Journal of Generalized Lie Theory and Applications

Hardy Spaces on Compact Riemann Surfaces with Boundary

A Zuevsky

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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.

Article information

J. Gen. Lie Theory Appl., Volume 9, Number S1 (2015), 8 pages.

First available in Project Euclid: 11 November 2016

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Zentralblatt MATH identifier

Half-order differentials Hardy spaces Riemann surfaces


Zuevsky, A. Hardy Spaces on Compact Riemann Surfaces with Boundary. J. Gen. Lie Theory Appl. 9 (2015), no. S1, 8 pages. doi:10.4172/1736-4337.S1-005.

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