Hardy and Rellich inequalities with Bessel pairs

TL;DR

This paper characterizes pairs of functions that satisfy Hardy and Rellich inequalities on various geometric and operator settings, extending classical inequalities and solving open problems in the field.

Contribution

It provides new characterizations for Hardy and Rellich inequalities involving Bessel pairs on diverse operators and geometries, including the Laplacian, sub-Laplacians, and groups.

Findings

Characterizations for Hardy inequalities with Bessel pairs on various operators.

Extension of results to $L^p$ spaces.

Examples on Laplacian, Baouendi-Grushin, and sub-Laplacians on groups.

Abstract

In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $\Omega \subset \mathbb{R}^n$ in order to have the following Hardy inequality \begin{equation*} \int_{\Omega} W(x) |\nabla u|_A^2 dx \geq \int_{\Omega} |\nabla d|^2_AH(x)|u|^2 dx, \,\,\, u \in C^{1}_0(\Omega), \end{equation*} where $d(x)$ is a suitable quasi-norm (gauge), $|\xi|^2_A = \langle A(x)\xi, \xi \rangle$ for $\xi \in \mathbb{R}^n$ and $A(x)$ is an $n\times n$ symmetric, uniformly positive definite matrix defined on a bounded domain $\Omega \subset \mathbb{R}^n$. We also give its $L^p$ analogue. As a consequence, we present examples for a standard Laplacian on $\mathbb{R}^n$, Baouendi-Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions $(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam \cite{GM_book}.