Hilbert transforms along Lipschitz direction fields: A lacunary model

We prove bounds for the truncated directional Hilbert transform in $L^p({\mathbb{R}}^2)$ for any $1<p<\mathrm{\infty }$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for $p=2$, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some $p$.