Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hilbert transforms and maximal functions along flat curves on the Heisenberg group

Lingxiao Zhang

Abstract

We establish the $L^p$ boundedness of Hilbert transforms and maximal functions along flat curves in the Heisenberg group. This generalizes the $\mathbb{R}^n$ result by Carbery, Christ, Vance, Wainger, and Watson. What is new about our result compared to the Heisenberg group generalization by Carbery, Wainger, and Wright is that we allow all three components of the curves to vary independently, we keep the original form of the conditions required in the $\mathbb{R}^n$ case, and our method is likely to be generalized to other stratified nilpotent groups.

Hilbert transforms and maximal functions along flat curves on the Heisenberg group

Abstract

We establish the boundedness of Hilbert transforms and maximal functions along flat curves in the Heisenberg group. This generalizes the result by Carbery, Christ, Vance, Wainger, and Watson. What is new about our result compared to the Heisenberg group generalization by Carbery, Wainger, and Wright is that we allow all three components of the curves to vary independently, we keep the original form of the conditions required in the case, and our method is likely to be generalized to other stratified nilpotent groups.
Paper Structure (9 sections, 30 theorems, 416 equations)

This paper contains 9 sections, 30 theorems, 416 equations.

Table of Contents

  1. Introduction
  2. Calderón-Zygmund theory
  3. Littlewood-Paley theory
  4. Key estimates: TEXT method
  5. Regularity of the density
  6. An estimation for the Jacobian
  7. Regularity of the Jacobian and the partition of unity functions
  8. Remarks on the omitted cases
  9. Boundedness of TEXT and TEXT

Key Result

Theorem 1.1

Suppose $\Gamma(t)=(t,\alpha(t))$ is a continuous odd curve in $\mathbb{R}^2$ with $\alpha'(0)=0$, and is $C^2$ for $t\in (0,\infty)$. Denote If $\alpha"(t)>0$ for $t>0$, and then $H_\Gamma$ is bounded on $L^p(\mathbb{R}^2)$ for $1<p< \infty$, and $M_\Gamma$ is bounded on $L^p(\mathbb{R}^2)$ for $1<p\leq \infty$.

Theorems & Definitions (75)

  • Theorem 1.1: CARBERYCHRIST
  • Theorem 1.2: VANCE
  • Theorem 1.3: Carbery, Wainger, and Wright CWW
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 65 more