Necessary and sufficient conditions for inner functions to be in \(Q_K(p, p - 2)\)-spaces
Let \(\mathbb D\) be the unit disc in \(\mathbb C\). For \(p\in (0,\infty),\, q\in (-2,\infty)\), \(Q_K(p,q)\) denotes the space of analytic functions on \(\mathbb D\) for which \[ \sup_{a\in\mathbb D} \int_{\mathbb D} | f^\prime(z)| ^p (1-| z|)^q K(g(z,a)) dA(z) <\infty, \] where \(K:[0,\infty) \to [0,\infty)\) is such that \(K(1)>0\) and \(g(z,a) = \log| (1-\overline az)/(a-z)| \) is the Green's function on \(\mathbb D\). In the paper the author studies the membership of inner functions in \(Q_K(p,q)\) spaces, in particular in the Möbius invariant \(Q_K(p,p-2)\) spaces. The main result of the paper is the following theorem which provides a complete characterization of inner functions in these spaces. Theorem. Let \(p\in (\frac12,\infty)\), and assume that there exist \(\alpha,\,\beta\in (1-\min\{p,1\}, \min\{p,1\})\) such that a nondecreasing function \(K\) satisfies \(r^{-\alpha}K(r)\) is decreasing and \(r^{-\beta}K(r)\) is increasing on \((0,1)\). Then an inner function belongs to \(Q_K(p,p-2)\) if and only if it is a Blaschke product associated with a sequence \(\{z_n\}\) satisfying \[ \sup_{a\in\mathbb D} \sum_n K(1-| \varphi_a(z_n)|) <\infty, \] where \(\varphi_a(z) = (a-z)/(1-\overline a z)\). The result of the paper generalizes several results of [\textit{F. Pérez-González} and \textit{J. Rättyä}, Proc. Edinb. Math. Soc., II. Ser. 52, No. 3, 751--770 (2009; Zbl 1179.30057)] as well as some of the results of [\textit{M. Essén} et al., J. Funct. Anal. 230, No. 1, 78--115 (2006; Zbl 1089.30049)].
scientific article; zbMATH DE number 6529019
scientific article