# Differentiating along rectangles with fixed shapes in a set of directions

## TL;DR

This paper investigates how certain shape-dependent rectangles can be used to differentiate functions in specific Orlicz spaces, revealing conditions under which differentiation is possible or not.

## Contribution

It introduces a shape-function framework for rectangles in the plane and analyzes their ability to differentiate various Orlicz spaces, including new examples and limitations.

## Key findings

- Certain shape-functions enable differentiation of L log L spaces.
- Fast-growing shape-functions may fail to differentiate L log^α L spaces.
- Examples demonstrate the impact of shape-function growth on differentiation capabilities.

## 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 $L\log^\alpha L(\R^2)$.