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

We extend the $L^1$ Stein-Weiss inequalities studied by De Nápoli 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 $L^1(\mathrm{\mid }x{\mathrm{\mid }}^{a}dx)$ data for all positive, non-integer exponents $a$. Second, in relation to integer exponents, while [4] showed that Stein-Weiss fails for $L^1(\mathrm{\mid }x\mathrm{\mid }dx)$ data, we prove a weaker Korn type Hardy-Sobolev inequality. These inequalities were previously inaccessible due to the growth of $\mathrm{\mid }x\mathrm{\mid }$, 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.