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Singular integrals along lacunary directions in $\mathbb{R}^n$

Natalia Accomazzo, Francesco Di Plinio, Ioannis Parissis

Abstract

A recent result by Parcet and Rogers is that finite order lacunarity characterizes the boundedness of the maximal averaging operator associated to an infinite set of directions in $\mathbb{R}^n$. Their proof is based on geometric-combinatorial coverings of fat hyperplanes by two-dimensional wedges. Seminal results by Nagel-Stein-Wainger relied on geometric coverings of n-dimensional nature. In this article we find the sharp cardinality estimate for singular integrals along finite subsets of finite order lacunary sets in all dimensions. Previous results only covered the special case of the directional Hilbert transform in dimensions two and three. The proof is new in all dimensions and relies, among other ideas, on a precise covering of the n-dimensional Nagel-Stein-Wainger cone by two-dimensional Parcet-Rogers wedges.

Singular integrals along lacunary directions in $\mathbb{R}^n$

Abstract

A recent result by Parcet and Rogers is that finite order lacunarity characterizes the boundedness of the maximal averaging operator associated to an infinite set of directions in . Their proof is based on geometric-combinatorial coverings of fat hyperplanes by two-dimensional wedges. Seminal results by Nagel-Stein-Wainger relied on geometric coverings of n-dimensional nature. In this article we find the sharp cardinality estimate for singular integrals along finite subsets of finite order lacunary sets in all dimensions. Previous results only covered the special case of the directional Hilbert transform in dimensions two and three. The proof is new in all dimensions and relies, among other ideas, on a precise covering of the n-dimensional Nagel-Stein-Wainger cone by two-dimensional Parcet-Rogers wedges.

Paper Structure

This paper contains 14 sections, 11 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lacunary sets of directions and associated frequency projections
  3. Lacunary sets of directions
  4. Nagel-Stein-Wainger frequency projections
  5. Parcet-Rogers frequency projections
  6. Some auxiliary results
  7. Directional weighted norm inequalities
  8. A maximal inequality for Nagel-Stein-Wainger cones
  9. The Chang-Wilson-Wolff reduction
  10. The proof of Theorem \ref{['thm:main']}
  11. The main splitting
  12. Concluding remarks and open questions
  13. Hilbert transform along lacunary-valued, Lipschitz-truncated fields
  14. Extensions to bi-parameter, non-translation invariant kernels

Key Result

Theorem 1

Let $n\ge 2$, $1<p<\infty$, and $\Omega \subset {\mathbb{S}^{n-1}}$ be a lacunary set of finite order. Then where the implicit constants depends on the dimension $n$, on $p$, and on the order of lacunarity of the set $\Omega$.

Figures (1)

  • Figure 2.1: The exterior of a Nagel-Stein-Wainger cone with axis $v$, corresponding to the operator $\mathrm{Id}-W_v$, covers the singularity $v^\perp$; a Parcet-Rogers wedge for a single $\sigma$ is also pictured.

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1.1
  • Definition 2.2
  • Lemma 2.5: Inclusion-Exclusion formula
  • proof
  • Proposition 3.2: Kurtz kurtz
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Proposition 3.6
  • ...and 6 more