Sharp estimates for commutators of bilinear operators on Morrey type spaces

Abstract

Denote by $T$ and $I_{\alpha }$ the bilinear Calderón-Zygmund operators and bilinear fractional integrals, respectively. In this paper, it is proved that if $b_1,b_2\in \mathrm{CMO}$ (the {\rm BMO}-closure of $C_c^{\infty }({\mathbb{R}}^n)$), $[\Pi \vec{b},T]$ and $[\Pi \vec{b},I_{\alpha }]$ $(\vec{b}=(b_1,b_2))$ are all the compact operators from ${\mathcal{M}}_{\vec{P}}^{p_0}$ (the norm of ${\mathcal{M}}_{\vec{P}}^{p_0}$ is strictly smaller than $2-$fold product of the Morrey norms) to $M_q^{q_0}$ for some suitable indexes $p_0,p_1,p_2$ and $q_0,q$. Specially, we also show that if $b_1=b_2$, then $b_1,b_2\in \mathrm{CMO}$ is necessary for the compactness of $[\Pi \vec{b},I_{\alpha }]$ on Morrey space.

27 pages. arXiv admin note: text overlap with arXiv:1612.01116

References in corpus (1)

• Characterization of CMO via compactness of the commutators of bilinear fractional integral operators