Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

摘要

We prove thatbis inLipβ(ω)if and only if the commutator[b,L-α/2]of the multiplication operator byband the general fractional integral operatorL-α/2is bounded from the weighted Morrey spaceLp,k(ω)toLq,kq/p(ω1-(1-α)q,ω), where0<β<1,0<α+β<n,1<p<n/(α+β),1/q=1/p-(α+β),0≤k<p/q,ωq/p∈A1,andrω>(1-k)/(p/(q-k)), and hererωdenotes the critical index ofωfor the reverse Hölder condition.