Maximal functions associated to filtrations (Q5927518)

Let \((X,\mu)\) and \((Y,\nu)\) be arbitrary measure spaces. To any sequence of measurable subsets \(\{Y_n \}\) of \(Y\) and any bounded linear operator \(T: L^p(Y) \to L^q(X)\) one can associate the maximal operator \(T^*f(x)=\sup_n |T(f \cdot \chi_{Y_n})(x)|\), where \(\chi_{Y_n}\) designates the characteristic function of \(Y_n\). It is proved that \(T^*\) is bounded from \(L^p\) into \(L^q\) provided that \(1 \leq p< q \leq \infty\) and the sets \(Y_n\) are nested. Classical theorems of Menshov and Zygmund are obtained as corollaries. Multilinear generalizations of this result are also established.