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

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.

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Correspondence to Javier Duoandikoetxea.

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The first author is supported by the grants MTM2014-53850-P of the Ministerio de Economía y Competitividad (Spain) and grant IT-641-13 of the Basque Gouvernment.

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Duoandikoetxea, J., Rosenthal, M. Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range. Potential Anal 53, 1255–1268 (2020). https://doi.org/10.1007/s11118-019-09805-8

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Keywords

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

Mathematics Subject Classification (2010)

  • 42B35
  • 42B25
  • 46E30
  • 42B20