{"paper":{"title":"An $L^p$-comparison, $p\\in (1,\\infty)$, on the finite differences of a discrete harmonic function at the boundary of a discrete box","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.AP","math.CA"],"primary_cat":"math.NA","authors_text":"Tuan Anh Nguyen","submitted_at":"2019-05-20T14:55:27Z","abstract_excerpt":"It is well-known that for a harmonic function $u$ defined on the unit ball of the $d$-dimensional Euclidean space, $d\\geq 2$, the tangential and normal component of the gradient $\\nabla u$ on the sphere are comparable by means of the $L^p$-norms, $p\\in(1,\\infty)$, up to multiplicative constants that depend only on $d,p$. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the $d$-dimensional lattice with multiplicative constants that do not depend on the size of the box."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.08151","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}