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Morrey-Type Spaces on Gauss Measure Spaces and Boundedness of Singular Integrals

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Abstract

In this paper, the authors introduce Morrey-type spaces on the locally doubling metric measure spaces, which means that the underlying measure enjoys the doubling and the reverse doubling properties only on a class of admissible balls, and then obtain the boundedness of the local Hardy–Littlewood maximal operator and the local fractional integral operator on such Morrey-type spaces. These Morrey-type spaces on the Gauss measure space are further proved to be naturally adapted to singular integrals associated with the Ornstein–Uhlenbeck operator. To be precise, by means of the locally doubling property and the geometric properties of the Gauss measure, the authors establish the equivalence between Morrey-type spaces and Campanato-type spaces on the Gauss measure space, and the boundedness for a class of singular integrals associated with the Ornstein–Uhlenbeck operator (including Riesz transforms of any order) on Morrey-type spaces over the Gauss measure space.

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References

  1. Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. 50, 201–230 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Adams, D.R., Xiao, J.: Regularity of Morrey commutators. Trans. Am. Math. Soc. 364, 4801–4818 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Adams, D.R., Xiao, J.: Morrey potentials and harmonic maps. Commun. Math. Phys. 308, 439–456 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Campanato, S.: Proprietá di inclusione per spazi di Morrey. Ric. Mat. 12, 67–86 (1963)

    MATH  MathSciNet  Google Scholar 

  6. Carbonaro, A., Mauceri, G., Meda, S.: H 1 and BMO for certain locally doubling metric measure spaces of finite measure. Colloq. Math. 118, 13–41 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Carbonaro, A., Mauceri, G., Meda, S.: H 1 and BMO for certain nondoubling metric measure spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8, 543–582 (2009)

    MATH  MathSciNet  Google Scholar 

  8. Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. (7) 7, 273–279 (1987). 1988

    MATH  MathSciNet  Google Scholar 

  9. Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Math., vol. 242. Springer, Berlin (1971)

    MATH  Google Scholar 

  10. Coifman, R.R., Weiss, G.: Extensions of hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  11. Duong, X.T., Xiao, J., Yan, L.X.: Old and new Morrey spaces with heat kernel bounds. J. Fourier Anal. Appl. 13, 87–111 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fabes, E.B., Gutiérrez, C., Scotto, R.: Weak type estimates for the Riesz transforms associated with the Gaussian measure. Rev. Mat. Iberoam. 10, 229–281 (1994)

    Article  MATH  Google Scholar 

  13. Forzani, L., Scotto, R.: The higher order Riesz transforms for Gaussian measure need not be weak type (1,1). Stud. Math. 131, 205–214 (1998)

    MATH  MathSciNet  Google Scholar 

  14. Gutiérrez, C.E.: On the Riesz transforms for Gaussian measures. J. Funct. Anal. 120, 107–134 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gutiérrez, C.E., Segovia, C., Torrea, J.L.: On higher order Riesz transforms for Gaussian measures. J. Fourier Anal. Appl. 2, 583–596 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. García-Cuerva, J., Mauceri, G., Sjögren, P., Torrea, J.L.: Higher-order Riesz operators for the Ornstein-Uhlenbeck semigroup. Potential Anal. 10, 379–407 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. García-Cuerva, J., Mauceri, G., Sjögren, P., Torrea, J.L.: Spectral multipliers for the Ornstein-Uhlenbeck semigroup. J. Anal. Math. 78, 281–305 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Harboure, E., Torrea, J.L., Viviani, B.: Vector-valued extensions of operators related to the Ornstein–Uhlenbeck semigroup. J. Anal. Math. 91, 1–29 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hedberg, L.I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36, 505–510 (1972)

    Article  MathSciNet  Google Scholar 

  20. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)

    Book  MATH  Google Scholar 

  21. Liu, L., Yang, D.: BLO spaces associated with the Ornstein-Uhlenbeck operator. Bull. Sci. Math. 132, 633–649 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu, L., Yang, D.: Characterizations of BMO associated with gauss measures via commutators of local fractional integrals. Isr. J. Math. 180, 285–315 (2010)

    Article  MATH  Google Scholar 

  23. Liu, L., Yang, D.: Boundedness of Littlewood-Paley operators associated with gauss measures. J. Inequal. Appl. 643948, 41 pp. (2010)

  24. Maas, J., Neerven, J., Portal, P.: Whitney coverings and the tent spaces T 1,q(γ) for the Gaussian measure. Ark. Mat. 50, 379–395 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Mauceri, G., Meda, S.: BMO and H 1 for the Ornstein-Uhlenbeck operator. J. Funct. Anal. 252, 278–313 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mauceri, G., Meda, S., Sjögren, P.: Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator. Rev. Mat. Iberoam. 28, 77–91 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mauceri, G., Meda, S., Sjögren, P.: A maximal function characterisation of the hardy space for the gauss measure. Proc. Am. Math. Soc. (to appear). arXiv:1006.5551

  28. Menárguez, T., Pérez, S., Soria, F.: The Mehler maximal function: a geometric proof of the weak type 1. J. Lond. Math. Soc. 61, 846–856 (2000)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Nazarov, F., Treil, S., Volberg, A.: The Tb-theorem on non-homogeneous spaces. Acta Math. 190, 151–239 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Pisier, G.: Riesz transforms: a simpler analytic proof of P.-A. Meyer’s inequality. In: Lecture Notes in Math., vol. 1321, pp. 485–501 (1988)

    Google Scholar 

  32. Peetre, J.: On the theory of \(\mathcal{L}_{p,\lambda}\) spaces. J. Funct. Anal. 4, 71–87 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pérez, S.: The local part and the strong type for operators related to the Gaussian measure. J. Geom. Anal. 11, 491–507 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  34. Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math. Sin. (Engl. Ser.) 21, 1535–1544 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Sawano, Y., Tanaka, H.: Equivalent norms for the Morrey spaces with non-doubling measures. Far East J. Math. Sci. 22, 387–404 (2006)

    MATH  MathSciNet  Google Scholar 

  36. Sawano, Y., Tanaka, H.: Predual spaces of Morrey spaces with non-doubling measures. Tokyo J. Math. 32, 471–486 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  37. Sjögren, P.: On the maximal function for the Mehler kernel. In: Lecture Notes in Math., vol. 992, pp. 73–82 (1983)

    Google Scholar 

  38. Sjögren, P.: Operators associated with the Hermite semigroup a survey. J. Fourier Anal. Appl. 3, 813–823 (1997). Special Issue

    Article  MATH  MathSciNet  Google Scholar 

  39. Urbina, W.: On singular integrals with respect to the Gaussian measure. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 18, 531–567 (1990)

    MathSciNet  Google Scholar 

  40. Tolsa, X.: BMO, H 1, and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319, 89–149 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, School of Information, Renmin University of China, Beijing, 100872, People’s Republic of China

    Liguang Liu

  2. Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Yoshihiro Sawano

  3. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Dachun Yang

Authors
  1. Liguang Liu
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  2. Yoshihiro Sawano
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  3. Dachun Yang
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Corresponding author

Correspondence to Dachun Yang.

Additional information

Communicated by Loukas Grafakos.

The first author is supported by National Natural Science Foundation of China (Grant No. 11101425). The second author is supported by Grant-in-Aid for Young Scientists (B) No. 21740104, Japan Society for the Promotion of Science. The third author is supported by National Natural Science Foundation (Grant No. 11171027) of China and Program for Changjiang Scholars and Innovative Research Team in University of China.

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Liu, L., Sawano, Y. & Yang, D. Morrey-Type Spaces on Gauss Measure Spaces and Boundedness of Singular Integrals. J Geom Anal 24, 1007–1051 (2014). https://doi.org/10.1007/s12220-012-9362-9

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  • DOI: https://doi.org/10.1007/s12220-012-9362-9

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