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

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Abstract

We study the spaces \(B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb {R}^{n})\) and \(F^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})\) 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 \(B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})\) and \(F^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})\) 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 \(B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})\) with the help of the so-called Peetre maximal functions. Our results do also hold for 2-microlocal function spaces \(B^{\boldsymbol{w}}_{{p(\cdot)},{q(\cdot)}}(\mathbb{R}^{n})\) and \(F^{\boldsymbol{w}}_{{p(\cdot)},{q(\cdot)}}(\mathbb{R}^{n})\) which are a slight generalization of generalized smoothness spaces and spaces of variable smoothness.

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Acknowledgements

The first author acknowledges the financial support provided by the DFG project HA 2794/5-1 “Wavelets and function spaces on domains”. Furthermore, the first author thanks the RICAM for his hospitality and support during a short term visit in Linz.

The second author acknowledges the financial support provided by the FWF project Y 432-N15 START-Preis “Sparse Approximation and Optimization in High Dimensions”.

We thank the anonymous referee for pointing the reference [18] out to us.

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  1. Mathematical Institute, Friedrich-Schiller-University Jena, 07737, Jena, Germany

    Henning Kempka

  2. Department of Mathematics, TU Berlin, Sec. MA 8-1 Str. des 17. Juni 136, 10623, Berlin, Germany

    Jan Vybíral

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  1. Henning Kempka
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Correspondence to Henning Kempka.

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Communicated by Arieh Iserles.

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Kempka, H., Vybíral, J. Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences. J Fourier Anal Appl 18, 852–891 (2012). https://doi.org/10.1007/s00041-012-9224-7

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  • DOI: https://doi.org/10.1007/s00041-012-9224-7

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