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 $a\otimes δ_0$, $δ_0$ being the Dirac delta, then the inequality $\|\mathbb{I}_α[f]\|_{L_{p,1}}\lesssim \|f\|_{L_1}$, $\frac{p-1}{p} = \fracα{d}$, holds true for functions $f\in\mathcal{W}\cap L_1$ with a uniform constant; here $\mathbb{I}_α$ is the Riesz potential of order $α$ and $L_{p,1}$ is the Lorentz space. This result implies as a particular case the inequality $\|\nabla^{m-1} f\|_{L_{\frac{d}{d-1},1}} \lesssim \|A f\|_{L_1}$, where $A$ is a canceling elliptic differential operator of order $m$.

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