Commutator estimates for Haar shifts with general measures

We study $L^p(\mu )$ estimates for the commutator $[H,b]$, where the operator $H$ is a dyadic model of the classical Hilbert transform introduced in \cite{arXiv:2012.10201,arXiv:2212.00090} and is adapted to a non-doubling Borel measure $\mu$ satisfying a dyadic regularity condition which is necessary for $H$ to be bounded on $L^p(\mu )$. We show that $\mathrm{\parallel }[H,b]{\mathrm{\parallel }}_{L^p(\mu )\to L^p(\mu )}\lesssim \mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}(\mu )}$, but to {\it 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 ${\widehat{A}}_p$ 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 $\mu$ for the study of different types of Haar shift operators.