Taiwanese Journal of Mathematics

AN HARDY ESTIMATE FOR COMMUTATORS OF PSEUDO-DIFFERENTIAL OPERATORS

Ha Duy Hung and Luong Dang Ky

Full-text: Open access

Abstract

Let $T$ be a pseudo-differential operator whose symbol belongs to the Hörmander class $S^m_{\rho,\delta}$ with $0\leq \delta\lt 1, 0\lt \rho\leq 1, \delta \leq \rho$ and $-(n+1)\lt m \leq - (n+1)(1-\rho)$. In present paper, we prove that if $b$ is a locally integrable function satisfying $$\sup_{{\rm balls}\; B\subset \mathbb R^n} \frac{\log(e+ 1/|B|)}{(1+ |B|)^\theta} \frac{1}{|B|}\int_{B} \Big|f(x)- \frac{1}{|B|}\int_{B} f(y) dy\Big|dx \lt \infty$$ for some $\theta\in [0,\infty)$, then the commutator $[b,T]$ is bounded on the local Hardy space $h^1(\mathbb R^n)$ introduced by  Goldberg [9].

As a consequence, when $\rho=1$ and $m=0$, we obtain an improvement of a recent result by Yang, Wang and Chen [21].

Article information

Source
Taiwanese J. Math., Volume 19, Number 4 (2015), 1097-1109.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133691

Digital Object Identifier
doi:10.11650/tjm.19.2015.5003

Mathematical Reviews number (MathSciNet)
MR3384681

Zentralblatt MATH identifier
1357.47051

Subjects
Primary: 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx] 42B35: Function spaces arising in harmonic analysis

Keywords
pseudo-differential operators Hardy spaces BMO spaces LMO spaces commutators

Citation

Hung, Ha Duy; Ky, Luong Dang. AN HARDY ESTIMATE FOR COMMUTATORS OF PSEUDO-DIFFERENTIAL OPERATORS. Taiwanese J. Math. 19 (2015), no. 4, 1097--1109. doi:10.11650/tjm.19.2015.5003. https://projecteuclid.org/euclid.twjm/1499133691


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References

  • J. Alvarez and J. Hounie, Estimates for the kernel and continuity properties of pseudo-differential operators, Ark. Mat., 28(1) (1990), 1-22.
  • P. Auscher and M. E. Taylor, Paradifferential operators and commutator estimates, Comm. Partial Differential Equations, 20(9-10) (1995), 1743-1775.
  • B. Bongioanni, E. Harboure and O. Salinas, commutators of Riesz transforms related to Schrödinger operators, J. Fourier Anal. Appl., 17(1) (2011), 115-134.
  • J. Cao, D.-C. Chang, D. Yang and S. Yang, Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems, Trans. Amer. Math. Soc., 365(9) (2013), 4729-4809.
  • S. Chanillo, Remarks on commutators of pseudo-differential operators, Contemp Math., 205 (1997), 33-37.
  • R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2), 103(3) (1976), 611-635.
  • G. Dafni, Local $VMO$ and weak convergence in $h^1$, Canad. Math. Bull., 45(1) (2002), 46-59.
  • X. Fu, D. Yang and W. Yuan, Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces, Taiwanese J. Math., 18(2) (2014), 509-557.
  • D. Goldberg, A local version of Hardy spaces, Duke J. Math., 46 (1979), 27-42.
  • J. Hounie and R. A. S. Kapp, Pseudodifferential operators on local Hardy spaces, J. Fourier Anal. Appl., 15(2) (2009), 153-178.
  • G. Hu, H. Lin and D. Yang, Commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type, Abstr. Appl. Anal., (2008), Art. ID 237937, 21 pp.
  • J.-L. Journé, Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón, Lecture Notes in Mathematics, 994. Springer-Verlag, Berlin, 1983.
  • L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc., 365(6) (2013), 2931-2958.
  • L. D. Ky, Endpoint estimates for commutators of singular integrals related to Schrödinger operators, arXiv: 1203.6335.
  • H. Lin, Y. Meng and D. Yang, Weighted estimates for commutators of multilinear Calderón-Zygmund operators with non-doubling measures, Acta Math. Sci. Ser. B Engl. Ed., 30(1) (2010), 1-18.
  • Y. Lin, Commutators of pseudo-differential operators, Sci. China Ser. A, 51(3) (2008), 453-460.
  • T. Ma, P. R. Stinga, J. L. Torrea and C. Zhang, Regularity estimates in Hölder spaces for Schrödinger operators via a $T1$ theorem, Ann. Mat. Pura Appl. (4), 193(2) (2014), 561-589.
  • C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128 (1995), 163-185.
  • L. Tang, Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators, J. Funct. Anal., 262(4) (2012), 1603-1629.
  • M. E. Taylor, Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, 100. Birkhäuser Boston, Inc., Boston, MA, 1991.
  • J. Yang, Y. Wang and W. Chen, Endpoint estimates for the commutator of pseudo-differential operators, Acta Math. Sci. Ser. B Engl. Ed., 34(2) (2014), 387-393.
  • D. Yang and S. Yang, Weighted local Orlicz Hardy spaces with applications to pseudo-differential operators, Dissertationes Math. (Rozprawy Mat.), 478 (2011), 78 pp.
  • D. Yang and S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications, Sci. China Math., 55(8) (2012), 1677-1720.
  • D. Yang and Y. Zhou, Localized Hardy spaces $H^1$ related to admissible functions on RD-spaces and applications to Schrödinger operators, Trans. Amer. Math. Soc., 363(3) (2011), 1197-1239.