  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.
