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Analysis of a Distinguished Laplacian on Solvable Lie Groups

✍ Scribed by Saverio Giulini; Giancarlo Mauceri

Publisher
John Wiley and Sons
Year
1993
Tongue
English
Weight
575 KB
Volume
163
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

We study a class of kernels associated to functions of a distinguished Laplacian on the solvable group AN occurring in the Iwasawa decomposition G = ANK of a noncompact semisimple Lie group G. We determine the maximal ideal space of a commutative subalgebra of L^1^, which contains the algebra generated by the heat kernel, and we prove that the spectrum of the Laplacian is the same on all L^p^ spaces, 1 ≀ p < ∞. When G is complex, we derive a formula that enables us to compute the L^p^ norm of these kernels in terms of a weighted L^p^ norm of the corresponding kernels for the Euclidean Laplacian on the tangent space. We also prove that, when G is either rank one or complex, certain Hardy‐Littlewood maximal operators, which are naturally associated with these kernels, are weak type (1, 1).

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