Let $R$ be a hyperbolic Riemann surface, $d_{\chi }$ a harmonic measure supported on the Martin boundary of $R$, and $H^{\infty }\left(d\chi \right)$ the subalgebra of $L^{\infty }\left(d\chi \right)$ consisting of the boundary values of bounded analytic functions on $R$. This paper gives a complete classification of the closed $H^{\infty }\left(d\chi \right)$-submodules of $L^p\left(d\chi \right)$, $1\le p\le \infty$ (weakly${}^{*}$ closed, if $p=\infty$, when $R$ is regular and admits a sufficiently large family of bounded multiplicative analytic functions satisfying an approximation condition. It also gives, as a corollary, a corresponding result for the Hardy...

We considerably improve our earlier results [Ann. Inst. Fourier, 24-4 (1974] concerning Cauchy-Read’s theorems, convergence of Green lines, and the structure of invariant subspaces for a class of hyperbolic Riemann surfaces.