Maximal functions, conjugations and multipliers between 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.