{"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 $W^{\\alpha, \\frac2\\alpha}(\\mathbb{C})$. 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^\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05799","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"}