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

Abstract

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

27 pages, 4 figures