Necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces

Evgeniya Burtseva, Lars‐Erik Persson and Natasha Samko

Mathematische Nachrichten, 2018, vol. 291, issue 11-12, 1655-1665

Abstract: 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 Rn).

Date: 2018
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