Almost Everywhere Convergence for Lebesgue Differentiation Processes Along Rectangles

In this paper, we study Lebesgue differentiation processes along rectangles R-k shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in L-p spaces. In particular, classes of examples of such processes failing to converge a.e., in L-8 are provided, for which R-k is known to be oriented along the slope k(-s) for s > 0, yielding an interesting counterpart to the fact that the directional maximal operator associated to the set {k(-s) : k ? N*} fails to be bounded in L-p for any 1 = p < 8.