Real-variable characterizations and their applications of matrix-weighted Besov spaces on spaces of homogeneous type

Abstract

In this article, the authors introduce matrix-weighted Besov spaces on a given space of homogeneous type, $({\mathcal {X}}, d,\mu ),$ in the sense of Coifman and Weiss and prove that matrix-weighted Besov spaces are independent of the choices of both approximations of the identity with exponential decay and spaces of distributions. Moreover, the authors establish the wavelet characterization of matrix-weighted Besov spaces, introduce almost diagonal operators on matrix-weighted Besov sequence spaces, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of matrix-weighted Besov spaces. As an application, the authors obtain the boundedness of Calderón–Zygmund operators on matrix-weighted Besov spaces. One novelty is that, by using the property of matrix weights and the boundedness of matrix-weighted Hardy–Littlewood maximal operators, all the proofs presented in this article are different from those on Euclidean spaces, the latter strongly rely on the fact that Schwartz functions on Euclidean spaces decay faster than any polynomial. Another novelty is that all the results of this article get rid of both the reverse doubling condition of the measure $\mu$ and the triangle inequality of the quasi-metric d under consideration, by fully using the geometrical properties of ${\mathcal {X}}$ expressed by dyadic reference points, dyadic cubes, and wavelets. Besides, all the results in this article are new even for Ahlfors regular spaces and RD-spaces.

Appendix A: Reverse Hölder inequality of matrix weights, matrix-weighted Hardy–Littlewood maximal operators, and Calderón reproducing formulae

In this Appendix, we explore some further properties of matrix weights, which are of independent interest, and recall the Calderón reproducing formulae. To be precise, in Sect. A.1, we establish an equivalent characterization of $A_p({\mathcal {X}}, {\mathbb {C}}^n)$-matrix weights. Using this, we establish the reverse Hölder inequality for matrix weights on ${\mathcal {X}} .$ In Sect. A.2, we prove that the matrix-weighted Hardy–Littlewood maximal operator $M_{W, p}$ is bounded from the vector-valued Lebesgue space $[L^p({\mathcal {X}})]^n$ to $L^p({\mathcal {X}})$ if and only if W is a Muckenhoupt $A_p$ matrix weight. In Sect. A.3, we recall the Calderón reproducing formulae.

1.1 Reverse Hölder inequality of matrix weights

In this subsection, we obtain the reverse Hölder inequality of matrix weights. Let us begin with establishing an useful equivalent characterization of $A_p({\mathcal {X}}, {\mathbb {C}}^n)$-matrix weights. To this end, we need two technical lemmas. The following lemma is just [55, Lemma 3.2].

Lemma A.1

Let $\alpha \in (0, \infty ) .$ Let ${\mathcal {H}}$ be an n-dimensional Hilbert space and $\{e_i\}_{i = 1}^n$ its orthonormal basis,  where $n \in {\mathbb {N}} .$ Then,  for any $T \in {\mathcal {L}}({\mathcal {H}}) ,$

$$\begin{aligned} \frac{1}{n} \sum _{i = 1}^n \left\| T e_i \right\| _{{\mathcal {H}}}^{\alpha } \le \left\| T \right\| _{{\mathcal {L}}({\mathcal {H}})}^{\alpha } \le \max \left\{ 1, n^{\alpha - 1} \right\} \sum _{i = 1}^n \Vert T e_i\Vert _{{\mathcal {H}}}^{\alpha }, \end{aligned}$$

where ${\mathcal {L}}({\mathcal {H}})$ denotes the space of all bounded linear operators from ${\mathcal {H}}$ to itself and,  for any $T \in {\mathcal {L}}({\mathcal {H}}) ,$

$$\begin{aligned} \Vert T \Vert _{{\mathcal {L}}({\mathcal {H}})} := \sup _{\Vert x\Vert _{{\mathcal {H}}} = 1} \Vert T x \Vert _{{\mathcal {H}}}. \end{aligned}$$

The following lemma when ${\mathcal {X}} = {\mathbb {R}}^d$ is just [55, (3.2)].

Lemma A.2

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ a matrix weight W satisfy that,  for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n ,$ $\rho _{p, B} \left( {\vec {z}} \right) < \infty$ and $\rho _{p', B}^* \left( {\vec {z}} \right) < \infty ,$ $\{A_B\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ be a family of reducing operators of order p for W,  and $\{A_B^{\#}\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ a family of dual reducing operators of order p for W. Then there exists a constant $C \in [1, \infty ) ,$ depending only on n,  such that

$$\begin{aligned} C^{-1} [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* \le \sup _{B \subset {\mathcal {X}}} \left\| A_B^{\#} A_B \right\| \le C [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^*. \end{aligned}$$

Proof

Let p,  W,  $\{A_B\}_{{\text {ball}} \, B \subset {\mathcal {X}}} ,$ and $\{A_B^\#\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ be the same as in the present lemma. By Remark 2.18 and Lemma 2.7, we conclude that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n ,$

$$\begin{aligned} \left( \rho _{p, B} \right) ^* \left( {\vec {z}} \right)&\sim \sup _{{\vec {u}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\} } \frac{|({\vec {z}}, {\vec {u}})|}{|A_B {\vec {u}}|} \sim \sup _{{\vec {u}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\} } \frac{|({\vec {z}}, A_B^{-1} {\vec {u}})|}{|{\vec {u}}|}\\&\sim \sup _{{\vec {u}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\} } \frac{|(A_B^{-1} {\vec {z}}, {\vec {u}})|}{|{\vec {u}}|} \sim \left| A_B^{-1} {\vec {z}} \right| , \end{aligned}$$

where $\rho _{p, B}$ is the same as in Definition 2.15 and $(\rho _{p, B})^*$ the same as in Definition 2.13 with $\rho$ replaced by $\rho _{p, B} .$ From this, (2.7), and (2.6), we infer that

$$\begin{aligned} {[}W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^*&\sim \sup _{{\text {ball}} \, B \subset {\mathcal {X}}, \, {\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\}} \frac{|A_B^{\#} {\vec {z}}|}{|A_B^{-1} {\vec {z}}|}\\&\sim \sup _{{\text {ball}} \, B \subset {\mathcal {X}}} \left( \sup _{{\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\}} \frac{|A_B^{\#} A_B {\vec {z}}|}{|{\vec {z}}|} \right) \sim \sup _{{\text {ball}} \, B \subset {\mathcal {X}}} \left\| A_B^{\#} A_B \right\| . \end{aligned}$$

This finishes the proof of Lemma A.2. $\square$

The following lemma gives an useful equivalent characterization of $A_p({\mathcal {X}}, {\mathbb {C}}^n)$-matrix weights, which goes back to [55, Lemma 1.3] when ${\mathcal {X}} = {\mathbb {R}}^d .$

Proposition A.3

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ and W be a matrix weight. Then $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n)$ if and only if

(A.1)

is finite. Moreover,  if $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ then there exists a constant $C \in [1, \infty ) ,$ depending only on n and p,  such that

$$\begin{aligned} C^{-1} [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* \le [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{\frac{1}{p}} \le C [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^*. \end{aligned}$$

Proof

Let p and $p'$ be the same as in the present lemma and W a matrix weight. To show the present lemma, we claim that, if, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n ,$

$$\begin{aligned} \rho _{p, B} ({\vec {z}})< \infty \ \text {and}\ \rho _{p', B}^* ({\vec {z}}) < \infty , \end{aligned}$$

(A.2)

where $\rho _{p, B}$ and $\rho _{p', B}^*$ are the same as in Definition 2.15, then

$$\begin{aligned} {[}W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{\frac{1}{p}} \sim [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^*, \end{aligned}$$

(A.3)

where the positive equivalence constants depend only on n and p. Indeed, by (A.2) and Lemma 2.17, we find that there exist a family $\{A_B\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ of reducing operators of order p for W and a family $\{A_B^{\#}\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ of dual reducing operators of order p for W. Let $\{{\vec {e}}_i\}_{i = 1}^n$ be an orthonormal basis of ${\mathbb {C}}^n .$ By Lemmas 2.8, A.1, and 2.25, (2.6), and (2.5), we conclude that, for any ball $B \subset {\mathcal {X}} ,$

From this and Lemma A.2, we deduce that

which completes the proof of (A.3).

Next, we prove that $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n)$ if and only if $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} < \infty .$ We first show the necessity. Assume that $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then, by Remark 2.20(iii), we find that (A.2) holds true, which, together with the above claim, further implies that

$$\begin{aligned} {[}W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} \sim \left( [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* \right) ^p < \infty . \end{aligned}$$

This finishes the proof of the necessity.

Now, we prove the sufficiency. To this end, assume that $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} < \infty .$ Then, by (A.1) and Definition 2.11(ii), we conclude that, for any ball $B \subset {\mathcal {X}} ,$ there exists $x \in B$ such that W(x) is invertible and

which, combined with Lemma 2.8, further implies that, for any ${\vec {z}} \in {\mathbb {C}}^n ,$

(A.4)

From (2.6), Lemma 2.8, and (A.1), we infer that, for any ${\vec {z}} \in {\mathbb {C}}^n ,$

which, together with (A.4), further implies that (A.2) holds true. Using this and the above claim, we conclude that

$$\begin{aligned} {[}W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* \sim [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{\frac{1}{p}} < \infty \end{aligned}$$

and, hence, $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ This finishes the proof of the sufficiency and, hence, Proposition A.3. $\square$

Remark A.4

Let $W := [w_{ij}]_{n \times n} .$ If $p \in (1, \infty )$ and $n = 1 ,$ then $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} = [w_{11}]_{A_p({\mathcal {X}})}.$

The following lemma was used in [65] and mentioned in [44, p. 1336] without a proof. For the convenience of the reader, we give the details of its proof here.

Lemma A.5

Let ${\mathcal {H}}$ be a finite-dimensional Hilbert space and $\rho$ a norm on ${\mathcal {H}} .$ Then,  for any $z \in {\mathcal {H}} ,$ $\rho (z) = \rho ^{**}(z) .$

Proof

If $z = \theta ,$ then, obviously, $\rho (z) = 0 = \rho ^{**}(z) .$

In what follows, fix $z \in {\mathcal {H}} {\setminus } \{\theta \} .$ We first show that $\rho (z) \ge \rho ^{**}(z) .$ Indeed, for any $u \in {\mathcal {H}} {\setminus } \{\theta \} ,$

$$\begin{aligned} \rho ^*(u) = \sup _{v \in {\mathcal {H}} {\setminus } \{\theta \}} \frac{|(u, v)|}{\rho (v)} \ge \frac{|(u, z)|}{\rho (z)}. \end{aligned}$$

From this, we deduce that, for any $u \in {\mathcal {H}} {\setminus } \{ \theta \} ,$

$$\begin{aligned} \frac{|(z, u)|}{\rho ^*(u)} \le \rho (z) \end{aligned}$$

and, hence,

$$\begin{aligned} \rho ^{**}(z) \le \rho (z). \end{aligned}$$

(A.5)

Next, we show that $\rho (z) \le \rho ^{**}(z) .$ To this end, for any $\lambda \in (1, \infty ) ,$ let $z_\lambda := [\lambda / \rho (z)] z$ and

$$\begin{aligned} B := \left\{ v \in {\mathcal {H}}:\ \rho (v) \le 1 \right\} . \end{aligned}$$

Then it is easy to prove that $z_\lambda \notin B$ and B is a convex, balanced, and closed set. From this and [57, Theorems 3.7 and 12.5], we infer that there exists $u_\lambda \in {\mathcal {H}} {\setminus } \{\theta \}$ such that

$$\begin{aligned} \sup _{v \in B} |(v, u_\lambda )| < (z_\lambda , u_\lambda ) = \lambda \frac{(z, u_\lambda )}{\rho (z)} \end{aligned}$$

and, hence,

$$\begin{aligned} \rho ^*(u_\lambda ) = \sup _{v \in {\mathcal {H}} {\setminus } \{\theta \}} \frac{|(u_\lambda , v)| }{\rho (v)} =\sup _{v \in B} |(v, u_\lambda )| < \lambda \frac{|(z, u_\lambda )|}{\rho (z)}. \end{aligned}$$

By this, we further conclude that

$$\begin{aligned} \rho (z) < \lambda \frac{|(z, u_\lambda )|}{\rho ^*(u_\lambda )} \le \lambda \sup _{v \in {\mathcal {H}} {\setminus } \{\theta \}} \frac{|(z, v)|}{\rho ^*(v)} = \lambda \rho ^{**}(z). \end{aligned}$$

Letting $\lambda \in (1, \infty )$ and $\lambda \rightarrow 1 ,$ we have $\rho (z) \le \rho ^{**}(z) .$ This, together with (A.5), then finishes the proof of Lemma A.5. $\square$

The following lemma when ${\mathcal {X}} = {\mathbb {R}}^d$ is a part of [55, Corollary 3.3].

Lemma A.6

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ and W be a matrix weight. Then $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n)$ if and only if $W^{1 - p'} \in A_{p'}({\mathcal {X}}, {\mathbb {C}}^n) .$ Moreover,  there exists a constant $C \in [1, \infty ) ,$ depending only on n and p,  such that,  for any $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$

$$\begin{aligned} C^{-1} [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{p' - 1} \le \left[ W^{1 - p'} \right] _{A_{p'}({\mathcal {X}}, {\mathbb {C}}^n)} \le C [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{p' - 1}. \end{aligned}$$

(A.6)

Proof

Let p,  $p' ,$ and W be the same as in the present lemma. To show the necessity, it suffices to prove that, for any $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ $W^{1 - p'} \in A_{p'}({\mathcal {X}}, {\mathbb {C}}^n)$ and

$$\begin{aligned} \left[ W^{1 - p'}\right] _{A_{p'}({\mathcal {X}}, {\mathbb {C}}^n)}^* \le [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* . \end{aligned}$$

(A.7)

Indeed, by Definition 2.15, we find that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\} ,$

(A.8)

and

and, hence,

$$\begin{aligned} \left( \rho _{p', B, W^{1 - p'}} \right) ^* \left( {\vec {z}} \right) = \sup _{{\vec {u}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\}} \frac{|({\vec {z}}, {\vec {u}})|}{\rho _{p', B}^*(z)} = \left( \rho _{p', B}^* \right) ^* \left( {\vec {z}} \right) . \end{aligned}$$

From this, (A.8), and (2.7), we infer that

$$\begin{aligned} \left[ W^{1 - p'}\right] _{A_{p'}({\mathcal {X}}, {\mathbb {C}}^n)}^* = \sup _{{\text {ball}} \, B \subset {\mathcal {X}}, \, {\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\} } \frac{\rho _{p, B} ({\vec {z}})}{(\rho _{p', B}^*)^* ({\vec {z}})}, \end{aligned}$$

(A.9)

where $\rho _{p, B}$ and $\rho _{p', B}^*$ are the same as in Definition 2.15 and $(\rho _{p', B}^*)^*$ is the same as in Definition 2.13 with $\rho$ replaced by $\rho _{p', B}^* .$ Using (2.7) and Lemma A.5, we conclude that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n ,$

$$\begin{aligned} \rho _{p, B} \left( {\vec {z}} \right)&= \left( \rho _{p, B} \right) ^{**} \left( {\vec {z}} \right) = \sup _{{\vec {u}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\}} \frac{|({\vec {z}}, {\vec {u}})|}{(\rho _{p, B})^*({\vec {u}})} \\&\le [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* \sup _{{\vec {u}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\}} \frac{|({\vec {z}}, {\vec {u}})|}{\rho _{p', B}^*({\vec {u}})} = [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* \left( \rho _{p', B}^* \right) ^* \left( {\vec {z}} \right) . \end{aligned}$$

This, combined with (A.9), then finishes the proof of (A.7) and, hence, the necessity.

Next, we show the sufficiency. To this end, assume that $W^{1 - p'} \in A_{p'}({\mathcal {X}}, {\mathbb {C}}^n) .$ By (A.7), we find that

$$\begin{aligned} {[}W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* = \left[ \left( W^{1 - p'}\right) ^{1 - p} \right] _{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* \le \left[ W^{1 - p'}\right] _{A_{p'}({\mathcal {X}}, {\mathbb {C}}^n)}^* < \infty , \end{aligned}$$

(A.10)

which further implies that $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n)$ and, hence, completes the proof of the sufficiency.

Finally, from (A.7) and (A.10), we deduce that, for any $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$

$$\begin{aligned} {[}W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* = \left[ W^{1 - p'}\right] _{A_{p'}({\mathcal {X}}, {\mathbb {C}}^n)}^*, \end{aligned}$$

(A.11)

which, together with Proposition A.3, further implies that

$$\begin{aligned} {[}W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{\frac{1}{p}} \sim \left[ W^{1 - p'} \right] _{A_{p'}({\mathcal {X}}, {\mathbb {C}}^n)}^{\frac{1}{p'}} \end{aligned}$$

and, hence,

$$\begin{aligned} \left[ W^{1 - p'} \right] _{A_{p'}({\mathcal {X}}, {\mathbb {C}}^n)} \sim [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{\frac{p'}{p}} \sim [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{p' - 1}. \end{aligned}$$

This finishes the proof of Lemma A.6. $\square$

Remark A.7

By (A.11) and Remark 2.20(ii), we find that, when ${\mathcal {X}} = {\mathbb {R}}^d$ and $n = 1 ,$ Lemma A.6 is just [25, Proposition 7.1.5(4)].

Before establishing the reverse Hölder inequality of matrix weights, we first recall the concept of $A_p({\mathcal {X}})$-weights on ${\mathcal {X}}$ (see, for instance, [42, 60]).

Definition A.8

Let $p \in [1, \infty ) .$ A locally integrable and nonnegative function w on ${\mathcal {X}}$ is called an $A_p({\mathcal {X}})$-weight if w satisfies that, for almost every $x \in {\mathcal {X}} ,$ $w(x) \in (0, \infty )$ and $[w]_{A_p({\mathcal {X}})} < \infty ,$ where

where $1/p + 1/p' = 1 .$ Define $A_\infty ({\mathcal {X}}) := \bigcup _{p \in [1, \infty )} A_p({\mathcal {X}})$ and

$$\begin{aligned} {[}w]_{A_\infty ({\mathcal {X}})} := \sup _{{\text {ball}} \, B \subset {\mathcal {X}}} \frac{1}{w(B)} \int _B M \left( w {{\textbf {1}}}_B\right) (x) \, d\mu (x), \end{aligned}$$

(A.12)

where w(B) is the same as in (1.2).

By an argument similar to that used in the proofs of [41, Proposition 2.2] and [60, p. 8, Lemma 12], we obtain the following lemma; we omit the details here.

Lemma A.9

Let $p \in [1, \infty )$ and $w \in A_p({\mathcal {X}}) .$ Then

• (i) there exists a constant $C_{({\mathcal {X}})} \in [1, \infty ) ,$ depending only on ${\mathcal {X}} ,$ such that

  $$\begin{aligned} {[}w]_{A_{\infty }({\mathcal {X}})} \le C_{({\mathcal {X}})} [w]_{A_p({\mathcal {X}})}, \end{aligned}$$

  where $[w]_{A_{\infty }({\mathcal {X}})}$ and $[w]_{A_p({\mathcal {X}})}$ are the same as in Definition A.8;

• (ii) for any ball $B \subset {\mathcal {X}}$ and any measurable set $E \subset B$ with $\mu (E) \in (0, \infty ) ,$

  $$\begin{aligned} w(B) \le [w]_{A_p({\mathcal {X}})} \left[ \frac{\mu (B)}{\mu (E)} \right] ^p w(E). \end{aligned}$$

The following weak reverse Hölder inequality is just [42, Theorem 1.1].

Lemma A.10

Let $w \in A_{\infty }({\mathcal {X}})$ and

$$\begin{aligned} r(w) := 1 + \frac{1}{6 [32 A_0^2 (4 A_0^2 + A_0)^2 ]^\omega [w]_{A_{\infty }({\mathcal {X}})}}, \end{aligned}$$

(A.13)

where $A_0 ,$ $\omega ,$ and $[w]_{A_{\infty }({\mathcal {X}})}$ are the same,  respectively,  as in Definition 2.1, (2.2),  and (A.12). Then,  for any ball $B \subset {\mathcal {X}} ,$

From the above two lemmas, we can further infer the following conclusion.

Corollary A.11

Let $p \in (1, \infty ) ,$ $w \in A_p({\mathcal {X}}) ,$ and

$$\begin{aligned} {\widetilde{r}}(w) := 1 + \frac{1}{6 [32 A_0^2 (4 A_0^2 + A_0)^2 ]^\omega C_{({\mathcal {X}})} [w]_{A_p({\mathcal {X}})}}, \end{aligned}$$

(A.14)

where $C_{({\mathcal {X}})}$ is the same as in Lemma A.9(i). Then there exists a constant $C_{({\mathcal {X}}, p)} \in [1, \infty ) ,$ depending only on ${\mathcal {X}}$ and p,  such that,  for any ball $B \subset {\mathcal {X}} ,$

Proof

Let p and w be the same as in the present corollary. By Lemma A.9(i), we find that there exists a constant $C_{({\mathcal {X}})} \in [1, \infty ) ,$ depending only on ${\mathcal {X}} ,$ such that

$$\begin{aligned} {[}w]_{A_{\infty }({\mathcal {X}})} \le C_{({\mathcal {X}})} [w]_{A_p({\mathcal {X}})}. \end{aligned}$$

(A.15)

Let r(w) be the same as in (A.13). By (A.13), (A.15), and (A.14), we conclude that

$$\begin{aligned} r(w) \ge 1 + \frac{1}{6 [32 A_0^2 (4 A_0^2 + A_0)^2 ]^\omega C_{({\mathcal {X}})} [w]_{A_p({\mathcal {X}})}} = {\widetilde{r}}(w). \end{aligned}$$

From this, the Hölder inequality, Lemmas A.10 and A.9(ii), and (2.2), we deduce that, for any ball $B \subset {\mathcal {X}} ,$

This finishes the proof of Corollary A.11. $\square$

Applying Proposition A.3 and an argument similar to that used in the proof of [65, Lemma 5.3], we obtain the following conclusion; we omit the details here.

Lemma A.12

Let $p \in (1, \infty )$ and $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then there exists a constant $C_{(n, p)} \in [1, \infty ) ,$ depending only on n and p,  such that,  for any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{ {\varvec{0}} \} ,$

$$\begin{aligned} \left[ w_{{\vec {z}}} \right] _{A_p({\mathcal {X}})} \le C_{(n, p)} [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}, \end{aligned}$$

where,  for any and $x \in {\mathcal {X}} ,$

$$\begin{aligned} w_{{\vec {z}}}(x) := \left| W^{\frac{1}{p}}(x) {\vec {z}} \right| ^p. \end{aligned}$$

(A.16)

The following proposition is the reverse Hölder inequality of matrix weights.

Proposition A.13

Let $p \in (1, \infty ) ,$ $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ and

$$\begin{aligned} r(W) := 1 + \frac{1}{6 [32 A_0^2 (4 A_0^2 + A_0)^2 ]^\omega C_{({\mathcal {X}})} C_{(n, p)} [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}}, \end{aligned}$$

(A.17)

where $C_{({\mathcal {X}})}$ and $C_{(n, p)}$ are the same as,  respectively, in Lemmas A.9(i) and A.12. Then there exists a constant $C_{({\mathcal {X}}, n, p)} \in [1, \infty ) ,$ depending only on ${\mathcal {X}} ,$ n,  and p,  such that,  for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{ {\varvec{0}} \} ,$

(A.18)

where $w_{{\vec {z}}}$ is the same as in (A.16).

Proof

Let p,  W,  and r(W) be the same as in the present proposition. By Lemma A.12, we find that there exists a constant $C_{(n, p)} \in [1, \infty ) ,$ depending only on n and p,  such that, for any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{ {\varvec{0}} \} ,$

$$\begin{aligned} \left[ w_{{\vec {z}}} \right] _{A_p({\mathcal {X}})} \le C_{(n, p)} [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}. \end{aligned}$$

(A.19)

From this and Corollary A.11, it follows that, for any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{ {\varvec{0}} \} ,$

(A.20)

where $C_{({\mathcal {X}}, p)}$ and ${\widetilde{r}}(w_{{\vec {z}}})$ are the same as, respectively, in Corollary A.11 and (A.14). By the definition of ${\widetilde{r}}(w_{{\vec {z}}})$ and (A.19), we conclude that same as, respectively,

$$\begin{aligned} {\widetilde{r}} \left( w_{{\vec {z}}} \right) \ge 1 + \frac{1}{6 [32 A_0^2 (4 A_0^2 + A_0)^2 ]^\omega C_{({\mathcal {X}})} C_{(n, p)} [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}} = r(W). \end{aligned}$$

From this, (A.20), and the Hölder inequality, we infer that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{ {\varvec{0}} \} ,$

This finishes the proof of Proposition A.13. $\square$

Remark A.14

When ${\mathcal {X}} = {\mathbb {R}}^d ,$ Goldberg (see, for instance, the proof of [24, Proposition 2.4]) gave a more precise version of Proposition A.13 in which the item $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}$ in (A.18) can be removed.

Now, we establish another reverse Hölder inequality of matrix weights on ${\mathcal {X}} ,$ which when ${\mathcal {X}} = {\mathbb {R}}^d$ was proved in the proof of [44, Lemma 3.5].

Proposition A.15

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ and

$$\begin{aligned} r^*(W) := 1 + \frac{1}{6 [32 A_0^2 (4 A_0^2 + A_0)^2 ]^\omega C_{({\mathcal {X}})} C_{(n, p)} C [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{p' - 1}}, \end{aligned}$$

(A.21)

where $C_{({\mathcal {X}})} ,$ $C_{(n, p)} ,$ and C are the same as,  respectively,  in Lemmas A.9(i), A.12, and A.6. Then there exists a constant $C_{({\mathcal {X}}, n, p)} \in [1, \infty ) ,$ depending only on ${\mathcal {X}} ,$ n,  and p,  such that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\} ,$

where,  for any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\}$ and $x \in {\mathcal {X}} ,$

$$\begin{aligned} w^*_{{\vec {z}}}(x) := \left| W^{-\frac{1}{p}}(x) {\vec {z}} \right| ^{p'}. \end{aligned}$$

Proof

Let p,  $p' ,$ W,  and $r^*(W)$ be the same as in the present lemma. By Lemma A.6, we find that $W^{1 - p'} \in A_{p'}({\mathcal {X}}, {\mathbb {C}}^n) .$ From this, Proposition A.13 with W and p replaced, respectively, by $W^{1-p'}$ and $p' ,$ and (A.6), we deduce that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\} ,$

(A.22)

where $r(W^{1 - p'})$ and $C_{({\mathcal {X}}, n, p)}$ are the same as in Proposition A.13 and C the same as in Lemma A.6. By (A.6) and the definition of $r(W^{1 - p'}) ,$ we conclude that

$$\begin{aligned} r \left( W^{1 - p'} \right) \ge 1 + \frac{1}{6 [32 A_0^2 (4 A_0^2 + A_0)^2 ]^\omega C_{({\mathcal {X}})} C_{(n, p)} C [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{p' - 1}} = r^*(W). \end{aligned}$$

From this, the Hölder inequality, and (A.22), we infer that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\} ,$

This finishes the proof of Proposition A.15. $\square$

1.2 Matrix-weighted Hardy–Littlewood maximal operators

In this subsection, we introduce the matrix-weighted Hardy–Littlewood maximal operator on spaces of homogeneous type and prove that this operator is bounded from $[L^p({\mathcal {X}})]^n$ to $L^p({\mathcal {X}})$ for any given $p \in (1, \infty ) .$ Let us begin with the concept of the norm of $[L^p({\mathcal {X}})]^n .$

Definition A.16

Let $p \in (0, \infty )$ and $[L^p({\mathcal {X}})]^n$ be the same as in (1.3). Then, for any ${\vec {f}} \in [L^p({\mathcal {X}})]^n ,$ its norm $\Vert {\vec {f}} \Vert _{[L^p({\mathcal {X}})]^n}$ is defined by setting

$$\begin{aligned} \left\| {\vec {f}} \right\| _{[L^p({\mathcal {X}})]^n} := \left[ \int _{\mathcal {X}} \left| {\vec {f}} (x) \right| ^p \, d\mu (x) \right] ^{\frac{1}{p}}. \end{aligned}$$

Now, we recall the concept of matrix-weighted Hardy–Littlewood maximal operators (see, for instance, [12, (1.3)], [24, (14)], and [44, p. 1337]).

Definition A.17

If $p \in (1, \infty )$ and W is a matrix weight, then the matrix-weighted Hardy–Littlewood maximal operator $M_{W, p}$ is defined by setting, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

The following theorem is the main result of this subsection.

Theorem A.18

Let $p \in (1, \infty )$ and W be a matrix weight. Then $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n)$ if and only if $M_{W, p}$ is bounded from $[L^p({\mathcal {X}})]^n$ to $L^p({\mathcal {X}}) .$

To show Theorem A.18, we need the following concept of auxiliary matrix-weighted Hardy–Littlewood maximal operators (see, for instance, [44, p. 1337], [12, (2.2)], and [24, (13)]).

Definition A.19

Let $p \in (1, \infty ) ,$ $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ and $\{A_B\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ be a family of reducing operators of order p for W. The auxiliary matrix-weighted Hardy–Littlewood maximal operator ${\widetilde{M}}_{W, p}$ is defined by setting, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

Remark A.20

In Definition A.19, let $\{ {\widetilde{A}}_B \}_{\text {ball } B \subset {\mathcal {X}}}$ be another family of reducing operators of order p for W. Then, by Remark 2.18, we find that, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

where the positive equivalence constants depend only on n. In this sense, ${\widetilde{M}}_{W, p}$ is independent of the choice of $\{A_B\}_{{\text {ball}} \, B \subset {\mathcal {X}}} .$

Applying Propositions A.13 and A.15 and an argument similar to that used in the proof of [24, Proposition 2.4] (see also [12, Proposition 2.1]), we obtain the following lemma; we omit the details of its proof here.

Lemma A.21

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ $\{A_B\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ be a family of reducing operators of order p for W,  and $\{A_B^\#\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ a family of dual reducing operators of order p for W. Then there exists a positive constant C,  depending only on ${\mathcal {X}} ,$ n,  p,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that,  for any ball $B \subset {\mathcal {X}} ,$

(A.23)

and

(A.24)

where r(W) and $r^*(W)$ are the same as,  respectively,  in (A.17) and (A.21).

Now, we prove that ${\widetilde{M}}_{W, p}$ is bounded from $[L^p({\mathcal {X}})]^n$ to $L^p({\mathcal {X}})$ for any given $p \in (1, \infty ) .$ From (A.24), the boundedness of M on $L^r({\mathcal {X}})$ with $r \in (1, \infty ]$ (see, for instance, [13] and [14, (3.6)]), and an argument similar to that used in the proof of [44, Lemma 3.5] (see also [24, Lemma 3.1]), we deduce the following conclusion; we omit the details here.

Lemma A.22

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ and $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then,  for any $r \in (1/[1 - \frac{1}{p' r^*(W)}], \infty ] ,$ there exists a constant $C \in [1, \infty ) ,$ depending only on ${\mathcal {X}} ,$ n,  p,  r,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that,  for any ${\vec {f}} \in [L^r({\mathcal {X}})]^n ,$

$$\begin{aligned} \left\| {\widetilde{M}}_{W, p} \left( {\vec {f}}\right) \right\| _{L^r({\mathcal {X}})} \le C \left\| {\vec {f}} \right\| _{[L^r({\mathcal {X}})]^n}. \end{aligned}$$

Next, we recall the concept of adjacent dyadic cube systems on spaces of homogeneous type. The following lemma is just [40, Theorem 4.1].

Lemma A.23

Let $\delta \in (0, 1)$ satisfy $96 A_0^6 \delta \le 1 .$ Then there exist a positive integer K,  a countable set of points,  $\{ z_{\alpha }^{k, t} :\ k \in {\mathbb {Z}},\ \alpha \in {\mathcal {A}}_k,\ t \in \{ 1, \ldots , K \} \} \subset {\mathcal {X}}$ with ${\mathcal {A}}_k ,$ for any $k \in {\mathbb {Z}} ,$ being a set of indices,  and a finite number of dyadic grids,  $\{ {\mathcal {D}}^t := \{ Q_{\alpha }^{k, t} :\ k \in {\mathbb {Z}},\ \alpha \in {\mathcal {A}}_k \} :\ t \in \{ 1, \ldots , K \} \} ,$ such that

• (a) for any $t \in \{ 1, \ldots , K \} ,$

  • (i) for any $k \in {\mathbb {Z}} ,$ $\bigcup _{\alpha \in {\mathcal {A}}_k} Q_\alpha ^{k, t} = {\mathcal {X}}$ and $\{ Q_\alpha ^{k, t} :\ \alpha \in {\mathcal {A}}_k \}$ is disjoint; 

  • (ii) if $k, l \in {\mathbb {Z}}$ and $k \le l ,$ then,  for any $\alpha \in {\mathcal {A}}_k$ and $\beta \in {\mathcal {B}}_l,$ either $Q_\beta ^l \subset Q_\alpha ^k$ or $Q_\beta ^l \cap Q_\alpha ^k = \emptyset ;$

  • (iii) for any $k \in {\mathbb {Z}}$ and $\alpha \in {\mathcal {A}}_k ,$ $B(z_\alpha ^{k, t}, (12 A_0^4)^{-1} \delta ^k) \subset Q_\alpha ^{k, t} \subset B(z_\alpha ^{k, t}, 4 A_0^2 \delta ^k) =: B(Q_\alpha ^{k, t}) ;$

• (b) there exists a constant $C \in (1, \infty ) ,$ depending only on $A_0$ and $\delta ,$ such that,  for any ball $B := B(x, r) \subset {\mathcal {X}} ,$ there exist $t \in \{ 1, \ldots , K \}$ and $Q_B \in {\mathcal {D}}^t$ such that $B \subset Q_B$ and ${\,\text {diam}\,}(Q_B) \le C r .$

In this subsection, $\{{\mathcal {D}}^t\}_{t=1}^K$ in Lemma A.23 is called an adjacent dyadic cube system. Moreover, let

$$\begin{aligned} {\mathcal {D}} := \bigcup _{t = 1}^K {\mathcal {D}}^t. \end{aligned}$$

(A.25)

The following lemma is just [40, Corollary 7.4].

Lemma A.24

• (i) For any $Q \in {\mathcal {D}} ,$ there exists a constant $C \in [1, \infty ) ,$ depending only on ${\mathcal {X}} ,$ such that $\mu (B(Q)) \le C \mu (Q) ,$ where B(Q) is the same as in Lemma A.23(a)(iii) with $Q_\alpha ^{k, t}$ replaced by Q; 

• (ii) For any ball $B \subset {\mathcal {X}} ,$ there exist $Q_B \in {\mathcal {D}}$ and a positive constant $C \in [1, \infty ) ,$ depending only on ${\mathcal {X}}$ and $\delta ,$ such that $B \subset Q_B$ and $\mu (Q_B) \le C \mu (B) .$

By Lemma A.24, we obtain the following conclusion.

Lemma A.25

Let $p \in (1, \infty )$ and $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then there exist a sequence $\{A_Q\}_{Q \in {\mathcal {D}}}$ of reducing operators of order p for W and a sequence $\{A_Q^{\#}\}_{Q \in {\mathcal {D}}}$ of dual reducing operators of order p for W.

Proof

Let p and W be the same as in the present lemma. By Definition 2.19, we find that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n ,$

$$\begin{aligned} \rho _{p, B} \left( {\vec {z}}\right)< \infty \ \text {and} \ \rho _{p', B}^* \left( {\vec {z}}\right) < \infty , \end{aligned}$$

where $1/p + 1/p' = 1$ and $\rho _{p, B}$ and $\rho _{p', B}^*$ are the same as in Definition 2.15. From this and Lemma A.24(i), we infer that, for any $Q \in {\mathcal {D}}$ and ${\vec {z}} \in {\mathbb {C}}^n ,$

$$\begin{aligned} \rho _{p, Q} \left( {\vec {z}}\right) \lesssim \rho _{p, B(Q)} \left( {\vec {z}}\right)< \infty \ \text {and} \ \rho _{p', Q}^* \left( {\vec {z}}\right) \lesssim \rho _{p', B(Q)}^* \left( {\vec {z}}\right) < \infty , \end{aligned}$$

which, combined with Lemma 2.17, further implies that there exist a sequence $\{A_Q\}_{Q \in {\mathcal {D}}}$ of reducing operators of order p for W and a sequence $\{A_Q^{\#}\}_{Q \in {\mathcal {D}}}$ of dual reducing operators of order p for W. This finishes the proof of Lemma A.25. $\square$

Now, we recall the concept of auxiliary dyadic matrix-weighted Hardy–Littlewood maximal operators (see, for instance, [44, p. 1337] and [24, p. 208]).

Definition A.26

Let $p \in (1, \infty ) ,$ $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ $\{A_Q\}_{Q \in {\mathcal {D}}}$ be a sequence of reducing operators of order p for W,  and K the same as in Lemma A.23. The auxiliary dyadic matrix-weighted Hardy–Littlewood maximal operator ${\widetilde{M}}_{W, p}^{{\mathcal {D}}^t}$ with $t \in \{1, \ldots , K\}$ is defined by setting, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

and the auxiliary dyadic class matrix-weighted Hardy–Littlewood maximal operator ${\widetilde{M}}_{W, p}^{{\mathcal {D}}}$ is defined by setting, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

where ${\mathcal {D}}$ is the same as in (A.25).

Remark A.27

In Definition A.26, let $\{ {\widetilde{A}}_Q \}_{Q \in {\mathcal {D}}}$ be another family of reducing operators of order p for W. Then, by Remark 2.18, we conclude that, for any $x \in {\mathcal {X}} ,$

and

where the positive equivalence constants depend only on n. In this sense, ${\widetilde{M}}_{W, p}^{{\mathcal {D}}^t}$ and ${\widetilde{M}}_{W, p}^{{\mathcal {D}}}$ are independent of the choice of $\{A_Q\}_{Q \in {\mathcal {D}}} .$

The following lemma shows the relations between ${\widetilde{M}}_{W, p}^{{\mathcal {D}}}$ and ${\widetilde{M}}_{W, p} .$

Lemma A.28

Let $p \in (1, \infty )$ and $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then there exists a constant $C \in [1, \infty ) ,$ depending only on ${\mathcal {X}} ,$ $\delta ,$ and n,  such that,  for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

$$\begin{aligned} C^{-1} {\widetilde{M}}_{W, p} \left( {\vec {f}}\right) (x) \le {\widetilde{M}}_{W, p}^{{\mathcal {D}}} \left( {\vec {f}}\right) (x) \le C {\widetilde{M}}_{W, p} \left( {\vec {f}}\right) (x). \end{aligned}$$

Proof

Let p and W be the same as in the present lemma. From Remark 2.20(iii) and Lemma A.25, we deduce that there exist two sequences $\{A_B\}_{{\text {ball}} \, B \subset {\mathcal {X}}}$ and $\{A_Q\}_{Q \in {\mathcal {D}}}$ of reducing operators of order p for W. By Remark 2.18 and Lemma A.24(i), we conclude that, for any $Q \in {\mathcal {D}}$ and ${\vec {z}} \in {\mathbb {C}}^n ,$

where B(Q) is the same as in Lemma A.23(a)(iii). From this and Lemma A.24, we infer that, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

(A.26)

Repeating the argument similar to that used in the proof of (A.26) and using Lemma A.24(ii), we find that, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

$$\begin{aligned} {\widetilde{M}}_{W, p} \left( {\vec {f}}\right) (x) \lesssim {\widetilde{M}}_{W, p}^{{\mathcal {D}}} \left( {\vec {f}}\right) (x). \end{aligned}$$

This finishes the proof of Lemma A.28. $\square$

By Lemmas A.22 and A.28, we obtain the following conclusion which when ${\mathcal {X}} = {\mathbb {R}}^d$ is just [24, Lemma 3.1] (see also [44, Lemma 3.5]).

Lemma A.29

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ and $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then,  for any $r \in (1/[1 - \frac{1}{p' r^*(W)}], \infty ) ,$ there exists a positive constant C,  depending only on ${\mathcal {X}} ,$ $\delta ,$ n,  p,  r,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that, for any ${\vec {f}} \in [L^p({\mathcal {X}})]^n ,$

$$\begin{aligned} \left\| {\widetilde{M}}_{W, p}^{{\mathcal {D}}} \left( {\vec {f}}\right) \right\| _{L^r({\mathcal {X}})} \le C \left\| {\vec {f}} \right\| _{[L^r({\mathcal {X}})]^n}. \end{aligned}$$

From an argument similar to that used in the proof of [54, p. 19], we deduce the following conclusion; we omit the details here.

Lemma A.30

Let $p \in (1, \infty ) ,$ $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ $\{A_Q\}_{Q \in {\mathcal {D}}}$ be a sequence of reducing operators of order p for W,  and $\{A_Q^{\#}\}_{Q \in {\mathcal {D}}}$ a sequence of dual reducing operators of order p for W. Then,  for any $Q \in {\mathcal {D}} ,$

$$\begin{aligned} \left\| \left( A_Q A_Q^{\#} \right) ^{-1} \right\| = \left\| \left( A_Q^{\#} A_Q \right) ^{-1} \right\| \le 1. \end{aligned}$$

The following lemma which when ${\mathcal {X}} = {\mathbb {R}}^d$ was used in [24, Lemma 3.3] (see also [44, Lemma 3.6]). For the convenience of the reader, we give the details here.

Lemma A.31

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ $\{A_Q\}_{Q \in {\mathcal {D}}}$ be a sequence of reducing operators of order p for W,  K the same as in Lemma A.23, and $t \in \{1, \ldots , K\} .$ Then there exists a positive constant C,  depending only on n,  p,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that,  for any $Q \in {\mathcal {D}}^t$ and a pairwise disjoint sequence $\{ Q_j \}_{j \in J} \subset {\mathcal {D}}^t$ with $\bigcup _{j \in J} Q_j \subset Q ,$

$$\begin{aligned} \sum _{j \in J} \mu (Q_j) \left\| A_Q A_{Q_j}^{-1} \right\| ^{p'} \le C \mu (Q). \end{aligned}$$

Proof

Let p,  $p' ,$ W,  K,  and t be the same as in the present lemma. From Lemma A.25, we deduce that there exist a sequence $\{A_Q\}_{Q \in {\mathcal {D}}}$ of reducing operators of order p for W. Let $\{{\vec {e}}_i\}_{i = 1}^n$ be an orthonormal basis of ${\mathbb {C}}^n .$ By Lemmas 2.8 and A.30, we conclude that, for any $Q, {\widetilde{Q}} \in {\mathcal {D}}^t ,$

$$\begin{aligned} \left\| A_Q A_{{\widetilde{Q}}}^{-1} \right\|&= \left\| A_{{\widetilde{Q}}}^{-1} A_Q \right\| \le \left\| A_{{\widetilde{Q}}}^{-1} \left( A_{{\widetilde{Q}}}^{\#}\right) ^{-1} \right\| \left\| A_{{\widetilde{Q}}}^{\#} A_Q \right\| ^{p'} \\&= \left\| \left( A_{{\widetilde{Q}}}^{\#} A_{{\widetilde{Q}}}\right) ^{-1} \right\| \left\| A_{{\widetilde{Q}}}^{\#} A_Q \right\| \le \left\| A_{{\widetilde{Q}}}^{\#} A_Q \right\| . \end{aligned}$$

This, together with Lemma A.1, (2.5), Lemma A.2, and Proposition A.3, further implies that, for any $Q \in {\mathcal {D}}^t$ and a pairwise disjoint sequence $\{ Q_j \}_{j \in J} \subset {\mathcal {D}}^t$ with $\bigcup _{j \in J} Q_j \subset Q ,$

$$\begin{aligned}&\sum _{j \in J} \mu (Q_j) \left\| A_Q A_{Q_j}^{-1} \right\| ^{p'}\\&\quad \le \sum _{j \in J} \mu (Q_j) \left\| A_{Q_j}^{\#} A_Q \right\| ^{p'} \sim \sum _{j \in J} \mu (Q_j) \sum _{i = 1}^n \left| A_{Q_j}^{\#} A_Q {\vec {e}}_i \right| ^{p'}\\&\quad \sim \sum _{i = 1}^n \sum _{j \in J} \int _{Q_j} \left| W^{-\frac{1}{p}}(x) A_Q {\vec {e}}_i \right| ^{p'} \, d\mu (x) \\&\quad \lesssim \sum _{i = 1}^n \int _Q \left| W^{-\frac{1}{p}}(x) A_Q {\vec {e}}_i \right| ^{p'} \, d\mu (x) \sim \mu (Q) \sum _{i = 1}^n \left| A_Q^{\#} A_Q {\vec {e}}_i \right| ^{p'} \\&\quad \sim \mu (Q) \left\| A_Q^{\#} A_Q \right\| ^{p'} \sim [W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^{\frac{p'}{p}} \mu (Q). \end{aligned}$$

This finishes the proof of Lemma A.31. $\square$

Applying (A.23), Lemmas A.30 and A.31, and an argument similar to that used in the proof of [44, Lemma 3.6], we obtain the following conclusion. For the convenience of the reader, we give the details here.

Lemma A.32

Let $p \in (1, \infty ) ,$ $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ $\{A_Q\}_{Q \in {\mathcal {D}}}$ be a sequence of reducing operators of order p for W,  K the same as in Lemma A.23, and $t \in \{1, \ldots , K\} .$ For any $Q \in {\mathcal {D}}^t$ and $x \in Q ,$ let

$$\begin{aligned} N_{Q, t}(x) := \sup _{\{ Q_\alpha ^{k, t} :\, k\in {\mathbb {Z}},\, \alpha \in {\mathcal {A}}_k,\, x \in Q_\alpha ^{k, t} \subset Q\}} \left\| W^{\frac{1}{p}} (x) A_{Q_{\alpha }^{k, t}}^{-1} \right\| . \end{aligned}$$

Then,  for any $r \in [0, p r(W)]$ with r(W) in (A.17),  there exists a positive constant C,  depending only on n,  p,  r,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that,  for any $Q \in {\mathcal {D}}^t ,$

$$\begin{aligned} \int _Q \left[ N_{Q, t}(x) \right] ^r \, d\mu (x) \le C \mu (Q). \end{aligned}$$

Proof

Let p,  W,  K,  t,  and r be the same as in the present lemma. We only consider the case $r \in (0, pr(W)]$ because the case $r = 0$ is trivial and we omit the details. From Lemma A.25, we infer that there exists a sequence $\{A_Q\}_{Q \in {\mathcal {D}}}$ of reducing operators of order p for W. For any $Q \in {\mathcal {D}}^t$ and $x \in Q ,$ let

$$\begin{aligned} N_{Q, t, m}(x) := \sup _{\{ Q_\alpha ^{k, t} :\, k \le m,\, \alpha \in {\mathcal {A}}_k,\, x \in Q_\alpha ^{k, t} \subset Q\}} \left\| W^{\frac{1}{p}} (x) A_{Q_{\alpha }^{k, t}}^{-1} \right\| . \end{aligned}$$

It is easy to show that, for any $Q \in {\mathcal {D}}^t$ and $x \in Q ,$ $N_{Q, t, m} (x)$ is increasing with respect to m and

$$\begin{aligned} N_{Q, t}(x) = \lim _{m \rightarrow \infty } N_{Q, t, m}(x). \end{aligned}$$

(A.27)

To prove the present lemma in the case $r \in (0, pr(W)] ,$ it suffices to show that there exists a positive constant C,  depending only on n,  p,  r,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that, for any $Q \in {\mathcal {D}}^t ,$

$$\begin{aligned} \int _Q \left[ N_{Q, t, m}(x) \right] ^r \, d\mu (x) \le C \mu (Q). \end{aligned}$$

(A.28)

Indeed, if (A.28) holds true, then, from (A.27), the fact that $N_{Q, t, m} (x)$ is increasing with respect to m,  the Levi lemma, and (A.28), we deduce that

$$\begin{aligned} \int _Q \left[ N_{Q, t} (x) \right] ^r \, d\mu (x) = \lim _{m \rightarrow \infty } \int _Q \left[ N_{Q,m}(x) \right] ^r \, d\mu (x) \le C \mu (Q), \end{aligned}$$

which is the desired conclusion.

Now, we prove (A.28). To this end, fix $Q := Q_{\alpha _0}^{k_0, t} \in {\mathcal {D}}^t .$ Let $m \in \{k_0, k_0 + 1, \ldots \}$ and ${\mathscr {C}} := (2C)^{1/p'} \in [1, \infty ) ,$ where $1/p + 1/p' = 1$ and C is the same as in Lemma A.31, and let

$$\begin{aligned} F_1 := \left\{ Q_\alpha ^{k, t} \subset Q :\ k \le m,\ \alpha \in {\mathcal {A}}_k,\ \left\| A_Q A_{Q_{\alpha }^{k, t}}^{-1} \right\| > {\mathscr {C}} \right\} \end{aligned}$$

and

$$\begin{aligned} D_1 := \{ R \in F_1 :\ R \text { is the maximal element of } F_1 \}. \end{aligned}$$

It is easy to show that $D_1$ is pairwise disjoint and $\bigcup _{R \in D_1} R \subset Q .$ From this, the definition of $D_1 ,$ and Lemma A.31, we infer that

$$\begin{aligned} \sum _{R \in D_1} \mu (R) \le \frac{1}{{\mathscr {C}}^{p'}} \sum _{R \in D_1} \mu (R) \left\| A_Q A_R^{-1} \right\| ^{p'} \le \frac{C}{{\mathscr {C}}^{p'}} \mu (Q) = \frac{1}{2} \mu (Q). \end{aligned}$$

(A.29)

Notice that, by Lemma A.30 and the definition of $D_1 ,$ we have, for any $x \in Q {\setminus } \bigcup _{R \in D_1} R$ and $Q_{\alpha }^{k, t}$ satisfying $x \in Q_{\alpha }^{k, t} \subset Q$ with $k \le m$ and $\alpha \in {\mathcal {A}}_k ,$

$$\begin{aligned} \left\| W^{\frac{1}{p}}(x) A_{Q_{\alpha }^{k, t}}^{-1} \right\|&\le \left\| W^{\frac{1}{p}}(x) A_Q^{\#} \right\| \left\| \left( A_Q^{\#} \right) ^{-1} A_Q^{-1} \right\| \left\| A_Q A_{Q_{\alpha }^{k, t}}^{-1} \right\| \\&= \left\| W^{\frac{1}{p}}(x) A_Q^{\#} \right\| \left\| \left( A_Q A_Q^{\#} \right) ^{-1} \right\| \left\| A_Q A_{Q_{\alpha }^{k, t}}^{-1} \right\| \\&\le {\mathscr {C}} \left\| W^{\frac{1}{p}}(x) A_Q^{\#} \right\| . \end{aligned}$$

From this, the Hölder inequality, and (A.23), we deduce that

(A.30)

where the implicit positive constants depend only on n,  p,  r,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} .$

If $D_1 = \emptyset ,$ then (A.30) is just (A.28) in this case. If $D_1 \ne \emptyset ,$ then, by the definition of $D_1$ and $M \in [1, \infty ) ,$ we find that, for any $R := Q_\alpha ^{k, t} \in D_1 ,$ $R \subsetneqq Q$ and, hence,

$$\begin{aligned} k \in [k_0+1, m]. \end{aligned}$$

(A.31)

For any $R \in D_1 ,$ let

$$\begin{aligned} F_R := \left\{ x \in R :\ N_{Q, t, m}(x) > N_{R, t, m}(x) \right\} . \end{aligned}$$

(A.32)

From this, the definition of $N_{Q, t, m} ,$ and Definition A.23(a)(ii), it follows that, for any $R \in D_1$ and $x \in F_R ,$

$$\begin{aligned} N_{Q, t, m}(x)&= \max \left\{ \sup _{\{ Q_{\alpha }^{k, t} :\, R \subsetneqq Q_{\alpha }^{k, t} \subset Q\}} \left\| W^{\frac{1}{p}} (x) A_{Q_{\alpha }^{k, t}}^{-1} \right\| ,\ N_{R, t, m}(x) \right\} \nonumber \\&= \sup _{\{Q_{\alpha }^{k, t} :\, R \subsetneqq Q_{\alpha }^{k, t} \subset Q\}} \left\| W^{\frac{1}{p}} (x) A_{Q_{\alpha }^{k, t}}^{-1} \right\| . \end{aligned}$$

(A.33)

For any $R \in D_1 ,$ by the definition of $D_1 ,$ we find that, for any $Q_{\alpha }^{k, t} \in {\mathcal {D}}^t$ satisfying $R \subsetneqq Q_{\alpha }^{k, t} \subset Q ,$

$$\begin{aligned} \left\| A_Q A_{Q_{\alpha }^{k, t}}^{-1} \right\| \le {\mathscr {C}}, \end{aligned}$$

which, together with (A.33) and Lemma A.30, further implies that, for any $x \in F_R ,$

$$\begin{aligned} N_{Q, t, m}(x)&\le \left\| W^{\frac{1}{p}}(x) A_Q^{\#} \right\| \left\| \left( A_Q^{\#} \right) ^{-1} A_Q^{-1} \right\| \sup _{\{Q_{\alpha }^{k, t} :\, R \subsetneqq Q_{\alpha }^{k, t} \subset Q\}} \left\| A_Q A_{Q_{\alpha }^{k, t}}^{-1} \right\| \nonumber \\&= \left\| W^{\frac{1}{p}}(x) A_Q^{\#} \right\| \left\| \left( A_Q A_Q^{\#} \right) ^{-1} \right\| \sup _{\{Q_{\alpha }^{k, t} :\, R \subsetneqq Q_{\alpha }^{k, t} \subset Q\}} \left\| A_Q A_{Q_{\alpha }^{k, t}}^{-1} \right\| \nonumber \\&\le {\mathscr {C}} \left\| W^{\frac{1}{p}}(x) A_Q^{\#} \right\| . \end{aligned}$$

(A.34)

Since $D_1$ is pairwise disjoint, it then follows that $\{F_R\}_{R \in D_1}$ is pairwise disjoint. From this, (A.34), the definitions of both $F_R$ and $D_1 ,$ the Hölder inequality, and (A.23), we infer that

which, combined with the fact that $D_1$ is pairwise disjoint, (A.30), and (A.32), further implies that there exists a positive constant ${\widetilde{C}} ,$ depending only on n,  p,  r,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that

$$\begin{aligned} \int _Q \left[ N_{Q, t, m}(x) \right] ^r \, d\mu (x)&= \int _{Q {\setminus } (\bigcup _{R \in D_1} R)} \left[ N_{Q, t, m}(x) \right] ^r \, d\mu (x) \nonumber \\&\qquad + \sum _{R \in D_1} \int _{F_R} \cdots +\sum _{R \in D_1} \int _{R {\setminus } F_R} \cdots \nonumber \\&\le {\widetilde{C}} \mu (Q) + \sum _{R \in D_1} \int _{R} \left[ N_{R, t, m}(x) \right] ^r \, d\mu (x). \end{aligned}$$

(A.35)

For any $R \in D_1 ,$ let

$$\begin{aligned} F_2^{(R)}:= & {} \left\{ Q_\alpha ^{k, t} \subset R :\ k \le m,\ \alpha \in {\mathcal {A}}_k,\ \left\| A_R A_{Q_{\alpha }^{k, t}}^{-1} \right\| > M \right\} , \\ D_2^{(R)}:= & {} \left\{ {\widetilde{Q}} \in F_2^{(R)} :\ {\widetilde{Q}} \text { is the maximal element of } F_2^{(R)} \right\} , \end{aligned}$$

and $D_2 := \bigcup _{R \in D_1} D_2^{(R)} .$ If $D_2 = \emptyset ,$ then, by an argument similar to that used in the estimation of (A.30), we obtain (A.28) in this case. If $D_2 \ne \emptyset ,$ then, by the definition of $D_2$ and $M \in [1, \infty ) ,$ we conclude that, for any ${\widetilde{Q}} := Q_\alpha ^{k, t} \in D_2 ,$ there exists $R \in D_1$ such that ${\widetilde{Q}} \in D_2^{(R)}$ and, hence, ${\widetilde{Q}} \subsetneqq R ,$ which, together with (A.31), further implies that $k \in [k_0+2, m] .$ Applying (A.35), an argument similar to that used in the estimation of (A.35), and (A.29), we find that

$$\begin{aligned}&\int _Q \left[ N_{Q, t, m}(x) \right] ^r \, d\mu (x)\\&\quad \le {\widetilde{C}} \mu (Q) + \sum _{R \in D_1} \left\{ {\widetilde{C}} \mu (R) + \sum _{{\widetilde{Q}} \in D_2^{(R)}} \int _{{\widetilde{Q}}} \left[ N_{{\widetilde{Q}}, t, m}(x) \right] ^r \, d\mu (x) \right\} \\&\quad \le {\widetilde{C}} \left( 1 + \frac{1}{2} \right) \mu (Q) + \sum _{{\widetilde{Q}} \in D_2} \int _{{\widetilde{Q}}} \left[ N_{{\widetilde{Q}}, t, m}(x) \right] ^r \, d\mu (x). \end{aligned}$$

If this process does not stop, then repeating the above procedure, we obtain a sequence $\{D_j\}_{j \in {\mathbb {N}}}$ of sets of cubes satisfying that, for any $j \in {\mathbb {N}} ,$

$$\begin{aligned} \int _Q \left[ N_{Q, t, m}(x) \right] ^r \, d\mu (x) {\le } {\widetilde{C}} \left( \sum _{i = 0}^{j - 1} \frac{1}{2^i} \right) \mu (Q) {+} \sum _{R \in D_j} \int _{R} \left[ N_{R, t, m}(x) \right] ^r \, d\mu (x) \end{aligned}$$

(A.36)

and, for any $R := Q_\alpha ^{k, t} \in D_j ,$ $k \in [k_0 + j, m] .$ Therefore, there exists a minimal $N \in {\mathbb {N}}$ such that $D_N = \emptyset .$ Then, by (A.36), we find that

$$\begin{aligned} \int _Q \left[ N_{Q, t, m}(x) \right] ^r \, d\mu (x) \le {\widetilde{C}} \left( \sum _{i = 0}^{N - 1} \frac{1}{2^i} \right) \mu (Q) \le 2 {\widetilde{C}} \mu (Q). \end{aligned}$$

This finishes the proof of (A.28) and, hence, Lemma A.32. $\square$

Applying Lemmas A.29 and A.32 and an argument similar to that used in the proof of [44, Lemma 1.3], we obtain the following boundedness of $M_{W, p}^{{\mathcal {D}}} ;$ we omit the details here.

Lemma A.33

Let $p \in (1, \infty )$ and $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then,  for any $r \in (1 - \frac{1}{p' r^*(W)}, p r(W)] ,$ there exists a positive constant C,  depending only on ${\mathcal {X}} ,$ $\delta ,$ n,  p,  r,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that,  for any ${\vec {f}} \in [L^r({\mathcal {X}})]^n ,$

$$\begin{aligned} \left\| M_{W, p}^{{\mathcal {D}}} \left( {\vec {f}}\right) \right\| _{L^r({\mathcal {X}})} \le C K \left\| {\vec {f}} \right\| _{[L^r({\mathcal {X}})]^n}, \end{aligned}$$

where K is the same as in Lemma A.23.

The following lemma establishes the relations between $M_{W, p}^{{\mathcal {D}}}$ and $M_{W, p} .$

Lemma A.34

Let $p \in (1, \infty )$ and $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then there exists a constant $C \in [1, \infty ) ,$ depending only on ${\mathcal {X}}$ and $\delta ,$ such that,  for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

$$\begin{aligned} C^{-1} M_{W, p} \left( {\vec {f}}\right) (x) \le M_{W, p}^{{\mathcal {D}}} \left( {\vec {f}}\right) (x) \le C M_{W, p} \left( {\vec {f}}\right) (x). \end{aligned}$$

Proof

Let p and W be the same as in the present lemma. By Lemma A.24(i), we find that, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n ,$ $x \in {\mathcal {X}} ,$ and $Q \in {\mathcal {D}}$ containing x, 

where B(Q) is the same as in Lemma A.23(a)(iii), which further implies that

On the other hand, by Lemma A.24(ii), we conclude that, for any ball $B \ni x ,$ there exists $Q_B \in {\mathcal {D}}$ such that, for any ${\vec {f}} \in [{\mathscr {M}}({\mathcal {X}})]^n$ and $x \in {\mathcal {X}} ,$

and, hence,

This finishes the proof of Lemma A.34. $\square$

From Lemmas A.33 and A.34, we deduce the following proposition which when ${\mathcal {X}} = {\mathbb {R}}^d$ is just [24, Theorem 3.2] (see also [44, Theorem 1.3]).

Proposition A.35

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ and $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then,  for any $r \in (1 - \frac{1}{p' r^*(W)}, p r(W)] ,$ there exists a positive constant C,  depending only on ${\mathcal {X}} ,$ $\delta ,$ n,  p,  r,  and $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)} ,$ such that,  for any ${\vec {f}} \in [L^r({\mathcal {X}})]^n ,$

$$\begin{aligned} \left\| M_{W, p} \left( {\vec {f}}\right) \right\| _{L^r({\mathcal {X}})} \le C K \left\| {\vec {f}} \right\| _{[L^r({\mathcal {X}})]^n}, \end{aligned}$$

where K is the same as in Lemma A.23.

Applying an argument similar to that used in the proof of [5, Theorem 2], we obtain the following conclusion; we omit the details here.

Lemma A.36

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ $[{\mathscr {M}}({\mathcal {X}})]^n$ be the same as in (1.3),  and $\vec {g} \in [{\mathscr {M}}({\mathcal {X}})]^n .$ Then

$$\begin{aligned} \left\| \vec {g} \right\| _{[L^{p'}({\mathcal {X}})]^n} = \sup _{\Vert {\vec {f}} \Vert _{[L^p({\mathcal {X}})]^n} = 1} \left| \int _{\mathcal {X}} \left( {\vec {f}}(x), \vec {g}(x) \right) _{{\mathbb {C}}^n} \, d\mu (x) \right| , \end{aligned}$$

where,  for any ${\vec {u}} := (u_1, \ldots , u_n), {\vec {v}} := (v_1, \ldots , v_n) \in {\mathbb {C}}^n ,$ $({\vec {u}}, {\vec {v}})_{{\mathbb {C}}^n} := \sum _{i = 1}^n u_i \overline{v_i} .$

From Lemma A.36, we deduce the following result.

Corollary A.37

Let $p \in (1, \infty ) ,$ $1/p + 1/p' = 1 ,$ W be a matrix weight,  $[{\mathscr {M}}({\mathcal {X}})]^n$ the same as in (1.3),  and $\vec {g} \in [{\mathscr {M}}({\mathcal {X}})]^n .$ Then

$$\begin{aligned} \left\| \vec {g} \right\| _{L^{p'}(W^{1 - p'}, {\mathcal {X}})} = \sup _{\Vert {\vec {f}} \Vert _{L^p(W, {\mathcal {X}})} = 1} \left| \int _{\mathcal {X}} \left( {\vec {f}}(x), \vec {g}(x) \right) _{{\mathbb {C}}^n} \, d\mu (x) \right| . \end{aligned}$$

Proof

Let p,  $p' ,$ W,  and $\vec {g}$ be the same as in the present corollary. By Lemma A.36, we find that

$$\begin{aligned} \left\| \vec {g} \right\| _{L^{p'}(W^{1 - p'}, {\mathcal {X}})}&= \left\| W^{-\frac{1}{p}} \vec {g} \right\| _{[L^{p'}({\mathcal {X}})]^n} \\&= \sup _{\Vert {\vec {f}} \Vert _{[L^p({\mathcal {X}})]^n} = 1} \left| \int _{\mathcal {X}} \left( {\vec {f}}(x), W^{-\frac{1}{p}}(x) \vec {g} (x) \right) _{{\mathbb {C}}^n} \, d\mu (x) \right| \\&= \sup _{\Vert W^{\frac{1}{p}} {\vec {f}} \Vert _{[L^p({\mathcal {X}})]^n} = 1} \left| \int _{\mathcal {X}} \left( W^{\frac{1}{p}} (x) {\vec {f}}(x), W^{-\frac{1}{p}} (x) \vec {g} (x) \right) _{{\mathbb {C}}^n} \, d\mu (x) \right| \\&= \sup _{\Vert {\vec {f}} \Vert _{L^p(W, {\mathcal {X}})} = 1} \left| \int _{\mathcal {X}} \left( {\vec {f}}(x), \vec {g}(x) \right) _{{\mathbb {C}}^n} \, d\mu (x) \right| . \end{aligned}$$

This finishes the proof of Corollary A.37. $\square$

The following equivalent characterization of $A_p({\mathcal {X}}, {\mathbb {C}}^n)$ when ${\mathcal {X}} = {\mathbb {R}}^d$ is just [24, Proposition 2.1] (see also [44, Proposition 3.1]).

Proposition A.38

Let $p \in (1, \infty )$ and W be a matrix weight. For any ball $B \subset {\mathcal {X}}$ and any ${\vec {f}} \in L^p(W, {\mathcal {X}}) ,$ define

(A.37)

Then $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n)$ if and only if

$$\begin{aligned} \sup _{{\text {ball}} \, B \subset {\mathcal {X}}} \left\| T_B \right\| _{{\mathcal {L}}(L^p(W, {\mathcal {X}}))} < \infty , \end{aligned}$$

here and thereafter,  for any ball B of ${\mathcal {X}} ,$ $\Vert T_B \Vert _{{\mathcal {L}}(L^p(W, {\mathcal {X}}))}$ denotes the operator norm of $T_B$ from $L^p(W, {\mathcal {X}})$ to itself. Moreover,  if $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) ,$ then

$$\begin{aligned} {[}W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* = \sup _{{\text {ball}} \, B \subset {\mathcal {X}}} \left\| T_B \right\| _{{\mathcal {L}}(L^p(W, {\mathcal {X}}))}. \end{aligned}$$

(A.38)

Proof

Let p and W be the same as in the present proposition. We first show the sufficiency. To this end, assume that $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ Then, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n ,$

$$\begin{aligned} \rho _{p, B} \left( {\vec {z}} \right)< \infty \ \text {and} \ \rho _{p', B}^* \left( {\vec {z}} \right) < \infty , \end{aligned}$$

(A.39)

where $1/p + 1/p' = 1$ and $\rho _{p, B}$ and $\rho _{p', B}^*$ are the same as in Definition 2.15 and, hence, $\rho _{p, B}$ is a norm on ${\mathbb {C}}^n .$ Let $\{{\vec {e}}_i\}_{i = 1}^n$ be an orthonormal basis of ${\mathbb {C}}^n .$ By the Hölder inequality, Lemma A.1, and (A.39), we conclude that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {f}} \in L^p(W, {\mathcal {X}}) ,$

$$\begin{aligned} \int _B \left| {\vec {f}}(x) \right| \, d\mu (x)&\le \int _B \left\| W^{-\frac{1}{p}}(x) \right\| \left| W^{\frac{1}{p}}(x) {\vec {f}}(x) \right| \, d\mu (x) \\&\le \left[ \int _B \left\| W^{-\frac{1}{p}}(x) \right\| ^{p'} \, d\mu (x) \right] ^{\frac{1}{p'}} \left[ \int _B \left| W^{\frac{1}{p}}(x) {\vec {f}}(x) \right| ^p \, d\mu (x) \right] ^{\frac{1}{p}} \\&\lesssim \left\| {\vec {f}} \right\| _{L^p(W, {\mathcal {X}})} \left[ \int _B \sum _{i = 1}^n \left| W^{-\frac{1}{p}}(x) {\vec {e}}_i \right| ^{p'} \, d\mu (x) \right] ^{\frac{1}{p'}} \\&\sim \left\| {\vec {f}} \right\| _{L^p(W, {\mathcal {X}})} \sum _{i = 1}^n \rho _{p', B}^* \left( {\vec {e}}_i \right) < \infty , \end{aligned}$$

where $\rho _{p', B}^*$ is the same as in Definition 2.15. From this, the fact that $\rho _{p, B}$ is a norm on ${\mathbb {C}}^n ,$ Lemma A.5, and Corollary A.37, we deduce that, for any ball $B \subset {\mathcal {X}} ,$

(A.40)

where $(\rho _{p, B})^*$ and $(\rho _{p, B})^{**}$ are the same as, respectively, in Definition 2.13 and Remark 2.14 with $\rho$ replaced by $\rho _{p, B} .$ By this, we conclude that

$$\begin{aligned} \sup _{{\text {ball}} \, B \subset {\mathcal {X}}} \left\| T_B \right\| _{{\mathcal {L}}(L^p(W, {\mathcal {X}}))} = \sup _{{\text {ball}} \, B \subset {\mathcal {X}},\, {\vec {z}} \in {\mathbb {C}}^n {\setminus } \{{\varvec{0}}\}} \frac{\rho _{p', B}^*({\vec {z}})}{(\rho _{p, B})^*({\vec {z}})} = [W]^*_{A_p({\mathcal {X}})}, \end{aligned}$$

which completes the proofs of both (A.38) and the sufficiency.

Next, we prove the necessity. To this end, assume that $\sup _{{\text {ball}} \, B \subset {\mathcal {X}}} \left\| T_B \right\| _{{\mathcal {L}}(L^p(W, {\mathcal {X}}))} < \infty .$ By this and (A.37), we find that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {f}} \in L^p(W, {\mathcal {X}}) ,$

$$\begin{aligned} \int _B \left| {\vec {f}}(x) \right| \, d\mu (x) < \infty \end{aligned}$$

(A.41)

and

$$\begin{aligned} \rho _{p, B} \left( \int _B {\vec {f}} (x) \, d\mu (x) \right)&= [\mu (B)]^{-\frac{1}{p}} \left\| T_B \left( {\vec {f}} \right) \right\| _{L^p(W, {\mathcal {X}})}\nonumber \\&\le [\mu (B)]^{-\frac{1}{p}} \left\| T_B \right\| _{{\mathcal {L}}(L^p(W, {\mathcal {X}}))} \left\| {\vec {f}} \right\| _{L^p(W, {\mathcal {X}})} < \infty . \end{aligned}$$

(A.42)

We claim that, for any ball $B \subset {\mathcal {X}} ,$ $\rho _{p, B}$ is a norm on ${\mathbb {C}}^n .$ Indeed, let $\{{\vec {e}}_i\}_{i = 1}^n$ be an orthonormal basis of ${\mathbb {C}}^n .$ By Lemma A.12, we conclude that, for any $i \in \{1, \ldots , n\} ,$ $w_{{\vec {e}}_i} \in A_p({\mathcal {X}})$ with $w_{{\vec {e}}_i}$ in (A.16). This, together with Lemma A.1, further implies that

$$\begin{aligned} \left\| W^{\frac{1}{p}}(\cdot ) \right\| ^p = \sum _{i = 1}^n w_{{\vec {e}}_i} (\cdot ) \in A_p({\mathcal {X}}) \end{aligned}$$

and, hence, $\Vert W^{1/p}(\cdot )\Vert ^p$ is locally integrable. From this, it follows that, for any ball $B \subset {\mathcal {X}} ,$ there exists $\lambda _B \in (0, 1)$ such that $\int _{\lambda _B B} h(x) \, d\mu (x) < \infty .$ By this, we find that, for any ball $B \subset {\mathcal {X}}$ and any ${\vec {z}} \in {\mathbb {C}}^n ,$

$$\begin{aligned} \left\| {{\textbf {1}}}_{\lambda _B B} {\vec {z}} \right\| _{L^p(W, {\mathcal {X}})}^p&= \int _{\mathcal {X}} \left| W^{\frac{1}{p}}(x) \left( {{\textbf {1}}}_{\lambda _B B} {\vec {z}} \right) \right| ^p \, d\mu (x)\\&\le \int _{\lambda _B B} \left\| W^{\frac{1}{p}}(x) \right\| ^p \, d\mu (x) \left| {\vec {z}} \right| ^p < \infty , \end{aligned}$$

which, combined with the definition of $\rho _{p, B}$ and (A.42) via ${\vec {f}} := {{\textbf {1}}}_{\lambda _B B} {\vec {z}} ,$ further implies that

$$\begin{aligned} \rho _{p, B} \left( {\vec {z}} \right)&= \left[ \mu \left( \lambda _B B \right) \right] ^{-1} \rho _{p, B} \left( \mu \left( \lambda _B B \right) {\vec {z}} \right) \\&= \left[ \mu \left( \lambda _B B \right) \right] ^{-1} \rho _{p, B} \left( \int _B {{\textbf {1}}}_{\lambda _B B} (x) {\vec {z}} \, d\mu (x) \right) < \infty . \end{aligned}$$

Then, by Remark 2.16, we conclude that, for any ball $B \subset {\mathcal {X}} ,$ $\rho _{p, B}$ is a norm on ${\mathbb {C}}^n .$ Using this, (A.41), Lemma A.5, and Corollary A.37 and repeating the estimation of (A.40), we conclude that

$$\begin{aligned} {[}W]^*_{A_p({\mathcal {X}})} = \sup _{{\text {ball}} \, B \subset {\mathcal {X}}} \left\| T_B \right\| _{{\mathcal {L}}(L^p(W, {\mathcal {X}}))} < \infty . \end{aligned}$$

This finishes the proof of Proposition A.38. $\square$

From Proposition A.38, we deduce the following conclusion which when ${\mathcal {X}} = {\mathbb {R}}^d$ is a part of [44, Corollary 3.2].

Corollary A.39

Let $p \in (1, \infty )$ and W be a matrix weight satisfying that there exists positive constant C such that,  for any ${\vec {f}} \in [L^p({\mathcal {X}})]^n ,$

$$\begin{aligned} \left\| M_{W, p} \left( {\vec {f}}\right) \right\| _{L^p({\mathcal {X}})} \le C \left\| {\vec {f}} \right\| _{[L^p({\mathcal {X}})]^n}. \end{aligned}$$

(A.43)

Then $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$

Proof

Let p and W be the same as in the present corollary and $T_B$ the same as in (A.37). Notice that, for any ball $B \subset {\mathcal {X}}$ and any $x \in B ,$ we have

which, together with (A.43), further implies that, for any ${\vec {f}} \in L^p(W, {\mathcal {X}}) ,$

and, hence, $\left\| T_B \right\| _{{\mathcal {L}}(L^p(W, {\mathcal {X}}))} \lesssim 1 .$ From this and Proposition A.38, we deduce that $[W]_{A_p({\mathcal {X}}, {\mathbb {C}}^n)}^* \lesssim 1$ and, hence, $W \in A_p({\mathcal {X}}, {\mathbb {C}}^n) .$ This finishes the proof of Corollary A.39. $\square$

Proof of Theorem A.18

By Proposition A.35 and Corollary A.39, we directly obtain the desired conclusion of the present theorem, which then completes the proof of Theorem A.18. $\square$

Remark A.40

Theorem A.18 when $n = 1$ coincides with [60, p. 5, Theorem 9], while Theorem A.18 when ${\mathcal {X}} = {\mathbb {R}}^d$ is a part of [44, Theorem 1.3 and Corollary 3.2].

1.3 Calderón reproducing formulae

The following wavelet reproducing formula is just [30, Theorem 3.4 and Corollary 3.5].

Lemma A.41

Let $\beta , \gamma \in (0, \eta )$ with $\eta$ in Lemma 2.28. Let $\{ \psi _\alpha ^k \}_{k \in {\mathbb {Z}},\, \alpha \in {\mathcal {G}}_k}$ be the same as in Lemma 2.28. Then,  for any $f \in \mathring{{\mathcal {G}}}_0^{\eta }(\beta ,\gamma )$ [resp. $(\mathring{{\mathcal {G}}}_0^{\eta }(\beta ,\gamma ))'],$

$$\begin{aligned} f = \sum _{k \in {\mathbb {Z}}} \sum _{\alpha \in {\mathcal {G}}_k} \left\langle f, \psi _\alpha ^k \right\rangle \psi _\alpha ^k \end{aligned}$$

in $\mathring{{\mathcal {G}}}_0^{\eta }(\beta ,\gamma )$ [resp. $(\mathring{{\mathcal {G}}}_0^{\eta }(\beta ,\gamma ))'].$

The following homogeneous continuous Calderón reproducing formulae are from [34, Theorem 4.16].

Lemma A.42

Let $\{Q_k\}_{k \in {\mathbb {Z}}}$ be an exp-ATI and $\beta , \gamma \in (0, \eta )$ with $\eta$ in Definition 2.30. Then there exists a sequence $\{{\widetilde{Q}}_k\}_{k=-\infty }^\infty$ of bounded linear operators on $L^2({\mathcal {X}})$ with kernels $\{{\widetilde{Q}}_k(x, y)\}_{k \in {\mathbb {Z}}}$ such that,  for any $f \in (\mathring{{\mathcal {G}}}_0^{\eta }(\beta ,\gamma ))' ,$

$$\begin{aligned} f = \sum _{k \in {\mathbb {Z}}} {\widetilde{Q}}_k Q_k (f) \end{aligned}$$

in $(\mathring{{\mathcal {G}}}_0^{\eta }(\beta ,\gamma ))' .$ Moreover,  there exists a positive constant C such that,  for any $k \in {\mathbb {Z}} ,$

• (i) for any $x, y \in {\mathcal {X}} ,$

  $$\begin{aligned} \left| {\widetilde{Q}}_k(x,y) \right| \le C P_\gamma \left( x, y; \delta ^k \right) , \end{aligned}$$

  where $P_\gamma ( x, y; \delta ^k )$ is the same as in (1.4) with $\varepsilon$ and r replaced,  respectively,  by $\gamma$ and $\delta ^k ;$

• (ii) for any $x, x', y \in {\mathcal {X}}$ with $d(x,x') \le (2A_0)^{-1} [\delta ^k + d(x,y)] ,$

  $$\begin{aligned} \left| {\widetilde{Q}}_k (x, y) - {\widetilde{Q}}_k (x', y) \right| \le C \left[ \frac{d(x, x')}{\delta ^k + d(x,y)} \right] ^\beta P_\gamma \left( x, y; \delta ^k \right) ; \end{aligned}$$

• (iii) for any $x\in {\mathcal {X}},$

  $$\begin{aligned} \int _{\mathcal {X}} {\widetilde{Q}}_k (x, y) \, d\mu (y) = 0 = \int _{\mathcal {X}} {\widetilde{Q}}_k (y, x) \, d\mu (y). \end{aligned}$$

To recall the homogeneous discrete Calderón reproducing formula obtained in [34, Theorem 5.11], let us begin with some concepts. We point out that these homogeneous reproducing formulae need the assumption $\mu ({\mathcal {X}}) = \infty$ which is equivalent to

$$\begin{aligned} {\,\text {diam}\,}{\mathcal {X}} := \sup \left\{ d(x, y) :\ x, y \in {\mathcal {X}} \right\} = \infty \end{aligned}$$

(see, for instance, Nakai and Yabuta [50, Lemma 5.1] or Auscher and Hytönen [3, Lemma 8.1]). Let $j_0 \in {\mathbb {N}}$ be sufficiently large such that $\delta ^{j_0} \le (2 A_0)^{-3} C_0 ,$ where $C_0$ is the same as in Lemma 2.26. Based on Lemma 2.26, for any $k \in {\mathbb {Z}}$ and $\alpha \in {\mathcal {A}}_k ,$ define

$$\begin{aligned} {\mathfrak {N}}(k, \alpha ) := \left\{ \tau \in {\mathcal {A}}_{k + j_0}: \ Q_\tau ^{k + j_0} \subset Q_\alpha ^k \right\} \end{aligned}$$

and $N(k, \alpha ) := \# {\mathfrak {N}}(k,\alpha ) .$ From Lemma 2.26, it follows that $\bigcup _{\tau \in {\mathfrak {N}}(k,\alpha )} Q_\tau ^{k + j_0} = Q_\alpha ^k$ and $N(k, \alpha ) \lesssim \delta ^{-j_0 \omega }$ with the implicit positive constant independent of k and $\alpha .$ We rearrange the set $\{Q_\tau ^{k+j_0}:\ \tau \in {\mathfrak {N}}(k, \alpha )\}$ as $\{Q_{\alpha }^{k, m}\}_{m = 1}^{N(k, \alpha )} .$ Also, for any $k \in {\mathbb {Z}} ,$ $\alpha \in {\mathcal {A}}_k ,$ and $k \in \{1, \ldots , N(k, \alpha )\} ,$ denote by $y_{\alpha }^{k, m}$ an arbitrary point in $Q_{\alpha }^{k, m}$ and by $z_\alpha ^{k, m}$ the “center” of $Q_{\alpha }^{k, m}.$

Lemma A.43

Let $\{Q_k\}_{k \in {\mathbb {Z}}}$ be an exp-ATI and $\beta , \gamma \in (0, \eta )$ with $\eta$ in Definition 2.30. For any $k \in {\mathbb {Z}} ,$ $\alpha \in {\mathcal {A}}_k ,$ and $m \in \{1, \ldots , N(k, \alpha )\} ,$ suppose that $y_{\alpha }^{k, m}$ is an arbitrary point in $Q_{\alpha }^{k, m}.$ Then there exists a sequence $\{{\widetilde{Q}}_k\}_{k \in {\mathbb {Z}}}$ of bounded linear operators on $L^2({\mathcal {X}})$ such that,  for any $f \in (\mathring{{\mathcal {G}}}_0^{\eta }(\beta ,\gamma ))' ,$

$$\begin{aligned} f(\cdot ) = \sum _{k \in {\mathbb {Z}}} \sum _{\alpha \in {\mathcal {A}}_k} \sum _{m=1}^{N(k,\alpha )}\mu \left( Q_{\alpha }^{k, m}\right) {\widetilde{Q}}_k \left( \cdot , y_{\alpha }^{k, m}\right) Q_k (f) \left( y_{\alpha }^{k, m}\right) \end{aligned}$$

in $(\mathring{{\mathcal {G}}}_0^{\eta }(\beta ,\gamma ))' .$ Moreover,  there exists a positive constant C,  independent of f and $\{ y_{\alpha }^{k, m}:\ k \in {\mathbb {Z}},\ \alpha \in$ ${\mathcal {A}}_k,\ m \in \{ 1, \dots , N(k,\alpha ) \} \} ,$ such that $\{{\widetilde{Q}}_k\}_{k \in {\mathbb {Z}}}$ satisfies (i), (ii), and (iii) of Lemma A.42.