{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.06198","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.06198/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}