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

Nadine Badr (1) , Emmanuel Russ (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

Let $\Gamma$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pf(x)=\sum_y p(x,y)f(y)$. Denote by $\nabla$ the discrete gradient. We give necessary and/or sufficient conditions on $\Gamma$ in order to compare $\left\Vert \nabla f\right\Vert_{p}$ and $\left\Vert (I-P)^{1/2}f\right\Vert_{p}$ 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.

Mots clés: interpolation; Graphs; discrete Laplacian; Riesz transforms; Littlewood-Paley inequalities; Sobolev spaces; interpolation.

Domaines: Equations aux dérivées partielles [math.AP]

Nadine Badr, Emmanuel Russ. Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs. Publicacions Matemàtiques, 2009, 53 (2), pp.273-328. ⟨hal-00243816⟩