Abstract:  In this lecture we present a new characterization of Hardy-Littlewood-Sobolev inequalities in $L^{1}$ norm for homogeneous linear differential operators with constant coefficients. Variants and applications are presented, in particular a version of $L^{1}$ Stein-Weiss inequality for vector fields. This is joint work with Jorge Hounie (UFSCar - Brazil) and (work in progress) with Pablo De Napoli (UBA - Argentina).