Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Atomic decompositions of Triebel-Lizorkin spaces with local weights and applications

  • 57 Accesses

Abstract

In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces F p,q s,w (ℝn with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish their atomic decompositions for s ∈ ℝ, p ∈ (0, 1] and q ∈ [p,∞). The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in (0, 1]. Finite atomic decompositions for smooth functions in F p,q s,w (ℝn are also obtained, which further implies that a (sub)linear operator that maps smooth atoms of F p,q s,w (ℝn uniformly into a bounded set of a (quasi-)Banach space is extended to a bounded operator on the whole F p,q s,w (ℝn. As an application, the boundedness of the local Riesz operator on the space F p,q s,w (ℝn is obtained.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    Bui, H., Weighted Hardy spaces, Math. Nachr., 103, 1981, 45–62.

  2. [2]

    Cao, W., Chen, J. and Fan, D., Boundedness of an oscillating multiplier on Triebel-Lizorkin spaces, Acta Math. Sin. (Engl. Ser.), 26, 2010, 2071–2084.

  3. [3]

    Chen, J., Yu, X., Zhang, Y. and Wang, H., Hypersingular parameterized Marcinkiewicz integrals with variable kernels on Sobolev and Hardy-Sobolev spaces, Appl. Math. J. Chinese Univ. Ser. B, 23, 2008, 420–430.

  4. [4]

    Frazier, M. and Jawerth, B., Decomposition of Besov spaces, Indiana Univ. Math. J., 34, 1985, 777–799.

  5. [5]

    Frazier, M. and Jawerth, B., A discrete transform and decompositions of distribution spaces, J. Funct. Anal., 93, 1990, 34–170.

  6. [6]

    Frazier, M., Jawerth, B. and Weiss, G., Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, 79, Amer. Math. Soc., Providence, R. I., 1991.

  7. [7]

    Goldberg, D., A local version of real Hardy spaces, Duke Math., 46, 1979, 27–42.

  8. [8]

    Han, Y., Paluszyński, M. and Weiss, G., A new atomic decomposition for the Triebel-Lizorkin spaces, Contemp. Math., 189, 1995, 235–249

  9. [9]

    Haroske, D. D. and Piotrowska, I., Atomic decompositions of function spaces with muckenhoupt weights, and some relation to fractal analysis, Math. Nachr., 281, 2008, 1476–1494.

  10. [10]

    Haroske, D. D. and Skrzypczak, L., Entropy and approximation numbers of embeddings of function spaces with muckenhoupt weights, I, Rev. Mat. Complut., 21, 2008, 135–177.

  11. [11]

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

  12. [12]

    Haroske, D. D. and Skrzypczak, L., Entropy and approximation numbers of embeddings of function spaces with muckenhoupt weights, II. general weights, Ann. Acad. Sci. Fenn. Math., 36, 2011, 111–138.

  13. [13]

    Haroske, D. D. and Skrzypczak, L., Entropy numbers of embeddings of function spaces with muckenhoupt weights, III. Some limiting cases, J. Funct. Spaces Appl., 9, 2011, 129–178.

  14. [14]

    Izuki, M. and Sawano, Y., Wavelet bases in the weighted Besov and Triebel-Lizorkin spaces with A p loc -weights, J. Approx. Theory, 161, 2009, 656–673.

  15. [15]

    Izuki, M. and Sawano, Y., Atomic decomposition for weighted Besov and Triebel-Lizorkin spaces, Math. Nachr., 285, 2012, 103–126.

  16. [16]

    Izuki, M., Sawano, Y. and Tanaka, H., Weighted Besov-Morrey spaces and Triebel-Lizorkin spaces, in: Harmonic analysis and nonlinear partial differential equations, 21–60, RIMS Kôkyûroku Bessatsu, Ser. B, 22, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010.

  17. [17]

    Liu, L. and Yang, D., Boundedness of sublinear operators in Triebel-Lizorkin spaces via atoms, Studia Math., 190, 2009, 163–183.

  18. [18]

    Rychkov, V. S., Littlewood-Paley theory and function spaces with A p loc weights, Math. Nachr., 224, 2001, 145–180.

  19. [19]

    Schmeißr, H.-J. and Triebel, H., Topics in Fourier Analysis and Function Spaces, John Wiley and Sons, Ltd., Chichester, 1987.

  20. [20]

    Schott, T., Function spaces with exponential weights, I, Math. Nachr., 189, 1998, 221–242.

  21. [21]

    Schott, T., Function spaces with exponential weights, II, Math. Nachr., 196, 1998, 231–250.

  22. [22]

    Tang, L., Weighted local Hardy spaces and their applications, Illinois J. Math. (to appear).

  23. [23]

    Tang, L., Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators, J. Funct. Anal., 262, 2012, 1603–1629.

  24. [24]

    Triebel, H., Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.

  25. [25]

    Yang, D. and Yang, S., Weighted local Orlicz Hardy spaces with applications to pseudo-differential operators, Dissertationes Math., 478, 2011, 1–78.

  26. [26]

    Yang, D. and Zhou, Y., Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms, J. Math. Anal. Appl., 339, 2008, 622–635.

  27. [27]

    Yang, D. and Zhou, Y., A boundedness criterion via atoms for linear operators in Hardy spaces, Constr. Approx., 29, 2009, 207–218.

Download references

Author information

Correspondence to Dachun Yang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11101425, 11171027) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120003110003).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Liu, L., Yang, D. Atomic decompositions of Triebel-Lizorkin spaces with local weights and applications. Chin. Ann. Math. Ser. B 35, 237–260 (2014). https://doi.org/10.1007/s11401-014-0824-1

Download citation

Keywords

  • Local weight
  • Triebel-Lizorkin space
  • Atom
  • Lusin-Area function
  • Riesz transform

2010 MR Subject Classification

  • 46E35
  • 47B06
  • 42B20
  • 42B35