Sobolev and Hardy-Sobolev spaces on graphs

TL;DR

This paper characterizes Sobolev and Hardy-Sobolev spaces on graphs using various methods and studies the boundedness of Riesz transforms, extending results from Riemannian manifolds to discrete graph settings.

Contribution

It provides equivalent characterizations of Sobolev and Hardy-Sobolev spaces on graphs and analyzes Riesz transform boundedness in this discrete context.

Findings

Equivalent characterizations of Sobolev spaces on graphs

Atomic decompositions and maximal functionals established

Boundedness of Riesz transforms on Hardy spaces proved

Abstract

Let $\Gamma$ be a graph. Under suitable geometric assumptions on $\Gamma$, we give several equivalent characterizations of Sobolev and Hardy-Sobolev spaces on $\Gamma$, in terms of maximal functionals, Haj{\l} asz type functionals or atomic decompositions. As an application, we study the boundedness of Riesz transforms on Hardy spaces on $\Gamma$. This gives the discrete counterpart of the corresponding results on Riemannian manifolds.