On a maximum principle for pseudocontinuable functions

Let $\theta$ be an inner function; $\alpha\in\mathbb C$, $|\alpha|=1$. Denote by $\sigma_\alpha$ the nonnegative singular measure whose Poisson integral is equal to $\operatorname{Re}\frac{\alpha+\theta}{\alpha-\theta}$. The Clark theorem allows us naturally to identity $H^2\ominus\theta H^2$ with $L^2(\sigma_\alpha)$. Let $U_\alpha$ be the unitary operator producing this identification. The main aim of this paper is to prove the following theorem.
Theorem. Let $f\in H^2\ominus\theta H^2$; $2<p\le+\infty$; $\alpha,\beta\in\mathbb C$; $|\alpha|=|\beta|=1$, $\alpha\ne\beta$. Suppose that $U_\alpha f\in L^p(\sigma_\alpha)$ and $U_\beta f\in L^p(\sigma_\beta)$. Then $f\in H^p$.
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