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 L1, which contains the algebra generated by the heat kernel, and we prove that the spectrum of the Laplacian is the same on all Lp spaces, 1 ≤ p < ∞. When G is complex, we derive a formula that enables us to compute the Lp norm of these kernels in terms of a weighted Lp 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).
Analysis of a distinguished Laplacean on solvable Lie groups
GIULINI, SAVERIO;MAUCERI, GIANCARLO
1993-01-01
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 L1, which contains the algebra generated by the heat kernel, and we prove that the spectrum of the Laplacian is the same on all Lp spaces, 1 ≤ p < ∞. When G is complex, we derive a formula that enables us to compute the Lp norm of these kernels in terms of a weighted Lp 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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