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

Research paper by **S. E. Boutiah, A. Rhandi, C. Tacelli**

Indexed on: **24 Nov '17**Published on: **24 Nov '17**Published in: **arXiv - Mathematics - Analysis of PDEs**

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#### 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.