Optimal Power-Weighted Birman--Hardy--Rellich-type Inequalities on Finite Intervals and Annuli

TL;DR

This paper establishes optimal power-weighted Hardy and Rellich inequalities on finite intervals and annuli, deriving both integral and differential forms, and extends these results to higher-order and multi-dimensional cases.

Contribution

It introduces new optimal power-weighted Hardy-Rellich inequalities in integral and differential forms on finite intervals and annuli, including higher-order and multi-dimensional extensions.

Findings

Derived optimal power-weighted Hardy inequalities on finite intervals.

Proved analogous inequalities in differential form with different optimal constants.

Extended inequalities to higher-order and multi-dimensional cases, including annuli.

Abstract

We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power-weighted Birman-Hardy-Rellich-type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one-dimensional Hardy-type result in differential form to derive an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman-Hardy-Rellich-type sequence of power-weighted higher-order Hardy-type inequalities for annuli. In the limit as the annulus approaches $\mathbb{R}^n\backslash\{0\}$, we recover well-known prior results on Rellich-type inequalities on $\mathbb{R}^n\backslash\{0\}$.