What is the maximizer of the Hardy-Littlewood-Sobolev fractional integration inequality Let $ 0 < \alpha < n $, and define
$$I_\alpha f (x): = \int_{ \mathbb R^n } \frac{f (y) }{  |x - y| ^\alpha }\, d y. $$
The Hardy-Littlewood-Sobolev fractional integration inequality states:
If $1 < p< q < \infty$ and  $ \frac{1}{p}-\frac{1}{q} =  1-\frac{ \alpha }{ n} $ then
$$\| I_ \alpha f \|_{ L^q ( \mathbb R ^n )} \le C_{p ,\alpha , n }\||f\|_{ L^ p(\mathbb R^n )}. \qquad(1)$$
Are there known maximizers (possibly up to a constant) of (1).
A reference or idea of proof would be very helpful.
A maximizer of (1) is a function $f$ for which (1) is an equality.
 A: Notice first that by duality, your inequality is equivalent to the Hardy-Littlewood-Sobolev inequality
$$
\iint \frac{f(x)\,g(y)}{|x-y|^\alpha}\,\mathrm d x\,\mathrm d y ≤ C_{p,\alpha,n}\, \|f\|_{L^p}\,\|g\|_{L^{q'}}.
$$
where $q' = \tfrac{q}{q-1}$.
The maximizers and the optimal constant are only explictly known in the case $p=q'$. In this case $p=\frac{2\,n}{2\,n-\alpha}$ and as proved by Lieb
$$
C_{p,\alpha,n} = \pi^{\alpha/2} \,\frac{\Gamma(\tfrac{n}{2}-\tfrac{\alpha}{2})}{\Gamma(n-\tfrac{\alpha}{2})} \left(\frac{\Gamma(n)}{\Gamma(\tfrac{n}{2})}\right)^{1-\alpha/n}
$$
and the optimizers are the functions of the form
$$
f(x) = \frac{C}{(a^2+|x-b|^2)^{(2n-\alpha)/2}}
$$
See e.g. Theorem 4.3 in the book Analysis by Lieb and Loss (and the remarks after).
In the other cases, the optimizers are known to exist, but I think they are not known explicitly, and the optimal constant is not known. However, there are bounds from above for the constant, and the minizers might be linked to the fast diffusion/porous media equation (see e.g. Hardy-Littlewood-Sobolev inequalities via fast diffusion flows)
A: Here is the theorem @LL 3.14 is referring to in his answer from "Analysis" by Lieb and Loss:

