{"paper":{"title":"A variation norm Carleson theorem for vector-valued Walsh-Fourier series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ioannis Parissis, Michael T. Lacey, Tuomas P. Hyt\\\"onen","submitted_at":"2012-09-15T09:36:54Z","abstract_excerpt":"We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hyt\\\"onen and Lacey. Given q \\geq 2 we show that, if the space X has tile-type t for all t>q, then the r-variation of the Walsh-Fourier sums of any function f \\in L^p ([0,1) ; X) belongs to L^p, whenever q<r \\leq \\infty and (r/(q-1))' < p < \\infty. We also show that if this conclusion is true for a variant of the variational Carleson operator then the space X necessarily has tile-type "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3383","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"}