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Transmission of harmonic functions through quasicircles on compact Riemann surfaces. (English) Zbl 1461.30051

Given a compact Riemann surface \(R\), a separating Jordan curve \(\Gamma\) such that \(R\setminus\Gamma = \Sigma_1\sqcup \Sigma_2\), and a function \(h\) on \(\Gamma\), it is natural to ask whether there exist elements of the Dirichlet spaces of \(\Sigma_1\) and \(\Sigma_2\) that have \(h\) as their boundary values. Of course, one cannot prescribe a completely arbitrary \(h\) on \(\Gamma\) if this question must make sense. The authors of the paper being reviewed prove that if \(\Gamma\) is a quasicircle, then \(h\) is the boundary value of an element of the Dirichlet space of \(\Sigma_1\) if and only if it is the boundary value of an element of the Dirichlet space of \(\Sigma_2\). The word “transmission”, in the title of the paper, refers to the resulting phenomenon of a harmonic function on \(\Sigma_1\) with finite Dirichlet seminorm determining a corresponding harmonic function in the Dirichlet space of \(\Sigma_2\). The authors also show that the resulting transmission map between the Dirichlet spaces of \(\Sigma_1\) and \(\Sigma_2\) is bounded.
The requirement that \(\Gamma\) be a quasicircle originates in an earlier work by the authors. In [J. Math. Anal. Appl. 448, No. 2, 864–884 (2017)], they showed that when \(R\) is the Riemann sphere, the transmission map exists and is bounded if and only if \(\Gamma\) is a quasicircle. The proof of this for a general compact Riemann surface uses sewing techniques for Riemann surfaces. Also, the proofs of some of the supporting results that the authors state along the way follow an approach that facilitates the use of sewing techniques.

MSC:

30C62 Quasiconformal mappings in the complex plane
30F15 Harmonic functions on Riemann surfaces
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