Domination of multilinear singular integrals by positive sparse forms

TL;DR

This paper proves a uniform domination of multilinear singular integrals by positive sparse forms, leading to new weighted boundedness results for the bilinear Hilbert transform and related operators.

Contribution

It introduces a novel domination principle for multilinear singular integrals, strengthening existing $L^p$ bounds and enabling a multilinear weighted theory with new weight classes.

Findings

Established uniform domination of trilinear multiplier forms by sparse forms.

Derived $L^{q_1}(v_1) imes L^{q_2}(v_2)$-boundedness for the bilinear Hilbert transform.

Recovered vector-valued bounds for bilinear Hilbert transforms using the domination principle.

Abstract

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.