A characterization for the boundedness of positive operators in a filtered measure space (Q2636905)

The authors characterize the \(L^p(\mu)\)-to-\(L^q(\mu)\) boundedness of positive operators of the form \(Tf=\sum_{i\in\mathbb{Z}}\alpha_i E_i f\), where the \(E_i=E(\;|\mathcal{F}_i)\) are an increasing sequence of conditional expectations, and \(\alpha_i\in L^\infty(\mathcal{F}_i)\) are given bounded positive functions. The characterizing condition is a slight weakening of the (obviously necessary) condition that \(\|1_F T(1_F)\|_{L^r(\mu)}\leq C\|1_F\|_{L^s(\mu)}\) for all \(F\in\bigcup_{i\in\mathbb{Z}}\mathcal{F}_i\) of finite measure and both \((r,s)=\{(q,p),(p',q')\}\). Previously, the \(L^p(\mu)\)-to-\(L^q(\nu)\) boundedness of such operators has been similarly characterized in the case of the dyadic conditional expectations of \(\mathbb{R}^d\), even for two different measures \(\mu\) and \(\nu\); thus the present result is a generalization in terms of the operators but not in terms of the measures. The proof is an adaptation of a dyadic argument of \textit{S.~Treil} [``A remark on two weight estimates for positive dyadic operators'', \url{arXiv:1201.1455}].