A bootstrapping approach to jump inequalities and their applications

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 Wróbel (arXiv:1708.04639 and arXiv:1804.07679), and also to operators of Radon type treated by Jones, Seeger, and Wright.

arXiv comment25 pages, small corrections

Cited by in corpus (7)

• Jump inequalities for translation-invariant operators of Radon type on ${\mathbb{Z}}^d$
• A variational restriction theorem
• Oscillation estimates for truncated singular Radon operators
• Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes
• Bootstrap methods in bounding discrete Radon operators
• $L^p$ norm of truncated Riesz transform and an improved dimension-free $L^p$ estimate for maximal Riesz transform
• Quantitative ergodic theorems for actions of groups of polynomial growth