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Vector-Valued Local Approximation Spaces

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Abstract

We prove that for every Banach space Y, the Besov spaces of functions from the n-dimensional Euclidean space to Y agree with suitable local approximation spaces with equivalent norms. In addition, we prove that the Sobolev spaces of type q are continuously embedded in the Besov spaces of the same type if and only if Y has martingale cotype q. We interpret this as an extension of earlier results of Xu (J Reine Angew Math 504:195–226, 1998), and Martínez et al. (Adv Math 203(2):430–475, 2006). These two results combined give the characterization that Y admits an equivalent norm with modulus of convexity of power type q if and only if weakly differentiable functions have good local approximations with polynomials.

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Acknowledgements

We would like to thank an anonymous referee for constructive comments that improved this paper.

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Author notes

  1. Jori Merikoski

    Present address: Department of Mathematics and Statistics, University of Turku, Turku, 20014, Finland

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68, Gustaf Hällströmin katu 2b, Helsinki, 00014, Finland

    Tuomas Hytönen & Jori Merikoski

Authors
  1. Tuomas Hytönen
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  2. Jori Merikoski
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Corresponding author

Correspondence to Jori Merikoski.

Additional information

Communicated by Winfried Sickel.

Both authors were partially supported by the ERC Starting Grant “AnProb” (Grant No. 278558) and the Finnish Centre of Excellence in Analysis and Dynamics Research (Grant No. 271983)

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Hytönen, T., Merikoski, J. Vector-Valued Local Approximation Spaces. J Fourier Anal Appl 25, 299–320 (2019). https://doi.org/10.1007/s00041-018-9598-2

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  • DOI: https://doi.org/10.1007/s00041-018-9598-2

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