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Necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces
Author(s) -
Burtseva Evgeniya,
Persson LarsErik,
Samko Natasha
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700356
Subject(s) - mathematics , differentiable function , hardy space , bounded function , pure mathematics , type (biology) , infinity , maximal function , power (physics) , mathematical analysis , ecology , physics , quantum mechanics , biology
We study one‐ and multi‐dimensional weighted Hardy operators on functions with Hölder‐type behavior. As a main result, we obtain necessary and sufficient conditions on the power weight under which both the left and right hand sided Hardy operators map, roughly speaking, functions with the Hölder behavior only at the singular point x = 0 to functions differentiable for x ≠ 0 and bounded after multiplication by a power weight. As a consequence, this implies necessary and sufficient conditions for the boundedness in Hölder spaces due to the corresponding imbeddings. In the multi‐dimensional case we provide, in fact, stronger Hardy inequalities via spherical means. We also separately consider the case of functions with Hölder‐type behavior at infinity (Hölder spaces on the compactified R n ).