Stein-Weiss, and power weight Korn type Hardy-Sobolev Inequalities in $L^1$ norm

TL;DR

This paper extends Stein-Weiss inequalities in the $L^1$ norm by replacing the cocanceling condition with a weaker assumption, and introduces Korn type Hardy-Sobolev inequalities for certain exponents, broadening the scope of these inequalities.

Contribution

It replaces the cocanceling condition with a weaker vanishing moment assumption and establishes new Hardy-Sobolev inequalities for $L^1$ data with growth conditions.

Findings

Extended Stein-Weiss inequalities to $L^1(|x|^{a} dx)$ data for positive, non-integer $a$

Proved Korn type Hardy-Sobolev inequalities for integer exponents where original inequalities fail

Provided a specific example demonstrating improved duality estimates for canceling operators

Abstract

We extend the $L^1$ Stein-Weiss inequalities studied by De N\'{a}poli and Picon [4] in two ways: First we address an open question posed by the authors about whether the cocanceling condition was necessary for some of their Stein-Weiss inequalities. We replace the cocanceling condition with a weaker vanishing moment assumption, and under this assumption extend the $L^1$ Stein-Weiss inequalities to $L^1(|x|^{a } dx)$ data for all positive, non-integer exponents $a$. Second, in relation to integer exponents, while [4] showed that Stein-Weiss fails for $L^1(|x| dx)$ data, we prove a weaker Korn type Hardy-Sobolev inequality. These inequalities were previously inaccessible due to the growth of $|x|$, and we demonstrate a specific example on $\mathbb{R}^2$ of where the original duality estimate by Bousquet and Van Schaftingen [2] for canceling operators can be improved.