A class of weighted Hardy type inequalities in $$\mathbb {R}^N$$

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