Inequalities for some maximal functions. II

Authors:
M. Cowling and G. Mauceri

Journal:
Trans. Amer. Math. Soc. **296** (1986), 341-365

MSC:
Primary 42B25; Secondary 42B10

DOI:
https://doi.org/10.1090/S0002-9947-1986-0837816-0

MathSciNet review:
837816

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Abstract | References | Similar Articles | Additional Information

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

**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**]>.

**[1]**Michael G. Cowling,*On Littlewood-Paley-Stein theory*, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 21–55. MR**639463****[2]**Michael Cowling and Giancarlo Mauceri,*Inequalities for some maximal functions. I*, Trans. Amer. Math. Soc.**287**(1985), no. 2, 431–455. MR**768718**, https://doi.org/10.1090/S0002-9947-1985-0768718-5**[3]**Allan Greenleaf,*Principal curvature and harmonic analysis*, Indiana Univ. Math. J.**30**(1981), no. 4, 519–537. MR**620265**, https://doi.org/10.1512/iumj.1981.30.30043**[4]**Christopher D. Sogge and Elias M. Stein,*Averages of functions over hypersurfaces in 𝑅ⁿ*, Invent. Math.**82**(1985), no. 3, 543–556. MR**811550**, https://doi.org/10.1007/BF01388869**[5]**Elias M. Stein,*Maximal functions. I. Spherical means*, Proc. Nat. Acad. Sci. U.S.A.**73**(1976), no. 7, 2174–2175. MR**0420116****[6]**E. M. Stein,*Some problems in harmonic analysis*, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–20. MR**545235****[7]**Elias M. Stein and Stephen Wainger,*Problems in harmonic analysis related to curvature*, Bull. Amer. Math. Soc.**84**(1978), no. 6, 1239–1295. MR**508453**, https://doi.org/10.1090/S0002-9904-1978-14554-6**[8]**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. MR**0304972**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0837816-0

Keywords:
Maximal function,
hypersurface,
Fourier transform

Article copyright:
© Copyright 1986
American Mathematical Society