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Jump inequalities via real interpolation

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Abstract

Jump inequalities are the \(r=2\) endpoint of Lépingle’s inequality for r-variation of martingales. Extending earlier work by Pisier and Xu (Probab Theory Relat Fields 77(4):497–514, 1988) we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from \(\mathbb {R}^{d}\) to \(\mathbb {Z}^{d}\) for jump estimates for Fourier multipliers.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Mariusz Mirek

  2. Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Mariusz Mirek

  3. Department of Mathematics, Princeton University, Princeton, NJ, 08544-100, USA

    Elias M. Stein

  4. Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Pavel Zorin-Kranich

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  1. Mariusz Mirek
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  2. Elias M. Stein
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  3. Pavel Zorin-Kranich
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Correspondence to Mariusz Mirek.

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Communicated by Loukas Grafakos.

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Mariusz Mirek was partially supported by the Schmidt Fellowship and the IAS Found for Math. and by the National Science Center, NCN Grant DEC-2015/19/B/ST1/01149. Elias M. Stein was partially supported by NSF Grant DMS-1265524. Pavel Zorin-Kranich was partially supported by the Hausdorff Center for Mathematics and DFG SFB-1060.

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Mirek, M., Stein, E.M. & Zorin-Kranich, P. Jump inequalities via real interpolation. Math. Ann. 376, 797–819 (2020). https://doi.org/10.1007/s00208-019-01889-2

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  • DOI: https://doi.org/10.1007/s00208-019-01889-2

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