  We give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev
inequality for self-adjoint $C_{0}$ semigroups and give a new probabilistic
representation of the classical fractional integral operators on $\R^n$ as
projections of martingale transforms. Using this formula we derive a new proof
of the classical Hardy-Littlewood-Sobolev inequality based on Burkholder-Gundy
and Doob's inequalities for martingales.
