Maximal Functions Associated to Filtrations

Let T be a bounded linear, or sublinear, operator from L p (Y) to L q (X). To any sequence of subsets Y j of Y is associated a maximal operator T f(x) = sup j jT (f Yj)(x)j. Under the hypotheses that q > p and the sets Y j 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 Schrr odinger operators.