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On the maximal directional Hilbert transform in three dimensions

Francesco Di Plinio, Ioannis Parissis

Abstract

We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_Ω$ along finite subsets of a finite order lacunary set of directions $Ω\subset \mathbb R^3$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.

On the maximal directional Hilbert transform in three dimensions

Abstract

We establish the sharp growth rate, in terms of cardinality, of the norms of the maximal Hilbert transform along finite subsets of a finite order lacunary set of directions , answering a question of Parcet and Rogers in dimension . Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.

Paper Structure

This paper contains 17 sections, 13 theorems, 117 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Maximal and singular integrals along sets of directions
  3. Techniques of proof
  4. Relation to the Hilbert transform along vector fields
  5. Plan of the article
  6. Lacunary sets of directions: definitions and notation
  7. Lacunary sets of directions of finite order
  8. Model operators
  9. Weighted norm inequalities for directional maximal operators
  10. Directional $A_p$ weights
  11. Extrapolation for $A_p ^\Omega$ weights
  12. Weighted inequalities for the lacunary directional maximal operator
  13. Proof of Theorem \ref{['thm:weightedM']}
  14. An almost orthogonality principle for the maximal Hilbert transform
  15. Proof of Theorem \ref{['th:main']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Omega\subset S^{2}$ be a finite order lacunary set PR2. Then for all $1<p<\infty$ The positive constant $C$ may depend on $1<p<\infty$ and on the lacunary order of $\Omega$ only.

Figures (2)

  • Figure 3.1: The Fourier support of the multipliers $K^{\circ}_{(1,2),\sigma},$$K^{-}_{(1,2),\sigma},$ and $K^{+}_{(1,2),\sigma}$.
  • Figure 3.2: Suppose $\omega$ belongs to the cell $S_{\bm{\ell}}$. The red line is the intersection with the sphere $S^2$ of the singularity $\xi\cdot \omega=0$ of $H_\omega$. The blue and yellow wedges are respectively $\Psi_{(1,2),\ell_{(1,2)}}$ and $\Psi_{(2,3),\ell_{(2,3)}}$ from \ref{['eq:wedgesp']}. As in the depicted octant $\xi_1$ and $\xi_3$ have the same sign, $\Psi_{(1,3),\ell_{(1,3)}}$ is not visualized.

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.2: Lacunary set
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.3
  • ...and 19 more