ANALYSIS OF A DISTINGUISHED LAPLACIAN ON SOLVABLE LIE-GROUPS

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 L(p) spaces, 1 less-than-or-equal-to p less-than-or-equal-to infinity. 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).