# A sharp estimate for the Hilbert transform along finite order lacunary sets of directions

## TL;DR

This paper establishes sharp bounds for the $L^p$ norms of the maximal directional Hilbert transform along lacunary sets of directions, showing they grow like the square root of the logarithm of the set size, and extends these bounds to certain vector fields.

## Contribution

It provides the first sharp estimates for the Hilbert transform along finite order lacunary directions and extends previous bounds to vector fields with lacunary range.

## Key findings

- $L^p$ norms grow like $( ext{log} ext{card}( heta))^{1/2}$ for lacunary directions.
- The truncated Hilbert transform along vector fields with lacunary range is $L^p$-bounded.
- Extends previous bounds by Demeter, Guo, and Thiele.

## Abstract

Let $D$ be a nonnegative integer and ${\mathbf{\Theta}}\subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<\infty$, of the maximal directional Hilbert transform in the plane $$ H_{{\mathbf{\Theta}}} f(x):= \sup_{v\in {\mathbf{\Theta}}} \Big|\mathrm{p.v.}\int_{\mathbb R }f(x+tv)\frac{\mathrm{d} t}{t}\Big|, \qquad x \in {\mathbb R}^2, $$ are comparable to $(\log\#{\mathbf{\Theta}})^\frac{1}{2}$. For vector fields $\mathsf{v}_D$ with range in a lacunary set of of order $D$ and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field $\mathsf{v}_D$, $$ H_{\mathsf{v}_D,1} f(x):= \mathrm{p.v.} \int_{ |t| \leq 1 } f(x+t\mathsf{v}_D(x)) \,\frac{\mathrm{d} t}{t}, $$ is $L^p$-bounded for all $1<p<\infty$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.