Weighted estimates for maximal functions associated to skeletons

We provide quantitative weighted estimates for the $L^p(w)$ norm of a maximal operator associated to cube skeletons in $\mathbb{R}^n$. The method of proof differs from the usual in the area of weighted inequalities since there are no covering arguments suitable for the geometry of skeletons. We use instead a combinatorial strategy that allows to obtain, after a linearization and discretization, $L^p$ bounds for the maximal operator from an estimate related to intersections between skeletons and $k$-planes.