\(H^ 1\)-\(BMO\) duality on Riemann surfaces (Q1209299)

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.