# Sharp estimates for commutators of bilinear operators on Morrey type spaces

## TL;DR

This paper establishes sharp estimates for the compactness of commutators of bilinear Calderón-Zygmund operators and bilinear fractional integrals on Morrey-type spaces, highlighting the role of CMO functions.

## Contribution

It proves the compactness of certain bilinear operator commutators on Morrey spaces and characterizes CMO functions as necessary for compactness when the functions are equal.

## Key findings

- Compactness of commutators on Morrey spaces is established.
- Characterization of CMO functions as necessary for compactness.
- Sharp estimates for bilinear operators on Morrey-type spaces.

## Abstract

Denote by $T$ and $I_{\alpha}$ the bilinear Calder\'{o}n-Zygmund operators and bilinear fractional integrals, respectively. In this paper, it is proved that if $b_{1},b_{2}\in {\rm CMO}$ (the {\rm BMO}-closure of $C^{\infty}_{c}(\mathbb{R}^n)$), $[\Pi \vec{b},T]$ and $[\Pi\vec{b},I_{\alpha}]$ $(\vec{b}=(b_{1},b_{2}))$ are all the compact operators from $\mathcal{M}^{p_{0}}_{\vec{P}}$ (the norm of $\mathcal{M}^{p_{0}}_{\vec{P}}$ is strictly smaller than $2-$fold product of the Morrey norms) to $M^{q_{0}}_{q}$ 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 $b_{1}, b_{2}\in {\rm CMO}$ is necessary for the compactness of $[\Pi\vec{b},I_{\alpha}]$ on Morrey space.