ANALYSIS ON SEMIDIRECT PRODUCTS AND HARMONIC MAPS

We study the analysis of a probability density K on a Lie group G, where G is a semidirect product of a compact group M with a nilpotent group N. To approximate analysis on G with analysis on N, it is natural to consider certain maps ("realizations") of G onto N. In this paper, we prove the existence of a realization of G in N which is K-harmonic (modulo the commutator subgroup of N). By utilizing this result and extending some ideas of Alexopoulos, we can prove the boundedness in L^p spaces of some new Riesz transforms associated with K, and obtain new regularity estimates for the convolution powers of K.