Maximal functions associated to filtrations

Let T be a bounded linear, or sublinear, operator from L-p(Y) to L-q(X). A maximal operator T*f(x) = sup(j) \T(f . chiY(j))(x)\ is associated to any s equence of subsets Y-j of Y. Under the hypotheses that q > p and the sets Y -j are nested, we prove that T* is also bounded. Classical theorems of Mens hov and Zygmund are obtained as corollaries. Multilinear generalizations of this theorem are also established. These results are motivated by applicat ions to the spectral analysis of Schrodinger operators. (C) 2001 Academic P ress.