Sobolev and Hardy-Sobolev spaces on graphs

Let $Γ$ be a graph. Under suitable geometric assumptions on $Γ$, we give several equivalent characterizations of Sobolev and Hardy-Sobolev spaces on $Γ$, in terms of maximal functionals, Hajł asz type functionals or atomic decompositions. As an application, we study the boundedness of Riesz transforms on Hardy spaces on $Γ$. This gives the discrete counterpart of the corresponding results on Riemannian manifolds.

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