\(H^ 1\)-\(BMO\) duality on Riemann surfaces
type: Item
instance of: scholarly article
title: \(H^ 1\)-\(BMO\) duality on Riemann surfaces
zbMATH Open document ID: 0798.30035
DOI: 10.1007/BF02566520
author: Hiroshige Shiga
published in: Commentarii Mathematici Helvetici
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/140286
review text: Let \(R\) be a Riemann surface whose universal covering surface is the unit disk \(\Delta\). Let \(\pi\) denote the projection map from \(\Delta\) onto \(R\). Denote by \(h^ 1(R)\) and \(\text{BMOH}(R)\) the Banach spaces of harmonic functions \(h\) on \(R\) for which the lift \(h\circ \pi\) is, respectively, the real part of a \(H^ 1\) (the usual analytic Hardy space) function or the real part of a Poisson integral of a function of bounded mean oscillation on the unit circle. A famous theorem of Charles Fefferman asserts that, when properly interpreted, one has the duality relation \(\text{BMOH}(\Delta)= (h^ 1(\Delta))^*\). In a previous paper the author showed that Fefferman's relation remains true when \(R\) is a bordered compact surface. On the other hand, he notes that an example of M. Heins shows that it does not remain true for all hyperbolic surfaces, so it becomes an interesting problem to find out for which kinds of surfaces it does hold. In the paper under review the author shows that Fefferman's relation holds for surfaces \(R\) which are ``\(\text{SO}_{\text{HB}}\)'' ends. Such surfaces are obtained by removing finitely many Jordan curves from an open surface which has no Green function. The proof is quite complicated.
Mathematics Subject Classification ID: 30F99
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 167710
zbMATH Keywords: bounded mean oscillation
reviewed by: Albert Baernstein II
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 167710