Domination of multilinear singular integrals by positive sparse forms

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 $A_{\frac{q+1}{2}}\cap R{H}_2$. 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.

25 pages. Version 2: added some references. Added an appendix where the main theorem is used to recover vector-valued bounds

References in corpus (3)

• On pointwise estimates involving sparse operators
• A sparse domination principle for rough singular integrals
• Sparse bilinear forms for Bochner Riesz multipliers and applications

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