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Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range

  • Javier DuoandikoetxeaEmail author
  • Marcel Rosenthal
Article
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Abstract

We prove that for operators satistying weighted inequalities with Ap weights the boundedness on a certain class of Morrey spaces holds with weights of the form |x|αw(x) for wAp. In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N. Samko for the Hilbert transform, by H. Tanaka for the Hardy-Littlewood maximal operator, and by S. Nakamura and Y. Sawano for Calderón-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Morrey spaces, we obtain new estimates even for the Hardy-Littlewood maximal operator. Moreover, we prove the necessity of certain Aq condition.

Keywords

Morrey spaces Muckenhoupt weights Hardy-Littlewood maximal operator Calderón-Zygmund operators 

Mathematics Subject Classification (2010)

42B35 42B25 46E30 42B20 

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Notes

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad del País Vasco/Euskal Herriko Unibertsitatea UPV/EHUBilbaoSpain
  2. 2.StuttgartGermany

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