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 – α /2 be the fractional integral of L for 0 < α < n . Suppose that b = ( b 1 , b 2 , …, b m ) is a finite family of locally integral functions, then the multilinear commutator generated by b and L – α /2 is defined by L – α /2 b f = [ b m , …, [ b 2 , [ b 1 , L – α /2 ]], …, ] f , where m ∈ ℤ + . When b 1 , b 2 , …, b m ∈ BMO or b j ∈ Λ   β   j(0 < β j < 1) for 1 ≤ j ≤ m , the authors study the boundedness of L – α /2 b . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)