Robust Near-Diagonal Green Function Estimates

Abstract

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