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

Abstract

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