Improved Hardy inequalities with a class of weights

TL;DR

This paper establishes conditions on potentials and weights to derive improved Hardy inequalities within weighted Sobolev spaces, utilizing a generalized vector field approach for both global and local cases.

Contribution

It introduces new conditions on potentials and weights to obtain improved Hardy inequalities, expanding the class of weights and potentials for which these inequalities hold.

Findings

Derived improved Hardy inequalities with general weight functions.

Provided local versions of the inequalities.

Used a generalized vector field method for proofs.

Abstract

\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}