Annales Academiæ Scientiarum Fennicæ

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 *L*^{p}
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.

Full document as PDF file

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

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