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

We prove $L^p$, $p\in (1,\mathrm{\infty })$ 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.