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

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 $\mathcal{R}=\{[a,b)\times [c,d):a<b,c<d\}$. 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.