Weighted Hardy's inequalities and Kolmogorov-type operators

Abstract

We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{φ^2} {|x|^2}dμ\leq\int_{\mathbb{R}^N}|\nabla φ|^2 dμ+C\int_{\mathbb{R}^N} φ^2dμ,\quad φ\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,μ}, \] with respect to a probability measure $d\mu$. Moreover, the optimality of the constant $c_{0,\mu }$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(Δu+\frac{\nabla μ}μ\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $d\mu$ allow the drift term to be of type $\frac{\nabla \mu }{\mu }=-|x{|}^{m-2}x$ with $m>0$.