Construction of Hardy Spaces on Riemann Surfaces with Boundaries

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. The line bundles of half-order differentials Δ₁ and Δ₂ are chosen so that the vector bundle $V^{X_{2}}_{\chi _{2}} \otimes {\Delta }_{2}$ on X ₂ would be the direct image of the vector bundle $V^{X_{1}}_{\chi _{1}} \otimes {\Delta }_{1}$. We then show that the Hardy spaces $H_{2, J_{1}(p)} (S_{1},V_{\chi _{1}} \otimes {\Delta }_{1})$ and $H_{2,J_{2}(p)} (S_{2},V_{\chi _{2}} \otimes {\Delta }_{2})$ are isometrically isomorphic. Using Theorems 3.1, 3.2 we then conjecture the existence a covariant functor from the category ${\mathcal {RH}}$ of finite bordered surfaces with vector bundle and signature matrices to the category of Kreı̆n spaces and isomorphisms which are ramified covering of Riemann surfaces.