Inequalities for some maximal functions. II
Authors:
M. Cowling and G. Mauceri
Journal:
Trans. Amer. Math. Soc. 296 (1986), 341365
MSC:
Primary 42B25; Secondary 42B10
MathSciNet review:
837816
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Abstract: Let be a smooth compact hypersurface in , and let be a measure on , absolutely continuous with respect to surface measure. For in denotes the dilate of by , normalised to have the same total variation as : for in denotes the maximal function . We seek conditions on which guarantee that the a priori estimate holds; this estimate entails that the sublinear operator extends to a bounded operator on the Lebesgue space . Our methods generalise E. M. Stein's treatment of the "spherical maximal function" [5]: a study of "Riesz operators", functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of . These depend on the geometry of and the relation between and surface measure on . In particular, we find that there are natural geometric maximal operators limited on if and only if is some number in , and may be greater than . This answers a question of S. Wainger posed by Stein [6]>.
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 M. Cowling, On LittlewoodPaleyStein theory, Rend. Circ. Mat. Palermo Suppl. 1 (1981), 2155. MR 639463 (83h:42024)
 [2]
 M. Cowling and G. Mauceri, Inequalities for some maximal functions. I, Trans. Amer. Math. Soc. 287 (1985), 431455. MR 768718 (86a:42023)
 [3]
 A. Greenleaf, Principal curvature and harmonic analysis, Indiana Math. J. 30 (1982), 519537. MR 620265 (84i:42030)
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 C. D. Sogge and E. M. Stein, Averages of functions over hypersurfaces in , Invent. Math. 82 (1985), 543556. MR 811550 (87d:42030)
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 E. M. Stein, Maximal functions: spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 21742175. MR 0420116 (54:8133a)
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 , Some problems in harmonic analysis, Proc. Sympos. Pure Math., vol. 35, part I, Amer. Math. Soc., Providence, R.I., 1979, pp. 320. MR 545235 (80m:42027)
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 E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 12391295. MR 508453 (80k:42023)
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 E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608378160
PII:
S 00029947(1986)08378160
Keywords:
Maximal function,
hypersurface,
Fourier transform
Article copyright:
© Copyright 1986
American Mathematical Society
