Hardy-Sobolev inequalities involving mixed radially and cylindrically symmetric weights

TL;DR

This paper investigates weighted Hardy-Sobolev inequalities with anisotropic weights in multi-dimensional space, establishing conditions for their validity and exploring extremal functions.

Contribution

It introduces new anisotropic weight conditions for Hardy-Sobolev inequalities and characterizes when extremal functions exist or do not exist.

Findings

Necessary and sufficient conditions for inequalities

Existence and nonexistence of extremal functions

Analysis of anisotropic weights in $\\mathbb{R}^d$

Abstract

We deal with weighted Hardy-Sobolev type inequalities for functions on $\mathbb{R}^d$, $d\geq 2$. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.