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Decompositions of non-homogeneous Herz-type Besov and Triebel-Lizorkin spaces

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Abstract

Decompositions of non-homogeneous Herz-type Besov and Triebel-Lizorkin spaces by atoms, molecules and wavelets are given. These results generalize the corresponding results for classical Besov and Triebel-Lizorkin spaces.

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References

  1. Barrios B, Betancor J. Characterizations of anisotropic Besov spaces. Math Nachr, 2011, 284: 1796–1819

    Article  MATH  MathSciNet  Google Scholar 

  2. Cao J, Yang D. Hardy spaces H p L (ℝn) associated to operators satisfying k-Davies-Gaffney estimates. Sci China Math, 2012, 55: 1403–1440

    Article  MATH  MathSciNet  Google Scholar 

  3. Deng D, Xu M, Yan L. Wavelet characterization of weighted Triebel-Lizorkin spaces. Approx Theory Appl, 2002, 18: 76–92

    MATH  MathSciNet  Google Scholar 

  4. Drihem D. Atomic decomposition of Besov-type and Triebel-Lizorkin-type spaces. Sci China Math, 2013, 56: 1073–1086

    Article  MATH  MathSciNet  Google Scholar 

  5. El Baraka A. An embedding theorem for Campanato spaces. Electron J Differential Equations, 2002, 66: 1–17

    Google Scholar 

  6. El Baraka A. Littlewood-Paley characterization for Campanato spaces. J Funct Spaces Appl, 2006, 4: 193–220

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Frazier M, Jawerth B, Weiss G. Littlewood-Paley Theory and the Study of Function Spaces. CBMS 79. Providence, RI: Amer Math Soc, 1991

    Google Scholar 

  10. Grafakos L, Liu L, Yang D. Boundedness of paraproduct operators on RD-spaces. Sci China Math, 2010, 53: 2097–2114

    Article  MATH  MathSciNet  Google Scholar 

  11. Hernández E, Yang D. Interpolation of Herz spaces and applications. Math Nachr, 1999, 205: 69–87

    Article  MATH  MathSciNet  Google Scholar 

  12. Ho K P. Littlewood-Paley spaces. Math Scand, 2011, 108: 77–102

    MATH  MathSciNet  Google Scholar 

  13. Izuki M. Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal Math, 2010, 36: 33–50

    Article  MATH  MathSciNet  Google Scholar 

  14. Izuki M. Boundedness of commutators on Herz spaces with variable exponent. Rend Circ Mat Palermo, 2010, 59: 199–213

    Article  MATH  MathSciNet  Google Scholar 

  15. Izuki M, Sawano Y. Wavelet bases in the weighted Besov and Triebel-Lizorkin spaces with A p loc-weights. J Approx Theory, 2009, 16: 656–673

    Article  MathSciNet  Google Scholar 

  16. Izuki M, Sawano Y. The Haar wavelet characterization of weighted Herz spaces and greediness of the Haar wavelet basis. J Math Anal Appl, 2010, 362: 140–155

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Izuki M, Sawano Y, Tanaka H. Weighted Besov-Morrey spaces and Triebel-Lizorkin spaces. In: Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kokyuroku Bessatsu, B22. Kyoto: Res Inst Math Sci (RIMS), 2010, 21–60

    Google Scholar 

  19. Kempka H. 2-Microlocal Besov and Triebel-Lizorkin spaces of variable integrability. Rev Mat Complut, 2009, 22: 227–25

    MATH  MathSciNet  Google Scholar 

  20. Kempka H. Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces. J Funct Spaces Appl, 2010, 8: 129–165

    Article  MATH  MathSciNet  Google Scholar 

  21. Kyriazis G. Decomposition systems for function spaces. Studia Math, 2003, 157: 133–169

    Article  MATH  MathSciNet  Google Scholar 

  22. Li B, Bownik M, Yang D, et al. Anisotropic singular integrals in product spaces. Sci China Math, 2010, 53: 3163–3178

    Article  MATH  MathSciNet  Google Scholar 

  23. Li X, Yang D. Boundedness of some sublinear operators on Herz spaces. IIIinois J Math, 1996, 40: 484–501

    MATH  Google Scholar 

  24. Liang Y, Sawano Y, Ullrich T, et al. A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Diss Math, 2013, 489: 1–114

    MathSciNet  Google Scholar 

  25. Liang Y, Sawano Y, Ullrich T, et al. New characterizations of Besov-Triebel-Lizorkin-Hausdorff spaces including coorbits and wavelets. J Fourier Anal Appl, 2012, 18: 1067–1111

    Article  MATH  MathSciNet  Google Scholar 

  26. Liang Y, Yang D, Yang S. Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates. Sci China Math, 2011, 54: 2395–2426

    Article  MATH  MathSciNet  Google Scholar 

  27. Lu S, Yabuta K, Yang D. The Boundedness of some sublinear operators in weighted Herz-type spaces. Kodai Math J, 2000, 23: 391–410

    Article  MATH  MathSciNet  Google Scholar 

  28. Lu S, Yang D. The decomposition of the weighted Herz spaces and its application. Sci China Ser A, 1995, 38: 147–158

    MATH  MathSciNet  Google Scholar 

  29. Lu S Z, Yang D, Hu G. Herz Type Spaces and Their Applications. Beijing: Science Press, 2008

    Google Scholar 

  30. Rauhut H, Ullrich T. Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type. J Funct Anal, 2011, 260: 3299–3362

    Article  MATH  MathSciNet  Google Scholar 

  31. Sawano Y. Wavelet characterization of Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Funct Approx Comment Math, 2008, 38: 93–107

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Stein E, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton University Press, 1971

    MATH  Google Scholar 

  34. Tang L, Yang D. Boundedness of vector-valued operators on weighted Herz spaces. Approx Theory Appl, 2000, 16: 58–70

    MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  36. Triebel H. Fractals and Spectra. Basel: Birkhäuser, 1997

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  38. Triebel H. Function Spaces and Wavelets on Domains. EMS Tracts in Math 7. Zürich: EMS Publ Houses, 2008

    Book  Google Scholar 

  39. Ullrich T. Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits. J Funct Spaces Appl, 2012, Article ID 163213, 47pp

    Google Scholar 

  40. Wojtaszczk P. A Mathematical Introduction to Wavelets. London Math Soc Stud Texts 37. Cambrige: Cambridge University Press, 1997

    Book  Google Scholar 

  41. Xu J. A discrete characterization of the Herz-type Triebel-Lizorkin spaces and its applications. Acta Math Sci, 2004, 24B: 412–420

    Google Scholar 

  42. Xu J. Equivalent norms of Herz type Besov and Triebel-Lizorkin spaces. J Funct Spaces Appl, 2005, 3: 17–31

    Article  MATH  MathSciNet  Google Scholar 

  43. Xu J. An admissibility for topological degree of Herz-type Besov and Triebel-Lizorkin spaces. Topol Methods Nonlinear Anal, 2009, 33: 327–334

    MATH  MathSciNet  Google Scholar 

  44. Xu J, Yang D. Applications of Herz-type Triebel-Lizorkin spaces. Acta Math Sci, 2003, 23B: 328–338

    Google Scholar 

  45. Xu J, Yang D. Herz-type Triebel-Lizorkin spaces (I). Acta Math Sin (Engl Ser), 2005, 21: 643–654

    Article  MATH  MathSciNet  Google Scholar 

  46. Yang D, Yang S. Local Hardy spaces of Musielak-Orlicz type and their applications. Sci China Math, 2012, 55: 1677–1720

    Article  MATH  MathSciNet  Google Scholar 

  47. Yang D, Yuan W. A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces. J Funct Anal, 2008, 255: 2760–2809

    Article  MATH  MathSciNet  Google Scholar 

  48. Yang D, Yuan W. New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces. Math Z, 2010, 265: 451–480

    Article  MATH  MathSciNet  Google Scholar 

  49. Yang D, Yuan W. Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means. Nonlinear Anal, 2010, 73: 3805–3820

    Article  MATH  MathSciNet  Google Scholar 

  50. Yang D, Yuan W. Relations among Besov-type spaces, Triebel-Lizorkin-type spaces and generalized Carleson measure spaces. Appl Anal, 2013, 92: 549–561

    Article  MATH  MathSciNet  Google Scholar 

  51. Yang D, Yuan W, Zhuo C. Fourier multipliers on Triebel-Lizorkin-type spaces. J Funct Spaces Appl, 2012, Article ID 431016, 37pp

    Google Scholar 

  52. Yuan W, Sickel W, Yang D. Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, 2005. Berlin: Springer-Verlag, 2010

    Book  MATH  Google Scholar 

  53. Yuan W, Sawano Y, Yang D. Decompositions of Besov-Hausdorff and Triebel-Lizorkin-Hausdorff spaces and their applications. J Math Anal Appl, 2010, 369: 736–757

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Hainan Normal University, Haikou, 571158, China

    JingShi Xu

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Xu, J. Decompositions of non-homogeneous Herz-type Besov and Triebel-Lizorkin spaces. Sci. China Math. 57, 315–331 (2014). https://doi.org/10.1007/s11425-013-4680-3

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  • DOI: https://doi.org/10.1007/s11425-013-4680-3

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