Invariant subspaces on open Riemann surfaces

Hasumi, Morisuke

Annales de l'Institut Fourier, Volume 24 (1974) no. 4, pp. 241-286.

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 spaces on $R$. A generalized Cauchy theorem and its converse for $R$ are proved in the course of establishing the main result. The theory of Green lines is also used effectively.

Soient $R$ une surface de Riemann hyperbolique, $d_{\chi }$ une mesure harmonique à support dans la frontière de Martin de $R$, et $H^{\infty }\left(d\chi \right)$ la sous-algèbre de $L^{\infty }\left(d\chi \right)$ formée des valeurs frontières de fonctions holomorphes bornées sur $R$. On donne une classification complète des $H^{\infty }\left(d\chi \right)$-sous-modules fermés de $L^p\left(d\chi \right)$, $1\le p\le \infty$ ($\sigma (L^{\infty },L^1)$-fermés, si $p=\infty$), lorsque $R$ est régulière et admet une famille suffisamment grande de fonctions analytiques multiplicatives bornées satisfaisant une condition d’approximation. On en déduit un résultat correspondant pour les espaces de Hardy sur $R$. Pour établir le résultat principal, on démontre et utilise un théorème de Cauchy généralisé et sa réciproque pour $R$. La théorie des lignes de Green est aussi utilisée effectivement.

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