New maximum principles for linear elliptic equations

Abstract

We prove extensions of the estimates of Aleksandrov and Bakel${}^{\prime }$man for linear elliptic operators in Euclidean space ${\mathbb{R}}^n$ to inhomogeneous terms in $L^q$ spaces for $q<n$. Our estimates depend on restrictions on the ellipticity of the operators determined by certain subcones of the positive cone. We also consider some applications to local pointwise and $L^2$ estimates.