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

Abstract

We extend the classical Kato's inequality in order to allow functions $u \in L^1_\mathrm{loc}$ 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.