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Science China Mathematics

, Volume 62, Issue 2, pp 283–350 | Cite as

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

  • Ziyi He
  • Liguang Liu
  • Dachun YangEmail author
  • Wen Yuan
Articles

Abstract

Assume that (X, d, μ) is a space of homogeneous type in the sense of Coifman and Weiss (1971, 1977). In this article, motivated by the breakthrough work of Auscher and Hytönen (2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al. (2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderón reproducing formulae on (X, d, μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.

Keywords

space of homogeneous type Calderón reproducing formula approximation of the identity wavelet space of test functions distribution 

MSC(2010)

42C40 42B20 42B25 30L99 

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11771446, 11571039, 11726621, 11761131002 and 11871100).

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical SciencesBeijing Normal UniversityBeijingChina
  2. 2.School of MathematicsRenmin University of ChinaBeijingChina

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