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Zeitschrift für Analysis und ihre Anwendungen

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Volume 39, Issue 4, 2020, pp. 461–473
DOI: 10.4171/ZAA/1668

Published online: 2020-10-22

Differentiating Orlicz Spaces with Rectangles Having Fixed Shapes in a Set of Directions

Emma D'Aniello[1] and Laurent Moonens[2]

(1) Università degli Studi della Campania Luigi Vanvitelli, Caserta, Italy
(2) Université Paris-Saclay, Orsay, France

In the present note, we examine the behavior of some homothecy-invariant differentiation basis of rectangles in the plane satisfying the following requirement: for a given rectangle to belong to the basis, the ratio of the largest of its side-lengths by the smallest one (which one calls its shape) has to be a fixed real number depending on the angle between its longest side and the horizontal line (yielding a shape-function). Depending on the allowed angles and the corresponding shape-function, a basis may differentiate various Orlicz spaces. We here give some examples of shape-functions so that the corresponding basis differentiates $L$ log$L(\mathbb R^2)$, and show that in some “model” situations, a fast-growing shape function (whose speed of growth depends on $\alpha > 0$) does not allow the differentiation of $L$ log$^{\alpha}L(\mathbb R^2)$.

Keywords: Maximal functions, differentiaton bases

D'Aniello Emma, Moonens Laurent: Differentiating Orlicz Spaces with Rectangles Having Fixed Shapes in a Set of Directions. Z. Anal. Anwend. 39 (2020), 461-473. doi: 10.4171/ZAA/1668