Nagoya Mathematical Journal

Pointwise multipliers for Campanato spaces on Gauss measure spaces

Liguang Liu and Dachun Yang

Full-text: Open access

Abstract

In this paper, the authors characterize pointwise multipliers for Campanato spaces on the Gauss measure space (Rn,||,γ), which includes BMO(γ) as a special case. As applications, several examples of the pointwise multipliers are given. Also, the authors give an example of a nonnegative function in BMO(γ) but not in BLO(γ).

Article information

Source
Nagoya Math. J., Volume 214 (2014), 169-193.

Dates
First available in Project Euclid: 25 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1395747184

Digital Object Identifier
doi:10.1215/00277630-2647739

Mathematical Reviews number (MathSciNet)
MR3211822

Zentralblatt MATH identifier
1293.42011

Subjects
Primary: 42B15: Multipliers
Secondary: 42B30: $H^p$-spaces 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Citation

Liu, Liguang; Yang, Dachun. Pointwise multipliers for Campanato spaces on Gauss measure spaces. Nagoya Math. J. 214 (2014), 169--193. doi:10.1215/00277630-2647739. https://projecteuclid.org/euclid.nmj/1395747184


Export citation

References

  • [1] A. Carbonaro, G. Mauceri, and S. Meda, $H^{1}$ and $\mathrm{BMO}$ for certain locally doubling metric measure spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), 543–582.
  • [2] A. Carbonaro, G. Mauceri, and S. Meda, $H^{1}$ and $\mathrm{BMO}$ for certain locally doubling metric measure spaces of finite measure, Colloq. Math. 118 (2010), 13–41.
  • [3] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
  • [4] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. (N.S.) 83 (1977), 569–645.
  • [5] L. Diening, Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), 657–700.
  • [6] E. B. Fabes, C. E. Gutiérrez, and R. Scotto, Weak-type estimates for the Riesz transforms associated with the Gaussian measure, Rev. Mat. Iberoam. 10 (1994), 229–281.
  • [7] L. Forzani and R. Scotto, The higher order Riesz transform for Gaussian measure need not be weak type $(1,1)$, Studia Math. 131 (1998), 205–214.
  • [8] J. García-Cuerva, G. Mauceri, P. Sjögren, and J. L. Torrea, Higher-order Riesz operators for the Ornstein–Uhlenbeck semigroup, Potential Anal. 10 (1999), 379–407.
  • [9] J. García-Cuerva, G. Mauceri, P. Sjögren and J. L. Torrea, Spectral multipliers for the Ornstein–Uhlenbeck semigroup, J. Anal. Math. 78 (1999), 281–305.
  • [10] C. E. Gutiérrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal. 120 (1994), 107–134.
  • [11] C. E. Gutiérrez, C. Segovia, and J. L. Torrea, On higher order Riesz transforms for Gaussian measures, J. Fourier Anal. Appl. 2 (1996), 583–596.
  • [12] E. Harboure, J. L. Torrea, and B. Viviani, Vector-valued extensions of operators related to the Ornstein–Uhlenbeck semigroup, J. Anal. Math. 91 (2003), 1–29.
  • [13] A. Hartmann, Pointwise multipliers in Hardy–Orlicz spaces, and interpolation, Math. Scand. 106 (2010), 107–140.
  • [14] S. Janson, On functions with conditions on the mean oscillation, Ark. Mat. 14 (1976), 189–196.
  • [15] L. D. Ky, New Hardy spaces of Musielak–Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory 78 (2014), 115–150.
  • [16] A. K. Lerner, Some remarks on the Hardy–Littlewood maximal function on variable $L^{p}$ spaces, Math. Z. 251 (2005), 509–521.
  • [17] H. Lin, E. Nakai, and D. Yang, Boundedness of Lusin-area and $g_{\lambda}^{\ast}$ functions on localized BMO spaces over doubling metric measure spaces, Bull. Sci. Math. 135 (2011), 59–88.
  • [18] L. Liu, Y. Sawano, and D. Yang, Morrey-type spaces on the Gauss measure spaces and boundedness of singular integrals, J. Geom. Anal., published electronically 2 October 2012. DOI 10.1007/s12220-012-9362-9.
  • [19] L. Liu and D. Yang, BLO spaces associated with the Ornstein–Uhlenbeck operator, Bull. Sci. Math. 132 (2008), 633–649.
  • [20] J. Maas, J. van Neerven, and P. Portal, Conical square functions and non-tangential maximal functions with respect to the Gaussian measure, Publ. Mat. 55 (2011), 313–341.
  • [21] J. Maas, J. Neerven and P. Portal, Whitney coverings and the tent spaces $T^{1,q}(\gamma)$ for the Gaussian measure, Ark. Mat. 50 (2012), 379–395.
  • [22] L. Maligranda and E. Nakai, Pointwise multipliers of Orlicz spaces, Arch. Math. (Basel) 95 (2010), 251–256.
  • [23] G. Mauceri and S. Meda, $\mathrm{BMO}$ and $H^{1}$ for the Ornstein–Uhlenbeck operator, J. Funct. Anal. 252 (2007), 278–313.
  • [24] G. Mauceri, S. Meda, and P. Sjögren, Endpoint estimates for first-order Riesz transforms associated to the Ornstein–Uhlenbeck operator, Rev. Mat. Iberoam. 28 (2012), 77–91.
  • [25] G. Mauceri, S. Meda and P. Sjögren, A maximal function characterization of the Hardy space for the Gauss measure, Proc. Amer. Math. Soc. 141 (2013), 1679–1692.
  • [26] T. Menárguez, S. Pérez, and F. Soria, The Mehler maximal function: A geometric proof of the weak type $1$, J. Lond. Math. Soc. (2) 61 (2000), 846–856.
  • [27] E. Nakai, Pointwise multipliers for functions of weighted bounded mean oscillation, Studia Math. 105 (1993), 105–119.
  • [28] E. Nakai, Pointwise multipliers on weighted BMO spaces, Studia Math. 125 (1997), 35–56.
  • [29] E. Nakai, A characterization of pointwise multipliers on the Morrey spaces, Sci. Math. Jpn. 3 (2000), 445–454.
  • [30] E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan 37 (1985), 207–218.
  • [31] E. Nakai and K. Yabuta, Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type, Sci. Math. Japon. 46 (1997), 15–28.
  • [32] S. Pérez, The local part and the strong type for operators related to the Gaussian measure, J. Geom. Anal. 11 (2001), 491–507.
  • [33] G. Pisier, “Riesz transforms: A simpler analytic proof of P.-A. Meyer’s inequality” in Séminaire de probabilités, XXII, Lecture Notes in Math. 1321, Springer, Berlin, 1988, 485–501.
  • [34] P. Sjögren, “On the maximal function for the Mehler kernel” in Harmonic Analysis (Cortona, 1982), Lecture Notes in Math. 992, Springer, Berlin, 1983, 73–82.
  • [35] P. Sjögren, Operators associated with the Hermite semigroup—a survey, J. Fourier Anal. Appl. 3 (1997), 813–823.
  • [36] S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Sc. Norm. Super. Pisa (3) 19 (1965), 593–608.
  • [37] D. A. Stegenga, Bounded Toeplitz operators on $H^{1}$ and applications of the duality between $H^{1}$ and the functions of bounded mean oscillation, Amer. J. Math. 98 (1976), 573–589.
  • [38] W. Urbina, On singular integrals with respect to the Gaussian measure, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 17 (1990), 531–567.
  • [39] K. Yabuta, Pointwise multipliers of weighted BMO spaces, Proc. Amer. Math. Soc. 117 (1993), 737–744.