{"paper":{"title":"Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CV","authors_text":"Huaying Wei, Michel Zinsmeister","submitted_at":"2025-06-05T02:30:40Z","abstract_excerpt":"Let $\\Gamma$ be a bounded Jordan curve and $\\Omega_i,\\Omega_e$ its two complementary components. For $s\\in(0,1)$ we define $\\mathcal{H}^s(\\Gamma)$ as the set of functions $f:\\Gamma\\to \\mathbb C$ having harmonic extension $u$ in $\\Omega_i\\cup \\Omega_e$ such that $$ \\iint_{\\Omega_i\\cup \\Omega_e} |\\nabla u(z)|^2 d(z,\\Gamma)^{1-2s} dxdy<+\\infty.$$ If $\\Gamma$ is further assumed to be rectifiable we define $H^s(\\Gamma)$ as the space of measurable functions $f:\\Gamma\\to \\mathbb C$ such that $$\\iint_{\\Gamma\\times \\Gamma}\\frac{|f(z)-f(\\zeta)|^2}{|z-\\zeta|^{1+2s}} d\\sigma(z)d\\sigma(\\zeta)<+\\infty.$$ Wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04564","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.04564/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}