In this paper, we characterize the class of distributions on a homogeneous Lie group $\mathfrak N$ that can be extended via Poisson integration to a solvable one-dimensional extension $\mathfrak S$ of $\mathfrak N$. To do so, we introduce the $\mathcal S'$-convolution on $\mathfrak N$ and show that the set of distributions that are $\mathcal S'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions. Moreover, we show that the $\mathcal S'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behavior. Finally, we show that such distributions satisfy some global weak-$L^1$ estimates.
