Commutators of Hilbert transforms along monomial curves

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 moreover show that the non-zero curvature of such monomial curves is important, since when considering Hilbert transforms associated with lines, the parallel versions of non-isotropic BMO space and testing BMO space have overlaps but do not have containment.