Probabilistic Approach to Fractional Integrals and the Hardy-Littlewood-Sobolev Inequality

We give a short summary of some of Varopoulos’ Hardy-Littlewood-Sobolev inequalities for self-adjoint \(C_{0}\) semigroups and give a new probabilistic representation of the classical fractional integral operators on \(\mathbb {R}^n\) 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.