Higher order Hardy-Rellich identities

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-Hénon 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 $\Vert {\Delta }_{\mathbb{H}}u\Vert$ in ${\mathbb{H}}^3$ and ${\mathbb{H}}^4$.