Derivatives of Inner Functions in Weighted Mixed Norm Spaces

For 0<p,q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p,q<\infty $$\end{document}, we characterize those radial weights ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} satisfying a two-sided doubling condition for which the asymptotic equation ‖Θ′‖Aωp,qq=∫01Mpq(r,Θ′)ω(r)dr≍∫01∫02π1-|Θ(reiθ)|1-rpdθq/pω(r)dr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Theta '\Vert _{A^{p,q}_\omega }^q= \int _0^1 M_p^q(r,\Theta ')\,\omega (r)\,\mathrm{d}r \asymp \int _0^1 \left( \int _0^{2\pi } \left( \frac{1-|\Theta (re^{i\theta })|}{1-r}\right) ^p \mathrm{d}\theta \right) ^{q/p} \omega (r)\, \mathrm{d}r \end{aligned}$$\end{document}is valid for all inner functions Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document}. 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 Aωp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{p,q}_\omega $$\end{document} are Blaschke products. Moreover, a condition which implies that the only inner functions whose derivative belongs to Aωp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{p,q}_\omega $$\end{document} are finite Blaschke products is proved.