A Note on Maximal Averages in the Plane

The study of directional maximal operators goes back at least to the works of Córdoba [Co1] and Strömberg [Str1] in the 70’s and is as recent as the results of Alfonseca, Soria and Vargas [ASV1], [ASV2], Karagulyan and Lacey [KL] and Bateman and Katz [Ba1], [BK1]. Progress in the area involved the work of many authors and we refer to [W], [ASV2], [KL] and references therein for historical background. The general situation considered in [ASV2] is the following. Suppose Ω = Ω0 ∪j Ωj ⊂ [0, 1] is a closed set of slopes in the plane, where Ω0 = {θ1 > θ2 > · · · > θj > · · · } is an ordered subset of Ω and Ωj = [θj−1, θj) ∩ Ω. Denote by BΩ be the collection of all rectangles so that its longest side has slope in Ω, and define MΩf(x) = sup x∈R∈BΩ 1 |R| ∫