Nuclear Embeddings of Besov Spaces into Zygmund Spaces

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Let \(d\in {\mathbb {N}}\) and let \(\Omega \) be a bounded Lipschitz domain in \({\mathbb {R}}^d\). We prove that the embedding \(I_d{:}B^d _{p,q}(\Omega ) \longrightarrow L_p (\log L)_a (\Omega )\) is nuclear if \(a<-1\) and \(1\le p,q\le \infty \), while if \(-1<a<0\), \(2<p<\infty \) and \(p\le q \le \infty \) the embedding \(I_d\) fails to be nuclear. Furthermore, if \(a=-1\), the embedding \(I_d{:}B^d _{\infty ,\infty }(\Omega ) \longrightarrow L_\infty (\log L)_{-1} (\Omega )\) is not nuclear.

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Correspondence to Thomas Kühn.

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Dedicated to the memory of Professor Jaak Peetre

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F. Cobos and T. Kühn have been supported in part by MTM2017-84058-P (AEI/FEDER, UE).

Communicated by Ruzhansky.

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Cobos, F., Edmunds, D.E. & Kühn, T. Nuclear Embeddings of Besov Spaces into Zygmund Spaces. J Fourier Anal Appl 26, 9 (2020) doi:10.1007/s00041-019-09709-6

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  • Besov spaces
  • Zygmund spaces
  • Nuclear embeddings

Mathematical Subject Classification

  • Primary 46E35
  • 47B10