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.

Synopsis:

This volume consists of seventeen peer-reviewed papers related to lectures on pseudodifferential operators presented at the meeting of the ISAAC Group in Pseudo-Differential Operators (IGPDO) held at the Middle East Technical University in Ankara, Turkey on August 13-19, 2007, and invited papers by experts in the field.Included in this volume are such topics as analysis and partial differential equations related to the Heisenberg group; global analysis and pseudo-differential analysis on non-compact manifolds and manifolds with singularities; Fourier integral operators and Colombeau algebras with applications to partial differential equations; exotic pseudo-differential operators and regularity results on quasi-elliptic operators and hypoelliptic operators; Stockwell transforms in time-frequency analysis; and, pseudo-differential operators on Lie groups with related results to sampling. This volume is a useful complement to the volumes "Advances in Pseudo-Differential Operators", "Pseudo-Differential Operators and Related Topics" and "Modern Trends in Pseudo-Differential Operators" published in the same series in, respectively, 2004, 2006 and 2007.

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