Improving $L^2$ estimates to Harnack inequalities

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 $\Omega \subset {\mathbb{R}}^n$ 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.

References in corpus (2)

• Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains
• On a class of weighted anisotropic Sobolev inequalities

Cited by in corpus (2)

• Heat kernel estimates for regional fractional Laplacians with multi-singular critical potentials in $C^{1,\beta }$ open sets
• Green kernel and Martin kernel of Schrödinger operators with singular potential and application to the B.V.P. for linear elliptic equations