Maximal functions, conjugations and multipliers between Toeplitz kernels

TL;DR

This paper explores the relationships between different characterizations of Toeplitz kernels, emphasizing the importance of symbol factorization and the connection of maximal functions with conjugation in the context of multipliers and model spaces.

Contribution

It provides a comprehensive analysis of the links between Riemann-Hilbert problems, maximal functions, and multipliers for Toeplitz kernels, highlighting the role of symbol factorization and conjugation.

Findings

Symbol factorization is crucial for deriving multipliers from model spaces to Toeplitz kernels.

Maximal functions are deeply connected with a natural conjugation on Toeplitz kernels.

The study clarifies the relationships between different characterizations of Toeplitz kernels.

Abstract

Toeplitz kernels can be defined by Riemann-Hilbert problems, by maximal functions, or by multipliers acting on model spaces. In this paper we study those different characterisations and their relations, highlighting, on the one hand, the crucial role played by symbol factorisation in obtaining multipliers from a model space onto a Toeplitz kernel, in particular isometric multipliers, and, on the other hand, a deep connection of maximal functions with a naturally defined conjugation on the Toeplitz kernel.