The Boundedness for Commutator of Fractional Integral Operator With Rough Variable Kernel

For b is an element of BMO(R-n) and 0 < alpha <= 1/2, the commutator of the fractional integral operator T-Omega,T-alpha with rough variable kernel is defined by [b, T-Omega,T-alpha] f(x) = integral(Rn) Omega(x, x - y)/vertical bar x - y vertical bar(n-alpha) (b(x) - b(y)) f(y)dy. In this paper the authors prove that the commutator [b, T-Omega,T-alpha] is a bounded operator from L2n/n+2 alpha (R-n) to L-2(R-n). The result obtained in this paper is substantial improvement and extension of some known results.