Boundedness of the differentiation operator in model spaces and application to Peller type inequalities

TL;DR

This paper investigates the boundedness of the differentiation operator on model subspaces of Hardy space with BMOA norm and applies the results to generalize Peller's inequality for rational functions.

Contribution

It establishes conditions for the boundedness of the differentiation operator in model spaces and extends Peller's inequality to a broader class of rational functions.

Findings

Boundedness criteria for the differentiation operator in model spaces.

Generalization of Peller's inequality for rational functions.

Application to radial-weighted Bergman spaces.

Abstract

Given an inner function $\Theta$ in the unit disc $\mathbb{D}$, we study the boundedness of the differentiation operator which acts from from the model subspace $K\_{\Theta}=\left(\Theta H^{2}\right)^{\perp}$ of the Hardy space $H^{2},$ equiped with the $BMOA$-norm, to some radial-weighted Bergman space. As an application, we generalize Peller's inequality for Besov norms of rational functions $f$ of degree $n\geq1$ having no poles in the closed unit disc $\overline{\mathbb{D}}$.