Pseudo-Differential Operators: Analysis, Applications and Computations

Operator Theory: Advances and Applications Volume 213 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Black...

Author: Luigi Rodino | M. W. Wong | Hongmei Zhu

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Operator Theory: Advances and Applications Volume 213 Founded in 1979 by Israel Gohberg

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

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany)

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

Pseudo-Differential Operators: Analysis, Applications and Computations Luigi Rodino Man W. Wong Hongmei Zhu Editors

Editors Luigi Rodino Dipartimento di Matematica Università di Torino Via Carlo Alberto, 10 10123 Torino Italy [email protected]

M.W. Wong and Hongmei Zhu Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada [email protected] [email protected]

2010 Mathematical Subject Classification: Primary: 22A10, 32A40, 32A45, 35A17, 35A22, 35B05, 35B40, 35B60, 35J70, 35K05, 35K65, 35L05, 35L40, 35S05, 35S15, 35S30, 43A77, 46F15, 47B10, 47B35, 47B37, 47G10, 47G30, 47L15, 58J35, 58J40, 58J50, 65R10, 92A55, 94A12; Secondary: 22C05, 30E25, 35G05, 35H10, 35J05, 42B10, 42B35, 47A10, 47A53, 47F05, 58J20, 65M60, 65T10, 94A12 ISBN 978-3-0348-0048-8 e-ISBN 978-3-0348-0049-5 DOI 10.1007/978-3-0348-0049-5 Library of Congress Control Number: 2011923066 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Q. Guo and M.W. Wong Adaptive Wavelet Computations for Inverses of Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

M. Pirhayati Spectral Theory of Pseudo-Differential Operators on S1 . . . . . . . . . . . . .

15

S. Molahajloo A Characterization of Compact Pseudo-Differential Operators on S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

B.-W. Schulze and M.W. Wong Mellin Operators with Asymptotics on Manifolds with Corners . . . . . .

31

B.-W. Schulze The Iterative Structure of the Corner Calculus . . . . . . . . . . . . . . . . . . . . . .

79

V.B. Vasilyev Elliptic Equations and Boundary Value Problems in Non-Smooth Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 C. Iwasaki Calculus of Pseudo-Differential Operators and a Local Index of Dirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 J. Delgado Lp Bounds for a Class of Fractional Powers of Subelliptic Operators . 137 V. Catan˘ a The Heat Kernel and Green Function of the Generalized Hermite Operator, and the Abstract Cauchy Problem for the Abstract Hermite Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

V.S. Rabinovich Local Exponential Estimates for h-Pseudo-Differential Operators with Operator-Valued Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

vi

Contents

R. DeLeo, T. Gramchev and A. Kirilov Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191

Y. Chiba Fuchsian Mild Microfunctions with Fractional Order and their Applications to Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . .

211

W. Ichinose The Continuity of Solutions with Respect to a Parameter to Symmetric Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 K. Benmeriem and C. Bouzar Generalized Gevrey Ultradistributions and their Microlocal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 L. Cohen Weyl Rule and Pseudo-Differential Operators for Arbitrary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

L. Galleani Time-Frequency Characterization of Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 P. Boggiatto, E. Carypis and A. Oliaro Wigner Representations Associated with Linear Transformations of the Time-Frequency Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 E. Cordero and F. Nicola Some Remarks on Localization Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Preface The ISAAC Group in Pseudo-Differential Operators (IGPDO) met again on July 13–18, 2009 at Imperial College London in England on the occasion of the Seventh Congress of the International Society for Analysis, its Applications and Computations (ISAAC). The special session for IGPDO turned out to be the largest session with over forty speakers filling out the entire schedule completely. Talks presented at the IGPDO session reflected the diversity of topics cutting across disciplines in the Analysis, Applications and Computations of Pseudo-Differential Operators. This volume contains eighteen peer-reviewed papers related to the talks given at the IGPDO session. Chapters 1–3 feature a chapter on the adaptive wavelet computations of inverses of pseudo-differential operators (Q. Guo and M.W. Wong) and two chapters on the pseudo-differential operators on the unit circle (M. Pirhayati; S. Molahajloo). The latter two chapters pave the way towards the discretization and numerical computations of pseudo-differential operators. Chapters 4–7 are on pseudo-differential operators and boundary value problems on manifolds with singularities, non-smooth domains and Riemannian manifolds (B.-W. Schulze and M.W. Wong; B.-W. Schulze; V.B. Vasilyev; C. Iwasaki). Chapters 8–11 are devoted to concrete partial differential equations that are of interest in physics and geometry (J. Delgado; V. Catan˘a; V.S. Rabinovich; R. DeLeo, T. Gramchev and A. Kirilov). Chapters 12–14 consist of chapters on microlocal analysis, hyperbolic equations and systems (Y. Chiba; W. Ichinose; K. Benmeriem and C. Bouzar). Chapters 15–18 are on topics related to Wigner transforms, Weyl transforms and localization operators (L. Cohen; L. Galleani; P. Boggiatto, E. Carypis and A. Oliaro; E. Cordero and F. Nicola). In an era of interdisciplinary studies in academia fuelled by research and development for societal and global needs, the role of pseudo-differential operators in the mathematical, physical, biological, atmospherical, geological and medical sciences is vital. Underpinning novel applications are deep understanding in the analysis and efficient numerical computations. It is expected that new developments in Analysis, Applications and Computations of Pseudo-Differential Operators will deepen our understanding of science in general and hence improve the knowledgebased well-being of the world. Future developments of IGPDO are geared in the direction of interdisciplinarity.

Adaptive Wavelet Computations for Inverses of Pseudo-Differential Operators Qiang Guo and M.W. Wong Abstract. For invertible pseudo-differential operators Tσ with symbols σ in S m , m ∈ R, we use biorthogonal wavelets to develop an adaptive algorithm to compute the Galerkin approximations of the solution u in the Sobolev space H m,2 of the equation Tσ u = f on R for every f in L2 (R). Mathematics Subject Classification (2000). 47G30, 65M60, 65T10. Keywords. Multiresolution analysis, scaling functions, wavelets, biorthogonal wavelets, vanishing moments, Sobolev spaces, pseudo-differential operators, Galerkin approximations, adaptive algorithms.

1. Introduction Wavelet methods are relatively recent developments with applications in pure and applied mathematics [1, 5]. Due to the localization properties that wavelets display both in space and frequency, the wavelet multiresolution analysis allows us to obtain an efficient sparse representation of a function, especially when the function exhibits singular behavior and large wavelet coefficients are near the singularity. Wavelet methods can distinguish smooth and singular regions automatically and hence lead to adaptive techniques based on multilevel methods. Reliable and efficient a posteriori error estimators have been derived for adaptive wavelet Galerkin schemes for elliptic partial differential equations, which are based on stable multiscale biorthogonal wavelet bases in, e.g., [3]. The developed adaptive refinement strategy guarantees an improvement for the approximate solution after the refinement step. We extend the adaptive strategy developed in [3] to compute inverses of pseudo-differential operators. The paper is organized as follows. In Section 2, we first recall the wavelet multiresolution analysis and describe the properties of biorthogonal wavelets. Then This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

L. Rodino et al. (eds.), Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, DOI 10.1007/978-3-0348-0049-5_1, © Springer Basel AG 2011

1

2

Q. Guo and M.W. Wong

the basics of pseudo-differential operators required for this paper are recalled. Residual estimates are given in Section 3 and a posteriori error estimates are derived in Section 4. Finally, an adaptive algorithm for the computations of inverses of nonsymmetric pseudo-differential operators is presented in Section 5.

2. Wavelets and pseudo-differential operators A multiresolution analysis (MRA) is a sequence of closed subspaces {Vj }j∈Z0 of L2 (R) such that Vj ⊂ Vj+1 , j ∈ Z, ∩j∈Z Vj = {0}, ∪j∈Z Vj = L2 (R), f ∈ Vj ⇔ D2 f ∈ Vj+1 ,

j ∈ Z,

and f ∈ V0 ⇔ T−k f ∈ V0 , k ∈ Z, where D2 and T−k are the dilation and the translation given, respectively, by (D2 g)(x) = g(2x),

x ∈ R,

and (T−k g)(x) = g(x − k), x ∈ R, for all measurable functions g on R. Let ϕ ∈ L2 (R). Then we consider the translations and dilations ϕj,k of ϕ given by ϕj,k (x) = 2j/2 ϕ(2j x − k), x ∈ R, for j, k ∈ Z. If for each fixed j ∈ Z, the sequence {ϕj,k : k ∈ Z} is an orthonormal sequence for Vj such that the sequence is uniformly stable in the sense that 1/2 cj,k ϕj,k ∼ |cj,k |2 k∈Z

k∈Z

2

uniformly with respect to j in Z, i.e., there exist positive constants C and C such that 2 1/2 1/2 2 2 C |cj,k | ≤ cj,k ϕj,k ≤ C |cj,k | , j ∈ Z, k∈Z

k∈Z

2

k∈Z

then we call ϕ a scaling function of the MRA. For j ∈ Z, we denote the orthogonal complement of Vj−1 in Vj by Wj . The raison d’ eˆtre for Wj−1 is that an element in Wj contains the details needed to pass from an approximation at level j − 1 to an approximation at level j. Let ψ ∈ W0 . Then the translations and dilations ψj,k of ψ are defined by ψj,k (x) = 2j/2 ψ(2j x − k),

x ∈ R,

Computations for Inverses of Pseudo-Differential Operators

3

for all j, k ∈ Z. If for each j in Z, the set {ψj,k : k ∈ Z} forms an orthonormal basis for Wj , then we call ψ a mother wavelet and ψj,k , j, k ∈ Z, the wavelets for the MRA. It is well known that Daubechies [4, 5] has constructed for L2 (R) orthonormal bases consisting of compactly supported wavelets that can be represented by polynomials of a fixed degree. The support of the Daubechies scaling function is [0, 2N − 1], where N is a positive integer. The length of the support increases linearly with the regularity. The corresponding mother wavelet then has compact support given by [1 − N, N ] and has N vanishing moments in the sense that ∞ xk ψ(x) dx = 0, k = 0, 1, 2, . . . , N − 1. −∞

If we denote for convenience W0 by V0 , then for all positive integers n, every element vn in Vn given by vn = cn,k ϕn,k , k∈Z

where each cn,k is a complex number, has an alternative multiscale representation given by the wavelets. More precisely, n vn = dj,k ψj,k , j=0 k∈Z

where each dj,k is a complex number. Equivalently, we can write Vn = ⊕nj=0 Wj . Now, we start with two biorthogonal MRAs of L2 (R). This means that {Vj }j∈Z and {V˜j }j∈Z are MRAs of L2 (R) such that the primal MRA {Vj }j∈Z and the dual MRA {V˜j }j∈Z can be equipped with, respectively, Riesz bases Φj = ˜ j = {ϕ˜j,k : k ∈ Z} with the property of biorthogonality to the {ϕj,k : k ∈ Z} and Φ effect that (ϕj,k , ϕ˜j,k ) = δk,k , k, k ∈ Z, where ( , ) is the inner product in L2 (R). Each of the primal scaling function ϕ and the dual scaling function ϕ˜ is assumed to have compact support such that the measure of ϕj,k and that of ϕ˜j,k are ∼ 2−j for all j, k ∈ Z. These biorthogonal bases also define projections Pj : L2 (R) → Vj and P˜j : L2 (R) → V˜j , which are uniformly stable in L2 (R). They are given by Pj v = (v, ϕ˜j,k )ϕj,k k∈Z

and P˜j v =

(v, ϕj,k )ϕ ˜j,k

k∈Z

for all v in L2 (R) and j = 0, 1, 2, . . . . The nestedness of the MRA spaces gives us the properties that Pj Pj+1 = Pj

4


and P˜j P˜j+1 = P˜j ˜ j given by for all j ∈ Z. Hence for j ∈ Z, the operators Qj and Q Qj = Pj+1 − Pj and ˜ j = P˜j+1 − P˜j Q are also projections. ˜ j are given by For j ∈ Z, the wavelet spaces Wj and W Wj = Vj+1 ∩ V˜j⊥ and ˜ j = V˜j+1 ∩ V ⊥ , W j ˜ j ) of Q ˜ j . The which are, respectively, the range R(Qj ) of Qj and the range R(Q ˜ j }j∈Z induce two multiscale decompositions of wavelet spaces {Wj }j∈Z and {W L2 (R) via ∞ ∞ v = P1 v + Qj v = Qj v, v ∈ L2 (R), j=1

j=0

where Q0 = P1 and v˜ = P˜1 +

∞

˜ j v, Q

v ∈ L2 (R).

j=1

˜ j are equipped Furthermore, we assume that for j ∈ Z, the wavelet spaces Wj and W with compactly supported biorthogonal Riesz bases denoted, respectively, by Ψj = {ψj,k : k ∈ Z} and ˜ j = {ψ˜j,k : k ∈ Z}. Ψ For all nonnegative integers n, we can introduce the canonical truncated projections Qn and Qn by Qn v =

n

(v, ψ˜j,k )ψj,k

j=0 k∈Z

and Qn v =

n

(v, ψj,k )ψ˜j,k

j=0 k∈Z

for all functions v in L2 (R). For s ∈ R, let H s,2 be the L2 -Sobolev space of order s defined to be the set of all tempered distributions u on R such that σ−s u ˆ ∈ L2 (R),

Computations for Inverses of Pseudo-Differential Operators where

σs (ξ) = (1 + ξ 2 )−s/2 ,

5

ξ ∈ R,

u ˆ is the Fourier transform of u and the Fourier transform fˆ of a function f in L1 (R) is defined by ∞ fˆ(ξ) = (2π)−1/2 e−ixξ f (x) dx, ξ ∈ R. −∞

The norm s,2 in H s,2 is given by 2

u s,2 =

∞

−∞

(1 + ξ 2 )s |ˆ u(ξ)|2 dξ,

Henceforth, we let λ = (j, k), where j is the level of resolution and k is the location. We let J be the index set given by J = {λ = (j, k) : j = 0, 1, 2, . . . , k ∈ Z}, and for λ = (j, k) in J, we define |λ| by |λ| = j. Then we have the following result. ˜ = {ψ˜λ : λ ∈ J} are Theorem 2.1. Suppose that Ψ = {ψλ : λ ∈ J} and Ψ 2 biorthogonal collections in L (R) such that the associated sequence {Qn}∞ n=0 of projections defined by n Qn v = (v, ψ˜j,k )ψj,k , v ∈ L2 (R), j=0 k∈Z

is uniformly bounded in the sense that there exists a positive constant C such that

Qnv s,2 ≤ C v s,2 , Then for all v ∈ H

n = 0, 1, 2, . . . .

s,2

, we have 1/2

v s,2 ∼ 22|λ|s |(v, ψ˜λ )|2 ,

s ∈ (−γ , γ),

λ∈J

where γ = sup{s ∈ R : ϕ ∈ H s,2 } and

γ = sup{s ∈ R : ψ ∈ H s,2 }.

It is worth pointing out that γ and γ are, respectively, less than or equal to the vanishing moments of ϕ and ϕ. ˜ For every real number m, we define S m to be the set of all functions σ in C ∞ (R × R) such that for all nonnegative integers α and β, there exists a positive constant Cα,β such that |(∂xα ∂ξβ σ)(x, ξ)| ≤ Cα,β (1 + |ξ|)m−β ,

x, ξ ∈ R.

6


Then we call σ a symbol of order m. Let σ ∈ S m . Then we define the pseudodifferential operator Tσ on the Schwartz space S on R by ∞ −1/2 (Tσ ϕ)(x) = (2π) eixξ σ(x, ξ)ϕ(ξ) ˆ dξ, x ∈ R, −∞

for all functions ϕ in S. The following result is well known. Theorem 2.2. Tσ can be extended to a bounded linear operator from H s,2 into H s−m,2 . A proof can be found in, for instance, the book [7]. Let σ ∈ S m . Suppose that there exist positive constants C and R such that |σ(x, ξ)| ≥ C(1 + |ξ|)m ,

|ξ| > R.

Then we say that the symbol σ is elliptic or the pseudo-differential operator Tσ is elliptic. The following result on spectral invariance [6] is well known. See also Theorem 4.9 in [2] in this connection. Theorem 2.3. Let σ ∈ S m be such that the pseudo-differential operator Tσ : H m/2,2 → H −m/2,2 is invertible. Then σ is elliptic and Tσ−1 is an elliptic pseudodifferential operator with symbol in S −m . The following estimate is useful to us. Theorem 2.4. Let σ ∈ S m be such that the pseudo-differential operator Tσ : H m/2,2 → H −m/2,2 is invertible. Then there exist positive constants C1 and C2 such that C1 Tσ u −m/2,2 ≤ u m/2,2 ≤ C2 Tσ u −m/2,2 ,

u ∈ H m/2,2 .

Proof. The “first” inequality follows from Theorem 2.2. By Theorems 2.2 and 2.3, there exists a positive constant C such that

u m/2,2 = Tσ−1 Tσ u m/2,2 ≤ C Tσ u −m/2,2 ,

u ∈ H m/2,2 .

This completes the proof.

The aim of this paper is to use adaptive wavelets to compute numerically the inverse of an invertible pseudo-differential operator Tσ : H m,2 → L2 (R), where σ ∈ S m and m = min(γ, γ ). This amounts to solving the pseudo-differential equation Tσ u = f (2.1) on R for u ∈ H m,2 for all functions f in L2 (R). To do this, we transform the equation (2.1) to the equation Tσ∗ Tσ u = Tσ∗ f

(2.2)


7

on R, where Tσ∗ denotes the formal adjoint of Tσ . Now, Tσ∗ Tσ is a pseudo-differential operator Tτ of order 2m and Tσ∗ f ∈ H −m,2 . Furthermore, Tτ is symmetric and there exist positive constants A and B such that A u 2m,2 ≤ (Tτ u, u) ≤ B u 2m,2 ,

u ∈ H m,2 .

(2.3)

The “second” inequality follows from Theorem 2.2. In fact, there exists a positive constant B such that (Tτ u, u) ≤ |(Tτ u, u)| ≤ Tτ u −m,2 u m,2 ≤ B u 2m,2,

u ∈ H m,2 .

On the other hand, we get from Theorems 2.2 and 2.3 a positive constant C such that

u 2m,2 = Tσ−1 Tσ u 2m,2 ≤ C Tσ u 2m,2 , u ∈ H m,2 . With slight abuse of notation, the problem (2.1) is then the same as solving for u in H m,2 to the equation Tσ u = f on R for every f in H −m,2 , where Tσ is a symmetric pseudo-differential operator of order 2m such that there exist positive constants A and B for which A u m,2 ≤ u Tσ ≤ B u m,2,

u ∈ H m,2 ,

where

u 2Tσ = (Tσ u, u). Remark 2.5. The existence of a positive constant A for which

u 2Tσ ≥ A u 2m,2 ,

u ∈ H m,2 ,

is a condition related to G˚ arding’s inequality on the symbol σ. See, e.g., the paper [8] in this connection. Adaptive wavelet methods in finding solutions to differential and integral equations can be found in [1, 3].

3. Residual estimates The problem of computing the inverse of Tσ : H m,2 → H −m,2 numerically is equivalent to the computation of subspaces VΛ of the form VΛ = span{ψλ : λ ∈ Λ} that are adapted to the unique solution u in H m,2 of the pseudo-differential equation Tσ u = f (3.1) on R for every function f in H −m,2 . To do this, we use the weak formulation of (3.1) to the effect of finding a solution uΛ in VΛ such that (Tσ uΛ , v) = (f, v),

v ∈ VΛ .

(3.2)

8


More precisely, for every tolerance eps, we seek a subset Λ of J such that the Galerkin approximation uΛ in VΛ defined by (3.2) satisfies the estimate

u − uΛ m,2 ≤ eps. This is to be achieved by successively upgrading Λ based on appropriate a posteriori estimates of a current Galerkin approximation uΛ . To this end, we define the residual term rΛ by rΛ = Tσ uΛ − f, which is the same as rΛ = Tσ (uΛ − u). So, by Theorem 2.4, we can find positive constants C1 and C2 such that C1 rΛ −m,2 ≤ u − uΛ m,2 ≤ C2 rΛ −m,2 for all subsets Λ of J. Thus, we can find positive constants C3 and C4 such that  1/2  1/2 C3  2−2m|λ| |(rΛ , ψλ )|2  ≤ rΛ −m,2 ≤ C4  2−2m|λ| |(rΛ , ψλ )|2  . λ∈J\Λ

J\Λ

Now, for λ ∈ J \ Λ, define δλ by δλ = 2−m|λ| |(rΛ , ψλ )|. Since uΛ ∈ VΛ , it follows that uΛ =

uλ ψλ ,

λ ∈Λ

where uλ = (uΛ , ψ˜λ ). So, for λ ∈ J \ Λ, δλ = 2

(Tσ ψλ , ψλ )uλ .

fλ −

−m|λ|

λ ∈Λ

Let µ be the Hölder exponent of ∂ γ ϕ. Then for all positive numbers ε and δ with δ < µ − 12 , we can choose positive numbers ε1 and ε2 such that 2(˜ r+1)

ε1

+ 2−δ/ε2 ≤ ε,

where r˜ is the vanishing moment of ϕ. ˜ For all λ in J and for every positive number ε, we define the tolerance set Jλ,ε by

min(|λ|,|λ |) Jλ,ε = {λ ∈ J : ||λ| − |λ || ≤ ε−1 d(supp(ψλ ), supp(ψλ )) ≤ ε−1 2 , 2 1 }.


9

Then we have the following lemma, which is Lemma 4.2 in [3]. Lemma 3.1. For λ ∈ J \ Λ, let eλ be defined by eλ = 2−m|λ| (Tσ ψλ , ψλ )uλ . λ ∈Λ\Jλ,ε

Then there exists a positive constant C5 such that 1/2 2 |eλ | ≤ C5 ε QΛ f −m,2 , λ∈J\Λ

where

QΛ f =

(f, ψλ )ψ˜λ .

λ∈Λ

We note that for λ ∈ J \ Λ,

δλ = 2−m|λ|

fλ −

+

λ ∈Λ∩Jλ,ε

λ ∈Λ\Jλ,ε

(Tσ ψλ , ψλ )uλ

≤ |dλ | + |eλ |, where dλ = 2

fλ −

−m|λ

λ ∈Λ∩Jλ,ε

(Tσ ψλ , ψλ )uλ

.

Let NΛ,ε be the set of all indices in the complement of Λ with influence set intersecting Λ. More precisely, NΛ,ε = {λ ∈ J \ Λ : Jλ,ε ∩ Λ = φ}. It can be shown that and NΛ,ε

NΛ,ε = ∪λ ∈Λ Jλ ,ε has at most a finite number of elements. Hence λ ∈ J \ (Λ ∪ NΛ,ε ) ⇒ Jλ ,ε ∩ Λ = φ.

Since

f ∈ H −m,2 ⇔

2−2m|λ| |fλ |2 < ∞,

λ∈J −2m|λ|

it follows that λ∈J\(NΛ,ε ∪Λ) 2 |fλ |2 can be made arbitrarily small by choosing Λ appropriately. Indeed, 2−2m|λ| |fλ |2 = 22m|λ| |fλ |2 − 2−2m|λ| |fλ |2 λ∈J\(NΛ,ε ∪Λ)

λ∈(NΛ,ε ∪Λ)

λ∈J

= ∼

f − QΛ∪NΛ,ε f 2−m,2 inf

f − v 2−m,2 ˜Λ∪N v∈V Λ,ε

≤ inf f − v 2−m,2 . ˜Λ v∈V

10


We can now lay out the basic assumptions to the effect that there are positive constants C6 and C7 such that C5 QΛ f −m,2 ≤ C6 f −m,2 and





1/2 2−2m|λ| |fλ |2 

λ∈J\Λ

≤ C7 inf f − v −m,2 v∈V˜Λ

for all subsets Λ of J.

4. A posteriori error bounds For λ ∈ J \ Λ, we define aλ by

−m|λ|

aλ = 2 (Tσ ψλ , ψλ )uλ

.

λ ∈Λ∩Jλ,ε

Theorem 4.1. Under the hypotheses of Lemma 3.1, we have   1/2    

u − uΛ m,2 ≤ C2 C4  a2λ  + C6 ε f −m,2 + C7 inf f − v −m,2   v∈V˜Λ  λ∈NΛ,ε  and  

1/2 a2λ 

λ∈NΛ,ε

≤

1

u − uΛ m,2 + C6 ε f −m,2 + C7 inf f − v −m,2 . ˜Λ C1 C3 v∈V

˜ ⊂ J. Then Theorem 4.2. Suppose that Λ ⊂ Λ  1/2 1 2  aλ  ≤

u ˜ − uΛ m,2 + C6 ε f −m,2 + C7 inf f − v −m,2 . ˜Λ C1 C3 Λ v∈V ˜ λ∈Λ∩N Λ,ε

˜ Then Proof. Let λ ∈ Λ. (Tσ uΛ , ψλ ) = (Tσ (uΛ − uΛ˜ ), ψλ ) + fλ . So, dλ (Λ, ε) ≤ 2−m|λ| |(Tσ (uΛ − uΛ˜ ), ψλ )| + |eλ |. Moreover, 1 1 2−2m|λ| |(Tσ (uΛ − uΛ˜ ), ψλ )|2 ≤ 2 Tσ (uΛ − uΛ˜ ) 2−m,2 ≤ 2 2 uΛ − uΛ˜ 2m,2 C C 3 1 C3 ˜ λ∈Λ\Λ

Computations for Inverses of Pseudo-Differential Operators So, by Lemma 3.1,  1/2  dλ (Λ, ε)2  ≤ ˜ λ∈Λ\Λ

11

1

u ˜ − uΛ m,2 + C5 ε QΛ f −m,2. C1 C3 Λ

Hence |aλ (Λ, ε)| ≤ |dλ (Λ, ε)| + 2−m|λ| |fλ |,

and the proof is complete.

5. An adaptive algorithm ˜ containing Λ, the solution in V ˜ apWe prove in this section that for a set Λ Λ proximates the actual solution better than the one in VΛ . To do this, we recall our assumptions spelled out at the end of Section 2 that the pseudo-differential operator Tσ is symmetric and there exist positive constants C8 and C9 such that C8 u m,2 ≤ u Tσ ≤ C9 u m,2,

u ∈ H m,2 ,

where

u 2Tσ = (Tσ u, u). Theorem 5.1. Let eps be a given tolerance. For θ∗ ∈ (0, 1), we define the number Ce by 1 1 − θ∗ Ce = + . C1 C3 2C2 C4 Let µ∗ be a positive number such that µ∗ Ce ≤

1 − θ∗ . 2(2 − θ∗ )C2 C4

Let ε be the positive number defined by ε=

µ∗ eps . 2C6 f −m,2

Suppose that Λ is a subset of J such that C7 inf f − v −m,2 ≤ ˜Λ v∈V

1 ∗ µ eps. 2

˜ of J such that Λ ⊂ Λ ˜ and Then for all subsets Λ  1/2  1/2  a2λ  ≥ (1 − θ∗ )  a2λ  , ˜ λ∈Λ∩N Λ,ε

λ∈NΛ,ε

there exists a number κ in (0, 1) such that

u − uΛ˜ Tσ ≤ κ u − uΛ Tσ .

12


Proof. We begin with the assumption that

u − uΛ m,2 ≥ By Theorems 4.1 and 4.2,

uΛ˜ − uΛ m,2 ≥ C1 C3

eps . Ce

1/2

− C6 ε f −m,2 − C7 inf f − v −m,2

a2λ

˜Λ v∈V

˜ λ∈Λ∩N Λ,ε

≥ C1 C3 {(1 − θ∗ )((C2 C4 )−1 u − uΛ m,2 − C6 ε f −m,2 − C7 inf f − v −m,2 ) − C6 ε f −m,2 − C7 inf f − v −m,2 } v∈V˜Λ

˜Λ v∈V

∗

−1

≥ C1 C3 ((1 − θ )(C2 C4 )

u − uΛ m,2 − (2 − θ ∗ )C6 ε f −m,2

− (2 − θ∗ )C7 inf f − v −m,2 ). v∈V˜Λ

So,

uΛ˜ − uΛ m,2 ≥ C1 C3

1 − θ∗

u − uΛ m,2 − (2 − θ∗ )µ∗ eps , C2 C4

and consequently

uΛ˜ − uΛ m,2

1 − θ∗ ∗ ∗ ≥ C1 C3 − (2 − θ )µ Ce u − uΛ m,2 C2 C4 C1 C3 (1 − θ∗ ) ≥

u − uΛ m,2 . 2C2 C4

Now,

uΛ˜ − uΛ 2Tσ = (Tσ uΛ˜ − Tσ uΛ , uΛ˜ − uΛ ) = uΛ˜ 2Tσ + uΛ 2Tσ − (Tσ uΛ˜ , uΛ ) − (Tσ uΛ , uΛ˜ ) = uΛ˜ 2Tσ + uΛ 2Tσ − (f, uΛ ) − (uΛ , f ) = uΛ˜ 2Tσ + uΛ 2Tσ − (Tσ uΛ , uΛ ) − (uΛ , Tσ uΛ ) = uΛ˜ 2Tσ − uΛ 2Tσ .

(5.1)

Also,

u − uΛ˜ 2Tσ = (Tσ u − Tσ uΛ˜ , u − uΛ˜ ) = u 2Tσ + uΛ˜ 2Tσ − (Tσ u, uΛ˜ ) − (Tσ uΛ˜ , u) = u 2Tσ + uΛ˜ 2Tσ − (f, uΛ˜ ) − (uΛ˜ , f ) = uΛ˜ 2Tσ + u 2Tσ − (Tσ uΛ˜ , uΛ˜ ) − (uΛ˜ , Tσ uΛ˜ ) = u 2Tσ − uΛ˜ 2Tσ .

(5.2)


13

Furthermore,

u − uΛ 2Tσ = (Tσ u − Tσ uΛ , u − uΛ ) = u 2Tσ + uΛ 2Tσ − (Tσ u, uΛ ) − (Tσ uΛ , u) = u 2Tσ + uΛ 2Tσ − (f, uΛ ) − (uΛ , f ) = u 2Tσ + uΛ 2Tσ − (Tσ uΛ , uΛ ) − (uΛ , Tσ uΛ ) = u 2Tσ − uΛ 2Tσ .

(5.3)

Therefore by (5.1)–(5.3),

uΛ˜ − uΛ 2Tσ = u − uΛ 2Tσ − u − uΛ˜ 2Tσ , or equivalently

u − uΛ˜ 2Tσ + uΛ˜ − uΛ 2Tσ = u − uΛ 2Tσ . Now,

uΛ˜ − uΛ Tσ ≥ C8 uΛ˜ − uΛ m,2 C1 C3 C8 (1 − θ∗ )

u − uΛ m,2 2C2 C4 C1 C3 C8 (1 − θ∗ ) ≥

u − uΛ Tσ . 2C2 C4 C9 ≥

(5.4)

Hence

u − uΛ˜ 2Tσ = u − uΛ 2Tσ − uΛ˜ − uΛ 2Tσ 2 C1 C3 C8 (1 − θ ∗ ) 2 ≤ u − uΛ Tσ −

u − uΛ 2Tσ 2C2 C4 C9 = κ2 u − uΛ 2Tσ , where

κ=

1−

C1 C3 C8 (1 − θ∗ ) 2C2 C4 C9

2 .

We can now give an adaptive algorithm as promised. An adaptive algorithm Given θ∗ ∈ (0, 1) and the desired accuracy eps, we proceed as follows: ∗

• Step 1: Compute ε = 2C6µfeps −m,2 . • Step2: Determine an index set Λ ⊂ J such that C7 inf f − v −m,2 < ˜Λ v∈V

1 ∗ µ eps. 2

• Step 3: Compute the Galerkin solution uΛ with respect to VΛ .

14

Q. Guo and M.W. Wong • Step 4: Compute ηΛ,ε =

1/2 a2λ

.

λ∈NΛ,ε

If ηΛ,ε < eps, then we stop and accept uΛ as a solution. Otherwise, go to the next step. ˜ such that Λ ⊂ Λ ˜ ⊂ J and • Step 5: Determine an index set Λ 1/2 a2λ ≥ (1 − θ∗ )ηΛ,ε , ˜ λ∈Λ∩N Λ,ε

˜ and go to Step 3 with Λ replaced by Λ.

References [1] A. Cohen, Numerical Analysis of Wavelet Methods, North-Holland, 2003. [2] A. Dasgupta, Ellipticity of Fredholm pseudo-differential operators on Lp (Rn ), in New Developments of Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkh¨ auser, 2009, 107–116. [3] S. Dahlke, W. Dahmen, R. Hochmuth and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numerical Math. 23 (1997), 21–47. [4] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909–996. [5] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. [6] E. Schrohe, Boundedness and spectral invariance for standard pseudodifferential operators on anisotropically weighted Lp -Sobolev spaces, Integral Equations Operator Theory 13 (1990), 271–284. [7] M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999. [8] M.W. Wong, Weak and strong solutions for pseudo-differential operators, in Advances in Analysis, World Scientific, 2005, 275–284. Qiang Guo Department of Mathematics Syracuse University Syracuse NY 13244-1150, USA e-mail: [email protected] M.W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto Ontario M3J 1P3, Canada e-mail: [email protected]

Spectral Theory of Pseudo-Differential Operators on S1 Mohammad Pirhayati Abstract. For a bounded pseudo-differential operator with the dense domain C ∞ (S1 ) on Lp (S1 ), the minimal and maximal operator are introduced. An analogue of Agmon-Douglis-Nirenberg [1] is proved and then is used to prove the uniqueness of the closed extension of an elliptic pseudo-differential operator of symbol of positive order. We show the Fredholmness of the minimal operator. The essential spectra of pseudo-differential operators on S1 are described. Mathematics Subject Classification (2000). Primary 47G30. Keywords. Pseudo-differential operators, Sobolev spaces, Fredholmness, ellipticity, essential spectra, indices.

1. Introduction In this paper the focus is on pseudo-differential operators on the unit circle S1 centered at the origin. For −∞ < m < ∞, let S m (S1 × Z) be the set all functions σ in C ∞ (S1 × Z) such that for all nonnegative integers α and β there exists a positive constant Cα,β for which |(∂θα ∂nβ σ)(θ, n)| ≤ Cα,β (1 + |n|)m−β ,

θ ∈ [−π, π], n ∈ Z.

Let σ ∈ S m (S1 × Z), −∞ < m < ∞. Then we define the pseudo-differential operator Tσ on L1 (S1 ) by (Tσ f )(θ) = einθ σ(θ, n)(FS1 f )(n), θ ∈ [−π, π], n∈Z

where −1

(FS1 f )(n) = (2π)

π −π

e−inθ f (θ) dθ,

n ∈ Z.

Basic properties of pseudo-differential operators with symbols in S m (S1 × Z), −∞ < m < ∞, can be found in [2, 3, 4, 6, 10, 9]. The basic calculi for the L. Rodino et al. (eds.), Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, DOI 10.1007/978-3-0348-0049-5_2, © Springer Basel AG 2011

15

16

M. Pirhayati

product and the formal adjoint of pseudo-differential operators with symbols in S m (S1 × Z) can be found in [9]. A symbol σ in S m (S1 × Z), −∞ < m < ∞, is said to be elliptic if there exist positive constants C and R such that |σ(θ, n)| ≥ C(1 + |n|)m ,

|n| ≥ R,

θ ∈ [−π, π].

The following theorem gives a parametrix for an elliptic pseudo-differential operator with symbol in S m (S1 × Z), ∞ < m < −∞, see [9]. Theorem 1.1. Let σ ∈ S m (S1 × Z), −∞ < m < ∞ be elliptic. Then there exists a symbol τ ∈ S −m (S1 × Z) such that Tσ Tτ = I + K

and

Tτ Tσ = I + R,

where K and R are infinitely smoothing in the sense that they are pseudo-differential operators with symbols in ∩m∈R S m (S1 × Z). Similar results for the symbol class S m (Rn × Rn ) of the pseudo-differential operators on Rn have been studied for example in [15]. In Section 2, we recall Lp -Sobolev spaces H s,p , −∞ < s < ∞, 1 ≤ p ≤ ∞, and we give some of the results in [7]. Then in Section 3, we consider bounded pseudodifferential operators Tσ on Lp (S1 ), 1 < p < ∞ with dense domain C ∞ (S1 ). The smallest and largest closed extension of Tσ are provided. The analogue of AgmonDouglis-Nirenberg [1], is given to prove that for an elliptic symbol σ of positive order m, the corresponding pseudo-differential operator has a unique closed extension with domain H m,p on Lp (S1 ). In Section 4, we focus on Fredholmness of pseudodifferential operator and its essential spectrum. Results on the Fredholmness of pseudo-differential operators on Rn can be found in [16, 13]. By using Theorem 2.9 in [7], we see that the minimal operator of an elliptic pseudo-differential operator of positive order is Fredholm. The essential spectra of the pseudo-differential operator and the minimal (maximal) operator are then provided. Similar results for the SG Pseudo-differential operator on Rn are given in [5, 8].

2. Lp-Sobolev spaces For −∞ < s < ∞, let Js be the pseudo-differential operator with symbol σs given by σs (n) = (1 + |n|2 )−s/2 , n ∈ Z. Js is called the Bessel potential of order s. Now, for −∞ < s < ∞ and 1 ≤ p ≤ ∞, we define the Lp -Sobolev space H s,p to be the set of all tempered distributions u for which J−s u is a function in Lp (S1 ). Then H s,p is a Banach space in which the norm · s,p is given by

u s,p = J−s u Lp(S1 ) ,

u ∈ H s,p .

It is easy to show that for −∞ < s, t < ∞, Jt is an isometry of H s,p onto H s+t,p .

Spectral Theory of Pseudo-Differential Operators on S1

17

The following theorem is known as Sobolev embedding theorem. Theorem 2.1. Let 1 < p < ∞ and s ≤ t. Then H t,p ⊆ H s,p and

u s,p ≤ u t,p ,

u ∈ H t,p .

Proposition 2.2. Let σ ∈ S m (S1 × Z), −∞ < m < ∞. Then Tσ : H s,p → H s−m,p is a bounded linear operator for 1 < p < ∞. Proposition 2.3. Let s < t. Then the inclusion operator i : H t,p → H s,p is compact for 1 ≤ p ≤ ∞. The results above can be found in [7].

3. Minimal and maximal operators Let σ ∈ S m (S1 × Z), m ∈ R. Then the formal adjoint of Tσ , denoted Tσ∗ is a linear operator on C ∞ (S1 ) such that (Tσ ϕ, ψ) = (ϕ, Tσ∗ ψ),

ϕ, ψ ∈ C ∞ (S1 ).

It can be proved that the formal adjoint of Tσ is a pseudo-differential operator of symbol of order −m (see [10]). The following proposition guarantee that the minimal operator of Tσ exists. Proposition 3.1. Let S m (S1 × Z), −∞ < m < ∞. Then Tσ : Lp (S1 ) → Lp (S1 ) is closable with dense domain C ∞ (S1 ) for 1 < p < ∞. ∞ 1 Proof. Let {ϕk }∞ k=1 be a sequence in C (S ) such that ϕk → 0 and Tσ ϕk → f for p 1 some f in L (S ) as k → ∞. We only need to show that f = 0. We have

(Tσ ϕk , ψ) = (ϕk , Tσ∗ ψ),

ψ ∈ C ∞ (S1 ), k = 1, 2, . . . .

Let k → ∞, then (f, ψ) = 0 for all ψ ∈ C ∞ (S1 ). By the density of C ∞ (S1 ) in Lp (S1 ), it follows that f = 0. Consider Tσ : Lp (S1 ) → Lp (S1 ) with domain C ∞ (S1 ). Then by Proposition 3.1, Tσ has a closed extension. Let Tσ,0 be the minimal operator of Tσ which is the smallest closed extension of Tσ . Then the domain D(Tσ,0 ) of Tσ,0 consists of all ∞ 1 functions u ∈ Lp (S1 ) for which there exists a sequence {ϕk }∞ k=1 in C (S ) such p 1 p 1 p 1 that ϕk → u in L (S ) and Tσ ϕk → f for some f ∈ L (S ) in L (S ) as k → ∞. ∞ 1 It can be shown that f does not depend on the choice of {ϕk }∞ k=1 in C (S ) and Tσ,0 u = f . We define the linear operator Tσ,1 on Lp (S1 ) with domain D(Tσ,1 ) by the following. Let f and u be in Lp(S1 ). Then we say that u ∈ D(Tσ,1 ) and Tσ,1 u = f if and only if (u, Tσ∗ ϕ) = (f, ϕ), ϕ ∈ C ∞ (S1 ). It can be proved that Tσ,1 is a closed linear operator from Lp (S1 ) into Lp (S1 ) with domain D(Tσ,1 ) containing C ∞ (S1 ). In fact, C ∞ (S1 ) is contained in the domain t t D(Tσ,1 ) of the true adjoint Tσ,1 of Tσ,1 . Furthermore, Tσ,1 (u) = Tσ (u) for all u in D(Tσ,1 ).

18

M. Pirhayati

It is easy to see that Tσ,1 is an extension of Tσ,0 . In fact Tσ,1 is the largest closed extension of Tσ in the sense that if B is any closed extension of Tσ such that C ∞ (S1 ) ⊆ D(B t ), then Tσ,1 is an extension of B. Tσ,1 is called the maximal operator of Tσ . The following theorem is an analogue of Agmon-Douglis-Nirenberg in [1]. Proposition 3.2. Let σ ∈ S m (S1 × Z), m > 0 be elliptic. Then there exist positive constants C and D > 0 such that C u m,p ≤ Tσ u Lp(S1 ) + u Lp(S1 ) ≤ D u m,p ,

u ∈ H m,p .

Proof. By the boundedness of Tσ in Proposition 2.2 and the boundedness of the inclusion operator in Theorem 2.1, there exists a positive constant D such that for all u ∈ H m,p ,

Tσ u Lp (S1 ) + u Lp (S1 ) ≤ D u m,p ,

u ∈ H m,p .

Since σ ∈ S m (S1 × Z) is elliptic, by Theorem 1.1, there exists a symbol τ ∈ S −m (S1 × Z) such that u = Tτ Tσ u − Ru,

u ∈ H m,p ,

where R is an infinitely smoothing operator in the sense that R is a pseudodifferential operator with symbol in ∩m∈R S m (S1 × Z). By using Proposition 2.2 again, Tσ u ∈ Lp (S1 ). Therefore, Tτ Tσ u ∈ H m,p , for all u ∈ H m,p , Moreover there exists a positive constant C such that

u m,p ≤ C( Tσ u Lp (S1 ) + u Lp(S1 ) ),

u ∈ H m,p .

We have the following result which we use in the next theorem. Lemma 3.3. Let s ∈ R and 1 < p < ∞. Then C ∞ (S1 ) is dense in H s,p . Proof. Let u ∈ H s,p . Then J−s u ∈ Lp (S1 ). Since C ∞ (S1 ) is dense in Lp (S1 ), there ∞ 1 p 1 exists a sequence {ϕk }∞ k=1 in C (S ) such that ϕk → J−s u in L (S ) as k → ∞. ∞ 1 Let ψk = Js ϕk , k = 1, 2, . . . . Then ψk ∈ C (S ), k = 1, 2, . . . , and

ψk − u s,p

= J−s ψk − J−s u Lp (S1 ) = ϕk − J−s u Lp (S1 ) → 0,

as k → ∞, which completes the proof.

The following theorem gives the domain of the minimal operator of an elliptic pseudo-differential operator with symbol of positive order. Theorem 3.4. Let σ ∈ S m (S1 × Z), m > 0, be elliptic. Then D(Tσ,0 ) = H m,p . Proof. Let u ∈ H m,p . Then by using the density of C ∞ (S1 ) in H m,p , there exists ∞ 1 m,p a sequence {ϕk }∞ and therefore in Lp (S1 ) k=1 in C (S ) such that ϕk → u in H as k → ∞. By Proposition 3.2, ϕk and Tσ ϕk are Cauchy sequences in Lp (S1 ). Therefore ϕk → u and Tσ ϕk → f for some f in Lp (S1 ) as k → ∞. This implies that u ∈ D(Tσ,0 ) and Tσ,0 u = f . Now assume that u ∈ D(Tσ,0 ). Then there exists ∞ 1 p 1 a sequence {ϕk }∞ k=1 in C (S ) such that ϕk → u in L (S ) and Tσ ϕk → f , for p 1 some f ∈ L (S ) as k → ∞. So, by Proposition 3.2, {ϕk }∞ k=1 is a Cauchy sequence


19

in H m,p . Since H m,p is complete, there exists v ∈ H m,p such that ϕk → v in H m,p as k → ∞. By Sobolev embedding theorem ϕk → v in Lp (S1 ) which implies that u = v ∈ H m,p . The following theorem shows that the closed extension of an elliptic pseudodifferential operator on Lp (S1 ) with symbol σ ∈ S m (S1 × Z), m > 0, is unique and moreover by Theorem 3.4, its domain is H m,p . Theorem 3.5. Let σ ∈ S m (S1 × Z), m > 0, be elliptic. Then Tσ,0 = Tσ,1 . Proof. Since Tσ,1 is a closed extension of Tσ,0 , by Theorem 3.4, it is enough to show that D(Tσ,1 ) ⊆ H m,p . Let u ∈ D(Tσ,1 ). By ellipticity of σ, there exists τ ∈ S −m (S1 × Z) such that u = Tτ Tσ u − Ru, where R is an infinitely smoothing operator. Since Tσ u = Tσ,1 u ∈ Lp(S1 ), by Proposition 2.2, it follows that u ∈ H m,p , which completes the proof.

4. Fredholm pseudo-differential operators A closed linear operator A from a complex Banach space X into a complex Banach space Y with dense domain D(A) is said to be Fredholm if • the range of A, R(A) is closed subspace of Y and • the null space of A, N (A) and the null space of the true adjoint of A, N (At ) are finite dimensional. The index of a Fredholm operator A is defined by i(A) = dim N (A) − dim N (At ) By Atkinson’s theorem, a closed linear operator A : X → Y with dense domain D(A) is Fredholm if and only if there exists a bounded linear operator B : Y → X such that K1 = AB − I : Y → Y and K2 = BA − I : X → X are compact operators. Let A : X → X be a closed linear operator with dense domain D(A) in the complex Banach space X. Then the spectrum of A, Σ(A) is defined by Σ(A) = C − ρ(A), where ρ(A) is the resolvent set of A given by ρ(A) = {λ ∈ C : A − λI is bijective}. The essential spectrum Σw (A) of A, which has been defined in [14] by Wolf given by Σw (A) = C − Φw (A), where Φw (A) = {λ ∈ C : A − λI is Fredholm}. Note that i(A − λI) is constant for all λ in a connected component of Φw (A). The essential spectrum Σs (A) of A in sense of Schechter [11] is defined by Σs (A) = C − Φs (A), where Φs (A) = {λ ∈ Φw (A) : i(A − λI) = 0}.

20

M. Pirhayati

For the properties of essential spectra see [12]. The following theorem gives a sufficient condition for Tσ : H s,p → H s−m,p to be a Fredholm operator. The proof can be found in [7]. Theorem 4.1. Let σ ∈ S m (S1 × Z), −∞ < m < ∞ be elliptic. Then for all −∞ < s < ∞ and 1 < p < ∞, Tσ : H s,p → H s−m,p is a Fredholm operator. In particular if σ ∈ S 0 (S1 ×Z), then the bounded linear operator Tσ : Lp (S1 ) → Lp (S1 ) is Fredholm. The following is an immediate corollary of Theorem 3.4 and Theorem 4.1. Corollary 4.2. Let σ ∈ S m (S1 × Z), m > 0 be elliptic. Then for 1 < p < ∞, Tσ,0 is a Fredholm operator on Lp (S1 ) with the domain H m,p . The following theorem gives the essential spectrum of an elliptic pseudodifferential operator of positive order. Theorem 4.3. Let σ ∈ S m (S1 × Z), m > 0 be elliptic. Then Σw (Tσ,0 ) = ∅. Proof. Let λ ∈ C. By Corollary 4.2, we need only to show that σ − λ is elliptic. The ellipticity of σ, implies that there exist constants C, R > 0 such that |λ| |σ(θ, n) − λ| ≥ C(1 + |n|)m − |λ| = (1 + |n|)m (C − ), θ ∈ [−π, π], (1 + |n|)m whenever |n| ≥ R. Since (1 + |n|)m → ∞ as |n| → ∞, there exists M > 0 such that C |σ(θ, n) − λ| ≥ (1 + |n|)m , |n| ≥ M, θ ∈ [−π, π], 2 which implies that σ − λ is elliptic. Let σ ∈ S m (S1 × Z), m ≥ 0. Then the following theorem is a result on the essential spectra of the bounded pseudo-differential operator Tσ with the domain H m,p on Lp (S1 ). Theorem 4.4. Let σ ∈ S m (S1 × Z), m ≥ 0. Then for Tσ on Lp(S1 ) with the domain H m,p , 1 < p < ∞, we have Σw (Tσ ) ⊆ {λ ∈ C : |λ| ≥ Li }, where Li = lim inf {( |n|→∞

inf θ∈[−π,π]

|σ(θ, n)|)(1 + |n|)−m }.

Proof. Let λ ∈ C be such that |λ| < Li . Then there exists > 0 such that |λ| + < Li . Since m ≥ 0, it follows that |λ| < (Li − )(1 + |n|)m . On the other hand, there exists a positive constant R such that inf {( inf |σ(n, θ)|)(1 + |n|)−m } > Li − . 2 |n|≥R θ∈[−π,π]


21

So, for |n| ≥ R, |σ(θ, n) − λ| ≥ |σ(θ, n)| − |λ| > (Li − − Li + )(1 + |n|)m 2 = (1 + |n|)m , θ ∈ [−π, π]. 2 Therefore, σ − λ is elliptic and hence Tσ − λI : Lp (S1 ) → Lp (S1 ) with domain H m,p is Fredholm. Thus, {λ ∈ C : |λ| < Li } ⊆ Φw (Tσ ), which implies that Σw (Tσ ) ⊆ {λ ∈ C : |λ| ≥ Li }.

We have the following theorem on the essential spectrum of a pseudo-differential operator of order 0 from Lp (S1 ) into Lp (S1 ). Theorem 4.5. Let σ ∈ S 0 (S1 × Z). Then for Tσ : Lp (S1 ) → Lp (S1 ), 1 < p < ∞, we have Σs (Tσ ) ⊆ {λ : |λ| ≤ Ls }, where Ls = lim sup{ sup |σ(θ, n)|}. |n|→∞

θ∈[−π,π]

Proof. Let λ ∈ C such that |λ| > Ls . Then there exists > 0 such that |λ| − > Ls , and there exists a positive number R such that |σ(θ, n)|} < Ls + . 2 θ∈[−π,π]

sup { sup

|n|≥R

For all |n| ≥ R, |σ(θ, n) − λ| ≥ |λ| − |σ(θ, n)| > L s + − Ls −

2

, θ ∈ [−π, π]. 2 Hence σ −λ is elliptic and by Theorem 4.1, Tσ −λI : Lp (S1 ) → Lp (S1 ) is Fredholm. Thus, {λ ∈ C : |λ| > Ls } ⊆ Φw (Tσ ), which is the same as Σw (Tσ ) ⊆ {λ ∈ C : |λ| ≤ Ls }. Since {λ ∈ C : |λ| > Ls } is a connected component of Φw (Tσ ), it follows that i(Tσ − λI) is a constant for all λ in {λ ∈ C : |λ| > Ls }. On the other hand, =

ρ(Tσ ) ∩ {λ ∈ C : |λ| > Ls } = ∅.

22

M. Pirhayati

Therefore, i(Tσ − λI) = 0 for all {λ ∈ C : |λ| > Ls }. This implies that Σs (Tσ ) ⊆ {λ ∈ C : |λ| ≤ Ls }.

We have the following spectral alternative for a pseudo-differential operator with symbol in S 0 (S1 × Z). Corollary 4.6. Let σ ∈ S 0 (S1 × Z) be such that lim sup( sup |n|→∞ θ∈[−π,π]

|σ(θ, n)|) = lim inf (

inf

|n|→∞ θ∈[−π,π]

|σ(θ, n)|) = L > 0.

Then Σw (Tσ ) = {λ ∈ C : |λ| = L}

or

Σs (Tσ ) ⊆ {λ ∈ C : |λ| = L}.

Proof. By Theorem 4.4 and Theorem 4.5, Σw (Tσ ) ⊆ {λ ∈ C : |λ| = L}. Suppose that Σw (Tσ ) = {λ ∈ C : |λ| = L}. Then there exists λ0 ∈ C such that |λ0 | = L and λ0 ∈ Φw (Tσ ). On the other hand, by Theorem 4.5, {λ ∈ C : |λ| > L} ⊆ Φs (Tσ ). Hence using the fact that Φw (Tσ ) is an open set and the index of Tσ − λI is constant on on every connected component of Φw (Tσ ) we get i(Tσ − λI) = 0 for all λ ∈ C with |λ| = L, which is the same as Σs (Tσ ) ⊆ {λ ∈ C : |λ| = L}, as asserted.

References [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623–727. [2] M.S. Agranovich, Spectral properties of elliptic pseudodifferential operators on a closed curve, Funktsional. Anal. i Prilozhen. 13 (1979), 54–56 (in Russian). [3] M.S. Agranovich, Elliptic pseudodifferential operators on a closed curve, Trudy Moskov. Mat. Obshch. 47 (1984), 22–67, 246 (in Russian); Trans. Moscow Math. Soc. (1985), 23–74. [4] B.A. Amosov, On the theory of pseudodifferential operators on the circle, Uspekhi Mat. Nauk. 43 (1988), 169–170 (in Russian); Russian Math. Surveys 43 (1988), 197– 198. [5] A. Dasgupta and M.W. Wong, Spectral theory of SG pseudo-differential operators on Lp (Rn ), Studia Math. 187 (2008), 185–197. [6] S. Molahajloo and M.W. Wong, Pseudo-differential operators on S1 , in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkh¨ auser, 2008, 297–306.


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[7] S. Molahajloo and M.W. Wong, Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on S1 , J. Pseudo-Differ. Oper. Appl. 1 (2010), 183–205. [8] F. Nicola and L. Rodino, SG pseudo-differential operators and weak hyperbolicity, Pliska Stud. Math. Bulgar. 15 (2002), 5–19. [9] M. Ruzhansky and V. Turunen, On the Fourier analysis of operators on the torus, in Modern Trends in Pseudo-Differential Operators, Operator Theory: Advances and Applications 172, Birkh¨ auser, 2007, 87–105. [10] M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries, Birkh¨ auser, 2009. [11] M. Schechter, On the essential spectrum of an arbitrary operator I, J. Math. Anal. Appl. 13 (1966), 205–215. [12] M. Schechter, Spectra of Partial Differential Operators, Second Edition, NorthHolland, 1986. [13] E. Schrohe, Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces, Ann. Global Anal. Geom. 10 (1992), 237–254. [14] F. Wolf, On essential spectrum of partial differential boundary problems, Comm. Pure Appl. Math. 12 (1959), 211–228. [15] M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999. [16] M.W. Wong, Fredholm pseudo-differential operators on weighted Sobolev spaces, Ark. Mat. 21 (1983), 271–282. Mohammad Pirhayati Department of Computer Science Islamic Azad University Malayer Branch Seyfie Park Malayer, Iran e-mail: [email protected]

A Characterization of Compact Pseudo-Differential Operators on S1 Shahla Molahajloo Abstract. We first prove that a pseudo-differential operator of symbol of order 0 is essentially normal. Then by using Gohber’s lemma and a result from [6], a necessary and sufficient condition for compactness of pseudo-differential operators on the unit circle is given. Mathematics Subject Classification (2000). Primary 47G30. Keywords. Pseudo-differential operators, compact operators, Calkin algebra, Fredholmness, essential spectra.

1. Introduction For −∞ < m < ∞, let S m (S1 × Z) be the set all functions σ in C ∞ (S1 × Z) such that for all nonnegative integers α and β there exists a positive constant Cα,β for which |(∂θα ∂nβ σ)(θ, n)| ≤ Cα,β (1 + |n|)m−β , θ ∈ [−π, π], n ∈ Z. Let σ ∈ S m (S1 × Z), −∞ < m < ∞. Then we define the pseudo-differential operator Tσ on L1 (S1 ) by (Tσ f )(θ) = einθ σ(θ, n)(FS1 f )(n), θ ∈ [−π, π], n∈Z

where (FS1 f )(n) = (2π)−1

π −π

e−inθ f (θ) dθ,

n ∈ Z.

Theorem 1.1. Let σ ∈ S m1 (S1 × Z) and τ ∈ S m2 (S1 × Z), −∞ < m1 , m2 < ∞. Then Tσ Tτ = Tλ , where (−i)µ λ∼ (∂nµ σ)(∂θµ τ ), µ! µ L. Rodino et al. (eds.), Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, DOI 10.1007/978-3-0348-0049-5_3, © Springer Basel AG 2011

25

26

S. Molahajloo

i.e., λ−

(−i)µ (∂nµ σ)(∂θµ τ ) ∈ S m1 +m2 −N µ!

µ C, µ µ where C is a positive constant. We write S(cl) (· · · ) for both S µ (· · · ) and Scl (· · · ). µ In particular, S(cl) (Rn ) denotes the class of symbols that are independent of x. The space Lµ(cl) (U ) of pseudo-differential operators of order µ on an open subset U of Rn is given by µ Lµ(cl) (U ) = {Op(a) : a ∈ S(cl) (U × U × Rn )},

where for all u ∈ C0∞ (U ),

ei(x−y)·ξ a(x, y, ξ)u(y) dy d¯ξ,

(Op(a)u)(x) = Rn

Rn

x ∈ Rn .

36

B.-W. Schulze and M.W. Wong

Here, d¯ξ = (2π)−n dξ and the double integral is interpreted as an oscillatory integral. Op(a) is a continuous operator from C0∞ (U ) into C ∞ (U ) and then can be extended to a continuous operator between compact and local versions of Sobolev spaces on U. We also consider pseudo-differential operators on a C ∞ manifold X equipped with a Riemannian metric and an associated measure dx. The corresponding space of operators of order µ is denoted by Lµ(cl) (X). The space L−∞ (X) of smoothing operators given by L−∞ (X) = Lµ (X) µ∈R ∞

can be identified with C (X × X) via L−∞ (X) C ↔ c ∈ C ∞ (X × X),

where (Cu)(x) =

c(x, y)u(y) dy,

x ∈ X,

X

for all u in C0∞ (X) and the function c is known as the kernel of the operator C. The space Lµ(cl) (X; Rl ) of parameter-dependent pseudo-differential operators with parameter λ ∈ Rl is built up with symbols µ a(x, y, ξ, λ) ∈ S(cl) (U × U × Rn+l ),

where U is a local chart of X. The space L−∞ (X; Rl ) of smoothing parameterdependent pseudo-differential operators with parameter λ ∈ Rl is defined by L−∞ (X; Rl ) = S(Rl , L−∞ (X)), where L−∞ (X) is endowed with the Fréchet topology from the identification with C ∞ (X ×X). We often use the fact that a space of symbols and a space of operators are Fréchet in a natural way. Let us now turn to the behavior of pseudo-differential operators near a singularity. The simplest case of a singularity is a smooth boundary, and we are therefore in the realm of the calculus of pseudo-differential boundary value problems. An aspect of the regularity of solutions to classical elliptic boundary value problems, e.g., the Dirichlet or Neumann problem for the Laplacian, is the smoothness up to the boundary when the right-hand sides and the boundary data are smooth. In this case we have Taylor asymptotics in the normal variable r ∈ R+ to the boundary as r → 0. So, pj = −j and mj = 0 for all j = 0, 1, . . . . However, this is true only for a very exceptional case, namely, the case when the operators have the transmission property at the boundary. See Boutet de Monvel [3] for more details and Schulze [44] for a new presentation of this calculus. There is also introduced the notion of the anti-transmission property, and in such a case we do not have Taylor asymptotics of solutions. A wellknown example in this category is the Dirichlet-to-Neumann operator with |ξ| as

Mellin Operators with Asymptotics on Manifolds with Corners

37

the homogeneous principal symbol up to some constant. More generally, pseudodifferential symbols a(x, ξ) which are smooth up to the boundary generically do not have the transmission property, as illustrated later by simple examples. Boundary value problems with such symbols have been studied systematically by Vishik and Eskin [51] and Eskin in his book [12]. See also Rempel and Schulze [31]. In this paper, we do not focus on this kind of problems, but recall only a few notions to introduce notation around the Mellin transform. The Mellin transform M is defined by ∞ (M u)(z) = rz−1 u(r)dr, 0

first for u ∈ C ∞ (R+ ) and z ∈ C whenever the integral exists and then later on for more general distributions. It can be shown that Mγ : rγ L2 (R+ ) → L2 (Γ1/2−γ ) is an isomorphism, where Mγ is the weighted Mellin transform given by (Mγ u)(z) = (M (r−γ u))(z + γ),

z ∈ Γ1/2−γ ,

for all u ∈ rγ L2 (R+ ). It can also be shown that the inverse Mγ−1 : L2 (Γ1/2−γ ) → rγ L2 (R+ ) is given by (Mγ−1 g)(r)

r−z g(z) d¯z,

=

r ∈ R+ ,

Γ1/2−γ

for all g in L2 (Γ1/2−γ ), where d¯z = (2πi)−1 dz. Since (M −1 zM u)(r) = −r(∂r u)(r),

r ∈ R+ ,

∞

for suitable functions u in C (R+ ), we can define the weighted Mellin pseudodifferential operator opγM (f ) in the R+ -direction corresponding to the Mellin symbol f (r, s, z) ∈ S µ (R+ × R+ × Γ1/2−γ ), which means that as a function of (r, s, ρ) ∈ R+ × R+ × R, f (r, s, 1/2 − γ + iρ) ∈ S µ (R+ × R+ × R). To wit, the action of opγM (f ) is given by ∞ ∞ − 1 −γ+iρ ) r (2 1 ds (opγM (f )u)(r) = f r, s, − γ + iρ u(s) d¯ρ s 2 s −∞ 0 for all r in R+ and all u in C0∞ (R+ ), where d¯ρ = (2π)−1 dρ. We often write opM (f ) = op0M (f ). For −∞ < s < ∞, let f s be the Mellin symbol defined by f s (z) = (1 + |z|2 )s/2 ,

z ∈ Γ1/2−γ .

(2.1)

38


Then we can define the weighted Mellin Sobolev space Hs,γ (R+ ) simply to be the completion of C0∞ (R+ ) with respect to the norm Hs,γ (R+ ) given by

u Hs,γ (R+ ) = opM (f s )(r−γ u) L2 (R+ ) ,

u ∈ C0∞ (R+ ).

(2.2)

By a cut-off function on the half-line R+ , we understand any real-valued function ω ∈ C0∞ (R+ ) such that ω is identically equal to 1 close to r = 0. For −∞ < s, γ < ∞, we also need in this paper the weighted cone space Ks,γ (R+ ) defined by Ks,γ (R+ ) = {ωu + (1 − ω)v : u ∈ Hs,γ (R+ ), v ∈ H s (R+ )},

(2.3)

where ω is any cut-off function on R+ . It can be shown that Ks,γ (R+ ) is well defined in the sense that the definition is independent of the choice of ω. The choice of the letter K is in deference to the German word “Kegel” for cone. Let a ∈ S µ (R+ × R), µ ∈ R. Then the pseudo-differential operator Op(f ) based on the Fourier transform is sometimes denoted by a(r, Dr ). Let f (r, z) ∈ S µ (R+ × Γ1/2−γ ) be a Mellin symbol. We write for simplicity f (r, ρ) = f (r, 1/2 − γ + iρ),

r ∈ R+ , ρ ∈ R.

Then the weighted Mellin pseudo-differential operator opγM (f ) in the R+ -direction has the same meaning as the Fuchs type operator f (r, −r∂r ). The motivation for passing from the Fourier pseudo-differential operator to the Mellin pseudodifferential operator in the normal direction is to control the calculus up to r = 0. However, we lose the form a(r, Dr ) and get instead f (r, −r∂r ), which appears unwelcome at a first glance. The good news though is that the Fourier representation can be transformed to the Mellin representation to the effect that r+a(r, Dr )e+ = f (r, −r∂r ) modulo a smoothing remainder, which can be controlled precisely up to r = 0. Here, e+ is the operator of extension by zero from R+ to R and r+ is the operator of restriction to R+ . Recall that a → r+ a(r, Dr )e+ is the quantization in Boutet de Monvel’s calculus, but if the symbol a has the transmission property at r = 0, then there is no need to reformulate the action via the Mellin transform. For symbols without the transmission property, the use of the Mellin formalism seems to be most natural. To illustrate the transformation in the general case, let us briefly look at oper0 ators with symbols a(ρ) ∈ Scl (Rρ ) of order 0, studied in Eskin [12], on L2 (R+ ) with Hilbert-Schmidt remainders. Generalizations to arbitrary order and to weighted cone spaces Ks,γ (R+ ) with smoothing Green remainders can be found in [32] and [39]. In this connection, the closed half-axis has the interpretation of an infinite cone with r = 0 as the conical singularity and r → ∞ as a conical exit to infinity.


39

0 A symbol a(ρ) ∈ Scl (Rρ ) has an asymptotic expansion

a(ρ) ∼

∞

−j a± j (iρ)

(2.4)

j=0

as ρ → ±∞, where a± j ∈ C, j = 0, 1, . . . . (The imaginary unit i before ρ in (2.4) is only for technical convenience.) Clearly the coefficients are uniquely determined ∞ by a, and for any choice of {a± j }j=0 , there is a symbol a with these coefficients. The transmission property and the anti-transmission property of a are then characterized by, respectively, the conditions − a+ j = aj

− and a+ j = −aj

for j = 0, 1, . . . . See [39] or [44]. The Mellin formalism is particularly simple for the operator r+ Opr (θ± )e+ = r+ Opr (χθ± )e+ mod L−∞ (R+ )∩L(L2 (R+ )), where θ± (ρ) is the characteristic function on R±,ρ and χ(ρ) is an excision function, i.e., χ(ρ) is a real-valued function which is identically equal to 0 on a neighborhood of ρ = 0 and is identically equal to 1 for large ρ. Then as has been shown in [12], r+ Opr (θ ± )e+ = opM (g ± ),

(2.5)

where

g ± (z) = (1 − e∓2πiz )−1 , z ∈ Γ1/2 . 0 Moreover, let us recall from [12] that for a(ρ) ∈ Scl (R), the operator A = r+ Opr (a)e+ : L2 (R+ ) → L2 (R+ )

(2.6)

is Fredholm if and only if a is elliptic, i.e., a(ρ) = 0,

ρ ∈ R,

a± 0

= 0 and the conormal symbol σc (A) of A given by − − + σc (A)(z) = a+ 0 g (z) + a0 g (z)

(2.7)

is nonzero for all z ∈ Γ1/2 . In addition, the index ind(A) of A can be computed and is given by z=1/2+∞ ind(A) = (2π)−1 (arg a)(ρ)|ρ=+∞ (2.8) ρ=−∞ + (arg σc (A))(z)|z=1/2−∞ . The conormal symbol σc (A) of A is a meromorphic function on C with simple poles at the integers, and − + − σc (A)(z) = 0 ⇔ z = (2πi)−1 {log |a+ 0 /a0 | + i arg (a0 /a0 ) + k}, ∞

k ∈ Z,

(2.9)

when a is elliptic. Finally, if χ is a function in C (C) such that χ(z) is identically equal to 0 for |Im z| < ε0 and χ(z) is identically equal to 1 for |Im z| > ε1 , where 0 < ε0 < ε1 , then 0 χ(z)σc (A)(z)|Γβ ∈ Scl (Γβ )

40


uniformly with respect to β on compact subsets of R. (Uniformity means that we can find a positive constant C such that for all β in a compact subset of R, χ(β + iρ)σc (A)(β + iρ) has an asymptotic expansion into symbols of order −j, j ∈ N, in ρ for |ρ| ≥ C.) The subalgebra A of L(L2 (R+ )) generated by 0 {A = r+ Opr (a)e+ : a ∈ Scl (R)}

is studied in [12]. In particular, compositions give rise to products in A modulo smoothing Mellin operators with meromorphic symbols. In [39] it is shown that A is a subalgebra of the cone algebra on R+ . See also [38]. The function σc (A) is a special Mellin symbol of the cone algebra with asymptotics. The cone in this case is the closed half-axis. In general, our cones have a non-trivial base X, e.g., a smooth compact manifold. The Mellin symbols in these cases are operator valued. Now, let U be an open subset of the complex plane C and let E be a Fréchet space. We denote by A(U, E) the space of all holomorphic functions on U with values in E. It is endowed with the natural Fréchet topology of uniform convergence on compact subsets of U . Definition 2.1. (i) A discrete Mellin asymptotic type is a sequence R = {(rj , nj )}j∈Z ⊆ C × N with |Re rj | → ∞ as |j| → ∞. (ii) Let πC R = {rj }j∈Z . Then a smoothing Mellin symbol f with discrete Mellin asymptotic type R is a meromorphic function f in A(C\ πC R, L−∞ (X)), where for each j ∈ Z, rj is a pole of order nj + 1, and the Laurent coefficient of (z −rj )−(k+1) is a finite-rank operator in L−∞ (X), 0 ≤ k ≤ nj , j ∈ Z. Moreover, if χ is any πC Rexcision function, i.e., χ ∈ C ∞ (C), χ(z) = 0 for dist (z, πC R) < ε0 , and χ(z) = 1 for dist (z, πC R) > ε1 , where 0 < ε0 < ε1 , then χf |Γβ ∈ S(Γβ , L−∞ (X)) uniformly with respect to β on compact subsets of R. (As usual, the line Γβ is treated as the real line R and the Schwartz space S(Γβ , L−∞ (X)) is to be taken as S(R, L−∞ (X)) with L−∞ (X) identified as C ∞ (X × X). Furthermore, we assume that the seminorms in S(Γβ , L−∞ (X)) are uniformly bounded with respect to β on compact subsets of R.) We denote the space of all smoothing Mellin symbols with discrete Mellin asymptotic type R by MR−∞ (X). µ (iii) MO (X) is the space of all h in A(C, Lµcl (X)) such that for all z in Γβ ,

h(z) ∈ Lµcl (X; Γβ ) uniformly with respect to β on compact subsets of R. (Uniformity here means that the family h(β + iρ) is bounded in the Fréchet topology of Lµcl (X; Rρ ) when β varies over compact subsets of R.)


41

(iv) Let µ ∈ R. Then MRµ (X) is the space of all Mellin symbols of order µ with discrete Mellin asymptotic type R defined by µ MRµ (X) = MO (X) + MR−∞ (X).

(2.10)

The sum in (2.10) is not direct. In the case when dim X = 0, we write MRµ for MRµ (X). As has been noted before, we have σc (A) ∈ MR0 , where R is the discrete Mellin asymptotic type defined by rj = j

and nj = 0

for all j in Z. Moreover, the conormal symbol σc (A) of the operator (1.2) given by σc (A)(z) =

µ

aj (0)z j ,

z ∈ C,

j=0 µ belongs to MO (X). A special case is the operator ∆X − z 2 for the Laplacian ∆X associated with a Riemannian metric on X. The analysis of the spaces MRµ (X), −∞ ≤ µ < ∞, is beautiful. Let us mention here a few results to motivate analogous results when X is replaced by a compact manifold B with edge. Details can be found in [38] or [42]. µ ˜ Proposition 2.2. Let f ∈ MRµ (X) and f˜ ∈ MR ˜ , and R and ˜ (X), where µ and µ ˜ R are, respectively, the orders and the discrete Mellin asymptotic types. Then the composition f f˜ is a Mellin symbol in MSµ+˜µ (X) of order µ + µ ˜ for some discrete Mellin asymptotic type S. µ Theorem 2.3. Let f ∈ MO (X) be such that

f (z) ∈ Lµ−1 cl (X, Γβ ), for some fixed β in R. Then

z ∈ Γβ ,

µ−1 f ∈ MO (X).

Definition 2.4. A symbol f in MRµ (X) is said to be elliptic if for some β ∈ R with πC R ∩ Γβ = ∅, f (z) ∈ Lµcl (X, Γβ ) is parameter-dependent elliptic for all z in Γβ . By virtue of Theorem 2.3, the condition for ellipticity in Definition 2.4 is independent of the choice of β. Clearly for dim X = 0, the condition of Definition 2.4 is the same as the ellipticity of the corresponding symbol on the line Γβ . Therefore the conormal symbol σc (A) given by − − + σc (A)(z) = a+ 0 g (z) + a0 g (z),

z ∈ Γ1/2 ,

occurring in (2.7) is elliptic as soon as the symbol a(ρ) is elliptic. Other examples of operators (1.2) with rµ σψ (0, x, r−1 ρ, ξ) = 0 µ for all nonzero (ρ, ξ) and conormal symbols σc (A) in MO (X) can be constructed. (See the definition of σψ (A) in the Introduction. In connection with the stratifications of singular manifolds to be studied later, the notation σ0 is used instead σψ .)

42


Theorem 2.5. Let f ∈ MRµ (X) be an elliptic Mellin symbol of order µ with discrete Mellin asymptotic type R. Then there is a Mellin symbol f −1 ∈ MS−µ (X) of order −µ with another discrete Mellin asymptotic type S which is the (two-sided ) inverse of f in the sense of the above-mentioned multiplication. Theorem 2.6. Let f (z) ∈ Lµcl (X, Γβ ), z ∈ Γβ , for some β ∈ R. Then there exists µ an element h in MO (X) such that h|Γβ = f

(2.11)

mod L−∞ (X; Γβ ). Theorem 2.6 shows two essential things. First, there are non-trivial elements µ in the space MO (X) for any µ ∈ R. Moreover, if f (z) is parameter-dependent elliptic, it follows that h(z) is elliptic in the above-mentioned sense. Another conµ sequence of Theorem 2.6 is that for any f in MR (X), we can use (2.10) to obtain a decomposition f = h + l, µ −∞ where h ∈ MO (X) and l ∈ MR (X) by applying Theorem 2.6 to f |Γβ for any β such that πC R ∩ Γβ = ∅. Theorem 2.6 is a result on a kernel cut-off method developed in [38] and then employed systematically in formulating the cone and edge algebras. An alternative proof can be found in [23]. µ Proposition 2.7. Let h ∈ MO (X) be elliptic. Then for −∞ < s < ∞,

h(z) : H s (X) → H s−µ (X),

z ∈ C,

(2.12)

is a holomorphic family of Fredholm operators with zero index. The set D of all z ∈ C where (2.6) is not bijective is discrete, and D ∩ {z ∈ C : c ≤ Re z ≤ c } is a finite set for all c and c with c ≤ c . Let us now formulate the asymptotic part of the cone algebra on the open stretched cone X ∧ = R+ × X, where X is a closed and compact C ∞ manifold of dimension n. A discrete asymptotic type P associated with weight data (γ, Θ) for a weight γ ∈ R and a weight interval Θ = (ϑ, 0] ⊆ R− ∪ {−∞} is a sequence P = {(pj , mj )}Jj=0 ⊆ C × N, where J ∈ N ∪ {+∞} is finite when ϑ > −∞, πC P = {pj }Jj=0 ⊂ {(n + 1)/2 − γ + ϑ < Re z < (n + 1)/2 − γ} and Re pj → −∞ as j → ∞ when J = +∞.

(2.13)


43

If ϑ is a finite weight, then we consider the space EP (X ∧ ) of singular functions with discrete asymptotic type P defined by   mj J   EP (X ∧ ) = ωcjk r−pj logk r : cjk ∈ C ∞ (X), 0 ≤ k ≤ mj , 0 ≤ j ≤ J ,   j=0 k=0

(2.14) where ω is a cut-off function on R+ . The space (2.14) is Fréchet in a natural way. Let AP (X) denote the space of all C ∞ (X)-valued meromorphic functions with poles at the points pj of order mj + 1, j = 0, 1, . . . , J. Let C be any smooth and closed curve in the complex plane oriented once in the counterclockwise direction and enclosing the set πC P in its interior. Then for all f in AP (X), we define the linear functional ζf on the space of all holomorphic functions A(C) on C by A(C) h → ζf , h = f (z)h(z) d¯z ∈ C. (2.15) C

ζf is an analytic functional on the complex plane carried by the compact set πC P . The analytic functional ζf is nothing but a linear combination of derivatives of order ≤ mj of Dirac distributions at the points pj with coefficients in C ∞ (X). It can be proved easily that the range of EP (X ∧ ) under the weighted Mellin transform Mδ , i.e., {(Mδ,r→z u)(z) : u(r) ∈ EP (X ∧ )} (2.16) for any δ ∈ R with Re pj < 1/2 − δ,

0 ≤ j ≤ J,

is a subspace of AP (X), and that the inverse is given by Mδ (EP (X ∧ )) f (z) → u(r) = ω(r)ζf , r−z ∈ EP (X ∧ ).

(2.17) ∧

Note that if χ is any πC P -excision function, then for every f ∈ Mδ (EP (X )), χf |Γβ ∈ S(Γβ , C ∞ (X)) uniformly with respect to β on compact subsets of R. (Uniformity here has the same meaning as that in Definition 2.1 (ii).) Let us now recall the definition of weighted Sobolev spaces on the infinite open stretched cone X ∧ . First, for −∞ < s < ∞, let f s (z) be any element in Lscl (X; Γ1/2 ) which is parameter-dependent elliptic and represents a bijective family of operators f s (z) : H s (X) → L2 (X), z ∈ Γ1/2 . Then we define the weighted Sobolev space Hs,γ (X ∧ ) to be the completion of C0∞ (R+ , C ∞ (X)) with respect to the norm Hs,γ (X ∧ ) given by

u Hs,γ (X ∧ ) = opM (f s )(r−γ+n/2 u) L2 (X ∧ ) , 2

∧

u ∈ C0∞ (R+ , C ∞ (X)),

(2.18)

where the measure for L (X ) is taken to be dr dx. Clearly, another choice of the order reducing family {f s : −∞ < s < ∞} gives rise to an equivalent norm in the

44


space Hs,γ (X ∧ ). Observe that Hs,γ (X ∧ ) = rγ Hs,0 (X ∧ ).

(2.19)

Another consequence of the definition is that H0,0 (X ∧ ) = r−n/2 L2 (R+ , L2 (X)). s Hcone (X ∧ )

(2.20)

Moreover, let be defined to be the set of all u ∈ × X)|X ∧ such ˜ of a that for all cut-off functions ω(r) on R+ , all diffeomorphisms χ1 : U → U n n+1 ˜ coordinate chart U on X to an open subset U of the unit sphere S in R , s Hloc (R

χ(r, x) = rχ1 (x),

r > 0,

and all ϕ ∈ C0∞ (U ), we have ((1 − ω)ϕu) ◦ χ−1 ∈ H s (Rn+1 ). We can now define as in (2.3) the weighted cone space Ks,γ (X ∧ ) by s (X ∧ )}, Ks,γ (X ∧ ) = {ωu + (1 − ω)v : u ∈ Hs,γ (X ∧ ), v ∈ Hcone

(2.21)

where ω is a cut-off function on R+ . Thus, we have a scale of Hilbert spaces in which K0,0 (X ∧ ) = r−g Ks,γ (X ∧ ). For −∞ < s, γ, g < ∞, we introduce the space Ks,γ;g (X ∧ ) given by Ks,γ;g (X ∧ ) = r−g Ks,γ (X ∧ ) for all s, γ, g ∈ R. In order to formulate subspaces with asymptotics, we first define s,γ the weighted cone space KΘ (X ∧ ) of order (s, γ) and flatness Θ = (ϑ, 0] by −1 s,γ KΘ (X ∧ ) = Ks,γ−ϑ−(j+1) (X ∧ ), j∈N

where each space in the intersection is equipped with the Fréchet topology. Now, for a discrete asymptotic type P associated with the weight data (γ, Θ) for finite Θ, we define the weighted cone space KPs,γ (X ∧ ) of order (s, γ) and with discrete asymptotic type P by s,γ KPs,γ (X ∧ ) = KΘ (X ∧ ) + EP (X ∧ ),

(2.22)

which is a direct sum of Fréchet spaces. In the case of an infinite weight interval Θ, we define KPs,γ by KPs,γ (X ∧ ) = proj lim KPs,γ (X ∧ ), k

(2.23)

k→∞

where Pk , k ∈ N, is the discrete asymptotic type associated with (γ, Θk ) for the finite weight intervals Θk = (−(k + 1), 0] given by Pk = {(p, m) ∈ P : Re p > (n + 1)/2 − γ − (k + 1)}.

Mellin Operators with Asymptotics on Manifolds with Corners Moreover, let SPγ (X ∧ ) be the space defined by SPγ (X ∧ ) = r−N KPN,γ (X ∧ )

45

(2.24)

N∈N

equipped with the Fréchet topology of the projective limit. An equivalent formulation of SPγ (X ∧ ) is given by SPγ (X ∧ ) = {ωu + (1 − ω)v : u ∈ KP∞,γ (X ∧ ), v ∈ SO (X ∧ )}, where ω is a cut-off function on R+ and SO (X ∧ ) = {u ∈ S(R, C ∞ (X)) : supp(u) ⊆ R+ × X}. An operator in the cone algebra will be formulated in connection with prescribed weight data g = (γ, γ − µ, Θ) for a weight γ, a weight shift µ determined by a pseudo-differential order, and a weight interval Θ where we observe poles in the complex plane of the Mellin covariable on the intersection of the left halfplanes determined by Γ(n+1)/2−γ and Γ(n+1)/2−(γ−µ) generated by asymptotics of functions and their images under the action of the operator. An operator G : Ks,γ (X ∧ ) → Ks−µ,γ−µ (X ∧ ) (2.25) ∧ is said to be a Green operator in the cone algebra on X if there are discrete asymptotic types P and Q associated with, respectively, the weight data (γ −µ, Θ) and (−γ, Θ), such that G and its formal adjoint G∗ induce continuous operators −γ G : Ks,γ;g (X ∧ ) → SPγ−µ (X ∧ ), G∗ : Ks,−γ+µ;g (X ∧ ) → SQ (X ∧ )

(2.26)

for all s, g ∈ R. The formal adjoint here refers to the non-degenerate sesquilinear pairing Ks,γ;g (X ∧ ) × K−s,−γ;−g (X ∧ ) → C obtained by extending (·, ·)K0,0 (X ∧ ) : C0∞ (X ∧ ) × C0∞ (X ∧ ) → C. We observe that the Green operator in (2.25) is always compact. Moreover, let fj ∈ MR−∞ (X), j ∈ N, and let ω(r) and ω (r) be cut-off funcj tions on R+ . Then for all discrete asymptotic types P associated with (γ, Θ) and all s in R, there exists a discrete asymptotic type Q associated with (γ − µ, Θ) such that the operators γ −n/2

r−µ+j ωopMj

(fj )ω : Ks,γ (X ∧ ) → K∞,γ−µ (X ∧ )

(2.27)

for γj ∈ R, γ − j ≤ γj ≤ γ with Γ(n+1)/2−γj ∩ πC R = ∅ are continuous and induce continuous operators γ−µ KPs,γ (X ∧ ) → SQ (X ∧ ).

Let g = (γ, γ − µ, Θ) be the weight data, where Θ = (−(k + 1), 0],

k ∈ N.

(2.28)

46


Then LM +G (X ∧ , g) is the set of all operators of the form M + G, where G is a Green operator in the cone algebra on X ∧ and M is a finite linear combination of operators of the form γ −n/2 r−µ+j ω opMj (fj )ω , where ω and ω are cut-off functions on R+ , fj ∈ MR−∞ (X), j and for 0 ≤ j ≤ k,

γ − j ≤ γj ≤ γ,

and Γ(n+1)/2−γj ∩ πC Rj = ∅. The operators in LM +G (X ∧ , g) furnish the asymptotic part of the cone calculus on X ∧ corresponding to the weight data g = (γ, γ − µ, Θ). 2.2. Edge spaces with asymptotics and smoothing operators A manifold B with smooth edge is a topological space with a subspace Y, the edge, such that both B \ Y and Y are C ∞ manifolds, and Y has a neighbourhood V with the structure of an X ∆ -bundle over Y for some compact C ∞ manifold X, where X ∆ = (R+ × X)/({0} × X) is the cone with base X. For convenience, we assume that V is trivial, i.e., V = X ∆ × Y. The extension of our considerations to the general case is straightforward. Let χ0 : U0 → Ω be a local chart on Y, where Ω is an open subset of Rq and q = dim Y . Let U be the restriction of the trivial bundle V to U0 given by U = X ∆ × U0 . Then we have an obvious isomorphism χ := IX ∆ × χ0 : U → X ∆ × Ω, where IX ∆ is the identity on X ∆ , and a diffeomorphism χreg : U \ Y → X ∧ × Ω.

(2.29)

To make the terminology and the setting transparent by means of an example, let B be the Euclidean space given by B = Rn+1+q = Rn+1 × Rqy . x ˜ It can be considered as a manifold with edge Y , where Y = Rqy . Then we can let X be the unit sphere S n in Rn+1 with center at the origin and as the neighbourhood V in the above definition we take, for instance, {˜ x ∈ Rn+1 : |˜ x| < 1} × Rq


47

instead of X ∆ × Rq = Rn+1+q . In other words, when we talk about V we have in mind a kind of tubular neighborhood of Y in B of finite diameter transversally to Y rather than the Cartesian product X ∆ × Y. Therefore we ask V to have the structure of an X ∆ -bundle, and the identification V = X ∆ × Y is only interpreted as a trivialization of V . Remark 2.8. The idea in the above example is to see Rn+1 × Rq as a manifold with edge, where Rn+1 is the model cone. Using polar coordinates, we can identify Rn+1 \ {0} with R+ × S n , but the cone itself with its tip is equal to Rn+1 . However, from the point of view of describing the edge singularity in terms of a local wedge {˜ x ∈ Rn+1 : |˜ x| < 1} × Rq , we replace the fiber Rn+1 by the unit ball in Rn+1 which is isomorphic to Rn+1 in the category of manifolds with conical singularities. This is why we invoke the concept of an X ∆ -bundle over the edge. Clearly, as noted before, we have in practice a kind of tubular neighborhood of the edge in mind, but in the local models it is not convenient to insist on the finite stretched cone like (0, 1) × X. This is because in transition maps we need a metric requirement of the kind that the distance to the edge remains invariant. In any case, {˜ x ∈ Rn+1 : |˜ x| < 1} × Rq is an admitted example as well. To see similar situations in other branches of geometry such as index theory, let us recall that in the first paper of Atiyah and Singer [2] on the index theorem, a tubular neighborhood of an embedded manifold is identified with a normal bundle, which is topologically the same. The final message is that passing from the X ∆ to ([0, 1) × X)/({0} × X) makes things extremely clumpsy. Remark 2.9. The manifolds with smooth edges form a category M1 with the natural notions of morphisms and isomorphisms, See, for instance, [45]. In local coordinates, we work with the splitting of variables (r, x, y) in X ∧ ×Ω. The wedge B given by B = X∆ × Ω is a manifold with edge Ω. Then the stretched manifold B associated withe the wedge B is given by B = R+ × X × Ω. It is a smooth manifold with boundary ∂B = X × Ω. The double 2B obtained by gluing together two copies of B along the common boundary ∂B is a C ∞ manifold given by 2B = R × X × Ω. This construction can be carried out similarly for a manifold B with edge in general. In other words, we can attach to B \ Y the trivial X-bundle X × Y over

48


Y and obtain in this way the stretched manifold B, which is a C ∞ manifold with boundary. Moreover, we have the double 2B, which is also a C ∞ manifold. Let Diffµdeg (B) denote the space of all operators A in Diffµ (B \ Y ) such that A can be represented by A = r−µ ajα (r, y)(−r∂r )j (rDy )α (2.30) j+|α|≤µ

in terms of coordinates (r, x, y) near the edge Y , where ajα ∈ C ∞ (R+ × Ω, Diffµ−(j+|α|) (X)),

j + |α| ≤ µ.

Besides the standard homogeneous principal symbol σ0 (A) and near the edge Y the reduced symbol σ ˜0 (A) given by σ ˜0 (A)(r, x, y, ρ, ξ, η) = rµ σψ (A)(r, x, y, r−1 ρ, ξ, r−1 η) for all (r, x, y) ∈ R+ × X × Ω and (ρ, ξ, η) ∈ R1+n+q , we have the homogeneous principal edge symbol σ1 (A) given by σ1 (A)(y, η) = r−µ ajα (0, y)(−r∂r )j (rη)α , y ∈ Ω, η ∈ Rq \ {0}, (2.31) j+|α|≤µ

as a family of continuous operators σ1 (A)(y, η) : Ks,γ (X ∧ ) → Ks−µ,γ−µ (X ∧ )

(2.32)

for all s ∈ R and a fixed γ ∈ R. Let us recall that the operators in (2.32) on the infinite stretched cone X ∧ belong to the class of SG pseudo-differential operators of order (µ, 0) as r → ∞. Details can be found in, e.g., [5], [6], [7], [8], [27], [28], [29], [30], [42]. The edge pseudo-differential algebra from [36] contains all operators A ∈ Diffµdeg (B) together with the parametrices of elliptic elements. We do not develop this program in this paper and refer to the works [38] or [42] of Schulze for details. More comments on the general background are given in Section 2.5 of this paper. In this section, we focus on the asymptotic part of the edge calculus. We assume that the operator A is elliptic with respect to σψ , i.e., σψ (A)(r, x, y, ρ, ξ, η) = 0 and σ ˜ψ (A)(r, x, y, ρ, ξ, η) = 0 for all nonzero (ρ, ξ, η) up to r = 0. Then we know that the operators in (2.32) are Fredholm under the condition that the conormal symbol σc (σ1 (A))(y, z) given by (σc (σ1 (A)))(y, z) =

µ

aj0 (0, y)z j : H s (X) → H s−µ (X)

(2.33)

j=0

is bijective for all z in Γ(n+1)/2−γ . This is the case for every γ outside some discrete set of real numbers because the set D(y) of all points z for which (σc (σ1 (A))(y, z) (2.33) is not bijective is a discrete set in the complex plane. It should be noted that in this qualitative discussion, we assume for the sake of convenience that (2.32) is


49

even a family of isomorphisms. Otherwise we can pass to a bijective 2 × 2 block matrix family of isomorphisms by adding extra entries of trace and potential type with respect to the edge when a certain topological obstruction on the operator A vanishes. See [38] or [47] for detailed discussions on these matters. Now, the edge pseudo-differential calculus tells us that the points z of D(y), which are just the poles pj (y) of (σc (σ1 (A))−1 (y, z) in the complex plane of some orders mj (y) + 1, contribute to the exponents of the asymptotics (1.1) of solutions to Au = f. In general, as the point y in the edge varies, the points pj (y), 0 ≤ j ≤ J, move in the complex plane C and the numbers mj (y), 0 ≤ j ≤ J, may have jumps, notwithstanding the numeration by j ∈ N is independent of y. Since the symbolic structures to express parametrices have to incorporate a priori all individual patterns of poles at the same time, such phenomena give rise to new problems in formulating the edge calculus to include the asymptotic information. The control of asymptotics with respect to y needs independent attention, as has been studied in the papers [40], [41] and [49]. The nutshell is that it is based on the notion of continuous asymptotics introduced in [33], [35] and [38], and we employ this concept systematically here. In order to keep the paper self-contained, we recall a few definitions and constructions in this theory. Let us first observe that the space EP (X ∧ ) in (2.14) of singular functions with discrete asymptotic type P defined by EP (X ∧ ) = {ω(r)ζf , r−z : f (z) ∈ AP (X)}

(2.34)

can be generalized immediately when we replace πC P by any compact subset K of {z ∈ C : (n + 1)/2 − γ + ϑ < Re z < (n + 1)/2 − γ}, AP (X) is defined by

AP (X) = A(C \ K, C ∞ (X))

and for all f in AP (X), the linear functional ζf on A(C) is defined by A(C) h → ζf , h = f (z)h(z) d¯z ∈ C,

(2.35)

C

where C is any smooth and closed curve inside {z ∈ C : (n + 1)/2 − γ + ϑ < Re z < (n + 1)/2 − γ} \ K oriented once in the counterclockwise direction and enclosing K in its interior. It is well known that the space A (K, C ∞ (X)) of C ∞ (X)-valued analytic functionals on the complex plane, carried by K, is characterized by the set of all linear functionals of the form (2.35). The singular functions ω(r)ζ, r−z for ζ ∈ A (K, C ∞ (X)) may be interpreted as a linear combination of scalar singular functions ω(r)r−z with discrete asymptotic type Pz = {(z, 0)} with the “density” ζ. This is just the motivation for continuous asymptotics.

50


We have until this point assumed that K ∩ Γ(n+1)/2−γ+ϑ = ∅. Let us now extend the notion of continuous asymptotics, still for finite ϑ, to any set K0 given by K0 = K ∩ {Re z > (n + 1)/2 − γ + ϑ}, where K is a compact subset of {Re z < (n + 1)/2 − γ}, i.e., K0 may touch the line Γ(n+1)/2−γ+ϑ . Then, by notation, we identify K0 with a continuous asymptotic type P associated with weight data (γ, Θ), and define the weighted cone space KPs,γ (X ∧ ) by s,γ KPs,γ (X ∧ ) = KΘ (X ∧ ) + EK (X ∧ ) (2.36) where EK (X ∧ ) = {ω(r)ζ, r−z : ζ ∈ A (K, C ∞ (X))}. The sum in (2.36) is not a direct sum unless K ⊂ {(n + 1)/2 − γ + ϑ < Re z < (n + 1)/2 − γ}. Moreover, it depends only on the set K0 = πC P. The space KPs,γ (X ∧ ) is Fréchet in a natural way and is independent of the choice of the cut-off function ω on R+ . We also note that ∞,γ EK (X ∧ ) ⊂ KΘ (X ∧ ) when K ⊂ {Re z ≤ (n + 1)/2 − γ + ϑ}. Remark 2.10. The space KPs,γ (X ∧ ) can be characterized as the subspace of all u ∈ Ks,γ (X ∧ ) such that for every β ∈ (γ, γ−ϑ), we can find a compact subset Kβ of {(n + 1)/2 − γ + ϑ < Re z < (n + 1)/2 − γ} and an element ζβ ∈ A (Kβ , C∞ (X)) for which u − ωζβ , r−z ∈ Ks,β (X ∧ ), where ω is any cut-off function on R+ . As in the case of discrete asymptotics, we can easily extend the definition to continuous asymptotic types P associated with (γ, Θ), where Θ = (−∞, 0]. Here, P is represented by a closed subset V of {Re z < (n + 1)/2 − γ} such that V ∩ {z ∈ C : c ≤ Re z ≤ c } is compact for all real numbers c and c with c ≤ c . Choosing any sequence {ϑj }j∈N with 0 > ϑ0 > · · · > ϑj > ϑj+1 > · · · and ϑj → −∞ as j → ∞, and letting Kj = V ∩ {z ∈ C : Re z > (n + 1)/2 − γ + ϑj },


51

we can associate a continuous asymptotic type Pj , as in the case for the set K0 . Instead of the previous compact set K, we may take for the present case the set V ∩ {z ∈ C : Re z ≥ (n + 1)/2 − γ + ϑj − β} for any positive number β. This gives us a sequence KPs,γ (X ∧ ), j ∈ N, of spaces j with continuous embeddings KPs,γ (X ∧ ) → KPs,γ (X ∧ ) j+1 j for all j ∈ N. We can then define KPs,γ (X ∧ ) by KPs,γ (X ∧ ) = proj lim KPs,γ (X ∧ ). j j→∞

The definition is independent of the specific choice of the sequence {ϑj }j∈N . Let πC P = V. Then a discrete or continuous asymptotic type P is said to satisfy the shadow condition if p ∈ πC P ⇒ p − k ∈ πC P for all k ∈ N with Re (p − k) > (n + 1)/2 − γ + ϑ. We employ the spaces (2.24) also for continuous asymptotic types P. We observe that analogues of the spaces of smoothing Mellin plus Green operators on X ∧ introduced at the end of Section 2.1 make sense also in the framework of continuous asymptotics. We give such constructions in the following section from the point of view of Mellin plus Green operator-valued edge symbols. In Chapter 3 we develop a similar calculus on B ∧ for a compact manifold B with smooth edge. Let us conclude this section with edge spaces with asymptotics and smoothing operators of the edge calculus. A crucial observation in this context is that for each λ ∈ R+ , the transformation κλ : C0∞ (X ∧ ) → C0∞ (X ∧ ) defined by (κλ u)(r, x) = λ(n+1)/2 u(λr, x), is unitary on r

−n/2

κλ : K

(r, x) ∈ X ∧ ,

(2.37)

L (R+ × X) and can be extended to isomorphisms 2

s,γ

(X ∧ ) → Ks,γ (X ∧ ), κλ : KPs,γ (X ∧ ) → KPs,γ (X ∧ )

(2.38)

for all s, γ ∈ R and all discrete or continuous asymptotic types P. A Hilbert space H is said to be endowed with a group action κ = {κλ }λ∈R+ if κλ : H → H is an isomorphism for every λ ∈ R+ , κλ κν = κλν ,

λ, ν ∈ R+ ,

and the map R+ λ → κλ h ∈ H is continuous for all h ∈ H. Now, let E be a Fréchet space written as a projective limit of Hilbert spaces Ej , j ∈ N, continuously embedded in E0 . Then we say that κ = {κλ }λ∈R+ is a group action on E if it is a group action on E0 which restricts to a group action on Ej for all j in N. It can be checked easily that the above-mentioned {κλ }λ∈R+ defines a group action on Ks,γ (X ∧ ). Moreover, for all s, γ ∈ R and all discrete or continuous

52


asymptotic types P , the Fréchet space KPs,γ (X ∧ ) admits a representation as a projective limit of Hilbert spaces Ej with E0 = Ks,γ (X ∧ ) such that {κλ }λ∈R+ is also a group action on E. Let H be a Hilbert space equipped with the group action κ = {κλ }λ∈R+ . Then for −∞ < s < ∞, we define the abstract edge space W s (Rq , H) to be the completion of S(Rq , H) with respect to the norm W s (Rq ,H) by ˆ(η) L2 (Rq ,H) ,

u W s(Rq ,H) = ηs κ−1 η u

u ∈ S(Rq , H),

(2.39)

where u ˆ = F u is the Fourier transform of u : Rq → H. Thus, we obtain a family s q W (R , H), −∞ < s < ∞, of Hilbert spaces, which clearly depend on the choice of the group action κ = {κλ}λ∈R+ . If a Fréchet space E is a projective limit given by E = proj lim Ej j→∞

is endowed with a group action, then we have the Hilbert spaces W s (Rq , Ej ) and hence we can define W s (Rq , E) by W s (Rq , E) = proj lim W s (Rq , Ej ). j→∞

W (R , E) is again a Fréchet space. Thus, the local edge spaces without and with asymptotics of type P are defined, respectively, as s

q

W s (Rq , Ks,γ (X ∧ )), W s (Rq , KPs,γ (X ∧ )).

(2.40)

Both the abstract edge spaces and the local edge spaces (2.40) are studied systematically in various expositions, e.g., [35] or [38]. In particular, it is important to note that s s Hcomp (Rq × X ∧ ) ⊂ W s (Rq , Ks,γ (X ∧ )) ⊂ Hloc (Rq × X ∧ )

(2.41)

for all s, γ ∈ R. This allows us to define weighted spaces s,γ s H[loc) (B) ⊂ Hloc (B \ Y )

for a manifold B with edge Y, locally near Y modelled on W s (Rq , Ks,γ (X ∧ )). In s,γ a similar manner, we obtain subspaces H[loc),P (B) of functions with asymptotics of type P. Moreover, we have [comp)-variants of the respective spaces. If B is compact, then we simply write H s,γ (B) and HPs,γ (B), respectively. The space s,γ H[loc) (B) consists of functions in L2loc(B \ Y ) which, locally near Y , are in W 0 (Rq , K0,0 (X ∧ )) = r−n/2 L2 (R+ × X × Rq ), where the L2 is with respect to the measure dr dx dy. Thus, there is a sesquilinear pairing 0,0 0,0 (·, ·) : H[comp) (B) × H[loc) (B) → C via the corresponding local “scalar” products. The restriction (·, ·) : C0∞ (B \ Y ) × C0∞ (B \ Y ) → C


53

induces a pairing s,γ −s,−γ H[comp) (B) × H[loc) (B) → C

(2.42) ∗

for all s, γ ∈ R. This then allows us to define the formal adjoint A of an operator A : C0∞ (B \ Y ) → C ∞ (B \ Y ) via (Au, v) = (u, A∗ v) for all u, v ∈ C0∞ (B \ Y ). For g = (γ, γ − µ, Θ), let L−∞ (B, g) denote the space of all operators C : C0∞ (B \ Y ) → C ∞ (B \ Y ) such that C and C ∗ extend to continuous operators s,γ ∞,γ−µ −s,−γ+µ ∞,−γ C : H[comp) (B) → H[loc),P (B), C ∗ : H[comp) (B) → H[loc),Q (B)

(2.43)

for all s ∈ R and for some asymptotic types P and Q, associated with the weight data (γ − µ, Θ) and (−γ, Θ), respectively. Remark 2.11. The operators L−∞ (B, g) play the role of smoothing operators in the edge pseudo-differential calculus on B. The space L−∞ (B, g)P,Q consisting of elements C with asymptotic types P and Q alluded to in (2.43) form a Fréchet space in a natural way. Thus, we can form the spaces S(Rl , L−∞ (B, g)P,Q ) and L−∞ (B, g; Rl ) of parameter-dependent smoothing operators with parameter λ in Rl by S(Rl , L−∞ (B, g)P,Q ) = L−∞ (B, g; Rl )P,Q and L−∞ (B, g; Rl ) =

L−∞ (B, g; Rl )P,Q .

P,Q

2.3. Mellin plus Green edge symbols µ It is desirable to extend the notion of scalar symbols in S(cl) (V × Rn ) introduced ˜ ˜ at the beginning of Section 2.1 to that of L(H, H)-valued symbols, where L(H, H) ˜ If is the space of all continuous operators between the Hilbert spaces H and H. absolute values in the estimates for the symbols are replaced only by the operator norms · L(H,H) ˜ , then such a generalization is completely trivial. In the edge ˜ in question are calculus, we employ the fact that the Hilbert spaces H and H endowed with, respectively, the group actions κ and κ ˜ defined in Section 2.2. Now, ˜ is the set of let µ ∈ R and let V be an open subset of Rp . Then S µ (V × Rq ; H, H) ∞ q ˜ such that all a(y, η) ∈ C (V × R , L(H, H)) sup (y,η)∈K×Rq

α β η−µ+|β| ˜ κ−1 ˜ < ∞ η {Dy Dη a(y, η)}κη L(H,H)

(2.44)

for all K V and multi-indices α ∈ Np , β ∈ Nq . In the case when V = Ω × Ω, where Ω is an open subset of Rq , we use the variables (y, y ) instead of y. The µ ˜ of classical symbols is defined in terms of asymptotic subspace Scl (V × Rq ; H, H)

54


expansions into symbols aµ−j (y, η) of twisted homogeneity µ − j, j ∈ N, in η for |η| > C, where C is a positive constant, i.e., aµ−j (y, λη) = λµ−j κ ˜ λ aµ−j (y, η)κ−1 λ for all |η| > C and all λ ≥ 1. As in the case of scalar symbols, the notation µ ˜ S(cl) (V × Rq ; H, H)

˜ by a Fréchet denotes the classical and the general symbol class. If we replace H µ ˜ ˜ ˜ space E = proj limj→∞ Ej with group action κ ˜ , then we define S(cl) (V × Rq ; H, E) by µ ˜ = proj lim S µ (V × Rq ; H, E ˜j ). S(cl) (V × Rq ; H, E) (2.45) (cl) j→∞

If H is also replaced by a Fréchet space E = proj limj→∞ Ej with group action κ, then µ ˜ a(y, η) ∈ S(cl) (V × Rq ; E, E) means that for all m ∈ N, there is an l = l(m) ∈ N such that µ ˜m ). a(y, η) ∈ S(cl) (V × Rq ; El , E µ ˜ the corresponding space of symbols with constant We denote by S(cl) (Rq ; E, E) coefficients. Examples of operator-valued symbols in the above definitions are, for instance, operators Mϕ of multiplication by functions ϕ ∈ C0∞ (R+ ) on the space Ks,γ (X ∧ ) with the group action κ defined in Section 2.2. More precisely, we have

Mϕ ∈ S 0 (Rq ; Ks,γ (X ∧ ), Ks,γ (X ∧ ))

(2.46)

for all s, γ ∈ R. Note that although Mϕ does not depend on η, the estimate of order 0 contains this variable. Another example is the multiplication operator Mϕη with ϕη (r) = ϕ(r[η]), r > 0, η ∈ Rq , (2.47) where Rq η → [η] ∈ R is any strictly positive function on C ∞ (Rq ) such that [η] = |η|,

|η| ≥ C,

for some positive constant C. In this case, we have 0 Mϕη ∈ Scl (Rq ; Ks,γ (X ∧ ), Ks,γ (X ∧ ))

(2.48)

for all s, γ ∈ R. The relations (2.46) and (2.48) remain true when we replace Ks,γ (X ∧ ) by KPs,γ (X ∧ ) for some asymptotic type P. Definition 2.12. For g = (γ, γ − µ, Θ), m ∈ R, and an open subset Ω of Rq , we m define the space RG (Ω × Rq , g)P,Q of Green symbols to be the set of all m g(y, η) ∈ Scl (Ω × Rq ; Ks,γ;e (X ∧ ), SPγ−µ (X ∧ )) (2.49) s,e∈R

Mellin Operators with Asymptotics on Manifolds with Corners such that g ∗ (y, η) ∈

−γ m Scl (Ω × Rq ; Ks,−γ+µ;e (X ∧ ), SQ (X ∧ )),

55

(2.50)

s,e∈R

where g ∗ (y, η) is the formal adjoint, P and Q are the asymptotic types associated with, respectively, the weight data (γ − µ, Θ) and (−γ, Θ). m We define RG (Ω × Rq , g) by m RG (Ω × Rq , g) =

m RG (Ω × Rq , g)P,Q .

(2.51)

P,Q

Green symbols are the essential elements of edge symbols or edge amplitude functions. The notation originally comes from the theory of boundary value problems. It turns out that a Green function of a classical elliptic boundary value problem, e.g., the Dirichlet problem for the Laplace equation on a smooth and bounded domain, is equal to a fundamental solution or a parametrix of the corresponding elliptic operator, plus a Green operator which is a pseudo-differential operator along the boundary. The Green symbols in local representations near the boundary act on the half-axis R+ , the inner normal to the boundary which is the substitute for X ∧ in this case. In boundary value problems with the transmission property at the boundary, the asymptotic types reflect the Taylor asymptotics. Green symbols and their associated operators in the edge calculus play a similar role, but the asymptotic types are more general and they are caused by the nature of elliptic regularity with asymptotics. Asymptotics are generated to a large extent by another ingredient of edge symbols – the so-called smoothing Mellin symbols. Recall that all features of edge symbols formulate a priori certain properties of the parametrices of elliptic operators including the structures that are responsible for the transformation or creation of asymptotics under the action of the associated pseudo-differential operators. Since this concerns all elliptic edge operators at the same time, no matter what the individual asymptotic data are, the “correct” definition itself is an important aspect of the edge pseudo-differential calculus. Let us extend Definition 2.1 to the case of continuous asymptotic types R for Mellin symbols. The space MR−∞ (X) of smoothing Mellin symbols with continuous asymptotic type R is defined to be the set of all f ∈ A(C \ V, L−∞ (X)), where V = πC R is a closed subset of C for which V ∩ {z ∈ C : c ≤ Re z ≤ c } is compact for all real numbers c and c with c ≤ c , such that for such c and c with c ≤ c , we can find a compact subset Kc,c of V and an element ζ ∈ A (Kc,c , L−∞ (X)) for which the function hc,c defined by hc,c (z) = (f (z) − Mσ,r→z (ω(r)ζw , r−w )(z)),

z ∈ C \ V,

(2.52)

belongs to A({c ≤ Re z ≤ c }, L−∞ (X)) and hc,c (z)|Γβ ∈ S(Γβ , L−∞ (X))

(2.53)

56


for all β ∈ (c, c ), uniformly with respect to β on compact subsets of R. (Uniformity is understood as in Definition 2.1 (ii). ζw , r−w means the pairing of the analytic functional ζ with respect to the complex variable w, and Mσ is the weighted Mellin transform for a weight σ such that 1/2 − σ > max{Re z : z ∈ Kc,c }.) MR−∞ (X)

The space is Fréchet in a natural way. In our terminology, R is essentially identified with the set V . Discussions in connection with continuous Mellin asymptotic types R may be formulated in terms of the corresponding sets V , but we prefer to adhere to the notation R rather than V when we talk about the asymptotic type associated with V. Definition 2.13. Let g = (γ, γ −µ, Θ), where Θ = (−(k + 1), 0] for some k ∈ N. The µ q space RM +G (Ω × R , g) of smoothing Mellin plus Green edge symbols is defined µ to be the set of all operator functions (m + g)(y, η), where g ∈ RG (Ω × Rq , g) and m is a finite linear combination of terms mjα (y, η) = r−µ+j ωη opMjα γ

for |α| ≤ j, 0 ≤ j ≤ k, fjα (y, z) ∈ C

∞

−n/2

(fjα )(y)η α ωη

(Ω, MR−∞ (X)), jα

(2.54)

weights γjα with

γ − j ≤ γjα ≤ γ and πC Rjα ∩ Γ(n+1)/2−γjα = ∅, and cut-off functions ω and ω on R+ . Without loss of generality, we assume that there is only one summand with j = 0. Then we observe that mjα (y, η) ∈ C ∞ (Ω × Rq , L(Ks,γ;e (X ∧ ), K∞,γ−µ;∞ (X ∧ ))) and

∞,γ−µ;∞ mjα (y, η) ∈ C ∞ (Ω × Rq , L(KPs,γ;e (X ∧ ), KQ (X ∧ ))) for all s, e ∈ R, arbitrary asymptotic types P and some resulting asymptotic types Q. Moreover, we have

mjα (y, λη) = λµ−j+|α| κλ mjα (y, η)κ−1 λ

(2.55)

for all λ ≥ 1 and |η| ≥ C for some positive constant C. Thus µ−j+|α|

mjα (y, η) ∈ Scl

(Ω × Rq ; Ks,γ;e (X ∧ ), K∞,γ−µ;∞ (X ∧ ))

(2.56)

and

∞,γ−µ;∞ mjα (y, η) ∈ Scl (Ω × Rq ; KPs,γ;e (X ∧ ), KQ (X ∧ )) (2.57) for all s, e ∈ R, arbitrary asymptotic types P and some resulting asymptotic types µ q Q. It follows altogether that every m + g ∈ RM +G (Ω × R , g) belongs to µ−j+|α|

µ µ ∞,γ−µ;∞ Scl (Ω × Rq ; Ks,γ;e (X ∧ ), K∞,γ−µ;∞ (X ∧ )) ∩ Scl (Ω × Rq ; KPs,γ;e (X ∧ ), KQ (X ∧ ))

for all s, e ∈ R, and all asymptotic types P with some resulting asymptotic types Q.


57

2.4. Edge operators in spaces with asymptotics We now employ the symbols in Definitions 2.12 and 2.13 to establish the asymptotic part of the edge calculus on a manifold B with smooth edge Y. We impose the same assumption as stipulated at the beginning of Section 2.2. To recapitulate, B is locally near Y nothing but X ∆ × Y . (The general case can be treated as well and the details are straightforward.) We choose an open cover of Y by coordinate neighborhoods U0,ι , ι ∈ I. Let {ϕι : ι ∈ I} be a partition of unity subordinate to the open cover. Then we choose functions ψι ∈ C0∞ (U0,ι ) that are equal to 1 on supp ϕι , ι ∈ I. Moreover, we fix charts χ0,ι : U0,ι → Ω, ι ∈ I, for some open subset Ω of Rq . (Without loss of generality, Ω can be taken to be independent of ι ∈ I.) We define Uι by Uι = X ∆ × U0,ι , and let χreg,ι : Uι \ Y → X ∧ × Ω be defined by χreg,ι (r, x, y) = (r, x, χ0,ι (y)),

(r, x, y) ∈ Uι \ Y.

Let g = (γ, γ − µ, Θ), where Θ = (−(k + 1), 0] for some k ∈ N. Then the asymptotic part of the edge calculus on B is the space LµM +G (B, g) of all operators A of the form A=

Aι + C,

(2.58)

(2.59)

ι∈I

with C ∈ L−∞ (B, g) and µ q Aι = (χ−1 reg,ι )∗ ωϕι Opy (aι )ψι ω , aι ∈ RM +G (Ω × R , g),

where ω and ω are cut-off functions on R+ . The space (2.58) belongs to the ingredients of the edge calculus on B. The remaining flat (non-smoothing) part is omitted here. Details can be found in [36], [38] or [42]. The space (2.58) is well known in the general edge calculus. However, as stated at the beginning of this paper, we are mainly interested in new asymptotic structures on manifolds with corners, which require more information than hitherto developed for the theory on manifolds with edges. In particular, there is a parameter-dependent analogue of (2.58), with parameter λ ∈ Rl , given by LµM +G (B, g; Rl ). (2.60) It consists of all operator families A(λ) defined by A(λ) = Aι (λ) + C(λ) ι∈I

with C(λ) ∈ L

−∞

(B, g; R ) and l

µ q+l Aι (λ) = (χ−1 reg,ι )∗ ωϕι Opy (aι )(λ)ψι ω , aι ∈ RM+G (Ω × Rη,λ , g),

where ω and ω are cut-off functions on R+ .

58


Every element A(λ) in (2.60) has a parameter-dependent principal edge symbolic structure, namely, σ1 (A)(y, η, λ), locally defined as the principal component of the respective classical operator-valued symbol in µ µ q+l s,γ RM+G (Ω × Rq+l (X ∧ ), K∞,γ−µ (X ∧ )). η,λ , g) ⊂ Scl (Ω × Rη,λ ; K

In the case when l = 0, we simply omit λ. As in the standard theory of pseudo-differential operators, we have a corresponding notion of properly supported operators. Every operator A ∈ LµM +G (B, g; Rl ) can be written in the form A = A0 + C, where A0 is a properly supported operator and C ∈ L−∞ (B, g; Rl ). Theorem 2.14. Let D ∈ L0M +G (B, g0 ), where g0 = (γ, γ, Θ). Suppose that σ1 (I + D)(y, η) : Ks,γ (X ∧ ) → Ks,γ (X ∧ ) is bijective for some s ∈ R. Then there is an H ∈

L0M +G (B, g0 )

(2.61) such that

(I +H)(I +D) = I, (I +D)(I +H) = I,

(2.62)

mod L−∞ (B, g0 ). 2.5. Comments on the general edge calculus Recall that when B is a manifold with edge Y, Diffµdeg(B) denotes the space of all A ∈ Diffµ (B \ Y ) such that A is of the form (2.30) in terms of coordinates (r, x, y) locally near the edge Y . An edge calculus is a pseudo-differential calculus on B, or more precisely, on B \ Y , that contains Diffµdeg (B) together with the parametrices of elliptic elements. The ellipticity is in terms of a principal symbolic hierarchy σ(A) = (σ0 (A), σ1 (A)),

(2.63)

where the components come from the strata (s0 (B), s1 (B)) defined by s0 (B) = B \ Y

and s1 (B) = Y.

The 0th component is the standard homogeneous principal symbol of A of order µ as a smooth function invariantly defined on T ∗ (B \ Y ) \ 0, where 0 denotes the zero section of the respective cotangent bundle. Locally near Y in the variables (r, x, y) ∈ R+ × X × Ω and the associated covariables (ρ, ξ, η) ∈ R1+n+q , we have the reduced symbol σ ˜0 (A) given by σ ˜0 (A)(r, x, y, ρ, ξ, η) = rµ σ0 (A)(r, x, y, r−1 ρ, ξ, r−1 η).

(2.64)

This is completely determined by σ0 (A) and is smooth up to r = 0. Moreover, as explained in Section 2.2, the principal edge symbol σ1 (A)(y, η),

(y, η) ∈ T ∗ Y \ 0

in (2.31), takes values in continuous operators (2.32), for all s ∈ R and all prescribed γ ∈ R. The weight γ is determined by the ellipticity. As soon as A is


59

elliptic with respect to σ0 , i.e., σ0 (A) = 0 as usual and σ ˜0 (A) = 0 up to r = 0, the operators (2.32) are Fredholm for all γ off some discrete set of real numbers, for any fixed y and for all η = 0. For the sake of simplicity, we assume now that (2.32) is a family of isomorphisms for some weight γ and all (y, η) ∈ T ∗ Y \ 0, i.e., A is (σ0 , σ1 )-elliptic, then the edge algebra in [36] gives us a parametrix P with σ0 (P ) = σ0−1 (A)

and σ1 (P ) = σ1−1 (A).

Otherwise, as noted before, we have to impose extra edge conditions, which are analogues of boundary conditions. In this theory, the parametrix P can be expressed as P = P0 + M + G, where P0 is a nonsmooth flat part on B \ Y and −1 M + G ∈ L−µ ) M +G (B, g

for g−1 = (γ − µ, γ, Θ). In this conclusion, the operator A itself is considered in connection with the weight data g = (γ, γ − µ, Θ), where the first two components indicate weights in the preimage and the image, respectively, under the action of A, while Θ determines a weight interval where we intend to control poles of the Mellin covariable in the complex plane, which lie on the left of the weight lines Γ(n+1)/2−γ and Γ(n+1)/2−(γ+µ) , to express asymptotics of solutions. To deepen the understanding of the role of the operator class LµM+G (B, g), we briefly sketch the idea on how to find a parametrix P of an elliptic A ∈ Diffµdeg (B). The arguments for the pseudo-differential case are similar. The first step is to invert σ0 (A) and to find, using the technical tools of the edge algebra, e.g., quantizations and index arrangements, an operator R inside the edge algebra of order −µ (and properly supported when B is not compact), which is again (σ0 , σ1 )-elliptic and satisfies the relations σ0 (R) = σ0−1 (A)

(2.65)

RA = I + D

(2.66)

and for some D ∈ with g0 = (γ, γ, Θ). Clearly, (2.65) is a consequence of (2.66). Indeed, L0M +G (B, g0 )

σ0 (RA) = σ0 (I + D) = σ0 (I) + σ0 (D) = 1 because σ0 (D) = 0. Moreover, for (y, η) ∈ T ∗ Y \ {0}, since σ1 (A)(y, η) and σ1 (R)(y, η) are bijective, we obtain the bijectivity of σ1 (R)(y, η)σ1 (A)(y, η) = σ1 (RA)(y, η) = σ1 (I + D)(y, η). Thus, the operator I + D is (σ0 , σ1 )-elliptic in the edge calculus. However, the non-smoothing part is already equal to I. Now, to complete the construction of the parametrix P , it suffices to apply Theorem 2.14, which gives us an operator H ∈ L0M +G (B, g0 ) such that (I + H)(I + D) = I

60


mod L−∞ (B, g0 ). This has the consequence that σ1 (I + H)(y, η) = σ1−1 (1 + D)(y, η) for all (y, η) ∈ T ∗ Y \ 0. Then P = (1 + H)R, as desired, i.e., we obtain PA = I mod L−∞ (B, g0 ). Since such a construction works from both sides, it follows that we also have AP = I −∞ mod L (B, g0 ).

3. The asymptotic part of the corner calculus 3.1. Spaces with double weights and iterated asymptotics A manifold M with second-order corners is a topological space with the following properties. 1. M contains a finite subset S of points, the corners of M , such that M \ S is a manifold with (smooth) edge. 2. Every point c in S has a neighbourhood V in M such that for some compact manifold B = B(v) with smooth edge, there is a homeomorphism χ : V → B ∆ , also called a singular chart, that restricts to an isomorphism χreg : V \ {c} → B ∧ in the category M1 of manifolds with smooth edges. See Remark 2.9. Another singular chart χ ˜ : V → B ∆ is said to be equivalent to χ if ∧ ∧ χ ˜reg ◦ χ−1 reg : B → B

is the restriction of an M1 -isomorphism R × B → R × B to B ∧ . In our discussions, we keep χ fixed. Then χreg gives us a corresponding splitting of variables (t, b) ∈ R+ × B = B ∧ close to a corner point. For simplicity, we assume that M only has one corner {c}. The general case can be dealt with in a completely similar manner. First we define an analogue Hs,(γ,δ) (B ∧ ) of the space Hs,γ (X ∧ ) as defined by (2.18), but now for a pair of weights (γ, δ) ∈ R2 . While the existence of an order reducing family f s ∈ Lscl (X; Γ1/2 ), −∞ < s < ∞, is more or less standard in the classical pseudo-differential calculus on a compact and smooth manifold X, analogous parameter-dependent reductions of orders and in fact large parts of the general pseudo-differential calculus on a manifold B with edge require more explanations. Of course, these topics and techniques exist in the literature and can be applied to introduce weighted corner spaces on B ∧ as in [43]. However, we prefer to give another approach by giving an alternative description of the


61

spaces Hs,γ (X ∧ ) based on what we call cylindrical Sobolev spaces on R × X. For −∞ < s < ∞, the cylindrical Sobolev space H s (R × X) on R × X consists of all elements u such that (ϕu ◦ κ−1 )(r, x) ∈ H s (R × X) for every chart κ : R × U → R1+n on R × X such that U is a coordinate neighborhood on X and κ(r, u) = (r, κ0 (u)), (r, u) ∈ R × U, where κ0 : U → Rn is a chart on X, and every ϕ ∈ C0∞ (U ). For all β in R and all functions v on R × X, let us define Sβ v by Sβ v(r, x) = e−(1/2−β)r v(e−r , x),

(r, x) ∈ R × X.

(3.1)

Then Sγ−n/2 : C0∞ (X ∧ ) → C0∞ (R × X) extends to an isomorphism Sγ−n/2 : Hs,γ (X ∧ ) → H s (R × X).

(3.2)

Thus, the weighted Sobolev space Hs,γ (X ∧ ) can be given an alternative definition as Hs,γ (X ∧ ) = {v(r, x) : e−((n+1)/2−γ)rv(e−r , x) ∈ H s (R × X)}.

(3.3)

If B is a manifold with edge Y , then R × B is a manifold with edge R × Y. Let us write R × B = (R × V ) ∪ (R × W ), where V is the neighbourhood of Y in B which is M1 -isomorphic to X ∆ × Y as explained at the beginning of Section 2.2, and W is an open subset of B \ Y such that V and W form an open cover of B. Let {ω, ϑ} denote a partition of unity subordinate to {V, W }. The variables in the cylinder R × B will be denoted by (t, b) and those in B ∧ by (t, b). Let W s (R × Y, Ks,γ (X ∧ )) (3.4) s denote the subspace of all v ∈ Hloc (R × X ∧ × Y ) such that for every chart

χ0 : U0 → Rq on Y and for every ϕ ∈ C0∞ (U0 ), we have s q s,γ ∧ (ϕv ◦ χ−1 (Xr,x )). 0 )(t, r, x, y) ∈ W (Rt × Ry , K

Definition 3.1. For all s, γ, δ ∈ R, the spaces H s,γ (R × B) and Hs,(γ,δ) (B ∧ ) are defined by H s,γ (R × B) = {ωf + ϑg : f ∈ W s (R × Y, Ks,γ (X ∧ )), g ∈ H s (R × 2B)},

(3.5)

and Hs,(γ,δ) (B ∧ ) = {u(t, b) : e−((n+2+q)/2−δ)t u(e−t , b) ∈ H s,γ (R × B)}.

(3.6)

62


(3.6) is an analog of (3.3). In the exponent on the right-hand side of (3.6), we have n + 1 + q = dim B. The same dimension, namely, n + 1 + q = dim 2B, is also on the right-hand side of (3.5). This is in conformity with the corresponding general definition of weighted spaces Hs,δ ((·)∧ ) for any closed and C ∞ manifold in parentheses such as 2B. The double 2B consists of two copies B± of B, where we assume that the original stretched manifold B is identified with B+ , and then supp ϑ ⊂ B+ \ Y. It follows from the definition that Hs,(γ,δ) (B ∧ ) = tδ Hs,(γ,0) (B ∧ )

(3.7)

for all s, γ, δ ∈ R. See also (2.19) in this connection. Moreover, H0,(0,0) (B ∧ ) = t−(n+1+q)/2 L2 (R+ , H 0,0 (B)).

(3.8)

For this, see also (2.20). Remark 3.2. The open stretched corner B ∧ = R+ × B is a manifold with edge Y ∧ = R+ × Y, and we have for all s, γ, δ ∈ R, s,γ Hs,(γ,δ) (B ∧ ) ⊂ H[loc) (R+ × B).

(3.9)

See Section 2.2. Now, for all s, γ, δ ∈ R, the space Hs,(γ,δ) (B ∧ ) can be defined equivalently in terms of the space W s,δ (R+ × Y, Ks,γ (X ∧ )) based on the same group action on Ks,γ (X ∧ ) as before, but with respect to the Mellin transform on R+ rather than the Fourier transform. The complex covariable of the Mellin transform in the t-direction is now denoted by v. Let us first formulate an abstract version. If H is a Hilbert space with group action κ = {κλ }λ∈R+ , then the weighted Mellin-Fourier edge space W s,δ (R+ × Rq , H) over R+ × Rq with values in H, of smoothness s and weight δ, is defined to be the completion of C0∞ (R+ × Rq , H) with respect to the norm W s,δ (R+ ×Rq ,H) given by −δ+d/2

u W s,δ (R+ ×Rq ,H) = v, ηs κ−1 u)(v, η) L2 (Γ1/2 ×Rq ,H) , v,η (Mt→v Fy→η t (3.10) for all u in C0∞ (R+ × Rq , H). See also the formula (2.39). The weight shift d/2 in this expression gives information about an “abstract” dimension, which comes from the space H and the dimension of the other variables involved. If we apply (3.10) to H = Ks,γ (X ∧ ), then we set d = n + 1 + q for n = dim X. Similarly, we define W s,δ (R+ × Rq , E) for any Fréchet space E endowed with a group action. For instance, we can let E = KPs,γ (X ∧ ) for some asymptotic type P. As in the case of the space in (3.4), we can define the space W s,δ (R+ × Y, Ks,γ (X ∧ )) to be s consisting of all u ∈ Hloc (R+ × X ∧ × Y ) such that s,δ (ϕu ◦ χ−1 (R+ × Rqy , Ks,γ (X ∧ )) 0 )(t, r, x, y) ∈ W

for all charts χ0 : U0 → Rq on Y and all ϕ ∈ C0∞ (U0 ). Then for all s, γ, δ ∈ R, Hs,(γ,δ) (B ∧ ) is defined by Hs,(γ,δ)(B ∧ ) = {ωf + ϑg : f ∈ W s,δ (R+ × Y, Ks,γ (X ∧ )), g ∈ Hs,δ ((2B)∧ )}, (3.11)


63

and the construction allows us to define subspaces HPedge (B ∧ ) with continuous asymptotics of type Pedge along the edge of B ∧ by s,(γ,δ)

HPedge (B ∧ ) = {ωf + ϑg : f ∈ W s,δ (R+ × Y, KPs,γ (X ∧ )), g ∈ Hs,δ ((2B)∧ )}. edge (3.12) We are now in the position to define iterated corner asymptotics. As announced at the beginning of this paper, the focus is on continuous asymptotics. The discrete case is treated in [43]. Let K be a compact subset of s,(γ,δ)

{v ∈ C : Re v < (d + 1)/2 − δ}, where d = dim B = n + 1 + q. Then we form the space EPedge (B ∧ ) by EPedge ,K (B ∧ ) = {ω(t)ζv , t−v : ζ ∈ A (K, HP∞,γ (B))} edge

(3.13)

for some fixed cut-off function ω on R+ . Then for all γ, δ ∈ R, we have EPedge ,K (B ∧ ) ⊂ H∞,(γ,δ) (B ∧ ).

(3.14)

Now, we choose a finite weight interval Θ = (ϑ, 0], independently of the one ins,(γ,δ) volved in Pedge , and we define HPedge ,Θ (B ∧ ) by HPedge ,Θ (B ∧ ) = {ωu0 +(1−ω)u1 : u0 ∈ ∩>0 HPedge s,(γ,δ)

s,(γ,δ−ϑ−)

(B ∧ ), u1 ∈ HPedge (B ∧ )}, (3.15) s,(γ,δ)

where ω is a cut-off function on R+ . Definition 3.3. For s, γ, δ ∈ R+ , the weighted corner space HPedge ,Pcorner (B ∧ ) with iterated asymptotic type (Pedge , Pcorner ) for an edge asymptotic type Pedge as in Section 2.2 and a corner asymptotic type Pcorner associated with the weight data (δ, Θ) is defined by s,(γ,δ)

HPedge ,Pcorner (B ∧ ) = HPedge ,Θ (B ∧ ) + EPedge ,K (B ∧ ). s,(γ,δ)

s,(γ,δ)

(3.16)

The meaning of the notation Pcorner in connection with the compact set K is similar to that of P in Section 2.2. See also the formula (2.36). The decomposition (3.16) is direct when K ⊂ {v ∈ C : (d + 1)/2 − δ + ϑ < Re v < (d + 1)/2 − δ}. In general, the set K may intersect the line Γ(d+1)/2−δ+ϑ , and then the only contribution to Pcorner comes from the set K0 given by K0 = K ∩ {v ∈ C : Re v > (d + 1)/2 − δ + ϑ}. The extension of the notion of continuous corner asymptotics to the case when Θ = (−∞, 0] again refers to a closed subset V of {v ∈ C : Re v < (d + 1)/2 − δ} that intersects the strip

{v ∈ C : c < Re v < c }

64


in a compact set for all real numbers c and c with c < c . Then for all sequences {ϑj }j∈N with the same properties as in Section 2.2 and all sets Vj , j ∈ N, given by Vj = V ∩ {v ∈ C : Re v ≥ (d + 1)/2 − δ + ϑj − β}, where β is some positive number, we have the spaces HPedge ,Pcorner ,j (B ∧ ) defined by s,(γ,δ)

HPedge ,Pcorner ,j (B ∧ ) = HPedge ,Θj (B ∧ ) + EPedge ,Vj (B ∧ ), s,(γ,δ)

s,(γ,δ)

j ∈ N,

(3.17)

with continuous embeddings HPedge ,Pcorner ,j+1 (B ∧ ) → HPedge ,Pcorner ,j (B ∧ ), s,(γ,δ)

s,(γ,δ)

j ∈ N.

(3.18)

Then we define HPedge ,Pcorner (B ∧ ) by s,(γ,δ)

HPedge ,Pcorner (B ∧ ) = proj lim HPedge ,Pcorner ,j (B ∧ ), s,(γ,δ)

s,(γ,δ)

(3.19)

j→∞

which defines Pcorner in terms of V . Now, let M be a manifold with corner c, i.e., M \{c} is a manifold with smooth edge, and c has a neighborhood V modelled on B ∆ for a compact manifold B with s,(γ,δ) smooth edge Y. In the case when M is compact, we define HPedge, ,Pcorner (M ) for all s, γ, δ ∈ R, by s,(γ,δ)

HPedge ,Pcorner (M ) s,γ = {ωf + (1 − ω)g : f ∈ HPedge ,Pcorner (B ∧ ), g ∈ H[loc),P (M \ {c})} edge s,(γ,δ)

(3.20)

for some cut-off function ω on M , i.e., ω is supported in a small neighbourhood of c and is identically equal to 1 close to c. Moreover, for M as in (3.20) and all s, γ, δ ∈ R, we define H s,(γ,δ) (M ) by s,γ H s,(γ,δ) (M ) = {ωf + (1 − ω)g : f ∈ Hs,(γ,δ)(B ∧ ), g ∈ H[loc) (M \ {c})}.

(3.21)

The space L−∞ M +G (M, g) of smoothing operators for a pair g of weight data given by g = ((γ, γ − µ, Θ), (δ, δ − µ, Θ)) on a manifold M with corner is defined to be the set of all operators C in L(H s,(γ,δ) (M ), H ∞,(γ−µ,δ−µ) (M )) such that C and its formal adjoint C ∗ induce continuous operators ∞,(γ−µ,δ−µ)

C : H s,(γ,δ) (M ) → HPedge ,Pcorner (M )

(3.22)

and ∞,(−γ,−δ)

C ∗ : H s,(−γ+µ,−δ+µ) (M ) → HQedge ,Qcorner (M )

(3.23)

for C-dependent asymptotic types Pedge , Pcorner and Qedge , Qcorner, associated with, respectively, the weight data (γ − µ, Θ), (δ − µ, Θ) and (−γ, Θ), (−δ, Θ), for all s ∈ R.


65

Remark 3.4. Analogous constructions make sense on a noncompact manifold M with corner v. In this case, we have the spaces s,(γ,δ)

s,(γ,δ)

H[comp)/[loc) (M ), H[comp)/[loc),Pedge ,Pcorner (M ),

(3.24)

−∞ and so on. Then L−∞ M +G (M, g) is the space of all operators C ∈ LM +G (M \ {v}, g1 ) for g1 = (γ, γ − µ, Θ)

such that C and C ∗ induce continuous operators ∞,(γ−µ,δ−µ)

s,(γ,δ)

C : H[comp) (M ) → H[loc),Pedge ,Pcorner (M ),

(3.25)

and C ∗ : H[comp)

s,(−γ+µ,−δ+µ)

∞,(−γ,−δ)

(M ) → H[loc),Qedge ,Qcorner (M )

(3.26)

for C-dependent asymptotic types Pedge , Pcorner and Qedge , Qcorner, associated with, respectively, the weight data (γ − µ, Θ), (δ − µ, Θ) and (−γ, Θ), (−δ, Θ), for all s ∈ R. 3.2. Mellin plus Green corner operators Let µ ∈ R and let g = (g1 , g2 ) be weight data with g1 = (γ, γ − µ, Θ1 ) and g2 = (δ, δ − µ, Θ2 ), where γ, δ ∈ R and for j = 1, 2, Θj = (−(kj + 1), 0],

kj ∈ N.

Then we are interested in studying in this section the space LµM+G (B ∧ , g) of Mellin plus Green corner operators of order µ. For simplicity, we assume that k = k1 = k2 . The general case, even for k1 = ∞ or k2 = ∞, is completely similar. The motivation for the space LµM +G (B ∧ , g) is quite similar to that of the simpler space LM +G (X ∧ , g) for a closed C ∞ manifold X, where we only have the weight data g1 = (γ, γ − µ, Θ) from the conical singularity as r → 0. Recall that these operator spaces are generated automatically when we construct parametrices of elliptic differential operators of Fuchs type on a manifold with conical singularities. Another role of these spaces is to a priori incorporate the wealth of individual asymptotic types that may occur in the elliptic regularity of solutions to elliptic equations. We let LµM +G (B ∧ , g) be the space defined by LµM+G (B ∧ , g) = LµM+G (B ∧ , g)corner + LµM +G (B ∧ , g)O ⊂ LµM +G (B ∧ , g1 ), (3.27) where the elements of each of the spaces are defined in this section. µ µ Let MM +G,O (B, g1 ) denote the space of all h(v) ∈ A(C, LM +G (B, g1 )) such that h(z) ∈ LµM +G (B, g1 ; Γβ ), z ∈ Γβ ,

66


uniformly for all β on compact subsets of R. (Uniformity again means that the estimates for the symbols can be made independently of β on compact subsets of R.) The main contributions of LµM +G (B ∧ , g)O are Mellin operators of the form t−µ opM

δ−d/2

(h),

(3.28)

where

µ h(t, v) ∈ C ∞ (R+ , MM +G,O (B, g1 )), which are independent of t for large t, i.e., for t ≥ C, where C is a positive constant.

Theorem 3.5. Let µ h(t, v) ∈ C ∞ (R+ , MM +G,O (B, g1 ))

and ν l(t, v) ∈ C ∞ (R+ , MM +G,O (B, h1 )),

where both are independent of t for large t, g1 = (γ − ν, γ − (µ + ν), Θ)

and

h1 = (γ, γ − ν, Θ).

Then we have t−µ opM

δ−ν−d/2

(h)t−ν opM

δ−d/2

(l) = t−(µ+ν) opM

δ−d/2

((T ν h)l),

(3.29)

µ+ν where (T ν h)l ∈ C ∞ (R+ , MM +G,O (B, g1 ◦ h1 )) can be expressed by a Mellin oscillatory integral ∞ ∞ ((T ν h)l)(t, v) = t−iτ (T ν h)(t, v + iτ )l(tt , v)dt /t d¯τ (3.30) −∞

0

and (T ν h)(t, v + iτ ) = h(t, v + ν + iτ ). µ Theorem 3.6. Let h(t, v) ∈ C ∞ (R+ , MM +G,O (B, g1 )) be independent of t for large −δ−d/2

t. Then the formal adjoint of t−µ opM (h) has the form t−µ opM where ∞ ∞ h∗ (t, v) = tiτ h∗ (tt , d + 1 − v + iτ)dt /t d¯τ δ−d/2

−∞

(T µ h∗ ), (3.31)

0

and ∗ denotes the pointwise formal adjoint of operators in LµM+G (B, g1 ). Theorems 3.5 and 3.6 are similar in structure as a corresponding result in [14]. However, the nature of the values of the holomorphic Mellin symbols here µ is fairly different from what is assumed in [14], namely, MO (X). Detailed proofs will be given in a separate paper. Also, Theorems 2.3 and 2.6 have analogs in the present context. µ Theorem 3.7. Let f (v) ∈ MM +G,O (B, g1 ) be such that

f (v) ∈ Lµ−1 M +G (B, g1 ; Γβ ),

v ∈ Γβ ,

µ−1 for some fixed real number β. Then we have f (v) ∈ MM +G,O (B, g1 ).


67

Theorem 3.8. Let f (v) ∈ LµM+G (B, g1 ; Γβ ) for some β ∈ R. Then there is an µ h ∈ MM +G,O (B, g1 ) such that h|Γβ = f (3.32) −∞ mod L (B, g1 ; Γβ ). As in Remark 2.11, the space L−∞ (B, g1 ) is a union of Fréchet spaces. Now, the space MR−∞ (B, g1 ) of smoothing Mellin symbols with continuous asymptotic type R is defined to be the set of all f ∈ A(C \ V, L−∞ (B, g1 )), where V = πC R is such that V ∩ {v ∈ C : c ≤ Re v ≤ c } is compact for all real numbers c and c with c ≤ c , and for all such c and c , we can find a compact subset Kc,c of V and a functional ζ in A (Kc,c , L−∞ (B, g1 )) for which hc,c defined by hc,c (v) = f (v) − Mσ,t→v (ω(t)ζw , t−w )(v)

lies in A({v ∈ C : c ≤ Re v ≤ c }, L

−∞

(3.33)

(B, g1 )) and

hc,c (v) ∈ S(Γβ , L−∞ (B, g1 )),

v ∈ Γβ ,

(3.34)

uniformly for all β in compact subsets of [c, c ]. (Uniformity is understood in the same sense as described in Definition 2.1 (ii), Mσ is again the weighted Mellin transform with weight σ such that 1/2 − σ > max{Re v : v ∈ Kc,c }.) The space MR−∞ (B, g1 ) is a union of Fréchet spaces. For functions ϕ and ψ, we write ϕ ≺ ψ if ψ is identically equal to 1 on supp ϕ. For the following definition, we choose cut-off functions and on the t-half-axis with ≺ , excision functions θ and θ such that θ ≺ θ and they are of the form θ = 1 − , θ = 1 − for another cut-off function . Definition 3.9. The space LµM +G (B ∆ , g) is defined to be the set of all operators A of the form A = Acorner + θAedge θ + C, (3.35) µ −∞ ∆ ∧ where C ∈ L (B , g) as in Remark 3.4, Aedge ∈ LM +G (B , g1 ) as in (2.58) applied to B ∧ as a noncompact manifold with smooth edge R+ × Y , and Acorner = t−µ opM

δ−d/2

for some h ∈

µ MM +G,O (B, g1 ),

(h) + {smoothing Mellin}

(3.36)

and {smoothing Mellin} is a finite linear combiδ −d/2

nation of operators of the form t−µ+j opMj (fj ), where fj ∈ MR−∞ (B, g1 ) for j continuous Mellin asymptotic types Rj with πC Rj ∩ Γ(d+1)/2−δj = ∅ and weights δ − j ≤ δj ≤ δ for 0 ≤ j ≤ k, where Θ = (−(k + 1), 0]. Without loss of generality, we assume that there is only one summand for j = 0, while for any j > 0 we may have several summands, which are natural in the case of continuous Mellin asymptotic types. The operators A in Definition 3.9 admit a pseudo-differential calculus with a pair σ(A) of principal symbols given by σ(A) = (σ1 (A), σ2 (A)),

(3.37)

68


and complete symbols or amplitude functions. We postpone the calculus to the following section, since we are mainly interested in a compact corner manifold rather than B ∆ . (In the noncompact case we have to deal with [comp)/[loc)-spaces, and localized or properly supported operators when we study compositions.) However, the symbols (3.37) are easier to be explained in local terms. First, we note that an element A in LµM+G (B ∆ , g) as an operator in the edge calculus on B ∧ is locally near the edge R+ × Y , in the splitting of variables (t, y, r, x) ∈ R+ × Ω × X ∧ , a classical pseudo-differential operator on R+ × Ω based on the Fourier transform in (t, y) with symbol µ s,γ a(t, y, τ, η) ∈ Scl (R+ × Ω × R1+q (X ∧ ), K∞,γ−µ (X ∧ )). τ,η ; K

Since we consider here only the asymptotic part of the theory, it follows that far from the edge R+ × Y the operator A is smoothing, and hence there is no relevant symbolic information. In other words, A ∈ L−∞ (R+ × int B) and σ0 (A) = 0 (See the notation in (3.46).) Similarly, as in Definition 2.13, the symbol has the form a(t, y, τ, η) = t−µ a ˜(t, y, tτ, η), where µ 1+q a˜(t, y, τ˜, η) = (m ˜ + g˜)(t, y, τ˜, η) ∈ RM +G (R+ × Ω × Rτ˜,η , g1 )

(3.38)

µ for a Green symbol g˜(t, y, τ˜, η) ∈ RG (R+ × Ω × R1+q , g1 ) which is smooth in t up to t = 0, while m(t, ˜ y, τ˜, η) is a finite linear combination of terms

m ˜ jα (t, y, τ˜, η) = r−µ+j ωτ˜,η opMjαr γ

−n/2

(fjα )(t, y)(˜ τ , η)α ωτ˜,η ,

(3.39)

for arbitrary |α| ≤ j, 0 ≤ j ≤ k, with Mellin symbols fjα (t, y, v) ∈ C ∞ (R+ × Ω, MR−∞ (X)), jα weights γjα for which γ − j ≤ γjα ≤ γ

and πC Rjα ∩ Γ(n+1)/2−γjα = ∅,

and cut-off functions ω and ω on R+ . The principal edge symbol σ1 (A) of A is defined by γ −n/2 σ1 (A)(t, y, τ, η) = t−µ r−µ+j ω|(tτ,η)| opMjαr (fjα )(t, y)(tτ, η)α ω|(tτ,η)| j=|α|

as an operator function Ks,γ (X ∧ ) → K∞,γ−µ (X ∧ ) for |(τ, η)| = 0. Here,

(3.40)

ω|˜τ ,η| (r) = ω(r|˜ τ , η|), and the same holds for ω . The principal edge symbol σ1 (A) satisfies the homogeneity relation to the effect that σ1 (A)(t, y, λτ, λη) = λµ κλ σ1 (A)(t, y, τ, η)κ−1 λ

(3.41)


69

for all λ ∈ R+ . We also note here that the reduced principal edge symbol σ ˜1 (A) is given by γ −n/2 σ ˜1 (A)(t, y, τ, η) = r−µ+j ω|(τ,η)| opMjαr (fjα )(t, y)(τ, η)α ω|(τ,η)| . j=|α|

The second component of (3.37) given by σ2 (A)(v) = h(0, v) + f0 (v),

(3.42)

is called the corner conormal symbol of A and is determined by the ingredients in (3.36). In the case when g2 = (δ, δ − µ, Θ), the complex variable v in (3.42) varies on the weight line Γ(d+1)/2−δ . The corner conormal symbol has analogous meaning as that of the cone theory. In the present case, it takes values in LµM +G (B, g1 ), which is defined by the formula (2.58), and is a family of continuous operators σ2 (A)(v) = H s,γ (B) → H ∞,γ−µ (B),

(3.43)

parametrized by v ∈ Γ(d+1)/2−δ for all s ∈ R. 3.3. Ellipticity in the asymptotic corner algebra Let M be a compact manifold with corner c, i.e., M \{c} is a manifold with smooth edge, and c has a neighbourhood V modelled on B ∆ for a compact manifold B with smooth edge Y. The space M is stratified with the strata {s0 (M ), s1 (M ), s2 (M )},

(3.44)

defined as the smooth manifolds by s2 (M ) = {c}, s1 (M ) = edge of (M \ {c}) and s0 (M ) = M \ {s1 (M ) ∪ s2 (M )}. Then M = s0 (M )∪s1 (M )∪s2 (M ), which is a disjoint union. We also write int(M ) instead of s0 (M ). Observe that M defined by M = s1 (M ) ∪ s2 (M ) is a manifold with conical singularity s2 (M ). As in (3.38), we have the subspace LµM +G (M, g) of

LµM +G (M

(3.45)

\ {v}, g1 ) consisting of all operators A of the form A = Acorner + θAedge θ + C.

Here, and are cut-off functions, i.e., they are supported in a small neighbourhood of {c} and identically equal to 1 close to c. Moreover, ≺ . θ and θ are excision functions, i.e., they are identically equal to 0 close to c, and θ ≺ θ . Furthermore, θ = 1 − and θ = 1 − for another cut-off function . In addition, Acorner ∈ LµM+G (B ∆ , g),

Aedge ∈ LµM +G (M \ {v}, g1 )

70


and C ∈ L−∞ (M, g). See Section 3.1 for a similar operator space. As usual in the pseudo-differential analysis on a stratified space in, say, [45], the strata contribute a principal symbolic hierarchy σ(A) spelled out as σ(A) = {σ0 (A), σ1 (A), σ2 (A)},

(3.46)

where σ0 (A) is the homogeneous principal symbol of A in the standard sense. By virtue of the fact that LµM +G (M, g) ⊂ L−∞ (int M ), we have σ0 (A) = 0 for every A ∈ LµM +G (M, g). We can now give the meaning of the other symbolic components. From the relation (3.45), the homogeneous principal symbol σ1 (A) of the edge calculus, namely, in the edge variables (t, y) ∈ R+ × Ω, where Ω is to be understood as local coordinates on Y , and the associated covariables (τ, η), is an operator function σ1 (A)(t, y, τ, η) : Ks,γ (X ∧ ) → K∞,γ−µ (X ∧ ),

(3.47)

for (τ, η) = 0. The function σ1 (A)(t, y, τ, η) is LM+G (X ∧ , g1 )-valued and can be written in the form σ1 (A)(t, y, τ, η) = t−µ σ ˜1 (A)(t, y, tτ, η)

(3.48)

locally near t = 0, for the so-called reduced edge symbol σ ˜1 (A)(t, y, τ˜, η) which is smooth up to t = 0 as an LM +G (X ∧ , g1 )-valued operator function. Moreover, σ2 (A)(v) is again the corner conormal symbol defined as in the preceding section. Theorem 3.10. An operator A in LµM+G (M, g) induces continuous operators A : H s,(γ,δ) (M ) → H s−µ,(γ−µ,δ−µ) (M )

(3.49)

and s,(γ,δ)

s−µ,(γ−µ,δ−µ)

A : HPedge ,Pcorner (M ) → HQedge ,Qcorner

(M )

(3.50)

for all s ∈ R and all pairs of asymptotic types (Pedge , Pcorner ) for some resulting pairs of asymptotic types (Qedge , Qcorner). Theorem 3.11. Let A ∈ LµM+G (M, g) and let B ∈ LνM+G (M, h) for g = ((γ, γ − µ, Θ), (δ, δ − µ, Θ)), ˜ δ˜ − ν, Θ)), h = ((˜ γ , γ˜ − ν, Θ), (δ, γ = γ˜ − ν and δ = δ˜ − ν. Then AB ∈ Lµ+ν M+G (M, g ◦ h) for ˜ δ˜ − (µ + ν), Θ)) g ◦ h = ((˜ γ , γ˜ − (µ + ν), Θ), (δ, and we have σ1 (AB) = σ1 (A)σ1 (B), σ2 (AB) = (T ν σ2 (A))σ2 (B), where (T ν f )(v) = f (v + ν).

(3.51)


71

Theorem 3.12. Let A ∈ LµM +G (M, g) for g = ((γ, γ − µ, Θ), (δ, δ − µ, Θ)). Let A∗ be the formal adjoint A defined by (Au, v) = (u, A∗ v) for all u, v ∈ C0∞ (int M ). Then we have A∗ ∈ LµM +G (M, g∗ ) for g∗ = ((−γ + µ, −γ, Θ), (−δ + µ, −δ, Θ)) and σ1 (A∗ ) = σ1 (A)∗ , σ2 (A∗ )(v) = (σ2 (A))∗ (d + 1 + µ − v), where ∗ denotes the formal adjoint in LµM +G (B, g1 ).

(3.52)

The structures of the corner calculus on M hitherto developed have similar applications to the characterization of parametrices as in the edge case illustrated in Section 2.5. Recall that on a manifold W with smooth edge, we have the space Diffµdeg (W ) of edge-degenerate differential operators of order µ. The latter space is Fréchet in a natural way. If M is a manifold with corner c, locally near c modelled on B ∆ for a manifold B with edge, then by definition, W = M \ {c} is again a manifold with edge. Let Diffµdeg(M ) denote the subspace of all operators A in Diffµdeg (M \{c}) that are locally near c expressed in the variables (t, b) ∈ R+ ×B as A = t−µ

µ

aj (t)(−t∂t )j ,

(3.53)

j=0 µ where the coefficients aj are in C ∞ (R+ , Diffµ−j deg (B)). Operators A ∈ Diffdeg (M ) are also called corner-degenerate. The principal symbolic hierarchy (3.46) of an operator A ∈ Diffµdeg(M ) first contains the homogeneous principal symbol σ0 (A) in the usual sense as a smooth function invariantly defined on T ∗ (s0 (M )) \ {0}. This is also referred to as the interior symbol. Close to s1 (M ) and s2 (M ), the principal symbolic hierarchy is degenerate in a typical way, and is hence also called corner-degenerate. In addition, close to the edge of M \ {c}, we have the “edge-reduced” interior symbol σ ˜0 (A) in accordance with the interpretation of A ∈ Diffµdeg(M \ {c}) in (2.64). If B is described locally near Y by (r, x, y) ∈ R+ × X × Ω and the edge of M \ {c} near c is described locally by (t, y), then

σ ˜0 (A)(t, r, x, y, τ, ρ, ξ, η) = rµ σ0 (A)(t, r, x, y, r−1 τ, r−1 ρ, ξ, r−1 η)

(3.54)

and σ ˜0 (A) is smooth up to r = 0. Finally, close to the corner c of M , we have the ˜ “corner-reduced” interior symbol σ ˜0 (A) defined by ˜˜0 (A)(t, r, x, y, τ, ρ, ξ, η) = tµ rµ σ0 (A)(t, r, x, y, t−1 r−1 τ, r−1 ρ, ξ, r−1 η) σ = tµ σ ˜0 (A)(t, r, x, y, t−1 τ, ρ, ξ, η)

(3.55)

˜˜0 (A) is smooth up to t = 0 and r = 0. Although σ and σ ˜0 (A) and σ ˜˜0 (A) are completely determined by σ0 (A), these derived objects play a role in the definition of ellipticity. The operator A is called σ0 -elliptic if σ0 (A) never vanishes as usual,

72


˜˜0 (A) does not vanish and if in addition σ ˜0 (A) does not vanish up to r = 0, and σ up to t = 0 and r = 0. The component σ1 (A) is the edge symbol of A, interpreted as an element of Diffµdeg (M \ {c}) as in (2.31). In local variables (t, y) ∈ R+ × Ω on the edge of M \ {c} near c, we have σ1 (A)(t, y, τ, η) as a family of continuous operators σ1 (A)(t, y, τ, η) : Ks,γ (X ∧ ) → Ks−µ,γ−µ (X ∧ ),

(3.56)

as in (3.47). By virtue of the corner-degenerate behavior of A, we can form the reduced edge symbol σ ˜1 (A) by means of σ ˜1 (A)(t, y, τ, η) = tµ σ1 (A)(t, y, t−1 τ, η)

(3.57)

which is smooth up to t = 0 as an operator function in L(Ks,γ (X ∧ ), Ks−µ,γ−µ (X ∧ )) for all s ∈ R. An operator A ∈ Diffµdeg (M ) is called (σ0 , σ1 )-elliptic if it is σ0 -elliptic in the former sense, and if it is σ1 -elliptic, i.e., σ1 (A) is a family of bijections for some fixed weight γ, and if also σ ˜1 (A)(t, y, τ, η) is a family of bijections, up to t = 0. On the nature of σ1 -ellipticity we can make similar comments as in Section 2.5. We may have included extra edge conditions, but for simplicity, we just demand the bijectivity without such conditions. Clearly, this depends on the chosen weight γ, and the discussion on the correct weights is a another story. The symbolic component σ2 (A) is also operator-valued. It has the meaning of the corner conormal symbol as in the case of σc (A) mentioned at the very beginning of the paper, namely, σ2 (A)(v) =

µ

aj (0)vj : H s,γ (B) → H s−µ,γ−µ (B),

(3.58)

j=0

s ∈ R. See the formula (3.43) for comparison. The operator A is said to be (σ0 , σ1 , σ2 )-elliptic or just elliptic if it is (σ0 , σ1 )-elliptic as before, and if (3.58) is a family of isomorphisms for all v ∈ Γ(d+1)/2−δ for some prescribed corner weight δ ∈ R. It is by no means obvious to tell for which weights (γ, δ) these ellipticity conditions hold, or whether a corner-degenerate σ0 -elliptic operator admits such weights at all. These questions on specific operators of interest are often topics of separate investigations in, for instance, [10] or [26]. Other concrete examples are studied in [17] and [18]. Nevertheless, the corner pseudo-differential calculus contains an ample supply of (σ0 , σ1 , σ2 )-elliptic elements. Similarly, as in the edge case, by corner calculus on M we understand a pseudo-differential calculus on s0 (M ) that contains Diffµdeg (M ) together with the parametrices of elliptic elements. The ideas to construct a parametrix, sketched in Section 2.5, and to multiply the given operator A by some elliptic operator R of opposite order with σ0 (R) = σ0−1 (A) make sense in the corner case as well. It reduces the problem to ellipticity and parametrices of operators of the form I + D for D ∈ L0M +G (M, g0 ) for g0 = ((γ, γ, Θ), (δ, δ, Θ)).


73

Definition 3.13. An operator I + D for D ∈ L0M+G (M, g0 ), g0 = ((γ, γ, Θ), (δ, δ, Θ)), is called elliptic if the edge symbol σ1 (I + D) : Ks,γ (X ∧ ) → Ks,γ (X ∧ ),

(3.59)

as an operator function in the variables in T ∗ (s1 (M )) \ 0 is everywhere bijective. Moreover, if the reduced edge symbol σ ˜1 (I + D)(t, y, τ, η) close to the corner point c is bijective up to t = 0, and if σ2 (I + D)(v) : H s,γ (B) → H s,γ (B),

(3.60)

is a family of isomorphisms for all v ∈ Γ(d+1)/2−δ and all s ∈ R. Theorem 3.14. Let I + D for D ∈ L0M +G (M, g0 ), g0 = ((γ, γ, Θ), (δ, δ, Θ)), be elliptic. Then there is an element H ∈ L0M +G (M, g0 ) such that I + H is a parametrix in the sense that (I + H)(I + D) − I, (I + D)(I + H) − I mod L

−∞

(3.61)

(M, g0 ).

As a consequence, we have σj (I + H) = σj−1 (I + D),

j = 1, 2.

If A is an elliptic operator with a nontrivial interior corner-degenerate symbol, then, as soon as we apply the corresponding complete corner algebra that includes compositions within the structure with the corresponding symbolic rules, we obtain the operator P = (1 + H)R, which is a parametrix of A. 3.4. Notes The focus of this paper is on the asymptotic part of the corner theory over M , namely, an algebra of smoothing, albeit singular operators on s0 (M ) with continuous asymptotics at M \ {c}. The non-smoothing complementary flat part is developed in [39] that includes operators with discrete asymptotics in the corner axis direction. A separate investigation should establish the necessary operator algebra aspects such as compositions for operators with continuous asymptotics and non-smoothing flat corner operators. Of course, the expected structures of such a generalization belong to the motivation for the approach as seen from the comments after Theorem 3.14. Another aspect, also addressed before, is that we have chosen to ignore the possibility of including trace and potential edge conditions. Such conditions are by no means superfluous in concrete applications. For instance, in the theory of pseudo-differential boundary value problems on a manifold with smooth boundary, which is a special case of a manifold with edge, say, with the transmission property at the boundary, it is natural to consider 2 × 2 block matrices of operators

74


A = (Aij )i,j=1,2 , as in [3]. While A11 just consists of the non-smoothing pseudodifferential operator “itself” plus a Green operator, or plus a smoothing Mellin plus Green operator in the calculus without the transmission property as in [12], [31] and [46], the entry A21 plays the role of a trace and A12 of a potential operator, A22 is simply a standard pseudo-differential operator on the boundary. We can characterize not only classical boundary value problems such as the Dirichlet and Neumann problem for Laplacians, together with their parametrices as elements of the respective 2 × 2 block matrix pseudo-differential algebra, but also operators appearing in mixed problems such as the Zaremba problem with jumping Dirichlet/Neumann conditions and their parametrices by a corresponding pseudodifferential algebra. Here, it is the edge algebra. Or after reducing the problem to the boundary, it is the algebra of pseudo-differential boundary value problems without the transmission property containing, e.g., the “Dirichlet-to-Neumann operator” which does not have the transmission property at the interface. This is just an occasion where smoothing Mellin operators come into play, even when the interface is smooth, and in an iterated version in the corner operator set-up when the interface has conical singularities. See, e.g., [18] for symbols with values in Boutet de Monvel’s calculus of boundary value problems. If the structure of an operator algebra is the focus of interest in an investigation, then it makes sense to drop for a while the trace and potential operators. In fact, those operators are very close to Green operators in the upper left corner. More precisely, in the edge case, it suffices to replace the symbol spaces in (2.49) and (2.50) in Definition 2.12 by, respectively, m Scl (Ω × Rq ; Ks,γ;e (X ∧ ) ⊕ Cj− , SPγ−µ (X ∧ ) ⊕ Cj+ )

(3.62)

and −γ m Scl (Ω × Rq ; Ks,−γ+µ;e (X ∧ ) ⊕ Cj+ , SQ (X ∧ ) ⊕ Cj− ),

(3.63)

for some j− , j+ ∈ N that describe the number of potential and trace entries. (Here, the group action on Cj± is assumed to be trivial, i.e., the identity for all λ). After such a change, the formal properties of the calculus like compositions and so on are completely analogous to the case j− = j+ = 0. This is worth to be checked in any new case, but it is more or less straightforward. In the 2 × 2 block matrix version of the edge or corner calculus, the ellipticity of the extra edge conditions means that σ1 (A) = σ1 (Aij )i,j=1,2 : Ks,γ (X ∧ ) ⊕ Cj− → Ks−µ,γ−µ (X ∧ ) ⊕ Cj+ ,

(3.64)

∗

is bijective for all points in T (s1 (M ))\{0} and also that locally near c the reduced edge symbol σ ˜1 (A)(t, y, τ, η) = tµ σ1 (A)(t, y, t−1 τ, η) is bijective up to t = 0. This is an analog of the Shapiro-Lopatinskij condition, which is well known in boundary value problems. In addition, σ2 (A) can be put in the form of a block matrix, and is required to be bijective on a given weight line in the complex plane.


75

In the following comments on more background and intents of the asymptotic corner analysis, we do not rule out the above-mentioned non-smoothing flat operators of the calculus and the scenario of 2 × 2 block matrices. However, let us give an idea why in principle we do not lose too much information when we restrict the calculus to the case j− = j+ = 0. If A is given in the block matrix as A = (Aij )i,j=1,2 and A : H s,(γ,δ) (M ) ⊕ H s,δ (M , J− ) → H s−µ,(γ−µ,δ−µ) (M ) ⊕ H s−µ,δ−µ (M , J+ ) (3.65) is elliptic and hence Fredholm, (J− , J+ ) = (Cj− , Cj+ ) or, in general, (J− , J+ ) is a pair of non-trivial vector bundles over M , then as in pseudo-differential algebras on singular spaces of lower rank [39] and [46], we can find elliptic operators R and L of the form R = t(1 + R11 , R21 ) : H s,(γ,δ) (M ) → H s,(γ,δ)(M ) ⊕ H s,δ (M , J− )

(3.66)

and L = (1+L11 , L12 ) : H s−µ,(γ−µ,δ−µ) (M )⊕H s−µ,δ−µ (M , J+ ) → H s−µ,(γ−µ,δ−µ) (M ), (3.67) where R11 , R21 and L11 , L12 belong to the block matrix-valued L0M +G (M, . . . )analogs of the operator spaces studied in Section 3.3, such that (3.66) and (3.67) are isomorphisms. Then the operator A˜ given by A˜ = LAR : H s,(γ,δ) (M ) → H s−µ,(γ−µ,δ−µ) (M )

(3.68)

is an element of the type of an upper left corner, elliptic and Fredholm. The transformation A → A˜ works in the opposite direction, too, and then, using the fact that the inverses L−1 and R−1 also belong to the calculus, we can reduce Fredholm property and the construction of parametrices to the case of upper left corners. Clearly, in the elliptic theory, the considerations immediately extend to the case of systems and even to operators acting on spaces of distributional sections of vector bundles. Note that the constructions of operators (3.66) and (3.67) rely on the fact that we can always find smoothing Mellin symbols f on the half-axis such that the operator 1 + ωopM (f )ω : L2 (R+ ) → L2 (R+ ) has a prescribed index, which can be computed by the formula (2.8). The fact that there are smoothing operators ωopM (f )ω in L−∞ (R+ ) on the half-axis that can be used to compute the indices by adding to a Fredholm operator (the identity operator here) is also typical in our Mellin plus Green operator scenario over a corner manifold M . See also Theorem 3.14. That is why the operators studied in Sections 3.2 and 3.3, albeit smoothing on s0 (M ), have a remarkably non-trivial symbolic structure and admit a notion of ellipticity.

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References [1] M.F. Atiyah and R. Bott, The index problem for manifolds with boundary, in Differential Analysis, Tata Institute Bombay, Oxford University Press, 1964, 175–186. [2] M.F. Atiyah and I.M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433. [3] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51. [4] D. Calvo, C.-I. Martin and B.-W. Schulze, Symbolic structures on corner manifolds, in Microlocal Analysis and Asymptotic Analysis, Keio University, 2005, 22–35. [5] A. Dasgupta, The Twisted Laplacian, the Laplacians on the Heisenberg Group and SG Pseudo-Differential Operators, Ph.D. Dissertation, York University, 2008. [6] A. Dasgupta, Ellipticity of Fredholm pseudo-differential operators on Lp (Rn ), in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkh¨ auser, 2009, 107–116. [7] A. Dasgupta and M.W. Wong, Spectral theory of SG pseudo-differential operators on Lp (Rn ), Studia Math. 187 (2008), 185–197. [8] A. Dasgupta and M.W. Wong, Spectral invariance of SG pseudo-differential operators on Lp (Rn ), in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications 205, Birkh¨ auser, 2010, 51–57. [9] G. De Donno and B.-W. Schulze, Meromorphic symbolic structures for boundary value problems on manifolds with edges, Math. Nachr. 279 (2006), 1–32. [10] N. Dines, Elliptic Operators on Corner Manifolds, Ph.D. Dissertation, University of Potsdam, 2006. [11] Y.V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Operator Theory: Advances and Applications 93, Birkh¨ auser, 1997. [12] G.I. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Transl. of Nauka, Moskva, 1973, Math. Monographs 52, American Mathematical Society, 1980. [13] H.-J. Flad, R. Schneider and B.-W. Schulze Asymptotic regularity of solutions of Hartree-Fock equations with Coulomb potential, Math. Meth. Appl. Sci. 31 (2008), 2172–2201. [14] J.B. Gil, B.-W. Schulze and J. Seiler, Cone pseudodifferential operators in the edge symbolic calculus, Osaka J. Math. 37 (2000), 219–258. [15] I.C. Gohberg and E.I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Math. USSR Sbornik 13 (1971), 603–625. [16] G. Harutyunyan and B.-W. Schulze, The relative index for corner singularities, Integral Equations Operator Theory 54 (2006), 385–426. [17] G. Harutyunyan and B.-W. Schulze, The Zaremba problem with singular interfaces as a corner boundary value problem, Potential Analysis 25 (2006), 327–369. [18] G. Harutyunyan and B.-W. Schulze, Elliptic Mixed, Transmission and Singular Crack Problems, European Mathematical Society, 2008. [19] L. H¨ ormander, The Analysis of Linear Partial Differential Operators, I, II, SpringerVerlag, 1983. [20] P. Jeanquartier, Transformation de Mellin et développements asymptotiques, Enseign. Math. 25 (1979), 285–308.


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[21] D. Kapanadze and B.-W. Schulze, Crack Theory and Edge Singularities, Kluwer Academic Publishers, 2003. [22] V.A. Kondratyev, Boundary value problems for elliptic equations in domains with conical points, Trudy Mosk. Mat. Obshch. 16 (1967), 209–292. [23] T. Krainer, Parabolic Pseudodifferential Operators and Long-Time Asymptotics of Solutions, Ph.D. Thesis, University of Potsdam, 2000. [24] X. Liu and B.-W. Schulze, Boundary value problems in edge representation, Math. Nachr. 280, (2007), 1–41. [25] L. Maniccia and B.-W. Schulze, An algebra of meromorphic corner symbols, Bull. Sci. Math. 127 (2003), 55–99. [26] C.-I. Martin and B.-W. Schulze, Parameter-dependent edge operators, Ann. Global Anal. Geom., to appear. [27] S. Molahajloo, A characterization of compact pseudo-differential operators on S1 , in Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, Birkh¨ auser, 2011, this volume, 27–32. [28] S. Molahajloo and M.W. Wong, Pseudo-differential operators on S1 , in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkh¨ auser, 2009, 297–306. [29] S. Molahajloo and M.W. Wong, Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on S1 , J. Pseudo-Differ. Oper. Appl. 1 (2010), 183–205. [30] M. Perhayati, Spectral theory of pseudo-differential operators on S1 , in PseudoDifferential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, Birkh¨ auser, 2011, this volume, 15–23. [31] S. Rempel and B.-W. Schulze, Parametrices and boundary symbolic calculus for elliptic boundary problems without transmission property, Math. Nachr. 105, (1982), 45–149. [32] S. Rempel and B.-W. Schulze, Mellin symbolic calculus and asymptotics for boundary value problems, in Seminar Analysis 1984/1985, Karl-Weierstrass Institut, Berlin, 1985, 23–72. [33] S. Rempel and B.-W. Schulze, Branching of asymptotics for elliptic operators on manifolds with edges, in Proc. Partial Differential Equations, Banach Center Publ. 19, PWN Polish Scientific Publisher, 1984. [34] B.-W. Schulze, Regularity with continuous and branching asymptotics for elliptic operators on manifolds with edges, Integral Equations Operator Theory 11 (1988), 557–602. [35] B.-W. Schulze, Ellipticity and continuous conormal asymptotics on manifolds with conical singularities, Math. Nachr. 136 (1988), 7–57. [36] B.-W. Schulze, Pseudo-differential operators on manifolds with edges, in Symp. Partial Differential Equations, Holzhau 1988, Teubner-Texte zur Mathematik 112, Leipzig, 1989, 259–287. [37] B.-W. Schulze, The Mellin pseudo-differential calculus on manifolds with corners, in Symp. Analysis in Domains and on Manifolds with Singularities, Breitenbrunn 1990, Teubner-Texte zur Mathematik 131, Leipzig, 1992, 208–289. [38] B.-W. Schulze, Pseudo-Differential Operators on Manifolds with Singularities, NorthHolland, 1991. [39] B.-W. Schulze, Pseudo-Differential Boundary Value Problems, Conical Singularities, and Asymptotics, Akademie Verlag, 1994.

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[40] B.-W. Schulze, The variable discrete asymptotics in pseudo-differential boundary value problems I, in Pseudo-Differential Calculus and Mathematical Physics, Advances in Partial Differential Equations, Akademie Verlag, 1994, 9–96. [41] B.-W. Schulze, The variable discrete asymptotics in pseudo-differential boundary value problems II, in Boundary Value Problems, Schr¨ odinger Operators, Deformation Quantization, Advances in Partial Differential Equations, Akademie Verlag, 1995, 9–69. [42] B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, 1998. [43] B.-W. Schulze, Operators with symbol hierarchies and iterated asymptotics, Publ. Res. Inst. Math. Sci. Kyoto 38, (2002), 735–802. [44] B.-W. Schulze, Boundary value problems with the transmission property, in PseudoDifferential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications 205, Birkh¨ auser, 2010, 1–50. [45] B.-W. Schulze, The iterative structure of the corner calculus, in Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, Birkh¨ auser, 2011, this volume, 79–103. [46] B.-W. Schulze and J. Seiler, The edge algebra structure of boundary value problems, Ann. Glob. Anal. Geom. 22 (2002), 197–265. [47] B.-W. Schulze and J. Seiler, Edge operators with conditions of Toeplitz type, J. Inst. Math. Jussieu 5 (2006), 101–123. [48] B.-W. Schulze and A. Volpato, Green operators in the edge calculus, Rend. Sem. Mat. Univ. Pol. Torino 66 (2008), 29–58. [49] B.-W. Schulze and A. Volpato, Continuous and variable discrete asymptotics, preprint. [50] J. Seiler, The cone algebra and a kernel characterization of Green operators, in Approaches to Singular Analysis, Adv. Partial Differential Equations, Operator Theory: Advances and Applications 125, Birkh¨ auser, 2001, 1–29. [51] M.I. Vishik and G.I. Eskin, Convolution equations in a bounded region, Uspekhi Mat. Nauk. 20 (1965), 89–152. [52] I. Witt, On the factorization of meromorphic Mellin symbols, in Approaches to Singular Analysis, Advances in Partial Differential Equations, Operator Theory, Advances and Applications, Birkh¨ auser, 2002, 279–306. [53] M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999. B.-W. Schulze Institut f¨ ur Mathematik, Universit¨ at Potsdam Am Neuen Palais 10 D-14469 Potsdam, Germany e-mail: [email protected] M.W. Wong Department of Mathematics and Statistics, York University 4700 Keele Street Toronto, Ontario M3J 1P3, Canada e-mail: [email protected]

The Iterative Structure of the Corner Calculus B.-W. Schulze Abstract. We give a brief survey on some new developments on elliptic operators on manifolds with regular geometric singularities. The material corresponds to an extended version of talks given by the author at the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27–31, 2008, at the MSRI, University of California at Berkeley, see also [56], and at the 7th ISAAC Congress, July 13–18, 2009, at Imperial College London. Mathematics Subject Classification (2000). Primary 35S35; Secondary 35J70. Keywords. Ellipticity on manifolds with singularities, stratified spaces, corner pseudo-differential operators, principal symbolic hierarchies.

1. Introduction Manifolds M with higher corners or edges of order k ∈ N are (in our notation) special stratified spaces, where k = 0 corresponds to smoothness, k = 1 to conical or edge singularities. Manifolds with singularities of order k form a category Mk . Ellipticity of operators will be expressed by a principal symbolic hierarchy σ = (σj )0≤j≤k with σ0 being the standard homogeneous principal symbol on the main stratum int M , while the components σj , j > 0, live on the other strata and are operator valued. Remark 1.1. (i) The half-axis R+ can be regarded as a manifold with conical singularity 0. (ii) The half-space R+ × Ω for an open set Ω ⊆ Rq is an example of a manifold with edge Ω and model cone R+ . (iii) Let X be a closed compact C ∞ manifold, then the quotient space X ∆ := (R+ × X)/({0} × X) which is an infinite cone with vertex v, represented by {0} × X, is a manifold with conical singularity v and base X. (iv) The wedge X ∆ × Ω with X and Ω as before is a manifold with edge Ω and model cone X ∆ . L. Rodino et al. (eds.), Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, DOI 10.1007/978-3-0348-0049-5_5, © Springer Basel AG 2011

79

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B.-W. Schulze

Remark 1.2. Consider a Riemannian metric of the form dr2 + r2 gX on the open stretched cone X ∧ := R+ × X where gX is a Riemannian metric on X. Then for the associated Laplace-Beltrami operator we obtain (for m = 2) j m ∂ −m A=r aj (r) −r , (1.1) ∂r j=0 with coefficients aj ∈ C ∞ (R+ , Diffm−j (X)). More generally, if we consider a Riemannian metric dr2 + r2 gX + dy 2 on the open stretched wedge X ∧ × Ω, then the associated Laplace-Beltrami operator has the form (for m = 2) j ∂ −m A=r ajα (r, y) −r (rDy )α (1.2) ∂r j+|α|≤m

with coefficients ajα ∈ C ∞ (R+ × Ω, Diffm−(j+|α|) (X)). The principal symbolic hierarchies are as follows. In the conical case we have σ0 (A) ∈ C ∞ (T ∗ (X ∧ ) \ 0), the usual homogeneous principal symbol of A (degenerate at r = 0), and σ1 (A)(z) =

m

aj (0)z j : H s (X) → H s−m (X),

(1.3)

j=0

the principal conormal symbol. In the edge case we have σ0 (A) ∈ C ∞ (T ∗ (X ∧ × Ω) \ 0), the usual homogeneous principal symbol of A (here edge-degenerate at r = 0), and j ∂ −m σ1 (A)(y, η) = r ajα (0, y) −r (rη)α : Ks,γ (X ∧ ) → Ks−m,γ−m (X ∧ ), ∂r j+|α|≤m

(1.4) the homogeneous principal edge symbol (the meaning of Ks,γ (X ∧ ) will be explained below; the notation “K” comes from “Kegel”). Note that there is a similarity between edge symbols and boundary symbols of differential operators on a manifold with smooth boundary. Consider, for instance, a differential operk α ator A = k+β≤m bkβ (r, y)Dr Dy on the half-space R+ × Ω, with coefficients ∞ bkβ ∈ C (R+ × Ω). Apart from σ0 (A)(r, y, , η) = bkβ (r, y)k η β (1.5) k+|β|=m

we have the principal boundary symbol σ1 (A)(y, η) = bkβ (0, y)Drk η β : H s (R+ ) → H s−m (R+ ),

s ∈ R.

k+|β|=m

Boundary symbols are homogeneous in the sense σ1 (A)(y, λη) = λµ κλ σ1 (A)(y, η)κ−1 λ

for all λ ∈ R+ .

(1.6)

The Iterative Structure of the Corner Calculus

81

Here κλ : H s (R+ ) → H s−m (R+ ) is a strongly continuous group of isomorphisms, defined by (κλ u)(r) := λ1/2 u(λr), λ ∈ R+ . A similar relation holds for edge symbols, based on κλ : Ks,γ (X ∧ ) → Ks−m,γ−m (X ∧ ), s, γ ∈ R, where (κλ u)(r, x) := λ(n+1)/2 u(λr, x), λ ∈ R+ . In the present paper we give an idea on how to formulate algebras of (pseudodifferential) operators on int M that contain the (for the nature of singularities typical) differential operators, together with the parametrices of elliptic elements. More details may be found in [53], [54], and in a new monograph in preparation [57], see also the references below.

2. The category Mk Stratified spaces of different kind occur in numerous fields of mathematics and also in the applied sciences. Here we single out specific categories of such spaces where certain elements of the analysis of PDE can be formulated in an iterative manner. General references on stratified spaces are Fulton and MacPherson [15], or Weinberger [68]. Definition 2.1. A topological space M (under some natural conditions on the topology in general) is said to be a manifold with singularities of order k ∈ N, k ≥ 1, if (i) M contains a subspace Y ∈ M0 such that M \ Y ∈ Mk−1 ; (ii) Y has a neighbourhood V ⊆ M which is a (locally trivial) cone bundle over Y with fibre X ∆ for some X ∈ Mk−1 . Transition maps X ∆→ X ∆ are induced by restrictions of Mk−1 -isomorphisms R × X → R × X to R+ × X. This gives rise to corresponding transition maps ˜ for the respective X ∆ -bundles over Y . X∆ × Ω → X∆ × Ω Remark 2.2. Mk is a category with a natural notion of morphisms and isomorphisms. The notation “manifold” for objects in Mk is only used here for convenience. In general those are no topological manifolds (for instance, X ∆ for a torus X = S 1 × S 1 ); it is also common to speak about pseudo-manifolds. Remark 2.3. Y =: Y k is called the minimal stratum of M . The space M \ Y k ∈ Mk−1 contains a space Y k−1 ∈ M0 , such that (M \ Y k ) \ Y k−1 ∈ Mk−2 , etc. This yields a representation M = Y 0 ∪Y 1 ∪ Y 2 ∪··· ∪ Yk as a disjoint union of strata Y j ∈ M0 . We set int M := Y 0 , called the maximal stratum of M, and dim M := dim(int M ). Moreover, M is locally near Y j modelled ∆ ∆ ∆ on an Xj−1 -bundle over Y j , for Xj−1 ∈ Mj−1 with trivialisations Xj−1 ×U j , U j ⊆ j RdimY open.

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Remark 2.4. (i) The same topological space M can be stratified in different ways. For instance, we have M = Rn ∈ M0 but also M ∈ M1 when we set Y 1 = {0}, Y 0 = Rn \ {0}. (ii) Definition 2.1 does not cover the most general case where our methods work in principle. For instance, in Rn we may single out an embedded singularity Y 1 := {0} ∪ S n−1 for the unit sphere S n−1 in Rn and define in that way a stratified space M. Then M \ Y 1 belongs to M0 , however, Y 1 itself is not in M0 for n > 1. Since the main focus of this exposition is to illustrate the iterative analytical constructions we set such generalisations aside. Of course, some simple modifications in terms of partitions of unity allow us to consider more general configurations. Remark 2.5. M ∈ Mk , L ∈ Ml implies M × L ∈ Mk+l . There are many other interesting properties of the categories Mk that we do not discuss in detail here. It would be desirable to develop the connection of our analysis on singular spaces with the work from topological side. For instance, D. Trotman informed me in Berkeley on his works with coauthors, cf. [4] jointly with Bekka, and [28] with King. Let us now consider an example. Assume X0 ∈ M0 , and form the space M := (· · · ((X0∆ × Ω1 )∆ × Ω2 )∆ × · · · )∆ × Ωk−1 )∆ × Ωk

(2.1)

for open manifolds Ωj ∈ M0 of dimension qj , j = 1, . . . , k. Then we have Y k = Ωk , Y k−1 = (R+ × Ωk−1 ) × Ωk , Y k−2 = (R+ × Ωk−2 ) × (R+ × Ωk−1 )×Ωk , . . . ,

(2.2)

Y = (R+ × Ω1 ) × · · · × (R+ × Ωk−1 ) × Ωk , 1

Y 0 = (R+ × X0 ) × (R+ × Ω1 ) × · · · × (R+ × Ωk−1 ) × Ωk . The space (2.1) has a typical corner singularity of order k and can be used for local descriptions of any M ∈ Mk . Later on we denote the local variables in Y k by yk , in Y k−1 by (rk , yk−1 , yk ), and so on, corresponding to charts on the respective smooth manifold. For M ∈ Mk in general we often write sj (M ) := Y j , j = 1, . . . , k,

and

(2.3)

M := M \ s0 (M ), M := M \ s0 (M ), . . . ,

(2.4)

using M ∈ Mk−1 , such that the main stratum s0 (M ) of M again makes sense, etc., or also M (j) := M ... (with j primes), and M (0) := M. Then sk−1 (M ) = sk (M ), . . . , s0 (M ) = s1 (M ), etc. The spaces M j form a chain M = M (0) ⊃ M (1) ⊃ · · · ⊃ M (k) = sk (M )

(2.5)

where M ∈ Mk−j for all j, and j

dim M (j) > dim M (l) for every 0 ≤ j < l ≤ k.

(2.6)


83

3. Corner-degenerate operators Let M ∈ Mk , k ≥ 1, and Y := Y k ; then Diffm deg (M ) denotes the set of all A ∈ m m Diffm (M \ Y ), (Diff (·) = Diff (·) in the smooth case) such that close to Y in deg deg the local variables (r, x) ∈ R+ ×X

for dim Y = 0,

we have A=r

−m

m

and (r, x, y) ∈ R+ ×X ×Ω for

∂ aj (r) −r ∂r j=0

dim Y > 0,

j for dim Y = 0,

with coefficients aj ∈ C ∞ (R+ , Diffm−j deg (X)), and j ∂ A = r−m ajα (r, y) −r (rDy )α ∂r

for dim Y > 0,

j+|α|≤m

with coefficients ajα ∈ C ∞ (R+ × Ω, Diffdeg symbolic hierarchies are iteratively defined by

m−(j+|α|)

(X)), respectively. The principal

σ(A) := (σ(A|M \Y ), σk (A))

(3.1)

where σ(A|M \Y ) is known by the steps before, while σk (A)(z) =

m

aj (0)z j

for dim Y = 0, z ∈ C,

(3.2)

j=0

and σk (A)(y, η) = r

−m

j+|α|≤m

j ∂ ajα (0, y) −r (rη)α for dim Y > 0, (y, η) ∈ T ∗ Ω\0. ∂r

(3.3) σk (A)(z) takes values in Diffm deg (X) for dim Y = 0, and σk (A)(y, η) takes values in ∧ Diffm deg (X ) for dim Y > 0. Remark 3.1. If we dissolve the information in (3.1) with respect to the stratification s(M ) := (s0 (M ), s1 (M ), . . . , sk (M ))

(3.4)

of M, we obtain k + 1 components of σ(A), namely, σ(A) = (σ0 (A), σ1 (A), . . . , σk (A)), ∞

∗

(3.5)

with σ0 (A) ∈ C (T (int M )\0) being the standard homogeneous principal symbol of A on the main stratum int M = s0 (M ) of M , while the other components σj (A) are operator valued and associated with Y j = sj (M ), where σj (A) is of analogous form as (3.3) for dim sj (M ) > 0 (in the latter case parametrised by T ∗ sj (M ) \ 0). The symbol σk (A) for dim sk (M ) = 0 acts between weighted spaces ∆ H s,γ(k−1) (Xk−1 ) where M is locally near sk (M ) modelled on Xk−1 . Moreover, σj (A) for 0 < j < k where dim sj (M ) > 0 is analogous to (3.3), depending on ∧ points in T ∗ sj (M ) \ 0 and acting between spaces of the form Ks,γ(j) (Xj−1 ) where

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∧ M is modelled on Xj−1 × U j locally near sj (M ), and γ(j − 1) is a (j − 1)-tuple of weights. The nature of the spaces H s,γ(k−1) and Ks,γ(j) will be illustrated below. For the moment, in order to possess operator families in connection with the symbols σj (A), j > 0, for A ∈ Diffm deg (M ) it suffices to interpret them as mappings ∧ on C0∞ (s0 (Xk−1 )) and C0∞ (s0 ((Xj−1 ))), respectively. m ˜ m+m ˜ ˜ ˜ Theorem 3.2. Let A ∈ Diffm deg (M ), A ∈ Diffdeg (M ); then AA ∈ Diffdeg (M ), ˜ = σ(A)σ(A) ˜ (with componentwise composition and the rule and we have σ(AA) m ˜ ˜ = T σ(A)σ(A) ˜ when dim sk (M ) = 0 where (T β f )(z) = f (z + β) for any σk (AA) real β).

The proof is simple; for j = 0 it is clear anyway. For k > 0 we first consider the case qk = dim sk (M ) > 0. We interpret the operators A and A˜ as operators in y with operator-valued coefficients, i.e., ˜bδ (y)D δ ; A= bβ (y)Dyβ , A˜ = (3.6) y then AA˜ = b(y, η) =

|β|≤m

|δ|≤m ˜

ϑ |ϑ|≤m+m ˜ cϑ (y)Dy .

|β|≤m

Setting ˜bδ (y)ηδ , c(y, η) = bβ (y)η β , ˜b(y, η) =

|δ|≤m ˜

|ϑ|≤m+m ˜

cϑ (y)η ϑ , (3.7)

it follows that c(y, η) = α∈Nq 1/α!(∂ηα b)(y, η)Dyα˜b(y, η) which is a finite sum. In the computation of σk only the term for |α| = 0 survives, since the definition of the principal symbol contains freezing of coefficients at r = 0, and η-differentiations produce powers of r such that the corresponding contribution to the edge symbol disappears. In a similar manner, rewriting the coefficients in A or A˜ as functions of r in the form a(r, ·) = a(0, ·) + ra1 (r, ·), and a ˜(r, ·) = a ˜(0, ·) + r˜ a1 (r, ·), respectively, for functions a1 (r, ·), a˜1 (r, ·) that are smooth up to r = 0 we see again that the contributions with the extra r-powers vanish. The composition rule for symbols σj for j < k can be verified in a similar manner. In the case dim sk (M ) = 0 we express our operators in terms of the Mellin transform ∞

M u(z) =

rz−1 u(r)dr

(3.8)

0

which has the property M ((−r∂r )u)(z) = zM u(z). For functions with compact support in r we obtain an entire function in z. This allows us to restrict M u(z) to any parallel Γβ := {z ∈ C : Re z = β} to the imaginary axis. For β = 1/2 − γ we obtain what we call the weighted Mellin transform Mγ u := M u|Γ1/2−γ . Setting opγM (h)u(z) := Mγ−1 h(r, z)(Mγ u)(z)

(3.9)

for a corresponding (in our case operator-valued) Mellin amplitude function h(r, z) we can write m ˜ ˜ ˜ A = r−m opγ− (h), A˜ = r−m opγM (h) (3.10) M


85

m ˜ j ˜ for h(r, z) = m ˜l (r)z l . Since the computation of σk j=0 aj (r)z , h(r, z) = l=0 a refers again to coefficients frozen at r = 0 for convenience we assume h = h(0, z), ˜ = ˜h(0, z) where σk (A)(z) = h(0, z), σk (A)(z) ˜ z). Now for the composition ˜ h = h(0, it follows that m ˜ ˜ ˜ ˜ ˜ = r−(m+m) ˜ AA˜ = r−m opγ− (h)r−m opγM (h) opγM ((T −m h)h) M

(3.11)

˜ ˜ which gives us σk (AA)(z) = h(z + m) ˜ h(z). The degenerate behaviour of operators in local “stretched variables” has a consequence also for the symbols (the latter notation will be clear in the following constructions). By Definition 2.1 a space M ∈ Mk , k ≥ 1, is locally near points of ∆ sk (M ) modelled on Xk−1 × Ωk , for an open set Ωk ⊆ Rqk , qk = dim sk (M ), and by stretched variables we understand a corresponding splitting over s0 (Xk−1 )∧ × Ωk into variables (rk , xk−1 , yk ). Denoting by (ρk , ξk−1 , ηk ) the corresponding covariables we obtain σ0 (A) as a function σ0 (A) = σ0 (A)(rk , xk−1 , yk , ρk , ξk−1 , ηk ),

(3.12)

and sk σ0 (A)(rk , xk−1 , yk , ρk , ξk−1 , ηk ) := rkm σ0 (A)(rk , xk−1 , yk , rk−1 ρk , ξk−1 , rk−1 ηk ), (3.13) is smooth up to rk = 0. The space Xk−1 ∈ Mk−1 is locally near sk−1 (Xk−1 ) ∆ modelled on Xk−2 × Ωk−1 for some Xk−2 ∈ Mk−2 and an open set Ωk−1 ⊆ qk−1 R , qk−1 = dim sk−1 (Xk−1 ), and we may decompose (xk−1 , ξk−1 ) into (xk−1 , ξk−1 ) = (rk−1 , xk−2 , yk−1 , ρk−1 , ξk−2 , ηk−1 )

(3.14)

for (rk−1 , xk−2 , yk−1 ) ∈ R+ × s0 (Xk−2 ) × Ωk−1 . This gives us σ0 (A) as a function σ0 (A) = σ0 (A)(rk−1 , rk , xk−2 , yk−1 , yk , ρk−1 , ρk , ξk−2 , ηk−1 , ηk ),

(3.15)

sk−1 sk σ0 (A)(rk−1 , rk , xk−2 , yk−1 , yk , ρk−1 , ρk , ξk−2 , ηk−1 , ηk ) −1 := (rk−1 rk )m σ0 (A) rk , rk−1 , xk−2 , yk−1 , yk , (rk−1 rk )−1 ρk , rk−1 ρk−1 , −1 ξk−2 , rk−1 ηk−1 , (rk−1 rk )−1 ηk

(3.16)

and

is smooth up to rk−1 = 0, rk = 0. This process can be continued, the next step concerns a neighbourhood of s0 (Xk−2 ) in variables and covariables (xk−2 , ξk−2 ) = (rk−2 , xk−3 , yk−2 , ρk−2 , ξk−3 , ηk−2 ),

(3.17)

and so on. At the end we obtain σ0 (A) =σ0 (A)(r1 , . . . , rk−1 , rk , x0 , y1 , . . . , yk−1 , yk , ρ1 , . . . , ρk−1 , ρk , ξ0 , η1 , . . . , ηk−1 , ηk ),

(3.18)

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and s1 · · · sk−1 sk σ0 (A)(r1 , . . . , rk−1 , rk , x0 , y1 , . . . , yk−1 , yk , ρ1 , . . . , ρk−1 , ρk , ξ0 , η1 , . . . , ηk−1 , ηk ) := (r1 . . . rk−1 rk )m σ0 (A)(r1 , . . . , rk−1 , rk , x0 , y1 , . . . , yk−1 , yk , r1−1 ρ1 , . . . , (r1 . . . rk−1 )−1 ρk−1 , (r1 . . . rk−1 rk )−1 ρk , ξ0 , r1−1 η1 , . . . , (r1 . . . rk−1 )−1 ηk−1 , (r1 . . . rk−1 rk )−1 ηk ) (3.19) is smooth up to r1 = 0, . . . , rk = 0. The functions {sk σ0 (A), . . . , s1 · · · sk−1 sk σ0 (A)}

(3.20)

are also referred to as the reduced interior symbols. Also the symbolic components σ1 (A), . . . , σk−1 (A) admit reduced variants, namely, {sk σ1 (A), . . . , s2 · · · sk σ1 (A)}, {sk σ2 (A), . . . , s3 · · · sk σ2 (A)}, ··· ,

(3.21)

{sk σk−2 (A), sk−1 sk σk−2 (A)}, sk σk−1 (A), while σk (A) = σk (A)(yk , ηk ) remains as it is, namely, an operator function ∧ ∧ C0∞ (int Xk−1 ) → C0∞ (int Xk−1 ) ∧ with int Xk−1 = (int Xk−1 )∧ being equipped with the variables (rk , xk−1 ). Let us now illustrate the definition in the case σk−1 (A), which is an operator function parametrised by points in T ∗ sk−1 (M \ sk (M )) \ 0. The reduced symbol refers to a ∧ neighbourhood close to sk (M ) in the representation of M \ sk (M ) as Gk−1 × Xk−2 with Gk−1 corresponding to a chart on sk−1 (M ) in the variables (rk , yk−1 , yk ) (cf. the notation in (2.2) where Gk−1 = (R+ × Ωk−1 ) × Ωk ). In these variables we have ∧ ∧ σk−1 (A)(rk , yk−1 , yk , ρk , ηk−1 , ηk ) : C0∞ (int Xk−2 ) → C0∞ (int Xk−2 )

(3.22)

∧ where int Xk−2 is equipped with the variables (rk−1 , xk−2 ). The degenerate behaviour of the operator A has the consequence that

sk σk−1 (A)(rk , yk−1 , yk , ρk , ηk−1 , ηk ) := rkm σk−1 (A)(rk , yk−1 , yk , rk−1ρk , ηk−1 , rk−1 ηk )

(3.23)

is smooth up to rk = 0. Let us continue this process one step more. Then it should be clear how it works in general. First the symbol σk−2 (A) is an operator function ∧ ∧ C0∞ (int Xk−3 ) → C0∞ (int Xk−3 )

parametrised by points in T ∗ sk−2 (M \ (sk (M ) ∪ sk−1 (M ))) \ 0. Close to sk (M ) the manifold sk−2 (M \ (sk (M ) ∪ sk−1 (M ))) admits local variables of the form


87

(rk , vk−1 , yk ) with covariables (ρk , θk−1 , ηk ). Then similarly as before we obtain sk σk−2 (A)(rk , vk−1 , yk , ρk , θk−1 , ηk ) := rkm σk−2 (A)(rk , vk−1 , yk , rk−1ρk , θk−1 , rk−1 ηk ) (3.24) as a smooth function up to rk = 0. However, close to sk−1 (M ) the variables vk−1 can be split up once again into vk−1 = (rk−1 , yk−1 ) with covariables θk−1 = (ρk−1 , ηk−1 ) which allows us to form sk−1 sk σk−2 (A)(rk−1 , rk , yk−1 , yk , ρk−1 , ρk , ηk−1 , ηk ) −1 := (rk−1 rk )m σk−2 (A) rk−1 , rk , yk−1 , yk , rk−1 ρk−1 , (rk−1 rk )−1 ρk , −1 rk−1 ηk−1 , (rk−1 rk )−1 ηk

(3.25)

as a smooth function up to rk−1 = rk = 0. It is now clear how to define (3.21) in general. Remark 3.3. A role of the principal symbols (3.5) including the reduced components (3.20) and (3.21) is to define an adequate notion of ellipticity of the operator A. Let us call the operator σ0 -elliptic if σ0 (A) never vanishes on T ∗ s0 (M ) \ 0, and if in addition sj · · · sk σ0 (A) does not vanish up to rj = · · · = rk = 0 for every j = 1, . . . , k. The right notion of ellipticity for the other components of (3.5) is an element of a more complete theory of the operators that we only sketch in this article. However, there is a non-trivial special case, namely, that we simply ask the operator functions σj (A) to be bijective between the above-mentioned weighted spaces for all points in T ∗ sj (M ) \ 0 for j = 1, . . . , k (when dim sk (M ) > 0, otherwise for all z on a suitable parallel to the imaginary axis), together with a similar bijectivity condition for all sj · · · sk σl (A), j = l + 1, . . . , k, and 1 ≤ l ≤ k − 1, up to rj = · · · = rk = 0. In general when A is σ0 -elliptic we cannot expect that the other symbolic components satisfy such bijectivity conditions. Under a suitable topological condition on σ0 (A), for instance, in the case k = 1 and dim sk (M ) > 0, we can generate extra edge conditions, a substitute of (Shapiro-Lopatinskij) elliptic boundary conditions, that fill up the Fredholm family σk (A) to a family of isomorphisms, provided that we (are able to) avoid some exceptional weights. The complete story is long and not the focus of the present paper. For dim sk (M ) = 0 things are a little easier; then we have bijectivity up to some discrete set of exceptional weights. Ellipticity under such aspects have been studied for the case k = 1 in many variants. For instance, in boundary value problems in smooth domains the role of elliptic boundary conditions is to have a bijective boundary symbol, cf. Boutet de Monvel [5]. However, the criterion on σ0 that makes this possible may be violated, cf. Atiyah, Bott [3] for a K-theoretic formulation of the respective topological obstruction in the case of differential operators, Boutet de Monvel [5] for operators with the transmission property at the boundary. A new algebra of boundary value problems that extends the one of Boutet de Monvel and admits arbitrary σ0 -elliptic operators with the transmission property (including those with nonvanishing topological obstruction) has been established by the author in [52], see

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also [55]. The approach of [52] unifies the concept of Shapiro-Lopatinskij elliptic conditions and global projection conditions, containing as special cases conditions of Atiyah-Patodi-Singer type. A generalisation to edge problems for smooth edges was given in the author’s joint paper [61] with Seiler, see also [60] which concerns boundary value problems without the transmission property. The discussion of all this material is worth to be continued, including the contributions of other authors; however, this is not the main topic here, and the problem of understanding the hierarchy of topological obstructions for general singularity orders and arbitrary σ0 -elliptic operators is addressed as one of the challenges in Section 4 below.

4. Problems and results The analysis of operators of the spaces Diffm deg (M ) on stratified spaces M ∈ Mk , k ∈ N, gives rise to a number of natural problems that are solved by works of several authors in this field or are open and still represent challenges for the future development. Let us give a list of such problems, and then some key words concerning results and references: (i) Construct a pseudo-differential calculus containing Diffm deg (M ) together with the parametrices of elliptic elements. (ii) Establish the Fredholm property and study the index of elliptic operators in weighted distribution spaces when M is compact. (iii) Understand ellipticity, parametrices, and Fredholm property in suitable weighted spaces when M has conical exits to infinity. (iv) Study parameter-dependent theories on M . (v) Characterise asymptotics of solutions to elliptic equations (discrete, continuous, variable/branching, or iterated), in simplest cases of the form u(r, ·) ∼

mj j

cjk (·)r−pj logk r

as r → 0,

(4.1)

k=0

with pj ∈ C, Re pj → −∞ as j → ∞ (if the expansion is infinite). (vi) Compute the points pj for interesting examples; those points appear as “nonlinear eigenvalues” of conormal symbols. (vii) Study various quantisations of corner-degenerate symbols, in particular, in terms of holomorphic/meromorphic operator functions with values in algebras of lower singularity order. (viii) Understand the hierarchy of topological obstructions appearing in the construction of elliptic operators with prescribed elliptic symbols σ0 . (ix) Study index theories, homotopy classifications, K¨ unneth formulas, etc., for the higher corner operator algebras.


89

Models from diverse applications with singular geometry: (i) Mixed problems, operators with/without the transmission property at the boundary (Zaremba problem, etc.). (ii) Boundary value problems in polyhedral domains with the induced metric from an ambient space, occurring in elasticity, mechanics (beams, shells, plates, . . . ). (iii) Crack problems with crack boundaries that are smooth or have singularities. (iv) Operators with singular potentials ∆ + V,

V =

N

cij |x(i) − x(j) |−1 ,

i,j=1 (j)

(j)

(j)

for the Laplacian ∆ in R3N , and x(j) = (x1 , x2 , x3 ) indicating the position of three-dimensional particles; the question is to describe the behaviour of solutions to (∆ + V )u = f, close to the singularities of the potential V, say, for smooth f. The operator algebras that we discuss here contain many special cases and substructures: (i) Singular integral operators with piecewise smooth coefficients (cf. Gohberg and Krupnik [20]). (ii) Mellin operators on the half-axis (cf. Eskin [13]). (iii) Operators on manifolds with conical exits to infinity (cf. Shubin [66], Parenti [39], Cordes [8]) (iv) Parameter-dependent operators (cf. Agranovich and Vishik [2]). (v) Boundary value problems without/with the transmission property at the boundary (cf. Vishik and Eskin [67], Eskin [13], Boutet de Monvel [5]). (vi) Totally characteristic operators (cf. Melrose [37], Melrose and Mendoza [38]). (vii) Edge-degenerate, and corner-degenerate operators (cf. Rempel and Schulze [42], Schulze [45], Mazzeo [36], Schulze [47]). (viii) Boundary value problems in the frame of the edge calculus (cf. Rempel and Schulze [40], Schulze [48], Schulze and Seiler [59]). Other authors of the research group of the University of Potsdam and guests during the past years contributed improvements and important new aspects, cf. Hirschmann [24], Witt [69], Gil [16], Gil and Mendoza [17], Seiler [64], [65], Krainer [30], [31], or the author’s joint papers with Schrohe [43], [44] Coriasco [9], Dines and Liu [11], Flad and Schneider [14], Wei [63], Martin [35]; other references are given below. There also appeared (or are in preparation) some monographs on these topics, in particular, [46], [48], [51], or, jointly with Egorov [12], Kapanadze [27], Harutyunyan [23], Volpato [62]. More details on the higher corner calculus will also be given in the author’s new monograph [57].

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5. Some typical tools of the higher corner calculus Let us first consider k = 0 which is the smooth case. On a C ∞ manifold M we have Lm cl (M ), the space of classical pseudo-differential operators of order m ∈ R (“classical” is here not essential, but for k > 0 we employ this assumption). For an A ∈ Lm cl (M ) we have the standard homogeneous principal symbol σ0 (A) ∈ C ∞ (T ∗ M \ 0). Clearly everything works for vector bundles as well. s s Let Hcomp (M ), Hloc (M ), s ∈ R, denote the standard Sobolev spaces, and s write H (M ) when M is compact or an Euclidean space. The spaces H s (Rn × Rq ) admit anisotropic reformulations that are useful for the singular cases, namely, H s (Rn × Rq ) = W s (Rq , H s (Rn )). Here W s (Rq , H) for some Hilbert space H which is endowed with a strongly continuous group {κλ }λ∈R+ of isomorphisms κλ : H → H means the completion of S(Rq , H) with respect to the norm { η2s κ−1 ˆ(η) 2H dη}1/2 , η = (1 + |η|2 )1/2 . η u In the case H = H s (Rn ) we set (κλ u)(x) = λn/2 u(λx). l Let Lm cl (M, R ) denote the space of parameter-dependent pseudo-differential operators with parameter λ ∈ Rl , l ∈ N, on an open C ∞ manifold M , with local amplitude functions a(x, ξ, λ) that are classical symbols in (ξ, λ) ∈ Rn+l , n = dim M , and L−∞ (M, Rl ) = S(Rl , L−∞ (M )), where L−∞ (M ) is identified with C ∞ (M × M ) via a fixed Riemannian metric. We employ the fact that for compact M there exist parameter-dependent elliptic order reducing isomorphisms Rm (λ) : H s (M ) → H s−m (M ) for every m and s. Let us now give an idea on how the respective parameterdependent operator spaces Am (M, g; Rl ), m ∈ R, are constructed in the case M ∈ M1 which corresponds to conical or edge singularities (and also contains the case of a manifold with smooth boundary). Here λ ∈ Rl is the parameter, and g = (γ, γ − m, Θ) are weight data for a weight γ ∈ R and a weight interval Θ = (θ, 0] for a −∞ ≤ θ < 0, where we control asymptotics. Let us forget about Rl for a while, and define weighted spaces, first for conical singularities. By Hs,γ (X ∧ ) on the open stretched cone X ∧ = R+ × X we denote the completion of C0∞ (X ∧ ) with respect to the norm ! "1/2 (2πi)−1

Rs (Imz)M u(z) 2L2(X) dz Γ n+1 −γ 2

where R (λ) ∈ is a parameter-dependent elliptic family, n = dim X, ∞ Γβ = {z ∈ C : Re z = β}, and M u(z) = 0 rz−1 u(r)dr is the Mellin transform on R+ . The space Hs,γ (X ∧ ) takes part in the definition of the space Ks,γ (X ∧ ), s namely, close to r = 0. Another ingredient close to r = ∞ is the space Hcone (X ∧ ). ∧ s Let us first define a version on R × X rather than X . The space Hcone (R × X) is defined to be the completion of C ∞ (R × X) with respect to the norm ! "1/2

r−s+n/2 Opr (p)(η 1 )u 2L2 (X) dr , s

Lscl (X; R)


91

r = (1 + r2 )1/2 , p(r, ρ, η) = p˜(rρ, rη), where p˜(˜ ρ, η˜) ∈ Lscl (X; R1+q ρ,˜ ˜ η ) is a 1 parameter-dependent elliptic family on X, and |η | is sufficiently large and fixed. Moreover, Opr (p)u(r) = ei(r−r )ρ p(r, ρ)u(r )dr d¯ρ, s s d¯ρ := (2π)−1 dρ. We set Hcone (X ∧ ) := Hcone (R × X)|X ∧ . For any cut-off function ω(r) we define s Ks,γ (X ∧ ) = ωHs,γ (X ∧ ) + (1 − ω)Hcone (X ∧ ),

and Ks,γ;g (X ∧ ) := r−g Ks,γ (X ∧ ), s, γ, g ∈ R. Observe that we have s s Hcomp (Rn × Rq ) ⊆ W s (Rq , Ks,γ (X ∧ )) ⊆ Hloc (Rn × Rq )

for all s, g ∈ R. The pseudo-differential background of cone and edge operator algebras are degenerate operators of the form r−m Opr,y (p), based on the Fourier transform, for p(r, y, ρ, η) = p˜(r, y, rρ, rη), where p˜(r, y, ρ˜, η˜) ∈ C ∞ (R+ × 1+q Ω, Lm cl (X; Rρ,˜ ˜ η )). It is useful to pass to Mellin quantisations, i.e., to operatorvalued symbols referring to the Mellin transform. Let us explain here the edge case, i.e., q > 0 (the conical case is simpler). To this end we define the space m MO (X; Rqη ), µ q 1+q consisting of all h(z, η) ∈ A(Cz , Lm cl (X; Rη )) such that h(β + iρ, η) ∈ Lcl (X; Rρ,η ) for every β ∈ R, uniformly in finite β-intervals. Here A(U, E) for an open set U ⊆ C and a Fréchet space E is the space of all holomorphic functions in U with values in E, in the topology of uniform convergence on compact subsets. An inversion process in the construction of parametrices of elliptic operators gives rise to symbols of the kind

f (y, z) ∈ C ∞ (Ω, MR−∞ (X)). Here R is an asymptotic type, in the most precise version y-wise discrete, otherwise a continuous asymptotic type. Let us give an idea of the discrete case. Then R is a sequence {(rj , nj )}j∈Z ⊂ C × N with |Re rj | → ∞ as |j| → ∞. The space −∞ MR (X) is defined to be the set of all f ∈ A(C \ πC R, L−∞ (X)), πC R := {rj }j∈Z , such that f is meromorphic with poles at the points rj of multiplicity nj + 1. Moreover, f (z) is strongly decreasing as |Im z| → ∞, i.e., if χ(z) is any πC Rexcision function (= 0 close to πC R, and = 1 when dist (z, πC R) > c for some c > 0) then χ(z)f (z)|Γβ ∈ S(Γβ , L−∞ (X)) for every β ∈ R, uniformly in compact β-intervals; here Γβ := {z ∈ C : Re z = β}. More generally we also employ so-called continuous asymptotic types R, represented by sets V ⊂ C such that V ∩ {c ≤ Re (z) ≤ c } is compact for every c ≤ c , cf. [46], [51]; V := πC R.

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Theorem 5.1. For every p(r, y, ρ, η) = p˜(r, y, rρ, rη), p˜(r, y, ρ˜, η˜) ∈ C ∞ (R+ × Ω, Lm ˜ ˜ (X; R1+q ˜) ∈ ρ,˜ ˜ η )) there exists an h(r, y, z, η) = h(r, y, z, rη) of the form h(r, y, z, η q ∞ m C (R+ × Ω, MO (X; Rη˜)) such that γ−n/2

Opr,y (p) = opM

Opy (h)

mod L−∞ (X ∧ × Ω)

(5.1)

for every γ ∈ R. The operator-valued amplitude functions a(y, η) = ω(r)r−m opMr

γ−n/2

(h)(y, η)˜ ω (r)+ smoothing (Mellin + Green) symbols (y, η) (5.2) with cut-off functions ω(r), ω ˜ (r) furnish the symbols of the edge pseudo-differential ˜ be Hilbert spaces calculus near r = 0. Those are symbols as follows. Let H and H with group actions {κλ }λ∈R+ and {˜ κλ }λ∈R+ , respectively. Then ˜ S m (Ω × Rq ; H, H) for m ∈ R and Ω ⊆ Rp open is defined to be the set of all a(y, η) ∈ C ∞ (Ω × ˜ such that Rq , L(H, H)) α β m−|β|

˜ κ−1 , ˜ ≤ cη η {Dy Dη a(y, η)}κη L(H,H)

uniformly on compact subsets of Ω, for all η ∈ Rq and all multi-indices α, β. The m ˜ of classical symbols is defined in terms of asympsubspace Scl (Ω × Rq ; H, H) ∞ totic expansions j=0 χ(η)a(µ−j) (y, η), where χ(η) is an excision function, and ˜ are of twisted homogeneity µ − j, i.e., a(m−j) (y, η) ∈ C ∞ (Ω × (Rq \ {0}), L(H, H)) a(m−j) (y, λη) = λµ−j κ ˜ λ a(m−j) (y, η)κ−1 λ , λ ∈ R+ . Parallel to such operator-valued symbols we have vector-valued spaces W s (Rq , H) for a Hilbert space H with group action {κλ }λ∈R+ , defined as the completion of 1/2 1/2 S(Rq , H) with respect to the norm η2s κ−1 ˆ(η) H dη . There is also a η u s s straightforward generalisation to spaces of the kind Wcomp (Ω, H) and Wloc (Ω, H), q respectively, over an open set Ω ⊆ R . Theorem 5.2. The above-mentioned operator functions a(y, η) of the form (5.2) belong to S m (Ω × Rq ; Ks,γ (X ∧ ), Ks−m,γ−m (X ∧ )) based on {κλ }λ∈R+ , defined by (κλ u)(r, x) = λ(n+1)/2 u(λr, x) for u ∈ Ks,γ (X ∧ ) and induce continuous operators s−m s Opy (a) : Wcomp (Ω, Ks,γ (X ∧ )) → Wloc (Ω, Ks−m,γ−m (X ∧ ))

(5.3)

for all s ∈ R. Remark 5.3. Observe that Theorem 5.1 and the continuity (5.3) show that r−m p(r, y, ρ, η) → a(y, η) → Opy (a) represents an operator convention (quantisation) for edge-degenerate symbols r−m p. In the author’s joint paper [18] with Gil and Seiler it has been proved


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that the first non-smoothing term in (5.2) is equivalent (mod Green operators) to another earlier quantisation of [45]. Let us now define smoothing Mellin plus Green symbols, already occurring in (5.2). Definition 5.4. (i) A smoothing Mellin symbol is an element gM in m Scl (Ω × Rq ; Ks,γ (X ∧ ), K∞,γ−m (X ∧ )) s∈R

which has an asymptotic expansion mod (X ∧ )) into symbols of the form r−m+j ω(r[η])opMjα γ

−n/2

# s∈R

S ∞ (Ω × Rq ; Ks,γ (X ∧ ), K∞,∞

(fjα )(y)η α ω ˜ (r[η]);

here η → [η] is a strictly positive function in C ∞ (Rq ) such that [η] = |η| for |η| > c for some c > 0, moreover, fjα (y, z) ∈ C ∞ (Ω, MR−∞ (X)), jα for some asymptotic types Rjα , and |α| ≤ j, πC Rjα ∩ Γ n+1 −γjα = ∅, γ − j ≤ 2 γjα ≤ γ. (ii) A Green symbol g(y, η) is defined by m g(y, η) ∈ Scl (Ω × Rq ; Ks,γ;g (X ∧ ), SP (X ∧ )), s,g∈R

and g ∗ (y, η) ∈

m Scl (Ω × Rq ; Ks,−m+γ;g (X ∧ ), SQ (X ∧ )),

s,g∈R

for continuous asymptotic types P, and Q, respectively. Concerning details on continuous asymptotics, see, for instance, [46], or [51]. For a smoothing Mellin symbol gM (y, η) we set σ1 (gM )(y, η) := r−m ω(r|η|)opM00 γ

−n/2

(f00 )(y)˜ ω (r|η|)

which is the homogeneous principal part of order m of the respective classical operator-valued symbol. Analogously, if g(y, η) is a Green symbol we set σ1 (g)(y, η) = g(m) (y, η) with g(m) being the homogeneous principal part of g of order m. Remark 5.5. Edge symbols a(y, η) are an analogue of boundary symbols from boundary value problems for operators with/without the transmission property at the boundary, cf. [5], [40], [58].

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Let M be a manifold with edge Y of dimension q > 0. Then the space of all edge pseudo-differential operators on M , referring to the weight data g = (γ, γ − m, Θ) for a weight γ ∈ R and a weight interval Θ = (ϑ, 0], −∞ ≤ ϑ < 0, (which indicates an interval on the left of γ, and γ − m, respectively, where we control asymptotics) is defined to be the subset Lm (M, g) ⊂ Lm cl (M \ Y ) of all operators A that are locally near Y of the form A = Opy (a) mod L−∞ (M, g) where a(y, η) is an edge amplitude function (5.2) while L−∞ (M, g) is defined by mapping properties to smooth functions with asymptotics. In order not to overload the explanations we omit some details on asymptotics; let us only note that the control of asymptotics in terms of Θ, for instance, in the discrete case (4.1), means that we observe exponents such that Re pj belong to the interval ((n + 1)/2 − γ + ϑ, (n + 1)/2 − γ) for functions in the preimage and to ((n + 1)/2 − γ − m + ϑ, (n + 1)/2 − γ − m) in the image. For a first understanding it suffices to imagine Θ = (−∞, 0]; then we may forget about Θ and write g = (γ, γ − m). The principal symbolic structure of operators A ∈ Lm (M, g), m ∈ R, g = (γ, γ − m, Θ) is defined by σ(A) = (σ0 (A), σ1 (A)) with σ0 (A) being the homogeneous principal symbol in the sense of Lm (M, g) ⊂ Lm cl (M \Y ), which is locally near r = 0 of the form σ0 (A) = r−m p˜(m) (r, x, y, rρ, ξ, rη) where p˜(m) (r, x, y, ρ, ˜ ξ, η˜) is the 1+q homogeneous principal symbol of the above family in C ∞ (R+ × Ω, Lm cl (X; Rρ,˜ ˜ η )). Moreover, for the case q > 0 we have σ1 (A)(y, η) = r−m opM

γ−n/2

(h0 )(y, η) + σ1 (gM + g)(y, η), ˜ y, z, rη), (y, η) ∈ T ∗ Ω \ 0, h0 (r, y, z, η) = h(0,

(5.4)

which is a family of linear continuous operators σ1 (A)(y, η) : Ks,γ (X ∧ ) → Ks−m,γ−m (X ∧ )

(5.5)

of homogeneity σ1 (A)(y, λη) = λm κλ σ1 (A)(y, η)κ−1 λ , λ ∈ R+ . s,γ s On M we form weighted spaces H[loc) (M ) ⊂ Hloc (M \ Y ), locally near Y modelled s q s,γ ∧ on W (R , K (X )), and we also have a [comp)-version of such spaces. If M is compact we simply write H s,γ (M ).

Theorem 5.6. Every A ∈ Lm (M, g), g = (γ, γ − m, Θ), M compact, induces continuous operators A : H s,γ (M ) → H s−m,γ−m (M ) (5.6) for any s ∈ R. The operator (5.6) is compact when σ(A) = 0. Definition 5.7. An operator A ∈ Lm (M, g), g = (γ, γ−m, Θ), is said to be elliptic if (i) A is elliptic as an operator in Lm cl (intM ), and if in addition locally near r = 0 the function p˜(m) (r, x, y, ρ, ˜ ξ, η˜) does not vanish for all (˜ ρ, ξ, η˜) = 0 up to r = 0; (ii) the operators (5.5) are bijective for all (y, η) ∈ T ∗ Y \ 0.


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Remark 5.8. The second condition of ellipticity concerning σ1 is stronger than necessary. It suffices to impose the Fredholm property together with a 2 × 2 block matrix extension of σ1 by extra trace and potential symbols to a family of isomorphisms, cf. also Remark 3.3. The extra symbols represent additional operators satisfying an analogue of the Shapiro-Lopatinskij condition, known from boundary value problems. Similarly as in the latter case this requires vanishing of a topological obstruction for σ0 (A) (concerning more details on that point, including the calculus when this topological obstruction does not vanish, see [61]). Theorem 5.9. An operator A ∈ Lm (M, g), g = (γ, γ−m, Θ), M compact, is elliptic with respect to (σ0 , σ1 ) if and only if (5.6) is Fredholm for some fixed s ∈ R. In general the ellipticity of A entails the existence of a parametrix in L−m (M, g−1 ) belonging to (σ0−1 , σ1−1 ). Remark 5.10. Parameter-dependent operators of the class Lm (M, g; Rl ) are defined in an analogous manner as for l = 0. There is then a notion of parameterdependent ellipticity. If Lm (M, g; Rl ) is parameter-dependent elliptic, M compact, then A(λ) : H s,γ (M ) → H s−m,γ−m (M ) (5.7) are isomorphisms for all λ ∈ Rl , |λ| sufficiently large, s ∈ R. Let Lm−1 (M, g; Rl ) := A ∈ Lm (M, g; Rl ) : σ(A) = 0 , and successively define Lm−j (M, g; Rl ) for every j ∈ N, g = (γ, γ − m, Θ). Theorem 5.11. For every s , s ∈ R and N ∈ N there exists a j ∈ N such that for A(λ) ∈ Lm−j (M, g; Rl ) we have

A(λ) L(H s ,γ (M ),H s ,γ−m (M )) ≤ cλ−N

(5.8)

for all λ ∈ R and some c > 0.

6. Higher corner operators We sketch a number of structures of the pseudo-differential operator calculus on a manifold with higher corners. For convenience we focus the consideration on the case k = 2. It will be fairly obvious that the concept is iterative and can be applied for higher corners as well, cf. [53]. This aspect is one of the main motivations of our approach. Another motivation is, of course, to express parametrices of elliptic elements within the calculus. This belongs to one of results of our theory. Special cases have been treated before, cf., [47], [54]. Other contributions to the higher corner calculus are [53], and the author’s joint papers with Maniccia [34], Krainer [32], Calvo and Martin [6], Calvo [7], Harutyunyan [21], [22], [23], Martin [35]. Let B ∈ M1 ; then a starting point are corner-degenerate families p(t, z, τ, ζ) = p˜(t, z, tτ, tζ)

(6.1)

˜ ∈ C ∞ (R+ × Σ, Lm (B, g; R1+d )), g = (γ, γ − m, Θ), p˜(t, z, τ˜, ζ) τ˜,ζ˜

(6.2)

where

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Σ ⊆ Rd open. Define m MO (B, g; Rdζ ) ⊂ A(C, Lm (B, g; Rdζ )) h(v, ζ)

(6.3)

h(δ + iτ, ζ) ∈ Lm (B, g; R1+d τ,ζ )

(6.4)

such that for every δ ∈ R, uniformly in compact δ-intervals. ˜ z, v, tζ) Theorem 6.1. For p(t, z, τ, ζ) as in (6.1) there exists a h(t, z, v, ζ) = h(t, ∞ m d ˜ ˜ for some h(t, z, v, ζ) ∈ C (R+ × Σ, MO (B, g; Rζ˜)) such that δ−b/2

Opt,z (p) = opM

Opz (h) mod L−∞ (R+ × Σ × B, g),

(6.5)

for b := dim B, for any δ ∈ R. Let us now define weighted spaces, first on B ∧ = R+ ×B for compact B ∈ M1 . The space Hs,(γ,δ) (B ∧ ) is defined to be the completion of C ∞ (R+ × int B) with respect to the norm ! "1/2 (2πi)−1

Rs (Im v)M u(z) 2H 0,γ−s (B) dv (6.6) Γ n+1 −δ 2

where R (λ) ∈ L (B, g; R) is an order reducing family of edge operators, b = s,γ dim B, g = (γ, γ −s, Θ). Moreover, we have the cone spaces Hcone (R×B) obtained ∞ as the completion of C (R × int B) with respect to the norm ! "1/2

t−s+dim B/2 Opt (p)(ζ 1 )u 2H 0,γ−s (B) dt s

s

˜ ∈ for a parameter-dependent elliptic family p(t, z, τ, ζ) = p˜(z, tτ, tζ), p˜(˜ τ , ζ) s d 1 L (M, g; Rζ˜), g = (γ, γ − s, Θ), and |ζ | sufficiently large and fixed. Then we set s,γ s,γ Hcone (B ∧ ) := Hcone (R × B)|B ∧ . (The notation “cone spaces” comes from the interpretation of t → ±∞ on R × B as conical exits to infinity.) Finally for any cut-off function ω(t) we set s,γ Ks,(γ,δ) (B ∧ ) = ωHs,(γ,δ)(B ∧ ) + (1 − ω)Hcone (B ∧ ),

and Ks,(γ,δ);g (B ∧ ) := t amplitude functions

−g

Ks,(γ,δ) (B ∧ ). Similarly as (5.2) we form operator-valued

a(z, ζ) = ω(t)t−m opMt

δ−n/2

(h)(z, ζ)˜ ω (t)+ smoothing (Mellin+Green) symbols (z, ζ) (6.7) m d s,(γ,δ) ∧ s−m,(γ−m,δ−m) ∧ belonging to S (Σ × R ; K (B ), K (B )). For M ∈ M2 with the minimal stratum Z ⊂ M, say, of dimension d > 0, we have the space of corner pseudo-differential operators Lm (M, g) for g = (g1 , g2 ), g1 = (γ, γ − m, Θ1 ), g2 = (δ, δ − m, Θ2 ),

(6.8)


97

consisting of all A ∈ Lm (M \ Z) that are locally near Z of the form A = Opz (a) mod L−∞ (M, g) where L−∞ (M, g) is defined by mapping properties to smooth functions with asymptotics. On Z we have weighted spaces s,γ H s,(γ,δ) (M ) ⊂ Hloc (M \ Z),

locally near Z modelled on W s (Rd , Ks,γ (B ∧ )). The principal symbolic structure of an A ∈ Lm (M, g) is given by (σ(A|M \Z ), σ2 (A)) where σ(A|M\Z ) is known from the case k = 1, and σ2 (A)(z, ζ) = t−m opMt

δ−b/2

(h0 )(z, ζ)

˜ z, v, tζ), which is a family of operators for h0 (t, z, v, ζ) = h(0, σ2 (A)(z, ζ) : Ks,(γ,δ) (B ∧ ) → Ks−m,(γ−m,δ−m) (B ∧ ) for (z, ζ) ∈ T ∗ Z \ 0, homogeneous in the sense σ2 (A)(z, λζ) = λm κλ σ2 (A)(z, ζ)κ−1 λ , λ > 0. If A or B is properly supported (such a property is defined in an analogous manner as in the smooth case) then AB belongs to the corner calculus again, and we have σ(AB) = σ(A)σ(B) with componentwise composition (for dim Z = 0 we take into account an order shift in the first component similarly as in Theorem 3.2). An operator A ∈ Lm (M, g) is said to be elliptic if A|M \Z is elliptic in the cal˜ σ1 (·, ζ) ˜ culus over M \ Z ∈ M1 , and if close to Z the symbolic components σ0 (·, ζ), d with parameter ζ˜ ∈ R \ {0} (substituting tζ) are parameter-dependent elliptic up to t = 0. The latter condition concerns an evident generalisation of Definition 5.7 to the case when we have an extra covariable ζ˜ which is also involved in edgedegenerate form and where the symbols (apart from a weight factor t−m ) also depend on t, smoothly up to t = 0. Theorem 6.2. Every A ∈ Lm (M, g), M compact, induces continuous operators A : H s,(γ,δ) (M ) → H s−m,(γ−m,δ−m) (M )

(6.9)

for all s ∈ R. The operator (6.9) is compact when σ(A) = 0. Theorem 6.3. Let A ∈ Lm (M, g), M compact; then the ellipticity of A entails the Fredholm property of (6.9) for all s. Moreover, if A is elliptic, M not necessarily compact, the operator A has a parametrix in L−m (M, g−1 ) belonging to σ −1 (A) (with componentwise inverses). Remark 6.4. The above-mentioned results on parameter-dependent operators in the case k = 1 have natural analogues for k = 2. They imply, in particular, the existence of order reducing operators in the calculus. Remark 6.5. Let X, Y, and Z be Riemannian manifolds with Riemannian metrics gX , gY , and gZ , respectively, and form the degenerate metric dt2 + t2 (dr2 + r2 gX + gY ) + gZ

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on the stretched corner R+ × (R+ × X × Y ) × Z. Then the associated LaplaceBeltrami operator belongs to the corner calculus for k = 2 on (X ∆ ×Y )∆ ×Z ∈ M2 . More generally, considering, for instance, M := (. . . ((X ∆ × Ω1 )∆ × Ω2 )∆ × · · · × Ωk−1 )∆ × Ωk for Riemannian manifolds X, Ω1 , . . . , Ωk , the corner metric 2 drk2 + rk2 (drk−1 + · · · + (dr22 + r22 (dr12 + r12 gX + gΩ1 ) + gΩ2 ) + · · · ) + gΩk ,

on R+ × (. . . (R+ × (R+ × X × Ω1 ) × Ω2 ) × · · · × Ωk−1 ) × Ωk gives rise to a Laplace-Beltrami operator belonging to Diff2deg (M ).

7. Concluding remarks The higher corner calculus that we presented here contains many technicalities that are derived from the program to cover all the substructures sketched in Section 3. A dominating aspect of our theory is to guarantee that the calculus is closed under the construction of parametrices of elliptic elements and that it reflects asymptotics of solutions and elliptic regularity in weighted spaces. What concerns the history of our approach, the above-mentioned information has been integrated from the very beginning, for instance, classical elliptic boundary value problems (BVPs) in the sense of Agmon, Douglis, and Nirenberg [1], the theory of pseudo-differential BVPs of Vishik and Eskin [67], [13], the calculus of Boutet de Monvel [5], details on singular integral operators and operators based on the Mellin transform on the half-axis [20], [13]. The symbolic structures have been invented in such a way that vanishing of principal symbols gives rise to compact operators (when the configuration is compact, otherwise after localisation). Clearly the Fredholm index of elliptic operators has been realised as something invariant under stable homotopies of elliptic principal symbols (through elliptic symbols). At some point the author together with Rempel [41] became aware of the similarity between the boundary symbolic calculus for BVPs without the transmission property, cf. [13], [40], and the theory of Kondratiev [29] where R+ is replaced by a cone with a non-trivial base X. The inclusion of edge problems was the next logical step in the development, and, after a preliminary work with Rempel [42], the paper [45] gave a first systematic edge pseudo-differential calculus. Another step of the development was the paper [47] where the theory has been extended to the case of manifolds with corners (locally modelled on a cone where the base has conical singularities). After that it took some time to develop more technical tools to make the approach really iterative, cf. [53], [54] (the paper [54] studies singularities modelled on cones where the base has edges). There are many aspects to be deepened and continued in future, for instance, on operator algebras where the symbols do not satisfy an analogue of the AtiyahBott condition for the existence of Shapiro-Lopatinskij elliptic edge conditions (for k ≥ 2; concerning the case of boundary value problems and edge problems, cf. [52],


99

and the author’s joint papers with Seiler [60], and [61]), moreover, on the nature of iterated and variable branching asymptotics of solutions, cf. [49], [50] for the case of boundary value problems, and the joint work with Volpato [62] for the case of edge problems, or the explicit computation of admissible weights or asymptotic data, cf. the author’s joint papers with Dines and Liu [11], [10], [33] for the case of corner or boundary value problems. Different schools on singular analysis apparently emphasize different types of degenerate operators (in stretched coordinates), and, although there are considerable intersections between the various attempts, it seems that there is no standard terminology on what is a corner-degenerate operator or a corner manifold. Therefore, we point out once again that our calculus is made for corner manifolds that include cones, wedges, cubes, higher polyhedra, etc., embedded in a smooth ambient space, and equipped with the induced (incomplete) corner metrics. Differential operators in the respective stretched coordinates are polynomials in degenerate vector fields of the form (∂/∂xj )j=1,...,n , r1 ∂/∂r1 , (r1 ∂/∂y1,l )l=1,...,q1 , r1 r2 ∂/∂r2 , (r1 r2 ∂/∂y2,l )l=1,...,q2 , . . . , r1 r2 . . . rk ∂/∂rk , (r1 r2 . . . rk ∂/∂yk,l )l=1,...,qk , (7.1) combined with weight factors (r1 . . . rk )−m for operators of order m, and with coefficients that are smooth in all variables up to rj = 0, j = 1, . . . , k. Here rj ∈ R+ , and (yj,l )l=1,...,qj is the variable on a qj -dimensional edge. This is exactly what we obtain as local descriptions of operators Diffm deg (M ), M ∈ Mk , cf. Section 2, or Remark 6.5. Moreover, if we are in the situation that an operator is given in a domain with polyhedral boundary, and the respective operator is expressed in Euclidean coordinates in Rn with smooth coefficients across the boundary (for instance, the standard Laplacian in Rn , then by repeatedly substituting polar coordinates (according to the order of singularity) we obtain also operators in our class. In such a case it is convenient to formulate everything in the variant of (pseudo-differential) boundary value problems, i.e., to replace the parameter-dependent operators, say, on a closed manifold X (as in Section 4) by the algebra of boundary value problems on X, (now for an X with boundary) with the transmission property at the smooth faces of the boundary. This aspect is systematically applied in [27], [23], and in numerous other papers mentioned before, jointly with Dines, Liu, Wei, and others.

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References [1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623–727. [2] M.S. Agranovich and M.I. Vishik, Elliptic problems with parameter and parabolic problems of general type, Uspekhi Mat. Nauk 19, 3 (1964), 53–161. [3] M.F. Atiyah and R. Bott, The index problem for manifolds with boundary, Coll. Differential Analysis, Tata Institute Bombay, Oxford University Press, Oxford , 1964, pp. 175–186. [4] K. Bekka and D. Trotman, Metric properties of stratified sets, Manuscripta math. 111 (2003), 71–95. [5] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51. [6] D. Calvo, C.-I. Martin, and B.-W. Schulze, Symbolic structures on corner manifolds, RIMS Conf. dedicated to L. Boutet de Monvel on “Microlocal Analysis and Asymptotic Analysis”, Kyoto, August 2004, Keio University, Tokyo, 2005, pp. 22–35. [7] D. Calvo and B.-W. Schulze, Edge symbolic structure of second generation, Math. Nachr. 282 (2009), 348–367. [8] H.O. Cordes, A global parametrix for pseudo-differential operators over Rn , with applications, Reprint, SFB 72, Universit¨ at Bonn, 1976. [9] S. Coriasco and B.-W. Schulze, Edge problems on configurations with model cones of different dimensions, Osaka J. Math. 43 (2006), 1–40. [10] N. Dines, Elliptic operators on corner manifolds, Ph.D. thesis, University of Potsdam, 2006. [11] N. Dines, X. Liu, and B.-W. Schulze, Edge quantisation of elliptic operators, Monatshefte f¨ ur Math. 156 (2009), 233–274. [12] Ju.V. Egorov and B.-W. Schulze, Pseudo-differential operators, singularities, applications, Oper. Theory: Adv. Appl. 93, Birkh¨ auser Verlag, Basel, 1997. [13] G.I. Eskin, Boundary value problems for elliptic pseudodifferential equations, Transl. of Nauka, Moskva, 1973, Math. Monographs, Amer. Math. Soc. 52, Providence, Rhode Island 1980. [14] H.-J. Flad, R. Schneider, and B.-W. Schulze Asymptotic regularity of solutions of Hartree-Fock equations with Coulomb potential, Math. Meth. in the Appl. Sci. 31, 18 (2008), 2172–2201. [15] W. Fulton and R. MacPherson, Categorical framework for the study of singular spaces, Memoirs of the AMS 243 (1981). [16] J.B. Gil, Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Math. Nachr. 250 (2003), 25–57. [17] J.B. Gil and G. Mendoza, Adjoints of the elliptic cone operators, Amer. J. Math. 125,2 (2003), 357–408. [18] J.B. Gil, B.-W. Schulze, and J. Seiler, Cone pseudodifferential operators in the edge symbolic calculus, Osaka J. Math. 37 (2000), 219–258. [19] I.C. Gohberg and E.I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Math. USSR Sbornik 13, 4 (1971), 603–625.


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[20] I.C. Gohberg and N.G. Krupnik, The algebra generated by the one-dimensional singular integral operators with piecewise continuous coefficients, Funk. Anal. i Prilozen. 4, 3 (1970), 26–36. [21] G. Harutjunjan and B.-W. Schulze, The relative index for corner singularities, Integr. Equ. Oper. Theory 54, 3 (2006), 385–426. [22] G. Harutjunjan and B.-W. Schulze, The Zaremba problem with singular interfaces as a corner boundary value problem, Potential Analysis 25, 4 (2006), 327–369. [23] G. Harutjunjan and B.-W. Schulze, Elliptic mixed, transmission and singular crack problems, European Mathematical Soc., Z¨ urich, 2008. [24] T. Hirschmann, Functional analysis in cone and edge Sobolev spaces, Ann. Global Anal. Geom. 8, 2 (1990), 167–192. [25] L. H¨ ormander, The analysis of linear partial differential operators, vol. 1 and 2, Springer-Verlag, New York, 1983. [26] P. Jeanquartier, Transformation de Mellin et développements asymptotiques, Enseign. Math. (2) 25 (1979), 285–308. [27] D. Kapanadze and B.-W. Schulze, Crack theory and edge singularities, Kluwer Academic Publ., Dordrecht, 2003. [28] H.C. King and D. Trotman, Poincaré-Hopf theorems on singular spaces, manuscript (2007). [29] V.A. Kondratyev, Boundary value problems for elliptic equations in domains with conical points, Trudy Mosk. Mat. Obshch. 16, (1967), 209–292. [30] T. Krainer, The calculus of Volterra Mellin pseudo-differential operators with operator-valued ymbols, Oper. Theory Adv. Appl. 138, Adv. in Partial Differential Equations “Parabolicity, Volterra Calculus, and Conical Singularities” (Albeverio, S. and Demuth, M. and Schrohe, E. and Schulze, B.-W., eds.), Birkh¨ auser Verlag, Basel, 2002, pp. 47–91. [31] T. Krainer, On the inverse of parabolic boundary value problems for large times, Japan. J. Math. 30, 1 (2004), 91–163. [32] T. Krainer and B.-W. Schulze, Long-time asymptotics with geometric singularities in the spatial variables, Contemporary Mathematics 364 (2004), 103–126. [33] X. Liu and B.-W. Schulze, Ellipticity on manifolds with edges and boundary, Monatshefte f¨ ur Mathematik 146, 4 (2005),295–331. [34] L. Maniccia and B.-W. Schulze, An algebra of meromorphic corner symbols, Bull. des Sciences Math. 127, 1 (2003), 55–99. [35] C.-I. Martin and B.-W. Schulze, Parameter-dependent edge operators, arXiv: 0908.2030.v1 [math.AP], 2009. [36] R. Mazzeo, Elliptic theory of differential edge operators I, Comm. Partial Differential Equations. 16 (1991), 1615–1664. [37] R.B. Melrose, Transformation of boundary problems, Acta Math. 147 (1981), 149– 236. [38] R.B. Melrose and G.A. Mendoza, Elliptic operators of totally characteristic type, Preprint MSRI 047 – 83, Math. Sci. Res. Institute, 1983. [39] C. Parenti, Operatori pseudo-differenziali in Rn e applicazioni, Annali Mat. Pura Appl. (4) 93 (1972), 359–389.

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[40] S. Rempel and B.-W. Schulze, Parametrices and boundary symbolic calculus for elliptic boundary problems without transmission property, Math. Nachr. 105 (1982), 45–149. [41] S. Rempel and B.-W. Schulze, Complete Mellin and Green symbolic calculus in spaces with conormal asymptotics, Ann. Global Anal. Geom. 4, 2 (1986), 137–224. [42] S. Rempel and B.-W. Schulze, Asymptotics for elliptic mixed boundary problems (pseudo-differential and Mellin operators in spaces with conormal singularity), Math. Res., vol.50, Akademie-Verlag, Berlin, 1989. [43] E. Schrohe and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities I, Adv. in Partial Differential Equations “Pseudo-Differential Calculus and Mathematical Physics”, Akademie Verlag, Berlin, 1994, pp. 97-209. [44] E. Schrohe and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities II, Adv. in Partial Differential Equations “Boundary Value Problems, Schr¨ odinger Operators, Deformation Quantization”, Akademie Verlag, Berlin, 1995, pp. 70-205. [45] B.-W. Schulze, Pseudo-differential operators on manifolds with edges, Symp. “Partial Differential Equations”, Holzhau 1988, Teubner-Texte zur Mathematik, vol. 112, Teubner, Leipzig, 1989, pp. 259–287. [46] B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, NorthHolland, Amsterdam, 1991. [47] B.-W. Schulze, The Mellin pseudo-differential calculus on manifolds with corners, Symp. “Analysis in Domains and on Manifolds with Singularities”, Breitenbrunn 1990, Teubner-Texte zur Mathematik, vol. 131, Teubner, Leipzig, 1992, pp. 208–289. [48] B.-W. Schulze, Pseudo-differential boundary value problems, conical singularities, and asymptotics, Akademie Verlag, Berlin, 1994. [49] B.-W. Schulze, The variable discrete asymptotics in pseudo-differential boundary value problems I, Advances in Partial Differential Equations (Pseudo-Differential Calculus and Mathematical Physics), Akademie Verlag, Berlin, 1994, pp. 9–96. [50] B.-W. Schulze, The variable discrete asymptotics in pseudo-differential boundary value problems II, Advances in Partial Differential Equations (Boundary Value Problems, Schr¨ odinger Operators, Deformation Quantization), Akademie Verlag, Berlin, 1995, pp. 9–69. [51] B.-W. Schulze, Boundary value problems and singular pseudo-differential operators, J. Wiley, Chichester, 1998. [52] B.-W. Schulze, An algebra of boundary value problems not requiring ShapiroLopatinskij conditions, J. Funct. Anal. 179 (2001), 374–408. [53] B.-W. Schulze, Operator algebras with symbol hierarchies on manifolds with singularities, Advances in Partial Differential Equations (Approaches to Singular Analysis) (J.Gil, D.Grieser, and Lesch M., eds.), Oper. Theory Adv. Appl., Birkh¨ auser Verlag, Basel, 2001, pp. 167–207. [54] B.-W. Schulze, Operators with symbol hierarchies and iterated asymptotics, Publications of RIMS, Kyoto University 38, 4 (2002),735–802. [55] B.-W. Schulze, Toeplitz operators, and ellipticity of boundary value problems with global projection conditions., Oper. Theory: Adv. Appl. 151, Advances in Partial


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B.-W. Schulze Institut f¨ ur Mathematik Universit¨ at Potsdam Am Neuen Palais 10 D-14469 Potsdam, Germany e-mail: [email protected]

Elliptic Equations and Boundary Value Problems in Non-Smooth Domains Vladimir B. Vasilyev Abstract. The theory of elliptic boundary value problems for pseudo-differential equations in domains with a non-smooth boundary can be constructed with the help of special factorization of the elliptic symbol which is located at a singular boundary point. Mathematics Subject Classification (2000). Primary 35S15; Secondary 42B20. Keywords. Elliptic equation, boundary value problem, pseudo-differential equation, wave factorization, Calderon-Zygmund operator, spectra, Fourier series.

1. Introduction In the paper [1] the authors studied elliptic boundary value problems with a parameter and parabolic problems with a smooth boundary for linear partial differential equations. Almost at the same time (1965) the paper by M.I. Vishik and G.I. Eskin appeared [14], and it was devoted in general to studying the same questions but in the “pseudo-differential” context. (The reader should consult G.I. Eskin [3] for further insight into this topic.) In the intervening years, the symbolic calculus for the algebra of the pseudo-differential boundary value problems and related index theory was developed. Concerning the theory of boundary value problems for pseudo-differential equations in non-smooth domains it seems that such a complete theory (including symbolic calculus + index theory) does not exist up to now.1 In the 1990s the author suggested the construction of such a theory with the concept of wave factorization[12, 13] for the elliptic symbol, and it is related to conical singularity and a trip to the multivariable complex domain (roughly speaking, it is the multivariable Wiener-Hopf method). In this way, particularly, the full solvability of the boundary value problem (the number of boundary conditions is defined by the index of wave factorization), the special algebraic condition 1 Following

the recommendation of the referee, the references [5, 6, 7, 8, 9, 10] have been added to provide another point of view on these problems.


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(Shapiro-Lopatinskii type) in a corner point, is necessary and sufficient. This condition for the Laplacian with the Dirichlet and Neumann data was verified by direct calculations. Further the author suggests using wave factorization to construct the theory of boundary value problems with a parameter (after[1]) taking the simplest 2D-equation of type a (Au)(x) = f (x, q), x ∈ C+ a as a model, where C+ = x ∈ R2 : x2 > a |x1 | , a > 0 , A is the elliptic pseudodifferential operator [2] with symbol A(ξ, q), ξ = (ξ1 , ζ2 ), depending on complex a parameter q, the solution u(x) we seek in the Sobolev-Slobodetsky space H s (C+ ). Under existing wave factorization for symbol A(ζ, q) one does not see serious obstacles for constructing an analogous theory as in the pure elliptic case [12, 13]; here we find that new boundary value problems arise.

2. Spaces and operators LetS (Rm ) be Schwartz space of infinitely differentiable, rapidly decreasing at infinity, functions. For function u (x) ∈ S (Rm ) its Fourier transform will be denoted by u˜ (ξ). On functions from S (Rm ) we define pseudo-differential operator A with symbol A (x, ξ) by the formula (Au) (x) = eix·ξ A (x, ξ) u ˜ (ξ) dξ, (2.1) Rm

where x · ξ denotes inner products of x and ξ, and at the right the inverse Fourier transform of product A (x, ξ) and u ˜ (ξ) is written (x is fixed here). Roughly speaking, the operator A can be defined by the kernel K (x, y) as the convolution operator by type K (x, x − y) u (y) dy,

(Au) (x) = Rm

where the integral is treated in a certain generalized sense. Such operators have been intensively studied by mathematicians since the mid-1960s, and then the symbolic calculus appeared, which permitted them to use symbols instead of operators. The convenient space scale for such operators is Sobolev-Slobodetsky space scale H s (Rm ); generally it consists of distributions. We say u ∈ H s (Rm ), if

u 2s = |˜ u (ξ)|2 (1 + |ξ|)2s dξ < +∞. (2.2) Rm m

Because S (R ) ⊂ H (R ) is dense, then originally (2.2) is defined on S (Rm ). Moreover, H s (Rm ) is Hilbert space, so there is an inner product: ∀u , v ∈ H s (Rm ) , m

s

2s

u ˜ (ξ) · v˜ (ξ) (1 + |ξ|) dξ,

(u, v) = Rm

where the over-line is complex conjugation.

Elliptic Equations and Boundary Value Problems

107

The pseudo-differential operators work well in the space scale H s (Rm ). We require some natural restrictions on symbol A (x, ξ): ∃ c1 , c2 > 0,

−α

c1 ≤ A (x, ξ) (1 + |ξ|) ≤ c2 , ∀x, ξ ∈ Rm , (2.3) and call α the order of pseudo-differential operator A. So, for example, if A (x, ξ) is polynomial on ξ with constant coefficients, then an order of such an operator is the degree of the polynomial. The operators with symbols (2.3) are bounded in the space H s (Rm ), namely A : H s (Rm ) → H s−α (Rm ) . But, it is important to note that such operators have a local property: if symbol A (x, ξ) is a smooth function on x, then at a nearby point x0 ∈ Rm , the operators with symbols A (x, ξ) and A (x0 , ξ) are almost the same; “almost” means up to small or compact perturbations. The last property permits us to transfer without force pseudo-differential operators from Euclidian space Rm onto smooth compact manifolds without boundary.

3. Equations and factorization A pseudo-differential equation is an equation of type (Au) (x) = f (x) , x ∈ D,

(3.1)

where D is a smooth manifold generally with boundary, which can be non-smooth. In particular, it may include the points of cones or wedges. The manifold is locally a part of Euclidean space Rm , and, depending on the point in which we localize the equation, we have different types of models of operators in model domains of Euclidian spaces, for which their invertibility implies normal solvability (Noetherian property, sometimes they say Fredholmness) of the original operator in (3.1). If the boundary of the manifold has a conical point x0 ∈ ∂D, then the model equation for (3.1) will be given by a (Ax0 u) (x) = f (x) , x ∈ C+ ,

(3.2) a C+ = x ∈ Rm : xm > a |x |, x = (x1 , . . . , xm−1 ), a > 0 , where Ax0 is a pseudo-differential operator with frozen pole A (x0 , ξ). Thus for studying normal solvability of equation (3.1) we have to find invertibility conditions for the model operator in cone (3.2). Its symbol will be denoted by A (ξ), briefly assuming the condition holds. a Definition. Symbol A (ξ) admits wave factorization with respect to cone C+ if it can be represented in the form

A (ξ) = A= (ξ) A= (ξ) ,

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where the factors A= (ξ),A= (ξ) satisfy the following conditions: A= (ξ) extends ∗

a holomorphically into the radial tube domain T (C+ ) over the cone ∗

a C+ = {x ∈ Rm : axm > |x |}

with estimate

∗

±1

±κ a ;

A= (ξ + iτ) ≤ c (1 + |ξ| + |τ |) , ∀τ ∈ C+ ∗

∗

∗

a a a the factor A= (ξ + iτ ) has analogous properties with change C+ −C+ ≡ C− . The number κ ∈ R is called the index of wave factorization. The existence of wave factorization and the number κ fully determine the solvability situations for equation (3.2). Let us introduce the integral operator G. It is a multivariable analogue of a well-known Cauchy type integral, which will be defined originally on functions u ∈ S (Rm ) by the formula −m/2 2 2 (Gu) (x) = lim u (y) |x − y | − a2 (xm − ym + iτ ) dy. τ →0+

Rm

Such an operator permits us to construct explicitly the solution of equation (3.2)), if the wave factorization a exists. Let us denote by H0s C+ the space of functions from H s (Rm ), which sup s a a port belongs to C+ , H C+ the space of distributions which admit continuation 2 0 + 2 s lf R , lf ∈ H R , f s = inf lf s , where the infimum is chosen from all l

continuation l. Theorem 1. If κ − s = δ, |δ| < 12 , then the equation (3.2) ahas the unique solution s−α a s f ∈ H0 C+ for arbitrary right-hand side u+ ∈ H C+ , −1 ˜ u ˜+ = A−1 = GA= l f .

The a priori

u+ s ≤ C f

+ s−α .

holds. Everywhere below we take m = 2 for simplicity. Theorem 2. If κ − s = n + δ, n > 0, n ∈ Z, |δ| < 12 , then the equation (3.2) for a a arbitrary right-hand side f ∈ H0s−α C+ has solutions u+ ∈ H s C+ , −1 −1 ˜ u ˜+ (ξ) = A−1 A= lf = QG2 Q n−1 k k ˜ + A−1 c ˜ (ξ − aξ ) (ξ + aξ ) + d (ξ + aξ ) (ξ − aξ ) k 1 2 1 2 k 1 2 1 2 = k=0

+

nδ k1 +k2 =0

ak1 k2 (ξ1 − aξ2 )

k1

(ξ1 + aξ2 )

k2

,


109

where ck , dk are arbitrary functions from H sk (R+ ), Q (ξ) is an arbitrary elliptic polynomial of order n, (α = n) , sk = s − κ + k + 12 , k = 0, 1, . . . , n − 1. The a priori estimate holds n−1 +

u+ s ≤ C f s−α + [ck ]sk + [dk ]sk + k=0

nδ

|ak1 k2 |

,

k1 +k2 =0

[·]s denotes the norm on the straight line. 1 Theorem 3. If κ − sa= n + δ, n < 0, n ∈ Z, |δ| < 2 , then equation (3.2) has s solution u+ ∈ H C+ iff

β1 β2

1 ∂ ∂ 1 ∂ ∂

− − A−1 l f (y) =0, =

a ∂y1 ∂y2 a ∂y1 ∂y2 y=0

β1 β2

1 ∂ ∂ 1 ∂ ∂ −1 − − A= l f (y)

=0, ay1 −y2 ≤0 a ∂y1 ∂y2 a ∂y1 ∂y2 ay1 +y2 =0

β1 β2

1 ∂ ∂ 1 ∂ ∂

− − A−1 l f (y) =

ay1 −y2 =0 = 0 . a ∂y1 ∂y2 a ∂y1 ∂y2 ay1 +y2 ≥0

4. Boundary value problems Let for simplicity n = 1, f = 0, a = 1. According to Theorem 2, the general solution of equation (3.2) has the form u ˜ (ξ1 , ξ2 ) = A−1 ˜0 (ξ1 − ξ2 ) + d˜0 (ξ1 + ξ2 ) , = (ξ1 , ξ2 ) c and one needs to determine the functions c˜0 , d˜0 . If the boundary conditions are given as restrictions of pseudo-differential operators on angle sides, then after linear change of variables, applying the Fourier transform and further applying the Mellin transform, we obtain a system of linear algebraic equations with the matrix  ˆ  ˆ 12 (λ) K11 (λ) K 1 0 ˆ 22 (λ) ˆ 21 (λ)  K  K 0 1 . K(λ) =   ˆ ˆ 1 0 M11 (λ) M12 (λ)  ˆ 22 (λ) ˆ 21 (λ) 0 1 M M The condition det K (λ ) = 0 is called the conical Shapiro-Lopatinskii condition. Let us consider the Dirichlet problem. Here Γ+ = x ∈ R 2 : x2 > | x1 | , and we are looking for the function u+ (x) ∈ H s (Γ+ ), satisfying the equation (Au+ ) (x) = 0, x ∈ Γ+ and boundary equation

u+ x2 =x1 = v1 , x1 >0

u+ x2 =−x1 = v2 . x1 0

+ [v2 ]

1 s− , κ−1 2

.

x2 =−x1 = v2 . x1 0, ξm > 0   & m 2 − |ξ |2 − k 2 , a2 ξ 2 − |ξ |2 − k 2 > 0, ξ < 0 = , ∓ a2 ξm m m  &    i |ξ |2 + k 2 − a2 ξ 2 , a2 ξ 2 − |ξ |2 − k 2 < 0 m

and is given by 2 ξm

2

− |ξ | + k = 2

'

m

&

2 |ξ |

+ 1 ξm + − − ' & 2 2 2 2 2 × a + 1 ξm − a ξm − |ξ | − k . a2

2 a2 ξm

k2

2. A diffraction problem of a spatial wave on a plane screen can be formulated from the mathematical point of view as follows: finding function u ∈ H 1 R3 , satisfying the Helmholtz equation ∆ + k 2 u (x) = 0, x ∈ R3 \Γ+ , Γ+ = x ∈ R3 : x3 = 0, x2 = | x1 | , and the boundary Dirichlet condition u (x ) = g (x ) , x ∈ Γ+ ,

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or Neumann ones ∂u (x ) = h (x ) , x ∈ Γ+ , (x3 = ±0) , ∂x3 1

1

g, h are given functions, g ∈ H 2 (Γ+ ) , h ∈ H − 2 (Γ+ ) , k ∈ C, k = k1 +i k2 , k2 > 0, x = (x1 , x2 ) . These two problems were reduced to solving equation (3.2) with symbol ± 1 A (ξ1 , ξ2 ) = ξ12 + ξ22 − k 2 2 (E. Meister, F.-O. Speck). 3. A problem on indentation of a wedge-shaped punch into an elastic half-space − 1 was reduced to equation (5) with symbol A (ξ1 , ξ2 ) = ξ12 + ξ22 2 also (V.A. Babeshko).

6. Calculations Here we consider two classical problems related to the Laplacian, which can be described completely by wave factorization. 1 1. Finding u+ ∈ H s C+ such that 1 (∆u+ ) (x) = 0, x ∈ C+ ,

and boundary conditions

u+ x2 =x1 = g1 (x2 + x1 ) , x1 >0

u+ x2 = − x1 = g2 (x2 − x1 ) x1 < 0

are valid. The determinant for such a problem is ∆A (λ) = 1 − 2 sin−2 (2π λ) sin2

π (2λ − 1) . 4

Theorem 6. If g1 , g2 ∈ H s (R+ ) , s > 12 , then there exists the unique solution 1 with a priori estimate of the Dirichlet problem in the space H s, 1 C+

u+

s, 1

≤ c ( [g1 ] s + [g2 ] s ) .

2. An analogous problem with mixed Dirichlet and Neumann conditions 1 (∆u+ ) (x) = 0, x ∈ C+ ,

∂ u+

u+

= g1 , = g2 , x2 =−x1 ∂ n x2 =x1 x1 < 0

x1 >0

n is a normal vector for straight line x2 = x1 . The determinant for such a problem is π ∆A (λ) = 1 − sin−2 (2π λ) sin (2λ − 1) . 2

Elliptic Equations and Boundary Value Problems 3

113

1

+ Theorem 7. If g1 ∈ H s− 2 , −1 (R+ ) , g2 ∈ H s− 2 (R ) , s > 32 , then there exists 1 a unique solution of this problem in the space H s, 1 C+ , and the a priori estimate

u+ s, 1 ≤ c [g1 ] 3 + [g2 ] 1 s−

2

,−1

s−

2

holds.

7. Appendix 1: Wave factorization and almost trigonometric series 1. In the theory of boundary value problems for pseudo-differential equations in domains with smooth boundary, one of the key points takes a special function, which helps factorizing the elliptic symbol and which is closely related to the Laplacian. This function is (m ≥ 3) ν(ξ , ξm ) =

|ξ | + iξm , |ξ | − iξm

ξ = (ξ1 , . . . , ξm−1 ).

Taking into account spherical coordinates in Rm−1 (ξ −→ (m, r)), and homogeneity of order 0 for function ν(ξ, ξm ), we have the function of two variables r, t, r + it , r > 0, ∞ < t < +∞, r − it which will be also homogeneous of order 0, and thus it can be treated as defined on the right arc of the unit circle in complex plane C, ν(r, +∞) = ν(0; +1), ν(r, −∞) = ν(0; −1). Now if we take polar coordinates (r, ϕ), instead of Descartes (τ, r) we obtain the function depending on ϕ only: ν(r, t) =

ν(ϕ) =

cos ϕ + i sin ϕ eiϕ = −iϕ , cos ϕ − i sin ϕ e

−π ≤ ϕ ≤ π.

The function ν(ϕ) is a reproducing function for a trigonometric (clas( π system π) sical Fourier series) in the sense that any function f (ϕ) ∈ L2 − ; can be 2 2 uniquely represented by convergent (at least in L2 -norm) Fourier series on powers of function ν(ϕ): +∞ f (ϕ) = cn ν n (ϕ). (7.1) n=−∞

Two functions |ξ |±iξm generate a pair for expansion of the elliptic Laplacian symbol 2 |ξ |2 + ξm = (|ξ | + |ξm ) (|ξ | − |ξm ) (7.2) by two factors; both of them admit a holomorphic continuation (under fixed ξ ) onto the upper and lower half-plane (without zeroes). Obviously, ν(ϕ) is the parametrically defined unit circle in C. As is well known, every pseudo-differential operator with symbol A(ξ) nondependent on spatial variable x can be represented by a product of Laplacian

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power and multidimensional singular integral, which symbol is a homogeneous of order 0 function (A.P. Calderon, A. Zygmund [2]). The “freezing coefficient” of an operator on boundary smooth points leads to a factorization problem for the elliptic symbol in one variable (related to normal direction), which is solved by factors (7.2), function ν(ϕ) and a Cauchy type integral. The image ν(ϕ) is a unit circle going twice in the positive direction. 2. This situation is changed cardinally, if the boundary is non-smooth, for example, there is a corner point at the boundary. In this case the model pseudo-differential a equation has been considered in the sector C+ = {x ∈ R2 : x2 ≥ a, a > 0}. The idea of factorization for elliptic symbols also works here, but we need to use multivariable complex analysis. So, particularly, wave factorization for the a Laplacian with respect to C+ looks as follows (m = 2) ' ' & & ξ12 + ξ22 = a2 + 1 · ξ2 + a2 ξ22 − ξ12 a2 + 1 · ξ2 − a2 ξ22 − ξ12 , (7.3) where the factors in (7.3) admit holomorphic continuation into radial tube domains ∗

over conjugate cones C a± under a convenient choice of square root branch. The interesting thought appears to introduce the function (its analogue of function ν): ' √ a2 + 1 · ξ2 + a2 ξ22 − ξ12 ' W (ξ1 , ξ2 ) = √ , a2 + 1 · ξ2 − a2 ξ22 − ξ12 which will be homogenous of order 0, and after taking into account polar coordinates it will look as (for simplicity a = 1) ' √ 2 sin ϕ + sin2 ϕ − cos2 ϕ ' W (ϕ) = √ 2 sin ϕ − sin2 ϕ − cos2 ϕ and one can consider the series of type +∞

cn W n (ϕ)

(7.4)

n=−∞

with reproducing function W (ϕ). The image of function W looks very interesting. At first, obviously, it is a closed curve on a complex plane, secondly, it has real pieces, and the pieces go up and down from the real axis. The quantitative picture is given as follows: 6

I @ i # • @ @ • - •c b! @" • @ R −i @

-


115

The curve goes twice in the positive direction (it is shown by arrows), √ √ 2−1 2+1 b= √ , c= √ . 2+1 2−1 It is remarkable that under a → ∞, the real piece of this curve (its segment [b, c]) compresses to point 1, but the complex piece converts to a unit circle. In more detail, 2 & √ 2 ± 2 W (ϕ) = 2 sin ϕ + sin ϕ − cos ϕ & √ 2 2 = 3 sin ϕ − cos ϕ ± 2 2 sin ϕ sin2 ϕ − cos2 ϕ, and for initial view, the series of type (7.4) is not worse than classical Fourier series (7.1) [4].

8. Appendix 2: Spectral properties of Calderon-Zygmund operators 8.1. Introduction The Calderon-Zygmund operator is an integral operator which is defined by the special kernel K(x), x ∈ Rm \ {0}, m ≥ 2; it is a function which is homogeneous of order −m, infinitely differentiable on unit sphere S m−1 and has vanishing mean value: 1) K(tx) = t−m K(x), ∀t > 0; ∞ m−1 2) K(x) ); ∈ C (S 3)

K(θ)dθ = 0. S m−1

Given kernel K(x) one constructs an integral operator, in which the integral is treated in the principal value sense (Ku)(x) ≡ v.p. K(x − y)u(y)dy ≡ lim K(x − y)u(y)dy. (8.1) Rm

ε→0 |x−y|>ε

Such integral operators (8.1) (singular integrals) were systematically studied by papers of A.P. Calderon and A. Zygmund (see [2]) as bounded operators in Lp (Rm )-spaces. They also considered generalizations (8.1), if the kernel is more complicated, namely it is the function K(x, y), x ∈ Rm , y ∈ Rm \ {0}, and under fixed x the kernel K(·, y) defines the Calderon-Zygmund operator according to formula (8.1). The last operators they called singular integrals with variable kernels. It seems, these operators have no attributes of interest; in general, they are convolutions which can be meant in a certain generalized sense. I do not know what are the original problems which A.P. Calderon and A. Zygmund would like to describe; I can describe my own wishes which moved me to study these

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operators. Now, it seems, I can make these three points. First, the studies of Calderon-Zygmund related to boundedness of operators (8.1) in spaces of integrable functions, and simultaneous studies of S.G. Mikhlin on solvability of equations with such operators made great progress in stating and developing the theory of pseudo-differential operators and boundary value problems. Second, the Calderon-Zygmund operator is really the convolution only, but the convolution theory gives the “mathematics” by which one describes the interaction “inputoutput” for linear system. Third, and last, the Calderon-Zygmund operator is a multidimensional analogue of the Hilbert transform which is widely used in digital signal and image processing. Now I will try to initiate the search for some interactions among these three aspects. 8.2. Discrete Calderon-Zygmund operator Let Zm be an integer lattice in Rm . Given a lattice they define functions of discrete argument (or discrete functions), and given a Calderon-Zygmund kernel they construct discrete singular convolution (Ku)(˜ x) = K(˜ x − y˜)u(˜ y ), x ˜ ∈ Zm , (8.2) y˜∈Zm \{0}

and convergence of the series may be treated as lim K(˜ x − y˜)u(˜ y ), N →∞

y˜∈CN ∩(Zm \{0})

where CN is a cube in Rm of size N ∈ Z : * + m CN = x ∈ R : max |xk | ≤ N . 1≤k≤n

In definition (8.1) the truncation at infinity is not essential because such operators at first were defined on infinitely differentiable functions with compact support, and after obtaining an Lp -estimate one can consider the limit case, taking into account that these functions are dense in Lp (Rm ). We will consider the operator (8.1) and (8.2) in space L2 (Rm ) and its discrete analogue 2 , because we will seriously apply the Fourier transform. Two of the key properties of the Fourier transform (which for the given function u(x), x ∈ Rm , we define by the formula (F u)(x) ≡ u ˜(ξ) ≡ u(x)eix·ξ dx, Rm

where x · ξ denotes the inner product in Rm ) are the following: 1) operator F of the Fourier transform is isometry on L2 (Rm ) (isomorphism + norm conservation);


117

2) operator F transforms the convolution of two functions, it is defined by the formula (u ∗ v)(x) ≡ u(x − y)v(y)dy, (8.3) Rm

to the product of their Fourier transforms: F (u ∗ v) = (F u) · (F v). It is easy to create a discrete Fourier transform F for discrete functions, and this Fourier transform will be denoted by Fd (see also [11]). We have (discrete functions we denote by subindex “d” too) u ˜d (ξ) = (Fd ud )(ξ) = ei˜x·ξ ud (˜ x), x ˜∈Zm

(ud ∗ vd )(˜ x) =

ud (˜ x − y˜)vd (˜ y ),

y˜∈Zm

and, respectively a discrete analogue of formula (8.3), Fd (ud ∗ vd ) = u ˜d (ξ) · v˜d (ξ). Let us note that u ˜d (ξ) is a 2π-periodic function defined in Rm , Q = [−π; π]m is a basic cube of periods. In general the discrete Calderon-Zygmund operator (8.2) reserves all properties of its continual analogue, and, it seems, this way will not lead to any interesting result. Further I will try to show that this matter is not so simple, and even something interesting can be found in this usual situation. 8.3. Operators and their symbols One of the basic principles of the so-called “symbolic calculus of pseudo-differential operators” is “freezing coefficients” principle or, in other words, “local principle”. Applying this principle to Calderon-Zygmund operators with “variable kernels” K(x, y) has led to a fine and full theory of pseudo-differential operators on manifolds and boundary value problems for such an operator on manifolds with boundary. But let us come back to Calderon-Zygmund operators. This is one of the simplest pseudo-differential operators, and if one “frees the pole x” for “variable kernel” K(x, y) and considers the kernel K(·, y) then it will be the operator of type (8.1). For such “singular convolutions” the Fourier transform is applicable, and the “convolution formula” holds. It can be written now as [F (Ku)] (ξ) = σ(ξ)˜ u(ξ). The function σ(ξ) is called the symbol of operator K [2] and the image of the symbol defines spectra of operator K. The symbol of operator K is a function σ : S m−1 → C which is infinitely differentiable on S m−1 . It is very hard for me to say anything on structure of the spectra, but from general analysis it

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follows only that this is a (non-empty) closed set. If we remember now the onedimensional analogue of the Calderon-Zygmund operator (it exists only alone, it is the Hilbert transform ), then its symbol is the function sign ξ, which is defined on a null-dimensional sphere (these are points ±1 of the real axis), the symbol is not defined at the origin. Let us note in the one-dimensional case a correspondence symbol – spectra is one-to-one, i.e., in other words given spectra the Hilbert transform is uniquely reconstructed. Since the spectra is treated as an observable phenomenon by physicists, the following question arises. Given spectra, can one reconstruct the Calderon-Zygmund operator? From a mathematical point of view we say the following: given spectra spec K of operator K, one can construct the infinitely differentiable (or continuous at least) function σ : S m−1 → spec K; if so then the operator K is easily constructed (roughly speaking by the inverse Fourier transform [2]). It is obvious that it is also possible up to automorphisms f : S m−1 → S m−1 , because any composition σ ◦ f gives the same. I will try to describe one such way, related to my earlier studies on boundary value problems [12, 13]. 8.4. Conical multiplier and spectral decomposition For simplicity we consider the plane case m = 2. Then S 1 is a unit circle. Let us Sa be a part of S 1 intersected by cone a C+ = x ∈ R2 : x2 > a|x1 |, a > 0 , where ma (x) is a function (multiplier) equal to 1 on Sa and 0 on another piece of S 1 . The question was formulated as follows: which kind of operator corresponds to such a multiplier in the Fourier image? As it was shown in [12], this is the convolution operator with the Bochner kernel [15], more precisely, it is an operator of type u˜(η1 , η2 ) dη1 dη2 def F (ma · u) = lim ≡ (Ga u ˜)(ξ). τ →0+ (ξ1 − η1 )2 − a2 (ξ2 − η2 + iτ )2 R2

In the case m = 2 the image σ : S 1 → spec K ≡ γ necessarily goes around a smooth curve on a complex plane. Let us take a partition of the curve by points

(vertices) λk ∈ γ, k = 1, 2, . . . , n, and on each arc λk−1 λk choose an arbitrary ˜ k . We do the same for S 1 , taking its partition for n (equal) pieces, and point λ ˜ (for near pieces of λ ˜ we take near pieces then each piece corresponds to a certain λ 1 1 of S ). Each arc of S will correspond to multiplier mak (x) (let us note that every Gak is a Calderon-Zygmund operator), and then the original operator K will be looked upon as K≈

n k=1

˜ k Ga ∆λk λ k


119

˜ k ), and as a limit (up to (recall that ak is angle size which can be connected to λ constants) has be obtained by the line integral over γ, K = λGλ dλ. γ

8.5. Additional remarks It is possible that given considerations will be useful for finding finite approximations of Calderon-Zygmund operators (multidimensional singular integrals) which are much in need for numerical modelling of all singular integral equations arising in different branches of physics and technics. Conserving the discrete Calderon-Zygmund operator introduced in 8.2, we note that as the lattice Zm is an infinite set, we need a finite approximation in this case too; here the important fact is that spectra of discrete and continuous operators are the same. m Let Zm with uniform step h for each variable. We will h be a lattice in R define “lattice kernel” uh (˜ x), x ˜ ∈ Zm h , and given kernel K(x) we will construct “lattice kernel” Kh (˜ x) restricting K(x) on Zm h \ {0}, and define discrete CalderonZygmund operator (Kh uh )(˜ x) = lim Kh (˜ x − y˜)uh (˜ y )hm (8.4) N→+∞

y˜∈QN h \{0}

m where QN h is a discrete cube of lattice Zh with side N h. The symbol σh (ξ) of such an operator can be treated as a multivariable discrete Fourier transform [2, 11] of its kernel Kh (˜ x) in principal value sense σh (ξ) = lim ei˜x·ξ Kh (˜ xhm . (8.5) N →+∞

y˜∈QN h \{0}

, -m Evidently, the symbol σh (ξ) is defined and continuous on −π , π \ {0} and h h a periodical function on Rm . Let us denote by L2 (Zm h ) the discrete analogue of L2 (Rm ). Applying the Fourier transform (8.5) to operator (8.4) leads to multiplication operator u ˜h (ξ) −→ σh (ξ)˜ uh (ξ)

(8.6)

m m of operator (8.4) acting , −πL2π(Z -mh ) → L2 (R ) L2 (Qh ), where Qh = h , h ) will coincide

in Fourier images, and the spectra (or operator (8.6) acting L2 (Qh ) → with image σh (ξ). What can we say if h tends to 0? It is fully evident that Qh \{0} will transform to Rm \ {0}, but what about σh (ξ) ? No less evident that σh (ξ) → σ(ξ) in the pointwise convergence sense.

(8.7)

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Some simple considerations can help us explain (8.7). Take u ∈ S(Rm ) (Schwartz class of infinitely differentiable rapidly decreasing at infinite functions) and write (8.1) in Fourier images σ(ξ)˜ u(ξ). “Discrete variant” of such an expression to “lattice function” uh (˜ x) constructing on u(x) leads to σh (ξ)˜ uh (ξ). So, as (8.1) and (8.4) under consideration will be bounded, then we have ∃c > 0, |σh (ξ)| ≤ c,

|σ(ξ)| ≤ c.

If so, we consider the difference ˜ |˜ |σ(ξ)˜ u(ξ) − σh (ξ)˜ uh (ξ)| ≤ |σ(ξ) − σh (ξ)| · |˜ u(ξ)| + |σh (ξ)| u(ξ) − u ˜h (ξ)|, which tend to 0 as h → 0, because σh and u ˜h are “cubature formulas” for corresponding integrals (if they exist). So, (8.7) holds exactly. Let Zm be an “integer number” lattice in Rm , Q = [−π, π] be a corresponding “unit” cube in dual space in the Fourier sense. One of the easily seen properties of symbol of operator (8.4) is the following. Proposition 1. σ1 (hξ) = σh (ξ). Let us note that σ1 is defined on Q \ {0} (period π), and σh is defined on Qh \ {0} (period πh−1 ). Further, because σh isn’t at 0, but under ξ = 0 the limit σh (ξ) exists as h → 0, then, it follows that the limit depends on the direction of vector ξ = 0 in which the limit exists. Thus, this direction can be determined by the point of the unit sphere S m−1 in which this radius-vector cuts the S m−1 . If one has fixed points ξ, then one has σ(ξ) = lim σh (ξ) = lim σ1 (hξ), h→0

h→0

and therefore has Proposition 2. For ∀ξ = 0 the σh (ξ) does not depend on h. The last proposition may be useful for expanding some approximate methods for solution of multidimensional singular integral equations with CalderonZygmund operators. But we have to take into account the hard work related to change of “infinite discrete” object by “finite discrete” ones; there are different possibilities, and the author thinks to consider these variants separately.


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References [1] M.S. Agranovich and M.I. Vishik, Elliptic problems with a parameter and general parabolic problems, Russian Math. Surveys 19 (1964), 53–161 (in Russian). [2] A. Bellow, C.E. Kenig and P. Malliavin, Editors, Selected Papers of Alberto P. Calderon with Commentary, Collected Works 21, AMS, 2008. [3] G.I. Eskin, Boundary Value Problems for Elliptic Pseudo-Differential Equations. AMS, Providence, RI, 1981. [4] G.H. Hardy and W.W. Rogosinski, Fourier Series, Cambridge University Press, 1944. [5] G. Harutjunjan and B.-W. Schulze, Elliptic Mixed, Transmission and Singular Crack Problems, European Mathematical Society, Z¨ urich, 2008. [6] D. Kapanadze and B.-W. Schulze, Crack Theory and Edge Singularities, Kluwer Academic Publishers, Dordrecht, 2003. [7] X. Liu and B.-W. Schulze, Ellipticity on manifolds with edges and boundary, Monatsh. Math. 146 (4) (2005), 295–331. [8] X. Liu and B.-W. Schulze, Boundary value problems in edge representation, Math. Nachr. 280 (5-6) (2007), 1–41. [9] E. Schrohe and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities I, in Pseudo-Differential Calculus and Mathematical Physics, Advances in Partial Differential Equations, Akademie Verlag, Berlin, 1994, 97–209. [10] E. Schrohe and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities II, in Boundary Value Problems, Schr¨ odinger Operators, Deformation Quantization, Advances in Partial Differential Equations, Akademie Verlag, Berlin, 1995, 70–205. [11] S.L. Sobolev, Introduction to the Theory of Cubature formulas, Moscow, Nauka, 1974 (in Russian). [12] V.B. Vasilyev, Wave Factorization of Elliptic Symbols: Theory and Applications, Kluwer Academic Publishers, 2000. [13] V.B. Vasilyev, Fourier Multipliers, Pseudo-Differential Equations, Wave Factorization, Boundary Value Problems, Editorial URSS, 2006 (in Russian). [14] M.I. Vishik and G.I. Eskin, Convolution equations in a bounded domain, Russian Math. Surveys 20 (1965), 89–152 (in Russian). [15] V.S. Vladimirov, Methods of Functions Theory of Many Complex Variables, Moscow, Nauka, 1964 (in Russian). Vladimir B. Vasilyev Bryansk State University Bezhitskaya 14 Bryansk 241036, Russia e-mail: [email protected]

Calculus of Pseudo-Differential Operators and a Local Index of Dirac Operators Chisato Iwasaki Abstract. We give the formula for a local index of Dirac operators on Riemannian manifolds of even dimension. Mathematics Subject Classification (2000). Primary 58J35; Secondary 58J40, 58J20. Keywords. Symbolic calculus, fundamental solution, Dirac operator, local index.

1. Introduction Let M be a compact Riemannian manifold without boundary of even dimension m = 2n and let E be a C(M )-module with compatible connection ∇. Suppose m Γ = im/2 c(X1 )c(X2 ) · · · c(Xm ), where c(Xj ) j=1 is a Clifford multiplication of m an orthonormal base of vector fields Xj j=1 . Then we have Γ2 = 1. The bundle E splits: E = E+ ⊕ E−, where E ± = {ϕ ∈ E : Γϕ = ±ϕ}. The Dirac operator splits D = D+ +D− , where D± = D|C ∞ (M,E ± ) according to the above decomposition. Then the index of D is defined by ind(D) = dim ker(D+ ) − dim ker(D− ). It is easy to see that the following equation holds: 2 2 tr(e−tD (x, x)|E + ) − tr(e−tD (x, x)|E − ) dvx , ind(D) = M


123

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where e−tD (x, y) is the kernel of the fundamental solution for the Cauchy problem, 2 2 that is, U (t)f = (e−tD )f (x) = M e−tD (x, y)f (y)dvy satisfies * d ( dt + D2 )U (t)f = 0 in (0, T) × M, U (0)f = f (x) on M. 2

We may call the following term a local index: 2 2 tr(e−tD (x, x)|E + ) − tr(e−tD (x, x)|E − ) dvx . The conclusion of this paper is that one can calculate the above local index by only the main term of the symbol of the fundamental solution, introducing a new weight of symbols of pseudodifferential operators. This paper gives the another proof of the theorem given by Patodi-Gilkey in other words. We note that D2 is an elliptic operator. But I use the method of construction of the fundamental solution for a degenerate parabolic operators which was studied in C. Iwasaki and N. Iwasaki[5]. This method is similar to that of C. Iwasaki[7]. The plan of this paper is as follows. In Section 2 and Section 3 we give definitions for Clifford Algebra, Clifford Bundle, compatible connections and the spin connection. In Section 4 we give the exact definition of the Dirac operator and the result of this paper. Section 5 is the main part of this paper in which we discuss the way to prove the result. The final section is devoted to the proof of the key Lemma which is stated in Section 5.

2. Clifford algebra Let V be a real vector space of finite dimension m with a inner product. Definition 2.1. (The Clifford algebra C(V ) of V ) The Clifford algebra C(V ) of V is the associative algebra with unit generated by the elements of V under the relation v · w + w · v = −2(v, w)1. Definition 2.2. A representation of the Clifford algebra C(V ) is an R-algebra homomorphism c : C(V ) −→ HomC (W, W ) into the algebra of C-linear transformations of a finite-dimensional complex vector space W . W is called a C(V )-module. We shall refer to the corresponding action ϕ · w = c(ϕ)(w)

for ϕ ∈ C(V ) and w ∈ W

as Clifford multiplication of w by ϕ. Example 1.

W = C(V ), c(v) = v·

Example 2.

W = ∧V (the exterior algebra), c(v) = (v∧) − ı(v).

Local Index of Dirac Operators

125

3. Clifford bundle and compatible connection Let M be a compact Riemannian manifold without boundary of even dimension in the rest of this paper. Definition 3.1. (1) The Clifford bundle C(M ) of M is a bundle over M , where the fibre over a point x ∈ M consists of the Clifford algebra C(T Mx) with the Riemannian metric for the real tangent vector space T Mx . (2) A representation of C(M ) is a fibre preserving algebra morphism c : C(M ) −→ Hom(E, E), where E is a smooth complex vector bundle over M . Without loss of generality one can assume that E has a Hermitian structure · · · , · · · such that for any x ∈ M c(v)s, s = −s, c(v)s s, s ∈ Ex ,

v ∈ Tx∗ (M ).

Definition 3.2. (1) Let E be a smooth vector bundle of fibre dimension N over M . A connection on E is a first-order partial differential operator ∇ : C ∞ (M ; E) −→ C ∞ (M ; T ∗ M ⊗ E) such that ∇(f s) = df ⊗ s + f ∇s,

f ∈ C ∞ (M ), s ∈ C ∞ (M ; E).

(2) A connection ∇ on a C(M )-module E is defined to be compatible with the C(M ) module structure, if it satisfies the following conditions: (i) vs, s = ∇v s, s + s, ∇v s for v ∈ T M, s, s ∈ C ∞ (M ; E). (ii) [∇, c(v)] = c(∇g v) for v ∈ C ∞ (M ; T X) with the Levi-Civita connection ∇g . This means ∇(c(v)s) = c(v)(∇s) + c(∇g v)s,

v ∈ C ∞ (M ; T X), s ∈ C ∞ (M ; E).

Example. E = ∧(T ∗ M ), ∇ = the Levi-Civita connection. Definition 3.3 (the spin connection ∇S ). Let a local frame {s1 , . . . , sN } of E on U . ∇S (sj ) =

N

S ωjk ⊗ sk ,

j=1 S where ω S = (ωjk ) denotes the connection 1-form which is an N × N matrix of 1-forms such that ωµS = ω S (Xµ ) is given by 1 σ ωµS = Γ cν cσ , cν = c(Xν ) 4 ν,σ µν

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with ∇gXµ Xα =

m

Γβµα Xβ .

β=1

Then the spin connection is compatible. In fact we have 1 σ ∇SXk c(X ) = Γkµ cµ cσ c 4 σ=µ

c(X )∇SXk

1 σ = Γkµ c cµ cσ , 4 σ=µ

where we use Γσkµ = −Γµkσ . By the above equalities we the following equalities hold: Γσkµ cµ cσ c = Γσkµ c cµ cσ + 2 Γσk cσ σ=µ

σ=µ,σ=,µ=

Γσkµ c cµ cσ

σ=µ

=

σ=

Γσkµ c cµ cσ

−2

σ=µ,σ=,µ=

So we have [∇SXk , c(X )] =

Γσk cσ = −

σ=

Γσk cσ .

σ= m

Γkσ cσ .

σ=1

The following facts can be found in B. Booß-Bavnbek and K.P. Wojciechowski [2], Proposition 3.4. There is exactly one irreducible C(V )-module ∆ and dim ∆ = 2n . We can write in a suitable local chart U E|U = ∆ ⊗ V0 , ∇|U = ∇S ⊗ 1 + 1 ⊗ ∇0 with a trivial bundle V0 .

4. Dirac operator and result Let M be a compact Riemannian manifold without boundary of even dimension m = 2n and let E be a C(M )-module with compatible connection ∇. Choose an orthonormal base of vector fields {X1 , . . . , Xm } on a contractible open set U . Let {w1 , . . . , wm } be the set of its dual 1-form. Note that we can choose a local frame for E over U so that the matrix c(Xj ) are constant (see [2]). Definition 4.1. (Dirac operator) Under the above assumption we define a Dirac operator D: m D= c(Xj )∇Xj . j=1

Local Index of Dirac Operators

127

E = ∧(T ∗ M ), ∇ = the Levi-Civita connection,

Example 1.

c(Xj ) = (wj ∧) − ι(Xj ), D = d + δ. M is a Kaehler manifold, E = ⊕nq=0 ∧0,q (M ), ∇ = the Levi-Civita

Example 2. connection,

c(Xj ) = (wj ∧) − ι(Xj ), D =

√

2(∂¯ + ∂¯∗ ).

Note that the following equations hold: c(Xj )c(Xk ) = −c(Xk )c(Xj ),

(j = k)

(1 ≤ j ≤ m).

2

c(Xj ) = 0

Theorem 4.2 (Patodi-Gilkey). If the connection is compatible, the we have {tr(e−tD (x, x)|E + ) − tr(e−tD (x, x)|E − )}dx ( ) 0 Ω/2 = (2πi)−m/2 det[ ] ∧ tr(e−Ω ) + 0(t), sinh(Ω/2) m 2

2

where Ω is the curvature form of ∇g , that is, Ω is an m × m matrix of 2-form defined by m 1 Ω= R(Xj , Xk )ω j ∧ ω k , 2 j,k=1

0

Ω is a matrix of 2-form defined by Ω0 =

m 1 0 R (Xj , Xk )ω j ∧ ω k 2 j,k=1

and [Φ]m means the m-form of Φ.

5. Method Proposition 5.1 (Lichnerowitz formula). Assume that the connection is compatible. Then we have m D2 = − {∇Xj ∇Xj − ∇∇gX Xj } + c(Xj )c(Xk )RE (Xj , Xk ) j

j=1

=−

m j=1

+

j 2. In [16] Chao Jiang Xu and Xusheng Zhu have studied the 2 2 operators Dx2 1 + x ˆ2k ˆ is in 1 Dx2 + c on R , where c > 0, k is a positive integer, x ∞ the class C (R), x ˆ = x for | x |≤ 2, xˆ is increasing, x ˆ = 4 sgn x if | x |≥ 4. They have proved some invertibility properties in the class of nonhomogeneous pseudodifferential operators. In a recent paper (cf. [11]) J. Delgado and A. Zamudio have studied the invertibility in the setting of H¨ ormander’s classes (cf. [14]) for the operators considered in [3]. In [10] some Lp estimates have been obtained for a This work has been partially supported by Universidad del Valle, Vicerrectoria Inv. Grant#7795.


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J. Delgado

class of fractional powers of subelliptic operators with loss of one derivative. In this paper we consider highly degenerate elliptic operators of type P (x, t, D) =

n

Dx2 i +

i=1

n

x ˆ2k i

i=1

m

Dt2i ,

i=1

where k ≥ 2 is an integer, x ˆ is as above, and n +m ≥ 2. This operator is subelliptic with index of subellipticity = k+1 and into the box {(x, t) : |xi | < 2, i = 1, . . . , n} we have a sum of squares of vector fields satisfying the Hörmander condition of order k + 1. In order to study Lp estimates in the context of H¨ ormander classes S(M t , G) we define a weight and a metric M (x, t, ξ, τ ) =

n i=1

ξi2

+

n i=1

x ˆ2k i

m

12 τi2

˜ + ξ

2δ

(1.2)

i=1

˜ = M −2/k (x, t, ξ, τ)ξ ˜ 2/k Gx˜,ξ˜(d˜ x, dξ)

n

dx2i +

i=1

˜ −2δ(1−1/k) + M −2/k (x, t, ξ, τ )ξ

m

dt2i

(1.3)

i=1 n

m

i=1

i=1

˜ −2 dξi2 + ξ

˜ = (1+ | ξ˜ |2 )1/2 and δ = where x ˜ = (x, t), ξ˜ = (ξ, τ); ξ

dτi2 ,

1 1+k .

The main results of this work are the following theorems on Lp boundedness. Theorem A. Let Ω ⊂ Rn+m be open, connected and relatively compact, if 0 ≤ < Q0 ˜ ∈ S(M −(n+m) , G).Then σ(x, t, D) is bounded from Lp (Ω) to x, ξ) 2(n+m) and σ(˜ p L (Ω), 1 < p < ∞, where Q0 = m + n. Q0 ˜ ∈ S(M −β , G) with 0 ≤ β < (n+m)0 . Theorem B. Let 0 ≤ < 2(n+m) and σ(˜ x, ξ) Then σ(x, t, D) is bounded from Lp (Ω) to Lp (Ω), for p in the interval

1 1

β

− ≤

2 p 2(n + m)0 .

The corresponding Lp boundedness has been established by the author in [10] for a sum of squares of vector fields satisfying the Hörmander condition of order 2, moreover some Lp bounds have been also obtained starting with a general class of second-order differential operators with index of subellipticity > 0. Here we extend the class of sum of squares of vector fields considered in [10]. We recall that Richard Beals has characterized in [1] and [2] Hörmander’s metrics g ensuring the Lp boundedness for operators in OpS(1, g) and 1 < p < ∞. The condition obtained by Beals for a split metric is Φz,θ (θ) ≤ C,

Lp Bounds for a Class of Fractional Powers

139

where Φw (θ) = gw (0, θ) and gw (z, θ) = gw (z, 0)+gw (0, θ). In our case this condition is equivalent to n − k1 n m 2 2 2k 2 2δ ˜ ˜ −2δ(1−1/k) ≤ C. |ξ| ξ + x ˆ τ + ξ ξ i

i=1

i

i=1

i

i=1 4

The left-hand side is of order ξ2− k+1 which is unbounded unless k = 1. Here we are considering k ≥ 2. Now we will introduce some basic properties of the Weyl-Hörmander calculus and of abstract Sobolev spaces defined with this theory.

2. Weyl–Hörmander Calculus Let u ∈ C ∞ (Rn ), x = (x1 , . . . , xn ) ∈ Rn , and let α = (α1 , . . . , αn ) ∈ Nn , we shall use the following notations: |α| = α1 + · · · + αn ; Dil u = (−i)l

∂lu αn 1 ; Dα u = Dnαn . . . D2α2 D1α1 u ; xα = xα 1 . . . xn . ∂xli

Definition 2.1. For a(x, ξ) ∈ S (R2n ) (x ∈ Rn and ξ ∈ Rn ), we define the classic quantization as the operator a(x, D) : S(Rn ) → S(Rn ) given by a(x, D)u(x) = (2π)−n ei(x−y)ξ a(x, ξ)u(y)dydξ. The Weyl quantization of a(x, ξ), is given by the operator aω : S(Rn ) → S(R ) defined by x+y ω −n a u(x) = (2π) ei(x−y)ξ a( , ξ)u(y)dydξ. 2 n

Definition 2.2. Let a(x, ξ), b(x, ξ) ∈ S(R2n ) we define (a#b)(X) = π −2n e−2iσ(X−Y1 ,X−Y2 ) a(Y1 )b(Y2 )dY1 dY2 , R2n ×R2n

where σ(X, Y ) = y · ξ − x · η for X = (x, ξ) and Y = (y, η). The operation # is defined in order to obtain a symbol for the composition aω ◦ bω , indeed one has aω ◦ bω = (a#b)ω . Definition 2.3. For X ∈ R2n let gX (·) be a positive definite quadratic form on R2n , we say that g(·) is a Hörmander metric if the following three conditions are satisfied: 1. Continuity – There exists constants C, c, c ∈ R such that gX (Y ) ≤ C, for X, Y ∈ R2n , c · gX+Y (T ) ≤ gX (T ) ≤ c · gX+Y (T ) for every T ∈ R2n .

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2. Uncertainty principle – For Y = (y, η) and Z = (z, ζ) we define σ(Y, Z) = z · η − y · ζ, and σ(T, W )2 σ gX (T ) = sup . W =0 gX (W ) We say that g satisfies the uncertainty principle if σ 1/2 gX (T ) ≥ 1, λg (X) = inf T =0 gX (T ) for all X ∈ R2n . 3. Temperancy – We say that g is temperated if there exists C > 0 and J ∈ N such that ±1 gX (·) ≤ C(1 + gYσ (X − Y ))J . gY (·) Remark 2.4. If g is of type gX (dx, dξ) =

n dx2i dξi2 + , ai (X) bi (X) i=1

where ai (X) and bi (X) are positive functions, then σ gX (dx, dξ)

=

n

bi (X)dx2i + ai (X)dξi2 .

i=1

Definition 2.5. We say that a strictly positive function M is a g-admissible weight or g-continuous if there exists C˜ > 0 and N ∈ N such that ±1 M (X + Y ) 1 ˜ ≤ C, if gX (Y ) ≤ , M (X) C˜ ±1 M (Y ) ˜ + g σ (X − Y ))N . ≤ C(1 Y M (X) Definition 2.6. For a H¨ ormander metric g and a g-admissible weight M , we denote by S(M, g) the set of all smooth functions a on R2n such that for any integer k there exists Ck ∈ R, such that for all X, T1 , . . . , Tk ∈ R2n |a(k) (T1 , . . . , Tk )| ≤ Ck M (X)

k 4

1/2

gX (Ti ).

i=1

For a ∈ S(M, g) we denote by a k,S(M,g) the minimum Ck satisfying the above inequality. Let Y ∈ R2n , r > 0 and let g be a H¨ ormander metric, we denote by UY,r the ball of radius r with center Y , that is UY,r = {X ∈ R2n | gY (X − Y ) ≤ r2 }.


141

Definition 2.7. Let Y ∈ R2n , r > 0 and let g be a H¨ ormander metric, the space of all confined symbols on UY,r , we denote by Conf(g, Y, r) the space S(R2n ) endowed with the semi-norms:

a k,Conf(g,Y,r) =

|∂T1 . . . ∂Tl a(X)|(1 + gYσ (X − UY,r ))k/2 ,

sup l≤k;X∈R2n

gY (Tj )≤1

where gYσ (X − UY,r ) =

inf

W ∈UY,r

gYσ (X − W ).

A family of symbols (aY ), indexed by Y (vectors in R2n ) is uniformly confined on UY,r , if the numbers aY k,Conf(g,Y,r) are bounded by constants Ck independents of Y . The condition of continuity for H¨ ormander’s metrics enable us to obtain a partition of unity. Theorem 2.8. Let g be a continuous metric on R2n . Then there exists r0 > 0 such that, for every r ∈ (0, r0 ], there exists a family of functions (ϕY )Y ∈R2n supported in UY,r and

sup ϕY k,S(1,g) < ∞

∀k ∈ N.

ϕY (X)|gY |1/2 dY = 1

∀X ∈ R2n ,

Y ∈R2

R2n

where |gY | is the determinant of gY . The partition of unity allows us to decompose each symbol a(x, ξ) into a family of symbols aY (X) = M (Y )−1 a(X)ϕY (X), supported in UY,r . In order to reconstruct a from the aY we have a(X) = M (Y )aY (X)|gY |1/2 dY. R2n

The symbols aY are uniformly confined in UY,r , more precisely we have ∀k, ∃C, ∀Y,

aY k,Conf (g,Y,r)≤ C a k,S(M,g) .

For a proof of Theorem 2.8 see [6]. We shall need also the following symmetric distance function ∆r . For X, Y ∈ R2n we define σ ∆r (X, Y ) = 1 + max{gX (UX,r − UY,r ), gYσ (UX,r − UY,r )},

where gYσ (UX,r − UY,r ) =

inf

W ∈UY,r

gYσ (Z − W ).

Z∈UX,r

Remark 2.9. There exists N and H such that for all X ∈ R2n 1/2 ∆−N dY < H. r (X, Y )|gY |

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Moreover if r2 ≤ C (C is the constant for the continuity of the metric), then there exists U0 , U2 > 0 and U3 , U1 such that for all X, Y ∈ R2n we have (gX (T )/gY (T ))± ≤ U0 ∆r (X, Y )U1 , ∀T ∈ R2n . (M (X)/M (Y ))± ≤ U2 ∆r (X, Y )U3 . The following additional condition on the metric will be useful to establish a comfortable definition of Sobolev spaces. Definition 2.10. A metric g is strongly temperate, if there exists a function (X, Y ) → d(X, Y ) > 0, defined on R2n × R2n , verifying the triangular inequality, such that there exists C > 0, n ≥ 0 and r > 0 with ±1 gX ≤ C(1 + d(X, Y ))n . gY 1 + d(X, Y ) ≤ C∆r (X, Y ). δ,δ Remark 2.11. The metric gX is strongly temperate, it can be deduced from ˜ 1−δ − ˜ d(X, Y ) = |ξ η 1−δ |,

where X = (x, ξ) and Y = (y, η). We say that a metric g is dominated by a strongly temperate metric, if there exists a strongly temperate metric g˜ and C > 0, N ≥ 0 such that

gX (·) ≤ g˜X (·). ±1 gY (·) ≤ C(1 + g˜Yσ (Y − Z))N . gZ (·)

Definition 2.12. If g is a metric dominated by a strongly temperate metric and M is a g-admissible weight we can associate a Sobolev space H(M, g) using the partition of unity (Theorem 2.8) furnish by the family (ϕY )Y ∈R2n . Let u ∈ S (Rn ) we say that u ∈ H(M, G) if 2 1/2 M (Y )2 ϕω dY < ∞, Y u L2 |gY | 1

2 1/2 and we define the norm u H(M,g) = ( M (Y )2 ϕω dY ) 2 . Y u L2 |gY |

Remark 2.13. It is shown in [11] that the metric G defined by (1.3) is a H¨ ormander metric, M defined by (1.2) is a G-admissible weight, and G is dominated by the strongly temperate metric g δ,δ on Rn+m × Rn+m . The action of the Weyl quantization on the Sobolev spaces is given by the following theorem (cf. [5]). Theorem 2.14. Let M and M1 be two g-admissible weights. For every a ∈ S(M, g), we have aω : H(M1 , g) → H(M1 /M, g). We can also identify H(1, g) (M = 1) with L2 (cf. [5]).


143

Theorem 2.15. For a H¨ ormander’s metric g we have H(1, g) = L2 . The following corollary ensure the existence of invertible symbols in Hörmander classes. Corollary 2.16. Let M be a g-admissible weight for a H¨ ormander metric g, there exists b ∈ S(M, g) and b ∈ S(M −1 , g) verifying b#b = b #b = 1. For every weight M1 , the operator bw is an isomorphism from H(M1 , g) onto H(M1 /M, g), w with inverse b . Definition 2.17. A g-weight m is called regular if m ∈ S(m, g). The dependence of H(M, g) with respect to the metric is given by the following theorem (cf. [5]): Theorem 2.18. If M is a regular weight for g1 and g2 , then H(M, g1 ) = H(M, g2 ). The following corollary is obtained immediately. Corollary 2.19. Let M be an admissible weight for the H¨ ormander metrics g1 and g2 such that g1 ≤ g2 . If M is a regular weight for g1 then H(M, g1 ) = H(M, g2 ). Remark 2.20. 1. Since g1,0 ≤ gρ,δ for every (ρ, δ) such that 0 ≤ δ ≤ ρ ≤ 1 (δ < 1), we obtain for all m ∈ R H m = H(ξm , g1,0 ) = H(ξm , gρ,δ ), because ξ is a regular weight for g1,0 , and H m denotes the classical Sobolev space of order m. 2. The weight M defined by (1.2) is regular with respect to G and the symbol P (x, t, ξ, τ) of the sublaplacian P belongs to S(M 2 , G) (cf. [11]). 3. There exists a constant C > 0 such that g1,0 ≤ CG, in fact one has that ˜ 2 ˜ −2 ≤ M − k2 ξ ˜ −2δ(1− k1 ) because M ≤ C ξ ˜ and k ≥ 2. 1 ≤ ( ξ ) k and ξ M

The operator P (x, t, D) being subelliptic we can associate it a subelliptic ˜s = distance on Rn+m (cf. [13]). This operator is a sublaplacian in the box B n+m {(x, t) ∈ R : |xi | < 2, i = 1, . . . , n} and is generated by the system of vector fields (P, Q): Pi = Dxi , i = 1, 2, . . . , n; Qjr = xˆkr Dtj , j = 1, 2, . . . , m; r = 1, . . . , n. This system satisfies the H¨ ormander condition of order = k + 1. The collection of balls B(˜ x, r) = {˜ y ∈ Rn+m ; (˜ x, y˜) < r}. corresponding to satisfies the local doubling condition on Rn+m with respect to the Lebesgue measure and verifies the following comparison with the Euclidean distance (cf. [13], [15]). Theorem 2.21. For every compact K ⊂ Rn+m there exists a constant C1 > 0 (depending on K) such that C1 −1 |˜ x − y˜| ≤ (˜ x, y˜) ≤ C1 |˜ x − y˜| k+1 , x ˜, y˜ ∈ K. 1

(2.1)

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J. Delgado

3. Lp Estimates In this section we establish some Lp bounds for a class of fractional powers of subelliptic operators. The Lp boundedness will be obtained from L∞ − BM O bounds and interpolation, in turn these bounds will be estimated from local L∞ − L∞ bounds. The starting point of this type of analysis is the classic paper [12] of m Charles Fefferman where he considers the Sρ,δ classes. These bounds were extended to Hörmander classes in [10], here we improved the main results contained in [10]. ˜ denotes the symbol of the operator Throughout this section p(x, t, ξ, τ) (= p(˜ x, ξ)) n n m 2 2k P (x, t, D) = Dx i + x î Dt2i . i=1

i=1

i=1

The next lemma will be essential for us in order to estimate an optimal fractional power which determines a suitable class for Lp boundedness. This lemma is an adaptation of Lemma 5.3 in [7] established for a system of vector fields satisfying the H¨ ormander condition of order 2. We consider Q0 = m +n where = k + 1, the index Q0 corresponds to the homogeneous dimension associated to the subelliptic ˜s . system (P, Q) in the box B Lemma 3.1. For the metric G and the weight M the following inequalities hold for every (x, t) ∈ Rn+m : Q0 sup R2s−Q0 M −2s (x, t, ξ, τ)dξdτ < ∞, if 0 < s < , (3.1) 2 R>1 M (x,t,ξ,τ )≤R

sup R2s−Q0 R>1

M −2s (x, t, ξ, τ)dξdτ < ∞,

if s >

Q0 . 2

M (x,t,ξ,τ )≥R

(3.2)

n Proof. We consider two cases. First, if i=1 x ˆ2k i ≥ 1 (elliptic zone) then we have −1 ˜ ˜ ≤ Cξ, ˜ C ξ ≤ M (˜ x, ξ)

n and both inequalities in the lemma hold. Now, if i=1 x ˆ2k i ≤ 1 (subelliptic zone), we define: 2 1 n m def 2 2 ˜

ξ = ξj + τj j=1

j=1

and def δa ξ˜ = (aξ1 , . . . , aξn , a τ1 , . . . , a τm ),

˜ = a ξ , ˜ and d(δa ξ) ˜ = an am dξ˜ = aQ0 dξ. ˜ Using the inequality We have δa ξ

˜ ≤ M (˜ ˜ c(1 + ξ ) x, ξ),

n we obtain for (x, t) such that i=1 x ˆ2k i ≤ 1, ˜ ξ˜ ≤ c−2s ˜ −2s dξ˜ . M −2s (˜ x, ξ)d

ξ

˜ M (˜ x,ξ)≤R

−1 R ˜ ξ≤c

Lp Bounds for a Class of Fractional Powers Taking ξ˜ = δR ξ , we have

˜ −2s dξ˜ = RQ0 −2s

ξ

Q0 2 ,

ξ −2s dξ .

ξ ≤c−1

−1 R ˜ ξ≤c

If 0 < s
0, ∀j} = B. We made a change of variables ξ = h(ξ) where ˜ = (ξ1 , . . . , ξn , τ1 , . . . , τm ), h(ξ) denoting by

eucl the Euclidean norm and using the fact 2s < m + n = Q0 we obtain

ξ −2s dξ = C ξ −2s dξ B

ξ ≤c−1

˜ −2s |τ1 · · · τm |k dξ˜

ξ

eucl

≤ Cm ˜ eucl ≤c−1 ξ

˜ −2s+km dξ˜ = C(c, s, Q0 ) < ∞.

ξ

eucl

≤ Cm ˜ eucl ≤c−1 ξ

˜ eucl for j = In the last inequality we have used the fact that |τj | ≤ ξ

1, . . . , m. This proves (3.1). For 2s > Q0 , we have −s n n m −2s −2s 2 2k 2 2δ ˜ ˜ ˜ M (˜ x, ξ)dξ ≤ c ξ + x ˆ τ + |ξ| dξ˜ i

A

˜ M (˜ x,ξ)≥R

with

i=1

i

i=1

i

i=1

5 n 12 n m 2 2k 2 2δ ˜ ˜ A = ξ ξi + x î τi + |ξ| ≥ cR . def

i=1

i=1

i=1

Using the change of variable ξ˜ = δR ξ , and having into account that = k + 1 we obtain −s n n m −2s Q −2s 2 2k 2k 2 2δ ˜ ξ˜ ≤ CR 0 M (˜ x, ξ)d ξ +R x ˆ τ + |ξ | dξ i

˜ M (˜ x,ξ)

A

i=1

i

i=1

i

i=1

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J. Delgado

where

  12 n n m   def 2 2δ A = ξ / |ξi |2 + R2k x ˆ2k |τ | + |ξ | ≥ c . i i   i=1

i=1

i=1

1 2k ˜ If R2k ni=1 x ˆ2k ξj = R( ni=1 xˆ2k i ≥ 1, we use the change of variable i ) ξj . If

n n m m 2k 2 2 R2k i=1 x ˆ2k ˆ2k i ≤ 1 we use the inequality 0 ≤ R i i=1 x i=1 |τi | ≤ i=1 |τi | . If 2s > Q0 then −s n n m 2 2k 2k 2 2δ ξi + R xî τi + |ξ | dξ i=1

A

i=1

i=1

n

≤ ! " 1 2 2δ 2 ( n i=1 ξi +|ξ | ) ≥c

−s ξi2

2δ

dξ ≤ C(s, Q0 ) < ∞.

+ |ξ |

i=1

The following lemma furnishes L∞ (Rn+m ) − L∞ (Rn+m ) estimates. Lemma 3.2. Let 0 ≤
n+m we have 2

− 12 , and q(x, t, ξ, τ ) ∈

≤ 3R for R > 1. Then, for

q(x, t, D)f ∞ ≤ C q l;S(M −(n+m) ,G) f L∞ , where the constant C is independent of q and f . Proof. Let R > 1 and q(x, t, ξ, τ) ∈ S(M −(n+m) , G) supported in ˜ 2δ ≤ 3R}, {(x, t, ξ, τ) ∈ Rn+m /R ≤ p(x, t, ξ, τ ) + ξ and a choice of a suitable . Then

˜ q(x, t, D)f (x, t) = (2π)−(n+m) ei(˜x−˜y)·ξ q(x, t, ξ, τ)f (˜ y )d˜ y dξ˜ = qˆx˜ (˜ x, y˜ − x˜)f (˜ y )d˜ y = qˆx˜ ∗ f (˜ x),

˜ = q(x, t, ξ, τ ) with respect to ξ, ˜ and we where qˆx˜ is the Fourier transform of qx˜ (ξ) have denoted x ˜ = (x, t), y˜ = (y, η). Thus |q(x, t, D)f (x, t)| ≤ ˆ qx˜ L1 f L∞ , (x, t) ∈ Rn+m . We will show that for each x˜ ∈ Rn+m and l >

n+m 2

ˆ qx˜ L1 ≤ C q l;S(M −(n+m) ,G) .


147

Let b > 0, applying the Cauchy-Schwarz inequality and Plancherel identity we have   12   12 n+m  n+m  ˜ 2 dξ˜ |ˆ qx˜ (˜ x)|d˜ y ≤ Cb 2  |ˆ qx˜ (˜ y )|2 d˜ y  ≤ Cb 2  |q(˜ x, ξ)| |˜ y| a suitable bn+m−2j = CR(n+m)−2j−

Q0 2

n+m . 2 Q j

0 + n+m

We can obtain the following inequality for 1

≤ CR 2 (2(n+m)+2j−Q0 ) = CR(n+m)+j−

Q0 2

.

The inequality holds if and only if −2j + obtaining

Q0 2(n+m)

−

1 2

Q0 j ≤j, n+m

≤ and the minimum 0 =

Q0 2(n+m)

− 12 .

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J. Delgado

Now we can state a theorem giving L∞ (Ω)−BM O (Ω) boundedness, for Ω ⊂ R open, connected and relatively compact. In particular this result extends Theorem 2.13 in [10] to a system satisfying the Hörmander condition of order k+1, ˜s . considering Ω ⊂⊂ B n+m

Theorem 3.3. Let 0 ≤
0 independent of f and an integer j such that

σ(x, t, D)f BMO ≤ C σ j;S(M −(n+m) ,G) f L∞ , f ∈ L∞ . Proof. Let f ∈ L∞ (Ω), x ˜0 ∈ Rn+m , and B = B(˜ x0 , r) ⊂ Ω. We wish to show that there exist an integer j and a constant C > 0 independent of f and B, such that 1 |σ(˜ x, D)f (˜ x) − gB |d˜ x ≤ C σ j;S(M −(n+m) ,G) f L∞ , (3.3) |B(˜ x0 , r)| B

where we have denoted g = σ(x, t, D)f. ˜ into two parts, σ = σ 0 + σ 1 , with σ 0 supported in We decompose σ(˜ x, ξ) ˜ + ξ ˜ 2δ ≤ 2r−1 , σ 1 supported in p(˜ ˜ + ξ ˜ 2δ ≥ 1 r−1 , and verifying the p(˜ x, ξ) x, ξ) 2 following inequalities

σ 0 l;S(M −(n+m) ,G) , σ 1 l;S(M −(n+m) ,G) ≤ Cl σ l;S(M −(n+m) ,G) , l ≥ 1.

(3.4)

The existence of the decomposition (3.4) is obtained in the following way, let 0 ≤ β ∈ C ∞ (R) equal to 1 if and only if |t| ≤ 12 , and supp β = {|t| ≤ 1} and ˜ x, ξ) ˜ = β(r(a(˜ ˜ + ξ ˜ 2δ )), β(˜ x, ξ) then ˜ = σ(˜ ˜ β(˜ ˜ x, ξ). ˜ σ 0 (˜ x, ξ) x, ξ) Applying Proposition 2.5 in [10] on Rn+m with σ ∈ S(M −(n+m) , G), M 2 = ˜ ˜ 2δ , M1 = M −(n+m) , λ = 0 and s = r we obtain p(˜ x, ξ) + ξ

σ 0 l;S(M −(n+m) ,G) ≤ Cl σ l;S(M −(n+m) ,G), for l ≥ 1. Letting σ 1 = σ − σ 0 , one can see that (3.5) is also valid for σ 1 . In order to obtain (3.3), we consider first σ 0 . We can write ∂xk σ 0 (x, t, D)f (x, t) = σx (x, t, D)f (x, t), ∂tk σ 0 (x, t, D)f (x, t) = σt (x, t, D)f (x, t), where σx is the symbol ˜ = ∂x σ 0 (˜ ˜ + iξk σ 0 (˜ ˜ σx (˜ x, ξ) x, ξ) x, ξ), k and σt is the symbol ˜ = ∂t σ 0 (˜ ˜ + iτk σ 0 (˜ ˜ σt (˜ x, ξ) x, ξ) x, ξ). k

(3.5)


149

We will only estimate σx , the analysis of σt being similar. Using a partition of unity, we can write ∞ ˜ ˜ σx (˜ x, ξ) = ρj (˜ x, ξ), j=1

˜ + ξ ˜ 2δ ∼ 2−j r−1 , and with ρj supported in p(˜ x, ξ)

ρj n+m;S(M −(n+m) ,G) ≤ C2−j r−1 σ n+m;S(M −(n+m) ,G) . Such a partition of unity is obtained from η : R → R defined by 0, if |s| ≤ 1 η(s) = 1, if |s| ≥ 2. Let ρ(s) = η(s) − η(2−1 s). Then supp ρ = {1 ≤ |s| ≤ 4}. One can verify that ∞ 1 = η(s) + ρ(2j s) , s ∈ R. j=1

˜ + ξ ˜ ) then Now, set s = r(p(˜ x, ξ) 2δ

˜ + ξ ˜ 2δ )) + 1 = η(r(p(˜ x, ξ)

∞

˜ + ξ ˜ 2δ )). ρ(r2j (p(˜ x, ξ)

j=1

˜ + ξ ˜ 2δ ) ≤ 1, we obtain The support of η being {|s| > 1} and r((˜ x, ξ) ∞ ˜ + ξ ˜ 2δ )) 1= ρ(r2j (p(˜ x, ξ) j=1

˜ + ξ ˜ 2δ ≤ r−1 }. for {p(˜ x, ξ) Taking into account that supp σ = supp σ 0 we have ∞ ˜ = ˜ + ξ ˜ 2δ )) · σ (˜ ˜ σ (˜ x, ξ) ρ(r2j (p(˜ x, ξ) x, ξ). j=1

Then one can choose ˜ = ρ(r2j (p(˜ ˜ + ξ ˜ 2δ )) · σ (˜ ˜ = αj (X)σ (X). ρj (˜ x, ξ) x, ξ) x, ξ) The estimation of the seminorms ρj l;S(M −(n+m) ,G) for l ≥ 1 is obtained as follows: since σ = ∂xk σ 0 + iξk σ 0 , we can write σ = σ1 + σ2 with σ1 ∈ S(M −(n+m) ϕ−1 1 , G) and

˜ G), σ2 ∈ S(M −(n+m) ξ, where we have written the metric G in the form 2 2 2 ˜ = dx + dt2 + dξ + dτ . GX (d˜ x, dξ) 2 2 2 ϕ1 Φ1 Φ2 ˜ g1,0 ) ⊂ S(ξ, ˜ G). ξk ∈ S(ξ,

150

J. Delgado

Applying Proposition 2.5 of [10] with s = 2j r, for each l, there exist constants A > 1, Cl such that for all j and every λ ∈ R |∂T1 · · · ∂Tl (σ1 (X) · αj (X))| 2λ j λ ≤ Cl A|λ| M −(n+m) (X)ϕ−1 1 (X)M (X)(2 r) σ1 l;S(M −(n+m) ϕ−1 ,G) . 1

Choosing λ = −1

σ1 · αj l;S(M −(n+m) ϕ−1 ,G)) ≤ Cl (2j r)−1 σ1 l;S(M −(m+n) ,G) 1

(3.6)

˜ since (ξ/M ) k · M −2 ≤ 1 and σ1 l;S(M −(n+m) ϕ−1 ,G) ≤ σ l;S(M −(m+n) ,G) . 1 Now, for σ2 we have for each l, there exists Cl such that for all λ: 1

|∂T1 · · · ∂Tl (σ2 (X) · αj (X))| ˜ 2λ (X)(2j r)λ σ

≤ Cl A|λ| M −(n+m) (X)ξM . ˜ 2 l;S(M −(m+n) ξ,G) ˜ ≥ 1 then If r > 1 we choose λ = 0, and using the fact that ξ |∂T1 · · · ∂Tl (σ2 (X) · αj (X))| ˜ ≤ Cl M −(n+m) (X)ξ σ

l;S(M −(n+m) ,G) k+1 ˜ 2δ ≤ p(˜ ˜ + ξ ˜ 2δ ≤ A2−j r−1 we get ξ ˜ ≤ C(2−j ) k+1 2 (r −1 ) 2 Since ξ x, ξ) and

σ2 · αj l;S(M −(m+n) ,G) ≤ Cl 2−j r−1 σ l,S(M −(m+n) ,G) . If r < 1 we choose λ =

(3.7)

− 1, then

k+1 2

|∂T1 · · · ∂Tl (σ2 (X) · αj (X))| ≤ CM −(n+m) (X)(2−j )

k+1 2

(r−1 )

k+1 2

(2j r)

k+1 2 −1

σ2 l;S(M −(m+n) ,G) .

Therefore

σ2 · αj l;S(M −(m+n) ,G) ≤ Cl 2−j r−1 σ l;S(M −(m+n) ,G) .

(3.8)

Thus, by (3.6), (3.7) and (3.8) we have

ρj l;S(M −(m+n) ,G) ≤ C2−j r−1 σ l;S(M −(m+n) ,G) .

(3.9)

Applying Lemma 3.2 and (3.9) there exists a constant C such that

∂xk σ 0 (˜ x, D)f L∞ ≤

∞

ρj (˜ x, D)f L∞

j=0

≤ Cr−1

∞

2−j σ n+m;S(M −(m+n) ,G) f L∞

j=0

≤ Cr−1 σ n+m;S(M −(m+n) ,G) f L∞ . Now, by the mean value theorem and the left inequality in (2.1) we have |σ 0 (˜ x, D)f (˜ x) − gB | ≤ C σ n+m;S(M −(m+n) ,G) f L∞ .

Lp Bounds for a Class of Fractional Powers Then 1 |B(˜ x0 , r)|

151

|σ 0 (˜ x, D)f (˜ x) − gB |d˜ x ≤ C σ l;S(M −(m+n) ,G) f L∞ .

(3.10)

B

This proves (3.3) for σ 0 . In order to study σ 1 , one considers a function φ on Rn+m , with 0 ≤ φ ≤ 10, 6 ⊂ {ξ˜ ∈ Rn+m : φ ≥ 1 on B(˜ x0 , r) and the Fourier transform φ6 verifying supp(φ) ˜ ≤ (C −1 r)b(−1) }, where b = k + 1. We write |ξ| , φ(˜ x) · σ 1 (˜ x, D)f (˜ x) = σ 1 (˜ x, D)(φf )(˜ x) + φ, σ1 (˜ x, D) f (˜ x) = I + II. (3.11) Estimation of I: By Corollary 2.16 applied to M (n+m) , there exist b ∈ S(M (n+m) , G)

and b ∈ S(M −(n+m) , G)

such that b#b = b #b = 1. Setting L = bw , and L−1 = bw , we decompose

σ 1 (˜ x, D)(φf ) = σ 1 (˜ x, D) · L L−1 · (φf ) .

(3.12)

Having into account that b ∈ S(M −(n+m) , G), ˜ − (n+m) δ 2 M −(n+m) ≤ ξ , and G ≤ gδ,δ , then ˜− b ∈ S(ξ

(n+m) δ 2

, gδ,δ ).

˜ is regular with respect to g1,0 and g1,0 ≤ gδ,δ we have Since that ξ ˜− H(ξ

(n+m) δ 2

˜− , gδ,δ ) = H(ξ

(n+m) δ 2

, g1,0 ) = H −

(n+m) δ 2

.

Therefore, L−1 : H −

(n+m) δ 2

−→ L2 .

(3.13)

Now, using the fact that the symbol of σ (˜ x, D) · L belongs to S(1, g), then there exists a constant C > 0 and an integer N such that 1

σ 1 (˜ x, D)(φf ) 2L2 ≤ C σ 1 2N ;S(M −(n+m) ,G) · L−1 (φf ) 2L2 . By (3.13)

L−1 (φf ) 2L2 ≤ C φf 2 − (n+m) δ . H

2

Recall that

φf 2 − (n+m) δ = J − H

2

(n+m) δ 2

(φf ) 2L2 ,

(3.14)

152

J. Delgado (n+m) δ

where J − 2 is the Bessel potential. Since this potential is a positive operator ˜ = ξ ˜ − (n+m) δ 2 (preserves positivity of functions) and a multiplier with symbol j(ξ) , we obtain

J −

(n+m) δ 2

(φf ) 2L2 ≤ f 2L∞ J −

(n+m) δ 2

≤ C1 f 2L∞ φ 2 ≤ C1 f 2L∞ (C ≤

(φ) 2L2

(n+m) δ

2 H− −1 b(n+m)

r)

C f 2L∞ |B(˜ x0 , r)|,

using in the last inequality the right inequality in (2.1). Thus

σ 1 (˜ x, D)(φf ) 2L2 ≤ C σ 1 2N;S(M −(n+m) ,G) f 2L∞ |B(˜ x0 , r)|. Applying the Cauchy-Schwarz inequality, we have   12 1 1 |σ 1 (˜ x, D)(φf )(˜ x)|d˜ x≤ |σ 1 (˜ x, D)(φf )(˜ x)|2 d˜ x |B(˜ x0 , δ)| |B| B

B

≤ C σ 1 N ;S(M −(n+m) ,G) f L∞ . This proves the estimate for I.

, In order to estimate II, we write φ, σ 1 (˜ x, D) f (˜ x) in the form θ(˜ x, D)f (˜ x),

where θ(˜ x, D) is given by the symbol ( ) ˜ 6 η ) σ 1 (˜ ˜ − σ 1 (˜ θ(˜ x, ξ) = ei˜x·˜η φ(˜ x, ξ) x, ξ˜ + η˜) d˜ η. Rn+m

We write ˜ = θ(˜ x, ξ)

∞

˜ θj (˜ x, ξ),

j=0

˜ supported in p(˜ ˜ + ξ ˜ 2δ ∼ 2j r−1 . with θj (˜ x, ξ) x, ξ) Since ( ) ˜ = 6 η ) σ1 (˜ ˜ − σ 1 (˜ θ(˜ x, ξ) ei˜x·˜η φ(˜ x, ξ) x, ξ˜ + η˜) d˜ η, |η|≤(C −1 r)b( −1)

the mean value theorem applied to the symbol σ with respect to ξ gives −1 θ ∈ S(M −(n+m) r− 2 (Φ−1 1 + Φ2 ), G), 1

and

θ

for l ≥ 1.

1

−1 l;S(M −(n+m) r − 2 (Φ−1 1 +Φ2 ),G)

≤ Cl σ l+1;S(M −(n+m) ,G),

(3.15)

(3.16)


153

A similar analysis used to estimate the seminorms ρj l;S(M −(m+n) ,G) gives the following estimates for the seminorms of θj j

θj l;S(M −(m+n) ,G) ≤ C2− 2 σ l+1;S(M −(m+n) ,G) , l ≥ 1. Applying Lemma 3.2 we obtain ∞ ,

φ, σ1 (˜ x, D)f L∞ ≤

θj (˜ x, D)f L∞ j=0

≤

∞

j

C2− 2 σ n+m+1;S(M −(n+m) ,G) f L∞

(3.17)

j=0

≤ C σ n+m+1;S(M −(n+m) ,G) f L∞ . Since φ ≥ 1 on B(˜ x0 , r), using (3.16) and (3.17) into (3.11) we have 1 1 1 |σ (˜ x, D)f (˜ x)|d˜ x≤ |φ(˜ x) · σ 1 (˜ x, D)f (˜ x)|d˜ x |B(˜ x0 , r)| |B(˜ x0 , r)| B

B

≤ C σ n+m+1;S(M −(n+m) ,G) f L∞ . This terminates the proof of the theorem.

As a consequence one can prove the following two theorems using real and complex interpolation respectively and the duality (H 1 ) = BM O (cf. [8], [9]), the proofs are completely similar to those of Theorems 2.8 and 2.10 in [10]. Q0 ˜ ∈ S(M −(n+m) , G). Then σ(x, t, D) Theorem 3.4. Let 0 ≤ < 2(n+m) and σ(˜ x, ξ) p p is bounded from L (Ω) to L (Ω), 1 < p < ∞. Moreover, there exists a constant C > 0 independent of f and an integer j such that

σ(x, t, D)f Lp ≤ C σ j;S(M −(n+m) ,G) f Lp . Q0 ˜ ∈ S(M −β , G) with 0 ≤ β < Theorem 3.5. Let 0 ≤ < 2(n+m) and σ(˜ x, ξ) (n + m)0 . Then σ(x, t, D) is bounded from Lp (Ω) to Lp(Ω), for p in the interval

1 1

β

− ≤ (3.18)

2 p 2(n + m)0 .

Moreover, there exists a constant C > 0 independent of f and an integer j such that

σ(x, t, D)f Lp ≤ C σ j;S(M −(n+m) ,G) f Lp . Acknowledgment I would like to thank the respectable anonymous referee for the valuable comments.

154

J. Delgado

References [1] R. Beals, Lp and H¨ older estimates for pseudodifferential operators: necessary conditions, in Harmonic Analysis in Euclidean Spaces, Proc. Symp. Pure Math., Williams Coll., Williamstown, Mass., 1978, Part 2, Proc. Symp. Pure Math. XXXV. Amer. Math. Soc., Providence, R. I, 1979, 153–157. [2] R. Beals, Lp and H¨ older estimates for pseudodifferential operators: sufficient conditions, Ann. Inst. Fourier (Grenoble) 29 (3) (1979), 239–260. [3] R. Beals, P. Greiner and B. Gaveau, Green’s functions for some highly degenerate elliptic operators, J. Funct. Anal. 165 (1999), 407–429. [4] R. Beals, P. Greiner and B. Gaveau, Erratum to Green’s functions for some highly degenerate elliptic operators, J. Funct. Anal. 165 (1999), no. 2, 407–429, J. Funct. Anal. 201 (1) (2003), 301. [5] J.-M. Bony and J.-Y. Chemin, Espaces fonctionnels associés au calcul de WeylH¨ ormander, Bull. Soc. Math. France 122 (1994), 77–118. [6] J.-M. Bony and N. Lerner, Quantification asymptotique et microlocalisation d’ordre supérieur I, Ann. Ecole Norm. Sup. (4) 22 (3) (1989), 377–433. [7] J.-Y. Chemin and C.-J. Xu, Inclusions de Sobolev en calcul de Weyl-H¨ ormander et champs de vecteurs sous-elliptiques, Ann. Ecole Norm. Sup. (4) 30 (6) (1997), 719–751. [8] R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homog` enes, Lectures Notes in Mathematics 242, Springer-Verlag, Berlin and New York, 1971. [9] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 (4) (1977), 569–645. [10] J. Delgado, Estimations Lp pour une classe d’opérateurs pseudo-différentiels dans le cadre du calcul de Weyl-H¨ ormander, J. Anal. Math. 100 (2006), 337–374. [11] J. Delgado and A. Zamudio, Invertibility for a class of degenerate elliptic operators, J. Pseudo-Differ. Oper. Appl. 1 (2010), 207–231. [12] C. Fefferman, Lp bounds for pseudo-differential operators, Israel J. Math. 14 (1973), 413–417. [13] C. Fefferman and D.H. Phong, Subelliptic eigenvalue problems, in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth, 1981. [14] L. H¨ ormander, The Analysis of Linear Partial Differential Operators, III, SpringerVerlag, 1985. [15] A. Nagel, E.M. Stein and S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103–147. [16] C. Xu and X. Zhu, On the inverse of a class of degenerate elliptic operator, Chinese J. Contemp. Math. 16 (3) (1995), 261–274. Julio Delgado Universidad del Valle Calle 13 100-00 Cali-Colombia e-mail: [email protected]

The Heat Kernel and Green Function of the Generalized Hermite Operator, and the Abstract Cauchy Problem for the Abstract Hermite Operator Viorel Catan˘a Abstract. Following Wong’s point of view (see [14], by Wong) we give a formula for the heat kernel of the generalized Hermite operator Lλ on R2 , λ ∈ R \ {0}. This formula is derived by means of pseudo-differential operators of the Weyl type, i.e., Weyl transforms, Fourier-Wigner transforms and Wigner transforms of generalized Hermite functions, which are the eigenfunctions of the generalized Hermite operators and form an orthonormal basis of L2 (R2 ) (see [2], by Catan˘ a). By means of the heat kernel, we give a formula for the Green function of Lλ , λ ∈ R \ {0}. Using the Green function and the heat kernel we give some applications concerning the global hypoellipticity of Lλ in the sense of Schwartz distributions, the ultracontractivity and the hyλ percontractivity of the strongly continuous one-parameter semigroup e−tL , t > 0, λ ∈ R \ {0}. We also give a formula for the one-parameter strongly continuous semigroup e−tA generated by the abstract Hermite operator A. The formula is derived by means of the abstract Weyl operators, the abstract Fourier-Wigner operator and the abstract Wigner operators. Mathematics Subject Classification (2000). Primary 47A75, 47D06; Secondary 47F05, 35Q40. Keywords. Fourier-Wigner transform, Wigner transform, Weyl transform, Hermite operator, heat kernel, Green function, strongly continuous one-parameter semigroup, global hypoellipticity, ultracontractivity, hypercontractivity.

1. The generalized Hermite operator Let λ ∈ R \ {0} be a fixed real constant and let differential operators on R2 , given by ∂ ∂ ∂ = |λ|−1/2 − i|λ|1/2 and λ ∂z ∂x ∂y

∂ ∂ , be linear partial ∂z λ ∂z λ

∂ ∂ ∂ = |λ|−1/2 + i|λ|1/2 . λ ∂x ∂y ∂z


155

156

V. Catan˘ a

Then we define the linear partial differential operator Lλ on R2 by 1 λ λ Lλ = − (Z λ Z + Z Z λ ), 2 where ∂ 1 Zλ = + z λ , z λ = |λ|1/2 x − i|λ|−1/2 y ∂z λ 2 and ∂ 1 λ Z = + z λ , z λ = |λ|1/2 x + i|λ|−1/2 y. 2 ∂z λ

(1.1)

λ

Let us observe that −Z is the formal adjoint of Z λ and Lλ is an elliptic partial differential operator on R2 , given by ∂2 ∂2 1 ∂ ∂ Lλ = −|λ|−1 2 − |λ| 2 + (|λ|x2 + |λ|−1 y 2 ) − i x|λ| − y|λ|−1 . (1.2) ∂x ∂y 4 ∂y ∂x We call the partial differential operator defined by (1.1) or (1.2) the generalized Hermite operator, because for |λ| = 1, λ ∈ R, Lλ is the usual Hermite operator 1 −∆ + (x2 + y 2 ), perturbed by the partial differential operator −iN , where 4 ∂2 ∂2 ∂ ∂ ∆= + and N =x −y . ∂x2 ∂y 2 ∂y ∂x λ

The operators Z λ , Z and Lλ for |λ| = 1 are studied by Thangavelu in his books [10], [11] by Wong in his papers [14], [15], and by Dasgupta-Wong in their joint paper [3]. Let us remark that the generalized Hermite operator Lλ can be factored as 2 2 1 1 λ −1 Dx − y + |λ| Dy + x , L = |λ| (1.3) 2 2 where 1 ∂ 1 ∂ Dx = and Dy = , i ∂x i ∂y and that his symbol is given by. 2 2 1 1 −1 σLλ (x, y, ξ, η) = |λ| ξ − y + |λ| η + x , (1.4) 2 2 for all (x, y), (ξ, η) in R2 . Thus, it follows from (1.4) that Lλ is an elliptic operator, but not globally elliptic. Also, it is not globally hypoelliptic in the sense of Shubin (see [9] Section 25). More details concerning the notion of global hypoellipticity can be found in the book [1] by Boggiatto, Buzano and Rodino. The paper is organized as follows. In Section 2 we recall some basic definitions (see the book [12], by Wong) and preliminary results (see the paper [2], by Catan˘a). In Section 3 we give a formula for the heat kernel of the generalized Hermite operator on R2 . This formula is derived by means of Weyl transforms, FourierWigner transforms and Wigner transforms of the generalized Hermite functions

Heat Kernel and Green Function of Generalized Hermite Operator

157

on R2 (see the paper [6], by Hulanicki and the paper [2] by Catan˘ a) which form an orthonormal basis of L2 (R2 ). By means of the heat kernel, we give in Section 4 a formula for the Green function of Lλ . Using the Green function and the heat kernel we give in Section 5 some applications concerning the global hypoellipticity of Lλ in the sense of Schwartz distributions. Also using the heat kernel and the Riesz-Thorin interpolation theorem, we prove in Section 6 the ultracontractivity λ and hypercontractivity of the strongly continuous one-parameter semigroup e−tL , t > 0, λ ∈ R \ {0} in the sense of Davies (see the books [4], by Davies and [8] by Reed and Simon). In Section 7, the last one we define some bilinear and linear operators which we called respectively the abstract Fourier-Wigner operator, the abstract Wigner operator and the abstract Weyl operator. By means of them we also define the abstract Hermite operator. Then we give a formula for the oneparameter strongly continuous semigroup generated by this operator. By means of this formula we can give an estimate for the solution of the (ACP) governed by the abstract Hermite operator, in terms of the norm of the initial data. Let us remark that the results in Sections 3–6 are also valid for the operator Lλ on R2n given by n 1 λ λ λ Lλ = − Zj Z j + Z j Zjλ (1.5) 2 j=1

where Zjλ =

∂ 1 + zλ , ∂zjλ 2 j

z λj = |λ|1/2 xj − i|λ|−1/2 yj ,

(1.6)

∂ 1 − zλ, λ 2 ∂zj

z λ = |λ|1/2 xj + i|λ|−1/2 yj ,

(1.7)

and λ

Zj = for j = 1, 2, . . . , n.

2. Basic definitions and preliminary results For two arbitrary functions f and g in the Schwartz space S(R) on R we define the Fourier-Wigner transform of f and g by +∞ p p V (f, g)(q, p) = (2π)−1/2 eiqy f y + g y− dy (2.1) 2 2 −∞ for all q, p ∈ R. It is well known that V (f, g) is a function in S(R2 ), the Schwartz space on R2 . It also can be proved that V : S(R) × S(R) → S(R2 ) is a bilinear mapping which is called the Fourier-Wigner transform and that it satisfies the Moyal identity, i.e., (V (f1 , g1 ), V (f2 , g2 )) = (f1 , f2 )(g1 , g2 ),

(2.2)

for all f1 , f2 , g1 , g2 ∈ S(R), where the notation (, ) is used to denote the inner product in L2 (R) or in L2 (R2 ).

158

V. Catan˘ a The Wigner transform W (f, g) of f and g in S(R) is defined by W (f, g) = V (f, g)∧ ,

(2.3)

where Fˆ is the Fourier transform of F , which is defined by Fˆ (ζ) = (2π)−n/2 e−iz·ζ F (z)dz, ζ ∈ Rn , Rn

for all F in S(R ), the Schwartz space on Rn . It is well known that +∞ p p W (f, g)(x, ξ) = (2π)−1/2 dp, e−iξp f x + g x− 2 2 −∞ n

(2.4)

for all x, ξ ∈ R and f, g ∈ S(R), and also that W : S(R) × S(R) → S(R2 ) is a bilinear operator which satisfies the Moyal identity that is (W (f1 , g1 ), W (f2 , g2 )) = (f1 , f2 )(g1 , g2 ).

(2.5)

In addition it is obvious that W (f, g) = W (g, f ),

f, g ∈ S(R).

(2.6)

Let σ ∈ Lp (R2 ), 1 ≤ p ≤ ∞. By means of the Wigner transform it can be defined the Weyl transform associated to the symbol σ by +∞ +∞ −1/2 (Wσ f, g) = (2π) σ(x, ξ)W (f, g)(x, ξ)dxdξ (2.7) −∞

−∞

for all f and g in S(R). It can be shown that Wσ : S(R) → S(R) is a linear continuous operator. For more details concerning these operators see the book [12], by Wong. Now, for k ∈ N let ek be the Hermite function on R, of order k, defined by √ x2 ek (x) = (2k · k! π)−1/2 e− 2 Hk (x) (2.8) for all x in R, where Hk is the Hermite polynomial of degree k given by k 2 2 d Hk (x) = (−1)k ex (e−x ) dx

(2.9)

for all x ∈ R. Together with Hulanicki (see [6]) let us introduce the generalized Hermite function eλk of order k on R defined by eλk (x) = |λ|−1/4 ek (|λ|−1/2 x)

(2.10)

for all x ∈ R and a fixed λ ∈ R \ {0}. For j, k ∈ N and a fixed λ ∈ R \ {0} we define the generalized Hermite function eλj,k of order (j, k) on R2 by eλj,k (x, y) = V (eλj , eλk )(x, y)

(2.11)

for all x, y ∈ R. Then we can formulate the following result, which is Theorem 2 in the paper [2], by Catan˘a.


159

Theorem 2.1. {eλj,k ; j, k = 0, 1, 2, . . .} is an orthonormal basis for L2 (R2 ). The spectral analysis of the generalized Hermite operator is supported by the following fact, which is Theorem 3 in the paper [2] by Catan˘ a. Theorem 2.2. Lλ eλj,k = (2k + 1)eλj,k , for all j and k in N. Remark 2.3. From Theorem 2.2 we see that for k = 0, 1, 2, . . . the number 2k + 1 is an eigenvalue of the generalized Hermite operator Lλ , and the generalized Hermite functions eλj,k , j = 0, 1, 2, . . . on R2 , are eigenfunctions of Lλ corresponding to the eigenvalue 2k + 1.

3. The heat kernel In the paper [2] by Catan˘ a (see Theorem 4) a formula for the strongly continuous λ one-parameter semigroup e−tL , t > 0 generated by Lλ in terms of Fourier-Wigner transform and Weyl transform is given. Now, we want to give a formula for the kernel Kλ (t, z, w) of the integral λ operator e−tL , t > 0, which we shall call the heat kernel of Lλ . Thus, by Theorem 2.1, we get e−tL f = λ

∞ ∞

e−(2k+1)t (f, eλj,k )eλj,k ,

(3.1)

k=0 j=0

for all f ∈ S(R2 ). So, we can rewrite equivalently (3.1) in the form e−tL f = λ

∞

e−(2k+1)t

∞

(f, eλj,k )eλj,k

(3.2)

j=0

k=0

and we try to realize

∞

(f, eλj,k )eλj,k as a twisted convolution of f with an appro-

j=0

priate function connected with the generating function of Laguerre polynomials. To this end let us recall that for two measurable functions f and g on C the twisted convolution f τ g is defined by (f τ g)(z) = f (z − w)g(w)eiτ [z,w] dw, (3.3) C

for all z ∈ C, where [z, w] = 2Im(zw) ¯ is the symplectic form of z and w. Now, we can state the following result. Theorem 3.1. For all nonnegative integers α, β, µ, ν and for all λ in R \ {0} we get eλα,β −1/4 eλµ,ν = (2π)1/2 δβ,µ eλα,ν , where δβ,µ is the Kronecker symbol.

(3.4)

160

V. Catan˘ a

Proof. Let ϕ, ψ ∈ S(R2 ). Then by (2.7) and (2.5) we get −1/2 λ (z)W (ϕ, ψ)(z)dz (We7 ϕ, ψ) = (2π) e7 λ α,β α,β C = (2π)−1/2 W (eλα , eλβ )(z)W (ϕ, ψ)(z)dz C −1/2 = (2π) W (ϕ, ψ)(z)W (eλβ , eλα )(z)dz

(3.5)

C

= (2π)−1/2 (W (ϕ, ψ), W (eλβ , eλα )) = (2π)−1/2 (ϕ, eλβ )(ψ, eλα ) = (2π)−1/2 (ϕ, eλβ )(eλα , ψ), ϕ, ψ ∈ S(R). Thus, by (3.5) we get We7 ϕ = (2π)−1/2 (ϕ, eλβ )eλα , λ

(3.6)

α,β

for all ϕ ∈ S(R). By (3.6) it follows that −1/2 λ λ We7 We7 (We7 λ λ ϕ = (2π) λ ϕ, eβ )eα µ,ν

α,β

µ,ν

= (2π)−1 (ϕ, eλν )(eλµ , eλβ )eλα = (2π)−1 (ϕ, eλν )δµ,β eλα = (2π)−1/2 Wδ

7 λ µ,β eα,ν

(3.7)

ϕ.

By the Weyl calculus (see the paper [5], by Grossmann, Loupias and Stein or Theorem 9.3 in the book [12] by Wong) we can write We7 We7 = Wω , λ λ

(3.8)

λ 8 λ ω ˆ = (2π)−1 e8 α,β 1/4 eµ,ν .

(3.9)

α,β

where

µ,ν

λ (z) = eλ (−z) and e8 λ (z) = eλ (−z), so they denote respecTo be specific, e8 µ,ν µ,ν α,β α,β λ λ tively the reflections of eα,β and eµ,ν . Now, let us observe that for all nonnegative integers α, β, µ and ν,

eλα,β −1/4 eλµ,ν = (2π)1/2 δβ,µ eλα,ν .

(3.10)

Indeed, by (3.7), (3.8) we get

Thus,

λ . ω = (2π)−1/2 δβ,µ e7 α,ν

(3.11)

λ . ω ˆ = (2π)−1/2 δβ,µ e8 α,ν

(3.12)

−1/2 8 λ λ . δβ,µ e8 eλα,β 1/4 e8 α,ν = (2π) µ,ν

(3.13)

So, by (3.9), (3.12)


161

Generally, ˜ = f˜ τ g˜ implies that h = f −τ g. h

(3.14)

Thus, by (3.14) we get eλα,β −1/4 eλµ,ν = (2π)1/2 δβ,µ eλα,ν .

(3.15)

Hence the proof is complete.

Now, we can state the following theorem which is the main step towards a formula for the heat kernel of Lλ , λ ∈ R \ {0}. Theorem 3.2. For all f ∈ L2 (R2 ) and all t > 0, e−tL f = k λ (t, ·) 1/4 f, λ

(3.16)

where k λ (t, z) =

1 1 − 1 |zλ |2 ctht · e 4 , 4π sht

(3.17)

for all z in C. Proof. Let f ∈ L2 (R2 ). Then, by (3.2) and Theorem 3.1 f −1/4 eλk,k =

∞

e−(2k+1)t (f, eλj,l )eλj,l −1/4 eλk,k

j,l=0

= (2π)1/2 = (2π)1/2

∞

(f, eλj,l )δl,k eλj,k

(3.18)

j,l=0 ∞

(f, eλj,k )eλj,k ,

j=0

for k = 0, 1, 2, . . .. Thus, by (3.2) and (3.18) e−tL f = (2π)−1/2 λ

−1/2

= (2π)

∞ k=0 ∞

e−(2k+1)t f −1/4 eλk,k (3.19) e−(2k+1)t eλk,k

1/4 f.

k=0

If we made a similar reasoning as in the proof of Theorem 23.3 in the book [12] by Wong we get ∞

1+r 1 1 − 14 1−r |z λ |2 eλk,k (z)rk = √ · e 1−r 2π k=0 r 1 1 − 1 |zλ |2 − 12 |zλ |2 1−r = √ · e 4 e 2π 1 − r

(3.20)

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V. Catan˘ a

for all 0 < r < 1 and z ∈ C, where z λ = |λ|1/2 q + i|λ|−1/2 p, if z = q + ip, q, p ∈ R. But the generating function for the Laguerre polynomials is given by the formula xr ∞ e− 1−r α k Lk (x)r = , |r| < 1 (3.21) (1 − r)α+1 k=0

(see Theorem 20.3 in the book [12], by Wong) where the series is uniformly and absolutely convergent on every compact subset of {r ∈ C; |r| < 1}. So, by (3.20), (3.21) we get ∞ r 1 1 − 1 |zλ |2 − 12 |zλ |2 1−r eλk,k (z)rk = √ · e 4 e 2π 1 − r k=0 (3.22) ∞ −1/2 0 1 λ 2 k − 14 |z λ |2 = (2π) Lk |z | r e . 2 k=0

Hence, by (3.22) eλk,k (z) = (2π)−1/2 L0k

1

1 λ2 |z λ |2 e− 4 |z | , z ∈ C,

2 z = q + ip, z λ = |λ|1/2 q + i|λ|−1/2 p, p, q ∈ R, and k = 0, 1, 2, . . . So, by (3.20) we get ∞ ∞ e−(2k+1)t eλk,k (z) = e−t eλk,k (z)(e−2t )k k=0

(3.23)

k=0

−2t t −t 1 1 −1/4|zλ |2 1+e−2t −1/4|z λ |2 et +e−t −1/2 1−e e −e = e−t (2π)−1/2 e = (2π) e 1 − e−2t et − e−t λ 2 1 = (2π)−1/2 e−1/4|z | ctht . (3.24) 2sht 1 −1/4|zλ |2 ctht Let us denote by k λ (t, z) = (2π)−1 e . Then, by (3.19) and (3.24) 2sht it follows λ e−tL f = k λ (t, ·) 1/4 f (3.25) as asserted and this completes the proof of the theorem.

Thus, for every f in L2 (R2 ), λ 1 (e−tL f )(z) = k λ (t, z − w)f (w)ei 4 [z,w] dw,

(3.26)

C

for all z ∈ C and t > 0. By (3.26) we see that the kernel of the integral operator λ e−tL which is also the heat kernel of Lλ is given by 1 1 1 − 1 |zλ −wλ |2 ctht i 1 [z,w] Kλ (t, z, w) = k λ (t, z − w)ei 4 [z,w] = · e 4 e4 (3.27) 4π sht for all z, w ∈ C and t > 0, where z λ = |λ|1/2 q + i|λ|−1/2 p, wλ = |λ|1/2 u + i|λ|−1/2 v, if

z = q + ip,

w = u + iv,

u, v, q, p ∈ R.


163

4. The Green function We know that the heat kernel of the operator Lλ is the kernel of the integral λ operator e−tL which is denoted by Kλ (t, ·, ·), and it satisfies the following relation λ (e−tL f )(z) = Kλ (t, z, w)f (w)dw, (4.1) C

for all z, w ∈ C, t > 0 and f ∈ L2 (R2 ). Thus, the kernel Gλ of the integral operator (Lλ )−1 which is also known as the Green function of Lλ is given by ∞ Gλ (z, w) = Kλ (t, z, w)dt, (4.2) 0

for all z and w in C. The following theorem give the explicit formula for Gλ . Theorem 4.1. The Green function of the operator Gλ is given by 1 1 1 λ |z − wλ |2 , Gλ (z, w) = e 4 i[z,w] K0 4π 4

(4.3)

for all z and w in C, where K0 is the modified Bessel function of order 0 defined by ∞ K0 (x) = e−xcht dt, x > 0. (4.4) 0

Proof. By (3.27) and (4.2) it follows ∞ ∞ 1 1 − 1 |zλ −wλ |2 ctht λ λ i[z,w] 1 4 G (z, w) = K (t, z, w)dt = e e 4 dt, 4π sht 0 0

(4.5)

for all z and w in C. We change the variable of integration by means of the formula u = ctht in the right-hand side of (4.5) and we get ∞ λ λ 2 1 1 1 Gλ (z, w) = ei 4 [z,w] (u2 − 1)−1/2 e− 4 |z −w | u du. (4.6) 4π 1 But

γ− 12 1 2 (u − 1) e du = √ Γ(γ)Kγ− 12 (µ) (4.7) π µ 1 (see [7], by Magnus, Oberhettinger and Soni), where Kν is the modified Bessel function order ν given by ∞ Kν (x) = e−xcht ch(νt)dt, x > 0. (4.8)

∞

2

γ−1 −µu

0

1 1 Thus, by (4.6), (4.7), (4.4) and (4.8) for γ = , µ = |z λ − wλ |2 and ν = 0 the 2 4 asserted formula is established.

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V. Catan˘ a

5. Global hypoellipticity The Green function can be used to prove that the generalized Hermite operator Lλ is globally hypoelliptic in the sense of Schwartz space of distributions. More precisely, we can state the following theorem. Theorem 5.1. The generalized Hermite operator Lλ , λ ∈ R \ {0} is globally hypoelliptic in the sense that u ∈ S (R2 ), Lλ u ∈ S(R2 ) ⇒ u ∈ S(R2 ). Proof. Let f = Lλ u. Then u(z) = ((Lλ )−1 f )(z) =

Gλ (z, w)f (w)dw C

= C

(5.1) g λ (w)e−i 4 [z,w] f (z − w)dw, 1

for all z in C, where

1 1 K0 |wλ |2 , w ∈ C, λ ∈ R \ {0}. 4π 4 n Let β ∈ N be a multi-index. Then, by (5.1) we get 1 β (∂ u)(z) = g λ (w)∂zβ e−i 4 [z,w] f (z − w) dw, g λ (w) =

(5.2)

(5.3)

C

for all z ∈ C. It can be proved that the interchange of differentiation and integration is permitted. To this end let us write

1

|g λ (w)| ∂zβ e−i 4 [z,w] f (z − w) dw = I1λ (z) + I2λ (z), (5.4) C

where

I1λ (z) =

and

|w|≤1

I2λ (z)

= |w|≥1

1

|g λ (w)| ∂zβ e−i 4 [z,w] f (z − w) dw

(5.5)

1

|g λ (w)| ∂zβ e−i 4 [z,w] f (z − w) dw.

(5.6)

We have the following estimates on the modified Bessel function of order 0 (see the paper [14] by Wong) |K0 (x)| ≤ Cη x−η ,

x>0

(5.7)

for every positive number η and some positive constant Cη . Thus, by (5.2), (5.7) and the hypotheses of Theorem 5.1 obviously follows sup |I1λ (z)| ≤ C1λ

(5.8)

sup |I2λ (z)| ≤ C2λ ,

(5.9)

z∈C

and z∈C


165

for the appropriate positive constants C1λ , C2λ . Now, if we take α and β be arbitrary multi-indices with α = 0, then we get |z α (∂ β u)(z)| ≤ 2|α| (J1λ (z) + J2λ (z)), for all z in C, where

J1λ (z) = and

C

J2λ (z) =

C

(5.10)

1

|w||α| |g λ (w)| ∂zβ e−i 4 [z,w] f (z − w) dw

(5.11)

1

|z − w||α| |g λ (w)| ∂zβ e−i 4 [z,w] f (z − w) dw.

(5.12)

It follows as in the case α = 0, that sup |J1λ (z)| < D1λ , for some positive constant z∈C

D1λ . We also can prove that sup |J2λ (z)| < D2λ , for some convenable positive conz∈C

stant D2λ , if we break C into |w| ≤ 1 and |w| ≥ 1 and if we use (5.2) and the estimated (5.7) for the Bessel function of order 0, K0 . Thus the proof is complete.

6. Ultracontractivity and hypercontractivity Let e−tA be a symmetric Markov semigroup on L2 (Ω, dx) (i.e., A ≥ 0 is a positive self-adjoint operator on L2 (Ω, dx)), where dx is a Borel measure on the locally compact, second countable Hausdorff space Ω. We say that e−tA is ultracontractive if e−tA is a bounded operator from L2 ∞ to L for all t > 0. We say that the strongly continuous one-parameter semigroups e−tA is hypercontractive if there exists t > 0, such that e−tA is a bounded linear operator from L2 to L4 (see the book [4] by Davies). The following theorem asserts that the strongly continuous one-parameter λ semigroup e−tL , t > 0, λ ∈ R \ {0} is ultracontractive. Theorem 6.1. For all t > 0, e−tL : L2 (R2 ) → L∞ (R2 ) is a bounded linear operator. λ

Proof. As a consequence of Theorem 3.2 (see 3.27) we have seen that the heat kernel Kλ of Lλ is given by 1 1 − 1 |zλ −wλ |2 ctht i 1 [z,w] 1 Kλ (t, z, w) = k λ (t, z − w)ei 4 [z,w] = e 4 e4 , (6.1) 4π sht for all z and w in C and t > 0. So, by (6.1) we get |Kλ (t, z, w)| ≤ Ct ,

(6.2)

for all z, w ∈ C and t > 0, where Ct =

1 . 4πsht

(6.3)

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V. Catan˘ a

Therefore, for all f ∈ L1 (R2 ),

λ |(e−tL f )(z)| =

Kλ (t, z, w)f (w)dw

C ≤ |Kλ (t, z, w)| |f (w)|dw ≤ Ct ||f ||1 ,

(6.4)

C

for all z ∈ C. Thus, by (6.4) ||e−tL f ||∞ ≤ Ct ||f ||1 . λ

∞

(6.5)

In addition to, for all f ∈ L (R ), |(e

−tLλ

2

λ

f )(z)| = K (t, z, w)f (w)dw

C λ 2 1 ≤ Ct ||f ||∞ e− 4 |w | ctht dw

(6.6)

C

= Ct ||f ||∞

4π 1 = ||f ||∞ , ctht cht

for all z ∈ C. Thus, by (6.6) 1 ||f ||∞ . (6.7) cht By using the Riesz-Thorin interpolation theorem and (6.5), (6.7) then we get p1 p−1 p 1 1 −tLλ ||e f ||∞ ≤ ||f ||p , (6.8) 4πsht cht ||e−tL f ||∞ ≤ λ

for all f ∈ Lp (R2 ) and 1 ≤ p ≤ ∞. Therefore, for p = 2 we give the ultracontractivity of the strongly continuous λ one-parameter semigroup e−tL , t > 0, λ ∈ R \ {0}. In consequence of the above proof we can give the following result. Corollary 6.2. For all t > 0, e−tL : Lp (R2 ) → L∞ (R2 ) is a bounded linear operator for 1 ≤ p ≤ ∞. Moreover, 1 ctht p 1 −tLλ ||e f ||∞ ≤ · ||f ||p , (6.9) 4π cht λ

for all f ∈ Lp (R)2 and 1 ≤ p ≤ ∞. Now, let us recall the following theorem (see Theorem 6 in the paper [2], by Catan˘a), which is needed in order to prove the hypercontractivity of the strongly λ continuous one-parameter semigroup e−tL , t > 0, λ ∈ R \ {0}. Theorem 6.3. For all t > 0, e−tL : Lp(R2 ) → L2 (R2 ) is a bounded linear operator and λ 1 1 1 ||e−tL f ||2 ≤ (2π) 2 − p ||f ||p , (6.10) 2sht p 2 for all f ∈ L (R ) and 1 ≤ p ≤ 2. λ


167

By Corollary 6.2 and Theorem 6.3 and using the Riesz-Thorin interpolation theorem (see Theorem 12.4 in the book [13] by Wong or Theorem 10.2 in the book [12] by Wong) we can obtain the following result. Theorem 6.4. For all t > 0, e−tL : Lp (R2 ) → Lq (R2 ) is a bounded linear operator for 1 ≤ p ≤ 2 and 2 ≤ q ≤ ∞. In addition to, we have that q−2 q−2 q2 q 2 ctht pq 1 1 (1− p )/q −tLλ ||e f ||q ≤ (2π) ||f ||p , (6.11) 4π cht 2sht λ

for all f ∈ Lp (R2 ) and 1 ≤ p ≤ 2, 2 ≤ q ≤ ∞. From Theorem 6.4 it follows for p = 2 and q = 4 that the strongly continuous λ one-parameter semigroup e−tL , t > 0, λ ∈ R \ {0} is hypercontractive.

7. The abstract Cauchy problem (ACP) for the abstract Hermite operator Let H and K be two separable complex and infinite-dimensional Hilbert spaces in which the inner products and the norms are denoted by (, )H , (, )K and || · ||H , || · ||K , respectively. Let us consider the bilinear operators V : H × H → K, W : H × H → K and the conjugate linear operator 9: K → K, such that for all f, g, f1 , g1 ∈ H and h ∈ K we have (V (f, g), V (f1 , g1 ))K = (f, f1 )H (g, g1 )H ,

(7.1)

(W (f, g), W (f1 , g1 ))K = (f, f1 )H (g, g1 )H ,

(7.2)

W (f, g) = W (g, f ) ˜ ˜ = h. h

(7.3) (7.4)

In addition we suppose that there exists a unitary linear operator F : K → K such that (h, V (f, g))K = (F (h), W (f, g))K , (7.5) for all f, g ∈ H and h ∈ K. Let σ ∈ K. Then we introduce a linear operator Wσ : H → H by (Wσ f, g)H = (σ, W (f, g))K ,

(7.6)

for all f, g ∈ H. We call the operators V, W, Wσ , the abstract Fourier-Wigner operator, the abstract Wigner operator respectively the abstract Weyl operator. Let {ej }, j = 0, 1, 2, . . . be an orthonormal basis of H such that {ejk }, j, k = 0, 1, 2, . . ., ejk = V (ej , ek ) is an orthonormal basis of K. Suppose now that we have a densely defined, positive and symmetric operator A : D(A) ⊂ K → K such that {ejk }, j, k = 0, 1, 2, . . ., are the eigenfunctions of the operator A, corresponding to the eigenvalues {λk }, k = 0, 1, 2, . . .. Suppose in

168

V. Catan˘ a

addition that λk ≥ 0, k = 0, 1, 2, . . . and that the series

e−λk t is convergent for

k≥0

every t > 0. We also suppose that Aejk = λk ejk , ∗

A ejk = λj ejk ,

(7.7) (7.8)

j, k = 0, 1, 2, . . ., where A∗ is the adjoint of the operator A. We shall call the operator A the abstract Hermite operator. We are interested to express the semigroup e−tA , t > 0 which the infinitesimal generator is −A, by means of the operators V and Wσ . Theorem 7.1. Let f ∈ D(A). Then for t > 0, e−tA f =

∞

e−λk t V (WF (f ) ek , ek ),

(7.9)

k=0

where the convergence of the series is in the norm of the Hilbert space K. Proof. By the above spectral properties of the operator A, we can write e−tA f =

∞ ∞

e−λk t (f, ejk )K ejk ,

(7.10)

k=0 j=0

for all f ∈ K. The series in (7.10) is convergent in the norm of the Hilbert space K. By the definitions of the operators, V, W and Wσ , we get (f, ejk )K = (f, V (ej , ek ))K = (F (f ), W (ej , ek ))K = (F (f ), W (ek , ej ))K

(7.11)

= (WF (f ) ek , ej )H , j, k = 0, 1, 2, . . . , f ∈ K. Similarly, (ejk , g)K = (g, ejk )K = (WF (g) ek , ej )H = (ej , WF (g) ek )H , j, k = 0, 1, 2, . . . , g ∈ K

(7.12)

By (7.11), (7.12) and Parseval’s identity we can give (e−tA f, g)K = =

=

∞ ∞ k=0 j=0 ∞ ∞

e−λk t (f, ejk )K (ejk , g)K e−λk t (WF (f ) ek , ej )H (ej , WF (g) ek )H

k=0 j=0 ∞ −λk t

e

k=0

(WF(f ) ek , WF (g) ek )H .

(7.13)


169

By (7.3), (7.5) and (7.6) we get (WF (f ) ek , WF(g) ek )H = (WF(g) ek , WF (f ) ek )H = (F (g), W (ek , W F (f ) ek ))K = (F(g), W (WF (f ) ek , ek ))K

(7.14)

= (g, V (WF (f ) ek , ek ))K = (V (WF (f ) ek , ek ), g)K . So, by (7.13), (7.14) and Fubini’s theorem we get (e−tA f, g)K = =

∞

e−λk t (V (WF(f ) ek , ek ), g)K

k=0 ∞

e

−λk t

V (WF (f ) ek , ek ), g

k=0

(7.15) ,

K

for all f, g ∈ K. By (7.15) we conclude that e−tA f =

∞

e−λk t V (WF (f ) ek , ek ),

(7.16)

k=0

for all f ∈ D(A). Now, let us remark by using (7.2) and (7.6) that the abstract Weyl operator Wσ : H → H is continuous. Moreover, we get ||Wσ ||∗ ≤ ||σ||K ,

(7.17)

where || · ||∗ denote the norm in the space L(H) of all the bounded linear operators from H to H. Then by (7.16), (7.17) we get ||e

−tA

f ||K ≤ = ≤ ≤

∞ k=0 ∞ k=0 ∞ k=0 ∞ k=0

e−λk t ||V (WF (f ) ek , ek )||K e−λk t ||WF (f ) ek ||H · ||ek ||H (7.18) e e

−λk t

−λk t

||WF (f ) ||∗ · ||ek ||H ||F (f )||K =

∞

e−λk t ||f ||K , f ∈ K

k=0

So, the series in the right-hand side of (7.9) is convergent in the norm of the Hilbert space K. Thus the proof is complete. By the above proof we can deduce the following result. Theorem 7.2. For t > 0, the Hermite semigroup e−tA initially defined on D(A) can be extended to a unique bounded operator from K to K, which we again denoted

170

V. Catan˘ a

by e−tA , and ||e−tA f ||K ≤

∞

e−λk t ||f ||K , ∈ K.

(7.19)

k=0

Remark 7.3. Let us consider the following (ACP)   du = −Au dt  u(0) = f, f ∈ K,

(7.20)

where u : [0, ∞) → K, u ∈ C 1 ([0, ∞)). Then u(t) = e−tA f , t > 0 is the solution of the (ACP) (7.20) and by Theorem 7.2 it follows that ∞ ||u(t)||K = ||e−tA f ||K ≤ e−λk t ||f ||K , t > 0. (7.21) k=0

Remark 7.4. Let we take H = L2 (R), K = L2 (R2 ), V : L2 (R) × L2 (R) → L2 (R2 ), W : L2 (R) × L2 (R) → L2 (R2 ), Wσ : L2 (R) → L2 (R), σ ∈ L2 (R2 ) to be the unique extensions to the bilinear respectively linear operators of the maps V : S(R) × S(R) → S(R2 ), W : S(R) × S(R) → S(R2 ) and Wσ : S(R) → S(R) given respectively by p p V (f, g)(q, p) = (2π)−n/2 eiqy f y + g y− dy, 2 2 R p p W (f, g)(x, ξ) = (2π)−n/2 e−iξp f x + g x− dp, 2 2 R and Wσ f, g = (2π)−n/2

σ(x, ξ)W (f, g)(x, ξ)dxdξ, R

R

for all f, g ∈ S(R), σ ∈ S(R2 ). In addition we take 9: L2 (R2 ) → L2 (R2 ), be the conjugate linear operator given by ˜ = h, ¯ h 1 ¯ ¯ be the Hermite operator on for all h ∈ L2 (R2 ). Then if we take A = − (ZZ +Z Z) 2 R2 , {ej }, j = 0, 1, 2, . . . the Hermite functions on R and {ej,k = V (ej , ek )}, j, k = 0, 1, 2, . . . the generalized Hermite functions on R2 then we recover respectively from Theorem 7.1 and Theorem 7.2, Theorem 5.1 and Theorem 6.2 in the paper 1 [15] by Wong. Also, if we take A = Lλ = (Z¯ λ Z λ + Z λ Z¯ λ ) be the generalized 2 Hermite operator on R2 , {eλj}, j = 0, 1, 2, . . . the generalized Hermite functions on R and {eλj,k = V (eλj , eλk )}, j, k = 0, 1, 2, . . . the generalized Hermite functions on R2 then it follows respectively from Theorem 7.1 and Theorem 7.2, Theorem 4 and Theorem 6 in the paper [2] by Catan˘ a.


171

References [1] P. Boggiatto, E. Buzano and L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie Verlag, 1996. [2] V. Catan˘ a, The heat equation for the generalized Hermite and generalized Landau operators, Integral Equations Operator Theory 66 (2010), 41–52. [3] A. Dasgupta and M.W. Wong, Essential self-adjointness and global hypoellipticity of the twisted Laplacian, Rend. Sem. Mat. Univ. Pol. Torino 66 (1) (2008), 75–85. [4] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. [5] A. Grossmann, G. Loupians and E.M. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble) 18 (1968), 343–368. [6] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math. 56 (1976), 165–173. [7] W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, 1964. [8] M. Reed and B. Simon, Fourier Analysis, Self-Adjointness, Academic Press, 1975. [9] M. A.Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, 1987. [10] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton University Press, 1993. [11] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Birkh¨ auser, 1998. [12] M.W. Wong, Weyl Transforms, Springer, 1998. [13] M.W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, 2002. [14] M.W. Wong, Weyl Transforms, the heat kernel and Green function of a degenerate elliptic operator, Ann. Global Anal. Geom. 28 (2005), 271–283. [15] M.W. Wong, The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 393–404. Viorel Catan˘ a University Politehnica of Bucharest Department of Mathematics I Splaiul Independent¸ei 313 RO-060042, Bucharest, Romania e-mail: catana [email protected]

Local Exponential Estimates for h-Pseudo-Differential Operators with Operator-Valued Symbols V.S. Rabinovich Abstract. We consider the class of h-pseudo-differential operators Ah u(x) = Oph (a)u(x) i = (2πh)−n a(x, ξ, h)u(y)e h (x−y)·ξ dydξ, u ∈ S(Rn , H1 ), R2n

where the symbol a has values in the space of bounded linear operators acting from a Hilbert space H1 into a Hilbert space H2 and satisfying additional estimates. We suppose that the symbol a(x, ξ, h) is analytically extended with respect to ξ to a tube domain Rn + iB, where B is a convex bounded domain in Rn containing the origin. The main result of the paper is local exponential estimates of solutions of h-pseudo-differential equations Oph (a)u = f in a domain of non-degeneracy of the operator-valued symbols. We apply this result to the tunnel effect for Schr¨ odinger operators with operator-valued potentials with applications to molecular physics and quantum waveguides. Mathematics Subject Classification (2000). Primary 35S05. Keywords. Semi-classical h-pseudo-differential operators with operator-valued symbols, tunnel effect in quantum physics.

1. Introduction We consider a class of h-pseudo-differential operators Ah u(x) = Oph (a)u(x) = (2πh)−n dξ Rn

Rn

(1) i

a(x, ξ, h)u(y)e h (x−y)·ξ dy, u ∈ S(Rn , H1 ),

This work was completed with the support of the CONACYT project 81615.


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with symbols a having values in the space of bounded linear operators acting from a Hilbert space H1 into a Hilbert space H2 and satisfying additional estimates. In formula (1) h > 0 is a small parameter, S(Rn , H1 ) is the space of vector-valued infinitely differentiable functions quickly decreasing at infinity with all derivatives. We suppose also that the symbol a(x, ξ, h) is analytically extended with respect to ξ to a tube domain Rn + iB where B is a convex bounded domain in Rn containing the origin. The main result of the paper is local exponential estimates of solutions of h-pseudo-differential equations Oph (a)u = f in a domain of non-degeneracy of the operator-valued symbols. We apply this result to the tunnel effect for Schrödinger operators with operator-valued potentials and give applications to some problems of molecular physics and quantum waveguides. Our approach is based on the construction of a local inverse operator in a domain where the symbol of h-pseudo-differential operator is non-degenerated, and on the formulas of commutation of h-pseudo-differential operators with exponential weights (see for instance [27], [36], [37]). It is well known that exponential estimate of solutions of scalar h-pseudodifferential equations is closely connected with the tunnel effect for Schrödinger operators which is formulated as follows: an eigenfunction uE corresponding to the energy E for the Schrödinger operator H = −h2 ∆+V is exponentially decreasing for h → 0 in the classically forbidden region Ω = {x : V (x) > E} (see for instance, [10], [29], [30], [34], [20] and references cited there). In turn the exponential estimates for operator-valued h-pseudo-differential operators can be applied to the investigation of exponential estimates for solutions of partial differential operators on Rn of the form Ah uh = fh where Ah u(x) = aα (x)(hDx )α Dxα , x ∈ Rn . (2) |α +α |≤m

That is only the part of differentiations in (2) depends on the small parameter h > 0. We also consider the exponential decreasing eigenfunctions of Schr¨ odinger operators with operator-valued potentials with applications to quantum physics. The exponential decay estimates are intensively studied. Note the well-known Agmon book [1] where the exponential estimates of the behavior of solutions of second-order elliptic operators have been obtained in terms of the special metric (Agmon metric), see also [18], [19]. Exponential estimates at infinity of solutions of pseudo-differential equations with analytical symbols have been studied in [27], [36], [37], [34], [29], [30]. Note also the papers [38], [39] where the relation between the essential spectrum of pseudo-differential operators and exponential decay behavior of their eigenfunctions at infinity have been established. In the recent paper [40] we considered the local exponential estimates of solutions of finite-dimensional h-pseudo-differential operators with applications to the tunnel effect for Schr¨ odinger, Dirac and Klein-Gordon operators. We applied the methods developed in that paper to operator-valued h-pseudo-differential operators. The paper is organized as follows. In Chapter 2 we give some auxiliary facts from the calculus of operator-valued h-pseudo-differential operators. The standard

h-Pseudo-Differential Operators with Operator-Valued Symbols

175

books on the pseudo-differential operators theory are [42], [43], [23], the theory of h-pseudo-differential operators is contained, for instance, in the books [10], [30], [32], [20], [24], operator-valued pseudo-differential operators are studied in [25], [26], [24], [36]. In presenting the theory of h-pseudo-differential operators with operator-valued symbols we follow our paper [36], where a convenient variant for applications of the theory of pseudo-differential operators with operator-valued symbols was offered. In Chapter 3 we consider the problem of local invertibility of operatorvalued h-pseudo-differential operators in the suitable functional spaces. Chapter 4 is devoted to the local exponential estimates of solutions of operator-valued hpseudo-differential equations. In Chapter 5 we consider the locally h-exponential (tunnel) estimates for Schrödinger operators with operator-valued potentials. We note that different aspects of the tunnelling in the quasi-classic approximation for Schr¨ odinger operators have been considered in [4], [8], [9], [7], [5], [6], [20], [28], [21], [10], [33], [29], [30]. In Chapter 4 we consider applications of results of Chapter 5 to the tunnel effect in molecular physics (see for instance [3] and reference cited there) and in the quantum waveguides (see [3], [14], [14], [16], [17], [5], [6]).

2. Locally inverse operators for h-pseudo-differential operators 2.1. Calculus of h-pseudo-differential operators with operator-valued symbols In this chapter we formulate (without proofs) main facts of the theory of h-pseudodifferential operators, following to our paper [36], where the convenient theory of pseudo-differential operators with operator symbols has been constructed. We will use the following notations: • If X, Y are Banach spaces then we denote by L(X, Y ) the linear space of all bounded linear operators acting from X in Y. In the case X = Y we will write L(X). • Let x = (x1 , . . . , xn ) ∈ Rn . Then we denote by ξ = (ξ1 , . . . , ξn ) ∈ Rn the vectors of the dual space with respect to the scalar product x · ξ = x1 ξ1 + · · · + xn ξn ; • Let α = (α1 , . . . , αn ) be a multiindex. Then |α| = α1 + · · · + αn , ∂xα = ∂ ∂xα11 . . . ∂xαnn , Dxj = −i ∂x , Dxα = Dxα11 . . . Dxαnn . If a is an operator-valued funcj (β)

2

tion on Rn × Rn we denote by a(α) = Dxβ ∂ξα a. We set ξ = (1 + |ξ| )1/2 , ξ ∈ Rn . • Let Ω be an open set in Rn , X be a Banach space. We denote by: (i) C ∞ (Ω, X) the set of infinitely differentiable on Ω functions with value in X; (ii) C0∞ (Ω, X) is subset of C ∞ (Ω, X) functions with a compact support in Ω;

176

V.S. Rabinovich (iii) if X is a Banach space then Cb∞ (Ω, X) is a subset of C ∞ (Ω, X) of the functions a such that |a|k = sup

∂xα a(x) X < ∞, |α|≤k

for every k ∈ N0 = N∪ {0} . (iv) S(Rn , X) is the subset of C ∞ (Rn , X) of functions with values in X such that |a|k = sup xk

∂xα a(x) X < ∞ x∈Rn

|α|≤k

for every k ∈ N0 = N∪ {0} . We omit X if X = C. • In what follows we consider separable Hilbert spaces only. Definition 1. Let H, H be Hilbert spaces and p : Rnξ → L(H, H ). We say that p is a weight function in the class O(H, H ) if there exist constants C > 0 and N > 0 such that the for every points ξ, η ∈ Rn N p(η)−1 p(ξ) ≤ C ξ − η , (3) L(H,H) N p(ξ)p−1 (η) ≤ C ξ − η . L(H ,H ) Definition 2. Let a : R2n × (0, 1] → L(H1 , H2 ), p1 ∈ O(H1 , H1 ), p2 ∈ O(H2 , H2 ). We say that a belongs to S(p1 , p2 ) if p2 (ξ)∂ β ∂ α a(x, ξ, h)p−1 (ξ) |a|l,p1 ,p2 = sup < ∞ (4) x ξ 1 L(H ,H ) |α+β|≤l

(x,ξ,h)∈R2n ×(0,1]

1

2

for every l ∈ N ∪ 0. The semi-norms |a|l define the Fréchet topology in S(p1 , p2 ). The operatorvalued functions in S(p1 , p2 ) are called the symbols. Definition 3. We correspond to the symbol a ∈ S(p1 , p2 ) the h-pseudo-differential operator −n Ah u(x) = Oph (a)u(x) = (2π) dξ a(x, hξ, h)u(y)ei(x−y)·ξ dy (5) Rn Rn i = (2πh)−n dξ a(x, ξ, h)u(y)e h (x−y)·ξ dy, u ∈ S(Rn , H1 ) Rn

Rn

where h ∈ (0, 1]. We denote the set of all h-pseudo-differential operators with symbols in S(p1 , p2 ) by OP Sh (p1 , p2 ). Definition 4. Let a : Rnx × Rny × Rnξ × (0, 1] → L(H1 , H2 ) and p1 ∈ O(H1 , H1 ), p2 ∈ O(H2 , H2 ) be the weight functions.


177

We say that a belongs to Sd (p1 , p2 ) if p2 (ξ)∂xα ∂yγ ∂βα a(x, y, ξ, h)p−1 (ξ) |a|l,p1 ,p2 = sup 1 L(H ,H ) |α+β+γ|≤l

(x,y,ξ,h)∈R3n ×(0,1]

1

0 the operator Ah is bounded from L2 (Rn , H1 ) into L2 (Rn , H2 ), and there exists C > 0, 2k1 , 2k2 ∈ 2N such that (β)

Ah L(L2 (Rn ,H1 ),L2 (Rn ,H2 )) ≤ C h|α| a(α) (x, ξ, h) . |α|≤2k1 ,|β|≤2k2

L(H1 ,H2 )

Proposition 6. Ah = Oph (a) ∈ OP Sh (p1 , p2 ) is bounded from S(Rn , H1 ) into S(Rn , H2 ). Hence the composition of pseudo-differential operators is correctly defined. Proposition 7. (i) Let A1h = Oph (a1 ) ∈ OP Sh (p1 , p2 ) and let A2h = Oph (a2 ) ∈ OP Sh (p2 , p3 ). Then A2h A1h = Oph (a2 a1 ) + hTh , where Th ∈ OP Sh (p1 , p3 ); (ii) Let Ah = Opd,h (a) ∈ OP Sd,h (p1 , p2 ). Then Ah = Oph (a# ) + hTh

(7)

where a (x, ξ) = a(x, x, ξ), Th ∈ OP Sh (p1 , p2 ). #

In what follows we will say that a# (x, ξ) = a(x, x, ξ) is the main symbol of the operator Ah = Opd,h (a) ∈ OP Sd,h (p1 , p2 ). • We say that A∗h is a formally adjoint operator for Ah ∈ OP Sh (p1 , p2 ) if for every functions u ∈ S(Rn , H1 ) and v ∈ S(Rn , H2 ) (Ah u, v)L2 (Rn ,H2 ) = (u, A∗h v)L2 (Rn ,H1 ) . Proposition 8. Let Ah = Oph (a) ∈ OP Sh (p1 , p2 ). Then A∗h ∈ OP Sh (p∗2 , p∗1 ) and A∗h = Oph (˜ a) + hTh , where a˜(x, ξ) = a(x, ξ)∗ , Th ∈ OP Sh (p∗2 , p∗1 ).

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Proposition 9 (Beals). Let Ah = Oph (a) ∈ OP Sh (IH1 , IH2 ) and suppose that Ah : Hh (Rn , IH1 ) → Hh (Rn , IH2 ) is invertible for h ∈ (0, h0 ]. Then A−1 ∈ h OP Sh (IH2 , IH1 ) for enough small h > 0. • In what follows we suppose that the weight functions p ∈ O(H1 , H2 ) satisfies the additional conditions. For all multiindices α, β there exists constants Cαβ > 0 such that p(α) (ξ))p−1 (ξ) ≤ Cαβ , (8) L(H2 ,H2 ) −1 p (ξ)p(α) (ξ) ≤ Cαβ . L(H ,H ) 1

1

Under these conditions p ∈ S(p, IH ), and p ∈ S(IH , p). • Let p ∈ O(H1 , H2 ) be a weight function satisfying (8). We introduce the h-Sobolev space Hh (Rn , p), h > 0 as the closure S(Rn , H) in the norm

u Hh (Rn ,p) = Oph (p)u L2 (Rn ,H2 ) . One can see that Oph (p) : Hh (Rn , p) → L2 (Rn , H2 ) is an isomorphism and Oph (p)−1 = Oph (p−1 ). Further, applying Propositions 7 we obtain Proposition 10. Ah = Oph (a) ∈ OP Sh (p1 , p2 ) is a bounded linear operator from Hh (Rn , p1 ) into Hh (Rn , p2 ) for enough small h ∈ (0, h0 ], and

Ah L(Hh (Rn ,p1 ),Hh (Rn ,p2 )) ≤ Ch |a|r where Ch > 0 and r ∈ N are independent on Ah . At least, applying Proposition 9 we obtain Proposition 11. Let Ah = Oph (a) ∈ OP Sh (p1 , p2 ) and Ah : Hh (Rn , p1 ) → Hhs (Rn , p2 ) be invertible for h ∈ (0, h0 ]. Then A−1 ∈ OP Sh (p2 , p1 ) for enough h small h > 0. • In what follows we denote by aIH an operator multiplication by a complexvalued function a. Note that the operator of multiplication by a ∈ Cb∞ (Rn ). is bounded in Hh (Rn , p), for every weight function p ∈ O(H1 , H2 ) satisfying conditions (8).

3. Local invertibility of h-pseudo-differential operators Definition 12. Let Ω be an open set in Rn . We say that Ah ∈ OP Sh (p1 , p2 ) where pj ∈ O(Hj , Hj ), j = 1, 2 acting from Hh (Rn , p1 ) into Hh (Rn , p2 ) is a locally invertible operator in Ω for enough small h > 0 if for every function φ ∈ C0∞ (Ω) there exist h0 > 0 and operators Lh,φ and Rh,φ such that Lh,φ Ah φIH1 = φIH1

(9)

for all h ∈ (0, h0 ]. The operators Lh,φ , Rh,φ are called the left and right locally inverse operators, respectively.


179

Theorem 13. Let Ah = Opd,h (a) ∈ OP Sd,h (p1 , p2 ) and Ω ⊂ Rn be an open set. Suppose that there exists inverse operator a(x, x, ξ)−1 : H2 → H1 for every (x, ξ) ∈ Ω × Rn , and p1 (ξ)a(x, x, ξ)−1 p−1 (ξ) sup < ∞. (10) 2

(x,ξ)∈Ω×Rn

L(H2 ,H1 )

Then the operator Ah : Hh (Rn , p1 ) → Hh (Rn , p2 ) is locally invertible in Ω for enough small h > 0. Proof. Fix φ ∈ C0∞ (Ω). Let ϕ ∈ C0∞ (Ω) and ϕ(x) = 1 for x ∈ Supp φ. Then, condition (10) implies that b(x, ξ) = ϕ(x)a−1 (x, x, ξ) ∈ S(p2 , p1 ). Applying Proposition 7 (i) we obtain Oph (b)Oph (a)φIH1 = (ϕ + Qh )φIH1 , (11) where Qh = hTh and Th ∈ OP Sh (p1 , p1 ). Proposition 10 yields that lim Qh L(Hh (Rn ,p1 )) = 0.

(12)

h→0

Because ϕφ = φ we obtain that for enough small h > 0 Oph (b)Oph (a)φIH1 = (IH1 + Qh )φIH1 . Formula (12) yields that there exists h0 > 0 such that Qh L(H(Rn ,p1 ) < 1 for all h ≤ h0 . Hence, setting Lh,φ = (I + Qh )−1 Oph (b) we obtain first equality in (9). Proposition 11 implies that (IH1 + Qh )−1 ∈ OP Sh (p1 , p1 ). Hence Lh,φ ∈ OP Sh. (p2 , p1 ). In the same way one can prove the existence of the operator Rh,φ ∈ OP Sh. (p2 , p1 ) such that second equality in (9) holds. Corollary 14. Let Ah = Opd,h (a) ∈ OP Sh (p1 , p2 ), and p1 (ξ)a(x, x, ξ)−1 p2 (ξ)−1 sup < ∞. L(H ,H ) (x,ξ)∈Rn ×Rn

2

(13)

1

Then there exists h0 > 0 and operators A−1 h ∈ OP Sh (p2 , p1 ) such that −1 A−1 h Ah = IH1 ; Ah Ah = IH2 ,

for every h ∈ (0, h0 ]. Hence, under condition (13) the operator Ah : Hh (Rn , p1 ) → Hh (Rn , p2 ) is invertible for enough small h ∈ (0, h0 ]. 3.1. Invertibility at infinity and Fredholm property of h-pseudo-differential operators Let χ ∈ C ∞ (R) and χ(x) = 0 if |x| ≤ 1 and χ(x) = 1 if |x| ≥ 2. For R > 0 we set χR (x) = χ(x/R) and φR = 1 − χR , BR = {x ∈ Rn : |x| < R} , and BR = n {x ∈ R : |x| > R} .

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Definition 15. We say that an operator Ah ∈ OP Sh (p1 , p2 ) acting from Hh (Rn , p1 ) into Hh (Rn , p2 ) is locally invertible at infinity for enough small h > 0 if there exist h0 > 0, R > 0 and operators Lh,R and Rh,R such that Lh,R Ah χR IH1 = χR IH1 , χR Ah Rh,R = χR IH2

(14)

for all h ∈ (0, h0 ). The operators Lh,R , Rh,R are called the left and right locally inverse operators, respectively. Theorem 16. Let Ah = Opd,h (a) ∈ OP Sd,h (p1 , p2 ) and there exist R0 > 0 such that the operator a(x, x, ξ) : H1 → H2 is invertible for every (x, ξ) ∈ BR × Rn , 0 and p1 (ξ)a(x, x, ξ)−1 p−1 (ξ) sup < ∞. 2 L(H2 ,H1 ) (x,ξ)∈BR ×Rn 0

Then for enough small h > 0 the operator Ah : Hh (Rn , p1 ) → Hh (Rn , p2 ) is locally invertible at infinity. Proof. Let ϕR ∈ Cb∞ (Rn ) such that Supp ϕR ⊂ BR and ϕR χR = χR . Condition 0 −1 (14) implies that bR (x, ξ) = ϕR (x)a(x, x, ξ) ∈ Sh (p2 , p1 ). Hence

Oph (bR )Oph (a)χR IH1 = (IH1 + Qh,R )χR IH1 . Formula (12) yields that there exists 0 < h0 such that Qh L(H(Rn ,p1 ) < 1 for all h ≤ h0 . Hence, setting Lh,R = (I + Qh,R )−1 Oph (bR ) we obtain first equality in (14). Proposition 11 implies that (I + Qh )−1 ∈ OP Sh (p1 , p1 ). Hence Lh,R ∈ OP Sh (p2 , p1 ). In the same way we prove that for enough small h > 0 there exists the operator Rh,R ∈ OP Sh (p2 , p1 ) such that the second equality in (14) holds. Definition 17. Let Ah = Opd,h (a) ∈ OP Sd,h (p1 ,p2 ). We say that Ah : Hh (Rn , p1 ) → Hh (Rn , p2 ) is a locally Fredholm operator if for every R > 0 there exist Bh,R , Dh,R ∈ L(Hh (Rn , p2 ), Hh (Rn , p1 )) such that Bh,R AφR IH1 = φR IH1 + Th,R ,

φR ADh,R = φR IH2 + Th,R

(15)

Th,R ,

where : Hh (R , p1 ) → Hh (Rn , p1 ), Th,R : Hh (Rn , p2 ) → Hh (Rn , p2 ) are compact operators. n

Theorem 18. Let Ah = Opd,h (a) ∈ OP Sd,h (p1 , p2 ). We suppose that: (i) the condition of Theorem 13 holds; (ii) the operator Ah is locally Fredholm. Then there exists h0 > 0 such that Ah : Hh (Rn , p1 ) → Hh (Rn , p2 ) is a Fredholm operator for every h ∈ (0, h0 ]. Proof. Let R, Lh,R and Rh,R be such that as in Theorem 16. We set Λh,R = Bh,R φR IH2 + Lh,R χR IH2 . Then,

Λh,R Ah = IH1 + Th,R + QR,h ,


181

where QR,h = Bh,R [φR , Ah ] + Bh,R [χR , Ah ] , and [φR , Ah ] , [χR , Ah ] are commutators of the corresponding operators. Proposition 7 implies that lim [φR , Ah ] L(Hh (Rn ,p1 ),Hh (Rn ,p2 ))

h→0

= lim [χR , Ah ] L(Hh (Rn ,p1 ),Hh (Rn ,p2 )) = 0. h→0

Because Bh,R , Lh,R ∈ S(p2 , p1 ) there exist h0 > 0 such that QR,h L(Hh (Rn ,p1 )) < 1 for all h ∈ (0, h0 ). Thus Λh,R = (IH1 + QR,h )−1 Λh,R ∈ S(p2 , p1 ) is the left regularizator of Ah . In the same way we will construct the right regularizator of Ah for h > 0 small enough.

4. h-pseudo-differential operators with analytical symbols and local exponential estimates 4.1. Operators and weight spaces Definition 19. Let B ⊂ Rn be a convex bounded domain containing the origin 0. We say that a double symbol a belongs to Sd (p1 , p2 , B) if: 1) the operator-valued function ξ → a(x, y, ξ, h) is extended analytically with respect to ξ into the tube domain Rn +iB for every (x, y, h) ∈ Rn ×Rn ×(0, 1], 2) for every multi-index α, β, γ there exist a constant Cαβγ such that p2 (ξ)∂ β ∂ γ ∂ α a(x, y, ξ + iη, h)p−1 (ξ) ≤ Cαβγ x y ξ 1 L(H ,H ) 1

2

for all (x, y, ξ + iη, h) ∈ Rn × Rn × (Rn + iB) × (0, 1]. We denote by OP Sd,h (p1 , p2 , B) the corresponding class of h-pseudo-differential operators with symbols a ∈ Sd (p1 , p2 , B). Definition 20. Let Ω be an open domain in Rn . We say that a positive C ∞ -function w(x) = ev(x) is a weight in the class R(Ω, B) if Supp v(x) ⊂ Ω and ∇v(x) ∈ B for every point x ∈ Ω. Proposition 21. Let Oph (a) ∈ Sd (p1 , p2 , B), w be a weight in R(Ω, B), and wh = exp hv , h > 0. Then wh−1 Opd,h (a)wh = Oph (˜ aw ) + hQh (16) where a ˜w (x, ξ, h) = a(x, x, ξ + i∇v(x), h) ∈ S(p1 , p2 ), and Qh ∈ OP Sh (p1 , p2 ). Proof. Let a ∈ S(p1 , p2 , B), a weight w = exp v ∈ R(Ω, B), and Opd (a) be a usual pseudo-differential operator (h = 1). It has been proved in [36] that w−1 Opd (a)w = Op(aw )

(17)

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where aw (x, y, ξ) = a(x, y, ξ + iθw (x, y)) ∈ S(p1 , p2 ), and

(18)

1

(∇v)((1 − t)x + ty) dt.

θw (x, y) =

(19)

0

Formulas (17), (18), (19) can be rewritten for for h-pseudo-differential operators as wh−1 Opd,h (a)wh = Opd,h (aw ). (20) Applying Proposition 5 (ii) we obtain formula (16). 4.2. Local exponential estimates We denote by Hh (Rn , p, w) where w is a C ∞ -weight the space of distributions with norm

u Hh (Rn ,p,w) = wu Hh (Rn ,p) . (21) Here we consider the local exponential estimate for solutions of h-pseudo-differential operators with analytical symbols of the form Ah uh = f,

(22)

where Ah ∈ OP Sd,h (p1 , p2 , B) is a pseudo-differential operator with a double symbol, f ∈ Hh (Rn , p2 ), and uh ∈ Hh (Rn , p1 ). Theorem 22. Let: (i) Ah = Opd,h (a), where a ∈ Sd (p1 , p2 , B), pj ∈ O(Hj , Hj ), j = 1, 2, (ii) Ω ⊂ Rn is an open domain, and w = ev ∈ R(Ω, B) . (iii) The operator-function a(x, x, ξ + i∇v(x)) is invertible for every (x, ξ) ∈ Ω × Rn , and p1 (x, ξ)a(x, x, ξ + i∇v(x))−1 p−1 (x, ξ) sup < ∞. (23) 2

(x,ξ)∈Ω×Rn

L(H2 ,H1 )

Then, for every φ ∈ Cb∞ (Ω) such that φ(x) = 1 on Supp v there exists C > 0 such that for enough small h > 0 v(x) v(x) h h φuh ≤ C uh Hh (Rn ,p1 ) + e Ah uh . (24) e Hh (Rn ,p1 )

Hh (Rn ,p2 )

Proof. The equation Ah uh = f h considered in the space with the weight wh = ev(x)/h is equivalent to the equation Ah,w ψh = ϕh , where ψh = wh uh ∈ Hh (Rn , p1 ), ϕh = wh f, Ah,w = ev(x)/h Ah e−v(x)/h . Note that the main symbol of the operator Ah,w is a(x, x, ξ + i∇v(x)). Let φ ∈ C0∞ (Ω) and φ(x) = 1 on the Supp v. Then Ah,w ψh = Ah,w φψh + Ah,w (1 − φ)ψh = ϕh .

(25)

Condition (23) and Theorem 13 imply that there exists an operator Lh,φ in OP Sh (p2 , p1 ) such that Lh,φ Ah,w φ = φ. Hence, applying Lh,φ to (25) we obtain φψh + Lh,φ Ah,w (1 − φ)ψh = Lh,φ ϕh .


183

Because the operator Lh,φ Ah,w ∈ OP Sh (p1 , p1 ), (1 − φ)ψh = (1 − φ)ev/h uh = (1 − φ)uh ∈ Hh (Rn , p1 ),and Lh,φ ϕh ∈ Hh (Rn , p1 ) we obtain, applying Proposition 5, that there exist h0 > 0 and C > 0 such that for all h ∈ (0, h0 ] the estimate

φψh Hh (Rn ,p1 ) ≤ C uh Hh (Rn ,p1 ) + ϕh Hh (Rn ,p2 ) (26) holds. Because ψh = e

v(x) h

uh , and ϕh = e

v(x) h

fh we obtain ((24)) from (26).

Corollary 23. Let conditions of Theorem 22 hold. Then there exists C > 0 such that the every solution uh ∈ Hh (Rn , p1 ) of the equation Ah uh = 0

φu

Hh (Rn ,p1 ,e

v(x) h

)

≤ C u Hh (Rn ,p1 )

for enough small h > 0.

Example 24. Let Rn = Rnx × Rnx . Let Ah be a differential operator on Rnx = Rnx × Rnx of the form Ah u(x) = aα (x)(hDx )α Dxα (27) |α +α |≤m

where aα ∈ Cb∞ (Rn ), acting from the Sobolev space Hhs (Rn ) into Hhs−m (Rn ), where Hhs (Rn ) is the Sobolev space with the norm

u H s (Rn ) = (1 + (hDx )2 + Dx2 )s/2 uh . L2 (Rn )

h

We consider the operator Ah as a bounded operator from Hhs (Rn ) into Hhs−m (Rn ). 2 2 We consider Λs (ξ ) = (1 + |ξ | + |Dx | )s/2 as an operator-valued function on Rn with values in L(H s (Rn ), L2 (Rn )). We set p1 (ξ) = Λs (ξ ), p2 (ξ) = Λs−m (ξ ). Then

Hhs (Rn ) = H(Rn , p1 ), Hhs−m (Rn ) = H(Rn , p2 ), and Ah : Hhs (Rn ) → Hhs−m (Rn ) is a h-pseudo-differential operator with operatorvalued symbol a ˆ(x , ξ ) = aα (x , x )(ξ )α Dxα ∈ S(p1 , p2 ). |α +α |≤m

In the evident way we can reformulate Theorems 13, 16, 18, 22 and their corollary for the case of differential operators where the part of derivatives depend on the small parameter h > 0. For example Theorem 22 is formulated as follows. Theorem 25. Let Ah be a differential operator of the form (27) acting from Hhs (Rn ) into Hhs−m (Rn ) and let: (i) Ω ⊂ Rn be an open domain, w(x ) = ev(x ) , v(x ) be a C ∞ -positive function on Rn with support in Ω , and ∇v(x ) ∈ Cb∞ (Supp v); (ii)

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The operator-valued function a ˆ(x, ξ + i∇v(x )) is invertible for every (x , ξ ) ∈ n Ω × R , and p1 (x, ξ)ˆ sup a(x, ξ + i∇v(x ))−1 p2 (x, ξ)−1 L(L2 (Rn ),L2 (Rn )) < ∞. (28) (x,ξ)∈BR ×Rn

Then, for every φ ∈ Cb∞ (Ω ) such that φ(x) = 1 on Supp v there exists C > 0 such that for enough small h > 0 v(x ) v(x ) h φuh ≤ C uh Hh (Rn ,p1 ) + e h Ah uh . (29) e n n Hh (R ,p1 )

Hh (R ,p2 )

5. Schrödinger operators with operator-valued potentials 5.1. Exponential estimates Let T be a positive self-adjoint operator on a Hilbert space H with a dense in H domain DT . Let T s , s ∈ R be the fractional powers of T defined by means of the spectral decomposition, and we denote by DT s , s ≥ 0 the subspace of H with the norm

ϕ DT s = T sϕ H < ∞, and by DT −s the space dual to DT s with respect to the scalar product u, vH in H. We consider the Schr¨ odinger operator Hh = −

h2 ∆x + L(x), x ∈ Rn , 2

(30)

on the Hilbert space L2 (Rn , H) of vector-functions with values in H, where L(x) : DT 1/2 → DT −1/2 is a bounded linear operator for every x ∈ Rn . We suppose that L(x) is symmetric on DT 1/2 . That is L(x)ϕ, ϕH = ϕ, L(x)ϕH , ϕ ∈ DT 1/2 . Moreover, we suppose that L(x) is strongly differentiable operator-valued function such that −1/2 β T ∂x L(x)T −1/2 L(H) ≤ Cβ (31) for every multiindex β. Let p(ξ) = (|ξ|2 + T )1/2 and Hh (Rn , p) be a Hilbert space of vector-functions n on R with values in H and the norm 1/2

u Hh (Rn ,p) = −h2 ∆ + T uL2 (Rn ,H) . Estimates (31) implies that the symbol σHh (x, ξ) = of the operator Hh belongs to S(p, p).

1 2

2

|ξ| + L(x)


185

The next theorem gives exponential estimates for solution of the equation Hh uh,λ = λuh,λ , where uh,λ ∈ Hh (R , p), λ ∈ R. n

Theorem 26. Let Ω ⊂ Rn and v(x) be a positive function such that ∂v(x) ∈ ∂xj ∞ n Cb (R ), and Supp v ⊂ Ω. We suppose that there exists δ > 0 such that for every x ∈ Ω and ϕ ∈ DT 1/2 . / 2 −λ − |∇v(x)| + L(x) ϕ, ϕ ≥ δ 2 T ϕ, ϕH , (32) H

Then for every function ψ ∈ Cb∞ (Ω) such that ψ(x) = 1 on Supp v v(x) ≤ C uh,λ Hh (Rn ,p) ψ(x)e h uh,λ n

(33)

Hh (R ,p)

for enough small h > 0. Proof. Condition (32) implies that for every ϕ ∈ DT 1/2 R σHh −λI (x, ξ + i∇v(x))ϕ, ϕH : ; 1 2 2 = |ξ| − |∇v(x)| − E + L(x) ϕ, ϕ 2 H : ; . / 1 2 ≥ |ξ| + δ 2 T ϕ, ϕ ≥ C |ξ|2 + T ϕ, ϕ 2 H H 2 1/2 2 = C (|ξ| + T ) ϕ H .

(34)

2

One can see that (|ξ| + T )1/2 is an isomorphism of DT 1/2 on H for every ξ ∈ Rn . 2 Let φξ = (|ξ| + T )1/2 ϕ, where ϕ ∈ DT 1/2 . Then . / 2 2 2 R (|ξ| + T )−1/2 σHh (x, ξ + i∇v(x))(|ξ| + T )−1/2 φξ , φξ ≥ C φξ H (35) H

for every ξ ∈ Rn . The estimate (35) yields that for every (x, ξ) ∈ Ω × Rn there 2 2 −1 exists the operator (|ξ| + T )1/2 σH (x, ξ + i∇v(x))(|ξ| + T )1/2 , and h 2 2 −1 1/2 sup (x, ξ + i∇v(x))(|ξ| + T ) ≤ C −1 . (|ξ| + T )1/2 σH h (x,ξ)∈Ω×Rn

Then the statement of Theorem 26 follows from Corollary 23.

L(H)

Remark 27. Theorem 26 can be considered as an analog of the tunnel effect for Schr¨ odinger operators with operator-valued potentials. Remark 28. Construction of the weight. (i) Let Ω be a bounded domain, and γ ∈ C0∞ (Ω) be a nonnegative function. Because |∇γ(x)| is a bounded function in Ω, one can find enough small ε > 0, such that for v = εγ estimate (32) holds. (ii) Let Ω be an unbounded domain, and γ ∈ C ∞ (Rn ) is a nonnegative function ∂γ with Supp γ ⊂ Ω, and ∂x ∈ Cb∞ (Ω), j = 1, . . . , n. We also can find ε > 0 j such for v = εγ condition (32) holds.

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5.2. Hamiltonians of molecular physics We consider the molecular Hamiltonian 2 2 H=− ∆x − ∆y + Φ(x, y) 2mµ 2me

(36)

Here x = {x1 , . . . , xK } are the positions of the K nuclei and y = {y1 , . . . , yN } the positions of the N electrons. The electrons have mass m and the nuclei have mass mµ , Φ(x, y) is the interaction potential. We suppose that the potential Φ ∈ Cb∞ (R3K )⊗L∞ (R3N ) and real-valued. Here n = 3K + 3N. After transformation to the atomic coordinates with = me = 1 we obtain the operator Hh = − where h =

&

m

mµ

h2 1 ∆x − ∆y + Φ(x, y), 2 2

is a small parameter.

For application of Theorem 26 we set H = L2 (R3N ), T = I − ∆y . Hence DT is the Sobolev space H 2 (R3N ) and DT 1/2 = H 1 (R3N ). The operator-valued potential L(x) is defined as 1 L(x) = − ∆y + Φ(x, y). 2 It is evident that L(x)ϕ, ϕH = ϕ, L(x)ϕH , for every ϕ ∈ DT 1/2 = H (R3N ), and (31) holds. 1/2 2 Let p(ξ) = |ξ| + T . The space Hh (R3N , p) for such p is the Sobolev 1

space Hh1 (Rn ) with the norm

u H 1 (Rn ) = (I − h2 ∆x − ∆y )1/2 u h

L2 (Rn )

< ∞.

We consider the exponential behavior of solutions of the equation Hh uh,λ = λuh,λ , in the space Hh1 (Rn ). Theorem 29. Let Ω ⊂ R3K , and the energy λ < µΩ,Φ = inf (x,y)∈Ω×R3N Φ(x, y). ∞ n Let v(x) be a positive function such that ∂v(x) ∂xj ∈ Cb (R ), and Supp v ⊂ Ω. We suppose that inf

x∈Ω

µΩ,Φ − λ − |∇v(x)|

2

= δ 2 > 0.

Then for every function ψ ∈ Cb∞ (Ω) such that ψ(x) = 1 on Supp v v(x) ψ(x)e h uh,λ 1 n ≤ C uh,λ H 1 (Rn ) Hh (R )

for a small enough h > 0.

h

(37)

(38)


187

Proof. Note that condition (37) provides the fulfillment of condition (32) of Theorem 26. 5.3. Quantum waveguides We consider the Dirichlet problem for a Schrödinger equation in the quantum waveguide h ∂2 1 − − ∆x + Φ(x, z) u(x, z) = λh u(x, z), (39) 2 ∂z 2 2 (x, z) ∈ D × R = Π, u |∂ = 0, where D is a bounded domain in Rn with an enough regular boundary, Φ ∈ ¯ ⊗ C ∞ (R) is a real-valued potential, h > 0 is a small parameter characterC (1) (D) b izing the thinness of the waveguide. We correspond to the Dirichlet problem (39) the operator Hh which can be realized as a pseudo-differential operator with operator-valued symbols σHh (y, ξ) = where

1 2 ξ + L(y), 2

1 L(y) = − ∆x + Φ(x, y) ϕ(x), x ∈ D, 2 ϕ |∂D = 0

is the operator of the Dirichlet problem in D, depending on the parameter y ∈ R. Let T be the operator of the Dirichlet problem for the Laplacian − 12 ∆x in the domain D, considered as unbounded operator on H = L2 (D) with domain DT = ϕ ∈ H 2 (D), ϕ |∂D = 0 . It is well known that T is a positively defined ˜ ˜ operator. Hence L(y) = T + Φ(y), where Φ(y)u(x) = Φ(x, y)u(x), x ∈ D and y ∈ R is a parameter. One can prove that condition (31) is satisfied. As above we set p(ξ) = ξ 2 + T, ξ ∈ R, and we introduce the Hilbert space Hh (R, p) as the space with norm 1/2 2 ∂u

u Hh (R,p) = − h + T u . ∂y 2 L (Π)

Theorem 30. Let Ω ⊂ R be an open set, and the energy λ < µΩ,Φ =

inf

Φ(x, y).

(x,y)∈D×Ω

Let v(y) be a positive smooth function on Ry such that v ∈ Cb∞ (Ry ), and Supp v ⊂ Ω. We suppose that 2 dv(y) inf µΩ,Φ − λ − = δ2 > 0 (40) dy (x,y)∈Ω×R3N

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Then for every function ψ ∈ Cb∞ (Ω) such that ψ(y) = 1 on Supp v v(y) ψ(y)e h uh,λ 1 n ≤ C uh,λ H 1 (Rn ) Hh (R )

(41)

h

for enough small h > 0. Proof. It is easy to check that condition (40) provides the fulfillment of condition (32) of Theorem 26.

References [1] S. Agmon, Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations, Princeton University Press, Princeton, 1982. [2] R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45–57. [3] V.V. Belov, S.Yu. Dobrokhotov and T.Ya. Tudorovskiy, Operator separation of variables for adiabatic problems in quantum and wave mechanics, Journal of Engineering Mathematics 55 (1-4) (2006), 183–237. [4] J. Br¨ uning, S. Yu. Dobrokhotov and E.S. Semenov, Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem, Regular and Chaotic Dynamics 11 (2006), 167–180. [5] S.Yu. Dobrokhotov and A.I. Shafarevich, “Momentum” tunnelling between tori and the splitting of eigenvalues of the Laplace-Beltrami operator on Liouville surfaces, Math. Phys. Anal. Geom. 2 (1999), 141–177. [6] S.Yu. Dobrokhotov and A.I. Shafarevich, Tunnel splitting of the spectrum of LaplaceBeltrami operators on two-dimensional surface with square-integrable geodesic flow, Funct. Anal. Pril. 34 (2000), 67–69. (in Russian); Funct. Anal. Appl. 34 (2000), 133–134. [7] S.Yu. Dobrokhotov, V.N. Kolokol’tsov and V.P. Maslov, Quantization of the Bellman equation, exponential asymptotics and tunnelling, in Idempotent Analysis, Editors: V.P. Maslov and S.N. Samborskii, Adv. Soviet Math. 13, Amer. Math. Soc., Providence, RI, 1992, 1–46. [8] S.Yu. Dobrokhotov and V.N. Kolokol’tsov, On the amplitude of the splitting of lower energy levels of the Schr¨ odinger operator with two symmetric wells, Teoret. Mat. Fiz. 94 (3) (1993), 426–434. (in Russian); Theoret. Math. Phys. 94 (1993), 300–305. [9] S.Yu. Dobrokhotov and V.N. Kolokol’tsov, The double-well splitting of the low energy levels for the Schr¨ odinger operator of discrete φ4 models on tori, J. Math. Phys. 36 (1995), 1038–1053. [10] M. Dimassi and J. Sj¨ ostrand, Spectral Asymptotics in the Semi-Classical Limit, London Math. Soc, Lecture Note Series 268, Cambridge University Press, 1999. [11] J. Dittrich and J. Kˇr´ıˇz, Bound states in straight quantum waveguides with combined boundary conditions, J. Math. Phys. 43 (2002), 3892–3915. [12] J. Dittrich and J. Kˇr´ıˇz, Curved planar quantum wires with Dirichlet and Neumann boundary conditions, J. Phys. A: Math. Gen. 35 (2002), L269–L275. [13] P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73–102.


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[14] P. Duclos, P. Exner and B. Meller, Open quantum dots: Resonances from perturbed symmetry and bound states in strong magnetic fields, Rep. Math. Phys. 47 (2001), 253–267. [15] P. Duclos, P. Exner and B. Meller: Exponential bounds on curvature induced resonances in a two-dimensional Dirichlet tube, Helv. Phys. Acta 71 (1998), 477–492. [16] P. Duclos, P. Exner and P. Stovıcek, Curvature induced resonances in a twodimensional Dirichlet tube, Ann. Inst. Henri Poincaré, 62 (1) (1995), 81–101. [17] [13] T. Ekholm and H. Kovarık, Stability of the magnetic Schr¨ odinger operator in a waveguide, Comm. Partial Differential Equations 30 (2005), 539–565. [18] R. Froese and I. Herbst, Exponential bound and absence of positive eigenvalue for N -body Schr¨ odinger operators, Comm. Math. Phys. 87 (1982), 429–447. [19] R. Froese, I. Herbst, M. Hoffman-Ostenhof and T. Hoffman-Ostenhof, L2 -exponential lower bound of the solutions of the Schr¨ odinger equation, Comm. Math. Phys. 87 (1982), 265–286. [20] B. Helffer, Semi-Classical Analysis for the Schr¨ odinger Operator and Applications, Lecture Notes in Mathematics 1336, Springer, Berlin, 1998. [21] B. Helffer and J. Sj¨ ostrand, Effect tunnel pour l’équation de Schr¨ odinger avec champ magnétique, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4 serie, 14 (1987), 625–657. [22] B. Helffer and J. Sj¨ ostrand, Multiple wells in the semiclassical limit, Math. Nachr. 124 (1985), 263–313. [23] L. H¨ ormander, The Analysis of Linear Partial Differential Operators, III, SpringerVerlag, New York, Berlin, Heidelberg, 2007. [24] V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer, Berlin, Heidelberg, New York, 1998. [25] S. Levendorskii, Asymptotic Distribution of Eigenvalues of Differential Operators, Kluwer Academic Publishers, 1990. [26] S. Levendorskii, Degenerate Elliptic Equations, Kluwer Academic Publishers, 1994. [27] Ya.A. Luckiy and V.S. Rabinovich, Pseudodifferential operators on spaces of functions of exponential behavior at infinity, Funct. Anal. Pril. 4 (1977), 79–80 (in Russian). [28] V.P. Maslov, Global exponential asymptotic behavior of solutions of the tunnel equations and the problem of large deviations, in International Conference on Analytical Methods in the Number Theory and Analysis, Moskow, 1981, Trudy Mat. Inst. Steklov. 163 (1984), 150–180 (in Russian). [29] A. Martinez, Microlocal exponential estimates and explication to tunnelling, in Microlocal Analysis and Spectral Theory, Editor: L. Rodino, NATO ASI Series, Series C: Mathematical and Physical Sciences 490, 1996, 349–376. [30] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Springer, New York, 2002. [31] A. Martinez, S. Nakamura and V. Sordoni, Phase space tunnelling in multistate scattering, preprint, UTMS 2001–1 January 15, 2001. [32] V.P. Maslov and M.V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Kluwer Academic Publishers, Dordrecht, 2002.

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[33] S. Nakamura, Tunnelling estimates for magnetic Schr¨ odinger operators, Comm. Math. Phys. 200 (1999), 25–34. [34] S. Nakamura, Agmon-type exponential decay estimates for pseudodifferential operators J. Math. Sci. Univ. Tokyo 5 (1998), 693–712. [35] G. Nenciu, On exponential decay of solutions of Schr¨ odinger and Dirac equations: bounds of eigenfunctions corresponding to energy in the gaps of essential spectrum, ´ Journées Equations aux Dériv´ ees Partielles (1994), Art. No. 7, 10 p. Full text djvu Reviews Zbl 0948.35506. [36] V.S. Rabinovich, Pseudodifferential operators with analytic symbols and some of its applications, in Linear Topological Spaces and Complex Analysis 2, Metu-T¨ ubitak, Ankara, 1995, 79–98. [37] V. Rabinovich, Pseudodifferential operators with analytic symbols and estimates for eigenfunctions of Schr¨ odinger operators, Z. Anal. Anwend. 21 (2) (2002), 351–370. [38] V. Rabinovich and S. Roch, Essential spectrum and exponential decay estimates of elliptic systems of partial differential equations, Applications to Schr¨ odinger and Dirac operators, Georgian Math. J. 15 (2) (2008), 1–19. [39] V.S. Rabinovich and S. Roch, Essential spectra of pseudodifferential operators and exponential decay of their solutions, applications to Schr¨ odinger operators, in Operator Algebras, Operator Theory and Applications, Operator Theory: Advances and Applications 181, Birkh¨ auser, 2008, 335–384. [40] V.S. Rabinovich, Local exponential estimates for h-pseudodifferential operators and tunnelling for Schr¨ odinger, Dirac, and square root Klein-Gordon operators, Russian J. Math. Phy. 16 (2) (2009), 300–308. [41] B. Simon, Semiclassical analysis of low lying eigenvalues, II. tunnelling, Ann. Math. 120 (1984), 89–118. [42] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Second Edition, Springer, 2001. [43] M.E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, New Jersey, 1981. V.S. Rabinovich National Polytechnic Institute ESIME Zacatenco Av. IPN 07738, Mexico D.F. Mexico e-mail: [email protected]

Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane Roberto DeLeo, Todor Gramchev and Alexandre Kirilov Abstract. We address some global solvability issues for classes of smooth nonsingular vector fields L in the plane related to cohomological equations Lu = f in geometry and dynamical systems. The first main result is that L is not surjective in C ∞ (R2 ) iff the geometrical condition – the existence of separatrix strips – holds. Next, for nonsurjective vector fields, we demonstrate that if the RHS f has at most infra-exponential growth in the separatrix strips we can find a global weak solution L1loc near the boundaries of the separatrix strips. Finally we investigate the global solvability for perturbations with zero-order p.d.o. We provide examples showing that our estimates are sharp. Mathematics Subject Classification (2000). Primary 35F05; Secondary 35S35. Keywords. Global solutions, separatrix strips, infra-exponential growth, pseudo-differential operators.

1. Introduction and main results We recall that Duistermaat and Hörmander, see [12], have demonstrated that a nonsingular smooth vector field X in an n-dimensional open manifold M is surjective if and only if it admits a global transversal section, namely a smooth hypersurface which is transversal to X at every point and cuts exactly once every of its integral trajectories. On the other hand, nonsurjective vector fields appear in the context of the geometry of foliations (see [17]) and dynamical systems (see [11, 20]). In particular, the issue of the global solvability of the cohomological equations of the type Xu = f is a challenging and difficult problem related to Geometry, Dynamical Systems (cf. [13], see also [18] on the solvability of systems of PDEs) and in the general theory of PDEs, see, e.g., [2, 3, 15, 21] on global solvability on tori, and [7, 10, 16] in the The first author was partially supported by INFN, Cagliari, Italy, the second one was partially supported by GNAMPA, INDAM and by a grant of MIUR, Italy, while the third author was partially supported by CNPq, Brazil.


191

192

R. DeLeo, T. Gramchev and A. Kirilov

Gelfand–Shilov spaces S µ (Rn ). Finally, we mention that the surjectivity in various functional spaces for linear partial differential operators of higher order have been extensively studied since 80’s (see [1, 4, 5] and the references therein). In this work we investigate in the framework of the general theory of PDEs the global solvability in the plane for smooth nonsingular vector fields which appear in the theory of foliations and in the study of cohomological equations in Geometry and Dynamical Systems. We also investigate the stability of the global solvability in weighted Sobolev spaces under perturbation with zero-order pseudo-differential operators. We consider smooth nonsingular real vector field in the plane Lu = p(t)∂t u + q(t)∂x u = f (t, x),

(1.1)

i.e., p and q are real-valued smooth functions which have no common zeros. One assumes that there is an integer N ≥ 2 and t1 < · · · < tN such that p(t) = 0 ⇐⇒ t = tj , j = 1, 2, . . . , N

(1.2)

with p (tj ) = 0,

j = 1, 2, . . . , N

(1.3)

and q admits at most one zero in ]tj , tj+1 [ for j = 1, 2, . . . , N − 1.

(1.4)

Note that the lines {t = tj }, j = 1, . . . , N , are characteristics for L. We also suppose that p and q are polynomials. Our results are true under weaker restrictions on p and q, but we prefer to exhibit the main novelties avoiding highly technical arguments and capturing particular cases of L of interest in geometry and dynamical systems (cf. [6] for foliations, see also [11] for a thorough discussion of its action on C ∞ (R2 )). For example, L0 u = (1 − t2 )∂t u − 2t∂x u (1.5) and more generally, Lλ,k u = (1 − t2 )∂t u + λtk ∂x u

(1.6)

for λ = 0, k ∈ N. The first main goal of the present work is to show that the existence of separatrix type phenomena for (1.1) is the only obstruction for the surjectivity in C ∞ (R2 ) of L. Moreover, we exhibit functional spaces associated to the separatrix strips where we can solve globally this cohomological equation in R2 and investigate the stability of this global solvability under perturbations of L with zero-order pseudo-differential operators in x. Definition 1.1. A strip Sj = {(t, x) : t ∈]tj , tj+1 [, x ∈ R}, with j ∈ {1, . . . , N − 1}

(1.7)

Global Solvability in Functional Spaces

193

is a separatrix for the vector field L above if all characteristic curves x = x(t; τ, y), starting at a point (τ, y) ∈ Sj satisfy either or

lim x(t; τ, y) = lim x(t; τ, y) = +∞, −

(1.8)

lim x(t; τ, y) = lim x(t; τ, y) = −∞.

(1.9)

t→t+ j t→t+ j

t→tj+1 t→t− j+1

We state the first new result of our article. Theorem 1.2. The following assertions are equivalent: i) the vector field L is not surjective in C ∞ (R2 ); ii) the vector field L admits a separatrix Sj , for some j ∈ {1, . . . , N − 1}; iii) there exists j ∈ {1, . . . , N − 1} and θj ∈]tj , tj+1 [ such that q(θj ) = 0 and q has opposite signs in ]tj , θj [ and ]θj , tj+1 [. In particular, the operators Lλ,k are not surjective in C ∞ (R2 ) if and only if k is odd. To illustrate the nonsurjectivity for simple example we point out that nonzero constants do not belong to L0 (C ∞ (R2 )). Direct calculations imply that L0 u = c has a weak solution u(t, x) = 2c ln |1+t| . We show for more general classes of RHS |1−t| ∞ 2 f ∈ C (R ) that every solution has singularity either at t = 1 or t = −1 (see Section 4 for more details). This example shows that in order to solve globally Lu = f one should allow some (weak) singularities of the type L1loc near the adjacent characteristics forming the separatrix strips. The second main novelty of this work is that, in order to find a global weak solution, in general the RHS f (t, x) should grow at most like O(eε|x| ), for |x| → ∞ uniformly in the separatrix strips Sj . Finally, we derive sharp estimates on the singularities of the global solutions u(t, x) of (1.1) near tj ∈ IL for large classes of smooth RHS f , where IL = {tj : Sj or Sj−1 is separatrix, j = 1, . . . , N }.

(1.10)

We point out that the part ii) of Theorem 1.2 implies that L is surjective in C ∞ (R2 ) if and only if IL is empty. In order to state the main result on the global solvability of (1.1) we introduce the subspace of the functions of infra-exponential growth in the x variable (e.g., cf. [19] where such growth plays an important role in theory of Fourier transform for hyperfunctions). . C ∞ (R : Expsl (R)) = {f ∈ C ∞ (R2 ) : ∀T > 0, ∀ε > 0, ∀α ∈ Z2+ , ∃C > 0 α s.t. |∂t,x f (t, x)| ≤ Ceε|x| , |t| ≤ T, x ∈ R}. (1.11)

We recall also the weighted Sobolev spaces H s1 ,s2 (Rn ) in Rn (see, e.g., [9]). . H s1 ,s2 (Rn ) = {f ∈ S (Rn ) : f s1 ,s2 = xs2 Ds1 f L2 < +∞} (1.12)

194


which ' measure the global regularity and the behaviour on ∞ in Rn , where x = 1 + x 2 . Theorem 1.3. Let L defined above be nonsurjective in C ∞ (R2 ). Then we can find a right inverse L−1 of L acting continuously # L−1 : C ∞ (R : Expsl (R)) −→ L1loc (R : Expsl (R)) C ∞ (R \ IL : Expsl (R)) (1.13) and L−1 : C(R : H s1 ,s2 (R)) −→ L1loc (R : H s1 ,s2 (R))

#

C(R \ IL : H s1 ,s2 (R)), (1.14)

with s1 , s2 ∈ R. Moreover, setting Ij =]tj , tj+1 [, for any ε > 0 we have   N 4  ≤ Cε,s1 ,s2 ,θ f

sup  |t − tj |ε L−1 j f (t, ·) H s1 ,s2 (R) C( I j :H s1 ,s2 (R)) . (1.15) t∈[−θ,θ]

j=1

Next, if f is a polynomial function with respect to x, i.e., f (t, x) =

k

f (t)x ,

=0

then L−1 f (t, x) =

k

g (t)x

(1.16)

=0

with gk (t) = O(lnk+1 |t − tj |) near t = tj , if Sj or Sj−1 is a separatrix g (t) = o(ln

k+1

|t − tj |) near t = tj , if Sj or Sj−1 is a separatrix,

(1.17) (1.18)

for = 0, . . . , k − 1. Finally, given a zero-order p.d.o. b(t, x, D) in x smoothly depending on t, and s1 , s2 ∈ R we can find ε0 = ε0 (L, s1 , s2 ) > 0 such that if max

sup

|α|≤[s1 ]+2 t∈[tj ,tj+1 ] |β|≤[s2 ]+2 (x,ξ)∈R2

x−α ξ−β |∂xα ∂ξβ b(t, x, ξ)| < ε0

(1.19)

then L + b(t, x, D) admits a right inverse which satisfies (1.14). The paper is organized as follows. Section 2 deals with the proof of Theorem 1.2 and exhibits some geometric features. We derive in the Section 3 precise estimates on suitable right inverses in the separatrix strips and proof a crucial gluing lemma. In Section 4 we obtain sharp results for L0 on the singular behaviour near the separatrix lines. In Section 5 we consider perturbations of the nonsurjective vector field L0 with a constant p.d.o. Finally, we discuss some possible generalizations in Section 6.


195

2. Separatrix strips and nonsurjectivity In this section we prove Theorem 1.2. We start by calculating the global “singular” characteristics of L after dividing by p(t), namely, rewriting formally Lu+bu = f to ˜ + 1 b(t, x, D)u Lu p(t)

=

f (t, x) p(t)

(2.1)

with ˜ Lu

= ∂t u +

q(t) ∂x u. p(t)

(2.2)

˜ different from t = tj , j = 1, . . . , N , are defined by The characteristics of L, q(t) , p(t)

x(t) ˙ =

x|t=τ = y

(2.3)

for some τ = tj , j = 1, . . . , N . We have Lemma 2.1. The function q(t)/p(t) has a global primitive ρ(t) such that ρ(t)

=

N

κj q(tj ) ln |t − tj | + ρ(t) ˜

(2.4)

j=1

where each κj ∈ R \ {0}, with j = 1, . . . , N , depends only on p(t) and ρ˜ ∈ C ∞ (R). Moreover, for each j ∈ {1, . . . , N − 1} fixed, we have κj κj+1 q(tj )q(tj+1 ) > 0

⇔ q admits a zero in ]tj , tj+1 [ of odd order

(2.5)

κj κj+1 q(tj )q(tj+1 ) < 0

⇔ q does not admit zero of odd order.

(2.6)

Proof. By the hypotheses (1.2), (1.3) on p and the decomposition of rational functions, there are nonzero real numbers κ1 , . . . , κN and r1 ∈ C ∞ (R) such that κj 1 = + r1 (t) p(t) j=1 t − tj

(2.7)

q(t) κj q(tj ) = + r2 (t) p(t) t − tj

(2.8)

N

which yields N

j=1

for some r2 ∈ C ∞ (R). The expression (2.4) follows by integration. We note that the hypothesis (1.2) implies q(tj ) = 0, and hence . cj = κj q(tj ) = 0, j = 1, . . . , N.

(2.9)

196

R. DeLeo, T. Gramchev and A. Kirilov Next, we present an important auxiliary result.

Lemma 2.2. Let x(t, y) be defined by x˙ =

λ(t − θ)k + q˜(t), (θ+ − t)(t − θ− )

x(θ) = y,

θ ∈]θ− , θ+ [,

(2.10)

with q˜ ∈ C ∞ ([θ− , θ+ ]). Then one can find r ∈ C ∞ ([θ− , θ+ ]) such that x(t, y) = y + c+ ln |t − θ+ | + c− ln |t − θ− | + r(t), where c± = ∓

λ(θ± − θ)k . θ+ − θ−

(2.11)

(2.12)

In particular, we observe that i) c+ c− > 0 ⇔ k is odd ⇔ c+ and c− have the same signal and λ > 0; ii) c+ c− < 0 ⇔ k is even ⇔ c+ and c− have different signals and λ < 0. Proof. The proof follows from the decomposition λ(t − θ)k λ(θ+ − θ)k λ(θ± − θ)k = + + q˜1 (t), (θ+ − t)(t − θ− ) (θ+ − θ− )(θ+ − t) (θ+ − θ− )(t − θ+ )

(2.13)

where q˜1 = 0 if k = 0, 1, and q˜1 is polynomial of degree k − 2, if k ≥ 2, and integration (from θ to t) of the RHS of (2.10). Now we present the main steps of the proof of Theorem 1.2. First, assume that Sj is a separatrix, for some j ∈ {1, . . . , N − 1}. In view of Lemmas 2.1 and 2.2, the characteristic curves of L, in Sj , can be written in the form: x(t, y) = y + cj ln |t − tj | + cj+1 ln |t − tj+1 | + Rj (t).

(2.14)

∞

with Rj ∈ C ([tj , tj+1 ]) and cj cj+1 > 0. We observe that cj cj+1 > 0 leads to lim x(t, y) = lim x(t, y) = sign (cj )∞,

t→t+ j

t→t− j+1

y ∈ R.

(2.15)

Clearly (2.15) implies that every smooth curve with endpoints on t = tj and t = tj+1 is hit at least twice by the characteristic curve (2.14) provided y $ 1 (respectively, −y $ 1) if cj > 0 (respectively, cj < 0), and therefore, the condition of Duistermaat-H¨ ormander for the surjectivity fails. Suppose now that there are no separatrix strips. Hence, p(t) and q(t) do not . change sign in [tj , tj+1 ], j = 0, 1, . . . , N , t0 = −∞, tN+1 = +∞ and fixing j, we note that the line segment x+νt = C, t ∈ [tj , tj+1 ] is transversal to L provided ν = 0 has the same sign as p(t)q(t) for some t ∈]tj , tj+1 [. So we have global piecewise smooth global transversal. Smoothing by mollifiers ε−1 ϕ(ε−1 t) near t = tj makes the curve smooth and still globally transversal provided 0 < ε % 1. The proof of Theorem 1.1 is complete.


197

Figure 1. Integral curves of L1,1 = (1 − t2 )∂t + t∂x and L1,2 = (1 − t2 )∂t + t2 ∂x , respectively. Clearly no global transversal exists for L1,1 while L1,2 is topologically equivalent to a constant vector field. Example. We focus on the vector fields Lλ,k defined in (1.6) and exhibit some geometric features. The integral trajectories of Lλ,k are given by the curves 0 1 ti 1 k1 x(t) = λ (−1) log |1 + t| − log |1 − t| − (2.16) 2 2 i i 0 we have sup |t − tj |ε |t − tj+1 |ε L−1 j f (t, ·) H s1 ,s2 (R) ≤ Cε,s1 ,s2 f C( I j :H s1 ,s2 (R))

t∈[tj ,tj+1 ]

iii) If f (t, x) =

(3.10)

k

=0 f (t)x , then

L−1 j f (t, x) =

k

g (t)x

(3.11)

=0

with gk (t) = gk (tµ )γµ lnk+1 k+1

g (t) = o(ln

1 |t−tµ | )

1 (1 |t−tµ |

+ o(1)) near t = tµ , γµ = 0, µ = j, j + 1,

near t = tµ , µ = j, j + 1, = 0, 1, . . . , k − 1.

(3.12) (3.13)

iv) Given a zero-order p.d.o. b(t, x, D) in x, smoothly depending on t, and s1 , s2 ∈ R, we can find ε0 = ε0 (L, s1 , s2 ) > 0 such that if max

sup

|α|≤[s1 ]+2 t∈[tj ,tj+1 ] |β|≤[s2 ]+2 (x,ξ)∈R2

x−α ξ−β |∂xα ∂ξβ b(t, x, ξ)| < ε0

(3.14)

then L + b(t, x, D) admits a right inverse which satisfies (3.9). Proof. We observe that for t close to tj we can write t |τ − tj | σj (τ ) L−1 f (t, x) = f (τ, cj ln + Mj (t, τ )) dτ j |t − tj | τ − tj θj

(3.15)

with σj ∈ C ∞ ([tj , θj ]), Mj ∈ C ∞ (∆j ), ∆j = {tj ≤ τ ≤ t ≤ θj }. Therefore, θj |τ − tj | σj (τ ) |∂xα L−1 f (t, x)| ≤ |∂xα f (τ, x + cj ln + Mj (t, τ )) | dτ j |t − t | τ − tj j t θj τ −t 1 ε|cj | ln t−t j ε|x| j ≤ Ce e dτ τ − tj t θj 1 1 = Ceε|x| dτ ε|c | j (t − tj ) (τ − tj )1−ε|cj | t 1 = Ceε|x| ((θj − tj )ε|cj | − (t − tj )ε|cj | ) ε|cj |(t − tj )ε|cj | = Ceε|x|

(θj − tj )ε|cj | (1 + O((t − tj )ε|cj | )). ε|cj |(t − tj )ε|cj |

(3.16)

Similarly, we derive that near tj+1 we have ε|x| |∂xα L−1 j f (t, x)| ≤ Ce

(tj+1 − θj )ε|cj+1 | (1 + O((tj+1 − t)ε|cj+1 | )). ε|cj+1 |(tj+1 − t)ε|cj+1 |

1 Clearly, (3.15), (3.16), (3.17) imply (3.6) provided 0 < ε < min{ |c1j | , |cj+1 | }.

(3.17)

200


As it concerns to item ii), taking into account the inequality |λ|−|s2 | xs2 x + λ−s2

sup

< +∞

(3.18)

x∈R,|λ|≥1

we observe that for α ∈ Z+ and s2 ∈ R we have for t near tj θj |τ −t | s2 α −1 1

· ∂x Lj f (t, ·) L2 ≤ C sup xs2 x + cj ln |t−tjj| −s2 |τ −t dτ j| x∈R

t

×

sup

·s2 ∂ α f (t, ·) L2

t∈[tj ,tj+1 ]

t

≤ C˜

ln|s2 |

θj

=

τ −tj t−tj

˜ C 1 ln|s2 |+1 |s2 | t − tj

1 sup ·s2 ∂ α f (t, ·) L2 τ −tj dτ t∈[tj ,tj+1 ]

sup

·s2 ∂ α f (t, ·) L2 .

(3.19)

t∈[tj ,tj+1 ]

Therefore we obtained (3.9) for s1 ∈ Z+ (summation in (3.19) over |α|). We conclude the general case for s1 by interpolation and duality arguments. Since the logarithmic singularity is weaker then any polynomial one, (3.19) yields (3.10) Next, we show a gluing lemma, which will imply that L−1 f (t, x) =

L−1 j f (t, x),

(t, x) ∈ Sj , j = 0, 1, . . . , N

(3.20)

is a right inverse satisfying the properties stated in Theorem 1.2. This gluing auxiliary assertion seems to be also a novelty “per se” and might be of an independent interest. Let Ω be an open domain in Rn and let δ > 0. Set Iδ =] − δ, δ[, Iδ+ =]0, δ[, − Iδ =] − δ, 0[, and ± Ω± δ = Iδ × Ω = {(t, x) : 0 < ±t < δ, x ∈ Ω},

Ωδ = Iδ × Ω = {(t, x) : |t| < δ, x ∈ Ω}.

(3.21) (3.22)

Consider the smooth vector field X = a(t, x)∂t +

n

aj (t, x)∂xj ,

(3.23)

x∈Ω

(3.24)

b = b(t, x) ∈ C ∞ (Ωδ )

(3.25)

j=1

having t = 0 as a characteristic, i.e., a0 (0, x) = 0, Let or, in the case Ω = R we allow n

b to be a zero-order p.d.o. in x (cf. [9]) depending smoothly on t ∈] − δ, δ[. (3.26)


201

We have Lemma 3.2. Let f ∈ C ∞ (Ωδ ) (respectively, f ∈ C( ] − δ, δ[ : H s1 ,s2 (Rn )) for some s1 , s2 ∈ R if Ω = Rn ). Suppose that u± ∈ C ∞ (Ω± δ )

(3.27)

u± ∈ C(Iδ± : H s1 ,s2 (Rn ))

(3.28)

(respectively, for some s1 , s2 ∈ R) satisfies Xu± + bu± = f Then

* u(t, x) =

u+ (t, x) u− (t, x)

in Ω± δ .

(3.29)

if (t, x) ∈ Ω+ δ if (t, x) ∈ Ω− δ

(3.30)

is a well-defined L1loc (Ω) (respectively, L1 (Iδ : H s1 ,s2 (Rn )) distributional solution of Xu = f in Ωδ provided u± ∈ L1 (Iδ± × K),

K ⊂⊂ Ω

(3.31)

(respectively, u± ∈ L1 (Iδ± : H s1 ,s2 (Rn )), if Ω = R )) and

(3.32)

n

a(t, x)u± (t, x)ϕ(t, x)dx = 0,

lim

t→0±

Rn

ϕ ∈ C0∞ (Ωδ )

(3.33)

Proof. Let ϕ(t, x) ∈ C0∞ (Ωδ ). We have to prove that u, X ∗ ϕ + b∗ ϕ = f, ϕ ∗

(3.34)

∗

where X (respectively, b ) stands for the adjoint of X (respectively, b). Taking into account (3.31), (3.32) and Lebesgue’s dominated convergence theorem we have u, X ∗ ϕ + b∗ ϕ = lim (Jε+ (u+ , ϕ) + Jε− (u− , ϕ)), ε→0

where Jε± (u± , ϕ) = ±

±δ

±ε

u± (t, x)(X ∗ ϕ(t, x) + b∗ (t, x, D)ϕ(t, x))dx dt.

(3.35)

(3.36)

Ω

Integration by parts, duality arguments, the Fubini theorem and (3.29) imply that ±δ Jε± (u± , ϕ) = ± (Xu± (t, x) + b(t, x, D)u± (t, x))ϕ(t, x)dxdt ±ε Ω + a(±ε, x)u± (±ε, x)ϕ(±ε, x)dx (3.37) Ω = f (t, x)ϕ(t, x) + a(±ε, x)u± (±ε, x)ϕ(±ε, x)dx. Ω± \Ω± ε δ

Ω

202


Next, using the hypothesis (3.33), we deduce that ± ± lim Jε (u , ϕ) = f (t, x)ϕ(t, x)dtdx Ω± δ

ε→0

and, plugging into the RHS of (3.34), we obtain, u, L∗ ϕ + b∗ ϕ = f (t, x)ϕ(t, x)dtdx +

Ω+ δ

=

Ω− δ

(3.38)

f (t, x)ϕ(t, x)dtdx

f (t, x)ϕ(t, x)dtdx.

(3.39)

Ωδ

This completes the proof of the lemma.

Combining Proposition 3.1 and Lemma 3.2 we derive the assertions for L−1 . As it concerns the perturbation with b(t, x, D), we reduce the equation in R2 to Lu + b(t, x, Dx )u = f on Sj , j = 0, 1, . . . , N . We are reduced to the study of the global solvability of u + L−1 b(t, x, D)u = L−1 f,

(t, x) ∈ Sj , j = 0, 1, . . . , N.

(3.40)

We apply the Picard type scheme uk = −L−1 b(t, x, D)uk−1 + L−1 f,

k ∈ N, u0 = 0

(3.41)

If j = 1, . . . , N , we use the results for H s1 ,s2 estimates of p.d.o. in Rn (e.g., cf. [9]) and choose ε0 so small that

b(t, x, D)L−1 L1 ([t1 ,tN ]:H s1 ,s2 )→L1 ([t1 ,tN ]:H s1 ,s2 ) < 1.

(3.42)

Using continuity arguments we can find δ > 0 (small enough) such that

L−1 b(t, x, D)L−1 L1 ([t− δ,tN +δ]:H s1 ,s2 )→L1 ([t1 −δ,tN +δ]:H s1 ,s2 ) < 1.

(3.43)

Since p(t) has no zeros for t > tN +δ and t ≤ t1 −δ we have the following estimates: there exists C = Cδ > 0 such that t

L−1 bu(t, ·) H s1 ,s2 ≤ Cδ

u(τ, ·) H s1 ,s2 dτ, (3.44) θj

for j = 0, t ≤ t1 − δ, j = N , t ≥ tN + δ. Combination of contraction and Gronwall inequalities (see [14]) imply the convergence of (3.41) and the existence of (L+b)−1 satisfying the last part of Theorem 1.2 ε Remark 3.3. We point out that the estimates for f ∈ C ∞ ( I j : Edec (R)) allows to µ extend solvability for L and L + b in Gelfand-Shilov spaces Sµ (R) in x, provided µ > 1. See [7] for global solvability and regularity results for some degenerate p.d.o. under similar subexponential decay conditions. We can show that, if the decay is superexponential the solution u loses this decay, unlike the solvability in Gelfand-Shilov spaces Sµµ , 1/2 ≤ µ ≤ 1, cf. see [10, 16] and the references therein.


203

4. The sharpness of the estimates for L0 We consider the model equation L0 u = f . Using the method of the characteristics, for t = ±1, one can write formally a right inverse of L0 in the following way

t

1 − τ2

1 . −1

L0 f = f τ, x + ln

dτ = G+ f + G− f, (4.1) 2 1 − τ2 1 − t 0 where . 1 G± f (t, x) = 2

t

0

1 − τ

1 + τ

1

f τ, x + ln

+ ln

dτ. 1−t

1+t 1±τ

(4.2)

ε We define in a natural way C ∞ (R : Egr (R)) as the inductive limit ε ε C ∞ (R : Egr (R)) = lim C ∞ ( [T, T ] : Egr (R)). T +∞

(4.3)

ε Observe that C ∞ (R : Egr (R)) is a vector subspace of C ∞ (R2 ) and, given ∞ ε ε f1 , f2 ∈ C (R : Egr (R)), we have f1 ·f2 ∈ C ∞ (R : Egr (R)). In particular, the projections π1 (t, x) = t and π2 (t, x) = x belong to this space and consequently, ε any polynomial function p belongs to C ∞ (R : Egr (R)). ∞ ε We introduce a topology on C (R : Egr (R)) by the following family of seminorms . ε ρj,k,T (f ) = sup{ |e−ε|x| ∂tα1 ∂xα2 f (t, x)|; |α1 | ≤ j, |α2 | ≤ k, |t| ≤ T, x ∈ R, } (4.4)

where T > 0 and j, k ∈ Z+ . Lemma 4.1. If a ∈ C 1 (R) and p ∈ N then, when t → 1, we have

t

1 a(1) p+1

1 − s

p 1 − s

a(s) ln

ds = ln

1 − t (1 + o(1)). 1−t 1−s p+1 0 Proof.

t t t

1 − s 1

1

p 1 − s

p 1 − s

ds = a(1) ds + ds a(s) lnp

ln a (s) ln 1

1−t 1−s 1−t 1−s 1 − t

0 0 0

a(1) p+1

1 − s

1 p = ln + o ln

1 − t

p+1 |1 − t|

a(1) p+1 1

ln =

1 − t (1 + o(1)). p+1 Lemma 4.2. If f is a monomial function with respect to x, i.e., f (t, x) = fj (t)xj , with fj ∈ C 1 (R) and j ∈ Z+ , then L−1 0 f (t, x) =

j =0

gj (t)x

(4.5)

204


with

f0 (±1)

1

gj0 (t) = ln

(1 + o(1)), 2 1 ∓ t

1 gj (t) = O( lnj+1− ), |1 ∓ t|

t → ±1

(4.6)

t → ±1.

(4.7)

Proof. From (4.1) and (4.2) we obtain

j

1 − τ

1 t 1

+ ln 1 + τ

G± f (t, x) = fj (τ ) x + ln

dτ

2 0 1−t 1+t 1±τ 0 1

j t

1 + τ j− 1

1 − τ

1 j

+ ln

= fj (τ ) ln

dτ x

1+t

2 0 1−t

1±τ =0

=

j

gj± (t) x

=0

where

t

1 − τ

1 + τ j− 1 1 j

gj± (t) = fj (τ ) ln

+ ln

dτ 2 0 1−t

1+t

1±τ

j− t

1 − τ j−−m 1 + τ 1 1 j j −

ln

= fj (τ ) lnm

1 + t 1 ± τ dτ. 2 m=0 m 1−t

0 Now, it follows from Lemma 4.1 that, near t = 1, we have

t

1 1 j j − m 1 − τ j−−m 1 + τ

gj− (t) = fj (τ ) ln

ln dτ

2 m=0 m 1−t 1+t 1−τ 0

f (1) lnj−−m

2

j− j

1

1+t 1 j j −

(1 + o(1)) = lnm+1

2 m=0 m m+1 1 − t

1

. = O lnj+1−

1 − t

Analogously, near t = −1, we have

j− t

1 − τ j−−m 1 + τ 1 1 j j −

ln

fj (τ ) lnm

1 + t 1 + τ dτ 2 m=0 m 1−t

0

j− m 2 1 j l fj (1) ln | 1−t | j−−m+1

1

= ln

1 + t (1 + o(1)) 2 m=0 m j − − m + 1

j+1− 1

= O ln

1 + t .

gj+ (t) =


205

In particular, for = 0, we have

1 t 1 fj (±1)

1

gj0± (t) = dτ = ln

(1 + o(1)). fj (τ ) 2 0 1±τ 2 1 ∓ t

The next assertion shows that we have sharp estimates on the singularities. Proposition 4.3. The following properties hold: there exists ε0 > 0 such that for all ε ∈]0, ε0 [ i) Given T > 0 and k ∈ Z+ , we have ε0 ε0 sup |(1 − t2 )ε L−1 0 f (t, ·)| ≤ CT ρ0,k,T (f ).

(4.8)

x∈R |α|≤k

ii) If f (t, x) =

k j=0

fj (t)xj , then L−1 0 f (t, x) =

k

gj (t)xj

(4.9)

j=0

with g0 (t) =

fk (±1) k+1 ln |1 ± t|(1 + o(1)), 2(k + 1)

gj (t) = O(lnk+1−j |1 ± t|),

t → ±1, t → ±1.

∞ ε iii) u = L−1 0 f is a weak solution of Lu = f for all f ∈ C (R : E (R)) such that ∀α ∈ Z+ , K ⊂⊂ R, there exist M > 0 such that

|∂xα u(t, x)| ≤ M |1 ± t|−ε ,

0 < |1 ± t| % 1, x ∈ K.

(4.10)

Proof. To prove i) we start by defining, for each T > 0, k ∈ Z+ and u ∈ C ∞ (R : E ε (R)) the following function: T

.

ε0 Pk,T (u) = sup e−ε0 |x| ∂xα u(t, x) dt. −T

x∈R |α|≤k

Thus, for any f ∈ C ∞ (R : E ε (R)), with 0 < ε < ε0 and t > 0 we have T

ε0 Pk,T (G− f ) = sup e−ε0 |x| ∂xα G− f (t, x) dt −T

2 3

−ε |x| α 1 t

1 1−τ 1+τ 0

sup e ∂x f (τ, x + ln | 1−t | + ln | 1+t |) dτ

dt 2 0 1−τ x∈R,

T

= −T

1 ≤ 2

x∈R, |α|≤k

|α|≤k T

−T

0

t

−ε |x| α

1 1−τ 1+τ 0

sup e ∂x f (τ, x + ln | 1−t | + ln | 1+t |) dτ

dt 1−τ

x∈R, |α|≤k

206

R. DeLeo, T. Gramchev and A. Kirilov ≤

1 ε ρ (f ) 2 0,k,T

T

−T

t

0

1

1+τ dτ dt sup

e−ε0 |x| exp(ε(|x| + ln | 1−τ | − ln | | )) 1−t 1+t 1−τ

x∈R,

|α|≤k

t

1 + τ −ε 1 − τ ε 1 1 ε−ε0 ε

≤ e ρ0,k,T (f )

1 + t 1 − t |1 − τ | dτ dt 2 −T 0 1 0 ≤ CTε0 ρε0,k,T (f )(1 − t)−ε . 2 By using the same arguments, when t < 0, we obtain an analogous estimate to G+ f, and consequently

T

ε0 ε0 ε0 2 −ε Pk,T (L−1 . 0 f (t, x)) ≤ CT ρ0,k,T (f )(1 − t )

To prove ii), we use the results in Lemmas 4.1 and 4.2 above to obtain j k k −1 −1 j L0 f (t, x) = L0 (fj (t)x ) = gj (t)x =0

=0

=0

  k k k  = gj (t) x = gj (t)x =0

=j

where

k k . gj (t) = gj (t) = O ln+1−j =j

=j

=0

1 1 = O lnk+1−j , when t → ±1. |1 ∓ t| |1 ∓ t|

To prove the statement iii), first, for 0 < t < 1, we have

1 − τ

1 + τ

1 t

α 1

α

|∂x (G− f (t, x))| ≤ ∂ f τ, x + ln

+ ln

dτ 2 0 x 1−t

1+t 1−τ

t

1 − τ ε 1 + τ −ε 1 1

≤ ρε0,α,T (f ) eε|x|

1 − t 1 + t |1 − τ | dτ 2 0

1 ≤ ρε0,α,T (f ) eε|x| ε−1 |1 − t|−ε . (4.11) 2 By using the same arguments, when −1 < t < 0, we obtain the same estimate to G+ f. Therefore 1 ε ρ (f ) eε|x| ε−1 |1 + t|−ε . 2 0,α,T Therefore, given f ∈ C ∞ (R : E ε (R)), α ∈ Z+ and K ⊂⊂ R, we set |∂xα (G+ f (t, x))| ≤

M = ε−1 ρε0,α,T (f ) sup eε|x| . x∈K

Thus, it follows from (4.1) and (4.2) that −ε |∂xα (L−1 , 0 f (t, x))| ≤ M |1 ± t|

0 < |1 ± t| % 1, x ∈ K.


207

Remark 4.4. Since the general solution of L0 u = f in [−1, 1] × R is given by u = ϕ(x + ln(1 − t2 )) + L−1 0 f (t, x),

(4.12)

with ϕ being a function (or distribution) of one variable, we observe that we have always singularity at t = −1 or t = +1. In view of the separatrix phenomena, we have not compensate both singularities in the general case, while we can “cancel” the singularity either at t = −1 or t = +1 If f ≡ c = 0, we exhibit, apart from u = 2c ln | 1+t |, two particular solutions: 1−t c u± (t, x) = ∓ x ± c ln |1 ∓ t|. (4.13) 2

5. Perturbation with nondegenerate p.d.o. The aim of this section is to show that if we perturb L0 with constants p.d.o., or more generally, a Fourier multiplier satisfying suitable nondegeneracy condi−1 tions, we can obtain L∞ without the smallness loc estimates in t for the (L0 + b) requirement on b. More precisely, we consider Lb u = (1 − t2 )∂t u − 2t∂x u + b(D)u = f (t, x)

(5.1)

where b(ξ) ∈ C(R) is real valued and bounded away from zero for ξ ∈ R.

(5.2)

Clearly (5.2) implies that one can find 0 < δ0 < δ1 such that either δ0 ≤ b(ξ) ≤ δ1 or −δ1 ≤ b(ξ) ≤ −δ0 , for ξ ∈ R.

(5.3)

Set u ˆ(t, ξ) = Fx→ξ u(t, ·) to be the partial Fourier transform in x, i.e., w(ξ) ˆ = e−ixξ w(x)dx. (5.4) R

Setting (formally) b(ξ) |1 + t| ln )= u ˆ(t, ξ) = exp(− 2 |1 − t| we obtain that

|1 − t| |1 + t|

b(ξ)/2 vˆ(t, ξ)

(5.5)

−b(ξ)/2 |1 − t| fˆ(t, ξ). (5.6) |1 + t| In view of the nondegeneracy condition (5.3) can write a right inverse of Lb 1 −1 which is L∞ ). Indeed, set loc in t (a better regularity than Lloc for L

Lˆ0 vˆ(t, ξ) =

Lˆb

−1

fˆ =

|1 − t| |1 + t|

b(ξ)/2

t

−sign (b)

|1−t2 |

i ln fˆ(s, ξ)e |1−τ 2 | ds. (1 − τ )|1 − τ |b(ξ)/2 (1 + s)|1 + s|−b(ξ)/2 (5.7)

208


Proposition 5.1. The operator L−1 acts continuously as L−1 in the spaces with 0 b subexponential decay. Furthermore, it acts continuously L−1 b

s : C(R : H s (R)) −→ L∞ loc (R : H (R))

(5.8)

and for every K > 0, s > 0, one can find C = CK > 0 such that C

L−1 ≤

f C([−K,K]:H s(R)) , b f L∞ ([−K,K]:H s (R)) δ0 for all f ∈ C(R : H s (R)) and δ0 > 0.

(5.9)

Proof. We have the crucial step is based on the estimates near t = ±1: ±b(ξ)/2 ˆ

Lˆ−1 b f (t, ·) L2 ≤ C0 f C([−K,K]:L2(R)) sup (|1 + sign (b)t|) ξ∈R

× | ≤

t

−sign (b)

1 ds| |1 − s|1+sign (b)/2 |1 + s|1−sign (b)/2

C

f C([−K,K]:L2(R)) |b|

(5.10)

where

 b(ξ)/2  |1 − t| C0 = sup  |1 + t| ξ∈R

t

−sign(b)

 1 (1 − τ )|1 − τ |b(ξ)/2 (1 + s)|1 + s|−b(ξ)/2

 2 ds ≤ . δ0

6. Final remarks First we observe that our results remain valid for vector fields of the type L = p(t)∂t + q(t, x)∂x provided q(t, x) is bounded for x, when x → ∞. The approach follows the same ideas, but the arguments of the proofs become more involved in view of the use of theorems on global behaviour of solutions of o.d.e. If q is not bounded, for x → ∞, we have more restrictive conditions on the growth of the RHS f . For example, if q(t, x) grows linearly in x (like SG first-order hyperbolic pseudo-differential operators cf. [9]), we have to require that the RHS f (t, x) grows less than every |x|γ , for every γ > 0. Next, we point out that if the RHS f decays to zero for x → ∞, the right inverses Lj f are continuous near t = tj . Next, as to possible multidimensional generalizations of the vector fields studied in the present work, we are also able to propose similar results for some classes of vector fields having smooth symmetries. E.g., consider the regular plane vector field L = (t2 − 15)(t2 + 15)∂x − (t2 − 25)(t2 − 9)t∂t . One can easily check that the rotations of L around the x axis in R3 with coordinates (t, x, y) gives rise to a regular vector field M having as separatrices the two cylinders y 2 + t2 = 9 and y 2 + t2 = 25. The cohomological equation M u = v hence is not solvable for


209

every smooth function v ∈ C ∞ (R3 ) because of the theorem of Duistermaat and Hörmander but our techniques can be used to find weak solutions. Finally, we point out to a natural problem related to the reduction of a perturbation L + b(t, x, D) to L by means of global conjugation formally J(t) ◦ (L + b) ◦ J −1 (t) = L, with J being a global p.d.o. or Fourier integral operator in x ∈ Rn depending smoothly on t ∈ R\IL , with singularities near t = tj , Sj or Sj+1 being separatrix strips. The example in Section 4 suggests that one should aim on estimates of J(t) in L1loc (R : B(Rn )), where B(Rn ) stands for some weighted Sobolev type space (cf. [8], [22], [23] and the references therein for global estimates in Rn for Fourier integral operators). Acknowledgment The authors are grateful to Adalberto Bergamasco and Michael Ruzhansky for useful discussions on arguments related to this article.

References [1] A. Albanese and P. Popivanov, Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions, Ann. Mat. Pura Appl. (4) 185 (2006), 395–409. [2] A. Bergamasco and P. Dattori da Silva, Solvability in the large for a class of vector fields on the torus, J. Math. Pures Appl. 86 (2006), 427–447. [3] A. Bergamasco and A. Kirilov, Global solvability for a class of overdetermined systems, J. Funct. Anal. 252 (2007), 603–629. [4] R.W. Braun, R. Meise and B.A. Taylor, The geometry of analytic varieties satisfying the local Phragmen-Lindelof condition and a geometric characterization of the partial differential operators that are surjective on A(R4 ), Trans. Amer. Math. Soc. 356 (2004), 1315–1383. [5] R.W. Braun, R. Meise and B.A. Taylor, Characterization of the linear partial differential equations that admit solution operators on Gevrey classes, J. Reine Angew. Math. 588 (2005), 169–220. [6] C. Camacho and A.L. Neto, Geometric Theory of Foliations, Birkh¨ auser, 1984. [7] M. Cappiello, T. Gramchev and L. Rodino, Sub-exponential decay and uniform holomorphic extensions for semilinear pseudodifferential equations, Comm. Partial Differential Equations 35 (2010), 846–877. [8] E. Cordero, F. Nicola and L. Rodino, Boundedness of Fourier integral operators on FLp spaces. Trans. Amer. Math. Soc. 361 (2009), 6049–6071. [9] H.O. Cordes, The Technique of Pseudodifferential Operators, LMS Lecture Note Series 202, Cambridge University Press, 1995. [10] A. Dasgupta and M.W. Wong, Weyl transforms and the heat equation for the subLaplacian on the Heisenberg group, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkh¨ auser, 2009, 33– 42. [11] R. DeLeo, Solvability of the cohomological equation for regular vector fields on the plane, to appear in Ann. Glob. Anal. Geom.

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[12] J. Duistermaat and L. H¨ ormander, Fourier integral operators II, Acta Math. 128 (1972), 183–269. [13] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. Math. (2) 146 (1997), 295–344. [14] D. Gourdin and T. Gramchev, Global in time solutions of evolution equations in scales of Banach function spaces in Rn , Bull. Sci. Math. 131 (2007), 761–786. [15] T. Gramchev, D. Dickinson and M. Yoshino, Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena, Proc. Edinb. Math. Soc. (2) 45 (2002), 731–759. [16] T. Gramchev, S. Pilipović and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkh¨ auser, 2009, 15–31. [17] A. Haefliger and G. Reeb, Varietés (non separées) ` a une dimension et structures feuilletées du plan, Enseignement Math. 3 (1957), 107–125. [18] M. Gromov, Partial Differential Relations, Springer Verlag, 1986. [19] A. Kaneko, Introduction to Hyperfunctions, Kluwer Academic Publishers, 1988. [20] S.P. Novikov, Dynamical systems and differential forms. Low-dimensional Hamiltonian systems, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math. 469 (2008), 271–287. [21] G. Petronilho and S.L. Zani, Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields, J. Differential Equations 244 (2008), 2372–2403. [22] M. Ruzhansky, On local and global regularity of Fourier integral operators, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkh¨ auser, 2009, 185–200. [23] M. Ruzhansky and M. Sugimoto, Global L2 -boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations 31 (2006), 547– 569. Roberto DeLeo and Todor Gramchev Dipartimento di Matematica e Informatica Universit` a di Cagliari Via Ospedale 72 I-09124 Cagliari, Italy e-mail: [email protected] [email protected] Alexandre Kirilov Departamento de Matem´ atica Universidade Federal do Paran´ a Caixa Postal 19081 81531-990 Curitiba Paran´ a, Brasil e-mail: [email protected]

Fuchsian Mild Microfunctions with Fractional Order and their Applications to Hyperbolic Equations Yasuo Chiba Abstract. Kataoka introduced a concept of mildness in boundary value problems. He defined mild microfunctions with boundary values. This theory has effective results in propagation of singularities of diffraction. Furthermore, Oaku introduced F-mild microfunctions and applied them to Fuchsian partial differential equations. Based on these theories, we introduce Fuchsian mild microfunctions with fractional order. We show the properties of such microfunctions and their applications to partial differential equations of hyperbolic type. By using a fractional coordinate transform and a quantised Legendre transform, degenerate hyperbolic equations are transformed into equations with derivatives of fractional order. We present a correspondence between solutions for the hyperbolic equations and those for the transformed equations. Mathematics Subject Classification (2000). Primary 35A27; Secondary 35L80, 35S99. Keywords. Boundary value problems, microlocal analysis, hyperbolic equations, hyperfunctions, F-mildness.

1. Introduction In microlocal analysis, the concept of Sato’s hyperfunctions is an effective method for analysing partial differential equations. Kataoka developed mild hyperfunctions for boundary value problems ([5], [6]). Though hyperfunctions do not always have boundary values, mild hyperfunctions are defined to be a sum of holomorphic functions with boundary values. Furthermore, Oaku introduced F-mildness in the theory of hyperfunctions and applied them to boundary value problems with Fuchsian partial differential equations ([8], [9]). As for them, the author constructs microlocal solutions for boundary value problems with hyperbolic equation ([3]). He uses a fractional coordinate transform and the quantised Legendre transform which are compatible with hyperfunction’s L. Rodino et al. (eds.), Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, DOI 10.1007/978-3-0348-0049-5_12, © Springer Basel AG 2011

211

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Y. Chiba

theory. In this method, hyperbolic equations are transformed into microdifferential equations with derivatives of fractional order. This construction of the solutions needs a fractional coordinate transform, which is not always compatible with microlocal analysis. For example, we cannot substitute t = tq (q: a positive and rational number) for a Heaviside function Y (t ) in the sense of hyperfunctions. Hence we newly define hyperfunctions with fractional power developed by Kataoka. Furthermore, we define these hyperfunctions with boundary values.

2. A definition of Fuchsian mild hyperfunctions with fractional order To begin with, we brush up a theory of F-mildness and its applications for boundary value problems with Fuchsian partial differential equations. From now on to the end, we set √ M = {(t, x) ∈ R×Rn }, M+ = {(t, x) ∈ M ; t ≥ 0}, N = {(t, x) ∈ M ; t = 0} and π : −1T ∗ N → N be a projection. Furthermore, B denotes a sheaf of hyperfunctions and C denotes a sheaf of microfunctions. · Definition 1 (Oaku). Let u(t, x) ∈ B({(t, x) ∈ M ; |t|+ |x− x| < r}). We call u(t, x) · an F -mild hyperfunction at (0, x) if and only if u(t, x) has an expression u(t, x) =

J

Fj (t, x +

√

−1Γj 0)

(1)

j=1

· on a domain {(t, x) ∈ M ; t > 0, t + |x − x| < r}. Here J is a positive integer and r > 0 is a positive number. Each Γj ∈ Rnx (j = 1, 2, . . . , J) is a convex cone and Fj (t, z) is holomorphic function on a neighbourhood of a domain · · D (x, Γj , r ) = {(t, z) ∈ R × Cn ; |t| + |z − x| < r , 0 ≤ t < r , Imz ∈ Γj }. B F denotes a sheaf of F-mild hyperfunctions. For the application of boundary value problems, we define F-mild microfunctions. For this purpose, we define ρmicrosupport of F-mild hyperfunctions. Definition √ 2. Let u be an F-mild hyperfunction. Then we define ρ-SS(u) as follows: x∗ = (t, x; −1ξ) is not included in ρ-SS(u) if and only if u has an expression (1) on a neighbourhood of x∗ with ξ ∈ Γj◦ := {ξ ∈ Rn ; y, ξ ≥ 0 for any y ∈ Γj }. For open connected sets U√⊂ N and ∆ ⊂ S n−1 , √ we associate a C-vector space {u ∈ BF (U ); ρ-SS(u)√∩ (U × −1∆ = ∅)} to U × −1∆. This correspondence defines a presheaf on −1S ∗ N . The sheaf associated with this presheaf is denoted by A ∗F . Definition 3 (F-mild microfunctions). The sheaf CNF|M+ is defined by F ∗F CNF|M+ = π −1 BN . |M+ /A

We call a germ of CNF |M+ an F-mild microfunction.

Fuchsian Mild Microfunctions and their Applications

213

In the case of 1-codimension, F -mild hyperfunctions are defined above. Oaku in [10] also mentions the definition of F -mild hyperfunctions in the higher-order case. He applied these F-mild hyperfunctions and F-mild microfunctions to Fuchsian partial differential equations. From now on, we set ∂t = ∂/∂t and ∂x = (∂/∂x1 , . . . , ∂/∂xn ). Theorem 4 (Oaku 85). Let P (t, x, ∂t , ∂x ) = tk ∂tm + A1 (t, x, ∂x )tk−1 ∂tm−1 + · · · + Ak (t, x, ∂x )∂tm−k + Ak+1 (t, x, ∂x )∂tm−k−1 + · · · + Am (t, x, ∂x ), where k and m are integers with 0 ≤ k ≤ m, ord Aj (t, x, ∂x ) ≤ j (1 ≤ j ≤ m) and ord Aj (0, x, ∂x ) = 0 (1 ≤ j ≤ k). We set aj (x) = Aj (0, x, ∂x ) (1 ≤ j ≤ k) and e(λ, x) = λ(λ − 1) · · · (λ − m + 1) + a1 (x)λ(λ − 1) · · · (λ − m + 2) + · · · + ak (x)λ(λ − 1) · · · (λ − m + k + 1). · Suppose that e(ν, x) = 0 for any integer ν ≥ m − k. If u is an F-mild micro· function defined on a neighbourhood of x satisfying P u = 0, ∂tν u(+0, x) = 0 (0 ≤ ν ≤ m − k − 1) · · on a neighbourhood of x, then u vanishes on a neighbourhood of x. Though Oaku considers Fuchsian operators, we will consider different kinds of operators including of irregular type. For this purpose, we can extend F-mildness to one with fractional order developed by Kataoka. Definition 5 (Fuchsian mild hyperfunctions with fractional order). A hyperfunc· · tion f (t, x) ∈ B({(t, x) ∈ M ; t > 0, t+ |x− x| < r}) is 1/-Fuchsian mild at (+0, x) with κ regularity if and only if the following conditions are satisfied. · (1) On a domain {(t, x) ∈ M ; t > 0, |x − x| < r}, f (t, x) can be expressed as f (t, x) =

J

1

Fj (t , x +

√

−1Γj 0),

(2)

j=1

˜ z) are holomorphic on a where Γj are open proper convex cone and Fj (t, · neighbourhood of D (x, Γj , ε) (j = 1, 2, . . . , J). (2) There exists a positive number C > 0 such that each Fj (t˜, x) can be holomorphically extended to a domain 1 · {(t˜, z) ∈ C × Cn ; |t˜| + |z − x| < ε , |t˜| < C|Imz| κ , Imz ∈ Γj }. Remark 6. In the case of = 1, a 1/-Fuchsian mild hyperfunction becomes an Oaku’s F-mild hyperfunction. Furthermore, in the case of = κ = 1, a 1/-Fuchsian mild hyperfunction coincides with a mild hyperfunction.

214

Y. Chiba

◦

We denote a sheaf of 1/-Fuchsian mild hyperfunctions with κ regularity F,1/,κ

by B N |M+ .

We define a sheaf A ∗F,1/ by the same way as in Definition 2. Let u be a 1/-Fuchsian mild hyperfunction. Then we define ρ1/ -SS(u) as follows: x∗ = √ (t, x; −1ξ) is not included in ρ1/ -SS(u) if and only if u has an expression (2) on a neighbourhood of x∗ with ξ ∈ Γj◦ . By changing ρ-SS(u) for ρ1/ -SS(u) in the above definition of an associated sheaf A ∗F , we have a sheaf A ∗F,1/ . Definition 7 (Fuchsian mild microfunctions with fractional order). We define a ◦

sheaf C

F,1/,κ

of 1/-Fuchsian mild microfunctions as ◦

C

F,1/,κ

◦

:= π −1 B F,1/,κ /A ∗F,1/ .

That is, a 1/-Fuchsian mild microfunction can be represented by a 1/-Fmild hyperfunction. Proposition 8. We have the following results about 1/-Fuchsian mild hyperfunctions. (1) O[t1/ ] operates 1/-Fuchsian mild hyperfunctions. Here O is a sheaf of holomorphic functions. (2) The operators t∂t and ∂x act 1/-Fuchsian mild hyperfunctions. Proof. Because of the expression (2) of 1/-Fuchsian mild hyperfunctions with defining functions Fj , we have the desired result. Lastly, we mention some properties of Fuchsian mild hyperfunctions with fractional order. Set X = Ct × Cnz and Y = {(t, z) ∈ X; t = 0}. Let EX/Y be a sheaf of microdifferential operators. A germ of EX/Y is expressed as A(t, z, ∂z ) with t as a holomorphic parameter. In the case that the regularity κ is equal to , a 1/-Fuchsian √

mild hyperfunction u(t, x) is represented as j=1 t(j−1)/ Fj (t, x + −1Γj 0) with a defining function Fj of a mild hyperfunction. Then we can show that a microdifferential operator A(t, x, ∂x ) act on a 1/-Fuchsian mild microfunction by applying the method of F -mild microfunctions. See the details in [9].

3. An application of Fuchsian mild hyperfunctions with fractional order to hyperbolic equations We consider the following partial differential operator on Rt × Rnx : P (t, ∂t , ∂x ) = ∂tm + a1 (t, ∂x )∂tm−1 + a2 (t, ∂x )∂tm−2 + · · · + am (t, ∂x ) with the principal symbol σ(P )(t, τ, ξ) =

m 4 j=1

(τ − t−1 αj (t)ξ)

(3)


215

j k at the origin ( = 2, 3, . . . ). Here aj (t, ∂x ) = |k|=0 ajk (t)∂x and each ajk (t) (k = 0, 1, . . . , j; j = 1, . . . , m) is analytic in a neighbourhood of t = 0. Furthermore, we assume hyperbolicity and the Levi condition on P : (i) Each αj (t) (j = 1, 2, . . . , m) is a real-valued function and αj (0) are mutually distinct. (ii) For 0 ≤ s < k − j, we assume ∂xs ajk (0) = 0, where k = 0, 1, . . . , j and j = 1, 2, . . . , m. To start with, u(t, x) is represented by a natural hyperfunction u 9(t, x) with supp(u) ⊂ {t ≥ 0}. We identify u 9(t, x) as a solution for tm P u 9(t, x) = 0 in a neighbourhood of t = 0. By using a fractional coordinate transform t t˜ = , (4) u 9(t, x) corresponds to a solution v(t˜, x) of Q(t˜, ∂t˜, ∂x )v(t˜, x) = 0, where Q is a partial differential operator whose coefficients have fractional power singularities with respect to t˜ and v(t˜, x) is a microfunction which is represented by a hyperfunction with support {t˜ ≥ 0}. We transform the operator Q by the quantised Legendre transform at ξ > 0: √ ∂t˜ → − −1w∂x , ∂x → ∂x , −1 β◦∗◦β : (5) √ √ t˜ → − −1∂w (∂x )−1 , x → x + −1∂w w(∂x )−1 . The operator Q is transformed to (β ◦ Q ◦ β −1 )(w, ∂w , ∂x ) = L(w, ∂w ) + R(w, ∂w , ∂x ), where L is a dominant part and R is a remaining term with fractional powers of ∂w . Concretely, L becomes an mth ordinary differential operator with polynomial coefficients which does not include neither ∂x nor fractional derivatives: m 4 √ m L = (constant) · (w − −1αj (0))∂w + (lower order). j=1

The equation we consider becomes (β ◦ Q ◦ β −1 )β[v] = 0. Here, β[v](w, x) is expressed at w = ∞ as β[v](w, x) = w−1 V (w−1/ , x), where V (z, x) is a microfunction with respect to (z, x) which is holomorphic at z = 0. The operator β ◦ Q ◦ β −1 has regular singularities at √ √ √ w = −1α1 (0), −1α2 (0), . . . , −1αm (0), ∞.

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We further assume the following condition: √ (iii) The characteristic exponents of the operator L at each w = −1αj (0) (j = 1, 2, . . . , m) are not integers. Furthermore, the characteristic exponent of the first degree α1 (t) ≡ 0. We have the following properties of the quantised Legendre transform β: √ 1 β(f (x)δ(t˜)) = ∂x f (x), β ◦ ∂t˜ ◦ β −1 = − −1w∂x . 2π ◦

The trace map C N |M+ u(t, x) → u(+0, x) ∈ CN is represented by the following: k t˜+ −( k +1) β u(+0, x) k = β(u(+0, x)∂t˜ δ(t˜)) Γ ( + 1) √ k = (− −1w∂x )−( +1) · (1/2π)∂x u(+0, x). Transforming t˜ = t /, we obtain √ − 1 k k −k (t+ )k ) = (− −1)−( +1) w−( +1) ∂x u(+0, x). k 2π Γ ( + 1) k

β(u(+0, x)

It follows that boundary values u(+0, x), ∂t u(+0, x), . . . , ∂tm−1 u(+0, x) become coefficients of w−1 , w−1−1/ , . . . , w−1−(m−1)/ . Furthermore, since the principal symbol σk/ (Rjk )(ξ) are linearly independent solutions of LU = 0, we have the invertibility of this matrix. Then we obtain the following theorem in the case of n = 1. Theorem 9 ([3]). For any j = 1, . . . , m and any microfunction u0 (x) at a point √ √ · p = (0, 0; ± −1) ∈ Rt × −1T ∗ Rx , we have a unique mild microfunction solution ◦ · u(t, x) ∈ C {t=0}|{t≥0} of a microlocal boundary value problem at p: ◦

P (t, ∂t , ∂x )u(t, x) = 0, t > 0 (in the sense of C

{t=0}|{t≥0} ),

u(+0, x) = u0 (x),

√ supp(ext(u)(t, x)) ∩ {t > 0} ⊂ {(t, x; −1(τ, ξ)); τ − t−1 αj (t)ξ = 0}.

(6)

Further, we have the equations ∂tk u(+0, x) = Rjk (∂x )u0 (x) (j = 1, 2, . . . , m; k = 0, 1, 2, . . . , m − 1), where Rjk (∂x ) is a microdifferential operator with fractional order at most k/. ◦ √ Here C {t=0}|{t≥0} is a sheaf on {t = 0} × −1T ∗ Rx of mild microfunctions ◦

([5]) and ext : C {t=0}|{t≥0} u(t, x) → u(t, x)Y (t) ∈ CRt ×Rx is the canonical extension to t ≤ 0. Example 10. In the case when P = ∂t2 − t2 ∂x2 , the operator P is transformed to 2 β ◦ Q ◦ β −1 = (w2 + 1)∂w + (7/2)w∂w + 3/2,

which becomes the Gauß hypergeometric operator after suitable linear transform.


217

In the last of this section, we show that the original equation P u = 0 can be characterised the transformed equation. We have already had a solution u(t, x) mild at t = 0. Using the fractional coordinate transform (4), u(t, x) becomes u((t˜)1/ , x) which has a singularity with fractional order at t˜ = 0. Thus our original problem P (t, ∂t , ∂x )u(t, x) = 0, t > 0 ({t ∈ R; t = 0} is non-characteristic for P ) is reduced to Q(t˜, ∂t˜, ∂x ) ext(u(t˜, x)) = 0 · in a neighbourhood of t = 0, x = x, where Q(t˜, ∂t˜, ∂x ) ≡ tm P (t, ∂t , ∂x ). Moreover, after the quantised Legendre transform, we can characterise our original problem by the following theorem in the case of κ = . Theorem 11. Let u(t, x) be a 1/-Fuchsian mild microfunction with regularity at √ · · (0, x). Then β[u](w, x) ∈ Cx Ow ({(w, x; −1dx); w ∈ C, Rew > 0, |x − x| < ε }) has the following properties: · (1) there exists F (w, x) ∈ Bx Ow ({(w, x); w ∈ C, Rew > 0, |x − x| < ε }) such that β[u](w, x) = [F (w, x)]. √ · (2) there exists G(ζ, x) ∈ Cx Oζ ({(ζ, x; −1dx); ζ ∈ C, |ζ| < ε , |x − x| < ε }) · such that we have β[u](w, x) = w−1 G(w−1/ , x) in a domain {(w, x); |x− x| < ε , | arg w| < π/2, |w| > ε− }. Conversely, if a pair (F, G) satisfies [F (w, x)] = [w−1 G(w−1/ , x)] as a section of · − Cx Ow ({|x − x| < ε , | arg w| < π/2, |w| > ε }), there exists a 1/-Fuchsian mild hyperfunction u(t, x) with regularity. Using an integral expression of the quantised Legendre transform with a defining function F (w, x) of β[u](w, x), we can show this theorem. Furthermore, we obtain the action of differential operators with fractional order on such F and G. 1/

Proposition 12. Operators ∂w , w∂w and ∂w theorem above. Proof. Firstly, we get +1 ζ ∂w F = ζ − ∂ζ G − ζ G

naturally act on F and G in the

ζ and w∂w F = ζ − ∂ζ G − G .

Secondly, by the definition of derivatives of fractional power order as the Riemann-Liouville integral, we have 1 F (s, x) 1/ ∂w F = ds, Γ (−1/) C1 (w − s)1+1/

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where C1 is a contour from s = ∞ enclosing s = w. Here we use an expansion ∞ F (s, x) = ζ k=0 ck ζ k in a neighbourhood of ζ = 0. By a transform ζ = w−1/ ζ, we obtain ∞ ζ +k 1/ ∂w F = Cζ ck ζ k+1 dζ , )1+1/ (1 − ζ C 2 k=0 where C is a suitable constant and C2 is a contour from ζ = 0 enclosing ζ = 1.

References [1] S. Alinhac, Branching of singularities for a class of hyperbolic operators, Indiana Univ. Math. J. 27 (1978), 1027–1037. [2] K. Amano and G. Nakamura, Branching of singularities for degenerate hyperbolic operators, Publ. Res. Inst. Math. Sci. Kyoto 20 (1984), 225–275. [3] Y. Chiba, A construction of pure solutions for degenerate hyperbolic operators, J. Math. Sci. Univ. Tokyo 16 (2009), 461–500. [4] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Math. Wiss. 292, Springer, 1990. [5] K. Kataoka, Micro-local theory of boundary value problems. I. Theory of mild hyperfunctions and Green’s formula, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 355–399. [6] K. Kataoka, Microlocal theory of boundary value problems. II. Theorems on regularity up to the boundary for reflective and diffractive operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 31–56. [7] K. Kataoka, Microlocal analysis of boundary value problems with regular or fractional power singularities, in Structure of Solutions of Differential Equations, Katata/Kyoto, 1995, World Scientific, 1996, 215–225. [8] T. Oaku, A canonical form of a system of microdifferential equations with noninvolutory characteristics and branching of singularities, Invent. Math. 65 (1981/82), 491–525. [9] T. Oaku, Microlocal boundary value problem for Fuchsian operators. I. F -mild microfunctions and uniqueness theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), 287–317. [10] T. Oaku, Higher-codimensional boundary value problem and F -mild hyperfunctions, in Algebraic Analysis, Academic Press, 1988, 571–586. [11] H. Yamane, Branching of singularities for some second or third order microhyperbolic operators, J. Math. Sci. Univ. Tokyo 2 (1995), 671–749. Yasuo Chiba School of Computer Science Tokyo University of Technology 1404-1 Katakura, Hachioji Tokyo 192-0982, Japan e-mail: [email protected]

The Continuity of Solutions with Respect to a Parameter to Symmetric Hyperbolic Systems Wataru Ichinose Abstract. In the preceding paper the initial problem to the Schr¨ odinger equations and the Dirac equations were studied with electromagnetic potentials depending on a parameter ≥ 0. It was proved that if electromagnetic potentials converge as → 0, then so do the solutions to the corresponding equations. In the present paper a generalization of the result on the Dirac equations is given to symmetric hyperbolic systems with coefficients depending continuously on a parameter. Mathematics Subject Classification (2000). Primary: 35L40. Keywords. Continuity of solutions, symmetric hyperbolic systems, parameter.

1. Introduction Let m, T > 0 be arbitrary constants, t ∈ [0, T ] and x = (x1 , . . . , xd ). Let (0 ≤ ≤ 1) be a parameter and consider ! a family of continuous electromagnetic " () () () () potentials V (t, x) 0≤≤1 ⊆ R and A (t, x) = A1 , . . . , Ad ⊆ Rd in [0, T ] × Rd . We consider the Dirac equations d ∂u 1 ∂ () i (t, x) = γj − Aj (t, x) u(t, x) ∂t i ∂xj j=1

0≤≤1

+ V () (t, x)Iu(t, x) + γd+1 mu(t, x),

(1.1)

where u = t (u1 , . . . , ul ) ∈ C l , γj (j = 1, . . . , d + 1) are l × l Hermitian constant matrices and I is the l × l identity matrix.

d αd 1 For a multi-index α = (α1 , . . . , αd ) we write |α| = j=1 αj , xα = xα 1 · · · xd , α α1 αd d ∂x = (∂/∂x1 ) · · · (∂/∂xd ) . Let S = S(R ) denote the Schwartz space of all Research partially supported by Grant-in-Aid for Scientific Research No.19540175, Ministry of Education, Culture, Sports, Science and Technology, Japanese Government.


219

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rapidly decreasing functions in Rd with the usual semi-norms. Let L2 = L2 (Rd ) be the space of all square integrable functions in Rd with inner product (·, ·) and norm · . For an integer M ≥ 0 we introduce the weighted Sobolev spaces

a a α α a := f + BM = BM (Rd ) := {f ∈ L2 ; f BM |α|=aM x f + |α|=a ∂x f < −a d 0 ∞} (a = 1, 2, . . . ). Let BM (R ) denote their dual spaces and set BM := L2 . If M = 1 the space B1a coincides with weighted Sobolev spaces B a (Rd ) introduced a in [15]. See also [6]. If M ≥ 2 the spaces BM are similar to the spaces introduced in [2]. In the preceding paper [9] we proved Theorem A. We assume that there exists an integer M ≥ 1 satisfying

() sup ∂xα Aj (t, x) ≤ Cα xM

(1.2)

0≤≤1

()

in [0, T ] × Rd for j = ' 0, 1, . . . , d and all α with constants Cα , where A0 (t, x) = −V () (t, x) and x = 1 + |x|2 . We also assume ()

(0)

lim ∂xα Aj (t, x) = ∂xα Aj (t, x)

(1.3)

→0

pointwise in [0, T ] × Rd for j = 0, 1, . . . , d and all α. Let f = t (f1 , . . . , fl ) ∈ a BM (Rd )l (a = 0, 1, . . . ) (resp. S(Rd )l ). Then, there exists a solution u (t) ∈ a−1 l 0 a l a l Et ([0, T ]; (BM ) ) ∩ Et1 ([0, T ]; (BM ) ) (resp. Et1 ([0, T ]; S l )), i.e., (BM ) -valued cona−1 l tinuous and (BM ) -valued continuously differentiable in [0, T ], to the Dirac equation (1.1) with u(0) = f and its solution is unique in −1 l Et0 ([0, T ]; (L2)l ) ∩ Et1 ([0, T ]; (BM ) ).

In addition, as tends to zero, u (t) converges to u0 (t) uniformly in [0, T ] in the a l topology of (BM ) (resp. S l ). In [9] the similar result to Theorem A for the Schrödinger equations 2 d ∂u 1 1 ∂ () i (t, x) = − Aj (t, x) u(t, x) + V () (t, x)u(t, x) ∂t 2m j=1 i ∂xj was also proved, which was applied to the theory of the Feynman path integral for the quantum electrodynamics in [8], where u(t, x) is a scalar function. In the present paper we consider symmetric hyperbolic systems as in [4, 10] () ∂u 1 ∂u (t, x) = γj (t, x) (t, x) ∂t i ∂xj d

()

iγ0 (t, x)

j=1

+

() γd+1 (t, x)u(t, x)

()

+ iγd+2 (t, x)u(t, x)

(1.4)

()

with a parameter (0 ≤ ≤ 1), where γj (t, x) (j = 0, 1, . . . , d + 1) are l × l ()

()

Hermitian matrices and γ0 (t, x) > 0 is assumed. We do not assume that γd+2 (t, x) is Hermitian. Our aim in the present paper is to give a generalization of Theorem A to (1.4). Following the proof of Theorem in [7], we can easily prove the existence

Symmetric Hyperbolic Systems

221

and uniqueness of solutions u (t) to (1.4) with u(0) = f . Hence, our main concern is to prove the continuity of solutions u (t) with respect to a parameter . In the theory of ordinary differential equations, a parameter being considered as an independent variable, the continuity of solutions with respect to a parameter can be proved from the continuity of solutions with respect to an initial value (cf. [12]). Another proof can be directly given by means of the successiveapproximations procedure (cf. [1, 3]). () () Consider the case that γ0 = I (0 ≤ ≤ 1) and γj (j = 1, 2, . . . , d + 2) are independent of t ∈ [0, T ]. Assume that the rhs of (1.4) are self-adjoint operators on L2 (Rd )l , which we write H , with a common core D. Then we know from Trotter’s theorem (cf. Theorem VIII. 21 in [14]) and Theorem VIII. 25 in [14] that if H ϕ tends to H0 ϕ in L2 (Rd )l for each ϕ ∈ D as → 0, then exp itH tends to exp itH0 strongly in L2 (Rd )l for each t ∈ [0, T ]. This result can be applied to the Dirac equations (1.1) in L2 (Rd )l from Theorem 4.3 in [17]. But, in general it a seems not easy to apply this result to our equations (1.4) in the spaces BM (Rd )l . In Chapter 10 of [11] a hyperbolic system with diagonal principal part   κ1 (t, X, Dx ) 0 ... 0   0 κ2 (t, X, Dx ) . . . 0 ∂u   i (t, x) =   u(t, x) .. .. . .   ∂t . . 0 . 0 0 . . . κl (t, X, Dx ) + B(t)u(t, x)

(1.5)

in [0, T ] is considered, where κ(t, x, ξ) (j = 1, 2, . . . , l) are real-valued continuous functions in [0, T ] × R2d , and j↓ B(t) = bjk (t, X, Dx ); 1, 2, . . . , l , k→ |∂ξα ∂xβ κj (t, x, ξ)| ≤ Cα,β ξ1−|α|

(j = 1, 2, . . . , l)

and |∂ξα ∂xβ bjk (t, x, ξ)| ≤ Cα,β ξ−|α|

(j, k = 1, 2, . . . , l)

in [0, T ] × R2d for all α and β are assumed. Here, κj (t, X, Dx ) is the pseudodifferential operator defined by 1 ix·ξ κj (t, X, Dx )f = e κ (t, x, ξ)dξ e−ix ·ξ f (x )dx (1.6) j d (2π) Rd Rd

d for f ∈ S(Rd ), where x · ξ = j=1 xj · ξj . In Theorem 7.1 of [11] the fundamental solution to (1.5) with u(0) = I is constructed in the form of the Fourier integral operator in [0, T0 ] for a sufficiently small T0 (0 < T0 ≤ T ), where solutions φj (t, x, ξ) (j = 1, 2, . . . , l) to the Hamilton-Jacobi equations ∂φ ∂t φ + κj t, x, = 0, φ(0, x, ξ) = x · ξ ∂x

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play a important role. Here ∂φ/∂x = (∂φ/∂x1 , . . . , ∂φ/∂xd ). We will be able to apply Kumano-go’s result above to (1.4) to prove a generalization of Theorem A. () But then, we need more limited assumptions on γj (t, x) than those in our main theorems stated in §2, we must assume that T0 is sufficiently small and we will need delicate analysis to consider the dependency of the fundamental solution with respect to a parameter . In the present paper we will prove a generalization of Theorem A to (1.4) by means of the abstract Ascoli-Arzelà theorem and the uniqueness of solutions to (1.4), following [7, 9]. Let us give a rough sketch. We may assume that we can write the rhs of (1.4) as the Weyl operator X + X (H (t)f ) (x) = H t, , Dx f 2 1 x + x ix·ξ −ix ·ξ := e dξ e h t, , ξ f (x )dx (1.7) (2π)d 2 for f ∈ S(Rd )l , where h (t, x, ξ) =

d

()

()

()

γj (t, x)ξj + γd+1 (t, x) + iγd+2 (t, x).

(1.8)

j=1

Let us take a real-valued χ ∈ S(Rd ) such that χ(0) = 1 and set h, (t, x, ξ) = χ( (xM + ξ))   d   () () () × γj (t, x)ξj + γd+1 (t, x) + iγd+2 (t, x)  

(1.9)

j=1

for a parameter (0 < ≤ 1) as in [7] and Chapter IV of [16]. We write H, (t) := H, (t, (X + X )/2, Dx). Let g(t) = g(t, ·) be an arbitrary function a l in Et0 ([0, T ]; (BM ) ) (a = 0, 1, . . . ) and consider symmetric hyperbolic systems ∂v (t, x) = H, (t)v(t, x) + g(t, x) (1.10) ∂t a l with v(0) = f ∈ (BM ) . Then we can easily prove by the successive-approximations a l procedure that there exists a solution v, (t) ∈ Et1 ([0, T ]; (BM ) ) satisfying t a )l ≤ Ca a )l + a )l dθ sup

v, (t) (BM

f (BM

g(θ) (BM (1.11) ()

iγ0 (t, x)

0≤≤1,0 0 and set

uη (t, x) = 0 (|x| ≤ 1) for 0 ≤ t ≤ T0 . So we can easily see from (2.8), (2.11) and (2.13) that uη (t, x) is in C ∞ ([0, T0 ] × R) with compact support and is the solution to (2.9) with u(0) = fη . Consequently from (2.10) we have

uη (T0 ) ≤ CT0 fη ≤ CT0 f .

(2.14)

It follows from (2.11)–(2.13) that ∞

fη exp e−T0 log |x| 2 dx

uη (T0 ) 2 = 2 1

= 2e

∞

T0

= 2eT0

1 ∞ 2

2

|fη (y)| y

eT0 −1

dy ≥ 2e

T0 2

∞

1+2τ T0 1 |χ(ηy)| y e −1 dy y 2

1 |χ(ηy)|2 dy. y

Consequently, lim uη (T0 ) 2 = ∞,

η→0

which contradicts (2.14).

Remark 2.1. In Remark 2 of [5] the solution u(t, x) to the equation (2.9) with u(0) = f where a(x) ∈ C ∞ (R), a(x) = 0 (x ≤ 0) and a(x) = x log x(log log x)η (x ≥ 10) for 0 < η ≤ 1 is considered. They stated without proof that there exists an initial function f ∈ S(R) satisfying limx→∞ |u(t, x)| = 1 for any t > 0. Our a(x) defined by (2.8) grows more slowly than that in [5] as x → ∞.

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Let us state a different result from Theorem 2.1, which gives a direct generalization of Theorem A. Theorem 2.3. For all α we assume (2.1), (2.2), (2.6) in [0, T ] × Rd and (2.7) pointwise in [0, T ] × Rd . We also assume in [0, T ] × Rd ()

sup |∂xα γj (t, x)| ≤ Cα < ∞

(2.15)

0≤≤1

for j = 1, 2, . . . , d and all α, and that there exist an integer M ≥ 1 and constants Cα satisfying () sup |∂xα γd+1 (t, x)| ≤ Cα xM (2.16) 0≤≤1

for all α. Then the same assertion as in Theorem A holds. Remark 2.2. In [4, 10, 11, 13, 16] the existence and uniqueness of solutions to (1.4) were studied under the assumptions of Theorem 2.3 with M = 0. Let us consider single hyperbolic second-order equations ∂t2 v(t, x) =

d

()

ajk (t, x)∂xj ∂xk v +

+c

()

bj (t, x)∂xj v

j=1

j,k=1 ()

d

()

(t, x)∂t v + d

(t, x)v

()

(2.17) ()

with a parameter (0 ≤ ≤ 1), where ajk (t, x) = akj (t, x) (j, k = 1, 2, . . . , d) are real valued. Corollary 2.4. We assume: (1) There exits a constant c > 0 such that () 0 < cI ≤ inf ajk (t, x); 0≤≤1

in [0, T ] × Rd . (2) We have

j↓ 1, 2, . . . , d k→

() sup ∂t ∂xα ajk (t, x) ≤ Cα

(2.18)

(2.19)

0≤≤1

in [0, T ] × Rd for all α and j, k = 1, 2, . . . , d with constants Cα . (3) Let ζ () (t, x) = () () ajk (t, x), bj (t, x), c() (t, x) and d() (t, x). Then we have

sup ∂xα ζ () (t, x) ≤ Cα < ∞ (2.20) 0≤≤1

in [0, T ] × Rd for all α and lim ∂xα ζ () (t, x) = ∂xα ζ (0) (t, x)

→0

(2.21)

pointwise in [0, T ] × Rd for all α. Let f ∈ B1a+1 (Rd ) (a = 0, 1, . . . ) (resp. S(Rd )) and g ∈ B1a (Rd ) (resp. S(Rd )). Then, there exists a solution v (t) in Et1 ([0, T ]; B1a)∩ Et2 ([0, T ]; B1a−1 ) (resp. Et2 ([0, T ]; S)) to (2.17) with v(0) = f and ∂t v(0) = g, and


227

its solution is unique in Et0 ([0, T ]; B11 )∩Et1 ([0, T ]; L2 )∩Et2 ([0, T ]; B1−1 ). In addition, as tends to zero, both v (t) and ∂t v (t) converge to v0 (t) and ∂t v0 (t) uniformly in [0, T ] in the topology of B1a (resp. S), respectively. Proof. We set uj = ∂xj v (j = 1, 2, . . . , d), ud+1 = ∂t v, ud+2 = v

(2.22)

as in [4, 10]. Then we have d

()

ajk (t, x)∂t uk =

k=1

d

()

ajk (t, x)∂t ∂xk v

k=1

=

d

()

ajk (t, x)∂xk ud+1

k=1

∂t ud+1 = ∂t2 v = =

()

(j = 1, 2, . . . , d),

ajk ∂xj ∂xk v +

j,k () ajk ∂xk uj

j,k

+

()

bj ∂xj v + c() ∂t v + d() v

j () bj u j

+ c() ud+1 + d() ud+2 ,

j

∂t ud+2 = ud+1 .

(2.23)

Then we can apply Theorem 2.3 with M = 1 to the symmetric hyperbolic system (2.23). Then we can prove Corollary 2.4.

3. The proof of main theorems We will first prove Theorem 2.1. Let M ≥ 1 be the integer in Theorem 2.1. For the a sake of simplicity we write BM as B a hereafter. It follows from Lemma 2.3 in [7] that there exist a constant µa ≥ 0 (a = 1, 2, . . . ) and a function wa (x, ξ) satisfying −1 |∂ξα ∂xβ wa (x, ξ)| ≤ Cα,β 1 + xaM + ξa (3.1) in [0, T ] × R2d for all α and β, and −1 aM Wa (x, Dx )f = Λ−1 + Dx a f a f := µa + X

(3.2)

for f ∈ S. It was proved in Lemma 2.4 of [7] that we have Ca−1 f Ba ≤ Λa f ≤ Ca f B a

(3.3)

with a constant Ca > 0. Lemma 3.1. The embedding map ia from B a+1 (Rd ) into B a (Rd ) is compact. Proof. Let Ω be a bounded open set in Rd . We define H01 (Ω) by the completion of the space of all C ∞ functions with compact support in Ω by the norm f +

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W. Ichinose

d

j=1 ∂xj f . We know well from Rellich’s theorem (cf. [13]) that the embedding map from H01 (Ω) into L2 (Ω) is compact. For f ∈ B 1 we have 2M 2 R |f (x)| dx ≤ x2M |f (x)|2 dx ≤ Const. f 2B 1 . |x|≥R

Rd

Hence we can easily prove that the embedding map i0 from B 1 (Rd ) into L2 (Rd ) is compact. It follows from Lemma 2.5 in [7] and (3.3) that Λa from B a+1 (Rd ) into 2 d a d B 1 (Rd ) and Λ−1 a from L (R ) into B (R ) are bounded operators, respectively. Since ia is written as ia = Λ−1 a ◦ i0 ◦ Λa ,

ia is also compact.

Noting the assumptions (2.4) and (2.6), we may assume without the loss of generality that we can write the rhs of (1.4) as (1.7). Let χ ∈ S(Rd ) be the realvalued function introduced in §1 and define h, (t, x, ξ) (0 ≤ ≤ 1, 0 < ≤ 1) by (1.9). For f ∈ (B a )l (a = 0, 1, . . . ) and g(t) ∈ Et0 ([0, T ]; (B a )l ) we consider symmetric hyperbolic systems (1.10). Then we can write (1.10) as t ! " () i v(t) − f = γ0 (θ)−1 H, (θ)v(θ) + g(θ) dθ. (3.4) 0

Since is positive, the function h, (t, x, ξ) and its all derivatives with respect to x and ξ are bounded in [0, T ] × R2d. So we see from Lemma 2.5 in [7] that H, (t) is a bounded operator from (B a )l into (B a )l . It follows from the assumptions (2.1) and (2.2) that () sup |∂xα γ0 (t, x)−1 | ≤ Cα < ∞ (3.5) 0≤≤1

in [0, T ] × R for all α. Hence we can find a solution v, (t) in Et1 ([0, T ]; (B a )l ) to (1.10) with v(0) = f from (3.4) by the successive-approximations procedure. For a complex number b let 'b denote the real part of b. We note that the solution v, (t) defined above to (1.10) belongs to Et1 ([0, T ]; (B a )l ). Hence we can prove d () v, (t), γ0 (t)v, (t) dt () ∂v, ∂γ () (t) + v, (t), 0 (t)v, (t) = 2' v, (t), γ0 (t) ∂t ∂t () ∂γ = 2' i v, (t), H, (t)v, (t) + 2' i v, (t), g(t) + v, (t), 0 (t)v, (t) . ∂t d

()

We assumed that γj (t, x) (j = 1, 2, . . . , d + 1) are Hermitian. Consequently, from (1.8), (1.9) and the definition of H, (t) we have H, (t) − H, (t)∗ = iγd+2 (t, x) + iγd+2 (t, x)∗ , ()

()

(3.6)


229

where H, (t)∗ denotes the formally adjoint operator in L2 (Rd )l . For the sake of simplicity let us write · (B a )l as · B a . Then, applying the assumptions (2.1), (2.2), (2.6) and (3.6) to the above, we get ! d () () v, (t), γ0 (t)v, (t) ≤ C v, (t), γ0 (t)v, (t) dt 1/2 " () + v, (t), γ0 (t)v, (t)

g(t)

(3.7) with a constant C independent of 0 ≤ ≤ 1 and 0 < ≤ 1, and so 1/2 ! 1/2 1 t " () () CT /2 v, (t), γ0 (t)v, (t) ≤e f, γ0 (0)f +

g(θ) dθ . 2 0 Hence from (2.1) and (2.2) we obtain sup

0≤≤1,0 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , such that " 1 ∀ε ≤ ε0 , |aε | ≤ c exp −kε− τ .

The ring C τ motivates the following, easy to prove, result. Proposition 5. If u ∈ G τ (Ω) and x ∈ Ω, then the element u (x) represented by (uε (x))ε is an element of C τ independent of the representative (uε )ε of u.

Generalized Gevrey Ultradistributions

239

A generalized Gevrey ultradistribution is not defined by its point values, in order to give a solution to this situation, set ! " −1 ΩτM = (xε )ε ∈ Ω]0,1] : ∃k > 0, ∃c > 0, ∃ε0 > 0, ∀ε ≤ ε0 , |xε | ≤ cekε τ . (4) Define in ΩτM the equivalence relation by xε ∼ yε ⇐⇒ ∀k > 0, ∃c > 0, ∃ε0 > 0, ∀ε ≤ ε0 , |xε − yε | ≤ ce−kε

−1 τ

.

(5)

9 τ = Ωτ / ∼ is called the set of exponential generalized Definition 4. The set Ω M points. The set of its compactly supported points is defined by ! " τ 9τ = x 9 Ω 9 = [(x ) ] ∈ Ω : ∃K a compact set of Ω, ∃ε > 0, ∀ε ≤ ε , x ∈ K . ε ε 0 0 ε c (6) 9 τc , then the generalized point Proposition 6. Let f ∈ G τ (Ω) and x 9 = [(xε )ε ] ∈ Ω value of f at x 9, i.e., f (9 x) = [(fε (xε ))ε ] is a well-defined element of the algebra of exponential generalized complex numbers C τ . The characterization of nullity of f ∈ G τ (Ω) is given by the following theorem. Theorem 7. Let f ∈ G τ (Ω) , then 9 τc . f = 0 in G τ (Ω) ⇐⇒ f (9 x) = 0 in C τ for all x 9∈Ω

4. Embedding of Gevrey ultradistributions We recall some definitions and results on Gevrey ultradistributions. A function f ∈ E σ (Ω) , if f ∈ C ∞ (Ω) and for every compact K of Ω, ∃c > 0, ∀α ∈ Zm +, sup |∂ α f (x)| ≤ c|α|+1 (α!)σ . x∈K

Obviously we have E (Ω) ⊂ E σ (Ω) if 1 ≤ t ≤ σ. It is well known that E 1 (Ω) = A (Ω) is the space of all real analytic functions in Ω. Denote by Dσ (Ω) the space E σ (Ω) ∩ C0∞ (Ω) , then D σ (Ω) is nontrivial if and only if σ > 1. The topological dual of Dσ (Ω) , denoted Dσ (Ω) , is called the space of Gevrey ultradistributions of order σ. The space Eσ (Ω) is the topological dual of E σ (Ω) and is identified with the space of Gevrey ultradistributions with compact support.

Definition 5. A differential operator of infinite order P (D) = aγ Dγ is called t

γ∈Zm +

a σ-ultradifferential operator, if for every h > 0 there exists c > 0 such that ∀γ ∈ Zm +, |aγ | ≤ c

h|γ| σ. (γ!)

(7)

240

K. Benmeriem and C. Bouzar The importance of σ-ultradifferential operators lies in the following result.

Proposition 8. Let T ∈ Eσ (Ω) , σ >

1 and γsupp T ⊂ K, then there exist a σultradifferential operator P (D) = aγ D , M > 0 and continuous functions γ∈Zm +

fγ ∈ C0 (K) such that sup γ∈Zm + ,x∈K

|fγ (x)| ≤ M

and

T =

a γ D γ fγ .

γ∈Zm +

The space S (ρ) (Rm ) , ρ > 1, see [5], is the space of functions ϕ ∈ C ∞ (Rm ) such that ∀b > 0, we have |β| |x|

ϕ b,ρ = sup |∂ α ϕ (x)| dx < ∞. (8) |α+β| α!ρ β!ρ b α,β∈Zm + Lemma 9. There exists φ ∈ S (ρ) (Rm ) satisfying φ (x) dx = 1 and xα φ (x) dx = 0, ∀α ∈ Zm + \{0}. Definition 6. The net φε = ε−m φ (./ε) , ε ∈ ]0, 1] , where φ satisfies the conditions of Lemma 9, is called a ρ-net of mollifiers. The space E σ (Ω) is embedded into G τ (Ω) by the standard canonical injection → G τ (Ω) , → [f ] = cl (fε )

I : E σ (Ω) f

(9)

where fε = f , ∀ε ∈ ]0, 1]. The following proposition gives the natural embedding of Gevrey ultradistributions into G τ (Ω) . Theorem 10. The map J0 : Eτ +ρ (Ω)

→ G τ (Ω) → [T ] = cl (T ∗ φε )/Ω

T

(10) ε

is an embedding. Proof. Let T ∈ Eτ +ρ (Ω) be such that supp T ⊂ K. Then we can find a (τ + ρ)-ultradifferential operator P (D) = continuous functions fγ with supp fγ ⊂ K, ∀γ ∈ Zm + , and such that T =

γ∈Zm +

a γ D γ fγ .

γ∈Zm +

sup γ∈Zm + ,x∈K

aγ D γ and

|fγ (x)| ≤ M,

Generalized Gevrey Ultradistributions We have

T ∗ φε (x) =

|γ|

aγ

γ∈Zm +

Let α ∈

Zm +,

(−1) ε|γ|

then

|∂ (T ∗ φε (x))| ≤ α

aγ

γ∈Zm +

1 ε|γ+α|

241

fγ (x + εy) Dγ φ (y) dy.

|fγ (x + εy)| Dγ+α φ (y) dy.

From (7) and the inequality (β + α)!t ≤ 2t|β+α| α!t β!t , ∀t ≥ 1,

(11)

we have, ∀h > 0, ∃c > 0, such that h|γ|

1 |∂ α (T ∗ φε (x))| ≤ c τ +ρ |γ+α| |fγ (x + εy)| Dγ+α φ (y) dy γ! ε m γ∈Z+

≤

cα!τ +ρ

γ∈Zm +

×

2(τ +ρ)|γ+α| h|γ| 1 b|γ+α| (γ + α)!τ ε|γ+α|

|fγ (x + εy)|

|Dγ+α φ (y)| dy, b|γ+α| (γ + α)!ρ

then for h > 12 , 1 α!τ +ρ

|∂ (T ∗ φε (x))| ≤ φ b,ρ M c α

γ∈Zm +

2

−|γ|

|γ+α| 2τ +ρ+1 bh 1 (γ + α)!τ ε|γ+α|

1 ≤ c exp k1 ε− τ , i.e.,

1 |∂ α (T ∗ φε (x))| ≤ c (α) exp k1 ε− τ , (12) τ +ρ+1 τ1 where k1 = τ 2 bh . Suppose that (T ∗ φε )ε ∈ N τ (Ω), then for every compact subset L of Ω, ∃c > 0, ∀k > 0, ∃ε0 ∈ ]0, 1] , 1 |T ∗ φε (x)| ≤ c exp −kε− τ , ∀x ∈ L, ∀ε ≤ ε0 . (13) Let χ ∈ Dτ +ρ (Ω) and χ = 1 in a neighborhood of K, then ∀ψ ∈ E τ +ρ (Ω) , T, ψ = T, χψ = lim (T ∗ φε ) (x) χ (x) ψ (x) dx. ε→0

Consequently, from (13), we obtain

(T ∗ φε ) (x) χ (x) ψ (x) dx ≤ c exp −kε− τ1 , ∀ε ≤ ε0 ,

which gives T, ψ = 0

242

K. Benmeriem and C. Bouzar In order to show the commutativity of the following diagram of embeddings Dτ −ρ+1 (Ω)

→ (

G τ (Ω) ↑ , Eτ +ρ (Ω)

we have to prove the following fundamental result. Proposition 11. Let f ∈ Dτ −ρ+1 (Ω), then f − (f ∗ φε )/Ω ∈ N τ (Ω) . ε

Proof. Let f ∈ D τ −ρ+1 (Ω) , then there exists a constant c > 0, such that |∂ α f (x)| ≤ c|α|+1 α!(τ −ρ+1) , ∀α ∈ Zm + , ∀x ∈ Ω. Let α ∈ Zm + , the Taylor’s formula and the properties of φε give (εy)β ∂ (f ∗ φε − f ) (x) = ∂ α+β f (ξ) φ (y) dy, β! α

|β|=N

where x ≤ ξ ≤ x + εy. Consequently, for b > 0, we have |∂ α (f ∗ φε − f ) (x)| |y|N

∂ α+β f (ξ) |φ (y)| dy ≤ εN β! |β|=N

(τ −ρ+1) N

≤ α!

ε

τ (τ −ρ+1)|α+β| |β|

β! 2

b

|β|=N

α+β

∂ f (ξ) |y||β| |φ (y)| dy. (α + β)!(τ −ρ+1) b|β| β!ρ

Let k > 0 and T > 0, then |∂ α (f ∗ φε − f ) (x)| −N

≤ α!(τ −ρ+1) (εN τ ) (k τ T ) × 2(τ −ρ+1)|α+β| (k τ bT )|β| N

|β|=N

α+β

∂ f (ξ) |y||β| |φ (y)| dy (α + β)!(τ −ρ+1) b|β| β!ρ

|α| N −N ≤ α!(τ −ρ+1) (εN τ ) (k τ T ) c φ b,ρ 2(τ −ρ+1) c |β| × 2(τ −ρ+1) k τ bT c|β| , |β|=N

Generalized Gevrey Ultradistributions hence, taking 2(τ −ρ+1) k τ bT c ≤

1 2a ,

243

with a > 1, we obtain −N

|∂ α (f ∗ φε − f ) (x)| ≤ α!(τ −ρ+1) (εN τ ) (k τ T ) |α| 1 |β| (τ −ρ+1) −N × c φ b,ρ 2 c a 2 N

|β|=N

≤ φ b,ρ c

|α|+1

α!

(τ −ρ+1)

(εN ) (k τ T )−N a−N . τ N

(14)

1

Let ε0 ∈ ]0, 1] such that ε0τ lnka < 1 and take T > 2τ , then 1 ln a 1 ε τ , ∀ε ≤ ε0 , Tτ −1 >1> k in particular, we have −1 −1 1 ln a 1 ln a 1 ετ Tτ − ετ > 1. k k Then, there exists N = N (ε) ∈ Z+ , such that −1 −1 ln a 1 ln a 1 1 τ τ ε 1. Finally, from (14), we have

1 |∂ α (f ∗ φε − f ) (x)| ≤ c exp −kε− τ ,

i.e., f ∗ φε − f ∈ N τ (Ω).

(16)

As in [4] and [1], we embed Dτ +ρ (Ω) into G τ (Ω) using the sheaf properties, then we have the following commutative diagram E τ −ρ+1 (Ω) → G τ (Ω) ↓ + . Dτ +ρ (Ω)

5. Regular generalized Gevrey ultradistributions To develop a local and a microlocal analysis with respect to a “good space of regular elements” one needs first to define these regular elements, the notion of singular support and its microlocalization with respect to the class of regular elements.

244

K. Benmeriem and C. Bouzar

τ,σ,∞ Definition 7. The space of σ-regular elements, denoted Em (Ω), is the space of ]0,1] ∞ (fε )ε ∈ (C (Ω)) satisfying, for every compact K of Ω, ∃k > 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , ∀α ∈ Zm + , ∀ε ≤ ε0 , 1 sup |∂ α fε (x)| ≤ c|α|+1 α!σ exp kε− τ . x∈K

Proposition 12. τ,σ,∞ 1) The space Em (Ω) is an algebra stable under the action of σ-ultradifferential operators. τ,σ,∞ τ,σ,∞ 2) The space N τ,σ,∞ (Ω) := N τ (Ω) ∩ Em (Ω) is an ideal of Em (Ω) . Now, we define the Gevrey regular elements of G τ (Ω) . Definition 8. The algebra of σ-regular generalized ultradistributions of order τ , denoted Gστ,∞ (Ω) , is the quotient algebra Gστ,∞ (Ω) =

τ,σ,∞ Em (Ω) . N τ,σ,∞ (Ω)

It is clear that E σ (Ω) → Gστ,∞ (Ω) and Gστ,∞ is subsheaf of G τ . This motivates the following definition. Definition 9. We define the Gστ,∞ -singular support of a generalized ultradistribution f ∈ G τ (Ω) , denoted by σ- sing suppg (f ) , as the complement of the largest open set Ω such that f ∈ Gστ,∞ (Ω´) . The following result is a Paley-Wiener type characterization of Gστ,∞ (Ω) . τ Proposition 13. Let f = cl (fε )ε ∈ GC (Ω) the set of generalized ultradistributions with compact support, then f is σ-regular if and only if ∃k1 > 0, ∃k2 > 0, ∃c > 0, ∃ε1 > 0, ∀ε ≤ ε1 , such that 1 1 |F (fε ) (ξ)| ≤ c exp k1 ε− τ − k2 |ξ| σ , ∀ξ ∈ Rm , (17)

where F (fε ) denote Fourier transform of fε . The algebra Gστ,∞ (Ω) plays the same role as the Oberguggenberger subalgebra of regular elements G ∞ (Ω) in the Colombeau algebra G (Ω). Theorem 14. We have τ −ρ+1 Gττ,∞ (Ω) . −ρ+1 (Ω) ∩ Dτ +ρ (Ω) = E τ +ρ Proof. Let S ∈ Gττ,∞ (Ω) −ρ+1 (Ω)∩ Dτ +ρ (Ω) , for any fixed x0 ∈ Ω we take ψ ∈ D with ψ ≡ 1 on neighborhood U of x0 , then T = ψS ∈ Eτ +ρ (Ω) . Let φε be a net of mollifiers with φˇ = φ and let χ ∈ D τ −ρ+1 (Ω) such that χ ≡ 1 on K = supp ψ. As [T ] ∈ Gττ,∞ −ρ+1 (Ω) , ∃k1 > 0, ∃k2 > 0, ∃c1 > 0, ∃ε1 > 0, ∀ε ≤ ε1 , 1 1 |F (χ (T ∗ φε )) (ξ)| ≤ c1 exp k1 ε− τ − k2 |ξ| τ −ρ+1 ,


245

then |F (χ (T ∗ φε )) (ξ) − F (T ) (ξ)| = |F (χ (T ∗ φε )) (ξ) − F (χT ) (ξ)|

B C

= T (x) , χ (x) e−iξx ∗ φε (x) − χ (x) e−iξx . As Eτ +ρ (Ω) ⊂ Eτ −ρ+1 (Ω) , then ∃L a compact subset of Ω such that ∀h > 0, ∃c > 0, and |F (χ (T ∗ φε )) (ξ) − F (T ) (ξ)| ≤c

h|α|

sup α∈Zm + ,x∈L

α!τ −ρ+1

α

∂ χ (x) e−iξx ∗ φε (x) − χ (x) e−iξx . x

We have e−iξ χ ∈ D τ −ρ+1 (Ω) and by (14), we obtain ∀k3 > 0, ∃c2 > 0, ∃η > 0, ∀ε ≤ η, |α|

sup α∈Zm + ,x∈L

c2

α!τ −ρ+1

1

α

∂ χ (x) e−iξx ∗ φε (x) − χ (x) e−iξx ≤ c2 e−k3 ε− τ , x

so there exists c = c (k3 ) > 0, such that −1 τ

|F (T ) (ξ) − F (χ (T ∗ φε )) (ξ)| ≤ c e−k3 ε

.

Let ε ≤ min (η, ε1 ) , then |F (T ) (ξ)| ≤ |F (T ) (ξ) − F (χ (T ∗ φε )) (ξ)| + |F (χ (T ∗ φε )) (ξ)| −1 τ

≤ c e−k3 ε

Take c = max (c , c1 ) , ε = ∃δ > 0, ∃c > 0 such that

+ c1 ek1 ε

−1 τ

.

τ

k1 (k2 − r) |ξ|

1

−k2 |ξ| τ −ρ+1

1 τ −ρ+1

|F (T ) (ξ)| ≤ ce−δ|ξ|

, r ∈ ]0, k2 [ and k3 =

1 τ −ρ+1

k1 r , then k2 − r

,

which means T = ψS ∈ E τ −ρ+1 (Ω). As ψ ≡ 1 on the neighborhood U of x0 , consequently S ∈ E τ −ρ+1 (Ω), which proves τ −ρ+1 Gττ,∞ (Ω) . −ρ+1 (Ω) ∩ Dτ +ρ (Ω) ⊂ E

We have E τ −ρ+1 (Ω) ⊂ E τ +ρ (Ω) ⊂ Dτ +ρ (Ω) and E τ −ρ+1 (Ω) ⊂ Gττ,∞ −ρ+1 (Ω) then E τ −ρ+1 (Ω) ⊂ Gττ,∞ −ρ+1 (Ω) ∩ Dτ +ρ (Ω) .

Consequently we have τ −ρ+1 Gττ,∞ (Ω) . −ρ+1 (Ω) ∩ Dτ +ρ (Ω) = E

246


6. Generalized Gevrey wave front The aim of this section is to introduce the generalized Gevrey wave front of a generalized ultradistribution f ∈ G τ (Ω) and to give its main properties. Definition 10. A point (x0 , ξ0 ) ∈ / W Fgτ,σ (f ) ⊂ Ω × Rm \ {0} if there exists φ ∈ σ D (Ω) , φ (x) = 1 neighborhood of x0 , and conic neighborhood Γ of ξ0 , ∃k1 > 0, ∃k2 > 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , such that ∀ξ ∈ Γ, ∀ε ≤ ε0 , 1 1 |F (φfε ) (ξ)| ≤ c exp k1 ε− τ − k2 |ξ| σ . The main properties of the generalized Gevrey wave front W Fgτ,σ are summarized in the following proposition.

Proposition 15. Let P (x, D) = aα (x) Dα be a partial differential operator with Gστ,∞ (Ω) coefficients, then

|α|≤m

W Fgτ,σ (P (x, D) f ) ⊂ W Fgτ,σ (f ) , ∀f ∈ G τ (Ω) . We need the following lemma to show the relationship between W Fgτ,σ (T ) and W F σ (T ) , when T ∈ Dτ +ρ (Ω). Lemma 16. Let ϕ ∈ Dρ (B (0, 2)) , 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 on B (0, 1) , and let φ ∈ S (ρ) , then ∃c > 0, ∃ν > 0, ∃ε0 > 0, ∀ε ∈ ]0, ε0 ] , ∀ξ ∈ Rm ,

1 1 ρ ρ

6

θε (ξ) ≤ cε−m e−νε |ξ| , m x where θε (x) = 1ε φ ε ϕ (x |ln ε|), and θ6 denotes the Fourier transform of θ. We have the following important result. Theorem 17. Let T ∈ Dτ +ρ (Ω) ∩ G τ (Ω) , then W Fgτ,τ −ρ+1 (T ) = W F τ −ρ+1 (T ) . Proof. Put σ = τ − ρ + 1 and let S ∈ Eτ +ρ (Ω) ⊂ Eσ (Ω) and ψ ∈ Dσ (Ω) , we have

. /

|F (ψ(S ∗ φε ))(ξ) − F (ψS)(ξ)| = S (x), ψ (x)e−iξx ∗ φ˘ε (x) − ψ (x)e−iξx , then ∃L a compact of Ω such that ∀h > 0, ∃c > 0, |F (ψ (S ∗ φε )) (ξ) − F (ψS) (ξ)|

h|α|

α −iξx ˘ε (x) − ψ (x) e−iξx

. ≤ c sup ∂ ψ (x) e ∗ φ

x α!σ α∈Zm + ,x∈L We have e−iξ ψ ∈ Dσ (Ω) , then, see (14), ∃c2 > 0, ∀k0 > 0, ∃η > 0, ∀ε ≤ η, |α|

1 c2

α −iξx ˘ε (x) − ψ (x) e−iξx

≤ c2 e−k0 ε− τ , sup ψ (x) e ∗ φ

∂ x σ m α∈Z+ ,x∈L α!

(18)

so there exist c > 0, ∀k0 > 0, ∃η > 0, ∀ε ≤ η, such that −1 τ

|F (ψS) (ξ) − F (ψ (S ∗ φε )) (ξ)| ≤ c e−k0 ε

.

(19)


247

Let T ∈ Dτ +ρ (Ω) ∩ G τ (Ω) and (x0 , ξ0 ) ∈ / W Fgτ,σ (T ), then there exist χ ∈ σ D (Ω) , χ (x) = 1 in a neighborhood of x0 , and a conic neighborhood Γ of ξ0 , ∃k1 > 0, ∃k2 > 0, ∃c1 > 0, ∃ε0 ∈ ]0, 1[ , such that ∀ξ ∈ Γ, ∀ε ≤ ε0 , −1 τ

|F (χ (T ∗ θε )) (ξ)| ≤ c1 ek1 ε

1

−k2 |ξ| σ

(20)

let ψ ∈ Dσ (Ω) equal 1 in neighborhood of x0 such that for sufficiently small ε we have χ ≡ 1 on supp ψ + B 0, |ln2ε| , and let ϕ ∈ Dσ (B (0, 2)) , 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 on B (0, 1) , then there exist ε0 < 1, such that ∀ε < ε0 ,

ψ (T ∗ θε ) (x) = ψ (χT ∗ θε ) (x) x 1 where θε (x) = m ϕ (x |ln ε|) φ . As χT ∈ Eτ +ρ (Ω), then see Proposition 14 [1] ε ε ψ (T ∗ θε ) (x) = ψ (χT ∗ θε ) (x) = ψ (χT ∗ φε ) (x) . Let ε ≤ min (η, ε0 ) and ξ ∈ Γ, we have |F (ψT ) (ξ)| ≤ |F (ψT ) (ξ) − F (ψ (T ∗ θε )) (ξ)| + |F (χ (T ∗ θε )) (ξ)| ≤ |F (ψχT ) (ξ) − F (ψ (χT ∗ φε )) (ξ)| + |F (χ (T ∗ θε )) (ξ)| then by (19) and (20) , we obtain −1

−1

1

|F (ψT ) (ξ)| ≤ c e−k0 ε τ + c1 ek1 ε τ −k2 |ξ| σ . τ k1 k1 r Take c = max (c , c1 ) , ε = , r ∈ ]0, k2 [ , k0 = , then ∃δ > 1 k2 − r (k2 − r) |ξ| σ 0, ∃c > 0, such that 1

|F (χT ) (ξ)| ≤ ce−δ|ξ| σ , which proves that (x0 , ξ0 ) ∈ / W F σ (T ) , i.e., W F σ (T ) ⊂ W Fgτ,σ (T ) . Suppose (x0 , ξ0 ) ∈ / W F σ (T ), then there exist χ ∈ Dσ (Ω) , χ (x) = 1 in a neighborhood of x0 , a conic neighborhood Γ of ξ0 , ∃λ > 0, ∃c1 > 0, such that ∀ξ ∈ Γ, 1

|F (χT ) (ξ)| ≤ c1 e−λ|ξ| σ .

(21)

Let also ψ ∈ Dσ (Ω) equal 1 inneighborhood of x0 such that for sufficiently small ε 2 we have χ ≡ 1 on supp ψ + B 0, |ln ε| , then there exist ε0 < 1, such that ∀ε < ε0 , ψ (T ∗ θε ) (x) = ψ (χT ∗ θε ) (x) . We have F (ψ (T ∗ θε )) (ξ) =

F (ψ) (ξ − η) F (χT ) (η) F (θε ) (η) dη.

248


Let Λ be a conic neighborhood of ξ0 such that, Λ ⊂ Γ. For a fixed ξ ∈ Λ, we have F (ψ (χT ∗ θε )) (ξ) = F (ψ) (ξ − η) F (χT ) (η) F (θε ) (η) dη A + F (ψ) (ξ − η) F (χT ) (η) F (θε ) (η) dη , B

where

! 1 " 1 1 A = η; |ξ − η| σ ≤ δ |ξ| σ + |η| σ

and

! 1 " 1 1 . B = η; |ξ − η| σ > δ |ξ| σ + |η| σ

We choose δ sufficiently small such that A ⊂ Γ and ψ ∈ Dσ (Ω), then ∃µ > 0, ∃c2 > 0, ∀ξ ∈ Rm ,

|ξ| < |η| < 2σ |ξ|. Since 2σ

1 |F (ψ) (ξ)| ≤ c2 exp −µ |ξ| σ .

Then ∃c > 0, ∃ε0 ∈ ]0, 1[ , ∀ε ≤ ε0 ,

F (ψ) (ξ − η) F (χT ) (η) F (θε ) (η) dη

A

1 λ 1

≤ c exp − |ξ| σ

exp −µ |η − ξ| σ F (θε ) (η) dη

. 2 A From Lemma 16, ∃c3 > 0, ∃ν > 0, ∃ε0 > 0, such that ∀ε ∈ ]0, ε0 ] , |F (θε ) (ξ)| ≤ c3 ε−m e−νε

1 ρ

1

|ξ| ρ

, ∀ξ ∈ Rm ,

then ∃c > 0, such that

F (ψ) (ξ − η) F (χT ) (η) F (θε ) (η) dη

A 1 1 1 λ 1 ≤ cε−m exp − |ξ| σ exp −µ |η − ξ| σ exp −νε ρ |η| ρ dη. 2 A We have ∃k > 0, ∀ε ∈ ]0, ε0 ] ,

1 1 1 ε−m exp −νε ρ |η| ρ ≤ exp kε− τ ,

so

1

F (ψ) (ξ − η) F (χT ) (η) F (θε ) (η) dη ≤ c exp kε− τ1 − λ |ξ| σ .

2 A

As χT ∈ Eτ +ρ (Ω) ⊂ Eσ (Ω) , then ∀l > 0, ∃c > 0, ∀ξ ∈ Rm , 1 |F (χT ) (ξ)| ≤ c exp l |ξ| σ ,

(22)

(23)


249

hence, we have

F (ψ) (ξ − η) F (χT ) (η) F (θε ) (η) dη

B 1 1 ≤c exp l |η| σ − µ |η − ξ| σ |F (θε )| dη B 1 1 1 1 −m σ ≤cε exp −µδ |ξ| exp (l − µδ) |η| σ − νε ρ |η| ρ dη, B

then, taking l − µδ = −a < 0 and using (22), we obtain for a constant c > 0,

1

F (ψ) (ξ − η) F (χT ) (η) F (θε ) (η) dη ≤ c exp kε− τ1 − µδ |ξ| σ . (24)

B

Consequently, (23) and (24) give ∃c > 0, ∃k1 > 0, ∃k2 > 0, 1 1 |F (ψ (T ∗ θε )) (ξ)| ≤ c exp k1 ε− τ − k2 |ξ| σ

(25)

which gives that (x0 , ξ0 ) ∈ / W Fgτ,σ (T ) , so W Fgτ,σ (T ) ⊂ W F σ (T ) which ends the proof. To extend the H¨ ormander’s result on the wave front set of the product of two distributions, define W Fgτ,σ (f ) + W Fgτ,σ (g) , where f, g ∈ G τ (Ω) , as the set (x, ξ + η) ; (x, ξ) ∈ W Fgτ,σ (f ) , (x, η) ∈ W Fgτ,σ (g) . Theorem 18. Let f, g ∈ G τ (Ω), such that ∀x ∈ Ω, (x, 0) ∈ / W Fgτ,σ (f ) + W Fgτ,σ (g) , then

W Fgτ,σ (f g) ⊆ W Fgτ,σ (f ) + W Fgτ,σ (g) ∪ W Fgτ,σ (f ) ∪ W Fgτ,σ (g) .

(26)

(27)

References [1] K. Benmeriem and C. Bouzar, Generalized Gevrey ultradistributions, New York J. Math. 15 (2009), 37–72. [2] K. Benmeriem and C. Bouzar, Ultraregular generalized functions of Colombeau type, J. Math. Sci. Univ. Tokyo 15 (4) (2008), 427–447. [3] J.F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland, 1984. [4] M. Grosser, M. Kunzinger, M. Oberguggenberger and R. Steinbauer, Geometric Theory of Generalized Functions, Kluwer Academic Publishers, 2001. [5] I.M. Gel’fand and G.E. Shilov, Generalized Functions, Volume 2, Academic Press, 1967. [6] G. H¨ ormann, M. Oberguggenberger and S. Pilipović. Microlocal hypoellipticity of linear differential operators with generalized functions as coefficients, Trans. Amer. Math. Soc. 358 (8), (2005), 3363–3383.

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[7] L. H¨ ormander. The Analysis of Linear Partial Differential Operators, I, Springer, 1983. [8] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Volume 3, Springer, 1973. [9] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes Math. 259, 1992. [10] L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, 1993. Khaled Benmeriem University of Mascara Bp 305 Route de Mamounia 29000 Mascara, Algeria e-mail: [email protected] [email protected] Chikh Bouzar Oran-Essenia University Oran 31000, Algeria e-mail: [email protected] [email protected]

Weyl Rule and Pseudo-Differential Operators for Arbitrary Operators Leon Cohen Abstract. The standard rules of association between functions and operators, ∂ such as the Weyl rule, deal with the operators x and D = 1i ∂x . We consider the generalization of these rules to arbitrary operators and in particular we discuss in detail the generalization when D is replaced with an arbitrary Hermitian operator. The physical and mathematical motivations for doing so are discussed and a number of examples are given. Mathematics Subject Classification (2000). Primary 35S05, 47G30. Keywords. Weyl transform, pseudo-differential operator, scale.

1. Introduction The fundamental idea of a correspondence rule, like the Weyl rule and similar rules, is to associate an ordinary function of two variables a(x, ξ), often called the symbol, with a corresponding operator Wa (x, D), where x and D are operators that satisfy the commutation rule [x, D] = xD − Dx = i

(1)

and where in the x representation D=

1 ∂ . i ∂x

(2)

The association is symbolized by Wa (x, D) ↔ a(x, ξ).

(3)

One approach to the Weyl rule, and indeed was the method that Weyl used [12], is to associate eiθx+iτ D ↔ eiθx+iτ ξ (4) This research was supported by the Office of Naval Research (N00014-09-1-0162).


251

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L. Cohen

where θ and τ are real variables. The association with an arbitrary symbol is then set because if one expands the symbol in terms of its Fourier transform 1 6 a(θ, τ ) = a(x, ξ) e−iθx−iτ ξ dxdξ (5) 4π 2 a(x, ξ) = 6 a(θ, τ ) eiθx+iτ ξ dθ dτ (6) then the association for a(x, ξ) is obtained by substituting eiθx+iτ D for eiθx+iτ ξ in Eq. (6), Wa (x, D) = 6 a(θ, τ ) eiθx+iτ D dθ dτ (7) 1 = a(x , ξ) eiθ(x−x )+iτ (D−ξ) dθ dτ dx dξ. (8) 2 4π This is the Weyl association, or rule. The Weyl transform [15] is the operation of Wa (x, D) on an arbitrary function, ϕ(x). It is straightforward to show that 1 x+τ Wa (x, D) ϕ(x) = a , ξ ei(x−τ )ξ ϕ(τ ) dτ dξ (9) 2π 2 Another rule is to take the association eiθx eiτ D ↔ eiθx+iτ ξ

(10)

which is often called the normal association and is the Kohn-Nirenberg procedure. The same argument as above leads to Ta (x, D) = 6 a(θ, τ )eiθx eiτ D dθ dτ (11) and

1 a (x, ξ) e−i(τ −x)ξ ϕ(τ ) dτ dξ. 2π A third rule is to take the association Ta (x, D) ϕ(x) =

eiτ D eiθx ↔ eiθx+iτ ξ

(12)

(13)

which is often called the anti-normal association. The association for a symbol is then Ra (x, D) = 6 a(θ, τ )eiτ D eiθx dθ dτ (14) and further Ra (x, D) ϕ(x) =

1 2π

a (τ, ξ) e−i(τ −x)ξ ϕ(τ ) dτ dξ.

Also, commonly taken is the so-called symmetrization rule [8] where 1 , iθx iτ D e e + eiτ D eiθx ↔ eiθx+iτ ξ . 2

(15)

(16)

Weyl Rule and Pseudo-Differential Operators

253

In fact, there are an infinite number of rules and a method to generate them was given in 1966 [1]. A review for getting these rules from various points of view is given in reference [5]. Our aim here is to study the generalization of the above rules when D is replaced with an arbitrary Hermitian operator, that we call E. The physical and mathematical motivations for doing so will be disused in the next section.1 As before we will write the association as Wa (x, E) ↔ a(x, ξ)

(17)

and we will consider the generalization of the above rules. For example, for the Weyl type rule and corresponding transform we will consider Wa (x, E) = 6 a(θ, τ) eiθx+iτ E dθ dτ (18) Wa (x, E) ϕ(x) = 6 a(θ, τ ) eiθx+iτ E ϕ(x)dθ dτ (19) and similarly for the other rules. In the next section we discuss the physical motivation for wanting to do this and subsequently we derive some of the main properties of these associations and transforms. Terminology. When we refer to the usual case, that is when we are considering the x and D operators as per Eq. (1) and (2), then we shall say that we are considering the (x, D) case. We shall use (x, E) to refer to the general case, that is when E is an arbitrary Hermitian operator. Also, all integrals go from −∞ to ∞ unless noted otherwise.

2. Motivation for generalizing to arbitrary operators The original motivation of Weyl and others was based on the then discovered idea that in quantum mechanics physical quantities are represented by operators rather than functions. The issue became how to write the operator corresponding to an ordinary function. Weyl came up with what is now known as the Weyl procedure and a number of others were suggested. All of the possible rules were derived in [1]. Furthermore, starting with Wigner [13], one attempted to write the quantum mechanical way of computing averages by the standard phase space method. In particular, if one makes the association A(x, D) ↔ a(x, ξ) then one attempts to write a(x, ξ) Pϕ (x, ξ) dxdξ = ϕ∗ (x) A(x, D) ϕ(x) dx 1 More

(20)

(21)

generally one can replace both x and D, by arbitrary Hermitian operators. We do not do so here for the sake of clarity and also because it is simpler.

254

L. Cohen

where x, ξ are position and momentum respectively, and where ϕ(x) is the state function that represents the system under consideration. In that case, D, as defined by Eq. (2), is the momentum operator. One approach to satisfy Eq. (21) is to recall the relationship of the characteristic function, M (θ, τ), and distribution, Pϕ (x, ξ), in standard probability theory M (θ, τ ) = eiθx+iτ ξ = eiθx+iτ ξ P (x, ξ)dxdξ (22) 1 Pϕ (x, ξ) = M (θ, τ )e−iθx−iτ ξ dθdτ. (23) 4π 2 Since the characteristic function is an average, we expect to find it by way of M (θ, τ ) = ϕ∗ (x)eiθx+iτ D ϕ(x) dx. (24) When Eq. (24) is substituted in Eq. (23) one obtains the Wigner distribution [9, 1, 4] 1 W (x, ξ) = ϕ∗ (x − 12 τ ) ϕ(x + 12 τ ) e−iτ ξ dτ. (25) 2π However, if for example one makes the association given by Eq. (10) or Eq. (13) one obtains different distributions. Similar consideration arose in time-frequency analysis where x is replaced by time, t and in that case D becomes the frequency ∂ operator, 1i ∂t and ϕ(t) is the signal [4]. Now consider the joint distribution of two arbitrary quantities denoted by (a, b) with corresponding operators (A, B). The characteristic function and distribution are related by 1 Pϕ (x, ξ) = M (α, β)e−iαa−iβb dαdβ (26) 4π 2 M (α, β) = eiαa+iβb = eiαa+iβb Pϕ (x, ξ)dadb. (27) Again, since the characteristic function is an average, we expect to calculate by way of M (α, β) = ϕ∗ (x)eiαA+iβB ϕ(x) dx (28) or by

M (α, β) =

ϕ(x)eiαA eiβB ϕ(x) dx

(29)

or other orderings[10, 11, 4]. Thus, one sees that the consideration of quantities such eiαA+iβB arise naturally. As previously mentioned, in this paper we will restrict ourselves to the case where the first operator remains x and the second operator is arbitrary which we call E. Hence, we write 1 Pϕ (x, ξ) = M (θ, τ )e−iθx−iτ ξ dθdτ (30) 4π 2

Weyl Rule and Pseudo-Differential Operators M (θ, τ) =

a(x, ξ)Pϕ (x, ξ) dxdξ =

255

ϕ∗ (x)eiθx+iτ E ϕ(x) dx

(31)

ϕ∗ (x) Wa (x, E) ϕ(x) dx.

(32)

The other main motivation is the pseudo-differential operators approach which indeed is closely related to the above [15, 16]. We illustrate with ordinary differential equations but the same idea holds for partial differential equations. Suppose we have a linear differential equation with constant coefficient, an , and with a driving term f (x), , P (D)ϕ(x) = an Dn + an−1 D n−1 + · · · + a1 D + a0 ϕ(x) = f (x). (33) The standard procedure is to define the Fourier transform of ϕ and f respectively 1 ϕ(x) = √ ϕ(ξ) 6 eiξx dξ (34) 2π 1 f (x) = √ f6(ξ) eiξx dξ (35) 2π and, by substituting into Eq. (33), to convert the differential equation into an algebraic one P (ξ)ϕ(ξ) 6 = f6(ξ). (36) The solution is then

6 1 f (ξ) iξx √ ϕ(x) = e dξ. (37) P (ξ) 2π However, if we have a differential equation with non-constant coefficients , P (x, D)u(x) = an (x)D n + an−1 (x)Dn−1 + · · · + a1 (x)D + a0 (x) ϕ(x) = f (x) (38) the above method does not work; nonetheless, it has been found useful to define the pseudo-differential operator, P (x, D), and associate it with the classical function P (x, ξ)2 . The association can be any one of the cases considered above and also an infinite number of others. Now, suppose we generalize the concept of a differential equation by considering an operator equation of the form , P (E)u(x) = an E n + an−1 E n−1 + · · · + a1 E + a0 u(x) = f (x) (39) where E is a Hermitian operator. To illustrate we take a specific example 1 d d E= x + x 2i dx dx

(40)

where E is called the scale operator [3]. Eq. (39) can be re-written in terms of D in which case we will have a differential equation with non-constant coefficients. 2 This

P (x, ξ) shoud not be confused with Pϕ (x, ξ).

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L. Cohen

However, if we use the eigenfunctions, γ(ξ, x), of the scale operator obtained by solving E γ(ξ, x) = ξγ(ξ, x) (41) then we can again convert Eq. (39) into an algebraic one. In particular, the eigenfunctions are 1 eiξ ln x √ γ(ξ, x) = √ , x ≥ 0. (42) x 2π If we define the scale transform and inverse by ∞ 1 Φ(ξ) = √ ϕ(x)γ ∗ (ξ, x) dx 2π 0 1 ϕ(x) = √ U (ξ) γ(ξ, x) dξ ; x ≥ 0 2π

(43) (44)

and similarly for f (x) and its scale transform, F (ξ), then Eq. (39) transforms into an algebraic equation P (ξ)Φ(ξ) = F (ξ) (45) and the solution is

1 F (ξ) u(x) = √ γ(ξ, x) dξ. (46) P (ξ) 2π Now, in a similar fashion one considers , P (x, E)u(x) = an (x)E n + an−1 (x)E n−1 + · · · + a1 (x)E + a0 (x) u(x) = f (x) (47) and this leads to rules of association for P (x, E) in total analogy with the P (x, D) case. We point out that what we have done above for the scale operator works for any other Hermitian operator since a Hermitian operator generates a complete and orthogonal set of eigenfunctions.

3. The eiθx+iτE association and transform We define the generalized operator transform associated with the symbol a(x, ξ) by Eq. (18) which we repeat here for convenience Wa (x, E) = 6 a(θ, τ) eiθx+iτ E dθ dτ. (48) Equivalently 1 Wa (x, E) = 4π 2

a(x, ξ) eiθ(x−x )+iτ (E−ξ ) dθ dτ dx dξ .

(49)

We now consider the operation of Wa (x, E) on an arbitrary function, ϕ(x), Wa (x, E) ϕ(x) = 6 a(θ, τ ) eiθx+iτ E ϕ(x) dθ dτ. (50)


257

To evaluate, one has to consider the simplification eiθx+iτ E . There are a number of special cases where eiθx+iτ E can be broken up into individual operators. The first case is when both operators commute with their commutator [14], [[ x, E ], x ] = [[ x, E ], E ] = 0

(51)

eiθx+iτ E = e−θτ [ x,E ]/2 eiτ E eiθx = eθτ [ x,E ]/2 eiθx eiτ E .

(52)

in which case

For future reference, we also note that eiτ E eiτ x = eθτ [ x,E ] eiθx eiτ E .

(53)

Another special case is when [3] [ x, E ] = α x

(54)

eiθx+iτ E = eiθµx eiτ E eiθx

(55)

in which case where µ =

1 , 1 − (1 + iτ α ) e−iτ α . iτ α

(56)

A general procedure that works in principle is now described. One solves the eigenvalue problem [2, 14], { θx + τ E } γ(λ, x) = λ γ(λ, x).

(57)

Since θx + τ E is Hermitian for θ and τ real the eigenvalue problem gives rise to a complete set of eigenfunctions, the γ’s. Hence, any function can be expressed in terms of the eigenfunctions f (x) = γ(λ, x)F (λ) dλ (58) with

γ ∗ (λ, x) f (x) dx.

F (λ) = Now

(59)

eiθx+iτ E f (x) =

eiλ γ(λ, x)F (λ) dλ

(60)

G(x, x ) f (x ) dx

(61)

eiλ γ ∗ (λ, x ) γ(λ, x) dλ.

(62)

and substituting for F (λ) we have

eiθx+iτ E f (x) = where G(x, x ) =

The drawback to this method is that one has to solve the eigenvalue problem.

258

L. Cohen

3.1. Example: Scale, E = 12 (xD + Dx) The scale operator, C, is defined by 1 1 E = C = (xD + Dx) = 2 2i

d d x + x dx dx

(63)

and has the following basic property [3] eiτ C f (x) = eτ /2 f (eτ x).

(64)

The commutation relation is [x, C] = ix and hence, using Eq. (55) with α = i, we have

(65) τ

eiθx+iτ C = eiθηx eiτ C eiθx = eiθηx eiθe with η=

τ

6 a(θ, τ) eiθηx eiθe

Wa (x, C) =

eiτ C

1 {(1 − τ ) eτ − 1}. τ

Therefore

x

x

(66) (67)

eiτ C dθ dτ.

(68)

Applying Eq. (64), we have τ

eiθηx eiθe

τ

x

eiτ C ϕ(x) = eiθηx eτ /2 eiθe x ϕ(eτ x)

and hence the transform is

Wa (x, C) ϕ(x) =

τ

6 a(θ, τ)eiθηx eτ /2 eiθe x ϕ(eτ x) dθ dτ

(69)

(70)

which simplifies to Wa (x, C) ϕ(x) =

τ /2

6 a(θ, τ )ei[2θxe

sinh(τ /2)]/τ

eτ /2 ϕ(eτ x) dθ dτ.

In obtaining Eq. (71) we have used the fact that 2 (η + eτ ) = (eτ − 1)/τ = eτ /2 sinh(τ /2). τ

(71)

(72)

3.2. Example: E = x + D For this case we use both methods described above, namely by the use of Eq. (52) and also by solving the eigenvalue problem as given by Eq. (61). Consider first the simplification of eiθx+iτ (x+D) . The commutator is [x, x + D] = [x, D] = i

(73)

eiθx+iτ (x+D) = ei(θ+τ )(x+τ /2) eiτ D

(74)

eiθx+iτ (x+D)ϕ(x) = ei(θ+τ )(x+τ /2) ϕ(x + τ ).

(75)

and using Eq. (52) we have and


259

Therefore

6 a(θ, τ )ei(θ+τ )(x+τ /2) eiτ D dθ dτ

Wa (x, E) =

and for the transform we obtain Wa (x, E) ϕ(x) = 6 a(θ, τ)ei(θ+τ )(x+τ /2) ϕ(x + τ ) dθ dτ = 6 a(θ, τ − x)ei(θ+τ −x)(x+τ )/2 ϕ(τ ) dθ dτ. Expressing this in terms of the symbol one obtains 1 x+τ Wa (x, E) ϕ(x) = a , ξ e−i(τ −x)ξ+i(τ −x)(τ +x)/2 ϕ(τ ) dτ dξ. 2π 2

(76)

(77) (78)

(79)

We now obtain the same result by the method leading to Eq. (61). The eigenvalue problem is {θx + τ E} u(λ, x) = λ u(λ, x) (80) or {(θ + τ )x + τ D} u(λ, x) = λ u(λ, x), (81) The eigenfunctions normalized to a delta function are easily obtained 2 1 u(λ, x) = √ ei[λx−(θ+τ )x /2]/τ ] (82) 2πτ and the calculation of G in Eq. (62) yields G(x, x ) = δ(x − τ − x)ei(θ+τ )[τ /2+x]. Therefore

G(x, x )ϕ(x ) dx

eiαx+iβE ϕ(x) = =

δ(x − τ − x)ei(θ+τ )[τ /2+x] ϕ(x ) dx

= ei(θ+τ )[τ /2+x]ϕ(τ + x)

(83)

(84) (85) (86)

which is the same as Eq. (75).

4. The eiθxeiτE association and transform For the association eiθx eiτ E ↔ eiθx+iτ ξ the correspondence between symbol and operator become Ta (x, E) = 6 a(θ, τ)eiθx eiτ E dθ dτ and for the transform we write Ta (x, E) ϕ(x) =

(87)

(88)

6 a(θ, τ ) eiθx eiτ E ϕ(x)dθ dτ.

(89)

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L. Cohen

Often the operation of eiτ E on a function can be carried out explicitly as in some of the examples we consider. However, in a more general sense eiτ E ϕ(x) can be carried out by the following procedure. Suppose we solve the eigenvalue problem E γ(λ, x) = λ γu(λ, x) then for an arbitrary function and inverse we have f (x) = γ(λ, x)F (λ) dλ with

F (λ) =

Therefore

γ ∗ (λ, x) f (x) dx.

eiτ E f (x) = eiτ E γ(λ, x)F (λ) dλ = eiτ λ γ(λ, x)F (λ) dλ = eiτ λ γ(λ, x) γ ∗ (λ, x ) f (x ) dλdx

or

G(τ, x, x ) =

(93) (94)

eiτ λ γ(λ, x) γ ∗ (λ, x ) dλ.

(96)

1 (xD 2

We have,

(92)

(95)

Substituting in Eq. (89) gives 6 a(θ, τ ) eiθx eiτ E ϕ(x)dθ dτ = 6 a(θ, τ)eiθx G(τ, x, x ) ϕ(x ) dx dθ dτ. 4.1. Example: Scale E = C =

(91)

G(β, x, x ) f (x )dx

eiτ E f (x) = with

(90)

(97)

+ Dx)

6 a(θ, τ)eiθx eiτ C dθ dτ

(98)

6 a(θ, τ ) eiθx eiτ E ϕ(x)dθ dτ 1 = a(x, ξ) e−iτ ξ eτ /2 ϕ(eτ x)dξ dτ. 2π

(99)

Ta (x, C) = and using Eq. (64) we obtain

Ta (x, C) ϕ(x) =

(100)

4.2. Example: E = x + D Since the commutator is [x, x + D] = [x, D] = i

(101)


261

we have, using Eq. (52), that eiθx eiτ (x+D) = eiτ and hence

2

6 a(θ, τ ) eiτ

Ta (x, E) =

/2 i(θ+τ )x iτ D

e

2

e

/2 i(θ+τ )x iτ D

e

e

(102)

dθ dτ.

For the transform we have 2 Wa (x, E) ϕ(x) = 6 a(θ, τ ) eiτ /2 ei(θ+τ )xϕ(x + τ ) dθ dτ 2 2 1 = a(x, ξ) ei(τ −ξ) /2−i(x−ξ) /2 ϕ(τ ) dτ dξ. 2π

(103)

(104) (105)

5. The eiτE eiθx association and transform For this case the association is eiτ E eiθx ↔ eiθx+iτ ξ

(106)

and the correspondence between symbol and operator is La (x, D) = 6 a(θ, τ )eiτ E eiθx dθ dτ. For the transform we have

(107)

6 a(θ, τ )eiτ E eiθx ϕ(x)dθ dτ.

La (x, D) ϕ(x) = 5.1. Example: Scale E = C =

1 (xD 2

The association is

(108)

+ Dx)

6 a(θ, τ )eiτ C eiθx dθ dτ

(109)

6 a(θ, τ ) eiτ C eiθx ϕ(x)dθ dτ

(110)

Ra (x, C) = and using Eq. (64) we have

Ra (x, C) ϕ(x) =

τ

6 a(θ, τ ) eτ /2 eiθe x ϕ(eτ x)dθ dτ 1 = a(eτ x, ξ) e−iτ ξ eτ /2 ϕ(eτ x)dξ dτ. 2π =

(111) (112)

5.2. Example: E = x + D Using Eqs. (52) and (53) we note that eiτ (x+D) eiθx = eiτ

2

/2 i(θ+τ )x iθτ iτ D

e

e

e

(113)

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L. Cohen

and therefore

6 a(θ, τ) eiτ

Ra (x, E) =

2

/2 i(θ+τ )x iθτ iτ D

e

e

e

dθ dτ.

For the transform we obtain, after some manipulation, that 2 Wa (x, E) ϕ(x) = 6 a(θ, τ) eiτ /2 ei(θ+τ )x eiθτ ϕ(x + τ ) dθ dτ 2 2 1 = a(τ, ξ) ei(τ −ξ) /2−i(x−ξ) /2 ϕ(τ ) dτ dξ. 2π

(114)

(115) (116)

6. Conclusion In conclusion we briefly discuss some of the issues in the development of the (x, E) case in analogy with the (x, D) case. We note that the (x, D) case arose in the early days of quantum mechanics because in quantum mechanics physical quantities are represented by operators. There were many methods proposed to associate operators with ordinary functions, among them the Weyl rule. For the (x, D) case, a general method to obtain all the correspondence rules is the generalized operator transform defined by [1, 5] Φ Aa (x, D) = 6 a(θ, τ )Φ(θ, τ ) eiθx+iτ D dθ dτ. (117) In Eq. (117) Φ(θ, τ ) is a two-dimensional function called the kernel [1, 7] and characterizes a specific transform and its properties. For example, if one takes Φ(θ, τ ) = 1 the Weyl rule is obtained and if one takes Φ(θ, τ) = e−iθτ /2 then the Kohn-Nirenberg procedure is obtained. The operation of AΦ a (x, D) on an arbitrary function, ϕ(x) is Φ Aa (x, D) ϕ(x) = 6 a(θ, τ − x)Φ(θ, τ − x) eiθ(τ +x)/2 ϕ(τ )dθ dτ (118) which in terms of the symbol is AΦ a

1 (x, D) ϕ(x) = 4π 2

τ +x a q+ , ξ Φ(θ, τ − x) 2

e−iθq−i(τ −x)ξ ϕ(τ ) dτ dq dξdθ.

(119)

Furthermore, to obtain the joint x, ξ the distribution, C(x, ξ), for the association AΦ a (x, D) ↔ a(x, ξ) the distribution should satisfy ϕ∗ (x) AΦ (x, D) ϕ(x)dx = a(x, ξ) C(x, ξ)dxdξ. a

(120)

(121)

One can show that one must take C(x, ξ) given by [1, 7] 1 C(x, ξ) = 2 ϕ∗ (x − τ /2)Φ(θ, τ ) eiθx −iθx−iτ ξ ϕ(x + τ /2)dθ dτ dx . (122) 4π


263

There are many relations that exist for the (x, D) case. For example, for two distributions characterized by Φ1 and Φ2 the corresponding distributions are related by 1 Φ2 (θ, τ ) −iθ(x−x )−iτ (ξ−ξ ) C2 (x, ξ) = e C1 (x, ξ) dθ dτ dx dξ (123) 2 4π Φ1 (θ, τ ) ∂ ∂ Φ2 i ∂x , i ∂ξ C1 (x, ξ). = (124) ∂ ∂ Φ1 i ∂x , i ∂ξ Can these results for the (x, D) case be carried out for the (x, E) case? The answer is only partially known and in fact considerable difficulties arise. Many relations cannot be obtained in a straightforward way because the commutation relation for (x, E) case is not simple as it is in the (x, D) case. For example, the equivalent of Eq. (124) is not known. For another example consider the derivation of Eq. (117) for the (x, D) case by what is called the Taylor series association, where the operator correspondence for xn ξ m is denoted by [5] Pnm ( x, D) ↔ xn ξ m .

(125)

Now expanding the symbol, a(x, ξ), in a Taylor series * n+m + ∞ 1 ∂ a(x, ξ) = a(x, ξ)| xn ξ m . n m x,ξ = 0 n!m! ∂x ∂ξ n,m=0 The operator becomes Aa ( x, D) =

∞ n,m=0

If one takes Pnm ( x, D) =

!

∂ n+m ∂xn ∂ξm

(126)

" a(x, ξ)|x,ξ = 0 n!m!

Pnm (x, D).

1 ∂ n+m iθx+iτ D

Φ(θ, τ ) e ↔ xn ξ m

n m n m i i ∂θ ∂τ θ,τ = 0

(127)

(128)

and substitutes into Eq. (127) one gets Eq. (117). Can this be carried over when we use general operators, that is for the case (x, E)? The answer is generally no in the form presented but clearly some formulation must exist. The reason for the difficulty is that differentiation of eiθx+iτ D is relatively easy but the differentiation of eiθx+iτ E is not. These issues are challenging and interesting and will be considered in a future paper.

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References [1] L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1966), 781–786, 1966. [2] L. Cohen, Expansion theorem for functions of operators, J. Math. Phys. 7 (1966), 244. [3] L. Cohen, The scale representation, IEEE Trans. Signal Processing 41 (1993), 3275– 3292. [4] L. Cohen, Time-Frequency Analysis, Prentice-Hall, New Jersey, 1995. [5] L. Cohen, The Weyl transform and its generalization, Rend. Sem. Mat. Univ. Pol. Torino, 66 (4) (2008), 1–12. [6] J.G. Kirkwood, Quantum statistics of almost classical ensembles, Phys. Rev. 44 (1993), 31–37. [7] H.W. Lee, Theory and application of the quantum phase-space distributionfunctions, Phys. Rep. 259 (1995), 147–211. [8] H. Margenau and R.N. Hill, Correlation between measurements in quantum theory, Prog. Theoret. Phys. 26 (1961), 722–738. [9] J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc. 45 (1949), 99–124. [10] M.O. Scully and M.S. Zubairy, Quantum Optics, Cambridge University Press, 1997. [11] M.O. Scully and L. Cohen, Quasi-Probability Distributions for Arbitrary Operators, in The Physics of Phase Space, Editors: Y.S. Kim and W.W. Zachary, SpringerVerlag, New York, 1987. [12] H. Weyl, The Theory of Groups and Quantum Mechanics, E.P. Dutton and Co., 1931. [13] E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), 749–759. [14] R.M. Wilcox, Exponential Operators and Parameter Differentiation in Quantum Physics, J. Math. Phys. 8 (1967), 962. [15] M.W. Wong, Weyl Transforms, Springer-Verlag, New York, 1998. [16] M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, New York, 1999. Leon Cohen Department of Physics and Astronomy City University of New York Hunter College 695 Park Ave. New York NY 10021, USA e-mail: [email protected]

Time-Frequency Characterization of Stochastic Differential Equations Lorenzo Galleani Abstract. Time-frequency analysis provides an effective description of the nonstationary random processes arising from random phenomena. These phenomena have typically a time-varying spectral content, which can be represented by using time-frequency distributions. The stochastic differential equation that models the nonstationary random process can be transformed in the time-frequency domain, and the properties of the resulting deterministic timefrequency equation clarify the nature of the nonstationary random process. We review the transformation to the time-frequency domain, and we prove the correctness of the obtained time-frequency equation. Mathematics Subject Classification (2000). Primary 60H10. Keywords. Time-frequency analysis, stochastic differential equations, random processes.

1. Introduction Signals in nature have typically a frequency content that changes with time. Human speech, for instance, is made by time-varying frequencies known as formants; the frequency of light intensifies in the red and orange regions of the visible spectrum during sunset; a variety of man-made and natural signals change their frequency due to the Doppler effect. To reveal the time-varying spectral content of these signals, it is necessary to use a representation that can tell what frequencies existed and when they existed. Time-frequency analysis is a field of signal processing which deals with the construction of such representations, known as time-frequency distributions [5, 12, 13]. Since signals are generated by systems, it is interesting to apply time-frequency analysis directly to systems, as is commonly done with the Fourier [20] This work was supported by the PRIN 2007 program.


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and Laplace transforms [21]. This coupling proves to be particularly effective for the study of random phenomena modeled by nonstationary random processes [10]. Consider the random process x(t), defined as the solution of the stochastic differential equation dn x(t) dx(t) + a0 x(t) = f (t). + · · · + a1 (1) dtn dt The coefficients a0 , . . . , an are deterministic constants, and the random process f (t) is a white Gaussian noise with zero mean and autocorrelation function an

Rf (t1 , t2 ) = E[f (t1 )f ∗ (t2 )] = N0 δ(t1 − t2 ).

(2)

The operator E is the expected value, δ(t) is the Dirac delta function, and the star sign indicates complex conjugation. The random system represented by Eq. (1) is used to model numerous stochastic phenomena, such as vibrations of structures [4], electric devices driven by noisy inputs [2], financial time series [18], and Brownian motion processes [11]. Equation (1) can be transformed in the time-frequency domain where a new equation is obtained [8, 9]. The advantages of studying the obtained time-frequency equation are manifold. First, we gain a deeper insight on the structure and properties of the nonstationary random process. Secondly, we expect to develop better techniques to design an equation that can modify the properties of a given random process to meet a set of given specifications. Furthermore, we expect to develop improved methods to identify the equation that models a given nonstationary random process. These applications are fundamental, since many results exist for stationary random processes, while only a few are available for the nonstationary case. In this chapter we review the basis of the transformation of a stochastic differential equation from the time domain to the time-frequency domain, and we prove that the solution to the equation in the time-frequency domain corresponds to the time-frequency distribution of the solution in the time domain, for the case of zero initial conditions. The chapter is organized as follows. First, in Section 2 we introduce the power spectrum for stationary random processes. Then, in Section 3 we describe the transformation of a stochastic differential equation from the time domain to the time-frequency domain. Finally, in Section 4 we prove the correctness of the solution obtained in the time-frequency domain.

2. Spectral analysis of stationary random processes Classical spectral analysis considers the case of a wide sense stationary (WSS) process, that is, of a random process x(t) with constant mean µx = E[x(t)]

(3)

and autocorrelation function that depends on the time difference only Rx (t1 , t2 ) = Rx (t1 − t2 ).

(4)

Time-Frequency Characterization

267

For WSS processes the autocorrelation function is often written with respect to the lag τ = t1 − t2 , that is Rx (τ ) = E[x(t)x∗ (t + τ )].

(5)

The power spectrum of a WSS process is defined as the Fourier transform of the autocorrelation function +∞ 1 Sx (ω) = √ Rx (τ )e−iτ ω dτ. (6) 2π −∞ For white Gaussian noise, whose autocorrelation function is given by Eq. (2), the power spectrum is flat N0 Sx (ω) = √ . (7) 2π The power spectrum for the class of random systems described by Eq. (1) can be obtained exactly. First, we rewrite Eq. (1) in polynomial notation Pn (D)x(t) = f (t),

(8)

Pn (D) = an Dn + · · · + a1 D + a0

(9)

where and d . dt Then, by defining the transfer function [1] D=

H(ω) =

1 Pn (iω)

(10)

(11)

we write the power spectrum of x(t) as [14] 2

Sx (ω) = |H(ω)| Sf (ω).

(12)

Since f (t) is a white Gaussian noise, we use Eq. (7) obtaining N0 2 Sx (ω) = √ |H(ω)| . 2π

(13)

This result holds under the hypothesis that the roots of the polynomial Pn (λ) have negative real part. If λk = αk + iβk (14) is the kth root of the equation Pn (λ) = 0,

(15)

it must hence be αk < 0,

k = 1, . . . , n.

(16)

This condition implies the strict stability of the random system represented by Eq. (8).

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3. From time to time-frequency Consider the initial condition problem Pn (D)x(t) = f (t),

t≥0

x(0) = x(1) (0) = · · · = x(n−1) (0) = 0, where (k)

x

dk x(t)

(0) = . dtk t=0

(17) (18)

(19)

Due to the transient, the random process x(t) solution to this problem is generally nonstationary [7]. If the stability condition of Eq. (16) holds, the process x(t) has a transient behavior at any finite time t and it reaches a steady state when t → ∞. To understand this concept, consider the Langevin equation [6, 22] dx(t) + βx(t) = f (t), dt

t ≥ 0,

(20)

which is a model for the velocity x(t) of a particle undergoing Brownian motion (we take the mass of the particle equal to one). The random process f (t) is a white Gaussian noise with autocorrelation function given by Eq. (2), and β is the friction. The variance of x(t) is given by ( ) N 0 2 1 − e−2βt , t ≥ 0. (21) σx2 (t) = E |x(t) − E[x(t)]| = 2β We see that the variance is a function of time, and consequently the random process x(t) is nonstationary. We also observe that the variance asymptotically reaches the value N0 lim σx2 (t) = . (22) t→∞ 2β This property holds if β > 0, a condition equivalent to the stability condition of Eq. (16). If x(t) is nonstationary, we expect also its frequency content to change with time. This fact can be proven by evaluating the time-frequency spectrum of x(t). Among the infinite possible time-frequency distributions, we use the Wigner spectrum, defined as [15–17] +∞ 1 W x (t, ω) = E[x∗ (t − τ /2)x(t + τ /2)]e−iτ ω dτ, (23) 2π −∞ which is the expected value of the Wigner distribution [23] +∞ 1 Wx (t, ω) = x∗ (t − τ /2)x(t + τ /2)e−iτ ω dτ. 2π −∞

(24)

The Wigner spectrum has several properties which make it an effective tool for the description of nonstationary random processes. It is directly connected to the

Time-Frequency Characterization autocorrelation function W x (t, ω) =

1 2π

+∞

−∞

Rx (t + τ /2, t − τ /2)e−iτ ω dτ.

269

(25)

The area of the Wigner spectrum at a given time t is the second-order moment of x(t) ( ) +∞ 2 E |x(t)| = W x (t, ω)dω. (26) −∞

If the process x(t) has zero mean, its variance equals the second-order moment, therefore +∞ σx2 (t) = W x (t, ω)dω. (27) −∞

For a WSS process, the Wigner spectrum equals the power spectrum, up to a constant 1 W x (t, ω) = √ Sx (ω). (28) 2π If f (t) is a white Gaussian noise and a(t) a deterministic function, the random process x(t) = a(t)f (t) (29) has a Wigner spectrum given by N0 2 |a(t)| . (30) 2π Moreover, it is possible to write a differential equation for the Wigner spectrum of the random process x(t) solution to Eq. (8). Such equation is given by [8,9] W x (t, ω) =

Pn∗ (A)Pn (B)W x (t, ω) = W f (t, ω),

(31)

where 1 ∂ 1 ∂ − iω, B= + iω (32) 2 ∂t 2 ∂t and the star sign indicates complex conjugation of the coefficients a0 , . . . , an . The forcing term of this equation is the Wigner spectrum of the forcing term f (t) of the time equation, Eq. (8), while the solution is the Wigner spectrum of the solution x(t). By replacing A and B from Eq. (32) we can rewrite Eq. (31) as A=

P2n (∂t )W x (t, ω) = W f (t, ω),

(33)

where ∂t =

∂ ∂t

(34)

and P2n (∂t ) = b2n ∂t2n + · · · + b1 ∂t + b0 .

(35)

The coefficients b0 , . . . , b2n depend on the coefficients a0 , . . . , an and on the frequency ω.

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When f (t) is a white Gaussian noise for t ≥ 0 and zero for t < 0, Eq. (33) becomes N0 P2n (∂t )W x (t, ω) = u(t), (36) 2π where u(t) is the Heaviside step function, defined as * 1, t ≥ 0 u(t) = . (37) 0, t < 0 The right-hand side of Eq. (36) is obtained by using Eq. (30). In [7] it is shown that, if the stability condition of Eq. (16) holds, the Wigner spectrum can be decomposed, at any time instant, into the sum of a stationary spectrum and a decaying spectrum W x (t, ω) = W S (t, ω) + W D (t, ω).

(38)

The stationary spectrum corresponds to the power spectrum, up to a constant 1 W S (t, ω) = √ u(t)Sx (ω), (39) 2π and the decaying spectrum is such that lim W D (t, ω) = 0.

(40)

t→∞

At t = 0 the stationary and decaying spectra are equal in magnitude and opposite in sign, therefore W x (0, ω) = 0. (41) Furthermore, by using Eqs. (38)–(40) we see that the Wigner spectrum asymptotically reaches the power spectrum, up to a constant 1 lim W x (t, ω) = √ Sx (ω). (42) t→∞ 2π

4. Equivalence of the solutions Consider the initial condition problem given in Eqs. (17)–(18). We now prove that the Wigner spectrum of the solution x(t) to this problem is identical to the solution of the initial condition problem P2n (∂t )W x (t, ω) = W f (t, ω) W x (0, ω) =

(1) W x (0, ω)

where (k) W x (0, ω)

= ··· =

(43)

(2n−1) Wx (0, ω)

∂ k W x (t, ω)

=

∂tk

.

= 0,

(44) (45)

t=0

To keep the notation simple, we use the calligraphic symbol W x (t, ω) for the Wigner spectrum arising from the time-frequency equation. We must therefore show that W x (t, ω) = W x (t, ω) (46)


271

for all t and ω values, where W x (t, ω) is the Wigner spectrum of x(t) in Eqs. (17)– (18). We start by proving that this identity holds when n = 1, then we extend the result to an arbitrary order n. 4.1. First-order equation Consider the first-order stochastic differential equation (D − λ)x(t) = f (t)

(47)

x(0) = 0,

(48)

with the initial condition where λ is a complex constant defined as λ = α + iβ. The solution of this initial condition problem is [19] t x(t) = h(t − t )f (t )dt .

(49)

(50)

0

If we take f (t) and h(t) to be zero for t < 0, then we can rewrite the solution x(t) by way of a convolution integral [24] +∞ x(t) = h(t − t )f (t )dt . (51) −∞

The function h(t) is the impulse response, or Green’s function, and it is the solution of the deterministic differential equation [1] (D − λ)h(t) = 0

(52)

h(0) = 1.

(53)

h(t) = u(t)e−λt .

(54)

with the initial condition We immediately obtain The Wigner spectrum of Eq. (51) is given by [3] +∞ W x (t, ω) = 2π Wh (t − t , ω)W f (t , ω)dt ,

(55)

−∞

where Wh (t, ω) is the Wigner distribution of the impulse response h(t). By replacing h(t) from Eq. (54) in Eq. (24) we have Wx (t, ω) =

1 sin [2t(ω + β)] u(t)e−2αt . π ω+β

(56)

Substituting this result in Eq. (55) yields +∞ sin [2(t − t )(ω + β)] 1 W x (t, ω) = 2π u(t − t )e−2α(t−t ) W f (t , ω)dt . (57) ω+β −∞ π

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We now evaluate the Wigner spectrum W x (t, ω) by solving Eqs. (17)–(18). By using Eqs. (31)–(32) we obtain the equation for the Wigner spectrum corresponding to Eq. (47) 2 2 3 ∂ ∂ 2 2 2 + 4α + 4α + 4(ω + β ) W x (t, ω) = 4W f (t, ω) (58) ∂t2 ∂t with the initial conditions (1)

W x (0, ω) = W x (0, ω) = 0. The impulse response g(t, ω) of Eq. (58) is the solution to [1] 2 2 3 ∂ ∂ 2 2 2 + 4α + 4α + 4(ω + β ) g(t, ω) = 0 ∂t2 ∂t g(0, ω) = 0,

g (1) (0, ω) = 1.

(59)

(60) (61)

We obtain g(t, ω) = From Eq. (51) we can write

1 sin [2t(ω + β)] u(t)e−2αt . 2 ω+β

W x (t, ω) = 4

+∞

−∞

g(t − t , ω)W f (t , ω)dt ,

which becomes, after replacing g(t, ω) from Eq. (62) +∞ sin [2(t − t )(ω + β)] 1 W x (t, ω) = 4 u(t − t )e−2α(t−t ) W f (t , ω)dt . 2 ω + β −∞

(62)

(63)

(64)

This result is identical to Eq. (57), and it hence proves the identity of Eq. (46) for the case n = 1. 4.2. Arbitrary order equation We first factor Eq. (17), obtaining an (D − λ1 ) · · · (D − λn )x(t) = f (t),

(65)

and then we rewrite it as a series of first-order equations an (D − λ1 )x1 (t) = f (t) (D − λ2 )x2 (t) = x1 (t)

(66) (67)

.. . (D − λn )xn (t) = xn−1 (t),

(68)

where xn (t) = x(t). (69) Since x(0) = xn (0) = 0, then xn−1 (0) = 0 in Eq. (68). By repeating this argument we obtain the initial conditions for Eqs. (66)–(68) x1 (0) = x2 (0) = · · · = xn (0) = 0.

(70)


273

The kth initial condition problem is therefore given by (D − λk )xk (t) = xk−1 (t) xk (0) = 0.

(71) (72)

When k = 1 the right-hand side is f (t). The corresponding equation in the Wigner spectrum domain is, from Eq. (58) 2 2 3 ∂ ∂ 2 2 2 + 4α + 4α + 4(ω + β ) W x (t, ω) = 4W f (t, ω) (73) k k k ∂t2 ∂t with the initial conditions (1)

W x (0, ω) = W x (0, ω) = 0.

(74)

The result of Section 4.1 shows that W xk (t, ω) = W xk (t, ω).

(75)

This identity holds for an arbitrary k, and hence also for k = n, which means, by using Eq. (69), that the identity of Eq. (46) holds for an arbitrary order n.

5. Conclusions We have shown how to transform a stochastic differential equation from the time domain to the time-frequency domain of the Wigner spectrum. In the timefrequency domain, the stochastic equation becomes a deterministic differential equation. By studying the properties of such equation, it is possible to characterize the nonstationary random process solution of the original stochastic equation. Furthermore, we have shown that the solution of the equation for the Wigner spectrum corresponds to the Wigner spectrum of the solution in the time domain, for the case of zero initial conditions. The extension of this result to the case of arbitrary initial conditions, as well as to stochastic partial differential equations is currently under investigation.

References [1] G. Birkhoff and G.C. Rota, Ordinary Differential Equations, Wiley, 1989. [2] L.O. Chua, C.A. Desoer and E.S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, 1987. [3] T.A.C.M. Claasen and W.F.G. Mecklenbrauker, The Wigner distribution – A tool for time-frequency signal analysis. Part I: Continuous time signals, Philips Journal of Research 35 (3) (1980), 217–250. [4] R.W. Clough, Dynamics of Structures, McGraw-Hill, 1993. [5] L. Cohen, Time-Frequency Analysis, Prentice Hall, 1995. [6] L. Cohen, The history of noise, IEEE Signal Processing Magazine 22 (6) (2005), 20–45.

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[7] L. Galleani, Decomposition of the instantaneous spectrum of a random system, Signal Processing 90 (3) (2010), 860–865. doi:10.1016/j.sigpro.2009.09.003. [8] L. Galleani and L. Cohen, The Wigner distribution for classical systems, Physics Letters A 302 (2002), 149–155. [9] L. Galleani and L. Cohen, Direct time-frequency characterization of linear systems governed by differential equations, IEEE Signal Processing Letters 11 (2004), 721– 724. [10] L. Galleani and L. Cohen, Nonstationary stochastic differential equations, in Advances in Nonlinear Signal and Image Processing, Editors: S. Marshall and G. Sicuranza, Hindawi, 2006. [11] C. W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag, 1983. [12] P. Loughlin, Special issue on applications of time-frequency Analysis, Editor: P. Loughlin, Proc. of the IEEE 84 (9), 1996. [13] P. Loughlin, J. Pitton and L. Atlas, Bilinear time-frequency representations: New insights and properties, IEEE Trans. Sig. Process. 41 (2) (1993), 750–767. [14] A. Papoulis and S.U. Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill, 2002. [15] W.D. Mark, Spectral analysis of the convolution and filtering of nonstationary stochastic processes,J. Sound Vib. 11 (1970), 19–63. [16] W.D. Mark, Power spectrum representation for nonstationary random vibration, in Random Vibration – Status and Recent Developments, Editor: D.H. Ielishakoff, Elsevier, 1986. [17] W. Martin and P. Flandrin, Wigner-Ville spectral analysis of nonstationary processes, in IEEE Trans. Acoustics, Speech, Signal Processing, ASSP-33 (6) (1985), 1461–1470. [18] T.C. Mills, The Econometric Modelling of Financial Time Series, Cambridge University Press, 1999. [19] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 2007. [20] A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill, 1962. [21] C.J. Savant, Fundamentals of the Laplace Transformation, McGraw-Hill, 1962. [22] M.C. Wang and G.E. Uhlenbeck, On the theory of the Brownian motion II, Rev. Modern Phys.17(2-3) (1945), 323–342. [23] E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), 749–759. [24] A.H. Zemanian, Distribution Theory and Transform Analysis, Dover, 1987. Lorenzo Galleani Dipartimento di Elettronica Politecnico di Torino Corso Duca degli Abruzzi 24 I-10129 Torino, Italy e-mail: [email protected]

Wigner Representations Associated with Linear Transformations of the Time-Frequency Plane Paolo Boggiatto, Evanthia Carypis and Alessandro Oliaro Abstract. In this paper we define a variation of the Wigner form depending on a linear transformation of the time-frequency plane and study the corresponding properties. This construction yields a natural geometric interpretation of the so-called “ghost frequencies” showed, among others, by the Wigner quadratic representation. In particular we prove that the representations in this new class which satisfy the support properties are just the so-called τ -Wigner forms. On the other hand for these new forms an “essential” re-adjustement of the supports is showed to be possible, whereas their features of moving almost arbitrarily the ghost frequencies is used to define representations with no interferences at all for a certain class of signals. Mathematics Subject Classification (2000). Primary 42B10, 47A07. Keywords. Time-Frequency representations, Wigner sesquilinear and quadratic form, interferences.

1. Introduction In this paper we study time-frequency representations that are generalizations of the classical Wigner transform. As is well known the Fourier transform contains all the information about a generic signal, but some of these, e.g., the time at which frequencies appear in the signal, are hidden in the complex phase. As a consequence one of the main aims of time-frequency analysis in the last 50 years has been to define suitable “2-variable” modifications of the Fourier transform in such a way that information about both time and frequency content of the signal are made explicit. They are functions or distributions Q(f )(x, ω) depending on the time x ∈ Rd and the frequency variables ω ∈ Rd ; they depend in a quadratic way on the signal f , and they have a physical interpretation as energy distribution of the signal in the time-frequency space Rdx × Rdω . Time-frequency representations L. Rodino et al. (eds.), Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications 213, DOI 10.1007/978-3-0348-0049-5_17, © Springer Basel AG 2011

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are widely studied in the literature, from both theoretical and applicative points of view; we refer for example to [5], [6], [11], [12], [13], [17], [20], [3], [14]. These forms moreover can be studied in connection with the theory of pseudodifferential operators, see for instance [19], [1], [15], [18]. One of the main problems in the analysis of signals is that in their representation very often some sort of artifacts (also known as “interferences” or “ghost frequencies”) appear in regions of the time-frequency plane where the signal contains in reality no energy. This is for example the case of the Wigner representation which is nevertheless one of the basic tools used in the context of signal theory. The desire to eliminate or at least reduce this drawback has led in literature to a variety of methods and alternative definitions of many different representations, each one with good features and disadvantages. In fact one of the main obstacles of time-frequency analysis is linked to the classical uncertainty principle, having as a consequence that the energy distribution of the frequencies of a signal cannot be concentrated in too small sets in the time-frequency plane. Then, due to the uncertainty principle in its various formulations, it is not possible to construct the best representation, since whenever one improves certain features other aspects become worse. This can be formalized in many ways, see for example [10], [11], [7]. One easy formulation is that, given a signal f (t), say in L2 (Rd ), a quadratic time-frequency representation Q(f )(x, ω) defined on the time-frequency plane (x, ω) ∈ R2d cannot satisfy at the same time the following properties (cf. [12]): – Positivity: Q(f )(x, ω) ≥ 0 for all (x, ω) ∈ R2d ; – Support property: Πx supp Q(f ) ⊂ C(supp f ) and Πω supp Q(f ) ⊂ C(supp fˆ), where Πx and Πω are the orthogonal projection on the x and ω space respectively, and C(E) indicates the convex hull of E ⊂ Rd . – Marginal distribution conditions: Q(f )(x, ω) dω = |f (x)|2 and Q(f )(x, ω) dx = |fˆ(ω)|2 . Rd

Rd

These conditions are very natural, having in mind that the values of the timefrequency representation Q(f )(x, ω) intend “ideally” to be a measure of the energy of the signal at time x and frequency ω, and the quantities |f (x)|2 and |fˆ(ω)|2 are the energy distributions of the signal with respect to time and frequency separately. We focus in this paper on time-frequency representations of Wigner type. Besides the classical Wigner distribution t t Wig(f ) = e−2πitω f x + f x− dt 2 2 other representations of Wigner type have been considered in the literature, such as, e.g., the τ -Wigner form, defined as Wigτ (f ) = e−2πitω f (x + τ t) f (x − (1 − τ )t) dt for τ ∈ [0, 1].

Wigner Representations Associated with Linear Transformations

277

The τ -Wigner representations have been studied in [12], and in [1] in connection with τ -Weyl pseudodifferential operators. Obviously the classical Wigner form is a particular case for τ = 12 . All these Wigner type representations satisfy the support property and the marginal distribution conditions, but not the positivity; Hudson Theorem in particular states that the classical Wigner is positive only on Gaussian type signals, cf. for example [9], [16]. Moreover, all the Wigτ present strong interferences, see, e.g., [6], [1]. We give in this paper a simple geometric interpretation of the presence of this kind of artifacts and we show that suitable modifications (more general than the Wigτ ) of the Wigner representations in dependance on a linear transformation can “move” the artifacts almost arbitrarily in the time-frequency plane (see Section 3). More precisely, in Section 2, we shall define WigU (f )(x, ω) = e−2πiωt f (ax + bt) f (cx + dt) dt, depending on the transformation U : (x, t) ∈ R2 → (ax + bt, cx + dt) ∈ R2 , where a, b, c, d ∈ R and ad−bc = 0. We show on one hand that the WigU which satisfy the support property with respect both to time and frequency are substantially of the kind of Wigτ , apart from a multiplication by a complex exponential; on the other hand, the freedom in the choice of the parameters in the WigU allows us to obtain new representations that satisfy the support properties with a certain degree of approximation but have the feature that the “ghosts” can be moved in different zones of the time-frequency plane. Moreover, the geometrical interpretation of the presence of the artifacts is used in Section 3 to define methods to obtain representations which show no interferences on some particular classes of signals.

2. The representation WigU and its properties One drawback of the classical Wigner distribution t t −2πiω·t Wig(f, g)(x, ω) = e f x+ g x− dt 2 2 Rd

(1)

in its applications to signal analysis is the question of the interferences, called also ghost frequencies. For example, if we consider a signal with only two frequencies ω1 and ω2 appearing in two disjoint intervals I1 , I2 , it is easy to verify that the graphical representation shows some frequencies which in reality do not exist. Geometrically these interferences are the results of the intersection of the lines x + 2t = k, x − 2t = k (where k indicates the end points of the intervals I1 , I2 ) in the plane (x, t). The situation is illustrated in Figure 1 of Section 3. We observe the same phenomenon in the case of the τ -Wigner distribution, with the difference that in this case the interferences split in the time-frequency plane, moving along the diagonal of the rectangle which is obtained considering the real frequencies (see [1]). The aim of our study is the investigation of a method to overcome the problem of the interferences. To this purpose we introduce a generalization of the Wigner distribution. Starting from the classical Wigner, noting that (t, x) →

278


x + 2t , x − 2t represents a linear transformation of the plane, we focus our study on a more generical transformation U : (t, x) ∈ R2 → (u, v) ∈ R2 defined as follows * u = ax + bt v = cx + dt

where a, b, c, d ∈ R and ad − bc = 0. Replacing the usual change of variables x + 2t , x − 2t with the transformation U we can introduce a generalization of the classical Wigner distribution, that we indicate WigU (limiting our study to the unidimensional case d = 1): WigU (f, g)(x, ω) = e−2πiωt f (ax + bt) g (cx + dt)dt. (2) R

Remark 1.

1 0 (a) For U = , WigU (f, g) coincides with the “Short-time Fourier trans1 −1 form” (STFT) Vg (f )(x, ω) (cf. [11]). (b) In the case, that we do not consider here, where the transformation U is allowed to depend on ω, we have that the Stockwell transform (cf. [4], [8]) is of type (2) with 1 0 U= , ω −ω 1

apart from a scale factor |ω| 2 . Before analyzing this new expression in connection with the problem of the ghost frequencies, we consider some properties required to an ideal time-frequency representation. We begin by showing that there exists a relationship between WigU and the Short-time Fourier transform (called also Gabor transform), similarly to the case of the classical Wigner distribution. Proposition 2. For all f, g ∈ L2 (R) we have WigU (f, g)(x, ω) = with b, d = 0 and g d (s) = g b

d s . b

1 2πiω a x b V e gd f b b

a−

bc d

x,

ω b

,

Proof. Using the change of variables s = ax + bt the proof follows easily.

(3)

As mentioned above, putting a = c = 1, b = 1/2 and d = −1/2, it is immediate to verify that (3) coincides with the formula Wig(f, g)(x, ω) = 2e4πixω Vg˜ f (2x, 2ω)

(4)

with g˜(x) = g(−x), that connects the classical Wigner representation to the STFT. Thanks to Proposition 2 we can use the continuity of the Gabor transform on Lp spaces to check the continuity of WigU . More precisely, proceeding as in [2], [14], we get the following proposition.


279

Proposition 3. Fix p, q such that q 2 and q p q. The application WigU : Lp (R) × Lp (R) → Lq (R2 ) is continuous. The next property shows the action of WigU on the Fourier transform of the signals. Proposition 4. Let f, g ∈ L2 (R), b = 0, d = 0, ad − bc = 0. We have

b 2πiω ( c + a )x d ω ad − bc ˆ

d b 2 WigU (f , gˆ)(x, ω) = e WigU f, g b − , bx . d ad − bc b d d2 (5) Proof. From Proposition 2 we have . / a 1 WigU (fˆ, gˆ)(x, ω) = e2πiω b x fˆ, µ ωb τ(a− bc )x gˆ d b d b . / sgn(b) 2πiω a x b = e f, τ− ωb µ(a− bc )x g b . d d |d| Observing that bc ω τ− ωb µ(a− bc )x g b (s) = e2πi(a− d ) b x µ(a− bc )x τ− ωb g b (s), d d d d a

dividing by 1b e2πiω b x , we have that

/ sgn(b) 2πiω c x . d WigU (fˆ, gˆ)(x, ω) = e f, µ(a− bc )x τ− ωb g b d d |d| : ; sgn(b) 2πiω c x d = e f, µ 1 [ ad−bc bx] τ ad−bc [− d ω ] g b2 d b d d ad−bc b |d| d2 b d ω ax sgn(b) 2πiω c 1 2πi( ad−bc bx − 2πiω )( ) d ad−bc b be d b e e = |d| b : ; × f, µ 1 [ ad−bc bx] τ ad−bc [− d ω ] g b2 d b d d ad−bc b d2 b

b 2πiω ( c + a )x d ω ad − bc d b =

e − , bx . WigU f, g b2 d ad − bc b d d2

Before continuing the investigation of the properties of WigU , we consider the last expression in the proof of Proposition 4 d ω ad − bc WigU f, g b2 − , bx . (6) ad − bc b d d2 If we perform the change of variables s = (ad−bc)bt , it is possible to rewrite (6) as d follows d ω ad − bc WigU f, g b2 − , bx ad − bc b d d2 d 1 d b −2πix dc ω −2πixt = e e f − ω+ t g t dt. (ad − bc)b b ad − bc ad − bc

280


In this last expression we recognize WigU , where U is given by the new triangular matrix: 1 d − b ad−bc U = . b 0 ad−bc Then from (6) it follows c d ω ad − bc de−2πix d ω WigU f, g b2 − , bx = WigU (f, g)(ω, x). (7) ad − bc b d (ad − bc)b d2 This formula permits to express a generic representation WigU as WigU with triangular matrix U , and shows another way of generalizing formula (4). Considering now Fourier transforms, from Proposition 4, we have a

sgn(b) e2πiω b x WigU (fˆ, gˆ)(x, ω) = WigU (f, g)(ω, x). sgn(d) ad − bc Exchanging the role of f, g and fˆ, gˆ, we finally have

(8)

a

sgn(b) e2πiω b x WigU (f, g)(x, ω) = WigU˜ (fˆ, gˆ)(ω, x), sgn(d) ad − bc with

˜= U

1 b

0

d − ad−bc b − ad−bc

(9)

.

The relation (9) is important to us as it will be useful in the following, in particular it will permit to verify for which matrices U WigU belongs to the Cohen class and satisfies the support property. We start by checking the properties on the support both in x and ω. (a) First we consider the time-variable x. If WigU is different from zero at one point (x, ω), we have, by (2), that there must exist t ∈ R such that f (ax + bt) = 0,

f (cx + dt) = 0,

that is ax + bt ∈ supp(f ), cx + dt ∈ supp(f ). In order that the support property with respect to x is satisfied we want that x belongs to the convex hull of the support of f . Expressing x as a linear combination of ax + bt and cx + dt, we require that there exist λ, µ ∈ R such that x = λ(ax + bt) + µ(cx + dt) ∈ C supp(f ),

(10)

with conditions λ+µ= 1

and

λ, µ 0.

Then we rewrite (10) as x = (λa + µc)x + (λb + µd)t and so it must be

  λa + µc = 1 λb + µd = 0  λ + µ = 1.

(11)


281

It is not difficult to check that this system has solutions of type (11) under one of the following conditions: 1. a 1, c > 1 and b = a−1 d; c−1 a−1 2. a 1, c < 1 and b = c−1 d; 3. a = 1, c = 1 and bd 0. (b) We consider now the case of the support property with respect to the ω-variable. Using relation (9) we can apply what we have just proved regarding support property with respect to the first variable (part (a)), considering the ma˜ instead of U . From the previous conditions 1 and 2, we have trix U 1 1. b ˜ Substituting in b = a−1 c−1 d the entries of the matrix U , we obtain 1 − 1b d b = . ad − bc 1 − 0 ad − bc We are then lead to the following additional conditions for the support property with respect to ω: 4. d = b − 1; 5. 0 < b 1. Summarizing we have obtained the following property. Proposition 5. The time-frequency representation WigU on L2 (R) satisfies the support property if (i) 0 < b 1, d = b − 1 and one of the following holds: (ii) a 1, c > 1 (or equivalently a 1, c < 1) and b = a−1 c−1 d; (ii ) a = c = 1 and bd 0. Remark 6. Putting b = τ in the condition (i) of the previous proposition, we obtain d = τ − 1 and the parameter τ varies in the interval (0, 1], exactly as it happens in the case of the Wigτ . The only difference here is that a and c can vary in all R. Furthermore we observe that, using (ii) and the substitution ax + τ t = x + τ s (t = 1−a τ x + s and dt = ds), it follows that: WigU (f, g)(x, ω) = e−2πiω

1−a τ x

Wigτ (f, g)(x, ω).

Therefore we can conclude that the WigU representations which satisfy support property with respect to both x and ω, are in fact Wigτ , apart from the multiplication by complex exponentials. However, even in the case where the transformation U does not satisfy the conditions of Proposition 5, we can remark that the displacement of the support is independent of the signal f . For this reason one can define a suitable fixed rescaling of WigU that will substantially re-localize the supports in their original

282


position in time and frequency. Precisely, considering a signal supported in the time-interval [k, k + α], with α > 0, the linear transformation U in the definition of WigU in (2) moves the point x = k to the point in the (x, t) plane given by the intersection of the two straight lines ax + bt = k and cx + dt = k, i.e., to a point d−b d−b whose x-coordinate is x = ad−bc k. Then the expression WigU (f ) ad−bc k, ω will re-localize the support of the signal with respect to the time-variable. Using the expression of WigU (fˆ, gˆ), obtained in (8), we can proceed in the same way for a signal whose Fourier transform is supported in the interval [h, h + β], with β > 0. By using the matrix U , the corresponding transformation of the ω-variable is given by ω = (b − d)h and the new expression d−b WigU (f ) k, (b − d)h (12) ad − bc re-localizes the supports both in time and frequency. Observe that the readjustement of the supports is not “perfect” in the case that the slopes of the lines ax + bt = k and cx + dt = k are both either positive or negative, and it can be easily checked that in this case (12) slightly enlarges the supports. We want to analyze now the conditions that ensure the covariance property for the WigU . Proposition 7. If

U =

1 1

τ τ −1

then Wig U is not covariant. As a consequence, in this case WigU does not belong to the Cohen class. Proof. At first we consider the case of a general translation τα , with α ∈ R, applied to f and g. It follows WigU (τα f, τα g)(x, ω) = e−2πiωt f (ax + bt − α)g(cx + dt − α)dt R (13) α α = e−2πiωt f a x − + bt g c x − + dt dt. a c R The covariance property requires that WigU (τα f, τα g)(x, ω) = WigU (f, g)(x − α, ω). Let us fix f, g ∈ S(R). Taking the inverse Fourier transform with respect to ω we then have from (13) that the following equality must hold for every f, g ∈ S(R) and x, t, α ∈ R: α α f α x− + bt g c x − + dt = f (a(x − α) + bt) g (c(x − α) + dt). a c The only possibility to satisfy the previous equality is then that a = c = 1, as we 2 can easily see by considering for example f (s) = g(s) = e−πs . Concerning the


283

case of a general modulation µα , the conclusion is analogous. In fact WigU (µα f, µα g)(x, ω) = e−2πiωt e2πiα(ax+bt) f (ax + bt)e−2πiα(cx+dt) g(cx + dt)dt =e

R −2πiα(c−a)x

WigU (f, g)(x, ω − (b − d)α)

and WigU satisfies covariance with respect to modulation if and only if b − d = 1. Therefore, the covariance property is satisfied only if WigU coincides with Wigτ . The next proposition shows when WigU is real (the proof is straightforward). Proposition 8. For f, g ∈ L2 (R) we have WigU (g, f )(x, ω) = WigV (f, g)(x, ω),

with V =

c a

−d −b

.

Remark 9. From the last proposition it follows that WigU (f ) = WigV (f ) and WigU is real if U = V . The conditions for the reality are then a=c

and

b = −d.

3. A geometrical interpretation of interferences and applications to signal coding We consider now the problem related to the interferences, to which we hinted in the previous sections. In particular the method that we are going to describe concerns a specific class of signals, precisely signals consisting of pure frequencies appearing in different time slots. These are particularly interesting in applications, as in principle every information can be “coded” by signals of this type by assigning a pure frequency to each of the symbols of a specific alphabet. As we have already observed, in the graphical representation of the Wigner transform on the timefrequency plane, one can notice false frequencies in the “middle” of any two real frequencies of the signal. One can see the same phenomenon using a τ -Wigner distribution, with the difference that the ghost frequencies are shifted with every τ , whereas the real frequencies appear for all τ at the right place, thanks to the support property (see [1]). Clearly the issue of the ghost frequencies represents a considerable problem in the physical interpretation of the Wigner distribution. Therefore we want to propose here a method which permits to overcome this question at least in the considered class of signals. The technique that we are going to show is applicable to a generical distribution WigU (f, g)(x, ω) = e−2πiω·t f (x + bt)g(x − bt)dt, (14) R

284


where, in the matrix U , a = c = 1 and d = −b = 0. Let f be a signal with two frequencies appearing in two disjoint bounded intervals [k, k + α] and [h, h + β], with α, β > 0, where we assume for the moment that α β and we suppose k + α < h. We consider the straight lines passing through the four extreme points of the intervals in the plane (x, t):  x ± bt = k    x ± bt = k + α x ± bt = h    x ± bt = h + β. It is immediate to verify that they intersect forming four diamonds D1 , D2 , D3 and D4 (see Figure 1). We then have that the quantity f (x + bt)f (x − bt) is supported

Figure 1 in the set D1 ∪ D2 ∪ D3 ∪ D4 . The two diamonds D1 and D3 correspond to the true frequencies, in fact they intersect the x-axis at the intervals [k, k + α] and [h, h + β], where the signal exists; D2 and D4 correspond on the contrary to the interferences. The idea of our method is to consider an horizontal strip S = {(x, t) : t ∈ [−M, M ]} and b > 0 such that the two diamonds D2 and D4 corresponding to the ghost frequencies lay outside S. In this way, limiting the integration in (14) to the interval [−M, M ] we totally remove the interferences from the representation of the signal. This is achieved if tB ≥ M , where tB is the


285

t-coordinate of the point B. On the other hand we have no loss of information about the true frequencies if the diamonds D1 and D3 are entirely inside the strip S and this is achieved if tA ≤ M , where tA is the t-coordinate of the point A. However, one can remark that with the change of variable bt = t2 in (14) the same effect is obtained considering the classical Wigner on a different strip. This leads us to introduce the transformation N t t WigN (f )(x, ω) := e−2πiωt f x + f x− dt (15) 2 2 −N which we apply to the previous signal f . Since tA = α and tB = h − (k + α), we have that there exists N such that tA N tB if and only if 2α h − k. In the general case (where not necessarily α ≥ β) one can easily see that this condition is generalized to h − k max{2α, β + α}. This machinery can be applied to signals with an arbitrary number of frequencies; we have found therefore the following condition in order to represent signals without ghost frequencies: “The silence time between any two consecutive frequencies of the signal has to be longer or equal to the transmission time of each of them”. We show now some graphical examples as application of our method. Example 1. Let f (s) = e4πis χ[3,6] (s) + e6πis χ[9,12] (s) be a signal with two frequencies ω1 = 2, ω2 = 3. The Wigner transform shows a ghost frequency in the middle of the two real frequencies (Figure 2).

Figure 2. Wig(f )(x, ω) The same happens in the case of the τ -Wigner distribution (let us fix τ = 0.8). As one can see in Figure 3, the ghost frequencies are split in the time-frequency plane. Figure 4 shows the representation of the signal without interferences obtained through an integration on a bounded interval [−N, N ]. It is possible in fact to apply the method with N = 3. Example 2. Let f (s) = e4πis χ[−8,−7] (s) + e10πis χ[−5,−3] (s) + e8πis χ[−1,0] (s) + e2πis χ[2,4] (s) + e8πis χ[6,8] (s) be a signal with five frequencies ω1 = 2, ω2 = 5,

286


Figure 3. Wig0.8 (f )(x, ω)

Figure 4. Wig N (f )(x, ω) ω3 = 4, ω4 = 1, ω5 = 4 appearing in intervals with different lengths. Figure 5 shows what happens if one applies the Wigner distribution and, as before, one can see the interferences. If we apply our method we overcome this issue and the graphical representation shows only the true frequencies (see Figure 6).

Figure 5. Wig(f )(x, ω)


287

Figure 6. Wig N (f )(x, ω) This last example shows that the ghosts in the Wigner transform on a signal with several frequencies can yield a representation which is not very clearly readable, whereas our method eliminates all artifacts without loss of information. Finally it is also remarkable that (15) is even simpler to use in computations than the usual Wigner transform. The price for this is of course the limited class of signals for which (15) gives the good results mentioned above.

References [1] P. Boggiatto, G. De Donno and A. Oliaro, Time-frequency representations of Wigner type and pseudo-differential operators, Trans. Amer. Math. Soc., 362 (2010), n. 9, 4955–4981. [2] P. Boggiatto, G. De Donno and A. Oliaro, Weyl quantization of Lebesgue spaces, Math. Nachr. 282 (12) (2009), 1656–1663. [3] P. Boggiatto, G. De Donno and A. Oliaro, A class of quadratic time-frequency representations based on the short-time Fourier transform, in Modern Trends in Pseudo-Differential Operators, Operator Theory: Advances and Applications 172, Birkh¨ auser, 2006, 235–249. [4] P. Boggiatto, C. Fern´ andez and A. Galbis, A Group representation related to the Stockwell transform, Indiana Univ. Math. J. 58 (5) (2009), 2277–2296. [5] L. Cohen, Time-frequency distributions – A review, Proc. IEEE 77 (7) (1989), 941– 981. [6] L. Cohen, Time-Frequency Analysis, Prentice Hall, New Jersey, 1995. [7] D.L. Donoho and P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (3) (1989), 906–931. [8] J. Du, M.W. Wong and H. Zhu, Continuous and discrete inversion formulas for the Stockwell transform, Integral Transforms Spec. Funct. 18 (7-8) (2007), 537–543. [9] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989. [10] G.B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (3) (1989), 207–238. [11] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston, 2001.

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[12] A.J.E.M. Janssen, Positivity and spread of bilinear time-frequency distributions, in Theory and Applications in Signal Processing, Editors: W. Mecklenbrauker and F. Hlawatsch, Elsevier Science, 1997, 1–58. [13] A.J.E.M. Janssen, Proof of a conjecture on the supports of Wigner distributions, J. Fourier Anal. Appl. 4 (6) (1998), 723–726. [14] E.H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys. 31 (3) (1990), 594–599. [15] A. Mohammed and M.W. Wong, Rihaczek transform and pseudo-differential operators, in Pseudo-Differential Operators: Partial Differential Equations and TimeFrequency Analysis, Fields Institute Communications 52, American Mathematical Society, 2007, 375–382. [16] J. Toft, Hudson’s theorem and rank one operators in Weyl calculus, in PseudoDifferential Operators and Related Topics, Operator Theory: Advances and Applications 164, Birkh¨ auser, Basel, 2006, 153–159. [17] M.W. Wong, Weyl Transforms, Springer-Verlag, 1998. [18] M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999. [19] M.W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, Basel, 2002. [20] M.W. Wong, Symmetry-breaking for Wigner transforms and Lp -boundedness of Weyl Transforms, in Advances in Pseudo-differential Operators, Operator Theory: Advances and Applications 155, Birkh¨ auser, 2004, 107–116. Paolo Boggiatto, Evanthia Carypis and Alessandro Oliaro Department of Mathematics University of Torino Via Carlo Alberto 10 I-10123 Torino, Italy e-mail: [email protected] [email protected] [email protected]

Some Remarks on Localization Operators Elena Cordero and Fabio Nicola Abstract. In this paper we present a new technique to prove optimal boundedness results for localization operators acting on Wiener amalgam spaces and, in particular, on Lp spaces, first obtained in [7] by another method. This new proof does not require the Schur-type Test introduced and employed there. We also briefly discuss the optimality of these results. Mathematics Subject Classification (2000). Primary 35S05, 46E30. Keywords. Short-time Fourier transform, modulation spaces, Wiener amalgam spaces, Lebesgue spaces, localization operators.

1. Introduction The name “localization operator” first appeared in 1988, when Daubechies [13] used these operators as a mathematical tool to localize a signal on the timefrequency plane. Since their first appearance, they have been extensively studied as an important mathematical tool in signal analysis and other applications (see [5, 6, 10, 27, 35] and references therein). But localization operators with Gaussian windows were already known in physics: they were introduced as a quantization rule by Berezin [1] in 1971 (the so-called Wick operators). We also recall their employment as approximation of pseudodifferential operators (wave packets) [12, 22]. Localization operators are also called Toeplitz operators (see, e.g., [14]) or short-time Fourier transform multipliers [20, 32]. We shall define them by means of a time-frequency representation: the short-time Fourier transform (STFT). Precisely, consider the linear operators of translation and modulation (so-called time-frequency shifts) given by Tx f (t) = f (t − x)

and Mω f (t) = e2πiωt f (t) .

(1.1)

The second author was partially supported by the Progetto MIUR Cofinanziato 2007 “Analisi Armonica”.


289

290

E. Cordero and F. Nicola

For a non-zero window function g in L2 (Rd ), the short-time Fourier transform (STFT) of a signal f ∈ L2 (Rd ) with respect to the window g is given by Vg f (x, ω) = f, Mω Tx g = f (t) g(t − x) e−2πiωt dt . (1.2) Rd

1 ,ϕ2 Definition 1.1. The localization operator Aϕ with symbol a ∈ S(R2d ) and wina d dows ϕ1 , ϕ2 ∈ S(R ) is defined to be 1 ,ϕ2 Aϕ f (t) = a(x, ω)Vϕ1 f (x, ω)Mω Tx ϕ2 (t) dxdω , f ∈ L2 (Rd ). (1.3) a

R2d

The definition extends to more general classes of symbols a, windows ϕ1 , ϕ2 , and functions f in natural way. If a = χΩ for some compact set Ω ⊆ R2d and 1 ,ϕ2 ϕ1 = ϕ2 , then Aϕ is interpreted as the part of f that “lives on the set Ω” in a 1 ,ϕ2 the time-frequency plane. This is why Aϕ is called a localization operator. If a 2d d a ∈ S (R ) and ϕ1 , ϕ2 ∈ S(R ), then (1.3) is a well-defined continuous operator 2 1 ,ϕ2 from S(Rd ) to S (Rd ). If ϕ1 (t) = ϕ2 (t) = e−πt , then Aa = Aϕ is the classical a ϕ1 ,ϕ2 anti-Wick operator and the mapping a → Aa is interpreted as a quantization rule [1, 28, 35]. In [5, 6] localization operators are viewed as a multilinear mapping 1 ,ϕ2 (a, ϕ1 , ϕ2 ) → Aϕ , a

(1.4)

acting on products of symbol and windows spaces. The dependence of the local1 ,ϕ2 ization operator Aϕ on all three parameters has been studied there in different a functional frameworks. The results in [5] enlarge the ones in the literature, concerning Lp spaces [35], potential and Sobolev spaces [2], modulation spaces [20, 29, 30]. Other boundedness results for STFT multipliers on Lp, modulation, and Wiener amalgam spaces are contained in [32]. On the footprints of [5], the study of localization operators can be carried to Gelfand-Shilov spaces and spaces of ultra-distributions [11]. Finally, the results in 1 ,ϕ2 [10] extend those of [5, 6], interpreting the definition of Aϕ in a weak sense, a that is 1 ,ϕ2 Aϕ f, g = aVϕ1 f, Vϕ2 g = a, Vϕ1 f Vϕ2 g, a

f, g ∈ S(Rd ) .

(1.5)

1 ,ϕ2 Aϕ a

In [7] we study the action of the operators on the Lebesgue spaces Lp p q and on Wiener amalgam spaces W (L , L ), when a is also assumed to belong to these spaces. We recall that a measurable function f belongs to W (Lp , Lq ) if the following norm  1 pq q

f W (Lp ,Lq ) =  |f (x)Tn χQ (x)|p  , (1.6) n∈Zd

Rd

where Q = [0, 1)d (with the usual adjustments if p = ∞ or q = ∞) is finite (see [25] and Section 2 below). In particular, W (Lp , Lp) = Lp . For example, one wonders what is the full range of exponents (q, r) such that for every a ∈ Lq the operator 1 ,ϕ2 Aϕ turns out to be bounded on Lr , for all windows ϕ1 , ϕ2 in reasonable classes. a

Some Remarks on Localization Operators

291

Partial results in this connection have been obtained in [3, 4, 34]. In [7] we have given a complete answer to this question and, more generally, to a similar question in the framework of Wiener amalgam spaces. There the approach was different from the previous ones. In particular, the techniques employed did not use Weyl operators results as in [5, 11]. Indeed, the 1 ,ϕ2 operator Aϕ was rewritten as an integral operator. Next, a Schur-type test for a the boundedness of integral operators on Wiener amalgam spaces was introduced and used to obtain boundedness results either on Lp or on W (Lp , Lq ). Here we obtain the same boundedness results without using the Schur-type test. Indeed, we show that the corresponding integral kernel is dominated by a convolution kernel in the Lebesgue space L1 (R2d ). Then, the convolution properties of Wiener amalgam spaces and Young’s Inequality give the desired result (see Theorem 3.3). As a particular case, one gets issues for Lebesgue spaces, which can be simplified as follows 1 ,ϕ2 Theorem 1.2. Let a ∈ Lq (R2d ), ϕ1 , ϕ2 ∈ S(Rd ). Then, Aϕ is bounded on a 1 1 1 r d L (R ), for all 1 ≤ q, r ≤ ∞, q ≥ | r − 2 |, with the uniform estimate 1 ,ϕ2

Aϕ f r ≤ Cϕ1 ,ϕ2 a q f r . a

Actually we will provide here an independent and simple prove of this result, both for the benefit of the reader who is only interested in Lebesgue spaces, and also as a preparation to the more technical arguments arising in the sequel. To end up, we underline that in [7] the boundedness results obtained on both Lebesgue and Wiener amalgam spaces were proved to be sharp. We shall recall these results in Section 4. In particular, the optimality for Lebesgue spaces reads: Let ϕ1 , ϕ2 ∈ C0∞ (Rd ), with ϕ1 (0) = ϕ2 (0) = 1, ϕ1 ≥ 0, ϕ2 ≥ 0. Assume that 1 ,ϕ2 for some 1 ≤ q, r ≤ ∞ and every a ∈ Lq (R2d ) the operator Aϕ is bounded on a Lr (Rd ). Then

1

1 1

≥ − . q r 2 Observe that similar methods can be applied to study the boundedness of localization operators on weighted Lp and Wiener amalgam spaces as well. Finally, we remark that a similar study has been carried out for the whole class of pseudodifferential operators in [8]. The paper is organized as follows. Section 2 is devoted to preliminary definitions and properties of the involved function spaces. In Section 3 we give the new proof of the boundedness results for localization operators. In Section 4 we sketch from [7] the proofs of the optimality of those results. Notation. Let us fix some notation we shall use later on (and have already used in this Introduction). We define xy = x · y, the scalar product on Rd . We define by C0∞ (Rd ) the space of smooth functions on Rd with compact support. The Schwartz

292


class is denoted by S(Rd ), the space of tempered distributions by S (Rd ). We use the brackets f, g to denote the extension to S (Rd ) × S(Rd ) of the inner product f, g = f (t)g(t)dt on L2 (Rd ). The Fourier transform is normalized to be fˆ(ω) = F f (ω) = f (t)e−2πitω dt. Throughout the paper, we shall use the notation A B, A B to indicate A ≤ cB, A ≥ cB respectively, for a suitable constant c > 0, whereas A , B if A ≤ cB and B ≤ kA, for suitable c, k > 0.

2. Time-frequency methods First we summarize some concepts and tools of time-frequency analysis, for an extended exposition we refer to the textbooks [22, 24]. 2.1. STFT properties The short-time Fourier transform (STFT) is defined in (1.2). The STFT Vg f is defined for f, g in many possible pairs of Banach spaces or topological vector spaces. For instance, it maps L2 (Rd ) × L2 (Rd ) into L2 (R2d ) and S(Rd ) × S(Rd ) into S(R2d ). Furthermore, it can be extended to a map from S(Rd ) × S (Rd ) into S (R2d ) and from S (Rd )×S(Rd ) into S (R2d ). The crucial properties of the STFT (for proofs, see [24] and [26]) we shall use in the sequel are the following. Lemma 2.1. Let f, g, ∈ L2 (Rd ), then we have (i) (STFT of time-frequency shifts) For y, ξ ∈ Rd , we have Vg (Mξ Ty f )(x, ω) = e−2πi(ω−ξ)y (Vg f )(x − y, ω − ξ).

(2.1)

(ii) (Orthogonality relations for STFT)

Vg f L2 (R2d ) = f L2(Rd ) g L2 (Rd ) .

(2.2)

(iii) (Switching f and g) (Vf g)(x, ω) = e−2πixω Vg f (−x, −ω).

(2.3)

(iv) (Fourier transform of a product of STFTs) (Vϕ 1 f Vϕ2 g)(x, ω) = (Vg f Vϕ2 ϕ1 )(−ω, x).

(2.4)

(v) (STFT of the Fourier transforms) Vg f (x, ω) = e−2πixω Vgˆ fˆ(ω, −x).

(2.5)

Proposition 2.2. Suppose p ≥ 2, f ∈ Lr (Rd ), g ∈ Lr (Rd ), with p ≤ min{r, r }. Then Vg f ∈ Lp(R2d ) and D E 1 E p p

Vg f p ≤ F 1 g r f r . (2.6) pp The following inequality is proved in [24, Lemma 11.3.3]:


293

Lemma 2.3. Let g0 , g, γ ∈ S(Rd ) such that γ, g = 0 and let f ∈ S (Rd ). Then 1 (|Vg f | ∗ |Vg0 γ|)(x, ω), |Vg0 f (x, ω)| ≤ |γ, g| for all (x, ω) ∈ R2d . Finally, we recall [7, Lemma 3.1]: Lemma 2.4. Let h ∈ C0∞ (Rd ), and consider the family of functions hλ (x) = h(x)e−πiλ|x| , 2

Then, if q ≥ 2,

Gλ q λ q − 2 ,

h

(2.7)

Gλ q λ q − 2 .

h

(2.8)

d

whereas, if 1 ≤ q < 2,

λ ≥ 1.

d

d

d

2.2. Function spaces For 1 ≤ p ≤ ∞, the FLp spaces, defined by ˆ = f }, FLp (Rd ) = {f ∈ S (Rd ) : ∃ h ∈ Lp (Rd ), h are Banach spaces equipped with the norm

f FLp = h Lp ,

ˆ = f. with h

(2.9)

The mixed-norm space L (R ), 1 ≤ p, q ≤ ∞, consists of all measurable functions on R2d such that the norm q1 pq

F Lp,q = |F (x, ω)|p dx dω (2.10) p,q

Rd

2d

Rd

(with obvious modifications when p = ∞ or q = ∞) is finite. The function spaces Lp Lq (R2d ), 1 ≤ p, q ≤ ∞, consists of all measurable functions on R2d such that the norm pq p1

F LpLq = |F (x, ω)|q dω dx (2.11) Rd

Rd

(with obvious modifications when p = ∞ or q = ∞) is finite. Notice that, for p = q, we have Lp Lp (R2d ) = Lp,p (R2d ) = Lp(R2d ). Wiener amalgam spaces. We briefly recall the definition and the main properties of Wiener amalgam spaces. We refer to [15, 17, 18, 19, 21, 25] for details. Let g ∈ C0∞ be a test function that satisfies g L2 = 1. We will refer to g as a window function. Let B one of the following Banach spaces: Lp , F Lp , Lp,q , Lp Lq , 1 ≤ p, q ≤ ∞. Let C be one of the following Banach spaces: Lp , Lp,q , Lp Lq , 1 ≤ p, q ≤ ∞. For any given temperate distribution f which is locally in B (i.e., gf ∈ B, ∀g ∈ C0∞ ), we set fB (x) = f Txg B . The Wiener amalgam space W (B, C) with local component B and global component C is defined as the space of all temperate distributions f locally in B

294


such that fB ∈ C. Endowed with the norm f W (B,C) = fB C , W (B, C) is a Banach space. Moreover, different choices of g ∈ C0∞ generate the same space and yield equivalent norms. If B = F L1 (the Fourier algebra), the space of windows for the Wiener amalgam spaces W (F L1 , C) can be enlarged to the so-called Feichtinger algebra W (FL1 , L1 ). Recall that the space C0∞ is dense in W (F L1 , L1 ). An equivalent norm for the space W (Lp , Lq ) is provided by (1.6). We use the following definition of mixed Wiener amalgam norms. Given a measurable function F on R2d we set

F W (Lp1 ,Lq1 )W (Lp2 ,Lq2 ) =

F (x, ·) W (Lp2 ,Lq2 ) W (Lp1 ,Lq1 )    pq1  q11   pq1 1 q 2 p2    2  p   dx  , , |F (x, ω)Tn χQ (ω)Tm χQ (x)| 2    d d m∈Zd

R

n∈Zd

R

where Q = [0, 1)d . The following properties of Wiener amalgam spaces will be frequently used in the sequel. Lemma 2.5. Let Bi , Ci , i = 1, 2, 3, be Banach spaces such that W (Bi , Ci ) are well defined. Then, (i) Convolution. If B1 ∗ B2 → B3 and C1 ∗ C2 → C3 , we have W (B1 , C1 ) ∗ W (B2 , C2 ) → W (B3 , C3 ).

(2.12)

(ii) Inclusions. If B1 → B2 and C1 → C2 , W (B1 , C1 ) → W (B2 , C2 ). Moreover, the inclusion of B1 into B2 need only hold “locally” and the inclusion of C1 into C2 “globally”. In particular, for 1 ≤ pi , qi ≤ ∞, i = 1, 2, we have p1 ≥ p2 and q1 ≤ q2 =⇒ W (Lp1 , Lq1 ) → W (Lp2 , Lq2 ).

(2.13)

(iii) Complex interpolation. For 0 < θ < 1, we have [W (B1 , C1 ), W (B2 , C2 )][θ] = W [B1 , B2 ][θ] , [C1 , C2 ][θ] , if C1 or C2 has absolutely continuous norm. (iv) Duality. Let p, q < ∞. If B , C are the topological dual spaces of the Banach spaces B, C respectively, then W (B, C) = W (B , C ).

(2.14)

(v) Hausdorff-Young. If 1 ≤ p, q ≤ 2 then

F (W (Lp , Lq )) → W (Lq , Lp ) (local and global properties are interchanged on the Fourier side).

(2.15)


295

(vi) Pointwise products. If B1 · B2 → B3 and C1 · C2 → C3 , we have W (B1 , C1 ) · W (B2 , C2 ) → W (B3 , C3 ).

(2.16)

Finally we establish a useful estimate for Wiener amalgam spaces, based on the classical Bernstein’s inequality (see, e.g., [33]), that we are going to recall first. Let B(x0 , R) be the ball of center x0 ∈ Rd and radius R > 0 in Rd . Lemma 2.6 (Bernstein’s inequality). Let f ∈ S (Rd ) such that fˆ is supported in B(x0 , R), and let 1 ≤ p ≤ q ≤ ∞. Then, there exists a positive constant C, independent of f , x0 , R, p, q, such that 1 1

f q ≤ CRd( p − q ) f p . (2.17) Proposition 2.7. Let 1 ≤ p ≤ q ≤ ∞. For every R > 0, there exists a constant CR > 0 such that, for every f ∈ S (Rd ) whose Fourier transform is supported in any ball of radius R, we have

f W (Lq ,Lp ) ≤ CR f p . Proof. Choose a Schwartz function g whose Fourier transform gˆ has compact support in B(0, 1), as window function arising in the definition of the norm in W (Lq , Lp ). Then, the function (Tx g)f , x ∈ Rd , has Fourier transform supported in a ball of radius R + 1. Therefore it follows from Bernstein’s inequality (Lemma 2.6) that

(Tx g)f q ≤ CR (Tx g)f p . Taking the Lp -norm with respect to x gives the conclusion. Modulation spaces. For their basic properties we refer to [16, 24]. Given a non-zero window g ∈ S(Rd ) and 1 ≤ p, q ≤ ∞, the modulation space p,q M (Rd ) consists of all tempered distributions f ∈ S (Rd ) such that the STFT, defined in (1.2), fulfills Vg f ∈ Lp,q (R2d ). The norm on M p,q is q/p 1/p p

f M p,q = Vg f Lp,q = |Vg f (x, ω)| dx dω . Rd

p

Rd

p,p

If p = q, we write M instead of M . M p,q is a Banach space whose definition is independent of the choice of the window g. Moreover, if g ∈ M 1 \ {0}, then Vg f Lp,q is an equivalent norm for M p,q (Rd ). Among the properties of modulation spaces, we record that M 2,2 = L2 , p1 ,q1 M → M p2 ,q2 , if p1 ≤ p2 and q1 ≤ q2 . If 1 ≤ p, q < ∞, then (M p,q ) = M p ,q . Modulation spaces and Wiener amalgam spaces are closely related: for p = q, we have 1/p p p

f W (F Lp ,Lp ) = |Vg f (x, ω)| m(x, ω) dx dω , f M p . (2.18) Rd

Rd

Finally, the next results will be useful in the sequel.

296


Proposition 2.8 ([9, Proposition 3.4]). Let g ∈ M 1 (Rd ) and 1 ≤ p ≤ ∞ be given. Then f ∈ M p (Rd ) if and only if Vg f ∈ M p (R2d ) with

Vg f M p , g M p f M p .

(2.19)

Lemma 2.9 ([5, Lemma 4.1]). Let 1 ≤ p, q ≤ ∞. If f ∈ M p,q (Rd ) and g ∈ M 1 (Rd ), then Vg f ∈ W (F L1 , Lp,q )(R2d ) with norm estimate

Vg f W (F L1,Lp,q ) f M p,q g M 1 .

(2.20)

We finally recall a characterization of modulation spaces. Let ψ ∈ S(Rd ) be such that supp ψ ⊂ (−1, 1)d and ψ(ω − k) = 1 for all ω ∈ Rd . (2.21) k∈Zd

Proposition 2.10. ([31]) Let 1 ≤ p, q ≤ ∞. An equivalent norm in M p,q is given by  1/q

f M p,q , 

ψ(D − k)f qLp  , k∈Zd

where ψ(D − k)f := F −1 (fˆTk ψ).

3. Sufficient boundedness conditions In what follows, we study the boundedness of localization operators on Lp and on Wiener amalgam spaces. We start by proving Theorem 1.2, concerning Lebesgue spaces. Proof of Theorem 1.2. It is well known that the result holds for q = ∞, r = 2 (see, e.g., [34], [35], or also [5, Theorem 1.1]). In fact, that case follows at once from (1.5), since the STFT defines an isometry L2 (Rd ) → L2 (R2d ). By the Riesz-Thorin interpolation theorem, it therefore suffices to prove the estimates 1 ,ϕ2

Aϕ f r a q f r , a ∈ Lq (R2d ), f ∈ Lr (Rd ), (3.1) a for 1 ≤ q ≤ 2, 1 ≤ r ≤ ∞ (see Figure 1 below). Moreover it is enough to prove (3.1) for Schwartz symbols a only, as one sees by a limiting argument which uses the density of S(R2d ) in Lq (R2d ). 1 ,ϕ2 Now, for a ∈ S(R2d ), the integral kernel of Aϕ reveals to be the Schwartz a function K(x, y) = a(t, ω)Mω Tt ϕ2 (x)M−ω Ttϕ1 (y)dtdω R2d = F2 a(t, y − x)Tt ϕ1 (y)Tt ϕ2 (x)dt, Rd


297

where F2 stands for the Fourier transform with respect to the second variable. By Hölder’s inequality, a linear change of variable and Hausdorff-Young’s inequality we obtain |K(x, y)| ≤ (F2 a) (·, y − x) q T−y ϕ1 T−x ϕ2 q = (F2 a) (·, y − x) q ϕ2 Tx−y ϕ1 q a(·, y − x) q ϕ2 Tx−y ϕ1 q . Hence, if we set B(z) = a(·, z) q ϕ2 T−z ϕ1 q , we have |K(x, y)| B(x − y). The desired estimate in (3.1) then will follow from Young’s inequality if we prove that

B 1 a q . This is achieved by using once again Hölder’s inequality: 1/q q B(z) dz a q

ϕ2 T−z ϕ1 q dz , (with obvious changes if p = ∞). The last integral is in fact absolutely convergent, for ϕ1 , ϕ2 ∈ S(Rd ). We now consider the case of symbols and functions in Wiener amalgam spaces. First, we present a preliminary result that is proved in [7]. It uses the following key observation (that comes out from the definition of STFT): if two window functions γ1 , γ2 ∈ L2 (Rd ) are supported in balls of radius R > 0, then the STFT Vγ1 γ2 (x, ω) has support contained in a strip B(y0 , 2R) × Rd , for some y0 ∈ Rd . We shall recall the proof for the benefit of the reader. First, let us denote by B(W (Lr , Ls )) the Banach space of bounded operators on W (Lr , Ls ). Proposition 3.1. Let a ∈ W (L1 , L∞ )(R2d ), ϕ1 , ϕ2 ∈ M 1 (Rd ). Then the operator 1 ,ϕ2 Aϕ is bounded on W (L2 , Ls )(Rd ) for every 1 ≤ s ≤ ∞, with the uniform a estimate 1 ,ϕ2

Aϕ

B(W (L2 ,Ls )) a W (L1 ,L∞ ) ϕ1 M 1 ϕ2 M 1 . a Proof. We have to prove the estimate 1 ,ϕ2 |Aϕ f, g| ≤ C a W (L1 ,L∞ ) f W (L2 ,Ls ) g W (L2 ,Ls ) ϕ1 M 1 ϕ2 M 1 , a

for every f ∈ W (L2 , Ls ), g ∈ W (L2 , Ls ). Using the weak definition (1.5), we can write ϕ1 ,ϕ2 Aa f, g = a(x, ω)Vϕ1 f (x, ω)Vϕ2 g(x, ω) dx dω. R2d

By Lemma 2.5 (ii), 1 ,ϕ2 |Aϕ f, g| ≤ a W (L1 ,L∞ ) Vϕ1 f Vϕ2 g W (L1 ,L∞ ) . a

298


Write f = k∈Zd f Tk χQ and g = h∈Zd gTh χQ . Moreover, choose ψ ∈ S(Rd ) satisfying (2.21) and write ϕ1 = l∈Zd ϕ1,l , ϕ2 = m∈Zd ϕ2,m , with ϕ1,l = ψ(D − l)ϕ1 and ϕ2,m = ψ(D − m)ϕ2 (with the notation in Proposition 2.10). We deduce 1 ,ϕ2 |Aϕ f, g| ≤ a W (L1 ,L∞ )

Vϕ1,l (f Tk χQ )Vϕ2,m (gTh χQ ) W (L1 ,L∞ ) . a m,l∈Zd k,h∈Zd

(3.2) By applying Lemma 2.1 (iv) and (v) we see that

F Vϕ (f Tk χQ )Vϕ2,m (gTh χQ ) (x, ω)

1,l = |VgTh χQ (f Tk χQ )(−ω, x)Vϕ2,m (ϕ1,l )(−ω, x)| = |VgTh χQ (f Tk χQ )(−ω, x)VϕG2 Tm ψ (G ϕ1 Tl ψ)(x, ω)|. Hence, by the observation stated just before the present proposition, the expression Vϕ1,l (f Tk χQ )Vϕ2,m (gTh χQ ) has Fourier transform supported in a ball in R2d whose radius is independent of k, h, m, l. Hence it follows from (3.2) and Proposition 2.7 that 1 ,ϕ2 |Aϕ f, g| a W (L1 ,L∞ )

Vϕ1,l (f Tk χQ )Vϕ2,m (gTh χQ ) 1 . (3.3) a m,l∈Zd k,h∈Zd

As a consequence of Cauchy-Schwarz’ inequality and Parseval’s formula, this last expression is ≤ a W (L1 ,L∞ )

Vϕ1,l (f Tk χQ )(x, ·) 2 Vϕ2,m (gTh χQ )(x, ·) 2 dx = a W (L1 ,L∞ ) ≤ a W (L1 ,L∞ ) ×

m,l∈Zd

Rd

m,l∈Zd k,h∈Zd

Rd

m,l∈Zd k,h∈Zd

Rd

f Tk χQ Tx ϕ1,l 2 gTh χQ Tx ϕ2,m 2 dx

f Tk χQ 2 gTh χQ 2

k,h∈Zd

Tk χQ Tx ϕ1,l ∞ Th χQ Tx ϕ2,m ∞ dx.

We will prove that

Tk χQ Tx ϕ1,l ∞ Th χQ Tx ϕ2,m ∞ dx Rd =

χQ Tx ϕ1,l ∞ χQ Tx+k−h ϕ2,m ∞ dx = vk−h,l,m , Rd

for a sequence v = vk,l,m ∈ l1 (Z3d ), satisfying

v l1 (Z3d ) ϕ1 M 1 ϕ2 M 1 .

(3.4)


299

Assuming (3.4), by Hölder’s and Young’s inequality, 1 ,ϕ2 |Aϕ f, g| a

a W (L1 ,L∞ )

vk−h,l,m gTh χQ 2 f Tk χQ 2

m,l∈Zd k,h∈Zd

a W (L1 ,L∞ )   s 1/s  1/s     × vk−h,l,m gThχQ 2  

f Tk χQ s2   m,l∈Zd

k∈Zd

h∈Zd

k∈Zd

≤ a W (L1 ,L∞ )   1/s  1/s   × vk,l,m  

gThχQ s2 

f Tk χQ s2  m,l∈Zd

k∈Zd

h∈Zd

k∈Zd

and the desired estimate follows by (3.4). Let us now prove (3.4). Indeed, vk,m,l =

χQ Tx ϕ1,l ∞

χQ Tx+k−h ϕ2,m ∞ dx k∈Zd

Rd

(3.5)

k∈Zd

, ϕ2,m W (L∞ ,L1 ) ×

χQ Tx ϕ1,l ∞ dx , ϕ1,l W (L∞ ,L1 ) ϕ2,m W (L∞ ,L1 )

(3.6)

ϕ1,l 1 ϕ2,m 1 ,

(3.7)

Rd

where the last estimate follows from Proposition 2.7. Then, by Proposition 2.10, we have vk,m,n

ϕ1,l 1

ϕ1,m 1 , ϕ1 M 1 ϕ2 M 1 , k,l,m∈Zd

l∈Zd

m∈Zd

which concludes the proof.

Remark 3.2. An argument similar to that in the proof of Proposition 3.1 (but 1 ,ϕ2 simpler) shows that, if a ∈ L∞ (R2d ) and ϕ1 , ϕ2 ∈ W (L∞ , L1 )(Rd ), then Aϕ is a 2 s d bounded on W (L , L )(R ) for every 1 ≤ s ≤ ∞. We are now ready to state the main contribution of this paper. Theorem 3.3. Let a ∈ W (Lp , Lq )(R2d ), ϕ1 , ϕ2 ∈ M 1 (Rd ). If

1

1 1

1 ≤ p, q, r, s ≤ ∞, ≥ − , q r 2

(3.8)

1 ,ϕ2 then the operator Aϕ is bounded on W (Lr , Ls )(Rd ), with the uniform estimate a 1 ,ϕ2

Aϕ

B(W (Lr ,Ls )) a W (Lp ,Lq ) ϕ1 M 1 ϕ2 M 1 . a

Figure 1 illustrates the range of exponents (1/r, 1/q) for the boundedness of

(3.9) 1 ,ϕ2 Aϕ . a

300

E. Cordero and F. Nicola 1 q

1

1 2

1 2

1

1 r

Figure 1. Estimate (3.9) holds for all pairs (1/r, 1/q) in the shadowed region, and 1 ≤ p, s ≤ ∞. Proof. By the inclusion relations for Wiener amalgam spaces, it suffices to show the claim for p = 1. For q = ∞, r = 2, the result was already proved in [5, Theorem 1.1]. Indeed, using the inclusion L1 ⊂ FL∞ and the inclusion relations for Wiener amalgam spaces, we have W (L1 , L∞ ) ⊂ W (F L∞ , L∞ ) = M ∞ . Hence, the symbol a is in the modulation space M ∞ and, consequently, the operator 1 ,ϕ2 Aϕ is bounded on M 2 = L2 . a 1 ,ϕ2 We shall show that the cases 1 ≤ q ≤ 2 yield the continuity of Aϕ on a r s W (L , L ) for every 1 ≤ r, s ≤ ∞, so that, by complex interpolation with the case (q, r) = (∞, 2), considered in Proposition 3.1 (see Lemma 2.5 (iii) and Figure 1) we 1 ,ϕ2 obtain the desired boundedness of Aϕ under the conditions (3.8), if s < ∞. The a remaining cases, when s = ∞ and q > 2 (and therefore r > 1) follows by duality, 2 ,ϕ1 1 ,ϕ2 ∗ for then W (Lr , Ls ) = W (Lr , Ls ) (Proposition 2.5 (iv)), and (Aϕ ) = Aϕ . a ¯ a Hence, let 1 ≤ q ≤ 2. 1 ,ϕ2 We rewrite Aϕ as the integral operator with kernel a K(x, y) = a(t, ω)Mω Tt ϕ2 (x)M−ω Tt ϕ1 (y)dtdω (3.10) R2d = F2 a(t, y − x)Tt ϕ1 (y)Tt ϕ2 (x)dt (3.11) Rd


301

= R

F (ϕ1 (y − ·)ϕ2 (x − ·))(ω)F a(ω, y − x)dω

(3.12)

Vϕ∗2 (Ty ϕ∗1 )(x, ω)F a(ω, y − x)dω,

(3.13)

d

= Rd

with the notation f ∗ (t) = f (−t). We estimate the kernel as follows: |K(x, y)| ≤ |Vϕ∗2 (Ty ϕ∗1 )(x, ω)Fa(ω, y − x)|dω d R = |Vϕ∗2 ϕ∗1 (x − y, ω)Fa(ω, y − x)|dω, Rd

since the STFT fulfills Vg (Ty f )(x, ω) = e−2πiωy Vg f (x − y, ω), see (2.1). For sake of simplicity, we set Φ := Vϕ∗2 ϕ∗1 . Moreover, we introduce the coordinate transformation τ F (t, u) = F (−u, t). So that the kernel K can be controlled from above by |K(x, y)| ≤ |τ Φ(ω, y − x)| |Fa(ω, y − x)|dω. Rd

Let us set B(t) :=

|τ Φ(ω, t)| |F a(ω, t)|dω and estimate its L1 -norm. Precisely,

B 1 = |τ Φ(ω, t)| |Fa(ω, t)|dωdt

Rd

Rd

Rd

≤ τ Φ W (Lq ,L1 ) F a W (Lq ,L∞ ) ≤ τ Φ M 1 a W (L1 ,Lq ) = Φ M 1 a W (L1 ,Lq ) ϕ1 M 1 ϕ2 M 1 a W (L1 ,Lq ) , where we used the inclusion FL1 ⊂ Lq , which holds locally and yields W (F L1 , L1 ) ⊂ W (Lq , L1 ) (Lemma 2.5, item (ii)) and the equality W (F L1 , L1 ) = M 1 and, finally, (2.19). Hence, we can write 1 ,ϕ2 |(Aϕ f )(x)| (|B ∗ | ∗ |f |)(x), a

with B ∗ (x) = B(−x) and Young’s Inequality combined with the Wiener convolution relations in Lemma 2.5 (i), gives the desired continuity on W (Lr , Ls ). Remark 3.4. Choosing r = s we get the boundedness results on Lr spaces, already stated in Theorem 1.2 for Schwartz windows.

4. Necessary boundedness conditions We first recall the following version of Lemma 2.4 for Wiener amalgam spaces. Lemma 4.1. With the notation of Lemma 2.4, we have, for q ≥ 2, Gλ W (Lp ,Lq ) λ q − 2 ,

h d

d

λ ≥ 1.

302


Proof. When p ≤ q, the result follows at once from Lemma 2.4, using the embedding Lq → W (Lp , Lq ). When p > q it suffices to apply Proposition 2.7 and Lemma 2.4 again. Proposition 4.2. Let ϕ1 , ϕ2 ∈ C0∞ (Rd ), χ ∈ C ∞ (Rd ), with ϕ1 (0) = ϕ2 (0) = χ(0) = 1, ϕ1 ≥ 0, ϕ2 ≥ 0, χ ≥ 0. If the estimate 1 ,ϕ2

χAϕ f r ≤ C a W (Lp ,Lq ) f W (Ls1 ,Ls2 ) , a

∀f ∈ S(Rd ), ∀a ∈ S(R2d ), (4.1)

holds for some 1 ≤ p, q, r, s1 , s2 ≤ ∞, then 1 1 1 ≥ − . q 2 r

(4.2)

Proof. We just give a sketch of the proof and we refer to [7] for details. We can assume q ≥ 2, otherwise the conclusion is trivially true. Consider a real-valued function h ∈ C0∞ (Rd ), with h(0) = 1, h ≥ 0. Let hλ (x) = h(x)e−πiλ|x| , 2

λ ≥ 1.

We test the estimate (4.1) on f = hλ and a(x, ω) = aλ (x, ω) = h(x)(F −1 hλ )(ω). Clearly, hλ W (Ls1 ,Ls2 ) is independent of λ. On the other hand, by Lemma 4.1 we have d d

aλ W (Lp ,Lq ) = h W (Lp ,Lq ) F −1 hλ W (Lp ,Lq ) λ q − 2 . Hence, if we prove that d 1 ,ϕ2

χAϕ f r λ− r , (4.3) aλ then (4.2) follows from (4.1), by letting λ → +∞. In fact (4.3) is a consequence of the point-wise estimate 1 ,ϕ2 |χ(x)(Aϕ hλ )(x)| 1, aλ

for |x| ≤ λ−1 ,

1 ,ϕ2 and λ ≥ λ0 large enough, which can be achieved from the expression of Aϕ in aλ (3.10) as an integral operator.

Theorem 4.3. Let ϕ1 , ϕ2 ∈ C0∞ (Rd ), with ϕ1 (0) = ϕ2 (0) = 1, ϕ1 ≥ 0, ϕ2 ≥ 0. If, for some 1 ≤ p, q, r, s ≤ ∞, the estimate 1 ,ϕ2

Aϕ f W (Lr ,Ls ) ≤ C a W (Lp ,Lq ) f W (Lr ,Ls ) , a

holds, then

1

1 1

≥ − . q r 2

∀f ∈ S(Rd ), ∀a ∈ S(R2d ), (4.4) (4.5)

Proof. We can suppose q ≥ 2 (otherwise (4.5) is trivially satisfied). We assume 2 ,ϕ1 1 ,ϕ2 ∗ r ≥ 2, the case 1 ≤ r < 2 follows by duality, for (Aϕ ) = Aϕ . a ¯ a ∞ d Let χ ∈ C0 (R ), χ ≥ 0, χ(0) = 1. Then, (4.4) implies that 1 ,ϕ2

χAϕ f W (Lr ,Ls ) ≤ C a W (Lp ,Lq ) f W (Lr ,Ls ) a

∀f ∈ S(Rd ), ∀a ∈ S(R2d ).


303

For functions u supported in a fixed compact subset we have u W (Lr ,Ls ) , u r , so that we have 1 ,ϕ2

χAϕ f r ≤ C a W (Lp ,Lq ) f W (Lr ,Ls ) , a

Then (4.5) follows from Proposition 4.2.

∀f ∈ S(Rd ), ∀a ∈ S(R2d ).

As a consequence of Theorem 4.3 and the Closed Graph Theorem we get the following result ([7]). Theorem 4.4. Let ϕ1 , ϕ2 ∈ C0∞ (Rd ), with ϕ1 (0) = ϕ2 (0) = 1, ϕ1 ≥ 0, ϕ2 ≥ 0. Assume that for some 1 ≤ p, q, r, s ≤ ∞ and every a ∈ W (Lp , Lq )(R2d ) the 1 ,ϕ2 operator Aϕ is bounded on W (Lr , Ls )(Rd ). Then (4.5) must hold. a

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