$L^ 2$-boundedness of spherical maximal operators with multidimensional parameter sets
HTML articles powered by AMS MathViewer
- by Young-Hwa Ha PDF
- Proc. Amer. Math. Soc. 105 (1989), 401-411 Request permission
Abstract:
For $s > 0$, let ${M_s}f(x) = \int _{|y| = 1} {f(x - sy)d\sigma (y)}$ be the spherical mean operator on ${R^n}$. For a certain class of surfaces $S$ in $R_ + ^{n + 1}$ with $\dim S = n - 2$ or $\dim S = n - 1$ with an additional condition, the maximal operator \[ \mathcal {M}f(x) = \sup \limits _{(u,s) \in S} |{M_s}f(x - u)|\] is shown to be bounded on ${L^2}({R^n})$. This extends (on ${L^2}({R^n})$) the theorem of Stein [7], where $S = \{ (0,s):s > 0\}$, and its generalizations to $\dim S = 1$ in Greenleaf [2] and Sogge and Stein [6].References
-
R. Courant and D. Hilbert, Methods of mathematical physics, Interscience, New York, 1962.
- Allan Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981), no. 4, 519–537. MR 620265, DOI 10.1512/iumj.1981.30.30043
- Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
- Alexander Nagel, Elias M. Stein, and Stephen Wainger, Hilbert transforms and maximal functions related to variable curves, 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. 95–98. MR 545242 R. Paley, A proof of a theorem on averages, Proc. London Math. Soc. 31 (1930), 289-300.
- Christopher D. Sogge and Elias M. Stein, Averages of functions over hypersurfaces in $\textbf {R}^n$, Invent. Math. 82 (1985), no. 3, 543–556. MR 811550, DOI 10.1007/BF01388869
- Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116, DOI 10.1073/pnas.73.7.2174
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- 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, DOI 10.1090/S0002-9904-1978-14554-6
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Stephen Wainger, Averages and singular integrals over lower-dimensional sets, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 357–421. MR 864376
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 105 (1989), 401-411
- MSC: Primary 42B25; Secondary 47B38
- DOI: https://doi.org/10.1090/S0002-9939-1989-0955460-X
- MathSciNet review: 955460