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

Abstract

We prove $L^p$ estimates, $p\in (1,\infty )$, 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. This paper also contains key technical ingredients for a companion paper with Christoph Thiele in which $L^p$ estimates are established for the full Hilbert transform. The operators considered here are singular integral variants of maximal operators arising in the study of planar differentiation problems.