{"paper":{"title":"Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Emmanuel Russ (LATP), Nadine Badr (LM-Orsay)","submitted_at":"2008-02-07T08:24:55Z","abstract_excerpt":"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 interpo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.0922","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}