Weighted compact commutator of bilinear Fourier multiplier operator

Abstract

Let T _σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that $\mathop {\sup }\limits_{k \in \mathbb{Z}} {\left\| {{\sigma _k}} \right\|_{{W^s}\left( {{\mathbb{R}^{2n}}} \right)}} < \infty $ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T _σ and CMO(ℝ ⁿ) functions is a compact operator from ${L^{{p_1}}}\left( {{\mathbb{R}^n},{\omega _1}} \right) \times {L^{{p_2}}}\left( {{\mathbb{R}^n},{\omega _2}} \right)$ to ${L^p}\left( {{\mathbb{R}^n},{\nu _{\vec \omega }}} \right)$ for appropriate indices p ₁, p ₂, p ∈ (1,∞) with $\frac{1}{p} = \frac{1}{{{p_1}}} + \frac{1}{{{p_2}}}$ and weights ω ₁,ω ₂ such that $\vec \omega = \left( {{\omega _1},{\omega _2}} \right) \in {A_{\vec p/\vec t}}\left( {{\mathbb{R}^{2n}}} \right)$.

Keywords

Bilinear Fourier multiplier / Commutator / Bi(sub)linear maximal operator / Compact operator

References

[1] Bényi A., Torres R. H.. Compact bilinear operators and commutators. Proc. Amer. Math. Soc., 2013, 141: 3609-3621

[2] Benyi A., Damián W., Moen K., Torres R.. Compactness of bilinear commutator: The weighted case. Michigan Math. J., 2015, 64: 39-51

[3] Bourdaud G., Lanze de Cristoforis M., Sickel W.. Functional calculus on BMO and related spaces. J. Func. Anal., 2002, 189: 515-538

[4] Bui A. T., Duong X. T.. Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers. Bull. Sci. Math., 2013, 137: 63-75

[5] Clop A., Cruz V.. Weighted estimates for Beltrami equations. Ann. Acad. Sci. Fenn. Math., 2013, 38: 91-113

[6] Coifman R. R., Meyer Y.. Au delà des opérateurs pseudo-différentiels. Astériaque, 1978, 57: 1-185

[7] Coifman R. R., Weiss G.. Extension of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 1977, 83: 569-645

[8] Fujita M., Tomita N.. Weighted norm inequalities for multilinear Fourier multipliers. Trans. Amer. Math. Soc., 2012, 364: 6335-6353

[9] Grafakos L., Liu L., Yang D.. Multilple weighted norm inequalities for maximal singular integrals with non-smooth kernels. Proc. Royal Soc. Edinb. Ser. A, 2011, 141: 755-775

[10] Grafakos L., Miyachi A., Tomita N.. On multilinear Fourier multipliers of limited smoothness. Canad. J. Math., 2013, 65: 299-330

[11] Grafakos L., Si Z.. The Hörmander multiplier theorem for multilinear operators. J. Reine. Angew. Math., 2012, 668: 133-147

[12] Grafakos L., Torres R. H.. Multilinear Calder´on-Zygmund theory. Adv. Math., 2002, 165: 124-164

[13] Grafakos L., Torres R. H.. Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J., 2002, 51: 1261-1276

[14] Hu G.. The compactness of the commutator of bilinear Fourier multiplier operator. Taiwanese J. Math., 2014, 18: 661-675

[15] Hu G., Lin C.. Weighted norm inequalities for multilinear singular integral operators and applications. Anal. Appl., 2014, 12: 269-291

[16] Hu G., Yi W.. Estimates for the commutators of bilinear Fourier multiplier. Czech. Math. J., 2013, 63: 1113-1134

[17] Jiao Y.. A weighted norm inequality for the bilinear Fourier multiplier operator. Math. Inequal. Appl., 2014, 17: 899-912

[18] Kenig C., Stein E. M.. Multilinear estimates and fractional integral. Math. Res. Lett., 1999, 6: 1-15

[19] Lerner A., Ombrossi S., Pérez C. New maximal functions and multiple weights for the multilinear Calder´on-Zygmund theorey. Adv. Math., 2009, 220: 1222-1264

[20] Miyachi A., Tomita N.. Minimal smoothness conditions for bilinear Fourier multiplier. Rev. Mat. Iberoamericana, 2013, 29: 495-530

[21] Tomita N.. A Hörmander type multiplier theorem for multilinear operator. J. Funct. Anal., 2010, 259: 2028-2044

[22] Zhou J., Li P.. Compactness of the commutator of multilinear Fourier multiplier operator on weighted Lebesgue space. J. Funct. Spaces, 2014