A uniform boundedness principle concerning inner functions (Q1113319)

Let \(L^{\infty}_ r\) and \(H^{\infty}\) denote, respectively, the bounded real-valued measurable functions on the unit circle T and the boundary values of bounded analytic functions in the unit disc. The author notes several parallels between the class \(U_ r\) of real unimodular (i.e., \(\pm 1\)-valued) functions in \(L^{\infty}_ r\) and the class I of unimodular (i.e., inner) functions in \(H^{\infty}\). For example, it is elementary that the closed convex hull of \(U_ r\) is the closed unit ball of \(L^{\infty}_ r\), while it is a deep result of Don Marshall and the closed convex hull of I is the closed unit ball of \(H^{\infty}\). The author provides a new proof of another such parallel due to \textit{J. Fernández} [Mich. Math. J. 35, No.2, 227-231 (1988)], a proof modelled on that of the Vitali-Hahn-Saks theorem in Dunford and Schwartz. Theorem. (Fernández) Let \(k_ n\in L^ 1(T)\), \(n=1,2,..\). and assume that \[ \sup_{n}| (2\pi)^{-1}\int_{T}k_ n\phi d\theta | <\infty,\quad all\quad \phi \in I. \] Then \[ \sup_{n}| (2\pi)^{- 1}\int_{T}k_ nf d\theta | <\infty,\quad all\quad f\in H^{\infty}. \]