Domination of multilinear singular integrals by positive sparse forms

We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving Lp-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the Lp-boundedness proved by Muscalu, Tao and Thiele, and entails as a corollary a novel rich multilinear weighted theory. A particular case of this theory is the Lq(v1) X Lq(v2)-boundedness of the bilinear Hilbert transform when the weight vj belong to the class Aq+1/2 ∩ RH2. Our proof relies on a stopping time construction based on newly developed localized outer-Lp embedding theorems for the wave packet transform. In the Appendix, we show how our domination principle can be applied to recover the vector-valued bounds for the bilinear Hilbert transforms recently proved by Benea and Muscalu.