Annals of Functional Analysis

Maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ on non-homogeneous space

Zheng Wang, Weiliang Xiao, and Taotao Zheng

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $(\mathcal{X},d,\mu)$ be a geometrically doubling metric space and assume that the measure $\mu$ satisfies the upper doubling condition. In this paper, the authors, by invoking a Cotlar type inequality, show that the maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ is bounded from $L^{p_{1}}(\mu)\times L^{p_{2}}(\mu)$ into $L^{p}(\mu)$ for any $p_{i}\in(1,\infty]$ and bounded from $L^{p_{1}}(\mu)\times L^{p_{2}}(\mu)$ into $L^{p,\infty}(\mu)$ for $p_{1}=1$ or $p_{2}=1$, where $p \in [1/2,\infty)$, $1/{p_{1}}+1/{p_{2}}=1/{p}$. Moreover, if $\vec{w}=(w_{1},w_{2})$ belongs to the weight class $A_{\vec{p}}^{\rho}(\mu)$, using the John-strömberg maximal operator and the John-strömberg sharp maximal operator, the authors obtain a weighted weak type estimate $L^{p_{1}}(w_{1}) \times L^{p_{1}}(w_{2}) \rightarrow L^{p,{\infty}}(v_{\vec{w}})$ for the maximal bilinear Calderón--Zygmund operators of type $\omega(t)$. By weakening the assumption of $\omega\in \mathrm{Dini}(1/2)$ into $\omega\in \mathrm{Dini}(1)$, the results obtained in this paper are substantial improvements and extensions of some known results, even on Euclidean spaces $\mathbb{R}^{n}$.

Article information

Ann. Funct. Anal., Volume 6, Number 4 (2015), 134-154.

First available in Project Euclid: 1 July 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Bilinear Calderón--Zygmund operator sharp maximal operator non-homogeneous space upper doubling measures weight


Zheng, Taotao; Wang, Zheng; Xiao, Weiliang. Maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ on non-homogeneous space. Ann. Funct. Anal. 6 (2015), no. 4, 134--154. doi:10.15352/afa/06-4-134.

Export citation


  • Á. Bényi, D. Maldonado, A.R. Nahmod and R.H. Torres, Bilinear paraproducts revisited, Math. Nachr. 283 (2010), no. 9, 1257–1276.
  • T.A. Bui and X.T. Duong, Hardy Spaces, regularized BMO Spaces and the boundedness of Calderón–Zygmund operators on non-homogeneous Spaces, J. Geom. Anal. 23 (2013), no. 2, 895–932.
  • J. Chen and D. Fan, Some bilinear estimates, J. Korean Math. Soc. 46 (2009), no. 3, 609–620.
  • J. Chen and G. Lu, Hömander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal. 101 (2014), 98–112.
  • L. Grafakos and R.H. Torres, Multilinear Calderón–Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164.
  • L. Grafakos and R.H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J. 51 (2002), no. 5, 1261–1276.
  • G. Hu, Y. Meng and D. Yang, Weighted norm inequalities for multilinear Calderón–Zygmund operators on non-homogeneous metric measure spaces, Forum Math. 26 (2014), no. 5, 1289–1322.
  • G. Hu and D. Yang, Weighted norm inequality for maximal singular integrals with nondoubling measure, Studia Math. 187 (2008), no. 2, 101–123.
  • T. Hytönen, A framework for non-homogeneous analysis on metric spaces, and the RBMO spaces of Tolsa, Publ. Mat. 54 (2010), no. 2, 485–504.
  • M. Lacey and J. Metcalfe, Paraproducts in one and several parameters, Forum Math. 19 (2007), no. 2, 325–351.
  • A.K. Lerner, S. Ombrosi, C. Pérez, R.H. Torres and R. Trujillo-Gonzalez, New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory, Adv. Math. 220 (2009), no. 2, 1222–1264.
  • G. Lu and P. Zhang, Multilinear Calderón–Zygmund operators with kernels of $\mathrm{Dini}'s$ type and applications, Nonlinear Anal. 107 (2014), 92–117.
  • D. Maldonado and V. Naibo, Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity, J. Fourier Anal. Appl. 15 (2009), no. 2, 218–261.
  • C. Muscalu, J. Pipher, T. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math. 193 (2004), no. 2, 269–296.
  • J. Orobitg and C. Pérez, $A_P$ weights for nondoubling measures in $\mathbb{R}^n$ and applications, Trans. Amer. Math. Soc. 354 (2002), no. 5, 2013–2033.
  • X. Tao and T. Zheng, Multilinear commutators of fractional integrals over Morrey spaces with non-doubling measures, Nonlinear Differ. Equ. Appl. 18 (2011), no. 3, 287–308.
  • X. Tolsa, BMO, $H^1$ and Calderón–Zygmund operators for non doubling measures, Math. Ann. 319 (2001), no. 1, 89–149.
  • K. Yabuta, Generalization of Calderón–Zygmund operators, Studia Math. 82 (1985), no. 1, 17–31.
  • T. Zheng, X. Tao and X. Wu, Bilinear Calderón–Zygmund operators of type $\upomega(t)$ on non-homogeneous space, J. Inequal. Appl. 2014, 2014:113.