Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups - Damek, Ewa; Dziubanski, Jacek; Jaming, Philippe; Pérez-Esteva, Salvador (Mathematica Scandinavica, 2009)
In this paper, we characterize the class of distributions on a homogeneous Lie group N that can be extended via Poisson integration to a solvable one-dimensional extension S of N. To do so, we introduce the S’-convolution on N and show that the set of distributions that are S’-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of L^1-functions. Moreover, we show that the S’-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behavior. Finally, we show that such distributions satisfy some global weak-L^1 estimates.