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Nuclear Embeddings of Besov Spaces into Zygmund Spaces

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Abstract

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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References

  1. 1.

    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988)

  2. 2.

    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976)

  3. 3.

    Cobos, F., Domínguez, O., Kühn, T.: On nuclearity of embeddings between Besov spaces. J. Approx. Theory 225, 209–223 (2018)

  4. 4.

    Cobos, F., Fernández-Cabrera, L.M., Manzano, A., Martínez, A.: Logarithmic interpolation spaces between quasi-Banach spaces. Z. Anal. Anwend. 26, 65–86 (2007)

  5. 5.

    Cobos, F., Segurado, A.: Description of logarithmic interpolation spaces by means of the \(J\)-functional and applications. J. Funct. Anal. 268, 2906–2945 (2015)

  6. 6.

    Diestel, J., Jarchow, H., Tongue, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)

  7. 7.

    Edmunds, D.E., Gurka, P., Lang, J.: Nuclearity and non-nuclearity of some Sobolev embeddings on domains. J. Approx. Theory 211, 94–103 (2016)

  8. 8.

    Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge (1996)

  9. 9.

    Evans, W.D., Opic, B.: Real interpolation with logarithmic functors and reiteration. Can. J. Math. 52, 920–960 (2000)

  10. 10.

    Grothendieck, A.: Produits Tensoriels Topologiques et Espaces Nucléaires, Memoirs Amer. Math. Soc. 16, Providence, (1955)

  11. 11.

    Gustavsson, J.: A function parameter in connection with interpolation of Banach spaces. Math. Scand. 42, 289–305 (1978)

  12. 12.

    Jameson, G.J.O.: Summing and Nuclear Norms in Banach Space Theory. Cambridge University Press, Cambridge (1987)

  13. 13.

    König, H.: Eigenvalue Distribution of Compact Operators. Birkhäuser, Basel (1986)

  14. 14.

    Maurin, K.: Abbildungen vom Hilbert-Schmidtschen Typus und ihre Anwendungen. Math. Scand. 9, 359–371 (1961)

  15. 15.

    Maurin, K.: Methods of Hilbert Spaces. PWN-Polish Scientific Publishers, Warsaw (1972)

  16. 16.

    Peetre, J.: New Thoughts on Besov Spaces. Duke University Mathematics Series, Durham (1976)

  17. 17.

    Persson, L.-E.: Interpolation with a parameter function. Math. Scand. 59, 199–222 (1986)

  18. 18.

    Pietsch, A.: \(r\)-Nukleare Sobolevsche Einbettungsoperatoren, in: Elliptische Differentialgleichungen II, Akademie-Verlag, Berlin (1971), pp. 203-215

  19. 19.

    Pietsch, A.: Operator Ideals. North-Holland, Amsterdam (1980)

  20. 20.

    Pietsch, A.: Eigenvalues and \(s\)-Numbers. Cambridge University Press, Cambridge (1987)

  21. 21.

    Pietsch, A., Triebel, H.: Interpolationstheorie für Banachideale von beschränkten linearen Operatoren. Studia Math. 31, 95–109 (1968)

  22. 22.

    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)

  23. 23.

    Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)

  24. 24.

    Triebel, H.: Function Spaces and Wavelets on Domains. European Mathematical Society Publishing House, Zürich (2008)

  25. 25.

    Triebel, H.: Nuclear embeddings in function spaces. Math. Nachr. 290, 3038–3048 (2017)

  26. 26.

    Yosida, K.: Functional Analysis. Springer, Berlin (1965)

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Author information

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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Keywords

  • Besov spaces
  • Zygmund spaces
  • Nuclear embeddings

Mathematical Subject Classification

  • Primary 46E35
  • 47B10