Averages of functions over hypersurfaces in \(\mathbb R^n\). (English) Zbl 0626.42009

Given a \(C^\infty\) hypersurface \(S\) in \(\mathbb R^n\), \(n\geq 3\), the ‘mean-value’ operators \(M_t\) are defined by \[ M_t(f)(x)=\int_{S}f(x-ty)\psi (y)\,d\sigma (y),\quad t>0, \] where \(\psi\) is a fixed function in \(C_0^\infty(S)\) and \(d\sigma\) is the induced Lebesgue measure on \(S\). The authors study \(L^p\) inequalities for the maximal operator \(\mathcal M\) defined by \((\mathcal Mf)(x)=\sup_{t>0}| M_t(f)(x)|.\) It is shown that if the Gaussian curvature of \(S\) does not vanish of infinite order at each point of \(S\), then \(\mathcal M\) is bounded in \(L^p\) in a certain range \(p>p_0(S)\).
Reviewer: B.P.Duggal


42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B25 Maximal functions, Littlewood-Paley theory
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[1] Greenleaf, A.: Principal curvature and harmonic analysis. Indiana Math. J.30, 519-537 (1982) · Zbl 0517.42029
[2] Hörmander, L.: The Analysis of Linear Partial Differential Operators, I. New York-Berlin: Springer 1983 · Zbl 0521.35001
[3] Milnor, J.: Morse Theory. Ann. Math. Stud. Princeton, New Jersey: Princeton Univ. Press 1973
[4] Randol, B.: On the asymptotic behavior of the Fourier transform of the indicator function of a convex set. Trans. Am. Math. Soc.139, 279-285 (1969) · Zbl 0183.26905
[5] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton, New Jersey: Princeton Univ. Press 1970 · Zbl 0207.13501
[6] Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Ann. Math. Stud. Princeton, New Jersey: Princeton Univ. Press 1970 · Zbl 0193.10502
[7] Stein, E.M.: Maximal functions: spherical means. Proc. Natl. Acad. Sci. USA73, 2174-2175 (1976) · Zbl 0332.42018
[8] Stein, E.M., Wainger, S.: Problems in harmonic analysis related to curvatures. Bull. Am. Math. Soc.84, 1239-1295 (1978) · Zbl 0393.42010
[9] Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton, New Jersey: Princeton Univ. Press 1971 · Zbl 0232.42007
[10] Svenson, I.: Estimates for the Fourier transform of the characteristic function of a convex set. Arkiv För Mathematik9, 11-22 (1971) · Zbl 0221.52001
[11] Zygmund, A.: Trigonometric Series. Cambridge-New York: Cambridge Univ. Press 1979 · JFM 58.0296.09
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