Kato's inequality when $\Delta u$ is a measure

Abstract

We extend the classical Kato's inequality in order to allow functions $u\in L_{\mathrm{loc}}^1$ such that $\Delta u$ is a Radon measure. This inequality has been applied by Brezis, Marcus, and Ponce to study the existence of solutions of the nonlinear equation $-\Delta u+g(u)=\mu$, where $\mu$ is a measure and $g:\mathbb{R}\to \mathbb{R}$ is an increasing continuous function.