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arXiv · 1905.02670

Differentiating along rectangles with fixed shapes in a set of directions

Emma D'Aniello·Laurent Moonens
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Abstract

In the present note, we examine the behavior of some homo\-thecy-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 \emph{shape}) has to be a fixed real number depending on the angle between its longest side and the horizontal line (yielding a \emph{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 $L\log L(\R^2)$, 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 $L\log^\alpha L(\R^2)$.

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Emma D'Aniello, Laurent Moonens. 2019-05-07. Differentiating along rectangles with fixed shapes in a set of directions. https://arxiv.org/abs/1905.02670

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