Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability. (MN-37) (Mathematical Notes, 37) - Softcover

6. Lectures on Vector Bundles Over Riemann Surfaces, by ROBERT C. GUNNING 7. Notes on Cobordism Theory, by ROBERT E. STONG 8. Stationary Stochastic Processes, by TAKEYUKI HIDA 9. Topics in Dynamics-I: Flows, by EDWARD NELSON 11. Boundary Behavior of Holomorphic Functions of Several Complex Variables, by E. M. STEIN 12. Lectures on Riemann Surfaces: Jacobi Varieties, by R. C. GUNNING 13. Topics in Algebraic and Analytic Geometry, by PHILLIP GRIFFITHS and JOHN ADAMS 14. Lectures on Complex Analytic Varieties: Finite Analytic Mappings, by R. C. GUNNING 15. Fourier Analysis on Local Fields, by M. H. TAIBLESON 16. Global Variational Analysis: Weierstrass Integrals on a Riemannian Manifold, by MARSTON MORSE 17. Introduction to Partial Differential Equations, by GERALD B. FOLLAND 18. Introduction to Ergodic Theory, by YA. G. SINAI, translated by V. SCHEFFER 19. Estimates for the ∂-Neumann Problem, by P. C. GREINER and E. M. STEIN 20. The Motion of a Surface by Its Mean Curvature, by KENNETH A. BRAKKE 21. Notes on Crystalline Cohomology, by PIERRE BERTHELOT and ARTHUR OGUS 22. On Uniformization of Complex Manifolds: The Role of Connections, by R. C. GUNNING 23. Introduction to Harmonic Analysis on Reductive P-adic Groups, by ALLAN J. SILBERGER 24. Lectures on Pseudo-Differential Operators: Regularity Theorems and Applications to Non-Elliptic Problems, by ALEXANDER NAGEL and E. M. STEIN 25. Non-Abelian Minimal Closed Ideals of Transitive Lie Algebras, by J. F. CONN 26. Exact Sequences in the Algebraic Theory of Surgery, by ANDREW RANICKI 27. Existence and Regularity of Minimal Surfaces on Riemannian Manifolds, by JON T. PITTS 28. Hardy Spaces on Homogenous Groups, by G. B. FOLLAND and E. M. STEIN 29. Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schödinger Operators, by SHMUEL AGMON 30. Formal Knot Theory, by Louis H. KAUFFMAN 31. Cohomology of Quotients in Symplectic and Algebraic Geometry, by FRANCES CLARE KIRWAN 32. Predicative Arithmetic, by EDWARD NELSON 33. Lectures on Counterexamples in Several Complex Variables, by JOHN ERIK FORNÆSS and BERIT STENSØNES 34. Lie Groups, Lie Algebras, and Cohomology, by ANTHONY W. KNAPP 35. Plateau's Problem and the Calculus of Variations, by MICHAEL STRUWE 36. Casson's Invariant for Oriented Homology 3-Spheres: An Exposition, by SELMAN AKBULUT and JOHN D. MCCARTHY 37. Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability, by DARYL GELLER

Many of the operators one meets in several complex variables, such as the famous Lewy operator, are not locally solvable. Nevertheless, such an operator L can be thoroughly studied if one can find a suitable relative parametrix--an operator K such that LK is essentially the orthogonal projection onto the range of L. The analysis is by far most decisive if one is able to work in the real analytic, as opposed to the smooth, setting. With this motivation, the author develops an analytic calculus for the Heisenberg group. Features include: simple, explicit formulae for products and adjoints; simple representation-theoretic conditions, analogous to ellipticity, for finding parametrices in the calculus; invariance under analytic contact transformations; regularity with respect to non-isotropic Sobolev and Lipschitz spaces; and preservation of local analyticity. The calculus is suitable for doing analysis on real analytic strictly pseudoconvex CR manifolds. In this context, the main new application is a proof that the Szego projection preserves local analyticity, even in the three-dimensional setting. Relative analytic parametrices are also constructed for the adjoint of the tangential Cauchy-Riemann operator.

Originally published in 1990.

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