Let $H^\infty(D)$ be the Hardy-space of bounded, holomorphic functions $\Phi:D\rightarrow \mathbb{C}$, defined on the unit disk $D\subset \mathbb{C}$. It is a standard result that $\Phi\in H^\infty(D)$ has radial limits $\Phi^*(\omega)=\lim_{r\rightarrow 1}\Phi(r\omega)$ for almost every $\omega \in S^1=\partial D$.
Questions.
- What can we say about the regularity of $\Phi^*\in L^\infty(S^1)$? What is the image of $H^\infty(D)$ under $\Phi\mapsto \Phi^*$ in $L^\infty(S^1)$?
- More concretely: Is there an example where $\Phi^*$ is continuous away from a single point and with existing left/right limits at that point?
In an attempt to refute 2) I've considered functions $\Phi(z)=\frac{1}{2\pi i} \int_{S^1} \frac{\varphi(\zeta)}{\zeta-z}d\zeta$, where $\varphi:S^1\backslash \{-1\}\rightarrow \mathbb{C}$ is continuously differentiable with existing (but possibly different) limits $a_\pm=\lim_{\theta\rightarrow \pm \pi}\varphi(e^{i\theta})$. I can show that$$ \Phi(-1+\epsilon)=\frac{(a_+-a_-)\ln\epsilon }{2\pi i} + \frac{a_++a_-}{2} - \frac{1}{2\pi i} \int_{-\pi}^\pi \psi(\theta) \log(e^{i\theta}+1-\epsilon) d\theta, $$ where $\psi(\theta) = \frac{d}{d\theta} \varphi(e^{i\theta})$ and $\log$ is the principal branch of the logarithm. Given a supposed example as in 2), we could set $\varphi=\Phi^*$ and arrive at a contriction if the previous display blows up for $\epsilon\rightarrow 0$. However, while the first term always blows up if $a_+\neq a_-$, it seems like this could in principle be cancelled by the integral (which can also blow up in the limit).