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With help of these logarithmic Laplacians, we build the $n$-th order Taylor expansion for fractional Laplacian with respect to the order and the Riesz operators: for $u \\in C^\\infty_c(\\mathbb{R}^N)$ and $x \\in \\mathbb{R}^N$, $$\n  (-\\Delta)^s u (x) = u(x) + \\sum^{n}_{m=1} \\frac{s^m}{m!}\\mathcal{L}_mu(x) + o(s^n) \\quad {\\rm as}\\ \\, s\\to 0^+ $$ and $$ \\big(\\Phi_s\\ast u\\big)(x) = u(x) + \\sum^{n}_{m=1}(-1)^m\\fra"},"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":"2307.06198","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2023-07-12T14:38:20Z","cross_cats_sorted":[],"title_canon_sha256":"d0e38331446fa36973070c444993d7cba99d069834eb5a31a602e6869d1cbcb7","abstract_canon_sha256":"b4959b84c17e5dd17b94bf6bf9d45955174567044cd8e89faff5911b9f81e060"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:49:51.311843Z","signature_b64":"u2Jv59d9XF3edKRMeqAgivA9KQ2nH1n6dx/0BDTMwX4ngB/wtYQiGkeVe9/kYu3cb3EVzfV+601pmNhSSukUDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ae3a46ff44fb02fd06ce905e92d8a2194c4b2d6d84245d3046f7ac43e539ca2","last_reissued_at":"2026-07-05T08:49:51.311369Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:49:51.311369Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On m-order logarithmic Laplacians and related propeties","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huyuan Chen","submitted_at":"2023-07-12T14:38:20Z","abstract_excerpt":"In this article, we study $m$-order logarithmic Laplacian $\\mathcal{L}_m$, which is a singular integro-differential operator with symbol $\\big(2\\ln |\\cdot|\\big)^m$ by the Fourier transform. 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