A class of weighted Hardy inequalities and applications to evolution problems

Abstract

We state the following weighted Hardy inequality: $$\begin{aligned} c_{o, \mu }\int _{{\mathbb {R}}^N}\frac{\varphi ^2 }{|x|^2}\, \mathrm{d}\mu \le \int _{{\mathbb {R}}^N} |\nabla \varphi |^2 \, \mathrm{d}\mu + K \int _{\mathbb {R}^N}\varphi ^2 \, \mathrm{d}\mu \quad \forall \, \varphi \in H_\mu ^1, \end{aligned}$$in the context of the study of the Kolmogorov operators: $$\begin{aligned} Lu=\Delta u+\frac{\nabla \mu }{\mu }\cdot \nabla u, \end{aligned}$$perturbed by inverse square potentials and of the related evolution problems. The function $$\mu $$ in the drift term is a probability density on $$\mathbb {R}^N$$. We prove the optimality of the constant $$c_{o, \mu }$$ and state existence and nonexistence results following the Cabre–Martel’s approach (Cabre and Martel in C R Acad Sci Paris 329 (11): 973–978, 1999) extended to Kolmogorov operators.