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$^{\alpha}L(\mathbb R^2)$.

中文翻译：

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在一组方向上具有固定形状的矩形的Orlicz空间的微分

在本说明中，我们研究了满足以下要求的平面中矩形的某些不变性微分基础的行为：对于给定的矩形属于该基础，其最大边长与最小边长之比（称为形状）（必须称为实数），具体取决于其最长边与水平线之间的角度（产生形状函数）。根据允许的角度和相应的形状函数，基础可以区分各种Orlicz空间。我们在这里给出一些形状函数的示例，以便相应的基础区分$ L $ log $ L（\ mathbb R ^ 2）$，并显示出在某些“模型”情况下，快速增长的形状函数（其速度为增长取决于$ \ alpha>