Canceling effects in higher-order Hardy-Sobolev inequalities

TL;DR

This paper extends Hardy-Sobolev inequalities to higher-order derivatives, revealing that the previously excluded power becomes admissible, thus broadening the understanding of boundary-weighted Sobolev norms.

Contribution

It demonstrates that the missing power in classical Hardy-Sobolev inequalities is permissible in higher-order cases, extending prior results and uncovering a canceling phenomenon.

Findings

The missing power is admissible in higher-order Hardy-Sobolev inequalities.

Extension of previous results by Castro and Wang, and Castro, Dávila and Wang.

Identification of a canceling phenomenon in boundary-weighted Sobolev norms.

Abstract

A classical first-order Hardy-Sobolev inequality in Euclidean domains, involving weighted norms depending on powers of the distance function from their boundary, is known to hold for every, but one, value of the power. We show that, by contrast, the missing power is admissible in a suitable counterpart for higher-order Sobolev norms. Our result complements and extends contributions by Castro and Wang [CW], and Castro, D\'avila and Wang [CDW1, CDW2], where a surprising canceling phenomenon underling the relevant inequalities was discovered in the special case of functions with derivatives in $L^1$.