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More generally, the heat operator is a contraction from L^p(B,\\mu_{s}) to L^q(B,\\mu_{s-t}) for t<s, provided that p and q satisfy (p-1)/(q-1) \\leq s/(s-t).\n  I give two proofs of this result, both very elementary."},"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":"0710.2137","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2007-10-10T23:05:34Z","cross_cats_sorted":["math.FA","math.MP"],"title_canon_sha256":"2fae8c1b25d63891f2ff3186df13fda70cf7e44357b9cd15382a83f571db87f8","abstract_canon_sha256":"2fa53085a89b92fc597078b64d28688fdba47c9a1727a7227ecc48fde63a1fba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:35.430143Z","signature_b64":"oHjF/4RyxfMX+vQVZXvgd1yxN7ahd6ryKO28DKXpKhiwJd+dQJDluZCkew+k1HC+E1XSubd1fw4feU0zGOYLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91c6a035f15b7b2d7319223a977f7a9caa91dadce154eef25e34461c72d86683","last_reissued_at":"2026-05-18T04:42:35.429285Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:35.429285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The heat operator in infinite dimensions","license":"","headline":"","cross_cats":["math.FA","math.MP"],"primary_cat":"math-ph","authors_text":"Brian C. 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