{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:P3ZAAYDTKBB5WRZPOXKQC2JCT5","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":"3a6ee5088f24e3299fef8182e02283a651bd34bbba3fa35c7dd728db93b60d5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-11-12T15:05:50Z","title_canon_sha256":"b0569c2a8b328ce9e600616f9e72aca8d90ab6304f0ff57b2dd7629fefdc8336"},"schema_version":"1.0","source":{"id":"1011.2937","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.2937","created_at":"2026-05-18T04:33:04Z"},{"alias_kind":"arxiv_version","alias_value":"1011.2937v2","created_at":"2026-05-18T04:33:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.2937","created_at":"2026-05-18T04:33:04Z"},{"alias_kind":"pith_short_12","alias_value":"P3ZAAYDTKBB5","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"P3ZAAYDTKBB5WRZP","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"P3ZAAYDT","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:81c887be5fe23b1274d60178afb8a1b519f19b61a0944bda9b70b3adc475adf4","target":"graph","created_at":"2026-05-18T04:33:04Z","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 $({\\mathcal X}, d, \\mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition and the non-atomic condition that $\\mu(\\{x\\})=0$ for all $x\\in{\\mathcal X}$. In this paper, we show that the boundedness of a Calder\\'on-Zygmund operator $T$ on $L^2(\\mu)$ is equivalent to that of $T$ on $L^p(\\mu)$ for some $p\\in (1, \\infty)$, and that of $T$ from $L^1(\\mu)$ to $L^{1,\\,\\infty}(\\mu).$ As an application, we prove that if $T$ is a Calder\\'on-Zygmund operator bounded on $L^2(\\mu)$, then its maximal operator is bounded on $L^p(\\mu)$ for ","authors_text":"Dachun Yang, Dongyong Yang, Suile Liu, Tuomas Hyt\\\"onen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-11-12T15:05:50Z","title":"Boundedness of Calder\\'on-Zygmund Operators on Non-homogeneous Metric Measure Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.2937","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":"sha256:6bfb0a62bd6e69b635de63ccc336f665e9279b2c4001cf9ca72aa7305f501a17","target":"record","created_at":"2026-05-18T04:33:04Z","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":"3a6ee5088f24e3299fef8182e02283a651bd34bbba3fa35c7dd728db93b60d5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-11-12T15:05:50Z","title_canon_sha256":"b0569c2a8b328ce9e600616f9e72aca8d90ab6304f0ff57b2dd7629fefdc8336"},"schema_version":"1.0","source":{"id":"1011.2937","kind":"arxiv","version":2}},"canonical_sha256":"7ef20060735043db472f75d50169229f77f74552af08d0d7181b40699019cff9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ef20060735043db472f75d50169229f77f74552af08d0d7181b40699019cff9","first_computed_at":"2026-05-18T04:33:04.027505Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:33:04.027505Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ii7OXAuzPte+Iv/KBEf7ytGBEhDj4M9NZyfJ+bfrTn1448SNv3SRjOv6tOzASGQlyVcnCinCDpHb/9mi2BrTCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:33:04.028170Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.2937","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6bfb0a62bd6e69b635de63ccc336f665e9279b2c4001cf9ca72aa7305f501a17","sha256:81c887be5fe23b1274d60178afb8a1b519f19b61a0944bda9b70b3adc475adf4"],"state_sha256":"729cc6b9a175be5599db457c24e95d9b33de8a597240b0b7de6c0e7c999f9063"}