Commutator estimates for Haar shifts with general measures

We study Lᵖ () estimates for the commutator H, b, where the operator H is a dyadic model of the classical Hilbert transform introduced in arXiv: 2012. 10201, arXiv: 2212. 00090 and is adapted to a non-doubling Borel measure satisfying a dyadic regularity condition which is necessary for H to be bounded on Lᵖ (). We show that \|H, b\|₋㵵 () ₋㵵 () \|b\|₁₌₎ (), but to characterize martingale BMO requires additional commutator information. We prove weighted inequalities for H, b together with a version of the John-Nirenberg inequality adapted to appropriate weight classes Aₚ that we define for our non-homogeneous setting. This requires establishing reverse H\"older inequalities for these new weight classes. Finally, we revisit the appropriate class of nonhomogeneous measures for the study of different types of Haar shift operators.