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Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs

Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs

Chapter:
(p.436) Chapter Seventeen Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs
Source:
Advances in Analysis
Author(s):
Fulvio Ricci, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159416.003.0017

This chapter turns to a special class of Gelfland pairs, here called “nilpotent Gelfland pairs” for simplicity's sake and denoted by (N, K). It considers implications derived from, in principle, whenever we have a finite family of homogeneous, self-adjoint, commuting, left-invariant differential operators on a homogenous nilpotent Lie group N. A natural situation to consider is the one where the given operators are characterized by the property of being invariant under the action of a given compact group K of automorphisms of N. The general strategy of proof here is based on a bootstrapping argument, whose steps are determined by the level of complexity of the pairs involved, where the “complexity” depends on the structure of N and the way K acts on two layers of the Lie algebra.

Keywords:   Gelfland pairs, spectral multipliers, convolution kernels, nilpotent Gelfland pairs, bootstrapping argument, Heisenberg fan

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