Differentiation properties of class L1([0,1)2) with respect to two different bases ofrectangles

The Lebesgue differentiation theorem claims that the integralaverages of f is an element of L-1([0,1)(2)) with respect to the family of axis-parallelsquaresconverge almost everywhere on [0,1)2. On the other hand, it isa well known result by Saks that there exist a function f is an element of L-1([0,1)(2)) such that its integral averages with respect to the family of axis-parallelrectanglesdiverge everywhere on [0,1)2. In this paper, we address thefollowing question: assume we have two different collections of rectan-gles; under which conditions does there exist a function f is an element of L-1([0,1)2) so that its integral averages converge with respect to one collection anddiverge with respect to another? More specifically, let C, D subset of(0,1]and consider rectangles with side lengths respectively in C and D. Weshow that if the sets C and Doccasionally become sufficiently "far"from each other, then such a function can be constructed. We also showthat in the class of positive functions our condition is necessary for sucha function to exist.