Inner functions and spaces of pseudocontinuable functions related to them

Let $\theta$ be an inner function; $\alpha\in\mathbb{C}$, $|\alpha|=1$. Then the harmonic function $\mathop{\mathrm{Re}}\frac{\alpha+\theta}{\alpha-\theta}$ is the Poisson integral of a singular measure $\sigma_\alpha$. The Clark theorem allows naturally to identify $H^2\ominus\theta H^2$ with $L^2(\sigma_\alpha)$. The aim of this paper is to investigate $L^p$-properties of this identification operator for $p\ne2$.