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

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.