Regularity of maximal functions and a Vitali covering lemma for the boundary

The classical Hardy-Littlewood maximal function theorem states that the Hardy-Littlewood maximal operator is a bounded operator on $L^p(\mathbb R^d)$ if and only if $1 < p \leq \infty$. The main ingredient to the proof is the Vitali covering lemma, which roughly states that any bounded set of balls has a pairwise disjoint subset which represents the volume of the original set well. In 1997 Juha Kinnunen proved the corresponding result for the gradient of the maximal function, i.e. that the $L^p(\mathbb R^d)$-norm of the gradient of the maximal function is controlled by the $L^p(\mathbb R^d)$-norm of the gradient of the function if $1 < p \leq \infty$. However, he provides no counterexample in the endpoint $p = 1$, and so in 2004 Hajłasz and Onninen formally posed the question if the endpoint gradient bound also holds. We present the current state of research on the problem and show a new variant of the Vitali covering lemma that concerns the perimeter instead of the volume.