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The Hardy Operator and the Gap Between L ∞ and BMO
Author(s) -
Lang Jan,
Pick Luboš
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005651
Subject(s) - bounded function , compact space , bounded mean oscillation , counterexample , mathematics , bounded operator , pure mathematics , compact operator , order (exchange) , banach space , space (punctuation) , hardy space , mathematical analysis , discrete mathematics , computer science , extension (predicate logic) , finance , economics , programming language , operating system
We study boundedness and compactness properties of the Hardy integral operator T f ( x ) = ∫ A x f from a weighted Banach function space X ( v ) into L ∞ and BMO. We give a new simple characterization of compactness of T from X ( v ) into BMO. We construct examples of spaces X ( v ) such that T ( X ( v )) is (a) bounded in L ∞ but not compact in BMO; (b) compact in BMO but not bounded in L ∞ ; (c) bounded in BMO but neither bounded in L ∞ nor compact in BMO; (d) bounded in L ∞ , compact in BMO and yet not compact in L ∞ . In order to obtain the last of the counterexamples we construct a new weighted Banach function space.