{"paper":{"title":"An inequality associated with $\\mathcal{Q}_p$ functions","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Fangqin Ye, Guanlong Bao","submitted_at":"2018-10-13T17:40:05Z","abstract_excerpt":"The M\\\"obius invariant space $\\mathcal{Q}_p$, $0<p<\\infty$, consists of functions $f$ which are analytic in the open unit disk $\\mathbb{D}$ with $$ \\|f\\|_{\\mathcal{Q}_p}=|f(0)|+\\sup_{w\\in \\D} \\left(\\int_\\D |f'(z)|^2(1-|\\sigma_w(z)|^2)^p dA(z)\\right)^{1/2}<\\infty, $$ where $\\sigma_w(z)=(w-z)/(1-\\overline{w}z)$ and $dA$ is the area measure on $\\mathbb{D}$. It is known that the following inequality $$ |f(0)|+\\sup_{w\\in \\D} \\left(\\int_\\D \\left|\\frac{f(z)-f(w)}{1-\\overline{w}z}\\right|^2 (1-|\\sigma_w(z)|^2)^p dA(z)\\right)^{1/2} \\lesssim \\|f\\|_{\\mathcal{Q}_p} $$ played a key role to characterize mult"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.05901","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"}