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This estimate implies a Gaussian type point-wise upper bound for the gradient of the Neumann heat kernel, which is applied to the study of the Hardy spaces, Riesz transforms, and regularity of solutions to the inhomogeneous "},"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":"1009.1965","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-09-10T09:41:29Z","cross_cats_sorted":[],"title_canon_sha256":"dec620530e8329bb87f167cc6688cfa4aed300568fa587edaea2d6368e3e9f1b","abstract_canon_sha256":"34d86d37015526c7c18fd35b2ad092e056feb955be8f8321146f56f709bc14cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:09.206058Z","signature_b64":"JDLoBfY4TgJ2OiazkgGRIvJrs4DVmpUY246op+UplgBXhk8WwqtShyrwhF59p1GNDuOXtkeT0i+GYJNUNqEaCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d84ae5e670b53822b5f731998ce276842a02cd411d186ac3ec1a1220c05b2870","last_reissued_at":"2026-05-18T04:40:09.205586Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:09.205586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gradient Estimate on the Neumann Semigroup and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang, Lixin Yan","submitted_at":"2010-09-10T09:41:29Z","abstract_excerpt":"We prove the following sharp upper bound for the gradient of the Neumann semigroup $P_t$ on a $d$-dimensional compact domain $\\OO$ with boundary either $C^2$-smooth or convex:\n  $$\\|\\nn P_t\\|_{1\\to \\infty}\\le \\ff{c}{t^{(d+1)/2}},\\ \\ t>0,$$ where $c>0$ is a constant depending on the domain and $\\|\\cdot\\|_{1\\to\\infty}$ is the operator norm from $L^1(\\OO)$ to $L^\\infty(\\OO)$. 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