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Sharp estimates for commutators of bilinear operators on Morrey type spaces. (English) Zbl 1445.42012
For the classical Calderon-Zygmund operator \(T\), R. R. Coifman et al. [Ann. Math. (2) 103, 611–635 (1976; Zbl 0326.32011)] showed that the commutator \[[b,T](f)=bT(f)-T(bf)\] is bounded on some \(L^p\), \(1 < p < \infty\), if and only if \(b\in BMO\). A. Uchiyama [Tohoku Math. J. (2) 30, 163–171 (1978; Zbl 0384.47023)] found that the commutator to compactness requires the symbol of commutator to be in \(CMO\), that is the closure of \(BMO\) of the space of \(C^\infty\) functions with compact suport. In recent many papers with respect to the commutators to some integral operators with multilinear setting on some integral spaces, especially, Y. Ding and T. Mei [Potential Anal. 42, No. 3, 717–748 (2015; Zbl 1321.42028)] studied the compactness of the linear commutator of bilinear operators from the product of Morrey spaces to Morrey spaces. T. Iida et al. [Positivity 16, No. 2, 339–358 (2012; Zbl 1256.42037)] introduced the multi-Morrey norms as follows: \[||(f_1,f_2)||_{{\mathcal M}^{p_0}_{\vec{P}}} :=\sup_Q|Q|^{1/p_0}\prod_{i=1}^2\Biggl{(}\frac{1}{|Q|}\int_Q|f(x)|^pdx\Biggr{)}^{1/p_i}<\infty,\] whose norm is strictly smaller than the 2-fold product of the Morrey spaces. In this paper, for \(T\) and \(I_\alpha\) the bilinear Calderón-Zygmund operators and bilinear fractional integrals, respectively, the authors prove that if \(b_1,\ b_2\in CMO\), \([\prod\vec{b},I_\alpha]\) \((\vec{b}=(b_1,b_2))\) are all compact operators from \({\mathcal M}_{\vec{P}}^{p_0}\) to \(M_q^{q_0}\) for some suitable indices \(p_0,\ p_1,\ p_2\) and \(q_0,\ q.\) Moreover, the authors show that if \(b_1=b_2\), then \(b_1,\ b_2\in CMO\) is necessary for the compactness of \([\prod\vec{b},I_\alpha]\) from \({\mathcal M}_{\vec{P}}^{p_0}\) to \(M_q^{q_0}\) for some suitable indices \(p_0,\ p_1,\ p_2\) and \(q_0,\ q\).
MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47B07 Linear operators defined by compactness properties
42B99 Harmonic analysis in several variables
47G99 Integral, integro-differential, and pseudodifferential operators
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
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