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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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