Two Weighted $BMO$ Estimates for the Maximal Bochner-Riesz Commutator

Abstract

In this note, the author prove that maximal Bochner-Riesz commutator $B^b_{\delta,\ast}$ generated by operator $B_{\delta,\ast}$ and  function $b\in BMO(\omega)$ is a bounded operator from $L^{p}(\mu)$ into $L^{p}(\nu)$, where $\omega\in(\mu\nu^{-1})^{\frac{1}{p}},\mu,\nu\in A_p$ for $1 < p <\infty$. The proof relies heavily on the pointwise estimates for the sharp maximal function of the commutator $B^b_{\delta,\ast}$.