Improved Hardy-Rellich inequalities

TL;DR

This paper explores how angular perturbations, especially magnetic fields, can improve Hardy-Rellich inequalities for perturbed Laplacians, revealing new bounds and properties of the best constants involved.

Contribution

It demonstrates that angular perturbations can enhance Hardy-Rellich inequalities and provides the first investigation of these inequalities under magnetic perturbations.

Findings

Angular perturbations improve Hardy-Rellich inequalities.

Optimal inequalities are established with non-attainable best constants.

Magnetic fields exemplify the diamagnetic phenomenon in this context.

Abstract

We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in \cite{LW1999} for the Hardy inequality, later by Evans and Lewis in \cite{EL2005} for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.