JournalszaaVol. 39, No. 4pp. 461–473

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

  • Emma D'Aniello

    Università degli Studi della Campania Luigi Vanvitelli, Caserta, Italy
  • Laurent Moonens

    Université Paris-Saclay, Orsay, France
Differentiating Orlicz Spaces with Rectangles Having Fixed Shapes in a Set of Directions cover

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Abstract

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 LL logL(R2)L(\mathbb R^2), and show that in some “model” situations, a fast-growing shape function (whose speed of growth depends on α>0\alpha > 0) does not allow the differentiation of LL logαL(R2)^{\alpha}L(\mathbb R^2).

Cite this article

Emma D'Aniello, Laurent Moonens, Differentiating Orlicz Spaces with Rectangles Having Fixed Shapes in a Set of Directions. Z. Anal. Anwend. 39 (2020), no. 4, pp. 461–473

DOI 10.4171/ZAA/1668