{"paper":{"title":"On the boundedness of generalized Ces\\`aro operators on Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Carlos Lizama, Luis S\\'anchez-Lajusticia, Pedro J. Miana, Rodrigo Ponce","submitted_at":"2013-04-05T06:34:33Z","abstract_excerpt":"For $\\beta>0$ and $p\\ge 1$, the generalized Ces\\`aro operator $$ \\mathcal{C}_\\beta f(t):=\\frac{\\beta}{t^\\beta}\\int_0^t (t-s)^{\\beta-1}f(s)ds $$ and its companion operator $\\mathcal{C}_\\beta^*$ defined on Sobolev spaces $\\mathcal{T}_p^{(\\alpha)}(t^\\alpha)$ and $\\mathcal{T}_p^{(\\alpha)}(| t|^\\alpha)$ (where $\\alpha\\ge 0$ is the fractional order of derivation and are embedded in $L^p(\\RR^+)$ and $L^p(\\RR)$ respectively) are studied. We prove that if $p>1$, then $\\mathcal{C}_\\beta$ and $\\mathcal{C}_\\beta^*$ are bounded operators and commute on $\\mathcal{T}_p^{(\\alpha)}(t^\\alpha)$ and $\\mathcal{T}_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1622","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"}