  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.
