Single annulus estimates for the variation-norm Hilbert transforms along Lipschitz vector fields

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.