Compactness of Commutators of Smoothing Pseudodifferential Operators and Pointwise Multiplication with Functions in the Space XMO

The space CMO is a proper subspace of BMO typically employed in harmonic analysis to prove compactness results for commutators. It has been shown, however, that there exists another space XMO , which is also a proper subspace of BMO but larger than CMO , so that the commutators of pointwise multiplications by its members with certain pseudodifferential operators of order zero (associated with singular integral operator of Calderón-Zygmund type) still produce compact operators on Lebesgue spaces. The purpose of this article is to proved the analogous result for smoothing pseudodifferential operators of finite order (associated with fractional integral operators).