Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range
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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.
KeywordsMorrey spaces Muckenhoupt weights Hardy-Littlewood maximal operator Calderón-Zygmund operators
Mathematics Subject Classification (2010)42B35 42B25 46E30 42B20
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- 1.Adams, D.R.: Morrey spaces, Lecture notes in applied and numerical harmonic analysis. Birkhäuser/Springer, Cham (2015)Google Scholar