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

$\Phi$-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

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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000344337300009} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84911979664} 

• http://mi.mathnet.ru/eng/smj2591
• http://mi.mathnet.ru/eng/smj/v55/i5/p1104

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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
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