Annales Academiæ Scientiarum Fennicæ
Volumen 45, 2020, 411-427
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.
Full document as PDF file
Copyright © 2020 by Academia Scientiarum Fennica