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Science China Mathematics

, Volume 61, Issue 9, pp 1647–1664 | Cite as

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

  • Jiecheng Chen
  • Jiawei Dai
  • Dashan Fan
  • Xiangrong Zhu
Articles
  • 39 Downloads

Abstract

In this paper, we study the boundedness of the Hausdorff operator Hϕ on the real line ℝ. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space Lp(ℝ) and the Hardy space H1(ℝ). The key idea is to reformulate Hϕ as a Calderón-Zygmund convolution operator, from which its boundedness is proved by verifying the Hörmander condition of the convolution kernel. Secondly, to prove the boundedness on the Hardy space Hp(ℝ) with 0 < p < 1; we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of Hp(ℝ) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H1(ℝ).

Keywords

Hausdorff operator Hardy space atomic characterization molecule 

MSC(2010)

42B25 

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11671363, 11471288 and 11601456). Also, the authors are indebted to the referees who gave them many helpful comments and suggestions.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Jiecheng Chen
    • 1
  • Jiawei Dai
    • 1
  • Dashan Fan
    • 2
  • Xiangrong Zhu
    • 1
  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  2. 2.Department of MathematicsUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

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