Failure of the matrix weighted bilinear Carleson embedding theorem

TL;DR

This paper demonstrates the failure of a natural matrix weighted bilinear Carleson embedding theorem under various conditions, highlighting the necessity of specific matrix weight conditions for the theorem to hold.

Contribution

It establishes that a uniform bound on the conditioning number of the matrix weight is necessary and sufficient for the bilinear embedding, and shows the failure of several natural formulations.

Findings

Failure of natural matrix weighted bilinear Carleson embedding formulations

Necessity and sufficiency of conditioning number bounds for embedding

Proof of matrix weighted redundancy condition

Abstract

We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A2 weight is also not sufficient. Indeed, a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding. We show that any improvement of a recent matrix weighted bilinear embedding, featuring a scalar Carleson sequence and inner products instead of norms must fail. In particular, replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products. Any formulation using norms, even in the presence of a scalar Carleson sequence must fail. As a positive result, we prove the so called matrix weighted redundancy condition in full generality.

Equations

$$\Vert f{\Vert }_{L_{\text{mathbbm}{C}^d}^2(W)}^2={\int }_{Q_0}\Vert W^{1/2}(x)f(x){\Vert }_{\text{mathbbm}{C}^d}^2\mathrm{d}x={\int }_{Q_0}⟨W(x)f(x),f(x){⟩}_{\text{mathbbm}{C}^d}\mathrm{d}x<\infty .$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}{\alpha }_Q⟨w{⟩}_Q^2\lesssim ⟨w{⟩}_K,\forall K\in \mathcal{D}(Q_0)$$

$$\frac{1}{w(K)}\sum _{Q\in \mathcal{D}(K)}{\beta }_Q\lesssim 1,\forall K\in \mathcal{D}(Q_0)\iff \sum _{Q\in \mathcal{D}(Q_0)}{\beta }_Q⟨g{⟩}_{Q,w}^2\lesssim \Vert g{\Vert }_{L_{\text{mathbbm}C}^2(w)}^2,$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}{A}_Q\lesssim 1,\forall K\in \mathcal{D}(Q_0)\iff \sum _{Q\in \mathcal{D}(Q_0)}⟨A_Q⟨f{⟩}_Q,⟨f{⟩}_Q{⟩}_{\text{mathbbm}{C}^d}\lesssim \Vert f{\Vert }_{L_{\text{mathbbm}{C}^d}^2}^2.$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}⟨W{⟩}_Q{A}_Q⟨W{⟩}_Q\lesssim ⟨W{⟩}_K,\forall K\in \mathcal{D}(Q_0)$$

$$M_{W}f(x)=\underset{Q\in \mathcal{D}(Q_0),x\in Q}{\sup }\Vert W^{1/2}(x)⟨W{⟩}_Q^{-1}⟨W^{1/2}f{⟩}_Q{\Vert }_{\text{mathbbm}{C}^d},$$

$$h_Q=-|Q{|}^{-1/2}{1}_{Q_{-}}+|Q{|}^{-1/2}{1}_{Q_{+}}$$

$$[w{]}_{A_2}=\underset{Q}{\sup }⟨w{⟩}_Q⟨w^{-1}{⟩}_Q.$$

$$h_Q=a_Q^w{1}_Q+b_Q^w{h}_Q^w$$

$$\sum _{Q\in \mathcal{D}(Q_0)}|⟨w^{1/2}f,h_Q{⟩}_{L^2}⟨w^{-1/2}g,h_Q{⟩}_{L^2}|\lesssim [w{]}_{A_2}\Vert f{\Vert }_{L_{\text{mathbbm}C}^2}\Vert g{\Vert }_{L_{\text{mathbbm}C}^2}.$$

$$\frac{1}{|K|}\sum _{Q\in K}{\alpha }_Q\lesssim 1,\forall K\in \mathcal{D}(Q_0),$$

$$\frac{1}{|K|}\sum _{Q\in K}{\alpha }_Q\lesssim 1,\forall K\in \mathcal{D}(Q_0)\Rightarrow$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}{\alpha }_Q\lesssim 1,\forall K\in \mathcal{D}(Q_0)$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}{\alpha }_Q⩽1,\forall K\in \mathcal{D}(Q_0)$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}{A}_Q\lesssim 1,\forall K\in \mathcal{D}(Q_0)$$

$$\sum _{Q\in \mathcal{D}(Q_0)}|⟨A_Q⟨W{⟩}_Q^{-1}⟨W^{1/2}f{⟩}_Q,⟨W^{-1}{⟩}_Q^{-1}⟨W^{-1/2}g{⟩}_Q{⟩}_{\text{mathbbm}{C}^d}|$$

$$\sum _{Q\in \mathcal{D}(Q_0)}\Vert A_Q^{1/2}⟨W{⟩}_Q^{-1}⟨W^{1/2}f{⟩}_Q{\Vert }_{\text{mathbbm}{C}^d}\Vert A_Q^{1/2}⟨W^{-1}{⟩}_Q^{-1}⟨W^{-1/2}g{⟩}_Q{\Vert }_{\text{mathbbm}{C}^d}$$

$$[W{]}_{C_2}=\underset{x}{\sup }\kappa (W(x))=\underset{x}{\sup }\Vert W(x){\Vert }_{\mathrm{op}}\Vert W^{-1}(x){\Vert }_{\mathrm{op}}=\frac{{\lambda }_{\max }(W(x))}{{\lambda }_{\min }(W(x))},$$

$$[W{]}_{A_2}=\underset{Q}{\sup }\Vert ⟨W{⟩}_Q^{1/2}⟨W^{-1}{⟩}_Q^{1/2}{\Vert }_{\mathrm{op}}^2.$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}{A}_Q\lesssim 1,\forall K\in \mathcal{D}(Q_0)$$

$$\frac{1}{|L|}\sum _{Q\in \mathcal{D}(L)}{B}_Q\lesssim 1,\forall L\in \mathcal{D}(Q_0).$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}⟨B_Q⟨W{⟩}_K^{-1/2}⟨W^{-1}{⟩}_Q^{-1/2}e,⟨W{⟩}_K^{-1/2}⟨W^{-1}{⟩}_Q^{-1/2}e{⟩}_{\text{mathbbm}{C}^d}$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}⟨B_Q⟨W^{-1}{⟩}_Q^{-1/2}⟨W{⟩}_K^{-1/2}e,⟨W^{-1}{⟩}_Q^{-1/2}⟨W{⟩}_K^{-1/2}e{⟩}_{\text{mathbbm}{C}^d}$$

$$\displaystyle\frac{1}{|{\color[rgb]{0,0,0}K}|}\sum_{Q\in\mathcal{D}({K})}\langle B_{Q}\langle W^{-1}\rangle^{-1/2}_{Q}{f},\langle W^{-1}\rangle^{-1/2}_{Q}{\color[rgb]{0,0,0}f}\rangle_{\mathbbm{C}^{d}}$$

$$\frac{1}{|K|}\sum _{Q\in \mathcal{D}(K)}⟨W^{-1}{⟩}_Q^{-1/2}{B}_Q⟨W^{-1}{⟩}_Q^{-1/2}\lesssim ⟨W{⟩}_K,\forall K\in \mathcal{D}(Q_0).$$

$$\sum _{Q\in \mathcal{D}(Q_0)}{\alpha }_Q\Vert ⟨W{⟩}_Q^{-1}⟨W^{1/2}f{⟩}_Q{\Vert }_{\text{mathbbm}{C}^d}\Vert ⟨W^{-1}{⟩}_Q^{-1}⟨W^{-1/2}g{⟩}_Q{\Vert }_{\text{mathbbm}{C}^d}$$

$$\mu (\mathcal{K})=\sum _{Q\in \mathcal{K}}{\alpha }_Q$$

$${\int }_0^{\infty }\left(\sum _{Q\in \{F(Q)>\lambda \}}{\alpha }_Q\right)\mathrm{d}\lambda ={\int }_0^{\infty }\mu (\{F(Q)>\lambda \})\mathrm{d}\lambda =\sum _{Q\in \mathcal{D}(Q_0)}F(Q){\alpha }_Q,$$

$$F(Q)=\Vert ⟨W{⟩}_Q^{-1}⟨W^{1/2}f{⟩}_Q{\Vert }_{\text{mathbbm}{C}^d}\Vert ⟨W^{-1}{⟩}_Q^{-1}⟨W^{-1/2}g{⟩}_Q{\Vert }_{\text{mathbbm}{C}^d}$$

$$\sum _{Q\in \{F(Q)>\lambda \}}{\alpha }_Q⩽\sum _{Q\in {\mathcal{J}}_{\lambda }}\sum _{Q^{\prime }\in \mathcal{D}(Q)}{\alpha }_Q⩽\sum _{Q\in {\mathcal{J}}_{\lambda }}|Q|.$$

Full text

work partially supported by ERC grant CHRiSHarMa no. DLV-682402

1. Introduction

The Carleson embedding theorem (CET) is a classical theorem in harmonic analysis with many applications to PDE. It states that a Carleson measure gives an $L^{2}$ embedding for a function. The Carleson embedding theorem first appeared in L. Carleson’s solution of the free interpolation problem [3] and later was used in his celebrated proof of the Corona Theorem [4].

In this paper we are primarily using the language of dyadic cubes. See for example [12] for the formulation therein and the illustration of its proof via Bellman functions. One may consider any underlying measure here, so the weighted version is just the same as the unweighted version with a very similar proof.

A weighted bilinear embedding theorem (BET) was an important tool in the early days of sharp weighted theory. Indeed, it was a crucial ingredient in the first sharp weighted estimates for classical singular operators [15]. It states that a Carleson measure gives rise to a bilinear estimate, featuring two different functions. In this note we are concerned with the failure of its matrix weighted analogs.

An unweighted CET with matrix Carleson measure holds trivially, derived from the scalar case. Other than in the scalar case, the extension to the weighted setting is not trivial. First versions imposed the so–called $A_{2}$ property of the matrix weight introduced by Treil–Volberg in [20], a condition that had been absent in the scalar case. Recently Culiuc–Treil [5] obtained the matrix weighted version of this theorem without any restriction on the weight, other than it being a matrix weight.

The so–called matrix $A_{2}$ conjecture asks for the exact growth of the matrix weighted norm estimate of the Hilbert transform acting on vector functions. The conjectured growth estimate is that of a linear dependence on the matrix $A_{2}$ characteristic of the weight. Motivated by this problem, we consider the question of a bilinear version of the matrix weighted Carleson lemma that might be useful for this task.

Recently one of the authors with Pott and Reguera proved an apparently weak version, featuring a matrix weight, but a scalar Carleson sequence instead of a matrix sequence and an estimate involving inner products instead of norms [17]. We show that this formulation is optimal in that any improvement of the statement fails. Indeed, the failure stems from the maximal excentricity of the matrix weight and not from any increase in the $A_{2}$ characteristic. We show that even assuming the $A_{2}$ condition, one can get an infinite bilinear embedding. We show that a bilinear estimate can be obtained in terms of the square root of the conditioning number and this condition is necessary.

The failure of BET is natural and requires only a very simple example. It is however an important notable difference to the scalar case. It is also in a contrast to a positive result on a matrix two–weighted $T1$ theorem by Bickel–Culiuc–Treil–Wick [2].

There is an array of difficulties encountered in the task to find various sharp weighted estimates in the matrix weighted case. Further, it is also very difficult to get definite negative answers. As of today, most optimal estimates elude us. We mention [9] for some interesting positive results in the matrix weighted setting. The first quantitative estimate for the Hilbert transform in the matrix weighted setting was given in [1] as well as upper and lower square function estimates. Some of these estimates in [1] were close to their scalar weighted analogs, but none of them matched the sharp scalar estimate. We now know that the estimate for the square function with matrix weight does not change its dependence on the $A_{2}$ characteristic as compared to the scalar case. The result is due to Hytönen–Petermichl–Volberg [8] and Treil [19] where both use a stopping time argument known as sparse domination. The classical sharp scalar weighted result by Huković–Treil–Volberg [7] was proved via Bellman functions and only required the simple Carleson Lemma, not the bilinear version. The estimate for the Hilbert transform is still open, with best to date estimate by Nazarov–Petermichl–Treil–Volberg [10], missing the sharp conjecture by a half power of the $A_{2}$ characteristic. For most known operator norms, the question of sharpness is unsettled, but there usually is just a raised power on the dependence of the $A_{2}$ charateristic, not complete failure of the estimate, such as what we see in the case of BET.

The first proof of a version of the scalar bilinear Carleson lemma is found in [16] and [15] and was rather complicated, using tools and a construction implicit in the seminal article by Nazarov–Treil–Volberg [13]. The argument features a rather cleverly gaged Bellman function and three conditions on the measure sequence rather than one. It was understood for some time by the experts that two of the arising conditions were redundant. We show that this redundancy is still true in the presence of a matrix weight and a matrix sequence. A previous result only allowed for scalar Carleson sequences. Since BET fails with matrix Carleson sequences, this redundancy result does not have this particular application, but it is an interesting estimate in its own right, useful for other matrix weighted tasks such as certain maximal function estimates. It can also be used in combination with CET to alter the testing condition, i.e. changing the Carleson sequence. This step was important in some scalar proofs in the non–homogenous setting, see [18] and [6].

2. Notation and detailed history

Let us say $Q_{0}=[0,1]$ is endowed with a dyadic filtration and let $\mathcal{D}$ be the dyadic grid. We call a $d\times d$ matrix–valued function $W$ a weight if $W(x)$ is positive semidefinite almost everywhere and if $W$ and $W^{-1}$ are locally integrable. One defines $L_{\mathbbm{C}^{d}}^{2}(W)$ to be the set of vector functions with

[TABLE]

When $d=1$ this becomes the classical weighted space $L_{\mathbbm{C}}^{2}(w)$. Let us denote by $\langle\cdot\rangle_{Q}$ the average of a scalar, vector or matrix function over the cube $Q$. By $\|\cdot\|_{\operatorname{op}}$ we mean the operator norm of the matrix.

The dyadic formulation of the Carleson embedding theorem reads as follows: let $(\alpha_{Q})$ be a sequence of non–negative scalars and let $w$ be a weight. Then for $f$ supported on $Q_{0}$

[TABLE]

There is no difference if the weight is identical to $1$, with the proof in [12] being exactly the same when switching to weighted averages and renormalizing appropriately. One can rewrite the assumption and conclusion renaming $\beta_{Q}=\alpha_{Q}\langle w\rangle^{2}_{Q}$ and $g=fw^{-1/2}$ with

[TABLE]

with $\langle g\rangle_{Q,w}=\frac{1}{w(Q)}\int_{Q}gw.$ There is as an immediate consequence a CET for matrix Carleson sequences $(A_{Q})$ and the matrix weight $W=\mathbf{1}$:

[TABLE]

Here, the left hand side inequality is understood in the sense of operators and its necessity ‘$\Leftarrow$’ is seen by testing the right hand inequality on functions $f=e\mathbf{1}_{K}$ for any vector $e$ and any $K\in\mathcal{D}(Q_{0})$. The deduction of the implication ‘$\Rightarrow$’ from the scalar case is easily observed by taking trace and using equivalence of norms at the cost of a dimensional constant. See [11] for details. A weighted version does not follow from the scalar case, though. The matrix weighted CET was proved only in 2015:

Theorem 2.1** (Culiuc, Treil).**

Let $W$ be a matrix weight of size $d\times d$ and $(A_{Q})$ a sequence of positive semidefinite matrices of size $d\times d$. Then for $f$ supported in $Q_{0}$

[TABLE]

The successful argument added a twist to the ‘Bellman function with a parameter’ invented in [11], that managed to make certain non–commutative obstacles disappear, which arise from differentiating functions with matrix variables.

Theorem 2.1 implies via a simple linearization argument the norm estimate below on a Doob type maximal function with matrix measure. The obtained norm estimate does not assume the $A_{2}$ condition. See for example [17] for an exposition of the well known argument as well as a motivation of the definition of this maximal function.

Theorem 2.2**.**

Let $W$ be a matrix weight of size $d\times d$ then $M_{W}:L^{2}_{\mathbbm{C}^{d}}\rightarrow L^{2}_{\mathbbm{C}}$ defined by

[TABLE]

is bounded.

We now give some background on the occurrance and motivation for the weighted bilinear Carleson lemma in the scalar case. In 1999, Nazarov–Treil–Volberg published their paper [13] on the necessary and sufficient conditions on the scalar weights $w$ and $v$ so that the operators $T_{\sigma}:L^{2}(w)\rightarrow L^{2}(v),h_{Q}\mapsto\sigma_{Q}h_{Q}$ are uniformly bounded. Here

[TABLE]

are the Haar functions and $\sigma_{Q}=\pm 1$. Wittwer then proved that $T_{\sigma}:L^{2}(w)\rightarrow L^{2}(w)$ has operator norm uniformly bounded by a linear function of the scalar $A_{2}$ characteristic of the weight

[TABLE]

This estimate is sharp, as it was shown to be true for classical operators such as the Hilbert transform [15]. In these early proofs the idea consisted of ‘disbalancing’ the Haar basis by finding coefficients $a^{w}_{Q}$ and $b^{w}_{Q}$

[TABLE]

so that the system $h^{w}_{Q}$ becomes orthonormal in $L^{2}(w)$ and similarly for $w^{-1}$. In its dualized form, the operators $T_{\sigma}$ are uniformly dominated by the sum

[TABLE]

When replacing the Haar functions by the equation (2.4) above, we obtain four sums. The sum featuring the two non–cancellative terms arising from the indicator functions being the most difficult. It is for this sum that the bilinear Carleson lemma came to life. The first version appeared in [13] in the two weight case with a rather complicated necessary condition on the two weights $w$ and $v$. A simplified version with easy to test conditions on the single weight $w$ appeared in [16] [15]:

[TABLE]

The conditions are not necessary. Indeed it has been known that two conditions are redundant:

[TABLE]

But not even the strongest condition is necessary for the conclusion of the bilinear embedding. The interest lied in the simpler testing condition and its successful application to sharp weighted estimates. The first proofs of this bilinear lemma were quite complicated.

Indeed, in 2008 Nazarov–Treil–Volberg [14] characterized two–weight estimates for individual multipliers $T_{\sigma}$ again in the flavor of a $T1$ theorem. There is a positive extension to the matrix case by Bickel–Culiuc–Treil–Wick [2].

Here is the best to date matrix weighted version of BET featuring a scalar sequence and an inner product:

Theorem 2.6** (Petermichl, Pott, Reguera).**

Let $(\alpha_{Q})$ be a sequence of non–negative scalars. Then for $f,g$ supported in $Q_{0}$

[TABLE]

We also study matrix analogs for the redundancy whose scalar version is the implication in (2). Here is the redundancy with scalar sequence and matrix weight, which was proved recently:

Theorem 2.7** (Petermichl, Pott, Reguera).**

Let $(\alpha_{Q})$ be a non–negative sequence and $W$ a matrix weight, then

[TABLE]

3. Main Results

In this paper we prove the sharpness of the bilinear Embedding Theorem 2.6, i.e. the failure of any improvement:

Theorem 3.1**.**

Let $(A_{Q})$ be a sequence of positive semidefinite matrices. The Carleson condition

[TABLE]

does not imply the existence of a constant so that

[TABLE]

for all matrix weights $W$ and functions $f,g$ supported in $Q_{0}$. The Carleson condition also does not imply

[TABLE]

for all matrix weights $W$ and functions $f,g$ supported in $Q_{0}$, not even if we only allow scalar sequences $A_{Q}=m_{Q}$.

We find that a natural bilinear Carleson embedding theorem only holds under a very strong condition on the matrix weight:

Definition 3.5**.**

Let $W$ be a matrix weight. We define the conditioning number of a matrix weight as

[TABLE]

where $\lambda_{\max}$ and $\lambda_{\min}$ return the maximal respectively minimal eigenvalue.

This is in a sharp contrast to the matrix $A_{2}$ condition in [20], the analog of the scalar version in equation (2.3):

Definition 3.6**.**

Let $W$ be a matrix weight. Then the matrix $A_{2}$ condition is

[TABLE]

See [20] for a list of elementary properties of this characteristic, such as $[W]_{A_{2}}\geqslant 1$. We prove:

Theorem 3.7**.**

Let $(A_{Q})$ be a sequence of positive semidefinite matrices and $W$ a matrix weight. Then

[TABLE]

The condition $[W]_{C_{2}}$ is necessary and the power $1/2$ is optimal.

Further, we deduce from Theorem 2.7 in full generality:

Theorem 3.8**.**

Let $(B_{Q})$ be a sequence of positive semidefinite matrices such that

[TABLE]

Then

[TABLE]

and

[TABLE]

for all vectors ${e}$ and all matrix weights $W$.

Notice that in Theorem 3.8 putting ${f}=\langle W\rangle^{-1/2}_{{K}}{e}$ the last inequality becomes

[TABLE]

that rewrites as the operator inequality resembling (2):

[TABLE]

4. Bilinear embedding theorem

We prove Theorems 3.1 and 3.7 in this section.

Theorem 3.7.

First, we can assume that the sequence consists of scalars, by switching to the Carleson sequence of maximal eigenvalues $\alpha_{Q}=\|A_{Q}\|_{\operatorname{op}}$ at the loss of a dimensional constant. Then we have to show the inequality

[TABLE]

In order to prove this inequality, let us define

[TABLE]

for any collection $\mathcal{K}$ of dyadic cubes. Let $F$ be any non–negative function defined on the dyadic cubes. Then let denote $\{F(Q)>\lambda\}$ the collection of cubes so that $F(Q)>\lambda$. It follows that

[TABLE]

which is the classical fact on Choquet integrals. Let us pose

[TABLE]

and let for any $\lambda>0$ the set $\mathcal{J}_{\lambda}$ denote the collection of maximal dyadic intervals for which $F(Q)>\lambda$. So the integrand above becomes

[TABLE]

Now let

[TABLE]

Observe that with

[TABLE]

and

[TABLE]

by Cauchy Schwarz for all $x$ and all $Q$ there holds

[TABLE]

Observe that

[TABLE]

and for $x\in Q$ we get

[TABLE]

So if $x\in Q$ with $F(Q)>\lambda$ then also $[W]^{1/2}_{C_{2}}\Phi(x)>\lambda$. So

[TABLE]

Integrating with respect to $\mathrm{d}\lambda$ gives

[TABLE]

Using the above estimate of the maximal function, Theorem 2.2 and an application of Cauchy Schwarz finishes the proof of estimate (4). ∎

Now, we prove Theorem 3.1, the failure of any improvement of Theorem 2.6.

Theorem 3.1.

Let us take the case $d=2$ and let $a$ and $b$ be orthogonal unit vectors. Let $W=aa^{\ast}+\varepsilon^{2}bb^{\ast}$ and thus $W^{-1}=aa^{\ast}+\varepsilon^{-2}bb^{\ast}$. Letting $f=W^{1/2}b\mathbf{1}_{Q_{0}}$ and $g=W^{-1/2}a\mathbf{1}_{Q_{0}}$ we get

[TABLE]

This gives us the order $\varepsilon$ for the right hand sides of inequalities (3.1) and (3.1). Further, we choose the Carleson sequence $A_{Q_{0}}=\mathbf{1}$ and $A_{Q}=\mathbf{0}$ for all other cubes. The Carleson intensity in inequality (3.2) is 1. We get for the sum in inequality (3.1) only one term:

[TABLE]

Letting $\varepsilon\to 0$, we have shown the failure of conclusion (3.1). The conditioning number of the weight $W$ is $[W]_{C_{2}}=\varepsilon^{2}$. Again letting $\varepsilon\rightarrow 0$ shows the necessity of $[W]^{1/2}_{C_{2}}$ occurring in Theorem 3.7. To see that the inequality (3.1) also fails, choose $A_{Q_{0}}=2^{-1}(a+b)(a+b)^{\ast}$ and $A_{Q}=\mathbf{0}$ for all other cubes. We see that

[TABLE]

showing the failure of conclusion (3.1). It is easy to replace the matrix Carleson sequence by the scalar sequence $\alpha_{Q_{0}}=1$ and $\alpha_{Q}=0$ for all other cubes as the expression (3.2) does not change. ∎

As we see, the bilinear Carleson Lemma fails violently - using constant weights with a high discrepancy in their eigenvalues is sufficient. It is natural that this quantity determines the growth of the estimate and not features measured by the matrix $A_{2}$ characteristic. Notice that $[W]_{A_{2}}=1$ for our example.

5. Redundant Carleson condition

In this section we prove Theorem 3.8. We sketch the proof of Theorem 2.7. Consider the matrix valued Bellman function of matrix variables $U,V$ and scalar variable $m$

[TABLE]

This function has domain $\mathbf{1}\leqslant V^{1/2}UV^{1/2}$ and $0\leqslant m\leqslant 1.$ There holds the size estimate

[TABLE]

Indeed, $0\leqslant U-V^{-1}\leqslant U-(m+1)^{-1}V^{-1}\leqslant U.$ The function $B$ is also concave: Dropping the linear dependence on $U$, its Hessian acting on the matrix difference $\Delta V$ and scalar $\Delta m$ is a positive multiple of

[TABLE]

Observe that

[TABLE]

If we add positive multiples of these non–negative terms to the Hessian, we can write it as a negative perfect square and therefore concavity follows. We also have

[TABLE]

One can check that the variables

[TABLE]

lie in the domain of $B$. We see that $m_{K}-|K|^{-1}\alpha_{K}=2^{-1}(m_{K_{-}}+m_{K_{+}})$. The usual Bellman dynamics argument gives the estimate on the operator sum:

[TABLE]

Iterating this argument gives the desired estimate for scalar sequences $(\alpha_{Q})$. We note that if the Carleson measure is not scalar, then the function to consider may be

[TABLE]

The concavity of this function is unclear to us. Now we argue that we can conclude anyways.

Theorem 3.8.

The required observation is the following:

[TABLE]

The implied constants may depend upon $d$. Applying Theorem 2.7 to the Carleson sequence $b_{Q}=\|B_{Q}\|_{\operatorname{op}}$ gives us

[TABLE]

and thus since $b_{Q}\mathbf{1}\geqslant B_{Q}$ we obtain

[TABLE]

from which it follows that

[TABLE]

Also

[TABLE]

implies in particular

[TABLE]

and with

[TABLE]

thus

[TABLE]

and so

[TABLE]

So we also obtain with $b_{Q}\mathbf{1}\geqslant B_{Q}$ that

[TABLE]

∎