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 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.

Banach-valued multilinear singular integrals

“…Several extensions and refinements of [30] have since appeared, see e.g. [1,4,8,9]. In general, as Corollary 1.2 demonstrates, Theorem 1.1 is outside the scope of the above references, although it does imply a strict subset of the ℓ p estimates of [30].…”

“…For our purposes here, we need a more nuanced notion than the usual, e.g. appearing in [9,19,25,26], fully discretized tree operator…”

T(1) theorem for dyadic singular integral forms associated with hypergraphs

“…As is well-known, the sparse domination also implies weighted estimated for the form in question. Once again, we merely adapt the setting from the paper [2] by Culiuc, Di Plinio, and Ou. Given the edge-set E with its collection of integers d = (d e ) e∈E defined as before, let p = (p e ) e∈E be an arbitrary tuple of exponents from [1, ∞] such that p e > d e for each e ∈ E and e∈E 1 pe = 1.…”

Weighted boundedness of the 2‐fold product of Hardy–Littlewood maximal operators

“…Consequently, for every , and , . It is worth mentioning that much more delicate weighted estimates for the bilinear Hilbert transform have been recently obtained in .…”

“…for every 1 > 1∕ , 2 > 1∕(1 − ) and 1 ∈ 1 , 2 ∈ (1− ) 2 . It is worth mentioning that much more delicate weighted estimates for the bilinear Hilbert transform have been recently obtained in [8].…”