Single annulus LP estimates for Hilbert transforms along vector fields

We prove L-P estimates, p is an element of (1, infinity), 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.