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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}_"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1304.1622","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-04-05T06:34:33Z","cross_cats_sorted":[],"title_canon_sha256":"731f6f4ab60a13ee0c584255afa0ffc0c9b6ad996954a5ccd3369b33f7b2dd2e","abstract_canon_sha256":"342d10a20bdb6578a0807fa4402dc743308198d6995c49f6a7c726ecb6640887"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:55.199451Z","signature_b64":"bOdlZb1CVZ+9V/S+qA/Xws3A/ogrcE2O2t26HSIRahrxwcmDcPtDH5162R5yUNWteoLYKbHfWAX9CTvLbgNgBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc73ab5c91f62817607985d34343900c34c2c9359e65d10d31aab43a2ab0affa","last_reissued_at":"2026-05-18T03:28:55.198867Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:55.198867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1304.1622","created_at":"2026-05-18T03:28:55.198966+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.1622v1","created_at":"2026-05-18T03:28:55.198966+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1622","created_at":"2026-05-18T03:28:55.198966+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZRZ2WXER6YUB","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZRZ2WXER6YUBOYDZ","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZRZ2WXER","created_at":"2026-05-18T12:28:09.283467+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ","json":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ.json","graph_json":"https://pith.science/api/pith-number/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/graph.json","events_json":"https://pith.science/api/pith-number/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/events.json","paper":"https://pith.science/paper/ZRZ2WXER"},"agent_actions":{"view_html":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ","download_json":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ.json","view_paper":"https://pith.science/paper/ZRZ2WXER","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.1622&json=true","fetch_graph":"https://pith.science/api/pith-number/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/graph.json","fetch_events":"https://pith.science/api/pith-number/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/action/storage_attestation","attest_author":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/action/author_attestation","sign_citation":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/action/citation_signature","submit_replication":"https://pith.science/pith/ZRZ2WXER6YUBOYDZQXJUGQ4QBQ/action/replication_record"}},"created_at":"2026-05-18T03:28:55.198966+00:00","updated_at":"2026-05-18T03:28:55.198966+00:00"}