Endpoint Variation and jump inequalities for rough singular integrals

TL;DR

This paper establishes weak type (1,1) bounds for variation and jump operators of rough singular integrals, resolving an open problem and confirming boundedness of related maximal operators.

Contribution

It proves weak type (1,1) bounds for variation and jump operators for rough singular integrals, addressing an open question in harmonic analysis.

Findings

Weak type (1,1) bounds for variation operators

Weak type (1,1) bounds for jump operators

Boundedness of maximal truncation operators

Abstract

In this article, we prove weak type $(1,1)$ bounds for the variation and jump operators corresponding to the family of truncations of singular integrals with rough kernels. This resolves an open question raised by Jones, Seeger and Wright (Trans. Amer. Math. Soc. (2008)). Moreover, as an immediate consequence of the variational estimate, we recover the weak type $(1,1)$ boundedness of the maximal truncation operator corresponding to singular integrals with rough kernels.