Hardy-Littlewood-Sobolev inequality states that if $1<p<q<\infty$, $1/r=1-1/p+1/q$, then we have
$$\left\|\frac{1}{|x|^{n/r}}\ast f\right\|\_{L^q(\mathbb R^n)}\le\|f\|\_{L^p(\mathbb R^n).}$$


Note that here $q=\infty$ is not allowed. My question is, is it possible to get some bounds for $q=\infty$, if we weaken the RHS norm to be in Lorentz space $L^{p,1}$:


$$\left\|\frac{1}{|x|^{n/r}}\ast f\right\|\_{L^q(\mathbb R^n)}\le\|f\|\_{L^{p,1}(\mathbb R^n).}$$


Any comments/references are welcome. Thanks.



Hardy-Littlewood-Sobolev inequality in Lorentz spaces
The Young inequality in Lorentz spaces covers these cases: if $p\_1,p\_2,p\in]1,\infty[$, $q\_1,q\_2,q\in[1,\infty]$,
 \begin{equation}
 \|f\ast g\|\_{L^{p,q}}\leq C\|f\|\_{L^{p\_1,q\_1}}
 \|g\|\_{L^{p\_2,q\_2}},\qquad p\_1^{-1}+p\_2^{-1}
 =1+p^{-1},\ q\_1^{-1}+q\_2^{-1}\geq q^{-1};
 \end{equation}
and if $p\_{1},p\_2\in ]1,\infty[$,
 $q\_1,q\_2\in[1,\infty]$, then
 \begin{equation}
 \|f\ast g\|\_{L^\infty}\leq C\|f\|\_{L^{p\_1,q\_1}}
 \|g\|\_{L^{p\_2,q\_2}},\qquad p\_1^{-1}+p\_2^{-1}
 =1,\ q\_1^{-1}+q\_2^{-1}\geq1.
 \end{equation}
Note that the second inequality contains your estimate.


The oldest reference I know for this is a paper by R.O'Neil, Convolution operators and $L(p,q)$ spaces, Duke Math. J. 30 (1963), 129-142.


