Resumen de Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups

E. Damek, Jacek Dziubanski , Jaming Philippe, Salvador Pérez-Esteva

• 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 L1-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-L1 estimates.