Weighted Hardy's inequality in a limiting case and the perturbed Kolmogorov equation

In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant in the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein-Uhlenbeck operator perturbed by a critical singular potential in two dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the limiting case of results which established in \cite{GGR(AA)} \cite{Hauer-Rhandi} in the non-critical cases, and are also considered as extensions of a result in \cite{Cabre-Martel} to the Kolmogorov operator case perturbed by a critical singular potential.