# Maximal operators on BMO and slices

## TL;DR

This paper investigates the behavior of maximal operators on BMO spaces, showing discontinuity of the Hardy-Littlewood operator on certain BMO variants, providing counterexamples for others, and exploring properties of slices of BMO functions.

## Contribution

It demonstrates the discontinuity of the Hardy-Littlewood maximal operator on morn, shows that some maximal operators are unbounded on morn, and analyzes slices of BMO functions.

## Key findings

- Hardy-Littlewood maximal operator is discontinuous on morn
- Counterexamples show strong and directional maximal operators are unbounded on morn
- Properties of slices of morn functions are characterized

## Abstract

We prove that the Hardy-Littlewood maximal operator is discontinuous on $\bmorn$ and maps $\vmorn$ to itself. A counterexample to boundedness of the strong and directional maximal operators on $\bmorn$ is given, and properties of slices of $\bmorn$ functions are discussed.