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.

Singular Integrals and Differentiability Properties of Functions, Stein refers to a paper of R. O'Neil for a treatment of fractional integration in the setting of Lorentz spaces. You might look there. $\endgroup$ – Matt Rosenzweig Mar 27 '16 at 1:19