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If $u$ is a harmonic mapping of the unit ball $B_n$ onto itself such that $u(0)=0$ and $\\|u\\|_p:=\\left(\\int_S|u(\\eta)|^pd\\sigma(\\eta)\\right)^{1/p}<\\infty$, $p\\ge 1$ then $|u(x)|\\le g_p(|x|)\\|u\\|_p$ for some smooth sharp function $g_p$ vanishing in $0$. Moreover we provide sharp constant $C_p$ in the inequality $\\|Du(0)\\|\\le C_p\\|u\\|_p$. Those two results extend some known result from harmonic mapping theory (\\cite[Chapter~VI]{ABR})."},"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":"1506.06410","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-21T20:17:25Z","cross_cats_sorted":[],"title_canon_sha256":"e32b28a9b6bf292af0c8d04185147f625c552983c66273dc03840afac25de4e9","abstract_canon_sha256":"82bd7eaa0158f58f081c9674c0ae450ec2ffcfc333f8379c39839ed5662dc017"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:45.289772Z","signature_b64":"cG4tPD99+v/taWzqrqfZLvt59EhdRlkh8SSara9nrHzhB+Ts9q2dB90ugPfMSL2Qg5b+5XMUBF8MwH6At+VMCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b420b40a176532e0cd1956018958d3b665f0ddb0ffac75bbfb186f927b5742a","last_reissued_at":"2026-05-18T01:41:45.289138Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:45.289138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Schwarz lemma for harmonic mappings in the unit ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Kalaj","submitted_at":"2015-06-21T20:17:25Z","abstract_excerpt":"We prove the following generalization of Schwarz lemma for harmonic mappings. 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