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

TL;DR

This paper investigates the conditions under which elliptic homogeneous linear equations with measure data are solvable in Lebesgue spaces, providing new characterizations and estimates for solutions across different p-values.

Contribution

It introduces a natural $(m,p)$-energy control framework for measure data and offers new $L^{1}$ estimates for elliptic operators, extending solvability results.

Findings

Characterization of solutions via $(m,p)$-energy control.

Sufficient conditions for solvability at $p= $.

New $L^{1}$ estimates for measures related to elliptic operators.

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.