We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving $L^p$-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the $L^p$-boundedness proved in \cite{ MTT} and entails as a corollary a rich multilinear weighted theory. In particular, we obtain $L^{ q_1}(v_1) \times L^{ q_2}(v_2)$-boundedness of the bilinear Hilbert transform when the weights $v_j$ belong to the class $A_{ \frac{ q+1}{ 2}}\cap RH_2$. Our proof relies on a stopping time construction based on newly developed localized outer-$L^p$ embedding theorems for the wave packet transform. In an 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.
Categories: math.CA