Hardy-Littlewood-Sobolev inequality for $p=1$

Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type $a\otimes \delta_0$, $\delta_0$ being the Dirac delta, then the inequality $\|\operatorname{I}_\alpha [f]\|_{L_{d/(d-\alpha),1}}\lesssim \|f\|_{L_1}$ holds true for functions $f\in\mathcal{W}\cap L_1$ with a uniform constant; here $\operatorname{I}_\alpha$ is the Riesz potential of order $\alpha$ and $L_{p,1}$ is the Lorentz space. As particular cases, this result implies the inequality $\|\nabla^{m-1} f\|_{L_{d/(d-1),1}} \lesssim \|A f\|_{L_1}$, where $A$ is a cancelling elliptic differential operator of order $m$, and the inequality $\|\operatorname{I}_\alpha f\|_{L_{d/(d-\alpha),1}} \lesssim \|f\|_{L_1}$, where $f$ is a divergence free vector field. Bibliography: 59 titles.

Hardy-Littlewood-Sobolev inequality, Bourgain-Brezis inequalities, cancelling differential operators