Hardy Spaces on Compact Riemann Surfaces with Boundary

Abstract

We consider the holomorphic unramified mapping of two arbitrary finite bordered Riemann surfaces. Extending the map to the doubles X 1 and X 2 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 ∆ 1 and ∆ 2 so that the vector bundle proven in the present work we then conjecture that there exists a covariant functor from the category  of finite bordered surfaces with vector bundle and signature matrices to the category of Kreĭn spaces and isomorphisms which are ramified covering of Riemann surfaces.