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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1210.4391","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-10-16T12:58:25Z","cross_cats_sorted":["math.AP","math.CA"],"title_canon_sha256":"f4f7edd9c8bdae902519bacde00c4590086be4d85deeddb702e69cc221fd23be","abstract_canon_sha256":"33365a5a4876b42410dd1063e92071ce63ee0e17e31f38f091ec1f66cb06bc87"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:06.112154Z","signature_b64":"QLJIxdDfem2O10jQY56oEeBMXola+k9lF1oTvPQvcTJM0QfWbR8idCwa/hpCzHPmi7/IDRff7uw1XiSU+d6HDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ade867bf3eaccccf6db1ee5503c8aed895002bf01df36cd6ef849c19fdfcd1ec","last_reissued_at":"2026-05-18T03:43:06.111366Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:06.111366Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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