Inner functions as improving multipliers
Let \textbf{D} be the open unit disk in the complex plane, \textbf{T} be the unit circle, and \(H(\)\textbf{D}\()\) be the space of holomorphic functions in \textbf{D}. A function \(\theta \in H(\)\textbf{D}\()\) is said to be inner if \(\theta\) is bounded and has radial limit \(\theta (\xi )\) of modulus \(1\) for almost every \(\xi \in\) \textbf{T}. The singular set \(\sigma (\theta )\) of \(\theta\) is the smallest closed set \(E \subset\) \textbf{T} such that \(\theta\) is analytic across \textbf{T}\(\setminus E\). Since a nontrivial inner function \(\theta\) is highly oscillatory near \(\sigma (\theta )\) one would expect that multiplying by \(\theta\) a function \(f \in H(\)\textbf{D}\()\) that is smooth in some sense on \textbf{T} will destroy smoothness. However, in some sense \(f \theta\) inherits the nice properties of \(f\) and even it can have an added smoothness. Inspired by this phenomenon, \textit{K.~Dyakonov} [J.~Funct.~Anal. 240, 429--444 (2006; Zbl 1116.46018)] introduced the following notion: Let \(X\) and \(Y\) be two classes of analytic functions in \textbf{D}, with \(X \subset Y\). An inner function \(\theta\) is said to be \((X,Y)\)-improving, if every function \(f \in X\) satisfying \(f \theta \in Y\) must actually satisfy \(f \theta \in X\). For \(0 < p < \infty\), the space \(Q_p\) consists of those functions \(f \in H(\)\textbf{D}\()\) such that \[ \sup_{|a| < 1} \int_{|z|<1} |f'(z)|^2 \left( \log \left|{1-\overline{a}z \over a-z}\right| \right)^p \, dA(z) < \infty , \] where \(dA(z)\) is the Lebesgue area measure. It is known that \(Q_p = B\) (the Bloch space) for all \(p > 1\), and \(Q_1 = BMOA\) (the space of those \(H^1\)-functions whose boundary values have bounded mean oscillation on \textbf{T}). For \(0 < \alpha \leq 1\), \(\Lambda^\alpha\) denotes the Lipschitz space, consisting of those functions \(f\) analytic in \textbf{D} and continuous on its closure such that \(|f(z)-f(w)| \leq C|z-w|\) \((z,w \in\) \textbf{D}\()\) for some \(C = C(f) > 0\). For \(1 \leq p < \infty\) and \(0 < s < 1\), the Besov space \(B_p^s\) is the class of functions \(f \in L^p(\)\textbf{T}\()\) for which \[ \left(\int_0^{2\pi} {dt \over |t|^{sp+1}} \int_{|\xi|=1} |f(e^{it}\xi) - f(\xi )|^p \, d\xi \right)^{1/p} < \infty. \] By \(AB_p^s\) it is denoted the subspace \(B_p^s \cap H(\)\textbf{D}\()\). For \(0 < p < \infty\), the Bergman space \(A^p\) is the set of all functions \(f \in H(\)\textbf{D}\()\) such that \(\int_{|z|<1} |f(z)|^p \, dA(z) < \infty\). In this paper, the author characterizes the \((X,Y)\)-improving inner functions for several pairs of spaces \((X,Y)\) and, in particular, improves some results of Dyakonov on the subject regarding Lipschitz spaces and Besov spaces. Specifically, the main results are the following: Theorem 1. Suppose that \(0<p<1\) and \(\theta\) is an inner function. Then the following are equivalent: (i) \(\theta \in Q_p\). (ii) \(\theta\) is \((Q_p,BMOA)\)-improving. (iii) \(\theta\) is \((Q_p,B)\)-improving. Theorem 2. (i) There exists an inner function \(\theta\) which is not \((Q_1,B)\)-improving. (ii) If \(0<p<1\) and \(q > p\), then any inner function in \(Q_p\) is \((Q_p,Q_q)\)-improving. (iii) If \(0<p<q<1\), then there exists an inner function which is \((Q_p,Q_q)\)-improving and does not belong to \(Q_p\). Theorem 3. Let \(1 \leq p < \infty\), \(0 < s < 1\) and \(\max \{0,s - {1 \over p}\} < \alpha < s\). If \(\theta\) is an inner function with \(\theta ' \in A^{1+(s - \alpha )p}\), then \(\theta\) is \((\Lambda^\alpha \cap B_p^s, \Lambda^\alpha )\)-improving. Theorem 4. Let \(1 \leq p < \infty\) and \(0 < s < {1 \over p}\), and let \(\theta\) be an inner function. Then \(\theta\) is \((AB_p^s \cap BMOA,BMOA)\)-improving if and only if \(\theta \in AB_p^s\).
scientific article; zbMATH DE number 5365111
scientific article