{"paper":{"title":"Remarks on a nonlinear nonlocal operator in Orlicz spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Arturo de Pablo, Ernesto Correa","submitted_at":"2018-09-04T13:18:48Z","abstract_excerpt":"We study integral operators $\\mathcal{L}u(x)=\\int_{\\mathbb{R^N}}\\psi(u(x)-u(y))J(x-y)\\,dy$ of the type of the fractional $p$-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincar\\'e inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel $J$ considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\\mathcal{L}u=f$ in a bounded domain $\\Omega$, and boundary condition $u\\equiv0$ on $\\Omega^c$; "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00937","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"}