#### Vol. 13, No. 2, 2020

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A bootstrapping approach to jump inequalities and their applications

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

Vol. 13 (2020), No. 2, 527–558
##### Abstract

The aim of this paper is to present an abstract and general approach to jump inequalities in harmonic analysis. Our principal conclusion is the refinement of $r$-variational estimates, previously known for $r>2$, to endpoint results for the jump quasiseminorm corresponding to $r=2$. This is applied to the dimension-free results recently obtained by the first two authors in collaboration with Bourgain, and Wróbel, and also to operators of Radon type treated by Jones, Seeger, and Wright.

##### Keywords
dimension-free estimate, jump inequality
##### Mathematical Subject Classification 2010
Primary: 42B25
Secondary: 42B20, 46B06