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

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