Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations

The International Workshop on Pseudo-Di?erential Operators: Complex Analysis and Partial Di?erential Equations was held at York University on August 4-8, 2008. The ?rst phase of the workshop on August 4-5 consisted of a mini-course on pseudo-di?erential operators and boundary value problems given by Professor Bert-Wolfgang Schulze of Universita t Potsdam for graduate students and po- docs. This was followed on August 6-8 by a conference emphasizing boundary value problems;explicit formulas in complex analysis and partialdi?erential eq- tions; pseudo-di?erential operators and calculi; analysis on the Heisenberg group and sub-Riemannian geometry; and Fourier analysis with applications in ti- frequency analysis and imaging. The role of complex analysis in the development of pseudo-di?erential op- 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 Po- son integral formula in the complex plane C. These formulas are instrumental in solving boundary value problems for the Cauchy-Riemann operator? and the Laplacian?onspeci?cdomainswith theunit disk andits biholomorphiccomp- 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 n n in C , which may be the unit polydisk or the unit ball in C .

Résumé
This volume is an outgrowth of the international workshop entitled 'Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations' held at York University on August 4†8, 2008. It consists of the expository paper based on the 6-hour minicourse given by Professor Bert-Wolfgang Schulze, and sixteen papers based on lectures given at the workshop and on invitations. While the focus is on the current developments of pseudo-differential operators in the context of complex analysis and partial differential equations, other topics related to the analysis, applications and computations of pseudo-differential operators are featured.

Contenu
Boundary Value Problems with the Transmission Property.- Spectral Invariance of SG Pseudo-Differential Operators on L p ? n .- Edge-Degenerate Families of Pseudo-Differential Operators on an Infinite Cylinder.- Global Regularity and Stability in S-Spaces for Classes of Degenerate Shubin Operators.- Weyl's Lemma and Converse Mean Value for Dunkl Operators.- Dirichlet Problems for Inhomogeneous Complex Mixed-Partial Differential Equations of Higher order in the Unit Disc: New View.- Dirichlet Problems for the Generalized n-Poisson Equation.- Schwarz, Riemann, Riemann-Hilbert Problems and Their Connections in Polydomains.- L p -Boundedness of Multilinear Pseudo-Differential Operators.- A Trace Formula for Nuclear Operators on L p .- Products of Two-Wavelet Multipliers and Their Traces.- Pseudo-Differential Operators on ?.- Pseudo-Differential Operators with Symbols in Modulation Spaces.- Phase-Space Differential Equations for Modes.- Two-Window Spectrograms and Their Integrals.- Time-Time Distributions for Discrete Wavelet Transforms.- The Stockwell Transform in Studying the Dynamics of Brain Functions.