Asymptotic average solutions to linear second order semi-elliptic PDEs: a Pizzetti-type Theorem

Abstract

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of {\it asymptotic average solutions} making pointwise solvable every Poisson equation $\mathcal{L}u(x)=-f(x)$ with continuous data $f$, where $\mathcal{L}$ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form.