Some results on BMOH and VMOH on Riemann surfaces (Q1174973)

Let \(R\) be a Riemann surface which allows a Green's function, and let \(G_ R(z,\lambda)\) denote the Green's function on \(R\) with singularity at \(\lambda\). Let \(\text{HD}(R)\) denote the collection of all (real valued) functions \(u(z)\) harmonic on \(R\) for which \(\iint_ R|\nabla u(z)|^ 2dx dy<\infty\), let \(\text{BMOH}(R)\) denote the collection of all (real valued) function \(u(z)\) harmonic on \(R\) for which there exists a finite constant \(M\) (depending on \(u)\) such that \(\iint_ R|\nabla u(z)|^ 2G_ R(z,\lambda)dx dy<M\) for each \(\lambda\in R\), and let \(\text{VMOH}(R)\) denote the collection of all (real valued) functions \(u(z)\) harmonic on \(R\) for which \[ \lim_{\lambda\to\partial R}\iint_ R|\nabla u(z)|^ 2 G_ R(z,\lambda) dx dy=0. \] It is shown that \(\text{HD}(R)\subset\text{VMOH}(R)\) whenever \(R\) is ``sufficiently nice'', but examples are given for which \(\text{HD}(R)\) is not even a subset of \(\text{BMOH}(R)\) (which always contains \(\text{VMOH}(R))\). The used method involves associating \(R\) with the fundamental region of an appropriate Fuchsian group \(\Gamma\) and associating the harmonic function \(u\) with a function \(f\) analytic in the unit disk \(D\), where \(f\) is additive automorphic relative to the Fuchsian group \(\Gamma\), which means that for each \(\gamma\in\Gamma\) there exists a constant \(A_ \gamma\) such that \(f(\gamma(z))=f(z)+A_ \gamma\) for each \(z\in D\). This is a variation of a method which has been used extensively to investigate \(\text{BMOA}(R)\).