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

Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = Σy p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ||∇f||p and‚ ||(I − P)1/2f||p uniformly in f for 1 < p < +∞. These conditions are different for p < 2 and p < 2. 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.