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 Mat. Sb., 2015, Volume 206, Number 1, Pages 5–28 (Mi msb8345)

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

A. Boivina, P. M. Gauthierb, P. V. Paramonovc

a University of Western Ontario
b Université de Montréal
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.
Bibliography: 22 titles.

Keywords: subharmonic function, Riemann surface, Green function, localization operator, automorphism group.

 Funding Agency Grant Number Natural Sciences and Engineering Research Council of Canada (NSERC) Ministry of Education and Science of the Russian Federation ÍØ-2900.2014.1 All the authors were partially supported by grants from NSERC (Canada). The research of the third author was also partially supported by the Programme for the Support of Leading Scientific Schools of the Russian Federation (grant no. ÍØ-2900.2014.1).

DOI: https://doi.org/10.4213/sm8345

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English version:
Sbornik: Mathematics, 2015, 206:1, 3–23

Bibliographic databases:

UDC: 517.574+517.545
MSC: 31A05, 30F99

Citation: A. Boivin, P. M. Gauthier, P. V. Paramonov, “Runge- and Walsh-type extensions of smooth subharmonic functions on open Riemann surfaces”, Mat. Sb., 206:1 (2015), 5–28; Sb. Math., 206:1 (2015), 3–23

Citation in format AMSBIB
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