Transference Methods in Analysis

Description

These ten lectures were presented by Guido Weiss at the University of Nebraska during the week of May 31 to June 4, 1976. They were a part of the Regional Conference Program sponsored by the Conference Board of the Mathematical Sciences and funded by the National Science Foundation. The topic chosen, ``the transference method'', involves a very simple idea that can be applied to several different branches of analysis. The authors have chosen familiar special cases in order to illustrate the use of transference: much that involves general locally compact abelian groups can be understood by examining the real line; the group of rotations can be used to explain what can be done with compact groups; $SL(2,\mathbf C)$ plays the same role vis-a-vis noncompact semisimple Lie groups. The main theme of these lectures is the interplay between properties of convolution operators on classical groups (such as the reals, integers, the torus) and operators associated with more general measure spaces. The basic idea behind this interplay is the notion of transferred operator; these are operators ``obtained'' from convolutions by replacing the translation by some action of the group (or, in some cases, a semigroup) and give rise, among other things, to an interaction between ergodic theory and harmonic analysis. There are illustrations of these ideas. A graduate student in analysis would be able to read most of this book. The work is partly expository, but is mostly ``self-contained''. Cover Title Copyright Contents Introduction Preface 1. Some Classical Examples of Transference 2. The General Transference Result 3. Multipliers Defined by the Action of Locally Compact Abelian Groups 4. Transference from the Integers and the Maximal Ergodic Theorem 5. Ergodic Flows and the Theory of H[sup(p)] Spaces 6. Integral Transforms with Zonal Kernels 7. Integral Transforms with Zonal Kernels (Continued) 8. Kernels Having Certain Invariance Properties with Respect to Representations of G. 9. The Group SL(2, C) 10. Some Aspects of Harmonic Analysis on SL(2, C) Bibliography Back Cover