Single annulus $L^p$ estimates for Hilbert transforms along vector fields

TL;DR

This paper establishes $L^p$ bounds for the Hilbert transform along a one-variable vector field restricted to functions with annular frequency support, advancing understanding of such transforms in harmonic analysis.

Contribution

It provides new $L^p$ estimates for the Hilbert transform along vector fields on annular frequency supports, complementing previous results for $p>2$ and supporting further work on full transforms.

Findings

Proved $L^p$ estimates for $p eq 2$ on annular frequency supports.

Extended the range of $p$ for which these estimates hold.

Provided technical tools for analyzing the full Hilbert transform.

Abstract

We prove $L^p$, $p\in (1,\infty)$ estimates on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when $p>2$ were proved by Lacey and Li in \cite{LL1}. This paper also contains key technical ingredients for a companion paper \cite{BT} with Christoph Thiele in which $L^p$ estimates are established for the full Hilbert transform.