A critical Hardy-Rellich inequality

TL;DR

This paper establishes a critical Hardy-Rellich inequality involving the gradient and Laplacian of functions in ty spaces, extending the understanding of inequalities in mathematical analysis.

Contribution

It proves a new critical Hardy-Rellich inequality with explicit constants, advancing the theoretical framework of functional inequalities.

Findings

Established a critical Hardy-Rellich inequality for all dimensions N

Derived explicit constant C_N in the inequality

Extended the inequality to functions vanishing outside compact sets

Abstract

In this work, we prove a critical version of a Hardy-Rellich type inequality. We show that for $N\geq 1$ there exists a constant $C_N>0$ such that \[ \int_{\mathbb R^N}\left|\nabla\left(\frac{u(x)}{|x|}\right)\right|^N\,\mathrm{d}x\leq C_N\int_{\mathbb R^N}\left|\Delta u(x)\right|^N\,\mathrm{d}x, \] for any $u\in C^\infty_c(\mathbb R^N\setminus\left\{0\right\})$.