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

Publication date: 1 October 2013

Abstract: 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.

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