# New Characterizations and Properties of Matrix $A_{\infty}$
Weights00footnotetext: 2020 Mathematics Subject Classification. Primary 46E30;
Secondary 47A56, 15A15, 42B35.Key words and phrases: matrix weight,
Muckenhoupt condition, reverse Hölder inequality, self-improvement.This
project is partially supported by the National Key Research and Development
Program of China (Grant No. 2020YFA0712900), the National Natural Science
Foundation of China (Grant Nos. 12122102, 12371093, and 12071197), the
Fundamental Research Funds for the Central Universities (Grant No.
2233300008), and the Research Council of Finland (Grant No. 346314).

Fan Bu, Tuomas Hytönen111Corresponding author, E-mail:
<EMAIL_ADDRESS>10, 2023/Final version., Dachun Yang and
Wen Yuan

Abstract We provide several new characterizations of $A_{p,\infty}$-matrix
weights, originally introduced by A. Volberg as matrix-valued substitutes of
the classical $A_{\infty}$ weights. In analogy with the notion of
$A_{p}$-dimension of matrix weights introduced in our previous work, we
introduce the concepts of the lower and the upper dimensions of
$A_{p,\infty}$-matrix weights, which enable us to obtain sharp estimates
related to their reducing operators. In a follow-up work, these results will
play a key role in the study of function spaces with $A_{p,\infty}$-matrix
weights, which extends earlier results in the more restricted class of
$A_{p}$-matrix weights.

##  1 Introduction

For a scalar-valued weight $w$, it is well known that its membership in the
Muckenhoupt class $A_{p}$ is the characterising condition for several
inequalities involving the boundedness of some operators (like the maximal
operator [28], the Hilbert transform [21], or more general singular integrals
[6]) in the weighted space $L^{p}(w)$. On the other hand, the weaker condition
$w\in A_{\infty}$ is often enough for various other estimates in these spaces
(like the comparison of singular integrals with the maximal operators [6]).

The study of the space $L^{2}(W)$ with a matrix weight $W$ goes back to Wiener
and Masani [37, §4]. Their work focused on the prediction theory for
multivariate stochastic processes. In order to address the problems related to
multivariate random stationary process and Toeplitz operators, Treil and
Volberg [34] introduced $A_{2}$-matrix weights and proved that the Hilbert
transform is bounded on $L^{2}(W)$ if and only if $W\in A_{2}$. Extensions to
$L^{p}(W)$ with $W\in A_{p}$ for general $p\in(1,\infty)$ were later found by
Nazarov and Treil [31] and Volberg [35]. For many other results on
$A_{p}$-matrix weights and $L^{p}(W)$, we refer the reader to [1, 2, 5, 7, 8,
9, 10, 11, 16, 24, 26, 27, 29, 30].

In the same work already mentioned, Volberg [35] also introduced an analogue
of the $A_{\infty}$ condition for matrix weights. To be precise, for matrix-
valued weights, this condition splits into a family of conditions
$A_{p,\infty}$, $p\in(0,\infty)$, but they all reduce to the classical
$A_{\infty}$, when the weight is scalar-valued. In analogy with results in the
scalar case, it was observed in [35] that the equivalence of certain matrix-
weighted Triebel–Lizorkin norms remains valid for this larger class of
weights.

Our interest in $A_{p,\infty}$ weights is motivated by a more general class of
similar applications in the theory of function spaces with matrix weights.
Following the development of matrix-weighted Besov spaces
$\dot{B}^{s}_{p,q}(W)$ by Frazier and Roudenko [13, 14, 32, 33] and Bu et al.
[4], and matrix-weighted Triebel–Lizorkin spaces $\dot{F}^{s}_{p,q}(W)$ by
Frazier and Roudenko [15] and Wang et al. [36] (after some results in the
special case $\dot{F}^{0}_{p,2}(W)$ already in [31, 35] and more recently in
[25]), the present authors [3] recently introduced and investigated a rather
general class of Besov-type and Triebel–Lizorkin-type spaces with matrix
weights. However, the theory of [3] was developed for $A_{p}$-matrix weights
only. Extending these results to the broader generality of
$A_{p,\infty}$-matrix weights motivated our systematic study of this
interesting class of weights in this article. Applications to Besov-type and
Triebel–Lizorkin-type spaces with $A_{p,\infty}$-matrix weights will be
considered in a follow-up work, but the present results should also have an
independent interest and many other applications involving matrix-valued
extensions of classical results dealing with $A_{\infty}$ weights.

The organization of the remainder of this article is as follows. In Section 2,
we recall some basic concepts and results from the theory of matrix weights.
In Section 3, we introduce a new definition of the matrix $A_{p,\infty}$ class
and prove its equivalence with Volberg’s definition, together with several
further characterisations. In Section 4, we explore a certain self-improvement
property of the original definition and obtain a matrix-valued substitute of
the classical identity $A_{\infty}=\bigcup_{q\in[1,\infty)}A_{q}$. Section 5
is concerned with reverse Hölder inequalities, which are another manifestation
of self-improvement. In Section 6, we introduce and study the concepts of the
upper and the lower dimensions of $A_{p,\infty}$ weights and use them to
estimate norms of the form $\|A_{Q}A_{R}^{-1}\|$, where $A_{Q}$ and $A_{R}$
are the _reducing operators_ related to $W\in A_{p,\infty}$ on different cubes
$Q,R\subset\mathbb{R}^{n}$. Such estimates for $A_{p}$-matrix weights played a
key role in the development of the theory of matrix-weighted Triebel–Lizorkin
spaces in [3, 15], and will be extensively used in our follow-up work, where
we extend this theory to the larger class of $A_{p,\infty}$-matrix weights. In
the final Section 7, we illustrate the new concepts by several examples, which
also show the sharpness of the results obtained in Section 6.

At the end of this section, we make some conventions on notation. Through the
whole article, we work on $\mathbb{R}^{n}$. The _ball_ $B$ of
$\mathbb{R}^{n}$, centered at $x\in\mathbb{R}^{n}$ with radius
$r\in(0,\infty)$, is defined by setting

$B:=\\{y\in\mathbb{R}^{n}:\ |x-y|<r\\}=:B(x,r);$

moreover, for any $\lambda\in(0,\infty)$, $\lambda B:=B(x,\lambda r)$. A
_cube_ $Q$ of $\mathbb{R}^{n}$ always has finite edge length and all edges of
cubes are always assumed to be parallel to coordinate axes, but $Q$ is not
required to be open or closed. For any cube $Q$ of $\mathbb{R}^{n}$, let
$c_{Q}$ be its center and $\ell(Q)$ its edge length. For any
$\lambda\in(0,\infty)$ and any cube $Q$ of $\mathbb{R}^{n}$, let $\lambda Q$
be the cube with the same center of $Q$ and the edge length $\lambda\ell(Q)$.
For any $r\in\mathbb{R}$, $r_{+}$ is defined as $r_{+}:=\max\\{0,r\\}$ and
$r_{-}$ is defined as $r_{-}:=\max\\{0,-r\\}$. For any $t\in(0,\infty)$,
$\log_{+}t:=\max\\{0,\log t\\}$. For any $a,b\in\mathbb{R}$, $a\wedge
b:=\min\\{a,b\\}$ and $a\vee b:=\max\\{a,b\\}$. The symbol $C$ denotes a
positive constant which is independent of the main parameters involved, but
may vary from line to line. The symbol $A\lesssim B$ means that $A\leq CB$ for
some positive constant $C$, while $A\sim B$ means $A\lesssim B\lesssim A$. Let
$\mathbb{N}:=\\{1,2,\ldots\\}$, $\mathbb{Z}_{+}:=\mathbb{N}\cup\\{0\\}$, and
$\mathbb{Z}_{+}^{n}:=(\mathbb{Z}_{+})^{n}$. We use $\mathbf{0}$ to denote the
_origin_ of $\mathbb{R}^{n}$. For any set $E\subset\mathbb{R}^{n}$, we use
$\mathbf{1}_{E}$ to denote its _characteristic function_. For any
$p\in(0,\infty]$, the _Lebesgue space_ $L^{p}(\mathbb{R}^{n})$ has the usual
meaning, and the _local Lebesgue space_ $L^{p}_{\mathrm{loc}}(\mathbb{R}^{n})$
is defined to be the set of all measurable functions $f$ on $\mathbb{R}^{n}$
such that, for any bounded measurable set $E$,

$\|f\|_{L^{p}(E)}:=\|f\mathbf{1}_{E}\|_{L^{p}(\mathbb{R}^{n})}<\infty.$

In what follows, we denote $L^{p}(\mathbb{R}^{n})$ and
$L^{p}_{\mathrm{loc}}(\mathbb{R}^{n})$ simply, respectively, by $L^{p}$ and
$L^{p}_{\mathrm{loc}}$. For any measurable function $w$ on $\mathbb{R}^{n}$
and any measurable set $E\subset\mathbb{R}^{n}$, let

$w(E):=\int_{E}w(x)\,dx$

and, for any measurable set $E\subset\mathbb{R}^{n}$ with $|E|\in(0,\infty)$,
let

$\fint_{E}w(x)\,dx:=\frac{1}{|E|}\int_{E}w(x)\,dx.$

The _Hardy–Littlewood maximal operator_ $\mathcal{M}$ is defined by setting,
for any $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$,

(1.1) $\mathcal{M}(f)(x):=\sup_{\mathrm{ball}\,B\ni x}\fint_{B}|f(y)|\,dy.$

Also, when we prove a proposition (or the like), in its proof we always use
the same symbols as in the statement itself of that proposition (or the like).

##  2 Background on matrix weights

In this section, we recall some basic concepts and results from the theory of
matrix weights. Let us begin with some basic concepts on matrices.

For any $m,n\in\mathbb{N}$, the set of all $m\times n$ complex-valued matrices
is denoted by $M_{m,n}(\mathbb{C})$ and $M_{m,m}(\mathbb{C})$ is simply
denoted by $M_{m}(\mathbb{C})$. The zero matrix in $M_{m,n}(\mathbb{C})$ is
denoted by $O_{m,n}$ and $O_{m,m}$ is simply denoted by $O_{m}$. For any
$A:=[a_{ij}]\in M_{m,n}(\mathbb{C})$, the conjugate transpose of $A$ is
denoted by $A^{*}$. For any $A\in M_{m}(\mathbb{C})$, let

(2.1) $\|A\|:=\sup_{\vec{z}\in\mathbb{C}^{m},\,|\vec{z}|=1}|A\vec{z}|.$

In what follows, we regard $\mathbb{C}^{m}$ as $M_{m,1}(\mathbb{C})$ and let
$\vec{\mathbf{0}}:=(0,\ldots,0)^{\mathrm{T}}\in\mathbb{C}^{m}$. The matrix
$A\in M_{m}(\mathbb{C})$ is called a _Hermitian matrix_ if $A^{*}=A$ and is
called a _unitary matrix_ if $A^{*}A=I_{m}$, where $I_{m}$ is the identity
matrix. For any $A:=[a_{ij}]\in M_{m,n}(\mathbb{C})$ and
$\vec{z}:=(z_{1},\ldots,z_{m})^{T}\in\mathbb{C}^{m}$,

$\displaystyle[A\ \vec{z}]:=\begin{bmatrix}a_{11}&\cdots&a_{1,n}&z_{1}\\\
\vdots&\ddots&\vdots&\vdots\\\ a_{m1}&\cdots&a_{m,n}&z_{m}\end{bmatrix}.$

Now, we recall the concepts of positive definite matrices and nonnegative
definite matrices (see, for instance, [20, (7.1.1a) and (7.1.1b)]).

######  Definition 2.1.

A matrix $A\in M_{m}(\mathbb{C})$ is said to be _positive definite_ if, for
any $\vec{z}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}$,
$\vec{z}^{*}A\vec{z}>0$, and $A$ is said to be _nonnegative definite_ if, for
any $\vec{z}\in\mathbb{C}^{m}$, $\vec{z}^{*}A\vec{z}\geq 0$.

######  Remark 2.2.

It is well known that any nonnegative definite matrix is always Hermitian
(see, for instance, [20, Theorem 4.1.4]).

From Remark 2.2 and [20, Theorem 5.6.2(d)], we immediately deduce the
following conclusion; we omit the details.

######  Lemma 2.3.

Let $A,B\in M_{m}(\mathbb{C})$ be two nonnegative definite matrices. Then
$\|AB\|=\|BA\|$ with the same norm $\|\cdot\|$ as in (2.1).

Let $A\in M_{m}(\mathbb{C})$ be a positive definite matrix and have
eigenvalues $\\{\lambda_{i}\\}_{i=1}^{m}$. Due to [20, Theorem 2.5.6(c)], we
find that there exists a unitary matrix $U\in M_{m}(\mathbb{C})$ such that

(2.2) $A=U\operatorname{diag}\,(\lambda_{1},\ldots,\lambda_{m})U^{*}.$

Moreover, by [20, Theorem 4.1.8], we find
$\\{\lambda_{i}\\}_{i=1}^{m}\subset(0,\infty)$. The following definition is
based on these conclusions and can be found in [19, (6.2.1)] (see also [18,
Definition 1.2]).

######  Definition 2.4.

Let $A\in M_{m}(\mathbb{C})$ be a positive definite matrix and have positive
eigenvalues $\\{\lambda_{i}\\}_{i=1}^{m}$. For any $\alpha\in\mathbb{R}$,
define

$A^{\alpha}:=U\operatorname{diag}\left(\lambda_{1}^{\alpha},\ldots,\lambda_{m}^{\alpha}\right)U^{*},$

where $U$ is the same as in (2.2).

######  Remark 2.5.

From [19, p. 408], we deduce that $A^{\alpha}$ is independent of the choices
of the order of $\\{\lambda_{i}\\}_{i=1}^{m}$ and $U$, and hence $A^{\alpha}$
is well defined.

Now, we recall the concept of matrix weights (see, for instance, [31, 34,
35]).

######  Definition 2.6.

A matrix-valued function $W:\ \mathbb{R}^{n}\to M_{m}(\mathbb{C})$ is called a
_matrix weight_ if $W$ satisfies that

  1. (i)

for any $x\in\mathbb{R}^{n}$, $W(x)$ is nonnegative definite;

  2. (ii)

for almost every $x\in\mathbb{R}^{n}$, $W(x)$ is invertible;

  3. (iii)

the entries of $W$ are all locally integrable.

A _scalar weight_ is a matrix weight with $m=1$.

Next, we recall the concept of reducing operators (see, for instance, [35,
(3.1)]).

######  Definition 2.7.

Let $p\in(0,\infty)$, $W$ be a matrix weight, and $E\subset\mathbb{R}^{n}$ a
bounded measurable set satisfying $|E|\in(0,\infty)$. The matrix $A_{E}\in
M_{m}(\mathbb{C})$ is called a _reducing operator_ of order $p$ for $W$ if
$A_{E}$ is positive definite and, for any $\vec{z}\in\mathbb{C}^{m}$,

(2.3)
$\left|A_{E}\vec{z}\right|\sim\left[\fint_{E}\left|W^{\frac{1}{p}}(x)\vec{z}\right|^{p}\,dx\right]^{\frac{1}{p}},$

where the positive equivalence constants depend only on $m$ and $p$.

######  Remark 2.8.

In Definition 2.7, the existence of $A_{E}$ is guaranteed by [16, Proposition
1.2] and [13, p. 1237]; we omit the details.

The following lemma is just [3, Lemma 2.11].

######  Lemma 2.9.

Let $p\in(0,\infty)$, $W$ be a matrix weight, and $E\subset\mathbb{R}^{n}$ a
bounded measurable set satisfying $|E|\in(0,\infty)$. If $A_{E}$ is a reducing
operator of order $p$ for $W$, then, for any matrix $M\in M_{m}(\mathbb{C})$,

$\displaystyle\|A_{E}M\|\sim\left[\fint_{E}\left\|W^{\frac{1}{p}}(x)M\right\|^{p}\,dx\right]^{\frac{1}{p}},$

where the positive equivalence constants depend only on $m$ and $p$.

Corresponding to the classical weight class $A_{p}(\mathbb{R}^{n})$ (see, for
instance, [17, Definitions 7.1.1 and 7.1.3]), we have the following concept of
$A_{p}$-matrix weights (see, for instance, [15, p. 490]).

######  Definition 2.10.

Let $p\in(0,\infty)$. A matrix weight $W$ on $\mathbb{R}^{n}$ is called an
$A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})$-_matrix weight_ if $W$ satisfies that,
when $p\in(0,1]$,

$\displaystyle[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})}:=\sup_{\mathrm{cube}\,Q}\mathop{\mathrm{\,ess\,sup\,}}_{y\in
Q}\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx<\infty$

or that, when $p\in(1,\infty)$,

$\displaystyle[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})}:=\sup_{\mathrm{cube}\,Q}\fint_{Q}\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p^{\prime}}\,dy\right]^{\frac{p}{p^{\prime}}}\,dx<\infty,$

where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. The
$A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})$-matrix weights reduce to
$A_{p}(\mathbb{R}^{n})$-weights when $m=1$.

If there is no confusion, we will denote
$A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})$ simply by $A_{p}$. The following lemma
is contained in [3, Lemma 2.15 and Corollary 2.16].

######  Lemma 2.11.

Let $p\in(1,\infty)$, $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, and matrix weight
$W\in A_{p}$. Then $\widetilde{W}:=W^{-\frac{p^{\prime}}{p}}\in
A_{p^{\prime}}$ and

(2.4)
$\displaystyle[W]_{A_{p}}\sim\left[W^{-\frac{p^{\prime}}{p}}\right]_{A_{p^{\prime}}}^{\frac{p}{p^{\prime}}},$

where the positive equivalence constants depend only on $m$ and $p$. Let $Q$
be a cube and $A_{Q}$ and $\widetilde{A}_{Q}$, respectively, the reducing
operators of order $p$ for $W$ and of order $p^{\prime}$ for $\widetilde{W}$.
Then, for any $M\in M_{m}(\mathbb{C})$,

(2.5)
$\left\|A_{Q}^{-1}M\right\|\sim\left\|\widetilde{A}_{Q}M\right\|\sim\left[\fint_{Q}\left\|W^{-\frac{1}{p}}(x)M\right\|^{p^{\prime}}\,dx\right]^{\frac{1}{p^{\prime}}},$

where the positive equivalence constants depend only on $m$, $p$, and
$[W]_{A_{p}}$.

##  3 Matrix $A_{\infty}$ weights

Corresponding to $A_{\infty}(\mathbb{R}^{n})$ (see, for instance, [17,
Definitions 7.3.1]), we introduce the concept of $A_{p,\infty}$-matrix weights
as follows. A priori, this is different from the definition given by Volberg
[35], but we will shortly prove that these definitions (together with several
further characterizations) are actually equivalent.

######  Definition 3.1.

Let $p\in(0,\infty)$. A matrix weight $W$ on $\mathbb{R}^{n}$ is called an
$A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$-_matrix weight_ if $W$ satisfies
that, for any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\log_{+}\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(\cdot)\right\|^{p}\,dx\right)\in
L^{1}(Q)$

and

$\displaystyle[W]_{A_{p,\infty}}$
$\displaystyle:=[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}$
$\displaystyle:=\sup_{\mathrm{cube}\,Q}\exp\left(\fint_{Q}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)<\infty.$

The $A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$-matrix weights go back to
$A_{\infty}(\mathbb{R}^{n})$-weights when $m=1$.

If there is no confusion, we will denote
$A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$ simply by $A_{p,\infty}$. The
following lemma gives the relations between scalar and matrix weights, which
follows immediately from their definitions; we omit the details.

######  Lemma 3.2.

Let $p\in(0,\infty)$. Let $w$ be a scalar weight and consider the matrix
weight $W:=wI_{m}$, where $I_{m}$ is the identity matrix. Then

  1. (i)

$W\in A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})$ if and only if $w\in
A_{\max\\{1,p\\}}(\mathbb{R}^{n})$;

  2. (ii)

$W\in A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$ if and only if $w\in
A_{\infty}(\mathbb{R}^{n})$.

We recall a well-known example of scalar weights (see, for instance, [17,
Example 7.1.7]).

######  Lemma 3.3.

For any $a\in\mathbb{R}$ and $x\in\mathbb{R}^{n}$, define the scalar weight
$w_{a}(x):=|x|^{a}$. Then the following three statements hold.

  1. (i)

$w_{a}\in A_{\infty}(\mathbb{R}^{n})$ if and only if $a\in(-n,\infty)$;

  2. (ii)

$w_{a}\in A_{1}(\mathbb{R}^{n})$ if and only if $a\in(-n,0]$;

  3. (iii)

For any $p\in(1,\infty)$, $w_{a}\in A_{p}(\mathbb{R}^{n})$ if and only if
$a\in(-n,n(p-1))$.

######  Remark 3.4.

Lemma 2.11 does not hold for $A_{p,\infty}$. Indeed, let $p\in(1,\infty)$ and
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$. By Lemma 3.3(i), we conclude for scalar
weights that

$\displaystyle w(\cdot):=|\cdot|^{n\frac{p}{p^{\prime}}}\in
A_{\infty}(\mathbb{R}^{n})\text{ but
}w^{-\frac{p^{\prime}}{p}}(\cdot)=|\cdot|^{-n}\notin
A_{\infty}(\mathbb{R}^{n}).$

Combined with Lemma 3.2, this implies for matrix weights that

$\displaystyle W:=wI_{m}\in A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})\text{
but }W^{-\frac{p^{\prime}}{p}}=w^{-\frac{p^{\prime}}{p}}I_{m}\notin
A_{p^{\prime},\infty}(\mathbb{R}^{n},\mathbb{C}^{m}).$

This defect leads to an essential difficulty that $\|A_{Q}^{-1}M\|$ does not
have a good enough equivalent form like (2.5) and hence a series of related
estimates no longer hold.

######  Lemma 3.5.

Let $p\in(0,\infty)$, $W\in A_{p,\infty}$, and $M\in M_{m}(\mathbb{C})$ be
nonzero. Then, for any cube $Q\subset\mathbb{R}^{n}$,

(3.1)
$\displaystyle\log\left[\left\|W^{-\frac{1}{p}}(\cdot)M\right\|^{p}\right]\in
L^{1}(Q)$

and, for any cube $Q\subset\mathbb{R}^{n}$,

(3.2)
$\displaystyle\left\|A_{Q}^{-1}M\right\|^{p}\sim\exp\left\\{\fint_{Q}\log\left[\left\|W^{-\frac{1}{p}}(x)M\right\|^{p}\right]\,dx\right\\},$

where the positive equivalence constants depend only on $m$, $p$, and
$[W]_{A_{p,\infty}}$.

###### Proof.

We first show

(3.3)
$\displaystyle\log_{+}\left[\left\|W^{-\frac{1}{p}}(\cdot)M\right\|^{p}\right]\in
L^{1}(Q).$

By $W\in A_{p,\infty}$ and Lemma 2.9, we find that, for any cube
$Q\subset\mathbb{R}^{n}$,

$\displaystyle\log_{+}\left[\left\|A_{Q}W^{-\frac{1}{p}}(\cdot)\right\|^{p}\right]\in
L^{1}(Q).$

From this and Lemma 2.3, we infer that, for any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\log_{+}\left[\left\|W^{-\frac{1}{p}}(\cdot)M\right\|^{p}\right]$
$\displaystyle\leq\log_{+}\left[\left\|W^{-\frac{1}{p}}(\cdot)A_{Q}\right\|^{p}\right]+\log_{+}\left[\left\|A_{Q}^{-1}M\right\|^{p}\right]$
$\displaystyle=\log_{+}\left[\left\|A_{Q}W^{-\frac{1}{p}}(\cdot)\right\|^{p}\right]+\log_{+}\left[\left\|A_{Q}^{-1}M\right\|^{p}\right]\in
L^{1}(Q).$

This finishes the proof of (3.3).

Next, we prove (3.1) and (3.2). For any cube $Q\subset\mathbb{R}^{n}$ and
almost every $x\in\mathbb{R}^{n}$,

$\displaystyle\left\|A_{Q}^{-1}M\right\|^{p}\leq\left\|A_{Q}^{-1}W^{\frac{1}{p}}(x)\right\|^{p}\left\|W^{-\frac{1}{p}}(x)M\right\|^{p},$

which, together with (3.3), further implies that

(3.4) $\displaystyle\left\|A_{Q}^{-1}M\right\|^{p}$
$\displaystyle\leq\exp\left\\{\fint_{Q}\log\left[\left\|A_{Q}^{-1}W^{\frac{1}{p}}(x)\right\|^{p}\right]\,dx\right\\}$
$\displaystyle\quad\times\exp\left\\{\fint_{Q}\log\left[\left\|W^{-\frac{1}{p}}(x)M\right\|^{p}\right]\,dx\right\\}.$

By Lemma 2.3, Jensen’s inequality, and Lemma 2.9, we have, for any cube
$Q\subset\mathbb{R}^{n}$,

$\displaystyle\exp\left\\{\fint_{Q}\log\left[\left\|A_{Q}^{-1}W^{\frac{1}{p}}(x)\right\|^{p}\right]\,dx\right\\}$
$\displaystyle\quad=\exp\left\\{\fint_{Q}\log\left[\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}\right]\,dx\right\\}$
$\displaystyle\quad\leq\fint_{Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}\,dx\sim\left\|A_{Q}A_{Q}^{-1}\right\|^{p}=1.$

This, combined with (3.4), further implies that, for any cube
$Q\subset\mathbb{R}^{n}$,

(3.5) $\displaystyle
0<\left\|A_{Q}^{-1}M\right\|^{p}\lesssim\exp\left\\{\fint_{Q}\log\left[\left\|W^{-\frac{1}{p}}(x)M\right\|^{p}\right]\,dx\right\\}.$

Using this and (3.3), we obtain (3.1).

On the other hand, from Lemmas 2.3 and 2.9 and the definition of
$[W]_{A_{p,\infty}}$, we deduce that, for any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\exp\left\\{\fint_{Q}\log\left[\left\|W^{-\frac{1}{p}}(x)M\right\|^{p}\right]\,dx\right\\}$
$\displaystyle\quad\leq\exp\left\\{\fint_{Q}\log\left[\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|^{p}\left\|A_{Q}^{-1}M\right\|^{p}\right]\,dx\right\\}$
$\displaystyle\quad=\exp\left\\{\fint_{Q}\log\left[\left\|A_{Q}W^{-\frac{1}{p}}(x)\right\|^{p}\right]\,dx\right\\}\left\|A_{Q}^{-1}M\right\|^{p}\lesssim[W]_{A_{p,\infty}}\left\|A_{Q}^{-1}M\right\|^{p},$

which, together with (3.5), further implies that (3.2). This finishes the
proof of Lemma 3.5. ∎

Now, we recall $\widetilde{A}_{p,\infty}$-matrix weights introduced by Volberg
(see [35, (2.2)]).

######  Definition 3.6.

Let $p\in(0,\infty)$. A matrix weight $W$ is called an
$\widetilde{A}_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$-_matrix weight_ if
there exists a positive constant $C$ such that, for any cube
$Q\subset\mathbb{R}^{n}$ and any $\vec{z}\in\mathbb{C}^{m}$,

$\displaystyle\exp\left(\fint_{Q}\log\left|W^{-\frac{1}{p}}(x)\vec{z}\right|\,dx\right)\leq
C\sup_{\vec{u}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}}|(\vec{z},\vec{u})|\left[\fint_{Q}\left|W^{\frac{1}{p}}(x)\vec{u}\right|^{p}\,dx\right]^{-\frac{1}{p}}.$

In what follows, we denote
$\widetilde{A}_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$ simply by
$\widetilde{A}_{p,\infty}$. The following proposition shows that Volberg’s
condition is equivalent to ours; it also gives several other equivalent
variants of both conditions.

######  Proposition 3.7.

Let $p\in(0,\infty)$ and $W$ be a matrix weight. For any cube
$Q\subset\mathbb{R}^{n}$, let $A_{Q}$ be the reducing operator of order $p$
for $W$. Then there exists a positive constant $C$, independent of cube $Q$,
such that the following conditions are all equivalent:

  1. (i)

$W\in\widetilde{A}_{p,\infty}$ in the sense of Definition 3.6;

  2. (ii)

$\displaystyle\exp\left(\fint_{Q}\log\left|W^{-\frac{1}{p}}(x)\vec{z}\right|\,dx\right)\leq
C\left|A_{Q}^{-1}\vec{z}\right|$ for any $\vec{z}\in\mathbb{C}^{m}$;

  3. (iii)

$\displaystyle\exp\left(\fint_{Q}\log_{+}\left|W^{-\frac{1}{p}}(x)A_{Q}\vec{v}\right|\,dx\right)\leq
C$ for any $\vec{v}\in\mathbb{C}^{m}$ with $|\vec{v}|=1$;

  4. (iv)

$\displaystyle\exp\left(\fint_{Q}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}U\right\|\,dx\right)\leq
C$ for any $U\in M_{m}(\mathbb{C})$ with $\|U\|=1$;

  5. (v)

$\displaystyle\exp\left(\fint_{Q}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\,dx\right)\leq
C$;

  6. (vi)

$\displaystyle\exp\left(\fint_{Q}\log_{+}\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|\,dx\right)\,dy\right)\leq
C$;

  7. (vii)

$W\in A_{p,\infty}$ in the sense of Definition 3.1;

  8. (viii)

$\displaystyle\exp\left(\fint_{Q}\log\left\|W^{-\frac{1}{p}}(x)M\right\|\,dx\right)\leq
C\left\|A_{Q}^{-1}M\right\|$ for any $M\in M_{m}(\mathbb{C})$.

###### Proof.

(i)$\Leftrightarrow$(ii): Both conditions (i) and (ii) involve an estimate of
the same left-hand side. Concerning the right-hand sides of these estimates,
by (2.3), we conclude that, for any cube $Q\subset\mathbb{R}^{n}$ and any
$\vec{z}\in\mathbb{C}^{m}$,

$\displaystyle\sup_{\vec{u}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}}|(\vec{z},\vec{u})|\left[\fint_{Q}\left|W^{\frac{1}{p}}(x)\vec{u}\right|^{p}\,dx\right]^{-\frac{1}{p}}$
$\displaystyle\quad\sim\sup_{\vec{u}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}}\frac{|(\vec{z},\vec{u})|}{|A_{Q}\vec{u}|}=\sup_{\vec{u}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}}\frac{|(\vec{z},A_{Q}^{-1}\vec{u})|}{|\vec{u}|}=\sup_{\vec{u}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}}\frac{|(A_{Q}^{-1}\vec{z},\vec{u})|}{|\vec{u}|}=\left|A_{Q}^{-1}\vec{z}\right|.$

Thus, the right-hand sides of (i) and (ii) are also comparable, and hence the
two conditions are equivalent.

(ii)$\Rightarrow$(iii): We follow an argument contained in the proof of [35,
Lemma 3.1]. For any cube $Q\subset\mathbb{R}^{n}$ and any
$\vec{v}\in\mathbb{C}^{m}$ with $|\vec{v}|=1$,

$\displaystyle\fint_{Q}\log_{+}\left|W^{-\frac{1}{p}}(x)A_{Q}\vec{v}\right|\,dx$
$\displaystyle\quad=\fint_{Q}\log\left|W^{-\frac{1}{p}}(x)A_{Q}\vec{v}\right|\,dx+\fint_{Q}\log_{+}\left[\left|W^{-\frac{1}{p}}(x)A_{Q}\vec{v}\right|^{-1}\right]\,dx$
$\displaystyle\quad=:I(\vec{v})+II(\vec{v}).$

By the assumption (ii), we obtain, for any $\vec{v}\in\mathbb{C}^{m}$ with
$|\vec{v}|=1$,

$I(\vec{v})\leq\log\left(C\left|A_{Q}^{-1}A_{Q}\vec{v}\right|\right)=\log\left(C\left|\vec{v}\right|\right)=\log
C,$

where $C$ is the same as in (ii). On the other hand, since
$W^{\frac{1}{p}}(x)$ and $A_{Q}$ are Hermitian matrices and by the
Cauchy–Schwarz inequality, we find that, for almost every $x\in\mathbb{R}^{n}$
and for any $\vec{v}\in\mathbb{C}^{m}$ with $|\vec{v}|=1$,

$1=\left|\vec{v}\right|^{2}=\left(W^{-\frac{1}{p}}(x)A_{Q}\vec{v},W^{\frac{1}{p}}(x)A_{Q}^{-1}\vec{v}\right)\leq\left|W^{-\frac{1}{p}}(x)A_{Q}\vec{v}\right|\left|W^{\frac{1}{p}}(x)A_{Q}^{-1}\vec{v}\right|.$

This, combined with the elementary inequality $\log_{+}t<t$ for
$t\in(0,\infty)$ and (2.3), further implies that, for any
$\vec{v}\in\mathbb{C}^{m}$ with $|\vec{v}|=1$,

$\displaystyle II(\vec{v})$
$\displaystyle\leq\fint_{Q}\log_{+}\left|W^{\frac{1}{p}}(x)A_{Q}^{-1}\vec{v}\right|\,dx=\frac{1}{p}\fint_{Q}\log_{+}\left[\left|W^{\frac{1}{p}}(x)A_{Q}^{-1}\vec{v}\right|^{p}\right]\,dx$
$\displaystyle\leq\frac{1}{p}\fint_{Q}\left|W^{\frac{1}{p}}(x)A_{Q}^{-1}\vec{v}\right|^{p}\,dx\sim\frac{1}{p}\left|A_{Q}A_{Q}^{-1}\vec{v}\right|^{p}=\frac{1}{p}.$

Combining the bounds for $I(\vec{v})$ and $II(\vec{v})$, we find that (iii)
holds and $C_{\eqref{W3}}\lesssim C_{\eqref{W2}}$, where the implicit positive
constant depends only on $m$ and $p$.

(iii)$\Rightarrow$(iv): For any $i\in\\{1,\ldots,m\\}$, let
$\vec{e}_{i}:=(0,\ldots,0,1,0,\ldots,0)^{T}\in\mathbb{C}^{m}$. Then, from the
Cauchy–Schwarz inequality, we infer that, for any $M\in M_{m}(\mathbb{C})$ and
$\vec{z}:=(z_{1},\ldots,z_{m})\in\mathbb{C}^{m}$,

$\left|M\vec{z}\right|\leq\sum_{i=1}^{m}\left|z_{i}\right|\left|M\vec{e}_{i}\right|\leq\sqrt{m}\left(\sum_{i=1}^{m}\left|z_{i}\right|^{2}\right)^{1/2}\max_{i\in\\{1,\ldots,m\\}}\left|M\vec{e}_{i}\right|$

and hence

$\displaystyle\|M\|\leq\sqrt{m}\max_{i\in\\{1,\ldots,m\\}}\left|M\vec{e}_{i}\right|.$

By this, we conclude that, for any cube $Q\subset\mathbb{R}^{n}$ and any $U\in
M_{m}(\mathbb{C})$ with $\|U\|=1$,

(3.6) $\displaystyle\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}^{-1}U\right\|$
$\displaystyle\leq\log_{+}\left[\sqrt{m}\max_{i\in\\{1,\ldots,m\\}}\left|W^{-\frac{1}{p}}(x)A_{Q}U\vec{e}_{i}\right|\right]$
$\displaystyle\leq\log\sqrt{m}+\sum_{i=1}^{m}\log_{+}\left|W^{-\frac{1}{p}}(x)A_{Q}U\vec{e}_{i}\right|$
$\displaystyle\leq\log\sqrt{m}+\sum_{i=1}^{m}\log_{+}\left|W^{-\frac{1}{p}}(x)A_{Q}\frac{U\vec{e}_{i}}{|U\vec{e}_{i}|}\right|,$

where the last step follows from the fact that
$|U\vec{e}_{i}|\leq\|U\|\,|\vec{e}_{i}|=1$. (If $U\vec{e}_{i}=0$, then the
$i$-th term in the sum vanishes and can simply be ignored.) Integrating both
sides of (3.6) and using the assumption (iii) with
$U\vec{e}_{i}/|U\vec{e}_{i}|$ in place of $\vec{v}_{i}$ for each
$i\in\\{1,\ldots,m\\}$, we obtain

$\fint_{Q}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}^{-1}U\right\|\,dx\leq\log\sqrt{m}+\sum_{i=1}^{m}C=\log\sqrt{m}+Cm,$

where $C$ is the same as in (iii). Thus, (iv) holds and
$C_{\eqref{W4}}\leq\log\sqrt{m}+C_{\eqref{W3}}m$.

(iv)$\Rightarrow$(v): This is obvious.

(v)$\Leftrightarrow$(vi): Let us first change the integration variable in (v)
to $y$. Taking both sides of (v) to power $p$ and using the usual rules, we
can further replace $\|W^{-\frac{1}{p}}(y)A_{Q}\|$ by
$\|W^{-\frac{1}{p}}(y)A_{Q}\|^{p}$. On the other hand, we have

$\left\|W^{-\frac{1}{p}}(y)A_{Q}\right\|^{p}=\left\|A_{Q}W^{-\frac{1}{p}}(y)\right\|^{p}\sim\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx.$

Observe that, if $A\leq cB$, where $c$ is a positive constant, then
$\log_{+}A\leq\log_{+}c+\log_{+}B$. From these three observations, the
equivalence (v)$\Leftrightarrow$(vi) easily follows.

(vi)$\Rightarrow$(vii): This is immediate from the definition and the estimate
$\log\leq\log_{+}$.

(vii)$\Rightarrow$(viii): This was proved in Lemma 3.5.

(viii)$\Rightarrow$(ii): This follows, with $C_{\eqref{W2}}\leq
C_{\eqref{W5}}$, from applying (viii) to $M=[O_{m,m-1}\ \vec{z}]$.

We have thus proved that
(i)$\Leftrightarrow$(ii)$\Rightarrow$(iii)$\Rightarrow$(iv)$\Rightarrow$(v)$\Leftrightarrow$(vi)$\Rightarrow$(vii)$\Rightarrow$(viii)$\Rightarrow$(ii),
and this shows that all these conditions are equivalent, which completes the
proof of Proposition 3.7. ∎

The following proposition gives an alternative way of computing the
$A_{p,\infty}$ characteristic, which is sometimes useful.

######  Proposition 3.8.

Let $p\in(0,\infty)$ and $W$ be a matrix weight. Assume, for any cube
$Q\subset\mathbb{R}^{n}$,

$\displaystyle\log_{+}\left(\int_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(\cdot)\right\|^{p}\,dx\right)\in
L^{1}(Q).$

Then

(3.7) $\displaystyle[W]_{A_{p,\infty}}$
$\displaystyle=\sup_{\mathrm{cube}\,Q}\sup_{H\in\mathcal{F}_{Q}}\left[\fint_{Q}\left\|W^{\frac{1}{p}}(y)H(y)\right\|^{p}\,dy\right]^{-1}$
$\displaystyle\quad\times\exp\left(\fint_{Q}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)H(y)\right\|^{p}\,dx\right)\,dy\right),$

where

$\displaystyle\mathcal{F}_{Q}:=$ $\displaystyle\Bigg{\\{}H:\ \mathbb{R}^{n}\to
M_{m}(\mathbb{C})\text{ measurable}:\
\fint_{Q}\left\|W^{\frac{1}{p}}(y)H(y)\right\|^{p}\,dy\neq 0,$
$\displaystyle\qquad\log_{+}\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)H(\cdot)\right\|^{p}\,dx\right)\in
L^{1}(Q)\Bigg{\\}}.$

###### Proof.

Taking $H=W^{-\frac{1}{p}}$, we find that the right-hand side of (3.7) is at
least as big as $[W]_{A_{p,\infty}}$. Now, we show that they are equal. By the
definition of $[W]_{A_{p,\infty}}$ and Jensen’s inequality, we have, for any
cube $Q\subset\mathbb{R}^{n}$ and any $H\in\mathcal{F}_{Q}$,

$\displaystyle\exp\left(\fint_{Q}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)H(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\leq\exp\left(\fint_{Q}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\qquad\times\exp\left\\{\fint_{Q}\log\left[\left\|W^{\frac{1}{p}}(y)H(y)\right\|^{p}\right]\,dy\right\\}$
$\displaystyle\quad\leq[W]_{A_{p,\infty}}\fint_{Q}\left\|W^{\frac{1}{p}}(y)H(y)\right\|^{p}\,dy.$

The proof of Proposition 3.7 is finished after dividing both sides by the last
factor and taking supremum over all functions $H\in\mathcal{F}_{Q}$ and all
cubes $Q$. ∎

As a consequence of the previous result, we obtain the following
distributional estimate.

######  Corollary 3.9.

Let $p\in(0,\infty)$, $W\in A_{p,\infty}$, and
$\\{A_{Q}\\}_{\mathrm{cube}\,Q}$ be a family of reducing operators of order
$p$ for $W$. Then there exists a positive constant $C$, depending only on $m$
and $p$, such that, for any cube $Q\subset\mathbb{R}^{n}$ and any
$M\in(0,\infty)$,

$\displaystyle\left|\left\\{y\in Q:\
\left\|A_{Q}W^{-\frac{1}{p}}(y)\right\|^{p}\geq
e^{M}\right\\}\right|\leq\frac{\log(C[W]_{A_{p,\infty}})}{M}|Q|.$

###### Proof.

Let cube $Q\subset\mathbb{R}^{n}$ and $M\in(0,\infty)$ be fixed. We make use
of Proposition 3.8 with

$\displaystyle H:=W^{-\frac{1}{p}}\mathbf{1}_{Q\cap
E_{Q}}+A_{Q}^{-1}\mathbf{1}_{Q\setminus E_{Q}},$

where

$\displaystyle E_{Q}:=\left\\{y\in Q:\
\left\|A_{Q}W^{-\frac{1}{p}}(y)\right\|^{p}\geq e^{M}\right\\}.$

We first prove that $H$ belongs to $\mathcal{F}_{Q}$ defined in Proposition
3.8. By the definition of $H$, we obtain

$\displaystyle\fint_{Q}\left\|W^{\frac{1}{p}}(y)H(y)\right\|^{p}\,dy=\frac{1}{|Q|}\left[|E_{Q}|+\int_{Q\setminus
E_{Q}}\left\|W^{\frac{1}{p}}(y)A_{Q}^{-1}\right\|^{p}\,dy\right],$

which, together with Definition 2.6(ii) and Lemma 2.9, further implies that

(3.8) $\displaystyle
0<\fint_{Q}\left\|W^{\frac{1}{p}}(y)H(y)\right\|^{p}\,dy\leq
1+\fint_{Q}\left\|W^{\frac{1}{p}}(y)A_{Q}^{-1}\right\|^{p}\,dy\sim
1+\left\|A_{Q}A_{Q}^{-1}\right\|^{p}\sim 1.$

Moreover, for any $y\in Q$,

$\displaystyle\log_{+}\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)H(y)\right\|^{p}\,dx\right)$
$\displaystyle=\mathbf{1}_{E_{Q}}(y)\log_{+}\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)$
$\displaystyle\quad+\mathbf{1}_{Q\setminus
E_{Q}}(y)\log_{+}\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}\,dx\right),$

where the first term is in $L^{1}(Q)$ by the definition of $W\in A_{p,\infty}$
and the second term is clearly in the same space as an indicator function
times a constant. These, combined with (3.8), further imply that
$H\in\mathcal{F}_{Q}$.

From Lemma 2.9 and the definitions of $H$ and $E_{Q}$, we infer that

$\displaystyle\exp\left(\fint_{Q}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)H(y)\right\|^{p}dx\right)\,dy\right)$
$\displaystyle\quad\sim\exp\left\\{\fint_{Q}\log\left[\|A_{Q}H(y)\|^{p}\right]\,dy\right\\}$
$\displaystyle\quad=\exp\left(\frac{1}{|Q|}\left\\{\int_{E_{Q}}\log\left[\left\|A_{Q}W^{-\frac{1}{p}}(y)\right\|^{p}\right]\,dy+\int_{Q\setminus
E_{Q}}\log\left(\left\|A_{Q}A_{Q}^{-1}\right\|^{p}\right)\,dy\right\\}\right)$
$\displaystyle\quad\geq\exp\left(\frac{1}{|Q|}\int_{E_{Q}}M\,dy\right)=\exp\left(\frac{|E_{Q}|M}{|Q|}\right),$

which, together with (3.8) and Proposition 3.8, further implies that

$\displaystyle[W]_{A_{p,\infty}}\gtrsim\exp\left(\frac{|E_{Q}|M}{|Q|}\right),$

from which the claimed bound follows by taking logarithms of both sides. This
finishes the proof of Corollary 3.9. ∎

The following variant is a restatement of [35, Lemma 3.1], there stated for
matrix weights satisfying the $\widetilde{A}_{p,\infty}$ condition of
Definition 3.6, which is equivalent to $A_{p,\infty}$ by Proposition 3.7.
Applying an argument similar to that used in the proof of Corollary 3.9, we
could give another proof of Lemma 3.10, but we omit the details.

######  Lemma 3.10.

Let $p\in(0,\infty)$, $W\in A_{p,\infty}$, and
$\\{A_{Q}\\}_{\mathrm{cube}\,Q}$ be a family of reducing operators of order
$p$ for $W$. Let $Q$ be a cube of $\mathbb{R}^{n}$ and $\\{Q_{j}\\}_{j\in
J}\subset Q$ a pairwise disjoint cube sequence. Let $M\in(0,\infty)$. If, for
any $j\in J$, $\|A_{Q}A_{Q_{j}}^{-1}\|^{p}\geq e^{M}$, then there exists a
positive constant $C$, depending only on $m$ and $p$, such that

$\displaystyle\sum_{j\in J}|Q_{j}|\leq\frac{\log(C[W]_{A_{p,\infty}})}{M}|Q|.$

##  4 Self-improvement properties

The following proposition is another equivalent characterization of
$A_{p,\infty}$-matrix weights. It can be seen as a self-improvement property
in the sense that the integrability of a logarithm, as in the equivalent
conditions of Proposition 3.7, already implies the integrability of a positive
power of the same function.

######  Proposition 4.1.

Let $p\in(0,\infty)$ and $W$ be a matrix weight. Then $W\in A_{p,\infty}$ if
and only if there exists $u\in(0,\infty)$ such that

(4.1)
$\displaystyle\sup_{\mathrm{cube}\,Q\subset\mathbb{R}^{n}}\fint_{Q}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|^{u}\,dx<\infty,$

where $A_{Q}$ is the reducing operator of order $p$ for $W$.

###### Proof.

We prove that (4.1) is equivalent to condition (v) of Proposition 3.7, which
is equivalent to $W\in A_{p,\infty}$ by the said proposition. From the
elementary inequality $\log_{+}t\leq(es)^{-1}t^{s}$ for any
$t,s\in(0,\infty)$, it is immediate that (4.1) implies condition (v) of
Proposition 3.7.

The converse implication is more delicate and involves an argument somewhat in
the spirit of the John–Nirenberg inequality for the exponential integrability
of BMO functions. Condition (v) of Proposition 3.7 is clearly equivalent to

$\displaystyle
M:=\sup_{\mathrm{cube}\,Q\subset\mathbb{R}^{n}}\fint_{Q}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\,dx<\infty.$

Let cube $Q\subset\mathbb{R}^{n}$ be fixed. Without loss of generality, we may
assume $Q$ is a dyadic cube. Let $\\{Q_{i}\\}_{i\in I}\subset Q$ be its
maximal dyadic subcubes such that, for any $i\in I$,

$\displaystyle\fint_{Q_{i}}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\,dx>2M.$

This, combined with the definition of $M$, further implies that

(4.2) $\displaystyle\sum_{i\in I}|Q_{i}|$
$\displaystyle<\frac{1}{2M}\sum_{i\in
I}\int_{Q_{i}}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\,dx$
$\displaystyle\leq\frac{1}{2M}\int_{Q}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\,dx\leq\frac{1}{2M}M|Q|=\frac{1}{2}|Q|.$

For any $i\in I$, let $\widehat{Q}_{i}:=\min\\{R\in\mathscr{Q}:\
Q_{i}\subsetneqq R\\}$. By the maximality, we obtain, for any $i\in I$,

(4.3)
$\displaystyle\fint_{Q_{i}}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\,dx\leq
2^{n}\fint_{\widehat{Q}_{i}}\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\,dx\leq
2^{n+1}M.$

We can then write

(4.4)
$\displaystyle\mathbf{1}_{Q}\log_{+}\left\|W^{-\frac{1}{p}}A_{Q}\right\|=\mathbf{1}_{Q\setminus\bigcup_{i}Q_{i}}\log_{+}\left\|W^{-\frac{1}{p}}A_{Q}\right\|+\sum_{i\in
I}\mathbf{1}_{Q_{i}}\log_{+}\left\|W^{-\frac{1}{p}}A_{Q}\right\|.$

From the definition of $\\{Q_{i}\\}_{i\in I}$ and Lebesgue’s differentiation
theorem (see, for instance, [17, Corollary 2.1.16]), we deduce that, for any
$x\notin\bigcup_{i\in I}Q_{i}$,

(4.5) $\displaystyle\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\leq 2M.$

On the other hand, for any $i\in I$ and $x\in Q_{i}$, we observe that

$\displaystyle\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\leq\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q_{i}}\right\|+\log_{+}\left\|A_{Q_{i}}^{-1}A_{Q}\right\|,$

where, for any $y\in Q_{i}$,

$\displaystyle\log_{+}\left\|A_{Q_{i}}^{-1}A_{Q}\right\|\leq\log_{+}\left\|A_{Q_{i}}^{-1}W^{\frac{1}{p}}(y)\right\|+\log_{+}\left\|W^{-\frac{1}{p}}(y)A_{Q}\right\|.$

Taking the average over $y\in Q_{i}$ and using (4.3) for the second term, we
obtain

$\displaystyle\log_{+}\left\|A_{Q_{i}}^{-1}A_{Q}\right\|\leq\fint_{Q_{i}}\log_{+}\left\|A_{Q_{i}}^{-1}W^{\frac{1}{p}}(y)\right\|\,dy+2^{n+1}M.$

Finally, we use again the elementary inequality $\log_{+}t\leq(ep)^{-1}t^{p}$
for any $t,p\in(0,\infty)$ and Lemmas 2.3 and 2.9 to find that

$\displaystyle\fint_{Q_{i}}\log_{+}\left\|A_{Q_{i}}^{-1}W^{\frac{1}{p}}(y)\right\|\,dy\lesssim\fint_{Q_{i}}\left\|A_{Q_{i}}^{-1}W^{\frac{1}{p}}(y)\right\|^{p}\,dy\sim\left\|A_{Q_{i}}^{-1}A_{Q_{i}}\right\|^{p}=1.$

Combining the previous few estimates, we have checked that there exists a
positive constant $\widetilde{M}$ such that, for any $i\in I$ and $x\in
Q_{i}$,

$\displaystyle\log_{+}\left\|W^{-\frac{1}{p}}A_{Q}\right\|\leq\log_{+}\left\|W^{-\frac{1}{p}}A_{Q_{i}}\right\|+\widetilde{M}.$

This, together with (4.4) and (4.5), further implies that

(4.6)
$\displaystyle\mathbf{1}_{Q}\log_{+}\left\|W^{-\frac{1}{p}}A_{Q}\right\|$
$\displaystyle\leq 2M\mathbf{1}_{Q\setminus\bigcup_{i\in I}Q_{i}}+\sum_{i\in
I}\mathbf{1}_{Q_{i}}\left(\log_{+}\left\|W^{-\frac{1}{p}}A_{Q_{i}}\right\|+\widetilde{M}\right)$
$\displaystyle\leq C\mathbf{1}_{Q}+\sum_{i\in
I}\mathbf{1}_{Q_{i}}\log_{+}\left\|W^{-\frac{1}{p}}A_{Q_{i}}\right\|,$

where $C:=\max\\{2M,\widetilde{M}\\}$ and $\sum_{i\in
I}|Q_{i}|\leq\frac{1}{2}|Q|$ by (4.2).

Each term on the right-hand side of (4.6) has the same form as the expression
on the left-hand side, and hence we can iterate this estimate. After
infinitely many iterations, we arrive at

$\displaystyle\mathbf{1}_{Q}\log_{+}\left\|W^{-\frac{1}{p}}A_{Q}\right\|\leq
C\sum_{k=0}^{\infty}\mathbf{1}_{\Omega_{k}}=C\sum_{k=0}^{\infty}(k+1)\mathbf{1}_{\Omega_{k}\setminus\Omega_{k+1}},$

where $\Omega_{0}=Q$ and $\Omega_{1}=\bigcup_{i\in I}Q_{i}$ are the cubes
constructed in the first step with $|\Omega_{1}|\leq\frac{1}{2}|\Omega_{0}|$
and, in general, $\Omega_{k}\subset\Omega_{k-1}$ is a union of cubes with
$|\Omega_{k}|\leq\frac{1}{2}|\Omega_{k-1}|\leq\cdots\leq 2^{-k}|Q|$. From
these, we infer that, if $u\in(0,\frac{\log 2}{C})$, then

$\displaystyle\fint_{Q}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|^{u}\,dx$
$\displaystyle\leq\fint_{Q}\exp\left(u\log_{+}\left\|W^{-\frac{1}{p}}(x)A_{Q}\right\|\right)\,dx$
$\displaystyle\leq\fint_{Q}\sum_{k=0}^{\infty}e^{uC(k+1)}\mathbf{1}_{\Omega_{k}\setminus\Omega_{k+1}}\,dx\leq\sum_{k=0}^{\infty}e^{uC(k+1)}2^{-k}<\infty.$

This finishes the proof of Proposition 4.1. ∎

Now, we can show two basic properties of $A_{p}$-matrix weights and
$A_{p,\infty}$-matrix weights. Recall that, when $m=1$, we have
$A_{p,\infty}(\mathbb{R}^{n},\mathbb{C})=A_{\infty}(\mathbb{R}^{n})$ and
$A_{p}(\mathbb{R}^{n},\mathbb{C})=A_{\max\\{1,p\\}}(\mathbb{R}^{n})$; hence
part (ii) of the following proposition recovers the classical identity
$A_{\infty}(\mathbb{R}^{n})=\bigcup_{q\in[1,\infty)}A_{q}(\mathbb{R}^{n})$ for
scalar weights and can be seen as a generalisation of this identity to the
case of matrix weights.

######  Proposition 4.2.

Let $p\in(0,\infty)$. Then the following two statements hold.

  1. (i)

If $q\in(p,\infty)$, then $A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})\subset
A_{q,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$. Moreover, there exists a
positive constant $C$, depending only on $m$, $p$, and $q$, such that, for any
matrix weight $W$,

$\displaystyle[W]_{A_{q,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}\leq
C[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}.$

  2. (ii)

It holds that $A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})\subsetneqq
A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})\subset\bigcup_{q\in(0,\infty)}A_{q}(\mathbb{R}^{n},\mathbb{C}^{m})$.
Moreover, there exists a positive constant $C$, depending only on $m$ and $p$,
such that, for any matrix weight $W$,

$\displaystyle[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}\leq
C[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})},$

where $C=1$ when $p\in(0,1]$.

###### Proof.

We first prove (i). Let us begin by recalling a useful estimate. By the proof
of [27, Lemma 2] (in which the two exponents, used in reverse roles in [27,
Lemma 2] compared to the present situation, satisfy $1\leq p<q<\infty$, but
its proof still works for any $0<p<q<\infty$), we obtain, for any
$x\in\mathbb{R}^{n}$ and almost every $y\in\mathbb{R}^{n}$,

(4.7)
$\displaystyle\left\|W^{\frac{1}{q}}(x)W^{-\frac{1}{q}}(y)\right\|^{q}\lesssim\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}.$

This, combined with Definition 3.1, further implies that

$\displaystyle[W]_{A_{q,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}\lesssim[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}.$

This finishes the proof of (i).

Now, we show (ii). We first prove $A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})\subset
A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$. To this end, we consider the
following two cases on $p$.

_Case 1)_ $p\in(0,1]$. In this case, for any cube $Q\subset\mathbb{R}^{n}$ and
almost every $y\in\mathbb{R}^{n}$, let

$\displaystyle
F(Q,y):=\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx.$

Then, for any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\exp\left(\fint_{Q}\log
F(Q,y)\,dy\right)\leq\mathop{\operatorname{ess\,sup}}_{y\in Q}F(Q,y)$

and hence
$[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}\leq[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})}$.
This finishes the proof of $A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})\subset
A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$ in this case.

_Case 2)_ $p\in(1,\infty)$. In this case, by Jensen’s inequality, we conclude
that

$\displaystyle\exp\left(\fint_{Q}\log
F(Q,y)\,dy\right)=\left\\{\exp\left(\fint_{Q}\log\left\\{[F(Q,y)]^{\frac{p^{\prime}}{p}}\right\\}\,dy\right)\right\\}^{\frac{p}{p^{\prime}}}\leq\left\\{\fint_{Q}[F(Q,y)]^{\frac{p^{\prime}}{p}}\,dy\right\\}^{\frac{p}{p^{\prime}}},$

which, together with (2.4), further implies that

$\displaystyle[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}\leq\left[W^{-\frac{p^{\prime}}{p}}\right]_{A_{p^{\prime}}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\frac{p}{p^{\prime}}}\sim[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})}.$

This finishes the proof of $A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})\subset
A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$ in this case.

Next, we show $A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})\neq
A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$. We choose
$q\in(n(p-1)_{+},\infty)$. From Lemma 3.3, we deduce that

$\displaystyle w(\cdot):=|\cdot|^{q}\in A_{\infty}(\mathbb{R}^{n})\setminus
A_{\max\\{1,p\\}}(\mathbb{R}^{n}).$

This, combined with Lemma 3.2, further implies that

$\displaystyle W:=wI_{m}\in
A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})\setminus
A_{p}(\mathbb{R}^{n},\mathbb{C}^{m}).$

This finishes the proof of $A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})\neq
A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$.

Finally, we prove
$A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})\subset\bigcup_{q\in(0,\infty)}A_{q}(\mathbb{R}^{n},\mathbb{C}^{m})$.
Let $W\in A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$. By Proposition 4.1 and
Lemmas 2.3 and 2.9, we find that there exists $u\in(0,\infty)$ such that

$\displaystyle\sup_{\mathrm{cube}\,Q\subset\mathbb{R}^{n}}\fint_{Q}\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right]^{u}\,dy<\infty.$

This, together with (2.4), (4.7), and Hölder’s inequality, further implies
that, for any $q\in(p,\infty)$ with $\frac{q^{\prime}}{q}\leq u$,

$\displaystyle[W]_{A_{q}(\mathbb{R}^{n},\mathbb{C}^{m})}$
$\displaystyle\sim\left[W^{-\frac{q^{\prime}}{q}}\right]_{A_{q^{\prime}}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\frac{q}{q^{\prime}}}$
$\displaystyle=\sup_{\mathrm{cube}\,Q\subset\mathbb{R}^{n}}\left\\{\fint_{Q}\left[\fint_{Q}\left\|W^{\frac{1}{q}}(x)W^{-\frac{1}{q}}(y)\right\|^{q}\,dx\right]^{\frac{q^{\prime}}{q}}\,dy\right\\}^{\frac{q}{q^{\prime}}}$
$\displaystyle\lesssim\sup_{\mathrm{cube}\,Q\subset\mathbb{R}^{n}}\left\\{\fint_{Q}\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right]^{\frac{q^{\prime}}{q}}\,dy\right\\}^{\frac{q}{q^{\prime}}}$
$\displaystyle\leq\sup_{\mathrm{cube}\,Q\subset\mathbb{R}^{n}}\left\\{\fint_{Q}\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right]^{u}\,dy\right\\}^{\frac{1}{u}}<\infty$

and hence $W\in A_{q}(\mathbb{R}^{n},\mathbb{C}^{m})$. This finishes the proof
of
$A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})\subset\bigcup_{q\in(0,\infty)}A_{q}(\mathbb{R}^{n},\mathbb{C}^{m})$
and hence (ii), which completes the proof of Proposition 4.2. ∎

##  5 Reverse Hölder inequalities

One of the most important properties of classical $A_{\infty}$ weights is the
reverse Hölder inequality that goes back to [6]. To establish a reverse Hölder
inequality for $A_{p,\infty}$-matrix weights, we need several technical
lemmas. The following conclusion is the reverse Hölder inequality of scalar
weights (see, for instance, [23, Theorem 2.3] or [22, Theorem 2.3(a)]).

######  Lemma 5.1.

Let $w$ be a scalar weight satisfying

$\displaystyle[w]_{A_{\infty}(\mathbb{R}^{n})}^{*}:=\sup_{\mathrm{cube}\,Q}\frac{1}{w(Q)}\int_{Q}\mathcal{M}(w\mathbf{1}_{Q})(x)\,dx<\infty,$

where $\mathcal{M}$ is the same as in (1.1). Then, for any

$\displaystyle
r\in\left[1,1+\frac{1}{2^{n+1}[w]_{A_{\infty}(\mathbb{R}^{n})}^{*}-1}\right]$

and any cube $Q\subset\mathbb{R}^{n}$,

$\fint_{Q}[w(x)]^{r}\,dx\leq 2\left[\fint_{Q}w(x)\,dx\right]^{r}.$

For any scalar weight $w$, from Lebesgue’s differentiation theorem (see, for
instance, [17, Corollary 2.1.16]), we deduce that, for any cube
$Q\subset\mathbb{R}^{n}$,

$\displaystyle\int_{Q}\mathcal{M}(w\mathbf{1}_{Q})(x)\,dx\geq\int_{Q}w(x)\,dx=w(Q)$

and hence $[w]_{A_{\infty}(\mathbb{R}^{n})}^{*}\in[1,\infty]$.

The following lemma is just [22, Proposition 2.2].

######  Lemma 5.2.

There exists a positive constant $\widetilde{C}$ such that, for any scalar
weight $w$,

$\displaystyle[w]_{A_{\infty}(\mathbb{R}^{n})}^{*}\leq\widetilde{C}[w]_{A_{\infty}(\mathbb{R}^{n})}.$

Next, we establish a useful relation between scalar weights and matrix
weights.

######  Lemma 5.3.

Let $p\in(0,\infty)$ and $W\in A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$.
Then, for any nonzero matrix $M\in M_{m}(\mathbb{C})$, the scalar function

$\displaystyle w_{M}:=\left\|W^{\frac{1}{p}}M\right\|^{p}$

satisfies $w_{M}\in A_{\infty}(\mathbb{R}^{n})$ with
$[w_{M}]_{A_{\infty}(\mathbb{R}^{n})}\leq[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}$,
and

$\displaystyle[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}:=\sup_{M\in
M_{m}(\mathbb{C})\setminus\\{O_{m}\\}}[w_{M}]_{A_{\infty}(\mathbb{R}^{n})}^{*}\leq\widetilde{C}[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})},$

where $\widetilde{C}$ is the same as in Lemma 5.2.

###### Proof.

Let $M\in M_{m}(\mathbb{C})$ be nonzero. By the definition of matrix weights,
we find that $w_{M}$ is a scalar weight. From Definition 2.6(ii), we infer
that, for any cube $Q\subset\mathbb{R}^{n}$, any $x\in Q$, and almost every
$y\in Q$,

$\displaystyle
w_{M}(x)\leq\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\left\|W^{\frac{1}{p}}(y)M\right\|^{p}$

and hence

$\displaystyle\fint_{Q}w_{M}(x)\,dx\leq\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\left\|W^{\frac{1}{p}}(y)M\right\|^{p},$

which further implies that

$\displaystyle\log\left(\fint_{Q}w_{M}(x)\,dx\right)+\log\left\\{[w_{M}(y)]^{-1}\right\\}\leq\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right).$

By this and $W\in A_{p,\infty}$, we conclude that $\log_{+}(w_{M}^{-1})\in
L^{1}_{\mathrm{loc}}$ and

$\displaystyle[w_{M}]_{A_{\infty}(\mathbb{R}^{n})}$
$\displaystyle=\sup_{\mathrm{cube}\,Q}\fint_{Q}w_{M}(x)\,dx\exp\left(\fint_{Q}\log\left\\{[w_{M}(y)]^{-1}\right\\}\,dy\right)$
$\displaystyle\leq\sup_{\mathrm{cube}\,Q}\exp\left(\fint_{Q}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle=[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}.$

This, combined with Lemma 5.2, further implies that $w_{M}\in
A_{\infty}(\mathbb{R}^{n})$ and

$\displaystyle[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}\leq\widetilde{C}\sup_{M\in
M_{m}(\mathbb{C})\setminus\\{O_{m}\\}}[w_{M}]_{A_{\infty}(\mathbb{R}^{n})}\leq\widetilde{C}[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})},$

which completes the proof of Lemma 5.3. ∎

A definition similar in form to
$[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}$ can be found
in [30, (1.5)] or [27, p. 3087]: For any
$\vec{z}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}$, define the scalar
weight $w_{\vec{z}}:=|W^{\frac{1}{p}}\vec{z}|^{p}$. For any matrix weight $W$,
define

$\displaystyle\widetilde{[W]}_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}:=\sup_{\vec{z}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}}\left[w_{\vec{z}}\right]_{A_{\infty}(\mathbb{R}^{n})}^{*}.$

The following conclusion shows that
$[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}$ and
$\widetilde{[W]}_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}$
are equivalent.

######  Proposition 5.4.

Let $p\in(0,\infty)$. Then there exists a positive constant $C$, depending
only on $m$ and $p$, such that, for any matrix weight $W$,

(5.1)
$\displaystyle\widetilde{[W]}_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}\leq[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}\leq
C\widetilde{[W]}_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}.$

###### Proof.

We first show the first inequality of (5.1). Let
$\vec{z}\in\mathbb{C}^{m}\setminus\\{\vec{\mathbf{0}}\\}$ and
$M_{\vec{z}}:=[O_{m,m-1}\ \vec{z}]$. Then $M_{\vec{z}}\in M_{m}(\mathbb{C})$
is nonzero and from (2.1), we deduce that, for any $x\in\mathbb{R}^{n}$,

$\displaystyle w_{M_{\vec{z}}}(x)=\left\|\left[O_{m,m-1}\
W^{\frac{1}{p}}(x)\vec{z}\right]\right\|^{p}=\left|W^{\frac{1}{p}}(x)\vec{z}\right|^{p}=w_{\vec{z}}(x).$

This, together with the definition of
$[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}$, further
implies that

$\displaystyle\left[w_{\vec{z}}\right]_{A_{\infty}(\mathbb{R}^{n})}^{*}=\left[w_{M_{\vec{z}}}\right]_{A_{\infty}(\mathbb{R}^{n})}^{*}\leq[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}},$

which completes the proof of the first inequality of (5.1).

Next, we prove the second inequality of (5.1). Let $M\in M_{m}(\mathbb{C})$ be
nonzero. Let $\\{\vec{e}_{i}\\}_{i=1}^{m}$ be any orthonormal basis of
$\mathbb{C}^{m}$. From [86, Lemma 3.2], we infer that, for any matrix $A\in
M_{m}(\mathbb{C})$,

$\displaystyle\|A\|\sim\left(\sum_{i=1}^{m}\left|A\vec{e}_{i}\right|^{p}\right)^{\frac{1}{p}}.$

This, combined with the definitions of $[w]_{A_{p}(\mathbb{R}^{n})}^{*}$ and
$\widetilde{[W]}_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}$,
further implies that, for any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle
w_{M}(Q)\sim\int_{Q}\sum_{i=1}^{m}w_{M\vec{e}_{i}}(x)\,dx=\sum_{i=1}^{m}w_{M\vec{e}_{i}}(Q)$

and

$\displaystyle\int_{Q}\mathcal{M}(w_{M}\mathbf{1}_{Q})(x)\,dx$
$\displaystyle\sim\int_{Q}\mathcal{M}\left(\sum_{i=1}^{m}w_{M\vec{e}_{i}}\mathbf{1}_{Q}\right)(x)\,dx$
$\displaystyle\leq\int_{Q}\sum_{i=1}^{m}\mathcal{M}(w_{M\vec{e}_{i}}\mathbf{1}_{Q})(x)\,dx$
$\displaystyle\leq\sum_{i=1}^{m}\left[w_{M\vec{e}_{i}}\right]_{A_{\infty}(\mathbb{R}^{n})}^{*}w_{M\vec{e}_{i}}(Q)$
$\displaystyle\leq\widetilde{[W]}_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}\sum_{i=1}^{m}w_{M\vec{e}_{i}}(Q),$

where $[w_{M\vec{e}_{i}}]_{A_{\infty}(\mathbb{R}^{n})}^{*}:=0$ if
$M\vec{e}_{i}=\vec{\mathbf{0}}$. Therefore,
$[w_{M}]_{A_{\infty}(\mathbb{R}^{n})}^{*}\lesssim\widetilde{[W]}_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}$.
This finishes the proof of Proposition 5.4. ∎

The following conclusion is a simple application of Lemma 5.3.

######  Proposition 5.5.

Let $p\in(0,\infty)$. Then

  1. (i)

for any $W\in A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$, one has
$[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}\in[1,\infty)$;

  2. (ii)

for any $W\in A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})$, one has
$[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})}\in[1,\infty)$.

###### Proof.

Let $M\in M_{m}(\mathbb{C})\setminus\\{O_{m}\\}$. Then, by Lemma 5.3 and [17,
Proposition 7.3.2(4)], we find that, for any $W\in
A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$,

$\displaystyle[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}\geq[w_{M}]_{A_{\infty}(\mathbb{R}^{n})}\geq
1.$

This finishes the proof of (i).

Now, we show (ii). Let $W\in A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})$. Using (i)
and Proposition 4.2(ii), we obtain, if $p\in(0,1]$, then

$\displaystyle[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})}\geq[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}\geq
1.$

On the other hand, if $p\in(1,\infty)$, then, from Hölder’s inequality, it
follows that, for any cube $Q\subset\mathbb{R}^{n}$ and $x\in Q$,

$\displaystyle\left\|W^{\frac{1}{p}}(x)\right\|$
$\displaystyle\leq\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|\left\|W^{\frac{1}{p}}(y)\right\|\,dy$
$\displaystyle\leq\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p^{\prime}}\,dy\right]^{\frac{1}{p^{\prime}}}\left[\fint_{Q}\left\|W^{\frac{1}{p}}(y)\right\|^{p}\,dy\right]^{\frac{1}{p}}$

and hence

$\displaystyle\fint_{Q}\left\|W^{\frac{1}{p}}(x)\right\|^{p}\,dx\leq[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})}\fint_{Q}\left\|W^{\frac{1}{p}}(y)\right\|^{p}\,dy.$

This further implies that $[W]_{A_{p}(\mathbb{R}^{n},\mathbb{C}^{m})}\geq 1$,
which completes the proof of (ii) and hence Proposition 5.5. ∎

Using Lemma 5.3 and the reverse Hölder inequality of scalar weights, we obtain
the following reverse Hölder inequality of $A_{p,\infty}$-matrix weights. The
case of $A_{p}$-matrix weights is well known, and the present proof is
analogous; both results are based on the facts that the inequality holds for
$A_{p,\infty}^{\mathrm{sc}}$-matrix weights (cf. [29, Lemma 2]), and that both
$A_{p}$ and $A_{p,\infty}$ are contained in $A_{p,\infty}^{\mathrm{sc}}$.

######  Proposition 5.6.

Let $p\in(0,\infty)$, $W\in A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})$, and
$M\in M_{m}(\mathbb{C})$. Then, for any

(5.2) $\displaystyle
r\in\left[1,1+\frac{1}{2^{n+1}[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}-1}\right]$

and any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\fint_{Q}\left\|W^{\frac{1}{p}}(x)M\right\|^{pr}\,dx\leq
2\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)M\right\|^{p}\,dx\right]^{r}.$

###### Proof.

If $M$ is a zero matrix, then the present lemma is obvious, so it is enough to
consider the nonzero matrix $M$. By Lemma 5.3, we find that the scalar weight

$\displaystyle w_{M}:=\left\|W^{\frac{1}{p}}M\right\|^{p}$

satisfies $w_{M}\in A_{\infty}(\mathbb{R}^{n})$ and
$[w_{M}]_{A_{\infty}(\mathbb{R}^{n})}^{*}\leq[W]_{A_{p,\infty}(\mathbb{R}^{n},\mathbb{C}^{m})}^{\mathrm{sc}}$.
From this and Lemma 5.1, we infer that, for any $r$ in (5.2) and any cube
$Q\subset\mathbb{R}^{n}$,

$\displaystyle\fint_{Q}[w_{M}(x)]^{r}\,dx\leq
2\left[\fint_{Q}w_{M}(x)\,dx\right]^{r}.$

This finishes the proof of Proposition 5.6. ∎

The following corollary is also useful.

######  Corollary 5.7.

Let $p\in(0,\infty)$, $W\in A_{p,\infty}$, and $\\{A_{Q}\\}_{Q\in\mathscr{Q}}$
be a sequence of reducing operators of order $p$ for $W$. Then the following
statements hold.

  1. (i)

There exists a positive constant $C$, depending only on $m$ and $p$, such
that, for any $r$ as in (5.2),

$\displaystyle\sup_{\mathrm{cube}\,Q}\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{pr}\,dx\right]^{\frac{1}{r}}\leq
C.$

  2. (ii)

There exists a positive constant $C$, depending only on $m$, $p$, and
$[W]_{A_{p,\infty}}$, such that, for any $r$ as in (5.2),

$\displaystyle\sup_{Q\in\mathscr{Q}}\left[\fint_{Q}\sup_{R\in\mathscr{Q},\,x\in
R\subset
Q}\left\|W^{\frac{1}{p}}(x)A_{R}^{-1}\right\|^{pr}\,dx\right]^{\frac{1}{r}}\leq
C.$

###### Proof.

From Proposition 5.6 and Lemma 2.9, we deduce that, for any $r$ in (5.2) and
any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{pr}\,dx\right]^{\frac{1}{r}}<2\fint_{Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}\,dx\sim\left\|A_{Q}A_{Q}^{-1}\right\|^{p}=1.$

This finishes the proof of (i).

Applying Lemma 3.10 and an argument similar to that used in the proof of [4,
Lemma A.32] (see also [26, Lemma 3.6]), we obtain (ii). This finishes the
proof of Corollary 5.7. ∎

We conclude this section by indicating an application of Corollary 5.7 to the
boundedness of certain pointwise multipliers acting on certain subspaces of
$L^{p}\ell^{q}$, the space of sequences $\\{f_{j}\\}_{j\in\mathbb{Z}}$ of
measurable functions on $\mathbb{R}^{n}$ such that

$\left\|\\{f_{j}\\}_{j\in\mathbb{Z}}\right\|_{L^{p}\ell^{q}}:=\left\|\left(\sum_{j\in\mathbb{Z}}|f_{j}|^{q}\right)^{\frac{1}{q}}\right\|_{L^{p}}<\infty,$

with usual modification if $q=\infty$.

For any $j\in\mathbb{Z}$, let $\mathscr{Q}_{j}$ be the collection of dyadic
cubes of edge length $2^{-j}$ and, for any nonnegative measurable function $f$
on $\mathbb{R}^{n}$ or any $f\in L^{1}_{\mathrm{loc}}$, let

(5.3)
$E_{j}(f):=\sum_{Q\in\mathscr{Q}_{j}}\left[\fint_{Q}f(x)\,dx\right]\mathbf{1}_{Q}$

be the related conditional expectation. Repeating an argument used in the
proof of [15, Corollary 3.8] with [15, Lemmas 3.2 and 3.3] replaced by
Corollary 5.7, we obtain the following conclusion; we omit the details.

######  Corollary 5.8.

Let $p\in(0,\infty)$, $q\in(0,\infty]$, $W\in A_{p,\infty}$, and
$\\{A_{Q}\\}_{Q\in\mathscr{Q}}$ be a sequence of reducing operators of order
$p$ for $W$. For any $j\in\mathbb{Z}$, let

$\displaystyle\gamma_{j}:=\sum_{Q\in\mathscr{Q}_{j}}\left\|W^{\frac{1}{p}}A_{Q}^{-1}\right\|\mathbf{1}_{Q}.$

Then there exists a positive constant $C$ such that, for any sequence
$\\{f_{j}\\}_{j\in\mathbb{Z}}$ of nonnegative measurable functions on
$\mathbb{R}^{n}$ or $\\{f_{j}\\}_{j\in\mathbb{Z}}\subset
L^{1}_{\mathrm{loc}}$,

$\left\|\left\\{\gamma_{j}E_{j}\left(f_{j}\right)\right\\}_{j\in\mathbb{Z}}\right\|_{L^{p}\ell^{q}}\leq
C\left\|\left\\{E_{j}\left(f_{j}\right)\right\\}_{j\in\mathbb{Z}}\right\|_{L^{p}\ell^{q}},$

where $E_{j}$ for any $j\in\mathbb{Z}$ is the same as in (5.3).

Corollary 5.8 extends [15, Corollary 3.8] from $A_{p}$ to the larger weight
class $A_{p,\infty}$. The said [15, Corollary 3.8] played a key role in the
theory of Triebel–Lizorkin spaces with $A_{p}$-matrix weights developed in
[15] and in its extension to Besov-type and Triebel–Lizorkin-type spaces in
[3]. Corollary 5.8 will be similarly used in our follow-up work, where this
theory is extended to the larger class of $A_{p,\infty}$ weights.

##  6 Upper and lower dimensions of $A_{p,\infty}$ weights

As observed in [3, 15, 33], several results in the theory of matrix-weighted
function spaces depend at the technical level on the possibility of replacing
a reducing operator $A_{Q}$ related to one cube $Q$ by a similar operator
$A_{R}$ for a different cube $R$; this, in turn, depends on estimates for the
norms $\|A_{Q}A_{R}^{-1}\|$. Several results may be obtained abstractly, by
postulating suitable upper bounds for the said norms, but it is also of
interest to be able to relate such bounds to the $A_{p}$ or the $A_{p,\infty}$
conditions of the matrix weight $W$. In [3], we introduced the concept of
$A_{p}$-dimension of a matrix weight and obtained the sharp control of
$\|A_{Q}A_{R}^{-1}\|$ in terms of this quantity when $W\in A_{p}$. Our goal in
this section is to extend these ideas to the larger class of
$A_{p,\infty}$-matrix weights.

First, we give some useful equivalences among three quantities on the
composition of reducing operators and its related matrix weight.

######  Proposition 6.1.

Let $p\in(0,\infty)$, $W\in A_{p,\infty}$, and
$\\{A_{Q}\\}_{\mathrm{cube}\,Q}$ be a family of reducing operators of order
$p$ for $W$. Then, for all cubes $Q,R\subset\mathbb{R}^{n}$,

$\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}$
$\displaystyle\sim\fint_{Q}\exp\left(\fint_{R}\log\left[\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\right]\,dy\right)\,dx$
$\displaystyle\sim\exp\left(\fint_{R}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right),$

where the positive equivalence constants depend only on $m$, $p$, and
$[W]_{A_{p,\infty}}$.

###### Proof.

By Lemmas 2.9, 2.3, and 3.5, we conclude that, for any cubes
$Q,R\subset\mathbb{R}^{n}$,

$\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}$
$\displaystyle\sim\fint_{Q}\left\|W^{\frac{1}{p}}(x)A_{R}^{-1}\right\|^{p}\,dx=\fint_{Q}\left\|A_{R}^{-1}W^{\frac{1}{p}}(x)\right\|^{p}\,dx$
$\displaystyle\sim\fint_{Q}\exp\left(\fint_{R}\log\left[\left\|W^{-\frac{1}{p}}(y)W^{\frac{1}{p}}(x)\right\|^{p}\right]\,dy\right)\,dx$
$\displaystyle=\fint_{Q}\exp\left(\fint_{R}\log\left[\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\right]\,dy\right)\,dx.$

From Lemmas 2.3, 3.5, and 2.9, we infer that, for any cubes
$Q,R\subset\mathbb{R}^{n}$,

$\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}$
$\displaystyle=\left\|A_{R}^{-1}A_{Q}\right\|^{p}\sim\exp\left(\fint_{R}\log\left[\left\|W^{-\frac{1}{p}}(y)A_{Q}\right\|^{p}\right]\,dy\right)$
$\displaystyle=\exp\left(\fint_{R}\log\left[\left\|A_{Q}W^{-\frac{1}{p}}(y)\right\|^{p}\right]\,dy\right)$
$\displaystyle\sim\exp\left(\fint_{R}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right).$

These finish the proof of Proposition 6.1. ∎

To obtain a sharp estimate on $\|A_{Q}A_{R}^{-1}\|$, we introduce the
following concepts of lower and upper dimensions.

######  Definition 6.2.

Let $p\in(0,\infty)$ and $d\in\mathbb{R}$. A matrix weight $W$ is said to have
_$A_{p,\infty}$ -lower dimension $d$_, denoted by
$W\in\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}(\mathbb{R}^{n},\mathbb{C}^{m})$,
if there exists a positive constant $C$ such that, for any cubes
$Q,R\subset\mathbb{R}^{n}$ with $Q\subset R$,

$\displaystyle\exp\left(\fint_{R}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)\leq
C\left[\frac{\ell(R)}{\ell(Q)}\right]^{d}.$

A matrix weight $W$ is said to have _$A_{p,\infty}$ -upper dimension $d$_,
denoted by
$W\in\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}(\mathbb{R}^{n},\mathbb{C}^{m})$,
if there exists a positive constant $C$ such that, for any
$\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\exp\left(\fint_{Q}\log\left(\fint_{\lambda
Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)\leq
C\lambda^{d}.$

In what follows, if there exists no confusion, we denote
$\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}(\mathbb{R}^{n},\mathbb{C}^{m})$
simply by $\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}$ and
$\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}(\mathbb{R}^{n},\mathbb{C}^{m})$
simply by $\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}$. We have the following
basic properties on $A_{p,\infty}$-lower dimensions.

######  Proposition 6.3.

Let $p\in(0,\infty)$. Then the following statements hold.

  1. (i)

For any $d\in(-\infty,0)$, one has
$\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}=\emptyset$.

  2. (ii)

$\bigcup_{d\in[0,n)}\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}=A_{p,\infty}$.

  3. (iii)

For any $d\in[n,\infty)$, one has
$\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}=A_{p,\infty}$.

  4. (iv)

For any $d_{1},d_{2}\in[0,n]$ with $d_{1}<d_{2}$, one has
$\mathbb{D}_{p,\infty,d_{1}}^{\mathrm{lower}}\subsetneqq\mathbb{D}_{p,\infty,d_{2}}^{\mathrm{lower}}$.

###### Proof.

We first prove (i). Let $d\in(-\infty,0)$. Assume there exists
$W\in\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}$. By Definition 2.6(iii), we
conclude that

$\displaystyle\left\|W^{\frac{1}{p}}(\cdot)A_{Q_{0,\mathbf{0}}}^{-1}\right\|^{p}\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n}),$

which, together with Lebesgue’s differentiation theorem (see, for instance,
[17, Corollary 2.1.16]), further implies that there exist $x_{0}\in
Q_{0,\mathbf{0}}$ such that $W^{\frac{1}{p}}(x_{0})$ is invertible and a
sequence of cubes $\\{Q_{k}\\}_{k=1}^{\infty}\subset Q_{0,\mathbf{0}}$ such
that the center of every $Q_{k}$ is $x_{0}$ and
$\lim_{k\to\infty}\ell(Q_{k})=0$ and such that

$\displaystyle\left\|W^{\frac{1}{p}}(x_{0})A_{Q_{0,\mathbf{0}}}^{-1}\right\|^{p}=\lim_{k\to\infty}\fint_{Q_{k}}\left\|W^{\frac{1}{p}}(x)A_{Q_{0,\mathbf{0}}}^{-1}\right\|^{p}\,dx.$

By this, Lemmas 2.9 and 6.1, and the assumption that
$W\in\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}$, we find that

$\displaystyle\left\|W^{\frac{1}{p}}(x_{0})A_{Q_{0,\mathbf{0}}}^{-1}\right\|^{p}$
$\displaystyle\sim\lim_{k\to\infty}\left\|A_{Q_{k}}A_{Q_{0,\mathbf{0}}}^{-1}\right\|^{p}$
$\displaystyle\sim\lim_{k\to\infty}\exp\left(\fint_{Q_{0,\mathbf{0}}}\log\left(\fint_{Q_{k}}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\lesssim\lim_{k\to\infty}\left[\frac{\ell(Q_{0,\mathbf{0}})}{\ell(Q_{k})}\right]^{d}=0$

and hence all entries of $W(x_{0})$ are $0$, which contradicts the
invertibility of $W^{\frac{1}{p}}(x_{0})$. Thus,
$\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}=\emptyset$. This finishes the proof
of (i).

Next, we show (ii). By Definition 6.2 with $Q=R$, we obtain, for any
$d\in[0,n)$, $\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}\subset A_{p,\infty}$
and hence

(6.1)
$\displaystyle\bigcup_{d\in[0,n)}\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}\subset
A_{p,\infty}.$

On the other hand, let $W\in A_{p,\infty}$. From Hölder’s inequality and
Proposition 5.6 with $r\in(1,\infty)$ as in (5.2), we deduce that, for any
cubes $Q,R\subset\mathbb{R}^{n}$ with $Q\subset R$,

$\displaystyle\exp\left(\fint_{R}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\leq\exp\left(\fint_{R}\log\left(\left[\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{pr}\,dx\right]^{\frac{1}{r}}\right)\,dy\right)$
$\displaystyle\quad\leq\left[\frac{\ell(R)}{\ell(Q)}\right]^{\frac{n}{r}}\exp\left(\fint_{R}\log\left(\left[\fint_{R}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{pr}\,dx\right]^{\frac{1}{r}}\right)\,dy\right)$
$\displaystyle\quad\leq
2^{\frac{1}{r}}\left[\frac{\ell(R)}{\ell(Q)}\right]^{\frac{n}{r}}\exp\left(\fint_{R}\log\left(\fint_{R}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\leq
2^{\frac{1}{r}}[W]_{A_{p,\infty}}\left[\frac{\ell(R)}{\ell(Q)}\right]^{\frac{n}{r}}$

and hence $W$ has $A_{p,\infty}$-lower dimension $\frac{n}{r}\in(0,n)$.
Therefore,

$\displaystyle
A_{p,\infty}\subset\bigcup_{d\in[0,n)}\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}.$

This, combined with (6.1), further implies (ii).

Now, we prove (iii). Let $d\in[n,\infty)$. By Definition 6.2 with $Q=R$, we
obtain $\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}\subset A_{p,\infty}$. On the
other hand, for any $W\in A_{p,\infty}$, from the definition of
$[W]_{A_{p,\infty}}$, we infer that, for any cubes $Q,R\subset\mathbb{R}^{n}$
with $Q\subset R$,

$\displaystyle\exp\left(\fint_{R}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\leq\left[\frac{\ell(R)}{\ell(Q)}\right]^{n}\exp\left(\fint_{R}\log\left(\fint_{R}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\leq[W]_{A_{p,\infty}}\left[\frac{\ell(R)}{\ell(Q)}\right]^{d}$

and hence $W\in\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}$. Therefore,
$\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}=A_{p,\infty}$, which completes the
proof of (iii).

The inclusion “$\subset$” in Claim (iv) follows directly from Definition 6.2.
That this is a strict inclusion “$\subsetneqq$” follows from the examples
given in Lemma 7.5 below. This completes the proof of Proposition 6.3. ∎

Now, we can give the following basic properties of $A_{p,\infty}$-upper
dimensions.

######  Proposition 6.4.

Let $p\in(0,\infty)$. Then the following statements hold.

  1. (i)

For any $d\in(-\infty,0)$,
$\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}=\emptyset$.

  2. (ii)

$\bigcup_{d\in[0,\infty)}\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}=A_{p,\infty}$.

  3. (iii)

For any $d_{1},d_{2}\in[0,\infty)$ with $d_{1}<d_{2}$,
$\mathbb{D}_{p,\infty,d_{1}}^{\mathrm{upper}}\subsetneqq\mathbb{D}_{p,\infty,d_{2}}^{\mathrm{upper}}$.

  4. (iv)

For any $p\in(0,1]$, $A_{p}\subset\mathbb{D}_{p,\infty,0}^{\mathrm{upper}}$.

Although we mostly deal with $A_{p,\infty}$ weights here, the last claim is
intentionally stated for $A_{p}$ weights only.

###### Proof.

We first prove (i). Let $d\in(-\infty,0)$. Assume that there exists
$W\in\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}$. By Lemma 3.5, we conclude that

$\displaystyle\log\left[\left\|W^{-\frac{1}{p}}(\cdot)A_{Q_{0,\mathbf{0}}}\right\|^{p}\right]\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n}),$

which, together with Lebesgue’s differentiation theorem (see, for instance,
[17, Corollary 2.1.16]), further implies that there exist $x_{0}\in
Q_{0,\mathbf{0}}$ such that $W^{\frac{1}{p}}(x_{0})$ is invertible and a
sequence of cubes $\\{Q_{k}\\}_{k=1}^{\infty}\subset Q_{0,\mathbf{0}}$ such
that the center of every $Q_{k}$ for any $k\in\mathbb{N}$ is $x_{0}$ and
$\lim_{k\to\infty}\ell(Q_{k})=0$ and such that

$\displaystyle\left\|W^{-\frac{1}{p}}(x_{0})A_{Q_{0,\mathbf{0}}}\right\|^{p}=\lim_{k\to\infty}\exp\left(\fint_{Q_{k}}\log\left[\left\|W^{-\frac{1}{p}}(y)A_{Q_{0,\mathbf{0}}}\right\|^{p}\right]\,dy\right).$

By this, Lemmas 2.3 and 2.9, and
$W\in\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}$, we find that

$\displaystyle\left\|W^{-\frac{1}{p}}(x_{0})A_{Q_{0,\mathbf{0}}}\right\|^{p}$
$\displaystyle\quad=\lim_{k\to\infty}\exp\left(\fint_{Q_{k}}\log\left[\left\|A_{Q_{0,\mathbf{0}}}W^{-\frac{1}{p}}(y)\right\|^{p}\right]\,dy\right)$
$\displaystyle\quad\sim\lim_{k\to\infty}\exp\left(\fint_{Q_{k}}\log\left(\fint_{Q_{0,\mathbf{0}}}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\lesssim\lim_{k\to\infty}\exp\left(\fint_{Q_{k}}\log\left(\fint_{Q_{x_{0}}}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\lesssim\lim_{k\to\infty}[\ell(Q_{k})]^{-d}=0,$

where $Q_{x_{0}}$ is centered at $x_{0}$, has edge length $\sim 1$, and
$Q_{0,\mathbf{0}}\subset Q_{x_{0}}$. Hence all entries of
$W^{-\frac{1}{p}}(x_{0})$ are $0$, which contradicts the fact that this is the
inverse matrix of $W^{\frac{1}{p}}(x_{0})$. Thus,
$\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}=\emptyset$. This finishes the proof
of (i).

Now, we show (ii). By Definition 6.2 with $\lambda=1$, we obtain, for any
$d\in[0,\infty)$, $\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}\subset
A_{p,\infty}$ and hence

(6.2)
$\displaystyle\bigcup_{d\in[0,\infty)}\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}\subset
A_{p,\infty}.$

It remains to prove
$A_{p,\infty}\subset\bigcup_{d\in[0,\infty)}\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}$.
Let $W\in A_{p,\infty}$ and $M\in(0,\infty)$, where $M$ is determined later.
For any cube $Q\subset\mathbb{R}^{n}$ and any $x\in\mathbb{R}^{n}$, let

$\displaystyle h_{Q}(x):=\begin{cases}M&\text{if }x\in Q,\\\ 1&\text{if
}x\in\mathbb{R}^{n}\setminus Q.\end{cases}$

For any cube $Q\subset\mathbb{R}^{n}$, by Proposition 3.8 with $Q$ and $H$
replaced, respectively, by $2Q$ and $h_{Q}A_{Q}^{-1}$, we obtain

$\displaystyle[W]_{A_{p,\infty}}$
$\displaystyle\geq\left[\fint_{2Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}[h_{Q}(x)]^{p}\,dx\right]^{-1}$
$\displaystyle\quad\times\exp\left(\fint_{2Q}\log\left(\fint_{2Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}[h_{Q}(y)]^{p}\,dx\right)\,dy\right)$
$\displaystyle=:I(Q)\times II(Q).$

From Lemma 2.9 and the definition of $h_{Q}$, it follows that, for any cube
$Q\subset\mathbb{R}^{n}$,

$\displaystyle[I(Q)]^{-1}$
$\displaystyle=\fint_{2Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}\,dx+(M^{p}-1)2^{-n}\fint_{Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}\,dx$
$\displaystyle\sim\left\|A_{2Q}A_{Q}^{-1}\right\|^{p}+M^{p}-1$

and

$\displaystyle II(Q)$
$\displaystyle=\left[\fint_{2Q}\left\|W^{\frac{1}{p}}(x)A_{Q}^{-1}\right\|^{p}\,dx\right]\exp\left(\fint_{2Q}\log\left\\{[h_{Q}(y)]^{p}\right\\}\,dy\right)$
$\displaystyle\sim\left\|A_{2Q}A_{Q}^{-1}\right\|^{p}\exp\left(\frac{\log(M^{p})}{2^{n}}\right).$

Thus, there exists a constant $C\in(0,1]$, depending only on $m$ and $p$, such
that, for any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle[W]_{A_{p,\infty}}$ $\displaystyle\geq
C\frac{\|A_{2Q}A_{Q}^{-1}\|^{p}}{\|A_{2Q}A_{Q}^{-1}\|^{p}+M^{p}-1}\exp\left(\frac{\log(M^{p})}{2^{n}}\right).$

Choosing $M\in(0,\infty)$ such that
$CM^{\frac{p}{2^{n}}}=2[W]_{A_{p,\infty}}$, then, for any cube
$Q\subset\mathbb{R}^{n}$, one has

$\displaystyle\left\|A_{2Q}A_{Q}^{-1}\right\|^{p}\leq
M^{p}-1<M^{p}=\left(\frac{2[W]_{A_{p,\infty}}}{C}\right)^{2^{n}}$

and hence

$\displaystyle\exp\left(\fint_{Q}\log\left(\fint_{\lambda
Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\lesssim\exp\left(\fint_{Q}\log\left(\fint_{2^{\lceil\log_{2}\lambda\rceil}Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\quad\sim\left\|A_{2^{\lceil\log_{2}\lambda\rceil}Q}A_{Q}^{-1}\right\|^{p}\leq\left(\frac{2[W]_{A_{p,\infty}}}{C}\right)^{2^{n}\lceil\log_{2}\lambda\rceil}$
$\displaystyle\quad\leq\left(\frac{2[W]_{A_{p,\infty}}}{C}\right)^{2^{n}(1+\log_{2}\lambda)}\sim\lambda^{2^{n}(1-\log_{2}C+\log_{2}[W]_{A_{p,\infty}})},$

which further implies that $W$ has $A_{p,\infty}$-upper dimension
$2^{n}(1-\log_{2}C+\log_{2}[W]_{A_{p,\infty}})\in[0,\infty)$. Therefore,

$\displaystyle
A_{p,\infty}\subset\bigcup_{d\in[0,\infty)}\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}.$

From this and (6.2), it follows that

$\displaystyle\bigcup_{d\in[0,\infty)}\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}=A_{p,\infty},$

which completes the proof of (ii).

Claim (iv) and the inclusion “$\subset$” in (iii) follow directly from
Definitions 6.2 and 2.10. That there is a strict inclusion “$\subsetneqq$” in
(iii) follows from the examples given in Lemma 7.5 below. This finishes the
proof of Proposition 6.4. ∎

For any matrix weight $W\in A_{p,\infty}$, let

$d_{p,\infty}^{\mathrm{lower}}(W):=\inf\\{d\in[0,n):\ W\text{ has
}A_{p,\infty}\text{-lower dimension }d\\}$

and

$d_{p,\infty}^{\mathrm{upper}}(W):=\inf\\{d\in[0,\infty):\ W\text{ has
}A_{p,\infty}\text{-upper dimension }d\\}.$

Let

$\displaystyle[\\![d_{p,\infty}^{\mathrm{lower}}(W),n):=\begin{cases}[d_{p,\infty}^{\mathrm{lower}}(W),n)&\text{if
}d_{p,\infty}^{\mathrm{lower}}(W)\text{ is }A_{p,\infty}\text{-lower dimension
of }W,\\\ (d_{p,\infty}^{\mathrm{lower}}(W),n)&\text{otherwise}\end{cases}$

and

$\displaystyle[\\![d_{p,\infty}^{\mathrm{upper}}(W),\infty):=\begin{cases}[d_{p,\infty}^{\mathrm{upper}}(W),\infty)&\text{if
}d_{p,\infty}^{\mathrm{upper}}(W)\text{ is }A_{p,\infty}\text{-upper dimension
of }W,\\\
(d_{p,\infty}^{\mathrm{upper}}(W),\infty)&\text{otherwise}.\end{cases}$

Finally, using the concept of $A_{p,\infty}$-dimensions, we obtain the
following _sharp_ estimate. For the sharpness, see Lemma 7.9 further below.

######  Proposition 6.5.

Let $p\in(0,\infty)$, $W\in A_{p,\infty}$, and
$\\{A_{Q}\\}_{\mathrm{cube}\,Q}$ be a family of reducing operators of order
$p$ for $W$. Let $d_{1}\in[\\![d_{p,\infty}^{\mathrm{lower}}(W),\infty)$ and
$d_{2}\in[\\![d_{p,\infty}^{\mathrm{upper}}(W),\infty)$. Then there exists a
positive constant $C$ such that, for any cubes $Q,R\subset\mathbb{R}^{n}$,

$\left\|A_{Q}A_{R}^{-1}\right\|^{p}\leq
C\max\left\\{\left[\frac{\ell(R)}{\ell(Q)}\right]^{d_{1}},\left[\frac{\ell(Q)}{\ell(R)}\right]^{d_{2}}\right\\}\left[1+\frac{|c_{Q}-c_{R}|}{\ell(Q)\vee\ell(R)}\right]^{d_{1}+d_{2}}.$

###### Proof.

Let us first consider the special case when $Q\subset R$. In this case, by
Lemma 6.1 and Definition 6.2, we find that

(6.3)
$\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}\sim\exp\left(\fint_{R}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)\lesssim\left[\frac{\ell(R)}{\ell(Q)}\right]^{d_{1}}.$

Next, we consider the special case where $R\subset Q$. In this case, using
some geometrical observations, we find that there exists
$\lambda\in[1,\infty)$ such that $\lambda\sim\ell(Q)/\ell(R)$ and
$Q\subset\lambda R$. These, together with Lemma 2.9 and Proposition 6.1,
further imply that

(6.4) $\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}$
$\displaystyle\sim\exp\left(\fint_{R}\log\left(\fint_{Q}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\lesssim\exp\left(\fint_{R}\log\left(\fint_{\lambda
R}\left\|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\right\|^{p}\,dx\right)\,dy\right)$
$\displaystyle\lesssim\lambda^{d_{2}}\sim\left[\frac{\ell(Q)}{\ell(R)}\right]^{d_{2}}.$

In the general case, we choose a third cube $S$ such that $Q\cup R\subset S$.
This clearly can be achieved with $\ell(S)\sim\ell(Q)+\ell(R)+|c_{Q}-c_{R}|$
by some geometrical observations. From this, (6.4), and (6.3), we deduce that

$\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}$
$\displaystyle\leq\left\|A_{Q}A_{S}^{-1}\right\|^{p}\left\|A_{S}A_{R}^{-1}\right\|^{p}\lesssim\left[\frac{\ell(S)}{\ell(Q)}\right]^{d_{1}}\left[\frac{\ell(S)}{\ell(R)}\right]^{d_{2}}$
$\displaystyle=\left[\frac{\ell(Q)\vee\ell(R)}{\ell(Q)}\right]^{d_{1}}\left[\frac{\ell(S)}{\ell(Q)\vee\ell(R)}\right]^{d_{1}+d_{2}}\left[\frac{\ell(Q)\vee\ell(R)}{\ell(R)}\right]^{d_{2}}$
$\displaystyle\sim\max\left\\{\left[\frac{\ell(R)}{\ell(Q)}\right]^{d_{1}},\left[\frac{\ell(Q)}{\ell(R)}\right]^{d_{2}}\right\\}\left[1+\frac{|c_{Q}-c_{R}|}{\ell(Q)\vee\ell(R)}\right]^{d_{1}+d_{2}}.$

This finishes the proof of Proposition 6.5. ∎

The following lemma is useful; since its proof is just some basic
calculations, we omit the details.

######  Lemma 6.6.

For any cubes $Q,R\subset\mathbb{R}^{n}$, any $x,x^{\prime}\in Q$, and any
$y,y^{\prime}\in R$,

$1+\frac{|x-y|}{\ell(Q)\vee\ell(R)}\sim
1+\frac{|x^{\prime}-y^{\prime}|}{\ell(Q)\vee\ell(R)},$

where the positive equivalence constants depend only on $n$.

Next, we recall the concept of dyadic cubes. For any $j\in\mathbb{Z}$ and
$k:=(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}$, let

$Q_{j,k}:=\prod_{i=1}^{n}2^{-j}[k_{i},k_{i}+1),$ $\mathscr{Q}:=\\{Q_{j,k}:\
j\in\mathbb{Z},\ k\in\mathbb{Z}^{n}\\},$ $\mathscr{Q}_{+}:=\\{Q_{j,k}:\
j\in\mathbb{Z}_{+},\ k\in\mathbb{Z}^{n}\\},$

and

$\displaystyle\mathscr{Q}_{j}:=\\{Q_{j,k}:\ k\in\mathbb{Z}^{n}\\}.$

For any $Q:=Q_{j,k}\in\mathscr{Q}$, we let $j_{Q}:=j$ and $x_{Q}:=2^{-j}k$.

Using Proposition 6.5 and Lemma 6.6, we directly obtain the following
estimates; we omit the details.

######  Lemma 6.7.

Let $p\in(0,\infty)$, $W\in A_{p,\infty}$, and
$\\{A_{Q}\\}_{\mathrm{cube}\,Q}$ be a family of reducing operators of order
$p$ for $W$. Let $d_{1}\in[\\![d_{p,\infty}^{\mathrm{lower}}(W),n)$ and
$d_{2}\in[\\![d_{p,\infty}^{\mathrm{upper}}(W),\infty)$. Then the following
three statements hold.

  1. (i)

There exists a positive constant $C$ such that, for any $Q,R\in\mathscr{Q}$,

$\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}\leq
C\max\left\\{\left[\frac{\ell(R)}{\ell(Q)}\right]^{d_{1}},\left[\frac{\ell(Q)}{\ell(R)}\right]^{d_{2}}\right\\}\left[1+\frac{|x_{Q}-x_{R}|}{\ell(Q)\vee\ell(R)}\right]^{d_{1}+d_{2}}.$

  2. (ii)

There exists a positive constant $C$ such that, for any $j\in\mathbb{Z}$ and
$Q,R\in\mathscr{Q}_{j}$,

$\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}\leq
C\left(1+2^{j}|x_{Q}-x_{R}|\right)^{d_{1}+d_{2}}.$

  3. (iii)

There exists a positive constant $C$ such that, for any cube
$Q,R\subset\mathbb{R}^{n}$ with $Q\cap R\neq\emptyset$,

$\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}\leq
C\max\left\\{\left[\frac{\ell(R)}{\ell(Q)}\right]^{d_{1}},\left[\frac{\ell(Q)}{\ell(R)}\right]^{d_{2}}\right\\}.$

##  7 Examples and sharpness

To show the sharpness of Proposition 6.5, corresponding to Definition 6.2, we
introduce the concepts of lower and upper dimensions of scalar weights.

######  Definition 7.1.

Let $p\in(0,\infty)$ and $d\in\mathbb{R}$. A scalar weight $w$ is said to have
_$A_{\infty}$ -lower dimension $d$_, denoted by
$w\in\mathbb{D}_{\infty,d}^{\mathrm{lower}}(\mathbb{R}^{n})$, if there exists
a positive constant $C$ such that, for any cubes $Q,R\subset\mathbb{R}^{n}$
with $Q\subset R$,

$\displaystyle\fint_{Q}w(x)\,dx\exp\left(\fint_{R}\log\left\\{[w(x)]^{-1}\right\\}\,dx\right)\leq
C\left[\frac{\ell(R)}{\ell(Q)}\right]^{d}.$

A scalar weight $w$ is said to have _$A_{\infty}$ -upper dimension $d$_,
denoted by $w\in\mathbb{D}_{\infty,d}^{\mathrm{upper}}(\mathbb{R}^{n})$, if
there exists a positive constant $C$ such that, for any $\lambda\in[1,\infty)$
and any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\fint_{\lambda
Q}w(x)\,dx\exp\left(\fint_{Q}\log\left\\{[w(x)]^{-1}\right\\}\,dx\right)\leq
C\lambda^{d}.$

The following lemma gives the relations between scalar and matrix weights,
which follows immediately from their definitions; we omit the details.

######  Lemma 7.2.

Let $p\in(0,\infty)$ and $d\in[0,\infty)$. Let $w$ be a scalar weight and
$W:=wI_{m}$, where $I_{m}$ is identity matrix. Then the following statements
hold.

  1. (i)

$W\in\mathbb{D}_{p,\infty,d}^{\mathrm{lower}}$ if and only if
$w\in\mathbb{D}_{\infty,d}^{\mathrm{lower}}$;

  2. (ii)

$W\in\mathbb{D}_{p,\infty,d}^{\mathrm{upper}}$ if and only if
$w\in\mathbb{D}_{\infty,d}^{\mathrm{upper}}$.

$A_{\infty}$-dimensions of scalar weights are analogues of the doubling and
the reverse doubling conditions, which reveal the intrinsic property of
$A_{\infty}$-lower and $A_{\infty}$-upper dimensions.

######  Proposition 7.3.

Let $p\in(0,\infty)$, $d\in\mathbb{R}$, and scalar weight $w\in A_{\infty}$.
Then

  1. (i)

$w\in\mathbb{D}_{\infty,d}^{\mathrm{lower}}(\mathbb{R}^{n})$ if and only if
there exists a positive constant $C$ such that, for any $\lambda\in[1,\infty)$
and any cube $Q\subset\mathbb{R}^{n}$,

(7.1) $\displaystyle\frac{w(Q)}{w(\lambda Q)}\leq C\lambda^{d-n};$

  2. (ii)

$w\in\mathbb{D}_{\infty,d}^{\mathrm{upper}}(\mathbb{R}^{n})$ if and only if
there exists a positive constant $C$ such that, for any $\lambda\in[1,\infty)$
and any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\frac{w(\lambda Q)}{w(Q)}\leq C\lambda^{d+n}.$

###### Proof.

By case $m=1$ of the definition of a reducing operator (2.3) and the estimate
of its inverse (3.2), we obtain

(7.2)
$\displaystyle\exp\left(\fint_{R}\log\left\\{[w(x)]^{-1}\right\\}\,dx\right)\sim\frac{|R|}{w(R)},$

which directly implies (ii) and the necessity of (i). Now, we show the
sufficiency of (i). From $w\in A_{\infty}(\mathbb{R}^{n})$, we infer that
there exists $p\in(1,\infty)$ such that $w\in A_{p}(\mathbb{R}^{n})$. By (7.2)
and (2.5), we find that, for any cube $R\subset\mathbb{R}^{n}$,

(7.3)
$\displaystyle\exp\left(\fint_{R}\log\left\\{[w(x)]^{-1}\right\\}\,dx\right)\sim\left\\{\fint_{R}[w(x)]^{-\frac{p^{\prime}}{p}}\,dx\right\\}^{\frac{p}{p^{\prime}}},$

where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. For any cubes
$Q,R\subset\mathbb{R}^{n}$ with $Q\subset R$, we have $R\subset\lambda Q$,
where $\lambda\sim\ell(R)/\ell(Q)$, which, combined with (7.3) and case $m=1$
of (7.1), further implies that

$\displaystyle\fint_{Q}w(x)\,dx\exp\left(\fint_{R}\log\left\\{[w(x)]^{-1}\right\\}\,dx\right)$
$\displaystyle\quad\sim\fint_{Q}w(x)\,dx\left\\{\fint_{R}[w(x)]^{-\frac{p^{\prime}}{p}}\,dx\right\\}^{\frac{p}{p^{\prime}}}$
$\displaystyle\quad\lesssim\fint_{Q}w(x)\,dx\left\\{\fint_{\lambda
Q}[w(x)]^{-\frac{p^{\prime}}{p}}\,dx\right\\}^{\frac{p}{p^{\prime}}}$
$\displaystyle\quad\sim\fint_{Q}w(x)\,dx\exp\left(\fint_{\lambda
Q}\log\left\\{[w(x)]^{-1}\right\\}\,dx\right)$
$\displaystyle\quad\sim\frac{|\lambda Q|}{|Q|}\frac{w(Q)}{w(\lambda
Q)}\lesssim\lambda^{d}\sim\left[\frac{\ell(R)}{\ell(Q)}\right]^{d}$

and hence $w\in\mathbb{D}_{\infty,d}^{\mathrm{lower}}(\mathbb{R}^{n})$. This
finishes the proof of Proposition 7.3. ∎

For any scalar weight $w$, let

$\displaystyle d_{\infty}^{\mathrm{lower}}(w):=\inf\\{d\in[0,n):\ w\text{ has
}A_{\infty}\text{-lower dimension }d\\}$

and

$\displaystyle d_{\infty}^{\mathrm{upper}}(w):=\inf\\{d\in[0,\infty):\ w\text{
has }A_{\infty}\text{-upper dimension }d\\}.$

######  Lemma 7.4.

Let $a\in(-n,\infty)$ and $b\in\mathbb{R}$. Define the scalar weight
$w_{a,b}(x):=|x|^{a}[\log(2+|x|)]^{b}$ for any $x\in\mathbb{R}^{n}$. Then
$w_{a,b}\in A_{\infty}(\mathbb{R}^{n})$ and

  1. (i)

for any $d\in(a_{-},\infty)$, there exists a positive constant $C$ such that,
for any $\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

(7.4) $\displaystyle\frac{w_{a,b}(Q)}{w_{a,b}(\lambda Q)}\leq C\lambda^{d-n};$

  2. (ii)

for any $d\in(-\infty,a_{-})$, the estimate (7.4) does not hold;

  3. (iii)

for $d=a_{-}$, the estimate (7.4) holds if and only if $a\in(0,\infty)$ or
$b\in[0,\infty)$;

  4. (iv)

for any $d\in(a_{+},\infty)$, there exists a positive constant $C$ such that,
for any $\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

(7.5) $\displaystyle\frac{w_{a,b}(\lambda Q)}{w_{a,b}(Q)}\leq C\lambda^{d+n};$

  5. (v)

for any $d\in(-\infty,a_{+})$, the estimate (7.5) does not hold;

  6. (vi)

for $d=a_{+}$, the estimate (7.5) holds if and only if $a\in(-\infty,0)$ or
$b\in(-\infty,0]$.

###### Proof.

By [3, Lemma 2.40 and Proposition 2.37], we obtain (i)-(iii). From [3,
Corollary 2.42], we deduce that, for any $\lambda\in[1,\infty)$ and any cube
$Q\subset\mathbb{R}^{n}$,

$\displaystyle\frac{w_{a,b}(\lambda Q)}{w_{a,b}(Q)}$
$\displaystyle\sim\lambda^{n}\left[\frac{|c_{Q}|+\lambda\ell(Q)}{|c_{Q}|+\ell(Q)}\right]^{a}\left\\{\frac{\log[2+|c_{Q}|+\lambda\ell(Q)]}{\log[2+|c_{Q}|+\ell(Q)]}\right\\}^{b}$
$\displaystyle\sim\lambda^{2n}\frac{w_{-a,-b}(Q)}{w_{-a,-b}(\lambda Q)}.$

This, together with (i)-(iii), further implies (iv)-(vi), which complete the
proof of Lemma 7.4. ∎

Applying Proposition 7.3 and Lemma 7.4, we obtain the following conclusion; we
omit the details.

######  Lemma 7.5.

Let $a\in(-n,\infty)$ and $b\in\mathbb{R}$. Let $w_{a,b}$ be the same as in
Lemma 7.4. Then $w_{a,b}\in A_{\infty}(\mathbb{R}^{n})$ and

  1. (i)

$d_{\infty}^{\mathrm{lower}}(w_{a,b})=a_{-}$ is $A_{\infty}$-lower dimension
of $w_{a,b}$ if and only if $a\in(0,\infty)$ or $b\in[0,\infty)$;

  2. (ii)

$d_{\infty}^{\mathrm{upper}}(w_{a,b})=a_{+}$ is $A_{\infty}$-upper dimension
of $w_{a,b}$ if and only if $a\in(-\infty,0)$ or $b\in(-\infty,0]$.

When $|x|\lesssim 1$, we have $w_{a,b}(x)\sim|x|^{a}$ and the disturbance term
$[\log(2+|x|)]^{b}$ has no effect. To solve this problem, we need the
following variant of Lemma 7.5.

######  Lemma 7.6.

Let $a\in(-n,\infty)$ and $b\in\mathbb{R}$. Let scalar weight
$\widetilde{w}_{a,b}(x):=|x|^{a}[\log(2+|x|^{-1})]^{b}$ for any
$x\in\mathbb{R}^{n}$. Then $\widetilde{w}_{a,b}\in A_{\infty}(\mathbb{R}^{n})$
and

  1. (i)

$d_{\infty}^{\mathrm{lower}}(\widetilde{w}_{a,b})=a_{-}$ is $A_{\infty}$-lower
dimension of $\widetilde{w}_{a,b}$ if and only if $a\in(0,\infty)$ or
$b\in(-\infty,0]$;

  2. (ii)

$d_{\infty}^{\mathrm{upper}}(\widetilde{w}_{a,b})=a_{+}$ is $A_{\infty}$-upper
dimension of $\widetilde{w}_{a,b}$ if and only if $a\in(-\infty,0)$ or
$b\in[0,\infty)$.

To prove Lemma 7.6, we need a technical lemma. Applying an argument similar to
that used in the proof of [3, Lemma 2.41], we obtain the following estimate;
we omit the details.

######  Lemma 7.7.

Let $a\in(-n,\infty)$, $b\in\mathbb{R}$, and $\widetilde{w}_{a,b}$ be the same
as in Lemma 7.6. Then, for any $x_{0}\in\mathbb{R}^{n}$ and $r\in(0,\infty)$,

$\displaystyle\fint_{B(x_{0},r)}\widetilde{w}_{a,b}(x)\,dx\sim(|x_{0}|+r)^{a}\left\\{\log\left[2+(|x_{0}|+r)^{-1}\right]\right\\}^{b},$

where the positive equivalence constants depend only on $n$, $a$, and $b$.

Lemma 7.7 remains true if we replace balls $B$ therein by cubes $Q$; we omit
the details.

######  Corollary 7.8.

Let $a\in(-n,\infty)$, $b\in\mathbb{R}$, and $\widetilde{w}_{a,b}$ be the same
as in Lemma 7.6. Then, for any cube $Q\subset\mathbb{R}^{n}$,

$\fint_{Q}\widetilde{w}_{a,b}(x)\,dx\sim[|c_{Q}|+\ell(Q)]^{a}\left(\log\left\\{2+[|c_{Q}|+\ell(Q)]^{-1}\right\\}\right)^{b},$

where the positive equivalence constants depend only on $n$, $a$, and $b$.

It is the time for us to prove Lemma 7.6.

###### Proof of Lemma 7.6.

We first show (i). Let $a\in(-n,\infty)$ and $b\in\mathbb{R}$. By [12, Lemma
2.3(v)], we find that $\widetilde{w}_{a,b}\in A_{\infty}(\mathbb{R}^{n})$.
This, combined with Proposition 7.3(i), further implies that, to prove (i), we
need only to show that

  1. (a)

for any $d\in(a_{-},\infty)$, there exists a positive constant $C$ such that,
for any $\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

(7.6) $\displaystyle\frac{\widetilde{w}_{a,b}(Q)}{\widetilde{w}_{a,b}(\lambda
Q)}\leq C\lambda^{d-n};$

  2. (b)

for any $d\in(-\infty,a_{-})$, (7.6) does not hold;

  3. (c)

for $d=a_{-}$, (7.6) holds if and only if $a\in(0,\infty)$ or
$b\in(-\infty,0]$.

Now, we prove (a). Let $d\in(a_{-},\infty)$. By Lemma 7.8, we conclude that,
for any $\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

(7.7) $\frac{\widetilde{w}_{a,b}(Q)}{\widetilde{w}_{a,b}(\lambda
Q)}\sim\lambda^{-n}\left[\frac{|c_{Q}|+\ell(Q)}{|c_{Q}|+\lambda\ell(Q)}\right]^{a}\left[\frac{\log(2+[|c_{Q}|+\ell(Q)]^{-1})}{\log(2+[|c_{Q}|+\lambda\ell(Q)]^{-1})}\right]^{b}.$

Applying an argument similar to that used in the estimation of [3, (2.37)], we
obtain, for any $\lambda\in[1,\infty)$,

$\frac{\widetilde{w}_{a,b}(Q)}{\widetilde{w}_{a,b}(\lambda
Q)}\lesssim\lambda^{(a_{-})-n}(1+\log_{2}\lambda)^{b_{+}}\lesssim\lambda^{d-n}.$

This finishes the proof of (a).

Next, we show (b). By (7.7), we obtain, for any $\lambda\in[1,\infty)$ and any
cube $Q\subset\mathbb{R}^{n}$ with $c_{Q}=\mathbf{0}$ and $\ell(Q)=1$,

$\frac{\widetilde{w}_{a,b}(Q)}{\widetilde{w}_{a,b}(\lambda
Q)}\gtrsim\lambda^{-a-n}\left[\frac{\log 3}{\log(2+\lambda^{-1})}\right]^{b};$

for any $\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$ with
$|c_{Q}|=\lambda$ and $\ell(Q)=1$,

$\frac{\widetilde{w}_{a,b}(Q)}{\widetilde{w}_{a,b}(\lambda
Q)}\gtrsim\lambda^{-n}\left[\frac{\lambda+1}{2\lambda}\right]^{a}\left[\frac{\log(2+(\lambda+1)^{-1})}{\log(2+(2\lambda)^{-1})}\right]^{b}\sim\lambda^{-n}$

and hence (7.6) does not hold if $d\in(-\infty,a_{-})$. This finishes the
proof of (b).

Finally, applying an argument similar to that used in the proof of [3, Lemma
2.40(ii)], we obtain (c). This finishes the proof of (i).

Now, we show (ii). By Lemma 7.8, we conclude that, for any
$\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\frac{\widetilde{w}_{a,b}(\lambda Q)}{\widetilde{w}_{a,b}(Q)}$
$\displaystyle\sim\lambda^{n}\left[\frac{|c_{Q}|+\lambda\ell(Q)}{|c_{Q}|+\ell(Q)}\right]^{a}\left[\frac{\log(2+[|c_{Q}|+\lambda\ell(Q)]^{-1})}{\log(2+[|c_{Q}|+\ell(Q)]^{-1})}\right]^{b}$
$\displaystyle\sim\lambda^{2n}\frac{\widetilde{w}_{-a,-b}(Q)}{\widetilde{w}_{-a,-b}(\lambda
Q)}.$

Applying this and (a)-(c), we obtain

  1. (d)

for any $d\in(a_{+},\infty)$, there exists a positive constant $C$ such that,
for any $\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

(7.8) $\displaystyle\frac{\widetilde{w}_{a,b}(\lambda
Q)}{\widetilde{w}_{a,b}(Q)}\leq C\lambda^{d+n};$

  2. (e)

for any $d\in(-\infty,a_{+})$, (7.8) does not hold;

  3. (f)

for $d=a_{+}$, (7.8) holds if and only if $a\in(-\infty,0)$ or
$b\in[0,\infty)$.

These, together with Proposition 7.3(ii), further imply (ii), which then
completes the proof of Lemma 7.6. ∎

Finally, we prove that the estimate of $\|A_{Q}A_{R}^{-1}\|$ in terms of the
upper and lower $A_{p,\infty}$-dimensions, as given in Proposition 6.5, is
_sharp_.

######  Lemma 7.9.

Let $d_{1}\in[0,n)$, $d_{2}\in[0,\infty)$, and $a,b,c\in\mathbb{R}$.

  1. (i)

If, for any $W\in A_{p,\infty}$ satisfying that
$d_{p,\infty}^{\mathrm{lower}}(W)=d_{1}$,
$d_{p,\infty}^{\mathrm{upper}}(W)=d_{2}$,
$W\in\mathbb{D}_{p,\infty,d_{1}}^{\mathrm{lower}}$, and
$W\in\mathbb{D}_{p,\infty,d_{2}}^{\mathrm{upper}}$, there exists a positive
constant $C$ such that, for any cubes $Q,R\subset\mathbb{R}^{n}$,

(7.9) $\displaystyle\left\|A_{Q}A_{R}^{-1}\right\|^{p}\leq
C\max\left\\{\left[\frac{\ell(R)}{\ell(Q)}\right]^{a},\left[\frac{\ell(Q)}{\ell(R)}\right]^{b}\right\\}\left[1+\frac{|c_{Q}-c_{R}|}{\ell(Q)\vee\ell(R)}\right]^{c},$

where $\\{A_{Q}\\}_{\mathrm{cube}\,Q}$ is a family of reducing operators of
order $p$ for $W$, then

$\displaystyle a\in[d_{1},\infty),\ b\in[d_{2},\infty),\text{ and
}c\in[d_{1}+d_{2},\infty).$

  2. (ii)

If, for any $W\in A_{p,\infty}$ satisfying that
$d_{p,\infty}^{\mathrm{lower}}(W)=d_{1}$,
$d_{p,\infty}^{\mathrm{upper}}(W)=d_{2}$,
$W\notin\mathbb{D}_{p,\infty,d_{1}}^{\mathrm{lower}}$, and
$W\in\mathbb{D}_{p,\infty,d_{2}}^{\mathrm{upper}}$, (7.9) holds, then

$\displaystyle a\in(d_{1},\infty),\ b\in[d_{2},\infty),\text{ and
}c\in(d_{1}+d_{2},\infty).$

  3. (iii)

If, for any $W\in A_{p,\infty}$ satisfying that
$d_{p,\infty}^{\mathrm{lower}}(W)=d_{1}$,
$d_{p,\infty}^{\mathrm{upper}}(W)=d_{2}$,
$W\in\mathbb{D}_{p,\infty,d_{1}}^{\mathrm{lower}}$, and
$W\notin\mathbb{D}_{p,\infty,d_{2}}^{\mathrm{upper}}$, (7.9) holds, then

$\displaystyle a\in[d_{1},\infty),\ b\in(d_{2},\infty),\text{ and
}c\in(d_{1}+d_{2},\infty).$

  4. (iv)

If, for any $W\in A_{p,\infty}$ satisfying that
$d_{p,\infty}^{\mathrm{lower}}(W)=d_{1}$,
$d_{p,\infty}^{\mathrm{upper}}(W)=d_{2}$,
$W\notin\mathbb{D}_{p,\infty,d_{1}}^{\mathrm{lower}}$, and
$W\notin\mathbb{D}_{p,\infty,d_{2}}^{\mathrm{upper}}$, (7.9) holds, then

(7.10) $\displaystyle a\in(d_{1},\infty),\ b\in(d_{2},\infty),\text{ and
}c\in(d_{1}+d_{2},\infty).$

###### Proof.

Due to similarity, we only show (iv). Let $x_{0}:=(1,0,\ldots,0)$ and matrix
weight $W:=wI_{m}$, where $I_{m}$ is identity matrix and, for any
$x\in\mathbb{R}^{n}$,

$w(x):=w_{1}(x)[w_{2}(x)]^{-1}:=\widetilde{w}_{-d_{1},1}(x)[\widetilde{w}_{-d_{2},1}(x-x_{0})]^{-1}.$

Now, we prove that $W$ satisfies all assumptions of (iv). We first show that
$d_{\infty}^{\mathrm{lower}}(w)=d_{1}$ and
$w\notin\mathbb{D}_{\infty,d_{1}}^{\mathrm{lower}}(\mathbb{R}^{n})$. To this
end, from Proposition 7.3(i), it follows that we need only to prove that

  1. (a)

for any $d\in(d_{1},\infty)$, there exists a positive constant $C$ such that,
for any $\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

(7.11) $\displaystyle\frac{w(Q)}{w(\lambda Q)}\leq C\lambda^{d-n};$

  2. (b)

for $d=d_{1}$, (7.11) does not hold.

Next, we show (a). Let $d\in(d_{1},\infty)$. By [12, Lemma 2.3(v)], Lemma
7.6(i), and Proposition 7.3(i), we conclude that, for any $i\in\\{1,2\\}$,
$w_{i}\in A_{1}(\mathbb{R}^{n})$ and, for any $\lambda\in[1,\infty)$ and any
cube $Q\subset\mathbb{R}^{n}$,

(7.12)
$\displaystyle\fint_{Q}w_{1}(x)\,dx\left\|w_{1}^{-1}\right\|_{L^{\infty}(\lambda
Q)}$ $\displaystyle=\frac{|\lambda Q|}{|Q|}\frac{w_{1}(Q)}{w_{1}(\lambda
Q)}\fint_{\lambda Q}w_{1}(x)\,dx\left\|w_{1}^{-1}\right\|_{L^{\infty}(\lambda
Q)}$ $\displaystyle\leq\lambda^{n}\frac{w_{1}(Q)}{w_{1}(\lambda
Q)}[w_{1}]_{A_{1}(\mathbb{R}^{n})}\lesssim\lambda^{d}.$

From $w_{1},w_{2}\in A_{1}(\mathbb{R}^{n})$ and the Jones factorization
theorem, we infer that $w\in A_{2}(\mathbb{R}^{n})\subset
A_{\infty}(\mathbb{R}^{n})$. This, combined with Lemma 3.2, further implies
that $W\in A_{p,\infty}$. By $w\in A_{2}(\mathbb{R}^{n})$, (2.5), (7.12), and
the definition of $[w_{2}]_{A_{1}(\mathbb{R}^{n})}$, we find that, for any
$\lambda\in[1,\infty)$ and any cube $Q\subset\mathbb{R}^{n}$,

$\displaystyle\frac{w(Q)}{w(\lambda Q)}$
$\displaystyle\sim\lambda^{-n}\fint_{Q}w(x)\,dx\fint_{\lambda
Q}[w(x)]^{-1}\,dx$
$\displaystyle\leq\lambda^{-n}\fint_{Q}w_{1}(x)\,dx\left\|w_{1}^{-1}\right\|_{L^{\infty}(\lambda
Q)}\fint_{\lambda Q}w_{2}(x)\,dx\left\|w_{2}^{-1}\right\|_{L^{\infty}(Q)}$
$\displaystyle\lesssim\lambda^{d-n}[w_{2}]_{A_{1}(\mathbb{R}^{n})}.$

This finishes the proof of (a).

Now, we show (b). By Corollary 7.8, we obtain, for any cube
$Q\subset\mathbb{R}^{n}$ with $c_{Q}=\mathbf{0}$ and $\ell(Q)<\frac{1}{2}$,

$\displaystyle
w(Q)\sim|Q|\fint_{Q}w_{1}(x)\,dx\sim[\ell(Q)]^{n-d_{1}}\log[2+\ell(Q)^{-1}].$

Thus, for any $\lambda\in(0,1)$ and any cube $Q\subset\mathbb{R}^{n}$ with
$c_{Q}=\mathbf{0}$ and $\ell(Q)=\frac{1}{3}$,

$\displaystyle\frac{w(\lambda
Q)}{w(Q)}\sim\lambda^{n-d_{1}}\frac{\log(2+3\lambda^{-1})}{\log 5},$

where

$\displaystyle\lim_{\lambda\to 0^{+}}\frac{\log(2+3\lambda^{-1})}{\log
5}=\infty$

and hence (7.11) does not hold if $d=d_{1}$. This finishes the proof of (b)
and hence $d_{\infty}^{\mathrm{lower}}(w)=d_{1}$ and
$w\notin\mathbb{D}_{\infty,d_{1}}^{\mathrm{lower}}(\mathbb{R}^{n})$. From
these and Lemma 7.2, it follows that $d_{p,\infty}^{\mathrm{lower}}(W)=d_{1}$
and $W\notin\mathbb{D}_{p,\infty,d_{1}}^{\mathrm{lower}}$. Applying an
argument similar to that used in the proofs of both
$d_{p,\infty}^{\mathrm{lower}}(W)=d_{1}$ and
$W\notin\mathbb{D}_{p,\infty,d_{1}}^{\mathrm{lower}}$, we obtain
$d_{p,\infty}^{\mathrm{upper}}(W)=d_{2}$ and
$W\notin\mathbb{D}_{p,\infty,d_{2}}^{\mathrm{upper}}$. Therefore, $W$
satisfies all assumptions of (iv).

Applying an argument similar to that used in the proof of [3, Lemma 2.47], we
obtain (7.10). This finishes the proof of Lemma 7.9. ∎

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Fan Bu

Laboratory of Mathematics and Complex Systems (Ministry of Education of
China), School of Mathematical Sciences, Beijing Normal University, Beijing
100875, The People’s Republic of China

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Tuomas Hytönen (Corresponding author)

Department of Mathematics and Statistics, University of Helsinki, (Pietari
Kalmin katu 5), P.O. Box 68, 00014 Helsinki, Finland

E-mail<EMAIL_ADDRESS>

Dachun Yang and Wen Yuan

Laboratory of Mathematics and Complex Systems (Ministry of Education of
China), School of Mathematical Sciences, Beijing Normal University, Beijing
100875, The People’s Republic of China

E-mails<EMAIL_ADDRESS>(D. Yang)

E-mails<EMAIL_ADDRESS>(W. Yuan)