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

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