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Sibirsk. Mat. Zh., 2014, Volume 55, Number 5, Pages 1104–1117 (Mi smj2591)  

This article is cited in 1 scientific paper (total in 1 paper)

$\Phi$-harmonic functions on discrete groups and the first $\ell^\Phi$-cohomology

Ya. A. Kopylovab, R. A. Panenkoa

a Sobolev Institute of Mathematics, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia

Abstract: We study the first cohomology groups of a countable discrete group $G$ with coefficients in a $G$-module $\ell^\Phi(G)$, where $\Phi$ is an $n$-function of class $\Delta_2(0)\cap\nabla_2(0)$. Developing the ideas of Puls and Martin–Valette for a finitely generated group $G$, we introduce the discrete $\Phi$-Laplacian and prove a theorem on the decomposition of the space of $\Phi$-Dirichlet finite functions into the direct sum of the spaces of $\Phi$-harmonic functions and $\ell^\Phi(G)$ (with an appropriate factorization). We prove also that if a finitely generated group $G$ has a finitely generated infinite amenable subgroup with infinite centralizer then $\overline H^1(G,\ell^\Phi(G))=0$. In conclusion, we show the triviality of the first cohomology group for the wreath product of two groups one of which is nonamenable.

Keywords: group, $N$-function, Orlicz space, $\Delta_2$-regularity, $\Phi$-harmonic function, $1$-cohomology.

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English version:
Siberian Mathematical Journal, 2014, 55:5, 904–914

Bibliographic databases:

UDC: 512.664.4+517.986.6
Received: 11.11.2013

Citation: Ya. A. Kopylov, R. A. Panenko, “$\Phi$-harmonic functions on discrete groups and the first $\ell^\Phi$-cohomology”, Sibirsk. Mat. Zh., 55:5 (2014), 1104–1117; Siberian Math. J., 55:5 (2014), 904–914

Citation in format AMSBIB
\Bibitem{KopPan14}
\by Ya.~A.~Kopylov, R.~A.~Panenko
\paper $\Phi$-harmonic functions on discrete groups and the first $\ell^\Phi$-cohomology
\jour Sibirsk. Mat. Zh.
\yr 2014
\vol 55
\issue 5
\pages 1104--1117
\mathnet{http://mi.mathnet.ru/smj2591}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3289114}
\transl
\jour Siberian Math. J.
\yr 2014
\vol 55
\issue 5
\pages 904--914
\crossref{https://doi.org/10.1134/S0037446614050097}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84911979664}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. Panenko, “$\Phi$-garmonicheskie funktsii na grafakh”, Sib. elektron. matem. izv., 14 (2017), 1–9  mathnet  crossref
  • Сибирский математический журнал Siberian Mathematical Journal
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