Boundedness and compactness of commutators associated with Lipschitz functions

Abstract

Let $\alpha \in (0,1]$, $\beta \in [0,n)$ and $T_{\Omega ,\beta }$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\mathrm{CMO}}_{\alpha }({\mathbb{R}}^n)$ is introduced and studied. In particular, the relationship between ${\mathrm{CMO}}_{\alpha }({\mathbb{R}}^n)$ and the Lipchitz space $Li{p}_{\alpha }({\mathbb{R}}^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{\Omega ,\beta }{)}_b^m$ on weighted Lebesgue spaces via functions in $Li{p}_{\alpha }({\mathbb{R}}^n)$, and an equivalent characterization of the compactness for $(T_{\Omega ,\beta }{)}_b^m$ via functions in ${\mathrm{CMO}}_{\alpha }({\mathbb{R}}^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators.

arXiv admin note: text overlap with arXiv:1712.08292