A critical Hardy-Rellich inequality
Hernán Castro
TL;DR
The paper addresses the critical Hardy-Rellich inequality in the case $p=N$, proving that for all $N\ge 1$ there exists a constant $C_N>0$ such that $\int_{\mathbb{R}^N} |\nabla(u/|x|)|^N dx \le C_N \int_{\mathbb{R}^N} |\Delta u|^N dx$ for $u\in C_c^{\infty}(\mathbb{R}^N\setminus\{0\})$. The authors develop a strategy that separates radial and angular contributions in spherical coordinates, bounding the angular part via a 1D Hardy-type inequality and the radial part via the operator $T_{2,1}$ on $\partial_{rr}u$, then use Calderón–Zygmund to relate $|D^2u|^N$ to $|\Delta u|^N$. A weighted extension with $|x|^a$ is established for $a<1$, and a corollary for $-N<a<1$ follows from weighted singular integral theory. The work clarifies the role of cancellations in critical Hardy-Rellich estimates, provides a framework for weighted versions, and raises open questions about optimal constants and fractional analogues.
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 \[ \int_{\mathbb R^N}\left|\nabla\left(\frac{u(x)}{|x|}\right)\right|^N\,\mathrm{d}x\leq C_N\int_{\mathbb R^N}\left|Δu(x)\right|^N\,\mathrm{d}x, \] for any $u\in C^\infty_c(\mathbb R^N\setminus\left\{0\right\})$.