  Let v be a planar Lipschitz vector field. We prove that the r-th
variation-norm Hilbert transform along v, composed with a standard
Littlewood-Paley projection operator P_k, is bounded from L^2 to L^{2, \infty},
and from L^p to itself for all p>2. Here r>2 and the operator norm is
independent of k\in \Z. This generalises Lacey and Li's result for the case of
the Hilbert transform. However, their result only assumes measurability for
vector fields. In contrast to that, we need to assume vector fields to be
Lipschitz.
