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This book provides a readable description of a technique, developed years ago but still current, for proving that solutions to certain (non-elliptic) partial differential equations only have real analytic solutions when the data are real analytic (locally).

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This book provides a very readable description of a technique, developed by the author years ago but as current as ever, for proving that solutions to certain (non-elliptic) partial differential equations only have real analytic solutions when the data are real analytic (locally). The technique is completely elementary but relies on a construction, a kind of a non-commutative power series, to localize the analysis of high powers of derivatives in the so-called bad direction. It is hoped that this work will permit a far greater audience of researchers to come to a deep understanding of this technique and its power and flexibility.

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1. What this book is and is not.- 2. Brief Introduction.- 3.Overview of Proofs.- 4. Full Proof for the Heisenberg Group.- 5. Coefficients.- 6. Pseudo-differential Problems.- 7. Sums of Squares and Real Vector Fields.- 8. \bar{\partial}-Neumann and the Boundary Laplacian.- 9. Symmetric Degeneracies.- 10. Details of the Previous Chapter. -11. Non-symplectic Strategem ahe.- 12. Operators of Kohn Type Which Lose Derivatives.- 13. Non-linear Problems.- 14. Treves´ Approach.- 15. Appendix.- Bibliography.

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From the reviews:

"The present book deals with the analytic and Gevrey local hypoellipticity of certain nonelliptic partial differential operators. ... this nice book is mostly addressed to Ph.D. students and researchers in harmonic analysis and partial differential equations, the reader being supposed to be familiar with the basic facts of pseudodifferential calculus and several complex variables. It represents the first presentation, in book form, of the challenging and still open problem of analytic and Gevrey hypoellipticity of sum-of-squares operators." (Fabio Nicola, Mathematical Reviews, Issue 2012 h)

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