One dimensional sharp discrete Hardy-Rellich inequalities

Abstract

In this paper, we establish discrete Hardy-Rellich inequalities on $\mathbb{N}$ with ${\Delta }^{\frac{\ell }{2}}$ and optimal constants, for any $\ell \ge 1$. As far as we are aware, these sharp inequalities are new for $\ell \ge 3$. Our approach is to use weighted equalities to get some sharp Hardy inequalities using shifting weights, then to settle the higher order cases by iteration. We provide also a new Hardy-Leray type inequality on $\mathbb{N}$ with the same constant as the continuous setting. Furthermore, the main ideas work also for general graphs or the ${\ell }^p$ setting.

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Cited by in corpus (1)

• Hardy inequalities for antisymmetric functions