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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 toan elliptic homogeneous linear differential operator $A(D)$ of order m. Ourmethod is based on $(m,p)-$energy control of $\mu$ giving a naturalcharacterization for solutions when $1\leq p < \infty$. We also obtainsufficient conditions in the limiting case $p=\infty$ using {new $L^{1}$estimates on measures for elliptic and canceling operators.

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