Kakeya sets and directional maximal operators in the plane

Abstract

Abstract. We completely characterize the boundedness of planar directional maximal operators on L p . More precisely, if Ω is a set of directions, we show that M Ω , the maximal operator associated to line segments in the directions Ω, is unbounded on L p , for all p < ∞, precisely when Ω admits Kakeya-type sets. In fact, we show that if Ω does not admit Kakeya sets, then Ω is a generalized lacunary set, and hence M Ω is bounded on L p , for p > 1.

Logarithmic L p Bounds for Maximal Directional Singular Integrals in the Plane

“…The more recent work [1] shows that bounds become independent of N M V p→p p 1, 1 < p < ∞, if and only if V is lacunary of finite order.…”

“…The fact that S L 2 →L 2 √ log N was proved in [6], and this is optimal for generic V. However, we strongly suspect that S L 2 →L 2,∞ (log log N) O (1) . If one proves this, then the conjectured bound MS L 2 →L 2,∞ √ log N (log log N) O(1) follows via the ChangWilson-Wolff inequality, as described in Section 3.…”

Directional discrepancy in two dimensions

“…This condition is also equivalent to the fact that Ω does not "admit Kakeya sets" (for details see [1], [15]). …”

“…Denote A Ω = {R ∈ A Ω : R ⊂ [0, 1] 2 }, or, alternatively, one may define A Ω = {R ∈ A Ω : diam(R) ≤ 1} with [0,1] 2 viewed as a torus. We have Theorem 1.2.…”

“…In addition, we also address a 'sibling' problem: studying the discrepancy with respect to collections B Ω,k of convex polygons in [0,1] 2 with at most k sides whose normals point in the directions defined by Ω (cf. [5], [7] for earlier results) and prove inequalities analogous to Theorems 1.1 and 1.2 (see Theorems 4.3 and 5.3 in the text).…”