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We obtain an explicit nonlocal kernel for the mean value formula for solutions of $(-\\triangle)^{s}f=0$ on a domain $D$ of $\\mathbb{R}^{n}$. When $D$ is Lipschitz we prove a Besov type regularity improvement for the solutions of $(-\\triangle)^{s}f=0$."},"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":"1307.7079","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-26T15:48:55Z","cross_cats_sorted":[],"title_canon_sha256":"c8143b18a4423eefe9e4601179a8d5cd258891f607b956edd6c9b6e623c01005","abstract_canon_sha256":"45903fb0cd10ec5ebc3c6a15c8f3a6e98d3af875fe577e5e1745e411309e112a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:25.397077Z","signature_b64":"WCuPkVbZ9rnRD3aqvjuuHUREgfI46tb02jY8uluNvi60H/IRrKPxA6cmauLEULTGzA71bO7XvvraqVNKwrVgDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad5d584b0df81ee04129f940368ff6a078e571991084ad2cdea18b9771d14766","last_reissued_at":"2026-05-18T03:17:25.396429Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:25.396429Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mean value formulas for solutions of some degenerate elliptic equations and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gast\\'on Beltritti, Hugo Aimar, Ivana G\\'omez","submitted_at":"2013-07-26T15:48:55Z","abstract_excerpt":"We prove a mean value formula for weak solutions of $div(|y|^{a}\\grad u)=0$ in $\\mathbb{R}^{n+1}=\\{(x,y): x\\in\\mathbb{R}^{n}, y\\in\\mathbb{R}\\}$, $-1<a<1$ and balls centered at points of the form $(x,0)$. We obtain an explicit nonlocal kernel for the mean value formula for solutions of $(-\\triangle)^{s}f=0$ on a domain $D$ of $\\mathbb{R}^{n}$. 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