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 $A:=(1+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })\mathrm{\Delta }+b\mathrm{\mid }x{\mathrm{\mid }}^{\alpha -1}\frac{x}{\mathrm{\mid }x\mathrm{\mid }}\cdot \mathrm{\nabla }-\mathrm{\mid }x{\mathrm{\mid }}^{\beta }$ satisfies $$k(t,x,y)\le c_1{e}^{{\lambda }_{0}t+c_2{t}^{-\gamma }}{\left(\frac{1+\mathrm{\mid }y{\mathrm{\mid }}^{\alpha }}{1+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}\right)}^{\frac{b}{2\alpha }}\frac{(\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }{)}^{-\frac{N-1}{2}-\frac{1}{4}(\beta -\alpha )}}{1+\mathrm{\mid }y{\mathrm{\mid }}^{\alpha }}{e}^{-\frac{\sqrt{2}}{\beta -\alpha +2}(\mathrm{\mid }x{\mathrm{\mid }}^{\frac{\beta -\alpha +2}{2}}+\mathrm{\mid }y{\mathrm{\mid }}^{\frac{\beta -\alpha +2}{2}})}$$ for $t>0,\mathrm{\mid }x\mathrm{\mid },\mathrm{\mid }y\mathrm{\mid }\ge 1$, 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.