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Our main result, Theorem 2.2, corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential $\\tilde I^{f}_{q}$. Moreover, for $f \\in L^{m}$ with $m>n$, we also obtain $C^{1,\\alpha}$ estimates, see Theorem 2.3 below. This improves one of the regularity results in [3], where a $C^{1,\\alpha}$ estimate was established de"},"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":"1904.13076","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-30T07:08:18Z","cross_cats_sorted":[],"title_canon_sha256":"aabf5cdfb2fe3b6d2e5e28204ae21263112a56d7e1e153f624fa23480afa47c7","abstract_canon_sha256":"cc8582ab5f771f67ed95b7d74e95850338556fc8ba44e19e975088ca291bf02d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:57.705013Z","signature_b64":"/+bBKUdUVEmXlkD8WEK5xiVF47/3jZ83JyVD69xTfhvzmVqrvutNzM6IVl2c0nMMps5wOhd4NIGQxRa0gAJ7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4217deba3b17267d7ddb3bd2b10238a9ba6dcd8382f834be78c75213f426e44d","last_reissued_at":"2026-05-17T23:45:57.704567Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:57.704567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gradient continuity estimates for the normalized $p$-Poisson equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Agnid Banerjee, Isidro H. Munive","submitted_at":"2019-04-30T07:08:18Z","abstract_excerpt":"In this paper, we obtain gradient continuity estimates for viscosity solutions of $\\Delta_{p}^N u= f$ in terms of the scaling critical $L(n,1 )$ norm of $f$, where $\\Delta_{p}^N$ is the normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.2, corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential $\\tilde I^{f}_{q}$. Moreover, for $f \\in L^{m}$ with $m>n$, we also obtain $C^{1,\\alpha}$ estimates, see Theorem 2.3 below. 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