{"paper":{"title":"Non-local Torsion functions and Embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Giovanni Franzina","submitted_at":"2018-01-23T10:26:25Z","abstract_excerpt":"Given $s \\in (0,1)$, we discuss the embedding of $\\mathcal D^{s,p}_0(\\Omega)$ in $L^q(\\Omega)$. In particular, for $1\\le q < p$ we deduce its compactness on all open sets $\\Omega\\subset \\mathbb R^N$ on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in $\\Omega$ in a suitable weak sense, for every open set $\\Omega$. The proofs make use of a non-local Hardy-type inequality in $\\mathcal D^{s,p}_0(\\Omega)$, involving the fractional torsion functi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.07469","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"}