Derivatives of Inner Functions in Weighted Mixed Norm Spaces

Abstract

For $$0<p,q<\infty $$ , we characterize those radial weights $$\omega $$ satisfying a two-sided doubling condition for which the asymptotic equation $$\begin{aligned} \Vert \Theta '\Vert _{A^{p,q}_\omega }^q= \int _0^1 M_p^q(r,\Theta ')\,\omega (r)\,\mathrm{d}r \asymp \int _0^1 \left( \int _0^{2\pi } \left( \frac{1-|\Theta (re^{i\theta })|}{1-r}\right) ^p \mathrm{d}\theta \right) ^{q/p} \omega (r)\, \mathrm{d}r \end{aligned}$$ is valid for all inner functions $$\Theta $$ . As a consequence of this result, we obtain a sharp condition which guarantees that the only inner functions whose derivative belongs to the weighted mixed norm space $$A^{p,q}_\omega $$ are Blaschke products. Moreover, a condition which implies that the only inner functions whose derivative belongs to $$A^{p,q}_\omega $$ are finite Blaschke products is proved.