{"paper":{"title":"The Sharp Constant in the Weak (1,1) Inequality for the Square Function: A New Proof","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alexander Volberg, Irina Holmes, Paata Ivanisvili","submitted_at":"2017-10-03T18:48:28Z","abstract_excerpt":"In this note we give a new proof of the sharp constant $C = e^{-1/2} + \\int_0^1 e^{-x^2/2}\\,dx$ in the weak (1, 1) inequality for the dyadic square function. The proof makes use of two Bellman functions $\\mathbb{L}$ and $\\mathbb{M}$ related to the problem, and relies on certain relationships between $\\mathbb{L}$ and $\\mathbb{M}$, as well as the boundary values of these functions, which we find explicitly. Moreover, these Bellman functions exhibit an interesting behavior: the boundary solution for $\\mathbb{M}$ yields the optimal obstacle condition for $\\mathbb{L}$, and vice versa."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.01346","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"}