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Remarks on Maximal Operators Over Arbitrary Sets of Directions
Author(s) -
Katz Nets Hawk
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609399005949
Subject(s) - mathematics , combinatorics , operator (biology) , cardinality (data modeling) , lemma (botany) , unit vector , norm (philosophy) , maximal operator , unit (ring theory) , upper and lower bounds , discrete mathematics , mathematical analysis , ecology , biochemistry , chemistry , mathematics education , poaceae , repressor , biology , computer science , transcription factor , political science , data mining , law , bounded function , gene
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 e s be the unit vector of slope s in R 2 . Then, following [ 7 ], we define the maximal operator on R 2 associated with the set Σ byM Σ 0 f ( x ) = sup s ∈ Σ1 2 ∫ | t | < 1| f ( x − t e s ) | d t .The history of the bounds obtained onM Σ 0is quite curious. The earliest study of related operators was carried out by Cordoba [ 2 ]. He obtained a bound of C √(1 + log N ) on the L 2 operator norm of the Kakeya maximal operator over rectangles of length 1 and eccentricity N . This operator is analogous toM Σ 0withΣ = { 1 N , 2 N , … , 1 } .However, for arbitrary sets Σ, the best known result seems to be C (1 + log N ). 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.