Illinois Journal of Mathematics

Bi-parameter Littlewood–Paley operators with upper doubling measures

Mingming Cao and Qingying Xue

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Abstract

Let $\mu=\mu_{n_{1}}\times\mu_{n_{2}}$, where $\mu_{n_{1}}$ and $\mu_{n_{2}}$ are upper doubling measures on $\mathbb{R}^{n_{1}}$ and $\mathbb{R}^{n_{2}}$, respectively. Let the pseudo-accretive function $b=b_{1}\otimes b_{2}$ satisfy a bi-parameter Carleson condition. In this paper, we established the $L^{2}(\mu)$ boundedness of non-homogeneous Littlewood–Paley $g_{\lambda}^{*}$-function with non-convolution type kernels on product spaces. This was mainly done by means of dyadic analysis and non-homogenous methods. The result is new even in the setting of Lebesgue measures.

Article information

Source
Illinois J. Math., Volume 61, Number 1-2 (2017), 53-79.

Dates
Received: 10 February 2017
Revised: 16 August 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1520046209

Digital Object Identifier
doi:10.1215/ijm/1520046209

Mathematical Reviews number (MathSciNet)
MR3770836

Zentralblatt MATH identifier
1388.42055

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Citation

Cao, Mingming; Xue, Qingying. Bi-parameter Littlewood–Paley operators with upper doubling measures. Illinois J. Math. 61 (2017), no. 1-2, 53--79. doi:10.1215/ijm/1520046209. https://projecteuclid.org/euclid.ijm/1520046209


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References

  • T. A. Bui and X. T. Duong, Hardy spaces, regularized $\mathit{BMO}$ spaces and the boundedness of Calderón–Zygmund operators on non-homogeneous spaces, J. Geom. Anal. 23 (2013), no. 2, 895–932.
  • T. A. Bui and M. Hormozi, Weighted bounds for multilinear quare functions, Potential Anal. 46 (2017), 135–148.
  • M. Cao, K. Li and Q. Xue, A characterization of two weight norm inequality for Littlewood–Paley $g_{\lambda}^*$-function, to appear in J. Geom. Anal. (2018).
  • M. Cao and Q. Xue, On the boundedness of bi-parameter Littlewood–Paley $g_{\lambda}^*$-function, J. Funct. Spaces 2016 (2016), Art. ID \bnumber3605639, 15 pp.
  • M. Cao and Q. Xue, A non-homogeneous local $\mathit{Tb}$ theorem for Littlewood–Paley $g_{\lambda}^{*}$-function with $L^p$-testing condition, to appear in Forum Math. (2018).
  • M. Cao and Q. Xue, $L^p$ boundedness of non-homogeneous Littlewood–Paley $g_{\lambda,\mu}^{*}$-function with non-doubling measures; available at \arxivurlarxiv:1605.04649.
  • X. Chen, Q. Xue and K. Yabuta, On multilinear Littlewood–Paley operators, Nonlinear Anal. 115 (2015), 25–40.
  • M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601–628.
  • Y. Ding, S. Lu and Q. Xue, Parametrized Littlewood–Paley operators on Hardy and weak Hardy spaces, Math. Nachr. 280 (2007), 351–363.
  • C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36.
  • Y. Han, M. Lee and C. Lin, Tb theorem on product spaces, Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 1, 117–141.
  • T. Hytönen, A framework for non-homogeneous analysis on metric spaces, and the $\mathit{RBMO}$ space of Tolsa, Publ. Mat. 54 (2010), no. 2, 485–504.
  • T. Hytönen, The vector-valued non-homogeneous $\mathit{Tb}$ theorem, Int. Math. Res. Not. IMRN 2 (2014), 451–511.
  • T. Hytönen and H. Martikainen, Non-homogeneous $T1$ theorem for bi-parameter singular integrals, Adv. Math. 261 (2014), 220–273.
  • A. K. Lerner, On some sharp weighted norm inequalities, J. Funct. Anal. 232 (2006), 477–494.
  • A. K. Lerner, On some weighted norm inequalities for Littlewood–Paley operators, Illinois J. Math. 52 (2008), no. 2, 653–666.
  • A. K. Lerner, On sharp aperture-weighted estimates for square functions, J. Fourier Anal. Appl. 20 (2014), no. 4, 784–800.
  • H. Martikainen, Boundedness of a class of bi-parameter square function in the upper half-space, J. Funct. Anal. 267 (2014), 3580–3597.
  • H. Martikainen and M. Mourgoglou, Square functions with general measures, Proc. Amer. Math. Soc. 142 (2014), no. 11, 3923–3931.
  • B. Muckenhoupt and R. L. Wheeden, Norm inequalities for the Littlewood–Paley function $g_\lambda^*$, Trans. Amer. Math. Soc. 191 (1974), 95–111.
  • F. Nazarov, S. Treil and A. Volberg, Cauchy integral and Calderón–Zygmund operators on nonhomogeneous spaces, Int. Math. Res. Not. IMRN 15 (1997), 703–726.
  • F. Nazarov, S. Treil and A. Volberg, Accretive system $\mathit{Tb}$-theorems on nonhomogeneous spaces, Duke Math. J. 113 (2002), no. 2, 259–312.
  • F. Nazarov, S. Treil and A. Volberg, The $\mathit{Tb}$-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151–239.
  • Y. Ou, A $T(b)$ theorem on product spaces, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6159–6197.
  • S. Shi, Q. Xue and K. Yabuta, On the boundedness of multilinear Littlewood–Paley $g_{\lambda}^*$ function, J. Math. Pures Appl. 101 (2014), 394–413.
  • E. M. Stein, On some function of Littlewood–Paley and Zygmund, Bull. Amer. Math. Soc. 67 (1961), 99–101.
  • X. Tolsa, $L^2$-boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), 269–304.
  • X. Tolsa, $\mathit{BMO}$, $H^1$, and Calderón–Zygmund operators for non doubling measures, Math. Ann. 319 (2001), 89–149.
  • X. Tolsa, Littlewood–Paley theory and the $T(1)$ theorem with non-doubling measures, Adv. Math. 164 (2001), 57–116.
  • Q. Xue and Y. Ding, Weighted estimates for the multilinear commutators of the Littlewood–Paley operators, Sci. China Ser. A 52 (2009), 1849–1868.
  • Q. Xue and J. Yan, On multilinear square function and its applications to multilinear Littlewood–Paley operators with non-convolution type kernels, J. Math. Anal. Appl. 422 (2015), 1342–1362.