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Normal operators constructed from generalized harmonic measures on open Riemann surfaces. (English) Zbl 0814.30025

Let \(R\) be an open Riemann surface. A bounded harmonic function \(u\) on \(R\) is called a generalized harmonic measure on \(R\) if the greatest harmonic minorant of \(u\) and \(1-u\) is equal to 0. Let \(\Gamma_ h\) be the Hilbert space of real harmonic differentials on \(R\) with finite Dirichlet norm, let \(\Gamma_{\widehat {hm}}\) be the completion in \(\Gamma_ h\) of the collection of differentials of generalized harmonic measures on \(R\), and let \(\Gamma_{\widehat {hwe}}\) be the subspace of \(\Gamma_ h\) consisting of harmonic differentials which have vanishing periods along almost all weakly dividing cycles. A normal operator \(\widehat {L}_ 1\) which is a generalization of \(L_ 1\)-operator introduced by L. Sario, is introduced. \(\widehat {L}_ 1 f\) is characterized by the properties: There exist a harmonic function \(u_{\widehat {hm}}\) on \(R\) with \(du_{\widehat {hm}}\in \Gamma_{\widehat {hm}}\) and a Dirichlet potential \(p\) on \(R\) such that \(\widehat {L}_ 1 f= u_{\widehat {hm}}+ p\) on an end \(R-V\) and \(\int_ c *\widehat {dL}_ 1 f=0\) for almost all weakly dividing cycles \(c\). An extremal property of \(\widehat {L}_ 1\)-operator is shown, further a modulus function obtained from \(\widehat {L}_ 1\)-operator is introduced, and an example related to it is given.

MSC:

30F15 Harmonic functions on Riemann surfaces
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