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

Ewa Damek (1) , Jacek Dziubanski (1) , Philippe Jaming (2) , Salvador Pérez-Esteva (3)

1 Instytut Matematyczny
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 UNAM-CUERNAVACA - Instituto de Matematicas, Unidad Cuernavaca

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.

Mots clés: homogeneous Lie groups; distribution; $ß'$-convolution; Poisson integrals

Domaines: Analyse classique [math.CA]

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⟩