Title: Directional maximal operators and lacunarity in higher dimensions.
Author(s): Javier Parcet and Keith Rogers.
Source: Amer. J. Math. 137 (2015) 1535-1557.
Date of publication: December, 2015.
Abstract: We introduce a notion of lacunarity in higher dimensions for which we can bound the associated directional maximal operators in Lp(Rn), with p>1. In particular, we are able to treat the classes previously considered by Nagel-Stein-Wainger, Sjogren-Sjolin and Carbery. Closely related to this, we find a characterization of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue-type differentiation of integrals in Lploc(Rn), replacing balls by tubes which point in these directions.