{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:KIV6BCK3OJLPP6DVLOZBJXACOD","short_pith_number":"pith:KIV6BCK3","canonical_record":{"source":{"id":"1110.0299","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-03T08:50:17Z","cross_cats_sorted":[],"title_canon_sha256":"b6f5b7e255e416f5c03ee3da8db117f4ef159271cea9b47d5856e3e03904cb49","abstract_canon_sha256":"cf5a632b77ca13f4af7977d55ffdb461e7c98789776b3c86873868b8238d72a1"},"schema_version":"1.0"},"canonical_sha256":"522be0895b7256f7f8755bb214dc0270fb7cf71935c696e334032c6e8416ffec","source":{"kind":"arxiv","id":"1110.0299","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.0299","created_at":"2026-05-18T04:11:50Z"},{"alias_kind":"arxiv_version","alias_value":"1110.0299v1","created_at":"2026-05-18T04:11:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.0299","created_at":"2026-05-18T04:11:50Z"},{"alias_kind":"pith_short_12","alias_value":"KIV6BCK3OJLP","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"KIV6BCK3OJLPP6DV","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"KIV6BCK3","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:KIV6BCK3OJLPP6DVLOZBJXACOD","target":"record","payload":{"canonical_record":{"source":{"id":"1110.0299","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-03T08:50:17Z","cross_cats_sorted":[],"title_canon_sha256":"b6f5b7e255e416f5c03ee3da8db117f4ef159271cea9b47d5856e3e03904cb49","abstract_canon_sha256":"cf5a632b77ca13f4af7977d55ffdb461e7c98789776b3c86873868b8238d72a1"},"schema_version":"1.0"},"canonical_sha256":"522be0895b7256f7f8755bb214dc0270fb7cf71935c696e334032c6e8416ffec","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:50.302605Z","signature_b64":"n6ez2OfOc/4jQNnWwo7fpNXy6ydQJ1SHyd3vfkpqMuFzStQLmmAzT5NPpUi3J8WWgxhxTgYro7kG1KDNLaFZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"522be0895b7256f7f8755bb214dc0270fb7cf71935c696e334032c6e8416ffec","last_reissued_at":"2026-05-18T04:11:50.302062Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:50.302062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.0299","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:11:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PICiR5LxDHT/tGR1hpJ0PDBV9BJApa/3egnFVjgiQV+/ueMI+wKxuGxa9eokDv8CSXI4zVK7FxFWeeUFF2zADA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T07:55:43.373191Z"},"content_sha256":"439ef421faf1e78e252ade35534a25397dc83508c584465b477323aa1ccc7751","schema_version":"1.0","event_id":"sha256:439ef421faf1e78e252ade35534a25397dc83508c584465b477323aa1ccc7751"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:KIV6BCK3OJLPP6DVLOZBJXACOD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On an Interesting Class of Variable Exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alexei Yu. Karlovich, Ilya M. Spitkovsky","submitted_at":"2011-10-03T08:50:17Z","abstract_excerpt":"Let $\\mathcal{M}(\\mathbb{R}^n)$ be the class of functions $p:\\mathbb{R}^n\\to[1,\\infty]$ bounded away from one and infinity and such that the Hardy-Littlewood maximal function is bounded on the variable Lebesgue space $L^{p(\\cdot)}(\\mathbb{R}^n)$. We denote by $\\mathcal{M}^*(\\mathbb{R}^n)$ the class of variable exponents $p\\in\\mathcal{M}(\\mathbb{R}^n)$ for which $1/p(x)=\\theta/p_0+(1-\\theta)/p_1(x)$ with some $p_0\\in(1,\\infty)$, $\\theta\\in(0,1)$, and $p_1\\in\\mathcal{M}(\\mathbb{R}^n)$. Rabinovich and Samko \\cite{RS08} observed that each globally log-H\\\"older continuous exponent belongs to $\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0299","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:11:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HpyWjFbw1LSu7l80y0fhd9opPtWo0vE2vQb+sGb82JxNAiZg/0ueIW/tKpsk6Ymz+D3wRV6BNkpAcpfmBMp3Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T07:55:43.373561Z"},"content_sha256":"4f68a8e71e139c11a67c56f125c990b17b15a5c1fde0bc78c3b0d3533ca06d76","schema_version":"1.0","event_id":"sha256:4f68a8e71e139c11a67c56f125c990b17b15a5c1fde0bc78c3b0d3533ca06d76"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KIV6BCK3OJLPP6DVLOZBJXACOD/bundle.json","state_url":"https://pith.science/pith/KIV6BCK3OJLPP6DVLOZBJXACOD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KIV6BCK3OJLPP6DVLOZBJXACOD/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":"2026-06-12T07:55:43Z","links":{"resolver":"https://pith.science/pith/KIV6BCK3OJLPP6DVLOZBJXACOD","bundle":"https://pith.science/pith/KIV6BCK3OJLPP6DVLOZBJXACOD/bundle.json","state":"https://pith.science/pith/KIV6BCK3OJLPP6DVLOZBJXACOD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KIV6BCK3OJLPP6DVLOZBJXACOD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:KIV6BCK3OJLPP6DVLOZBJXACOD","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":"cf5a632b77ca13f4af7977d55ffdb461e7c98789776b3c86873868b8238d72a1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-03T08:50:17Z","title_canon_sha256":"b6f5b7e255e416f5c03ee3da8db117f4ef159271cea9b47d5856e3e03904cb49"},"schema_version":"1.0","source":{"id":"1110.0299","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.0299","created_at":"2026-05-18T04:11:50Z"},{"alias_kind":"arxiv_version","alias_value":"1110.0299v1","created_at":"2026-05-18T04:11:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.0299","created_at":"2026-05-18T04:11:50Z"},{"alias_kind":"pith_short_12","alias_value":"KIV6BCK3OJLP","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"KIV6BCK3OJLPP6DV","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"KIV6BCK3","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:4f68a8e71e139c11a67c56f125c990b17b15a5c1fde0bc78c3b0d3533ca06d76","target":"graph","created_at":"2026-05-18T04:11:50Z","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":"Let $\\mathcal{M}(\\mathbb{R}^n)$ be the class of functions $p:\\mathbb{R}^n\\to[1,\\infty]$ bounded away from one and infinity and such that the Hardy-Littlewood maximal function is bounded on the variable Lebesgue space $L^{p(\\cdot)}(\\mathbb{R}^n)$. We denote by $\\mathcal{M}^*(\\mathbb{R}^n)$ the class of variable exponents $p\\in\\mathcal{M}(\\mathbb{R}^n)$ for which $1/p(x)=\\theta/p_0+(1-\\theta)/p_1(x)$ with some $p_0\\in(1,\\infty)$, $\\theta\\in(0,1)$, and $p_1\\in\\mathcal{M}(\\mathbb{R}^n)$. Rabinovich and Samko \\cite{RS08} observed that each globally log-H\\\"older continuous exponent belongs to $\\math","authors_text":"Alexei Yu. Karlovich, Ilya M. Spitkovsky","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-03T08:50:17Z","title":"On an Interesting Class of Variable Exponents"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0299","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":"sha256:439ef421faf1e78e252ade35534a25397dc83508c584465b477323aa1ccc7751","target":"record","created_at":"2026-05-18T04:11:50Z","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":"cf5a632b77ca13f4af7977d55ffdb461e7c98789776b3c86873868b8238d72a1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-03T08:50:17Z","title_canon_sha256":"b6f5b7e255e416f5c03ee3da8db117f4ef159271cea9b47d5856e3e03904cb49"},"schema_version":"1.0","source":{"id":"1110.0299","kind":"arxiv","version":1}},"canonical_sha256":"522be0895b7256f7f8755bb214dc0270fb7cf71935c696e334032c6e8416ffec","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"522be0895b7256f7f8755bb214dc0270fb7cf71935c696e334032c6e8416ffec","first_computed_at":"2026-05-18T04:11:50.302062Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:50.302062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n6ez2OfOc/4jQNnWwo7fpNXy6ydQJ1SHyd3vfkpqMuFzStQLmmAzT5NPpUi3J8WWgxhxTgYro7kG1KDNLaFZCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:50.302605Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.0299","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:439ef421faf1e78e252ade35534a25397dc83508c584465b477323aa1ccc7751","sha256:4f68a8e71e139c11a67c56f125c990b17b15a5c1fde0bc78c3b0d3533ca06d76"],"state_sha256":"53eacaf47d34723f7af0a5903cc741a7fc6dddcbbd2882a43313559c411d7df2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ojb3Ocgu+3ntSO5iTG6pZFI+GUMSBFm8uB60QX9qsYVDb9SC2twKwOcotQF6XfVxzHulg3NN5vvXuZt380lOBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T07:55:43.375594Z","bundle_sha256":"59c79af2a166c6c2066a029c7e45d00ab4fa9a3c8417cd55aa215499cd815e92"}}