Inner factors of analytic functions of variable smoothness in the closed disk

Abstract:

Let $p(\zeta )$ be a positive function defined on the unit circle $\mathbb {T}$ and satisfying the condition \begin{equation*} |p(\zeta _2)-p(\zeta _1)|\le \frac {c_0}{\log \frac {e} {|\zeta _2-\zeta _1|}}, \quad \zeta _1,\zeta _2\in \mathbb {T}, \end{equation*} $p_-=\min _{\zeta \in \mathbb {T}}p(\zeta )$. Futhermore, let $0<\alpha <1$, $r\ge 0$, $r\in \mathbb {Z}$, and assume that $p_->\frac {1}{\alpha }$. Define a class of analytic functions in the unit disk $\mathbb {D}$ as follows: $f\in H^{p(\,\cdot \,)}_{r+\alpha }$ if \begin{equation*} \sup _{0<\rho <1}\,\sup _{0<|\theta |<\pi } \int ^{2\pi }_0 \bigg |\frac {f^{(r)}(\rho e^{i(\lambda +\theta )})-f^{(r)}(\rho e^{i\lambda })} {|\theta |^{\alpha }}\bigg |^{p(e^{i\lambda )}}\,d\lambda <\infty . \end{equation*} The following main results are proved.

Theorem 1. Let $f\in H^{p(\,\cdot \,)}_{r+\alpha },$ and let $I$ be an inner function, $f/I\in H^1$. Then $f/I\in H^{p(\,\cdot \,)}_{r+\alpha }$.

Theorem 2. Let $f\in H^{p(\,\cdot \,)}_{r+\alpha },$ and let $I$ be an inner function, $f/I\in H^{\infty }$. Assume that the multiplicity of every zero of $f$ in $\mathbb {D}$ is at least $r+1$. Then $fI\in H^{p(\,\cdot \,)}_{r+\alpha }$.