{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:BRBQWLGZRDAKTYRPDWZIKXA777","short_pith_number":"pith:BRBQWLGZ","schema_version":"1.0","canonical_sha256":"0c430b2cd988c0a9e22f1db2855c1fffc15477c0be5bcf28db3ee5a3d48c9ec3","source":{"kind":"arxiv","id":"1608.02009","version":1},"attestation_state":"computed","paper":{"title":"A quasiconformal composition problem for the Q-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jie Xiao, Pekka Koskela, Yi Ru-Ya Zhang, Yuan Zhou","submitted_at":"2016-08-05T20:11:23Z","abstract_excerpt":"Given a quasiconformal mapping $f:\\mathbb R^n\\to\\mathbb R^n$ with $n\\ge2$, we show that (un-)boundedness of the composition operator ${\\bf C}_f$ on the spaces $Q_{\\alpha}(\\mathbb R^n)$ depends on the index $\\alpha$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $\\alpha$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Cons"},"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":"1608.02009","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-05T20:11:23Z","cross_cats_sorted":[],"title_canon_sha256":"09068a45538c369d86734ac7fecffe3696f56835b68d0117d43252983a27ef73","abstract_canon_sha256":"2dece618e55cd8642d87d40cee98e31dbb9e71c7bddf7346c6cf24cec4c68630"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:41.162397Z","signature_b64":"ZQxpLnG0bDlJuLV1vNfI1188NYhBfnCn3tIMB+2/lUPfPdSvSFykO8t4c6V2e/nmEmV5p3uMUt8udQGKNGVYAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c430b2cd988c0a9e22f1db2855c1fffc15477c0be5bcf28db3ee5a3d48c9ec3","last_reissued_at":"2026-05-18T01:09:41.162006Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:41.162006Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A quasiconformal composition problem for the Q-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jie Xiao, Pekka Koskela, Yi Ru-Ya Zhang, Yuan Zhou","submitted_at":"2016-08-05T20:11:23Z","abstract_excerpt":"Given a quasiconformal mapping $f:\\mathbb R^n\\to\\mathbb R^n$ with $n\\ge2$, we show that (un-)boundedness of the composition operator ${\\bf C}_f$ on the spaces $Q_{\\alpha}(\\mathbb R^n)$ depends on the index $\\alpha$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $\\alpha$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Cons"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02009","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1608.02009","created_at":"2026-05-18T01:09:41.162057+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.02009v1","created_at":"2026-05-18T01:09:41.162057+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.02009","created_at":"2026-05-18T01:09:41.162057+00:00"},{"alias_kind":"pith_short_12","alias_value":"BRBQWLGZRDAK","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"BRBQWLGZRDAKTYRP","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"BRBQWLGZ","created_at":"2026-05-18T12:30:07.202191+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/BRBQWLGZRDAKTYRPDWZIKXA777","json":"https://pith.science/pith/BRBQWLGZRDAKTYRPDWZIKXA777.json","graph_json":"https://pith.science/api/pith-number/BRBQWLGZRDAKTYRPDWZIKXA777/graph.json","events_json":"https://pith.science/api/pith-number/BRBQWLGZRDAKTYRPDWZIKXA777/events.json","paper":"https://pith.science/paper/BRBQWLGZ"},"agent_actions":{"view_html":"https://pith.science/pith/BRBQWLGZRDAKTYRPDWZIKXA777","download_json":"https://pith.science/pith/BRBQWLGZRDAKTYRPDWZIKXA777.json","view_paper":"https://pith.science/paper/BRBQWLGZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.02009&json=true","fetch_graph":"https://pith.science/api/pith-number/BRBQWLGZRDAKTYRPDWZIKXA777/graph.json","fetch_events":"https://pith.science/api/pith-number/BRBQWLGZRDAKTYRPDWZIKXA777/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BRBQWLGZRDAKTYRPDWZIKXA777/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BRBQWLGZRDAKTYRPDWZIKXA777/action/storage_attestation","attest_author":"https://pith.science/pith/BRBQWLGZRDAKTYRPDWZIKXA777/action/author_attestation","sign_citation":"https://pith.science/pith/BRBQWLGZRDAKTYRPDWZIKXA777/action/citation_signature","submit_replication":"https://pith.science/pith/BRBQWLGZRDAKTYRPDWZIKXA777/action/replication_record"}},"created_at":"2026-05-18T01:09:41.162057+00:00","updated_at":"2026-05-18T01:09:41.162057+00:00"}