# Variations on the Boman covering lemma

## TL;DR

This paper investigates variants of the Boman covering lemma and their connection to the boundedness of maximal operators, revealing new implications and weaker conditions under which the lemma holds.

## Contribution

It establishes that strong type bounds of the uncentered maximal operator imply generalized Boman covering lemmas, and explores weaker conditions for the lemma's validity.

## Key findings

- Strong type (q,q) of uncentered maximal operator implies generalized Boman covering lemma.
- Boundedness of centered maximal operator entails weak versions of the lemma.
- Unbounded uncentered maximal operator does not prevent the lemma from holding for all p.

## Abstract

We explore some variants of the Boman covering lemma, and their relationship to the boundedness properties of the maximal operator. Let $1 < p < \infty$ and let $q$ be its conjugate exponent. We prove that the strong type $(q,q)$ of the uncentered maximal operator, by itself, implies certain generalizations of the Boman covering lemma for the exponent $p$, and in turn, these generalizations entail the weak type $(q,q)$ of the centered maximal operator. We show by example that it is possible for the uncentered maximal operator to be unbounded for all $1 < s < \infty$, while the conclusion of the lemma holds for every $1 < p < \infty$; thus, the latter condition is much weaker. Also, the boundedness of the centered maximal operator entails weak versions of the lemma.