Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities

We prove upper and lower bounds of the heat kernel for the operator $\mathrm{\Delta }-\mathrm{\nabla }(\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }})\cdot \mathrm{\nabla }$ in \mathbb{R}^{n}\setminus\{0} where $\alpha >0$. We obtain these bounds from an isoperimetric inequality for a measure ${\mathrm{e}}^{-\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}}dx$ on ${\mathbb{R}}^n\setminus \{0\}$. The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.