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

In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^alpha)Delta +b|x|^{alpha-1}frac{x}{|x|}cdot nabla -|x|^eta$ 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}{2alpha}} frac{(|x||y|)^{-frac{N-1}{2}-rac{1}{4}(eta-alpha)}}{1+|y|^alpha} e^{-frac{sqrt{2}}{eta-alpha+2}left(|x|^{frac{eta-alpha+2}{2}}+ |y|^{frac{eta-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 $gamma=frac{eta-alpha+2}{eta+alpha-2}$, in the case where $N>2, alpha>2$ and $eta>alpha -2$. The proof is based on the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted space.