Optimal Power-Weighted Birman--Hardy--Rellich-type Inequalities on Finite Intervals and Annuli
Fritz Gesztesy, Michael M. H. Pang
TL;DR
This work establishes sharp power-weighted Hardy inequalities on finite intervals and on multidimensional annuli, revealing that integral- and differential-form constants are distinct and providing explicit extremals. It develops an iteration scheme to generate Birman–Hardy–Rellich-type sequences of higher-order inequalities, both in one dimension and for annuli, and connects these to spherical-harmonic decompositions in higher dimensions. The results are shown to recover classical Rellich-type inequalities in the appropriate limits as annuli expand to ${\mathbb R}^n\setminus\{0\}$. The findings offer a unified framework linking 1D and nD Hardy–Rellich-type inequalities with explicit optimal constants and extremals, advancing spectral-analytic applications on bounded and unbounded domains.
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 $\mathbb{R}^n\backslash\{0\}$.