Characterizes positive critical Hardy weights for Laplacians on weighted graphs and identifies an optimal Hardy weight for fractional Laplacians under suitable assumptions, with examples on Cayley, curvature, and fractal graphs.
A new proof of G romov's theorem on groups of polynomial growth
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Establishes affine rigidity for Lipschitz harmonic functions on nilpotent groups and quasi-isometric invariance of the Lipschitz harmonic space on polynomial growth groups under adapted smooth Abelian-centered measures.
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Positive Criticality and Optimal Hardy Inequality for Fractional Laplacians
Characterizes positive critical Hardy weights for Laplacians on weighted graphs and identifies an optimal Hardy weight for fractional Laplacians under suitable assumptions, with examples on Cayley, curvature, and fractal graphs.
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The Lipschitz Liouville Property, Affine Rigidity, and Coarse Harmonic Coordinates on Groups of Polynomial Growth
Establishes affine rigidity for Lipschitz harmonic functions on nilpotent groups and quasi-isometric invariance of the Lipschitz harmonic space on polynomial growth groups under adapted smooth Abelian-centered measures.