One dimensional sharp discrete Hardy-Rellich inequalities

TL;DR

This paper establishes new sharp discrete Hardy-Rellich inequalities on the natural numbers for higher orders, providing optimal constants and extending to general graphs and ^p spaces.

Contribution

The paper introduces the first sharp discrete Hardy-Rellich inequalities for   and higher, with optimal constants, and extends the approach to graphs and ^p spaces.

Findings

Established sharp inequalities for   with optimal constants

Derived a new Hardy-Leray type inequality on 

Extended methods to general graphs and ^p spaces

Abstract

In this paper, we establish discrete Hardy-Rellich inequalities on $\mathbb{N}$ with $\Delta^\frac{\ell}{2}$ and optimal constants, for any $\ell \geq 1$. As far as we are aware, these sharp inequalities are new for $\ell \geq 3$. 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.