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

Authors

  • Ewa Damek
  • Jacek Dziubanski
  • Philippe Jaming
  • Salvador Pérez-Esteva

DOI:

https://doi.org/10.7146/math.scand.a-15105

Abstract

In this paper, we characterize the class of distributions on a homogeneous Lie group $\mathfrak N$ that can be extended via Poisson integration to a solvable one-dimensional extension $\mathfrak S$ of $\mathfrak N$. To do so, we introduce the $\mathcal S'$-convolution on $\mathfrak N$ and show that the set of distributions that are $\mathcal S'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions. Moreover, we show that the $\mathcal S'$-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.

Downloads

Published

2009-09-01

How to Cite

Damek, E., Dziubanski, J., Jaming, P., & Pérez-Esteva, S. (2009). Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups. MATHEMATICA SCANDINAVICA, 105(1), 31–65. https://doi.org/10.7146/math.scand.a-15105

Issue

Section

Articles