A bootstrapping approach to jump inequalities and their applications

TL;DR

This paper introduces a general approach to jump inequalities in harmonic analysis, refining variational estimates to end-point results and applying them to dimension-free operators and Radon-type operators.

Contribution

It provides a novel abstract method for jump inequalities, extending known results to critical endpoint cases and broadening their applications.

Findings

Refined $r$-variational estimates to endpoint $r=2$

Applied results to dimension-free harmonic analysis operators

Extended jump inequalities to Radon-type operators

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 end-point results for the jump quasi-seminorm corresponding to $r=2$. This is applied to the dimension-free results recently obtained by the first two authors in collaboration with Bourgain and Wr\'obel (arXiv:1708.04639 and arXiv:1804.07679), and also to operators of Radon type treated by Jones, Seeger, and Wright.