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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 $$\\left(\\int_{0}^{\\infty} u("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1309.5636","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-22T19:02:06Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"31aa57e1814e68dcbe711afd076a1ba8c9dfff776419b3b9a89a76a6b0a5ff7a","abstract_canon_sha256":"db13c56469ae9010975c44280588be097bad0e4a6b5f79700f81469087695428"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:35.713784Z","signature_b64":"NSQCItXLvQmJPuxYD8eHTlYJGWK++PCWwQPnb61l16TfdXAJrXNTCrFz2Q2zlxVVL4gVWe23y7gfvuxZrzIWAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72ea25975ba95791e7da16b0738bd8757c32419b7b04ea5edc1b6b6fd4b82950","last_reissued_at":"2026-05-18T03:12:35.713123Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:35.713123Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On weighted strong type inequalities for the generalized weighted mean operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Ondrej Hutn\\'ik","submitted_at":"2013-09-22T19:02:06Z","abstract_excerpt":"The generalized weighted mean operator $\\mathbf{M}^{g}_{w}$ is given by $$[\\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 $g^{-1}$. 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