  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.
