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

TL;DR

This paper investigates conditions on graphs that allow for the comparison of discrete gradient norms and fractional powers of Markov operators, establishing Littlewood-Paley inequalities and interpolation results for Sobolev spaces on graphs.

Contribution

It introduces new conditions for comparing Sobolev norms on graphs and develops Littlewood-Paley inequalities and interpolation results in this setting.

Findings

Established necessary and sufficient conditions for norm comparison on graphs.

Proved Littlewood-Paley inequalities for Sobolev spaces on graphs.

Developed interpolation results for Sobolev spaces in the graph context.

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 Calder\'on-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.