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On $\\mathbb{R}$, the best bound for Lipschitz functions is $ \\operatorname{Lip} ( Mf) \\le (\\sqrt2 -1)\\operatorname{Lip}( f).$ In higher dimensions, we determ"},"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":"1009.1359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-09-07T18:53:57Z","cross_cats_sorted":[],"title_canon_sha256":"2de1b287ddff61a3636b2180ef4f8a4ddc3350237055d7940ac6906a83f86907","abstract_canon_sha256":"8197e240307e4dba8d7f929d1618c43e2f729a33bd469e6937ecb0803e7b2b5f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:15.664658Z","signature_b64":"oK8uYn14yFMSuH5SGfBUDwXwGYB+q5b7F2gMW2TuizRpjQYfCeni20qmBY+p/nevBjG4NkBY2+vJ1Ou+4ZgpBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6e30d3d8edeaa0d57510c3b733300d37b7e558b4b01af2a8bf0eb9202bb4781","last_reissued_at":"2026-05-18T04:41:15.663884Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:15.663884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"J. 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