Single annulus estimates for the variation-norm Hilbert transforms along Lipschitz vector fields

TL;DR

This paper establishes boundedness of variation-norm Hilbert transforms along Lipschitz vector fields in the plane, extending previous results that only required measurability, with implications for harmonic analysis.

Contribution

It proves boundedness of the r-th variation-norm Hilbert transform along Lipschitz vector fields, generalizing prior results that assumed only measurability.

Findings

Boundedness from L^2 to weak L^2

Boundedness on L^p for p>2

Norm independence of scale parameter k

Abstract

Let v be a planar Lipschitz vector field. We prove that the r-th variation-norm Hilbert transform along v, composed with a standard Littlewood-Paley projection operator P_k, is bounded from L^2 to L^{2, \infty}, and from L^p to itself for all p>2. Here r>2 and the operator norm is independent of k\in \Z. This generalises Lacey and Li's result for the case of the Hilbert transform. However, their result only assumes measurability for vector fields. In contrast to that, we need to assume vector fields to be Lipschitz.