{"paper":{"title":"Two weight norm inequalities for the $g$ function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Kangwei Li, Michael T Lacey","submitted_at":"2013-09-23T15:20:09Z","abstract_excerpt":"Given two weights $\\sigma, w$ on $\\mathbb R ^{n}$, the classical $g$-function satisfies the norm inequality $\\lVert g (f\\sigma)\\rVert_{L ^2 (w)} \\lesssim \\lVert f\\rVert_{L ^2 (\\sigma)}$ if and only if the two weight Muckenhoupt $A_2$ condition holds, and a family of testing conditions holds, namely \\begin{equation*} \\iint_{Q (I)} (\\nabla P_t (\\sigma \\mathbf 1_I)(x, t))^2 \\; dw \\, t dt \\lesssim \\sigma (I) \\end{equation*} uniformly over all cubes $I \\subset \\mathbb R ^{n}$, and $Q (I)$ is the Carleson box over $I$. A corresponding characterization for the intrinsic square function of Wilson also"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5839","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"}