Advertisement

Estimates for the Kakeya Maximal Operator on Radial Functions in Rn

  • Anthony Carbery
  • Eugenio Hernández
  • Fernando Soria

Abstract

For a real number N > 1, the Kakeya maximal operator K N is defined on locally integrable functions fof R n as
$$ {K_N}f\left( x \right) = \mathop {\sup }\limits_{x \in R \in {B_N}} \frac{1}{{\left| R \right|}}\int {_R} \left| {f\left( y \right)} \right|dy $$
where B N denotes the class of all rectangles in R n of eccentricity N, that is, congruent with any dilate of the rectangle [0,1]n-1x [0, N], and where x007C;Ax007C; represents the Lebesgue measure of the set A.

Keywords

Maximal Operator Maximal Function Radial Function Singular Integral Operator Weak Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Carbery, E. Romera and F. Soria, Radial weights and mixed norm inequalities for the disc multiplier, Preprint.Google Scholar
  2. [2]
    A. Cordoba, The Kakeya maximal function and the spherical summation multiplier, Amer. J. Math 99(1977), 1–22.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    C. Fefferman, A note on the spherical summation multiplier, Israel J. Math. 15(1973), 44–52.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    A. Garsia, “Topics in almost everywhere convergence,” Markham Publising Co., 1970.Google Scholar
  5. [5]
    M. de Guzmán, “Differentiation of Integrals in Rn,” Springer-Verlag, 1975.Google Scholar
  6. [6]
    S. Igari, Kakeya’s maximal function for radial functions, preprint.Google Scholar
  7. [7]
    P. Sjögren and F. Soria, Weak type (1,1) estimates for some extension operators related to rough maximal functions, Preprint.Google Scholar
  8. [8]
    E. M. Stein, “Singular Integrals and Differentiability properties of functions,” Princeton U. Press, 1970.zbMATHGoogle Scholar
  9. [9]
    J. Strömberg, Maximal functions associated to rectangles with uniformly distributed direccions, Annals of Math. 107(1978), 309–402.CrossRefGoogle Scholar
  10. [10]
    S. Wainger, Applications of Fourier Transforms to averages over lower dimensional sets, Proc. Symp. in Pure Math. XXXV, part I(1979), 85–94.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Tokyo 1991

Authors and Affiliations

  • Anthony Carbery
    • 1
  • Eugenio Hernández
    • 2
  • Fernando Soria
    • 3
  1. 1.University of SussexFalmer, BringtonUK
  2. 2.Universidad Autónoma de MadridMadridSpain
  3. 3.Univ. Autónoma de Madrid and The Institute for Advanced StudyPrincetonUSA

Personalised recommendations