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Atomic decompositions of Triebel-Lizorkin spaces with local weights and applications

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Abstract

In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces F s,w p,q (ℝn with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish their atomic decompositions for s ∈ ℝ, p ∈ (0, 1] and q ∈ [p,∞). The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in (0, 1]. Finite atomic decompositions for smooth functions in F s,w p,q (ℝn are also obtained, which further implies that a (sub)linear operator that maps smooth atoms of F s,w p,q (ℝn uniformly into a bounded set of a (quasi-)Banach space is extended to a bounded operator on the whole F s,w p,q (ℝn. As an application, the boundedness of the local Riesz operator on the space F s,w p,q (ℝn is obtained.

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Authors and Affiliations

  1. Department of Mathematics, School of Information, Renmin University of China, Beijing, 100872, China

    Liguang Liu

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    Dachun Yang

Authors
  1. Liguang Liu
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  2. Dachun Yang
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Correspondence to Dachun Yang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11101425, 11171027) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120003110003).

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Liu, L., Yang, D. Atomic decompositions of Triebel-Lizorkin spaces with local weights and applications. Chin. Ann. Math. Ser. B 35, 237–260 (2014). https://doi.org/10.1007/s11401-014-0824-1

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  • DOI: https://doi.org/10.1007/s11401-014-0824-1

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