Estimates for the Kakeya Maximal Operator on Radial Functions in Rn

  • Anthony Carbery
  • Eugenio Hernández
  • Fernando Soria


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.


Maximal Operator Maximal Function Radial Function Singular Integral Operator Weak Type 
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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

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