{"paper":{"title":"Borderline Weak Type Estimates for Singular Integrals and Square Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos Domingo-Salazar, Guillermo Rey, Michael T. Lacey","submitted_at":"2015-05-07T18:25:16Z","abstract_excerpt":"For any Calder\\'on-Zygmund operator $ T$, any weight $ w$, and $ \\alpha >1$, the operator $ T$ is bounded as a map from $ L ^{1} (M _{ L \\log\\log L (\\log\\log\\log L) ^{\\alpha } } w )$ into weak-$L^1(w)$. The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman-Fefferman, P\\'erez, and Hyt\\\"onen-P\\'erez, on the $ L (\\log L) ^{\\epsilon }$ scale. Also, for square functions $ S f$, and weights $ w \\in A_p$, the norm of $ S$ from $ L ^p (w)$ to weak-$L^p (w)$, $ 2\\leq p < \\infty $, is bounded by $ [w] _{A_p}^{1/2} (1+\\log [w] _{A_ \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01804","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"}