{"paper":{"title":"A note on the spaces of variable integrability and summability of Almeida and H\\\"ast\\\"o","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Henning Kempka, Jan Vybiral","submitted_at":"2011-02-08T13:09:00Z","abstract_excerpt":"We address an open problem posed recently by Almeida and H\\\"ast\\\"o in \\cite{AlHa10}. They defined the spaces $\\ellqp$ of variable integrability and summability and showed that $\\|\\cdot|\\ellqp\\|$ is a norm if $q$ is constant almost everywhere or if $\\esssup_{x\\in\\R^n}1/p(x)+1/q(x)\\le 1$. Nevertheless, the natural conjecture (expressed also in \\cite{AlHa10}) is that the expression is a norm if $p(x),q(x)\\ge 1$ almost everywhere. We show, that $\\|\\cdot|\\ellqp\\|$ is a norm, if $1\\le q(x)\\le p(x)$ for almost every $x\\in\\R^n.$ Furthermore, we construct an example of $p(x)$ and $q(x)$ with $\\min(p(x)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1597","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"}