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Boundedness of a composition of dyadic paraproducts

Ana Čolović

TL;DR

The article resolves the boundedness problem for the composition of dyadic paraproducts of type $(0,1,0,1)$, providing necessary and sufficient testing conditions for $P_b^{(0,1)}\circ P_d^{(0,1)}$ and showing norm equivalence to a model operator on the upper half-plane. It introduces a transplantation framework to $\mathcal{H}$, defines a model operator $T^{(0,1,0,1)}_{b,d}$ with matching Gram matrices, and reduces boundedness to three testing conditions $A$, $B$, and $C$. The authors prove the necessity of these conditions via forward/backward testing on weighted Carleson tiles and establish sufficiency by careful decomposition of the associated bilinear form into three controllable parts, ultimately bounding the operator by the product $\|b\|_{BMO}\|d\|_{BMO}$. A direct $H^1$–$BMO$ duality argument provides the sharp $BMO$-norm control of the overall bound, completing the characterization. This work settles the remaining case in the dyadic paraproduct composition problem and links the dyadic setting to two-weight-type testing phenomena on the upper half-plane.

Abstract

We resolve the question of the boundedness of the composition of dyadic paraproducts, first posed by Pott, Reguera, Sawyer, and Wick in~\cite{PotCarSawWic}, by providing necessary and sufficient conditions for their boundedness.

Boundedness of a composition of dyadic paraproducts

TL;DR

The article resolves the boundedness problem for the composition of dyadic paraproducts of type , providing necessary and sufficient testing conditions for and showing norm equivalence to a model operator on the upper half-plane. It introduces a transplantation framework to , defines a model operator with matching Gram matrices, and reduces boundedness to three testing conditions , , and . The authors prove the necessity of these conditions via forward/backward testing on weighted Carleson tiles and establish sufficiency by careful decomposition of the associated bilinear form into three controllable parts, ultimately bounding the operator by the product . A direct – duality argument provides the sharp -norm control of the overall bound, completing the characterization. This work settles the remaining case in the dyadic paraproduct composition problem and links the dyadic setting to two-weight-type testing phenomena on the upper half-plane.

Abstract

We resolve the question of the boundedness of the composition of dyadic paraproducts, first posed by Pott, Reguera, Sawyer, and Wick in~\cite{PotCarSawWic}, by providing necessary and sufficient conditions for their boundedness.

Paper Structure

This paper contains 7 sections, 1 theorem, 62 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background on the Upper Half-Plane
  4. Boundedness of Type (0,1,0,1) Composition
  5. Necessity of the Conditions
  6. Proof of sufficiency
  7. Control of the bounds by the $BMO$ norms

Key Result

Theorem 1.2

Let $b=\{b_I\}_{I\in \mathcal{D}},$$d=\{d_I\}_{I\in \mathcal{D}}.$ For $I\in \mathcal{D},$ let For $I,J\in \mathcal{D}, J \subsetneq I,$ let $\delta^{(I,J)}=$ The operator $P_{b,d}^{(0,1,0,1)}$ is bounded on $L^2(\mathbb{R})$ if and only if the following conditions hold: The norm of $\|\Pi_b\Pi_d:L^2(\mathbb{R})\to L^2(\mathbb{R})\|\simeq A+B+C.$ Moreover, $(A + B + C) \lesssim \|b\|_{BMO} \|

Theorems & Definitions (3)

  • Definition 1.1
  • Theorem 1.2
  • proof