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REMARKS ON MAXIMAL OPERATORS OVER ARBITRARY SETS OF DIRECTIONS

Published online by Cambridge University Press:  01 November 1999

NETS HAWK KATZ
NETS HAWK KATZ
Affiliation:
Department of Mathematics, Edinburgh University, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ
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Abstract

Throughout this paper, we shall let Σ be a subset of [0, 1] having cardinality N. We shall consider Σ to be a set of slopes, and for any s∈Σ, we shall let es be the unit vector of slope s in ℝ2. Then, following [7], we define the maximal operator on ℝ2 associated with the set Σ by

formula here

The history of the bounds obtained on [Mfr ]0Σ is quite curious. The earliest study of related operators was carried out by Cordoba [2]. He obtained a bound of C√(1+logN) on the L2 operator norm of the Kakeya maximal operator over rectangles of length 1 and eccentricity N. This operator is analogous to [Mfr ]0Σ with

formula here

However, for arbitrary sets Σ, the best known result seems to be C(1+logN). This follows from Lemma 5.1 in [1], but a point of view which produces a proof appears already in [8]. However, in this paper, we prove the following.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 31 , Issue 6 , November 1999 , pp. 700 - 710
Copyright
© The London Mathematical Society 1999

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