7513_ot164titelei 1 2.12.2005, 10:55 Uhr Operator Theory: Advances and Applications Vol. 164 Editor: I. Gohberg Editorial Office: S. T. Kuroda (Tokyo) School of Mathematical P. Lancaster (Calgary) Sciences L. E. Lerer (Haifa) Tel Aviv University B. Mityagin (Columbus) Ramat Aviv, Israel V. Olshevsky (Storrs) M. Putinar (Santa Barbara) Editorial Board: L. Rodman (Williamsburg) D. Alpay (Beer-Sheva) J. Rovnyak (Charlottesville) J. Arazy (Haifa) D. E. Sarason (Berkeley) A. Atzmon (Tel Aviv) I. M. Spitkovsky (Williamsburg) J. A. Ball (Blacksburg) S. Treil (Providence) A. Ben-Artzi (Tel Aviv) H. Upmeier (Marburg) H. Bercovici (Bloomington) S. M. Verduyn Lunel (Leiden) A. Böttcher (Chemnitz) D. Voiculescu (Berkeley) K. Clancey (Athens, USA) D. Xia (Nashville) L. A. Coburn (Buffalo) D. Yafaev (Rennes) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) Honorary and Advisory R. G. Douglas (College Station) Editorial Board: A. Dijksma (Groningen) C. Foias (Bloomington) H. Dym (Rehovot) P. R. Halmos (Santa Clara) P. A. Fuhrmann (Beer Sheva) T. Kailath (Stanford) B. Gramsch (Mainz) H. Langer (Wien) J. A. Helton (La Jolla) P. D. Lax (New York) M. A. Kaashoek (Amsterdam) M. S. Livsic (Beer Sheva) H. G. Kaper (Argonne) H. Widom (Santa Cruz) 7513_ot164titelei 2 2.12.2005, 10:55 Uhr Pseudo-Differential Operators and Related Topics Paolo Boggiatto Luigi Rodino Joachim Toft M.W. Wong Editors Birkhäuser Verlag . . Basel Boston Berlin 7513_ot164titelei 3 2.12.2005, 10:55 Uhr Editors: Paolo Boggiatto Joachim Toft Dipartimento di Matematica School of Mathematics and Systems Università di Torino Engineering Via Carlo Alberto, 10 Växjö University 10123 Torino SE-351 95 Växjö Italy Sweden e-mail: [email protected] e-mail: [email protected] Luigi Rodino M. W. Wong Dipartimento di Matematica Department of Mathematics and Statistics Università di Torino York University Via Carlo Alberto, 10 4700 Keele Street 10123 Torino Toronto, Ontario M3J 1P3 Italy Canada e-mail: [email protected] e-mail: [email protected] 2000 Mathematics Subject Classification: Primary 35A17, 35L15, 35L40, 35L45, 35L60, 35L80, 35P05, 35S05, 35S30, 42C15, 43A80, 46F05, 47G30, 53D12, 58J40, 60G12, 60G15, 81R30; Secondary 17B45, 35B65, 35C20, 35J50, 35L30, 42C40, 44A05, 45C05, 46G10, 53D05, 53D50, 58B15, 58D30, 58J30, 60G57 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA 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 3-7643-7513-2 Birkhäuser Verlag, 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. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-7513-2 e-ISBN: 3-7643-7514-0 ISBN-13: 978-3-7643-7513-3 9 8 7 6 5 4 3 2 1 www.birkhauser.ch 7513_ot164titelei 4 2.12.2005, 10:55 Uhr Contents Preface ......................................................................vii M.S. Agranovich Strongly Elliptic Second Order Systems with Spectral Parameter in Transmission Conditions on a Nonclosed Surface ......................1 A. Ascanelli and M. Cicognani Well-Posedness of the Cauchy Problem for Some Degenerate Hyperbolic Operators ...................................................23 M. Cappiello and L. Zanghirati Quasilinear Hyperbolic Equations with SG-Coefficients .................43 I. Kamotski and M. Ruzhansky Representation of Solutions and Regularity Properties for Weakly Hyperbolic Systems ....................................................53 M. Ruzhansky and M. Sugimoto Global Calculus of Fourier Integral Operators,Weighted Estimates, and Applications to Global Analysis of Hyperbolic Equations ...........65 G. Garello and A. Morando Lp-Continuity for Pseudo-Differential Operators ........................79 V.S. Rabinovich Fredholm Property of Pseudo-Differential Operators on Weighted H¨older-Zygmund Spaces ................................................95 M.W. Wong Weyl Transforms and Convolution Operators on the Heisenberg Group 115 M. de Gosson Uncertainty Principle, Phase Space Ellipsoids and Weyl Calculus ......121 K. Furutani Pseudo-Differential Operator and Reproducing Kernels Arising in Geometric Quantization ...............................................133 J. Toft Hudson’s Theorem and Rank One Operators in Weyl Calculus .........153 A. Khrennikov Distributions and Pseudo-Differential Operators on Infinite-Dimensional Spaces with Applications in Quantum Physics .........................161 N. Teofanov Ultradistributions and Time-Frequency Analysis .......................173 vi Contents O. Christensen Frames and Generalized Shift-Invariant Systems .......................193 P. Wahlberg The Wigner Distribution of Gaussian Weakly Harmonizable Stochastic Processes ...................................................211 E. Cordero, F. De Mari, K. Nowak, and A. Tabacco Reproducing Groups for the Metaplectic Representation ...............227 Preface Asasatelliteconferencetothe FourthCongressofEuropeanMathematicsheldat StockholmUniversityin2004,theInternationalConferenceonPseudo-Differential Operators and Related Topics was held at V¨axjo¨ University in Sweden from June 22toJune25,2004.TheconferencewassupportedbyVa¨xjo¨University,theFIRB Research Group on Microlocal Analysis of Universit`a di Torino, and the Inter- national Society for Analysis, its Applications and Computation (ISAAC). The conferencewaswellattendedbyabout50mathematiciansfromBulgaria,Canada, Denmark,England,Finland,Germany,Italy,Japan,Mexico,SerbiaandMontene- gro, Russia and Sweden. The conference covered a broad spectrum of topics related to pseudo-differential operators such as partial differential equations, quantization, Wigner transforms andWeyltransformsonLiegroups,mathematicalphysics,time-frequencyanalysis and stochastic processes. The speakers were enthusiastic about the prospect of publishing articles based on their presentations in a volume to be published in ProfessorIsraelGohberg’sprestigiousseriesentitled “OperatorTheory:Advances and Applications”. All contributions from speakers have been carefully refereed and the articles col- lected in this volume give a representative flavour of the mathematics presented at the conference. This volume is a permanent record of the conference and a valuable complement to the volume “Advances in Pseudo-Differential Operators” published in the same series in 2004, which is devoted to the Special Session on Pseudo-Differential Operators at the Fourth ISAAC Congress held at York Uni- versity in August 2003. OperatorTheory: Advances andApplications,Vol.164,1–21 (cid:1)c 2006Birkh¨auserVerlagBasel/Switzerland Strongly Elliptic Second Order Systems with Spectral Parameter in Transmission Conditions on a Nonclosed Surface M.S. Agranovich Abstract. WeconsideraclassofsecondorderstronglyellipticsystemsinRn, n≥3,outsideaboundednonclosedsurfaceSwithtransmissionconditionson S containing a spectral parameter. Assuming that S and its boundary γ are Lipschitz, we reduce the problems to spectral equations on S for operators of potential type. We prove the invertibility of these operators in suitable Sobolev type spaces and indicate spectral consequences. Simultaneously, we prove the unique solvability of the Dirichlet and Neumann problems with boundary data on S. Mathematics SubjectClassification (2000). Primary 35P05; Secondary 35J50, 45C05. Keywords. Strong ellipticity, transmission condition, spectral equation, Lip- schitz surface, surface potential, variational approach, Wiener–Hopf method. 1. Introduction 1.1. Statement of the Problems We consider the second order system of partial differential equations L (∂)u(x):=L (∂)u(x)+ω2u(x)=0 (1.1) ω 0 in Rn \S, where S is an (n−1)-dimensional surface with (n −2)-dimensional boundaryγ.More precisely,we assumethat S is a partofa closedsurface1 Γ;the latter consists of two open parts S = Γ+ and Γ− without common points and of theircommonboundaryγ.By∂ wedenotethe“vector”ofpartialderivatives∂ = j ∂/∂ ,j =1,...,n.TheoperatorL ishomogeneouswithrespecttodifferentiations j 0 and has constant coefficients. The numerical parameter ω = ω +iω belongs to 1 2 TheworkwassupportedbythegrantofRFFINo.04-01-00914. 1Aclosedsurfaceisacompactsurfacewithoutboundary. 2 M.S. Agranovich the closed upper half-plane (ω ≥ 0) and is nonnegative if it is real. At infinity, 2 the solutions are subjected to natural radiation or decay conditions depending on ω. The assumptions are formulated more precisely below in Subsection 1.2. We assume that n ≥ 3 to avoid the consideration of logarithmic potentials. The surfaceΓdividesthecomplementofitselfintoaboundedsimplyconnecteddomain Ω+ and an unbounded domain Ω−. The superscripts + and − will also be used to denote the boundary values of functions on the inner and outer sides of Γ, respectively. By ν =ν(x) we denote the unit outward normal at points x∈Γ. Ourmaingoalistoconsidertwospectralproblemsfor(1.1)withtransmission conditions on S containing a spectral parameter λ. The correspondingspaces will be specified later. Problem I. u+ =u−, Tu−−Tu+ =λu± on S. (1.2) Problem II. Tu+ =Tu−, Tu± =λ[u−−u+] on S. (1.3) Here Tu is the conormal derivative, see (1.8) below. In the simplest case of the Helmholtz equation, Tu is the normal derivative ∂ u. We wish to describe ν somespectralpropertiesoftheseproblems.Wewillseeshortlythattheyareclosely connectedwiththenon-spectralDirichletandNeumannproblemsforsystem(1.1): The Dirichlet problem. u± =f on S. (1.4) The Neumann problem. Tu± =g on S. (1.5) The surfaces Γ and γ are assumed to be either C∞ or Lipschitz; Γ is con- nected, while γ can consist of several components. The normal ν(x) is defined almost everywhere in the Lipschitz case. In the case of a closed surface S = Γ, Problems I and II and some other problemsfortheHelmholtzequationwereposedbythephysicistsKatsenelenbaum andhiscollaboratorsSivovandVoitovichinthe60s.Seethebook[1]anditsrevised English edition [2]. In [1], a mathematical supplement [3] is contained, written by the author of the present paper, with the analysis of these and similar problems in acoustics andelectrodynamics by tools of the theories of surface potentials and pseudo-differential operators. The surface was assumed to be smooth. The initial results were obtained by the author in collaboration with his graduate student Golubeva; see also her note [4]. Conditions (1.2) and (1.3) can be interpreted as related to a half-transparentscreen. Similar problems with boundary and transmission conditions on a closed Lipschitz surface for the Helmholtz equation were considered in [5] and for the Lam´e system in elasticity theory (and n = 3) in [6]. The general case of systems (1.1) was considered in [7] and [8]. In [8], systems with variable coefficients were included into consideration. More precisely, in the last three papers the surface is Strongly Elliptic Second Order Systems 3 assumed to be either smooth or Lipschitz. Of course,no theory of elliptic pseudo- differential operators exists in the case of a Lipschitz surface, but there is an extensivetheoryofclassicalsurfacepotentialsandnon-spectralproblems;see[7,8] and numerous references therein. InSection2,werecallsomedefinitionsandtechnicaltoolsfrom[7,8]related to the case of a closed surface. We also add some supplementary material. In particular, we introduce the hypersingular operator and Caldero´n projections for general systems (1.1) following, e.g., the paper [9] on the Laplace and Helmholtz equations. Non-spectralproblems(Dirichlet,Neumann,andmoregeneral)withdataon a nonclosed surface for the Helmholtz equation and the Lam´e system were first consideredbyStephan[10,11]andthenbyCostabelandStephan[12].Inelasticity theory, a non-closed surface has the meaning of a crack. The Lam´e system mod- els an isotropic medium. Cracks in anisotropic elastic media were considered by Duduchava, Natroshvili and Shargorodski [13] and by some other authors. More- over, non-spectral problems in elasticity were considered with much more general conditions on S and in much more general spaces than in the present paper, see also, e.g., [14] and references therein. These authors followed Vishik and Eskin (e.g., see [15], [16]) and Eskin [17] andusedtheWiener–HopfmethodassumingthatΓandγ aresufficientlysmooth. In Section 3, the main in the present paper, we will show that it is possible to consider Lipschitz surfaces Γ and γ using the simplest Sobolev type spaces H±1/2(S) and H(cid:1)±1/2(S). (The spaces are defined in Subsection 2.5.) Instead of the Wiener–Hopf method, a modification of the classical variational approach is used(seeourPropositions3.2and3.4forthecaseofpureimaginaryω).We prove the unique solvability ofthe Dirichlet andNeumann problems for generalsystems (1.1) and the invertibility of the potential type operators on S corresponding to these problems. It seems to us that these non-spectral results are of interest even for the Helmholtz and Lam´e equations. Exactly the same invertible operators occur in the spectral equations on S correspondingtoourspectralproblemsIandII.Thusthesimplestspectralresults become available in the Lipschitz case; see Subsection 3.4. In Section 4, we briefly mention some further results; they will be published elsewhere. 1.2. Exact Statement of the Assumptions (See [7] for details.) The operator L (∂) is an m × m matrix. Replacing ∂ by 0 ξ =(ξ ,...,ξ ), we obtain the principal symbol of the operator −L (∂): 1 n 0 (cid:2)(cid:2) L (ξ)= A ξ ξ , A =(ars). (1.6) 0 jk j k jk jk Here the A are real matrices satisfying the symmetry condition ars =asr. The jk jk kj matrix L (ξ) is assumed to be positive definite for ξ (cid:3)= 0, which is the strong 0 ellipticity condition for the operator −L(∂). Besides, as in [7, 8], we impose the