A note on Lebesgue solvability of elliptic homogeneous linear equations with measure data

Abstract

In this work, we present new results on solvability of the equation $A^{*}(D)f=\mu$ for $f \in L^{p}$ and positive measure data $\mu$ associated to an elliptic homogeneous linear differential operator $A(D)$ of order m. Our method is based on $(m,p)-$energy control of $\mu$ giving a natural characterization for solutions when $1\leq p < \infty$. We also obtain sufficient conditions in the limiting case $p=\infty$ using {new $L^{1}$ estimates on measures for elliptic and canceling operators.