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Nuclear and Compact Embeddings in Function Spaces of Generalised Smoothness

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Abstract

We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain Ω ⊂ ℝd. This covers, in particular, the well-known situation for spaces of Besov and Triebel–Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in ∓r spaces, which we could recently extend to the vector-valued setting needed here.

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Authors and Affiliations

  1. Institute of Mathematics, Friedrich Schiller University Jena, 07737, Jena, Germany

    D. D. Haroske & H.-G. Leopold

  2. CMUC, Department of Mathematics, University of Coimbra, Largo D. Dinis, 3000-143, Coimbra, Portugal

    S. D. Moura

  3. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614, Poznań, Poland

    L. Skrzypczak

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  1. D. D. Haroske
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  2. H.-G. Leopold
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  3. S. D. Moura
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  4. L. Skrzypczak
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Correspondence to D. D. Haroske.

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Dedicated to Oleg V. Besov on the occasion of his 90th birthday

Both authors were partially supported by the German Research Foundation (DFG), Grant no. Ha 2794/8-1.

The author was partially supported by the Center for Mathematics of the University of Coimbra-UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.

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Haroske, D.D., Leopold, HG., Moura, S.D. et al. Nuclear and Compact Embeddings in Function Spaces of Generalised Smoothness. Anal Math 49, 1007–1039 (2023). https://doi.org/10.1007/s10476-023-0238-y

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  • DOI: https://doi.org/10.1007/s10476-023-0238-y

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