{"paper":{"title":"The Sobolev space $W_2^{1/2}$: Simultaneous improvement of functions by a homeomorphism of the circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Vladimir Lebedev","submitted_at":"2025-11-11T05:18:06Z","abstract_excerpt":"It is known that for every continuous real-valued function $f$ on the circle $\\mathbb T=\\mathbb R/2\\pi\\mathbb Z$ there exists a change of variable, i.e., a self-homeomorphism $h$ of $\\mathbb T$, such that the superposition $f\\circ h$ is in the Sobolev space $W_2^{1/2}(\\mathbb T)$. We obtain new results on simultaneous improvement of functions by a single change of variable in relation to the space $W_2^{1/2}(\\mathbb T)$. The main result is as follows: there does not exist a self-homeomorphism $h$ of $\\mathbb T$ such that $f\\circ h\\in W_2^{1/2}(\\mathbb T)$ for every $f\\in \\mathrm{Lip}_{1/2}(\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.07840","kind":"arxiv","version":4},"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"}