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More precisely, we show for any $q>1$ that \\begin{equation*}\n  f\\in L^{\\frac{qp}{p-1}}_{\\rm loc}\n  \\quad\\Longrightarrow\\quad\n  \\nabla u\\in L^{qp}_{\\rm loc}. \\end{equation*} The qualitative result is accompanied by a local quantitative estimate."},"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":"2503.05903","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-03-07T19:53:26Z","cross_cats_sorted":[],"title_canon_sha256":"e8d373cdf02343c29dd170ec94b9dc5cd129770bcad1995d54959376a276c78c","abstract_canon_sha256":"de41fc9fdf60d7f260c6c0a9a3b3b8889be290f71be97b14bd91e245b02322ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:26:48.833068Z","signature_b64":"5y2go1QVJBcBxFaUbDzw7kJhMYamNPho6L9iOjstsY6yuRdeCnv4kEUHSfpyiHKYRLr/X0rfp8vqC6vlwDy8BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91ab4a7dbb4961ab2c880e060cced31368c92feed10ad50086c8f353cdb8c94a","last_reissued_at":"2026-07-05T10:26:48.832127Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:26:48.832127Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gradient estimates for the fractional $p$-Poisson equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Frank Duzaar, Kristian Moring, Naian Liao, Verena B\\\"ogelein","submitted_at":"2025-03-07T19:53:26Z","abstract_excerpt":"We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\\left( - \\Delta_p\\right)^s u = f$. 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