{"paper":{"title":"Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Joseph Feneuil","submitted_at":"2026-06-03T21:53:52Z","abstract_excerpt":"In the $D$-dimensional Vicsek graph, we prove that the Riesz-like inequality $ \\|\\nabla f\\|_p \\leq C \\|\\Delta^\\gamma f\\|_p $ holds for every $p\\in(1,\\infty)$ and every $ 0<\\gamma<\\gamma^*(p):=\\frac{1}{D+1}+\\frac{D-1}{D+1}\\,\\frac{1}{p}, $ while it fails whenever $p\\in(1,\\infty)$ and $\\gamma^*(p)<\\gamma<1$. Thus, the validity of the inequality remains open only at the critical exponent $\\gamma=\\gamma^*(p)$. This provides the first example of an $L^p$-bounded ``super-Riesz transform'', namely an operator of the form $\\nabla \\Delta^{-\\gamma}$ with $\\gamma$ strictly larger than the Euclidean thresh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05475","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.05475/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}