TY - JOUR AU - Damek, Ewa AU - Dziubanski, Jacek AU - Jaming, Philippe AU - PĂ©rez-Esteva, Salvador PY - 2009/09/01 Y2 - 2024/03/29 TI - Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups JF - MATHEMATICA SCANDINAVICA JA - Math. Scand. VL - 105 IS - 1 SE - Articles DO - 10.7146/math.scand.a-15105 UR - https://www.mscand.dk/article/view/15105 SP - 31-65 AB - 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. ER -