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Characterization of compactness of commutators of bilinear singular integral operators. (English) Zbl 1410.42017

This work extends to bilinear Riesz transforms a commutator characterization of BMO and CMO obtained in the work of R. R. Coifman et al. [Ann. Math. (2) 103, 611–635 (1976; Zbl 0326.32011)] showing that commutators of Riesz transforms and BMO functions are bounded on \(L^p\)-spaces, and later of S. Janson [Ark. Mat. 16, 263–270 (1978; Zbl 0404.42013)] and of A. Uchiyama [Tohoku Math. J. (2) 30, 163–171 (1978; Zbl 0384.47023)] showing that boundedness of such commutators characterizes membership in CMO, the space of BMO limits of smooth, compactly supported functions.
One defines bilinear Riesz transforms \(R^k_u\) by \[ R^k_y (f,g)(x)=p.v.\int\int_{\mathbb{R}^{2n}}\frac{ x_k-y_k}{(|x-y|^2+|x-z|^2)^{n+1/2}}\,f(y) g(z) dy\, dz \] and \(R^k_z (f,g)(x)\) is defined similarly with the roles of \(y\) and \(z\) interchanged. One defines commutators \([b, R^k_y]_1=R^k_y(bf,g)-b R^k_y(f,g)\), \([b, R^k_y]_2=R^k_y(f,bg)-b R^k_y(f,g)\), and corresponding commutators with \(y\) and \(z\) interchanged. The main theorem states that if \(1 < p_1,p_2<\infty\) where \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}<2\) then each of the commutators \([b, R^k_u]_i\) (\(k=1,\dots, n\), \(u=y\) or \(z\), and \(i=1\) or \(2\)) is bounded from \(L^{p_1}\times L^{p_2}\to L^p\) if and only if \(b\in \text{CMO}\).
The sufficiency of \(b\in \text{CMO}\) for boundedness was established by Á. Bényi and R. H. Torres [Proc. Am. Math. Soc. 141, No. 10, 3609–3621 (2013; Zbl 1281.42010)] and R. H. Torres et al. [J. Anal. 26, No. 2, 227–234 (2018; Zbl 1402.42021)]. Thus, the result boils down to the necessity of \(b\in \text{CMO}\).
The key steps in the proof amount to a reduction to the approach used by Uchiyama in the linear case. Compactness of the commutator requires \(b\in \text{BMO}\). The subspace CMO can be characterized by three extra conditions, failure of any of which leads to a contradiction with the hypothesis that the bilinear commutator is compact from \(L^{p_1}\times L^{p_2}\to L^p\). These conditions are the vanishing mean condition (\(\lim_{a\to 0}\sup_{|Q|=a} \frac{1}{|Q|}\int_Q |b-b_Q|=0\)), a corresponding condition as \(a\to\infty\), and the condition that for any cube \(Q\), the mean oscillation of \(b\) over \(Q+y\) tends to zero as \(|y|\to\infty\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47B07 Linear operators defined by compactness properties
42B35 Function spaces arising in harmonic analysis
47G99 Integral, integro-differential, and pseudodifferential operators
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References:

[1] B\'enyi, \'Arp\'ad; Torres, Rodolfo H., Compact bilinear operators and commutators, Proc. Amer. Math. Soc., 141, 10, 3609-3621 (2013) · Zbl 1281.42010
[2] Chaffee, Lucas, Characterizations of bounded mean oscillation through commutators of bilinear singular integral operators, Proc. Roy. Soc. Edinburgh Sect. A, 146, 6, 1159-1166 (2016) · Zbl 1362.42024
[3] Chaffee, Lucas; Torres, Rodolfo H., Characterization of compactness of the commutators of bilinear fractional integral operators, Potential Anal., 43, 3, 481-494 (2015) · Zbl 1337.42010
[4] Chen, Yanping; Ding, Yong; Wang, Xinxia, Compactness of commutators of Riesz potential on Morrey spaces, Potential Anal., 30, 4, 301-313 (2009) · Zbl 1163.42004
[5] Coifman, R. R.; Rochberg, R.; Weiss, Guido, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2), 103, 3, 611-635 (1976) · Zbl 0326.32011
[6] Grafakos, Loukas; Torres, Rodolfo H., Multilinear Calder\'on-Zygmund theory, Adv. Math., 165, 1, 124-164 (2002) · Zbl 1032.42020
[7] Janson, Svante, Mean oscillation and commutators of singular integral operators, Ark. Mat., 16, 2, 263-270 (1978) · Zbl 0404.42013
[8] Lerner, Andrei K.; Ombrosi, Sheldy; P\'erez, Carlos; Torres, Rodolfo H.; Trujillo-Gonz\'alez, Rodrigo, New maximal functions and multiple weights for the multilinear Calder\'on-Zygmund theory, Adv. Math., 220, 4, 1222-1264 (2009) · Zbl 1160.42009
[9] Li, Ji; Wick, Brett D., Weak factorizations of the Hardy space \(H^1(\mathbb{R}^n)\) in terms of multilinear Riesz transforms, Canad. Math. Bull., 60, 3, 571-585 (2017) · Zbl 1372.42018
[10] P\'erez, Carlos; Torres, Rodolfo H., Sharp maximal function estimates for multilinear singular integrals. Harmonic analysis at Mount Holyoke, South Hadley, MA, 2001, Contemp. Math. 320, 323-331 (2003), Amer. Math. Soc., Providence, RI · Zbl 1045.42011
[11] Stein, Elias M.; Weiss, Guido, Introduction to Fourier analysis on Euclidean spaces, x+297 pp. (1971), Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J. · Zbl 0232.42007
[12] Tang, Lin, Weighted estimates for vector-valued commutators of multilinear operators, Proc. Roy. Soc. Edinburgh Sect. A, 138, 4, 897-922 (2008) · Zbl 1152.42306
[13] TX Rodolfo H. Torres, Qingying Xue, and Jingquan Wang, Compact bilinear commutators: the quasi-Banach space case, J. Anal., to appear. · Zbl 1402.42021
[14] Uchiyama, Akihito, On the compactness of operators of Hankel type, T\^ohoku Math. J. (2), 30, 1, 163-171 (1978) · Zbl 0384.47023
[15] WZT Dinghuai Wang, Jiang Zhou, and Zhidong Teng, Characterizations of weighted \(BMO\) space and its application, arXiv:1707.01639. · Zbl 1489.47064
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