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We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function."},"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":"2505.17219","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-05-22T18:48:51Z","cross_cats_sorted":[],"title_canon_sha256":"e75e804a708743c192d52064ebf58215b6ab23374cd0fd7403af606df6ec249e","abstract_canon_sha256":"2cbf93d8b0af20cbb4964e6652125751932035b6622ed5cf6a6fd347077e8eb9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:07:49.080781Z","signature_b64":"kyRgq7wvnff6MXw3TdgcI0SrMQgyHPBn+ki2Oe1tcx/kYTYX86TwlzlrQU9D3bMxB8S3mK3gEdyjZIrM/bqIBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3525fdb3813b03c81dd40c3ae7e93a304b8874428b8d1bbfd124ed748e08a490","last_reissued_at":"2026-07-05T11:07:49.080159Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:07:49.080159Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Compactness of the $L_p$ dual Minkowski problem in $\\mathbb{R}^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Christos Saroglou, Karoly J. 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