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Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces

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Abstract

We will prove that for 1 < p < ∞ and 0 < λ < n, the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator \(M_\gamma^c\) equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. When p = 1 and 0 < λ < n, it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator \(M_\gamma^c\) equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. Moreover, the same results are true for the truncated uncentered Hardy-Littlewood maximal operator. Our work extends the previous results of Lebesgue spaces to Morrey spaces.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. 11871452), the Project of Henan Provincial Department of Education (No. 18A110028), and the Nanhu Scholar Program for Young Scholars of XYNU.

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Correspondence to Mingquan Wei.

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Zhang, X., Wei, M., Yan, D. et al. Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces. Front. Math. China (2020). https://doi.org/10.1007/s11464-020-0812-6

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Keywords

  • Hardy-Littlewood maximal function
  • truncated Hardy-Littlewood maximal function
  • Morrey norms
  • weak Morrey norms

MSC

  • 42B20
  • 42B25