Orlicz-Hardy spaces associated with operators

Article

Abstract

Let L be a linear operator in L2 (ℝn) and generate an analytic semigroup {etL}t⩾0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0,∞) be of upper type 1 and of critical lower type po(ω) ∂ (n/(n+θ(L)),1] and ρ(t) = tt1/ω−1(t−1) for t ∈ (0,∞). We introduce the Orlicz-Hardy space Hω, L (ℝn) and the BMO-type space BMOρ, L (ℝn) and establish the John-Nirenberg inequality for BMOρ, L (ℝn) functions and the duality relation between Hω, L ((ℝn) and BMOρ, L* (ℝn), where L* denotes the adjoint operator of L in L2 (ℝn). Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMOρ, L* (ℝn) and the molecular characterization of Hω, L (ℝn); the latter is used to establish the boundedness of the generalized fractional operator Lργ from Hω, L (ℝn) to HL1 (ℝn) or Lq (ℝn) with certain q > 1, where HL (ℝn) is the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = tp for t ∈ (0,∞) and p ∈ (n/(n + θ(L)), 1].

Keywords

Orlicz function Orlicz-Hardy space BMO duality molecule fractional integral 

MSC(2000)

42B30 42B35 42B20 42B25 

References

  1. 1.
    Auscher P, Duong X T, McIntosh A. Boundedness of Banach space valued singular integral operators and Hardy spaces. Unpublished manuscript, 2005Google Scholar
  2. 2.
    Deng D, Duong X T, Sikora A, et al. Comparison of the classical BMO with the BMO spaces associated with operators and applications. Rev Mat Iberoamericana, 24: 267–296 (2008)MATHMathSciNetGoogle Scholar
  3. 3.
    Deng D, Duong X T, Yan L. A characterization of the Morrey-Campanato spaces. Math Z, 250: 641–655 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Duong X T, Xiao J, Yan L. Old and new Morrey spaces with heat kernel bounds. J Fourier Anal Appl, 13: 87–111 (2007)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Duong X T, Yan L. New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Comm Pure Appl Math, 58: 1375–1420 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Duong X T, Yan L. Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J Amer Math Soc, 18: 943–973 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dziubański J, Zienkiewicz J. H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloq Math, 98: 5–38 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hofmann S, Mayboroda S. Hardy and BMO spaces associated to divergence form elliptic operators. Math Ann, in pressGoogle Scholar
  9. 9.
    Yan L. Littlewood-Paley functions associated to second order elliptic operators. Math Z, 246: 655–666 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Yan L. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc, 360: 4383–4408 (2008)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Duong X T, Yan L. New Morrey-Campanato spaces associated with operators and applications. Unpublished manuscript, 2006Google Scholar
  12. 12.
    Viviani B E. An atomic decomposition of the predual of BMO(ρ). Rev Mat Iberoamericana, 3: 401–425 (1987)MATHMathSciNetGoogle Scholar
  13. 13.
    Serra C. Molecular characterization of Hardy-Orlicz spaces. Rev Un Mat Argentina, 40: 203–217 (1996)MATHMathSciNetGoogle Scholar
  14. 14.
    Coifman R R, Weiss G. Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Math, Vol. 242. Berlin: Springer, 1971MATHGoogle Scholar
  15. 15.
    Tang L. New function spaces of Morrey-Campanato type on spaces of homogeneous type. Illinois J Math, 51: 625–644 (2007)MATHMathSciNetGoogle Scholar
  16. 16.
    Harboure E, Salinas O, Viviani B. A look at BMOϕ(ω) through Carleson measures. J Fourier Anal Appl, 13: 267–284 (2007)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    McIntosh A. Operators which have an H functional calculus. Miniconference on operator theory and partial differential equations. Proc Centre Math Appl Austral Nat Univ, 14: 210–231 (1986)MathSciNetGoogle Scholar
  18. 18.
    Albrecht D, Duong X T, McIntosh A. Operator theory and harmonic analysis. Instructional Workshop on Analysis and Geometry. Proc Centre Math Appl Austral Nat Univ, 34: 77–136 (1996)MathSciNetGoogle Scholar
  19. 19.
    Ouhabaz E M. Analysis of Heat Equations on Domains. London Math Soc Mono, 31. Princeton, NJ: Princeton University Press, 2004Google Scholar
  20. 20.
    Coulhon T, Duong X T. Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss. Adv Differential Equations, 5: 343–368 (2000)MATHMathSciNetGoogle Scholar
  21. 21.
    Yosida K. Functional Analysis. 6th ed. Berlin: Spring-Verlag, 1978MATHGoogle Scholar
  22. 22.
    Blunck S, Kunstmann P C. Weak type (p, p) estimates for Riesz transforms. Math Z, 247: 137–148 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hofmann S, Martell J. L p bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ Mat, 47: 497–515 (2003)MATHMathSciNetGoogle Scholar
  24. 24.
    Auscher P, Tchamitchian P. Square root problem for divergence operators and related topics. Astérisque, 249: 1–172 (1998)Google Scholar
  25. 25.
    Davies E B. Heat Kernels and Spectral Theory. Cambridge: Cambridge University Press, 1989MATHGoogle Scholar
  26. 26.
    Yang D, Zhou Y. Some new characterizations on spaces of functions with bounded mean oscillation. Math Nachr, in pressGoogle Scholar
  27. 27.
    Duong X T, McIntosh A. Singular integral operators with non-smooth kernels on irregular domains. Rev Mat Iberoamericana, 15: 233–265 (1999)MATHMathSciNetGoogle Scholar
  28. 28.
    Coifman R R, Meyer Y, Stein E M. Some new functions and their applications to harmonic analysis. J Funct Anal, 62: 304–315 (1985)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Stein E M. Singular Integral and Differentiability Properties of Functions. Princeton, NJ: Princeton University Press, 1970Google Scholar
  30. 30.
    Varopoulos N, Saloff-Coste L, Coulhon T. Analysis and Geometry on Groups. Cambridge: Cambridge University Press, 1993Google Scholar
  31. 31.
    Stein E M. Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton University Press, 1993MATHGoogle Scholar
  32. 32.
    Taibleson M H, Weiss G. The molecular characterization of certain Hardy spaces. Astérisque, 77: 67–149 (1980)MATHMathSciNetGoogle Scholar
  33. 33.
    Hu G, Yang D, Zhou Y. Boundedness of singular integrals in Hardy space on spaces of homogenous type. Taiwanese J Math, in pressGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems, Ministry of EducationSchool of Mathematical Sciences, Beijing Normal UniversityBeijingChina

Personalised recommendations