{"paper":{"title":"Weak type (1, 1) inequalities for discrete rough maximal functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mariusz Mirek","submitted_at":"2013-05-02T20:38:30Z","abstract_excerpt":"The aim of this paper is to show that the discrete maximal function $$\\mathcal{M}_{h}f(x)=\\sup_{N\\in\\mathbb{N}}\\frac{1}{|\\mathbf{N}_{h}\\cap[1, N]|}\\Big|\\sum_{n\\in \\mathbf{N}_{h}\\cap[1, N]}f(x-n)\\Big|,\\ \\ \\mbox{for $x\\in\\mathbb{Z}$},$$ is of weak type $(1, 1)$, where $\\mathbf{N}_{h}=\\{n\\in\\mathbb{N}: \\exists_{m\\in\\mathbb{N}}\\ n=\\lfloor h(m)\\rfloor\\}$ for an appropriate function $h$. As a consequence we also obtain pointwise ergodic theorem along the set $\\mathbf{N}_{h}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.0575","kind":"arxiv","version":2},"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"}