Multiparameter maximal functions along dilation-invariant hypersurfaces. (English) Zbl 0578.42018

Summary: Consider the hypersurface \(x_{n+1}=\prod^{n}_{1}x_ i^{\alpha_ i}\) in \(R^{n+1}\). The associated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles \(\{0\leq x_ i\leq h_ i,i=1,...,n\}\). We prove that this operator is bounded on \(L^ p\) for \(p>1\). An analogous result is proved for a quadratic surface in \(R^ 3\).


42B25 Maximal functions, Littlewood-Paley theory
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[1] Walter Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766 – 770. · Zbl 0143.34701
[2] Alexander Nagel and Elias M. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 (1984), no. 1, 83 – 106. · Zbl 0546.42017
[3] A. Nagel, E. M. Stein, and S. Wainger, Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 3, 1060 – 1062. · Zbl 0391.42015
[4] Alexander Nagel and Stephen Wainger, \?² boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multiple parameter group, Amer. J. Math. 99 (1977), no. 4, 761 – 785. · Zbl 0374.44003
[5] Peter Sjögren, Fatou theorems and maximal functions for eigenfunctions of the Laplace-Beltrami operator in a bidisk, J. Reine Angew. Math. 345 (1983), 93 – 110. · Zbl 0512.43005
[6] -, Admissible convergence of Poisson integrals in symmetric spaces, Preprint no. 1985-4, Chalmers Univ. of Technology and Univ. of Göteborg, Göteborg, Sweden.
[7] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[8] Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239 – 1295. · Zbl 0393.42010
[9] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. · Zbl 0232.42007
[10] Robert S. Strichartz, Singular integrals supported on submanifolds, Studia Math. 74 (1982), no. 2, 137 – 151. · Zbl 0501.43007
[11] James T. Vance Jr., \?^{\?}-boundedness of the multiple Hilbert transform along a surface, Pacific J. Math. 108 (1983), no. 1, 221 – 241. · Zbl 0462.44001
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