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

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

hal-00120271, version 1

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

Mathematica Scandinavica 105 (2009) 31-65

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.

1: Instytut Matematyczny
Uniwersytet Wroclawski
2: Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO)
Université d'Orléans – CNRS : UMR7349
3: Instituto de Matematicas, Unidad Cuernavaca (UNAM-CUERNAVACA)
UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO

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http://hal.archives-ouvertes.fr/hal-00120271
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From: Philippe Jaming <>
Submitted on: Wednesday, 13 December 2006 18:17:15
Updated on: Wednesday, 2 September 2009 10:38:02

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%% hal-00120271, version 1
%% http://hal.archives-ouvertes.fr/hal-00120271
@article{damek:hal-00120271,
    hal_id = {hal-00120271},
    url = {http://hal.archives-ouvertes.fr/hal-00120271},
    title = {{Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups}},
    author = {Damek, Ewa and Dziubanski, Jacek and Jaming, Philippe and P{\'e}rez-Esteva, Salvador},
    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.}},
    keywords = {homogeneous Lie groups, distribution, ${\ss}'$-convolution, Poisson integrals},
    language = {English},
    affiliation = {Instytut Matematyczny , Math{\'e}matiques - Analyse, Probabilit{\'e}s, Mod{\'e}lisation - Orl{\'e}ans - MAPMO , Instituto de Matematicas, Unidad Cuernavaca - UNAM-CUERNAVACA},
    pages = {31-65},
    journal = {Mathematica Scandinavica},
    volume = {105},
    note = {Research partially financed by: E.D., J.D., Ph.J.: European Commission Harmonic Analysis and Related Problems 2002-2006 IHP Network (Contract Number: HPRN-CT-2001-00273 - HARP).E.D., J.D. : European Commission Marie Curie Host Fellowship for the Transfer of Knowledge "Harmonic Analysis, Nonlinear Analysis and Probability" MTKD-CT-2004-013389. J.D : Polish founds for science 2005--2008 (research project 1P03A03029). S.P.-E. : Conacyt-DAIC U48633-F },
    audience = {international },
    year = {2009},
    pdf = {http://hal.archives-ouvertes.fr/hal-00120271/PDF/homo61006.pdf},
}