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Square roots of the Bessel operators and the related Littlewood–Paley estimates

Volume 263 / 2022

Yanping Chen, Xuan Thinh Duong, Ji Li, Wenyu Tao, Dongyong Yang Studia Mathematica 263 (2022), 19-58 MSC: Primary 42B20; Secondary 42B25. DOI: 10.4064/sm190922-19-11 Published online: 22 November 2021

Abstract

Let $\Delta _{\lambda }$ and $S_{\lambda }$, $\lambda \in \mathbb {R}_+:=(0,+\infty )$, be the two Bessel operators studied by Muckenhoupt–Stein (1965). We prove that the square root of the Bessel operators and the corresponding “gradient” operators are equivalent in $L^p$ spaces for $1 \lt p \lt \infty $. Moreover, by using holomorphic functional calculus, we establish the characterizations of boundedness on $L^p$ spaces associated with Bessel operators in terms of the Littlewood–Paley $g$-function with respect to the square root of the Bessel operator. Also, we study boundedness properties of Littlewood–Paley $g$-function associated with the square root of the Bessel operator on the odd $\rm {BMO}$ space $\rm {BMO}_+$ and the atomic Hardy space $H^1$.

Authors

  • Yanping ChenSchool of Mathematics and Physics
    University of Science and Technology Beijing
    Beijing 100083, China
    e-mail
  • Xuan Thinh DuongDepartment of Mathematics
    Macquarie University
    Sydney, NSW 2109, Australia
    e-mail
  • Ji LiDepartment of Mathematics
    Macquarie University
    Sydney, NSW 2109, Australia
    e-mail
  • Wenyu TaoSchool of Mathematics and Physics
    University of Science and Technology Beijing
    Beijing 100083, China
    e-mail
  • Dongyong YangSchool of Mathematical Sciences
    Xiamen University
    Xiamen 361005, China
    e-mail

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