Averaging on $n$-dimensional rectangles

TL;DR

This paper studies the differentiation properties of translation invariant bases of rectangles in n-dimensional space, identifying the largest Orlicz space they can differentiate, and improving existing analytical techniques.

Contribution

It advances the understanding of differentiation bases in higher dimensions by refining methods to determine the maximal Orlicz space they differentiate.

Findings

Identifies $L ext{log}^{n-1}L(R^n)$ as the largest Orlicz space differentiated by certain bases.

Improves analytical techniques from previous studies by Stokolos (1988, 2008).

Provides new insights into the structure of translation invariant differentiation bases in $ extbf{R}^n$.

Abstract

In this work we investigate families of translation invariant differentiation bases $B$ of rectangles in $R^n$, for which $L\log^{n-1}L(R^n)$ is the largest Orlicz space that $B$ differentiates. In particular, we improve on techniques developed by A.~Stokolos 1988 and 2008.