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

In the present paper we prove upper and lower bounds of the heat kernel for the operator{\Delta-\nabla({|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}^{-{|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.