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

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.