{"paper":{"title":"A Discrete Quadratic Carleson Theorem on $ \\ell ^2 $ with a Restricted Supremum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ben Krause, Michael Lacey","submitted_at":"2015-12-22T00:29:55Z","abstract_excerpt":"Consider the discrete maximal function acting on $\\ell^2(\\mathbb Z)$ functions \\[ \\mathcal{C}_{\\Lambda} f( n ) := \\sup_{ \\lambda \\in \\Lambda} \\left| \\sum_{m \\neq 0} f(n-m) \\frac{e^{2 \\pi i\\lambda m^2}} {m} \\right| \\] where $\\Lambda \\subset [0,1]$. We give sufficient conditions on $\\Lambda$, met by certain kinds of Cantor sets, for this to be a bounded sublinear operator. This result is a discrete analogue of E. M. Stein's integral result, that the maximal operator below is bounded on $L^2(\\mathbb R)$. \\[ \\mathcal{C}_2 f(x):= \\sup_{\\lambda \\in \\mathbb R} \\left| \\int f(x-y) \\frac{e^{2\\pi i \\lamb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06918","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"}