{"paper":{"title":"On weighted strong type inequalities for the generalized weighted mean operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Ondrej Hutn\\'ik","submitted_at":"2013-09-22T19:02:06Z","abstract_excerpt":"The generalized weighted mean operator $\\mathbf{M}^{g}_{w}$ is given by $$[\\mathbf{M}^{g}_{w}f](x)= g^{-1}\\left(\\frac{1}{W(x)}\\int_{0}^{x}w(t)g(f(t))\\,\\mathrm{d}t\\right),$$ with $$W(x)=\\int_{0}^{x} w(s)\\,\\mathrm{d}s, \\quad \\textrm{for} x \\in (0, +\\infty),$$ where $w$ is a positive measurable function on $(0,+\\infty)$ and $g$ is a real continuous strictly monotone function with its inverse $g^{-1}$. We give some sufficient conditions on weights $u,v$ on $(0,+\\infty)$ for which there exists a positive constant $C$ such that the weighted strong type $(p,q)$ inequality $$\\left(\\int_{0}^{\\infty} u("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5636","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"}