Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$

TL;DR

This paper establishes a deep connection between Hardy's inequality and Sobolev-Lorentz embedding inequalities in n, revealing new optimal bounds, attainability properties, and the role of extremal functions, thus advancing the theoretical understanding of these fundamental inequalities.

Contribution

It completes previous results by establishing optimal embedding inequalities for Sobolev-Lorentz spaces in the range p<q<, and shows Hardy's inequality is equivalent to a Sobolev-Marcinkiewicz embedding, with new insights into attainability.

Findings

Hardy inequality is equivalent to Sobolev-Marcinkiewicz embedding inequality.

Optimal embedding inequalities are established for the range p<q<.

The Sobolev-Marcinkiewicz embedding is attained by 'ghost' extremal functions, unlike Hardy inequality.

Abstract

We investigate connections between Hardy's inequality in the whole space $\mathbb{R}^n$ and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, (1977)] and [G. Talenti, An inequality between $u^*$ and $|{\rm{grad}} u^*|$, (1992)] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm $\|\nabla\,\cdot\,\|_{p,q}$ also in the range $p < q<\infty$, which remained essentially open since the work of Alvino. Attainability of the best embedding constants is also studied, as well as the limiting case when $q=\infty$. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev-Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called "ghost" extremal functions of [Brezis-V\'azquez, Blow-up solutions of some nonlinear elliptic problems, (1977)], in striking contrast with the Hardy inequality, which is never attained. In this sense, our functional approach seems to be more natural than the classical Sobolev setting, answering a question raised by Brezis and V\'azquez.