{"paper":{"title":"Radial fractional Laplace operators and Hessian inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fausto Ferrari, Igor E. Verbitsky","submitted_at":"2012-03-14T17:17:16Z","abstract_excerpt":"In this paper we deduce a formula for the fractional Laplace operator\n  $(-\\Delta)^{s}$ on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with $(-\\Delta)^{s}$, and apply it to a problem related to the Hessian inequality of Sobolev type: $$\\int_{\\mathbb{R}^n}|(-\\Delta)^{\\frac{k}{k+1}} u|^{k+1} dx \\le C   \\int_{\\mathbb{R}^n} - u \\, F_k[u] \\, dx, $$ where $F_k$ is the $k$-Hessian operator on $\\mathbb{R}^n$, $1\\le k < \\frac{n}{2}$, under some restrictions on a $k$-convex function $u$. In particular, we show that the class of $u$ for whic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3149","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"}