{"paper":{"title":"Square function estimates for conical regions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Shengwen Gan, Shukun Wu","submitted_at":"2022-03-23T03:02:23Z","abstract_excerpt":"We prove square function estimates for certain conical regions. Specifically, let $\\{\\Delta_j\\}$ be regions of the unit sphere $\\mathbb{S}^{n-1}$ and let $S_j f$ be the smooth Fourier restriction of $f$ to the conical region $\\{\\xi\\in\\mathbb{R}^n:\\xi/|\\xi|\\in\\Delta_j\\}$. We are interested in the following estimate\n  $$\\Big\\|(\\sum_j|S_jf|^2)^{1/2}\\Big\\|_p\\lesssim_\\epsilon \\delta^{-\\epsilon}\\|f\\|_p.$$\n  The first result is: when $\\{\\Delta_j\\}$ is a set of disjoint $\\delta$-balls, then the estimate holds for $p=4$. The second result is: In $\\mathbb{R}^3$, when $\\{\\Delta_j\\}$ is a set of disjoint "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2203.12155","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2203.12155/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}