Robust Near-Diagonal Green Function Estimates

Abstract

Abstract We prove sharp near-diagonal pointwise bounds for the Green function $G_\Omega (x,y)$ for nonlocal operators of fractional order $\alpha \in (0,2)$. The novelty of our results is two-fold: the estimates are robust as $\alpha \to 2-$ and we prove the bounds without making use of the Dirichlet heat kernel $p_\Omega (t;x,y)$. In this way, we can cover cases, in which the Green function satisfies isotropic bounds but the heat kernel does not.