Explorations in harmonic analysis. With applications to complex function theory and the Heisenberg group
Harmonic analysis is a venerable part of modern mathematics. Its roots began, perhaps, with the late eighteenth-century discussions of the wave equation. Using the method of separation of variables, it was realized that the equation could be solved with a data function of the form \(\varphi(x) = \sin jx\) for \(j\in\mathbb Z\). It was natural to ask, using the philosophy of superposition, whether the equation could then be solved with data on the interval \([0,\pi]\) consisting of a finite linear combination of the \(\sin jx\). With an affirmative answer to that question, one is led to ask about infinite linear combinations. This was an interesting venue in which physical reasoning interacted with mathematical reasoning. Physical intuition certainly suggests that any continuous function \(\varphi\) can be a data function for the wave equation. So one is led to ask whether any continuous \(\varphi\) can be expressed as an (infinite) superposition of sine functions. Thus the fundamental question of Fourier series was born. Fourier series live on the interval \([0,2\pi)\), or, even more naturally, on the circle group \(\mathbb T\). The Fourier analysis of the real line (i.e., the Fourier transform) was introduced at about the same time as Fourier series. But it was not until the mid-twentieth century that Fourier analysis on \(\mathbb R^N\) came to fruition. Meanwhile, abstract harmonic analysis (i.e., the harmonic analysis of locally compact abelian groups) had developed a life of its own and the theory of Lie group representations provided a natural crucible for noncommutative harmonic analysis. The point here is that the subject of harmonic analysis is a point of view and a collection of tools, and harmonic analysts continually seek new venues in which to ply their wares. In the 1970s, E.M.~Stein and his school introduced the idea of studying classical harmonic analysis -- fractional integrals and singular integrals -- on the Heisenberg group. This turned out to be a powerful device for developing sharp estimates for the integral operators (the Bergman projection, the Szegö projection, etc.) that arise naturally in the several complex variables setting. It also gave sharp subelliptic estimates for the \(\overline{\partial}_b\) problem. It is arguable that modern harmonic analysis (at least linear harmonic analysis) is the study of integral operators. Zygmund and Stein have pioneered this point of view, and Stein's introduction of Heisenberg group analysis validated it and illustrated it in a vital context. Certainly the integral operators of several complex variables are quite different from those that arise in the classical setting of one complex variable. And it is not just the well-worn differences between one-variable analysis and several-variable analysis. It is the nonisotropic nature of the operators of several complex variables. There is also a certain noncommutativity arising from the behavior of certain key vector fields. In appropriate contexts, the structure of the Heisenberg group very naturally models the structure of the canonical operators of several complex variables, and provides the means for obtaining sharp estimates thereof. The purpose of the present book is to explain this rich circle of ideas. The author intends to do so in a context for students. The harmonic analysis of several complex variables builds on copious background material. The author provides the necessary background in classical Fourier series, leading up to the Hilbert transform. That is the entree into singular integrals. Passing to several real-variables, we meet the Riesz fractional integrals and the Calderón-Zygmund singular integrals. The aggregate of all the integral operators encountered thus far provides motivation for considering pseudodifferential operators. The material on Euclidean integral operators that is described up to this point is a self-contained course in its own right. It serves as an introduction to analysis on the Heisenberg group. In this new arena, the author first provides suitable background material on the function theory of several complex variables. This includes analyticity, the Cauchy-Riemann equations, pseudoconvexity, and the Levi problem. All of this is a prelude to the generalized Cayley transform and an analysis of the automorphism group of the Siegel upper half-space. From this venue the Heisenberg group arises in a complex-analytically natural fashion. Just to put the material presented here into context: The author develops the ideas of integral operators up through pseudodifferential operators not because he is going to use pseudodifferential operators as such. Rather, they are the natural climax for this study. These ideas are of particular interest because they put into context and explain the idea of ``order'' of an integral operator (and of an error term). In addition, when the author later makes statements about asymptotic expansions for the Bergman kernel, the pseudodifferential ideas will help students to put the ideas into context. The pseudodifferential operator ideas are also lurking in the background when the author discusses subelliptic estimates for the \(\overline{\partial}\)-problem in the last section of the book. In addition, the author presents some of the ideas from the real-variable theory of Hardy spaces not because he is going to use them in the context of the Heisenberg group. Rather, they are the natural culmination of a study of integral operators in the context of harmonic analysis. Thus Chapters 1--5 of this book, Ontology and history of real analysis, The central idea: The Hilbert transform, Fractional and singular integrals, Several complex variables, constitute a basic introspection to harmonic analysis. Chapters 6--8, Pseudoconvexity and domains of holomorphy, Canonical complex integral operators, Hardy spaces old and new, provide a bridge between harmonic analysis and complex function theory. And Chapters 9--10, Introduction to the Heisenberg group, Analysis on the Heisenberg group, are dessert: They introduce students to some of the cutting-edge ideas about the Siegel upper half-space and the Heisenberg group. Analysis on the Heisenberg group still smacks of Euclidean space. But now the author is working in a step-one nilpotent Lie group. So dilations, translations, convolutions, and many other artifacts of harmonic analysis take a new form. Even such a fundamental idea as fractional integration must be rethought. Certainly one of the profound new ideas is that the critical dimension for integrability is no longer the topological dimension. Now we have a new idea of homogeneous dimension, which is actually one greater than the topological dimension. And there are powerful analytic reasons why this must be so. The author develops the analysis of the Heisenberg group in some detail, so that he may define and calculate bounds on both fractional and singular integrals in this new setting. The applications are provided to the study of the Szegö and Poisson-Szegö integrals. The book concludes with Chapter 11, \textit{A Coda on Domains of Finite Type}, which is the next development in this chain of ideas, and is the focus of current research . The author provides considerable background here for the punch line, which is analysis on the Heisenberg group. The author introduces in this graduate text a few didactic tools to make the reading stimulating and engaging for students: {\parindent5mm \begin{itemize}\item[1.] Each chapter begins with a Prologue, introducing students to the key ideas that will unfold in the text that follows. \item[2.] Each section begins with a Capsule, giving a quick preview of that unit of material. \item[3.] Each key theorem or proposition is preceded by a Prelude, putting the result in context and providing motivation. \item[4.] At key junctures the author includes an Exercise for the reader to encourage the neophyte to pick up a pencil, do some calculations, and get involved with the material. \end{itemize}} These devices will break up the usual dry exposition of a research monograph and make this text more like an invitation to the subject. In total, this is an ambitious introduction to a particular direction in modern harmonic analysis. It presents harmonic analysis in vitro -- in a context in which it is actually applied: complex variables and partial differential equations. This will make the learning experience more meaningful for graduate students who are just beginning to forge a path of research. We expect the readers of this book to be ready to take a number of different directions in exploring the research literature and beginning his or her own investigations.
scientific article; zbMATH DE number 5574611