Commutator estimates for Haar shifts with general measures

We study $L^p\left(\mu \right)$ estimates for the commutator $[H,b]$, where the operator $H$ is a dyadic model of the classical Hilbert transform introduced in [9], [10] and is adapted to a non-doubling Borel measure μ satisfying a dyadic regularity condition which is necessary for $H$ to be bounded on $L^p\left(\mu \right)$. We show that ${\Vert [H,b]\Vert }_{L^p\left(\mu \right)\to L^p\left(\mu \right)}\lesssim {\Vert b\Vert }_{\mathrm{BMO}\left(\mu \right)}$, 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 ${\hat{A}}_p$ that we define for our non-homogeneous setting. This requires establishing reverse Hölder 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.