Peak sets for analytic Hölder classes

A closed subset $E$ of the unit circle $\mathbb T$ is called a peak set for the analytic Hölder class $A^\alpha$, $0<\alpha<1$, if there exists a function $f$ in $A^\alpha$ such that $f|E\equiv1$ and $|f(z)|<1$ for $z\in\operatorname{clos}{\mathbb {D}}\setminus E$. It is proved that $E$ E is a peak set for $A^\alpha$ if and only if there exists a nonnegative Borel measure $\mu$ on $\mathbb T$ such that $|\frac{d\mu}{dt}(e^{it})+i\tilde\mu(e^{it})|^{-1}$ coincides almost everywhere on $\mathbb T$ with a function in $\Lambda_\alpha$ vanishing on $E$. A sufficient condition for a set to be a pick set is also obtained.