Hardy spaces on compact Riemann surfaces with boundary

zbMATH Open1371.30036arXiv0911.3908MaRDI QIDQ501758

Publication date: 10 January 2017

Published in: Journal of Generalized Lie Theory and Applications (Search for Journal in Brave)

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

D e l t a_{1}

and

D e l t a_{2}

so that the vector bundle

V_{c h i_{a} d}^{X_{2}} o t i m e s D e l t a_{2}

on

X_{2}

would be the direct image of the vector bundle

V_{c h i_{a} o}^{X_{1}} o t i m e s D e l t a_{1}

. We then show that the Hardy spaces

H_{2, J_{1} (p)} (S_{1}, V_{c h i_{a} o} o t i m e s D e l t a_{1})

and

H_{2, J_{2} (p)} (S_{2}, V_{c h i_{a} d} o t i m e s D e l t a_{2})

are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation

c h i_{a} d

of the fundamental group

p i o d (X_{2}, p_{0})

given a matrix representation

c h i_{a} o

of the fundamental group

p i o d (X_{1}, p'_{0})

. %% On the basis of the results of cite{vin} and Theorem ef{theorem_1} proven in the present work we then conjecture that there exists a covariant functor from the category

c a l R H

of finite bordered surfaces with vector bundle and signature matrices to the category of Kreu{i}n spaces and isomorphisms which are ramified covering of Riemann surfaces.

Full work available at URL: https://arxiv.org/abs/0911.3908

zbMATH Keywords

Riemann surfaces Hardy spaces half-order differentials

Mathematics Subject Classification ID

Compact Riemann surfaces and uniformization (30F10) (H^p)-spaces (42B30) Hardy spaces (30H10)