have multiplets of leading symbols (with internal compatibility conditions) which determine the operator modulo someone of “lower order” (in the case of the cone of interior and conormal order). In more complex situations such as boundary value problems or higher singularities we get symbol hierarchies with components that are also operator-valued and have again a symbolic structure. Remember that the ellipticity of a classical boundary value problem (say of Boutet de Monvel’s type) is the bijectivity of the interior principal symbol and of a so-called boundary symbol, which is just the Shapiro-Lopatinskij condition, cf. BOUTETDE MONVEL [B9], REMPEL/SCHULZE [R8]. The correct notion of a leading symbol is also necessary for the index theory, for example the (stable) homotopy classification of the ellipticity. Here we have necessarily homotopies through operators where all symbol components are bijective. ’This determines a group of equivalence classes which is, for instance, for the stretched conce C, different from the usual one in the Atiyah-Singer index theorem, [S9]. Note that the equivalence beformulated in K-theoretic terms, cf. SCHULZE
7
Introduction
tween ellipticity and Fredholm property depends on the choice of spaces. Although there are many reasons to find the Xe.y spaces natural we also might take other spaces, based, for instance, on LP norms, p ?= 2. On the other hand, we have the dependence on the weight y anyway. Also the index of the operator then may depend on the space. Under the ellipticity for different y the index in general depends on y. Note that in the theory of boundary value problems without the transmission property the ellipticity also depends of the Sobolev space smoothness s, cf. VISIK/ESKIN Cvl]-[v4]. It turns out in this case that s plays the role of a hidden weight, cf. 1.2.2. Theorem 16 below. An idea of the operator calculus with symbolic structure is to represent the operators explicitly in terms of an operator convention symbol of A
+A,
where A is first applied, for instance, to C” functions with compact support, outside the singularity. Then there follows a discussion to which distribution spaces A admits a natural extension. The answer is that A extends to all “reasonable spaces”, up to certain natural exceptions. The exceptions may concern, for instance, y, which is required to avoid the poles of the Mellin symbols, cf. the discussion below. The operator convention usually refers to the complete symbols. For the complete symbols which are given in local terms it is interesting to have the analogue of the Leibniz rule under compositions. Over int C we have the same as in the standard yDO calculus. So it suffices to look at (1) close to t = 0. Since we have two leading symbol components, we also should have two notions of complete symbols with the corresponding composition rules. Let us first reformulate (1) in terms of the Mellin tranSfonn
s
m
Mu(z) =
tz-’u(t) dt,
z E C.
0
For u E C,”(R+)we obtain M u ( z ) E d ( C ) (= the space of entire functions). It is well-known that M extends by continuity to an isomorphism
M : L2(R+)+L2({Re z = 1/2}).
s
Then (M-lg) ( 1 ) = 1/2ni t-‘g(z) dz, where the integral is taken over r,,,= = {Rez = 1/2}. Now we obviously have M -IzM = - ta/at, defined on the subspace of all u E L2(R+)with talatu E L2(R+).This suggests the notion of a Mellin yDO op,(a)u = M-’a(t, z)M, where a(t, z) belongs to some space of symbols. We are interested in this idea in many variants. (1) shows that one should insert, for instance, operator-valued symbols, in this case P
a (2, z) =
1aj(t)z’ E SS (C, Diff “X)). j=O
Then (1) equals (5). Set
8
Introduction
-c
Then a(f, z) f ko;k(A) ( z ) as t +0. The Taylor expansion can be replaced by a convergent sum by choosing a cut-off function w ( t ) (i.e. w E C,”(R+),w = 1 close to t = 0) and a sequence of constants cj -+ OJ increasing sufficiently fast, i.e.
c m
a ~ ( tz) , =
t k W ( C k t > u i k ( A )(z)w(ckt),
k=O
where aF(t,z) = a(t, z) - aM(t,z) is flat at t = 0 of infinite order (the cut-off function on both sides will be convenient in the calculus, for the moment it would suffice one of the cut-off factors). Thus
where F = opM(ap)is a flat operator. o L k ( A )( z ) is called the Mellin symbol of A of conormal order - k. The expression (7) shows us a further crucial idea for dealing with operators near singularities. (7) contains a “Mellin expansion” with Mellin symbols which are uniquely determined by A and which determine A modulo a flat and a Green operator. The Green operators will be explained below. Here they disappear. uMM(A)= { g i k ( A ) } k E N is the complete Mellin symbol of A. If B is another differential operator of order v, of analogous form as A, then C = AB belongs to our class, again, and
for p = 0, 1, 2, ... (8) is called the Mellin translation product. It is the Leibniz rule on the level of the operator-valued Mellin symbols. For the complete compressed symbol G b ( . ) (t, x, Z,5) in local coordinates we get g b ( c ) ( t , x, 5
0=
1 c zo>o:%(A)(t, r, x,
The sum is taken over all
01
a,,
DE,= -ia/atj,
= (ao,cc’), =m
xk,
t?(tat)*a:‘gb(B)(t, x, 5
5).
and
to= Z,
x0 = 1
If we construct a parametrix of an elliptic differential operator A we reach a class of rgDO-s over int C with a corresponding specific behaviour close to aC. The precise construction is just the content of the systematic calculus. It will be developed in Chapter 2. The idea is analogous as for standard yDO-s. We first invert the leading symbols and then determine step by step the complete symbol by using the Leibniz rule. A non-standard element is an adequate notion of the negligible operators which are the remainders if we prescribe the complete symbols. They are called Green operators. Over int C that are smoothing operators but close to aC they map to functions with “conormal asymptotics” u(t, x )
- 12 (jk(X)t-Pilogkt m
m
j=O
k=O
as
t
+o,
9
Introduction
where pj E @, Repj + - w as j + a, mj E N, and t j k E Lj, 0 5 k 5 mj, Lj c Cm(X)being finite-dimensional subspaces. The adjoints have an analogous mapping property. It is convenient to call the sequence
p = { ( P j , mj, Lj)}jE N an asymptotic type. Then we have subspaces % i ( C ) of W ( C ) with such asymptotic types. The global t direction transversal to aC is kept fixed. It turns out that a diffeomorphism x : C + C (smooth up to the boundary) induces a pull-back x * : %;(C) +%$p(C) with some other asymptotic type x*P, depending on P and x. Here n, N P = n, N ~ * ifP P satisfies a natural “shadow condition”, nc being the projection to 42 x IN.The invariant interpretation of % i ( C ) is that for every choice of a diffeomorphism V [0,1) x aC, V being a tubular neighbourhood of aC, we have an asymptotic type P:It represents the system [PI = { x * P } of all pull-backs under diffeomorphisms x : [0,1) x aC-+[O,l) X aC, and %,S,(C) is the subspace of those u E X s ( C )which belong to %;(C) for P E [PI in the coordinates responsible for P. Then for abbreviation we omit the brackets, again, For the operator convention with the Mellin transform we have to fix the global normal direction t anyway. So this should not lead to confusions. %;(C) is a nuclear FrBchet space and the Green operators G are characterized by G: X S ( C )+%;(C), G*: %.(C) +%z(C), for all s E R with certain asymptotic types P, Q depending on G (but not on s). In the parametrix construction for A we solve in particular step by step the equations a i k ( A B )= O
O i k ( A B ) = 1 for k = O ,
for k + O
and obtain a sequence hk(Z) = @ i k ( B )( z ) of meromorphic operator functions, where the coefficient in the Lauren? expansion at ( z - q ) - ( k +l), k 2 0, for every pole q, is a finite-dimensional operator with kernel in C m ( XX X ) . This leads to a corresponding structure axiom for the operator-valued Mellin symbols in the calculus. For the parametrix B of A we now should expect m
Hk = t k W ( C k t ) opdh&)w(ckt) I
F a flat, G a Green operator, cf. (7). Examples show that hk for k > 0 may have poles at the integration line Re z = 1/2 that refers to the inverse Mellin transform. Thus we have to modify Hk and take instead Hk (y&)= t
+
” w ( c k t ) opu ( T -“hk) ?- ” w ( C k t )
for an appropriate choice of constants yk, (TQh)( z ) = h ( z + e ) . The calculus then shows that different Y k only lead to a change by Green operators, and that then B
10
Introduction
gives indeed a parametrix of A in the sense that A B - Z, BA - Z are both Green operators. A special case are the operators of Fuchs type on R+ P
A
=
C ak(t) (- t a / a t ) k , k=O
ak E Cm(R+).Here the Mellin symbols are scalar meromorphic functions. This case is a convenient model for studying the main effects. Therefore in our book in Chapter 1 we elaborate this in detail. The calculus over R+XX then may be regarded as a “conification”ofthe standard 1pD0 calculus on X by means of the R, theory. The approach leads to a number of interesting questions. It turns out that not only the “totally characteristic” operators allow Mellin expansions. If we are given, for instance, a scalar classical 1pD0 symbol a(t, r ) on R of order zero (tbeing the covariable to t E R) then the operator
(9) r+op(a)e+: Lz(R+)+L2(R+) (e+ u being the extension of u by 0, r+ the restriction to R + , op(a)v = = ~ ~ e ’ ( ‘ - % zt) ( t u(t’)dt’dt) , has a Mellin expansion in the sense m
r+op(a)e+-
w(cjt)tjop’$hj) u(cjt) =flat + Green,
j-0
for an appropriate choice of constants cj, yj and uniquely determined meromorphic hj which are independent of the choice of cj, y j . Here opL(h) = tYopW(T-Yh)t-Y. m
For instance, if a(t, z) = a ( t )
- j = o aj(it)-jfor z+
fa,then
hj(z) = Iafg+(z) + a;g-(z)lJ;(z) g+(z) = (1 - e-zniz)-l, g-(z) = 1 - g+(z), fo(z) = 1, x ( j - z)]-l for j L 1.
with
&(z) = [(l - z ) . ...
Analogous Mellin expansions exist for symbols of arbitrary orders. The key word of this discussion is the pseudo-differential calculus without the transmission property (cf. REMPEL/SCHULZE “1, [Rll]), which is another exciting chapter of the Mellin operator calculus. The Mellin expansions also exist for systems of equations. The parametrix constructions are analogous as above for the cone. They completely avoid factorizations of symbols into “+” and “ - ” symbols as it is custom in the Wiener Hopf technique. This is useful for systems and other situations, where factorizations break down. In the calculus of boundary value problems without the transmission property the leading Mellin symbol ho(Y, z) = a,+(y)g+(z)+ a,(Y)g-(z)
(10) is of interest also in the index theory (for simplicity here we talk about the case of order zero). It comes from the homogeneous principal part ao(y, f ) of the 1pD0 symbol a(x, F), y = (x’, 0), x = (x’, t ) , t being normal to the boundary. Then for F,=(O, f l ) , where 0 stands for F’ = 0, we have a: = a:(y) = ao(y, F*). The function ho can be interpreted as a connection of the values of ao(y, 8 at the south and
Introduction
11
north poles ( y , Sc) of the cosphere (of the domain, restricted to the boundary) along the conormal interval connecting the poles. For this reason in [R2] we also spoke about the conormal symbol. The conormal symbol takes part in the ellipticity (it is then a connection through isomorphisms) and hence in the homotopy classification of elliptic boundary problems and the analogue of the Atiyah-Singer index theorem. Note that the form of h,(z) was given in ESKIN[E4]. It was later used by many authors for solving concrete problems of singular behaviour. In particular it plays a role in the solution of mixed elliptic boundary problems, cf. REMPEL/ SCHULZE [R4]. The question of a Mellin representation of (non-totally charakteristic) lyDO-s is reasonable also in higher dimensions. For instance, if we restrict A E L!!,(R"+') to R"+I\{O}we may ask whether this restriction belongs to the cone algebra in the above sense with respect to R,X S".The answer is positive. The first ideas in this [P7].They are elaborated in detail in the monodirection go back to PLAMENEVSKIJ graph [PSI and in more general situations also in [Rll]. The positive answer may already be conjectured in virtue of the parametrix construction for an elliptic differential operator in R"+', written in polar coordinates. The parametrix then follows by a Mellin expansion (under the additional Mellin symbol ellipticity) but the parIn other words the study of ametrix has a standard lyDO symbol on the whole R"+l. cone algebras is not finished with the Mellin calculus. There are interesting subalgebras with more precise structures in the mentioned sense. For instance, in the zero order case on R+ the operators (9) generate such a subalgebra. Another important question is the concrete position of the exponents in the asymptotic expansions (or of the poles in the Mellin symbols that determine the asymptotics). This is usually an extra problem which is not solved automatically by the general calculus. But there are exceptions, where fortunately this information can be derived explicitly, namely for the mentioned subalgebra generated by the operators (9) and the higher-order analogues. This leads to the explicit asymptotics for mixed elliptic boundary problems in all dimensions, cf. the discussion in the beginning. Now let us return once again to the elliptic regularity with asymptotics. It was formulated above for the cone in the sense that when the data of the problem belong to a Sobolev space with asymptotics then the solutions belong to an analogous space with shifted smoothness index. The resulting asymptotic type is generated by the given one and that from the zeros of the leading Mellin symbol (= the poles in the parametrix). This behaviour is in sharp contrast to the case of standard lyDO-s on closed compact manifolds or of standard boundary value problems. In the latter situation the spaces do not contain any data which are contributed by the concrete operator, except of the order. The asymptotics for boundary value problems in the classical context may be illustrated by the C" regularity, for instance, for the Dirichlet problem for the Laplacian. If the data (right hand side and boundary conditions) are C" up to the boundary, then the solution also are C" up to the boundary. The asymptotics here coincide with the Taylor expansion, i.e., p j = - j and mj = 0 for all j . In other words the spaces are the same for all those problems. Of course we could try to remove the problem in the general situation by a definition and to pass to the inductive limit !Xis of all XeS,over P.This is then respected by all constructions. We could regard P also as a further index running over all asymptotic types, which is "shifted" together with s under the actions. This point of 2 Schulze, Operators engl.
12
Introduction
view is in fact justified and will play a role in the interpretation of the approach. In particular the length of the asymptotic expansions (as the length of the weight interoal) can be introduced as an additional scale index, interpreted as the conormal order. Then the smoothing operators improve the interior order s and the conormal order. On the other hand we are now at a really delicate point of the solvability theory of our singular problems. We have already mentioned the boundary value problems without the transmission property where the Mellin symbols depend on the variable y on the boundary. For manifolds with edges the calculus contains similarly coneoperator-valued Mellin symbols depending on y E (edge}. But then the zeros of the leading Mellin symbols depend in general on y . Thus the poles of the Mellin image of the solutions depend on y, including the multiplicities. The associated exponents p j in the asymptotics of the solutions constitute clouds of points in the complex plane that can be rather chaotic and irregular with varying y. In simple cases we have trajectories with intersections, where different exponents suddenly coincide. So we have to understand the structure of the branching behaviour of the exponents p i , the jumping of the mj and the discontinuities of the tjkwith varying parameters. It turns out that the concept of continuous asymptotics is adequate for analyzing this phenomenon. The idea is to extend the notion of asymptotics by means of “continuously distributed densities” of exponents which smear out the discrete exponents over a set in a controlled way. This will be performed in terms of analytic functionals sitting in the complex plane for every fixed y. They will then depend on y as C” functions. Then we can talk about C“ functions of analytic functionals which are point-wise discrete and of finite orders. The y dependent point-wise dis[RlO], crete objects of our solutions are actually of this sort, cf. REMPEL/SCHULZE SCHULZE [SlO]. Although this idea is simple it requires from the very beginning a formulation of the calculus also for the Sobolev spaces with continuous conormal asymptotics. This is a program on its own, already on R+. A part of the theory in Chapter 1 is devoted to this theory. At the same time it illustrates the variety of locally convex distribution spaces which may be used to formulate an “elliptic regularity”. The Sobolev spaces with continuous asymptotics constitute a family of spaces with an interesting functional analysis and they are worth to be studied also independently. The aspect of adequate function spaces is a central part of the operator theory also for the higher singularities obtained from conifications and edgifications of given ones. Clearly this is not independent of the symbolic structures for the corresponding operator classes. The conification means that we take locally a cone R+XM over a base M with singularities and then perform a Mellin operator calculus with M-operator-valued Mellin symbols, where M-operators are those which are already constructed for M. The Mellin symbols are a result of a parameter-dependent variant of the proper Mcalculus. In particular the usual cone calculus employs parameter-dependent yDO-s over X, where the parameter plays the role of the Mellin covariable. Similarly we need for the corners a parameter-dependent cone theory. The comer operators then are totally characteristic with respect to several directions. Note that the parameter-dependent cone theory also leads to branching asymptotic data, cf. SCHULZE [S27].
13
Introduction
The edgification is roughly speaking that we multiply M by Rq and then construct a pseudo-differential calculus over R4 with M-operator-valued symbols, now with respect to the Fourier transform. It turns out that for the Fredholm theory we need extra boundary (and potential) conditions with respect to the edge Rq (the “edge conditions”) which satisfy an analogue of the Shapiro-Lopatinskij condition. This leads to an additional edge symbolic level which is a generalization of the boundary symbolic [B9], and of level for boundary value problems in the sense of BOUTETDE MONVEL REMPEL/SCHULZE [W], [Rll]. Boundary value problems fit into the concept of operators on manifolds with edges, where in this case the edge is the boundary and the local model close to the boundary is Rq x R,, where R, is just the cone with trivial (i.e. zero-dimensional) base. The operators in edge algebras are matrices
where A is a proper edge operator, G is a wDO with Green-operator-valued symbol (in the case of [B9] and a Green + Mellin operator in the case of [W], [Rll]), T a trace operator which represents a boundary condition with respect to the edge, K is in some sense dual to a trace operator, it represents a potential which maps distributions on the edge to distributions on the manifold, and Q is a 1pD0 on the edge. The operators G are generated in the calculus, for instance, in compositions of matrices without such parts, and also in the parametrix constructions. For the existence of elliptic edge conditions G, T, K, Q to an elliptic A we have again a topological obstruction as for the “usual” boundary value problems. If we pass to conifications of this and then again to edgifications with new edge conditions and so on we see how complex the calculus becomes, with more and more additional data and operator-valued symbolic levels, based on the parameterdependent version of the preceding calculus. In particular we get symbol hierarchies with a rich internal structure which accompany the operator classes and determine the ellipticity, the stable homotopy classification and the index theory. The analytic functionals are vector or operator-valued for the higher singularities and the continuous asymptotics become necessary already for simple differential operators which are close to the corner modelled by R, X R, XX 3 (t, s, x) of the form
Note for completeness that the typical dgferential operators over an edge, locally modelled by Rq X (R, X M ) , M a manifold with singularities, are of the form
C
aa(y)
( Y , t ) E ~q x
R+,
la1 5 P
where a , ( y ) E Cm(Rq,Diff@-l*I(R,xM)) and Diff’(R+ XM) is the class of typical differential operators for the cone R, X M ; cf. (1) for M = X. For a motivation, see, [SlO]. for instance, SCHULZE The additional conditions with respect to boundaries or edges also have the function to relate theories on spaces with boundaries and without boundaries to each 2’
14
Introduction
other. For instance if X is a compact C" manifold with boundary, then we have over (X, ax) Boutet de Monvel's theory of operators of the form (11). The conification of this gives us the corresponding theory of boundary value problems on manifolds with conical singularities, cf. REMPEL/SCHULZE [R7]. It contains again operator matrices of the form (lo), now over R+XX, and the subclass of right lower comers Q is just the conification of the standard pDO-s over ax, i.e., the cone algebra on R + x a X that was discussed above. Now the right lower corners are ob[R4], tained by reductions of elliptic problems to the boundary, cf. REMPEL/SCHUUE [R8]. This is a classical point of view which is also involved in the index theory and [B9], REMPEL/SCHUUE the stable homotopy classification, cf. BOUTETDE MONVEL [R2], [R8]. It illustrates at the same time a specific subclass which is elliptic without the extra edge conditions, i.e., where (11) is elliptic for G = T = K = Q = 0. Under other aspects TELEMAN [T4], [TS] also arrived at such operators (in the context of elliptic complexes). The number of additional conditions (modulo the difference of trace and potential conditions) depends in general (in contrast to the classical case of [B9]) on the Sobolev space smoothness, cf. [E4], [R4]. The aspect of boundary conditions and the nature of right lower comers in the operator algebras leads to the necessity to study spaces with singularities which are no C manifolds but have branchings close to the exceptional points. This has been admitted anyway as we have alowed, for instance, bases of cones to have several connection components. Now we see that the calculus requires this automatically as the supporting spaces of the right lower comer operators Q,,when we start with simpler spaces. For instance for the calculus of boundary value problems over a cube in R3we need at the same time the calculus of operators over the one-dimensional skeleton of edges. In general, for the calculus on a piecewise linear manifold we need the theories over all lower dimensional skeletons. This illustrates from a more geometric point of view the hierarchies of objects that were mentioned above in the more analytical context of Fredholm operators. The point of view of operator algebras with symbolic structures is useful also for elliptic complexes. The notion of leading symbols admits the definition of ellipticity of a complex. Then the standard things such as parametrix constructions and stable homotopies are also possible for elliptic complexes, cf. ATIYAH/BO?T [A7], PILLAT/ SCHULZE [PSI, SCHUUE[S12]. As we have seen a strategy of dealing with the singular problems including the functional analysis of the associated spaces is to perform the pseudo-differential calculus in several variants, namely in terms of the Mellin and the Fourier transform and with operator-valued symbols which have again a known symbolic structure. The operator-valued aspect is not understood in the naive sense. It will be formulated by a subtle notLon of symbol estimates for the amplitude function a ( y , r ) E Sp(Rq1 x R42; E, E ) or e S#((R+)qlx Rq2; E, E ) , based on groups xA, CAof operators acting on E and g, respectively, or on families of global order reducing operators. The elements of this calculus will be given in Chapter 3, concerning the edge operators. At the same time the material is organized in such a way that we also see the method of iterating the procedure for dealing with higher singularities. This leads to a unifying point of view for the various sorts of singular problems which may be a starting point of further applications and concrete questions. The remaining part of this book follows this line. In particular we study boundary value problems and variops combinations of the basic constructions.
Introduction
15
Since the size of this book was to be limited, many interesting questions remain for future presentation. This concerns in particular iterations such as problems of quarter plane type, crack problems, mixed elliptic problems where the jumping manifold has singularities, or parabolic and other non-elliptic problems and nonlinear problems. Moreover the whole area of questions is by no means finished in the literature, although there are groups in several centers of the analysis in the world with a long tradition and experience. In recent years the interest on operators in domains or on manifolds with singularities increased considerably, not only because of the concrete applications and the numerical aspects but also of the relations to the structure mathematics. In particular the index theory seems to be far from being in a final state. The analytical knowledge should be finally so complete as it was used about wDO-s in the analogous situation for the classical AtiyahSinger index theorem, say in the K-theoretic set-up. If we expect such a synthesis of methods and ideas from different parts of mathematics also for the singular problems, then the completion of this index theory still lies in the future. Our interest in this book is more the analytical content of the calculus. There is a variety of sources in mathematics that have contributed under different aspects. It is not possible here to give a complete appreciation of the merits in all the areas. So we only mention some textbooks and papers. The topics are (among others) the functional analysis of distribution spaces such as Sobolev spaces and their modifications, classical integral transform, special functions and asymptotics of distributions, Tauber theorems, elements of complex function theory. Phragmen-Lindeloif theorems, analytic functionals (cf. GELFANDISCHILOW [Gl], TITCHMARSH [T8], OLVER[Ol], T ~ E B E[Tll], L SCHAPIRA [S22], ATIYAH[A6], JEANQUARTIER [Jl]), *-algebras of operators, abstract Fredholm theory and symbolic structures (cf. DIXMIER[D2], DOUGLAS[D31, CORDESIHERMAN [C7], BROWN/DOUGLAS/ FILLMORE [BlO], KASPAROV [Kl], GRAMSCH[GS], the standard calculus of pseudo-differential operators, of pseudo-differential boundary value problems, singular integral operators, Wiener Hopf techniques (cf. KOHNINIRENBERG [K3], HORMANDER [H3], [H4], TREVES[T9], ATIYAH [A4], GOCHBERG/KRUPNIK [G2], WIDOM[Wl], VISIK/ESKIN[Vl]-[VS], BouTET DE MONVEL [B9], DYNIN [D61, [D71), boundary value problems for yD0-s without the transmission property, vD0-s based on the Mellin transform (cf. VISIdEskin yVl]-[VS], ESKIN[E4], REMPEL/SCHULZE [R2], [R9], LEWIS/PARENTI [L2], SCHULZE [SlS]), operators on manifolds (with and without boundary) with conical points, edges, corners, Fredholm property in weighted Sobolev spaces, asymptotics of solutions, degenerate ellipticity (cf. KONDRAT’EV [K5], PLAMENEVSKU [P8], KONDRAT’EV/~LEYNIK [K6], GRISVARD [G8], REMPEL/SCHUUE [R9], MELROSE [MlO], SCHULZE [S24]), parameter-dependent operator theories, Agmon’s condition and generalizations (cf. AGRANOVI~YVISIK [A2], SEELEY [S19], SUBIN[S27], GRUBB[GlO], REMPEL/SCHULZE [R3], SCHULZE [SlS]), index theory around the Atiyah-Singer theorem (cf. ATIYAH/SINGER [A8], A T I Y A H / B [A7], O ~ PALAIS[Pl], BOUTET DE MONVEL [B9]),
16
Introduction
(viii) differential geometry, index theory on manifoldr with singularities and on noncompact manifolds; singularities of differentiable mappings (cf. CORDES[CS], RABINOWE [Rl], CHEEGER [Cl], TELEMAN [T4], [TS], [T6], [T7], LOCKHART/ [A3]), MCOWEN[L4], MULLER[M14], ARNOLD/VAREENKO/GUSEIN-ZADE (ix)
numerical aspects, models in applied sciences, non-linear problems (BAFICHERA [F3], MOROZOV [M13], LEGUILLON/SANCHEZ-PALENCIA [Lll).
B U ~ K A[Bl],
Our exposition may be regarded as a part of the program to establish a calculus for singularities of any order by repeated conifications and edgifications. It is planned another volume [S17] which continues this approach and contains more applications and examples. Acknowledgement: The author is grateful to T. HIRSCHMANN (Karl-Weierstrass-Institute, Berlin) for many suggestions to the content and improvements of technical de(Karl-Weierstrass-Institute, Berlin) and M. LORENZ tails, further to M. BARANOWSKI (Technical University, Chemnitz) for contributing details and valuable discussions.
1.
The Conormal Asymptotics on IR,
1.1.
The Mellin Transform
1.1.1.
Basic Properties
Section 1.1. contains classical material on the Mellin transform and related questions. The exposition is not completely self-contained. The elementary assertions are often exercises for the reader or may easily be found in the literature. The Mellin transform is defined by the integral formula m
k(z)
= Mu(z) =
dt J tZu(t) 7 , 0
z =x
+ iy a complex variable. First assume
u E Cr(R+), R, = { t E R: t > 0). Then
k(z) is an entire function in z E C and satisfies the estimates lri(z>l 5 CN(1
+ (zI)-Na'R='
for every N E N with a constant cN > 0, and a > 1, supp u E [a-', a ] . As usual N denotes the set of non-negative integers [O, 1,2, ...}. On MC;(R+) we have for arbitrary 71 E C
(M-'h) ( t ) =
2ni
I
r L h ( z )dz,
r, := {z E C: Re z = Re v} .
(2)
r T
The Mellin transform satisfies the following simple identities
M ( t - P u ) (2)
= ( M u ) (2 - p ) ,
(4)
M((1og f l u ) (z)
= (&Mu) (2) 9
(5)
M ( u ( t p ) )(2)
= e-'(Mu) (e-'z),
(6)
u E C,"(R+).They can be extended for various extensions of M to larger distribution spaces if the corresponding operations are well-defined on both sides. First we extend M to L z spaces with weight y
LZ'Y(R+):= tYLZ(R+), y f w ,
LZ.O(R+)= L2(R+). The composition
(7)
18
R+
1. Conormal Asymptotics on m
M,,u(Y)=
I
dt
-,t
t1/2-~+iy~(t)
0
Mo = M .
Set x = log t and ( @ , , u ) ( ~ ) = e ( ~ ~ ~ - y ) ~ u(Yg)(O= (e~),
Then
@?
g(-O.
obviously extends to an isomorphism Oy:L2.Y(Rt)+ L2(R)
and
m
(YM,,u)(0= J e-ixE(@yu)(x) d x = (FOylo(0, 0
where F is the Fourier transform
(Fu) (0 =
1e-ixfu(x)dx.
Rf) From F: L 2 ( R x ) ~ L 2 ( and YM,,= Fay
(9)
we then obtain the following 1. Proposition. The Mellin transform in the sense (8), u E Corn@+), extenh to an iso-
morphism
My:L2*Y(R+) +LZ(Rez = 112 - y ) , y E R, and m
(~;'h)(t)=J-
2n
J t-(1/2-ytiy)h(1/2-y+iy)dy. -m
(10) can also be written in the form ( M ; ' h ) ( t ) =-
1
1
t-"h(z) dz. 2 m rl/2We use the notation G(z) = M,,u(z) for various y when y is known and fixed. For y = 0 we set as above Mo = M.Note that from
1.1. Mellin Transform
19
This is Parseval's theorem for the Mellin transform. Also the other standard identities for the Fourier transform have an analogue for the Mellin transform. In view of the property (4) in most cases it suffices to consider y = 0. It may happen that Mu(z), u E Lz(R+),extends to a holomorphic function h ( z ) in a domain R S @, r,,,c d As mentioned we also consider h(z) as the Mellin transform of u. Often R consists of a strip - m 5 IX < Rez < B j0 3 , or h (z) is a meromorphic function in @.
2. Theorem. The following conditions are equivalent (i) h(z) is the Mellin transform of some function u E L2(R+),supp u E [O,a ] , a > 0 a constant, (ii) h ( z ) is holomorphic in Re z > 1/2, the function h d ( y ) : y + h(1/2 + 6 + iy) belongs to LZ(Ry)for all 6 > 0, and there are constants a, c > 0 such that
IlhallLzW,5 cad for all 6 E R+
If (i) and (ii) are satisfied, then hd tends to a limit ho in Lz(R) as 6 + +0, and IlhdllL2rn) = ( 2 W 2lltdu(t)llm+, > 6 E R+ . For details of the proof cf. [Jl]. Let Zu(t) = 1-1 u ( t - ' ) .
Then Z induces an isometric isomorphism
I : Lz(R+)+LZ(R+), With
(Th) (z) := h(1- z) we then have MZu(z) = TMUUZ). Set
L&(R+)= { u E Lz(R+):supp u bounded], L&(R+)= { u E Lz(R+): dist (0, supp u ) > 0). Then Z induces a bijection between L:b,(R+)and L&,(R+).This yields an analogue of Theorem 2 for L&,(R+)which refers to the half plane Re z < 112. It is clear that u E L:b,(R+)n L$,(R+) implies u E L2*y(R+) for all y E R. Then Mu E SQ (C)(= the space of entire functions) and Mu(z)lr,,E L2(r,,)for all E R. is called a cut-offfunction (with reThroughout this book a function w E Cm(R+) spect to t = 0) if supp w is bounded and w = 1 near t = 0. Then x = 1 - w is an excision function (with respect to t = 0), i.e. vanishing near t = 0 and x E 1 for t > c, c > 0 some constant. For u E L2(R+)and a cut-off function w we may write u = wu + (1 - w ) u , where w u E L&(R+), (1 - w ) u E L&,(R+).Then Mu(z) = M ( w u ) ( 2 ) + M((1 - w ) u ) (z), z E r,,, is a decomposition of Mu(z), where M(wu)(z) has a holomorphic extension to Rez < 1/2, and M((1 - w ) u ) (z) to Re z > 112, and the behaviour is as in Theorem 2.
20
1. Conormal Asymptotics on R,
Denote by L@+)
the space of all u E L2(R,) for which
(i) M u ( z ) is holomorphic in Re z > 112, (ii) M ~ ( z ) l +~ ,E, L2(rllz ~ + y ) for all Y E R+, (iii) if y > ~ ~ , y - ) y then ~ , Mu(1/2+y+iy)+Mu(1/2+y0+iy)
for all yo E R, .
in L2(Ry)
Similarly define L;d,(R+)by the analogous condition with respect to Rez < 1/2. Then L&(Rt) = L;&) > L;b,)(Rt) = Lfd)(R+) be the Schwartz space on R and with proper inclusions. Indeed, let Y(R)
Yo(R+)= { u = U I R , : u E Y(R), supp u s R+}. Then
yo(R+)c L:,)(R+) n L$)(Rt). Now let us come to another extension of the Mellin transform. Set rt 8' = {u E a'(R+): u = r+u for some u E Z'(R)} , where r+ is the operator of restriction to R,. r+Z' is a commutative algebra with the convolution *,as product
< u1* u2, Q, > := < U l ( S ) c3 U 2 ( f ) , Q,(st) > , Q, E Cr(R+),u l , u2 E rt Z'. The Dirac measure a1 at to
*.
f = 1 is
the unit with respect
The Mellin transform on r+Z' is defined by M ~ ( z ) : = < u , r + t ~ - ~ u>=, r + u E r + Z ' . M u ( z ) is holomorphic for Re z > q with some q = q ( u ) E W and independent of the choice of u with u = r+u. Denote by a+(@) the space of all functions h ( z ) which are holomorphic in Re z > q for some q = q ( h ) E R and lh(z)l 5 c(1
+I
~ l ) ~ a ~ ~ ~
with constants c, a > 0 , m E R. If h l , h2 are equal in some half plane Re z > const we identify h , and h 2 . Then So (C!) is a commutative algebra with respect to the addition and multiplication of functions in Re z > const. +
3. Theorem. r+ Z' 3 u +Mu defines an isomorphism
M : r+Z'+dt(C!) of algebras, Ma,= 1. For a proof cf. [Jl]. This theorem was only given for completeness. We will not use the extension of M in this form. We shall choose instead suitable norms such that M extends by continuity in a natural way. The corresponding spaces on R, cannot always be compared by inclusions, similarly as L2.Y for different y. Of course, the various extensions coincide on intersections. For instance for u E L$,,Y(R+):= t YL&(R+)c r+ Z' we obtain M u ( z ) = M y u ( z ) for Rez > 112 - y.
21
1.1, Mellin Transform
The operator tdldt induces an isomorphism r + Z’+ r + Z’. Moreover u+ t - p u , p E @, defines an isomorphism t - P : r + Z ‘ + r + Z ’ , and u E r + Z ‘ implies (log t ) u E r+ f’.The formulas (3), (4), (3, (6) remain in force over F Z ’ . Let J; u E C;(R+) and set m
in the sense of r + Z ’ , and m
According to Theorem 3 we can express (14) in the Mellin image, namely
Thus, with h ( z ) = f”(z), op,[f] u ( t ) = M - l { h ( z ) M u ( z ) } =: op,(h) u ( t ) .
(16)
(16) has the form of a pseudo-differential operator associated with the Mellin transform and the symbol h ( z ) . The function h ( z ) is called the Mellin symbol and op,(h) the corresponding Mellin operator. We have op,(hlhz) = opM(hl)op,(h2). (16) can be extended to a continuous operator op,(h):
L2(R+)+L2(R+).
(17)
Clearly we may admit arbitrary functions h on r1,* which are multipliers in Lz(rl,2). Mellin operators will be used in many variants in this book, for example as restrictions to subspaces of Lz(R+)and more general Mellin symbols h ( z ) . But for the moment we want to discuss operators in L2(R+). Let us express the adjoint with respect to the L2(R+)scalar product m
(u, v ) L ~ R + )=
.f u(t)v(t) dt.
0
We have opMlfl*= O P ~ V [ * ~ If[*W , := ( I . ) ( t ) , or in terms of h ( z ) = f”(z) opy(h)* = op,(h*),
h*(z) := F(1 - 2).
The Mellin operators have a nice behaviour with respect to X I : LZ(R+) +L2(R+),
( x A u )( 1 ) := 11’2u(1t), 1 E R + .
(18)
22
1. Conormal Asymptotics on R,
Observe that (20) is unitary for every A E R+ and x,x,,
= x,.
We have
M(x,u)(z) = f”Z-*Mu(z)
and x, O
P x i 1~= O P , ~I
A E R,
The transform
(M%,u) (1/2 + iy) = A1/2(Mu)(1/2 + iAy) Thus
12; op,(h) with
cil = 0py(hl),
A E R+
hA(1/2+ iy) := h(1/2 + iAy).
The conjugate of op,(h) with (11) has the form (26) Zop,(h)Z-’ = ophf(h’), h’(z) := h(1- z) , cf. (12). Clearly I-’ = I. There is a straightfoward generalization of the Mellin operators to the weighted L z spaces. Set oph(h) = M i 1 h ( z ) M , , h (z) given on
rlI2
- y,
y
E R.
Then
opy,(h): LZsY(R+)-,L2J(R+) if h ( z ) is a multiplier in L2(r1,2 - Y ) . It can easily be proved that oph(h) = rYopy(T-Yh)t-Y,
(27)
(28) Moreover observe that for the adjoints in the sense of the pairing between L2,Yand LZ.-v (T%)(z):=h(z+6).
(OPh(h))* = 0PifY,Y(h*), with h*(z) as in (19). Let us stop here formal consideration of Mellin operators. They will be studied in more detail in 1.3. with individual information on the asymptotics of distributions. Let us return once again to Theorem 2. There is a relation between the Mellin transform and the Hawdorff moment problem that consists in finding a function u on (0,l) for which 1
S t k u ( t ) d t = p k forall
EN
0
with a prescribed sequence of constants { p k J kN, E the m o m e m of u. If we talk about u E L2(0,1) the solution is equivalent to the following problem. Given a sequence {pk}kE N, find h ( z ) as in Theorem 2 with a = 1, such that h ( k ) = p k - l , k = l , 2 , 3) . . .
23
1.1. Mellin Transform
In this form the moment problem has been studied by TALENTI[T2] under the aspect of ill-posedness. Here we only want to mention without proofs the following well-known properties.
4. Theorem. The solution of the Hausdorff moment problem is unique. There is a solution u E Lz(O,l) for given {pk}ksN ifand only if
with a constant c, and Ampk:= j=O
function u E L2(0,1)satish
where rr cannot be replaced by a smaller constant.
More details and references may be found in [T2]. Note in particular that induces a continuous operator u -+ {pk}k p : LZ(0,1)-+ l ’ , {0} = kerp, but p - l : im p
-+
1.1.2.
Ibll = T I , Lz(O,l) is not continuous.
The Spaces %qY(R+),Xq”(R+)
Let Hs(R),s E R, be the classical Sobolev space of all u E Y’(R)with
IIuIIH~(R)=
{J l(1+ I r l 2 Y ( ~ ~ )dT}’l2 ( ~ ) l<2m,
Y ( R ) being the dual of the Schwartz space Y(R), and F : 9‘(R,) dimensional Fourier transform. Then for s E N we have H’(R)={uE9’(R):
(;)JuEL2(R),
j=o,
-+
9’(R,) the one-
}
..., S .
Define
Hs(R+)= { u = r + u : v E Hs(R)}, H@,)
= {u E
H“R): supp u 2 W,) .
(1)
We consider Hi&) in the induced topology and H”(R,) = Hs(R)/H&) in the quotient topology. It is well-known that Ct(lR) is dense in Hs(R)for all s E R. Thus H”(1R) could equivalently be defined as completion of Ct(R) with respect to (J.(IH8. The property 1.1.1.(3) suggests the definition of an analogue of the Sobolev spaces based on - t dldt instead of dldt.
1. Definition. Xs(R+),s E R, is the completion of C,”(R+)with respect to
24
1. Conormal Asymptotics on R+
For y E R we denote by %“(lR+)
I dz I indicates the Lebesgue
the completion of C,”(R+)with respect to
measure on the corresponding line.
The formula 1.1.1.(9) shows that u E %$Y(R+)is equivalent to
f ( 1 + IF12>slF@Yu(R12 dF< i.e.,
m,
aYinduces an isomorphism
q:
%S*Y(IR+)
HS(lR),
4
in other words
%&Y(R+) = {t-1’2+yu(logt ) : u ( x ) E H“R)}. The substitution is understood as extension of the function pull-back to distributions. We see in particular that
ty3tS(IR+) For s E N we have
=%sv~(lR+),
S,
{
(
y ER.
ZS(R+)= u E L2(R+): - t -
:ty
(3)
u E L 2 ( R + ) , j = 0,
... , s
d . 2. Remark. t - induces continuous operators dt
d
%‘*Y(R+)+%‘-‘ny(R+)
t--’
dt
’
for every s, y E R.In particular
d
1-: dt
%=-,”(It+) +%-*y(R+)
is continuous.
Another simple observation is the following 3. Remark. The multiplication by a cut-onfunction w defines continuous operators w:
%&?(It+) -+%JY(R+)
for all s, y E R.
Let us introduce the following notation. If E is a complete locally convex vector space which is a module over an algebra A we denote by [ a ] & a E A, the completion of aE = { a e : e E E } in E . Below we shall prove that ZGY(R+)is a module over C;(R+). Then we have in particular the closed subspace [w]%JY(lR,) c %SY(R+) for every cut-off function w.
25
1.1. Mellin Transform
We have canonical continuous embeddings %s'.y(R+)
-+%&Y(R+), s' 2 s
(4)
[ w ] %d.f(R+)--.* [ w ] %Sy(R+), s' 2 s,
(5)
y' 2 y .
%SY(R+) is a complex separable Hilbert space with the scalar product
(u,V ) ~ C S V ( R +=)
J
rllz-
(1 + I ~ I z > s & u ( z )M,v(z)ldzl.
Here MY is understood as extension by continuity of the Mellin transform with the weight y to an isomorphism MY: z S Y ( R + ) (FHs)(rln- y ) +
>
F = Fv+y being the Fourier transform on the real 4 axis, which is identified with r,,,- 3 1/2 - y + iv, and y = Im z is the dual variable. If y is fixed we also write in this case ii instead of MYu.(u, u ) ~ o ( for ~ + )u, u E C t (R+)extends to a non-degenerate sesquilinear form %%Y(R+) x 2 - q -Y(R+)+c2 . This admits the identification %-S.-'(R+) = (%%Y(R+))'.
(equivalence of norms), where the supremum can be taken over all v E C,"(R+) or v E X-S-Y(R+) which gives the same. The embeddings (4), (5) are not compact for s > t, but %qe(R+)n %%-c(R+) 4 %'(R+)
(7)
is compact for s > t and arbitrary & > 0 . The space on the left of (7) is equipped with the intersection norm (cf. Definition 4, below). Note that the transform 1.1.1.(11) induces isomorphisms I : %SY(R+)--,%S-Y(R+) -
for all s, y
E
R. In particular
I : %e"(R+) =_ %7e"(R+), s E R .
It is useful also to define the spaces
X&Y(R+)= { u E a'@+): w u E %SY(P(+),
(1 - w ) u E HS(W+)}, s, y
E
R,
w being some cut-off function. It is obvious that the definition is independent of the choice of w. The space X S Y(R+)may be understood in terms of sums of spaces.
4. Definition. Let F be a vector space and E o , El be vector subspaces. Define
Eo + El
:= {uO
+~
1 U: O
E
Eo,
~1
E El}.
26
1. Conormal Asymptotics on R+
If F is a topological Hausdorff vector space and Eo, El are normed spaces, we define IIuIIEotE1 =
u=uo+u]
( ~ ~ u O ~ ~ Ellu111E,)> o+
It is well-known that when Eo, El are Banach spaces we also get Banach spaces Eo+ El, E o n El. If L := {(u, - u ) : u E Eo n El}, then Eo + El
1 Eo@ EIIL
with the quotient topology on the right. Eo + El, Eo n El may be equipped with natural Hilbert space structures if Eo, El are Hilbert spaces. Now it is easily seen that
X$Y(R+)= [ w ] W Y ( R ++) [ l - w ] H ; ( R + ) .
(9)
for any cut-off function w. C;(R+) is dense in XSY(R+). 5. Remark. Throughout this book the notion of a sum of spaces will be used also for locally convex
vector spaces Eo, El, that are not necessarily normed spaces. Every couple vo, v1 of semi-norms induces a semi-norm v, v ( u ) :=
inf (vo(uo)+ v l ( u l ) )
u=uo+q
on the sum. Eo + El then is conridered in the locally convex topology induced by the set of all those v.
In particular $Ei are Frdchet spaces and vi runs over a countable system of n o m on Ei, i = 0, 1, then v runs over a countable system of norms on the sum Eo -k El which is a Frdchet space. are Hilbert n o m then the v also are Hilbert norms.
If the vi
Similarly we consider intersections of locally convex vector spaces in the corresponding intersection topology. If Eo, El are Frbchet spaces then Eo n El is also is a Frbchet space. Let us return to the ! W y spaces. } ~defined in the previous section acts as a group of continuous The group { x ~ R+ operators on %~Y(R+), where
I I X A I I ~ ( ~ ~ ~ ( R + ) , ~=~ A", ~ ~ ( R1+ E) )Rt. This follows immediately from (2) and the formula 1.1.1. (22). Now we fix a cut-off function w and define the operator a(c, E ) : u ( t ) + tew(ct)u ( t ) , E,
c E R,. Then we obviously have ( a ( c , E ) U ) ( t ) = c-"(x,a(l, E ) X I l U ) ( t ) .
Thus a ( c , E ) : X&Y(R,)+XSY(R,)
is continuous and Ila(c, E ) I I ~ ( ~ ~ ~ ( R + ) , ~ ~ Y ( R + ) ) 5 ksy,e c - ~ , (10) where ksY,eis the norm of a(1, E ) in 2(slesy(Rt), %&Y(R+)).It is obvious that k8,Y,e s constant uniformly in y E [ - 6, 6'1 for every 6, 6' 2 0. From this it follows
27
1.1. Mellin Transform 6. Lemma. For every E
> 0 we have
Ila(c, &)lle(X..Y(R+).X..y(R+)) +o uniformly in y E [-8, 6’1 for every 8,6’ 2 0.
c
+
C0
Throughout this book we denote by
.nZ,the operator of multiplication by p ,
(11)
p some function. It is often convenient also to write p, i.e. pu = ,/u,u.
7. Proposition. .hq: C;(R+) tinuous operator
+
C,”(R+),p E Corn@+), extends by continuiiy to a con(12)
Atq: !X*y(R+)+X*y(R+)
for every s, y E R, and p +Atq induces a continuous embedding
(the right hand side is considered in the topology of the projective limit). An analogous statement holds for XSY(R+). ProoJ: In view of Remark 3 we may restrict ourselves to the subspace Ct(E+)oof those p with p(0) = 0. This space is. of codimension 1 in C;(R+) and has {cw : c E C} as a complement. Moreover, it is clear that it suffices to prove (12) for y = 0. Now let p E Ct(E+)o, u E C;(R+) and write
m
With
z = 1/2 + iy,
w
=
112 + iy’,
b(y - y’) := f t w - z - l p ( t ) dt,
we
obtain
0
M ( p u ) (y’) = b(y - y’) t(1/2 + iy) Uy, d y = (27r-I dy. Let us now apply 1.1.3. Proposition 1, which may be obtained independently. Then
]b(Y-Y‘)ISciv(p)(l + I Y - Y ’ ~ ) - ~ ,Y, Y ’ E R for every N E N and cN(p)+O as p+O. Set g ( y ) = C(1/2 +iy), f ( y ) = = 6(1/2 + iy), v E C; (R,).Then with (. , . ) s y ~ ( R + )=: (., .)o we get
3
Schulze, Operators engl.
28
1. Conormal Asymptotics on R+
with K ( y , y') = b ( y - y') (1 + lyl)-" (1 + 1 ~ " ) ~ . Applying the well-known inequality (1+lcrl)(l+ISI)-'~1+In-BI, a , / 3 ~ Rweget , IK(y, y')( 5 cN(Q))(1 + ly-y'l)-N(l + l y - ~ ' l ) " ' .
For N large enough this yields
I IK(Y, Y')l dy 5
CICN(P),
J IK(y, y')l dy' 5 c2cN(P))
with constants cl, c2 that are independent of y' and y, respectively. Thus with
II IISYR+) =: II.lls f
or with some constant c > 0 independent of u, u, p. Then the formula (6) yields the assertion (12). The continuity (13) is obvious. The discussion of XqY(R+) is completely analogous, since p E C;(R+) means that we have anyway a finite interval. 0 8. Remark. %"(R+)=
n W(R+)c C"(R+)n LZ(R+)is no algebra underpoint-wise multiplicaSER
tion, for u ( t ) = t-1/4w(t) E %"(R+)but u z 6 LZ(R+).On the other hand (12) is continuous alsofor lI2(R+)= wt 1/2R"(R+), w being an arbitrary cut-offfunction, and Q, 4.nZ,yields a continuous embedding
q~E w%".
w % = = - 1 / * ( ~ + ) .--,
n ~ ( W ( R + ) ,!w(R+))
SYER
and the same for the XJY spaces.
This follows by the same arguments as Proposition 7. 9. Remark. M(%"(R+)) is not closed under differentiations (d/dy)N. For later references we set
T ( R + )= [ u E %"(R+): logNt.u ( t ) E %"(R,)
for all N E N),
cf: 1.1.1. (5). This is a Fdchet space in a natural semi-nom system and the Mellin image is just the Schwartz space on [Rez = 1/2}. Thus T ( R + )is nuclear.
Let a(z) E Cm(I'li2- ?), m E R,and consider opL(a), cf. 1.1.1. (27). Assume that (14) c l ( l + ~ z ~ ) m ~ ~ a ( z+lzl)" ) ~ ~ c forall 2(l ZE~,/?-~ with certain constants clr c2 > 0. Then the second inequality of (14) implies the continuity of opL(a): WY(R+)-,%s-my(R+) (15) for all s E R. The first inequality of (14) yields that (15) is an isomorphism, where opY,(a)-' = opt(a-1). The action of opL(a) on X.Y(R+) may be first defined on C;(R+) and then extended by continuity.
29
1.1. Mellin Transform
As an example consider rn E N and
c m
u(2)
=
uj zj
j=O
with constants uo, ..., a,,,. Then (14) is satisfied if the polynomial u ( z ) has no zeros on r,,z-y. Then the isomorphism
c m
opL(u) =
uj ( - 2 ; ) ' :
!Wy(R+)+%s-my(R+)
j=O
can be interpreted as an assertion on the unique solvability of the equation Fuj(-t$)Ju=
j=O
f
in our class of spaces. We shall return to this aspect below in a systematic calculus. Then the uj may also depend on t, and we obtain Fredholm operators. This consideration suggests to ask for substitutions of the Mellin transform for solving equations of the sort
c aj(-tQ+)Ju=f m
j=O
in adequate function spaces, e E R. It turns out that the equations (17) have a completely different behaviour for --OD < e < 1 and 1 < e < -OD. First let e < 1 and I
Then dsldt = t - Q + 0 on R, , and we can define the transform m
F(Q) U(Z):= e-is(f)rU(t)2 - 0 dt, 0
u E Lz,Q'*(R+).
With the substitution
(EP)u ) (s) = u ( t ( s ) ) we obviously have F(Q) FEP), F the Fourier transform. Thus
(18)
F(Q)(tQD,u) (z) = zF@)u(z),
1 d D, = 7-, for all u E L2-Q/2(R+) for which tQD,uE L2*Q/Z(R+), and hence we are re1 dt duced to the simple case e = 0, where we may apply the usual calculus of ordinary differential equations. For e > 1 we set 1 , l - P
s(t)=el-e
.
30
1. Conormal Asymptotics on R+
This defines a diffeomorphism s: (0, m) +(O, l), s ( t ) +O for t -+O, s ( t ) 41 for t +m. We have
and then
E‘,P’ where
( d4 u ) tQ-
(EiP’u) (s)
(s) = s-
d (E‘P’u)( 8 ) , ds
:= u ( t ( s ) ) .
) ME?) we obtain With the transform M ( Q =
and hence the case
1.1.3.
e > 1 may be reduced to e = 1 that was discussed above.
Examples and Special Formulas
In this section we give some explicit formulas of special Mellin transforms. A large collection of Mellin transforms may be found in [C3]. First consider the function u(t) = ~
( t1 -)p logkt, p
E
C , R e p < 112,
(1)
where ~ ( tdenotes ) the characteristic function of the interval (0,l). Then C(z) = Mu(z)=
(-l)kk! (z-p)k+l
’
i.e., C(z) ist meromorphic with a pole at z = p of order k + 1 (remember that holomorphic or meromorphic extensions of M u ( z ) from f1,2to some domain 0 c C are also denoted as Mellin transform of u ) . The expression (2) for arbitrary k follows from that for k = 0 by differentiating with respect to the complex variable p . Letx(z)EC”(C), O s x ( z ) s l , x ( z ) = l for I z - p l > d , ~ ( z ) = O f o r l z - p 1 < 6 / 2 , 6 > 0 fixed. Then C(z)x(z)lr, E L Z ( Q for all
e E R.
If o ( t ) is a cut-off function and u ( t ) = w ( t ) 1-p logkt, R e p < 112,
then M ( u - u ) (z) is an entire function. Thus C(z) is also meromorphic with a pole at z = p of order k + 1. Moreover
(1 + IzI)s~(z)x(z)Ir,E Lz(rQ) for all s,
e E R.In particular w ( t ) t-P
iogk t E w
y~+ = ) saR
XS(R+).
31
1.1. Mellin Transform
The following simple proposition is left to the reader as an exercise.
1. Proposition. Let Q, E C,"(E+) and h (z) = MQ,(z). Then (i) h(z) is meromorphic in the whole complex plane with simple poles at most at z = -k, k E N , (ii) i f x ( z ) E Cm(@),~ ( z =) 0 for Iz + kl < 1/6, ~ ( z =) 1 for Iz + kl > 216 and all k E W,then Jx(z)h(z)l 5 cN,'(p) (1 + ] z I ) - for ~ all z E''I and all e E R, N E N with constants cN,,(q) 2 0, (iii) if{^,^]^^ is a sequence tending to zero in Corn@+) then cN,,(pj)+0 for all e E R, NEN, (iv) l e f C k = { Z : I Z + k l = 6 ) , k E N , 6 < 1 , then
Another well-known example of a Mellin transform is the Euler
r function
m
r(z)
t2-'
=
e-'dt.
(3)
0
The standard properties can be found in many elementary text books, see for instance, [L6]. r(z) is a meromorphic function with simple poles at z = - k, k E N and residues
k!
The
r function satisfies the functional equation
r ( z + 1) = z r ( z ) or more generally
+ k + 1)= (z + k) (z + k - 1).... (Z + 1)r ( z + 1). Moreover T(n + 1) = n! for all n = 1 , 2 , ... , and T(z) * 0 for all z E @. r(z
The
*
r function has the property
6.
in particular r(1/2)= Let us also mention the formula
a E @\R+,0 < R e z < 1, loga = log la1 + i a r g a , 0 < arga < 2.n 1.1.4.
Holomorphic Functions in a Strip
The Mellin transform of distributions on R + often leads to holomorphic functions in strips (1) S ( a , B ) = {a< Re z < p ] ,
32
1. Conormal Asymptotics on R+
a < P. It is useful to have estimates on boundary. Let us set
r,, a < e < p, in terms of the values on the
S [ a , P ]= {a5 Re z 5 P }
and S[a,p) = {a5 Rez < p } , S ( a , B ] = {a< Rez 5 P } .
1. Lemma. Let y l , yz E R,y1 d 15. Then L2,@(R+) E Lz.’l(R+)n LZsn(R+) foraJleEW, y 1 S e 6 y 2 ,and IIUIIi2.Q(R+) d IIU11?,2*Y1(R+) + IIUIIiZ’y(R+)
for aJ1 u E L2sP(R+). ProoS. We have m
IIUIIL~.Y(R+)= j lu(t)I2 r Z dt, y
Y E
R,
0
and then the assertion immediately follows from t - 2 0 5 t - 2 ~ 1 + t-2n. 0 By &(a),0 S C open, we denote the space of all holomorphic functions in 0.We consider d(0)in the topology of uniform convergence on all compact subsets of a. Then d(0)is a FrBchet space. Let us apply the Mellin transform to the space L2vY1(R+)n LZ*n(R+),y1 < y z . Then it is a simple exercise to check that we get functions in d(S(1/2 - y2,1/2 - n)). 2. Corollary. For y1 < yz the space
M(L2-~1(R+) n L2sn(R+))
(2)
can be characterized as the subspace of all h E d(S(1/2 - y 2 , 1/2 - yl)) for which j Ih(z)I2l d z l ~c with some constant c = c(h) 2 0 for all e with y1 5 e 5 y z . These h rill -
satkfi the estimates rl12-
Ih(z)121dzld
rill- y1
Ih(z)lZldZI+
rill-
3. Lemma. Let h(z) E d(S(a,B))and Ih (z)I2 I dz I 5 a for a < e < rR
with some constant a > 0. Then for every E > 0 a for Z E S ( a + E , p - E ) .
Ih(z)121dzl, Y 1 s e 4 Y z .
33
1.1. Mellin Transform
Proo$ We have
where z = x + iy, a + E < x < B - E , and C, denotes the rectangle x + E - ib, x + E + ib, x - E + i b , x - E - ib, b > ( y ( .Let J be one of the sides of the rectangle parallel to the imaginary axis. Then
by the Schwarz inequality. For dw,
m(b)=
Ib being the interval x - E + i b , x + E + i b , we obtain for Ib( e lyl + 1
Thus m
J
(lm(b)l2+ lm(-b)I2) dbs2rrEa2.
lYl+l
Hence there is a sequence { bj} tending to j - a . In this way we obtain
for all j . Then the result follows for j -
+ m such that I m (bj)l2 + I m (- bj)I2 +0 as
00.
0
4. Lemma. Let h(z) E d(S(0,l)) be bounded and continuous in the closed strip O s R e z 5 1 . Then
Ih(iyIl6 0 0 , I h ( l + i y ) l s a, for all y E R implies
Ih (x + iy)l=
Proof. Let
E
a;
for all y E R , 0
> 0, 1 E R , and set
h,(z) = eez2+lzh (Z).
Then it is easily seen that h,(z)+O
for I1mzl-m
and Ih,(iy)l
s (10,
Ihe(l + i y ) l a~l e e + l
x 5 1.
34
1. Conormal Asymptotics on
R+
~~~
In view of the principle of Phragmen-Lindelof we get lhe(z)l 5 max(ao, a, ee++"), i.e. (h(x + iy)l 5 e-C(x2--*)max(aoe-"+", a,e(l-X)+"++" 1 for every fixed x, y . For E - 0 we obtain Jh(x+ iy)J 6 max(a0e-", ale'-"),
e = eA.
The smallest possible value of the right hand side is obtained for aoe-x= ale1-", i.e. for e = ao/al. For this e we get the desired inequality. 0 Let h(z) be holomorphic near r,. Set (1 + l~12)'Ih(Z)~21d~l}1~Z
(3)
for r~ R . 5. Definition. d r ( S ( a , B ) ) for r E R is the space of all h E d ( S ( a ,B)) with llhllr,p < c for all e E S ( a + E , - E ) and E > 0 with a + E < B - E , c = c ( h , E ) > 0 being a constant. d ' ( S [ a , B ] ) is the subspace of all h E d r ( S ( a ,B)) for which
lim
Ilhllr,a+e<m,
e++O
lim
Ilhllr,p-e<
e++O
Similarly we can define spaces associated with the half open strips S [ a ,B ) , S ( a , B ] . ~ . ~ ~where r , o , e runs over all rational reals in the strip. The same is true of the spaces over S [ a , B ) , S ( a , B ] , S [ a , B ] , and d ' ( S [ a , B ] ) is even a Hilbert space with the norm
d r ( S ( a , B ) ) is a Frbchet space with the norm system ~
(IIhII:,a + IIh II:,p)1/2. Let us restrict the consideration in the sequel to open or closed strips. The halfopen cases can easily be added by the reader.
6. Proposition. The restriction map to sub-strips induces continuous operators (4) B" d'(S(a, B ) ) , (5) d ' ( s [ a ' , B'1) -+ 1W a ,PI) for r' 2 r, a' 5 a, B 6 B'. For a' < a, 0 < B' the operator ( 5 ) is compact when r' > r and Hilbert-Schmidt when rf > r + 1.
d'(S(a',
+
Proof. The continuity of (4), (5) is obvious. Now let - m < a'' < a', B' < B" < m and 8"))be a function with let ms(z) E 99 (S(a", c,(l
+ l I m z l ) s 6 Ims(z)l 5 cz(l + ( I m z ( ) s ,
Fl(l + l I m z l ) - s s lms(z)-ll 5 F2(1+ IImzl)-s
35
1.1. Mellin Transform
for all z E S[d,B']. We can set, for instance, ms(z) = ( a + z), for any a E R with a + z 0 in the given strip. Set g(z) = rn'(z) h ( z ) . Let Co = &S(lr', B'), C1 = as(&,j9). Then for z E C1 we have
*
1 g(z) = 2x1
w-z
1 ms(w>g(w) dw, dw = - mS(w)-I 2x1 w-z
i.e., mS(z)-l g(z) =
Ks(z, w) ms(w) g(w) dw G
with 1 &(z, w) = -(w - z)-I mS(z)-l mS(w)-l. 25r 1
For t = r + s, r' = r + 2s we have the equivalence of norms
IlmS(z)-l g(z)ll'~(c,)-
Ilms(w)g(W)IIo(G) llh ( w ) I I ~ ~ w , F I ) , IIh(Z)lld'(sl.,B1).
Thus the assertion reduces to the question of compactness or Hilbert-Schmidt property of f(w)+
(wl (z) = j K,(z, w) f(w) dw
as operator L2(Co)+Lz(C1).From I (w - z)-I I 5 constant for (w - z) E CoX C1it follows IK,(z, w)lz ldwl ldzl< a for s > 1/2. This yields the Hilbert-Schmidt property. The kernel KS(z,w) is continuous and tends to zero for s > 0 when ( z J-+ m or I w ( + a . L e t y c ( z , w ) b e i n C " ( C 0 x C l ) , 0 ~ y c ( z , w ) 5 1p, c = l f o r l z l + I w l < d 2 , y e = 0 for J z J+ J w J> c. Then for every E > O we find a c > O with JI(1- yc)KsIJ <E (with the norm in 5?(L2(Co),Lz(C,)). Thus tpcKS+ K, in 5?(L2(Co),Lz(Cl)) as c+ a.Now y c K , can be approximated by finite-dimensional operators. This yields the compactness for s > 0. 0
7 . Lemma. For evely fixed p E C and d l , dz E R , c E R, , there are constants c l , c2 > 0 such that
for d1 5 Re z 5 d2, IIm zI 2 c , where c is so large that there are no poles ofr(z b1 s Re z 5 d2, lImzl B c. A proof of this result may be found in [TS], Section 4.4.
+ p ) in
8. Remark. It can be proved that I'(z + p ) l I ' ( z ) behaves like a classical ryD0 symbol for Im z k 00 on Re z = a for every a E R,i.e., there is an asymptotic expansion --.)
-
r(z+ p ) / r ( z )- z p C akz-' k=O
with coefficients ak depending on p (cJ [Ol], ChapterIV, $5.1.).
36
1. Conormal Asymptotics on R+
Recall that the space S:,(R) of classical symbols of order p on R (with “constant coefficients? is defined as the set of all a ( z ) E Cm(R)with
lD$l(z)l s c ( l + 1zI)”k
(6)
forallzER, kEN,withconstantsc(k)>O,and a ( z ) -
c
afP-Jfort+fm.
i-0
N
Set bf (a) = Iuf
I for
j E IN. Further let
~ ( z I@‘+(%) ) uj’
a x ( z )= a ( % )j-0
+ @-(t)a,:} with the characteristic function @* of R* and an excision function x . Then a,(z) E S:;(” ‘)(It) and we can evaluate the best constants on the right of the symbol estimates ID;a,(z)ls constant (1 + IT^)"--(^+ l)-rfor all r, N E N . They form a system of semi-norms on S:,(R) which is countable and denoted by { ~ ~ ( a ) } ~ , ~ . Together with b,! ( a ) we get a semi-norm system on S:!(R) under which it is a FrBchet space. Now denote by %‘P(S(ol,/I)) for a < /I the subspace of all h ( z ) E d ( S ( a , p ) )for which h,(z) = h ( e + iz) satisfies h , ( t ) E S:,(R) for all e, a < e < B, and a+c
Ck(hp)<
sup
9
bf(h,)<
(7)
a+e
for every E > O with or+ e < /I - E , and all j , k~ N . Then it is not hard to show that %‘%’l(or,/I)) is a Frbchet space with the seminorm system h -+ {ck(hp,), bJ ( l ~ , , ) } ~ , N, with el = or + 2-“ for r even, e, = = /I - 2-” for r odd, and a certain N > 0 sufficiently large. Below this space will be employed to define more subtle spaces of Mellin symbols in the operator calculus on R+.
1.1.5.
Analytic Functionals
The conormal asymptotics that will be studied below in Section 1.2. may be interpreted in the Mellin image in terms of anabtic functionab. Here we shall consider some elementary properties. Let g(z) be a meromorphic function in C with poles Pk E C of multiplicities mk + 1 , k E N . Choose a closed piecewise smooth curve c c c \ {Pk} surrounding p l , . .., p N clockwise, where the other points remain in the exterior. Then
U
d(c)3 h - + ( Z ; h )
:== 6 1
g ( z ) h ( z )dz
defines a linear continuous functional on SQ (C). With
then follows
(1)
37
1.1. Mellin Transform
where k ! ( - l ) k 5.J k is the Laurent coefficient of g at ( z -
l).
This is an imme-
diate consequence of Cauchy's integral formula
in the case p
=pj
which implies
The topological dual d'(rr) of the space d ( V ), U S C open, is called the space of analytic functionals on U. 1. Definition. Let K c C be a compact set. An element A E d ' ( C ) is said to be carried by K if for every open neighbourhood U of K there is a constant Cu such that I(1,h)l 5 Cusup lh(z)l forall ZE
u
(3)
hE ~ ( C ) .
Denote by d ' ( K ) the subspace of all 1E d ' ( C ) carried by K . Taking the best constants Cu in the estimates as semi-norms on d ' ( K ) we get a locally convex topology in which d ' ( K ) is a nuclear Frkchet space. If K is a compact set, C \ K connected, and C a piecewise smooth curve surrounding K clockwise, then (1) defines an element 1E d ' ( K ) for every given function g ( z ) E d (C \ K ) . This induces a linear operator p: d ( C \ K ) + d ' ( K )
(4)
with ker p = d ( C ) . Let us check that (4) is surjective. Set f ( z , w >= M t - + , ( t - " ~ ( t N ,f o ( z , w > = ( z - w > - ' ,
(5)
M is defined first for 1 Re w < - and then extended by translation to arbitrary w via the formula 1.1.1.(4). 2 From 1.1.3.(2) we obtain that w E C , o a fixed cut-off function. The Mellin transform
f ( z ,w ) - f o k w ) E
(CA
for every fixed w E C. Now let 1E d ' ( K ) and define g ( z ) = ( A , , f ( z , w > )€ d ( @ \ K )
(6)
(the pairing is understood with respect to the variable w). ( 6 ) induces an operator
e: d'(K)+d(C\
K)
and we want to show that pe = 1 , i.e. 1 = p g . Set g o ( z )= (A,,fO(z, w ) ) , Then we have also A1 = p g o , since g - go is in d ( C ) , i.e.
( A ~h,) =
2x1
1h ( z )go(z) dz, ,
h E &(@I.
= pg.
38
1. Conormal Asymptotics on
R,
I
1 In view of Cauchy’s integral formula h ( w ) = - h ( z ) f O ( z w, ) dz we get 2ni
forall h e & ( ( @ ) ,i.e. A=l,. Note that f ( z , w ) and f o ( z ,w ) are fundamental solutions of the Cauchy-Riemann operator 3. The functions g(z) and g o ( z ) are the corresponding “potentials” of 1. The reader may easily verify that both p and e are continuous. Then ep:
d (C \ K ) +d (C \ K )
is a continuous projection. Set &,(@\ K ) = im ep. Then we have a direct decomposition d ( C \ K ) = d ( C ) + dI(C \ K ) ,
(7)
where d (C) is canonically identified with a corresponding closed subspace of d (C \ K ) . Moreover p induces an isomorphism p: d , ( C \ K )1 d ’ ( K )
(8)
with the inverse e. The bijection (8) admits an equivalent description of the Frkchet space structure in d ’ ( K ) by means of a norm system over d l ( C \ K ). Remember that we have assumed that C \ K is connected. Let C, c { z : dist ( z , K ) < 2 - 3 be a smooth curve surrounding K . Then ~-+lle4Lzcc,,,
1Ed’(K)
(9)
is a norm over d ‘ ( K ). If j runs over IN then we get a countable norm system which is equivalent to the previous one. If V c C is a set we denote by Vc the complement of the union of all unbounded connected components of C \ Then Vc is closed and = V‘. If J is an index set and ( V J I E aI system of sets we define
r.
2 . Theorem. Let Ki be compact sets with Kf = K i , i = 1 , 2 . Then d’(K1 + K2) = d’(K1)+ d’(K2) in the sense of sums of Frkchet spaces.
The proof is an immediate consequence of the solvability of the classical Cousin U = U1n U2.The result says that for every problem applied to Ui = C\Ki, h ~ d ( U there ) exist h i E d ( U i ) with h = hl - h 2 . In particular if K 1n K 2 = 0 the decomposition is direct, i.e., every CE d ’ ( K ) admits a unique decomposition t =& + C2 with tiE s9’(Ki). This decomposition induces continuous projections restKi:d ’ ( K )-+d ’ ( K i ) , i = 1 , 2 .
(11)
39
1.1. Mellin Transform
Let us mention a special case if, for instance, there is a disk {lz - zll 5 6 ) = D with K1cintD, K2nD=0.ThenforC~d’(K)wehave
A = rest,
C = restK,
given by ( A , h ) =1/27ri S ( e O ( z ) h ( z ) d z , h ~ d ( C ) . aD
Now let us return to the example mentioned in the beginning. If K = { p l } u ... u {pN) and I Ed’(K) given by (2) then we may insert in particular t-” which is holomorphic in w E C for every fixed t E R+. Then u ( t ) := (
Iw,t - ” ) E Crn(R+)
and
The precise meaning of the asymptotics will be given in Section 1.2. The Mellin transform applied to u ( t ) w ( t ) yields M ( u w ) (z) = (I,, f ( z , w)) E
(C \ K)
which is a meromorphic function with the same Laurent coefficients at (z - p j ) - ( k +l ) , 0 5 k 5 mj, as for h in (1).In Section 1.2.2. we shall also allow in (12) more general C E d’(C) and then obtain the notion of continuous cononnal agmptotim. In order to have further examples let us set K = {O) and define an operator a: d(@)+d(C\{O)) by a(g(z)) = g(z-’). Then t p o a: sQ(@)-*d’((O}) gives us a rich set of functionals carried by the origin. The functions (12) have asymptotics as in (13) where N = 1 , p 1 = 0 , and in general m , = m . The mapping (8) also gives us explicit examples. If p is a (signed) measure in C, K = supp p compact, then g(z)= ffo(Z,W)dp(w)Ed(@\W Let Kc = K. The carrier of tpg may be smaller than K. There is a well-defined system of maximal open sets U c C for which K1 = C \ U satisfies K; = K, and g(z) extends to some g, E d (C \ K,) . By the Liouville theorem we always have K1 9 0 for p 9 0. The set K, is not necessarily unique. For instance if C is a smooth curve in C which connects two points zo 9 z, then h + h (z) dz,
h E d (C)
C
defines an analytic functional carried by C. But it only depends of zo, z, and not on the concrete choice of C. Note that by the Hahn-Banach theorem every A E sO‘(C) may be written in the form (A,h)=Jh(z)dv(z), for some signed measure
Y
hEd(@),
with compact support.
40
1. Conormal Asymptotics on R+
In our applications we also need the spaces of vector-valued analytic functionals 1 ' ( K ) Q,E, E being a locally convex vector space. The above considerations have a straightforward extension to this case. The spaces E are often Frkchet spaces or projective or inductive limits of Frkchet spaces. In this connection it is useful to have the following 3 . Lemma. Let E, F, G be Frdchet spaces. Then ( E + F ) Q,G
E@,G
+ FQ,G.
(14)
Proof. Every v E ( E + F ) Q,G admits a respresentation as a converging sum m
where LiE C , lAil < m , ui+O in E + F, gi+ 0 in G.There exist sequences ei+O in E , fi+ 0 in F, such that ui = ei + fi. Then m
m
Liei@gi E EQ,G,
v' =
v"
lihQgi E FQ,G,
=
i=l
i=l
v = v' + v". This yields a continuous embedding ( E + F ) @,G G EQ,G + F@,G. Conversely, the continuous embeddings E , F 4 E + F induce continuous embeddings E @ , G , FQ,G 4 ( E + F)@%G and thus also EQ,G + FQ,G G ( E + F ) @,G. 0 The formula (14) obviously extends to the case when G is a projective or inductive limit of Frkchet spaces.
4. Corollary. Under the conditions of Theorem 2 we have
1 ' ( K 1 + Kz) @*E
= d'(K1) Q,E
+ d'(KZ) @,E,
E being a Frdchet space (or a projective or inductive limit of Frdchet spaces). Let 0 be an open set in some Euklidean space and E = Cm(a). Then 1 ' ( K ) Q,Cm(a) may be interpreted as the space Cm(R,dse'(K)) of C" functions over R with values in 1 ' ( K ) . Every g o , z) E Cm(O,1(C \ K ) ) gives rise to a function ( ( y ) E Cm(a,1 ' ( K ) ) . A case of particular interest is that ( ( y ) has the form (2) for every fixed y E 0,where N , mj, pi may depend on y . An example is g ( y , 2) = ( a ( y )- z1-l M y ) - z ) - l for a , b E Cm(a),a ( y ) , b ( y ) E K for all y . The changing of multiplicities of the poles with varying y corresponds to a "branching of asymptotics" that will be discussed in detail in Section 2.3.2. This phenomenon is one of the main motivations for introducing asymptotics associated with non-discrete analytic functionals. We denote by ~4'(@})(~),p E @, m E N , the subspace of all (E d'(b})of the form
41
1.1. Mellin Transform
Aj E 6:. Then d'(b))(") 2 Cm+l.A function CE Cm(R,d'(C)),R 5 R"open, is called point-wise discrete and of finite order if for every y E R there is a set of pairs P ( ~ ( Y )=) {oIr(Y) mr(y))lr= I , ..., ~ ( y c) C X N such that 9
5. Definition. Cm(R,d'(C))* is the subspace of all (E Cm(R,Set(@)) which are pointwise discrete and of finite order and for which the following condition is satisfied. I f K c c C , K , c c R are compact sets then sup ( 1 + m ( y ) ) < m, where the sum YE
is taken over all m ( y ) , for which there
Ki
c
b a pair ( p ( y ) ,m ( y ) )E P ( ( ( y ) ) with p ( y ) E
K. Moreover let Cm(R,d ' ( K ) ) ' , K c c C, be the subspace of all (E Cm(R,d'(C))* such that ( ( y ) is carried by K for all y E R. 6. Theorem. The dlfferentiation
D;,first defined on Cm(0,
&I(@)),
induces a linear op-
erator
0;:C m ( R , d ' ( K ) ) ' + Cm(R,d'(K))'
(15)
for every multi-index a, and every K c c C .
For the proof we first need some notation. If g ( z ) is a meromorphic function we denote by P ( g ) the subset of C X N with ( p , m ) E P ( g ) e p is pole of g of multiplicity m + 1. Let AZK(R,C)* be the set of all g ( y , z ) E Cm(R,1(C \ K ) ) such that (a) g ( y , z ) extends to a meromorphic function in C for every y E R (b) let K 1 c c R and
where the sum is taken over all 1 + m ( y ) for which ( p ( y ) ,m ( y ) )E P ( g ( y , .)); then N ( g ; K,) < m for all K 1 c c 0. If g is independent of y we set N ( g ) = (1 + m ) .
1
@, m ) E P ( g )
7. Lemma. Let {fk(z)}k be a sequence of meromorphic functions with poles in K and converges to a function f' E d ( C \ K ) for N c f k ) 5 N for all k. Assume that f&\K k+ m . Then f ' can be extended to a meromorphic function f; i.e. f Ic, = f ' , and N ( f ) 5 N.
Proof. Let p,, k r . .., pNvk),k be the poles of f k , counted with their multiplicities. Define the sequence of polynomials bk(z) = ( z - P1.k ) * ... .( z - P N v k ) , k) '
The set of these polynomials belongs to a bounded subset of the space of all polynomials of degree 5 N , since the pj,k vary in a compact set. Then we find a subsequence { b k j } j E which converges to a polynomial b of degree 5 N . Thus the sequence of entire functions h j ( z ) = f k , ( z )bk,(z)tends to a limit h E d ( C ) and hence we can set f = h / b . 0
42
1. Conormal Asyrnptotics on R,
8. Lemma. g ( y , z)
E
&,(a,
C)' implies (D;g) ( y , z)
E A K ( R C)* ,
and
N(D;g; K , ) 5 2IaIN(g;K 2 ) for every multi-index a and euery disk K , = {ly - yls 6} contained in R,K 2 = {ly 5 6 + E } for euery E > 0 sufficiently small. Proof It suffices to consider la1 = 1 . Then the result follows by induction. For instance let a = ( l , O , . . . ,O), y = ( y l , y ' ) , y' = ( y 2 ,... , y n ) . Then Lemma 8 can be applied to the sequence f d z ) = k ( f ( Y i - l/k,Y',2) - f ( Y i , Y ' ,
2))
for fixed y = ( y , ,y') E R and k E N . Then we get the result for every y and also the desired estimate of N(D;g; K,).0 Proof of Theorem 6. The operator ping
e, applied for every fixed y E 0,induces a map-
e: C m ( R , d ' ( K ) )+./nK(R,C)* * which commutes with D;. Thus (15) is an immediate consequence of Lemma 8. 0
1.2.
Spaces with Conormal Asymptotics
1.2.1.
The Spaces with Discrete Asymptotics
This section studies spaces of distributions on R + with so-called discrete conormal asymptotics for t +0 and t + m . They have the form mi
with a sequence [(p], m j ) ) j , c C X N that constitutes a (discrete) asymptotic type. For t + O ( t + m) the sum with respect t o j is taken over different subsets of Z. The asymptotics may be considered separately for t + 0 and t+ m , but it is convenient to unify the notations. Now let us pass to the precise definitions. 1. Definition. Denote by 3 the set of all countable subsets P c C x W for which n c P n [ a < R e z < , 9 ] isfinite for all a , B E R , nc: C x N + C . Moreover let%?', y E R,be the subset of all P E 5%with nc P n rl,,- = 0 . The elements of S V are called discrete asymptotic types with respect to the weight y .
Let P = { ( p j , m j ) } j E zQ, = { ( q k , n k ) } k e Z E % . Then we write P S Q iff ncPGncQ and mj 5 nk whenever p j = qk for certain j , k . Moreover we set T'p={(Pj+e,mj)}j,z,
eE@.
(2)
Thus W =T - y a 0 . The asymptotic expansions will often be considered only for those p j that belong to a strip in C parallel to the imaginary axis. Since we need open, closed, and halfopen strips of variable size, we want to introduce some abbreviations.
43
1.2. Spaces with Conormal Asymptotics
Let 3:= gmbe the set of all pairs A = (6,s’)with - 6,6’ E R, or 6 = 6’ = 0. Moreover Yll (YOl ; gl0)denotes the set of all couples A = [a, 6’1, - 6,6‘ E R, (A = (6,6’], - ~ E R , ,~ ’ E R or + 6 = 6 ’ = 0 ; A = [ 6 , 6 ‘ ) , - ~ E W + , ~ ’ E R or , 6 = 6 ’ = 0 ) . We also admit 6 = --m or 6’ = m . Write
A l s A 2 for
-fils - d 2 ,
6:s S;,
A l , A z E 3afl,a, = 0 , l . An element A E gaflis called finite if both 6 and 6’ are finite. The exception 6 = 6’ = 0 was made to have the case “no asymptotic information’’. For A = [0, 01 we talk about trivial A . In the following we often shall tacitly assume A to be non-trivial. The trivial case is usually obvious and can separately be considered. If A E Yapwe denote by int A E 3 the couple obtained by replacing all brackets by (,). Introduce the strips
(112 + 6 < Re z < 112 + a’} {1/2 + 6 5 Re z < 1/2 + 6’) S A = { {1/2 + 6 < Re z s 112 + S’} (1/2 + 6 s Re z 5 1/2 + a’} For y E R we set Sl; = T-VSA, S! = SA. Here TQA= ( z + e : Z E A } ,
for for for for
A = (6,s’)E gm, A = [6, 6’) E gl0, A = (6,6’] E gal, A = [a, 6’1 E Yll.
(3) (4)
A & @ ,e E @ . For any P E 2,y E R , A E gap,we define the subset Pz of P by PI; = {(pi, mj)E P: pj E S:}
and PA:= P!. The intuitive meaning of Pl; is that we only consider that part of the asymptotics that belongs to the ”weight interval” A around the reference weight y. From Definition 1 it follows that PI; is a finite set if A is finite. Let us define the asymptotics first in the image under the Mellin transform M , cf. 1.1.3. Set
llhlls,~ =
L
(1 + Iz12)slh(z)lzIdzl}l’z,
(5)
IdzJ being the Lebesgue measure on I‘,. Clearly ( 5 ) depends only on ReB. A function x E Cm(@)is called an A-excision function for some A G @, if 0 s x 5 1, x(z) = 0 for all z in an open neighbourhood of 6, ~ ( z=)1 outside another open neighbourhood of 6. Let s, y E R, P = {(pj,mj)}E W ,A E 3 non-trivial, and set
sQSp.5 = ( h E sQ(Sl;\ n, P ) : h has poles at pj of multiplicity mj+ 1 for all for every n,P-excision function x and pi E Sl;,sup llxhlls,fl< B E
(6)
SI,
A , = (6 + E , 6’-
E)
for all E > 0 with 6 + E < O , 6‘- E > O } .
It is obvious that So:,; is a Frechet space with the norm system ll~hII,,~, where x runs over all ncP-excision functions and E Sl;, together with the semi-norms, given by h + l G k l , with Gk being the coefficient in the Laurent series of h at ( z - pj)-(k+l), p j E Si,0 s k 5 mj. 4
Schulze, Operators engl
44
1. Conormal Asymptotics on R,
The simple consideration that a countable system of semi-norms suffices is left for different P only differ by finite-dimensional to the reader. It is clear that d>,> subspaces provided A is finite. By definition we have a>,> c My!Xs.y(R+)for all P E W ,A E ~ . 2. Definition. Lets, y E R , P E W , A E 3 non-trivial. Then !XSpY(R+), ir defined as the preimage of d$,>under the weighted Mellin transform M y , equipped with the induced Frkchet space structure. For trivial A we set !X>'(R+), = !Xs*Y(R+). For abbreviation we also set
!X>Y(R+)= !X$y(R+)(-m,q. If P = 0 we write 0 as subscript.
We have natural continuous embeddings %> "(R+)d,4 !X>Y(R+)d
(7)
for A'z A , s'z s, P ' s P obtained by the restriction map in the Mellin image. Then
X>'(R+) = c lim !X$Y(R+)A As3
is a FrCchet space. It may be equipped with a countable system of Hilbert norms. 3. Remark. .%;'Y(R+)A(regarded as projective limits over all s) are nuclear Frichet spaces for all non-trivial A E 3.
Indeed, we have !x;.Y(R+),:
=
n!x>Y(R+), nX>Y(R+),. =
StR
S E Z
The second equality follows immediately from the corresponding property for finite A and P = 0. In the latter case this is obvious if we look at the characterization of %$Y(R+)din the Mellin image. In the operator calculus below we shall often assume that - 6 = 6' = k E W for the weight strips. This case will be indicated by 9 = (- k , k ) , i.e., !X>Y(R+)B = !X>Y(R+)(-k,k, for
k
> 0,
= !Xs.Y(R+)
for k = 0.
(8)
Fix a cut-off function w ( t ) and let w ( p , k , t ) = w ( t ) t - P logkt, w ' ( p , k , t ) = w(t-I) t - P logkt, (p, k ) E C x
IN.
For finite 4 E .W, Q = { ( q i ,n i ) } i = ,.._,, Re qj < 112 - y for 1 s j 5 r, Re qj > 112 - y for r + 1 5j 5 M with a certain r, we denote by Z;(R+) the vector space spanned by { w ( q j , k , t ) : 1 s j 5 r, 0 5 k 5 m j } u { w ' ( q j , k,t ) : r + 1 5 j 5 M, 0 s k 5 mj} .
45
1.2. Spaces with Conormal Asymptotics
For arbitrary P E a)’, A E gUsfinite, we set
Z$(W+),: = Yb(R+) with Q = Pl;. It is then obvious that
Y$(IR+),= %$y(R+), for every s E R and finite A
E 3.
(9)
Moreover
%;Y(R+),= %“d(R+), + Y$(R+)d
(10)
is a direct decomposition. Next we define %$Y(R+), for arbitrary A E gUs, or,p = 0 , l . First let A E !Yl0 ( A E gal), A finite, and introduce the space d2.5 of all h E d ( i n t S:) for which Ilh < 00 for all B E int S$ and
a>,>is obviously a Frkchet space with the semi-norms of 1 ( i n t Sl;) together with Il.llss, B E int S; and that defined by the limit (11) for A E glO((l2)for A E Pol).For A E !Il1,A = [ 6 , 6’1, we simply define sP&,yd as the subspace of h E d(int Sl;) where llhlls,s< 00, B E int Sl;,and both limits ( l l ) , (12) are finite, which makes sense for 6 0 or 6’ =k 0. For 6 = 6’ = 0 we set = M,,%s*Y(IR+) = F(Hs(Re z = 1/2 - y ) ) . Then Se;,,yd is a Hilbert space for A E . ! Y ~ ~finite, with a “natural” Hilbert space
*
norm. Now let %zY(R+), be the subspace of all u E %% y(R+)for which Myu E sQz,,yd.It will be considered in the topology induced by the bijection
M y :%;y(w+),
a;,;.
+
4. Definition. An element u E Xs*y(R+) is called flat of order -6 at t = 0 and 6’ at t = with respect to the reference weight y if u E %>y(R+),b,b,l, - 6,6’ E R+. w e talk about flatness of order -6 - 0 at t = 0 , 6’ - 0 at t = 00 with respect to y if u E %;”(IR+)(a,y,, -6,6’ E IR+.In an analogous manner we can define flatness for the harf open weight intervals. 5. Definition. For A E YUsfinite, a, p
= 0 , 1 , P E W,we
%$“Y(R+),= %Sdy(R+),+ % $ @ + ) A , equipped with the topology of the sum. In particular
%$Y(R+)[O,O] = .7eS*Y(R+) For y = 0 we shall omit the corresponding subscript. 4’
set
46
1. Conormal Asymptotics on R+
The definition says that the elements u E %$Y(IR+), the sense that
cc mi
u(t) -
j
tkw(Pj,
k=O
k , 1) -
1 1
have asymptotic expansions in
mi
c:kw’(P/, k , 1) E xJd‘(R+)d,
(13)
k=O
i.e., the difference is flat according to Definition 4. The sums are taken over those < 112 - y } (prE ncP1; n {Rez > 112 - y } ) . This is independent of the choice of w but P$ is uniquely determined by (13). For A E Y this is obvious. For closed or half-open weight intervals there may be points in ncP$ with real parts being equal 1/2 - y + 6 or 1/2 - y + 6‘. The corresponding consideration is a simple exercise. To illustrate the situation we want to discuss an example. Let us take
j ( l ) for which pj E ncP1; n [Re z
% ~ o ( w + ) [ ,= ~ o{ ul E L*(R+>:wt-’u E LZ(W+)}
and P = {( - 1/2,0)} ; then P ~ l , = o lP and YO,(R+)= (cwt1’2: c E C}.
ZO,(W+)is a one-dimensional vector space with Y!(R+)n %$o(R+)l,,o, = [O}. Then the sum W 0 ( ~ + ) , l , O 1=
~$o(R+)[l,O] + ZO,(R+)
(taken in LB’(R+)and equipped with the topology of the sum) is a Hilbert space conand Y;(R,) as closed subspaces. So we have a well-defined taining %$O(IR+)~l,ol projection
W 0(R+)[l,O]
+
YO,@+)
that determines the coefficient c in the “asymptotics”. In general we have also complementary projections rm1; : %;Y(lR+), +%eS,”(IR+),, as:: %S;Y(R+)A + Z;(IR+),, s, y E IR, P Ea’, A E gwp finite, 01, = 0 , l . It is clear that there are continuous embeddings (7) for A 2 A‘, A , A‘ E gap,s z s‘, P’5 P. This allows to define the projective limits slespY(R,),*,q = g%spY(R+),a,d’lr 6 5 0, d’20
%$Y(R+)(,,d,]= l&l%$Y(R+)[a,af], 6’ 2 0, dz0
where the subscript on the left may be replaced by [6, m] and Note that
[m,
6’1, respectively.
tA!X;Y(R+), = %y.l;(R+)d in a natural way, P E W , A E gM,where T-AP= {(pj - 1,mj)} E W + A for P E 5%’. This shows that a simple shift of the weights admits to consider a fixed y. From now on
47
1.2. Spaces with Conormal Asymptotics
we often discuss the case y = 0. Set %;@+)A
= .%$O(R+)A,
A E gap-, P E
a',
and in particular %;(R+) = %;(R+)(m,m). We also use the notation
%:p jA + = @ %:Y(R+)~, P E l Y
where we omit A ( y ) again for A = (a, m ) ( y = 0). Every P E Ro allows a decomposition P = Q u R , where ncQc p e z < 1 / 2 } , ncR c {Rez > 1 / 2 } . It can easily be proved that
%;m+)= X p + )+ %s,(R+) in the sense of sums of FrCchet spaces (cf. 1.1.2. Remark 5 ) .
(15)
6. Definition. Let P I ,P2E 5%.Denote by PI + PzE 5% that element which is characterized bY nc(P, + PJ = ncPl u ncPz and
Note that the sum can be described in terms of patterns of poles of meromorphic functions under multiplication. Indeed, let h, E So (C \ ncP), P = {(pj, z , be meromorphic with poles at pj of multiplicity mj + 1, j~ Z, and hp 9 0 on C \ ncP. Then h,, h, belongs to PI + PZ. The equality (15) holds for arbitrary P,Q, R E Rowhenever P = Q + R. The asymptotics will often be considered separately for t -+O and t + m.
7 . Definition. Let 9 be the set of all R E Rowith ncR c {Re z < 1 / 2 } . Similarly define 9 v : = T-V9 as subsets of RV, y E R (such that 9 = Po),and 9-" = U9 V . V
The elements of X i @ + ) ,P E9, are obviously flat of infinite order at m , and those of %eS,(R+),Q E Ro\ 9 for ncQn r,,,= 0, flat of infinite order at zero. Incidentally we set
Yp(R+) = X;(R+) , P E 9.
(16) As mentioned above the space Yp(R+) is nuclear and FrCchet. We have a canonical identification
.Yo(R+) Yo(R+):= { u E .Y(R): supp u & R+], Y(W) being the Schwartz space on the real axis. In the case Po := {(-A
O>}j E N
we have Ypo(R+)= 3'(R+)IR+.The asymptotics for t +0 of the type Po correspond to the Taylor expansion. So we also talk about Taylor asymptotics. Note that the duals satisfy YO@+)'= {UlR,: uEY'(R)} I Ypo(R+)' = { u E Y'(R+): supp u z R+}.
48
1. Conormal AsvmDtotics on R,
In [ R l l ] ,Section 1.2.1., there was given a characterization of Yp(R,.)'for arbitrary P E P. The finite linear combinations of derivatives of the Dirac measure at the origin are replaced by elements of The isomorphism 1.1.2. (8) transforms the asymptotics for t -+ 0 to correspondingones for t 00. In other words I induces an isomorphism -+
I : %",R+) for every R
---f
%s,.(lR,)
= {(pi, m j ) ] j , z ~ 3 1with 0
R'={(l-pj,
mj>Ij,z.
A similar transformation can be formulate- for the weighted spaces with asymptotics. 8. Remark. The condition on the asymptotic types that the p j satisfy R e p j < 112 - y (> 112 - y ) for t +0 ( t + m) is not necessary for the calculus. This property was only imposed for notational convenience. We could have assumed as well different weights at t = 0 and t = and a description in the Mellin image after cutting the distributions at t = 0 and t = Q), respectively. Q)
Let P = { ( p i , mj)Ij bers
E 9 and
consider the space X, of sequences of complex num-
IrjklOjkgmj,jcN
in the topology of component-wise convergence. Let Lj = {z: )z - pjl = E ] and E > 0 so small that Lj surrounds no other point of nCP.Fix u E %;(R+) and consider the analytic functional
(434,h
) : 2ni = I /Mu(z)h(z)dz,
h~d(@).
Lj
(17) represents the corresponding part of the Laurent expansion of p j in a unique way. Write the Laurent expansion in the form
Mu(z) =
2
Mu at the point
( - l ) k k ! Z ; . k ( z - p j ) - ( k + lvj(z), )+
k=O
vj being holomorphic near pi. (17) admits to recover the coefficients f& by inserting polynomials h ( z ) that have a zero at pi of order k, 0 5 k 5 mJ.The formula 1.1.3. ( 2 ) shows that then the f& are just the coefficients of the q m p t o t i c s of u. Denote by
[M:S(ei(R,) + x p the linear operator that maps u to the sequence of coefficients in the asymptotics. Then we have an analogue of a Bore1 theorem that asserts in its classical form that for every sequence of coeffcients there is a C" function that has just these Taylor coefficients. 9. Theorem. The sequence
*
0 +!XS,(R+) !XS,(R+) X, +0 (18) is exact for every s E R. The space S(eS,(R+)can be written as a sum of Frdchet spaces
49
1.2. Spaces with Conormal Asymptotics
XS,(R+)= X",R+) + XF(R+). An analogous result holds for P E 3 1 O (i.e. with agmptotics also for t
(19) -+
m).
I&
Proof. Let P = ( ( p j , mj)}j,and E @, 0 s k 5 m j , j E Z, be an arbitrary sequence. We shall prove that for some choice of constants cj the sum m
mi
converges in the space %;(It+). By definition we find a sequence of weight intervals A N = (6N,1 5 3E 3, A N + 1 2 A N , such that
Write u ( t ) = U N ( t ) + u ; ( t ) , where u)N(t) is the sum over all j , k for which E > 0 such that
pJ > 112 + aN.For pJ < 112 + d N we find an V ( t ) := w ( t ) &k
f-PJ-EIOgktE XqY(R+)
for all y E [aN,db]. Choose c > 0 @(C,
SO
that w ( c t ) w ( t ) = w ( c r ) . Then
E ) V ( t ) = feU(Cf) U ( t ) =
w ( d ) &k t-pJIOgkl
(cf. the notations after 1.1.2. Remark 5). Applying 1.1.2. Lemma 6 we get IIw ( c t ) 6Jk t-" logktllWYY(R+)
0
as c -+a, uniformly in y E [aN,6 3 . We also could write w ( c t ) &k
t-PJlogkt-+O
in
provided c, = c," inas c ---* m. In view of (10) the sum (20) converges in Xi(W+)dN creases sufficiently fast. Now for c, = c; we have convergence for all N. From (21) we get convergence in Xi(R+).This proves the surjectivity of 5: The other properties for the exactness of the sequence are obvious. Next observe that our method yields even a sequence cJ for which (20) converges in %S,(R+)for all s E R. Indeed if we denote by cJ(s) the sequence that was just constructed for the given s we obtain for cJ := c , ( j ) the convergence for all s. Thus for every f E X i ( R + ) we find an u E !XF(R+) with f = uo + u for some uo E XS,(R+). From the continuous embeddings X;(R+) !X;(R+), %; (R+) -+!X;(R+) we obtain the continuity of -+
zL(R+)+ XF (R+)+xeS,(R+). This embedding is clearly injective. On the other hand the above consideration has proved the surjectivity. Thus we get (19). 0 10. Remark. !%S,(R+)has in XeS,(R+)no topological complement unless P is finite. In other words 9 a continuous right inverse of 5.
we did not construct in the proof of Theorem
We say that P E Y t y satisfies the shadow condition if ( p , m) E P implies ( p + j , m ) E P for R e p < 112 - yand ( p - j , m ) E P for R e p > 112 - yfor all j E I N . Let x : R+ R+ be a diffeomorphism which extends to a diffeomorphism %: R +R with C(0) = 0 and let x ( t ) = t for t 2 c with some constant c > 0. -+
50
1. Conormal Asymptotics on R+
11. Theorem. Let P E RY satisfv the shadow condition. Then the pull-back under x induces an isomorphism x*:
%$Y(R+)d+%$Y(R+),
for all s E R,A
E
Yap,CC,
= 0,
(22)
1.
Proof. First remark that it suffices to show the assertion for any finite A, since then it also follows for the projective limits involved in the definition for infinite A. Moreover we only have to show that the pull-back induces the corresponding continuous operators, since the injectivity (and then also the surjectivity from (%*)-’) follows from the known property of the pull-backs of distributions on R+. We also may assume that P E 9 Y because of the conditions on x. Finally it suffices to consider A = [a, 01, since A = (S,O]then follows by taking the projective limit. By definition we have X$”(R+)d= %iy(R+)d+ zYp(R+)d,
A
= [d, 01.
Then it sufices to show that x * : %i”(R+)d X;”(R+)d,
(23)
x * : q,(R+), +X“p(R+)’4.
(24) is spanned by functions of the
Let us begin with the discussion of (24). form u ( T ) = w ( r ) T-PlogkE p E @,
Let first k = 0. Write
k E IN.
T= x ( t ) , Then
( x * u )( t ) = u ( x ( t ) ) = w ( x ( t ) ) ( x ( t ) ) - P . For x we may apply the Taylor expansion at 0 N- 1
x(r) =
C cjtj+ tNxN(t),
co = 0,
j-0
x N ( t )being C” up to t = 0. Then
x ( t ) - p = ( c l t ) - p {. . . } - p .
For {. .. } - p we can apply the well-known formula
with certain dj( p , t ) which are smooth up to t = 0. It follows that u ( x ( t ) ) belongs to %$Y(R+)d for a finite partial sum contributes to modulo %JOY(R+),, and the
51
1.2. Spaces with Conormal Asymptotics
*
remainder also belongs to X;y(R+),.The case k 0 follows by differentiating the transformation law with respect to p . For (23) we shall employ 1.2.2. Theorem 16 and the well-known invariance of the standard Sobolev spaces under pull-backs. This yields the invariance also of XqS(R+).Moreover we have
(2 6) A = [a, 01. Then we obtain (23) by combining the above discussion with the invariance of XqS(R+).0 w.!qjYY(R+), = WtY-s-dX~s(R+)
12.Theorem. Let P E 5 t y satish the shadow condition and let ip E Corn@+) or Ct(R+).Then .nZ, (the operator of multiplication by ip) induces continuous operators ip(t-l) E
A,: %Spy@+), +XSpY(R+)d
for all s E R,A
E 5+ ’
a,B = 0,1.
Proot Similarly as in the proof of Theorem 11 it suffices to consider closed finite weight intervals A . Again we may take A = [6,0] if we multiply by q ( t ) .The multiplication by i p ( t r 1 ) may be reduced to the case of p ( t ) by applying the transformation I. Using again the decomposition of Definition 5 we may consider XS,Y(R+), and separately. The equation
WX;~(R+), = wt-dXq”(R+) shows that the argument for X$OY(R+),follows from 1.1.2. Proposition 7. The consideration for y:@+), is trivial and left to the reader. It shows that A P u E Ce;(R+), mod X;Y(R+), for u E Z;(R+),, . 0
1.2.2.
The Spaces with Continuous Asymptotics
This section deals with an analogue of the class of spaces with discrete asymptotic types. The intuitive meaning of the continuous asymptotic types is that the exponents in the asymptotics are distributed over a region in the complex plane with a “density’’ that may be described in terms of analytic functionah.
1. Definition. A closed subset V S C is called a carrier of asymptotics if (i) V n {a5 Re z s B } is compact for all a, B E R, (ii) V c= K Let Y be the set of all carriers of asymptotics and Y Y : = { V E Y: v n r , , 2 - y = 0 ) .
Denote by Yl; the set of all V E Y” such that (i) V =
U b, V, compact,
ieZ
(ii) the numbers ZE
y},
satisfy uj+ < ti for all j
E Z,
u,=sup{Rez:
zj=inf{Rez:
ZE
y}
52
1. Conorrnal Asymptotics on
R+
rj++m as j + - w . Moreover set Yd= U Y;. We have ncP E Yz for every P E !W. (iii)
o;.-+-m
as
j+m,
V
Without loss of generality we can choose the numeration of V, for VE Y$ in such a way that V ; : c { R e z > 1 / 2 - y } for j < O . Q c { R e z < 1 / 2 - y } for j r 0 , The condition (ii) says that for every j there is a strip S(aj, pj) (cf. 1.1.4. (1)) with Q c S(aj,Bj), V\ vj n S ( a j , pj) = 0. 2. Remark. V,,VlE YV implies V, + VlE YV. Moreover every V E YY can be written V = V,+ Vlfor certain V,,V, E Ydy.
(1s
The spaces YtS;'(W+) associated with V E Y V will be introduced as projective limits of spaces LJtS;V(R+)d,A E 17, related to the strip Sl;.We shall start with the assumption V n as; = 0. This will be dropped below. For s, y E W, V E Y, A E 17, Vn 3s; = 0, we denote by 13,; the space of all h E 1 ( S ; \ V) with sup llxhI(8,a< 03 for every V-excision function x and A,, E > 0, BE
Sl,
as in 1.2.1. ( 6 ) . Then 1 S y , ;d is a Frkchet space with the norm system llxhIlra, where x runs over all V excision functions and E Sl;,together with the semi-norm system induced from 1(S;\ V). Note that the meaning of depends of the type of the subscript V . In particular d;,;c l g dfor P E 3, W = ncP, where the inclusion is proper. 1;,ydis a closed subspace in the induced topology. For VE ' I r Y we have dsV.ydc M,.!X".Y(IR+).
lvd
3. Definition. Let s, y E R, V E YY,A E 17 finite, V n aSl; = 0. Z'hen Yt7eS,Y(R+),, is defined as the preimage of 1 under the weighted Mellin transform My, equipped with the induced Frichet space structure.
In view of V n aSl; = 0 we can find a curve L c Sl;\ V surrounding V n Sl;.Fix the orientation clock-wise. Then h +(t[u"], h)
:=A 2x1 /P(z) h ( z ) dz,
(1)
h E 1( C ) , represents an element in 1 ' ( V n Sl;)(independent of the choice of L). It is obvious that the associated operator (MY:Ytje4yY(R+)A+ 1 ' ( V n Sl;)
is continuous. In 1.1.5. it was described a procedure to associate with every compact set K E C a subspace of d(C\K) consisting of all g(z) = (4,f ( z , w ) ) , A E ~ ' ( K Here ). we want to perform an analogous construction in the Mellin preimage, where we consider both t - w o ( t )and t - w w ( t - l ) . For VE Y Y we have a unique decomposition V = V, + V, with V, c {Re z < 1/2 - y } , V, c (Re z > 112 - y } . Let V n 3s; = 0. The decomposition of 1.1.5. Theorem 2 1 ' ( V n Sl;)= d ' ( V , n Sl;)+ d ' ( V , n Sl;) (2)
53
1.2. Spaces with Conormal Asymptotics
is direct, i.e., every 1E d ’ ( V n Sl;)can be written as 1= a unique way. We then defme the space
YYy(R+),= {(Al,
t-”w(t))
+ A?, 1,E Se’(V, n Sl;),in
+ (,I2,t r w w ( t - l ) ) : 1 E d ’ ( V n Sl;)},
equipped with the (nuclear Frkhet) topology induced by the bijection Se’( V n Sl;)
‘Z;(R+),. (3) The cut-off function w is fixed. Clearly we have a direct decomposition of Frkchet spaces X;Y(R+),= XS,Y(R+),+ ZYy(R+),, (4) A E 3 finite (cf. also 1.2.1. (10)). The composition of (Mywith (3) gives us a projection -+
as; : XSVY(R+), %?;(It+),. -+
Assume that A
E
3 is finite and let V, V, , V2E ‘ I r Y be so that Definition 3 applies.
4. Theorem. W e have
X$Y(R+),= XS,Y(R+),+ X;Y(R+)A,
(5)
%eS,’(R+), = X;;(R+>A+ xeS,p+Li for V = Vl + V,, where the sums h o l h in the sense of Frgchet spaces.
(6)
ProoJ: YYy(R+), 4 X?“(R+), yields together with (4) a continuous operator -+ xZY(R+), + XFsY(R+)A. It is surjective because of (4) for s = m. Since it is also injective, we get the isomorphism ( 5 ) . (6) is an immediate consequence of (4) and 1.1.5. Theorem 2. 0
Now we want to extend Definition 3 to arbitrary V E Yy, A = (6, 6’) finite, and then pass to - 6, 6’ 00. We use the following “decomposition method”. First observe that V, = V n 3 can be written as V, = Vl + V2,where V, = V,, with compact sets -+
n
fez
V,, E Yo, V,, c Sl;, a(KJ)< z ( V L J 1), + and z(V,,) + 1/2 - y + 6‘ for j + 03, i = 1 , 2 . Here
a(KJ) 112 - y +
a(A) = sup {Re z: z E A } , z ( A ) = inf{Re z: z E A }
+ 6 for j
-+
-00,
(7)
for a set A E @. V, may be obtained in the form V, = V, n F,, where F, is a “fence”, consisting of a countable union of closed strips {/3,sRe z s B,, +*}, PI, < A,+ for all j , &-+1/2-y+6 for j + - m , &+1/2-y+6’ for j + m , i = l , 2 , and Sl;= Fl u F2. By construction there exist sequences Alk E 3, Ark< A for all k E IN, i = 1, 2, with Sl;= U SdyIk,V, n as;, = 0 for all k E N, i = 1, 2. Definition 3 yields ke N
Then the spaces X;ry(R+)d,k.
x;yR+),
:=
gXeSylY(R+),,k keN
is a FrCchet space which is independent of the concrete choice of the sequence Ark. The sum ( 6 ) is a FrCchet space, and it only depends on V but not on the concrete choice of the V,,, Ark. Moreover we have again the decomposition (5).
54
1. Conormal AsvmDtotics on R,
The proof is a simple consequence of 1.1.5. Theorem 2, which can be modified first to finite sums for carriers of asymptotics with a positive distance to 3s;.Then we may pass to projective limits. The obvious details will be dropped here. Note that a part of the proof consists of that of Theorem 8, below. 5. Remark. For s' 2 s, V' E V, A'
z A, we have canonical continuous embeddings
.%$"(R+)I'G .%S,'(R+), . is a nuclear Frkchet space, A E 3:
For completeness we will give a definition of %>"(lR+), for arbitrary A E !YED, a,/3 = 0, 1, although these spaces play no essential role in this book. 6.Definition. Define %>Y(R+)d for V E V ,A equipped with the topologv of the sum.
E gap, s, Y E IR, by
the formula
(9,
In particular it makes sense to talk about the spaces %>"(R+)[a,ol, %e$V"(R+)[o, r). For 6 = 6' = 0 we simply have %$y(R+)ro,ol = %%y(R+). For abbreviation we set !X",R+), = %>o(W+)d. Then %SyY(R+)d = fY%;-yv(R+)d,
T - W = {z - y :
ZE
A E 3,)
U,B = 0,
1,
lq.
Moreover define %&Y('BY(R+)d
=
@ %>Y(R;)d,
(8)
VEYV
where A ( y ) will also be omitted when A = (- m, m) ( y = 0). The notation with dot in 1.2.1. (14) indicates the discrete asymptotics. Similarly as in 1.2.1.(8) we set !Xe$VY(R+)e = !X>y(R+)(-kk, for k E N \ {0}, =%.y(R+) for k = O , 9 = ( - k , k ) . (9) Note that the space %>y(R+),V E 7 0 , can also be described as follows. For every u E %>"(R+)there exists a sequence E d ' ( K j ) ,j E Z, with compact sets
K j c { R e z < 1 / 2 - y } for j z 0 , c{Rez>1/2-y} sup{Rez: Z E K ~ } - , - ~ as j + m , inf{Rez: z e K j } + = m as j - , - m ,
for j < O ,
and a sequence of constants cj > 0, such that m
u(t)=
C
j=O
-m
(49
t-.> o ( c j t ) +
C
j = -1
(z;.,tWw>W ( ( c j t ) - ' )
converges in %;sy(R+)and u ( t ) - u ( t ) E !X",R+). Here V 2 tion we have not necessarily Kin Kj = 0 for i + j , and the
U K j . In this descripiEZ
4 are not always unique.
55
1.2. Spaces with Conormal Asymptotics
7 . Remark. The procedure of extending Definition 3 from V n aSl; = 0 for finite A E 3; to arbitrary VE V, A E 3, will often be applied in this book. For references below we shall call it the "decomposition method".
Let us now give further interpretations on the intuitive meaning of the continuous asymptotics. Let V E T ; , V = U V,, be as in Definition 1. Choose a sequence {ej}j with a,+ < ej < zj for all j E Z and a smooth curve Lj c {ej< Re z < ejsurrounding V, clock-wise. Define
This yields a sequence of continuous operators
GM,,: X;)'(R+) - + d ' ( y ) . (1 1) Let Xy= X d'(V,)be equipped with the topology of componentwise convergence. jr2
The sequence (11) then induces a continuous operator [My: X7e$VY(R+) +Xv. 8 . Theorem. The sequence 0 -+ XS,Y(R+) -+ X;"(R+)
TM Xy
+
0
(12)
is exact, s E R,V E Tl;.
ProoJ: As in the proof of 1.2.1. Theorem 9 the essential step is to show that [M,, is surjective. For notational convenience we first consider the case that V c {Rez < 1/2 - y } . Without loss of generality we may assume y = 0. Let aj E a'(V,) be an arbitrary sequence. We then prove that there exists a sequence of constants cj such that m
converges in %$(It+). As in the case with discrete asymptotics we find by definition a sequence of weight intervals AN = (aN,8;) E 3, A N + 2 A N , such that
Write u ( t ) = u N ( t ) + u k ( t ) , where u k ( t ) is the sum over all j vor which V, c {Rez > 1/2 + aN}. For V, c {Rez < 1/2 + 6,) we have for some E > 0 s u f i ciently small v,(t)
for all
= w(t)
(a,,
t-w-c)
E
x~Q(R+)
e E a[, a;].
Choose c > 0 $0 large that w ( c t ) w ( t ) = w ( c t ) . Then a ( c , E ) v j ( t )= t e w ( c t ) v j ( t )= w ( c t ) ( A j , t - " ) .
Applying 1.1.2. Lemma 6 we get
IIw(c~)( a j , t
- W ) I I ~ -0 ~ ~ ~ + ~
as c + 03, uniformly in @ E a[, a',]. This implies w ( c t ) (aj, t - " ) -0 in %7e",(IR+),,
56
1. Conormal Asymptotics on R,
as c 4m. In view of (4) the sum (13) converges in !XeS,(W+),, provided cj = c r increases sufficiently fast. Now for ci = c j we have convergence for all N. From (14) we get convergence in %“,R+). This proves the surjectivity of (M in the mentioned case. For arbitrary V we can do the same for t + 00, i.e., replace w(cjt) by w ( ( c j t ) - ’ ) in the corresponding sum for the carriers of asymptotics on the right of the weight line. This gives us the result in general. 0 It is clear that for every finite A E 3 with V n 3s; = 0 we have continuous complementary projections rm; : !XS;y(R+) +!X;y(!R+)d, as;: !XS,y(R+) + Y;(R+)d. Thus every u E 3f$VY(R+) has an asymptotic expansion m
-m
( w ‘ ( t ) = w ( t - ’ ) ) in the sense that (15) holds for every finite A E 3 with V n as; = 0. Here Aj E d’( 5 ) are unique analytic functionals applied to the holomorphic function trW. The expansion (16) explains why we talk about the continuous asymptotics. The exponents in the asymptotic term (AjZi,t-”) w are distributed over the carrier 5 with the “density” Aj.
9. Remark. We may also define continuous asymptotic types of the sort P = ((5,L J ) ) J2E, V = U V, E Vg,Lj c d’(vj)a finite-dimensional subspace. Then for A E 3, V n aSl; = 0 we can introduce the space
%%Y(Rt)A = { u E %;Y(R,)d: CjM,u E Lj for all j with Vj c S $ } . This concept extenak that of the discrete arymptotic types in a more immediate way.
We do not go into further details here. After the results of the preceding section it is spaces. clear how to arrange an adequate theory of such !X7eSpY(R+)d 10. Remark. Let P E 5%y, then %7es,(R,) is a closed subspace of every %S,(R,) with ncP C V.
Let us point out once again that the functionals A, in the asymptotic expansion (16) are uniquely determined by u. In the case V = z C P for some P E 9 % y that means that the rjk in 1.2.1. (1) are unique. On the other hand for u E %s;cY,(lR+) we get expansions of the form
for t --* 0 and t + 00, respectively. Often we are interested in the asymptotics only for t +0 or t += phism 1.1.2. (8) induces isomorphisms
m.
The isomor-
I : !XS;Y(R+)+!X$--y(R,)
for every V E YY with W = 11 - z + y : z - y E V) E Y-V. In particular the asymptotic behaviour for t +0 transforms into someone for t + m and conversely. Let us introduce carrier classes for the asymptotics at t = 0.
1.2. Spaces with Conormal Asymptotics
51
11. Definition. Denote by 2‘‘the subclass of all V E 7 r V with V n {Re z > 1/2 - y] = 0, and set 2: = 7rl;n 9 7 . For y = 0 the upper subscript will also be omitted.
The elements of X i Y ( R + )BEB: , are flat of infinite order at m and those of I X i ” ( R + )flat of infinite order at zero. We have in particular analogues of the spaces 1.2.1. (16) namely
~ B ( R=+Xg(R+), ) B E 2, (18) which are nuclear Frechet spaces. Up to now we did not say anything on an asymptotic interpretation of u E X7eSyY(R+) for arbitrary V E V. Let us give some brief comment to this case. For simplicity set y = 0. Fix a finite A = (6, 6’) E 3 and write V=Vl+V,,
V,=Vn$,
V,=Vn(@\S,).
Then XS,(R+)= Xbl(R+)+ XG2(R+),where Xi2(R+)c X;(R+)(a+&, a, - E ) for every - 6 > E > 0. Further we have %bl(R+)= X:(R+) + G I ( R + ) ( a - c , r + & )
for every E > O . In other words every U E X ; ~ ( R can + ) be written in the form u = f + g with f E Xi@+), g(t) =
w’(t) =
( A , t - ” ) w ( t ) + (A’, t - ” ) w ’ ( t ) ,
(19)
w ( t - ’ ) , for certain AEsQ’(Vn
$ n { R e z s 1/2}), A ’ E d ’ ( V n $ n { R e z z 1/21).
(20)
Thus we get the following. 12. Remark. Let Y E Yo, A = (6, 6’) E 3finite. u E X$(R+). Then there exist functionals (20) such that
u - g E Xb(R+)ca+e, d’-e,
for every
E
> 0, -6 > E .
The functionals A, A’ are not unique unless V n as, = 0. The non-uniqueness is caused by contributions carried by V n as, since ZW(R+)(a - e , a‘ + E ) c X:(R+)(a+ E , 6’ - e) for every W c as,. On the other hand we get the above notion of asymptotics when V n as, = 0. Let VE 7 r v , V = V, + V2, V, c {Rez < 1/2 - y } , V2c {Re z > 1/2 - y } . We say that V satisfies the shadow condition if T-jV, G V,, TiV2E V, for all j E N. 13. Theorem. Let Y E 7 r y satis& the shadow condition and x : R++R+ be as in 1.2.1.
Theorem 11. Then the pull-back under x induces an isomorphism x * : X;Y(R+),
for all s E R, A
E
+
X;’(w+),
(21)
cc, B = 0, 1.
Proof. As in the proof of 1.2.1. Theorem 11 we may use the decomposition (4), here when V n a S $ = 0 . Let V = V , + V , , V 1 c { R e z < 1 / 2 - y } , V 2 c { R e z > 1 / 2 - y } . Assume for simplicity that T J V 1n as$= 0 for all j E N. The case without this assumption can be treated by decomposition arguments similarly as in the above decomposition method. The simple details are left to the reader. The arguments for
58
1. Conormal Asymptotics on IR,
%SgV(R+), were already given in the proof of 1.2.1. Theorem 11. Now let 1 E d f ( V l n SI).Applying 1 to 1.2.1. (25) with respect to the variable p we immediately get the desired result. 0
14.Theorem. Let YE YY satisfy the shadow condition and let ~ , ( t - lE ) Corn( E+). Then A$,induces continuous operators Ap:%>Y(R+), for all s E R, A
E 5'@,
-
Q, E
Corn(,+)
or
%>Y(R+),
a,B = 0, 1.
ProoJ As in the proof of Theorem 13 let V n as; = 0, T-W, n = 0 (for the multiplication by p ( t ) , and TJV,n as: = 0 for p(t-l)), j~ N. Then general case then follows by decomposition arguments. We can apply again (4) and treat %$y(R+)d, ZYy(R+), separately. The first part was discussed in 1.2.1. Theorem 12. The second part follows by inserting the Taylor expansion of Q, ( t ) which yields shifts of carriers of asymptotics. In particular wd (A, t - " ) = w (4 t - " + j ) , j E N . 0 Now we want to introduce another class of spaces that will play a role in several applications. First let e E R and set
HSQ(R+) = { u E HS(Rt): tp(l- 0) u E HS((R+)}, w being a cut-off function. This is a Banach space with the norm
and different w yield equivalent norms. We have continuous embeddings
H""(]R+)
G HS,p(R+)
e 5 e'.
for s 5 $,
15. Definition. Let P E P, B E b y , y, s E R,A
E
, cc = 0, 1. Define
X$,Y(R,), = [w]%$,Y(R,), + [ l - w] HS.'(R+),
Xs;?(R+),= [ w ) %s;Y(R+),+ 11 - w ] H"6'(R+), 6' being the second component of A . All spaces are equipped with the topologies of the corresponding sums. For A = [0, 01 the spaces are independent of the asymptotic types. Then we get XS-Y(IR,)
:= [ w ] %ssy(R+)
+ [ l - w ] H"(R+).
(22)
It can easily be proved that this definition is correct, i.e. independent of the choice of w. Note that the end points of the weight intervals A have another interpretation as for the %SpY(R+),spaces. The meaning of the left end point is as before with respect to the weight y, and the right one is independent of y. For9=(-k,0], k E N w e s e t
and analogously for continuous asymptotic types.
59
1.2. Spaces with Conormal Asymptotics
Let us introduce the notation
d E !.YE*,
OL = 0,
1.
As above y will also be omitted for y = 0. Now we shall formulate a result which shows that the XSY(R+)-spaces are a natural modification of the “usual” Sobolev spaces on the half axis. Let S E R , x ( s ) = m a x { k ~ N :k < l s l - 1/2), and
.Ts = {linear span of tjw(t) for j
= 0,
. ..,,x ( s ) )
for s > 1/2, .Ts= {O] for s 5 112, o a fixed cut-off function. Further set
as= {linear span of (d/dt)Ja, for j = 0, ... , x ( s ) ) for s < - 1/2, as= {0}for s 2 - 1/2, with 6, being the Dirac measure at t = 0. 16. Theorem. Let s E R, then there are canonical isomorphisms
Xsss((R+)+ .Ts for s 2 0 , s - XSs(R+) for s s 0 ,
HS(R ) = +
{
for s 2 0 , XSs(R+)+ as for s 5 0, s
* 1/2 mod Z , * 1/2 m o d Z .
The isomorphism for HS(R+)follows by identifying distributions on R+ and that for Hi@+) by duality. The identifications are continuous (in both directions).
A proof may be found in [ R l l ] , Section 2.3.1. and in [S17], Section 2.1.2. Theorem 16 shows that the Sobolev smoothness s in HS(R+),H @ + ) plays a two-fold role close to the boundary. It describes the smoothness itself but also a weight in the sense of the totally characteristic spaces which are defined in terms of the Mellin transform. The Mellin operator calculus in Section 1.3. will show that it is natural to accept two scale parameters (s,y ) at the spaces, i.e., to distinguish s from the “conormal order” y. The operators with Mellin symbols of order y and con(s - y, y - e ) (modulo asymptotic terms which ormal order e induce shifts (s, y ) are smooth in the interior), where s is treated in an analogous manner as for lyDO-s on closed compact manifolds whereas y may be restricted by natural weight conditions which come from the positions of poles of the Mellin symbols. Another observation is that it is well-motivated also to allow spaces of the sort
-
X s y ( R ++ ) %pa@+), e
P E W ,and similarly with P replaced by VE V‘. This type of spaces was systematically used in [ R l l ] . In Section 3.3.5. we shall return to this aspect, again. 5
Schulze. Operators ens1
60
1. Conormal Asymptotics on
1.2.3.
Mellin Symbols and Actions
R,
In 1.1.1.(16) there were introduced MellinoperatorsopM(a),actingonR+.Nowwedescribe classes of Mellin symbols a for which the action transforms the discrete and the continuous conormal asymptotics in an adequate way. The notations are parallel to those for the spaces in the preceding sections. 1. Definition. Let p E R, A E 3 non-trivial, R = ((pi, z. Denote by Em$,; the space of all a(z) E d(Sj;\ ncR) which are meromorphic with poles at all pi E S; n ncR of multiplicities mi + 1, further (xa) (e + iy) E SCl(Ry) for all e E Sj; and every n,R-excision function x, and
SUP
a+e
Ck(hp) < O o l
sup
bf (h,)
< 00
a+e
for h,(y) := (xa) (e + iy), a := 112 - y + 6, B := 1/2 - y + 6’, A = (6, a’), for all E > 0 with a + E < B - E, and all j , k E N (cf. also 1.1.4. (7)). For trivial A we set for any R = Sp cl (Re z = 1/2 - y )
.
(1) Here and in the sequel we use systematically the spaces S p ( . ..), S$(...) of the standard calculus of pseudo-differential operators (lyDO-s). For instance, S”(D x RN), R S R” open ,u E R,denotes the subspace of all a (x, E) E Cm(RX RN)such that %Rfl*y &A =
E m ” 3 . Y
015 c(l + ItI)”-lpl for all multi-indices a E IN”, B E I” and (x, t)E K X RNfor every K cc D with constants c = c(a, B, K ) > 0. For the subspace of x independent a it is also custom to lD:Df a(x,
write Sp(RN).Below constant coefficients will also be indicated by the subscript denotes the subspace of all a(x, 0 (a(E)) which have const. S$(D x RN)(S&(WN)) an asymptotic expansion into elements which are positively homogeneous of degree p - j in E for 2 const. The details may be found in every textbook on lyDO-s such as [H4], [S27], [R81; . .. We tacitly consider the occurring spaces of amplitude functions in the Frfxhet topologies that follow in a natural way from the definitions (cf. also the considerations in Section 1.2.4. after Definition 3). By definition a(z) E m$,Y,has a Laurent expansion at every pj E Sj; n ncR m.
m
This induces a system of linear mappings
4: m;,.
+cm1+1,
(2)
h[a] := (vjo,. . ., vj,,,,). Moreover we have linear mappings L
%$,Y,
d (Sj; \ ncR),
+
4,x ) : m$,Y, s$(N +
(3) (4)
for every e E Sj; and every ncR-excision function x, i(e, x)a := (xa)lr. We consider the space in the weakest locally convex topology under which 42), (3), (4) are continuous. Then m$,; is a Frbchet space and
61
1.2. Spaces with Conormal Asymptotics
is a nuclear Frkchet space (for A non-trivial). Set
mr, =l&mK,;, A C 9
i.e. 'Bll,= %;,'-,, (this space is independent of y). Throughout this book we often will employ definitions of spaces that suggest natural locally convex topologies. We then usually mention properties such as the FrBchet space structure, nuclearity, continuous embeddings. The proofs, if dropped or briefly sketched, are simple exercises for the reader. We also want to define Mellin symbols that reflect the continuous asymptotics.
2. Definition. Let p, y E R, A E 3 finite, non-trivial, Y E Y,V n 3s: = 0. Denote by mt; the space of all a ( z ) E So (S; \ V) with (xu) (e + iy) E S:,(Ry) for all e E St; and every V-excision function x, and Ck(hp) a+e
<
sup
bj'(h,) < m
a+e
for h,(y) = ( x u ) (e + iy), 01 = 1/2 - y + 6, p = 1/2- y + a', A = (6, a'), for all E > 0 with 01 + E < - E, and all j , k E N. For trivial A we set again mt,; = %"Y. The emply V will be indicated by 0.
The definition leads to a system of linear mappings 1:
Em$,;
i(e,
+So(s:\v),
XI:
+
G(%)
(5) (6)
for every e E S: and every V-excision function x. We then consider the space 'iOtt3 in the weakest locally convex topology for which all mappings (5), ( 6 ) are continu-
ous. It is then a Frkchet space and
% , y * v = lim a$,; c
lr€R
is a nuclear Frkchet space (for non-trivial A ) . Note that when L c S: is a curve surrounding S; n V then h-+(([a],h):=-L 2ni l a ( z ) h ( z ) d z ,
h E So(@), represents an element in Se'(S2 n V). The mapping (: %$,; -+So'(S: n V)
is then continuous. The definition of %k;, d E 3 finite, can be extended to arbitrary V E Y (i.e. without the condition V n as: = 0) by the same scheme as in the definition of 9t7eS,v(R+)A in Section 1.2.2.
62
1. Conomal Asymptotics on R,
First we set = V n 4,i = 1,2. Then by Definition 2 we get the spaces "np,rp, , i = 1 , 2 , k E N. The projective limits for k +00 yield the corresponding spaces with subscripts A
= (6,
6'). The next step is to set
in the topology of the sum and finally
mg = ag,(-m,m) = lglmt; A€9
in the topology of the projective limit. This space is independent of y. We could also define spaces a;,;, for general A E Yes, a,B = 0,1. The details will be dropped. In this book we consider Mellin symbols mainly for infinite A or trivial A . The calculus can of course be arranged with spaces of Mellin symbols that are only given in adequate weight strips according to the associated conormal orders (explained below in Section 1.3.1.). This would yield a proper generalization, though obvious. So we only discuss the mentioned simpler case. ' instead of Another simple generalization concerns non-classical symbols, i.e. S S:!. This is also left to the reader as an exercise. Define sg(a) =
V when U E ~ X ; , Y E Y,
sg'(a)=R
when
Observe that s g ( P a ) = T-Ysg(a),
U E ~ ; ,R
E%.
y E R ,
and the same with dot. Here ( P a ) (z) = a ( z + y ) . Moreover
alr,,*-E 5 W . Y for sg(a) n r,,,-
=0.
3 . Remark. Point out once again that we have applied the "decomposition method", here in a variant for Mellin .rymbols, cf: 1.2.2. Remark I . The starting point was V E Y, A E 3 finite, with V n 3s: = 0 . This extends then to arbitrary V E Y, A E 3.
The scheme will be generalized below in many variants and referred to as the decomposition method for Mellin symbols. For abbreviation we also use the notations
4. Proposition. % is ? a closed I$ subspace of mm.,. Remember that ncR E Yd. If a E %$, V E Yd, V =
U V, (cf. 1.2.2. Definition 1) jsZ
for every V, we get an analytic functional
63
1.2. Spaces with Conormal Asymptotics
h E SQ (C), Lj being a curve surrounding V, and no further components of V. Consider
in the topology of the component-wise convergence. Denote by
I:
Sm’,’X,
the continuous operator defined by the sequence of the G. The discrete analogue of X y for R E R is the space of sequences of complex numbers XR={ck: Osksmj, ~ E Z ) . In this case
I
may be interpreted as a continuous operator
I : rm:
+XR.
5. Theorem. Let p E R, V E Yd,V = U V,. Then the sequence
o+m~+w’,J+Xv+o is exact. I f R E R then 0 +m$
+m$ J+ X R 0‘
is aho exact. The proof of this result is analogous to that of 1.2.2. Theorem 8. So it will be dropped. As a corollary of the proof together with extra considerations based on the decomposition method it follows 6. Theorem. For evely p E R,R E R, V E Y we have
n:=nr,+m,m, Em’, = mr,+ a;”. Moreover
w’, =w r,+
for
V = V, + V,,
V , , v, E Y.
The same relations also hold with the subscripts A E 3:
Another obvious property is 7. Proposition.
a E TD1:, a E %It’,,
* ab E,;% :: b E BY,* ab E i lX;:> b
for every p, v E R, R, Q E R, V, W EY. Moreover a E := ii(1- .T) E ‘5R& (wb) with
R * = {(I- F j y V*={l-z:
mj)}jez
ZEVj.
for R = {bj,m j > } j e ~ i
(n’,)implies
a*(z)
64
1. Conormal Asymptotics on R,
For references below we shall briefly define certain algebras of sequences. Let &! be an algebra of functions a ( z ) which are holomorphic in C \ V, where V = V ( a ) has the property that V n { a5 Re z 5 @}is compact for all a,B E R.Sums and products of a, b E &! are defined in C \ ( V ( a )u V ( b ) ) .Then X &! is an algebra with jaN the product
h = {hj}j N, f
=
E
X &!. The sum
jeN
h + f is defined component-wise. We
will often employ that sort of algebras, also in the case that A2 consists of U-valued functions, U being an algebra which is a locally convex vector space. Incidentally we also use the product
In order to have a convenient notation let us introduce. 8. Definition. # M is called the Mellin translation product; #w the conjugate Mellin
translation product. %Remark. h ( z ) = ( h j ( z ) ) j e N ~X J n i m p l i e s h ’ ( z ) : = ( h j ( l - r ) ] j ,X ~ ~M . jEN
JEN
Remember that the conjugate Mellin symbols are associated with the transform ( Z u ) ( t ) = t - l u ( t - ] ) , cf. 1.1.1.(11),(12). Let h = { h j }E X A2 and set jEN
W,(h) = { z E C : ho(z)= 0 , dist (z, V(ho))2 E } ,
E
> 0.
In the case of U-valued ho the condition ho(z)= 0 is to be replaced by ho(z) is not invertible in U. 10. Proposition. Let 1 := {1,0,0, have h 0 E &! whenever
...} E &! and assume
that for h = {hi} E
X
A we
jEN
W J h ) n { a 5 Re z 5 p } isfinite for all a, @ E R,E > 0. Then there is an f E X &! with h # M f = f jeN an f ’ E X A2 with h #w f ’ =f ’ # w h = 1.
#M
h = 1 and
jaN
Proof. f o ( z ):= h;’(z) is defined on C \ (V(ho)u { z : ho(z)= 0}), and by the condition f o E &!. Then it suffices to solve
1 hj(z - k ) f k ( z )= O
for 12 1.
j + k= 1
Assume that fo, .. . , fr-
A ( z ) =fo(z - 0
E &!
c
is already determined. Then
j+k=l kL1-I
hj(z - k) h ( z ) E
Clearly f = {A}is also a left inverse of h. The conclusions for #w are analogous. 0
65
1.2. Spaces with Conormal Asymptotics
As mentioned in 1.1.2. the Mellin symbols give rise to Mellin operaforsop,(a), cf. also 1.1.2. (15). We shall admit Mellin symbols a E 'Bm'.O and more generally a(t, r,
z) E Cm(R+XR,,
Clearly Cm(R+X R , , Define
S$(R+X R ,
'BKO)
op,(a)u(t)=
z = 1/2 + iy.
"tKo),
it-(l/2+iy)a
X
{Re z = 1/2}), cf. (1).
[
(t, r,++ii)
-m
d y = (2n)-' dy,
u E Ct(R+).
drl
/ r l / 2 + i y u ( r ) T
dy,
The operators of this form are an analogue of the pseudo-differential operators. The role of the Fourier transform now plays the Mellin transform. For this reason we also speak about Mellin lyDO-s. The Mellin lyDO-s can be compared with the usual ones in various ways. First we have. 11. Proposition. Let a(t, r, z ) E Cm(R+XR+,Zm"o), p E R. Then op,(a) is an operator in L$(R+) with a complete symbol b ( t , z) = a(f, t, -i&) modSE1-'(R+). An analogous statement holds for the version with non-classical symbols. ProoJ The following calculations are formal for the moment and will be justified in terms of oscillatory integrals in the following section. We have for u E Ct(R+)
-m
m m
Now set x = log t, x'
= log t,
and denote by x : R +R+ the diffeomorphism defined m
by
x(x) = t = ex. Then
( x * u ) (x') = u(ex'),
m
op,(a) u ( t ) = (x*)-'
ei(X-x')r -m
-m
X Z(ex, ex', z) ( x * u ) (x') dx' dz with C(t, r, z) = a(t, r, 112 - iz) F2. Thus op,(a) = (x*)-IAx*, where A is a lyDO on R with the amplitude function Z(ex, e", z). Since the push-forward of a (classical) lyDO under a diffeomorphism is a (classical) lyDO, again, we get the first part of the assertion. It remains to express the complete symbol of op,(a). A standard formula (cf., e.g., [S27]) says that A has the complete symbol Z(ex, ex, z) mod S$- '(R). Thus
b (t, z)
= a ( f , t, - ix'(x)
z) mod S:l- '(R+),
where x'(x) = ex = t. The latter relation is a consequence of the behaviour of complete symbols of lyDO-s under coordinate transformations. The analogous arguments apply for non-classical symbols. 0 12. Remark. It can be proved that every A E Lg,(R+) is of the form A = opy(a) for some a(t, r, z) E Cm(R+X R + ,5lWo). An analogous result is true of Lp(R+), where EmKO t to be replaced by Sp on {Re z = 1/2}.
66
1. Conormal Asymptotics on R,
Let us introduce the notation (13) af(op,(a)) (r, z) = a @ ) ( t ,t, -iz), where a(#)(t, r, -iz) denotes the homogeneous principal part of a(t, r, -iz) of order p. Incidentally we use the notation a:(op,(a))(t, z) = a ( ” ) ( t ,t, -itz), cf. 1.3.3. For the moment this plays no role.
(14)
13. Definition. Denote by S(p)(TzR+\O) the space of all p ( t , t)E Cm(R+x(R\O)) for which alsop(t-’, z ) C”(R+X(R\O)) ~ andp(1,Az) = A @ p ( f ,z) f o r a N A E R + .
We consider S(’)(T;R+\O) in the Frkchet topology that immediately follows by the definition. The notation T;R+ indicates the “compressed cotangent bundle” of R+ (compressed with respect to t = 0, t = m, cf. [M9]). This interpretation is not essential for the moment. We only have to keep in mind that the homogeneous principal symbol of op,(a) in the sense of L;,(R+) is p ( t , tz) if p ( t , z) = a:(op,(a))(r,
z)E S ( ” ( T ~ R + \ O ) .
(15)
From Proposition 11 it is obvious that (15) holds when a(t, r, z ) E C,”(R+ XR,,WO).
Let us discuss another relation between Mellin yDO-s and usual yDO-s. Define the space SfO,,,,(R2 X R) = { a ( t , r, z)E S;,(RZx R): SUPPSl a E [O, C l l x [O, czl, c1, CZ > 01 . (16) Set op,(a) = r+ op(a)e+, op(a) being the standard yDO action on IR with the amplitude function a, e+ the operator of extension by zero to R-, r+ the operator of restriction to IR+. Every a ( t , r, z)E Sfo,,,,(RZX IR) can also be interpreted as an eleand then we have an associated Mellin yDO op,(a). ment in C,”(a+Xa+, Bfl~o) 14. Theorem. There is a buection
such that for all u E C,”(R+)
op,(a) u ( t ) = op,(b) u ( t )
with
b = ya
15. Lemma. The function
(: ) (9
F(t, r ) = - - 1 log-’ - ,
f,
r E R+
belongs to C”(R+x R+).
Proof. It suffices to show that the function f ( x ) = x log-’(l + x) is C” close to x = O , since the assertion outside a neighbourhood of x = O is obvious. Set g ( x ) = log(1 + x ) and consider the Taylor expansion
61
1.2. Spaces with Conormal Asymptotics
m
Set a1= ( - l ) ' ( l + Z)-l, then f ( x ) has the Taylor expansion m
f(x)-
z p k X k
for x + O ,
k=O
where
B1 = - a 1 , p2 = -a2 + a : , p3 = -a3 - a: + 2a1a2,...
In particular we see that ( , fa )k( o ) = k ! p k ,
kE-]N. 0
(18) Proof of Theorem 14. Let a(?,r, 1 / 2 + iy) E S,$),,,(R* X R) and define op,(a) by the formula (12). Then
with ao(t,r, z)
:= a(7, r,
-iz). From the definition of F(t, r ) we obtain
log t - log r = ( t - r ) F-'(t, r ) r - l . With A := F - ' ( f , r ) f l z we get op,(a) u ( t ) =
r-l
dz = F d A i.e.,
ei('-r)db(t,r, A) u ( r ) d r d l
with b ( t , r, A) = F(t, r ) ao(t,r, rF(t, r ) A ) .
Let us show that b(t, r, A ) E S,$),JR2x R). In view of Lemma 15 we already know that b E Sp(W+ XR+XR).It is obvious that al(t, r, z) = F(t, r ) ao(t, r, z) belongs to S@(R* X R), since a. vanishes of infinite order at t = r = 0. Then al also vanishes of infinite order at t = r = 0. It is clear that ID: D', D: al(6 r, z)I 5
pklm(t9
r ) (1 + IzI>'-
for every k, I,m E N with certain p k / m E Cm(Rz)that vanish of infinite order at t = r = 0. By straightforward calculations we see that
ID: D', D? b(t, r, All 5 p;ilm(t,r ) ( 1 + IrF(t, r)A()"-'" with pih being of analogous behaviour as
e-'= ( c + 1 c2 1,
le-'AI>(c+ IAl)-ls1
5 ( c + le-'Al)(c+
IAl)-'se-'
pk[m.
From the elementary inequalities
for p t 1 , for O < es 1 ,
A E R , e E R, , we then obtain the symbol estimates I D : D ' , D ? ~ ( ~ , ~5 , A@klm(trr)(l+ )I 1A1)~--"
68
1. Conormal AsvmDtotics on R,
for all k,I , m E N with &I,, being of analogous behaviour as Q)kl,,,. I n other words we have obtained the mapping y for which (17) holds. The calculations can also be carried out in the reverse direction, i.e., we get our assertion. 0 Remember that in 1.1.2.(15) we have established the continuity
%yR+)+%s-qR+) for a E Em”,o, s E R.
(19)
op,(a):
16. Proposition. Let a ( t , r, z) E Corn@+ x 8 + ,Em@*o). Then op,(a) operators (19) for all s E R.
induces continuous
ProoJ Let us first assume that u is independent of r. Write a ( t , z) in the form a ( t , Z) = wl(t) ao(z) + tal(t, z), w1 being a cut-off function, w1 = 1 on supp a, ao(z) = a(0, z), and al(t, z) E Corn@+, ‘3Rfi-O) according to the Taylor formula. We already know that op,(uo) has the desired continuity property. 1.1.2. Remark3 shows that it suffices to consider a2(t,z) = t u l ( f ,z). Our operator is then of the form 1 Au(r)=t - 2 a 2 ( t , ~ ) u ”dz, (~) 2x1 rli2 u E Ct(R+), u“ = Mu.Applying the Mellin transform on both sides we obtain
I
m
1
11
MAu(w)=t W - l t-zuaz(t,z) G(z) d z d t 2x1 o r1,2
1 Here
b(w - z, z) i ( z ) dz.
m
dt b(t,z)=[ t c u z ( t , z ) t , J = w - z , Re(=0. 0
Let us show that for every N
EN
there is a constant
lb(c,z)l 5 eN(1 + IcI)-N(l + lzl)’ on Re 5 = 0, Re z = 1/2. We have m
m
=
rllt ~ - 1 ( - r ~ ) a 2 ( t , ~ ) d t 0
m
N
=... = C N j t C - l ( - r % ) a2(t,z)dt, 0
eN= eN(az)such that (20)
69
1.2. Spaces with Conormal Asymptotics
i.e.,
m
N
P N ( O= 1 ajP, j=1 such a way that for a constant cN > 0 for any polynomial
aj= 1 . For every N we can choose PN in
IpN((-’)Id C N ( l + on Re (= 0. We then obtain from (21)
From (20) it follows that K ( z , w ) := b(w - z,z) (1 + J z J ) - S ( l + tw1y-9
satisfies for N > lp - SI + 1 with some constant cN> 0, independent of z, w E r,,,(the inteIndeed, we have grals are taken over rIl2).
It was used that (1 + loll) (1 + IBI)-’ d 1 + 101that
cf. 1.1.2.(6), II.(Js= I(.1 3Ls(R+).
(f,u ) =~
For f = A u and v E Ct(R+)we get
MAu(w) 6(w) dw r112
for a,B E R. Now remember
70
1. Conormal Asymptotics on R,
Thus
5 ( 2 n ) - ’ I ~ I K ( z , w ) ( l+
5 (270-l x
{ss
I~I)”’“~)(~+IZI)~U”(~)I(~~II~~I
IK(z, w)I ((1 + IwlY”sl~(w)l)21dzlIdwl}”*
{ISIK(z,w)l((l+ Izl)”lJ(z)l)21dzlldwl}1~2
5 ( 2 W ’ C~eNIIVll,,-sll~lls.
From (22) we then obtain
IIA4ls-# 5 ~ C N e N I I ~ I I s with some fixed C and hence the desired continuity of A in the case of r independent Mellin symbols. In particular we see that
llAIIS,s-r := IIAIle(~:W - P ) 5 e(a) for the original operator A = op,(a) with a constant @(a)with e(a)+O as a + 0 in C;(R+, W~o), s fixed. For proving the continuity for Mellin symbols that also depend on r we first remind of the following general result. 17. Proposition. Let E , F be Frichet spaces, then every u E E@,F can be written as an absolutely converging sum m
u=
C Aiei@h, i= 1
where
C [Ail < m and ei+O
in E , f i + O
in F as i-,
m
.
A proof may be found for instance in [S3],Chapter 111, 5 7. The symbol a(t, r, z ) can be regarded as an element of
c;([o,B1,
rmP*O)
@ZC,”([O,B’l)
for certain @,/I’ > 0. Applying Proposition 17 we get a representation m
a(t, r, z > =
C Aiei(t, z)h(r),
i-0
where eiE C ~ ( [ O , B ] , ’ 9 V o )C;([O,B’]) ,h~ tend tozerofor i+m and xlAil< m . The first part of the proof shows that op, induces a continuous operator op,:
c;([o,B],m q + r e ( 3 P ( R + ) , sles-JJ(w+)).
From 1.1.2. Proposition 7 it follows that the multiplication by f~ C;([O,B’]> induces a continuous operator Corn( [O,B’I)
92 (sle.(R+),XS@+) 1.
+
71
1.2. Spaces with Conormal Asyrnptotics
It suffices now to establish the convergence of w
OP,(~)=
C
Ai
op,(ei)
~
f
,
i=O
in S?(!Xs(R+),!Xss-P(R+)),which follows from
c 14 m
Ilop,(a)
IloPM(e1) 1Is.s - P l l ~ f , I l S , S
5
IIs.s-/i
1-0
c
and llopy(el)Ils,s-P+O, l l ~ ~ , l l s , s - + Ofor i + m and 14 < m. Now let us return to the symbol spaces EDI;, Ern;, V E V ,R
E Ao.
18. Proposition. Let ,u E R and a ( z ) E W;, V E Yo. Then op,(a) induces continuous operators O P M ( ~ ):
x;(R+),
for every B E V", s E R, A
-+
E
xi;"@+)^
3. If a ( z ) E Fm; then op,(a)
(23) induces continuous opera-
tOrS
OPM(~ : )%",(a+>, -*X>i'i;lrR(R+)~ foreueryPEAo, S E R ,A
E
~
ProoJ: The application of op,(a) is defined in the Mellin image by multiplying M u ( z ) by a ( z ) . The properties of the spaces XSg(R+),, !%>((a+), are characterized in terms of the Mellin image by 1.2.2. Definition 3 and the decomposition method. For the discrete asymptotics our statement is then completely obvious. For the continuous asymptotics it is obvious as well under the condition that B n as, = V n as, = 0. It is now an exercise in terms of the decomposition method and sums and projective limits of spaces to extend this to arbitrary B, V. The simple details are left to the reader. Let us only sketch some elements. By decomposing B and using the sequence of weight intervals A l k we may reduce the assertion to B, n as,, = 0. For fixed i and A l k we decompose a ( z ) in the form a,(z) + az(z), where sg (al) n as,, = 0 sg (a2) n S,,= 0, Jlk= (alk+ E , S:, - E ) for A l k = (alk,a:,), E > 0 fixed and sufficiently small, For a l we already know how to proceed whereas for a2 we apply once again a decomposition az = a; + a; where sg(a',) = sg(a2) n F', s g ( a 3 = sg(az) n F" with appropriate fences of strips, cf. the constructions in Section 1.2.2. It is then clear that both for a; and a; we get continuous mappings of !X;(R+),,, in !Xi;P(R+),;k and !X;-P(R+),f, where A;,(AYk) runs over a system of weight intervals tending to At,. Then we get our result by passing to projective limits. 0 Let us discuss now an example (cf. also 1.1.2.(16)). Denote by Diff*((R+)the space of all differential operators A on R+ of order p with coefficients in Cm(R+)and P
j=O
72
1. Conormal Asymptotics on R,
with certain aj, bk E Cm(E+)and a fixed cut-off function w . (The definition is independent of the choice of the cut-off function w . ) Let a,(?>
- c tkaF1, b j ( t ) - c tkby1 m
m
k=O
k=O
be the Taylor expansion of the coefficients aj, bj at t = 0. Define
c
=
a;?(A) (z) =
ajk'zj,
a,.(A)
= {flifk(A)}k€N>
Then
an;l(A), a&i)
bikl(l - z)/
j=O
]=O
@M(A)
c P
Ir
o i k ( A ) (z)
E
= {Gf%Q}k€N.
rnz
for all k and
aL(A)-' E Em,@,
&#(A)-'
E
rn,.
for certain finite asymptotic types R , Q E 2. It is a simple exercise to check the following
19. Proposition. Every A E Diff"(R+) induces continuous operators A : %s(R+)+%s-@(R+), A : %",(R+)+2;; "(R,)
for every PI E R' with some Pz E 2 O , P25 PI,and every s E R.
Diffm(R+)=
u
Diff@(R+)
Ir
is an algebra filtered by the orders, and a, ( A m = ahfu1# M ahf(B),
a,.(AB)
=
a&)
#M' a,.(B)
for A , B E Diffm(R+). Clearly A E Difffl(R+) also induces continuous operators
A : %;(R+)+2;-r(R+),
SER, VE
TO.
In Section 1.3.1. we examine the Fredholm property of (24). 1.2.4.
Mellin Pseudo-Differential Operators
For our purposes below we have to examine the Mellin 1pD0-s in more detail. We shall discuss for the moment operators on R, . An analogous calculus will be developed below also on (R,)" and with operator-valued symbols, cf. Sections 2.1.3., 2.1.4. and SCHULZE[S15]. Remember that for p ( t , r, z ) E Cm(lR+X R+,Em'.o) the operator A = op&) may formally be written in the form
13
1.2. Spaces with Conormal Asymptotics m
m
where a ( t , r , y ) : = p ( t , r , 112 + iy) E S:,(R+xR+xR). We shall interprete (1) in the sense of oscillatory integrals. Set E ( t , r,y) = ( ~ ) 1 ’ 2 + i y .
N E IN.For
Q
E
S 2 ( R t x R + XR), u E C;(Rt) we get
m
=
m
.f ./ E ( t , r , ~ {)( ‘ L ) N a ( tr, , ~~)
(f ’4) drdy.
(3)
0
-m
In virtue of (‘L)NSfl2 Sfl-Nthe right hand side of (3) makes sense for every a E S’ provided ,u - N 5 2. Now let 8(y) E C,”(R), 8 ( y ) = 1 near y = 0. Then we have the following 1. Proposition. Let a E P ( R + XR, XR), u E C;(R+), and set m
m
E > 0. Then A,u(t)-* A u ( t ) us E + O for Au being defined by the right hand side of (3), p-N~-2,t~W+fixed.
Proof Integrating by parts yields m
A,u(t) =
m
./ ./ E ( t , r , ~ I)( ‘ L > N 8 ( ~~ ~( >tr , ~ )
~ ( 7 r-9 )
-m
drdy.
0
Choosing N sufficiently large, by the Lebesgue theorem we may pass on the right to the limit E+O which exists and equals the right hand side of (3). 0 From this it follows that the limit does not depend on N for N large enough and on the concrete choice of the cut-off function 8. The extension of A to arbitrary a E 9 preserves the usual rules of treating integrals such as integration by parts. They will be used without further comments and may be justified by considerations analogous to the proof of Proposition 1. In view of 1.2.3. Proposition4 we know that the Mellin yDO-s are another representation of standard yDO-s. ( 1 ) only represents a specific operator Convention (the “Mellin operator convention”), which is a rule to pass from an amplitude func-
1. Conormal Asymptotics on R,
14
tion to an operator, different from that with the Fourier transform. All essential aspects of standard vDO-s can be developed with the Mellin convention. 2. Remark. Mellin wD0-s may be associated with any weight y, i.e. with M yinstead of M 1.1.1. Propositionl). In other words ifwe replace E ( t , r , y ) by
= Mo
(cf:
we get another calculus, i.e. with another operator convention. It is completely analogous to the case y=o.
In the context of Mellin vDO-s we also want to control the properties including t = 0, t = OJ.Then we need assumptions on the amplitude functions close to t = 0, t=m.
. .
for ail k ; l , j E N,0 s t , r 2 T, T E R+,Y E R,with constants c = c ( k , l , j , T ) > 0 (ii) a'(t, r , y ) = a ( t - l , r-1, - y ) E Cm(R+xR, xR), and a' satisfies analogous estimates as a . Scl(R+X R+X R) denotes the subspace of classical symbols of S'@+ X R+X R). Moreover S$(E+XR+XR) denotes the space of all a E S"(R+xR+xR)for which t - N o ( t ) r - M o ( r ) a ( t r, , y ) , t - N o ( t ) r - M o ( r ) a ' ( t ,r , y ) E SQ+XR+XR) for all N, M E N , o a cut-off function, and S$,c,(R+x R+x R) = S$(R+x 8, x R) n s~,(R+ x R+x R). In an analogous sense we use the notations S ( R +XR), ... for symbols depending only on ( t , y ) .Further let Cr(R+) be the space of a l l q ( 1 ) E Cm(R+)with p(t-l) E cm(R+). S i m i Z a r l y w e u s e n o t a t i o n s s u c h a s ~ ~ ( E + c:(@, ~ ~ + ) , ~ ) , =p 1 , 2 , E being a topological vector space, and so on.
Note that for (Iu)( 2 ) = t-' u ( t - ' ) the operator A' = IAI-' is of analogous form as (1) with the amplitude function a'. We will often assume the amplitude functions to have bounded supports in ( t , r ) and t , respectively, since by conjugating with I we X R) defined by can interchange the role of 0 and OJ.Remember that S~o~,cl(Rz 1.2.3.(16) can be identified with the subspace of all a ( t , r , y ) E S$,cl@+ X F + XR) with bounded supports in ( t , r ) . The spaces P ( R +XR+X R), 9 ( R +X R+x R), S$(R+xR+R) and also those with one factor R, (R+) are FrBchet spaces with systems of semi-norms given by the best constants in the corresponding estimates. The subspaces of classical symbols may also be endowed with adequate FrBchet structures (stronger than the induced ones). Let us explain this, for instance, for S;,(R+XR). If p ( t , y ) belongs to this space we have an asymptotic expansion m
p(t,y)
- z ~ ( , - ~ ~ ( with t , y uniquep(,-J,E ) Cm(R+X(R\O)),homogeneous i n y of j=O
order p - j . Fix k E N . Then we have the semi-norms of pb,, ...,p ( , - k ) in Cm(R+x (R \ 0)) that may be regarded as semi-norms over S$(R+XR), and further k
p r - j in S,- k-l(R+XR), also interpreted as semi-norms
the semi-norms of p j=O
75
1.2. Spaces with Conormal Asymptotics
over S$(R+x R), where p,, - E Cm(R+X R), p,, -i= p(,,? for lyl norms for all k yield a Frhchet structure on S:l(R+ X R+X R).
e 1. These semi-
4. Proposition. The spaces S p ( R + X a + X R ) (S!!l(R+XR+XR), S $ ( 2 i + x R + X R ) , S$,cl(R+ X R+X R)) are closed under asymptotic sums in the sense that when aj E SPj, pj + - m as j + m (and pj = p - j for classical symbols) we find an a E 9 in the correN
sponding class, p
= max { p j ) , so
that the order of a -
aj tends
to - m as N+
m,
and
j=O
a ir unique modulo S”of the corresponding class. The proof is practically the same as for the standard symbol spacesSP(R+ x R+X R ) . The asymptotic sum a
-
m
aj may be obtained as convergent series j=O
with a sequence of constants cj > 0 tending to 00 sufficiently fast as j + m, x being an excision function in the covariable y . As for standard pDO-s we may calculate the distributional kernels of the operators in terms of the amplitude functions. If we have in (1) an amplitude function a ( t , r , y ) E S ~ ( R + x R + x R p) , s - 2 , then m
0
with m
K A ( t ,r, e ) =
e - 1 / 2 - i Y a (rt , y ) dy,
(4)
-m
m
Note that for every N E W there is a p ( N ) (to be chosen sufficiently negative) such E C ~ ( R + X R +when ) a(t,r,y)ES~(M(R+XW+XR).
m
This extends the formula (4) to arbitrary orders of a. Let B be another Mellin pDO with the kernel K B 6
Schulze, Operators eagl.
76
1. Conormal Asymptotics on R+ m
0
with m
m
This is only a formal calculation. We do not automatically know whether the integral over e converges. 5. Definition. A Mellin yD0 (1) is called properly supported i f K A ( f , r, t l r ) has a proper support in R+xR+ (i.e,, a;’P n supp K A is compact for every compact set P c IR,, 3ti: R, X R++R+ being the canonical projection to the ithfactor, i = 1,2).
It is obvious that (7) makes sense when A or B is properly supported. 6. Definition. Denote by M L I ( R + ) ( M L $ ( R + ) ML’(R+), , ML:l(R+)) the space of all operators A + G, where A is a properly supported Mellin yD0 with an amplitude function a ( t , r, y ) E P(R+ x R, x R) (s$(R+ x R+x R), S@+ x R+x R), s$(R+ x R+x R)) and G an operator with kernel in Cm(R+XlR,), whereasfor R, the operator G is required
to induce continuous mappings
G: sles(R+)+ !?tern@+), G* : XS(R+)+!?tm(W+)
(8)
for all s E R . It can be proved that
ML”R+)
= LP(R+),
ML$(R+)= L$(R+),
cf. also 1.2.3. Remark 12. The notation with M is used to emphasize the Mellin operator convention in (1). The operators in M J ~ - ~ ( R are+the ) standard smoothing operators over R+ whereas M L - - ( R + ) consists of all G with (8). This is a consequence of 1.2.3. Proposition 16. It is clear that
ML#(R+)c MLP(R+) and the same with the subscripts cl. 7. Proposition. Let A E ML’(R+) be properly supported. Then A induces continuous op-
erators A : C;(R+)+ C,”(R+), A : Cm(lR+)-$ Cm(R+).
1.2. Spaces with Conormal Asymptotics
77
Let us reformulate (4) in another way. By inserting
and integrating by parts we get m
for every N E W.
(&)a E S"-N N
Since
for a E P , the integral on the right of (9) becomes
smoother the larger N is. In view of (log e)-NE Cm(R+\{l}) we thus obtain the first part of the following 8. Proposition. For every a ( t , r , y ) E S ~ ( R + X R + X we R ) have
R) foreveryyEC,"(R+),y = 1 closetog= 1. I f a ( t , r , y ) E S ~ ( R + x W + xweobtain KA(t,r, e)lQ+l E cm*(R+xR+x(R+\OI)),
(1 - y ( e ) ) K A ( t , r , e )E c S ~ + X R + , P ( R + ) ) . Conversely let Km(t,r, e ) E c ~ ( RX + R +, P(R+)) (Cz(IR+xii,, P(E+)). Then ( 5 ) yielak an amplitude function a ( t , r , y ) in S " ( R + X R + X R()S m ( R + X R + X RAnalo)). gous statements hold for r- (t-) independent amplitude functions (the corresponding variable then disappears in the kernek). Remember that the space P(R+)was introduced in 1.1.2. Remark 9. Proof The space W ( R + )is defined as the inverse image under M of (1 + Iy12)-d2 L2(Rez = 112). Here {Rez = 112) is identified with Ry.Thus for every s there is an N such that the integral on the right of (9) belongs to .!XS(R+).If we multiply by (1 - y @ ) )(log e)-Nwe get again a function in XS(R+).The same is true of the kernel for ( a l a y ) N a ( tr, , y ) . Thus (9) shows that even (1 - y ( e ) )l o g N e . K ( t , r , e ) EC " ( R + x R + ,XS(R+))
for all s and all N . This yields (11). It is obvious that smoothness in t , r up to t = r = 0 is preserved. The converse is a simple consequence of the formulas (41, (5). 0 9. Corollary. Every Mellin y D 0 ( 9 ) with an amplitude function a ( t , r, y ) E P(R+x R+ x R) (S$(R+X 8, X R)) belongs to MLp(R+) (MLf,(R+)).The analogous statement holak for the objects over R+.
For notational convenience we will consider for a moment the symbols in the form a ( t , iy). Let us also work, for instance, with ]Ti+3t (the constructions also hold of course for the open half axis). 6'
78
1. Conormal Asymptotics on R,
10. Proposition. For every a(?,r, iy) E S@+ al(t, r , z ) E Cm(R+x E+, SQ (C)) so that
XR+XR)
there exists a function
a(t,r,iy)- al(t,r,z)1~,,2-yESI(-1(QTi+~lR+~]R)
for every y E R. An analogous statement holdr for r- (or tions as well as for classical ones.
t-)
(12)
independent amplitude func-
v(e) as in Proposition 8. From the kernel
ProoJ Fix a function
m
we can pass to another symbol m
which is an entire function in z and C” of t , r E R,. For y = 0 we obtain (12) from XR, X R). For arbitrary y the differProposition 8, the difference is even in S-m(R+ XR, X R ) ence (12) equals mod
sm(a+
m
I
m
de e-y)v(e)KA(t,r,e)-=
e1/2+iy(l-
Q
0
I
o
e1/2+iyq(e)
de logeKA(t,r,e),
with @(e):= (log e ) - l ( l - p )v ( e )E Corn@+). By formula (9) we know that loge KA is the kernel belonging to slay a E S h - l . This immediately yields the assertion. 0 We shall now develop some elements of the symbolic calculus for Mellin yDO-s. The results will be obtained first for the operators over R+ and then in the corresponding form for R+. Let AEMLP((IR+)be properly supported. Then A may be applied to the function f y (t ) = t-(1’2
+
iY),
cf. Proposition 7. Set
Then m
m
y
ER
fixed,
19
1.2. Spaces with Conormal Asymptotics
11. Definition. A n a ( t , y ) E Sfi(R+xR)with A - op,(a) E M L - - ( R + ) is called a complete Mellin symbol of A E MLP(R+). Similarly an a ( t , y ) E P ( R +XR) with A - op,(a) E ML-"(R+) is called a complete Mellin symbol of A E MLP(R+). The construction of Proposition 10 gives many complete symbols if we have a given one.
12. Proposition. Let A E MLP(R+) be properly supported. Then (13) is a complete symbol of A. ProoJ We may assume that A is written in the form (1). Using 1.1.1.(9) we can write op,(a) u = M-IaMu
=
O-'F1aFFOu,
0 := O0, aF(x,x ' , y ) = a(eX,ex', -y). f;'(t)
0-l = 0-l
It is clear that (Ofy) (x) = e-ixY and
eixY. Thus
f;' op,(a) f y = 0-l {eixYF1aFFe-ix'Y}.
(15)
We get by the standard formula for complete symbols of usual wDO-s that the function on the right of (15) is a symbol in our class which was the point to be proved, since we already have (14). 0 Clearly every A E MLfi(R+)has a complete symbol aA(t,y), since it equals a properly supported operator mod ML-"(R+). Another consequence of the standard calculus of VDO-s is 13. Remark. If a i ( t , y ) are Complete symbols of A E ML'(R+), - a 2 ( t , y )E S " ( R + x R ) . Thus A + u A ( t , y ) induces an isomorphism
i = 1,2, then a , ( t , y )
ML"(R+)/ML-"(R+) P ( R +XR)/S"(R+ XR), and the same with subscripts cl.
14. Proposition. Let a ( t , r , y ) E S#(R+x R +XR) be of the form
aN E P.Then the operator (1) can be represented by an amplitude function in Sfi- N. ProoJ We obviously have m
and
(-f log-+)
E S f i - N .Thus it suffices to show that
;(
-
C y R , XR,).
80
1. Conormal Asymptotics on R,
First it is clear that this function is C” outside the diagonal of R+x R, . Set f ( x ) = x-N logN(1 + x ) ,
1
1 + x = -.
t
Applying the Taylor expansion of logN(l + x ) at x
c
=0
we get a series
m
logN(l + x ) = x N
cjxj
co = 1.
with
j=o
*
Thus f ( 0 ) 0 and hence the inverse exists at the origin. We also have to check the derivatives of f-’ at x = 0. We have
I
‘1
( f - ’ ( ~ ) ) ‘ = Nx-’ xN log-N(l + X ) - xN+’ = Nx-’
From x log-’(l
+ x)- l +1x
{
+X) ] l+x
1 - x log-’(l
- 1 + c ; x + c;x’
+ ...
(1 + x ) -
l+x
XN
log-N(l+ x ) .
for x - 0
then it follows that
f - ’ ( O ) also does exist. Thus we obtain inductively (16). 0 15. Proposition. For every a ( t , r , y ) E S”(R+X R + XR) and giuen N EN there exkt symbob b , ( t , y ) E S”(R+xR),aN(t,r , y ) E S ’ - N ( R + ~ R + X Rsuch ) thar
dr Here
b d f , y )=
a).( ck! -
N-’ k=O
1 (1 i ay
-r-
aar)k
a(t, r , y ) I r n r .
Proof. Let a ( t , r , y ) E S”(lR+XR+XR) and consider the Taylor expansion near the diagonal r = t
where a N , o ( tr, , y ) E S’(R+ xR+XR).Then
c
N-1
h ( t )=
AkU(f)
+RNU(f),
k=O
where R N is the operator associated with ( r - t)NaN,o(t,r , y ) , and
By Proposition 14 the remainder R N is an operator with an amplitude function in S’
- N.
81
1.2. Spaces with Conormal Asymptotics
Set
:(
Fk(r,t ) = log-*(:)
k
- 1)
(cf. 1.2.3. Lemma 15) and
for some function b ( l , r ) . Then A k has the amplitude function Qk(t,
r , Y ) = Fk(r,t ) (TkD,ka)( t , Y ) .
We can write N-1
A k = OPM(ak) =
opM(akf) + R N f=O
with another remainder
R N
of analogous structure as the above one and
U k l ( t , r , y ) = Ff(?, r ) D;(T'Uk) ( t , y ) =
F f ( t ,t ) T f F k D yk ( T k U ) ( t , y ) .
The procedure can be iterated and we then obtain
N-l
N-l
with a remainder which has an amplitude function in S'-N and ao(l,Y>
:= ao(4 t , y )
akO(l,y)
= D:(Tka)(i,y)
ak,O(f,y)
=
T'FkDf+' y (Tka)(t,JJ)
akl k f l ( t , y ) = TkNFkN-ITkN-IFkN-l... T k z P l ~ y k l + + k ~ ( T kal) ( t , y )E
S'-
(kl+
+
~ N ) ( RXR). +
Since Fk depends on r l l , the expressions T'Fk are constants, independent of t , which can be expressed by the Taylor coefficients in the proof of 1.2.3. Lemma 15. Thus a k , k f l ( f r Y ) = ckl kNDykl+ + k N ( T k l a()t , y ) with certain constants ckl
kN.
In view of
stants d, our result can be summarized as follows.
82
1. Conormal Asvmptotics on
R,
a
There is a system of differential operators p,(Dy, -r-),ar where p,,,(q, e ) denotes a polynomial, homogeneous of order m in (7, e ) E RZ,such that for every N there is an M with
These pm are independent of p . They are uniquely determined by the relations (18) if we insert all polynomials a(t,r,y) =
1 a j ( t ,r ) p ,
aj E c ~ ( R + x R + ) .
j=O
1
Then an elementary calculation shows that p,,,(v, e) = q m p ,m m!
E N.
0
16. Theorem. Let C ? A ( f , Y ) E Sr(R+XR) be a complete symbol of A E ML"((R+),written Then in theform (1) with a ( t , r , y ) E S@(R+xR+xR).
Zf A E MLr(R+), a ( t , r , y ) E Sr@+ XR+XR), then there is a complete symbol C?A(t,y) E SP(R+XR). An analogous statement holds for the subclass of classical Mellin 1DO-S. Proof: The first statement is an immediate consequence of Proposition 15. From a ( t , r, y ) E P(R+
x R+ XR) we get
(+
$) k
&)k(
-r
a ( t , r, y )
Ire,
E S'-~(R+XR).
Then the asymptotic sum can be carried out within P ( R +XR), cf. Proposition 4. This sum is obviously a complete symbol of A E M L r ( R + ) .The procedure obviously preserves classical symbols. 0
As an analogue of Remark 13 we easily obtain 17. Remark. There is a canonical isomorphism
MLp(R+)/ML-"(R+)f P ( R +XR)/Sm(R+XR),
and the same with the subscript cl.
Let us express the formal adjoint of (1) with respect to the scalar product of L2(R+). For u , v E C,"(R+)we get
83
1.2. Spaces with Conormal Asymptotics
Thus
m
m
with a*(r, t , y ) = a ( t , r , y ) . Similarly we can calculate the transposed operator ‘A which has the amplitude function al(r, t , y ) = a ( t , r, - y ) . In particular from (14) it follows m
‘Au(r) =
m
II
-m
E ( t , r , y ) UA(r9 - y ) U ( Y )
0
dr r dy.
From Theorem 16 it follows an asymptotic formula for a complete symbol u,(r,y) of ‘A. Applying the transposing twice we get the operator (14) in the form
For the Mellin transform of Au we get dr r1’2+iyurA(r, -y) u(r)r Let us apply this to a Mellin yDO B with a complete symbol a B ( t , y ) Composing . with (14) then yields ~ ~ u (= t[ [)
dr ~ ( tr , y ) U . ( t , y ) urrs(r, -y> u(r) T d y
(21)
18. Theorem. Let A E MLP(R+), B E ML’(R+), (MLr(R+),ML’(R+)) and A or B be (MLf’+”(R+)) and properly supported. Then AB E MLP+ ”(R+)
ProoJ The proof is an immediate consequence of the formulas (211, (20) and Theorem 16 (cf. also the analogous well-known situation for standard vDO-s, e.g. in
[S27]). 0 Next we want to discuss the classes MLE,(R+), ML$(B+) once again under the aspect of recovering the homogeneous components of complete symbols. For A E ML$(R+) we already know that u A ( t , y )is unique modulo S ” ( R + X R ) . Thus there is a unique sequence a@-J’(t,y) of C” functions on R+X(R\{O}), homogeneous in y of order p - j , with
84
1. Conormal Asymptotics on R,
- c X ( Y ) a"-"(t,y), m
UA(4Y)
j=O
x
being an excision function on R,,.Since for A tive a A ( t , y )E #@+ xR) we get a'@"'(t,y)
E
E M L @ ( a + there )
is a representa-
Cm(R+x(R\{O}))
for allj. For references below we set # ; ( A ) (C, z) = a y t , - t-1 z),
(22)
cf. also 1.2.3.(13) with the same meaning (in 1.2.3. the symbol was written with dependence of i y ) . Now assume that we have defined a method to produce a(@)by means of the action of A . Then Al : = A - opy(xa(@)) E M L @ - - ' ( R(ML@--'(R+)) +) admits to recover and so on. Thus a representative a A ( t , y )may also be obtained by calculating step by step the homogeneous components. We shall see in Section 1.3.1. that there exist order reducing operators that allow to restrict the consideration to ,u = 0. So let us assume ,u = 0. First remind of the recovering procedure for classical tyDO-s in R" with respect to the phase function (x - x') t. (x, F) denotes points in R"x R"= T*R", and
Fu(F) =
s e-ixfu(x)dx
parametrized by the Fourier transform. Define an operator family RA(xo,to), 1 E R + , (xo, 60) E T*R"\O, 2(x(Rl(xo, ~ o ) u ) ( x ) : = 1 " 4 e i " x ~ ~ u ( 1 1 ~x,,)),
Then (R;'(X,,~~)v ) (x) = ,1-n'4
UE
C;(Rn).
e-'~"'''"+~)fOu(IZ-1/*x + xo).
In view of IIRi(xo?50) UIILZ(RT = II~IILz(RT for all 1 E R+and xo,6,the operators Rl(xo,to) extend by continuity to continuous operators in I,*@"). 19. Lemma. For every u E L2(R")we have
R,(x0,to)u - 0
weakly in Lz(Wn)
asA-w.
ProoJ: Set for abbreviation Rl
= Rl(xo,
to). First let
u , u E C;(Rn). Then
85
1.2. Spaces with Conormal Asymptotics
Now let u E L2(R"), v E C;(R"). Choose a sequence uj E C;(R"), uj+ u in Lz(R") and write
I(~~~,~)l~lI(~~(~-~j)~~>I+I(~~~j~~)I~Il~ For j and A sufficiently large the right hand side is less than an aribtrary E > 0. Thus (Riu, v)+O for A + m and u E L2(R"), v E C;(R"). If also v E L2(R")we choose Vk E C;(R"), vk+ v in L2(R")and then obtain the assertion by an analogous argument. 0 Now let A be an operator in L:,(R") which is compactly supported, i.e. A = pA'ry for some A' E L~l(R") and certain q, v E C;(R"). Let d 0 ) ( x ,0 be the homogeneous principal symbol of A . 20. Proposition. For every fixed ( x o , to) E T*R"\O and u E C;(R") we have R i' ( x o , 50)( A + C) RA( x o , 50) u + a(0)(xo,to) u in Lz(Rn)for A + m , C being an arbitrary compact operator in L2(R").
A proof of Proposition 20 was given in [R8],Section 2.3.4.1. Here we only sketch the method. First we can prove by straightforward calculation that
(FR,u)(5) =
e-i*(c-&)(Fu)
(A-'l2(5
- Ato))
and for a ( x , 0 E S:,(Rn x R") (R;'F-'u(x,
& F R ~ u ) ( x=) ( F - ' u ( A - ' / ~ x+ x O , A(A-1/2(+ & ) ) F u ) ( x ) .
If C is a compact operator in L2(R") we have in view of Lemma 19 in
R;'CRAu+O
L2(R")
Thus C may be neglected at all and we also may replace a (x, 0 by is an excision function (i.e. x E Cm(R"),x = 0 in an open neighbourhood of 5 = 0 , x = 1 outside another neighbourhood of the origin). Then we get by homogeneity
for
A-
03.
~ ( 5a(O)(x, ) F), where x
R;'F-'x(& a ( O ) ( ~ , 0 F R ~ u ( x ) = F-'x(A(A-"y+ 50)) a'O'(A-"2x+ xo,A-1/25+ 50) F u ( x ) .
Now
x(A(A-'/25+
50))
-
u(O)(A-'/2x+ xo, # l - 1 / 2 5 +
50)-
a(O)(xo,50)
for every fixed x , 5 and A 03. The precise proof then follows by an extra consideration which shows even the asserted convergence. We want to apply this to n = 1 and a classical Mellin 1pD0 A = M-'a(t, r, y ) M. By Proposition 15 we may replace a ( f ,r, y ) by a (f, y ) := a (t, f, y ) modulo a lower order operator which only affects the lower order components of a,. Denote the new operator by A, again. Then
Au = @-lF-laFF@u, aF(x,& := a(ex, -[), u E C;(R+). Fix to E R+, yo E R \ 0 , and set xo = log fo , to = - y o , further (with @ = 0 0 Of 1.1.1.(9)) 4(to,
Yo) = @-'Ri(xo, -50) @,
A E R+,
86
1. Conormal Asymptotics on R,
Then S;'(to,Yo)AS,~(to,Yo)u = @-'R;'(xo, - t o ) F - ' a ~ F R ~ ( x o-50) , @u. 21. Proposition. Let A be giuen by (1) with a ( t , r , y ) E St,(R+XR+XR)and assume that a(t, r , y ) = 0 for (t, r ) 6 K with some K c c R +X R + . Let a(O)(t,r, y ) be the homogeneous principal symbol of a(t, r, y ) and set ucO)(t, y ) = ucO)(t,1, y ) . Then S;'(to, YO) & 4 ( t o , YO)u+a"'(to,
in L2(R+)as A+ = 1 close to t o .
co for
YO)u
every u E Ci(R+),p, being an arbitrary function in Ci(R+), p,
Proof. We can write A l = pAp, = vOPM(a1) p, + C1with some az(t,y ) and compact operator in LZ(R+).Then
c1acting as
S;'(to,Yo)p,Ap,SA(to,Yo) = S;'M-lPaiMSI(to,yo) u + S ; l ~ t o , y o ) C I S ~ ~ t o , y o(23) )u for large A , since supp &(to, yo) u is contained in any given neighbourhood of to for A large enough. We know that the second item on the right of (23) tends to zero in L2(R+)as A+ a. Thus it suffices to consider the first one which is of the form
@-1R
-1 A (x0,
-tO)F-'(Pal)FFRA(xO, -50) @u. Set u = @u. From Proposition 20 it follows that R;l(xO,
- t o ) F-'(~al)F~RA(xOi - t o ) v*(alF)(o)(XO,
-60)
in LZ(R)as A + W . In view of the isomorphism 0:Lz(R+)+L2(R) (which induces an isomorphism Ci(R+)+ Ci(R)) we then obtain that S;'(to,Yo) M-'p,alMSA(to, Yo) u+ in L z ( R + ) 0 .
@-l(alF)(o)(xo,
-to)
@u = a'O'(t0, Yo) u
For references below we need a natural Frechet structure in the space ML:,(K+). Define the mappings ut-j: A d S@-J)(R+x(R\O)),
where u;-'(A) ( t , y ) is the component of a complete symbol a, of A of homogeneity p - j , S'l-n(K+x (R\ 0)) being the space of all p ( f , y )E Cm(R+ x (R \ 0)) with A t - ' , - y ) E Cm(K+x (R\O)) which are homogeneous of order p - j in y , considered in the corresponding FrCchet topology. Fix k E N . Then we have the seminorms of $ - ' ( A ) , j = 0 , ... , k, in the corresponding spaces that may be regarded as semi-norms on ML!!,(K+).Fix an excision function x . Then
87
1.2. Spaces with Conormal Asymptotics
(cf. 1.2.3. Proposition 16). (24) is a FrCchet space in a canonical way. The corresponding semi-norms applied to A ( x , k ) , A * ( x , k ) may also be regarded as seminorms over ML;,(K+).If we do this for all k E N we get a countable semi-norm system on ML;,(K+) under which it is a FrCchet space, independently of the choice of X.
Let us consider the classes MLP(Kt), ML;,(K+) from another point of view. Define the subclasses of flat operators
M L ; ( K + ) s M L q K + ) , ML;, ,,(K+)SML;l(K+) as follows. 22.Definition. Denote by M L ; " ( K + ) the space of all G E ML--(R+)which induce continuous operators G : Xs(R+)+Xz(R+),
G*: %'(R+) +Xz(R+)
for all s E W.Let ML;(R+) (ML;, ,@+)) be the space of all A + G where A is giwn by (1) with a(t, r , y ) E S$(K+Xa+xR)( S ~ , , , ( ~ + X ~ + x Rand )) GE ML;"(w+).
The constructions for MLP((B+), ML:,(R+)make sense within the subclasses of flat objects. The proofs of the following statements are left to the reader as exercises. 23. Proposition. Every A E ML;(K+) (ML$,,,(R+))admits a decomposition A = A . + A , , where A . E ML$(R+)(ML;, ,,(K+)) is properly supported and Al E M L i a ( K + ) .
Let A E ML;(K+) (ML;, ,@+)) be properly supported. Then the formula (13) gives rise to complete symbols a, (t,y ) E S$(K+X R) (S;, ,,(K+ x R)) with asymptotic expansions (19). The construction is not quite symmetric with respect to adjoints. opM(aA)*is not flat, but there is an A , E M L - " ( K + ) so that op,(a,)*+ A l E ML;(K+). 24.Theorem. For ML$(R+) (ML;,,,(K+))there holds an analogue of Theorem 18. 25. Remark. Every A E M L $ ( K + ) induces continuous operators A, A*: X ( R + ) -+ Si-''((pi+) for every s E R.
26. Definition. A E LP(R+) is called frat at t = 0 (of infinite order) if there is an 2 E Lp(R) with supp Tu,supp A"su E K+for all u E %'(It). Denote by L;(R+)the space of all A E LP(lR+)for which A and IAI-' arefrat at t = 0 (of infinite order). In a similar way we may define L$, cl(K+).
Remember that ( I u ) ( 2 ) = t-'u(t-') is a transform with ILp(R+)I-' the same for classical operators. As a simple consequence of 1.2.3. Theorem 14 we get 27.Theorem. For every p E R we have
ML$(R+)= L $ ( K + ) and the same with the subscripts cl.
= LP(R+)and
88
1. Conormal Asymptotics on R+
1.2.5.
Green and Flat Operators on R,
The spaces with conormal asymptotics lead in a natural way to operators that are smoothing of some conormal order. We call them Green operators, a notation which comes from Boutet de Monvel's algebra [B9], [RS].In the simplest case a Green operator is of the form, m
OPc(g) u ( t ) =
g(t, 8) U(S)ds
(1)
0
with a kernel g E S(E+) NnS(K+), and u E Lz(R+).The adjoint operator has an analogous structure with the kernel g*(t, s) = g(s, t ) . 1. Definition. A n A E 5?(L2(R+),L2(W+)) is called a Green operator of the class
!XG(R+),, A E 3, if there are carriers of asymptotics B, C E Yo, depending on A, such that A, A* induce continuous operators
respectively. A is called a Green operator of the class %,(R+); if there are discrete asymptotic types P, Q E 9to,depending on A, such that A , A* induce continuous operators
respectively. We also write %G(R+)= %G(W+), , %,(R+)*= %,(It+):
for A
= (-03,
03).
Moreover we write 9 instead of A when 9 = (- k, k), k E N , where %G(R+)(O, 0) := ML--(K+) cf. 1.2.4. Definition 6 . Note that the classes of Green operators depend on the choice of the scalar pro. choice may destroy the mapping properties (3) and (4), reduct in L z ( R + )Another spectively. The scalar product is assumed to be fixed once and for all as in 1.1.1.(18). On the other hand we also might relax the conditions and only talk about the mapping properties of the operators itself. This would yield all essential results of our calculus. In view of the nuclearity of the spaces !Xi (It+), a continuous operator (2) can be represented by a kernel 3
a(t, s) E 2;@ + ) A
@*
LZ(R+).
Let J denote the antilinear operator of complex conjugation. From (3) it follows
m, t ) E 2;(R+),NnL2(R+) and thus a(t, $1 E LZ(R+> @n
Jx;(It+),.
89
1.2. Spaces with Conormal Asymptotics
c=
Clearly JX;(R+)A= X ; ( R + ) A , {I:z E C) . are locally convex vector spaces with continuous embeddings If Y, g, P, F G Yo, g G go,we set
so
2?onP@,,
Y B r %= r@,, YA (5) equipped with the topology of the intersection, cf. 1.1.2. Remark 5 . To,goare = L*(R+)we obtain kept fixed for various .7, g. In the present case for P =
%(R+)A= I o p d a ) : a E X i ( R + ) A @ i - J X ; ( R +B,) AC, E Yo} and similarly
WR+);= Iopda): a E XF
@r J X z @ + ) A ,
P, Q E 3'1,
(6)
(7)
A E 3. 2. Remark. I f can be prooed that .!XF (R+)d@3r.!Xi (R+),, = 2;(R+),@,,.!Xi (R+),for all P,R E 5to, A E 31
A proof was given in an analogous situation in [Rll], Section 1.2.2. The equality @r= @,, for discrete asymptotic types extends the well-known result that when u(t, s) can be differentiated separately in t, s then also the mixed derivatives do exist (in spaces of C- functions). In particular Y(R+)@rY(R+)= Y(R+> @,,Y(R+),or more generally
YP(R+) @i- Y R (R+) = YP(R+) @n Y'Q(R+) 9
P, R E 9, cf. 1.2.1. Definition 7 and formula 1.2.1.(16). The analogous equality for the continuous asymptotics could not yet be proved (it seems to be false). 3. Proposition. %G(R+)A, A E 3, is a *-algebra of operators in L 2 ( R + ) . Every A E %!G(R+)A induces continuous operators A : Xs(R+) 2; (R+)A A*: Xs(R+)4X;(R+),j +
I
for every s E R, B, C E Yo depending on A (not on s). Analogous assertions hold for %G(R+);. Proof. The *-algebra property is obvious. The continuous extensions of A , A* to Xs(R+)follows immediately from (6) and (7), respectively. 0 4. Proposition. Every A
E
%G(R+)A ( E %(It+);)
induces compacr operators
A, A * : Xs(R+)+Xs(R+),S E R .
n
Proof. From Proposition 3 it follows that A : Xs(R+)+Xs'"(R,) Xdv -'((R+) is continuous for some E > 0 and every s' > s. The composition with the embedding of the intersection into Xs(R+)is then compact, cf. 1.1.2.(7). In an analogous way we argue for A*. 0 5. Proposition. Let G E %,(R+), , A E 3, and assume that 1+ G : Xs(R+)+Xs(R+)
90
1. Conormal Asymptotics on R+
is invertible forfixed s E R.Then (8) is invertible for all s E R and (1 + C)-I for some G1E WG(R+), . Moreover G E '&(It+): implies G1 E WG(R+)~.
=1
+ G1
Proof. (1 + G ) u = 0, u E W m ( R + ) implies , u E F ( R + ) because of the mapping properties of Green operators. Thus the kernel of (8) is trivial for all s E R . Since ind (1 + G) = 0 in W(IR+),cf. Proposition 4, we get the bijectivity for all s E R. Now we have G1 = 1 - (1 + G)-l E Se(XS(R+), W((Rt)) and (1 + G ) (1 + GI) = 1 which implies G + G1 + GG1= 0, G1 = - G - GGI. From (2) for the operator G it follows the analogous mapping property for G1. For the adjoints we can argue in an analogous way, i.e. G, is a Green operator, too. Clearly the dotted subclass is preserved under the operation G+ G1,0
N be a finite-dimensional subspace of !XE (R,)for some B E Yo. Then the orthogonal projection P : L *(R+) +N belongs to a,(&). An analogous statement holdr for the discrete agmptotics as well as for the classes with finite weight intervals.
6. Proposition. Let
Proof The mapping property (2) holds by definition. In view of P* = P we get the same for the adjoint. This argument applies also for discrete asymptotic types. 0 S?(Xs(R+), !X-c(R,)) is calledflat of order -6 at t = 0 and
7.Definition.An A E
SER
6' at t = Q), 0 5 - 6, 6' 5
Q)
, if A induces continuous operators
A, A* : XS(lR+) +2;'(R+)~J,
for all s E R . In a similar manner we can define flatness with respect to open and half open weight intervals (cf. 1.2.1. Definition 4).
8 = (- k, k ) , k E N, p E R, is the set of all A which are flat of order k at t = 0 and t = m.
8. Definition.
Note that
WR+) :=
n
WR+),
=M
E ML$(R+)
L ~c , ( ~ + ) ,
8=(-k, k)
ks N
cf. 1.2.4. Definition 22. Moreover it is clear that %~(R+),is a *-subalgebra of ML;,(K+) filtered by the orders, cf. 1.2.4. Theorem 18, and 92g(R+)(o,o, = ML;,(K+). 9. Remark. Let a(?,z ) E C,"(R+, Ems), then
wtaopM(a) t'W E R$(R+)a
for a,B 2 0,B = ( - k, k ) , k = a + B E N
This will be obtained below in the proof of Theorem 14.
91
1.3. Mellin Expansions of Operators
further %$+ G @ + )
=
n%$+
C(w+),
k
and similarly with dots.
Throughout the calculus we also could use the one-sided Green and flat classes, i.e. without any information on the adjoint operators. We also could admit different flatness orders at t = 0 and t = m. Our convention was made for notational convenience. For every A E %$+ c(R+)awe have a well-defined homogeneous principal symbol
o!(A) (t, z) E S(”(T;W+\O),
(9)
(cf. 1.2.4.(22) and 1.2.3. Definition 13) which is flat of order k at t = 0 and t = m. Denote the subspace of symbols in S(@)(T:K+\O) that are flat of order k by S”(7-g
R+\O),.
(10)
Let (10) be equipped with the induced Fr6chet space structure. 1.2.4. Theorem 18 implies the following
11. Proposition. %+; ,(R+), , 9 = (- k, k ) , k E N , is a *-algebra,filtered by the orders, and %G(R+), is a two-sided ideal. A E %$+ G(W+)B,B E %+; G(R+)e implies
o!+’(AB) (t, z) = 4 X A ) (t, z) (t, r ) , = a ! ( A ) (t, z). o;(A*) (t, z) An analogous statement holds for the dotted subclass. 12.Defintion. An operator 1 + A with A E %$+ JR+),, k E IN \ {0}, is called elliptic if 1 + ui(A)(t, z) does not vanish for all t E E+(including t = m) and all z 90. 13.Proposition.Let 1 + A , A E %+: G(R+)8, be elliptic, 1 5 k 5 . Then there is a parametrix of the form 1 + B for some B E %$+ G(R+),in the sense that 1 - (1 + A ) (1 + B ) , 1 - (1 + B) (1 + A) E %G(R+)e.
(11)
An analogous assertion holds for the dotted subclass. ProoJ It is obvious that (1 + o!(A))-l = 1 + p for some p E S(O)(T:K+\O),. Let B” E %$(R+),be the operatcr associated with x ( z ) p ( t ,z), x being an excision function in z. Then (1 + A ) (1 + B ) = 1 + C, where C E %;:,(R+),. Now we can apply a for:; G(R+)B such that mal Neumann series argument. There exists an operator Cl E % (1 + C) (1 + C,) = 1 + G for some G E %,(R+),. Indeed we only have to verify that j E N, there is an F E %+; c(R+),such that for every sequence 4 E %;;JG(R+),, N- 1
F-
C FjE%k;NG(R+)8
j=O
for all N . But this follows by analogous arguments as in the standard calculus of
yDO-s. We write F
-
7 Schulze, Operators engl
m
m
Fj. Now it suffices to set C, j=O
- C (-1)W. j=1
Then B follows
92
1. Conormal Asymptotics on R,
from 1 + B = (1 + B", (1 + CJ, and 1 + B is a right parametrix of 1 + A . In an analogous manner we find a left parametrix. Then 1 + B is a parametrix in the sense of (11). 0
'Ton
14.Theorem. Let h E %RC, V E T ap ,e 0 , and w , w1 be arbitrary cut-offfunctions. Then wPop,(h) w1- w opu(TBh) t40, E %G(R+). (12) M o r e o v e r i f h E % R ( ; , V E ' T - Y , y E R , u , p z O , a + p = k ~ Nthen , (13)
wt"OPM(Ph) tpw, E %$+G(R+)B for 9 = (- k, k ) . Analogous results hold for h E %*R,: stead.
with the dotted Green classes in-
Proof. Applying 1.2.3. Theorem 6 we can write h = ho + h l , where hoe Em;,, h l E "Ill;". Denote by A the operator in (12). Then A = A. + A l , where Ai is associated with hi. Now let u E Ct(R+) and set v = (21~9-lw l u . Then t-'hi(z) C(z) dz - w
Aiu = wtp
rl12
j
t-'hi(z
+ 0) iT(z + p ) dz
rliz
Applying Cauchy's integral formula we obtain
Aiu = wtfl j t-'h,(z) iT(z) dz, C
where C is a smooth curve surrounding V n {1/2 < Rez < 1/2 + p } . Here we have used that hi(z) u ( z ) is holomorphic in {1/2 < Rez < 1/2 + p } \ V . Since ho(z) u ( z ) is holomorphic in the whole strip we get Aou 0. The function A l u is obviously an element in X;(R+), W = T - S V n {1/2 - p < Rez < 1/2} (cf. the construction of 1.2.2.) and it is a simple exercise to verify that A l extends to a continuous operator Al: Xs(R+)+Xk(R+) for every S E R. For the adjoint we can argue in the same way, in other words we have obtained (12), cf. Definition 1. For proving (13) we write again h = ho + hl as above and show that
AF := wt' op,(T'ho) taw, E %$(R+)B, AG := wfuopW(TVzl)tho, E % G ( R + ) B . By the arguments that lead to (14) we get AF=
WtkOPu(P-phO) w 1 = w ~p~(Z'~+''-'ho) Pol.
Thus AF satisfies the conditions of Definition 8. The Mellin symbol hl belongs to %Rim. Choose E, 6, 0 < E < 6 < p, and write V = V ( E )+ V(6) for certain V ( E )E Y e - Y + p n Y - Y , V ( ~ ) ' E' T - ' - y + p n 7 r - Y . Then hl may be written as hl = f+ g, with certain f~ %,)R;; g E %R&; and we get A0 = wt" opu(TYf)ta~l+ wt" opu(T"g)t'wl.
Using analogous arguments as above we can write AG = A, G E %,(R+) and
+ Ad + G
with some
93
1.3. Mellin Expansions of Operators
A, = ~ t O P ,~( T " +- ~ - ' ~~) ~ ' W I , Ad = wtk-' Op,(Td+"-'g)t6W1. A,, A,, induce continuous operators A,: %'(R+) + % z ( R + ) ( - k + e , m ) , -46:
%'@+I
+%g(R+)(-k+d,m)
for all s E R. Since %z(R+)(-k+e,m) 4 Xz(R +)(-k + d ,the m )operator , AG is continuous in the sense
(15) where C E Y o is some carrier of asymptotics which may be chosen independently of AG: %'(R+)
6. Since 6, 0 < 6 <
+
%: (R+)(-k+ 8. m ) ,
is arbitrary, we get the continuity
A G : %'(R+)
+
.7ez (R+)(-k, m)
t
SE
R.
Here we have used that %; (R + )(-k m), is the projective limit of the spaces on the right of (15) for 6 -0. For the adjoint we can do the same, i.e., A G is a Green operator. 0 15. Theorem. Let h E %$, V E Yo, y E R and wi, Gibe arbitrary cut-off functions, i = 1,2. Then w1
op,(h)w2 - 61 OPhf(h)62E %+ G(R+),
(16)
w1
OP,(h) (1 - w d E %+
(17)
G(R+).
An analogous result holds for h E %:,
Proof We have
R E 5to, with the dotted Green class instead.
94
1. Conormal Asymptotics on
R+
is continuous as well as op,(hi;)
( ~ 1 -
4): n s ( R + +nL-’(Rt), ) w : XL-’(IR+) +!XL-’(R,)
which shows that (wl - Gl) op,(ho)w E %$(R+).The second item on the right of (18) can be treated in the same way. The proof for the discrete asymptotics is completely analogous. The proof of (17) follows by similar arguments and is left to the reader. U
16. Proposition. Let A E %;(R+I8, 8 = (- k, k ) , k E N, m(z) E %R$, p, v E R, wl, w2 be arbitrary cut-ofl functions. Then w1 opM(m)w2A, Awl opM(m)w2E %$”(R+),,
Aw‘, opdm)w; E %gt”(R+h,
w’l op,(m)w;A,
u p ) = w,(t-’). Prooj For every /.? 2 0 we have w1 op,(Tam)t~w,
= wlt~opM(m)w,,
since m is holomorphic, cf. the calculations in the proof of Theorem 14. This shows that Mellin actions with holomorphic symbols preserve flatness of any order, and hence we get the first couple of relations. The second one follows by conjugating with I.
17. Proposition. For every k E N the sequence 0 -+%$;k(R+), -%$+G(R,)8*S@)(TgR+ \
O), +o
is exact and splits. An analogous statement holdr for the classes with dot.
The splitting property means that there is a linear operator opt:
s‘”(TgRt
\
O), +%$+G(R+),
which is a right inverse of at. Proof of Proposition 17. First remember that the image of %$, C(R+),under at belongs to the space (10). It is clear that the kernel is %$;;(R,),. Let us show that at is surjective. Fix an element p(t, iy) in S(”(TgRt\O)B and let a(t, iy) E Sg@+ X R ) have p as homogeneous principal part. We can choose a in such a way that it is flat of order k at t = 0, t = a.Applying 1.2.4. Proposition 10, we get another classical symbol a,(t, z) with the same homogeneous principal part which is holomorphic in z. By construction it is also flat of order k. Remark 9 and its analogue for t = 00 shows that the associated Mellin 1pD0 is flat in the sense of Definition 8. The chain of mappings p + a + al +opM(al) defines a splitting of the exact sequence. 0 Using Proposition 17 and the operators opt-’, j E IN,we can define a sequence of symbol mappings
UP-”:
%$, G(R+)e+ S(””(TZR+ \
O),
95
1.3. Mellin Expansions of Operators
in the following way. We set A,, = A, @(A)
=
ai(A,,), A,
-1=
A, - opi(ai(A)),
a t - " ( A ) = ai-'(A,,-l), A,-? =A,-, - O P ~ - ' ( U ~ - ' ( A , - J ) , a t - * ) ( A ) = a5-2(A,-2), Then A,-j
E
...
%;;jG(R+), for all j . Set 1
X S(p-J)(T;R+\ 0), ,
SymbK'(T;E+ \ O), =
j=O
considered in the topology of component-wise convergence. The sequence of constitutes a linear mapping up': %+;
a t -Is
G(R+)ff + Symb'.'(T;a+ \ 0), .
As an immediate consequence of Proposition 17 we get 18. Proposition. For every k, I E IN the sequence
+%$+G(R+)ff Symb@.'(T;a+\ 0), +O is exact and splits. An analogous statement holds for the classes with dots. 0 +%;;$+')(It+),
A right inverse of af' is given by opb.'(p,, ..., p,-')=opg(p,,)+Opg-'(p,,-,)+
... +opg-'(p,-/).
The assertions remain valid if we replace the subscripts F + G by F alone. We want to define in %;+G(R+)ff and %$+G(R+)i natural locally convex topologies. Using Definition 10 and the natural locally convex topologies in the classes of Green operators, given by the identification (6), (7) for A = (- k, k ) , it suffices to define a topology for the flat operators. In the class of {flat-Green}-operators we then take the topologies of the sums. Proposition 18 suggests to use the isomorphism \ 0),, %;(R+), %;-('+')(R+),@ Symb~~'(T;R+ where we fix a suitable topology in the first item on the right. Every A E %$-('+ ')(It+), induces continuous operators
A, A*: Xs(R+)+X~-""'(R+)ff, S E W . Denote by 55';,$+'+' the subspace of all A
E
(19)
n55'(Xs(R+),X S - p + ' +(It+)) ' for which S
A, A* induce continuous operators (19). This is a locally convex vector space in a natural semi-norm system (it is a Frkchet space). %;-('+')(R+), CZF - p ,+,/ + l is not a closed subspace, but we consider it in the induced topology. This yields a topology in %;(R+), depending on 1. Denote the space in this topology by %$'(It+),. Then we have continuous embeddings %$'+ '(R+)e
-+
%$'(R+)ff
96
1. Conormal AsvmDtotics on R+
Now we introduce in %$(R+),the projective limit topology for 1 + m. It can easily be verified that it is a Frkchet space, and it is independent of the involved concrete choice of splittings in the exact symbol sequences. Note that %G(lR+), is not a Frkchet space but an inductive limit
~ G ( R +=) B @ Xi (R+)B @r J X z (R+), 8, CE ro
of Frbchet spaces. In order to prove the convergence of a series of operators in this space we need that all operators belong to the corresponding subspace with fixed B, C and we have convergence there, where only a countable system of semi-norms is involved. In our applications we often may apply the following k E N \ {0)fixed, 19. Remark. Assume that ( G j ) j E is a sequence of Green operators in %G(R+)B, and aj,Bj non-negative reals, aj a,Bj -+ m as j -+ m. Then there are B, C E Y o such that -+
w t a j G j t hE % ~ ( R + ) , 8 , J % ~ ( R + ) ,
for all j , w being a cut-off function. An analogous remark holds for the dotted subclass.
20. Theorem. Let h E Em;, V E T o , a,B > 0, n functions wl, w2 and let c z 1. Then
=
a + /lE IN, n > k E IN. Fix cut-off
A, := w,(ct)t"opW(Toh)tow,(ct) + O in %$+ c(lR+)e as c -+m. An analogous assertion holdr for the classes with dots. Proof. Choose a cut-off function w with w ( t ) w i ( c t )= w i ( c t ) for all c 2 1, i = 1 , 2 . Write a = a + a',B = b + B' with a, b > 0 and a',B' 2 0, 01' + B' = k. Then A, = w,(ct)t"A:tbw2(ct), A : = wtu' op,(TBh)tbw,
where A : E %$+ o(lR+)8 by Theorem 14. As in the proof of Theorem 14 we may write. h = h o + h , , hoEQt$, h l ~ Q t ~ m , a n d A , = A c , o + A c , l , A ~ = A ~ , o + A ~ , l , A c , i , A longing to hi,
Ac,i = wl(ct)t"A:,itbo2(ct), i = 0, 1 ,
4 0A:,o , E W4R+)e,
A , 1 A:,i E %(R+)B. From 1.1.2. Lemma 6 we obtain that w,(ct)t", w2(ct)tbtend to zero in the norm of operators in Xs(lR+).Since A : , , is a Green operator, it induces continuous mappings 3
A>,(?: X,'(lR+)+%;(It+)$),
where
Xi(It+)$), 1 E IN,is a chain of Hilbert spaces with Xi(R+)B= li&qR+)$), I
B E Y o depending on A. Then A$', = t'wl(ct)AQ,oz(ct)tb:Xs(R+)+!Xi(lR+)$)
tends to zero as c + m in the norm of operators. An analogous statement is true of the adjoints. Thus for every s E Z, 1 E IN the sequences of operators A c , l ,A;,, tend to zero in the operator norms between corresponding Hilbert spaces. This follows also
97
1.3. Mellin Expansions of Operators
for all s E R, 1 E IN.Thus we obtain that A , +0 in %G(R+)B.Next we show that Ac,O-0 in %$(R+)8. It is obvious that all homogeneous components p,, - j , c of the complete symbol of AGOtend to zero in SymbP.'(T; R+\O), 0 5 j s 1. Applying the splitting property in Proposition 18 the operator A G Ocan be written as A G O = A2A', where A 2 6 ° ~ % g - " + " ( W + ) Band , A ~ 6 ° = o p s K 1 ( P K C , . . . , p ~+O- l Cas ) c + m . Theoperator A k f ) has also the factors t"wl(ct),tbw,(ct) on the corresponding sides. From 1.1.2. Lemma 6 we then get A2ho+O, (Akh')*+O in the sense of (19) for all S E Z , I E N . This implies the convergence for all s E R, I E IN,Thus A, +0 in the given topology. 0
1.3.
The Mellin Expansion of Operators
1.3.1.
Operators of Lower Conormal Order
Our next objective is to extend the operator classes of the preceding section by Mellin operators. First we study finite Mellin sums. In 1.3.2. we pass to infinite Mellin expansions. We shall have both the objects with discrete and continuous asymptotics. Remember once again that we set opL(h) = tyOpM(T-yh)t-y, h E EDZ;,
V E YY,cf. 1.2.2. Definition 1.
%#(R+);, p E R, 9 = (- k, k ) , k E IN \ (O}, is the set of all operators A = S + S' + F + G, (2)
1. Definition.
where F E %$(R+)@, G E iRG(R+);(cf. Section 1.2.5.) and
c {y
k - 1. ..
s
=
wtjop2(h,)w,
(3)
j=O
S'
=I
}
wt'opj;;l(f ; ) w I - ' ,
l=O
(4)
with arbitraly hj E f,' E Em;;, R j E W j , R ; E Ryi, 0 2 y, E - j , 0 2 y; 2 - 1 for . .. , k - 1, w an arbitrary cut-offfunction. Moreover define
j , 1 = 0,
For unifying notations we set for 9 = (0,O) Wc(R+);= %'(R+)B = h!fJ!dG(R+).
(6)
Remember that by definition %$' = EDZ;,(-m,m), cf. Section 1.2.3. In view of 1.2.5. Theorem 15 the class %p(R+); is independent of the choice of the cut-off function w. Define
c?J(A) (z)= h , ( z ) ,
c?$(A) (z)
=f
;(1 - z),
(7)
98
1. Conormal Asymptotics on R+
- 1 , and call oJ(A) (a&!(A)) the Mellin symbols of A belonging to the conormal orders - j ( - l) with respect to t = 0 ( t = 03). Set
j , I = 0,. .. , k
u&l) = (aR(A),...) a & k + l ( A ) ; &A),
...
)
U&!+’(A))
(8)
( k is kept fixed in this notation). Write c-ordA=-(l+l) c’-ordA = - ( I +
aJ(A)=O, Osjsl,
for
1) for
a,&)
= 0,
0 s j s 1.
c-ord, c’-ord is an abbreviation for conormal order. By definition
A E W”(R+)i* c-ord A , c’-ord A 2 - k . By the recovering procedure of the Mellin symbols (cf. Theorem 4 below) we shall see that this is a correct definition. In particular
A E %+:
c(W+)i * c-ord A
= c’-ord A =
-k .
2. Proposition. Every A E W”(R+); induces continuous operators
A : B;(R+)@ +xi“R+)@ for every P E Rowith some Q = Q(P,A ) E 5%’ and all s E R.
Prooj First observe that the multiplication by tors t Yw:
x$(R+)8
x;-vp(ol(R+)( - k, m)
tYw, y
2 0, induces continuous opera-
7
s E R. Here P(o,is the subset of P with ncP(o) c {Re z < 1/2}. An analogous assertion holds for P ‘ ( l - w ) . Then the continuity is an immediate consequence of 1.2.3. Proposition 18 and 1.2.5. Definition 1, Definition 8. 0
Now we shall study the symbolic structure belonging to operators in W p ( R + ) i . We shall see that analogous definitions then apply to W’(R+)B,the corresponding class with respect to the continuous asymptotics, which will be considered below. We have already introduced the Mellin symbols (7), (8), the latter ones play the role of the “complete Mellin symbol”. In Section 1.2.3. have been defined the interior 1pD0 symbols (in the “compressed sense”) a! (op$(hj)) (z) := ag(hj)(z) := hj,(#)(-iz),
hi,(,,)being the homogeneous principal part of order p of hj(-iz). Set
c c
k- 1
ags) (t, a) =
wZ(t)Pa:(hj) (t),
j=O
k-1
a’d(S’)(t, z)
=
w*(r-’)t-’a:(
f ;) (-z)
I=O
Both symbols belong to S ( @ ) ( c R\+ 0), cf. 1.2.5. From 1.2.5. (9) we get a symbol ag(F)(t, z) E S(fi)(Z‘$R+ \ 0). Define
og(A)(t, z)
=
ag(S)(t, z)
+ ag(S’) ( t , z) + ag(F) ( t , z).
99
1.3. Mellin Expansions of Operators
It also belongs to S(p)(T;R+\ 0). Set
2 e g f f = a.,., x
a!&,
cf. 1.2.3. (10). Then a,,,(A)E 2a!&for A
E Rp(R+);.
3 . Definition. b”(R+);, k E N \ {O}, is the subspace of all ( ( h j } ,{hj], p } E 2 ~ $ , x* S(~)(T:R+ ~ \ 0) for which
where f j ( z ) = h ; ( l -
2).
ker (uM,a!) = % ;: L(R+);. (16) Proof First we shall prove that a, is well-defined. The surjectivity is clear by definition of Wp(R+);. Set u(t, w ) = t-”w(t), w a fixed cut-off function, Re w < 1/2, C(E,zo)= ( w : Iw - zoI = E } , E > 0. From 1.1.5. we know that 1 M,,, u ( c W ) = f ( z , W ) + 2-w with a function f ( z , w ) which is holomorphic both in z and w. Let A
=
w,tjopgh)w],
w1 a cut-off function, 0 2 y 2 -j, h
M,,,Au(t,
W ) = h(z
+j)
E %Rfsy.*, then
z+j-w
where f’(z, w ) is holomorphic both in z and w in Re z < 112, Re w < 1/2. Now let A be given by (2). We claim that k-1 j-0
z+j-w
(17)
100
1. Conormal Asymptotics on R,
with another f"(z, w ) which is holomorphic both in z and w for 112 - k < Re z < 1/2, IIm zI > c sufficiently large, Re w < 112. Indeed, for A = S this is certainly true. On the other hand if A = S' + F + G we have f" alone. This is obvious when G = 0, and for A including G we read off the assertion from the mapping properties of G (the Mellin image of Gu for each u E L2(R+)is holomorphic in z as desired and also in w as we insert a holomorphic family of L2functions). Now fix z j , 1/2 - k < Re zj < 1/2, IIm zjl sufficiently large. Then
is holomorphic in w for lzj + j - w I < E,
E
> 0 sufficiently small. Thus
1 [ M t , , w l A w l u ( t , w)dw, 2x1
h j ( z j + j ) = --
C = C(E,z j + j ) .
In other words we can recover hj in a small disc around zj + j in terms of the action of A on u ( t , w). As a meromorphic function hj is uniquely defined everywhere. In an analogous manner we can proceed with the Mellin symbols belonging to t = 00. This proves the assertion concerning a,. Next consider a$. If p ( t , iy) E S ( p ) ( T ; R +\ 0) is given we find an a ( t , iy) E Sp(R+XR) such that a(t,iy) - x ( y ) p ( t , i y ) ~Sp--I(R+xR),x an excision function (iy is written only for technical reasons). We may replace a(t, iy) by b(t, z)Ir1,* where b(t, z ) is given by 1.2.4. Proposition 10 ( b is there called all. Consider the Taylor expansion at zero k-1
t J a j ( z )+ tka(k)(t,2 ) .
b(t, z ) = i=O
belongs to 9Ip(Rt);. For the second item on the right of (18) we use 1.2.5. Remark 9. a$(A1) obviously coincides with p close to t = 0. Similarly we find an operator A2 with the analogous property with respect to t = 00, Then PO = p - a$(A1) - @$(A2)belongs to S("(T;R+ \ 0) and has compact support with respect to t E R+. For the surjectivity of a$ it suffices to find an A . in our class with a:(Ao)= p o . Let ao(t, iy) be associated with po as above a with p , a o € C,"(R+,%Rho). If p(t) E C,"(W+) is a function with pao= ao, then ao(t,iy) p(r) E C,"(R+xR+,%Rpvo) and A o : =opM(aopl)E %ll,(R+)is of the desired sort, cf. 1.2.5. Remark 9. From %p(R+);c L;,(R,) we immediately obtain that 05 is a well-defined map in the sense of (121, i.e., a$(A)is uniquely determined. Here we have used 1.2.4. Proposition 20 to n = 1 and all strictly positive f. Thus if a$(A)vanishes we have necessarily A E L;;'(Xt+). This implies that a,$(A), # $ ( A ) are in %R!a'-l-*. Moreover A - (S+ S') E %ll,-1(lR+)8,where S, S' are associated with the corresponding Mellin symbols. In other words we have obtained (15). The construction shows at the same time that (13) is also surjective, since aj above might be replaced by hi without changing a$(A1)close to t = 0 and similarly for t = m (note that we have necessarily aj - hj E cf. 1.2.4. Proposition 10). Then (16) follows from %R;sms*,
101
1.3. Mellin Expansions of Operators
%$;L(R+); = %+; JR+);n %v-l(R+);.0 Note that another method of recovering the leading Mellin term was given in [R2], Section 1, Proposition 1.9. The above arguments show that ( l l ) , (12), (13) induce even surjective operators on the subclasses %#(R+)@,,,, 8, = (- k - m, k + m ) , m E W,
%w+);m+ @b,*@, %P(IR+)~,,,+ S("( T ; E+\O), %7"(R+)s,+@jYR+); 3
and the kernels are the intersections of %@(R+);,,, with (14), (15) and (16), respectively. In particular we may set k = 1, m = 0. Then we get
5 . Corollary. The triple u = ( u k , uL, a;) induces a surjective linear operator
c: %qR+); -+ %"R+);, , A
= (-1,
l), and ker u = ( A E %P-l(R+);: c-ord A
= c'ord
A
= -1).
(19) is a consequence of
%W1(R+);n%$;L(W+); 6. Theorem. A E %"(It+);,
= { A E % ~ - ~ ( R + )c-ordA=c'ordA ;:
=
-1).
B E %"(R+);implies A B E %@+"(R+); and
1 7 0 , ..., k - 1,
a;+"AB)
(f,
z) = uS(A)(1, z)a',(B)( t , z).
(22)
ProoJ Let A be given by (2), B = T + T ' + H + C, HE G E W,(iR+);, and T, T' be defined in an analogous manner as S,S' for A. Since A E LCl(IR+), B E L:l(R+), we obtain (22) by applying the usual composition rule for the homogeneous principal symbols of classical VDO-s and then replacing z by ? - I t . In order to prove that
(S+S'+F+G)(T+ T'+H+C)EW+~(W+);, we have to evluate the compositions ST, ST', ... separately. From 1.2.5. Proposition 11 we already know that ( F + G ) ( H + C ) E %$',k(R+);. We want to show that
S(H+ C),S ' ( H + C),( F + G ) T , ( F + C ) T ' E % ; I ; ( I R + ) ; ,
(23)
SC, S'C, GT, GT' E %G(R+);,
(24)
where
102
1. Conormal Asymptotics on R,
moreover
ST', S'TE %;:;(It& and
ST, S'T' E %#+ "(Rt); with the composition rules (20), (21). Let us begin with (24) and consider SC. From 1.2.5. Definition 1 and Proposition 2 we know that
SC: LZ(R+) +%; (R+)B is continuous for some P E 91°. We have to show the analogous continuity for (SC)*= C*S*. First it is obvious that
c
k-1
S* =
wop;yh;)fjw
j=O
and hence, by Proposition 2, S*: L2(lR+)+%-"(It+) is continuous. Moreover C*E 91G(It+);induces by 1.2.5. Proposition 3 a continuous operator C*: X-"(lR+) + X ; ( R + ) Bwith some Q e 9 l 0 . Thus SC satisfies the second condition of 1.2.5. Definition 1, i.e., we have proved (24). For proving (23) we consider, for instance, the composition SH. Applying 1.2.3. Theorem 6 the operator S can be written in the form S = So + s,,where the Mellin symbols of So are in %R$, those of S1 in 9Z;smvo.The operator H induces a continuous map H:LZ(R+)+%;p(R+)8 and S,: +%;@+I8 is continuous by Proposition 2 for some P E91°. Thus S I H satisfies the first condition of 1.2.5. Definition 1. Applying again Proposition 2 to S: and the continuity of H * : %-(It+) +%;(EL+) we see that S,H also satisfies The operator Socan be written the second condition, in other words & H E 91G(R+)e. in the form ..k - -1
So =
c
j=O
c
k. - 1
~
wtJop,(hj,,)w =
wop,(TJhj,,)tjw,
j-0
h , , E a$. Then SOHE X;+Y(Rt)e follows from 1.2.5. Proposition 16. The relation (25) will be left to the reader as an exercise. From 1.2.5. Theorem 15 it is clear anyway that we can choose the cut-off functions involved in S, S', T, T ' in such a way that the compositions in (25) vanish at all. It remains to consider (26) and to show the composition rules of the Mellin symbols. Let us discuss ST, the second item follows by conjugating with the bijection I. It suffices to show that for h E mhn', g E n;2*
E
= wfPopk(h) wtqopk(g) w = wfP+qopf*I((T-qh)g) w
+R
with some R E %;+,yG(IR+); and 0 z cz -(p + q ) . From 1.2.5. Theorem 14 it follows that w f P o p k ( h ) w l q= wfP+qop&(T-qh)wmod910(Rt)'. Moreover we always find a l in the mentioned interval such that both T-qh and g belong to %;R: L * (since the Mellin symbols are meromorphic). Applying once again 1.2.5. Theorem 14 we can write WtP+qOpb(T-qh)W = WtPtqOp$,(T-qh)W mOd%G(]R+)', w1
oPk(g)W = W I oP$,(g)wmod%G(R+)',
103
1.3. Mellin Expansions of Operators
w being a cut-off function with wwl = w. Thus
E
=
wtp+qopf;l(T-4h)wopf;l(g)wmodW;I',(R+);.
Here we have used the composition properties that were already established. Now w in the middle of the latter equation can be removed by 1.2.5. Theorem 15. Since opf;l(T-%) opf;l(g) = opL((T-qh)g), our theorem is completely proved.
7. Proposition. A E %~(lR+); implies A* E %F(R+);,ZAZ-l E %Rp(R+);,and U&'(A*) ( z ) =
oi'(A)*( z - I ) ,
aG(A*) (z)= aG(A)*(Z + I ) , where h*(z) = F(1- z),
(27) 2 = 0,
. .. , k - 1 ,
(t, z) = 4x4)(t, r ) u;'(ZAz-l) ( z ) = o&4) (1 - z ) , US(A.1
(28)
3
a&ZAZ-l) ( z ) = ai'(A)(1 - z ) , 1 = 0 , . .. , k - 1,
aS(ZAz-1)(t, t)= aS(A)(r', z) . Proof The assertions are basically obvious and may be regarded as exercises. Let us check, for instance, the Mellin symbols of A *. They follow by taking the formal adjoints of S,S' in (2). We have (wtj 0P2(hj)W)* = w op;yhf)th,
h f ( z ) = &(l - Z),
cf. 1.1.1. (19) and wopi'j(hf)tjw
= wtJop;'j(T-Jh~)wmod
W,(R+)*,
cf. 1.2.5. Theorem 14. This shows the form of the Mellin symbols at zero of A*. Those for t = 03 follow in a similar way. 8. Corollary. Wm(R+);is a *-algebra filtered by the orders, and %g(R+)B + W,(R+);
and %,(It+);
are two-sided ideals in %-(R+);.
9. Definition. An operator A
* *
E Wp(R+);,
9 = (- k, k ) , k E N \ {0}, is culled elliptic
if
(i) a;(~) (t, z) 0 for all (r, z) E R+x (R\ 0 ) , (ii) uL(A)( z ) 0, &(A) ( z ) O for all z E r,,,.
*
Note that the ellipticity is preserved under the canonical mapping
Bp(R+)i4 W'(R+); (29) for 9' 2 8. It is also preserved under compositions of elliptic operators and adjoints. 10. Definition. An operator B E %-N(R+);is called a paramefrix of A E Wp(R+); $ AB - Z, BA - Z E WG(R+);.
104
1. Conormal Asymptotics on W +
11. Theorem. Let A E W@(lR,)i, then the following conditions are equivalent (i) A : Xs(Rt) +X s-@(Rt) is a Fredholm operator for afixed
SE
R,
(ii) A is elliptic. is elliptic there is a parametrix B E W-@((wt)i. Moreover A u = f E X i ( R + ) f f , R E .%O, u E X-m(R+),implies u E X$+@((lR+)ff for some P E X o depending on R and A. I n particular ker A c X;o(R+),for some Po E .%O, and ind A = dim ker A - dim ker A * is independent of s E W.
If A
Proof. First assume that A is elliptic. Let us construct a parametrix B E %-@(Itt);in terms of the symbols that we determine from g:(AB - I ) = 1 , d&'(AB - I ) = a&!(AB- I ) = 610, 1 = 0, . . ., k - 1. We obtain p(t , z) := o!(A)-l(t, T) E S-@(T;R+\ O), h b (z ) := &(A)-' ( z ) E %ifo**. ho(z):= & A ) - ' ( z ) , Then the Mellin translation formulas (20), (21) lead to equations for the lower order Mellin symbols, namely
G r ( A )( z - q)h,(z)
= 0,
r+q=I
# $ ( A ) ( z + q ) h b (z) = 0 r+q=l
for 1 = 1, . . ., k - 1. This yields h j ( z )= a2(B)( z ) , h j ( z ) = on;!(B)( z )E %if* in a unique way. As a consequence of the compatibility conditions for the compooi(A))we immediately obtain nents of (oM(A), {[hi},{ h i } ,P I E @-@(R+)i. Using the surjectivity of (13) we obtain an operator
o,@(B") = p , ud(B")= h j , j = 0,
B"E W2-'(R+)iwith
dZ(B")= hg,
... , k - 1. From (16) it follows that AB" - I = R E 8;: c ( ~ t ) ; .
In other words B" is a right parametrix of A in the sense of a {flat + Green} remainder of order -1. The operator I + R is obviously elliptic. From 1.2.5. Proposi, : G(Rt)i with ( I + R ) ( I + R,) - I E %G(R+)ff. Thus tion 2 3 we get an R 1E 3 B = B ( I + R,) is a right parametrix. Analogous arguments yield the existence of a left parametrix in W-p(Rt);, i.e., B is at the same time left and right parametrix. The existence of B implies the Fredholm property of A. Indeed B induces continuous operators Xs(R+) -+Xst@(Rt)for all S E Rand A B = I + G : Xs(Rt) -+Xs(Rt), where G is of Green type, i.e. compact in the X s spaces (cf. 1.2.5. Proposition 4). From abstract
105
1.3. Mellin Expansions of Operators
functional analysis then it follows the Fredholm property of A. Now we easily obtain the asserted elliptic regularity with asymptotics. Indeed let u E X-=(R+)and Au = f X;(R+),. ~ Then we get BAu = (I + G ) u = B f E Xb+’(R+), for some Q E a0.From 1.2.5. Definition 1 it follows Gu E X>(R+), for another Q’ E a0. Thus u E XSof#(R+)@ + XZ(R+),c XeSp+”(R,), with P = Q + Q’ E a0. In particular the kernel of A belongs to Xrs(R+);.Thus it is independent of s. The same is true of kerA*, i.e., indA is independent of s. It remains to prove that the ellipticity is necessary for the Fredholm property. It is convenient first to reduce the order of the operator. 12. Lemma. For every y E W,9 = (- k, k ) , k RP E W(R+);.
EN
\ {0}, there exists an elliptic operator
Proof. First we choose an arbitrary a(t, iy) E SP(ii?+XW) (in the notations of 1.2.4. Proposition 10) for which the homogeneous principal part of order y does not vanish up to t = 0 and t = m (cf. also 1.2.4. Definition 3). Using 1.2.4. Proposition 10 we find an a l ( t , z) E Cm(R+,sQ(@)) such that a ( t , iy) - a l ( t , z) E SP-I(R+xR). Since q ( 0 , z), a l ( a , z) have non-vanishing limits for 1 Im zI we find a p E R such that al(O, z + p ) , a l ( a , z + B ) is non-vanishing for all z ~ f ~ Set / ~ . P ( t , z) = a(t, z + p). Then RP := 0pM(P)E MLP(R+)belongs to %P(R+)ifor every k (9 = ( - k , k ) ) , and it is elliptic as desired. 0 -+
Let us continue the proof of Theorem 11. If A is a Fredholm operator then A . = R-PA : Xo(R+) Xo(W+> is also Fredholm, R-P being as in Lemma 12 for -y. Here we have used the Fredholm property of R-P. In orther words without loss of generality we may assume that ord A = 0. By Definition 1 the operator A is of the form A = w OP,(U)W + IWop,(b)WZ + F1+ F + G , (30) +
where F1 denotes the remaining Mellin sums, and F E %>(R+),, G E gG(R+);, a, b E 9t::’. 13. Lemma. Let A E 9l0(R+);,a(z) = & ( A ) (z), a’(z) = b ( l - z) = a $ ( A ) (z). Then for every zo E r,,, there exist sequences h(zo),f;(zo) E Cz(R+),A E R+,with = llf;(zo)llL2(R+~ = 1 for all 1, (i) Ilh(ZO)llL2~R+) (ii) h(z0), f ; ( z o ) weakly tend to zero in Lz(iR+)
CIS
1
+
m
,
(iii) IIAh(z0) - a ( z o ) ~ ( z o ) ~ ~ L ~ (0R, + , -+
IIAf;(zO) - a’(ZO)f;(ZO)lIL2(R+) 0 ,
for 1
-+
w.
Proof. We choose an arbitrary f~ Cz(R+),supp f c (0, 11, IlfllL2(R+) mine h(zo) ( t ) in such a way that
(Mh(zo)) (112 + iy) = A 1 / 2 ( M f(1/2 )
+ i1(y - y o ) ) ,
=
1 and deter-
106
1. Conormal Asymptotics on R,
y = Im z, yo = Im zo. This is obviously satsified for f i ( z o ) (2) = J 2 T I ~ - 1 / 2 t (1 - ~2 - iyof (2 l/9 cf. 1.1.1. (24), (25). Note that supp fi(zo)c (0, 6) for every 6 > 0 and A 2 A ( 6 ) . Then we set f ; ( z o )= Zfi(zo),cf. 1.1.1. (11). The property (i) is an immediate consequence of Parseval's theorem for the Mellin transform, cf. Section 1.1.1., and the isometry of Z in L2(R+). The proof that fi weakly tends to zero for A + CQ follows by analogous arguments as in 1.2.4. Lemma 19, here applied to M$ So it may be dropped. Now let us set fi = f i ( z o ) and consider Afi, A being written in the form (30). Since G is compact in L2(R+)we have Gfi +O in L2(R+)for A +CQ because of the property (ii). Moreover (Fl + F ) f i +O in L2(R+) since both Fl and F are flat at t = 0 of order 2 1 and the support of fi shrinks to a small neighbourhood of t = 0. The obvious details may be dropped here. Moreover Zw op,(b)wZfi = 0 for A sufficiently large. For large A we have wfi = fi. Thus it suffices to show that 9
IIw opM(a)fi - a(zO)fillLZ(R+) +o for A + m. Since (1 - w ) op,(a)w is of the type flat + Green (cf. 1.2.5. Theorem 15) we also may drop w in front of op,. Then
IloPM(a)h - a(ZO)hlliZ(R+) = (25r1-l II a ( z ) ( M h )( z ) - O l ( Z 0 ) (Mh) ( z ) Ili2(r,,,)
s l[a(1/2 + iy) - a(1/2 + iyo)]A1/Z(Mf)(1/2 + iA(y - yo))12dy =(2n)-' s l [ ~ ( 1 / 2 + i y ~ - i A - ~ ) - a ( 1 /+iyo)](Mf)(1/2+iy)12dy+0 2 m
= (2n)-l
-m
-m
-m
as A - W . 0 Continuation of the proof of Theorem 11: The Fredholm property of A : L2(R+)+L2(R+)implies the existence of a continuous operator B : Lz(R+)+L2(R+)such that BA = 1 + K, where K is compact. Thus
IIuIIL~(R+)
I1(BA - K )u I~L~(R+) 5 II BAu I L z ( R +) 5 c IIAu IILz( R +) + II KZJIIL~(R+)
=
+ II Ku I~LZ( R +) (3 1)
I
u E LZ(R+),c > 0 a constant. Now we fix zo E
rl,,and insert u = f i ( z o ) .Then
1 = I1 fi(zo)~IL~(R+) 5 c IIAfi(zo)IIL~(R+) + II Kfi(zo)I I L ~ ~ + ) . In view of Lemma 13 for every E > 0 there is a A. E W such that
11 Afi(zo)~ ~ L z ( R 5+ ) I1 a(zo)fi(zo)~ILz( R +) + E 5 la(zo)l + 8 , I1Kfi(zo)IILZ*+) S for all A 2 lo.Thus 1 5 c la(zo)l + 28 for every E > 0 and hence a(zo) 0. In an anais arbitrary, we obtain that (ii) of logous manner we argue for a ' ( z ) . Since zo E rlI2 Definition 9 is necessary for the Fredholm property of A . It remains to show that (i) is also necessary.
*
1.3. Mellin Expansions of Operators
107
In view of the compatibility conditions of Definition 3 we already obtain from (ii) in Definition 9 that & ( A ) (t, z) 9 0 for all z 0 and 0 s t s E, 5 t s m for E > 0 sufficiently small. Now observe that A as an element in %O(R+);also belongs to MLtl(R+).Using 1.2.3. Proposition 11 it suffices to show that the homogeneous principal symbol of the corresponding lyDO in L:,(R+) is non-vanishing for t 0. Let us identify for a moment the t half axis with { x > O} and use the notations of 1.2.4. Proposition 20 for n = 1. Fix X ~ R+ E and t o e R and choose functions QI, v E C:(R+) with q ( x ) = 1, y ( x ) = 1 in a neighbourhood of xo. For 1 sufficiently large we have yrRA(xo,t0)q= RA(xo,t0)p.Moreover there is a compact operator K 1 in L2(R+)with
*
*
AVRn(X0, t0)P = VARA(X0,t0)P + KlRA(xo,to>QI,.
Using the notations in the estimate (31) for u = y R n ( x o ,t0)pwe obtain 0 < c = lIUIILZ(R+)=
- K)uIILz(R+) 5 IIBVARIP)llLz(R+) + IIK2RIqIILz(R+) with another compact operator K , , RA= RA(xo,to). Since llK2RAllLz(R+) + O as 1 + m, cf. 1.2.4. Lemma 19, we obtain for every E > 0 a 1 =A(&) > 0 such that for A 2 a(&) C SIIBvARIPIILz(R+) + E = I1 BRAR;'vAV~AQIIILZ(R+) +E
5 C~~IR;'VAWRIQIIIL~(R+) +E 5 clll a(O'(x0, t o > ~ 1 1 + 1 2~ 5 ClllPIl la(O)(xo,t 0 ) l + 2E,
provided 1 is large enough. Here we have used 1.2.4. Proposition 20. Since c = 11rpl1 for large 1,we immediately obtain a(0)(xo,to) 0 , in other words a ! ( A ) is non-vanishing as required in Definition 9, (i). Thus the proof of Theorem 11 is finished. 0
*
Now let us draw a number of conclusions from Theorem 11.
14. Proposition. Let A
E
%p(R+)i,9 = (- k, k ) , k E N \ { 0 } , and
A : XS(R+)+X"p(R+) (32) be an isomorphism for some s. Then (32) is an isomorphism for all s E R, and A-1 E %-P(R+)i. Prooj From Theorem 11 we know that A is necessarily elliptic, and then kerA, cokerA are independent of s. Thus (32) is an isomorphism for all s. In view of the ellipticity of A we find a parametrix B E %-p(R+)i,where B = 0. Let us show that B can be chosen in such a way that it is an isomorphism. Indeed, there are finite-dimensional subspaces N* c X,mS(R+)iof the same dimension with N+= ker B, N - 63 im B = XS(R+),where B : Xs+p(R+)+Xs(R+).Clearly N may be chosen independently of s. Now let P: Xo(R+)+ N + be the orthogonal projection to .N+. Then P E % , ( W + ) ~ , cf. 1.2.5. Proposition 6. Choose an isomorphism Q: N + + N - . Then B1:= B ( I - P) + QP: Xs+p(R+)+Xs(R+) is an isomorphism. Since - BP + QP E %,(lR+)i, the operator B1 E %-p(R+); also is a parametrix of A . Moreover AB1= I + G : Xs(R+)+Xs(R+) . 1.2.5. Proposition 5 we find a is an isomorphism, and G E % , ( R + ) ~Applying G1 E %G(R+)iwith ( I + G)-I = I + GI. Thus A-' = Bl(I + GI) E %-"(R+)i.0 8
Schulze, Operators engl.
108
1. Conormal Asymptotics on R,
Let us continue now the discussion of the example at the end of Section 1.2.3. It is obvious that every A E DiffP(R+) has the form k-1
k-1
A=
(l-w)t-Jop,(h;)(l-w)+F+G
wtjop,(hj)w+ j=O
(33)
j=O
w being an arbitrary cut-off function, k E W \ {0}, hj = u d ( A ) , h ; = u$(A),
F E % $ ( R + ) ~ G€WG(R+);, , Thus DiffP(R+) c
n W(R,);
B=(-k,k).
=: W(R+)*.
ksN
For the subclass Diff’(R+) we can apply all assertions of Theorem 11. In particular if A E Diffw(R+) is elliptic then there is a parametrix B E %-$(R+);with Mellin symbols that are not necessarily polynomials in z. Here k E N \ (0)is arbitrary and finite. In the following section we show that there exists even a parametrix in %-P(W+)*, i.e. including k = a. In the class DiffP(R+) we can easily construct examples of elliptic operators. For instance
is elliptic for every p E C \ r,,,. The asymptotic types of solutions to A u = f then contain explicitly the complex number p , where Rep < 1/2 only affects t = 0 and Rep > 1/2 only t = CQ. The operators in DiffP(R+) lead to interesting considerations on the nature of being Fuchs type (or totally characteristic) of an operator on the half axis. First of all it is obvious that every differential operator
can be written in the form
This suggests that the totally characteristic form is closely related to the “usual one” up to a weighr factor. Under an ellipticity condition we can express a parametrix of A” on R+in terms of standard yDO-s and smoothing operators up to t = 0, cf. [B9],[R8]. On the other hand we have a parametrix construction for A in the sense of Mellin operators, according to Theorem 11. Since both parametrices concern the same operator they must coincide up to negligible terms. In other wor_ds, we also get Mellin regresentations of the yDO-s occurring in parametrices of A. The reinterpretation A +A for differential operators does not rely on the ellipticity. So it is reasonable to expect that such a reinterpretation also exists for arbitrary yDO-s on R + . It turns out that this is true indeed, through it needs certain preparations on the action of vDO-s close to t = 0. A calculus in this sense was elaborated in [Rll].
109
1.3. Mellin Expansions of Operators
We shall return to this aspect in 1.3.4. and 3.3.6. in connection with boundary value problems. In the simplest form (namely order zero) such ylDO action have been mentioned in the introduction. It can be proved that operators of the form
wlr+ o p ( a ) e + w 2 , a ( t , z) E S:,(R x W) with arbitrary cut-off function w l , w 2 , generate a subalgebra of %O(R+)*which is closed under parametrix constructions (close to t = 0). The Mellin translation product of the associated Mellin symbols is compatible with the Leibniz rule between the ylDO symbols modulo Mellin symbols in Em;sm**. This shows that Wo(IR+)' contains many subalgebras with concrete interpretations. One of them is generated by those a ( t , z) that have the transmission property with respect to t = 0, cf. [B9],[R8]. Another subalgebra consists of those operators of Definition 1 for which hj,
. f ;E n;sm.'.
Next we will formulate the analogous calculus in the set-up of spaces and Mellin symbols with continuous conormal asymptotics. We do not necessarily assume that the carriers of asymptotics V of the Mellin symbols have the property r,,,-,n V = 0 for some y E IR unless the conormal order is zero. It is obvious that a calculus with such a condition would automatically lead to the more general case, since sums like h l ( z )+ h2(2 + e ) for arbitrary Mellin symbols h l , h2 and e E W are of the more general form. Therefore, we define the notion of decomposition data. 15. Definition. Let h E Em;, Y E Y, j E N. Denote by dec (h, j ) the set of all finite sequences r = { ( h v , e V ) :~ = l , . . . , N l , (34) N arbitrary, with
h, E Em;",
V, E Ya,
0 2 ev2 - j
for all v, h =
c V
h,,,. V =
1V,. (35) V
Let us call {ev)the sequence of weights of y. With every h E Em; we associate the expression
where op$( .) on the right is understood in the sense of (1). We hope that our notation will not cause any confusion. The meaning of op%(.) depends on the nature of e and will be indicated if necessary. Clearly h E Em;, V E YQ,0 2 e 2 - j , implies y = (h, e ) E dec (h, j ) , and then opL(h) = op%(h) where op& corresponds to (36), op$ to (1). Thus (36) is an extension of the earlier notation. 16. Proposition. For every j y E dec (h, j ) with N 5 2.
EN
\
{O}, h E Em';, V E Y, there exist decomposition data
Proof Let e, e' be arbitrary reals, 0 2 e > e ' 2 -j. Then V can be written as V = Vl + V2 with Vl E Ye, V, E Ye'. It suffices to set V, = V \ S:,2, V, = V n for A = (-6, a), A12 = (-6/2, 6/2) and 6 > 0 sufficiently small. Then we can apply 1.2.3. Theorem 6.
110
1. Conormal Asymptotics on R,
17. Dewtion. Denote by 91P(R+)a,p E R, 9 = (- k, k ) , k E N \ {O), the set of all operators (2) where F E91!k(R+)ff,G E 91G(R+)~, and S,S' are given by the formulas (3), (4) with arbitrary hi E %,; f i E Em",,, b, V; E Y, yj E dec (hj, j ) , y ; E dec ( f i , I ) , j , 1 = 0, . .. , k - 1, w an arhitrary cut-off function. Moreover define w ~ ( R +=)
n WR+)~.
(37)
keN
Remember that 91"(R+)efor 19 = (0,O) was already defined by (6) above.
18. Proposition. Every A E %P(R+),induces continuous operators A : !Xi(R+)e+!X;-'(R+)e
for every B E Yo with some C = C(B, A ) E Yo, and all s E R. Proof The assertion is an immediate consequence of 1.2.5. Definition 8 and Proposition 3, and 1.2.3. Proposition 18. 0 19. Proposition. For any decomposition data y, E dec (h, j ) , h E !Ill;,j E N, and cutofffunctions wi, Gi,i = 1 , 2 , we have
w,tJop&(h)w, - G,tjop&(h)G, E %;+,JR+).
Proof: Without loss of generality we may assume wi = Gi= w with some fixed cut-off function w, since the change of the cut-off functions in the expressions w,tjop$(h,)w, only contributes terms as asserted, cf. 1.2.5. Theorem 15. In other words we have to discuss a difference of the sort
Eet us choose reals e, e' with 0 2 e > e' 2 - j and decompositions h, = f v + f k, hx = g, + g:, where f,,gx,:E:mE f i,gk E 9t:bp.' This can be done in such a way that
f, E %:;@v
I
n mh@,f E
n
m $ @ y
a$@',
:
g, Emd,:% n ,m :: g E 9t:$ n mtsp'. Applying 1.2.5. (12) we get modulo 91G(R+)the following equations
wtjzopg(h,)w
=
wtjop%(rn)w + wtjop$(m')w,
V
wtjxop>(h",)w = wtjopG(n)w
+ wtJop$(n')w.
x
Here m =
x V
m
f,,
1f:,
ml=
V
n=l
g x , x
n'= c g : . B y definition we have
+ m' = n + n' = h and then f:= m - n = -(m'
x
- n'). This yields modulo fRG(R+)
c = w t j op$( f ) w - w d op$( f ) w . Since obviously f :E:nrE nm ,': we can apply once again 1.2.5. (12) and obtain c E %G(R+).0 Proposition 19 shows that the Mellin symbols of A E 91fl(R+)eare independent of the choice of the decomposition data and cut-off functions.
111
1.3. Mellin Expansions of Operators
For analyzing the resulting asymptotic type C in Proposition 18 it is convenient to fix the decomposition data, according to Proposition 16. Let us discuss, for instance, the part belonging to the Mellin symbols at zero. We set Yo = Kho,
O)}
1
j z 1,
Yj={(hj,l,ej.l),(hj.2rej,z)}r
where hj = hj,
+ h j , z , hj,
ej, = j / 2 , ej,
E ‘3Rb,v,v = 1, 2, and =j -
1/ 16 ,
v,
=
V n Kj,”,
v = 1, 2,
Then (3) induces continuous operators
B(,, = B n {Re z < 1/2}. This only depends on the decompositions hj = hj, + hj,*.
ej,”and 5,”but not of the concrete
20. Remark. Denote by W(R+)8,V, v = 1, 2, the subspace of all A E W”(R+)8for which s g ( u i J ( A ) )5 K , , , s g ( a G ( A ) ) E KJ,”:= 11 - Z: z E K,,”}, f o r a l l j = O , ..., k - 1 . Then
(39)
RP(R+), = WR+)s.1+ XK(R+)8.2
in the sense of vector spaces.
With the class %’(R+)ewe can perform a similar calculus as above for the dotted subclass. All definitions and propositions have immediate analogues for the continuous asymptotics. We will not repeat everything but only sketch the points where we have modifications. 2 1. Definition. Set
eke = m:o , and let
k- 1
x
x m:,,
j=l
k E N \ {O},
be the subspace of all { { h i } , { h i } ,p } E 26L,, x S(”)(TIlR+\ 0)
for which the compatibility condition (10) is satisfied. Let c
A
= (-1,
2 m p x S””(T$+
\ O),
l), be the subspace of all { h , , hb, p } satk;fying (10) for j = 1 = 0.
112
1. Conormal Asymptotics on R,
The definition of a,, 05 applies as for the discrete asymptotics, and we have the
"5) define surjective operators %'(Rt)a -+@k,a,
22. Theorem. a,, a!, (a,, a,: (15
%"R,)a
:
-+
S"'(T$R+ \ 0) ,
05): %'(R+)a+@'(R+)a with the kernels
ker a , ker a5
= %+ :
ker(u,, 43
= %$;'&+)a,
c(Rt)e,
= %"'(R+)8,
and respectively. Moreover the triple a = (a", a", a;) induces a surjective linear operator a: %'(Rt)a +@'(Rt),,
A
= (-1,
l), with k e r a = { A E % " - ~ ( R , c-ordA )~: = c'-ordA
=
-I}.
23.Theorem. %"(R+)ais a *-algebra filtered by the orders, and %~+c(R+)a and %c(R+)aare two-sided ideals. On symbolic level we have the formulas (20), (21), (22), (27), (28). Moreover Z%@(R+)aZ-l = %'(R+)ewith the same transformations of the symbols as in Proposition 7. Prooj The arguments are similar as in Theorem 6. The new specific point is the composition of Mellin operators WtP opd,(h)w, wtqopgg)W, p, q E N , and y E dec (h, p ) , 6 E dec (g, q ) , h E %!&, g E %is. For notational convenience we will write if we talk about equations modulo flat + Green operators. From 1.2.5. Theorem 15 it follows that wtPop&(h)w2tqopd,(g)w wtPopb(h)wtqopd,(g)w. We may restrict ourselves to
-
-
y = {(hi, e j ) : j = 1,2}, 0 2
ej2 -P,
6 = I(&, &): k = 1,2}, 0 2 Ak 2 - 4 , because of the Propositions 16, 19. Again by Proposition 19 we can pass to decomposition data where el = R1 = 0, ez = Rz = - E for some E, 0 < E < 1. Then our composition is of the form w {tp op,(hl) + t P- opM(Tshz)rE}w { t q opy(gl) + r e op,(T8gz)te)}o. This is equivalent to w op&((T-qh)g)w with a certain 7 E dec ((T-qh)g, p + q ) if we show the following equivalences (I
wtP - a op,( Taa)t a u t @b- op,( Tsb)t ''w
where b E %;:tI
(E
dec ( ( F a )b, p
+ q),
and cc, B
-@,
w op&(f)w + w opL("T)w
L0
- w op&((T-qa)b)w ,
with p - a, q - B 2 0, a E
- w op&(f + J ) w
(40) -a,
(41)
113
1.3. Mellin Expansions of Operators
f7
for every f, f " "ta ~ms, F E dec (f, r ) , FE d e c ( x r ) with some 9 E dec (f+ r ) . (41) is certainly true, since we can always pass to decomposition data by equivalence with the same sequences of weights, cf. Proposition 19. In other words it remains to show (40). Let us replace (40) for a moment by
(42)
W t Y OpM(3wf 'opM(b")f ' 0 .
B=
If 0 we can remove w in the middle by 1.2.5. Theorem 15. So let decompositions N
a> 0. Choose
N
and write for (42)
=
I
w t ~ [ ~ o p , ( $ ) w t ~ ' + k o p , ( b ;tpw )
B=
with pi + pj, pi, pi 2 0 to be fixed. We find the decompositions of 6 g i n such a way that
< Re z < 1/2 + B) + ( i - l)B/N < Rez < 112 + ( i + 1)P/N}, sg(b;.) n {1/2 - B< Re z < 112) G - {1/2 - B+ ( j - l)B/N < Re z < 112 + ( j + l)B//N}. Thus for N 2 4 we can choose the Pi, Bj such that sg(4) n {1/2
E - {1/2
4 ~ m z ~ n ~ m ~ h. E- B4 z,O n " t m J , .
Hence we obtain from 1.2.5. Theorem 14
-
OpM(&)WtBt+'jopM(6) t p ~Wt"+piopM(T-pi&)w opy( TBjh)t @ +4w wtY+pi opM(T-pi&) op,( Tpjq)t @ @jw = wtY+Bi op,( T-pi&Taj&)t!'+Bjw = WtP-'+B.0pM( T-Di + maiTBj + Bbj)f q + a - Biw
w ~ Y
+
I
with ui = T-"& bj = T-Bh. Now T-Bj + @ai TPj pb. = T4 + a - pi(( T-4ai)bI.)' +
since Bj +
=q
+ a - Pi. This is exactly what we need.
0
24. Remark. The canonical mapping
W"(R+)w+%"(R+)e for %' t % is an algebra homomorphism compatible with the operations *, I.
114
1. Conormal Asymptotics on R+
25. Definition. An operator A E %@(IR+), is called elliptic if the conditions (i), (ii) of Definition 9 are satisfied. B E %--"(R+),is called a parametrix of A if
AB - I , BA - I E %G(R+)a. 26, Remark. The notion of a parametrix always refers to the algebra. In the smaller dotted subclass we assume the remainders AB - I, BA - I to be dotted Green operators. In the more general situations below we understand parametrices in a similar way. In ambiguous cases we talk about parametrices within the given class. Then the algebra property automatical& ensures that the remainders belong to the corresponding spaces of Green operators.
27. Theorem. Let A E %@(It+), be given. Then the conditions (i), (ii) in Theorem 11 are equivalent. If A is elliptic there is a parametrix B E %--"(R+),. Moreover Au = f E Xg(IR+),, C E Yo,u E K m ( R + )implies , u E X;+-"(iR+),for some D E T odepending on C and A. In particular ker A c X;o(R+),for some DoE T oand ind A = dim ker A - dim ker A* is independent of s E W. Proof. The method of proving Theorem 27 does not contain any specific novelty compared with Theorem 11, except perhaps that the inverses of the leading Mellin But this is quite simple, since it suffices to take the inverse on symbols are in %%';Po. C \ {z: dist (z, sg(h) > E } for any E > 0, where it is meromorphic, h = o L ( A ) or aO,(A). For E +0 we might obtain more and more poles with increasing multiplicibut this does not destroy the property h-' E %%';Po. 0 ties when h E "tho\ %%';o~',
The higher-dimensional generalizations of our calculus as well as the infinite Mellin sums require natural locally convex topologies in the spaces %@(R+)iand %@(R+),. Let us consider the class with continuous asymptotics. The discrete case then is completely analogous and the details are left to the reader. In Section 1.2.5. was already defined an adequate locally convex topology in the space %$+ G @ + ) B
= %$(W+)t3
(43)
+ a7,(R+)B*
The topology of %@((w+), will be defined in such a way that (43) is a closed sub%% of Mellin ':, space. First remind of the locally convex topologies in the space %Rho, \ 0) (cf. Section 1.2.3.). Then the symbol spaces a$,, symbols and of S(@)(T;R+ and @@(It+), have also canonical locally convex topologies. From Theorem 22 we get an exact sequence (OM,a!)
0 +%$;L(R+)e +%@(R+)B
'W((R+)#
+o.
(44)
This sequence does not split for k > 1, since the lower order Mellin terms require individual decompositions according to the asymptotic types. So we do not find a by right inverse of (aM,a;). But if we replace %@(R+), and W(R+), by @W+),*u
= (GM, a;) %-"(R+),,",
v = 1 , 2 (cf. Remark 20) then the corresponding sequence splits.
Indeed, we find a mapping O P ~ :@j'(R+)~,v +%'(R+)s,v
115
1.3. Mellin Expansions of Operators
with (aM,a;) Op, = 1. It suffices to set Op,({{hj}, { h i } ,p } ) = Al
c
k-1
A1 = w
t'op$'(hj)W
+ zw
c
+ A o , with
k-1
t'op$qh;)wz-',
I=0
j=O
where h;'(z) := h ; ( l - z ) , ej,"given by (38), A . an operator satisfying 1.2.5. Definition 8 and
@;(A,)(t, t)+ aS(A1) (t, = P ( t , d The construction of A. is evident. Thus we have vector space isomorphisms
.
WR+)B," W;L(R+>, @@"(R+)ff,", 1 , 2 . This gives via the bijection a locally convex topology in %"(R+)8,v and by (39) then also in %fl(R+)ff. An equivalent description of the topology of %"(R+)ff,v can be obtained as follows. From Theorem 22 we get an exact sequence Y =
0 '%$+c(W+)ff
- % m + ) B , " a @ L , f f ,-0 "
which splits. This yields a vector space isomorphism
%"(R+)ff," +%$+G-(R+)ff @@L,ff,". (45) In 1.2.5. it was introduced a natural locally convex topology in the space %$+ c(R+)ff. Then the topology of %p(R+)ff,, equals the weakest one (locally convex) under which both (45) and a; are continuous. 1.3.2.
The Mellin Expansions
The spaces 1.3.1.(5) and (37) will be regarded as the projective limits of %@(R+); and %"(R+)B, respectively, over k E N, 9 = (- k , k ) . Let us set @& = @if,,-,,,),@W+) = @"(R+)(-m,-),
and similarly with dots. The goal of this section is to generalize the results of Section 1.3.1. to the case t9 = (- a , m ) . For brevity we mainly discuss the classes with continuous asymptotics. The case with discrete asymptotics is completely analogous but simpler and then left to the reader. Assertions for the projective limits will be expressed as for the classes with subscripts 9,where k = m. The operators A in %"(a+) have by definition expansions into Mellin actions like wrJopb(hj) w in the sense that the difference of A and a finite Mellin sum of length N is flat and Green, where - N may be thought as the conormal order of the remainder, tending to -m. That is why we talk about Mellin expansions of operators. The idea of Mellin expansions is one of the supporting principles for the operator spaces in our book. Such operators transform the spaces with asymptotics in an adequate way, and the solvability is formulated in such spaces. It is an interesting question which operators do belong to this class, when they are given a priori by an-
116
1. Conormal Asymptotics on R,
other analytic expression such as r+op(a)e+, cf. the Introduction, formula (9). We shall return to this aspect below in Section 1.3.4.
1. Proposition. 1.3.1. Propositions2, 18, hold including k
= m.
Proof: In view of W(R+)c %fl(R+)8 for all k E IN we can apply 1.3.1. Proposition 18 to A E W#((IR+). Thus A : Xi(R+)8+X7es,-’(R+)8
(1)
is continuous for every k , B E To,and C = C ( B ,A ) E Yo is obviously independent of k. Thus A also induces continuous operators (1) for k = m. The same consideration applies for the dotted class. 0 In this section we use the notation uM(A) for A E Wp(R+) in the sense of 1.3.1. (8) with k = m, i.e., the meaning of U, is determined by the length of the Mellin expansion of the operator.
05,
2. Theorem. a,
(a,, a;) define surjective operators
a,:
W’(R+)+$&,
a; :
B’(R+) + S(”(T$+\O),
05) : WR+) @”‘R+) +
with the kernels ker a,
= %+ @+),
ker a;
= W-l(R+),
and
05)
ker(a,,
= %h(R+),
respectively. An analogous result holds for the dotted subsclass. Proof: It suffices to show the surjectivities of the symbol mappings. The assertions on the kernels follow from Wp(R+) c %!’(R+)8for all k , ker
=
nker k
=
nW+
G ( ~ + ) 8=
%+:
G ( ~ + )
k
and similarly for ker a;, ker(u,, a:). The surjectivity of a, is a consequence of the following
3. Lemma. Let hj EB;,, V, E 7, j E IN, V0e V , be an arbitraly sequence and yj = {(hj,v,ej,J : v = 1,2} be a sequence of decomposition data such that
j+ej,,+a as j + m , v=1,2. Then there exists a sequence of constants cj such that the sum -ej,v+oJ,
m
w ( c j t ) tjop$(hj)w(cjt)
S= j-0
converges in %’(It+)
as well as the sum of the a ~ o i n t s .
1.3. Mellin Expansions of Operators
117
ProoJ Set Mj,”(cj)= w ( c j t ) dop>v(hj,.)w(cjt). Then S = S, + S2 where
c m
S” =
Y,”(Cj),
v = 1,2.
j=O
It suffices to show the convergence separately for v = 1,2. Fix k E IN \ (0) and write m
S, = S i + S :, S:
=
Mj,Jcj). j=k
By the assumptions on the weights ej,”we know from 1.2.5. Theorem20 that S” converges in 92$+ G(R+)a provided cj is increasing sufficiently fast as j + m. Indeed, for the convergence we only have to check a countable semi-norm system in ill$+G(R+)a, cf. 1.2.5. Remark 19. An appropriate choice of the cj follows by a diagonal argument. Since k E N is arbitrary, we may apply once again the diagonal argument and obtain the convergence in the projective limit W#(R+).It remains to u f ) . Here we may follow the lines of the proof of prove the surjectivity of (aM, 1.3.1. Theorems 4 and 22. First we establish the surjectivity of uf by replacing 1.3.1. (18) by m
:=
C w ( c j t )tjopM(aj)w ( c j t ) j=O
and a j ( z )E Em$ associated with b(t, z ) in the given way. If cj+ m sufficiently fast we get as above the convergence of the series in %P(R+) for all k (note that wt’ opM(aj)w = utJop$(aj) w for all reals r j 2 0 with j - r j 2 0 , cf. the proof of 1.2.5. Theorem 14). Moreover m
p l ( t , iy): =
C a ; ( w ( c j t ) tjopM(uj)w ( c j t ) )
j=O
converges in S(P)(TzE+\O) for a suitable choice of the cj (this is the argument of the classical Bore1 theorem). Then p ( t , iy) - p l ( t , iy) is flat of order m at t = 0. In the same way we proceed for t = 00 and find an operator A 2 E !V(R+) such that po = p - uf(A1)- a f ( A 2 )E S@)(T:R+\O) is flat of order m at t = 0, f = m. Now we can pass to a F0(t,z ) by the arguments of 1.2.4. Proposition 10, F0(t,z ) being holomorphic in z and still flat, jio(r, iy) E S!!!(R+XR) having p o ( t , iy) as homogeneous principal part. The associated Mellin IJDO A . belongs to %$(R+)and then A = A . + A l + A 2 has the property $ ( A ) = p . At the same time we see that (aM, uf) also is surjective, since aj may be replaced by any other sequence of Mellin symbols hj with the only property that the compatibility condition 1.3.1. (10) holds for allj. Here we might pay with another order of decrease of the cj which does not affect the result. 0 4. Theorem. Wm(R+) = U W ( R + ) is u *-algebra, filtered by the orders, and Ir
a;+G(R+),91G(R+)are two-sided ideals in 9IZm(R+). Moreover the operators admit
the conjugation with I in W(R+). For the symbols we have the same rules as in 1.3.1. Theorem 6 and 1.3.1. Proposition I. An analogous result holds for the dotted subclass. ProoJ By definition we have W(R+)c W(R+), for all k~ N, J!, E R. Then 1.3.1.
118
1. Conormal Asymptotics on R,
Theorem 23 implies A E W'(R+), B E % " ( R + ) A a B E %""@+)a for all k E N , i.e. A B E W+"(R+).In addition we see that the symbolic rules 1.3.1. (20), (21), (22) hold, since they are true even for all finite k . The assertions on the * operation and the involution I are obvious as well. From 1.3.1. Theorem6 we obtain the same for the dotted subclass. 0
5. Defhition. An operator A E W(R+) is called elliptic if it satisfies the conditions (i), (ii) of 1.3.1. Definition 9. B E %-#(IR+) is called a parametrix of A if A B - I , B A - I E %G(R+).
For A, B in the dotted subclasses we have A B - I , BA - I E WG(R+)*. 6. Theorem, For A E W(R+) the conditions (i), (ii) of 1.3.1. Theorem 11 are equivalent. -If A is elliptic there is a parametrix B E %-fi(R+).Moreover Au = f E %",(R+), C E Yo,U E % - " ( W + ) ,im p liesu E % ~ + p (R t) forsomeDE YOdependingonCandA.In for some Do E Yo, and ind A = dim ker A - dim ker A* particular ker A c Xi&) is independent of s E R . If A E W(R+)' is elliptic there is a parametrix B E V ~ ( R + ) ' .Moreover A u = f E X i @ + ) ,R E a0, u E Wm(R+),implies u E %;+p(R+)for some P E a0depending on R and A. In particular ker A c %y0(Rt)for some PoE so. Prooj The equivalence of ellipticity and Fredholm property is a consequence of 1.3.1. Theorem 11. The ellipticity allows to determine the Mellin symbols b, = oiq(B), bb = u;T(B), q E N , by solving the equations d i ' ( A B ) = a;!(AB) = 81,0for all 1 E N. By the constructions of the proof of Lemma 3 we find an operator B"E %-p(R+)with b, = g&,(B"), bb = aiT(B")for all q. Theorem 4 now imAs in the proof of 1.3.1. Theorem 27 we plies that AB"= 1 + # with I?E%$+GIR+). can choose B" in such a way that 1 + H is elliptic. From 1.2.5. _Proposition13, we get :, c(R+),and then B := B(1+ H ) E Wp(R+) is a parametrix 1 + H of 1 + I?, H E % obviously a parametrix of A . The arguments for the elliptic regularity are analogous as for 1.3.1. Theorem27. 0 Remember that the classes of smoothing operators in W(R+) and Rp(Rt)' are %G(R+)and 91G(R+)',respectively, where
%G(R+)*
5
%;(R+) @n%z(R+).
(4)
P,QeIR0
We have the analogues of 1.2.5. Proposition 3, 4, 5, 6 for 8 = (-
m, a).
7. Theorem. Let A E %'(R+)a, 8 = (- k , k ) , 1 5 k 5 a,and A : Xs(R+)+W-~(R+) be an isomorphism for a fixed s E IR. Then (5) is an isomorphism for all s E R, and A-' E %-fl(Rt)8.The analogous result holdr within the dotted subclass.
1.3. Mellin Expansions of Operators
119
Proof. The bijectivity of (5) implies the ellipticity, cf. 1.3.1. Theorems 11 and 27, and ker A , coker A a_re independent of s. Thus (5) is bijective for all s. Moreover we find a parametrix B E %-'(R+)8(%-p(R+);). It can obviously be chosen as a bijective operator from X s - - r ( R + )to Xs(Ri), _since ind B = 0 and ker B , coker B"c%rs(R+)8(Xrs(R+);),so that B := B + G becomes bijective for some finite-dimensional Green operator. Now BA = 1 + G, G E Rc, is again bijective, and (1 + G)-' = 1 + GI for another Green operator GI,cf. 1.2.5. Proposition 5 and its 00). Thus A-1 = (1 + GI) B E %7-fl(R+)e (%-"(R+);).0 analogue for (a,
1.3.3.
Operators with Mellin Expansions at Zero
The calculus of operators with Mellin expansions induces a calculus, where the Mellin expansions only refer to t = 0. In this section we shall formulate the corresponding theory. We will also talk about actions in spaces with weights. This is another generalization of the results in 1.3.1. and 1.3.2. If we are interested in the Mellin expansions only for t+O we shall employ the S Hfb,(R+),which are XssY(R+)close to t = 0 and HS(R+)close to spaces Xs*y(R+) t = a,cf. 1.2.2. (22), and the subspaces with asymptotics. As usual y will also be omitted when y = 0. By definition we have Xo
xs* -y(R+) c, +
s, y E R,
which admits an identification
XS.Y(R+)'=
-y(R+).
The formal adjoint A* of some continuous operator A : Xs,Y(R+)+X'*Q(R+)
will be defined with respect to (.,
A* :
.)L~. Then
X-', -Q(R+) -+ X-', -Y(R+).
1. Definition. An operator A
E
ne(xs(~+), x-(R+))
S€R
ir called a Green operator of the class 2c(R+)8,19 = (- k , 01, 1 5 k 5 00, $ A , A* induce continuous operators
A : Xs(lR+) + Xi (It+),,,
A* : XS(R+) + Xz (R+)w,
9' = (- k , m), for all s E R with certain B , C E do.Let 2,(R+); be the subclass defined bY
A : Xs(R+)+XF(R+)g, A* : Xs(R+)+Xi (R+)w,
120
1. Conormal Asymptotics on pi,
for all S E W and certain P , QEP, depending on A. For k = 0 we define = 2,(R+); by the mapping properties A , A * : X'(R+)+X,"(R+)p,m).
Remember that ( - k , a) indicates flatness at in the sense of 1.2.2. Definition 15. Every A E e,(R+), has a kernel g(t, t') E C"(W+XW+) with x ( t , t') g(t, t') E Y(R x W)JR+xR+ for every x E Cm(R+ x R + ) with x = 0 for t < ~ / 2or t' < d 2 , x = 1 for t > E and t ' > E for some E > 0. Now let us introduce further classes of operators with a prescribed behaviour for t+ 03. Let L;l(R)o be the space of all A E L$(R) which are of the form A u ( t )=
s
ei('-'')'a(t, t', t)u(t') dt' dz,
(1)
where a ( t , t ' , t)E 9 ( W x R)8,S;,(W)),,,,,, S~l(W)c,,,, being the space of classical t, 1' independent amplitude functions, in the corresponding induced Frkchet topology. Note that the amplitude functions which are admitted in (1) can be described as a ( t , t ', t)E Cm(Rx R x R)with ItNt'MD~,.D,ka(t,t ' , z ) I 5 c(1 + IzI)"-k
for all cc E N2,.k E N, N , MEN, t , t ' , z E R, with constants c = c(a,k , N , M). For L:(R), we can'easily derive the standard elements of a yDO calculus such as rules for compositions, adjoints, complete (i.e. t' independent) symbols, asymptotic sums, ..., where everything refers in an obvious manner globally to R. In particular L,m(W)o consists of the operators with kernels in the Schwartz space Y(W X R). Let Lg1(R),be the subspace of all A E L;l(R) which are invariant under the translations Tp: W+R, P ( t )= t + e, for all e E R. An operator A E LE1(R)),can always be written in the form A u ( t )=
s ei"-")'a(z)
u(t') dt'dz
with some a ( z ) E S~l(R)c,,,,.We set a (z) = a! ( A )(z).
(2)
This is a correct definition. Indeed, let Z(z) E S!!l(R)c,,,, be another complete symreprebol of A with constant coefficients. Then ao(z) = a(z) - Z(z) E S-m(lR)const sents the vanishing operator.We then have 0 = J K o ( t- t ' ) u ( r ' ) dt' for all u E C,"(R) for K o ( t )= J e1f7ao(z)dz.If this vanishes for all t it also vanishes at t = 0. Since then
Ko(-r'> u ( t ' ) d t ' = 0 for all
uE
C;(R),
we get KO= 0, in other words a. = 0.
2 . Definition. L$(W), denotes the space of all operators A = A o + A,, where Ao E L;i(R)o, Ac E L;i(Nc.
121
1.3. Mellin Expansions of Operators
It can easily be proved that the decomposition of A unique. This admits the definition
E L$(R), as
@!(A1 ( r ): = a! (A,) (t)
a sum A.
+ A , is (3)
for A E L$(R), 3. Definition. ML't,(R+),k the space of all operators of thefonn
A
=
wAn,S + ~ A o f + xACf
(4)
with arbitrary AME MLEl(R+), A. E A, E L!!!(R),, and o,6 E Corn@+), w , S = 1 close to r = 0 , X , ~ Cm(R+) E with x , f = 0 for t < ~ / 2 x, , f = 1 for t > E with some E > 0. 2$(R+)@, 9 = (- k , 01, 0 5 k 5 a, is the subspace of all A E ML$(R+),for which cJ: 1.2.5. Definition 8. Moreover we set AM belongs to W$(R+)(-kk,, 2$+ G ( R + ) B
+ 2G(R+)B
= 2$(R+)B
(5)
and analogously with dots. Any A
E M L $ ( R + ) ,induces
continuous operators
A : Xs(R+)+Xs-@(R+)
(6)
for all s E R. It is a nice exercise in terms of pDO-s and the theory of 1.2.4. to show that the elements of the calculus such as compositions, adjoints, ... , can be established also for ML:(R+), . In particular ML:(R+), is a graded algebra, and
ML:(R+)o:
=
{the operators (8) with A, = 0)
(7)
is a two-sided ideal. For A E MLgl(R+),we can recover A, E L$(R)cin a unique way by inserting functions with support tending to m, such that the remaining part dies down. This yields a symbol map
4: ML:l(W+)l--+
%(R)CO"St
with &A) = a;(A,). Moreover we have ML$(R+), c L$(R+) and thus a homogeneous principal symbol map a;: ML;,(R+), +P ( T * R + \ O ) ,
a @ ( A ) ( t , t )being the homogeneous principal symbol of A of order ,u and the corresponding space of homogeneous symbols. Clearly o $ ( A ) ( t , z ) + ~ ~ ; ( A , ) ( for t ) t + m and o $ ( A ) ( t , t - ' t )is C"up to t = 0 . Let @'(T*R+\O), be the subspace o f a l l p ( t , t ) E C"(R+X(R\O)) for which , for all A > 0, ( t , t)E R+x (R \ O), (i) p ( t , Ar) = A @ p ( t t) (ii) p ( r , t - ' t ) E c ~ ( R + x ( R \ O ) ) , (iii) there is some p y ' ( z ) E Cm(R\O) satisfying (i) such that for p o ( t ,t)= p ( t , r ) - x p ? ) ( t ) with x as in Definition 3
SG)(T*R+\O)
ItN a:po(tl t)1 5 CN,k
for all
and every N , k E N with constants
t E Hi+, cN,k
17)= 1
> 0.
122
1. Conormal Asymptotics on R,
Then a; defines a surjective map a;: MQ!,(W+),+
S$)(T'lTC+\O)I.
We can easily characterize the subspace
S$")c Sy(T*R+\O), x S~,(R),,",, of all pairs { p v ( t ,z),p,(z)} for which there is an A E ML;,@+), with p w = $ ( A ) , p c = a;(A),namely by the compatibility condition p w ( t ,z) - p $ ) ( z ) + O for t+ m, if p',"'(z)is the homogeneous principal part of p c ( z ) of order p . The space Sp) is FrCchet in a natural way. The kernel of the surjective symbol map a= (a;,c7;>: MLgl(R+)l+Sp)
(8)
is hfL$-l(R+)o, cf. (7). Clearly for A E ML@+), , B E MLt;(R+),
(9)
a(AB) = a ( A )a ( B ) ,
a(A*) = ( a ( A ) ) *
(10)
with component-wise multiplications in (9) and component-wise adjoints in (10) that we can read off from the formulas in 1.2.4. 4. Definition. P(R+);, p
A
=
S+ F +
E
W,8 = ( - k , 01, 1 5 k 5 m , is the set of all operators
G,
(1 1)
where F + G E 2$+G(R+)iand k-1
S=
wtjOp2(hj)w j=O
%r;v~',
with arbitrary hj E 0 2 yj 2 - j , and w an arbitrary cut-offfunction.f!'(R+)8 is the set of all (11) with F + G E i!$+ G(R+)eand hi E Ern,: , yj E dec (hj,j ) . For k = 0 we simp& set P(R+); = ~ Y ( I R +=) ~ML;,(R+),. Any A E 2"(R+)8(P(R+);) induces continuous operators
respectively, for all s E R, where the resulting asymptotic types only depend on those of the argument functions and on A but not on s. We set a z ( A ) = hj and
M A ) = Ia2(A)Io,- ~ j s k - l . ' _
The symbol map c7 also makes sense over f!p(R+)e. Let @t(R+)8(@;(R+)i) be the subspace of all { { h j } ,p v , pc} E X Sy)(a$, X S?)) for which pv satisfies the compatibility conditions of 1.3.1.(10) that belong to the origin.
123
1.3. Mellin Expansions of Operators
5.Theorem. uM,u, (aM,u) defne surjective linear operators UM:
~'(~+)B*%,tJs,
u:
E'(R+)8'S\fl),
(UM,fl>:ep(R+)8'@:(R+)8,
1 5 k 5 m, with the kernekr
el.,+c(R+)8, = er - y ~ +n)m ~ y y ~ + ) ,, ,
ker uM ker u
=
ker(u,, u) = f;,;',(R+)8
nML;;'(R+),,.
An analogous result holds for the classes with dots. Proof. The proof is an obvious modification of that of 1.3.1. Theorems 4, 22 and 1.3.2. Theorem 2. 0 We have an obvious analogue of 1.3.1. Corollary 5 for the Soperator spaces. A slight modification of the corresponding assertions of the previous sections yields 6. Theorem. A E B E i?Y(R+)8 implies A B E Pfl+Y(R+)8, A* E f!fi(R+)8, for the Mellin symbols of A B and A* we have the formulas 1.3.1. (20) and 1.3.1.(27), respectively, whereas for u it follows (7) and (8), respectively. A n analogous result holds for the classes with dots.
7. Definition. A
E
f!P(R+)*,9 = (- k, a),15 k 5
, is called elliptic if
(i) $ ( A ) ( & t - ' z ) f O fo r a ll(t, t ) ~ W + x ( R \ 0 ) , (ii) u ! ( A ) (t)f 0 for all Z E R , (iii) & A ) (z) f 0 for all z E rlI2. 8. Definition. An operator B E 2-'(R+)8 is called a parametrix of A
AB - I,
BA - I
E
E
f'(R+)a if
&(R+)B.
For A, B in the dotted subclass we require the Green remainders to be in the dotted subclass.
9.Theorem. For A
E
2p(R+)8,1 5 k 5
, p E R , the following conditions are equivalent
(i) (6) is a Fredholm operator for a fixed (ii) A is elliptic.
SE
R.
If A is elliptic there is a parametrix B E f?-P(R+)*.Moreover A u = f E XS,,(W+),, u E X m ( R + ) ,imply u E Xi: where the asymptotic type of u depends on that o f f and A , but not on s. For A E P'(R+); we get B E i?-"((R+);, and f E X",(R+);, u E X m ( R + )imply , u E XS,:'(R+);, where again the asymptotic type of u is independent of s. Proof. Theorem 9 is an analogue of earlier results of this type, cf. 1.3.2. Theorem 6. The only new point concerns (i) (ii) with respect to the ellipticity condition (ii) in Definition 7. So we want to show that the Fredholm property of (6) implies 9 Schulze. Operators e n d
124
1. Conormal Asymptotics on W,
&A) (z) f 0 for all z E R . The other elements of the proof are left to the reader. First note that for every p E R there exists an elliptic operator RP E P(Rt); (cf. for analogous considerations 1.3.1. Lemma 12). Then it suffices to consider AR-P. Thus without loss of generality we may assume p = 0. The Fredholm property ensures the existence of a parametrix B : L2(R+)+LZ(Rt),with BA = 1 + K , K compact. Thus
llullLVR,) 5 c IlAull LZ(R+) + IIKull LZ(R+) with a constant c > 0, cf. 1.3.1.(31). The operator A is of the form
(15)
A = w ( S + F ) G + x A o i + x A , i + G, S the Mellin sum, F the flat Mellin operator, G a Green operator, and Ao, A, as in (4). Let us assume for a moment that A. = 0. Then
A q = w (S + F ) Gq + xA,X"Q, + Gq. Without loss of generality we assume that #:(A) (z) 2 0 for all t;otherwise we consider instead the operator AA*. Let toE R be fixed and choose a sequence of func- ~ ~ the Dirac tions pk(t)EY(R), k E N , with (@k(z))2+d,-ro as k + m , 1 3 ~ being = qk(t- N - l ) , N E N , we have &.,(z) = eir(Ntl)&(z). measure at t o .For qkN(t) For every fixed EN it follows fpkN+O weakly in L2(R) as N + m . For every E > 0 there is an N,(k) E N such that IIAPlkN - xAci%vII L?R+) < E for all N e N o ( k ) . Here we have used that G is compact. Now (15) for u = qkNIR+ implies for some constant c > 0, c
~ NR +I IILZ(R+)5 IlA~mllLZ(R+) s IIxACV~NIILZ(R+)+ E S IIAcqmll L ~ ( R+) E
(16) for all N e N o ( k )where in addition N o ( k ) is chosen so large that the first inequality in (16) also holds. Since A, has constant coefficients we get A,q,
= &A)
(z) &,(z)
= #!(A)
(z) eir(N+ l) &(z)
and from Parseval's equation
I l ~ , ~ ) w v l l ~=~ (2W''' ( ~ ) IlO;(A> (z) eir"+l) @ k ( z ) l l ~ z ( ~=) llg:(A) (z) @,dz)ll~2(~), (17) since ei7(N+ l) is of module 1. Now we have Ila!(A)(d @dz)ll$(R)= ( g ! ( A 1 2 W , @2k(z))+4%l)2(~o) for k + m . Then (16), (17) yield d ( A ) (z,) > 0 . If A. is non-vanishing we only use the fact that it is a small operator for large 2. Then by replacing e by 2e in the above calculation and choosing w such that xA,fis non-zero only for the large t , we get the desired estimate in general. 0 The proof of the necessity of @!(A) (z) f 0 for the Fredholm property also could be derived from the necessity of the Mellin ellipticity close to zero in the context of t1'2Wfl(R+)8t-"*. In fact, from 1.1.2. we know that (Qil12u)(x) = u(ex) induces an isomorphism
XS'"(R,) --+ HS(R).
125
1.3. Mellin Expansions of Operators
Now 0 opL2(h) w
: sres,l/z(R+)~sre,-,,’/2(R+)
for h E Emt:/2 has a local parametrix close to t = 0 (in an obvious sense) iff h (z) f 0 on Re z = 0, cf. 1.3.1. Theorem 11. Now M1/Z= Y - y - ’ F @ l , i.e. z, wopgz(h)w = wM;;2hMl,zw = W @ ; ~ F - ’ Y / ~ ’ V - ’ F @ ~ , ~ ~ . Thus the yDO with constant coefficients F - ’ Y h Y - ’ F = F-’hAF with h A ( n = h(-iF), has a local parametrix close to x = --a, iff hA(& f 0 for all F E R. In the applications the operators with Mellin expansions are often multiplied by a weight function, according to the “interior order”. Let e E R and g@(t)E Cm(R+) be a strictly positive function with
with certain constants 0 < c1 < c2 <
R”’(R+)a
=
Q)
. Let us set
g‘-”2e”(R+),g-’,
(19)
y E IR, and analogously with dots, 9 = (- k, 0); 0 5 k 5 Q) . The operators A E Rp*y(W+)e(@ ‘(It+);) then induce continuous mappings
A : X~Y(R+)e+X~-”’Y-p(R+)B and
A : X j Y(R+)e+Xb-H ”-”(R+)e, Y with certain Q respectively, for all s E R and B E 9’ with certain C E 9 ‘ - p (PE P E P V - C ) where the resulting asymptotic types depend on the original ones and on A. It is obvious that the above definitions and results have straightforward generalizations to the S”*Yclasses. In particular we then talk about Mellin symbols of conorma1 orders p - j , j = 0, ..., k - 1 , and the condition (iii) in Definition 7 is to be replaced by
a&(A) ( z ) f 0 on z E r1,, - ‘. (20) In compositions we have to assume that the weights fit together. Note that the conditions on the behaviour of yDO symbols of operators in M J ~ $ ( Rfor + ) t~+ -a, can also be modified by demanding asymptotic expansions with respect to t of the form p ( t , z)
-
m
t-jpj(z) for t +
03,
with coefficients p j ( z )
j=O
E S~,(lR),,,,,. The essential elements of the calculus in this section can easily be generalized to this case. It will be explicitly elaborated in higher dimensions in Section 2.2.4.
1.3.4.
Examples and Remarks (The Mellin Calculus of Standard wDO-s on lR+)
In this section we want to describe subspaces of Rky(R+) of a particularly simple structure. Proofs will be omitted here. The details may be found in [Rll] and [SS]. 9‘
126
1. Conormal Asymptotics on R,
Let us first assume p
= y = 0.
We consider the algebra of operators generated by
r+ op(a) e+ =: op,(a) E 9 ( L Z ( R + ) )
(1)
for arbitrary a(r) E S~l(R)const (cf. the notations from (9) of the Introduction). Set g + ( z )= (1 - e-2niz)-l > g - ( z ) = (1 - e2nir)-l. (2) These are meromorphic functions with simple poles at the real integers. Further g*(z + k) = $(z) for all k E Z, and 0 for r+ + m 1 for r + - w '
(3)
for all B E R. From g + ( z )+ g - ( z ) = 1 it follows the converse behaviour of g - ( z ) . Now let
there follow unique coefficients uf E C , j e IN, From the definition of S:l(lR)const with m
i = 4 - 1 . Define (6) a&) ( z ) := ( a f g + ( z )+ a j g - ( z ) }& ( z ) E a;; with R j = ( ( r k , O ) } k . Z , where rk runs over Z \ { 1 , 2 , . . . , J } for j r l , and Ro = { (k, O)},, z . As usual we further set aM(a)= { ~ d ( a ) }z~. Note , that the space
m y := ( a d ( a ) :a E S0,,(R),,,,,} is of complex dimension 2. The sequence of Mellin symbols hj(z)= a i ( a )( z ) , j E N , can be used to produce operators
c m
A
=
W(Cjt)
r' 0P$(hj) w(c,t)
j=O
with constants cj+
m
- y j - + m , j + yj+w
, increasing sufficiently fast, and reals yj withj r - yj t 0 and as j + m , cf. 1.3.2. Lemma 3.
1. Theorem. Let wl, w2 be arbitrary cut-off functions. Then
~I{oP,(~ )A 1 ~
2
(1 - 01){oP,(~) - A 1 wi E %+G(R+)*
,
(cf: 1.3.3.(5) with 9 = (-m,O)). In other words op,(a)
=A
+F +G
with a frat operator F and a Green operator G with discrete asymptotics. 2. Remark. The proof of Theorem 1 employs in particular that OPdW
= ophf(g*),
127
1.3. Mellin Expansions of Operators
where P ( z ) is the characteristicfunction ofR+, &(t) = 1- 8+(t)(cf. ako [E4],6 15). On the left of (9) it was used that vD0 actions in L' can also be associated with non-smooth ymbols with constant coeflcienis.
3. Corollary. (1) induces continuous operators op,(a): W R + ) + W ( R + ) , op,(a): W(R+)-,X",Rt) for every s E R and P E .Po with some Q E Podepending on P and a.
(10) (11)
4.Theorem. Let a, b E S~,(R),,,,,.Then a i ( a b )( z )=
o i r ( a )(z - q ) aiq(b)(z)
mod
o s : ;%
r+q=j
for all j
EN
(cJ 1.2.3.(10)).
In the description of complete Mellin symbols we can also write @ d a b )= g d a ) # M M b ) . This suggests to consider the space of all operators
A
= op,(a)
+m +g
with a E S%R))const, g E EG(R+)*, m
m=
C o ( c j t )t j o p & q )o ( c j t ) j=O
with a cut-off function o,arbitrary sequences f i E %is-* ', fo E %g;m3 O, *, and reals y j , where f j has no pole on T l , z - v j j, 2 - yj 2 0, - y j + w , j + yj+w for j + m , and cj+ 00 so fast that the sum converges in 2-m(R+)*, cf. 1.3.2. Lemma 3. As we know from Section 1.2.5. the concrete choice of data w, cj, yj only affects m mod 2G(R+)'. In view of (7) the operators (12) form a subspace of i?O(Rt)*.In particular m + g E f!i;G(R+)'. 5.Theorem. Let A, A be of the form (12) with certain a, a"€ S~,(R),,,,,, i.e. A + r, J =op,(iT) + Fwith r, F E 2;TG(R+)'. Then
= op,(a)
AAI= opv(aiT)+ rl for another rl E 2:; G(R+)*.Further A* = op,(a*) + r2 with a* = 5,r, E 2 i ; G(R+)*. Clearly the symbolic rules of 1.3.3. Theorem 6 remain in force. In other words we have obtained an interesting subalgebra of P(R+)* which is linked in a particular way to standard VDO-s on R+ . Remember that ESKINin [E4] has described a more crude version, i.e. with a control only of the leading Mellin symbols. Our algebra contains {op,(a): a E S~,(R+),,,,,}and Theorem 4 gives rise which appears in compositions op,(a) op,(iT). to a minimal subspace of %;sm** Further the asymptotic types of the Green operators are more special than those allowed in 2G(R+)*.In particular we get a nice survey on the algebra generated by a single op,(a).
128
1. Conormal Asymptotics on R,
On the other hand we can pass to the larger algebra 2i(lR+)c 2O(R,)* consisting of all operators (12) where a(t, z) E S!,(R) depends on t and has a behaviour for t+ m as the 1pD0 symbols in 2O(R+)of Section 1.3.3. In this case we set
where a(t, z)
- c tkaLk1(z)for
t+O,
~ [ ~ ] (EzS~l(R)const. ) Let a, 6 al be of this
k=O
kind and a l = a #, Z,i.e.
Then the substitute of Theorem 4 is m
aM(a#, b ) = uM(a),# a,(b)
mod
)( Em,-6"-'. j=O
6.Theorem. Let A = op,(a) + r, 2= o p , ( a +-Fwith a, ZES;,(R) depending on t as mentioned and r, FE 2:; G(R+)*.Then AA = op,(a #, 3 + rl for another rl E 2:; c(W+)*. Further A * = op,(a*) + rz with a*(t, z)
-c
1
o:a:a(t, z),
r2 E 2 ~ ;G ( ~ + ) * .
7. Definition. A = op,(a) + r E 2i(lR+) is called elliptic if (i) q0,(t, z) 0 for all t and z =k 0 (a,,, being the homogeneous principal part of a(t, z) of order 0 ) , (ii) ut(A) (z) 0 for all z E R, (iii) aL(A) ( z ) 0 for all z E r1,, .
* * *
A E 2i(lR+) is elliptic in this sense iff it is elliptic in the sence of the larger class 2O(lR+)*,cf. 1.3.3. Definition 7. 8. Theorem. For A E $,(R+) the following conditions are equivalent (i) A : Xs(R+)+Xs((R+)is a Fredholm operator for all s E W, (ii) A is elliptic. I f A is elliptic there is a parametrix B E 2i(lR+) of the form B = op,(b) + r, for some r, E 2i: G(R+)* and b,o, = a(0:. Further the assertion on elliptic regularity of 1.3.3. Theorem 9 holdr.
Note that the only new point in Theorem 8 compared with 1.3.3. Theorem 9 is that there is a parametrix in the smaller class. The construction of a parametrix can be carried out successively by inverting the symbol components. First we choose a symbol b with a # , b 1. An obvious Lemma then says that there is an elliptic #E 2i(R+) that has the 1pD0 symbol b . From (15) it follows that A B = 1 + r, r E 2iT G(R+)*,and 1 + r is again elliptic. By solving successively a , ( l + r ) #,,,a& + r'J = 1 we find an r' E G(lR+)*with (1+ r ) (1 + r') = 1 + g, g E EG(R+)*. Then B = B ( l + r') E 2i(R+)is as desired.
-
129
1.3. Mellin Expansions of Operators
Another method is to solve a&) quence of Mellin symbols of B in the form a;@) (z)
E
{
%RL-*I) k=O
=
1 immediately. This yields the se-
+ %.*R;
(16)
We obtain a rather explicit information on the asymptotics of solutions u of Au = f E Xb(W+),Q E .!Po. This is interesting in particular for A = op,(a). The leading term of the asymptotics is then determined by the zeros of ak(a) = a: g+(z) + a; g-(z), where a: is formed for a(0, z), cf. (5). The lower order terms follow easily from the Mellin symbolic rules. 9. Lemma. The zeros o f a + g + ( z )+ a-g-(z) for a*& z=-
1
2n i
are given by
{log la+/a-I+ i arg (a+/a-)} + k , k E Z.
Note that
ok(B)(z)
=
{ a ; g+(z)
+ a; g-(z)}-l
= { (a:)-'g+(z)
+ (a;>-'g-(Z)} + h(z),
for some h E %X;sm*', cf. (16). Since g*(z) are meromorphic with simple poles at the real integers, the specific contribution to the poles of a k ( B ) comes from the term h of order - m . Let a (t)E S~,(pO,,,,,. It is convenient to regard a as a mapping a : R + C ,z+ a (z) . This is a C" curve V in the complex plane with endpoints a: = lim a ( t ) , cf. Fig. 7. r-i-
Fig. I
The straight connection 3 is just given by the values of a,'g+(z) runs over r1,,. A consequence of Theorem 8 is that
A
= op,(a):
+ a;g-(z)
when z
LZ(R+)+LZ(R+)
is a Fredholm operator iff 0 6 V u 3. For every ~ E % X ; ~ ~we~ ~ get ~ * by r,,,3 z+ a; g+(z) + a; g-(z) +f(z) another connection A. The operator
F = w o p M ( f )w : L2(R+)+LZ(R+)
130
1. Conormal Asymptotics on
IR,
is not compact unless f = 0. A + F: LZ(R+).+L2(R+)is a Fredholm operator iff 0 @ 8 u A. All this is classical knowledge from the theory of singular integraloperators, as well as the formula ind(A + F) = winding number of IS1.+@},where V is thought to be parametrized by 0 5 Q, 5 n and A by n 6 Q, 5 2n. A case of particular interest is a: = a,, i.e., V is a closed curve. In this case we talk about the weak transmission property of a (or op,(a)). It is quite instructive to look at homotopies of operators A(A) + F(A) = op,(a) (A) + wopM(fl (A) w ,
(17)
0 s As 1, where a(z,A), f ( z , A ) depend continuously on A. 10. Proposition. For every elliptic operator A + F = op,(a) + w opM(fl w there exists a homotopy (17) throughellipticoperatorssuchthatA = A(O),F = F(0) andA(1) hastheweak transmission property whereas F(1)= 0. Next we want to examine the conditions under which (10)induces continuous mappings (18) op,(a) : X",(R+) X",R+) for T = {(-j,O)}jeN, i.e. for the "Taylor asymptotics" at zero. There are many ways of reformulating op,(a) by Mellin actions. In this case it is convenient to look at the difference +
N
w op,(a) w j=O
wtJopM(aJ(a)) w =: G o ( N )+ G , ( N ) .
Then Go(N) has a kernel in C;(R+x R,) with k ( N ) - + m as N + m, and G , ( N ) E i!,(R+)* induces an operator Lz(R+)+ Corn( R+) (i.e. G , ( N ) = op,(g) for some g(t, s) E YT(R+) @,$'Q(R+), Q E 9'). Thus the property (18) is governed by the corresponding property of w op,(a?(a>) w .
11. Defmition. a(z) E St,(R)const is said to have the transmission property if af = a,: for all j E N. We talk about the transmission properly of a(t, z) E S:l(R) if ark](,) has this property for all k E IN (c$ the above notation in (14)). In view of (2) we obtain a$(a) ( z ) = affi(z). Since f i is holomorphic on the left of we get the weight line rllZ
12. Theorem. op,(a) induces continuous operators (18) iff a(t, z) has the transmission property. In particular we get op,(a) : Cm(R+)n Xs(R+).+ Cm(R+)n Xs(R+).Further op,(a) : X",R+) .+X",R+) is continuousfor arbitrary P E 9' which satisfies the shadow condition.
As an exercise the reader may invent an analogue of the transmission property for operators of the class &O(R,)' in general, and similarly for W''(R+)' below in case of non-zero orders and weights. The Mellin expansions of yDO-s on R+ can also be used to define actions on weighted spaces Xs.Y(R+).
131
1.3. Mellin Expansions of Operators
13. Theorem. Let a ( t , z) E S:,(R) befixed (with the behaviour for t--, as required for 1pD0 symbols in the class Co(R+)). Then for every y E R with y -(k + 1)/2 for all k E N there exists a finite-dimensional operator G, E C,(R+)such that op,(a) - G , induces continuous operators
*
op,(a) - G,: Xs*Y(R+) +Xs*Y(R+), op,(a) - G,: X “ p ( R + ) + X a Y ( R + )
for every s E R and P E 9
Y
with some Q E 9, depending on P and a.
It is clear that for y > - 1/2 we may set Gy= 0. For y < - 112 we employ (19). A sufficiently smooth kernel can always be modified by a finite-dimensional one such that the remaining kernel is of a sufficiently high flatness at t = 0 (i.e., we remove a finite Ta lor expansion). The actions wopM(u:(a))w can be replaced by w o p b ( u i ( a ) )o where we get an error of the asserted sort. Green operators can also be modified by finite-dimensional ones that the remaining operator has the desired extension to spaces with weight y. Theorem13 leads to the question to what extent an operator convention like a + r + op(a)e+=op,(a) is justified at all. In (8) we have replaced op,(a) by another operator which is defined independently on the extension of distributions to the negative half line. This gave us the continuous actions (lo), (11). But it looks like a “fortunate accident” in the zero order case that op,(a) in L * ( R + )can be reformulated by Mellin expansions. For higher order symbols the procedure is much less clear (at least at a first glance). The method is to shrink the domain first, i.e. to take C;(R+) instead of L * ( R + )and then to reformulate the standard action op,(a) : C,”(R+)+ C”(R+).The resulting expressions show the spaces to which op(a) can be extended. This is again based on the Mellin symbols. The final results are then quite analogous to those for order zero.
Y
14. Definition. Let a ( z ) E S~I(R)c,,,t, P E R , and define aj’ E @
by a(z)
m
- C (i z)p-jaf for z+
f 00, larg(i z)l s d 2 . Then
j=O
is called the Mellin symbol of a for the cononnal order p - j. If a ( t , z) E S $ ( R ) we define analogously as in (14). u!&J(a)= u$-’(aIk]) with
C
/+k=j
Fora(z)€Sgl(R)c,,,twe have o$-,-’(a)Ermr,~JwithR,j=((Tk,O)}kEZ,r k = j - , U - k fOrkEN, T k = - k f o r - l - k E N . a E S!!l(R)c,,,, has by definition the transmission property if p E Z and af = a; for all j. The transmission property of a E Sg,(R) is defined by the corresponding conditions for all (cf. also the above Definition 11). In that case we get again much simpler systems of poles of u$-’(a). 15. Theorem. Let a ( t , z) E S$(R)(and assume the behaviour for 1as for 1pD0 symbols in Cp(R+)). Let y E R, - 1/2 < y < p + 112, y 1/2 mod Z. Then op,(a) : C ; ( R + ) + Cm(R+)extends to continuous operators op,(a): XS,Y(R+)~XS-p,Y-r(R+) (2 1)
*
132
1. Conormal Asymptotics on
R,
for all s E R and (21) induces continuous operators op,(a) :
X;y(R+)+
"(R+)
for every P E 9, with some Q E 9,- p, depending on P and a. Under the transmission property we get a natural analogue of Theorem 12. 16. Theorem. Under the conditions of Theorem15 the operator (21) belongs to &"'y(R+)*(cf. 1.3.3. (19) with 9 = (-m,O]) and
o$-j(op,(a)>
=
o$-j(a) for all j
E N.
It is clear that the conditions of Theorems 15, 16 ensure that I',,,-,contains no pole of d$(a). The orders p 5 -112 are not included, though op,(a) may be defined on Xs.Y(R+)(e.g. when s, y 2 0, p 5 0). In general we cannot expect a mapping X&Y(R+)+X"-fl.y-p(IR+),since the image contains singular functions with exponents -pj, Repj > 112 - y + p . On the other hand the weights which are too negative are excluded. For obtaining operators A E @Y(R+)' with o$-'(a) = c$-j(A) we can apply 1.3.2. Lemma 4, modified by weight shifts. We can set m
A
=
C w(cjt)t ~ + j o p ~ ( a ~ - j -wj (ca, )t )) j=O
with c,+ m sufficiently fast, j + yj 2 y 2 yj for all j, - y j + 05, j + y,+ a.Then for every choice of A there exists an operator F E &:(lR+) (i.e. flat of infinite order at t = 0) with op,(a) - { A + F ) E gBgG(R+)'gA (23) for certain B, A E R. The following theorem says that the remaining kernel can be chosen in a particular way. E W, 112 + y - p E Z, 112 + y 6 - N. Then there exbts a finite-dimensional operator G, E gBi!G(R+)' g A for appropriate B, A E R (depending on p, y ) such that op,(a) - G, E fPY(R+)* (i.e., op,(a) - G, has a canonical extension from Ct(IR+) as an operator in our class). This implies in particular the corresponding mapping properties of op,(a) - G,.
17. Theorem. Let a be as in Theorem 15 and y
This result can be interpreted as an operator convention, i.e. a rule to determine an operator in Wy(R+)'for the given symbol a. The method of proof is quite simple. First we can write op,(a) = A + F + G where gfiGgAE i!G(R+)' for certain B , A E R and A , F are as asserted. The calculation of (23) shows that B, A only depend on p , y . Thus it suffices to write G as Go + G, with GoE gY-"i!,(R+)' g-y, G, finite-dimensional. Since G = g-B(.?g-A for G"E 2G(R+)*the formal adjoint of G induces continuous +X;, -'(R+) for some P E FA. For every y there exists a operators G*: X&B(R+) projection l7:X;,-'(R+)+X; -"(R+)where Q E F y , Q f P, and 1 - l7 of finite dimension. Then l7C* is a mapping to X ; m * - Y ( Rand + ) hence (nG*)* is defined on Xssy(R+),s E R,and maps to XF--B(R+) for some P"E F B . For the same Leason as above there is a projection fi: x F , - s ( R + ) + X ~ , Y - p ( R + &E) , 9 " - p , Q 5 P, with 1 - fi finite-dimensional. Then Go = fi(l7G*)* is as asserted.
-
133
1.3. Mellin Expansions of Operators
18. Remark. The Definition I and Theorem 8 have obvious generalizations to the case of arbitrary orders. 19. Remark. The whole calculus of wD0-s on R, in the MeNin approach including the construction of parametrices extends immediately 10 rystems of equations. Note in particular that we do not need any kind of factorizations of symbols.
The rgDO-s on R+ occur in the boundary symbolic calculus for boundary value problems. The localized situation is the half space R: = (x : xn > 0}, x = (xl, . . ., x,). For any given p ( x , 5) E S;,(R”) with p(,(x, 5) as homogeneous principal part of order p we consider
5‘, t n ) := P(,)(x,0 I x n 0 for every fixed x’ = (x,, . .. ,x,- J, and 5‘ = (5,, . . . ,5,- * 0. For simplicity let us neglect x’ and write a(F’, z), t = 5,. The function a(F’, z) is homogeneous in (t’, z) of order p and belongs to S!!,(R),,,,, for every 5’ * 0. The coefficients a; (5’) are a(x’,
=
homogeneous polynomials in
of order j . In fact, we can write
a((’,z) = ~ z ~ ~ a ( ~ kzl ~ ).- ~ ” ,
Set e = ( t 1 - l . Then the coefficients are (up to the powers of i) the Taylor coefficients of a(e5‘, k 1 ) at e = 0, namely
Thus the Mellin symbols of Definition 14 are polynomials in F of order j . This is a remarkable structure with many analytical consequences, cf. also 3.3.3. ( 5 ) . In Chapter3 we shall explain the 1pD0 actions in the half space (cf. also 3.3.6.). Let us finish this section with some examples. For a ( [ ‘ , z) = 1519 p E R, 5 = (5’, z),
we obtain a:(f’) = (ki)-F,
a:
(5’) = 0 for j odd, for j even,
a,?([’) = (ki)-p+jcj,,lt’12
with constants cj,. The symbol Another example is
151N
has the transmission property iff p E 2 2 .
a ( z ) = (e + iz)-k for k E N, e E R, fixed. In this case it follows
a i k - ’ ( a ) (z) = eJhjk(z),
where hjk
= djk
for k
=0
hjk(Z) = (-I)’(
and k+j-l
) fl ( r - z)-’ j+k r= 1
for
k > 0.
134
1. Conormal Asymptotics on R,
The form of the Mellin symbols for a (z) = (e - i z ) - k ,k E N, follows from the formulas for adjoints, cf. 1.3.1. Proposition 7. More generally for arbitrary p E R we have
with hi, Jz)
=
(7)r(l - z)/r(l -
z
+ j - p ) . From the rule for adjoints (which can
be applied quite formally) we get the Mellin symbols for (e - iz)!’. Using the standard properties for the r function it follows for p E lN that u&-’( (e i 7 ) ’ ) (z) is a finite sequence of polynomials in z as it ought to be anyway for differential opera-
tors.
*
2.
Operators on Manifolds with Conical Singularities
2.0.
Preliminary Remarks
Let X be a closed compact C” manifold and LC(X) be the class of standard gDO-s over X of order g, L$(X) the subclass of classical gDO-s. We then have the wellknown theory of Lp(X), Lgl(X) in the sense of algebra properties, the symbolic structures, adequate Sobolev spaces Hs(X), the concept of ellipticity and so on. In the preceding chapter we have established a theory of Mellin gDO-s on R+ that contains analogous elements. The idea of the present chapter is now to perform a ”conflcation” of Lgl(X) by means of the operator spaces W’(R+), W(R+)’,. .. , in order to obtain corresponding operator spaces over R, X X as higher-dimensional analogues of those over W,. They will be employed then as the local models of gDO-s on manifolds with conical singularities. A “manifold” M with conical singularities may be thought as a compact topological space with exceptional points u l , . .. ,u N , such that M\ { u , , . ..,u N ) is a C” manifold, and M can be identified locally near any vertex uj with R, XXj/{O} X Xj for some closed compact C” manifold X j , where we also keep in mind the local R, actions 1 ( t , x ) = (At, x), ( t , x) E R+X X j , 1 E R,. To be more precise, every uj has an open neighbourhood Ujwhich admits a diffeomorphism
which induces a homeomorphism lpj: U j + [ O ,
1) x Xj/{O) x
xj
and pj 0 61 = A l p j , where a1 denotes the local R, action on U,\ {uj}. X : = IJXj is called the base of the cone. There exists a compact C” manifold C with boundary aC 3 X (the stretched manifold associated with M) such that M 1 C/aC and there is a tubular neighbourhood V z [0, 1) x X of aC and a diffeomorphism
2
constituted by the pj. Note that this definition does not imply that M is a topological manifold close to the uj, cf. Fig. 8. For instance, X j may be a torus or have several connected components. In particular M may be a “net” which consists of a finite number of branching points ul , . ..,uN and a finite number of one-dimensional connections between ui, uj, where ( i , j ) runs over a subset of IN x IN. Our analysis will be performed on the stretched manifold C. This allows to identify all uj with one vertex u. Of course, we then obtain also other spaces M Iwith the same base.
136
2. Operators on Manifolds with Conical Singularities V
Fig. 8
Outside the singularities the calculus will be the usual one. Thus we often may localize the constructions close to the singularities. It is then convenient to pass to the stretched manifold
c, = R, xx,
(1)
where for sake of symmetry we will observe t = 0 and t = 03 at the same time. In other words C1 is interpreted as a stretched manifold with respect to t = 0 andt = m. C1 also might be identified with [0,1]X X.
In Fig.9 we have indicated the projections to two variants of manifolds with conical points. There are many reasons for considering differential and pseudo-differential operators on manifolds with conical singularities. If we define on X ^ = R, X X
Fig. 9
2.1. Spaces with Conormal Asymptotics for the Cone
137
the Riemannian metric dt2 + t 2 g , g being a Riemannian metric on X,then we have a geometric cone over X . The associated Laplace Beltrami operator is of the form t-"
(- q,
Aj(t) j=O
t
where A j ( t ) E Cm(R+,Diffe-j(X)), p = 2 (cf. formula (2) of the Introduction). The weight factor t-" is often unessential in the calculus and will then be dropped. We also may admit g to depend on t , g = g ( t ) being C" up to t = 0. Another example of an operator of the form (2) follows byjnserting polar coordinates (t, x) = 2,2~ R"+ \ {0}, into a differential operator A ( & &) in R"+ of order p with smooth coefficients. Here X = S".These examples emphasize the role of the t variable. The Mellin calculus also will refer to t . The differential operators on C which are of the form
A = j = O Aj(t)(-t-$r
(3)
close to aC in the coordinates (t, x) E V, A j ( t ) being as mentioned, are called of Fuchs type or total& characteristic. They can be written as polynomials in vector fields on C that are tangent to X . This is an invariant interpretation which is independent of the choice of the normal direction t to aC. The operators of this sort play an important role also for the higher singularities such as edges and comers. The local model of an edge is Rq X M (the "edgification" of M), where M is a manifold with conical singularities, and q is the dimension of the edge. In particular when the base of M is of dimension zero then we get the local model of a manifold with boundary, namely Rq x K+. In other words boundaries are edges where the base of the cone is trivial. Comers may locally be thought as cones E+x M/{O} x M , where M has conical singularities. In that case we have edges of dimension 1 emanated from the corners. By iterating conifications and edgifications of given spaces with singularities we can generate higher singularities. The program then is the adequate procedure with operator algebras with symbolic structures. The conification of L g , ( X ) , X being of the class @",is the fist step, where the other factor comes from the calculus on R+. The Mellin symbols will now be operator-valued, with values in L $ ( X ) . We can repeat to some extent all essential elements of the calculus of Chapter 1, but there are also specific difficulties. One might hope then that the higher iterations are straightforward, for instance, for higher order comers. Unfortunately this is more difficult. Nevertheless the theory for the cone will be a guide for the comer calculus and the higher singularities which justifies the attention to this case.
2.1.
Spaces with Conormal Asymptotics for the Cone
2.1.1.
The Spaces with Continuous and Discrete Asymptotics
Let X be a closed compact @" manifold, n = dimX. The standard calculus of pDO-s on X admits a simple parameter-dependent variant. If A is a set in a metric vector space with metric I. I then the amplitude functions in local coordinates
138
2. Operators on Manifolds with Conical Singularities
x E U,U being a coordinate neighbourhood on X , also depend on 1.In our case we assume A = R' for some I E N and treat 1E A as an extra covariable. Sp(0 X R";A ) , R E Rmopen, denotes the space of all p ( y , 5, A) E Cm(RX R" X A ) such that ID;D$,PO? E, 1115 c ( l +
IEl + I ~ I ) ~ - - ' ~ ~ 01 E INm, /3 E R"+',y E K c c R , (5, 1)E R"X R', c = c(01,
for all multi-indices /l K,) > 0. Let S:,(R x R"; A ) denote the subspace of classical parameter-dependent amplitude functions. For instance, x ( f , A )([El2 + 1112)Nz belongs to S$(R X R";A), x being an excision function in R"+'. Denote by Lp(X;A)(L$(X;A)) the space of all parameter dependent VDO-s a(1) over X which are defined locally by means of amplitude functions p ( x , x ' , f , 1 ) E S q u x U x R " ; A ) ( S : , ( u xU X R " ; A ) ) and a standard partition of unity construction. Remember that in [S27] there was elaborated a parameter dependent theory of VDO-s. Our definition is a bit stronger. We impose the symbol estimates also with respect to the derivatives in A. But the essential elements of [S27] have a natural analogue here. In Section 2.3.1. we return to the calculus in more detail. Here we only want to employ one particular classical parameter-dependent operator bp(1) for every p E R with the homogeneous principal symbol ( 15l2+ l11*)N2,such that
b"1): H"(x)-H"-c(x) is an isomorphism for all 1E A and s E R. Here H s ( X ) denotes the classical Sobolev space on X with smoothness order s E R. The existence of such a b y l ) is a classical result and may also be found in [S27]. First it follows that any bp(1) E L:,(X; A ) with the mentioned homogeneous principal symbol induces isomorphisms as required for 111 suficiently large. Then by replacing A by 111 + lAll it becomes an isomorphism for all 1E R provided Ifl I is sufficiently large. Thus we can set P(1) =
WJl+IJl!).
Now let us introduce the cone Sobolev spaces in terms of the order reducing families b.(A). It is convenient first to consider x^.The general case of a manifold with conical singularities follows by simple localization arguments. 1. Definition. Let X"= R+X X , s, y E R , then with respect to the norm
{
.f
II~llm~~x*)= llW(z)(M,+,u)(z, rl12- v(n)
z = Im z, y ( n ) = y
!Xq
Y(X") b the completion of C;(X")
.)II~~cm ldzl]1'2,
(1)
- n / 2 . Further we set !Xko(X^) = !Xs(X^).
It can easily be checked that the space ! X k Y ( X ^ ) is independent of the concrete choice of the order reducing family bs(z),i.e., another choice gives rise to an equivalent norm. can also be described in local terms over lR+ x R , R being a The spaces 39 "(X1) coordinate neighbourhood on X , and then reproduced globally by a partition of unity with respect to an open covering of X.Let, for instance, R = R". Define 2%"(R+ X R")as the closure of C;(R+ X R")with respect to
2.1. Spaces with Conormal Asymptotics for the Cone
139
Here [fl denotes a strictly positive function of SE R" with [5] = 151 for 151 2 c > 0 and F the Fourier transform in R".Then we get by globalizing a norm which is equivalent to (1). cf.1.1.1.(9), Fl = Fl,xo+Fo being the one-dimensional FouUsing YMfin,= rier transform, we obtain m
Thus @fin):
X*y(R+XRn)+HS(Rn+l)
is an isomorphism, i.e.
x * ~ ( R + x R ~ {t-1/2+y(n)~(logr, )= x): u(x0, X ) E H ~ ( R : ~ + ~ ) } , and similarly for X instead of
(3)
W".This shows in particular that
X*Y(X^)= tcIw(X").
(4)
Moreover we see that W ( x " )= t-"2L2(R+ XX),
where L2(R+X X) refers to dt dx, dt being the Lebesgue measure on R+, dx a positive smooth density on X . The choice of the reference weight line r,,+1),2 for y = 0 is motivated by the Lz norm in R:+ \ {0}which is in polar coordinates i?= (r, x) E R+x S"
dx a positive smooth density on S". Thus the Xospaces on manifolds with conical singularities correspond to the ordinary L2 spaces whereas Lz over the stretched manifold C contains a weight shift. For the abstract calculus the choice of a "natural weight" is unimportant. We could take as well any other weight. As an obvious analogue of a corresponding statement in Section 1.1.2. we get from (2) 2. Proposition. For s E N we have
XS>"2(R+XR")= { u E L2(R+XR"): D;(ta,)ku E LZ[R+XR") forall l a l + k s s } .
(6)
From this it follows easily an analogous characterization of X8 Y(R+X R") for arbitrary ~ E RS,E N . 10 Schulze, Operators engl.
140
2. Operators on Manifolds with Conical Singularities
3. Remark. The local identflcation (6) suggests a relation between the totally characteristic Sobolev spaces and the "usual"ones, up to a weight. It exists, indeed, and may be found, for instance, in [Dl], Appendix A, (cf: also [S17];1.1.2. Theorem 17).
For future references we want to introduce the spaces X'"(R+X&c)r (7) Y(R+XRComp)denotes the subspace of all u E 2%y(R+ xR") for which u ( t , x ) = 0 for all x E D\ K , K = K ( u ) a compact subset of 0. X q Y(R+XRloc) is the space of all u E LB'(R+X 0)with pu E X&"(R+X D,,,,) for every p E C;(g). If x : D + R is a diffeomorphism the pull-back x* of distributions induces bijections between y(R+x Dcomp and X%"(R+X ficom, An analogous assertion holds for the space XsY(X") with respect to a diffeomorphism x : X + X . These assertions follow immediately from the above relations to the standard Sobolev spaces, cf. formula (3). The spaces X'*Y(X*)can be equipped with scalar products under which they are Hilbert spaces. We only need explicitly the .%"@") scalar product .7e"'(R+X&mp)r
D f Wnbeing an open set.
m
(u,V ) @ ( X ~ =
J J u ( t , X ) v(t, X ) t" dt dx. x o
Then (., .)$ extends to a sesquilinear form X8Y(X1)X the identification (2% Y(X"))' = X-4 - y ( x " ) , s, y E pi. We obviously have continuous embeddings
(8) X 8
-y(Xa)+C that admits
(9)
XS',Y(X^)4 X%Y(X") for every s, s' E R , s' 2 s, y E R. The embedding
xi'.a(x") n~ s ' . yx*) Y+
Y-
~8
~(x")
(10)
is compact for s' > s and any E > 0. These properties are the analogues of the corresponding statements for R+ in Section 1.1.2. We return to the Xs,Y spaces below in Section 2.1.6. with proofs in an abstract setting. For the moment they are exercises. The proof of (10) will use the following simple proposition, cf. Section 2.1.4. below for the techniques of the proof. 4. Proposition. For every p E R and x, x'
b"z)
E
ER
there exists a famib
$2 (@, L W ) )
such that for evelyjixed e E R , x 5 e 5 x' (i) b@(e+ it) E L$(X; Rr), (ii) b@(e+ iz) is an order reducing f a m i b in the above sense, where t plays the role of the parameter 1. Let us reformulate the local norm expressions (2). We have
=
fl [5]"(1 + ~t~2>'[f]1'2~FMu(1/2 - y(n) + i[fl
t,
f ) I 2 dtdz,
2.1. Spaces with Conormal Asyrnptotics for the Cone
141
where as above y ( n ) = y - n/2. Consider the Mellin transform of the function w ( r ) = t-fin)u, w E L*(R+).Then
(MgAw)(1/2
+ iz) = A1lZ(Mw)(112 + i h ) ,
A E R+ ,
cf. 1.1.1.(25), and thus
f (1 + I ~ I ~ ) w / ~ I(112 ( M+~~) [ Ez)lz I dt =
II~[~,WII&(~+).
The left hand side equals
f (1 + l z 1 2 ) " [ ~ 1 / 2 1 ( ~(112 u > - y ( n ) + i[Q z)l*ctz and g[fl( t - Y ( n ) u )
= t-fin)/[flg [€I
111~;~(t-fi")v)ll$(~+) = I l t Y ( n ) - Y ( n ) / [ f 1 2 [flv ll@
K")(R+).
This yields the following
5 . Proposition. For every s, y E R we have I I U l l k Y ( R , x R ~=
f [5lZs11
t y ( n ) ( l - Ifl-9,-
[fl( F u )
(6 .)II&Y(")(R+) dl.
Proposition 5 plays no essential role in the sequel. It only will be used below as an example for spaces defined in terms of group actions. 6.Theorem. Let Q, E Cr(E+) and A, be the operator of multiplication by q. Then A, induces a continuous operator &,: %&Y(X^)+%*Y(X")
for all s, y E R. Moreover
Q, +
Jn, induces a continuous embedding
n WYX-)),
c;(K+)+
&YER
with e(. ..) being considered in the norm topology.
This result will be proved in a more general version in Section 2.1.6. by the same technique as 1.1.2. Proposition 7. Instead of (1) we can write
IIu~~*v(x~
=
{
f
llb'(z)(My(n)u)(z, .)II;2(x,
rlI2- d n )
Idzl}'li,
Md being the weighted Mellin transform. Set ( x : " ' u ) ( t , x) =A(n+')/*u(At, x ) .
From M d ( x Y ) u )(z, x ) = A ( n + 1 ) / 2 - z M dx~) (then ~ , it follows I I x Y ) ~ l I x ~ ~= ( x~, y l I ~ l 1 3 5 ~ ( x ~
(12)
for all A E R+. Thus x y ) , A E R+, acts as a group of continuous operators on 2%y ( X " ) , and IIxY)IIP(*v(x.)) = A'.
(8) shows that x Y ) is unitary on X o ( X " ) . 10'
(13)
142
2. Operators on Manifolds with Conical Singularities
Let similarly as in Section 1.1.2. (a@,E ) u ) ( t , X) = tew(cr) u(r, x ) , 6,
c E R,
. From (a(c, E ) u ) (t, X) = ~-~(x(cn)a(l, E ) (x(cn))-lu ) (t, x)
then it follows that the norm of a(c, E ) : Xss Y(X^)+Xq Y(XA)satisfies Ila(c,
6 ks,y, K C , l ~ (Thus ~ ~we~ obtain ~ ~ )
e>llsmy(X-))
with ksy,'= Ila(1, ~
) l
.
7.Proposition. Let E > 0 befixed, then the norm of a(c, E ) in 2 ( X *Y(X")) ten& to zero as c + a , uniformly in Y E [-a, 6'1 for every 6, 6' 2 0 andfixed S E R.
8. Proposition. Let QI E C;(E+) be fixed, and pA(t)= p(At), A E R+. Then the operator of multiplication by Q I ~satisfies IIJ4,Ilfe(3eJr, 5 c with a constant c = c(p) independent of
A, and c(p) +O
as QI+ 0 in
C;(E+).
Proox We have ( A , u ) (t, x) = ( X : " ) & ~ , ( X : " ) ) - ~ u ) (t, X )
for all A E R, . From Theorem 6 and (13) then it follows
\lJ%il y1(mr, 6 Il./tt,, IIte(x4y) 5 const for all 1 E R+. The second statement is a consequence of Theorem 6 . 0 Definition 1 suggests to consider the order reducing family as an operator-valued Mellin symbol. If we interpret b S ( z )as a function b s ( z ) on f,,,- we can talk about the weighted Mellin action op$")(V) = tY(n)M-lbS(z - y ( n ) ) Mt-Y("),
(14) This corresponds to the notations of Chapter 1, where for n = dimX > 0 it contains a corresponding weight shift by - n / 2 . Then (14) induces isomorphisms o p p (b": X7e'.Y(X")+xr- s. Y(X^)
(15)
for all s, r E R.In particular op$")(bS):
2 4
Y(Xa)*%O.
Y ( X ^= ) pWL2 ( R + X X ) . .
A
Observe that for b s ( z ) as in Proposition 4 u Opp(bs) u = opd("(bS) M
for every y, 6 E R and u E C ; ( X " ) . The technique for proving this is the same as in 1.2.5. Theorem 14. The arguments will be repeated once again below in 2.1.5. in the analogous situation with operator-valued Mellin symbols. Now u E C;(Xa) implies that f"(z, .) = b"(z)Mu(z, .) E d ( C , L 2 ( X ) ) and
{ rl12
IIullm~=
- y(n)
llb"(z)Mu(z,.)IItzcx,ldz1}1'2
143
2.1. Spaces with Conormal Asymptotics for the Cone
for all y varying in a finite interval of the real axis and suitable x , x'. Since then IIUllX%Y=
IIM-'fSIIfW(X7>
(16)
we get for every yo, y1 E W and all y with yo 5 y 5 y1 IIuII4csY(x35
(17)
c(IIull4csyo(x, + I I u l l w ~ l ~ x 7 )
with a constant c > 0, independently of y . This follows from (16) by expressing the Lz-norm as usual and t-Y 5 c ( t - M + t-Yi), t E R, . In addition we see that c can be chosen independently of s for s varying in a finite interval. 9. Proposition. Let so, sl, y o , y1 E R , s, = 9so + (1 - 9)sl, yr = 9'yo + ( 1 - 9')y l , 0 5 9,8' 5 1 . Then there is a constant c > 0 , independent of 9,such that Il~lIWe.Ye(x75
c
c
IluIIWPv,(x)
sj=O.l
for all u E C t ( X " ) . Proox In (17) we may insert y = y 8 , and s = s8 for all 9, a', 0 5 9, 9' 5 1 , with c independent of 9, 9'.Then it suffices to show that II~IIwe.n 5 C ( I I 4 I r n , Y 8 + IIullWl.YJ, i = 0,1, with another constant c, 0 5 9 5 1 . But this is an immediate consequence of the isomorphism %qy(X^)
-+
Hs(R x X ) ,
u ( t , x ) +e(l'z- m)* u (e*, x), xo= log t , and the well-known interpolation property of the standard Sobolev spaces, cf. [ T l l ] . The definition of Xq ?-spaces can easily be extended to general stretched manifolds C belonging to manifolds with conical singularities. Over X^we also have the analogue of the Xs-y-spacesof Section 1.1.2., cf. also the definitions at the end of this section.
10. Definition. Denote by R(X) the set of all sequences
(18)
P = {bj,mj,Lj))jez
forwhich n c x n P = { ( p j , m j ) } j E Z (~cRt 1.2.1. Definitionl) and L j c Cm(X)afinitedimensional subspace. Moreover we set
W n ) ( x=) { P E R( X ) :
nCP n r,,,- v(n) = 0},
y E R,y ( n ) = y - n/2, nCP = b j } jz .E As usual we talk about discrete asymptotic types. Moreover set
P'(n)(X)= { P E RY(")(X): n,P c {Rez < 1/2 - y ( n ) } } . P E R)'(n)(X)is said to satisfy the "shadow condition" if ( p j , m j , L,) (pj - k , mj, L j ) E P for
(pj
+ k , m,, L j ) E P
E
P implies
Repj < 112 - y ( n ) ,
for Repj > 1/2 - y ( n ) for all k E N .
144
2. Operators on Manifolds with Conical Singularities
In %(X) we have 5 natural semi-ordering s , where-P 5 Q means that ncP 5 n, Q, and m 5 6 ,M E M whenever (p, m ,M ) E P , (p, m", M ) E Q. For y E R we define the strips Sf"),A E gap,C C , =~ 0,1, in the same way as in Section 1.2.1., ,"l;(n) =
T-dZSY
A'
If P E %(X) is given and y E R, A E Yap,we define the subset P f n )of P by the condition that
n,(P\ Pl;("))n Sl;(") = 0. In other words we have everything as in 1.2.1. with respect to the weight line Re z = ( n + 1)/2. If h ( z ) is a g'(X)-valued function on rp= {Re z = p } we define
z = Imz.
11. Definition. Let s, y E W, V E W, A space of all
h E 99 ( S p\
E
9, Vn as$(")= 0. Then SeS,(X)l; is the sub-
v,H"X))
such that sup {llxhlls,s: E S$y)}< 00 for every V-excision function in 1.2.1.(6), and
for every f
E Se (C),L
(20)
x
and A , ,
E
> 0, as
being a smooth curve in S$")\ V surrounding S$(") n V.
(21) gives rise to a linear operator [: $;((X);+$'(Vn
Sl;'")) BrCrn(X).
Moreover we have by (20) an embedding SQ L(X)l; c Se (SS;'"'\ V, Hs((x)).We endow dS,(X)$ with the weakest locally convex topology under which all these mappings are continuous as well as the norms Ilxhl18,pfor all /J E Sj;'") and all x . Then IS,(X): is a Frtchet space. Similarly as in 1.2.2. the space SeS,(X)l; will be the starting point for the definition of Sobolev spaces over the cone with continuous conormal asymptotics. We also shall discuss the spaces with discrete asymptotics. For P = { p j , m j , L i ) }E%(X) we denote by SesS,(X)l; the subspace of all h E dS,(X)l;, V = n,Pl;("), which are meromorphic in Sl;("), with poles at all p , E Sf") of multiplicities mj + 1 , where the coefficients at (z - p j ) - ( k +l ) in the Laurent expansion at pi belong to Lj for all k , 0 s k s mi. It is easy to see that 99;(X)yd is a closed subspace of SpF(X)l; in the induced topology for every V E K V n = 0, with n&") C V. For V E V'(") the weighted Mellin transform My(")defines an injective operator
My("): SeS,(X)l;+9eS~Y(XA).
2.1. Spaces with Conormal Asymptotics for the Cone
145
12. Definition. Let s, y E R,P E W n ) ( X ) A, E 3 non-trivial. Then X;"(X^), is defined as the subspace of all u E Xs,Y(X^)for which My(")u E d ; ( X ) ; , equipped with the corresponding Frdchet topologv. Similarly for V E F("), V n as$(") = 0, we define %S;"(X^), c %s.Y(X^) by My(")u E dSy(X): with the corresponding Frdchet topology. For trivial A we write %eSpY(X^), = %S;Y(X^) = %ssY(X^). P = 0 or V = 0 will be indicated by 0. We then talk aboutflatness of order A .
The scheme of extending Definition 12 by the decomposition method to arbitrary V E W nis) completely analogous to Section 1.2.2. So it will be dropped but tacitly used below. According to the notations in 1.2.1. and 1.2.2. we set
%;(x^),= %;o(X-)A,
PE a-wyx)
and similarly for V E V w 2 moreover ,
%;Y(x^)= gl%;yX^)Ar
PEW(")(X)
A € 9
and similarly for V E Y)'(").The projective limits over A correspond to A = (- m, 00). Also in the notations below A will often be omitted in this case. Further we set
%:;(x*); = I& X;Y(X^), (inductive limit over P E %y(")(X)),
%:,Y(x^),= I& XS;Y(X^), (inductive limit over V E V"'")). y will also be dropped for y = 0. For finite Q E % ~ ( " ) ( X ) ,Q = {(g,, n,, M , ) } J = l , , N , Reg, < 112 - y(n) for 1 5j5 r, Reg, > 1/2 - y(n) for r + 1 5 j 5 N , we denote by Z ; ( X ^ ) the vector space spanned by '&(XI
ck
w(q,, k, t ) ,
GdX)
w'(q,, k , l ) ,
E M,, 0 5 k 5 m, and 1 S j 5 r for w , r + 1 5 j 5 N for w' (cf. also the notations in 1.2.1.). For arbitrary P E W ( " ) ( X ) A, E 9Esfinite, we set
Z i ( X ^ ) A= Z b ( X " ) , Q = Pf"). Then
Z i ( X ^ ) , c %;.Y(X^), for every finite A
E 3. Moreover
%$'(X^),
= X:'(X^),
+ z&(x-)A
(22)
is a direct decomposition. We can easily define XsdY(X^), for every A E YEP, C C ,=~ 0,1, as the subspace of ~ ~for , ~ 1 in the parts of as;(") which those u E %:Y(X^),,,, for which ( I M , , ( , , ) U<~ 00 correspond to [ or ] in the weight interval. Then (22) yields a definition of %;Y(X^)A for finite A E gas.This extends to arbitrary A E Yapby a projective limit of spaces with finite weight intervals. For continuous asymptotic types we can do the same, but we mainly need the variants with V n as;'") = 0. In that case it suffices to set
%";(X^),
= X"dY(X^),
+ %;'Y(x-)A,,
(23)
146
2. Operators on Manifolds with Conical Singularities
A E gb,,where A , E 3, A , > A , and the end points of A , have a distance less than E to the end points of A with e so small that also V n as$:) = 0. The space (23) is then independent of the concrete choice of E . Another possibility is, of course,
X$Y(X^), = Xsd'(X^), + X;*"(X^).
(24)
It is equivalent to (23) under the mentioned condition and extends the definition of %3y(X^),to arbitrary V E 'W"),A E Yap,01, B = 0 , l . For V E W") we can write V = V,+ V, with VlE By("),Vz E Yy(") \ By("), Vz n rlI2 - y ( n ) = 0. Let A E be finite and V n as;(") = 0. Then we have the direct decomposition d ' ( V n S$"))@J?(X>
= d ' ( V , n S;("))
@*Cm(X)+ &(Vz n S;(")> @Jm(X).
Let us define the space
Z;(X^),
=
{ ( A l , r w w ( r ) ) + ( A 2 , r w w ( t - l ) ) : Aed9'(VnSy(n))@,$m(X)},
where A = A, + Az corresponds to the latter decomposition, and w is a fixed cut-off function. We consider Z $ ( X ^ ) , in the nuclear Frbchet topology induced by the bijection with a'(V n S;'"') COZCm(X). It is obvious that then %$Y(X^),
= X>'(X^),
+ Z$(X^),
(25)
is a direct decomposition, A E finite, V n aSl(")= 0. Now let us turn to a number of further simple properties of our spaces. For s' L s, A' 2 A we have canonical continuous embeddings
%$y(X^),. 4 X>y(X^), for P' 5 P, %$y(X^),.
4 %$"(X^),
for
V' s V.
%;gY(X^),%;9Y(X^) are nuclear Fr6chet spaces for A E ~ For . every AER we have
t"y(x^),
= Xy&."X^),,
t"$Y(X^),
=
Xy$qx^),,
where T - A P = { ( p j - A , m j , L , > j j E Z P , being given by (18), and T-"= ZE
iq.
{z-A:
The transformation
I("): u (r, x) -,t-" - u (t- l, x)
(26)
induces isomorphisms I("): XS.Y(X^)+%%-Y(x-) which restrict to isomorphisms
I("): %:,y(X^); -,%:8-yx-);f and the same without dots. Here A' follows from A by interchanging the end points. The resulting asymptotic types can easily be expressed by the original ones.
147
2.1. SDaces with Cononnal AsvmDtotics for the Cone
13. Remark. (11) induces a group of linear continuous operators on the spaces lreJpy(X^),, lreiY'(X^), for every choice of the data P , A , .,. involved in the definition of the spaces.
As mentioned for every A ' L A we have canonical embeddings X$y(X^),. 4 X>"(X^), and the same with V . These embeddings composed with the canonical projections from (22) and (25) yield continuous projections for every finite A as;: !X$y(XA),.- Z;(X^),, as;: !X$y(X^),,,-+ ZY,(X^),, where for V we impose the mentioned conditions with respect to the weight intervals. The projections represent the notion of discrete and continuous conormal asymptotics for the cone, in the sense that
rml; = id - asl; is a map into Xgy(X^),.If we apply this to finite A tending to A' = (- m, 00) we get proper asymptotic expansions (for V this means that we assume V E Yf")). The dependence on the choice of w only causes flat contributions. Let P = { ( p j , mj, Lj)}je E W ( " ) ( X )and let X,(X) be the space of all sequences
{tk(x)l, s ks m,,j e z with rjk€ Lj, 0 s k d mi, in the topology of the component-wise convergence. Fix u E X$Y(X^)and consider the vector-valued analytic functional
1
(&[El, h ) = - M u ( z ) h ( z ) dz, 2n1 c,
where Cj = {z: Iz - pjl = E } and E > 0 so small that Iz - pil > E for all i * j . Then 134is Lj-valued and represents the part of the Laurent expansion of My(,,) u at the l), 0 s k 5 mj, in a unique way. In other words point pi with the powers (z (28) admits to recover the bk for all j , k . Thus we get a linear operator SMy(n):
2$y(XA)+X~(X)
which (uniquely) produces the coefficients of the asymptotic expansions of u into expressions t-Pj logktfor t +0 and t + 00, and ( is continuous, s E R . The analogue of 1.2.1. Theorem 9 for the cone X A is as follows. 14. Theorem. Let P E W ( " ) ( X )be fixed. Then the sequence
o+
X>'(X^)+ X y ( X - ) -
is exact for every s E R. The space
CMm
X p ( X ) +0
X$"(X^) can be written as a sum of FrPchet spaces
!X>Y(X^) = !Xgyx^)+ X y ( X ^ ) . For V E Y$n), V = U 5, u E Xgy (XA), we define
(30)
148
2. Operators on Manifolds with Conical Singularities
where Ci surrounds erators
5 in the usual way. Then we get a sequence of continuous op-
: %$Y(X^) -,d'(5)BnCrn(X).
(3 1)
Let X , ( X ) = X d'(V,) BZCrn(X)be equipped with the topology of component-wise jeZ
convergence. The sequence (3 1) then induces a continuous operator
(My(,) : 9e$"(X^) -,X , ( X ) . It shows that the analytic functionals occurring in the asymptotic expansions are unique. 15. Theorem. Let
o-,
-
V E Y$(n), then the sequence
%;y(x-)
---*
%$Y(X^)
W(")
X,(X)
-,0
k exact. The space .X$y(XA) can be written as a sum of Frdchet spaces
Xg;'(X^)= sle;'(x^)+ %;~y(x-). This decomposition also holds for arbitrary V E Yy(")as well as for the corresponding spaces with the subscripts A E 3. Moreover
%$'(X^), = .X$;(x^),+ %$;(x-),
(32)
whenever V , , V, E Yy(")with V = Vl + V,. The proofs of Theorem 14 and Theorem 15 are completely analogous to those of the corresponding results in the Sections 1.2.1. and 1.2.2. and so will be dropped. Note that here we have to use Proposition 7. Another element of the proof is 1.1.5. Theorem 2, together with 1.1.5. Lemma 3, here applied to G = C m ( X ) . 16. Remark. XSpy(X^),P E W(")(X),is a closed subspace of X$V(X^)for every RcPs
v.
V E W n with )
For the spaces with asymptotics in the cone we can make an analogue of 1.2.2. Remark 9, where here Li c d ' ( V i )BnCm(X)would be of finite dimension (or another closed subspace). This yields a more precise notion of the continuous asymptotics but we do not use it in this book. For future references we want to express the spaces %>y(XA)Aand %$Y(X^), as projective limits of Hilbert spaces
and analogously for P . For simplicity we only consider A E 3, the other types of weight intervals are analogous. First consider the continuous asymptotics. It sufm), follow by takfices to look at finite A . The infinite cases (6, a),(-00, a'), (-a, ing projective limits which can be carried out in terms of the sequences of Hilbert spaces for the finite case. The decomposition method reduces the situation to (32), where V , , V, are asymptotic types with sufficiently many gaps in the sense that
2.1. Spaces with Conormal Asymptotics for the Cone
there are A i k as in 1.2.2.,tending to A for k+ have constructed X$,y(X^)$:k we obtain
m,
149
with Vi n as$:) = 0 for all k . If we
Thus we are reduced to the situation of Definition 12. Here we may employ (25), such that we only have to construct the Hilbert spaces !?t:y(X^),O, ZL(X^)i1 (the numeration with j E Z at every step is independent of the previous one). Set A j = (Sj, 6)) with 6, = 6 + I j l - l , 6; = 6' - 1jl-I for A = (6,s'). Then X:y(X^)$l may be defined as the closure of X:"(X^), with respect to the norm
Finally, by definition,
Z;(XA),
d ' ( K ) @nCm(X), K
=
V n Xi;'").
Then we may write
Z'Y,(X^)$'
= d'(K)'" @HH'(X),
where H J ( X )is the standard Sobolev space over X of smoothness j with some adequate choice of a scalar product, and mH denotes the Hilbert space tensor product. d ' ( K ) O denotes the closure of d ' ( K ) with respect to the norm 1.1.5.(9). In this construction we have used several times that the sum of Hilbert spaces has again a natural Hilbert space structure. The construction of (33) shows once again that the Hilbert norm system than can be used to define the Frbchet space structure of %$y(X^), is rather complex. It may be a challange to the reader to keep in mind all the steps that have to be checked in proofs of continuity of operators between such spaces. In the comer and edge calculus we have even to iterate the constructions. On the other hand we want to emphasize at this point that we can reduce all this to formal manipulations in functional analytic terms such as projective limits and sums of spaces. Then the proofs are only simple exercises in terms of norm systems, after considerations in a starting situation, where the asymptotic types do not intersect the boundary of the weight strip. The latter case then is often obvious. In other words we may often limit the complexity of considerations by the hint that things go in the mentioned standard way. The analogue of ( 3 3 ) for P instead of V is simpler, for %?;(X^), is of finite dimension and hence we have simpler Hilbert spaces as before. 17. Theorem. Let P E W ( " ) ( X )satkfi the shadow condition, and let p ( t ) E p(t-') E Ct(a+). Then .At, induces continuous operators !?t>'(X*)A
%$y(x^)A
C t ( R + )or (3 4)
for all A E gha.If V E W"'") satirfies the shadow condition we obtain the analogous assertion for %>y(X^)A. Proof: In view of the definition of X$'(X^), for non-finite A in terms of a projective limit over spaces with finite A it suffices to consider the finite case. For simpli-
150
2. Operators on Manifolds with Conical Singularities
city we restrict ourselves to A E 3. The other types of weight intervals may be treated in an analogous manner. Without loss of generality we may discuss p(t) E Corn@+). The second case follows then from the first one by conjugating with I("),cf. (26). Applying (22) it suffices to prove that
(35) is an immediate consequence of Theorem6, since for instance for A = [a, 6'1 E 311 X:Y(x^), = ~ ~ . Y - ~ (nXXs>Y-d'(X^). ^)
By a projective limit argument then it follows (35) also for the half-open weight intervals. N- 1
Now we write p(t) =
cjt'w"(t)+ t N p N ( z )where , pdf) E C;(R+), w" being a
j=O
cut-off function with Gp = p, Gw = w . Then u ( t , x ) = [(x)
c
t-P
logkfw(t) implies
N-1
p(t) u ( t , x ) =
Cjl(X)
t-(P-'~logkt
w(t)
+ P N ( t ) t N U ( t , x).
j=O
The first sum in the latter expression belongs to Z;(X^), modulo X : y ( X A ) dMore. , N is large enough. Then pNtNu(L,x) E X;y(XA), over t N u ( t ,x) E X $ y ( X A ) when because of the above statement. The continuity of (34) is obvious. The arguments for X>y(X^),are analogous. By decomposition arguments we may restrict ourselves to the case when V n as;(") = 0. Then (25) shows that it suffices to discuss pu for u E Y?Y,(X^),, K = V n ,#3;@). If [E sQ'(K) &Cm(X) then ([,t-")w(t)tJG(t)=
([,r-(W-'~)w(r).
Modulo flat terms we obtain ([, 0-J) w E Xy(X-),
for all j . The continuity of A, is again simple. 0 18. Remark. Let V E 'Wn) and @ ( t ,x) E C,"(B+X X ) or @ ( t - l ,x) E C,"(B+X X ) . Then there b a n other W EV nsuch ) that
4,: Yey(x-)A-a y ( x - ) , b continuous, where V = W when V satbfies the shadow condition. Similarly for every P E W ( " ) ( X ) there is some Q E W ( " ) ( X )such that
Jttv.. S S P. Y (x^)A-a;qx-)d b continuous. If P s a t w e s the shadow condition then nc P = nc Q,nc of a sequence of triples @, m , L ) to the sequence of first two components.
being the projection
This follows easily from the constructions of the proof of Theorem 17 together with Corn@+ X X ) = Corn@+) @.,Cm(X)and the representation of elements of projective tensor products as converging series of functions like p j ( t )v j ( x ) , cf. 1.2.3. Proposi-
2.1. Spaces with Conormal Asymptotics for the Cone
151
tion 17. (It applies here, since we may replace for every concrete choice of Q, the space Cr(R+) by the Frkchet subspace of all functions that are supported by a fixed interval.) For the calculus below we need a number of variants of the spaces with asymptotics. First there is an immediate extension to manifolds M with conical singularities. Let C be the associated stretched manifold. Denote by w a function in Cm(C)with w = 1 in a tubular neighbourhood U 2 [0, 1) X X of X, w = 0 outside another tubular neighbourhood. Here X is a closed compact manifold. Different conical points of M correspond to different connected components of X , cf. the remarks in Section 2.0. 19. Definition. Let s, y E R and define %*Y(C) = [w]%&Y(V ) + [ l - w]Hfoc(intC), where %&Y(U)= { n * u : u E %*Y(X^)}, and n: int U+X^ defined by int U z (0, 1) X X G R, XX = X^ (CJ: also the notations after 1.1.2. Remark 3). Moverover for P E 9y(n)(X),B E g y ( " ) we set
+ [ l - W ] Hsoc(intC) , = [w]!Xiy(U)A+ [ l - W ] Hfoc(intC) ,
%S,"(C>A= [w] %S,'(U)A %;"(C)A
%S,Y(U), being defined as pull-back of %S,y(X^)Aunder x and similarly %S,"(U),, A = [6, 01 or = (6, 01, 6 h - m. The spaces are equipped with the topologies of the corresponding sums. It is clear that [w] %S,y(U)Aonly depends on the left end point of A E 5'@,and the same for B. Therefore we use incidentally the notations
2 = 2 0 = {(S, 01: =
0262
-m},
{[S, 01: 0 2 6 2 a}
(36) (37)
as the sets of weight intervals that are responsible only for t +0. 20. Remark. Another trivial generalization of the spaces may be obtained i f we distinguish between the various connected components of X and admit different asymptotic Wpes and weight internab over them. The functional analysis of this section has a straighrfonvard extension to the spaces in the Definitions 19.
Now let us pass to a version of spaces on X^ where the behaviour at infinity is modified, compared with that of the %&''-spaces. Let us first introduce the space H S ( X ^ ) .For every coordinate neighbourhood R of X we can identify R +x R in the local coordinates (t, x ) with a conical set dc R:", where (t, x ) are th? polar coordinates to 2. This yields a diffeomorphism 01: R + x R + R with x(At, x ) = &(t, x) for all A > 0, where A on the right is the canonical R, action in Rfl+I.Now let H s ( X ^ ) be the space of all u E a'(X^)such that for every R and every Q, E Cr(R) (1 - w ) p u E (1 - w ) x*(HS(R"+l)I,-), t"/*wQ,uE H"R x
a)(,,,,
w ( r ) being a cut-off function, x * the pull-back under x and H s ( R x R). = HS(RX
Rn)lRx for R being regarded as a subset of R". The space Hs(X^) can
152
2. Operators on Manifolds with Conical Singularities
easily be equipped with a norm under which it is a Banach space (even a Hilbert space with an associated scalar product). For e E R we set
H"Q(X^>= g;QH"X^).
(38)
Here gf'(t) E Cm(R+) is a strictly positive function which equals 1 close to t = 0 and t Qfor f > const. (38) is a Banach space with respect to the norm llgf'4IHYx-,= Il4lH~P(x-) 9
and the norms for different choices of gf' are equivalent. We have continuous embeddings
HS'sp'(X^)G H J Q ( X ^ ) for s 5 s',
e 5 e'.
21. Definition. Let P E P ' ( n ) ( X ) ,B E BY("), y , s E R , A E
X$Y(X^),= [ w ] Z $ Y ( X ^ ) ,
+ [l - w]HSd'(X^),
XyyX^),
+ [l - w ] H $ d ' ( X ^ ) ,
= [ w ] ZS,Y(X^),
01
=0,l.
Define
6' being the second component of A.
All spaces are equipped with the topologies of the corresponding sums. For A = [0, 01 all spaces are independent of the asymptotic types. We mainly need the case e = 0,
X $ Y ( X ^ ) := [ w ] Z"Y(X^)+ [l - w ] H " X ^ ) ,
(3 9)
22. Remark. For the spaces .!YC&v(X^) we have an obvious analogue of Proposition I.
The definitions of our spaces lead to natural embeddings such as
X5'3V'(X^) 4 X " ' ( X ^ ) for s' > s, y' > y and to the standard program of further functional analytic discussions. In order to limit the size of the book we drop most of these things. They can easily be added by the reader. For references below let us mention the following. = t-"'*LZ(R+ X X ) gives rise to a sesquilinear form The scalar product of XOvO(X^)
XS*Y(X^)x X $-Y(x-)+c which admits to write
X"Y(X^)' = X - " - Y ( x - ) , s, y E R (the prime indicates the dual space). 23. Remark. There is an analogue of Proposition 9 for the norms of X s?-spaces instead of 2 S . V . The space C z ( X ^ ) is dense in XJ.Vfor every s, y E R.
153
2.1. Spaces with Conormal Asymptotics for the Cone
2.1.2.
Operator-Valued Mellin Symbols
The next step in the conification of L z l ( X ) are the spaces of Mellin symbols for the cone. The spaces L i I ( X ) and L i I ( X ;A ) (cf. also the notations in the beginning of 2.1.1.) have natural FrCchet topologies. The explicit definition for L$(X) will be given below in Section 2.1.4. in a more general context (cf. 2.1.4. (18)). The case L:(X; A ) is completely analogous and will also be used here. (For the moment the construction of adequate FrCchet topologies could be regarded as an exercise.) Denote by a countable system of semi-norms for Lg1(X;A ) . In concrete cases A will be indicated, here A = R. Let us first introduce the discrete asymptotic types of cone Mellin symbols. 1. Definition. Denote by % ( X
X
X ) the set of all sequences
P = {(pi, mj, N j ) } j e z
(1)
with {(pi, rnj)ljEzE 5% and finite-dimensional subspaces N j c L-"(X) of finite-dimensional operators, j E Z . Moreover let 9LY(")(Xx X ) = ( P E % ( X x X ) : nc n r,,,- y ( n ) = 0) , y E R, y ( n = y - n / 2 . Set ncP = {pi} z , nc N P = { ( p j , m,) } z . For 1 E C we z. set TAP= { ( p j + A, mj, Nj)}je
2. Definition. Let p, y E R, A E Z non-trivial, R space of all
E
%(X
X
X ) . Denote by 9 l i ( X ) l ; the
a ( z ) E &(Sf")\ ncR, L $ ( X ) )
(2)
for which (i) a ( z ) is meromorphic with poles at all p j E S$") n ncR of multiplicities mi Laurent expansions mi
with
V,k
E
Nj,
+ 1 and
m
0 5 k 5 m j , V j ; E L $ ( x ) , 1 E N,
(ii) aA,(t) := ( x u ) (e + i t ) E Lg,(X; IR,) f o r all e E S$" and every ncR-excision function x , and sup{Aj(ap,,): e E SdYjn)} < oJ with A, = ( 6 + E, 6' - E ) for a f f E > 0, 6 + ~ < 0 6, ' - ~ > 0 ,A = ( 6 , 6 ' ) , a n d a l l j E N . The class L $ in this definition was used for convenience and could be replaced by L". The property of being classical admits to separate the (parameter-dependent) homogeneous principal symbol of a ( z ) and (locally) also the homogeneous lower order terms. For Cj = { z : Iz - pjl < E } and E > 0 so small that Cj surrounds no other P k , k S j , the application
h E SQ (C), represents an element in d'(pj})("'j) C 3 N j , where mj indicates the order of the analytic functionals, cf. 1.1.5. So the definition of 9l$(X);induces a system of linear mappings
154
2. Operators on Manifolds with Conical Singularities
&:
%:(X)$
1:
%:(X)Z + d ( S $ ( " ) \mcR,
+d'(bj])(m~)@ Nj, L$(X)),
4e, x ) : W J X ) $ + L W ; R,)
(4)
(5)
(6)
for every e E Sy(")and every n,R-excision function ,y. The space %:(X)$ will be endowed with the weakest locally convex topology under which all mappings (4), (S), (6) are continuous. It is then a FrCchet space (since a countable set of e, x suffices) and
Em," ( X ) $ = pJ Em: ( X ) $ PER
is a nuclear Frkchet space. Set
This is just the space %:(X)(-w,w), and it is independent of y. Moreover we define
(8) ~"'(X= ) L f I ( X ; r~,, - y(n)) . Let us regard this space as the case when A is trivial, i.e. A = [0, 01, and R disappears. As for the R, calculus of Chapter 1 we are mainly concerned with A = (- 00, m) or, incidentally, with (8). So the classes with subscript A will be neglected from now on. Many assertions have obvious analogues for arbitrary A, y. 3. Definition. Let p, y E R, A E Y be non-trivial, V E Y, V n aSl;(")= 0. Denote by Emoll;(X)$ the space of all a(z) E d(S$(")\
V, L $ ( X ) )
(9)
for which
(i)
d (C)
3
a ( z )h ( z ) dz
h + ( ([a], h ) =
(10)
L
represents an element in d'(S$(")n V) B nL-"(X), L c S1;'")being a cume surrounding s$(") n V, (ii) aRJz) = k a ) (e + iz) E L:,(X; R) for all e E S1;'") and every V-excbion function x, further sup (Aj(aR,x): e E S:(")}< for all E > 0 as in Definition 2, and all j E N. For trivial A we set again Emol;(d)Z = %P.y(X). V = 0 will be indicated by 0 .
The definition gives rise to linear mappings
d'(sl;(") n V) BnL - " ( X ) ,
I:
Em;(X)l+
1:
Em;(x);+ d ( s y \ v, LEI(X)),
4, x ) : %oll;@)Z every e E S$") and
+ G ( X ; R,) for every V-excision function ,y. We consider %l;(X); in the weakest locally convex topology for which these mappings are continuous. It is then a FrBchet space. %m;"(X); is a nuclear Frkchet space.
155
2.1. Spaces with Conormal Asymptotics for the Cone
Note that we also can define a class Em$JX); by the same conditions as in Definit follows ition 3 except of (i). For a E EDz;,('X); [ [ a ] E d ' ( S f " )n V) @On L!!,(x).
It is clear that
m;"
( X ) ; = m;,"(X);.
(11)
The definition of %;(X);, A E 3 finite, extends to arbitrary YE Y by the decomposition method by an obvious generalization of the procedure of Section 1.2.3. In particular we have in m;(X); the corresponding locally convex topology under which it is a Frtchet space. Again we mainly consider the projective limit
which is independent of y. Let us set
-
m&(X)* = lim
m$(X), EOZkYX)' =
R E%(X x X )
'm$(X).
lim R E%%X
x X)
We write %(a>= V (sg'(a) = R ) for a E roZ$(X) (W$(X)). The spaces EDZ:,(X), EDZ&(X)' are invariant under translations a(z) +(TOa) (z) = a(z
+ /9),
B E @,
TBm:s'(X)= 'iIW&"- p(X) , here sg(T42) = T-hg(a) and the same with dots. Similarly as in Section 1.2.3. we have for V E Y,, V =
U 5, the space jeZ
&(X) =
x d ' ( 5 )@On C"(X x X)
jcZ
in the topology of component-wise convergence (where Cm(XX X) is interpreted as L-"(X)) and for R E %(X X X ) the space of sequences
XR(X)= {G~(x)E N,: 0 5 k 5 m j , j
E Z} ,
where Nj is the j t hspace involved in R,cf. Definition 1. This gives rise to continuous operators
I: W(X) I : %(x)
+
11 Schulze. Operators engl.
Xdx)
I
156
2. Operators on Manifolds with Conical Singularities
4. Theorem. Let p E R, V E Yd,V = U vj. Then the sequence
0 is exact.
-+
%$(X)+%r,(X) + X,(X)
If R E 3( X x X )
0
+
the analogous sequence
0 +%$(X) -+%r,(X) +X,(X)
+o
is also exact.
The proof follows by analogous arguments as in the scalar case of Chapter 1. This yields also
5. Theorem. For every p E R,R E 3 ( X x X), V E Y we have
rmrm = mm + %Rrn(x),
(16)
%r,(x)= %$(X) + m;-(x).
(17)
Moreover
9 t F ( X ) = %$,(X) + %t2(X) for V = V, + V2, V, E Y. (18) Note that analogous decompositions can be proved in a more abstract context, cf. GRAMSCHKABALLO [G6] and the references given there. Theorem 5 holds in the corresponding form also for the classes %;(x);, 9t;(X);. Let us remind of the decomposition for a E %g(X) when Vn asp)= 0. With the function f(z, w) from 1.1.5. (4) we can form
where L1 is a curve as in (10) with el < dist ( L , , V) < e2 for el, e2 > 0 sufficiently small. We want to verify that a, E %;"(X) and "a1 = ([all. (19) Let L be another curve as in (10) with e2 < dist (V, L ) < e3, e3 also being small. Then for h E 99 (C)
=
2ni
I
L1
a(w) h(w) dw.
Here we have used that
I
1
h(w) = 1/2ni (z - w)-' h(z) dz = 1/2ni f(z, w) h ( z ) dz L
L
This shows (19). We also can insert the function h(z) =f(G,z) in ([a,], G E {z: dist(V, z) t Q}. Then al($
1 2ni
=-
I
I
1 al(z)f(G, z) dz = - a(w)f(G, w) dw. 2ni L1
157
2.1. SDaces with Conormal Asvmptotics for the Cone
Thus a,(@>belongs to L-"(X)for every @ E S;'") \ V ( E ,may be chosen so small as we want). Now a, E %;" (X); is a consequence of the definition of a , , the finite order increase of a as I Im zI + 00, and the strong decrease of f when IIm z J+ a. Then a. = a - a1E %$(X);, i.e., a = a. + a, is the desired decomposition. For a E %;(X);, R E Ry(")(XX X),we obtain automatically a, E %", (X);. Theorem 5 in general follows by decomposition arguments. It is clear that a1 is not unique. If we take f , ( z , w ) = M,,,(t-"o(ct)) instead of f ( z , w ) we get another a, = a,(c) for every fixed choice of c > O , with a - al(c) E %g(X)l;. Then a l ( c ) +0 in Ern;" (X):for c + 00. This is the main observation for the proof of Theorem 4 (cf. the analogous arguments in the proof of 1.2.2. Theorem 8). 6. Proposition. Let p, v E R, then @or the point-wise compositions) a E %$(X),
b E %Y,(X) j a b E %$"(X)
for every Q, R E R(X x X ) with some resulting SEa(X x X ) where nc x N S = nc NR + nc Q Similarly a E % c ( X ) , b E mb(X) * ab E %$'+Y,(X) for every V, W E T. Moreover a E %;(X) ( E W t ( X ) )implies a * ( z ):= a ( * ) ( n+ 1 - 2 ) E %;.(X)
(E
%X&(X)),
where (*) indicates the formal adjoint WOO,taken point-wise, with some R * E R (Xx X), nc NR * = {( n + 1 - d;., m j ) } , V* = {n + 1 - 5: z E V) .
Proof. The proof is straightforward. So we only want to sketch the arguments. First let us consider the case of continuous asymptotics. It is clear that it suffices to show that a E mr,(X);, b E rmY,(X)l; =$ ab E %r,:yw(x); for any y E R and finite A . By using the decomposition arguments for the carriers of asymptotics it suffices to assume that Vn aSl;(")= W n aSl;("'= 0. This is the situation of Definition 3. Now it is obvious that the point-wise compositions of the operator functions a, b satisfy again the conditions (9) and (ii) of Definition 3 with the order p + v. For (i) we use Theorem 5 in the version for finite weight intervals and write a = a. + a , , b = bo + b l , where a. E mg(X);, bo E Wo(X):,a, E %;" (X);, bl E %;*((x);. Then ab = C oibj and aoboE %$+ '(X);, further aibj E;% ; ,+,,(X); for i + j > 0. From (11) then it follows that ab E %{',Y,(X);. Next let a E %$(X),6 E m;(X).Then it suffices to check (i) of Definition 2. Let p j E ncR and write a ( z ) in the form ( 3 ) close to p j . Similarly for qj E ncQ we have n.
b(z)=
f
Sjk(Z
- qj)-(k+
k=O
1)
+ I=O
(i',(z - qj)'
(20)
*
with obvious notations. For the compositions we have to discuss either qj p j or qj = p j . In the first case we take instead of (20) the Taylor expansion of b ( z ) near p j , namely m
b(z)=
C PjLz -
I=O
11.
PjI1,
(21)
158
2. Operators on Manifolds with Conical Singularities
Bjr E L$(X).Then we have to compose (3) either with (20) or (21). The coefficients at the negative powers of z - p j are in both cases of the form V j k p or &k, where #I belongs to L$(X).In virtue of the conditions about v j k these compositions have the properties as required. Thus we obtain in fact our result also for the discrete asymptotic types. The assertion on a* is obvious. 0 Note that the finite-dimensional subspaces of Cm(Xx X) in S E a ( X x X) depend not only on those in Q and R but also on finite parts of the Taylor expansion of the factors near the resulting poles. Now we come to a first discussion of Mellin-Fourier yDO-s that will be studied in more detail in the following section. For every a(t, r, z ) E C,"(R+XR+,!OthY(X)), z = 1/2 - y ( n ) + i t we can form the operators
opp)(a)u ( 2 )
= tY(")
Mi!+,(T-Y(")o) (2, r, z)M,,, r-y(")u ( r ),
(22)
y E R, u e C,"(XA),that we call Mellin-Fourier VDO-s. In this context it is adequate to restrict the consideration to y = n / 2 , since the general case follows by a simple translation of the weight line. As usual we write op; = op,.
By analogous arguments as for 1.2.3. Proposition 11 we obtain 7. Proposition. Let a(t, r, z ) E Cm(R+XR+, !OtKn'*(X)), p E R. Then op,(a) b a WOO in L$(X*),and the complete symbol over a*, R being a coordinate neighbourhood on X, equals
a(t, t , x , - i h ; ~ ) m o d S ~ ~ ' ( Q ^ x R " + ' ) (23) when a(t, r, x, ( n + 1)/2 + it, [) denotes a complete symbol of a in the sense of a parameter-dependent yD0, dependent on the extra variables r, t. Let a")(t, t, x, -it, 0 be the homogeneous principal part in (z, a(t, t, x, -it, 0,a being given by (23), and set
5>
of order p of
is the homogeneous principal symbol of op,(a) as an operator in L f ( X ^ ) .For abbreviation we also write a f ( a ) and $ ( a ) instead of (24) and (25), respectively. It is convenient to adopt the notion of the compressed cotangent bundle T t C belonging to the stretched manifold associated with a manifold M with conical singularities. T; C is defined as the real C"-vector bundle over C which is dual to the vector bundle belonging to the locally free C"(C) module of C" vector over C that are tangent to aC. In local coordinates ( t , x) close to aC those vector fields are of the form
a
n
a
uo(t, x ) t -+ vj(t, x) -(where (1, x) at j = l axj
a normal coordinate to the boundary) TZC are locally of the form
uk E
n
1
~ ~ ( t , x ) t - l d t + czj(t,x)dxj, j=l
Nk E
C" up to t = 0.
has the usual meaning, in particular t is C" up to
t = 0,
and the C" sections of
2.1. Spaces with Conormal Asyrnptotics for the Cone
159
On TZC we have a natural C" structure and a canonical R, action on the fibres. Thus we can talk about the subspace
S"'(T:C \ 0) c C"(T$C\O)
(26) of all C" functions on T$ C minus the zero section which are positively homogeneous of order p. In particular in local coordinates (t, x, z, 6 close to t = 0 that means p ( t , x, z, 0 E C" up to t = 0, lz, 51 0, and
*
P ( l , x, 1z,A t ) =A'p(t, x, 5
n
for all 1 E R, . This can be applied in particular to x^,where T $ X ^ is compressed with respect to t = 0, t = 00. It corresponds to T$C1, with C, as in Section 2.0. (24) gives rise to a mapping 6 ' : m w . n J 2(X)--j S"'(
T: x^ \ 0) ,
(27) interpreted as the totally characteristic principal symbolic map on the associated operators. We shall later on return to this aspect, where we use the natural Frkchet topology of S(')(T f r \ 0) . The compressed cotangent bundles T$ also make sense over [0, 11 x R or 8, x R for any open R 5 Rn as well as S(")(T:. ..). This will be used in the following section. b
8.Theorem. Let s, Y E R, A E gmP,a,B = 0, 1, and a(z) E m:(X), R E Ry(")(XX X). Then op$")((a) induces continuous operators op$")(a): % y ( X - ) , --j %;-"'(x"), (28) for every P E W ( " ) ( X ) with some Q E W ( " ) ( X ) depending on P and a. Similarly for a(z) E Rt(X), V E Yy("), we get continuous operators op$")(a): %J,Y(X^),
+
X~~~f(X"),
(29)
for every B E Y y(").
Proof: Without loss of generality we consider the case y(n) = 0. For simplicity we only discuss the case A E 3. The arguments for the other types of weight intervals are analogous. Let us now remind of the proof of the continuity of A = op,(a) in the % q n J 2 spaces. From 2.1.1. (1) we obtain with t = Im z I I A ~ l l h ~= - ~J~llb"-"(z) n M(M-'a(z) Mu)Il',z ldzl =
J IIao(~)b"(~)~~11',2Id~I
with ao(z) = b"-'(z) a(z)b-"(z) and integrations over f,,,.Since ao(z) is a parameter-dependent 1pD0 of order zero we have CA
= SUP
IIaO(z)llm < 00
z E rl/2
I
and hence (lao(z)bS(z)Mu112,2 5 C, llbS(z)Mu11',2for all z E fl,,, which yields
5 c,
IIAU11~1-&"/2
I ~ U ~ ~ ~ 4 " / 2 .
For proving (29) we fust observe that the action corresponds to a multiplication of holomorphic functions in (S, \ B ) n (S, \ V) E S, \ ( B + V) with point-wise ac-
160
2. Operators on Manifolds with Conical Singularities
tion of operators along X. By the decomposition method involved in the definition of Emg"'z(X) and of the spaces it may be assumed that B n as, = V n as, = 0. Then for proving (29) we have to check that h + L 2ni / a ( z ) M u ( z ) h ( z ) d z , h e & ( ( @ )
(30)
is Se'(S, n ( B + V)) Q, C"(X)-valued, C being a curve surrounding S, n ( B + V) and that
Ila+ d z ) a ( z )M ~ ( ~ ) 1 1 2 s- , const ,~
(31) for all B E S,, x D ( z )being a D-excision function (Il.llsa was defined by 2.1.1. (19)). (31) follows by the same scheme as above. So it remains to discuss (30). In virtue of Theorem 5 and 2.1.1. Theorem 15 we can write
+ a1 , u = uo + u1,
a = a0
a0
E Em$(X),
a1
uo E s X e p ( X - ) ,
E
IIxe(z) M U ( Z ) I I : ~
Em;-(x), u1
E a;,"l2(XA).
Then we have to consider (30) with the products a,(z)Muj(z) under the integral, i, j = 0 , 1. But all combinations have obviously the asserted property, and it is clear that the corresponding semi-norm estimates hold. Thus we obtain (29). For proving (28) it remains to verify that the discrete asymptotics are preserved. Then we get automatically the continuity, since the spaces sXeJp"'* (XA),are closed in %:$' ( X - ) , . Consider an arbitrary p E S,. Then the Laurent expansion of a ( z ) i ( z ) at p follows by composing the Laurent expansions of a ( z ) and i ( z ) , namely
cf. Definition 2 and 2.1.1. Definition 12, where m = m(a, p ) , n = n(u, p ) , and vjk = 0 for p 6 ncR. This shows that the coeficient at ( z - p)' of the composition is a vj'k , C v i I$., where the fourth variant linear combination of vectors vjk l,, vjk disappears for - m - n - 2 5 r s - 1, i.e., vjk runs over a finite-dimensional space of smoothing operators or ( , over a finite-dimensional subspace of Cm(X). Thus the resulting coefficients constitute indeed finite-dimensional spaces in C"(X) as required in 2.1.1. Definition 10.
(L,
As in the scalar case the Mellin operators for symbols with asymptotics usually occur with cut-off factors and weight shifts. Let us finish this section by some identities about Mellin operators. 9. Proposition. Let
A
= w opP)(h)o = W r y ( " )
opM(T-~(") h)t-Y(%,
h E Em;(X), V E Vn). Then the formal aq'joint with respect to the (. , .)aooc-)-scalar product is
A* = ~ o p L ~ ) ( " ) ( h * h) *~( ,z ) = h(*)(n+ 1 - f),
(32)
( - y ) ( n ) = - y - n / 2 , h(*) being the formal aq'joint operator with respect to (., . ) L ~ ( m . Moreover we have the identities
2.1. Spaces with Conormal Asymptotics for the Cone
161
p ) { w o p p ( h ) w ]( p ) - l = "' Opfn;Y'(")(T-"h')o'
(33)
where w ' ( t ) = w ( t - ' ) , (T-"h')( z ) = h ( n + 1 - z ) , and
%y {w o p p ( h ) w ) (%y)-' =
w1
opp(h)wl
(34)
with (wJ (1) = w(At), A E R + .
ProoJ: Let u, v E Ct(R+XX), then (Au, V)RO(X-) = =
J t"(Au(t, .), ~ ( t ,
dt
.))L~(X)
t " ( u ( t , .), t-"A[*lt"v(t,.))Lz(m
dt
with A [*I as the formal adjoint with respect to (. , .)Lz(x-), A [*I = - Y(n)(op,( T-Y(")h)) I*ltY(")w = wt-Y(") Op,(TY(n)h*)tY(")W,
Thus
h * ( z ) = h'*'(l - 5 ) .
A * = t-"A [*It" = ut-Y(n)- Op,(TY(n)+ n(T-"h*))tY(")+ "w
and - y ( n ) - n = - y + n/2 - n = - y - n/2 = (- y ) ( n ) . Next we show (33). From 2.1.1. (26) and 1.1.1. (26) we get with
e = y(n)
I (")w opR (h ) w ( I (")) - 1 = t - "lot op, (T- Qh) t -put -"I- 1 = t-" - PW'IOpy(T-Ph)I-lW't"+ P Q
= 1-"-
Qw' Opy((T-Qh)')w'f"+ Q ,
From (T-Qh)'=TQh'= TQ+"(T-"h') and -(n + e ) = - y ( n ) - n = ( - y ) ( n ) then it follows (33). Let us finally consider (34), cf. 2.1.1. (11). Set again e = y ( n ) . Then
%y0 op~(h)w(x(;')-' = xiw op&(h)wx;' = %*WfQ Op,(T-Qh)t-Qw%;' = =
rQwAxlop,(T-~h).x;'t-Qwi qtQopM(T-Qh)r-Qwl = wAop$,(h)w,.
Here we have used 1.1.1. (23).
2.1.3.
Mellin-Fourier Pseudo-Differential Operators I
The theory of 1.2.4. allows a generalization to a calculus of wDO-s on R+ x R" (or R+XO, O 5 R" open) with respect to the Mellin transform along R+ and the Fourier transform along R". In the first part we formulate the corresponding definitions and results. They play an analogous role for the cone theory as those of 1.2.4. for the operator theory on R+. In the second part we discuss a variant which is global along X, in the sense of a Mellin 1pD0 calculus with symbols that are yDO-s along X. Set
E ( t , x, t', x', r, 5) = (t/f')-1/2-irei("-"')F. Let a(t, x, f', x', r, 5) be in the symbol space SP((R+XR")2 covariable to (1, x). Consider operators A of the form
(1) X
qr,++,'), (t, 0 being the
162
2. Operators on Manifolds with Conical Singularities m
m
III
Au(t, x) =
-m
U[
m
0
-m
m
E ( t , x, t', x', z, 0 a(t, x, t', x', r, 0 u ( t ' , x')dx'U[,Uz, -m
dt' t
(2)
= (25r)-" d[, Ur = (25r)-l dz, and let first u E C,"(R+XR"). If an operator is given by (2) we also write
A
= OP,(U),
This notation points out the Mellin action whereas the lyD action in the other variables is tacitly carried out. For convenience we shall refer systematically to the weight line r,,,.As in Section 1.2.4. an important aspect will be that we also control the amplitude functions up to t = t' = 0 (and = 00). Formal adjoints will be taken with respect to the L z scalar product based on the measure dt dx. As earlier we assume that the reader is familiar with the standard calculus of wDO-s (cf., e.g., [S27], [H4], p81) and we will freely use here the constructions such as oscillatory integrals, asymptotic expansions of symbols and so on. In particular, there is a straightforward extension of 1.2.4. Proposition 1 to the integrals (2) here by doing the same along R"as in the usual WDO-calculus. In other words we obtain a differential operator L of first order with the property LE = E, ( Z ) PG P - l , and then Au (1, x) =
sE
(t, x, t ', x',
r, 0 {('L)Na (t,x, t ', x', r, 0 u (t ', x') (2')
- l} dx' U [ d t ' Uz, (3)
provided the order of a is < N - (1 + n ) . The integral sign on the right of (3) is an abbreviation for the integrations in (2). Let us sketch the construction of the operator L. First remember that the proof of 2.1.2. Theorem 7 contains a transformation of op,(a) induced by the diffeomorphism x : R +R+,x ( s ) = t = es, s = logt. If we set t ' = es', ia= -112 - iz, then~ u ( t , x =) (x*)-lJ E " ( s , x , s ' , x ' , a , ~ ) a ( e s , x , e S ' ' , x-' ,a + i/2,5) x ( x * u ) (s, x) dx'dfds do,
(4) E"(s, x, s', x', a,0 = ei@--u+i(x-")4The interpretation of the complex argument in a ( ..., - a + i/2, [) is that we simply regard a as a function on the corresponding parallel to the real axis in_the complex plane, according to r e - u + i/2. Now there is a differential operator L of first order in the variables s, x, s', x', a,[with cf., for instance, [H3], [S27]. Moreover E" is the pull-back of E under the mapping J7: (R x Rn)ZX R1 x R" +(R, x_R")2x R' X R", given by ii(s,-x, s', x', z, 0 = (2, x, t', x', - u + i/2, 0. Thus L il*E = x'*E leads to L = (X"*)-lh-*.In other words we obtain
zE"=
1. Proposition. Let 8 (r,0 E C,"(R"+I),B(z, 0 = 1 close to (z, 0 = 0 and set a, = Baa, 8,(z, 0 = B(m, ~ 0 E> ,0. Denote by A, the operator obtained by inserting a, instead of a. Then for every u E C;(R+ x R")
A,u(t)+Au(t)
as E + O ,
where Au is defined by the right hand side of (3), N suflciently large.
Here and in the sequel we omit the proofs when the arguments are completely analogous to the corresponding statements in 1.2.4. The symbol spaces
2.1. Spaces with Conormal Asymptotics for the Cone
163
S'((R+x x R"+l), P ( R +x R" X R"+l) and also with the subscript cl (classical symbols) will be used in the sense analogous to 1.2.4. Definition 3, where we have to add the other variables. For instance, (i) in 1.2.4. Definition 3 is to be replaced X R"+l)and by a(t, x, t', x ' , z, 8 E Cm((R+XR")Z
forall k,k', j E N , x , x ' , B E N " , O ~ tt ,' s T , T E ] R + , ~ E R , ~ E R " , X E K C ~ R " , with constants c = c(k, k', ...) > 0 , and the same for t-' or (f')-l instead o f t or t'. Moreover we have the analogues of the symbol spaces S$, S$,cl(flatness with respect to the variable(s) on R+, including infinity). It is clear that we have a corresponding version of 1.2.4. Proposition 4 in the context here. To every operator (2) with a E S"((R+XR ") X R"+l ) we can form the dktn'butional kernel KA(t,x, t', x', t / t ' , x - x ' ) so that (for p sufficiently negative)
and
Incidentally we also write K ( a ) instead of KA. For arbitrary ,u these formulas may be interpreted in the distributional sense. The singular support of KA E LB'(R+X R" X R+X R")
(7) belongs to diag (R+X R") which corresponds to B = 1, [= 0. This is an immediate consequence of (4) and from the property of distributional kernels of standard yDO-s that have the singular support on the diagonal of R X R". Accordingly we talk (under proper support of KAclose to diag (R+x R")) about proper support of A. It is clear that everything can be performed for arbitrary open 0 E R". This will tacitly be used from now on. 2. Definition. ML'(R+ x 0 ) (ML$(R+ x a ) , ML@+ x n ) , ML$(R+x 0 ) ) is the space of all operators A + G, where A is an operator of the form (2) with an amplitude function X R"+'), s'm+ x m 2 a(t, x, t', x', r, 0 in S"(R+XO)ZX R"+l)(S$(R+XO)Z x R"+1), S ! ! ! ( ( ~ + XX~ R"+l)) )~ and G an operator with kernel in Cm(R+x 0 x R+X 0)for the classes over R+X 0,whereas for R+X 0 we demand that G induces continuous operators G, G*: !Xq""(R+XOc,mp)4!Xm."'2(R+ XOloc) '
(8)
for all s E R, cf. 2.1.1. (7). The *refers to formal adjoints with respect to the LZ(R+x 0 ) scalar product.
164
2. ODerators on Manifolds with Conical Sinaularities
Remember that the weight shift n/2 comes from 2.1.1. (5). Note that every A E MLP(R+X n ) induces a continuous operator C;(R+xR) +C"(R+xR). If A is properly supported then we get continuous operators A : C;(R+Xll) +C,"(R+xR), C"(W+xO) +C"(R+xR). Definition 2 says in particular what we call ML-"(R+ x 0 ) and ML-"(R+ xn).The class ML-"(R+ XL?) consists of all integral operators with kernels in Cm((R+X12)2). It contains ML-"(R+ Xa).A result of Theorem 4 below will be that ML-"(R+ X R) is the space of all operators G for which G,G* extend to continuous operators (8) for all s E R. Similarly to the standard calculus of VDO-s in the definition of ML"(R+XL?), ML;,(R+x a ) we could set G = 0, since the operators G can be genX R"+l). erated by G = opM(ao)where a. E S-"((R+ In fact let G be the integral operator with the kernel g(t, x, t', x') E Cm((R+XL?)*). Choose a function B(z, 0 E C,"(R"+l),8(z, 0 2 0, B(T, 0 dzd(= 1. Then
I
ao(r,x, t', x', T,
5) = (t/r')1'2+ire-i(x-x')E g(t, x, t', x ' ) t ' W ~5)
belongs to S-"((R+xO)2 x R"+l)and it is obvious from (2) that G = op,(ao). On the other hand if we apply that to a kernel g E Cm((R+XO)2) for which the associated integral operator belongs to ML-"(R+XO) we do not obtain in general a. E S-"((R+XLt)2 X R"+l). This suggests certain extensions of MLp(R+ XR), where the amplitude functions are "conormal" close to t = r' = 0 and a. But we neglect this possibility. 3. Remark. We have MLP(R+Xn)= LP(R+XD), ML$(R+XR) = L$(R+XO), but M indicates another operator convention than for the L" classes. We also could starl with the weighted Mellin trandorm, cJ 1.2.4.Remark 2. This would yield the same classes but with operator conventions (and symbolic rules under the algebra operations and coordinate traqformations) depending on the weight y.
4. Theorem. Any A E MLp((w+X l 2 ) induces continuous operators A : 9t?~"~2(R+~Rcomp) + 9 t ? s - h n / 2 (R+x 4 0 3
(9)
for all s E R. Prooj The arguments are completely analogous as for 1.2.3. Proposition 16. The generalization with the extra x, x' variables and the Fourier transform refers to 2.1.1. (2) and the duality to the Lz(lR+x a ) scalar product for subspaces that are localized in the x-variables. The obvious details may be dropped. Let us return to the distributional kernels (7) under the aspects that have been discussed in 1.2.4. in the special case of R + . We are mainly interested in the case R+X 0,since in virtue of Remark 3 the case of R+XO is basically the standard one. So we formulate the propositions in the first case and remark once and for all that over R+x a we can do the same. Denote by C:(R+XO) the space of all q(t, x) E C"(R+xa) with q(r-l, x) E C"(R+xO).In an analogous sense we use the notation Cm*((R+xO)2) and the same for vector-valued functions, or functions of (r, x) ((r, x, r', x')) and further variables varying in an open set.
165
2.1. Spaces with Conormal Asymptotics for the Cone
5. Definition. P * ( R +XR") C the subspace of all u E %m*n/z(R+ XR") for which (MFu)(z, 5) E 9(R;lA), M = M,+(l12+irl being the Mellin transform, F = F,+c the Fourier transform, 9(R"+l ) the Schwartz space on {Rez = 1/2} x R". We consider P 2 ( R +x W") in the FrBchet topology induced by the one-toone correspondence to 9(R"+l ) . 6. Proposition. Let a(t, x, t', x', z, restricts to a function KA(f, x, t', x', /%
0 E Sw(@+ XL?)2 X Rn+l ) ,
nl(,
()+(l,O)
E cm*((R+ xR)2
then KA defined by (S),
((R+'R") \ {(I, o)})). (10)
Moreover
(1 - y(B, O)KA(t, x, t', x', B, 0E c:@+X W , P Z ( R +XR")) (11) for every y E Ct(R+XR"), y = 1 close to (B, 0= (1, 0). The formula ( 5 ) induces an isomorphism S--((Zi+ xL!)~x R"+l ) 1 C;((ii+ xLI)~,P 2 ( R +xRn)).
(12)
Proof. Let us sketch some elements of the proof, although the arguments are analogous to 1.2.4. Proposition 8. First it is clear that the "non-trivial" part concerns the respectively. So we may restrict ourselves to (t, x, t', x') covariables (z, 0 and (B, 0, independent amplitude functions, From (4) we get x*A (x*)-' = A , where 2 is a standard vDO on R X 0.Remember that this transform corresponds to F-'M-'a(z, 5)MF = @-lFF;l(YaYy-l)FoF@
with (Fov)(a) = e-'""v(s) ds, and @ = O0, Y as in Section 1.1.1., i.e. @ = x*. Set G(u, 5) = a ( - a , 0. The kernel K(B, 0 Cefined by ( 5 ) for a with B = t / t ' , 5 = x - x', is related via the diffeomorphism x to K(C0,0, the kernel for the standard vDO A" with the amplitude function G and to= s 1s'. This yields immediately (lo), using the property that the singular support of K is the origin (lo,0 = 0. It is clear that an amplitude function of order -a is in the Schwartz space with respect to the covariables. Then the bijection (12) follows immediately from (5) and the definition R n proof ) . of (11) follows by a straightforward extension of that of of ~ ~ ~ z ( W + x The 1.2.4. (11). Note that a(t, x, t', x', z,
0 E Sw((R+xi2)' X W"")
implies for analogous reasons
(1 - y(B))KA(r,X, t', x', B, 0E Cm*((R+xW,P'z(R+xRn)) for every y E C;(R+), y = 1 close to B = 1. In other words if KA is given and
- W(b))KA
(13) a decomposition with fixed y of the mentioned sort, then we get an associated decomposition KA
= Y(B)KA + ( l
a = a. + al , a. E S - - ( . . .), a ] E S"(. ..) with (R+X 0 ) l x R"+'in the brackets. In the following we write for convenience also iz instead of functions.
(14) t
in the amplitude
166
2. Operators on Manifolds with Conical Sinaularities
7. Proposition, For wery a(t, x, t', x ' , iz, E S P ( ( R + xR)2x R"+l) there exis$ a function X R",d(@,)) (15) u 1 ( t , x , t ' , x ' , z , 5 ) CZ((R+XQ)* ~ such that f o r every y E R
a(t, x, t', x', iz, 5> - al(t, x, t', x', y
+ iz, f ) E SP - l ((R + X Q ) 2
X
R"+l)
and in particular X Rn+l). a(t, x, t', x', iz, 0 - al(t, x, t', x', 1/2 + i7, 5) E S-m((R+XQ)2
An analogous statement holds in the ( f ' , classical symbols.
XI)-
and
(1,
x)-independent case as well as for
u1 follows by the decomposition (14) with iz instead of z. We may also choose p = p(e, 0 as in Proposition 6. Then we get a decomposition A =op,(a)
= A o + A l , A. = opM(ao)E ML-"(R+xQ), A l = op,(al) properly supported.
E
ML@+xQ), and A l
8. Definition. An a(t,x, z, f ) E P ( R +XO x R"+l) with A - op,(a) E ML-"(R+ xQ) is called a complete Mellin symbol of A E MLP(R+x n ) . Similarly an a(t, x, z, F) E Sfl(R+xQX R"+l) with A - op,(a) E ML-"(R+ XQ) is called a complete Mellin symbol of A E ML'(R,xQ).
Set f , 7 t
( t x ) = t-l/2-ireixt 9
and let A E ML"(R+xQ) be properly supported. Then
0 =f
S@(R+XR X R"+l) (16) is a complete Mellin symbol of A. It is of course not clear that it belongs to S@+ XQ x R"+l)when A E ML'(R,xQ). The following theorem and the subsequent remark show that (16) can be modified by a element in S-"(R+xR x R"+l) such that the resulting symbol is smooth in t up to t = 0, t = m. The arguments are analogous as for 1.2.4. Theorem 16. u,(r, x, z,
,:A&E
9. Theorem. Every A E ML#((w+XQ) (MLP(R+XQ)) has a complete Mellin symbol uA(t,x, z, 8 E S"(R,XR X Rntl) (S@+xQ X R"+l)).I f A is written in the form (2) then
A E ML !,(. ..) implies a, E Sf,(...). 10. Remark. There is a canonical isomorphism
ML"(R+Xfl)/ML-"(R+xO) 1 S'"(R+xflx R"+')/S--(R+xflX R"+I) and the same with the subscripi cl and for R+ X R It is induced by A + 0,. Let us give some more comment to the calculations for proving Theorem 9. Consider a neighbourhood of t = t' = 0. One of the arguments is the Taylor expansion at r = t', x = x' of the amplitude function
167
2.1. Spaces with Conormal Asymptotics for the Cone
+ r N ( k x, t’, x’, z,
0
(18)
9
a = (k, E), r, a remainder which is flat of order N at (r’, x’) = (t, x ) . Assume for a moment that we are given an amplitude function p ( t , x, t’, 1 - t ‘ / t , x - x ’ , z,
0
which depends on 1- t ’ / t , x - x ’ as a polynomial. Let us pass to the distribution K ( p ) (t, x, t‘, x’, 1- t‘/t, x - x’, B, 0 by the formula ( 5 ) with p instead of a. By inserting /3 = tlt‘, (= x - x’ and multiplying by @), p E C;(R+), p = 1 close to B = 1, we get another distribution (19) Ki = w @ ) Kb)(t, X, t‘, x’, 1 - B-’, I, B, 0 . Applying MB4,Fc+r to (19) we get an amplitude function F(t, x, t’, x ‘ , z, 0,where the extra arguments are switched to the (z, 0 dependence, and K O (t, x, t’, x ’ , B, 0 equals (19). j7 is an amplitude function in our class, since KO= Ki(t, X, t’, x ’ , B, 0 - K @ ) (2, X, t’, x’, log B, I, B, E Cm((R+ x O ) ~F, 2 ( R +xRn))
0
and
K @ ) 0,x,
t’, x’,
log B, I, 0 = Kb’)(t, x,
t’, x’,
B, 0
with p’(t, x, t’, x ‘ , z, 0 = p ( t , x, t’, x’, a,, D r , z, 0,i.e. j 7 = p ’ rnodS-”((R+xLl)* x Rn+l). Here we have used Proposition 6 in the variant for R+X R It is clear that smoothness up to 0 in t, 1’ is preserved in all construction if we start with a(t, x, r’, x’, z, 0 E S p ( ( R + x 0 ) 2 x R“+l) and take for p one of the summands in (18) or the remainder, the latter one written in the form
with smooth f N , # . In other words (18) for a E s”((R+xa)2 X Rn+l)can be replaced by p,(i‘, X, Z,0 + p N ( t , X, t’, XI, Z‘, 0 E S-”((R+XO)’ X R””), U ( t , X, t ’ , X’, Z,0 -
1
lald N - 1
where p a € S’-IuI(R+xR X R”+l),la)5 N - 1, and p N e S.-N((R+Xf2)2X R”+l).It is now elementary to identify the terms in the sum (17) with the p a .
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2. Operators on Manifolds with Conical Singularities
This is the consequence of an obvious combination of the methods for proving 1.2.4. Theorem 18 and the symbolic formulas for standard ryDO-s (Leibniz rule and expression for the adjoint). Another straightforward extension of standard ryDO techniques gives us the coordinate invaeance of the Mellin-Fourier ryDO-s un_der diffeomorphisms of alone. Let 0,B 5 R" be open sets and xl: B + B a diffeomorphism. Set x ( t , x) = (t, x l ( x ) ) .Then we have induced mappings x * : C:(R+Xd)
Ct(W+ XB), Cm(R+X d )
4
4
C"(R+XB),
The push-forward of an A E MLN(R+x n ) under x is defined by %*A=(%*)-'Ax: C;(R+Xd)+C"(R+Xd).
a
Denote the coordinates in T*B and T*d by (x, and (2, F), respectively. (xl, t(dxl)-l) induces a map T*B + T*d, 2 = x l ( x ) , f = f(dxl)-l&dx being the differential of x. Then we also have (x, '(dx)-l): T*(R+xR) + T * ( R + x d )
in an obvious way. Let r ( y , x ) = x ( y ) - x ( x ) - dx(x) ( y - x ) , and pa(x, q ) = D;eir(y,X)V(y=x, a any multi-index. It is easy to verify that p,(x, q ) is a polynomial in q of degree
5 la1/2. 12. Theorem,. x induces buections x * : MLP(R+XB)+ML"(R+Xd), %*: ML"R+XB) +MLP(R+Xd). I f a ( t , x, r, 8 is a complete Jymbol of A then any complete symbol a" of the wmptotic expansion
x
(2 1)
x = %,A admits
= x1-y2).
As mentioned the proof of (20), (22) is a straightforward generalization of a corresponding classical result on ryDO-s. The bijection (21) is left to the reader as an exercise. Note that the invariance of ML-m(R+XB)is a consequence of the invariance of the spaces WY(R+ x Bcomp), W Y(R+X Bloc)under the diffeomorphisms x, cf. Section 2.1.1. 13. Definition. Let X be a closed compact C" manifold, n = dim X. Then MLN(R+ x X ) (ML@+xX)) is the space of all AEL'(R+XX) such that x.(pAy~) E MLP(R+ X B ) (MLr(R+XB)) for every chart x1: U +0,U being a coordinate neighbourhood of X, B E W" open, x(t, x ) = (t, x l ( x ) ) , and every p, E Ct(U).The subspaces of classical operators will be indicated as usual by subscripts cl.
Note that a(t, t', z ) E C;(R+ x R + ,!Dtkn/z(X)) implies opM(a)E ML$(R+XX). An immediate consequence of Theorem 4 and 2.1.1. Theorem 6 is
169
2.1. Spaces with Conormal Asymptotics for the Cone
14. Theorem. Every A
E MLp(R+ X X )
induces continuous operators A : z&n12(XA)+xS-Knl2(XA)
for all s E R. For A = opM(a),a(t, x, t ' , x',
8 E S;l((R+xL?)2x US(A)(t, x, t,0 = a Y t , x, t, x, -t, 0 , t,
R"+l),we set
(23) where a") is the homogeneous principal part of a of order p. (23) is the analogue of 1.2.3. (13). In view of Remark 3 we have also A E LgI(R+xn) with the homogeneous principal symbol a f ; ( A (1, ) x, t,0 = aS(A) (2, x, 4 (24) cf. 2.1.2. Proposition 7 and 2.1.2.(21), (22). Note that from Theorem 9 it follows a;(A>(t,x, t,0 = a Y t , x, - 5 [) for any complete symbol a(t, x, t,0 of A, where (p) indicates the homogeneous principal part of order p. This gives us a well-defined mapping
n
3
a;: M L ; l ( R + x a ) + s ~ ~ ~ ( T g ( R + x a ) \ o ) , (25) where S(4) over the compressed cotangent bundle Tg(R+ X {manifold})\ 0 was defined in the previous section. From Theorem l l it follows that (25) has natural properties with respect to compositions and adjoints. Theorem 9 shows that (25) commutes with multiplications of operators by functions ql(t, x) E C ; ( R + x a ) or q ( x ) E C t ( 0 ) from both sides. a! induces in a natural way a principal symbol map a;: MLgl(R+x x ) + s @)(Tg(R+x x ) \ 0 ) . It is obviously sujective, and ker a; = ML$x x) . From Theorem 11 and Theorem 12 it follows
(2 6)
'(a+
15. Theorem. A E ML@+ X X ) , B E ML'(R+ X X ) implies AB E ML'+ and A* E MLp(R+X X ) , where
a$"(AB)
=
"(R+x X )
a!(A) a ; ( B ) ,
a!(A*) = a { ( A ) . Now let us briefly discuss Mellin-Fourier pseudo-differential operators that are flat of infinite order at t = 0, t = m. 16. Definition. Denote by ML ;" (a + X X ) the subspace of all G E ML-"(R+ X X ) which induce continuous operators
G, G*: W d 2 ( X A )+%'z(XA) for all s E R. Let ML@+
X x )
(ML$,,,(R, X X ) ) be the space of all A
+ G, where
N
A
A j , Aj = ~ f ( q ~ ) i A " j ~ ) ~ ) ,
= j= 1
xi: Uj +aj as in Definition 13, where LUl, ..., UN]is an open coveriig of X by coordinate neighbourhooh, qj, wj E Cr(aj),Aj = op,(ZJ with Zj E S&((R+ XOj)*X R"+l) (cf. the notations in the beginning of this section).
170
2. Operators on Manifolds with Conical Singularities
For the classes ML;,,,,(a+ XX) we have obvious analogues of 1.2.4. Proposition 23, Theorem 24. An immediate generalization of 1.2.4. Definition 26 yields the classes L;(R+XX), L;,,,(R+XX). As in the one-dimensional case then it follows
17. Theorem. We have ML;(E+ XX) = L$(R+XX) for all p E R, and the same with the subscripts cl.
Note that there exists also an immediate analogue of 1.2.3. Theorem 14 for R+X R". Let us now make some further remarks about the structure of Mellin-Fourier VDO-s.The subclass SC(R"+l)cOnst of amplitude functions a(z, 0 with "constant coefficients" leads to a subclass of operators that we call ML@+ x R"),o,st.By definition A E MLr(R+x R") belongs to it if A = op,(a) + G, where a has constant coefficients and G E ML-"(R+ XR"). Applying Proposition 7 to ao(iz, 0 we find another al(z, 0 such that ao(iz, 0 - u1(1/2 + iz, 0 E S-", ul(z, 0 being holomorphic in z. Since amplitude functions in S-" give rise to operators in ML-", we obtain ML@+ x R"),onst= ML$(R+x W")h0,, where the subscript hol indicates the class of all A = op,(h) + G, where h E S~(Rn+l)const is holomorphic, G E ML-"(R+XR"). Now let a(t, x, t', x', ir, F) E S p ( ( R + X R")zX R"+l) be arbitrary. For simplicity assume that u = u' + a", where a' has constant coefficients, a'' compact support in (R+xRn)2.Then a(t, x, t', x',
iz, 5) E C ~ ~ ( ( R + X R ~ ) ~ ) ~ , , S ~ ( R " + ~ ) ~ ~ ~ ~ ~ .
In other words more general amplitude functions may be generated by tensor products
PO)t u b ) adz, 0 $07 W'), v,@ E C;(Rn). Now we can pass again from uo to al , where we get a change on operator level by ML--(R+xR"). If we finally replace p(t), @(t') by p, @ E Corn@+),
m
c m
@(t? =
pjt"w(Cjt')
+ @o(t')
j-0
with a cut-off function w and a sequence of constants cj + m, increasing so fast that the sums in (28), (29) converge, and pol @o flat at 0 of infinite order, we obtain the more elementary Mellin symbols ajk = w(c,t)tjy(x) a1(1/2 + iz, 0 @ ( x ' ) t ' k w ( c k t ' ) . The associated actions, together with ML-m(a+ x R") and ML$(R+ x R") generate by construction ML@+ XR"). Similarly as in 1.2.5. Theorem 14 the t' powers can be commuted through op,, in other words OPdajd
= w(cjt>t'+ktu(X)op,(al)
V(X? w ( C k t ' )
*
(30)
2.1. Spaces with Conormal Asymptotics for the Cone
171
This gives us indeed a very simple structure of MLr(R+ XR") from the point of view of elementary actions like (30) (besides analogous ones at 00) and negligible and flat operators, the latter ones being identified by Theorem 17. An analogous description holds for classical operators and also globally along X. The spaces ML@+ XX), MLg,(R+XX) can be equipped with Frkchet topologies by a standard scheme, that we apply in our book in many variants. Then we can summarize the above observations as the following 18. Theorem. Every A
A
=S
E L?&+
XX) can be written as
+ S' + F + G,
where F E L$@+ XX), G E ML-"(R+ XX), and m
s = 1 w(cjt)tJop,(aj)W ( C j t ' ) j=O
for a sequence aj E Em$""(X), cj+ 00 suflciently fast, S' = Z(").?(Z("))-' with g o f analogous structure as S, where the convergence refers to the natural Frichet topologies in ML" and ML:,, respectively.
Note that the constructions for Theorem 18 make the phenomenon more transparent from another point of view, namely that the smoothness of amplitude functions up to t = 0, is preserved under all operations in the calculus modulo the remainders in ML-". 2.1.4.
Mellin-Fourier Pseudo-DifferentialOperators I1
The idea of the conification of L:!(X) is to apply the R+-calculus of Chapter I with Lg,(X)-valued Mellin symbols. It is desirable to be able to iterate the method by inserting, for instance, cone operator-valued symbols and so on. So we can try to apply this to general operator-valued symbols, where some structure conditions are satisfied. The shape of this more axiomatic approach will be obtained by an appropriate reformulation of the material of the preceding section. Let 'Z = { E S } ) , , be the scale of Sobolev spaces over X, i.e. E s = Hs(X). We know from the discussion in the beginning of Section 2.1.1. that for every p E IR there exists a b"(z) E Lgl(X; R,)with the following properties (i) bfi(z): E" +E"-' is an isomorphism for every s E R,Z E IR, (ii) b-p(z) := (b"(z))-l E L:"(X; Rr). If necessary W(Z) can be chosen to be formally self-adjoint. In fact, b:(z) = bfi12(t)( b " / 2 ( ~ )would )* be such a choice. Let us employ in this section a modified notation of the conified Sobolev spaces, namely
X%Y(BR+, E s ) = X%Y(XA), for Es = H"(X), (1) where !Py(R+,E") is the completion of Corn@+, Em)with respect to the norm IIUIIS~.Y(R+,E= 3
y(n) = y - n/2. 12 Schulze. Operators engl.
(2)
172
2. Operators on Manifolds with Conical Singularities
Let %;omp(R+, E s ) be the space of all u E %$Y(R+,E s ) with compact support with respect to t E R+ and %fo,(R+,E s ) the space of all u E 9‘(R+, E s ) with qu E %;omp(R+, Es)for every q E C,”(R+). Clearly the definition is independent of y. We have b@(Imz ) E %&Y(X) for all y E R, and op$”)(b”) induces isomorphism op$”)(b@):%s”(R+,E’) +.7eS-@~Y(R+, E”@) for all s E R.We also could define IIUllR*y(R+, E?
= IIoP~lj”)(b”)uIIXo.Y(R+, E@).
(3)
As before we prefer to consider mainly the case y = n/2. Now let a(t, t ‘ , z) E Cm(R+x R +, %@-n/2(X)),z E rl,, . Then we can pass to the operator-valued distributional kernel m
~ ( t t’,,
f @ - 1 / 2 - i r a ( t , r’, z) ~z
B) =
(4)
-m
which is in one-to-one correspondence to a by m
a(t, r’, z) = J B1/2+irK(t, t’, B)B-’d@. 0
These relations are formal for the moment, but we can give them a precise meaning. First remind of a well-known estimate for parameter-dependent WDO-s.Set
Then a(z) E L@(X;R,) satisfies for every v 2 p I l a ( ~ ) I l ~ ( E5: ~C-P~( G P, v )
(7)
for all Z E R,with a constant c = c(s, 1,v ) , cf. [S27], Chapter 11, Section 9.2. This result easily *extendsto operator functions which smoothly depend on the extra variables 1, r’, where the constant c can be chosen uniformly on compact subsets of R+xR+. We shall see that (4)can be interpreted as m
K ( t , r’,
( aaa)NJ B-1/2-ir(l/2+ iz)-Na(t, t‘, z) d7
B ) = -B-
for N sufficiently large, where the differentiation is understood in the sense of 9’(R+,5?(Es, E””)), v 2 p and (8) is then independent of the choice of N. In the further discussion we want to neglect for a moment the t, t’ dependence. Consider the relations
173
2.1. Spaces with Conormal Asymptotics for the Cone
for C" operator functions a : R, 3 L := 2(B1, B2),where Biare Hilbert spaces and L is equipped with the operator-norm topology. The functions a ( z ) in consideration have the property (11)
110,"a(t>llL 5 Ck(1 + Izl>Q
for all k E N, where e = @ ( a )is a constant. Let us fix a Hilbert space Bo and choose isomorphisms bi: B i + B o . Then a +ao = b2ab;' induces a bijection L + L o = ~ ( B oBo). , Set
IbOIlh=llaoa~ll~. We consider L in the norm induced from Lo by the bijection L -+ Lo. It is equivalent to the original one. The estimates (11) are obviously equivalent to llDr ao(r)llh
s Ck(1 + IzlY
for all k E W, aO(z)= b,a(z)b;'. Thus the abstract considerations may be performed in terms of Lo, and later on we can tacitly replace Lo by L. We can introduce the space Xs(R+,Lo)as the completion of C,"(R+, Lo)with respect to the norm
{/
Y'*
l l f I l ~ ( ~ +=, h ) (1 + l ~ 1 2 ) s l l a ( ~ ) a * (dz ~)ll~
I
where a ( z ) = (Mf) (1/2 + iz). Moreover let &"(It, Lo) be the subspace of those a(z) E 9'(R, Lo) for which
Then the Mellin transform extends by continuity from C,"(R+,Lo) to an isomorphism M : Xs(R+,Lo)+ AS((R,Lo), S E R . The same follows for L . In particular for L = % ( E S , E S - " ) we get the desired interpretation of (4), ( 5 ) for every fixed t, t'. As in the calculus on R+ we can define scalar Mellin operators with symbols h ( z ) E %ao. Then
op,(h):
XS(R+,L ) +W - q R + , L )
(12) is continuous for all s. For h ( z ) = h p ( z ) = (112 + iz)", (12) is an isomorphism with the inverse op,(h-"). For p = N we have op,(hN) = ( - p a / a p ) N with the inverse oP,(h-N). This is just the interpretation of (8). It can be used to justify formal manipulations such as integrations by parts in (4) also when a is not decreasing in z. It will be applied below without further comments. The Schwartz space on R of L-valued functions 9 ( R , L ) is defined by the estimates
for all j , k E N. It can also be characterized by
12.
174
2. Operators on Manifolds with Conical Singularities
for all j , k E N . We have 9 (R, L ) c fi(R, L ) for all s E R. Set
T(R+,L ) = M - - ' 9 ( R , L ) .
(13)
Then f ( B ) E T(R+,L ) is equivalent to
for all j , k E N, cf. 1.1.1.(3), (5). 1. Lemma. a(z) E Li1(X;pi,) implies
K ( B ) E Cm(R+\ {l), 9 ( E s , E"")) for every v 2 p, s E lR. Proof. Set v (B) = log B, then ~ $ - 1 / 2 -ir
Let KN(/3)=
=-
(B)B-l/2 -ire
/?-1/2-i7Dya(z) Uz. Then integration by parts yields
K(B) = V(B)-NKN(B), (14) N E IN arbitrary. We have D y a ( z ) E LcTN(X;R,).Thus we get the estimates (7) with p - N instead of p. This shows that for every M E N we find an N = N ( M ) such that KN(B)E CM(R+,9 ( E s , E"")). From v ( ~ ) - ~Cm(R+ E \ (1)) we obtain K ( B ) E CM(R+\ {l),9 ( E s , E s - " ) ) for every M. 0 2. Remark. The proof of Lemma 1 shows at the same time that for all s E R
x ( B ) K(B) E Crn(R+, 2 ( E S ,Em)) for every x E C"(R+)with x ( B ) = 1 for B 6 [E, E - ' ] with arbitrary E, 0 < E < 1, and x vanishing at B = 1 of infinite order. In fact, first we obtain x ( B ) K(B) E Cm(R+,9 ( E s , E"")) for all s. But the formula (14) shows that we can choose v so negative as we want when N is large enough for obtaining the convergence of integrals of S?(Es, ES-")-valued functions. 3. Theorem. Let a (z)E Lg1(X;R,) and x be as in Remark 2. Then
x ( B ) K ( B ) E T(R+,9 ( E S ,Ern)). Moreover
L - - ( x ; R,) = M n T ( R + ,~ ( E sE,- ) ) . S
Prooj From (14) it follows V(B)NK(B) =
J B-1/2-irD;Na(z)u.c
for every N E N. For every v, FER we find an N E N such that
D y a ( z ) E Af((R,9 ( E s , E"")).
175
2.1. SDaces with Conormal Asymptotics for the Cone
In fact, D;a(z)
E
L$-'"(X; R,) implies the estimates c(l+
I ~ D ; U ( ~ ) ~ ~ ~ ( E S5, E S - ~
IZI)'-~-"
for 0 2 v 2 ,u - N. Thus
s (1+
I z 1 2 ) " ~ D ~ a ( z ) l l ~ ( Ed~z s, E c o. -nrs t
s (1+
1z1')'(1
+ IzlZ)fi-N-Ydz<
provided N is so large that 3 + p - N - v < - 1. For v 2 0 the conclusion is analogous. Thus
u ( B ) ~ K ( BE).7ef(R+,9 ( E s , E"")) for every given S; v, provided N enough we have even that
(-B$)'v(B)"K(B)
=
= N(S; v )
is large enough. If N
=
N ( $ v, ko)is large
sB-1'2-ir(1/2+ it)kD;a(z) dzE Z5(R+,~ ( E SE s, - - ' ) )
for all k 5 ko.For f = 0 this means in particular
Trivial estimates show that (17) also holds for xK instead of K. Now we have to show that N may be in fact an arbitrary natural number if we insert xK. To this end we first observe that we have this for xoxK, when xo is a function of analogous quality as x, but xo = 0 in an open neighbourhood of B = 1 . This is again very simple. So it remains to consider ( 1 - xo)xK.But this is trivial again, since for every 0 s j < N
uJ(1- x 0 ) x = uJ- N ( l - B)N-'(l - X o ) U N ( l - B ) ' - N x , and ( 1 - B)J-Nx is close to B = 1 of analogous quality as x whereas u J - ~ ( ~ B)"-J(l - xo) is in C;(R+) which does not destroy the estimates. Thus the first part of the theorem is proved. For the second part we observe that L-"(X; R,) can be characterized as the Schwartz space Y'(R,, Cm(XX X ) ) of Cm(XX X)-valued functions on R. From the definition of T(R+, . ..) it is obvious that this coincides with the right hand side of (16). 0 Now we want to summarize some properties of L!!,(X) in the form of axioms that are required for the values of a yDO calculus with operator-valued symbols. To this end we set
Y'=L$(X), F E R . (a) Y"is a vector subspace of 9 '=
(18)
n9 ( E s , E s - - c ) , S
(b) Y@+lG Y', (c) there is a bilinear mapping YfiX Y v+ Y"+", called the composition, where I E Yoplays the role of the identity, (d) there is an involution *: Y@+ YP,called the formal adjoint (with respect to the E O scalar product), (e) Y-"= T m ,
176
2. Operators on Manifolds with Conical Singularities
(f) if Aj E YP-j, j E N, is an arbitrary sequence, then there exists an A E Yfl such that for every N E N N-1
A-
AjEY”N j=O
( A is then unique mod Y--, and A is called the asymptotic sum of the A 1Aj).
-
In addition we have a principal symbolic structure. In abstract terms we set
EP = S(”(T*X \ 0), where the space on the right is the set of all p ( x , 5> E C”(T*X \ 0) with p(x,A5> = A@p(x,5> for all 1 E lR+ and all (x, F) E T*X\O (that are clearly local coordinates, but the R+ action along the fibres is invariant). For every p there is a space of principal symbols Zp with a Frkchet space structure and a surjective linear mapping u’: Y’+Z’
with Y#- = ker U P , there is a linear mapping op’: Z’+Y’ with IS’op’ = identity, and op’ is continuous with respect to the topology induced by !8’ (op’ is called an operator convention), there is a bilinear mapping CN x 6 ”+C p + ” (the “composition”) with aP+”(AB)= @ ( A ) a”(B)for every A E Yfi,B E Yv, there is an involution *: Z’+P with @’(A*) = @(A)*. Let us introduce in !P a Frkchet topology. Every A composition
= A. E Y p
admits a unique de-
’
A0 = A1 + o p ’ ( ~ ’ ( A o ) ) , A1 E Yfi- . An analogous decomposition follows for Al and so on, Aj
= Aj+ 1
+ Opfi-’(U’-j(Aj)),
Aj+ 1 E Y”-j-
’
for all j E N. Thus the fixed choice of the 0pP-j for all j gives us well defined sequences of linear mappings 1.: yv+zfl-j, q : y @ - + Z @ - j by Aj(A)= A j , q ( A ) = @‘-’(Aj), j E N. The weakest locally convex topology in Y’ under which all these mappings are continuous, makes Y p to a Frkchet space. If op’ is organized as usual by applying F-laF in local coordinates and forming a sum over expressions pF-’aFcp, pcp = cp, where cp runs over a partition of unity with reis independspect to an open finite covering of X,then the Frkchet topology of Y@ ent of the concrete choice of the local coordinates, the partition of unity and the functions p. Then we obtain the following property (fJ let A j ( e ) E Cm(Rm,YP-J),j E N, be an arbitrary sequence, moreover let cjJe) E Cm(Rm)be an arbitrary sequence, j E N, y E N”. Then there is a se-
2.1. Spaces with Conormal Asymptotics for the Cone
177
quence A j ( e ) E Cm(lRm, Y p - j ) with A j ( e ) - x j ( e )E Cm(Rm, Ym) for all j such m
that
1 cj,,D$&)
converges in Cm(Rm,Y.) for all y E Nm.
j=O
Clearly for the convergence only the semi-norms from the mappings Aj play a role, since f& vanishes on YP-J for j > k. The proof of (fi) in the case (18) can be regarded as an exercise. Now let us return to the order reducing families bp(z). For every y, 6 E R, k E N, we have ( D t b Y ( z ) ) b d ( t ) , b’(z)Dtbd(z)E I ~ ; , + ’ - ~ ( R,). X; This is a consequence of
(19)
L W ;R r ) L W ; R,)G J%+ ” ( X ;k), D , ~ L : ~ (R,) x; E L:~R,). Remember that bN(z) as an element of L $ ( X ; R,) satisfies the estimates (7), u 2 p. Let I l . l l s r be the norm of 2 ( E s , E f ) . For abbreviation we denote the norm in E s by 11. IIS. Let us introduce parameter-dependent norms 11.11 s; ,in E s by IIulls;r:= IlbS(t)~IIo, u E E s . Then we get parameter-dependent norms in 2 ( E s , E S - @3) a IlaIIs,s - p ; * := IPS- fl(z)ab-s(z)110.0.
If b””(t) E L:,(X; R,), S E R, is another order reducing family with the properties listed in the beginning, there are constants c’, c” (depending on s) with
-
CtllbS(z)uIlo5 IlbYz)uIlo 5 c”ll&z)uIlo
for all u E E’. In this sense we can talk about equivalence of parameter-dependent norms. Similarly we have a notion of equivalence of parameter-dependent operator norms. In the sequel we assume that the bS(z)are fixed once and for all, for all s E R. Then bp(z)b’(z) is equivalent to b p + ” ( t )in the sense of equivalence of the associated norms. Our choice of reductions of orders implies the following proposition which is left to the reader as an exercise. 4. Proposition. For every s, s‘,s“ E R, s’ 5 s 5 s“, and every a E 9’we have
lla II4 s - p ; 7 5 c max Illa IIb.b - p ; 7 , IlaIId,.S,* - p ; 7 1 with a constant c = c ( d , s“) independent of Z E R and of the operator a. The order reducing families can be used to define spaces of operator-valued amplitude functions for a Mellin lyDO calculus. Remember that 7? denotes the scale of Sobolev spaces over X. From now on we set for abbreviation QP =
(R+)P,
-
Q P = (R+)P,
p EN.
For every a(r, z) E Cm(QPX W,Y ” ) we set m
~ ( a(r,)
=
J b-1” -m
-i7a
(r, z) dz,
178
2. Operators on Manifolds with Conical Singularities
provided it is defined by an extension of the inverse Mellin transform. Then m
a(r, z)= ~ B 1 ’ Z + i ‘ K ( a ) ( r , B ) B - l d ~ .
(21)
0
5. Definition. Sp(@ X R; Z), p = 1,2, denotes the space of all a(r, z) E C”,@ x R,Yp) with (i) D : D t a ( r , z ) E C m + ( @ x R , YH-k)forallccENP, k E N ,
(ii) for all s E R we have
IlD:Dfa (r, z) lls,s - + k ; r 5 c (c
for all 01 E I N P , k E N , and all (r, z) E @‘ X R wizh a constant c depending on cc, k,s, (iii) K ( a ) (r, B ) has the property that x(B>K(a>(r, B ) E c:(@, Y(R+,Y ? - ~ N for every x as in Remark 2. In an analogous manner we can define the spaces Sp(QP x R;Z), where in particular (ii) holds for (r, z) E K X R for every compact K cc QP and c may depend also on K. Remember that Y? - - = nY?(Es;E m ) is endowed with the corresponding Frkchet S
topology. K ( a ) (r, B ) is well-defined for analogous reasons as in the beginning of this section. The condition (iii) is called the scale axiom (for the distributional kernels). In the following we only consider the spaces over @, those over QP are analogous. The definition of SF(@ X R; Z) induces a natural semi-norm system on this space. First we have the mappings SV(@ x
R;Z) + Cm,(@ x R,Yp - k )
from (i) for all a,k, secondly Sp(@
X
R; Z) + C:(@, F(R+,Y?--))
from (iii) for all x, and finally the best constants from (ii). Then Sp(@ x R; Z) becomes a Frkchet space. Note that (ii) can be replaced by the system of estimates
I(D:D;(bs-”+k(t)a(r,z)b-s(z))llo,o 5c for all cc E Np, k E N , (r, z) E @’x R. It can easily be proved that a E S#(...), a ’ ~ , ! i ‘ p ’ ( ...) implies a a ’ E S p + ’ ’ (...), and D : D t a E S H - k ( . . . ) . Let S h ( . ..),on,t be the subspace of all a E SF(. ..) which are independent of r. Then for Z = { H S ( X ) } , R
L$,(X;R) E Sp(@ X R; Z)co,,t
(22)
L-=(X; R) = s--(@ x R;e > c o n s t .
(23)
and
179
2.1. Spaces with Conormal Asymptotics for the Cone
In particular bp(z) E Sp(@ X R; Y),,,,, for every ,u E R.The inclusion for ,u 9 - m is proper. The subspace of S p ( @ x R; Z)constwhich corresponds to Lg,(X;R) can be characterized by additional axiomatic requirements. They will be discussed below. For the moment we want to continue with the Mellin lyDO calculus with amplitude functions in the spaces of Definition 5. 6. Proposition. The formula (20) induces an isomorphism S--(@
X
R; Z) +C;(&
T ( R + ,5Y”)).
This follows for analogous reasons as Theorem 3, cf. also the formula (16). Now let us write for a moment iz instead of z. 7. Proposition. For every a (r, iz) E Sp(@
s4(cZ,YN)),
al(r, z ) E G(@,
X
R; Y ) there exists an
z = e + iz,
such that for every e E R -
a(r, it) - al(r, e + iz) E S p - l ( Q p
X
R;Z ) ,
(24)
in particular a(r, i t ) - al(r, 1/2 + iz) E s--(@ x R; Y ) .
(25) Prooj The typical part of the proof concerns r independent amplitude functions. So let us suppose this for convenience. Choose a function E C:(R+), w(B) = 1 close to p = 1. By Definition 5 we have
yrv)
(1 - v U ) ) K ( a )(B) E n R + 2-3 Applying (21) to (1 - y ) K ( a ) we get an ao(1/2 + iz) E S-m( ...), cf. Proposition 6, and hence 9
s p1’”’‘w(B)K(a) Q3)B-l dp m
a1(1/2+ i t ) =
0
has the property a(iz) = a1(1/2+ iz) mod S--(. ..). Thus we have proved (25). Now m
al(z>=
I B ’ w w K ( a )(B) B-’@
is obviously holomorphic in z and Yp-valued. For (24) we can repeat the arguments of the proof of 1.2.4. Proposition 10. 0 Let us discuss a generalization of Proposition 7 with respect to the analogues of the operator-valued Mellin symbols of Section 2.1.2.
8.Definition. m;(X; Y ) , V E Y“, is the space of all a ( z ) E a(@ \ V, Yp) such that R ; Z),,,,, for all e E R (26) for every @“ curve c C \ V with a diffeomorphic projection + r, parallel to the real axis and \ r, being a bounded set. a l f Q ( z )is to be interpreted as the value of a on which only depenh on t = Im z. Furthermore the conditions which express (26) in tenns of the properties of Definition 5 are required uniformly on every finite e -interval. a IfQ(z) E Sp(@
c
X
c
180
2. Operators on Manifolds with Conical Singularities
For V E V we define % f ( X ; Z) = T~%!&(X; Z) and
% f ( X ; Z) := %r,(X;
e )+ %l;2(x; Z) ,
V;. E V, V = Vl + V2.Here we use a natural Frkchet topology in %l;(X; Z) when V V, and the sum is to be understood as usual, including the Frkchet space structures. We do not perform here a discussion like that in Section 2.1.2. for the subspaces % f ( X ) c %l;(X; C) in the case Z = {HS(X)}sER E
but there exists, of course, a corresponding generalization. We also could talk about spaces of the sort % f ( X ; Z): as the analogues of %l;(X)l;, and subspaces with "discrete asymptotics". 9.Proposition. For evely a(z) E %l;(X; Z) there exists an al(z) E %g(X; Z) such that a (z) - a1(z) E rm;- ( X ; Z).
ProoJ We may assume V E Yo. As in the proof of Proposition 7 we set m
1 K ( a ) (B) = - B-'a(z) dz, al(2) = 2ni r,,2 0
f
B ' w W K ( a ) (B) B-'dB.
Then it suffices to ensure that
is in S-" on every curve & 3 z. In virtue of the scale axiom the difference (27) is certainly Y-""-valued. On the other hand the arguments which show the strong decrease 0 for IzI+ m including the derivatives in z apply also for z varying along
4,
Note that in view of
a;" ( X ) = %;"
( X ; Z) for
Z = {HS(X)}sER
the proof of Proposition 9 is an alternative method for proving 2.1.2. Theorem 5 . 10. Theorem. For every sequence aj E Sp-J(@ x
R;Z),
j E IN, there exists an a
N-1
E P(@X
R;Z) with a -
1aj6 SN-~(@X
R;Z) for every N E N .
j=O
-
Clearly a is unique modulo S-" (. ..), and we write as usual a C a j . The proof of Theorem 10 is not evident at a first glance, since any excision in the covariable z would destroy the smoothness of remainders along the fibre spaces I?. Here we only want to sketch the proof. More details will be given below in the Sections 3.2.3., 3.2.4. in a slightly more general situation. On the other hand, in the cone theory we are interested in amplitude functions that belong to L;;-'(X; W,)for every r. In this subclass we can apply excisions of local amplitude functions in (F,z) and then obtain an asymptotic sum even in this subclass. Therefore the general case does not affect the essence of the cone calculus.
181
2.1. Spaces with Conormal Asymptotics for the Cone
Proof of Theorem 10 (the idea). Let v(B)E Ct(R+), w(B) = 1 close to 3/ = 1. Set yl(cB)= ~ ( -1c ( 1 - B ) ) , c 2 1 a constant. The idea is to choose a sequence of 4 E 9-J with aj - Cj E S-” such that
1
K(Zj) (r, B)I
~ , + r { ~ l ( c j ~ )
j=O
converges in Sfl for a sequence of constants c j , increasing sufficiently fast. Let us drop again the r dependence which is not the typical point in the proof. For every M E W , V E R there , is a j o E N such that
P(B) for all j
K ( a j )(B) E CiM(R+, y”)
z j o .By applying the Taylor expansion at B = 1 we get a decomposition
w(B) K ( a j ) (B) = v(B) pM(B) ~ ( % ( raj> ) + R M ( a j ) (B), where PM,PM are polynomials in the corresponding variables of order M, and R M ( a j )is flat at B = 1 of order M.For all j sufficiently large we get V(B)
w
~ ( h ( raj)) E c ~ ( R +
~ M ( B )
9
with the given v , and &(aj) (B) E Cf(R+, Y”). Now we start with sequences Mk E N , vk E R , with Mk+ m , vk+ - 0 3 , and choose j = j k so large that the given decomposition applies for j 2 j k . This yields intervals L k , j k + l - 11 for which all j E L k , j k + 1 - 11 fit to Mk, vk. Set a > = MB-.,RMk(aj)(B),j E [ j k , j k + l - 11. Then by construction K ( a j ) ( B ) is flat at B = 1 of an order that tends to m as j + 00. Then an argument of the type for proving the Bore1 theorem (that asserts that for any given sequence of constants there is a @“ function with these constants as Taylor coefficients at a point) yields the convergence
after an appropriate choice of c j . An extra consideration shows before that integral expressions as in (28) estimate the semi-norms for our space Sfl. Here we also use the estimates (6) for the order reducing families. Now we also have the functions
ay(r)E yO(R, yk), K ( a y ) = W ( B ) p M k ( ~K(FMkaj) ) 3
jE[jk,jk+l
- 11. Using the above axiom ( f i ) there are Cy E y(R, Y ” k ) with a: - c?;
9(R,
such that the sum over [lvl(cjB)K(Cy) (B)ll:,-,dB converges, too. Then we can set Cj = aJ + Cy and obtain the convergence of the original sum. If necessary we drop finite partial sums for every fixed semi-norm for having sufficiently negative orders of the terms in the remaining sums. The details of the proof show that the choice of iiy is possible in such a way that it fits to all semi-norms and to the constants cj which are determined before by a diagonal argument. 0 E
Now we are in the position to establish the standard elements of a Mellin 1pD0 calculus on R+ with P-valued amplitude functions in the sense of Section 1.2.4. with completely analogous definitions and results.
182
2. Operators on Manifolds with Conical Singularities
11.Definition. MLp((B+ ; Y ) is the space of all operators A + G,where A = op,(a) with some a E S#(@ x IR, Y ) and an operator G which induces continuous mappings G, G*: 2%"'(R, , E') -+ Xm* ""(R +, E m )* ML#(R, ; C) is the space of all A
12. Theorem. Any A
E
= op,(a)
with a E S p ( Q 2 x IR; Y ) .
ML"((K+; C) induces continuous operators
A : 2 4 " 2 ( W + , ES)-+2'-p-"12(R+, ES-p) for all s E R. Similarly A
E MLp(R+ ; C)
induces continuous operators
A : 2&,mp(IRt, Es)+2;o;"((W+, E"") for all s E R . 13. Definition. An a(t, z) E S@(R+XR; C) is called a complete symbol of A E ML'(B+ ; C) i f A - op,(a) E ML--((K, ; C) . Similarly we define a complete symbol of A E MLJ'(R,; C). 14. Theorem. Every A E MLp((K+; C) has a complete symbol aA(t,z) E P(R+ XR; C), and i f A =op,(a) + G with a~ P ( @ X R ; C ) , G E M L - " ( R + ; C) then
An analogous assertion holds for the classes over IR+ . The proof follows in an analogous manner as the corresponding 1.2.4.Theorem 16, cf. also the remarks to 2.1.3.Theorem 9, in particular the technique to switch tlt' dependence to some z dependence, which shows once again the smoothness of remainders up to t, f' = 0 (or m). Note that we also have analogues of 1.2.4.Remark 17 and Theorem 18. It is obvious that for C = {Hs(X)},,R,
ML#(K+x X) E ML#(R+; C) with equality for ,u = - a . MLp(R+ XX) can be defined as in Definition 11, where the amplitude function a(t, t', z) belongs to L't,,(X;@ X R). Here @ is treated as a compact set. More precisely L't,,(X;@ X IR) consists of all operator families a(t, t ' , z), (t, t ' , t)E @ X IR, which are obtained as sums of operators voau(t,t', z) pu, where U belongs to an open finite covering of X,the system of pu E C;(U) is a subordinated partition of unity, vu E C : ( U ) , vopu= pu, and (xl)* vUau(t, t ' , z) pu (xl as in 2.1.3. Definition 13) is a yDO in Rn, based on the Fourier transform, with an amplitude function p ( t , t', x, x', z, E) E Cm(@ x RZn X El:,;') satisfying the symbol estimates lD;,,..D!.[P(t,
t ' , x, x', z,
016 c(1 + 1%
tl)p-lpl
for all multi-indices CI, fi and all (t, t ' , x, x') E K , K cc@ x It2", (z, 0 E IR"+l, c = c(cc, fi, K ) a constant, and the same for the functions, obtained by replacing t or t' by t-' or (t')-'. In addition p is assumed to allow an asymptotic expansion into
183
2.1. Spaces with Conormal Asymptotics for the Cone
functions p j that are positively homogeneous in (z, 0 of order p - j (for lz, 51 2 const). All general constructions to the classes S N ( p X R ; Z),MLN(K+; Z) such as passing to complete symbols, compositions and so on can be carried out within MJ!,$(R+ x X ) . In particular Proposition 7 restricts to amplitude functions in L:,(X; p X R ) . Note that the above scheme of introducing in L $ ( X ) a FrBchet space structure can easily be modified to obtain a natural Frkchet topology in L:,(X; p x R ) . In an analogous manner we also can define
L!!,(X; @ x RN), p , N E N, (2 9) with an N-dimensional space of parameters z. The extra parameters can be useful for many reasons. Below they play the role of covariables in another symbolic level with either a Fourier or a Mellin operator convention. For the moment they can be employed to establish a parameter-dependent variant of ML!!,(K+ x X) namely A4L:l(R+ x x; W).
(30)
It follows by applying the definition of MLP(K+ ; Y ) to amplitude functions a ( t , t', z, ?) E L:,(X; x R, x R?) with respect to t, t', as before, whereas M L - " ( K + x X ; R?) consists of Y(R,-, M L - " ( R + x X ) ) . In other words every A ( ? ) E M L $ , ( K + x X ;R?) has the form A ( ? ) = opM(a)(?)+ G(?) with a as mentioned and G(?) E M L - " ( W + x X ; It?), The parameter-dependent operator-valued amplitude functions satisfy obvious analogues of Theorem 3 and Proposition 6 , namely
Theorem.
15.
a(r, z, ?) E L;,(X; Q' x R:,
Let
?)
and
K ( a ) (r, p, ?)
m
=
J p-l'*-ira(r, z, q
~ z Then .
-m
xw
K ( a )(r, 8, ?) E Y ( R ~c; , (Qz, where x is as in Remark 2, and L--(X;
@ x R:
?) =
TW+,TI)),
M,.+,Y(Rf, Cm+(p; .T(R+, T))).
The method of proving Proposition 7 also easily extends to amplitude functions in
L$(X;p x R:, ?).
z = e + iz, such that for every
e EW a(r, iz, ?) - al(r, e + iz, ?) E L:;
and in particular
l(X;
p x Rt,
?)
184
2. Operators on Manifolds with Conical Singularities
We want to apply Proposition 16 to a special parameter-dependent wDO b$(it, 23 with the homogeneous parameter-dependent principal symbol (I ZI + 15l2+ l?lz)"'z. Such a b' clearly exists, since for every homogeneous principal symbol the above construction by local expressions (together with an excision in ( t , [, T ) ) gives an associated operator. By Proposition 16 there is a b:(z, ?) such that b { ( e + it, F) has the same homogeneous parameter-dependent principal symbol, for every fixed e E IR.In virtue of the parameter-dependent ellipticity there is a constant c1 > 0 such that
+ it, 8 : H s ( X ) + H s - - ' ( X ) is an isomorphism for It, ?I 2 c1 and e = 1/2. This is true in particular for b:(e
t cl. Fix a with I?, I 2 c1 and set b';(z) = b{(z, Z,). b';(z) is holomorphic in z and bijective as operator in 2 ( H S ( X ) ,H"p(X)) for all z with I1/2-Re zI < e , e > 0 sufficiently bmall. Then for every 6 > O we find a constant c(6) such that b r ( z ) := b t ( c ( 6 ) z ) is bijective in I1/2 - Re zI < 6. If we are given arbitrary x, x' E R and choose 6 so large that 1/2 - 6 < x , x' < 1/2 + 6, then b'(z) is just as asserted in 2.1.1. Proposition 4. 17. Remark. The order reducing families bp(z) obtained in this way have the p r o p e q that b#(e i t ) by(@+ i t ) b-@+")(e+ i t ) and b-F(&+ i t ) b'(e iz) are uniformly fin t) bounded in % ( L z ( x ) )for all p, v E R,uniformly in e, $for 112 - 6 + e < e, @ < 112 + 6 - E for all E > 0 .
+
+
2.1.5.
Green and Flat Cone Operators
In this section we introduce the Green and flat operators for the cone. They constitute a subalgebra of the cone algebra consisting of all operators for which the Mellin symbols vanish. Most of the statements are completely analogous to those in 1.2.5. The proofs will be omitted then without further comments and left to the reader as exercises. 1. Definition. An A E 2 ( X o ( X " )X, O ( X " ) )is called a Green operator of the class %,(X*),, A = (6, 6') E 3, if A, A* induce continuous operators
A : X o ( X " )+X;(X"), ,
(1)
A*: X o ( X ^ ) + X ~ ( X " ) ,
(2)
for certain B, C E 7r - n / 2 , depending on A . Define the subclass %G(Xa)iby the condition A : X o ( X A+ ) 2;( X - ) , ,
(3)
A * : Xo(X")+ XG(X"),
(4)
depending on A. for certain P,Q E 5%R-n/z(X)
The adjoint operators refer to a scalar product in X o ( X " )that we keep fixed once and for all. Write %G(x")
= %G(x")(-m,
m)
>
%,(x")*=
Often it suffices to consider the case A % G ( x " ) 8 = % G ( x " ) ( - k k),
= (-
%G(x");-m,
m)
.
k, k ) , k E N \ {0). We then set
%G(x*); = %G(XA);-k,k).
2.1. Spaces with Conormal Asymptotics for the Cone
185
In order to unify the notations we also write %G(X*)(O,O)=
%G(r);o,O)
=
f - " 2 M L - m ( K + X X )t"*.
Thus the Green classes are defined including k = 0. Every kernel g E X o ( X " )@, Xo(XA)induces a Hilbert-Schmidt operator opG(g) u(t, x ) = ( S ( l , x, t', X I ) , Ju(t', x?)@'(X.)
in X o ( X " ) g. is uniquely determined by opG(g). Then A = opG(g)+g induces isomorphisms %G(X"),
lim
2
~
% z ( X " ) , @rJXF(X")A
B,C€T-"z
and lim , X ; ( X " ) , @zJXE(X")A
%G(X");
P, Q E 5 V d 2 ( X )
for all A E 3. This follows in an analogous manner as the corresponding relations in 1.2.5. In (6) it was used that
X;(X^), C3JzX;(IA), 1 2;(XA), @rSq(X^), for discrete asymptotic types P,R. The operator J in (5), (6) might be omitted, of
(c
being the complex conjugate), course, since for instance J % ~ ( X " ) ,= % F ( X * ) but with J we can read off immediately the asymptotic types of the adjoint operators. Note that the identifications (9,( 6 ) give rise to natural locally convex topologies in the spaces %,(X"), and 9lG(X");.
2. Proposition. %G(X*),, A E 3, is a *-algebra of operators in X o ( X * ) .Every A E %G(X-), induces continuous operators A:
%S(X")
+2;( X J , ,
A*: Xs(XA)+X;(Xa), for every s E R , where B, C E T M are 2the same as in (l),(2). A n analogous assertion holds for 3. Proposition. Every A
E W G ( X " ) ,,
A E 3, induces compact operators
A , A*: Xs(X")+%s(X*),
SE
R.
The same is true, of course, for A in the dotted subclass.
4.Proposition. Let G E %,(X*), and assume that 1 + G: Xs(X")+Xs(X") is invertible for fixed s E R . Then (7) is invertible for all s E R and (1 +
1 + GI
for some GIE % G ( X " ) A Moreover . G E X G ( X " ) ; implies GIE '9IG(X");.
(7)
186
2. Operators on Manifolds with Conical Singularities
5. Proposition. Let N be a finite-dimensional subspace of Xi( X - ) for some B E Y - n / 2 . Then the orthogonal projection P : Xo(X-)+N belongs to WG(X").A n analogous statement holds for the discrete asymptotics as well as for the classes with finite weight intervals. Se(Xs(Xa),Xs-'(X")) is called frat of order - 6 at
6. Definition. An operator A E SER
t = 0 and 6' at
t = m ,0 5
- 6, 6' 5 03, if A induces continuous operators
A, A*: Xs(X")+Xi-'(X"),j for all s E R, A = [6, 6'1. In a similar manner we can also define flatness with respect to open and halfopen weight intervals. Observe that when A satisfies Definition 6 then A' = Z(n)A(Z(n))-l is flat of order 6' at t = 0 and of order - 6 at t = m, P")being defined by 2.1.1. (26). 7. Definition. W$(XJ8, 9 = (-k,k ) , k E N , p E R, is the set of all A E rd2(ML{,(R+ X X ) ) td2 which are flat of order k at t = 0 and t = m .
The notation in Definition 7 means, of course, that any A A = t-d2Aotd2with some A . E ML;,(R+ x X ) . We have
E W:(Xa)e
is of the form
w:(x-):= kn 3 : ~ " )= ~ ML;, c l ( ~ + X I , eN cf. 2.1.3. Theorem 17. Moreover, by definition, W~(X^)(O,O) = tP2ML{,(R+X X ) t"2.
W;(X^), is a *-subalgebra of r d 2 M L ; ( R +X X ) t d Z , filtered by the orders. 8. Remark. Let a(t, z ) E C:(R+, Em$(X)), then
taopid2(a)
R$(X")#
for a,/I 2 0, k = a + /I E N.
Note that for every A"E M L { , ( R + X X ) there is a complete symbol al(t, z ) E C z ( R + ,Fm$(X)). This is a consequence of the Z independent variant of 2.1.4. Proposition 16 and the existence of a complete symbol by 2.1.4. Theorem 14, applied to the subclass ML"(R+X X ) . A
=
rd2op,(al)
td2
+ G E W:(X")e,
G E t-dZML-"(R+X X ) tdZ,does not necessarily imply that G or t-d2 opM(a,)td2 are flat. In fact we can choose an arbitrary a. E C",(K+, Fm;"(X)), ao(O,z ) 0 . Then
*
G~= t-"/2op,(ao) tM2 E W,(X*>*nt-d2ML-"
2.1. Spaces with Conormal Asyrnptotics for the Cone
9. Definition. For every 0 5 k
s
187
we set
%$+ C(X"),= %$(X")8+ ~ G ( X A ) , , %$+G
For 8 = (-
W ;
m, m )
For every A
=
^h+% G f i 7 ; .
we also omit the corresponding subscript.
E %$+ G(X")B we
have a well-defined homogeneous principal symbol
cr;(A) (t, x, z,0 E W ( T ; x^ \O)
(8) which is flat of order k at t = 0 and t = m . Denote the subspace of symbols in S(@)(T; x^ \ 0) that are flat of order k by
W)( T ; x^ \ O),
,
(9)
equipped with the induced FrCchet space structure.
An analogous statement holdr for the dotted subclass.
1l.Definition. An operator 1+ A , with A E %$+ c(X^),, 15 k s m, is called elliptic if 1+ a t ( A )(t, x, z,8 does not vanish for all (1, x ) E x^ (including t = m) and all (z,0 ER"+'\O. 12.Proposition. Let 1 + A , A E %$+ ,(X"),, be elliptic, 15 k 5 . Then there exists a parametrix of the form 1 + B for some B E %$+ G(X-)ein the sense that 1- (1 + A ) ( 1 + B ) , 1- (1+ B ) (1+ A ) E %G(X"),.
An analogous statement holds for A in the dotted subclass with %,(X*);.
n
13.Theorem. Let h E Em$(X), VE Y d 2yLd2- f l , B 2 0 , and w , o1be arbitrary cuto f ffunctions. Then wtaop;Mz(h)wl - w o p i d 2 ( P h )t Moreover i f h ~ E m ' ; ( X ) V, E Y . - d 2 - Y ,
h,E
%,(X").
(10)
~ E Rcc,/lzO, , o l + B = k E N , then
WtaopiM2(TYh)tpwl E %$+ G ( X * ) a .
(11)
Analogous results hold for h E Em&(X)*with the dotted Green classes.
14.Theorem. Let h E Em$:(X), p E Pi, and wi,Gi be arbitrary cut-offfunctions, i = 1,2. Then 01 opid2(h)w2 - Gl opn;"JZ(h)G2E%$+G(X"), ~1
opiM2(h)(1-o2)~%$+G(X~).
An analogous result holds for h E Emgso(X)*with %$+ G(Xn)*. 13 Schulze, Operators engl.
188
2. Operators on Manifolds with Conical Singularities
15. Proposition. Let A cut-off functions. Then
E %;(X"),,
m ( z ) E W $ ( X ) ,,u, Y E
R, and wlr w2 be arbitrary
w2 E %$+"(X^),,
w1 op;""(m)
wzA, Awl op;""(m)
w ; op;""(m)
w ; A , A w ; opid2(m)w ; E %$'"(XI),,
w i ( t ) = wi(t-1).
16. Proposition. For every k E N the sequence
o+n$;',(x~),+W$+,(x"),
AS(')(T$
r \O),+O
is exact and splits. An analogous statement holds for the classes with dot.
The splitting property means that there is a linear operator opf: S'r'(T;X^ \o)B+W$+c(x"), which is a right inverse of o f . This follows in an analogous manner as in the proof of 1.2.5. Proposition 17. We then obtain a sequence of (non-canonical) symbol mappings .(r b - : Wr,+ (X ^), + S(@- "( T; \ O),
r
JJ
by posing o r ) ( A ) = a f ( A , ) , A, = A , A,-l = A , , - opf(of(A)), a'9-1)(A) = of-'(A,-,), ...
depending on the choice of the operator mappings opf-j. Set I
equipped with the topology of the component-wise convergence. Then the sequence of at constitutes a linear mapping - J J
a$'': W $ + G ( x " ) , ~ s y m b , " ( ~ ; r\o),. 17. Proposition. For every k, 1 E N the sequence 0 +3,F -+( GI +
1)
af ' (X"),+%$+G(XA)e -SyrnbK'(T;X^
\O),+O
b exact and splits. An analogous statement holds for the classes with dots, and also for F as subscript instead of F + G.
Then we can use again the isomorphism %$(X"), %$-('+')(X"), @Symbkl(T; x^ \O),, Moreover we have continuous operators A, A*: ~ 3 ( X - ) ~ r e s , - ' + ' + ' ( X 3 e ,S E R for all A E % $ - ( ' + ' ) ( X * ) , . This gives us a natural locally convex topology in
%$(Xa),, and then also in W$+o(XA),, %$+ J X " ) ; , via Definition 9 and (3, (6).
189
2.1. Spaces with Conormal Asymptotics for the Cone
18.Remark. Assume that (GjIj b a sequence of Green operators in %G(X*)a, k E W \ {O} fixed. and aj,4 non-negative reals, aj+a, bj+ as j + a . Then there are B, C E Y - d 2such that wt"lGjt'1~E % z ( X " ) a 81-J%z(X")a for all j , where o b a cut-off function. An analogous remark h o l h for the dotted subclass.
19.Theorem. Let h E Emt(X), Y E Y-B-n/2,a,/3 > 0, n = a + B E N, n cut-offfunctions wl, wz and let c L 1 . Then
> k E N.Fix
Ac:= wl(ct) t"op~""(TBh)tPw,(ct)+O in %$+ c(R+)e as c+
m
. An analogous assertion holak for the classes with dots.
Now we will briefly describe the analogous operator classes on C, where C is the stretched manifold belonging to a manifold M with conical singularities, X being the base of the cone. The corresponding spaces were already defined in Section 2.1.1.
20. Definition. We set (with the notations of 2.1.1. Definition 19)
+
%c(C), = w%G(U)AW (1- ~ ) L - ~ ( i n t C ) ( lw- ) ,
where !TIc( U), is induced in an obvious way by int U+ X".Similarly we define the subclass with dot. Moreover %$(C), = w%$(U)gW + (1 - w ) L:,(int C) (1 - w),
%$( U)e being induced by int U+
%$+ c(C)a,
X^.We also use the notations
%$+ c(c>;
in the analogous sense as above.
Clearly only the left components of A = (6, 6') or 8 = (- k, k ) are involved. So we may set A = (6,O) or 8 = (- k, 0). The theory of this section has a straightforward generalization to C instead of Xa.The obvious details will be tacitly used below. 2.1.6.
Appendix I (Mellin Sobolev Spaces and Reductions of Orders)
Now we shall return to a number of technical points of the preceding sections. At the same time we want to continue the axiomatic consideration for the function spaces modelled in terms of scales of spaces and parameter-dependent reductions of orders. The approach of Section 2.1.4. is a hint that manything can be formalized. Let us summarize some properties of the scales 8 = (ES}ssRthat have been employed for defining the conified spaces. 8 runs over a system (E of scales of Hilbert spaces with (i) continuous embeddings Ed 4 E" for every s' 2 s, (ii) E" = E" is dense in every ES,
n
S€R
(iii) the EO scalar product (., .)o induces non-degenerate sesquilmear pairings (., .)a: ES X E-'+@ for all s, which admit the identification E-s = (E")', (iv) if 8 = {P}, +!= {fi)E C5 and a E r)P(ES,b - p ) for some p E R then S
Ilalh s - /i 5 c max Illall&s'- p Ilallf.f - A whenever s' 5 s 5 s" with a constant c = c(s', s"). 9
190
2. Operators on Manifolds with Conical Singularities
Let us set P(Z,
e)= n Z ( P , F-+) SER
and P ( Z ) = 5!?”(Z, Z). In view of (iv) the space 5?fi(Z, Assume that we are given a function
e)has a natural Frbchet structure.
b+(t)E C”(R, P ( Z ) ) for every p E R with the following properties , all T E R , further b-+(r) (i), b’(7): Es+,??-fi is an isomorphism for every ~ E R and = (b’ ( 7 ) )- , b O(r)-identity, < m for every p, Y E R,t~ R (ii), supllb’(r) b*(7)(b-(’+Y)(z))110,0 ( 4 0 Ilb-’(z)llo,o 5 41 + I7l)-”, Ilb’(7)llKo 5 c ( l + 171)’ for every p B 0, with a constant c = c@) > 0 (cf.2.1.4.(7)). A system of functions bP(7) E C”(R, !P(Z)), p E R,also satisfying the conditions (i), - (iii), is called equivalent to {b”(.t))CER,if g-+(r)b#(s), b-”(r) G(z) are uniformly (in 7 ) bounded in 9?(Eo) for all p E R,cf.2.1.4. Remark 17. Let s, y E R and define 2 4 YR+, Z) as the closure of C;(R+, F )with respect to the norm
z = Im z, M the Mellin transform. The family { ~ I ( T ) } . ~of reductions of orders for the scale Z is fixed once and for all. Of course, the spaces 2%Y(R+, Z) remain unchanged if we pass to an equivalent system of reductions of orders. In the abstract setting there is no reason to introduce a weight shift in the notations as it was done in 2.1.4.(2). In concrete cases it may be more adequate to integrate in (1) over I‘l,* - , yo = y + q with some frxed q , depending on Z. We set as usual W(R+, Z) = W0(R+,Z) . Note that br(t) may be interpreted as an abstract Mellin symbol which induces isomorphisms
,,
opL(b): WY(R+, Z)+!P-KY(R+, Z) for all ~ E R . The spaces 24Y(R+, Z) are natural generalizations of those in the Sections 1.1.2., 2.1.1., and they admit an analogous theory. Let us mention some points that are more or less exercises. First of all the 2 4 Y(R+, Z) form a scale of Hilbert spaces, and (u, u
I
) =~(27~)-l (b’(z)M~(z),b-*(r)Mu(Z))oIdZI r112
R0= Xo(R+,Z), extends to a non-degenerate sesquilinear form %4Y(R+,Z ) x E & - Y ( R +Z)+C ,
which admits the identification
E.-Y(R+, Z) = ( 2 4 Y(R+, Z))’. Moreover
where the supremum is taken over all
IJ E E 4 -Y(R+,
a ) ,u + 0.
191
2.1. Spaces with Conormal Asymptotics for the Cone
Let us also introduce Sobolev spaces based on the Fourier transform. Denote by HS(R,8 ) the closure of CZ(R, E") with respect to the norm
s E R , F the Fourier transform on R. H'(R, 8 ) is a Hilbert space with a natural scalar product. The spaces R ( R , 8 ) admit an analogous functional analytic discussion as the X*Y(R+, 8 ) . Similarly as in the scalar case we have for ( a Y u )(x) = e[1'2-y)xu0 ex l l b Y ~ ) M y u ( z ) lldzl= l~ r112-
Thus
IlWO (Fay) u(Oll> dF.
induces an isomorphism
3:2%Y(R+,8 )+H"(R,8 ) or equivalently X * y ( R + , 8 ) =[t-1'2+Yu(logt): u ( x ) E H ~ ( R , ~ ) } .
This shows in particular
2%Y(R+,8 ) = tyXs(R+, 8 ) , and often it suffices to consider the case y = 0. 1. Theorem. The operator .nC, of multiplication by o, E C;@?+), first defined on C;(R+,F), extends by continuity to a continuous operator
AZ,: WY(R+, 8 ) + X * y ( R + ,8 )
(6)
for all s, y E R.Moreover o, +At, induces a continuous embedding
CXR+)c,
n 9e(X4Y(R+,8 ) ) .
S€R
An analogous result is true of the multiplication operators &,, o, E CZ(R), in HS(R,8 ) .
Proof. Let us begin with the spaces Hs(R,8 ). The scheme of the proof is analogous to that of 1.1.2. Proposition 7 , but here we have an extra discussion for the families of order reducing operators. First we have with u^ = Fu ~ ( o , ( xu) ( x ) ) ( t l )= ~ { o , ( x ]' ) e * W O~ t(tl)} =
{e-lx%p(x) elxWO U F } dx =
B(5-
7) u^(& US,
p([ - q ) = eix(f-n)o,(x) dx. Here I@([- q)lsc , ( o , ) ( l + I f - ~ Now let u, v E CZ(R, F )and write
l ) for - ~every N E N with
cN(o,)+O as o,+O
in CZ(R).
Using (iii), we obtain
C A t , tl) Ic
c
( l + ~ t ~ ) - s ( l + l t l ~for Y s20, (1 + 1t1)'(1 + l ~ l ) - for ~ SI 0.
Let us discuss s 2 0. The case s = 0 is completely analogous. From (7) it follows I(.nz,u,
V)HO(R.'oI
Ic f l
IS(t- tl)l (1 f Itl)-'(l + lql)sllb"(t) c(Oll0 IlWtl) G(tl)ll~dtdtl
S cfl
K(E, tl) IlbYO U^(Bllidtdtl
fl K(E, t l ) Ilb-s(tl) v^(tl)ll~dfdr],
where
K ( t , a):= IS([- tl)l (1 + l t I ) - v + l t l l ) s IC C N ( Q , ) ( l + It-
tll)-N+r
( c denotes different constants). Thus we get as in the proof of 1.1.2. Proposition 7 I(-Iylqu? v ) h p W . Z ) ~ 5 c c N ( ( P )
((U((Hs((R.l)(Iu((H-'(R,n.
Using the analogue of (2) for the Hs(R, Z) spaces we obtain
Ib%ullHS(R.Zl IccN('?)
IIUIIB(R,'o.
This proves the W-version of our theorem. The X*?-version follows in an analogous manner, where we first use that the operator of multiplication by w is continuous, w being a cut-off function, and then apply the calculations as in 1.1.2. Proposition 7 for Q, with p(0) = 0. 0
2. Remark. Let w be a fixed cut-offfunction. Then [ w ] X*"(R+,Z) + [l - W ] 2%"(R+, Z)
= X s y ( R +Z) ,
in the sense of sums of Hilbert spaces. Moreover
[PI %*'(R+, 8)1 [PI
HS(R,Z)
in a canonical way. The method of proving the continuity (6) also applies to q(t) = w(t) to for shows that there are continuous embeddings 1:
e E R + . Then ( 5 )
[ w ] Xd*f(R+,Z) 4 [ w ] X*Y(R+,Z)
(9)
for s' 2 s, y' 2 y . Here we also have employed (iii), for the order reducing families.
Thus r~~~ -
Ilb'(4 &u(z
+ e)ll> d z <
(10)
for all e 2 0 , u E [ w ] X d - f ( R +Z). , Moreover, similarly as in the discussion around 1.1.1. Theorem 2, it follows
M p E $(Re z > 1/2 - y', E d ) .
2.1. Spaces with Conormal Asymptotics for the Cone
193
Let A = (6, 6') E 3 be finite, s E R.Denote by da(SA, Z) the subspace of all h E d(SA, Es) for which the supremum of
over all Q E SA8is finite, A , = (6 + e, 6'- e), for all E > 0 with 6 + e < O , 6'- e > O . If A = [a, 6'1 E gI1we denote by da(S,,Z) the subspace of all h E ds(S(,,, #), Z) for which lim Ilhllj1/2+a+c, lim I l h l l j 1 / 2 + ~ - a < e-
L++O
+o
Similarly we can introduce the spaces d J ( S AZ) , for A E gM, or, fi = 0,1, cf. Section 1.2.1. for , is a Hilbert the analogous definitions in the scalar case. For A = [6, 6'1 E gl1finite d S ( S AZ) space with the norm Illhll:1/2+a+ Ilhll:1/2+XI
1/2 '
Now let us impose further conditions that are fulfilled in our applications. The first one concerns the scales, namely (i), for s' > s the embeddings E f c, Es are compact. The second one is an analogue of 2.1.1. Proposition 4, (iv), for every x, x' E R , x < x ' , E > 0, there exist functions W ( z )E d(x - e < Re z < x'
+e,9(S))
for all p E R,such that {b'(e + ir)},,f,is an order reducing family for the scale Z satisfying (i), - (iii), for every fixed Q E [x, x'] and {b'(e + iz)},,ea is equivalent to (b'(@+ iz)JreRfor all
e, P E [x, %'I. 3.Theorem. Under the conditions (i),, (iv), the canonical embedding (9) is compact for s' > s, y'
> y and every cut-ofl function o.Moreover 11:
[PI H'(R, Z) 4 [PI HYR, Z)
is compact for s' > s and every
Q, E
(11)
C:(R).
4. Lemma. Let A, A' E gl1be finite, SAc int Sx . Then the restriction map d"'S,., is compact for s'
Z)+dysA,Z)
> s.
Proof. Let C, = as,. , C1= as, t = s + r with s' - s = 2r > 0. Set g(z) = b f ( z )h ( z ) , bf(z) being an order reducing holomorphic family as in (iv), for x = 1/2 + d2, % ' = 112 + S;, A' =[6,,6;],andlet hEdO"(Sd.,Z).ThenforzEClweget
194
2. Operators on Manifolds with Conical Singularities
as operator L*(Co,Eo)+L*(CI, Eo). For r > 0 the operators br(w)-’ are compact in EO and satisfy the estimates Ilb’(w)-’)lo,oI~ (+ lIwIY-‘,
W E CO,
and the same for b’(z)-l with z E C1. Then Kr(z,w ) is compact for every (z, w ) E C1x Co and IlUz,
w)IIo,o
I~ ( + 1 lwl)-r(l
+ Izl)-‘
with another constant c > 0. By analogous arguments as in the proof of 1.1.4. Proposition 6 we then obtain that (12) is compact for r > 0.
Proof of Theorem 3. The subspace w C ; ( R + , F ) is dense in [ ~ ] 2 ~ * f ( R Z),+ and , u E wC; X (R+, E m ) implies that the norms (10 are finite for all e E R. If u, E wCT (R+, E ”) is a seY’(R+,Z) then (10) converges for all e e 0. For e = 0 this is the quence converging in [w]gs’~
This implies the definition and for p > 0 it follows from the continuity of the operator Mu,,. convergence of the holomorphic functions Mfuv E So (Re z > 1/2 - y’, Ef) which have boun- of the quality of the Mellin image of [w]2<~ f(R+, Z). In particular dary values on TI,* [w]%?’*)’(R+, Z) can be identified with a closed subspace Soi(S,,, Z) , of Sof(&, Z), where S,,. = {1/2 - y’ IRe z I112 - y’ + e’}, e’ > 0. Similarly the Mellin image of [w] 2 4 y(R+,Z) corresponds to a closed subspace So:(S,,, Z) of Sos(S,, Z),where S, = {1/2 - y 5 Re z I1/2 - y + e } . Let us assume e > 0. Then the diagram
[w]XS’3Y’(R+, Z)-[w]WY(R+,
1MY,
Z)
.1 MY
obviously commutes, where the Mellin images are canonically identified with the holomorphic functions in the corresponding strips. My, and My are isomorphisms. Thus L = M i ’ 7 M f . Since 7 is compact by Lemma 4, we get the compactness of the embedding L . For proving the compactness of i 1 we remember that [p] HS’(R,Z) may be identified with a Z) for every cut-off function w with w(t)p(logt) closed subspace %:y’, of [w]XS’*f(R+, = p(log t ) and every y’ E pi. The identification is given by 2:f= (t-1’2+)’u(logt): UE[Q)]Hf(R,Z)).
Similarly we identify [p]Hs(R, Z) with 2;’for some y E R, y < y ’ . Then i 1 translates to A1q- ”1, where L : 2 : +2; is induced by (9). Since L is compact as we already know we get the compactness of the composition with J & - ~ for , this multiplication is continuous on the corresponding space. 0 5. Corollary. The embedding ~ S ~ . Y + E ( R + , Z ) ~ ~ ~ , Y - E ( R +c) ,Z)~~~Y(R+,
is compact for s‘ > s and E > 0. In fact, after the above discussion we have the compact embeddings
[w]23”y+rC*[w]2$*Y,[ l - w ] ~ ” ~ y - “ [ l - w W ] ~ ~ ~ . Then the compact embedding (11) together with [w]2 8 ’ . Y + a + [p] 2 S ‘ + [1 - 01 z s ” . Y - 8 = 2s‘. Y + for p E C;(R+), p = 1 on supp w
8
2s’. Y - 8
supp (1 - w ) , yields the assertion.
195
2.1. Spaces with Conorrnal Asymptotics for the Cone
2.1.7.
Appendix I1 (Polar Coordinates in yDO-s)
Our next goal is a discussion of operators in polar coordinates in RZ+' \ [O}. Let x E Sn= {I21= 1). Occasionally x will also denote local coordinates (xl, ... ,x , ) in a coordinate neighbourhood I/ of S". Let x : R, x S"+ RZ" \ (01
(1)
be defined by ( t , x ) + 2 , ?=[?I, x=?//l?l. Set ( 6 , ~(2) ) = p(12), (xntp) ( 1 , x ) = W @ t , x ) , 1 E R,, g, E Cm(R"+\ [O)), It is then trivial that for the pull-back x*
tp E
C"(R+ X Sn).
xAn*g,= x*6,g, for every 1 > 0. Clearly (2) induces corresponding relations on other function and distribution spaces. Remember that x* induces an isomorphism
n*: LZ(R"+I)?
t-"/ZLZ(R+xS").
(2)
(3)
Another simple observation is that ( 6 Y ' g , ) ( 2 ) = 1("+11'2(6Ag,)(2) is a family of unitary operatory on I ~ ~ ( R , + ~ ) . Let V be a subspace in Lz(Rn+I) which is invariant under the action of 61 for all 1 E R+, for instance, V = C;(R"+l \ [O)). Further assume that the operators in consideration map V into
Lz(R"+l). An operator A : V--+L2(R"+') (4) is called homogeneous of order p E C , if A6* = 1 ' bAA for all 1 E R+ . In an analogous manner we x S"). In view of (2) the space V, = x* Y is define the homogeneity of operators in t-"/2L2(R+ invariant under x,, 1 E R+, and the operator n*A: Vo-. t - " / Z L Z ( R + x S n ) , defined by ty+n*A(x*)-lW is homogeneous or order p when A has this property. Let us consider some examples. For simplicity let V = C;(R"+'\ {O}). The operator a/a2j is homogeneous of order 1, for a / a f j p ( l x ) = 1@/32,g,) (12). Moreover the operator of multiplication by Fk is of order -1. Thus
is homogeneous of order zero. It can easily be proved that
The operator A, of multiplication by f E Cm(Rnfl \ [O)) is homogeneous of order p iff f(12)= A-"f(?). Then f can be written as
f(2)= 121-r f
(*).
In particular we have x*2,
=
twj(x), j
=
1, ... , n
+ 1,
(5)
with wj E Cm(Sn).This corresponds to the classical formulas for polar coordinates, for instance, for n + 1 = 2,
196
2. Operators on Manifolds with Conical Singularities
n*Zl= t cos x , n*Z2= t sin x , x E S', where n* is usually dropped for convenience. Let us form the matrix
E being the unit n x n matrix. Then ' d ( n - I ) = t-lT-IE, with Id being the transposed of the differential. The standard behaviour of ryDO-s under diffeomorphisms says that (1) induces a bijection
n* : L'(R"+I \ {O))+L"(R, X Sn) and the same for classical lyDO-s (we hope that the letter L for spaces of ryDO-s will not be mixed up with the notation in (3)). Moreover we have the following 1. Proposition. Let Z(2, f ) be a complete symbol of a lyD0 A E V(Rn+l\ {O)). Then any complete symbol a ( t , x , t,0 of A = z*A, the pull-back under x (x being local coordinates on S")admits the asymptotic expansion
- 1a!z(~)(z,
a(r, x , t ,E)I(,.~)=
0,
t - l ~ - l nt-l=lpa(x,
with a polynomial Pa in [f (tr,f ) of degpe 5 lal/2and coefficients in C"(U), U being the coordi,f) (2,f). nate parch on Sn, ~ 7 ( ~ ) (=2(af$ Remember that t-lal Pa follows in the form
a a ( t ,X , t ,f ) = 0 ;e i M %r , D I f = * with h ( Z , F , t , f ) = (X(z^)-X(Z)-dX(?)(F-Z),(t,F)), (121, Z/l2l),
(6)
x=&-'.
In this case x(?)=
with = 3121.Since D k i ( ? ) is homogeneous in IF1 of order =1 21,k2(2), ... , x.+ i of the form 1 - ( y ( for i = 1, - ( y ( for i > l , we obtain that ( D ~ x ( ? ) , ( t , f ) ) l i c is I2l-IY'f (I?lz, 8. An analogous consideration on homogeneity for ( D i d x ( 5 ) (t,f)) yields the scheme of homogeneities (1 - Iyl, -I?[, ... , - 1 ~ 1 ) and hence the asserted special form of the am. 2. Definition. I,@"+' \ {O)) denotes the subspuce of all X E L;l(Rn+ \ { O } ) with theJo1lowing property. For every coordinate neighbourhood U of S"and a complete Jymbol Z(2,f ) of A on
~ ^ = { Z E R " + ~ \ { O: )Z / / ( ~ ( E U } the homogeneous components of order p - j in Z(t, x ,
F) -
(7) const of the arymptotic expansion
m
Z(,-,,(t,
x , F) (where (1, x ) stanak for 2) are functions
j=O
a;,-J,(t,X,F) E c - ( R + x u xW+1). Let L$(o-)be the set of all A E Lfl(oA)such that for every open Uoc Uwith ODc U we have A ~ Q=, A"lafor some A"E EtI(Rn+'\ (0)). Further Ef,(X^), X being a closed compact C" manifold, X^ = R+xX, denotes the subspace of all A E Lf,(X^) such that for every coordinate
197
2.1. Spaces with Conormal Asymptotics for the Cone
neighbourhood V of X and any diffeomorphism-XI : V+ U with a coordinate neighbourhood U c Sn the push-forward under x : R+X V+ U^ with x ( i , x ) = i,yl(x) leads to x * ( A ( ~ + ~ ~ ) E LS(l7). Note that there is a canonical embedding i: L$(R"+') c+L$(R"+'\{O}).
Clearly
e$(Rn+l \ (0)):= [A E L$(R"+\ [O}): A - ~ E L - - ( R ~ + ~ \ [ O ) forsome ) dfL$(R"+')} is a proper subspace. The asymptotic expansion of Proposition 1 yields for A"E L$(R"+ \ {0}) a ( i , x , t,8
- i-' C a ( , , - h ( i x, , i t , 8
(10)
j=o
with a(,-J,E C"(R+ x U X R2") homogeneous in (= (it,8,10 2 const, of order p - j. In fact the asymptotic expansion of Proposition 1 can be interpreted as
:1 cT!+C?~(?,
F) denotes the
unique homogeneous extension of
c?:;)-
J,(2,F) from large
5 * 0, then the homogenous terms in the latter asymptotic expansion are just i-#
[i!
,,-
iJz(*) j ( iT-' , '
IF1 to
0 Pa(x,01 = f-flbv(i,x, 0,
where v indicates the homogeneity, v 5 p - j - lal/2. Then if x(0 denotes an excision function in (we can define a(,,-k) as the (finite) sum over all Xb, where v = p - k. 3. Definition. n$(X^), X being a C" manifold, denoies ihe subspace of all A E L$(X^) for which a complete symbol p ( t , x , t,.f) over U^= R+ x U, U any coordinate neighbourhood of X, has an asymptotic expansion m
p ( i $x , T,8
-C
P ( p -J](t,x , i t ,
8
(11)
j=O
wiih homogeneous p(,,- Jl(i,x , 0 in ( = (it,8,I(\ t const, of order p X U X Rn+l).Further we set a " ( X ^ ) = L-"(X-).
-j ,
belonging to Cm(R+
In an analogous meaning we use the notation IZ$(U^) for any coordinate neighbourhood U of X, and IZ-"( U-)= L?( U-). Instead of (11) we can equivalently demand asymptotic expansions of the form m
where r,, - are classical amplitude functions of order p - j of analogous quality as the p(,,above (i.e. smoothness up to i = 0). In fact we can use the finite Taylor expansion at i = 0,
where the P $ ) - ~ , are still of order p - j. Then we define r,,-j by m
J,
198
2. Operators on Manifolds with Conical Singularities
Thus by rearranging the asymptotic sums we can achieve the maximal powers tj in front of the summands of order p j which is a unique representation modulo terms of Crder -a. After the above discussion of polar coordinates in wDO-s from X : W - 3 U- we get mappings t p X* : i L$(0-)-3 nZl(u-). (13)
-
They are not canonical, i.e. depend on excisions for any concrete operator. If x1: Y-, U is a diffeomorphism, x ( t , x) := t x l ( x ) , we get the corresponding push-forwards
x*: lL$(V-)+L$(iJ-), XI:
n w - ) n:l(LI^) +
which commute with (13) modulo operators in U--(U*). The latter property is_a simple corolare lary of the rule of substitution of variables in a wDO. Since the classes L$(U-), nEl(U^) also preserved under multiplications by functions in C,"(U) we can globalize (13) by using a fixed open covering of X by coordinate neighbourhoods and a fixed partition of unity. This yields a mapping of the corresponding classes over X-, and we denote it again by tfln*, namely t'nl: L:l(XA)-+ n$(x-). (14) The choice of the open covering of X and of the partitiw of unity is fixed, such that (14) is well-defined. Another choice gives the same modulo n:,-I . (11) suggests to apply the Mellin 1pD0 convention op, instead of op,. Set f i , o ( t , x , z , ~ ) = r ; - j ( t , x , 7where , ~ , r i - j ( ..., 7 , 8 : = r r - j ( ..., -7,R, I = 1 / 2 + i r . T h e n
cf. 1.2.3. Proposition 11 (op, refers to the combination t7 in I,,-,). We have seen that (15) follows from the formula under coordinate transformations, namely x : R X U-, R+x U, x = ko,xl), xo(s) = t = e', xl(x) = x. In 1.2.3., 2.1.3. we have only expressed the leading term of the difference. But we can, of course, carry out the complete formula for the remainder. It shows that OP,(~,,O)= oP,(b) with
and @ k associated with x as mentioned. It is easy to see that @k is a polynomial in t7 of degree kl2. The asymptotic sum for b can be carried out by excision in [ = (t7, [). Thus we have proved that op,(r,, - j ) = 0PuCfi.o) + OPv(bj.1) mod L-"(U^), where bj,l = r,,-j - b is a classical amplitude function, depending on (t7, 0, of order p - j - 1, which is smooth up to t = 0 in the mentioned sense. By the same procedure we get OP.(bj, 1) = OPMU,, 1) + OPy(bj. 2) mod L-"(u^),
oP,(bj.t) where ord & ( t , X, I , &
-1
= O p ~ c fk,), + OPv(bj, k + I)
mod L-m(U^)9
4 , k = ord bj,k = p - j - k, and we have smoothness up to f j , d t , X, z , n .
Then
k-0
op,(r,,-j) =opMCfi)modL-"(U-).
I=
0. Let
2.1. Spaces with Conormal Asymptotics for the Cone
199
E L:,(U^) with
w ( t ) A w ( t )E ML$(u^)for any
4. Definition. ME:(U^) is the subspace of all A cut-offfunction w .
Similarly as above by rearranging the asymptotic sum of a complete Mellin symbol h(r, x , T,8 of wAw (in the sense of 2.1.3. Definition 8) we can assume that m
h(t,x,7,5)-
1t’h,-,(r,x,t,8 J=o
with amplitude functions h,-, of order I./ - j which are smooth up to r = 0. By a globalization we can define ML:,(X^) for any C” manifold X. Now the above Mellin reformulation yields a (non-canonical) mapping
which can be globalized
8: n:,(x-) + ML:!(X-), where the Mellin symbols in (17) for closed compact X can be thought to be operator-valued. 8 is non-canonical, since it contains asymptotic expansions with excisions that are not the same for all operators. Nevertheless, 8 induces a linear map
n:l(x-)lam(xA) M L :,(X--)lML-“(X-) +
(18)
and the same over U. By removing negligible Mellin symbols we can always pass to holomorphic ones. Thus we get in particular the 5. Theorem. There is a non-canonical mapping
n$: L$(X^)+ M L $ ( X ^ ) ,
n&:= 8Pn*, where for every 26 L$(X^) m
n$x-
tj opM(h,-,)
(20)
j-0
is understood in the sense of L$(X*)), (19) can be chosen in such a way that for all j . h#-,(t,z) E Cm(R+,!lX$-J(X))
(-
6. Remark. Theorem 5 can easily be generalized to yrD0-s on X^ with symbols which have in the coordinates i = ( t , x) the more general osymptotics
IT(X
IF)
- CIil -P!a(x, h
with(7jasbefore(oforderIr)andpiEC, Repj+-m
asj+m.
In Section 2.2.2. we shall see that asymptotic sums like (20) can be carried out in the form m
w(c,t) t - ” + ’ o p d h , - , ) w(c,t) modL$,,,(~+xX), 1-0
w being a cut-off function, c,+
m sufficiently fast as j + m, cf. also 2.1.3. Theorem 17. The mapping (19) can be interpreted as a sort of operator conwntion that associates with a given homogeneous principal symbol an operator. It means that when we first pass to
200
2. Operators on Manifolds with Conical Singularities
A"E L$(X-)
we find modulo lower order terms another representation A where
tNA E ML;l(XA) is based on the Mellin transform. We then obtain the continuity
wt'Aw: %"(X^) !P-N(X^).
Thus a standard wDO (based on the Fourier transform) has a "conormal soul" with respect to the conical singularity t = 0. Below we shall return to this discussion under the aspect of more general singularities, namely edges, where we impose certain restrictions which correspond locally to the subspaces of (9). In the following for simplicity we mainly discuss X = S". Considerations on localizations, coordinate invariance and so on yield analogous constructions for general X. It is desgable to say more about the negligible operators that were dropped in the representation of A E LEl(Rn+ \ {0}) in polar coordinates and also to give another interpretation of the operator-valued Mellin symbols. To this end we want to sketch an idea from [Rll]. More details may be found there, in Section 4.2.3. Let us return once again to the above homogeneity. By conjugation this induces a notion of homogeneity of distributions in R"+ \ {O}. In fact let u E C;(R"+l\ {0}), f~ C"(R"+' \ {O]) and f ( A 3 = A'f ( x ) for all A E R+ . Then (f(AF),u(?))=
R"+ 1
f(AF)u(?)dF=
I f(Y3u(A-'y3A-(n+1)d~=A-n--'
( f , b U ) .
R"+l
A distribution f E 9'(Rn+ \ {O}) is called homogeneous of order I
(f,dl-l
u) =AZ+
"
+
1
(f,u)
E@
if (21)
for all u E C;(R"+ \ {O}), 1 E R+ . Denote by 9'(Rn+ \ {O})(z) the space of all those f.Moreover define D'(R"+l)(') as the space of all f ~ a ' ( R " + ' ) for which (21) holds for all u E Ct(Rn+l), and set D'(Rnfl)(')={ f 9'(R"+')('): ~ f l R n + l \ ( q E C"(R"+'\ {O})}. (22) We take D'(Rn+l)(z)in the topology induced by the canonical mappings into 9'(Rn+l)(') and Cm(Rn+\ {O}). Let for example
with constants a,. Then f~ D'(R"+')(-N-"-'). Examples of homogeneous distributions also follow from the Mellin transform. If u ( t , x ) E V(R+)@,9'(Sn) (Z'(0) being as usual the subspace of all distributions in 0 with ( I , x ) E Se(C,LD'(S")). Moreover compact support) then (Mf-'u) t-'(M,.+'u) for every fixed I
E @.
u(t,x)==
( I ,x ) E
9'(Rn+ \ (O})(-z)
Thus the formula 1
1
t - 2 ( M f - - . z u ) ( z , xdz )
(24)
can be interpreted as a decomposition of u into homogeneous distributions. Let f e C"(Rn+l\ {0}), u E C,"(R+x S n )and define the pull-back offunder a in the distributional sense by (f,a*u) = ( a t f , u ) , where the pairing on the right refers to dt dx. Then (nrf)( t , x ) = t"(a*f) ( t , x ) . This induces a pull-back a; : 9 ( R n + \ {0})--.,D'(R+ x Sn).
201
2.1. Spaces with Conormal Asymptotics for the Cone
Now let fy) E Cm(Rnfl\ [0}) be homogeneous of order z , i.e. f';)(q= 121zh(2/12() for some E Cm(Sn).Then
fl
m
(n2f, u)
=
I I t""fl(x)u(t,x) d t dx. 0
(25)
S"
We want to construct a continuous operator q ( z ) : Cm(Sn) +D'(R"+
(26)
by a regularization of the homogeneous extension of fl at the origin. This can be done on the level of polar coordinates. For every p, E Cr(R+)we have
-
I t'p,(t) d t = ( - l ) N
0
m
N
t ' + N p ( m ( t )dt
( r + k)-' k=l
0
:
for every N E N. This shows that the distribution t extends to a meromorphic family (in r) of distributions E 9'(R) with simple poles at r = - k for k = 1 , 2 , ... . Thus T ( r + l)-I t represents a holomorphic family of distributions E 9'(R). The same procedure applies to (25) where we may assume u to be of the form u = u ( t x ) , f =t x . Then r ( z + n + l)-If?) E So (C, 9'(Rn+I)), where fy) is understood as the meromorphic extension off?) which is first defined for Re L + n + 1 > 0. Now
:
q(~)f~:=r(z+n+l)-~f';)
belongs to D'(Rn+ for every z E C.In particular q(z)fl for z = - N - n - 1 equals (23) with coefficients a. that are linear forms on Cm(Sn),applied to fl . Note that (26) is an isomorphism for all z E C with z + n + 1 I$ -N. Moreover the described regularization can be generalized to homogeneous extensions of distributions E a'(S").The Fourier transform F in R"+ induces an isomorphism F: D'(Rn+l)(Z)+D'(RfI+I ) ( - Z - n - l )
(27)
and the same for 9 instead of D. A proof of (27) and further details on homogeneous distributions may be found in [H4], vol. 1. Now let us remind of the formula 1.2.4. (13) for a (complete) Mellin symbol of an operator A . It also applies in the operator-valued sense in the spirit of Section 2.1.4. In other words a Mellin symbol of A E MLJ'(R+;Z), C = [HS(S")},,R, follows by h ( t , z ) p = tzA((t')-zp,), R e z = 112, p, E Hs(Sn).Of course, such a formula only makes sense when A is properly supported in t . But it yields an ansatz for other sorts of operators that are not known to be Mellin operators. Let us disregard the concrete nature of A and consider a continuous operator XI: Dr(Rn+I)(-Z)+ D'(RfI+l)(-Z) (28)
for R e z = 1/2. Then a ( t independent) analogue of a Mellin symbol h ( z ) acting on Cm(Sn)3 p, follows by p,+t-zp,+Alt-zp, = t-lry+ry,
i.e. ry = h ( z ) p, E Cm(Sn), Al =?*XI. In particular let us choose Al as follows. Denote by r' the operator of restriction to S", r': D'(Rn+l)(z)-,Cm(Sn).Let a ( f ) E Cm(Rn+'\(0)) be homogeneous of order p, 1/2 + p 6 -N. Then we form first D'(Rn+l)(')3f= q ( - z ) p , + r ' F q ( - z ) p , =
@E C"(S").
(29)
202
2. Operators on Manifolds with Conical Singularities
Further let ul = ulsn and apply the chain of mappings @+ul@+q(z-n-l
+ p ) ( u l @ ) ~ ~ F ~ ~ ~ ~ l ~ ( z - n - l + p ) ( u l @(30) ).
Then (29), (30) yield together a continuous operator (28), and it is justified to call the mapping h , ( z ) : p + p , y~ being r' of the last distribution in (30), the Mellin symbol of_Al. The construction shows that h,(_z) extends to a holomorphic operator function in z . Now A = IF[-'' x T( - z + n,+ 1) T(z + p ) A , extends to a lyDO in Rni \ {0}with the homogeneous principal symbol a(E) and vanishing lower order symbols. In [Rll] it was proved that h ( z ) = T(-z + n + 1) T(z + p ) h l ( z )E ' 9 t ~ s ( S n In ) * . particular we see that the r-factors cause a rather special pattern of ~ 0 1 3 . Thus any classical lyDO A as in Thecrem 8 can be treated by evaluating step by step the Mellin symbols of the coefficients u', - k([) in
where the sum over j is the Taylor expansion for 12)-+ 0, a', - k(F) homogeneous in F+ 0 of order p - k for all j . In this way any classical yrDO 2 as in Theorem 5 for 1/2 + p -N can be represented by a Mellin sum
with certain h j ( z )E 5Dl:,(Sn)', more precisely o(n*$ w equals (31) modulo XSn) and an operator with kernel in [w]%;''(R+ xS")* @,[w]%>@(R+XS")' for a certain Q E R.This information is much more precise than that of Theorem 5 not only on the smoothing operators that have to be neglected but also with respect to the natural meromorphic structure of the resulting Mellin symbols. The order p for all hj is caused from the Taylor expansion of the original symbol for t+ 0; then we have no r dependence of Mellin symbols but a flat remainder. Note that Mellii representations of standard lyDO-s were earlier obtained in the work of PLAMENEVSKU, cf. [P7], [PSI, but by other methods.
2.2.
The Mellin Expansions for the Cone
2.2.1.
Operators with Finite Mellin Expansions
Our next objective is to extend the algebra of 2.1.5. Proposition 10 by finite sums of Mellin operators. The constructions are analogous to those in Section 1.3.1. We only need the case B = ( - k , k ) , k E N . Of course, we also could talk about arbitrary weight intervals. 1.Definition. Denote by W(X");,p E R, 8 = (- k, k ) , k E N \ {O}, the set of all operators A=S+S'+F+G, where F E %$(Xa)8,G E WG(Xa);and k-1
wd op$(hj) w ,
S= j=o
(1)
203
2.2. Mellin Expansions for the Cone
with arbirrary hj E "tll,(X),f i E 'iIRmll,,( X ) , R j E %YJ(X x X ) , R E %"t(X x X ) , - n / 2 2 fi 2 -j - d2, - d 2 2 y ; 2 - I - n d for j , I = 0, . ..,k - 1 , w an arbitrary cut-off function.
Moreover we set
oG(A) ( z ) = h ; ( z ) :=f ; ( n + 1 - z ) ,
..., k - 1, and call (6) ((7)) the Mellin symbols of A belonging to the conorma1 orders - j (-I) with respect to t = 0 ( t = a). Set
j , I = 0,
'(A), &(A), . . ., 02' ' ( A ) ) (8) U,(A) = ( & ( A ) , ..., ( k is kept fixed in this notation). By the recovering procedure of the Mellin symbols we shall see that this is a correct definition. Write for a ; ( A ) = O ,
c-ordA=-(I+1) c'-ordA
=
-(I
+ 1)
Osjsl,
for a&A) = 0, 0 5 j 5 1.
It is clear that by definition
A EW(XA);
c-cord A, c'-ord A 2 - k ,
A E %$+ c(X");* c-ord A
= c'-ord A = - k .
In view of the results of 2.1.5. we could fix w in Definition 1. The class W ( X " ) ; then is independent of w .
2. Proposition. Every A E %p(X"); induces continuous operators A : !7t;(XA)ff+!7t;-"(X")ff for every s E R, R E %-"z(X) with some Q E %-dz(X) depending on A and P.
The proof is an immediate consequence of 2.1.5. Definitions 1 and 7 and 2.1.2. Remember that in virtue of %.(X"); E r-dzML:l(iiT+ X X ) rdZ
(9)
and 2.1.3. Theorem 4,every A E W ( X " ) ; induces in particular continuous operators
A : !7ts(X^)-.Xs-"(XA) for all s E R . The inclusion (9) suggests that when A E W ( X * ) ; is written in the form A 14
= op,(a)
Schulzc, Operators engl
+ G,
(10)
204
2. Operators on Manifolds with Conical Singularities
a(t, z) E C",(a+, L;,(X; RJ), G E t-"12ML-"(R+XX)t"*, the Mellin symbols of A , say for t = 0, are simply (l/j!)a J / a P a ( t , z) I f = o with the additional property to be extendable in z to elements in W:,(X)*. But this is not true, even for j = 0, since Wm(X"); belongs to ML--(R+ X X) , and in particular the operators with Mellin symbols in W;sm(X)' are of this type. 3. Remark. (10) is compact for A
E
W-'fl^); and c-ord A 5 - 1, c'-ord A 5 - 1.
4.Remark. Let Afbe the operators of multiplication byJ f E Cz(a+ X X ) . Then A every k.
If f ( f ,x ) -
f E
no(XA);for
m
t J q j ( x ) is the Taylor expansion o f f af t = 0 then a d ( A f )= A , and j=O
a3Af) vanishes for all j . Similarly we can define the operators of multiplication by functions in I(")C;(K+XX) (I("))-] and also obtain elements in 9l0(XA);.
Next we will discuss the symbolic structure of the operators in WP(X"); in more detail. Analogous constructions then apply to the corresponding class with continuous asymptotics below. In virtue of (9) and the definitions in 2.1.3. we have for every A E W(X7; a symbol a i ( A ) ( t , x , z ,~ E S ( " ) ( T ~ X ^ \ O ) . (11) If A is given in the form (1) then cf. 2.1.5. (8), where k-1
af(S'>(t,x, z,
F) =
c
w * ( t - ' ) t - ' o : ( f ; ) ( x , -z,
6).
I=O
Occasionally, we call ai(A) the totalZy characteristic interior symbol of A . Set
x W;,(X)',
k-1
a&(X);
=
Wt:(x)* x
j =1
9 = ( - k , k ) , and 2a&(X);
= a&(X);
x a&(x);
(13)
(cf. the notation 2.1.2. (14)). Then aM(A)belongs to the space (13).
5. Definition.
qP(X");, 9 = (- k, k ) ,
{ { h j } , {/I;}, p } E 26&(x); x
wheref;(z) = h ; ( n + 1 - z).
k E N \ { O } , is the subspace of all tuples
S q T ; X^ \ 0 ) for which
2.2. Mellin Expansions for the Cone
--
at) define linear surjective operators
6. Theorem. ,a , a!, (a,
a! :
nqx-); nqxl);
(UM,US):
%'(x*);'6'(xA);
a , :
205
26$(X)i, S("( T ; x^ \ O),
with the kernels
ker a , ker a;
=
yx");, = %;;',(x");. =w p-
ker(a,, a:)
The proof of Theorem 6 follows by analogous arguments as in 1.3.1. Theorem 4. Similarly we get
-
7. Corollary. The triple a = (a", a$, US) induces a surjective linear operator u : nqx");
A
=
(-1, l ) , and
eqx");,
keru= ( A E ! W - ' ( X " ) ; : c-ordA=c'-ordA= -1). 8.Theorem. A
E W(X");,B
a&m)
E %"(X*)i implies AB E W + " ( X " ) iand
( z )= p+q=1
a&! ( A ) ( z + q ) a 2 ( B ) (z),
(16)
I = O , ..., k - 1 , aS"(AB) ( 6 x, 7, 5) = aS(A) ( 2 , x, 7, 5) a;(B)(1, x, 7, 5 ) . (17) This follows again by the same scheme as 1.3.1. Theorem 6. As an analogue of 1.3.1. Proposition 7 we obtain 9. Proposition. A E W ( X * ) ; implies A*
E
nqx");,A ' . *-
Z'"'A (Z(n))-l
E
w"(x");,
where
a i ' ( A * ) ( z )= a i ' ( A ) * ( z- I ) ,
a&4*)(z)
=
a i ! ( A ) * ( z+ I )
(18) I = 0 , ..., k - 1, and h * ( z ) = h ( * ) ( n + 1 - T ) ((*) indicates the adjoint, taken pointwise), further
aS(A*)(t, x, 7, 5) = a W ) ( t ,x, 7, 51,
ai'(A')(z) l = O , ..., k - 1,
=
u i ' ( ~( )n + 1 - z),
aS(A')(t,x, 7, 0 = ag(A)(t-', x, 14'
(19)
a,&i')
-7,
0.
(z) = a & ~ () n
+ 1 - z),
206
2. Operators on Manifolds with Conical Singularities
10. Corollary. nm(X"); is a *-algebra, filtered by the orders, and %+; o ( X * ) ; , %,(X*); are two-sided ideals.
Remember that the notation algebra is always understood in the sense that the operations have to be admissible for two operators, for example the sum is allowed only when the difference of the orders is an integer (since we are talking about classical operators). 11. DeAnition. A n operator A E % @ ( X A ) i ,8 = (- k, k ) , k 2 1 , is called elliptic if (i) @;(A )(t, x, t,5) f 0 for all t, x, t,F, 0 5 t 5 , It,FI f 0, (ii) the families of operators
a L ( A ) ( z ) ,&(A ) ( z ) : H"(X)-+H"@(X) (20) are buective for all z E r,,+ and fixed s E R . Clearly the second condition is satisfied for all s E R if it holds for s = so, since a L ( A )( z ), & (A ) ( z ) are families of classical elliptic vDO-s on X . If A is elliptic then A* and A' also are elliptic (A*, A' being as in Proposition 9). 12.Remark. .(fA E W(X");satisfies the condition (i) of Definition 11 then there is a countable set with ej-+ fm as j - t k m such that the operator families (20) are boective for all
{ejIjEzc R zE
r,, and all e E R\
In fact, & A ) ( z ) , &(A) ( z ) are meromorphic operator functions with values in L : , ( X ) . By removing operator-valued symbols in %t;:(X)* we get even holomorphic L:,(X)-valued functions h ( z ) , h ' ( z ) , cf. 2.1.2. Theorem 5. Let us consider now, for instance, h ( z ) . The homogeneous principal symbol of h ( z ) (in the non-parameterdependent sense) is independent of z. As a bijective operator (20) it is necessarily elliptic for all Re z = ( n + 1)/2, since the Fredholm property between Sobolev spaces over X is equivalent to the ellipticity (cf., e.g., [R8]). Thus h ( z ) is a Fredholm operator for all z E @. Now we can apply 2.2.5. Theorem 1to the holomorphic family h ( z ) of Fredholm operators and obtain a countable set of zj where h ( z ) is not bijective. Since h ( z ) is in addition parameter-dependent elliptic, it is bijective for I Im zI sufficiently large, uniformly in every finite strip c1 5 Re z 5 c2. Thus in those strips there are only finitely many zj. 13. D e f i t i o n . An operator B E % - @ ( X * ) iis called a parametrix of A E %@(X");i f
A B - I , BA - I E %G(X");. 14. Theorem. Let A E %@(X^);be interpreted as an operator
A : Xs(X") +S"- (X") @
(2 1)
for fixed s E W.Then the following conditions are equivalent (i) (21) is a Fredholm operator, (ii) A is elliptic. If A is elliptic there is a parametrix B E % - @ ( X - ) i .Moreover Au = f E XS,(X*)e,R E 5?-"'z(X), u E X m ( X A ) imply , u E X;'@(X*)efor all s E W with some P E 5?-"'2(X) depending on A and R. In particular ker A c sle;o(X")e
2.2. Mellin Expansions for the Cone
207
for some PoE .5%e-"z(X),ker A is independent of s, as well as
ind A = dim ker A - dim ker A * .
(22)
15. Remark. A s a corollary of Theorem 14 we obtain the (weaker) assertion that when A is elliptic Au= f e W ( X " ) ,U E W " ( X ' ) = U E W + ' ( X * ) for all s E R. 16. Remark. Any operaior A E W ( X " ) ; can be interpreted as an operator in Xo(X")which is un9 ( A ) = W ( X * ) . Then A equals the clobounded for p > 0 , with dense domain 9 ( A ) c XO(X^), sure of A I G(w.,in the sense of operaiors in the Hilbert space. The same is true of A*. If A is elliptic = L z ( M ) , M being the manifold with conical singularities then (22) is the "L2-index"of A (Xo(X") belonging to the stretched object X *, cf. 2.0. Fig. 8 , 9).
Proof of Theorem 14. The proof follows by the same scheme as that of 1.3.1. Theorem 11. Let us restrict ourselves here to the elements that require additional considerations. First we have to show that ho(z) = o L ( A ) - l ( z )belongs to '3Xi$o(X)'. The inverse of &A) ( 2 ) is certainly holomorphic and L ,'(X)-valued for large 1Im 1 , and it is again a parameter-dependent lyDO along parallels to the imaginary axis. But this inverse extends to a meromorphic operator function in C with the pattern of poles and Laurent coefficients as required in 2.1.2. (7). This is a consequence of 2.2.5. Theorem 1. Now the lower order Mellin symbols of B are also in our class because of 2.1.2. Proposition 6. The same can be done for the Mellin symbols at m. Now we get again a right parametrix of l? with {flat + Green} remainder which yields a right parametrix B after 2.1.5. Proposition 12. This is just a parametrix in 92-"(X"); and we get the Fredholm property by 2.1.5. Proposition 3, and in the standard manner the elliptic regularity. It remains to establish the necessity of the ellipticity for A to be a Fredholm operator. Since both the Fredholm property and the ellipticity remain untouched under compositions of elliptic operators we may reduce the situation to order zero by composing with an elliptic operator of order - p . The latter one always exists by an obvious analogue of 1.3.1. Lemma 12. Let us explicitly formulate the corresponding version of 1.3.1. Lemma 13. Every AE%O(X"); can be written in the form A = t-R/2AotR/z,A o = op,(o) + op,(a') + F + G with a ( z ) , a'(z) E %::(X)* and t-"l2(F + G) tM2E %$+ o(X"):, cf. also 2.1.5. Theorems 13 and 14. 17. Lemma. For every zo E TI,, the sequences h, f ; E Ct(R+), 1 E R + , from 1.3.1. Lemma 13 have the property that for every u ( x ) E L z ( X ) and ul(t, x) =h(t)u ( x ) , U X t , x) =f;(t) 4 x 1
I P o 4 - hao(z0) vll LZ(X)-, 0, u o u ; - f;ab(zo) vlIL2(x^)
+
aslt-m.
0
208
2. Operators on Manifolds with Conical Singularities
Proof. Analogous considerations as in 1.3.1. Lemma 13 allow to replace A. either by opM(ao)or op,(ab). Let us consider, for instance, op,(ao). Then, for zo = 112 + izo, IloPhf(ao) u, - ao(z0) ulllLz(x7 = (2nI-l Ilao(z) (Mud (z, x) - ao(z0) (Mu,) ( 5 x)I12Lz(r,,zx x ) m
=
(27t)-I
j II[ao(1/2 + iz) - a0(1/2 + izo)]v ( x ) A1'2(Mf) (112 + iA(z - ~ o ) ) l l i z ( dz ~ -m
m
=
(2n)-'
j II[ao(1/2 + izo- iA-lz) - ao(1/2 + izo)]u ( x ) (Mf)(1/2 + iz)ll&x) dz -m
-0
as A + m . O
Now let us come to the proof of the bijectivity of ao(z), a;(.). The Fredholm property of A implies the Fredholm property of A. in L 2 ( X * ) . Then there is a Bo E 9 ( L 2 ( X " ) )with BOAo= 1 + K, K compact, and hence
Il4ILz(x? 5 c IIAO4IL*(X.) + IIKullLz(x-) (23) for all u E L2 (X " ) with a constant c > 0. Now insert uA from Lemma 17. Since u, weakly tends to zero in L z ( X " ) for A+m, we obtain Ku,+O in L z ( X " ) for A+ In view of (23) for every E > 0 there is a A. E N such that
m.
5 c IIAouAllL2(X7 + E for all A 2 Lo. Using Lemma 17 we get IIuAlILz(x7 = llvllL2(x,
IIAouAllL2(X, 5 llfiao(z0)
VIIL2(X7+
&
= Ilao(z0)
UIIL2(x)
+E
when A 2 A1 with some Al . This implies for A 2 lo+ A1
5 c Ilao(z0) VIIL2(x) + (1 + c) 6 for all u E L 2 ( X ) .Since E > 0 is arbitrary, we get the bijectivity of ao(zo)as operator on L 2 ( X ) . The consideration for ah is analogous. Let us set bo(zo)= ao(zo)-', v = bdzd u . Then Ilbo(zo) U I I ~ Z ( 5 ~ c llull L ~ ( m and hence IIbo(zo)IIB(Lz(x)) 5c = IIBollecLqx,,.Thus the inverse of ao(zo)is uniformly bounded for lImzol+ m . From that we conclude that the parameter-dependent homogeneous principal symz = Im z , does not vanish for IT, [I = 1 . bol of ao(zo) as a function of (z, 0, The compatibility conditions in Definition 5 show that ot(Ao)(t, x, z, 0 f 0 for all It,El f 0 and 0 s t s E , 5 t 5 for E > 0 sufficiently small. Next we use that %"(X"); S t - " / 2 M L ~ , ( ~ + xtd2 X )c L ; @ + x X ) . In view of 2.1.3. Remark 3 and 2.1.3. (24) it suffices to show that the homogeneous principal symbol of the corresponding wDO in L:,(R+ X X ) does not vanish for t f 0. But this follows by an obvious analogue of the last part of the proof of 1.3.1. Theorem 11, where we use again 1.2.4. Lemma 19. 0 ll4l'2(x)
Now we want to make some further remarks to the asymptotics of solutions, say for t +0. The construction of the parametrix shows that the main contribution to the asymptotics comes from the poles of the inverse of the principal Mellin symbol, except of a contribution from the Green part in the original operator. The "lower order terms" are constituted by the poles of the lower order Mellin symbols, composed with translations of the inverse of the principal Mellin symbol. The coeflcienfs in the
209
2.2. Mellin Expansions for the Cone
asymtotics follow from the corresponding Laurent expansions and are of the quality as expressed in 2.1.1. Definitions 10 and 12, u(f, x )
- c 2 c k ( x ) t-pJlogkf m
m
j = O
k=O
as
t+O,
c k ( x ) E Cm(X).
(24)
The position and multiplicity of poles of the inverse principal Mellin symbol is a priori not known, except of a general description of their distribution in the complex plane, which follows from Section 2.2.5. in abstract functional analytic terms. The problem of determining these data concretely is analogous to that of evaluating the eigenvalues and multiplicities of an operator, where here the eigenvalues are nonlinearly involved. In other words, for every concrete operator the eigenvalues remain an extra problem. In the literature this induced a lot of papers only devoted to this question. The coefficients ( j k ( X ) are not invariant under a diffeomorphism of E,. We always keep the t direction fixed close to t = 0 (a).The coefficients of solutions of Au = f depend in general on the global behaviour on R, XXof the right hand side f . This is intuitively clear also from the calculation of the asymptotics of oop,(h) X (1 - w ) f close to t = 0, for some h E 9 t ; p ( X ) * , cf. 2.1.5. Theorem 14. Let us discuss a simple class of examples which belong to the motivations for the general constructions.
18.Definition. Denote by DiffP(X^) the space of all differential operators on X^ of order p with C l coefficients (over R, X U,U any coordinate neighbourhood on X ) for which
with arbitrary ak, bk E Cm(K+,Difffl- k(X)), 0 5 k 5 p , and any cut-off function w ( t ) , w‘(t) = w(t-1).
Consider the Taylor expansion of ak at t
c
=0
m
ak(t, X , 0,)
tJakj(x,0,)
j=O Ir
a k j z k . Similarly define the coefficients b k j ( x , 0,)belonging to
and set h j ( z ) = k=O
P
bk(t, X , 0,)and h j ( z ) =
b k j ( n+ 1 - z ) ~ .
k=O
It is then obvious that =
..N - 1
1wtjopM(hj)w + c w ’ t - j o p , ( h j )
N-1
1.
A
j=O
~
~
w‘
+ FN
j=O
with some F N ~ % $ ( X A ) eSNN, = ( - N , N ) for every NEN\{O}, F N ~ D i f f ” ( X ” ) .
210
2. Operators on Manifolds with Conical Singularities
Thus we may set h j ( z ) = u$(A) ( z ) , h i ( z ) = a&A) ( z ) , and we have hj, h ; E % $ ( X ) , EN,
(2 5) Diffp(X*)c W(X")*. The Mellin symbols hi, hi are polynomials in z with Diffp(X)-valued coefficients. Assume that, for instance, ho(z)= ao(x, 0,)- zp, hb(z) = ab(x, 0,)- zp, ao, ab E Diff"(X).
Definition ll(i) implies the ellipticity of the differential operators ao, a&on X . Moreover (ii) is satisfied if zp is no eigenvalue of a. and a&for all z E f,,+ 1)/2. Thus (ii) in Definition 11 is a condition on the "non-linear"eigenvalues of elliptic operators on X , where here the non-linearity in z is polynomial. As mentioned in Section 2.0. we get operators in w Diff@(R+ X S")w if we insert polar coordinates x"= (t, x) into a differential operator A"(2,Df)of order in Wn+l_ Clearly not every element in w Diffp(R+x S")w can be interpreted as such an A close to x" = 0. In general it is an interesting question to look at the subclass of wW(R+x S"); w which corresponds to standard VDO-s A" in Rn+ close to x" = 0, and to what extent the ellipticity of A" corresponds to the ellipticity of the associated wAw close to t = 0 in the sense of Definition 11. We shall return to this point in more detail in Section 2.2.3. Note that the results of PLAMENEVSKIJ [P7], [PSI can be interpreted in principle in the sense that every classical wDO in It"+ close to x" = 0 induces an operator in wW(R+X Sn);w for every 9 = (- k , k ) , k E N. More complete information, also for boundary value problems, may be found in [Rll]. Next we formulate the analogous calculus for Mellin symbols and spaces with continuous conormal usymptoticr. We shall follow the lines of the second part of Section 1.3.1. 19. Definition. Let h E %l;(X), V E Y, j E N. Denote by dec,(h,j), the set of allfinite sequences y = {(hv,eV)},, I ,.,,,N , N arbitrary, with
-
h, E %l;v(X),
VvE YQ?,- n/2 2 ev2 - j - n/2 for all v , V
V
Set N
where op9 on the right is understood in the usual sense. of Y .
{el,...,eN}are
the weights
20. Proposition. For every j E N \ {O}, h E Sml;(X), Y E Y, there exist decomposition data y E dec,(h, j ) with N s 2.
211
2.2. Mellin Exuansions for the Cone
The proof is analogous as for 1.3.1. Proposition 16, where here we use 2.1.2. Theorem 5. The proof also shows that for every sequence of weights {el,. . . , e N } , -n/2 2 - e v 2 - j - n/2, j 2 1, with ei ej for at least one pair ( i , j ) there is a y E dec,(h, j ) with these weights, for every h E m;(X).
*
E R, 9 = (- k , k ) , k E N \ {O}, is the set of aN operators (1) where F E 91$(XA),, G E 9IG(X^),, and S, S’ are given by the formulas (2), (3) with arbitrary hj,f ; E 9t:s(X), yj E dec.(hj,j), y ; E dec,( f ’, I ) f o r j , 1 = 0 , . .. ,k - 1, w an arbitrary cut-offfunction. The definition for 9 = (0,O)was given by (4).
21. Definition. !P(X^),,
We set
The standard notations such as a d ( A ) , a&A), a;(A), a ; ( A ) , for W(X^),and will be used, again.
... also make sense
22. Proposition. Every A E W@(X^),induces continuous operators A : %7e”,(X^)a+ %$-@(X”), for every s E R , B E 7r - d 2 with some C = C(B,A ) E 7r - d 2 .
The proof is analogous as for 1.3.1. Proposition 18. The corresponding technical tools were given in 2.1.1., 2.1.2., 2.1.5.
23. Proposition. Let y , E dec,(h,j) be decomposition data for h E m$(X)and w i , Gi be arbitrary cut-offfunctions, i = 1,2. Then w l t j o p ~ ( h ) w-z ~ l r j o p ~ ( h ) G ~ ~ W ~ + G ( ~ ^ ) . The proof follows by the same scheme as for 1.3.1. Proposition 19. Define
K,,l=C \ Kj.2 =
n+l [7 (4+ &)< Re
[n +F l -
-
z
<
(i+ i)
5 Re z 2
24. Remark. Denote by %”(X-)e,,, v
=
1,2, the subspace of all A
Sg ( u i / ( A ) ) E Kj,v,
sg(u&A))S K ; . ” : = { n + 1 - z : z E KP,”}, j , p = O ,..., k - 1 . Then
%qx^)e = %’(x^)B,l + %qX^)e,J in the sense of vector spaces.
E
%@(X-),for which
212
2. Operators on Manifolds with Conical Singularities
25. Definition. Set k- 1
@W), = EmkO(X)x
x Emm:,(X)
j=l
and let
@'(X^), c 26&(X)B x S("(T:X^\O) be the subspace of all tuples { { h j } ,{ h ; ) , p } which satisfL the compatibility conditions (14).
26. Theorem. u,, a!, (aM, u!) define linear surjective operators UM :
%' (x-1 B
a; :
%'(.r)B+
(UM, us):
%'(x")B'@'(x^)>,
+
2@'&(X)B, S(")(TgX^\o),
with the kernels
n:+ G(X^)ff,
ker U,
=
ker a!
= %#
-
yx-)&
ker ( U M , a!) = 3 : ; !.JX^),. Moreover the triple u = (a", a;., a!) induces a linear surjective operator u:
%'(X^)
@ +@#
(x-)",
(28)
A = ( - l , l ) , with keru={AE%@-1(X")8: c-cordA = c ' - o r d A = -1). The proof is a straightforward generalization of that of 1.3.1. Theorem 22. 27. Theorem. nm(X*), is a *-algebrafiltered by the orders, and %+;
C(x^)B,
%G(x^)B
are two-sided ideals. On symbolic level we have the formulas (15), (16), (17), (18), (19). Moreover I(")W ' @^)a( I("))- 1 = %'(X^), with the same transformations of the symbols as in Proposition 9.
The proof is an obvious modification of that for 1.3.1. Theorem 23. 28. Corollary. A E %"(XA)B,B E %"(X-),implies A B - BA E %'+"-l(X")a, and the same for the dotted subclass.
Note that for A , B E %m(XA)Bwe have c-ord AB = max [(c-ordA ) + (c-ord B), - k } and the same for c'-ord.
213
2.2. Mellin Expansions for the Cone
29. Remark. Let &f be the operator of multiplication by f, f E C i ( R + X X ) . Then .NfA - A M f € Rp-'(X^)8is not compact as o p e r a t o r ~ s ( X ^ ) ~ ~ s - ' (unless X ^ ) f ( t , x ) vanishes for t = 0. f (0, x) = 0 implies c-ord Af 5 - 1.
30. Definition. An operator A E %P(X^),is called elliptic ifthe conditions (i), (ii) of Definition ll are satisfied. B E %-P(X^),is called a parametrix of A if
AB - I , BA - I E %,(X"),. 31. Theorem. For A E %P(X^),we have (i)* (ii) of Theorem 14. I f A is elliptic there is a parametrix B E %-fl(X-),. Moreover A u = f E X7eS,(X^),, C E YVnl2, u E E"(X^),imply u E Xi+P(X-), for all s E R with some D E Y n l 2 depending on A and C. In particular ker A c X&(X^), for some Do E Y d 2 ker,A is independent of s, and we have the formula (22).
It is clear that the remarks after Theorem 14 about elliptic operators in the dotted class have straightforward extensions to the general class %"(X^),.So we may drop the details. Let us add some further properties that hold both for %P(X^),and the dotted subclass.
32. Proposition. Let A E W ( X ^ ) , be elliptic and ind A = 0. Then there exists a finitedimensional P E %,(x^),such that A
+ P : Xs(X^)+Xs-fl(X^)
is invertible for all s E R.An analogous result holds for A E %P(X^);. Proof, Let M = ker A , then M c X;(X-),is finite-dimensional and independent of s. Similarly & = kerA* c XG(X^), is independent of s, and d i m M = dim&. Choose projections Po : P(X^)+M , P1: P(X^)+ which are Green operators by 2.1.5. Proposition 5. They extend (or restrict) to projections Po: XS(X^)+M, PI:
Xs-!J(x-)+&.
Now choose an arbitrary linear operator Q: M+&. of the desired sort. 0
Then P:= PIQPois obviously
Remember that when (21) is invertible for fixed s it is invertible for all s E R. Then A is necessarily elliptic. 33. Proposition. Let A E %P(X^), and assume that (21) is invertible. Then A-' E %-fl(X^),. An analogous result holds for A E %P(X^)i. Proof, Let B E %-'(X^)a be a parametrix of A. Then ind B = 0, and by Proposiis invertition 32 we find a Green operator P such that B + P : XS-"(X^)+!Xs(X^) ble. Thus (B + P ) A = I + G : Xs(X^) -+ Xs(X^)
is invertible, G being a Green operator. Applying 2.1.5. Proposition4 we get ( I + G ) - l = I GI with another Green operator. Thus A-' = ( I + GI) (B + P ) E %-P(X^),since %"(X^), is closed under compositions.
+
214
2. Operators on Manifolds with Conical Singularities
34. Proposition. Let A E %P(X^), be elliptic. Then there exist numbers N , M E N and an invertible operator
with K E %;(X^), 63 C", T EC N 63 %Is(XA)B.Then s4-l is of analogous structure with a parametrix B E %-P(XA)Bin the lefr upper corner. An analogous result holdr for the dotted subclass.
The proof will be given below in more abstract terms in Section 2.2.5. Now we define natural locally convex topologies in the spaces %"(X")B,%"(X-)i. Let us restrict ourselves to the case with continuous asymptotics. The discrete one is completely analogous. In 3.1.4. we have defined a topology in %$+ o(X^), that will be used here. Moreover it is clear that the symbol space @P(X^), has a canonical locally convex topology. From Theorem 26 we have an exact sequence
O-+%$+
us, o(x^)B-,%"(x-)B-@"(x-)B'o. (OM,
In an analogous manner we obtain exact sequences with the subscript v of Remark 24, v = 1,2. These sequences then split and we get vector space isomorphisms %"(X-)B," 2 %;;L(X-)fJ
@ @"(X-)ff,U,
where BP(XA)e,,:=(aM,a;) % f l ( X - ) , , . This gives us topologies in %"(XA),," and via (27) also in % p ( X ^ ) B . 2.2.2.
Operators with Asymptotic Mellin Expansions
This section extends the results of 2.2.1. to the injhite "weight interval"9 = (-m, m). The spaces 2.2.1.(5) and (26) will be equipped with the corresponding projective limit topologies. Remember that we have continuous embeddings
W ( X A ) ) ,4 . %"(X^), for 8' 2 9, and the same with dots. Set
@L(X)= @ $ ( X ) p . , q , W X - )= @"(X--)(--,q and analogously with dots. As in the one-dimensional case we mainly discuss the case with continuous asymptotics. The discrete one is completely analogous. Assertions for the projective limits may be formulated as for the weight subscripts 9 = ( - k , k ) , where k = m . 1. Proposition. 2.2.1. Propositions2, 22 hold including k = m. The proof is obvious, cf. 1.3.2. Proposition 1. In this section we use the notation g M ( A )for A E in the sense of 2.2.1.(8) with k = m. 2. Theorem. 2.2.1. Theorems6, 26 hold including k = m. The proof follows in the analogous manner as for 1.3.2. Theorem 2. In particular we have an analogue of 1.3.2. Lemma 3, namely
215
2.2. Mellin Expansions for the Cone
3. Lemma. Let hi E EmFj(X), V, E V,j E N, VoE T Mbe2an , arbitrary sequence and yj = { ( h j , " , e j , J :v = 1,2} be an arbitrary sequence of decomposition data in dec,(hj,j) such that -ej,y--+w,
j+ej,"+m
as j + W ,
v=1,2.
Then there exists a sequence of constants cj such that the sum m
S=
w ( c j t )d op;(hj) w ( c j t ) j=O
converges in W ( X ^ ) as well as the sum of the adjoints, w being afixed cut-offfunction.
The tools are here 2.1.5. Theorem 19 and 2.1.5. Remark 18. The surjectivity of the symbol map (a,+,,a;) in Theorem 2 is similar as in 1.3.2. Theorem 2, here by using 2.1.5. Theorem 13 and 2.1.3. Proposition 7 or 2.1.4. Proposition 7, restricted to L;,(X;R)-valued amplitude functions. In particular we have obtained that every A E %P(X^) is of the form
c
A = j = O w ( c j t ) t j o p ; ( h j ) w ( c j t ) +I [ n ) { ~ w ( c , t ) t / o p ~ ~ ; ) w ( c , t ) } ( Z ( n+) F) -+l G with unique h j ,f E EmtS(X),and decomposition data yj E dec,(hj, j ) , y ; E dec,(fi, 1) where the weights satisfy the growth conditions of Lemma3, cj+m sufficiently fast, and F + G E 92$+ c(X^). It is clear that
2
j=N+1
w(cjz) tjop2(hj)w ( c j t )E %$+c(x^)B,
with aN = (-- N , N ) , cf. 2.1.5.(11), and the same for the sum over 12 N + 1 of Mellin operators at m. Then 2.2.1. Proposition 23 shows that any change of the decomposition data and of the constants cj under which the sums remain convergent only affects the infinite sums by operators in %$+ c(X^). Clearly we may also change the cut-off functions by 2.2.1. Proposition 23. 4. Theorem. nm(X^)=
uW(X^)is a *-algebra,filtered by the orders, and 9;+G(X^), P
%,(X^) are two-sided ideals in nm(X^).Moreover the operators admit the conjugation with I @ )in W(X^).For the symbols we have the same rules as in 2.2.1. Theorem 8, Proposition 9. An analogous result holdr for the dotted subclass. The proof is an obvious modification of those of the corresponding results in Section 2.2.1.
5 . Definition. An operator A E W ( X ^ ) is called elliptic if it is elliptic in the sense of the larger classes W(X^)>, forjinite 9, cf. 2.2.1. Dejinitionsll, 30. B E !l-r(X^) is called a parametrix of A E W(X^)if AB - I ,
BA - I E %,(X^).
For A , B in the dotted subclasses we demand AB - Z, BA - I
E %c(X^)*.
216-
2 . Operators on Manifolds with Conical Singularities
6. Theorem. For A E W(X-) the conditions (i), (ii) of 2.2.1. Theorem 14 are equivalent. If A is elliptic there is a parametrix B E %-F(X-).Moreover Au = f E sle:(X-), C E T d u2E ,X"(X^), implies u E !Xi: "X*) for some D E T - d depending z on C and A. In particular we have ker A c X&(X ) for some Do f T-d2,and ind A is independent of s E W. If A E %P(X*)* is elliptic there is a parametrix B E %-p(X*)*. Moreover Au = f E X7eJ,(XA)>,R E %-d2(X), u E !K"(X-), implies u E S p P ( X " ) for some PE%-"~(X) depending on R and A. Moreover kerA c %zo(X*) for some Po E %-"2(X).
The proof is again evident after the corresponding 1.3.2. Theorem6 and 2.2.1. Theorems 14, 31. Remember once again that the classes of smoothing operators in %P(X-) and W(X*)' are %,(X^) and RG(XA)*, respectively, n,(X-)
=
@
%;(x*)c3rx;(x-)
B.CE T d 2
E,(X^)*..
x;(x-)@zx;(x-)
& l P. Q E % - d 2 ( X )
and we have analogoues of 2.1.5. Propositions2, 3 , 4 for 8 = (-m,m). 'Smoothing' in this interpretation, of course, includes the conormal order. The Green operators are interpreted as ones of conormal order -a.Smoothing in the "usual" sense, i.e., with respect to the Sobolev space index are also the operators in Wm(X-) and Wm(XA)*, where the Mellin symbols are just in %iSm(X)and %;>(X)',
respectively.
I. Remark. 2.2.1. Propositions32, 22, 34 have corresponding analogues for % = (-m, m). Let us finally note that %"(Xl>, %"(X?* contain many interesting subalgebras, cf. also the remarks after 1.3.1. Proposition 14. One of them may be defined by the condition
aZ(A), a$(A)
E roZ:;
'j(X)
for all j E N and a fixed r E IN.This is an immediate consequence of the composition rules for Mellin symbols in 2.2.1. Theorem 8 and 2.1.2. Proposition 6. These subalgebras are closed under parametrix construction and forming inverses in case of invertibility in the totally characteristic Sobolev spaces. If %:(X^) (iR:(X*)*) denotes the corresponding subspace of %@(X^)(%P(XA)*),then it is closed in the induced topology.
2.2.3.
Operators on Manifolds with Conical Singularities
There is an immediate generalization of the calculus of 2.2.1., 2.2.2. to manifolds with conical singularities. We shall formulate here the corresponding theory including the aspect of spaces with weights and of distributional sections in vector bundles. Let M be a compact "manifold" with conical singularities and C the stretched manifold. Remember that C is a compact C" manifold with boundary aC X, X
217
2.2. Mellin Expansions for the Cone
being the base of the cone. We fix a tubular neighbourhood V of aC with int V g (0,l) x X c,X-. Outside V we can perform the standard calculus of yDO-s on C" manifolds. Remember that M is not necessarily a C manifold, cf. Section 2.0. We may allow, for instance, that X has several connection components. If Y is a C" manifold (with or without boundary) we denote by Vect(Y) the set of all complex C" vector bundles over Y. Isomorphic bundles will be identified. Restrictions of bundles to subspaces will often be denoted by the same letters. For E E Vect(C) we have the standard Sobolev spaces H:,,,(int C , E), H;,,(int C , E) of distributional sections in E. Moreover El x* E' for some E' E Vect(X), n: V 1[0,1) X X+ X. The canonical projection X - + X will also be denoted by n. Then the pull-back induces a bijection II*:Vect(X) +Vect(X^). For E E Vect(X^), E = x* E', we denote by Xs(X^,E), s E R,the closure of C;(X^, E) with respect to the norm
,
IJ
= II~llw(x-,~)
IIb&(Imz)(Mu)(z,.)II&,~)
rl.+ 1)12
ldzl
Yf2 ,
where bi.(z), Z E R, is a parameter-dependent classical yDO on X acting between the distributional sections of E', which induces isomorphisms b>(z): Hs(X, E')+L2(X, E')
(1)
for all z E R. The existence of the order-reducingfamily (1) follows for analogous reasons as for the case of trivial bundles, cf. the discussions in the beginning of Section 2.1.1. The L2 spaces of sections in E' refer to a Riemannian metric on X and a Hermitean metric in E' that are fixed once and for all. We can define the spaces Xs.Y(XA,E) for s,y E R,by the obvious analogue of 2.1.1. Definition 1, or alternatively Xs9Y(X", E) = tYXs(X^,E).
(2)
We adopt here all conventions of 2.1.1. such as XS*O= Xsand so on. For standard notations and results on yDO-s on manifolds in spaces of distributional sections of vector bundles we refer to [A8], [Pl], [R8]. Let S(X, E'), E' E Vect(X), be the set of all sequences 2.1.1.(18) with nc P E 5% and finite-dimensional subspaces Lj c Cm(X,E'), further W ( X , E') 3 P nc P E W .Denote by W(X, E') the subset of asymptotic types belonging to t = 0. For simplifying notations the union of the sets S ( X , E'),SY(X, E') and P ( X , E') over E' E Vect(X) will be denoted again by S ( X ) , W(X) and P ( X ) , respectively. The spaces X;'(X^),, !?I?$~(X.')~of Section 2.1.1. have obvious generalizations to the case of distributional sections in vector bundles E E Vect(X^). In other words we get the spaces
-
X$'(X^, E)d, XS;y(X^,E)d
(3)
for all s, y E R, A E Yap,a,B = 0,1, P E Sy(")(X,E'), V E YY("), y(n) = y - n/2, E' = Elx being the unique E' E Vect(X) with E = n*E'. It is clear that the constructions of Section 2.1.1. can be carried out in an analogous form also for the spaces (3). The corresponding results will tacitly be used here.
218
2. Operators on Manifolds with Conical Singularities
Moreover we can form the spaces %%Y ( V, E ) , 2:Y ( V, E ) , ,Xi '( V, E),, defined as the pull-backs of %eJ '(X E), ... under int V+ (0,l) X X 4 X ", V a tubular neighbourhood of aC in C. Then for E E Vect(C) we obtain the spaces A,
%eJY(C,E ) = [ w ] %eJ'(V, E )
+ [l - w ] H;,,(int
C, E),
%S,'(C, E ) = [ w ] %;y(V, E ) d + [ l - w ] Hi0,(int C, E),
(4)
(5)
P E 9fin)(X,E'), A E gp, B = 0,1, and the same with the subscripts B E W n )w. is a cut-off function on C, i.e. in Cm(C),supp w c V, w = 1 close to aC. Set
%;e:sy(c, E ) ; = g q y c , E),,,
(6)
%;,Y(C, E),,
(7)
=
lim %eS-,'(C, E),,
-t
with the inductive limits over all P E Wn)(X,E') and B E By("), respectively. For E', F' E Vect(X) we denote by E' W F' the tensor product of n:E' and x:F', ni being the canonical projection of X X X to the ith factor, i = 1,2. Let %(X X X, F'W ( E l * ) ) be the set of all sequences 2.1.2.(1) with nC,,Pe a, Nj being a finitedimensional subspace of Cm(Xx X, F' IXI ( E l ) * ) . Let %(X x X) be the union of all % ( X X X, F' W (El)*) over all E', F' E Vect(X). Moreover we defme RY(X X X, . ..), W ( X X X) by the condition nc N PE 3''. Let L"X; E', F') be the space of yDO-s on X of order p acting between the distributional sections of E', F' E Vect(X), and L;,(X; E', F') the set of classical operators. Every A E L'(X; E', F') induces continuous operators A : Hs(X, E l ) + Hs-p(X, F')
for all s E R . We can easily extend the definitions of 2.1.2. concerning the Mellin symbols with discrete and continuous asymptotics and obtain the classes Em;(X; E', F'), Em;(X; E', F'), p E R,
R E %(X
X ; F'IXI (El)*), V E Y. Moreover we set Em;(X; E', F'), Em{,(X; E', F'). = X
g R
Em;,(X; E F') I,
=d lim Em;(X;
E', F') ,
V
where the inductive limits are taken over all R E %(X X X; F' H ( E l ) * ) and V E V, respectively. We also have the subspaces Emtl(X; E', F')*, Emt8Y(X;E', F') as obvious analogues of the corresponding spaces in 2.1.2., and the same with subscripts R E Ry(")(XX X, F' H ( E l ) * ) and V E Yy("),respectively. Finally, we pass to the unions over all E', F' E Vect(X) of the mentioned Mellin symbol spaces and then drop again E', F in the brackets. The results of Section 2.1.2. have a straightforward extension to the present case.
219
2.2. Mellin Expansions for the Cone
Incidentally we use the notations sg'(h)
=
R , sg(h) = V
(8)
for h E B$and E B:, respectively. Now we obtain continuous operators (9) wop$")(a)w: .%$Y(C,E)d+.%b-lr*Y(C, for every a E B:sY(X; E', E"')' and P E P n ) ( X ,E') with some resulting Q E W n ) ( X ,El'), for arbitrary A E &, = 0 , l ; s, y E R . The interpretation of (9) is that everything is carried out first over (0,l) X X , where the cut-off is supported and then transformed to the corresponding tubular neighbourhood of aC in C, where the action then makes sense for the spaces defined by (5). For the continuous asymptotic types we get wop$")(a) w : .%iy(C, E)d+.%L-ky(C, E")), (10)
) another D E LBfi"), and all when a EB!&Y(XX X ; E', EI') for every B E W nwith A , s, y as mentioned. Analogously to Section 2.1.2. we also have the space B K Y(X; E', F') of operatorvalued amplitude functions, here in the variant of L:,(X; E', F')-valued symbols, i.e. Bk "(Xi E', F') = L:I(X; E', F'; fi,, - y ( n ) ) , on the right being the space of parameter-dependent classical yDO-s on X with the parameter T E f,,,- f i n ) , and the action refers to the distributional sections of E', F. There is a straightforward extension of the classes ML:,(R+X X ) to ML$(R+ X X ; E, F ) , E, F E Vect(E+X X ) , where the Mellin amplitude functions belong to C;((R+)', B","/'(X; E', F')) . An alternative would be a simple modification of 2.1.3. Definition 13, i.e. a definition in local terms by a partition of unity. A E ML:,(K+ X X ; E, F ) induces continuous operators A : .%&'"'(X",E)-+.%'-'."*(X^,
F)
and G E ML;-(R+ X X ; E, F) is characterized by
G : .%', "'(X", E ) +.%", "/'(X", F), G*: %~"'(X", F)+.%m~"2(X", E ) for all s E R. Every A E MLgl(R+X X ; E, F ) has a homogeneous yDO symbol
crG(A): z*E+ n'F, n: T*X " \ O+ X " being the canonical projection, where
(11)
a f ( A )(t, x, T, 6 = af;(A)(t, x, t - ' ~ 0 , has a smooth extension up to t = 0 and t = m . In contrast to the notations of symbols in 2.1.2.-2.1.4. we prefer here to use c; instead of u;. Denote by SP)(T*X^\O; E, F ) (12) the space of all bundle morphisms (11) which admit the mentioned smooth extensions after replacing T by t - ' ~ . 15
Schulzc, Operators engl.
220
2. Ouerators on Manifolds with Conical Singularities
I. Definition. %$(C;E, &, 9 = (- k, OJ, k E N u {a), p E R, E, RE Vect(C) denotes the set of all A E L!!,(int C; E, E ) for which wAw induces an element in t-"12ML~,(E,XX; E, E) tdZ which isflat of order k at t = 0 (cJ:2.1.5. Definitions 6, 7), w being a cut-off function supported by a tubular neighbourhood of aC in C. Clearly E, 8 on R, X X are to be interpreted as the pull-backs of the restrictions E', ? !, to aC 2 X under the projection E,XX+ X. Denote by
S(d)(T*C\O; E, E) the space of all bundle morphisms 6: n*E+ n*E, n: T*C\ O+ C,which are positively homogeneous of order p, and which induce close to aC an element in S t ) ( T*x^ \ 0; E, E") . We say that a is invertible in Sp) if it is first invertible over int C,but 6 with z replaced by t-lz is invertible up to t = 0.
2.Definition. 91G(C;E,E")a,a = ( - k , O ) , k E N u {m}, E,EeVect(C), denotes the set of all A E.%(X~(C, E ) , Z o ( C ,E)) which induce continuous operators A : So(C, E ) + X i ( C , E)a, A * : ZO(C,E ) + Z g ( C ,
for certain B, D E V - n / zdepending , on A. The subclass 9IG(C;E, E"); is defined by the analogous mapping properties, where B and D are replaced by P E P d 2 ( X ,E"') and Q E Fd2(X, E'), respectively. The operators in 9'tG(...)a are called Green operators (with discrete asymptotics in case of the dotted class). 2.1.5. Propositions 2, 3, 4, 5 have straightforward generalizations to the Green operators of Definition 2. We adopt analogous notations as in Section 2.1.5. such as %$+G(c;
E, E)a = %$(C;E, E)a + %G(C;E, E)a
and so on, It is clear that the operators in 9't;tG(...)a can be composed within the class and G ( C ;E, E")a we have the symbol map that the adjoints belong to the class. On a$.+ a;. Then
o+%$iL(c;E, @)a+W$,G(C;
E, E ) a A S i p ) ( T * C \ O ; E, E)a+O
is exakt and splits, Sip)(... ) a being the subspace of S"(. ..) for which the morphisms are flat of order k at aC after replacing z by t-%. The same follows for the classes with dots. This can be used to introduce in %$, an adequate locally convex topology, by the same scheme as in Section 2.1.5. It is obvious that the Mellin operators (9), (10) have also symbols a:(wop$')(a) w ) E Si!')(T*C\O;
E, E).
A simple modification of-2.2.1. Definitions 1, 21, gives us the operator spaces W ( X " ;E, &)a, W ( X ^ ;E, E ) ; for every 9 = ( - 4 k ) , k E N u {m} (for k = as the intersection over the spaces with finite k), E, E E Vect(X").
22 1
2.2. Mellin Expansions for the Cone
3.Defintion. W(C; E,&, 9 = ( - k , O ] , k E I N u { m } , E,E€Vect(C), denotes the set of all A E L:,(int C; E, E) for which WAOinduces an element in %#(X";E, Z)o(k. k), w being a cut-offfunction, supported by a tubular neighbourhood of aC in C. A n analogous definition applies for W(C; E, ,!?);. For k = the subscript 9 will also be dropped. Set
k-1
6$,(X; E', E"')e = Smt:(X;
E',
E"') x )( Sm:,(X;
E',
E"'),
j = 1
9 = (- k, 01, and denote by
@(C; E, E)e c @$,(x; E',
E"') x sg")(T*C\
0; E, E)
the subspace of all sequences {{ h j },p v } for which the compatibility conditions 2.2.1.(14) to t = 0 are satisfied, with p ( t , x, r, F) = pv(t, x, t-lr, t) close to t = 0. Analogous notations are used with dots.
4.Theorem. uM = { u i } , at, (aM,u;) define linear surjective operators :
W(C; E, E)e+@$,(X; E',
a; :
%"(C;E, E)e+Sg")(T*C\O;
F)e,
X=X,
E, E),
(uM,u;): W(C; E, E)e+@"(C; E, E)e, 9 = (- k, 01, k E u { a } ,k 2 1, with the kernels %$+ c ( C ; E, ,!?)a, %"-l(C; E, E)e and %$; L(C; E, E)e, respectively. Moreover the pair u = (a",, u:) induces a linear sur-
jective operator 6 : W(C;
A
= (-l,O],
E, E ) e + 6 " ( C ; E, E)",
(13)
with
keru={AE%2'-1(C;E,E),: c-ordA= -1). Analogous assertions hold with doe.
5. Theorem. A
El, E)e, B E %"(C; E, E1)e implies AB E W"+"(C; E, E)e and
E %"C;
G%W( z ) =
(14)
1
G'(A)
(2
- 4)u i q ( B )( z ) ,
p+q=l
l = O , ..., k - 1,
u;+'(AB) = uc(A)u',(B). Moreover A * E Wr(C; 6 E J e with the analogues of 2.2.1.(18), (19). The same assertions hold with doe.
The Theorems 4, 5 follows by obvious modifications of the proofs of the corresponding theorems in 2.2.1. and 2.2.2.
6.Definition. A E W(C; E, E)s, 9 = (- k, 01, k (i) $(A): n*E+n* F is invertible in S f ) , (ii) the family of operators
E IN u {a}, k2
u$(A)(z): Hs(X, E')+Hs-@(X, E"') r,,+ 1),2 and fixed s E R .
is bijective for all z E 1s
1 , is culled elliptic, if
(15)
222
2. Operators on Manifolds with Conical Singularities
7. Definition. An operator B E %-”(C;6 E ) , is called a parametrix of A E
%’(C; E, E))Bif
AB - I E %G(C;E, E)8, BA - Z E %G(C;E, E ) 8 . For A in the dotted class we require B to be in the dotted class, and then A B - I, BA - I are Green operators with discrete asymptotics. 8.Theorem. Let A E % ” ( C ; E , & , P E R , 8 = ( - k , O ] , k E N u { m } , k Z 1 , E , E E Vect(C) , be regarded as operator A : !Xs(C, E)+!Xs-’(C, E) (16) for fixed s E R . Then the following conditions are equivalent (i) (16) is a Fredholm operator, (ii) A is elliptic. I f A is elliptic there is a parametrix B E %-’(C; E)8. Moreover we have the elliptic regularitiy in the Sobolev spaces with asymptotics as an im-mediate analogue of that in 2.2.1. Theorem 31 and 2.2.2. Theorem 6. A E %”(C;E, E ) ; implies the same for the classes with discrete asymptotics. The proof of Theorem 8 is obvious after the proofs of the corresponding theorems in the Sections 2.2.1., 2.2.2. 9.Remark. Let A l , A2 E W(C; E, E)@be elliptic and &Ai)
=&
4 2 ) , U:(AJ = $(A2).
Then ind A l = ind A 2 .
In fact, from (13), (14) we know that A l - A 2 E %”-‘(C; E, E)8-and c-ordA = - 1 . Thus A l - A 2 defines a compact operator W ( C , E)+!Xs-”(C, E ) . We will not repeat here all the propositions that have been formulated in 2.2.1., 2.2.2. as corollaries around the ellipticity. They can easily be added by the reader in the present case. Let us also remember that the method of introducing locally convex topologies in the spaces %”(. ..)a and %’(. ..); applies, again. The topologies for k = 00 then follow by the corresponding projective limits over those for finite k . Sometimes it is useful to know the existence of a reduction of orders within the class. 10. Theorem. For every V E R , E€Vect(C) there exists an elliptic operator R’ E %’(C; E, E)’ which induces isomorphisms
R’: !Xs(C,E)+!XS-”(C, E ) ,
for all s~ R,A
= (6,0),0 5
(17)
-6 5 m ,
ProoJ First we easily find for every v E R an elliptic operator in %’(C; E, E)’ . In fact, it suffices to produce the pair ( h ( z ) ,p,) E W(C; E, E ) ; , A = (- 1,0], for which the components satisfy the conditions of Definition 6, and then to use the surjectivity of (13) for 8 = (-m,O].
223
2.2. Mellin Expansions for the Cone
It is not hard to construct first pv. This trivial step is left to the reader. From that which is a parameter-dependent we can pass to an operator function on r(n+l),2 yDO h,(iz) E L’,,(X;E‘, E‘; r,,, 1)/2) for which the parameter-dependent homogeneous principal symbol just coincides with p , when pw is linked to p by replacing 7 by t-’7 and taking the trace at t = 0. Now a standard procedure as in 2.1.4. Proposition 16 yields an hl E mt;(X;E‘, El)* which is associated again with p this way on every parallel to the imaginary axis. As a holomorphic family of Fredholm operaE’) which is on these parallels also parameter-dependent tors Hs(X, E‘) +Hs-”(X, elliptic we get isomorphisms for large I Im zI uniformly in every strip c1 < Re z < c2 for every fixed cl, c2 E R , cf. Section 2.2.5. Thus h , ( z ) is bijective on every parallel r, except of a discrete subset of exceptional points { @ j } j G z , m as ljl+ m. Thus there is a 5 for which h ( z ) := h,(z + e”, is bijective on r,,+ This is just the desired principal Mellin symbol. Denote by 7“ an operator in W(C; E, E)’ (for every fixed choice of Y ) with aL(T”)= h ( z ) , cr;(T”)= p v . Set R ” = T’~2(T”/2)*. Then R“ is formally selfadjoint, elliptic, and thus has index zero. By Theorem 8 we have ker E , ker (@)* c !Xrs(C, E)’ . Thus there is a finite-dimensional G E !RG(C;E, E)’ such that R” := R”. + G defines isomorphisms for all s E R. Since both the subspaces with discrete and continuous asymptotics are preserved by the action of R”, we also get the bijections (18), (19). Clearly ( R ” ) - l belongs again to W-“(C; E, E ) as a consequence of the general fact, that the inverse can be carried out within the class. We shall return below once again to the aspect of order reducing operators within the class, in the context of paramesr-dependent cone operators. Note that the c o r ~ p o sition by R” from the right (or R’ from the left, when indicates the bundle E ) induces bijections W(C; E, E“); 4W+”(C; E, E);
-
for all p E R and the same without dots, 9 = (- k, 01, k E IN u { a } . Let us now turn to a simple generalization of the theory of cone operators. Consider for simplicity 8 = (-a), 01. Let gy E Cm(C)be a function, which is strictly positive on int C and gY= t Y for 0 < t < E , where t is the global normal coordinate in a tubular neighbourhood V = (0,l) x aC of aC, 0 < E < 1. Then we can pass to the weighted cone operator spaces gf%z”(C;E, E) g-Y
for arbitrary y, operators
Y”E
ss;x F(C; E, E)
(20) R , and the same with dots. Every A in (20) induces continuous =:
A : %*“C,E)+%’-”>F(C,E) for every s E W. All the above results immediately extend to the spaces with weights. A minor modification concerns the composition, where only operators of the form A = gFAog-Yl, gYIBogY are composed, and for the formal adjoint we have
*:
SJ!z”;%F(C; E, E)-+%z”.-%-Y(C; E ) .
We can also distinguish between different weights for the different connection components of aC. This is another trivial generalization of the cone theory.
224
2. Operators on Manifolds with Conical Singularities
The operators in (20) can be called to be of conormal order y - p. In particular the Mellin symbols would be
uL-f-'(A), j E N . An operator A = gvAog-Y in (20) is called elliptic if $(Ao) is invertible in Sg" and U L - ~ ( A( )z ) bijective in the sense of (15) for all z E r,,, 1 ) , 2 - Y . The further details on the class (20) may easily be added by the reader. The shape of a differential operator J(I,Di) in R:" of order p in polar coordinates (2, x ) = I shows that p = y - p is a natural weight in the image. Let us set Rk Y(C;E,
E") = g " - P W ( C ; E, E") g - y ,
(21)
y E R , and the same with dots. The subclasses of Green operators will be denoted by 82 Y . As usual y will also be omitted for y = 0.
11.Definition. DiffP(C; E, E") is the space of-all differential operators of order p on C, acting between the distributional sections of E, E E Vect(C) , which are in a tubular neighbourhood V 1 [0, 1) X X of aC = X of the form
akE Cm([O,l), DiffP-k(X; E',
E"'))
By definition we then have DiffP(C; E, E") g-P G R"P(C; E, E")'. It is clear that even DiffP(C; E, E") g-C S ' RkY(C; E, E"))' for every y E R , since the conjugation by tial operators untouched, for
a
(-rz-y)kt-y=t-(-t&)
k
gy
keeps the totally characteristic differen-
.
In view of the flatness properties it is justified to set for all y E IR
R$ "(C;E, E") = R$(C; E, E) = %:(C; E, E). We have otj-Cop,(h) w E JV Y(C; E, E"))' for all h E %$(X;E',
E"')
and all j E IN, y E R.
It is interesting to know to what extent in general an operator A operators in RP,'(. . .)' for other A E R .
E Rk y ( . ..)'
induces
12.Theorem. L e t A E RkY(C; E, E")* and assume that u&(A)( z ) E 9tm(r~-'-"'2 (X)*for a given A E R . Then there exist weights q, j r IR~ an$ a finite-dimensional operator G E gf%G(C; E, E)' g-? such that A - G E Rk '(C; E, E)' .
2.2. Mellin Expansions for the Cone
225
ProoJ: Let us assume for simplicity that the bundles E, k are trivial and of fibre dimension 1. Every A E SSp* '(C)* can be written as m
A
=
1 w ( c j t ) tY-p+jopz(hj)t - y w ( c j t ) + F + G j=O
(22)
with a sequence of hj E EmtSYj(X)*,- n/2 2 yj 2 - n/2 - j , F E R$(C), G E R$ '(C) . Let us first modify G by a finite-dimensional Green operator G(') such that G - G(') E 8% '(C) . By definition G induces continuous operators G : 2%y ( C) -+ 2 ;Y ~ - #( C) , G*: 2%'-"(C)
22 - y ( C ) with discrete asymptotic types P E W - p - n / z ( X ) , Q E . F Y - " I ~ ( X )The . formal adjoint refers to the X0(C)-scalar product. For every 1 there exist minimal asymptotic types P ( 1 ) E P, Q ( 1 ) E Q with finite @(A), ncQ(A),for which Po = P\ P ( 1 ) E P " - " - ~ ~ ( X Qo ) , = Q\ Q ( 1 ) E . F " - ~ ( ( X ) . Then we can construct direct decompositions
+
= + NP('), 2 2 - Y ( c=) 2". p0 (c)+ NQ(A) 2;sy-qc)
2;;'-p(c)
-1
with finite-dimensional subspaces NP(')and NQU), respectively. The projections , induce a projection along N p ( l )NQ(')
2,; - (C) B mJ 2 ;- Y( C) +2;;- "( C) B xJ 2 2 -'( C) (23) with a finite-dimensional kernel, which can be identified with G,) . If we identify G with an element on the left of (23) then G - G(') is just as required. In the sum (22) we have by construction y - y j + w , y - p + j + yj+ ~0 as j + Q),cf. Section 2.2.2. Thus there is an N = N ( 1 ) such that m
Therefore it remains to change the first N Mellin terms by Green operators in order to obtain also elements in Rk'(C). This can easily be done by the method of 2.1.5. Theorem 13. Write wJ = w ( c J f ) , and set = PhJ.Then for u E Cr(int C), supp u in the tubular neighbourhood of X , we obtain a finite-dimensional operator GJ with wJfJ- p t Y + y/ op,( T ( Y + Y d h ; ) t - ( Y + Ydw, - 0, tJ- p t ' + 21 op,( T('+ A,)%)
6
(24) for any choice of AJ with E For j > 0 and fixed 1 we always find A, with - n/2 2 1, 2 - n12 - j of this kind, whereas for j = 0 this is a condition to ho, namely that ho is holomorphic near r(n+l),z-y+A, in other words ho EE~;~Y-"-"~(X)*. The commutation relation (24) applies to all u which are flat enough at t = 0 , i.e. of the form u = t d u , , u1 E Cr(C), and 6 - ( y + yJ) 2 0, 6 - (A 1,) 2 0, j = 0, ..,,N ( 1 ). For analogous reasons there is a 8, to be chosen . the standard commutalarge enough, such that t8GJtdmaps into X m s n / z ( C )Now tion arguments for Mellin actions (cf.2.1.5. Theorem 13) show that ffGJtahas a ker1,)..., . , N ( 1 ) . Thus we can set G = G(')+ nel in 2 ~ ~ f , . z ( C ) ' B ' n 2 ~ ~ fj z=( C X t-(*+'J)wJU =
GJ
6 "t;t+'~(X)*.
+
226
2. Operators on Manifolds with Conical Singularities
13.Remark. Let A E
n WY(C; E, E)* such that a;(g'A)
satisfies (i) of Definition 6 . Then there
YER
is a countable set { Q ~ } , ,of~ reals with lejl+m
a L ( A ) ( z ) : H"X, E')+H"-r(X, is buective on
r,,,
-
as l j l + m
such that
E3
for all y E R \ { ej)jE and all s E R.
If A is as in Remark 13 and
e # ej for all j
then A induces Fredholm operators
AY: Xe.Y(C;E ) + X S - " Y - " ( C ; E") (25) for all s E IR. The index ind AY of (25) is independent of s but it depends in general on Y.
In view of Remark 9 the index only depends on a & ( A ) and #;(A). As a corollary of the abstract Fredholm theory we obtain that there is an A l with a&(Al)= a&(A) and ind A l = ind A , but a;(Al) is a bit different from a ; ( A ) close to aC such that
a)a;(gpA1)(f, x , f - 1
t,F ) I t = o = 0
for all k E IN,k 2 1, where the derivatives refer to the first argument. Then, because of the compatibility conditions between a; and the sequence of a!&], we obtain a&-'(A1) = 0 f o r j 2 1. In other words, for calculating the index it suffices to assume that only the leading Mellin symbol is non-vanishing. Now let AYE $V"Y(C;E , E")* be of the form
AY
= w f - p opj;j")(h)w
+ F,
(26)
h E Sm:;ly(X;E', E)*,F E .R$(C;E , E"). In other words we assume u&-j(A) to be nontrivial only for j = 0 . Because of the discrete asymptotics of h we can form
A* = w t - p opP'(h) w
+ F'
E W*(C; E , E")'
(27)
for every A E IR for which A(n) = A - n/2 does not belong to n, sg'(h). This holds up to a discrete set of reals as in Remark 13. If AY is elliptic in P a y ( ...)* then A A is elliptic in X'sA( ...)* up to another discrete set of reals (possibly greater than that for which A* is well-defined, because of the zeros of a & ( A ) on certain parallels to the imaginary axis). If A is of this sort, then we have a Fredholm operator
A': XssA(C, E)+X'-p,*-p(C,E). (28) Let A > y t O , and set 6 = A - y , A = ( - 6 , 0 ] . We want to compare indAv and ind A*. First it is clear that AY also induces a Fredholm operator AY: Xe:;Y(C,E ) ; - + X ~ ; p ~ y - p ( C , E " ) ~ Fith the same index. For simplifying-notations we want to pass to the operators AY = gMAV, 24 . = gpA*,and assume E , E to be trivial and of fibre dimension 1. Let 0 < E < 6 and denote by X:;(C);,, the subspace of u E X;:(C); for which the Mellin image of w u is holomorphic in the strip
( n + 1)/2 - A + E < Re z < ( n + 1)/2 - y . Then Y 2 also induces Fredholm operators BY:= 2:: ~ ~ ; ( C ) ; , , ~ X ~ g p , Y ( C ) ; , , ,
221
2.2. Mellin Expansions for the Cone
provided E is small enough. This is due to the holomorphy and invertibility of h in a small strip around Re z = ( n + 1)/2 - 1, and ind AY = ind BY. It is clear that there are projections nl
%g(c);,,-%(esoy(c)A = x q c ) , %is-
Y(
c,;,
n2
-C
9;- '(C),
xs- K A( C) .
Then B1 := ZZ, BVZZ, has the same index as AA.Thus it suffices to compare ind B*, ind BY, both acting as operators
BA,BY: X;;(C);,
+
".'(C);,,.
2.1.5. Theorem 13 shows that the difference B A- BY is a Green operator GvAwith discrete asymptotics. The image consists of functions with the asymptotic terms Cjk(x)t-p, logkt, 0 5 k 5 m j ,
and the kernel of functions with the asymptotic terms e,,(x)
t-4'
logxt, 0 5 x 5 n , .
Here (pi,mi, L j ) E {(p, m,L ) E sg'(h): p E { ( n + 1)/2 - 1 < Re z < ( n + 1)/2 - y } } , (q,, n,, M,)E {(q, n, M )E sg'(h-'): q E { ( n + 1)/2 - A < Re z < ( n + 1)/2 - y ) } . Set PYA = C (dim Lj)(mj + 1) , NvA= 1 (dim M,) (n, + 1) where the sum is taken over the elements where p j , q, are in the indicated strip. Then BY= B A+ G,A shows that ker BY
+ im GvA, = ker B A+ ker GvA,
ind BY
= ind B A
im BY i.e.
= im B A
+ (NvA- PYA).
It is obvious that the assumption y 2 0 plays no role, since we can pass to operators in spaces with other weights by conjugating with appropriate weight factors without changing the index. Thus we have proved the following 14. Theorem. Let AYE @Y(C; E , @*, A* E XJ'**(C;E , .@* be of the form (26) and (27), respectively, 12 y , a&(AY) = &(AA) = h, a:(Ay) = $(AA)). Moreover define the numbers PYA, NyA as above according to the poles and zeros of h in the strip ( n + 1)/2 - 1 < Re z < ( n + 1)/2 - y . Then (25), (28) satkfv
ind AY - ind A* = NvA - PYA. We want to illustrate Theorem 14 by an example. Let A E DiffP(C; E ,? !,) g-p, interpreted as an operator (25) or likewise (28). Let 1> y and consider the continuous ~ i Y Ainduces an incluembedding i y ~!:W A ( C ,E ) 4 %svY(C,E ) . Then A A= A Y L , and sion i y A :ker A A4 ker A)'. Thus for decreasing y we may expect more and more solutions. ker A A and ker AY are characterized by asymptotic expansions
228
2. Operators on Manifolds with Conical Singularities
with Repj > (n + 1)/2 - A and Repj > (n + 1)/2 - y , respectively. The extra exponents in the strip (n + 1)/2 - A < Re z < (n + 1)/2 - y (2 9) come from the zeros of u$,(A) in (29) that are unessential for the weight A but not for y . The solutions for y+ - 00 become more and more singular at t = 0 and dim kerAy+ m if the number of zeros in Re z > (n + 1)/2 - A is infinite. Let us consider another example which shows explicitly the asymptotic type of the leading Mellin symbol. In the Introduction, formula (2), we have mentioned the Laplace-Beltrami operator associated with the Riemannian metric dt2 + t2g on R, x X . Denote this operator by A (g). Then
uk(A(g>)( z ) = z2 - (n - 1) z + A k ) , (3 0) A(g) being the Laplace-Beltrami operator on X for g, and n = dimX. It is clear that (30) is not bijective at precisely those z E C where z2
- (n - 1)
+a=o
for some eigenvalue A of A(g), i.e., where A(g) u = Au has non-trivial solutions. It follows n-1 n-1 zr =2+ -a If A runs over the eigenvalues of A(g) then (31) yields the system of points where (30) is not bijective and the eigenspaces are just the finite-dimensional subspaces of Cm(X)that occur as further data in the asymptotic types. We may admit here that g = g(t) depends explicitly on t as a C" function up to t = 0. Then for g we have to insert g(0) for evaluating (31). The Taylor expansion of g at t = 0 shows that the leading Mellin symbol and hence the leading asymptotic data only depend on g(0). The eigenvalues of A(g) are known in many concrete cases. Let X = S" be the unit sphere in R"l, g the Riemannian metric which is induced by the standard metric in R"+l. Then the eigenvalues of A(g) are A, = - k(k + n - l), k E N. (A proof may be found, for instance, in TRIEBEL [TlO], $31.) 2.2.4.
Operators on R, X X in the XssY Spaces
This section considers a variant of the cone theory on X^ = R, X X for the spaces Xs.Y(X^),similarly to the theory of 1.3.3. The Xs*Y spaces were introduced by 2.1.1.(39) and the subspaces with asymptotics by 2.1.1. Definition 21. First we want to give some preliminary constructions on pDO-s in R" or R" \ IOI, Au(x): = op(a) u ( x ) : = j ei(X-d)4z(x,x',0 u(x')dx'df, (1) a(x, x', 5) E P(R2" x Rm).The global behaviour of pDO-s in R" has been investi[CS] or gated by several authors in the literature, cf. SUBIN[S27], Chapter 1, CORDES SCHROHE [S4]. Our approach here is close to the theory of [C5], [S4].
229
2.2. Mellin Expansions for the Cone
Let us check the behaviour of (1) under affne transformations T Rm+Rm, T x = Tox + f = y with an invertible matrix To and f E R”. For u ( y ) = u ( T - ’ y ) the push-forward B = T*A has the form Bv(y)
= s s e l ( Y - Y ’ ) ~ b ( y , y ’ , t l ) u ( ydy‘dq, ’)
b(Y,Y’,tl) = a ( T - ’ y , T-ly’, Totl), t l = T-’ 0
5.
An amplitude function a ( x ,x’, 5) is called homogeneous in x , x’ of order (e,e’) for large 1x1, lx’l if a(Ax, x’, 5) = AQa(x,x’, l )
(2)
for all x , x’ E Rm,1x1 2 c, a(x,Ax’,[) = A ~ ‘ a ( x , x ’ , @ (3) for all x , x’ E R“, 1x’I 2 c, 1 2 1. (Below we shall see that only the sum e + e’ is interesting for the constructions). Analogously we define for a ( x , @ homogeneity in x of order e for large 1x1. The homogeneity is invariant under linear transformations. 1. Definition. S”,(Q,p’)(Rzm x Rm),p , e, e’ E R, denotes the space of all a ( x , x’, 5) E C”(RZmx R m )with
lDzD$
a ( x , x’, 5)1 5 c ( 5)” -
x )Q- lgl( x ’ ) ~-‘ la’l
(4)
for all multi-indices a, a’,p E N” and a constant c = c(a,a’,p ) > 0 , for all (5) = (1+ 151z)1’2.We denote by S#.(Q)(Rm x Rm)the subspace of all ( x ,x’, 5) E R3’”, a E S”,(Q,O)(RZm x Rm)which are independent of x’. The best constants in (4) define a Frechet topology on the space Similarly as in the standard case we can define asymptotic expansions of amplitude where p j , ej,eJ+ - m as j + m. There is mod S ” ~ ( - ” *a-un”) functions a, E S”J.(QJ.Q;) 9 9 ‘ Q - Q ’ ) .
N
ique a E Sp~(Q~Q’), p = max { p j } , E
S#N,(QNpQh)
e = max {ej), e’ = max {QJ],
for all N and p N ,e#,eb+
such that a -
1 aj j=O
-m
as N +
m.
We can obtain a in the
form
with a sequence of constants cj tending to m sufficiently fast as j + 00, x an excision function, i.e. x E C”(R), x = 0 close to 0, x = 1 outside a neighbourhood of 0, ( x , x’,F) = (1 + IxI2 + lx’I2+ 1512)1/2. The sum then converges in the topology of Sp,(Q.p’). Now let us turn to the list of standard elements of a wDO calculus in R“ with amplitude functions in Spi(Q,Q’) classes. The proofs may be found in [C5], ~41. First (1) can be defined for u E Y-’(Rm).The integral over x’ is carried out first and the result is integrated then over 5 as a Lebesgue integrable function for every fixed x . This defines a continuous operator op(a): Y-’(Rm)+9(Rm).
230
2. Operators on Manifolds with Conical Singularities
The space of operators with a E S"-(-". -") coincides with the space of integral opLet us denote this by L-m.(-m)(Rm). Further let erators with kernels in 9'(RZm). L'v(Q)(Wm)be the space of all op(a) + C with some a E S*,(Q), C E L-",(-"). For any given a E S'.(Q*Q')(RZm X Wm) there exists an ti E S**(Q+ @')(Itm X Itm) such that op(a) = o p ( 3 mod L-",(-"),and we have the usual formula
a E S"*(Q), b E S",(y)implies op(a) op(b) = op(c) C E L-"*(-"),where
+ C for
certain c E S p + " , ( Q + Y ) and
The latter proposition shows that in most cases it suffices to consider x' independent amplitude functions.
2. Definition. Let Hs(Rm),s E IR, be the standard Sobolev space and g:(x), y E R,be a strictly positive function in Cm(IRm)with g:(x) = lxly for 1x1 2 const > 0. Set H"y(Rm) = gJx) HS(IR") in the topology induced by the bijection Hs(Rm)+H".y(IRm).
(8)
Clearly the definition is independent of the concrete choice of the function g:. We have continuous embeddings
Hs'*y'(Rm) 4 Hs*y(Rm), (9) for s' 2 s, y' 2 y , which are compact for s' > s, y' > y . Further there is a natural identification H%v(Rm)'= H-8, - y ( R m ) via the Lz(Wm)scalar product. 3 . Proposition. Let p ( x ) E Cm(IRm)and p(Ax) = AQp(x)for all A 2 1 and 1x1 2 c, with a constant co > 0. Then the multiplication by Q, induces continuous operators
N v :Hs-y(Rm)+ Hs*"-Q(IRm) for all s, y E R. It is obvious that the assertion holds for s E N. Then we can pass to s E Z by duality. The general case follows by an interpolation argument with respect to s. An analogous result holds for functions Q, which admit an asymptotic expansion m where Q , @ - ~ is homogeneous of order e - j for 1x1 z cj. This space can be equipped with a natural Frkchet topology (which does not rely on the concrete cj), and then when
N,+O in 5!?(Hs~~(Rm), HssY-4(Rm)) 0 in this space.
Q, +
(10)
231
2.2. Mellin Expansions for the Cone
4. Proposition. op(a) for a E S"*(Q)(R" X R") induce continuous operators op(a): Hs,Y(Rm)+ Hs-"Y-Q(Rm) for all s, y E R.
(11)
An operator A = op(a) + C, a E S@v(Q)(R" x R"),C E L-"*(-")(lR")is called elliptic if a(x, 0 +: 0 for all ( x ,0 E R" X Rm and Ix, 51 2 c for some constant c > 0 , and if
la-'(x,[)I 5 c l ( f ) - ' ( x ) - Q for Ix,512 c with some constant c1 > O .
5. Theorem. A n elliptic A induces Fredholm operators A : Hss Y (R") +Hs - P , Y - P(Rm)
(12) for all s,y E R. Further there is a parametrix B in the sense that A B - Z, BA - Z E L-".'-"' (R"). We are interested in special subclasses of L"a(Q)(Wm). The only objective here is to describe operators which are so simple as possible for certain precautions at infinity for a (parameter-dependent) cone calculus. It will play a role in Chapter3 as an aspect of an operator-valued edge symbolic calculus.
6. Definition. S3("(R" x R") denotes the set of all a ( x ,f ) E S:l(Rmx Rm)for which there is a sequence a p -k ( X , 5) E S:,(RmX R"), k E IN,where a p -k is homogeneous in x of order e - k for large 1x1, and N
a(x,R-
C a Q - k ( x , O E S ~ ~ ( Q -(R" N - lx) R")
(13)
k=O
for all N E N ; an analogous condition is required for b ( x , f ) := a ( &x ) .
Note that a p- E S"*(Qk)(Rm X R"). An analogous definition makes sense for x , x'-dependent amplitude functions.
7. Definition. L:i(')(R"') is the space of all operators A a E S$(@)(Rm x It") and C E L-"~(")(Rm).
= Al
+ C where A l = op(a) for
where a(,,(x, 5) E Cm((Rm \ {0}) X R m )is the unique function, homogeneous in x of order e, which coincides with ap(x,8 for large 1x1, cf. (13). Further (T;(A)( x ,0 E Cm(ItmX (R" \ ( 0 ) ) )denotes the homogeneous principal symbol of order p of A . Let Z p l e be the space of all pairs
be, P A E Cm(Rmx (Rm\ {OD) X Cm((Rm \ (01) X R")
(15)
with (p,,p,) = ( ( T ~ ( Aa):,( A ) ) for a certain A E L~i(")(R"). The components satisfy the compatibiliw condition that for large 1x1 the homogeneous principal part of p c in 5 of order p coincides with the homogeneous principal part of p c in x of order e. Let w E Cr(Rr), w ( x ) = 1 close to 0, x E Cm(R;"),x ( 5 ) = 0 close to 0, x = 1 for we have large IFI. For (p,(x, R , p c ( x ,0)E Z",Q ~ ( X , F > = ~ ( X ) X ( S ) P , ( ~ , R + ( ~ - ~ ( X ) ) P ~ S~i(@)(R"'XR'"). ( X , ~ E (16)
232
2. Operators on Manifolds with Conical Singularities
Thus (a;, a:) can also be interpreted as a surjective mapping S~i(")(Rm X R") +'pQ. The component-wise multiplication induces a bilinear map p,Q )( p . d + ~ ! 4 + V 3 Q + ~ Most of the following results are obvious modifications of standard facts on rgDO-s and they are either sketched or left as exercises. 8. Theorem. A E L$(Q)(R"'), B E Lii(a)(R"')implies A B E L $ +v,(Q+d)(Rm) and we have (7) for the corresponding (complete) symbols, in particular
a;+'(AB)
= a;(A)
@"(AB)
=
a;(B).
Moreover
a;(A)o!(B).
The formal a&oint A* of A belongs to L~i'Q'(Rm), we have the usual asymptotic formula for the complete symbol of A*, in particular oF(A*) (x, 0 = aC(A) (x, f ) and moreover
@(A*) (x, F) = Cx-4) (x, 0. Note that the sequence
(R")
O + ~ ~ l - ' , ( ~ - l ) ( ~ m ) + ~cl ~ , ( ~ )
4 ) ,ZP,p
(UC,
0
+
(19)
is exact. The space Lgi(P)(Rm) can be equipped with a natural Frbchet space structure by the same scheme as in Section2.1.4. A special case of Proposition 4 is the continuity A : Hs,v(Rm)+p -P , Y - q R m ) (20) for A E L!!i(')(R"), s, Y , e E R. I f A E L : ~ - ~ . ((R") Q-~ ( -) aft(A)= @ ; ( A )= 0 ) then (12) is compact. An operator A E L~i'"(R") is called elliptic if (i) cr;(A) (x, F) + 0 for all x E R", F E R" \ {0}, (ii) @ ( A ) (x, 0 9 0 for all x E R"\ {O}, F E R", and the homogeneous principal part in F of order p of o : ( A ) (x, 5) is +O for all x ~ R " \ { 0 } , FER"\{O}. An operator B E L,P-(-Q)(R") is called a parametrix of A if AB - I , BA - I E L-"*'-"' (R"). Note that the ellipticity is invariant under linear transformations of R". By inverting the components of ( p v , p c )E P Qsatisfying (i), (ii) we get an element ( qru,qc) in C!'--Pcf. (16). Now we can construct an operator B E LJ"Q)(Rm) with ( a i P ( B ) uLQ(B)) , = ( q v , qc). In view of Theorem 8 and a! of AB - I , BA - I vanish. Together with Proposition 4 we then obtain 9. Theorem. I f A E L$(Q)(Rm) is elliptic it induces a Fredholm operator
A : Hs.V(Rm)+Hs - P , V - Q(Rm)
(2 1) where A B - I , for all s , y , e E R, and there is a parametrix B E LiP-(-P)(Rm) BA - I E L-"p(-")(Rm).The Fredholm property of (21) forfixed s, y implies the ellipticity.
233
2.2. Mellin Expansions for the Cone
In particular kerA c Y ( R m )Since . indA = dim kerA - dim kerA*, the index is independent of s, y . Note that L;i(O)(R"') contains the subclass of yDO-s L3(0)(Rm)const of the form A = A,,,,, + A _ , where Aeon,, is of the form (1) with a (5) independent of x, x', and A _ , E L-m-(-m)(Rm). Then a special case of Theorem 9 is that an elliptic differential operator A of order p in R" with constant coefficients induces a Fredholm operator Hs(R")---* H S - P ( R m )iff the complete symbol does not vanish for all [ E R". The specific point of the above discussion for us is the behaviour of the operators close to m. It can be generalized to X A= R, X X with respect to t + 0 0 , X being a closed compact C" manifold of dimension n = m - 1. Let { U j } j = be an open covering of X by coordinate neighbourhoods which are diffeomorphic to coordinate neighbourhoods of S". The diffeomorphisms Uj+ extends by homogeneity to diffeomorphisms
oj
uj = u,:
0; = {2E R"+1 \ {O} : 21 121E U j } , x j ( l t , x) = I x j ( t , x ) for all 1 E R,. xi: R, x
+
10. Definition. Lfi(Q)(XA)is the space of all A E Lg,(X^) such that for every p(x), @(x)E C,"(Uj)and cut-offfunctions w ( t ) , G ( t )
xj+(l-w)pAQ(l-G)~L'di(Q)(R"+l)for j = 1 , ..., N . It can easily be proved that this is a correct definition, i.e. independent of the concrete choice of data { Uj}, xi,and so on. Let us give the arguments for the invariance of o : ( A ) ( x , 0 under a homogeneous diffeomorphism x : Ua+ U^,where we only have to look at the points far from the origin. A is of the form
Au(2)=
r)
ei("f')Fa:(A) (2, u(Z')d?'dr+ A o ,
(22)
where A. belongs to L$(Q- I ) . The reader may verify that L:i" - ') is preserved under +. It remains then to examine the integral on the right of (22). For evaluating a:(x,A) we can apply the asymptotic formula for complete symbols of yDO-s under push-forward namely
a("):= a;a, y"= x ( 2 ) , fi the covariable of y",
@,Cy, 7 j ) = D ; e i h ( ~ ~ ~ l l hi = ( 2i, ,Z; 7 j ) = ( x ( 3 - x ( 2 ) - x ' ( 2 ) ( ? - 2),7 j ) The invariance follows from cJXx*A) ( x ( 3 , 7j) = @ ( A ) G , ' x ' ( 2 ) $1, since the lower order terms of (23), i.e. those for a Z 0, are also of lower order with respect to the homogeneity in 1y"l. This comes from the factors @. and the homogeneity of x . Clearly, the precise arguments have to consider the further remainders that are involved in the general proof of (23) but they do not contribute to the terms of highest order in ly"[. The coordinate invariance of more general type on non-compact manifolds has been discussed also by SCHROHE [S4].
234
2. Operators on Manifolds with Conical Singularities
Now we want to combine Definition 10 with the properties of cone operators close to zero. Let us point out once again that in X " close to zero we prefer the coordinates ( 1 , x), x varying on the base X whereas for large t the local coordinates on X" are induced by a diffeomorphism xl: U-+ U, U a coordinates neighbourhood on X , d a coordinates neighbourhood on S", and x : ( t 1, a) x U -+ d = { 3 E IR"+ l : 3/13l = xl(x) E 0,151= t > tl} . (24) Then x(&, x) = k(t,x ) for all (t, x) E ( t l , m) x U, 1 L 1. The concrete choice of tl is unimportant; we may set, for instance, tl = 1, For simplicity from now on we only consider e = 0. The general case is completely analogous. The idea for the operator classes is to impose the properties of ML'(B+ X X ) or of ' W ( X " ) ; close to t = 0 and those of L$('O)(X^)for large t . If we use a cut-off function w ( t ) we interprete it as usual or likewise in the coordinates C, i.e. as ~(131). Define the spaces 9 ( X X ^=) [W]Xrn7Y(X")+ [ l - w ] Y ' ( r ) , (25) where Y'(X ):= Y'(R+) gnC r n ( X ) ,9'(E+)= Y(R) I K+ ,
Y'gx-),= [ w ] x;*yxa.), + [l - w ] Y'(X),
(26)
Y ' i ( X " ) , = [ w ] x ; ~ Y ( x A )+, [ l - w ] 9 ( X )
(27)
for B E W"), P E W n ) ( X ) ,A
E 3, and
q 8 ( X " ) , = @Y'g(X"),,
Y';a(x");
B
= l&Y'i(X"),
(28)
P
with the inductive limits over all B E LWn)and P E P ( " ) ( X ) ,respectively. For y = 0 we also omit the upper subscripts at the spaces. Let p E R and set ML$'O)(X) := wML'd,(X)w + (1 - w ) L$'O'(X") (1- w ) + M L - r n ( X I ) l , (29) cf.2.1.3. Definition 13, where M L - " ( X ) , denotes the space of all operators C with @" kernels which induce continuous operators
c, c*: x*q x " )+Pyx") for all s E R (C* refers to L 2 ( X " ) ) . We are more interested in the subclasses with asymptotics. 11.DeAnition. c!'(XA)8, B = ( - k , 01, k E N , is the space of all operators oftheform A
+ (1 - ~ ) A 1 ( 1 - W ) + G
= WAOW
for arbitrary A. E 3'(X381, B1= (- k, k ) , A l E L$'o'(X*), and G with @" kernel which induces continuous operators G,G*:x 8 ( x " ) + 9 a a ( x " ) 8 (30) for all s E R. The class of those G will also be denoted by c!g(X*),.
In an analogous manner we define f!"(X*);, the corresponding subclasses with dots (i.e. A. E W ( X " ) ; ,, and the spaces on the right of (30) are to be replaced by the dotted ones).
235
2.2. Mellin Expansions for the Cone
Note that
2p(Xa), E g - " / 2 M L c i ( o ) ( X ) g n / 2 . Finally, according to 1.3.3. (19), we can consider
S k q X J , = gY-W(X"),g-Y
(3 1)
which is again a trivial generalization. So we want to discuss the further aspects for the 2 classes. (The notations specialized for dim X = 0 do not exactly coincide with those of 1.3.3. The classes here are slightly more general, cf. also the remark at the end of 1.3.3.). The main difference to 2.2.1., 2.2.2. is the modified symbolic level for t + m which is involved in the ellipticity. First there are the Mellin symbols at zero aM(A)(2) = { od(A)( z ) } and the homogeneous principal symbol of order p
a ; ( A ) ( t , x , t , t ) for ( t , x ) E K + x X , ( t , O + o , which has the property that a;(A) (t, x, t-lt, Finally we have
a9 ( A1 ( x , F) := a9 ( A 1) ( x ,
0 is C" up to t = 0.
a,
where A l is as in Definition 11 and f is interpreted as fibre variable in P X ^ over ( t , x), cf. the constructions after Definition 10. The homogeneous principal symbols a;(A) of operators A E 2fi(X^),run over a space S t ) ( T* x^ \ 0), of functions on TCX" \ 0, homogeneous of order p with respect to the lR+ action along the fibres. The subscript b indicates smoothness for t +0 after replacing z by r-%, whereas the subscript 1 indicates the behaviour of a; for operators in L$(O'(X") for r+ m . We say that a is invertible in Sp)(...)lif it is first invertible for t > 0 as usual, but for t replaced by t-ltit is also invertible up to t=O.
Let us drop here the discussion of exact symbol sequences as the analogues of 1.3.3. Theorem 5 or 2.2.1. Theorem 26. Of course, it can be done, and it yields naturespectively. We have natural rules for ral locally topologies in i?p(X"),and f!@(X");, compositions of A E 2 " ( X J 8 , B E f?"(X"),and formal adjoints, where aM(AB)follows as usual as the Mellin translation product of a , ( A ) and a,@), further a;+ ' ( A B ) = aC(A)a ; @ ) , a: (AB)
= a: ( A ) a: ( B ) .
The symbolic rules for A* are analogous to those in Theorem 8 and 2.2.1. Proposition 9. Let us explicitly formulate the version of ellipticity in the 2' classes. 12. Definition. A
E 2p(X"),,
9 = (- k, 01, k E N \ {0), is called elliptic
if
(i) $ ( A ) is invertible in Sp)(T* x^ \O)l, (ii) &A) ( x , F"> f 0 on T*X^ in Fof order p is f 0 on T*X^\O, and the homogeneous principal part of &A) in f o f order p does not vanish for F+ 0 , (iii) ab(A)(2): B ( X ) +H"@(X) is bijective for all z E r,,+ 16
Schulze, Operators engl.
236
2. Operators on Manifolds with Conical Singularities
13. Definition. An operator B E e-j'(XA), is called a parametrix of A E e'(x"), if A B - Z, BA - Z E e $(X " ), .
For A , B in the dotted subclass we obtain the Green remainders in the dotted subclass.
14.Theorem. Let A
E e" (X " ), be
elliptic. Then
A : Xs(X")+Xs-f'(Xa)
(32)
is a Fredholm operator for every s E R . The Fredholm property for a fixed s E R implies the ellipticity. There is aparametrix B E e-'(x")8, and A u = f E X:&X")8,u E X m ( X " )
imply u E Xi:' ( ( X A ) 8 where the asymptotic type of u depends on that o f f and A but not on s. Analogous assertions hold for the dotted subclasses. The proof is an obvious combination of the above constructions for L!!j(O)(X")and those from Section 2.2.1. All assertions hold including 8 = (-a,01. It is clear that ker A c Yas(x")asuch that ind A is independent of s. We can generalize Theorem 14 also to the spaces g ;yX S (X " ), y E R , and those with asymptotics. The simple modification is left to the reader. For the E p classes of cone operators we can draw analogous further conclusions as for the 9l' classes of Section 2.2.1. In particular we have
15. Proposition. Let A E ~ P ( X * ) and , assume that (32) is invertible. Then A-' E e-#(X"),. An analogous result holds for A E ei'(X^);. The proof is based again on an analogue of 2.2.1. Proposition 32. 16. Remark. The cone operators of the classes P as well as the dotted ones satis& the analogue of 2.2.1. Proposition 34. Here the %is(X")a spaces are to be replaced by Ya8(X*)a.
17. Remark. There is a straighrfonvard extension of the theory in this section to operators acting between dirtributional sections of vector bundles over X*.cf also Section 2.2.3.
2.2.5.
Appendix (Meromorphic Operator Functions)
In the previous sections we have seen that the regularity with asymptotics for the solutions of elliptic equations near conical singularities follows from functional analytic properties of families of Fredholm operators. This section presents some auxiliary material in this context. It will not be formulated here in its most general form. Fredholm families, in particular holomorphic and meromorphic ones, may be subject of independent investigations. For this we [G4], [G5], GRAMSCHXABALLO [G6]. refer, for instance, to the papers of GRAMSCH The following observation has contributed to the axioms for the operator-valued Mellin symbols with discrete asymptotics. Let H I , Hzbe Hilbert spaces, U 5 @'a connected open set and A ( z ) : H , +Hz , z E U,a holomorphic family of Fredholm operators. Assume t h g there is a TE U such that A(,?) is an isomorphism. Then for every open subset V c Uwith V c c U,there exists a scalar non-vanishing holomorphic function a ( z ) over Vwith a ( z ) = O o A ( z ) is no isomorphism, z E V. In particular for r = 1 there is a countable subset { P , ) , ~ , c Vsuch that A ( z ) is an isomorphism for z f p,. Let r = 1 und Ul = { z E U : A ( z ) invertible}. Then A - l ( z ) , z E U1, can be extended to a meromorphic family of operators over U,and the operators C,, in the Laurent expansion at p j EU\U1
237
2.2. Mellin Expansions for the Cone
r.
are finite-dimensional, where dim Cjk5 N = N ( V) for all j with p j E We shall give here a proof in the more general situation of operator families. They play a role for parameter-dependent cone operators. 1. Theorem. Let Y be a compact topological space, U G C open, connected, H 1 , Hz Hilbert spaces, and A : Y X U -Y 2 (H I, Hz) a continuous operator function which is Fredholm for every fi, z) and holomorphic in z for each fixed y . Assume that for every y E Y there is a i= i ( y ) E U such that A& ?(y)): H 1 - Hz is an isomorphism. Then A-'(y, z ) defined on {(y, z ) E Y X U : A(y, z ) invertible} can be extended to Y X U as a continuous (in y ) family of meromorphic operator functions, where dimCjk(Y) 5 N ( K ) , 0 S k 5 m j ( y ) , m,(y) 5 M ( K )
for a l l y E Y and all p j = p j ( y ) in a compact subset K c U,with constants N ( K ) , M ( K ) < m . In addition there is a continuous finite-dimensional subbundle E ( K ) c Y x Hz over Y (i.e., the transition functions are continuous) with
im Cj&) E E ( K ) , for all y E Y
(1)
and all pj E K ( E ( K ) , denotes the fibre over y ) ,
ProoJ: Without loss of generality we may assume H = H 1 = H z . Let V, be a connected open U.Restrictions to K of holomorphic functions in an open neighbourhood of K set, K := are called holomorphic on K . Consider the operator family A as a homomorphism of Hilbert bundles A: Y X K x H + Y x K X Hwhich is holomorphic in Z E K . It is known (cf.[R8], Section 1.1.3.4., and Lemma 7 below) that there exists a subspace M c H of finite codimension such that mA: Y X K X H - Y X K X M
vcc
is a surjective operator family, m : H - M denotes the orthogonal projection. Then L := ker(mA) is a finite-dimensional vector bundle over Y X K , holomorphic in z , and the orthogonal projection I : Y X K x H+ L constitutes a family of operators which is holomorphic along K . The fibre dimension N of L coincides with dimM1, M 1 being the orthogonal complement of M . Denote by LL the family of orthogonal complements of the fibres of L. Then the operator family A can be identified with a matrix
L1 YXKXM B P . @ + @ , A=(T L YxKxMl withB=mA(l-I), P=mAl, T = ( l - m ) A ( l - l ) , ible and B-' is holomorphic in z . Then 1
YxKxM
:): Y x K x M L @
1 -B-'P Jz=(o
):
Ll
+
Ll
@*@,
L
L
are holomorphic isomorphisms and we have
16'
Q = ( l - m ) A l . T h e f a m i l y Bisinvert-
YxKxM @
YxKxM'
238
2. Operators on Manifolds with Conical Singularities
Thus A is an isomorphism at precisely those points b, z ) E Y x K , where the finite-dimensional morphism
Q - TE-'P: L-+ Y X K X M 1 is an isomorphism. Let R : L-+ Y x K x C Nbe a trivialization of L , holomorphic in z . Set
F = ( Q - TE-'P)R-': YX K X C N - + Y X K X MI,
F being a matrix function which is holomorphic in z . Then
A-'
= J;'
(E-0 ' F-' )
(JzJ,)-l
at all points b, z ) , where F is an isomorphism. Let
be the Laurent expansion of F-' at z = p j . Then
with
Now @JJ,-z) and Ckb, z ) are holomorphic in z . Applying the Taylor expansion at z = p j to cr(y, z ) , Cj+kb,z ) , we get the Laurent expansion of z ) , where the resulting coefficients obviously have the asserted properties. In particular (1) is a consequence of the fact that the part where the operators are no isomorphisms, is concentrated in the finite-dimensional right lower corner. 0
2.Remark. Theorem 1 shows that the behaviour of the poles of A b , z ) - l is analogous to that of the poles of families of scalar meromorphicfunctions. In particular for every yo E Y there is a countable system of poles pj(vo)E U,j E Z . The numeration of them is completely independent of that for another yo. The multiplicities mj(yo) + 1 are not constant in general for varying yo, CJ Section 1.1.5. Next assume that R is a C manifold. 3. Definition. (i) n0(R)denotes the system of all subsets S c R X C such that for every open RO c R w i t h $ c c R a n d e v e r y c o m p a c t K o c c C t h e r e i s a r = r s E C - ( ~ , s 4 ' ( @ ) )(cf. ' 1.1.5. Definition 5) with Sn(R0 X KO)C ((Yo,Po): POE ~P(((YO)))]. (ii) n,(R)denotes the system of all subsets S c R X C X N such that for every no,KO as in (i)
(iii) Let H be a Hilberi space and G,(H) =
u G k ( H ) ,where G k ( H )is the set of all k-dimensional
kt- N
subspaces of H.
n(0,H ) denotes the system of all subsets S c R X
C X N X G,(H) such that being the projection to R X C X N ) and for every Ro, KOas in (i) there is a finite-dimensional @" subbundle E = E (ao,KO)c Ro X H such that (yo, po, mo , L )
rr,
NSc n,(R) (nn
2.2. Mellin Expansions for the Cone
239
x KOx N) imply L E Em.Let n(R,H") be the subsysE S andCyo,p o , mo) E nnxc NS (0, tem of those S which are contained in R X C X N x H" for some linear (not necessarily closed) subspace H" S H . 4. Definition. C"(R X C, %(HI, H2))'denotes the space of all operator functions A@, z) with (i)ACy, z ) is c" in R x C forsome $(A) E Ilo(R), (ii) ACy, z ) extends for every fixed y E R to a meromorphic operator function and S(A) E n(R,H2), where S(A) is the set of all Cy, p , m, L ) such that p is pole of A at y of multiplici@ m t 1, and the image of the Luurent coefficient of ( z - ~ ) - ( ~ + l0) 5 , k 5 m, i s in L.
\so
5.Theorem. Let R c R" be an open set and TCy, z ) E Cm(Rx C, %(HI, H2))*, then (DJT)Cy, z ) , first defined on R X C \so(T), extends to a function in Cm(RX C, % ( H I , H2))'for every multi-index a.
Proof: First observe that the sets S E n0(R) have the following property. If (yo,po) E S, there is a e > 0 such that Cyo, z ) B Sfor every z with 0 < dist(z, po) < 29, further there is an open neighbourhood Ro of yo such that Cy, z) S for all z with ~ / s2 dist(z, po) s e and all y E 0,. Let p o E R be fixed, po = poCyo) a pole of TCyo,.) and e > 0, Ro as mentioned, KO= (z: lz-poI
e Bi:=( z : Iz - zo( = e - ei},i = 1,2,and J E C , e > s e } .Let 0 < E~ < e 2 , e - E~ >j-,
IJ-poI>e-e,.
Set
y E 0,Iz - pol > e - s2. Then
Indeed
and 1 W'
The constructions in 1.1.5. may easily be generalized to operator-valued merornorphic families. In particular TI@,z ) and TCy, z) represent families of operator-valued analytic functionals
and similarly c[TJCy), B = { z : Iz -pol = e - E ) , el 5 E = e2. The equation (2) shows that TICy,I ) - TCy, z ) is holomorphic in [ z : Iz -pol < e - E ~ for } Thus ([TI] Cy) = ([TI Cy) for all y E 0.Now TI& z) is obviously a c" family of fiall y E 0,. nite-dimensional operators with values and ranges of the Laurent coefficients in Ey, y E Ro, cf. Definition 3 (iii). Now our assertion follows if we apply a straightforward generalization of 1.1.5. Theorem 6 to c" families of finite-dimensional merornorphic operator functions.
240
2. Operators on Manifolds with Conical Singularities
6.Remark. 1.1.5. Lemma 8 gives a control on the multiplicities of the poles of DJT, moreover So(T) = So(D;T), CJ Definition 4(i), and the bundles E, belonging to Oo,KO in the sense of Definition 3(iii), are independent of 01. For future references we now come to some further simple properties of Fredholm operatorvalued functions. The calculus of yDO-s on manifolds with singularities (in particular of boundary value problems) often leads to the following situation. Let H I , H2 be Hilbert spaces and A : H I + H2 a Fredholm operator. Choose M, N E N such that
is invertible with certain finite-dimensional operators T, K, Q (it is clear that then N - M = ind A). Then a trivial lemma says that 1-' is of the form
where B is a parametrix of A , i.e. BA - I, AB - I compact. Such a result is needed in concrete cases under extra properties of the finite-dimensional operators T,K, Tl , K1 (for example to have smooth kernels when Hi are L2 spaces or to belong to another specific algebra of operators). If the matrices like (3) play the role of "boundary symbols" then we have to do the same for continuous (or c")families over a parameter space Y. Without loss of generality we assume that Y is connected. I . Lemma. Let Y be a compact topological space, A : Y-+ 9 ( H 1 ,H2) a continuous operator function which is Fredholm for every y. Then there exist numbers M, N E N and continuous operator families T b ) : H1+CN, KCy): C'+H2, Qb): C M - + C N , such that db) is invertible for a l l y E Y and d-'(y) ir continuous, too. Y is a c" manifold with A(v) being c" in y, then T, K, Q can be chosen as c" families, and K 1 ( y ) is c" in y. ProoJ Let us first show that there exists a closed subspa_cef12 c H2 of finite codimension such that P2ACy): H1+ H2 is surjective for ally, P2:H 2 + H 2 being the orthogonal projection. Indeed, for y = yo we find a finite-dimensional operator K and an M b o ) E N such that H1 (Abo), KOlo)): @ +H2
v
@Mh)
is surjective. Since ACy) is continuous, there is an olen neighbourhood VCyo)of yo such that ( A C y ) , K b o ) ) is surjective for all Y E VCyo). Let HCyo) be the orthogonal complement of KCyo)(C'h)). The construction can be carried out for every yo E Y. This yields an open covering of Y by open sets VCy) and a system Of spaces H P ) . Choose a finite subcovering {V(YO),..., V(y,)} of Y. Then the space H2 = H2(yo)n ...n H 2 ( y v )has the asserted property. Now it is obvious that for some constant operator K : CM+ H 2 , M = codim H 2 , the operator Hl (ACy),K ) : @ -*H2 (5) CM
is surjective for all y E Y . ( 5 ) is a continuous Fredholm family. We then know that ker(AQ), K ) is a finite-dimensional continuous vector subbundle of Y x ( H , @ CM). Let P1b): Y x ( H I @ CM)+ ker(ACy), K ) be the orthogonal projection and Pb):ker(Ab), K) Y X C Na trivialization. Then we may set TCy) = Po11 PiCy) R , Qb)= Pb) Pi@)R ' ,
-+
2.2. Mellin Expansions for the Cone
241
where R : Y X HI + Y X ( H I@ CM),R': Y X CM+ Y X ( H I@)'C are the canonical embeddings. If A ( $ ) is a @" Fredholm family then &($) is also C". Now it is well-known, that for a continuous (C") family of invertible operators between Hilbert spaces the family of the inverses is again continuous (C"). Thus Lemma 7 is proved. 0 8. Remark. Let H be a Hilbertspace. 3 a n ideal of operators in 2(H,H ) . If 1 + C : H + H is invertible and C E 3 then (1 + C)-l = 1+ C1 with another Cl E 3.
Indeed, if B = (1 + C)-', then C1= B - 1 satisfies (1 + Cl) (1 + C) = 1, i.e. C1= - C -C,CE!Y. As noted in the beginning we often need more concrete information on d - ' ( y ) . Therefore we shall construct d - l ( y ) in a more explicit way. We start with an invertible operator family (3) over Y and express d - ' ( y ) . Fer simplicity let us talk about C" families. Usually we know that for A ( y ) there is a C" family B ( y ) of parametrices of A ( y ) . Then we can pass to a family
of invertible operators, where A - = N - M = ind A . Assume without loss of generality that M t M, otherwge we add the identical matrix of suitable dimension to (6). Then by adding the identical (M- M ) X ( M - M)-matrix to (3) we get a family of isomorphisms, again. It is clear that it suffices to construct the inverse of the latter operator family, in other words we may set N = N, M = M. Now
is invertible for all y, and G is a C" family of compact operators. Let us choose an invertible operator family of the form
where fil : C N + C N is invertible. This is possible, since the invertible matrices form an open dense subset in the space of all N X N-matrices. Then if D1 is a small perturbation of D 1 we do not destroy the invertibility of the whole operator matrix. Assume that we have already constructed VI. Then
where D2 is necessarily an invertible N X N matrix. Since
(:
1 -C2D;' D;l
;)-I=(()
)=:
2
we get 1 - l = 2VI3,which is a composition of operators that are known. It remains to show how V 1looks like. To this end we observe that 1 -C,fi;'
(0
1
)+;Is,
1+G 0 0 fiJ
:)=(
(7)
242
2. Operators on Manifolds with Conical Singularities
= G - C1b;' S1 being Fmpact. Since the factors on both sides of V are_invertible in our class, we know that 1 + G: H I + H I is invertible. Now we use that (1 + G)-' = 1 + G' with some compact operator G , cf. Remark 8. Thus the formula (7)admits easily to express V1. Let us also mention another standard method to invert operator functions within a given class. Consider for simplicity single operators. The case of families is then obvious.
9. Lemma. Let H be a Hilbert space, B c 9 ( H , H ) a Banach subalgebra. Let A E B and A : H + H be invertible. Then A-' E B.
Proof. Let I: be the spectrum of A ; Z c C is bounded. For A B Z,111 large enough, we have a convergent series
since B is closed. From
C being a sufficiently large curve around
2.3.
Z,and (8) we obtain A-I E B. 0
The Parameter-DependentCone Calculus
It is necessary for some applications to study the cone operators of the Sections 2.2.1.-2.2.3. depending on parameters. They may occur as extra variables and covariables in a further operator level based on the Mellin or the Fourier transform, or as parameters in homotopies of operators. In those cases we have to expect nonconstant asymptotic data. In particular the point-wise discrete asymptotics may change with varying parameters. This will be studied in the following sections. 2.3.1.
The Cone Algebra with Parameters
Our first objective is to study a parameter dependence in form of further couariables. Remember that we have used already standard vDO-s L I ( X ; A ) , L g 1 ( X ; A ) depending on a parameter A E A := R'(cf. the beginning of Section 2.1.1.). X is not necessarily compact. For the cone operators I will play an analogous role. For sirnplicity we restrict ourselves to A = R'; we also might consider conical sets in a vector space. The cone operators of the Sections 2.2.1., 2.2.2. form a subclass of P Z M L P ( R + X X ) td2. So we have first to formulate the concept of Mellin-Fourier vDO-s in the parameter-dependent case. A further step is then to introduce the parameter-dependent Green and Mellin operators. First we set ML-"(R+xX;A)= 9(A,ML-"(R+xX)),
where Y'(...) denotes the Schwartz space over A of ML-"(. ..)-valued functions. This refers to a canonical topology of ML-"(. ..) which is defined as follows. The operators in ML-"(a+X X ) are characterized by the mapping properties
243
2.3. Parameter-Dependent Cone Calculus
A , A*: %s,"'2(X^) +%m*"'2(X^)
for all s E R. This induces embeddings
1, i*:
ML-"(R+ x X ) +
nS(%s-*2(XA),
SER
%m*"'2(X^)).The space on the right is Frkchet. Now ML-"(R+ X X ) is equipped with the weakest locally convex topology under which 1, .1 are continuous. It is then also a Frkchet space. In an analogous manner we get a topology in ML-"(R, X X ) . 1. Definition. MLp(R+xR;A)(ML;l(R+xa;A), M L p ( R + x R ; A ) ,...) is thespace of all operator families A ( A ) + G(A), where A ( A ) h of the form 2.1.3. (2) with an amplitude function a(t, x, t', x', z, &A) in Sp((R+XR)' X R"+l X R1)(Scl(( A! R+XR)' x Rn+lx R'),...) (cf. 2.1.3. Definition 2) and G(A) an operator with kernel in S ( A , Cm((R+XR)'))for the classes over R+XR, whereas for R+x R we demand that G, G* belong to S ( A ,.Y?(%s*"'2(R+ XRComp), %"vd2(R+XRl,,,)))
for all S E
The calculus of 2.1.3. has a straightforward extension to the parameter-dependent case. Most of the elements may be left to the reader as exercises. The cut-off procedure of 2.1.3. Proposition6 for every fixed 1 has the obvious consequence that the parameter-dependent negligible kernels are Schwartz functions of A. In other words 2.1.3.(11), (12) is to be replaced by
(1 - w(B,C))K"(f,X,t',x',B,r,n)ESO(A,CZ'(R+xRn)2,.T"'2(R+xR"))) and
S m ( ( R + x Rx) 2R"+'x R:)
2
S ( A , CZ'((R+xR)2,.T"'2(R+XR"))),
respectively. Also 2.1.3. Proposition 7 has the corresponding parameter-dependent analogue. The notations a!, a; from 2.1.3.(23), (24) are in the parameter-dependent case to be replaced by
respectively, A = op,(a), a ( t ,x , t', x', z, [, A ) E S;,((R+XR)*x R"+ x Rl), a @ )being the homogeneous principal part of order p in (z, [, A).
2. Definition. Let X be a closed compact C" manifold, n = dimX. Then ML"(R+ x X ; A ) ( M L p ( E + X X ; A ) )h the space of all A ( A ) + G(A) E L p ( R + X X ; A ) such that x * ( p A ( A )w ) E M L " ( W + x R ; A ) ( M L " ( R + x R ) )for every chart xl: U + R , U being a coordinate neighbourhood of X , x ( t , x ) = ( t , x l ( x ) ) , and every p, w E C t ( U ) further G(A) has a kernel in S ( A , Cm((R+X X ) ' ) ) for the class over R+x X , whereas for R+X X we demand that G, G* belong to S(A,5?(%~d2(X"),%m~"'2(X"))) for all s E R . As usual cl indicates the subclasses of classical operators. Clearly an A ( 1 ) E M J ! . ; ~ ~ ) ( . . . induces ;A) for every fixed A,, an operator in the corresponding class without parameters.
244
2. Operators on Manifolds with Conical Singularities
3. Definition. %$(X^;A),, 8 = ( - k , k ) , k~ N, p ER, is the set of all A t - d Z M L $ ( R +x X ; A ) td2 which are flat of order k at t = 0 and t = 00 for all I E A .
E
Next we want to introduce parameter-dependent Green operators. There are two variants, according to the discrete and the continuous asymptotics. For the moment we restrict ourselves to the continuous asymptotics. The point-wise discrete case needs more comment and will be studied in detail below. In 2.1.5. has been introduced ZG(X^),,A E Y,including an adequate locally convex topology, cf. 2.1.5. (5). Now we set @ Y(A,%i(x^),@ r J % z ( X ^ ) d ) ,
%G(X^;A), =
(3)
B. CE rY-ni2
8 = ( - k , k ) , 05 k 5
m.
4. Remark. BG(XA; A ) Acould also be defined by a parameter-dependent analogue of 2.1.5. Defnition 1, where we would impose a strong decrease of the operator norms
G(A): sle5(XA)-+sle;(XA)$3, G * ( l ) : sle5(X^)+sleL(X^)$3 for all s, r E R,j E Z, where (j) indicates the closure with respect to the j i h norm in a norm system which describes the corresponding topology. cf: 2.1.1. (33). This may be used for proving assertions of the type that 9 2 ~ ( X ^A; ) Ais a two-sided ideal in another space of parameter-dependent operators.
Note that a parameter-dependent Green operator G ( A ) of the class (3) contains I-independent carriers of asymptotics B , C. This is exactly the picture that we obtain in calculations with parameter-dependent Mellin operators to be introduced next. If we have in mind applications with discrete asymptotics for every fixed A we admit a bounded variation of the imaginary parts of the poles in every finite strip parallel to the imaginary axis in the complex plane. The definition of parameter-dependent Mellin symbols is based on L$(X; A ) similarly as L $ ( X ) is involved in the non-parameter dependent case. We endow L $ ( X ;A ) with the natural Frkchet topology, introduced in an analogous manner as for L$(X).A method may be to identify LEI(X;A)with the subspace of L$(X X R') of operators that are invariant under translations along R' and then to carry over the topology induced by L$(X X R'). 5. Definition. Let p, y E pi, A Wf,(X; A ) : the space of all
E Y, be
non-trivial, VE Y, V n &Sf") = 0. Denote by
a(z,A) Ed(S1;(")\ V , L ; l ( X ; A ) ) for which
(0
d (c) 3 h+ ([[a1 (I), h ) =
&
I
a(z, A ) h ( z ) dz
represents an element in d'(S1;'") n V) BnL--(X;A ) , L c S:(") being a curve surrounding S$(")n V ,
245
2.3. Parameter-Dependent Cone Calculus
(ii) & a ) (e + i z, A ) E L!!](X;]R, x A ) for all For trivial A we set
e E S;'") and all V-excision functions x .
rnr,(X;A ) ; = rnP.Y(X;A ) = L!!,(X; r,,,- ?(") x A ) .
(5) Similarly as in the case without A we get in m ; ( X ; A ) : a natural locally convex topology under which it is a FrCchet space. The definition of m ; ( X ; A ) ; , A E 3 finite, extends to arbitrary V again by an obvious generalization of the decomposition method of Section 1.2.3. 'At the same time we get the FrCchet topology in "t;(X; A ) ; for general V. Then we set rtnf,(X; A ) = c lim %l;(X; A ) ; A€J
which is independent of y , and rnt,(X;A ) =
5 mgx;A ) , VE
r
rn:;yx;A) = @ rnt(X;A). Y E T-fi")
6. Theorem. For every p E R, V E V we have
rn;(x;A) = rn$(X;A)+rnirn(X;A), r n ' ; ( X ; A ) = rn';1(X;A) + r n ; 2 ( X ; A ) for
v= v,+ v,.
The method of proof is an evident generalization of that for 2.1.2. Theorem 5. The parameter-dependence makes no additional problem (cf. also 1.1.5. Lemma 3 and the comments after 2.1.2. Theorem 5). 7. Remark. The method of proving 2.1.4. Proposition I shows that ZmKY(X;A) = n;(x;A)+ w - m ' y ( x ; A )
for every y E R.
Note that there is also a straightforward generalization of 2.1.2. Proposition 7 to the ) ) , op$(a) ( A ) case with parameters. If a ( ? ,r, z,A) E Cm(R+x R + , ~ f l * B + ' " * ( X ; Athen belongs to L:,(XA;A ) with the complete parameter-dependent symbol a ( ? ,t,x, -itz,[,A) mod S"-'(RAx Itn+' x Rl),in local coordinates on X . Set a;(op$(a))(r,x,z,5,A)
=aYt,t,x,
a:(op$(a)) ( t , x , z, ( , A )
=
-it,F,A>,
a(P)(t,t , x, -i tr, 5, A ) .
In particular a; may be interpreted as a mapping a;:
rn~.'"Z(X;A)-,S'P)(T$~xA\O),
(7)
where \ 0 indicates (z, [, A ) 9 0 and the space on the right of (7) is an obvious generalization of that in 2.1.2. (27). (7) also makes sense as a mapping 0;:
r n : s ( x ; A ) - + S ' q T $ X ^ x A \O)
(8)
246
2. Ooerators on Manifolds with Conical Singularities
2 0 ,,and w l r w 2 be arbitrary 8. Theorem. Let h E %Rml;(X;A),V E T dn 2T d 2 - @ cut-off functions. Then
w1tsopiMz(h)(A)w2- w1 opid2(Tah)(A)taw2EWG(X^;A). Zfh~%ml;(X;A),V ~ c l ca ~n d~c r , B Z O , a + B = k E N , then
wltaopid2(h)(A)tawz E % ! & + G ( ~ ^ ; ~ ) B , 9 = (- k , k ) . Moreover w1
opidz(h) (A)0 2 -
opidz(h)( A ) G2 E 8!&+ G(x*;A),
for any other cut-offfunctions GI, G2, and w1
Opp(h)(A)(l-
02)E 8 $ + G ( X A ; A ) .
2.2.1. Definition 19 can immediately be generalized to %ml;(X;A) 3 h ( z , A ) , where the sets V, and the ev are assumed to be independent of A. This yields a corresponding notion dec,(h,j) of decomposition data for parameter-dependent Mellin symbols. We then have
9. Proposition. For every j E IN \ {0},h E %Rl;(X;A ) , V E Y, there exist decomposition data y E dec,(h,j) with N 5 2. The proof employs the second equation of Theorem 6 . 2.2.1. Definition 21 extends in an obvious manner to the parameter-dependent objects. In other words W’(X^; A ) B ,9 = (- k , k ) , k E N \ {0},is defined as the set of all operator families A ( A ) = S(A) + S’(A) + F (A) + G(A),
(9)
( F + G)( A ) E %!kt G ( X A ; A ) Band ,
c
k- 1
S(A) =
W t J opyhj) (A)
w
j=O
for arbitrary hj(A)E %R;,(X; A ) , yj E dec,(hj, j ) , S’(A) of analogous form as S(A). For 9 = (0,O)we set
= I(,)
#(A) I(,) with $(A) being
%%l’^;A)(o,o)= t - d 2 M L : , ( R + x x ; A td2. )
Further W@((X-;A)=
nw(x-;A),.
keN
Since A (A) E %@(X*;A ) , implies A (A,) E Wp(X*)Bfor every fixed loE A , the Mellin symbolic levels follow immediately from the non-parameter-dependent case. In other words
241
2.3. Parameter-Dependent Cone Calculus
a$(A) (2, A) := o&4 (A)) (2) and similarly for a;. For of(A),a;(A) we do not use the definition of Wfl(X^),applied for every fixed A. This would lead to A independent homogeneous principal symbols. We take here the definition of parameter-dependent homogeneous principal symbols induced from t-dZML:,(R+x X ; A ) t d Z2 Wfl(X^;A),,
both in the compressed or the I ~ D sense. In other words we have the parameter-dependent interior symbol mappings a;: WZ”(X^;A),-S‘”’(T:X^x A\O), a;: nZ”(x-;A),-S‘fl’(T*X^ XA\O),
where a ; ( A ) (... ,z, [,A) = a;(A)(... ,t-lz, (,A), cf. ( l ) ,( 2 ) . Set
x rm;&x;A),
k- 1
G & ( X ; A ) , = rmt;O(X;A) x
j = 1
and let @ ’ ” ( X ^ ; A ) , c 2 0 & ( X ; A ) x, S ‘ ” ( T : F x n \ o ) be the subspace of all tuples ( ( h j ) ,{ h ; } , p } which satisfy the compatibility conditions G ( h j ) (z, 5, A ) =
1 9 7 ~ P
( ZX,’z, 5, A ) I I =
07
j = 0, ... , k - 1 , and analogously for h i , cf. also 2.2.1. ( 1 4 ) .
10. Theorem. a,
a; and (aM, a!) define linear surjective operators from Wfl(X^;A ) ,
to 2 6 & ( X ; A ) , , S(fl)(T:X^x A \O)
and GP((X^;A),,
respectively, 9 = (- k , k ) , 1 5 k 5 m, with the kernels
W $ + G ( X A ; A ) , , W f l - - ’ ( X ^ ; A ) , and W $ ; ; ( X ^ ; A ) , , respectively. Moreover we have the parameter-dependent analogue of 2.2.1. (28).
The proof of Theorem 10 for 9 = (- m , m ) is based on the corresponding version for 9 = (- k , k ) with finite k and the locally convex topology in WP(X^;A),. This follows in an analogous manner as in 2.2.1. Then we set
n
w ~ ( x - ; A ) ( -= ~ , ~ )W X - ; A ) ( - ~ , ~ ) keN
in the projective limit topology and show the surjectivity of the Mellin symbol mappings again by convergent series as in 2.2.2. The parameter-dependence only needs obvious modifications of the arguments.
248
2. Operators on Manifolds with Conical Singularities
11. Remark. The differentiation with respect to 1 induces continuous operators
D f : %’(A’*; A),++ R’-”’(X^;A),+ for all ) E W‘.
12. Theorem. %m(X”; A),+,8 = (- k,k), 1 5 k 5 , is a *-algebrafiltered by the orders, and %+; c(X*; A ) , , W C ( X ^ ;A), are two-sided ideals. On symbolic level we have the parameter-dependent analogues of 2.2.1. (15)-(19). Moreover the conjugation with I(”)pre-
serves the class %-(X^; A ) , with the analogues of the symbolic rules of 2.2.1. Proposition 9. Let us introduce the function p ( p , v ; 1 ) = (1 + 111)” for v z 0,
= (1
+ IAl)@-”
where p , v E W,v 2 p . Denote the norm of operators in .Y?(X’(X3,X‘(X^)>by
for v 5 0,
(10)
ll.lls,r.
13. Theorem. A ( 1 ) E %P(X^;A), satisfies the estimates
ll~(~)lIs,,-” 5 CP(P,V,A) for all A E W’, s E R, and evev v 2 p , with a constant c = c(s, v ) > 0 .
(1 1)
Proof. We have by definition %’(x^; A ) , C = t-“‘*ML@+ xX; A ) t Ill2 and it suffices to show that even the operator families T ( 1 )E ML”(R+XX;A) satisfy the estimates
IIT (1)Ilre(W.%rJ.%J - ’.“/Z(X) 5 CP (P7 ;1). But this follows by a straightforward generalization of the arguments in SUBIN’S monograph [S27] for Theorem 9.1 in Chapter 2. 0 14. Definition. An operator family A ( 1 ) E %”(X^; A),, 8 = (- k,k), 15 k 5 m, is called parameter-dependent elliptic if
0)
a ! ( A ) ( t , x , z , ( , A ) + O forall
t,x
O ~ t s m ,lz,(,AI+O,
(12)
(ii) the families of operators &(A) ( z , 1 ) , &(A)
(2,A): Hs(X)-+H’-”(X)
are buective for all z E r,,+ 1)12 and all 1 E A for fixed s E R.
As usual the bijectivity (13) for fixed s implies the bijectivity for all s. The families a L ( A )(z,A), &(A) ( z , A ) of yDO-s over X belong to L:,(X; r,,, 1)12 X A).From (12) it follows that they are parameter-dependent elliptic in the sense of the parameterdependent standard WDO-sover X,cf. also [S27], Section 9.1. Let a ( z ,1) = &A) (z,A) or = a L , ( A )(z,A), and let (ii) of Definition 14 be satisfied. Further let a(z,A) E %R:,(X)* for every fixed 1. Then a-l(z,A) extends from r(n+1)12 to some element in %RJ‘(X)* for every fixed A with a pattern of poles for varying 1as being assumed in the definition of %R:,(X; A ) . This is a consequence
2.3. Parameter-Dependent Cone Calculus
249
of 2.2.5. Theorem 1. In an analogous manner we can prove that in general a ( z , A ) E E ~ ; ~ ( X ; Aand ) a ( z , A ) : H S ( X ) + H S - p ( X ) bijective for all Z E f(n+1),2, A E A , implies a-’(z, A ) E Em;:(X; A). 15. Remark. For every p E R, 1 5 k 5 m, there exists an A (A) E %:(X^;A ) @ ,which is parameterdependent elliptic in the sense of Definition 14 and o J ( A ) ( 2 , A), o J ( A )( z , A ) E 91is(X)*for every fixed A E A and all j E N .
In fact, first we can easily construct a parameter-dependent elliptic operator A (A) with Mellin symbols of analogous sort as those for the operators b” at the end of Section 2.1.4. such that A ( A ) only depends on IA(.Then we can pass to parameterdependent holomorphic Mellin symbols by applying Theorem 6 and remove the remainders. This does not destroy the ellipticity for large 111. If we replace A by 1 LoI + 11I with IA. I sufficiently large, we get a family which satisfies Definition 14 for all A. 16. Theorem. Let A (A) E % ” ( X A ; A ) 89, = (- k , k ) , 1 5 k 5 m, be parameter-dependent elliptic. Then there is a B(A) E %2-”(X^;A)e which is a parametrix of A ( A ) in the sense
A(A)B(A)-Z, B(A)A(R)-zEW~(X^;n),. Moreover there is a constant c > 0 such that
A (A): Xs(X^)+Xs- @(A‘-)
(14)
is an isomorphism for all A, [ A ]2 c, and all s E R. ProoJ: The parametrix construction follows the lines of the proofs of the corresponding theorems in 2.2.1., 2.2.2. and contains no essential new elements in the parameter-dependent case. It remains to show the bijectivity of (14) for large IAI. From G(1) = B(A)A(A)- Z E and the strong decrease for [A[+ m of the ), s, r E R,it follows that Z + G(A) is invertible operator norms in 9 ( X S ( X ^ Xr(X-)), for large IA(in @(X-) as well as in XS(X^)for all s E W,since kernels and cokemels are independent of s. Let us show that N ( A ) = (G(A) + Z ) - l - Z E 91G(X-;A)fffor 111 2 const. To this end we write N ( 1 ) = -G(A) ( I + G(A))-l. Then both the mapping properties as well as the A behaviour of N ( A ) are inherited from G(A), cf. Remark 4. The same can be done for the adjoints, i.e., N ( A ) is as required. Thus A = ( I + N(A))B(A) is the inverse of (14). 17. Remark. The theory of this section has a straightforward generalization to operators on manifolds with conical singularities in the sense of 2.2.3. In other wordr we have the classes W ( C ; E, F ; A ) @ ,and in particular an analogue of Theorem 16.
Thus for every E E Vect (C) and every p that
E
R there exists an A E P ( C ; E, E)’ such
A : X s ( C ,E ) + X s - ” ( C , E )
is an isomorphism for all s E R. It suffices to choose a parameter-dependent elliptic family A (A) in W(C; E, E ; A ) which is point-wise (i.e. with respect to A) discrete, then to fix lo,lAol sufficiently large, and to set A = A (A,,).
250
2. Operators on Manifolds with Conical Singularities
C" Dependence of the Point-Wise Discrete Class 2.3.2. Next we talk about parameter-dependence of cone operators, where the parameter y varies in an open set Q E Rq. Here we neglect any specific behaviour for lyl + 00 but deal with operator families which are point-wise discrete in the sense of the dotted cone operators. For simplicity we consider the case R + x X ;operators over manifolds with conical singularities in general follow easily by the usual localization. First we have an obvious notion of C" families of cone operators based on the locally convex topologies in the spaces %fl(XA)a,cf. 2.2.1., 2.2.2. Then A ( y ) E C"(Q, % f l ( X A ) B implies )
G / ( A ) ( Y , z ) := ( Y ) ) ( z )E Cm(Q,mc,(x)> for a certain V, E 7r independent of y E 0. Moreover G ( y ) E Cm(Q,%G(X^)a)means
G E C"(Q, XE ( X A ) a@ r JSrez ( X - ) a ) for certain y independent carriers of asymptotics B, C E V-n'2. We have the spaces C"(Q, X7e$By(X^)d), B E Wn),A E 3,
and
Note that the decomposition properties for the spaces with continuous asymptotics as well as for the spaces of Mellin symbols have corresponding parameter-dependent analogues. For instance 2.1.1. (32) implies C"(Q, X y ( x ^ ) d )= C"(Q,
X(esyly(x-)d) + CW(Q,X7esy2y(x-)d)
(3)
whenever V, , V2E Yy@)with V = V, + V 2 .Moreover C"(Q, X ; q x - ) d ) = C"(0, X y ( X ^ ) d ) + C"(Q, Y e $ Y ( X ^ ) d )
(4)
for all VE V("). From 2.1.2. (18) it follows C " ( 4 rrnl;(X))
=
m;*(x))
C"(Q, , q ( X > )+ W Q ,
(5)
for V,, V2E 7r, V = V, + V2.Here we have used 1.1.5. Lemma 3. Further C"(Q, rrng(x))= C"(Q, m;"(x))+ C"(0,rnll,(X)). (6) The notion of C" families of cone operators with continuous asymptotics admits to define C" with a point-wise discrete behaviour. We can consider several variants. Below we then restrict ourselves to one of the cases. 1. Definition. C"(Q, % # ( X ^ ) i ) , B = ( - k , k ) , 0 5 k 5 00, is the subspace of a12 A E C"(Q, %fl(X-),) with A ( y ) E %fl(X^)i for every fixed y E 0.
The operator families in Cw(Q,%fl(X^);) have Mellin symbols and Green operators that are of a corresponding point-wise discrete character. The results of Section 2.2.5. suggest to consider subclasses of C"(Q, %fl(X^)i) with some control of the variation of multiplicities of poles and of the Laurent coeffi-
251
2.3. Parameter-Dependent Cone Calculus
cients with varyingy. This will be done below. For illustrating the character of our definitions we also generalize C"(l2, % p ( X ^ ) l ) . Denote by %fl(X)i' the subclass of those A E %c(X"), for which the carriers of asymptotics of Mellin symbols as well as of the Green operators are discrete sets in the complex plane. This means that the defining analytic functionals are carried by sets of the same types as for %p(X^)i but we drop the condition that the involved poles are of finite multiplicities. Remember that in the scalar case (i.e. dim X = 0) analytic functionals of that sort have been constructed in Section 1.1.5. (cf. 1.1.5. (8) for K = (Po}).
2. Definition. C"(0, %p(X^)i' is the subspace of all A E C"(0, %p(X^)8)with A(y) E %P(X^)i*for every fixed y E R Then, of course,
C"@* %qX-);)c C"(l2,
nqx-);.).
Now let us pass to a subclass Cm(Q %qX-),)*c C"(l2,
%qx-);)
in which we have the most precise information. Let us first generalize 1.1.5. Definition 5 in connection with 2.2.5. Definition 3. If H = L2(X),H" = C"(X),we get the sets n(n,Cm(X)). 3. Definition. C"(Q sP'(@) BXC"(X))* ir; the subspace of all (E C"(0, Se'(C)
BXC"(X))with (i) the carrier of ( ( y ) is contained in some V = V ( 0 E Y, for all y E 0, (ii) ((y) is for all y E l2 of the form
h E SO(@), and the set s ( 0 of all {(Y,Pj(Y), mj(Y), [njZik(X, ~ ) l k =,.... o m,(y)): Y E Q j E Z} belongs to fI(l2, C"(X)), [. , .] being the linear span of functions of x forfixed Y. Similarly as in 1.1.5. we denote by P(((y)) the set of pairs (Pj(y), mj(y)}j= ,,.,,(,,) which occur in S ( 0 for the pointy. By Cm(0,sP'(K) BXC ( X ) ) '
(8)
we denote the subspace of all (E C"(0, a'(@) BxC"(X))*carried by the compact set K for all y E R The elements (E C"(0, sP'(C) BXCm(X))*have the following property. Let yo E 0 be fixed and set (Pl(yo), . .. ,pN(yo)}= xcP(((yo)) n S: for any fixed y E R, and finite A E 3: Further let {ql(yo),..., qM(y0)) = x.,P(((y,)) n 3s;. Then for every E > 0 with dist (pj(yo), 3s;) > 2 ~ ,j = 1, . .. , N (9) we find an open neighbourhood
noof yo with noccn such that
N
M
j =1
k=l
N, :=
u {Z: E/2 < dist (pj(yo), Z)} n { s; \ u
(Z:
diSt(&(yo), 2 ) d S/2}
}
contains no element of x,P(((y)) for any other y E 0,. By construction Cm(X)),A , := (6 + E, 6' - E ) for A = (6, 6'). element in Cm(a0,SO'(S;) 0% 17
Schulze, Operators engl.
(10)
(In, is an
252
2. Operators on Manifolds with Conical Singularities
(clock-wise oriented) and N
M , = N , n U { z : lz-Pj(Yo)I>E).
(12)
j=l
Let A ( D , C; Cm(X))*denote the subspace of all C"(X)-valued functions g ( y , z ) , y E D,z E @, for which there is a [E C"(D,d ' ( C ) C3JnCm(X))*such that for any fixed Y E 0 d Y , z) E \ n c P ( t ( y ) ) )BnC r n ( X )
a(@
is meromorphic with poles at p j ( y ) of multiplicities m j ( y ) + 1 for all
j = 1, ..., N ( y ) , and further (C(Y)W> f
kw ) ) -
1
I d Y , w)f(z, w ) d W l , , M , E C"(fl0, d ( M e ) an ~"(X)) L.
for every system of data y E R, finite A E Y,yo E D,Do, E > 0 with (9) and the other mentioned properties. Note that for [ E (8) it follows a generalization of the space @)' that was defined in Section 1.1.5. 4. Definition. C"(0,d'(Sdy)BnCm(X))' is the set of all functions [(y):
D ' d ' ( S l ; ) BnC r n ( X )
such that (i) of Definition 3 holdr, (ii) [ ( y ) is for all y E D of the form (7) with some finite set I p l ( y ) , ...,P ~ ( ~ ) ( Yc) Sl; } and certain Ajk(x,y) E Cm(X);P ( [ ( y ) ) denotes the set of all pairs { ( p j ( y ) ,rnj(y)): j = 1, ... > N ( Y ) ) , (iii) for every yo E R and every E > 0 with (11) for N = N ( y ) there is an open neighbourhood Do of y o , noCCD, such that MY)
u { z : ~ /
j= 1
contains no element of n c P ( [ ( y ) )for all y (iv) the function
y
E
4,with
E
no,
the sum over all j with p j ( y ) E n c P ( [ ( y ) )n Sze, belongs to
&(Do, @; Crn(X))*. A(0,S; Cm(X))*denotes the space of all
y dependent families of meromorphic C"(X)-valued functions g ( y , z ) in the strip Sl; for which there is a [E C"(D, d ' ( S l ; )BnCm(X))*such that g ( y , z ) modulo Cm(D, d((S1)BnCm(X))has the form (13), y E D,where the sum is taken over all p j ( y ) E Sl;.
253
2.3. Parameter-Dependent Cone Calculus
5 . Defmition. Let R & Rg be open, s, y E R, A E 3finite. Then Cm(R,%;;(XA),)*denotes the subspace of all u ( y ) E Cm(R,%$"(X^)d) such that for some decomposition u = uo + u1 with uo E Cm(R,%>y(X")d), u1 E Cm(R,%T;y(XA)d),c& (4), the Mellin image of u1 belongs to &(a, S;'"); Cm(X))*. Clearly the condition is independent of the concrete choice of uo, u l . For A' 2 A we have
s Crn(R,%;;(X-),)*.
Crn(R,%;;(X^)J
This allows to define the dotted spaces also for infinite A , for instance, A = (-00, m), as
Let us set
Crn(R,%Sy'(X^)J
=
Crn(R,%;;(X-)J
=
cyo, %>Y(X^)J
Then VE
ur
n Crn(R,%;;(XA)J.
Cm(q% y ( X - ) J .
(15)
~(n)
As usual y will also be omitted for y = 0. If we introduce in Cm(R,%$y(XA)d)* a locally convex topology we can take in (15) the topology of the inductive limit and in (14) of the projective limit. The topology of Cm(R,%$y(X^)A)' is defined as follows. First we have a canonical embedding Crn(R,%$'(X^)J
4
Crn(R,%(e$YY(X")J.
(16)
Further u + u (y o ) induces a linear mapping
Crn(R,% y ( X " ) J
+
%;;(x-);
(17)
. we endow Cm(R,%;Y(X^)A)'with (cf. Section 2.1.1.) for every y o ~ R Then the weakest locally convex topology under which (16) and all (17) are continuous. The intuitive meaning of the functions u E Cm(R,%;;(XJd)* is that the asymptotics for t +O (and t + a) are of variable and branching character. Let us write u = u ( t , x, y ) , and consider, for instance, t +O, and A = (-m,m). Then the definition implies that for every y there exists an asymptotic type P ( y ) = { ( p j ( y ) ,m j ( y ) , L j ( y ) ) } j E E P'(n)(X),y ( n ) = y - n / 2 , cf. 2.1.1. Definition 10, such that m
u ( t , x,
y)
m,(y)
- C kC= O A ~ ~ (y)t-pj(y) x, logkt
for
t +O
(18)
j=O
in the sense of the notions of Section 2.1.1. The exponents p j ( y ) have to be numerated separately for every fixed y . They may constitute clouds of points with a rather chaotic distribution of the values of m j ( y ) .In simple cases the p j ( y ) form trajectories, where the m j ( y ) make jumps at the intersection points and the Ajk(x,y ) E L j ( y ) also make jumps. The character of the variation of the asymptotic coefficients A, was described in 2.2.5. Definition 3 (iii). For instance, let a, b E Cm(R),A, p E Cm(X)and set go(-% Y, z ) = A ( x ) ( a ( y >- z ) - ' ( b ( y ) - z ) - l . 17'
254
2. Operators on Manifolds with Conical Singularities
I I I
I I I
I
I
I
I I I
e
I I
I % -d.(n)
Fig. 10
The constructions of Section 1.1.5. show that there is another function g ( x , y, z) with a strong decrease as I Im z I + 00 and g o b , y , z) - g ( x , Y , z) E Crn(O,a(@) @n Crn(X>>.
Thus g ( x , y, z) has the same poles as go(x, y, z). Assume that the values of a ( y ) , b ( y ) are in Rez < 112 - y ( n ) . Then there is a u(t, x, y ) E Crn(O,XmsY(X^)) such that g = My(”)u for all X E X , y e n . The form of g shows even that u(t, x, y) E Cm(O,%;;”(XA))*. Let El = I Y E 0:a ( y ) = b(yN, Then the asymptotics for t +O are
Z2 = { Y E 0:a ( y )
- -A(x)t-“(”)log t for y Z1, u(t, x, y ) - A ( ~ ) t - ~ + ( yA) ( ~ ) t - ~ ( yfor ) &. u(t, x, y )
E
* Wy)}. (19)
(20) Although this is a very simple example we see how the character of the asymptotics If, for instance, 0 = R,and the values of a, b run jumps if we pass from Z1 to Z2. over circles in the complex plane, then the jumps are at the interesection points. In Fig. 10 we see two such configurations with intersections. If we look at (19), (20) it is not obvious that it allowed to differentiate with re(t, x , y ) has again point-wise discrete asymptotics. spect to y and that then (0;~) But this is true, indeed, and we then obtain higher logarithmic terms in general. YE
6. Theorem. The differentiation
0;:Cm(O,%5,y(XA),) +Crn(O,S 5 , y ( X A ) d ) induces continuous operators
0;:Crn(O,X>”(X^),j)*+ Crn(O,X>”(X^),j)’ for all multi-indices a E W.
255
2.3. Parameter-Dependent Cone Calculus
Proof: This result is an easy combination of 1.1.5. Theorem 6 and 2.2.5. Theorem 1, in particular the element of the proof which reduces the situation to a finite-dimensional effect. 0
7 . Defdtion. An operator family G ( y ) E C"(0, 'iRG(XA)A),A E 7!, C"(0, X G ( X ^ ) , ) * if it induces continuous operators D , " G ( y ) ,D,"G*(y): c"(0, .%'(X^)) + cm(O,% I s ( X ^ ) ~ ) *
belongs to
for all a E NQ.
It can easily be checked that the basic statements on Green operators of Section 2.1.5. have analogues in the present situation. In particular we obtain automatically continuous operators D , " G ( y ) ,D,"G*(y): c"(0, %?'(X-))+ c"(0, X:(X^)A)*
for all s E R. The next step is to introduce the dotted Mellin symbol classes. The definitions are similar to the dotted spaces with asymptotics. First we have an obvious analogue of Definition 3 for Cm(XX X) instead of Cm(X).In other words we get
Crn(O,d ' ( C ) @Jn C"(X x X))..
(21) Here C " ( X x X) is interpreted as L-"(X). The coefficients Ajk in (7) are to be replaced by Ajk(x, x', y ) , and the set n(0,Cm(Xx X ) ) can be expressed with respect to the reference space H = Lz(X X X),cf. 2.2.5. Definition 3. Then we also get a corresponding space A(0,C; C " ( X X X))*and an analogue of the mapping (10). Similarly we obtain C"(0, d ' ( S l ; )BnCm(XX X))*for every y E R and finite A E 3 which yields the space A(0,Sl;;Cm(Xx X ) ) * . Now we set for V E 7r C"(0, !Vt;"(X)l;)* = Cm(O,"Ot;"(X):> n &(a, S;(");Cm(Xx X))*.
The restriction map induces embeddings 8)'
4
C"(0,
m;"
(X)l;)*
for all A' 2 A. Then we define the y independent space
cyn, m;yx))*= n c-(o, rm;"(x)l;)*. A E 2
By definition we have embeddings CY0, Em;, (X>l;)' 4 C W , %m;"
and restriction maps h C"(0,
(X)l;),
h ( y o ) ,which give rise to
Ern;" (X)l;)' +"Oti8"(X)l; = lim
%Rim(X);
R E Z X X )
for every yo E 0. We endow the space C"(0,"Ot;" (X)l;)' with the weakest locally convex topology under which all these mappings are continuous. Then we also get the corresponding topologies in (22) as projective limits and
-
C"(0, "Ot~sm(X)>* = lim C"(0, YE
as inductive limit.
"y
roZ;" (X))*
256
2. Operators on Manifolds with Conical Singularities
8. Definition. Let 0 E Rq be open, p E R.Then Cm(QEm:,(X))' denotes the subspace of all h ( y ) E Cm(O,9t:s(X)) of the form h = ho + hl with ho E Cm(O,Em$(X)), hl E Cm(O,Em;sm(X))*, cJ: also (6). Further Cm(O,Em$sY(X))*denotes the subspace of all h which belong to Cm(O,Emt:(X)). According to
cy0, rmf,(X))* = Crn(O,Em,-6"(X))*+ C(0,Em$(X)) we take this space in the topology of the sum and Cm(O,%t:(X))* duced topology.
in the in-
9. Remark. The diflerentiation
0;:CYn, EDI:,(X))
+
%:m)
Crn(.n,
induces continuous operators 0;: Cm(Q 'iDt[,(X))* -+ Cm(n,EDI:,(X))'
for all a E Nq.
The definitions have a number of consequences which are more or less straightforward. For every h E Cm(O,Emt:(X))' we can form the operator family op$")(h) ( y ) which induces continuous operators opj;S"'(h) (y): CY0, x;;(xA))*-+ Crn(O,xs,,KY(X))* for every s E R. The concept of decomposition data in the sense of 2.2.1. Definition 19 also applies to the dotted Mellin symbol families.
1O.Defition. Let h E Cm(O,Em'&(X))*,j E IN. Then dec,(h, j ) denotes the set of allfinite sequences Y = {OV, ev)lv, ,,,,N , N arbitrary. with - n12 2 ev2 - j - n12 for all v , h, E Crn(O,n;,Q.+ "12(X))*, h =
c
h,.
V
c N
We write oph(h) =
op%(h,), y E dec,(h, j ) , where opb is understood as usual.
v= 1
Note that there is an analogue of 2.2.1. Proposition 20 for the dotted Mellin symbol families. It is based on
cyo,Em;s(x))* = Crn(O,rm;sY(x))* + C m ( qm$:(x))* (23) for any choice of y, 6 E W with y f 6. Thus with every h E Cm(O,Emcl,,(X))*,y E dec,(h, j ) , we can associate the continuous actions w , t j o p h ( h ) (y) 0 2 : cy0, xs,,(x)>*+ Crn(O,xs,;yX)y wl, w2 being arbitrary cut-off functions.
(24)
11.Proposition. Let h E Cm(O,Em:,(X))*, and y, q E dec,(h, j ) , wi,Gi be arbitrary cutoff functions, i = 1 , 2 . Then w , t j o p ; ( h ) (y) 02 - G1tjop;$(h) (y) G2 E Crn(O,9Ir,(X^))+ P(0,%,(X*))*.
257
2.3. Parameter-DependentCone Calculus
Note that also 2.1.5. Theorems 1 3 , 14 have corresponding analogues in the present situation. can also be orThe parameter-dependent Mellin actions for h E Cm(D,n:,(X))* ganized as follows. Let j E N \ {O}, and fix yo E 0.Then there is a eoE [- n/2, j - n/2] with h C y ) E '9t!&po+ "'(X). for all y in an open neighbourhood Doof yo. If we do that for every point in D we find a locally finite covering of D by open sets f & , f i k c C o , and & E [ - n n / 2 , j - n / 2 ] , such that h ( J ' ) E ~ ~ ~ + " ' ( Xfor ) * allyEf&. and consider the operator family Let {vk}be a partition of unity belonging to (Dk}
w,tic
Q)kCy)
k
opz(h)Cy) w2: C m ( 4xs,,(x))*+ C m(q%",'(X))'.
(25)
Then (24) equals (25) modulo Cm(D,%$(X*))+ Cm(D,!XG(X*))*.For unifying the procedures we do not rely on the latter possibility of defining Mellin actions with parameters but it looks a bit simpler, since we only employ the point-wise discrete character and the moderate variation of the poles under varying y . 12. Definition. Cm(D,%#(X"),)*,,u E R,8 = (- k, k ) , k E N \ {0}, is the set of all opera tor families
ACy) = SCy) + S'b)+ FCy) + GCy), where FCy) E Cm(D,% $ ( X J 6 ) , GCy) E Cm(D,%G(Xa)6)', and
c
k-1
SCy) =
wtjop;(hj) w
j=O
with arbitrary hj E Cm(D,EDz:,(X"))', yj E dec,(hj,j), and some cut-ofl function w, and S'(y) = P"S"((y) I(,) with S"(y) being of analogous sort as S(y).
The results of the Sections 2.2.1., 2.2.2. have a straightforward generalization to the present parameter-dependent case. Every A E Cm(Q %.(X"),)* induces continuous operators
A : C-(D,sles,,(X76)'+ Crn(D,res,;'(X")J
(27)
for every s E R , 9 = (- k, k), 1 5 k 5 m . The parameter-dependent symbolic levels are
6:cm(q%'(X7,)+
(72,
(7J:
cyo, S@)(TZx^ \O))
Crn(0,%'(x"),) + Crn(D, rmf:(X>)*
k-1
x
)( Crn(D,rn&(X))* j=1
The method of introducing locally convex topologies in the cone operator spaces has an analogue also for Cm(D,%#(X"),)*, 1 5 k 5 00. Then
0;:Cm(D,%#(X*)6)'+ Cm(D,%'(XA),)* is continuous.
(28)
258
2. Operators on Manifolds with Conical Singularities
13. Theorem. A E Cm(Q %"(X"),)*, B E CW(Q%'(X"),)* implies A B E Cm(Q %"+"(X"),)*, A * E Cm(L?,%"(X")~)*,I(")AI(")E Cm(ll,%"(X"),)', 9 =(-k,k), k E (IN\ {O})U{m}, where the operations are understood point-wise, and we have the same symbolic rules as in 2.2.1. Theorem 8 and Proposition 9. An operator family A E Cm@,%"(X")8)' is called elliptic if A(y) satisfies 2.2.1. Definition 1 for every y e a . B E Cm(Q %-"(X38)' is called a parametrix of A if AB - I, BA - I E Cm(O,ill&"),)*, cf. Definition 7. 14.Theorem. Let A E Cm(0,% p ( X ^ ) , ) * , 9 = ( - k , k ) , k E (N \ {0}) u { w } , be elliptic. Then we have point-wise the results of 2.2.1. Theorem 14, but we can choose a parametrix B E Cm(O, %-p(X"),)*. Thus Au = f E Cm(O,%;s(Xa)8)*, u E Cm(ll,T m ( X " ) )implies u E Cm(L?,% ; i P ( X J 8 ) * ,for every s E W.Analogous results hold for the classes of Definitions 1, 2. The proofs of the Theorems 13, 14 are left to the reader as exercises. For The%Jt;:(X))* as orem 14 one has to use in particular that the bijectivity of h (y) E @"(a, implies h-l(y) E Cm(Q operator family H s ( X ) + H s - " ( X ) for all z E r(n+1)/2 %Jt;:(X))' (cf. also Section 2.2.5.). Analogous results hold for the versions Cm(12,W(X");), C"(Q W(X");'),based on the corresponding spaces
Cm(L?,X;,(X");) and
Cm(0,%;&X");*)
respectively, as well as on the adequate C" families of Mellin symbols. Finally an evident modification of Theorem 14 also holds for Cm(12,W(Xa),). As mentioned in the beginning all definitions and results on the branching asymptotics extend to arbitrary manifolds with conical singularities. Let C be the corresponding stretched manifold. Then we have the spaces
cmm%SYC).4)*, A
= (401,
Cm(Q %:$Yc))*,
6 s 0, as well as the corresponding spaces of Green operators, further C m ( 4W C ) d * ,
cm(a%YC))*,
8 = (- k, 01, k E (N \ {0}) u corresponding analogue.
2.3.3.
{m}
, and so on. In particular Theorem 14 has the
Branching of Asymptotics Near an Edge
We want to apply the notion of varying branching asymptotics to differential operators on manif o l h with edges of dimension q . Here we consider the local situation of a wedge of the form
cxa,
(1)
R S Rq open,
C a (stretched) manifold with conical singularities, cf. Section 2.0. The base X of the cone is assumed to be a closed compact C" manifold. We also might admit X to be compact, C", with boundary. This would be a proper wedge and the calculus would apply boundary value problems for the cone, which has an analogous structure as the theory of 2.2.,
but with Boutet de Monvel algebra valued Mellin symbols along X,cf. [R7].The consideration for boundaryless X will show that this generalization is then straightforward.
2.3. Parameter-Dependent Cone Calculus
259
Let DiffP(C X 0) be the space of all differential operators over (int C) X 0 of order p which are close to the edge R in the coordinates (r, x, y ) E X " X R of the form
with Nub)E C"(0, DiffP-lul(X")), cf.2.2.1. Definition 18. Since we are interested in the behaviour for r + O , the part for t+ 00 will be neglected. A problem is the elliptic regularity in spaces with asyrnptotics if we do not assume constant positions of zeros of the principal Mellin symbols with varying y along the edge 0. We shall solve it here under the assumption that the solutions are C" along 0 and the operator elliptic along X^,close to the edge, cf. the definitions below. The corresponding Sobolev space analogue is more complicated and will not be presented here. It needs a stronger notion of ellipticity anyway (cf. also Chapter 3). First we want to give a motivation for studying operators of the form (2). Assume that we are given a differential operator
d(f,Y, &, 0,) =
I.l+
c
a,d5, y ) D@;
181f P
in RZt X 0 with c" coefficients. If we restrict d to Rnc \ (0} 2 R, X S" we can interpret A" as an operator on the "wedge" R, X S " X 0. By inserting polar coordinates (t, x ) = 5, x = i?//lfl, r = 121, it follows
with certain caP(x, 0,) E Difflfll-p(Sn).Thus
6= 1.1
181
1 asu(r,x , y ) r - l o l 1 &(x, 1815
+
y ) . After rearranging the last sum we obtain
aPu(r,x, y ) := a,&,
A"=
0,)
p=0
P
1 Na(y)(tDy)u
t-'
1.1 B P
with certain y dependent differential operators of the form
with KupE Cm(i?,xO, Diff'-I"I-P(S")). The weight factor t-' is not essential here and will be omitted in the calculus. Then we get just (2) for X = Sn. Let us choose cut-off functions wl(r), w2(r) which are also regarded as functions on C x 0. If A E DiffP(C x 0) then
can be interpreted as an operator on X x 0, and we have
w,N,(.v) cf.2.3.2. (26).
w2 E
C(0,% P - l q X - ) ) * ,
(4)
260
2. Operators on Manifolds with Conical Singularities
1. Dewtion ‘W(0)’denotes the set of all formal series of the form
where m
with A’, E wlCm(fl, %”-lRl(X*))*w2 for all j , a, c-ord A’,(y)
- j - la1 for all j ,
a, y,
(7)
(cf Section 2.2.1.). By omitting the dots in (7) we get by definition Fmfi(0). Then F m P ( 0 ) ’ c Fm”(0). (3) can be interpreted as an element of
if we write wlNR(y)w2 in the form
FmP(0)‘
c wl(cjt)tjN’,(y) w2(c,t) m
wlNR(y)w2 =
mod C(0,%$-lu1(XA))
j-0
with a sequence of constants cj, increasing sufficiently fast for j + m , and N’, obtained by Taylor expansion of the coefficients with respect to t at t = 0 (cf. Section 2.2.2.). Clearly the sum with respect to j may be taken finite, j = 0, ..., p, since the remaining sum is simply the remainder of the Taylor formula which is of the required flatness order. Set
U”(A)O,t])=A051), UA(Fm’(0))
= {U’(A):
A E Fm’(0)},
and similarly with dots. The following remarks hold for the classes without and with dots. uA(Fmm) is obviously an algebra with respect to point-wise composition (i.e. (y, t])-wise and composition of formal power series in t]) and also with respect to the Leibniz product
1
a 0 r t ] ) # b 0 ! t ] ) = CR a l ( ~ : a 0 1 , t ] ) ) a , * b ~ y , t ] ) .
Both with respect to point-wise and to # composition we may talk about invertibility in a* mm). The elements in Fm,(0) := {AE Fmm(0): Ah E Cm(O,%,(X”)) for all j , a ) (8) are called Green elements of Fmm(0). a, of (8) is a two-sided ideal in aA(Fmm(0)) both with respect to point-wise and # multiplication. In W(0)we have also an algebra structure with the composition m
m
m
m
that treats Dy as a usual differentiation in y. It is then clear that UA( A B ) = ( A1# UA( B ) for all A, B E Fm”(0).Moreover Fm,(0) is a two-sided ideal in Fm3-”(0).
2.Definition. An element A E Fmm(0is) calledproper if there exist sequences NU,a ) ,MU,a ) E N tending to m asj+ laJ+m, such that A’,,,) = tN(i.dNj&) tM(i.e) for all j , a, with certain Nj, E Cm(O,%fi-lal(X”)).Let us call NU, a ) , MU,a ) the flatness orders of A’, and (Ah)*, respectively.
261
2.3. Parameter-Dependent Cone Calculus
For our purposes it would be sufficient to assume NG,a)= MG, a)for all j , a,but the maximal numbers with the mentioned properties are different in general. They describe halfplanes of holomorphy in the Mellin image after the actions of A; and (A',)*, respectively. Without loss of generality we may assume ~ ( ja) ,5
~ ( j&),; MU, a)5 ~ ( j&);
whenever j + la15 J + JEl.In this case we talk about monotonic flatness orders. Observe thatfor every A E mmthere exists an A which is proper with monotonic flatness orders and A - A E !IDG. The proper elements in Bm form a subalgebra W. As mentioned all assertions hold analogously also for the dotted subclasses. 3. Delinition. An element A E F ~ ) " ( Ris) called elliptic along X Afor small t E Cm(R,%"(X")) is elliptic for a l l y E R and small t, which means
b boectiue on
r("+ and aE(aA(A)(Y,
a,,(A)(y, 0 )
0 ) )( t , x, T,0 # 0 for 0 5 t 5 E with some E > 0.
4.Delinition. An element P E W r ( R ) is called a parametrix of A cut-off functions G,, G2with GIG2= GIand Gl(AP- l ) & ,
if
E
B @ ( Rfor ) small
t
if there are
Gi(PA-l)(j,E'IDG(O).
For A E W'(O)* we use an analogous definition with 'ZBG(0)*. For abbreviation let us talk about ellipticity along X ^ and parametrix. It always means that for small t .
5.Remark. Let A E %@(R) and P E TJl-"(R) be a Earametrix. Then we can ahvayspass to a proper parametrix. r f A is proper then Gl(AP - 1) G2,Gl(PA - 1) Gz E loG(R) are also proper. 6. Theorem. A E 'W(R) is elliptic along X ^ iff it has a parametrix P E W r ( l 2 ) . An analogous result holds for the dotted class.
Proof. The existence of a parametrix P is equivalent to the existence of a solution of a d ( P ) # 0, ( A 1 = 1 mod UA
close to t
= 0,
(9)
where
with operator families P$J) being of the required orders. In fact from (9) it follows GIP!(y)A;(y) G2= GImod %,(X*) for every y E O,i.e., P!(y) is locally close to 0 a parametrix in the sense of the cone algebra W ( X "). Thus the existence of P implies the ellipticity along X ". Conversely the ellipticity of A along X " leads to a parametrix P by solving (9). The first step is to construct a point-wise parametrix of aA(Ao)(y,q ) , cf. ( 6 ) , i.e. a parametrix in %-"(X") for every fixed (y, q ) E T*O. It is clear that we find a P;(y) E C"(O, %-@(X")) which is a parametrix of A\(y) for each y E 0,cf. 2.3.2. Theorem 14. Now we make the ansatz
and solve the equation
for small t .
262
2. Operators on Manifolds with Conical Singularities
Equations of this sort in the proof mean the corresponding equivalence after multiplying by cut-off functions from both sides. We obtain
P$J)A%J)
=
-
C
P;ol)AO,cV) mod%dX"),
a+B=y 181< I Y I
y
#
0, and hence we may take
Since PE(y) E Cm(Q%-@(X")) and P;(y) E Cm(O,% - p - l B l ( X * ) ) , get pO,(y)E Cm(O,%@-l~l(X")). Moreover c-ordPO,(y) = c-ordP;(y)
+ c-ordAt(y) + c-ordP!(y)
A:(y)
=
E
Cm(O,% p - l a l ( X " ) ) ,
we
-Is[ - la1 = -1yI
as required. Next we solve the equation P ( y , q ) # A ( y , q ) = 1 modulo Green terms by the an-
satz
A being written in the form (5). This gives rise to
modulo Green terms, i.e.
Po(y,q) was already established, and we have to check the orders of P b ( y ) in P ' ( y , q ) = Pb(y) qb. This can be done by induction. We have P1(y, q ) in the form
c B
i.e.
cys are constants determined by D;qd = cyaqd-Y. From
ordP: = - y - 161, ordA! = p c-ordPf= - k - l 6 l ,
- lal,
ordP:
c-ordAl,= - j - l a l ,
= -p
- [el,
c-ordP:=
-lei
then it follows immediately ord P; = - p - 181, c-ord Pb = - 1 - IpI. The construction for the dotted class is analogous. 0 Now we consider a differential operator A E Diff@(CX 0).As noted above w, Diff'(C x 0)w2 c W(0)'.
263
2.3. Parameter-Dependent Cone Calculus
A', E Cm(O,% ~ - ~ a ~ ( X ^ ) )c-ordA',(y) *, i- j - la1 for all j , a,y . A E D W ( C x 0)is called elliptic along X^ close to the edge if (10) is elliptic in the sense of Definition 3.
I . Theorem. Let A E DiffP(C x 0)be elliptic along X^ close to the edge. Then Au
=f
E Cm(O,Xis(C))', s E R,
and u E Cm(O,X?(C)) imply wu E C=(O, Xi:.(C))'forsome cut-offfunction w. Ifthe asymptotic &pe o f f is given the resulting one for u can explicitly be expressed in terms of that o f f and the components of d A ) ( Y , z) = { ~ ( A ( Y () Z) ) I , ~ N .
Proof: For abbreviation we denote (10) again by A ( y , 0,).Applying Theorem 6 to (10) we find a parametrix P E !E-.(fl)*which is proper, cf. Remark 5 . It suffices to consider the equation Au = f E Cm(O,Xis(X*))*. By composing from the left by P we get modulo cut-offs (with the same abbreviations as in the proof of Theorem 6 ) PAu
= (1
+ G ) u = w,
(11)
where G E !En,(0)' is proper. Write P and G in the form
Then we know from the preceding section that P ; ( y ) : C"((1, X:s(x-))*-.3
c-(q X::fi(x-))*,
Gh(y) : Cm(O,X - - ( X ^ ) )-+ Cm(O,.%rs(XA)).
for all j , B, I, y, and in addition P:(y)D,8f(y) = t N ( k 8 a , w ( t ) f i ( ~ ) I G h ( y ) D Ju ( y ) = t M ( L y h ( t ) u l ( y )
11 tending to i$ln$y as k + + m and 1 + IyI --+ for certain N ( k , 8 ) 5 N(k, B) for k + 5 k + Is[,and similarly for M(...), f l E CyO, X::r(X^))*,
u1 E
C-(O,
m
respectively, and
n;s(x^))*
It is clear that the concrete asymptotic types in f l , u1 can explicitly be calculated, though it is sufficiently complicated. For fi we have first to employ 2.3.2. Theorem 6 and 2.3.2. (27). The nature of the asymptotic type of Gb follows from theorems of the type 2.3.2. Proposition 11 that have repeatedly to be applied for evaluating (11). The asymptotic types in P are known if we calculate those of P : ( y ) . They are determined by the inverse of aL(uA(A)( y , 0)) (I).Now let u E Cm(O,%'(X^)) for some r E R.Choose an L E W and set M
Then we obtain (modulo cut-off's)
c M
where PLf
=
k181=0
c t'(hY)wuI
L+.
t N ( k f i w f i ,GLu =
llvl=O
and RL is a remainder in B-'(O)* with
264
2. ODerators on Manifolds with Conical Sinaularities
flatness order 2 p ( L ) of the components where p ( L + 1) a p ( L ) for all L, and p ( L ) + m as L + m. Further RL is a finite sum
RL(Y,4) =
C
RL,~(Y)D,P,
1PId L + P
RL,@E Cm(O,! R - l ( X * ) ) for all L, Q. We obviously have PLf E C"(4 %?i:"(X*))* and the asymptotics of PLf and PL+ coincide in ( ( n + 1)/2 - N(L, 0) < Re z < ( n + 1)/2], interpreted as a strip in the Mellin image. Moreover GLu E C"(0, %i(X*))', where the asymptotics of GLu and GL+1 u coincide in the strip { ( n + 1)/2 - M(L, 0) < Re z < ( n + 1)/2]. Finally RLu E C ( 0 ,rer+l ( X * ) ) is flat of order p ( L ) . From (12) it follows
+ RL ( C m ( 4%r(X"')]E Cm[4z;g(x-))*
u E C"(0, %::'(X*))*
+ R L I C Y 4 %?"(X*))I,
(13) Y = min {s + p, r + I}. Let z be an arbitrary element in the semi-norm system that defines the topology of C"(0, .!%?;g(X-))',Then there is a k E N such that n only refers to the values of Mu (the Mellin image) in the strip ( ( n + 1)/2 - k < Re z < ( n + 1)/2}. In view of k(L)+ m as L + m from (13) for L large enough we obtain n(u)< m. This yields u E C"(0, %?&(X*))*. In particular we have u E C"(0, S r + l ( X * ) ) when r + 1 5 s + p. By iteration of the conclusion we , i = m i n ( s + p , r + 2 ) and so on. This yields finally U E get u ~ C " ( 0.!%?:g(XA))', C"(0, 2;:'(X*))*. 0 Let us discuss a bit more the intuitive idea of Theorem 7. If A E Diff'(C x 0)is elliptic on (int C) x 0 in the usual sense and in addition elliptic along X* close to the edge then we would like to have a result which says that f~ {some edge Sobolev space of orders with asymptotic~]implies the same for the solution u, with transformed asymptotics and order s + p. This requires further investigations. The edge asymptotics of distributions in edge Sobolev spaces depend on the order, and the whole functional analytic background has to be developed first. This will be done in Chapter 3. Moreover it is adequate to pose additional trace and potential conditions with respect to the edge for obtaining Fredholm operators, cf. Section 3.3.4. On the other hand Theorem 7 is more general and gives some insight into the relations between conormal orders of the coefficients of DJ and the step by step procedure for evaluating the lower (conormal) order terms in the parametrix. The fact that the point-wise discrete branching (or smoothly varying) asymptotics are inherited by the solution from the right hand side is by no means evident even in the case of C" along the boundary. The patametrix is no local operator and, as an "integral operator" with respect to y, it might smear the y dependent distribution of discrete asymptotic exponents over large sets in the complex plane. A characteristic element of the structure of o,,('lB*(0))is that the Mellin symbols are polynomials in 7. We shall see below in 3.3.2. that this is a rather typical behaviour also in the Sobolev space description. For A ( y , q ) E uA(?W(0)) we have
(cf. (5)). Let us identify for a moment the complete Mellin symbols with formal power series in q and a further variable A, where the exponent of A indicates the conormal order. Then m
265
2.3.Parameter-Dependent Cone Calculus This yields
and thus
Note that the sum over y is finite. The combination of the Leibniz product and the Mellin translation product could have been used above for determining the (unique) sequence of Mellin symbols of a parametrix P of A. The “operations” associated with such sequences of Mellin symbols only contain differentiations in y which are connected with a corresponding conormal order. The higher the order of differentiations the more on the left in the complex Mellin plane are located the associated contributions to the asymptotics. The picture becomes more and more complicated with growing order of differentiations but the point-wise discrete behaviour of the asymptotics is preserved. We want to construct examples, where the branching of asymptotics can be observed explicitly. It sufices to look at the leading Mellin symbol a b ( A : ( y ) )(z) in the notation of (6)and to have cases where the points of non-bijectivity are just of the branching behaviour. Let us suppose y E 0 c R’.If g is a Riemannian metric on X we set B ( g ) = ( t a i a t y + (n - i)ta/at + a(g),
(14)
A(g) being the Laplace-Beltrami operator on metrics g,, g2 we can consider the equation
X associated with g. Then for two Riemannian
=f.
(15)
( B ( g 2 )+ t 2 a ; ) (B(gl) + t2a+
The leading Mellin symbol of (15) equals (z2 - (n - 1)z
+ A(g2))(z2 - (n - l)z + A(g,)).
(16)
We can assume that g,, g2 depend explicitly on y, for instance, in the form gi(y)= pi(y)gwith some fixed Riemannian metric g and strictly positive pi E Cm(0).From the solutions 2.2.3. (31) there follow the non-bijectivity points of (16) by
A being any eigenvalue of A(g). It is obvious now that the example can be arranged in such a way that Z L ~ Y O = ) Z Z , * ( Y O ) for Some YO but Z I , + ( Y ) Z Z , + ( Y ) for all Y * Y O , I Y -YOI < e for sufficiently small. In fact, we can assume pl(yo) = ~ Z ( Y O ) , pl(y) p2(y) for y * Y O , lY - Yo1 < 8. In Section 3.3.6.we shall obtain further examples that follow from another point of view.
*
*
3.
Operators on Manifolds with Edges
A "manifoldnwith edges (i.e. a "wedge") is a topological space M containing a subspace Y such that (i) M \ Y is a C" manifold of some dimension p , N
(ii) Y =
U Yj for some N,
j-1
& n 5 = 0 for i + j , where Yj is a closed compact C"
manifold of dimension qj < p , (iii) every y E Yj has a neighbourhood U c M for which there is a homeomorphism X:
U+KjXR
(1)
for an open R E RU and K j = [0, 1) x X j \ {0} X X j for some closed compact C" manifold X j (dim Xi= p - qj - l),where x restricts to diffeomorphisms
U \ Y, + (0, 1) x X j x R, U n Yj +a, and if o c M, j?: o + R j X 6 is of analogous sort, y E 0, then Kj = Rj, and the transition mapping x : K j x (0n 6)+Rj X (nn 6) is assumed to satisfy x(At, x, y ) = Ax(t, x, y ) whenever 1 > 0, 0 < t < 1, At < 1. This is required for j = 1, ..., N. Clearly the bases X j of the cones Kj for different j may be different. It is not our aim to perform here a deeper geometric discussion of the wedges M. Intuitively the 5 are edges of dimensions qj and the local models are (cone} X edge, where the bases of the cones are closed compact C" manifolds. The definition can easily be generalized to the case when X j is compact, C", with smooth boundary axj. This corresponds to manifolds with edges and boundaries. We are not necessarily concerned with usual C manifolds. M may be, for instance, obtained by gluing together a finite number of unit balls in Rm along their boundaries by choosing identifying diffeomorphisms y j : S m - + S"- l, j = 1, ... , M. Simpler examples are given in Fig. 11. The space in (i) is the surface of a lense shaped space, (ii) means the surface of the cylinder over the disk\ (smaller disk}, (iii) is a bounded domain in R" with C" boundary, (iv) means the cone over a torus, multiplied by a sphere S4. (v) means a system of one-dimensional intervals connecting a finite number of points in R", multiplied by S'J. Of course, Sq might be replaced by any other closed compact C" manifold Y. Further examples follow by considering a closed compact C" manifold M and taking for Y the union of disjoint closed compact C" manifolds Yj, dim 5 < dimM. Similarly to the case of conical singularities it is useful to pass to the stretched manifolds, where we take instead of (1)
[O, 1) x xj x 0. Often we then only look at (0, 1) x X j X R.
267
3. Operators on Manifolds with Edges
(ii)
(iv)
(V
1
Fig. 11
The calculus in this chapter is devoted to the analysis of adequate spaces of yrDO-s over (the stretched) wedges, acting in adapted Sobolev spaces, and to the concept of ellipticity. The main analytic difficulty is that the edges cause degenerate operators with a typical edge degeneracy. For instance the Laplace-Beltrami operator on (R+XX) x R4 with respect to the metric dt + r 2 g + dy2, g being a Riemannian metric on X, is of the form
with p
= 2,
certain A j E Cm(R + ,Diff’-J(X)), and B E Diff#((Rq), in this case
4
D:;. Edge degeneracy means that the differentiation with respect to y occurs
B= i- 1
in the combination tDy close to t = 0. The weight factor t-” in front of the operator plays a more significant role in the edge theory than for the cone, cf. 2.0. (2). This type of degenerate operators is also of independent interest. For sake of simplicity the analysis for wedges will be performed for the case when the base X of the cone is a sphere. The general case shall be dropped in this book. It may be found in [S17]. 18 Schulzc, Operators cngl
268
3. Operators on Manifolds with Edges
3.1.
Preliminary Constructions
3.1.1.
Motivation of the Approach
The general approach for a theory of operators on manifolds with edges may start with a reformulation of Sobolev space norms and differential operators in Rn+ x Rq 3 ( u , ~ ) Then . we regard (Rn+\ {0}) X I R q as a special local model of a manifold with q-dimensional edge, based on the cone It"+ \ {0} 2 R+X S". Let us fix a function [ q ]E Cm(RQ) with [q]> 0 for all q = (m,... , qq) and [ q ] = Iql for 1~12 1. Let h(v, 7 ) be a function of (v, q ) E R"+' X Wq and define
(1)
(x?)h)(v, ?#l)=1("+')/2h(lv, q ) , 1>0. Moreover we set
(x(")(rl)h)(v, 7)= b[;/h ) (v, 3) * For fixed n we also set for abbreviation xn = x y ,
(2)
x ( q ) = x'"'(q).
+L2(R"+1) is a group of unitary operators on L2(Rn+l) Note that .q: L2(Rn+') with the standard scalar product (f, g) = f(u)g(v) du. For the partial Fourier transform F,,, we have x,Fu-,f(v, Y ) = Fu+vxil f(v, Y ) . We define the norm on the Sobolev space Hs(R"+ X
IIUIIHS((R~+~~R~=
Rq), s E 112
{J(IvIz+[~12>sIFu(v,q)12 dvdv}
F being the Fourier transform in 1. Lemma. Set
(3)
It'+"+q.
,
R,by
269
3.1. Preliminary Constructions
The underlying idea of reformulating the Sobolev space norm in R'+"+qis to regard a hypersurface as an "edge".Then the norm far from it remains untouched if we modify the "cone space" Hs(R"+') close to the "vertex" 0 E R"+ in an appropriate way, cf. formula (16) below. Anisotropic descriptions of the standard Sobolev spaces are useful also for higher order singularities. The Sobolev space norms correspond to the homogeneity of differential operators in It1+" + q with constant coeficients
which is a family of differential operators in R"+ parametrized by q. Then it is easy to see that
a(Aq) = A'XA a(q)x;' for all A E R + . Thus a ( q ) = [ q ] " x ( q ) a ( q / [ q ] ) x - ' ( q ) for all q
(5) E Rq.
2. Lemma. Let a(7): R;
-+
n~(HS((R"+~),
~ s - q ~ n + i ) )
seR
be a continuous operator function which is homogeneous in the sense ( 5 ) for all q E Rq, )qI 2 c with a constant c > 0 . Then Au
= Fq-!,y(a(q)Fy-v
(6)
A 2 1,
u),
fimt defined on Cr(R;, Hm(R"+ l)) , extends to a continuous operator A : HS(R1+"+q) +HS-'(R1+"+q),S E R . Proof. Let
Il.lls be the norm in HS(R"+').From Lemma 1 then it follows
I I A u I I ~ J - P (=~ ~ + ~ + ~ l ~ x ~ ' ~ ~ ~ F y ~ , , ~ F ~dq~ y a ~ ~ ~ F y ~
J [a1 - 11%- ' ( ? ) a ( q ) F y 41:- 'dq = J [vlZsIla(~~[tll>x-'(tl)Fy-bvullf-, di. =
z(s
')
-bq
(7)
Using Ila(ql[q])f(q)ll:-,, 5 const Ilf(q)ll: with a constant independent of q the right hand side of (7) can be estimated by
ullf dq = C O ~ S ~ I I U I I Z H ~ ( ~ , + "0 +~~. const J [q]2sl[x-1(q)Fy+,, Let us give a more general example of an operator family satisfying the conditions with of Lemma 2. Consider a function a(v, q ) E Cm(R1+"+q) a (Av, A q ) = L'a (v, q ) (8) for all 12 1 and Iv, q1 2 const. Let a ( q ) be the associated family of yDO-s with respect to the v-variable
a ( q )f ( u ) = 18'
JJ e'("-")'a(v,
q )f ( u ' ) du' d v .
270
3. Operators on Manifolds with Edges
Then n ( q ) has obviously the property (6). Now let us insert ~ ( A u ’ ) .Then a(Aq)f(A.) = J ei(u-U‘)va(v, Aq)f(Au’) du’ Uv
uv
=
J ei(u-l-W*a(v, Aq)f(u’)A-(n++”
=
J ei(hJ-u ‘ ) l - ‘ v a ( v , Aq)f(v’)A-(n+ 1) du‘ dv
-
1ei(lU-%I (Av,A q ) f ( u ’ ) du’ Uv
= A#
du’
fl ei(Au”-u’)va(v, q ) f ( v ’ )du’ Uv
for 1 2 1, 1qIz const. Thus a(Aq)x,= Apxla(q) for A 2 1, 1711 2 const. 3. Remark. Let a(u, d , Y , 9 ) E Cm((Rn+\ {0})2x R1’ “ + q ) satisfv (8) for every fixed u, u’ for all 1t 1, IY, 91 t const, and
*0
a ( A u , v ’ , v , t ] ) = a(u,A’u‘,v,q) = a(u,u‘,v, 8)
for all 1,A‘ E R+. Then the operator family a ( 9 )f ( u ) = j ~ e i ~ U - u ’ ~ ’ a ( u , u ‘ , ~ , ~ ) f ( u ~ d u ’ c t ~ ,
(9)
f E Ct(R”+* \ (O}), satirfies the homogeneity condition (5) for 1b 1, 191 2 const.
Now let us reformulate (4) in another way. By inserting polar coordinates ( t , x) = u in lRnt \ {0}, t E R+,t = Ivl, x E Sn, we obtain
with certain caf E Difflal-f(Sn)(Diffj( ...) denotes the space of all differential operators of order j on the manifold in the brackets). For (4) we then obtain
bapl(x,D,)= aapcgf(x,0,). Define the operator family t-pb(tq) acting in the variables ( t , x) by
which corresponds, of course, to a (q). Set WY(Rnt\ {O)) = XssY(Rt x Sn), cf. 2.1.1.(39). 4. Definition. WS(RQ, Xs-?’(lRn+ \ {0})), s, y E R, is the closure of Ct((Rnt \ {0})x I R g ) with respect to the norm
3.1. Preliminary Constructions
271
This is motivated by Lemma 1 and gives a natural generalization of the standard Sobolev spaces. Ws(Rq,Xs,y(Rn+\ {O})) will be regarded as a wedge Sobolev space, where the “wedge” in this case is (W+X Sn)x Wq. Compared with the usual Sobolev spaces we have here an extra weight y which can be chosen independently of s. Let us set for a moment E S . Y = X & Y ( R ~ +(0)). ~\
and the uniform bound for the norm follows, since q / [ q ]runs over a compact set. Thus (13) may be estimated by cIIuIIws. 0 One might expect that the operators (9) also lead to continuous operators (11). But for yDO-s the substitution of polar coordinates is more complicated, cf. 2.1.7. We shall return below to this question. The idea is to establish first an operator convention, i.e. to choose a Mellin operator in the ( t , x ) coordinates which is modulo a smoothing operator the same as the given one and then to apply analogously the homogeneity. 6. Remark. The method of Definition 4 shows a general principle of obtaining spaces along an “edge”Rq, modelled over another space E, for instance a Banach space. It also depends on a group x AE C(R+,Z&(E)) of operators which b kept fixed. Then the closure of Ct(Rq, E ) with respect to the norm IlullWS(Rq,n=
{I[ ~ 1 2 s l l x - ’ ( q )
Fy-+vull~dt]}1’2
x ( q ) = xi,,), defines an abstract ”wedge Sobolev space” WS(Rq,E ) for every s E R.
(14)
272
3. Operators on Manifolds with Edges
This aspect will be discussed in more detail in the following section. Note that
-
-
11. IIE 11 11 ;* 11.11 WYR4E) 11 .11
(15)
E) I
i.e., equivalence of norms in E implies equivalence of the associated norms of the wedge spaces. 7 . Proposition. For every E > 0 there are constants cl(e), c ~ ( E>) 0 such that
{o}))
C 1 ( & ) IIUIIHs(Rlfn+p)5 111111WS(Rq,Xg*v(Rn+'\
for all u E C,"(R1+
+
Q)
5 CZ(&) [ l ~ l l H W 1 + n + 9
with suppuc{(u,y): I u l t e } ; ~ E R .
Proox First it is clear that there are constants El(&), & ( E ) > 0 such that EIIlfIIH*(R"+l) 5 kfllXS*YR"+l \{O))
5
E2~~f~~Hs(Rn+1)
for a l l f e C;(Rn+') with suppfE [ u : IuI t e}. If u E C,"(R1+n+Q) is as assumed then Fy,,,u(u, a ) = f ( u , a ) has a support with respect to u in l u l 2 E for all E Rq. Now f([a]-' u, a ) is supported by Iu( t for all a E Rq with another el > 0. Thus we get the asserted estimate by the calculations that lead to (15). In other words if ~ ( uE) C"(R"++),X = 0 for IuI < 1 / 2 , = ~ 1 for 1u1> 1, then ly]Hs(R1+nc'7) = [XI W(Wq,X~y(Rn+'\{0})). (16) This equation indicates a principle for the whole calculus. Outside the singularities we want to preserve the standard operator classes (including the standard ellipticity) and also the standard Sobolev spaces. Then we are able to localize the constructions and to define the analogous objects globally by using open coverings and partitions of unity. Let us mention another identity between Sobolev spaces and wedge spaces. If 0 E Rq is a domain, H ; ( f i ) denotes the closure of C,"(0)with respect to the Hs(RP) norm. In particular we have the spaces H @ + ) =: E and H ; ( v ) , RJ+'=((u,y): u > O } . Set ( x l u ) ( u ) = A 1 ' * u ( l u ) ,AER,. Then
-
Ws(Rq,H;(R+)) = HA(RJ+'). (17) This is also a consequence of Lemma 1, since the wedge space norm coincides with the Sobolev space norm, here restricted to C,"(R$+l).By duality we then get
Ws(RQ, Hs(R+))= Hs(R4,+')
(18)
for all s E R. 3.1.2.
Abstract Sobolev Spaces with Respect to an Edge
Let us now discuss more systematically the idea of 3.1.1. Remark 6. Let E be a Banach space and Ze,(E)the space of all continuous operators in E, equipped with the strong operator topology. We fix a function
R++se(E), E C(R+, ZdE)) with values in the invertible operators, with x l x p = x l p for all A, e E R+.
(1)
2 73
3.1. Preliminary Constructions
A standard example is E = LZ(R"+') and ( x J ) ( v )= A("+')'*f(Au). In this case { x ~ R} + ~is even a group of unitary operators. In general we do not expect that IIxAllecnis uniformly bounded for varying A E W+ (as usual S ( E ) is equipped with the norm topology). Set
1. Lemma. Let {xi}nER+ be a group of operators, x AE C ( R + , S u ( E ) ) .Then there are constants M , c > 0 such that Ilxlllecn Ic K Y
for all A E R,.
(3)
ProoJ The continuity of x1 implies that for every fixed b > 0 for all A E [e-b,eb],
JJxA uJJ5 E m
u E E , m = m ( u , b ) a constant. From the Banach-Steinhaus theorem then it follows llxllletn 5 B for all 1 E [e-b,eb]with a constant B ( b ) . Now we get by induction IIx1llfecn 5 B" for A E [e-nb,enb]or B1+[b-'lloE41 11% A 11g(E) < =
- Bl+b-'lb41. 5
Ilx;'Ilecw 5 Beoo8B)b'1'o~i - BAb-"08B for A 2 1. In other words the 5 Thus assertion is satisfied for c = B , M = b-' log B . 0 2. Definition. Let E be a Banach space and { x ~ R+ } ~be a continuous group in g U ( E ) . Then WS(Rq, E ) , s E R, is defined as the completion of Y(R4, E ) = Y(R4)m n E with respect to the norm
{s [TIZsllx-'(l)Fy~,u(~)II~d~} 1'2
IlullWa(Rq,E)=
9
(4)
x ( ~ =) x L V lMoreover . H S ( R q , E ) denotes the closure of Y ( R q , E ) with respect to rhe
norm 1111IIHyRq,E) =
{I [ t l l z s I I,~ (tl) ~ ~ I1$ dv} '".
(5)
WS(Rq,E ) is also called the wedge Soboleu space, modelled over E. Clearly the space Ws(Rq,E ) depends on the choice of the transformation group xl. But it is usually kept fixed, so we do not indicate it explicitly in the notation. An exception is xA= identity for all A with the corresponding space HS(Rq,E ) . Different groups { x l } , {GI} lead in general to non-equivalent wedge spaces. Note that we obtain the spaces W8(Rq,E ) , HS(R'J,E ) also as the completions of Ct(R4, E ) with respect to the corresponding norms. We also may define the spaces as the closures of other dense subspaces. For example let .7(Rq,E ) be the space of all u E Y ( R 4 E ) with l l ~ Y + I l ~ ( T ) I l E5 C"T1-N
for every N E N with a constant c, > 0. Take F(Rq, E ) in the FrCchet topology, given by the best constants in the estimates. Then .7(Rq,E) is dense both in Ws(Rq,E ) and Hs(Rq,E ) .
274
3. Operators on Manifolds with Edges
In virtue of Lemma 1 the operator T = F-1 v-y
1)FY-v
(6)
~ - 1 (
defines a topological isomorphism T : .7(Rq, E ) +.7(RQ,E ) . 3. Proposition. The operator T extends by continuity to an isometric isomorphism
T Ws(RQ, E ) +HS(Rq,E ) .
(7) Proof. First it is obvious that Ws(Rq, E ) is the closure of .7(Rq,E ) with respect to the norm
IIT - IIW * W , = IIlJ IIH Y W , E ) (9) for all u E S(R@, E ) . Since S(Rq,E ) is dense in Hs(Rq,E ) , it follows (9) for all u E Hs(R'J,E). 0 More generally the wedge spaces modelled by { E ,x A } , { E ,gn}are isomorphic for any choice of continuous groups in S'JE). Indeed, if
F: WS(Rq,E ) +Hs(Rq,E ) is the isomorphism of Proposition 3 related to 12;, then we get the isomorphism F-l
T WS(Rq, E)+
'@(Rq,
E).
4. Proposition. Let E l , E2 be Banach spaces with associated continuous groups xi,l in S r ( E i ) ,x i ( q )= xi,[v1,i = 1,2. Further let A : El +E2 be a linear continuous operator satisfying
kT1(v) Axl(fl)ll9(E1,E2)5 c A [ l f l l '
(10)
for all 7 E Rq with a constant cA > 0 and some p E R. Then A induces a continuous operator 2:Ws(Rq,El)+ WS-#(Rq, E2), for all s E R, (11)
and the norm of 2 in S'(WS(R4, El),Ws-p(Rq,Ez))is 5 C A .
Proof. The proof is straightforward. A" will be defined first for u ( y ) E C;(Rq, El) by applying A for every fixed y . Then Au E C,"(Rq,E2). Now it follows from
J l ~ ~ I l & - q p , E z=) J [VI~'"')IJJ~;'(~) F A u ( v ) ~ ~dlt Z, =
J [s12~s-p)Il~;1(~)~~l(tl)~;1(tl)F~(rl)ll?E, d?t
S c:
[tllzs~~x;'(?)~u(71)II~, d? = C:llUII&(Rq,E,)*0
Let us consider an example to Proposition4. Let E be a space of functions over R"+'\ (0)with the action (xnf)( v ) = A ( k + 1 ) ' 2 f ( A t J )
(12)
3.1. Preliminary Constructions
275
within E such that { x A }E C ( R + , s 0 ( E ) Define ). the space EQ= {lvlQf:f~ E } with ~ - the Q ~ operator ~ ~ of ~ multiplication . by 1vl-Q defines an the norm llgllER = ~ [ ~ v Then isomorphism .htp,-Q: E'+ E . On EQ we also have an action of x1 with Thus
~ii,~,-~: W s ( R qE)? , WS-Q(R4, E). In particular
Ws+Q(R~,IvlQXs~y(Rn+'\{0})) = WS(Wq,Xs~Y(R"+l\ {O})).
(13)
5 . Proposition. Let A : E l + E2 be as in Proposition 4 and let A be surjective. Assume that there is a right inverse R : E2+El with Ik;'(q)
RxZ(q)ll%'e(E2.E1) 5 c R [ q l - l ' .
Then R": Ws(R4,E2)+ Ws+"(R9,E l ) is a right inverse of ( l l ) , in other words (11) is surjective for all s E R. Prooi The continuity of R" fojlows from Proposition 4. Now let u ( y ) E Ct(Rq,E2). Then xR"u(y) = u ( y ) , i.e., R is a right inverse of A on a dense subspace of Ws(Rq,E2).From this it follows immediately that A"R" is the identity on Ws(R',E2).
Let us construct an example to the situation in Propositions 4, 5. .Set El = Hs(R), E2 = Hs((R+), and let A = r+ be the operator of restriction to R + . Further let u ( 1 t ) , 1 E R+. Then ( x l u )( t ) = Ilx-'(tt)
r+x(tl)lIP(E1.E*) 5
1
for all q . Thus r+ induces a restriction operator (now again denoted by r + )
r+: WS(Rq,Hs(R))+ WS(Rq, Hs((R+)) (14) for all S E R. (We know that independently from 3.1.1. Lemma 1 and formula (18).) Assume now sz 0 , s + 1/2 mod Z, and construct a right inverse e : : H s ( R + ) 4Hs(R)with Ilx-'(q) e:
X(q)IIY"(E2.El)
5 c1
with c1 independent of q . First it is clear that H i @ + ) (the closure of Cr(R+)in HS(R))admits an extension operator (e:),, : H @ + )
+
Hs((lR)
so that x ( q ) - l ( e : ) o x ( q )is bounded in Y?(H@+), Hs((R)),uniformly in q , namely the canonical embedding. Let so = [s + 1/21 (= the largest integer less or equal than s + U2). Then there is a direct decomposition H"(R+)
=
H;(R+) a3 E
(15)
276
3. Operators on Manifolds with Edges
with some subspace E"cC,"(w,) of dimension so. It can be chosen in the form
w being a cut-off function. Set p ( t ) = w ( t ) for t 2 0, = w ( - t ) for t 5 0. Then Q, E Cr(R),
F;
and
(C a j t ' w ( t ) ) := C ajtJQ,(t)
gives us an extension of i i E, ~ it suffices to set
E" of the
desired sort. Now for u = uo + u', uo E Hi(&),
e: u = (e:)Ouo+ Z:k.
This induces then a right inverse of (14). With the notion of W s spaces we can derive also other standard properties of Sobolev spaces. For instance, we have Ws(R"-l,C) = HS(Rn-l), where the transformation group on C acts as the identity. Set El = Hs(R,), Ez = C , x l ( q ) u ( t ) = [ q ] 1 ' 2 u ( 1 [ q ]For ) . s > 1/2 the restriction operator r': H"(R,)+C satisfies the conditions of Proposition 4 with p = 112. It follows that WS(R"-1,HS('+)) = Hs(R:)+Hs-1'2(R"-1) = Ws-1'2(Rn-1,@) is continuous, s > 1/2. Now let us return to the situation of Definition 2. A subspace El c E is called invariant under ( x A }if {q}induces a group of isomorphisms R++!2?(El). In general we allow that El is a space with a stronger topology than that induced by E . The assumption of Proposition 4 is satisfied with p = 0 and we then obtain continuous embeddings Ws(R', El) G Ws(R4,E ) (17) for all s E R. 6. Lemma. Let E = Eo El3 El be the direct sum of invariant Banach subspaces E o , E l , and let Bi: E + Ei be the corresponding continuous complementary projections. Then
W s ( R 4E, ) = Ws(R4,Eo) El3 Ws(Rq,El) with induced continuous complementary projections
ni: Ws(Rg,E ) + Ws(Rq,E J , i = 0 , l . The proof is obvious. In the applications we also may have the following situation. Let Eo, E be BaEach spaces, Eo G E a continuous embedding, and Eo invariant. Moreover let E c E, be a finite-dimensional subspace (not necessarily invariant) and Eo n E = {O]. Define a subspace
Ys(RP,E") c Ws(R4,E ) by
CYS(R4, E) = T-l{HS(Rq) €9 E"} ,
277
3.1. Pteliminarv Constructions
in the topology induc_ed by the bijection T-' of the topology of Hs(Rq)@ E". If we fix a base e l , ... , eN in E we obtain
-
Cj = Fvj. A more concise description is u E YS(R~, E)
~ u ( q =) x ( q )
for some
v E H ~ ( I REl. ~,
(18)
7 . Lemma. We have WS(Rq,Eo + E") = WS(Rq,Eo) + 7rs(Rq,E") (+ indicates the Banach space sums, being direct) and complementary projections Bo: Eo + E"+ Eo, $: Eo + E"+ E" induce complementary projections
no: Ws(Rq,Eo + E") + Ws(Rq,Eo)
5: WS(R4,Eo + E")+ 'P(Rq, E"). Proof. Since T - l : HS(Rq,E ) + WS(RQ, E ) is a bijection, it is clear that T-I HS(Rq,Eo) = Ws(Rq,Eo) is direct to T-' HS(Rq,E") = 'P(Rq, E"). NOWB0,s" induces complementary projections 98 : HS(Rq,Eo
+ E")
+
HS(Rq,Eo), @: HS(Rq,Eo + E") +HS(Rq,E")
and we then set no= T-'& T, 5 = T-' @ T . 0
In the applications E play: the role of a weighted Sobolev space, Eo of a subspace with a better weight and E of a space spanned by singular functions. Consider as an example E = Hs(R+),s 2 0 , s f 112 mod Z, q = n - 1, and the decomposition (15) with E" given by (16). Then Eo = H",E+),and our decomposition means Hs(R:) = H i ( p + ) @ 'P(R"-', E"),
{1
so - 1
Fy+,,'P(Rn-l,E")
=
a j [ q ] 1 ' 2 ( r [ q ] ) j ~ ( t [ q ] ) G j (aqj)e: C , u j ~ H S ( R n - l )
j=o
This proves in particular the well-known fact that (for s > 1/2, s f 1/2 mod Z)
is a surjective operator with the kernel HS,(p+),r' being the restriction to 1 = 0. A right inverse is given by the expressions in (19). The definition of W(RQ,E ) can be extended in a canonical way to more general E , which are not necessarily Banach spaces. Let E ( k ) ,k E N , be a sequence of Banach spaces with continuous embeddings j ( k +l): E ( k + 4 E ( k )for all k , where each E ( k )is invariant under the action of x, and satisfies the conditions of Definition 2. We do not assume that the constants c and M of Lemma 1 for E = E ( k )are independent of k . Endow E(") = E ( k )with the k
Frkchet topology of the projective limit.
278
3. ODeraton on Manifolds with Edges
For u E E ( k + l )we have % - l ( q ) j ( k + " ~ ( 8u )=j ( k + 1
) ~
for all q E Rq ( ~ - ~ (onq the ) left refers to E(k)).Thus the conditions of Proposition 4 are satisfied with p = 0 and we get continuous embeddings j ( k + 1): W(RP,E(k+1)) w(Rq, EW) for all k. Now we define
w ( R ~ , E(-)) =
nw ( R ~ ,
~ ( k ) )
k
with the Frbchet topology of the projective limit. Let E , E,, E" be Frtchet spaces,an_d E = Eo + Ebe a direct sum with continuous projections 8,: E+ E,, 8: E-* 6 , E not necessarily finite-dimensional. Let Eo allow the construction of W(R4,E,) as mentioned, but E is assumed not necessarily invariant under x,. Then V(Rq, E"> = T-'HS(Rq, Z)
is again a Frtchet space, and there is an immediate analogue of Lemma 7. In a similar way we can form inductive limits of W spaces with respect to invariant families of spaces, lim W(Rq,E(") =: W ( R q lim Eck)) d
d
The obvious details are left to the reader. The index sets in such constructions are not necessarily countable. Definition 2 suggests a list of functional analytic discussions such as continuous embeddings
WS'(R@,E ) G W(R4, E ) for s'z s, (20) duality and so on. We want to make here only some few remarks under certain special assumptions which are satisfied in our applications. E is assumed to be a Hilbert space which is continuously embedded into another Hilbert space EO with the scalar product (. , .),, {xi} is a group of unitary operators in EO, and E is invariant under x,. At the same time we also consider the dual E' of E , equipped with the norm
uto
We can define Ilp,llEfirst for p, E EO, where lIp,IIE Illpll~,and then E' follows by completion of JT' with respect to (21). Standard arguments show that (. , .)o extends to a sesquilinear form E x E ' + C , and
(i.e. equivalence of norms). The action of {xA}on E induces by duality an action { x i } on E'. It is obvious that the constants c, M for E, x, of Lemma 1 are the same as for E', x;.
279
3.1. Preliminary Constructions
8. Definition. If E, I?', E', {x,} is as mentioned we call {E, Eo, E'; x,} a Hilbert space triple with unitary actions on EO. From now on we tacitly assume that the spaces E in consideration belong to such a triple with a reference space I?' where x, is unitary. Then for every s 2 0 we get a triple of spaces W ( R 4 ,E
)
G
WO(Rq,Eo) G Ws(Rq, E ' ) ,
(23) where the embeddings are continuous. This suggests to look for a corresponding unitary group x, on W(Rq,Eo). The choice should be in such a way that W(RP x Rq, E ) = W(RP,W(R4, E ) ) , (24) where the space on the right is based on x, whereas the left hand side is formed by an expression analogous to (4) for R p + q and x,. It turns out that one has to set
(25) YEIR', A E R+, where xA acts as before on the values of u in E . Then x, E C(R+,% ( W ( R q ,0 ) ) . (XAU)
( Y ) = xdq'2u(AY),
9. hoposition. Let s 2 0 , then IW(R4, El, W(R4, E O ) ,W-s(Rq,E'); XAI
is a Hilbert space triple with unitary actions on w(Rq, Eo) The proof is straightforward. The identity (24) follows by an obvious generalization of the calculations for 3.1.1. Lemma 1. In particular the scalar product of W(Rq, Eo) is given by
As a corollary of Proposition 9 we obtain 10. Proposition. (27) extendr to a non-degenerate sesquilinear form W(R4, E ) x W-s(Rq, E ' ) + @ , and we have W-S(Rq,E') = ( WS(Rq,E))'. The following norms are equivalent llullWs(Rq,E)
- suPI(u,
v)~-O,~,Eo)~/~~v~~W-sCR~,E.)
U
with the supremum over v E W-S(Rq,E') , v f 0,
llvll W - W q , G)- SUP I(k V)@(Rq, U
@)
I/llull WYR", E )
with the supremum over u E CNr(R4, E ) , u f 0.
11.Definition. W~omp(R'Jr E ) denotes the space of all u E Ws(Rq,E ) with compact support with respect to y E Wq, further WSoc(Rq,E ) the space of all u E W(Rq, E ) with pu E W(Rq, E ) for all Q, E C;(Rq).
280
3. Operators on Manifolds with Edges
The spaces W&mpOoc)(Rq, E ) will be taken in analogous locally convex topologies as the comp(1oc)-versions of standard Sobolev spaces. If a c Rq is an open bounded set with smooth boundary we define W(0,E ) = {uln: u E W(Rq, E ) } . Similarly as in Definition 11 we also can introduce the spaces W~,,mpOoc~(~, E ) . Incidentally if we want to point out that comp, loc refers only to the variable in L! (for example when E is a distribution space on another manifold) we employ (28) W(0compr E), W(a1oc9 E ) instead of the above notations. Let us finally mention another generalization of the W(R9, E ) spaces. Instead of [qp in Definition 2 we could insert a weight function e ( v )which is strictly positive and for which
is finite for all u E Y(lR4,E ) . An example is e(q>= [rl”logk(l + [vl), k E N , s E R. 12. Definition. W(Q)(Rq, E ) is the closure of Y(lRq,E ) with respect to the norm (29). Moreover let H(Q)(Rq, E ) be the completion of Y(lRq,E ) with respect to
We call a weight function e ( q ) admissible
if (29), (30) arejinite overY(Rq, E ) .
A straightforward calculation yields
13.Proposition. Let Ei.x i ( q ) be as in Proposition 4 and A ( q ) : El+ E2 be a measurable family of linear continuous operators satisbing llxil(q) A ( q ) X 1 ( 7 ] ) I l B ( E I , E z ) 5 c A e A ( ? ) for all q E Wq. Further let e ( q ) and (elea)( 9 ) be admissible weight functions. Then A induces a continuous operator
2:WQ)(Rq,El)+ W(Q‘Q4)(lRqI E2) and fhe operator norm of A” is 5 c A .
3.2.
Pseudo-Differential Operators with Operator-Valued Symbols
Pseudo-differential operators on manifolds with edges may expected to be to some extent vDO-s along the edges with symbols acting as operators in the transversal cone spaces. On the other hand one should be able to localize the theory outside the edge, where the calculus has to coincide with the standard one with scalar symbols. This suggests also a reformulation of the standard calculus in anisotropic terms, according to the anisotropic description of Sobolev spaces in lR1+n+Q, cf. 3.1.1. Lemma 1. An abstract theory should contain both the calculus of WDO-swith operator-valued symbols and a correspondence to some “isotropic” background. The present chapter develops the elements of such a theory in general. Later on it will be applied to the concrete situation of operators on manifolds with edges.
281
3.2. vDO-s with Operator-Valued Symbols
3.2.1.
Amplitude Functions and Continuity of Pseudo-DifferentialOperators
Let E , E" be Banach spaces and {xA}E C(R+,& ( E ) ) , {Ci} E C(R+, Y?c(E")) be fixed groups of operators in E and E", respectively. As usual we set x ( q ) = x,,,~and the same with tilda. 1. Definition. Denote by Sr(aX Rq; E, E"), 0 E RPopen, p E R , the space of all
a ( y , 7)E C(aX Rq,W E , 8)) for which
IIC-W (D;D+(Y, tt)) ~ ( V ) I I ~ s( ~c[d--181 . ~ (1) for all multi-indices a E Np, B E Nq and y E K CCQ, q E Rq, with a constant c = c ( a , b, K ) > 0. S@(aX Rq; E, E") is a FrCchet space with the best constants c in the estimates as semi-norms. It may happen that E" runs over a system of spaces Ek,k E N , with continuous embeddings Ek+l4 Ekfor all k . Then E = Ekis a FrCchet space in a natural
n k
way and we write
sqa x ~
4
E,; E ) =
nsqa x w;E, ,PI, k
also equippe? with the corresponding FrCchet topology. Clearly it is assumed that CA acts on all Ek.
2. Proposition. Let 9 ( R q ;E, E")const be the subspace of those a E P(0x which are independent of y, taken in the induced FrPchet topology. Then SN(0 X Rq; E,
E") = Cm(a)BX9(Rq; E, E")co,s,
.
The proof follows from
P(a x Rq; E, E") = Cm(O,B(Rq; E, It is clear that
P(a X I R q ; E, E") S P'(aX Rq; E, E") when p' 2 p . Moreover
&'''(ax Rq;
E,) P(0X
Rq;
E l , E") 5 9+ '(aX
Rq;
with the point-wise composition of operator functions, and
D;D$(Y, 9 ) E sp--18l(a x ~
9
E,; E")
for a ( y , q ) E P(a x Rq; E, E"), 01 E I N', b E Nq. Let us consider some simple examples. 3. Example. Let o(t) be a cut-offfunction, a,( q ) = 0(ct [q 1) (tq)" :
01 E
"7, 01
f
x". ' ( X ") +x". ' ( X *) ,
0, and
E l , E2)
Rq;
E, E")
282
3. Operators on Manifolds with Edges
where the operator is understood as multiplication by the corresponding function of t, c > 0 being a constant. Then
a h ) E So@'; and a,(q)+O
xq xq y,
Y)const
in this symbol space (IS c+
m,
s, y E R.
Indeed we have a&q) = xAuc(q)x;'
for
a L 1 , [ q ]2 1.
Thus for A = [ q ]
aC(tll[ql> = %-'(a>a,(q) x ( v ) = w(ct) tl"l(tll[ttl>". The operator of multiplication by o ( c t ) tl"l tends to zero in S?(W v, W ') for c+ 00 (cf.2.1.1. Remark 22) and (q/[q])"remains uniformly bounded. The derivatives with respect to q can be checked in the same way. 4.Example. Let E = E = Lz(R+),p ( t ) E C;(E+) and a& q ) = AZq be the operator of multiplication by p. Then a E So(Wq; Lz(IR+),Lz(R+))co,,,but it is no classical symbol in the sense of 3.2.2. Definition 11 below. Indeed, since a does not depend on y , q , we only have to check the condition (1) for o l = b = O . We have x - ' ( v ) Q I x ( v )= P,(t[ttl-')
which is an operator family over L2(R+),uniformly bounded in q ,
1I ~ ( t [ v l - 'u)( t ) l zd t s c s lu(t)lzd t with c = s ~ p l p ( t [ q ] - ' ) 1 ~ . 7
5. Proposition. Let p ( t ) E C;(a+), then AZq E So(Rq;Xq'(X-), y(X3)con,tfor all s, ~ E RandAZ,+O , in So(...)constforp+O in C;(K+).
Proof. We only have to repeat the proof of 2.1.6. Theorem 1 for the operator of multiplication by p ( t [ q ] - l ) and to observe that the norm is uniformly bounded in qER'.o
The reader may check the orders of derivatives of p, which are involved in the associated semi-norms of Aqin So(...),on,t. They depend on s. With every a ( y ,y', q ) E S@(OX O x R'J;E, E) we associate a 1pD0 Op(a) u ( y ) =
fl ei(Y-J")qa(y,y',q ) u ( y ' ) dy' d q ,
(2)
d q = (27~)7dq, u E C;(O, E ) . Then Op(a): C;(O, E ) + Cm(O,E )
(3) is continuous. The definition of Op(a) is based on classical oscillatory integral arguments that we do not repeat here. For details cf. [H3],[S27],[R8].
283
3.2. wDO-s with Operator-Valued Symbols
6. Theorem. Let W ( R qE, J , s E R, be modules over C;(Rq) with /IP)uIIwJ(R?E,)
s c i ( p )IIUllwJ.cRq,Ei)
(4)
and c i ( p ) + O as p+O in C;(Rq), i = 1 , 2 . Then Op(a) for a(y, y', a ) E 9(R2q x Rq; E l , E 2 ) , p E R,extends to a continuous operator (5) Op(a): wScomp(Rq, E d + w;;wq,E2) for all s E R. An analogous result holds for a(y, y', a ) E B ( R 2 X Rq, E l , E2) and W~omp,,oc spaces over open a 5 Rq.
Let us first prove a more particular result which holds without the assumption that the W sspaces are C;(Rq)-modules; for convenience we consider in the proofs the case 0 = Rq. 7 .Lemma. The operator (3) for a ( a ) E P(R4; E l ,
has continuous extensions
Op(a): WS(R4, E l ) + WS-'(Rq, E2) for all s E R,where
tIOp(a)lle(w: W3-r) 5
SUP VERq
[a]-" IbiYv) a ( a ) ~ l ( a ) l l e ( E ~ . E ~ ) .
Proof: By definition we have for 3.1.2. (6)
11 OP (a) II&-@(Rq, E2) = llTop ( a ) 11 $-p(Rq, E2)' Moreover TOp(a)u= F-'x;l(q)FF-'a(a) Fu = F-'x;'(9)a(~)x1(a)FF-lx;l(a)Fu = {F-l%il(a) a(a) %(a>Fl
Tu.
Set u = Tu. Then u11>-p(Rq,E2)
= llF-lxil(a) =
5
a(?) x d v ) Fullk-w(Rq,E2)
1[a12~"-"~IIxi1(a) a(a) SUP VERq
xl(a)
drt
[al-2'llxi1(a) a(a) xl(a>llH(E,,E2) J [aI2"11~~112E, da
= SUP [91-2'llx;1(a) a ( a ) ~ l ( a ) l l k ( E l . E 2 ) I I ~ ~ I I > ~ ~0 ,E,). VERq
Proof of Theorem 6. Let us first assume that a b , y ' , a ) = 0 for y 6 K,, y' B K,,where K, = { l y l s c } , c > 0 some constant. According to Proposition 2 the function a & y', a ) can be written in the form m
a ( ~Y', , 7)
=
C ajpj(Y) P:(Y') aj(tl),
(6)
j=O
clAjl
whereljtjE, pj+O,p;+OinC;(Kc), Alp,the operator of multiplication by p. Then
aj+OinS~(...),o,,t.Denote by
rn
C aj./n,, OP (aj)
j=O
19 Schulze. Operators ewl.
(7)
284
3. Operators on Manifolds with Edges
converges in S? ("Mr"(R4,E l ) , W -"(Rq, E,)) . Indeed, we have (with obvious notations)
c
since [Aj[< and the other norm factors tend to zero. Here we have used that by assumption c;(p,), c : ( p i ) 40 for all s and Lemma 7. For u E Ct(R4, El) it follows Op ( a ) u =
c AjAZvj Op
(aj)Jul,;u E Cr(W4, E,).
The estimate (8) shows that (7) extends by continuity to an operator in 9 ( W ,W - " )It. is now obvious that for arbitrary a(y, y', g) we get a continuous extension in the sense of (5). 0 8. Theorem. Let {E, Eo, E'; xA} be a Hilbert space triple with unitary actions on Eo (cf: 3.1.2. Definition 8). Then Wr(R4, E ) , Wr(Rq, E o ) , W(R4, E') are Ct(lRq)-modules and the norms of A, tend to zero in the corresponding spaces when p+O in Cr(Rq).
Prooj Consider, for.instance, the assertion for E . We write A = A, as a qDO with g-independent amplitude function,
3.2. wDO-s with ODerator-Valued Symbols
Using the inequality (1 + l a [ )(1
+ lbl)-’
285
5 1 + la - bl for a, b E R we thus obtain
for every N E N with some other constant cN(p). Choosing N large enough we immediately obtain (11). Since cN(p)+0 as Q, -+0 in CF(Rq), our theorem is completely proved. 0 As a consequence of the proof we see that la1s N
for all Q, E Cr(R4) with supp Q, S K C C N with ~ a constant c that only depends on K, 19’
286
3. Operators on Manifolds with Edges
max { IM - sI , IM + sI } + q < N , with the constant M of 3.1.2. Lemma 1. The same follows for E' whereas for E = E o we may set M = O . From now on we assume that W(W,E) are modules over C;(Rq) for all occurring E with the property as in Theorem 6. It can be proved that this is true anyway for any Banach space E, provided that X, E C(R+, S?,,(E)), cf. [ H Z ] .
E">
9. Proposition. Let a & y', q ) E S#(R*qX Rq;E, be given and a & y', q ) = 0 for ly - y ' l < E with some E > 0. Then Op ( a ) induces continuous operators
OP ( a ) : K,mp(Rq,0 - WEARq,g )
(13)
for all s E R.
Proof. By classical arguments we get
s
= =
Since
ei(y-Y')qa(y,y', q ) u ( y ' ) dy'dq
{[(y-y')"l-'D;ei(y-Y"~}a(y,y',q)u(y') dy'dq
s ei(Y-Y')q[(y'
- y ) ' ] - ' D ; a ( y , y ' , q ) u ( y ' ) dy'dq.
[ ( y ' - ~ ) ~ i - l ~ ; a~ (' y, ,~ ) E s P - ~ ~ ~ ( RE,E) *QxR~;
for all
(x
E Wq
we get our assertion from Theorem 6. 0
Let us give a further example of operators with operator-valued symbols in our sense. Consider a classical (scalar) symbol p ( x , y, F, q ) E S!!,(R:' X q ) . For simplicity assume that p has compact support with respect to x. Set
E
= Hs(R,"),
E"= H"P(R,"),
( x A u )(x) = Am'2u(Ax)for u in E or
10. Proposition. Under the mentioned assumptions the operator family
a ( y , tl) u ( x ) =
sl ei(X-X')%@,Y , F, 7) u ( x 3 dx' dF
sat~esa(y,q)ESP(R,QRR::;E,El).
'
Proof. Let us first assume that p does not depend on x. Then x;' a ( y , q ) x , u(x) =
ei(X-X')tp(y, AF, q ) u(x') dx' dF
which follows by the same calculations as in 3.1.1. after formula ( 8 ) . For every N E IN we have N
3.2. yDO-s with Operator-Valued Symbols
287
Thus N
x;' a(y, q ) x Au(x) =
fl eicx-')ep,-j(y, At, q ) u(x') dx' U E + fl ei(X-X')ErN(y, 15,q ) u(x') dx' U F .
j=O
(15) The mentioned constants in the above homogeneity can be organized in such a way that
5, v/[vI) Pp-j(Y, [~15,[qIf~/[qlI)= [~I'-'P#-~(Y, for all q E Rq,5~ Rm.Since the lyDO-s in Rmwith the symbols p,,-'(y, 5, q / [ q ] )are , when q runs over Rq, y over a comuniformly bounded in 2 ( H s ( R m )H"-'(R")) pact subset of Rq, the operators in the sum in (15) satisfy our symbol estimate (1) for 01 = B = 0. For the remainder we argue as follows. Let [5, q] be a strictly positive function which equals 15,q) for IF, 01 2 const. Then the lyDO in Rm associated with "715, ql-"t+NtlrN(y,[qlc q ) is uniformly bounded in 2(Hs(Rm))for all q E Rq and y varying on a compact subset of Rq. This operator composed with that for "~15,q]F-(Nt')is just the remainder for 1= [ q ] .But the latter factor is of the sort as treated before. The symbol estimates for arbitrary 01, B follow in the same way. Note that (15) shows that (14) is classical in the sense of 3.2.2. Definition 11. The argument for the remainder shows that it is also allowed to set N = 0. If we do so we also obtain a(y, q ) +O in SF(R4 x Rq; E, ,&?),,,,, for p 4 0 in S!!,(Rmtq x Rm+q).Then a tensor product argument together with an analogue of Proposition 5 for Sobolev spaces yields the assertion also when p depends on x. 0 Note that p(x, y, 5, q ) E S-"(R"+qX Rmtq)implies H"(Rm)) a(y, q ) E S-"(Rq x Rq; Hs(Rm), for all s E R (cf. also the notation in the beginning of this section).
3.2.2.
The Standard Elements of the Calculus
Now we want to continue the discussion of the spaces Sp(R X Rq; E, E"), 0 S RP open. Usually we have p = q or p = 2q. In the latter case we also take the open sets in the form R2 = R X R,R C RP open. For a E SJ'we write p = ord a.
1. Proposition. Let a j E S p j ( 0 2 X Rq; E, E), j E IN, be a sequence, p j + --oo + 00. Then there exists an a E S'(R2 X Rq; E, ,&?), p = max {pi),such that
as
j
(
ord a - x a j :j
)
+-m
as N + w .
If n" is another such symbol in S' then a - n" E S-". The proof is an obvious modification of that for scalar symbols, so it may be omitted.
288
3. Operators on Manifolds with Edges
-C m
AS usual we write a
a j . I f x is an
excision function, i.e. x E c ~ ( R ~~) ,( 7=)o
j=O
close to
v = 0, x ( v ) = 1 for 171 2 const, then
converges in SP(02 X Rq; E, 1 )if cj is a sequence of constants with cj + 00 sufficiently fast, and a 1 a j . Note that the form (1) of obtaining asymptotic sums is not very efficient in concrete calculations with operator-valued ymbols, since operators associated with amplitude functions in S-"(O* X RQ;E, E) are very "large". They have no particular "regularity"in direction of the fibre spaces E, For this reason we shall introduce in Section 3.2.3. a more precise calculus in terms of scales of spaces %?= { E 9 } , , and a certain "scale axiom". (1) will be replaced by a manipulation on the level of distributional kernels. It may be performed already in the present case and the corresponding result then also applies for the scales. Let us first define the analogues of the standard yDO-s in the operator-valued setting.
-
2. Definition. Denote by LP(O; E, E) the space of all operators A 3.2.1. (2)) with a E SP(B2X R; E, E). Set Lm= U LP, L-" = LP. For A E LP we also write ,u = ord A.
= op(a)
(cf.
n
Ir
Note
that
E S-"(02 x
P
for every
k ( y , y ) E Cm(02,5!?(E,E"))
RQ;E, g ) such that
J HY, Y? u(Y') d f =
there
is an
a ( y , y', 9 )
Y', a ) u(Y') dy' d v , ei(y-Y')va.(~,
u E C;(O, E). In fact, it suffices to set d y , Y',
v ) = e-i(Y-Y')qk(y,Y ' ) e(v>
with an arbitrary e E C;(Rq), J e(s) U v = 1. Every A E L"O; E, Z) has a distributional kernel k(y,y', y - y') which is 5!? (E, E)-valued and given by k(Y,Y',
n = s eika(y, Y', v ) 0 ,
(2)
where the interpretation refers again to oscillatory integral arguments. Then ~ ( Y , Y ' v, ) =
J e-ilqk(x Y ' , o d t .
(3)
Let us examine some useful properties of (2). First it is clear that the integral (2) exists in the standard sense when p is negative enough. In fact we know by defmition that IlX"-'(tl) a ( y , Y ' ,
and
v ) x(v)lle(+ 2) 5 c 171'
I l ~ ( ~ ) l l r5e CR ~ ~ llx-'(71)1l~(~) 5 CM with different constants c and y, y' varying in a compact set. Then
289
3.2. IJDO-swith Operator-Valued Symbols
6
s Il~(lf)lllp(wII~-'~tf)
sc
Q ( Y , Y', r)x(l)llecE,hllx-'(?l)llrecRd?t
J [ q ] p + M + B d q< m
provided p + M + A < -4. An analogous argument shows that for every j
E
IN there is a p ( j ) such that
W Y ,Y', 0E C"(f12, CJ(Rf,2 ( E , E")N (4) for all p s p ( j ) . Now a ( y , y ' , q ) E S p implies (1 + 1q12)-Na(y,y', q ) E S p - Z N . From D i eih = q* eW 'i it follows the existence of a differential operator 9Fwin [with constant coefficients with Lilyw eitq = (1 + IqIZ)Neilq.Thus
k ( y , Y',
o = 9 ~J'"
+ Iq12)-Na(Y,Y', v ) d q
(5) coincides with (2) when p is sufficiently negative and it also makes sense for those ,u for which p - 2 N + M + n? < - q. The mentioned oscillatory integral arguments show that ( 5 ) is then independent of N. Thus (5) provides a definition of k ( y , y', 0 for arbitrary orders, where 9 f wis to be taken in the distributional sense, N sufficiently large. As for scalar WDO-sthe distributional interpretation of 9FM is necessary only in a neighbourhood of [ = 0. In fact, we can easily show that for the restriction ko of k to Qz x (Rq\ {0}) we have eiCq(l
k o ( Y , ~ 'O, E C"(f12 X (Rq\ {O)), 2 ( E , El).
(6)
*
Namely from 0;eiCq = r" eitq it follows for [ 0 1 1 1- 2 N g z A eiCn = eik with some differential operator 9iNin q with constant coefficients. From (5) it follows by integrating by parts k ( y , y',
0=
1[1-2@
J eiCq9iN(1+J ~ I ~ ) - N Qy',( ~q, ) d q .
(7)
Now 0;(1 + I q J Z ) - NEaS p - 2 N - l r r l and (4) show that the integral on the right of (7) is in CJ(* with respect to [ E Rqand j ( R ) + m for fl+ m. Since N is fixed, we obtain (6). We can show even more, namely 3. Proposition. Let a ( y , y', q ) E S'(f12 X Rq; E, E") and k ( y , y', 0 be defined by (2). Then for every p E Ct(Rq), yl(O = 1 in Q neighbourhood of [ = 0 , we have
(1 - vcn>k(Y, Y',
n E Crn(f12)Q,
W R f )@nZ(KE"),
(8)
Y(Rq) being the Schwartz space. Moreover (2) induces an isomorphism
S-"(RZx Rq; E, E") + Cm(f12)Q n 9(Rz) Qz2 ( E ; E") .
(9)
The proof will be dropped. It is of the same type which was formulated in many variants before, cf. 2.1.3. Proposition 6, 2.1.4. Theorem 15.
290
3. Operators on Manifolds with Edges
The formula (2) defines a space of distributional kerneb belonging to amplitude functions a E S p ( 0 2 x Rq; E, E ) that will be denoted by T ” ( 0 2x Rq; E, E ) . In other words we have by definition an isomorphism F - l : Sfl((nzX R :; E, @)T, T”(flZX Rf;E, E), (10)
F = .Fr,,,the Fourier transform in R*. We endow TP with the Frkchet topology induced by (10). It is convenient also to use the space S[l,(12R2 X Rq; E, E) of amplitude functions which are defined in the same way as P but where x,, CA are replaced by the identical operators. In other words the symbol estimates are of the form (1 1) l l q , , q a ( Y , Y ’ , r)lle(E.EJ5 c[rl’-’@’ with the usual co-ments on the constants c. This gives rise to an analogous space Tr1,(flZx Rq; E, E ) of distributional kernels with the corresponding Frkchet topology. Now we fix a function ry E Ct(R‘J), ry(O = 1 close to I =0. Let us denote the left hand side of (2) also by K ( a ) ( y , y’, 0. For every constant c > 0 we then have a ( y , Y ’ , r ) - ( 4 - 7 V(CO K ( a ) )( Y , Y’, a ) E S-”(fl2x R‘;E, E“> > a E S”(RZx Rq; E, R). This is a consequence of (9).
4.Theorem. Let E, E” be Hilbert spaces. Let aj E Spj(flz x Rq; E, E), j E N, be a sequence, pj 4- 00 as J + m . Then there is a sequence of amplitude functions aj,oE S - ” ( 0 2 x Rq; E, E ) and a sequence of constants cj such that m
K
:=
C Y ( c j O K ( a j - aj,o>( X Y ’ , o
j=O
converges in T@((O* X Rq; E, E), p = max { p i } . If a denotes the amplitude function belonging to K via (10) then a
-
m
j=O
aj.
Proof. In virtue of 3.1.2. Lemma 1 and the form of the symbol estimates 3.1.2. (1) it is obvious that we have continuous embeddings
for every p E R and the constants M,M belonging to {a}, {Cl}.Thus we also get the corresponding continuous embeddings for the T spaces. For proving the asserted convergence it suffices to consider the semi-norm systems in the spaces with subscript (1). Indeed, we may write K=
N
m
j=O
j=N+1
C y ( ~ j OK(&j) + C
~(cjO K(Ej) = K N
+ KIN,
Zj := aj - aj,o.Without loss of generality we assume pN+ < p N . If K h converges in T::, p N = max { p j } ,then it also converges in T ” N + M + a and hence also in TP for N j z N t 1
so large that p N + M + # 5 p. Thus we may ignore in the definition of the seminorm systems the group actions. Next observe that for any given semi-norm n on the space T:,N, it suffices to prove n(y(cOK(Ej))+O
as c + m
for j 2 j l ( n )
(12)
291
3.2. yDO-s with Operator-Valued Symbols
after the appropriate choice of u ~ ,Then ~ . we can ensure the convergence of our sum with respect to nfor c + m sufficiently fast. Since nruns over a countable system, a diagonal argument then yields a sequence cj which fits for all semi-norms. For abbreviation we consider from now on the case of y, y’ independent amplitude functions. The arguments then apply after obvious modifications to the general case and will be dropped then. From (4) we obt$in a sequence MjeN with M j + m as j+m such that K ( a j ) E CMj(Rq, 9 ( E , E ) ) for j 2 j o with some fixed j o large 0 we obtain enough. Let @ E Ct(Rq), @y = y. By Taylor expansion of K ( a j )at
c=
rn
K ( a j ) (0=
1.1
1
CN,- 6*-- pa(^) aj(lt) dV + Ky(0, 1
5
Here Pa(~) are polynomials in v of order I&[, and N j is chosen in such a way that the integrals baj = J Pa(q)a j ( q ) dq converge in 9 ( E , ? !,) for 0 5 1011 5 N j - 1, but N j + m as j + m, and that K y ( 0 is at least continuous in C. Such a choice of N j is obviously possible. From (9) we get
C
a j , o ( ~ ) = ~ ~ - q { @ la1 d( N, O- 1 6 * b a j } E S - m ( . . . ; ~ , E ) ,
c=
and K ( Z j )(0is flat at 0 of order N j . It remains to show that for an appropriate K ( Z j )(0 converges in T? As mentioned we choice of constants cj the sum 1 v(cj() always may remove a finite partial sum such that it suffices to deal with Th, with v so negative as we want. Further it is allowed to fix n and to make a choice of cj = cj(n). The symbol estimates for SYl, in the case of constant coefficients are 110; a(V)llL 5 c [ v 1 ” - ’ 4
for all a E “7, c = c(01) constants, L := 9 ( E , g). This can be replaced by I l V q a(V)IlL 5 c [VI’+”O
for all a,p E Nq with
1011
+ vo g
and any fixed vo E N. In particular
is a semi-norm system for SX, (. . with v = - y o , where a,fi runs over the indicated set of multi-indices. We want to pass to semi-norms of the form a
+
{ J IITFDfK(a) (011; dc}”’
(14)
and show that they are stronger than (13) up to a fixed loss of order, only depending on q. Observe that in (13) we can interchange the order of application of @, D ; , up to equivalence. Let f ( s )be an operator function of the form f ( q ) = r,~y’D?q n D 9 ... q“D$+1 a ( q )
with certain multi-indices yj E INq and f being of sufficient decrease for (71+ m. Then
f(vt)= iq
J f i ( i )d i
ii< rl
292
3. Operators on Manifolds with Edges
2 93
3.2. yrDO-s with Operator-Valued Symbols
as c + w. In an analogous manner we can deal with the ( derivatives, where the ~ F > E. arizing powers of c in the derivatives of y ( c 0 are compensated by c - when The case q > 1 is completely analogous. This yields (15) which was the remaining point of the proof. Note that the intuitive idea of the latter proof is very simple. Namely the kernels for amplitude functions with decreasing orders become modulo C" kernels more and more flat on the diagonal (or on (= 0). Outside the diagonal they are C" anyway. Now the asymptotic procedure of Theorem 4 is of the same structure as the proof of Borel's theorem which asserts that for any given series of coefficients bj E C there is a C" function with just these Taylor coefficients at a given point, say ( = 0. Such a function would be of the form m
with cj + 05 increasing sufficiently fast. This is, of course, also the idea of proving the convergence (1) related to lql 4 w . But in the following section we shall see that the kernel cut-off near the diagonal has some advantages for evaluating the regularity of remainders as mappings E +E between the fibre spaces. 5. Definition. S@)(RX (Rq\ {O]); E, E") a E C"(R x (Rq \ [O]), S?(E, E")) with
denotes
the
space
of
all
(16) a (y , 1 7 11 = 1' % a ( Y , tl 1x;' for all 1 E R+,y E 0,q E Rq\ {0]. Moreover we denote by S(.)(R X Rq; E, h?) the space of all a E C"(R x Rq, S?(E,2))satisfjing (16) for all 1 2 1 and all q E Rq, lql 2 const, with a constant depending on a. Analogous notations will be used when a depenh cn ( y , y ' ) E R2.The spaces of ( y , y ' ) independent a will be denoted by S("(R4 \ { O } ; E, E ) and S(r)(Rq;E, E), respectively. 6. Lemma. S(p)(R X
Rq;
E, E") c S"(R X R*;E, E"),
xS@)(R x (Rq\ {O)); E, E") c S'(R x Rq; E, E") for any excision function ~ ( q (i.e. ) X E Cm(Rq), x (q1 > const).
=0
close to q
= 0,
x
(18) = 1 for
ProoJ Let us first show that
0:S(P)(Ox (Rq \ {O}); E, E") c S(fl-lal)(Rx (Rq \ {O]); E, E"). Set ah(y, q ) = 1hI-l { a ( y , q + b ) - a ( y , q)], h E Rq \{O], Ihl small. Then for every fixed a E R+ it follows ah(,% 1q) = a"-'2AaMO:q ) x ; I for h ' = h l l . From this we easily obtain the above relation for I a I = 1. Then we can proceed by induction. It is also obvious that derivatives in y preserve the relation (16). If a E S(F)(RX (Rq \ {O}); E, E") then Xa obviously satisfies the symbol estimates 3.2.1. (1) for 01 = B = 0. The above calculation shows immediately the same for all a,B. This proves (18). Then (17) is an obvious consequence. 0
294
3. Operators on Manifolds with Edges
7. Remark. Let ao: E
,i? be a continuous operator. Set
-+
for 12 1, 1111 z const.
In fact, we have [ q ] = l q l for Iqlzconst. and hence [Aq]”=A[q]fifor 1 2 1, 1 1 12 const. Moreover k(q)= kIvr,x”(1q)= kAk[,,land the same for xA, 1 2 1, 1 1 12 const. This shows (20). The groups xi, kAdo not act necessarily smoothly in 1. For instance (xiu)( t ) = A1’*u(l?),u E L2(R+),is not a C” action on L*(R+).So we do not know whether (19) belongs to S(”)(a x Rq; E, E ) . 8. Lemma. S(P)(RX Rq; E , E) is not empty.
ProoJ: Choose a function rp E C,”(R) with rp(0) = 1, supp rp c [Set for a moment
E, E]
for 0 < E < 1/2.
is well-defined, since rp is in C,”(R+)with respect to 6 E R+ for every fixed 1.We have for every u E E m
m
m
m
3.2. VDO-s with Operator-Valued Symbols
295
Analogous estimates apply for the derivatives of a,. Then ae(IV1) E
S(’)(Rq\{O}; E * E ) >
a , ( [ q ] ) E W)(Rq; E , E).
Now let us assume
s
E
to be variable and write pdinstead of p. Moreover let
m
pch 0,
p,(t)
d t = 1.
-m
9. Proposition. For every fixed 1 and u E E it follows
a,(R)u+ao(A)u in the norm of
for E + O ,
E.
Proof We have
(’ ’)
a,(A) u - ao(1) u = l-lpe - [a0(6)- ao(2)l IJ d6 0
and II(ac(1)
-ao(A)) ~ 1 1 i~A 5 - ’ P e ( F ) l [ ( a o ( d ) - d n ) ) UllgdS 0
6 supII(ao(6) - ao(1)) UIIE, dal,
(’ ’)
where Z, denotes the support of A-’ pc - in 6 for fixed 1.Here we have used that J 1-’p,((l- 6)/1)d6 = 1. Since a o ( l )u is continuous in 1,and Z, shrinks to 1 as E - 0 we get our assertion. 0 10. Remark. It is often intezesting to know whether S@)(Rq;E , l?) contains an a ( q ) which induces E for eueryjixed E Rg.In concrete cases this can be proved, indeed, cf: 3.2.5. Proposition 3 below. an isomorphism a ( 7 ) : E
Let us make a further _remark on homogeneous operator-valued functions. Let o ( q ) E S@)(Rq\ [O}; E , E ) . By introducing polar coordinates q = (e, p) E R, X S 4 - I in Rq\ [0} we obtain
In view of the growth conditions on the norms of Is, and x p for there exists a po E R such that for every p 2 po and e E R, there is a constant c(e, p ) such that al(p) = a(1, p).
e+ 0, e+
m
296
3. Operators on Manifolds with Edges
for every e E R+,and it is obvious that A ( @ rp) , E S@+l)(Rq\ {0}; E , E“). By definition we have
The operation above.
Q+
A is a counterpart of the calculation in the proof of Lemma 6
11. Definition. An a ( y , y ’ ,q ) E S,((02 x Rq; E , E“, is called classical if there is a seE S@-’](aZ x Rq;E , E“) with Q C a, -,). By $‘l(02 x Rq; E , E ) we dequence E , E ) denotes the space of all note the space of d l classical Q(Y,Y’,8). Further Lfl(a; A E L,((n; E , E) of the form A = Op ( a ) with some Q(Y,Y’, q ) E Sf,(a2 X Rq; E , E ) .
-
12. Remark. For any given a ( y , q ) E Sg,(fl X Rq;E , E) we can recover in a unique way the sequence of homogeneous components a(,,-,)(y, q ) E S(”-J?(flX (Rq\ {O)); E , E)
for which a
- 1Xa(, -n,x an excision function. In particular we have
-
a,,(y,q) = liml-rZ;’a(y,lq)xA A+
(23)
in the operator norm of S?(E,E”)).
In fact, we have ~ ( yq ), = a(,)(y,[ q ] )+ r ( y , q ) with r E Sf1-’. Now it suffices to observe that a(,)(y,?I> = lim A-’cil a(,)(y,[Aql) %A A+-
which is trivial and 0 = lim A-NZX”;’r ( y , A q )x1 which follows from I+-
lim l l A - ( , - l ) c ~ l r ( y , A q ) 5 ~ ~const ll I+-
as a consequence of 3.2.1.(1). Let us now return to the yDO-s A E ZYQI; E , E), cf. Definition 2. The relations (8), (9) show that for every A E L”(l2;E , E ) there is an A. E P(0;E,E“) which is properly supported and A - A. E L-”(O; E , E”) (properly supported means that the projectionsofsuppk(y,y’,y-y’)for k ( y , y ’ , O = K ( a ) ( y , y ’ , OA=Op(a), , toOyand a,,,are proper, i.e., the preimages of compact subsets are compact). In fact it suffices to pass to ao(y,y’,q ) = x ( y , y ’ ) ~ ( y , y q’ ,) where x E Cm(a2),x = 1 in an open neighbourhood of diag 0,supp x proper. In that case A . = Op (ao) is just of the desired sort. After the calculations in the beginning of this section around the pseudo-locality of 3.2.1. Proposition 9 the reader may check that this is true indeed. Note that the distributional kernel k ( y , y ’ , y - y’) is defined for (y,y’) E a2with respect to the first couple of arguments but for y , y’ E Rq with respect to (= y - y’. If necessary we restrict y , y ’ also in ( to 0.For = IRq we can pass to a properly Proposition 3. In that case supported kernel also by the cut-off by means of ~ (in 0 the associated ao(y,y‘, q ) has even a holomophic extension to q E Cq. It is easy to see that when A E L”(Rq; E , E ) is properly supported, we obtain instead of 3.2.1.(5) continuous operators
297
3.2. vDO-s with Operator-ValuedSymbols
A : W&mp(Rq, E ) + W&,&(Rq,E"),
WSoc(Rq,E ) ---* W;; "(RQ,E)
for every s E R. They extend the continuous operators
A : Ct(Wq, E ) + C;(lRq, 81, C"(Rq, E ) + C"(R4,
E")
Let A E Lfl((R4;E , 8)be properly supported and fix f~ E . Then eiYvfE C"(Wq, E ) for fixed r ] , and A(eiYqA E C"(R4, 8). f+ a ( y , r ] ) f : = e-iY?A eiyqf
(24)
defines a function a ( y , r ] ) E C"(Rq X Rq, 9 ( E ,I?)). Now as in the calculus of scalar lyDO-s for u E C"(Rq, E ) it follows ~ u ( y=)
J eiYva(y,
r ] ) ~ ( r ] dr]. )
Indeed, u ( y 9 = j eiY'qG(r])dr] implies Au(y)=
s A eiY'vG(r])dr] s eiy'J(e-'YVAeiYv)G ( v ) =
br].
An a ( y , r ] ) E Sfl(RX Rq; E , E", is called a complete symbol of A A - Op(a) E L-=(a;E , 8).
E Lfl((Q; E,
8) if
It is not our aim to repeat here all the classical elements of the lyDO calculus. For future references we only formulate some standard theorems, here in the version of operator-valued symbols that can be obtained in an analogous manner as for scalar lyDO-s. The proofs can be found in [H3], [S27], [T9]. 13. Theorem. Let A E L@((R;E, B) be given by 3.2.1.(2) and properly supported. Then A has a complete symbol uA(y,r ] ) E Sp(a x Rq; E , E). It admits the usymptotic expansion
Observe that the expansion on the right of (25) can also be obtained in the following way. Let a ( y , y ' ,7) E Sfl(12zX Rq;E , @ and apply the Taylor expansion
Then A = Op(a) can be written in the form
Using the identity D ; ei(Y-Y')4= ( y - y')" ei(Y-Y')vand integrating by parts we get
~N(y,y',r ] ) is flat of order N in y - y' on the diagonal, so we may apply the trick with integration by parts to Op ( a N )and obtain that
298
3. Operators on Manifolds with Edges
OP ( a d = OP with cNE S'-N(R2q x Rq; E , E). Thus N-1
ordOp(a) Ial=o
1 1 D ; a ~ a ( y , y ' , 7 ) l r = Y . ) + - m as N + m . a*
14. Theorem. Let A E L'(Rq; F, E"), B E L"(R4;E , F ) , and A or B be properly supported. Then AB E LP+"(R'J;E , E). U both A and B are properly supported then C = AB is also properly supported and for the corresponding complete symbols we have
Assume that E , E" belong to Hilbert space triples { E , EO, E'; x A } ,{E,E0,p ;gA}in the sense of 3.1.2. Definition 8. Define formal adjoints by
15. Theorem. A E L"(Rq; E , E") implies A* E LW(RQ;p,E') and any complete symbol a,. of A* admits the asymptotic expansion
where u:"(y, 7) is defined by (aaf,g ) p = (f, a:"g)e for every fixed y , 7 and f E E , E.
gE
Now let x : 0-0' be a diffeomorphism. Denote the coordinates in T * R and T * f l ' by ( y , 7) and ( x ,0, respectively. Let A E L"R; E , E),
A : C;(f2, E ) + Cm(R,E").
-
Then we get an operator A'
= %*A defined A
X'
C;(R', E )
C;(R, E ) -
by the composition
Cm(R,E") A Cm(R',E"),
(29)
where x * ( x * ) is the pull-back (push-forward)under x . Let dx: TO+ TO' be the differential of x, dx = (x, x ' ) , and tx,:Rf +R4, be the transposed of x'. Then we have the following 16. Theorem. A diffeomorphkm x :
R+ R' induces a push-fonvard
x * : L'(R; E,E")+LJ'(a'; E , E )
by (29). For the complete symbols dA(y,q ) and bAr(x, f ) of A and A' = x* A, respectively, we have
aA'(x,0I x -
x(y)
1 - C2 a
@:#A)
( Y , 'x'(Y>0 Qa(Y, 0,
where Qa(y, f ) = D: eih(y.r.fiIz=y, h ( y , z, 0 = { x ( z ) - x ( y ) - x ' ( y ) (z - y ) } . f .
Let us conclude this section with a compactness result, cf. also [L7].
(30)
299
3.2. yDO-s with Operator-Valued Symbols
17. Proposition. Let a ( y , q ) E S'(R4 X Rq; E , E"), E , E" being separable Hilbert spaces, and Y < 0. Moreover let supp a be compact in y and a ( y , q): E +E" compact for a l l y , q. Then o p ( a ) : WS(R4, E) + WyRq, E") is a compact operator for all s E R.
Pro08 Let us first assume that supp a is compact with respect to (y, 7). Choose a . . . , N and define partition of unity for RQx Rq of product form { p i ~ j } lj. = &(Y, V ) =
1pi(Y>vj(qt>a ( y i , q j ) ,
where y i E supp pi, qj E supp y j . Let s > 0 , M E W be given. The partition of unity may be chosen so fine on supp a that llx-'(q> q
W Y , 11) - O(Y, ?)) m)ll9w,fi < 6
for all ( y , q ) E RQX Rq and lczl 5 M. Then (cf. 3.2.1. Lemma 7 and 3.2.1.(12)) IIOP(Z - a ) Il%(W: rs)5 C.5
(3 1)
provided M = M ( s ) is large enough. Here C is a constant depending on the support of a . Let u E Ws(Rq,E). Then O P ( U~( Y > =
-
1p i ( Y ) a ( y i , q j ) J vj(q)~ ( 9d t)
represents O p ( 3 as composition
Ws(Rq,E )
E --j E+ Ws(Rq,i?),
where the middle factor is the compact operator a ( y i ,q i ) .Hence Op (3is compact and (31) implies for s-0 that Op(a) is also compact. The assumption that the support of a ( y , q ) is compact in Rq x Rq may be removed by using the fact that any pseudo-differential operator of strictly negative order may be approximated as operator on these spaces by an operator Op(a'), supp a' compact in Rq x Rq. 0
3.2.3.
The Scale Axiom
The calculus of yDO-s with operator-valued symbols can be enriched by further axioms which lead to more precise results for the remainders in asymptotic procedures and to a more realistic abstract model for the applications. The example at th_e end of Section 3.2.1. shows that in the "naive" version of the theory with single spaces E, E, .. we have neglected information that is in practice often known, namely that the spaces run over scales Z = with certain standard functional analytic properties. We shall now study the yDO calculus under this additional aspect. Remember that in Section2.1.4. we have already discussed such ideas in the context of the Mellin transform and of symbolic structures based on reductions of orders. Here we prefer to continue the approach of the preceding sections with groups R. Let e be a system of scales 8 = {ESJssR of Hilbert spaces Es which satisfy the following conditions
.
w.1) there are continuous embeddings E"' c+ E S for every s' )= s, (E.2) E m = Es is dense in every E:
n
SER
20
Schulze, Operator8 engl.
300
3. Operators on Manifolds with Edges
(E.3) the E o scalar product (. , .)o induces non-degenerate sesquilinear pairings (. , .)o: E s X E-”C
for all s which admit the identification E-8 = ( E S ) ’ , (E.4) there is a group [ x i } E C(R+,.%e,(Es))such that for all s
{E”,EO, E-8; x l }
(1)
*
is a Hilbert space triple with unitary actions, cf. 3.1.2. Definition 8, (E.5) if Z = [ E s } , = [fi}E @ and a E 2 ( E s ,I?-#) for some p E R then 110
n
S€R
lls.s - 5 c max { 110 lls~.s~ -
@,
lla l l s ~ ~-. s,} ~ ~
(2)
whenev_er s’ s s 5 s”, with a constant c = c(s’,s”). Here e(EJ,E‘).
ll.lls,, denotes the norm in
ep(Z) = WJZ, a). In view of (E.5) the space (3) has a natural Frbchet topology. Any A E P ( Z , Z) has a formal adjoint A*, i.e. (Au, u ) ~ :( u ,A*u)@, first defined for u E E m , u E %-. This extends in a unique way to an A* E Se@(Z, a). 1. Remark. We could modify our system of requirements by allowing dual scales Z* E @ which are not necessarily equal to Z. This would imply A* E W (Z*, Z*). For simplicity we will not formulate this aspect in general. But in the applications we shall tacitly employ it, for instance, for Z = {X.. “(X^)}R ,
e=
{XS.
q r ) }E , J
y , PE R fixed. Here the dualities refer to Xoqo(X^)(+ Eo or E*O in general).
Next assume that for every pair Z,
YqZ,
e~@ and every p E R there is given a subspace
e)c W ( Z , e)
(4)
with the following properties for all p , v E R
e),
Yp+’(Z,f)2 Yp(Z, A E Y#(Z,Z’), B E Yy(Z’,2) irnply BA E Yp+”(Z, further I Ep ( Z , Z), AEYp(5 implies A* E W(Z, Z), YTm( Z, Z) = 9--( Z, f), where - m indicates the intersections over p E R, of the corresponding cksses of order p. (AS) A E Y p ( Z, Z) imylies S;’ Ax1 E Y p ( Z, f)for all rZ E R+, (A.6) if A,E Y @ - ’ ( ZZ), , j c IN,is an arbitrary sequence there exists an A E Y@(Z, f) with
(A.l) (A.2) (A.3) (A.4)
e),
e)
N
A - ~ A , ~ Y p - ( ~ + l ) ( Z , ? ) f o rN a lEl N . j-0
Then A is unique modulo !Fm(Z,9). Below we shall impose further natural axioms as they are known for standard vDO-s. Let us pass to the requirements on the symbolic structures that are fulfilled in the concrete cases. (S.l) To every Y ” ( Z ,f)there belongs a space of “principal symbols” P ( Y , topology and a surjective linear mapping up:
with
Y q e , e)+zqZ,
Y p - l( Z,
e) = ker
up,
e)
e)with a Frbchet (5)
301
3.2. vDO-s with Operator-Valued Symbols (S.2) there is a linear mapping 0p’: Z’(Z, 2)- Yfi(Z,d )
being a-right inverse of @, and (6) is continuous with respect to the topology in Yfi(Z,Z) induced by (4) (ope is called an operator convention), ( S . 3 ) there is a bilinear mapping D(Z,Z’) x Z”(Z’, d ) + Z f i + ’ ( Z , d ) (written as composition) such that A E Y p ( Z , Z’), B-E Y’(Z’,?) implies a’+”(BA)= a’(B) oP(A), (S.4) there is an involution *: P(Z,Z ) + P ( Z , Z) with a”(A*)= a@(A)*. The procedure of Section 2.1.4. yields a system of linear mappings Aj: Y q Z , d)+e+-J(Z,d ) , q : Yfi(Z,d ) + P - J ( Z ,d )
(7)
and we take P ( Z , d ) in the weakest locally convex topology under which all these mappings are continuous. (A.7) P ( Z , $?)is a Frbchet space in this topology. (A.63 Let U S R,” be open and A j ( x )E C“(U, Y ’ - j ( Z , g)),j E W,be an arbitrary sequence. ~ ~ sequence , ~ ~ of ~ functions m in Cm(U).Thecthere Moreover let ( ~ ~ , ~ be} an~arbitrary is a sequence A ; ( x ) E C”(U, Yfi-j(Z,a)) with A j ( x )- A J ( x )E Cm(U,F ’ ” ( 8 , Z ) ) for all j, such that m
C cj,B(x)D$A;(x)
j=O
converges in C”(U, Yp(Z,2)) for all p E Nm. From now on we assume that all the mentioned axioms hold. 2. Definition.L e t 0 E RP be open, p E R. By S’(D x
Rq;
Z, d ) we denote the subspace of all
a ( y , q ) E Cm(aX Rq, Y r ( Z , d ) ) With (0 D , ” D { a ( y , q ) E C m ( f 2 x R qYp-lbl(Z,$?)) , n S~-181(aXRq;E~,,-’+IsI) for all a E NP,/IE Wq and all s E R, (ii) K ( a ) ( y , 0 = $ eicqa(y,q)Uq(defined in the sense of oscillatory integrals) satisfies the following = 1 close to [ = 0 , we have scale axiom: for every v([) E C;(Rq),
The notation P(aX Rq; Z,?) was used for abbreviation, though these spaces also depend on the operator classes !?‘#(a, Z). But we keep them fiied once and for all. x R‘J;Z, Z) are operator functions with values in Y-@, i.e. Note that the elements in sm(a “smoothing” along the fibre spaces. The definition gives rise to a system of natural mappings Lap:
sqa x R P ; ~ , d-cya )
x
RP,
Y M - ~ ~ z ,
n
d ) ) sr-ln(a x
E S , ~ - f i + i ~ i )
(8)
S
according to (i) for all a E NP,9, E Wq, and further wk:
sqa x ~
4
Z,; d1-n c - ( ~ , Y ( R ~ , z ( E ~ , E - ) )
(9)
S
defined by a + ( l - v ( Z k n ) K ( a ) , EN. Then Sp(R X Rg; Z, $?)is a Frbchet space in the projective limit topology with respect to these mappings. It is independent of the choice of the function v. 20’
302
3. Operators on Manifolds with Edges
The case of 3.2.1. Proposition 10 fits to our definition. In fact, we can set
e = {H"R")}$
Z=
E
R.
Define Yp = YP(Z,Z) as the space of all A E L;,(R:) with complete symbols p ( x , 0 E Y(R~)@,S~,(Rm),on,, further Y-"= Se-" = Se-"(Z). Then the above axioms (E.k), ( A j ) , (S.0 for all k, j, I , including j = 6', are fulfilled. There also exists a parameter-dependent version with parameters ( y , v ) E fl X Rq,based on amplitude functions P(x,.Y,F,
E
C"(a> @nYP(R?) @nS%RmX
R'9cona.
Then for the operator family 3.2.1.(14) we have a ( y , 71) E S '(R
X
Rq; Z, Z).
The property (i) of Definition 2 employs 3.2.1. Proposition 10, where we use that differentiations with respect to t] diminish the order of p in all covariables, i.e. including [. For (ii) we may argue similarly as in the kernel cut-off theorems of the type 2.1.4. Theorem 15.
z)
3. Definition. Let a 5 Rq be open, p E R, then L#(fl; Z, denotes the space of all operators = Op(a), cJ: 3.2.1.(2), with a ( y , y ' , t ] )E S'(f12 X Rq; Z, Z).
A
By definition we have
L-"(fl; Z; 2) consists of all G: Wt,,,(fl, E s ) + Wir(i2,%") for which G E z(w:omp(fl,
E5),WiXfl, Em)),
G' E 5?W:,,,(fl, PI,WiXfl, E")) for all s E R. In view of 3.2.1. Theorem6 every A E L'(0; Z, e) induces continuous operators A:
w;,,,(fl,
Er) -+W;;'(fl,
E"'- ')
(10)
(11)
for all s,r E R.For simplicity we restrict ourselves to r = s. From K ( a ) ( y , 0 = F;!.(a(y, 11) we get a bijection F-l:
SP(R2 X R:; Z,
2) + T"(f12X Rf; Z, e)
(12)
to a space T " ( 0 2x Rq;Z, 9) that we consider in the induced Frkchet topology, fl E Rp open.
4.Theorem. Let aLe S-j(f12 x R'; Z, i?), j E W, be a sequence. Then there exists an a E S P ( f l * X RQ;Z, 8)such that for every N E N N
a
- 1 a j E S r - ( N + l ) ( f l 2 x RP; Z, 2) j=O
and a is unique modulo S-"(f12X Rq;Z,
2).
Proof: The proof follows the lines of that from 3.2.2. Theorem 4 with the appropriate modifications. First note that for the same reasons here we may neglect the groups (xA},{%,,}, such that for notational convenience we drop them. Furthermore for simplicity we consider the y, y' independent case which is the specific one. Now we shall again pass to the kernels and show that there are bj E S r - J with aj - bj E S-" such that for some choice of constants c,
303
3.2. lyDO-s with Operator-Valued Symbols m
K=
1v ( c j O K ( b j ) t O
j=O
converges in the space T’. We tacitly use some technique from the proof of 3.2.2. Theorem 4 such as the repeated removing of finite partial sums of a length depending on the choice of a fixed semi-norm n in the space T’. Thus we will have no additional difficulty with the extra gain of regularity along the fibre spaces that is required in the contribution S”lfll(O x R Q ;Es,& - p + I p I ) in (8). The C”( ... , Y@-IfllQ..)) part will also make no troubles, since for every fixed semi-norm n1 of Cm(... , Yp-lfll(Z, Z)) that relies on th_e Aj in the corresponding topology of Yp-lpl, cf. (7), and any sequence E P - J... ( ; Z, Z) we have nl(ap-j)= 0 for all j t j , with some jl
=il(nl).
So let us concentrate on the second contribution on the right of (8). Tken we have to deal with the semi-norms n of the form 3.2.2. (14) where L stands for S?(ES,E r ) , where r = r(B). Let us form aj by the same procedure as in 3.2.2. Theorem 4 and consider the sum
e))
e))
where Cj,oE C,”(Rq,Y@-j(Z, is defined by aj,o- Cj,oE C,”(Rq, !Fm(Z, and a convergence condition that we shall derive from (A.6’). As in the proof of 3.2.2. Theorem 4 we obtain the convergence
with respect to n, where cj is an appropriate choice of constants. But then we can arrange also in such a the convergence of (13) since in spite of the factors [FDF and gj(cjO we find way that
C
tu(cjO K(Cj.0) (O
j tjo(n)
also converges with respect to R. This can even be done in a manner that the choice of Zj,ofits to all n in question, since the convergence of all (14) is ensured before for all n. Then we set bj = Cj + Cj,o. Now (13) converges automatically with respect to the semi-norms from (9), since every such semi-norm is non-zero only on a finite number of summands of (13). 0
e)
Any A E L@(O;Z, can be wrjtten as a sum A = A. + C, whe? A. E L’(O; Z, @) is properly supported and C_EL-”(O; Z, Z). An a(y, q ) E-S~(OX Rq; Z, Z) is called a complete symbol of A E L’(i2; Z, Z) if A - Op(a) E L-”(O; Z, Z). The following two theorems may be regarded as exercises.
5.Theorem. Let A E L’LO; Z, 2) be properb supported. Then A has a compleie symbol a ( y , q ) E P ( O X Rq; Z, Z) with the aqympiotic expansion 3.2.2. (25) (valid in the sense of the symbol classes of Definition 2). 6. Theorem. The analogues of 3.2.2. Theorems 14, 15, 16 are valid also in the present set-up of gD0-s in the sense of Definition 3 .
3.2.4.
Amplitude Functions with Reductions of Orders
We shall now study an alternative variant for a pDO calculus with operator-valued symbols, based on reductions of orders. It will be formulated in terms of axioms that we can read off also from a corresponding reformulation of standard WDO-s.
304
3. Operators on Manifolds with Edges
Some elements in the context of Mellin wDO-s have been discussed already in Section 2.1.4. The motivation for the second approach is that it is not always convenient to deal with a localized theory but with someone being global along a C” manifold. Remember that in the example at the end of Section 3.2.1. it was important to have a group {xA} acting on Hs(Rm).If we replace R”, for instance, by a closed compact C” manifold X then we loose this structure. Nevertheless it may be reasonable to talk of wDO-s anlong RP with amplitude functions taking values in S?(H3(X),H’(X)). In other words we should look first at a reformulation of Sobolev spaces H 5 ( X X RQ)which substitutes 3.1.1. Lemma 1, and then at U ( X x Rq), written as a class of wDO-s with symbols, acting along X globally. This will show at the same time from a new paint of view the role of parameter-dependent wDO-s. In the beginning of Section 2.1.1. we have mentioned a family bF(q), q E Rq, of ryDO-s in L$(X;Rq) for which b”(q): H 5 ( X )+Hs-”(X) (1) is an isomorphism for all s E R. Then we can define H 5 ( X X RP) as the closure of C c ( X X RQ) with respect to the norm
IIuIIH~(xxRP)
=[
~ l l b ” ( ~ ) ( ~ u~) (+v q, .)I ;z(x,
dq
Y’’
(2)
cf. also 2.1.1. Definition 1. In this way we obtain a definition of H 5 ( X X Rq) which is equivalent to the usual one (based, for instance, on localizations, i.e. x*(Q,u) E H5(RmX Rq) for every coordinate neighbourhood II of X, x : U +Rq being a chart, Q, E Cc(U)). The spaces L’(X; Rq), L&(X;Rq) have natural Frtchet topologies, and we can form P ( X ; 0 X Rq) := C”(0) B nP ( X ; Rq) (3) and similarly Q ( X ; 0 X Rq), 0 E RP open. This yields operator families a(y, q )E
C”UJ x Rq,W X ) )
for which
0;D{ a ( y , q ) E C“(0 x Rq, L”-IBI(X))
(4)
for all a E NP,b E Nq, and Ilb”-”+’lB’(q) (Dy”D{
11)) b-S(V)lle(Lz(x,, 5 c
(5)
for all a E NP,3/ E Nq, s E R, (y,q ) E K X Rq, K cc0, c = c ( a , b, s, K ) a constant. Further we have an analogue of the scale axiom of 3.2.3. Definition 2. In the abstract setting we use analogous notations as in 3.2.3. such as C for a system of scales, where cow we ass_ume all properties ( E . i ) except of (E.4). Further we deal with the spaces W(Z, Z), ,E”(Z, Z) with all properties (A.j),(S.k),except of (AS). The role of the groups { x A }play now order reducing families with properties as they are known in the special case of (1). It is assumed that for every Z E CE there is given a family b”(q)E C“(Rq, P ( Z ,
a))
(6)
with the following properties (B.l) b”(q): E x + E S - ” is an isomorphism for every s, p E R, q E R*, b - ” ( q ) = ( b ( q ) ) - * , b o ( q ) equals the identical operator, (B.2) D{ b ( q ) E C”(Rq, Y”-lfll(Z, a))for every B E “7, (B.3) for every y, 6, v, S E R, B E Nq, v z p : = y + 6 -[!I, we have
with a constant c = c(s, p, v ) > 0, p ( ...) defined by 2.1.4.(6),
%(ES,E‘),
Il.ll%,being the norm in
305
3.2. wDO-s with Operator-Valued Symbols
(B.4) bp(q) satisfies the scale axiom, cf. 3.2.3. Definition 2 (ii). Let us introduce in E s the parameter-dependent norms
IIUII~:~:=I I ~uiiE0 W
(7)
and in S?(Es, E"') Ilallsr;'t:= Ilfi(tl) ab-"(tt)lliecE0,E4,
(8)
where b;(q) denotes the o@er reduction belonging to 9 = (8.).Then a further condition is (B.5) for every a E Sp(Z, Z) we have llall ss - p ; 't 5 c Im= lla I1 .f.d - p ; 8 , lla 1 I s",E - p ; 7 )
for all s,s', s" E R with s' 5 s 5 s" for a constant c = c(s', s") independent of q. Note that (B.3) implies in particular
R C R p be open, p E R. By Sp(R x a ( y , q ) E cm(o x RP,P(Z, 9))
1. Definition. Let
with (i) Dy"D{ a ( y , 0 ) E Cm(Rx for all a E NP, B E Nq, (ii) for all s E R we have IID;D:a(Y>
Rq,
Yp-lfll(Z,
v)llss-p+l+8l;'t
Rq;
Z,
2) we denote the subspace of all
e)) (11)
5c
for all a E NP, B E Nq and all (y, 4)E K X Rq, K c c R , with a constant c = c(a, P, K ) > 0, (iii) a ( y , q ) satkfies the scale axiom, cfi 3.2.3. Definition 2, (ii).
For simplicity here we have used the same notations for the spaces of amplitude functions as in the preceding section. We hope this will not cause confusions; in any concrete case it will ) on reductions of orbecome clear whether we are in the context of wDO-s based on ( x ~ or ders. Usually we are interested in the cases p = q or p = 2q. For p = 2q we also write (y, y') in the arguments. Here we use in parThe definition induces apatural Frbchet topology in Sp(R X Rq; Z, ticular (B.5). If Sp(Rq; Z, Z),,,,, denotes the subspace of all y-independent a in the induced topology then
e).
S#(R X R a ;Z,
9)= Cm(R)8, S p ( R q ;Z, ~),,,,t.
Remember that the symbol estimates in (ii) say that
llp-p+1+81(v) (Dy"D: a ( y , 4))b - s ( ( t l ) l l ~ ~ ~6Oc. .~ ~ This condition can be replaced by the system of estimates
IlDy"D: (b"S-p+%)
r l ) b - S ( ~ ) l l l ~ ( ~5o c, ~ ~ )
without changing the spaces S@(...). Note that b"q)
E
S"(R X
Rq; Z,
Z),
p ER
It can easily be proved that a, E Sp(R X RP; Z ', a l a z E S p + " ( 0 X R ' 7 Z, ;
e).
e),a2 E SV(RX Rq;Z, Z') imply
(12)
306
3. Operators on Manifolds with Edges
In particular S"(R X Rq; Z,
g) = {k'ao: a.
E So(OX Rq; Z,
g)]
So(R X Rq; Z,
f)}.
= {sob@: a. E
2. Theorem. Let R 2 RP be open, p _ R, ~ and a, E &'@-'(a X Rq; Z, there exists an a E S'(R X Rq; Z, Z) such that for every N E W
f),j E N,be a sequence. Then
N
a
- 1 a, E S"-(N+')(aX Rq; Z, f) ,=o
holds, and a is unique mod S-=(RX Rq; Z,
9).
Proof: The proof follows by similar arguments as for 3.2.3. Theorem 4. The main difference here is the other semi-norm system for the space P. But the method of defining a convergent sum of kernels does not employ that in an essential way. For every fixed semi-norm from (1 1) we can remove a finite partial sum such that the orders of all remaining terms are negative enough. Because of the growth properties of the order reducing families (B.3) it is allowed, up to a finite loss of order, to replace the norms in (11) by non-parameter depending ones. Thus the specific novelty disappears and we can proceed as before. 0 3. Definition. Ws(Rq, Z), s E R, denotes the completion of Ct(R4, E") with respect to the norm
Many constructions of 3.1.2., 3.2.1.-3.2.3. have analogues in the present set-up. In particular we have continuous embeddings W"'(Rq,Z)
G
W"(Rq, Z) for s' 2 s .
The scalar product (u, ~ ) w ~ ( R s , Q J= ( ~ y - , u , F y - , u ) ~ ~ d t l induces a non-degenerate sesquilinear form WS(Rq,Z) x W-s(Rq, Z) + C which admits the identification Ws(Rq, Z) = (WS(Rq,Z))' 4. Proposition. WS(Rq,Z) is a C,"(Rq) module. Moreover
-/U,+O for
Q,
in S?(Ws(R*,Z))
+O in Ct(Rq).
The proof is practically the same as that for 2.1.6. Theorem 1. Following the standard scheme for Sobolev spaces we also can define the spaces W~omp,loc(R, Z) for any open R E Rq.
5. Dewtion. Let R S Rq be open, p E R, the? L*(R; Z, A = Op(a) with a ( y , y', q ) E S"(R2X RP;Z, Z)).
f) denotes the space of all operators
307
3.2. yDO-s with Operator-ValuedSymbols
Every A E L'(R; %, ?) can be writtep as a sum A perly supported and C E L-"(R; %, %).
= A,,
+ C, where A. E L'(R; Z, ?) is pro-
2) induces continuous operators A : w;,,,(R, a) + w;&"R,9)
6. Theorem. Every A E L'(R; %, for all s E R.
ProoJ: The assertion is obvious for A = Op(a) and a E S'(Rq; %, ~"),,,,,.Using (12) and Proposition 4 it follows then in general by analogous considerations as for 3.2.1. Theorem 6. 0
9) be properly supported. Then A has a complete ymbol in SJ'(0 X Rq; Z, Z) with the asymptotic expansion 3.2.2. (25), valid in the sense of Theorem 2. Furthermore the analogues of 3.2.2. Theorems 14, 15, 16 hold also in the present setting of vD0-s based on reductions of orders.
7.Theorem. Lei A E LP(R; %,
The proof is again an exercise.
Edge Spaces with Asymptotics The abstract wedge Soboleu spaces Ws(Rq, E ) of Section 3.1.2. will be applied for E = X$Y(X^),cf. 2.1.1. (39), and the subspaces with cwymptotics. The definition refers to (xn u ) (r, x) = ,i(n+1)'2 u ( l t , x ) , n = dimX. (1) The Hilbert spaces !X;"(X^)t', of 2.1.1. (33), B E B y ( " ) . or the discrete analogues !X;y(X^)$' P E P"'")(X),give rise to the Hilbert spaces 3.2.5.
XS,Y(X^)$'
%S,"(x-)y+ [l - w ] H"X^)
:= [ w ]
(2) and analogously to XS,Y(X^)Lj3. These spaces are independent of the choice of the cut-off function w. Moreover x, satisfies the assumption of 3.1.2. Lemma 1 for all s, y, B, (P), A, j. This yields well-defined spaces W S Y ( X " x Rq)
WS(Rq,X $ Y ( X ^ ) ) , WS,Y(X^X Rq)$' := Ws(Rq, X;"(X^)$'), :=
(3) (4)
and analogously with P. According to the abstract notations of 3.1.2. we set
W ; y ( X ^ x Rq),, = lim WJBY(X^XRq)$" t
(5)
jsN
and the same with P. We also adopt earlier conventions such as
W&Y^ x Wq),,
=
W $ O ( X ^ x Rq),, ,
(6)
w;yx-x Rq = W$BY(X^X Rq)(-_,o,,
(7)
W : l ( X ^ X Rq)A= lim W ; y ( X ^ X Rq),,,
(8)
d
B
W::(X^X
Rq);
=
-
lim W"pY(X^X Rq),, P
with the inductive limits over B E B y ( " ) and P E P"")(X), respectively, and so on.
(9)
308
3. Operators on Manifolds with Edges
For studying the Sobolev spaces with asymptotics in more detail we first want to establish some properties of the spaces (3). First there are continuous embeddings W d s ' ( X ^ X W4) 4 W4Y(X^
x
W4)
(10)
for s' 2 s, y' 2 y. For the proof we apply 3.1.2. Proposition 4 to the continuous embedding A : Xs'~''(X^) +X$Y(X^),where lI~-'(q)Ax(q)llS CA 5 CA[q]"-" for all 1.
The first inequality follows from the fact that A is the identity on X " f ( X ^ ) .Then the induced operator A" is just the embedding (10). For E o = Wo(XAX Rq), E = ' V Y ( X ^ X Rq), s, y 2 0, we can form 3.1.2. (21) which leads to the identification
W-$-Y(x-xRP) = W*V(X^X I R q ) ' .
(11)
A simple consideration (left to the reader) yields (11) also for arbitrary s,y E R. m
Note that
~ ~ V ~ ~ =& O t-" ( ~Ilu(i, - ~ y ) l l i ~dt ( ~and 0
is a Hilbert space triple with unitary actions on X o ( X ^ )(under the standard scalar X X ) ) , we get an associated triple of wedge Soproduct from X O ( X ^=) t-n'ZLZ(R+ bolev spaces, cf. 3.1.2. (26). 1. Theorem. The spaces W"V(X^X Rq) are modules over C,"(R+XR4). The multiplication by rp induces continuous operators
.Atp: W$Y(X^X Rq) + W".Y(X^X Rq)
for all s, y E R, and rp +.Atp defines a continuous embedding
n
C ; ( R + ~ R ~ - -~, ( W Y X x- ~ 4 ) ) S€R
for all ~ E R Analogous . statements hold for Corn@+) and C,"(Rq) instead of C;(R+ x R4). ProoJ From 2.1.1. Theorem 6 it follows that XJY(X^)is a C;(E+)-module and the operator of multiplication by rp tends to zero in Se(X$Y(X^),X$Y(X^))for rp +O in C;(R+), for all s,y. Now the group x1 acts on Xs*Y(X^)by the formula (1). Thus ( x ; ' r p x l u ) ( t ) = r p ( A - ' f ) u ( t ) , rp E C,"(R+), u E X'"*Y(X^). Applying 2.1.1. Proposition 8 (which is true in the analogous form also for Xs!Y)and 3.1.2. Proposition 4 we obtain that the multiplication by rp also induces a continuous operator in Ws*'(X^ X Rq) that tends to zero for rp +0. Now let y E C,"(Rq).Since (13) is a Hilbert space triple with unitary actions, the assumptions of 3.2.1. Theorem 8 are satisfied and hence the multiplication by y inx Wq) tending to zero for y +0. duces a continuous operator in Ws*"(X^
309
3.2.lyDO-s with Operator-Valued Symbols
Finally let x E C;(R+ XR9 and suppx c { ( y ,t ) : IyI2 + t 2 5 c}. Then there are sequences t ~ C~; ( {€l y l s c}), p i e C r ( { t5 c } ) such that vi-+O, pi+0 and
Then
W~II
C IAiI IWJ I W ~ ~ I I
(15)
9
where 11.11 denotes the norm of continuous operators in W s * " ( X ^X RQ).This proves the first assertion. The second one follows by using the property of the tensor product that the semi-norms of x may be obtained by i n f C IAiIp(vi)q ( p i ) > where the infimum is taken over all representations of x by (14), p being an arbitrary continuous semi-norm on C;( { l y l s c}), q someone on C;( { t 5 c } ) , From (15) then it follows the second statement of our theorem. 0 According to the general notations we set for any open R E RQ
w:bymp(xx a) = { u E w"yx^ x RQ):supp,u
c c a},
(16)
W;g(x-x a)= { u E 9 ' ( R ,.xs.qx-)):pu E W S , ' ( X ^ x RQ) for all
p E C;(R)}.
(17)
Here supp, denotes the projection of supp u to a. Further pu is interpreted as extension by zero to RQ. Let us mention a number of equivalent norms on the space WssY(X^x Rq). Choose an isomorphism PS*Y:XS*Y(X^)+XO(X^). Then IIPs*YfIlxo(X-, is an equivalent norm on XsvY(X^)= E . Denote it by I(.(IE and the original one by II.IIi. Applying 3.1.1. (15) we get the equivalence of the associated wedge space norms.
2 . Proposition. Set
310
3. Operators on Manifolds with Edges
3. Proposition. There exist% an as*"(q)E W(Rq; Xs*'(X^), XO(X^)),,,,, such that O P ( ~ ~ .Ws,Y(X^ ~): x Rq) + W ( X - x Rg) is an isomorphism.
Proof: Assume that we have constructed a 1qD0 a. on X^ which defines an isomorphism a o :Xs*y(X^)+Xo(X^).Then the operator family &(A) given by 3.2.2.(21) is certainly C" in A E R + , i.e., the smoothing convolution of the proof of 3.2.2. Lemma8 is not necessary in this case. In other words a o ( [ q ] ) belongs to S@)(Rq; Xs*y(X^), XO(X^))and it is an isomorphism Xs.Y(X^)+X o ( X ^ )for all q E Rq. For ,u = s we get by definition as9Y(q) as desired. Now the construction 4 0 can be performed by an analogue of the method for proving 2.2.3. Theorem 10 in the framework of 2.2.4. or alternatively by using a parameter-dependent analogue of the theory of 2.2.4. and a conclusion like that of 2.3.1. Remark 17. Note that weight shifts can be (if desirable) imposed afterwards by the arguments of Remark 13 below. The details are simple and left to the reader as an exercise. Let x : 8, xRq+ K, X R q be a diffeomorphism of the form x = (id, x ) with the identity id on 8, and a diffeomorphism x on Rq which equals the identity outside a compact set. 4. Theorem. The pull-back under x induces continuous operators
x*:
x R4)+ W S . Y ( X ^ x RP), and the same for the comp, loc spaces, for all s, y E R. WS.Y(X^
(20)
Proof: By Remark 3 we have A := Op(assy)E LS(R*;Xs*Y(X^), X o ( X ^ ) ) and A-' = Op((aSsY)-') E X O ( X ^ )Xs.'(X^)). , In view of the special form of x we may also write x. Now x*:
WS.Y(X^
can be written as x*
=
x Rq)+ x* WS.'(X^ x R'I) (x*A-lx*)%*A with the isomorphisms (18) and
x * : W(XA X R4)-
-
w ( X ^X
Rq),
(2 1)
cf. (12). From 3.2.2. Theorem 16 it follows that x*A-'x* E L-S(Rq;XO(X^), XsvY(X^)).Thus it induces a continuous mapping x*A-lx*: W-&JX^ x WP)+
W;;)cy(xx Rq),
cf. 3.2.1. Theorems 6, 8. The assumption on x implies that x * K ' xt has even constant coefficients outside a compact set i n Rq, i.e., it also induces a continuous operator W ( X ^ X Rq)+ Ws.Y(X" X Rq). This proves (20). It is then obvious that we get the same for the comp, loc spaces. 0 5 . Proposition, Let s' > s, y' > y , and
Cr(E+XR4). Then the embedding [a]WS'*Y'(X^x R4)+[p] W S - ' ( X ^ x Wq) Q,
E
is compact (cf: the notation after 1.1.2. Remark 3).
Proof: Let us first assume that Q, = y'e for p' E C,"(Wq), e E C,"@+), further choose a p E C,"(Rq) with yp' = w'. The embedding b : [e]Xs',y'(X^)+[e]Xs,y(X^)
3.2. WDO-swith Operator-Valued Symbols
311
is compact, cf. 2.1.6. Theorem3. Thus the symbol a ( y , q )= y ~ ( y ) [ q ] ~ is - ~of' bnegative order, compact operator-valued and of compact support in y. Thus it satisfies We the conditions of 3.2.2. Proposition 7, with E = [el xS'~Y'(X^),E"= [el XS."(XA). - ~6= ' ) Op(b). Our embedding in queshave Op(a) = w Op([q]"-"'b) = w O P ( [ ~ ] ~ 6, tion is
~ 6[w'] : WS'"''(X- X Rq)+[~p'] W r s * y ( Xx- Rq).
(22)
Let u E [w'] Ws',Y'(X^X Rq). Then
I l ~ ~ ~ l l L =- J~ [~ ~v 1, 2~ s l l ~ - ' ( ~ ) ~ ~ ( ~ ) 1 1 ~ d ~ =
.f ~ t l 1 2 s ' l l ~ ~ 1 " s ' ~ ~ ' ~ ~ ~ ~ ~ ~ ~ ~ ~ l l ~ ~ t l
5
J [ r 1 2 S ' l l ~ - ' ( t t ) ~ ~ =l l ~ d t l
IIwu11~qRv*E).
In view of
J
[ ~ l " - " ' x - ' (F~b) ~ ( qIIi) dtl I l O ~ ( b ) ~ l l L .=~ v ,[~1~"11 ~ and 3.2.2. Proposition 17 we thus obtain the compactness of (22). It is a simple exercise to show that the norm of (22) tends to zero when w'+O in Ct(Rq),e+O in Ct(a+). Then a tensor product argument yields the assertion in general. Now let M be a compact "manifold" with edges, cf. the notations in the beginning of this chapter. Consider for simplicity the case of one edge Y of dimension q (closed, compact, C", oriented). Every y E Y has a neighbourhood U such that there is a homeomorphism a0: U - T X R/{O} X X X R,R S Rq open. In all analytical considerations we pass to the stretched manifold W associated with M.This is a compact C" manifold with boundary, and there is defined a continuous mapping p o : W+ M which restricts to a diffeomorphism W\ a W+ M\ Y, every w E a W has a neighbourhood N such that there is a diffeomorphism 6: N + X ^ X R for open R E Rq, and the transition mappings x (an 6) + x (0 n 6) commute with the canonical R+ actions for small A E R, . Further the system of "charts" 6, is compatible with the system of 6 in the sense that the diagrams
x.-
a
-
a.
-
N-XA
1Po
u-x-
xR .1 no x R/{O} x x x R
commute, no being the canonical projection to the quotient space. We assume that the transition mappings x (0n 6) +X^ x (Rn fi) are only y dependent for sgall t , i.e. of the form (x, t , y ) +(x, 1, ~ ( y )for ) a diffeomorphism x : R n fi+ D n 0,t being small. On W we have the charts of the type 6: N + X ^ X R close to a W and further x : V+ G for open neighbourhoods V c int W, G S R"+' + q open, n = dimX. Let U be a covering of W by neighbourhoods of these two sorts and talk about edge and interior neighbourhoods, respectively.
x.-
312
3. Operators on Manifolds with Edges
6. Definition. Ws*Y(W)denotes the space of aN u ~! 3 ' ( i n W) t with - (S*)-'p,u E W:&p(X^ x a)for every edge neighbourhood and aN p, = 6* $, $ E C i ( X * X a), and (x*)-' p u E H ~ , , , ( G ) for every interior neighbourhood V, and all p, E Ci(y).
(It would be more adequate to talk about WsS"(int W) instead of Ws.'(W)but we hope our simplified notation will not cause confusions). Clearly the definition contains the fixed atlas, but it is independent of the concrete choice under an evident definition of equivalence. Thus the definition is correct. Note that
(23)
Ws.Y(W) c H;,,(int W).
Next let us return again to the local situation and to the subspaces with constant discrete q m p t o t i c s
W " P ( X ^ x Rq),
=
WJ(R4,xy(x-),),
(24)
A E gm,a = 0, 1 (cf. 2.1.1.(36), (37)), P E Py(n)(X). It is clear that X$y(X^),pis a subspace of X$y(X^),for s' 2 s, A' 2 A , which is invariant under the action of x A .This leads to continuous embeddings
W $ Y ( X ^ x R')& 4 W > Y ( X ^ x Rq),
(25)
and in particular
W $ Y ( X ^ x Rq),
4 W S * Y ( X ^x
Rq).
In Section 3.1.2. around Lemma 7 we have already considered in abstract terms the 1),2 + d = 0. Then analogous situation. Let P E Wn),A = [6, 01 finite, ncP n r(,+
E := X$Y(X^),= Eo + E" with
Eo = X y ( x - ) d = X q y - y x - ) ,
E"
=
ZF(X^), c X;*y(x-),,
This defines a "potential operator" LP'kJ =
Op(Pk.":
H"(R4)+
W $ Y ( X ^ x Rq),.
313
3.2.1pDO-swith Operator-Valued Symbols
F y + , ( L p , k - A(9) ~ ) = [ql("+1)'2(t[~l)-P logk(t[ql)w(t[V1)1(x) 3(71),
3(q)= FV-,v, v E HS(Rq). According to 3.1.2. Lemma 7 we obtain 7. Proposition. Let P E P"'"),A = [a, 01 E gl be finite, n, P n
r(,+ +
d
= 0. Then
we
have a direct decomposition W$"(X^XRq)d= Ws~Y+'(X^XRq)+ {+LP~k~AHs(Rq)},
(3 1)
where the sum is taken over ail (p,m , L ) with (28) and 0 5 k 5 m,1 running over a base of L. The intuitive meaning of the decomposition (31) is that every u(t,x,y) E W $ y ( X ^x Rq) admits a finite asymptotic expansion u(t,x,y)
- 1Uvj
(32)
with a remainder in the space with the better weight y - 6 , {U} being a sequence of potential operators, vj E Hs(Rq). By construction the asymptotic terms belong to WS~Y(X^ x W4). Note that they are not of the form t-Plogktw(t)A(x)f ( y )
(33)
with certain f E Hs(Rq), 1 E C m ( X ) .For instance the Fourier transform of (33) for k = 0 in Rq direction is t - p w ( t [q] ) 1(x) [?)I(" - p 8 71). This shows that the smoothness in y depends on Rep and it decreases for decreasing Rep. +
8. Remark. The decomposition (31) depends on the choice of the cut-offfunction w . I f w , is another H"(Rq)-+Ws*"(X^ X RQ). one and LTksAthe associatedpotential operator then L?k*A- LP*k*A: The potential operators also depend on the choice of the function q [ q ]. I f [ q l Ois another one, then for the associated potential operator L t we have -+
'3'
LP_.?*::=LtksA- LP.kA E S-"(Rq;C, x".Y(X^)),
Set L!?:,' = Op( r"P_.?qwith T?,k.I(q) = { [ q ] ~ + 1 ) / 2 ( t [ q ] o > - ~ l o g k ( t [ q ] o ) - [q]'"+"'2(t[q])-"log~(t[q])} w ( t ) @ A .
Since
lP_.2A - Il"_,_k.I E S"(R4; @, Xm>"(X^)), we get LF51-
L?5A1:HS(R'I)+
W","(X^x
R4)
for all s E R.Thus we can make the following 9. Remark. If [q10 is an arbitrary function with analogous properties as [ q ] ,then for the associated potential operator L t k ~ Awe have { L c k s A -L p , k - A } v =t - P l o g k t w ( t ) l ( x )f(y)modW"*"(X^XRq) for every u E Hs(Rq),where f E H"(R*).In other words the errors in this case are in fact of the form (33), modulo smooth flat functions which are accepted as remainders anyway.
3 14
3. Operators on Manifolds with Edges
The asymptotic behaviour of the functions in W " p ( X ^ X Rq) induces a system of natural trace operators TPskJ on this space. As we have seen the Fourier image of an asymptotic term is lP*k,A(q) o^(q),v E Hs(Wq). Now the Laurent expansion of the Mellin transform of P k % ( q ) = x ( q ) lTk'o^(q) consists of ( z - P ) ~ + 'multiplied by a coefficient. Since M , , , x ( q ) = [ q ] ( " + 1 ) / 2 - 2 M(for , - z the Mellin transform M with or without weight) this coefficient is cA(x) [ 7 1 ] ( " + ~ ) ' ~ - P oc^ (aqconstant. ), A(x) stands for a vector in the finite-dimensional space L c C"(X), running over a base. It can be removed after a pairing with a vector A' in the dual base. In other words the composition TPsksA:u+ w := Fy-11 u+ F;Ly {Laurent coefficient of Mw at (z I, paired with A'} defines a trace operator T P k,. A : w>Y(X^ x Rq), +Hs - ( n + 1)/2 + R e p (wq)
(34) (35)
for every triple p , k , 1 as in (27) with (28). Remember that we have mentioned such trace operators also in 3.1.2. in the special case of Sobolev spaces on R: (where n plays the role of q + 1). That are just differentiations in normal direction composed with the restriction to the boundary. The connection with the Mellin transform was already explained in 1.1.3. Proposition 1 . Let us define modified trace operators BPvk.': ~ + c F ; . ! , ~ [ q l - ( ~ + ~{Laurent ) / ~ + P coefficient of Mw at ( z - P ) ~ + ' , paired with A'}, w = Fy+,,u, c a constant. They induce continuous operators BPvk,l: W$y(X^ X Rq),+HS(Rq). Under an appropriate choice of c the composition LP-k*ABp,k.A equals the projection to the span of L P ~ k ~ A H s ( whereas Rq) BPpk,ALP-k.A equals the identity on H8(RQ). The wedge Sobolev spaces with discrete asymptotics also admit the variant of the infinite weight interval
W$y(X^ X Rq) = lim W>?(X^ X Rq)4r. c rsN
Here A , = [d,, 01 is a sequence with zcP n I',,,+ n / 2 + ,,,= 0, hr+ - m as r+ m. Then we get infinite asymptotic expansions of the form (32) with remainders in WS."(X^ x R4). v p . k , A E Hs((Rq) be an arbitrary sequence where p runs over z c P , A over a base of L ( p ) and 0 5 k 5 m ( p ) , for all (p, m ( p ) , L ( p ) ) E P . Then there exists an u E Ws*Y(X^X Wq) such that for every r E N
10. Theorem. Let
u-
CL
~ * ~E W;Y(X^ A v ~ x ~~ q ~) , , ~
with the sum over all p , k , A for which 1 / 2 + n/2 - ( y - 6,) < Rep < 1/2 + n/2 - y . Alternative& we could say that Bp.k , A U
= Vp,k,l
forall p, k,A.
315
3.2. wDO-swith Operator-Valued Symbols
The proof is based on the following Lemma. Denote by Lf;:a the analogue of the operator (30) formed with w(ct) instead of w ( t ) , c 2 1 a constant. 11. Lemma. Let u E Hs(Rq) be fixed. Then
Lf;:'u+O
in W > ' ( X - x R q ) 4
as
c+m.
The proof of this Lemma is a simple exercise and left to the reader, cf. also 3.2.1. Example 3. Now u can be written as a convergent sum =
1Lf&,t,A))
Op. k, A
with constants c ( p , k,A)+ m increasing sufficiently fast for Rep+ - m . An appropriate choice of the constants follows by a diagonal argument after establishing the convergence with respect to one of the countable many semi-norms. Observe that
a
-* XS.Y(X^) +XS - 1.y- 1 at *
(x-)
can be interpreted as an element of S'(1Rq; XssY(X^),Xsl * Y - '(X-)), for x-'(s)
a ata x ( s ) = [slat,
a
cf. 3.2.1. (1). Thus - induces continuous operators at
2. Wps.Y(X" x at .
R4)+
Ws-1.Y-1(XAx R4).
The idea of 1.1.3. Proposition 1 of expressing traces of derivatives in t at t = 0 also applies to the wedge spaces. For u E W f Y ( X AX Rq)4,P = { ( - j , 0, L j ) } j N, we can calculate the residues in the Mellin image of Fy+,,u at the points z = - k for all k E N with 112 + n/2 - ( y - 6) < - k, y > 112 + n/2. In particular we get in this generalized sense
for all k with 1/2 + n/2 - y < - k. It follows 12. Remark. u E Wa*"(XA X R4) implies u E Ca@+
X X X Rq)
and u vanishes of infinite order at
t=O.
Let
w(t)
be a cut-off function, 0 d w ( t ) d 1 for all
t,
and set
+ 1 - w ( t ) , B E R.
g @ ( t )= t @ w ( t )
13. Remark. For ga(t[q])= ( t [ q ] ) f l w ( t [ q ] )+ (1 - w ( t [ q l ) ) , E R, we have
n
~ W VEI r) smq; x r . y ( ~ - ) , x r , y + a ( ~ - ) ) c o n a t with ga(t[ql)-l
n s:,(R~;xr,~+~(x-),xr.~(x-))~~~~. r
21 Schulze, Operators cngl.
(36)
316
3. Operators on Manifolds with Edges
The associated wD0-s induce weight shift isomorphisms Op(gb(t[q])): W'(R9, X ' v Y ( X ^ ) ) yW'(RP, X'*Y+P(X^))
for all s,r, y E R, with the inverse Op((gP(t[q))-'), in particular for s = r Op(gB(t[q])): Ws*Y(X^ X Rg)?
Ws*Y+B(XLX R*).
(37)
The subspaces with asymptotics are preserved under (37) up to a T-0 shift of the carriers of asymptotics.
Next we want to discuss in more detail the wedge Sobolev spaces with continuous asymptotics ( 5 ) and (7), respectively. We have repeatedly to use spaces over sums of spaces Ei. Let us first prove an abstract result on sums. 14. Lemma. Let F be a topological Hausdoflspace where xA acts as a group of linear mappings and let Eo, El be Banach spaces, embedded in F, and x AE C(R+,4P,(Ei)) for i = 0 , l . Then for E = Eo + El it follows x AE C(R+,4Pu(E))(cf. 1.1.2. Definition 4 and 3.1.2. Lemma 1). Further
Ws(R', E) = Ws(R', Eo) + WS(R', El)
(38)
(this sum refers, for instance, to al(Rq, EO+ El)).
Proof. The continuity of x A in A as an Su(Ei)-valuedfunction can be expressed as - xduillE, +0 for A +Ao and every A. E R+, ui E Ei. For every linear operator A on F which induces elemehts in 4P(E,), i = 0 , 1, we have for u = uo + ul, u iE Ei IIA(U0
+ U b l l E S IlA~OlIE+ I I 4 l l E s llAuOllEo + llAulllE1 5 IIA I ~ Z C E ~ ~ I + ~ UI OI A~ ~I E~~~ C E ~ ~ I I ~ ~ ~ ~ E ~
s (llAIbCE0)
+ llAIlr?CE&
(IluOll&
+ IlulllEl).
This shows by taking inf over all u = uo + ul, u iE Ei, that llAullE
5 (llAhEo)
llAIIZCEl)) llUllE
and hence
[IA
h E )
5 IIA
IlZ(EcJ +
[IA
IlZ(E1).
This proves in particular xAE C(R+,S,(E)) for all A and in addition - xlo) ullE +O for A +Ao, u E E . Next observe that for f E Wa(Rq,E,)
I l f 11 W'(R4 E ) 6 Ilf 11 W'(R4 Ei), i = 0 , 1, i.e., we have continuous embeddings WS(Rq, Ei) G W8(R', E) , i = 0 , 1. Now a trivial general result says that if Banach spaces Wi are continuously embedded into a Banach space W then it follows a continuous embedding Wo+ Wl 4 W. In other words
317
3.2. tpDO-s with Operator-Valued Symbols
W'(R', Eo) + W"(R', El) 4 W(R', E ) is continuous. For proving the equality we first apply the isomorphisms
(39)
0p([qlS): WS(Rq,G) + WO(R4, G ) for G = E or Ei,i = 0, 1. Assume that we already know that WO(Rq,Eo) + Wo(Rq,El) = Wo(Rq,E ) .
(40)
Then we obtain W"(R', E ) = Op ([q]-') {Wo(R', Eo) + Wo(R', El)}. Now we use the fact that if T i : H i +Liare continuous operators with TOh = T'h for all h E H o n H l , then these operators induce a unique T : H o + H 1 +Lo + L 1 such that TI Hi = T i , i = 0, 1. The assumptions are fulfilled in particular for T i = Op ([ql-"): WO(R',Ei) + W'(R', Ei). Thus
Op ([q]-'): Wo(R', Eo) + Wo(R', El) + WS(R', Eo) + W"(R', El)
is continuous. In other words it follows the continuous embedding Ws(Rq,E ) c 'W'(R9 Eo) + W'(Rq,El). It remains to show (40). To this end we observe that the functions of the form N
h=
1 &hi, Si= l,,
hie E
i= 1
with disjoint measurable sets M i €Rq are dense in Wo(Rq, E ) . Thus for every f~ WO(R4, E ) and 8 > 0 there exists an h of this sort such that Now
IIh -fIIk0(~9,0 5 &. N
I I ~ I I Z ~=Oj1 ~ ~IPj, hi112X0mq.m. ~ =1 For every 6 > 0 there are vectors e j i E Ei with hj = ejo+ ejl and
IIejoIIio + IIej1II2E15 IIhjIIi + 6. N
Thus for hi=
1 Sj eji we get j=l
llhO1lko(Rq,E,,)
+ llhlllko(Rq,El) 5 llh11b(R +9 &, E )
for 6 = &IN. Thus we get sequences hi(&) E WO(RQ, Ei)with h0(&)+ hl(&) f in Wo(Rq,E ) for E +O. This proves (40). 0 --+
The equality (38) extends to more general spaces Ei by natural operations such as projective limits and it will be used below also in such cases. A = (6,0] finite, were defined by Remember that the space XJ,'(X^),, B E BYy("), the decomposition method as the projective limit of sums 2 '1
318
3. Operators on Manifolds with Edges
with "many" gaps (such where Aik = (&, 01, 6ik +6 as k + for i = 1 , 2 , Bi E that the sets Bi look like "fences"), B = B1+ B2,Bi n f1,,+ n/2+aik = 0 for all i, k,
X;"(X^)z= c lirn { [ w ]X;"(X^)$J+ [l - w ] H W ) } j e 2
cf. 2.1.1. (33) and the construction after that formula, here adapted to the case of half open weight intervals in 2. In view of
Ws(Rq, X4,y(X-)d) =c lim { WS(Rq,XJ,:(X^),,,>+ WS(Rq,X;;(X-),,J) k
for X'JBiy(XA)dik = lim XJ,y(X^)y:or with the alternative notations
7-
the investigation of wedge spaces with continuous asymptotics leads to the discussion of the case when
cf. 2.1.1.(25), where ZY,(X^)d= {(Clv, t - " ) w ( t ) :
CE d'(B$)@JYrCrn(X)}
with B , = B n ( ( n + 1 ) / 2 - ( y - 6 ) < R e z < ( n + 1 ) / 2 - y } ,
and
W$B"(X"X Rq)A = Ws(R', Xa"(X^),)+ Ys(R'J,Z%(X^),j),
(44)
cf. the notations in 3.1.2. after Lemma 7 and the easy extension of Lemma 14 to Ys spaces, Ws(Rq, X ~ y ( X " )=A W ) a " ( X ^ X Rq). (44) is a direct decomposition. Since X J Y ( X - )=~ lim XqY-a-c(X"),it follows t
C>O
WaY(X^X Rq)A= @ W*y-a-a( x - x
R4).
a>O
This is the flat remainder space of the asymptotic behaviour. The specific part is
Ys(Rq, %V,(X^),). For every u(t, x, y) E (45) we have
(45)
x - ~ ( v Fy-.,, ) u E &(Rq, Z % ( X - ) , ) ,
fis(R4,,E) = Fy-.,,HJ(R;, E). Remember that Z&YA)A is a nuclear Frkchet space. Special elements are of the form
Fy+*u(t, x, rl) = ~tll'"+l"Z(C,,(~~tll)-w) o(t[sl)u^(a)
(46)
319
3.2.lyDO-s with Operator-ValuedSymbols
for certain u E Hs(Rq),(E d'(B,) 9*Cm(X),but in contrast to the special case of the discrete asymptotics the space (45) is not finitely generated over Hs(Rq).The analogues of the above potential operators can be defined for every 1. Set &I> =X(lt)(IW, t - 9 (E
w(0,
(47) S:,(Rq;C , X4y(X^))const. The asso-
s&'(B,) 9, Cm(X).Then (47) belongs to S
ciated potential operator defines a mapping
Lc = Op (It): Hs(Rq)+ WJBY(X"XRq) . For the influence of the choice of w and q +[ q ]we can make analogous remarks as for the discrete asymptotics. They are left to the reader. Let us also have a look to certain trace operators. Choose a projection
n: W i Y ( X ^ XRq), + YS(Rq,C ; ( X ^ ) , ) (cf. (42), for instance, induced by a projection X i Y ( X ^ ) + , C 6 ( X ^ ) , ) and let 1'E Cm(X),h E d (C). Then
T : u + F-' ( x ( q ) F M n u ( q ) , h @A') is an analogue of (34). Here F = Fy+,,, M = M,,,, h is paired with the function with respect to z by an integration along a curve surrounding B,, and 1' is paired by the L 2 ( X ) scalar product. Then T : WJBY(X^xRP), + H S - ( n + 1)/2+ b ( R q ) , b=max{Rez: Z E B ) . Note that the elements of W i Y ( X ^ XRq), admit an analogous procedure of reconstructing analytic functionals as the functions with asymptotics in the cone theory. In fact, let u E W i Y ( X ^ xRq),. Then x - ~ ( v F~-.,, ) uE
A s ( R ~x;Y(x^),) , = lim A s ( R ~ X ,J , Y ( X ^ ) : J ~ ) . t
i
Now
M f + z w ( t ) x - l ( q )F Y + ? u As(Rq, ~ M,,,[olX~y(X^),) . yields an element $u can- be paired as mentioned with h ~ d ( @ ) This E HS(Rq,d'(B,) 9, Cm(X)).In other words an alternative way of expressing "trace operators" is to form the mapping S : W S , Y ( X ^ X Rq), +HS(Rq, d'(B,) c3, C r n ( X ) ) , S = F;!.?,!? For B, = Bo + B1 with sup {Rez: z E B,) < inf{Re z : z E Bo} we obviously have a direct decomposition Hs(Rq,d'(B,) mZ Cm(X))= HS(Rq,d ' ( B i )8 Cm(X)).
1
i=o,1
By integrating in the pairing with h E d ( C ) over curves surrounding only the component Bi (similarly as in 2.1.1. (31)) we get operators Sj: WiY(X^ X Rq), +HS(Rq, d'(Bi) c3= Cm(X)). (48) Now let A = (-00, 01 and B E Bf"), B = IJ Bi (cf. 1.2.2. Definition 11). Then we get ie N
the continuous operators (48) for all i E N.
320
3. Operators on Manifolds with Edges
eiE HJ(Rq,sO’(Bi)63% Cm(X)) there exists a u E WJBy(X-X Rq) with Siu = ei for all i E N. I f 6 C another element with Siu = Sicfor (X”X Rq). all i then u - 6 E WJom 15. Theorem. For every sequence
Theorem 15 is the continuous analogue of Theorem 10, cf. also 2.1.1. Theorem 15. The proof is left to the reader as an exercise. The interpretation of the asymptotic terms as images of H3(Rq)under potential operators makes it natural also to admit potential symbols with non-constant coeficients. So let us generalize (29) or (47) and consider lAY, Y’, tl) = x ( v ) ( U Y , Y’), t - ” ) w ( t ) with [(y, y’) E Cm(RZq, sS’(B@)63% Cm(X)), where we assume independence of y, y’ for lyl, ly’J2 c with some constant c > 0. We then have
and we can define continuous operators
L( = Op (It): Hs(Rq) + WJ“(XnXRq) . Particular cases are ~ ( yy’)1Ppk5”(q) , or ~ ( yy‘)lc(q) , with the symbols (29) and (47), respectively, for y(y, y’) E Ct(Rzq). It is interesting also to consider (49) 5(Y, Y’) E Cm(R2q,ss’(B@)63% Crn(X))* > cf. 2.3.2. (8). This leads to a definition of wedge Sobolev spaces with branching discrete asymptotics. As in the general lyDO calculus it is convenient from time to time to switch to equivalent symbols with dependence either on y or y’ by the formula
or
cf. 3.2.2. Theorem 13 or its analogue for (51). 16. Lemma. For every 6 > 0 there exists an N
EN
such that
induces an operator
op(lN): HS((Rq)
wpJ.y+d(x” x Rq).
An analogous statement is true of (51). Proof We have
3.2. yDO-s with Operator-Valued Symbols
321
where f(v) := [v""'"lZo^(71), 8(7]) = [v]-'"'"/2x(?7), ma(Y) = l / a ! 3 ; ( C w ( y ,y ' ) , r-") w(t)l,,,,, rn,(y, y ' ) = ( 5 ( M ( y y, ' ) , t - " ) w ( t > being the remainder of the Taylor expansion at y = y' which is flat of order N on the diagonal, kME Cm(R2q,a'(&)@,, Cm(X)).Now we use 0;ei(Y-Y')S= (y - y')" ei(Y-Y')n.Then integration by parts yields for la1 = N Op(I,)u= srei(y-Y')n(-0n)"6(v)m,(y,y ' ) f ( y ' ) d y ' d v . Since 6 ( v ) mN(y,y') = ( C ( N ) ( yy,' ) , ( t [ v ] ) - " )w ( t [ q ] ) the differentiation in leads to flat terms from the derivatives of w or to extra powers of t which causes flatness of order N. Choosing N large enough we obtain the desired mapping property of the remainder. The second assertion follows in an analogous way. Now we come to the analogue of the Theorems 1, 4 for the subspaces with asymptotics. 17. Theorem. Let P E 3"(")(X) satkfi the shadow condition, i.e.,
T-jP
P for all j E N .
(52)
Then the spaces W>y(X^XRq)A,A E ga, a = 0, 1, are modules over Ct(Rt xRq). The multiplication by p induces a continuous operator Jn, E 9?(WJ-,y(X-xRq)A) and p +Atwa continuous embedding
c;(R+xR~)+
n ~ P ( W S , Y ( X - RX ~ ) J .
seR
Ct(a+)or Ct(R4) instead of C t ( & XRq). Proof. Consider, for instance, the case A = ( 6 , 0 ] E go.In view of Theorem 1 we al-
An analogous statement holdr for
ready have the assertion for the subspace
c1vc~0yx-xR ' ) ~=
n
WSY+P(X-X RQ).
oSQ<-d
Since
W>Y(X^XRq)A= WJ0Y(X^XRq)A+ YS(Rq,Z'T(X^)A)
it suffices to consider YJ(Rq,Z';(XA)A),spanned by the functions (30). Set for abbreviation m(p, k ; t ) = t - P logkt First consider the multiplication by p E C;(E+), p ( t ) L A k A v = J e i ~ { p ( r ) x ( v ) m ( pk, ; t ) A ( x ) w ( t ) } c ( v )d v , u E Hs(Rq). Applying the Taylor expansion we can write N- 1
(53)
322
3. Operators on Manifolds with Edges
Thus (53) equals N- 1
cj ~ e i ~ w , ( t ) t j x ( q ) r n k( p; t, ) A w ( t ) u " ( q ) U q j=O
+ rpoJeimol(r)rNx(q)m(p, k ; t > ~ w ( tU)Q N- 1
=
C cj,y-j.k~
vj
+ Q,oLp-"~s"vN,
j=O
where fij(q)= [g]-jv"(q),j = 0, . .., N. For N large enough we have
poLP - N,kA VN E ws y &(X^x R'I). Here we have used Theorem 1. Moreover (52) implies L P - J * k Aj v E Ys((IRq,2?>(X^)d).It is now a simple exercise to verify that the multiplication by cp even induces a continuous operator Jn,tending to zero for Q, 4 0 . Now let w E Ct(Rq) and consider
V L P I ~ ~je'Y"{~(y)x(v)m(p, V= k ; t ) I w ( t ) } f i ( q )U q . Write m(p, k ; 1 ) = (Tw, r W )with an = e-i%(y') dy' then it follows yLPek*u=
Choose an N
appropriate
(E
Se'(lp)). With
ei(Y-y)rlx(q){Lt r Z )wAy(y) u(y3 dy' U q .
G(q)
(55)
> 1 and write
with appropriate ya(y') E CZ(Rq), yN(y, y') E C;(R2g), yN being flat at y = y' of order N. For abbreviation we now set f ( q ) = [q](n+1)'2v^(q)Aand 6 ( q ) = [ q ] - ( " l)I2 + %(a).Then v f i k *u =
J ei(y-y')q~ ( q( 4)
t-z)w
323
3.2. wDO-s with Operator-Valued Symbols
=
(c, r - 9
j
w(”(t)
+
1
Cj”
(z; (- 2) (- 2 - 1 ) ...
v= 1
(-2-v+
l)t-Z-”)w(j-”)(t).
Thus (58) is a sum of expressions of the form
l ei(y-y”qd(v){ ( h v ( z ) r )t - ’ - ” + j ) where h v ( z ) is a polynomial in
2,
w(j-”)}
and j 2 v, j
r , ( v ) [ s ] - j l y a ( y ’ )f ( y ’ ) dy’ dq,
+ 12 0. This can be also written as
eiY”d(v> [(hvr)t - ’ - ” + j > ~ ( j - ” ) r} d v ) h I - j =
n lyaf
( v ) dv
ei*X(q) {(hvZ;t r - ” - j ) w ( J - ” ) } f K j , , (dv v) ,
(59)
n j , , ( q ) = r l ( v )[ v ] - j lyau (v)1. The functionals
C belonging to the given asymptotic type P may obviously be multiplied by polynomials without loosing the property to belong to P. Then the functions (59) for j = v actually belong to YS(Rq,2?;(X-)A) whereas for j > v we have a derivative of w being flat at t = 0. This leads to contributions in Ws”(X^XRq). Note that we have also used that 1y.u E Hs(Rq) and that r l ( v )[ q ] - jis of order 5 0. The contribution from lyN in the formula can be treated in the same way as in the proof of Lemma 16, i.e., it belongs to WqY-d(XAx Rq) for N large enough. An elementary consideration in terms of the involved semi-norm system yields Atv +0 as y +0. The general case of (t, y ) dependent factors follows again by a tensor product argument, similarly as in the proof of Theorem 1 . fq
Note that the shadow condition was used to ensure that the asymptotics admit multiplications by tjw, j E N. The same condition appears in the analogue of Theorem 17 with continuous asymptotics. In other words we have 18. Theorem. Let B E By(“) satbfy the shadow condition, i.e.,
Then the space Wiy(X^xItq)*, A = (6, 01, b a module over Corn( R+XIRq). We have A t w ~ S?(W>Y(X^X Rq)A)and&q -0 in rhbspace f o r p +O in C;(R+xRq). An
n
SER
analogous statement h o l k for C,“(R + ) or Ct(R4) instead of Corn( R + xR’I).
Proof: Let T-jB n r(,+ 1)/2 - ( y - d) = 0 for all j E N. The general case follows by decomposition arguments and playing with the weights which is left as an exercise. The proof of Theorem 17 shows that without changing essential things we may replace C @ 1 E sQ’(b}) @n Cm(X) by any CE sQ’(K)@n Cm(X)where K is a component of Byd.In other words we get the result by an obvious generalization of the proof for the discrete case. 0 Now we want to prove an analogue of Theorem 4.
324
3. Operators on Manifolds with Edges
19. Theorem. Let%be as in Theorem 4, A = (6, 01, and let P E P"'")(X), B E B y ( " ) s a t m the shadow conditions ( 5 2 ) and (60),respectively. The pull-back underx induces operators
x*: WJpY(X^X Rq)A+ W%y(X^XR'),, and
x': W y ( x ^ x R4)A + wJByx-xRqA,
respectively.
Proof. Let us first consider the discrete case. In view of Theorem 4 it suffices to prove that x*: 'V'(Rq, Z:(X")A) + W$y(X^X R q ) d .
Let u = Op ( l p s k l ) v, v E Hs((Rq), i.e. u = j eiY"6(q>(5; g(t, z))f('(tl)d t f ,
cf. the abbreviations in the proof of Theorem 17. By assumption x is induced by a diffeomorphism x : Rj +R$. Let 8 = x - l , y = 8(R. Then
s
x 4 t , r")= eice(n-y')96(tl) ( I , g ( t , z ) ) f ( y ? dy' dtl =
~ei(e(fi-B(r))~6(q)(5; g(t, z))f(8(9?)Dl(f? dy'dq,
Dl(y") = Idetd8(y"')l. Choose a function pl(Y; 9')E Cm(R2q)with p1 = 1 in an open neighbourhood of diag (Rq x Rq), p1 = 0 for 19- f ' l > E with some E > 0. Further let po = l - p l . With p(y3 :=f(S(y3>we set hi(r, y3 =
ei(e(n-e(~))~6(q)(5; g)q(y"')Dl(y"')pi(Y;y") dy'dq,
(8(n
i = 0 , 1. As in the standard calculus of VDO-s we write (Y; ?')ij= - 8(y"'))q close to the diagonal. Then i j = 8-'(Y; 9')q with a matrix 8-' of functions in Cm(R2q)which is non-singular close to the diagonal. Choosing pl in such a way that 8 is non-singular on supp p1 we can write
9 = 1ei(F-r)*6(v)(5; g) D(Y; Y')pl(Y; 9')~ ( 9 dY' ' ) dij, = )detO(Y; Y')ID,(Y').By definition we have hdt,
D(Y; f')
6(tl) (5; g) = (5; ( t
9')711>-') w ( f [W9'1711).
(61)
20. Lemma. The function h(t, Y ) =
J ei(y,y')q(z;t W Y ,
~ 3 7 1{ )w ( t W Y , Y ' ) V I )
x w ( t [VDI P ( Y , Y ' ) d Y ' ) dY' d?l belongs to W4m(X^XRg).Here p := D p l . Proof: It suffices to observe that
(5; t [ S ( Y , Y ' h l )
w P ( Y , Y ' h l ) - w ( t [ql)Ip(Y,Y')
belongs to S-("+l)"(R2q X Rq; C,Xm*m(X^)) with a behaviour of dependence on y, y' such that the associated operator maps H'(Rq) to Ws(XAXRq). But this is obvious. The flatness is a consequence of the difference of cut-off functions.
325
3.3. WDO-son Manifolds with Edges
In other words in considering hl we may neglect in (61) in the cut-off the 8 factor at @ We could do the same for the other factor in (61). Instead of this we only observe that the 1pD0 associated with
( 5 , (r [8(y, y?tll)-') w ( t [ t l I ) p ( y , y ? E S-(n+1)/2(R24 x Rq;@, X Y X " ) ) has the mapping property that we need, namely to map point-wise to the desired asymptotic type and to have the nice y, y' dependence for large y , y ' . In other X R!),. For dealing with ho(t, 7 )we may argue similarly as words hl(t, y) E W%y(X^ in the proof of Theorem 17 where the flatness of the integrand on the diagonal was used to prove flatness of the associated WDOsymbols in sense of the weight after applying integrations by part. Thus we obtain our result for the discrete asymptotics. The continuous case follows in an analogous manner, after the usual decomposition arguments and inserting for [ the corresponding general analytic functionals. 0 21. Definition. Let W be as in Definition 6 and P E 9 y y c n ) ( X()B E B y ( " ) ) satirfv the shadow condition (52) ((60)). Then W"p(W),(W$y(W),) ford E 1denotes the subspace of all u E W * y ( W ) with (8*)-l q u E W"p(X^xRq), (E Wiy(X*xRq),) for every edge neighbourhood and all q = a*@, @E C , " ( r X 0).
In view of the Theorem 17, 18, 19 this is a correct definition. Note that for A E J1 the spaces W$py( W),, W i y W), ( (as well as the local analogues) can also be defined, similarly as for the cone spaces. This generalization is obvious for discrete asymptotic types as well as for B n r,,+ 1)12 - ( y - d) = 0 which are the only cases that are occasionally needed. The above considerations have corresponding analogues that will tacitly be used. Analogously to the notations in the cone theory (cf. 2.1.1. (40), (41)) we set
3.3.
Pseudo-Differential Operators on Manifolds with Edges
3.3.1.
Pseudo-DifferentialOperators of Wedge Type I
The typical differential operators on a wedge close to an edge are assumed to be of a form that follows by introducing polar coordinates (1, x) = 2 E R:+ \ {0} 1 It+X S" in a differential operator
aa(2, y ) E Cm(Rn+ l+q). In Section 2.3.3. we have seen that in the
(t, x)
coordinates (1) takes the form
326
3. Operators on Manifolds with Edges
where
and AmkE Cm(R+XRq, DW"'"l-k(X)), X = S". If we admit X to be arbitrary we get just the typical differential operators on wedges in general. From now on we content ourselves with the case X = S". According to our philosophy the construction of a class of 1pD0-son {cone}X RP should reflect the idea of looking for those operators that constitute the parametrices of the typical differential operators. The analysis will be performed for pseudo-differential symbols a(%, y, E q ) E Sf,(Rn+lX 8 X R;,i'+q),
R E R; open. S(')(T*(R"+ X 0)\O) denote2 the space of all p(%, y, E q ) E Cm(Rn+I X 0 X (Rn+l+q \ {0}))with p ( 2 , y, At, Aq) = A#p(%, y, q ) for all A > 0, ( 2 , y ) ~ R " + ~ x( R E q, ) * O . Denote by S("(cT*(X^x 0)\O) the space of all functions t-Pq(t, x, y, lz, F, Cq) which are of the form
P G , Y, sl t l ) l i = n ( C x ) , e = , - I T - l t ,
(4)
cf. the notations from 2.1.7. (1) and 2.1.7. Proposition 1. By definition there is a canonical isomorphism S@)(cT*(X^xR) \O) +S(fi)(T*(R"+lx 8 )\O). (5) A (stretched) wedge globally close to an edge Y can be identified with a bundle
r
over Y with the fibre x^ = R+XX. The wedge itself is then 8\ ({O} X X).The reader X Y; In general the transition functions are asmay imagine first a trivial bundle T sumed to be induced from those of an associated real vector bundle over Y with fibre Rn+l, where the transformations of the fibres (with a fixed base) have determinant 1. Let V denote the X^ bundle which is formed by by removing {O) X X in each fibre. We now have an evident notion of homogeneous functions of the class S'JqcT*V\ 0) (6) (namely by demanding to belong to S(fi)(cT*(X^xR) \O) for every trivialization X - x LI of V over R c Y). Finally for a (stretched) wedge W globally which has a neighbourhood of Y of the form vl 1 < we have the space (7) S(")(cT*(intW) \ 0) of all C" functions p(w, x ) on T*(int W) \(I that are homogeneous of order p with respect to the R+ actions in the fibre variable and which are of the form (6) close to Y. 1. Definition. We say that p ( w , x ) E S(P)(cT*(intW) \ 0) is non-vanhhing up to the edge i f p 9 0 on T*(int W) \ 0 and if localb close to the edge Y in the coordinates (2, y, t,q ) the function is non-vanishing including 2 = 0 for all y E Y and (t,q) 0.
*
Clearly this is close to Y just the condition of non-vanishing of the image of p under (5).
327
3.3. wDO-s on Manifolds with Edges
Denote by S y ( c T * ( X - x a)\O) (8) the preimage under ( 5 ) of those p ( 2 , y , E q ) that are independent of 2. The invariance under the transition functions for V gives rise to the space Sy'(cT*V\O) c S'"(cT*V\O) (9) of elements that belong to (8) over all trivializations X^x 0.There are canonical restriction mappings (freezing of coefficients) r,: S"'(cT*(X-x
r, : S("(cT*V\
a)\O)
+Sy(cT*(X^x
a)\ O ) ,
0 ) + S("(cT* V\ 0) ,
(10) (11)
induced by (12)
r A P ( 2 v Y ~ E V ) : = p ( o ,Y v h ) .
2. Definition. C"(cX^X a) denotes the subspace of all u E C"(X^X 0)for which n*u E Cm(R"+X 0). C"(OY^X a) is a FrBchet space in a canonical way. It is obvious that
C"(CX^X a)c S"(cT*(X-x
a)\O)
(13) and Q, E C"(cX^x a),p E Sy)(cT*(X^x 0)\ 0 ) implies pp E S(fi)(cT*(X^xa)\ 0). Let us mention as an exercise
3. Proposition. S"'(CT*(X^X
a)\ 0) = C"(CX^X a)@* S$'(cT*(X--x a)\ 0 ) .
(14)
Most of the following considerations will refer to X^X 0 for a coordinate neighbourhood 0 of the edge Y. The invariance under transition diffeomorphisms for V will be obvious. Then we can easily pass to the corresponding global constructions. The pull-back of symbols in S$(R"+ X R x R" q) under n X id: X^X 0 +Rn+l X +
+
leads to the notion of a global complete symbol on X^X a in the (t, x, y)-coordinates. Remember that for an open neighbourhood U on Sn we set U A =R+x U and
@'= {ZERn+'\ {0}: 2ll2l E U}. Let us fur an open covering U = { U,,. .. , U,} of S n= X by coordinate neighbourhoods Uj.Then U^:= {UJr}is an open covering of X-.We assume that the coordinate diffeomorphisms between intersections UJ*n U; from the coordinates of U; to those of U; extend to affine transformations of R"+lafter inserting the images under n. If U is a coordinate neighbourhood of X we denote by S$(c8^X R X Rn+l+q) the space of all p ( 2 , y, F, q ) E S g ( 8 ^ x a X R"+l+q)such that there is a FE S$(Rn+lX a X Rn+l+q) with p = 51II-* x n x R n + l + q .
328
3. Operators on Manifolds with Edges
Let S;l(c!?x 0 x Rn+' be the subspace of all p are independent of 2 for large 121.
E
S;l(cd-X R X Rn+ '9) which
4. Definition. A global complete symbol
(15) P = bjlj = 1, ..., N (with respect to U) g a systempj E Si(_coT_X R x R"+'+q)such thatpjl 0; n 0; transforms to pkI UJrn U; mod S--(c(Uj n U;) X 0 X R"+l + q ) under the 1pD0 substitution rule for complete symbols with respect to the admitted diffeomorphisms (i.e. extendable to affine transformations), j , k = 1, ..., N. Although this notion is somehow tautological in the present situation we want to fix it, since it is an ingredient in analogous considerations for more general cone bases. Moreover in polar coordinates we have to keep in mind in fact different local coordinates in Uj for different j . Denote by S$(CU^ x n x R"+l+q)l
(16)
the set of all complete symbols (15) with p j E SgI(coIX R X R"" 'q)l for ..., N. We want to reformulate the complete symbols close to the edge in terms of polar coordinates such that the associated lyDO actions coincide modulo smoothing operators. This will be expressed by non-canonical symbol mappings induced by
j = 1,
n: u-+o-, for any U-EU ^.
n(t,x)=2,
(1 7)
5. Definition. S!!l,,(~AXR X R"' l' q, denotes the subspace of all p(t, x, y, 0 E S$(R+X U X R X R"+ 'q ) which have an asymptotic expansion P (', x, y, 0
-
m
j-0
P(p -Jf)(t, x, y, 0
(18)
with t = ( t % & t t l ) , P(p-~)(t,x,Y , O E C m ( ~ X f l x R " l + q ) ,
(19)
~ @ - ~ f )X (, tY,, 10 = J p ~ @ - ~ ]X,( tY,, 0 for aN 1 L 1, 161 L const. Z'he excisions in cawing out the asymptotic sum (18) are taken with respect to In an analogous manner as 2.1.7. (13) it follows 6. Proposition. There is a non-canonical mapping R*: Sfl(coAX0 X Rntl + q ) + S$,(v^X
such that
-
0 x Rn+l + q )
(20)
n* OPV(P)( Y , 1 ) t-'oPv(n*P) (x11, (21) where indicates equivalence modulo parameter-dependent smoothing operators (y, 1 be. ing the parameters), and n* on the left of (21) indicates the pull-back of 1pD0-s under 3t, opv is understood as 1pD0 with respect to the corresponding Fourier transform in the 2and (r, x)-spaces, respective&.
-
329
3.3. vDO-s on Manifolds with Edges
As in 2.1.7. "non-canonical" means that the mapping is to be organized for every concrete symbol by individual excisions. The precise parameter dependence of the smoothing operator family which remains uncertain in (21) is not important for the moment, but it is involved indirectly in the final operator convention with Mellin actions along the t axis. This will be introduced below. Now let us remind of the parameter-dependent Mellin symbol class
mqx;~ q =)Q ( X ; r,,,- ,,(") x RQ), cf. 2.3.1. (9, or the local analogue rl,,-
(22)
R" X Rq) (23) which is defined as the space of classical amplitude functions h ( x , 112 - y ( n ) + iz, 5, 71) in the frequency variables (z, t,tl), t = Im z, z E rl,,- ,(,), x E U. S W
X
y(n)
X
With every h (z, 71) E WKY(X;Rq) we associate the weighted Mellin action := My(:) h(z, t7)My(n) = tY(")
(24) which is correctly defined on functions, say with bounded support in t. The letter M indicates that the parameter is multiplied by t. The local form of op, for h E S:l(U x r,/,- y ( n ) x R"+q)is opp)(h) (71)
op,(h(., tq))t-Y(")
O P ~ Y(tl)~ =) My(!,)F-'h(x, 2, 5, ttl)FMy,n) (25) with the Fourier transform F in R".This is defined for functions with compact support in X E U. The notations extend in a natural way to amplitude functions h ( t , x, Y , 2, F tl) E cm(R+xfl)@zS:dUx r m - y ( n ) - x R" q ) =: S:,(Fx R x r,,,- ,,(") x R"+'J),R S Rq open, or similarly to h E Cm(R+XR)@,m.Y(X; Rq). We then write opg)(h) ( y , v ) . +
7. Proposition. For every y E R there exbts a non-canonical mapping
m,* : s;,(cPx R x R"+1
+
4) +
sgl(V^x
x
r,,,-
y(n)
X R"+q)
such that
-
OP,(P) ( Y , tl) t-'opP(h) ( Y , 71) for h = m: p, and being as above. It*
-
Proox By Proposition 6 it suffices to reformulate the right hand side of (21). Set a. = a(t, X, Y, 0 = Z*P, C= (@, F W , and write fOO,x, Y , 2, E, ttl) = ao(t, x, Y, -z, 5, m),z = 112 - y(n) + iz. Then op,(ao) (Y, 71) - OPP)(fO)( Y , v )
- oPv(a1) ( Y , 71)
for some al E S$
330
3. Operators on Manifolds with Edges
Then it is allowed to argue as for y ( n ) = 0, modulo smoothing remainders. For every j E N we set
Act, x, Y, z, 5, ttl) = ajo, x, Y, - r, 5, ttl) for X
z
3
= 112 - Y ( n ) + ir,
0 E S$rd(u^X R X Rn+l+q).Then f j ( t , x, y, z, 6,1)E SE,-j(u^X R and for some a j + lE S$,-i-'(TAX R X R"+l+q) it follows OPv(aj>( Y , I I ) - OPP'(fj) (Y, tl) = oP,(aj+ 1) (Y, tl) .
aj(t, x, y,
rl,2-y(n) X
Rn+q),
m
Then it suffices to define h
- cfj,with excisions in (r,F, q ) . 0 j-0
Let p = bj}= 1, ..., be a complete global symbol in the sense of Definition 4 and (pj} a partition of unity on X belonging to U. Further let vj E C;( V j ) ,pjvj = pj for all j . Define N
OP,(P)
=
C pjoPv(Pj)
(2 6 )
vj.
i-1
Here pj, vj are fixed once and for all. 8. Corollary. For evev y E R there exists a non-canonical mapping m; : {complete symbols of order p }
R+x R ) @I, Em;(X; Rq)
+ Cm(
such that R* OPV(P) ( Y , tl)
- t-# 0PP"'h)( Y , tl)
(2 7)
for h = m;p. In fact, from Proposition 7 the assertion is obvious for (22) instead of Em;(X; Rq). But a kernel cut-off argument as in 2.1.4. Proposition 7 admits to pass to
Em;(X; Rq). 9. Definition. Let p = lpi} be a global complete symbol in the sense of Definition 4, p = ordp, and h E Cm(R+X R ) @ I n E m K Y ( X ; Rq).Then p is called y-equivalent to t-rh if
(27) holds.
Remark. The systems of pairs { p , t-fih] for p ~ S $ ( c L l *X O X R"+'+q), h E Cm(R+x O ) 0 , W"v(X; RP) which are y-equivalent behave well under natural operations such asformal compositions or formal a4oints with respect to given weights. In the first case the complete global symbols are paired by the Leibniz products local& and the Mellin symbols by the analogue of the Leibniz product for the Mellin calculus, cf. 2.1.3. Theorem 11. In the second case it refers to the vD0 rule for formal a4oints of complete symbols in the vD0 calculus based on the Fourier and Mellin transform. respectiveb. 10.
11. Corollary. For every y E R there exists a non-canonical mapping
S")(cT*(X*xO)\O) such that
a*OP,cYp(,,) (Y, tt)
+C"(fi+XO)@,Wf,(X; Rq)
- t-'op@"'f
1(Y, tt),
331
3.3. vDO-s on Manifolds with Edges
x ( E q ) being an excision function, f the image of pep,
under ( 2 8 ) . Here op,cyP(,,,) is defined analogously to (26) where p j are the local representatives of ~ p (. ~ , 12. Remark. (28) can be chosen in such a way that it induces a mapping
S y ) ( c T * ( X ^ x a )\ 0) .+ Cm(0)8,m $ ( X ; Rq).
(29)
Note that h ( y , z, q ) E Cm@)63’n’BkY(X;Rq) implies op$’(h) ( y , Aq) = A ’ X A ( t P opP’(h) ( y , q)) x;’
(30) for all A E R,, ( x n u )(2, x) = A(“+ 1)’2 u(At, x). An analogous homogeneity is fulfilled E S!?(cT*(X^x 0 )\ 01, for Ir 1 2 const. for OP,OIP(,,))( Y , 711, P(,,) Let us apply Corollary 11 to p(,,,E S $ ) ( c T * ( X ^ x a)\ 0) with an f ( y , z, q ) E Cm(a,‘BKY(X; Rq)),y-equivalent to xp(,,,, cf. Remark 12. Then 2-N
*
(31) n* OP&(,,)) ( Y , 7) t-’ O P P ( f ) ( Y , 7) for Iq(2 const. The operators on both sides of (31) can be extended by homogeneity in the sense of (30) to all q 0. Let w, wl, w2 be cut-off functions with (32)
ww, = 0 , wwz = w 2 .
The operator function d ( Y , II)
= w(t
I?ll)t-~ OPL3.f) ( Y , V ) W l ( t 1111)
+ (1 - w ( t IaIN n* OPv(P(,,))( Y , 11) (1 - w2(t lal))
(33)
is homogeneous of order p, i.e. d(Y, AV)
d ( Y , rt) x;’
=
(34)
a
for all A E R,, ( y , q ) E x (Rq \ {O}). It is equivalent to n* opV(P(,,J( y , 7). For abbreviation we often set
7)= (1 - w ( t [rl))a* OP,(P) (Y, 7)(1 - w2(t [rll)) where p = { p j } is a complete global symbol, U,(Y,
1
(35)
Then p = b,}+a ( y , q ) for any choice of h, t-rh being y-equivalent to p , can be interpreted as a parameter dependent operator convention which defines for p a family of continuous operators a(y,q): X * q x ^ ) +XS-”@(X^)
(39)
for every s E R, e = y - p. The weight y is arbitrary, bu_t fixed. In 3.2.3. were defined the classes of amplitude fun_ctionsS ” ( a X Rq;‘Z, ‘Z) with respect to the scales ‘Z, and operator spaces Y p ( ’ Z , ‘Z). For our applications it is convenient to modify the point of view. We have more concrete information that admit to avoid to check 22
Schulze, Operators engl.
332
3. Operators on Manifolds with Edges
explicitly the validity of the scale axiom (though it could be done). Moreover we distinguish between scales and dual scales. The role of C, f? play here ZY and CQ,respectively, with C " = {E"')sGR,
E"'=X"(X''),
(40)
e = y - p, and the operator spaces are yv;Y,Q
(41)
= gQ(",MLp(x^) g - Y ( n ) ,
cf. 2.22.4.(29). 13. DeFition. Let R G Rq be open, v, y,
e E R,y - Y 1 e. Denote by Sv;"*Q(Rz x Wq)
the subspace of all a ( y , y', q ) E Cm(RZx Rq, !P;Y,Q) with D ; , . D { a ( y , y ' , q ) E C"(Rz X Rq, y " - l p l ; y , ~ ) rl &"-Ifll(.fl2
(42) x Rq; EJY,~ s - v + l S l ~ ~ (43) )
for all 01 E W, B E Nq and all s E R. The second space on the right of (43) refers to the actions ( x A u ) ( t ,x) - ,I("+ l)'zu(,It, x) on the spaces EJa, cf. 3.2.1. Definitioh 1. The space of all a which only depend on ( y , q ) will be denoted by Sv;yQ(O X Rq). If a satisfies the relations (43) with S;;lfll instead we get by definition the classes S;iy74. 14. Proposition. Let p = ( p j } E S;,(cU^ X R (38) belongs to S';"Q(RX Rq).
X
R"+'+q)l. Then the operator function
ProoJ: By definition the operator family is a finite sum of expressions (44) + ~ ' (101) t OPyGj) ( Y , 01 w ; C ~ [qI) > w(t [q])t-' op@Y(h"j)(Y, 7)ol(t . h . = p . h,vj, . jTj = p j p j y j (for abbreviation n* is dropped now). It suffices to prove the assertion for (44) for every j . For convenience we set again h = hj,,p = jTj. It is clear that (44) satisfies (42) for Y = p. The same is true of the derivatives with respect to y. For the condition (43) we mainly check the derivatives with respect to q, those with respect to y are simple. Thus for simplicity we omit in this proof the variable y at all. Set
, ,
wol = o,owz= w z , where the arguments in the cut-off functions are t [ q ] .
We have repeatedly to check estimates of the form Ilx-'(q) { D t b ( V ) }~ ( ~ ) l l e ( E q ~ , E * + a 5 - ~c p [) ~ l ' - l ~ '
2 6 and all s E R. Denote this system of estimates for s E R by Est(y, B, 6) for 6' 5 6. a ( ? ) certainly satisfies Est(p, 0,O). Since for (y,
e is fixed). Note that Est(p, B, 6) * Est(p, B, 6') a p ( v )= w ( D { m )w 1 + w ' ( D { n ) oi
(46)
333
3.3. lyDO-s on Manifolds with Edges
is of analogous structure (in particular Y@-lDl;y*Q-valued), (46) satisfies Est(p - 1/31, /3, 6) for all 6 5 IpI. For the remaining terms we first consider ak := alaqk, 1 5 k 5 q. If = (0, ... , 1, ... , 0) has 1 at the k*hplace we write a(k)= a,,-. We obtain a k a ( q >= a(k)(q)+ f h+)
with
f ( q ) := (akw)mwl + (a,w’) n u ; , g(q) = wmakw,
+ w’nakw;.
We want to show that f ( q ) , g ( q ) are Y-m;ysQ-valued. Here we use that pyv;Y.Q@ c y-m;v.e
, @ = const outside a compact set, and p@ = 0. This can whenever p, @ E C m ( K + )p, be applied to g ( q ) , for w&w, = 0 , ( 1 - w ) a , ( l - w2) = 0. Further we write
f ( q ) = (a,w) m u , + (a,w’) mw, + (akw’) n o ; - ( = (a,w’) nw; - (a& m w l .
a d ) mw,
Let p, @ E C;(R+), p@= p, p & ( l - w ) = a k ( l - w ) . Then by inserting t [ q ]in. p, @ we can write
f ( s >= ( a k w 7 p n @ 4 - ( a d ) pm@wl + f o ( t l ) , where fo(q) is Y-“; y. *-valued. By construction we know that f,= pn@ - p m @ is !Fm; Y* ‘valued, i.e. f ( V ) = ( a d ) v n @ w l - w2) + hh) with Y-”;”Q-valuedf,. Since a k ( l - w ) = 0 on supp(1 - w1 - w 2 ) it follows that f ( q ) itself is Y-”;”Q-valued.For the present step we only need that all occurring functions are Y”-l;y,Q-valued.The differentiations of the cut-offs yield a gain of a factor t (on the right or on the left). Thus the functions g, f o ,f 1 , h and then also f satisfy Est ( p - 1, /3, 1 ) for all /3 with 1/31 = 1 . By analogous considerations we can treat the higher derivatives. 0 15. Definition. Cm(cX1 x
a),denotes the subspace of all u E Cm(cX1 x a) which have
an asymptotic expansion m
with u - E~ C m ( c X ”x a), u-~(&, x, y ) = A - k j ( t , x, y ) f o r all 1 2 1, t 2 const. The asymptotics (47)f o r t + m are interpreted in the same sense asf o r classical symbols of order 0 where t plays the role of a covariable.
This gives rise to a natural FrCchet topology in Cm(cX1 x a),. In an analogous manner we can define Cm(cX1 x a2),. 16. Remark. The assumptionp E S:,(cll^ X B X Rn+’+q)lin Proposition 14 was made for convenience. We could impose as well for t+ m analogous properties as in Section 2.2.4. for the classes L;{O’. But this behaviour will be cutted off anyway in the global theory. 22’
334
3. Operators on Manifolds with Edges
For instance, we could consider { p p j } for { p j }E S:,(cU^ X 0 X R n + i + q ) l , Q, E Cm(cX1 X or more generally complete symbols {q}which are for t .--, 00 like classical symbols in t of order 0. They would have the "L;,(")-behaviour"for t + m . Examples are given by the following simple
17.Proposition. The multiplication Jn, by q(t, x, y , y ' ) E Cm(cX" X a2),acting as a y, y' dependent family of operators in E",V , s, y E R , induces an element in So;y* )'(a2 X Rq)for every y E R. Prooj The y , y ' dependence is not the essential point, so for simplicity it will be dropped. Since we have no q dependence, we only have to check the symbol estimates
IIx-Yv) Q,(t, X ) x h ) l l B ( E & q 5 c , %-'(a)p ( t , x) x ( q ) = p ( t [ q ] - ' ,x ) . In the special case p ( t , x) = 0 for t > const the assertion follows easily by a modification of the proof of 2.1.6. Theorem 1, where we have to insert t [ q ] - * instead of t . It suffices there to observe that the constant c N ( p )is uniformly bounded in q . The same argument for every fixed s also applies , sufficiently large. Thus it remains to if Q, only satisfies / p ( t , x)l 5 const t - N ( s )N(s) check the estimates when q(1t, x) = 1-jq(t, x) for 1 2 1 , t 2 const. But this is again simple, since it is allowed now to insert those u E Ess V with u = 0 for t 5 const. Then we may deal with the Hs-V(X")estimates that are certainly true for s E N , and then by duality and interpolation for all s E R. 0 Set
YQ= [ o ] X m * @+( [l r )- w ] Y ( X " )
e E R , cf. Section 2.2.4. The space (48) can be written as a projective limit of Hilo), j E N , 9'.ti+ l ) 4 YQ, '0 for all j , bert spaces YQ, YQ= + lim YQ. 03.
(49)
J
It is not hard to invent an appropriate choice of the YQ,Q. This is left to the reader as an exercise. Spaces of amplitude functions with operator-valued symbols which map into Y Qare interpreted in the sense of the notations after 3.2.1. Definition 1.
(with the point-wise formal arljoint *). We write analogously M: v- Q(nx Rq) for the corresponding y' independent operator families. M',";yv @((n x (Rq \ { 0 } ) )denotes the space of all
335
3.3. wDO-s on Manifolds with Edges
for every j , k E W.rfwe omit the subscript cl in ( S O ) , (51) then tJgtkbelongs again to the corresponding classes of order v - (j + k ) . Similarly g b , q ) E M");"Q(Rx (Rq\ (0))) implies t j g b , q ) tk E M ( " - Jk):y,*(R x (Rq\ {O})) forallj, k e N . 20. Remark. Let g v - j E M b - j ; Y 3 P ( f 2X2 Rq) be an arbitrary sequence. Then there exists a g N E
M2Y.e(R2 X Rq)
which
is
unique
mod Mi"; y,e(R2 X Rq)
such
that
g
-
gv-j j=O
E M L - ( ~ + ' ) ; " Q ( R ~ forevery XR~) NEW.
The construction of g follows by a standard excision construction with respect to the covariables q. It refers to a natural FrCchet topology of Mz which follows immediately from Definition 18. There is a canonical mapping g;
: M;
O-3
-
Q(n*x RQ)
M y y Rx
(R4 \
{O})) .
It follows from 3.2.2. (23) after restricting gb, y', q ) to y
(54) = y'
21. Proposition. Let h ( t , y , y', q ) E C"@+ x 0')@n Em-". y ( X ; Rq) . Then t - P w ( t [ q ] ) opg'(h) & y ' , q ) G ( t [ q ] )E M8Y,@(f2'x
R4).
(55)
Further h o b , q ) E Cm(f2)BnEm-".Y(X;Rq) implies t-J'w(tlq1) opg)(h0)0:q ) G(tlq1)E M$);y-@(f2 x (Rq\
[O})).
(56)
Here w , w" are arbitraiy cut-off functions, e = y - p .
Proof: First consider (55). For simplicity assume that h does not depend on y , y ' ; the general case follows by an obvious generalization. The Mellin symbol is then h (t, z, tq) . By applying the Taylor expansion at t = 0 with respect to the first argument it follows N
h ( t , Z, t t l ) =
C tjhj(z, t v ) + h,,(t,
Z, tit>
j=O
with a remainder h,, of analogous quality as h but flat of order N at t = 0. Then t-Pw(t[vl) tjOPY(,")(hJ) (7)o"(t[vl)
is homogeneous of order p - j for Iq I 2 const and has the mapping properties as demanded in Definition 18. Further the term with h,, can be written as t-"w(t[ttl) tNOP$'Cn (11) G(t[?l)
(57)
336
3. Operators on Manifolds with Ednes
with tNf = h,, . We may assume that f vanishes for large t , since the cut-offs remove the part for large t anyway. Let w1 be another cut-off function with ow1= o.Then (57) has the form
w,(t[rl) t N { t - W t [ 1 1 >OPg’m (1)G(t[11)1* The factor w l ( t [ q ] )tN causes a symbol order p - N if we prove that of order p . But the latter property follows from
c
I...} is a symbol
m
f(ti
2,
1 )=
&pk(t)fk(z, 11,
k-0
pk+O in
c
C;(K,), fk-0 in 1 0 2 - ” s Y ( X ; RQ), f-Ppk(f)
< m, and
o(t[711) O P g ’ W (1)G(t[Vl)+O
in 9(.. . ; EJ.y, YQ), and the analogous property for the formal adjoints. This proves that ( 5 5 ) is indeed a classical symbol as required in Definition 18. The assertion for (56) is obvious. 0 Most of the above constructions around global complete symbols p and the y-equivalence to f-yh extend immediately to the case of dependence on 0:y‘) E R2 instead of y E 0.In particular Proposition 7, Corollary 8 have the corresponding extension to this case. In the following we shall tacitly use the obvious modifications of notations and facts such as global complete symbols of the classes Sc,(cU^X R2 x R”+l + q ) l , the ( y , y ’ ) dependent variant of (38) and Proposition 14. We shall see below that usual lyDO techniques along R allow to reduce the objects to y dependent ones. But for the moment it is convenient to admit dependence also on y’.
22. Definition. M Y ; ~ ~ Qx(Rg) R Z for v, y, space of all operator families of the form
e E R, p = y - e, p - v E N ,
aCv, Y’, 1 )= av0: Y’, 1 ) + U M O : Y’, 1 ) + 805 Y’,
denotes the
v),
where
n’ OP,(P) 05 Y’, 1 )a m l ) with a complete global symbol p = Ipj} E S:,(cU^ X R2X Rn+l+Q)l, c$ (16), avOs Y’, 71) = w’(t[tlI)
(58)
aMO: Y’, 1 )= d t [ ? l ) t-” OPCiYh) o? Y’, 1 )@l(t[tll)
(59)
for some h E Cm(R+X R 2 ) Q, 102”,Y(X; Rq), 1-’h being y-equivalent to p , and gb,y’, 1 ) E M: y,*(R2 x Rq). The subspace of y’ independent elements is denoted by MY;Y-Q(R X
RQ). M(”);y. @(RX (Rq\ {0}))denotes the space of all operator families of the form a,,O!
1 )= dO: 1 ) + gcy, 11,
(60) where d(y, 1 ) is of the form (33), here with v instead of p and gb,1)E M$);”Q(Rx (RQ \
IOI))
*
In the sequel for abbreviation we often omit R* in (33) and (36), respectively. Note that M-”; Y. Q(02 x R‘) = Mom;Y. Q(02 x WQ). (61)
337
3.3. VDO-s on Manifolds with Edges
23. Remark. Let p = bj}be a complete global symbol of order v and t-'h y-equivalent to p. Define a,(y, y', q ) , aM(y,y', q ) by (35) and (36), respective&, with cut-offfunctions w , wi satkfying (32). Define analogously civ(y, y', q ) , y', q ) with cut-offfunctionsG,Gi, GG, = G , GGz = G2. Then a,+ a M =ci,+ & m o d M ~ y ~ P (X0Rq). 2
An analogous remark holds if we interchange w , wi in ( 3 9 , (36) or if we consider the M(v);Y, P (ax (Rq\ { 0 } ) )class. 24. Remark. From 2.3.1. Remark 7 we obtain Cm(R+ X 0 2 ) @,%p.Y(X; Rq)= [ Cm(R+ x 0 2 ) @ , % $ ( X ; Rq)} + (Cm(R+XO2)@,,%-=*Y(X; R q ) } , cf: also 1.1.5. Lemma 3. Proposition 21 then shows that in the Definition 22 we may assume without loss of genera&
f E Cm(0)@,%ll,(X; Rq) and h E Cm(R+ X Oz) 8,%ll,(X; Rq), respectively.
3.3.2.
Pseudo-Differential Operators of Wedge Type I1
From 3.3.1. Proposition 14 and Definition 18 it follows that M';"@(R2 x Rq) c s'; y, P(az X Rq). We shall consider in this section the corresponding vDO-s along
a.
W c m ; Y * Px( X 0) " for y, p E IR b the set of all operators ww-::mp(x* x a), w;:(x-x a)) for which C * E ~(w:o;llpp(x-~ a),
1. Definition.
CE
n
seR
seR
Wpldcc-y(X" X a))(cJ ako 3.2.5. (16), (17)). Further WY,;y * Q ( Xx" a) denotes the space of all operators G + C, G = Op(g) for certain g E MY,;y* P(a2 x Rq), C E W i m y,; Q ( X A x a).The operators in WY,;y , P ( Xx^ a) are called Green wedge operators Cfor the trivial weight interval).
2.Definition.Forv,y,e~lR,y - e - v E N , wedefine
W v ; y , ~ ( Xa) " =~ {Op(a) + C: a E M ~ ; ~ x Rq), , Q C( E~W~ ~ " ; " , Q ( Xa)]. ^X (1) The operators in w';y. Q ( X "X a) are called wedge vDO-s Cfor the trivial weight interval). From 3.2.1. Theorem 6, 2.1.1. Theorem 6 it follows that every A induces continuous operators
A : Wt,',p(X"
X
0)- W&"sQ(X^ X 0)
E W';"P(X^ X
0) (2)
for all s E R (cf. 3.2.5. (16), (17)). In the following sections we will investigate subclasses of the wedge vDO-s with asymptotic information for a given weight interval 9 = (- k, 01. Of course, we could do this at once and regard Wv;yl as the special case when k = 0. But we want to divide the calculus in a similar manner as for the cone operators. Certain yDO aspects are independent of the asymptotics whereas the asymptotics then occupy the attention to functional analytic refinements. For dealing with the operators in W v ; " Q ( Xxa a)we need the analogues of stand. ard properties of amplitude functions in a VDO calculus, here for M': y. Q(R2X Rq) Q
338
3. Operators on Manifolds with Edges
The amplitude functions have a symbolic structure, again, namely with respect to the interior and to the edge. Set for abbreviation
Z;:= S‘”’(cT*(X^x 8 )\ O), ,
(3) where the subscript 1 indicates the subspace of those p E S(”)(cT*(X^x 8 )\ 0) with p = p f o r large t for some EES y ) ( c P ( X “x 8 )\ 0). Further define
I;,,:=s y ( c P ( x - x O)\O), 7. P := Y. Q(8 x (RQ \ {O})). Note that Z;,,,c Z;. E;
]M(y);
(4) (5)
3. Proposition. There exists a canonical surjective mapping
,
a;, :
,
Z?v3
Q
+
Z.,,
(6)
with ker a;, = M‘,.);v, e(8x (Rq \ { O } ) ) . Further there is a non-canonical mapping
,
,
op;, : EL, +Z; y, with a;, 0 op;, = id.
,
,,
Q
(7)
This will be obtained as a corollary of Proposition 5 below.
4. Proposition. There C a canonical surjective mapping a;: M“;Yle(8 x~
4
+) E ;
(8)
with ker a; = MV-’;RQ(8 x Rq) + M $ v v Q ( 8x Rq) and a non-canonical mapping
op;: E ; +Mv;VvQ(8X RQ) with a;
0
(9)
op; = id.
Proof. By definition a E M”;v*Q(8 x Rq) is of the form a = a , + g, a , E M”;”Q (8 x Rq), g E M;yaQ(8x Rq). The Green elements g are smoothing; thus a ; ( g ) = 0. Moreover
IM”;”Q(R x R4) ldiagnc cya)@*LS,(X^;R4). (10) Then a; of an element in IM”;~~Q(8 x Rq) is nothing else than the (ydependent) parameter-dependent homogeneous principal symbol-of order Y , and it is unique. Now, if we are given a p(,,)E Z;,we can choose a (5, q)-excision function x and form from xp(”)a global complete symbol p = bj}where the local representatives of p(”)are just the homogeneous principal parts of order Y of pi. Now we always find a Mellin symbol h, such that t-”h is y-equivalent to p . If we form 3.3.1. ( 3 9 , (36) we can set op;(p(,)) = a , + aM. 0 Let us consider r,, 0 a;: M Y ; Y s Q ( 8 2 x Rq)-Z;,,,, (with an obvious extension of a; to M”;V~Q(8* x Rq)) and
M:
x
O(n2X Rg) := ker a;,,.
339
3.3. IDDO-son Manifolds with Edaes
5. Proposition. There is a canonical sujective mapping g; : M’; Y. “ 0 2 x ] R q ) +q Y. e
(13)
+
with kero; = IM’,-’;y-Q(R X Rq) lM:y,e(R X I R q ) for the restriction of (13) to the y’-independent subclass. Further there is a non-canonical mapping op; : with a;
0
z;
v. e +
]MV:
v, @(a2 x Rq)
(14)
op; = id.
Proof. By definition a(y, y‘, 7)E l M Y ; Y , Q is of the form a(y, Y’, 7)= w‘(t[7l) O P , ~ ) ( YY’, , 7)w X t [ 7 l ) + w(t[7l) t-”op$’(h) (Y, Y‘, 8 ) @ l ( t [ V l ) + dY,Y’, 8 ) with a complete symbol p = bj}, a Mellin symbol h such that t-’h is y-equivalent to p, and a g E M’;” (Remember that x* at opv is omitted for abbreviation). Let us write pj = mj, pj. where pi, = r, {of the homogeneous principal part of pj of q), and pj,o the remainorder v , restricted to y = y ‘ } , x an excision function in der. Then for 7 9 0 we can pass to the operator family
,+
Q.
,
(c
N
O P , ~ ”(Y, ) 7):=
C qj oP,bj,,)
(Y, 7)v j ,
j=1
cf. 3.3.1. (26). Further set h,
= hl
Then
a;(a> (Y, 7):= w’(t[vl) OP,(PA) (Y, 7)w;(t[7l) + w(t[71) t-’oPP’(h,) (Y, 7)wi(t[tll) + g,”)(Y,7)
(15)
with g(”)= a;(g), cf.3.3.1. (54) is obviously an element in M(”);”Q. In particular it satisfies the homogeneity condition (16)
G ( a ) (Y, 17)= n”%a;(a) (Y, 7)x i ’
for all A E R, , y E a, 7 E Rq \ {0}. It is clear that the mapping (13) defined by (15) is canonical. Another choice of the cut-offs only affects the Green remainders in an adequate way. The form of ker a; is obvious. By definition every element in M(’);yvQ is of the form (15). In order to define opi it suffices to form
w’(t[al) OPvbA) (Y, 7)w;(t[71)
+ w(t[7l)
(r,
t-‘oPcP(h,> (Y, 7)W l ( t [ 7 1 ) + Xl(7) g(”)(Y,7).
(17)
Here x is a 7)-excision function, x1 an 7-excision function. Then (17) is just the image of an element in l M ( ” ) ; Y . Q under op;. Proof of Proposition 3. As we have seen in the proof of Proposition 5 we can define a; of any element of Z; by composing (14) and (8). Now a;,, follows by restricting the result to 0.(11) follows by composing (9) and (13) applied to Zb,, c 2;. 0 6. Remark. Every a ( y , a ) E My;ys@(L’ ( c j 3.3.1. Definition 13), a,, E Mi;V,Q(L’ pal part ofal ( c j also 3.2.2. (23)).
In fact we can set a.
= op;(a;(a)).
+ ao, a1E S:,;v.e(L’X Rq)
X RI) can be written as a = a1 X Rq). Then a l ( a ) coincides with
the homogeneous princi-
340
3. Operators on Manifolds with Edges
7. Proposition. There are the natural inclusions
MY;”Q(R2 X Rq) E - Mv’;”Q(R2 X Rq) for
Y’
- v EN ,
(18)
Proof (18) is obvious. (19) is also obvious as far as it concerns the y , y’ derivatives or operator families in M; y * Q . So it remains to consider a , + a M of 3.3.1. Definition 22. It is clear that w f ( t [ t l l )0: OP,(P) (Y,Y’, 7)4 ( t [ t l l ) + w ( t [ t l l )t - q OPP(h) (Y,Y’,tl) wl(t[?l)
belongs to M’-lPl;y,Q.Moreover it is easy to see that {D8,w’(t[tll)}OP,(P) W X t [ t l l )+ { D : w ( t [ t l l ) }t-’op$’(h)
Wl(f[?I)
+ w‘(t[tll) OP,(P) { D : w ; ( t [ t l I ) }
+ w ( t [ q ] )t-’op$”’h)
{D;wl(t[v])E } M;-lPl;Y*Q
(cf. also the arguments in the proof of 3.3.1. Proposition 14). 0 E M’-j;y*Q(R2 X Wq),j e N, be an arbitrary sequence. Then 8. Proposition. Let x Rq) such that there exist3 an a E M”;y.Q(RzX Rq) which is unique mod M-”;Y*Q(R2 N
a(y,y’,?) -
1U , - ~ ( ~ , ~ ’ , ~ ) E M V - ( N + ~ x) ; Y ~ ~ ( R ~ ~
q
)
j=O
for all N
E N.
Proof By definition there is a sequence of pairs ( p v - j , t - Y + j h V - j ) ,where p v - j is a global complete symbol of order Y - j , and h v - j a Mellin symbol, t - ’ + j h , - j being y-equivalent to p Y - j . Such sequences of pairs allow to define their asymptotic sums in the sense that there exists a (p, r ’ h ) such that for every N E N the difference
c N
(p, P h ) -
( p v - j , t - Y + i h V - j ) (being again of this kind) has order
j=O
With (p, t-’h) we can associate an a ,
Y
- ( N + 1).
+ aME Mv;Y’Qby 3.3.1.(38). Then there are
N
g v - j E M L - j ; Y *with Q a,+
a M - z ( ~ , - ~ - g ~ ~- M ~~ ) - ( ~ + ~ ) : y , Q fNoE r aN1.1 j=O
Let xl(q)be an ?-excision function and cj a sequence of constants, growing sufficiently fast as j + 05. Then m
g ( Y , Y ’ , v )= C
~ l ( t l / c j ) g v - j ( ~ , ~ ‘ , t l ) ~ ~ ~ ~ ~ ~
j=O
defines the asymptotic sum of the g,-j (cf. also 3.3.1. Remark20). Then a := a , + uM + g has obviously the desired property. 0
341
3.3. wDO-s on Manifolds with Edaes
a
Proposition. For every a ( y , y', q ) E M v ; " Q ( f lXz RQ) there exists b ( y , q ) E Mv;y*Q(R X Rq) such that Op(a) - Op(b) E W i m y*Q(X^ ; x 8).
9.
Proof Applying the notations of the proof of 3.2.2. Theorem13 we obtain that aNE MY;Y~Q(ilZ X Rq) is flat on diagn of order N . Integration by parts yields Op(aN) = Op(iTN)where iTN E M Y - N ; Y , Q ( 0X2Rq), cf. Proposition 19. From Proposition 8 we find
and hence C:= Op(a) - Op(b) E W ' - N Y . Q ( X ^ X 0)for all N . 0 By analogous considerations we obtain 10. Proposition. For every A E Wv;Y.Q(XA X 0)there exists a properly supported (with respect to a) operator A l E Wv;Y.Q(X^ X 0)such that A - A l E Wim;y.@(XA X 0). Note that p(y) E C;(n),A E W";YsQ(XA x 0)implies PA, Ap, E W";Y.Q(X^ x a). The technique of 3.2.2. Theorem 13 also yields
11. Proposition. Let 9,y E C,"(L?), supp Q, n supp y = 0. E W,";Y,Q(X^ x 0)for every a E M v ; ~ . ~ x( R Rq2).
Then
Op(qay)
12. Proposition. a E Mv;YvQ(R2 x Wq) implies for the point-wise formal a&oint a* E Mu; - Y ( f 1 2 x Rq), implies analogously a. E M(');K Q(ax (R*\ {0))) atv,E ]M'; - Y (ax (RQ\ {0)))and -@I
-Qs
a:(a*) = (@:(a))*, @;(a*)= (cr;(a))*. Here the * f o r the y-symbolic level indicates complex conjugation whereas for the A-symbolic level we take the point-wise formal adjoints, again.
This result is left to the reader as an exercise. We have to use in particular 2.1.3. Theorem 11, 3.3.1. Remarks23, 10.
13. Proposition. a E Mv;@,a(nz x Rq), b E ]M"; y,Q(az X Rq) implies ab E lMv+ v ' ; y * a ( R 2 X Rq) with respect to the point-wise multiplication; analogously b,V,) E EM@'); ys Q(n x (R4 \ (0))) implies a(V ) E M(");p . 6 (0x (R4 \ I01 )I, a(v ) b(v') E M ( Y + v'); Y . d(L?x (Rq\ {O))), and a;+ " ( a b ) = u ; ( a ) a ; ( b ) ,
(20)
a;+"(ab) = u ; ( a ) a f ; ( b ) .
(2 1)
Proof First we show that if a or b belongs to the class with subscript G then ab E M;+ "';y,a. This is obvious if both a and b are of this type. Assume now, for instance, that b E M$;Y,Q.Further suppose for a moment that a = a" + a , where h ( y , z,q ) in a M ,p in a, do not depend explicitly on t. Then a is a classical symbol, i.e., ab is also classical. The Green property follows from the mapping properties of the composition and its adjoint. This argument also yields the desired mapping properties in the t dependent case. The only point is to show that the composition is again a classical symbol. In order to illustrate the idea let us assume for a moment
342
3. Operators on Manifolds with Edges
that a consists of the multiplication by p(t, x ) for some Q, E Cm(cX1), Q, = 0 for t > const. (cf. 3.3.1. Definition 15). Applying the Taylor expansion at t = 0 we get N+ ..
Q,(?,X ) =
-
1
C
tjQ,j(X)
+ tNp(N+l)
j=O
with I ) E Cm(R+ XX), (1 - W ) Q , ~ +E ~C - ( C X - ) ~for any cut-off function w, cf. 3.3.1. Definition 15. In view of 3.3.1. Remark 19 b E lM$y * e implies t j p j b =: g+, E I M g - j ; y x e . Further Q,(~+ ,) b has the mapping properties expressed in 3.3.1.(50), (51), but with dropped subscript cl. By 3.3.1. Remark 19 we obtain tNp(,+ 1) b of this type again with order m
cf. 3.3.1. Remark20, and hence
v ' - N . This shows that q b j=O
pg E i M ; ; y * * .
Now if h and p in aMand a,, respectively, are t dependent, we can apply the analogous arguments by Taylor expansion at ? = 0. This shows that indeed ab E IM;' "'; Next we consider {wt-'opQ'(h)
01
+ w' op,(p)
w ; } {Gt-" op$)(K)
GI
+ 47 o p , O
G ; } , (22)
where the arguments in the cut-off functions are always r [ q ] . For abbreviation we have omitted n* at the op, actions. By 3.3.1. Remark 23 we may choose the cut-off functions in a convenient way. So we assume that w , G' = 0 and w2 = G2. Then there remain the compositions y(n) $ 6 w t - opg)(h) ~ w1 G6-U' OPM (23) ( lr w'oP,(P)w;G'oP,mw;,
(24)
w' op,(p) w;G?-v' op$)(K)
61.
(25)
Applying 3.3.1. Remark24 it is allowed to assume h E Crn(R+XLP)@,rmV,(X; Rq), KE
Crn(R+xa2) @.,rm;(x; R4).
By assumption (cf. 3.3.1. Definition 22) we have e - Y 2 6, y - v' 2 e, in particular y - Y' - e 2 0. The composition (23) is of the form W?-'+'(")
op,(fl
= Wt-v-u'+y(n)
OpM(7) t-Y("'61
?Y-v'-Pwl
OPM(f1) w1 oPM(.f) t-Y("' G l ,
(26)
In the latter equation we have used with f := T-Q(")h,1-T-"(")K,fi= T-"+"'+Qf. the commutation argument of 2.1.5. Theorem 13. Next we show that 0 OpMYIcfi) w1 opM(7) Gl
- w OpUcfZ)
(27)
for a Mellin symbol fz of order Y + v' and analogous quality as the original ones, indicates a remainder belonging to the G class after the corresponding where
-
343
3.3. cyDO-s on Manifolds with Edges
weight shifts as in (26). For calculating f 2 we can use the analogue of 2.1.3. Theorem 11, here applied in a parameter-dependent version. We obtain fi(t,Y,Y’, z, t?)
- 1 -1 k!D:fl,(f,Y,Y’,z, trl) (-taf)ko,(t[711)j(t,y,y‘,z,t q ) , k=O
t=Imz. The operation contains a multiplication from the left by w . So we may neglect the ] ) k > 0 , since w , = 1 on supp w . Thus we may take f i derivatives ( - t a f ) k ~ l ( t [ qfor in the form
, f ) for the kth summand on the right, f = t q , we have ord If we write f k ) ( t , y , y ’z, + Y‘ - k in (t,f ) . The asymptotic sum can be carried out with respect to excisions in (t,f ) ; remainders are then in the admitted G class after applying the actions. Having the asymptotic sum we can finally pass to a symbol in Cm(R+xOz) @,nWo’+”’(X; Rq),where again a remainder of the G class appears. In other words we have proved (27). Next we turn to (25) and write
fk) =Y
w ’ o p , ( p ) w ; G ~ - ~ ’ o p ~ ~ ( h “= ) Ga,, + ao, a , = w ‘ op,(p) w ;Gt -”’ opp(h-) GI w ; , ao= w ‘ o p , ( p ) w ‘ , ~ t - “ ’ o p ~ ’ ( h ” ) G , w 2 .
-
We easily obtain that a. E l M ~ m ; a, Y ~ pw’op,(p)w;J , o p , ( p ” ) ~ ’ ~The . latter equivalence follows from GJ,= J and w ;Gt -v’ opp(h-) GI w ;
- w ; G op,(p? GI w ; .
Thus, by standard yDO calculus
(24) + a ,
-
0’
OP,@) ~
‘ OP,O 2 w‘2 - w’ O
#aw ; ,
P , ~
where p # p” denotes the Leibniz product, taken for all localizations. Here we have The anused that w ’ w ; = w‘. Summing up we obtain that (22) belongs to ML+”’;y,d. alogous calculation for the homogeneous operator classes is left to the reader as an exercise. The symbolic rule (20) is an immediate consequence of (10). Next observe that (21) for a und b in the lMG classes is a special case of the corresponding rule of composition of leading homogeneous parts of classical operatorvalued symbols. More generally (21) follows immediately for classical symbols a , b of the corresponding orders. Using Remark 6 we can write a = al + ao, b = b, + bo with a , E s:iq(az x Rq), bl E s;;”@(a2 x RP), a0 E lM;;QqR*x R q), bo E IM$Y*e(R2 x Rq).The elements of the form
zo = w(t[?tl)
OP,@)
w,(t[rl) + w’(t[rll) t-”opa”’(h) w;(t[711).
(28)
M; e. d(0 X Rq) are characterized byvanishing of h ( f , y , y ’ ,z, q ) I d i a gR at t = 0 and of the homogeneous principal part of pIdiagR on the edge (i.e. at Z = 0 in local terms). lM~:e,d(Rz X RQ) is generated by these elements together with M; qa2 x W4).
344
3. Operators on Manifolds with Edges
In other words a,, = Zo+ 8 for some 8 E M;P~a(D2 X Rq) and similarly for bo. Since the Green symbols are classical we can add them to a l and b l , respectively, i.e. assume a. = Zoo,bo = 6 . Now it is easy to check that the composition of operator families where one factor belongs to the Mo class belongs again to hio.Thus the edge symbol o;”’(ab) equals a;+”‘(albl)= a;(al) cri(bl) = # ; ( a ) oz(b). 0 Recall that we have defined the canonical mappings a;:
I M v ; yqa2
x R4) +I;,
g;: MV;Y.P(aZx
Let
]R4)+z;YvP.
ZV;Y . P c ZV x ZV; Y , P P
A
be the subspace of all pairs ( p , , p A ) for which p v = $,(a), p A = a;(a) for some a E Mv;”sQ(f22 x Wq). A pair ( p v , p A )E Z; X Z; obviously belongs to Cv;Y,*iff rApv = C T ; , ~cf.~ 3.3.1. ~ , (11) and the formula (6). We obtain a surjective mapping ysp
(a;,a;): MV;YqaZx R”+6’;Y.P
with ker(a;, a;)
= MV-l;Y~@(RZ x Rq).
The latter relation is an immediate consequence of Propositions 4, 5. One can easily construct a non-canonical mapping opv: CV;Y . P + M V ; Y . P with (a;, a;) 0 o p v =identity on
6’;Y.P.
For A = Op(a) + C E Wv;P,Y(XA x a), C E W ~ ~ ; Y * Qx( a), X ^ we set 14. Definition.
a E Mv;Y~Q(f22 x Wq),
a’,(A):= a;(a), a;(A):= a;(a). From the standard VDO calculus it follows that this is a correct definition. In fact if a, a’ E IMV;Y-Q(O2 x RQ)give rise to the same A then the complete 1pD0 symbols in local 2,y-coordinates of any U^ X R c X^ X 0,U^ E U ^, are of order - 00 (observe x ^a)).Thus a - a‘ E M;Y.P(Qz X Rq). But then an anthat W v ; Y v 4 ( X AX 0)c J ~ : ~ ( X Y , P ( X ^x 0) alogous argument in the operator-valued framework c L:l(fl; Es-y,E m * @yields )) a - a’ E Mi”; Y ; “(a2 x Rq). By construction a;, a; vany-P. ish over More pedantically the notations should distinguish between the symbolic mapWe hope our simplification will not cause confupings over Mv;YsQand Wv;YvQ. sions. From Definition 14 there follow the surjective mappings
wi
a; :
Wv;Y.Q(x^ x n)+s;,
a; :
W’; Y.P(X^ x 0)+q Y . P
(a”Y ’
b yA ) :
(X^x 0)+,y;Y.0,
WKY,P
345
3.3. vDO-s on Manifolds with Edges
and the kernels can be characterized analogously as above for the M Y ; Y v Q spaces. In particular ker(a;, a;) = w'-l;YsQ(X^x a). From the abstract WDOcalculus with operator-valued symbols we have the notion of being properly supported with respect to R. 15. Proposition. Every A E Wv;YvQ(X^ X 0)can be written as A = A l + G where A l E w';Y,Q(X^X 0)is properly supported with respect to 0 and G E WGm;y , p ( X ^X a).
Proof Let x ( y , y ' ) E Cm(nx a), x = 1 in an open neighbourhood of d i a g a , suppx x Rq),A = Op(a), we write proper. For a E Mv;Y.Q(R2 A l = O pk a ),
G = Op( (1 - x ) a ) .
Then A l is properly supported and it belongs to wY;Yj4(X^X a).By using the explicit calculations of 3.2.2. Theorem 13 we can write for every N E N
where here Op(aN+l)E W Y - ~ - ' ; Y ~ Q ( Xx ^a), and 0;a;,(xa) ly, = y E M"-IaI;~~Q(12 X Rq), cf. Proposition 7. We obtain even A l - A E W -m;v.Q(X^ X a) = W;m;"34(X^X a). For A E W;y2@(X^ X 0)this follows as in the abstract calculus whereas for a ( y , q ) = ( a , + a M )( y , q ) it follows from D f a ; , k a ) l , . = , E M-";"Q(R X Rq) for a =# 0. This shows also G E W ~ m ; y ~ 4X( X a). ^ 0 16. Theorem. Let A E W v Y; . P ( X ^
L*(a,X o ( X A ) )Then . A* E Wv;
-Q*
X a)and A* be the formal adjoint with respect to -"(XA X 0)and
a;(A*) = aL(A)*, o l ( A * ) = a i ( A ) * . The * f o r a, is simply the complex conjugate.
The proof follows easily from 3.2.2. Theorem 15 combined with the concrete information in our subclass, in particular from Proposition 12. The simple details are left to the reader. 17. Theorem. A E Wv;Q;d(X^ x a), B E W v ' ; y , Q ( XxAa)and A or B properly supported (with respect to 0)implies AB E W v +''; Y s 6 (X^ x 0)and
a;"'(AB) = a ; ( A ) o : ( B ) ,
If A or B
al'"(AB)
=
a;(A) aZ(B).
is a Green operator then also the composition.
Proof. The assertion is an easy consequence of the calculations that lead to 3.2.2. Theorem 14 (where we always observe that the remainders in asymptotic formulas give rise to operators in our class) and of Proposition 15. 0
Our constructions have natural properties with respect to coordinate changes. This admits to define the corresponding global objects. Let V be the bundle over Y with s, y E R, by fibre X^, cf. the beginning of 3.3.1. Then we have the spaces Ws,Y(V),
346
3. Operators on Manifolds with Edges
demanding the restrictions to Vln X A X D to belong to W$;(X-X 0).This is invariant under the admitted transition mappings. Denote by W-m;“sQ(V) the space of all operators
ne ( w y ~wm9Q(y)) ), G* n e ( w-Q(v), w-, -m). GE
SER
with
seR
18. Definition. W”;Y.Q(V) denotes the space of all operators A + G , where A over Vln 1 X^ X D induces an operator in Wv;”s*(XA x D)for every coordinate neighbourhood D of Y , G E W-m:YzQ(V). The local coordinates in Vln are always assumed to be as in the beginning of 3.3.1.
The definition is correct. This follows from the first statement of
x : X^ x R+X^ X a‘ be a transition diffeomorphism as mentioned in 3.3.1. Then there are natural pull-backs x*: w”;Y.P(xx a)+W”;Y.Q(XA x a’), (29)
19. Theorem. Let
x;:
S‘qcT*(X- x R)\O)+S(qcT*(X-x
x;:
~ ( v ) Y; .
D’)\O),
e ( T * a \ 0) +M(v);Y.P(T*D‘ \ o),
(30) (31)
where (30) is understood as in 3.3.1. (S), in (31) we have a derived pull-back of ( y ,q ) dependent operator families, and (y‘, q‘) = (xoy,(‘dXo)-’q ) , xo : 0- 0’ being the diffeomorphism induced by x . Further
a’,(x*A)= x*,aY,(A), a$(x*A>= x::cr>(A). This result is left to the reader as a useful exercise. Finally let W be the stretched manifold for a manifold M with edge Y. In 3.2.6. we have defined the spaces WS,’(
w),
s, y E R
as the subspaces of H;,,(int W) which coincide close to aW with elements of Ws*y(y).This allows to define the class W-m;YsQ(W) by the above scheme and also Ws;Y-Q(w) 3 A + G , G E W--;Y,Q(W), A E Li,(int W), where A close to awcoincides with some element of Ws;Y*Q(V). We can also introduce the global classes
S(”(cT * w \ O),
hi‘”’; Y , * ( T * Y \ 0)
with obvious notations and the global principal symbolic mappings 0;:
W’~Y’Q(w)-,S(”)(CT*W\O),
(32)
0;:
W’; ’.Q(W)+]M‘”; V,P(T*y\ O),
(33)
ker(a’,, 0;)= Wv-*;Y~Q(w). We are more interested in subclasses of operators with asymptotic information and a more relevant Mellin symbolic structure. So we stop the global discussion here and return to the global aspects below in connection with our subclasses.
347
3.3. WDO-son Manifolds with Edges
3.3.3.
Subalgebras with Asymptotics
In 2.2.4. (27) we have defined the space 9’ls(XA),, 9 = ( - k , O ] . It can equivalently be written as
for A = ( - k , m) (cf. 1.2.2. Definition 11, 2.1.1. Definition 12 including the extension by the decomposition method). Y’Pays(X-), is also an inductive limit over B E W”)of Hilbert spaces !~I?;~(X-)$’, and we set
cf. 2.1.1. (33). Since x1 E C(R+,!i9u(%’;y(X-)$?)) for all B , s, y , j , k , we can define the symbol spaces P ( R x Rq; E , S ; ( X A ) , ) for any Hilbert space E with xi E C(R+,S U ( E ) )(cf. the definitions in the beginning of 3.2.1.), as well as the subspaces of classical symbols.
1. Definition. Ell$yvp(R2x Wq), for 9 = (- k, 01, y - e - v E IN, R 5 Rq open, denotes the space of all g ( y , y’, v )
C=W
RP,
n
~ ( ~ S . Y ( X - ) , ~ = . Q ( X - ) ) )
ssR
such that for certain B E W”), D E g(-y)(”) and all s E R (9 d Y , Y ’ , tl) E mazx Rq; xs9’(X-),9’i(X-),), (ii) g* 01,y ’ , v ) E SIl(R2x Rq; X s - Q ( X ^ )9, ’ ; y ( X A ) , ) , g*(. ..) being the point-wise formal adjoint. The sets B, D are called the carriers of asymptotics of g. Further we denote by Ell$‘);y~Q(R X (Rq\ (0)) the space of all g ( y , o)
cv
with g*(y,v )
cw
nWWX-),WX-),)) x (w\ {OH, n -Q(X-),W(X-),N
x
( ~ q \
{OD,
SER
WS~
saR
and B, D as above, where d Y , A v ) = A%€!(Y, tl) x;’ (3) for all A E R, , ( y , v ) E R x (R‘I\ {O)). The elements in aY; y* (Ell$“; y. are called Green symbols (homogeneous Green symbols) for the wedge calculus. Q ,
2.Remark. More pedantically we could use notations like 83 y*p(02 X Rg)@,. .., since the classes depend on the weight interval 8 = (- k,01, k E N w {m} . But 8 is jixed once and for all. So we usually omit corresponding subscripts here and in the future classes of objects that contain 8.
Now let us remind of 2.2.1. Definition 21. We need an analogous definition with parameters. If h ( y , y ’ ) E C“(R2, 9t:,(X)) is given and j E W \ {0}, then there exist decomposition data e E dec,(h, j ) with N s 2 , independently of y , y ’ E R. This follows from 2.3.1. Proposition 9. 23 Schulzc, Operators cngl
348
3. Operators on Manifolds with Edges
An analogous result holds for dec;(h, j ) , defined by the same data as dec,(h,j) with the only modification y ( n ) 2 eV2 - j - y ( n ) for all v (cf.2.2.1. Definition 19). = dec:(h,j)
3.Definition. L e t v , y , e E l R , p = y - e , , u - v E N , k E N \ { O } ; 8 = ( - k , O ] . Then RV;~.@(R2 x Wq) denotes the subset of all a ( y , y', 7)E MV;y.Q(R2 X Rq) of the form a ( y , Y ' , 7)= a&, Y ' , 7) + a&,
Y', tl) + m ( y , Y ' , 7) + g(y, Y', 71,
(4) where a,, aM are defined by the same expressions as in 3.3.1. Definition 22 with p E S;,(cll- X R2 x R n + l + q ) l but , now h E Cm(R+xR2)@nEm;(X; Rq) (qf also 3.3.1. Remark 24), p, t-'h being y-equivalent, further g(y, y', 7)E %'J K e(llzX Rq), and m = 0 for k ( v ) : = k - ,u + v < 1 ,
for k ( v ) 2 1 , where hja(y,y', z ) E C"(R2, Em;s"(X)), E dec;(hj,, p - v + j ) for p = y - e, further w , w l , w z are cut-offfunctions as in 3.3.1. Definition 22. I f a is independent of y' we get by definition 3"; yv Q(R X Rq) . %("); x Q(Rx (IRq \ {0})) denotes the space of all operator families of the form
(6) a(v)(Y,7)= d(Y, 7)+ C(Y, 7)+ g(Y>71, where d ( y , q ) is of the form 3.3.1. (33) and in addition f ( y , z, 7)E Cm(ll,EmL(X; Rq)), further g ( y , 7)E %$);Y3e(R X (Rq\ [O})), and m = 0 for k ( v ) < 1, k(v)- 1
C ( Y , 7)= w ( t l ~ l ) t - "
C
j=o
2' (I$,
I
OP$(hja)(y)Va w ( t l V l )
(7)
for k ( v ) 2 1, where hjaE Cm(R,Em;sm(X)), and .sjaas above.
Let us further denote by %2p';$(l12x Rq) the subspace of all a & y', q ) of the form (4) with a, = 0, aM = 0 . Analogously we define the M + G classes for R instead of O2 or the corresponding subclasses %$)+ ,(ax (Rq\ {O}) of %(")(Rx (Rq \ {O))).
The following remark on the structure of %"; y. Q(llz x Rq) follows easily from the Mellin techniques of the cone theory. 4.Remark. Let p , h be as in Definition 3 and 3, Gi other cut-offfunctionswith GGl = G, GG2 = G2. Define C,, EM analogously in terms of G, Gi. Then a,+aM=C~+CMmodR~y'P(n2xHZ9).
An analogous remark holds if we interchange w , wi in the expressions for a,, aM or 7. Q(Rx (Rq \ {0})) class. Changing of w and of the decompoif we consider the 3'"); sition data also leads to remainders in the corresponding Green classes. By definition we have c p ; A Q(fp x R4) c M'; Y. e ( p x R4). (8) The cone operator theory implies
R ~ Y ; $ ( R Z X R q ) c M ~ ~ . ~R4). (R2x
(9)
349
3.3. VDO-s on Manifolds with Edges
All assertions of the above theory of 1pD0-s with 1M';"Q-valued symbols remain in force for W;"Qas soon as we have checked that the subclasses are closed under the operations (algebraic and asymptotic ones or concerning coordinate changes). The program of this section is just to establish these refinements. Let us denote by W i y ( X xL?loc)8,B E . ! B y ( " ) , the subspace of all u E W?o:(X"x for which Q,U extends to an element in W i y ( X " X Rq)@for all Q, E Cr(0). Further W$,Y(X"x~ c o m p ) 8denotes the subspace of all u E W;y(xlXRq)8 which have compact support in a with respect to the variable y E 0. Let U i m ; y * Q ( X " Xa)be the subspace of all operators C ES?(W&&,p(XX a), W;:(X"x a)) for which there exist elements B E .!BQ('), D E . ! B ( - y ) ( ' ) such that
a)
cES?(W;,y,,(X^Xa), w;-Q(X"x aloe)),
(10)
C*E e(w:o;nep(X"x a),w;,-y(X"x a,oc))
(11) for all S E R. The operators in U ; m ; y 3 Q ( X " X0 ) will also be called negligible edge Green operators.
5.Definition. U ' ; y ~ p ( xa) l x for 9 = ( - k , 01, k E N \ {O}, v, y, e E R,y - e - V E N , denotes the set of all Op(a) + C for a E W ; ~ , XQ Rq), ( ~ ~C E U ; " ; ~ . ~ ( X "a). X Further UZY; (U; y, denotes the subsets with a E %$: $ ( E 3%y. @). The operators in U";y. Q(X"X 0 ) are called wedge I ~ D O - Sthose , in U%y3 Q ( X Ax 0 )Green operators (with continuous asymptotics in the weight interval 9). Q )
Since U";y. Q(X"X a)c w";7, Q ( XX 0 ), every A operators 3.3.2. (2.).
E U"; y,
@ ( XX 0 ) induces continuous
6. Proposition. Let A E U'; ys 4(X" x a), then for every B E . B y ( " ) there exists a D E dQcn) such that A induces continuous operators A : W$B"(X"Xi2comp)8+ W S g " l Q ( X " X
f210c)8
for all s E lR.
a)
Proof. For C E U ; m ; Y , Q ( X " X the assertion holds by definition. For simplifying notations let us now set R = Rq and assume that all amplitude functions are independent of y , y' for Iyl, Iy'l sufficiently large. Then we may drop the comp, loc subscripts. If g b , y', ?) E %%y.Q then Op(g):
W S V ( X " X R4)+
W S ; " , Q ( x - XlRq
is continuous with some D E This is an immediate consequence of Definition 1, 3.2.1. Theorem 6 and the properties of the Sobolev spaces with asymptotics. In fact, first we have the representations
W ~ Q ( Rqlff X X= W ( R ~ X,~ Q ( X "=)c lim ~ ) W(Rq, X i Q ( X " ) y ' ) , j
where X i e ( x l ) t lare Hilbert spaces, obtained as
cf. 3.2.5. (2). The whole construction can be organized to be compatible with the 23'
350
3. Operators on Manifolds with Edges
spaces in (2), such that there are canonical continuous embeddings XS,@(X")2> c, X;Q(X")yfor all s, e, D , j . Further the spaces satisfy the assumptions of 3.2.1. Theorem 6. This yields first Op(g): WJY(xIxRq)+ Wa-"*Q(Rq, XEsQ(x^)r) for all j, and then, by taking the projective limit over j, Op(g): W'Y(x^XR')+
W'-*Q(R", X>Q(X")e) c W;-"'Q(X" X a).
Clearly X;@(X"),on the right might be replaced also by Sg(X"),.Next let a(Y, Y', a ) = U,(Y, Y', a ) + a& Y', a). Then a(Y, Y', a ) belongs to SS1(R2@ X Rg; XS, '(X")P, Xi- "'(X")v) "5
with e' = y - v , for all j , D = PB.Here we tacitly assume that the construction of the spaces Xiy(X")$l,2%;- O'(X")$' fits together in the sense, that the whole schemes of asymptotic data involved in the decomposition procedure of 3.2.5. (2) as well as of curves surrounding corresponding sets in the complex plane (that are involved in the norm contributions by the analytic functionals) are translations of each other under 2'". Then it follows again the continuity "9
Op(a): Ws(Rq,XJ,"(X")t')+WS-"(Rq, Xi-"*Q'(X")t3) and hence Op(a): WS,Y(X"X]RQ),+
w;-",Q'(X"x Rq),.
From e 5 e' there follow the continuous embeddings W;-".Q'(XA x Rp)ec, W;- "vQ(X^X Rq), and hence the desired continuity. It remains to consider amplitude functions of the form (5). We claim that Op(m): WS,"(X"x Rq),+ Ws-"(Rq, X;Q'(xl)B) for an appropriate D E &"(n) the form
(12) G %'(" e'I,= y - v . By definition m is a sum of terms of
w(1[a1) t - * + j t a OPhfCf) 0: Y 3 a*t-W[al)
for certain f~ C"(Q2, %R;c(X)), 6 E R, 1011 5 j. Except oftranslations connected with the exponents of t the picture is point-wise as in 2.1.2. Theorem 8 for the continuous asymptotics. Since the translations from the decomposition data as well as from j only cause trivial modifications of the arguments, we want to assume for simplicity j = 0, 6 = y ( n ) , f~ C"(O*, %R;"(X)), V E 3fi").Let us call this amplitude function again m. For establishing the relation between B and D it suffices to employ 2.1.2. Theorem 8. It follows D = P ( B + V). If we replace B by the larger B + Vwe can regard (12) as the composition of the continuous embedding WS,"(. ..), + WS,:,,(...), and of Op(m), interpreted as operator on WS,:,,(...),.Thus without loss of generality we can set B = B + V. Then we get similarly as above m(y, y', a ) E S',,(R2qx
Rq;
XS,y(X")m,X>Q'(X")m)
for all j , D = PB. This proves (12). Summing up, the various contributions of 4 y , Y', a ) = g ( Y , Y ' , a ) + {a, + a d ( Y , Y ' , a ) + m ( x Y', a ) map the space W j v ( X ^X Rq), continuously into
Ws-"(Rq, Xi;Q(X")8) + WS-"(Rq, X S , ; " ~ Q ( x+l Ws-"(Rq, )~) XiiQ(x^),).
(13)
351
3.3. wDO-s on Manifolds with Edges
D, is independent of D , D2 = PB,and D, is a sum of various sets in W ncontaining ) B and the asymptotic types of the Mellin symbols in m,shifted by the translations from the decomposition data. As a corollary of 3.2.5. Lemma 14 we get a continuous embedding of the space (13) into WL-”*Q(X”x R‘J)B with D = D1 + D2 + D 3 . 0 7. Remark. Let t 9 = ( - k , O ] , m z k ( v ) = k - p + v z l , E dec;(h,a, p - v + j ) , j = k ( v ) , ..., m . Then
hjaECm(K+xR2)~,~~~(X),
Another result of this rype is t1%$: $(a2x R’1) t m c %L-+(;!lm); ,. Y. @(a2 x Rq)
for arbitrary I, m E N,where M + G on the right can be replaced by G for v - ( I + m ) < 1 - k according to
%$$oG
k ( v ) < 1.
= %;jY3@ for
+ p,
This is an immediate consequence of relations of the type 2.1.5. (10) and of the observation that free t powers cause a corresponding decrease of order in the sense of operator-valued amplitude functions. Another corollary of cone operator techniques (cf. the corresponding modification of 2.1.5. Theorem 14) is 8.Remark. Let hja, eja be as in Remark I, now for j = 0, ..., k - 1 . Set
9.Proposition. There are natural inclusions
w’;(0’x R‘J)
8”; Y. P ( 0 2 x R‘J) s for v‘ - v E N , further 09 ,D:5Jp; Y. ~
Y. P
x R‘J)E 3”-181; Y. @(a2 x R‘J)
( 0 2
Y Y
for all a E W ,B E Nq.In particular
D;y,D:8‘&T$(02 for all a,/3 with
X
R‘J) E 8L-1f11;”*Q(02 X R‘J)
2 k.
Proof. (15) is obvious as well as (16), as far as it concerns the y, y’ derivatives or ~ Pproof . of D:{a, + a M }E 9lv-1flJ;y,Q is practically the operator families in 8 ~ yThe same as the corresponding part of 3.3.2. Proposition 7. Here we only have to insert the Il)l;-valued h, a property which is preserved under differentiations. It remains to consider (5).
352
3. Operators on Manifolds with Edges
Let us drop for simplicity y , y'. Then for 1 5 r 5 q
obviously belongs to W-'; 7 . Q . For dealing with g ( v ) we observe that D,,,w(t[v]) = ( t ( a / a t ) w ) (t[v])D,,[v]. Here we gain a power of z which causes the decrease of the order of the amplitude function with respect to by 1 Further ( a / a t ) w has compact support on IR, for every fixed 9. Applying Remark 8 we obtain g ( v ) E %;-l;y.Q. The assertion for the higher derivatives in 11 follows by induction. 0 10. Proposition. Let a,-j E %v-j;V*p(i22 X Rq), j E W,be an arbitrary sequence, where the carriers of asymptotia of the involved Green part are independent of j . Then there is an a E 3';h @(aZ x IRq) which is unique mod 32," y* e; ( f 1 2 X Wq) such that
for all N
E IN.
Proof: The first step of the proof concerning ( p , - j , f - +jhV-j ) is the same as that of 3.3.2. Proposition 8. The only modification here is to preserve Ern;-valued Mellin symbols. By kernel cut-off arguments we know that h in (p, f-"h) of the proof of 3.3.2. Proposition 8 can be chosen of this sort. Let a, + aME % " ; Y , Q be formed by means of (p, r ' h ) . Then there is an r, E with a,
+ aM - (a, - r,) ~ % ~ - l ; y * Q ,
and successively we find r, -
E
%k$$such that for every N (I
N
a,+aM- C ( a y - j - r v - j ) ~ % ~ - ( N + ' ) ; ~ , ~ . j=O
From Remark 7 we know that r, be defined as g
-
E
%Y,-j; X Q
for k ( Y ) - j < 1. Now let g E 8;- k ( v ) ; ys
m
rv-j with excisions in the covariable 9. j = k(v)
Then it is obvious that k(v) -1
a = a,+ a M +
C
rVmj+g
j=O
satisfies (17) for all N. Further it is obvious that a is unique mod
%;m;Y*Q.
0
353
3.3. yDO-s on Manifolds with Edges
11. Proposition. For every a ( y , y', a ) E RY;P.Q(l.?*x Wq) there exk& u b ( y , a ) E R";.Q(R x RQ)such that OP(U)- O p ( b ) E U,m;"3P(X" X a). ProoJ We apply the method of proving 3.2.2. Theorem 13 and obtain (with the notations there) that aN E %";"Q(RZ x Rq) is flat of order N on diagR. Integration by parts yields O p ( a N )= Op(a;Y), where ZN E R"-N;Y,P(122X Rq), cf. Proposition 9. According to Proposition 10 we find
and hence C := Op(a) - Op(b) E W N ; " * Q ( X " XO) for every N E N . The explicit form of the amplitude functions c that belong to % " - N Y , Q ( f 2 2 X RQ)for every N shows that C = Op(c) certainly has the mapping property (10). The formal adjoints Y* P(X"x0). can be treated analogously. Thus C E aim; By analogous considerations we obtain
12. Proposition. For every A E 3";x p(x^ X 0)there exists a properly supported (with respect to 0)operatorAl E U " ; Y . @ ( X " X a) that A - A , E ~ ~ " ; ~ ~a~) (. X " X
n
13.Proposition. Let A E N';y-P(X^x 0)and rp, t,u E Ci(a),supp rp supp t,u = 0. Then Y, "X" x a). rpAw E Clearly r p ~ C ; ( a ) , A E ' U " ; ~ , @ ( X " Ximplies O) PA,A ~ ~ E U " ; ~ . Q ( X " X L ? ) . Our next objective is to apply the symbolic constructions with a;, a; of 3.3.2. to the subsets (8). By restriction we first have the principal interior symbolic mapping
%";Y*Q(a x RQ)+z;.
0;:
(18)
Further we now define
z;
Y.
P := %z(");
Y. "(a
x (R4 \ {O})).
This notation reflects our system of abbreviations. The fixed weight strip 9 = (- k, 01 is dropped anyway, and Z2YtP of 3.3.2. corresponds to k = 0. The map a; of 3.3.2. restricts to
%":nP(a x RQ)j z 2 Y . e
(19) which is the principal edge symbolic mapping. If we define Z";y,Q in an analogous manner as the corresponding space of pairs of symbols in 3.3.2. we obtain g;:
(g;, 0;):
R';"P(ax ]R1) + , p Y . Q
(20)
14.Proposition. The mappings (18), (19), (20) are sujectiue and ker a; - ~ V - ~ ; V . Q + ~ ~ Y . P , -
+ %;-I;KP,
ker u;
= 3";n P
ker(a;, a:)
=%"-l;Y.@,
where %:Y** := ker a;,*, c& 3.3.2. (6), (all symbol spaces are taken ouer R there are non-canonical mappings
X
R4).Further
354
3. ODerators on Manifolds with Edaes
op;: Zi+g";Y,Q, op;:
z2 YP Q+
op": ZCY,Qj g with 0;0 op; = id, spaces.
U>
BY Y, Q C Y , Q
0
op: = id, (u;, u;) 0 op" = id over the corresponding symbol
Prooi The surjectivity of (18) follows as in the proof of 3.3.2. Proposition 4. The only modification is that we assume here h to be %;-valued. This could have been done anyway. The resulting expression a, + aM defines just op;, applied to p , E Z;. The surjectivity of cs; also follows from the existence of op;, defined on pA E Z ~ y ~ p , which is of the form 3.3.2. (17) plus the part from (7), namely k(v) - 1
m ( y , 7) = w ( t [ V I H - ~
C
tj
C op%(hja)( Y ) q a w ( t [vI).
lalsj
j=O
(21)
Finally op;, op; can be combined to an operator convention op" by defining a, + aM + xl(q)g(u)+ m ( y , q ) as the image of (PI, p A )E Z";Y,Q. The assertion on the kernels follows immediately from Definition 3. 0 The amplitude functions in %";y*Q(RZ X Rq) for k ( v ) L 1 have a Mellin symbolic structure which is part of the asymptotic information of the associated operators. Set for fixed p, v, y, 8 = (- k, 01 from Definition 3 Z$Y = cya)€31R% y ( x )
and
for v < p and j = 0, ..., k ( v ) - 1 or Y = p and j = 1, . .., k - 1.We want to introduce Mellin symbolic mappings for the conormal orders v - j
us:
gI";"Q(,(j x R4)+Z$Y,
(22)
x Rq) j Z 5 - j . (23) Rq) ( by 0 2applying first the non-ca(22), (23) can be extended then also to % " ; ~ ~X @ X Rq)+%";"-Q(Rx Rq), nonical mapping from Proposition 11 %";"Q(R2 a ( y , y', q ) b ( y , q ) , i.e. ub-j(u) := a h - j ( b ) (where the non-canonical data disappear automatically). Now let ~ K Y > P ( ~ J
a ( y , 7)= ap(y, 7)+ aM(Y, 9 ) + m ( y , q ) + g(Y, 4)I
with aM(y, 9 ) = u(t [ql)t-' op?)(h)wl(t [91) 3
h(t, Y, 5 q ) E Crn(R+XR)@,%(X; R 9 , k(u)
-1
C
m ( y 9 7)= w ( t [ v I ) ~ - ~
j=O
tj
C o~%(hja)( y ) v U w ( t[vI).
lalij
ThejthTaylor coeficient of h(t, y, z, tq) in t at t = 0 is a polynomial in q of order j with coefficients in Cm(R)€3 ELRL(X). Denote it by ub-j(a, + aM) ( y , q ) . Then we set
3 . 3 . ~ - D O -on s Manifolds with Edges
@Y,-j(a)(Y, Z ,
a ) = Qk'(a;w
+ a d ( Y , Z) + hj(Y7 z ) P j ( V ) .
355 (24)
This yields (22), (23). Note that akj vanishes by definition on WG;y9p. Analogous Mellin symbols can be defined over %(');y.Q(R x (Rq\ {0))) = Z;YsQ. Let us consider in particular a&:
q v . e
+ X l G YM.
(25)
It is then obvious that the diagram %#;.e(n x RP) SZ;;Y,Q
commutes. In other words the principal Mellin symbolic level is uniquely determined by the principal edge symbolic level. The same follows, of course, for Y < p, p-vEN.
15. Proposition. a E %";Y.Q(R X Rq) implies for the point-wise formal adjoint Rq); analogously u(")E %("):Y*e(R x (Rq\ {0})) implies a:", E%(Y);-&-Y (ax (Rq \ {0))),and
a* E % " ; - Q * - Y ( ~ x
a ;(o* ) = (ab(a))*, o;(a*) = ( u ; ( a ) ) * ,
(26)
oY,(a*) = T"(aY,(a))*.
(27)
This is left to the reader as an exercise. In particular (27) is the weighted (and parameter-dependent) analogue of the first formula of 2.2.1. (18). 16. Proposition. a E %";e,d(0x Wq), b E %"';y.Q(a X Rq) implies ab E % " + " ' ; Y * 6 x (ax Rq) with respect to the point-wise multiplication, analogously a(")E %(");QJ(O x Rq \ {0})),b,,) E %("');~~@(L! x (Rq\ {0}))implies a(v)b(g, E %("+ #);Ysd x (0x (Rq\ {O})). Moreover a;+"(ab) = a ; ( a ) a:(b), a;+"'(ab)
=
a;(a) a i ( b ) ,
o ~ " ' ( a b=)( T " a Y , ( a ) ) a & ( b ) . The classes with subscripts G or M + G form two-sided ideals.
(28) (29) (30)
Proof The symbolic rules (28), (29) are a special case of 3.3.2. Proposition 13. (30) follows from the Mellin symbolic rules of cone operators. The proof that a or b in the class with subscript G implies ab of this type again follows by analogous arguments as the corresponding fact from 3.3.2. Proposition 13. The same is true of the compositions between operators of the type a, + a N . The considerations in evaluating 3.3.2. (22) (cf. Remark 4) are formulated in terms of Em;-valued Mellin symbols, and the occurring remainders with subscript G belong even to the present Green classes. This is a consequence of the common cone operator calculations. The composition between operators of the types M + G also follows from the cone theory, i.e., the result is of this type, again. Compositions of the form { w ' o p , ( p ) w ; } m with m of the form ( 5 ) may be neglected after a convenient choice of the cut-off-func-
356
3. Operators on Manifolds with Edges
tions (cf. Remark 4). There remain the compositions {wt-”op@)(h)wl}m,(or with the converse order). Again simple cone calculations show that the result is of the type M + G. The classes of homogeneous operator families can be treated in an analogous manner. 0 17. Remark. Let us indicate for a moment we weight interval % = (- k, 01 at the classes of DejiniX IRq)a. The symbolic structure and the mapping properties of tion 3, i.e. use the notation %v;)’,Q(i22 X Rq)8 a natural locally convex topology. Green amplitude functions admit to introduce in 91y;v~p(D2 For 8‘ = (- k’, 01, k’ 2 k, we then have canonical continuous embeddings of the space for 8’ into that for %. In 91v;vqi22 x
nv)(-m,o, = n A”;V-(R~x
~
q
)
~
ktN
we can introduce the projective limit topology. We obtain a class of amplitude functions of analogous structure as (14), where ( 5 ) is to be replaced by a converging sum m ( y , Y’, tl) = t - ”
1 W(Cjt [ttllt’ l 41a JoP*(hjcJ
( A Y ’ ) tl”W(cjt [tll)
j=o
with constants cj increasing sufficiently fast as j + m ( c j the analogous ideas of Section 2.2.2.). All elements of the present theory with finite weight intervals have an analogue for % = (- m, 01.
For limiting the size of the exposition we will not carry out it here explicitly. Let us now return to the operators Op(a). 18. Definition. For A x (X^Xa),we set
= Op(a)
+ C E Uv:Y-Q(X^x a), a E !RV;v,p(a x Rq), C E U,”;y.Q
a;(A):= aY,(a), @ ; ( A ) := a;(a), aL-j(A) := aL-j(a) for k ( v ) 2 I, j
= 0,..., k ( v ) -
I.
By definition we obtain an analogue of Proposition 14. We mainly need the case of (bb,a;):
U”;Y,p(X^Xa) +Z”;Y-Q.
This is surjective, we have ker(a’,, a:)
a)
= U”-l;Y~p(XAX
and there is a non-canonical mapping 0pV: ZCV,Qj U v ; V , Q ( X ^ Xa)
(32)
with (a;, a;) 0 opv= id onZv;VgQ. The Mellin symbolic structure will be of interest for us mainly when v = p. So it can be neglected for the moment.
19.Theorem. Let A E U ~ ; V , Q ( Xa) ^ ~ and A* be the formal adjoint. Then A * E U”;-Qs-Y(X“Xa) and
a’,(A*) = a’,(A)*, ai(A*)= o;(A)*, aL(A*) = T”(U‘,(A))* (cf: also Proposition 15). The proof is a simple exercise.
357
3.3. vDO-s on Manifolds with Edges
E Uv;Q,a(X n), A ~B E U”‘;V~Q(X^X 0)and A or B properly supported (with respect to 0)implies AB E 9Iv+v’;V*a(X^x 0)and
20. Theorem. A
ar’“’(AB) = ab(A)a C ( B ) , a>’”(AB) = a:(A) a i ( B ) . I f A or B belongs to the class with subscript G ( M + G), then also the composition, After the tools of the calculus the proof is straightforward. It follows in particular that for A = Op(a) + C, B = Op(b) + 0,a and b only ( y , v)-dependent, C, D negligible edge Green operators, AB = Op(a # b ) + K where a # b is an amplitude function with an asymptotic expansion as in 3.2.2. (26) and K another negligible edge Green operator. Now let p = v = e - 6, p’ = v’ = y - e. Then we obtain
a&+p’(AB)= {Tr’a&(A)}a $ ( B ) . (33) As usual TO indicates the translation by /3 in the complex Mellin plane. It is interesting in this case to express the analogue of the Leibniz product for the complete Mellin symbols aM(A ) (Y, 5 I) := {a&-j(A) (.Y> z, 9 ) )j
UM(B)(Y,Z, I ) ’ = { & ’ ( B ) ( Y >
Z,
=
0, ..,, k - 1 3
I ) } j = o,..., k - 1 .
Let us identify for a moment complete Mellin symbols with finite formal power series in L k-1
h ( ~5,
17
A) =
1Lp-jhj(Y9 5 11.
j=O
For
c
k-1
f ( Y , z, I,
=
Lp‘-’fi(y, 5 a )
j=O
we set
( h # & M f ) ( Y ,5 I,L>=
c
1 -a- ! -lqh(Y,
z, I , L)#Ma;f(Y, 2, I , 2)
x { T p ‘ - ’ D ; h j ( y ,5 71)}
a , ” f i ( Z,~ ,I).
This is a finite sum, since the dependence on 11 is polynomial. The motivation for the latter sum is, of course, the Mellin translation product that we know from the cone theory. Now
@M(AB)(Y,Z, I ) = { ~ & + ” - ‘ ( A B ) ( Y , zI ,) I r = o,..., k - 1 follows by ar+P’-r
W)
c
( Y , z, I) =
1
(34)
D ; u & - ~ ( A()Y , z + p’ - i, I)
(a(+ i+ j = r
x
aaal” y
M
-i
( B ) ( Y , 2, I).
Note that for the complete Mellin symbol a&*)
(35) (for example in the case
p = v = y - e ) one can derive a simple expression in terms of the complete Mellin
symbol aM(A),namely
358
3. Operators on Manifolds with Edges
r = 0, . .. ,k - 1. Here h*( ... ,z, ...) = h(*)(... , n + 1- Z; ...), where (*) indicates the point-wise formal adjoint operator function (cf. 22.1. Proposition 9). 3.3.4.
Edge Boundary Value Problems
Now we turn to the edge boundary value problems and to the concept ofellipticiry of wedge wDO-s. Analogously to the theory of boundary value problems in the standard sense we shall pose additional edge trace and potential conditions. The formal scheme is the same as for boundary value problems in BOUTETDE MONVEL’S algebra [B9] or more generally in VihK/ESKI”S work [Vl], ..., [VS] and REMPEL/SCHULZE [Rll]. In the elliptic theory a condition to A E 91p;y,p(XAx0 ) for y - e = p is that the edge symbol d ; ( A ) (y, 71) is a family of Fredholm operators. The idea is then to enrich d; ( A ) by further finite-dimensional operator functions which constitute together with u $ ( A ) a family of isomorphisms. This is just an analogue of the classical Shapiro-Lopatinskij condition. The nature of the additional data follows from the properties of ker d; ( A )(y, q ) , ker d; ( A *) (y, q ) . Such constructions are part of a more axiomatic approach for higher edges and corners. For notational convenience we often assume here that the X*-bundle V belonging to the stretched manifold W is trivial, i.e. V = X * x Y. The general case can be treated in an analogous manner. We shall talk of the condition E, if the symbol component of quality d is elliptic (i.e. a family of isomorphisms) and about F, if we only have a family of Fredholm operators. 1. Definition. An operator A E U””*p(X^X a),p = y - e, satisfies the condition E, ifa;(A)E Z; has an inverse in Xi’’(c$ (3.3.2. (3)), E,,” i f ~ : , ~ (:= A r) A o ; ( A )E Z;,Ahas an inverse in X;,; (cf. 3.3.1. ( l l ) , 3.3.2. (4)), E M if a$(A)(y, z) := H s ( O +H”-’(X) is an isomorphism for all y E 0 and all zE - y ( n ) , s E R, FA if
a;(A)(y, q ) : X*Y(X^) +X”-”“(X^) is a Fredholm operator for all y E 0,q E Rq \ {0}, s E R.
(1)
As we know the condition EM (FA) for s = so implies EM (FA)for all s E R.Another consequence of the cone calculus is
E,,A>EM-FA, cf. Section 2.2.4. Set ’
(2)
d(Y, 71) = (Y, 7). (3) Under the condition FA the index of (3) is finite and independent of s. We have
kerd(y, 71) = W X A h , kerd*(y, 7) = Y;:(X*)B. In view of d ( y , Aq) = A”xAd(y,q ) x ; ’ for 1E R,, ind d(y, q ) only depends on y, y / 171.
(4)
359
3.3. yDO-s on Manifolds with Edges
For a n y compact topological space Q and Hilbert spaces H 1 , H2 we can consider continuous functions f : Q +X ( H 1 ,H2), X ( H 1 , H2) being the space of Fredholm operators, in the topology induced by Y?(Hl, H2). Let K ( Q ) be the K group over Q (cf. [AS], [A8]), consisting by definition of equivalence classes [E, F ] of complex vector bundles E, F E Vect ( Q ) . Then the family of pairs (kerf(x), cokerf(x)), x E Q, can be canonically indentified with an element in K ( Q ) , called the index off
(5) ind, f~ K ( Q ) . The construction can be applied in particular to f = d, H 1 = XsY(X^), H2 = Xs-”Q(X^), and Q equal to
s*Q = {( JJ, a):
y E Q,
E sq- I}
,
for any open Rl, fil CCR, i.e. S*6, = S*R lfil, S*R being the cosphere bundle induced by the cotangent bundle T * R and a fixed Riemannian metric in it. Then dl(y, a ) := d(y, ~ ) ( ~ fisi a, Fredholm family, parametrized by the compact space S*fil. Let us carry out the construction of (5) in our concrete situation (cf. also 2.2.5. Lemma 7). First there is a finite-dimensional subspace J(61) cW
X -18
(6)
with im d(y, a ) + N-(6,) = Xs-”@(X^ for ) all (y, a ) E S*filand all s E R.The open set R plays the role of a coordinate neighbourhood on the edge. For global constructions along the edge it is allowed to shrink R a bit and then to assume that R coincides with 0,. In addition we may assume 30, to be of the class C”. In other words for simplicity we suppose that N- = N - ( n l )can be chosen independently of 0,. Let N- =dim N- and choose a C” family of isomorphisms c-(y, a ) : C N - + N - , (y, a ) E S*fi Then
(7)
(with d = d,) is a surjective family of Fredholm operators. The family of kernels N+(Y, a ) = ker(d(y, a ) , C-(Y, a)) @ CN-}. It is constitutes a finite-dimensional subbundle N+ of S * 6 x {Yls(X^)e isomorphic to some G+EVect(S*O).A n y isomorphism b : N++G+ corresponds to an element of d 8 N*,, N*,being the dual bundle. For analytical reasons we real8 Then b = (c+, c‘), according to ize N*,as a subbundle of S * 6 x ( 9 ~ ~ ( X ^@) CN-). the projections of N+ to Xs-Y(X^)or CN-.In other words the isomorphism b at the point (y, 7) acts like b(Y, a ) : (f(Y>a h dY, 71))
(C+(Y,a), f(Y, h ) ) + (C‘(Y7 a), €!(Y, a ) ) . Thus (8) can be completed to an isomorphism +
(c+)(y, being the fibre of G+ over (y, a). Clearly ind d ( y, a ) = dim (c+),, - N- . 4)
),
(9)
360
3. Operators on Manifolds with Edges
The construction of (10) could have been carried out with N- + M instead of N- for @ CM instead of (trivial bundles Q X Ck over a space Q are any M E IN and also denoted by Ck).Then the pair (G+, CN-)represents an equivalence class
e+
4
[GI,,CN-]=[GI++C@M,@N-+M
and indsa d = [
]E K(S*d),
e+,CN-].
We shall assume that
indsadEp*K(d), where p : S*d+d is the canonical projection and p * : K(d) + K ( S * d ) the K group homomorphism, generated by the pull-backs of vector bundles under p . For M large enough there are G * ~ V e c t ( d ) with G + @ C M = p * G + , S*d x @ N - + M = p * G - , where, of course, G- =dx CN-+M. If R is diffeomorphic to a ball (i.e. contractible) we have G+=fiX CN+for some N+EN.The notations G* are often used for convenience (the global analogues can be non-trivial anyway). According to (4) we extend e ( y , q ) by homogeneity
for all A E R+. This gives in particular extensions of c*(y, q), c'(y, q ) by homogeneity of order p. e ( y , q ) is now a family of isomorphisms
Xs"Q(XX^)
X.Y(X^) e(y,v):
@
(12)
@
+
(G-)y
(G+)y
for all ( y , q ) E R x (R4\ 0) = T * R \ {0}(the latter notation will be justified by the transformation behaviour of the index under diffeomorphisms of R). Now let ~ ( qbe) an excision function. Then
r ( y , r ) = X(V)C'(Y, q ) E
SiM x lR4; C N - ,CN+)
with respect to {xA}on Xs*Y,X S Qand the trivial actions on CN*.The relations (13), (14) together with the above information are the motivation of the following
2. Definition. We set %!!;Q(R2 x RP) = SEI( 0 2 x Further %t:;y(R2 X R4) denotes
R4;C,
Y:s(X-),).
the space
of all
b ( y , y', q ) E
nSgl(R2x
lRq;
S
X'?Y(X^),C) for which the point-wise formal adjoint b * ( y , y ' , q ) belongs lo %!!-Y(R2 X R4) (cf. also thefirst bracket on the right of (13)). Independence ony' will be indicated by R instead of Rz.
361
3.3. wDO-s on Manifolds with Ednes
Remember that the weight interval is kept fixed once and for all. By definition the operation * induces bijections *.. %?Y(R2 x RP) +%!;-Y(R2 x Rq),
*: %!Q(R2 x RP) +%? -yo2 x R4). The elements of %TYare called (scalar) trace symbols, those in % " : Y (scalar) potential symbols. The vector-valued ones follows by replacing C by CN*.This is just the nature of (13) and (14), respectively. We can also write
b ( y , y', q ) E CN+63 %ll,;Y(R2 x Rq), c ( y , y', q ) E %!Q(R2 x
(15)
63 C N - . (16) Since the symbol spaces of Definition 2 are classical, we have the spaces of homogeneous elements of order ,u %$):Y(Rx (Rq \ {0})) , %?@(Rx (Rq \ (0))). W4)
For b E %$);", c E %?);@it follows
4') = I%c(Y,
= A " ~ ( Y ,7)~;'~ C ( Y ,
b(y,
for all A E W+,( y , q ) E R x
(IRq
\ {O}). If x ( q ) is an excision function then
(1 )%$); Y c 8"; + Y , x(q)%Y;Y c%!J'
(17)
(with obvious notations). Further there are canonical mappings g;:
+
%KY
+%2(lr):y
> %ce
where in the case of y, y' dependence we first set y
KOE Sn ER
(B&mp(n)
(18)
+%(lr);e -
= y'.
An operator
, W 2@(X^ X Rd)
is called a negligible potential operator. The formal adjoint of KO then belongs to 5?(Ws-@((X^X Rcomp), H;,(R)). An operator To E S?(W$Y(X^xRcomp),
n
n
seR
seR
H;",,(O))is called a negligible trace operator if its formal adjoint belongs to
n
g(ff:omp(n),
W2-y(X^x~ l o c ) ) .
S€R
3. Definition. Let v, y, e E R, y - e - Y E IN, kEW\{O}, B";Y.e(X^x0 ) denotes the space of operators
8 = ( - k , 01. Then
")
1=(" T R ' where A is an m X 1 matrix of operators in %";Y.Q(X^X a),and T = Op(b) + To, K = Op ( c ) + KOfor certain (15), (16) (with v instead of p), matrices (N+X 1, rn X N - ) of negligible trace and potential operators To and KO,respectively, and an N+ X N- matrix R of yD0-s in L:,(O);m, 1, N+ being arbitrary. Bim;Y,e(X^ x 0 ) denotes the space of a1 V with the entries Ao, KO,To, R o , where A. E %~m;Y;e(X"X 0 )63 (C" 63 C'), KO and To matrices of negligible potential and trace 63 CN-). operators, respectively, and Ro E L-"(R) 63 (CN+ Further we define B$Y;$(Bz y.e) as the subset of those 1 E B";Y,Q for which the left upper corners A belong to %O$;, (%; Y . e ) .
362
3. Operators on Manifolds with Edges
Observe that
For being able to treat the Green operators in the left upper comers and the other X Rq)of all operatorentries T, K,R in a unified way we define the class %>y,P(Oz valued symbols
where g E %2y,;y*Q, b E %:Y, c E %?Q,r E Si, over Qz X Rq. As soon as we discuss algebraic operations such as compositions, formal adjoints, ..., continuity properties, symbolic structures, we often assume for convenience m = 1 = 1, N+=N - = l . In the applications we shall switch automatically to the given dimensions which needs only trivial modifications. The ellipticity will require arbitrary N* and m = I , where the case m > 1 is obvious, again, after the calculus for m=l. 4. Proposition. Any 1E 23";"Q(X^X 0)induces continuous operators
W Z m p ( X ^ X 0) 1:
+
a3
Hkmp(fl) for all s E R,further wy(x--x 1:
WsY P loe ( X ^ X O ) a3 H,",-,
.(acomp)B
CB
-+
'(fa
w;-" * Q ( X ^ X O,oc)B a3
Hiomp(0) H L ''(0) for every B E By(") with some D E B@(n), for all s E R. Proof. (21) is a consequence of 1= Op ( a ) + V for 'Z E B ; m ; " * Q ( X ^ X a),
(with obvious notations) and 3.2.1. Theorems 6, 8, 3.2.5.(3), (16), (17). In 3.3.3. Proposition 6 it was obtained (22) for the left upper corner A of 1. (22) for the entries T, R of (19) follows from (21). For K we can argue in a completely analogous manner as for the Green operators in the proof of 3.3.3. Proposition 6. 0 5. Remark. The operators Op ( c ) for c E % ~ v * *X (R0q) 2have C" kernels over X"X 0.The finite orders are only caused by a typical singularity close to t = 0. In fact if q,(t, x, y ) E C;(X^X O ) , i = 1,2, then
b of infinite flatness oder in t at t = 0 which causes a true order v = --, cf: 3.3.1. Remark 1 9 and formula (29) below.
363
3.3.lyDO-s on Manifolds with Edges
The following propositions are obvious generalizations of the corresponding ones of Section 3.3.3. (Propositions 9, 10, 11, 12, 13). So the proofs will be dropped.
6.Proposition. There are natural inclusions %:”P(aZx R4) E %pya2x R4) for v‘ - Y E IN,further
D;y,D{ %2Y*@(a2 X R‘J)E %;-IBI;y,Q(az X Rq) for all a E N2q,fl E Nq. E % ; - j ; y , p ( f 1 2 X Rq), j E IN, be an arbitrary sequence. Then 7. Proposition. Let x Wq) such that there is an a E %:y,p(Ozx Rq) which is unique mod%2,”;Y.p((RZ N
a-
C a, -
E 3;- (N+ 1); Y, ~
x
(23)
( 0 2 Rq)
j=O
for all N E N. 8. Proposition. For every a ( y , y’, 9) E %:y,p(122x WQ) there exists a b ( y , 71) E %:Y~Px ( RRq) such that Op ( a ) - Op ( b ) E B , m ; y ~ p ( X ^ x
a).
9. Proposition. For every SS E Bu;V-p(X^X0 ) there exists a properly supported (with re0). spect to f2) operator SS1 E B”;Y.p(X^xa) such that A - A l E B)Gm;y~P(X^X 10. Proposition. Let SS E B v ; ~ p ( X ^0) x and p, p E C,”(O), supp p n supp p = 0. Then pdp E B;m;Y*P(XAX a).
Next we want to extend the symbolic structures of the previous section to B”;”@(X^X0). Denote by B 2 y , pthe set of all operator families
y E 0,71 E Rq \ {O}, with d E Z2yzp, b E %$“;Y, c E %?);Q,r E S ( ” ( 0 X (Rq \ {0}))(i.e., r is C“ and homogeneous in 4 of order Y ) . On Bv;Y*Q(X^x a)we define the principal symbolic mappings
a;:8”;Y.qX-x a) +z;, BKY9P(X^X0)-+B;Y.P,
where a;(&):= a;@) and
:= a;(b), a;(K) := a;(c) in the notations of Definition 3 and (18), and with a;(?“) o;(R)being the homogeneous principal symbol of R of order v. Further we have the Mellin qymbob aL-j(SS) defined as those of the left upper comers. 24
Schulze. Operators engl.
364
3. Operators on Manifolds with Edges
Denote by B";Y,Qthe space of all pairs ( p v , p A )E~i x B>Yvp for which (pv, a,,) E Sv,when a,, denotes the left upper comer of p,,. Then we get a principal
symbolic map @:=(a;, a;):
B";Y.P(x^Xa) +B";Y,P.
(25)
11. Proposition. ker 0"= BV-l;~~P(XAX a). Proof The result for the left upper comers is just 3.3.3. (31). For the remaining entries it suffices to employ the definition which says that these vDO-s are classical and hence those of lower order have vanishing leading homogeneous symbols. 0 12. Remark. ( 2 5 ) is surjective and there is a non-canonical mapping (an *operator convention?
opv:
BTKP
-+BCY~P(X"X0)
such that a" 0 opv= id on B';"*.
In fact, op"on the left upper corners is just 3.3.3. (32). For the right lower comers opv is nothing else than a choice of a yrDO on 0 for a given homogeneous principal symbol. Finally, ) (17)) and to pass to the asfor the trace and potential parts it suffices to multiply by ~ ( 7(cf. sociated yrDO along 0 with operator-valued symbols.
13. Proposition. Let d j E Bv-j;y,Q(XAX a),j E W,be an arbitrary sequence. Then there exists an s4 E Bv;y,p(X^xa) with N
for every N E N. s4 is unique modB)3,";Y*P(X^X a). As usual we writes4
-
j=O
Proof For the left upper corners the assertion is an immediate consequence of 3.3.3. Proposition 10. For the remaining entries it follows immediately from Proposition 7. 0 14. Remark. If we indicate for a moment ihe weighi intervals 8 = ( - k , 01 as subscripts, e.g. %2y-a(02 X IRq)a, B";Y~Q(X"X 0 ) a , . .., we can also look ai k = m. Apart from the lefi upper comers of 3.3.3. Remark 17 we have here the remaining entries of B ~ y ~ P ( Xll)~..m,ol ^X ihar can be introO),-,,o,, ducedassO= g + V f o r ~ = O p ( g ) , g ~ % ~ y ~ P ( R 2 X R P ) (~T~- ,Eo l~, ~ ~ y ~ P ( X - X where the corresponding classes are defined in terms of projecirve limits over k. All elements of the calculus for finiie k 2 1 can be formulaied also for k = w.
The consideration of formal adjoints both of symbols and of operators is straightforas ward. This operation is possible in particular for B>,YdP:=a;(B;nQ(X^X 0))
*.. BV;V,Q +BV;-P,-V A, G A,G
Y
where we have to take transposed matrices. This is the only new element compared with the formal adjoints of edge symbols of the previous section. We then obtain 15. Theorem. s4 E Bv;y-Q(X^x a) implies 1*E B";-Q--Y(X"Xa) and
a;(a*) = a;(&)*, o;(s4*) = 0;(s4)* and also the Mellin symbolic rule under * as in 3.3.3. Theorem 19.
(26)
365
3.3. pDO-s on Manifolds with Edges
16. Theorem. 1E B ” ~ Q ~ d ( XR), ^ Xd E V”‘Y.Q(XAX 0)and d or 9properly supported with respect to R implies 1d E %”+ v‘;y,d(X^x R). For the symbols it follows
a;+ ”’(193) = o;(d)a;(a),
(27)
a;+”‘(dd) = a;(d)a;(a). (28) The Mellin symbolic rule is the same as for left upper comers. r f 1 or d belongs to the class with subscript G ( M + G ) , then also the composition. or d E B;m;y,Q the composition belongs to Proof. First it is clear that for 1E Bim;p*d B;”;Y,d,cf. Proposition 4, and all symbol components vanish. For d E B;@,’, d E B$Y’pit is obvious that 13 E B23;;’” ‘ ; x ~and that the A-symbols compose. This follows by a slight modification of the consideration for left upper comers which shows that the formal properties of BGare analogous to those of 21G. But this is true at all also for the remaining compositions. For instance, the elements of the proof of 3.3.2. Proposition 13 which are based on 3.3.1. Remark 19 now employ in the analogous consideration that g ( y , y ’ , q ) E %:Q-d(Rz X Rq) implies
for all j , k E IN.In other words there is no essential difference to the proof of 3.3.3. Proposition 16, Theorem 20. 0 17.Remark. An aspect of the proof of Theorem 16 is that a E B2Q.d,b E B i : y . Qimplies a , 3.3.3. Proposition 16 for the lefr upper comers. abE B ; + y ’ ; v ~cJ:
We need the operator classes B’;”Qand the symbol classes also globally on a stretched manifold W belonging to a manifold with edges. The typical part concerns a neighbourhood of aK So we first consider the X A bundle V over K cf. the beginning of 3.3.1. It is convenient to write for a moment
C;(X^ X 0)= S(’)(cT*(X^x
a)\ O),
(cf. 3.3.2. (3)) and B: Y * Q ( XXA0)instead of B: sponding base spaces.
Y,p
(cf. (24)) for indicating the corre-
18. Theorem. Let x : X*x R +X-X R’ be a transition diffeomorphism as assumed in 3.3.1. Then there are natural pull-backs
x * : B3’;YqX-x 0)-+B”Y;Yqx-xa’), x ; : C ; ( x - x a)+ C ; ( x - x R’),
(30) (31)
Proof. So E B”;Y,@(X^x 0)is defined as 1= Op(a) + Y?,a ( y , y’, q ) being an operaX Rq) (cf. 3.3.3. Definition 3) tor-valued symbol with left upper comer in %’;”Q(RZ and the other entries in %J;Y.Q(Rz X Rq) (cf. Definition 3 and the following nota24’
366
3. Operators on Manifolds with Edges
tions), V E B3,m;y*p(X^X 0). The operators in Bl,m;y~e(X^x 0)are defined by the mapping properties, namely to induce continuous operators
8:w$&p(xAx 0)@ H:omp(fl) ww/.Bmdp(xAx 0 l o c ) f f @ H,”,,(a> v*: w&;(x^x 0)@H;omp(a) + w;; -yx-x 0,0c)8 @ H,m,,(0) 3
-+
s E R (cf. also Proposition 4 and the remarks before). It is clear that all involved
x 0) spaces are invariant under the transformations. Thus x*: B;m;yzp(X^ +BGm;YsQ(X”X 0‘). For dealing with Op(a) it suffices to use the following result. Let E, E“ be Banach spaces and y~:= ( x , xo): E X 0 +E X a’, @ = (g,xo): E x 0 + g x 0’be bundle isomorphisms, i.e. xo: +a’ a diffeomorphism, x y : Ey +E,,, 5 : + Ey, isomorphisms, of the class C” in y ; y‘ = x o ( y ) .Define y ~ * : C,”(0’, E ) + Cr(0, E ) by ( w * u ) ( y ) = ( x j ’ u ) (xa(y)), and analogously @*: C ” ( 4 g ) +C”(a’, E“). For A E L W l ; E, E ) we set A’ = @*Ap*. Then A ’ E L”(0’;E, E ) . The proof will only be sketched here. The simple (since standard) details are left to the reader as an exercise. The scheme is analogous as for 3.2.2. Theorem 16. Let us write ( A u ) ( y ) = Jei(y-z)qa(y, z;q)u(z) d z d q ,
a
u E C t ( 0 , E l . For u = v*u, y’ = x o ( y ) , z’ = xo(z) it follows
( , j j * ~ W *( y~’)) = J ei(e(y’)-P(z’))v xptu,)a(e(y’>, e(z’), v ) x &
41‘)dz dq,
where y = e ( u ‘ ):= x;‘(y’), z = e(z’). Inserting dz = Do(z’) dz’ with the corresponding determinant Do(z’)from the diffeomorphism, and ( p ( y ’ )- e(z‘))q = ( y ’ - z ’ ) ~ ‘ (close to the diagonal) such that q = @(y’, z’)q’, dq = C(y’, z’) dq’ with @ ( y ’ , z’), C ( y ’ , z’) as further standard ingredients, from the transformation of the oscillatory integral, we get with D ( y ’ , z’) = Do(z’) C ( y ’ , z’) ( ~ ’ u( )y ’ ) =
J ei(Y‘-z‘)tI2p ( y , ) a ( e ( y ’ )e(z’>, , @ ( Y ’ , z’)v’)x;&j
D ( Y ’ , z’) ~ ( z ’ dz’ ) Qv
Here D ( y ’ , y ’ ) = 1 and @ ( y ’ , y ’ ) = ‘(dxdy’)). Now we get the new amplitude function by
modulo terms of order - m , cf. 3.2.2. Theorem 13. We apply this to classical operators and the transformation of the homogeneous principal part Then
a(”,(Y,v ) := a(”)(Y,Y, v ) .
vh;’
U { ” ) ( Y ’ > 7’)= gyaa,”,(Y,
for Y’ = XO(Y) I
v’ = ‘(dxo(Y’))-’.
Let us now continue the proof by applying the abstract remarks in the present situation. We may discuss left upper corners and the remaining part with X;YvQ-valued symbols separately. The latter part, denoted by g,consists of classical
3.3. wDO-s on Manifolds with Edges
367
operators. The transformed amplitude functions follow as described. From the invariance of spaces with asymptotics and Definition 2, Propositions 6, 7, we obtain the desired invariance of the associated operators. Moreover we obtain the transformation of the leading part. In the present notation it means
y ' = x o ( y ) , q ' = ' ( d x 0 ( y ' ) ) - ' q , where x', is the pull-back of spaces over the given point y . (34) is just the pull-back formula for edge symbols. We shall see that it has the analogous form for SQ in general. a).We may apply It remains to consider the left upper comers A EU";Y.@(X^X again the above consideration. For identifying the resulting operator-valued symbol u ' ( y ' , a') in %";Y,Q(Q' X Wq) we have to apply that %2";Y>P(fp
x R 4) c C"(R2 x RP, g'i!"(X^), g - y ) ,
(cf. 2.2.4. Definition 11) and that x: e " ( X ^ ) , k*,)-' = i!"(XA), (which is easy after the above discussion), further we use 3.3.3. Propositions 9, 10. The formula cr;(X*A) ( y ' , a') = x: ai(A)( y , q ) h',)-'is rather obvious now, since in the above formula for a' we have to look at the term with cc = 0, y' = z' and to apply the construction of 3.3.2. (13). Finally (31) coincides with the corresponding statement of 3.3.2. Theorem 19. 0 There is an obvious extension of 3.2.5. Definition 6 to the case of stretched manifolds W where the bundle V is non-trivial. Our discussions have also used that a diffeomorphism x : X A X ^ induces an isomorphism x*: X & Y ( X ^ )-,X.Y(X^)
-
for every s, y E W.The transition diffeomorphisms of X^ -bundles V give rise to y dependent x with a cocycle property. Let V, be the fibre of V over y. Then we have the system of spaces X&Y(Vy)which form a X . Y ( X ^ ) bundle over Y. We want to introduce the adequate notion of global (constant) discrete asymptotic types and also of continuous asymptotic types. Fix p E @, R e p < 1/2 - y(n), k E N, and let F c C m ( X )be a finite-dimensional vector space. Then x* induces an isomorphism
x*: [r-plogkrw(r)@A:A E F} +{t-Plogktw(t)@A': A'E F'} for some other finite-dimensional vector space F' c C m ( X )which is isomorphic to F. The isomorphism F + F' is independent of p , k. Thus for every P E .Py(n)(X)there is another P ' E .P"(n)(X)such that x* induces an isomorphism x * : X s , Y ( x - ) d X$PY(X^)d,
-
A
E
?=, GL = 0, 1. For the continuous asymptotics we get x * : X i Y ( X ^ ) d + X>'$BY(X^)d,
B E B y ( " ) . 95") denotes the set of all sequences P = { ( p j ,mj, L j ) } j E for which { ( p j , m j ) } j p E . P Y ( " ) (cf. 1.2.1. Definition 7) and Lj is a finite-dimensional vector bundle over Y with fibres in C m ( X ) ,associated with V in the sense that the isomorphisms X & y ( X ^ )x (an a')- X * Y ( X A ) x (0n a')
368
3. Operators on Manifolds with Edges
transform the fibres Lj, +Ljs, according to F +F I . There is an evident extension of the definition of the shadow condition of 2.1.1. Definition 10. Let Y = IRq and (p, m, L ) E !PYV("), k 6 m - 1, Y = Wq. Then we can form potential symbols ip.kqy, 7) = X(7)t-P iogktw(t) B a ( y ) ,
where A : Rq + Cm(X)is a section of the bundle L belonging to (p, k ) . If D,*A(y) is bounded, uniformly in y E Rq, for every cc E "7, then O p ( 1 p q : HS(R'I) + w*qx*xRP) is continuous. Let P satisfy the shadow condition. We define W f Y ( X * X R9),, A = (6,0], as W % y ( X * XR9)A+ {the linear span of all for (p, m, L ) E P, 112 - y ( n ) + 6 < Rep < 112 - y ( n ) , 0 5 k 6 m - 1 , A im Op (lPskA) running over a base of sections in L with the mentioned boundedness}. Take Wfy(X*x Rq), in the natural (obvious) locally convex topology. Further regard W$By(X*xRq)Aas in the previous section, B E gy("). 3.2.5. Theorems 17, 18 have the corresponding analogues for the spaces W f y ( X * xRQ)A, W > y ( X * X R9)4of the present section. Let x be as mentioned and A = (6,0]. Then we get continuous operators x * : Wfy(x-x RqA+ WfU(X*X Rq)4 with some other P
x*:
E SF") over Rq,
further W y ( x A xR9)4 + W i Y ( X * X R9),
for every B E . B y ( " ) . If P over W9 is the local representative of a global asymptotic type then P' is just the transformed one in other local coordinates. In other words we can easily generalize 3.2.5. Definition 21 to
Wfg(W)A for stretched manifolds W and P E !Pvn). Moreover we get a definition of Wiy(W),for B E B y ( " ) . Now we are in the posiwith U";y*Q(W) as the subclasses of left upper tion to introduce the classes B":"Q(W) the space of all corners. First we denote by Bim;y*Q(W) VE
n 2 ( W S ~ ~ We)H ~ YG, - ) , w2p(w8e H
~ YG,)) ,
SER
for certain G,EVect (Y) for which
is the space of all operators So + V, V E Bi";y*g(W), where 19.Definition. Bv;y*Q(W) the left upper corner A of So belongs to L;,(int W) and the restriction of So to a neighboura), hood of some w E 3 W identified with X * x Q, Q c Y open, belongs to Bu;y,Q(X*x with respect to fixed trivializations of the bundles G* I , further
is an operator with C" kernel for arbitrary pie C,"(int W), i = 1 , 2 , (c$ also Remark 5 ) .
369
3.3. rgDO-s on Manifolds with Edges
As above it would be more rigorous to use the notation B”;y,Q(intW). We hope our shorter notation will not cause confusions. Note that the bundles Gi occurring in the definition make it necessary to extend the invariance of Theorem 18 in an obvious manner by adding cocycles for G*. Furthermore, Definition 19 has an immediate generalization to m X 1 matrices in the left upper corners (cf. Definition 3) as well as to operators acting between distributional sections of non-trivial bundles, after establishing the corresponding invariance under additional cocycles. If necessary this slight extension of Definition 19 will tacitly be used. The above local theory of operators of the B”;”,Q classes has an obvious global analogue. We want briefly to formulate the most important things. 20. Proposition. Every 99 E B V ; y , Q ( W ) induces continuous operators 99: Ws,”(W)@HS(Y, G-)+W””*Q (w) @ H“-’(Y, G+),
99 : Ws;sY(W), @ Hs( r, G-)
+ W”,-”*@( W),
@ I f s - ”( r, G+)
(35) (36)
for all s E R, G+ being the bundles over Y belonging to 99.
The relations (33) allow to introduce the symbol spaces Ci(W), B2”“v)
(37)
defined by the corresponding local behaviour over neighbourhoods X-X Q and the invariance under transition functions of the mentioned sort, and for Z;(W) in addition the structure of functions in Cm(T *(int W) \ 0) which are homogeneous of order v with respect to the canonical R+ actions in the fibres of the cotangent bundle. Remember that V is the X^-bundle which is associated with W . We then have the canonical symbol mappings a;: B”;y*Q(W) -+Z;(W),
where B’;“@(w>c Zt(w> x Bf;:v*Q(V) is defined as the space of all pairs @,, p,,) with pv = a; ( d ) ,pA = u: (99) for some 99 E B”;fi@(W). The system of mappings r,,: Z; +Z;,,, (cf. 3.3.1. ( l l ) , 3.3.2. (4)) given in local terms is invariant. Thus we have a space Z i , A ( v ) of functions which are over X^ x Q elements in ZX,, relative to X ^ X 0.Then r,, induces a global mapping rA: ZyW(W) +Z;,A(v). (39) Furthermore, the mapping 3.3.2. (6) restricted to left upper corners of matrices in B:”@ induces a global mapping a;,,,: B:Y.Q(V) _ * Z ; , A ( v )
’
Then (pv, p,,) E B’;”P(W) can be characterized by the conditions pv E Zi(w), p A € B:”.Q(v) and r A p v = a ; , A p , , . 21. Theorem. (38) is surjectiue and ker by = 8”l;”,@(w>.Further there exists a non-ca+B”;”Q(W)with u“ 0 op’ = identify on B“;”Q(W). nonical mapping op”: B“;”*Q(W)
370
3. Operators on Manifolds with Edges
22. Theorem. Let V E B”-l;ysp(W),,u = y - e. Then
V: W$Y(W)@ Hs(Y, G-)-+ WS-HQ(W) @ H”-”(Y, G+) is a compact operator for all s E R (G, being the bundles belonging to
V).
59 is defined as an operator of the form V = Vv+ VG, where VGE 2%;- ’ ; Y , Q ( W) and Vv= Op ( a ) @ 0, where a is a symbol of the form alp+ aM+ m
Pro05
as in 3.3.3. (4) with
IJ
= ,u - 1. The
operator Op ( a ) induces a continuous mapping
V v : W%Y(W) + ~ s - P + l , Q + (W), l
and Ws-@+ltQ+l(W) G WS-KQ(W) is compact as a consequence of 3.2.5. Proposition 5 . Further VG induces continuous operators
VG: wqY(W)@HS(Y, G-)~ ~ s - C + l ~ p + e ( W )G+) $HS-P+l(Y, for some E > 0. The gain of weight E in the first component follows from the asymptotics in the image under Green and potential operators, since the carriers of asymptotics have a strictly positive distance to the reference weight line. Now it suf(W) + c@ H s - p + l ( Y , G ,) fices again to apply the compactness of W s - p + l s Q 4 Ws-&Q(W)@Hs-@(Y, G+).
2 3 . Proposition. Let d j E B3’-’;”Q(W), j E IN, be an arbitrary sequence. Then there such that exists an d E IxV*EQ(W)which is unique mod!B3,”;Y~P(W) N
1-x d Q i ~ B 3 ’ - ( N + 1 ) ; n ~ ( W ) j-0
for every N
E N.
The local operations of taking formal adjoints on symbolic level lead to global ones, namely *. B K Y , P ( W > +Bv:-e2-Y(W) defined component-wise. Further we have natural compositions. 24Theorem. d EB”;Y*Q(W) implies d * EB*;-Q~-Y(W) and the global form of (26) holh. Further d E !Bv’~d~*(W) implies dd E Bv+v‘;d~Q(W) brovided the G+ bundle of d equals the G- bundle of d ) and the global forms of the symbolic rules (27), (28) hold. It is useful to have a global shift of weights both for spaces and operators. To this end we look at GP := Op (gp(t [q])) of 3.2.5. Remark 13. The operator-valued symbol gp(t [ q ] )acts as operator of multiplication by a function which equals the identity for t > const with a constant independent of y. Thus if 3 is a cut-off function with w”(t)w(t[q])= w ( t [ q ] )for all t, q (which exists) the operator GB induces isomorphisms
G!: [GIW$Y(X^XRq)+[GI W4Y+P(XAx Rq), Gf: [l - w“] Ws(XAXRq) [l- 31 Ws(X^XR‘J), --+
(40)
(41)
where Ge is simply the identity. The same is true for operators @(A) defined by gP(r [q,A]), where [q,A] has the analogous meaning as [ r ]but with the “covector” (q,A), A E R. Let us regard 3 also
371
3.3.1pDO-son Manifolds with Edges
as a function on Cm(W) with w“ = 1 close to a W, w” = 0 outside a neighbourhood of the form N (cf. the notations after the proof of 3.2.5. Proposition 5). Choose an open covering {OjIj= 1, ,,., of Y by coordinate neighbourhoods Oj and a subordinated partition of unity {pj] = ..., L . Further let y j E Ct(Oj),p j y j = pj for all j . Then we have the operators yjGP(A)pj: W&r(X^X O j )+ W&z+4(X^X Oj)
for all j and we can form L
c“fl(A) =
1 y j G P ( l ) p j : WSY(X^XY)
--*
W$Y+B(X-XY ) ,
(42)
j = 1
where corresponding push-forwards of operators to Y are tacitly involved. Now the standard technique for parameter-dependent VDO-s (cf. Section 2.3.1) shows that (42) is an isomorphism for IA 12 [lo I for some A, E R.Let us fix [ AI sufficiently large, A = A ] , and set again GP:= GS(A,). Then GO induces isomorphisms analogous to (40), (41) with Rq replaced by Y. Since [GI W S Y ( X ^ x Y) is supported by a neighbourhood N and the action of GD outside N is the identity, we can construct an operator GS: W$Y(W)+ W S Y + f l ( W )
(43)
(for simplicity again denoted by GP) by completing N to W and extending the operator by the identity outside of N. It is clear that (43) is still an isomorphism for all SE
R.
25. Theorem. Let P E Py(*)(X),B E BY@“‘ s a t k b the shadow conditions 3.2.5.(52) and (60), respectively. Then (43) induces komorphisms Ga:
W>’(w)d
+%‘“>Y+S(w)~,
GS: W>Y(W)A -+ W BS Y + ( W@ )A for all s, y E R and all A . Further
(44) (45)
Proof. (44), (45) are obvious. (46) is an immediate consequence of the calculations on compositions of operators of the B classes. 0 The material of 2.3.2., 2.3.3. suggests the question whether there is a natural class Bj’;”Q(w)* c B@;”P(W)in which the involved Mellin symbols hja(y, y ’ , z ) (cf. 3.3.3. (5)) belong to C”(02, ‘m;sm(X))*(cf. 2.3.2. (22)), the smoothing Green operators map to spaces with point-wise discrete asymptotic data, described by pointwise discrete analytic functionals (cf. 2.3.2. Definition 4) and so on. An adequate theory of such a subclass is possible, indeed. It will be published in a separate paper of the author. Here it would require further sections. For limiting the size of the exposition we drop this calculus, except for a “trivial” variant, where all the involved asymptotic data are y, y ’ independent.
372
3. Ouerators on Manifolds with Edges
26. Definition. U'*Y,P(X^X a)(*) denotes the subset of all A where C induces continuous operators
= A.
+ C E I ' ; y ~ Q ( X ^ a), X
c: w ;:mp( x^x a) w ; q x ^ x .nlOC), c*: w ~ o ~ p ( xa) ^ x+ w ; * - y x - x aloe) +
for certain P E P ( " ) ( X ) , Q E P ( - ~ ) ( " ) ( Xfurther ), A. = Op(a) with a of the form 3.3.3. (4), where hj& y', z ) E Cm(R2,%;,"(X)) with R j E 5%( X X X ) , and g ( y , Y', ?I) being characterized as element in 32yvQ(R2 X RQ)with
sy,(n2x RQ;X * Y ( X ^ ) , q ( x ^ ) B ) , g * ( y , Y', ?I) E sxa2x Rq; x $ - v ^ q) (, X - ) B ) ,
g ( y , Y', ?I)
E
PI E W'(")(X>,Q1E P( - Y) ( ") ( X)(all these asymptotic data depend on A). It is clear now that R";e(f12X Rq)(') has to be defined as the set of all b ( y , y', 1) in RQ)with
R';Q(02 x
b ( y , Y', 1)E
s$(a2 X R';
@,
yCep(X^)~)
for some P E W " ) ( X ) ,depending on b, and 3TY(02 X Rq)'') 3 b by the condition b* E RYY(f22x Rq)(')(cf. Definition 2). This leads to an obvious modification of Definition 3, in other words we get the subclass
g':Y.P(x^xa)(*) c B';Y,Q(x^xa)
(47)
of "edge boundary value problems" with constant asymptotic data. It can easily be proved that (30) induces
x * : ~ v ; ~ . e ( xa)(*) ^ x + B V Y . P ( X ^a')(*) X
(48)
which admits the dotted variant of Definition 19, namely the subclass B v ; Y , P ( ~ )B(v*; y) , e(w).
(49)
For A E B";y*Q(w)(*) then it follows the continuity A : W>Y(W),$HS(Y, C-)+ ~ ~ ~ " ~ Q ( w > ~ ~ G+) H s - " ( Y ,
(50)
for every P E Pv")with some Q E P)ev("), depending on P and A, for all s E R. 21.Remark. The %classes with (*) are closed under the operations in Theorem 24 and under the weight shift (46).
3.3.5.
Ellipticity, Fredholm Property, Asymptotics of Solutions
In this section we assume for notational convenience that the X^ bundle V associated with the stretched manifold W is trivial, i.e. V = X A X Y. The version for nontrivial V could be elaborated as well. It is left to the reader as an exercise. (For instance, in the formula (1) below X^ has to be replaced by X$Y(Vy),V, being the fibre of V over y.)
373
3.3.lyDO-s on Manifolds with Edges
1. Definition. An operator 1E % w : ” Q ( w ) , p = y - Q, is called elliptic if (i) uf:(d) (w, ,y,) 0 up to the edges, w E W, x 0 , (I$ 3.3.1. Definition l), ~ ; ( d( y), 7): X$V(X^)@ (G-)y + Xs-p.@(X^) @ (G+)y (ii) (1) is an isomorphism for all ( y , 71) E T *Y \ 0 and a given s E IR ( ( G * ) ybeing the fibre of G+ overy). We also talk about the ellipticity of the pair ( u f : ( d ) ,u$(d)) E B’;”Q.
*
*
Remember that the bijectivity of (1) for one s = so implies the same for all s E R. Clearly (i) is the same as E, of 3.3.4. Definition 1. 2. Remark. The ellipticity is preserved under the operations of 3.3.4. Theorem 24, i.e.. adjoints and compositions of elliptic operators are elliptic. Further the weight shifr conjugation 3.3.4. (46) preserves the ellipticity.
3. Definition. An operator d E % - p . @ . Y ( w)is called a parametrix of d E Bfi~”Q(w) if dgj - 1 E 813,“:”‘ (w) > (2)
m - 1 E s,m-”v(w). (3) (Remember that a weight interval 9 = (- k , 01 is involved in the definition of Green operators as well as in the whole operator class). 4. Theorem. Let d E B””*Q(w), Q = y - p, be elliptic. Then d induces Fredholm operators 1:w s . y ( w ) @ H S ( Y , G-)+w”-lr,@(w)@Hs-qr, C,) (4) for all s E R. There exists a parametrix d E B - f l ; @ , Y ( w).Further
d u = f E w::(w), @H‘(r, G,) , u E w-mxv( w)@ H-”( r, G-) ,
r E W,implies u E W::wy(w), @ H‘+@(Y,C-)
(5) (6) (7)
5. Remark. The elliptici@ of 1 is also necessary for the Fredholm propem of (4). The proof will be dropped here. 6. Remark. The concept of the class W;.Q(w)admits a natural extension to operators
1:W & Y ( W , E ) @ H s (Gx _ ) ~ W S - L . P ( W , F ) @ H S - r (G,) Y, (8) with vector bundles E, F over W. The notion of ellipticity can easily be generalized to this situation and we then have corresponding analogues of Theorem 4 and Remark 5 .
The proof of Theorem 4 will be given in several steps. First we have to look at the inverses of the symbol components of d.It is obvious that the non-vanishing of at (d)E Zf: up to the edges implies the same of uf:( & ) - I E Z.; In 3.3.4. we have introduced the edge symbol classes B ; y - pconsisting of 2 x2-matrices of operator families parametrized by T*Y \O with Z;ysp in the left upper corner. Let us define aXn: B2y*e+ZyW,*, uh: Bf;:Y,P+Z>Y by applying 3.3.2. ( 6 ) , 3.3.3. (22) to left upper comers.
374
3. Ouerators on Manifolds with Edges
7 . Proposition. Assume that a ( y , r ] ) E B2;Y,einduces isomorphisms a ( y , r ] ) : X k Y ( X ” ) @ ( G - ) y + X s - p x Q ( X A ) @ (G+),,
for all ( y , r ] ) E T * Y \O.
(9)
Then a-’(y, r ] ) E BAp;PsY, and
ProoJ
We first construct a-’(y, q ) locally in a neighbourhood of any T * Y \ 0. The resulting operator functions for different neighbourhoods coincide over intersections. Hence we get the inverse globally over T*Y \ O . We shall see then that there is no problem to identify the operator family as an element E B , p ; e , ythen the composition rule for edge in our classes. If we expect that symbols in 3.3.4. Theorem 16 and 3.3.2.(20) show that the relation (10) is neceswith sary. So we start the construction of a-’(y, r ] ) with a left upper comer in S;p:Q.Y (10) as the a,;-symbol. Let us write ( y o , q0)E
where in particular a1,(y, r ] ) :
X $ Y ( X ^ ) --+x”-p.Q(x^).
(1 3)
Consider the operators for fixed ( y , r ] ) , Ir] I = 1 (and a fixed Riemannian metric on Y). The elements aI2, aZ1and azz are finite-dimensional. Since (12) is an isomorphism, (13) is necessarily a Fredholm operator. From 3.3.4. (2) it follows that
a$(a)(y, z ) : H y X ) + H ” - r ( X ) is an isomorphism for all z E r,,,- y ( n ) , s E R. Then we get (11) as a necessary conse, cf. 2.2.4. Dequence, cf. 3.3.3. (30). By construction (13) belongs to g Q f ! ” ( X ^ )g-Y finition l l , gd being in C m ( X ^ ) ,gd > 0, gd = td close to t = 0, gd = 1 for t 2 const. Remember that the definition of P ( X ^ ) , is based on 2.2.4. Definition 10 with a special choice of local charts. Here we can modify equivalently the description by considering expressions like wAowl (1 - o ) A l ( l - w z ) + G with cut-off functions w, wi satisfying 3.3.1. (32). It is obvious that the material of 2.2.4. from Definition 12 to Remark 17 has a straightforward generalization to the weighted classes gQf!fl(X^),g - y , cf. also the considerations at the end of Section 1.3.3. For instance, the ellipticity of an operator A E g Q 2 ” ( X ^ ) 8 g - vis by definition the ellipticity of g-QAgr in the sense of 2.2.4. Definition 12. Observe that the notion of o;-symbols of 2.2.4. refers to “polar coordinates” (t, x) close to t = 0. Here we have a more special situation because of the nature of S;.It admits to employ the Euclidean coordinates which are involved in the description of G ; , , ( U ~ J E S;,,in the present context. The invertibility of $,(all) here is compatible with the condition (i) of 2.2.4. Definition 12 after removing the factor t-” close to t = 0. It follows that all E g Q f ! ” ( X ^ )g-Y , is elliptic. In fact, (i) was just discussed. The Mellin ellipticity (iii) follows from the Fredholm property of (13) which is known from the cone theory. Further a:(all) is mrtainly non-zero since this coincides in the coordinates 2,y, E r ] with cr;,,(all) ( y , [, q ) for r ] 9 0, cf. 3.3.1. (21).
375
3.3. wDO-s on Manifolds with Edges
The next step of constructing the inverse of (12) is to choose a parametrix d~ g - y 2 - p ( X - ) 8 g Qof all which is associated with a;,A(ull)-l as a,-symbol in the mentioned Euclidean coordinates. d follows from 2.2.4. and the underlying cone theory. From 2.3.2. (modified for the 2 - N ( X A ) , class) we know that d ( y , q ) can be chosen as a C" function of y, q, 1711 = 1, in an open neighbourhood of the given point ( y , 7). The formal consideration after 2.2.5. Remark 8 together with the concrete constructions for 3.3.4. (10) yields an isomorphism e ( y , 7): x ~ - K Q ( x CB"(G+), ) + X S Y ( X ^ )CB (G"_), with vector spaces (Gd, that can be interpreted as the fibres of bundles G+, G-EVect (Y) with indsuall = [G+, G-1, cf. 3.3.4. (S), Q = S*Y being the cosphere bundle induced by T*Y. The arguments about the dimensions M,N in 2.2.5.(6) apply, again, such that we may assume (Gt),= (G*),.Using e ( y , q), which plays the role of 2.2.5.(6) for inverting the given operator function, we can carry out the construction after 2.2.5. Remark 8 for obtaining a - ' ( y , 7). This is a procedure in terms of algebraic operations which preserve C" in the parameters and also the nature of elements in B ~ y ~ Y I S0* c R ,Y being the given neighbourhood of y (cf. 3.3.4. Remark 17). As usual (cf. 3.3.4. (11)) we can extend a-'(y, q ) by homogeneity - p to all q + 0. Then 0 we obtain a - ' ( y , q ) E Bip;QsY.
Proof of Theorem 4. The ellipticity of d implies a; (d)-' EZ;' and a; (So)-' E B;P;Q,v,cf. Proposition 7. From (10) it follows (0;( d ) - l ,a; ( d ) - l ) E B-';Q9V. Applying 3.3.4. Theorem 21 we find an operator BoE B--";Q*Y with a; (a)-'= a,p (So), a; (&)-I = (8,). 3.3.4. Theorem 24 implies that 908 E Bo;Q,Q and u\ (d8,- 1) = 0, at (&do- 1) = 0. Thus 3.3.4. Theorem 21 yields that V0 -- d8,- 1 E B-1;Q.Q. From 3.3.4. Theorem 24 we obtain ( - ~ ~ V ~ E B for - ~ every ~ Q ~j EQN . Applying
3.3.4. Proposition 13 we get a V1 E B3-';Q>Q such that Vl
-
"
(-l)jVd, in other j=l
words (1 + Vo)(1 + VJ = 1 + V with V E B-";Q*Q. This yields for d = O0(l + Vl) % = dQd- 1 E 8 -";Q,Q = B-"; F,Q (14) G , (cf. 3.3.4. (20)). Thus SB is a right parametrix. In an analogous manner we can construct a left parametrix. It coincides with 8 modulo B~";Qvy. Thus we have proved that the ellipticity implies the existence of a parametrix. The Fredholm property now follows in a standard way. The parametrix 8 defines a continuous operator 8:W s - k Q ( W ) C B H s - p ( YG,,) + W $ Y ( W ) @ H S ( Y , G-) for every s E R,and (14) induces compact operators
V: W S - b Q ( W ) @ H S - p ( YG+) , + W s - p * Q ( W ) @ H S - " ( Y G+), , (cf. 3.3.4. Theorem 22). Thus (4) is a Fredholrn operator. We have &$ - 1 E Bim;RYand
e:=
e: W s Y ( W )CB H s ( G - )
+
WZy(W)@ @ H"(G-)
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3. Operators on Manifolds with Edges
for ?very S E R . Multiplying (5) from the left by B we obtain (1 + V)u = B f E W:,’”Q(W)e @ H r + p ( Y , G - ) and hence (7). 0 Now we can draw analogous conclusions as in the standard elliptic theory. 8. Corollary. Let d
E B@;ysQ(W),
p = y - e, be elliptic. Then
k e r d c Wz”(W)e@ H”(Y, G - )
(15)
and the index of (4) is independent of s.
In fact (15) follows from (5), ( 6 ) * (7). The index of d coincides with dimkers? - d i m k e r d * (cf. 3.3.4. Theorem 24). Since d * E B”;-Q,-“( W) is also elliptic, we know again that k e r d * is independent of S.
9. Remark. Let d,99’ E W;Y.Q(W) be elliptic and
ind d
O$ (d) = O$
(d’), 05 (d)= dj (90’). Then
= ind d’.
This is an immediate consequence of 3.3.4. Theorem 21, 22. Let us now come to further considerations on the role of weights and the choice of edge trace and potential operators. First it is clear that u: ( d )E Z; is completely independent of any weight y. Moreover the results of Section 3.3.3. show that for every p E Z: and every y E R there exists an A E Ui’;”Q(W),e = y - p, with uf ( A ) = p v . Let 3.3.4. (1) be satisfied and consider indFyg$(A)E K ( S * Y ) . Let p : S*Y + Y be the canonical projection and p * : K ( Y ) + K ( S * Y ) the pull-back of K groups. Let A E U”’”Q(W) and assume that 3.3.4. (1) is Fredholm for all ( Y , s) E S*y.
10.Proposition. A necessary and sufficient condition for the existence of an elliptic 1E B!’;y,Q( W) with A as the left upper corner is indFrb$(A) € p * K ( Y ) .
(16)
Proof. The existence of an elliptic s? E B”;”Q(W) ensures by definition the bundles G* EVect (Y) such that (1) is bijective. The constructions in the beginning of the preceding section then show that indsr c$(So) = [P*G+,p*G-] E p * K ( Y ) . On the other hand the arguments yield at the same time that (16) is also sufficient. 0
The Fredholm property of 3.3.4.(1) depends on p(,,):= r A $ ( A ) as well as on the leading Mellin symbol aU,(A), cf. 3.3.4.(2). Remember once again that for t?=(-k,O] U $ ( A ) ( Y , s ) = a ( y , s) = w(tlsl)t-!’oPP(f)(Y, s)w1(tIsI) + (1 - w(tlsl))oP,(P(,))(Y, s)(l- wdtlrll)) t - 1
377
3.3. ylDO-s on Manifolds with Edges
G ( a ) ( y ,z ) = f ( y , z, 0) + ho(Y, z ) : H“X) +H”-r(X) (18) is independent on q. The Fredholm property of 3.3.4. (1) and the global index element indsr cr$(A)are independent of the h, for j 2 1, g ( y , q), and of the concrete q dependence of J since changes of these data only cause compact operators. In particular (16) is a property of r,, a t ( A ) alone, provided a ( ~q ), is Fredholm at all. This proves the following
11.Proposition. Let A E W ” ” Q ( W ) ,A ’ E W ~ . Q ’ ( Wand ) r,,a;(A) = r,,u’,(A’). I f both for A and A’ the operator families 3.3.4. (1) with respect to the corresponding weights are Fredholm for all ( y , q ) E S * Y then (16) for A is equivalent to the analogous property for A’. It may happen that cr$ ( A )( y , q ) is independent of y E Y (for example if Y = S1 and the symbols are rotation symmetric). Then a $ ( A ) = @,(a) is independent of y. Thus (18) is bijective at a point z E C independently of y. Let us now assume that ho E aR:sy(X)’. Then, if 3.3.4. (1) is a Fredholm family parametrized by S*K (18) is also a Fredholm family parametrized by C which is bijective except of a pattern of points bj}jaz, with p j d rIl2-,,(,,) for all j , and I p i } j n {cl < Re z < c2) finite for all c l , c2 E IR.Further cr$(a)-l extends to a meromorphic Fredholm family in aRi[(X)*. This is exactly the picture of the cone theory under the assumption of the ellipticity (cf. 2.2.1.(20) and the proof of 2.2.1. Theorem 14).
12. Proposition. Let pwE C’, satisfy (i) of Definition 1 and let r, pv be independent of y E Y (cJ: 3.3.4. (39)). Then for every y E R there exists an A E % p ; ” Q ( W ) , e = y - p, with pv = o’,(A), such that u $ ( A )( y , q ) defines a Fredholm family 3.3.4. (1) for all ( y , q ) E T*Y \O. I f w e choose d $ ( A ) in such a way that all hi in the representation (17) vanish, then it induces Fredholm families
a;(A)(y,q): X q X - ) forall(y,q)ET*Y\O, mark 12.
(x-1,
(19) a n d a l 1 6 E R \ { e j } j E Z with certain realse,asin2.2.1. Re+ X S - K d - p
ProoJ: Applying 3.3.1. Corollary 8 we can always form an element a l ( y , q ) E % p ; ; ” P ( R X Rq) with $(a,) = p v , where the associated Mellin symbol is holomorphic in the sense 3.3.1. (62). Then a ( y , r ] ) := a $ ( a l )( y , q ) is of the form (17) where all h, vanish. We may also set g = 0. From 2.2.1. Remark 12 it follows that (18) is bijective on Re z = 112 - 6 ( n ) for all 6 E IR \ {e,}, z for certain reals e,. Thus a ( y , q) is elliptic in the sense of the class ga-pf!”(XA)8g-a for all these 6 and hence (19) is Fredholm. Now if we prescribe an arbitrary y = 6 it may happen that the bijectivity of (18) is violated on finitely many points p o , . . ., pm E f,,, - ?(,,). Let us choose an fo E aR;sm(X)*such that f”:=f ( 1 + f o ) is holomorphic and satisfies the bijectivity (18) on f1,2-y(n).The construction of f o is quite elementary. The singularities at po, ... ,pm correspond to the poles of the meromorphic operator family f - ’ ( z ) at thzse points. Since f is associated with pv in an analogous way as f ( f - f E %i8m(X)*)we can form (17) with f instead of f and obtain the desired edge symbol for the weight y.
378
3. Operators on Manifolds with Edges
If the Fredholm family (19) satisfies (16) for fixed 6 = a0, then also for all 6 E R \ { e j } j Ez, cf. Proposition 11. The index may change if we pass from do to a1 (say 61 < 6), when 112 - aO(n)< Re z < 112 - 6 , ( n ) contains some ej (cf. also 2.2.3. Theorem 14). According to Proposition 10 we can form elliptic operators daE 23#;a-d-#(W)by constructing extra trace and potential conditions along the edge by the methods of Section 3.3.4. The number of conditions (more precisely the fibre dimensions of G- , G+) depend on the dimensions of cokernel and kernel of (19), i.e. on 6. Let us consider an example. Denote by Diff#(W) the subspace of all P E Diff# (int W) which are close to a W in the coordinates 12,y ) E U X R restrictions of differential operators of order p in IRn++'X R to UoX R for every open U,,Goc U (cf. also the notations in the beginning of 3.3.1.). Any P E Diff") can cf. 3.3.1. (15), where all pi also be described by? global complete symbol p = bj}, are polynomials in (6,r ] ) of order p. Differential operators, in local coordinates being of the form 3.3.1.(1), can be written in the (t, x ) coordinates in the form 3.3.1. (2). Thus we get canonical local representatives of the Mellin symbols h for which 3.3.1. (27) holds, in this case with equality. In view of the holomorphy, the weight y is arbitrary. Nevertheless, p determines a system of exceptional weights ej where the Fredholm property of (19) breaks down. Now let us remind of the sub(W)(*)c 93M;y.Q (W). The continuity 3.3.4. (50) suggests the elliptic reguclass B#;"Q larity in spaces with discrete asymptotics. 13.Theorem. Let 1E 23#;">Q(W)(*)be elliptic and u $ ( A ) - I ( y , z ) E Cm(K" t i # ( X ) )for some R E 5'i (XX X) independent of y (with A as the left upper corner of 1). Then there exists a parametrix d E 23-f';Q~"(W)(*)of 1,and
W y (W), @ H'(K
(20)
W ; ; t P , Y ( W ) 8 @ H r + P ( GY -, )
(2 1)
G+) for some Q E W n ) ( X ) ,r E IR, and ( 6 ) implies SQU = f E
U E
for a P E Py(n)(X)depending on 1and Q. Proof. The steps in the proof of Theorem 4 show that all properties which characterize the subclasses with (*) are preserved, cf. 3.3.4. Remark 27. The proof of the dotted analogue of Proposition 7 employs in particular the condition that the asymptotic information of d $ ( A ) - l is independent of y . Another new element is to carry out the asymptotic sum Vl C (-1)jV;. We have then to use the corresponding version of 3.3.4. Proposition 23 for the ( 0 ) classes. It can be formulated under the condition that the d j have the same asymptotic data for all j . This is satisfied here (for appropriate "maximal asymptotic data"). 0
-
Note that the traditional edge theory in the literature deals with differential operators in the context of discrete asymptotics of Theorem 13 (X is usually assumed to have a non-empty C" boundary). This is, of course, a subject of independent interest with many concrete (often rather difficult) problems. For instance, the appropriate choice of y or the concrete form of P have to be studied separately for any concrete operator (this requires the calculation of the asymptotic type of b L ( A ) - l (y, 2)). Also the nature of the coefficients in fhe asymptotics is worth to be analyzed in more detail. Let us make some general remarks to this point.
379
3.3.1pDO-son Manifolds with Edges
As we know from Section 3.2.5. the discrete asymptotics for u E WQpy(W), can locally be written in the form ~ ( tX,, Y )
F-'
1;
Ajm(X) cjm(tt)
[vI'"
}
"'Yt [qI)-'j log"' ( t [?I> w ( t [ttD ,
(22)
F-1 = F-1 ,,--.y, vjmE Hs(R4), Ajm E Cm(X), and {(Pimj)}j, = ncXNP, where the sum is taken over all j , k with 112 - y ( n ) - k < Repj < 112 - y ( n ) , where 8 = (- k, 01, and 0 5 m s mj. Here means equality modulo remainders in WJoy(X"xRg),. The form (22) for the asymptotics is not quite canonical. For instance i6 we change [ q ] then we get an error in W ; s y ( X A XRq),. Therefore, it is more natural to write the
-
asymptotic expansions in the form
where gjm(t,x, q ) are the items under the sum on the right of (22), and /IjrnE Cm(X),
fimE H"(R9) (cf. 3.2.5. Remark 9).
It is convenient to introduce a notation for the splitting (23) of asymptotic expansions. Let us call the first sum the singular, the second one the regular part of the asymptotics. The definition is correct in the sense that the singular part is unique modulo some regular part. The singular part is invariant under coordinate changes modulo regular terms. Now consider once again the elliptic regularity with discrete asymptotics which says that (21) holds under the mentioned conditions. Set s = r + p. For simplicity It is a let us disregard the component in Hs((Y, G-)and talk about u E WJpy(W),. reasonable question to what extent the coefficients of the asymptotic5 of u in a neighbourhood U of a point yo E Y are determined by the coefficients of the asymptotics off in U and of the local behaviour of So there. The proof of Theorem 4 gives immediately the answer, namely 14. Proposition. The singular part of the asymptotics of (21) in a neighbourhood U 3 yo of the edge is uniquely determined (modulo a regular remainder) by the singularpart of the asymptoticr o f f in U and of the restriction of So to a neighbourhood N of the form [0, E ) X X X U in local coordinates, E > 0 . The regular part of the asymptotics of u depencis on the global behaviour o f f and So. 15. Remark. The notion singular and regular parts of asymptotic expansions can be &fined also for the continuous asymptotics. Then we get a corresponding analogue of Proposition 14.
The elliptic theory of this section has a number of straightforward consequences. First it is easy to generalize the whole calculus to systems in the sense of DouglisNirenberg ellipticiv So =
j - I.
....N
vectors of reals (sl,...,sN),( t l , ..., tN), (yl, ... , yN), (el,... , eN)with si- tj = yi - ej for all i, j . We shall not carry out in detail anything
with
d i j E B J i - ' J ; y i - Q j ( W ) and
of this material here. Let us only mention that the ellipticity conditions are formulated for the matrices of leading orders si- t j , i.e. the Douglis-Nirenberg ellipticity of (u2- '](Soij)) and the bijectivity of 25
Scbulze, Operaton en&
380
3. Operators on Manifolds with Edges N ( ~ j - ' j( 1 i j ) )
( y , 7): @
i=l
N
{X"*"(X^) @ G - , i }+ @ {X'J*Qj(XA) @G+,j] 1-1
on T*Y \O. Then we get the Fredholm property of N
N
1:@ ( W " " ( W ) @ H s ' ( Y ,G-,i)} + @ i-1
{ W ' j * Q j ( W ) @ H ' j ( YG ) +,j)},
J=1
the parametrix within the class and asymptotics of solutions. The ellipticity conditions can also be relaxed by demanding either surjectivity or iqjectivity. Then we get by definition the case of overdetermined and underdetermined systems, respectively, with existence of right (left) parametrices within the classes. For solving the problems with "Douglis-Nirenberg orders" si - t j it is useful (but not necessary) to know order reducing operators in the classes. An A E U";y*Q(W), e = y - p, is called order reducing if it induces isomorphisms A : W"Y(W)9 Ws-"@(W) (24) for all s E R.The author is convinced that for every p, y E R such an A does exist. But it is not yet constructed, except for certain special cases. Let, for instance, G be an open set in R",C compact, Y = aG of class C". Then G can be interpreted as a manifold with edge Y and the "model cone" R+. In this case coincides with the stretched manifold W. We have over G Boutet de Monvel's algebra of wDO boundary value problems for wDO-s with the transmksion properly with respect to the boundary, cf. [B9], [R8].This class contains order reducing operators for every integer p. The translation into Russian of [R8]contains a proof, based on a parameter-dependent variant of Boutet de Monvel's class. The technique is similar as for the proof of 2.3.1. Theorem 16. The construction of (24) also should follow from a parameter-dependent variant of the present edge theory. The special case of boundary value problems suggests many further constructions for general manifolds with edges. This opens a large program of useful considerations which are by no means all straightforward, for instance, the index theory and an analogue of the Atjah-Singer index theorem. Below we shall briefly sketch some problems in this direction. Observe that in the class W""*Q(W)there exists the principle of reducing a problem to the edge Y, analogously to the classical reduction to the boundary in the theory of boundary value problems. Consider, for instance, two elliptic operators of the form
c
in W;"Q(W).They contain different edge trace conditions for a given A E Y P ; K Q ( W ) . From Theorem 4 we get a parametrix goof d oin ' B 3 - p ; b Y ( W ) . It has the form of a row matrix 9 0
KO)*
= (Po,
Then APo- 1, AKo-O, ToPo-O, ToKO- 1 where modB;";ypY(W).It follows
-
means equivalence
3.3.lyDO-s on Manifolds with Edges
381
and V is also elliptic. In particular R = TIKois a classical elliptic lyDO along Y. It is the reduction of d1to the edge (by means of do).Note that ind V = indd, - inddo = ind R. It is easy to express a parametrix dl of d 1in terms of B0and of a parametrix R(-')of R, namely
(Po- KoR'-"T1Po, KoR(-')). If we have instead of (25) general matrices 9 1=
(26)
we can perform by simple algebraic manipulations an analogous rein B";Y-Q(W), duction of d 1to Y by means of do(cf. [R8], Section 3.2.1.3.) and express B1in terms of b0and of the reduced object on Y. Let us now make some remarks on the choice of spaces WSY for the Fredholm property of edge operators. First let W = and consider the local situation in R: = {x : x,, > 0}, x = ( X I , ... , x,,).It is then custom to realize the operators in the form A : Hs(R:) +H""(R:) (27) when A is a lyDO with the transmission property, p = ord A, A = r + Op (a) e + , where e + extends distributions from R: to R"by zero, r + restricts to R: . In order to avoid comp, loc spaces it was assumed that a E S$(R: X Rq) is independent of x for large 1x1. Op(a) is used here in the meaning F;AX a(x, OF,,,. Remember that Boutet de Monvel's calculus also contains Green operators; they are dropped for the moment. It is assumed in (27) that s > 1/2. This formalism has some features that may be critized. The operator convention a +r + Op(a)e+ looks a bit artificial. It seems to be justified only under the transmission property. Further the restriction on s is unpleasant, since the operators should shift all real Sobolev space orders as it is the the case in the interior. Alternative operator conventions are contained in the edge theory by establishing a relation S;,(Rn x Rn) +%t';Y.Q(Rn-1x Rn-1), e = y - p, with operator-valued symbols, acting along the cone R+3xn, where afterwards the action along y = (xl, ..., x,,- 1) E R"- is carried out. At the same time the corresponding spaces are given, namely W ~ Y ( R + X R "= - ~WS(Rfl-l, ) !XSy(R+)). Remember that XqS(R+)= Hi(R+) for s > -1/2, cf. 1.2.2. Theorem 6. Thus WSS(R+XRfl-')= WS(Rn-l,Hi @+>I = Hi (R:)
(28)
for s > -112, H ; ( R : ) being identified with the closure of C,"(R:) in Hs(Rn) (cf. also the discussion after 3.1.2. Proposition 5). This shows that the WSYspaces point out a twofold role of s in H;(R:) close to x,, = 0, namely as smoothness and as weight.. The calculus in the W4Y description treats the interior smoothness separately and shifts it for all real s, whereas s > -1/2 turns out to be a weight condition. 25'
382
3. Operators on Manifolds with Edges
On the other hand we have, of course, HS(R:) 9 H;(]R:) for s > 112, such that we do not recover at once an action like (27) with the new operator convention. It is therefore justified to look for an extension of the edge theory which produces in the case of classical boundary value problems actions between the classical Sobolev spaces Ha(R:). Let us choose a projection ps: Hs(R:)
-
+
Hi(R:)
which is induced by a fixed decomposition 3.1.2. (15) and the associated splitting of 3.1.2. Lemma 7. Then (27) for A with the transmission property can be splitted into A = A o + G , Ao:=ps-,ApS, and G = A - A. is an operator of Green type in Boutet de Monvel's class. Remember that cpGy has a kernel in C"(R: XR:) for every cp, ry E C;(R:). Under this aspect the difference between A and A. is not too suspicious. G only governs the mapping between the terms with (Taylor-) asymptotics associated with poles on the right of the weight lines corresponding to s and s - p, respectively. To be more precise we have the decompositions Hs(R:) = Hi(%) a3 Ys((Rn-l, Es), being given by 3.1.2.(16), YS(Rn-l, $) by 3.1.2.(19), and the same for s - p (which is non-trivial only for s - p > 1/2). Then A can be identified with a matrix
with KO= p s - , A ( l - ps), To = (1 - ps-,)Aps, Ro = (1 - ps-,)A(l - ps). The weight in the form (28) is r,,,- s line which is responsible for the-description of whereas the poles of Mu for v E E, are just the real integers in 1/2 - s < Re z < 1/2, i.e. on the right of f ' 1 1 2 - s (cf. also 1.1.3. Proposition 1). Let us now switch again to the notations of the edge theory, n = dimX, q = dim Y and so on. Our discussion has shown that it is also reasonable to organize an edge lyDO calculus in spaces of the form
H;(E)
W"Y(X^X RQ)a3 YSpy(XAxRq),
(29) where P = {(pj, mj, L j ) } j= 1, .._, is a finite discrete asymptotic type with Repj > 1/2 - y(n), Lj c Cm(X)finite-dimensional, j = 1, ...,N,and YSpY(X^xRq) spanned by the functions
F i L y{ A j m ( x > f i j m ( v ) IttI'"' ')/'(t[vI)-'J logrn(1ItlD w ( t [vD) for all Ajm E Lj, vjmE Hs(RQ),0 5 m 5 mi,j = 1, ... , N. Note that there is a natural identification between (29) and W$"(X^X Rq)A,where 112 - v ( n ) > Repj for all j and A = (S,O], 6 = y - v. Under the latter point of view it seems to be no essential difference between the versions of edge theories in spaces 7y4" or 1v47 a3 YSpY But the new aspect here is that the asymptotic types P,Q (in domain and range, respectively) are kept fixed and independent of further individual information on the poles of Mellin symbols. We will not go into further details here. The calculus with
3.3. qDO-s on Manifolds with Edges
383
extra singular terms on the right of the reference weight lines has been elaborated in [Rll] for spaces with discrete asymptotics. DE MONVEL’S The shape of the index theory of boundary value problems in BOUTET paper [B9] (as an extension of the concept of ATIYAHISINGER [A7]) gives the scheme for analogous considerations for edge problems of the B’;V,Qclasses. We have seen that [W]. they generalize Boutet de Monvel’s class as well as that of REMPEL/SCHULZE In B’;V,Q(in the version of Remark 6) we can introduce a notion of stable homotopy equivalence (cf. also [R8], Section 3.2.1.1.). For compositions it is more convenient to pass to B’:= (G-Q$1)B’;Y*Q(GY$1)
with the weight shift isomorphisms of 3.3.4. Theorem 25. Ellipticity in B”is defined as ellipticity of the corresponding operator in B’;”Q.The notion of principal symbols (o:, o;) can be introduced also for the images under the weight shifts. Denote by Ell’ (W) the set of stable equivalence classes of elliptic operators (or likewise of elliptic pairs of principal symbols). Then the index of the associated Fredholm operator induces a mapping ind: Ell’ (W) +Z (cf. Remark 9). It can be proved that there is a natural bijection
EllP(W) y+E1l0(W)=:E11(W) -
(for Ello(W) the weight shifts disappear anyway). Now it is a natural problem to look for another (topological) description of the group Ell (W) and of the index homomorphism ind: Ell(W) + Z . This is still unsolved in general. Remember that the analogue of Ell for elliptic yDO-s on closed compact C” manifolds X is just K(T*X\O), the K group over the cotangent bundle \{zero section}. If X is C”, compact, with boundary, then we have to take K(T*(intX) \O), cf. [B9]. The same is true of the algebra of [W]. A description of stable homotopy classes for manifolds with conical singularities may be found in [S9]. An adequate solution for manifolds with edges seems to require a deep knowledge both of the analytical content and of the algebraic topology. Let us finally consider an example. Let g be a Riemannian metric on X and B ( g ) be the operator 2.3.3. (14) on XA.The Laplace-Beltrami operator on XAx B for the metric dt2 + t2g + dy2 (dy2 being the Euclidean metric, induced by R4)equals
More generally in (30) we shall allow that g smoothly depends on t, y (C”up to t = 0). The operator (30) certainly satisfies the ellipticity condition (i) of Definition 1. For (ii) it is necessary that 4(A)(Y, 1)= t-2{B(go)- t2lr1I2}, g o = g l r = o ,
384
3. Operators on Manifolds with Edges
defines a Fredholm operator for every y and q
* 0. The operator (31) is Fredholm iff
( y , z) = zz - ( n - l ) z + A(go): H s ( X ) + H s - 2 ( X )
(cf. 2.2.3. (30)) has no point of non-bijectivity on the weight line f,,+ - (cf. also 3.3.4. (2); is satisfied here). The points of non-bijectivity follow again by 2.2.3.(31), where A ( y ) is an eigenvalue of A(go). This yields a system of (pdependent) exceptional weights where (31) is not Fredholm. Explicit examples fob lows by 2.3.3. (17), e.g. for X = Sn and the induced metric from R"+ For the admissible weights the criterion of Proposition 10 applies (in its local form), i.e., for every yo and an admissible weight y there is an open neighbourhood 4 3 yo and an 4)with A in the left upper comer. The number elliptic operator SQ E !B2;",Y-2(X^x of additional trace and potential conditions depends on the dimensions of kernel and cokemel of (31). These dimensions also depend on y. The behaviour is analogous to that in 2.2.3. Theorem 14. 3.3.6.
Examples and Remarks (WDO-swithout the Transmission Property)
We have repeatedly mentioned that the theory of operators on manifolds with edges contains the special case of boundary value problems. The edge is then the boundary and the cone degenerates to R+, interpreted as the inner normal. This section gives supplementary remarks and examples to this case. The proofs only require simple modifications of the general technique and are lea to the reader as exercises. The edge calculus for boundary value problems does not rely on the transmission property. wDO-s with violated transmission property appear in many concrete applications such as mixed and crack problems. On the other hand the boundary value problems may play the role of an operator-valued symbolic structure for a cone theory with a cone base X and ax 0. This can be again the starting point for a wedge theory with a further edge Y. The problems (i)z, (i& of the Introduction are of this kind, cf.Fig.4, 5 . The cone in both cases consists of {(t,x ) : t E R,, 0 5 x = a ) with 01 = T[ for (i)2, a = 27r for (ii)z. Locally the boundary value problems are considered in R: = {(y, t ) E R": y E R"- l , t > 0 ) . In this section we assume n > 1. For analogous reasons as in Section 1.3.4. it is instructive fist to look at zero order symbols a ( x , 5) E S:,(Rn x R")), x = ( y , t ) , l = (q, z). For convenience we assume that a(x, 6 is independent of x for large ( x1. As usual we denote by
*
e+: L2(R:)+L2(Rn)
the operator of extension by zero to t 5 0 and by r+: LZ(Rn)+L*(R:) the restriction to r > 0 (for simplicity we do not indicate the dimension n explicitly and use the notations at the same time for n = 1).
385
3.3. pDO-s on Manifolds with Edges
Similarly to 1.3.4. (1) we can form Op,(a) := r+ Op(a) e+: L2(R:)+L2(R:) (1) with Op(a) = F - ' a ( x , 6 F, F being the Fourier transform in R". A question is now again what can we say about the structure of operators in the algebra generated by (1) when a runs over the mentioned space of symbols. Our answer contains, of course, that Op,(a) E Uo;o~o(R+xR"-') (in the notations of 3.3.3. Definition 5 with X " = R +, R = R"-l, 9 = (- m,O]). But here we have more concrete information. Let OP&) (Y, v ) = r + F ; f , d y , t, I, z) F,+re+. Then op,(a) ( y , q ) E So(R"-'X R"-'; L2(R+),L2(R+)) with respect to x,: u ( t ) +A1'2u(At) (q is unitary in L2(R+)),and OP&) = F;Ay op,(a) 0, ?I) Fy,-.q. Assume for a moment that a is independent of t . Then op,(a) ( y , q ) E S;,(R"--' x R"-'; L2(R+),L2(R+)).As explained in 1.3.4. the (Y, q)-dependent Mellin symbol aJ(a) ( y , q, z ) of a ( y , q, z) is a homogeneous polynomial in q of order j and a @" function of y. If a also depends on t we get polynomials in q of order j , not necessarily homogeneous. In particular ak(a>(x z ) = a i ( y ) g + ( z )+ a i ( Y ) g ( z ) , (2) where a:(Y)=a(o)(Y, k11, ~ ) I I = o , ~ = o , ~ = ~ I , (3) qo)(x, 0 being the homogeneous principal part of a ( x , 6 of order zero. a(,,(x, 8 is uniquely determined by qo,(x, 6 I = i.e. by the values on the cosphere bundle. a ; ( y ) are just the values on the "north and south pole", respectively, over the boundary t = 0. Let us fix a diffeomorphism (- 1,1) r1,2, c(z) = z with Im ((z) + k 00 as z+ T 1 and regard [- 1,1] 3 z as the corresponding piece of the conormal bundle of the boundary. Then #,(a) (y, ((z)) is a smooth connection of a i ( y ) parametrized by this conormal interval, cf. Fig. 12. For this reason in [R2] the leading Mellin symbol was called the conormal symbol.
c:
+
conormal i n t e r v a l
cosphere of t h e domain over the boundary I j l = 1
cosphere of t h e boundary 171= 1
Fig. 12
386
3. Operators on Manifolds with Edges
1.Definition. l?i(R:)(*) denotes the space of all operators A
= Op,(a)
+ M + G,
where a ( x , 0 E SO,,(Rnx R") is x independent for large 1x1, and M + G E U$$:(R+ xR"-')(')(cf. 3.3.3. Definition 3, where M denotes the Mellin, G the Green part (cf. 3.3.4. Definition 26).
Throughout this section we talk about the infinite weight interval 8 = (-m,O]. We could impose weaker conditions with respect to the x dependence of a (e.g. as in 1.3.4. in the one-dimensional case); our assumptions were made for convenience. Remember that the Mellin symbols hj,(y, y', z ) occurring in the definition of M (cf. 3.3.3. (7)) are for the class with subscript ( 0 ) functions in C"(R"-' X R"], Wm) 5 for certain Rj E 5%independent of y , y' (cf. 1.2.1. Definition 1 and the beginning of Section 1.2.3.). We set a i ( 4 (x,
n
= a,o,(x,
ac;l'(~) (Y, 71, z ) =
n,
1 oiWkI)(Y, 71,z ) + ac;l'(~) (Y, 71,z ) ,
I+ k=j
m
where arkl(y, 0 is defined by a ( y , t, 5)
- 1 tkaIkl(y,F) for t+O
and
k=O
ai,'(alk9(Y, 71,z ) = + (Y, 71) g+(z)+ - (Y, 71) g-(z>}fr(z) (4) withfi(z) as in 1.3.4. (4). Remember that a,$(M) was defined by 3.3.3. (24) and Definition 18. As noted above a\kl**(y,71) is a homogeneouspolynomial in 71 of order 1. The Mellin symbol ac;l'(A)( y , 71,z ) is altogether a polynomial in 71 of order j . Since the contributions from M are rapidely decreasing for 1 Irn z1- m we may interprete also d $ ( A ) (y, [(z)) as a connection of a $ ( y ) , parametrized by z E [ - 1, 11. Let us use the notation uM(a) = with
d a >( y , 71,z>=
1 ac;l'(alkl)(Y, 71,z ) .
I+k=j
2.Theorem. Let A = Op,(a_) + R, A"= Op,(a") + R E ~ ~ ( R ! $ *R,) , R"EU%$~(R!$*). Then AA-E $(R:)(') and AA = Op,(a #, a") + R 1 ,where a #, a'denotes a symbol evaluated by the Leibniz product of complete symbols and R 1E U%$$(R:)(*). Further A* = Op,(a*) + R2 where a* follows by the rule of formal a4oints for complete symbols, R 2 E US$t(R+)'". We have also a nice control of the Mellin symbols. Consider, for instance, the compositions. Then we obtain 3.3.3. (34), (35) with p = p' = 0. Another relation here is m
where #,,M is the tensor product between the Mellin translation composition in z and the Leibniz rule in y , cf.3.3.3. (35). 99[71]jdenofesthe space of polynomials in q of degree j , with coefficients in SQ.
3.3. VDO-s on Manifolds with Edges
387
Note that the very special pattern of poles of the Mellin symbols (4) imply more precise remainders in ( 5 ) than indicated. The operators A = Op,(a) + M + G E f!;(R:)(*) have a principal boundary symbol ~ ? ( A ) ( Ytl>:=a:(a>(~, , o ) + o : ( M + G>(Y,v ) , where a : ( M ) was defined in 3.3.3. (19) and U;(a)(Y, ?I):=oP,(a(o)l,=o)(Y,711, 71 =# 0. This is a family of operators
a;(A) (Y, 71): L2(R+)--+LZ(IR+) with a: ( A ) (y, 1 q ) = xAa:( A ) (y, 71) x;’ for all 1 E R, . The boundary symbols behave under the algebraic operations as they should do, for instance , a;(M(Y, 71)= 4 ! i ( A ) ( Y , v ) ~ ; ( m J71). The operator class %(It:)(’) is preserved under push-forwards with respect to diffewhich extend to diffeomorphisms R“+R” that preserve omorphisms x : R:+R: the t direction for small I t l . It follows the usual transformation rule for the complete interior symbols. Now let W be a compact @“ manifold with boundary Y. Then we can define the space %(W)(*) by the standard constructions of globalizing the local operator spaces, using an open covering of W, a partition of unity, and so on. Finally we can pass to a space B;(W)(*) of matrices
where A E i!;(W)(*)and K , T, Q are operators of the types that have been introduced in 3.3.4. in the (*)-variant from the subclass 3.3.4. (47). Let us consider also Bz(W) = Bo;oso(W) obtained by replacing M + G in the left upper comers by operators in 1%’$%(W)in general and admitting K, T to be of the class Bo(W).In other words we drop the ( 0 ) conditions for M + G , K,T and allow instead the general continuous asymptotic types. Then, of course, %?(: W)(’) c B;(W) and the elements of the above calculus hold for B;(W). 1E B;(W) induces continuous operators 1:W ( W ) @ H s ( ( y G , - ) + W ( W ) @ H S ( Y , G,) (7) for all s E R. Remember that WO(w, = L2(W). Denote as usual by W;(W) ( P e p o ) , W;(W) ( B E ~ O the ) subspaces with asymptotics. The operators 1E B:(W)(*) then induce 1: W;( W) @ Hs( G-) + W;( W) @ Hs( G,) (8) for every P E 9’with some Q E Po,depending on P,1.Those in B;( W) induce operators between the spaces with continuous asymptotics. This follows easily from the corresponding mapping properties along R, which was explained in 1.3.4. In particular we can explicitly observe the shift of asymptotic data P+ Q. There is a “universal part” from the lyDO itself associated with the poles of the functions g*(z) and then another “individual part” from the Mellin and Green operators which act smoothing along R, .
388
3. Operators on Manifolds with Edges
The various symbolic levels for the entries induce the corresponding ones for the matrices altogether, in particular ot(SS)(x, 0 = a ; ( A ) ( x , 8 E S(O)(PW\O), aL(sQ)(Y, z) = &A)
(Y, z) E
WY, m?s*),
(Y, 7) € T+Y\O.
As we know the operators in St(W)can be composed within the class, provided the bundles G+ belonging to the factors fit together. Further we have the adjoints in the class. The operations are compatible with the corresponding ones on principal symbolic level (for all symbolic components). Further the ( 0 ) classes are preserved under the operations. 3. Definition. A E S : ( W) is called elliptic i f (i) a;(d)(x,0 0 for all (x, R E T+W\O, (ii) aO,(d)( y , q ) is bijective as operator family (9) for all ( y , q ) E PY\
*
0.
Note that (ii) implies aL(SS)(y, z) 9 0 for all z E I',,,, y E Y. It means in the above interpretation that a&(&) (y, [(z)) is a connection of a i ( y ) and a ; ( y ) through nonvanishing functions along the conormal piece [-I, I] 3 t. 4. Theorem. For SS E mi( W) the following conditions are equivalent (i) (7) is a Fredholm operator for an s E R, (ii) SQ is elliptic. lfd is elliptic, there is a parametrix B E S;(W) . Further we have the elliptic regularity and ifaL(SS)is independent from 3.5.5. Theorem 4, for 8 = (-00, 01. l f d E Si(W)(*) of y then we find B in St(W)(*) and we get the elliptic regularity in the sense of 3.3.5.
Theorem 13. Remember in particular that the elliptic regularity for the
( 0 )
class says that
S S U = ~ E W J , ( W ) @ H ~ ( ( Y , GU + )E, W - " ( W ) @ H - " ( Y , C-) implies u E W"p(w)@ Hs(Y, G-) for every Q E Powith some P E Po.In the present situation we have very concrete information on the transform Q- P. The Mellin calculus on R+shows that the new asymptotic contributions come from the zeros of a;(&) (y, z). Let us assume that the left upper corner of SS is locally Op,(a) for some a E S:,(RnX R"). Then @&(So) equals (2). From 1.3.4. Lemma 9 we know the zeros, indeed. Now the transformation P+ Q follows from the complete Mellin symbol of L7l which can be determined by inverting (5). This gives a step by step procedure for the sequence a i ( B ) with the explicit position of poles. Note that the latter procedure is completely independent of the condition that a;(&) does not depend on y. This restriction in Theorem 4 was made, since the general calculus with variable (or branching) asymptotics was not elaborated in this book. The precise regularity result in the variable case requires first the definition of corresponding spaces and then the elements of the calculus, including Green op-
389
3.3.ruDO-s on Manifolds with Edges
erators with variable discrete asymptotic types. After the material of 2.3.3. and Chapter 3 this is, of course, in a sense to be expected. It will be presented in a separate paper. 5.Remark. The MeNin approach of boundary value problems for vD0-s (of any order) as well as the whole edge theoly in general has a straightforward extension to systems of equations or operators acting between distributional sections of vector bundles. In particular for constructing parametrices we do not need factorizations of symbols as it was used, for instance, in [E4] (cf. also 1.3.4.Remark 19).
Let 99 (A), 0 5 A 5 1 , be a family of operators in B:(W), continuously depending on A (in the topology of operators, which is canonically defined by the usual scheme in terms of symbols and operator norms in Sobolev spaces). If 99 (A) is elliptic for all A then ind 99 (0) = ind 99 (1). The homotopies through elliptic operators may also be regarded as homotopies of symbol triples
{@:( oL(&(A)), w))@, :(~(4)losAsl, where the first two components are non-vanishing functions on the "cage" formed by S* W v N, S* W being the unit cosphere bundle induced by T'W\ 0 and N the unit conormal interval bundle Y X [ - 1, 11 3 ( y , z) . Similarly to the case of dimension 1 (cf. 1.3.4.) we can geometrically obtain homotopies, where the values of u: on north and south pole coincide for A = 1 and aL is the connection by homogeneity zero. (The existence of such homotopies follows, since S * W u N and S*W u NIN are of the same homotopy type.) The latter case is by definition the weak transmirsion property. In [R2]it was proved that the elliptic operators in PV(W) are stable homotopic (through elliptic operators) to operators in Boutet de Monvel's algebra. The theory of this subclass may be found in [B9], [R8].It is a calculus of pseudo-differential boundary value problems where the wDO has the transmission property with respect to the boundary. For the zero order wDO-s it means in local ( y ) = a?], - ( y ) for all j , k E W and all y (cf. (4) and the corresponding determs finition in 1.3.4.). +
6.Remark. There is an immediate analogue of 1.3.4. Theorem 13 for higher dimensions, where Gy is here a Green operator with finite-dimensional boundary symbol.
A calculus of boundary value problems for higher order yrDO-s without the transmission property is also included in the material of the preceding sections. The only specific point now is that the Mellin symbols of an operator A E B p ; @( W) with given a:(A) = a should be linked to a in the canonical way as it was explained for the homogeneous components at the end of Section 1.3.4. In other words let a ( x , 0 E S:,(R" X R"), a ( x , 0 ucp-,)(x, 0x(5>, where a(,,- ,,(x, 0 is homogeneous in
-
r
F of order p - r, x(R an excision function, further ~ ( # - ~ )1,( 0 y ,-
tka[:- r)(y,0 k
as t +0. Then we obtain the coefficients a$!-*r),j ( y , q ) from 1.3.4. (24) and the Mellin symbols
(here we use g*(z
+ m) = g'(z)
for every m
E Z),
390
3. Operators on Manifolds with Edges g G - r - j ( a ( f i - r J ( y ,?I
z)=
C
~ ~ - r - ' ( a \ ~ - r~ ) 9) (~ ~ 1~ ,
k+l=j
g ~ - " ( a ) (09~ z)= ,
C
g ~ - ~ - j ( a ( , i - r )V, ) (z), ~,
r+j=m
m E W.Further the Green operators should be restricted to the subclass where the asymptotic types are generated from those in the present particular Mellin symbols under the operations of the calculus. This yields not only a higher dimensional analogue of 1.3.4. Theorem 17 (cf. also Remark 6) but an explicit description of the shift of asymptotic data in the elliptic regularity, similarly as explained above for the zero order case. In other words we then have to look again at the zeros of u&(qPJ( y , z). Let p be an open c" manifold which is a neighbouring manifold of W (i.e., W is contained in as a compact c" manifold of the same dimension, with boundary r). Let A E L;!(W) and consider the operator
w
r+Ae+: Ct(int W) + C"(int W),
(10)
where e+ denotes the extension to @by zero, r+ the restriction to int W. Then the mentioned analogue of 1.3.4. Theorem 17 yields an operator Gy with finite dimensional and smoothing boundary symbol (and a singular behaviour near Y which is determined by the poles of the Mellin symbols of A relative to Y) such that r+Ae+- G E
Yv P (
W)"),
e =y -p
(11)
for every y E R \ {ej}jEwhere ej is a discrete sequence of reals with lei[+ m as Ijl (more precisely, (11) is obtained by extension from C;(int W) to W y ( W ) ) . Clearly the ej are explicitly known. We want to finish the discussion by some examples. In the beginning of this section we did remind of the mixed problems of the type (i)*from the Introduction. They can be treated by an edge calculus as mentioned, but we can also employ the method of reducing the problem to the boundary. The formal scheme was briefly sketched in the previous section (cf. 3.3.5. (25)). Let 0 be a bounded domain in R"+ 3 2, 3 0 of the class c",and let 3 0 = X+ uX- with compact @" manifold X + with c" boundary Y = X+ nX- , Y c 3 0 being of codimension 1. Consider an elliptic differential operator D in 0 with coefficients in C-(fi), ord D = 2 , and let -j m
be an elliptic boundary value problem, To being a trace operator of the form
with c" coefficients (rz denotes the operator of restriction to the set Z). Let TI = (T+, T - ) be a second pair of trace operators,
3.3. wDO-s on Manifolds with Edges
391
with @" coefficients, satisfying the Shapiro-Lopatinskij condition with respect to D over X , . Then
represents an elliptic mixed boundary value problem for A . If T- are the Dirichlet, T+ the Neumann conditions, we get the Zaremba problem. If P are given by different vector fields, nowhere tangent to 30, we obtain the problem with jumping oblique derivatives. For simplicity we choose To in the form that T-=rx_To (we then talk about the normalized reduction to the boundary). Now we use d oto reduce dl to the boundary. dobelongs to Boutet de Monvel's class and we find a parametrix do= (Po,KO)of doin this class. Now we consider the composition dido which is modulo negligible operators a matrix of operators of triangular form, with 1 in the left upper, TIKoin the right lower comer. For algebraic reasons it is evident that it suffices to deal with TIKo.Then we can pass to the original problem by taking the composition back (i.e. composing from the right by a stabilization of d o ,cf. [R4]for more details on the algebraic manipulations). TIKocan be regarded as a lyDO on 3 0 with a discontinuity along Y. If we denote by Ki the restriction of KO to C;(int X,) then (in view of the special relation between T and To) we have T - K ; = 1 over X - modulo an operator with kernel in Crn(X-x . Consider TIKo for a moment on C;(a0\ Y) and denote by e*: C;(intX*) + C ; @ a \ Y) the extension operators by zero to the corresponding opposite sides. Then TIKocan be identified with
x-)
where the second row is equivalent to (0, 1) modulo smoothing operators. In other words we have again a triang_ularmatrix with the essential part T+KOe+. Let us set W = X+ and let W c a 0 be a neighbouring manifold of W. It can be obtained by adding a small open tubular neighbourhood of Y in X?to W. Then
is a classical elliptic lyDO on @with r+Ae+= T+KOe+in the notations of (10). Now we can pass to (11) and solve a boundary value problem on W in the class W Y , @( W)(*) after adding appropriate boundary (and potential) conditions along Y. The arising Fredholm operator gives rise to a new Fredholm operator for the original problem sS1 consisting of a matrix
with derived extra conditions .!Tl, X1, Q1 along Y. Formally the procedure is practically the same as in [R4].We do not carry out all elements here in detail. Above all we want to have a look at the operator A for D = A (the Laplacian) for the Zaremba problem and with jumping oblique derivatives.
3. Operators on Manifolds with Edges
392
Denote the local coordinates in R"+ by x' = (xl, ..., xn+ where 0 corresponds to R:+ = { x,, + > 0) and Y to xn+ = x,, = 0. Write the jumping oblique derivatives in the form
'
where r* are the restrictions to {xn+ = 0, x, S 0}. Because of the Shapiro-Lopatinskij condition we assume that y, 6 are nowhere vanishing. For simplicity we further assume that the coefficients are functions of y = (xl, ...,x,,- '). Let x = (xl, . .., x,,) and 5 = (q,z), q = (ql, ..., v,,-~), z the covariable to x,. Then the operator A (obtained by reducing Z"+ to {xn+ = 0} all by means of T)has the symbol
-
Thus A is a 1pD0 of order zero. The coefficients (3) follow in the form
According to 1.3.4. Lemma 9 we form the function
and the zeros of the leading Mellin symbol 1 z(y)=2niIlogIh(y)l+iargh(y)}+k,
~ E Z .
(16)
Note that logIh(y)( = 0 when a,B, y , 6 are real-valued. The behaviour of the solutions z b ) can be illustrated by supposing, for example, that the (complex) vector fields Tc and T coincide in an open neighbourhood U of a point yo. Then we obtain the usual elliptic regularity which is reflected by the solutions z(y) E Z for y E U.Outside U we have P 9 T in general. Travelling on a curve R 3 A +y(A) c R"- l( Y) that intersects a U we observe moving solutions z b ) in the complex plane, cf. Fig. 13, where we wrote zk(A)= z(y(A)) + k, k E Z . For purposes below Fig. 13 contains a second variant wk(A). The asymptotics of solutions u' of the original problem are linked to the asymptotics of solutions u for the problem on {x,,+ = 0, x,, > 0} by applying the "inuerse reduction to the boundary". This is an unspecific step which does not contribute more than standard data. For instance, u contains expressions of the type Ko(e+u),KObeing a potential operator in Boutet de Monvel's class (cf. also 3.3.5. (26)). This final step transforms the asymptotics for x,, = t +O for every fixed yo to other ones for (x: + x: + J1" = r 0 in the normal plane to Y over yo. The new asymptotic coeficients are then functions of the angle variable p, 0 5 p 5 2n. We content ourselves here with these remarks but keep in mind that the behaviour of u' for r+O with re-
-
+
3.3. wDO-s on Manifolds with Edges
393
Fig. 13
spect to exponents and multiplicities is the same as that of u for t+ 0. The complete discussion of larger classes of examples will be given in [S17]. Let us now discuss another mixed boundary problem dZfor A with the trace condition T2consisting of (12) over X+ and the Dirichlet condition over X- . The operator that follows by reducing T2 to the boundary (where now d ois just the Dirichlet problem for A) has in local coordinates the symbol n-1
C
~ ( Y , o = aiitli+ar+iyIFI. i- 1
The principal Mellin symbol follows from the formula 1.3.4. (20) for j = 0, ,u = 1. Using r ( l - z) = -zr(-z) we obtain 4 4 ( a ) ( y , z) = -zIa;(y)g+(z)
+ aig-(z)l,
where a i ( y ) = & a( y) + iy(y). Thus we have to set in this case
The zeros are given by (16) and in addition by z = 0 for &(y) = 0. The Zaremba problem corresponds to a = 0, y = 1. Then the solutions of (16) are z=1/2+k, keZ. Note that the exponents of the asymptotics for the plane Zaremba problem can also be obtained by a direct calculation. The Laplacian in RZin polar coordinates (r, cp) is of the form a2
A=-+ar2
1 a2 -+-1 -.a r2 acp2 r ar
394
3. Operators on Manifolds with Edges
Let us consider the solutions of Au we set /I = 1/2 + k, k E Z, then
u(r,cp)=O for cp=O,
= 0 of
the form u (r, cp) = rP sin (&I), /I2= p 2 . If
au
-(r,cp)=O
av
for p = n .
Thus u(r, cp) solves the Zaremba problem in the upper half plane with vanishing data and we obtain the asymptotics as r+O with exponents 112 + k. Clearly the singularity near r = 0 is restricted in the rigorous theory by the required Sobolev class. Moreover we only admit weights 4 1/2 + k for k E Z. Finally we want to construct examples of mixed boundary value problems where we can read off a branching behaviour of the asymptotics of analogous quality as in 2.3.2., 2.3.3. Our method will show at the same time that this is no accident but rather typical. Choose two couples of jumping oblique derivative conditions P,S* and denote by z k ( y ) ,w k ( y ) , k E Z, the associated values which follow from (16). Then we can easily construct a picture like in Fig. 13. In other words both
Z1={ y E R " - ' : z ' ( y ) = w ' ( y ) } , Z2 = { y E R"-': z '(Y) ~ ' ( y ) }
*
are non-empty in general. Now consider the mixed problem for
(oA -1 A )
(:)=(;)
in
~ : + 1
(this corresponds to the bipotential equation) and pose for U; (U;) the jumping conditions P (S*).An easy calculation then shows that the reduction to the boundary (in the mentioned normalized form) gives rise to a triangular matrix of symbols where a ( y , 0 and b ( y , 0 are the diagonal elements, a being associated with P by (14), b with S*by the analogous relation. The points z E C where the leading Mellin symbol is not invertible follow as the zeros of the determinant a ( y , 0 b(y, 0.They are simple and different for y E C2and double for y E El. Thus if we pass from Z1to Z2we get jumps of the asymptotics where the logarithmic terms disappear (cf. also 2.3.2. (19), (20)).
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Index
Abstract wedge Sobolev space 271,306 Adjoint Mellin symbol 21,63, 157 Adjoint of a wedge lyDO 345,356,364 Adjoints of Mellin operators 21,22, 83 Adjoints of operators with Mellin expansions 103,112,123,205,212,215 A-excision function 43 Amplitude functions 138 Amplitude functions based on reductions of orders 305 Amplitude functions, operator-valued 293 Amplitude functions with weight at m 229 Analytic functional 36 Analytic functionals associated with meromorphic functions 39 Anisotropic description of standard Sobolev spaces 268 Asymptotic expansion (continuous) 56,147 Asymptotic expansion (discrete) 46, 147 Asymptotic expansion for the complete lyDsymbol 297 Asymptotic expansion in wedge Sobolev spaces 313,314 Asymptotics of solutions 104, 114,216,222 Asymptotic sum of operators 176,300 Asymptotic sum of operator-valued amplitude functions 287 Asymptotic sums of operator-valued distributional kernels 290,303,373,378 Asymptotic sums of symbols 75, 180,302 Asymptotic sum of wedge boundary problems 364, 370 Asymptotic sum of wedge symbols 340, 341 Asymptotic sum of wedge symbols with asymptotics 352, 363 Asymptotic type 9,42 Asymptotic type for the cone 143,217 Asymptotic types for operator-valued Mellin symbols 153,218 Asymptotic type of a concrete leading Mellin symbol 228 Base of the cone 135 Binomial coefficients 50 Borel theorem for the continuous asymptotics 55, 148
Borel theorem for the discrete asymptotics 48,147 Boundary conditions along the edge 361 Boundary symbol 6,393 Boundary value problems 387 Boundary value problem without transmission property 12 Branching of discrete asymptotics 40,253 Branching of asymptotics for concrete examples 265,394 Branching of exponents of the asymptotics 12 Carrier classes for r --* 0 57 Carrier of an analytic functional 37 Carrier of asymptotics 51 Classical operator-valued amplitude functions 296 Classical pseudo-differential operators with operator-valued symbol 296 Classical symbols on R 36 Coefficients of the asymptotics for the wedge 379 Commutators in the cone algebra 212 Commutators of Mellin operators with powers o f t 92,187 Commutators of Mellin operators with powers o f t in the parameter-dependent case 246 Compact embeddings 34 Compact embeddings of Mellin Sobolev spaces 25,193 Compactness of Green operators 89, 185 Compactness of lower order wedge pDO-s 370 Compactness of pseudo-differential operators in abstract wedge Sobolev spaces 299 Complete Mellin symbol 79, 82,98, 166,203 Complete Mellin symbols as formal power series 264 Complete (pseudo-differential) symbol 298, 303,307 Complete symbol in the operator-valued Mellin calculus 182 Compositions of Mellin-Fourier cyDO-s 167
403
Index Composition of Mellin symbols for the cone 157 Composition of Mellin ylDO-s 83 Compositions of operators with Mellin expansions 101, 112, 117,205,212,215 Composition of wedge operators 345, 365 Composition of wedge ylDO-s 357 Compressed cotangent bundle 5,66, 158 Cone Sobolev space 138 Conical singularity 2 Conification of a calculus 135 Conified Sobolev spaces 171 Conification of the standard yDO calculus 12 Conjugate Mellin translation product 64 Conjugation of cone Mellin operators with group actions 161 Conormal asymptotics 8 , 4 2 Conormal order 8, 12,59, 98,203 Conormal symbol 385 Continuity of cone Mellin operators in spaces with asymptotics 159 Continuity of wedge ylDO-s in spaces with asymptotics 362, 369 Continuous asymptotics 56 Convolution 20 Cousin problem 38 Crack problem 3, 384 Cut-off function 19 Decomposition data of Mellin symbols 109, 2 10 Decomposition method 55,62 Decomposition of a distribution into homogeneous ones 200 Degenerate differential operators 29 Differentiation with respect to parameters in the class of point-wise discrete cone operators 257 Dirac measure 20 Discrete asymptotic type 42 Discrete conormal asymptotics 42 Distributional kernel of a Mellin yDO 75, 163 Distributional kernel of a pseudo-differential operator with operator-valued amplitude function 288, 301 Duality of weighted Mellin Sobolev spaces 25, 119,140,190 Duality of weighted wedge Sobolev spaces 308 Dual spaces with respect to a reference scalar product 278
Douglis-Nirenberg ellipticity of wedge wD0-s 379 Edge conditions with constant discrete asymptotics 372 Edge conditions with continuous asymptotics 361 Edgification 137 Ellipticity along X^ 261 Elliptic operator 6 , 9 1 , 103, 114, 118, 123, 128,187,206,213,221,232,235,373,388 Elliptic regularity with asymptotics 11, 104, ii4,123,206,213,222,236,373,37a, 388 Euler r-function 31 Exact symbol sequence 94,99,112,116,123, 188,205,221,346,356,369 Excision function 19 Factorization of symbols 10, 133,389 Families of cone operators with point-wise discrete asymptotics 257,260 Families of Mellin symbols with point-wise discrete asymptotic types 255 Families of point-wise discrete analytic functionals 252 Flatness of some order 45 Flat operator 87,90, 186,220 Formal adjoints of cone Mellin operators 160 Formal adjoint operator 175, 300 Fourier transform 18 Fourier transform of homogeneous distributions 201 FrCchet topology in parameter-dependent classical pseudo-differential operators 244 FrCchet topology for classical ylDO-s 176 Fredholm family 237 Fredholm operator 6, 104,123, 128,206, 216,222,226,231,232,236,359,373,388 Fredholm property, necessity of ellipticity 104,123,206 Freezing of coefficients 327 Fuchs type 5,10,108,137 Global behaviour of cyDO-s in Rm228 Global complete symbol 328 Global edge boundary value problem 368 Global obstruction for the existence of elliptic edge conditions 376 Global symbol spaces for the wedge 369 Global wedge symbol classes 346 Global wedge wDO-s 346 Green operator for the cone 9, 184, 220
404
Index
Green operator of Boutet de Monvel's class 88,381 Green operator with asymptotics for the wedge 349 Green operator with continuous asymptotics 88,118 Green operator with discrete asymptotics 88, 118 Green symbol with asymptotics for the wedge 347 Green wedge operator 337 Group action (one-parameter) in a Banach space 273 Hausdorffmoment problem 23 Hilbert space tensor product 149 Hilbert space triple with unitary actions on the reference space 279,300 Holomorphic functions in a, Frbchet space of 32 Homogeneity of an operator with respect to the canonical R+action 195 Homogeneity of operator-valued functions 293 Homogeneous distribution 200 Homogeneous Green symbol with asymptotics for the wedge 349 Homogeneous principal symbol 83, 169, 158 Homotopies of elliptic boundary value problems 389 Homotopies of elliptic edge boundary value problems 383 Index 104,213,226,359,376,383 Index of a Fredholm family 359 Index of cone operators for different weights 227 Interior cyDO symbol 98 Interior cyDO symbol for the wedge 338 Intersection norm 26 Invariance of conormal asymptotics under diffeomorphisms 50,57,324 Invariance of wedge Sobolev spaces under diffeomorphisms 310,324 Invertibility within a given operator class 241 Invertibility within the class id + Green operator 89, 185 Invertibility within the class of cone operators 213 Invertibility within the class of operators with Mellin expansions 107, 118 Invertibility within the class of operatorvalued edge symbols 374
Kernel cut-off 78,165,174,289 K group 359 Laplace-Beltrami operator 5,137,228,265, 267,383 Laurent coefficients as coefficients of the asymptotics 48 Laurent expansion of a meromorphic Fredholm family 236 Leibniz rule 8, 386 Locally convex topology for Green and flat operators 95 Locally convex topology for operators with Mellin expansions 115,214 Lz-spaces with weight 17 Manifold with conical singularities 135,216 Manifold with edges 266 Mellin convolution 20 Mellin expansion 10, 115, 126 Mellin expansion for the cone 8,202,211 Mellin expansion for the cone with parameters 246 Mellin-Fourier pseudo-differential operator 158,161 Mellin operator 27,97 Mellin operator convention 73,199 Mellin pseudo-differential operator 71,79 Mellin symbol 8,21, 386, 389 Mellin symbolic mapping for the wedge 355 Mellin symbols for the cone with continuous asymptotics 156 Mellin symbols for the cone with discrete asymptotics 154,203 Mellin symbols for a pseudo-differential symbol on R+126,131,386 Mellin symbols with continuous asymptotics 62 Mellin symbols with discrete asymptotics 61, 203 Mellin transform 7, 17, 30 Mellin translation product 8, 64, 101,205, 221,386 Mellin translation product for wedge cyDO-s 357 Meromorphic family of Fredholm operators 237 Mixed boundary problem 3,11,384,390 Moments of a function 22 Monotonic flatness orders 261 Negligible operator 8 Negligible wedge Green operator 349
405
Index Negligible wedge potential operator 361 Negligible wedge trace operator 361 Non-canonical symbol mapping 328 Non-linear eigenvalues of elliptic operators on the base 210 Normalized reduction to the boundary 391 One-sided Green operator 91 Operator algebra with symbolic structure 300 Operator convention 7, 176, 301,337,338, 344,354,364,369 Operator-valued amplitude function 281, 301 Operator-valued amplitude functions that are point-wise isomorphisms 295 Operator-valued Mellin amplitude functions 178,179 Operator-valued symbol with asymptotics for the wedge 348 Order-reducing isomorphisms 138,140,142, 172,177,184,222,304 Oscillatory integral 73, 162 Overdetermined systems of wedge lyDO-s 380 Parameter-dependent amplitude functions 138 Parameter-dependent ellipticity 248 Parameter-dependent Green Operators 244 Parameter-dependent holomorphic Fredholm family 237 Parameter-dependent Mellin-Fourier pseudo-differential operators 243 Parameter-dependent norms in Sobolev spaces 177,305 Parameter-dependent operator-valued Mellin symbols 245 Parameter-dependent pseudo-differential operators 138 Parametrix 9, 91, 104, 114, 118, 123, 128, 187,206,213,215,222,232,236,373,378, 388 Parametrix within the point-wise discrete class 258 Parseval's theorem for the Mellin transform 19 Point-wise discrete C" functions of analytic functionals 41 Polar coordinates 195 Polar coordinates in differential operators 259 Polar coordinates in standard yDO-s 196, 328
Potential amplitude function for the wedge 360 Potential operator along an edge 13, 312 Principal boundary symbol 387 Principal edge symbolic mapping 353, 356, 364,369 Principal interior symbolic mapping for the wedge 353,356,364,369 Principal symbolic structure 176, 300 Problem with jumping oblique derivatives 392 Projective tensor products of FrCchet spaces 40,70 Properly supported operator 76, 163, 296, 341,345,353 Pseudo-differential operator based on the Mellin transform 21 Pseudo-differential operators in Rm with weight at 230 Pseudo-differential operator with operatorvalued amplitude function 288, 302,306 Pseudo-differential operators without the transmission property 10,389 Pull-back of the continuous asymptotics under diffeomorphisms 57 Pull-back of the discrete asymptotics under diffeomorphisms 50 Push-forward of a lyDO under a diffeomorphism 298
-
Recovering of the principal Mellin symbol 86,104 Recovering of the principal yDO symbol 85 Recovering of the lyDO symbol at m 124 Reduction of an edge boundary value problem to the edge 381 Reduction of orders on a manifold with conical singularities 222,249 Reduction to the boundary 381,391 Reference weight 43 Reference weight for the cone theory 139 Regularity of solutions with point-wise discrete asymptotics 263 Regular part of the edge asymptotics 379 Regularization of homogeneous distributions 201 Scale axiom 301 Scales of Hilbert spaces 300 Schwartz space on R 47 Schwartz spaces of operator-valued functions 174
406
Index
Schwartz space t(R+) for the Mellin transform on R, 28 Schwartz space P'*(R, x R")related to the Mellin transform on R, 165 Screen problem 3 Shadow condition 9 , 4 9 , 5 7 , 143, 321, 323 Shapiro-Lopatinskij condition 6, 13, 358 Shift of weights for wedge wDO-s 371 Singular functions in wedge Sobolev spaces 312,379 Singular part of the edge asymptotics 379 Sobolev space in R" with weight at 230 Sobolev space on a manifold with conical singularities 151, 218 Sobolev space on R 23 Sobolev space on R, ,based on the Mellin transform 23 Sobolev space with asymptotics 47, 54 Sobolev space with asymptotics for the cone 145 Stretched manifold 2, 135 Stretched manifold for a wedge 311 Sum of abstract wedge Sobolev spaces 316 Sum of asymptotic types 47 Sum of carriers of asymptotics 38 Sum of locally convex vector spaces 26 Symbol estimates with group actions 281 Symbol estimates with reductions of orders 305 Symbol hierarchy 6 Symbolic structure of operators with Mellin expansions 101 Taylor asymptotics 47, 130 Totally characteristic operator 5, 108, 137 Totally characteristic principal symbol 158, 169,204 Totally characteristic Sobolev space 5 Trace amplitude function for the wedge 361 Trace operator along the edge 13,314 Traces at t = 0 in the Mellin image 31 Translation of asymptotic types 42 Translations of Mellin symbols 62
Transmission problem 3 Transmission property 130,131,389 Typical differential operators on a wedge 325 Underdetermined systems of wedge wDO-s 380 Variable multiplicities of poles in meromorphic Fredholm families 238 Variable point-wise discrete asymptotics 253 Vector-valued analytic functional 40 Vector-valued Sobolev spaces 273 Weak transmission property 130,389 Wedge amplitude functions with asymptotics for the infinite weight interval 356 Wedge pseudo-differential operator 337 Wedge Sobolev space 273,307 Wedge Sobolev space with asymptotics 307 Wedge Sobolev space with singular terms on the right of the weight line 382 Wedge !#DOwith constant discrete asymptotics 372 Weighted Mellin action 97, 329 Weighted Mellin transform 17 Weighted Sobolev spaces for the cone 138, 217 Weighted Sobolev space for the Mellin transform 23 Weighted Sobolev space for the wedge 307 Weight function 125,223 Weight line 17 Weight shift isomorphism 146 Weight shifts in wedge Sobolev spaces 316, 371 Weight strip 43 Wiener-Hopf technique 10 Zaremba problem 3, 391, 393 Zaremba problem in the upper half plane 394 Zeros of Mellin symbols in the complex plane 129
Symbol Index
Cm(L?,51R;m(X)Y) 255 255 Cm(L?,%"(X-)~)* 257 C"(0, W(X^))* 257 Cm(cX" x 0) 327 C"(CX^ x a), 333 l?'(R:)(*) 386 ('!f W)(*) 387 $(R+)" 128 C "(a,'B&(X-))'
Diff"(R+) 72 Diff'(R+) 71, 108 dec(h,j) 109, 210 Diff#(x^) 209 Diff'(C;E, E', 224
g @ ( t ) 125 gy 223
gT 230 g*(z)
10, 126
I 9 3 43 34 43 indA 104, 118, 227 I(") 146 indpf 359
408
Symbol Index
21 21 opj$(.) 22, 109, 185 OPG(.) 88 opt’ 95 op@ 176,301,354,356,364 Op(a) 282, 385 OPMt.1
oPM(.)
409
Symbol Index
op$)((h) (Y, a) 329 op:., 338 op; 338, 354 op; 339,354 op,(a) 126 PA 43 P; 43 PI -t P2 47 9 47 9" 47 W(X) 143, 217 9 Y ( X , E ' ) 217 p ( p , v ; A ) 172, 248 9$") 367, 372
Q P 177
W+ 17 r + V 20 r+ 20, 390 5% 42 5%y 42 rml; 46, 147 B(X) 143, 218 %p(X) 143, 218 . % ( X x x ) 153 5%vc")(XX X ) 153 .%(X,E') 217 5%Y(X,E') 217
% ( X x X , F ' m ( E ' ) * ) 218 r, 327, 369 %;y*Q 353 %;yo' x Rq) 347 Y. Q (0 x (R@\{q)) 347 %";Y.Q(R' X Rq) 348 %(");y*Q(Q X (Rq\ {q)) 348 Wi$Pc(Qx2RQ) 348 %$'iy8(Q X (Rq\ (4)) 348 WP(Q' x RQ) 360 %$ '(a2 X Wq) 360 %!$;"(a X (RQ\{q)) 361 R?);Q(Q x (Rq\ {q)) 361 W;@(f2' x Rq) 362 %f0(Q2 X Rq)(*) 372 %~Y(Rzx R'J)(*)) 372
9(R) 20
Ybm
20
Y(R) 23 S(a,B) 31
SIsB1 32 SA 43
Sl; 43
YP(R+)47 ~ B ( R +63 ) S ' ( R x W N ) 60 S@(WN) 60 S$(RN) 60 S$(Rx RN) 60 sg(a) 62, 219 sg'(a) 62, 219 S("](TEX+\O) 66 S'(K+)k x a) 74 S$(E,)t x K, x IR) 74 S$(R+)kx R) 74 S$scI(K+)k x R) 74 S'(lR+)k x R) 74 S"')(TZX+\O), 91 Symb'~'(TZX+\O), 95 e,. 99 @'(lR+)i 99 as,, 111 W(R+), 111 @& 115 @I@(R+)115 S(g)(T*a+\O), 121 P ( R x R"; A ) 138 S$(Q x R n ; A ) 138 S("(TXC\O) 159 SP((K+ X lR")z X R"+') 163 S@(W+ XlR" X IR"+l) 163 Y ( R , A ) 173 S.(@ x IR; Y) 178 S(")(T;F\O) 159, 169, 187 5" 195 @$(X)i 204 @J'(X*); 204 @Q(X)e 212 @'(X^)e 212 @f,(X) 214 W ( X * ) 214 S'g'(T *x^\ 0; E, F ) 220 S(g)(T*C\ O;E,E) 220 @r,(X; E'$') 221 @"(C;E,E)e 221 S C . ( Q . P ~x( R R m~ )~ 229 SatP)(Wm X W") 231 9 Y ( X ^ ) 234 Sg(X*)A 234, 347 9 $ ( X A ) , 234 3'is(X*)A 234, 347 9'p,YB(X^)J 234 @ $ ( X ; A ) e 247
Ta 22, 43 .7(W+) 28 TQP 46 T $ x + 66 TZC 158 .7""(W+ X W") 165 .7(7(R+,L.) 174 T'(Q2 X Wq; E, E) 290 T&,(Q2x Rq; E, 290 T'(f2' x Rq;Z, Z ) 302
F)
410
Symbol Index Xp(X)
X,(X)
w~(R~,x".~(R"+l\(o})) 270
147, 155 148, 155
21, 26, 268, 272, 307, 385 22 %y' 141 %(?I) 268 Zc 42, 143 Z C ~ N143, 153 now) 238 h'1(0) 238 n(a,m 238 n(n,~m)239 aE(oPdu)) 66 $(oP~(a)) 66 a?! 95 a,'(A) (2) 98, 203 a&A) (2) 98,203 a&) 98, 221 aE(h) 99 U g ( A ) 99, 169, 204, 219, 243 121, 158, 338, 353, 363 a: 121, 231 a$ 158, 219, 245 ot(A) 169, 219, 231, 245 Zp 176 ap 176, 300, 364 z"'*@231
x;
ai(A)- 260 P 8 , Z ) 300 335, 339, 353, 358, 361 S; 338 Xi," 338 Z2y,p 338, 353 o ; , ~ 338, 369 a$ 354 ,pi''.@ 353 E$v 354 Zff 354 af$-'(A) 363 C ; ( X A X n ) 365 .Z;(W) 369 f l ~ j a ) 386 a z ( a ) 386 aY 18 Y 18 Yp 175 Y"(Z,*) 300 yDO 1 w 19 w' 331 [fl 139 ll.lls.a 34, 43, 144 #M 64 #M. 64 Or 89, 185 [?I] 139, 268 1I.IIs;r 177 Il.II~.t;r 177, 305 IXI 218 Il.IlS,I 248 # 260 9, 40 OH 149, 185
