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Harmonic analysis in phase space: G.B. Folland G.B.

The phrase "harmonic analysis in phase space" is a concise if somewhat inadequate name for the area of analysis on R^n that involves the Heisenberg group, quantization, the Weyl operational calculus, the metaplectic representation, wave packets, and related concepts: it is meant to suggest analysis on the configuration space R^n done by working in the phase space R^n x R^n. The ideas that fall under this rubric have originated in several different fields - Fourier analysis, partial differential equations, mathematical physics, representation theory, and number theory, among others. As a result, although these ideas are individually well known to workers in such fields, their close kinship and the cross-fertilization they can provide have often been insufficiently appreciated. One of the principal objectives of this monograph is to give a coherent account of this material, comprising not just an efficient tour of the major avenues but also an exploration of some picturesque byways.

Here is a brief guide to the main features of the book. Readers should begin by perusing the Prologue and perhaps refreshing their knowledge about Gaussian integrals by glancing at Appendix A.

Chapter 1 is devoted to the description of the representations of the Heisenberg group and various integral transforms and special functions associated to them, with motivation from physics. The material in the first eight sections is the foundation for all that follows, although readers who wish to proceed quickly to pseudodifferential operators can skip Sections 1.5-1.7.

The main point of Chapter 2 is the development of the Weyl calculus of pseudodifferential operators. As a tool for studying differential equations, the Weyl calculus is essentially equivalent to the standard Kohn-Nirenberg calculus - in fact, this equivalence is the principal result of Section 2.2 - but it is somewhat more elegant and more natural from the point of view of harmonic analysis. Its close connection with the Heisenberg group yields some insights which are useful in the proofs of the Calderon-Vaillancourt @,0) estimate and the sharp Garding inequality in Sections 2.5 and 2.6 and in the arguments of Section 3.1. Since my aim is to provide a reasonably accessible introduction rather than to develop a general theory (in contrast to Hormander [70]), I mainly restrict attention to the standard symbol classes S^{m}_{ho,\delta}. Moreover, I assume that the relevant estimates on symbols and their derivatives hold uniformly on all of R^n rather than on compact sets. This simplification makes the theory cleaner without restricting its generality in an essential way, as the study of localized symbols can generally be reduced to the study of global ones by standard tricks involving cutoff functions.

Chapter 3 grew out of my attempt to understand the Cordoba-Fefferman paper [33] on wave packet transforms in the context of the Weyl calculus. What has resulted, in Sections 3.1 and 3.2, is a new approach to their results which shows that, for some purposes, their Gaussian wave packets can be replaced by arbitrary (nonzero) even Schwartz class functions.

Chapter 4 is devoted to the metaplectic representation. It is more comprehensive than most other accounts in the literature, but it is still only an introduction. An exhaustive discussion of the many facets and applications of this beautiful representation and its siblings and children would require a book by itself.

Finally, Chapter 5 is my own retelling of some recent work of R. Howe, whom I wish to thank for permission to include his results. It ties together a number of strands from the previous chapters and provides, in my opinion, a satisfying conclusion to the book.

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