Jump inequalities for translation-invariant operators of Radon type on $\mathbb{Z}^d$

TL;DR

This paper establishes strong jump inequalities for a broad class of Radon-type operators on integer lattices, extending variational estimate results to the discrete and ergodic contexts.

Contribution

It introduces strong jump inequalities for Radon-type operators in discrete and ergodic settings, providing the $r=2$ endpoint of variational estimates.

Findings

Proved strong jump inequalities for Radon-type operators.

Extended variational estimate endpoints to discrete and ergodic frameworks.

Enhanced understanding of operator behavior in these mathematical settings.

Abstract

We prove strong jump inequalities for a large class of operators of Radon type in the discrete and ergodic theoretical settings. These inequalities are the $r=2$ endpoints of the $r$-variational estimates studied in arXiv:1512.07523.