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

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