Solvability of elliptic homogeneous linear equations with measure data in weighted Lebesgue spaces

Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $\mu$ be a vector-valued Borel measure and $w$ be a positive locally integrable function on ${\mathbb{R}}^N$. In this work, we present sufficient conditions on $\mu$ and $w$ for the existence of solutions in the weighted Lebesgue spaces $L_w^p$ for the equation $A^{\ast }(D)f=\mu$, for $1\le p<\mathrm{\infty }$. Those conditions are related to a certain control of the Riesz potential of the measure $\mu$. We also present sufficient conditions for the solvability when $p=\mathrm{\infty }$ adding a canceling condition on the operator. Our method is based on a new weighted $L^1$ Stein-Weiss type inequality on measures for a special class of vector fields.