Hardy Spaces on Compact Riemann Surfaces with Boundary

TL;DR

This paper establishes an isometric isomorphism between Hardy spaces on finite bordered Riemann surfaces via holomorphic unramified mappings, extending the classical theory to more general surfaces with boundary.

Contribution

It constructs explicit isometric isomorphisms between Hardy spaces on different Riemann surfaces using vector bundles and fundamental group representations, extending the theory to bordered surfaces.

Findings

Hardy spaces on different bordered Riemann surfaces are isometrically isomorphic.

Explicit isometric isomorphisms are constructed using vector bundles and fundamental group representations.

Conjecture of a covariant functor from bordered surfaces with vector bundles to Krea1n spaces.

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 $\Delta_1$ and $\Delta_2$ so that the vector bundle $V^{X_2}_{\chi_\ad} \otimes \Delta_2$ on $X_2$ would be the direct image of the vector bundle $V^{X_1}_{\chi_\ao} \otimes \Delta_1$. We then show that the Hardy spaces $H_{2, J_1(p)} (S_1,V_{\chi_\ao} \otimes \Delta_1)$ and $H_{2,J_2(p)} (S_2,V_{\chi_\ad} \otimes \Delta_2)$ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation $\chi_\ad$ of the fundamental group $\piod(X_2, p_0)$ given a matrix representation $\chi_\ao$ 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\u{\i}n spaces and isomorphisms which are ramified covering of Riemann surfaces.