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

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

2008-02-07

arxiv.org/abs/0802.0922v1

Publicacions Matem\`atiques 53, 2 (2009) 273-328

Mathematics

Analysis of PDEs

Scientific paper

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 $12$. 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.

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