Hardy-Littlewood-Sobolev inequality in Lorentz spaces 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.
 A: 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.
A: The right substitute for $L^{\infty}$ is the space $BMO$, in the following sense: Singular integral operators in $L^1$ is not bounded from $L^1$ to itself, but is bounded from $H^1$ to $L^1$. Hence, by duality, these operators are bounded from $L^\infty$ to $BMO$.
