A new approach to weighted Hardy-Rellich inequalities: improvements, symmetrization principle and symmetry breaking

TL;DR

This paper develops new weighted Hardy-Rellich inequalities using Bessel pairs, explores symmetry and symmetry breaking phenomena, and partially resolves an open question about conditions for inequalities to hold for all functions versus radial functions.

Contribution

It introduces a novel approach using Bessel pairs to establish optimal weighted Hardy-Rellich inequalities and analyzes their symmetry properties, addressing an open problem in the field.

Findings

Sharpened Hardy-Rellich inequalities with weight conditions.

Identified criteria for symmetry and symmetry breaking in inequalities.

Partially answered an open question on the equivalence of inequalities for all and radial functions.

Abstract

We investigate necessary and sufficient conditions on the weights for the Hardy-Rellich inequalities to hold, and propose a new way to use the notion of Bessel pair to establish the optimal Hardy-Rellich type inequalities. Our results sharpened earlier Hardy-Rellich and Rellich type inequalities in the literature. We also study several results about the symmetry and symmetry breaking properties of the Rellich type and Hardy-Rellich type inequalities, and then partially answered an open question raised by Ghoussoub and Moradifam. Namely, we will present conditions on the weights such that the Rellich type and Hardy-Rellich type inequalities hold for all functions if and only if the same inequalities hold for all radial functions.