Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences

By H. Kempka, Jan Vyb'iral. Journal of Fourier Analysis and Applications, 2011

83 citations recorded by the source corpus.

We study the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb {R}^{n})$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$\end{document} of Besov and Triebel-Lizorkin type as introduced recently in Almeida and Hästö (J. Funct. Anal. 258(5):1628–2655, 2010) and Diening et al. (J. Funct. Anal. 256(6):1731–1768, 2009). Both scales cover many classical spaces with fixed exponents as well as function spaces of variable smoothness and function spaces of variable integrability. The spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$\end{document} have been introduced in Almeida and Hästö (J. Funct. Anal. 258(5):1628–2655, 2010) and Diening et al. (J. Funct. Anal. 256(6):1731–1768, 2009) by Fourier analytical tools, as the decomposition of unity. Surprisingly, our main result states that these spaces also allow a characterization in the time-domain with the help of classical ball means of differences. To that end, we first prove a local means characterization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$\end{document} with the help of the so-called Peetre maximal functions. Our results do also hold for 2-microlocal function spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{\boldsymbol{w}}_{{p(\cdot)},{q(\cdot)}}(\mathbb{R}^{n})$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F^{\boldsymbol{w}}_{{p(\cdot)},{q(\cdot)}}(\mathbb{R}^{n})$\end{document} which are a slight generalization of generalized smoothness spaces and spaces of variable smoothness.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5165ba54b346a78ca0c5270b180edcc5d7b12fad

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In Applied Mathematics

  1. Wen Yuan
  2. Dachun Yang
  3. Ciqiang Zhuo
  4. Jun Liu
  5. Yoshihiro Sawano
  6. Yangyang Zhang
  7. Songbai Wang
  8. Yuan Zhou

In Mathematical Physics

  1. Wen Yuan
  2. Dachun Yang
  3. Ciqiang Zhuo
  4. Yoshihiro Sawano
  5. Jun Liu
  6. Yangyang Zhang
  7. Songbai Wang
  8. Yuan Zhou

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