Let be a positive Radon measure on which satisfies the growth condition for all balls , where is a fixed number . For , let be the fractional integral operator defined by . And for and suitable , , define the multilinear commutator
The authors give the following: Let and . Then if ( space introduced by Tolsa for a non-homogeneous space), , there exists such that . When , this was given by W. Chen and E. Sawyer [Ill. J. Math. 46, No. 4, 1287–1298 (2002; Zbl 1033.42008)]. The authors also give a weak type endpoint estimate for this multilinear commutator when , for (Orlicz type function space).