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

Abstract

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 $\Vert \nabla f{\Vert }_p$ and $\Vert (I-P{)}^{1/2}f{\Vert }_p$ uniformly in $f$ for $1<p<+\infty$. 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.