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 $Δ_1$ and $Δ_2$ so that the vector bundle $V^{X_2}_{χ_\ad} \otimes Δ_2$ on $X_2$ would be the direct image of the vector bundle $V^{X_1}_{χ_\ao} \otimes Δ_1$. We then show that the Hardy spaces $H_{2, J_1(p)} (S_1,V_{χ_\ao} \otimes Δ_1)$ and $H_{2,J_2(p)} (S_2,V_{χ_\ad} \otimes Δ_2)$ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation $χ_\ad$ of the fundamental group $\piod(X_2, p_0)$ given a matrix representation $χ_\ao$ of the fundamental group $\piod(X_1, p'_0)$. %% On the basis of the results of \cite{vin} and Theorem \ref{theorem_1} proven in the present work we then conjecture that there exists a covariant functor from the category ${\cal RH}$ of finite bordered surfaces with vector bundle and signature matrices to the category of Kre\uın spaces and isomorphisms which are ramified covering of Riemann surfaces.