{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:PKBOPMMUJ2P7JPVA3TKYIL37ZU","short_pith_number":"pith:PKBOPMMU","canonical_record":{"source":{"id":"1709.00380","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-01T16:03:02Z","cross_cats_sorted":[],"title_canon_sha256":"e53fa3c521323d9bdc28e309f4eee5ffcf94d5b3a3a824daa9cc0ac118b032f3","abstract_canon_sha256":"8acb05a9295532f085b94a0531e9405435186454afa0dbeb58dc9ad6ad552e28"},"schema_version":"1.0"},"canonical_sha256":"7a82e7b1944e9ff4bea0dcd5842f7fcd075605fe20f5689f6f928ee5a21b9113","source":{"kind":"arxiv","id":"1709.00380","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.00380","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"arxiv_version","alias_value":"1709.00380v2","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.00380","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"pith_short_12","alias_value":"PKBOPMMUJ2P7","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PKBOPMMUJ2P7JPVA","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PKBOPMMU","created_at":"2026-05-18T12:31:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:PKBOPMMUJ2P7JPVA3TKYIL37ZU","target":"record","payload":{"canonical_record":{"source":{"id":"1709.00380","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-01T16:03:02Z","cross_cats_sorted":[],"title_canon_sha256":"e53fa3c521323d9bdc28e309f4eee5ffcf94d5b3a3a824daa9cc0ac118b032f3","abstract_canon_sha256":"8acb05a9295532f085b94a0531e9405435186454afa0dbeb58dc9ad6ad552e28"},"schema_version":"1.0"},"canonical_sha256":"7a82e7b1944e9ff4bea0dcd5842f7fcd075605fe20f5689f6f928ee5a21b9113","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:36.397720Z","signature_b64":"OJllUhKfgp0RP0u3OKWVpSDcuEBXN0syI9b/buP6kFEiPj5J3AzWkkjfVoK1ZCeentzCx+a9BwSQfXNxXIcsBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a82e7b1944e9ff4bea0dcd5842f7fcd075605fe20f5689f6f928ee5a21b9113","last_reissued_at":"2026-05-18T00:08:36.397013Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:36.397013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.00380","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:08:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dU6MLKGn2+9IUXf2j0/nkccaO93ynGov9RyigFEXxzmERcE1NZHqobI/AturS/zV0DMlzGfxM09WTTP43xwsCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T16:51:37.462236Z"},"content_sha256":"fa0afaa1d8060eb1c573ab8400a2449b510e93699c6b31bb102a2c51e7aaf134","schema_version":"1.0","event_id":"sha256:fa0afaa1d8060eb1c573ab8400a2449b510e93699c6b31bb102a2c51e7aaf134"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:PKBOPMMUJ2P7JPVA3TKYIL37ZU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weights of reasonable growth and their application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mateusz Krukowski","submitted_at":"2017-09-01T16:03:02Z","abstract_excerpt":"In the paper, we introduce the concept of weight with reasonable growth on a locally compact group $G$. We verify that these weights form a natural class to work with, by examining the most common examples. We proceed with the discussion of the $L^p-$conjecture. With the use of Riesz-Thorin-Stein-Weiss interpolation, we establish that \\mbox{$L^p_{\\omega}(G)\\star L^p_{\\omega}(G)\\subset L^p_{\\omega}(G)$}, $p>1$ implies that $L^p_{\\omega}(G)\\star L^q_{\\omega}(G)\\subset L^q_{\\omega}(G)$ for $q$ which lies between $p$ and $p'$. At last, we confirm the $L^p_{\\omega}-$conjecture for weights of $(p,q)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00380","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:08:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wk292s+x7eO5SoShp03vVzRcV4fTdBqDSjA+lslnDsLvKeDZDKxOh16BviZ2CTTKeRgXzTOszALmj5IJL7EdDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T16:51:37.462596Z"},"content_sha256":"bdab2396531f0375e0dea627fc195fa4968ec8f65a1b6749ba7cf56ecb6dd436","schema_version":"1.0","event_id":"sha256:bdab2396531f0375e0dea627fc195fa4968ec8f65a1b6749ba7cf56ecb6dd436"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PKBOPMMUJ2P7JPVA3TKYIL37ZU/bundle.json","state_url":"https://pith.science/pith/PKBOPMMUJ2P7JPVA3TKYIL37ZU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PKBOPMMUJ2P7JPVA3TKYIL37ZU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T16:51:37Z","links":{"resolver":"https://pith.science/pith/PKBOPMMUJ2P7JPVA3TKYIL37ZU","bundle":"https://pith.science/pith/PKBOPMMUJ2P7JPVA3TKYIL37ZU/bundle.json","state":"https://pith.science/pith/PKBOPMMUJ2P7JPVA3TKYIL37ZU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PKBOPMMUJ2P7JPVA3TKYIL37ZU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:PKBOPMMUJ2P7JPVA3TKYIL37ZU","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":"8acb05a9295532f085b94a0531e9405435186454afa0dbeb58dc9ad6ad552e28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-01T16:03:02Z","title_canon_sha256":"e53fa3c521323d9bdc28e309f4eee5ffcf94d5b3a3a824daa9cc0ac118b032f3"},"schema_version":"1.0","source":{"id":"1709.00380","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.00380","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"arxiv_version","alias_value":"1709.00380v2","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.00380","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"pith_short_12","alias_value":"PKBOPMMUJ2P7","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PKBOPMMUJ2P7JPVA","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PKBOPMMU","created_at":"2026-05-18T12:31:37Z"}],"graph_snapshots":[{"event_id":"sha256:bdab2396531f0375e0dea627fc195fa4968ec8f65a1b6749ba7cf56ecb6dd436","target":"graph","created_at":"2026-05-18T00:08:36Z","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":"In the paper, we introduce the concept of weight with reasonable growth on a locally compact group $G$. We verify that these weights form a natural class to work with, by examining the most common examples. We proceed with the discussion of the $L^p-$conjecture. With the use of Riesz-Thorin-Stein-Weiss interpolation, we establish that \\mbox{$L^p_{\\omega}(G)\\star L^p_{\\omega}(G)\\subset L^p_{\\omega}(G)$}, $p>1$ implies that $L^p_{\\omega}(G)\\star L^q_{\\omega}(G)\\subset L^q_{\\omega}(G)$ for $q$ which lies between $p$ and $p'$. At last, we confirm the $L^p_{\\omega}-$conjecture for weights of $(p,q)","authors_text":"Mateusz Krukowski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-01T16:03:02Z","title":"Weights of reasonable growth and their application"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00380","kind":"arxiv","version":2},"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:fa0afaa1d8060eb1c573ab8400a2449b510e93699c6b31bb102a2c51e7aaf134","target":"record","created_at":"2026-05-18T00:08:36Z","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":"8acb05a9295532f085b94a0531e9405435186454afa0dbeb58dc9ad6ad552e28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-01T16:03:02Z","title_canon_sha256":"e53fa3c521323d9bdc28e309f4eee5ffcf94d5b3a3a824daa9cc0ac118b032f3"},"schema_version":"1.0","source":{"id":"1709.00380","kind":"arxiv","version":2}},"canonical_sha256":"7a82e7b1944e9ff4bea0dcd5842f7fcd075605fe20f5689f6f928ee5a21b9113","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a82e7b1944e9ff4bea0dcd5842f7fcd075605fe20f5689f6f928ee5a21b9113","first_computed_at":"2026-05-18T00:08:36.397013Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:36.397013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OJllUhKfgp0RP0u3OKWVpSDcuEBXN0syI9b/buP6kFEiPj5J3AzWkkjfVoK1ZCeentzCx+a9BwSQfXNxXIcsBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:36.397720Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.00380","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa0afaa1d8060eb1c573ab8400a2449b510e93699c6b31bb102a2c51e7aaf134","sha256:bdab2396531f0375e0dea627fc195fa4968ec8f65a1b6749ba7cf56ecb6dd436"],"state_sha256":"d5d339e021efbd78e348f45155276e812e69d1cdf3530daf772c9fcb2f27c98a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IFT5Yw9tQnmAh1KzybBqWyejzKnOSn1TeACsZE9+cnkKwGmvI9YzQVtglA+g0Fcc6BonyTFcnPaeBwlkktpPCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T16:51:37.464535Z","bundle_sha256":"cbabd264be7b8d4a3f5012bc9ed380048599d2cb6a5d27dcc95b47526f168856"}}