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Namely, if $\\rho$ is a nonnegative weight such that $-\\Delta_{p}\\rho\\geq0$, then the Hardy inequality $$c\\int_{M}\\frac{\\abs{u}^{p}}{\\rho^{p}}\\abs{\\nabla \\rho}^{p} dv_{g} \\leq \\int_{M}\\abs{\\nabla u}^{p} dv_{g}, \\quad u\\in\\Cinfinito_{0}(M)$$ holds. We show concrete examples specializing the function $\\rho$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1210.5723","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-21T12:42:30Z","cross_cats_sorted":[],"title_canon_sha256":"d0da6e775a11574c1d7b4d35bd577ee94e7d37d9e6a4519bb075458a5d761014","abstract_canon_sha256":"1877e16d63aba4282f5f1e6aa7f0f8da96d1553581226e6bc3361628e92ace10"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:05.559416Z","signature_b64":"+AbFNSf1O3cAnkM2Y92RMsHP1Z7G1Xi1PnjV+WLkBizAp0yf+jl7XiDvlp7Nu7/5/nKJzDvLFxxQdpwpEQhMCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed83e2429ac17395b616bad90d32eba767036fae499f3d697ce9b4236a233fdb","last_reissued_at":"2026-05-18T03:28:05.558832Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:05.558832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hardy inequalities on Riemannian manifolds and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lorenzo D'Ambrosio, Serena Dipierro","submitted_at":"2012-10-21T12:42:30Z","abstract_excerpt":"We prove a simple sufficient criteria to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second-order differential operator $\\Delta_{p}u := \\Div(\\abs{\\nabla u}^{p-2}\\nabla u)$. Namely, if $\\rho$ is a nonnegative weight such that $-\\Delta_{p}\\rho\\geq0$, then the Hardy inequality $$c\\int_{M}\\frac{\\abs{u}^{p}}{\\rho^{p}}\\abs{\\nabla \\rho}^{p} dv_{g} \\leq \\int_{M}\\abs{\\nabla u}^{p} dv_{g}, \\quad u\\in\\Cinfinito_{0}(M)$$ holds. 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