Runge- and Walsh-type extensions of smooth subharmonic functions on open Riemann surfaces

In this paper we study several settings of the $C^m$-subharmonic extension problem on open Riemann surfaces. The problem is completely solved (for all $m\in[0,+\infty)$) 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 $C^m$-subharmonic extensions, automorphic with respect to some appropriate groups of automorphisms of an open Riemann surface.
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