{"paper":{"title":"On the pointwise convergence of the sequence of partial Fourier Sums along lacunary subsequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Victor Lie","submitted_at":"2012-04-30T03:27:35Z","abstract_excerpt":"In his 2006 ICM invited address, Konyagin mentioned the following conjecture: if $S_n f$ stands for the $n$-th partial Fourier sum of $f$ and ${n_j}_j\\subset \\N$ is a lacunary sequence, then $S_{n_j} f$ is a.e. pointwise convergent for any $f\\in L\\log\\log L$. In this paper we will show that $| \\sup_{j} |S_{n_j}(f)| |_{1,\\infty}\\leq C |f|_{1} \\log\\log (10+\\frac{|f|_{\\infty}}{|f|_1})\\:.$ As a direct consequence we obtain that $S_{n_j}f \\rightarrow f $ a.e. for $f\\in L\\log\\log L\\log\\log\\log L$. The (discrete) Walsh model version of this last fact was proved by Do and Lacey but their methods do no"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.6542","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"}