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

Given an inner function $\mathrm{\Theta }$ in the unit disc $\mathbb{D}$, we study the boundedness of the differentiation operator which acts from from the model subspace $K\mathrm{\_}\mathrm{\Theta }={\left(\mathrm{\Theta }{H}^2\right)}^{\perp }$ 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 $n\ge 1$ having no poles in the closed unit disc $\bar{\mathbb{D}}$.