A reprise of the NTV conjecture for the Hilbert transform

TL;DR

This paper presents a new proof of the NTV conjecture for the Hilbert transform, utilizing a different decomposition and control approach based on Sawyer's potential theorem, building on prior foundational work.

Contribution

It introduces a novel proof technique for the NTV conjecture, modifying key components like the bilinear form decomposition and energy control methods.

Findings

New proof of the NTV conjecture for the Hilbert transform

Modified decomposition of the bilinear form

Alternative control of functional energy using Sawyer's potential theorem

Abstract

We give a slightly different proof of the NTV conjecture for the Hilbert transform that was proved by T. Hyt\"onen, M. Lacey, E.T. Sawyer, C.-Y. Shen and I. Uriarte-Tuero, building on previous work of F. Nazarov, S. Treil and A. Volberg. After modifying the decomposition of the main bilinear form, we give a new proof of control of functional energy that is based on the potential Theorem 1 of [Saw3], rather than the Poisson Theorem 2 that is used in all other proofs in the literature. This approach was pioneered in the first version of Sawyer and Wick [SaWi] on the ArXiv. Then we alter the bottom-up corona construction, the size functional, the straddling lemmas, and the use of recursion of admissible collections of pairs of intervals, from M. Lacey [Lac]. However, the essence of control of the stopping form remains as in the fundamental work of Lacey.

Equations

$${\mathfrak{N}}_H\left(\sigma ,\omega \right)\equiv \underset{f\in L^2\left(\mathbb{R};\sigma \right)}{\sup }\frac{1}{{\Vert f\Vert }_{L^2\left(\sigma \right)}}{\Vert H\left(f\sigma \right)\Vert }_{L^2\left(\omega \right)},$$

$${\mathcal{A}}_2\left(\sigma ,\omega \right)\equiv \underset{I\in \mathcal{I}}{\sup }\left(\frac{1}{\left|I\right|}{\int }_I{\mathbf{s}}_I{\left(x\right)}^{2}d\omega \left(x\right)\right)\left(\frac{1}{\left|I\right|}{\int }_I{\mathbf{s}}_I{\left(y\right)}^{2}d\sigma \left(y\right)\right),$$

$${\mathfrak{T}}_H\left(\sigma ,\omega \right)\equiv \underset{I\in \mathcal{I}}{\sup }\frac{1}{\sqrt{{\left|I\right|}_{\sigma }}}{\Vert H{1}_I\sigma \Vert }_{L^2\left(\omega \right)},$$

$${\mathcal{A}}_2^{\mathrm{hole}}\left(\sigma ,\omega \right)$$

$$\mathrm{P}\left(I,\mu \right)$$

$$A_2\left(\sigma ,\omega \right)\equiv \underset{I\in \mathcal{I}}{\sup }\frac{{\left|I\right|}_{\sigma }}{\left|I\right|}\frac{{\left|I\right|}_{\omega }}{\left|I\right|},$$

$$A_2^{\mathrm{offset}}\left(\sigma ,\omega \right)\equiv \underset{\left(Q,Q^{\prime }\right)\in \mathcal{N}}{\sup }\left(\frac{1}{\left|Q^{\prime }\right|}{\int }_{Q^{\prime }}d\omega \right)\left(\frac{1}{\left|Q\right|}{\int }_{Q}d\sigma \right),$$

$$c\le \frac{\sqrt{{\mathcal{A}}_2^{\mathrm{hole}}\left(\sigma ,\omega \right)}+{\mathfrak{T}}_H\left(\sigma ,\omega \right)+{\mathfrak{T}}_H\left(\omega ,\sigma \right)}{{\mathfrak{N}}_H\left(\sigma ,\omega \right)}\le C.$$

$$\mathrm{P}\left(J,\nu \right)\equiv {\int }_{\mathbb{R}}\frac{\ell \left(J\right)}{{\left(\ell \left(J\right)+\left|y-c_J\right|\right)}^2}d\nu \left(y\right).$$

$${\left|{⟨H_{\sigma }{1}_{F\setminus I^{\prime }},{△}_J^{\omega }g⟩}_{\omega }\right|}^2$$

$${\left|{⟨H_{\sigma }{1}_{F\setminus I^{\prime }},\sum _{J^{\prime }\subset J}{a}_{J^{\prime }}{h}_{J^{\prime }}^{\omega }⟩}_{\omega }\right|}^2$$

$$\mathrm{P}(J,\mu 1_{K\setminus I})\lesssim {\left(\frac{\ell \left(J\right)}{\ell \left(I\right)}\right)}^{1-2\epsilon }\mathrm{P}(I,\mu 1_{K\setminus I}).$$

$${⟨H_{\sigma }f,g⟩}_{\omega }=\sum _{I,J\in \mathcal{D}}{⟨H_{\sigma }{△}_I^{\sigma }f,{△}_J^{\omega }g⟩}_{\omega },$$

$${⟨H_{\sigma }f,g⟩}_{\omega }$$

$${\mathsf{B}}_{\mathrm{below}}\left(f,g\right)={\mathsf{B}}_{\mathrm{neigh}}\left(f,g\right)+{\mathsf{B}}_{\mathrm{far}}\left(f,g\right)+{\mathsf{B}}_{\mathrm{para}}\left(f,g\right)+{\mathsf{B}}_{\mathrm{stop}}\left(f,g\right),$$

$${\mathcal{E}}_2{\left(\sigma ,\omega \right)}^2\equiv \underset{{\bigcup ^{\cdot }}_{r=1}^{\infty }{I}_r\subset I}{\sup }{\left(\frac{\mathrm{P}\left(I_r,1_{I\setminus I_r}\sigma \right)}{\ell \left(I_r\right)}\right)}^2\mathsf{E}{\left(I_r,\omega \right)}^2\frac{{\left|I_r\right|}_{\omega }}{{\left|I\right|}_{\sigma }}\text{ and its dual }{\mathcal{E}}_2\left(\omega ,\sigma \right),$$

$$\mathfrak{W}\left(\sigma ,\omega \right)\equiv \underset{I,J\mathrm{adjacent}:2^{-\tau }\ell \left(I\right)\le \ell \left(J\right)\le 2^{\tau }\ell \left(I\right)}{\sup }\left|{\int }_J\left(H_{\sigma }{1}_I\right)d\omega \right|.$$

$${\mathsf{B}}_{\mathrm{below}}={\mathsf{B}}_{\mathrm{neigh}}+{\mathsf{B}}_{\mathrm{far}}+{\mathsf{B}}_{\mathrm{para}}+{\mathsf{B}}_{\mathrm{stop}},$$

$${\mathcal{P}}_{\mathrm{below}}\equiv \left\{\left(I,J\right)\in {\mathcal{D}}_{\mathrm{good}}\times {\mathcal{D}}_{\mathrm{good}}:J{\subset }_{\tau }I\right\}$$

$${\mathsf{B}}_{\mathrm{below}}\left(f,g\right)$$

$$\left|{\mathsf{B}}_{\mathrm{neigh}}\left(f,g\right)\right|\le C_{\epsilon }\sqrt{A_2^{\mathrm{offset}}\left(\sigma ,\omega \right)}{\Vert f\Vert }_{L^2(\sigma )}{\Vert g\Vert }_{L^2(\omega )}.$$

$${⟨H_{\sigma }\left(1_{\theta \left(I_J\right)}{△}_I^{\sigma }f\right),{△}_J^{\omega }g⟩}_{\omega }=E_{\theta \left(I_J\right)}^{\sigma }{\Delta }_I^{\sigma }f\cdot {⟨H_{\sigma }\left(1_{\theta \left(I_J\right)}\right),{△}_J^{\omega }g⟩}_{\omega },$$

$${\mathsf{B}}_{\mathrm{neigh}}\left(f,g\right)=\sum _{I,J\in {\mathcal{D}}_{\mathrm{good}}\text{ and }J{\subset }_{\tau }I}\left(E_{\theta \left(I_J\right)}^{\sigma }{\Delta }_I^{\sigma }f\right){⟨H_{\sigma }\left(1_{\theta \left(I_J\right)}\sigma \right),{\Delta }_J^{\omega }g⟩}_{\omega }.$$

$$\left|⟨H_{\sigma }\left(1_{\theta \left(I_J\right)}\right),{\Delta }_J^{\omega }g{⟩}_{\omega }\right|$$

$$\lesssim {\Vert {\Delta }_J^{\omega }g\Vert }_{L^2\left(\omega \right)}\sqrt{{\left|J\right|}_{\omega }}\cdot 2^{-\left(1-2\epsilon \right)s}\mathrm{P}\left(I,1_{\theta \left(I_J\right)}\sigma \right).$$

$$A(I,s)$$

$$\le \sum _{+,-}{2}^{-\left(1-2\epsilon \right)s}|E_{I_{\mp }}^{\sigma }{\Delta }_I^{\sigma }f|\mathrm{P}(I,1_{I_{\mp }}\sigma )\sum _{J:2^s\ell \left(J\right)=\ell \left(I\right):J\subset I_{\pm }}{\Vert {\Delta }_J^{\omega }g\Vert }_{L^2\left(\omega \right)}\sqrt{{\left|J\right|}_{\omega }}$$

$$\lesssim \sum _{+,-}{2}^{-\left(1-2\epsilon \right)s}|E_{I_{\mp }}^{\sigma }{\Delta }_I^{\sigma }f|\mathrm{P}(I,1_{I_{\mp }}\sigma )\sqrt{{\left|I_{\pm }\right|}_{\omega }}\sqrt{\sum _{J\in {\mathfrak{C}}_{\mathcal{D}}^{\left(s\right)}\left(I_{\pm }\right)}{\Vert {\Delta }_J^{\omega }g\Vert }_{L^2\left(\omega \right)}^2},$$

$$\sum _{I\in \mathcal{D}}\sum _{J\in {\mathfrak{C}}_{\mathcal{D}}^{\left(s\right)}\left(I\right)}{\Vert {\Delta }_J^{\omega }g\Vert }_{L^2\left(\omega \right)}^2\le {\Vert g\Vert }_{L^2(\omega )}^2.$$

$$\left|E_{\theta \left(I_J\right)}^{\sigma }{\Delta }_I^{\sigma }f\right|\le \sqrt{E_{\theta \left(I_J\right)}^{\sigma }{\left|{\Delta }_I^{\sigma }f\right|}^2}\le {\Vert {\Delta }_I^{\sigma }f\Vert }_{L^2\left(\sigma \right)}{\left|\theta \left(I_J\right)\right|}_{\sigma }^{-\frac{1}{2}},$$

$$A(I,s)$$

$${\left|I_{\mp }\right|}_{\sigma }^{-\frac{1}{2}}\mathrm{P}(I,1_{I_{\mp }}\sigma )\sqrt{{\left|I_{\pm }\right|}_{\omega }}\lesssim \frac{\sqrt{{\left|I_{\mp }\right|}_{\sigma }}\sqrt{{\left|I_{\pm }\right|}_{\omega }}}{\left|I\right|}\lesssim \sqrt{A_2^{\mathrm{offset}}\left(\sigma ,\omega \right)}.$$

$$\sum _{I\in \mathcal{D}}A(I,s)\lesssim \sqrt{A_2^{\mathrm{offset}}\left(\sigma ,\omega \right)}{2}^{-\left(1-2\epsilon \right)s}\sqrt{\sum _{I\in \mathcal{D}}{\Vert {\Delta }_I^{\sigma }f\Vert }_{L^2\left(\sigma \right)}^2}\sqrt{\sum _{I\in \mathcal{D}}\sum _{J\in {\mathfrak{C}}_{\mathcal{D}}^{\left(s\right)}\left(I\right)}{\Vert {\Delta }_J^{\omega }g\Vert }_{L^2\left(\omega \right)}^2}$$

Full text

After modifying the decomposition of the main bilinear form, the proof alters the bottom-up corona construction, the size functional, the straddling lemmas, and the use of recursion of admissible collections of pairs of intervals, from M. Lacey [Lac]. However, the essence of control of the stopping form remains as in the fundamental work of Lacey. The alternate approach of Sawyer and Wick [SaWi] to controlling functional energy is also described, which differs from that previously in the literature, [LaSaShUr3] and [Hyt2].

1. 1 Introduction

2. 2 Reprise of the proof of the $NTV$ conjecture

3. 2.1 First steps and outline of the decomposition of forms

4. 2.2 Control of the disjoint and comparable forms by testing and offset Muckenhoupt characteristics

5. 2.3 Control of the neighbour form by the offset Muckenhoupt characteristic

6. 3 Reduction of the home form by coronas

7. 3.1 Control of the far form by way of functional energy

8. 4 Reduction of the diagonal form by the NTV reach

9. 4.1 Control of the paraproduct form by the testing characteristic

10. 4.2 Control of the stopping form by the Poisson-Energy characteristic

11. 4.2.1 The Stopping Child Lemma

12. 4.2.2 Dual tree decomposition

13. 4.2.3 Completion of the proof

1. Introduction

F. Nazarov, S. Treil and A. Volberg formulated the two weight question for the Hilbert transform [Vol], that in turn led to the famous NTV conjecture:

Conjecture 1**.**

[Vol*]*Given two positive locally finite Borel measures $\sigma$ and $\omega$ on the real line $\mathbb{R}$, the Hilbert transform $Hf\left(x\right)=\mathop{\mathrm{p}v}\int_{\mathbb{R}}\frac{f\left(y\right)}{y-x}dy$ is bounded from $L^{2}\left(\sigma\right)$ to $L^{2}\left(\omega\right)$, i.e. the operator norm

[TABLE]

is finite uniformly over appropriate truncations of $H$, if the two tailed Muckenhoupt characteristic,

[TABLE]

is finite, where $\mathbf{s}_{I}\left(x\right)=\frac{\ell\left(I\right)}{\ell\left(I\right)+\left|x-c_{I}\right|}$, and if the testing characteristic

[TABLE]

is finite as well as its dual $\mathfrak{T}_{H}\left(\omega,\sigma\right)$.

In a groundbreaking series of papers including [NTV1],[NTV2] and [NTV3], Nazarov, Treil and Volberg used weighted Haar decompositions with random grids of $\left(r,\varepsilon\right)-\mathop{\mathrm{g}ood}$ grids, introduced their ‘pivotal condition’, and proved the above conjecture under the side assumption that this pivotal condition held. Subsequently in [LaSaUr2], it was shown that the pivotal condition was not necessary for the norm inequality to hold in general, a necessary ‘energy condition’ was introduced as a substitute, and a hybrid merging of these two conditions was shown to be sufficient for use as a side condition. Eventually, these three authors with C.-Y. Shen established the NTV conjecture in the absence of common point masses in the measures $\sigma$ and $\omega$ in a two part paper; M. Lacey, E. T. Sawyer, C.-Y. Shen and I. Uriarte-Tuero [LaSaShUr3] and M. Lacey [Lac].

The final assumption of no common point masses was removed shortly after by T. Hytönen [Hyt2] who replaced the two tailed Muckenhoupt characteristic with a variant involving holes,

[TABLE]

and who also simplified some aspects of the proof. The reason for introducing the holes is that if $\sigma$ and $\omega$ share a common point mass, then the classical Muckenhoupt characteristic

[TABLE]

fails to be finite - simply let $I$ in the $\sup$ above shrink to a common point mass. The $\mathop{\mathrm{h}oled}$ Muckenhoupt condition trivially implies the more elementary $\mathop{\mathrm{o}ffset}$ Muckenhoupt condition

[TABLE]

where $\mathcal{N}$ is the set of neighbouring pairs $\left(Q,Q^{\prime}\right)$ of intervals of comparable side length, i.e. $\left(Q,Q^{\prime}\right)\in\mathcal{D}$ for some dyadic grid $\mathcal{D}$ where $2^{-r}\ell\left(Q^{\prime}\right)\leq\ell\left(Q\right)\leq 2^{r}\ell\left(Q^{\prime}\right)$, $Q\cap Q^{\prime}=\emptyset$, and $\overline{Q}\cap\overline{Q^{\prime}}\not=\emptyset$. Here $r$ is the goodness parameter in [NTV3] and [LaSaShUr3].

The purpose of this paper is to give a modified proof of the above results for the Hilbert transform, and to highlight where the various characteristics enter into the proof. Our proof is somewhat simplified in that it avoids the recursion of admissible collections of pairs of intervals that was used in Part II [Lac], and the recent approach to controlling functional energy due to Sawyer and Wick [SaWi], is discussed in the setting $p=2$. However, apart from some reorganization using nonlinear bounds, the essence of control of the stopping form remains as in the fundamental work of M. Lacey [Lac], and the control of functional energy is similar in spirit to that in [LaSaShUr3], and can in fact be imported as a black box from [LaSaShUr3] if one wishes.

Theorem 2**.**

Let $\sigma$ and $\omega$ be positive locally finite Borel measures on $\mathbb{R}$. The Hilbert transform $H_{\sigma}$ is bounded from $L^{2}\left(\sigma\right)$ to $L^{2}\left(\omega\right)$ if and only if the two tailed Muckenhoupt characteristic with holes, and both of the testing characteristics, are finite. More precisely, there are universal positive constants $c,C$ such that for all pairs $\left(\sigma,\omega\right)$ of positive locally finite Borel measures,

[TABLE]

Acknowledgement 3**.**

We thank Brett Wick for fruitful discussions on both the far and stopping forms.

2. Reprise of the proof of the $NTV$ conjecture

We assume the Haar supports of all functions $f\in L^{2}\left(\sigma\right)$ and $g\in L^{2}\left(\omega\right)$ are contained, along with their two children, in the $\left(r,\varepsilon\right)$-$\mathop{\mathrm{g}ood}$ grid $\mathcal{D}_{\mathop{\mathrm{g}ood}}$, where $J\in\mathcal{D}_{\mathop{\mathrm{g}ood}}$ if $\mathop{\mathrm{d}ist}\left(J,\partial K\right)\geq\frac{1}{2}\ell\left(J\right)^{\varepsilon}\ell\left(K\right)^{1-\varepsilon}$ for all $K\supset J$ with $\ell\left(K\right)\geq 2^{r}\ell\left(J\right)$. See [NTV3] and [LaSaShUr3] for the reduction of the norm inequality to good functions with goodness parameters $r\in\mathbb{N}$ and $\varepsilon>0$.

We will make use of the following bound for sums of Haar coefficients, which is proved e.g. in [LaSaUr2] and [LaSaShUr3]. For any interval $J$ with center $c_{J}$, and any finite measure $\nu$, define the Poisson integral,

[TABLE]

Lemma 4** (Monotonicity and Energy Lemma).**

Suppose that $J\subset I^{\prime}\subset 2I^{\prime}\subset F$ are dyadic intervals. Then

[TABLE]

We will also need the following critical Poisson Decay Lemma from [Vol].

Lemma 5** (Poisson Decay Lemma).**

Suppose $J\subset I\subset K$ are dyadic intervals and that $d\left(J,\partial I\right)>2\ell\left(J\right)^{\varepsilon}\ell\left(I\right)^{1-\varepsilon}$ for some $0<\varepsilon<\frac{1}{2}$. Then for a positive Borel measure $\mu$ we have

[TABLE]

2.1. First steps and outline of the decomposition of forms

Now we can begin the proof of Theorem 2. Following [NTV3] and [LaSaShUr3], we expand the Hilbert transform bilinear form $\left\langle H_{\sigma}f,g\right\rangle_{\omega}$ in terms of the Haar decompositions of $f$ and $g$,

[TABLE]

and then assuming the Haar supports of $f$ and $g$ lie in an $\left(r,\varepsilon\right)-\mathop{\mathrm{g}ood}$ grid $D_{\mathop{\mathrm{g}ood}}$ (see [NTV3] for the origin of goodness, and also [LaSaShUr3] for its use here), we decompose the double sum above as follows,

[TABLE]

where $\tau\in\mathbb{N}$ is fixed below, larger than the goodness parameter $r$, and $J\subset_{\tau}I$ means that $J\subset I$ and $\ell\left(J\right)\leq 2^{-\tau}\ell\left(I\right)$. The first two forms are symmetric, and using a CZ-energy corona decomposition with parameter $\Gamma>1$, we subsequently decompose the below form $\mathsf{B}_{\mathop{\mathrm{b}elow}}\left(f,g\right)$ into another four forms,

[TABLE]

in which there is control of both averages of $f$ and a Poisson-Energy characteristic $\Gamma\left(\sigma,\omega\right)$ in each corona. Altogether, we will eventually have eleven forms in our decomposition of the inner product $\left\langle H_{\sigma}f,g\right\rangle_{\omega}$. It turns out that all of the forms, save for the far, paraproduct, and stopping forms, are controlled directly by the parameter $\Gamma$ and the testing and holed Muckenhoupt characteristics. Control of the far, paraproduct, and stopping forms will require finiteness of the energy characteristics introduced in [LaSaUr2],

[TABLE]

where the supremum is taken over all pairwise disjoint subdecompositions of an interval $I$ into dyadic subintervals $I_{r}\in\mathcal{D}\left[I\right]$. The holed Muckenhoupt characteristic is needed only to control the far form, while the smaller offset Muckenhoupt characteristic suffices elsewhere.

It may help to make a couple of comments on the broad structure of the proof. The testing characteristics are used to control forms in which the measures $\sigma$ and $\omega$ ‘see each other’ in inner products $\left\langle H_{\sigma}\mathbf{1}_{K}\bigtriangleup_{I}^{\sigma}f,\bigtriangleup_{J}^{\omega}g\right\rangle_{\omega}$ with $J\subset K$ (such as comparable and paraproduct forms), while the holed and offset Muckenhoupt characteristics are used to control forms in which the measures $\sigma$ and $\omega$ ‘do not see each other’ in inner products $\left\langle H_{\sigma}\mathbf{1}_{K}\bigtriangleup_{I}^{\sigma}f,\bigtriangleup_{J}^{\omega}g\right\rangle_{\omega}$ with $K\cap J=\emptyset$ (such as disjoint, neighbour, far and stopping forms).

In some of these cases, there arise parts of the form in which the coronas associated to the measures are far apart, and it is then necessary to derive additional geometric decay in the distance between coronas, much as in the well-known Colar-Stein Lemma.

A main tool for this is the Poisson Decay Lemma, which extracts geometric decay from goodness of intervals in the Haar supports of the functions, as pioneered in [NTV3]. This decay is used in controlling the disjoint, outer and stopping forms. Additional sources of needed geometric decay arise from the coronas themselves as follows. There are three coronas at play in the proof:

1. (1)

a top-down Calderón-Zygmund corona that derives geometric decay from a $\sigma$-Carleson condition, that holds automatically from the definition of the stopping times, 2. (2)

a top-down Poisson-Energy corona that also derives geometric decay from a $\sigma$-Carleson condition, but that is no longer automatic from the definition, and instead requires the energy condition, as pioneered in [LaSaUr2], 3. (3)

and a bottom-up Energy corona that derives geometric decay from an ‘$\omega$-energy Carleson condition’, that again holds automatically from the definition of the stopping times, as pioneered in [Lac].

Since the first two corona constructions both yield a $\sigma$-Carleson condition, it is convenient to bundle them together in a single CZ-PE corona construction, see (3.1) for the associated stopping times. This corona construction has two additional important properties, control of averages of $f$ that is crucial for controlling the paraproduct form, and control of a Poisson-Energy characteristic that is crucial for controlling the stopping form. The $\sigma$-Carleson condition is used in the far, paraproduct and stopping forms while the ‘$\omega$-energy Carleson condition’ is used in the stopping form.

2.2. Control of the disjoint and comparable forms by testing and

offset Muckenhoupt characteristics

The disjoint form $\mathsf{B}_{\mathop{\mathrm{d}isjoint}}\left(f,g\right)$, and the comparable forms $\mathsf{B}_{\mathop{\mathrm{c}omparable}}\left(f,g\right)$ and $\mathsf{B}_{\mathop{\mathrm{c}omparable}}^{\ast}\left(f,g\right)$, are controlled in [SaShUr12, Section 4]111the proof given there is for the $Tb$ theorem, and the reader can note that only the offset Muckenhoupt condition is used using only the testing characteristics, the offset Muckenhoupt characteristic, and the following weak boundedness characteristic,

[TABLE]

Similar results are in [NTV3], [LaSaShUr3], and [Hyt2]. Note that it follows from [Hyt2, Lemma 2.4] that $\mathfrak{W}\left(\sigma,\omega\right)\leq A_{2}^{\mathop{\mathrm{o}ffset}}\left(\sigma,\omega\right)$.

The forms $\mathsf{B}_{\mathop{\mathrm{b}elow}}\left(f,g\right)$ and $\mathsf{B}_{\mathop{\mathrm{a}bove}}\left(f,g\right)$ are symmetric, and so we need only bound the below form $\mathsf{B}_{\mathop{\mathrm{b}elow}}\left(f,g\right)$. To acccomplish this we decompose the below form $\mathsf{B}_{\mathop{\mathrm{b}elow}}$ further into neighbour, far, paraproduct and stopping forms,

[TABLE]

which are detailed as we progress through the proof.

Let

[TABLE]

be the set of pairs of $\mathop{\mathrm{g}ood}$ dyadic intervals $\left(I,J\right)$ with $J$ at least $\tau$ levels below and inside $I$. We begin by splitting the $\mathop{\mathrm{b}elow}$ form into home and neighbour forms, where $\theta K$ denotes the dyadic sibling of $K\in\mathcal{D}$,

[TABLE]

2.3. Control of the neighbour form by the offset Muckenhoupt

characteristic

The neighbour form is easily controlled by the $A_{2}^{\mathop{\mathrm{o}ffset}}$ condition using Lemma 4 and the fact that the intervals $J$ are good, namely we claim

[TABLE]

To see this, momentarily fix an integer $s\geq\tau$. We have

[TABLE]

and thus we can write

[TABLE]

Now we use the Poisson Decay Lemma. Assume that $J\subset_{\tau}I$. Let $\frac{\ell\left(J\right)}{\ell\left(I\right)}=2^{-s}$ with $s\geq\tau$ in the pivotal estimate in Lemma 4 with $J\subset I$ to obtain

[TABLE]

where we have used (2.1).

Now we pigeonhole pairs $\left(I,J\right)$ of intervals by requiring $J\in\mathfrak{C}_{\mathcal{D}}^{\left(s\right)}\left(I\right)$, i.e. $J\subset I$ and $\ell\left(J\right)=2^{-s}\ell\left(I\right)$, and we further separate the two cases where $I_{J}=I_{\pm}$, the right and left children of $I$. We have

[TABLE]

where we have used Cauchy-Schwarz in the last line, and we also note that

[TABLE]

Using

[TABLE]

and the offset Muckenhoupt condition, we can thus estimate,

[TABLE]

since $\mathrm{P}(I,\mathbf{1}_{I_{\mp}}\sigma)\lesssim\frac{\left|I_{\mp}\right|_{\sigma}}{\left|I\right|}$ shows that

[TABLE]

An application of Cauchy-Schwarz to the sum over $I$ using (2.4) then shows that

[TABLE]

Now we sum in $s\geq\tau$ to obtain,

[TABLE]

3. Reduction of the home form by coronas

In order to control the home form, we must pigeonhole the pairs of intervals $\left(I,J\right)\in\mathcal{P}_{\mathop{\mathrm{b}elow}}$ into a collection of pairwise disjoint corona boxes in which both averages of $f$ are controlled, and a Poisson-Energy characteristic is controlled. Then we split the home form into two forms according to this decomposition, which we call the diagonal and far forms. But first we need to construct the corona decomposition.

Fix $\Gamma>1$ and a large dyadic interval $T$. Define a sequence of stopping times $\left\{\mathcal{F}_{n}\right\}_{n=0}^{\infty}$ depending on $T$, $\sigma$ and $\omega$ recursively as follows. Let $\mathcal{F}_{0}=\left\{T\right\}$. Given $\mathcal{F}_{n}$, define $\mathcal{F}_{n+1}$ to consist of the maximal intervals $I^{\prime}\in\mathcal{D}_{\mathop{\mathrm{g}ood}}$ for which there is $I\in\mathcal{F}_{n}$ with $I^{\prime}\subset I$ and

[TABLE]

where

[TABLE]

is the energy functional introduced in [LaSaUr2], and $\mathsf{P}_{\mathcal{D}\left[I\right]}^{\sigma}=\sum_{K\in\mathcal{D}:\ K\subset I}\bigtriangleup_{K}^{\sigma}$. Set $\mathcal{F}\equiv\bigcup_{n=0}^{\infty}\mathcal{F}_{n}$, which we refer to as the CZ-PE stopping times for $T$, $\sigma$ and $\omega$, and which we note consists of $\mathop{\mathrm{g}ood}$ intervals, i.e. $\mathcal{F}\subset\mathcal{D}_{\mathop{\mathrm{g}ood}}$. Denote the associated coronas by $\mathcal{C}_{\mathcal{F}}\left(F\right)$ and the grandchildren at depth $m\in N$ of $F$ in the tree $\mathcal{F}$ by $\mathfrak{C}_{\mathcal{F}}^{\left(m\right)}\left(F\right)$, with $\mathfrak{C}_{\mathcal{F}}^{\left(1\right)}\left(F\right)$ abbreviated to $\mathfrak{C}_{\mathcal{F}}\left(F\right)$. We will consistly use calligraphic font $\mathcal{C}$ to denote coronas, and fraktur font $\mathfrak{C}$ to denote children. Finally, we define

[TABLE]

The point of introducing the corona decomposition $\mathcal{D}\left[T\right]=\overset{\cdot}{\bigcup}_{F\in\mathcal{F}}\mathcal{C}_{\mathcal{F}}\left(F\right)$ is that we obtain control of both the averages $E_{I}^{\sigma}\left|\mathsf{P}_{\mathcal{D}\left[F\right]}f\right|\equiv\frac{1}{\left|I\right|_{\sigma}}\int_{I}\left|\mathsf{P}_{\mathcal{D}\left[F\right]}f\right|d\sigma$ of projections of $f$, and the Poisson-Energy functional

[TABLE]

in each $\mathop{\mathrm{g}ood}$ corona $\mathcal{C}_{\mathcal{F}}^{\mathop{\mathrm{g}ood}}\left(F\right)\equiv\mathcal{C}_{\mathcal{F}}\left(F\right)\cap\mathcal{D}_{\mathop{\mathrm{g}ood}}$, i.e.

[TABLE]

In particular this inequality shows that the Poisson-Energy characteristic

[TABLE]

of $\sigma$ and $\omega$ with respect to the stopping times $\mathcal{F}$, is dominated by the parameter $\Gamma$ chosen in (3.1). For future reference we note that two more refinements of the characteristic will appear in connection with the Stopping Child Lemma and the control of the stopping form below.

**Consequences of the energy condition: **

If we assume the finiteness of the energy characteristic (2.2) (which is often referred to as the energy condition), and if we take $\Gamma>4\mathcal{E}_{2}\left(\sigma,\omega\right)$ in (3.1), we obtain a $\sigma$-Carleson, or $\sigma$-sparse, condition for the CZ-energy stopping times $\mathcal{F}$,

[TABLE]

which can then be iterated to obtain geometric decay in generations,

[TABLE]

In addition we obtain the quasi-orthogonality inequality, see e.g. [LaSaShUr3] and [SaShUr7], both of which apply here since $E_{F}^{\sigma}\left(\left|\mathsf{P}_{F}f\right|\right)=E_{F}^{\sigma}\left(\left|f-E_{F}^{\sigma}f\right|\right)\leq 2E_{F}^{\sigma}\left(\left|f\right|\right)$,

[TABLE]

The finiteness of the energy characteristic $\mathcal{E}_{2}\left(\sigma,\omega\right)$ will be needed to enforce (3.3), that is in turn needed to control the far, paraproduct and stopping forms, at which point we can appeal to the following simple inequality from [LaSaUr2]222In [LaSaUr2], this inequality is stated with a one-tailed Muckenhoupt characteristic $\mathcal{A}_{2}\left(\sigma,\omega\right)$ on the right hand side, but as the reader easily observes, the display at the top of page 319 in the proof given in [LaSaUr2] holds for the smaller characteristic $A_{2}^{\mathop{\mathrm{o}ffset}}\left(\sigma,\omega\right)$.,

[TABLE]

that controls $\mathcal{E}_{2}\left(\sigma,\omega\right)$ by the testing characteristic for the Hilbert transform and the offset Muckenhoupt characteristic. Unfortunately this simple inequality fails for most other Calderón-Zygmund operators in place of the Hilbert transform, including Riesz transforms in higher dimensions, see [Saw] and [SaShUr11], and this failure limits the current proof to essentially just the Hilbert transform and similar operators on the real line.

Now we can pigeonhole the pairs of intervals arising in the sum defining the below form. Given the corona decomposition of $\mathcal{D}$ according to the Calderón-Zygmund stopping times $\mathcal{F}$ constructed above, we define the analogous decomposition of $\mathcal{P}_{\mathop{\mathrm{b}elow}}$,

[TABLE]

Then we consider the corresponding decomposition of the home form into diagonal and far forms,

[TABLE]

We next decompose the $\mathop{\mathrm{f}ar}$ form into corona pieces using $\mathcal{P}_{\mathop{\mathrm{f}ar}}^{F,G}\equiv\left[\mathcal{C}_{\mathcal{F}}\left(F\right)\times\mathcal{C}_{\mathcal{F}}\left(G\right)\right]\cap\mathcal{P}_{\mathop{\mathrm{b}elow}}$,

[TABLE]

Now for $m>\tau$ and $F\in\mathcal{F}$ we define

[TABLE]

3.1. Control of the far form by way of functional energy

Here we will control the $\mathop{\mathrm{f}ar}$ form $\mathsf{B}_{\mathop{\mathrm{f}ar}}\left(f,g\right)$ first by the local testing and functional energy characteristic. The $\mathop{\mathrm{f}ar}$ form is given by

[TABLE]

which we write with the dummy variable $G$ replaced by $F$,

[TABLE]

Given any collection $\mathcal{H}\subset\mathcal{D}$ of intervals, and a dyadic interval $J$, we define the corresponding Haar projection $\mathsf{P}_{\mathcal{H}}^{\omega}$ and its localization $\mathsf{P}_{\mathcal{H};J}^{\omega}$ to $J$ by

[TABLE]

Definition 6**.**

Given any interval $F\in\mathcal{D}$, we define the $\left(r,\varepsilon\right)$-Whitney* collection $\mathcal{M}_{\left(r,\varepsilon\right)-\mathop{\mathrm{d}eep}}\left(F\right)$ of $F$ to be the set of dyadic subintervals $W\subset F$ that are maximal with respect to the property that $W\subset_{r,\varepsilon}F$. Let $\mathcal{D}\left[W\right]\equiv\left\{J\in\mathcal{D}:J\subset W\right\}$.*

Clearly the intervals in $\mathcal{M}_{\left(r,\varepsilon\right)-\mathop{\mathrm{d}eep}}\left(F\right)$ form a pairwise disjoint decomposition of $F$.

Definition 7**.**

Let $\mathfrak{F}\left(\sigma,\omega\right)$ be the smallest constant in the ‘functional energy’ inequality below, holding for all $h\in L^{2}\left(\sigma\right)$ and all collections $\mathcal{F}\subset\mathcal{D}$, and where $x$ denotes the identity function on $\mathbb{R}$:

[TABLE]

There is a similar definition of the dual constant $\mathfrak{F}^{\ast}\left(\omega,\sigma\right)$. The Intertwining Proposition will control the following Intertwining form,

[TABLE]

whose difference from $\mathsf{B}_{\mathop{\mathrm{f}ar}}\left(f,g\right)$ is

[TABLE]

This difference form is easily controlled using the proofs of the disjoint and comparable forms above,

[TABLE]

Definition 8**.**

A collection $\mathcal{F}$ of dyadic intervals is $\sigma$-Carleson* if*

[TABLE]

The constant $C_{\mathcal{F}}$ is referred to as the Carleson norm of $\mathcal{F}$.

We now show that the functional energy inequality (3.7), together with local interval testing, suffices to prove the following Intertwining Proposition, [SaShUr7] and [SaShUr12], when $\mathcal{F}$ is $\sigma$-Carleson.

Let $\mathcal{F}$ be any subset of $\mathcal{D}$. For any $J\in\mathcal{D}$, we define $\pi_{\mathcal{F}}^{0}J$ to be the smallest $F\in\mathcal{F}$ that contains $J$. Then for $s\geq 1$, we recursively define $\pi_{\mathcal{F}}^{s}J$ to be the smallest $F\in\mathcal{F}$ that strictly contains $\pi_{\mathcal{F}}^{s-1}J$. This definition satisfies $\pi_{\mathcal{F}}^{s+t}J=\pi_{\mathcal{F}}^{s}\pi_{\mathcal{F}}^{t}J$ for all $s,t\geq 0$ and $J\in\mathcal{D}$. In particular $\pi_{\mathcal{F}}^{s}J=\pi_{\mathcal{F}}^{s}F$ where $F=\pi_{\mathcal{F}}^{0}J$. In the special case $\mathcal{F}=\mathcal{D}$ we often suppress the subscript $\mathcal{F}$ and simply write $\pi^{s}$ for $\pi_{\mathcal{D}}^{s}$. Finally, for $F\in\mathcal{F}$, we write $\mathfrak{C}_{\mathcal{F}}\left(F\right)\equiv\left\{F^{\prime}\in\mathcal{F}:\pi_{\mathcal{F}}^{1}F^{\prime}=F\right\}$ for the collection of $\mathcal{F}$-children of $F$.

Proposition 9** (The Intertwining Proposition).**

Suppose that $\mathcal{F}$ is $\sigma$-Carleson. Then

[TABLE]

Proof.

We write the left hand side of the display above as

[TABLE]

where

[TABLE]

Note that $g_{F}$ is supported in $F$, and that $f_{F}$ is constant on $F$. The intervals $I$ occurring in this sum are linearly and consecutively ordered by inclusion, along with the intervals $F^{\prime}\in\mathcal{F}$ that contain $F$. More precisely, we can write

[TABLE]

where $F_{m}=\pi_{\mathcal{F}}^{m}F$ for all $m\geq 1$. We can also write

[TABLE]

where $I_{k}=\pi_{\mathcal{D}}^{k}F$ for all $k\geq 1$, and by convention we set $I_{0}=F$. There is a (unique) subsequence $\left\{k_{m}\right\}_{m=1}^{N}$ such that

[TABLE]

Recall that

[TABLE]

Assume now that $k_{m}\leq k<k_{m+1}$. Using a telescoping sum, we compute that for

[TABLE]

we have

[TABLE]

Now $f_{F}$ is constant on $F$ and

[TABLE]

where

[TABLE]

Now we write

[TABLE]

Then local interval testing and quasi-orthogonality, together with the fact that $f_{F}$ is a constant on $F$ bounded by $\alpha_{\mathcal{F}}\left(F\right)$, give

[TABLE]

Now $\mathbf{1}_{F^{c}}f_{F}$ is supported outside $F$, and each $J$ in the Haar support of $g_{F}=\mathsf{P}_{\mathcal{C}_{\mathcal{F}}\left(F\right)}^{\omega}g$ is either in $\mathcal{N}_{r}\left(F\right)=\left\{J\in\mathcal{D}\left[F\right]:\ell\left(J\right)\geq 2^{-r}\ell\left(F\right)\right\}$, in which case the desired bound for term $I$ is straightforward, or $J$ is $\left(r,\varepsilon\right)$-deeply embedded in $F$, i.e. $J\subset_{r,\varepsilon}F$, and so $J\subset_{r,\varepsilon}W$ for some $W\in\mathcal{M}_{\left(r,\varepsilon\right)-\mathop{\mathrm{d}eep}}\left(F\right)$. Thus with $\mathcal{C}_{\mathcal{F}}^{\flat}\left(F\right)\equiv\mathcal{C}_{\mathcal{F}}\left(F\right)\setminus\mathcal{N}_{r}\left(F\right)$ we have

[TABLE]

The second factor is bounded by $C\left\|g\right\|_{L^{2}\left(\omega\right)}$, and we use the second line in the Energy Lemma 4 on the first factor to bound it by,

[TABLE]

using the definition of the functional energy characteristic $\mathfrak{F}\left(\sigma,\omega\right)$, and the dyadic $\sigma$-maximal function inequality $\left\|\Phi\right\|_{L^{p}\left(\sigma\right)}\lesssim\left\|M_{\sigma}f\right\|_{L^{p}\left(\sigma\right)}\lesssim\left\|f\right\|_{L^{p}\left(\sigma\right)}$. This completes the proof of the Intertwining Proposition 9.

Thus we have the following control of the $\mathop{\mathrm{f}ar}$ form,

[TABLE]

Finally then, to control the quadratic functional energy characteristic $\mathfrak{F}\left(\sigma,\omega\right)$, we can of course appeal to the bound in [LaSaShUr3] as a black box, to conclude that,

[TABLE]

thus completing our treatment of the $\mathop{\mathrm{f}ar}$ form.

Alternatively, we can use the following somewhat simpler argument introduced in [SaWi]. Consider the refined functional energy inequality,

[TABLE]

in which the sum in $W\in\mathcal{M}_{\left(r,\varepsilon\right)-\mathop{\mathrm{d}eep}}\left(F\right)$ is restricted to the corona $\mathcal{C}_{\mathcal{F}}\left(F\right)$. This smaller functional energy also serves to control the Intertwining Proposition by an inspection of (3.1) above. Using

[TABLE]

where $\overset{\cdot}{\bigcup}$ denotes that the sets $\mathcal{M}_{\left(r,\varepsilon\right)-\mathop{\mathrm{d}eep}}\left(F\right)\cap\mathcal{C}_{\mathcal{F}}\left(F\right)$ are pairwise disjoint, together with the orthogonality of the projections $\left\{\mathsf{P}_{\mathcal{C}_{\mathcal{F}}\left(F\right)\cap\mathcal{D}\left[W\right]}^{\omega}x\right\}_{\begin{subarray}{c}F\in\mathcal{F}\\ W\in\mathcal{M}_{\left(r,\varepsilon\right)-\mathop{\mathrm{d}eep}}\left(F\right)\cap\mathcal{C}_{\mathcal{F}}\left(F\right)\end{subarray}}$ in $L^{2}\left(\omega\right)$, this refined functional energy inequality is equivalent to the boundedness from $L^{2}\left(\sigma\right)$ to $L^{2}\left(\omega\right)$ of the operator $Th\left(x\right)\equiv\int_{\mathbb{R}}K\left(x,y\right)h\left(y\right)d\sigma\left(y\right)$ with kernel,

[TABLE]

where $Z$ denotes the identity map $Z\left(x\right)=x$ on $\mathbb{R}$.

Now we enlarge the kernel $K\left(x,y\right)$ by removing the indicator $\mathbf{1}_{R\setminus W^{\ast}}\left(y\right)$ from the right hand side and replacing the projection $\mathsf{P}_{\mathcal{C}_{\mathcal{F}}\left(F\right)\cap\mathcal{D}\left[W\right]}^{\omega}Z\left(x\right)=\sum_{J\in\mathcal{C}_{\mathcal{F}}\left(F\right)\cap\mathcal{D}\left[W\right]}\bigtriangleup_{J}^{\omega}Z\left(x\right)$ by the ‘absolute’ projection,

[TABLE]

Then boundedness of the positive linear operator $\widehat{T}h\left(x\right)\equiv\int_{\mathbb{R}}\widehat{K}\left(x,y\right)h\left(y\right)d\sigma\left(y\right)$, where

[TABLE]

is sufficient for the refined functional energy inequality.

Now the kernel $K$ is

1. (1)

essentially decreasing in the variable $y$ away from $x$, i.e.

[TABLE] 2. (2)

roughly constant in the variable $y$ on intervals away from $x$, i.e.

[TABLE] 3. (3)

and reverse Hölder in the variable $x$ on intervals away from $y$, i.e. for every $1<r<\infty$, there is $C_{r}<\infty$ such that,

[TABLE]

Properties (1) and (2) are easy consequences of the formula for the Poisson kernel, see [SaWi, (7.10) and (7.11)] for details, and see [SaWi, implicit in the proof of Theorem 51] for property (3). Now as shown in [SaWi, Section 11, the appendix], this reduces boundedness of $\widehat{T}$ to two dual testing conditions, which can then be controlled as in [SaWi, Subsections 7.3 and 7.4], and we leave these details to the reader (they simplify considerably in the case $p=2$).

4. Reduction of the diagonal form by the NTV reach

We first apply the clever ‘NTV reach’ of [NTV3], which splits the diagonal form

[TABLE]

into a paraproduct and stopping form,

[TABLE]

4.1. Control of the paraproduct form by the testing characteristic

We control the local paraproduct form $\mathsf{B}_{\mathop{\mathrm{p}ara}}^{F}\left(f,g\right)$ by the testing condition for $H$. Indeed, from the telescoping identity for Haar projections, see e.g. [LaSaUr2], we have

[TABLE]

where $I^{\natural}\left(J\right)$ is the smallest $I\in\mathcal{C}_{\mathcal{F}}\left(F\right)\cap\Lambda_{f}^{\sigma}$ (where $\Lambda_{f}^{\sigma}$ is the Haar support of $f)$, that contains $J$ and satisfies $J\subset_{r}I$.

Thus from Cauchy-Schwarz and the bound on the coefficients $\lambda_{J}\equiv E_{I^{\natural}\left(J\right)_{J}}^{\sigma}f-E_{F}^{\sigma}f$ given by $\left|\lambda_{J}\right|\lesssim\alpha_{\mathcal{F}}\left(F\right)$, we have

[TABLE]

Then summing in $F$ we obtain the following bound for the paraproduct form $\mathsf{B}_{\mathop{\mathrm{p}ara}}\left(f,g\right)\equiv\sum_{F\in\mathcal{F}}\mathsf{B}_{\mathop{\mathrm{p}ara}}^{F}\left(f,g\right)$,

[TABLE]

by quasi-orthogonality in (3.4) and orthogonality in Haar projections $\mathsf{P}_{\mathcal{C}_{\mathcal{F}}\left(F\right)}^{\omega}g$.

4.2. Control of the stopping form by the Poisson-Energy characteristic

To control the stopping form, it suffices to assume the Haar supports of $f$ and $g$ are contained in a fixed large, but finite subset $\digamma$ of the grid $\mathcal{D}$, and to uniformly control over these subsets, each local stopping form

[TABLE]

Indeed, then quasi-orthogonality in (3.4) and orthogonality in Haar projections will be used to finish control of the stopping form. In fact we will prove

[TABLE]

where $\mathrm{P}\mathsf{E}_{\mathcal{F}}\left(\sigma,\omega\right)$ is defined in (3.2). The key technical estimate needed is the following Stopping Child Lemma, which is a reformulation of the ‘straddling’ lemmas in M. Lacey [Lac, Lemmas 3.19 and 3.16].

4.2.1. The Stopping Child Lemma

Let$\,\mathcal{F}$ be a collection of $\mathop{\mathrm{g}ood}$ stopping times. Suppose $F\in\mathcal{F}$ is fixed for the moment and let $\mathcal{A}\subset\mathcal{C}_{\mathcal{F}}\left(F\right)$ be a collection of $\mathop{\mathrm{g}ood}$ stopping times with top interval $F$. For $A\in\mathcal{A}$ we define the ‘stopping child’ bilinear form,

[TABLE]

where $\left(I,J\right]\equiv\left\{K\in\mathcal{D}:J\subset K\subsetneqq I\right\}$ denotes the tower in the grid $\mathcal{D}$ with endpoints $I$ (not included) and $J$ (included), and $\Lambda_{f}^{\sigma}\left[S\right]$ is the smallest interval in the Haar support $\Lambda_{f}^{\sigma}$ of $f$ that contains $S$, and $\varphi_{J}^{S}\equiv\sum_{I\in\left(S,A\right]:\ J\subset_{\tau}\Lambda_{f}^{\sigma}\left[S\right]}\left(E_{I_{J}}^{\sigma}\bigtriangleup_{I}^{\sigma}f\right)\mathbf{1}_{A_{0}\setminus S}$. The presence of the indicator $\mathbf{1}_{F\setminus S}$ suggests the name ‘stopping child’ bilinear form. We also define a refined Poisson-Energy characteristic by,

[TABLE]

where $\Lambda_{g}^{\omega}\equiv\left\{J\in\mathcal{D}:\bigtriangleup_{J}^{\omega}g\neq 0\right\}$ is the Haar support of $g$ in $L^{2}\left(\omega\right)$, and

[TABLE]

where $\mathcal{M}_{\mathop{\mathrm{g}ood}}^{\mathop{\mathrm{t}rip}}\left(S\right)$ is the collection of maximal $\mathop{\mathrm{g}ood}$ subintervals $I$ of $S$ whose triples are contained in $S$. This characteristic depends on $\mathcal{A}$, $A$ and $\Lambda_{g}^{\omega}$, as well as on $\sigma$ and $\omega$, and is a variation on the crucial size function introduced by M. Lacey in [Lac]. Set

[TABLE]

and note the trivial inequality

[TABLE]

where $\mathrm{P}\mathsf{E}_{\mathcal{F}}^{F}\left(\sigma,\omega\right)$ and $\mathrm{P}\mathsf{E}_{\mathcal{F}}\left(\sigma,\omega\right)$ are defined in (3.2).

Lemma 10** (Stopping Child Lemma).**

(a reformulation of [Lac, Lemmas 3.19 and 3.26]) Let $f\in L^{2}\left(\sigma\right)$, $g\in L^{2}\left(\omega\right)$ have $\mathop{\mathrm{g}ood}$ Haar supports, along with their children, let $\mathcal{F}$ be a collection of $\mathop{\mathrm{g}ood}$ stopping times, and let $\mathcal{A}\subset\mathcal{C}_{\mathcal{F}}\left(F\right)$ be a collection of $\mathop{\mathrm{g}ood}$ stopping times with top $F$. Set

[TABLE]

Then for all $A\in\mathcal{A}$ we have the nonlinear bound,

[TABLE]

Proof.

By the telescoping property of martingale differences, together with the bound $\alpha_{A}\left(S\right)$ on the averages of $\mathsf{P}_{\mathcal{C}_{\mathcal{A}}\left(A\right)}^{\sigma}f$ in the $\mathop{\mathrm{g}ood}$ intervals in the tower $\left(\Lambda_{f}^{\sigma}\left[S\right],A\right]$, and the goodness assumption on $f$, we have

[TABLE]

From the Monotonicity Lemma and the fact that

[TABLE]

where $\mathcal{N}_{\tau}\left(S\right)\equiv\left\{J\subset S:\ell\left(J\right)\geq 2^{-\tau}\ell\left(S\right)\right\}$ is the set of ‘$\tau$-nearby’ dyadic intervals in $S$, we have

[TABLE]

Since

[TABLE]

for $K\in\mathcal{W}_{\mathop{\mathrm{g}ood}}^{F,\mathop{\mathrm{t}rip}}\left(S\right)\cup\mathcal{N}_{\tau}\left(S\right)$, we have with $\widehat{\mathsf{P}}_{S;K}^{\omega}\equiv\sum_{\begin{subarray}{c}J\in\Lambda_{g}^{\omega}:\ J\subset_{\tau}\Lambda_{f}^{\sigma}\left[S\right]\\ J\subset W\end{subarray}}\bigtriangleup_{J}^{\omega}$,

[TABLE]

The corresponding bound for the form $\left|\mathsf{B}_{\mathcal{A}}^{A}\right|_{\mathop{\mathrm{s}traddle}}^{\mathop{\mathrm{n}ear}}\left(f,g\right)$ is similar, but easier since there are at most $2^{\tau+1}$ intervals $J$ in $\Lambda_{g}^{\omega}\cap\mathcal{N}_{\tau}\left(S\right)$, and this completes the proof of the Stopping Child Lemma.

4.2.2. Dual tree decomposition

To control the local stopping forms $\mathsf{B}_{\mathop{\mathrm{s}top}}^{F}\left(f,g\right)$, we need to introduce further corona decompositions within each corona $\mathcal{C}_{\mathcal{F}}\left(F\right)$. These coronas will be associated to stopping intervals $\mathcal{U}\left[F\right]\subset\mathcal{C}_{\mathcal{F}}\left(F\right)$, whose construction uses a dual tree decomposition originating with M. Lacey in [Lac]. For the sake of generality, we present the decomposition in the setting of trees that are not necessarily dyadic.

Definition 11**.**

Let $\mathcal{T}$ be a tree with root $o$.

1. (1)

Let $P\left(\alpha\right)\equiv\left\{\beta\in\mathcal{T}:\beta\succeq\alpha\right\}$ and $S\left(\alpha\right)\equiv\left\{\beta\in\mathcal{T}:\beta\preceq\alpha\right\}$ denote the predessor and successor sets of $\alpha\in\mathcal{T}$. 2. (2)

A geodesic $\mathfrak{g}$ is a maximal linearly ordered subset of $\mathcal{T}$. A finite geodesic $\mathfrak{g}$ is an interval $\mathfrak{g}=\left[\alpha,\beta\right]=P\left(\beta\right)\setminus S\left(\alpha\right)$, and an infinite geodesic is an interval $\mathfrak{g}=\mathfrak{g}\setminus P\left(\alpha\right)$ for some $\alpha\in\mathfrak{g}$. 3. (3)

A **stopping time333This definition of stopping time used in the theory of trees is a slight variant of what has been used above, but should cause no confusion.* $T$ for a tree $\mathcal{T}$ is a subset $T\subset\mathcal{T}$ such that*

[TABLE] 4. (4)

A sequence $\left\{T_{n}\right\}_{n=0}^{N}$ of stopping times $T_{n}$ is decreasing if, for every $\beta\in T_{n+1}$ with $0\leq n<N$, there is $\beta^{\prime}\in T_{n}$ such that $S\left(\beta\right)\subset S\left(\beta^{\prime}\right)$. We think of such a sequence as getting further from the root as $n$ increases. 5. (5)

For $T$ a stopping time in $\mathcal{T}$ and $\alpha\in\mathcal{T}$, we define

[TABLE]

where the interval $\left[\beta,\alpha\right)=\emptyset$ unless $\beta\prec\alpha$. In the case $\left[T,\alpha\right)=\emptyset$, we write $\alpha\preceq T$, and in the case $\left[T,\alpha\right)\not=\emptyset$, we write $\alpha\succ T$. The set $\left[T,\alpha\right)$ can be thought of as the set of points in the tree $\mathcal{T}$ that ‘lie between’ $T$ and $\alpha$ but are strictly less than $\alpha$. 6. (6)

For any $\alpha\in\mathcal{T}$, we define the set of its children $\mathfrak{C}_{T}\left(\alpha\right)$ to consist of the maximal elements $\beta\in\mathcal{T}$ such that $\beta\prec\alpha$.

We define the dual integration operator $I^{\ast}$ on $\mathcal{T}$ by $I^{\ast}\mu\left(\alpha\right)\equiv\sum_{\beta\in\mathcal{T}:\ \beta\preccurlyeq\alpha}\mu\left(\beta\right)$. Here is the dual stopping time lemma that abstracts that of M. Lacey in [Lac].

Lemma 12**.**

Let $\mathcal{T}$ be a tree with root $o$, and suppose there is a uniform bound on the number of children of any tree element. Suppose further that $\mu:\mathcal{T}\rightarrow\left[0,\infty\right)$ is nontrivial with finite support, and let $T_{0}$ be the stopping time consisting of the minimal tree elements in the support of $\mu$. Fix $\Gamma>1$. If there is no element $\alpha\in\mathcal{T}$ with $I^{\ast}\mu\left(\alpha\right)>\Gamma\sum_{\beta\in\mathcal{T}:\ \beta\prec\alpha}I^{\ast}\mu\left(\beta\right)$, we say that $\mu$ is $\Gamma$-irreducible. Otherwise, there is a unique increasing sequence $\left\{T_{n}\right\}_{n=0}^{N+1}$, with $T_{N+1}=\left\{o\right\}$, of stopping times $T_{n}$ such that for all $n\in\mathbb{N}$ with $n\leq N$,

[TABLE]

Moreover, for $1\leq n\leq N+1$, this unique sequence $\left\{T_{n}\right\}_{n=0}^{N+1}$ satisfies

[TABLE]

Proof.

If $T_{n}$ is already defined, let $T_{n+1}$ consist of all minimal points $\alpha\in\mathcal{T}$ satisfying $I^{\ast}\mu\left(\alpha\right)>\Gamma\sum_{\beta\in T_{n}:\ \beta\prec\alpha}I^{\ast}\mu\left(\beta\right)$, provided at least one such point $\alpha$ exists. If not then set $N=n$ and define $T_{N+1}\equiv\left\{o\right\}$. It is easy to see that $\left\{T_{n}\right\}_{n=0}^{N+1}$ is an increasing sequence of stopping times that satisfies (4.6), and is unique with these properties. Moreover (4.7) also holds since for $\alpha\in T_{n}$ we have

[TABLE]

and the same argument proves the second line in (4.7), since $\gamma\in S\left(\alpha\right)\setminus\left\{\alpha\right\}\setminus\bigcup_{\delta\in T_{n-1}}S\left(\delta\right)$ was not chosen by the stopping criterion in the first line of (4.6), and hence

[TABLE]

4.2.3. Completion of the proof

Momentarily fix $\theta>0$ in Lemma 12, which will be chosen at the very end of the proof. For $F\in\mathcal{F}$, we consider the dyadic tree $\mathcal{T}\equiv\mathcal{D}$, and let $\mathcal{U}\left[F\right]=\mathcal{U}_{\Lambda_{g}^{\omega}}\left[F\right]$ denote the collection of intervals constructed in Lemma 12 with $o=F$, $\Gamma=1+\theta$ and $\mu_{g}:\mathcal{D}\rightarrow\left[0,\infty\right)$ defined by\ \mu_{g}\left(J\right)=\left\{\begin{array}[]{ccc}\left\|\bigtriangleup_{J}^{\omega}x\right\|_{L^{2}\left(\omega\right)}^{2}&\text{ for }&J\in\Lambda_{g}^{\omega}\\ 0&\text{ for }&J\notin\Lambda_{g}^{\omega}\end{array}\right. for $J\in\mathcal{D}$. If $\mu_{g}$ is irreducible, the ensuing arguments are greatly simplified, as the reader can easily check. Note that $\mathcal{U}\left[F\right]$ depends only only the $\omega$-Haar support $\Lambda_{g}^{\omega}$ of $g$, and not on $g$ itself. We will sometimes write just $\mathcal{U}\left[F\right]$ when $\Lambda_{g}^{\omega}$ is understood. Define $\mathop{\mathrm{M}IN}\left(\Lambda_{g}^{\omega}\right)$ to be the minimal intervals $J\in\mathcal{C}_{\mathcal{F}}^{\mathop{\mathrm{g}ood}}\left(F\right)$ for which $\bigtriangleup_{J}^{\omega}x\neq 0$. From (4.7), we have tight control of the $\mathop{\mathrm{g}ood}$ $\omega$-projections of $g$ in the coronas,

[TABLE]

for $K\in\mathcal{C}_{\mathcal{U}\left[F\right]}^{\mathop{\mathrm{g}ood}}\left(U\right)\setminus\mathop{\mathrm{M}IN}\left(\Lambda_{g}^{\omega}\right)$, where $\mathcal{D}^{\mathop{\mathrm{n}o}\mathop{\mathrm{t}op}}\left(U\right)\equiv\mathcal{D}\left(U\right)\setminus\left\{U\right\}$, and the projections are restricted to $\mathop{\mathrm{g}ood}$ intervals. We also have geometric decay in grandchildren from iterating the first line in (4.6),

[TABLE]

Definition 13**.**

Fix $f\in L^{2}\left(\sigma\right)$ and let $\mathcal{F}$ be the stopping times constructed using the criterion (3.1), that depends only on $f$, $\sigma$ and $\omega$. For $g\in L^{2}\left(\omega\right)$ let $\mathcal{U}\left[F\right]=\mathcal{U}_{\Lambda_{g}^{\omega}}\left[F\right],F\in\mathcal{F}$ be as constructed above, depending only on $g$ and $\omega$. Then for $F\in\mathcal{F}$, define the form,

[TABLE]

We will prove the bound (4.2) for $f$ and $g$. Denote the collection of pairs $\left(I,J\right)$ arising in the sum defining $\mathsf{B}_{\mathop{\mathrm{s}top}}^{F}\left(f,g\right)$ by $\mathcal{P}^{F}$, so that

[TABLE]

We use the corona decomposition associated with the stopping times $\mathcal{U}$, namely

[TABLE]

to obtain the decomposition of the stopping form $\mathsf{B}_{\mathop{\mathrm{s}top}}^{F}\left(f,g\right)$ into ‘diagonal’ and ‘far’ stopping forms,

[TABLE]

where

[TABLE]

We will control this far stopping form $\mathsf{B}_{\mathop{\mathrm{f}ar}\mathop{\mathrm{s}top}}^{F}\left(f,g\right)$ using the Stopping Child Lemma, in which we will derive geometric decay from (4.9) and the Poisson-Energy characteristic $\mathrm{P}\mathsf{E}_{\mathcal{A}}^{A,\mathop{\mathrm{t}rip}}\left(\Lambda_{g_{t}}^{\omega};\sigma,\omega\right)$ with

[TABLE]

To reduce notational clutter, we write $\mathcal{U}_{g}$ instead of $\mathcal{U}_{\Lambda_{g}^{\omega}}$, even though it depends only on the $\omega$-Haar support of $g$. Indeed, we then have for $U\in\mathcal{U}_{g}$ and $t\geq 1$,

[TABLE]

where

[TABLE]

and $g_{t}\equiv\sum_{V\in\mathfrak{C}_{\mathcal{U}_{g}}^{\left(t\right)}\left(U\right)}\mathsf{P}_{\mathcal{C}_{\mathcal{U}_{g}}\left(V\right)}^{\omega}g\equiv\mathsf{P}_{\mathcal{C}_{\mathcal{U}_{g}}^{\left(t\right)}\left(U\right)}^{\omega}g$, and where

[TABLE]

i.e. we gain geometrically in $t$ when passing from the Haar support of $g_{t}$ to that of $g$.

We now introduce the size functional of Lacey [Lac], denoted here by

[TABLE]

Recalling the previous characteristics from (3.2) and (4.3), we have

[TABLE]

Thus we have from the Stopping Child Lemma applied to each $U\in\mathcal{U}_{g}$,

[TABLE]

by Cauchy-Schwarz, boundedness of the dyadic maximal function $M_{\sigma}$ on $L^{2}\left(\sigma\right)$, and orthogonality in both $f$ and $g$.

Finally it remains to control the diagonal stopping form $\mathsf{B}_{\mathop{\mathrm{d}iag}\mathop{\mathrm{s}top}}^{F}\left(f,g\right)$, where

[TABLE]

Note that the intervals $J$ arising in $\mathsf{B}_{\mathop{\mathrm{d}iag}\mathop{\mathrm{s}top}}^{F}\left(f,g\right)$ actually lie in $\mathcal{C}_{\mathcal{U}_{g}}^{\mathop{\mathrm{n}o}\mathop{\mathrm{t}op}}\equiv\mathcal{C}_{\mathcal{U}_{g}}\setminus\left\{U\right\}$ by the requirements that $\ell\left(J\right)\leq 2^{-\tau}\ell\left(I\right)$ and $I\in\mathcal{C}_{\mathcal{U}_{g}}\left(U\right)$. In particular, the minimal stopping intervals $U\in\mathcal{U}_{g}$ do not contribute to the sum $\sum_{U\in\mathcal{U}_{g}}$ defining $\mathsf{B}_{\mathop{\mathrm{d}iag}\mathop{\mathrm{s}top}}^{F}\left(f,g\right)$.

Next we note that

[TABLE]

For $F\in\mathcal{F}$, set

[TABLE]

which is finite because of our assumption that the Haar supports of $f$ and $g$ are restricted to a fixed finite subset $\digamma$ of the grid $\mathcal{D}$. We next use (4.8), (4.10) and orthogonality of the projections $\mathsf{P}_{\mathcal{C}_{\mathcal{U}}\left(U\right)}^{\mu}$ to deduce that

[TABLE]

where $\theta>0$ is as in (4.8). Note that (4.8) applies here precisely because the tops of the coronas are missing in $\mathcal{C}_{\mathcal{U}_{\Lambda_{g}^{\omega}}}^{\mathop{\mathrm{n}o}\mathop{\mathrm{t}op}}\left(U\right)$. Indeed, for $K\in\mathcal{C}_{\mathcal{F}}^{\mathop{\mathrm{g}ood}}\left(F\right)$, it suffices to show,

[TABLE]

If $\Lambda_{g_{U}^{\mathop{\mathrm{n}o}\mathop{\mathrm{t}op}}}^{\omega}\cap\mathcal{D}\left[K\right]$ is empty, there is nothing to prove, and if $K\supset U$, the first line in (4.8) yields (4.12). In the remaining case $K\in\mathcal{C}_{\mathcal{U}_{\Lambda_{g}^{\omega}}}^{\mathop{\mathrm{n}o}\mathop{\mathrm{t}op}}\left(U\right)\setminus\mathop{\mathrm{M}IN}\left(\Lambda_{g}^{\omega}\right)$, the second line in (4.8) yields (4.12).

Since $\Theta_{\digamma}^{F}<\infty$, we conclude that $\Theta_{\digamma}^{F}\leq\frac{2C}{1-\theta}\leq 4C$ provided $0<\theta<\frac{1}{2}$. Hence we have the inequality,

[TABLE]

using (4.10), which when combined with (4.2.3), completes the required control (4.2) of the stopping form $\mathsf{B}_{\mathop{\mathrm{s}top}}^{F}\left(f,g\right)$ when the Haar supports of $f$ and $g$ are in a fixed finite subset $\digamma$ of the grid $\mathcal{D}$. Since such functions are dense in $L^{2}\left(\sigma\right)$ and $L^{2}\left(\omega\right)$ as $\digamma$ ranges over all finite subsets $\digamma$ of $\mathcal{D}$, the proof of (4.2) is complete.

Proof of Theorem 2.

Collecting all of the above form estimates proves Theorem 2.