{"paper":{"title":"The pointwise convergence of Fourier Series (II). Strong $L^1$ case for the lacunary Carleson operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Victor Lie","submitted_at":"2019-02-10T17:08:13Z","abstract_excerpt":"We prove that the lacunary Carleson operator is bounded from $L \\log L$ to $L^{1}$. This result is sharp.\n  The proof is based on two newly introduced concepts: 1) the \\emph{time-frequency regularization of a measurable set} and 2) the \\emph{set-resolution of the time-frequency plane at $0-$frequency}.\n  These two concepts will play the central role in providing a special tile decomposition adapted to the interaction between the \\emph{structure} of the lacunary Carleson operator and the corresponding \\emph{structure} of a fix measurable set.\n  Another key insight of our paper is that it provid"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.03630","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"}