On weak type inequalities for rare maximal functions in ℝⁿ

The study of one-dimensional rare maximal functions was started in [4, 5]. The main result in [5] was obtained with the help of some general procedure. The goal of the present article is to adapt the procedure (we call it “dyadic crystallization”) to the multidimensional setting and to demonstrate that rare maximal functions have properties not better than the Strong Maximal Function.