Commutator estimates for Haar shifts with general measures

We study Lᵖ(μ) 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 μ satisfying a dyadic regularity condition which is necessary for H to be bounded on Lᵖ(μ). We show that \|[H, b]\|Lᵖ(μ) → Lᵖ(μ) \|b\|BMO(μ), 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\"{o}lder inequalities for these new weight classes. Finally, we revisit the appropriate class of nonhomogeneous measures μ for the study of different types of Haar shift operators.