Revista Matemática Complutense

, Volume 29, Issue 2, pp 245–270 | Cite as

Characterizations of variable exponent Hardy spaces via Riesz transforms

  • Dachun Yang
  • Ciqiang ZhuoEmail author
  • Eiichi Nakai


Let \(p(\cdot ):\ \mathbb R^n\rightarrow (0,\infty )\) be a variable exponent function satisfying that there exists a constant \(p_0\in (0,p_-)\), where \(p_-:=\hbox {ess inf}_{x\in \mathbb R^n}p(x)\), such that the Hardy–Littlewood maximal operator is bounded on the variable exponent Lebesgue space \(L^{p(\cdot )/p_0}(\mathbb R^n)\). In this article, via investigating relations between boundary values of harmonic functions on the upper half space and elements of variable exponent Hardy spaces \(H^{p(\cdot )}(\mathbb R^n)\) introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize \(H^{p(\cdot )}(\mathbb R^n)\) via the first order Riesz transforms when \(p_-\in (\frac{n-1}{n},\infty )\), and via compositions of all the first order Riesz transforms when \(p_-\in (0,\frac{n-1}{n})\).


Hardy space Variable exponent Riesz transform  Harmonic function 

Mathematics Subject Classification

Primary 42B30 Secondary 47B06 42B35 42B25 



The authors would like to express their deep thanks to Professor Yoshihiro Sawano for his several useful conversations on the subject of this article and careful reading on the manuscript of this article.


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Copyright information

© Universidad Complutense de Madrid 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijingPeople’s Republic of China
  2. 2.Department of MathematicsIbaraki UniversityMitoJapan

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