Preface

Summary

The intent of these notes is to provide a detailed and comprehensive treatment of harmonic and subharmonic function theory on hyperbolic space in Rⁿ. Although our primary emphasis will be in the setting of the unit ball with hyperbolic metric ds given by

we will also consider the analogue of many of the results in the hyperbolic half-space ℍ. Undoubtedly some of the results are known, either in the setting of rank one noncompact symmetric spaces (e.g. [38]), or more generally, in Riemannian spaces (e.g. [13]). An excellent introduction to harmonic function theory on noncompact symmetric spaces can be found in the survey article [47] by A. Koranyi. The 1973 paper by K. Minemura [57] provides an introduction to harmonic function theory on real hyperbolic space considered as a rank one noncompact symmetric space. Other contributions to the subject area in this setting will be indicated in the text.

With the goal of making these notes accessible to a broad audience, our approach does not require any knowledge of Lie groups and only a limited knowledge of differential geometry. The development of the theory is analogous to the approach taken by W. Rudin [72] and by the author [84] in their development of Möbius invariant harmonic function theory on the hermitian ball in ℂⁿ. Although our primary emphasis is on harmonic function theory on the ball, we do include many relevant results for the hyperbolic upper half-space ℍ, both in the text and in the exercises. With only one or two exceptions, the notes are self-contained with the only prerequisites being a standard beginning graduate course in real analysis.

In Chapter 1 we provide a brief review of Möbius transformation in Rn. This is followed in Chapter 2 by a characterization of the group of Möbius self-maps of the unit ball in Rⁿ. As in [72] we define a family of Möbius transformations of satisfying, and for all. Furthermore, for every, it is proved that there exists and an orthogonal transformation A such that.