Table Of ContentOperator Theory: Advances and Applications
Vol. 205
Editor:
I. Gohberg
Editorial Office:
School of Mathematical Sciences V. Olshevski (Storrs, CT, USA)
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Ramat Aviv A.C.M. Ran (Amsterdam, The Netherlands)
Israel L. Rodman (Williamsburg, VA, USA)
J. Rovnyak (Charlottesville, VA, USA)
B.-W. Schulze (Potsdam, Germany)
F. Speck (Lisboa, Portugal)
Editorial Board: I.M. Spitkovsky (Williamsburg, VA, USA)
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L.E. Lerer (Haifa, Israel) B. Silbermann (Chemnitz, Germany)
B. Mityagin (Columbus, OH, USA) H. Widom (Santa Cruz, CA, USA)
Pseudo-Differential Operators:
Complex Analysis and
Partial Differential Equations
International Workshop, York University, Canada,
August 4–8, 2008
B.-W. Schulze
M.W. Wong
Editors
Birkhäuser
Basel · Boston · Berlin
Editors:
Bert-Wolfgang Schulze M. W. Wong
Institut für Mathematik Department of Mathematics and Statistics
Universität Potsdam York University
Am Neuen Palais 10 4700 Keele Street
14469 Potsdam Toronto, Ontario M3J 1P3
Germany Canada
e-mail: [email protected] e-mail: [email protected]
2000 Mathematics Subject Classification: Primary 30G30, 31A30, 32A25, 33C05, 35, 42B10,
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31A10, 32A07, 32A26, 32A30, 32A40, 42C40, 45E05, 46F05, 65T50
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Contents
Preface.......................................................................vii
B.-W. Schulze
Boundary Value Problems with the Transmission Property................1
A. Dasgupta and M.W. Wong
Spectral Invariance of SG Pseudo-Differential Operators on Lp(Rn)......51
J. Abed and B.-W. Schulze
Edge-DegenerateFamilies of Pseudo-Differential Operators
on an Infinite Cylinder..................................................59
T. Gramchev, S. Pilipovi´c and L. Rodino
Global Regularity and Stability in S-Spaces for Classes of Degenerate
Shubin Operators.......................................................81
M. Maslouhi and R. Daher
Weyl’s Lemma and Converse Mean Values for Dunkl Operators..........91
H. Begehr, Z. Du and N. Wang
Dirichlet Problems for Inhomogeneous Complex Mixed Partial
Differential Equations of Higher Order in the Unit Disc: New View.....101
U¨. Aksoy and A.O. C¸elebi
Dirichlet Problems for Generalized n-PoissonEquations................129
A. Mohammed
Schwarz, Riemann, Riemann–Hilbert Problems and Their
Connections in Polydomains............................................143
V. Catana˘, S. Molahajloo and M.W. Wong
Lp-Boundedness of Multilinear Pseudo-Differential Operators...........167
J. Delgado
A Trace Formula for Nuclear Operators on Lp..........................181
V. Catana˘
Products of Two-Wavelet Multipliers and Their Traces.................195
S. Molahajloo
Pseudo-Differential Operators on Z.....................................213
J. Toft
Pseudo-Differential Operators with Symbols in Modulation Spaces......223
vi Contents
L. Cohen
Phase-Space Differential Equations for Modes..........................235
P. Boggiatto, G. De Donno and A. Oliaro
Two-Window Spectrograms and Their Integrals........................251
C.R. Pinnegar, H. Khosravani and P. Federico
Time-Time Distributions for Discrete Wavelet Transforms..............269
C. Liu, W. Gaetz and H. Zhu
The Stockwell Transform in Studying the Dynamics of Brain
Functions..............................................................277
Preface
The International Workshop on Pseudo-Differential Operators: Complex Analysis
and Partial Differential Equations was held at York University on August 4–8,
2008. The first phase of the workshop on August 4–5 consisted of a mini-course
on pseudo-differential operators and boundary value problems given by Professor
Bert-Wolfgang Schulze of Universita¨t Potsdam for graduate students and post-
docs. This was followed on August 6–8 by a conference emphasizing boundary
valueproblems;explicitformulasincomplexanalysisandpartialdifferentialequa-
tions; pseudo-differential operators and calculi; analysis on the Heisenberg group
and sub-Riemannian geometry; and Fourier analysis with applications in time-
frequency analysis and imaging.
The role of complex analysis in the development of pseudo-differential oper-
ators can best be seen in the context of the well-known Cauchy kernel and the
related Poisson kernel in, respectively, the Cauchy integral formula and the Pois-
son integral formula in the complex plane C. These formulas are instrumental
in solving boundary value problems for the Cauchy-Riemann operator ∂ and the
LaplacianΔonspecificdomainswiththeunitdiskanditsbiholomorphiccompan-
ion, i.e., the upper half-plane, as paradigm models. The corresponding problems
in several complex variables can be formulated in the context of the unit disk
in Cn, which may be the unit polydisk or the unit ball in Cn. Analogues of the
Cauchy kernel and the Poissonkernel and their ramifications to express solutions
of boundary value problems in several complex variables can be looked upon as
singularintegraloperators,whicharedefactoequivalentmanifestationsofpseudo-
differential operators. It is the vision that bringing together experts in explicit
formulas for boundary value problems in complex analysis working with kernels
and specialists in pseudo-differentialoperatorsworkingwith symbols shouldbuild
synergybetweenthe twogroups.Thefunctionalanalysisandreal-lifeapplications
ofpseudo-differentialoperatorsarealwaysamongtopprioritiesinouragendaand
these are well represented in the workshopand in this volume.
OperatorTheory:
Advances andApplications,Vol.205, 1–50
(cid:2)c 2009Birkh¨auserVerlagBasel/Switzerland
Boundary Value Problems with the
Transmission Property
B.-W. Schulze
Abstract. Wegiveasurveyonthecalculusof (pseudo-differential) boundary
valueproblemswiththetransmisionpropertyattheboundary,andellipticity
in theShapiro–Lopatinskij sense. Apart from theoriginal resultsof thework
of Boutet de Monvel we present an approach based on the ideas of the edge
calculus. In a final section we introduce symbols with the anti-transmission
property.
Mathematics SubjectClassification (2000). 35J40, 58J32, 58J40.
Keywords. Pseudo-differential boundary value problems, transmission and
anti-transmission property,boundarysymbolic calculus, Shapiro–Lopatinskij
ellipticity, parametrices.
1. Introduction
Boundaryvalue problems (BVPs)for elliptic (pseudo-)differential operatorshave
attracted mathematicians and physicists during all periods of modern analysis.
While the definition of ellipticity of an operator on an open (smooth) manifold is
very simple, such a notion in connection with a (smooth or non-smooth) bound-
aryismuchlessevident. Duringthe pastfew yearsthe interestinBVPsincreased
again considerably, motivated by new applications and also by unsolved prob-
lems in the frame of the structural understanding of ellipticity in new situations.
Several classical periods of the development created deep and beautiful ideas, for
instance, in connection with function theory, potential theory, with boundary op-
erators satisfying the complementing condition, cf. Agmon, Douglas, Nirenberg
[1], or pseudo-differential theories from Vishik and Eskin [29], Eskin [7], Boutet
de Monvel [4]. Other branches of the development concern ellipticity with global
projection conditions (analogues of Atiyah, Patodi, Singer conditions, cf. [3]), or
elliptic theories on manifolds with geometric singularities, cf. the author’s papers
[18] or [19].
2 B.-W. Schulze
After allthat it is not easyto imagine how many basic and interesting prob-
lemsremainedopen.Apartofthenewdevelopmentsisconnectedwiththeanalysis
onconfigurationswithsingularitiesthatincludesboundaryvalueproblems.Inthat
contextitseemstobe desirableto seethe pseudo-differentialmachineryofBoutet
de Monvel and also of Vishik and Eskin from an alternative viewpoint, using the
achievementsofthe coneandedgepseudo-differentialcalculusasispointedoutin
[16],[21],andin the author’sjointpaper withSeiler [25],see alsothe monographs
[22], or those jointly with Egorov [6], Kapanadze [12], Harutyunyan [10].
Our exposition just intends to emphasize such an approach, here mainly
focused on operators with the transmission property at the boundary from the
work of Boutet de Monvel. We also introduce symbols with the anti-transmission
property at the boundary. Together with those with the transmission property
they spanthe spaceofall(classical)symbolsthataresmoothuptothe boundary.
A pseudo-differential calculus for such general symbols needs more tools from the
edge algebra than developed here.
The present paper is the elaborated version of introductory lectures, given
during the International Workshop on Pseudo-Differential Operators, Complex
Analysis and Partial Differential Equations at York University on August 4–8,
2008, in Toronto.
2. Interior and Boundary Symbols for Differential Operators
LetX beaC∞ manifoldwithboundaryY =∂X.Moreover,let2X bethedouble,
definedbygluingtogethertwocopiesX±ofX toaC∞manifoldalongthecommon
boundaryY.LetusfixaRiemannianmetricon2X andconsiderY intheinduced
metric. There is then a tubular neighbourhood of Y in 2X that can be identified
with Y ×[−1,1], with a splitting of variables x = (y,t), where t is the variable
normal to the boundary and y ∈ Y. We assume that (y,t) belongs to X =: X
+
for 0≤t≤1 and to X− for −1≤t≤0 .
If M is a C∞ manifold (with or without boundary), by Diffμ(M) we de-
note the set of all differential operators of order μ on M with smooth coefficients
(smooth up to the boundary when ∂M (cid:4)=∅).
Local descriptions near Y will refer to charts
χ:U →Ω×R
for open U ⊆2X, U ∩Y (cid:4)=∅, and open Ω⊆Rn−1, and induced charts
χ:U ∩Y →Ω
on Y and
χ± :U± :=U ∩X± →Ω×R±
onX± neartheboundary.ConcerningthetransitionmapsΩ×R→Ω(cid:2)×R,(y,t)→
(y˜,t˜), for simplicity we assume that the normal variable remains unchanged near
the boundary, i.e., t=t˜for |t| sufficiently small.The map y →y˜correspondsto a
diffeomorphism Ω→Ω(cid:2).
