Commutator estimates for Haar shifts with general measures

Abstract

We study $L^p(\hat{I}¼)$ 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 $\hat{I}¼$ satisfying a dyadic regularity condition which is necessary for $H$ to be bounded on $L^p(\hat{I}¼)$. We show that $\Vert [H,b]{\Vert }_{L^p(\hat{I}¼)\to L^p(\hat{I}¼)}\lesssim \Vert b{\Vert }_{\mathrm{BMO}(\hat{I}¼)}$, 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 ${\hat{A}}_p$ 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 $\hat{I}¼$ for the study of different types of Haar shift operators.