## Maximal functions associated to filtrations.(English)Zbl 0974.47025

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.

### MSC:

 47B38 Linear operators on function spaces (general) 47G10 Integral operators
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### References:

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