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

We prove that b is in Lip(beta)(omega) 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 weighted Morrey space L-p,L-k(omega) to L-q,L- 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 <= k < p/q, omega(q/p) is an element of A(1), and r(omega) > (1 - k)/(p/(q - k)), and here r(omega) denotes the critical index of omega for the reverse Holder condition.