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.

Contenuti:

Phase-space Weyl calculus and global hypoellipticity of a class of degenerate elliptic partial differential operators.- On 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 Lp(Rn).- 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 H�lder 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 S1.- On pseudo-differential operators on the group Su(2).- Sampling and pseudo-differential operators.

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