di?erential operators in particular will be developed hand in glove with appli- tions andcomputation inthe physical,biologicaland medicalsciences.This theme will play an important role in the forthcoming volumes on pseudo-di?erential - erators originating from IGPDO. The Editors OperatorTheory: Advances andApplications,Vol.189, 1-14 c 2008Birkh auserVerlagBasel/Switzerland Phase-Space Weyl Calculus and Global Hypoellipticity of a Class of Degenerate Elliptic Partial Di?erential Operators Maurice de Gosson Abstract. In a recent series of papers M.W. Wong has studied a degenerate elliptic partial di?erential 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 - proach highlights the relationship of Wong's constructions with the quantum mechanics of charged particles in a uniform magnetic ?eld. Using Shubin's classes of pseudodi?erential symbols we prove global hypoellipticity results for arbitrary phase-space operators arising from elliptic operators on con- uration space. Mathematics Subject Classi?cation (2000). Primary 47F30; Secondary 35B65, 46F05. Keywords. Degenerate elliptic operators, hypoellipticity, phase space Weyl calculus, Shubin symbols.

Zusammenfassung

This volume consists of peer-reviewed papers related to lectures on pseudo-differential operators presented at the meeting of the ISAAC Group in Pseudo-Differential Operators (IGPDO) held on August 13-18, 2007, and invited papers by experts in the field.

Inhalt
Phase-Space Weyl Calculus and Global Hypoellipticity of a Class of Degenerate Elliptic Partial Differential Operators.- Classes of Degenerate Elliptic Operators in Gelfand-Shilov Spaces.- Weyl Transforms and the Heat Equation for the Sub-Laplacian on the Heisenberg Group.- Construction of the Fundamental Solution and Curvature of Manifolds with Boundary.- Operators with Corner-Degenerate Symbols.- Ellipticity of Fredholm Pseudo-Differential Operators on L p (? n ).- Hyperbolic Systems with Discontinuous Coefficients: Generalized Wavefront Sets.- Generalized Fourier Integral Operators on Spaces of Colombeau Type.- On Local and Global Regularity of Fourier Integral Operators.- Type 1,1-Operators Defined by Vanishing Frequency Modulation.- Regularity for Quasi-Elliptic Pseudo-Differential Operators with Symbols in Hölder Classes.- Multi-Anisotropic Gevrey Regularity of Hypoelliptic Operators.- Modified Stockwell Transforms and Time-Frequency Analysis.- Localization Operators for Two-Dimensional Stockwell Transforms.- Pseudo-Differential Operators on .- On Pseudo-Differential Operators on Group SU(2).- Sampling and Pseudo-Differential Operators.