Summary: The main result of this paper is the commutator theorem: if the processes \(\mu=(\mu_t)\) and \(\lambda= (\lambda_t)\) are \(A_p\) weights, then the commutator operator from \(L^p_M(\mu)\) into \(L^p(\lambda)\) defined by \([T,M_b]f= \{b_tE((Tf)\mid{\mathcal J}_t)- F(T(bf)\mid{\mathcal J}_t)\}\) is bounded when \(b=(b_t)\in\text{BMO}_v\), where \(v=(\mu\lambda^{-1})^{1/p}\). Moreover, we give a sharp function theorem and its several applications.