{"paper":{"title":"Factorizations and Hardy-Rellich-Type Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fritz Gesztesy, Lance Littlejohn","submitted_at":"2017-01-31T06:49:27Z","abstract_excerpt":"The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\\alpha,\\beta} := - \\Delta + \\alpha |x|^{-2} x \\cdot \\nabla + \\beta |x|^{-2}$, $\\alpha, \\beta \\in \\mathbb{R}$, $x \\in \\mathbb{R}^n \\backslash \\{0\\}$, $n \\in \\mathbb{N}$, $n \\geq 2$, and its formal adjoint, denoted by $T_{\\alpha,\\beta}^+$, we show that nonnegativity of $T_{\\alpha,\\beta}^+ T_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08929","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"}