Variation for singular integrals on Lipschitz graphs: L^p and endpoint estimates

TL;DR

This paper establishes boundedness of r-variation and oscillation operators for Calderón-Zygmund singular integrals with odd kernel on Lipschitz graphs, covering L^p, weak-L^1, and BMO spaces, including endpoint estimates.

Contribution

It proves new boundedness results for variation and oscillation operators on Lipschitz graphs, extending understanding of singular integrals in these geometric contexts.

Findings

Boundedness of r-variation and oscillation operators in L^p(H) for 1<p<∞

Weak-L^1 boundedness from finite Radon measures to H"older spaces

Operators map bounded H-measurable functions to BMO(H)

Abstract

Let 0<n<d be integers and let H denote the n-dimensional Hausdorff measure restricted to an n-dimensional Lipschitz graph in R^d with slope strictly less than 1. For r>2, we prove that the r-variation and oscillation for Calder\'on-Zygmund singular integrals with odd kernel are bounded operators in L^p(H) for 1<p finite, from L^1(H) to weak-L^1(H), and from the space of bounded H-measurable functions to BMO(H). Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in R^d to weak-L^1(H).