New York Journal of Mathematics
Volume 22 (2016) 933-942


Laurent Moonens

Differentiating along rectangles, in lacunary directions

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Published: September 9, 2016
Keywords: Lebesgue differentiation theorem, Hardy-Littlewood maximal operator, lacunary directions
Subject: Primary 42B25; Secondary 26B05

We show that, given some lacunary sequence of angles θ=(θj)j∈N not converging too fast to zero, it is possible to build a rare differentiation basis B of rectangles parallel to the axes that differentiates L1(R2) while the basis Bθ obtained from B by allowing its elements to rotate around their lower left vertex by the angles θj, j∈N, fails to differentiate all Orlicz spaces lying between L1(R2) and Llog L(R2).


This work was partially supported by the French ANR project "GEOMETRYA'' no. ANR-12-BS01-0014.

Author information

Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS UMR8628, Université Paris-Saclay, Bâtiment 425, F-91405 Orsay Cedex, France.