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Abstract
The Hilbert transforms associated with monomial curves have a natural nonisotropic 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 nonisotropic 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 nonisotropic BMO space is contained in the testing BMO space. We moreover show that the nonzero curvature of such monomial curves is important, since when considering Hilbert transforms associated with lines, the parallel versions of nonisotropic BMO space and testing BMO space have overlaps but do not have containment.
Original language  English 

Pages (fromto)  295311 
Number of pages  17 
Journal  Studia Mathematica 
Volume  257 
Issue number  3 
DOIs  
Publication status  Published  2021 
Keywords
 BMO
 commutator
 Hilbert transform
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Harmonic analysis and dispersive partial differential equations
Li, J., Guo, Z., Kenig, C. & Nakanishi, K.
31/01/17 → …
Project: Research

Harmonic analysis: function spaces and partial differential equations
Duong, X., Hofmann, S., Ouhabaz, E. M. & Wick, B.
11/02/19 → 10/02/22
Project: Other