# On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces

## TL;DR

This paper explores the relationships between weak and strong type inequalities for modified maximal operators on specific non-doubling metric measure spaces, characterizing possible configurations of these inequalities.

## Contribution

It describes necessary conditions for the sets of p-values where inequalities hold and illustrates configurations with constructed non-doubling spaces.

## Key findings

- Characterization of admissible configurations of inequality sets
- Necessary conditions for these sets in non-doubling spaces
- Examples illustrating each configuration

## Abstract

In this article we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$, the sets of all $p \in [1, \infty]$ for which the weak and strong type $(p,p)$ inequalities hold for the centered and non-centered modified Hardy--Littlewood maximal operators, $M^{\rm c}_k$ and $M_k$, $k \geq 1$. For any fixed $k$ we describe the necessary conditions that $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$ must satisfy in general and illustrate each admissible configuration with a properly chosen non-doubling metric measure space. We also give some partial results related to an analogous problem stated for varying $k$.