New Developments in Pseudo-Differential Operators: ISAAC Group in Pseudo-Differential Operators

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Author: Luigi Rodino | Man Wah Wong

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I ••

11

Operator Theory: Advances and Applications Vol. 189

Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv Israel

Editorial Board: D. Alpay (Beer Sheva, Israel) J. Arazy (Haifa, Israel) A. Atzmon (Tel Aviv, Israel) J.A. Ball (Blacksburg, VA, USA) H. Bart (Rotterdam, The Netherlands) A. Ben-Artzi (Tel Aviv, Israel) H. Bercovici (Bloomington, IN, USA) A. Böttcher (Chemnitz, Germany) K. Clancey (Athens, GA, USA) R. Curto (Iowa, IA, USA) K. R. Davidson (Waterloo, ON, Canada) M. Demuth (Clausthal-Zellerfeld, Germany) A. Dijksma (Groningen, The Netherlands) R. G. Douglas (College Station, TX, USA) R. Duduchava (Tbilisi, Georgia) A. Ferreira dos Santos (Lisboa, Portugal) A.E. Frazho (West Lafayette, IN, USA) P.A. Fuhrmann (Beer Sheva, Israel) B. Gramsch (Mainz, Germany) H.G. Kaper (Argonne, IL, USA) S.T. Kuroda (Tokyo, Japan) L.E. Lerer (Haifa, Israel) B. Mityagin (Columbus, OH, USA)

V. Olshevski (Storrs, CT, USA) M. Putinar (Santa Barbara, CA, USA) A.C.M. Ran (Amsterdam, The Netherlands) L. Rodman (Williamsburg, VA, USA) J. Rovnyak (Charlottesville, VA, USA) B.-W. Schulze (Potsdam, Germany) F. Speck (Lisboa, Portugal) I.M. Spitkovsky (Williamsburg, VA, USA) S. Treil (Providence, RI, USA) C. Tretter (Bern, Switzerland) H. Upmeier (Marburg, Germany) N. Vasilevski (Mexico, D.F., Mexico) S. Verduyn Lunel (Leiden, The Netherlands) D. Voiculescu (Berkeley, CA, USA) D. Xia (Nashville, TN, USA) D. Yafaev (Rennes, France)

Honorary and Advisory Editorial Board: L.A. Coburn (Buffalo, NY, USA) H. Dym (Rehovot, Israel) C. Foias (College Station, TX, USA) J.W. Helton (San Diego, CA, USA) T. Kailath (Stanford, CA, USA) M.A. Kaashoek (Amsterdam, The Netherlands) P. Lancaster (Calgary, AB, Canada) H. Langer (Vienna, Austria) P.D. Lax (New York, NY, USA) D. Sarason (Berkeley, CA, USA) B. Silbermann (Chemnitz, Germany) H. Widom (Santa Cruz, CA, USA)

New Developments in Pseudo-Differential Operators ISAAC Group in Pseudo-Differential Operators (IGPDO), Middle East Technical University, Ankara,Turkey, August 2007

Luigi Rodino M.W. Wong Editors

Birkhäuser Basel · Boston · Berlin

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

2000 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, 92C55, 94A12

Library of Congress Control Number: 2008936167

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-8968-0 Birkhäuser Verlag AG, Basel - Boston - Berlin 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. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-8968-0 e-ISBN 978-3-7643-8969-7 987654321 www.birkhauser.ch

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

vii

M. de Gosson Phase-Space Weyl Calculus and Global Hypoellipticity of a Class of Degenerate Elliptic Partial Differential Operators . . . . . . .

1

T. Gramchev, S. Pilipović and L. Rodino Classes of Degenerate Elliptic Operators in Gelfand–Shilov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

A. Dasgupta and M.W. Wong Weyl Transforms and the Heat Equation for the Sub-Laplacian on the Heisenberg Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

C. Iwasaki Construction of the Fundamental Solution and Curvature of Manifolds with Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

J. Abed and B.-W. Schulze Operators with Corner-Degenerate Symbols . . . . . . . . . . . . . . . . . . . . . . . . . .

67

A. Dasgupta Ellipticity of Fredholm Pseudo-Differential Operators on Lp (Rn ) . . . . . 107 M. Oberguggenberger Hyperbolic Systems with Discontinuous Coefficients: Generalized Wavefront Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 C. Garetto Generalized Fourier Integral Operators on Spaces of Colombeau Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 M. Ruzhansky On Local and Global Regularity of Fourier Integral Operators . . . . . . . . 185 J. Johnsen Type 1,1-Operators Defined by Vanishing Frequency Modulation . . . . . 201

vi

Contents

G. Garello and A. Morando Regularity for Quasi-Elliptic Pseudo-Differential Operators with Symbols in H¨ older Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 C. Bouzar and A. Dali Multi-Anisotropic Gevrey Regularity of Hypoelliptic Operators . . . . . .

265

Q. Guo, S. Molahajloo and M.W. Wong Modified Stockwell Transforms and Time-Frequency Analysis . . . . . . . . 275 Y. Liu Localization Operators for Two-Dimensional Stockwell Transforms . . . 287 S. Molahajloo and M.W. Wong Pseudo-Differential Operators on S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297

M. Ruzhansky and V. Turunen On Pseudo-Differential Operators on Group SU(2) . . . . . . . . . . . . . . . . . . . 307 A. Mohammed and M.W. Wong Sampling and Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . 323

Operator Theory: Advances and Applications, Vol. 189, vii–viii c 2008 Birkh¨ auser Verlag Basel/Switzerland

Preface On the occasion of the Sixth Congress of the International Society for Analysis, its Applications and Computation (ISAAC), the ISAAC Group in Pseudo-Differential Operators (IGPDO) met again at the Middle East Technical University in Ankara, Turkey on August 13–18, 2007. Papers related to the talks given at the special session on pseudo-differential operators and invited papers from experts in the field are peer-reviewed for inclusion in this volume. This volume consists of seventeen papers on pseudo-differential operators and related topics. Results on twisted Laplacians and their extensions, and the subLaplacian on the Heisenberg group are arranged together in the first three papers. Global analysis on manifolds with boundaries, pseudo-differential analysis on manifolds with singularities, and the ellipticity and Fredholmness of pseudo-differential operators are presented in the next three papers. Following these are three papers on hyperbolic equations and systems, Fourier integral operators and Colombeau algebras. Then there are three papers on exotic pseudo-differential operators and regularity results on quasi-elliptic operators and hypoelliptic operators. Two papers on the Stockwell transforms in time-frequency analysis are then given. The last three papers are devoted to pseudo-differential operators on the unit circle, the torus, SU(2) and related results on sampling. This volume should be useful to graduate students and researchers in mathematical analysis with applications in sciences and engineering. It is a valuable complement to the volumes “Advances in Pseudo-Differential Operators”, “PseudoDifferential Operators and Related Topics” and “Modern Trends in Pseudo-Differential Operators” published in the same series edited by Professor Israel Gohberg in, respectively, 2004, 2006 and 2007. It should also be read in conjunction with the volume “Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis”, Editors: Luigi Rodino, Bert-Wolfgang Schulze and M.W. Wong, Fields Institute Communications Series 57, American Mathematical Society, 2007 and the volume “Pseudo-Differential Operators: Quantization and Signals”, Editors: Luigi Rodino and M.W. Wong, Fondazione C.I.M.E., Firenze, Lecture Notes in Mathematics 1949, Springer, 2008. In an era of explosive advances in quantitative sciences and information technologies, it is envisaged that mathematical analysis in general and pseudo-

viii

Preface

differential operators in particular will be developed hand in glove with applications and computation in the physical, biological and medical sciences. This theme will play an important role in the forthcoming volumes on pseudo-differential operators originating from IGPDO.

The Editors

Operator Theory: Advances and Applications, Vol. 189, 1–14 c 2008 Birkh¨ auser Verlag Basel/Switzerland

Phase-Space Weyl Calculus and Global Hypoellipticity of a Class of Degenerate Elliptic Partial Differential Operators Maurice de Gosson Abstract. In a recent series of papers M.W. Wong has studied a degenerate elliptic partial differential operator related to the Heisenberg group. It turns out that Wong’s example is best understood when replaced in the context of the phase-space Weyl calculus we have developed in previous work; this approach highlights the relationship of Wong’s constructions with the quantum mechanics of charged particles in a uniform magnetic field. Using Shubin’s classes of pseudodifferential symbols we prove global hypoellipticity results for arbitrary phase-space operators arising from elliptic operators on configuration space. Mathematics Subject Classification (2000). Primary 47F30; Secondary 35B65, 46F05. Keywords. Degenerate elliptic operators, hypoellipticity, phase space Weyl calculus, Shubin symbols.

Introduction In a recent series of interesting papers [25, 26, 27] M.W. Wong (also see Dasgupta and Wong [5]) discusses various properties of the partial differential operator 1 (1) W = − (ZZ + ZZ) 2 where Z and Z are the vector fields on R2 defined by 1 1 ∂ ∂ + z , Z= + z. Z= (2) ∂z 2 ∂z 2 Writing z = x + iy the operator W has the explicit form ∂ ∂ 1 −y W = −Δ − i x (3) + (x2 + y 2 ) ∂y ∂x 4

2

M. de Gosson

where Δ is the usual Laplace operator in the x, y variables. It turns out that Wong’s results are only the “tip of an iceberg” because the operator (3) can be viewed as the phase-space version of the Hermite operator −Δ + x2 obtained by using a quantization procedure we have introduced in previous work. To understand this, let us make the two following independent observations: • The operator W has a well-known meaning in physics. Consider in fact an electron with mass m and charge e placed in a strong uniform magnetic field B directed along the z axis: B = (0, 0, Bz ). The Hamiltonian operator is, in a particular choice of gauge (see Section 2), 2 ∂ ∂ 2 mωL Δ + iωL y −x (x2 + y 2 ) (4) H=− + 2m ∂x ∂y 2 where ωL is the “Larmor frequency”. This operator reduces to Wong’s operator (3) if x and y are swapped and units are chosen so that = 1, m = 1/2, and ωL = 1; the spectrum of the operator (4) is well known; it consists of the sequence “energy levels” EN = (2N + 1)ωL (see Subsection 2.1); the spectrum of W is thus given by the sequence of numbers, EN = 2N + 1 for N = 0, 1, . . . , a fact which Wong rediscovers in [25] using complicated calculations involving the Wigner formalism and special function theory. • On the other hand, a straightforward calculation shows that we can rewrite the operator W more compactly as 2 2 1 1 ∂ ∂ − y + −i + x ; (5) W = −i ∂x 2 ∂y 2 setting p = −y this operator becomes 2 2 1 1 ∂ ∂ + p + i + x W = −i ∂x 2 ∂p 2

(6)

which makes apparent that W is obtained from the harmonic oscillator Hamiltonian H(x, p) = 12 (p2 + x2 ) using the “quantization rules” x −→ X = i

1 ∂ 1 ∂ + x , p −→ P = −i + p ∂p 2 ∂x 2

(7)

we have studied and exploited in [10, 11] in connection with the phase space Schr¨ odinger equation of Torres-Vega and Frederick [20, 21]; notice that X and P satisfy the same commutation relation [X, P ] = i as satisfied by the operators x and −i∂/∂x. This paper consists of two parts. In the first part (Sections 1–2) we will focus on the “phase space Weyl calculus” aspect of Wong’s operator and its generalizations; we also briefly review our previous definitions and results from [10, 11]. In the second part of this paper (Section 3) we show that our phase-space Weyl calculus together with Shubin’s pseudodifferential calculus (Shubin [19], Chapter IV)

Phase-Space Weyl Calculus and Global Hypoelliticity

3

allows us not only to recover (in a trivial way) the global hypoellipticity of Wong’s operator W , but to prove that every operator = a i ∂ + 1 x, − i ∂ + 1 p A ∂p 2 ∂x 2 ∂ ∂ + 12 x,−i ∂x + 12 p) in any positiveobtained by replacing formally (x, p) by (i ∂p definite quadratic form a = a(x, p) is globally hypoelliptic (we will in fact prove a 2n-dimensional statement, allowing the symbol a to be defined on R2n ).

Notation We denote by S(Rn ) the Schwartz space of all smooth complex-valued functions on Rn which decrease, together with their derivatives, faster than the inverse than any polynomial when |x| → ∞. The dual S (Rn ) of S(Rn ) is the space of tempered distributions on Rn . Operators acting on functions (or distributions) defined on Rn will be denoted by capital letters A, B, C, . . . while operators acting on functions (or distributions) defined on Rn defined on the symplectic space (R2n , σ) will denoted by covering B, C, . . . Functions on Rn will usually be denoted capital letters with a tilde: A, by lower-case Greek letters ψ, φ, . . . while functions on R2n are denoted by uppercase Greek letters Ψ, Φ, . . . We will use standard multi-index notation: if α = αn 1 (α1 , . . . , αn ) is a sequence of non-negative integers we write xα = xα 1 · · · xn if α |α| α1 αn x = (x1 , . . . , xn ), and Dx = (−i) ∂x1 · · · ∂xn with |α| = α1 + · · · + αn . We will denote by σ the standard symplectic form on the vector space Rn × n R ≡ R2n : σ(z, z ) = p · x − p · x for z = (x, p), z = (x , p ). The symplectic group of (R2n , σ) will be denoted by Sp(n): it is the group of all linear automorphisms s of R2n such that σ(sz, sz ) = σ(z, z ) for all z, z ∈ R2n .

1. Phase-space Weyl calculus For proofs and a detailed exposition we refer to de Gosson [11] (the phase space calculus was introduced in de Gosson [9, 10] following a suggestion in the paper by Grossmann et al. [14]). 1.1. Definitions Let A : S(Rn )−→S (Rn ) be a continuous linear operator. In view of Schwartz’s kernel theorem there exists a distribution K ∈ S (Rn × Rn ) such that Af (x) = K(x, y)f (y)dy Rn

(the integral being interpreted as a partial distribution bracket). The Weyl symbol of A is by definition (see, e.g., [11], Theorem 6.12, or Wong’s book [24]) the

4

M. de Gosson

tempered distribution a given by a(x, p) = e−ip·y K(x + 12 y, x − 12 y)dy. Rn

Defining the twisted Weyl symbol aσ as being the symplectic Fourier transform of a, that is n 1 aσ (z) = e−iσ(z,z ) a(z)dz 2π R2n the operator A is given by the Bochner integral n 1 aσ (z0 )T (z0 )dz0 ; (8) A= 2π R2n here T (z) is the Heisenberg–Weyl operator defined by T (z0 )ψ(x) = ei(p0 ·x− 2 p0 ·x0 ) ψ(x − x0 ) 1

(9)

if z0 = (x0 , p0 ). The operator A can be (at least formally) shown to act on ψ ∈ S(Rn ) by the formula n 1 1 (x + y), p ψ(y)dpdy. Aψ(x) = eip·(x−y) a 2π 2 Rn ×Rn : S(R2n )−→S (R2n ) by the formula We now associate to A an operator A n = 1 A aσ (z0 )T(z0 )dz0 2π R2n

(10)

where T(z0 ) acts on S (R2n ) via i T(z0 )Ψ(z) = e− 2 σ(z,z0 ) Ψ(z − z0 ).

(11)

Observe that the operators T(z0 ) satisfy the same canonical commutation relations as the Heisenberg–Weyl operators T (z0 ) namely T(z1 )T(z0 ) = eiσ(z1 ,z0 ) T(z0 )T(z1 )

(12)

hence they correspond as we will see below to some unitary representation of the Heisenberg (not on L2 (Rn ) but on a closed subspace of L2 (R2n )). For φ ∈ S(Rn ), ||φ||L2 = 1, we define an operator Uφ : L2 (Rn ) −→ L2 (R2n ) by the formula π n/2 W (ψ, φ)( 12 z) (13) Uφ ψ(z) = 2 where W (ψ, φ) is the cross-Wigner transform of the pair (ψ, φ): n 1 W (ψ, φ)(x, p) = e−ip·y ψ(x + 12 y)φ(x − 12 y)dy. 2π Rn We will call Uφ the wave-packet transform with window φ; is essentially the shorttime Fourier transform Vφ used in time-frequency analysis and defined by e−2πiω·t ψ(t)φ(t − x)dt. Vφ (x, ω) = Rn


5

When φ is a Gaussian both Uφ and Vφ are closely related to the Bargmann transform [1]. We have (de Gosson [11], Theorem 10.6 p. 312): Theorem 1. Let φ ∈ S(Rn ). Then: (i) Uφ is an isometry of L2 (Rn ) on a closed subspace Hφ of L2 (R2n ); (ii) We have Uφ∗ Uφ = I on L2 (Rn ) and the operator Pφ = Uφ Uφ∗ is the orthogonal projection in L2 (R2n ) onto the space Hφ ; (iii) The intertwining formulae φ = Uφ A T(z0 )Uφ = Uφ T (z0 ) , AU

(14)

hold for all φ ∈ S(Rn ). In particular (iii) implies that we have 1 ∂ 2 xj + i ∂pj Uφ ψ = Uφ (xj ψ) 1 ∂ ∂ p − i j 2 ∂xj Uφ ψ = Uφ (−i ∂xj ψ)

(15a) (15b)

is the phase-space for all ψ ∈ S(Rn ). This motivates the following notation: if A operator with symbol a we will write = a( 1 x + i ∂ +, 1 p − i ∂ ). A 2 ∂p 2 ∂x 1.2. The composition property We will need the following composition property: Proposition 2. (i) Assume that the compose AB of the Weyl operators exists and B exists as well and we have A B = AB. is a Weyl operator. Then A n n has kernel (ii) If the Weyl operator R has kernel KR ∈ S(R × R ) then R 2n 2n KR ∈ S(R × R ). Proof. (i) See de Gosson [11], Proposition 10.13, p. 320. (ii) The Weyl symbol r of R is related to the kernel KR of R by the formula e−ip·y K(x + 12 y, x − 12 y)dy r(x, p) = Rn

hence r ∈ S(R ) if KR ∈ S(Rn × Rn ) from which follows that we also have rσ ∈ S(R2n ) where rσ is the symplectic Fourier transform of r. We have, using (10), n i 1 RΨ(z) = rσ (z0 )e− 2 σ(z,z0 ) Ψ(z − z0 )dz0 2π R2n that is, setting u = z − z0 : n i 1 RΨ(z) = rσ (z − u)e− 2 σ(z,z−u) Ψ(u)du. 2π R2n 2n

6

M. de Gosson

is thus given by the formula Since σ(z, z − u) = −σ(z, u) the kernel of R n 1 i KR (z, u) = e 2 σ(z,u) rσ (z − u). 2π The function (z, u) −→ rσ (z − u) being in S(R2n × R2n ) so is KR .

(16)

1.3. Symplectic covariance Since we are in the business of Weyl operators, let us study the symplectic covariance properties of the corresponding phase space operators. Recall that the symplectic group Sp(n) has a double covering which can be faithfully represented by a group of unitary operators acting on L2 (R2n ); that group is called the metaplectic group and we will denote it by Mp(n). The standard “metaplectic covariance formula” for Weyl calculus reads as follows: for s ∈ Sp(n) let S be any of the two operators in Mp(n) corresponding to s. Then if A has Weyl symbol a the operator SAS −1 has Weyl symbol a ◦ s−1 . In de Gosson [8] we proved that if s has no eigenvalue equal to one, then

n i 1 i±ν 2 exp Ms z T (z)dz (17) S= 2π 2 | det(s − I)| R2n where MS = MsT is the symplectic Cayley transform defined by MS = 12 J(s + I)(s − I)−1 and ν an integer (the “Conley–Zehnder index”) that need not preoccupy us here; in addition we showed that every S ∈ Mp(n) can be written as the product of exactly two operators of the type above. In view of formula (10) the operator S determines naturally a phase-space operator n

1 i i±ν (18) S = exp Ms z 2 T(z)dz 2π 2 | det(s − I)| R2n satisfying the second intertwining relation (14) in Theorem 1. Proposition 3. Let s ∈ Sp(n) and = a( 1 x + i ∂ , 1 p − i ∂ ) A 2 ∂p 2 ∂x = (a ◦ s−1 )( 1 x + i ∂ , 1 p − i ∂ ). B 2 ∂p 2 ∂x = SA S−1 where S ∈ Mp(n) is any of the two metaplectic operators We have B corresponding to s. Proof. It is an immediate consequence of the usual symplectic covariance formula ∂ ∂ (a ◦ s−1 )(x, −i ∂x ) = Sa(x, −i ∂x )S −1

for Weyl operators (see de Gosson [11], Chapter 10, §10.3.3). Alternatively, it follows from the metaplectic covariance relation ST(z)S−1 = T(sz).


7

2. Alternative quantizations 2.1. Some physical considerations For the physical background we refer the reader to Landau and Lifshitz [17] or Messiah [18]. Consider an electron placed in a strong uniform magnetic field B directed along the z axis: B = (0, 0, Bz ); if A is a vector potential defined by B = ∇r × A (r = (x, y, z)) the Hamiltonian function is e 2 1 p− A (19) H= 2m c with p = (px , py , pz ). Choosing the vector potential A such that A = 12 (r × B) (“symmetric gauge”) and disregarding the unessential component pz , the Hamiltonian H takes the particular form 2 1 2 mωL (px + p2y ) − ωL Lz + (x2 + y 2 ) 2m 2 with ωL = eBz /2mc (“Larmor frequency”) and Lz = xpy − ypx is the angular moment in the z direction. The corresponding quantum operator is given by 2 ∂ mωL ∂ 2 Δ + iωL y −x + (x2 + y 2 ) (20) Hsym = − 2m ∂x ∂y 2

Hsym =

where Δ is the Laplacian in the x, y variables. As already observed in the Introduction this operator reduces to Wong’s operator (3) if units are chosen appropriately. Suppose now we choose the vector potential as A = (−Bz y, 0, 0) (this is called the “Landau gauge” in Physics); then the Hamiltonian function takes the simple form

2 1 eBz HLan = px + y + p2y 2m c and the corresponding the quantum operator is ∂ 1 2 Δ − iωy + mω 2 y 2 (21) 2m ∂x 2 where ω = eBz /2mc is the “cyclotron frequency”. It is easy to determine the spectrum of HLan (which is the same as that of Hsym since a change of gauge does not affect the spectrum). Noticing that x does not appear explicitly in the function i HLan the momentum px is thus a conserved quantity; setting ψ(x, y) = e px x φ(y) it is easy to check that the eigenvalue problem HLan ψ = Eψ reduces to HLan = −

−

2 d2 φ 1 + mω 2 (y − y0 )2 = Eψ 2m dy 2 2

where y0 = −px c/eBz ; but this is just the eigenvalue problem for a translated harmonic oscillator with mass m and frequency ω, whose spectrum consists of the sequence EN = (N + 12 )ω , N = 0, 1, 2, . . . .

8

M. de Gosson

Choosing appropriate units, we recover the spectrum of Wong’s operator W as announced in the Introduction. It turns out that the harmonic oscillator operator (“Hermite operator”) 2 ∂ 2 + 2m ∂x2 and HLan are

HHer =

1 mω 2 x2 (22) 2 all obtained from the Hamiltonian

as well as the operators Hsym function 1 2 1 H= p + mω 2 x2 2m 2 by applying different quantization rules: ∂ ); • HHer corresponds to the standard prescription (x, p) −→ (x, −i ∂x ∂ 1 ∂ 1 • Hsym corresponds to the rule (x, p) −→ (i ∂p + 2 x, −i ∂x + 2 p); ∂ ∂ • HLan corresponds to the rule (x, p) −→ (i ∂p + 12 x, −i ∂x ). Let us generalize this discussion to more general operators.

2.2. Extension; Bopp quantization As seen above, there is a certain arbitrariness in the definition of our phase-space associated with A. If we replace the “quantum translation operWeyl operator A ator” T(z0 ) defined by (11) by any operator satisfying the canonical commutation Asrelations (12) we will obtain another operator having similar properties as A. sume for instance that we define (choosing units in which = 1), 1 T (z0 )Ψ(z) = ei(p0 ·x− 2 p0 ·x0 ) Ψ(z − z0 )

which amounts to extending the Heisenberg–Weyl operator (9) in a trivial way by allowing them to act on phase-space functions. The corresponding phase-space Weyl operator is obtained by replacing definition (10) by the expression n 1 A = aσ (z0 )T (z0 )dz0 . 2π R2n This choice corresponds to the quantization rules ∂ 1 ∂ + xj , pj −→ −i xj −→ X = i ∂pj 2 ∂xj

(23)

studied in de Gosson [9]; setting y = −p this quantizes the harmonic oscillator Hamiltonian into the operator ∂ (24) W = −Δ + x2 − 2ix ∂y in place of Wong’s operator (3) (see de Gosson [10] for a discussion of other possible quantization compatible with the canonical commutation relations, and their interpretation in terms of classical phases). It is noteworthy that the quantization rules (x, p) −→ (X, P ) listed above obey the canonical commutation rules [X, P ] = i and thus correspond to different (but of course isomorphic) representations of the Heisenberg group (the


9

∂ rule (x, p) −→ (x, −i ∂x ) corresponds to the usual Schr¨ odinger representation). There are of course other choices. One easily verifies that for any quadruple of real numbers (α, β, γ, δ) such that αδ − βγ = 1 the operators

X = αx + iβ

∂ ∂ , P = γp + iδ ∂p ∂x

(25)

satisfy [X, P ] = i and therefore define a bona fide quantization rule for which the (α,β,γ,δ) statements in Theorem 1 remain true for an adequate redefinition Uφ of the transform Uφ for which the intertwining relations (14) should be replaced by ∂ (α,β,γ,δ) (α,β,γ,δ) αx + iβ ψ = Uφ (xj ψ) Uφ ∂p ∂ (α,β,γ,δ) (α,β,γ,δ) ∂ ψ = Uφ (−i ∂x ψ). γp + iδ Uφ j ∂x Applying the rules (25) to the harmonic oscillator Hamiltonian one obtains a whole class of degenerate elliptic operators: 2 ∂ 1 2 ∂2 ∂ 1 (α,β,γ,δ) 2 ∂ A − δγp =− +β − i αβx δ + (α2 x2 + γ 2 p2 ). 2 ∂x2 ∂p2 ∂p ∂x 4 We notice that the choice α = γ = 1, β = −δ = 1/2 yields the so-called “Bopp quantization” (Bopp [3]) rules XBopp = x + i

∂ ∂ , PBopp = p − i 2 ∂p 2 ∂x

which play an important role in deformation quantization. Still, physically, the phase-space quantization choice = 1 x + i ∂ , P = 1 p − i ∂ X 2 ∂p 2 ∂x

(26)

has particular symmetry properties which makes it more attractive; it seems to play a role in the study of quantum gravity (Isidro and de Gosson [15, 16]).

3. Hypoellipticity in the Schwartz space S(Rn ) Let us now study the question of global hypoellipticity for a class of phase-space operators generalizing those of Wong. 3.1. Global hypoellipticity and Shubin symbols Let A be a partial differential operator (or more generally, a pseudodifferential operator). One says that A is hypoelliptic (in the usual sense) if Aψ ∈ C ∞ (Rn ) implies that ψ ∈ C ∞ (Rn ). Recall that the partial differential operator aα (x)Dxα , aα ∈ C ∞ (Rn ) A(x, D) = |α|≤m

10

M. de Gosson

(or, more generally, a classical pseudodifferential operator) is said to be elliptic if its principal symbol am (x, p) = aα (x)pα |α|=m

has the property that am (z) = 0 if and only if z = 0. An elliptic operator is hypoelliptic (in the usual sense), as is easily seen by constructing an approximate inverse, or parametrix (see for instance Shubin [19] or Trèves [22]). More precisely B (resp. B ) is called a left (resp. right) parametrix if BA = I + R (resp. AB = I + R ) where R and R are smoothing operators, that is R, R : S (Rn ) −→ C ∞ (Rn ) (equivalently, R and R have smooth kernels). The hypoellipticity of an elliptic operator easily follows using the existence of a left parametrix; assume in fact that Aψ = φ is in C ∞ (Rn ). Then ψ = Bφ − Rψ is also in C ∞ (Rn ): we have Bφ ∈ C ∞ (Rn ) because φ ∈ C ∞ (Rn ) and, on the other hand Rψ ∈ C ∞ (Rn ) because R is smoothing. For our purposes the notion of ordinary hypoellipticity as just described is rather useless, because it gives no possibility of controlling the behaviour at infinity. It is preferable to use the notion of global hypoellipticity as introduced by Shubin [19], Corollary 25.1, p. 186 (also Boggiatto et al. [2], p. 70). We will say that a linear operator A : S (Rn ) −→ S (Rn ) is globally hypoelliptic if ψ ∈ S (Rn ) and Aψ ∈ S(Rn ) =⇒ ψ ∈ S(Rn ).

(27)

Shubin [19] (Chapter IV, §23) has introduced very convenient classes of glob1 ,m0 (R2n ) (m0 , m1 ∈ R and 0 < ρ ≤ 1) be the ally hypoelliptic operators. Let HΓm ρ complex vector space of all functions a ∈ C ∞ (Rn ) for which there exists R ≥ 0 such that for |z| ≥ R the following estimates hold: C0 |z|m0 ≤ |a(z)| ≤ C1 |z|m1 |Dzα a(z)|

−ρ|α|

≤ Cα |a(z)||z|

(28a) (28b)

with C0 , C1 , Cα ≥ 0. The main properties we will need are summarized in the following Theorem: 1 ,m0 (R2n ) and A the Weyl operator with symTheorem 4 (Shubin). Let a ∈ HΓm ρ bol a. 1 ,−m0 (R2n ) such that (i) There exists a Weyl operator B with symbol b ∈ HΓ−m ρ BA = I + R1 and AB = I + R2 where R1 , R2 have kernels in S(Rn × Rn ) (ii) The operator A is globally hypoelliptic.

(Note that (ii) immediately follows from (i).) We will call the operator B a Shubin parametrix of A.


11

In the context of Wong’s operator W the following example is crucial: Example 5. The Hermite operator −Δ + |x|2 is globally hypoelliptic: it suffices to note that the Weyl symbol of −Δ + |x|2 is a(z) = |z|2 and thus trivially satisfies the estimates (28) with m0 = m1 = 2, ρ = 1. In next subsection we generalize this example. 3.2. Main result We claim that: 1 ,m0 Theorem 6. Assume that the Weyl symbol a of A is in HΓm (R2n ) (hence A is ρ a globally hypoelliptic pseudodifferential operator). Then the phase-space operator is globally hypoelliptic: A

∈ S(R2n ) =⇒ Ψ ∈ S(R2n ). Ψ ∈ S (R2n ) , AΨ Proof. Let B be a Shubin parametrix of A: BA = I + R where R is an operator = Ψ we have Ψ = BΨ − RΨ. Clearly with kernel in S(Rn × Rn ). Writing AΨ 2n 2n BΨ ∈ S(R ) hence BΨ ∈ S(R ) so it suffices to show that RΨ ∈ S(R2n ) for is in S(Rn × Rn ), all Ψ ∈ S (R2n ); for this it suffices to show that the kernel of R but this follows from Proposition 2, (ii). Quadratic forms on R2n are very interesting objects: they can be viewed as the Hamiltonian functions generalizing the harmonic oscillator. We recall the following very useful symplectic diagonalization result, which goes back to Williamson [23]: for every real positive-definite symmetric matrix there exists such that s ∈ Sp(n) Λ 0 the diagonal sT M s = D where D is a diagonal matrix of the type 0 Λ entries of Λ consisting of the moduli ω1 , . . . , ωn of the eigenvalues of JM (these are precisely of the type ±iωj since JM is equivalent to the antisymmetric matrix M 1/2 JM 1/2 ). Theorem 6 has the following interesting consequence: Corollary 7. Let a be a positive-definite quadratic form on R2n : a(z) = with M = M T > 0.

1 2Mz

·z

(i) The associated phase space operator = a( 1 x + i ∂ , 1 p − i ∂ ) A 2 ∂p 2 ∂x obtained by the quantization rule (26) is globally hypoelliptic. = SA S−1 with symbol (ii) There exists s ∈ Sp(n) such that the operator B −1 b = a ◦ s is given by the formula

n 2 ωj 1 ∂ 1 ∂ 2 B= + 2 p − i ∂x . 2 x + i ∂p 2 j=1

12

M. de Gosson

1 ,m0 Proof. (i) It suffices to show that a ∈ HΓm (R2n ) for some m1 , m0 , ρ. Writing ρ T S M S = D with S and D as above, and ordering the entries of D so that ω1 ≤ · · · ≤ ωn we have ω1 2 ωn 2 |z| ≤ a(z) ≤ |z| . 2 2

We have |sz| ≤ ||s|| · |z| and |s−1 z| ≤ ||s−1 || · |z| hence there exist constants C0 , C1 such that C0 |z|2 ≤ a(z) ≤ C1 |z|2 which is condition (28a) with m0 = m1 = 2. Let α be a multi-index; if |α| > 2 then |Dzα a(z)| = 0. Suppose |α| ≤ 2; in view of the homogeneity of a there exists Cα > 0 such that |Dzα a(z)| ≤ Cα |z|2 |z|−|α| = Cα a(z)|z|−|α| so that condition (28b) holds with ρ = 1. The statement (ii) is an obvious consequence of the fact that if a(z) = 12 M z · z then b(z) = a(s−1 s) =

n ωj j=1

2

(x2j + p2j ).

(Of course, in the first part of the proof of Corollary 7 we could have used standard diagonalization of the symbol a by orthogonal matrices.)

4. Concluding remarks A question which poses itself is whether the global hypoellipticity of Section 3 can be generalized to other functional spaces than the Schwartz space S(R2n ). As has been pointed out to me by Franz Luef (NuHAG, Vienna), this might very well be the case if one considers Feichtinger’s [6, 7] weighted modulation spaces Mvp,q (also see Gröchenig [13], Chapter 11 for a study of these spaces). The main interest of modulation spaces comes from the fact that they allow a simultaneous control of both local regularity and decay at infinity. Besides their intrinsic interest in Functional Analysis, they play an important role not only in time-frequency analysis (for which they were originally designed), but also in the study of the regularity of the solutions of Schr¨ odinger’s equation as we have shown in [12]. We will come back to this important question in a forthcoming paper. In that context let us mention that Dasgupta and Wong [5] have obtained interesting regularity results in Gelfand–Shilov spaces; their work certainly deserves to be extended to the operators we considered in Corollary 7. Another fact which is certainly worth to be scrutinized is the following. As we pointed out several times in this paper, a change of phase-space quantization seems to correspond to a change of gauge. Is there any “universal rule” behind this property which we only checked for physical operators associated with a magnetic field?


13

References [1] V. Bargmann, On a Hilbert Space of Analytic Functions and an Associated Integral Transform. Part II. A Family of Related Function Spaces Application to Distribution Theory, Communications on Pure and Applied Mathematics 20(1) (1967), 1–101. [2] P. Boggiatto, E. Buzano and L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie Verlag, 1996. [3] F. Bopp, La mécanique quantique est-elle une mécanique statistique particulière? Ann. Inst. H. Poincar´ e 15 (1956), 81–112. [4] E. Cordero and K. Gr¨ ochenig, Symbolic calculus and Fredholm property for localization operators, J. Fourier Anal. Appl. 12(3) (2006), 371–392. [5] 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), 43–53. [6] H.G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92 (1981), 269–289. [7] H.G. Feichtinger, Modulation spaces on locally compact abelian groups,Technical Report, University of Vienna, 1983. [8] M. de Gosson, On the Weyl Representation of Metaplectic Operators. Letters Math. Phys.72 (2005), 129–142. [9] M. de Gosson, Extended Weyl Calculus and Application to the Phase-Space Schr¨ odinger Equation, J. Phys. A: Math. and General 38 (2005), L325–L329 [10] M. de Gosson, Symplectically Covariant Schr¨ odinger Equation in Phase Space, J. Phys.A: Math. Gen. 38 (2005), 9263–9287 [11] M. de Gosson, Symplectic Geometry and Quantum Mechanics, “Operator Theory: Advances and Applications” (Subseries: “Advances in Partial Differential Equations”), Vol. 166 Birkh¨ auser, Basel, 2006 [12] M. de Gosson, Semi-classical propagation of wavepackets for the phase space Schr¨ odinger equation; interpretation in terms of the Feichtinger algebra, J. Phys. A: Math. and Theo. [in press] [13] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal., Birkh¨ auser, Boston, MA, 2000. [14] A. Grossmann, G. Loupias, E.M. Stein, An algebra of pseudo-differential operators and quantum mechanics in phase space, Ann. Inst. Fourier, Grenoble 18(2) (1968), 343–368. [15] J. Isidro and M. de Gosson, Abelian gerbes as a gauge theory of quantum mechanics on phase space, J. Phys. A: Math. and Theo. 40 (2007), 3549–3567. [16] J. Isidro and M. de Gosson, A gauge theory of quantum mechanics, Modern Physics Letters A, 22(3) (2007), 191–200. [17] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, Pergamon Press, 1997. [18] A. Messiah, Quantum Mechanics (in two volumes), North-Holland Publ. Co., 1991; translated from the French; original title: Mécanique Quantique, Dunod, Paris, 1961. [19] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, 1987. [original Russian edition in Nauka, Moskva, 1978.] [20] G. Torres-Vega and J.H. Frederick, Quantum mechanics in phase space: New approaches to the correspondence principle. J. Chem. Phys. 93 (12) (1990), 8862–8874.

14

M. de Gosson

[21] G. Torres-Vega and J.H. Frederick, A quantum mechanical representation in phase space, J. Chem. Phys. 98(4) (1993), 3103–3120. [22] F. Trèves, Introduction to Pseudo Differential and Fourier Integral Operators, University Series in Mathematics, Plenum Publ. Co., 1981. [23] J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math. 58 (1936), 141–163 [24] M.W. Wong, Weyl Transforms. Springer-Verlag, 1998. [25] M.W. Wong, Weyl transforms and a degenerate elliptic partial differential equation, Proc. R. Soc. A 461 (2005), 3863–3870. [doi:10.1098/rspa.2005.1560] [26] M.W. Wong, Weyl transforms, the heat kernel and Green Function of a degenerate elliptic operator, Ann. Global Anal. and Geom. 28 (2005), 271–283. [27] M.W. Wong, The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 393–404. Maurice de Gosson Max-Planck-Institut f¨ ur Mathematik Pf. 7280 D-53072 Bonn e-mail: [email protected] Current address: University of Vienna Faculty of Mathematics, NUHAG A-1090 Vienna, Austria


Classes of Degenerate Elliptic Operators in Gelfand–Shilov Spaces Todor Gramchev, Stevan Pilipović and Luigi Rodino Abstract. We propose a novel approach for the study of the uniform regularity and the decay at infinity for Shubin type pseudo-differential operators which are globally hypoelliptic but not necessarily globally and even locally elliptic. The basic idea is to use the special role of the Hermite functions for the characterization of inductive and projective Gelfand–Shilov spaces. In this way we transform the problem to infinite dimensional linear systems on S Banach spaces of sequences by using Fourier series expansion with respect to the Hermite functions. As applications of our general results we obtain new theorems for global hypoellipticity for classes of degenerate operators in tensorized generalizations of Shubin spaces and in inductive and projective Gelfand–Shilov spaces. Mathematics Subject Classification (2000). Primary 47F30; Secondary 46F05, 35B65. Keywords. Harmonic oscillator type operators, S spaces, global hypoellipticity, Gelfand–Shilov spaces.

1. Introduction The aim of this paper is to investigate the uniform regularity and the type of exponential decay at infinity for the solutions to linear partial differential equations P u = f globally defined in Rn , where P is an operator with polynomial coefficients of Shubin type (cf. Shubin [26]) without assuming global ellipticity. We focus our attention on tensor products of harmonic oscillators. In fact, the globally elliptic Shubin operators generalize the Schr¨ odinger harmonic oscillator operator (1.1) H = −Δ + |x|2 , T.G and L.R. were partially supported by GNAMPA–INDAM, S.P. was partially supported by the Ministry of Sciences of Serbia, project 144016.

16

T. Gramchev, S. Pilipović and L. Rodino

appearing in QuantumMechanics. The spectrum of H in L2 (Rn ) is discrete with n eigenvalues λ = λk = j=1 (2kj + 1), k = (k1 , . . . , kn ) ∈ Zn+ while eigenfunctions are the Hermite functions n √ −n/4 −1/2 u(x) = Hk (x) = π (kj !) hkj ( 2xj ) exp(−|x|2 /2), (1.2) j=1

where hr (t) stands for the rth Hermite polynomial, e.g., cf. Magnus et al [18]. We refer to the books of Helffer [13] and Boggiatto et al [4] for further generalizations and references concerning spectral theory of globally elliptic operators. We may distinguish two different aspects. The first is the behaviour at infinity, which in (1.2) is super-exponential of order 2, i.e., for positive constants C, ε: |u(x)| ≤ Ce−ε|x| , 2

x ∈ Rn .

(1.3)

The main interest here comes historically from Quantum Mechanics, where the exponential decay of eigenfunctions has been intensively studied, see for instance Agmon [2]. As a second aspect, returning to the harmonic oscillator (1.1), we point out the fact that the Hermite functions Hk (x), k ∈ Zn+ , (1.2) are not only entire functions, but for every γ ∈]0, 1/2[ there exists Aγ > 0 such that |∂zα Hk (z)| ≤ A|α|+|k|+1 (α!)1/2 e−ε|z| , γ 2

z ∈ Cn , |Imz| < γ|Rez|, α ∈ Zn+ ,

(1.4)

with ε = (1 − 2γ)/2, see [6]. The estimates (1.4) lead in a natural way to the idea that the appropriate functional framework to study (1.3) and (1.4) simultaneously is given by the spaces of Gelfand–Shilov type (cf. the classical book of Gelfand and Shilov [10], see also Mitjagin [19], Avantaggiati [3], Pilipovic [20] and references in these papers). We recall that f belongs to the inductive (respectively, projective) Gelfand–Shilov space Sνμ (Rn ) (respectively, Σμν (Rn )), μ > 0, ν > 0, μ + ν ≥ 1 (respectively, μ + ν > 1), iff f ∈ C ∞ (Rn ) and there exists A > 0 (respectively, for every A > 0) such that we can find C > 0 satisfying the following estimates |∂xα f (x)| ≤ CA−|α| (α!)μ e−A|x| for all x ∈ R , α ∈ n

Zn+

1/ν

(1.5)

or, equivalently

sup |xβ ∂xα f (x)| ≤ CA−|α|−|β| (α!)μ (β!)ν ,

x∈Rn

α, β ∈ Zn+ .

(1.6)

This definition gives the smallest scale of nontrivial inductive Gelfand–Shilov μ type space S1−μ (Rn ), μ ∈]0, 1[ while Σμν (Rn ) contains nonzero functions iff μ+ ν > 1/2

1/2

1. Especially if σ = μ = 1/2, then Σ1/2 = {0}. Note that S1/2 ⊂ Σσσ ⊂ Sσσ , σ > 1/2. We call the elements of their respective duals (Sνμ (Rn )) and (Σμν (Rn )) tempered inductive and projective μ-ultradistributions. The bounds (1.5), (1.6) with μ ≤ 1 grant that f extends to Cn as an entire function, with uniform estimates in neighbourhoods of the real axis, see [10] for

Degenerate Elliptic Operators

17

precise statements. In particular, for the Hermite functions in (1.2), we can read 1/2 u ∈ S1/2 (Rn ). We also mention that projective spaces Σμμ and their further extensions (cf. [20] for another definition in critical case σ = 1/2) are used as models in chaos expansions with applications to stochastic differential equations [22]. Recently, Cappiello et al [6], [7] have proved the Sνμ −regularity of eigenfunctions to Shubin type partial differential operators in Rn P = cαβ xβ Dxα , (1.7) |α|+|β|≤m

where m is a positive integer, provided P is globally elliptic, namely, there exist C > 0 and R > 0 such that β α 2 2 m/2 c x ξ , |x| + |ξ| ≥ R. (1.8) αβ ≥ C(1 + |x| + |ξ| ) |α|+|β|≤m

Global ellipticity in the previous sense implies both local regularity and asymptotic decay of the solutions, namely we have the following basic result (see for example Shubin [26]): P u = f ∈ S(Rn ) for u ∈ S (Rn ) implies actually u ∈ S(Rn ). This is what is meant by global hypoellipticity in S(Rn ), and same definition we shall use replacing the Schwartz class S(Rn ) by the Gelfand–Shilov space. The main theorem in [6] states that P is also globally hypoelliptic in the Gelfand–Shilov spaces Sνμ (Rn ), μ, ν > 0, μ ≥ 1/2, ν ≥ 1/2. In this paper we limit our attention the symmetric Gelfand–Shilov spaces Sμμ , (Rn ), μ ≥ 1/2 and Σμμ (Rn ), μ > 1/2. One is led naturally to ask whether we can weaken the global ellipticity condition (1.8) for linear operators with polynomial coefficients. One of the prime motivation for our investigations is the second-order degenerate Shubin type operator in R2 considered by Wong [27]–[29]. 1 W = Dx2 1 + Dx2 2 + (x21 + x22 ) − x2 Dx1 + x1 Dx2 4 2 2 x2 x1 1 ˜ = Dx 1 − + Dx2 + = − (Z Z˜ + ZZ) 2 2 2

(1.9)

where, ∂ 1 ∂ 1 + z¯, Z˜ = − z, Dxk = i−1 ∂xk , ∂z 2 ∂ z¯ 2 1 1 ∂ ∂ = (∂x1 + i∂x2 ), = (∂x1 − i∂x2 ), z = x1 + ix2 . ∂z 2 ∂ z¯ 2 We mention also that Popivanov [24] has studied for global hypoellipticity in S(Rn ) another class of second-order degenerate Shubin type operators. In particular, it is shown in [29] that W is globally hypoelliptic, namely Z=

u ∈ S (R2 ), W u ∈ S(R2 ) ⇒ u ∈ S(R2 ).

(1.10)

18


Later on, Dasgupta and Wong [9] established global hypoellipticity for W in the 1/2 critical inductive Gelfand–Shilov class S1/2 (R2 ). We propose a novel approach for dealing with classes of degenerate Shubin type operators and establishing global hypoellipticity both in the Schwartz class and in Gelfand–Shilov classes. As a particular case, we recapture the aforementioned global hypoellipticity results in [29], [9]. The crucial ingredient of our techniques is the use the framework of the so-called S spaces as well as the special role of the Hermite functions for the characterization of spaces of Gelfand–Shilov type ultradifferentiable functions (see Antosik et al [1], Reed and Simon [25], Kashpirovskij [16], Avantaggiati [3], Pilipović [20], [21], Holden, et al. [14], Kami´ nski et al [15], Langenbruch [17]), Budinˇcević et al [5]. The paper is organized as follows. In Section 2, we recall Shubin class of spaces ([26]) and (as a novelty) characterize them through Hermite expansions. Moreover, we recall the characterization of Gelfand–Shilov type spaces through Hermite expansions and we derive new characterizations of these spaces through estimates of powers of harmonic oscillators. In Section 3 we introduce tensor-valued harmonic oscillators, characterize them through Hermite expansions, define tensorized generalization of Sobolev– Shubin spaces and show the main global hypoellipticity results. Section 4 deals with the degenerate operator W in [29], namely, we reduce W by means of a time frequency analysis conjugation operator to the one dimensional harmonic oscillator (a particular case of tensorized operators studied in the previous section). Hence, as an outcome we give a new proof and extensions to the projective Gelfand–Shilov spaces of the global hypoellipticity results in [29], [9]. Moreover, by intrinsic characterizations of W −1 , we show that the loss of derivatives for W is 2 in the scale of Sobolev–Shubin spaces.

2. Hermite expansions for Shubin and Gelfand–Shilov type spaces Let s ∈ R. We denote by Qs (Rn ) the spaces introduced by Shubin [26], namely, Qs (Rn ) := {u ∈ S (Rn ) : uQs := (1 + |x|2 − Δ)s/2 uL2 (Rn ) < +∞}.

(2.1)

Actually, this space can be defined through another equivalent sequence of norms, see [26], Ch. IV, 25.3, showing that the Hilbert structure induces the dual pairing of Qs and Q−s . Moreover, every A ∈ Gm ρ (cf. [26], Ch. IV) defines a continuous mapping A : Qs (Rn ) → Qs−m (Rn ) and A : Qs (Rn ) → Qs−m−ε (Rn ), ε > 0, is a compact one. We propose a novel result, a characterization of Qs be means of the Hermite functions Hj (x), j ∈ Zn+ .


19

We recall that, given a tempered distribution v ∈ S (Rn ), or more generally, a tempered μ-ultradistribution v ∈ (Sμμ (Rn )) , respectively (Σμμ (Rn )) , we can write α ∈ Zn+ ,

vα = (v, Hα ),

(2.2)

where the dual pairing (v, Hα ) is understood in the sense of tempered distributions or tempered μ ultradistributions. Here α ∈ Zn+ ,

Hα (x) = Hα1 (x1 ) . . . Hαn (xn ), Hk (t) = π −1/4 k!−1/2 e−t

2

√ /2 hk ( 2t),

(2.3)

k ∈ Z+ ,

(2.4)

where 2

hk (t) = (−1)k et

/2

d dt

k

[k/2]

(e−t

2

/2

)=

(−1)r 2−r

r=0

k! tk−2r r!(k − 2r)!

k ∈ Z+ , (2.5)

is the kth Hermite polynomial, e.g., see [18]. Clearly,

[k/2]

Hk (t) = π

−1/4

−1/2

k!

(−1)r 2k/2−2r

r=0

[k/2]

= π −1/4 k!1/2

(−1)r 2k/2−2r

r=0

2 k! tk−2r e−t /2 r!(k − 2r)!

2 1 tk−2r e−t /2 r!(k − 2r)!

k ∈ Z+ .

(2.6)

For more details cf. Holden et al [14], see also Pilipovic [20], [15], Budinˇcević at all [5], Langenbruch [17]. Following [14] (see also Reed and Simon [25]) we introduce an ordering of all n-dimensional multi-indices αj = (αj1 , . . . , αjn ) satisfying 0 ≤ j ≤ k ⇒ |αj | ≤ |αk |.

(2.7)

Now define Hj (x) = Hαj (x), j ∈ Z+ . Here is the first main result of our paper. Theorem 2.1. Let v be a tempered distribution, i.e., v ∈ S (Rn ), so that v= vα Hα (x)

(2.8)

α∈Zn +

or

∞

v˜j Hj (x) with v˜j = vαj ,

j = 0, 1, . . . ,

(2.9)

Then the following statements are equivalent i) v ∈ Qs (Rn ). 1/2 ii) {vα }α∈Zn+ ls := |vα |2 < α >s < +∞.

(2.10)

v=

j=0

iii)

{˜ vj }∞ j=0 ls/n

:=

α∈Zn + ∞

1/2

|˜ vj | < j >

j=0

2

s/n

< +∞.

(2.11)

20


Let now μ ≥ 1/2, respectively μ > 1/2. Then the following properties are equivalent: a) v ∈ Sμμ (Rn ), respectively v ∈ Σμμ (Rn ); b) there exist κ and C > 0, respectively, for every κ > 0 there exists C > 0 such that |vα | ≤ Ce−κ|α|

1/(2μ)

α ∈ Zn+ .

,

(2.12)

c) there exist κ and C > 0, respectively, for every κ > 0 there exists C > 0 such that |˜ vj | ≤ Ce−κj

1/(2nμ)

j ∈ Z+ .

,

(2.13)

d) there exists C > 0 and A > 0, respectively, for every C > 0 there exists A > 0 such that (|x|2 − Δ)s/2 uL2 (Rn ) ≤ AC s ssμ ,

s ∈ N;

(2.14)

Proof. We will prove the equivalence of i) and ii) since the equivalence of ii) and iii) is clear. Since v = α∈Zn vα Hα in S (Rn ) and +

(|x|2 − Δ)s/2 Hα = (λα )s/2 Hα ,

λα =

n

(2αj + 1), α ∈ Zn+ ,

j=1

we have (|x|2 − Δ)s/2 v, Hα = v, (|x|2 − Δ)s/2 Hj = vα λs/2 α Hα . By the continuity, this implies (|x|2 − Δ)s/2 v =

(λα )s/2 vα Hα

α∈Zn +

in S (Rn ) and thus in L2 (Rn ). This gives that (|x|2 − Δ)s/2 has a left inverse in L2 (Rn ), i.e. (|x|2 − Δ)−s/2 (|x|2 − Δ)s/2 = I. We have vQs = (1 + |x|2 − Δ)s/2 vL2 = (1 + |x|2 − Δ)s/2 (|x|2 − Δ)−s/2 (|x|2 − Δ)s/2 vL2 ≤ B(s, Dx )(|x|2 − Δ)s/2 vL2 , where B(s, Dx ) is of zero order. Moreover, by the use of the Hermite expansion for v it follows that the operator norm of B satisfies BL2 →L2 ≤ sup

α∈Zn +

1 1 + n j=1 (2αj + 1)

s/2

=

1+

1 n

s/2

Degenerate Elliptic Operators which leads

21

s/2 1 1+ (|x|2 − Δ)s/2 vL2 n 1/2 s/2 1 = 1+ λsα vα2 n n

vQs ≤

α∈Z+

s 1/2 ∞ n ≤ 2s/2 (2αi + 1) vα2 α∈Zn +

i=1

s/2 n √ s 2 ≤ (4 2) (1 + αi ) vα2 )1/2 α∈Zn +

i=1

√ = (4 2)s {vα }α∈Zn+ ls

(2.15)

We obtain the opposite implication in the same way. The proof of equivalences of a), b) and c) is well known. (cf. [15], [17], [5]). So we will prove the equivalence of b) and d) in the inductive case. Assume b). If u = α∈Zn aα Hα ∈ Sμμ (Rn ), then by (2.12) we get + s/2 n (|x|2 − Δ)s/2 uL2 (Rn ) = (2α + 1) a H j α α 2 n α∈Zn +

≤ C sup

L (R )

j=1

n

α∈Zn +

(2αj + 1)s/2 e−(κ−ε)|α|

1/(2μ)

j=1

−ε|α|1/(2μ) . e × α∈Zn +

(2.16)

2

Now, calculating sup in (2.16), we obtain (2.14). Suppose now that d) holds. We will prove b). By similar calculations as above we obtain that d) yields s ∞ n |vα |2 (2αi + 1) ≤ A2 . (2.17) 2s s2μs C n i=1 α∈Z+

This implies

s/2 |vα | (2αi + 1) ≤A s sμs C i=1

n

(2.18)

for all α ∈ Zn+ , s ∈ N. We conclude the proof by observing that there exists κ > 0 such that n s/2 1/(2μ) ( i=1 (2αi + 1)) sup ≤ eκ|α| (2.19) 2s s2μs C s∈N for all α ∈ Zn+ , which implies (2.12), i.e., b) holds. The proof is complete.

22


3. Tensor products of multidimensional harmonic oscillators Consider the operator R=

n

Ri , where Ri = x2i −

1

∂2 , i = 1, . . . , n ∂x2i

(3.1)

mn n 1 and Rm = Rm 1 . . . Rn , m ∈ Z+ . This operator is not hypoelliptic in the sense of [26], Definition 25.1. The next theorem is a simple consequence of Theorem 2.1, d) and the wellknown inequality

s1 ! . . . sn ! ≤ (s1 + · · · + sn )! ≤ C s1 +···+sn s1 ! . . . sn ! for some C > 0. Theorem 3.1. Let u ∈ S(Rn ). Then u ∈ Σμμ (Rn ), μ > 1/2 (respectively, u ∈ Sμμ (Rn ), μ ≥ 1/2) iff s/2 R uL2 sup < +∞ As s!σ s∈Zn + for every (respectively, for some) A > 0. Let us introduce somewhat more general version of Rm which will be instrumental in characterizing the tensorized Shubin spaces. We take a positive integer d ≤ n such that Rn is decomposed into a direct sum of d linear subspaces Rn = Rn1 ⊕ · · · ⊕ Rnd ,

n1 + · · · + nd = n.

(3.2)

αk ∈ Zn+k , k = 1, . . . , d,

(3.3)

Given α ∈ Zn+ , by (3.2) we can write α = (α1 , . . . , αd ),

and similarly for x ∈ Rn we write x = (x1 , . . . , xd ). We consider the operator Pm;λ (x, Dx ) =

d

(|xk |2 − ΔRnk )mk + λ

k=1

= (|x1 |2 − ΔRn1 )m1 ◦ . . . ◦ (|xd |2 − ΔRnd )md + λ,

(3.4)

with λ ∈ C Note that Pm;λ is of order 2|m| = 2m1 + . . . + 2md but it is not even locally elliptic if d ≥ 2. Indeed, its principal symbol (locally in Rn ) is given by 0 Pm;λ (x, ξ) =

d

(|ξ k |2 )mk .

(3.5)

k=1

The operator Pm;λ (x, Dx ) is not globally hypoelliptic in the sense of [26], Definition 25.1. Moreover, the homogeneous operator defined by the principal part (3.5) is not even locally hypoelliptic as constant coefficients operator if d ≥ 2. The spectral properties and the action in L2 (Rn ) based spaces of the model operators Pm;λ (x, Dx ) are analyzed completely in the next theorem, which is the second main result of the paper.


23

Theorem 3.2. Let u be a tempered or tempered μ-ultradistribution in Rn . We have d k mk Pm;λ (2|j | + 1) + λ uj Hj (x) (3.6) (x, Dx )u = j∈Zn +

k=1

where, recall, j = (j 1 , . . . , j d ) ∈ Zn+1 ⊕ · · · ⊕ Zn+d . Suppose now that Reλ ≥ 0. Then 2 n Pm;λ (x, Dx )u ∈ L (R )

(3.7)

is equivalent to j∈Zn +

d

2 (2|j | + 1) k

mk

+λ

|uj |2 < +∞.

(3.8)

k=1

Moreover, if (Rn ) u ∈ Q2|m|

(3.9)

then (3.8) holds. Finally, the operator Ps (x, Dx )u =

d

(|xk |2 − ΔRnk )sk /2 u

(3.10)

k=1

is well defined for all s = (s1 , . . . , sd ) ∈ Rd by the spectral properties of the powers of the harmonic oscillator, is self-adjoint with the spectrum defined by spec Ps = {

d

(2|j k | + 1)sk /2 : j = (j 1 , . . . , j d ) ∈ Zn+1 × · · · × Zn+d }.

(3.11)

k=1

with the Hermite functions forming an orthonormal L2 (Rn ) base, i.e., Ps (x, Dx )Hj =

d

(2|j k | + 1)sk /2 Hj (x).

(3.12)

k=1

Finally, Ps (x, Dx ) acts continuously in the spaces of tempered μ-ultradistributions by the formula d (2|j k | + 1)sk /2 uj Hj (x). (3.13) Ps (x, Dx )u = j∈Zn +

k=1

and is globally Sμμ (Rn ), (respectively, Σμμ (Rn )) hypoelliptic for every μ ≥ 1/2 (respectively, μ > 1/2).

24


Proof. One observes that Pm;λ (x, Dx )Hj (x) d |xk |2 − ΔRnk )mk Hj k (xk ) + λHj (x) = k=1

=

d

(3.14)

d (2|j k | + 1)mk Hj k (xk ) + λHj (x) = (2|j k | + 1)mk + λ Hj (x)

k=1

k=1

which leads to (3.6) and the equivalence between (3.7) and (3.8). As it concerns the implication (3.9) =⇒ (3.8) it follows from the fact that one can find c1 > 0 such that d (2tk + 1)mk ≤ c1 (t1 + · · · + td + 1)|m| (3.15) k=1

for all tk ≥ 0, k = 1, . . . , d. The validity of (3.11) and (3.12) follows from the spectral theory. The global hypoellipticity part follows from the fact that the equation Ps (x, Dx )u = f , in view of the representations, uj Hj (x), f = fj Hj (x) u= j∈Zn +

j∈Zn +

and (3.13), is equivalent to d

(2|j k | + 1)sk /2 uj = fj ,

j ∈ Zn+ ,

(3.16)

k=1

Therefore, uj =

d

−1 (2|j | + 1) k

sk /2

fj ,

j ∈ Zn+ ,

(3.17)

k=1

and we may apply the characterization of Sμμ (Rn ) and Σμμ (Rn ) in the first part of this section. The proof is complete. Note that (3.9) implies (3.8). This leads naturally to the definition of the following generalization of the Shubin spaces: Qs (Rn1 ⊕ · · · ⊕ Rnd ) := {u ∈ S (Rn ) : s.t. Ps (x, Dx )uL2 < +∞},

(3.18)

for s = (s1 , . . . , sd ) ∈ R . d

Remark 3.3. We note that if d = 1 the space Qs ((Rn1 ⊕ · · · ⊕ Rnd )) coincides with the usual Shubin space. Proposition 3.4. We have Qs (Rn1 ⊕ · · · ⊕ Rnd ) = S(Rn1 ⊕ · · · ⊕ Rnd ) = S(Rn ). s∈Rd

The proof is a direct consequence of the definition and Theorem 2.1.


25

4. Reduction of W to lower-dimensional harmonic oscillators In this section we transform the operator W in [29] and reduce the problem of the global hypoellipticity in the Schwartz class and the Gelfand–Shilov spaces for W to the one-dimensional harmonic oscillator in R2 , which is a particular example of the tensorized harmonic oscillators in the preceding section. Define ˜ = KW K, W (4.1) ˜ where z → K(z) is the operator of conjugation: K(z) = z¯, z ∈ C. Clearly, W W = ˜. WW By [30], if j, k ∈ Z2+ , x, y ∈ R and (Hj )j∈Z+ is the Hermite basis in L2 (R), then y y ej,k (x, y) = V (Hj (x), Hk (y)) = F −1 (Hj (· + )Hk (· − ))(x), 2 2 is an orthonormal basis of L2 (R2 ). Recall, W ej,k (x, y) = (2k + 1)ej,k (x, y), j, k ∈ Z+ , x, y ∈ R. It is well known (see Pilipović and Teofanov [23], Gr¨ ochenig and Zimmermann [12], Cordero et al [8], Cappiello et al [7], and the references therein) that the Wigner and Fourier transform continuously map Sσσ (Rn ) × Sσσ (Rn ) into Sσσ (R2n ). Moreover, we have: Proposition 4.1. ˜ ej,k = (2j + 1)ej,k , j, k ∈ Z+ . W

(4.2)

Proof. It is shown in [29] that ˜ j,k = i(2k + 2)1/2 ej,k+1 . Zej,k = i(2k)1/2 ej,k−1 , Ze

(4.3)

Observing that Hj (−x) = (−1)j Hj (x), x ∈ R, j ∈ Z+ , we have y y ek,j (x, y) = V (Hk (x), Hj (y)) = F Hk · + Hj · − (x) 2 2 y y (x) = F −1 Hk − · + Hj − · − 2 2 y y Hk · − (x) = (−1)j+k F −1 Hj · + 2 2 = (−1)j+k ej,k (x, y), x, y ∈ R. Using this we have ˜ ej,k = KW ((−1)j+k ek,j ) = K((−1)j+k (2j + 1)ek,j ) = (2j + 1)ej,k . W The proof is complete.

Theorem 4.2. Denote by J a mapping L2 (R2 ) → L2 (R2 ) defined by J(Hi (x)Hj (y)) = ei,j (x, y), i, j ∈ N0 , x, y ∈ R,

(4.4)

26


i.e., J

ai,j Hi (x)Hj (y) = ai,j ei,j (x, y),

i,j∈Z2+

(4.5)

i,j∈Z2+

for

h(x, y) =

ai,j Hi (x)Hj (y) ∈ L2 (R2 ).

i,j∈Z2+

Then we have: a) J is an automorphism of L2 (R2 ) onto itself: y y −1 eixξ1 f (ξ1 + , ξ1 − )dξ1 , x, y ∈ R, J(f )(x, y) = (2π) 2 2 R and J −1 (g)(p, q) =

e−it

p+q 2

R 2

g(t, p − q)dt, p, q ∈ R.

(4.6)

(4.7)

b) With the notation R(x) = x + Dx2 and Iy the identity mapping, we have ˜. J(Ix × R(y))J −1 = W, J(R(x) × Iy )J −1 = W For any k = (k1 , k2 ) ∈ Z2+ , ˜ k1 W k2 . J(Rk1 (x) × Rk2 (y))J −1 = W Proof. a) It is enough to prove the assertion for am,n Hm (x)Hn (y) ∈ L2 (R2 ) ∩ S(R2 ). f (x, y) = (m,n)∈Z2+

We have

y y am,n em,n (x,y) = ar,s F Jf (x,y) = Hr ξ1 + Hs ξ1 − (x) 2 2 (m,n)∈Z2+ (r,s)∈Z2+ τ τ −1 Hr ξ1 + Hs ξ1 − (η)Hr (η)Hs (τ )dηdτ ar,s F = 2 2 R R 2 2 (m,n)∈Z+

=

−1

(r,s)∈Z+

× Hr (x)Hs (y) τ τ F −1 ar,s Hr ξ1 + Hs ξ1 − (η)Hr (η)Hs (τ )dηdτ 2 2 R R 2 2

(m,n)∈Z+

(r,s)∈Z+

× Hr (x)Hs (y) τ τ (η)Hr (η)Hs (τ )dηdτ Hr (x)Hs (y) F −1 f ξ1 + ,ξ1 − = 2 2 R R (m,n)∈Z2+ y y (x). = F −1 f ξ1 + ,ξ1 − 2 2 This proves the part a).


27

The assertion b) is an easy consequence by noticing that J(Sσσ ) = Sσσ , J(Σσσ ) = Σσσ , σ ≥ 1/2, respectively, σ > 1/2. This completes the proof.

We may now easily recapture the results of [29] and [9] for global hypoellipticity of W . It is enough to observe that operator J in (4.5) is an automorphism in S(R2 ), Sμμ (R2 ), μ ≥ 1/2, Σμμ (R2 ), μ > 1/2, with extensions to the duals and the inverse J −1 defined by (4.7). It follows from Theorem 4.2., b), that W and are S(R2 ), S μ (R2 ), Σμ (R2 ), globally hypoelliptic iff Ix × R(y), R(x) × Iy are W μ μ globally hypoelliptic in the same frames. This is granted by the last statement in Proposition 3.4 for s = (0, 2) and s = (2, 0) in R2 . Remark 4.3. We point out that Theorem 4.2 can be extended over R2n in the two ways. First, by introducing Wprod (x1 , x2 , Dx1 , Dx2 , . . . , x2n−1 , x2n , Dx2n−1 , Dx2n ) defined by W (x1 , x2 , Dx1 , Dx2 ) . . . W (x2n−1 , x2n , Dx2n−1 , Dx2n ). ˜ prod in a similar way, and secondly, introducing and W 1 Wsum (x, D) = −Δ + |x|2 + (x2k−1 Dx2k − x2k Dx2k−1 ) 4 n

(4.8)

k=1

˜ sum = KWsum K, where K is the operator of conjugation in Cn . and W As an alternative approach with respect to the conjugation formula in Theorem 4.2, b) we also can show that the operator W and its polynomial extensions are globally reducible to tensorized harmonic oscillators via a global Fourier integral operator Ev(x) = exp(iϕ(x, η))ˆ v (η) dη (4.9) R2

where the phase function is given by x1 x2 + x2 η1 + x1 η2 , (4.10) 2 More details and other generalizations will be presented in a forthcoming paper [11]. ϕ(x, η) = η1 η2 +

We conclude this section by proposing some remarks on the Green function W −1 with the kernel (see [29]) ∞ ξ 2 + η2 ), K0 (t) = K1 (ξ, η) = (4π)−1 K0 ( e−t cosh s ds, t > 0. 4 0 Proposition 4.4. The function K0 (t) has the following properties: i) for every ε ∈]0, 1/2[ there exists Aε > 0 such that K0 (t) ≤ Aε (1 + ln t)e−(1/2−ε)t ,

t>0

(4.11)

28


ii) for every ε ∈]0, 1[ there exists Bε > 0 such that K0 (t) ≥ Bε (1 + ln t)e−(1/2−ε)t ,

t > 0.

(4.12)

iii) for every ε ∈]0, 1[ there exists Cε > 0 such that tk−1 |K0 (t)| ≥ Cεk (k − 1)!e−(1−ε)t , (k)

t > 0.

(4.13)

iv) there exists C > 0 such that (τ 2 + η 2 )|k|/2 |K1 (τ, η)| ≤ C |k|+1 |k||k| , (τ, η) ∈ R2 , k = (k1 , k2 ) ∈ Z2+ . (k)

(4.14)

Proof. We have cosh t ≤ et while for every ε ∈]0, 1[ we can find με > 0, νε > 0 such that νε + (1/2 − ε)et ≤ cosh t ≤ (1/2 + ε)et + με ,

t > 0.

(4.15)

t

Therefore, after the change of the variables re = ξ, we obtain +∞ +∞ −t(1/2−ε)er −tνε −νε t K0 (t) ≤ e dr = e ξ −1 e−(1/2−ε)ξ dξ

(4.16)

t

0 +∞

K0 (t) ≥

e

−t(1/2+ε)er −tμε

dr = e

−με t

+∞

ξ −1 e−(1/2+ε)ξ dξ

(4.17)

t

0

The asymptotic analysis of the integral above for t 0 and t → +∞ yield the desired estimates in i) and ii). Similarly, we obtain that +∞ (s) |t(s−1) K0 (t)| = e−t cosh r (cosh r)s dr 0

=e

−νε t

+∞

ξ s−1 e−(1/2−ε)ξ dξ

(4.18)

t

and we use estimates for the Euler gamma function. This gives iii) (and smoothing effects of K1 ). 2 2 2 . Let k = (k1 , k2 ) ∈ Z+ . We have iv) Put g(τ, η) = (4π)−1 K0 τ +η 4 ∞ 2 2 τ +η e− 4 (−1/2)k1 +k2 (cosh tk1 +k2 (−1/2)k1 +k2 τ k1 η k2 (4π)g (k) (τ, η) = 0

+ cosh tk1 +k2 −1 τ k1 η k2 −1 + cosh tk1 +k2 −1 τ k1 −1 η k2 + cosh tk1 +k2 −2 τ k1 −1 η k2 −1 )dt. Put u = et and write s = (u2 + 1)/u. We continue ∞ τ 2 +η2 |k| (k) e−s 4 (−2) (4π)g (τ, η) = 1 |k| k1 k2 s τ η + s|k|−1 τ k1 η k2 −1 + s|k|−1 τ k1 −1 η k2 + s|k|−2 τ k1 −1 η k2 −1 du/u. ∞ We will use 1 (u2 + 1)3 /u4 du < ∞ and the inequality (for m > 3) 2 2 sup sm−3 e−s(τ +η )/4 ≤ s

(m − 3)m−3 . em−3 (τ 2 + η 2 )m−3


29

We continue 2|k| (4π)|g (k) (τ, η)| ≤ C

4|k|−3 (|k| − 3)|k|−3 k1 k2 |τ | η e|k|−3 (τ 2 + η 2 )|k|−3 +

(4|k|−4 (|k| − 4)|k|−4 k1 k2 −1 |τ | η e|k|−4 (τ 2 + η 2 )|k|−4

+

4|k|−4 (|k| − 4)|k|−4 k1 −1 k2 |τ | η e|k|−4 (τ 2 + η 2 )|k|−4

+

4|k|−5 (|k| − 5)|k|−5 k1 −1 k2 −1 |τ | η . e|k|−5 (τ 2 + η 2 )|k|−5

Now we use

1 |τ |p |η|q ≤ p+q . 2 2 p+q 2 (|τ | + |η| ) (|τ | + |η|2 ) 2 With this and suitable C > 0 we have (4.14). Combining the preceding results, we obtain the main global hypoellipticity results in [29], [9] with some generalizations in the projective Gelfand–Shilov spaces and in the Sobolev–Shubin type spaces in R2 . Proposition 4.5. We have i) W −1 : Qs (R2 ) → Qs (R2 ), s ∈ R, is continuous and the loss of derivatives for W equals 2. ii) Let σ ≥ 1/2, respectively, σ > 1/2. Then W : Sσσ (R2 ) → Sσσ (R2 ), respectively, W : Σσσ (R2 ) → Σσσ (R2 )

(4.19)

is an automorphism. iii) Let for some m ∈ Zn+ , W m u = f ∈ Sσσ (R2 ), σ ≥ 1/2 (respectively, ∈ Σσσ (R2 ), σ > 1/2). Then u ∈ Sσσ (R2 ), σ ≥ 1/2 (respectively, ∈ Σσσ (R2 ), σ > 1/2). Proof. We only explain that the second part of i) follows from Proposition 4.4 iii) with k = 1 and the definition of K1 (see also Proposition 4.4 iv)). Acknowledgment The research was initiated during the stay of S.P. as a visiting professor in the University of Cagliari in January 2007 and was concluded while T.G. and L.R. visited the University of Novi Sad in December 2007. The authors thank the inviting institutions for the support and the excellent condition for research.

30


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[20] S. Pilipović, Generalization of Zemanian spaces of generalized functions which elements have series expansion, SIAM J. Math. Anal. 17 (1986), 477–484. [21] S. Pilipovic, Tempered ultradistributions, Boll. Unione Mat. Ital. VII. Ser. B 2 (1988), 235–251. [22] S. Pilipović, and D. Seleˇsi, Expansion theorems for generalized random processes, Wick products and applications to stochastic differential equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), 79–110. [23] S. Pilipovic and N. Teofanov, Pseudodifferential operators on ultramodulation spaces, J. Funct. Anal. 208 (2004), 194–228. [24] P. Popivanov, A link between between small divisors and smoothness of the solutions of a class of partial differential equations, Ann. Global Anal. Geom. 1, (1983), 77–92. [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics I Academic Press, 1972. [26] M. Shubin, Pseudodifferential Operators and Spectral theory. Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. [27] M.W. Wong, Weyl transforms and a degenerate elliptic partial differential equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), 3863–3870. [28] M.W. Wong, The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 393–404. [29] M.W. Wong, Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Ann. Global. Anal. Geom. 28 (2005), 271–283. [30] M.W. Wong, Weyl Transforms Springer-Verlag, Berlin, 1998. Todor Gramchev Dipartimento di Matematica e Informatica Universit` a di Cagliari Via Ospedale 72 I-09124 Cagliari, Italy e-mail: [email protected] Stevan Pilipović Department of Mathematics and Informatics University of Novi Sad Trg D. Obradovića 4 21000 Novi Sad, Serbia e-mail: [email protected] Luigi Rodino Dipartimento di Matematica Universit` a di Torino Via Carlo Alberto 10 I-10123 Torino, Italy e-mail: [email protected]


Weyl Transforms and the Heat Equation for the Sub-Laplacian on the Heisenberg Group Aparajita Dasgupta and M.W. Wong Abstract. Using Weyl transforms depending on Planck’s constant, we give a new formula for the solution of the initial value problem governed by the sub-Laplacian on the one-dimensional Heisenberg group. Estimates for the solution in terms of the initial value are given. Mathematics Subject Classification (2000). Primary 47F05, 47G30; Secondary 35H20. Keywords. Heisenberg group, sub-Laplacian, twisted Laplacian, Fourier– Wigner transforms, Weyl transforms, Hermite functions, heat kernel, semigroup, Sobolev spaces.

1. The Heisenberg group If we identify R2 with the complex plane C via R2 (x, y) ↔ z = x + iy ∈ C and let H = C × R, then H becomes a non-commutative group when equipped with the multiplication · given by 1 (z, t) · (w, s) = z + w, t + s + [z, w] , (z, t), (w, s) ∈ H, 4 where [z, w] is the symplectic form of z and w defined by [z, w] = 2 Im(zw). In fact, H is a unimodular Lie group on which the Haar measure is just the ordinary Lebesgue measure dz dt. This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

34

A. Dasgupta and M.W. Wong

Let h be the Lie algebra of left-invariant vector fields on H. A basis for h is then given by X, Y and T , where X=

1 ∂ ∂ + y , ∂x 2 ∂t

Y =

and T = The sub-Laplacian L on H is defined by

∂ 1 ∂ − x , ∂y 2 ∂t

∂ . ∂t

L = −(X 2 + Y 2 ). A simple computation gives

1 ∂ ∂ ∂2 ∂ L = −Δ − (x2 + y 2 ) 2 + x −y , 4 ∂t ∂y ∂x ∂t

where

∂2 ∂2 + . ∂x2 ∂y 2 Details on the Heisenberg group H and the sub-Laplacian L can be found in [9, 10, 11]. The aim of this paper is to give a new formula for the strongly continuous one-parameter semigroup e−uL , u > 0, generated by L. Moreover, we use the Sobolev spaces L2s (H), s ∈ R, in [1] to estimate e−uL f L2s (H) , u > 0, in terms of f L2(H) for all f in L2 (H), and to give an estimate for e−uL f L2 (H) in terms of f L2s(H) . The function F on H × (0, ∞) given by Δ=

F (z, t, u) = (e−uL f )(z, t),

(z, t) ∈ H, u > 0,

is in fact the solution of the initial value problem ∂F (z, t) ∈ H, u > 0, ∂u (z, t, u) = −(LF )(z, t, u), (z, t) ∈ H,

F (z, t, 0) = f (z, t),

(1.1)

for the sub-Laplacian L. Using the same techniques in Section 2 of [1], we get for all f ∈ L2 (H) and u > 0, ∞ (e−uL f )(z, t) = (2π)−1/2 e−itτ (e−uLτ f τ )(z) dτ, (z, t) ∈ H, (1.2) −∞

where Lτ , τ > 0, is the τ -twisted Laplacian given by ∂ 1 2 ∂ 2 2 −y Lτ = −Δ + (x + y )τ − i x τ 4 ∂y ∂x and f τ is the function on C given by τ −1/2 f (z) = (2π)

∞

−∞

eitτ f (z, t) dt,

z ∈ C,

The Heat Equation for the Sub-Laplacian

35

provided that the integral exists. In fact, f τ (z) is the inverse Fourier transform of f (z, t) with respect to t evaluated at τ. In this paper, the nonzero parameter τ can be looked at as Planck’s constant. To put the main results in perspective, let us note that there exists an explicit formula for the heat kernel of L. To recall, for positive time u, the heat kernel of L is the kernel Ku of the integral operator e−uL such that e−uL f = Ku ∗H f for all suitable functions f on H, where (Ku ∗H f )(z, t) = Ku ((z, t) · (w, s)−1 )f (w, s) dw ds, H

(z, t) ∈ H,

provided that the integral exists. In fact, the formula for Ku , u > 0, given by ∞ 2 1 τ 1 e− 4 τ |z| coth(τ u) dτ, (z, t) ∈ H, e−itτ Ku (z, t) = 2 8π −∞ sinh(τ u) dates back to the independent works of Gaveau [4] and Hulanicki [6] in the seventies. Recent works and references on the heat kernel and the heat equation for the sub-Laplacian L can be found in [8, 9, 10, 11, 12, 13] among others. Although the explicit formula for the heat kernel of L is interesting in its own right, it is not a tractable tool for obtaining the estimates for the solution of the initial value problem (1.1) that we have in mind. Since the symbol σ(L) of L is given by 2 2 1 1 + η − xτ σ(L)(x, y, t; ξ, η, τ ) = ξ + yτ 2 2 for all (x, y, t) and (ξ, η, τ ) in R3 , it is easy to see that L is a nowhere elliptic partial differential operator on R3 . So, the standard techniques of strongly continuous oneparameter semigroups generated by strongly elliptic operators in [2, 3, 5, 7, 17] and others are not applicable to the sub-Laplacian L. For the main results on the estimates in this paper, we use a formula for e−uL in terms of the τ -Weyl transforms and the τ -Fourier–Wigner transforms of Hermite functions, which we recall in, respectively, Section 2 and Section 3. A result on the L2 -boundedness of τ -Weyl transforms is instrumental in obtaining the estimates. Basic information on the classical Fourier–Wigner transforms, Wigner transforms and Weyl transforms can be found in [14] among others. The τ -Weyl transforms and a result on the L2 -boundedness of the τ -Weyl transforms are given in Section 2. The τ -Fourier–Wigner transforms of Hermite functions are recalled in Section 3. A formula for e−uLτ f, u > 0, for every function f in L2 (H) is given in Section 4. This formula is an extension of the formula obtained in [16] for τ = 1 and gives a formula for e−uL , u > 0, immediately using the inverse Fourier transform as indicated by (1.2). In Section 5, we use the family L2s (H), s ∈ R, of Sobolev spaces with respect to the center of the Heisenberg group in [1] to obtain Sobolev estimates for e−uL f, u > 0, in terms of f L2(H) , and estimates for e−uL f L2 (H) , u > 0, in terms of the Sobolev norms f L2s(H) of f in L2s (H).

36


2. τ -Weyl transforms Let f and g be functions in L2 (R). Then for τ in R \ {0}, the τ -Fourier–Wigner transform Vτ (f, g) is defined by ∞ p p dy (2.1) eiτ qy f y + g y− Vτ (f, g)(q, p) = (2π)−1/2 |τ |1/2 2 2 −∞ for all q and p in R. In fact, Vτ (f, g)(q, p) = |τ |1/2 V (f, g)(τ q, p),

q, p ∈ R,

where V (f, g) is the classical Fourier–Wigner transform of f and g. A proof can be found in [1]. It can be proved that Vτ (f, g) is a function in L2 (C) and we have the Moyal identity stating that Vτ (f, g)L2 (C) = f L2 (R) gL2(R) ,

τ ∈ R \ {0}.

(2.2)

We define the τ -Wigner transform Wτ (f, g) of f and g by Wτ (f, g) = Vτ (f, g)∧ .

(2.3)

Then we have the following connection of the τ -Wigner transform with the usual Wigner transform. Theorem 2.1. Let τ ∈ R \ {0}. Then for all functions f and g in L2 (R), Wτ (f, g)(x, ξ) = |τ |−1/2 W (f, g)(x/τ, ξ),

x, ξ ∈ R,

where W (f, g) is the classical Wigner transform of f and g. It is obvious that Wτ (f, g) = Wτ (g, f ),

f, g ∈ L2 (R).

(2.4)

Let σ ∈ Lp (C), 1 ≤ p ≤ ∞. Then for all τ in R \ {0} and all functions f in the Schwartz space S(R) on R, we define Wστ f to be the tempered distribution on R by ∞ ∞ (Wστ f, g) = (2π)−1/2 σ(x, ξ)Wτ (f, g)(x, ξ) dx dξ (2.5) −∞

−∞

for all g in S(R), where (F, G) is defined by F (z)G(z) dz (F, G) = Rn

for all measurable functions F and G on Rn , provided that the integral exists. We call Wστ the τ -Weyl transform associated to the symbol σ. It is easy to see that if σ is a symbol in the Schwartz space S(C) on C, then Wστ f is a function in S(R) for all f in S(R). We have the following estimate for the norm of the Weyl transform Wσˆτ in terms of the Lp norm of the symbol σ when σ ∈ Lp (C), 1 ≤ p ≤ 2.


37

Theorem 2.2. Let σ ∈ Lp (C), 1 ≤ p ≤ 2. Then Wσˆτ : L2 (R) → L2 (R) is a bounded linear operator and Wσˆτ ∗ ≤ (2π)−1/p |τ |−(1/2)+(1/p) σLp (C) , where Wσˆτ ∗ is the operator norm of Wσˆτ : L2 (R) → L2 (R). Proof. Let f and g be functions in S(R). Then ∞ ∞ σ ˆ (x, ξ)Wτ (f, g)(x, ξ) dx dξ (Wσˆτ f, g) = (2π)−1/2 −∞ −∞ ∞ ∞ σ ˆ (x, ξ)W (f, g)(x/τ, ξ) dx dξ = (2π)−1 |τ |−1/2 −∞ −∞ ∞ ∞ = (2π)−1 |τ |1/2 σ ˆ (τ x, ξ)W (f, g)(x, ξ) dx dξ. −∞

∞

But σ ˆ (τ x, ξ) = |τ |−1 σ 1/τ (x, ξ),

x, ξ ∈ R,

where σ1/τ is the dilation of σ with respect to the first variable by the amount 1/τ . More precisely, σ1/τ (q, p) = σ(q/τ, p), So, (Wσˆτ f, g) = (2π)−1/2 |τ |−1/2

∞

−∞

∞

−∞

q, p ∈ R.

σ 1/τ (x, ξ)W (f, g)(x, ξ) dx dξ

f, g), = |τ |−1/2 (Wσ 1/τ where Wσ is the classical Weyl transform with symbol σ 1/τ . 1/τ τ Thus, it follows from Theorem 21.1 in [15] that Wσˆ : L2 (R) → L2 (R) is a bounded linear operator and Wσˆτ ∗ ≤ |τ |−1/2 (2π)−1/p σ1/τ Lp (C) = (2π)−1/p |τ |−(1/2)+(1/p) σLp (C) .

3. Fourier–Wigner transforms of Hermite functions For τ ∈ R \ {0} and for k = 0, 1, 2, . . . , we define eτk to be the function on R by eτk (x) = |τ |1/4 ek ( |τ |x), x ∈ R. Here, ek is the Hermite function of order k defined by 2 1 ek (x) = k √ 1/2 e−x /2 Hk (x), x ∈ R, (2 k! π) where Hk is the Hermite polynomial of degree k given by k 2 d k x2 /2 Hk (x) = (−1) e (e−x ), x ∈ R. dx

38


For j, k = 0, 1, 2, . . . , we define eτj,k on R2 by eτj,k = Vτ (eτj , eτk ). {ej,k

The following theorem gives the connection of {eτj,k : j, k = 0, 1, 2, . . . } with : j, k = 0, 1, 2, . . . }, where ej,k = V (ej , ek ),

j, k = 0, 1, 2, . . . .

A proof can be found in [1]. Theorem 3.1. For τ ∈ R \ {0} and for j, k = 0, 1, 2, . . . , τ eτj,k (q, p) = |τ |1/2 ej,k q, |τ |p , q, p ∈ R. |τ | Theorem 3.2. {eτj,k : j, k = 0, 1, 2, . . . } forms an orthonormal basis for L2 (R2 ). Theorem 3.2 follows from Theorem 3.1 and Theorem 21.2 in [14] to the effect that {ej,k : j, k = 0, 1, 2, . . . } is an orthonormal basis for L2 (R2 ). Theorem 3.3. For j, k = 0, 1, 2, . . . , Lτ eτj,k = (2k + 1)|τ |eτj,k . Theorem 3.3 can be proved using Theorem 3.1 and Theorem 22.2 in [14] telling us that for j, k = 0, 1, 2, . . . , ej,k is an eigenfunction of L1 corresponding to the eigenvalue 2k + 1.

4. A formula and an estimate for e−uLτ , u > 0 Let τ ∈ R\{0}. Then a formula for e−uLτ , u > 0, is given by the following theorem. Theorem 4.1. Let f ∈ L2 (C). Then for u > 0, ∞ e−uLτ f = (2π)1/2 e−(2k+1)|τ |u Vτ (Wfτˆ eτk , eτk ), k=0

where the convergence of the series is understood to be in L2 (C). Proof. Let f ∈ L2 (C). Then for u > 0, ∞ ∞ e−uLτ f = e−(2k+1)|τ |u (f, eτj,k )eτj,k ,

(4.1)

k=0 j=0

where the series is convergent in L2 (C). Now, by Plancherel’s theorem, (2.1) and (2.3)–(2.5), f (z)Vτ (eτj , eτk )(z) dz = fˆ(ζ)Vτ (eτj , eτk )∧ (ζ) dζ (f, eτj,k ) = C C τ τ ˆ = (4.2) f (ζ)Wτ (ej , ek )(ζ) dζ = (2π)1/2 (Wfêτk , eτj ) C

for j, k = 0, 1, 2, . . . .


39

Similarly, for j, k = 0, 1, 2, . . . , and g in L2 (C), we get (eτj,k , g) = (g, eτj,k ) = (2π)1/2 (Wgˆτ eτk , eτj ) = (2π)1/2 (eτj , Wgˆτ eτk ).

(4.3)

So, by (4.1)–(4.3), Fubini’s theorem and Parseval’s identity, (e−uLτ f, g) = 2π

= 2π

∞ k=0 ∞

e−(2k+1)|τ |u

∞

(Wfτˆ eτk , eτj )(eτj , Wgˆτ eτk )

j=0

e−(2k+1)|τ |u (Wfτˆ eτk , Wgˆτ eτk ).

(4.4)

k=0

By Plancherel’s theorem and (2.3)–(2.5), (Wfτˆ eτk , Wgˆτ eτk ) = (2π)−1/2 gˆ(z)Wτ (eτk , Wfτˆ eτk )(z) dz C −1/2 = (2π) Wτ (Wfτˆ eτk , eτk )(z)ˆ g(z) dz C = (2π)−1/2 Vτ (Wfτˆ eτk , eτk )(z)g(z) dz

(4.5)

C

for k = 0, 1, 2, . . . . Thus, by (4.4), (4.5) and Fubini’s theorem, (e−uLτ f, g) = (2π)1/2 1/2

= (2π)

∞

e−(2k+1)|τ |u (Vτ (Wfτˆ eτk , eτk ), g)

k=0 ∞

e

−(2k+1)|τ |u

Vτ (Wfτˆ eτk , eτk ), g

(4.6)

k=0

for all f and g in L2 (C) and u > 0. Thus, by (4.6), e−uLτ f = (2π)1/2

∞

e−(2k+1)|τ |u Vτ (Wfτˆ eτk , eτk )

k=0

for all f in L2 (C) and u > 0. e

For all τ in R \ {0}, we have the following estimate for the L2 norm of f, u > 0, in terms of the Lp norm of f .

−uLτ

Theorem 4.2. Let τ ∈ R \ {0}. Then for all functions f in Lp (C), 1 ≤ p ≤ 2, 1 f Lp(C) . e−uLτ f L2 (C) ≤ (2π)−(1/p)+(1/2) |τ |−(1/2)+(1/p) 2 sinh(|τ |u) Proof. By Theorem 4.1, the Moyal identity (2.2) and the fact that eτk L2 (R) = 1,

k = 0, 1, 2, . . . ,

we get e−uLτ f L2 (C) ≤ (2π)1/2 e−|τ |u

∞ k=0

e−2k|τ |u Wfτˆ eτk L2 (R) ,

u > 0.

(4.7)

40


Applying Theorem 2.2 to (4.7), we get e

−uLτ

−(1/p)+(1/2)

f L2 (C) ≤ (2π)

−(1/2)+(1/p) −|τ |u

|τ |

e

∞

e

−2k|τ |u

f Lp(C)

k=0

= (2π)−(1/p)+(1/2) |τ |−(1/2)+(1/p)

1 f Lp(C) , 2 sinh(|τ |u)

as asserted.

5. Sobolev estimates for e−uL , u > 0 Let s ∈ R. Then as in [1], we define L2s (H) to be the set of all functions f in L2 (H) such that ∞ C

−∞

|τ |2s |f τ (z)|2 dτ dz < ∞.

For every f in L2s (H), we define the norm f L2s (H) by ∞ f 2L2s (H) = |τ |2s |f τ (z)|2 dτ dz. −∞

C

L2s (H)

Then it can be shown easily that inner product ( , )L2s (H) is given by (f, g)L2s (H) = C

∞

−∞

is an inner product space in which the |τ |2s f τ (z)g τ (z) dτ dz

for all f and g in L2s (H). Theorem 5.1. Let s ≥ 1. Then for u > 0, e−uL : L2 (H) → L2s (H) is a bounded linear operator and cs e−uL f L2s (H) ≤ s f L2 (H) , f ∈ L2 (H), 2u where cs = sup (|τ |s /sinh |τ |). τ ∈R\{0}

Proof. Let u > 0. Then, by Fubini’s theorem, Plancherel’s theorem, (1.2) and Theorem 4.2 with p = 2, ∞ −uL 2 e f L2s (H) = |τ |2s |(e−uL f )τ (z)|2 dτ dz C −∞ ∞ 2s −uL τ 2 = |τ | |(e f ) (z)| dz dτ −∞ C ∞ 2s −uLτ τ 2 = |τ | |(e f )(z)| dz dτ −∞ C ∞ |τ |2s e−uLτ f τ 2L2 (C) dτ = −∞


41

|τ |2s τ 2 f L2 (C) dτ 2 −∞ sinh (|τ |u) 1 ∞ |τ |2s τ 2 = |f (z)| dz dτ 4 −∞ sinh2 (|τ |u) C ∞ 1 |τ |2s ˇ(z, τ /u)|2 dz dτ, = | f 4 u2s+1 −∞ sinh2 (|τ |u) C

1 = 4

∞

where fˇ is the inverse Fourier transform of f with respect to t. So, using a simple change of variable and letting sup (|τ |2s /sinh2 |τ |),

Cs = we get e

−uL

f 2L2s (H)

Cs ≤ 2s 4u

τ ∈R\{0}

∞

−∞

Cs 2 ˇ |f (z, τ )| dz dτ = 2s f 2L2 (H) 4u C

and this completes the proof. The following result complements Theorem 5.1.

Theorem 5.2. Let s ≤ −1. Then for u > 0, e−uL : L2s (H) → L2 (H) is a bounded linear operator and c−s e−uL f L2 (H) ≤ −s f L2s (H) , f ∈ L2s (H), 2u where c−s = sup (|τ |−s sinh |τ |). τ ∈{0}

The proof of Theorem 5.2 is very similar to that of Theorem 5.1 and is hence omitted.

References [1] A. Dasgupta and M.W. Wong, Weyl transforms and the inverse of the sub-Laplacian on the Heisenberg group, in Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Editors: L. Rodino, B.-W. Schulze and M.W. Wong, Fields Institute Communications, American Mathematical Society, 2007, 27– 36. [2] E.B. Davies, One-Parameter Semigroups, Academic Press, 1980. [3] A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, 1969. [4] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95–153. [5] E. Hille and R.S. Phillips, Functional Analysis and Semi-groups, Third Printing of Revised Version of 1957, American Mathematical Society, 1974.

42


[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] G.M. Iancu and M.W. Wong, Global solutions of semilinear heat equations in Hilbert spaces, Abstr. Appl. Anal. 1 (1996), 263–276. [8] J. Kim and M.W. Wong, Positive definite temperature functions on the Heisenberg group, Appl. Anal. 85 (2006), 987–1000. [9] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993. [10] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Birkh¨ auser, 1998. [11] S. Thangavelu, An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups, Birkh¨ auser, 2004. [12] S. Thangavelu, A survey of Hardy type theorems, in Advances in Analysis, Editors: H.G.W. Begehr, R.P. Gilbert, M.E. Muldoon and M.W. Wong, World Scientific, 2005, 39–70. [13] J. Tie, The non-isotropic twisted Laplacian on Cn and the sub-Laplacian on Hn , Comm. Partial Differential Equations 31 (2006), 1047–1069. [14] M.W. Wong, Weyl Transforms, Springer-Verlag, 1998. [15] M.W. Wong, Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Ann. Global Anal. Geom. 28 (2005), 271–283. [16] M.W. Wong, The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 393–404. [17] K. Yosida, Functional Analysis, Reprint of Sixth Edition, Springer-Verlag, 1995. Aparajita Dasgupta and M.W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto Ontario M3J 1P3, Canada e-mail: [email protected] [email protected]


Construction of the Fundamental Solution and Curvature of Manifolds with Boundary Chisato Iwasaki Abstract. The method of construction of the fundamental solution for the heat equations of the initial boundary value problem on manifolds with boundary, which is applicable to calculate traces of operators, is discussed. This shows a relation of the singularity of the fundamental solution and the curvature of manifolds. We may say that this is an extension of a local version of Gauss-Bonnet-Chern theorem for manifolds with boundary. Mathematics Subject Classification (2000). Primary 58J35; Secondary 58J40, 58J20. Keywords. Symbolic calculus, fundamental solution, curvature, local index.

1. Introduction In this paper we give, by means of symbolic calculus of pseudo-differential operators, an extension theorem of a local version of Gauss-Bonnet-Chern theorem on manifolds with boundary given in C. Iwasaki [11]. Let M be a Riemannian manifold and let χ(M ) be the Euler characteristic of M . Let dv and dσ be a volume element of M and that of its boundary ∂M respectively. The Gauss-Bonnet-Chern theorem which was proved by S. Chern [2], [3] states, broadly speaking, that topological quantity can be represented by the geometric quantities. Gauss-Bonnet-Chern Theorem. Let M be a Riemannian manifold of dimension n. (1) For M without boundary we have Cn (x, M )dv. χ(M ) = M

(2) If M has boundary,

χ(M ) =

Cn (x, M )dv +

M

Dn−1 (x)dσ. ∂M

44

C. Iwasaki The precise definitions of Cn (x, M ) and Dn−1 (x) are given in Section 3 and Section 5 respectively. We note that if n is odd, 1 Dn−1 (x) = Cn−1 (x, ∂M ) 2 holds. Analytical proofs are based on the following formula (cf. V.K. Patodi [15]): n χ(M ) = (−1)p trep (t, x, x)dv, (1.1) M p=0

where ep (t, x, y) denotes the kernel of the fundamental solution Ep (t) for the heat equation for Δp on differential p-forms Ap (M ) = Γ(∧p T ∗ (M )) and tr ep (t, x, x) means the trace of operator ep (t, x, x) on ∧p Tx∗ (M ). If M has no boundary, Ep (t) is the fundamental solution for the Cauchy problem, that is, Ep (t)f (x) = ep (t, x, y)ϕ(y)dvy , ϕ ∈ Ap (M ) M

satisfies for 0 < T < ∞

d + Δp Ep (t) = 0 dt Ep (0) = I

in (0, T ) × M, in M.

If M has boundary ∂M , Ep (t) satisfies the following equations instead of the above equations d + Δp Ep (t) = 0 in (0, T ) × M, dt on (0, T ) × ∂M, Bp Ep (t) = 0 Ep (0) = I

in M,

with some boundary operator Bp (see Section 4). So, we may say that a local version of Gauss-Bonnet-Chern theorem holds, if we have n √ (−1)p trep (t, x, x) = Cn (x, M ) + 0( t) (1.2) p=0

as t tends to 0. The author has proved (1.2) for x ∈ M \∂M and has also shown for x ∈ ∂M n

1 (−1)p trep (t, x, x) = 2Dn−1 (x) √ + O(1) t p=0

as t → 0

(1.3)

in [11]. The author has shown a way to constructing the fundamental solution of which the main term is enough to show (1.2) and (1.3). It is constructed by technique of pseudo-differential operators of new weights on symbols.

Fundamental Solution and Curvature

45

In this paper, a generalization of a local version of Gauss-Bonnet-Chern theorem on manifolds with boundary is obtained. Before stating our theorems, we introduce some notations. We denote I and I0 the set of index I = {I = (i1 , i2 , . . . , ir ) : 0 ≤ r ≤ n, 1 ≤ i1 < · · · < ir ≤ n}, I0 = {I = (i1 , i2 , . . . , ir ) : 0 ≤ r ≤ n − 1, 1 ≤ i1 < · · · < ir ≤ n − 1} and

a 0 = 0 if a < b, or b < 0, = 1. b 0 In the rest of this paper fix an integer such that 0 ≤ ≤ n.

Theorem 1.1 (Main Theorem). Let M be a Riemannian manifold of dimension n with boundary and let Ep (t) be the fundamental solution on Γ(∧p T ∗ (M )). Given arbitrary constants {kj }j=+1,...,n define the constants fp as follows: n n−p ! n−p (0 ≤ p ≤ n). (1.4) kj fp = (−1)p + n−j n− j=max{p,+1}

(1) If x ∈ M \∂M , we have n n n 1 fp trep (t, x, x) = C (x)t− 2 + 2 + 0(t− 2 + 2 + 2 )

as t → 0,

p=0

where C (x) is given in Definition 3.4. (2) If x ∈ ∂M , we have n n 1 n fp trep (t, x, x) = 2D−1 (x)t− 2 + 2 − 2 + 0(t− 2 + 2 )

as t → 0,

p=0

where D−1 (x) is given in Definition 5.3. (3) We have n fp trep (t, x, x)ψ(x)dv M p=0

=

D−1 (x)ψ(x)dσ t− 2 + 2 + 0(t− 2 + 2 + 2 ) n

C (x, M )ψ(x)dv + M

n

1

∂M

as t → 0 for any ψ(x) ∈ C ∞ (M ). Remark 1.2. Assume = n. Then fp = (−1)p for all p as in (1.2) and C = Cn , D−1 = Dn−1 . So statements (1) and (2) of Main Theorem are a local version of Gauss-Bonnet-Chern theorem proved in [11]. The statement (3) is the GaussBonnet-Chern theorem. Remark 1.3. If M has no boundary, D−1 vanishes. So (1) of the Main Theorem has been obtained in C. Iwasaki [12].

46

C. Iwasaki

Remark 1.4. Assume kj = 0 for all j. Then fp = (−1)p n−p n− (0 ≤ p ≤ ), fp = 0( + 1 ≤ p ≤ n). In this case, under the assumption M has no boundary, (1) of the Main Theorem coincides with the result given in [5]. Since Bp is a coercive boundary operator for Δp , it is well known that ep (t, x, y) is regular either x = y or t > 0. On the diagonal set, ep (t, x, y) has singularity with respect to t as follows: t → 0. ep (t, x, x) ∼ c0 (x)t− 2 + c1 (x)t− 2 + 2 + · · · + · · · n In [11] it is proved that the singularity of p=0 (−1)p tr ep (t, x, x) at any point x in M vanishes by an algebraic theorem on linear spaces stated in H.L. Cycon-R.G. Froese-W. Kirsch-B. Simon [4]. This theorem is given in Section 2 of this paper, in a form suitable for our discussion. Our point is that one can prove the Main Theorem by only calculating the main term of the symbol of the fundamental solution, introducing a new weight of symbols of pseudo-differential operators. The plan of this paper is the following. In Section 2 we state some algebraic theorems, which are proved in [12]. In Section 3 the asymptotic expansion of the fundamental solution for the Cauchy problem is discussed. This argument is similar to one of [12]. Our boundary operator Bp will be explained in Section 4. See [11] for the detailed argument. Section 5 is devoted to the proof of (2) and (3) of the Main Theorem, which is an essential part of this paper. n

n

1

2. Algebraic properties for the calculation of the trace Let V be a vector space of dimension n with an inner product and let ∧p (V ) ∗ be its anti-symmetric p tensors. Set ∧ (V ) = np=0 ∧p (V ). Fix an orthonormal basis {v1 , . . . , vn } of V . Let a∗i be a linear transformation on ∧∗ (V ) defined by a∗i v = vi ∧ v and let ai be the adjoint operator of a∗i on ∧∗ (V ). We note that {a∗i , aj }1≤i,j≤n satisfy the following relations. ai aj + aj ai = 0, a∗i a∗j + a∗j a∗i = 0,

(2.1)

ai a∗j + a∗j ai = δij . Definition 2.1. Set βφ = 1, βj = a∗j aj − aj a∗j for 1 ≤ j ≤ n and βI = βi1 · · · βik for I = (i1 , . . . , ik ) ∈ I. It is clear that a∗k ak =

1 (1 + βk ), 2

βj βk = βk βj ,

βj2 = 1.


47

The following assertion is essentially due to [15]. Proposition 2.2. We have for any I = (i1 , . . . , ik ) ∈ I the following assertions: (1) If p < k tr[βI aj1 aj2 · · · ajp a∗h1 a∗h2 · · · a∗hp ] = 0. (2) Suppose p = k and {j1 , j2 , . . . , jk } = {i1 , i2 , . . . , ik } or {h1 , h2 , . . . , hk } = {i1 , i2 , . . . , ik }. Then tr[βI aj1 aj2 · · · ajp a∗h1 a∗h2 · · · a∗hp ] = 0. (3) Let π, σ be elements of the permutation group of degree k. Then we have tr[βI a∗iπ(1) aiσ(1) a∗iπ(2) aiσ(2) · · · a∗iπ(k) aiσ(k) ] = 2n−k sign(π) sign(σ). ∗ p Definition 2.3. Let Ψp be the projection of ∧ (V ) on ∧ (V ) and let Γ0 = 1, Γk = I∈I,(I)=k βI .

The following theorem is a key algebraic argument of this paper. Theorem 2.4. Let fp (0 ≤ p ≤ n) be constants of the form n n−p ! n−p p kj fp = (−1) + n−j n− j=max(+1,p)

with any constants kj ( + 1 ≤ j ≤ n). Then the following equation holds n

fq Ψq = (−1) 2−n Γ +

q=0

n

αp Γp

p=+1

with αp ( + 1 ≤ p ≤ n) defined by p

αp = (−1)

2

−n

! p p p + kj . j j=+1

Especially (1) If = n, then

n

fq Ψq = (−1)n Γn

q=0

holds if and only if fp = (−1)p for any p. (2) If αp = (−1)p 2 ( + 1 ≤ p ≤ n), (−1)p n−p , (0 ≤ p ≤ ); n− fp = 0, ( + 1 ≤ p ≤ n). (3) If αp = (−1) 2−n p ( + 1 ≤ p ≤ n), 0, (0 ≤ p ≤ n − − 1); fp = n−+p p (−1) n− ; (n − ≤ p ≤ n). −n p

48

C. Iwasaki

3. The proof of statement (1) of the Main Theorem We give a rough review of a proof of the statement of (1) of the Main Theorem. This argument is similar to that of [12]. Let M be a smooth Riemannian manifold of dimension n with a Riemannian metric g. Let X1 , X2 , . . . , Xn be a local orthonormal frame of T (M ) in a local path U . And let ω 1 , ω 2 , . . . , ω n be its dual. The differential d and its dual ϑ acting on Γ(∧p T ∗ (M )) are written as follows,using the Levi-Civita connection ∇ (see Appendix A of [14]): d=

n

ϑ=−

e(ω j )∇Xj ,

j=1

n

ı(Xj )∇Xj ,

j=1

where we use the following notation. Notation. e(ω j )ω = ω j ∧ ω, ı(Xj )ω(Y1 , . . . , Yp−1 ) = ω(Xj , Y1 , . . . , Yp−1 ). Let the function cki,j be defined for 1 ≤ i, j, k ≤ n by the identities: ∇Xi Xj =

n

cki,j Xk

k=1

and let R(X, Y ) be the curvature transformation, that is " # R(X, Y ) = ∇X , ∇Y − ∇[X,Y ] . Set

n

R(Xi , Xj )Xk =

Rmkij Xm 1 ≤ i, j, k, ≤ n.

m=1

The Laplacian Δ = dϑ + ϑd on Weitzenb¨ ock’s formula; Δ = −{

n

∇Xj ∇Xj −

j=1

n

n p=0

∇(∇Xj Xj ) +

j=1

Γ(∧p T ∗ (M )) can be expressed by n

e(ω))ı(Xj )R(Xi , Xj )}.

i,j=1

We use the following notations in the rest of this paper: a∗j = e(ω∗),

ak = ι(Xk ).

aI = ai1 ai2 · · · aip ,

a∗I = a∗ip · · · a∗i1 ,

ω I = ω i1 ∧ ω i2 ∧ · · · ∧ ω ip

for I = (i1 , i2 , . . . , ip ) ∈ I.

From the fact that our connection is the Riemannian connection the coefficients cki,j satisfy n (cki,j − ckj,i )Xk . (3.1) cki,j = −cji,k , [Xi , Xj ] = k=1


49

By the above notations we have for ϕJ ω J ∈ A∗ (M ) ∇Xj (ϕJ ω J ) = (Xj ϕJ )ω J − ϕJ GJ (ω J ) using ∇Xj (ω i ) = −

n

cij,k ω k ,

where Gj =

k=1

n

∗ cm j,k ak am .

k,m=1

We have also R(Xi , Xj )(ϕJ ω J ) = −

n

ϕJ Rmkij a∗k am (ω J ).

k,m=1

Then we have the representation in a local chart U n n (Xj I − Gj )2 − cji,i (Xj I − Gj ) − Δ=− j=1

i,j=1

n

Rmkij a∗i aj a∗k am

i,j,k,m=1

∗

∗

∗

on A (M ). Here I is the identity operator on ∧ (T (M )). Take a local coordinate {x1 , . . . , xn } of U . Let {ξ1 , . . . , ξn } be its dual. By the above argument we have Lemma 3.1. Let us denote the symbol of Xj by αj , that is, αj = σ(Xj ). Then the symbol of Δ is given by σ(Δ) = −

n

(αj I − Gj )2 + R + r1 ,

j=1

where R=

r1 =

n

Rmkij a∗i aj a∗k am ,

i,j,k,m=1 n

i{

k,j=1

n ∂ ∂ ∂ ∂ αj αj − αj Gj }I + cjk,k (αj I − Gj ). ∂ξk ∂xk ∂ξk ∂xk j,k=1

A = (Aij ) denotes a matrix whose elements are Aij = a∗i aj (1 ≤ i, j ≤ n). m Definition 3.2. A subset K m of S1,0 is given by K m = {p(x, ξ : A); polynomials with respect to ξ and Ai,j , (i, j = 1, 2, . . . , n) of order m with coefficients in B(Rn )}. Here B(Rn ) is the set of smooth functions in Rn whose all partial derivatives are bounded. We define a pseudo-differential operator P = p(x, D : A) acting on A∗ (M ) of a symbol σ(P ) = p(x, ξ : A) = I,J pI,J (x, ξ)a∗I aJ ∈ K m as follows. pI,J (x, D)ϕK a∗I aJ (ω K ). p(x, D : A)(ϕK ω K ) = I,J

50

C. Iwasaki

In our case, by Lemma 3.1 the symbol of Δ is of the form σ(Δ) = r2 + r1 , where n r2 = − (αj I − Gj )2 + R. j=1 j

It is clear that rj belongs to K for j = 1, 2. If we review the results in [11], we see the following facts. The above representation of Δ signifies that the fundamental solutionEp (t)(ep (t, x, y)) have a common representation for any p. So we write for short E(t)(e(t, x, y)). In our notation, Ep (t) = Ψp E(t). It is sufficient to construct an asymptotic solution for the fundamental solution locally because we can reduce the construction to solving an integral equation on the manifold M of Volterra type. Our fundamental solution is of the form E(t) = U (t) + V (t), where U (t) is the fundamental solution for the Cauchy problem and V (t) is smooth with respect t off the boundary. See details in Section 5. Now consider the following Cauchy problem: d + Δ U (t) = 0 in (0, T ) × Rn , dt in Rn . lim U (t) = I t→0

U (t) constructed in [11] is an integral operator U (t)ϕ(x) = u(t, x, y)ϕ(y)dvy , ϕ ∈ A∗ (M ) M

and U (t) has an expansion. U (t) ∼

uj (t, x, D),

(3.2)

j=0

where uj (t, x, D) are pseudo-differential operators with parameter t. The following fact is obtained in p. 255 of [11]. The main part of U (t) is obtained as a pseudo-differential operator with symbol u0 (t, x, ξ) = e−r2 t . So we have u ˜0 (t, x, x) = (2π)−n u0 (t, x, ξ)dξ (3.3) Rn n √ 1 √ = detge−tR 1 + 0( t) . 2 πt We shall calculate tr (βI u ˜0 (t, x, x))dx = for I ∈ I, (I) = r.

1 √ 2 πt

n

√ tr (βI e−tR )dv(1 + 0( t)),

(3.4)

Fundamental Solution and Curvature Using e

−tR

=

∞ (−R)k

k!

k=0

51

t , k

and by Proposition 2.2 we have m tr βI (−R)m tm! + 0(tm+1 ), if −tR tr (βI e )= r+1 0(t 2 ), if

r = 2m ; r is odd.

(3.5)

We have the following proposition for the terms of the right-hand side of the above equation. Proposition 3.3. For I = (i1 , i2 , . . . , ir ) ∈ I (r = 2m) sign(π) sign(σ) tr(βI (−R)m ) = 2n−r−m π,σ∈Sr

×Riπ(1) iπ(2) iσ(1) iσ(2) · · · Riπ(r−1) iπ(r) iσ(r−1) iσ(r) . Definition 3.4. (1-i) If is odd, C (x, M ) = 0 (1-ii) If is even ( = 2m), C (x, M ) =

CI (x, M ),

I∈I,(I)=

where CI (x, M ) =

1 √ 2 π

n

m 1 1 sign(π) sign(σ) m! 2 π,σ∈S

× Riπ(1) iπ(2) iσ(1) iσ(2) · · · · · · Riπ(−1) iπ() iσ(−1) iσ() for I = (i1 , i2 , . . . , i ) ∈ I. By (3.4), (3.5) and Proposition 3.3, we have the following equation for (I) = r √ n r n r 2n−r t− 2 + 2 CI (x, M ) detg + 0(t− 2 + 2 +1 ), if r = 2m ; tr (βI u˜0 (t, x, x)) = n r 1 if r is odd. 0(t− 2 + 2 + 2 ), (3.6) Similarly we have j

tr (βI u ˜j (t, x, x))dx = 0(t− 2 + 2 + 2 ). n

r

(3.7)

By (3.6) and (3.7) we have n r 1 1 tr (βI u(t, x, x)) = tr (βI u ˜0 (t, x, x)) √ + 0(t− 2 + 2 + 2 ) detg n r n r 1 2n−r t− 2 + 2 CI (x, M ) + 0(t− 2 + 2 + 2 ), = n r 1 0(t− 2 + 2 + 2 ),

if r = 2m ; if r is odd.

(3.8)

52

C. Iwasaki

By Theorem 2.4 we obtain n tr fp Ψp u(t, x, x) = (−1) 2−n p=0

tr (βI u(t, x, x))

I∈I,(I)= n

+

αp tr (Γp u(t, x, x)).

p=+1

By (3.6) and (3.7) it is easy to see tr (Γp u(t, x, x)) = 0(t− 2 + 2 + 2 ) for any p ≥ + 1. n

1

(3.9)

Applying (3.8) and (3.9), we have n n n 1 fp Ψp u(t, x, x) = C (x, M )t− 2 + 2 + 0(t− 2 + 2 + 2 ) x ∈ M. tr p=0

4. The construction of the fundamental solution with boundary The main part of the construction of the fundamental solution or its asymptotics lies in constructing these in a local chart (cf. C. Iwasaki [10]). So it is enough to show a method of construction of the fundamental solution in Rn+ . The study in [10] is applicable for the construction of the fundamental solution for our initialboundary value problem. In this case we introduce the symbol class Js according m to [11], as we used K m instead of S1,0 in Section 3. First of all, we write down the boundary operator Bp for Ap (M ) in a local chart. Take a local patch Ω near ∂M such that ∂M is defined by {ρ = 0} in Ω and M ∩ Ω ⊂ {ρ ≥ 0}. Assume that ω n = cdρ with some function c on M . Note that Xj (1 ≤ j ≤ n − 1) are tangential to ∂M . The coefficients of the Levi-Civita connection satisfy both (3.1) and the following relations: 1 ≤ i, j ≤ n − 1. cni,j = cnj,i Choose local coordinates {x1 , . . . , xn } in Ω such that M ∩ Ω = {(x , xn ); x ∈ U, xn ≥ 0}, Γ = ∂M ∩ Ω = {(x , 0); x ∈ U} and d Xn |Γ = . dxn The boundary operator Bp is as follows:

ϕ ∈ Dom(ϑ), where Dom(ϑ) = {ϕ =

dϕ ∈ Dom(ϑ),

ϕJ ω J : ϕJ |Γ = 0 if n ∈ J}.

J

Note that Dom(ϑ) is the set of all ψ of A∗ (M ) which satisfy (dϕ, ψ)dv = (ϕ, ϑψ)dv, for any ϕ ∈ A∗ (M ). M

M


53

We can write our boundary condition in the following form: J ϕJ |Γ = 0 if n ∈ J, B ϕJ ω |Γ = 0, n∈J /

where

B = Xn −

cjn,k a∗k aj −

1≤j,k≤n−1

cnj,k a∗j ak .

(4.1)

1≤j,k≤n−1

Definition 4.1. (1) Set

γ = γ(x : A) = Gn |Γ =

(cjn,k |Γ )a∗k aj ,

1≤j,k≤n

b = b(x : A) = −

(4.2)

(cnj,k |Γ )a∗j ak +

(cjn,k |Γ )a∗k aj

(4.3)

j=n or k=n

1≤j,k≤n−1

(2) Set P = a∗n an , Q = an a∗n = I − P. We note that by (4.1) B=

d − γ + b. dxn

Let the fundamental solution be of the form E(t) = U (t) + V (t), where U (t) is the fundamental solution for the Cauchy problem. It is sufficient to construct an operator V (t) which satisfies the following problem in Rn+ . d + Δ V (t) = 0 in (0, T ) × Rn+ , dt PV (t) = −PU (t)

on (0, T ) × Rn−1 × {xn = 0},

BQV (t) = −BQU (t)

on (0, T ) × Rn−1 × {xn = 0},

lim V (t) = 0

in Rn+ .

t→0

We introduce the following symbol classes Fs as follows. Definition 4.2. Fs is the set of all finite sums of functions of the following form td (xn ) r(x , ξ , ξn : A) (r ∈ K s+2d+ ). We have by Lemma 3.1 r2 |Γ = q2 = (ξn + iγ)2 + β(x , ξ : A) ∈ F2 , where β = β(x , ξ : A) = −

n−1

(αj |Γ )I − Gj |Γ

j=1

2 + R|Γ .

54

C. Iwasaki

Let {w ˜j,k } be symbols defined in Definition 7 of [10] for integers j and nonpositive integers k as follows: j+1 ω 1 1 √ √ , if j ≥ 0, hj w ˜j,0 (t, ω) = √ π 2 t 2 t j+1 ∞ −j−1 ω 1 1 −(σ+ 2√ )2 (−σ) t √ e w ˜j,0 (t, ω) = − √ dσ, if j ≤ −1, (−j − 1)! π 2 t 0 For k ≤ −1 1 w ˜j,k (t, ω; b) = − √ π

1 √ 2 t

j+k+1

∞

e

√ −k−1 ω −(σ+ 2√ )2 +2b tσ (−σ) t

(−k − 1)! 0 ω × hj (σ + √ )dσ, if j ≥ 0; 2 t j+k+1 ∞ 1 1 (−τ )−j−1 √ w ˜j,k (t, ω; b) = √ dτ (−j − 1)! π 2 t 0 ∞ √ −k−1 ω −(σ+τ + 2√ )2 +2b tσ (−σ) t dσ, if j ≤ −1, × e (−k − 1)! 0

d j −σ where hj (σ) = {( dσ ) e }eσ . 2

2

Definition 4.3. For a pair (j, k) of integer j and nonpositive integer k we define a function v˜j,k (t, xn , yn ; b, γ) = eγ(xn−yn ) w ˜j,k (t, xn + yn ; b), with b and γ given in Definition 4.1. An operator Vj,k (t; b, γ) corresponding to v˜j,k is defined as follows for a function ϕ(yn ) defined on R1+ . ∞ (Vj,k (t; b, γ)ϕ)(xn ) = v˜j,k (t, xn , yn ; b, γ)ϕ(yn )dyn . 0

The above series of integral operators Vj,k has nice properties. See [11]. One of the properties is the following one. Proposition 4.4. Vj,k (t; b, γ) satisfies for k ≤ −1 2 $ ∂ ∂ ¯1 , − −γ in (0, T ) × R Vj,k (t; b, γ) = 0 + ∂t ∂xn ∂ ¯1 , − γ Vj,k (t; b, γ) = Vj+1,k (t; b, γ) in (0, T ) × R + ∂xn BVj,k (t; b, γ) = Vj,k+1 (t; b, γ) + Zj,k lim (Vj,k (t; b, γ)ϕ)(xn ) = 0

t→+0

¯1 , in (0, T ) × R +

in xn > 0, ¯ 1+ ), for ϕ = ϕJ ω J , ϕJ ∈ C(R J


55

where Zj,k is an operator defined as ∞ Zj,k ϕ = [b, exp{γ(xn − yn )}]w ˜j,k (t, xn + yn ; b)ϕ(yn )dyn . 0

Definition 4.5. (1) Js is the set of all finite sums of the functions of the following form g(t, xn , yn , x , ξ : A) = td (xn ) q(x , ξ : A)˜ vj,k (t, xn , yn ; b(x : A), γ(x : A))

× e−β(x ,ξ :A)t (d, , j, k ∈ Z, d ≥ 0, ≥ 0, k ≤ 0, q ∈ K m (Rn−1 ) with m = s + 2d + − j − k). For a symbol g(t, xn , yn , x , ξ , A) ∈ Js we define an integral-pseudo-differential operator G as follows: ∞ g(t, xn , yn , x , D : A)ϕ(·, yn )dyn . (Gϕ)(t, x , xn : A) = 0

So the kernel g˜ of operator G is given by g˜(t, x , xn , y , yn : A) = (2π)−(n−1)

ei(x −y

)•ξ

g(t, xn , yn , x , ξ : A)dξ .

Rn−1

˜ s is the set of all matrices g whose elements belong to B([0, T ] × [0, ∞) × (2) R m (Rn−1 )) and satisfy the following estimate for any α, β, a, b, k [0, ∞); S1,0 ∂ α ∂ β ∂ a ∂ b ∂ k g ∂ξ ∂x ∂xn ∂yn ∂t s+1+2k+a+b √ |α| 1 (xn + yn )2 −|α| 2 ≤ Cα,β min(|ξ | − c0 |ξ | t , t ) √ exp −δ 4t t for any δ < 1 and some c0 > 0. By the argument of page 276 of [11] an asymptotic EN (t) of the fundamental solution for the mixed problem is obtained of the form for any integer N EN (t) = U0 + U1 + · · · + UN + V1 + V0 + · · · + V−N , where Uj = uj (t; x, D) are pseudo-differential operators given in (3.2) and Vj are integral operators whose symbols vj (t, xn , yn , x , ξ : A) ∈ Jj . Theorem 7.2 of [11] asserts that d ˜ −N + Δ EN (t) = 0 mod R in (0, T ) × Rn+ , dt ˜ −N −2 PEN (t) = 0 mod R on (0, T ) × Rn−1 × {xn = 0}, ˜ −N −1 mod R on (0, T ) × Rn−1 × {xn = 0}. BQEN (t) = 0 Especially we have

v1 (t, xn , yn , x , ξ : A) = 2Q˜ v1,−1 (t, xn , yn ; b, γ)e−β(x ,ξ :A)t .

(4.4)

56

C. Iwasaki

In the rest of this paper v˜j (t, x , xn , y , yn : A) denotes the kernel of the integral operator Vj , that is, v˜j (t, x , xn , y , yn : A) = (2π)−(n−1) ei(x −y )•ξ vj (t, xn , yn , x , ξ : A)dξ . Rn−1

5. The proof of Statements (2) and (3) of the Main Theorem n We will calculate tr ( p=0 fp Ψp Vj (t, x, x))(j ≤ 1). We prepare some lemmas and the definition of D−1 (x). ˆ Let R(W, Z, X, Y ) be the Riemannian curvature tensors induced on ∂M . From the equation of Gauss we have ˆ i , Xj , Xk , X ) + cnk,j cn,i − cn,j cnk,i , R(Xi , Xj , Xk , X ) = R(X 1 ≤ i, j, k, ≤ n − 1 on ∂M. So it is clear that on ∂M Rijk a∗i aj a∗k a = 1≤i,j,k,≤n−1

−

ˆ ijk a∗i aj a∗k a R

cni,j cnk, a∗i aj a∗k a +

1≤i,j,k,≤n−1

cnj,j cnk, a∗k a .

1≤j,k,≤n−1

Proposition 5.1. We have by the Bianchi’s identity ˆ ijkh a∗i aj a∗k ah R 1≤i,j,k,h≤n−1

= Proof.

1 2

1≤i,j,k,h≤n−1

ˆ ikjh a∗i aj a∗k ah − 1 R 2

(5.2)

ˆ kjki a∗j ai . R

1≤i,j,k≤n−1

By the properties of the curvature transformation we have ˆ ijkh a∗i aj a∗k ah R 1≤i,j,k,h≤n−1

=

ˆ ihkj a∗ ah a∗ aj R i k

1≤i,j,k,h≤n−1

=

1≤i,j,k≤n−1

ˆ ihjj a∗ ah R i

1≤i,j,h≤n−1

ˆ ihkj a∗ aj a∗ ah R i k

1≤i,j,k,h≤n−1

+

ˆ ihkj a∗ aj ah a∗ + R i k

1≤i,j,k,h≤n−1

=−

(5.1)

1≤i,j,k,≤n−1

ˆ ikkj a∗ aj . R i

+

1≤i,j,h≤n−1

ˆ ihjj a∗ ah R i


57

From the above equation it is easy to see ˆ ijkh a∗ aj a∗ ah R 2 i k 1≤i,j,k,h≤n−1

=

1≤i,j,k,h≤n−1

=

îjkh − R ˆ ihkj )a∗i aj a∗k ah + (R

ˆ ikjh a∗i aj a∗k ah − R

1≤i,j,k,h≤n−1

îjkk + R ˆ ikkj )a∗i aj (R

1≤i,j,k≤n−1

ˆ ikjk a∗i aj . R

1≤i,j,k≤n−1

Lemma 5.2. For any N ∈ N we have hN ∈ K−1 such that 2 −R e−βt = (I + hN )Πn−1 j=1 exp{(αj I − Gj ) t}e where R = R|∂M = 1≤i,j,k,≤n Rijk a∗i aj a∗k a |∂M .

˜ −N −1 , mod R

t

We give the definition of D−1 (x) on x ∈ ∂M . Definition 5.3. For x ∈ ∂M D−1 (x) is defined as follows: D−1 (x) = DI (x), I∈I0 ,(I)=−1

(i) If is odd ( = 2m + 1), n−1 m 1 1 1 1 √ DI (x) = 2 2 π m! 2

sign(π) sign(σ)

π,σ∈S−1

î î i i i ······R ×R π(1) π(2) σ(1) σ(2) π(−2) iπ(−1) iσ(−2) iσ(−1) for I = (i1 , i2 , . . . , i−1 ) ∈ I0 . (ii) If is even ( = 2m), n−1 m−1 dm−k−1 1 k 1 1 √ DI (x) = 2 2 π k! 2 k=0

î i i i ··· ×R π(1) π(2) σ(1) σ(2)

sign(π) sign(σ)

π,σ∈S−1

î ···R π(2k−1) iπ(2k) iσ(2k−1) iσ(2k)

× cnπ(2k+1),σ(2k+1) cnπ(2k+2),σ(2k+2) · · ·

· · · cnπ(−1),σ(−1)

for I = (i1 , i2 , . . . , i−1 ) ∈ I0 with (−1)k √ . k!(k + 12 ) π Now the calculation of tr βI v˜1 (t, x , 0) is obtained as follows. dk =

Lemma 5.4. Let gˆ be the Riemannian metric induced on ∂M and ε be a positive constant. (1) Let I = (i1 , i2 , . . . , il−1 , n) ∈ I and I0 = (i1 , i2 , . . . , il−1 ) ∈ I0 .

58

C. Iwasaki (1-i) We have tr βI v˜1 (t, x , 0, x , 0)

n 1 n = (−1) 2n−+1 t− 2 + 2 − 2 DI0 (x ) det g(x , 0) + 0(t− 2 + 2 ).

(1-ii) We have for any ψ(x , xn ) ∈ C ∞ (Rn ) ε tr βI v˜1 (t, x , xn , x , xn ) ψ(x , xn )dxn 0 n n 1 = (−1) 2n− t− 2 + 2 ψ(x , 0)DI0 (x ) det gˆ(x ) + 0(t− 2 + 2 + 2 ). (2) Let I ∈ I0 , (I) = or I ∈ I, (I) ≥ + 1. (2-i) We have n tr βI v˜1 (t, x , 0, x , 0) = 0(t− 2 + 2 ). (2-ii) We have for any ψ(x , xn ) ∈ C ∞ (Rn ) ε n 1 tr βI v˜1 (t, x , xn , x , xn ) ψ(x , xn )dxn = 0(t− 2 + 2 + 2 ). 0

Proof.

By (4.4) and Lemma 5.2 we have n−1 √ 1 √ Qw ˜1,−1 (t, xn , xn : b)e−β(x ,η : tA) dη v˜1 (t, x, x) = 2 2π t Rn−1 n−1 √

1 √ =2 (1 + 0( t)) det gQw ˜1,−1 (t, xn , xn : b)e−tR . 2 πt

By 1 b w ˜1,−1 (t, 0, 0 : b) = √ + √ 2 πt t we have tr βI v˜1 (t, x , 0, x , 0) =

∞

e

√ −σ2 +2b tσ

0

1 √ 2 πt

n−1

√ ∞ 1 (b t)j dσ = √ 2 t j=0 Γ( 2j + 1)

√ (1 + 0( t)) det g

√ ∞ ∞ 1 (b t)j (−tR )k × √ tr βI Q . k! t Γ( j2 + 1) k=0 j=0

On the other hand, applying Proposition 16 in [10] we have ε v˜1,−1 (t, xn , xn : γ, b)ψ(x)dxn 0 ε = w ˜1,−1 (t, xn , xn : b)ψ(x)dxn 0 √ i ∞ ∂ 1 ( t)j k ∼ b ψ(x , 0)2−i . 4 j=0 Γ( j2 + 1) ∂xn k,i≥0,k+i=j

(5.3)


59

So the following equality holds: n−1 ε √ 1 √ tr βI v˜1 (t, x, x) ψ(x)dxn = (1 + 0( t)) det gˆ (5.4) 2 πt 0 ⎛ ⎞ √ i ∞ ∞ ∂ ( t)j (−tR )m ⎠ 1 ⎝ k −i . × tr βI Q b ψ(x , 0)2 j 2 ∂xn m! Γ( 2 + 1) k,i≥0,k+i=j m=0 j=0 Owing to Theorem 3.1 of [11] and its corollary we have ⎞ ⎛ √ j i ∞ ∞ m ∂ ( t) (−tR ) ⎠ tr ⎝βI Q bk ψ(x , 0)2−i j ∂x m! Γ( + 1) n 2 m=0 j=0 k,i≥0,k+i=j ⎛ ⎞ √ ∞ ∞ (b t)j (−tR )k ⎠ = tr ⎝βI Q ψ(x , 0) j k! Γ( + 1) 2 j=0 k=0 ⎛ ⎞ √ j i ∞ ∞ m ∂ ( t) (−tR ) ⎠ + tr ⎝βI Q bk ψ(x , 0)2−i . j ∂x m! Γ( + 1) n 2 m=0 j=0 k≥0,i≥1,k+i=j

From Proposition 2.2 and the assumption (I) ≥ we have ⎞ ⎛ √ j ∞ ∞ (−tR )k (b t) ⎠ tr ⎝βI Q j k! Γ( + 1) 2 j=0 k=0 ⎞ ⎛ j k (−R b ) ⎠ t 2 − 12 + 0(t 2 ) = tr ⎝βI Q j k! Γ( 2 + 1) j+2k=−1 and ⎛

⎞ √ j i ∞ ∞ m ∂ ( t) (−tR ) ⎠ tr⎝βI Q bk ψ(x , 0)2−i = 0(t 2 ). j ∂x m! Γ( + 1) n 2 m=0 j=0 k≥0,i≥1,k+i=j

It is sufficient to calculate only ⎛ X,I = tr ⎝βI Q

j+2k=−1

⎞ j

k

b (−R ) ⎠ Γ( 2j + 1) k!

for the proofs of both (1) and (2) because by (5.3), (5.4) and the above equation we have n−1 n 1 n 1 √ tr βI v˜1 (t, x , 0, x , 0) = t− 2 + 2 − 2 det g(x , 0)X,I + 0(t− 2 + 2 ) 2 π

60 and

C. Iwasaki

ε

tr βI v˜1 (t, x, x) ψ(x)dxn

0

1 = 2

1 √ 2 π

n−1

t− 2 + 2 n

n 1 det gˆ(x )X,I ψ(x , 0) + 0(t− 2 + 2 + 2 ).

We calculate X,I dividing cases into (I) and (II) according to . (I) If = 2m + 1 is odd, then j must be even. Set j = 2μ. We can write 1 b2μ (−R )k = tr βI Q(b2 − R )m . tr βI Q X2m+1,I = Γ(μ + 1) k! m! μ+k=m

On the other hand, we have, by (5.1) and the definition of b that is given in (4.3) ˆ ijkh a∗ aj a∗ ah + R Aijkh a∗i aj a∗k ah mod K 1 b2 − R = i k i=n or j=n

1≤i,j,k,h≤n−1

with some functions Aijkh . Applying Proposition 5.1, we have 1 ˆ ikjh a∗i aj a∗k ah + R b2 − R = Aijkh a∗i aj a∗k ah mod K 1 . 2 i=n or j=n 1≤i,j,k,h≤n−1

(5.5)

˜ I,J Then it holds with some functions αI,J and α ⎞m ⎛ 1 ˆ ikjh a∗i aj a∗k ah ⎠ R (b2 − R )m = ⎝ 2 1≤i,j,k,h≤n−1 + αI,J a∗I aJ n∈I or n∈J,(I)=2m,(J)=2m

+

α ˜ I,J a∗I aJ .

(I)≤2m−1,(J)≤2m−1

Noting Q =

an a∗n

=1−

we have with some functions γI,J ⎛ ⎞m 1 ˆ ikjh a∗ aj a∗ ah ⎠ R = −a∗n an ⎝ i k 2 1≤i,j,k,h≤n−1 + γI,J a∗I aJ .

a∗n an ,

Q(b2 − R )m

(I)≤2m,(J)≤2m

Owing to Proposition 2.2 we have m!X2m+1,I = tr βI Q(b2 − R )m ⎛ 1 = −tr ⎝βI a∗n an 2

1≤i,j,k,h≤n−1

ˆ ikjh a∗i aj a∗k ah R

m

⎞ ⎠.

(5.6)


61

If I = (i1 , i2 , . . . , i−1 , n), then by a method similar to Lemma 6.2 of [11] we have by Proposition 2.5 m 1 n− m!X2m+1,I = −2 sign(π) sign(σ) 2 π,σ∈S−1

î î i i i ···R . ×R π(1) π(2) σ(1) σ(2) π(−2) iπ(−1) iσ(−2) iσ(−1) Finally we have

1 1−n √ DI0 . (5.7) 2 π If I ∈ I0 or (I) ≥ + 1, then we have by Proposition 2.2 and (5.6) we have X2m+1,I = 0. X2m+1,I = −2n−+1

(II) If = 2m is even, in this case we have m−1 X2m,I = tr βI Q k=0

Set ˆ= R

b2m−2k−1 (−R )k . Γ(m − k + 12 ) k!

ˆ ikj a∗ aj a∗ a . R i k

1≤i,j,k,≤n−1

By (5.5) it holds that 1ˆ R − b2 + Aijkh a∗i aj a∗k ah mod K 1 . 2 i=n or j=n Then we can write with some functions αI,J 1 ˆ − b2 k + (−R )k = R αI,J a∗I aJ mod K 2k−1 . 2 −R =

n∈I or n∈J,(I)=2k,(J)=2k

So we have with some functions α ˜I,J m−1 k=0

m−1 ˆ k b2m−2k−1 (−b2 )j ( 1 R) b2m−2k−1 (−R )k 2 = k ! Γ(m − k + 12 ) k! Γ(m − k + 12 ) k +j=k j! k=0 + α ˜ I,J a∗I aJ mod K 2m−2 n∈I or n∈J, (I)=2m−1, (J)=2m−1

=

m−1 k=0

+

dm−k−1 b2m−2k−1

n∈I or n∈J, (I)=2m−1, (J)=2m−1

ˆ k ( 12 R) k!

α ˜ I,J a∗I aJ mod K 2m−2

62

C. Iwasaki

with

k

dk =

j=0

(−1)j (−1)k √ . 3 = j!Γ(k − j + 2 ) k!(k + 12 ) π

From Proposition 2.2 and the assumption (I) ≥ we have m−1 1 ˆ k ∗ 2m−2k−1 ( 2 R) dm−k−1 b X2m.I = −tr βI an an k!

(5.8)

k=0

since we obtain m−1 m−1 ˆ k b2m−2k−1 (−R )k ( 1 R) ∗ = −a Q a dm−k−1 b2m−2k−1 2 n n 1 k! Γ(m − k + 2 ) k! k=0 k=0 + γ˜I,J a∗I aJ (I)≤2m−1,(J)≤2m−1

with some constants γ˜I,J . If I = (i1 , i2 , . . . , i−1 , n) we have X2m.I = −2

n−

m−1

2m−2k−1

(−1)

k=0

×

k 1 dm−k−1 2 k!

ˆπ(1)π(2)σ(1)σ(2) · · · sign(π) sign(σ)R

π,σ∈S−1 n ˆ π(2k−1)π(2k)σ(2k−1)σ(2k) cn ···R π(2k+1),σ(2k+1) cπ(2k+2),σ(2k+2) · · ·

· · · cnπ(−1),σ(−1) . In this case we have

1 1−n √ DI0 . 2 π If I ∈ I0 or (I) ≥ + 1, then by (5.8) and Proposition 2.2 we have X2m,I = 2n−+1

(5.9)

X2m,I = 0. By (5.7) and (5.9) it holds that X,I = (−1) 2n−+1

1 1−n √ DI0 2 π

if (I) = (i1 , i2 , . . . , i−1 , n). So the proof is complete.

Corollary 5.5. Let ψ be a smooth function defined in Rn . For any integer N we have n fp tr(Ψp v˜1 )(t, x, x) p=0

=

0(tN ), n 1 n 2t− 2 + 2 − 2 D−1 (x ) det gˆ(x , 0) + 0(t− 2 + 2 ),

if if

xn = 0; xn = 0.


63

For any positive constant ε we have ε n fp tr(Ψp v˜1 )(t, x , xn , x , xn )ψ(x , xn )dxn 0 p=0

n n 1 = t− 2 + 2 D−1 (x ) det gˆ(x , 0)ψ(x , 0) + 0(t− 2 + 2 + 2 ).

By Theorem 2.4 and Lemma 5.4 we have the assertion. n In order to calculate the trace of p=0 fp Ψp Vj (j = 0, −1, −2, . . . , −N ) we have the following lemma which is obtained by the similar method to Lemma 5.4. Proof.

˜ of the operator that corresponds to h Lemma 5.6. If h ∈ Js , then for the kernel h has the following estimates: For any integer N and for any smooth function ψ defined in Rn n if xn = 0; 0(tN ), ˜ fp tr(Ψp h)(t, x, x) = −n + 2 − s2 2 ), if xn = 0. 0(t p=0 and for any positive constant ε ε n n s 1 ˜ fp tr(Ψp h)(t, x , xn , x , xn )ψ(x , xn )dxn = 0(t− 2 + 2 − 2 + 2 ). 0 p=0

Corollary 5.7. Let ψ be a smooth function defined in Rn . For any integer N we have for any j ≤ 0 n 0(tN ), if xn = 0; fp tr(Ψp v˜j )(t, x, x) = j −n + − 0(t 2 2 2 ), if xn = 0. p=0 and for any positive constant ε ε n j n 1 fp tr(Ψp v˜j )(t, x , xn , x , xn )ψ(x , xn )dxn = 0(t− 2 + 2 − 2 + 2 ). 0 p=0

Proof of (2) and (3) of the Main Theorem. If we study the asymptotic behavior of the fundamental solution, it is sufficient to consider the fundamental solution locally by an argument similar to that employed for the case M is closed, which was proved, for example in [11]. In a local patch we have e(t, x, x)dv = u˜(t, x, x)dx + v˜(t, x, x)dx, u ˜(t, x, y) = det g(x)u(t, x, y), where u(t, x, y) is obtained in Section 3. By Corollary 5.5 and Corollary 5.7 we obtain Lemma 5.8. Let ψ be a smooth function defined in Rn . For any integer N we have n 0(tN ), if xn = 0; fp tr(Ψp v˜)(t,x,x) = −n + 2 − 12 −n + 2 2 2 2t D−1 (x ) detˆ g (x ,0) + 0(t ), if xn = 0. p=0

64

C. Iwasaki

and for any positive constant ε ε n fp tr(Ψp v˜)(t, x , xn , x , xn )ψ(x , xn )dxn 0 p=0

n n 1 = t− 2 + 2 D−1 (x ) det gˆ(x , 0)ψ(x , 0) + 0(t− 2 + 2 + 2 ).

From Lemma 5.8 and ep (t, x, x) = Ψp (˜ u(t, x, x) + v˜(t, x, x)) √

1 det g

it is easy to show for any N n n 1 fp tr ep (t, x, x) = √ fp tr(Ψp u˜)(t, x, x) + 0(tN ) x ∈ M \∂M, det g p=0 p=0 =

n

fp tr(Ψp u)(t, x, x) + 0(tN ) x ∈ M \∂M,

p=0

= C (x)t− 2 + 2 + 0(t− 2 + 2 + 2 ) x ∈ M \∂M, n

n

n

1

fp tr ep (t, x, x) = 2D−1 (x)t− 2 + 2 − 2 + 0(t− 2 + 2 ) x ∈ ∂M. n

1

n

p=0

and

n

fp tr ep (t, x, x)ψ(x)dv

M p=0

=

C (x)ψ(x)dv +

M

So the proof is complete.

n n 1 D−1 (x )ψ(x )dσ t− 2 + 2 + 0(t− 2 + 2 + 2 ).

∂M

References [1] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, SpringerVerlag, 1992. [2] S. Chern, A simple intrinsic proof of the Gauss Bonnet formula for closed Riemannian manifolds, Ann. Math. 45 (1944), 747–752. [3] S. Chern, On the curvature integral in a Riemannian manifold, Ann. Math. 46 (1945), 674–684. [4] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators, Texts and Monographs in Physics, Springer, 1987. [5] P. G¨ unther and R. Schimming, Curvature and spectrum of compact Riemannian manifolds, J. Diff. Geom. 12 (1977), 599–618. [6] E. Getzler, The local Atiyah-Singer index theorem, in Critical Phenomena, Random Systems, Gauge Theories, Editors: K. Sterwalder and R. Stora Les Houches, NorthHolland, 1984, 967–974.


65

[7] E. Getzler, A short proof of the local Atiyah-Singer index theorem, Topology 25 (1986), 111–117. [8] P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish, Inc., 1984. [9] P.B. Gilkey, The boundary integrand in the formula for the signature and Euler characteristic of a Riemannian manifold with boundary, Adv. Math. 15 (1975), 334– 360. [10] C. Iwasaki, The asymptotic expansion of the fundamental solution for initialboundary value problems and its application, Osaka J. Math. 31 (1994), 663–728. [11] C. Iwasaki, A proof of the Gauss-Bonnet-Chern theorem by the symbol calculus of pseudo-differential operators, Japanese J. Math. 21 (1995), 235–285. [12] C. Iwasaki, Symbolic calculus of pseudo-differential operators and curvature of manifolds, in Modern Trends in Pseudo-Differential Operators, Editors: J. Toft, M.W. Wong and H. Zhu, Birkh¨ auser, 2007, 51–66. [13] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry I, II, John Wiley & Sons, 1963. [14] S. Murakami, Manifolds, Kyoritsusshuppan, 1969 (in Japanese). [15] V.K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Diff. Geom. 5 (1971), 233–249. Chisato Iwasaki Department of Mathematics University of Hyogo 2167 Shosha Himeji Hyogo 671-2201, Japan e-mail: [email protected]


Operators with Corner-Degenerate Symbols Jamil Abed and Bert-Wolfgang Schulze Abstract. We establish elements of a new approach to ellipticity and parametrices within operator algebras on a manifold with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaces. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The “full” calculus is voluminous; so we content ourselves here with some typical aspects such as symbols in terms of order reducing families, classes of relevant examples, and operators near a corner point. Mathematics Subject Classification (2000). Primary 35S35; Secondary 35J70. Keywords. Pseudo-differential operators, weighted spaces, edge- and cornerdegenerate symbols, ellipticity near corner singularities.

Introduction This paper is aimed at studying operators with certain degenerate operator-valued amplitude functions, motivated by the iterative calculus of pseudo-differential operators on manifolds with higher singularities. Here, in contrast to [38], [39], we develop the aspect of symbols, based on “abstract” reductions of orders which makes the approach transparent from a new point of view. To illustrate the idea, let us first consider, for example, the Laplacian on a manifold with conical singularities (say, without boundary). In this case the ellipticity does not only refer to the “standard” principal homogeneous symbol but also to the so-called conormal symbol. The latter one, contributed by the conical point, is operator-valued and singles out the weights in Sobolev spaces, where the operator has the Fredholm property. Another example of ellipticity with different principal symbolic components is the case of boundary value problems. The boundary (say, smooth), interpreted as an edge, contributes the operator-valued boundary (or edge) symbol which is responsible for the nature of boundary conditions (for instance, of Dirichlet or Neumann

68

J. Abed and B.-W. Schulze

type in the case of the Laplacian). In general, if the configuration has polyhedral singularities of order k, we have to expect a principal symbolic hierarchy of length k + 1, with components contributed by the various strata. In order to characterise the solvability of elliptic equations, especially, the regularity of solutions in suitable scales of spaces, it is adequate to embed the problem in a pseudo-differential calculus, and to construct a parametrix. For higher singularities this is a program of tremendous complexity. It is therefore advisable to organise general elements of the calculus by means of an axiomatic framework which contains the typical features, such as the cone- or edge-degenerate behaviour of symbols but ignores the (in general) huge tail of k − 1 iterative steps to reach the singularity level k. The “concrete” (pseudo-differential) calculus of operators on manifolds with conical or edge singularities may be found in several papers and monographs, see, for instance, [29], [33], [32], [5]. Operators on manifolds of singularity order 2 are studied in [34], [38], [17], [7]. Theories of that kind are also possible for boundary value problems with the transmission property at the (smooth part of the) boundary, see, for instance, [28], [13], [9]. This is useful in numerous applications, for instance, to models of elasticity or crack theory, see [13], [8]. Elements of operator structures on manifolds with higher singularities are developed, for instance, in [37], [1]. The nature of such theories depends very much on specific assumptions on the degeneracy of the involved symbols. There are worldwide different schools studying operators on singular manifolds, partly motivated by problems of geometry, index theory, and topology, see, for instance, Melrose [18], Melrose and Piazza [19], Nistor [24], Nazaikinskij, Savin, Sternin [20], [21], [22], and many others. We do not study here operators of “multi-Fuchs” type, often associated with “corner manifolds”. Our operators are of a rather different behaviour with respect to the degeneracy of symbols. Nevertheless the various theories have intersections and common sources, see the paper of Kondratyev [14] or papers and monographs of other representatives of a corresponding Russian school, see, for instance, [26], [27]. Let us briefly recall a few basic facts on operators on manifolds with conical singularities or edges. Let M be a manifold with conical singularity v ∈ M , i.e., M \ {v} is smooth, and M is close to v modelled on a cone X Δ := (R+ × X)/({0} × X) with base X, where X is a closed compact C ∞ manifold. We then have differential operators of order μ ∈ N on M \ {v}, locally near v in the splitting of variables (r, x) ∈ R+ × X of the form j μ ∂ A := r−μ aj (r) −r (0.1) ∂r j=0 with coefficients aj ∈ C ∞ (R+ , Diffμ−j (X)) (here Diffν (·) denotes the space of all differential operators of order ν on the manifold in parentheses, with smooth coefficients). Observe that when we consider a Riemannian metric on R+ × X := X ∧ of the form dr2 + r2 gX , where gX is a Riemannian metric on X, then the associated Laplace-Beltrami operator is just of the form (0.1) for μ = 2. For such operators we have the homogeneous principal symbol σψ (A) ∈ C ∞ (T ∗ (M \{v})\0),

Operators with Corner-Degenerate Symbols

69

and locally near v in the variables (r, x) with covariables (ρ, ξ) the function σ ˜ψ (A)(r, x, ρ, ξ) := rμ σψ (A)(r, x, r−1 ρ, ξ) which is smooth up to r = 0. If a symbol (or an operator function) contains r and ρ in the combination rρ we speak of degeneracy of Fuchs type. It is interesting to ask the nature of an operator algebra that contains Fuchs type differential operators of the from (0.1) on X Δ , together with the parametrices of elliptic elements. An analogous problem is meaningful on M . Answers may be found in [33], including the tools of the resulting so-called cone algebra. As noted above the ellipticity close to the tip r = 0 is connected with a second symbolic structure, namely, the conormal symbol σc (A)(w) :=

μ

aj (0)wj : H s (X) → H s−μ (X)

(0.2)

j=0

which is a family of operators, depending on w ∈ Γ n+1 −γ , Γβ := {w ∈ C : Rew = 2 β}, n = dim X. Here H s (X) are the standard Sobolev spaces of smoothness s ∈ R on X. Ellipticity of A with respect to a weight γ ∈ R means that (0.2) is a family of isomorphisms for all w ∈ Γ n+1 −γ . 2

The ellipticity on the infinite cone X Δ refers to a further principal symbolic structure, to be observed when r → ∞. The behaviour in that respect is not symmetric under the substitution r → r−1 . The present axiomatic approach will refer to “abstract” corners represented by r → 0. The considerations are based on specific insight on families of reductions of orders in given scales of spaces (in the simplest case H s (X), s ∈ R, when the corner is a conical singularity). In order to motivate our general constructions we briefly recall the form of corner operators of second generation. First, a differential operator on an open stretched wedge R+ × X × Ω (r, x, y), Ω ⊆ Rq open, is called edge-degenerate, if it has the form j ∂ −μ ajα (r, y) −r (rDy )α , (0.3) A=r ∂r j+|α|≤μ

ajα ∈ C ∞ (R+ × Ω, Diff (X)). Observe that (0.3) can be written in the form A = r−μ Opr,y (p) for an operator-valued symbol p of the form p(r, y, ρ, η) = p˜(r, y, rρ, rη) and p˜(r, y, ρ˜, η˜) ∈ C ∞ (R+ × Ω, Lμcl (X; R1+q ρ,˜ ˜ η )), ei(r−r )ρ+i(y−y )η p(r, y, ρ, η)u(r , y )dr dy d¯ρd¯η. Opr,y (p)u(r, y) = μ−(j+|α|)

Here Lμcl (X; Rlλ ) means the space of classical parameter-dependent pseudo-differential operators on X of order μ, with parameter λ ∈ Rl , that is, locally on X the operators are given in terms of amplitude functions a(x, ξ, λ), where (ξ, λ) is treated as an (n + l)-dimensional covariable, and we have L−∞ (X; Rl ) := S(Rl , L−∞ (X)) with L−∞ (X) being the (Fréchet) space of smoothing operators on X.

70


Let Diffμdeg (M ) for a manifold M with edge Y denote the space of all differential operators on M \ Y of order μ that are locally near Y in the splitting of variables (r, x, y) ∈ R+ × X × Ω of the form (0.3). If we replace in the definition the edge covariable η by (η, λ) ∈ Rq+l (q = dim Y ) we obtain parameter-dependent families of operators in Diffμdeg (M ). Similarly as (0.1) an operator of the form j μ ∂ −μ aj (t) −t A := t ∂t j=0 is called corner degenerate if aj ∈ C ∞ (R+ , Diffμ−j deg (M )), j = 0, 1, . . . , μ. The corner μ conormal symbol σc (A)(z) = j=0 aj (0)z j , z ∈ Γ dim M +1 −δ for a corner weight 2 δ ∈ R, is just a parameter-dependent family in Diffμdeg (M ) with parameter Imz on the indicated weight line. The program to study such operators close to the tip t → 0 (see [1], [7]) is just a concrete realisation of the present theory. This paper is organised as follows. In Chapter 1 we introduce spaces of symbols based on families of reductions of orders in given scales of (analogues of Sobolev) spaces. Chapter 2 is devoted to the specific effects of an axiomatic calculus near the tip of the corner. The corner axis is represented by a real axis R r, and the operators take values in vector-valued analogues of Sobolev spaces in r. As indicated above, our results are designed as a step of a larger concept of abstract edge and corner theories, organised in an iterative manner. The full calculus employs the one for r → ∞ as a counterpart of our Mellin operators on R+ near r = 0. However, the continuation of the calculus in that sense needs more space than available in the present note.

1. Symbols associated with order reductions 1.1. Scales and order reducing families Let E denote the set of all families E = (E s )s∈R of Hilbert spaces with continuous ) embeddings E s → E s , s ≥ s, so that E ∞ := s∈R E s is dense in every E s , s ∈ R and that there is a dual scale E ∗ = (E ∗s )s∈R with a non-degenerate sesquilinear pairing (., .)0 : E 0 × E ∗0 → C, such that (., .)0 : E ∞ × E ∗∞ → C, extends to a non-degenerate sesquilinear pairing E s × E ∗−s → C for every s ∈ R, where supf ∈E ∗−s \{0}

|(u,f )0 | f E ∗−s

|(g,v)0 | g E s

are equivalent norms in the spaces E s and E ∗−s , respectively; moreover, if E = (E s )s∈R , E = ) s s−μ := s )s∈R are two scales in consideration and a ∈ Lμ (E, E) ), (E s∈R L(E , E for some μ ∈ R, then sup as,s−μ < ∞ s∈[s ,s ]

and supg∈E s \{0}


71

for every s ≤ s ; here .s,˜s := .L(E s ,E s˜) . Later on, in the case s = s˜ = 0 we often write . := .0,0 . Let us say that a scale E ∈ E is said to have the compact embedding property, if the embeddings E s → E s are compact when s > s. has a formal adjoint a∗ ∈ Lμ (E∗ , E ∗ ), obtained Remark 1.1. Every a ∈ Lμ (E, E) ∗ ∗∞ . by (au, v)0 = (u, a v)0 for all u ∈ E ∞ , v ∈ E is Fréchet in a natural way for every μ ∈ R. Remark 1.2. The space Lμ (E, E) Definition 1.3. A system (bμ (η))μ∈R of operator functions bμ (η) ∈ C ∞ (Rq ,Lμ (E,E)) is called an order reducing family of the scale E, if bμ (η) : E s → E s−μ is an isomorphism for every s, μ ∈ R, η ∈ Rq , b0 (η) = id for every η ∈ Rq , and (i) Dηβ bμ (η) ∈ C ∞ (Rq , Lμ−|β| (E, E)) for every β ∈ Nq ; (ii) for every s ∈ R, β ∈ Nq we have max

|β|≤k

sup bs−μ+|β| (η){Dηβ bμ (η)}b−s (η)0,0 < ∞

η∈Rq s∈[s ,s ]

for all k ∈ N, and for all real s ≤ s . (iii) for every μ, ν ∈ R, ν ≥ μ, we have sup bμ (η)s,s−ν ≤ cηB

s∈[s ,s ]

for all η ∈ Rq and s ≤ s with constants c(μ, ν, s), B(μ, ν, s) > 0, uniformly bounded in compact s-intervals and compact μ, ν-intervals for ν ≥ μ; moreover, for every μ ≤ 0 we have bμ (η)0,0 ≤ cημ for all η ∈ Rq with constants c > 0, uniformly bounded in compact μintervals, μ ≤ 0. Clearly the operators bμ in (iii) for ν ≥ μ or μ ≤ 0, are composed with a corresponding embedding operator. −1 are equivalent In addition we require that the operator families (bμ (η)) to b−μ (η), according to the following notation. Another order reducing family (bμ1 (η))μ∈R , η ∈ Rq , in the scale E is said to be equivalent to (bμ (η))μ∈R , if for every s ∈ R, β ∈ Nq , there are constants c = c(β, s) such that s−μ+|β|

b1

(η){Dηβ bμ (η)}b−s 1 (η)0,0 ≤ c,

bs−μ+|β|(η){Dηβ bμ1 (η)}b−s (η)0,0 ≤ c, for all η ∈ Rq , uniformly in s ∈ [s , s ] for every s ≤ s . Remark 1.4. Parameter-dependent theories of operators are common in many concrete contexts. For instance, if Ω is an (open) C ∞ manifold, there is the space Lμcl (Ω, Rq ) of parameter-dependent pseudo-differential operators on Ω of order μ ∈ R, with parameter η ∈ Rq , where the local amplitude functions a(x, ξ, η)

72


are classical symbols in (ξ, η) ∈ Rn+q , treated as covariables, n = dim Ω, while L−∞ (Ω, Rq ) is the space of Schwartz functions in η ∈ Rq with values in L−∞ (Ω), the space of smoothing operators on Ω. Later on we will also consider specific examples with more control on the dependence on η, namely, when Ω = M \ {v} for a manifold M with conical singularity v. Example. Let X be a closed compact C ∞ manifold, E s := H s (X), s ∈ R, the scale of classical Sobolev spaces on X and bμ (η) ∈ Lμcl (X; Rqη ) a parameter-dependent elliptic family that induces isomorphisms bμ (η) : H s (X) → H s−μ (X) for all s ∈ R. Then for ν ≥ μ we have bμ (η)L(H s (X),H s−ν (X)) ≤ cηπ(μ,ν) for all η ∈ Rq , uniformly in s ∈ [s , s ] for arbitrary s , s , as well as in compact μ- and ν-intervals for ν ≥ μ, where π(μ, ν) := max(μ, μ − ν) with a constant c = c(μ, ν, s , s ) > 0. Observe that supξ∈Rp all η ∈ Rq .

(1.1)

ξ,η

ξν

μ

≤ ηπ(μ,ν) for

Remark 1.5. Let bs (˜ τ , η˜) ∈ Lμcl (X; R1+q τ˜,˜ η ) be an order reducing family as in the above example, now with the parameter (˜ τ , η˜) ∈ R1+q rather than η, and of order s ∈ R. Then, setting bs (t, τ, η) := bs (tτ, tη) the expression ! 12 [t]−s Opt (bs )(η 1 )u2L2 (X) dt for η 1 ∈ Rq \ {0}, |η 1 | sufficiently large, is a norm on the space S(R, C ∞ (X)). Let s Hcone (R × X) denote the completion of S(R, C ∞ (X)) in this norm. Observe that this space is independent of the choice of η 1 , |η 1 | sufficiently large. For reference s;g s (R × X) := t−g Hcone (R × X), g ∈ R, below we also form weighted variants Hcone and set s;g s;g (R+ × X) := Hcone (R × X)|R+ ×X . (1.2) Hcone s;g As is known, cf. [13], the spaces Hcone (R × X) are weighted Sobolev spaces in the calculus of pseudo-differential operators on R+ ×X with |t| → ∞ being interpreted as a conical exit to infinity.

Another feature of order reducing families, known, for instance, in the case of the above example, is that when U ⊆ Rp is an open set and m(y) ∈ C ∞ (U ) a strictly positive function, m(y) ≥ c for c > 0 and for all y ∈ U , the family bs1 (y, η) := bs (m(y)η), s ∈ R, is order reducing in the sense of Definition 1.3 and equivalent to b(η) for every y ∈ U , uniformly in y ∈ K for any compact subset K ⊂ U . A natural requirement is that when m > 0 is a parameter, there is a constant M = M (s , s ) > 0 such that bs (η)b−s (mη)0,0 ≤ c max(m, m−1 )M

for every s ∈ [s , s ], m ∈ R+ , and η ∈ R . q

(1.3)


73

We now turn to another example of an order reducing family, motivated by the calculus of pseudo-differential operators on a manifold with edge (here in “abstract” form), where all the above requirements are satisfied, including the latter one. Definition 1.6. (i) If H is a Hilbert space and κ := {κλ }λ∈R+ a group of isomorphisms κλ : H → H, such that λ → κλ h defines a continuous function R+ → H for every h ∈ H, and κλ κρ = κλρ for λ, ρ ∈ R, we call κ a group action on H. (ii) Let H = (H s )s∈R ∈ E and assume that H 0 is endowed with a group action κ = {κλ }λ∈R+ that restricts (for s > 0) or extends (for s < 0) to a group action on H s for every s ∈ R. In addition, we assume that κ is a unitary group action on H 0 . We then say that H is endowed with a group action. If H and κ are as in Definition 1.6 (i), it is known that there are constants c, M > 0, such that κλ L(H) ≤ c max(λ, λ−1 )M (1.4) for all λ ∈ R+ . Let W s (Rq , H) denote the completion of S(Rq , H) with respect to the norm 12 2 uW s(Rq ,H) := η2s κ−1 u ˆ (η) dη ; H

η u ˆ(η) = Fy→η u(η) is the Fourier transform in Rq . The space W s (Rq , H) will be referred to as edge space on Rq of smoothness s ∈ R (modelled on H). Given a scale H = (H s )s∈R ∈ E with group action we have the edge spaces W s := W s (Rq , H s ), s ∈ R. If necessary we also write W s (Rq , H s )κ . The spaces form again a scale W := (W s )s∈R ∈ E. For purposes below we now formulate a class of operator-valued symbols κ,˜κ (1.5) S μ (U × Rq ; H, H) endowed with group actions κ = for open U ⊆ Rp and Hilbert spaces H and H, ˜ = {˜ κλ }λ∈R+ , respectively, as follows. The space (1.5) is defined to be {κλ }λ∈R+ , κ such that the set of all a(y, η) ∈ C ∞ (U × Rq , L(H, H)) sup (y,η)∈K×Rq

α β η−μ+|β| ˜ κ−1 ˜ 0. (ii) For every s, μ, ν ∈ R, ν ≥ μ, we have bμ (η)L(W s ,W s−ν ) ≤ cηπ(μ,ν)+M(s)+M(s−μ)

(1.8)

for all η ∈ Rq , with a constant c(μ, s) > 0, and M (s) ≥ 0 defined by κλ L(H s ,H s ) ≤ cλM(s) for all λ ≥ 1. Proof. (i) Let us check the estimate (1.7). For the computations we denote by j : H −μ → H 0 the embedding operator. We have for u ∈ W 0 μ 2 b (η)uW 0 = jp(ξ, η)(F u)(ξ)2H 0 dξ −1 −1 2 = κ−1

ξ,η jκ ξ,η κ ξ,η p(ξ, η)κ ξ,η κ ξ,η (F u)(ξ)H 0 dξ −1 −1 2 2 ≤ κ−1

ξ,η jκ ξ,η L(H −μ ,H 0 ) κ ξ,η p(ξ, η)κ ξ,η κ ξ,η (F u)(ξ)H −μ dξ −1 2 2 ≤c κ−1

ξ,η p(ξ, η)κ ξ,η L(H 0 ,H −μ ) κ ξ,η (F u)(ξ)H 0 dξ ≤c sup ξ, η2μ u2W 0 . ξ∈Rp

Thus bμ (η)L(W 0 ,W 0 ) ≤ c supξ∈Rp ξ, ημ ≤ cημ , since μ ≤ 0.


75

(ii) Let j : H s−μ → H s−ν denote the canonical embedding. For every fixed s ∈ R we have 2 bμ (η)u2W s−ν = ξ2(s−ν) κ−1

ξ jp(ξ, η)(Fx→ξ u)(ξ)H s−ν dξ −s 2 = ξ2(s−ν) κ−1 ξs κ−1

ξ jp(ξ, η)κ ξ ξ

ξ (Fx→ξ u)(ξ)H s−ν dξ 2 2 = sup ξ−2ν κ−1 jp(ξ, η)κ ξ2s κ−1

ξ L(H s ,H s−ν )

ξ

ξ Fx→ξ u(ξ)H s dξ. ξ∈Rp

For the first factor on the right we obtain s s−ν ) κ−1

ξ jp(ξ, η) κ ξ L(H ,H −1 s−μ ,H s−ν ) κ s s−μ ) ≤ κ−1

ξ jκ ξ L(H

ξ p(ξ, η)κ ξ L(H ,H s s−μ ) ≤ cκ−1

ξ p(ξ, η)κ ξ L(H ,H

with a constant c > 0. We employed here that κ−1

ξ jκ ξ L(H s−μ ,H s−ν ) ≤ c for all ξ ∈ Rp . Moreover, κ−1

ξ p(ξ, η)κ ξ L(H s ,H s−μ ) −1 −1 ≤ κ−1

ξ κ ξ,η L(H s−μ ,H s−μ ) κ ξ,η p(ξ, η)κ ξ,η L(H s ,H s−μ ) κ ξ,η κ ξ L(H s ,H s )

≤ cξ, ημ κ ξ,η ξ−1 L(H s−μ ,H s−μ ) κ ξ,η−1 ξ L(H s ,H s ) M(s−μ)+M(s) ξ, η ≤ cξ, ημ . ξ As usual, c > 0 denotes different constants (they may also depend on s); the numbers M (s), s ∈ R, are determined by the estimates κλ L(H s ,H s ) ≤ cλM(s) for all λ ≥ 1. We obtain altogether that bμ (η)L(W s ,W s−ν ) ≤ c sup

ξ∈Rn

ξ, ημ ξ, η M(s−μ)+M(s) ξν ξ

≤ cηπ(μ,ν)+M(s−μ)+M(s) .

It can be proved that the operators in Proposition 1.8 also have the uniformity properties with respect to s, μ, ν in compact sets, imposed in Definition 1.3. 1.2. Symbols based on order reductions We now turn to operator-valued symbols, referring to scales s )s∈R ∈ E. E = (E s )s∈R , E = (E For purposes below we slightly generalise the concept of order reducing families by replacing the parameter space Rq η by H η, where

H := {η = (η , η ) ∈ Rq +q : q = q + q , η = 0}.

(1.9)

76


In other words for every μ ∈ R we fix order-reducing families bμ (η) and ˜bμ (η) respectively, where η varies over H, and the properties of in the scales E and E, Definition 1.3 are required for all η ∈ H. In many cases we may admit the case H = Rq as well. for open U ⊆ Rp , μ ∈ R, we denote the set of Definition 1.9. By S μ (U × H; E, E) such that all a(y, η) ∈ C ∞ (U × H, Lμ (E, E)) Dyα Dηβ a(y, η) ∈ C ∞ (U × H, Lμ−|β| (E, E)), (1.10) and for every s ∈ R we have sup

max

|α|+|β|≤k

˜bs−μ+|β| (η){Dyα Dηβ a(y, η)}b−s (η)0,0

(1.11)

y∈K,η∈H,|η|≥h s∈[s ,s ]

is finite for all compact K ⊂ U , k ∈ N, h > 0. denote the subspace of all elements of S μ (U × H; E, E) that Let S μ (H; E, E) are independent of y. Observe that when (bμ (η))μ∈R is an order reducing family parametrised by η ∈ H then we have (1.12) bμ (η) ∈ S μ (H; E, E) for every μ ∈ R. is Fréchet with the semi-norms Remark 1.10. The space S μ (U × H; E, E) a → max sup ˜bs−μ+|β| (η){Dα Dβ a(y, η)}b−s (η)0,0 |α|+|β|≤k

y

η

(1.13)

(y,η)∈K×H,|η|≥h s∈[s ,s ]

parametrised by compact K ⊂ U , s ∈ Z, α ∈ Np , β ∈ Nq , h > 0, which are the best constants in the estimates (1.11). We then have = C ∞ (U, S μ (H; E, E)) = C ∞ (U )⊗ ˆ π S μ (H; E, E). S μ (U × H; E, E) We will also employ other variants of such symbols, for instance, when Ω ⊆ Rm is an open set, := C ∞ (R+ × Ω, S μ (H; E, E)). S μ (R+ × Ω × H; E, E) In order to emphasize the similarity of our considerations for H with the case H = Rq we often write again Rq and later on tacitly use the corresponding results for H in general. in η of order μ and Remark 1.11. Let a(y, η) ∈ S μ (U × Rq ) be a polynomial E = (E s )s∈R a scale and identify Dyα Dηβ a(y, η) with Dyα Dηβ a(y, η) ι with the embedding ι : E s → E s−μ+|β| . Then we have bs−μ+|β| (η) Dyα Dηβ a(y, η) b−s (η)0,0 ≤ |Dyα Dηβ a(y, η)|b−μ+|β| (η)0,0 ≤ cημ−|β| η−μ+|β| = c for all β ∈ Nq , |β| ≤ μ, y ∈ K ⊂ U , K compact (see Definition 1.3 (iii)). Thus a(y, η) is canonically identified with an element of S μ (U × Rq ; E, E).

Operators with Corner-Degenerate Symbols Proposition 1.12. We have := S −∞ (U × Rq ; E, E)

77

= C ∞ (U, S(Rq , L−∞ (E, E))). S μ (U × Rq ; E, E)

μ∈R

Proof. Let us show the assertion for y-independent symbols; the y-dependent case is then straightforward. For notational convenience we set E = E; the general case is analogous. First let a(η) ∈ S −∞ (Rq ; E, E), which means that a(η) ∈ C ∞ (Rq , L−∞ (E, E)) and bs+N (η){Dηβ a(η)}b−s (η)0,0 < c

(1.14)

for all s ∈ R, N ∈ N, β ∈ N and show that q

sup ηM Dηβ a(η)s,t < ∞

(1.15)

η∈Rq

for every s, t ∈ R, M ∈ N, β ∈ Nq . To estimate (1.15) it is enough to assume t > 0. We have ηM Dηβ a(η)s,t = b−kt (η)bkt (η)ηM Dηβ a(η)b−s (η)bs (η)s,t

(1.16)

for every k ∈ N, k ≥ 1, it is sufficient to show that the right-hand side is uniformly bounded in η ∈ Rq for sufficiently large choice of k. The right-hand side of (1.16) can be estimated by b−t (η)0,t b(1−k)t (η)0,0 bkt (η)Dηβ a(η)b−s (η)0,0 bs (η)s,0 . Using bkt (η)Dηβ a(η)b−s (η)0,0 ≤ c, which is true by assumption and the estimates

bs (η)s,0 ≤ cηB , b−t (η)0,t ≤ cηB , with different B, B ∈ R and b(1−k)t (η)0,0 ≤ cη(1−k)t (see Definition 1.3 (iii)) we obtain altogether ηM Dηβ a(η)s,t ≤ cηM+B+B

+(1−k)t

for some c > 0. Choosing k large enough it follows that the exponent on the right-hand side is < 0, i.e., we obtain uniform boundedness in η ∈ Rq . To show the reverse direction suppose that a(η) satisfies (1.15), and let β ∈ Nq , M, s, t ∈ R be arbitrary. We have bt (η)Dηβ a(η)b−s (η)0,0 ≤ bt (η)η−M t,0 η2M Dηβ a(η)s,t η−M b−s (η)0,s . (1.17) Now using (1.15) and the estimates

bt (η)η−M t,0 ≤ cηA−M , η−M b−s (η)0,s ≤ cηA −M , with constants A, A ∈ R, we obtain

bt (η)Dηβ a(η)b−s (η)0,0 ≤ cηA+A −2M . Choosing M large enough we get uniform boundedness of (1.17) in η ∈ Rq which completes the proof.

78


and μ ≤ 0. Then we have Proposition 1.13. Let a(y, η) ∈ S μ (U × Rq ; E, E) a(y, η)0,0 ≤ cημ for all y ∈ K ⊂ U , K compact, η ∈ Rq , with a constant c = c(s, K) > 0. Proof. For simplicity we consider the y-independent case. It is enough to show that a(η)uE 0 ≤ cημ uE 0 for all u ∈ E ∞ . Let j : E −μ → E 0 denote the embedding operator. We then have a(η)uE 0 =a(η)b−μ (η)jbμ (η)uE 0 ≤a(η)b−μ (η)L(E 0 ,E 0 ) jbμ (η)uE 0 ≤ cημ uE 0 .

μ ∈ R, satisfies the estiProposition 1.14. A symbol a(y, η) ∈ S μ (U × Rq ; E, E), mates (1.18) a(y, η)s,s−ν ≤ cηA q for every ν ≥ μ, for every y ∈ K ⊂ U , K compact, η ∈ R , s ∈ R, with constants c = c(s, μ, ν) > 0, A = A(s, μ, ν, K) > 0 that are uniformly bounded when s, μ, ν vary over compact sets, ν ≥ μ. s−μ → Proof. For simplicity we consider again the y-independent case. Let j : E s−ν E be the embedding operator. Then we have a(η)s,s−ν = j˜b−s+μ (η)˜bs−μ (η)a(η)b−s (η)bs (η)s,s−ν ≤ j˜b−s+μ (η)0,s−ν ˜bs−μ (η)a(η)b−s (η)0,0 bs (η)s,0 . Applying (1.11) and Definition 1.3 (iii) we obtain (1.18) with A = B(−s + μ, −s + ν, 0) + B(s, s, 0), together with the uniform boundedness of the involved constants. Also here it can be proved that the involved constants in Propositions 1.13, 1.14 are uniform in compact sets with respect to s, μ, ν. Proposition 1.15. The symbol spaces have the following properties: ⊆ S μ (U × Rq ; E, E) for every μ ≥ μ; (i) S μ (U × Rq ; E, E) α β μ q μ−|β| for every α ∈ Np , β ∈ Nq ; (U × Rq ; E, E) (ii) Dy Dη S (U × R ; E, E) ⊆ S μ q ν q μ+ν (U × R ; E, E0 ) ⊆ S for every μ, ν ∈ R (U × Rq ; E, E) (iii) S (U × R ; E0 , E)S (the notation on the left-hand side of the latter relation means the space of all (y, η)-wise compositions of elements in the respective factors). Proof. For simplicity we consider symbols with constant coefficients. Let us write · := · 0,0 , etc. means (1.10) and (1.11); this implies (i) a(η) ∈ S μ (Rq ; E, E) ˜bs−μ +|β| (η){Dηβ a(η)}b−s (η) = ˜bμ−μ (η)˜bs−μ+|β| (η){Dηβ a(η)}b−s (η) ≤ cημ−μ ˜bs−μ+|β| (η){Dηβ a(η)}b−s (η) ≤ c˜bs−μ+|β| (η){Dηβ a(η)}b−s (η).

We employed μ − μ ≤ 0 and the property (iii) in Definition 1.3.


79

(ii) The estimates (1.10) can be written as ˜bs−(μ−|β|) (η){Dηβ a(η)}b−s (η) ≤ c which just means that Dηβ a(η) ∈ S μ−|β| (Rq ; E, E). a (iii) Given a(η) ∈ S μ (Rq ; E0 , E), ˜(η) ∈ S ν (Rq ; E, E0 ) we have (with obvious meaning of notation) s−ν+|γ| (η){Dηγ a ˜(η)}b−s (η) ≤ c, ˜bs−μ+|δ| (η){Dηδ a(η)}b−s ˜b0 0 (η) ≤ c

for all γ, δ ∈ Nq . If α ∈ Nq is any multi-index, Dηα (a˜ a)(η) is a linear combination ˜(η) with |γ| + |δ| = |α|. It follows that of compositions Dηδ a(η)Dηγ a ˜bs−(μ+ν)+|α| (η)Dηδ a(η){Dηγ a ˜(η)}b−s (η) −s+ν−|γ| s−ν+|γ| = ˜bs−(μ+ν)+|α| (η)Dηδ a(η)b0 (η)b0 (η)Dηγ a ˜(η)b−s (η) s−ν+|γ| ≤ ˜bt−μ+|α|−|γ|(η)Dηδ a(η)b−t (η)Dηγ a ˜(η)b−s (η) 0 (η) b0

(1.19)

for t = s − ν + |γ|; the right-hand side is bounded in η, since |α| − |γ| = |δ|.

Remark 1.16. Observe from (1.19) that the semi-norms of compositions of symbols can be estimated by products of semi-norms of the factors. 1.3. An example from the parameter-dependent cone calculus We now construct a specific family of reductions of orders between weighted spaces on a compact manifold M with conical singularity v, locally near v modelled on a cone X Δ := (R+ × X)/({0} × X) with a smooth compact manifold X as base. The parameter η will play the role of covariables of the calculus of operators on a manifold with edge; that is why we talk about an example from the edge calculus. The associated “abstract” cone calculus according to what we did so far in the Sections 1.1 and 1.2 and then below in Chapter 2 will be a contribution to the calculus of corner operators of second generation. It will be convenient to pass to the stretched manifold M associated with M which is a compact C ∞ manifold with boundary ∂M ∼ = X such that when we squeeze down ∂M to a single point v we just recover M . Close to ∂M the manifold M is equal to a cylinder [0, 1) × X (t, x), a collar neighbourhood of ∂M in M . A part of the considerations will be performed on the open stretched cone X ∧ := R+ × X (t, x) where we identify (0, 1) × X with the interior of the collar neighbourhood (for convenience, without indicating any pull backs of functions or := 2M be the double of M operators with respect to that identification). Let M (obtained by gluing together two copies M± of M along the common boundary is a closed compact C ∞ manifold. ∂M, where we identify M with M+ ); then M On the space M we have a family of weighted Sobolev spaces H s,γ (M ), s, γ ∈ R, that may be defined as s (M \ {v})}, H s,γ (M ) := {σu + (1 − σ)v : u ∈ Hs,γ (X ∧ ), v ∈ Hloc

80


where σ(t) is a cut-off function (i.e., σ ∈ C0∞ (R+ ), σ ≡ 1 near t = 0), σ(t) = 0 for t > 2/3. Here Hs,γ (X ∧ ) is defined to be the completion of C0∞ (X ∧ ) with respect to the norm 12 1 μ 2 bbase (Im w)(Mu)(w)L2 (X) dw , (1.20) 2πi Γ n+1 2

−γ

bμbase (τ )

∈ Lμcl (X; Rτ ) is a family of reductions of order on n = dim X, where X, similarly as in the example in Section 1.1 (in particular, bsbase (τ ) : H s (X) → H 0 (X) = L2 (X) is a family of isomorphisms). Moreover, M is the Mellin trans∞ form, (Mu)(w) = 0 tw−1 u(t)dt, w ∈ C the complex Mellin covariable, and Γβ := {w ∈ C : Re w = β} for any real β. From t H (X ∧ ) = Hs,γ+δ (X ∧ ) for all s, γ, δ ∈ R it follows the existence of a strictly positive function hδ ∈ C ∞ (M \ {v}), such that the operator of multiplication by hδ induces an isomorphism δ

s,γ

hδ : H s,γ (M ) → H s,γ+δ (M )

(1.21)

for every s, γ, δ ∈ R. Moreover, again according to the example in Section 1.1, now for any smooth we have an order reducing family ˜b(η) in the scale of Sobolev compact manifold M s spaces H (M ), s ∈ R. More generally, we employ parameter-dependent families ; Rq ). The symbols a(η) that we want to establish in the scale a ˜(η) ∈ Lμcl (M s,γ H (M ) on our compact manifold M with conical singularity v will be essentially (i.e., modulo Schwartz functions in η with values in globally smoothing operators on M ) constructed in the form σ + (1 − σ)aint (η)(1 − σ ˜˜ ), (1.22) a(η) := σaedge (η)˜ ˜˜ (t) on the half-axis, supaint (η) := a ˜(η)|intM , with cut-off functions σ(t), σ ˜ (t), σ ported in [0, 2/3), with the property ˜ σ ˜≺σ≺σ ˜ (here σ ≺ σ ˜ means the σ ˜ is equal to 1 in a neighbourhood of supp σ). The “edge” part of (1.22) will be defined in the variables (t, x) ∈ X ∧ . Let us choose a parameter-dependent elliptic family of operators of order μ on X p˜(t, τ˜, η˜) ∈ C ∞ (R+ , Lμcl (X; R1+q τ˜,˜ η )). Setting p(t, τ, η) := p˜(t, tτ, tη) (1.23) we have what is known as an edge-degenerate family of operators on X. We now employ the following Mellin quantisation theorem. μ Definition 1.17. Let MO (X; Rq ) defined as the set of all h(z, η) ∈ A(C, Lμcl (X; Rq )) such that h(β + iτ, η) ∈ Lμcl (X; R1+q τ,η ) for every β ∈ R, uniformly in compact βintervals (here A(C, E) with any Fréchet space E denotes the space of all E-valued


81

holomorphic functions in C, in the Fréchet topology of uniform convergence on compact sets). μ Observe that also MO (X; Rq ) is a Fréchet space in a natural way. Given an ∞ f (t, t , z, η) ∈ C (R+ × R+ , Lμcl (X; Γ 12 −γ × Rq )) we set ∞

opγM (f )(η)u(r)

:= R

0

t t

−( 12 −γ+iτ ) 1 dt f t, t , − γ + iτ, η u(t ) d¯τ, 2 t

−1

d¯τ := (2π) dτ , which is regarded as a (parameter-dependent) weighted pseudodifferential operator with symbol f , referring to the weight γ ∈ R. There exists an element ˜ z, η˜) ∈ C ∞ (R+ , M μ (X; Rq )) (1.24) h(t, O η ˜ such that, when we set ˜ z, tη) h(t, z, η) := h(t,

(1.25)

opγM (h)(η) = Opt (p)(η)

(1.26)

we have mod L

−∞

(X

∧

; Rqη ),

for every weight γ ∈ R. Observe that when we set

˜ z, tη) p0 (t, τ, η) := p˜(0, tτ, tη), h0 (t, z, η) := h(0, we also have opγM (h0 )(η) = Opt (p0 )(η) mod L−∞ (X ∧ ; R+ ), for all γ ∈ R. ˜˜ (t) such that ω ˜˜ ≺ ω ≺ ω Let us now choose cut-off functions ω(t), ω ˜ (t), ω ˜. Fix the notation ωη (t) := ω(t[η]), and form the operator function γ− n

ωη (t) aedge (η) := ωη (t)t−μ opM 2 (h)(η)˜ ˜˜ η (t) + m(η) + g(η). (1.27) + t−μ 1 − ωη (t) Opt (p)(η) 1 − ω Here m(η) and g(η) are smoothing Mellin and Green symbols of the edge calculus. The definition of m(η) is based on smoothing Mellin symbols f (z) ∈ M −∞ (X; Γβ ). Here M −∞ (X; Γβ ) is the subspace of all f (z) ∈ L−∞ (X; Γβ ) such that for some ε > 0 (depending on f ) the function f extends to an l(z) ∈ A(Uβ,ε , L−∞ (X)) where Uβ,ε := {z ∈ C : |Rez − β| < ε} and l(δ + iτ ) ∈ L−∞ (X; Rτ ) for every δ ∈ (β − ε, β + ε), uniformly in compact subintervals. By definition we then have f (β + iτ ) = l(β + iτ ); for brevity we often denote the holomorphic extension l of f again by f . For f ∈ M −∞ (X; Γ n+1 −γ ) we set 2

m(η) := for any cut-off functions ω, ω ˜.

γ− n ωη t−μ ωη opM 2 (f )˜

82


In order to explain the structure of g(η) in (1.27) we first introduce weighted spaces on the infinite stretched cone X ∧ = R+ × X, namely, s;g (X ∧ ) Ks,γ;g (X ∧ ) := ωHs,γ (X ∧ ) + (1 − ω)Hcone

(1.28)

for any s, γ, g ∈ R, and a cut-off function ω, see (1.20) which defines Hs,γ (X ∧ ) and the formula (1.2). Moreover, we set Ks,γ (X ∧ ) := Ks,γ;0 (X ∧ ). The operator families g(η) are so-called Green symbols in the covariable η ∈ Rq , defined by μ g(η) ∈ Scl (Rqη ; Ks,γ;g (X ∧ ), S γ−μ+ε (X ∧ )),

(1.29)

μ (Rqη ; Ks,−γ+μ;g (X ∧ ), S −γ+ε (X ∧ )), g ∗ (η) ∈ Scl

(1.30)

∗

for all s, γ, g ∈ R, where g denotes the η-wise formal adjoint with respect to the n scalar product of K0,0;0 (X ∧ ) = r− 2 L2 (R+ × X) and ε = ε(g) > 0. Here S β (X ∧ ) := ωK∞,β (X ∧ ) + (1 − ω)S(R+ , C ∞ (X)) for any cut-off function ω. The notion of operator-valued symbols in (1.29), (1.30) (rather than Hilbert spaces) refers to (1.5) in its generalisation to Fréchet spaces H with group actions (see Remark 1.7) that is in the present case given by κλ : u(t, x) → λ

n+1 2 +g

u(λt, x), λ ∈ R+

(1.31)

n = dim X, both in the spaces Ks;γ,g (X ∧ ) and S γ−μ+ε (X ∧ ). The following theorem is crucial for proving that our new order reduction family is well defined. Therefore we will sketch the main steps of the proof, which is based on the edge calculus. Various aspects of the proof can be found in the literature, for example in Kapanadze and Schulze [12, Proposition 3.3.79], Schrohe and Schulze [31], Harutyunyan and Schulze [9]. Among the tools we have the pseudo-differential operators on X ∧ interpreted as a manifold with conical exit to infinity r → ∞; the general background may be found in Schulze [36]. The calculus of such exit operators goes back to Parenti [25], Cordes [3], Shubin [42], and others. Theorem 1.18. We have σaedge (η)˜ σ ∈ S μ (Rq ; Ks,γ;g (X ∧ ), Ks−μ,γ−μ;g (X ∧ ))

(1.32)

for every s, g ∈ R, more precisely, Dηβ {σaedge (η)˜ σ } ∈ S μ−|β| (Rq ; Ks,γ;g (X ∧ ), Ks−μ+|β|,γ−μ;g (X ∧ ))

(1.33)

for all s, g ∈ R and all β ∈ Nq . (The spaces of symbols in (1.32), (1.33) refer to the group action (1.31)). Proof. To prove the assertions it is enough to consider the case without m(η)+g(η), since the latter sum maps to K∞,γ;g (X ∧ ) anyway. The first part of the Theorem is known, see, for instance, [9] or [4]. Concerning the relation (1.33) we write σ = σ{ac (η) + aψ (η)}˜ σ σaedge (η)˜

(1.34)


83

with γ− n 2

ac (η) := t−μ ωη opM

(h)(η)˜ ωη ,

˜˜ η ) aψ (η) := t−μ (1 − ωη )Opt (p)(η)(1 − ω and it suffices to take the summands separately. In order to show (1.33) we consider, for instance, the derivative ∂/∂ηj =: ∂j for some 1 ≤ j ≤ q. By iterating the process we then obtain the assertion. We have σ = σ{∂j ac (η) + ∂j aψ (η)}˜ σ = b1 (η) + b2 (η) + b3 (η) ∂j σ{ac (η) + aψ (η)}˜ with

! ˜˜ η ) σ (h)(η)∂j ω ˜ η + (1 − ωη )Opt (p)(η)∂j (1 − ω ˜, ! n γ− ˜˜ η ) σ ˜˜, ωη opM 2 (∂j h)(η)˜ ωη + (1 − ωη )Opt (∂j p)(η)(1 − ω ! γ− n ˜˜ η ) σ (∂j ωη )opM 2 (h)(η)˜ ωη + (∂j (1 − ωη ))Opt (p)(η)(1 − ω ˜. γ− n 2

b1 (η) := σt−μ ωη opM b2 (η) := σt−μ b3 (η) := σt−μ

˜η ≡ 0 In b1 (η) we can apply a pseudo-locality argument which is possible since ∂j ω ˜ on supp ωη and ∂j (1 − ω ˜ η ) ≡ 0 on supp (1 − ωη ); this yields (together with similar considerations as for the proof of (1.32)) b1 (η) ∈ S μ−1 (Rq ; Ks,γ;g (X ∧ ), K∞,γ−μ;g (X ∧ )). Moreover we obtain b2 (η) ∈ S μ−1 (Rq ; Ks,γ;g (X ∧ ), Ks−μ+1,γ−μ;g (X ∧ )) since ∂j h and ∂j p are of order μ − 1 (again combined with arguments for (1.32)). Concerning b3 (η) we use the fact that there is a ψ ∈ C0∞ (R+ ) such that ψ ≡ 1 on ˜ ˜ − ψ ≡ 0 on supp ∂j ω and (1 − ω ˜ ) − ψ ≡ 0 on supp ∂j ω. Thus, when supp ∂j ω, ω we set ψη (t) := ψ(t[η]), we obtain b3 (η) := c3 (η) + c4 (η) with ! γ− n ˜, c3 (η) := σt−μ (∂j ωη )opM 2 (h)(η)ψη − (∂j ωη )Opt (p)(η)ψη σ ! n γ− ˜˜ η ) − ψη ] σ c4 (η) := σt−μ (∂j ωη )opM 2 (h)(η)[˜ ωη − ψη ] − (∂j ωη )Opt (p)(η)[(1 − ω ˜. Here, using ∂j ωη = (ω )η ∂j (t[η]) which yields an extra power of t on the left of the operator, together with pseudo-locality, we obtain c4 (η) ∈ S μ−1 (Rq ; Ks,γ;g (X ∧ ), K∞,γ−μ;g (X ∧ )). To treat c3 (η) we employ that both ∂j ωη and ψη are compactly supported on R+ . Using the property (1.26), we have γ− n ˜ c3 (η) = σt−μ (∂j ωη ) opM 2 (h)(η) − Opt (p)(η) ψη σ ∈ S μ−1 (Rq ; Ks,γ;g (X ∧ ), K∞,γ−μ;g (X ∧ )).

84


) Definition 1.19. A family of operators c(η) ∈ S(Rq , s∈R L(H s,γ (M ), H ∞,δ (M ))) is called a smoothing element in the parameter-dependent cone calculus on M associated with the weight data (γ, δ) ∈ R2 , written c ∈ CG (M, (γ, δ); Rq ), if there is an ε = ε(c) > 0 such that c(η) ∈ S(Rq , L(H s,γ (M ), H ∞,δ+ε (M ))), c (η) ∈ S(Rq , L(H s,−δ (M ), H ∞,−γ+ε (M ))); ∗

for all s ∈ R; here c∗ is the η-wise formal adjoint of c with respect to the H 0,0 (M )scalar product. The η-wise kernels of the operators c(η) are in C ∞ ((M \ {v}) × (M \ {v})). However, they are of flatness ε in the respective distance variables to v, relative to the weights δ and γ, respectively. Let us look at a simple example to illustrate the structure. We choose elements k ∈ S(Rq , H ∞,δ+ε (M )), k ∈ S(Rq , H ∞,−γ+ε (M )) and assume for convenience that k and k vanish outside a neighbourhood of v, for all η ∈ Rq . Then with respect to a local splitting of variables (t, x) near v we can write k = k(η, t, x) and k = k (η, t , x ), respectively. Set c(η)u(t, x) := k(η, t, x)k (η, t , x )u(t , x )tn dt dx with the formal adjoint c∗ (η)v(t , x ) :=

k (η, t , x )k(η, t, x)v(t, x)tn dtdx.

Then c(η) is a smoothing element in the parameter-dependent cone calculus. By C μ (M, (γ, γ − μ); Rq ) we denote the set of all operator families ˜˜ ) + c(η) a(η) = σaedge (η)˜ σ + (1 − σ)aint (η)(1 − σ

(1.35)

Lμcl (M \{v}; Rq ),

where aedge is of the form (1.27), aint ∈ while c(η) is a parameterdependent smoothing operator on M , associated with the weight data (γ, γ − μ). Theorem 1.20. Let M be a compact manifold with a conical singularity. Then the η-dependent families (1.22) which define continuous operators a(η) : H s,γ (M ) → H s−ν,γ−ν (M )

(1.36)

for all s ∈ R, ν ≥ μ, have the properties: a(η)L(H s,γ (M),H s−ν,γ−ν (M)) ≤ cηB

(1.37)

for all η ∈ Rq , and s ∈ R, with constants c = c(μ, ν, s) > 0, B = B(μ, ν, s), and, when μ ≤ 0 (1.38) a(η)L(H 0,0 (M),H 0,0 (M)) ≤ cημ for all η ∈ R, s ∈ R, with constants c = c(μ, s) > 0. ˜˜ ) as we see from Proof. The result is known for the summand (1 − σ)aint (η)(1 − σ the example in Section 1.1. Therefore, we may concentrate on σ : H s,γ (M ) → H s−ν,γ−ν (M ). p(η) := σaedge (η)˜


85

To show (1.37) we pass to σaedge (η)˜ σ : Ks,γ (X ∧ ) → Ks−ν,γ−ν (X ∧ ). Then Theorem 1.18 shows that we have symbolic estimates, especially μ κ−1

η p(η)κ η L(Ks,γ (X ∧ ),Ks−μ,γ−μ (X ∧ )) ≤ cη .

We have p(η)L(Ks,γ (X ∧ ),Ks−ν,γ−ν (X ∧ )) ≤ p(η)L(Ks,γ (X ∧ ),Ks−μ,γ−μ (X ∧ )) , and −1 s,γ ∧ s−μ,γ−μ (X ∧ )) p(η)L(Ks,γ (X ∧ ),Ks−μ,γ−μ (X ∧ )) = κ η κ−1

η p(η)κ η κ η L(K (X ),K s−μ,γ−μ (X ∧ ),Ks,γ (X ∧ )) ≤ κ η L(Ks−μ,γ−μ (X ∧ ),Ks−μ,γ−μ (X ∧ )) κ−1

η p(η)κ η L(K

μ+M +M s,γ ∧ s,γ ∧ κ−1 .

η L(K (X ),K (X )) ≤ cη

Here we used that κ η , κ−1

η satisfy estimates like (1.4). For (1.38) we employ that κλ is operating as a unitary group on K0,0 (X ∧ ). This gives us 0,0 ∧ 0,0 ∧ p(η)L(K0,0 (X ∧ ),K0,0 (X ∧ )) = κ−1

η p(η)κ η L(K (X ),K (X ))

μ 0,0 ∧ −μ,−μ (X ∧ )) ≤ cη . ≤ κ−1

η p(η)κ η L(K (X ),K

Theorem 1.21. For every k ∈ Z there exists an fk (z) ∈ M −∞ (X; Γ n+1 −γ ) such 2 that for every cut-off functions ω, ω ˜ the operator γ− n 2

A := 1 + ωopM

(fk )˜ ω : H s,γ (M ) → H s,γ (M )

(1.39)

is Fredholm and of index k, for all s ∈ R. Proof. We employ the result (cf. [35]) that for every k ∈ Z there exists an fk (z) such that n := 1 + ωopγ− 2 (fk )˜ A ω : Ks,γ (X ∧ ) → Ks,γ (X ∧ ) (1.40) M is Fredholm of index k. Recall that the proof of the latter result follows from a corresponding theorem in the case dim X = 0. The Mellin symbol fk is constructed in such a way that 1+fk (z) = 0 for all z ∈ Γ 12 −γ and the argument of 1+fk (z)|Γ 1 −γ 2 varies from 1 to 2πk when z ∈ Γ 12 −γ goes from Imz = −∞ to Imz = +∞. The choice of ω, ω ˜ is unessential; so we assume that ω, ω ˜ ≡ 0 for r ≥ 1 − ε with some := X Δ as a union of [0, 1 + ε ) × X /({0} × ε > 0. Let us represent the cone M 2 − and (1 − ε , ∞) × X =: M + . Then X) =: M 2 n

= 1 + ωopγ− 2 (fk )˜ = 1. A| ω , A| M M− M+

(1.41) ε Moreover, without loss of generality, we represent M as a union [0, 1 + 2) × ∞ X /({0} × X) ∪ M+ where M+ is an open C manifold which intersects [0, 1 +

86


× X /({0} × X) =: M− in a cylinder of the form (1 − 2ε , 1 + 2ε ) × X. Let B denote the operator on M , defined by ε 2)

γ− n 2

B− := A|M− = 1 + ωopM

(fk )˜ ω , B+ := A|M+ = 1

(1.42)

We are then in a special situation of cutting and pasting of Fredholm operators. by setting We can pass to manifolds with conical singularities N and N − ∪ M+ , N = M− ∪ M + N =M , respectively, by and transferring the former operators in (1.41), (1.42) to N and N and A to belong to M ± and M± to corresponding gluing together the ± pieces of A operators B on N and B on N . We then have the relative index formula − indB indA − indB = indA (1.43) and M are the same as B and N where B and N (see [23]). In the present case A are the same as A and M . It follows that − indB = indB − indA. indA (1.44) From (1.43), (1.44) it follows that indA = indB = indA.

Theorem 1.22. There is a choice of m and g such that the operators (1.22) form a family of isomorphisms a(η) : H s,γ (M ) → H s−μ,γ−μ (M )

(1.45)

for all s ∈ R and all η ∈ Rq . Proof. We choose a function p(t, τ, η, ζ) := p˜(tτ, tη, ζ) similarly as (1.23) where p˜(˜ τ , η˜, ζ) ∈ Lμcl (X; R1+q+l τ˜,˜ η ,ζ ), l ≥ 1, is parameter-dependent elliptic with parameters τ˜, η˜, ζ. For purposes below we specify p˜(˜ τ , η˜, ζ) in such a way that the parameter-dependent homogeneous principal symbol in (x, τ˜, ξ, η˜, ζ) for (˜ τ , ξ, η˜, ζ) = 0 is equal to μ

(|˜ τ |2 + |ξ|2 + |˜ η |2 + |ζ|2 ) 2 . We now form an element μ ˜ h(z, η˜, ζ) ∈ MO (X; Rq+l η ˜,ζ )

analogously as (1.24) such that h(t, z, η, ζ) := ˜ h(z, tη, ζ) satisfies opγM (h)(η, ζ) = Opt (p)(η, ζ) l mod L−∞ (X ∧ ; Rq+l η,ζ ). For every fixed ζ ∈ R this is exactly as before, but in this way we obtain corresponding ζ-dependent families of such objects. It follows ! γ− n ˜ σ := t−μ σ ωη opM 2 (h)(η, ζ)˜ ωη + χη Opt (p)(η, ζ)χ ˜η σ σbedge (η, ζ)˜


87

with ˜˜ η (t). χη (t) := 1 − ωη (t), χ ˜η (t) := 1 − ω Let us form the principal edge symbol γ− n 2

σ∧ (σbedge σ ˜ )(η, ζ) = t−μ ω|η| opM

! (h)(η, ζ)˜ ω|η| + χ|η| Opt (p)(η, ζ)χ ˜|η|

for |η| = 0. The latter is interpreted as a family of continuous operators σ∧ (σbedge σ ˜ )(η, ζ) : Ks,γ;g (X ∧ ) → Ks−μ,γ−μ;g (X ∧ )

(1.46)

which is elliptic as a family of classical pseudo-differential operators on X ∧ . In addition it is exit elliptic on X ∧ with respect to the conical exit of X ∧ to infinity. In order that (1.46) is Fredholm for the given weight γ ∈ R and all s, g ∈ R it is necessary and sufficient that the subordinate conormal symbol ˜ )(z, ζ) : H s (X) → H s−μ (X) σc σ∧ (σbedge σ is a family of isomorphisms for all z ∈ Γ n+1 −γ . This is standard information from 2 the calculus on the stretched cone X ∧ . By definition the conormal symbol is just ˜ z, 0, ζ) : H s (X) → H s−μ (X). h(0,

(1.47)

˜ + iτ, 0, ζ) is parameter-dependent elliptic on X with Since by construction h(β 1+l parameters (τ, ζ) ∈ R , for every β ∈ R (uniformly in finite β-intervals) there is a C > 0 such that (1.47) becomes bijective whenever |τ, ζ| > C. In particular, choosing ζ large enough it follows the bijectivity for all τ ∈ R, i.e., for all z ∈ Γ n+1 −γ . Let us fix ζ 1 in that way and write again 2

˜ tη, ζ 1 ). p(t, τ, η) := p˜(tτ, tη, ζ 1 ), h(t, z, η) := h(z, We are now in the same situation we started with, but we know in addition that (1.46) is a family of Fredholm operators of a certain index, say, −k for some k ∈ Z. With the smoothing Mellin symbol fk (z) as in (1.40) we now form the composition γ− n 2

σbedge (η)˜ σ (1 + ωη opM

(fk )˜ ωη )

(1.48)

which is of the form γ− n 2

σbedge (η)˜ σ + ωη opM

(f )˜ ωη + g(η)

(1.49)

for another smoothing Mellin symbol f (z) and a certain Green symbol g(η). Here, by a suitable choice of ω, ω ˜ , without loss of generality we assume that σ ≡ 1 and σ ˜ ≡ 1 on supp ωη ∪ supp ω ˜ η , for all η ∈ Rq . Since (1.48) is a composition of parameter-dependent cone operators the associated edge symbol is equal to γ− n 2

(fk )˜ ω|η| ) : Ks,γ (X ∧ ) → Ks−μ,γ−μ (X ∧ ) (1.50) which is a family of Fredholm operators of index 0. By construction (1.50) depends only on |η|. For η ∈ S q−1 we now add a Green operator g0 on X ∧ such that ˜ )(η)(1 + ω|η| opM F (η) := σ∧ (σbedge σ

F (η) + g0 (η) : Ks,γ (X ∧ ) → Ks−μ,γ−μ (X ∧ )

88


is an isomorphism; it is known that such g0 (of finite rank) exists (for N = N dim ker F (η) it can be written in the form g0 u := j=1 (u, vj )wj , where (·, ·) is the K0,0 (X ∧ )-scalar product and (vj )j=1,...,N and (wj )j=1,...,N are orthonormal systems of functions in C0∞ (X ∧ )). Setting ˜ g(η) := σϑ(η)|η|μ κ|η| g0 κ−1 |η| σ with an excision function ϑ(η) in Rq we obtain a Green symbol with σ∧ (g)(η) = |η|μ κ|η| g0 κ−1 |η| and hence σ∧ (F (η) + g(η)) : Ks,γ (X ∧ ) → Ks−μ,γ−μ (X ∧ ) is a family of isomorphisms for all η ∈ Rq \ {0}. Setting * + γ− n γ− n aedge (η) := t−μ ωη opM 2 (h)(η)˜ ωη + χη Opt (p)(η)χ ˜η 1 + ωη opM 2 (fk )˜ ωη + |η|μ ϑ(η)κ|η| g0 κ−1 |η|

(1.51)

we obtain an operator family σaedge (η)˜ σ = F (η) + g(η) as announced before. Next we choose a parameter-dependent elliptic aint (η) ∈ Lμcl (M \{v}; Rqη ) such that its parameter-dependent homogeneous principal symbol close to t = 0 (in the splitting of variables (t, x)) is equal to μ

(|τ |2 + |ξ|2 + |η|2 ) 2 . Then we form ˜˜ ) a(η) := σaedge (η)˜ σ + (1 − σ)aint (η)(1 − σ ˜ with σ, σ ˜, σ ˜ as in (1.22). This is now a parameter-dependent elliptic element of the cone calculus on M with parameter η ∈ Rq . It is known, see the explanations after this proof, that there is a constant C > 0 such that the operators (1.45) are isomorphisms for all |η| ≥ C. Now, in order to construct a(η) such that (1.45) are isomorphisms for all η ∈ Rq we simply perform the construction with (η, λ) ∈ Rq+r , r ≥ 1 in place of η, then obtain a family a(η, λ) and define a(η) := a(η, λ1 ) with a λ1 ∈ Rr , |λ1 | ≥ C. Let us now give more information on the above-mentioned space C μ (M, g; Rq ), g = (γ, γ − μ), of parameter-dependent cone operators on M of order μ ∈ R, with the weight data g. The elements a(η) ∈ C μ (M, g; Rq ) have a principal symbolic hierarchy σ(a) := (σψ (a), σ∧ (a))

(1.52)

where σψ (a) is the parameter-dependent homogeneous principal symbol of order μ, defined through a(η) ∈ Lμcl (M \ {v}; Rq ). This determines the reduced symbol σ ˜ψ (a)(t, x, τ, ξ, η) := tμ σψ (a)(t, x, t−1 τ, ξ, t−1 η)


89

given close to v in the splitting of variables (t, x) with covariables (τ, ξ). By construction σ ˜ψ (a) is smooth up to t = 0. The second component σ∧ (a)(η) is defined as γ− n 2

σ∧ (a)(η) := t−μ ω|η| opM

(h0 )(η)˜ ω|η|

˜˜ |η| ) + σ∧ (m + g)(η) + t−μ (1 − ω|η| )Opt (p0 )(η)(1 − ω where σ∧ (m + g)(η) is just the (twisted) homogeneous principal symbol of m + g as a classical operator-valued symbol. The element a(η) of CG (M, g; Rq ) represents families of continuous operators a(η) : H s,γ (M ) → H s,γ−μ (M )

(1.53)

for all s ∈ R. Definition 1.23. An element a(η) ∈ C μ (M, g; Rq ) is called elliptic, if ˜ψ (a) (i) σψ (a) never vanishes as a function on T ∗ ((M \ {v}) × Rq ) \ 0 and if σ does not vanish for all (t, x, τ, ξ, η), (τ, ξ, η) = 0, up to t = 0; (ii) σ∧ (a)(η) : Ks,γ (X ∧ ) → Ks−μ,γ−μ (X ∧ ) is a family of isomorphisms for all η = 0, and any s ∈ R. Theorem 1.24. If a(η) ∈ C μ (M, g; Rq ), g = (γ, γ − μ) is elliptic, there exists an element a(−1) (η) ∈ C −μ (M, g −1 ; Rq ) g −1 := (γ − μ, γ), such that 1 − a(−1) (η)a(η) ∈ CG (M, g l ; Rq ), 1 − a(η)a(−1) (η) ∈ CG (M, g r ; Rq ), where g l := (γ, γ), g r := (γ − μ, γ − μ). The proof employs known elements of the edge symbolic calculus (cf. [36]); so we do not recall the details here. Let us only note that the inverses of σψ (a), σ ˜ψ (a) and σ∧ (a) can be employed to construct an operator family b(η) ∈ C −μ (M,g −1 ;Rq ) such that σψ (a(−1) ) = σψ (b), σ ˜ψ (a(−1) ) = σ ˜ψ (b), σ∧ (a(−1) ) = σ∧ (b). This gives us 1 − b(η)a(η) =: c0 (η) ∈ C −1 (M, g l ; Rq ), and a formal Neumann series argument allows b(η) to a left parametrix a(−1) (η) by set us to improve ∞ j ting a(−1) (η) := j=0 c0 (η) b(η) (using the existence of the asymptotic sum in C 0 (M, g; Rq )). In a similar manner we can construct a right parametrix, i.e., a(−1) (η) is as desired. Corollary 1.25. If a(η) is as in Theorem 1.24, then (1.53) is a family of Fredholm operators of index 0, and there is a constant C > 0 such that the operators (1.53) are isomorphisms for all |η| ≥ C, s ∈ R. Corollary 1.26. If we perform the construction of Theorem 1.24 with the parameter (η, λ) ∈ Rq+l , l ≥ 1, rather than η, Corollary 1.25 yields that a(η, λ) is invertible for all η ∈ Rq , |λ| ≥ C. Then, setting a(η) := a(η, λ1 ), |λ1 | ≥ C fixed, we obtain a−1 (η) ∈ C −μ (M, g −1 ; Rq ).

90


Observe that the operator functions of Theorem 1.20 refer to scales of spaces with two parameters, namely, s ∈ R, the smoothness, and γ ∈ R, the weight. Compared with Definition 1.9 we have here an additional weight. There are two ways to make the different view points compatible. One is to apply weight reducing isomorphisms h−μ : H s,γ (M ) → H s,γ−μ (M ) (1.54) in (1.21). Then, passing from a(η) : H s,γ (M ) → H s−μ,γ−μ (M )

(1.55)

to

bμ (η) := h−γ+μ a(η)hγ : H s,0 (M ) → H s−μ,0 (M ) (1.56) we obtain operator functions between spaces only referring to s but with properties as required in Definition 1.9 (which remains to be verified). Remark 1.27. The spaces E s := H s,0 (M ), s ∈ R, form a scale with the properties at the beginning of Section 1.1. Another way is to modify the abstract framework by admitting scales E s,γ rather than E s , where in general γ may be in Rk (which is motivated by the higher corner calculus). We do not study the second possibility here but we only note that the variant with E s,γ -spaces is very similar to the one without γ. Let us now look at operator functions of the form (1.56). Theorem 1.28. The operators (1.56) constitute an order reducing family in the spaces E s := H s,0 (M ), where the properties (i)–(iii) of Definition 1.3 are satisfied. Proof. In this proof we concentrate on the properties of our operators for every fixed s, μ, ν with ν ≥ μ. The uniformity of the involved constants can easily be deduced; however, the simple (but lengthy) considerations will be left out. (i) We have to show that Dηβ bμ (η) = Dηβ {h−γ+μ a(η)hγ } ∈ C ∞ (Rq , L(E s , E s−μ+|β| )) for all s ∈ R, β ∈ Nq . According to (1.22) the operator function is a sum of two contributions. The second summand ˜˜ ) (1 − σ)h−γ+μ aint (η)hγ (1 − σ is a parameter-dependent family in Lμcl (2M; Rq ) and obviously has the desired property. The first summand is of the form σh−γ+μ {aedge (η) + m(η) + g(η)}hγ σ ˜. From the proof of Theorem 1.20 we have σ ∈ S μ−|β| (Rq ; Ks,γ;g (X ∧ ), Ks−μ+|β|,γ−μ;g (X ∧ )) Dηβ σaedge (η)˜ for every β ∈ Nq . In particular, these operator functions are smooth in η and the derivatives improve the smoothness in the image by |β|. This gives us the ˜ . The C ∞ dependence of m(η) + g(η) in desired property of σh−γ+μ aedge (η)hγ σ η is clear (those are operator-valued symbols), and they map to K∞,γ−μ;g (X ∧ )


91

anyway. Therefore, the desired property of σh−γ+μ {m(η) + g(η)}hγ σ ˜ is satisfied as well. (ii) This property essentially corresponds to the fact that the product in consideration close to the conical point is a symbol in η of order zero and that the group action in K0,0 (X ∧ )-spaces is unitary. Outside the conical point the boundedness is as in the example in Section 1.1. (iii) The proof of this property close to the conical point is of a similar structure as Proposition 1.8, since our operators are based on operator-valued symbols referring to spaces with group action. The contribution outside the conical point is as in the same example in Section 1.1. Remark 1.29. For E s := Hs,0 (M ), s ∈ R, E = (E s )s∈R , the operator functions bμ (η) of the form (1.56) belong to S μ (Rq ; E, E) (see the notation after Definition 1.9).

2. Operators referring to a corner point 2.1. Weighted spaces Let E = (E s )s∈R ∈ E be a scale and (bμ ())μ∈R , ∈ R, be an order reducing family (see Definition 1.3 with q = 1). We define a new scale of spaces adapted to the Mellin transform and the approach of the cone calculus. In the following definition the Mellin transform refers to the variable r ∈ R+ , i.e., M = Mr→w . Definition 2.1. For every s, γ ∈ R we define the space Hs,γ (R+ , E) to be the completion of C0∞ (R+ , E ∞ ) with respect to be norm 12 1 s 2 s,γ b (Imw)(M u)(w)E 0 dw (2.1) uH (R+ ,E) = 2πi Γ d+1 2

−γ

for a d = dE ∈ N. The Mellin transform M in (2.1) is interpreted as the weighted Mellin transform Mγ− d . 2

The role of dE is an extra information, given together with the scale E. In the example E = (H s (X))s∈R for a closed compact C ∞ manifold X we set dE := dim X. Observe that when we replace the order reducing family in (2.1) by an equivalent one the resulting norm is equivalent to (2.1). By virtue of the identity rβ Hs,γ (R+ , E) = Hs,γ+β (R+ , E) for every s, γ, β ∈ R, it is often enough to refer the considerations to one particular weight, or to set dE = 0. (2.2) For simplicity we now assume (2.2).

92


Let us consider Definition 1.9 for the case U = R, q = 1, and denote the covariable now by ∈ R. Set := S μ (R × R × R; E, E)| S μ (R+ × R+ × R; E, E) R+ ×R+ ×R

and := {a(r, r , w) ∈ C ∞ (R+ × R+ × Γδ , Lμ (E, E)) S μ (R+ × R+ × Γδ ; E, E) : a(r, r , δ + i) ∈ S μ (R+ × R+ × Rρ ; E, E)} for any δ ∈ R. The subspaces of r -independent ((r, r )-independent) symbols are (S μ (R; E, E )) and S μ (R+ × Γδ ; E, E) (S μ (Γδ ; E, E)), denoted by S μ (R+ × R; E, E) respectively. we set Given an element f (r, r , w) ∈ S μ (R+ × R+ × Γ 12 −γ ; E, E) ∞ −( 1 −γ+i) r 1 2 γ 1 dr − γ + i u(r f r, r , ) d. (2.3) opM (f )u(r) = 2π r 2 r 0 Let, for instance, f be independent of r . Then (2.3) induces a continuous operator ∞ ). opγM (f ) : C0∞ (R+ , E ∞ ) → C ∞ (R+ , E opγM (f )

−1 Mγ,w→r f (r, w)Mγ,r →w .

In fact, we have = transform Mγ induces a continuous operator

(2.4)

The weighted Mellin

Mγ : C0∞ (R+ , E s ) → S(Γ 12 −γ , E s ) for every s ∈ R. The subsequent multiplication of Mγ u(w) by f (r, w) gives rise to s−μ )), and then it follows easily that opγ (f )u ∈ an element in C ∞ (R+ , S(Γ 12 −γ , E M s−μ ). We now formulate a continuity result, first for the case of symbols C ∞ (R+ , E with constant coefficients. the operator (2.4) extends to a Theorem 2.2. For every f (w) ∈ S μ (Γ 12 −γ ; E, E) continuous operator opγM (f ) : Hs,γ (R+ , E) → Hs−μ,γ (R+ , E) (2.5) for every s ∈ R. Moreover, f → opγM (f ) induces a continuous operator → L(Hs,γ (R+ , E), Hs−μ,γ (R+ , E)) S μ (Γ 12 −γ ; E, E)

(2.6)

for every s ∈ R. Proof. We have opγM (f )u2Hs−μ,γ (R ,E) + 1 1 s−μ −1 ˜ − γ + i (Mγ u) − γ + i 2E 0 d = b ()Mγ Mγ f 2 2 R 1 1 s−μ −s s ˜ − γ + i b ()b ()(Mγ u) − γ + i 2E 0 d ()f = b 2 2 R ≤ c2 u2Hs,γ (R+ ,E)

Operators with Corner-Degenerate Symbols with

c = sup ˜bs−μ ()f ∈R

93

1 − γ + i b−s ()L(E 0 ,E 0 ) 2

which is finite for every s ∈ R (cf. the estimates (1.10)). Thus we have proved the continuity both of (2.5) and (2.6). In order to generalise Theorem 2.2 to symbols with variable coefficients we impose conditions of reasonable generality that allow us to reduce the arguments to a vector-valued analogue of Kumano-go’s technique. Given a Fréchet space V with a countable system of semi-norms (πι )ι∈N that defines its topology, we denote by ∞ (R+ × R+ , V ) CB

the set of all u(r, r ) ∈ C ∞ (R+ × R+ , V ) such that sup πι (r∂r )k (r ∂r )k u(r, r ) < ∞ r,r ∈R+

∞ for all k, k ∈ N. In a similar manner by CB (R+ , V ) we denote the set of such functions that are independent of r . Moreover, we set μ := C ∞ (R+ × R+ , S μ (Γ 1 ; E, E)) SB (R+ × R+ × Γ 12 −γ ; E, E) B 2 −γ μ := C ∞ (R+ , S μ (Γ 1 −γ ; E, E)). (R+ × Γ 12 −γ ; E, E) and, similarly, SB B 2 μ the operator opγ (f ) inTheorem 2.3. For every f (r, w) ∈ SB (R+ × Γ 12 −γ ; E, E) M duces a continuous mapping

opγM (f ) : Hs,γ (R+ , E) → Hs−μ,γ (R+ , E), and f → opγM (f ) a continuous operator μ → L(Hs,γ (R+ , E), Hs−μ,γ (R+ , E)) SB (R+ × Γ 12 −γ ; E, E)

for every s ∈ R. Parallel to the spaces of Definition 2.1 it also makes sense to consider their “cylindrical” analogue, defined as follows. Definition 2.4. Let (bs (η))s∈R , be an order reducing family as in Definition 1.3. For every s ∈ R we define the space H s (Rq , E) to be the completion of C0∞ (Rq , E ∞ ) with respect to the norm ! 12 uH s (Rq ,E) := bs (η)(F u)(η)2E 0 dη . Rq

94


Clearly, similarly as above, with a symbol a(y, y , η) ∈ S μ (Rq × Rq × Rq ; E, E) (when we impose a suitable control with respect to the dependence on y for large |y |) we can associate a pseudo-differential operator Opy (a)u(y) = ei(y−y )η a(y, y , η)u(y )dy d¯η. In particular, if a = a(η) has constant coefficients, then we obtain a continuous operator Opy (a) : H s (Rq , E) → H s−μ (Rq , E) for every s ∈ R. In the case of variable coefficients we need some precautions on the nature of symbols. This will be postponed for the moment. We are mainly interested in the case q = 1. Consider the transformation (Sγ u)(y) := e−( 2 −γ)y u(e−y ) 1

from functions in r ∈ R+ to functions in y ∈ R. We then have the identity 1 − γ + i = (F Sγ u)() (Mγ u) 2 with F being the one-dimensional Fourier transform. This gives us ! 12 1 bs (η)(F Sγ u)(η)2E 0 dη = Sγ uH s (R,E) = uHs,γ (R+ ,E) , 2π R i.e., Sγ induces an isomorphism Sγ : Hs,γ (R+ , E) → H s (R, E). Remark 2.5. By reformulating the expression (2.3) we obtain 1 1 1 dr opγM (f )u(r) = e( 2 −γ+i)(log r −log r) f r, r , − γ + i u(r ) d. 2π 2 r

Substituting r = e−y , r = e−y gives us 1 1 γ opM (f )u(r) = ei(y−y ) e( 2 −γ)(y−y ) f (e−y , 2π 1 e−y , − γ + i)u(e−y )dy d = Opy (gγ )v(y) 2

with v(y) := u(e−y ) and gγ (y, y , ) := e( 2 −γ)(y−y ) f (e−y , e−y , 12 − γ + i). In other words, if χ : R+ → R is defined by χ(r) = − log r =: y, we have (χ∗ v)(r) = v(− log r) or ((χ−1 )∗ u)(g) = u(e−y ) and 1

opγM (f ) = χ∗ Opy (gγ )(χ−1 )∗ . Thus Opy (gγ ) is the operator push forward of opγM (f ) under χ.


95

2.2. Mellin quantisation and kernel cut-off The axiomatic cone calculus that we develop here is a substructure of the gen of the form eral calculus of operators with symbols in a(r, ρ) ∈ S μ (R+ × R; E, E) μ a(r, ρ) = a ˜(r, rρ), a˜(r, ρ˜) ∈ S (R+ × Rρ˜; E, E) (up to a weight factor and modulo smoothing operators) with a special control near r = 0 via Mellin quantisation. By Rq ) we denote the space of all Schwartz functions in η ∈ Rq with L−∞ (R+ ; E, E; values in operators ∞ ). C0∞ (R+ , E −∞ ) → C ∞ (R+ , E We then define Rq ) = {Opr (a)(η) + C(η) : Lμ (R+ ; E, E; −∞ Rq )}. (R+ ; E, E; a(r, ρ, η) ∈ S μ (R+ × R1+q ρ,η ; E, E), C(η) ∈ L

Our next objective is to formulate a Mellin quantisation result of symbols a(r, ρ, η) = a ˜(r, rρ, rη), a ˜(r, ρ˜, η˜) ∈ S μ (R+ × R1+q ρ,˜ ˜ η ; E, E)

(2.7)

(see Remark 1.10). μ q ) we denote the set of all h(z, η˜) ∈ A(C,S μ (Rq ;E, E)) (E, E;R Definition 2.6. By MO η ˜ η ˜ such that h(β + iρ, η˜) ∈ S μ (Rρ × Rqη˜; E, E)

for every β ∈ R, uniformly in compact β-intervals. For q = 0 we simply write μ (E, E). MO ˜ Theorem 2.7. For every symbol a(r,ρ,η) of the form (2.7) there exists an h(r,z, η˜) ∈ μ ∞ q ˜ C (R+ ,MO (E, E;R )) such that for h(r, z, η) := h(r, z, rη) and every δ ∈ R we have opδM (h)(η) = Opr (a)(η) Rq ). modulo operators in L−∞ (R+ ; E, E; This result in the context of operator-valued symbols based on order reductions is mentioned here for completeness. It is contained in a joint work (in preparation) of the second author with C.-I. Martin (Potsdam) and N. Rablou (G¨ ottingen). It extends a corresponding result of the edge symbolic calculus, see [6, Theorem 3.2]. More information in that case is given in [15, Chapter 4]. Here we adapt some part of this approach to realise the kernel cut-off principle that allows us to recognize how many parameter-dependent meromorphic Mellin symbols exist. denotes the space of all h(ζ,η) ∈ A(C,S μ (Rqη ;E, E)) Definition 2.8. S μ (C × Rq ;E, E) such that h(ρ + iδ, η) ∈ S μ (R1+q ρ,η ; E, E) for every δ ∈ R, uniformly in compact δ-intervals.

96


is a generalisation of M μ (E, E), however, Clearly the space S μ (C × Rq ; E, E) O with an interchanged role of real and imaginary part of the complex covariable. we consider a so-called kernel cut-off To produce elements of S μ (C × Rq ; E, E) operator → S μ (C × Rq ; E, E) V : C0∞ (R) × S μ (R1+q ; E, E) into V (ϕ)a (ζ, η) ∈ transforming an arbitrary element a(ρ, η) ∈ S μ (R1+q ; E, E) for any ϕ ∈ C ∞ (R). It will be useful to admit ϕ to belong to the S μ (C × Rq ; E, E) 0 ∞ space Cb (R) := {ϕ ∈ C ∞ (Rθ ) : supθ∈R |Dθk ϕ(θ)| < ∞ for every k ∈ N}. We set ρ, (2.8) V (ϕ)a (ρ, η) := e−iθρ˜ϕ(θ)a(ρ − ρ˜, η)dθd¯˜ interpreted as an oscillatory integral (see also [16]). We now prove the following result: Theorem 2.9. The kernel cut-off operator V : (ϕ, a) → V (ϕ)a defines a bilinear and continuous mapping → S μ (R1+q ; E, E), V : Cb∞ (R) × S μ (R1+q ; E, E) (2.9) and V (ϕ)a (ρ, η) admits an asymptotic expansion ∞ (−1)k k Dθ ϕ(0)∂ρk a(ρ, η). V (ϕ)a (ρ, η) ∼ k!

(2.10)

k=0

Proof. First note that the mapping (ϕ, a) → ϕ(θ)a(ρ − ρ˜, η), μ ∞ q Cb∞ (R) × S μ (R1+q ρ,η ; E, E) → C (Rρ,η , Sb (Rθ × Rρ˜; E, E))

:= C ∞ (Rθ , S μ (Rρ˜; E, E)) is bilinear and continuous. For for Sbμ (Rθ × Rρ˜; E, E) b the proof of the continuity of (2.9) it suffices to verify that V (ϕ)a (ρ, η) ∈ and then to apply the closed graph theorem. By virtue of S μ (R1+q ; E, E) β β Dρ,η a) (ρ, η) V (ϕ)a (ρ, η) = V (ϕ)(Dρ,η for every β ∈ N1+q we only have to check that for every s ∈ R ˜bs−μ (ρ, η) V (ϕ)a (ρ, η)b−s (ρ, η) 0 0 ≤ c L(E ,E )

(2.11)

for all (ρ, η) ∈ R , with a constant c = c(s) > 0. We regularise the oscillatory integral (2.8) V (ϕ)a (ρ, η) = e−iθρ˜θ−2 {(1 − ∂θ2 )N ϕ(θ)}aN (ρ, ρ˜, η)dθd¯˜ ρ 1+q

for

aN (ρ, ρ˜, η) := (1 − ∂ρ2˜ ){˜ ρ−2N a(ρ − ρ˜, η)}. The function (2.12) is a linear combination of terms ρ−2N )(∂ρk a)(ρ − ρ˜, η) for 0 ≤ j, k ≤ 2. (∂ρj˜˜

(2.12)

Operators with Corner-Degenerate Symbols We have

ρ−2N ) (2.13) e−iθρ˜θ−2 (1 − ∂θ2 )N ϕ(θ)(∂ρj˜˜ ! (∂ρk a)(ρ − ρ˜, η)dθd¯˜ ρ b−s (ρ, η)L(E 0 ,E 0 ) ˜bs−μ (ρ, η)˜b−s+μ (ρ − ρ˜, η)˜bs−μ (ρ − ρ˜, η) e−iθρ˜θ−2 (1 − ∂ 2 )N ϕ(θ) = θ

˜s−μ

b

97

(ρ, η)

(∂ρj˜˜ ρ−2N )(∂ρk a)(ρ − ρ˜, η)}b−s (ρ − ρ˜, η)bs (ρ − ρ˜, η)b−s (ρ, η)dθd¯˜ ρL(E 0 ,E 0 ) ≤c ˜bs−μ (ρ, η)˜b−s+μ (ρ − ρ˜, η)L(E 0 ,E 0 ) ˜bs−μ (ρ − ρ˜, η)(∂ρj˜˜ ρ−2N ) ρ. (∂ρk a)(ρ − ρ˜, η)b−s (ρ − ρ˜, η)L(E 0 ,E 0 ) bs (ρ − ρ˜, η)b−s (ρ, η)L(E 0 ,E 0 ) d¯˜ For the norms under the integral we apply the Taylor expansion bs (ρ − ρ˜, η) =

M 1 m s (∂ρ b )(ρ, η)(−ρ˜)m m! m=0 ρ˜ M+1 1 + (1 − t)M (∂ρM+1 bs )(ρ − t˜ ρ, η)dt. M! 0

This yields bs (ρ − ρ˜, η)b−s (ρ, η)L(E 0 ,E 0 ) M 1 ˜ ρm (∂ρm bs )(ρ, η)b−s (ρ, η)L(E 0 ,E 0 ) ≤ m! m=0 ˜ ρM+1 1 + (1 − t)M (∂ρM+1 bs )(ρ − t˜ ρ, η)b−s (ρ, η)L(E 0 ,E 0 ) dt. M! 0

By virtue of (1.12), Proposition 1.15 and Proposition 1.13 we obtain (∂ρm bs )(ρ, η)b−s (ρ, η)L(E 0 ,E 0 ) ≤ cρ, η−m . Moreover, using Definition 1.3 (iii), it follows that ρ, η)b−s (ρ − t˜ ρ, η)bs (ρ − t˜ ρ, η)b−s (ρ, η)L(E 0 ,E 0 ) (∂ρM+1 bs )(ρ − t˜ ≤ c(∂ρM+1 bs )(ρ − t˜ ρ, η)b−s (ρ − t˜ ρ, η)L(E 0 ,E 0 ) bs (ρ − t˜ ρ, η)L(E s ,E 0 ) b−s (ρ, η)L(E 0 ,E s ) ≤ ρ − t˜ ρ, ηB1 (s) ρ, ηB2 (s) with certain Bi (s), i = 1, 2. We thus obtain bs (ρ − ρ˜, η)b−s (ρ, η)L(E 0 ,E 0 ) ≤ c˜ ρM+1 ( sup ρ − t˜ ρ, η−(M+1)+B1 (s) )ρ, ηB2 (s) . |t|≤1

By Peetre’s inequality for L ≥ 0 we have sup|t|≤1 ρ − t˜ ρ, η−L ≤ c˜ ρL ρ, η−L . Thus choosing M so large that −(M + 1) + B1 (s) ≤ 0, −(M + 1) + B1 (s) + B2 (s) ≤ 0,

98


it follows that bs (ρ − ρ˜, η)b−s (ρ, η)L(E 0 ,E 0 )

(2.14)

≤ c˜ ρM+1 ˜ ρM+1−B1 (s) ρ, η−(M+1)+B1 (s)+B2 (s) ≤ c˜ ρA(s) for A(s) := 2(M + 1) − B2 (s). In a similar manner we can show that ˜bs−μ (ρ, η)˜b−s+μ (ρ − ρ˜, η)L(E 0 ,E 0 ) ≤ c˜ ρA(s)

(2.15)

for some A(s) ∈ R. Applying (2.14) and (2.15) in the estimate (2.13) it follows that j s−μ −s ˜ b |∂ρ˜˜ (ρ, η) V (ϕ)a (ρ, η)b (ρ, η)L(E 0 ,E 0 ) ≤ c ρ−2N |˜ ρA(s)+A(s) d¯˜ ρ. 0≤j≤2

(2.16) Since N ∈ N can be chosen as large as we want, it follows that the right-hand side of (2.16) is finite for an appropriate N . This completes the proof of (2.11). The relation (2.10) immediately follows by applying the Taylor expansion of ϕ at 0. Theorem 2.10. The kernel cut-off operator V : (ϕ, a) → V (ϕ)a defines a bilinear and continuous mapping → S μ (C × Rq ; E, E). V : C0∞ (R) × S μ (R1+q ; E, E) Proof. Writing V (ϕ)a (ρ, η) =

e−iθρ ϕ(θ)

(2.17)

eiθρ a(ρ , η)d¯ρ dθ

we see that V (ϕ)a (ρ, η) is the Fourier transform of a distribution ϕ(θ) eiθρ a(ρ , η)d¯ρ ∈ S (Rθ , Lμ (E, E)) with compact support. This extends to a holomorphic Lμ (E, E)-valued function in ζ = ρ + iδ, given by V (ϕ)a (ρ + iδ, η) = V (ϕδ )a (ρ, η) for ϕδ (θ) := eθδ ϕ(θ). Theorem 2.9 gives us V (ϕ)a (ρ + iδ, η) ∈ S μ (R1+q ; E, E) for every δ ∈ R. By virtue of the continuity of δ → ϕδ , R → C0∞ (R) and of the continuity of (2.9) it follows that (2.17) induces a continuous mapping → S μ (Iδ × Rq ; E, E), V : C0∞ (R) × S μ (R1+q ; E, E) Iδ := {ζ ∈ C : Imζ = δ} which is uniform in compact δ-intervals. The closed graph theorem gives us also the continuity of (2.17) with respect to the Fréchet topology of S μ (C × Rq ; E, E).


99

2.3. Meromorphic Mellin symbols and operators with asymptotics As an ingredient of our cone algebra we now study meromorphic Mellin symbols, μ (see Definition 2.6 for q = 0). starting from MO (E, E) μ and h|Γ ∈ S μ−ε (Γβ ; E, E) for some ε > 0 entails Theorem 2.11. h ∈ MO (E, E) β μ−ε h ∈ MO (E, E).

Proof. The ideas of the proof are similar to the case of the cone calculus with smooth base X and the scales H s (X) s∈R (see, e.g., the thesis of Seiler [40]). μ f (w) ∈ M ν (E, E0 ); then for pointwise Proposition 2.12. Let h(w) ∈ MO (E0 , E), O μ+ν composition we have h(w)f (w) ∈ MO (E, E).

Proof. The proof is obvious. μ MO (E, E)

Definition 2.13. An element h(w) ∈ is called elliptic, if for some β ∈ R s−μ are invertible for all s ∈ R and h−1 (β + iρ) ∈ the operators h(β + iρ) : E s → E E). S −μ (Rρ ; E, μ be elliptic. Then, (E, E) Theorem 2.14. Let h ∈ MO

s−μ h(w) : E s → E

(2.18)

is a holomorphic family of Fredholm operators of index zero for s ∈ R. There is a set D ⊂ C, with D ∩ {c ≤ Re w ≤ c } finite for every c ≤ c , such that the operators (2.18) are invertible for all w ∈ C \ D. E). Applying a version Proof. By assumption we have g := (h|Γβ )−1 ∈ S −μ (Γβ ; E, of the kernel cut-off construction, now referring to parallels of the imaginary axis rather than the real axis, with a function ψ ∈ C0∞ (R+ ), ψ ≡ 1 near 1, we obtain a continuous operator E) → M −μ (E, E) V (ψ) : S −μ (Γβ ; E, O

E). Setting h(−1) (w) := V (ψ)g we obtain where V (ψ)g|Γβ = g mod S (Γβ ; E, −μ E), and from Proposition 2.12 it follows that h(−1) (w) ∈ MO (E, −∞

0 0 h(w)h(−1) (w) ∈ MO (E, E), h(−1) (w)h(w) ∈ MO (E, E)

and E), h(−1) (w)h(w)|Γ − 1 ∈ S −∞ (Γβ ; E, E), h(w)h(−1) (w)|Γβ − 1 ∈ S −∞ (Γβ ; E, β (2.19) for every β ∈ R, and hence h(w)h(−1) (w) = 1 + m(w), h(−1) (w)h(w) = 1 + l(w) −∞ MO (E, E),

for certain m(w) ∈ fixed w ∈ C the operators

l(w) ∈

−∞ MO (E, E).

(2.20)

For every s ∈ R and every

∞ , l(w) : E s → E ∞ s → E m(w) : E

100


are continuous. Therefore, since the scales have the compact embedding property, from (2.20) we obtain that h(−1) (w) is a two-sided parametrix of h(w) for every w, i.e., the operators (2.18) are Fredholm. Since h(w) ∈ A(C, Lμ (E s , E s−μ )) is continuous in w ∈ C we have ind h(w1 ) = ind h(w2 ) for every w1 , w2 ∈ C. However, since h(w) = 0 consists of invertible operators on the line Γβ it follows that ind h(w) = 0 for all w ∈ C. Finally, from the relations (2.18) we see that for every c ≤ c there is an L(c, c ) > 0 such that the operators (2.18) are invertible for all w ∈ C with |Im w| ≥ L(c, c ), c ≤ Re w ≤ c . Then a general result on holomorphic Fredholm families tells us that the strip c ≤ Re w ≤ c contains at most finitely many points where (2.18) is not invertible. Those points just constitute the set D, it is also independent of s, since ker h(w) is independent of s as we easily see from (2.20) and the smoothing remainders; then vanishing of the index shows that the invertibility holds exactly when ker h(w) = 0. Theorem 2.15. The ellipticity of h with respect to Γβ as in Definition 2.13 entails the ellipticity with respect to Γδ for all δ ∈ R such that Γδ ∩ D = ∅. In that sense Definition 2.13 is independent of the choice of β. Proof. Let us apply the kernel cut-off operator V (ψε ), where ψε ∈ C0∞ (R+ ) is of the form ψε (t) = ψ(εt), ε > 0, for some cut-off function ψ. Then, setting −μ (E, E) V (ψε )(h−1 (β + iρ)) =: fε ∈ MO

E) and fε |Γ → h−1 (β + iρ) as ε → 0 in the topology we have fε |Γβ ∈ S −μ (Γβ ; E, β −μ E). This shows us that fε1 |Γ is pointwise invertible when ε1 > 0 is of S (Γβ ; E, β sufficiently small. Let us set h(−1) (w) = fε1 (w). According to Proposition 2.12 we 0 have g(w) := h(−1) (w)h(w) ∈ MO (E, E) and by construction g|Γβ = 1 + l for some l ∈ S −∞ (Γβ ; E, E). −∞ Then Theorem 2.11 yields g = 1 mod MO (E, E). It follows that

h−1 |Γδ h|Γδ = 1 + lδ for some lδ ∈ S −∞ (Γδ ; E, E) and hence, since h|Γδ is pointwise invertible, h(−1) |Γδ = (1 + lδ ) (h|Γδ )−1 . which yields

−1

(2.21) (h|Γδ ) = (1 + lδ )−1 h(−1) |Γδ . −∞ From Proposition 1.12 we know that lδ ∈ S(Γδ , L (E, E)) and it is also clear that (1 + lδ )−1 = 1 + mδ for some mδ ∈ S(Γδ , L−∞ (E, E)). Then Proposition 1.15 E). (iii) shows that (h|Γδ )−1 ∈ S −μ (Γδ ; E, A sequence R = {(pj , mj , Lj )}j∈Z is called a discrete asymptotic type of Mellin symbols, if pj ∈ C, mj ∈ N, and is a finite-dimensional subspace of finite rank operators; moreover, Lj ⊂ L−∞ (E, E) πC R := (pj )j∈Z is assumed to intersect the strips {w ∈ C : c1 ≤ Rew ≤ c2 } in


101

denote the space of all functions a finite set, for every c1 ≤ c2 . Let MR−∞ (E, E) −∞ m ∈ A(C \ πC R, L (E, E)) which are meromorphic with poles at the points pj of multiplicity mj + 1 and Laurent coefficients at (w − pj )−(k+1) belonging to Lj for every δ ∈ R, uniformly in for 0 ≤ k ≤ mj , and χ(w)m(w)|Γδ ∈ S(Γδ ; E, E) compact δ-intervals, where χ is any πC R-excision function. Moreover, we set := M μ (E, E) + M −∞ (E, E). M μ (E, E) (2.22) O

R

MRμ (E0 , E),

Theorem 2.16. Let h ∈ f orders μ, ν ∈ R, then we have hf ∈ type P .

R

∈ MSν (E, E0 ) MPμ+ν (E, E)

with asymptotic types R, S and with some resulting asymptotic

Proof. The proof of this result is analogous to the one in the “concrete” cone calculus, see [33]. Proposition 2.17. For every m ∈ MR−∞ (E, E) there exists an m(−1) ∈ MS−∞ (E, E) with another asymptotic type S such that 1 + m(w) 1 + m(−1) (w) = 1. For the proof we employ the following Lemma. Lemma 2.18. Let E be a Banach space, U ⊆ C open, 0 ∈ U , and let h ∈ A(U, L(E)) be an element such that h(w) = 0 on a closed space of finite codimension F ⊆ E. Moreover, let a1 , . . . , aN ∈ L(E) be operators of finite rank, for some N ∈ N \ {0}. Then there is a δ > 0 such that the meromorphic L(E)-valued function f (w) = 1 + h(w) +

N

aj w−j

j=1

is invertible for all 0 < |w| < δ. Proof of Proposition 2.17. First observe that if m ∈ L−∞ (E, E) is an operator such that 1 + m : Es → Es is invertible for all s ∈ R, we can define an operator g ∈ L0 (E, E) such that (1 + m)(1 + g) = 1. This gives us 1 + m + g + mg = 1, and m, mg ∈ L−∞ (E, E) implies g = −m(1 + g) ∈ L−∞ (E, E). Moreover, our operator function 1 + m is holomorphic in C \ πC R. Then g = (1 + m)−1 − 1 is holomorphic in C \ D with values in L−∞ (E, E), where D ⊆ C is a countable set such that {w ∈ C : c1 ≤ Rew ≤ c2 } ∩ {w ∈ C : dist(w, πC R) > ε} ∩ D is finite for every c1 ≤ c2 and ε > 0.If χ(w) is any D-excision function, ∞ then we have a representation (1 + m)−1 = j=0 (−1)j mj as a convergent series of functions with values in L(E s , E s ) uniformly in w ∈ C for c ≤ Rew ≤ c , |Imw| ≥ C for every c ≤ c and C > 0 sufficiently In a similar manner ∞ large. j j we obtain convergence of all w-derivatives of (−1) m in a set of such a j=0 structure. Thus, from g = −m(1 + g) = −m(1 + m)−1 and the Schwartz property of m for large |Imw|, uniformly in finite strips c ≤ Rew ≤ c , we obtain the same

102


of g itself. It remains to show that g is meromorphic with poles at the points of D, that D has no accumulation points at πC R, and that the Laurent coefficients are of the desired kind, namely, to belong to L−∞ (E, E) and to be of finite rank. Let us verify that there are no accumulation points of the singularities of (1 + m(w))−1 . Let w0 be a pole of m, i.e., w0 ∈ πC R. Then we can write 1 + m(w) = 1 + m0 (w) +

K

bk (w − w0 )−k

k=1

with suitable K ∈ N, m0 holomorphic in a neighbourhood of w0 and L−∞ (E, E)valued, with finite rank operators bk . Note that m0 ≡ −1. Setting n(w) := K −k we have k=1 bk (w − w0 ) 1 + m(w) = (1 + m0 (w))(1 + (1 + m0 (w))−1 n(w)). Since m0 is holomorphic near w0 and 1+m0 (w) a Fredholm family, the singularities of (1 + m0 (w))−1 form a countable discrete set; therefore there is a δ > 0 such that (1 + m0 (w))−1 exists for all w such that 0 < |w − w0 | < δ. Moreover, (1 + N m0 (w))−1 n(w) can be written in the form h(w) + j=1 aj (w − w0 )−j with a suitable h which is holomorphic near w0 and finite rank operators aj , 1 ≤ j ≤ N . ) The operator (1 + m0 (w))−1 n(w) vanishes on the space F := K k=1 ker bk which )N is of finite codimension. Setting M := j=1 ker aj it follows that h(w)u = 0 for all u ∈ M ∩ F ; the latter space is also of finite codimension. Lemma 2.18 then shows that 1 + (1 + m0 (w))−1 n(w) is invertible in 0 < |w − w0 | < δ for a suitable δ > 0. μ be elliptic, then there is an f ∈ M −μ (E, E) with (E, E) Theorem 2.19. Let h ∈ MO S asymptotic type S such that hf = 1. μ be as in the proof of Theorem 2.14. Then we have (E, E) Proof. Let h(−1) (w) ∈ MO the relation (2.20). By virtue of Proposition 2.17 there exists a g ∈ MP−∞ (E, E) for some asymptotic type P such that (1 + m(w))(1 + g(w)) = 1. This yields E), according to Theh(w)f (w) = 1 for f := h(−1) (1+g) which belongs to MS−μ (E, −μ ˜ orem 2.16. In a similar manner we find an f ∈ M (E, E) such that f˜(w)h(w) = 1. S This implies f = f˜.

is said to be elliptic, if there is a β ∈ R such that Definition 2.20. A g ∈ MRμ (E, E) −1 −μ (g|Γβ ) ∈ S (Rρ ; E, E). is elliptic, there is an f ∈ M −μ (E, E) such that Theorem 2.21. If g ∈ MRμ (E, E) S gf = 1. Proof. Applying a kernel cut-off argument to (g|Γβ )−1 we can find an h(−1) ∈ −μ E). By definition we have (E, E) such that h(−1) |Γβ − (g|Γβ )−1 ∈ S −∞ (Γβ ; E, MO μ −∞ Then h(−1) g0 |Γ = g = g0 + g1 for certain g0 ∈ MO (E, E), g1 ∈ MR (E, E). β −∞ −∞ (−1) 1 mod S (Γβ ; E, E) implies h g0 = 1 mod MO (E, E) (see Theorem 2.11). If


103

follows that h(−1) g = 1 + m for some m ∈ MR−∞ (E, E) with an asymptotic type R (see Theorem 2.16). Thus Proposition 2.17 gives us g −1 = (1 + m)−1 h(−1) ∈ E) with some asymptotic type S. MS−μ (E, Parallel to the spaces of Mellin symbols (2.22) we now introduce subspaces of Hs,γ (R+ , E) with discrete asymptotics. We consider a sequence P := {(pj , mj )}0≤j≤N

(2.23)

with N ∈ N ∪ {+∞}, mj ∈ N, 0 ≤ j ≤ N . A sequence (2.23) is said to be a discrete asymptotic type, associated with weight data (γ, Θ) (with a weight γ ∈ R and a weight interval Θ = (ϑ, 0], −∞ ≤ ϑ ≤ 0), if d+1 d+1 − γ + ϑ < Re w < − γ}, 2 2 and πC P is finite when ϑ is finite, and Re pj → −∞ as j → ∞ when ϑ = −∞ and N = +∞. We will say that P satisfies the shadow condition, if (p, m) ∈ P implies d+1 (p − j, m) ∈ P for all j ∈ N with d+1 2 − γ + ϑ < Re (p − j) < 2 − γ. If Θ is finite we define the (finite-dimensional) space πC P := {pj }0≤j≤N ⊂ {w ∈ C :

SP (R+ , E) :=

mj N

ω(r)cjk r−pj logk r : cjk ∈ E ∞ , 0 ≤ k ≤ mj , 0 ≤ j ≤ N

!

j=0 k=0

with some fixed cut-off function ω on the half-axis. We then have SP (R+ , E) ⊂ H∞,γ (R+ , E). Moreover, we set 1

s,γ HΘ (R+ , E) := ← lim {ωHs,γ−ϑ− m+1 (R+ , E) + (1 − ω)Hs,γ (R+ , E)} −− − m∈N

endowed with the Fréchet topology of the projective limit, and s,γ (R+ , E) + SP (R+ , E) HPs,γ (R+ , E) := HΘ

as a direct sum of Fréchet spaces. In order to formulate spaces with discrete asymptotics of type P in the case Θ = (−∞, 0] we form Pk := {(p, m) ∈ P : Re p > d+1 2 − γ − (k + 1)} for any (R+ , E) together with k ∈ N. From the above construction we have the spaces HPs,γ k continuous embeddings (R+ , E) → HPs,γ (R+ , E), k ∈ N. HPs,γ k+1 k We then define HPs,γ (R+ , E) := ← lim Hs,γ (R+ , E) −− Pk k∈N

in the corresponding Fréchet topology of the projective limit.

(2.24)

104


Remark 2.22. The relation u ∈ HPs,γ (R+ , E) with P being associated with (γ, Θ), Θ = (−∞, 0], is equivalent with the existence of (unique) coefficients cjk ∈ E ∞ , 0 ≤ k ≤ mj , such that for every l ∈ R+ there is an N = N (l) ∈ N with mj N ω(r) u(r, x) − cjk r−pj logk r ∈ Hs,γ+l (R+ , E). j=0 k=0

Similarly as in the “concrete” cone calculus (see [33]) we have the following continuity result: the operator (2.5) restricts to a continuous Theorem 2.23. For every f ∈ MRμ (E, E) operator s−μ,γ opγM (f ) : HPs,γ (R+ , E) → HQ (R+ , E) for every s ∈ R and every asymptotic type P with some resulting Q. The case of Mellin symbols with variable coefficients is also of interest in the and corner calculus. It is then adequate to assume f (r, w) ∈ C ∞ (R+ , MRμ (E, E)) γ ω in combination with cut-off functions ω(r), ω ˜ (r). to consider operators ωopM (f )˜ s,γ s−μ,γ Those induce continuous operators HP (R+ , E) → HQ (R+ , E) as well.

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Ellipticity of Fredholm Pseudo-Differential Operators on Lp(Rn) Aparajita Dasgupta Abstract. Based on the works of Grieme and Schulze [8], it is shown in this paper that if a pseudo-differential operator with symbol in S m , −∞ < m < ∞, is Fredholm on Lp (Rn ), 1 < p < ∞, then the pseudo-differential operator is elliptic. The basic idea is to construct an isometric operator Rλ , λ ∈ R \ {0}, on Lp (Rn ) in order to prove the ellipticity of the Fredholm pseudo-differential operator with symbol in S 0 . This result is then generalized for arbitrary symbol classes. Mathematics Subject Classification (2000). Primary 35S05, 47G30; Secondary 47A53. Keywords. Pseudo-differential operators, SG pseudo-differential operators, Sobolev spaces, Fredholm operators and elliptic pseudo-differential operators.

1. Introduction For m ∈ (−∞, ∞), let S m be the set of all functions in C ∞ (R2n ) such that for all multi-indices α and β, there exists a positive constant Cα,β for which |(Dxα Dξβ σ)(x, ξ)| ≤ Cα,β (1 + |ξ|)m−|β| ,

x, ξ ∈ Rn .

Let σ ∈ S m . Then we define the corresponding pseudo-differential operator Tσ on the Schwartz space S by −n/2 ˆ dξ, x ∈ Rn . (Tσ φ)(x) = (2π) eix·ξ σ(x, ξ)φ(ξ) Rn

It can be proved easily that Tσ : S → S is a continuous linear mapping. In fact, as is shown in Theorem 11.9 in the book [20] by Wong, Tσ can be extended to a bounded linear operator from H s,p into H s−m,p for −∞ < s < ∞ and 1 < p < ∞. This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

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The aim of this paper is to prove that if Tσ : H s,p → H s−m,p is Fredholm for some s and p with −∞ < s < ∞ and 1 < p < ∞, then σ must be elliptic in the sense that there exist positive constants C and R such that |σ(x, ξ)| ≥ C(1 + |ξ|)m ,

x, ξ ∈ Rn .

That Fredholmness does not follow from ellipticity can be seen from the Laplacian Δ on Rn . It is well known that Δ is elliptic, but Δ is not a Fredhlom operator from H 2,p into Lp (Rn ) for 1 < p < ∞. In this section we give an introduction to the SG pseudo-differential operators, which are also known as pseudo-differential operators of global type. They are also known as pseudo-differential operators with exit behavior, as in [7, 16]. A discussion of SG pseudo-differential operators and related topics can also be found in [3, 4, 10, 11, 12, 13, 16] and the references therein. The main techniques that we use in this paper come from the study of SG pseudo-differential operators in the Ph.D. thesis [8] of Grieme, which we now recall. Let m1 , m2 ∈ (−∞, ∞). Then S m1 ,m2 is the set of all functions in C ∞ (R2n ) such that for all multi-indices α and β, there exist positive constants Cα,β for which m −|α| m −|β| |(Dxα Dξβ σ)(x, ξ)| ≤ Cα,β x 2 ξ 1 , x, ξ ∈ Rn , where . denotes the function on RN given by z = (1 + |z|2 )1/2 , z ∈ RN , for every positive integer N. For all σ ∈ S m1 ,m2 we denote 1 ,m2 pm (σ) = α,β

sup (x,ξ)∈R2n

|Dxα Dξβ σ(x, ξ)|x

−m2 +|α|

ξ

−m1 +|β|

! ,

for all multi-indices α and β. A function in S m1 ,m2 is said to be a SG symbol of orders m1 , m2 . It is clear that if σ ∈ S m1 ,m2 , then σ ∈ S m1 , where S m1 is the class of symbols of classical pseudo-differential operators studied exclusively in the book [18] by Wong. For σ ∈ S m1 ,m2 , we then define the pseudo-differential operator Tσ with symbol σ by ˆ (Tσ φ)(x) = (2π)−n/2 eix·ξ σ(x, ξ)φ(ξ)dξ, x ∈ Rn , Rn

for all functions φ in the Schwartz space S, where ˆ = (2π)−n/2 φ(ξ) e−ix·ξ φ(x)dx, ξ ∈ Rn . Rn

It can be easily shown that Tσ : S → S is a continuous linear mapping. Further information and formal properties of SG pseudo-differential operators can be found in [6]. The aim of this paper is to prove ellipticity of Fredholm pseudodifferential operators on Lp (Rn ). In Section 2, the Lp -Sobolev spaces for SG pseudo-differential operators is discussed. In Section 3, the elliptic and Fredholm SG pseudo-differential operators are recalled and the Fredholmness of elliptic SG pseudo-differential operators is given. Detail works on this can be found

Ellipticity of Fredholm Pseudo-Differential Operators on Lp (Rn )

109

in [6]. In Section 4, an isometric operator Rλ on Lp (Rn ) is introduced in order to prove the ellipticity of the Fredholm pseudo-differential operator Tσ , with symbol σ ∈ S 0 . Later this result is generalized for symbols in arbitrary symbol classes S m , m ∈ (−∞, ∞), which again in turn used to prove the ellipticity of the Fredholm SG pseudo-differential operators on Lp (Rn ). Similar results for SG pseudo-differential operators on L2 (Rn ) can be found in [8].

2. Sobolev spaces For s1 , s2 ∈ (−∞, ∞) let Js1 ,s2 be the Bessel potential of orders s1 , s2 defined by Js1 ,s2 = Tσs1 ,s2 , where σs1 ,s2 (x, ξ) = x

−s2

−s1

ξ

, (x, ξ) ∈ Rn .

Obviously then σs1 ,s2 ∈ S −s1 ,−s2 . It can be shown easily that for −∞ < s1 , s2 < ∞, the mapping Js1 ,s2 : S → S is a bijection and = J−s1 ,0 J0,−s2 Js−1 1 ,s2

(2.1)

and Js−1 is a pseudo-differential operator of orders −s1 , −s2 . 1 ,s2 For 1 < p < ∞ and −∞ < s1 , s2 < ∞, we define the Lp -Sobolev space H s1 ,s2 ,p of orders s1 , s2 by ! H s1 ,s2 ,p = u ∈ S : J−s1 ,−s2 u ∈ Lp (Rn ) . Then H s1 ,s2 ,p is a Banach space in which the norm .s1 ,s2 ,p is given by us1 ,s2 ,p = Js1 ,s2 uLp (Rn ) , u ∈ H −s1 ,−s2 ,p , where .Lp (Rn ) is the norm in Lp (Rn ). Obviously, H 0,0,p = Lp (Rn ). Then we have, Theorem 2.1. Let σ ∈ S m1 ,m2 , −∞ < m1 , m2 < ∞. Then for 1 < p < ∞ and −∞ < s1 , s2 < ∞, Tσ : H s1 ,s2 ,p → H s1 −m1 ,s2 −m2 ,p is a bounded linear operator.

3. Elliptic and Fredholm SG pseudo-differential operators Let σ ∈ S m1 ,m2 , −∞ < m1 , m2 < ∞. Then σ is said to be elliptic if there exists positive constants C and R such that |σ(x, ξ)| ≥ Cx

m2

ξ

m1

, |x|2 + |ξ|2 ≥ R.

Then the pseudo-differential operator Tσ with the elliptic symbol σ is called the elliptic operator. Theorem 3.1. Let σ ∈ S m1 ,m2 , −∞ < m1 , m2 < ∞, be elliptic. Then there exists a symbol τ in S −m1 ,−m2 such that Tσ Tτ = I + S

and

Tτ Tσ = I + R,

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where R and S are infinitely smoothing ) in the sense that they are SG pseudodifferential operators with symbols in k1 ,k2 ∈R S k1 ,k2 . The SG pseudo-differential operator Tτ in Theorem 3.1 is known as a parametrix of Tσ . Now let us first recall from the book [14] by Schechter, 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 R(A) of A is a closed subspace of Y, the null space N (A) of A and the null space N (At ) of the true adjoint At of A are finite dimensional. For a Fredholm operator A, the index i(A) of A is defined by i(A) = dim N (A) − dim N (At ). The following theorem which is known as the Atkinson theorem also gives another equivalent definition of the Fredholm operators. Theorem 3.2. Let A be a closed linear operator from a complex Banach space X into a complex Banach space Y with a dense domain D(A). Then A is Fredholm if and only if we can find a bounded linear operator B : Y → X, a compact operator K1 : X → X and a compact operator K2 : Y → Y such that BA = I + K1 on D(A) and AB = I + K2 on Y. All the results on Fredholm operators hitherto can be found in the book [14] by Schechter. The first result in this section is the following theorem which can be found in [6]. Theorem 3.3. Let σ ∈ S m1 ,m2 , m1 , m2 > 0, be elliptic. Then for 1 < p < ∞, Tσ,0 is a Fredholm operator on Lp (Rn ) with domain H m1 ,m2 ,p . Furthermore, if σ ∈ S 0,0 is elliptic, then the bounded linear operator Tσ : Lp (Rn ) → Lp (Rn ) is Fredholm.

4. Ellipticity of Fredholm pseudo-differential operators Here we want to prove that the Fredholm pseudo-differential operators is elliptic on Lp (Rn ) and for that we need some technical preparations. Definition 4.1. Let λ > 0, x0 , ξ0 ∈ Rn ∪{0} and τ ∈ R+ ∪ {0}. For every 1 < p < ∞ we define an operator, Rλ : Lp (Rn ) → Lp (Rn ) by, (Rλ (x0 , ξ0 )u)(x) = λτ n/p eiλxξ0 u(λτ (x − x0 )). Then it is easy to prove, Theorem 4.2. The operator Rλ (x0 , ξ0 ) : Lp (Rn ) → Lp (Rn ) is an isometrical invertible operator and the inverse is given by −τ

(Rλ−1 (x0 , ξ0 )u)(x) = λ−τ n/p e−iλ(x0 +λ

x)ξ0

u(x0 + λ−τ x).

Ellipticity of Fredholm Pseudo-Differential Operators on Lp (Rn ) Proof. Fix x0 , ξ0 ∈ Rn and Rλ := Rλ (x0 , ξ0 ). Let u ∈ Lp (Rn ). Then Rλ upp = |Rλ u(x)|p dx = λτ n |u(λτ (x − x0 ))|p dx. Rn

111

(4.2)

Rn

Substituting y = λτ (x − x0 ) in (4.2), we have Rλ up = up . So Rλ is an isometric operator on Lp (Rn ). Now let Rλ u(x) = v(x). Then from the definition (4.1), v(x) = λτ n/p eiλxξ0 u(λτ (x − x0 )).

(4.3)

Again substituting λ (x − x0 ) = y in (4.3), we have τ

−τ

u(y) = λ−τ n/p e−iλ(x0 +λ Then

y)ξ0

−τ

Rλ−1 v(y) = λ−τ n/p e−iλ(x0 +λ

v(x0 + λ−τ y).

y)ξ0

v(x0 + λ−τ y).

Theorem 4.3. For all u ∈ Lp (Rn ) and v ∈ Lp (Rn ), where 1/p + 1/p = 1, 1 < p < ∞ and τ > 0, (Rλ (x0 , ξ0 )u, v)2 → 0 as λ → ∞. Proof. Fix x0 , ξ0 ∈ Rn and let Rλ := Rλ (x0 , ξ0 ). Furthermore, let u, v ∈ C0∞ (Rn ). Then we have τ n/p |(Rλ u, v)| ≤ λ |u(λτ (x − x0 ))v(x)|dx Rn = λτ n/p λ−τ n |u(y)v(x0 + λ−τ y)|dy n R −τ n/p =λ |v(x0 + λ−τ y)u(y)|dy Rn −τ n/p ≤λ sup |v(y)| |u(y)|dy. (4.4) x∈Rn

Rn

C0∞ (Rn ).

So then (Rλ u, v) → 0 as λ→ ∞ for all u, v ∈ Lp (Rn ), so for > 0 there exists J ∈ N such that,

Now C0∞ (Rn ) is dense in

|(Rλ u, v) − (Rλ uj , vj )| ≤ /2,

for all j ≥ J, where uj , vj ∈ C0∞ (Rn ) and u ∈ Lp (Rn ), v ∈ Lp (Rn ). Then again using the fact that (Rλ uj , vj ) → 0 as λ → ∞ for uj , vj ∈ C0∞ (Rn ) the result follows immediately. Lemma 4.4. For every Tσ , σ ∈ S m1 ,m2 , Rλ−1 (x0 , ξ0 )Tσ Rλ (x0 , ξ0 ) = Tσλ

(4.5)

σλ (x, η) = σ(x0 + λ−τ x, λξ0 + λτ η).

(4.6)

holds with

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Moreover if σ ∈ S 0,0 , λ ≥ 1 and 0 ≤ τ ≤ 1/2, then the estimate |∂xα ∂ξβ σλ (x, η)| ≤ Cpα,β (σ)

η|β| |β|

|ξ0 |

λ−τ |α| λ−(1−2τ )|β|

(4.7)

holds for all x, η ∈ Rn and a suitable constant c > 0. Here pα,β denotes the corresponding semi-norm on S 0,0 . Proof. Fix x0 , ξ0 ∈ Rn and let Rλ := Rλ (x0 , ξ0 ). We first write Rλ−1 Tσ Rλ u(x) in the form −τ −τ eiλ(x0 +λ x−y)ξ σ(x0 +λ−τ x, ξ)eiλyξ0 u(λτ (y −x0 ))dydξ. e−iλ(x0 +λ x)ξ0 (2π)−n Then substituting z := λτ (y − x0 ), the above expression takes the form −τ −τ n −n λ (2π) eiλ (ξ−λξ0 )(x−z) σ(x0 + λ−τ x, ξ)u(z)dzdξ.

(4.8)

Now if we set η := λ−τ (ξ − λξ0 ) in (4.8), the expression above turns to (2π)−n ei(x−z)η σ(x0 + λ−τ x, λξ0 + λτ η)u(z)dzdξ.

(4.9)

Then by (4.6) the expression in (4.9) equals to Tσλ u(x). Hence (4.5) holds. Now then by using (4.4), chain rule and Peetre’s inequality we have, |∂xα ∂ξβ σλ (x, η)| = |∂xα ∂ξβ σ(x0 + λ−τ x, λξ0 + λτ η)| = |(∂xα ∂ξβ σ)(x0 + λ−τ x, λξ0 + λτ η)λ−τ |α| λτ |β| | −|β| −τ |α| τ |β|

≤ cpα,β (σ)|ξ0 ||β| λξ0 + λτ η ≤ cpα,β (σ)|ξ0 |−|β| η|β| λ

(4τ −2)|β| 2

λ

λ

λ−τ |α| .

(4.10)

Hence (4.7) is proved.

The next lemma is an immediate consequence of Lemma 4.4 and Taylor’s formula. Lemma 4.5. Let σ ∈ S 0,0 , λ ≥ 1, and 0 < τ < 1/2. Then • For |α| + |β| > 0, ∂xα ∂ξβ σλ (x, η) → 0 as λ → ∞, uniformly on every compact K ⊂ R2n . • If λ → ∞, then σλ (x, η) − σλ (0, 0) → 0 uniformly on every compact K ⊂ R2n . Lemma 4.4 and Lemma 4.5 are also valid for σ ∈ S m , m ∈ (−∞, ∞), moreover for σ ∈ S 0 (4.7) is also true. And in that case pα,β is the corresponding semi-norm on the space S 0 .


113

Remark 4.6. Let σ ∈ S 0 and 0 < τ < 1/2. Then Aλ := {σλ : λ ≥ 1} ⊆ C ∞ (R2n ) is a bounded set in C ∞ (R2n ). Now since in C ∞ (R2n ) every bounded sequence has a convergent subsequence so without any loss of generality let {σkn } be a convergent subsequence of {σλn } and σλkn → σ∞

as n → ∞ in C ∞ (R2n ) under the topology of corresponding semi-norms given by pN (σλ ) =

sup

sup {|Dα σλ (x)| : λ ≥ 1} ,

0≤|α|≤N x∈KN

where Ki ⊆ Ki+1 are compact sets in R2n and R2n = ∪∞ i=1 Ki . Then from Lemma 4.5 we have ∂xα ∂ηβ σ∞ = 0, in every compact set K ⊆ R2n . Hence σ∞ is a constant in R2n . Lemma 4.7. Let σ ∈ S 0 and 0 < τ < 1/2. Moreover, let σ∞ ∈ C and λk → ∞ such that σλk → σ∞ ∞ 2n in C (R ). Then Tσ Rλk u → σ∞ u (4.11) Rλ−1 k in Lp (Rn ) for all u in Lp (Rn ). Proof. Fix x0 , ξ0 ∈ Rn and let Rλk := Rλk (x0 , ξ0 ). Now since Tσ Rλk ≤ Rλ−1 Tσ Rλk = Tσ , Rλ−1 k k so it is sufficient to show (4.11) on a dense subset of Lp (Rn ). Let u ∈ S. Then |eixη σλ (x, η)ˆ u(η)| ≤ cp0,0 (σ)|ˆ u(η)|.

(4.12)

By the dominated convergence theorem and (4.11) we have (Rλ−1 Tσ Rλk )u(x) → σ∞ u(x), k

(4.13)

pointwise for all x ∈ Rn as λk → ∞. Moreover for all l ∈ N, using (4.5), (4.6) and (4.7) and by integration by parts, Tσ Rλk u(x)| ≤ C |x Rλ−1 k 2l

for all λ ≥ 1. Hence

Tσ Rλk )|p ≤ C x |(Rλ−1 k

Now if, 2lp > n then x

−2lp

−2lp

(4.14) .

(4.15)

∈ L1 (Rn ). So

|(Rλ−1 Tσ Rλk − σ∞ )u(x)| ≤ C1 x k

−2lp

.

(4.16)

Now then (4.11) follows from (4.14), (4.17) and the dominated convergence theorem. The following theorem is one of the main theorems of this paper.

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Theorem 4.8. Let σ ∈ S 0 be such that Tσ : Lp (Rn ) → Lp (Rn ), 1 < p < ∞, is a Fredholm operator. Then there are constants C > 0 and R > 0 such that, |σ(x, ξ)| ≥ C for all x ∈ Rn and |ξ| ≥ R. Proof. Since Tσ is a Fredholm operator so there exist operators S ∈ B(Lp (Rn )) and K ∈ K(Lp (Rn )) such that STσ = I + K, K is compact. Consider the set, 1 n n M = ξ ∈ R : ∃x ∈ R : |σ(x, ξ)| ≤ . 2S Now if M is bounded then the proof is obvious. Suppose M is not bounded. Then there exists a sequence (xk , ξk ) in R2n such that |ξk | → ∞ and |σ(xk , ξk )| ≤ 1/2S. Without loss of generality we get an σ∞ ∈ C such that, 1 (4.17) σ(xk , ξk ) → σ∞ , |σ∞ | ≤ 2S as |ξk | → ∞. Let

ξk λk := |ξk |, Rk := Rλk xk , . |ξk |

(4.18)

Then from Lemma 4.4

(4.19) Rk−1 Tσ Rk = Tσk τ τ (4.20) σk (x, η) = σ(xk + λk x, ξk + λk η). Now since (4.7) holds uniformly for all (x0 , ξ0 ) ∈ Rn × {ξ : |ξ| = 1}, we have |∂xα ∂ηβ σk (x, η)| ≤ cpα,β (σ)η

|β| −τ |α| −(1−2τ )|β| λk λk

(4.21)

We denote

σk (0, 0) = σ(xk , ξk ) = σk∞ . Then using Taylor’s formula it can be shown, σk (x, η) − σk∞ → 0

(4.22) (4.23)

uniformly for (x, η) in a compact set K ⊂ R as k → ∞. Now let u ∈ S. Using the dominated convergence theorem and proceeding similarly as Lemma 4.7 we have, −2lp |(Rk−1 Tσ Rk u(x) − σk∞ u(x))|p ≤ Cx , (4.24) 2n

where l ∈ N. So x gence theorem,

−2lp

∈ Lp (Rn ) when 2lp > n. Again by the dominated conver-

Rk−1 Tσ Rk u(x) → σk∞ u(x) in Lp (Rn ) as k → ∞. By (4.22) and (4.26), we have

(4.25)

Rk−1 Tσ Rk u(x) → σ∞ u(x)

(4.26)

in L (R ) as k → ∞. For u ∈ L (R ), u = 0, p

n

p

n

0 < up = (STσ + K)Rk up ≤ STσ Rk up + KRk up ≤ Rk−1 Tσ Rk up S + KRk up .

(4.27)


115

Now by Lemma 4.7 and using the fact that K is a compact operator and Lemma 4.3 it follows that KRk up → 0 as k → ∞. Then from (4.28) up ≤ S|σ∞ |up .

(4.28)

Then, (4.18) and (4.29) yield the contradiction 1 1 ≤ |σ∞ | ≤ , S 2S which completes the proof.

The above theorem (4.8) can be generalized for all values of m ∈ (−∞, ∞). Theorem 4.9. Let σ ∈ S m , −∞ < m < ∞ and let Tσ : H s,p → H s−m,p a Fredholm operator. Then Tσ is an elliptic operator. Proof. Tσ : H s,p → H s−m,p , J−s : H s,p → Lp (Rn ) and Jm−s : H s−m,p → Lp (Rn ) are bounded linear operators. Here Js , the Bessel potential, is a pseudo-differential operator with symbol σs ∈ S −s with σs (ξ) = ξ−s . Let Jm−s Tσ Js = Tτ .

(4.29)

Then Tτ : Lp (Rn ) → Lp (Rn ), where τ ∈ S . Now since Js is one-one and onto so Js is Fredholm and elliptic for all s ∈ (−∞, ∞). So by Theorem 3.3 Tτ is elliptic. By (4.30) and the fact that Js , s ∈ Rn , is bijective it follows immediately that Tσ is elliptic. 0

The following theorem which is a simple consequence of Theorem 3.3 and the main theorems of this paper, Theorem 4.9 and Theorem 4.8, proves the ellipticity of the Fredholm SG pseudo-differential operators on Lp (Rn ). Theorem 4.10. Let σ ∈ S m1 ,m2 , m1 , m2 ∈ (−∞, ∞) and let Tσ : H s1 ,s2 ,p → H s1 −m1 ,s2 −m2 ,p is a Fredholm operator. Then Tσ is elliptic. Moreover if, σ ∈ S 0,0 then Tσ : Lp (Rn ) → Lp (Rn ) is Fredholm if and only if it is elliptic.

References [1] F.V. Atkinson, The normal solubility of linear equations in normed spaces, Mat. Sbornik N.S 28(70) (1951), 3–14. (in Russian) [2] P. Boggiatto, E. Buzano and L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie-Verlag, 1996. [3] M. Cappiello and L. Rodino, SG-pseudo-differential operators and Gelfand–Shilov spaces, Rocky Mountain J. Math. 36 (2006), 1117–1148. [4] H.O. Cordes, The Technique of Pseudodifferential Operators, Cambridge University Press, 1995.

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[5] S. Coriasco and L. Rodino, Cauchy problem for SG-hyperbolic equations with constant multiplicities, Ricerche Mat. Suppl. Vol. XLVIII (1999), 25–43. [6] A. Dasgupta and M.W. Wong, Spectral theory of SG pseudo-differential operators on Lp (Rn ), Studia Math. 187 (2008), 185–197. [7] Y.V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Birkh¨ auser, 1997. [8] U. Grieme, Pseudo-Differential Operators with Operator-Valued Symbols on NonCompact Manifolds, Ph.D. Thesis, Universit¨ at Potsdam, 1998. [9] V.V. Grushin, Pseudo-differential operators on Rn with bounded symbols, Funct. Anal. Appl. 4 (1970), 202–212. [10] G. Harutyunyan and B.-W. Schulze, Elliptic Mixed, Transmission and Singular Crack Problems, EMS Tracts in Mathematics 4, European Mathematical Society, 2008. [11] F. Nicola, K-theory of SG-pseudo-differential algebras, Proc. Amer. Math. Soc. 131 (2003), 2841–2848. [12] F. Nicola and L. Rodino, SG pseudo-differential operators and weak hyperbolicity, Pliska Stud. Math. Bulgar. 15 (2002), 5–19. [13] C. Parenti, Operatori pseudo-differentiali in Rn e applicazioni, Ann. Mat. Pura Appl. 93 (1972), 359–389. [14] M. Schechter, Spectra of Partial Differential Operators, Second Edition, NorthHolland, 1986. [15] M. Schechter, Principles of Functional Analysis, Second Edition, American Mathematical Society, 2002. [16] B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, 1998. [17] M. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, 1987. [18] M.W. Wong, Fredholm pseudo-differential operators on weighted Sobolev spaces, Ark. Mat. 21 (1983), 271–282. [19] M.W. Wong, Spectral theory of pseudo-differential operators, Adv. in Appl. Math. 15 (1994), 437–451. [20] M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999. [21] M.W. Wong, M-elliptic pseudo-differential operators on Lp (Rn ), Math. Nachr. 279 (2006), 319–326. Aparajita Dasgupta Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada e-mail: [email protected]


Hyperbolic Systems with Discontinuous Coefficients: Generalized Wavefront Sets Michael Oberguggenberger Abstract. We study linear hyperbolic systems of pseudo-differential equations with nonsmooth, possibly discontinuous symbols and distributional data. This note initiates an approach whereby solutions are constructed in the dual of the Colombeau algebra of generalized functions. We compute the generalized wave front set of the solution to a transport equation with discontinuous propagation speed and delta functions as initial data. The generalized wave front set turns out to have a more refined and informative structure than the wavefront set of the corresponding distributional limit. Mathematics Subject Classification (2000). Primary 35D05, 35S10, 46F30; Secondary 35D10, 35L45. Keywords. Pseudo-differential equations, algebras of generalized functions, duality methods, propagation of singularities.

1. Introduction In this note, we study symmetric hyperbolic (n × n)-systems of differential or pseudo-differential equations ∂t u(t, x) = A(t, x, Dx ) u(t, x) + f (t, x), u(0, x) = g(x),

x ∈ Rd .

x ∈ Rd , t > 0, (1.1)

We are interested in a framework that allows distributional data and non-smooth, e.g., discontinuous coefficients simultaneously. Simple examples show that in such a situation, solutions in the sense of distributions may fail to exist. For this reason, we take recourse to the theory of generalized functions of Colombeau [1, 2]. The matrix of operators in (1.1) is either given by d A(t, x, Dx ) = Aj (t, x)Dxj + A0 (t, x), (1.2) j=1

where the entries of A0 , . . . , Ad are elements of the Colombeau algebra G([0, ∞) × Rd ), or by a matrix of first-order pseudo-differential operators with Colombeau symbols.

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It has been known since about two decades that hyperbolic systems of the type (1.1), (1.2) have unique solutions in the Colombeau algebra G([0, ∞)×Rd ), see [15, 17, 18]; the case of a scalar first-order hyperbolic pseudo-differential equation was settled by [12]. By now, microlocal ellipticity is well established for operators with Colombeau coefficients or symbols [7, 8, 14]. However, the propagation of singularities along bicharacteristics for solutions in G([0, ∞) × Rd ) has been a notoriously difficult problem. Positive results have been obtained for solutions in G([0, ∞) × Rd ) to systems with classical, smooth coefficients [8] and in the special case of a transport equation in conservative form [9, 13]. The general question still remains open. In [4], Garetto introduced the continuous dual of a Colombeau algebra and proved microlocal elliptic regularity results in [5]. The dual framework suggests itself as ideally suited for addressing the problem of propagation of singularities for (1.1) in the Colombeau setting. Indeed, one may use the Colombeau algebra for accommodating regularizations of the coefficients (or symbols) of the operators, while keeping the initial data and the forcing terms as distributions (now viewed as elements of the dual of the Colombeau algebra). It is the purpose of this note to show that this program works (and does so with much more ease than remaining on the level of the algebras). Thus, we will first show that system (1.1) is solvable in the dual space, the and second, indicate space of C-linear, continuous functionals L(Gc ([0, ∞)×Rd ), C) in an explicit example of a transport equation that the Colombeau generalized wave front set can be calculated and exhibits a finer structure than the classical wave front set of the distributional limit of the solution. The plan of the paper is as follows. In Section 2, we begin by recalling basic results from Colombeau algebras and their duals. In Section 3, we discuss the Fourier transform of C-linear functionals, the definition of their wave front set, and the notion of generalized pseudo-differential operators acting on them. It is worthwhile to point out that there are two types of regularity for elements of the dual: to belong to the Colombeau algebra G or to belong to the subalgebra G ∞ of regular elements. In Section 4 we show that system (1.1) is solvable in the dual of the Colombeau algebra under certain hypotheses on the symbols and data. We discuss the question of uniqueness and show that G-regular solutions are unique. Section 5 is devoted to an explicit calculation of the wave front set of a (generic) scalar transport equation with discontinuous coefficient and initial data given by a delta function. We exemplify the solution concept and show that the fiber of the wave front set above the kink in the characteristic curve carrying the singularities is smaller than the wave front set of the distributional limit and reflects the shape of the kink. The general question of propagation of singularities along bicharacteristics remains open; we hope that this note will spark interest in the duality methods and lead to progress on the problem.

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2. Colombeau theory: Basic notions In this section we collect various basic notions from the theory of generalized functions of Colombeau [1, 2] which we need in this paper. For more details we refer, e.g., to [10]. Let Ω be an open subset of Rd . The starting point of the theory as we use it here are families (uε )ε∈(0,1] of smooth functions uε ∈ C ∞ (Ω) for 0 < ε ≤ 1. The following subalgebras are singled out: Moderate families, denoted by EM (Ω), are defined by the property: ∀K Ω ∀α ∈ Nd0 ∃p ≥ 0 : sup |∂ α uε (x)| = O(ε−p ) as ε → 0.

(2.1)

x∈K

Null or negligible families, denoted by N (Ω), are defined by the property: ∀K Ω ∀α ∈ Nd0 ∀q ≥ 0 : sup |∂ α uε (x)| = O(εq ) as ε → 0.

(2.2)

x∈K

The null families form a differential ideal in the collection of moderate families. The Colombeau algebra is the factor algebra G(Ω) = EM (Ω)/N (Ω). Restrictions of the elements of G(Ω) to open subsets of Ω are defined on representatives. Thus the notion of support of an element is meaningful; actually, Ω → G(Ω) is a sheaf of differential algebras on Rd . We denote the compactly supported Colombeau generalized functions by Gc (Ω). The space of compactly supported distributions E (Ω) is imbedded in G(Ω) by convolution: ι : E (Ω) → G(Ω), ι(w) = class of ((w ∗ ϕε )|Ω )ε∈(0,1] ,

(2.3)

where

ϕε (x) = ε−d ϕ (x/ε) (2.4) is obtained by scaling a fixed test function ϕ ∈ S(Rd ) of integral one with all moments vanishing. By the sheaf property, this can be extended in a unique way to an imbedding of the space of distributions D (Ω) in G(Ω). This imbedding renders C ∞ (Ω) a faithful subalgebra. In fact, given f ∈ C ∞ (Ω), one can define a corresponding element of G(Ω) by the constant imbedding σ(f ) = class of (fε )ε∈(0,1] with fε ≡ f for all ε. Then ι(f ) = σ(f ) in G(Ω). Families (rε )ε∈(0,1] of complex numbers such that |rε | = O(ε−p ) as ε → 0 for some p ≥ 0 are called moderate, those for which |rε | = O(εq ) for every q ≥ 0 are of Colombeau generalized numbers is obtained by termed negligible. The ring C factoring moderate families of complex numbers with respect to negligible families. coincides with the ring of constants in the differential When Ω is connected, C algebra G(Ω). Regularity theory for linear equations has been based on the subalgebra G ∞ (Ω) of regular generalized functions in G(Ω). It is defined by those elements which have a representative satisfying ∀K Ω ∃p ≥ 0 ∀α ∈ Nd0 : sup |∂ α uε (x)| = O(ε−p ) as ε → 0. x∈K

(2.5)

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Observe the change of quantifiers with respect to formula (2.1); locally, all derivatives of a regular generalized function have the same order of growth in ε > 0. It holds that that (see [18]) G ∞ (Ω) ∩ D (Ω) = C ∞ (Ω). For K Ω, α ∈ Nd0 , p ≥ 0, denote by VK,α,p ⊂ G(Ω) the set of elements u of G(Ω) such that sup |∂ α uε (x)| = O(εp ) as ε → 0 x∈K

for some – and hence all – representatives (uε )ε∈(0,1] of u. The so-called sharp topology on G(Ω) is obtained by taking the collection of all VK,α,p as a base of of Colombeau generalized numbers neighborhoods of zero. Similarly, the ring C may be topologized by means of the neighborhoods Vp , p ≥ 0, defined by those turns into an moderate elements that satisfy |rε | = O(εp ). With this topology, C ultrametric, complete topological ring and G(Ω) becomes an ultrametric, complete Further, Gc (Ω) topological differential algebra and a topological module over C. can be endowed with the structure of an inductive limit of topological C-modules. The theory of topological C-modules has been developed in detail by [3]. of continDuality theory. Following [3, 4], we introduce the dual L(Gc (Ω), C) uous C-linear forms on Gc (Ω) (with values in C). As usual, we denote the action Equipped of an element u of the dual on an element ψ ∈ Gc (Ω) by u, ψ ∈ C. d with the corresponding weak topology, L(Gc (Ω), C), Ω ⊂ R , forms a sheaf of topological C-modules on Rd . by means of the The Colombeau algebra G(Ω) is imbedded in L(Gc (Ω), C) assignment u, ψ = class of uε (x)ψε (x)dx ε∈(0,1]

where (uε )ε∈(0,1] and (ψε )ε∈(0,1] are representatives of u and ψ. What is more in C by important, D (Ω) can be directly imbedded in L(Gc (Ω), C) w, ψ = class of (w, ψε )ε∈(0,1] for w ∈ D (Ω), ψ ∈ Gc (Ω). More generally, moderate families of distributions A family of distributions is called mod(wε )ε∈(0,1] define elements of L(Gc (Ω), C). erate, if for all compact subsets K of Ω: ∃j ∈ N0 ∃p ≥ 0 : |wε , ϕ| ≤ ε−p

sup

|∂ α ϕ(x)|

(2.6)

x∈K,|α|≤j

for all ϕ ∈ D(Ω) with support in K (and sufficiently small ε). Elements of defined in this way (by a representing family of distributions) are L(Gc (Ω), C) called basic functionals. Observe that, in particular, functionals defined by elements of G(Ω) are basic; the action on a test function does not depend on the choice of representative.


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are sheaf imbeddings. ThereThe imbeddings G ∞ (Ω) → G(Ω) → L(Gc (Ω), C) fore, given u ∈ L(Gc (Ω), C), it makes sense to consider the largest open subset ω of Ω on which u is defined by an element of G(ω). The complement of this set is called the G-singular support of u. In the same way, the G ∞ -singular support can be introduced. Finally, the theory can be carried over to generalized functions on the closed half-space [0, ∞)×Rd as well. The representatives of the elements of the Colombeau algebra G([0, ∞)×Rd ) satisfy estimates (2.1), respectively (2.2) on compact subsets of [0, ∞) × Rd , i.e., up to the boundary {t = 0}. Similarly, Gc ([0, ∞) × Rd ) denotes the elements of G([0, ∞) × Rd ) whose support is a compact subset of the closed half-space [0, ∞) × Rd . The restriction of an element u ∈ Gc ([0, ∞) × Rd ) to the boundary {t = 0} is defined on representatives by u|{t=0} = class of (uε (0, ·))ε∈(0,1] . The space of C-linear functionals on [0, ∞) × Rd is defined as the dual space d L(Gc ([0, ∞) × R ), C).

3. Fourier transform and pseudo-differential operators can be identified The space of compactly supported functionals from L(Gc (Ω), C) It is tempting to define the Fourier transform of an element with L(G(Ω), C). as a map from Rd into C by u ∈ L(G(Rd ), C) ξ → u, eξ ,

eξ (x) = e −ixξ .

However, we need to be a bit more precise about the range of the Fourier transform. Thus we introduce the subalgebra Eτ (Rd ) of EM (Rd ) as those families (uε )ε∈(0,1] of smooth functions which satisfy ∀α ∈ Nd0 ∃p ≥ 0 : sup ξ−p |∂ α uε (ξ)| = O(ε−p ) as ε → 0;

(3.1)

ξ∈Rd

as usual, we let ξ = (1 + |ξ|2 )1/2 . The ideal Nτ (Rd ) in Eτ (Rd ) is constituted by the families with the property ∀α ∈ Nd0 ∃p ≥ 0 ∀q ≥ 0 : sup ξ−p |∂ α uε (ξ)| = O(εq ) as ε → 0. ξ∈Rd

The algebra Gτ (Rd ) = Eτ (Rd )/Nτ (Rd ) is called the algebra of tempered Colombeau generalized functions. There exists a representing Now let w be a basic functional in L(G(Rd ), C). family of compactly supported distributions (wε )ε∈(0,1] and a compact set K ⊂ Rd such that an estimate of type (2.6) holds for all ϕ ∈ C ∞ (Rd ). It follows that the family of smooth functions ξ → wε , eξ , ε ∈ (0, 1], satisfies property (3.1). Actually, one may take the same exponent p from (2.6) for all α. This will serve as the definition of the Fourier transform.

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is the Definition 3.1. The Fourier transform of a basic functional w ∈ L(G(Rd ), C) d element F w ∈ Gτ (R ) defined by the representing family (ξ → wε , eξ )ε∈(0,1] . We are ready to define the generalized wave front set of a basic functional. First, let v ∈ Gτ (Rd ) and ξ0 ∈ Rd \ 0. Then ξ0 is called a G-regular direction for v, if there is a representative (vε )ε∈(0,1] and an open cone Γ ⊂ Rd \ 0 about ξ0 such that (3.2) ∀l ≥ 0 ∃p ≥ 0 : supξl |vε (ξ)| = O(ε−p ) as ε → 0. ξ∈Γ

If instead we have that ∃p ≥ 0 ∀l ≥ 0 : supξl |vε (ξ)| = O(ε−p ) as ε → 0,

(3.3)

ξ∈Γ

then ξ0 is called a G ∞ -regular direction for v. (x0 , ξ0 ) ∈ Ω × Rd \ 0. Definition 3.2. Let w be a basic functional in L(Gc (Ω), C), Then w is called microlocally G-regular, respectively microlocally G ∞ -regular at (x0 , ξ0 ), if there is a cut-off function φ ∈ D(Ω), φ(x0 ) = 0, such that ξ0 is a G-regular direction, respectively G ∞ -regular direction for F (φw). The collection of all (x0 , ξ0 ) at which w is not microlocally regular forms the G-wave front set WFG (w), respectively G ∞ -wave front set WFG ∞ (w). It is clear that in case w is defined by an element of G(Ω), then WFG (w) is empty and WFG ∞ (w) coincides with the previously defined generalized wave front set WFg (w) (see [14, 16]). Also, if w is given by a single distribution, both WFG (w) and WFG ∞ (w) coincide with the distributional wave front set WF(w), because F (φw) does not depend on ε. The following result is due to [5] who proved it by means of a pseudo-differential characterization of the generalized wave front set. For the purpose of illustration, we will provide a direct proof. Then the projection Proposition 3.3. Let w be a basic functional in L(Gc (Ω), C). of WFG (w) to Ω coincides with the G-singular support of w, and the projection of WFG ∞ (w) yields the G ∞ -singular support of w. Proof. We begin with the global argument. Suppose that vε = F (φwε ) satisfies the estimate (3.2) on Γ = Rd \ 0. We have to produce a representative (uε )ε∈(0,1] of an element u ∈ G(Ω) such that u(x)ψ(x)dx = φw, ψ for all ψ ∈ Gc (Ω). To this end, we take ρ ∈ D(Rd ), ρ(x)dx = 1, set in C ρε (x) = e 1/ε ρ(e 1/ε x) and define uε = (φwε ) ∗ ρε . Clearly, uε is smooth. Due to the sequence of estimates (3.2) on the Fourier transform of φwε , we have that ∂ α uε is bounded for all α ∈ Nd0 and sufficiently small ε (depending on α), and ∂ α uε L∞ (Rd ) ≤ ∂ α (φwε )L∞ (Rd ) ρL1 (Rd ) = O(ε−p ) as ε → 0


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for some p ≥ 0. Thus (uε )ε∈(0,1] is moderate and qualifies as a representative of an element u ∈ G(Ω). It follows from the mean value theorem that the family ψε ∗ ρˇε is a representative of ψ as well, whereˇdenotes inflection. Thus uε (x)ψε (x)dx − φwε , ψε = φwε , ψε ∗ ρˇε − ψε is a negligible family of numbers, as desired. In other words, if the G-wave front set of a basic functional is empty, then it is given by an element of G(Ω). The microlocal version is obtained by the usual argument, piecing together finitely many cones and convolving the corresponding finitely many cut-off functions. The assertion about the G ∞ -singular support is proved in the same way. Generalized pseudo-differential operators. Recall that the H¨ ormander class S m (Ω×Rd ) of symbols of order m is defined as the set of functions a ∈ C ∞ (Ω×Rd ) such that (m) (3.4) |a|K,α,β = sup ξ−m+|α| |∂ξα ∂xβ a(x, ξ)| < ∞ x∈K,ξ∈Rd

for all compact subsets K of Ω and all α, β ∈ Nd0 . Equipped with the seminorms (3.4), S m (Ω × Rd ) is a Fréchet space. We will later make use of time-dependent symbols t → a(t) which are given by C ∞ -maps of t ∈ [0, T ] with values in S m (Ω × Rd ), equipped with the seminorms (m)

(m)

|a|T,k,K,α,β = sup |∂tk a(t)|K,α,β .

(3.5)

t∈[0,T ]

A family of symbols (aε )ε∈(0,1] from S m (Ω × Rd ) will be termed moderate, if ∀K Ω ∀α, β ∈ Nd0 ∃p ≥ 0 : |aε |K,α,β = O(ε−p ) as ε → 0. (m)

Similarly, the family is called negligible, if (m)

∀K Ω ∀α, β ∈ Nd0 ∀q ≥ 0 : |aε |K,α,β = O(εq ) as ε → 0. The space Sm (Ω × Rd ) of generalized symbols is the factor space of moderate modulo negligible symbols. In a similar way, generalized amplitudes are defined as generalized symbols on Ω × Ω × Rd and represented by families of the form aε (x, y, ξ), ε ∈ (0, 1]. Given a generalized amplitude a and an element u of Gc (Ω), the integral vε (x) = e i(x−y)ξ aε (x, y, ξ)uε (y)dy d−ξ Rd

Ω

can be interpreted as an oscillatory integral (inserting the operator ξ−N Dy N enforces convergence) and is easily seen to produce a family (vε )ε∈(0,1] of smooth functions that belong to EM (Ω). Changing representatives of a or u results in a family that differs from (vε )ε∈(0,1] by an element of N (Ω). Therefore, the assignment v = Au yields an operator A : Gc (Ω) → G(Ω) which is formally denoted by Au(x) = e i(x−y)ξ a(x, y, ξ)u(y)dy d−ξ Rd

Ω

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and called a generalized pseudo-differential operator. The operator A is continuous with respect to the sharp topologies. If A is properly supported (this being implied by the existence of amplitudes aε (x, y, ξ) whose support with respect to (x, y) is a proper subset of Ω × Ω independently of ε), then A maps Gc (Ω) into itself. The formal transpose of A is also a generalized pseudo-differential operator given by T e i(x−y)ξ a(y, x, −ξ)u(y)dy d−ξ. A u(x) = Rd

Ω

For proofs of all these statements we refer to [7]. Following [5], the action of a properly supported generalized pseudo-differential operator can be extended to the dual: → L(Gc (Ω), C), A : L(Gc (Ω), C) Au, ψ = u, AT ψ. The extended operator is continuous (both for the weak and the strong topologies). Finally, we recall [7] that the generalized symbol of a properly supported, generalized pseudo-differential operator A with a regular amplitude can be defined by e i(x−y)ξ χ(x, y)a(x, y, ξ + η)d−y dη σA (x, ξ) = Rd

Ω

with a suitable proper cut-off function χ. Here an amplitude a ∈ Sm (Ω × Ω × Rd ) is called regular if it is represented by a family satisfying ∀K Ω × Ω ∃p ≥ 0 ∀α, β ∈ Nd0 : |aε |K,α,β = O(ε−p ) as ε → 0 (m)

(with p not depending on α, β). Then σA belongs to Sm (Ω × Rd ) and Au(x) = e ixξ σA (x, ξ)F u(ξ)d−ξ Rd

for all u ∈ Gc (Ω). The symbolic calculus was recently extended to the full class of generalized pseudo-differential operators by [6]. This concludes the brief recall of pseudo-differential operators in the Colombeau setting. Everything that has been said can be (and has been) generalized to m with 0 ≤ ρ < δ ≤ 1. operators based on the H¨ ormander class Sρ,δ

4. Symmetric hyperbolic systems: Existence theory In this section we construct solutions in the dual of the Colombeau algebra to the initial value problem ∂t u = Au + f, (4.1) u(0, ·) = g. Here n, n g ∈ L(Gc (Rd ), C) f ∈ L(Gc ([0, ∞) × Rd ), C) and A = A(t, x, Dx ) will be a time-dependent (n × n)-matrix of first-order generalized pseudo-differential operators. We look for solutions n. u ∈ L(Gc ([0, ∞) × Rd ), C)


125

To this end, we will first have to clarify the solution concept. The matrix of pseudodifferential operators will be given by representing amplitudes aε (t, x, y, ξ) which are C ∞ -maps of t ∈ [0, ∞) with values in S m (Rd × Rd × Rd ), moderate in the seminorms (3.5), that is, ∀T > 0 ∀k ∈ N0 ∀K Ω ∀α, β ∈ Nd0 ∃p ≥ 0 : |aε |T,k,K,α,β = O(ε−p ) as ε → 0. (m)

We may identify two such amplitudes if their difference is negligible in the seminorms (3.5). Assuming that the operator A = A(t, x, Dx ) is properly supported at each fixed t ≥ 0, an inspection of the relevant proofs in [7] shows that n → L(Gc ([0, ∞) × Rd ), C) n A : L(Gc ([0, ∞) × Rd ), C) and so does its formal transpose AT . By a solution to system (4.1) we mean an n such that element u ∈ L(Gc ([0, ∞) × Rd ), C) u, ∂t ψ + u, AT ψ + f, ψ + g, ψ(0, ·) = 0 (4.2) n for all ψ ∈ Gc ([0, ∞) × Rd) . Here the first three brackets denote duality on in C [0, ∞) × Rd , the fourth on Rd . The concept is motivated by a formal integration by parts. To arrive at the existence result, a number of hypotheses will be needed. (H1) (Basic Sobolev structure) The data f, g are basic functionals, given by representing families (fε )ε∈(0,1] , (gε )ε∈(0,1] such that fε ∈ C([0, ∞) : Hs (Rd )) and gε ∈ Hs (Rd ) for some s ∈ R with the corresponding seminorms moderate, i.e., sup fε (t)Hs (Rd ) = O(ε−p ),

gε Hs (Rd ) = O(ε−p ) as ε → 0

t∈[0,T ]

for some p ≥ 0 and all T ≥ 0 (p may depend on T ). (H2) (Smooth time-dependence of symbols) A is a matrix of generalized firstorder pseudo-differential operators given by families of properly supported amplitudes aε (t, x, y, ξ) which are C ∞ -maps of t ∈ [0, ∞) with values in S 1 (Rd × Rd × Rd ). (H3) (Logarithmic estimates) For all T ≥ 0, k ∈ N0 , K Rd × Rd , α, β ∈ Nd0 : (1)

|aε |T,k,K,α,β = O(| log ε|) as ε → 0. (H4) (Symmetric hyperbolicity) A + A is a matrix of generalized pseudo-differential operators of order 0, where A = A¯T denotes the formal adjoint. (H5) (Compact supports) The symbols aε (t, x, y, ξ) as well as the functions fε (t, x), when restricted to any strip [0, T ] × Rd , have compact support with respect to the variable x, independently of ε; the same holds of the data gε (x). Remark 4.1. (i) The basic structure required in (H1) is inherent in our method of proof. (ii) What concerns assumption (H2), smooth dependence on t is already required in the classical theory. By what has been said about properly supported,

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regular pseudo-differential operators in Section 3, the formal transpose AT also possesses a family of defining amplitudes smoothly depending on t, as do the symbols of A and AT . Note that the logarithmic estimate (H3) entails, in particular, that A is a matrix of regular pseudo-differential operators. (iii) The logarithmic estimate (H3) is needed in the energy estimate and cannot be dropped as is seen from simple examples even of ordinary differential equations [18]. However, it is required only up to orders |α|, |β| ≤ d + 1 and k = 0 if moderate estimates of type O(ε−p ) hold for the other values of k, α, β. (iv) The compactness of the supports is convenient; it allows to employ the simple arguments from [19]. In the scalar case, it can be removed by the techniques of [12]. Note that due to the formulas in Section 3, the compactness assumption on the amplitudes of A entails the one on its transpose AT and the symbols σA and σAT . (v) The Sobolev index s may well be negative, so data given by distributions or by moderate families of distributions are admitted. Theorem 4.2. Under the hypotheses (H1) through (H5), the initial value problem n. (4.1) has a solution u ∈ L(Gc ([0, ∞) × Rd ), C) Proof. Choosing representatives of the data and the symbols in (4.1), it is well known that the system ∂t uε = Aε uε + fε , (4.3) uε (0, ·) = gε has a unique classical solution uε ∈ C([0, ∞) : Hs (Rd )) ∩ C 1 ([0, ∞) : Hs−1 (Rd )); to see this, we may use the compactness of the supports and follow the arguments in [19]. We are going to show that the family (uε )ε∈(0,1] qualifies as a representative of a solution u ∈ L(Gc ([0, ∞) × Rd ), C). 2 d First, at fixed 0 ≤ t ≤ T the operators Aε + AT ε map L (R ) continuously into itself with an operator norm Cε which is an affine function of (1)

|σAε |T,0,K,α,β

(1)

and |σATε |T,0,K,α,β

for |α|, |β| ≤ d+1 where K is a compact set containing the supports of the symbols. Following the arguments in [19], we easily arrive at the energy estimate t fε (τ )2Hs (Rd ) dτ exp(Cε t) uε (t)2Hs (Rd ) ≤ gε 2Hs (Rd ) + 0

for 0 ≤ t ≤ T . By assumption, Cε = O(| log ε|); thus the family (uε )ε∈(0,1] satisfies moderate estimates in the space C([0, ∞) : Hs (Rd )). In particular, it forms a moderate family of distributions and hence defines a basic functional u ∈ L(Gc ([0, ∞) × Rd ), C). Now let ψ ∈ Gc ([0, ∞) × Rd ). Its representatives ψε , ε ∈ (0, 1], belong to |s| C([0, ∞) : H0 (Rd )), in particular. The action of the functional u on ψ is given in

Hyperbolic Systems: Generalized Wavefront Sets terms of representatives by

∞

127

uε (t), ψε (t, ·)dt

0 |s|

where the brackets denote duality of Hs and H0 (or simply integration if s ≥ 0). We use that uε solves (4.3) and compute ∞ ∞ T uε (t), (∂t + Aε )ψε (t, ·)dt + fε (t), ψε (t, ·)dt + gε , ψε (0, ·) 0 0 ∞ ∞ (−∂t + Aε )uε (t), ψε (t, ·)dt + fε (t), ψε (t, ·)dt = 0, = 0

0

thus u is a solution in the sense of (4.2).

Remark 4.3. Due to the hypotheses on the supports of the data and the operators, the solution u also has compact support on every strip [0, T ] × Rd . We turn to the question of uniqueness of solutions. We do not know whether the current formulation of our solution concept is strong enough to have uniqueness; but we may enforce uniqueness in the following way. The notion of the Colombeau space GE based on a locally convex topological vector space E can be found in [3]. If we take the space E = C([0, ∞) : Hs (Rd )) ∩ C 1 ([0, ∞) : Hs−1 (Rd )) for such a construction, the arguments above show that problem (4.1) will have a solution in GE in the sense that the equations hold in GF with F = C([0, ∞) : Hs−1 (Rd )). The energy estimates will show that such a solution is unique. This assertion actually encodes the fact that a negligible perturbation (measured in the Hs -norm) of the data leads to a negligible perturbation of the classical solution. Uniqueness in the sense inherent to the solution concept in L(Gc ([0, ∞) × can be inferred for G-regular solutions. A C-linear functional on Gc ([0, ∞)× Rd ), C) Rd ) is G-regular if it is given by an element of G([0, ∞) × Rd ). Proposition 4.4. Under the hypotheses (H1) through (H5), there is at most one G-regular solution with compact support on every strip [0, T ] × Rd to the initial value problem (4.1). Proof. The difference w = u − v of two G-regular solutions satisfies w, ∂t ψ + w, AT ψ = 0 for all ψ ∈ Gc ([0, ∞) × Rd ) n . Consider the final value problem in C ∂t θ = −AT θ + w, ¯ θ(T, ·) = 0.

(4.4)

Due to the regularity and support assumption, the representatives wε of w belong to C k ([0, ∞) : Hs (Rd )) for every k ≥ 0 and s ∈ R. Due to hyperbolicity, we may mimic the proof of Theorem 4.2 backward in time and obtain solutions θε which also belong to every space C k ([0, T ] : Hs (Rd )) and in addition – by Remark 4.3 – have compact support in the strip [0, T ]× Rd. Since each θε is smooth and vanishes

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identically at t = T , we may extend it by zero for t > T and obtain an element θ ∈ Gc ([0, ∞) × Rd ) in this way which solves (4.4). Therefore, T T 0 = w, ∂t θ + w, A θ = |w(t, x)|2 dxdt 0

T

Rd

This implies that in C. 0 Rd w(t, x)ψ(t, x)dxdt vanishes in C for every ψ ∈ d n Gc ([0, T ] × R ) . Since T is arbitrary, w is zero as a functional and u = v. Remark 4.5. G-regular solutions exist: if the data f, g are G-regular with compact support in the appropriate sense as above, then the solution to (4.1) is G-regular.

5. Propagation of singularities: An example In this section we study an example of a transport equation with discontinuous coefficient and distributional data that demonstrates the advantage of the solution concept in the dual of the Colombeau algebra. The point is that the coefficient is regularized to produce an element of the Colombeau algebra, while the initial data enter in a non-regularized way as members of the dual. The model equation we study can be written formally as ∂t u(t, x) + H(t − 1)∂x u(t, x) = 0, u(0, x) = δ(x)

(5.1)

where H is the Heaviside function and δ the Dirac measure. For 0 < t < 1, we have ∂t u = 0, while (∂t + ∂x )u = 0 for t > 1 – thus heuristically the solution should be of the form δ(x), 0 ≤ t < 1, u(t, x) = (5.2) δ(x − t + 1), 1 < t. Note that the insertion of u in (5.1) a priori has no meaning in the theory of distributions, due to the product of ∂x u with the jump function at (t, x) = (1, 0). We shall construct a solution in the Colombeau dual space L(Gc ([0, ∞) × R), C), show that its restriction to {(t, x) ∈ [0, ∞) × R : t = 1} actually is given by (5.2) and compute its microlocal behavior at (t, x) = (1, 0). We begin by regularizing the coefficient. Let ρ ∈ D(R), supp ρ = [−1, 1], ρ > 0 on (0, 1) and symmetric, ρ(x)dx = 1 and put (t−1)/ε λε (t) = (5.3) ρ(s)ds = μ t−1 ε t

−∞

with μ(t) = −∞ ρ(s)ds. Then λε (t) = 0 for t ≤ 1 − ε, λε (t) = 1 for t ≥ 1 + ε and strictly increasing between 1−ε and 1+ε. In particular, λε (t) → H(t−1) as ε → 0. The family (λε )ε∈(0,1] is moderate and thus defines an element λ ∈ G([0, ∞) × R). We interpret the initial value problem (5.1) as the equation ∂t u + λ∂x u = 0,

u(0, ·) = δ

(5.4)


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in the sense of formula (4.2). Theorem 4.2 is not required, in L(Gc ([0, ∞) × R), C) because we can construct a solution explicitly. Put ⎧ 0, 0 ≤ t ≤ 1 − ε, ⎪ t ⎨ t Λε (t) = (5.5) λε (s)ds = 1−ε λε (s)ds, 1 − ε ≤ t ≤ 1 + ε, ⎪ 0 ⎩ t − 1, 1 + ε ≤ t. At the level of representatives, equation (5.4) reads ∂t uε + λε ∂x uε = 0,

uε (0, ·) = δ

(5.6)

and makes sense in C ([0, ∞) : D (R)), say. The unique distributional solution is given by uε (t, x) = δ(x − Λε (t)). 1

Clearly (uε )ε∈(0,1] is a moderate family of distributions and thus defines an element which solves (5.4). of L(Gc ([0, ∞) × R), C), If ψ ∈ Gc ([0, 1) × R), then the support of its representatives is contained in 1 [0, 1 − η] × R for some η > 0. It follows from (5.5) that uε , ψε = 0 ψε (t, 0)dt for sufficiently small ε. This shows that the restriction of u to [0, 1) × R is equal to the distribution 1 ⊗ δ. Similarly, we can compute the restriction to (1, ∞) × R and arrive at the assertion that the restriction of the generalized solution to {(t, x) ∈ [0, ∞) × R : t = 1} is indeed given by (5.2). What is more, the distributional limit (in D ([0, ∞) × R)) of the solutions uε as ε → 0 is the same, more precisely, it is given by the distribution v(t, x) = (H ⊗ δ)(1 − t, x) + (H ⊗ δ)(t − 1, x − t + 1); this follows from the pointwise convergence properties of Λε . Remark 5.1. (The wave front set of the distributional limit) The classical wave front set of the limiting distribution v is given by WF(v) = W0 ∪ W1 ∪ W2 where W0 = {(t, 0, 0, ξ) : 0 ≤ t < 1, ξ = 0} W1 = {(1, 0, τ, ξ) : (τ, ξ) = (0, 0)}

(5.7)

W2 = {(t, t − 1, −τ, τ ) : 1 < t, τ = 0}. Indeed, the assertions for t = 1 are standard. To see that the fiber over the point (1, 0) contains no regular direction, we just observe that ∂t (∂t + ∂x )v(t, x) = −δ(t − 1) ⊗ δ (x). It is clear that the wave front set of this latter distribution is equal to {(1, 0, τ, ξ) : (τ, ξ) = (0, 0)} and is contained in WF(v). Thus the classical wave front set contains no structural information on the singular behavior near the point (1, 0). However, the result shows that the wave front sets of the distributions ∂x v and H(t − 1) are not in favorable position; so their product in the sense of H¨ ormander does not exist.

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We are going to calculate the G- and the G ∞ -wave front sets of the solution and demonstrate that they exhibit a refined structure. u ∈ L(Gc ([0, ∞) × R), C) Proposition 5.2. The G-singular support of the solution u ∈ L(Gc ([0, ∞) × R), C) to (5.4) is equal to S = {(t, 0) : 0 ≤ t ≤ 1} ∪ {(t, t − 1) : t ≥ 1}. Proof. It is clear that the support of u equals S, which in turn contains the Gsingular support. So let (t0 , x0 ) ∈ S and assume there is a neighborhood V of (t0 , x0 ) on which u is given by an element u0 ∈ G(V ). Let K be a compact subset of V containing (t0 , x0 ) in its interior. By assumption, there is p ≥ 0 such that sup |u0ε (t, x)| = O(ε−p ) as ε → 0. (t,x)∈K

Choose points (tε , xε ) on the support of uε that converge to (t0 , x0 ) as ε → 0. Take ϕ ∈ D(R2 ) with support in [−1, 1] × [−1, 1], ϕ ≥ 0 and ϕ ≡ 1 on [−1/2, 1/2] × [−1/2, 1/2] and put ψε (t, x) = ϕ(ε−p−1 (t − tε ), ε−p−1 (x − xε )). Clearly, (ψε )ε∈(0,1] is a moderate family of smooth functions with support in K for sufficiently small ε, hence defines an element ψ ∈ Gc ([0, ∞) × R). In addition, the support of ψε is contained in the square Rε with center (tε , xε ) and side-length 2εp+1 , and ψε ≡ 1 on the sub-square Qε with with center (tε , xε ) and side-length εp+1 . By (5.5), the slope of the curve x = Λε (t) is bounded by 1. Therefore, the portion of the curve for tε − εp+1 ≤ t ≤ tε + εp+1 remains inside Qε . We conclude that ∞ tε +εp+1 /2 uε , ψε = ψε (t, Λε (t))dt ≥ dt = εp+1 tε −εp+1 /2

0

while

u0ε (t, x)ψε (t, x)dtdx ≤ V

ε−p dtdx ≤ 4ε−p ε2p+2 = 4εp+2 ,

Rε

a contradiction.

The G ∞ -singular support of u is equal to S as well, since it contains the G-singular support. We shall now compute the G-wave front set of the generalized solution in We are going to see that the non-regular directions above L(Gc ([0, ∞) × R), C). the point (1, 0) are confined to the cone spanned by the normals to the supporting tangents of S. to Proposition 5.3. The G-wave front set of the solution u ∈ L(Gc ([0, ∞) × R), C) (5.4) equals WFG (u) = W0 ∪ W1/2 ∪ W2 where W0 , W2 are as in (5.7) and W1/2 = {(1, 0, τ, ξ) : ξ < 0, 0 ≤ τ ≤ −ξ} ∪ {(1, 0, τ, ξ) : ξ > 0, −ξ ≤ τ ≤ 0}.


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Proof. It was shown above that the restriction of u to {(t, x) ∈ [0, ∞) × R : t = 1} is given by the distribution (5.2). We remarked in Section 3 that the wave front set of a distribution coincides with its G-wave front set. Therefore, the non-regular directions above the points (t0 , x0 ) = (1, 0) are described by the sets W0 and W2 . It remains to study the fiber above (1, 0). The upper bound. Take χ, φ ∈ D(R) with χ(1) = φ(0) = 1. Then 1 0 F χ(t)φ(x) δ(x − Λε (t)) (τ, ξ) = δ(x − Λε (t)), χ(t)φ(x) e −itτ −ixξ ∞ χ(t)φ(Λε (t)) e −itτ −iΛε (t)ξ dt = 0 ∞ χ(t) e −itτ −iΛε (t)ξ dt = 0

where we chose φ such that φ(Λε (t)) = 1 for t ∈ supp χ, which is clearly possible. The prospective regular directions are given by R = {(τ, ξ) = (1, β) : −1 < β < ∞} ∪ {(τ, ξ) = (−1, β) : −∞ < β < 1}. We consider only the first part of R and take any proper sub-cone of the form Γ = {(τ, ξ) : τ > 0, β0 τ ≤ ξ ≤ β1 τ } with −1 < β0 < 0 and β1 > 0. Recalling that 0 ≤ λε ≤ 1, we have for (τ, ξ) ∈ Γ that 1 + λε (t) τξ ≥ 1 + β0 > 0. Thus the equation s = t + Λε (t) τξ

(5.8)

can be solved for t; denote the solution by t = Mε (s, τ, ξ). Finally, when t varies in the compact support of χ, t + Λε (t) τξ remains in a compact set K independently of ε for (τ, ξ) ∈ Γ. Thus we can perform the change of coordinates (5.8) in the integral and arrive at ∞ −1 χ(t) e −itτ −iΛε (t)ξ dt = χ(Mε (s, τ, ξ)) e −isτ 1 + λε (Mε (s, τ, ξ)) τξ ds. K

0

Integrating m-times by parts, we see that the second integral is a finite sum of terms of the form −k 2 (l) ξ m−1 1 m −isτ χ(j) (Mε ) 1 + λε (Mε ) τξ e ds (5.9) l λε (Mε ) τ τ K

where we wrote Mε = Mε (s, τ, ξ) for short and used the observation that the mth derivative of Mε (s, τ, ξ) with respect to s is a sum of products of powers of (1 + λε (Mε ) τξ )−1 , derivatives of λε , and the factor (ξ/τ )m−1 . We already know that 1 + λε (t) τξ is bounded away from zero by 1 + β0 . Each derivative of λε is globally bounded by some negative power of ε. Further, τ ≥ min 1/β1 , 1/|β0 | |ξ|

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for (τ, ξ) ∈ Γ. Thus ξ/τ is bounded and τ ≥ c(1 + |ξ| + |τ |) on Γ for some constant c > 0 and τ ≥ 1, say. In conclusion, given m ∈ N0 , there is p ≥ 0, C > 0 such that ∞ χ(t) e −itτ −iΛε (t)ξ dt ≤ Cε−p (1 + |ξ| + |τ |)−m 0

for (τ, ξ) ∈ Γ. Since Γ may be chosen as to contain any given sub-cone of the first component of R, we see that every direction in the first component of R is G-regular. A similar argument applies to the second component. Thus the set W1/2 gives an upper bound for the non-regular directions over the point (1, 0). The lower bound. The prospective singular directions are given by D = {(τ, ξ) = (−α, 1) : 0 ≤ α ≤ 1} ∪ {(τ, ξ) = (α, −1) : 0 ≤ α ≤ 1}. We shall consider the first component of D. Given α ∈ (0, 1) and ε ∈ (0, 1], there is a unique tε ∈ [1 − ε, 1 + ε] such that λε (tε ) = α. In fact, we compute from (5.3) that tε = 1 + εμ−1 (α) −1 where μ is the inverse of μ restricted to [−1, 1]. We shall use the fact that the Fourier transform of the distribution χ(t) δ(x − Λε (t)) behaves like ξ −1/2 in the direction (−αξ, ξ) as ξ → ∞, with χ as above. This will follow from [11, Theorem 7.7.5] (method of stationary phase). We write tα − Λε (t) = tλε (tε ) − Λε (t) = tε λε (tε ) − Λε (tε ) + fε (t)

(5.10)

with fε (t) = (t − tε )λε (tε ) + Λε (tε ) − Λε (t). Then

fε (t) = λε (tε ) − λε (t),

fε (t) = − 1ε ρ

t−1 ε

so

fε (tε ) = 0, fε (tε ) = 0, fε (tε ) = − 1ε ρ μ−1 (α) = 0 and fε (t) = 0 except when t = tε . Thus the hypotheses of [11, Theorem 7.7.5] are satisfied and we conclude that there is a constant Cε > 0 such that ∞ 1/2 χ(t) e ifε (t)ξ dt − 2πi/ξfε (tε ) χ(tε ) ≤ Cε ξ −1 l≤2 sup |∂tl χ| (5.11) 0

−1/2 and bε for for ξ > 0. Writing aε for the absolute value of 2πi/fε (tε ) l Cε sup |∂t χ| and observing that the first term on the right-hand side of (5.10) does not depend on t, we infer from inequality (5.11) that ∞ ∞ aε bε (5.12) χ(t) e itαξ−iΛε (t)ξ dt = χ(t) e itfε (t)ξ dt ≥ √ − , ξ ξ 0 0 valid for each ε ∈ (0, 1] and ξ > 0. If (−αξ, ξ) were a G-regular direction for F (χu), inequality (3.2) should hold, in particular, for l = 1. This would mean that there are constants p ≥ 0, C > 0 and ε0 > 0 such that ∞ χ(t) e itαξ−iΛε (t)ξ dt ≤ Cε−p 1 + |ξ| 0


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for all ε ≤ ε0 and all ξ > 0. This clearly contradicts the lower estimate (5.12), fixing ε = ε0 and letting ξ → ∞. Thus the set of non-regular directions contains all (−α, 1) with 0 < α < 1. The directions with α = 0 and α = 1 are also contained in it, since this set is closed. The second part of D is treated in the same way, hence the set W1/2 provides a lower bound for the non-regular directions over the point (1, 0) as well. Proposition 5.4. The G ∞ -wave front set of the solution u ∈ L(Gc ([0, ∞) × R), C) to (5.4) equals WFG ∞ (u) = W0 ∪ W1 ∪ W2 . Proof. By an argument analogous to the one at the beginning of the proof of Proposition 5.3, the non-G ∞ -regular directions above the points (t0 , x0 ) = (1, 0) are described by the sets W0 and W2 . Thus we only have to look at the fiber above the point (1, 0). We will show that W1 ⊂ WFG ∞ (∂t (∂t + ∂x )u); all the more, W1 ⊂ WFG ∞ (u). Observe that the support of ∂t (∂t + ∂x )u consists of the single point (1, 0), so the cut-off is superfluous and we simply calculate Fε (τ, ξ) = −F ∂t (∂t + ∂x )δ(x − Λε (t)) (τ, ξ) 1 0 = − ∂t (∂t + ∂x )δ(x − Λε (t)), e −itτ −ixξ 1+2ε −itτ −iΛε (t)ξ e θ t−1 dt. = τ (τ + ξ) ε 1−2ε

Here we used the fact that, as a distribution, ∂t (∂t + ∂x )δ(x − Λε (t)) vanishes for |t − 1| > ε. The insertion of the cut-off term θ takes care of that, where θ ∈ D(R), θ ≡ 1 on [−1, 1], θ ≡ 0 outside [−2, 2]. Next, we insert the explicit expression t (s−1)/ε (t−1)/ε s ρ(r)dr ds = ε ρ(r)dr ds Λε (t) = 1−ε

−∞

−1

−∞

and change coordinates to see that 2 t s Fε (τ, ξ) = τ (τ + ξ) ε θ(t) e −i(1+εt)τ −iεξ −1 −∞ ρ(r) dr ds dt. −2

Choosing any direction τ = αξ with α ∈ R, α = 0, we have that 2 t s Fε (αξ, ξ) = ξ 2 α(α + 1) ε e −iαξ θ(t) e −iεαξt−iεξ −1 −∞ ρ(r) dr ds dt. −2

Now suppose there is p ≥ 0 such that for all l ∈ N0 supξl |Fε (αξ, ξ)| = O(ε−p ) as ε → 0. ξ>0

2 √ Choosing ξε = 1/ ε, the integral converges to −2 θ(t)dt, so ξε l |Fε (αξε , ξε )| ≈ ε1−(l+2)/2 = ε−l/2

(5.13)

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contradicting (5.13) when l/2 > p, as long as α = −1. Thus all directions τ = αξ with α = 0, α = −1 are not G ∞ -regular. The set of non-regular directions is closed; thus there is no G ∞ -regular direction above (1, 0). We can extract further information on how the wave front set depends on the regularity of the coefficient λ. Recall that a family (rε )ε∈(0,1] is termed a slow scale net if ∀p > 0 : |rε |p = O(ε−1 ) as ε → 0. In place of the definition (5.3) of λε with ε as regularization parameter (i.e., scale 1/ε) we could also choose another scale rε → ∞ and put (t−1)rε ρ(s)ds. λε (t) = −∞

It is straightforward to see that λ defined in this way belongs to G ∞ ([0, ∞) × R) if and only if (rε )ε∈(0,1] is a slow scale net. In fact, one can show [14] that a bounded family of smooth functions defines an element of G ∞ if and only if all its derivatives satisfy local slow scale bounds. Proposition 5.5. If the coefficient λ defined as above belongs to G ∞ ([0, ∞) × R), to then the generalized wave front sets of the solution u ∈ L(Gc ([0, ∞) × R), C) (5.4) satisfy WFG (u) = WFG ∞ (u) = W0 ∪ W1/2 ∪ W2 . Proof. The proof proceeds along the same lines as the one of Proposition 5.3. When we arrive at the expression (5.9), we observe that all factors are bounded by slow scale nets, hence so is their product. In particular, ε−1 gives an asymptotic upper bound, and we conclude that ∞ χ(t) e −itτ −iΛε (t)ξ dt ≤ Cε−1 (1 + |ξ| + |τ |)−m 0

for (τ, ξ) ∈ Γ and all m ∈ N0 . Thus every direction in the set R is G ∞ -regular and so WFG (u) ⊂ WFG ∞ (u) ⊂ W0 ∪ W1/2 ∪ W2 . On the other hand, the argument leading to the contradiction with (5.12) does not depend on the choice of scale. Therefore, the directions (−αξ, ξ) with 0 ≤ α ≤ 1 are not G-regular for F (χu), hence W0 ∪ W1/2 ∪ W2 ⊂ WFG (u) as desired.

Acknowledgments The author gratefully acknowledges support by the Laboratoire Analyse, Optimisation, Contrˆ ole of the Université des Antilles et de la Guyane during the completion of this work. Thanks are due to an anonymous referee whose careful reading helped eliminating some of the author’s lapses.


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References [1] J.F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland, Amsterdam, 1984. [2] J.F. Colombeau, Elementary Introduction to New Generalized Functions, NorthHolland, Amsterdam, 1985. [3] C. Garetto, Topological structures in Colombeau algebras: topological C-modules and duality theory, Acta. Appl. Math. 88 (2005), 81–123. [4] C. Garetto, Topological structures in Colombeau algebras: investigation of the duals of Gc (Ω), G(Ω), and GS (Ω), Monatsh. Math. 146 (2005), 203–226. [5] C. Garetto, Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity, New York J. Math. 12 (2006), 275-318. [6] C. Garetto, Generalized Fourier integral operators on spaces of Colombeau type, in New Developments in Pseudo-Differential Operators, Editors: Luigi Rodino and M.W. Wong, Birkh¨ auser, 2008, 141–188. [7] C. Garetto, T. Gramchev and M. Oberguggenberger, Pseudodifferential operators with generalized symbols and regularity theory, Electron. J. Diff. Eqns. 2005 (116) (2005), 1–43. [8] C. Garetto and G. H¨ ormann, Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities, Proc. Edinb. Math. Soc. 48 (2005), 603–629. [9] C. Garetto and G. H¨ ormann, On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets, Bull. T. CXXXIII Acad. Serbe Sci. Arts, Cl. Sci. Math. Nat., Sci. Math. 31 (2006), 115–136. [10] M. Grosser, M. Kunzinger, M. Oberguggenberger and R. Steinbauer. Geometric Theory of Generalized Functions with Applications to General Relativity, Kluwer Acad. Publ., Dordrecht, 2001. [11] L. H¨ ormander, The Analysis of Linear Partial Differential Operators I, 2nd Ed., Springer-Verlag, Berlin, 1990. [12] G. H¨ ormann, First-order hyperbolic pseudodifferential equations with generalized symbols, J. Math. Anal. Appl. 293 (2004), 40–56. [13] G. H¨ ormann and M.V. de Hoop, Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients, Acta Appl. Math. 67 (2001), 173–224. [14] G. H¨ ormann, M. Oberguggenberger and S. Pilipović, Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients, Trans. Amer. Math. Soc. 358 (2006), 3363–3383. [15] F. Lafon and M. Oberguggenberger, Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case, J. Math. Anal. Appl. 160 (1991), 93–106. [16] M. Nedeljkov, S. Pilipović and D. Scarpalézos, The Linear Theory of Colombeau Generalized Functions, Longman Scientific & Technical, Harlow, 1998.

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[17] M. Oberguggenberger, Hyperbolic systems with discontinuous coefficients: Generalized solutions and a transmission problem in acoustics, J. Math. Anal. Appl. 142 (1989), 452–467. [18] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Longman Scientific & Technical, Harlow, 1992. [19] M.E. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey 1981. Michael Oberguggenberger Arbeitsbereich f¨ ur Technische Mathematik Universit¨ at Innsbruck Technikerstraße 13 A-6020 Innsbruck, Austria e-mail: [email protected]


Generalized Fourier Integral Operators on Spaces of Colombeau Type Claudia Garetto Abstract. Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudo-differential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distributional data. Mathematics Subject Classification (2000). Primary 35S30; Secondary 46F30. Keywords. Fourier integral operators, Colombeau algebras.

1. Introduction This work is part of a program that aims to solve linear partial differential equations with non-smooth coefficients and highly singular data and investigate the qualitative properties of the solutions. A well-established theory with powerful analytic methods is available in the case of operators with (relatively) smooth coefficients [21], but cannot be applied to many models from physics which involve non-smooth variations of the physical parameters. These models require indeed partial differential operators where the smoothness assumption on the coefficients is dropped. Furthermore, in case of nonlinear operations (cf. [25, 29, 35]), the theory of distribution does not provide a general framework in which solutions exist. An alternative framework is provided by the theory of Colombeau algebras of generalized functions [4, 19, 35]. We recall that the space of distributions D (Ω) is embedded via convolution with a mollifier in the Colombeau algebra G(Ω) of generalized functions on Ω and interpreting the non-smooth coefficients and data as elements of the Colombeau algebra, existence and uniqueness has been established This work was completed with the support of FWF (Austria), grants T305-N13 and Y237-N13.

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for many classes of equations by now [1, 2, 3, 5, 23, 27, 30, 33, 35, 37, 38, 40]. In order to study the regularity of solutions, microlocal techniques have to be introduced into this setting, in particular, pseudo-differential operators with generalized amplitudes and generalized wave front sets. This has been done in the papers [15, 16, 17, 22, 24, 26, 28, 32, 39], with a special attention for elliptic equations and hypoellipticity. The interest for hyperbolic equations, regularity of solutions and inverse problems (determining the non-smooth coefficients from the data is an important problem in geophysics [8]), leads in the case of differential operators with Colombeau coefficients, to a theory of Fourier integral operators with generalized amplitudes and generalized phase functions. This has been initiated in [18] and has provided some first results on propagation of singularities in the dual of the Colombeau algebra Gc (Ω). We recall that within the Colombeau L(Gc (Ω), C) algebra G(Ω), regularity theory is based on the subalgebra G ∞ (Ω) of regular generalized functions, whose intersection with D (Ω) coincides with C ∞ (Ω). Since two different regularity theories coexist in the G ∞ (Ω) ⊆ G(Ω) ⊆ L(Gc (Ω), C), dual: one based on G(Ω) and one based on G ∞ (Ω). This work can be considered as a compendium of [18], in the sense that collects (without proof) the main results achieved in [18] and studies the composition between a generalized Fourier integral operator and a generalized pseudodifferential operator in addition. We can now describe the contents in more detail. Section 2 provides the needed background of Colombeau theory. In particular, topological concepts, generalized symbols and the definition of G- and G ∞ -wave front set are recalled. In Subsection 2.5 we elaborate and state in full generality the notion of asymptotic expansion of a generalized symbol introduced for the first time in [15] and we prove a new and technically useful characterization. Section 3 develops the foundations for generalized Fourier integral operators: oscillatory integrals with generalized phase functions and amplitudes. They are then supplemented by an additional parameter in Section 4, leading to the notion of a Fourier integral operator with generalized amplitude and phase function. We study the mapping properties of such opera and we present tors on Colombeau algebras, the extension to the dual L(Gc (Ω), C) suitable assumptions on phase function and amplitude which lead to G ∞ -mapping properties. The core of the work is Section 5, where, by making use of some technical preliminaries, we study in Theorem 5.10 the composition a(x, D)Fω (b) of a generalized pseudo-differential operator a(x, D) with a generalized Fourier integral operator of the form Fω (b)(u)(x) = eiω(x,η) b(x, η)3 u(η) d−η. Rn

The final Section 6 collects the first results of microlocal analysis for generalized Fourier integral operators obtained in [18, Section 4]. A deeper investigation of the microlocal properties of generalized Fourier integral operators is current topic of research.

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2. Basic notions: Colombeau and duality theory This section gives some background of Colombeau and duality theory for the techniques used in the sequel of the current work. As main sources we refer to [12, 13, 15, 16, 19]. 2.1. Nets of complex numbers Before dealing with the major points of the Colombeau construction we begin by recalling some definitions concerning elements of C(0,1] . A net (uε )ε in C(0,1] is said to be strictly nonzero if there exist r > 0 and η ∈ (0, 1] such that |uε | ≥ εr for all ε ∈ (0, η]. The regularity issues discussed in Sections 3 and 4 will make use of the following concept of slow scale net (s.s.n). A slow scale net is a net (rε )ε ∈ C(0,1] such that |rε |q ≤ cq ε−1 . ∀q ≥ 0 ∃cq > 0 ∀ε ∈ (0, 1] Throughout this paper we will always consider slow scale nets (rε )ε of positive real numbers with inf ε∈(0,1] rε = 0. A net (uε )ε in C(0,1] is said to be slow scale-strictly nonzero is there exist a slow scale net (sε )ε and η ∈ (0, 1] such that |uε | ≥ 1/sε for all ε ∈ (0, η]. 2.2. C-modules of generalized functions based on a locally convex topological vector space E The most common algebras of generalized functions of Colombeau type as well as the spaces of generalized symbols we deal with are introduced and investigated under a topological point of view by referring to the following models. Let E be a locally convex topological vector space topologized through the family of seminorms {pi }i∈I . The elements of ME := {(uε )ε ∈ E (0,1] : ∀i ∈ I ∃N ∈ N

pi (uε ) = O(ε−N ) as ε → 0},

(0,1] Msc : ∀i ∈ I ∃(ωε )ε s.s.n. pi (uε ) = O(ωε ) as ε → 0}, E := {(uε )ε ∈ E (0,1] M∞ : ∃N ∈ N ∀i ∈ I E := {(uε )ε ∈ E

NE := {(uε )ε ∈ E (0,1] : ∀i ∈ I ∀q ∈ N

pi (uε ) = O(ε−N ) as ε → 0}, pi (uε ) = O(εq ) as ε → 0},

are called E-moderate, E-moderate of slow scale type, E-regular and E-negligible, respectively. We define the space of generalized functions based on E as the factor space GE := ME /NE . := EM /N , is obtained The ring of complex generalized numbers, denoted by C by taking E = C. C is not a field since by Theorem 1.2.38 in [19] only the elements which are strictly nonzero (i.e., the elements which have a representative strictly nonzero) are invertible and vice versa. Note that all the representatives of u ∈ C are strictly nonzero once we know that there exists at least one which is strictly nonzero. When u has a representative which is slow scale-strictly nonzero we say that it is slow scale-invertible.

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For any locally convex topological vector space E the space GE has the strucsc := Msc The C-module GE ture of a C-module. E /NE of generalized functions of slow ∞ scale type and the C-module GE := M∞ /N E of regular generalized functions are E subrings of GE with more refined assumptions of moderateness at the level of representatives. We use the notation u = [(uε )ε ] for the class u of (uε )ε in GE . This is the usual way we adopt to denote an equivalence class. The family of seminorms {pi }i∈I on E determines a locally convex C-linear topology on GE (see [12, Definition 1.6]) by means of the valuations vpi ([(uε )ε ]) := vpi ((uε )ε ) := sup{b ∈ R :

pi (uε ) = O(εb ) as ε → 0}

and the corresponding ultra-pseudo-seminorms {Pi }i∈I , where Pi (u) = e−vpi (u) . For the sake of brevity we omit to report definitions and properties of valuations and ultra-pseudo-seminorms in the abstract context of C-modules. Such a theoretical presentation can be found in [12, Subsections 1.1, 1.2]. We recall that on the valuation and the ultra-pseudo-norm obtained through the absolute value C ∞ in C are denoted by vC and | · |e respectively. Concerning the space GE of regular generalized functions based on E the moderateness properties of M∞ E allows to define the valuation ∞ ((uε )ε ) := sup{b ∈ R : ∀i ∈ I vE

pi (uε ) = O(εb ) as ε → 0} ∞

∞ ∞ and leads to the ultra-pseudo-norm PE (u) := e−vE (u) . which extends to GE The Colombeau algebra G(Ω) = EM (Ω)/N (Ω) can be obtained as a C-module of GE -type by choosing E = E(Ω). Topologized through the family of seminorms pK,i (f ) = supx∈K,|α|≤i |∂ α f (x)| where K Ω, the space E(Ω) induces on G(Ω) a metrizable and complete locally convex C-linear topology which is determined by

the ultra-pseudo-seminorms PK,i (u) = e−vpK,i (u) . From a structural point of view Ω → G(Ω) is a fine sheaf of differential algebras on Rn . The Colombeau algebra Gc (Ω) of generalized functions with compact support is topologized by means of a strict inductive limit procedure. More precisely, setting GK (Ω) := {u ∈ Gc (Ω) : supp u ⊆ K} for K Ω, Gc (Ω) is the strict inductive limit of the sequence of locally convex topological C-modules (GKn (Ω))n∈N , where (Kn )n∈N is an exhausting sequence of compact subsets of Ω such that Kn ⊆ Kn+1 . We endow GK (Ω) with the topology induced by GDK (Ω) where K is a compact subset containing K in its interior. For more details concerning the topological structure of Gc (Ω) see [13, Example 3.7]. Regularity theory in the Colombeau context as initiated in [35] is based on the subalgebra G ∞ (Ω) of all elements u of G(Ω) having a representative (uε )ε belonging to the set ∞ EM (Ω) := {(uε )ε ∈ E[Ω] : ∀K Ω ∃N ∈ N ∀α ∈ Nn

sup |∂ α uε (x)| = O(ε−N ) as ε → 0}. x∈K


141

G ∞ (Ω) can be seen as the intersection ∩KΩ G ∞ (K), where G ∞ (K) is the space of all u ∈ G(Ω) having a representative (uε )ε satisfying the condition: ∃N ∈ N ∀α ∈ Nn , supx∈K |∂ α uε (x)| = O(ε−N ). The ultra-pseudo-seminorms PG ∞ (K) (u) := e−vG∞ (K) , where vG ∞ (K) := sup{b ∈ R : ∀α ∈ Nn

sup |∂ α uε (x)| = O(εb )} x∈K

equip G ∞ (Ω) with the topological structure of a Fréchet C-module. ∞ ∞ ∞ (Ω) := Finally, let us consider the algebra Gc (Ω) := G (Ω) ∩ Gc (Ω). On GK ∞ {u ∈ G (Ω) : supp u ⊆ K} with K Ω, we define the ultra-pseudo-norm ∞ −vK (u) ∞ ∞ ∞ (Ω) (u) = e PGK where vK (u) := vD (u) and K is any compact set conK (Ω) taining K in its interior. At this point, given an exhausting sequence (Kn )n of com∞ pact subsets of Ω, the strict inductive limit procedure equips Gc∞ (Ω) = ∪n GK (Ω) n with a complete and separated locally convex C-linear topology (see [13, Example 3.13]). 2.3. Topological dual of a Colombeau algebra A duality theory for C-modules had been developed in [12] in the framework of topological and locally convex topological C-modules. Starting from an investi gation of L(G, C), the C-module of all C-linear and continuous functionals on G, it provides the theoretical tools for dealing with the topological duals of the and L(Gc (Ω), C) Colombeau algebras Gc (Ω) and G(Ω). In the paper L(G(Ω), C are endowed with the topology of uniform convergence on bounded subsets. This is determined by the ultra-pseudo-seminorms PB ◦ (T ) = sup |T (u)|e , u∈B

where B is varying in the family of all bounded subsets of G(Ω) and Gc (Ω) respectively. For general results concerning the relation between boundedness and ultra-pseudo-seminorms in the context of locally convex topological C-modules we refer to [13, Section 1]. For the choice of topologies illustrated in this section Theorem 3.1 in [13] shows the following chains of continuous embeddings: G ∞ (Ω) ⊆ G(Ω) ⊆ L(Gc (Ω), C),

(2.1)

Gc∞ (Ω) ⊆ Gc (Ω) ⊆ L(G(Ω), C), ⊆ L(Gc (Ω), C). L(G(Ω), C)

(2.2)

(2.3) In (2.1) and (2.2) the inclusion in the dual is given via integration u → v → u(x)v(x)dx (for definitions and properties of the integral of a Colombeau Ω generalized functions see [19]) while the embedding in (2.3) is determined by the is a sheaf we can define the support inclusion Gc (Ω) ⊆ G(Ω). Since Ω → L(Gc (Ω), C) of a functional T (denoted by supp T ). In analogy with distribution theory, from can be identified with the set of Theorem 1.2 in [13] we have that L(G(Ω), C) having compact support. functionals in L(Gc (Ω), C)

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By (2.1) it is meaningful to measure the regularity of a functional in the with respect to the algebras G(Ω) and G ∞ (Ω). We define the dual L(Gc (Ω), C) G-singular support of T (singsuppG T ) as the complement of the set of all points x ∈ Ω such that the restriction of T to some open neighborhood V of x belongs to G(V ). Analogously replacing G with G ∞ we introduce the notion of G ∞ -singular support of T denoted by singsuppG ∞ T . This investigation of regularity is connected with the notions of generalized wave front sets considered in Subsection 2.8 and and L(G(Ω), C) which have a will be focused on the functionals in L(Gc (Ω), C) is basic if there exists a “basic” structure. In detail, we say that T ∈ L(Gc (Ω), C) net (Tε )ε ∈ D (Ω)(0,1] fulfilling the following condition: for all K Ω there exist j ∈ N, c > 0, N ∈ N and η ∈ (0, 1] such that ∀f ∈ DK (Ω) ∀ε ∈ (0, η]

|Tε (f )| ≤ cε−N

sup

|∂ α f (x)|

x∈K,|α|≤j

and T u = [(Tε uε )ε ] for all u ∈ Gc (Ω). is said to be basic if there exists In the same way a functional T ∈ L(G(Ω), C) (0,1] a net (Tε )ε ∈ E (Ω) such that there exist K Ω, j ∈ N, c > 0, N ∈ N and η ∈ (0, 1] with the property ∀f ∈ C ∞ (Ω) ∀ε ∈ (0, η]

|Tε (f )| ≤ cε−N

sup

|∂ α f (x)|

x∈K,|α|≤j

and T u = [(Tε uε )ε ] for all u ∈ G(Ω). and Lb (G(Ω), C) of basic functionals are C-linear Clearly the sets Lb (Gc (Ω), C) and L(G(Ω), C) respectively. In addition if T is a basic subspaces of L(Gc (Ω), C) is basic. We functional in L(Gc (Ω), C) and u ∈ Gc (Ω) then uT ∈ L(G(Ω), C) recall that nets (Tε )ε which define basic maps as above were already considered in [9, 10] with slightly more general notions of moderateness and different choices of notations and language. 2.4. Generalized symbols For the convenience of the reader we recall a few basic notions concerning the sets of symbols employed in the course of this work. More details can be found in [15, 16] where a theory of generalized pseudo-differential operators acting on Colombeau algebras is developed. m Definitions. Let Ω be an open subset of Rn , m ∈ R and ρ, δ ∈ [0, 1]. Sρ,δ (Ω × Rp ) denotes the set of symbols of order m and type (ρ, δ) as introduced by H¨ ormander in [20]. The subscript (ρ, δ) is omitted when ρ = 1 and δ = 0. If V is an open m (V ) as the set of all a ∈ C ∞ (V ) such that for all conic set of Ω × Rp we define Sρ,δ K V, sup ξ−m+ρ|α|−δ|β| |∂ξα ∂xβ a(x, ξ)| < ∞, (x,ξ)∈K c

1 where K := {(x, tξ) : (x, ξ) ∈ K t ≥ 1}. We also make use of the space Shg (Ω × p 1 p R \ 0) of all a ∈ S (Ω × R \ 0) homogeneous of degree 1 in ξ. Note that the c


143

assumption of homogeneity allows to state the defining conditions above in terms of the seminorms sup |ξ|−1+α |∂ξα ∂xβ a(x, ξ)| x∈K,ξ∈Rp \0

where K is any compact subset of Ω. m The space of generalized symbols Sρ,δ (Ω × Rp ) is the C-module of GE -type m p obtained by taking E = Sρ,δ (Ω × R ) equipped with the family of seminorms (m)

|a|ρ,δ,K,j =

sup

sup |∂ξα ∂xβ a(x, ξ)|ξ−m+ρ|α|−δ|β| ,

x∈K,ξ∈Rn |α+β|≤j

(m)

K Ω, j ∈ N. (m)

The valuation corresponding to |·|ρ,δ,K,j gives the ultra-pseudo-seminorm Pρ,δ,K,j . (m) Sm (Ω × Rp ) topologized through the family {P }KΩ,j∈N of ultra-pseudoρ,δ

ρ,δ,K,j

m (Ω×Rp ) we use the notation seminorms is a Fréchet C-module. In analogy with Sρ,δ Sm (V ) for the C-module GS m (V ) . ρ,δ

ρ,δ

m Sρ,δ (Ωx ×Rpξ ) has the structure of a sheaf with respect to Ω. So it is meaningful to talk of the support with respect to x of a generalized symbol a (suppx a). m (Ω × Rp ) (cone supp a) as the comWe define the conic support of a ∈ Sρ,δ plement of the set of points (x0 , ξ0 ) ∈ Ω × Rp such that there exists a relatively compact open neighborhood U of x0 , a conic open neighborhood Γ of ξ0 and a representative (aε )ε of a satisfying the condition

∀α ∈ Np ∀β ∈ Nn ∀q ∈ N

sup ξ−m+ρ|α|−δ|β| |∂ξα ∂xβ aε (x, ξ)| = O(εq ) as ε → 0.

x∈U,ξ∈Γ

(2.4) By definition cone supp a is a closed conic subset of Ω×Rp . The generalized symbol a is 0 on Ω \ πx (cone supp a). Slow scale symbols. In the paper the classes of the factor space GSscm (Ω×Rp ) are ρ,δ called generalized symbols of slow scale type. For simplicity we introduce the nom,sc m m tation Sρ,δ (Ω × Rp ). Substituting Sρ,δ (Ω × Rp ) with Sρ,δ (V ) we obtain the set m,sc sc S (V ) := G m of slow scale symbols on the open set V ⊆ Ω × (Rp \ 0). ρ,δ

Sρ,δ (V )

Generalized symbols of order −∞. Different notions of regularity are related to the sets S−∞ (Ω × Rp ) and S−∞,sc (Ω × Rp ) of generalized symbols of order −∞. The space S−∞ (Ω × Rp ) of generalized symbols of order −∞ is defined as the Cmodule GS −∞ (Ω×Rp ) . Its elements are equivalence classes a whose representatives (aε )ε have the property |aε |K,j = O(ε−N ) as ε → 0, where N depends on the order m of the symbol, on the order j of the derivatives and on the compact set K ⊆ Ω. S−∞,sc (Ω × Rp ) is defined by substituting O(ε−N ) with O(λε ) in the previous estimate, where (λε )ε is a slow scale net depending as above on the order m of the symbol, on the order j of the derivatives and on the compact set K ⊆ Ω. It (m)

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follows that (aε )ε is G ∞ -regular, in the sense that |aε |K,j = O(ε−1 ) (m)

as ε → 0 for all m, j and K Ω. Generalized microsupports. The G- and G ∞ -regularity of generalized symbols on Ω × Rn is measured in conical neighborhoods by means of the following notions of microsupports. l (Ω × Rn ) and (x0 , ξ0 ) ∈ T ∗ (Ω) \ 0. The symbol a is G-smoothing Let a ∈ Sρ,δ at (x0 , ξ0 ) if there exist a representative (aε )ε of a, a relatively compact open neighborhood U of x0 and a conic neighborhood Γ ⊆ Rn \ 0 of ξ0 such that ∀m ∈ R ∀α, β ∈ Nn ∃N ∈ N ∃c > 0 ∃η ∈ (0, 1] ∀(x, ξ) ∈ U × Γ ∀ε ∈ (0, η] |∂ξα ∂xβ aε (x, ξ)| ≤ cξm ε−N . (2.5) The symbol a is G ∞ -smoothing at (x0 , ξ0 ) if there exist a representative (aε )ε of a, a relatively compact open neighborhood U of x0 , a conic neighborhood Γ ⊆ Rn \ 0 of ξ0 and a natural number N ∈ N such that ∀m ∈ R ∀α, β ∈ Nn ∃c > 0 ∃η ∈ (0, 1] ∀(x, ξ) ∈ U × Γ ∀ε ∈ (0, η] |∂ξα ∂xβ aε (x, ξ)| ≤ cξm ε−N . (2.6) We define the G-microsupport of a, denoted by μ suppG (a), as the complement in T ∗ (Ω) \ 0 of the set of points (x0 , ξ0 ) where a is G-smoothing and the G ∞ microsupport of a, denoted by μ suppG ∞ (a), as the complement in T ∗ (Ω) \ 0 of the set of points (x0 , ξ0 ) where a is G ∞ -smoothing. Continuity Results. By simple reasoning at the level of representatives one proves that the usual operations between generalized symbols, as product and derivation, are continuous. In particular the C-bilinear map m m (Ω × Rp ) → Sρ,δ (Ω × Rp ) : (u, a) → a(y, ξ)u(y) Gc (Ω) × Sρ,δ

(2.7)

l is continuous. If l < −p each b ∈ Sρ,δ (Ω×Rp ) can be integrated on K ×Rp , K Ω, by setting

b(y, ξ) dy dξ := bε (y, ξ) dy dξ . K×Rp

K×Rp

ε

Moreover if suppy b Ω we define the integral of b on Ω × Rp as b(y, ξ) dy dξ := b(y, ξ) dy dξ, Ω×Rp

K×Rp

where K is any compact set containing suppy b in its interior. Integration defines a continuous C-linear functional on this space of generalized symbols with compact support in y as it is proven in [18, Proposition 1.1, Remark 1.2].


145

m,sc m 2.5. Asymptotic expansions in Sρ,δ (Ω × Rp ) and Sρ,δ (Ω × Rp ) In this subsection we elaborate and state in full generality the notion of asymptotic expansion of a generalized symbol introduced for the first time in [15]. We also provide a technical result which will be useful in Section 5. We begin by working on moderate nets of symbols and we recall that a net (Cε )ε ∈ C(0,1] is said to be of slow scale type if there exists a slow scale net (ωε )ε such that |Cε | = O(ωε ).

Definition 2.1. Let {mj }j∈N be sequences of real numbers with mj −∞, m0 = m. (i) Let {(aj, ) }j∈N be a sequence of elements (aj, ) ∈ MS mj (Ω×Rp ) . We say that ρ,δ ∞ the formal series j=0 (aj, ) is the asymptotic expansion of (a ) ∈ E[Ω×Rn ], (a ) ∼ j (aj, ) for short, iff for all r ≥ 1 r−1 mr a − aj, ∈ MSρ,δ (Ω×Rp ) . j=0

(ii) Let {(aj, ) }j∈N be a sequence of elements (aj, ) ∈ Mscmj . We say that Sρ,δ (Ω×Rp ) ∞ the formal series j=0 (aj, ) is the asymptotic expansion of (a ) ∈ E[Ω×Rn ], (a ) ∼sc j (aj, ) for short, iff for all r ≥ 1 r−1 aj, ∈ Msc a − S mr (Ω×Rp ) . j=0

ρ,δ

Theorem 2.2. (i) Let {(aj, ) }j∈N be a sequence of elements (aj, ) ∈ MS mj (Ω×Rp ) with mj ρ,δ

m (Ω×Rp ) such that (a ) ∼ −∞ and m0 = m. Then, there exists (aε )ε ∈ MSρ,δ j (aj, ) . Moreover, if (a ) ∼ j (aj, ) then (aε − aε )ε ∈ MS −∞ (Ω×Rp ) . with mj (ii) Let {(aj, ) }j∈N be a sequence of elements (aj, ) ∈ Mscmj p

Sρ,δ (Ω×R )

−∞ and m0 = m. Then, there exists (aε )ε ∈ Msc m (Ω×Rp ) such that (a ) ∼sc Sρ,δ sc j (aj, ) . Moreover, if (a ) ∼sc j (aj, ) then (aε − aε )ε ∈ MS −∞ (Ω×Rp ) . Proof. The proof follows the classical line of arguments, but we will have to keep track of the -dependence carefully. We consider a sequence of relatively compact open sets {Vl } contained in Ω, such that for all l ∈ N, Vl ⊂ Kl = Vl ⊂ Vl+1 and 4 ∞ p l∈N Vl = Ω. Let ψ ∈ C (R ), 0 ≤ ψ(ξ) ≤ 1, such that ψ(ξ) = 0 for |ξ| ≤ 1 and ψ(ξ) = 1 for |ξ| ≥ 2. (i) We introduce bj, (x, ξ) = ψ(λj,ε ξ)aj, (x, ξ), where λj,ε will be positive constants with λj+1,ε < λj,ε < 1, λj,ε → 0 if j → ∞. We can define bj, (x, ξ). (2.8) a (x, ξ) = j∈N

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This sum is locally finite and therefore (aε )ε ∈ E[Ω × Rp ]. We observe that |α| ∂ α (ψ(λj,ε ξ)) = ∂ α ψ(λj,ε ξ)λj,ε , supp (∂ α ψ(λj,ε ξ)) ⊆ {ξ : 1/λj,ε ≤ |ξ| ≤ 2/λj,ε }, and that 1/λj,ε ≤ |ξ| ≤ 2/λj,ε implies λj,ε ≤ 2/|ξ| ≤ 4/(1 + |ξ|). We first estimate bj,ε . Fixing K Ω and α ∈ Np , β ∈ Nn , we obtain for j ∈ N, ∈ (0, 1], x ∈ K, ξ ∈ Rp , α |α−γ| (mj ) α β |∂ξ ∂x bj, (x, ξ)| ≤ |∂ α−γ ψ(λj,ε ξ)aj,ε |ρ,δ,K,γ,β ξmj −ρ|γ|+δ|β| λ γ j,ε γ≤α (mj ) ≤ c(ψ, γ)4|α−γ|ξ−|α−γ| |aj,ε |ρ,δ,K,γ,β ξmj −ρ|γ|+δ|β| γ≤α

≤ Cj,α,β,K,ε ξmj −ρ|α|+δ|β| , where Cj,α,β,K,ε :=

j c(ψ, γ)4|α−γ| |aj,ε |ρ,δ,K,γ,β .

(m )

γ≤α

Since (Cj,α,β,K,ε )ε is a moderate net of positive numbers, we have that (bj,ε )ε ∈ MS mj (Ω×Rp ) . At this point we choose λj,ε such that for |α + β| ≤ j, l ≤ j ρ,δ

Cj,α,β,Kl ,ε λj,ε ≤ 2−j .

(2.9)

m (Ω×Rp ) . Since Our aim is to prove that a (x, ξ) defined in (2.8) belongs to MSρ,δ there exists Nj ∈ N and ηj ∈ (0, 1] such that

Cj,α,β,Kl ,ε ≤ ε−Nj for l ≤ j and |α + β| ≤ j, we take λj,ε = 2−j εNj on the interval (0, ηj ]. We observe that ∀K Ω, ∃l ∈ N : K ⊂ Vl ⊂ Kl , ∀α0 ∈ Np , ∀β0 ∈ Nn , ∃j0 ∈ N, j0 ≥ l : |α0 | + |β0 | ≤ j0 ,

mj0 + 1 ≤ m, (2.10)

and we write (a ) as the sum of the following two terms: j 0 −1

bj, (x, ξ) +

j=0

+∞

bj, (x, ξ) = f (x, ξ) + s (x, ξ).

j=j0

For x ∈ K, we have that |∂ξα0 ∂xβ0 f (x, ξ)|

≤

j 0 −1

j |bj,ε |ρ,δ,K,α ξmj −ρ|α0 |+δ|β0 | 0 ,β0

(m )

j=0

≤

j 0 −1 j=0

(mj ) |bj,ε |ρ,δ,K,α 0 ,β0

ξm−ρ|α0 |+δ|β0 | ,

Generalized Fourier Integral Operators where

j 0 −1

147

(m )

j |bj,ε |ρ,δ,K,α 0 ,β0

∈ EM . ε

j=0

We now turn to s (x, ξ). From the estimates on bj,ε and (2.9), we get for x ∈ K and ∈ (0, 1], |∂ξα0 ∂xβ0 s (x, ξ)| ≤

+∞

Cj,α0 ,β0 ,Kl ξmj −ρ|α0 |+δ|β0 |

j=j0

≤

+∞

−1 2−j λ−1 ξmj +1−ρ|α0 |+δ|β0 | ≤ j,ε ξ

j=j0

+∞

−1 2−j λ−1 ξm−ρ|α0 |+δ|β0 | . j,ε ξ

j=j0

Since ψ(ξ) is identically equal to 0 for |ξ| ≤ 1, we can assume in our estimates that ξ−1 ≤ λj,ε , and therefore from (2.10), we conclude that |∂ξα0 ∂xβ0 s (x, ξ)| ≤ 2ξm−ρ|α0 |+δ|β0 | , for all x ∈ K, ξ ∈ Rp and ε ∈ (0, 1]. In order to prove that (a ) ∼ j (aj, ) we fix r ≥ 1 and we write a (x, ξ) −

r−1

aj, (x, ξ) =

j=0

r−1

(ψ(λj,ε ξ) − 1)aj, (x, ξ) +

j=0

+∞

ψ(λj,ε ξ)aj, (x, ξ)

j=r

= g (x, ξ) + t (x, ξ). Recall that ψ ∈ C ∞ (Rp ) was chosen such that ψ − 1 ∈ Cc∞ (Rp ) and supp(ψ − 1) ⊆ {ξ : |ξ| ≤ 2}. Thus, for 0 ≤ j ≤ r − 1, supp(ψ(λj,ε ξ) − 1) ⊆ {ξ : |λj,ε ξ| ≤ 2} ⊆ {ξ : |ξ| ≤ 2λ−1 r−1,ε }. As a consequence, for fixed K Ω and for all ∈ (0, 1], |∂ξα ∂xβ g (x, ξ)|

r−1 α |α | (mj ) mj −mr +ρ|α | ≤ |aj,ε |ρ,δ,K,α−α λj,ε c(ψ, α )2λ−1 ,β r−1,ε α j=0 α ≤α

· ξmr −ρ|α|+δ|β| , (m )

j where, from our assumptions on (aj,ε )ε and (λj,ε )ε , the nets (|aj,ε |ρ,δ,K,α−α ,β )ε −1 mj −mr +ρ|α | and (2λr−1,ε )ε are both moderate. Repeating the same arguments mr used in the construction of (a ) we have that (t ) belongs to MSρ,δ (Ω×Rp ) . It is clear that (aε )ε is uniquely determined by j (aj,ε )ε modulo MS −∞ (Ω×Rp ) .

(ii) In the slow scale case one easily sees that (bj,ε )ε ∈ Mscmj

Sρ,δ (Ω×Rp )

since there exists a slow scale net ωj (ε) and ηj ∈ (0, 1] such that Cj,α,β,Kl ,ε ≤ ωj (ε)

. Moreover,

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for l ≤ j and |α + β| ≤ j, we can take λj,ε = 2−j ωj−1 (ε) on the interval (0, ηj ]. It follows that (aε )ε ∈ Msc m (Ω×Rp ) and that both the nets (gε )ε and (tε )ε belong to Sρ,δ . Msc mr S (Ω×Rp ) ρ,δ

Proposition 2.3. (i) Let {(aj, ) }j∈N be a sequence of elements (aj, ) ∈ MS mj (Ω×Rp ) with mj ρ,δ

−∞ and m0 = m. Let (aε )ε ∈ E[Ω × Rp ] such that for all K Ω, for all α, β there exists μ ∈ R and (Cε )ε ∈ EM such that |∂ξα ∂xβ aε (x, ξ)| ≤ Cε ξμ ,

(2.11)

for all x ∈ K, ξ ∈ Rp , ε ∈ (0, 1]. Furthermore, assume that for any r ≥ 1 and K Ω there exists μr = μr (K) and (Cr,ε )ε = (Cr,ε (K))ε ∈ EM such that μr → −∞ as r → +∞ and r−1 aε (x, ξ) − aj,ε (x, ξ) ≤ Cr,ε ξμr (2.12) j=0

for all x ∈ K, ξ ∈ R , ε ∈ (0, 1]. Then, (aε )ε ∼ j (aj, ) . , the nets (Cε )ε and (Cr,ε )ε of slow scale (ii) (i) holds with (aj, ) ∈ Mscmj Sρ,δ (Ω×Rp ) type and (aε )ε ∼sc j (aj, ) in the sense of Definition 2.1(ii). p

The proof of Proposition 2.3 requires the following lemma. Lemma 2.4. Let K1 and K2 be two compact sets in Rp such that K1 ⊂ Int K2 . Then there exists a constant C > 0 such that for any smooth function f on a neighborhood of K2 , the following estimate holds: 2 α α sup |D f (x)| ≤ C sup |f (x)| sup |f (x)| + sup |D f (x)| . x∈K1

|α|=1

x∈K2

x∈K2

x∈K2

|α|=2

Proof of Proposition 2.3. (i) ByTheorem 2.2 we know that there exists (bε )ε ∈ m (Ω×Rp ) such that (bε )ε ∼ MSρ,δ j (aj,ε )ε .We consider the difference dε = aε − bε . From (2.11) and the moderateness of (bε )ε we have that for all α, β and K Ω there exist (Cε )ε and μ such that

|∂ξα ∂xβ dε (x, ξ)| ≤ Cε ξμ ,

(2.13)

for all x ∈ K, ξ ∈ Rp and ε ∈ (0, 1]. Combining (bε )ε ∼ j (aj,ε )ε with (2.12) we obtain that for all r > 0 and for all K Ω there exists (Cr,ε (K))ε ∈ EM such that |dε (x, ξ)| ≤ Cr,ε (K)ξ−r ,

x ∈ K, ξ ∈ Rp , ε ∈ (0, 1].

Set dξ,ε (x, θ) = dε (x, ξ + θ). Then, ∂θα ∂xβ dξ,ε (x, θ)|θ=0 = ∂ξα ∂xβ d(x, ξ), and applying Lemma 2.4 with K1 = K ×0 and K2 = K ×{|θ| ≤ 1}, where K ⊂ IntK ⊂ K Ω,

Generalized Fourier Integral Operators we obtain sup x∈K

2 |∂ξα ∂xβ dε (x, ξ)| ≤ C

|α+β|=1

·

sup x∈K ,|θ|≤1

|dε (x, ξ + θ)| +

−r

≤ CCr,ε (K ) sup ξ + θ |θ|≤1

(K)ξ−r , ≤ Cr,ε

sup x∈K ,|θ|≤1

sup x∈K ,|θ|≤1

149

|dε (x, ξ + θ)|·

|∂ξα ∂xβ dε (x, ξ

+ θ)|

|α+β|=2 Cr,ε (K ) sup ξ + θ−r + Cε (K ) sup ξ + θμ (K,2) |θ|≤1

|θ|≤1

(2.14)

Cr,ε (K)

where ∈ EM . By induction one can prove that for all r > 0, for all K Ω and for all α ∈ Np , β ∈ Nn , there exists a moderate net (cε )ε such that the estimate |∂ξα ∂xβ dε (x, ξ)| ≤ cε ξ−r is valid for all x ∈ K, ξ∈ Rp and ε ∈ (0, 1]. This means that (dε )ε ∈ MS −∞ (Ω×Rp ) and therefore (aε )ε ∼ j (aj,ε )ε . (ii) It is clear that when we work with nets of slow scale type then (dε )ε ∈ and (a ) ∼ Msc −∞ p ε ε sc j (aj, ) . S (Ω×R ) m (Ω×Rp \0) Remark 2.5. Proposition 2.3 can be stated for nets of symbols in MSρ,δ sc and MS m (Ω×Rp \0) . The proof make use of (2.13) when |ξ| ≤ 1 and (2.14) when

|ξ| > 1.

ρ,δ

Definition 2.6. Let {mj }j∈N with mj −∞ and m0 = m. m (i) Let {aj }j∈N be a sequence of symbols aj ∈ Sρ,δj (Ω × Rp ). We say that the m formal series j aj is the asymptotic expansion of a ∈ Sρ,δ (Ω × Rp ), a ∼ (a ) of a and, for every j, j aj for short, iff there exist a representative representatives (aj, ) of aj , such that (a ) ∼ j (aj, ) . m ,sc (ii) Let {aj }j∈N be a sequence of symbols aj ∈ Sρ,δj (Ω × Rp ). We say that m,sc (Ω × Rp ), the formal series j aj is the asymptotic expansion of a ∈ Sρ,δ a ∼ j aj for short, iff there exist a representative (a ) of a and, for every j, representatives (aj, ) of aj , such that (a ) ∼sc j (aj, ) . 2.6. Generalized pseudo-differential operators m Let Ω be an open subset of Rn and a ∈ Sρ,δ (Ω × Rn ). The generalized oscillatory integral (see [15]) ei(x−y)ξ a(x, ξ)u(y) dy d−ξ := ei(x−y)ξ aε (x, ξ)uε (y) dy d−ξ + N (Ω), Ω×Rn

Ω×Rn

ε

defines the action of the pseudo-differential operator a(x, D) with generalized symm bol a ∈ Sρ,δ (Ω × Rn ) on u ∈ Gc (Ω). The operator a(x, D) maps Gc (Ω) continuously to into G(Ω) and can be extended to a continuous C-linear map from L(G(Ω), C) ∞ If a is of slow scale type then a(x, D) maps G (Ω) continuously into L(Gc (Ω), C). c

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G ∞ (Ω). Pseudo-differential operators with generalized symbol of order −∞ are reg to G(Ω) if a ∈ S−∞ (Ω × Rn ) ularizing, in the sense that a(x, D) maps Lb (G(Ω), C) ∞ −∞,sc n to G (Ω) if a ∈ S (Ω × R ). Clearly, all the previous results and Lb (G(Ω), C) can be stated for pseudo-differential operators given by a generalized amplitude m a(x, y, ξ) ∈ Sρ,δ (Ω × Ω × Rn ). For a complete overview on generalized pseudodifferential operators acting on spaces of Colombeau type we advise the reader to refer to [14, 15, 16] 2.7. Generalized elliptic symbols One of the main issues in developing a theory of generalized symbols has been the search for a notion of generalized elliptic symbol. This is obviously related to the construction of a generalized pseudo-differential parametrix by means of which to investigate problems of G- and G ∞ -regularity. In the sequel we recall some of the results obtain in this direction in [15, 16], which will be employed in Section 5. We work at the level of representatives and we set ρ = 1, δ = 0. We leave to the reader the proof of the next proposition which is based on [15, Section 6]. Proposition 2.7. Let (aε )ε ∈ MS m (Ω×Rn \0) such that (e1) for all K Ω there exists s ∈ R, (Rε )ε ∈ EM strictly nonzero and η ∈ (0, 1] such that |aε (x, ξ)| ≥ εs ξm , for all x ∈ K, |ξ| ≥ Rε and ε ∈ (0, η]. Then, (i) for all K Ω, for all α, β ∈ Nn there exist N ∈ N, (Rε )ε ∈ EM strictly nonzero and η ∈ (0, 1] such that |∂ξα ∂xβ aε (x, ξ)| ≤ ε−N ξ−|α| |aε (x, ξ)| for all x ∈ K, |ξ| ≥ Rε and ε ∈ (0, η]; (ii) (i) holds for the net (a−1 ε )ε ; (iii) if (aε )ε ∈ MS m (Ω×Rn \0) with m < m then (e1) holds for the net (aε + aε )ε . Let (aε )ε ∈ Msc S m (Ω×Rn \0) such that (e2) for all K Ω there exists (sε )ε with (s−1 ε )ε s.s.n., (Rε )ε s.s.n. and η ∈ (0, 1] such that |aε (x, ξ)| ≥ sε ξm , for all x ∈ K, |ξ| ≥ Rε and ε ∈ (0, η]. Then, (iv) for all K Ω, for all α, β ∈ Nn there exist (cε )ε , (Rε )ε s.s.n and η ∈ (0, 1] such that |∂ξα ∂xβ aε (x, ξ)| ≤ cε ξ−|α| |aε (x, ξ)| for all x ∈ K, |ξ| ≥ Rε and ε ∈ (0, η]; (v) (i) holds for the net (a−1 ε )ε ; (vi) if (aε )ε ∈ Msc with m < m then (e2) holds for the net (aε + aε )ε . m n S (Ω×R \0)


151

Proposition 2.8. Let (aε )ε be a net of elliptic symbols of S m (Ω × Rn \ 0). (i) If (aε )ε ∈ MS m (Ω×Rn \0) fulfills condition (e1) then there exists (pε )ε ∈ MS −m (Ω×Rn \0) such that for all ε ∈ (0, 1] pε aε = 1 + rε , where (rε )ε ∈ MS −∞ (Ω×Rn \0) . (ii) If (aε )ε ∈ Msc S m (Ω×Rn \0) fulfills condition (e2) then there exists (pε )ε ∈ Msc −m n S (Ω×R \0) such that for all ε ∈ (0, 1] pε aε = 1 + rε , where (rε )ε ∈ Msc S −∞ (Ω×Rn \0) . Proof. As in [15, Proposition 6.4] we define pε as ξ ψj (x), a−1 (x, ξ)ϕ Rj, j where ψj is a partition of unity subordinated to a covering of relatively compact subsets Ωj of Ω, (Rj,ε )ε is the radius corresponding to Ωj and ϕ is a smooth function on Rn such that ϕ(ξ) = 0 for |ξ| ≤ 1 and ϕ(ξ) = 1 for |ξ| ≥ 2. From Proposition 2.7 we have that (e1) yields (pε )ε ∈ MS −m (Ω×Rn \0) and (e2) yields (pε )ε ∈ Msc S −m (Ω×Rn \0) . Let K Ω. By construction, for all x ∈ K, pε (x, ξ)aε (x, ξ) = 1 +

j0 j0 ξ ξ ϕ )ψj (x) − 1 = 1 + ) − 1 ψj (x), ϕ Rj, Rj, j=0 j=0

and the following estimates hold for all l > 0 and α ∈ Nn \ 0: ξ ξ supξl |ϕ ) − 1| ≤ sup ξl |ϕ ) − 1| ≤ cϕ 2Rj,ε l , R Rj, j, ξ=0 |ξ|≤2Rj, ξ ξ supξl |∂ξα ϕ )|(Rj,ε )−|α| ≤ sup ξl |∂ξα ϕ )|(Rj,ε )−|α| Rj, Rj, ξ=0 Rj,ε ≤|ξ|≤2Rj, ≤ cϕ 2Rj,ε l (Rj,ε )−|α| . We deduce that (pε aε − 1)ε belongs to MS −∞ (Ω×Rn \0) under the hypothesis (e1) on (aε )ε and that (pε aε − 1)ε belongs to Msc S −∞ (Ω×Rn \0) under the hypothesis (e2) on (aε )ε . 2.8. Microlocal analysis in the Colombeau context: Generalized wave front sets in L(Gc (Ω), C) In this subsection we recall the basic notions of microlocal analysis which involve the duals of the Colombeau algebras Gc (Ω) and G(Ω) and have been developed in [14]. In this generalized context the role which is classically played by S (Rn ) is given to the Colombeau algebra GS (Rn ) := GS (Rn ) . GS (Rn ) is topologized as in is endowed with the topology of uniform Subsection 2.2 and its dual L(GS (Rn ), C)

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convergence on bounded subsets. In the sequel Gτ (Rn ) denotes the Colombeau algebra of tempered generalized functions defined as the quotient Eτ (Rn )/Nτ (Rn ), where Eτ (Rn ) is the algebra of all τ -moderate nets (uε )ε ∈ Eτ [Rn ] := OM (Rn )(0,1] such that ∀α ∈ Nn ∃N ∈ N

sup (1 + |x|)−N |∂ α uε (x)| = O(ε−N )

x∈Rn

as ε → 0

and Nτ (Rn ) is the ideal of all τ -negligible nets (uε )ε ∈ Eτ [Rn ] such that ∀α ∈ Nn ∃N ∈ N ∀q ∈ N

sup (1 + |x|)−N |∂ α uε (x)| = O(εq ) as ε → 0.

x∈Rn

Theorem 3.8 in [12] shows that we have the chain of continuous embeddings GS (Rn ) ⊆ Gτ (Rn ) ⊆ L(GS (Rn ), C). Moreover, since for any u ∈ Gc (Ω) with supp u ⊆ K Ω and any K Ω with K ⊂ Int K one can find a representative (uε )ε with supp uε ⊆ K for all ε ∈ (0, 1], we have that Gc (Ω) is continuously embedded into GS (Rn ). and L(G(Ω), C). The Fourier The Fourier transform on GS (Rn ), L(GS (Rn ), C) n transform on GS (R ) is defined by the corresponding transformation at the level of representatives, as follows: F : GS (Rn ) → GS (Rn ) : u → [(5 uε )ε ]. F is a C-linear continuous map from GS (Rn ) into itself which extends to the dual in a natural way. In detail, we define the Fourier transform of T ∈ L(GS (Rn ), C) given by as the functional in L(GS (Rn ), C) F (T )(u) = T (F u). is embedded in L(G (Rn ), C) by means As shown in [14, Remark 1.5] L(G(Ω), C) S of the map : T → u → T ((uε | )ε + N (Ω)) . → L(G (Rn ), C) L(G(Ω), C) Ω S we have from [14, PropoIn particular, when T is a basic functional in L(G(Ω), C) sition 1.6, Remark 1.7] that the Fourier transform of T is the tempered generalized function obtained as the action of T (y) on e−iyξ , i.e., F (T ) = T (e−i·ξ ) = (Tε (e−i·ξ ))ε + Nτ (Rn ). The notions of GGeneralized Wave front sets of a functional in L(Gc (Ω), C). have been wave front set and G ∞ -wave front set of a functional in L(Gc (Ω), C) introduced in [14] as direct analogues of the distributional wave front set in [20]. They employ a subset of the space GSscm (Ω×Rn ) of generalized symbols of slow scale m (Ω × Rn ) (see [16, Definition 1.1]) and a suitable notion of type denoted by S sc slow scale micro-ellipticity [16, Definition 1.2]. In detail, (x0 , ξ0 ) ∈ WFG T (resp. 0 (Ω × (x0 , ξ0 ) ∈ WFG ∞ T ) if there exists a(x, D) properly supported with a ∈ S sc


153

Rn ) such that a is slow scale micro-elliptic at (x0 , ξ0 ) and a(x, D)T ∈ G(Ω) (resp. a(x, D)T ∈ G ∞ (Ω)). Proposition 3.14 in [14] proves When T is a basic functional of L(Gc (Ω), C), that one can limit to classical properly supported pseudo-differential operators in the definition of WFG T and WFG ∞ T . More precisely, Char(A) (2.15) Wcl,G (T ) := AT ∈G(Ω)

and Wcl,G ∞ (T ) :=

Char(A)

(2.16)

AT ∈G ∞ (Ω)

where the intersections are taken over all the classical properly supported operators A ∈ Ψ0 (Ω) such that AT ∈ G(Ω) in (2.15) and AT ∈ G ∞ (Ω) in (2.16). WFG T and WFG ∞ T are both closed conic subsets of T ∗ (Ω) \ 0 and, as proved in [14, Proposition 3.5], πΩ (WFG T ) = sing suppG T and πΩ (WFG ∞ T ) = sing suppG ∞ T. Characterization of WFG T and WFG ∞ T when T is a basic functional. We will employ a useful characterization of the G-wave front set and the G ∞ -wave front set valid for functionals which are basic. It involves the sets of generalized functions ∞ GS ,0 (Γ) and GS,0 (Γ), defined on the conic subset Γ of Rn \ 0, as follows: GS ,0 (Γ) := {u ∈ Gτ (Rn ) : ∃(uε )ε ∈ u ∀l ∈ R ∃N ∈ N supξl |uε (ξ)| = O(ε−N ) as ε → 0}, ξ∈Γ ∞ (Γ) := {u ∈ Gτ (Rn ) : ∃(uε )ε ∈ u ∃N ∈ N ∀l ∈ R GS,0

supξl |uε (ξ)| = O(ε−N ) as ε → 0}. ξ∈Γ

Theorem 3.13 in [14] shows that: Let T ∈ L(Gc (Ω), C). (i) (x0 , ξ0 ) ∈ WFG T if and only if there exists a conic neighborhood Γ of ξ0 and a cut-off function ϕ ∈ Cc∞ (Ω) with ϕ(x0 ) = 1 such that F (ϕT ) ∈ GS ,0 (Γ). (ii) (x0 , ξ0 ) ∈ WFG ∞ T if and only if there exists a conic neighborhood Γ of ξ0 and ∞ (Γ). a cut-off function ϕ ∈ Cc∞ (Ω) with ϕ(x0 ) = 1 such that F (ϕT ) ∈ GS,0

3. Generalized oscillatory integrals: Definition This section is devoted to a notion of oscillatory integral where both the amplitude and the phase function are generalized objects of Colombeau type. In the sequel Ω is an arbitrary open subset of Rn . We recall that φ(y, ξ) is a phase function on Ω × Rp if it is a smooth function on Ω × Rp \ 0, real-valued,

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positively homogeneous of degree 1 in ξ with ∇y,ξ φ(y, ξ) = 0 for all y ∈ Ω and ξ ∈ Rp \0. We denote the set of all phase functions on Ω×Rp by Φ(Ω×Rp ) and the set of all nets in Φ(Ω×Rp )(0,1] by Φ[Ω×Rp ]. The notations concerning classes of symbols have been introduced in Subsection 2.4. The proofs of the statements collected in this section can be found in [18]. In the paper [18] the authors deal with generalized m symbols in Sρ,δ (Ω× Rp ) as well as with regular generalized symbols. This last class of symbols is modelled on the subalgebra G ∞ (Ω) of regular generalized functions and contains the generalized symbols of slow scale type as a submodule. Even though many statements of Section 3, 4 and 6 hold for regular symbols as well, for the sake of simplicity and in order to have uniformity of assumptions between m (Ω × Rp ) and phase functions and symbols, we limit in this work to consider Sρ,δ m,sc (Ω × Rp ) of generalized symbols of slow scale type. the smaller class Sρ,δ Definition 3.1. An element of MΦ (Ω × Rp ) is a net (φε )ε ∈ Φ[Ω × Rp ] satisfying the conditions: 1 (Ω×Rp \0) , (i) (φε )ε ∈ MShg (ii) for all K Ω the net 2 ξ inf p ∇φε y, y∈K,ξ∈R \0 |ξ| ε is strictly nonzero. On MΦ (Ω × Rp ) we introduce the equivalence relation ∼ as follows: (φε )ε ∼ (ωε )ε 1 (Ω×Rp \0) . The elements of the factor space if and only if (φε − ωε ) ∈ NShg × Rp ) := MΦ (Ω × Rp )/∼. Φ(Ω will be called generalized phase functions. × Rp ). We shall employ the equivalence class notation [(φε )ε ] for φ ∈ Φ(Ω p When (φε )ε is a net of phase functions, i.e., (φε )ε ∈ Φ[Ω × R ], Lemma 1.2.1 in [20] shows that there exists a family of partial differential operators (Lφε )ε such that t Lφε eiφε = eiφε for all ε ∈ (0, 1]. Lφε is of the form p j=1

∂ ∂ + bk,ε (y, ξ) + cε (y, ξ), ∂ξj ∂yk n

aj,ε (y, ξ)

(3.1)

k=1

where the coefficients (aj,ε )ε belong to S 0 [Ω× Rp ] and (bk,ε )ε , (cε )ε are elements of S −1 [Ω × Rp ]. The following technical lemma is crucial in proving Proposition 3.3. Lemma 3.2. (i) Let ϕφε (y, ξ) := |∇φε (y, ξ/|ξ|)|−2 . If (φε )ε ∈ MΦ (Ω × Rp ) then 0 (Ω×Rp \0) . (ϕφε )ε ∈ MShg

(ii) If (φε )ε , (ωε )ε ∈ MΦ (Ω × Rp ) and (φε )ε ∼ (ωε )ε then 0 (Ω×Rp \0) (∂ξj φε )ϕφε − (∂ξj ωε )ϕωε ε ∈ NShg


155

for all j = 1, . . . , p and (∂yk φε )|ξ|−2 ϕφε − (∂yk ωε )|ξ|−2 ϕωε ε ∈ NS −1 (Ω×Rp \0) hg

for all k = 1, . . . , n. Proposition 3.3. (i) If (φε )ε ∈ MΦ (Ω×Rp ) then (aj,ε )ε ∈ MS 0 (Ω×Rp ) for all j = 1, . . . , p, (bk,ε )ε ∈ MS −1 (Ω×Rp ) for all k = 1, . . . , n, and (cε )ε ∈ MS −1 (Ω×Rp ) . (ii) If (φε )ε , (ωε )ε ∈ MΦ (Ω × Rp ) and (φε )ε ∼ (ωε )ε then L φε − L ω ε =

p

∂ ∂ + bk,ε (y, ξ) + cε (y, ξ), ∂ξj ∂yk n

aj,ε (y, ξ)

j=1

k=1

(aj,ε )ε

(bk,ε )ε

∈ NS 0 (Ω×Rp ) , ∈ NS −1 (Ω×Rp ) and (cε )ε ∈ NS −1 (Ω×Rp ) for where all j = 1, . . . , p and k = 1, . . . , n. As a consequence of Propositions 3.3 we can claim that any generalized phase × Rp ) defines a generalized partial differential operator function φ ∈ Φ(Ω Lφ (y, ξ, ∂y , ∂ξ ) =

p

∂ ∂ + bk (y, ξ) + c(y, ξ) ∂ξj ∂yk n

aj (y, ξ)

j=1

k=1

and {bk }nk=1 , c are generalized symbols in S0 (Ω × Rp ) whose coefficients −1 p m (Ω×Rp ) continuously and S (Ω×R ), respectively. By construction, Lφ maps Sρ,δ m−s m (Ω×Rp ), where s = min{ρ, 1−δ}. Hence Lkφ is continuous from Sρ,δ (Ω× into Sρ,δ m−ks p p R ) to Sρ,δ (Ω × R ). Before stating the next proposition we recall a classical lemma valid any symbol φ ∈ S 1 (Ω × Rp \ 0). {aj }pj=1

Lemma 3.4. For all α ∈ Np and β ∈ Nn , ∂ξα ∂yβ eiφ(y,ξ) = eiφ(y,ξ) cα1 ,...,αk ,β1 ,...,βk ∂ξα1 ∂yβ1 φ(y, ξ) . . . ∂ξαk ∂yβk φ(y, ξ). k≤|α+β|, α1 +α2 +···+αk =α β1 +β2 +···+βk =β

It follows that ∂ξα ∂yβ eiφ(y,ξ) = eiφ(y,ξ) aα,β (y, ξ), where aα,β ∈ S |β| (Ω × Rp \ 0) and (|β|)

|aα,β |K,j ≤ c

sup

sup

y∈K,ξ=0 |γ+δ|≤|α+β|+j

ξ−1+|γ| |∂ξγ ∂yδ φ(y, ξ)|,

(3.2)

where the constant c depends only on α, β, and j. From (3.2) we have that (φε )ε ∈ MS 1 (Ω×Rp \0)

⇒

(aα,β,ε )ε ∈ MS |β| (Ω×Rp \0)

or more in general that the net (aα,β,ε )ε has the “ε-scale properties” of (φε )ε .

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× Rp ). The exponential Proposition 3.5. Let φ ∈ Φ(Ω eiφ(y,ξ) 1 is a well-defined element of S0,1 (Ω × Rp \ 0).

Proof. From Lemma 3.4 we have that if (φε )ε ∈ MΦ (Ω × Rp ) then (eiφε (y,ξ) )ε ∈ 0 (Ω×Rp \0) . When (φε )ε ∼ (ωε )ε , the equality MS0,1 eiωε (y,ξ) − eiφε (y,ξ) = eiωε (y,ξ) 1 − ei(φε −ωε )(y,ξ) = eiωε (y,ξ)

p

ei(φε −ωε )(y,θξ) ∂ξj (φε − ωε )(y, θξ)iξj ,

j=1

with θ ∈ (0, 1), implies that sup y∈K,ξ∈Rp \0

|ξ|−1 eiωε (y,ξ) − eiφε (y,ξ) = O(εq )

(3.3)

for all q ∈ N. At this point writing ∂ξα ∂yβ (eiωε (y,ξ) − eiφε (y,ξ) ) as ∂ξα ∂yβ eiωε (y,ξ) 1 − ei(φε −ωε )(y,ξ) + α β + ∂ξα ∂yβ eiωε (y,ξ) − ∂ξα−α ∂yβ−β ei(φε −ωε )(y,ξ) α β α 0 such that for all g ∈ CK

(|β |)

(m)

|aβ ,ε |K ,0 |bε |K ,|β| sup |∂ β Fωε (bε )(g)(x)| ≤ c max β ≤β

x∈K

sup

|∂ γ g(y)|,

y∈K,|γ|≤h

holds, we conclude that when [(aβ ,ε )ε ] and [(bε )ε ] are symbols of slow scale type then the map Fω (b) : Gc∞ (Ω) → G ∞ (Ω ) is continuous. (iv) If suppx b Ω from the first assertion we have that Fω (b) ∈ Gc (Ω ). Under the assumptions of (ii) for the phase function ω we have that t Fω (b) maps G(Ω ) continuously into G(Ω) and therefore Fω (b) can be extended to a map from to L(G(Ω ), C). L(G(Ω), C) Remark 5.2. Taking Ω = Rn and noting that Gc (Ω ) ⊆ Gc (Rn ), it is clear that Fω (b) maps Gc (Rn ) into Gc (Rn ) when suppx b Ω . In addition, t Fω (b) : G(Rn ) → G(Rn ) → L(G(Rn ), C). and Fω (b) : L(G(Rn ), C) In the sequel we assume Ω = Ω ⊆ Rn . Our main purpose is to investigate the composition a(x, D) ◦ Fω (b), where a(x, D) is a generalized pseudo-differential operator and Fω (b) a generalized Fourier integral operator as in (5.1). This requires some technical preliminaries. Technical preliminaries The proof of the following lemma can be found in [6, Lemmas A.11, A.12]. Lemma 5.3. Let a ∈ C ∞ (Ω × Rn \ 0) and ω ∈ C ∞ (Ω × Rn \ 0). Then, σ α σ ∂xβ ∂ηγ+σ a(x, ∇x ω(x, η)) · ∂x ∂η (a(x, ∇x ω(x, η)) = σ σ ≤σ

|β+γ|≤|α| |σ |≤|σ |

σ σ−σ α · Pη,σ Pxβγ (x, η), (x, η)∂η

where

σ Pη,σ (x, η) = 1

σ Pη,σ =

δ1 ,...,δq s1 ,...,sq

if s ,...,s

cδ11 ,...,δqq ∂ηδ1 ∂xs1 ω(x, η) . . . ∂ηδq ∂xsq ω(x, η)

σ = 0,

otherwise,

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with q = |σ |,

q j=1

α Pxβγ =1 α = Pxβγ

|δj | = |σ | and

if ,...,sr δ1 dsδ11 ,...,δ ∂x ∂xs1 ω(x, η) . . . ∂xδr ∂xsr ω(x, η) r

γ = 0,

otherwise,

δ1 ,...,δr s1 ,...,sr

with |γ| = r and

r

j=1

|δj | + |β| = |α|.

Proposition 5.4. n 1 (Ω × R (h1) Let (ωε )ε ∈ MShg \ 0) such that ∇x ωε = 0 for all ε ∈ (0, 1] and for all K Ω η inf ∇x ωε x, x∈K,η∈Rn \0 |η| ε

is strictly non-zero. (i) If (aε )ε ∈ MS m (Ω×Rn \0) then (aε (x, ∇x ωε (x, η)))ε ∈ MS m (Ω×Rn \0) ; (ii) if (aε )ε ∈ NS m (Ω×Rn \0) then (aε (x, ∇x ωε (x, η)))ε ∈ NS m (Ω×Rn \0) . n (h2) Let (ωε )ε ∈ Msc \ 0) such that ∇x ωε = 0 for all ε ∈ (0, 1] and for 1 (Ω × R Shg all K Ω η ∇x ωε x, inf x∈K,η∈Rn \0 |η| ε

is slow scale strictly non-zero. sc (iii) If (aε )ε ∈ Msc S m (Ω×Rn \0) then (aε (x, ∇x ωε (x, η)))ε ∈ MS m (Ω×Rn \0) . n 1 (Ω × R \ 0) with (ωε )ε and (ω )ε satisfying the (h3) Finally, let (ωε − ωε )ε ∈ NShg ε hypothesis (h1) above. (iv) If (aε )ε ∈ MS m (Ω×Rn \0) then (aε (x, ∇x ωε (x, η)) − aε (x, ∇x ωε (x, η)))ε ∈ NS m (Ω×Rn \0) . Proof. From Lemma 5.3 it follows that ∂xα ∂ησ (aε (x, ∇x ωε (x, η)) is a finite sum of terms of the type

∂xα ∂ησ aε (x, ∇x ωε (x, η))gα ,σ ,ε (x, η),

where (gα ,σ ,ε )ε is a net of symbols in S |σ |−|σ| (Ω × Rn \ 0). Note that (gα ,σ ,ε )ε depends on (ωε )ε and is actually a finite sum of products of derivatives of (ωε )ε . One can easily prove that n 1 (Ω × R \ 0) (ωε )ε ∈ MShg

(ωε )ε ∈

Msc 1 (Ω Shg

× R \ 0) n

⇒

(gα ,σ ,ε )ε ∈ MS |σ |−|σ| (Ω×Rn \0) ,

⇒

(gα ,σ ,ε )ε ∈ Msc S |σ |−|σ| (Ω×Rn \0)

(5.2)

and that the following ∀α , σ ∈ Nn ∀K Ω ∃(λε )ε ∈ R(0,1] ∀x ∈ K ∀η ∈ Rn \ 0 ∀ε ∈ (0, 1]

|∂xα ∂ησ aε (x, ∇x ωε (x, η))| ≤ λε ∇x ωε m−|σ |

(5.3)


165

holds, with (λε )ε ∈ EM if (aε )ε ∈ MS m (Ω×Rn \0) , (λε )ε slow scale net if (aε )ε ∈ Msc S m (Ω×Rn \0) and (λε )ε ∈ N if (aε )ε ∈ NS m (Ω×Rn \0) . Now, let us consider (∇x ωε (x, η))ε . We have that ⇒

(h1)

∀K Ω ∃r > 0 ∃c1 , c2 > 0 ∃η ∈ (0, 1] ∀x ∈ K ∀|η| ≥ 1 ∀ε ∈ (0, η] ηc1 εr ≤ |∇x ωε (x, η)| ≤ c2 ε−r η,

⇒

(h2)

∀K Ω ∃(με )ε s.s.n ∃η ∈ (0, 1] ∀x ∈ K ∀|η| ≥ 1 ∀ε ∈ (0, η] ημ−1 ε ≤ |∇x ωε (x, η)| ≤ με η.

Under the hypothesis (h1), combining (5.2) with (5.3) we obtain the assertions (i) and (ii). Moreover, from the second implications of (5.2) and (5.3) we see that (h2) yields (iii). It remains to prove that if (h3) holds and (aε )ε is a moderate net of symbols then (aε (x, ∇x ωε (x, η)) − aε (x, ∇x ωε (x, η)))ε ∈ NS m (Ω×Rn \0) . If suffices to write ∂xα ∂ησ (aε (x, ∇x ωε (x, η)) − aε (x, ∇x ωε (x, η))) as the finite sum ∂xα ∂ησ aε (x, ∇x ωε (x, η))(gα ,σ (ωε ) − gα ,σ (ωε )) α ,σ

+

[∂xα ∂ησ aε (x, ∇x ωε (x, η)) − ∂xα ∂ησ aε (x, ∇x ωε (x, η))]gα ,σ (ωε ). (5.4)

α ,σ

An inspection of Lemma 5.3 shows that the net (gα ,σ (ωε ) − gα ,σ (ωε ))ε belongs to NS |σ |−|σ| (Ω×Rn \0) and from the hypothesis (h1) on (ωε )ε it follows that the first summand in (5.4) is an element of NS m−|σ| (Ω×Rn \0) . We use Taylor’s formula on the second summand of (5.4). Therefore, for x varying in a compact set K and for ε small enough we can estimate

|∂xα ∂ησ aε (x, ∇x ωε (x, η)) − ∂xα ∂ησ aε (x, ∇x ωε (x, η))| by means of n

ε−N ∇x ωε (x, η) + θ(∇x ωε (x, η) − ∇x ωε (x, η))m−|σ

|−1

|∂xj (ωε − ωε )(x, η)|

j=1

≤ εq ∇x ωε (x, η) + θ(∇x ωε (x, η) − ∇x ωε (x, η))m−|σ

|−1

η,

where θ ∈ [0, 1]. Since, taking ε small and |η| ≥ 1 the following inequalities εr η, 2 |∇x ωε (x, η) + θ(∇x ωε (x, η) − ∇x ωε (x, η))| ≤ ε−r η

|∇x ωε (x, η) + θ(∇x ωε (x, η) − ∇x ωε (x, η))| ≥ εr η − εr+1 η ≥

hold for some r > 0, we conclude that α σ ∂x ∂η aε (x, ∇x ωε (x, η)) − ∂xα ∂ησ aε (x, ∇x ωε (x, η)) ε ∈ NS m−|σ | (Ω×Rn \0) . Thus, from (gα ,σ (ωε ))ε ∈ MS |σ |−|σ| (Ω×Rn \0) we have that the second summand of (5.4) belongs to NS m−|σ| (Ω×Rn \0) and the proof is complete.

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1 Corollary 5.5. If a ∈ Sm (Ω × Rn \ 0) and ω ∈ Shg (Ω × Rn \ 0) has a representative satisfying condition (h1) of Proposition 5.4 then a(x, ∇x ω(x, η)) ∈ Sm (Ω × Rn \ 0). 1,sc (Ω × Rn \ 0) has a representative satisfying If a ∈ Sm,sc (Ω × Rn \ 0) and ω ∈ Shg condition (h2) of Proposition 5.4, then a(x, ∇x ω(x, η)) ∈ Sm,sc (Ω × Rn \ 0). 1 Let ω ∈ Shg (Ω × Rn \ 0) have a representative satisfying (h1). We want to investigate the properties of Dzβ eiω(z,x,η) |z=x , (5.5)

where ω(z, x, η) := ω(z, η) − ω(x, η) − ∇x ω(x, η), z − x. We make use of the following technical lemma, whose proof can be found in [7, Proposition 15]. Lemma 5.6. Let ω ∈ C ∞ (Ω × Rn \ 0) and ω(z, x, η) as above. Then, for |β| = 0,we have

Dzβ eiω(z,x,η) = eiω(z,x,η) (∇z ω(z, η) − ∇x ω(x, η))β +

n1,j1

cj1 (∇z ω(z, η) − ∇x ω(x, η))θj1

j1

γ

∂z j1 ,j2 ω(z, η)

j2 =1

+

n2,j1

cj1

j1

δj1 ,j2

∂z

ω(z, η) ,

j2 =1

where cj1 , cj1 are suitable constants, |γj1 ,j2 | ≥ 2, |δj1 ,j2 | ≥ 2 and

n1,j1

θj1 +

n2,j1

γj1 ,j2 =

j2 =1

δj1 ,j2 = β.

j2 =1

It follows that n1,j1 n2,j1 γj ,j δj ,j Dzβ eiω(z,x,η) |z=x = cj1 ∂x 1 2 ω(x, η) + cj1 ∂x 1 2 ω(x, η), (5.6) j1

j2 =1

j1

j2 =1

with

n1,j1

j2 =1

n2,j1

γj1 ,j2 =

δj1 ,j2 = β.

j2 =1

Moreover, from |γj1 ,j2 | ≥ 2, |δj1 ,j2 | ≥ 2 we have |β| ≥ 2n1,j1 and |β| ≥ 2n2,j1 . Since the constants cj1 , cj1 do not depend on ω, we can use the formula (5.6) in estimating the net (Dzβ eiωε (z,x,η) |z=x )ε . Proposition 5.7. 1 (i) If ω ∈ Shg (Ω × Rn \ 0) then (5.5) is a well-defined element of S|β|/2 (Ω × Rn \ 0). (ii) If ω ∈ S1,sc (Ω×Rn \0) then (5.5) is a well-defined element of S|β|/2,sc (Ω×Rn \0). hg

Generalized Fourier Integral Operators Proof. From (5.6) we have that n 1 (Ω × R (ωε )ε ∈ MShg \ 0)

⇒

n (ωε )ε ∈ Msc S 1 (Ω × R \ 0)

⇒

hg

167

(Dzβ eiωε (z,x,η) |z=x )ε ∈ MS |β|/2 (Ω×Rn \0) , (Dzβ eiωε (z,x,η) |z=x )ε ∈ Msc S |β|/2 (Ω×Rn \0) .

1 (Ω×Rn \0) entails Noting that (ωε − ωε )ε ∈ NShg

n 1,j1 j2 =1 n2,j1

n1,j1 γ ∂xj1 ,j2 ωε (x, η)

−

j2 =1 n2,j1 δj ,j ∂x 1 2 ωε (x, η)

j2 =1

−

j2 =1

γ ∂xj1 ,j2 ωε (x, η)

∈ NS |β|/2 (Ω×Rn \0) , ε

δj ,j ∂x 1 2 ωε (x, η)

∈ NS |β|/2 (Ω×Rn \0) , ε

we conclude that the net Dzβ eiωε (z,x,η) |z=x − Dzβ eiωε (z,x,η) |z=x ε belongs to NS |β|/2 (Ω×Rn \0) . By combining Corollary 5.5 with Proposition 5.7 we obtain the following statement. Proposition 5.8. Let α ∈ Nn and ∂ξα a(x, ∇x ω(x, η)) α iω(z,x,η) Dz e b(z, η) |z=x . (5.7) α! 1 (i) If a ∈ Sm (Ω × Rn \ 0), ω ∈ Shg (Ω × Rn \ 0) has a representative satisfying condition (h1) and b ∈ Sl (Ω × Rn \ 0), then hα ∈ S l+m−|α|/2 (Ω × Rn \ 0) for all α. 1,sc (ii) If a ∈ Sm,sc (Ω × Rn \ 0), ω ∈ Shg (Ω × Rn \ 0) has a representative satisfying condition (h2) and b ∈ S l,sc (Ω × Rn \ 0), then hα ∈ S l+m−|α|/2,sc (Ω × Rn \ 0) for all α. hα (x, η) =

Our next task is to give a closer look to eiω(x,η) . 1 (Ω × Rn \ 0) have a representative satisfying condition Proposition 5.9. Let ω ∈ Shg (h1). Then for any positive integer N there exists pN ∈ S−2N (Ω × Rn \ 0) such that

eiω(x,η) =

iω(x,η) pN (x, η)ΔN , x + r(x, η) e

(5.8)

where r ∈ S−∞ (Ω × Rn \ 0). 1 (Ω × Rn \ 0) is of slow scale type and has a representative satisfying If ω ∈ Shg condition (h2) then pN and r are of slow scale type. Proof. Let (ωε )ε be a representative of ω satisfying (h1). We leave to the reader to prove by induction that iωε (x,η) = aε (x, η)eiωε (x,η) , ΔN x e

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where (aε )ε ∈ MS 2N (Ω×Rn \0) with principal part given by a2N,ε = (−1)N |∇x ωε (x, η)|2N . From (h1) we have that ∇x ωε = 0 for all ε ∈ (0, 1] and for all K Ω there exist r > 0 and ε0 ∈ (0, 1] such that |∇x ωε (x, η)| ≥ εr |η| for all x ∈ K, η = 0 and ε ∈ (0, ε0 ]. Hence, εr |∇x ωε (x, η)| ≥ η, 2 for |η| ≥ 1, x ∈ K and ε ∈ (0, ε0 ]. It follows from Proposition 2.7(iii) that (aε )ε is a net of elliptic symbols of S 2N (Ω × Rn \ 0) such that for all K Ω there exist s ∈ R, (Rε )ε strictly nonzero and ε0 ∈ (0, 1] such that |aε (x, η)| ≥ εs η2N , for x ∈ K, |η| ≥ Rε and ε ∈ (0, ε0 ]. By Proposition 2.8(i) we find (pN,ε )ε ∈ MS −2N (Ω×Rn \0) and (rε )ε ∈ MS −∞ (Ω×Rn \0) such that pN,ε aε = 1 − rε for all ε. Therefore, e

iωε (x,η)

=

pN,ε (x, η)ΔN x

(5.9)

+ rε (x, η) eiωε (x,η) .

This equality at the representatives’ level implies the equality (5.8) between equiv1 alence classes of S0,1 (Ω × Rn \ 0). Now, let ω be a slow scale symbol with a representative (ωε )ε satisfying condition (h2). From Proposition 2.7(vi) we have that (aε )ε ∈ Msc S 2N (Ω×Rn \0) is a net of elliptic symbols such that for some (sε )ε inverse of a slow scale net, (Rε )ε slow scale net and ε0 ∈ (0, 1] the inequality |aε (x, η)| ≥ sε η2N , holds for all x ∈ K, |η| ≥ Rε and ε ∈ (0, ε0 ]. Proposition 2.8(ii) shows that (5.9) sc is true for some (pN,ε )ε ∈ Msc S −2N (Ω×Rn \0) and (rε )ε ∈ MS −∞ (Ω×Rn \0) . Main Theorems The make use of the previous propositions in proving the main theorems of this section: Theorems 5.10 and 5.11. 1 Theorem 5.10. Let ω ∈ Shg (Ω × Rn \ 0) have a representative satisfying condition m n (h1). Let a ∈ S (Ω × R ) and b ∈ Sl (Ω × Rn \ 0) with suppx b Ω. Then, the operator a(x, D)Fω (b) has the following properties: into L(Gc (Ω), C); (i) maps Gc (Ω) into G(Ω) and L(G(Ω), C) (ii) is of the form eiω(x,η) h(x, η)3 u(η) d−η + r(x, D)u, Rn


169

where h ∈ S l+m (Ω × Rn \ 0) has asymptotic expansion given by the symbols hα defined in (5.7) and r ∈ S−∞ (Ω × Rn \ 0). Proof. From Proposition 5.1(iv) is clear that Fω (b) maps Gc (Ω) and L(G(Ω), C) into themselves respectively. We obtain (i) combining this results with the usual mapping properties of a generalized pseudo-differential operator. We now have to investigate the composition a(x, D)Fω (b)u(x) = ei(x−z)θ a(x, θ)Fω (b)u(z) dz d−θ Ω×Rn i(x−z)θ iω(z,η) − = e a(x, θ) e b(z, η)3 u(η) d η dz d−θ Ω×Rn Rn = ei((x−z)θ+ω(z,η)) a(x, θ)b(z, η) dz d−θ u 3(η) d−η, Rn

Ω×Rn

for u ∈ Gc (Ω). The last integral in dz and d−θ is regarded as the oscillatory integral ei(x−z)θ a(x, θ)b(z, η)eiω(z,η) dz d−θ, (5.10) Ω×Rn

∈ Gc (Ωz ). with b(z, η)e In the sequel we will work at the level of representatives and we will follow the proof of Theorem 4.1.1 in [31]. iω(z,η)

Step 1. Let(σε )ε such that σε ≥ cεs for some c, s > 0 and for all ε ∈ (0, 1]. We take ϕ ∈ C ∞ (Rn ) such that ϕ(y) = 1 for |y| ≤ 1/2 and ϕ(y) = 0 for |y| ≥ 1 and we set x − z x − z bε (z, η) + 1 − ϕ bε (z, η). bε (z, η) = bε (z, x, η) + bε (z, x, η) = ϕ σε σε We now write the integral in dz and d−θ of (5.10) as ei((x−z)θ+ωε (z,η)) aε (x, θ)bε (z, x, η) dz d−θ Ω×Rn ei((x−z)θ+ωε (z,η)) aε (x, θ)bε (z, x, η) dz d−θ := I1,ε (x, η) + I2,ε (x, η) + Ω×Rn

and we begin to investigate the properties of (I2,ε )ε . Proposition 5.9 provides the identity eiωε (z,η) = pN,ε (z, η)ΔN + r (z, η) eiωε (z,η) , ε z where (pN,ε )ε ∈ MS −2N (Ω×Rn \0) and (rε )ε ∈ MS −∞ (Ω×Rn \0) , and allows us to write (I2,ε )ε as iωε (z,η) N i(x−z)θ e Δz e pN,ε (z, η)bε (z, x, η) aε (x, θ) dz d−θ Ω×Rn 1 2 + ei(x−z)θ aε (x, θ)bε (z, x, η)rε (z, η)eiωε (z,η) dz d−θ := I2,ε (x, η) + I2,ε (x, η) Ω×Rn

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2 2 The net (I2,ε )ε ∈ MS −∞ (Ω×Rn \0). Indeed, I2,ε (x, η) = Rn Rn gε (x, η, z, θ) dz d−θ, where gε (x, η, z, θ) = ei(x−z)θ aε (x, θ)bε (z, x, η)rε (z, η)eiωε (z,η) and the following holds: for all K Ω, for all α ∈ Nn and d > 0 exist N ∈ N and ε0 ∈ (0, 1] such that (iθ)α ≤ ε−N θm η−d . g (x, η, z, θ) dz ε Rn

This is due to the fact that suppz bε ⊆ Kb Ω for all ε and (rε )ε ∈ MS −∞ (Ω×Rn \0) . Step 2. By construction bε (z, x, η) = 0 if |x−z| ≤ σε /2 for all ε ∈ (0, 1]. By making use of the identity ei(x−z)θ = |x − z|−2k (−Δθ )k ei(x−z)θ we have 1 I2,ε (x,η) iωε (z,η) N −2k k i(x−z)θ Δz |x − z| (−Δθ ) e pN,ε (z,η)bε (z,x,η) aε (x,θ)dz d−θ = e Ω×Rn iωε (z,η) k N i(x−z)θ −2k = e (−Δθ ) aε (x,θ)Δz e |x − z| pN,ε (z,η)bε (z,x,η) dz d−θ. Ω×Rn

It follows that for x ∈ K Ω and ε small enough 1 −Na −Np −Nb −2k −|γ| |I2,ε (x, η)| ≤ cε (σε ) cγ σε ≤ ε−N

Rn

|γ|≤2N

Rn

θm−2k+2N dθ η−2N +l

θm−2k+2N dθ η−2N +l .

Hence, given d > 0 and taking N, k such that −2N +l < −d and m−2k+2N < −n 1 we obtain that (I2,ε )ε is a moderate net of symbols of order −∞ on Ω × Rn \ 0. Summarizing, 1 2 ei((x−z)θ+ωε (z,η)) aε (x, θ)bε (z, η) dz d−θ = I1,ε (x, η) + I2,ε (x, η) + I2,ε (x, η), Ω×Rn

1 2 )ε and (I2,ε )ε belong to MS −∞ (Ω×Rn \0) . where (I2,ε

Step 3. It remains to study I1,ε (x, η) = Ω×Rn

ei((x−z)θ+ωε (z,η)) aε (x, θ)bε (z, x, η) dz d−θ.

We expand aε (x, θ) with respect to θ at θ = ∇x ωε (x, η) and we observe that (θ − ∇x ωε (x, η))α ei(x−z)(θ−∇x ωε (x,η)) = (−1)|α| Dzα ei(x−z)(θ−∇x ωε (x,η)) .


171

By integrating by parts we obtain e−iωε (x,η) I1,ε (x, η) = eiω ε (z,x,η) ei(x−z)(θ−∇x ωε (x,η)) bε (z, x, η)aε (x, θ) dz d−θ Ω×Rn

1 = ∂ α aε (x, ∇x ωε (x, η))· α! ξ |α| 0. We define the sets Wτ1ε ,η = {θ ∈ Rn : |θ| < τε |η|},

Wτ2ε ,η = Rn \ Wτ1ε ,η .

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Set now χε (θ) := χ(θ/τε ). By construction we have that χε (θ/|η|) = 1 on Wτ1ε ,η , 1 and supp(1 − χε (·/|η|)) ⊆ Wτ2ε ,η . We write Rα,ε (x, η) as supp χε (·/|η|) ⊆ W2τ ε ,η α iω ε (z,x,η) Dz e bε (z, x, η) e−i(x−z)θ rα,ε (x, η, θ)χε (θ/|η|) dz d−θ Ω×Rn α iω ε (z,x,η) Dz e bε (z, x, η) e−i(x−z)θ rα,ε (x, η, θ)(1 − χε (θ/|η|)) dz d−θ + Ω×Rn

1 2 := Rα,ε (x, η) + Rα,ε (x, η). 1 . We make use of the identity We begin by estimating the net Rα,ε

e−i(x−z)θ = (1 + |η|2 |x − z|2 )−N (1 − |η|2 Δθ )N e−i(x−z)θ which yields Dzα eiωε (z,x,η) bε (z, x, η) e−i(x−z)θ · Ω×Rn · (1 + |η|2 |x − z|2 )−N (1 − |η|2 Δθ )N rα,ε (x, η, θ)χε (θ/|η|) dz d−θ.

1 (x, η) = Rα,ε

By the moderateness of the net (ωε ) and Taylor’s formula we have the inequality |∇x ωε (x, η) − ∇z ωε (z, η)| ≤ cε−M |η|1 |x − z|,

(5.11)

valid for z ∈ Kb , |x − z| ≤ σε and σε small enough such that ∪ε∈(0,1] {z + λ(x − z) : z ∈ Kb , |x − z| ≤ σε , λ ∈ [0, 1]} ⊆ K Ω. Clearly M depends on the compact set K . By Lemma 5.6 we have that for all x and z as above, |η| ≥ 1 and ε ∈ (0, ε0 ], the estimate β iω (z,x,η) |β| ≤ c ε−M η 2 (1 + |η|2 |x − z|2 )Lβ Dz e ε holds for some Lβ ∈ N and M ∈ N. Hence, recalling that σε ≥ cεs for some c, s > 0, we are led from the previous considerations to α iω (z,x,η) |α| D e ε ≤ Cε−N ηl+ 2 (1 + |η|2 |x − z|2 )Lα , b (z, x, η) (5.12) ε z valid for |η| ≥ 1, ε small enough and N depending on Kb , α and the bound cεs of σε . Before considering (1−|η|2 Δθ )N rα,ε (x, η, θ)χε (θ/|η|) it is useful to investigate 1 the quantity |∇x ωε (x, η) − tθ| for x ∈ K Ω and θ ∈ W2τ . We recall that there ε ,η exists r > 0, c0 , c1 > 0 and ε0 ∈ (0, 1] such that c0 εr |η| ≤ |∇x ωε (x, η)| ≤ c1 ε−r |η|, 1 then |θ| < 2τε |η|, we for all x ∈ K, η = 0 and ε ∈ (0, ε0 ]. Since, if θ ∈ W2τ ε ,η 1 obtain, for all x ∈ K, θ ∈ W2τε ,η and t ∈ [0, 1], the following estimates: −r |∇x ωε (x, η) − tθ| ≤ |∇x ωε (x, η)| + |θ| ≤ (1 + 2τε c−1 )|∇x ωε (x, η)| 0 ε −r |∇x ωε (x, η) − tθ| ≥ (1 − 2τε c−1 )|∇x ωε (x, η)|. 0 ε

Generalized Fourier Integral Operators It follows that assuming τε ≤

εr 4c−1 0

173

the inequality

c0 r 3 1 3 ε |η| ≤ |∇x ωε (x, η)| ≤ |∇x ωε (x, η) − tθ| ≤ |∇x ωε (x, η)| ≤ c1 ε−r |η| (5.13) 2 2 2 2 1 , t ∈ [0, 1] and ε small enough. We make use holds for x ∈ K, η = 0, θ ∈ W2τ ε ,η 2 N of (5.13) in estimating (1 − |η| Δθ ) rα,ε (x, η, θ)χε (θ/|η|) and we conclude that for all N ∈ N there exists N such that |(1 − |η|2 Δθ )N rα,ε (x, η, θ)χε (θ/|η|) | ≤ ε−N ηm−|α| , (5.14) 1 and ε ∈ (0, ε0 ]. A combination of (5.12) with for all x ∈ K, η = 0, θ ∈ W2τ ε ,η (5.14) entails |α| 1 −N −N m+l− 2 |Rα,ε (x, η)| ≤ ε η dθ (1 + |η|2 |y|2 )Lα −N dy. Rn

1 W2τ ε ,η

Therefore, choosing N ≥ Lα + 1 |Rα,ε (x, η)| ≤ cε−N

n+1 2

−N

we obtain

(τε )n ηm+l−

≤ c1 ε−N1 ηm+l−

|α| 2

|α| 2

|η|n

Rn

z−n−1 dz|η|−n

,

for x ∈ K and |η| ≥ 1. The case |η| ≤ 1 requires less precise estimates. More precisely, it is enough to see that from Lemma 5.6 we have that for all α there exists some d ∈ R such that α iω (z,x,η) D e ε bε (z, x, η) ≤ Cε−M ηd z for all x ∈ K, z ∈ Kb and |x − z| ≤ σε . Thus, 1 |Rα,ε (x, η)| ≤ cεN2 ηd−l−

|α| 2

ηm+l−

|α| 2

≤ c2 εN2 ηm+l−

|α| 2

,

when |η| ≤ 1. In conclusion, there exists (CK,ε )ε ∈ EM such that 1 |Rα,ε (x, η)| ≤ CK,ε ηm+l−

|α| 2

for all x ∈ K, η ∈ Rn \ 0 and ε ∈ (0, 1]. 2 Step 5. Finally, we consider Rα,ε (x, η). By Lemma 5.6 we can write Dzα eiωε (z,x,η) bε (z, x, η)

as the finite sum eiωε (z,x,η)

bα,β,ε (z, x, η),

β

where, by making use of the hypotheses on bε and σε , the following holds: ∀β ∈ Nn ∃mβ ∈ R ∀γ ∈ Nn ∃(μβ,γ,ε )ε ∈ EM ∀x ∈ Ω ∀z ∈ Ω ∀η ∈ Rn \ 0 ∀ε ∈ (0, 1] |∂zγ bα,β,ε (z, x, η)| ≤ μβ,γ,ε ηmβ ,

174

C. Garetto

with bα,β,ε (z, x, η) = 0 for |x − z| ≥ σε . Hence, we have 2 Rα,ε (x, η) = β

=

β

eiωε (z,x,η) bα,β,ε (z, x, η)e−i(x−z)θ rα,ε (x, η, θ)(1 − χε (θ/|η|)) dz d−θ

Ω×Rn

Rn

e−ixθ rα,ε (x, η, θ)(1 − χε (θ/|η|))

eiρε (z,x,η,θ) bα,β,ε (z, x, η) dz d−θ, Ω

where ρε (z, x, η, θ) = ω ε (z, x, η) + zθ. Since χε (θ/|η|) = 1 for θ ∈ Wτ1ε ,η , we may limit ourselves to consider θ ∈ Wτ2ε ,η , i.e., |θ| ≥ τε |η|. We investigate now the properties of the net (ρε )ε . We have ∇z ρε (z, x, η, θ) = θ + ∇z ωε (z, η) − ∇x ωε (x, η), and therefore (5.11) yields |θ + ∇z ωε (z, η) − ∇x ωε (x, η)| ≤ |θ| + ε−M σε |η| ≤ |θ|(1 + ε−M σε τε−1 ) for θ ∈ Wτ2ε ,η , |x − z| < σε , z ∈ Kb and ε small enough. We now take σε so small that ε−M σε ≤ τ2ε . From (5.11) and the previous assumptions we obtain 1 |θ|. 2 strictly nonzero and (λ2,ε )ε ∈ EM such

|θ + ∇z ωε (z, η) − ∇x ωε (x, η)| ≥ |θ| − ε−M σε |η| ≥ |θ| − ε−M σε τε−1 |θ| ≥

In other words, there exists (λ1,ε )ε ∈ EM that λ1,ε |θ| ≤ |θ + ∇z ωε (z, η) − ∇x ωε (x, η)| ≤ λ2,ε |θ|, 2 for θ ∈ Wτε ,η , |x − z| < σε , z ∈ Kb and ε ∈ (0, 1]. Consider now iρε (z,x,η,θ) pN,ε (z, x, η, θ) = e−iρε (z,x,η,θ)ΔN . z e γ γ Noting that ∂z ρε (z, x, η, θ) = ∂z ωε (z, η) for |γ| ≥ 2, and making use of the previous estimates on |∇z ρε (z, x, η, θ)|, one can prove by induction that iρε (z,x,η,θ) iρε (z,x,η,θ) N 2N e = e |∇ ρ (z, x, η, θ)| + s (z, x, η, θ) , (−1) ΔN z ε N,ε z where (sN,ε )ε has the following property: ∃l ∈ [0, 2N ) ∀γ ∈ Nn ∃(sγ,N,ε )ε ∈ EM

|∂zγ sN,ε (z, x, η, θ)| ≤ sγ,N,ε |θ|l , (5.15)

for |η| ≥ 1, θ ∈ Wτ2ε ,η , |x − z| < σε and z ∈ Kb . It follows that 1 1 |pN,ε (z, x, η, θ)| ≥ 2N |θ|2N − s0,N,ε |θ|l = |θ|2N 2N − s0,N,ε |θ|l−2N 2 2 1 2N ≥ 2N +1 |θ| , (5.16) 2 1

for θ ∈ Wτ2ε ,η , |x − z| < σε , z ∈ Kb and |η| ≥ λN,ε := τε−1 (22N +1 s0,N,ε ) 2N −l . Moreover, we have that for all γ ∈ Nn there exists (aγ,N,ε )ε ∈ EM such that |∂zγ |∇z ρε (z, x, η, θ)|2N | ≤ aγ,N,ε |θ|2N ,

(5.17)


175

for |η| ≥ 1, θ ∈ Wτ2ε ,η , |x−z| < σε and z ∈ Kb . This allows us to prove by induction that −2N |∂zγ p−1 , N,ε (z, x, η, θ)| ≤ bγ,N,ε|θ| (5.18) for θ ∈ Wτ2ε ,η , |x − z| < σε , z ∈ Kb and |η| ≥ λγ,N,ε . The assertion (5.18) is clear for γ = 0 by (5.16). Assume now that (5.18) holds for |γ | < N and take |γ| = N . From p−1 N,ε pN,ε = 1 we obtain γ γ−γ ∂zγ p−1 (z, x, η, θ)p (z, x, η, θ) = − pN,ε (z, x, η, θ) ∂zγ p−1 N,ε N,ε N,ε (z, x, η, θ)∂z γ

∀γ ∈ Nn ∃(bγ,N,ε )ε ∈ EM ∃(λγ,N,ε )ε ∈ EM

γ 0,

(C2)

|∂yα ∂ξ φ(y, ξ)| ≤ Cα , |∂y ∂ξβ φ(y, ξ)| ≤ Cβ ∀(y, ξ) ∈ Rn × Rn , 1 ≤ |α|, |β| ≤ 2n + 2.

∀(y, ξ) ∈ Rn × Rn ;

Note that condition (C1) is a global version of the local graph condition, 0 . which is necessary even for local L2 -bounds for operators with amplitudes in S1,0 The importance of condition (C2) is that now we must take only mixed derivatives with respect to y and ξ, so the phase functions for the canonical transforms in smoothing problems satisfy this condition. Indeed, if φ(y, ξ) = y · ψ(ξ), where ψ is homogeneous of order one for large ξ and | det Dψ(ξ)| ≥ C > 0, then condition (C2) is satisfied for large frequencies. An additional argument is required for small frequencies and it can be found in [50]. It can be noted that conditions (C1) and (C2) are considerably weaker than those appearing in the analysis of SG-pseudo-differential operators (see Cordes [13]), SG-Fourier integral operators, (see Coriasco [14]), or for operator of Shubin type (see Boggiatto, Buzano and Rodino [5]). We consider first operators of the form ei(x·ξ+φ(y,ξ)) a(x, ξ)u(y)dξ dy. T u(x) = Rn

Rn

For such operators we have the following theorem Theorem 7.1 ([50]). Let φ(y, ξ) satisfy conditions (C1), (C2). Let a(x, ξ) satisfy one of the following conditions: (1) [Calder´ on-Vaillancourt] ∂xα ∂ξβ a(x, ξ) ∈ L∞ (Rnx × Rnξ ), α, β ∈ {0, 1}n . (2) [Cordes] ∂xα ∂ξβ a(x, ξ) ∈ L∞ (Rnx × Rnξ ), |α|, |β| ≤ [n/2] + 1.

(3) [Cordes] ∃λ, λ > n/2 : (1 − Δx )λ/2 (1 − Δξ )λ /2 a(x, ξ) ∈ L∞ (Rnx × Rnξ ). (4) [Childs] difference conditions to be found in [50]. (5) [Coifman-Meyer] ∂xα ∂ξβ a(x, ξ) ∈ L∞ (Rnx × Rnξ ), |α| ≤ [n/2] + 1, β ∈ {0, 1}n. (6) [Coifman-Meyer] ∃2 ≤ p < ∞: ∂xα ∂ξβ a(x, ξ) ∈ Lp (Rnx × Rnξ ), |α| ≤ [n(1/2 − 1/p)] + 1, |β| ≤ 2n. Then T is L2 (Rn )-bounded. In brackets we list the names of the authors of the corresponding results for pseudo-differential operators. This theorem follows from a more general statement for the L2 -boundedness of Fourier integral operators with symbols in Besov spaces that appeared in [50].

On Local and Global Regularity of Fourier Integral Operators

195

There are also global L2 -boundedness theorems for adjoint operators, as well as for Fourier integral operators with general amplitudes. We refer to [50] for details of all such statements, as well as for results in weighted L2 (Rn ) spaces. The global calculus of operators (7.1) as well as global boundedness theorems in weighted Sobolev spaces can be found in [51].

8. Fourier integral operators in Colombeau’s spaces In this section we will discuss properties of Fourier integral operators in the Colombeau’s spaces of new generalised functions. For details on these spaces we refer to, e.g., [11] or [34]. The theory of pseudo-differential operators in Colombeau’s spaces has been developed in [18] and [19]. Elements of the corresponding theory of Fourier integral operators has been laid down in [20]. At the same time, hyperbolic partial differential equations in Colombeau’s spaces have been studied in [28], [25] by the energy methods, yielding relevant extensions of L2 type results to the setting of new generalised functions. In this section we will present corresponding Lp -results in the setting of Colombeau’s generalised functions, for Fourier integral operators, with subsequent corresponding implications for solutions to hyperbolic equations. Let X be a bounded open set. Consider families (u ) of functions u ∈ Lp∞ (X), 0 < ≤ 1, where Lp∞ (X) stands for the space of function in Lp (X) for which all derivatives also belong to Lp (X). One can single out several important families of such functions dependent on their behaviour with respect to . Thus, the class of moderate families ELp (X) is defined as the collection of families satisfying ∀α ≥ 0 ∃N ≥ 0 : ||∂ α u ||Lp = O(−N ) as → 0. The class of null families NLp (X) is defined as a subclass of ELp (X) satisfying the condition that ∀N ≥ 0 : ||u ||Lp = O(N ) as → 0. The Colombeau’s algebra GLp (X) is then defined as Lp−∞

4

GLp (X) := ELp (X)/NLp (X).

= s∈R Lps are embedded in GLp (X) by the mapping ι(u) = Distributions [(u ∗ (ρ )] , where ρ (x) = −n ρ(x/) is the standard Friedrichs mollifier. Subsequently, one can define the corresponding classes of generalized pseudodifferential operators. Thus, the generalized symbol is defined as a family (a ) of usual symbols a ∈ S m such that ∀k, l ∃N : sup sup (1 + |ξ|)−m+|α| ∂ α ∂ β a (x, ξ) = O(−N ) as → 0. |α|≤k,|β|≤l x,ξ∈Rn

ξ

x

Then A : GLp (X) → GLp (X) is a generalised pseudo-differential operator with generalised symbol (a ) if, on the representative level, it acts as (u ) → (a (t, x, Dx )u ) .

196

M. Ruzhansky

To formulate the results, also the notion of the slow scale is required. A generalised symbol (a ) is said to be of log-type up to order (k, l) (or to be slow scale) if sup sup (1 + |ξ|)−m+|α| ∂ξα ∂xβ a (x, ξ) = O(log(1/)) as → 0. |α|≤k,|β|≤l x,ξ∈Rn

In [28], Lafon and Oberguggenberger considered partial differential operators of the form n A= aj (t, x)∂xj + b(t, x). j=1

They investigated the Cauchy problem for operator Dt + A and showed the existence of solutions in GL∞ (X) provided that b and ∂xk aj are of log-type. Moreover, if aj and b are constant for large x, then the solution is also unique, and in [35] an example of the non-uniqueness was given if this condition breaks. In [25], Hörmann considered the Cauchy problem for more general pseudo-differential operators A of log-type, for which he showed existence and uniqueness in GL2 (X) (also giving some estimates on k and l in the log-type assumption). In particular, the nonuniqueness effect disappears in GL2 (X) compared to GL∞ (X). These results may provide some hints on the behaviour of relevant Fourier integral operators in Colombeau’ spaces. Let us consider generalised Fourier integral operators of the form T u(x) = eiφ(x,y,ξ) a(x, y, ξ)u(y)dyd−ξ. X

Rn

Let us define the regular Colombeau’s algebra. The class of regular families RLp (X) is defined as a class of functions satisfying the condition ∃N ≥ 0 ∀α ≥ 0 : ||∂ α u ||Lp = O(−N ) as → 0. Then the regular Colombeau algebra GL∞p (X) is defined by GL∞p (X) := RLp (X)/NLp (X). In [20], Garetto, H¨ ormann and Oberguggenberger showed that if φ is a nondegenerate phase function then T maps GL∞ (X) to itself continuously. If the phase function φ is a generalised family satisfying the slow scale assumption (e.g., logtype), and a = (a ) is a regular family of amplitudes, then T maps locally continuously the space of regular Colombeau’s functions GL∞∞ (X) to itself. The following theorem extends this to Colombeau’s algebras over any Lp (X), 1 ≤ p ≤ ∞. Theorem 8.1. Let 1 ≤ p ≤ ∞. If φ is a non-degenerate (generalised) phase function then T maps GLp (X) to itself continuously. If the phase function φ is a generalised family satisfying the slow scale assumption (e.g., log-type), and a = (a ) is a regular family of amplitudes, then A maps locally continuously the space of regular Colombeau’s generalised functions GL∞p (X) to itself.


197

Consequently, propagators for strictly hyperbolic Cauchy problems are continuous in GLp (X), and in GL∞p (X) under the log-type assumption on the phase, for all 1 ≤ p ≤ ∞. The proof is based on the extension to the Colombeau’s setting of the eikonal and transport equations, modulo controllable errors with respect to , and on estimates for generalised Fourier integral operators in GLp (X). Details of proofs and exact losses of regularity in Colombeau’s spaces will appear in [46].

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[54] A. Seeger, C.D. Sogge and E.M. Stein, Regularity properties of Fourier integral operators, Ann. Math. 134 (1991), 231–251. [55] C.D. Sogge, Fourier Integrals in Classical Analysis, Cambridge University Press, 1993. [56] E.M. Stein, Lp boundedness of certain convolution operators, Bull. Amer. Math. Soc. 77 (1971), 404–405. [57] E.M. Stein, Harmonic Analysis, Princeton University Press, Princeton, 1993. [58] M. Sugimoto, L2 -boundedness of pseudo-differential operators satisfying Besov estimates I, J. Math. Soc. Japan 40 (1988), 105–122. or Matematik [59] M. Sugimoto, On some Lp -estimates for hyperbolic equations, Arkiv f¨ 30 (1992), 149–162. [60] T. Tao, The weak-type (1, 1) of Fourier integral operators of order −(n − 1)/2. J. Aust. Math. Soc. 76 (2004), 1–21. [61] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 2, Plenum Press, 1982. Michael Ruzhansky Department of Mathematics Imperial College London United Kingdom e-mail: [email protected]


Type 1,1-Operators Defined by Vanishing Frequency Modulation Jon Johnsen Abstract. This paper presents a general definition of pseudo-differential operators of type 1, 1; the definition is shown to be the largest one that is both compatible with negligible operators and stable under vanishing frequency modulation. Elaborating counter-examples of Ching, H¨ ormander and Parenti– Rodino, type 1, 1-operators with unclosable graphs are proved to exist; others are shown to lack the microlocal property as they flip the wavefront set of an almost nowhere differentiable function. In contrast the definition is shown to imply the pseudo-local property, so type 1, 1-operators cannot create singularities but only change their nature. The familiar rule that the support of the argument is transported by the support of the distribution kernel is generalised to arbitrary type 1, 1-operators. A similar spectral support rule is also proved. As no restrictions appear for classical type 1, 0-operators, this is a new result which in many cases makes it unnecessary to reduce to elementary symbols. As an important tool, a convergent sequence of distributions is said to converge regularly if it moreover converges as smooth functions outside the singular support of the limit. This notion is shown to allow limit processes in extended versions of the formula relating operators and kernels. Mathematics Subject Classification (2000). Primary 35S05. Keywords. Exotic pseudo-differential operators, type 1, 1, pseudo-local, spectral support rule, regular convergence, flipped wavefront sets.

1. Introduction Pseudo-differential operators are generally well understood as a result of extensive analysis since the mid 1960s; but there is an exception for operators of type 1, 1. d (Rn × Rn ), which is sometimes These have symbols in the Hörmander class S1,1 called exotic because of the operators’ atypical properties.

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d Recall that a symbol a(x, η) ∈ C ∞ (R2n ) belongs to S1,1 (Rn × Rn ) if it for all multiindices α, β satisfies the estimates

|Dηα Dxβ a(x, η)| ≤ Cα,β (1 + |η|)d−|α|+|β| .

(1.1)

For such a symbol, a(x, D)u = OP(a)u = Au is defined at least for u in the Schwartz space S(Rn ) by the usual integral, whereby the Fourier transformation ∧ is denoted F u(ξ) = u(ξ) = Rn e− i x·ξ u(x) dx, ei ξ·η a(x, η)F u(η) dη. (1.2) a(x, D)u(x) = (2π)−n Rn

The purpose of the present article is to suggest a general definition of operators with type 1, 1-symbols; that is, to define a(x, D)u for u in a maximal subspace D(A) such that S(Rn ) ⊂ D(A) ⊂ S (Rn ). (1.3) Seemingly this question has not been addressed directly before. But as a fundamental contribution, L. H¨ ormander [10, 11] used H s -estimates to extend type 1, 1-operators by continuity from S(Rn ) and characterised the possible s up to a limit point. For other questions it seems necessary to have an explicit definition of type 1, 1-operators. Consider, e.g., the pseudo-local property, sing supp Au ⊂ sing supp u for all u ∈ D(A).

(1.4)

In the proof of this, it is of course of little use just to know the action of A on u ∈ S(Rn ), as both sets are empty for such u. And to apply the fact that the distribution kernel K(x, y) of A is C ∞ for x = y one would have to know more on A and its domain D(A) than just (1.3). To give a brief account of the present contribution, let ψ ∈ C0∞ (Rn ) denote an auxiliary function for which ψ = 1 in a neighbourhood of the origin. Then the frequency modulated versions of u ∈ S (Rn ) and of a(x, η) with respect to x are given for m ∈ N by um = ψ(2−m D)u = F −1 (ψ(2−m ·)F u) a

m

= ψ(2

−m

Dx )a =

−1 Fξ→x (ψ(2−m ξ)Fx→ξ a(ξ, η)).

(1.5) (1.6)

Therefore a(x, D) is said to be stable under vanishing frequency modulation if for every u in its domain am (x, D)um −−−−→ a(x, D)u m→∞

in D (Rn ).

(1.7)

Whilst classical pseudo-differential operators have this property, the purpose is to ∞ (Rn × Rn ) show that (1.7) can be used as a definition of a(x, D)u when a ∈ S1,1 n is given; hereby D(a(x, D)) consists of the u ∈ S (R ) for which the limit exists independently of ψ. The limit in (1.7) serves as a substitute of the usual extensions by continuity from S(Rn ). In this introduction it is to be understood in (1.7) that, for all u ∈ S (Rn ), am (x, D)um = OP(am (x, η)ψ(2−m η))u,

(1.8)

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where the right-hand side is in OP(S −∞ ). The expression am (x, D)um itself is ∞ brief, but problematic if taken literally since also am (x, η) ∈ S1,1 . However, using m n m m that supp F (u ) R , it will later be seen that a (x, D)u can be defined via (1.8) and that this is compatible with (1.7); thenceforth am (x, D)um will be a short and safe notation. The definition is discussed in detail below, and shown to imply that type 1, 1-operators are pseudo-local (cf. (1.4) and Theorem 6.4). In comparison they do not in general preserve wavefront sets, for following C. Parenti and L. Rodino [19] a version of a well-known example due to C.H. Ching is shown to flip the wavefront set WF(wθ ) = Rn × (R+ θ) into Rn × (R+ (−θ)) for some wθ , that when the order d ∈ ]0, 1] is an almost nowhere differentiable function. Moreover the following well-known support rule is extended to arbitrary ∞ (Rn × Rn )) with distribution kernel K (cf. Theorem 8.1), a(x, D) ∈ OP(S1,1 supp a(x, D)u ⊂ supp K ◦ supp u for all u ∈ D(a(x, D)). (1.9) Here supp K ◦ supp u := x ∈ Rn ∃y ∈ supp u : (x, y) ∈ supp K , whereby supp K is thought of as a relation on Rn that maps, or transports, every set M ⊂ Rn to the set (supp K) ◦ M of everything related to an element of M . There is an analogous result which seems to be new, even for classical symbols ∞ a ∈ S1,0 . It gives a spectral support rule, relating frequencies ξ ∈ supp F (Au) to those in supp F u: if only u ∈ D(A) is such that (1.7) holds in the topology of S (Rn ), then (cf. Theorem 8.4) Ξ=

supp F (a(x, D)u) ⊂ Ξ, ξ + η (ξ, η) ∈ supp Fx→ξ a, η ∈ supp F u .

(1.10) (1.11)

This is highly analogous to (1.9), for Ξ = supp K ◦ supp F u, where K is the kernel of the conjugated operator F a(x, D)F −1 . There is a forerunner of (1.10)–(1.11) in [15], where it was only possible to cover the case F u ∈ E (Rn ), as the information on D(a(x, D)) was inadequate without the definition in (1.7). The spectral support rule (1.10) often makes it possible to by-pass a reduction to elementary symbols, that were introduced by R. Coifman and Y. Meyer [5] in order to control spectra like supp F a(x, D)u in the Lp -theory of general pseudodifferential operators. Use of (1.10)–(1.11) simplifies the theory, for it would be rather inconvenient to add in (1.7) an extra limit process resulting from approximation of a(x, η) by elementary symbols. Both (1.9) and (1.10) are established as consequences of the formula relating an operator A to its kernel K ∈ D (Rn × Rn ), Au, v = K, v ⊗ u .

(1.12)

It is shown below (cf. Theorems 7.4 and 8.1) that also the right-hand side makes sense as it stands for u ∈ D (Rn ), although K and v ⊗ u are distributions then, as long as v is a test function such that sing supp K sing supp v ⊗ u = ∅. (1.13) supp K supp v ⊗ u Rn × Rn ,

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That (1.13) suffices for (1.12) follows from the extendability of the bilinear form ·, · in distribution theory to pairs (u, f ) fulfilling analogous conditions. This simple extension of u, f has the advantage that u, f ν → u, f when u or f has compact support and f ν ∈ C ∞ (Rn ) are such that f ν −−−−→ f ν→∞

both in D (Rn ) and in C ∞ (Rn \ sing supp f ).

(1.14)

Such sequences (f ν ) are below said to converge regularly to f ; they are easily obtained by convolution. In these terms, ·, · is stable under regular convergence if one entry is in E . This set-up is convenient for the derivation of (1.12)–(1.13) for type 1, 1operators. Indeed, the kernel Km of the approximating operator am (x, D)um equals K ∗ F −1 (ψm ⊗ ψm ) conjugated by the coordinate change (x, y) → (x, x − y), so that Km converges regularly to K; whence (1.12) results in the limit m → ∞. Based on this the support rules (1.9)–(1.10) follow in a natural way. However, the simple criterion in (1.13) and its stability under regular convergence, that might be known, could be useful also for other questions. The main contributions in this paper consist first of all of the definition (1.7) and the spectral support rule (1.10) ff; secondly of the proofs of pseudolocality (1.4) and the support rule (1.9) as well as the extension of the kernel formula (1.12)–(1.13). Moreover, a(x, D)u is shown to be compatible with the usual pseudo-differential operators (cf. Sections 4–5). In addition there are various improvements of known results on type 1, 1operators. This overlap is elucidated (in parenthetic remarks) in the next section. 1.1. On known results for type 1, 1-operators The pathologies of type 1, 1-operators were revealed around 1972–73. On the one 0 hand, C.H. Ching [4] gave examples of symbols a ∈ S1,1 for which the correspond2 n 2 ing operators are unbounded from L (R ) to L (K) for every K Rn (they can moreover be taken unclosable in S (Rn ), as shown in Lemma 3.2 below). On the other hand, E.M. Stein (1972-73) showed C s -boundedness1 for s > 0 and orders d = 0. Afterwards C. Parenti and L. Rodino [19] discovered that some type 1, 1operators do not preserve wavefront sets (cf. Section 3.2 where this result of [19] is extended to all d ∈ R, n ∈ N). The pseudo-local property of type 1, 1-operators was also claimed in [19], but not backed up by adequate arguments; cf. Remark 6.5 below. (The question is therefore taken up in Theorem 6.4, where the first full proof is given.) 1 Noted

by Y. Meyer [17], with reference to lecture notes at Princeton 1972/73. E.M. Stein stated the C s -result in [24, VII.1.3]; at the end of Ch. VII its origins were given as “Stein [1973a]” (that is Singular integrals and estimates for the Cauchy-Riemann equations, Bull. Amer. Math. Soc. 79 (1973), 440–445) but probably should have been “Stein [1973b]”: “Pseudo-differential operators, Notes by D.H. Phong for a course given at Princeton University 1972-73”.

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Around 1980, Y. Meyer [17, 18] obtained the famous property that a composition operator u → F (u), for a fixed C ∞ -function F with F (0) = 0, acting on u ∈ Hps (Rn ) for s > n/p, can be written F (u) = au (x, D)u for a specific u-dependent symbol au ∈ Littlewood–Paley partition of unity, au (x, η) =

∞

0 S1,1 .

mj (x)Φj (η),

mj (x) = 0

j=0

1

(1.15)

Namely, when 1 =

∞ j=0

Φj is a

F ( Φk (D)u(x) + tΦj (D)u(x)) dt. k<j

(1.16) This gave a convenient proof of the fact that u → F (u) maps Hps (Rn ) into itself d for s > n/p. Indeed, this follows as Y. Meyer for general a ∈ S1,1 , using reduction to elementary symbols, established continuity a(x,D)

Hps+d (Rn ) −−−−→ Hps (Rn )

for s > 0, 1 < p < ∞.

(1.17)

(In Section 9.2 these results are deduced from the definition in (1.7), and continuity on Hps of u → F ◦ u is added in a straightforward way in Theorem 9.4.) It was also realised then that type 1, 1-operators show up in J.-M. Bony’s paradifferential calculus [1] of non-linear partial differential equations. In the wake of this, T. Runst [20] treated the continuity in Besov spaces s s Bp,q for p ∈ ]0, ∞] and in Lizorkin–Triebel spaces Fp,q for p ∈ ]0, ∞[ , although the necessary control of the frequency changes created by a(x, D) was not quite achieved in [20]. (This flaw was explained and remedied in [15] by means of a less general version of (1.10).) Around the same time G. Bourdaud proved that a given type 1, 1-operator a(x, D) : C0∞ (Rn ) → D (Rn ) of order 0 is L2 -bounded if and only if its adjoint a(x, D)∗ : C0∞ (Rn ) → D (Rn ) is also a type 1, 1-operator; cf. [2], [3, Th 3]. Except for a limit point, L. H¨ ormander characterised the s ∈ R for which d a given a ∈ S1,1 is bounded H s+d → H s ; cf. [10, 11] and also [12] where a few improvements are added. As a novelty in the analysis, an important role was shown to be played by the twisted diagonal T = { (ξ, η) ∈ Rn × Rn | ξ + η = 0 }.

(1.18)

∧

E.g., if the partially Fourier transformed symbol a(ξ, η) := Fx→ξ a(x, η) vanishes in a conical neighbourhood of a non-compact part of T , i.e., if ∧

∃C ≥ 1 : C(|ξ + η| + 1) ≤ |η| =⇒ a(x, η) = 0,

(1.19)

then a(x, D) : H s+d → H s is continuous for every s ∈ R. Moreover, continuity for all s > s0 was shown to be equivalent to a specific asymptotic behaviour of ∧ a(ξ, η) at T . For operators with additional properties, a symbolic calculus was also developed together with a sharp G˚ arding inequality; cf. [10, 11, 12].

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s For domains of type 1, 1-operators, the scale Fp,q (Rn ) of Lizorkin–Triebel spaces was recently shown to play a role, for it was proved in [14, 15] that for all d p ∈ [1, ∞[ , every a ∈ S1,1 gives a bounded linear map a(x,D)

d (Rn ) −−−−→ Lp (Rn ). Fp,1

(1.20)

This is a substitute of boundedness from Hpd (or of Lp -boundedness for d = 0), s s s s Fp,1 for 1 < p < ∞. Inside the Fp,q and Bp,q scales, (1.20) gives as Hps = Fp,2 maximal domains for a(x, D) in Lp , for it was noted in [15, Lem. 2.3] that already d d to D and from Bp,q to D for every Ching’s operator is discontinuous from Fp,q n q > 1. Continuity was proved in [15] for s > max(0, p − n), 0 < p < ∞, as a map a(x,D)

s+d s Fp,q (Rn ) −−−−→ Fp,r (Rn ) for

r ≥ q, r >

n n+s .

(1.21)

Moreover, (1.19) was shown to imply (1.21) for every s ∈ R, r = q. Analogously s . (In Section 9.1 it is shown how the techniques behind (1.21) apply in for Bp,q the present set-up, and as a special case (1.17) is rederived in this way; cf. Theorem 9.2.) d (Rn × As indicated, a general definition of a(x, D)u for a given symbol a ∈ S1,1 n R ) seems to have been unavailable hitherto. L. H¨ ormander [10, 11] estimated Au for arbitrary u ∈ S(Rn ) in the H s -scale, which of course gives4a uniquely defined bounded operator A : H s+d → H s ; and an extension of A to s>s0 H s+d (Rn ) for some limit s0 or possibly even s0 = −∞, depending on a. R. Torres [25] also estimated Au for u ∈ S(Rn ), using the framework of M. Frazier and B. Jawerth [6, 7]. This gave unique extensions by continuity to s+d s (Rn ) → Fp,q (Rn ) for all s so large that, for all multiindices γ, maps Fp,q n n (1.22) 0 ≤ |γ| < max 0, − n, − n − s =⇒ A∗ (xγ ) = 0. p q (As noted in [25], this is related to the conditions on the twisted given 4 s diagonal by L. H¨ ormander.) This approach will at most define A on Fp,q (Rn ). In addition it was shown in [15, Prop. 1] that every type 1, 1-operator A extends to the space F −1 E (Rn ). (Extension to F −1 E is also considered in Section 4 in connection with compatibility questions.) Clearly F −1 E contains 4 4 s all polynomi , so this develals |α|≤k cα xα , and these do not belong to H s , nor to Fp,q opment only emphasizes the need for a general definition of type 1, 1-operators, without reference to spaces other than S (Rn ). 1.2. Remarks on the construction As indicated above, the extension of an operator a(x, D) of type 1, 1 from the Schwartz space S(Rn ) to a larger domain D(a(x, D)) in S (Rn ) can roughly be made as follows: −1 ∧ (a(ξ, η)ψm (ξ)), ψm = ψ(2−m ·) for a cut-off Introducing am (x, η) = Fξ→x ∞ n function ψ ∈ C0 (R ) with ψ = 1 around the origin, then a(x, D)u is defined

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when u ∈ S (Rn ) is such that aψ (x, D)u = limm→∞ OP(am (x, η)ψm (η)) exists in D (Rn ) and does not depend on ψ. And in the affirmative case, a(x, D)u := aψ (x, D)u = lim OP(am (x, η)ψm (η))u. m→∞

(1.23)

Fundamentally, the role of am (x, η) is to make the domain of a(x, D) as large as possible: since a(x, η) is less special than am (x, η), the demands on the pair (a, u) would be stronger if only the OP(a(x, η)ψm (η))u were required to converge. And the domain of a(x, D) would possibly also be smaller, had not the same ψ been used twice to form am (x, η)ψm (η). Finally, to take the limit in S (Rn ) instead might also exclude some u from D(a(x, D)). (However, the D -limit makes it more demanding to justify compositions b(x, D)a(x, D) of type 1, 1-operators.) Although (1.23) is an unconventional definition, it is not as arbitrary as it may ∞ seem. In fact, cf. Theorem 5.9 below, the resulting map a → a(x, D), a ∈ S1,1 , can be characterised as the largest extension of (1.2) that both gives operators stable under vanishing frequency modulation and is compatible with OP on S −∞ . ∞ For δ < ρ it is even compatible with the classes OP(Sρ,δ ) in a certain local sense, termed strong compatibility below. In addition to this, there are at least three simple indications that the defini∞ tion is reasonable. First of all, if the symbol a(x, η) is classical, say a ∈ S1,0 , then m m the usual S -continuous extension of OP(a) fulfils OP(a (x, η)ψ (η))u → OP(a)u as a consequence of standard facts (cf. Proposition 5.4 below). Secondly, the definition also gives back the usual product au, when a(x) is a ∞ independent of η. In fact a ∈ Cb∞ (Rn ) then, and since am (x)ψm (η) ∈ symbol in S1,1 −∞ n n (R × R ), every u ∈ S gives the following S ∧

OP(am (x)ψm (η))u = am · F −1 (ψm u) = am um −−−−→ au. m→∞

(1.24)

So despite the apparent asymmetry in OP(am ψm )u, where only the symbol is subjected to frequency modulation, the definition is consistent with the product au. However, the expression am (x, D)um , that enters (1.7), is symmetric in this sense. Thirdly, continuity properties of a(x, D) can be conveniently analysed using Littlewood–Paley techniques applied to both the symbol a and the distribution u. This is facilitated because the Fourier multiplication by ψm occurs in both entries of am (x, D)um . Indeed, one can take ψm to be the first m + 1 terms in a Littlewood–Paley partition of unity 1 = ∞ j=0 Φj ; then bilinearity gives a direct transition to the paradifferential splitting that has been used repeatedly for Lp continuity results since the 1980s. The reader is referred to Section 9 for details. Remark 1.1. Analogously to (1.23), there is an extension of the pointwise product 1 1 1 1 n (u1 , u2 ) → u1 u2 , where uj ∈ Lloc pj (R ) for j = 1, 2 with p1 , p2 , p1 + p2 ∈ [0, 1], to n n the pairs (u, v) in S (R ) × S (R ) for which there is a ψ-independent limit π(u, v) := lim um v m m→∞

(1.25)

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This general product π(u, v) was introduced and extensively analysed with paramultiplication in [13]; e.g., the convergence in (1.24) follows directly from [13, Prop. 3.6]. By (1.24) one recovers π(a, u) from (1.23) when the symbol a(x, η) is independent of η. (An open question for π(·, ·) is settled in Theorem 6.7, where partial associativity is proved from the fact that multiplication by C ∞ -functions commutes with vanishing frequency modulation.) The definition sketched in (1.7) was used rather implicitly in recent works of the author [14, 15]. In the present article, the purpose is to introduce the definition of a(x, D)u in (1.23) systematically and to show that it is consistent with (1.2). Section 2 gives a review of notation and some preparations, whereas in Section 3 the special properties of type 1, 1-operators are elaborated. Section 4 deals with preliminary extensions of type 1, 1-operators, using cut-off techniques. The general definition of a(x, D) is given in Section 5, where it is proved to be consistent with the usual one if, say a(x, η) coincides (for η running through an open set d d , or Sρ,δ with ρ > δ. SecΣ ⊂ Rn ) with an element of the classical symbol class S1,0 tion 6 contains the proof of the pseudo-local property. As a preparation, extended action of the bracket ·, · from distribution theory is studied in Section 7, with consequences for distribution kernels. A control of supp a(x, D)u is proved in Section 8, as is the spectral support rule in a general version. Finally Section 9 deals with continuity in the Sobolev spaces Hps and a quick review of the consequences for composite functions.

2. Notation and preparations The distribution spaces E , S and D , that are dual to C ∞ , S and C0∞ respectively, have the usual meaning as in, e.g., [9]. OM (Rn ) stands for the space of slowly increasing functions, i.e., the f ∈ C ∞ (Rn ) such that to every multiindex α there are Cα > 0, Nα > 0 such that |Dα f (x)| ≤ Cα (1+|x|)Nα for all x ∈ Rn . In addition Cb∞ denotes the Frechét space of smooth functions with bounded derivatives of any order. The Sobolev space Hps (Rn ) with s ∈ R and 1 < p < ∞ is normed by uHps = F −1 ((1 + |ξ|2 )s/2 F u)p , whereby up = ( Rn |u|p dx)1/p is the norm of Lp (Rn ); similarly · ∞ denotes that of L∞ (Rn ). That a subset M of Rn has compact closure is indicated by M Rn . As usual c denotes a real constant specific to the place of occurrence. With the short-hand ξ = (1 + |ξ|2 )1/2 , a symbol a(x, η) is said to be in d Sρ,δ (Rn × Rn ) if a ∈ C ∞ (R2n ) and for all multiindices α, β there exists Cα,β ≥ 0 such that |Dξα Dxβ a(x, ξ)| ≤ Cα,β ξd−ρ|α|+δ|β| .

(2.1)

Here it is assumed that the order d ∈ R and 0 < ρ ≤ 1, 0 ≤ δ ≤ 1 with δ ≤ ρ, which is understood throughout unless further restrictions are given.

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Along with this there is a pseudo-differential operator a(x, D) defined on every u in the Schwartz space S(Rn ) by the Lebesgue integral ∧ −n a(x, D)u(x) = OP(a)u(x) = (2π) ei x·η a(x, η)u(η) dη. (2.2) Rn

Here η is the dual variable to y ∈ Rn (u is seen as a function of y), while ξ is used for the dual variable to x. If ψ ∈ C0∞ (Rn ) and ψ = 1 near 0, then ψm = ψ(2−m ·) gives: d (Rn ×Rn ) when a(x, η) itself Lemma 2.1. am (x, η) = ψm (Dx )a(x, η) belongs to Sρ,δ m d does so, and then a(x, η) = limm→∞ a (x, η)ψm (η) holds in Sρ,δ (Rn × Rn ) when d ≥ d + δ and d > d. d is bounded with respect to x, the first part results from Proof. Since a ∈ Sρ,δ ∨ |Dxβ Dηα am (x, η)| ≤ |ψ(y)||Dxβ Dηα a(x − 2−m y, η)| dy ≤ Cα,β ηd−ρ|α|+δ|β| . (2.3) d+δ Since ψ(0) = 1 the mean value theorem gives am → a in Sρ,δ ; and for any d > d d one has am (x, η)ψm (η)−am (x, η) → 0 in Sρ,δ ; whence am (x, η)ψm (η)−a → 0.

It is straightforward to show from (2.2) that the bilinear map d (Rn × Rn ) × S(Rn ) −→ S(Rn ) OP : Sρ,δ

(2.4)

is continuous. Hereby S(Rn ) has a Fréchet space structure with seminorms ψ |S, N = sup{ |xN Dβ ψ(x)| | x ∈ Rn , |β| ≤ N };

(2.5)

d (Rn × Rn ) is a Fréchet space with the least Cα,β in (2.1) as seminorms. whilst S1,1 4 d ∞ For fixed symbol a in S1,1 := d S1,1 the map (2.4) induces a continuous operator a(x, D) : S(Rn ) → S(Rn ), that cannot in general be extended to a continuous map S (Rn ) → D (Rn ); this is well known cf. Section 3 below. The next lemma extends [9, Lem. 8.1.1] from u ∈ E (Rn ) to general u ∈ n S (R ). The extension is irrelevant for the definition of wavefront sets WF(u), but useful for calculations. It is hardly a surprising result, but without an adequate reference a proof is given here. Recall that V ⊂ Rn is a cone if R+ V ⊂ V . Throughout R± = { t ∈ R | ±t > 0 }. First the singular cone Σ(u) is defined as the complement in Rn \ {0} of those ∧ ξ = 0 contained in an open cone Γ ⊂ Rn \ {0} fulfilling that u is in Lloc 1 over Γ and ∧

CN := supηN |u(η)| < ∞,

N > 0.

(2.6)

Γ

Then Σ(u) = ∅ when u ∈ S(Rn ), and only then (the unit sphere Sn−1 is compact). Lemma 2.2. Whenever u ∈ S (Rn ), then Σ(ϕu) ⊂ Σ(u) for all ϕ ∈ C0∞ (Rn ), and WF(u) ⊂ sing supp u × Σ(u).

(2.7)

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J. Johnsen ∧

∧

Proof. It is well known that S ∗ S ⊂ OM , so ϕu(ξ) 5 = (2π)−n u, ϕ(ξ − ·) is C ∞ . 5 < Given a cone Γ disjoint from Σ(u), it suffices to show that supΓ1 ηN |ϕu(η)| ∞ on every closed cone Γ1 ⊂ Γ ∪ {0} with supremum independent of Γ1 . When ξ ξ = 0 is fixed in Γ1 , then |ξ| ∈ Γ1 ∩ Sn−1 and this set has distance d > 0 to Rn \ Γ, so for 0 < θ < 1 one has η ∈ Γ in the cone Vθ = { η = 0 | |ξ − η| < θd|ξ| }. ∧ Supposing u = 0 in B(0, 14 ), one can take χ0 + χ1 = 1 on Sn−1 such that ξ ξ χ0 ∈ C ∞ (Sn−1 ), χ0 (ζ) = 1 for |ζ − |ξ| | < d3 and χ0 (ζ) = 0 for |ζ − |ξ| | ≥ d2 . Then η = 0 gives ∧

ϕ(ξ − η) = ϕ0 (η) + ϕ1 (η),

for

∧

η ϕj (η) := χj ( |η| )ϕ(ξ − η),

(2.8)

and both terms are in C ∞ (Rn \ {0}) with respect to η, by stereographic projection ∧ and the chain rule. Now supp ϕ0 ⊂ V2/3 ⊂ Γ and u ∈ Lloc 1 (Γ) with rapid decay in ∧ Γ so that one can estimate u, ϕ0 by means of an integral, ∧ ∧ ∧ η ξN | u, ϕ0 | ≤ 2N ξ − ηN |χ0 ( |η| )||ϕ(ξ − η)|ηN |u(η)| dη Γ (2.9) ∧ N ≤ 2 CN +n+1 ϕ·N ∞ χ0 ∞ η−n−1 dη < ∞. In Rn \ V1/3 one has |ξ| ≤ d3 |ξ − η| and |η| ≤ (1 + 3d )|ξ − η|, so using the seminorms in (2.5), ∧

ξN | u, ϕ1 | ≤ c

sup η ∈V / 1/3 ,|α|≤M

∧

η ξN ηM |Dηα (χ1 ( |η| )ϕ(ξ − η))|

≤ c(1 + d3 )N +M (

∧

Dα χ1 ∞ ) ϕ |S, N + M < ∞.

(2.10)

|α|≤M

Finally one can take χ ˜ ∈ C0∞ (Rn ) such that χ ˜ = 1 for |η| ≤ 14 with support in ∧ ∧ B(0, 12 ), then ξN | u, χ ˜ | ≤ c χ ˜ |S, N + M ϕ |S, N + M follows as in (2.10). N All bounds are uniform in ξ and in Γ1 , hence sup 5 < ∞. This proves ) Γ · |ϕu| Σ(ϕu) ⊂ Σ(u), so (2.7) holds as WF(u) = {(x,ξ) | ξ ∈ ϕ(x)=0, ϕ∈C ∞ Σ(ϕu)}. 0

Remark 2.3. In Lemma 2.2 equality obviously holds in (2.7) if the singular cone is a ray, i.e., if Σ(u) = R+ ζ for some ζ ∈ Rn . In such cases WF(u) can be easily determined.

3. Special properties of type 1, 1-operators Many of the pathological properties of type 1,1-operators can be obtained from simple examples of the form aθ (x, η) =

∞ j=1

2jd e− i 2

j

x·θ

χ(2−j η),

(3.1)

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211

whereby χ ∈ C0∞ (Rn ) with supp χ ⊂ { η | 34 ≤ |η| ≤ 54 }; and θ ∈ Rn is fixed. d since the terms are disjoint. Clearly aθ is in S1,1 Such symbols were used by C.H. Ching [4] and G. Bourdaud [3] for d = 0, |θ| = 1 to show L2 -unboundedness. Refining their counter-examples, L. H¨ ormander linked continuity from H s with s ≥ −r, r ∈ N0 , to the property that θ is a zero of χ of order r. This is generalised to θ ∈ Rn here because (3.1) with |θ| = 1 enters the proof that type 1, 1-operators do not always preserve wavefront sets. And by consideration of arbitrary orders d ∈ R the counter-examples get interesting additional properties; cf. Remark 3.5 ff. ∧ From the definition of aθ (x, η) in (3.1) it is clear that u ∈ C0∞ (Rn ) gives ∞ ∧ −n (2π) (3.2) ei x·η 2jd χ(2−j η + θ)u(η + 2j θ) dη. aθ (x, D)u = j=1

Then the adjoint bθ (x, D) : S (Rn ) → S (Rn ) of aθ (x, D) fulfils, for all v ∈ S(Rn ), u, bθ (x, D)v = aθ (x, D)u, v ∞ ∧ −n jd −j ∧ j u(ξ) 2 χ(2 ¯ ξ)v(ξ − 2 θ) dξ, = (2π)

(3.3)

j=1

∞

∧

¯ −j ξ)v(ξ − 2j θ). This gives a convenient way to so F bθ (x, D)v(ξ) = j=1 2jd χ(2 s calculate the H -norm of bθ (x, D)v, for when this is finite the disjoint supports of the χ(2−j ·) imply (2π)n bθ (x, D)v2H s ∞ = j=1

=

∧

(1 + |ξ|2 )s 4jd |χ(2−j ξ)v(ξ − 2j θ)|2 dξ

2j−1 |d| that for η ∈ B(2k θ, r), k > J, −N cN 4 N 2j r kd −jd j (k−j)d cN . 2 2 |5 ϕv(η−2 θ)| ≤ 2 − ≤ 1− 2kN 2k 2k 2JN (1 − 2d−N ) n

J<jJ

This estimate is uniform in k > J; hence sing supp wθ = Rn . So by Lemma 2.2 and Remark 2.3, WF(u) = sing supp wθ × Σ(wθ ) = Rn × (R+ θ), i.e., (3.18) is obtained. Remark 3.4. It is clear from (3.20) that a2θ (x, η)w(θ, d; x) = w(θ , 0; x) for θ = θ+2θ , |θ | = 1 = |θ|. But θ can point in any direction in Rn , so type 1, 1-operators can make arbitrary directional changes in wavefront sets (as noted in [19]). Remark 3.5. There is an amusing reason why the counter-example wθ in Proposition 3.3 is singular on all of Rn , i.e., why sing supp wθ = Rn . In fact wθ (x) = ∞ −jd i 2j t v(x)f (x · θ) where f (t) = e , and this is for 0 < d ≤ 1 a wellj=1 2 known variant of Weierstrass’ nowhere differentiable function (a fact that could have substantiated the argument for formula (19) in [19]). That the theory of type 1, 1-operators is linked to this classical construction seems to be previously unobserved. ∞ j Remark 3.6. To elucidate Remark 3.5, f (t) = j=1 2−jd ei 2 t is investigated here. ∞ Clearly f ∈ S (R) for all d ∈ R, as the Fourier transformed series 2π j=1 2−jd δ2j converges there. By uniform convergence f is for d > 0 a continuous 2π-periodic and bounded function. Nowhere-differentiability for 0 < d ≤ 1 is an easy (maybe not widely known) exercise in distribution theory: supp F f is lacunary, so any choice of χ ∈ ) ∧ ∧ ∧ S(R) such that χ(1) = 1 and supp χ ⊂ ] 21 , 2[ will give supp χ(2−k ) supp δ2j = ∅ only for j = k, which entails k χ(z)(f (t − 2−k z) − f (t)) dz; (3.25) 2−kd ei 2 t = 2k χ(2k ·) ∗ f (t) = R

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so if f were differentiable at t0 , G(h) := h1 (f (t0 + h) − f (t0 )) would be in C(R) ∩ L∞ (R), and the contradiction d > 1 would follow since by majorised convergence k ∧ 2(1−d)k ei 2 t0 = − G(−2−k z)zχ(z) dz − −−−→ f (t0 ) · (−D)χ(0) = 0. (3.26) k→∞

Moreover, if m < d ≤ m + 1 for m ∈ N it follows by termwise differentiation that f ∈ C m (R), but with f (m) nowhere differentiable; so sing supp f = R for all d > 0. For d ≤ 0 one has f ∈ / Lloc 1 , for f is invariant in S under translation by 2π, loc so if f ∈ L1 is assumed, f, ϕ = f ϕ dt holds for ϕ in C0∞ (R) as well as in S, since | f ϕ dt| ≤ c supt∈R (1 + |t|2 )|ϕ(t)| follows from the fact that (1 + r2 )−1 f (r) is in L1 : 2π ∞ 2(p+1)π |f (r)| |f (r)| 2 dr = dr ≤ |f | dr (3.27) 1+(2pπ)2 < ∞. 2 1 + r2 2pπ 0 R 1+r p=0 p∈Z

Therefore the convolution in (3.25) is given by the integral also in this case. By taking t outside a Gδ -set G of measure 0, f is continuous (in R \ G) at t, whence 2−kd = | 2k χ(2k r)(f (t − r) − f (t)) dr| − −−− → 0. (3.28) k→∞

In fact, for ε > 0 the part with |r| < δ is < ε for some δ > 0, but sup|r|≥δ (1 + |t − r|2 )2k |χ(2k r)| = O(2−k ), so since L1 (R) by (3.27) contains r → (f (t − r) − f (t))/(1 + |t − r|2 ) the limit 0 results. Thence the contradiction d > 0. cos(bj t) , To complete the picture, Weierstrass’ original function W (t) = ∞ j=0 aj where b ≥ a > 1, is nowhere differentiable by the same argument. One only has to j j take supp F χ ⊂ ] 1b , b[ , F χ(1) = 1, for in F cos(bj ·) = 2π 2 (δb + δ−b ) the last term ∧ is removed by χ(b−j ·), so that χ(b−j D) yields a second microlocalisation of W . As k in (3.25)–(3.26) it follows that ( ab )k ei b t0 → 0 for k → ∞, contradicting that b ≥ a. A further study of nowhere differentiable functions by means of microlocalisation can be found in [16]. d (R); for 0 < d < 1 Remark 3.7. As a precise account of the regularity, f ∈ B∞,∞ this Besov space consists of Hölder continuous functions of order d. Indeed, the d is from the left part of (3.25) seen to equal 1, when χ is taken norm of f in B∞,∞ ∞ as Φ1 in a suitable Littlewood–Paley partition of unity 1 = j=0 Φj . Moreover, d s (R) when d ∈ R, 0 < p < ∞, for the definition in [26] of Fp,q gives, f ∈ Fp,∞;loc ∧

1 when v ∈ S(R) with supp v ⊂ B(0, 20 ), 1/p j jd p vf Fp,∞ (sup 2 |Φj (D)(vf )|) dt d = = sup |v(t)ei 2 t |p = vp . (3.29) j

j

d s the Fp,∞ -regularity is sharp. That ϕf ∈ Fp,∞ (R) for ϕ ∈ ϕ d ∞ = v vf ∈ Fp,∞ when ϕ ∈ C0 (R) has support in R \ {v =

These are identities, so C0∞ (R) results from ϕf 0}; one can reduce to this with a partition of unity on ϕ and translation of v in each term.

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Remark 3.8. Returning to Remark 3.5, note that wθ (x) = v(x)f (x · θ) is almost nowhere differentiable for 0 < d ≤ 1, since v has isolated zeroes. If d ≤ 0 n −kd then w(θ, d; ·) ∈ / Lloc v(x) = 1 (R ) for else one can derive the contradiction 2 kn k 2 χ(2 ·) ∗ [vf ( ·, θ )] → 0 by modifying the corresponding part of Remark 3.6. As in Remark 3.7 it follows that d (Rn ) for all p ∈ ]0, ∞[ , d ∈ R. w(θ, d; x) ∈ Fp,∞;loc

(3.30)

If w(θ, ˜ d; x) is defined as w(θ, d; x) except with a further factor 1/j in each summand, similar arguments yield that w ˜ ∈ H s now for s ≤ d as well as the other properties in Proposition 3.3, with a nowhere differentiable series for 0 < d < 1. d Moreover, w(θ, ˜ d; ·) is in Fp,q;loc (Rn ) as soon as q > 1 for every p ∈ ]0, ∞[ . Hence the counter-examples with unclosable graphs and violated microlocal properties d are both related to Lizorkin–Triebel spaces Fp,q with arbitrary q > 1; cf. (1.20) and Section 3.1.

4. Preliminary extensions Throughout Fy→η etc. will denote partial Fourier transformations, that are all homeomorphisms on S (R2n ). In general the Fourier transformation in all variables ∧ is written F u or u, except that for a symbol a ∈ S (R2n ), ∧

a(ξ, η) := Fx→ξ a(ξ, η).

(4.1)

Transformation of coordinates via (x, y) → (x, x − y), that has matrix M = I 0 −1 = M , is indicated by f ◦ M . I −I As a preparation some well-known formulae are recalled: ∞ Proposition 4.1. Let a ∈ S1,1 (Rn × Rn ) and u, v, f, g ∈ S(Rn ). Then i x·η

∧

e a(x, D)u, v = (2π) n a(x, η), v(x) ⊗ u(η) = K, v ⊗ u . I 0 −1 −1 (a)(x, x − y) = Fη→y (a(x, η)) ◦ . K(x, y) = Fη→y I −I ∧ ∧ ∧ Fa(x, D)f, F g = (2π)−n a(ξ − η, η)f (η)g(ξ) dξdη.

(4.2) (4.3) (4.4)

Here . . . dξdη is valid as an integral for a ∈ S(R2n ), but should be read as the ∞ . scalar product on S × S for general a ∈ S1,1 Proof. By Fubini’s theorem, (4.2) holds for a ∈ S(R2n ), when K is given as in (4.3). The bijection a ↔ K extends to a homeomorphism on S (R2n ). So by d+1 d density of S in S1,1 , as subsets of S1,1 hence of S , the identities in (4.2) hold d for all a ∈ S1,1 . Formula (4.4) results from (4.2) for u = f , v = F 2 g, since ∧

∧

∧

(F 2 g) ⊗ f = Fξ→x (g ⊗ f ).

218

J. Johnsen ∧

The partially Fourier transformed symbol a(ξ, η) is closely related to the distribution kernel K(x, y) of a(x, D) as well as to the kernel K(ξ, η) of the conjugation F a(x, D)F −1 of a(x, D) by the Fourier transformation on Rn . Indeed, ∧ modulo simple isomorphisms, a gives both K and the frequencies in K: ∞ (Rn × Rn ), and K, K and M are as above, Proposition 4.2. When a ∈ S1,1 ∧

∧

F K(ξ, η) = a(ξ + η, −η) = a ◦ M t (ξ, η) = (2π)n K(ξ, −η).

(4.5)

−1 (e− i x·η a(x, −η)), since F −1 commutes with Proof. (4.3) implies that K = Fη→y reflections in η and y. Then (4.5) follows by application of F and (4.4).

The right-hand side of (4.2) is inconvenient for the definition of type 1, 1operators, as in general both entries of K, v ⊗ u have singularities (in some cases this can be handled, cf. Section 7). However, it is a well-known fact that also in case ρ = 1 = δ the kernels only have singularities along the diagonal. d (Rn × Rn ) the kernel K(x, y) is C ∞ for x = y. Lemma 4.3. For every a ∈ S1,1

Proof. For N so large that d + |β| + |α| − 2N < −n, α β |z|2N Dxβ Dzα Fη→z (a(x, η)) = Fη→z (ΔN η (η Dx a(x, η)))

(4.6)

is a continuous function, so any derivative of K is so for x = y.

Instead the middle of (4.2) gives a convenient way to prove that every type 1, 1-operator extends to F −1 E (Rn ), i.e., to the space of tempered distributions with compact spectrum. This result was first observed in [15], but the following argument should be interesting for its simplicity. When v ∈ C0∞ (Rn ) and u ∈ F −1 C0∞ (Rn ) then (4.2) gives ∧

a(x, D)u, v = v(x) ⊗ u(η),

ei x·η ∞ (2π)n a(x, η) E ×C .

(4.7)

This suggests to introduce A : F −1 E (Rn ) → C ∞ (Rn ) given by ∧

Au(x) = u,

ei x, · (2π)n a(x, ·) .

(4.8)

∞

That Au is in C is a standard fact used, e.g., in the construction of tensor products on E (Rn )×E (Rn ); cf. [9, Th. 5.1.1]. By definition of the tensor product ∧ of arbitrary v, u ∈ E (Rn ) in (4.7), they should act successively on the C ∞ -function, ∞ so for v in C0 (Rn ), ∧ i ·, · −n a · (2π) = v, Au = v(x)Au(x) dx. (4.9) v ⊗ u, e Rn −1

This and (4.7) gives Au = a(x, D)u for every u ∈ F C0∞ = S ∩ F −1 E ; hence a(x, D) and A are compatible. Therefore a(x, D) extends to a map a(x, D) : S(Rn ) + F −1 E (Rn ) → C ∞ (Rn )

(4.10)

by setting a(x, D)u = a(x, D)v + Av when u = v + v for v ∈ S and F v ∈ E (if 0 = v + v , clearly v = −v is in F −1 C0∞ , hence gives identical images, i.e., a(x, D)v + Av = 0).

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219

By the duality of E and C ∞ , the right-hand side of (4.8) should be calculated by multiplying a(x, η) by a cut-off function χ(η) equalling 1 on a neighbourhood of supp F u. The resulting symbol χ(η)a(x, η) is clearly in d S −∞ (Rn × Rn ) := Sρ,δ (Rn × Rn ). (4.11) d,ρ,δ

A systematic exploitation of localisations χ(η)a(x, η) is found in the next section. 4.1. Extension by spectral localisation For type 1, 1-operators, this section gives a first extension, based on cut-off techniques and arguments from algebra. The latter are trivial, but important for several compatibility questions that are treated here. Let SΣ (Rn ) denote the closed subspace of distributions with spectrum in a given open set Σ ⊂ Rn , i.e., ∧ SΣ (Rn ) = u ∈ S (Rn ) supp u ⊂ Σ . (4.12) Clearly the intersection SΣ (Rn ) := S(Rn ) ∩ SΣ (Rn ) is dense in SΣ (Rn ). ∞ As a basic assumption in this section, a(x, η) ∈ S1,1 should have the properties of a more ‘well-behaved’ symbol class S as η runs through a given open set Σ ⊂ Rn . It would then be natural, and necessary, to extend a(x, D) to every u ∈ SΣ (Rn ) by letting it act as an operator with symbol in the class S. To turn this idea into a definition, an arbitrary linear subspace S ⊂ S (R2n )∩ ∞ C (R2n ) will in the following be called a standard symbol space if, for every b ∈ S, the integral in (2.2) gives an operator OP(b) : S → S which extends to a continuous linear map OP(b) : S (Rn ) → S (Rn ). (4.13) (Such an extension is unique, so the notation need not relate OP(b) to the choice of S. To avoid confusion, the type 1, 1 operator under extension is usually denoted d with (ρ, δ) = (1, 1); whilst OP(b) could be a(x, D).) An example could be S = Sρ,δ the extension to S of b(x, D) given by the adjoint of b∗ (x, D) : S(Rn ) → S(Rn ), that in its turn is defined from the adjoint symbol b∗ (x, ξ) = ei Dx ·Dξ ¯b(x, ξ). ∞ Using this, a(x, D) can be extended if the symbol a ∈ S1,1 is locally in a n standard symbol space S in an open set Σ ⊂ R . Specifically this means that for every closed set F ⊂ Σ there exists a cut-off function χ ∈ Cb∞ (Rn ), not necessarily supported by Σ, such that χ ≡ 1 on a neighbourhood of F,

χ(η)a(x, η) ∈ S.

(4.14)

Instead of a(x, η)χ(η), the slightly more correct a(1 ⊗ χ) is often preferred in the sequel. ∞ (Rn × Rn ) that is locally in S in an open Proposition 4.4. For each symbol a ∈ S1,1 n n set Σ ⊂ R , there is defined a map SΣ (R ) → S (Rn ) by

a(x, D)u = OP(a(1 ⊗ χ))(u),

(4.15)

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J. Johnsen

which has the same value for all χ ∈ Cb∞ (Rn ) satisfying (4.14) for F = supp F u. The map is compatible with a(x, D) : S(Rn ) → S(Rn ). Proof. Let u ∈ SΣ (Rn ). By (4.14) and (4.13), OP(a(1 ⊗ χ)) is defined on S u. If χ1 is another such function, a(x, η)(χ(η) − χ1 (η)) is in the vector space S and equals 0 for η in some open set Σ1 ⊃ supp u, so that by density of SΣ1 in SΣ 1 , 0 = OP(a(1 ⊗ (χ − χ1 )))u.

(4.16)

Therefore (4.15) is independent of the choice of χ, so the map OP(a(1 ⊗ χ))u is defined; it equals a(x, D)u for every u ∈ S ∩ SΣ by (2.2). The compatibility in Proposition 4.4 gives of course a map on the algebraic subspace S(Rn ) + SΣ (Rn ) ⊂ S (Rn ); cf. (4.18). But more holds: ∞ Theorem 4.5. For every a ∈ S1,1 (Rn × Rn ) that in an open set Σ ⊂ Rn is locally in a standard symbol space S (cf. (4.13)), the operator a(x, D) extends to a linear map a(x,D)

S(Rn ) + SΣ (Rn ) −−−−→ S (Rn )

given by

(4.17)

a(x, D)u = a(x, D)v + OP(a(1 ⊗ χ))v

SΣ (Rn );

(4.18)

Cb∞ (Rn )

hereby χ ∈ can be any whenever u = v + v for v ∈ S(R ), v ∈ function fulfilling (4.14) for F = supp F v . The extension is uniquely determined by coinciding with (4.13) on SΣ (Rn ). n

Proof. For uniqueness, let OP(a) be any extension agreeing with (4.13) on SΣ (Rn ). Then linearity gives, for any splitting u = v + v and χ as in the theorem, that OP(a)u = OP(a)v + OP(a)v = a(x, D)v + OP(a)v = a(x, D)v + OP(a(1 ⊗ χ))v . (4.19) To show that (4.18) actually defines the desired map, suppose u = v+v = w+ w for some v, w ∈ S and v , w ∈ SΣ . Applying a(x, D) to v − w and OP(a(1 ⊗ χ)) to w − v , with χ taken so that χ ≡ 1 on a neighbourhood of F = supp F v ∪ supp F w ⊃ supp F (w − v ), it follows from the compatibility in Proposition 4.4 and linearity that, for χ = χ1 = χ2 ,

a(x, D)v + OP(a(1 ⊗ χ1 ))v = a(x, D)w + OP(a(1 ⊗ χ2 ))w .

(4.20)

By Proposition 4.4 one can then pass to arbitrary χ1 , χ2 equalling 1 around supp F v , respectively supp F w , without changing the left and right-hand sides. This means that (4.18) gives a map, for a(x, D)v + OP(a(1 ⊗ χ))v is independent of the splitting u = v + v and of the corresponding choice of χ; thence linear dependence on u follows too. Theorem 4.5 gives a basic extension of type 1, 1-operators, that could have ∞ been a definition (justified by the given arguments). When a ∈ S1,1 happens to be in S too, then χ ≡ 1Rn and v = 0 yields a(x, D)u = OP(a)u, so the definition (4.18) gives back the S -continuous operators with symbols in S.

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221

Before these questions are pursued, the construction’s dependence on S and Σ is investigated. ∞ Proposition 4.6. Let S and S˜ be standard symbol spaces, and let a ∈ S1,1 be locally n ˜ Then the in S in some open set Σ ⊂ R and also locally in S˜ in an open set Σ. induced maps

a(x, D) : S + SΣ → S ,

a(x, D) : S + SΣ →S

(4.21)

are compatible when either Σ has the property that χ in (4.14) for every F can be ˜ has the analogous property. taken with supp χ ⊂ Σ, or Σ Proof. One can reduce to the case S = S˜ by introducing the subspace S + S˜ ⊂ S (R2n ): for every b ∈ S, ˜b ∈ S˜ the definition by the usual integral shows that OP(b + ˜b) = OP(b) + OP(˜b) on S(Rn ).

(4.22)

Here OP(b + ˜b) extends to a continuous, linear map S (Rn ) → S (Rn ) since the ˜ both OP(b + ˜b), right-hand side does so. If b + ˜b = b1 + ˜b1 for b1 ∈ S, ˜b1 ∈ S, OP(b1 + ˜b1 ) extend to S , where they coincide as they do so on S. Hence every b + ˜b in S + S˜ gives an unambiguously defined operator on S (Rn ), as required in (4.13); i.e., S + S˜ is a standard space. Let u = v + w = v˜ + w ˜ for some v, v˜ ∈ S, w ∈ SΣ and w ˜ ∈ SΣ . By the ˜ ϕ ≡ 1 on a last assumption there exists, e.g., ϕ ∈ Cb∞ (Rn ) such that supp ϕ ⊂ Σ, ˜ ˜ neighbourhood of F and a(1 ⊗ ϕ) ∈ S. In particular 1 − ϕ = 0 around F˜ so u = ϕ(D)(v + w) + (1 − ϕ(D))(˜ v + w) ˜ = v˜ + ϕ(D)(v − v˜) + ϕ(D)w.

(4.23)

n While v = v˜ +ϕ(D)(v − v˜) is in S(Rn ), the term ϕ(D)w is in SΣ∩ ˜ (R ). By taking Σ ˜ ψ = 1 in a neighbourhood of supp F ϕ(D)w ⊂ Σ ∩ Σ, it is clear that one gets

a(x, D)u = a(x, D)v + OP(a(1 ⊗ ψ))ϕ(D)w by application of Theorem 4.5 both for SΣ and SΣ˜ .

(4.24)

As a simple application for Σ = Rn , every u ∈ F −1 E (Rn ) is in SΣ = S (Rn ); and a is locally in S −∞ since b(x, η) = a(x, η)χ(η) is in S −∞ for every χ ∈ C0∞ (Rn ), in particular when χ = 1 around supp F u. Therefore Theorem 4.5 yields a unique extension of a(x, D) to a linear map S(Rn ) + F −1 E (Rn ). (Proposition 4.6 shows ∞ or let Σ depend on u that one can replace the reference to S −∞ by, e.g., S1,0 without changing the image a(x, D)u.) This approach is more elementary than (4.7) ff. In addition it gives that a(x, D) maps F −1 E (Rn ) into OM (Rn ). Recall that every a(x, D) in OP(S −∞ (Rn × Rn )) is a map S → OM (cf. [23, Cor. 3.8]), since if u ∈ S , then (1 + |x|2 )−N u ∈ H −N for some N > 0 and every commutator [Dα a(x, D), (1 + |x|2 )N ] is by inspection in OP(S −∞ ). These facts imply the next result.

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∞ Corollary 4.7. Every operator a(x, D) with symbol in S1,1 extends uniquely to a map a(x, D) : S(Rn ) + F −1 (E (Rn )) → OM (Rn ), (4.25)

which is given by Theorem 4.5 with S = S −∞ . Notice that the corollary’s statement is purely algebraic, since continuity properties are not involved in (4.25). Similarly one has another extension result. ∞ d Proposition 4.8. If a ∈ S1,1 is locally in the symbol class S1,0 (Rn × Rn ) in an open n cone V ⊂ R (i.e., tη ∈ V for all t > 0 and η ∈ V ), then (4.18) yields a unique extension (4.26) a(x, D) : S + SV → S . ∞ If some a ∈ S1,1 satisfies the hypotheses of Proposition 4.8, it follows from Proposition 4.6 that the two extensions in (4.25)–(4.26) are compatible with one another. ∞ Example.By Corollary 4.7, the domain of every a(x, D) in OP(S1,1 ) contains polyα nomials |α|≤k cα x , as their spectra equal {0}, and, e.g., also the C ∞ -functions α

xα ei x·z = (2π)n F −1 (D η δz (η));

sin x1 x1

. . . sinxnxn = π n F −1 1[−1;1]n .

(4.27)

5. Definition by vanishing frequency modulation The full extension of type 1, 1-operators is given here by means of a limiting procedure. d (Rn × Rn ), d ∈ R, and suitable To define a(x, D)u in general for a ∈ S1,1 n u ∈ S (R ), it is convenient for an arbitrary ψ ∈ C0∞ (Rn ) with ψ ≡ 1 in a neighbourhood of the origin to introduce the following notation, with ψm (ξ) := ψ(2−m ξ), ∧

um = F −1 (ψm u) = ψm (D)u, m

a (x, η) =

−1 Fξ→x (ψm (ξ)Fx→ξ a(ξ, η))

(5.1) = ψm (Dx )a(x, η).

(5.2)

This is referred to as a frequency modulation of u and of a(x, η) with respect to x; the full frequency modulation of a will be am (x, η)ψm (η), i.e., am (1 ⊗ ψm ). Since ∞ am is in S1,1 by Lemma 2.1, the compact support of ψm gives that am (1 ⊗ ψm ) ∈ S −∞ .

(5.3)

Hence OP(am (1 ⊗ ψm )) is defined on S (Rn ), and since limm→∞ am (1 ⊗ ψm ) = a d+1 holds in S1,1 , it should be natural to make a tentative definition of a(x, D) as a(x, D)u = lim OP(am (1 ⊗ ψm ))u. m→∞

(5.4)

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Some difficulties that might appear in this connection are dealt with in the formal Definition 5.1. The pseudo-differential operator a(x, D)u is defined as the limit in d (5.4) for those a ∈ S1,1 (Rn × Rn ) and u ∈ S (Rn ) for which the limit • exists in the topology of D (Rn ) for every ψ ∈ C0∞ (Rn ) equalling 1 in a neighbourhood of the origin, and • is independent of such ψ. To show that a(x, D) extends the operator defined on S(Rn ) by (2.2), it suffices to combine Lemma 2.1 with (2.4). As shown below, Definition 5.1 also gives back both the usual operator OP(a) if a is, e.g., of type 1, 0 and the extensions in Section 4. ∞ and As an elementary observation, by using the definition for a fixed a ∈ S1,1 by the calculus of limits, the operator is defined for u in a subspace of S (Rn ). This will be denoted by D(a(x, D)), or D(A) if A := a(x, D), in the following. Clearly D(A) ⊃ S(Rn ), so A is a densely defined and linear operator from n S (R ) to D (Rn ) (borrowing terminology from unbounded operators in Hilbert spaces). This description cannot be improved much in general, for by Lemma 3.2, a(x, η) can be chosen so that A with D(A) = S(Rn ) is unclosable. But one has ∞ Proposition 5.2. For a, b in S1,1 (Rn × Rn ) the following properties are equivalent:

(i) (ii) (iii) (iv)

a(x, η) = b(x, η) for all (x, η) ∈ R2n ; a(x, D) = b(x, D) as operators from S (Rn ) to D (Rn ); a(x, D)u = b(x, D)u for every u ∈ S(Rn ); the distribution kernels fulfil Ka = Kb .

d d ↔ OP(S1,1 ); and the operator In particular the map a → a(x, D) is a bijection S1,1 a(x, D) is completely determined by its action on the Schwartz space.

The last property is perhaps not obvious from the outset, because, in general, there is neither density of S ⊂ D(a(x, D)) nor continuity of a(x, D) to appeal to. However it follows at once, as it is straightforward to see that (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) =⇒ (i). The following notion is very convenient for the analysis of a(x, D): Definition 5.3. A standard symbol space S on Rn × Rn is said to be stable under vanishing frequency modulation if in addition to (4.13), (i) bm (1 ⊗ ψm )(x, η) = ψ(2−m Dx )b(x, η)ψm (η), is in S for every b ∈ S, m ∈ N, and every ψ ∈ C0∞ (Rn ) equalling 1 near the origin, (ii) for every u ∈ S (Rn ), and ψ as above, OP(bm (1 ⊗ ψm ))u → OP(b)u

in D (Rn ) for m → ∞.

For short S and the operator class OP(S) are then said to be stable.

(5.5)

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Note that (i) requires the operator class OP(S) to be invariant under full frequency modulation; whereas (ii) requires OP(S) to be invariant under vanishing frequency modulation in the sense that the limit gives back the original operator OP(b). d ∞ Although S1,1 is not a standard space, OP(S1,1 ) is also said to be stable, as ∞ . Other stable spaces (5.5) holds by definition for every u in D(b(x, D)), b ∈ S1,1 exist as well: d Proposition 5.4. Every Sρ,δ (Rn × Rn ) with ρ > δ for ρ, δ ∈ [0, 1] is a stable symbol space. Moreover, (5.5) holds in the S -topology.

d , Proof. By Lemma 2.1 condition (i) is satisfied, and lim bm (1 ⊗ ψm ) = b in Sρ,δ ∗ d > d + δ. As OP(b) is the adjoint of b (x, D) = OP(exp(i Dx · Dη )b), each ϕ ∈ S(Rn ) gives

( OP(bm (1 ⊗ ψm ))u − OP(b)u | ϕ ) = ( u | OP(ei Dx ·Dη (bm (1 ⊗ ψm )− b))ϕ ) −−−−→ 0, m→∞

d d → Sρ,δ for ρ > δ. since passage to adjoint symbols b → b∗ is continuous Sρ,δ

(5.6)

Proposition 5.4 makes the definition of a(x, D) by vanishing frequency modulation look natural. To analyse the consistency questions in general, it is recalled that OP(a) is defined on S(Rn ) by the integral (2.2) if a is in a standard space S or ∞ in S = S1,1 . And for a standard space S, OP(a) extends uniquely to a continuous linear map on S (Rn ). Let now a → OP(a) be an arbitrary assignment such that OP(a), for each ∞ a ∈ S1,1 , is a linear operator from S (Rn ) to D (Rn ). Then the maps OP and OP n are compatible on a standard symbol space S if D(OP(a)) = S (R ) for every ∞ a ∈ S ∩ S1,1 and OP(a)u = OP(a)u

for all u ∈ S (Rn ).

(5.7)

∞ Moreover, OP and OP are called strongly compatible on S if, whenever a is in S1,1 n n and belongs to S locally in some open set Σ ⊂ R , it will hold that SΣ (R ) ⊂ D(OP(a)) and

OP(a)u = OP(a(1 ⊗ χ))u

for all u ∈ SΣ (Rn ). ∧

(5.8)

Hereby χ ∈ Cb∞ (Rn ) should fulfil (4.14) for F = supp u and a(1 ⊗ χ) ∈ S. (The right-hand side of (5.8) makes sense because of χ, but it does not depend on χ since S is standard.) Taking Σ = Rn and χ ≡ 1, strong compatibility clearly implies compatibility. As an example Corollary 4.7 shows that, if the preliminary extension of Section 4.1 is written OP, then OP(a) is strongly compatible with OP on S −∞ . More generally Theorem 4.5 gives strong compatibility of OP with OP on every standard symbol class S.

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The following theorem shows that a(x, D) given by Definition 5.1 contains every extension provided by Theorem 4.5 when S is stable. ∞ Theorem 5.5. Let a ∈ S1,1 and Σ ⊂ Rn be an open set such that a locally in Σ d belongs to a stable symbol class S (such as Sρ,δ for ρ > δ). Then every u ∈ S(Rn )+ n SΣ (R ) belongs to the domain D(a(x, D)) given by Definition 5.1. Moreover,

a(x, D)u = OP(a)v + OP(a(1 ⊗ χ))v

whenever u is split as u = v + v for v ∈ S(R ), v ∈ fulfils (4.14) for F = supp F v . n

SΣ (Rn ),

(5.9) and χ ∈

Cb∞ (Rn )

Proof. Let u ∈ SΣ . Since a is locally in S in Σ one can take χ as in the theorem, so that a(1 ⊗ χ) ∈ S. Using that S in particular is a standard space, approximation ∧ of u from C0∞ gives OP(am (x, η)(1 − χ(η))ψm (η))u = 0. Now (5.5) applies, since S is stable; and multiplication by χ(η) and ψm (Dx ) commute in S (Rn × Rn ), so OP(a(1 ⊗ χ))u = lim OP(ψm (η)χ(η)ψm (Dx )a(x, η))u m

= lim OP(am (1 ⊗ ψm ))u = a(x, D)u.

(5.10)

m

This shows that SΣ (Rn ) ⊂ D(a(x, D)). And for u ∈ S(Rn ) it is seen already from (2.4) that am (x, D)um → a(x, D)u in S(Rn ) for m → ∞. Since a(x, D) is linear by Definition 5.1, it follows that every u in S + SΣ belongs to D(a(x, D)) and that (5.9) holds. In particular the last statement that (5.9) is independent of v, v and χ is implied by this (and by Theorem 4.5). Remark 5.6. It is noteworthy that Theorem 5.5 resolves a dilemma resulting from ∞ application of a(x, D) ∈ OP(S1,1 ) to u ∈ F −1 (E (Rn )): then a(x, D)u can be calculated by using both Corollary 4.7 and Definition 5.1. But by taking S = S −∞ , Theorem 5.5 entails that the two methods give the same result. ˜ are unnecessary It follows from Theorem 5.5 that the assumptions on Σ and Σ in Proposition 4.6 in case S is stable (this emphasizes the advantage of using vanishing frequency modulation). As a reformulation of Theorem 5.5 one has ∞ (Rn × Rn ) by Definition 5.1 Corollary 5.7. The operator a(x, D) given for a ∈ S1,1 is strongly compatible with OP on every stable symbol space S. In particular ∞ a(x, D)u = OP(a)u holds for every u ∈ S (Rn ) when a ∈ Sρ,δ (Rn × Rn ) for some δ < ρ.

As a special case a(x, D) gives back OP(a) on S −∞ . This may also be shown by verifying (5.7) directly, but one can only simplify (5.10) slightly by taking χ ≡ 1 on Rn . The various consistency results obtained in this section can be summed up thus: ∞ . Then a(x, D)u Corollary 5.8. Let a(x, D) be given by Definition 5.1 for a ∈ S1,1 n equals the integral in (2.2) for u ∈ S(R ) or the extension in Corollary 4.7 for every u ∈ F −1 E (Rn ); and it coincides with the extension of OP(a) to S (Rn ) if a d is in Sρ,δ (Rn × Rn ) for some ρ > δ.

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To characterise the operators provided by Definition 5.1, it is convenient to ignore that the compatibility of a(x, D) is strong (cf. Corollary 5.7). Indeed, the map a → a(x, D) is simply the largest compatible extension stable under vanishing frequency modulation: Theorem 5.9. The operator a(x, D) given by Definition 5.1 is one among the op∞ erator assignments a → OP(a), a ∈ S1,1 (Rn × Rn ) with the properties that (i) OP(·) is compatible with OP on S −∞ (cf. (5.7)); (ii) each operator OP(b) is stable under vanishing frequency modulation, i.e., ∞ OP(b)u = limm→∞ OP(bm (1 ⊗ ψm ))u for every u ∈ D(OP(b)), b ∈ S1,1 . OP is such a map, then OP(a) ⊂ a(x, D) for every And moreover, whenever ∞ a ∈ S1,1 . m (1 ⊗ ψm )) by (i) is defined on all of S . Note that (ii) makes sense as OP(b Proof. Let a → OP(a) be any map fulfilling (i) and (ii); such maps exist since a → a(x, D) was seen above to have these properties. If u ∈ D(OP(a)) it follows from (i) that m (1 ⊗ ψm ))u. OP(am (1 ⊗ ψm ))u = OP(a (5.11) Here the right-hand side converges to OP(a)u by (ii); since ψ is arbitrary this means OP(a)u = a(x, D)u. Hence OP(a) ⊂ a(x, D). This section is concluded with a few remarks on the practical aspects of Definition 5.1. From the integral in (2.2), one would at once infer the following alter egos for the full frequency modulation of a(x, D): if χ ∈ C0∞ (Rn ) fulfils χ = 1 around supp ψm , then ∧

am (x, D)um = OP(am (x, η)χ(η))F −1 (ψm u) = OP(am (x, η)ψm (η))u.

(5.12)

However, these identities hold also for more general cut-off functions χ. ∞ Lemma 5.10. For every a ∈ S1,1 , u ∈ S and every ψ ∈ C0∞ with ψ = 1 near the origin, the formula (5.12) holds for all m and all χ ∈ Cb∞ for which χ = 1 in a ∧ neighbourhood of F = supp(ψm u) and a(1 ⊗ χ) ∈ S −∞ .

Proof. The last part of (5.12) follows from (2.2) if u is a Schwartz function, hence for all u ∈ S (Rn ) since both am (1 ⊗ ψm ) and am (1 ⊗ χ) belong to S −∞ . Since ∞ um ∈ F −1 E and am (x, η) ∈ S1,1 , Corollary 5.8 shows that am (x, D)(um ) can be calculated by the extension in Section 4.1; then Theorem 4.5 gives the left-hand side of (5.12). In view of (5.12), one could alternatively have defined a(x, D)u as a limit of am (x, D)um . This would be an advantage in as much as the expression am (x, D)um is a natural point of departure for Littlewood–Paley analysis of a(x, D)u (as explained later, cf. (9.8)); it would also make a and u enter in a more symmetric fashion. But as a drawback the resulting definition of a(x, D) would then have two

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steps, the first one being an extension to F −1 E as in Section 4.1. In comparison the limit in (5.4) only refers to S −∞ , cf. (5.3), which made it possible to state Definition 5.1 directly; cf. (1.7). Formula (5.12) is so self-suggesting that it is convenient to write am (x, D)um without further explanation, instead of the slightly tedious OP(am (1⊗ψm ))u, that enters Definition 5.1. (This is permitted as the two expressions are equal for every choice of the auxiliary function ψ, cf. Lemma 5.10.) Since Definition 5.1 is based on a limit of am (x, D)um , it is useful to relate the distribution kernel Km (x, y) of u → am (x, D)um to the kernel K(x, y) of a(x, D). The symbol of am (x, D)um is am (1 ⊗ ψm ) ∈ S −∞ , cf. (5.12), so (4.3) and the definition of am give, for all u, v ∈ S(Rn ), −1 am (x, D)um , v = Fη→y (am (1 ⊗ ψm ))(x, x − y), v(x) ⊗ u(y)

(5.13) = F −1 (ψm (ξ)ψm (η)Fx→ξ a(ξ, η))) ◦ M, v ⊗ u . 0 = I on R2n , and M = II −I = M −1 , formula (5.13) shows

−1 −1 Because F Fη→y Fξ→x that Km (x, x − y) = F −1 ((ψm ⊗ ψm )F (K ◦ M ))(x, y).

(5.14)

This can be restated as follows: ∞ Proposition 5.11. When a ∈ S1,1 and ψ ∈ C0∞ (Rn ) equals 1 in a neighbourhood of the origin, then the distribution kernel Km (x, y) of u → am (x, D)um , cf. (5.12), is the function in C ∞ (Rn × Rn ) given by

Km (x, y) = F −1 (ψm ⊗ ψm ) ∗ (K ◦ M )(x, x − y),

(5.15)

which is the conjugation by ◦M of the convolution of K(x, y) by the Schwartz function 4nm F −1 ψ(2m x)F −1 ψ(2m y). Naturally, this result will be useful for the discussion in the next section.

6. Preservation of C ∞ -smoothness It is well known that every classical pseudo-differential operator A = a(x, D) is pseudo-local, sing supp Au ⊂ sing supp u for every u ∈ D(A).

(6.1)

In the context of type 1, 1-operators, the requirement u ∈ D(A) should be made explicitly as the domain D(A) in many cases will be only a proper subspace of S (Rn ). It could be useful to call Ω := Rn \ sing supp u the regular set of u, for this set has the important property that regularisations of u converge (not just in S (Rn ) but also) in the topology of C ∞ (Ω). This fact could well be folklore, but references seem unavailable, and since it is the crux of the below proof of pseudo-locality, details are given for the reader’s convenience.

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Lemma 6.1 (The Regular Convergence Lemma). Let u ∈ S (Rn ) and set ψk (ξ) = ψ(εk ξ) for some sequence εk 0 and ψ ∈ S(Rn ). Then ψk (D)u → ψ(0) · u

for

k→∞

(6.2)

in the Fréchet space C ∞ (Rn \ sing supp u). If F −1 ψ ∈ C0∞ (Rn ) the conclusion holds for all u ∈ D (Rn ), if ψk (D)u is replaced by (F −1 ψk ) ∗ u. In the topology of S (Rn ) the well-known property (6.2) is easy, for test ∧ ∧ against ϕ¯ ∈ S(Rn ) reduces (6.2) to the fact that ψ(εk ·)ϕ → ψ(0)ϕ in S(Rn ). The main case is of course ψ(0) = 1. For ψ(0) = 0 one obtains the occasionally useful fact that ψk (D)u → 0 in C ∞ over the regular set of u. Proof. Let K Rn \ sing supp u =: Ω and take a partition of unity 1 = ϕ + χ with ϕ ∈ C ∞ (Rn ) such that ϕ ≡ 1 on a neighbourhood of K and supp ϕ Ω. This ∞ n gives −1a splitting ψk (D)u = ψk (D)(ϕu) + ψk (D)(χu) where ϕu ∈ C0 (R ). Since F ψ dx = ψ(0), ∨ ∨ α D [ψ k ∗ (ϕu) − ψ(0)ϕu] = ψ(y)[Dα (ϕu)(x − εk y) − Dα (ϕu)(x)] dy =−

n j=1

1

∨

ψ(y)∂xj Dα (ϕu)(x − θεk y)εk yj dθdy.

(6.3)

0

Using the seminorms in (2.5) in a crude way, |∂j Dα (ϕu)(x − θεk y)εk yj | ≤ εk |y| ϕu |S, |α| + 1 0,

(6.4)

∨

so consequently Dα (ψ k ∗ (ϕu)) → ψ(0)Dα u uniformly on K. u For ψk (D)(χu) it is used that continuity of S(Rn ) − → C gives c, N > 0 such that ∨ ∨ x − y yN β | χu, Dxα ψ k (x − ·) | ≤ c (6.5) Dy (χ(y)Dα ψ( sup )). n+|α| εk y∈Rn , |β|≤N ε k

Here 0 < d := dist(K, supp χ) ≤ |x − y| for x ∈ K, y ∈ supp χ, so every −1 l l N negative power of εk fulfils ε−l ≤ k ≤ (1 + εk |x − y|) /d . Moreover, (1 + |y|) N −1 cK (1 + |x − y|/εk ) for εk < 1. So evaluation of an S-seminorm at F ψ yields supK |Dα (F −1 ψk ∗ (χu))| ≤ Cεk 0. When F −1 ψ ∈ C0∞ , clearly (F −1 ψk ) ∗ (χu) = 0 around K eventually. In the sequel the main case is the one in which ψ itself has compact support, so the proof above is needed. 6.1. The pseudo-local property The following sharpening of Lemma 6.1 shows that, in certain situations, one even has convergence f ψk (D)u → f u in S. To obtain this in a general set-up, let x ∈ Rn be split in two groups as x = (x , x ) with x ∈ Rn , x ∈ Rn .

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Proposition 6.2. Suppose u ∈ S (Rn ) has sing supp u ⊂ { x = (x , x ) | x = 0 } and that f u ∈ S(Rn ) for every f in the subclass C ⊂ Cb∞ (Rn ) consisting of the f for which |x | is bounded on supp f and supp f ∩ { x | x = 0 } = ∅. Then ∧

→ ψ(0)f u f F −1 (ψk u) −−−− k→∞

in S(Rn ),

(6.6)

for every sequence ψk = ψ(εk ·) given as in Lemma 6.1. Proof. For f ∈ C it is straightforward to see that there is a δ such that inf{ |x | | x ∈ supp f } ≥ δ > 0.

(6.7)

One can then take ϕ ∈ C such that ϕ ≡ 1 where |x | ≥ δ/2, hence on K = supp f . Mimicking the proof of Lemma 6.1, compactness of K is not needed since ϕu is in S by assumption. Instead of (6.4) one should estimate xN |∂j Dα (ϕu)(x − θεk y)εk yj |,

(6.8)

but (1 + |x|)N ≤ (1 + |x − θεk y|)N (1 + |y|)N when εk < 1, so it follows mutatis mutandis that for an arbitrary seminorm, ∨

f ψ k ∗ (ϕu) − ψ(0)f u |S, N ≤ cεk 0. And because χ = 1 − ϕ fulfils d = dist(K, supp χ) ≥

δ 2

(6.9)

> 0, one gets as in (6.5),

∨

f ψ k ∗ (χu) |S, N ≤ cεk .

(6.10)

Indeed, xN ≤ (1 + |y|)N (1 + |x − y|/εk )N and now factors like yN are harmless as (1 + |y|)N ≤ (1 + |y |)N (1 + |y |)N ≤ (1 + |x |)N (1 + |y − x|)N (1 + |y |)N , (6.11) where |y | < δ on supp χ whilst |x | is bounded on supp f .

For distribution kernels there is a similar result, but in this case it is well known that one need not assume rapid decay: let f ∈ Cb∞ (R2n ) have its support disjoint from the diagonal Δ = { (x, x) | x ∈ Rn } and bounded in the x-direction, that is Δ ∩ supp f = ∅ (6.12) ∃R > 0 : (x, y) ∈ supp f =⇒ |x| ≤ R. Then f (x, y)K(x, y) is in S(R2n ) whenever K is the kernel of a type 1, 1-operator. Indeed, (1 + |(x, y)|)N ≤ (1 + |x|)N (1 + |y|)N ≤ (1 + |x|)2N (1 + |y − x|)N

(6.13)

and here |x| is bounded on supp f , so by setting z = x − y in (4.6) one obtains that (x, y)N Dxα Dyβ (f K) is bounded for all N , α, β.

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J. Johnsen Invoking Proposition 6.2 this gives

∞ Proposition 6.3. If a ∈ S1,1 (Rn × Rn ) has kernel K and Km is the approximating kernel given by (5.15), then it holds for every f ∈ Cb∞ (R2n ) with the property (6.12) that f Km −−−−→ f K in S(R2n ). (6.14) m→∞

Proof. The class C of Proposition 6.2 contains f (x, x − y), and Proposition 5.11 gives f (x, x − y)Km (x, x − y) = f (x, x − y)F −1 (ψm ⊗ ψm ) ∗ (K ◦ M ).

(6.15)

The right-hand side tends to (f K) ◦ M in S(R ) according to Proposition 6.2, so it remains to use the continuity of ◦M in S(Rn ). 2n

It can now be proved that operators of type 1, 1 are pseudo-local. The argument below is classical up to the appeal to (6.18). In case A is S -continuous, this formula follows at once from the density of S in S . However, in general A is not even closable, but instead the limiting procedure of Definition 5.1 applies via the approximation in Proposition 6.3. ∞ (Rn × Rn ) the operator A = a(x, D) has the Theorem 6.4. For every a ∈ S1,1 pseudo-local property; that is sing supp Au ⊂ sing supp u for every u ∈ D(A).

Proof. Let ψ, χ ∈ C0∞ (Rn ) have supports disjoint from sing supp u such that χ ≡ 1 around supp ψ. Then χu ∈ C0∞ (Rn ) so that also (1 − χ)u is in the subspace D(A) and ψAu = ψA(χu) + ψA(1 − χ)u. (6.16) ∞ n Here ψA(χu) ∈ C0 (R ) since A : S → S, while u → ψA(1 − χ)u is seen at once to have kernel ˜ K(x, y) = ψ(x)K(x, y)(1 − χ(y)). (6.17) The function f (x, y) = ψ(x)(1 − χ(y)) fulfils (6.12), for Δ contains no contact ˜ ∈ S(R2n ) point of { f = 0 } because dist(supp ψ, supp(1 − χ)) > 0. Therefore K as seen after (6.12). This strongly suggests that, with χ1 = 1 − χ, ˜ for all ϕ ∈ C0∞ (Rn ). ψAχ1 u, ϕ = ϕ ⊗ u, K (6.18) And it suffices to prove this identity, for by definition of the tensor product it ˜ entails that ψAχ1 u = u, K(x, ·) which is a C ∞ -function of x ∈ Rn . m Now if Am := OP(a (x, η)ψm (η)) and Km is its kernel, one can take ul ∈ C0∞ (Rn ) such that ul → u in S . Applying Definition 5.1 to A, the S -continuity of Am gives ψAχ1 u, ϕ = lim Am χ1 u, ψϕ = lim lim Km , (ψϕ) ⊗ (χ1 ul ) . m→∞

m→∞ l→∞

(6.19)

Here Km ∈ C ∞ (R2n ) by Lemma 5.11, so for the right-hand side one finds, since u → ϕ ⊗ u is S -continuous and f Km ∈ S(R2n ), (6.20) ψ(x)ϕ(x)χ1 (y)ul (y)Km (x, y) d(x, y) −−−→ ϕ ⊗ u, (ψ ⊗ χ1 )Km . l→∞

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˜ in S(R2n ) by Proposition 6.3, the proof is As (ψ ⊗ χ1 )Km = f Km → f K = K complete. Remark 6.5. Theorem 6.4 was anticipated by C. Parenti and L. Rodino [19], although they just appealed to the fact that K(x, y) is C ∞ for x = y. This does not ˜ ·) , quite suffice as ψAχ1 u should be identified with a C ∞ -function, e.g., u, K(x, n for u ∈ D(A) \ S(R ); which is non-trivial in the absence of continuity and the usual rules of calculus. 6.2. A digression on products The opportunity is taken here to settle an open problem for the generalised pointwise product π(u, v) mentioned in Remark 1.1. First the commutation of pointwise multiplication and vanishing frequency modulation is discussed. Let u ∈ S (Rn ) and f ∈ OM (Rn ) be given and ψm = ψ(2−m ·) for some arbitrary ψ ∈ S(Rn ) with ψ(0) = 1. Approximating f u in two ways in S , Bm u := ψm (D)(f u) − f ψm (D)u → 0 for

m → ∞.

(6.21)

This commutation in the limit is not, however, a direct consequence of pseudodifferential calculus, for the commutator Bm has amplitude bm (x, y, η) = (f (y) − β a(x, y, η)| ≤ f (x))ψ(2−m η), which is in the space of symbols with estimates |Dηα Dx,y −N n n for all N > 0, K R ×R . As such OP(bm (x, y, η)) is only defined Cα,β,K,N η on E (Rn ). However, (6.21) is seen at once to hold in C ∞ (Rn \ sing supp u), by using Lemma 6.1 on both terms. The next results confirms that Bm u → 0 even in C ∞ (Rn ), despite the singularities of u. The idea is to use Lemma 6.1 once more to get a reduction to f ∈ C0∞ (Rn ), so that Bm → 0 in the globally estimated class OP(S −∞ (Rn × Rn )): Proposition 6.6. When u ∈ S (Rn ), f ∈ OM (Rn ), and ψ ∈ S(Rn ) with ψ(0) = 1, it holds true that limm→∞ (ψm (D)(f u) − f ψm (D)u) = 0 in the topology of C ∞ (Rn ). Proof. When χ ∈ C0∞ (Rn ) equals 1 on a neighbourhood of a given compact set K ⊂ Rn , then K is contained in the regular set of (1 − χ)u, so it follows as above from Lemma 6.1 that supK,|α|≤l |Dα (ψm (D)(f (1 − χ)u) − f ψm (D)(1 − χ)u)| → 0 for m → ∞. It now suffices to cover the case in which K ⊂ supp u ⊂ supp f Rn . Then Bm has symbol bm (x, η) = (ei Dx ·Dη − 1)f (x)ψm (η) ∈ S −∞ (Rn × Rn ).

(6.22)

However, u ∈ H t for some t < 0, and Bm ∈ B(H t , H s ) for all s > 0, whence Dα Bm u∞ ≤ cBm uH s ≤ cBm uH t for s > l + n/p. (6.23) |α|≤l

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It remains to show that the operator norm Bm → 0. Using direct estimates as in, e.g., [10, Prop. 2.2], it is enough to show for all N > 0, α, β that lim sup ηN |Dxβ Dηα bm (x, η)| → 0 for

m → ∞.

m→∞ η∈Rn

(6.24)

But Dηα , Dxβ commute with ei Dx ·Dη , so it suffices to treat α = 0 = β for general f and ψ, i.e., to show that uniformly in x ∈ Rn |(ei Dx ·Dη − 1)f (x)ψm (η)| ≤ c |Dxβ Dηα Dx · Dη f (x)ψm (η)| |α|+|β|≤2n+2 (6.25) −m(|α|+1) −N η ). = O(2 The estimate to the left is known, and follows directly from [10, Prop. B.2]. Altogether supx∈K,|α|≤l |Dα Bm u| → 0 for m → ∞, as claimed.

Besides being of interest in its own right, Proposition 6.6 gives at once a natural property of associativity for the product π in Remark 1.1. Theorem 6.7. The product (u, v) → π(u, v) is partially associative, i.e., when (u, v) ∈ S (Rn ) × S (Rn ) is in the domain of π(·, ·) so is (f u, v) and (u, f v) for every f ∈ OM (Rn ) and f π(u, v) = π(f u, v) = π(u, f v). Proof. For every ϕ ∈ C (K) = ψ ∈ C0∞ (Rn ) supp ψ ⊂ K , K Rn

(6.26)

∞

(f u)m v m , ϕ − f · um v m , ϕ = v m , ((f u)m − f · um )ϕ −−−−→ 0, m→∞

(6.27)

for by Banach–Steinhauss’ theorem it suffices that ((f u)m − f · um )ϕ → 0 in C ∞ (K), which holds since (f u)m − f · um → 0 in C ∞ (Rn ) according to Proposition 6.6. Hence f π(u, v) = π(f u, v); the other identity is justified similarly.

7. Extended action of distributions To prepare for Section 8 it is exploited that the map (u, f ) → u, f is defined also for certain u, f in D (Rn ) that do not belong to dual spaces. This bilinear form is moreover shown to have a property of stability under regular convergence. 7.1. A review First it is recalled that the product f u is defined for f, u ∈ D (Rn ) if sing supp f sing supp u = ∅.

(7.1)

In fact, R is covered by Y1 = R \sing supp u and Y2 = R \sing supp f ; in Y1 there is a product (f u)Y1 ∈ D (Y1 ) given by (f u)Y1 , ϕ = f, uϕ for ϕ ∈ C0∞ (Y1 ), and similarly u, f ϕ , ϕ ∈ C0∞ (Y2 ) defines a product (f u)Y2 ∈ D (Y2 ); and for ϕ ∈ C0∞ (Y1 ∩ Y2 ) both products are given by the C ∞ -function f (x)u(x) so they coincide on Y1 ∩ Y2 ; hence f u is well defined in D (Rn ) and given on ϕ ∈ C0∞ (Rn ) n

n

n

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233

by the following expression, where the splitting ϕ = ϕ1 + ϕ2 for ϕj ∈ C0∞ (Yj ) is obtained from a partition of unity, f u, ϕ = f, uϕ1 + u, f ϕ2 .

(7.2)

This follows from the recollement de morceaux theorem, cf. [21, Thm. I.IV] or [9, Thm. 2.2.4]; by the proof of this, (7.2) does not depend on how the partition is chosen. Remark 7.1. Therefore, when F1 , F2 are closed sets in Rn given with the properties sing supp u ⊂ F1 , sing supp f ⊂ F2 and F1 ∩ F2 = ∅ (so that Rn is covered by their complements) one can always take the splitting in (7.2) such that ϕ1 ∈ C0∞ (Rn \ F1 ), ϕ2 ∈ C0∞ (Rn \ F2 ). Secondly f → u, f for u ∈ D (Rn ) is a well-defined linear map on the subspace of f ∈ D (Rn ) such that (7.1) holds together with supp u ∩ supp f Rn .

(7.3)

In fact u, f := f u, 1 is possible: f u is defined by (7.1) and is in E by (7.3), so by [9, Th 2.2.5] the map ψ → f u, ψ extends from C0∞ (Rn ) to all ψ ∈ C ∞ (Rn ), uniquely among the extensions that vanish when supp ψ ∩ supp f u = ∅; hence it is defined on the canonical choice ψ ≡ 1, and for all ϕ ∈ C0∞ (Rn ) equal to 1 around supp f u, u, f = f u, 1 = f u, ϕ . (7.4) These constructions have been quoted in a slightly modified form from [9, Sect. 3.1]. The definition implies that (f, u) → f u is bilinear; it is clearly commutative and is partially associative in the sense that ψ(f u) = (ψf )u = f (ψu) when ψ ∈ C ∞ (Rn ) while f , u fulfill (7.1). This also yields supp f u ⊂ supp f ∩ supp u.

(7.5)

When applying cut-off functions, partial associativity entails (χf )(ϕu) = f u when χ, ϕ equal 1 around supp f ∩ supp u. Therefore test against 1 gives that ϕu, χf = u, f . 7.2. Stability under regular convergence The product f u is not continuous, for f = 0 is the limit in D of f ν = e−ν|x| ∈ C ∞ and for u = δ0 it is clear that f ν u = δ0 → 0 = f u. As a remedy it is noted that f u is separately stable under regular convergence; cf. Lemma 6.1. This carries over to the extended bilinear form ·, · under a compactness condition: 2

Theorem 7.2. Let u, f ∈ D (Rn ), f ν ∈ C ∞ (Rn ) fulfil limν f ν = f in both D (Rn ) and C ∞ (Rn \ F ) for F = sing supp f . When u, f have disjoint singular supports, cf. (7.1), then (7.6) f ν u → f u in D (Rn ) for ν → ∞. ) If moreover supp u supp f is compact and χ ∈ C0∞ (Rn ) equals 1 around this set, lim χu, f ν = lim u, χf ν = u, f .

ν→∞

ν→∞

(7.7)

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J. Johnsen

) Here one can take χ ≡ 1 on Rn if a compact set contains supp u or ν supp(f ν u). The conclusions hold verbatim when F ⊂ Rn is closed and sing supp f ⊂ F ⊂ (Rn \ sing supp u). Proof. To show (7.6) for a general F , note that (7.2) applies to the product f ν u of f ν ∈ C ∞ and u ∈ D . Using Remark 7.1 and that f ν → f in C ∞ (Rn \ F ), one has f ν ϕ2 → f ϕ2 in C0∞ (Rn \ F ); the other term on the right-hand side of (7.2) converges by the D -convergence of the f ν . Therefore f ν u, ϕ → f, uϕ1 + u, f ϕ2 = f u, ϕ . By the definition of u, f above, when χ is as in the theorem, then the just proved fact that f ν u → f u in D leads to (7.7) since u, f = f u, 1 = f u, χ = lim f ν u, χ . ν→∞

(7.8)

4 When ν supp(f ν u) is precompact and χ = 1 on a neighbourhood, then 0 = f ν u, 1 − χ can be added to (7.8), which yields limν u, f ν by the extended definition of ·, · . Remark 7.3. In general (7.7) cannot hold without the cut-off function χ. E.g., for n ≥ 2 and x = (x , xn ) one may take f = 1{xn≤0} and u = 1{xn ≥1/|x |2 } , so that u, f = 0. Setting f ν = 2nν ϕ(2ν ·) ∗ f for ϕ ∈ C0∞ with ϕ ≥ 0, ϕ = 1, and supp ϕ ⊂ { (y , yn ) | 1 ≤ yn ≤ 2, |y | ≤ 1 }, it holds for x ∈ Σν = { 0 ≤ xn ≤ 2−ν } that xn − 2−ν yn ≤ 0 on supp ϕ so that f ν (x) = ϕ(y)f (x − 2−ν y) dy = ϕ dy = 1. (7.9) Hence supp u ∩ supp f ν is unbounded, so u, f ν is undefined (hardly just a tech ν ν nical obstacle as u, f = uf dx = ∞ would be the value). 7.3. Consequences for kernels Although it is on the borderline of the present subject, it would not be natural to omit that Theorem 7.2 gives an easy way to extend the link between an operator and its kernel: Theorem 7.4. Let A : S (Rn ) → S (Rn ) be a continuous linear map with distribution kernel K(x, y) ∈ S (Rn × Rn ). Suppose that u ∈ S (Rn ) and v ∈ C0∞ (Rn ) satisfy (7.10) supp K supp v ⊗ u Rn × Rn , sing supp K sing supp v ⊗ u = ∅. (7.11) Then Au, v = K, v⊗u , with extended action of ·, · . When A is a continuous linear map D (Rn ) → D (Rn ), this is valid for u ∈ D (Rn ), v ∈ C0∞ (Rn ) fulfilling (7.10)–(7.11).

Type 1, 1-Operators

235

Proof. By the conditions on u and v, the expression K, v ⊗ u is well defined. By mollification there is regular convergence to u of a sequence uν ∈ C ∞ (Rn ); this gives v ⊗ uν −−−− → v ⊗ u in S (Rn × Rn ) and C ∞ (Ω) (7.12) ν→∞

when Ω = (Rn × Rn ) \ (supp v × sing supp u) = R2n \ sing supp(v ⊗ u). Applying Theorem 7.2 on R2n , the cut-off function may be taken as κ(x)χ(y) for some κ, χ ∈ C0∞ (Rn ) such that κ equals 1 on supp v and κ ⊗ χ = 1 on the compact set supp K ∩ supp(v ⊗ u). This gives K, v ⊗ u = lim (κ ⊗ χ)K, v ⊗ uν = lim A(χuν ), v = A(χu), v . (7.13) ν→∞

ν→∞

−m

For χ = ψ(2 ·) and ψ = 1 near 0, the conclusion follows from the continuity of A since ψ(2−m ·)u → u in S . The D -case is similar. Remark 7.5. The conditions (7.10)–(7.11) are far from optimal, for (v⊗u)K4acts on m 1 if its just an integrable distribution, that is if (v ⊗u)K belongs to DL 1 = m W1 2n on R . Similarly (7.11) is not necessary for (v⊗u)·K to make sense; e.g., it suffices that (x, ξ) ∈ / WF(K) ∩ (− WF(v ⊗ u)) whenever (x, ξ) ∈ R2n . More generally the existence of the product π(K, v ⊗ u) would suffice; cf. Remark 1.1. The above result applies in particular to the pseudo-differential operators A d corresponding to a standard symbol space S, such as S1,0 (Rn × Rn ). So does the next consequence. Corollary 7.6. When A is as in Theorem 7.4, it holds for every u ∈ S (Rn ) that supp Au ⊂ supp K ◦ supp u. (7.14) Hereby supp K ◦ supp u = x ∈ Rn ∃y ∈ supp u : (x, y) ∈ supp K , which is a closed set if supp u Rn . The result extends to u ∈ D (Rn ) when A is D continuous. Proof. Whenever v ∈ C0∞ (Rn ) fulfils supp v Rn \ supp K ◦ supp u, then supp K supp(v ⊗ u) = ∅. (7.15) For else some (x, y) ∈ supp K would fulfill y ∈ supp u and x ∈ supp v, in contradiction with the support condition on v. By (7.15) the assumptions of Theorem 7.4 are satisfied, so Au, v = K, v ⊗u = 0. Hence Au = 0 holds outside the closure of supp K ◦ supp u. Remark 7.7. The argument of Corollary 7.6 is completely standard for u ∈ C0∞ , cf. [9, Thm 5.2.4] or [22, Prop 3.1]; a limiting argument then implies (7.14) for general u. However, the proof above is a direct generalisation of the C0∞ -case, made possible by the extended action of ·, · in Theorem 7.4. This method may be interesting in its own right; e.g., it extends to type 1, 1-operators also when these are not S -continuous, cf. Section 8.

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8. Kernels and transport of support Using the preceding section, the well-known support rule is here extended to operators of type 1, 1. As a novelty also a spectral support rule is deduced. 8.1. The support rule for type 1,1-operators As analogues of Theorem 7.4 and Corollary 7.6 one has: ∞ Theorem 8.1. If a ∈ S1,1 (Rn × Rn ) has kernel K, then a(x, D)u, v = K, v ⊗ u whenever u ∈ D(a(x, D)), v ∈ C0∞ (Rn ) fulfill (7.10)–(7.11). And for all u ∈ D(a(x, D)) the support rule holds, i.e., supp Au ⊂ supp K ◦ supp u.

Proof. a(x, D)u = limm→∞ Am u where Am = OP(am (1 ⊗ ψm )) ∈ OP(S −∞ ); its kernel Km is given by Proposition 5.11. However, Km need not fulfil (7.10) together with u, v, but by use of convolutions and cut-off functions one can find uν in C0∞ (Rn ) such that uν → u in S (Rn ) and in C ∞ (Rn \sing supp u) for ν → ∞. Then Theorem 7.4 gives Au, v = lim lim Am uν , v = lim lim Km , v ⊗ uν . m→∞ ν→∞

m→∞ ν→∞

(8.1)

To control the supports, one can take a function f fulfilling (6.12) by setting f (x, y) = g(x)h(x−y) for some g ∈ C0∞ (Rn ) with g = 1 on supp v and h ∈ C ∞ (Rn ) such that h(y) = 0 for |y| < 1 while h(y) = 1 for |y| > 2. Then Km = f Km + (1 − f )Km , where the f Km tend to f K in S according to Proposition 6.3. The supports of (1 − f )Km (v ⊗ uν ), m, ν ∈ N, all lie in the precompact set B(0, R) × B(0, R+2) when B(0, R) ⊃ supp v, so since u, v are assumed to fulfil (7.10)–(7.11), Theorem 7.2 gives Au, v = lim lim f Km , v ⊗ uν + lim lim (1 − f )Km , v ⊗ uν m→∞ ν→∞

m→∞ ν→∞

= f K, v ⊗ u + (1 − f )K, v ⊗ u = K, v ⊗ u . Now the support rule follows by repeating the proof of Corollary 7.6.

(8.2)

8.2. The spectral support rule Although it has not attracted much attention, it is a natural and useful task to determine the frequencies entering x → a(x, D)u(x). But since ∧

F a(x, D)u = F a(x, D)F −1 (u)

(8.3)

∧

the task is rather to control how the support of u is changed by F a(x, D)F −1 , i.e., by the conjugation of a(x, D) by the Fourier transformation. ∞ Even for A ∈ OP(S1,0 ) this has seemingly not been carried out before. How−1 ever, since F AF : S (Rn ) → S (Rn ) is continuous for such A, it is straightforward to apply Theorem 7.4 and Corollary 7.6 to the distribution kernel ∧

K(ξ, η) = (2π)−n a(ξ − η, η) of F AF

−1

; cf. Proposition 4.2.

(8.4)

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237

This yields at once the following general result: ∞ Theorem 8.2. If a ∈ S1,0 (Rn × Rn ) and K is as above, then

supp F a(x, D)u ⊂ supp K ◦ supp F u

u ∈ S (Rn ).

for every

(8.5)

Here the right-hand side is closed if supp F u Rn . Here (8.5) may also be written explicitly as in (1.10)–(1.11). It is easily gen∞ (Rn × Rn ) with δ < ρ. For eralised to standard symbol spaces S such as Sρ,δ elementary symbols in the sense of [5] the spectral support rule (8.5) follows at once, but as it stands Theorem 8.2 seems to be a new result even for classical type 1, 0-operators. The reader is referred to [15, Sect. 1.2] for more remarks on Theorem 8.2, in particular that it makes it unnecessary to reduce to elementary symbols in the Lp -theory (which is implicitly sketched in Section 9 below). To extend the above to type 1, 1-operators, the next result applies to the conjugated operator F a(x, D)F −1 instead of Theorem 7.4. ∧

∞ Theorem 8.3. Let a ∈ S1,1 (Rn × Rn ) and denote by K(ξ, η) = (2π)−n a(ξ − η, η) the distribution kernel of F a(x, D)F −1 ; and suppose u ∈ D(a(x, D)) ⊂ S (Rn ) is such that, for some ψ as in Definition 5.1,

a(x, D)u = lim am (x, D)um m→∞

S (Rn ),

holds in

(8.6)

∧

and that v ∈ C0∞ (Rn ) satisfies ∧ ∧ supp K supp v ⊗ u Rn × Rn ,

sing supp K

∧

∧

sing supp v ⊗ u = ∅. (8.7)

Then it holds, with extended action of ·, · , ∧

∧

∧

∧

Fa(x, D)F −1 (u), v = K, v ⊗ u .

(8.8)

Proof. For u ∈ D(a(x, D)) the left-hand side of (8.8) makes sense by (8.6); and the right-hand side does so by (8.7), cf. Section 7. The equality follows from (8.6): Letting ψm = ψ(2−m ·) there is some ν such that ψν = 1 on a neighbourhood of supp ψ, so ψm+ν ψm = ψm for all m. Then 1 ⊗ ψm and ψm (ξ − η)ψm (η) equal 1 on the intersection of the supports in (8.7) for all sufficiently large m, so ∧

∧

∧

∧

K, v ⊗ u = ψm (ξ − η)ψm+ν (η)K(ξ, η), v(ξ)ψm (η)u(η) .

(8.9)

Here Km = ψm (ξ − η)ψm+ν (η)K(ξ, η) is the kernel of F Am F −1 , when Am = OP(am (1 ⊗ ψm+ν )); cf. Proposition 4.2. Clearly Am has symbol in S −∞ . ∧ Moreover, mollification of ψm u = F um gives a sequence (F um )k of functions ∞ n in C0 (R ), that all have their supports in a fixed compact set M . Invoking regular convergence, cf. Lemma 6.1, it follows that ∧

∧

v ⊗ (F um )k −−−− → v ⊗ Fum k→∞ ∧

in S (R2n ) and C ∞ (Ω) ∧

R2n \ Ω = supp v × sing supp F um = sing supp(v ⊗ Fum ).

(8.10) (8.11)

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J. Johnsen ∧

Since all supports are contained in supp v × M , Theorem 7.2 applied on R2n and the S -continuity of Am imply ∧

∧

∧

Km , v ⊗ (F um ) = lim Km , v ⊗ (F um )k = FAm F −1 (F um ), v . k→∞

(8.12)

According to Lemma 5.10 the factor ψm+ν can here be removed from the symbol of Am , so it is implied by (8.9), (8.12) and the explicit assumption of S -convergence in (8.6) that ∧

∧

∧

K, v ⊗ u = FAm um , v = lim am (x, D)um , F 2 v = a(x, D)u, F 2 v ,

(8.13)

m→∞

since F 2 v ∈ S. Transposing F , formula (8.8) results.

The assumption of S -convergence in (8.6) cannot be omitted from the above proof, although in the last line am (x, D)um , F 2 v is independent of m. E.g., j m (m) 2j cos x is in S (R), but since F ( j=0 (−1) the (2j) x ) = 2πδ0 + c2 δ0 + · · · + cm δ0 power series converges to cos x in D but not in S as cosine isn’t a polynomial. Moreover, if F v = 1 around 0 for some F v ∈ C0∞ ([−1, 1]), one clearly has 2π = j m 2j 2 j=0 (−1) (2j) x , F v for every m as the derivatives of F v vanish at the origin. And yet cos, F 2 v = 2π 12 (δ1 + δ−1 ), F v = 0 = 2π. The next result extends [15, Prop. 1.4] from the case of u ∈ C ∞ (Rn ) with ∧ supp u Rn to almost arbitrary distributions u ∈ D(a(x, D)); but the proof is significantly simpler here. Instead of (8.5), the explicit form given in (1.10)–(1.11) is preferred for practical purposes. ∞ (Rn × Rn ) and suppose Theorem 8.4 (The Spectral Support Rule). Let a ∈ S1,1 ∞ n u ∈ D(a(x, D)) is such that, for some ψ ∈ C0 (R ) equalling 1 around the origin, the convergence of Definition 5.1 holds in the topology of S (Rn ), i.e.,

a(x, D)u = lim am (x, D)um

in

m→∞

S (Rn ).

(8.14)

∧

Then (8.5) holds, that is with Ξ = supp K ◦ supp u one has Ξ= When u ∈ F

−1

supp F (a(x, D)u) ⊂ Ξ, ∧ ∧ ξ + η (ξ, η) ∈ supp a, η ∈ supp u .

(8.15) (8.16)

E (R ) then (8.14) holds automatically and Ξ is closed for such u. n

∧

Proof. That Ξ = supp K ◦ supp u has the form in (8.16) follows by substituting ζ = ξ + η. Using Theorem 8.3 instead of Theorem 7.4, the proof of Corollary 7.6 can now be repeated mutatis mutandis; which gives the inclusion in question. The redundancy of (8.14) for u ∈ F −1 E follows since, by Lemma 5.10, one can for large m write am (x, D)um = OP(am (1 ⊗ χ)ψm )u for a fixed cut-off function χ. Then Proposition 5.4 gives S -convergence, for multiplication by 1 ⊗ χ commutes with ψm (Dx ) and a(1 ⊗ χ) ∈ S −∞ . That Ξ is closed then is straightforward to verify.

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239

Remark 8.5. The set Ξ in (8.16) need not be closed if supp F u is non-compact, for ∧ supp a may contain points arbitrarily close to the twisted diagonal. E.g., if n = 1 ∧ ∧ and supp u = [2, ∞[ whilst supp a consists of the (ξ, η) such that η ≥ −ξ − 1ξ for ξ ≤ −1, and η ≥ 2 for ξ ≥ −1, then (ξk , ηk ) = (−k, k + 1/k) fulfils Ξ ξk + ηk = 1 / Ξ. k 0, although 0 ∈ That a(x, D)u should be in S in (8.14) is natural in order that F a(x, D)u makes sense before its support is investigated. One could conjecture that the condition of convergence in S is redundant, so that it would suffice to assume a(x, D)u is an element of S . But it is not clear (whether and) how this can be proved.

9. Continuity in Sobolev spaces As a last justification of Definition 5.1 its close connection to estimates in Sobolev spaces will be indicated. 9.1. Littlewood–Paley decompositions d For the purposes of this section, one may for a ∈ S1,1 (Rn × Rn ) consider the limit aψ (x, D)u = lim OP(ψ(2−m Dx )a(x, η)ψ(2−m η))u.

(9.1)

m→∞

By the definition, u is in D(a(x, D)) if aψ (x, D)u exists for all ψ and is independent of ψ, as ψ runs through C0∞ (Rn ) with ψ = 1 around the origin. From aψ (x, D) there is a particularly easy passage to the paradifferential decomposition used by J.-M. Bony [1]. For this purpose, note that to each fixed ψ there exist R > r > 0 satisfying ψ(ξ) = 1 for |ξ| ≤ r;

ψ(ξ) = 0 for

|ξ| ≥ R ≥ 1.

(9.2)

Moreover it is convenient to let h stand for an integer such that R ≤ r2h−2 . To obtain a Littlewood–Paley decomposition from ψ, define ϕ = ψ − ψ(2·). Then it is clear that ϕ(2−k ·) is supported in a corona, supp ϕ(2−k ·) ⊂ ξ r2k−1 ≤ |ξ| ≤ R2k , for k ≥ 1. (9.3) ∞ −k The identity 1 = ψ(ξ) + k=1 ϕ(2 ξ) follows by letting m → ∞ in the telescopic sum, (9.4) ψ(2−m ξ) = ψ(ξ) + ϕ(ξ/2) + · · · + ϕ(ξ/2m ). Using this one can localise functions u(x) and symbols a(x, η) to frequencies |η| ≈ 2j for j ≥ 1 by setting ∧

uj = ϕ(2−j D)u,

−1 aj (x, η) = ϕ(2−j Dx )a(x, η) = Fξ→x (ϕ(2−j ξ)a(ξ, η)). (9.5)

u0 = ψ(D)u,

a0 (x, η) = ψ(Dx )a(x, η)

(9.6)

Similarly localisation to balls |η| ≤ R2j are written, now with upper indices, as uj = ψ(2−j D)u,

−1 aj (x, η) = ψ(2−j Dx )a(x, η) = Fξ→x (ψ(2−j ξ)Fx→ξ a(ξ, η)). (9.7)

240

J. Johnsen

Moreover, u0 = u0 and a0 = a0 . In addition both, e.g., aj = 0 and aj = 0 should be understood when j < 0. (In order not to have two different meanings of sub- and superscripts on functions, the dilations ψ(2−j ·) are simply written as such; and the corresponding Fourier multiplier as ψ(2−j D).) Note that ak (x, D) = OP(ψ(2−k Dx )a(x, η)) etc. However, returning to (9.1), the relation (9.4) applies twice, whence bilinearity gives am (x, D)um = OP([a0 (x, η) + · · · + am (x, η)][ψ(η) + · · · + ϕ(2−m η)])u =

m

(9.8)

aj (x, D)uk .

j,k=0

Of course the sum may be split in three groups in which j ≤ k − h, |j − k| < h and k ≤ j − h, respectively. In the limit m → ∞ this gives the decomposition (1)

(2)

(3)

aψ (x, D)u = aψ (x, D)u + aψ (x, D)u + aψ (x, D)u,

(9.9)

whenever a and u fit together such that the three series below converge in D (Rn ): (1) aψ (x, D)u

=

∞

(2)

k=h j≤k−h ∞

(3)

∞

aψ (x, D)u =

aψ (x, D)u =

aj (x, D)uk =

∞

ak−h (x, D)uk

(9.10)

k=h

ak−h+1 (x, D)uk + · · · + ak−1 (x, D)uk + ak (x, D)uk k=0 (9.11) + ak (x, D)uk−1 + · · · + ak (x, D)uk−h+1 aj (x, D)uk =

j=h k≤j−h

∞

aj (x, D)uj−h .

(9.12)

j=h

Also (9.11) has a brief form, namely (2)

aψ (x, D)u =

∞

((ak − ak−h )(x, D)uk + ak (x, D)(uk−1 − uk−h )).

(9.13)

k=0

One advantage of the decomposition is that the terms of the first and last series fulfil a dyadic corona condition; whereas the in second the spectra are in general only restricted to balls: d Proposition 9.1. If a ∈ S1,1 (Rn × Rn ) and u ∈ S (Rn ), and r, R are chosen as in (9.2) for each auxiliary function ψ, then every h ∈ N such that R ≤ r2h−2 gives r k 5R k k−h 2 ≤ |ξ| ≤ 2 (x, D)uk ) ⊂ ξ supp F (a (9.14) 4 4 r 5R k 2 . (9.15) supp F (ak (x, D)uk−h ) ⊂ ξ 2k ≤ |ξ| ≤ 4 4

Type 1, 1-Operators

241

(2)

Moreover, for aψ (x, D), supp F ak (x, D)(uk−1 − uk−h )+ (ak − ak−h )(x, D)uk ⊂ ξ |ξ| ≤ 2R2k (9.16) If a satisfies (1.19) this support is contained in r ξ h+1 ≤ |ξ| ≤ 2R2k 2 C

(9.17)

for all k ≥ h + 1 + log2 (C/r). This proposition follows straightforwardly from the spectral support rule in Theorem 8.4, with a special case explained in [15], so further details should hardly be needed here. (The constants are different in [15], since h only fulfils R < r2h−1 , that suffices in general.) In addition one can estimate each series, using the Hardy–Littlewood maximal operator M uk (x). This gives, e.g., for ν = 1, when the Fefferman–Stein inequality is used in the last step, Rn

∞

p |2 a

sk k−h

2

(x, D)uk (x)|

2

1 ∞ 1 p 2 (s+d)k 2 dx ≤ c(a) |2 M uk (·)| p

k=h

k=0

≤ c c(a)uHps+d ;

(9.18)

d here c(a) is a continuous seminorm on a ∈ S1,1 . Similar estimates are obtained for ν = 2 and ν = 3. The reader is referred to [15] for brevity here. (Although the s set-up was more general there with Besov spaces Bp,q and Triebel–Lizorkin spaces s s Fp,q , it is easy to specialise to the present Hp -framework, mainly by setting q = 2 s . One difference in the framework of [15] is that certain in the treatment of the Fp,q ˜ j amount to special ˜ functions Φj enter the expressions aj,k (x, D)uk there, but the Φ choices of χ in the above formula (5.12), hence may be removed when convenient.) Combining such estimates with Proposition 9.1 it follows in a well-known way that for ν = 1 and ν = 3, (ν)

aψ (x, D)uHps ≤ cuHps+d

for s ∈ R, 1 < p < ∞.

(9.19)

For ν = 2 this holds for s > 0, because the coronas are replaced by balls. (1) (2) (3) Consequently u → aψ (x, D)u = aψ (x, D)u + aψ (x, D)u + aψ (x, D)u is a bounded linear operator Hps+d (Rn ) → Hps (Rn ) for s > 0 and aψ (x, D)uHps ≤ C(a)uHps+d ,

(9.20)

d where C(a) is a continuous seminorm on a ∈ S1,1 . Moreover, if a fulfils (1.19), then the last part of Proposition 9.1 leads to continuity for all s ∈ R. Moreover, density of the Schwartz space S(Rn ) in Hps+d (Rn ) yields that aψ (x, D) is independent of ψ, for they all agree with OP(a)u whenever u ∈ S(Rn ); cf. (9.9), (9.8) and (2.4). So by Definition 5.1 it follows that a(x, D)u is defined on every u ∈ Hps+d with s > 0.

242

J. Johnsen More precisely one has

d Theorem 9.2. Let a(x, η) be a symbol in S1,1 (Rn × Rn ). Then for every s > 0, 1 < p < ∞ the type 1, 1-operator a(x, D) has Hps+d (Rn ) in its domain and it is a continuous linear map

a(x, D) : Hps+d (Rn ) → Hps (Rn ).

(9.21)

This extends to all s ∈ R when a fulfils the twisted diagonal condition (1.19). In [15] a similar proof was given for Besov and Lizorkin–Triebel spaces, i.e., s s and Fp,q . But this contains the above Theorem 9.2 in view of the wellfor Bp,q s (Rn ) for 1 < p < ∞, which through a reduction known identification Hps (Rn ) = Fp,2 to s = 0 results from the Littlewood–Paley inequality. The reader is referred to the more general continuity results in [15], which also cover the H¨ older–Zygmund s . classes because of the identification C s = B∞,∞ s However, a little precaution is needed because S(Rn ) is not dense in B∞,q . s Even so a(x, D) is defined on and bounded from B∞,q for s > d (and s = d, q = 1 cf. (1.20) ff and [15, (1.6)]), which may be seen from the Besov space estimates of [15] and the argument preceding Theorem 9.2 as follows. By lowering ) −1 s one can s F E is dense; arrange that q < ∞, in which case it is well known that B∞,q s if it is so for all u ∈ F −1 E . whence aψ (x, D)u is independent of ψ for all u ∈ B∞,q This last property is a consequence of the fact that F −1 E is in the domain of a(x, D); cf. Theorem 5.5 and Remark 5.6. Remark 9.3. It is evident that the counter-example in Proposition 3.3 relied on an extension of continuity of a2θ (x, D) to a bounded operator H s+d → H s for arbitrary s < d. Moreover, this extension has not previously been identified with the definition of a2θ (x, D) by vanishing frequency modulation. However, by the density of S, it follows from the last part of Theorem 9.2 that these two extensions are identical on H s for all s ∈ R, whence the operators in Definition 5.1 lack the microlocal property in the treated cases. 9.2. Composite functions Finally it is verified that the formal definition of type 1, 1-operators by vanishing frequency modulation also plays well together with Y. Meyer’s formula for composite functions. Consider the map u → F ◦ u given by F (u(x)) for a fixed F ∈ C ∞ (R) and a real-valued u ∈ Hps00 (Rn ) for s0 > n/p0 , 1 < p0 < ∞. Then u is uniformly continuous and bounded on Rn as well as in Lp (Rn ) for p0 ≤ p ≤ ∞. Note that with the notation of the previous section, and in particular (9.4), one has in Lp (Rn ) u0 + u1 + · · · + um = um = 2mn F −1 ψ(2m ·) ∗ u −−−−→ u. m→∞

(9.22)

Type 1, 1-Operators

243

Assuming that F (0) = 0 when p < ∞, then v → F ◦ v is Lipschitz continuous on the metric subspace Lp (Rn , B) for every ball B R, 1 F (w(x)) − F (v(x)) = F (v(x) + t(w(x) − v(x))) dt · (w(x) − v(x)) (9.23) 0

F ◦ w − F ◦ vp ≤ sup |F | · w − vp .

(9.24)

B

Since um ∞ ≤ F −1 ψ1 u∞ , one can take B so large that it contains u(Rn ) and um (Rn ) for every m, so since uk − uk−1 = uk it follows from the Lipschitz continuity that with limits in Lp , p0 ≤ p ≤ ∞, F (u(x)) = lim F (um (x)) = F (0) + lim m→∞

= F (0) +

m→∞

∞ k=0

1

F (u

k−1

m

(F (uk (x)) − F (uk−1 (x)))

k=0

(9.25)

(x) + tuk (x)) dt · ϕ(2

−k

D)u(x).

0

1 Setting mk (x) = 0 F (uk−1 (x) + tuk (x)) dt it is not difficult to see that mk ∈ ∨ C ∞ (Rn ) with bounded derivatives of any order because Dβ uk = 2k(n+|β|) ϕ(2k ·)∗u; and since 2k ≈ |η| on supp ϕ(2−k ·) that au (x, η) :=

∞

0 mk (x)ϕ(2−k η) ∈ S1,1 (Rn × Rn ).

(9.26)

k=0

That the corresponding type 1, 1-operator au (x, D) gives important information on the function F (u(x)) was the idea of Y. Meyer [17, 18], and it is now confirmed that his results remain valid when the operators are based on Definition 5.1: Theorem 9.4. When u ∈ Hps00 (Rn ) for s0 > n/p0 , 1 < p0 < ∞ is real valued, and F ∈ C ∞ (R) with F (0) = 0, then the type 1, 1-operator au (x, D) is a bounded linear operator au (x, D) : Hps (Rn ) → Hps (Rn )

for every

s > 0, 1 < p < ∞.

(9.27)

Taking s = s0 , p = p0 one has au (x, D)u(x) = F (u(x)), and the map u → F ◦ u is continuous on Hps00 (Rn , R). 0 Proof. The continuity on Hps follows from Theorem 9.2 since au ∈ S1,1 . As the s proof of this theorem shows, the operator norm b(x, D) in B(Hp ) is estimated 0 1 1 by a seminorm c(b) on b ∈ S1,1 ⊂ S1,1 , and (9.26) converges in S1,1 , so one has in s B(Hp ) that m mk (x)ϕ(2−k η) −−−−→ au (x, D). (9.28) OP k=0

m→∞

244

J. Johnsen

By (9.25) this implies that in the space Lp0 m au (x, D)u = lim OP mk (x)ϕ(2−k η) u m→∞

=

∞

k=0

mk (x)ϕ(2

−k

(9.29) D)u = F ◦ u.

k=0

Hence u → F ◦ u is a map Hps00 → Hps00 , which is continuous since for v → u F (v(x)) − F (u(x)) = au (x, D)(v − u)+[av (x, D) − au (x, D)]u +[av (x, D) − au (x, D)](v − u) → 0.

(9.30)

Indeed, by continuity of au (x, D) the first term tends to 0, and by the Banach– Steinhauss theorem the two other terms do so if only av (x, D) → au (x, D) in 0 B(Hps00 ), i.e., if av → au in S1,1 . However, the non-linear map u → au is contins0 0 uous from Hp0 to S1,1 , for (1 + |η|)|α|−|β| |Dηα Dxβ (av (x, η) − au (x, η))| is at each η estimated uniformly by terms that may have v − u∞ as a factor or contains 1 supx∈Rn 0 |F (l) (v k−1 (x) + tvk (x)) − F (l) (uk−1 (x) + tuk (x))| dt, which tends to 0 by the uniform continuity of F (l) on a sufficiently large ball. Among the merits of the theorem, note that for non-integer s it is non-trivial to prove that F (u(x)) is in Hps when u is so. When needed the reader may derive s s and Fp,q from the estimates in [15]. Moreover, contisimilar results for the Bp,q nuity of u → F ◦ u is as shown a straightforward consequence of the factorisation au (x, D)u, but this was not mentioned in [17, 18, 12]. Remark 9.5. As a small extension of the above, it may be noted that when F is bounded on R, then the assumption on u can be relaxed to u ∈ Lp0 for 1 ≤ p0 ≤ ∞, for F (u(x)) is defined, and the linearisation formula (9.25) still holds as u → F ◦ u is Lipschitz continuous on Lp0 (Rn , R) in this case (however, the symbol au (x, η) has much weaker properties). Acknowledgement My thanks are due to the anonymous referee for requesting a more explicit comparison with the existing literature.

References [1] J.-M. Bony, Calcul symbolique et propagations des singularités pour les équations ´ Norm. Sup. 14 (1981), 209–246. aux dérivées partielles non linéaires, Ann. scient. Ec. [2] G. Bourdaud, Sur les opérateurs pseudo-diff´ erentiels ` a coefficients peu reguliers, Thèse, Univ. de Paris-Sud, 1983. [3]

, Une algèbre maximale d’opérateurs pseudo-différentiels, Comm. Partial Differential Equations 13 (1988), no. 9, 1059–1083.

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245

[4] C.H. Ching, Pseudo-differential operators with nonregular symbols, J. Differential Equations 11 (1972), 436–447. [5] R.R. Coifman and Y. Meyer, Au del` a des op´ erateurs pseudo-diff´ erentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978. [6] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777–799. [7] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34–170. [8] G. Garello, Microlocal properties for pseudodifferential operators of type 1, 1, Comm. Partial Differential Equations 19 (1994), 791–801. [9] L. H¨ ormander, The Analysis of Linear Partial Differential Operators, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1983, 1985. , Pseudo-differential operators of type 1, 1, Comm. Partial Differential Equa[10] tions 13 (1988), no. 9, 1085–1111. , Continuity of pseudo-differential operators of type 1, 1, Comm. Partial Dif[11] ferential Equations 14 (1989), no. 2, 231–243. , Lectures on Nonlinear Differential Equations, Mathématiques & applica[12] tions, vol. 26, Springer-Verlag, Berlin, 1997. [13] J. Johnsen, Pointwise multiplication of Besov and Triebel–Lizorkin spaces, Math. Nachr. 175 (1995), 85–133. , Domains of type 1, 1 operators: a case for Triebel–Lizorkin spaces, C. R. [14] Acad. Sci. Paris Sér. I Math. 339 (2004), no. 2, 115–118. , Domains of pseudo-differential operators: a case for the Triebel–Lizorkin [15] spaces, J. Function Spaces Appl. 3 (2005), 263–286. , Simple proofs of nowhere-differentiability for Weierstrass’ function and cases [16] of slow growth, Preprint R-2008-02, Aalborg University, 2008. [17] Y. Meyer, Régularité des solutions des équations aux dérivées partielles non linéaires (d’après J.-M. Bony), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math. 842, Springer, Berlin, 1981, 293–302. , Remarques sur un théorème de J.-M. Bony, in Proceedings of the Seminar [18] on Harmonic Analysis (Pisa, 1980), 1981, 1–20. [19] C. Parenti and L. Rodino, A pseudo differential operator which shifts the wave front set, Proc. Amer. Math. Soc. 72 (1978), 251–257. [20] T. Runst, Pseudodifferential operators of the “exotic” class L01,1 in spaces of Besov and Triebel-Lizorkin type, Ann. Global Anal. Geom. 3 (1985), no. 1, 13–28. [21] L. Schwartz, Théorie des Distributions, Revised and Enlarged ed., Hermann, Paris, 1966. [22] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987, Translated from the 1978 Russian original by Stig I. Andersson. [23] X. Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. [24] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.

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[25] R.H. Torres, Continuity properties of pseudodifferential operators of type 1, 1, Comm. Partial Differential Equations 15 (1990), 1313–1328. [26] H. Triebel, Theory of Function Spaces, Monographs in mathematics, 78, Birkh¨ auser Verlag, Basel, 1983. Jon Johnsen Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G DK-9220 Aalborg Øst, Denmark e-mail: [email protected]


Regularity for Quasi-Elliptic Pseudo-Differential Operators with Symbols in H¨ older Classes Gianluca Garello and Alessandro Morando Abstract. After proving a result of continuity for pseudo-differential operators whose symbols belong to H¨ older spaces and satisfy a decay of quasihomogeneous type, the authors construct a suitable symbolic calculus and a parametrix for quasi-elliptic operators; these tools are applied to the study of quasi-elliptic linear partial differential equations with H¨ older coefficients. Mathematics Subject Classification (2000). Primary 35S05; Secondary 35A17. Keywords. Non regular pseudo-differential operators, Besov spaces, Sobolev spaces.

1. Introduction It is well known that pseudo-differential operators of classical type, defined by a(x, D)u = (2π)−n ei x,ξ a(x, ξ)ˆ u(ξ) dξ, u ∈ S(Rn ), (1.1) ˆ = F u is the Fourier transform of u, with symbols where x, ξ = nj=1 xj ξj and u m in the H¨ ormander classes Sρ,δ : |∂ξα ∂xβ a(x, ξ)| < Cα,β (1 + |ξ|)m−ρ|α|+δ|β| ,

Cα,β > 0, x, ξ ∈ Rn ,

(1.2)

are not in general L bounded, for p = 2, when m = 0, 0 ≤ δ < ρ < 1; see, e.g., Feffermann [1]. In the present paper we deal with symbols a(x, ξ) which take values in Hölder spaces of quasi-homogeneous type in the x variable, and whose derivatives satisfy a decay of type (1.2), with respect to a quasi-homogeneous weight ξM precisely defined in the next (2.1). The pseudo-differential operators thus obtained satisfy in natural way the Marcinkiewicz-Lizorkin Lemma for Fourier multipliers, see [9, §IV.6], thanks to the quasi-homogeneous structure of their symbols; hence their Lp continuity follows, when m = 0. p

248

G. Garello and A. Morando

Lp continuity for pseudo-differential operators with symbols in Besov, H¨ older and Sobolev weighted classes are considered by the authors themselves also for more general weight functions, but always in the case δ = 0, see, e.g., [4], [5]. About this subject let us notice that also the smooth quasi-homogeneous symbol classes are defined by Lascar [6] with ρ = 1 and δ = 0, precisely only a “logarithmic” increase of ∂x a(x, ξ) is considered there. We then need also a variant of Lascar calculus for pseudo-differential operators. Namely we obtain asymptotic expansion for adjoint and compositions of operators and we construct a parametrix in the quasi-elliptic case. Applications to quasi elliptic partial differential equations with weighted Hölder coefficients may be then obtained. The plan of the paper is the following: In §2 the quasi-homogeneous weights older ξM are introduced and the related quasi-homogeneous Sobolev and H¨ s,p s,M and B∞,∞ defined, by means of a generalization of the dyadic parclasses, HM tition of unity, see [12]. In the same section we introduce the quasi-homogeneous symbol classes and prove the continuity results. In §3 we improve the Lascar’s pseudo-differential calculus in [6], for adjoint and composition. We then define the quasi-elliptic symbols and provide results of regularity for quasi-elliptic linear partial differential operators with H¨ older coefficients. A number of applications of pseudo-differential operators with non regular symbols in the study of regularity to non linear partial differential equations are given in the literature of the last twenty years, see, e.g., [10], [11] and the references given there. We hope in the next future in similar applications of our techniques.

2. Symbol classes and continuity for pseudo-differential operators In the following M = (m1 , . . . , mn ) is a weight vector with positive integer components, such that min mj = 1 and 1≤j≤n

|ξ|M :=

n

2m ξj j

12 ,

ξ ∈ Rn

(2.1)

j=1

is called quasi-homogeneous weight function on Rn . 1 := m11 , . . . , m1n and ξ2M := (1 + |ξ|2M ). Clearly We set m∗ := max mj , M 1≤j≤n

the usual Euclidean norm |ξ| corresponds to the quasi-homogeneous weight in the case M = (1, . . . , 1) (in this case, we also set ξ2 := 1 + |ξ|2 ). By easy computations, see, e.g., [3] we obtain the following Proposition 2.1. For any weight vector M there exists a suitable positive constant C such that ∗

i) C1 ξ ≤ ξM ≤ Cξm , ξ ∈ Rn , ii) |ξ + η|M ≤ C(|ξ|M + |η|M ), ξ, η ∈ Rn ;

Quasi-Elliptic Pseudo-Differential Operators in H¨ older Classes

249

iii) (quasi-homogeneity) for any t > 0, |t1/M ξ|M = t|ξ|M , where t1/M ξ = (t1/m1 ξ1 , . . . , t1/mn ξn ); 1 1− M ,α

iv) ξ γ ∂ α+γ |ξ|M ≤ Cα,γ ξM

, for any α, γ ∈ Zn+ and ξ = 0.

The detailed proof, that we omit in the following, can be found in [5] and in the references given there. Proposition 2.2 (Quasi-Homogeneous Dyadic Decomposition). For some K > 1 let us consider the cut-off function φ(t) ∈ C0∞ ([0, +∞]) such that ≤ 1, φ(t) = 0 ≤ φ(t) 1 , φ(t) = 0, when t > K. Set now ϕ0 (ξ) = φ 2−1/M ξ M −φ(|ξ|M ) 1 for 0 ≤ t ≤ 2K and (2.2) ϕ−1 (ξ) = φ (|ξ|M ) , ϕh (ξ) = ϕ0 2−h/M ξ . Then for any α, γ ∈ Zn+ a positive constant Cα,γ,K exists such that: ! K,M := ξ ∈ Rn , |ξ|M ≤ K supp ϕ−1 ⊂ C−1 ! 1 h−1 2 ≤ |ξ|M ≤ K2h+1 , h ≥ 0; supp ϕh ⊂ ChK,M := ξ ∈ Rn ; K ∞ ϕh (ξ) = 1, for all ξ ∈ Rn ; h=−1 ∞

ϕh (D)u = u,

with convergence in S (Rn );

(2.3)

(2.4)

(2.5)

h=−1

γ α+γ 1 ξ ∂ ϕh (ξ) ≤ Cα,γ,K 2− M ,αh ,

ξ ∈ Rn , h = −1, 0, . . . .

(2.6)

Moreover for any fixed ξ ∈ R the sum in (2.4) reduces to a finite number of terms, independent of the choice of ξ itself. n

We say the sequence ϕ := {ϕh }∞ h=−1 , defined in (2.2), quasi-homogeneous n partition of unity. In the following for brevity we set u h = ϕh (D)u for u ∈ S (R ) and h = −1, 0, . . . . For each l ≥ −1 we also set φl := ϕh . Moreover, we will use h≤l

the short notation ξM ∼ 2h to mean that ξ belongs to the quasi-homogeneous 1 h−1 annulus ChK,M := {ξ ∈ Rn : K 2 ≤ ξM ≤ K2h+1 }. Following the arguments in [12, §10.1] we can introduce now the classes of quasi-homogeneous Sobolev and H¨ older functions and state their properties in suitable way. For any s ∈ R and u ∈ S (Rn ) we say that: s,M if • u belongs to the quasi-homogeneous H¨ older space B∞,∞ := uϕ B s,M ∞,∞

sup 2sh uh L∞ < ∞

(2.7)

h=−1,...

is satisfied for some quasi-homogeneous partition of unity ϕ; s,p , 1 < p < ∞ if • u belongs to the quasi-homogeneous Sobolev space HM s −1 s,p := D ·sM F uLp < ∞. uHM M uLp := F

(2.8)

250


Different choices of the partition of unity ϕ in (2.7) give raise to equivalent norms, s,M noted by · B∞,∞ s,M . When M = (1, . . . , 1) and s > 0, B older∞,∞ is the usual H¨ Zygmund space. s,p Both the spaces have a Banach structure and HM admits S(Rn ) as dense subspace. Proposition 2.3. Let us consider s ∈ R and p ∈]1, +∞[; then, for every quasihomogeneous dyadic partition of unity ϕ := {ϕh }∞ h=−1 , there exists a constant C > 1 such that the inequalities ∞ ∞ 12 12 sh 1 sh 2 2 s,p ≤ C 4 |ϕh (D)u| ≤ uHM 4 |ϕh (D)u| p (2.9) C p L L h=−1

h=−1

are verified for any u ∈ S (R ). n

Proposition 2.3 is a straightforward generalization of [12, Theorem 2.5.6], where characterizations of classical Sobolev spaces are considered. The main difference consists of using a vector-valued version of the MarcinkiewiczLizorkin Lemma for Lp continuity of Fourier Multipliers [9, §IV.6], instead of the Michlin-H¨ ormander Lemma used in [12, Theorem 2.5.6]. The following Proposition gives a dyadic characterization of quasi-homogeneous Sobolev and H¨ older spaces Proposition 2.4. Let us consider a sequence of Schwartz distributions {uh }∞ h=−1 ⊂ K,M n S (R ) and a constant K > 1 such that supp u ˆh ⊂ Ch for any h ≥ −1. Set now u := ∞ u . h h=−1 The following properties are satisfied: r,M sup 2rh uh L∞ < ∞ ⇒ u ∈ B∞,∞ , r ∈ R, h≥−1 rh and (2.10) 2 uh L∞ . r,M ≤ C1 sup uB∞,∞ h≥−1

and

4sh uh

∞

s,p uHM

s,p ∈ L p R n ; 2 ⇒ u ∈ HM , h=−1 ∞ sh 2 1/2 ≤ C2 p. h=−1 4 |uh |

s∈R

(2.11)

L

Here p ∈]1, ∞[ and the constants C1 , C2 are independent of the sequence {uh }∞ h=−1 . Proof. The proof of (2.11) follows plainly from [8, Propositions 1.1, 1.2], where the result is stated for classical Sobolev spaces. To show that with the assumption r,M in (2.10) u ∈ B∞,∞ let us take a smooth partition of unity ϕ = {ϕk }+∞ k=−1 ; since K,M supp u ˆh ⊂ Ch we can find a positive integer N0 such that for all k ≥ −1: ϕk (D)u

L∞

≤

k+N 0 h=max{−1,k−N0 }

ϕk (D)uh L∞ .

(2.12)


251

Moreover we can estimate as follows: ϕk (D)uh L∞ ≤ F −1 ϕk L1 uh L∞ (2.13) ≤ F −1 ϕk L1 2−rh supq=−1,0,... 2rq uq L∞ . +∞ It can be proved that the sequence F −1 ϕk L1 k=−1 is upper bounded; hence, from inequalities (2.12), (2.13) we obtain ϕk (D)uL∞ ≤ C supq=−1,0,... 2rq uq L∞

k+N 0

2−rh

h=max{−1,k−N0 }

≤ C1 2−kr supq=−1,0,... 2rq uq L∞ ,

k = −1, 0, . . . ,

where C, C1 are positive constants independent of u and k. This proves that r,M and gives the estimate in (2.10). u ∈ B∞,∞ With straightforward computations we obtain the following Proposition 2.5. When s > 0, r > 0 and p ∈]1, +∞[ there exist positive constants C1 , C2 such that the properties (2.10), (2.11) are true for all the sequences of n ˆh ⊆ BhK,M := {ξ ∈ Rn : Schwartz distributions {uh }∞ h=−1 ⊂ S (R ) with supp u |ξ|M ≤ K2h+1 }, h = −1, 0, . . . . m will be the class of functions Definition 2.6. Given m ∈ R and δ ∈ [0, 1], SM,δ ∞ n n a(x, ξ) ∈ C (R × R ) such that for all multi-indices α, β ∈ Zn+ the following holds true: 1 1 m− α, M +δ β, M

|Dxβ Dξα a(x, ξ)| ≤ Cα,β ξM

,

∀x, ξ ∈ Rn ,

(2.14)

with some positive constant Cα,β . Exploiting the estimates in Proposition 2.1 (i) one checks that: max{mm∗ ,m}

m SM,δ ⊂ S1/m∗ ,δm∗

.

(2.15)

The above inclusion establishes a relation between quasi-homogeneous classes and m the usual H¨ ormander’s symbol classes Sρ,δ , with ρ ≤ m1∗ < 1. r,M m SM,δ is defined to be Definition 2.7. For r > 0, m ∈ R and δ ∈ [0, 1], B∞,∞ the set of measurable functions a(x, ξ) such that for every α ∈ Zn+ , the following inequalities hold true 1 m− α, M

|Dξα a(x, ξ)| ≤ Cα ξM

1 m− α, M

r,M ≤ Cα ξ Dξα a(·, ξ)B∞,∞ M

,

+δr

∀x, ξ ∈ Rn ;

(2.16)

∀ξ ∈ Rn .

(2.17)

,

m m m m Later on, we will set SM := SM,0 , SM,M := SM,1 (same notations for non regular r,M m classes B∞,∞ SM,δ ).

252

G. Garello and A. Morando Repeating the arguments of [4] (cf. also [10]), we are lead to the following:

Lemma 2.8. Let r > 0 and δ ∈ [0, 1] be fixed. There exists a sequence of positive numbers {ck }k∈Zn , satisfying ck < ∞, such that for every symbol a(x, ξ) ∈ k∈Zn r,M 0 SM,δ the following decomposition B∞,∞ a(x, ξ) = ck ak (x, ξ),

(2.18)

k∈Zn

holds true, with absolute convergence in L∞ (R2n ). Moreover, for each k ∈ Zn we have: +∞ ak (x, ξ) = dkh (x)ψhk (ξ), (2.19) h=−1 k ∞ r,M ∞ n where {dkh }∞ h=−1 and {ψh }h=−1 are sequences in B∞,∞ and C0 (R ) respectively, satisfying the following assumptions:

i) There exists a positive constant L > 0 such that: |dkh (x)| ≤ L,

hrδ r,M ≤ L2 dkh B∞,∞ ,

∀x ∈ Rn , h = −1, 0, . . . , ∀k ∈ Zn ;

(2.20)

ii) There exist positive constants C and K > 1 such that for all k ∈ Zn and h = −1, 0, . . . : supp ψhk |ξ γ Dα+γ ψhk (ξ)|

⊆ ChK,M ; ≤ C2− α, M h , 1

∀ξ ∈ Rn .

Starting from the previous lemma and following the arguments of Taylor [10, Thm. 2.1.A], we obtain the following r,M m SM,δ , then for all Proposition 2.9. If r > 0, m ∈ R, δ ∈ [0, 1] and a(x, ξ) ∈ B∞,∞ s ∈](δ − 1)r, r[ and p ∈]1, +∞[ the following s+m,p s,p a(x, D) : HM → HM

a(x, D) :

s+m,M B∞,∞

→

s,M B∞,∞

(2.21) (2.22)

are linear continuous operators. If δ < 1 the mapping property (2.22) is still true for s = r Proof. We can reduce to prove the result for m = 0; moreover, in view of Lemma 2.8 it will be enough proving the statement for an elementary symbol. Thus, let us consider a zeroth order symbol of the form a(x, ξ) =

∞ h=−1

dh (x)ψh (ξ),


253

r,M where {dh } and {ψh } are sequences in B∞,∞ and C0∞ (Rn ) respectively, fulfilling the assumptions of Lemma 2.8. Then, using a quasi-homogeneous partition of unity {ϕl }, the corresponding operator a(x, D) can be decomposed as (N0 )

a(x, D)u = T1

(N0 )

u + T2

(N0 )

u + T3

u

where (N0 )

T1

∞

u :=

h−N 0

dh,l uh ,

(2.23)

h=N0 −1 l=−1 (N0 )

T2

h+N 0 −1

∞

u :=

dh,l uh ,

(2.24)

h=−1 l=max{−1,h−N0 +1} (N0 )

T3

∞

u :=

l−N 0

dh,l uh ,

(2.25)

l=N0 −1 h=−1

and N0 is a sufficiently large positive integer; for all h, l = −1, 0, . . . , we have set uh = ψh (D)u and dh,l = ϕl (D)dh . Therefore, Proposition 2.9 becomes a consequence of Lemmas 2.10–2.12 below; in all of them, we make the same hypotheses of Proposition 2.9. Lemma 2.10. For all s ∈ R and p ∈]1, +∞[ the following (N0 )

T1

s,p s,p : HM → HM ,

(2.26)

(N ) T1 0

s,M s,M : B∞,∞ → B∞,∞

(2.27)

are linear continuous operators. h Proof. Since d h,l uh is supported on ξM ∼ 2 , for −1 ≤ l ≤ h−N0 , by Proposition 2.4, we compute: ⎛ h−N 2 ⎞ 12 ∞ 0 (N ) s,p ≤ C ⎝ T1 0 uHM 4sh dh,l uh ⎠ . (2.28) l=−1 h=N0 −1 p L

On the other hand, h−N 0

dh,l (x) = φh−N0 (D)dh (x) = (F −1 φh−N0 ∗ dh )(x),

l=−1

hence

h−N 0 dh,l (x) ≤ F −1 φh−N0 L1 dh L∞ ≤ Cdh L∞ , l=−1

∀x ∈ Rn .

254


Using the previous inequality to estimate the right-hand side of (2.28) and accounting the boundedness of {dh } in L∞ , we find 12 ∞ (N0) sh 2 ≤ CuH s,p . s,p ≤ C T1 uHM 4 |uh | M p h=−1 L

This shows the mapping property (2.26). To prove property (2.27), we use estimate (2.10) in Proposition 2.4, together with the boundedness of dh,l L∞ , to get h−N 0 (N0 ) sh T1 uB∞,∞ s,M ≤ C sup 2 dh,l uh h=−1,0,... l=−1

≤C

sup h=−1,0,...

L∞

2 uh L∞ ≤ CuB∞,∞ s,M .

sh

Lemma 2.11. For all s > (δ − 1)r and p ∈]1, +∞[ the following (N0 )

T2

(N ) T2 0

s+r(1−δ),p

s,p : HM → HM

(2.29)

s,M s+r(1−δ),M : B∞,∞ → B∞,∞

(2.30)

are linear continuous operators. h+1 Proof. Since d }, h,l uh is supported in a quasi-homogeneous ball {ξM ≤ K2 as long as max{−1, h − N0 + 1} ≤ l ≤ h + N0 − 1 and s + (1 − δ)r > 0, we can (N ) estimate T2 0 uH s+(1−δ)r,p by M

⎛ 2 ⎞ 12 h+N −1 ∞ 0 ⎟ ⎜ s+(1−δ)r C 4 h d u ⎠ ⎝ h,l h l=max{−1,h−N0 +1} h=−1

.

(2.31)

Lp

hrδ−rl r,M ≤ 2 On the other hand, the inequalities dh,l L∞ ≤ 2−rl dh B∞,∞ yield

h+N h+N 0 −1 0 −1 dh,l (x)uh (x) ≤ 2r(δh−l) |uh (x)| l=max{−1,h−N0 +1} l=max{−1,h−N0 +1}

≤ CN0 2r(δ−1)h |uh (x)|,

(2.32)

∀x ∈ Rn ,

where CN0 > 0 depends only on N0 , r and δ. Using the latter inequalities to majorize (2.31), we find the mapping property (2.29). To prove (2.30), we only need to use the dyadic characterization of positive order H¨ older spaces in Proposition 2.5, instead of the same for Sobolev spaces; then using again the estimates dh,l L∞ ≤ 2hrδ−rl and (2.32), we manage as before to

Quasi-Elliptic Pseudo-Differential Operators in H¨ older Classes find (N ) T2 0 uB s+(1−δ)r,M ∞,∞

255

h+N 0 −1 (s+(1−δ)r)h ≤ C sup 2 dh,l uh h=−1,0,... l=max{−1,h−N0 +1 }

L∞

≤C

2 uh L∞ ≤ CuB∞,∞ s,M , sh

sup h=−1,0,...

with positive constants C depending only on r, s, δ and N0 . Lemma 2.12. For all s < r and p ∈]1, +∞[ the following s+(δ−1)r,p

(N0 )

T3

(N ) T3 0

: HM :

s+(δ−1)r,M B∞,∞

s,p → HM

(2.33)

→

(2.34)

s,M B∞,∞

are linear continuous operators. If δ < 1 then (N0 )

T3

rε,M r,M : B∞,∞ → B∞,∞

(2.35)

is also a continuous operator for any 0 ≤ δ < ε. l Proof. Since for each index l ≥ N0 −1, d h,l uh is supported on an annulus ξM ∼ 2 , as long as −1 ≤ h ≤ l − N0 , applying again the dyadic characterization of Sobolev spaces gives ⎛ l−N 2 ⎞ 12 ∞ 0 (N ) s,p ≤ C ⎝ 4sl dh,l uh ⎠ T3 0 uHM h=−1 l=N0 −1 p L ⎧ ⎫ 12 2 l−N ∞ ⎨ ⎬ 0 sl ≤C 4 |dh,l ||uh | . ⎩ ⎭ h=−1 l=N0 −1 p L

Then, using once more the estimates dh,l L∞ ≤ 2 , the argument of the last Lp -norm above can be bounded by ⎧ 2 ⎫ 12 l−N ∞ ⎬ ⎨ 0 2(s−r)(l−h) 2(s+(δ−1)r)h |uh | . ⎭ ⎩ rδh−rl

l=N0 −1

h=−1

Provided that s − r < 0, by Young’s inequality, the latter quantity is estimated in its turn by ∞ 12 (s+r(δ−1))h 2 C 4 |uh | , h=N0 −1

with C :=

∞ l=N0 −1

2(s−r)l .

256

G. Garello and A. Morando From previous calculations, we derive 12 ∞ (N0 ) (s+(δ−1)r)h 2 s,p T3 uHM ≤ C 4 |uh | h=−1

≤ CuH s+(δ−1)r,p , M

Lp

which shows the mapping property (2.33). To establish (2.34) it is now enough to use the first part of Proposition 2.4, and repeat the previous calculations to find (N0 )

T3

s,M ≤ C uB∞,∞ sup 2sl

l≥N0 −1

≤ C sup

l−N 0

l−N 0

l≥N0 −1 h=−1

≤C

sup h=−1,0,...

since supl≥N0 −1

l−N 0

dh,l L∞ uh L∞

h=−1

2(s−r)(l−h) 2(s+(δ−1)r)h uh L∞

2(s+(δ−1)r)h uh L∞ ≤ CuB s+(δ−1)r,M , ∞,∞

2(s−r)(l−h) is finite, as long as s − r < 0.

h=−1

To prove (2.35), let ε > δ be arbitrarily fixed; using (2.10) we find: (N0)

T3

r,m uB∞,∞ ≤ C sup 2rl

l≥N0 −1

≤ C sup

l−N 0

dh,l L∞ uh L∞

h=−1 l−N 0

l≥N0 −1 h=−1

2hrδ uh L∞

≤ C sup 2hrε uh L∞ sup h≥−1

since

sup

l−N 0

l≥N0 −1 h=−1

l≥N0 −1

(2.36) l−N 0

2hr(δ−ε) ≤ CuB rε,M ∞,∞ ,

h=−1

2hr(δ−ε) is finite, as δ − ε < 0; in (2.36) C stands for suitable

different positive constants, depending only on r, ε and δ.

Remark 2.13. For δ = 0, the statement of Proposition 2.9 provides the already known Sobolev and H¨ older continuity properties for operators with non regular r,M m SM , see [5]. symbols in B∞,∞ On the extremely opposite border case δ = 1, Proposition 2.9 tells that r,M m operators with non regular symbols in B∞,∞ SM,1 , r > 0, possess the mapping properties (2.21), (2.22) for every s ∈]0, r[. m r,M m Since, of course, the inclusions SM,δ ⊂ B∞,∞ SM,δ are true for all positive r, as a consequence of Proposition 2.9 we have the following m Corollary 2.14. If a(x, ξ) ∈ SM,δ , for 0 ≤ δ < 1, then the mapping properties (2.21), (2.22) are true for all s ∈ R.


257

3. Symbolic calculus and parametrix for quasi-elliptic pseudo-differential operators Proposition 3.1. Symbolic calculus. For m, m ∈ R, δ, δ ∈ [0, 1], μ, ν ∈ Zn+ the following are true: 1 1 m− μ, M +δ ν, M

(μ)

m ⇒ σ(ν) := Dξμ Dxν σ ∈ SM,δ σ ∈ SM,δ

;

(3.1)

m+m m m , τ ∈ SM,δ σ ∈ SM,δ ⇒ σ · τ ∈ S M,max{δ,δ } .

(3.2)

Proof. Let α, β ∈ Zn+ be arbitrarily fixed. Then 1 1 m− α+μ, M +δ β+ν, M

(μ)

|Dξα Dxβ σ(ν) (x, ξ)| = |Dξα+μ Dxβ+ν σ(x, ξ)| ≤ CξM

,

with positive constant C = Cα,β,μ,ν , shows (3.1). Leibnitz’s rule also implies: αβ α−μ β−ν Dx σ(x, ξ)||Dξμ Dxν τ (x, ξ)| |Dξα Dxβ (σ · τ )(x, ξ)| ≤ μ ν |Dξ ≤

0≤μ≤α , 0≤ν≤β 1 1 m− α−μ, M +δ β−ν, M

Cα,β,μ,ν ξM

0≤μ≤α , 0≤ν≤β 1 1 m+m − α, M +max{δ,δ } β, M

1 1 m − μ, M +δ ν, M

ξM

≤ CξM

which shows (3.2).

m Definition 3.2. Asymptotic expansion. Let σ ∈ SM,δ . We say that a sequence mj {σj }j≥0 of symbols σj ∈ SM,δ is an asymptotic expansion of σ if {mj }j≥0 is a decreasing sequence with m0 = m, lim mj = −∞, and for any positive integer N : j→+∞ mN σ− σj ∈ SM,δ .

In this case, we write σ ∼

j ε0 > ε1 > · · · > εj → 0 , as j → +∞.

258


We define the symbol σ(x, ξ) by σ(x, ξ) =

+∞ j=0

1

ψ(εjM ξ)σj (x, ξ).

m0 and Then one can repeat the arguments of [13] to prove that σ ∈ SM,δ σ ∼ j σj .

In order to develop a functional calculus for pseudo-differential operators m it is convenient to introduce with symbols in the quasi-homogeneous classes SM,δ a class of apparently more general operators involving multiple symbols. m Definition 3.4. For m ∈ R and δ1 , δ2 ∈ [0, 1], SM,δ will be the class of functions 1 ,δ2 ∞ n n n a(x, y, ξ) ∈ C (R × R × R ) such that for all multi-indices α, β, γ ∈ Zn+ there exists a positive constant Cα,β,γ such that 1 1 1 m− α, M +δ1 β, M +δ2 γ, M

|Dξα Dxβ Dyγ a(x, y, ξ)| ≤ Cα,β,γ ξM

,

x, y, ξ ∈ Rn . (3.3)

m we define the operator A = a(x, D, y) by When a(x, y, ξ) ∈ SM,δ 1 ,δ2 Au(x) = (2π)−n ei x−y,ξ a(x, y, ξ)u(y) dy dξ , u ∈ S(Rn ).

(3.4)

Under the above conditions, for v ∈ S(Rn ) −n Au , v = (2π) ei x−y,ξ a(x, y, ξ)u(y)v(x) dx dy dξ

(3.5)

is defined as an ordinary oscillatory integral. It is easily verified that, for fixed u, the expression (3.5), viewed as a functional of v, defines a distribution Au ∈ S (Rn ). Therefore a linear operator A : S(Rn ) → S (Rn ) is defined; we will formally write it as the integral (3.4). m Proposition 3.5. If a(x, y, ξ) ∈ SM,δ with 0 ≤ δ2 < m1∗ then the operator (3.4) 1 ,δ2 m m belongs to OpSM,δ with δ = max{δ1 , δ2 }; in fact A = q(x, D), where q ∈ SM,δ has the asymptotic expansion i|α| Dα Dα a(x, y, ξ)| y=x . q(x, ξ) ∼ (3.6) α! ξ y α≥0

Proof. A formal calculation gives Au(x) = (2π)−n with q(x, ξ) = (2π)−n

u(ξ)dξ , ei x,ξ q(x, ξ)3

ei x−y,ξ−η a(x, y, ξ)dξdy =

and ˆb(x, η, ξ) := (2π)−n

3b(x, η, ξ + η)dη ,

e−i y,η a(x, x + y, ξ)dy.

(3.7)


259

The hypotheses on a(x, y, ξ) imply 1 1 m− α, M +δ β, M +δ2 ν

|Dξα Dxβ ˆb(x, η, ξ)| ≤ Cν,α,β ξM

η−ν M

for all α, β ∈ Zn+ , ν ∈ Z+ and suitable Cα,β,ν > 0. A Taylor’s expansion of ˆb(x, η, xi + η) in the last argument about ξ gives N 1 m− m ˆ ∗ +δ2 ν −ν αˆ α (iD ) b(x, η, ξ)η ≤ CηN b(x, η, ξ + η) − ξ M ηM sup ξ + tηM α! |α| 0, 0 ≤ δ < 1 is M -quasi-elliptic if there exist two positive constants C, K such that |p(x, ξ)| > Cξm when |ξ|M > K. (3.8) M,

260


m Proposition 3.7. Let p(x, ξ) ∈ SM,δ be M -quasi-elliptic, for m ∈ R and 0 ≤ δ < −m 1 ; then we can find q(x, ξ) ∈ S ∗ M,δ such that m

q(x, D)p(x, D) = I + R1 (x, D),

R1 (x, ξ) ∈ S −∞

p(x, D)q(x, D) = I + R2 (x, D),

R2 (x, ξ) ∈ S −∞ .

(3.9)

Proof. Let K be the constant in (3.8) and choose ψ(ξ) ∈ C ∞ (Rn ) be equal to 0 for |ξ|M ≤ K and equal to 1 for |ξ|M ≥ 2K. Set now q0 (x, ξ) =

ψ(ξ) . p(x, ξ)

(3.10)

Since the derivatives of ψ(ξ) vanish for great ξ, using the chain rule and observing 1 1 −m− α, M +δ β, M

that |∂xα ∂ξβ (p(x, ξ)−1 )| < Cα,β ξM −m q0 (x, ξ) ∈ SM,δ . As long as 0 ≤ δ
K, we obtain that < 1, Proposition 3.6 applies and thus 1

m∗ . r0 (x, ξ), r˜0 (x, ξ) ∈ SM,δ

(3.11)

Let now s(x, D) be the formal expansion r0 (x, D) − r0 (x, D)2 + · · · and set 0 , we obtain q(x, D) = (I − s(x, D))q0 (x, D). Since s(x, D) ∈ OpSM,δ q(x, D)p(x, D) = I + R(x, D),

R(x, ξ) ∈ S −∞ .

(3.12)

˜ D) such that p(x, D)˜ In the same way we can build q˜(x, D) and R(x, q (x, D) = ˜ D). Since (q(x, D)p(x, D))˜ I + R(x, q (x, D) = q(x, D)(p(x, D)˜ q (x, D)), q(x, D) = q˜(x, D)mod S −∞ follows, then (3.9) holds. In the lack of a suitable parametrix of M -quasi-elliptic operators with non smooth symbols, for studying the regularity of solutions to M -quasi-elliptic equations with non smooth coefficients, following the approach in Taylor [10, §1.3], we introduce the next symbol decomposition. Let φ be a fixed C ∞ function such that φ(ξ) = 1 for ξM ≤ 1 and φ(ξ) = 0 1 1 1 for ξM > 2. For a given ε > 0 we set φ(ε M ξ) := φ(ε m1 ξ1 , . . . , ε mn ξn ). The following quasi-homogeneous version of [10, Lemma 1.3.A] can be readily proved. Lemma 3.8. For all r > t ≥ 0 and ε ∈]0, 1] the following estimates hold true − β, M Dxβ φ(ε M D)uB∞,∞ r,M ≤ Cβ ε uB∞,∞ r,M , 1

u − φ(ε

1 M

1

t D)uB∞,∞ r−t,M ≤ Cε u r,M , B∞,∞ 1 M

r,M . u − φ(ε D)uL∞ ≤ Cεr uB∞,∞

with positive constants Cβ , C independent of ε.

(3.13) (3.14) (3.15)


261

Proof. To prove (3.13), we show that {ε M ,β ξ β φ(ε M ξ)}0 0, the latter inequalm ities prove that a# (x, ξ) ∈ SM,δ (cf. Definition 2.6).

262

G. Garello and A. Morando Since

h

ϕh = 1, we can write ∞ hδ ab (x, ξ) = (I − φ(2− M D))a(x, ξ)ϕh (ξ). h=−1

For an arbitrary multi-index α = 0, we find: ∞ α hδ Dξα ab (x, ξ) = (I − φ(2− M D))Dξα−ν a(x, ξ)Dξν ϕh (ξ). ν ν≤α

h=−1

Using now (3.15) to estimate the L∞ norm of ab (x, ξ), we find: ∞ α hδ Dξα a(·, ξ)L∞ ≤ (Id − φ(2− M D))Dξα−ν a(·, ξ)L∞ |Dξν ϕh (ξ)| ν ≤

∞ α ν≤α

ν

ν≤α

2

−hδr

D

h=−1

α−ν

h=−1

1 m−δr− α, M

−h ν, M a(·, ξ)B∞,∞ r,M 2 ≤ CξM 1

.

The latter provides an estimate of type (2.16) with m − δr instead of m. To check the validity of an estimate of type (2.17), with m − δr instead of m, we use (3.14) r,M with t = 0 to estimate the B∞,∞ norm of Dξα ab (x, ξ). r,M m Proposition 3.10. If the symbol a(x, ξ) ∈ B∞,∞ SM , m ∈ R, is M -quasi-elliptic, m then for any 0 < δ ≤ 1, a (x, ξ) ∈ SM,δ satisfies the same condition.

Proof. By the M -quasi-ellipticity there exist positive constants C1 , R1 such that r,M ∞ the same |a(x, ξ)| > C1 ξm M when |ξ|M > R1 . From the embedding B∞,∞ → L r,M . For any R0 > 0 we can find a positive integer estimate follows for a(·, ξ)B∞,∞ h0 , which increases together with R0 , such that ψh (ξ) = 0 as long as |ξ|M > R0 and h = −1, . . . , h0 − 1. We can then write: ∞ δ a (x, ξ) = φ 2−h M Dx a(x, ξ)ϕh (ξ), |ξ|M > R0 . (3.16) Set for brevity φ 2

h=h0 δ −h M

· = φh (·)

By means of (3.16), the Cauchy-Schwarz inequality and (3.15) in Lemma 3.8, when |ξ|M > R0 we can estimate |a (x, ξ) − a(x, ξ)|2 2 ∞ = (φh (Dx ) − I) a(x, ξ)ϕh (ξ) h=h0 =

∞

h+N 0

(φh (Dx )) − I) a(x, ξ)ϕh (ξ), (φk (Dx ) − I) a(x, ξ)ϕk (ξ)

h=h0 k=h−N0

=

N0

∞

(φh (Dx ) − I) a(x, ξ)ϕh (ξ), (φh+t (Dx ) − I) a(x, ξ)ϕh+t (ξ)

t=−N0 h=h0

≤

N0

∞

t=−N0 h=h0

(φh (Dx )) − I) a(·, ξ)∞ |ϕh (ξ)| (φh+t (Dx ) − I) a(·, ξ)∞ |ϕh+t (ξ)|

Quasi-Elliptic Pseudo-Differential Operators in H¨ older Classes ≤ C2 ≤

N0

∞

263

2−hδr 2−(h+t)δr a(·, ξ)2B r,M

t=−N0 h=h0 ∞ 2 −2hδr C a(·, ξ)2B r,M h=h0 2 ∞,∞

∞,∞

≤ C 2 2−2h0 δr a(·, ξ)2B r,M , ∞,∞

where C denotes different positive constants depending only on δ, N0 and r. −h0 δr Since a(·, ξ)B∞,∞ r,M < cξm < M , let us fix R0 large enough to have: C2 C1 2c .

Then for |ξ|M > max{R0 , R1 }

|a (x, ξ)| ≥ |a(x, ξ)| − |a (x, ξ) − a(x, ξ)| > C1 ξm M −

C1 m 2 ξM

C1 m 2 ξM

=

(3.17)

follows and the proof is concluded.

As an application of the previous considerations on smooth and non-smooth pseudo-differential symbol classes with quasi-homogeneous decay let us consider the linear partial differential operator of quasi-homogeneous order m ∈ Z+ r,M A(x, D)u := aα (x)Dα , aα ∈ B∞,∞ , r > 0. (3.18) 1

α, M ≤m

r,M m Since the symbol A(x, ξ) belongs to B∞,∞ SM , for given 0 < δ < 1 we can split it

A(x, ξ) = A (x, ξ) + Ab (x, ξ), m−δr m r,M A (x, ξ) ∈ SM,δ , Ab (x, ξ) ∈ B∞,∞ SM,δ .

(3.19)

Consider now the linear partial differential equation A(x, D)u = f

(3.20)

and split it by means of (3.19). Assuming now that A(x, D) is M -quasi-elliptic, −m then A (x, D) admits a left parametrix B(x, ξ) ∈ SM,δ , when 0 < δ < m1∗ , see Propositions 3.10, 3.7. Applying now B(x, D) to both sides of (3.20) and using Proposition 3.7 we obtain: R(x, ξ) ∈ S −∞

u = B(x, D)f − R(x, D)u − B(x, D)Ab (x, D)u, s−m,p HM

(3.21)

s−δr,p Assume now that f ∈ (resp. f ∈ and consider u in HM s−δr,M (resp. in B∞,∞ ), then by means of the continuity properties in Proposition 2.9 s,p s,M it follows that u ∈ HM (resp. u ∈ B∞,∞ ), when (δ − 1)r + m < s < r + m (resp. (δ − 1)r + m < s ≤ r + m), 0 < δ < m1∗ . s−m,M B∞,∞ )

We have then proved the following Theorem 3.11. Let A(x, D)u = f be a linear partial differential equation with H¨ older coefficients introduced by means of (3.18). Assume that A(x, D) is M multi-quasi-elliptic. Then it follows for any solution u: s−δr,p s−m,p , f ∈ HM u ∈ HM s−δr,M s−m,M u ∈ B∞,∞ , f ∈ B∞,∞

for any 0 < δ
0, j = 1, . . . , n} .

D =

D1α1

...

Dnαn , Dj

=

The space C0∞ (Ω) is the space of functions u ∈ C ∞ with compact support in Ω. The space of distributions on Ω is denoted D (Ω) . Definition 2.1. Let A be a finite subset of Rn+ , the Newton’s polyhedron of A, denoted Γ(A), is the convex hull of {0} ∪ A. A Newton’s polyhedron Γ is always characterized by Γ= α ∈ Rn+ , q, α ≤ 1 , q∈A(Γ)

where A (Γ) is a finite subset of Rn {0}. ) α ∈ Rn+ , q, α ≤ 1 be a Newton’s polyhedron, Γ Definition 2.2. Let Γ= q∈A(Γ)

is said to be regular, if qj > 0, ∀j = 1, . . . , n; ∀q = (q1 , . . . , qn ) ∈ A (Γ) We associate with a regular Newton’s polyhedron Γ the following elements V (Γ) = s0 = 0, s1 , . . . , sm(Γ) the set of vertices of Γ |ξ|Γ = |ξ|ν , ξ ∈ Rn , where |ξ|ν = |ξ1 |ν1 . . . |ξn |νn ν∈V(Γ)

k (α, Γ) = inf t > 0, t−1 α ∈ Γ = max α, q , α ∈ Rn+ q∈A(Γ)

μ (Γ) = max q∈A(Γ) 1≤ j≤n

qj−1

called the formal order of Γ

A differential operators with complex constant coefficients P (D) = has its complete symbol aα ξ α . P (ξ) = α

α

aα D α

Multi-Anisotropic Gevrey Regularity

267

Definition 2.3. The of P , denoted Γ(P ), is the convex hull Newton’s polyhedron of the set {0} ∪ α ∈ Zn+ : aα = 0 . Define the weight function |ξ|P =

|ξ α | , ∀ξ ∈ Rn ,

α∈V(P )

where V (P ) = V (Γ (P )) is the set of vertices of Γ (P ) . Recall d(ξ) := dist(ξ, N (P )), where N (P ) := {ζ ∈ Cn : P (ζ) = 0} Definition 2.4. The differential operator P (D) is said hypoelliptic in Ω, if singsupp P (D)u = singsupp u, ∀u ∈ D (Ω) The characterization of hypoelliptic differential operators with constant coefficients is du to L. Hörmander. The following result, see Theorem 4.1.3 of [6], gives some characterizations of the hypoellipticity. Theorem 2.5. Let P (D) be a differential operator with constant coefficients, the following properties are equivalent: i) The operator P (D) is hypoelliptic. d ii) ∃C > 0, ∃d > 0, |ξ| ≤ Cd(ξ), ∀ξ ∈ Rn , |ξ| large. |Dα P (ξ)| → 0, ∀α = 0. iii) If ξ ∈ Rn , |ξ| → +∞, then |P (ξ)| |Dα P (ξ)| −ρ|α| ≤ C |ξ| iv) ∃C > 0, ∃ρ > 0, , ∀ξ ∈ Rn , |ξ| large. |P (ξ)| The connection between an hypoelliptic operator and its Newton’s polyhedron is given by the following proposition. Proposition 2.6. The Newton’s polyhedron of an hypoelliptic differential operator is regular.

Proof. See [3].

Remark 2.7. The converse is not true, = Dx2 − Dy2 has a regular Newton’s polyhedron with vertices {(0, 0) , (2, 0), (0, 2)} , but the operator is not hypoelliptic. We introduce multi-quasielliptic polynomials which are a natural generalization of the classical quasi-elliptic operators. These operators were characterized first by V.P. Mikha¨ılov [8], then studied by J. Friberg [3] and finally far developed by S.G. Gindikin and L.R. Volevich [4]. Definition 2.8. The polynomial P (ξ) = aα ξ α is said to be multi-quasielliptic, if α

i) its Newton’s polyhedron Γ (P ) is regular. ii) ∃C > 0 such that |ξ|P ≤ C(1 + |P (ξ)|), ∀ξ ∈ Rn

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C. Bouzar and A. Dali

Proposition 2.9. A multi-quasielliptic operator P (D) is hypoelliptic.

Proof. See [3] or [4].

Remark 2.10. The converse is not true. Indeed, consider the following polynomial P (ξ, η) = iξ 5 + iξη 4 − 4iξ 4 η − 4iξ 2 η 3 + 6iξ 3 η 2 + iξ 3 + iξη 2 + ξ 4 η 2 + η 6 − 4ξ 3 η 3 − 4ξη 5 + 6ξ 2 η 4 + η 2 ξ 2 + η 4 , which is hypoelliptic thanks to Theorem 4.1.9 of [6]. We have 4 P(1,1) (ξ, η) = η 2 ξ 4 + η 4 − 4ξ 3 η − 4ξη 3 + 6ξ 2 η 2 = η 2 (ξ − η) The q = (1, 1)-quasiprincipal part of P (ξ, η) degenerates on the straight ξ = η, hence the polynomial P (ξ, η) is not multi-quasielliptic, see [4].

3. Multi-anisotropic Gevrey vectors The multi-anisotropic Gevrey spaces were explicitly defined by L. Zanghirati in [10] for studying the multi-anistropic Gevrey regularity of multi-quasielliptic differential operators by the method of elliptic iterates. Definition 3.1. Let Ω be an open subset of Rn , Γ a regular Newton’s polyhedron and s ≥ 1. Denote Gs, Γ (Ω) the space of functions u ∈ C ∞ (Ω) such that ∀K ⊂ Ω, ∃C > 0, ∀α ∈ Zn+ , sup |Dα u(x)| ≤ C |α|+1 k(α, Γ)sk(α,Γ)

(3.1)

x∈K

Example. If Γ is the regular Newton’s polyhedron defined by n m−1 α ≤ 1, m ∈ R Γ= α ∈ Rn+ : j j + , j j=1

then

u ∈ C ∞ (Ω) , ∀K ⊂ Ω, ∃C > 0, ∀α ∈ Zn+ , Gs, Γ (Ω) = |Dα u (x)| ≤ C |α|+1 α, qs α,q m m ,..., and m := maxmj , i.e., Gs, Γ (Ω) is the classical where q := j m1 mn anisotropic Gevrey space Gs, q (Ω). If m1 = m2 = · · · = mn , we obtain the classical isotropic Gevrey space Gs (Ω) .

Definition 3.2. Let Γ be the regular Newton’s polyhedron of P (D) and s ≥ 1, the space of Gevrey vectors of P (D), denoted Gs (Ω, P ), is the space of u ∈ C ∞ (Ω) such that, ∀K compact of Ω, ∃C > 0, ∀l ∈ N, l sμ(Γ) P u ∞ ≤ C l+1 (l!) L (K) Remark 3.3. We can take lslμ(Γ) instead of (l!)sμ(Γ) .


269

We recall a result of L. Zanghirati [10] and C. Bouzar and R. Cha¨ıli [1] which gives the multi-anisotropic Gevrey regularity of Gevrey vectors of multiquasielliptic operators. Theorem 3.4. Let Ω be an open subset of Rn , s > 1 and P a linear differential operator with complex constant coefficients with regular Newton’s polyhedron Γ. Then the following assertions are equivalent: i) P is multi-quasielliptic in Ω ii) Gs (Ω, P ) = Gs, Γ (Ω)

4. Multi-anisotropic Gevrey hypoellipticity of hypoelliptic operators In this section, P =

aα Dα is an hypoelliptic differential operator with complex

α

constant coefficients. Definition 4.1. A finite set H ⊂ Rn+ is said a polyhedron of hypoellipticity of P, if 1. ∀ν ∈ H, ∃C > 0, ∀ξ ∈ Rn , |ξ ν | ≤ C (1 + d (ξ)) 2. H has vertices with rational components. 3. H is regular. Remark 4.2. If ν belongs to the convex hull of H, i.e., ν = and

λi βi , where βi ∈ H

i∈I I f ini ν

λi = 1, λi ≥ 0, then |ξ| ≤ C (1 + d(ξ)) , ∀ξ ∈ Rn , therefor it is natural to

i∈I

assume that H is convex. Definition 4.3. Denote σ be the smallest natural integer such that σV (H) ⊂ 2Nn0 , and define the differential operator QH (D) , by QH (D) = Dσα α∈V(H)

Proposition 4.4. The operator QH (D) is multi-quasielliptic. Proof. The Newton’s polyhedron of the differential operator QH has vertices with even positive integer components. Then |ξ σα | = |ξ|QH , |QH (ξ)| = α∈V(H)

hence 1 + |ξ|QH ≤ (1 + |QH (ξ)|) , ∀ξ ∈ Rn

270

C. Bouzar and A. Dali Let v ∈ C0∞ (Rn ) , s ∈ Z+ and ε > 0, then 2 s 2 |||v|||s,ε := (1 + εd (ξ)) |3 v (ξ)| dξ Rn

The following result is Lemma 4.4.3 of [6]. Lemma 4.5. Let u be a solution of the equation P u = 0 defined in the ball Bε = {x ∈ Rn : |x| < ε} , and let ϕ ∈ C0∞ (B1 ) and the integer s ≥ 1. Then 2 2 (α) ε−2|α| P (α) (D) (ϕε u) ≤ C ε−2|α| (D) u dx, (4.1) P s,ε

α=0

Bε

α=0

where C is independent of ε and u. Remark 4.6. In the lemma ϕε denotes ϕε (x) := ϕ

x ε .

Thanks to this lemma, we obtain the following result. Lemma 4.7. Let β ∈ Zn+ ∩ σH, then there exists a constant C > 0, such that for every solution u of P u = 0 in Bε and ε ∈ ]0, 1[ , we have 2 2 ε−2|α| P (α) (D) Dβ u dx ≤ C ε−2|α| P (α) (D) u dx ε2σ α=0

α=0

Bε

2

Bε

Proof. Let β ∈ Zn+ ∩ σH, from (1) of definition 4.1, we have β ξ ≤ C σ (1 + d (ξ))σ , hence ∃C > 0, ∀ε ∈ ]0, 1[ , ∀ξ ∈ Rn , εσ ξ β ≤ C σ dσ,ε (ξ)

(4.2)

−n

Multiplying (4.2) by (2π)

|3 v (ξ)| and integrating with respect to ξ, we obtain β 2 2σ D v dx ≤ C 2 |||v|||2 ε (4.3) σ,ε

Let ϕ ∈ C0∞ (B1 ) equals 1 in B 21 and apply the estimate (4.3) to v = P (α) (D)(ϕε u), then 2 2 (α) 2σ ε (D) Dβ (ϕε u) dx ≤ C 2 P (α) (D) (ϕε u) P σ,ε 2 2 ε2σ ε−2|α| P (α) (D) Dβ (ϕε u) dx ≤ C 2 ε−2|α| P (α) (D) (ϕε u) , α=0

σ,ε

α=0

consequently Lemma 4.6 gives 2 2 ε2σ ε−2|α| P (α) (D) Dβ (ϕε u) dx ≤ C ε−2|α| P (α) (D) (u) dx α=0

α=0

Bε

Multi-Anisotropic Gevrey Regularity As ϕε (x) = ϕ

271

x

= 1 in B 2ε , then 2 2 ε2σ ε−2|α| P (α) (D) Dβ u dx ≤ C ε−2|α| P (α) (D) (u) dx ε

α=0

α=0

Bε

Bε

2

Proposition 4.8. Let Ω be a bounded open set in Rn and β exists a constant C > 0, such that for every u solution δ ∈ ]0, 1[ , we have 2 (α) β −2σ −2|α| δ −2|α| (D) D u dx ≤ Cδ δ P α=0

Ωδ

∈ Zn+ ∩ σH, then there of P u = 0 in Ω and 2 (α) (D) u dx, P

Ω

α=0

where Ωδ = {x ∈ Ω : dist (x, ∂Ω) > δ} Proof. The proof is obtained from the precedent lemma and follows the same reasoning as the proof of Theorem 4.4.2 of [6]. Corollary 4.9. Let P (D) an hypoelliptic operator, then ∃C > 0 such that for every solution of P (D) u = 0 in Ω,∀ε ∈ ]0, 1[ , ∀j = 1, 2, . . . , we have 2 ε−2|α| QH (D) P (α) (D) u ε2σ 2 L (Ωεj )

0=α∈Nn 0

2 ≤C ε−2|α| P (α) (D) u 2 L

0=α∈Nn 0

(4.4) (Ωε(j−1) )

The principal result of this section is the following theorem. Theorem 4.10. Let u be a solution of the hypoelliptic equation P (D) u = 0 in Ω, then for every ω ⊂⊂ Ω, there is a constant C > 0, such that ∀j ∈ N, we have j ≤ C (j+1) j σj (4.5) QH (D) u 2 L (ω)

Proof. Since ρ = ρ (ω, ∂Ω) > 0, then there exists δ ∈ ]0, ρ[ such that ω ⊂ Ωδ ⊂ Ω. δ Take ε = , j ∈ N, and let us show by induction on j the following estimate j 2 ε2jσ+2m ε−2|α| QjH (D) P (α) (D) u < C 2(j+1) , (4.6) 2 0=α∈Nn 0

L (Ωjε )

where m is the order of P . As every solution u of an hypoelliptic equation is C ∞ , then there exists C > 0 such that (4.6) is satisfied for j = 0. Suppose that (4.6) is true for j ≤ l (l ≥ 0), we have to prove that it remains true for j = l + 1. Since v = QlH (D) u is also a

272

C. Bouzar and A. Dali

solution of equation P (D) u = 0, then from Corollary 4.10, we obtain 2 (α) ε2σ(l+1)+2m ε−2|α| Ql+1 (D) P (D) u 2 H L (Ωε(l+1) ) 0=α 2 ≤ Cε2σl+2m ε−2|α| P (α) (D) QlH (D) u 2

(4.7)

L (Ωεl )

0=α

By the induction hypothesis, we have 2 ε2σl+2m ε−2|α| P (α) (D) QlH (D) u 2

2(l+1)

L (Ωεl )

0=α

consequently, we obtain 2 (α) ε2σ(l+1)+2m ε−2|α| Ql+1 (D) u 2 H (D) P L

0=α

≤ C1

2(l+2)

(Ωε(l+1) )

hence ∀j ∈ N, we have 2 ε2σj+2m ε−2|α| P (α) (D) QjH (D) u 2

L (Ωεj )

0=α

,

≤ C2

2(j+1)

≤ C2

,

(4.8)

The estimate (4.8) with |α| = m gives ∀j ∈ N, 2 j 2(j+1) ≤ ε−2σj C2 , QH (D) u L2 (Ωεj )

δ as ε = , then j 2 j QH (D) u 2

L (Ωεj )

hence

2σj j 2(j+1) ≤ C2 ≤ C 2(j+1) j 2σj , δ

j QH (D) u

L2 (Ωjε )

≤ C (j+1) j σj

We denote Gs, H (Ω) the multi-anisotropic Gevrey space associated with H and by μH and μQ the respective formal orders of the Newton’s polyhedrons H and Γ (QH ), then we have the following relations Γ (QH ) = σH and μQ = σμH The principal result of this paper is the following theorem. Theorem 4.11. Every solution u ∈ D (Ω) of the hypoelliptic equation P (D) u = 0 σ ,H is a function of G μH (Ω) . Proof. Theorem 4.11 says that every u solution of the hypoelliptic equation P u = 0 σ is a Gevrey vector of the operator QH , i.e., we have u ∈ G μQ (Ω, QH ) . From Theorem 3.4 and as the operator QH is multi-quasielliptic, then we have u ∈ σ 1 , Γ(QH ) , σH (Ω) , and consequently u ∈ G μH (Ω) . A simple computation shows G μQ σ ,H s, σH sσ, H that in general G (Ω) = G Ω, hence u ∈ G μH (Ω) .


273

Remark 4.12. It is interesting to compare the result of the theorem with the microlocal Gevrey regularity result obtained in [2].

References [1] C. Bouzar and R. Cha¨ıli, Gevrey vectors of multi-quasielliptic systems, Proc. Amer. Math. Soc. 131(5) (2003), 1565–1572. [2] C. Bouzar and R. Cha¨ıli, A Gevrey microlocal analysis of multi-anisotropic differential operators, Rend. Sem. Mat. Univ. Pol. Torino, 64(3) (2006), 305–318. [3] J. Friberg, Multi-quasielliptic polynomials, Ann. Sc. Norm. Sup. Pisa. Cl. di. Sc. 21 (1967), 239–260. [4] S.G. Gindikin and L.R. Volevich, The Method of Newton Polyhedron in the Theory of Partial Differential Equations, Kluwer, 1992. [5] G.H. Hakobyan, Estimates of the higher order derivatives of the solution of hypoelliptic equations, Rend. Sem. Mat. Univ. Pol. Torino, 61(4) (2003), 443–459. [6] L. H¨ ormander, Linear Partial Differential Operators, Springer-Verlag, 1969. [7] L. H¨ ormander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, Second Edition, Springer-Verlag, 1990. [8] V.P. Mikha¨ılov, The behavior at infinity of a class of polynomials, Proc. Steklov. Inst. Mat. 91 (1967), 59–80. [9] L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, 1993. [10] L. Zanghirati, Iterati di una classe di operatori ipoelliptici e classi generalizzate di Gevrey, Suppl. Boll. U.M.I. (1980), 177–195. C. Bouzar Department of Mathematics Oran-Essenia University Oran, Algeria e-mail: [email protected] A. Dali Institut des Sciences exactes Centre Universitaire de Bechar, Algeria e-mail: [email protected]


Modified Stockwell Transforms and Time-Frequency Analysis Qiang Guo, Shahla Molahajloo and M.W. Wong Abstract. We give results complementary to those in the paper [4] from the perspective of time-frequency analysis [1] to the effect that high frequencies can be amplified and low frequencies diminished. Time-frequency spectra for the chirp, the sum of the cosine functions and the Gaussian-modulated sinusoidal pulse are presented for comparisons. The property of the absolutely referenced phase information of modified Stockwell transforms is given in terms of Riesz transforms. Mathematics Subject Classification (2000). Primary 65R10, 92A55, 94A12; Secondary 47G10, 47G30. Keywords. Stockwell transforms, Stockwell spectra, signals, resolution of the identity formula, absolutely referenced phase information, Riesz transforms.

1. Introduction

∞ Let ϕ ∈ L1 (R) ∩ L2 (R) be such that −∞ ϕ(x) dx = 1. Then, as a hybrid of the Gabor transform and the wavelet transform, the Stockwell transform Sϕ f of a signal f in L2 (R) with respect to the window ϕ is defined by ∞ e−ixξ f (x)ϕ(ξ(x − b))dx (Sϕ f )(b, ξ) = (2π)−1/2 |ξ| −∞

for all b in R and ξ in R \ {0}. Putting the Stockwell transform in perspective, we note that for all f ∈ L2 (R), b ∈ R and ξ ∈ R \ {0}, (Sϕ f )(b, ξ) = (f, ϕb,ξ )L2 (R) , where ( , )L2 (R) is the inner product in L2 (R) and ϕb,ξ = (2π)−1/2 Mξ T−b Dξ1 ϕ. This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

276

Q. Guo, S. Molahajloo and M.W. Wong

Here, Mξ , T−b and Dξ1 are the modulation operator, the translation operator and the dilation operator defined, respectively, by (Mξ h)(x) = eixξ h(x), (T−b h)(x) = h(x − b) and (Dξ1 h)(x) = |ξ|h(ξx) for all x in R and all measurable functions h on R. Applications of Stockwell transforms abound and can be found in [6, 7]. In the paper [4] are first introduced the modified Stockwell transforms Sϕs parametrized by s, where 1 ≤ s ≤ ∞. These modified Stockwell transforms include the classical Stockwell transforms when s = 1 and are reminiscent of the wavelet transforms as explained in Remark 1.2 when s = 2. ∞ To recall, let ϕ ∈ L1 (R)∩L2 (R) be such that −∞ ϕ(x) dx = 1. Let f ∈ L2 (R). Then for 1 ≤ s ≤ ∞, we define the modified Stockwell transform Sϕs f of f by (Sϕs f )(b, ξ) = (f, ϕb,ξ s )L2 (R) ,

b ∈ R, ξ ∈ R \ {0},

where −1/2 Mξ T−b Dξs ϕ ϕb,ξ s = (2π)

and the dilation operator Dξs here is defined by (Dξs h)(x) = |ξ|1/s h(ξx)

(1.1)

for all x in R and all measurable functions h on R. More explicitly,

(Sϕs f )(b, ξ) = |ξ|−1/s (Sϕ f )(b, ξ),

b ∈ R, ξ ∈ R \ {0},

(1.2)

where s is the conjugate index of s, i.e., 1s + s1 = 1, and ∞ s −1/2 1/s (Sϕ f )(b, ξ) = (2π) |ξ| e−ixξ f (x)ϕ(ξ(x − b))dx −∞

for all b in R and ξ in R \ {0}. In view of (1.2), we see that modified Stockwell transforms modulate frequencies in such a way that low frequencies are amplified and high frequencies reduced. This is illustrated in Figure 1 for s = 1, s = 2 and s = 8 when we use the Gaussian window ϕ given by 2 ϕ(x) = (2π)−1/2 e−x /2 , x ∈ R. We note in particular that very low frequencies, which are almost undetected by the Stockwell transform, can be seen using the modified Stockwell transform with s = 2 and clearly manifested if the modified Stockwell transform with s = 8 is used. With the new normalizing factor for the dilation operator in (1.1), we have the following result.

Modified Stockwell Transforms and Time-Frequency Analysis

277

Theorem 1.1. For all f in L2 (R),

(Sϕs f )(b, ξ) = (2π)−1/2 e−ibξ |ξ|(1/2)−(1/s ) (Ωψ f )(b, 1/ξ),

b ∈ R, ξ ∈ R \ {0},

where ψ(x) = eix ϕ(x), x ∈ R, and Ωψ is the wavelet transform corresponding to the mother wavelet ψ given by ∞ x−b 1 (Ωψ f )(b, a) = √ f (x)ψ dx, b ∈ R, a ∈ R \ {0}, a −∞ a for all functions f in L2 (R). Details on wavelet transforms can be found in [2, 8] among others. Remark 1.2. If we let s = 2, then we get the modified Stockwell transform Sϕ2 , which comes closest to the wavelet transform. This is particularly so for applications for which the amplitude spectra alone are of interest, but the applicability of the computational techniques available for wavelet transforms is limited and complicated by converting the scaling variable to the frequency variable via the equation a = 1/ξ. More precisely, the numerical implementation to carry out a = 1/ξ may introduce disturbing oscillations. After the presentation of the results in [4] by M.W. Wong at the meeting of the National Centre for Medical Device Developments (NCMDD)1 held at IBM Hawthorne in New York in February 2008, it was pointed out by the scientists in the audience that a variation of the modified Stockwell transforms to diminish low frequencies and amplify high frequencies should also be of interest. That this is possible can be done easily by looking at the modified Stockwell transforms for 0 < s < 1 instead of 1 ≤ s ≤ ∞. This is done in Section 2 by comparing the modified Stockwell transform of the chirp for 0 < s < 1 with that for 1 ≤ s ≤ ∞ given in [4]. This comparison is further illuminated in Section 3 with signals given by the sum of cosines and the Gaussian-modulated sinusoidal pulse. The property of the absolutely referenced phase information of the modified Stockwell transforms for 0 < s ≤ ∞ and its connection with Riesz transforms are given in Section 4. At the NCMDD meeting, it was also suggested that dilation of the window ϕ by some function w of the frequency ξ should have good applications in time-frequency analysis. The study of variants of the Stockwell transforms in this direction led by M.W. Wong and his group has led to new research projects in mathematical analysis and applications.

1 NCMDD

is a new initiative and joint venture in which York University and IBM Markham in the Greater Toronto Area in Canada are key players. It is expected that the Southlake Regional Medical Centre, located in Newmarket, Ontario, Canada, will be an industrial partner.

278


2. A variation

∞ Let ϕ ∈ L1 (R) ∩ L2 (R) be such that −∞ ϕ(x) dx = 1. Then for 0 < s < 1, we define the modified Stockwell transform Sϕs f of a signal f by ∞ e−ixξ f (x)ϕ(ξ(x − b))dx (Sϕs f )(b, ξ) = (2π)−1/2 |ξ|1/s −∞

for all b in R and ξ in R \ {0}. Equivalently, (Sϕs f )(b, ξ) = (f, ϕb,ξ s )L2 (R) , where

−1/2 ϕb,ξ Mξ T−b Dξs ϕ. s = (2π) Here, the modulation operator Mξ and the translation operator T−b are the same as before, but Dξs is the dilation operator with s now in (0, 1) instead of [1, ∞]. Alternatively,

(Sϕs f )(b, ξ) = |ξ|−1/s (Sϕ f )(b, ξ),

b ∈ R, ξ ∈ R \ {0}.

It is clear that the modified Stockwell transforms Sϕs , 0 < s < 1, amplify the high frequencies and diminish the low frequencies. That this is in fact the case can be illustrated by Figure 2 for the Gaussian window and for s = 1/2, s = 1/4 and s = 1/8, which should be looked at side by side with Figure 1. Remark 2.1. Theorem 1.1 remains valid for the modified Stockwell transforms Sϕs with 0 < s < 1, but now the exponent in |ξ|(1/2)−(1/s ) is always greater than 1/2. Thus, the numerical implementations described in Remark 1.2 can be carried out without disturbing oscillations. The resolution of the identity formula for the modified Stockwell transforms Sϕs in Theorem 2.4 of [4] for 1 ≤ s ≤ ∞ can now be extended to the full range 0 < s ≤ ∞ in exactly the same way as in [4]. More precisely, we have the following result, which is an analog of Theorem 2.3 in [3] for modified Stockwell transforms. ∞ Theorem 2.2. Let ϕ ∈ L1 (R) ∩ L2 (R) be such that −∞ ϕ(x) dx = 1 and ∞ |ϕ(ξ ˆ − 1)|2 dξ < ∞, |ξ| −∞ where ϕ(ξ) ˆ = (F ϕ)(ξ) = (2π)−1/2

∞

e−ixξ ϕ(x) dx,

−∞

ξ ∈ R.

Then for 0 < s ≤ ∞ and for all f and g in L2 (R), we get ∞ ∞ db dξ 1 b,ξ (f, ϕb,ξ , (f, g)L2 (R) = s )L2 (R) (ϕs , g)L2 (R) cϕ −∞ −∞ |ξ|1−(2/s ) where

cϕ =

∞

−∞

|ϕ(ξ ˆ − 1)|2 dξ. |ξ|

Modified Stockwell Transforms and Time-Frequency Analysis Stockwell Spectrum

Time Series 1

0 100 frequency

amplitude

0.5

0

−0.5

200 300 400

0

0.5

1 time

1.5

500

2

0

0.5

1 time

1.5

2

Modified Stockwell Transform with s=8 0

100

100 frequency

frequency

Modified Stockwell Transform with s=2 0

200 300 400 500

279

200 300 400

0

0.5

1 time

1.5

500

2

0

0.5

1 time

1.5

2

Figure 1. Modified Stockwell Spectra of the Chirp, 1 ≤ s ≤ ∞ Modified Stockwell Spectrum with s=1/2

Time Series 1

0 100 frequency

amplitude

0.5

0

−0.5

200 300 400

0

0.5

1 time

1.5

500

2

0

0

100

100

200 300 400 500

1 time

1.5

2

Modified Stockwell Spectrum with s=1/8

0

frequency

frequency


0.5

200 300 400

0

0.5

1 time

1.5

2

500

0

0.5

1 time

1.5

2

Figure 2. Modified Stockwell Spectra of the Chirp, 0 < s < 1

An immediate corollary is that for 0 < s ≤ ∞, every signal f in L2 (R) can be reconstructed from its time-frequency spectrum Sϕs f via the formula 1 f= cϕ

∞ −∞

∞

−∞

(Sϕs f )(b, ξ)ϕb,ξ s

db dξ . |ξ|1−(2/s )

280


3. Two signals We first give in this section the modified Stockwell transforms of the signal given by the sum of cosines for selected values of s in 1 ≤ s ≤ ∞ and in 0 < s < 1. Then we do the same for the Gaussian-modulated sinusoidal pulse. Time Series

Stockwell Spectrum

1.5

0

1

0.1 frequency

amplitude

0.5 0

0.2 0.3

−0.5 0.4 −1 0

50

100

150

200

0.5

250

0

50

100

time

0

0.1

0.1 frequency

frequency

200

250

Modified Stockwell Spectrum with s=8

Modified Stockwell Spectrum with s=2 0

0.2 0.3

0.2 0.3 0.4

0.4 0.5

150 time

0

50

150

100

200

0.5

250

0

50

150

100

200

250

time

time

Figure 3. Modified Stockwell Spectra of Cosines, 1 ≤ s ≤ ∞ TIme Series


1.5

0

1

0.1 frequency

amplitude

0.5 0

0.2 0.3

−0.5 0.4 −1 0

50

100

150

200

0.5

250

0

50

100

time

200

250

Modified Stockwell Spectrum with s=1/8 0

0.1

0.1 frequency

frequency

Modified Stockwell Spectrum with s=1/4 0

0.2 0.3 0.4 0.5

150 time

0.2 0.3 0.4

0

50

100

150 time

200

250

0.5

0

50

100

150

200

250

time

Figure 4. Modified Stockwell Spectra of Cosines, 0 < s < 1 It is again manifested that frequencies are amplified and diminished as in the case of the chirp. The same is true for the Gaussian-modulated sinusoidal pulse.

Modified Stockwell Transforms and Time-Frequency Analysis 5

Time Series

281

Stockwell Spectrum

x 10 0

0.8 1

0.4

frequency

amplitude

0.6 0.2 0

2 3

−0.2 4

−0.4 −0.6

5 −3

5

−1

0 time

1

2

3

−3

−2

−1

−5

x 10

5


x 10

x 10

0

0

1

1 frequency

frequency

−2

2 3

0 time

1

2

3 −5

x 10


2 3 4

4

5

5 −3

−2

−1

0 time

1

2

−3

3

−2

−1

−5

x 10

0 time

1

2

3 −5

x 10

Figure 5. Modified Stockwell Spectra of the Pulse, 1 ≤ s ≤ ∞ 5

TIme Series

x 10


0 0.8 1 frequency

0.4 0.2 0

2 3

−0.2 4

−0.4 −0.6

5 −3

5

x 10

−2

−1

0 time

1

2

3

−3

−2

−1

−5

x 10

5


x 10

0

0

1

1 frequency

frequency

amplitude

0.6

2 3 4

0 time

1

2

3 −5

x 10


2 3 4

5

5 −3

−2

−1

0 time

1

2

3

−3 −5

x 10

−2

−1

0 time

1

2

3

Figure 6. Modified Stockwell Spectra of the Pulse, 0 < s < 1

−5

x 10

282


4. Absolutely referenced phase information The property of absolutely referenced phase information for the Stockwell transform can be formulated mathematically in the following theorem. ∞ Theorem 4.1. Let ϕ ∈ L1 (R) ∩ L2 (R) be such that −∞ ϕ(x) dx = 1. Then for all functions f in L1 (R) ∩ L2 (R), ∞ (Sϕ f )(b, ·) db = fˆ. −∞

Theorem 4.1 can be found in the original paper [7] and a rigorous proof can be found in [9]. By Theorem 4.1 and (1.2), we have the following version of the property of absolutely referenced phase information for modified Stockwell transforms, which is Theorem 8 in [4]. ∞ Theorem 4.2. Let ϕ ∈ L1 (R) ∩ L2 (R) be such that −∞ ϕ(x) dx = 1. Then for all functions f in L1 (R) ∩ L2 (R), ∞ (Sϕs f )(b, ·) db = F −1 | · |−1/s F f, F −1 −∞

where 0 < s ≤ ∞ and

1 s

+

1 s

= 1. (a) A Signal

1

(c) Modified Stockwell Spectrum (s=1)

Frequency (Hz)

0.5

61

0.4 0.3 0.2 0.1 0

0

0

20

40

60

80

Time (s) Amplitude

(b) Modified Fourier Spectrum (s=1)

Amplitude

2

2 1

(d) Riesz Transform (s=1)

Figure 7. Images for s = 1

100

120


283

Remark 4.3. For 0 < s ≤ ∞, the function F −1 | · |−1/s F f is known as the Riesz transform, also known as the Riesz potential, of the signal f . Chapter V of the book [5] by Stein contains the mathematical analysis of Riesz transforms. Thus, Theorem 4.2 offers a new dimension for an understanding of Riesz transforms in terms of modified Stockwell transforms. We display in Figure 7 the modified Stockwell transform, the modified Fourier transform and the Riesz transform of a signal for s = 1. In this case when the modified Stockwell transform coincides with the Stockwell transform, the modified Fourier transform and the Riesz transform coincide with, respectively, the Fourier transform and the original signal. The corresponding images for s = 2 and s = 1/2 are given in, respectively, Figure 8 and Figure 9.

Amplitude

(a) A Signal

2 1

(b) Modified Fourier Spectrum (s =2)

(c) Modified Stockwell Spectrum (s=2)

Frequency (Hz)

0.5

16

0.4 0.3 0.2 0.1 0

0

0

20

40

60

80

100

120

Time (s) Amplitude

0.4 0.2 (d) Riesz Transform (s=2)

Figure 8. Images for s = 2

We see from the modified Stockwell spectrum in Figure 8 that indeed the low frequencies are amplified and high frequencies diminished. We also note that the Fourier transform is modified and the Riesz transform of the signal is now different from the original signal. Since 0 < s < 1 in Figure 9, the high frequencies are amplified and low frequencies diminished in the modified Stockwell spectrum, as confirmed by the figure.

284


1

(c) Modified Stockwell Spectrum (s=1/2)

0.5 Frequency (Hz)

(b) Modified Fourier Spectrum (s= 1/2)

Amplitude

(a) A Signal

2

1654

0.4 0.3 0.2 0.1 0

0

0

20

40

60

80

100

120

Amplitude

Time (s) 60 40 20 (d) Riesz Transform (s=1/2)

Figure 9. Images for s = 1/2

5. Conclusions and future research The results in this paper complement those in [4] and are intended to reflect on the feedback on the presentation of the results in [4] by M.W. Wong at the NCMDD meeting held at IBM Hawthorne in New York in February 2008. More importantly, one of the goals of [4] and this paper is to document the Stockwell transform and its variants as new hybrids of the Gabor transform and the wavelet transform in time-frequency analysis. Most importantly, it is envisaged that new areas of mathematical analysis with applications can be cultivated.

References [1] L. Cohen, Time-Frequency Analysis, Prentice-Hall, 1995. [2] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. [3] J. Du, M.W. Wong and H. Zhu, Continuous and discrete inversion formulas for the Stockwell transform, Integral Transforms Spec. Funct. 18 (2007), 537–543. [4] Q. Guo and M.W. Wong, Modified Stockwell transforms, Mem. Accademia Sci. Torino, to appear. [5] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1971.


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[6] R.G. Stockwell, Why use the S transform?, in Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Editors: L. Rodino, B.W. Schulze and M.W. Wong, Fields Institute Communications Series 52, American Mathematical Society, 2007, 279–309. [7] R.G. Stockwell, L. Mansinha and R.P. Lowe, Localization of the complex spectrum: the S transform, IEEE Trans. Signal Processing 44 (1996), 998–1001. [8] M.W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, 2002. [9] M.W. Wong and H. Zhu, A characterization of the Stockwell spectrum, in Modern Trends in Pseudo-Differential Operators, Editors: J. Toft, M.W. Wong and H. Zhu, Operator Theory: Advances and Applications 172, Birkh¨ auser, 2007, 251–257. Qiang Guo, Shahla Molahajloo and M.W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada e-mail: [email protected] [email protected] [email protected]


Localization Operators for Two-Dimensional Stockwell Transforms Yu Liu Abstract. We start with an introduction about the Stockwell transforms and the resolution of the identity formula. Localization operators corresponding to the Stockwell transforms are then defined. It is proved that under suitable conditions on the symbols, the localization operators are, in descending order of complexity, paracommutators, paraproducts and Fourier multipliers. Mathematics Subject Classification (2000). Primary 47G10, 47G30. Keywords. Stockwell transform, localization Operator, paracommutators, paraproducts, Fourier multiplier.

1. Introduction 1.1. The 1D Stockwell transforms The Stockwell transform is a hybrid of the Gabor transform and the wavelet transform. For a signal f in L2 (R), the one-dimensional Stockwell transform Sϕ f with respect to the window ϕ in L1 (R) ∩ L2 (R) is the time-frequency representation of f given by ∞ −1/2 (Sϕ f )(b, ξ) = (2π) |ξ| e−ixξ f (x)ϕ(ξ(x − b)) dx, b, ξ ∈ R. −∞

More transparently, (Sϕ f )(b, ξ) = (f, ϕb,ξ )L2 (R) ,

where

ϕb,ξ = (2π)−1/2 Mξ T−b Dξ ϕ

and ( , )L2 (R) is the inner product in L2 (R). Here, the modulation operator Mξ , the translation operator T−b and the dilation operator Dξ are, respectively, defined by (Mξ h)(x) = eixξ h(x),

(T−b h)(x) = h(x − b) and (Dξ h)(x) = |ξ|h(ξx),

This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

288

Y. Liu

for all x in R and all measurable functions h on R. The Stockwell transform Sϕ f can be looked at as a modified Gabor transform of f with a window, which is designed in such a way that the window is wide for low frequency and narrow for high frequency. Besides the modulation with respect to frequency ξ, a distinguishing feature of the Stockwell transform from the well-known wavelet transform is the normalizing factor in the dilation operator, which is |·| instead of |·|1/2 . This feature is responsible for the property of the absolutely referenced phase information, which accounts for the fact that the Stockwell transform is very much akin to the Fourier transform. See Theorem 1.3 in this connection. Applications of the Stockwell transform abound in the literature. See, for instance, [5, 13] and [6, 17] for applications in, respectively, geophysics and medical imaging. In an attempt to reconstruct a signal f from its Stockwell spectrum {(Sϕ f )(b, ξ) : b, ξ ∈ R}, we have the following result in [4]. Theorem 1.1. Let ϕ ∈ L2 (R) be such that ϕL2 (R) = 1 and ∞ |ϕ(ξ ˆ − 1)|2 dξ < ∞. |ξ| −∞ Then for all signals f and g in L2 (R), ∞ ∞ 1 db dξ , (f, ϕb,ξ )L2 (R) (ϕb,ξ , g)L2 (R) (f, g)L2 (R) = cϕ −∞ −∞ |ξ|

where

(1.1)

(1.2)

∞

|ϕ(ξ ˆ − 1)|2 dξ, |ξ| −∞ and ∧ denotes the Fourier transform defined by Fˆ (ζ) = (2π)−N/2 e−ix·ζ F (x) dx cϕ =

RN 1

N

for all F in L (R ). Remark 1.2. Theorem 1.1 is known as the Plancherel formula or the resolution of the identity formula for the one-dimensional Stockwell transform. The integrability condition (1.1) is the admissibility condition for a function ϕ in L2 (R) to be a window. An important corollary of Theorem 1.1 is that every signal f can be reconstructed from its Stockwell spectrum by means of the inversion formula ∞ ∞ db dξ 1 . (f, ϕb,ξ )L2 (R) ϕb,ξ f= cϕ −∞ −∞ |ξ| That the admissibility condition (1.1) is a necessary condition for the inversion formula for the Stockwell transform can be seen by letting f = g = ϕ in (1.2). Details can be found in [3] and [14]. There is another inversion formula for the one-dimensional Stockwell transform, which is a contribution in [13] and is therein referred to as the property of absolutely referenced phase information. It is formulated in the following theorem.

Localization Operators for Two-Dimensional Stockwell Transforms

289

Theorem 1.3. Let ϕ ∈ L1 (R) ∩ L2 (R) be such that ∞ ϕ(x) dx = 1. −∞

Then for all signals f in L2 (R), fˆ =

∞

−∞

(Sϕ f )(b, ·) db.

1.2. The 2D Stockwell transforms The two-dimensional Stockwell transform is defined in the following way. For a signal f in L2 (R2 ), the one-dimensional Stockwell transform Sϕ f with respect to the window ϕ in L1 (R2 ) ∩ L2 (R2 ) is the time-frequency representation of f given by (Sϕ f )(b, ξ) = (2π)−1 |ξ1 ||ξ2 | e−ixξ f (x)ϕ(ξ(x − b)) dx, b, ξ = (ξ1 , ξ2 ) ∈ R2 . R2

More transparently, (Sϕ f )(b, ξ) = (f, ϕb,ξ )L2 (R2 ) , where ϕb,ξ = (2π)−1 Mξ T−b Dξ ϕ and ( , )L2 (R2 ) is the inner product in L2 (R2 ). Here, the modulation operator Mξ , the translation operator T−b and the dilation operator Dξ are, respectively, defined by (Mξ h)(x) = eixξ h(x),

(T−b h)(x) = h(x − b) and (Dξ h)(x) = |ξ1 ||ξ2 |h(ξx)

for all x in R and all measurable functions h on R2 . Applications of two-dimensional Stockwell transforms can be found in [5], [9] and others. In an attempt to reconstruct a signal f from its Stockwell spectrum {(Sϕ f )(b, ξ) : b, ξ ∈ R2 }, we have the following result in [8]. 2

Theorem 1.4. Let ϕ ∈ L2 (R2 ) be such that ϕL2 (R2 ) = 1 and |ϕ(ξ ˆ − (1, 1))|2 dξ < ∞. |ξ1 ||ξ2 | R2

(1.3)

Then for all signals f and g in L2 (R2 ), db dξ 1 (f, g)L2 (R2 ) = , (1.4) (f, ϕb,ξ )L2 (R2 ) (ϕb,ξ , g)L2 (R2 ) cϕ R2 R2 |ξ1 ||ξ2 | 2 ˆ where cϕ = R2 |ϕ(ξ−(1,1))| dξ, and ∧ denotes the Fourier transform defined by |ξ1 ||ξ2 | −N/2 ˆ F (ζ) = (2π) e−ix·ζ F (x) dx RN 1

N

for all F in L (R ).

290

Y. Liu

Remark 1.5. An corollary of Theorem 1.4 is that every signal f can be reconstructed from its Stockwell spectrum by means of the inversion formula 1 db dξ f= . (f, ϕb,ξ )L2 (R2 ) ϕb,ξ cϕ R2 R2 |ξ1 ||ξ2 | That the admissibility condition (1.3) is a necessary condition for the inversion formula for the Stockwell transform can be seen by letting f = g = ϕ in (1.4). Details can be found in [8] . There is another inversion formula for the two-dimensional Stockwell transform, which is a contribution in [13]. It is formulated in the following theorem. Theorem 1.6. Let ϕ ∈ L1 (R2 ) ∩ L2 (R2 ) be such that ϕ(x) dx = 1. R2 2

2

Then for all signals f in L (R ),

fˆ = R2

(Sϕ f )(b, ·) db.

1.3. The localization operators Let U be the upper half-plane given by U = {(b, a) : b ∈ R, a > 0}. Then with respect to the binary operation · on U defined by (b1 , a1 ) · (b2 , a2 ) = (b1 + a1 b2 , a1 a2 ),

(b1 , a1 ), (b2 , a2 ) ∈ U,

U becomes a non-abelian groupin which (0,1) is the identity element and the inverse element of (b, a) is − ab , a1 for all (b, a) in U . In fact, it is a non-unimodular Lie group on which the left Haar measure and the right Haar measure are given, respectively, by db da db da . dμ = and dν = a2 a It is commonly known as the affine group in the literature. 2 Let H+ (R) be the subspace of L2 (R) defined by 2 (R) = {f ∈ L2 (R) : supp(fˆ) ⊆ [0, ∞)}, H+

where supp(fˆ) is the support of the Fourier transform fˆ. The Fourier transform F3 of a function F in L2 (Rn ) to be used in this paper is the one defined by F3 (ξ) = lim (2π)−n/2 e−ix·ξ χR (x)F (x) dx, R→∞

Rn

where x · ξ is the inner product of x and ξ in Rn , χR is the characteristic function on the ball {x ∈ Rn : |x| ≤ R}, and the convergence is understood to occur in 2 L2 (Rn ). We define H− (R) to be the subspace of L2 (R) by 2 H− (R) = {f ∈ L2 (R) : supp(fˆ) ⊆ (−∞, 0]}.


291

2 2 H+ (R) and H− (R) are known as the Hardy space and the conjugate Hardy space respectively. It is convenient at this point to mention that the results in this section are 2 stated in the context of the left Haar measure dμ and the Hardy space H+ (R) only. 2 2 (R)) be the set of all unitary operators on H+ (R). It is a Now, we let U (H+ 2 group with respect to the usual composition of mappings. Let π : U → U (H+ (R)) be the mapping defined by x−b 1 (π(b, a)f )(x) = √ f , x ∈ R, a a 2 (R). Then it can be proved that π is an irreducible for all (b, a) in U and all f in H+ 2 2 and unitary representation of U on H+ (R). The raison d’être for H± (R) is that 2 the unitary representation π : U → U (L (R)) is not irreducible. 2 In fact, π is a square-integrable representation of U on H+ (R) in the sense 2 that there exists a nonzero function ϕ in H+ (R) such that ∞ ∞ db da |(ϕ, π(b, a)ϕ)L2 (R) |2 2 < ∞. (1.5) a 0 −∞ 2 (R) for which ϕL2 (R) = 1 and (1.1) is valid an We call any function ϕ in H+ 2 (R). We admissible wavelet for the square-integrable representation π of U on H+ have the following characterization of admissible wavelets.

Theorem 1.7. The set AW (π) of all admissible wavelets for the square-integrable 2 (R) is given by representation π of U on H+ ∞ 2 |ϕ(ξ)| ˆ 2 dξ < ∞ . (R) : ϕL2 (R) = 1, AW (π) = ϕ ∈ H+ |ξ| 0 The proof of Theorem 1.1 follows from the fact that the integral in (1.1) is ∞ ˆ 2 equal to 2π 0 |ϕ(ξ)| |ξ| dξ. A proof can be found in Daubechies [3], Wong [14] and others. Let ϕ ∈ AW (π). Then we define the wavelet transform Wϕ f of a function f 2 in H+ (R) by (Wϕ f )(b, a) = (f, π(b, a)ϕ)L2 (R) , (b, a) ∈ U. Then we have the following resolution of the identity formula, which can also be found in Daubechies [3] and Wong [14]. Theorem 1.8. let ϕ ∈ AW (π) and let cϕ be the constant defined by ∞ 2 |ϕ(ξ)| ˆ dξ. cϕ = 2π |ξ| 0 2 Then for all f and g in H+ (R), ∞ ∞ db da 1 (f, g)L2 (R) = (f, π(b, a)ϕ)L2 (R) (π(b, a)ϕ, g)L2 (R) 2 . cϕ 0 a −∞

292

Y. Liu

Let ϕ ∈ AW (π) and let F be a measurable function on U . Then we define 2 2 the localization operator LF,ϕ : H+ (R) → H+ (R) associated to the symbol F and the admissible wavelet ϕ by ∞ ∞ db da 1 (LF,ϕ f, g)L2 (R) = F (b, a)(f, π(b, a)ϕ)L2 (R) (π(b, a)ϕ, g)L2 (R) 2 cϕ 0 a −∞ for all f and g in L2 (R). It is shown in Wong [15] that if F (b, a) = β(b)α(a),

(b, a) ∈ U,

where α and β are suitable functions on (0, ∞) and (−∞, ∞) respectively, then the localization operator LF,ϕ is in fact a paracommutator studied in Janson and Peetre [7], Peng [10, 11] and Peng and Wong [12]. Furthermore, in Wong [15], it is proved that if F is a suitable function of b only, then LF,ϕ is given in terms of a paraproduct in the sense of Coifman and Meyer [2], and if F is a function of a only, then LF,ϕ is a Fourier multiplier. The aim of this paper is to extend the results hitherto described to the twodimensional Stockwell transform. Under conditions similar to those in Wong [15], we show that the localization operators are given by paracommutators, paraproducts and Fourier multipliers .

2. Localization operators for two-dimensional Stockwell transforms Let F ∈ Lp (R2 × R2 ), 1 ≤ p ≤ ∞, then for all f ∈ L2 (R2 ), we define LF,ϕ f by 1 db dξ (LF,ϕ f, g)L2 (R2 ) = F (b, ξ)(Sϕ f )(b, ξ)(Sϕ g)(b, ξ) cϕ R2 R2 |ξ1 ||ξ2 | for g in L2 (R2 ). The following two results are the Stockwell transform analog of Proposition 12.1 and Theorem 14.5 in the book [14] by Wong. Theorem 2.1. Let ϕ ∈ L2 (R2 ) be an admissible wavelet for the Stockwell transform. Let F ∈ Lp (R2 × R2 ), 1 ≤ p ≤ ∞, and LF,ϕ : L2 (R2 ) → L2 (R2 ) be the localization operator associated to the symbol F and the admissible wavelet ϕ ,then there exists a unique bounded linear operator LF,ϕ : L2 (R2 ) → L2 (R2 ) such that 1

LF,ϕ∗ ≤ (1/cϕ ) p F p , where ∗ is the norm of bounded operators over L2 (R2 ). Theorem 2.2. Let ϕ ∈ L2 (R2 ) be an admissible wavelet for the Stockwell transform. Let F ∈ Lp (R2 × R2 ), 1 ≤ p ≤ ∞, and LF,ϕ : L2 (R2 ) → L2 (R2 ) be the localization operator associated to the symbol F and the admissible wavelet ϕ then is LF,ϕ in the Schatten–von Neumann class Sp , especially LF,ϕ is compact, and p1 1 LF,ϕSp ≤ F p , cϕ where Sp is the norm in Sp .


293

We give in the following sections explicit formulas for the localization operators LF,ϕ : L2 (R2 ) → L2 (R2 ) under different conditions imposed on the symbol F . These results are extensions of results obtained in the paper [15] for the standard wavelet transforms on the affine group U.

3. Paracommutators Let F be a function on R2 × R2 given by F (b, ξ) = β(b)α(ξ),

b, ξ ∈ R2 ,

where α and β are suitable functions on R2 . Then using Plancherel’s theorem and Fubini’s theorem, we get for all f and g in L2 (R2 ), 1 db dξ (LF,ϕ f, g)L2 (R2 ) = F (b, ξ)(Sϕ f )(b, ξ)(Sϕ g)(b, ξ) cϕ R2 R2 |ξ1 ||ξ2 | dξ 1 α(ξ) β(b)(Sϕ f )(b, ξ)(Sϕ g)(b, ξ) db = cϕ R2 |ξ1 ||ξ2 | R2 2π ˆ − ζ)fˆ(ζ)ˆ = A(ζ, η)β(η g (η) dζ dη, (3.1) cϕ R2 R2 where

A(ζ, η) = R2

α(ξ)ϕ(ξ ˆ −1 (ζ − ξ))ϕ(ξ ˆ −1 (η − ξ))

dξ , |ξ1 ||ξ2 |

for all ξ and η in R , where ξ −1 = (ξ1−1 , ξ2−1 ), for all ξ = (ξ1 , ξ2 ) ∈ R2 . Thus, the localization operator LF,ϕ is a paracommutator with Fourier kernel A and symbol β. 2

4. A Paraproduct connection If the symbol F is independent of the second variable, i.e., it is given by F (b, ξ) = β(b), b, ξ ∈ R2 . In order to simplify the computations that follow, let us introduce a notation. Let ξ is some point in R2 . Then we define the mollifier uξ of a function u in L2 (R2 ) by uξ (x) = |ξ1 ||ξ2 |ei(ξ1 x1 +ξ2 x2 ) u(ξx), x ∈ R2 . An easy computation gives 5ξ (ζ) = u u ˆ(ξ −1 (ζ − ξ)),

ξ ∈ R2 ,

Where ξ −1 = (ξ1−1 , ξ2−2 ). Let ψ be the function on R2 defined by ψ(x) = ϕ(−x),

x ∈ R2 .

294

Y. Liu

Then, we get (LF,ϕ f, g)L2 (R2 ) =

2π cϕ

R2

R2

R2

ˆ − ζ) 5ξ (ζ)5 ϕξ (η)fˆ(ζ)β(η ψ

dξ dζ gˆ(η) dη. |ξ1 ||ξ2 |

Since

(u ∗ v)∧ = 2πˆ uvˆ 2 for sufficiently nice functions u and v on R , it follows from Plancherel’s formula that for all f and g in L2 (R2 ), 1 dξ 2 2 (LF,ϕ f, g)L (R ) = ((β(ψξ ∗ f )) ∗ ϕξ )(x) g(x) dx (4.1) cϕ R2 |ξ1 ||ξ2 | R2 and hence 1 (LF,ϕ f )(x) = cϕ

R2

((β(ψξ ∗ f )) ∗ ϕξ )(x)

dξ , |ξ1 ||ξ2 |

x ∈ R2 .

Further manipulations of (8.1) using Fubini’s theorem give 1 β(y)pψ (f, g)(y) dy (LF,ϕ f, g)L2 (R2 ) = cϕ R2 for all f and g in L2 (R2 ), where pψ (f, g) is the paraproduct of f and g with respect to ψ given by dξ , y ∈ R2 . pψ (f, g)(y) = (ψξ ∗ f )(y)(ψξ ∗ g)(y) |ξ ||ξ | 2 1 2 R

5. Fourier multipliers Let F be a function on R2 × R2 given by F (b, ξ) = α(ξ),

b, ξ ∈ R2 .

Then, by (3.1), we obtain for all f and g in L2 (R2 ),

=

(LF,ϕ f, g)L2 (R2 ) 2π dξ dζ dη. α(ξ)5 ϕξ (ζ)5 ϕξ (η)fˆ(ξ)ˆ g (η) cϕ R2 R2 R2 |ξ1 ||ξ2 |

Let φ(x) = e−|x| /2 , x ∈ R2 . For all positive numbers ε, let Iε be the number defined by 2π dξ dζ dη, (5.1) Iε = α(ξ)5 ϕξ (ζ)5 ϕξ (η)fˆ(ζ)ˆ g (η)φε (ζ − η) cϕ R2 R2 R2 |ξ1 ||ξ2 | where φε is the Friedrich mollifier of φ. Then 2π dξ ˆ Iε = α(ξ)5 ϕξ (ζ)f (ζ)((5 ϕξ gˆ) ∗ φε )(ζ) dζ. cϕ R2 |ξ1 ||ξ2 | R2 2

Now, there exists a sequence {εj }∞ j=1 of positive numbers such that εj → 0, as j → ∞ and ϕξ gˆ (5 ϕξ gˆ) ∗ φεj → 2π5


295

in L2 (R2 ) and almost everywhere on R2 as j → ∞. Thus, 4π 2 dξ Iεj → α(ξ)5 ϕξ fˆ(ζ)5 g (ζ) ϕξ (ζ)ˆ dζ. cϕ R2 |ξ1 ||ξ2 | R2 On the other hand, since the Fourier transform of the function e− compare (3.1) and (5.1), we obtain |b|2 2 1 db dξ I = . α(ξ)e− 2 (Sϕ f )(b, ξ)(Sϕ g)(b, ξ) cϕ R2 R2 |ξ1 ||ξ2 |

|·|2 2 2

is φˆ ,

Thus, using Lebesgue’s dominated convergence theorem, we see that Iεj → (LF,ϕ f, g)L2 (R2 ) as j → ∞. Therefore for all f and g in L2 (R2 ), (LF,ϕ f, g)L2 (R2 ) = (Tm f, g)L2 (R2 ) , where Tm is the Fourier multiplier with symbol m given by dξ 4π 2 , ξ ∈ R2 . α(ξ)|5 ϕξ (ζ)|2 m(ζ) = cϕ R2 |ξ1 ||ξ2 |

References [1] J.-P. Antoine, R. Murenzi, P. Vandergheynst and S.T. Ali, Two-Dimensional Wavelets and their Relatives, Cambridge University Press, 2004. [2] R.R. Coifman and Y. Meyer, Au Del` a des Opérateurs Pseudo-Différentiels, Astérisque 57, 1978. [3] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. [4] J. Du, M.W. Wong and H. Zhu, Continuous and Discrete Inversion Formulas for the Stockwell transform, Integral Transforms Spec. Funct., 18 (2007), 537–543. [5] M.G. Eramian, R.A. Schincariol, L. Mansinha and R.G. Stockwell, Generation of aquifer heterogeneity maps using two dimensional special texture segmentation techniques, Math. Geology 31 (1999), 327–348. [6] B.G. Goodyear, H. Zhu, R.A. Brown and J.R. Mitchell, Removal of phase artifacts from fMRI data using a Stockwell transform filter improves brain activity detection, Magn. Reson. Med. 51 (2004), 16–21. [7] S. Janson and J. Peetre, Paracommutators-boundedness and Schatten-von Neumann properties, Trans. Amer. Math. Soc. 305 (1988), 467–504. [8] Y. Liu and M.W. Wong , Inversion Formulas for Two-Dimensional Stockwell Transforms, in Pseudo-Differential Operators: Partial Differential Equations and TimeFrequency Analysis, Editors: L. Rodino, B.-W. Schulze and M.W. Wong, Fields Institute Communications Series, American Mathematical Society, 2007, 323–330. [9] L. Mansinha, R.G. Stockwell and R.P. Lowe, Pattern analysis with two-dimensional spectral localization: applications of 2-dimensional S-transforms, Physica A 239 (1997), 286–295. [10] L. Peng, On the compactness of commutators, Ark. Mat. 26 (1988), 315–325.

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[11] L. Peng, Wavelets and paracommutators, Ark. Mat. 31 (1993), 83–99. [12] L. Peng and M.W. Wong, Compensated compactness and paracommutators, J. London Math. Soc. 62(2) (2000), 505–520. [13] R.G. Stockwell, L. Mansinha and R.P. Lowe, Localization of the complex spectrum: the S-transform, IEEE Trans. Signal Processing 44 (1996), 998–1001. [14] M.W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, 2002. [15] M.W. Wong, Localization operators on the affine group and paracommutators, in Progress in Analysis, Editors: H.G.W. Begehr, R.P. Gilbert and M.W. Wong, World Scientific, 2003, 663–669. [16] M.W. Wong and H. Zhu, A characterization of Stockwell spectra, in Modern Trends in Pseudo-Differential Operators, Editors: J. Toft, M.W. Wong and H. Zhu, Birkh¨ auser, 2007, 251–257. [17] H. Zhu, B.G. Goodyear, M.L. Lauzon, R.A. Brown, G.S. Mayer, L. Mansinha, A.G. Law and J.R. Mitchell, A new multiscale Fourier analysis for MRI, Med. Phys. 30 (2003), 1134–1141. Yu Liu Department of Mathematics and Statistics York University 4700 Keele Street Toronto Ontario M3J 1P3, Canada e-mail: [email protected]


Pseudo-Differential Operators on S1 Shahla Molahajloo and M.W. Wong Abstract. Pseudo-differential operators on the unit circle S1 with center at the origin are defined in terms of symbols on S1 × Z. Results on the boundedness and compactness of these pseudo-differential operators on L2 (S1 ) are presented. In addition, we prove a result on the Lp -boundedness for 1 < p < ∞. Mathematics Subject Classification (2000). Primary 47G30; Secondary 42A45. Keywords. Pseudo-differential operators, L2 -boundedness, Lp -boundedness, L2 -compactness, Hilbert–Schmidt operators, Fourier multipliers, Littlewood– Paley theory.

1. Introduction For m ∈ (−∞, ∞), let S m be the set of all functions σ in C ∞ (Rn × Rn ) such that for all multi-indices α and β, there exists a positive constant Cα,β for which |(Dxα Dξβ σ)(x, ξ)| ≤ Cα,β (1 + |ξ|)m−|β| ,

x, ξ ∈ Rn .

A function σ in S m is called a symbol. Let σ be a symbol. Then we define the pseudo-differential operator Tσ on the Schwartz space S by eix·ξ σ(x, ξ)ϕ(ξ) ˆ dξ, x ∈ Rn , (Tσ ϕ)(x) = (2π)−n/2 Rn

for all functions ϕ in S, where −n/2

ϕ(ξ) ˆ = (2π)

Rn

e−ix·ξ ϕ(x) dx,

ξ ∈ Rn .

It is easy to prove that Tσ maps S into S continuously. A much deeper fact is that if σ ∈ S 0 , then Tσ can be extended to a bounded linear operator from Lp (Rn ) into Lp (Rn ) for 1 < p < ∞. See, for instance, Theorem 10.7 in the book [13] by Wong. Although the phase space Rn × Rn is used in defining symbols of pseudodifferential operators on Rn , a more appropriate setting is, loosely speaking, the This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

298

S. Molahajloo and M.W. Wong

3 when we are interested in pseudo-differential operators on G, phase space G × G 3 is the set of all irreducible and unitary representations where G is a Lie group and G n 5 n of G. In the case of R , R can be identified with Rn and hence the phase space Rn × Rn used for pseudo-differential operators on Rn is consistent with this big picture. In this paper the focus is on pseudo-differential operators on S1 with symbols defined on the phase space S1 × Z, where S1 is the unit circle centered at 51 . To be more precise, let σ be a measurable the origin and Z is the dual group S function on S1 × Z. Then for every function f in L1 (S1 ), we define the function Tσ f on S1 by (Tσ f )(θ) = σ(θ, n)fˆ(n)einθ , θ ∈ [−π, π], n∈Z

where

π 1 ˆ f (n) = e−inθ f (θ) dθ, n ∈ Z. 2π −π Among the most interesting problems are the ones on finding good conditions on σ to produce bounded and compact pseudo-differential operators Tσ : Lp (S1 ) → Lp (S1 ), 1 < p < ∞. Some results in this direction are given in this paper. Related works can be found in [1, 2, 3, 5, 6, 7] among others. We begin in Section 2 with an Lp formula for 1 ≤ p < ∞. This is then used in Section 3 to prove that a pseudo-differential operator Tσ : L2 (S1 ) → L2 (S1 ) is Hilbert–Schmidt if and only if σ ∈ L2 (S1 × Z). In Section 4, necessary conditions and sufficient conditions on symbols σ are given for the corresponding pseudo-differential operators Tσ : L2 (S1 ) → L2 (S1 ) to be bounded linear operators. Sufficient conditions on symbols σ are given in Section 5 to guarantee that the corresponding pseudo-differential operators Tσ : L2 (S1 ) → L2 (S1 ) are compact. We give in Section 6 sufficient conditions on symbols σ to give bounded linear operators Tσ : Lp (S1 ) → Lp (S1 ), 1 < p < ∞. The results in this paper are valid for the multi-dimensional torus using multiple Fourier series in, e.g., the book [10] by Stein and Weiss. For the sake of simpler notation and greater transparency, we have chosen to work only with the unit circle S1 with center at the origin.

2. An Lp formula, 1 ≤ p < ∞ For j ∈ Z, we let ej be the function defined by ej (θ) = eijθ ,

θ ∈ [−π, π].

Then we have the following formula. Theorem 2.1. Let σ ∈ Lp (S1 × Z), 1 ≤ p < ∞. Then Tσ ej pLp(S1 ) = σpLp (S1 ×Z) . j∈Z

Pseudo-Differential Operators on S1 Proof. For j ∈ Z, we get (Tσ ej )(θ) =

σ(θ, n)e3j (n)einθ ,

299

θ ∈ [−π, π].

(2.1)

n∈Z

But 1 e3j (n) = 2π

π

e −π

−inθ ijθ

e

1 dθ = 2π

π

e

i(j−n)θ

dθ =

−π

1,

n = j,

0,

n = j.

(2.2)

So, by (2.1) and (2.2), (Tσ ej )(θ) = σ(θ, j)eijθ = σ(θ, j)ej (θ), Hence

j∈Z

Tσ ej pLp (S1 ) =

j∈Z

θ ∈ [−π, π].

(2.3)

π −π

|σ(θ, j)|p dθ = σpLp (S1 ×Z) ,

as asserted.

3. Hilbert–Schmidt operators Let us begin with the fact that a bounded linear operator A on a complex and separable Hilbert space X is a Hilbert–Schmidt operator ⇔ there exists an orthonormal basis {ϕk }∞ k=1 for X such that ∞

Aϕk 2X < ∞,

k=1

where X denotes the norm in X. Moreover, if A is a Hilbert–Schmidt operator on X, then the Hilbert–Schmidt norm AHS of A is given by A2HS =

∞

Aϕk 2X ,

k=1

where {ϕk }∞ k=1 is any orthonormal basis for X. So, using Theorem 2.1 and the fact 2 1 that {(2π)−1/2 ek }∞ k=−∞ is an orthonormal basis for L (S ), we have the following result. Theorem 3.1. The pseudo-differential operator Tσ : L2 (S1 ) → L2 (S1 ) is a Hilbert– Schmidt operator ⇔ σ ∈ L2 (S1 × Z). Moreover, if Tσ : L2 (S1 ) → L2 (S1 ) is a Hilbert–Schmidt operator, then Tσ HS = (2π)−1/2 σL2 (S1 ×Z) .

300


4. L2 -boundedness The proof of Theorem 2.1 gives a necessary condition on a measurable function σ on S1 × Z for Tσ : L2 (S1 ) → L2 (S1 ) to be a bounded linear operator. To wit, let σ be a measurable function on S1 × Z and let f ∈ L2 (S1 ). Then for n ∈ Z, we can use (2.3) to obtain (Tσ en )(θ) = σ(θ, n)en (θ), θ ∈ [−π, π], and hence π 2 Tσ en L2 (S1 ) = |σ(θ, n)|2 dθ. −π

So, if Tσ : L2 (S1 ) → L2 (S1 ) is a bounded linear operator, we can get a positive constant C such that Tσ en 2L2 (S1 ) ≤ Cen 2L2 (S1 ) , which is then the same as

n ∈ Z,

π

−π

|σ(θ, n)|2 dθ ≤ 2πC,

n ∈ Z.

Therefore a necessary condition for Tσ : L2 (S1 ) → L2 (S1 ) to be a bounded linear operator is π sup |σ(θ, n)|2 dθ < ∞. (4.1) n∈Z

−π

The following example shows that (4.1) is not sufficient for Tσ : L2 (Rn ) → L2 (Rn ) to be a bounded linear operator. Example 4.1. Let σ be the function on S1 × Z defined by σ(θ, n) = e−inθ ,

θ ∈ [−π, π], n ∈ Z,

and let f be the function on S defined by ∞ 1 inθ e , f (θ) = n n=1 1

θ ∈ [−π, π].

Then it is easy to see that f ∈ L2 (S1 ), but Tσ f ∈ / L2 (S1 ). The following theorem gives sufficient conditions for a pseudo-differential operator Tσ : L2 (S1 ) → L2 (S1 ) to be a bounded linear operator. Theorem 4.2. Let σ be a measurable function on S1 × Z. Suppose that we can find a positive constant C and a function w in L1 (Z) such that |ˆ σ (m, n)| ≤ C|w(m)|,

m, n ∈ Z,

(4.2)

π 1 e−imθ σ(θ, n) dθ. 2π −π Then Tσ : L2 (S1 ) → L2 (S1 ) is a bounded linear operator and

where

σ ˆ (m, n) =

Tσ ∗ ≤ CwL1 (Z) , where ∗ is the norm in the C ∗ -algebra of all bounded linear operators on L2 (S1 ).

Pseudo-Differential Operators on S1

301

Proof. Let f ∈ C 2 (S1 ). Then we have 2 π 2 inθ ˆ Tσ f L2 (S1 ) = σ(θ, n)f (n)e dθ

−π n∈Z π

=

2 ˆ(n)ei(m+n)θ dθ. σ ˆ (m, n) f

−π n∈Z m∈Z

So,

Tσ f 2L2 (S1 )

2 ikθ ˆ σ ˆ (k − n, n)f (n)e dθ

π

=

−π n∈Z k∈Z

2 ˆ(n) eikθ dθ. σ ˆ (k − n, n) f

π

=

−π k∈Z

(4.3)

n∈Z

2 1 Using (4.3) and the orthogonality of the functions {ej }∞ j=−∞ in L (S ), 2 2 ˆ σ ˆ (k − n, n)f (n) Tσ f L2 (S1 ) = 2π k∈Z n∈Z

≤ 2π

k∈Z

Using (4.2), we get Tσ f 2L2 (S1 )

≤ 2πC

2

k∈Z

2 |ˆ σ (k − n, n)||fˆ(n)| .

(4.4)

n∈Z

2 ˆ |w(k − n)| |f (n)| ≤ 2πC 2 |(|w| ∗ |fˆ|)(k)|2 ,

n∈Z

k∈Z

where |w| ∗ |fˆ| is the convolution of |w| and |fˆ|. Finally, using Young’s inequality, we have (4.5) Tσ f 2 2 1 ≤ 2πC 2 w2 1 fˆ2 2 = C 2 w2 1 f 2 2 1 . L (S )

2

1

L (Z)

2

L (Z)

L (Z)

L (S )

1

Since C (S ) is dense in L (S ), it follows that (4.5) holds for all functions f in L2 (S1 ). Remark 4.3. In order to justify the interchange of the two sums in (4.3), we note that by Fubini’s theorem, it is sufficient to prove that |ˆ σ (k − n, n)||fˆ(n)| < ∞. k∈Z n∈Z

Since f ∈ C (S ), it follows that 2

1

|fˆ(n)| ≤ O(n−2 ) as |n| → ∞. Hence fˆ ∈ L1 (Z). Using an argument in the proof of Theorem 4.2, we have |ˆ σ (k − n, n)||fˆ(n)| ≤ CwL1 (Z) fˆL1 (Z) < ∞. k∈Z n∈Z

302


Remark 4.4. The sufficient conditions on the symbol σ to ensure the boundedness of Tσ : L2 (S1 ) → L2 (S1 ) in [5, 6] require a certain number of derivatives of σ with respect to θ. By means of Theorem 4.2 in this paper and Bernstein’s theorem, we see that all we need is that the symbol σ is in the Lipschitz class Λα , α > 12 . See Section 3 in Chapter VI of the book [14] by Zygmund for results in this direction.

5. L2 -compactness Let σ be a measurable function on S1 × Z such that Tσ : L2 (S1 ) → L2 (S1 ) is compact. Since en → 0 weakly in L2 (S1 ) as |n| → ∞, it follows from the compactness of Tσ : L2 (S1 ) → L2 (S1 ) that Tσ en L2 (S1 ) → 0 as |n| → ∞. By (2.3), we see that π |σ(θ, n)|2 dθ → 0 (5.1) −π

as |n| → ∞. That the condition (5.1) is not enough for Tσ : L2 (S1 ) → L2 (S1 ) to be compact can be illustrated by the following example. Example 5.1. Let σ be the function on S1 × Z defined by 1 −inθ , n > 1, ln n e σ(θ, n) = 0, n ≤ 1, for all θ in [−π, π]. If we let f be the function on S1 defined by ∞ 1 inθ e , f (θ) = n n=1

θ ∈ [−π, π],

then f ∈ L2 (S1 ). But (Tσ f )(θ) =

∞

1 , n ln n n=2

θ ∈ [−π, π].

So, Tσ f is not even in L2 (S1 ). The following theorem gives a sufficient condition for the L2 -compactness of pseudo-differential operators on L2 (S1 ). Theorem 5.2. Let σ be a measurable function on S1 × Z. Suppose that we can find a function w in L1 (Z) and a function C on Z such that lim C(n) = 0

|n|→∞

and |ˆ σ (m, n)| ≤ C(n)|w(m)|, Then Tσ : L (S ) → L (S ) is compact. 2

1

2

1

m, n ∈ Z.

(5.2)


303

Proof. For all positive integers N , we define the function σN on S1 × Z by σ(θ, n), |n| ≤ N, σN (θ, n) = 0, |n| > N, for all θ in [−π, π] and n in Z. Then for N = 1, 2, . . . , π N π |σN (θ, n)|2 dθ = |σ(θ, n)|2 dθ < ∞ n∈Z

−π

−π

n=−N

and hence by Theorem 3.1, TσN : L (S ) → L (S1 ) is a Hilbert–Schmidt operator. Let τN = σ − σN . Then by the definition of σN , we have 0, |n| ≤ N, τN (θ, n) = σ(θ, n), |n| > N, 2

1

2

for all θ in [−π, π]. Let ε be a given positive number. Then there exists a positive integer N0 such that |C(n)| < ε whenever |n| > N0 . So, for N ≥ N0 , 2 π π 2 2 inθ ˆ (TσN − Tσ )f L2 (S1 ) = |(TτN f )(θ)| dθ = τN (n, θ)f (n)e dθ. −π

−π n∈Z

Now, we use the same argument in the derivation of (4.4) to get 2 |ˆ τ (k − n, n)||fˆ(n)| (TσN − Tσ )f 2L2 (S1 ) ≤ 2π = 2π Using (5.2), we get (TσN −

Tσ )f 2L2 (S1 )

≤ 2π

k∈Z

n∈Z

k∈Z

|n|>N

2 ˆ C(n)|w(k − n)| |f (n)|

k∈Z

≤ 2πε

2 ˆ |ˆ σ (k − n, n)||f (n)| .

2

|n|>N

k∈Z

2 ˆ |w(k − n)| |f (n)| .

n∈Z

Using the same argument in the the derivation of (4.5), we have (TσN − Tσ )f 2L2 (S1 ) ≤ ε2 Bf 2L2 (S1 ) , where B = w2L1 (Z) . Thus, for N ≥ N0 , Tσ − TσN ∗ ≤

√ Bε.

In other words, Tσ is the limit in norm of the sequence {TσN }∞ N =1 and so must be compact.

304


6. Lp -boundedness The following lemma gives sufficient conditions on a function σ on Z to ensure that the corresponding pseudo-differential operator Tσ : Lp (S1 ) → Lp (S1 ), 1 < p < ∞, is a bounded linear operator. Pseudo-differential operators Tσ , where σ is a function on Z only, are often known as Fourier multipliers. Lemma 6.1. Let σ be a measurable function on Z and let k be the smallest integer greater than 12 . Suppose that there exists a positive constant C such that |(∂ j σ)(n)| ≤ Cn−j ,

n ∈ Z,

for 0 ≤ j ≤ k, where ∂ is the difference operator given by j j j−l j (∂ σ)(n) = (−1) σ(n + l), n ∈ Z, l j

l=0

and n = (1 + |n|2 )1/2 , n ∈ Z. Then for 1 < p < ∞, Tσ : Lp (S1 ) → Lp (S1 ) is a bounded linear operator and there exists a positive constant B, depending on p only, such that Tσ f Lp(S1 ) ≤ BCf Lp(S1 ) ,

f ∈ Lp (S1 ).

(6.1)

Remark 6.2. The condition on the number k of “derivatives” in Lemma 6.1 can best be understood in the context of the multi-dimensional torus in which k should be the smallest integer greater than d/2, where d is the dimension of the torus. The proof of Lemma 6.1 entails the use of the Littlewood–Paley theory in Fourier series in, e.g., Chapter XV of the book [14] by Zygmund. See, in particular, Theorem 4.14 in Chapter XV of [14] in this connection. Extensions of Lemma 6.1 to the context of compact Lie groups are attributed to Weiss [11, 12] and the Littlewood–Paley theory for compact Lie groups can be found in Stein [9]. Analogs ormander [4] and Stein [8]. of Lemma 6.1 for Rn can be found in the works of H¨ The following theorem gives sufficient conditions for the Lp -boundedness of pseudo-differential operators on S1 . The ideas for the result and its proof come from Theorem 10.7 in the book [13] by Wong. Theorem 6.3. Let σ be a measurable function on S1 × Z and let k be the smallest integer greater than 12 . Suppose that we can find a positive constant C and a function w in L1 (Z) such that ˆ )(m, n)| ≤ C|w(m)|n−j , |(∂nj σ

m, n ∈ Z,

(6.2)

for 0 ≤ j ≤ k, where is the partial difference operator with respect to the variable n in Z. Then for 1 < p < ∞, Tσ : Lp (S1 ) → Lp (S1 ) is a bounded linear operator. Moreover, there exists a positive constant B depending only on p such that ∂nj

Tσ ∗ ≤ BCwL1 (Z) f Lp(S1 ) , where ∗ here denotes the norm in the Banach algebra of all bounded linear operators on Lp (S1 ).


305

Proof. Let f ∈ Lp (S1 ). Then by Fubini’s theorem, (Tσ f )(θ) = fˆ(n)σ(θ, n)einθ n∈Z

=

σ ˆ (m, n)eimθ einθ fˆ(n)

n∈Z

=

m∈Z

eimθ

m∈Z

=

σ ˆ (m, n)fˆ(n)einθ

n∈Z

e

imθ

(Tσm f )(θ)

m∈Z

for all θ in [−π, π], where 1 σm (n) = σ ˆ (m, n) = 2π

π

e−imθ σ(θ, n) dθ,

m, n ∈ Z.

−π

By (6.2), ˆ )(m, n)| ≤ C|w(m)|n−j , |(∂ j σm )(n)| = |(∂nj σ

m, n ∈ Z,

for 0 ≤ j ≤ k. Therefore by Lemma 6.1, there exists a positive constant B depending only on p such that Tσm f Lp(S1 ) ≤ BC|w(m)| f Lp (S1 ) ,

m ∈ Z.

(6.3)

Then by (6.3) and Minkowski’s inequality, we get Tσ f

Lp (S1 )

=

π

p p1 imθ e (Tσm f )(θ) dθ

−π m∈Z

≤

m∈Z

=

π

−π

|(Tσm f )(θ)|p dθ

p1

Tσm f Lp (S1 )

m∈Z

≤ BCwL1 (Z) f Lp(S1 ) , and this completes the proof.

Acknowledgment We are grateful to Professor M.S. Agranovich for very useful comments on the results and the presentation of this paper, and to Professor Ville Turunen for pointing out the preprint [6] to M.W. Wong at the conference on pseudo-differential operators held at V¨ axj¨ o University in Sweden on June 23–27, 2008.

306


References [1] M.S. Agranovich, Spectral properties of elliptic pseudodifferential operators on a closed curve, (Russian) Funktsional. Anal. i Prilozhen. 13 (1979), 54–56. [2] M.S. Agranovich, Elliptic pseudodifferential operators on a closed curve, (Russian) Trudy Moskov. Mat. Obshch. 47 (1984), 22–67, 246. [3] B.A. Amosov, On the theory of pseudodifferential operators on the circle, (Russian) Uspekhi Mat. Nauk. 43 (1988), 169–170; Translation in Russian Math. Surveys 43 (1988), 197–198. [4] L. H¨ ormander, Estimates for translation invariant linear operators in Lp spaces, Acta Math. 104 (1960), 93–139. [5] M. Ruzhansky and V. Turunen, On the Fourier analysis of operators on the torus, in Modern Trends in Pseudo-Differential Operators, Editors: J. Toft, M.W. Wong and H. Zhu, Birkh¨ auser, 2007, 87–105. [6] M. Ruzhansky and V. Turunen, Quantization of pseudo-differential operators on the torus, preprint, arXiv:0805.2892. [7] J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer-Verlag, 2002. [8] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [9] E.M. Stein, Topics in Harmonic Analysis Related to Littlewood–Paley Theory, Third Printing with Corrections, Princeton University Press, 1985. [10] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. [11] N.J. Weiss, Multipliers on compact Lie groups, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 930–931. [12] N.J. Weiss, Lp estimates for bi-invariant operators on compact Lie groups, Amer. J. Math. 94 (1972), 103–118. [13] M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999. [14] A. Zygmund, Trigonometric Series, Third Edition, Volumes I & II Combined, Cambridge University Press, 2002. Shahla Molahajloo and M.W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto Ontario M3J 1P3, Canada e-mail: [email protected] [email protected]


On Pseudo-Differential Operators on Group SU(2) Michael Ruzhansky and Ville Turunen Abstract. In this paper we will outline elements of the global calculus of pseudo-differential operators on the group SU(2). This is a part of a more general approach to pseudo-differential operators on compact Lie groups that will appear in [17]. Mathematics Subject Classification (2000). Primary 35S05; Secondary 22E30. Keywords. Pseudo-differential operator, Lie groups, SU(2).

1. Introduction The paper is devoted to outline a global approach to pseudo-differential operators on matrix group SU(2), without resorting to local charts. This can be done by presenting functions on the group by Fourier series obtained from the representations of the group. Due to non-commutativity, the Fourier coefficients become matrices of varying dimension. A pseudo-differential operator can be presented as a convolution operator-valued mapping on the group. The corresponding Fourier coefficient matrices provide a natural global symbol of the pseudo-differential operator. Consequently, a global calculus and full symbols of pseudo-differential operators can be obtained on the 3-dimensional sphere, with further geometric implications. Let us first recall and fix the notation for the standard notions of the Fourier analysis. In view of our applications, in the formulae below we will emphasize the 5n of Rn is isomorphic to Rn , thus keeping the notation fact that the dual group R n 5 of R . The Fourier transform 5n ) (f → f3) : S(Rn ) → S(R is defined by f3(ξ) =

f (x) e−i2πx·ξ dx,

Rn

The first author was supported by a Royal Society grant and by the EPSRC Grant EP/E062873/01.

308

M. Ruzhansky and V. Turunen

and the Fourier inversion formula is f (x) =

5n R

f3(ξ) ei2πx·ξ dξ.

Operator A : S(Rn ) → S(Rn ) is a pseudo-differential operator of order m ∈ R if (Af )(x) = σA (x, ξ) f3(ξ) ei2πx·ξ dξ, 5n R

5n ) of A satisfies symbolic inequalities where the symbol σA ∈ C (R × R α β ∂ξ ∂x σA (x, ξ) ≤ CAαβm ξm−|α| , ∞

n

5n (see, e.g., Kohn and Nirenfor all multi-indices α and β and all x ∈ Rn and ξ ∈ R berg [11], H¨ ormander [9]). In this case we write A ∈ Ψm (Rn ) and σA ∈ S m (Rn ). On a compact manifold M without boundary the space Ψm (M ) of pseudodifferential operators can be defined via localisations. One disadvantage of this definition is that it often destroys the geometric and other global structures of M . The main problem is that there is no canonical way to write the phase of a pseudo-differential operator when working in local coordinates. However, if one fixes a connection on the manifold, one can define the phase in a global way in terms of the connection (see, e.g., Widom [32], Safarov [18], Sharafutdinov [21]). If G is a Lie group, we may construct another global representation of pseudodifferential operators on G and their calculus by relying on the globally defined group structure. Moreover, such analysis can be extended to symmetric spaces, where we have a transitive action G × M → M of a Lie group G on a manifold M . In this way we obtain the calculus on M as a “shadow” from the calculus on G. For example, for M = Sn−1 we can take G = SO(n), and for the complex sphere M = {x ∈ Cn : xCn = 1} we can take G = SU(n). A version of pseudo-differential operators on SU(2) for functions with finite Fourier series (i.e., for trigonometric polynomials) was analysed in [8]. The important starting point for our analysis is the global Fourier analysis on the Lie group G. Two important ingredients that are needed on G are (aG ) the Fourier transform; (bG ) multiplication and its relation to the Fourier transform. Analogies of these operations on Rn are (aRn ) x → eix·ξ (ξ ∈ Rn ) and the Fourier integral; (bRn ) multiplication and the Fourier transform related by αf (−ix)

= ∂ξα f3.

On torus Tn = (R/2πZ)n we have the following analogies: (aTn ) x → eix·ξ (ξ ∈ Zn ) and the Fourier series; (bTn ) multiplication and the Fourier transform are related by

where for n = 1, we define x(α)

(α) f = %α f3, x ξ −ix α = e − 1 , and similarly in higher dimensions.

On Pseudo-Differential Operators on Group SU(2)

309

The relation between the toroidal and Euclidean representations of pseudodifferential operators in one dimension (i.e., on the circle T1 ) was given by Agranovich ([1, 2, 3]), who showed that A ∈ Ψm (T1 ) if and only if it can be written as (Af )(x) = σA (x, ξ) f3(ξ) eix·ξ , ξ∈Z1 ∞

where the toroidal symbol σA ∈ C (T1 × Z1 ) of A satisfies α β % ∂ σA (x, ξ) ≤ CAαβm ξm−|α| , ξ x and where the difference operator %ξ is defined by (%ξ σ)(ξ) = σ(ξ + 1) − σ(ξ). For Ψ (T ) the natural analogy holds with partial difference operators %α ξ . This result has been extended to a more comprehensive analysis of pseudo-differential operators and the corresponding toroidal representations of Fourier integral operators in all dimensions in [14] and [15]. We also mention the related paper by Amosov [4] on pseudo-differential operators on the circle, and more general characterisations of pseudo-differential operators on the torus by McLean [13]. We note that such discrete analysis has applications for numerical mathematics (e.g., [19], [20], [30]). m

n

2. Calculus on compact Lie groups G A global description and calculus of pseudo-differential operators is possible on general compact Lie groups G and homogeneous spaces of the type G/K with K∼ = Tn . The main ingredients of this analysis are (aG ) the Fourier transform based on the usual Haar measure μG on G together with irreducible unitary representations and corresponding Fourier series; (bG ) multiplication and the Fourier series related by the exponential mapping, using Taylor polynomials in deformed exponential coordinates; (cG ) global pseudo-differential operator calculus exploiting the Fourier series. Let us now outline these elements in more detail. (aG ) Fourier transform Let Cc×d denote the space of complex matrices with c rows and d columns, and let U(d) ⊂ Cd×d be the set of d-dimensional unitary matrices. Let D(G) = C ∞ (G) 3 consists of equivalence classes denote the test function space. The unitary dual G [ξ] of irreducible unitary representations ξ of G. In the sequel, from each class 3 we choose one representative ξ : G → U(dim(ξ)), so that the unitary [ξ] ∈ G, matrices ξ(x) = ξmn (x) m,n define dim(ξ)2 functions ξmn ∈ C ∞ (G). Fourier coefficient f3(ξ) ∈ Cdim(ξ)×dim(ξ) of f ∈ D(G) is defined by f (x) ξmn (x) dμG (x) f3(ξ) = f (x) ξ(x) dμG (x) = , G

G

m,n

310


so that the Fourier series becomes f (x) = dim(ξ) Tr f3(ξ) ξ ∗ (x) , 3 [ξ]∈G

where ξ ∗ (x) = ξ(x)∗ , and Tr is the trace functional (i.e., the sum of the diagonal elements in a matrix). We can also notice the useful relation f ∗ g(ξ) = f3(ξ) g3(ξ). If A : D(G) → D(G) is a linear continuous map, then it can be written as dim(ξ) Tr σA (x, ξ) f3(ξ) ξ ∗ (x) , Af (x) = 3 [ξ]∈G

where the (matrix) symbol of A is defined as the mapping (x, ξ) → σA (x, ξ) ∈ dim(ξ)×dim(ξ) C , where ξmk (x) (Aξnk )(x) σA (x, ξ) = ξ(x) (A(ξ ∗ ))(x) = . k

m,n

(bG ) Multiplication and the Fourier series The Taylor expansion of a smooth function f : G → C nearby the neutral element I ∈ G becomes 1 x(α) (∂x(α) f )(I), f (x) ≈ α! |α|≤N

with x ≈ x in, e.g., the exponential coordinates. We can then define the difference operators by the relation (α)

α

(α) f (ξ) =: %α f3(ξ). x ξ (cG ) Pseudo-differential operators The following theorem relates pseudo-differential operators with their globally defined symbols: Theorem 2.1 ([27]). A ∈ L(D(G)) belongs to Ψm (G) if and only if σA ∈ S m (G) =

∞

Skm (G).

k=0

In this theorem, the symbol classes Skm (G) are defined in the following recursive way: Definition 2.2. Symbol σA ∈ S0m (G) if and only if α β %ξ ∂x σA (x, ξ) ≤ CAαβm ξm−|α| , where the norm is the usual operator norm, see Section 5 for details. Here, the weight is ξ = (1 − λξ )1/2 ,


311

where λξ ≤ 0 is the eigenvalue of the bi-invariant Laplacian Δ : C ∞ (G) → C ∞ (G) on the eigenspace spanned by the matrix element functions ξmn ∈ C ∞ (G). Furm thermore, σA ∈ Sk+1 (G) if and only if σA ∈ Skm (G), σ∂j σA − σA σ∂j ∈ Skm (G), m+1−|γ|

(G),

m+1−|γ|

(G),

(%γξ σ∂j ) σA ∈ Sk (%γξ σA ) σ∂j ∈ Sk

where |γ| > 0 and 1 ≤ j ≤ dim(G). Then we define ∞ S m (G) = Skm (G). k=0

We note that the proof of Theorem 2.1 is based on the commutator characterisation of pseudo-differential operators (see, e.g., [5], [7], [6], [26]; see also [25] for a different characterisation on the sphere, studying smoothness of operator-valued mappings). It is well known that if A ∈ ΨmA and B ∈ ΨmB then AB, BA ∈ ΨmA +mB , but the commutator satisfies [A, B] = AB − BA ∈ ΨmA +mB −1 . The commutator characterisation uses this property to characterise pseudo-differential operators: Theorem 2.3. A ∈ Ψm (M ) if and only if [Dk+1 , [Dk , . . . [D1 , A] · · · ]] ∈ L(H m (M ), H 0 (M )) for every sequence of smooth vector fields Dk on M , where H m (M ) is the Sobolev space of order m ∈ R on M . Details of these constructions with applications to the analysis on Lie groups will appear in [17]. We also note that this applies also to non-invariant pseudodifferential operators (compared to, e.g., [22], [23]).

3. Calculus on G = SU(2) Let us now explain the constructions of the previous section in more detail in the case of the matrix group SU(2). In this case the special unitary group of complex rotations G = SU(2) = {x ∈ C2×2 : x∗ = x−1 , det(x) = 1} acts transitively on the unit sphere in C2 , so the analysis also applies for pseudodifferential operator on spheres in C2 . The group SU(2) is especially important since G ∼ = S3 , the 3-sphere with the quaternionic structure. In particular, we obtain the global calculus of pseudo-differential operators on S3 , which allows further extensions to general closed simply connected 3-manifolds by the pullback by the Ricci flow (details of this will appear in [16]).

312

M. Ruzhansky and V. Turunen Let us define basic rotations ωj (t) ∈ G (j = 1, 2, 3) by it/2 cos 2t − sin 2t e cos 2t i sin 2t , , i sin 2t cos 2t sin 2t cos 2t 0

0 e−it/2

.

The Euler angles φ, θ, ψ of x ∈ G satisfy x = x(φ, θ, ψ) := ω3 (φ) ω2 (θ) ω3 (ψ), where −π < φ ≤ π, 0 ≤ θ ≤ π, −2π < ψ ≤ 2π. The Euler angles define local coordinates for G whenever 0 < θ < π (with natural interpretation when φ = π or ψ = 2π). Let us now define the principal ingredients for the analysis. (aG ) Fourier transform The Haar integral on G = SU(2) in the Euler angles is given by f dμG f → G

=

1 16π 2

2π

π

π

f (x) sin(θ) dφ dθ dψ. −2π

0

−π

The unitary dual of G = SU(2) is 1 3∼ G = tl ∈ Hom(G, U(2l + 1)) : l ∈ N , 2 where U(d) ⊂ Cd×d is the unitary matrix group of dimension d, and functions tlmn ∈ C ∞ (G) are products of exponentials and Legendre functions. The Fourier transform f → f3 on G = SU(2) becomes l 3 3 f (x) tl (x) dμG (x), f (l) = f (t ) := G

with the inverse Fourier transform given by (2l + 1) Tr f3(l) (tl )∗ . f= l∈N/2

For details of these formulae see, e.g., Vilenkin [31] and Zelobenko [33]. We also 3 note that the Fourier transform f3(ξ) of f ∈ D(G) is defined for every ξ ∈ [ξ] ∈ G, 3 i.e., f (ξ) : Vξ → Vξ is linear on the representation vector space Vξ of the irreducible 3 then f3(ξ) and representation ξ : G → Aut(Vξ ) of G. If ξ ∼ η (i.e., [ξ] = [η] ∈ G) 3 f (η) are related by intertwining. (bG ) Taylor polynomials Lie group G = SU(2) has Lie algebra g = su(2) = {X(z) : z ∈ R3 }, where 1 iz1 − z2 iz3 X(z) = . −iz3 2 iz1 + z2


313

If {ej }3j=1 is the standard basis of R3 and Xj = X(ej ) (Pauli matrices) then we have commutator relations: [X1 , X2 ] = X3 ,

[X2 , X3 ] = X1 ,

[X3 , X1 ] = X2 .

Now, let us describe the vector fields in the basis of the Lie algebra. For Y ∈ su(2), let AY : C ∞ (SU(2)) → C ∞ (SU(2)) be the left-invariant differential operator defined by

d AY f (x) = f (x exp(tY )) . dt t=0

Let us denote Aj := AXj . For a multi-index β ∈ N3 , define the left-invariant differential operator ∂ β := Aβ1 1 Aβ2 2 Aβ3 3 . Let us also define creation, annihilation, and neutral operators by the following formulae: ∂+ := iA1 − A2 , ∂− := iA1 + A2 , ∂0 := iA3 . In Euler angles these operators can be expressed as ∂ 1 ∂ cos(θ) ∂ ∂+ = e−iψ i − + , ∂θ sin(θ) ∂φ sin(θ) ∂ψ 1 ∂ cos(θ) ∂ ∂ + − ∂− = eiψ i , ∂θ sin(θ) ∂φ sin(θ) ∂ψ ∂ . ∂0 = i ∂ψ Note that the invariant Laplace operator on G is given by Δ = A21 + A22 + A23 . Let t := zR3 /2. The Rodrigues formula states that we have sin(t) 1 0 exp(X(z)) = cos(t) + X(z) . 0 1 t Define now functions xij ∈ C ∞ (G) by x11 (x) x= x21 (x)

x12 (x) x22 (x)

∈ G.

These functions (or their linear combinations) will play the role of coordinates. We can now define the Taylor polynomial x(α) ∈ C ∞ (G) (α ∈ N3 ) by setting x(α) = (x12 )α1 (x21 )α2 (x11 − x22 )α3 .

314


The corresponding difference operators %α ξ are then defined by (α) f (ξ). 3 %α ξ f (ξ) := x In the Taylor expansion formula, Taylor polynomials are multiplied by the partial differential operators of corresponding orders. For this purpose, we define ∂ (α) = cαβ ∂ β , |β|≤|α|

so that we have ∂ (α) x(β) |x=I = α! δα,β . Now the Taylor expansion for a function f ∈ D(G) at the neutral element I ∈ G becomes 1 f (x) ∼ x(α) (∂ (α) f )(I). α! α≥0

Let us finally briefly recall the irreducible unitary representations for the group SU(2). Group SU(2) acts naturally on space C[z1 , z2 ] by the formula T : SU(2) → GL(C[z1 , z2 ]),

(T (x)f )(z) := f (zx).

Invariant subspaces Vl ⊂ C[z1 , z2 ] of homogeneous 2l-degree polynomials, where l ∈ N/2, give the restriction Tl : SU(2) → GL(Vl ),

(Tl (x)f )(z) = f (zx).

If T∞ is an irreducible representation of SU(2) on a vector space V∞ then T∞ is equivalent to Tl , where dim(V∞ ) = 2l + 1. Operators Tl (x) can be viewed as matrices and now we will recall the matrix elements of Tl , which can be written as l (cos(θ)), tlmn (φ, θ, ψ) = e−i(mφ+nφ) Pmn

where l Pmn (x)

=

− x)(n−m)/2 (1 + x)(m+n)/2

(1 clmn

d dx

l−m

with constants clmn

(−1)l−n in−m = 2−l (l − n)! (l + n)!

"

# (1 − x)l−n (1 + x)l+n ,

= (l + m)! . (l − m)!

Multiplication of two elements tlm n and tlmn can be expressed as

tlm n

tlmn

=

l+l

ll (l+k)

ll (l+k)

Cm m(m +m) Cn n(n +n) tl+k (m +m)(n +n) ,

k=|l−l | ll (l+k)

with the Clebsch–Gordan coefficients Cm m(m +m) . In particular, let us denote 1/2 1/2 t−1/2,−1/2 t−1/2,+1/2 t−− t−+ 1/2 = 1/2 =t 1/2 t+− t++ t t +1/2,−1/2

+1/2,+1/2


315

and x± := x ± 1/2 for x ∈ R. Then we have the following multiplication formulae + − (2l + 1)tlmn t−− = (l − m + 1)(l − n + 1) tlm− n− − (l + m)(l + n) tlm− n− , + − (2l + 1)tlmn t++ = (l + m + 1)(l + n + 1) tlm+ n+ − (l − m)(l − n) tlm+ n+ , + − (2l + 1)tlmn t−+ = (l − m + 1)(l + n + 1) tlm− n+ − (l + m)(l − n) tlm− n+ , + − (2l + 1)tlmn t+− = (l + m + 1)(l − n + 1) tlm+ n− − (l − m)(l + n) tlm+ n− .

4. Differences for symbols on SU(2) Let us define functions q+ := t−+ ,

(4.1)

q− := t+− ,

(4.2)

q0 := t++ − t−− .

(4.3)

The key property of these functions is that each q+ , q− , q0 ∈ D(G) vanishes at the neutral element I ∈ G. Let σ = s3, which means that σ : D(G) → D(G) is the left convolution operator with convolution kernel s ∈ D (G): σf = s ∗ f. Let us define “difference operators” %+ , %− , %0 acting on symbols σ by %+ σ := q + s,

(4.4)

%− σ := q − s,

(4.5)

%0 σ := q 0 s.

(4.6)

These difference operators will play the role of derivatives in the definition and calculus of pseudo-differential operators. In general, they depend on the structure 3 of the group G. For example, in the Euclidean space we have of the dual space G n n ∼ 5 R = R , so their analogues are simply the differential operators ∂ξα . For the 5n ∼ torus, we have T = Zn , and the corresponding difference operators %α have been ξ

analysed, e.g., in [14] and [26]. We can now define the conditions on globally defined pseudo-differential symbols on SU(2) in terms of these difference operators. First let us show that a pseudo-differential operator can be globally parametrised by a matrix symbol σA (x, ξ), where x ∈ SU(2), ξ is a discrete variable corresponding to the dual of SU(2), and the size of the matrix σA (x, ξ) increases with ξ. In fact, using the structure of the dual of SU(2), we can identify ξ with a half-integer in N/2, so that σA (x, ξ) becomes a matrix of the size (2ξ + 1) × (2ξ + 1). In general, the matrix symbol of a continuous linear operator A : D(G) → D(G) will be written 3 If [ξ] = [η] then as σA (x, ξ), and it is defined for every x ∈ G and ξ ∈ [ξ] ∈ G. σA (x, ξ) and σA (x, η) are related and allow a geometric interpretation. We refer

316


to [16] for further details, and will only give here the description based on the identification of the dual of SU(2) with the set N/2 of half-integers. By using the Fourier series, we can conclude that a continuous linear operator A : D(G) → D(G) belongs to Ψm (G) if and only if it can be written as (2ξ + 1) Tr σA (x, ξ) f3(ξ) tξ (x)∗ (Af )(x) = ξ∈N/2

=

ξ ξ

(2ξ + 1)

m=−ξ n=−ξ

ξ∈N/2

where the symbol ξ

⎛ tξmn (x) ⎝

ξ ∗

σA (x, ξ) = t (x) (A(t ) )(x) =

ξ

ξ

⎞ σA (x, ξ)mk f3(ξ)kn ⎠ ,

k=−ξ

tξmk (x)

(Atξnk )(x)

k=−ξ

= σA (x, ξ)mn m,n

m,n

satisfies symbol inequalities that will be described below. With a natural embedding D(S2 ) → D(G), we also note that every pseudo-differential operator B ∈ Ψm (S2 ) has an extension A ∈ Ψm (G) such that A|D(S2 ) = B, see [27]. Let us consider the following example: for ξ ∈ N/2 and a (2ξ + 1) × (2ξ + 1) matrix A, let Amn denote the matrix element on the mth row and nth column, where |m|, |n| ≤ ξ such that m, n ∈ {±ξ, ±(ξ − 1), ±(ξ − 2), · · · }. The symbols of the first-order partial differential operators ∂+ , ∂− , ∂0 are then given by − (ξ − n)(ξ + n + 1), if m = n + 1, σ∂+ (x, ξ)mn = 0, otherwise. − (ξ + n)(ξ − n + 1), if m = n − 1, σ∂− (x, ξ)mn = 0, otherwise. n, if m = n, σ∂0 (x, ξ)mn = 0, otherwise, 1, if m = n, σI (x, ξ)mn = 0, otherwise, where I = (f → f ) : D(G) → D(G) is the identity operator. Moreover, we have the following properties: %+ σΔ = −σ∂− ,

%− σΔ = −σ∂+ ,

%0 σΔ = −σ∂0 ,

where Δ is the invariant Laplacian of G, σI = %+ σ∂+ = %− σ∂− = %0 σ∂0 , and 0 = %+ σI = %− σI = %0 σI . Furthermore, %α σ∂η (x, ξ) = 0, if α, η ∈ {+, −, 0} are such that α = η.


317

5. Symbol inequalities In this section we study inequalities describing symbols of pseudo-differential operators on G = SU(2). For a vector v = (vj )nj=1 ∈ Cd we use the Euclidean norm vCd given by d |vj |2 , v2Cd := j=1

and for a matrix M ∈ Cd×d the corresponding operator norm M = M Cd×d := sup M vCd : v ∈ Cd , vCd ≤ 1 . We notice that we have the operator norm equality

! 3 . a(ξ) : ξ ∈ G f → f ∗ aL(L2 (G)) = sup 3

In order to describe matrix symbols of standard pseudo-differential operators on SU(2), we need to check conditions of Theorem 2.1. First, σA ∈ S0m (G), i.e., it satisfies α β % ∂ σA (x, ξ) ≤ Cαβ ξm−|α| (5.1) ξ x for all multi-indices α, β, all x ∈ SU(2), and all ξ ∈ N/2. On G = SU(2), the weight ξ becomes 1/2 . ξ = 1 + ξ + ξ 2 m+1−|γ|

(G) is automatically satisfied due We note that condition (%γξ σ∂j )σA ∈ Sk to the example presented at the end of the previous section. We will now show that in order to satisfy conditions of Theorem 2.1, matrices σA (x, ξ) must have a certain “rapid off-diagonal decay” property. By the example in the previous section, σ∂0 (x, ξ)ij = i δij , so that [σ∂0 (x, ξ), σA (x, ξ)]ij = (σ∂0 (x, ξ)ik σA (x, ξ)kj − σA (x, ξ)ik σ∂0 (x, ξ)kj ) k

=

(i δik σA (x, ξ)kj − σA (x, ξ)ik k δkj )

k

= (i − j) σA (x, ξ)ij . Let us iterate such a commutator p ∈ Z+ times, to obtain that the symbol (x, ξ) → (i − j)p σA (x, ξ)ij i,j must belong to S m (G) ⊂ S0m (G), regardless of p. Thus the pseudo-differential symbol σA must have “rapid off-diagonal decay”: sup ξ−m i − jp |σA (x, ξ)ij | < ∞ ξ,i,j

– notice that in a matrix aij i,j , the distance from the location of the matrix element aij to the diagonal is |i − j|.

318


What about the commutators [σ∂+ , σA ] and [σ∂− , σA ]? Due to the symmetries of the symbols of ∂± , studying the case of σ∂+ will be enough. By the example in the previous section, σ∂+ (x, ξ)ij = − (ξ − i)(ξ + i + 1) δi+1,j , so that [σ∂+ (x, ξ), σA (x, ξ)]ij σ∂+ (x, ξ)ik σA (x, ξ)kj − σA (x, ξ)ik σ∂+ (x, ξ)kj = k

− (ξ − i)(ξ + i + 1) δi+1,k σA (x, ξ)kj = k

+σA (x, ξ)ik

(ξ − k)(ξ + k + 1) δk+1,j

(ξ − m)(ξ + m + 1) σA (x, ξ)i+1,j + (ξ − n + 1)(ξ + n) σA (x, ξ)i,j−1 .

=−

At first sight, this commutator may look baffling: is this some sort of weighted difference operator acting on σA “along the diagonal”? We have to understand this first-order commutator condition properly before we may consider higherorder iterated commutators. So how to understand the behaviour of [σ∂+ (x, ξ), σA (x, ξ)]? We claim that this is a variant of the “rapid off-diagonal decay” property. Let us explain what we mean by this. Briefly: ∂0 = iA3 , so the A3 -symbol commutator condition means “rapid off-diagonal decay”. “Badly behaving” operators ∂+ , ∂− are linear combinations of left-invariant vector fields A1 , A2 , which are conjugates to A3 (and hence essentially similar in behaviour); then the idea is to exploit the diffeomorphism invariance of Ψm (G). We know from the local theory that the pseudo-differential operator class Ψm (G) is diffeomorphism-invariant: if φ : G → G is a diffeomorphism and A ∈ Ψm (G), then also Aφ ∈ Ψm (G), where Aφ f = A(f ◦ φ) ◦ φ−1 . Especially, let us consider the inner automorphisms φu = (x → u−1 xu) : G → G, where u ∈ G. Such φu : G → G is a diffeomorphism that maps one-parametric 3 then subgroups to one-parametric subgroups. If ξ ∈ [ξ] ∈ G f (u−1 xu) ξ(x) dμG (x) = f (x) ξ(uxu−1 ) dμG (x) f ◦ φu (ξ) = G G = ξ(u) f (x) ξ(x) dμG (x) ξ(u)∗ = ξ(u) f3(ξ) ξ(u)∗ , G


319

which is similar to f3(ξ) by transform ξ(u). Recall the left-invariant vector fields A1 , A2 , A3 out of which operators ∂+ , ∂− , ∂0 were defined as linear combinations. Thus we are interested in commutators [σAj (x, ξ), σA (x, ξ)], and we already know the good behaviour of the case A3 , since ∂0 = iA3 . Now, σA1 (x, ξ) = tξ (v1 )∗ σA3 (x, ξ) tξ (v1 ), σA2 (x, ξ) = tξ (v2 )∗ σA3 (x, ξ) tξ (v2 ), where vj ∈ G and tξ : SU(2) → U(2ξ + 1) is the usual irreducible unitary matrix representation. Hence " # [σA1 (x, ξ), σA (x, ξ)] = tξ (v1 )∗ σA3 (x, ξ)tξ (v1 ), σA (x, ξ) " # = tξ (v1 )∗ σA3 (x, ξ), tξ (v1 )σA (x, ξ)tξ (v1 )∗ tξ (v1 ). We may assume that here A ∈ Ψm (G) is left-invariant, i.e., A = (f → f ∗a), so that σA (x, ξ) = 3 a(ξ) for every ξ ∈ N/2 (in these symbol commutator properties, we may always fix x ∈ G, even if A is not left-invariant). Now ((x, ξ) → 3 a(ξ)) ∈ S m (G), m and due to the diffeomorphism-invariance of Ψ (G), also (x, ξ) → tξ (v1 ) 3 a(ξ) tξ (v1 )∗ must belong to S m (G), thus “decaying rapidly off-diagonal”. Thereby we already understand the behaviour of the commutator # " σA3 (x, ξ), tξ (v1 )σA (x, ξ)tξ (v1 )∗ , and this is similar (by transform tξ (v1 )∗ , see above) to [σA1 (x, ξ), σA (x, ξ)] . The same kind of reasoning applies to the commutator [σA2 (x, ξ), σA (x, ξ)] . Thus the symbol commutator condition is satisfied if we require that (x, ξ) → tξ (u) σA (x, ξ) tξ (u)∗ “decays rapidly off-diagonal” for every u ∈ G.

6. Global calculus Properties (aRn ) and (bRn ) can be used to construct the global calculus of the introduced matrix symbols. For example, we have the well-known composition formula for pseudo-differential operators A and B in Rn : 1 σAB (x, ξ) ∼ (∂ α σA )(x, ξ) (∂xα σB )(x, ξ), α! ξ α≥0

320


see, e.g., [11], [10], [12]. On the torus Tn , properties (aTn ) and (bTn ) allow to describe the composition of pseudo-differential operators in terms of toroidal symbols in an analogous way: σAB (x, ξ) ∼

1 (α) (%α ξ σA )(x, ξ) (∂x σB )(x, ξ). α!

α≥0

This formula was obtained in [29] and was further extended to compositions with toroidal Fourier integral operators in [14]. In the case of SU(2), we can use properties (aG ) and (bG ) to show that it also has the form σAB (x, ξ) ∼

1 (α) (%α ξ σA )(x, ξ) (∂x σB )(x, ξ). α!

α≥0

In fact, this formula can be also viewed in the following way. Let Qα σA (x) = π(y → qˇα (y) sA (x)(y)) with qˇα ∈ C ∞ (G) satisfying qˇα (exp(z)) = z α for z in a small neighbourhood of the origin on the Lie algebra g of G. Then we have the following elements of the calculus: Theorem 6.1 (M.E. Taylor [24]). σA∗ (x) ∼

1 Qα ∂xα σA (x)∗ , α!

α≥0

σAB (x) ∼

1 (Qα σA (x)) ∂xα σB (x). α!

α≥0

∞ Theorem 6.2. Let A ∈ Ψm (G) with expansion A ∼ j=0 Aj , Aj ∈ Ψm−j (G), and assume that x → σA0 (x)−1 is a symbol of B0 ∈ Ψ−m+1−ε (G) (for some ε > 0). ∞ Then A is elliptic with a parametrix B, σB ∼ k=0 σBk , where σB0 (x) = σA0 (x)−1 , σBN (x) = −σB0 (x)

−k N −1 N

(Qγ σAj (x)) ∂xγ σBk (x).

k=0 j=0 γ:j+k+|γ|=N

For further details of these results we refer to [24], [27], [28], and for the relation to introduced symbol classes we refer to [17]. Acknowledgements The authors wish to express their gratitude to Matania Ben-Artzi for valuable comments leading to an improvement of the manuscript.


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References [1] M.S. Agranovich, Spectral properties of elliptic pseudodifferential operators on a closed curve, Funktsional. Anal. i Prilozhen 13 (1979), 54–56 (in Russian). [2] M.S. Agranovich, On elliptic pseudodifferential operators on a closed curve, Trans. Moscow Math. Soc. 47 (1985), 23–74. [3] M.S. Agranovich, Elliptic operators on closed manifolds, Itogi Nauki i Tehniki, Ser. Sovrem. Probl. Mat. Fund. Napravl, 63 (1990), 5–129 (in Russian). (English translation in Encyclopaedia Math. Sci. 63 (1994), 1–130.) [4] B.A. Amosov, On the theory of pseudodifferential operators on the circle, Uspekhi Mat. Nauk 43 (1988), 169–170 (in Russian). (English translation in Russian Math. Surveys 43 (1988), 197–198.) [5] R. Beals, Characterization of pseudo-differential operators and applications, Duke Math. J. 44 (1977), 45–57. [6] H.O. Cordes, On pseudodifferential operators and smoothness of special Lie group representations, Manuscripta Math. 28 (1979), 51–69. [7] J. Dunau, Fonctions d’un opérateur elliptique sur une variété compacte, J. Math. Pures et Appl. 56 (1977), 367–391. [8] F. Geshwind and N.H. Katz, Pseudodifferential operators on SU(2), J. Fourier Anal. Appl. 3 (1997), 193–205. [9] L. H¨ ormander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 501–517. [10] L. H¨ ormander, The Analysis of Linear Partial Differential Operators III-IV, Berlin: Springer-Verlag, 1985. [11] J.J. Kohn and L. Nirenberg, On the algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269–305. [12] H. Kumano-go, Pseudo-Differential Operators, MIT Press, Cambridge, Mass.London, 1981. [13] W. McLean, Local and global description of periodic pseudodifferential operators, Math. Nachr. 150 (1991), 151–161. [14] M. Ruzhansky and V. Turunen, On the Fourier analysis of operators on the torus, in Modern Trends in Pseudo-Differential Operators, Editors: J. Toft, M.W. Wong and H. Zhu, Operator Theory: Advances and Applications 172, Birkh¨ auser, 2007, 87–105, [15] M. Ruzhansky and V. Turunen, On pseudo-differential and Fourier integral operators on the torus, in preparation. [16] M. Ruzhansky and V. Turunen, Pseudo-differential operators on spheres, in preparation. [17] M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries, Birkh¨ auser, to appear. [18] Yu. Safarov, Pseudodifferential operators and linear connections, Proc. London Math. Soc. 74 (1997), 379–416. [19] J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.

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[20] J. Saranen and W.L. Wendland, The Fourier series representation of pseudodifferential operators on closed curves, Complex Variables, Theory Appl. 8 (1987), 55–64. [21] V.A. Sharafutdinov, Geometric symbol calculus for pseudodifferential operators I Siberian Adv. Math. 15 (2005), 81–125. [Translation of Mat. Tr. 7 (2004), 159–206]. [22] H. Stetkær, Invariant pseudo-differential operators, Math. Scand. 28 (1971), 105–123. [23] R. Strichartz, Invariant pseudo-differential operators on a Lie group, Ann. Scuola Norm. Sup. Pisa 26 (1972), 587–611. [24] M.E. Taylor, Noncommutative Microlocal Analysis. Mem. Amer. Math. Soc. 52 (1984). [25] M.E. Taylor, Beals–Cordes-type characterizations of pseudodifferential operators, Proc. Amer. Math. Soc. 125 (1997), 1711–1716. [26] V. Turunen, Commutator characterization of periodic pseudodifferential operators, Z. Anal. Anw. 19 (2000), 95–108. [27] V. Turunen, Pseudodifferential Calculus on Compact Lie Groups and Homogeneous Spaces, Ph.D. Thesis, Helsinki University of Technology, 2001. [28] V. Turunen, Pseudodifferential calculus on the 2-sphere, Proc. Estonian Acad. Sci. Phys. Math. 53(3) (2004), 156–164. [29] V. Turunen and G. Vainikko, On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anw. 17 (1998), 9–22. [30] G. Vainikko, An integral operator representation of classical periodic pseudodifferential operators, Z. Anal. Anw. 18 (1999), 687–699. [31] N. Vilenkin, Special Functions and the Theory of Group Representations, Trans. Math. Monographs 22, Amer. Math. Soc., 1968. [32] H. Widom, A complete symbolic calculus for pseudodifferential operators, Bull. Sci. Math. 104 (1980), 19–63. [33] D. Zelobenko, Compact Lie groups and their Representations. Trans. Math. Monographs 40, Amer. Math. Soc., 1973. Michael Ruzhansky Department of Mathematics Imperial College London United Kingdom e-mail: [email protected] Ville Turunen Institute of Mathematics Helsinki University of Technology Finland e-mail: [email protected]


Sampling and Pseudo-Differential Operators Alip Mohammed and M.W. Wong Abstract. Discrete formulas for pseudo-differential operators based on the Shannon–Whittaker sampling formula and the Poisson summation formula are given. Mathematics Subject Classification (2000). Primary 42B10, 47G30. Keywords. Fourier transforms, pseudo-differential operators, Shannon– Whittaker sampling formula, Poisson summation formula.

1. Introduction Pseudo-differential operators, first introduced by Kohn and Nirenberg [18] and immediately modified by H¨ ormander [15], have been used with great success in partial differential equations. The usefulness in quantum physics has been manifested via the representation of pseudo-differential operators envisaged by Weyl [26], and studied in detail in [16] and [27] among others. Fuelled by wavelets, pseudodifferential operators have been studied extensively for the past two decades as localization operators [5, 6, 9, 10, 11, 12, 29] and wavelet multipliers [17, 19, 30, 31, 32]. Applications of pseudo-differential operators to sciences and engineering entail computations. Prompted by the success stories of the fast Fourier transform [7, 8, 25], it is reasonable to seek computational techniques based on the Fourier transform. Among the various manifestations of the pseudo-differential operators in the literature hitherto described, the original representation in [18], studied extensively in the book [28], is most intimately and explicitly tied to the Fourier transform. The aim of this paper is to give discrete formulas for these pseudodifferential operators based on sampling related to the Fourier transform. They This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

324

A. Mohammed and M.W. Wong

are the Shannon–Whittaker sampling formula and the Poisson summation formula. Accounts of these sampling formulas abound and can be found in, e.g., Chapter 8 of [13] and Part C of Chapter 7 in [24]. We first recall in Section 2 the pseudo-differential operators studied exclusively in the paper [18] and the book [28]. Its relations with the Rihaczek distribution in time-frequency analysis [21] recently studied in [3, 4] and [20] are described. The Shannon–Whittaker sampling formula is recalled in Section 3 and used to give a sampling formula for pseudo-differential operators. The Poisson summation formula is recalled in Section 4 and a sampling formula based on it is derived. This is an amazingly simple and beautiful formula and its connections with recent developments in magnetic resonance imaging (MRI) are its raison d’être.

2. Pseudo-differential operators Let m ∈ (−∞, ∞). Then we denote by S m the set of all functions σ in C ∞ (Rn ×Rn ) such that for all multi-indices α and β, there exists a positive constant Cα,β for which |(Dxα Dξβ σ)(x, ξ)| ≤ Cα,β (1 + |ξ|)m−|β| , x, ξ ∈ Rn . We call a function in S m a symbol of order m. Let σ ∈ S m . Then we define the pseudo-differential operator Tσ corresponding to the symbol σ by −n/2 eix·ξ σ(x, ξ)ϕ(ξ) ˆ dξ, x ∈ Rn , (Tσ ϕ)(x) = (2π) Rn

for all ϕ in the Schwartz space S(Rn ), where the Fourier transform ϕˆ is defined by −n/2 e−ix·ξ ϕ(x) dx, ξ ∈ Rn . ϕ(ξ) ˆ = (2π) Rn

It can be proved easily that Tσ maps S(Rn ) into S(Rn ) continuously. Let σ ∈ S(Rn × Rn ). Then for all functions ϕ and ψ in S(Rn ), (Tσ ϕ)(x)ψ(x) dx (Tσ ϕ, ψ)L2 (Rn ) = Rn = (2π)−n/2 eix·ξ σ(x, ξ)ϕ(ξ)ψ(x) ˆ dx dξ Rn Rn = (2π)−n/2 σ(x, ξ)eix·ξ ϕ(ξ)ψ(x) ˆ dx dξ Rn Rn = (2π)−n/2 σ(x, ξ)R(ϕ, ψ)(x, ξ) dx dξ, Rn

(2.1)

Rn

where R(ϕ, ψ) is the Rihaczek transform of ϕ and ψ introduced in [20] and is defined by R(ϕ, ψ)(x, ξ) = eix·ξ ϕ(ξ)ψ(x), ˆ x, ξ ∈ Rn . n n It is obvious that R(ϕ, ψ) is a function in S(R × R ) and the formula (2.1) can be used to define pseudo-differential operators with symbols in the space S (Rn × Rn )

Sampling and Pseudo-Differential Operators

325

of all tempered distributions on Rn × Rn . To wit, let σ ∈ S (Rn × Rn ). Then the pseudo-differential operator Tσ with symbol σ is defined by (Tσ ϕ)(ψ) = (2π)−n/2 σ(R(ϕ, ψ)),

ϕ, ψ ∈ S(Rn ).

3. The Shannon–Whittaker sampling formula We begin with the classical Shannon–Whittaker sampling theorem. Theorem 3.1. Let f ∈ L2 (R) be such that fˆ(ξ) = 0,

|ξ| > π,

and

∞

|f (k)| < ∞.

k=−∞

Then f (x) =

∞

f (k) sinc π(x − k),

x ∈ R,

k=−∞

where sin x , x ∈ R, x and the convergence of the series is uniform on every compact subset of R. sinc x =

In signal analysis, a function f in L2 (R) for which the support of fˆ is contained in [−L, L] is known as a bandlimited signal or, more precisely, an Lbandlimited signal. By Theorem 3.1, we have the following immediate result by suitable scaling. Theorem 3.2. Let f be an L-bandlimited signal such that ∞ f kπ < ∞. L k=−∞

Then f (x) =

∞ k=−∞

f

kπ L

sinc (Lx − kπ),

x ∈ R,

where the convergence of the series is uniform on every compact subset of R. Remark 3.3. Theorem 3.2 says that an L-bandlimited signal f is determined πby its : k ∈ Z . We call the number L the values on the uniformly spaced sample kπ L sample spacing and the number L the sample rate or the Nyquist rate. Sampling at π L a rate lower than π is called undersampling and sampling at a rate higher than L π is called oversampling. With oversampling, we can obtain a more rapidly convergent Shannon–Whittaker sampling formula in which the sinc function is replaced by another suitable function. Details are given on pages 279–281 of [13].

326


Theorem 3.4. Let σ ∈ L2 (R × R). Then for all functions f in L2 (R) such that ∞ < ∞, we get fˆ(ξ) = 0 whenever |ξ| > L, and k=−∞ f kπ L ∞ kπ (Tσ f )(x) = (2π)−1/2 f (ˇ σ (x, ·) ∗ T−kπ/L DL sinc)(x) L k=−∞

for almost all x in R, where σ ˇ is the inverse Fourier transform of σ with respect to the second variable, T−kπ/L and DL are the translation operator and dilation operator defined by kπ (T−kπ/L h)(x) = h x − , x ∈ R, L and x ∈ R,

(DL h)(x) = h(Lx), for all measurable functions h on R.

Proof. For a fixed x in R, if we denote by Mx the modulation operator defined by y ∈ R,

(Mx h)(y) = eixy h(y),

and by σ ˆ the Fourier transform of σ with respect to the second variable, then we get by Theorem 3.2 ∞ −1/2 (Tσ f )(x) = (2π) eixξ σ(x, ξ)fˆ(ξ) dξ −∞ ∞ −1/2 (Mx σ(x, ·))(ξ)fˆ(ξ) dξ = (2π) −∞ ∞ −1/2 (T−x σ ˆ (x, ·))(y)f (y) dy = (2π) −∞ ∞ = (2π)−1/2 σ ˆ (x, y − x)f (y) dy −∞ ∞ σ ˇ (x, x − y)f (y) dy = (2π)−1/2 = (2π)−1/2

−∞ ∞

−∞ −1/2

= (2π)

= (2π)−1/2

∞

σ ˇ (x, x − y)

∞ k=−∞ ∞

f f

k=−∞

for almost all x in R, as required.

kπ L kπ L

k=−∞ ∞

−∞

f

kπ L

sinc (Ly − kπ) dy

σ ˇ (x, x − y) sinc (Ly − kπ) dy

(ˇ σ (x, ·) ∗ T−kπ/L DL sinc)(x)


327

Remark 3.5. The interchange of the order of integration and summation in the proof of Theorem 3.4 can be justified. Indeed, using the assumptions that σ ∈ L2 (R × R) and ∞ f kπ < ∞, L k=−∞

we can apply Schwarz’ inequality, Plancherel’s theorem and the fact that the sinc function is in L2 (R) to obtain ∞ ∞ f kπ |ˇ σ (x, x − y))| |sinc (Ly − kπ)| dy L −∞ k=−∞ 1/2 ∞ 1/2 ∞ ∞ kπ 2 2 ≤ |ˇ σ (x, y)| dy |sinc y| dy L−1/2 f L −∞ −∞ k=−∞

∞ ∞ π 1 ˆ kπ fL (x) = f (x + 2kL) = f eikπx/L . 2L L k=−∞

k=−∞

It follows from the Poisson summation formula that if f is a function in L2 (R) such that f (x) = 0, |x| > L, then > ∞ π 1 ˆ kπ f f (x) = eikπx/L , x ∈ [−L, L], 2L L k=−∞

where the convergence of the series is uniform on [−L, L]. Theorem 4.2. Let σ ∈ L2 (R × R) be such that σ ˇ (x, ·) ∈ L1 (R) for almost all x in R. Then for all functions f in L2 (R) such that f (x) = 0,

|x| > L,

Sampling and Pseudo-Differential Operators we get

> (Tσ f )(x) =

329

∞ π 1 ˆ kπ kπ ikπx/L σ x, f e 2L L L k=−∞

for almost all x in R. Proof. From the proof of Theorem 3.4, we have ∞ −1/2 σ ˇ (x, x − y)f (y) dy, (Tσ f )(x) = (2π) −∞

x ∈ R.

So, using the Poisson summation formula in Theorem 4.2, we get ∞ ∞ 1 ˆ kπ σ ˇ (x, x − y)eikπy/L dy f (Tσ f )(x) = 2L L −∞ k=−∞ ∞ ∞ 1 ˆ kπ = σ ˇ (x, z)eikπ(x−z)/L dz f 2L L −∞ k=−∞ > ∞ kπ π1 kπ ikπx/L ˆ σ x, f = e 2L L L k=−∞

for almost all x in R, as asserted.

For applications in imaging, n-dimensional analogues of the Poisson summation formula and the corresponding sampling formula for pseudo-differential operators can be formulated. First, we have the following n-dimensional Poisson summation formula. Theorem 4.3. For positive numbers L1 , L2 , . . . , Ln , let f ∈ L1 (Rn ) be such that ∞ απ ˆ f < ∞. L α ,α ,...,α =−∞ 1

n

2

Then at each point x of continuity of fL , where fL (y) =

∞

f (y1 + 2α1 L1 , y2 + 2α2 L2 , . . . , yn + 2αn Ln ),

y ∈ Rn ,

α1 ,α2 ,...,αn =−∞

we get fL (x) =

n π n/2 1 2 L j=1 j

∞ α1 ,α2 ,...,αn

n απ fˆ eiαj πx/Lj . L =−∞ j=1

The corresponding sampling formula for pseudo-differential operators is given in the following theorem. Theorem 4.4. Let σ ∈ L2 (Rn × Rn ) be such that σ ˇ (x, ·) ∈ L1 (Rn )

330


for almost all x in Rn . Then for all functions f in L2 (Rn ) such that f (x) = 0,

x∈ /

n

[−Lj , Lj ],

j=1

we get (Tσ f )(x) =

n π n/2 1 2 L j=1 j

∞ α1 ,α2 ,...,αn

n απ απ eiαj πx/Lj σ x, fˆ L j=1 L =−∞

for almost all x in R . n

5. Conclusions Two sampling formulas for pseudo-differential operators are given in this paper. Fast computation is the motivation for these formulas. The first is based on the Shannon–Whittaker sampling formula and works for bandlimited signals. The second formula entails the use of the Poisson summation formula and is tailored for time-limited signals in, e.g., MRI. The results are presented in the framework of modern mathematical analysis with precise conditions to guarantee convergence and interchange of order of integration in different circumstances. No attempt, however, is made to present the results under more general hypotheses. In the case of the discrete formula for pseudo-differential operators built on the Poisson summation formula, related results can be found in the work [22] on the Fourier analysis of operators on the torus, which in turn can be traced to the original works of Agranovich [1, 2]. The numerical analysis of pseudo-differential operators can be found in [23] among others.

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