A class of weighted Hardy inequalities and applications to evolution problems

Abstract

\begin{abstract} We state the following weighted Hardy inequality \begin{equation*} c_{o, \mu}\int_{{\R}^N}\frac{\varphi^2 }{|x|^2}\, d\mu\le \int_{{\R}^N} |\nabla\varphi|^2 \, d\mu + K \int_{\R^N}\varphi^2 \, d\mu \quad \forall\, \varphi \in H_\mu^1 %\qquad c\le c_\mu, \end{equation*} in the context of the study of the Kolmogorov operators \begin{equation*} Lu=\Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u \end{equation*} perturbed by inverse square potentials and of the related evolution problems. The function $\mu$ in the drift term is a probability density on $\R^N$. We prove the optimality of the constant $c_{o, \mu}$ and state existence and nonexistence results following the Cabr\'e-Martel's approach \cite{CabreMartel} extended to Kolmogorov operators. \end{abstract}