Differentiation properties of class ${L}^1([0,1]^2)$ with respect to two different basis of rectangles

TL;DR

This paper investigates the differentiation properties of functions in L^1([0,1]^2) with respect to two different bases of rectangles, establishing conditions under which divergence or convergence of integral averages occurs for each basis.

Contribution

It introduces conditions on the sets of rectangle side lengths that determine when a function can diverge on one basis and converge on another, providing both sufficiency and necessity results.

Findings

Constructs functions with divergence on one basis and convergence on another under certain conditions.

Shows that the 'distance' between sets of side lengths influences differentiation properties.

Establishes necessary conditions for positive functions to exhibit these divergence and convergence behaviors.

Abstract

It is a well-known result by Saks \cite{Saks1934} that there exists a function $f \in L^1(\mathbb{R}^2)$ so that for almost every $(x,y)\in \mathbb{R}^2$ \[ \lim_{\substack{\mathrm{diam} R\rightarrow 0, \\ (x,y) \in R \in \mathcal{R}}}\left|\frac{1}{|R|}\int_R f(x,y)\, dxdy\right|=\infty, \] where $\mathcal{R}=\{[a,b)\times [c,d)\colon a<b, c<d\}$. In this note we address the following question: assume we have two different collections of rectangles; under which conditions there exists a function $f \in L^1(\mathbb{R}^2)$ so that its integral averages are divergence with respect to one collection and convergence with respect to another? More specifically, let $\mathcal{D}, \mathcal{C} \subset (0,1]$ and consider rectangles with side lengths in $\mathcal{D}$ and respectively in $\mathcal{C}$. We show that if the sets $\mathcal{D}$ and $\mathcal{C}$ are sufficient ``far" from each other, then such a function can be constructed. We also show that in the class of positive functions our condition is also necessary for such a function to exist.