Abstract
In this paper we study several settings of the -subharmonic extension problem on open Riemann surfaces. The problem is completely solved (for all ) for so-called Runge-type extensions. Several (in some sense sharp) sufficient conditions and counterexamples are found also for Walsh-type extensions. As applications, these results allow us to prove the existence of -subharmonic extensions, automorphic with respect to some appropriate groups of automorphisms of an open Riemann surface. Bibliography: 22 titles.
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