It is known that $L^1$ Sobolev and Stein-Weiss inequalities follow a slightly different pattern to their $L^p$ counterparts. For instance, they serve as weaker replacement inequalities for the failure of $L^1$ inequalities between differential operators. Recent work by P. De Napoli and T. Picon has shown that the canceling/cocanceling framework introduced by J. Van Schaftingen can be used to extend the $L^1$ Stein-Weiss inequalities beyond their expected range. Inspired by these ideas, we investigate if the cocanceling constraint can be weakened in a subcritical setting, and we also explore how one may further extend these inequalities to a critical case.
