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

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 $$H_{\mathbf{\Theta }}f(x):=\underset{v\in \mathbf{\Theta }}{\sup }|p.v.{\int }_{\mathbb{R}}f(x+tv)\frac{\mathrm{d}t}{t}|,x\in {\mathbb{R}}^2,$$ 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 ${\mathsf{v}}_D$, $$H_{{\mathsf{v}}_D,1}f(x):=p.v.{\int }_{|t|\le 1}f(x+t{\mathsf{v}}_D(x))\frac{\mathrm{d}t}{t},$$ 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.

20 pages, 2 figures. Submitted. Changes: clarified the definition of D-lacunary set and streamlined the notation

References in corpus (1)

• Lipschitz linearization of the maximal hyperbolic cross multiplier

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• Singular integrals along lacunary directions in ${\mathbb{R}}^n$
• Almost-orthogonality principles for certain directional maximal functions