Zeitschrift für verallgemeinerte Lügentheorie und -anwendungen

Hardy Spaces on Compact Riemann Surfaces with Boundary

Abstract

Zuevsky A

We consider the holomorphic unramified mapping of two arbitrary finite bordered Riemann surfaces. Extending the map to the doubles X1 and X2 of Riemann surfaces we define the vector bundle on the second double as a direct image of the vector bundle on first double. We choose line bundles of half-order differentials Δ1 and Δ2 so that the vector bundle 2 χ2 2 V X ⊗Δ on X2 would be the direct image of the vector bundle 1 1 χ ⊗Δ2 V X . We then show that the Hardy spaces 2, 1( ) ( 1, χ1 1) H J p S V ⊗Δ and 2, 2 ( ) ( 2, χ2 2) H J p S V ⊗Δ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation χ2 of the fundamental group π1(X2, p0) given a matrix representation χ1 of the fundamental group π1(X1, p'0). On the basis of the results of Alpay et al. and Theorem 3.1 proven in the present work we then conjecture that there exists a covariant functor from the category ï��ï�� of finite bordered surfaces with vector bundle and signature matrices to the category of KreÄ­n spaces and isomorphisms which are ramified covering of Riemann surfaces.