Acta Mathematica Sinica

, Volume 18, Issue 3, pp 447–454

The Fefferman-Stein-Type Inequality for the Kakeya Maximal Operator II

ORIGINAL ARTICLES

DOI: 10.1007/s10114-002-0184-8

Cite this article as:
Tanaka, H. Acta Math Sinica (2002) 18: 447. doi:10.1007/s10114-002-0184-8
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Abstract

Let Kδ, 0 < δ≪1, be the Kakeya maximal operator defined as the supremum of averages over tubes of the eccentricity δ. The (so-called) Fefferman-Stein-type inequality: \( {\left\| {K_{\delta } f} \right\|}_{{L^{d} {\left( {{\text{R}}^{d} ,w} \right)}}} \leqslant C_{d} {\left( {\frac{1} {\delta }} \right)}^{{{{\left( {d - 2} \right)}} \mathord{\left/ {\vphantom {{{\left( {d - 2} \right)}} {2d}}} \right. \kern-\nulldelimiterspace} {2d}}} {\left( {\log {\left( {\frac{1} {\delta }} \right)}} \right)}^{{a_{d} }} {\left\| f \right\|}_{{L^{d} {\left( {{\text{R}}^{d} ,K_{\delta } w} \right)}}} \)is shown, where Cd and αd are constants depending only on the dimension d and w is a weight. The result contains the exponent (d−2)/2d which is smaller than the exponent (d−2)(d−1)/d(2d−3) obtained in [7].

Keywords

Maximal functions Weighted inequalties 

MR (2000) Subject Classification

42B25 

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  1. 1.Department of MathematicsGakushuin UniversityTokyoJapan

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