{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:77NXVZW2ST7IIT76OR5J3FY274","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a82d8682bea62bf25968f69ef585284e75879291d38c16a63e40393787b1f5b2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-10T12:04:21Z","title_canon_sha256":"488d39c12b01898d31d88cdde202f8cb6bbcbebbd95e90787f1a2b3210a3185e"},"schema_version":"1.0","source":{"id":"1804.03469","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.03469","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"arxiv_version","alias_value":"1804.03469v1","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.03469","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"pith_short_12","alias_value":"77NXVZW2ST7I","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"77NXVZW2ST7IIT76","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"77NXVZW2","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:e55caceb091e96ddd5ffc555849f644b898a7a41bb404fd238d930ef47c6baaa","target":"graph","created_at":"2026-05-18T00:18:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove that if $1 \\leq p, q \\leq \\infty$, then the spaces $L_p +L_q$ and $L_p \\cap L_q$ are isomorphic if and only if $p = q$. In particular, $L_2 +L_{\\infty}$ and $L_2 \\cap L_{\\infty}$ are not isomorphic which is an answer to a question formulated in the paper S. V. Astashkin and L. Maligranda, \\textit{$L_p + L_{\\infty}$ and $L_p \\cap L_{\\infty}$ are not isomorphic for all $1 \\leq p < \\infty, p \\neq 2$}, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2181--2194.","authors_text":"Lech Maligranda, Sergey Astashkin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-10T12:04:21Z","title":"$L_p +L_q$ and $L_p \\cap L_q$ are not isomorphic for all $1 \\leq p, q \\leq \\infty$, $p \\neq q$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03469","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:af71b5deaafbadfc08f3a6720a27e7bb3f1e53a1d00bbc760eb7f40b3609f600","target":"record","created_at":"2026-05-18T00:18:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a82d8682bea62bf25968f69ef585284e75879291d38c16a63e40393787b1f5b2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-10T12:04:21Z","title_canon_sha256":"488d39c12b01898d31d88cdde202f8cb6bbcbebbd95e90787f1a2b3210a3185e"},"schema_version":"1.0","source":{"id":"1804.03469","kind":"arxiv","version":1}},"canonical_sha256":"ffdb7ae6da94fe844ffe747a9d971aff093f41ba4d6bd49283b6935a703279e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ffdb7ae6da94fe844ffe747a9d971aff093f41ba4d6bd49283b6935a703279e5","first_computed_at":"2026-05-18T00:18:48.090801Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:18:48.090801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"obRYWh8Nd1h+g9IR6x1hl3lPO1Swc25IOAheY2k9XVancco2QxkbW/etY4oGoL/TM0DtMNMV0uTpoZo4Ql8CBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:18:48.091574Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.03469","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:af71b5deaafbadfc08f3a6720a27e7bb3f1e53a1d00bbc760eb7f40b3609f600","sha256:e55caceb091e96ddd5ffc555849f644b898a7a41bb404fd238d930ef47c6baaa"],"state_sha256":"3d233a04f72cd01b339608f022ce16699c502b745217511b6d393045f2572cc2"}