This paper has two purposes. First, we explain and describe the Besicovitch covering lemma, and we provide a new proof. Applications are given, particularly to the ideas of Nagel and Stein about Fatou theorems through approach regions which are not nontangential. Second, we examine the strong maximal function and give a new, simple, geometric proof of its $L^p$ boundedness.
Steven G. Krantz. "The Besicovitch covering lemma and maximal functions." Rocky Mountain J. Math. 49 (2) 539 - 555, 2019. https://doi.org/10.1216/RMJ-2019-49-2-539