A class of weighted Hardy type inequalities in R^N

In the paper we prove the weighted Hardy type inequality \begin{equation} \int_{{\mathbb R}^N}V\varphi^2 \mu(x)dx\le \int_{\R^N}|\nabla \varphi|^2\mu(x)dx +K\int_{\R^N}\varphi^2\mu(x)dx, \end{equation} for functions $\varphi$ in a weighted Sobolev space $H^1_\mu$, for a wider class of potentials $V$ than inverse square potentials and for weight functions $\mu$ of a quite general type. The case $\mu=1$ is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators \begin{equation*} Lu=\Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u \end{equation*} perturbed by singular potentials