# Boundedness of classical operators on rearrangement-invariant spaces

## TL;DR

This paper investigates the boundedness of classical harmonic analysis operators on rearrangement-invariant spaces, providing sharp characterizations of optimal domain and range spaces and illustrating with specific examples.

## Contribution

It offers a comprehensive analysis of operator boundedness on rearrangement-invariant spaces, including sharpness and optimal space characterizations, which were previously not fully understood.

## Key findings

- Characterization of optimal domain and range spaces for classical operators
- Complete solutions when a rearrangement-invariant partner space exists
- Examples demonstrating sharp results with common function spaces

## Abstract

We study the behaviour on rearrangement-invariant spaces of such classical operators of interest in harmonic analysis as the Hardy-Littlewood maximal operator (including the fractional version), the Hilbert and Stieltjes transforms, and the Riesz potential. The focus is on sharpness questions, and we present characterisations of the optimal domain (or range) partner spaces when the range (domain) is fixed. When a rearrangement-invariant partner space exists at all, a complete characterisation of the situation is given. We illustrate the results with a variety of examples of sharp particular results involving customary function spaces.