Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms

TL;DR

This paper establishes upper bounds for the heat kernel of a class of elliptic operators with unbounded coefficients, using log-Sobolev inequalities and semigroup ultracontractivity in weighted spaces.

Contribution

It provides explicit kernel estimates for elliptic operators with unbounded diffusion, drift, and potential terms, extending previous results to more general unbounded coefficient cases.

Findings

Derived explicit heat kernel upper bounds for the operator A

Connected log-Sobolev inequalities with ultracontractivity properties

Applicable to operators with unbounded coefficients in high dimensions

Abstract

In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^\alpha)\Delta +b|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -|x|^\beta$ satisfies $$ k(t,x,y) \leq c_1e^{\lambda_0 t+ c_2t^{-\gamma}}\left(\frac{1+|y|^\alpha}{1+|x|^\alpha}\right)^{\frac{b}{2\alpha}} \frac{(|x||y|)^{-\frac{N-1}{2}-\frac{1}{4}(\beta-\alpha)}}{1+|y|^\alpha} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}\left(|x|^{\frac{\beta-\alpha+2}{2}}+ |y|^{\frac{\beta-\alpha+2}{2}}\right)} $$ for $t>0,\,|x|,\,|y|\ge 1$, where $b\in\mathbb{R}$, $c_1,\,c_2$ are positive constants, $\lambda_0$ is the largest eigenvalue of the operator $A$, and $\gamma=\frac{\beta-\alpha+2}{\beta+\alpha-2}$, in the case where $N>2,\,\alpha>2$ and $\beta>\alpha -2$. The proof is based on the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted space.