TITLE: Generalized Hardy-Littlewood-Sobolev Inequality
QUESTION [6 upvotes]: 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 $(\sharp)$?

REPLY [3 votes]: 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.