Estimates for Multilinear Commutators of Generalized Fractional Integral Operators on Weighted Morrey Spaces

Let L be the infinitesimal generator of an analytic semigroup on L-2 (R-n) with Gaussian kernel bounds, and let L-alpha/2 be the fractional integrals of L for 0 < alpha < n Assume that (b) over right arrow = (b(1), b(2),...,b(m)) is a finite family of locally integrable functions; then the multilinear commutators generated by (b) over right arrow and L-alpha/2 are defined by L-(b) over right arrow(-alpha/2) f = [b(m),..., [b(2), [b(1), L-alpha/2]],...]f. Assume that b(j) belongs to weighted BMO space, j = 1, 2,...,m; the authors obtain the boundedness of L-(b) over right arrow(-alpha/2) on weighted Morrey spaces. As a special case, when L = -Delta is the Laplacian operator, the authors also obtain the boundedness of the multilinear fractional commutator I-alpha(b) on weighted Morrey spaces. The main results in this paper are substantial improvements and extensions of some known results.