{"paper":{"title":"Weighted $L^p$ bounds for the Marcinkiewicz integral","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guoen Hu, Meng Qu","submitted_at":"2016-02-01T14:49:53Z","abstract_excerpt":"Let $\\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\\mathcal{M}_{\\Omega}$ be the higher-dimensional Marcinkiewicz integral associated with $\\Omega$. In this paper, the authors proved that if $\\Omega\\in L^q(S^{n-1})$ for some $q\\in (1,\\,\\infty]$, then for $p\\in (q',\\,\\infty)$ and $w\\in A_{p}(\\mathbb{R}^n)$, the bound of $\\mathcal{M}_{\\Omega}$ on $L^p(\\mathbb{R}^n,\\,w)$ is less than $C[w]_{A_{p/q'}}^{2\\max\\{1,\\,\\frac{1}{p-q'}\\}}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00549","kind":"arxiv","version":1},"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"}