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

Abstract

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 $\Vert {\mathbb{I}}_{\alpha }[f]{\Vert }_{L_{p,1}}\lesssim \Vert f{\Vert }_{L_1}$, $\frac{p-1}{p}=\frac{\alpha }{d}$, holds true for functions $f\in \mathcal{W}\cap L_1$ with a uniform constant; here ${\mathbb{I}}_{\alpha }$ is the Riesz potential of order $\alpha$ and $L_{p,1}$ is the Lorentz space. This result implies as a particular case the inequality $\Vert {\nabla }^{m-1}f{\Vert }_{L_{\frac{d}{d-1},1}}\lesssim \Vert Af{\Vert }_{L_1}$, where $A$ is a canceling elliptic differential operator of order $m$.

41 pages; second version contains several new corollaries and more citations; third version corrects an error in section 3