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Our main result states that $$\\log\\partial_z\\phi \\in W^{\\alpha, \\frac2\\alpha}(\\mathbb{C})$$ whenever $\\alpha>\\frac12$. Our method relies on an $n$-dimensional result, which asserts the compactness of the commutator $$[b,(-\\Delta)^\\frac{\\beta}{2}]:L^\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1507.05799","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-07-21T12:09:19Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"1a9bf406cb7f005af564cfec0e0ac941f542e260ca01c9bb1f4a8fd3df0dbea7","abstract_canon_sha256":"0f97aec1b6a3ad6e90658b7909830411f2ef9f2ca402ac8973e16e97b6cdb071"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:32.429584Z","signature_b64":"+mPBPLv6hP97SL7Uh9QHo4LjCYWtkq0znBDarsw3idQOK8pl4TzzRKN/mtsVisMCHrDizeaW13buPj4ceVSTBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b90aca3b09e96026257b24c8c7dae81ed75deb037f361e5b3a161c91cb9b7a5","last_reissued_at":"2026-05-18T01:36:32.429098Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:32.429098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Beltrami equations with coefficient in the fractional Sobolev space $W^{\\theta, \\frac2{\\theta}}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CV","authors_text":"Albert Clop, Antonio Luis Bais\\'on, Joan Orobitg","submitted_at":"2015-07-21T12:09:19Z","abstract_excerpt":"In this paper, we look at quasiconformal solutions $\\phi:\\mathbb{C}\\to\\mathbb{C}$ of Beltrami equations $$ \\partial_{\\overline{z}} \\phi(z)=\\mu(z)\\,\\partial_z \\phi (z). $$ where $\\mu\\in L^\\infty(\\mathbb{C})$ is compactly supported on $\\mathbb{D}$, $\\|\\mu\\|_\\infty<1$ and belongs to the fractional Sobolev space $W^{\\alpha, \\frac2\\alpha}(\\mathbb{C})$. 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