Remarks on Maximal Operators Over Arbitrary Sets of Directions

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.