Characterizations for inner functions in certain function spaces

Abstract

For $\frac{1}{2}<p<\infty$, $0<q<\infty$ and a certain two-sided doubling weight $\omega$, we characterize those inner functions $\Theta$ for which $$\Vert {\Theta }^{\prime }{\Vert }_{A_{\omega }^{p,q}}^q={\int }_0^1{\left({\int }_0^{2\pi }|{\Theta }^{\prime }(r{e}^{i\theta }){|}^{p}d\theta \right)}^{q/p}\omega (r)dr<\infty .$$ 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.

14 pages, minor changes

References in corpus (1)

• Embedding theorems for Bergman spaces via harmonic analysis