Abstract
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.
Bañuelos is supported in part by NSF Grant # 0603701-DMS.
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Acknowledgments
David Applebaum would like to thank both the London Mathematical Society and Purdue University for the financial support which enabled this project to get off the ground during the summer of 2012. Both authors would like to thank Krzysztof Bogdan for inviting them to the 6th international conference on stochastic analysis at Bȩdlewo in September 2012 where we able to make much progress.
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Applebaum, D., Bañuelos, R. (2015). Probabilistic Approach to Fractional Integrals and the Hardy-Littlewood-Sobolev Inequality. In: Mityushev, V., Ruzhansky, M. (eds) Analytic Methods in Interdisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 116. Springer, Cham. https://doi.org/10.1007/978-3-319-12148-2_2
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