  In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can
be related by duality. In this paper, we investigate how to relate these
inequalities using the flow of a fast diffusion equation in dimension $d\ge3$.
The main consequence is an improvement of Sobolev's inequality when $d\ge5$,
which involves the various terms of the dual Hardy-Littlewood-Sobolev
inequality. In dimension $d=2$, Onofri's inequality plays the role of Sobolev's
inequality and can also be related to its dual inequality, the logarithmic
Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equation.
