The Heisenberg Group and Its Relatives in the Work of Elias M. Stein
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The Heisenberg Group and Its Relatives in the Work of Elias M. Stein Gerald B. Folland1 Received: 17 October 2019 © Mathematica Josephina, Inc. 2019
Abstract We survey the work of Elias M. Stein in the field of analysis on the Heisenberg group and other nilpotent Lie groups, together with its applications to complex analysis in several variables and partial differential equations. Keywords Harmonic analysis · Heisenberg group · Homogeneous group · Nilpotent Lie group Mathematics Subject Classification Primary 43A80 · Secondary 32V20 · 35B65 · 42B20 · 42B37
1 Introduction A substantial part of Elias M. Stein’s research, beginning around 1970 and continuing through the rest of his life, had to do with analysis on the Heisenberg group and more general noncommutative nilpotent Lie groups, as well as analysis on other manifolds for which such groups provide model cases and analytic tools. The purpose of this article is to offer a brief survey of this work. In order to keep the scope within reasonable bounds, I adopted two general principles. First, the central focus is on the Heisenberg group. The level of detail provided for results in more general settings is, so to speak, a decreasing function of the distance from the Heisenberg group, dropping to zero when the connection with nilpotent groups becomes negligible or nonexistent. Second, although many other people have contributed to the research in this area, I do not discuss any papers of which Stein is not the author or co-author. The only exceptions to this rule are a few citations of papers that provide some essential background material for Stein’s work.
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Gerald B. Folland [email protected] Department of Mathematics, University of Washington, Seattle, WA 98195, USA
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G. B. Folland
Stein and his collaborators frequently announced their results via notes in the Bulletin of the American Mathematical Society or the Proceedings of the National Academy of Sciences before publishing a more complete account elsewhere. In this survey I have generally cited only the fully detailed papers, passing over the research announcements except in a few cases where there is a reason for mentioning them specifically.
2 Background In this section we review some concepts, terminology, and notation that will be needed for the rest of the paper. Let g be a real Lie algebra. Its lower central series is the descending sequence {g(k) } of ideals defined by g(1) = g and g(k) = [g, g(k−1) ] for k > 1. g is nilpotent of step k if g(k) = {0} = g(k+1) . We shall be interested in the following nested family of subclasses within the class of nilpotent Lie algebras: homogeneous ⊃ graded ⊃ stratified ⊃ H-type ⊃ Heisenberg. Here are the definitions: A homogeneous Lie algebra is a nilpotent Lie algebra g equipped with a one-parameter family {δr : r > 0} of automorphisms of the form δr = r A (= exp(A log r )) where A is a diagonalizable linear transformation of g with positive eigenvalues; we shall call the automorphisms in such a family dilations. A nilpotent Lie algebra g equipped with
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