Higher order Hardy-Rellich identities

TL;DR

This paper develops Hardy-Rellich identities for polyharmonic and radial Laplacian operators with weights in Euclidean and Riemannian spaces, leading to new inequalities and extensions of classical Rellich results.

Contribution

It introduces a unified approach to derive Hardy-Rellich identities with weights on Euclidean and Riemannian manifolds, extending and improving existing inequalities.

Findings

Extended Rellich inequalities to weighted cases.

Derived new Rellich inequalities involving logarithmic weights.

Established Rellich identities on hyperbolic space.

Abstract

In this paper, we show Hardy-Rellich identities for polyharmonic operators $\Delta^m$ and radial Laplacian $\Delta_r^m$ in $\mathbb{R}^n$ with Hardy-H\'enon weight $|x|^\alpha$ for all $m, n\in \mathbb{N}, \alpha\in \mathbb{R}$. Moreover, the iterative method is applied to give Hardy-Rellich equalities with general weights on Riemannian manifolds. These identities provide naturally an alternative approach to obtain and improve Hardy-Rellich type inequalities. As example of application, we extend several Rellich inequalities of Tertikas-Zographopoulos (Adv. Math. 2007) to the weighted case; using equality with weights involving logarithmic, we show another new weighted Rellich estimate between integrals of $\Delta u$ and $|\nabla u|$; we establish also a Rellich identity involving the Laplace-Beltrami operator $\Delta_\mathbb{H}$ and the radial Laplacian $\Delta_{\rho, \mathbb{H}}$ of the hyperbolic space $\mathbb{H}^n$, which yields in particular brand-new Rellich inequalities for $\|\Delta_\mathbb{H} u\|$ in $\mathbb{H}^3$ and $\mathbb{H}^4$.