Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs (Q2271350)

Let \(\Gamma\) be a graph endowed with a reversible Markov kernel \(p\) and let \(Pf(x)=\sum_{y}p(x,y)f(y)\). Sufficient and necessary conditions are given such that the norms \(\|\nabla f\|_p\) and \(\|(I-P)^{1/2}f\|_p\) can be estimated with each other for \(1<p<\infty\), where \(\nabla\) is the discrete gradient. Different conditions are obtained for \(p<2\) and \(p>2\). Littlewood-Paley inequalities and interpolation theorems for Sobolev spaces are also proved.