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

Abstract : In this paper, we characterize the class of distributions on an homogeneous Lie group $\fN$ that can be extended via Poisson integration to a solvable one-dimensional extension $\fS$ of $\fN$. To do so, we introducte the $ß'$-convolution on $\fN$ and show that the set of distributions that are $ß'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions. Moreover, we show that the $ß'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behaviour. Finally, we show that such distributions satisfy some global weak-$L^1$ estimates.
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https://hal.archives-ouvertes.fr/hal-00120271
Contributor : Philippe Jaming <>
Submitted on : Wednesday, December 13, 2006 - 6:17:15 PM
Last modification on : Monday, May 6, 2019 - 10:28:08 AM
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Ewa Damek, Jacek Dziubanski, Philippe Jaming, Salvador Pérez-Esteva. Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups. Mathematica Scandinavica, 2009, 105, pp.31-65. ⟨hal-00120271⟩

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