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Analysis of a Distinguished Laplacian on Solvable Lie Groups
Author(s) -
Giulini Saverio,
Mauceri Giancarlo
Publication year - 1993
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19931630115
Subject(s) - mathematics , lie group , laplace operator , pure mathematics , subalgebra , lie algebra , heat kernel , simple lie group , euclidean space , adjoint representation , symmetric space , (g,k) module , combinatorics , algebra over a field , mathematical analysis , adjoint representation of a lie algebra , lie conformal algebra
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).