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It is shown, by Morrey's Lemma and isoperimetric inequality, that every $(K_1, K_2)$-quasiregular mapping satisfies a H\\\"older condition with exponent $\\alpha$ on compact subsets of its domain, where \\begin{align} \\alpha=\\begin{cases} 1/K_1, & \\text{for } K_1>1, \\\\ \\text{any positive number less than } 1, & \\text{for } K_1=1 \\text{ and } K_2>0, \\\\ 1, & \\text{for } K_1=1 \\text{ and } K_2=0, \\\\ 1, & \\text{for } K_1<1,\\\\ \\end{cases} \\end{align} Differentiability almost everywhere of $(K_1, K_2)$-quasiregular mappings is also derived."},"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":"1812.07779","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-19T06:51:40Z","cross_cats_sorted":[],"title_canon_sha256":"f13daba53390aff6e0164c9aca83766563cb495beda1e08a87215386e1d010a4","abstract_canon_sha256":"8044811deb058d4626f6155b8723469f080a1fa5ffab3740abfb17234429adf8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:46.308270Z","signature_b64":"OmrVzyqclXHvirAKv2rX8UULUe2YbRd5uaUuojAeF3ZQX8+1216Zlh+Q6fXfKKQgQ7USkTB44vnNDh9Cneo7Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e56633609caf884c8faf11827e56761c02d65eff46872e17ee7c3b686d036dfa","last_reissued_at":"2026-05-17T23:57:46.307761Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:46.307761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"H\\\"older Continuity and Differentiability Almost Everywhere of $(K_1, K_2)$-Quasiregular Mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chao Liu, Hongya Gao, Junwei Li","submitted_at":"2018-12-19T06:51:40Z","abstract_excerpt":"This paper deals with $(K_1, K_2)$-quasiregular mappings. It is shown, by Morrey's Lemma and isoperimetric inequality, that every $(K_1, K_2)$-quasiregular mapping satisfies a H\\\"older condition with exponent $\\alpha$ on compact subsets of its domain, where \\begin{align} \\alpha=\\begin{cases} 1/K_1, & \\text{for } K_1>1, \\\\ \\text{any positive number less than } 1, & \\text{for } K_1=1 \\text{ and } K_2>0, \\\\ 1, & \\text{for } K_1=1 \\text{ and } K_2=0, \\\\ 1, & \\text{for } K_1<1,\\\\ \\end{cases} \\end{align} Differentiability almost everywhere of $(K_1, K_2)$-quasiregular mappings is also derived."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.07779","kind":"arxiv","version":3},"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":"1812.07779","created_at":"2026-05-17T23:57:46.307839+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.07779v3","created_at":"2026-05-17T23:57:46.307839+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.07779","created_at":"2026-05-17T23:57:46.307839+00:00"},{"alias_kind":"pith_short_12","alias_value":"4VTDGYE4V6EE","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"4VTDGYE4V6EEZD5P","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"4VTDGYE4","created_at":"2026-05-18T12:32:05.422762+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/4VTDGYE4V6EEZD5PCGBH4VTWDQ","json":"https://pith.science/pith/4VTDGYE4V6EEZD5PCGBH4VTWDQ.json","graph_json":"https://pith.science/api/pith-number/4VTDGYE4V6EEZD5PCGBH4VTWDQ/graph.json","events_json":"https://pith.science/api/pith-number/4VTDGYE4V6EEZD5PCGBH4VTWDQ/events.json","paper":"https://pith.science/paper/4VTDGYE4"},"agent_actions":{"view_html":"https://pith.science/pith/4VTDGYE4V6EEZD5PCGBH4VTWDQ","download_json":"https://pith.science/pith/4VTDGYE4V6EEZD5PCGBH4VTWDQ.json","view_paper":"https://pith.science/paper/4VTDGYE4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.07779&json=true","fetch_graph":"https://pith.science/api/pith-number/4VTDGYE4V6EEZD5PCGBH4VTWDQ/graph.json","fetch_events":"https://pith.science/api/pith-number/4VTDGYE4V6EEZD5PCGBH4VTWDQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4VTDGYE4V6EEZD5PCGBH4VTWDQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4VTDGYE4V6EEZD5PCGBH4VTWDQ/action/storage_attestation","attest_author":"https://pith.science/pith/4VTDGYE4V6EEZD5PCGBH4VTWDQ/action/author_attestation","sign_citation":"https://pith.science/pith/4VTDGYE4V6EEZD5PCGBH4VTWDQ/action/citation_signature","submit_replication":"https://pith.science/pith/4VTDGYE4V6EEZD5PCGBH4VTWDQ/action/replication_record"}},"created_at":"2026-05-17T23:57:46.307839+00:00","updated_at":"2026-05-17T23:57:46.307839+00:00"}