{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:VKZDD4ICPSXN2PULTK7OD7JZKL","short_pith_number":"pith:VKZDD4IC","schema_version":"1.0","canonical_sha256":"aab231f1027caedd3e8b9abee1fd3952d7356a01872aa4969a728621b942f01b","source":{"kind":"arxiv","id":"1102.1597","version":2},"attestation_state":"computed","paper":{"title":"A note on the spaces of variable integrability and summability of Almeida and H\\\"ast\\\"o","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Henning Kempka, Jan Vybiral","submitted_at":"2011-02-08T13:09:00Z","abstract_excerpt":"We address an open problem posed recently by Almeida and H\\\"ast\\\"o in \\cite{AlHa10}. They defined the spaces $\\ellqp$ of variable integrability and summability and showed that $\\|\\cdot|\\ellqp\\|$ is a norm if $q$ is constant almost everywhere or if $\\esssup_{x\\in\\R^n}1/p(x)+1/q(x)\\le 1$. Nevertheless, the natural conjecture (expressed also in \\cite{AlHa10}) is that the expression is a norm if $p(x),q(x)\\ge 1$ almost everywhere. We show, that $\\|\\cdot|\\ellqp\\|$ is a norm, if $1\\le q(x)\\le p(x)$ for almost every $x\\in\\R^n.$ Furthermore, we construct an example of $p(x)$ and $q(x)$ with $\\min(p(x)"},"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":"1102.1597","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-02-08T13:09:00Z","cross_cats_sorted":[],"title_canon_sha256":"59b2717b41642d811cc82c1c38166c925388a467aeaaf517c9c13c538387f3c9","abstract_canon_sha256":"9953d34795a4c09bd84b56b2ace765f9d1a3a6090e250cb137dbc8714db0b1a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:27.643872Z","signature_b64":"/2POV1jluTQ8YR46G9DINeEcbVXLJ4UthrT4HWZ4GfWQxKVNZSTsnD7i6pOQzCaYRe/hueccGQ3FrQVNhiEuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aab231f1027caedd3e8b9abee1fd3952d7356a01872aa4969a728621b942f01b","last_reissued_at":"2026-05-18T04:05:27.643312Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:27.643312Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the spaces of variable integrability and summability of Almeida and H\\\"ast\\\"o","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Henning Kempka, Jan Vybiral","submitted_at":"2011-02-08T13:09:00Z","abstract_excerpt":"We address an open problem posed recently by Almeida and H\\\"ast\\\"o in \\cite{AlHa10}. They defined the spaces $\\ellqp$ of variable integrability and summability and showed that $\\|\\cdot|\\ellqp\\|$ is a norm if $q$ is constant almost everywhere or if $\\esssup_{x\\in\\R^n}1/p(x)+1/q(x)\\le 1$. Nevertheless, the natural conjecture (expressed also in \\cite{AlHa10}) is that the expression is a norm if $p(x),q(x)\\ge 1$ almost everywhere. We show, that $\\|\\cdot|\\ellqp\\|$ is a norm, if $1\\le q(x)\\le p(x)$ for almost every $x\\in\\R^n.$ Furthermore, we construct an example of $p(x)$ and $q(x)$ with $\\min(p(x)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1597","kind":"arxiv","version":2},"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":"1102.1597","created_at":"2026-05-18T04:05:27.643433+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.1597v2","created_at":"2026-05-18T04:05:27.643433+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.1597","created_at":"2026-05-18T04:05:27.643433+00:00"},{"alias_kind":"pith_short_12","alias_value":"VKZDD4ICPSXN","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"VKZDD4ICPSXN2PUL","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"VKZDD4IC","created_at":"2026-05-18T12:26:44.992195+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/VKZDD4ICPSXN2PULTK7OD7JZKL","json":"https://pith.science/pith/VKZDD4ICPSXN2PULTK7OD7JZKL.json","graph_json":"https://pith.science/api/pith-number/VKZDD4ICPSXN2PULTK7OD7JZKL/graph.json","events_json":"https://pith.science/api/pith-number/VKZDD4ICPSXN2PULTK7OD7JZKL/events.json","paper":"https://pith.science/paper/VKZDD4IC"},"agent_actions":{"view_html":"https://pith.science/pith/VKZDD4ICPSXN2PULTK7OD7JZKL","download_json":"https://pith.science/pith/VKZDD4ICPSXN2PULTK7OD7JZKL.json","view_paper":"https://pith.science/paper/VKZDD4IC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.1597&json=true","fetch_graph":"https://pith.science/api/pith-number/VKZDD4ICPSXN2PULTK7OD7JZKL/graph.json","fetch_events":"https://pith.science/api/pith-number/VKZDD4ICPSXN2PULTK7OD7JZKL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VKZDD4ICPSXN2PULTK7OD7JZKL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VKZDD4ICPSXN2PULTK7OD7JZKL/action/storage_attestation","attest_author":"https://pith.science/pith/VKZDD4ICPSXN2PULTK7OD7JZKL/action/author_attestation","sign_citation":"https://pith.science/pith/VKZDD4ICPSXN2PULTK7OD7JZKL/action/citation_signature","submit_replication":"https://pith.science/pith/VKZDD4ICPSXN2PULTK7OD7JZKL/action/replication_record"}},"created_at":"2026-05-18T04:05:27.643433+00:00","updated_at":"2026-05-18T04:05:27.643433+00:00"}