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 w ∈ Ap. 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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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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DOI: https://doi.org/10.1007/s11118-019-09805-8