Taiwanese Journal of Mathematics

COMPACTNESS OF THE COMMUTATOR OF BILINEAR FOURIER MULTIPLIER OPERATOR

Guoen Hu

Full-text: Open access

Abstract

Let $b_1, b_2 \in {\rm CMO}(\mathbb{R}^n)$ and $T_{\sigma}$ be the bilinear Fourier multiplier operator with associated multiplier $\sigma$ satisfies the Sobolev regularity that $\sup_{\kappa\in \mathbb{Z}}\|\sigma_{\kappa}\|_{W^{s_1,\,s_2}(\mathbb{R}^{2n})}\lt \infty$ for some $s_1,\,s_2\in (n/2,\,n]$. In this paper, it is proved that the commutator defined by $$T_{\sigma,\,\vec{b}}(f_1,\,f_2)(x) = b_1(x) T_{\sigma}(f_1,\,f_2)(x) - T_{\sigma}(b_1 f_1,\,f_2)(x) + b_2(x) T_{\sigma}(f_1,\,f_2)(x) - T_{\sigma}(f_1,\,b_2 f_2)(x)$$ is a compact operator from $L^{p_1}(\mathbb{R}^n) \times L^{p_2}(\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ when $p_k \in (n/s_k,\,\infty)$ $(k=1,\,2)$, $p \in (1,\,\infty)$ with $1/p = 1/p_1 + 1/p_2$.

Article information

Source
Taiwanese J. Math., Volume 18, Number 2 (2014), 661-675.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706407

Digital Object Identifier
doi:10.11650/tjm.18.2014.3676

Mathematical Reviews number (MathSciNet)
MR3188524

Zentralblatt MATH identifier
1357.42005

Subjects
Primary: 42B15: Multipliers 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Keywords
bilinear Fourier multiplier commutator CMO$(\mathbb{R}^n)$ compact operator

Citation

Hu, Guoen. COMPACTNESS OF THE COMMUTATOR OF BILINEAR FOURIER MULTIPLIER OPERATOR. Taiwanese J. Math. 18 (2014), no. 2, 661--675. doi:10.11650/tjm.18.2014.3676. https://projecteuclid.org/euclid.twjm/1499706407


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References

  • A. Bényi and R. H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc., 141 (2013), 3609-3621.
  • G. Bourdaud, M. Lanze de Cristoforis and W. Sickel, Functional calculus on BMO and related spaces, J. Func. Anal., 189 (2002), 515-538.
  • A. T. Bui and X. T. Duong, Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers, Bull. Sci. Math., 137 (2013), 63-75.
  • M. Carrozza and A. Passarelli Di Napoli, Composition of maximal operators, Publ. Mat., 40 (1996), 397-409.
  • R. R. Coifman and Y. Meyer, Nonlinear Harmonic Analysis, Pperator Theory and PDE, Beijing Lectures in Harmonic Analysis, Beijing, 1984, pp. 3-45; Ann. of Math. Stud. 112, Princeton Univ. Press, Princeton, NJ, 1986.
  • R. Coifman and G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645.
  • M. Fujita and N. Tomita, Weighted norm inequalities for multilinear Fourier multipliers, Trans. Amer. Math. Soc., 364 (2012), 6335-6353.
  • L. Grafakos, A. Miyachi and N. Tomita, On multilinear Fourier multipliers of limited smoothness, Canad. J. Math., 65 (2013), 299-330.
  • L. Grafakos and Z. Si, The Hörmander multiplier theorem for multilinear operators, J. Reine. Angew. Math., 668 (2012), 133-147.
  • L. Grafakos and R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math., 165 (2002), 124-164.
  • L. Grafakos and R. H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J., 51 (2002), 1261-1276.
  • G. Hu and C. Lin, Weighted norm inequalities for multilinear singular integral operators and applications, Anal. Appl., to appear, arXiv:1208.6346.
  • G. Hu and W. Yi, Estimates for the commutators of bilinear Fourier multiplier, Czech. Math. J., to appear.
  • C. Kenig and E. M. Stein, Multilinear estimates and fractional integral, Math. Res. Lett., 6 (1999), 1-15.
  • A. Miyachi and N. Tomita, Minimal smoothness conditions for bilinear Fourier multiplier, Rev. Mat. Iberoamericana, 29 (2013), 495-530.
  • N. Tomita, A Hörmander type multiplier theorem for multilinear operator, J. Funct. Anal., 259 (2010), 2028-2044.
  • A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J., 30 (1978), 163-171.
  • K. Yosida, Function Analysis, Springer-Verlag, Berlin, 1995.