Stein-Weiss inequality in $L^{1}$ norm for vector fields

TL;DR

This paper explores the limit case of the Stein--Weiss inequality for vector fields in the $L^{1}$ norm, characterizing certain vector fields and extending div-curl inequalities and two-weight inequalities with general weights.

Contribution

It introduces a characterization for vector fields related to cocanceling operators at the $p=1$ limit and extends existing inequalities to broader contexts.

Findings

Characterization of vector fields associated with cocanceling operators at $p=1$

Recovery of div-curl inequalities from the literature

Extension of two-weight inequalities with general weights

Abstract

In this work, we investigate the limit case $p=1$ of the classical Stein--Weiss inequality for the Riesz potential.We present a characterization for a special class of vector fields associated to cocanceling operators introduced by Van Schaftingen in arXiv:1104.0192. As an application, we recover some div-curl inequalities found in the literature. In addition, we discuss a two-weight inequality with general weights in the scalar case, extending the previous result of Sawyer to this case.