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

Averaging on $n$-dimensional rectangles