Inner functions and \(l^p\)-multipliers (Q1966234)

For \(1\leq p\leq \infty\), let \(A_p^+(D)=\{f:f\) analytic in the unit disk \(D=\{z\in{\mathbb C}: |z|<1\}\), \(f(z)=\sum_{n=0}^{\infty} \widehat f(n)z^n\) and \(\widehat f=\{\widehat f(n)\); \(n=0,1,\ldots\}\in l^p\}\). An analytic function \(m\) in \(D\) is called an \(l^p\)-multiplier if for every function \(f\in A_p^+(D)\) the product \(m\cdot f\in A_p^+(D)\). The family of \(l^p\)-multipliers is denoted by \(M_p^+(D)\). The author studies the following problem: what inner functions belong to \(M_p^+(D)\)? A function \(I\), analytic in \(D\), is said to be inner if \(|I(z)|\leq 1\), \(z\in D\), and \(|I(e^{it})|=1\) almost everywhere. It is well known that any inner function admits a factorization \(I=\lambda BS\), where \(\lambda\) is a constant, \(|\lambda|=1\), \(B\) is a Blaschke product with zeroes \(\{z_n\}\), where \(\sum_{n=1}^{\infty} (1-|z_n|)<\infty\), and \(S\) is a singular inner function, that is, an inner function without zeros in \(D\) and \(S(0)>0\). Every singular inner function is of the form \[ S(z)=\exp\Biggl(-\int_{\mathbb T}\frac{\xi+z}{\xi-z} d\mu(\xi)\Biggr), \] where \(\mu\) is a positive singular measure on the circle \(\mathbb T=\{z\in{\mathbb C}:|z|=1\}\). The closed support of the measure \(\mu\) is called the spectrum of \(S\). The author shows in several theorems that if the spectrum \(E\) of \(S\) is insufficiently massive (for example, if \(E\) is a porous set) then \(S\not\in M_p^+(D)\) for all \(p\), \(p\neq 2\), \(1\leq p\leq\infty\). Related to the Vinogradov-Verbitskii theorem [\textit{S. A. Vinogradov}, Dokl. Akad. Nauk. SSSR 254, 1301-1306 (1980; Zbl 0482.42004) and \textit{I. Eh. Verbitskij}, Funkts. Anal. Prilozh. 14, No. 3, 67-68 (1980; Zbl 0455.46006)] it is shown for a Blaschke product \(B\) with zeros \(\{z_n\}\), \(\lim z_n=1\) and \(z_n\in\overline {\mathbb C}^+\cap S_{\alpha}=\{z\in{\mathbb C}: \text{Im } z\geq 0\}\cap\{z\in{\mathbb C}:|1-z|\leq \alpha(1-|z|)\}\), \(\alpha>1\), that if \(B\in M_p^+(D)\) for some \(p\), \(p\neq 2\), \(1\leq p\leq\infty\), then \[ \sum_{n:|1-z_n|<\varepsilon} |1-z_n|=O(\varepsilon),\quad \varepsilon\to +0. \]