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Commutators of Hilbert transforms along monomial curves

Rosemarie Bongers, Zihua Guo, Ji Li, Brett D. Wick

Abstract

The Hilbert transforms associated with monomial curves have a natural non-isotropic structure. We study the commutator of such Hilbert transforms and a symbol $b$ and prove the upper bound of this commutator when $b$ is in the corresponding non-isotropic BMO space by using the Cauchy integral trick. We also consider the lower bound of this commutator by introducing a new testing BMO space associated with the given monomial curve, which shows that the classical non-isotropic BMO space is contained in the testing BMO space. We also show that the non-zero curvature of such monomial curves are important, since when considering Hilbert transforms associated with lines, the parallel version of non-isotropic BMO space and testing BMO space have overlaps but do not have containment.

Commutators of Hilbert transforms along monomial curves

Abstract

The Hilbert transforms associated with monomial curves have a natural non-isotropic structure. We study the commutator of such Hilbert transforms and a symbol and prove the upper bound of this commutator when is in the corresponding non-isotropic BMO space by using the Cauchy integral trick. We also consider the lower bound of this commutator by introducing a new testing BMO space associated with the given monomial curve, which shows that the classical non-isotropic BMO space is contained in the testing BMO space. We also show that the non-zero curvature of such monomial curves are important, since when considering Hilbert transforms associated with lines, the parallel version of non-isotropic BMO space and testing BMO space have overlaps but do not have containment.

Paper Structure

This paper contains 6 sections, 11 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Sufficient condition for boundedness: Proof of Theorem \ref{['theorem:parabolic_sufficient']}
  3. Necessary condition for boundedness: Proof of Theorem \ref{['theorem:parabolic_necessary']}
  4. The non-parabolic setting
  5. Curves with torsion
  6. References

Key Result

Theorem 1.1

If $b \in BMO_{\gamma}$, and $1<p<\infty$ then the commutator $[b, H_{\gamma}]$ is bounded from $L^p(\mathbb{R}^2)$ to itself, and

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Theorem 2.1
  • proof
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 10 more