Christ, Michael; Kiselev, Alexander Maximal functions associated to filtrations. (English) Zbl 0974.47025 J. Funct. Anal. 179, No. 2, 409-425 (2001). 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. Reviewer: Boris Rubin (Jerusalem) Cited in 3 ReviewsCited in 209 Documents MSC: 47B38 Linear operators on function spaces (general) 47G10 Integral operators Keywords:maximal operators; \(L^p\)-estimates; characteristic function × Cite Format Result Cite Review PDF Full Text: DOI Link References: [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.; 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.; 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.; T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations, in press. · Zbl 0966.35027 [8] Zygmund, A., Trigonometric Series (1977), Cambridge Univ. Press: 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.