Carleson measures associated with families of multilinear operators
We investigate the construction of Carleson measures from families of multilinear integral operators applied to tuples of $L^\infty $ and BMO functions. We show that if the family $R_t$ of multilinear operators has cancellation in each variable, then for BMO functions $b_1, \dots , b_m$, the measure $|R_t(b_1, \dots , b_m)(x)|^2 dxdt/t$ is Carleson. However, if the family of multilinear operators has cancellation in all variables combined, this result is still valid if $b_j$ are $L^\infty $ functions, but it may fail if $b_j$ 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].