Characterizations for inner functions in certain function spaces

TL;DR

This paper characterizes inner functions based on the integrability of their derivatives in specific weighted function spaces, providing new criteria for their classification.

Contribution

It introduces novel characterizations for inner functions in weighted spaces, extending previous results to broader parameter ranges and derivatives belonging to Bergman spaces.

Findings

Characterizations for inner functions with derivatives in weighted spaces

Extension of results to cases where p ≥ q

Additional criteria for derivatives in Bergman spaces

Abstract

For $\frac12<p<\infty$, $0<q<\infty$ and a certain two-sided doubling weight $\omega$, we characterize those inner functions $\Theta$ for which $$\|\Theta'\|_{A^{p,q}_\omega}^q=\int_0^1 \left(\int_0^{2\pi} |\Theta'(re^{i\theta})|^p d\theta\right)^{q/p} \omega(r)\,dr<\infty.$$ Then we show a modified version of this result for $p\ge q$. Moreover, two additional characterizations for inner functions whose derivative belongs to the Bergman space $A_\omega^{p,p}$ are given.