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Hardy spaces for ball quasi-Banach function spaces

Volume 525 / 2017

Yoshihiro Sawano, Kwok-Pun Ho, Dachun Yang, Sibei Yang Dissertationes Mathematicae 525 (2017), 1-102 DOI: 10.4064/dm750-9-2016 Published online: 20 July 2017


This article unifies the theory for Hardy spaces built on Banach lattices on ${\mathbb{R}}^n$ satisfying certain weak conditions on indicator functions of balls. The authors introduce a new family of function spaces, named the ball quasi-Banach function spaces, to define Hardy type spaces. The ones in this article extend classical Hardy spaces and include various known function spaces, for example, Hardy–Lorentz spaces, Hardy–Herz spaces, Hardy–Orlicz spaces, Hardy–Morrey spaces, Musielak–Orlicz–Hardy spaces, variable Hardy spaces and variable Hardy–Morrey spaces. Among them, Hardy–Herz spaces are shown to naturally arise in the context of any function spaces above. The example of Hardy–Morrey spaces shows that the absolute continuity of the quasi-norm is not necessary, which is used to guarantee the density of the set of functions having compact supports in Hardy spaces for ball quasi-Banach function spaces, but the decomposition result on these Hardy-type spaces never requires this absolute continuity of the quasi-norm. Moreover, via assuming that the powered Hardy–Littlewood maximal operator satisfies certain Fefferman–Stein vector-valued maximal inequality as well as it is bounded on the associate space, the atomic characterizations of Hardy type spaces are obtained. Although the results are based on the rather abstract theory of function spaces, they improve and extend the results for Orlicz spaces and Musielak–Orlicz spaces. Moreover, local Hardy type spaces and Hardy type spaces associated with operators in this setting are also studied.


  • Yoshihiro SawanoDepartment of Mathematics
    and Information Sciences
    Tokyo Metropolitan University
    1-1 Minami Ohsawa, Hachioji
    Tokyo 192-0397, Japan
  • Kwok-Pun HoDepartment of Mathematics
    and Information Technology
    The Hong Kong Institute of Education
    10 Lo Ping Road, Tai Po
    Hong Kong, China
  • Dachun YangSchool of Mathematical Sciences
    Beijing Normal University
    Laboratory of Mathematics
    and Complex Systems
    Ministry of Education
    Beijing 100875
    People’s Republic of China
  • Sibei YangSchool of Mathematics and Statistics
    Gansu Key Laboratory of Applied Mathematics
    and Complex Systems
    Lanzhou University
    Lanzhou, Gansu 730000
    People’s Republic of China

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