{"paper":{"title":"The Rearrangement-Invariant space $\\Gamma_{p,\\phi}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.FA","authors_text":"Amiran Gogatishvili, Ron Kerman","submitted_at":"2012-10-16T12:58:25Z","abstract_excerpt":"Fix $b\\in (0,\\infty)$ and $p\\in (1,\\infty)$. Let $\\phi$ be a positive measurable function on $I_b:=(0,b)$. Define the Lorentz Gamma norm, $\\r_{p,\\phi}$, at the measurable function $f:\\R+\\to\\R+$ by $\\rph(f):=[\\int_0^bf^{**}(t)^p\\phi(t)dt]^{\\frac1p}$, in which $f^{**}(t):=t^{-1}\\int_0^tf^{*}(s)ds$, where $f^*(t):=\\mu_f^{-1}(t)$, with $\\mu_f(s):=|\\{x\\in I_b: |f(x)|>s\\}|$.\n  Our aim in this paper is to study the rearrangement-invariant space determined by $\\rph$. In particular, we determine its K\\\"othe dual and its Boyd indices. Using the latter a sufficient condition is given for a Cald\\'eron-Zyg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.4391","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"}