Stabiliti and Identity of analytic functions of Hardy classes

Abstract

Let $E$ be a subset of the unit disc $U$ of the complex plane $\CC$. Recall that $H^p(U)$ is the space of all holomorphic functions $g$ on $U$ for which $\|g\|_{H^p}$ $<$ $\infty$. Put \begin{equation} C_p(\epsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, \|g\|_p\leq 1, |g(\zeta)| \leq \epsilon \forall \zeta\in E\}, \end{equation} for positive $\epsilon$ and $R$ in $(0, 1)$. It can be seen that $C_p(\epsilon, R)$ is bounded from above by $(1-R^2)^{-1/p}$. \begin{theorem} If $\bar E$ $\subset$ $U$ then there exists $\epsilon_0>0$ such that for $0<\epsilon <\epsilon_0$ there is correspondingly a finite Blaschke product $B_{\epsilon}(z)$ whose zeros are in $\bar E$ satisfying \begin{eqnarray*} \max_{|z|\leq R}|B_\epsilon (z)|\leq C_p(\epsilon, R)\leq C\max_{|z|\leq R}|B_\epsilon (z)|^{1/2}, \end{eqnarray*} where $C$ is a positive constant that depends only on $R$ and $p$. Moreover we have \begin{eqnarray*} \sup_{z\in E}|B_{\epsilon}(z)|\leq \epsilon. \end{eqnarray*} \end{theorem}