Robust near-diagonal Green function estimates

TL;DR

This paper establishes sharp, robust near-diagonal bounds for the Green function of fractional order nonlocal operators, valid as the order approaches 2, without relying on heat kernel estimates, thus broadening applicability.

Contribution

It provides the first near-diagonal Green function bounds that are stable as the fractional order approaches 2, without using heat kernel estimates.

Findings

Green function bounds are robust as $oldsymbol{ ext{alpha} o 2-}$

Bounds are obtained without heat kernel estimates

Applicable to cases with isotropic Green function bounds but not heat kernel bounds

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.