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 $\mathbb{ H}^4$.
Categories: math.AP