{"paper":{"title":"Nonlocal gradient, the nonlocal Laplacian and maximum principles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guillermo Garc\\'ia-S\\'aez, Jos\\'e Camilo Rueda-Ni\\~no, Jos\\'e Carlos Bellido","submitted_at":"2026-07-14T18:10:19Z","abstract_excerpt":"We study the nonlocal $\\rho$-Laplacian, defined as the composition of the nonlocal divergence and gradient operators associated with a general radial kernel $\\rho$: $\\Delta_\\rho u=\\mbox{div}_\\rho\\left(D_\\rho u\\right)$. Our first main contribution is to establish a precise connection between this operator and the class of integro-differential elliptic operators studied by Fern\\'andez-Real and Ros-Oton (\\cite{FernandezRos}), identifying explicit conditions on the kernel $\\rho$ that guarantee membership in this class. Our second main contribution concerns maximum and comparison principles for the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13161","kind":"arxiv","version":1},"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/2607.13161/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"}