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The \(p\)-harmonic boundary for quasi-isometric graphs and manifolds. (English) Zbl 1257.31008

Summary: Let \(p\) be a real number greater than one. Suppose that a graph \(G\) of bounded degree is quasi-isometric with a Riemannian manifold \(M\) with certain properties. Under these conditions we will show that the \(p\)-harmonic boundary of \(G\) is homeomorphic to the \(p\)-harmonic boundary of \(M\). We will also prove that there is a bijection between the \(p\)-harmonic functions on \(G\) and the \(p\)-harmonic functions on \(M\).

MSC:

31C12 Potential theory on Riemannian manifolds and other spaces
31C20 Discrete potential theory
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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References:

[1] Ilkka Holopainen, Rough isometries and p-harmonic functions with finite Dirichlet integral , Rev. Mat. Iberoamer. 10 (1994), 143-176. · Zbl 0797.31008
[2] Ilkka Holopainen and Paolo M. Soardi, \(p\)-harmonic functions on graphs and manifolds , Manuscr. Math. 94 (1997), 95-110. · Zbl 0898.31007
[3] Masahiko Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds , J. Math. Soc. Japan 37 (1985), 391-413. · Zbl 0554.53030
[4] —, Rough isometries and the parabolicity of Riemannian manifolds , J. Math. Soc. Japan 38 (1986), 227-238. · Zbl 0577.53031
[5] Yong Hah Lee, Rough isometry and \(p\)-harmonic boundaries of complete Riemannian manifolds , Potential Anal. 23 (2005), 83-97. · Zbl 1082.31005
[6] Michael J. Puls, Graphs of bounded degree and the \(p\)-harmonic boundary , Pacific J. Math. 248 (2010), 429-452. · Zbl 1228.43004
[7] L. Sario and M. Nakai, Classification theory of Riemann surfaces , Die Grundl. Math. Wissen. 164 , Springer-Verlag, New York, 1970. \noindentstyle · Zbl 0199.40603
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