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Curved commutators in the plane

Kangwei Li, Henri Martikainen, Tuomas Oikari

Abstract

We complete the $L^p$ boundedness theory of commutators of Hilbert transforms along monomial curves by providing the previously missing lower bounds. This optimal result now covers all monomial curves while previous results had significant geometric restrictions. We also, for the first time, develop the corresponding necessity theory for curves with non-vanishing torsion.

Curved commutators in the plane

Abstract

We complete the boundedness theory of commutators of Hilbert transforms along monomial curves by providing the previously missing lower bounds. This optimal result now covers all monomial curves while previous results had significant geometric restrictions. We also, for the first time, develop the corresponding necessity theory for curves with non-vanishing torsion.
Paper Structure (5 sections, 14 theorems, 230 equations)

This paper contains 5 sections, 14 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry behind the factorization
  4. Approximate weak factorization
  5. Curves with non-vanishing torsion

Key Result

Theorem 1.1

Let $b\in L_{\operatorname{loc}}^1(\mathbb{R}^2;\mathbb{C})$ and $H_\gamma f(x) := \int_{\mathbb{R}} f(x-\gamma(t)) \frac{\,\mathrm{d} t}{t}$ be the Hilbert transform along the curve where $\beta_2>\beta_1>0$. Let $1 < p < \infty$ and suppose that $[b, H_\gamma]$ is a bounded operator on $L^p.$ Then $b\in \operatorname{BMO}_\gamma(\mathbb{R}^2)$ -- in fact, we have the quantitative commutator low

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.8
  • proof
  • Lemma 2.14
  • proof
  • Lemma 2.17
  • ...and 17 more