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

Francesco Di Plinio, Ioannis Parissis

Abstract

Let $D$ be a nonnegative integer and ${\mathbfΘ}\subset S^1$ 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 $$ H_{\mathbfΘ} f(x):= \sup_{v\in {\mathbfΘ}} \Big|\mathrm{p.v.}\int_{\mathbb R }f(x+tv)\frac{\mathrm{d} t}{t}\Big|, \qquad x \in {\mathbb R}^2, $$ are comparable to $(\log\#{\mathbfΘ})^\frac{1}{2}$. 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.

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

Abstract

Let be a nonnegative integer and be a lacunary set of directions of order . We show that the norms, , of the maximal directional Hilbert transform in the plane are comparable to . For vector fields with range in a lacunary set of of order and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field , is -bounded for all . These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.

Paper Structure

This paper contains 22 sections, 8 theorems, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Some conventions concerning lacunary sets of directions
  4. A representation of the Hilbert transform in terms of cone projections
  5. Maximal functions and cone frequency projections in $D$-lacunary directions
  6. Maximal Fourier multipliers and square functions
  7. A single annulus pointwise estimate
  8. A Fefferman-Stein inequality for directional maximal functions
  9. Proof of Theorem \ref{['t.D-lacunary:main']}
  10. Initial reductions
  11. Proof of \ref{['D-lacunary:SFE']}
  12. Proof of Theorem \ref{['t.Liplac']}
  13. Preliminary reductions
  14. Reduction of \ref{['e.T2:main1']} to a lacunary vector-valued estimate
  15. Proof of Proposition \ref{['p.tvv']}
  16. ...and 7 more sections

Key Result

Theorem 1

Let ${\mathbf \Theta}\subset S^1$ be a $D$-lacunary set of directions. The maximal directional Hilbert transform $H_{\mathbf \Theta} f(x)\coloneqq \sup_{v \in {\mathbf \Theta}} |H_{v} f(x)|$ obeys the bounds with constants $c,C>0$ depending only on $D,p$.

Figures (2)

  • Figure 1: Lacunary sets and cones
  • Figure 2: A figure for the proof of Proposition \ref{['p.tvv']}. The dark shaded rectangles represent the approximate frequency supports of $\Delta_{\theta,j,\ell,k}$.

Theorems & Definitions (17)

  • Definition 1: successor
  • Definition 2: $D$-lacunary sets
  • Theorem 1
  • Corollary 1.1
  • Theorem 2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • ...and 7 more