Generalized Hardy-Littlewood-Sobolev Inequality The Hardy-Littlewood-Sobolev Inequality says that 
$$\text{for $p,q,r\in (1,+\infty)$ such that }\quad 
1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$}
$$
$$
\exists C, \forall u\in L^p(\mathbb R^n),\quad 
\Vert{u\ast\vert\cdot\vert^{-n/q}}\Vert_{L^r(\mathbb R^n)}\le C
\Vert u\Vert_{L^p(\mathbb R^n)}.
$$
Setting $v_q(x)=\vert x\vert^{-n/q}$, we see that $v_q\in L^q_w(\mathbb R^n)$, which is also the Lorentz space
$L^{q,\infty}(\mathbb R^n)$ (the latter space is a Banach space when $q\in (1,+\infty)$).
Is there a generalization of Young's inequality such as 
$$
\exists C, \forall u\in L^p(\mathbb R^n),
\forall v\in L^{q,\infty}(\mathbb R^n),\quad
\Vert{u\ast v}\Vert_{L^r(\mathbb R^n)}\le C
\Vert u\Vert_{L^p(\mathbb R^n)}\Vert v\Vert_{L^{q,\infty}(\mathbb R^n)},
$$
with $p,q,r$ satisfying $(\sharp)$?
 A: See Corollary 2.15 in the paper 

V.F. Kovalenko, M.A. Perelmuter, Yu.A. Semenov, Schrödinger Operators with $L^{l/2}_w (R^l)$-Potentials, J. Math. Phys., Vol. 22, No. 5, 1981, pp. 1033-1044

link to the paper on journal website
A: The weak type Young inequality
$$
\Vert{u\ast v}\Vert_{L^r(\mathbb R^n)}\le C_{p,q,r}
\Vert u\Vert_{L^p(\mathbb R^n)}\Vert v\Vert_{L^{q,\infty}(\mathbb R^n)},
\quad 1<p,q,r<\infty,
\quad \frac1p+\frac1q = 1 + \frac1r
$$
is in fact a direct consequence of the Hardy-Littlewood-Sobolev inequality.
By the Riesz rearrangement inequality (a special case of the Brascamp-Lieb-Luttinger inequlaity) we have
$$
|\iint f(x) g(x-y) h(y) dxdy| \leq |\iint f^*(x) g^*(x-y) h^*(y) dxdy|,
$$
where $f^*$ is the symmetric non-increasing rearrangement of a function $f$.
Now,
$$
f^*(x) \leq C_n |x|^{-n/q}\|f\|_{q,\infty}
$$
and the claim follows by duality.
This argument has the advantage that it gives the optimal constant provided that the optimal constant in the HLS inequality is known (thus at least in the case $p=r'$), as opposed to interpolation arguments.
A: Inequalities like this are discussed in this paper of Nakanishi: http://epubs.siam.org/doi/pdf/10.1137/S0036141000369083
(See section two.) 
As others have mentioned already, one can get estimates like this from the standard estimates plus interpolation. 
