The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff's range

Author:
Hitoshi Tanaka

Journal:
Proc. Amer. Math. Soc. **133** (2005), 763-772

MSC (2000):
Primary 42B25

DOI:
https://doi.org/10.1090/S0002-9939-04-07623-3

Published electronically:
August 20, 2004

MathSciNet review:
2113926

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let , , be the Kakeya (Nikodým) maximal operator defined as the supremum of averages over tubes of eccentricity . The (so-called) Fefferman-Stein type inequality:

is shown in the range , where and are some constants depending only on and the dimension and is a weight. The result is a sharp bound up to -factors.

**1.**A. Córdoba,*The Kakeya maximal function and the spherical summation multiplier*, Amer. J. Math.,**99**(1977), 1-22. MR**56:6259****2.**D. Müller and F. Soria,*A double-weight inequality for the Kakeya maximal function*, Fourier Anal. Appl., Kahane Special Issue (1995), 467-478. MR**96k:42026****3.**C. Sogge,*Concerning Nikodým-type sets in -dimensional curved spaces*, J. Amer. Math. Soc.,**12**(1999), 1-31. MR**99h:42037****4.**T. Tao,*From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE*, Notices Amer. Math. Soc.**48**(2001), no3, 294-303. MR**2002b:42021****5.**H. Tanaka,*Some weighted inequalities for the Kakeya maximal operator on functions of product type*, J. Math. Sci. Univ. Tokyo,**6**(1999), 315-333. MR**2001e:42029a****6.**H. Tanaka,*A weighted inequality for the Kakeya maximal operator with a special base*, Tokyo J. Math.,**23**(2000), 255-267. MR**2002g:42027****7.**H. Tanaka,*The Fefferman-Stein type inequality for the Kakeya maximal operator*, Proc. Amer. Math. Soc.,**129**(2001), 2373-2378. MR**2002e:42024****8.**H. Tanaka,*The Fefferman-Stein type inequality for the Kakeya maximal operator II*, Acta Mathematica Sinica, English Series,**18**(2002) no3, 447-454. MR**2003j:42026****9.**T. Tao, A. Vargas, L. Vega,*A bilinear approach to the restriction and Kakeya conjectures*, J. Amer. Math. Soc.,**11**(1998), 967-1000. MR**99f:42026****10.**A. M. Vargas,*A weighted inequality for the Kakeya maximal operator*, Proc. Amer. Math. Soc.,**120**(1994), 1101-1105. MR**94f:42023****11.**T. Wolff,*An improved bound for Kakeya type maximal functions*, Rev. Mat. Iberoamericana,**11**(1995), 651-674. MR**96m:42034****12.**T. Wolff,*Recent work connected with the Kakeya problem*, Prospects in Mathematics (Princeton, NJ, 1996), 129-162, Amer. Math. Soc., Providence, RI, 1999.MR**2000d:42010**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
42B25

Retrieve articles in all journals with MSC (2000): 42B25

Additional Information

**Hitoshi Tanaka**

Affiliation:
Department of Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan

Email:
htanaka@ms.u-tokyo.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-04-07623-3

Received by editor(s):
October 22, 2003

Published electronically:
August 20, 2004

Additional Notes:
This work was supported by the Fūjyukai Foundation.

Communicated by:
Andreas Seeger

Article copyright:
© Copyright 2004
American Mathematical Society