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

In this work, we present new results on solvability of the equation $A^{\ast }(D)f=\mu$ 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 $1\le p<\mathrm{\infty }$. We also obtain sufficient conditions in the limiting case $p=\mathrm{\infty }$ using {new $L^1$ estimates on measures for elliptic and canceling operators.