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Abstract

Let $L$ be a strictly elliptic second order operator on a bounded domain ${\mit \Omega } \subset {{\mathbb R}}^{n}$. Let $u$ be a solution to $Lu=\mathop {\rm div}\vec {f}$ in ${\mit \Omega } $, $u=0$ on $\partial {\mit \Omega } $. Sufficient conditions on two measures, $\mu $ and $\nu $ defined on ${\mit \Omega } $, are established which imply that the $L^{q}({\mit \Omega } ,d\mu )$ norm of $| \nabla u| $ is dominated by the $L^{p}({\mit \Omega } ,dv)$ norms of $\mathop {\rm div}\vec {f}$ and $| \vec {f}| $. If we replace $| \nabla u| $ by a local Hölder norm of $u$, the conditions on $\mu $ and $\nu $ can be significantly weaker.