Improved Hardy inequalities with a class of weights

\begin{abstract} In the paper we state conditions on potentials $V$ to get the improved Hardy inequality with weight \begin{equation*} \begin{split} c_{N,\mu}\int_{\R^N}\frac{\varphi^2}{|x|^2}\mu(x)dx&+ \int_{\R^N}V\,\varphi^2\mu(x)dx \\&\le \int_{\R^N}|\nabla \varphi|^2\mu(x)dx +K_1 \int_{\R^N} \varphi^2\mu(x)dx, \end{split} \end{equation*} for functions $\varphi$ in a weighted Sobolev space and for weight functions $\mu$ of a quite general type. Some local improved Hardy inequalities are also given. To get the results we use a generalized vector field method. \end{abstract}