A note on the Kakeya maximal operator

Abstract

Abstract. In this paper we obtain an upper bound for the L 2 norm for maximal operators associated to arbitrary finite sets of directions in R 2 .

A remark on maximal operators along directions in ${\Bbb R}^2$

“…Also, for lacunary sequences the result holds with β = 0 (see [4] and [7]) and therefore one can take an infinite sequence in this case. In 1995, Barrionuevo [3] obtained the following result:…”

Recent work connected with the Kakeya problem

“…An alternate proof of the two dimensional Carleson-Sjolin result along these lines was given by Cordoba [18]. In three or more dimensions, progress on this problem was initiated by Bourgain (see [10]) who obtained a numerology between partial results which however does not show that (2) would imply (37). For a recent improvement in the numerology see [38] and [59].…”

“…On the other hand, a number of related questions concerning logarithmic factors have been solved only recently or are still open. In particular we should mention the results of Barrionuevo [2] and Katz [26], [27] on the question of maximal functions defined using families of directions in the plane.…”

“…Let Π j = T j × [0, 1] ⊂ R n × R, and letΠ j =T j × [2,3]. By the first step of the proof there are functions u j and f j , u j = f j , with f j supported on Π j , f j ∞ ≤ 1, and | ∂u j ∂t | ≥ const on a subset Y j ⊂Π j which satisfies |Y t j | ≈ δ n−1 for each t ∈ (2, 3).…”

On unboundedness of maximal operators for directional Hilbert transforms

“…They proved the boundedness of the norms of these operators in L p , 1 < p < ∞, for a lacunary U . Upper bounds of such operators depending on the cardinality #U of the set U are considered in the papers [4], [5], [6], [13], [14], [19], and the definitive estimates are due to N. Katz ([13], [14]), where he obtained a logarithmic order for the norms of two different maximal operators depending on #U . Various generalizations of these results were considered in series of papers ( [1], [2], [3], [10], [17]).…”