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In this work, we investigate the limit case $p=1$ of the classicalStein--Weiss inequality for the Riesz potential.We present a characterizationfor a special class of vector fields associated to cocanceling operatorsintroduced by Van Schaftingen in arXiv:1104.0192. As an application, we recoversome div-curl inequalities found in the literature. In addition, we discuss atwo-weight inequality with general weights in the scalar case, extending theprevious result of Sawyer to this case.

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