{"paper":{"title":"Square functions and the Hamming cube: Duality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Alexander Volberg, Fedor Nazarov, Paata Ivanisvili","submitted_at":"2017-06-06T18:59:36Z","abstract_excerpt":"For $1<p\\leq 2$, any $n\\geq 1$ and any $f:\\{-1,1\\}^{n} \\to \\mathbb{R}$, we obtain $(\\mathbb{E} |\\nabla f|^{p})^{1/p} \\geq C(p)(\\mathbb{E}|f|^{p} - |\\mathbb{E}f|^{p})^{1/p}$ where $C(p)$ is the smallest positive zero of the confluent hypergeometric function ${}_{1}F_{1}(\\frac{p}{2(1-p)}, \\frac{1}{2}, \\frac{x^{2}}{2})$. Our approach is based on a certain duality between the classical square function estimates on the Euclidean space and the gradient estimates on the Hamming cube."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01930","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"}