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Continuous embeddings of Besov-Morrey function spaces

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Abstract

We study embeddings of spaces of Besov-Morrey type, , and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation for a Muckenhoupt \(\mathcal{A}_\infty\) weight w, with w α (x) = |x|α, α > −d, as a typical example.

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References

  1. Edmunds, D. E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996

    Book  MATH  Google Scholar 

  2. Haroske, D., Triebel, H.: Entropy numbers in weighted function spaces and eigenvalue distribution of some degenerate pseudodifferential operators I. Math. Nachr., 167, 131–156 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kühn, Th., Leopold, H.-G., Sickel, W., et al.: Entropy numbers of embeddings of weighted Besov spaces. Constr. Approx., 23, 61–77 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kühn, Th., Leopold, H.-G., Sickel, W., et al.: Entropy numbers of embeddings of weighted Besov spaces II. Proc. Edinburgh Math. Soc. (2), 49, 331–359 (2006)

    Article  MATH  Google Scholar 

  5. Kühn, Th., Leopold, H.-G., Sickel, W., et al.: Entropy numbers of embeddings of weighted Besov spaces III. Weights of logarithmic type. Math. Z., 255, 1–15 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Haroske, D. D., Piotrowska, I.: Atomic decompositions of function spaces with Muckenhoupt weights, and some relations to fractal analysis. Math. Nachr., 281, 1476–1494 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Haroske, D. D., Skrzypczak, L.: Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, I. Rev. Mat. Complut., 21(1), 135–177 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Haroske, D. D., Skrzypczak, L.: Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, II. General weights. Ann. Acad. Sci. Fenn. Math., 36, 111–138 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Haroske, D. D., Skrzypczak, L.: Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases. J. Funct. Spaces Appl., 9(2), 129–178 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Haroske, D. D., Skrzypczak, L.: Spectral theory of some degenerate elliptic operators with local singularities. J. Math. Anal. Appl., 371(1), 282–299 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bui, H.-Q.: Weighted Besov and Triebel spaces: Interpolation by the real method. Hiroshima Math. J., 12(3), 581–605 (1982)

    MathSciNet  MATH  Google Scholar 

  12. Bui, H.-Q.: Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures. J. Funct. Anal., 55(1), 39–62 (1984)

    Article  MathSciNet  Google Scholar 

  13. Bui, H.-Q., Paluszyński, M., Taibleson, M. H.: A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Studia Math., 119(3), 219–246 (1996)

    MathSciNet  MATH  Google Scholar 

  14. Bui, H.-Q., Paluszyński, M., Taibleson, M. H.: Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case q < 1. J. Fourier Anal. Appl., 3(Spec. Iss.), 837–846 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Roudenko, S.: Matrix-weighted Besov spaces. Trans. Amer. Math. Soc., 355, 273–314 (2002)

    Article  MathSciNet  Google Scholar 

  16. Bownik, M.: Atomic and molecular decompositions of anisotropic Besov spaces. Math. Z., 250(3), 539–571 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bownik, M., Ho, K. P.: Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces. Trans. Amer. Math. Soc., 358(4), 1469–1510 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Morrey, C. B., Jr.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc., 43(1), 126–166 (1938)

    Article  MathSciNet  Google Scholar 

  19. Peetre, J.: On the theory of L p,λ spaces. J. Funct. Anal., 4, 71–87 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Differential Equations, 19(5–6), 959–1014 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mazzucato, A. L.: Besov-Morrey spaces: function space theory and applications to non-linear PDE. Trans. Amer. Math. Soc., 355(4), 1297–1364 (electronic) (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sawano, Y.: Wavelet characterizations of Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Funct. Approx. Comment. Math., 38, 93–107 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sawano, Y.: A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Acta Mathematica Sinica, English Series, 25(8), 1223–1242 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sawano, Y.: Brezis-Galloüet-Wainger type inequality for Besov-Morrey spaces. Studia Math., 196, 91–101 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sawano, Y., Tanaka, H.: Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math. Z., 257(4), 871–905 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sawano, Y., Tanaka, H.: Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces for non-doubling measures. Math. Nachr., 282(12), 1788–1810 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tang, L., Xu, J.: Some properties of Morrey type Besov-Triebel spaces. Math. Nachr., 278(7–8), 904–917 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Math. 2005, Springer, Berlin, 2010

    Book  MATH  Google Scholar 

  29. Sawano, Y., Sugano, S., Tanaka, H.: Identification of the image of Morrey spaces by the fractional integral operators. Proc. A. Razmadze Math. Inst., 149, 87–93 (2009)

    MathSciNet  MATH  Google Scholar 

  30. García-Cuerva, J., Rubio de Francia, J. L.: Weighted norm inequalities and related topics. In: North-Holland Mathematics Studies, Vol. 116, North-Holland, Amsterdam, 1985

    Book  MATH  Google Scholar 

  31. Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. In: Princeton Mathematical Series, Vol. 43, Princeton University Press, Princeton, 1993

    MATH  Google Scholar 

  32. Torchinsky, A.: Real-variable methods in harmonic analysis. In: Pure and Applied Mathematics, Vol. 123, Academic Press Inc., Orlando, FL, 1986

    MATH  Google Scholar 

  33. Triebel, H.: Theory of Function Spaces, Birkhäuser, Basel, 1983

    Book  Google Scholar 

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

    Book  MATH  Google Scholar 

  35. Triebel, H.: Theory of Function Spaces III, Birkhäuser, Basel, 2006

    MATH  Google Scholar 

  36. Rychkov, V. S.: Littlewood-Paley theory and function spaces with A locp weights. Math. Nachr., 224, 145–180 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Izuki, M., Sawano, Y.: Wavelet bases in the weighted Besov and Triebel-Lizorkin spaces with locp weights. J. Approx. Theory, 161(2), 656–673 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Dorothee D. Haroske

  2. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Ul. Umultowska 87, 61-614, Poznań, Poland

    Leszek Skrzypczak

Authors
  1. Dorothee D. Haroske
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  2. Leszek Skrzypczak
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Correspondence to Dorothee D. Haroske.

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The first author is partly supported by Heisenberg grant Ha 2794/1-2

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Haroske, D.D., Skrzypczak, L. Continuous embeddings of Besov-Morrey function spaces. Acta. Math. Sin.-English Ser. 28, 1307–1328 (2012). https://doi.org/10.1007/s10114-012-1119-7

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  • DOI: https://doi.org/10.1007/s10114-012-1119-7

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