Weak $(1,1)$ estimate for maximal truncated rough singular integral operator

TL;DR

This paper proves that the maximal truncated rough singular integral operator is of weak type (1,1), resolving a long-standing open problem about its endpoint behavior, which was previously unknown.

Contribution

It establishes the weak (1,1) boundedness of the maximal truncated rough singular integral operator, a significant advancement in harmonic analysis.

Findings

Proves weak (1,1) boundedness of the operator

Completes the understanding of endpoint behavior for this class of operators

Addresses a problem posed since Calderón and Zygmund's foundational work

Abstract

In their seminal work (Amer. J. Math. 78: 289-309, 1956), Calder\'on and Zygmund introduced the maximal truncated rough singular integral operator and established its $L^p$-boundedness for $1 < p < \infty$. However, the endpoint case $p = 1$ remained an open problem. This paper resolves this problem. More precisely, we prove that the maximal truncated rough singular integral operator is of weak type $(1,1)$.