Estimates for some Kakeya-type maximal operators
HTML articles powered by AMS MathViewer
- by Jose Barrionuevo PDF
- Trans. Amer. Math. Soc. 335 (1993), 667-682 Request permission
Abstract:
We use an abstract version of a theorem of Kolmogorov-Seliverstov-Paley to obtain sharp ${L^2}$ estimates for maximal operators of the form: \[ {\mathcal {M}_\mathcal {B}}f(x) = \sup \limits _{x \in S \in \mathcal {B}} \frac {1}{{|S|}}\int _S {|f(x - y)|dy} \] . We consider the cases where $\mathcal {B}$ is the class of all rectangles in ${{\mathbf {R}}^n}$ congruent to some dilate of ${[0,1]^{n - 1}} \times [0,{N^{ - 1}}]$; the class congruent to dilates of ${[0,{N^{ - 1}}]^{n - 1}} \times [0,1]$ ; and, in ${{\mathbf {R}}^2}$ , the class of all rectangles with longest side parallel to a particular countable set of directions that include the lacunary and the uniformly distributed cases.References
-
J. Barrionuevo, Ph.D. thesis, University of Rochester, 1990.
- Michael Christ, Javier Duoandikoetxea, and José L. Rubio de Francia, Maximal operators related to the Radon transform and the Calderón-Zygmund method of rotations, Duke Math. J. 53 (1986), no. 1, 189–209. MR 835805, DOI 10.1215/S0012-7094-86-05313-5
- Antonio Cordoba, The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977), no. 1, 1–22. MR 447949, DOI 10.2307/2374006
- Antonio Córdoba, Geometric Fourier analysis, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vii, 215–226 (English, with French summary). MR 688026
- Antonio Córdoba, Maximal functions, covering lemmas and Fourier multipliers, 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. 29–50. MR 545237
- A. Córdoba, The multiplier problem for the polygon, Ann. of Math. (2) 105 (1977), no. 3, 581–588. MR 438022, DOI 10.2307/1970926
- A. Cordoba and R. Fefferman, On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 2, 423–425. MR 433117, DOI 10.1073/pnas.74.2.423 C. A. Fefferman, A note on spherical summation multipliers, Israel J. Math 15 (1973).
- Young-Hwa Ha, $L^2$-boundedness of spherical maximal operators with multidimensional parameter sets, Proc. Amer. Math. Soc. 105 (1989), no. 2, 401–411. MR 955460, DOI 10.1090/S0002-9939-1989-0955460-X L. Hörmander, The analysis of linear partial differential operators, Vol. I, Springer-Verlag, Berlin and New York, 1983. A. N. Kolmogorov and G. Seliverstov, Sur la convergence des séries de Fourier, C.R. Acad. Sci. Paris 178 (1925). —, Sur la convergence des séries de Fourier, Rend. Accad. Naz. Lincei 3 (1926).
- 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. MR 466470, DOI 10.1073/pnas.75.3.1060 E. M. Stein, Lecture Notes, Princeton Univ., 1978-1979 academic year.
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 J. Strömberg, Maximal functions associated to uniformly distributed families of directions, Ann. of Math. 108 (1978). —, Weak estimates on a maximal function with rectangles in certain directions, Ark. Mat. 15 (1977).
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 335 (1993), 667-682
- MSC: Primary 42B25; Secondary 47B38
- DOI: https://doi.org/10.1090/S0002-9947-1993-1150012-9
- MathSciNet review: 1150012