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

2013/10/01 by David Applebaum, Applebaum, David, Rodrigo Bañuelos +2
Mathematics · #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations #math.PR

paper · pdf · doi:10.48550/arxiv.1310.0182

arxiv created 2013/10/01 · arxiv updated 2013/10/02

Abstract

We give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev inequality for self-adjoint C₀ semigroups and give a new probabilistic representation of the classical fractional integral operators on \Rⁿ 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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