Weighted Hardy's inequalities and Kolmogorov-type operators

We give general conditions to state the weighted Hardy inequality $$c{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{\mathrm{\mid }x{\mathrm{\mid }}^2}d\mu \le {\int }_{{\mathbb{R}}^N}\mathrm{\mid }\mathrm{\nabla }\phi {\mathrm{\mid }}^{2}d\mu +C{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu ,\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^N),c\le c_{0,\mu },$$ with respect to a probability measure $d\mu$. Moreover, the optimality of the constant $c_{0,\mu }$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential $$Lu+Vu=(\mathrm{\Delta }u+\frac{\mathrm{\nabla }\mu }{\mu }\cdot \mathrm{\nabla }u)+\frac{c}{\mathrm{\mid }x{\mathrm{\mid }}^2}u$$ for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $d\mu$ allow the drift term to be of type $\frac{\mathrm{\nabla }\mu }{\mu }=-\mathrm{\mid }x{\mathrm{\mid }}^{m-2}x$ with $m>0$.