Boundedness of multilinear commutators of generalized fractional integrals

Let L be the infinitesimal generator of an analytic semigroup on L-2(R-n) with Gaussian kernel bound, and let L-alpha/2 be the fractional integral of L for 0 < alpha < n. Suppose that (b) over right arrow = (b(1), b(2), ..., b(m)) is a finite family of locally integral functions, then the multilinear commutator generated by (b) over right arrow and L-alpha/2 is defined by L((b) over right arrow)(-alpha/2)f = [b(m), ..., [b(2), [b(1,) L-alpha/2]], ...,] f, where m is an element of Z(+). When b(1), b(2), ..., b(m) is an element of BMO or b(j) is an element of Lambda over dot(beta j) (0 < beta(j) < 1) for 1 <= j <= m, the authors study the boundedness of L-(b) over right arrow(-alpha/2). (C) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.