  We prove bounds for the truncated directional Hilbert transform in
$L^p(\mathbb{R}^2)$ for any $1<p<\infty$ 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 $p$.
