STUDIA MATHEMATICA 211 (1) (2012) Carleson measures associated with families of multilinear operators by Loukas Grafakos (Columbia, MO) and Lucas Oliveira (Porto Alegre, RS) Abstract. We investigate the construction of Carleson measures from families of multilinear integral operators applied to tuples of L ∞ and BMO functions. We show that if the family Rt of multilinear operators has cancellation in each variable, then for BMO functions b1,...,bm, the measure |Rt (b1,...,bm)(x)| 2 dxdt/t is Carleson. However, if the family of multilinear operators has cancellation in all variables combined, this result is still valid if bj are L ∞ functions, but it may fail if bj are unbounded BMO functions, as we indicate via an example. As an application of our results we obtain a multilinear quadratic T (1) type theorem and a multilinear version of a quadratic T (b) theorem analogous to those by Semmes [Proc. Amer. Math. Soc. 110 (1990), 721–726]. 1. Introduction. A positive measure dμ(x, t) on R n+1 + is called a Car- leson measure if (1.1) kdμk C = sup Q⊂R n 1 |Q| dμ(T (Q)) < ∞, where |Q| denotes the Lebesgue measure of the cube Q, T (Q)= Q×(0,l(Q)] denotes the tent over Q, and l(Q) is the side length of Q. Carleson measures arose in the work of Carleson [2], [3] and turned out to be tools of funda- mental importance in harmonic analysis; we mention for instance the great role they play in the study of the Cauchy integrals along Lipschitz curves [5], the T (1) theorem [6], and in the study of the Kato problem [1]. There is a natural connection between Carleson measures, families of linear operators acting L 2 , and BMO functions. Precisely, let {R t } t>0 be a family of integral operators (1.2) R t (f )(x)=  R n K t (x, y)f (y) dy 2010 Mathematics Subject Classification : Primary 42B99; Secondary 42B25. Key words and phrases : Carleson measures, multilinear operators. DOI: 10.4064/sm211-1-4 [71] c  Instytut Matematyczny PAN, 2012