Maximal Functions Associated to Filtrations

Let T be a bounded linear, or sublinear, operator from Lp(Y) to Lq(X). A maximal operator T*f(x)=supj |T(f·χYj)(x)| is associated to any sequence of subsets Yj of Y. Under the hypotheses that q>p and the sets Yj are nested, we prove that T* is also bounded. Classical theorems of Menshov and Zygmund are obtained as corollaries. Multilinear generalizations of this theorem are also established. These results are motivated by applications to the spectral analysis of Schrödinger operators.