Improving $L^2$ estimates to Harnack inequalities

TL;DR

This paper develops advanced $L^2$ estimates for elliptic operators with singular potentials, leading to optimal Sobolev inequalities, ultracontractive semigroup generation, and boundary Harnack inequalities with sharp heat kernel bounds.

Contribution

It introduces new $L^2$ estimates for operators with singular potentials, enabling boundary Harnack inequalities and precise heat kernel estimates.

Findings

Established optimal Sobolev inequalities for ${ m extbf{ extit L}}$

Proved the generation of an intrinsic ultracontractive semigroup

Derived a parabolic Harnack inequality up to the boundary and sharp heat kernel estimates

Abstract

We consider operators of the form ${\mathcal L}=-L-V$, where $L$ is an elliptic operator and $V$ is a singular potential, defined on a smooth bounded domain $\Omega\subset \R^n$ with Dirichlet boundary conditions. We allow the boundary of $\Omega$ to be made of various pieces of different codimension. We assume that ${\mathcal L}$ has a generalized first eigenfunction of which we know two sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator ${\mathcal L}$, we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack inequality up to the boundary as well as sharp heat kernel estimates.