{"paper":{"title":"On the endpoint regularity of discrete maximal operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Emanuel Carneiro, Kevin Hughes","submitted_at":"2013-09-06T04:09:43Z","abstract_excerpt":"Given a discrete function $f:\\Z^d \\to \\R$ we consider the maximal operator $$Mf(\\vec{n}) = \\sup_{r\\geq0} \\frac{1}{N(r)} \\sum_{\\vec{m} \\in \\bar{\\Omega}_r} \\big|f(\\vec{n} + \\vec{m})\\big|,$$ where $\\big\\{\\bar{\\Omega}_r\\big\\}_{r \\geq 0}$ are dilations of a convex set $\\Omega$ (open, bounded and with Lipschitz boudary) containing the origin and $N(r)$ is the number of lattice points inside $\\bar{\\Omega}_r$. We prove here that the operator $f \\mapsto \\nabla M f$ is bounded and continuous from $l^1(\\Z^d)$ to $l^1(\\Z^d)$. We also prove the same result for the non-centered version of this discrete maxi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1535","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":""},"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"}