Hardy-Littlewood-Sobolev Inequalities via Fast Diffusion Flows

We give a simple proof of the $\lambda =d-2$ cases of the sharp Hardy-Littlewood-Sobolev inequality for $d\ge 3$, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for $d=2$ via a monotone flow governed by the fast diffusion equation.