Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces

Abstract

We prove that $b$ is in $Lip_{\bz}(\bz)$ if and only if the commutator $[b,L^{-\alpha /2}]$ of the multiplication operator by $b$ and the general fractional integral operator $L^{-\alpha /2}$ is bounded from the weighed Morrey space $L^{p,k}(\omega )$ to $L^{q,kq/p}({\omega }^{1-(1-\alpha /n)q},\omega )$, where $0<\beta <1$, $0<\alpha +\beta <n,1<p<n/(\alpha +\beta )$, $1/q=1/p-(\alpha +\beta )/n,$ $0\le k<p/q,$ ${\omega }^{q/p}\in A_1$ and $r_{\omega }>\frac{1-k}{p/q-k},$ and here $r_{\omega }$ denotes the critical index of $\omega$ for the reverse Hölder condition.

12 pages; Classical Analysis and ODEs (math.CA), Functional Analysis (math.FA)