# Weighted estimates for maximal functions associated to skeletons

## TL;DR

This paper establishes quantitative weighted bounds for a maximal operator linked to cube skeletons in multi-dimensional space, using a novel combinatorial approach instead of traditional covering arguments.

## Contribution

It introduces a new combinatorial method to derive weighted $L^p$ estimates for skeleton-based maximal functions, differing from classical techniques.

## Key findings

- Established weighted $L^p$ bounds for skeleton maximal operators
- Developed a combinatorial approach suitable for skeleton geometry
- Connected intersection estimates with $k$-planes to maximal function bounds

## Abstract

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.