Mathematica
Volumen 45, 2020, 411-427

# DIFFERENTIATING ORLICZ SPACES WITH RARE BASES OF RECTANGLES

## Emma D'Aniello, Laurent Moonens and Joseph M. Rosenblatt

Università degli Studi della Campania "Luigi Vanvitelli", Dipartimento di Matematica e Fisica
Viale Lincoln n. 5, 81100 Caserta, Italia; emma.daniello 'at' unicampania.it

Université Paris-Sud, CNRS UMR8628, Université Paris-Saclay
Laboratoire de Mathématiques d'Orsay, Bâtiment 307
F-91405 Orsay Cedex, France; laurent.moonens 'at' math.u-psud.fr

University of Illinois at Urbana-Champaign, Department of Mathematics
1409 W. Green Street, Urbana, IL 61801-2975, U.S.A.; rosnbltt 'at' illinois.edu

Abstract. In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space. We also make a simple observation showing that the maximal operator associated to rectangles oriented in a fixed sequence of directions, is either bounded on all Lp spaces for 1 < p ≤ ∞, or fails to be bounded on any of them (adding the case p = ∞ to a dichotomy obtained previously by Bateman).

2010 Mathematics Subject Classification: Primary 42B25; Secondary 26B05.

Key words: Lebesgue's differentiation theorem, rectangular differentiation bases, directional maximal operators.

Reference to this article: E. D'Aniello, L. Moonens and J. M. Rosenblatt: Differentiating Orlicz spaces with rare bases of rectangles. Ann. Acad. Sci. Fenn. Math. 45 (2020), 411-427.

https://doi.org/10.5186/aasfm.2020.4523