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Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

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Let A be an expansive dilation on \({{\mathbb R}^n}\) and w a Muckenhoupt \({\mathcal A_\infty(A)}\) weight. In this paper, for all parameters \({\alpha\in{\mathbb R} }\) and \({p,q\in(0,\infty)}\), the authors identify the dual spaces of weighted anisotropic Besov spaces \({\dot B^\alpha_{p,q}(A;w)}\) and Triebel–Lizorkin spaces \({\dot F^\alpha_{p,q}(A;w)}\) with some new weighted Besov-type and Triebel–Lizorkin-type spaces. The corresponding results on anisotropic Besov spaces \({\dot B^\alpha_{p,q}(A; \mu)}\) and Triebel–Lizorkin spaces \({\dot F^\alpha_{p,q}(A; \mu)}\) associated with \({\rho_A}\) -doubling measure μ are also established. All results are new even for the classical weighted Besov and Triebel–Lizorkin spaces in the isotropic setting. In particular, the authors also obtain the \({\varphi}\) -transform characterization of the dual spaces of the classical weighted Hardy spaces on \({{\mathbb R}^n}\).

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Authors and Affiliations

  1. Department of Mathematics, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Baode Li

  2. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    Marcin Bownik

  3. Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Dachun Yang & Wen Yuan

Authors
  1. Baode Li
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  2. Marcin Bownik
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  3. Dachun Yang
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  4. Wen Yuan
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Corresponding author

Correspondence to Dachun Yang.

Additional information

B. Li is supported by the National Natural Science Foundation of China (Grant No. 11001234) and the Start-up Funding Doctor of Xinjiang University (Grant No. BS090109), M. Bownik is supported by National Science Foundation of US (Grant No. DMS 0653881) and D. Yang is supported by the National Natural Science Foundation (Grant No. 10871025) of China and Program for Changjiang Scholars and Innovative Research Team in University of China.

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Li, B., Bownik, M., Yang, D. et al. Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces. Positivity 16, 213–244 (2012). https://doi.org/10.1007/s11117-011-0119-7

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