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 Lqfalse(v1false)×Lqfalse(v2false)‐boundedness of the bilinear Hilbert transform when the weight vj belong to the class Aq+12∩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.
A. We prove that bilinear forms associated to the rough homogeneous singular integralswhere Ω ∈ L q (S d −1 ) has vanishing average and 1 < q ≤ ∞, and to Bochner-Riesz means at the critical index in R d are dominated by sparse forms involving (1, p) averages. This domination is stronger than the weak-L 1 estimates for T Ω and for Bochner-Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative A p -weighted estimates for Bochner-Riesz means and for homogeneous singular integrals with unbounded angular part, extending previous results of Hytönen-Roncal-Tapiola for T Ω . Our results follow from a new abstract sparse domination principle which does not rely on weak endpoint estimates for maximal truncations.
Consider the discrete cubic Hilbert transform defined on finitely supported functions f on Z byWe prove that there exists r < 2 and universal constant C such that for all finitely supported f, g on Z there exists an (r, r)-sparse form Λ r,r for whichThis is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.It is known [12,25] that this operator extends to a bounded linear operator on ℓ p (Z) to ℓ p (Z), for all 1 < p < ∞. We prove a sparse bound, which in turn proves certain weighted inequalities. Both results are entirely new. By an interval we mean a setWe say a collection of intervals S is sparse if there are subsets E S ⊂ S ⊂ Z with (a) |E S | > 1 4 |S|, uniformly in S ∈ S, and (b) the sets {E S : S ∈ S} are pairwise disjoint.
In this paper we prove the weighted martingale Carleson embedding theorem with matrix weights both in the domain and in the target space.2010 Mathematics Subject Classification. Primary 42B20, 60G42, 60G46.
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