{"paper":{"title":"A sublinear version of Schur's lemma and elliptic PDE","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Igor E. Verbitsky, Stephen Quinn","submitted_at":"2017-02-09T02:30:04Z","abstract_excerpt":"We study the weighted norm inequality of $(1,q)$-type,\n  \\[ \\Vert \\mathbf{G}\\nu \\Vert_{L^q(\\Omega, d\\sigma)} \\le C \\Vert \\nu \\Vert, \\quad \\text{ for all } \\nu \\in \\mathcal{M}^+(\\Omega), \\] along with its weak-type analogue, for $0 < q < 1$, where $\\mathbf{G}$ is an integral operator associated with the nonnegative kernel $G(x,y)$. Here $\\mathcal{M}^+(\\Omega)$ denotes the class of positive Radon measures in $\\Omega$; $\\sigma, \\nu \\in \\mathcal{M}^+(\\Omega)$, and $||\\nu||=\\nu(\\Omega)$.\n  For both weak-type and strong-type inequalities, we provide conditions which characterize the measures $\\sigma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02682","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":""},"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"}