Commutators of Hilbert transforms along monomial curves

TL;DR

This paper investigates the boundedness of commutators of Hilbert transforms along monomial curves within non-isotropic BMO spaces, revealing the importance of curvature in these operators.

Contribution

It establishes upper and lower bounds for these commutators using novel non-isotropic BMO and testing BMO spaces, highlighting the role of curvature.

Findings

Upper bound of commutator in non-isotropic BMO space

Introduction of testing BMO space associated with monomial curves

Curvature of curves affects BMO space containment

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.