Jump inequalities via real interpolation

Mariusz Mirek; Elias M. Stein; Pavel Zorin-Kranich

doi:10.1007/s00208-019-01889-2

Jump inequalities via real interpolation

Mariusz Mirek, Elias M. Stein, Pavel Zorin-Kranich

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

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 R^d to Z^d for jump estimates for Fourier multipliers.

Original language	English (US)
Pages (from-to)	797-819
Number of pages	23
Journal	Mathematische Annalen
Volume	376
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-019-01889-2
State	Published - Feb 1 2020

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00208-019-01889-2

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author = "Mariusz Mirek and Stein, {Elias M.} and Pavel Zorin-Kranich",

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N2 - 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 Rd to Zd for jump estimates for Fourier multipliers.

AB - 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 Rd to Zd for jump estimates for Fourier multipliers.

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Jump inequalities via real interpolation

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