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

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

2006-12-13

arxiv.org/abs/math/0612368v1

Mathematica Scandinavica 105 (2009) 31-65

Mathematics

Classical Analysis and ODEs

Scientific paper

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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