Carleson measures associated with families of multilinear operators

Loukas Grafakos ¹   ; Lucas Oliveira ²

¹ Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A.
² Departamento de Matemática Universidade Federal do Rio Grande do Sul Porto Alegre, RS 91509-900, Brazil

Studia Mathematica, Tome 211 (2012) no. 1, pp. 71-94

Résumé

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].

Détail

DOI : 10.4064/sm211-1-4

Keywords: investigate construction carleson measures families multilinear integral operators applied tuples infty bmo functions family multilinear operators has cancellation each variable bmo functions dots measure dots dxdt carleson however family multilinear operators has cancellation variables combined result still valid infty functions may fail unbounded bmo functions indicate via example application results obtain multilinear quadratic type theorem multilinear version quadratic theorem analogous those semmes proc amer math soc