Michael Christ, Alexander Kiselev
Maximal functions associated to filtrations
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ABSTRACT.  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 $T^*f(x) = \sup_j |T(f\cdot\chi_{Y_j})(x)|$.
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 Schr\"odinger operators.