# Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs

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* Corresponding author 1 LM-Orsay - Laboratoire de Mathématiques d-Orsay 2 LATP - Laboratoire d-Analyse, Topologie, Probabilités

Abstract : Let $\Gamma$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pfx=\sum y px,yfy$. Denote by $abla$ the discrete gradient. We give necessary and-or sufficient conditions on $\Gamma$ in order to compare $\left\Vert abla f ight\Vert {p}$ and $\left\Vert I-P^{1-2}f ight\Vert {p}$ uniformly in $f$ for $12$. The proofs rely on recent techniques developed to handle operators beyond the class of Calderón-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest.

Keywords : interpolation Graphs discrete Laplacian Riesz transforms Littlewood-Paley inequalities Sobolev spaces interpolation.