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

The Lebesgue differentiation theorem claims that the integral averages of \(f\in L^{1}([0,1)^2)\) $f\in L^1({[0,1)}^2)$ with respect to the family of axis-parallel squares converge almost everywhere on \([0,1)^2\) ${[0,1)}^2$ . On the other hand, it is a well known result by Saks that there exist a function \(f \in L^{1}([0,1)^2)\) $f\in L^1({[0,1)}^2)$ such that its integral averages with respect to the family of axis-parallel rectangles diverge everywhere on \([0,1)^2\) ${[0,1)}^2$ . In this paper, we address the following question: assume we have two different collections of rectangles; under which conditions does there exist a function \(f \in L^{1}([0,1)^2)\) $f\in L^1({[0,1)}^2)$ so that its integral averages converge with respect to one collection and diverge with respect to another? More specifically, let \({\varvec{C}}, {\varvec{D}} \subset (0,1]\) $\mathit{C},\mathit{D}\subset (0,1]$ and consider rectangles with side lengths respectively in \({\varvec{C}}\) $\mathit{C}$ and \({\varvec{D}}\) $\mathit{D}$ . We show that if the sets \({\varvec{C}}\) $\mathit{C}$ and \({\varvec{D}}\) $\mathit{D}$ occasionally become sufficiently “far” from each other, then such a function can be constructed. We also show that in the class of positive functions our condition is necessary for such a function to exist.