Nonelliptic Partial Differential Equations, 2011 Analytic Hypoellipticity and the Courage to Localize High Powers of T Developments in Mathematics Series, Vol. 22

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.

The exposition is generous and relaxed, allowing the reader to come to terms to the technique at their own pace. The main difficulty, localization, is approached directly from the beginning with simple examples Numerous applications are included There is no similar book There are other techniques for proving analytic hypoellipticity, but each has its own difficulties. While this is elementary but not simple, once the few basic formulas are established the rest is combinatorial in nature, and not conceptually difficult Includes supplementary material: sn.pub/extras