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

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 $$k(t,x,y)\le c_1{e}^{{\lambda }_{0}t+c_2{t}^{-\gamma }}{\left(\frac{1+|y{|}^{\alpha }}{1+|x{|}^{\alpha }}\right)}^{\frac{b}{2\alpha }}\frac{(|x||y|{)}^{-\frac{N-1}{2}-\frac{1}{4}(\beta -\alpha )}}{1+|y{|}^{\alpha }}{e}^{-\frac{\sqrt{2}}{\beta -\alpha +2}\left(|x{|}^{\frac{\beta -\alpha +2}{2}}+|y{|}^{\frac{\beta -\alpha +2}{2}}\right)}$$ 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.

15 pages