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