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.


47B38 Linear operators on function spaces (general)
47G10 Integral operators
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[1] Christ, M.; Kiselev, A., Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results, J. amer. math. soc., 11, 771-797, (1998) · Zbl 0899.34051
[2] M. Christ, and, A. Kiselev, WKB asymptotics of generalized eigenfunctions of one-dimensional Schrödinger operators with slowly decaying potentials, J. Funct. Anal, to appear. · Zbl 0985.34078
[3] Kiselev, A., Interpolation theorem related to a.e. convergence of integral operators, Proc. amer. math. soc., 127, 1781-1788, (1999) · Zbl 0918.42023
[4] Menshov, D., Sur LES series de fonctions orthogonales, Fund. math., 10, 375-420, (1927) · JFM 53.0267.03
[5] Paley, R.E.A.C., Some theorems on orthonormal functions, Studia math., 3, 226-245, (1931) · Zbl 0003.35201
[6] H. Smith, and, C. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations, in press. · Zbl 0972.35014
[7] T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations, in press.
[8] Zygmund, A., Trigonometric series, (1977), Cambridge Univ. Press Cambridge · Zbl 0367.42001
[9] Zygmund, A., A remark on Fourier transforms, Proc. Cambridge philos. soc., 32, 321-327, (1936) · JFM 62.0468.02
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