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 $V_{\chi_2}^{X_2}⊗Δ_2$ on $X_2$ would be the direct image of the vector bundle $V_{\chi_1}^{X_1}⊗Δ_2$. We then show that the Hardy spaces $H_{2,J_1(p)}(S_1,V_{χ_1}⊗Δ_1$ and $H_{2,J_2(p)}(S_2,V_{χ_2}⊗Δ_2$ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation $χ_2$ of the fundamental group $π_1(X_2, p_0)$ given a matrix representation $χ_1$ of the fundamental group $π_1(X_1, p'_0)$. 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 $\mathcal{RH}$ 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.