{"paper":{"title":"Spectral asymptotic formula of Bessel--Riesz commutator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dmitriy Zanin, Fedor Sukochev, Ji Li, Zhijie Fan","submitted_at":"2024-11-22T13:36:19Z","abstract_excerpt":"Let $R_{\\lambda,j}$ be the $j$-th Bessel--Riesz transform, where $n\\geq 1$, $\\lambda>0$, and $j=1,\\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{\\lambda,j}]$, the commutator of $R_{\\lambda,j}$ with multiplication operator $M_f$, based on building a preliminary result that the endpoint weak Schatten norm of $[M_f,R_{\\lambda,j}]$ can be characterised via homogeneous Sobolev norm $\\dot{W}^{1,n+1}(\\mathbb{R}_+^{n+1})$ of the symbol $f$. Specifically, the asymptotic coefficient is equivalent to $\\|f\\|_{\\dot{W}^{1,n+1}(\\mathbb{R}_+^{n+1})}.$ Our main strategy is to re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14928","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/2411.14928/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"}