Dual Spaces of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Abstract

Let A be a general expansive matrix on $\mathbb {R}^n$ and $H_A^{p(\cdot ),q}(\mathbb R^n)$ be the anisotropic variable Hardy–Lorentz space associated with A, where $p(\cdot ):\ \mathbb R^n\rightarrow (0,\infty ]$ denotes a variable exponent function satisfying the globally log-Hölder continuous condition and $q\in (0,\infty )$. In this article, the authors give the appropriate dual space of $H_A^{p(\cdot ),q}(\mathbb {R}^n)$ with full range $p(\cdot )$ by introducing some anisotropic variable $\eta$-type Campanato spaces. As an application, the Carleson measure characterizations of these $\eta$-type Campanato spaces are established. To achieve these, the authors first deduce several equivalent characterizations of the anisotropic variable $\eta$-type Campanato space, and then introduce the anisotropic variable tent-Lorentz spaces and establish their atomic decomposition. All these results are new even for the isotropic Hardy–Lorentz space $H^{p,q}(\mathbb {R}^n)$ and the isotropic variable Hardy–Lorentz space $H^{p(\cdot ),q}(\mathbb {R}^n)$.

1 Introduction

As is well-known, a very famous duality theorem is that the bounded mean oscillation function space BMO$({\mathbb {R}^n})$ is the dual space of the Hardy space $H^1({\mathbb {R}^n})$, which is due to Fefferman and Stein [9] and can be regarded as a milestone in the development of the real-variable theory of Hardy-type function spaces. This celebrated duality theorem was later extended to the well-known Hardy space $H^p({\mathbb {R}^n})$, with $p\in (0,1]$, by Taibleson and Weiss [40] in 1980. Indeed, they showed that the Campanato space, introduced by Campanato [4], is the dual space of $H^p({\mathbb {R}^n})$. Nowadays, the Campanato space plays a significant role in partial differential equations and harmonic analysis, and has been systematically studied and developed so far. For instance, Nakai [28,29,30] extended the Campanato spaces into the spaces of homogeneous type and studied fractional integral operators and singular integral operators on Campanato spaces or their predual spaces; Rafeiro and Samko [33, 35, 36, 38] considered the boundedness of some important operators, such as fractional operators and Riesz potential operators, from variable exponent Morrey spaces to variable exponent Campanato spaces. Moreover, Bonami et al. [2] established a bilinear decomposition theorem for multiplications of functions in $H^p(\mathbb R^n)$ and the Campanato space $\mathfrak {C}_{1/p-1}$, which is very useful in the estimate of div-curl lemma. Some applications of Campanato spaces with variable growth condition to the Navier–Stokes equation were presented by Nakai and Yoneda in [32]; For more progress on Campanato spaces, we refer the reader to [3, 14, 16, 17, 31, 41, 49].

Let $p(\cdot ):\ \mathbb R^n\rightarrow (0,\infty )$ be a variable exponent function satisfying the so-called globally log-Hölder continuous condition,

In 2012, Nakai and Sawano [31] introduced the variable Hardy space $H^{p(\cdot )}({\mathbb {R}^n})$ and proved that the dual of $H^{p(\cdot )}({\mathbb {R}^n})$ is the variable Campanato space under the assumption $0< p_-\le p_+\le 1$ (see [31, Theorem 7.5]), which generalized the corresponding result of Taibleson and Weiss [40] to the variable setting. Very recently, Huang and Wang [18] extended this dual result to all exponents $0< p_-\le p_+<\infty$, which hence gives a complete answer the open question proposed by Izuki et al. in [19, Section 9.3]. In addition, Yan et al. [48] and Jiao et al. [21, 45], respectively, investigated the variable weak Hardy space ${H^{p(\cdot ),{\infty }}({\mathbb {R}^n})}$ and the variable Hardy–Lorentz space ${H^{p(\cdot ),q}({\mathbb {R}^n})}$. Due to their fine intrinsic structure, the function spaces in the variable exponents setting have proved very useful for applications into fractional calculus, harmonic analysis, partial differential equations and variational integrals with nonstandard growth conditions (see book [6] and also [10, 20, 34, 37, 39, 46, 47]).

On another hand, recall that the variable Lorentz space was introduced by Kempka and Vybíral [22] in 2014, which includes both the variable Lebesgue space and the classical Lorentz space as special cases. Then based on the variable Lorentz space, the anisotropic variable Hardy–Lorentz space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ associated with a general expansive matrix A was introduced and studied by Liu et al. [25, 26], in which they charactered ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ in terms of various maximal functions, (finite) atoms and Littlewood–Paley functions. Also, via the real interpolation, Liu et al. [26] further proved that the anisotropic variable Hardy–Lorentz space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ serves as the intermediate space between the anisotropic variable Hardy space ${H_A^{p(\cdot )}({\mathbb {R}^n})}$ and the space ${L^{\infty }({\mathbb {R}^n})}$. We should point out that the space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ includes the isotropic variable Hardy–Lorentz space ${H^{p(\cdot ),q}({\mathbb {R}^n})}$, the isotropic Hardy–Lorentz space $H^{p,q}(\mathbb {R}^n)$ and the anisotropic Hardy space $H_A^p({\mathbb {R}^n})$ as special cases. Recently, W. Wang and A. Wang given the duality of the anisotropic variable Hardy–Lorentz space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, where $p(\cdot )\in C^{\log }({\mathbb {R}^n})$ with $p_+\in (0,1]$ and $q\in (0,1]$; see [43, Theorem 4.6]. Besides this, Jiao et al. [21] investigated the dual space of the classical variable Hardy–Lorentz space ${H^{p(\cdot ),q}({\mathbb {R}^n})}$ with $p_+\in (0,1]$ and $q\in (0,\infty )$. For more progress on anisotropic Hardy-type spaces, we refer the reader to [11,12,13, 15, 23, 42, 44].

However, when $p_+\in (1,\infty )$ and $q\in (0,\infty )$, the dual space of ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, even for the isotropic case ${H^{p(\cdot ),q}({\mathbb {R}^n})}$, is still unclear so far. Observe also that the dual space of ${H^{p(\cdot ),q}({\mathbb {R}^n})}$ obtained in Jiao et al. [21] takes two completely different forms to define for the two cases: $q\in (0,1]$ and $q\in (1,\infty )$. Thus, it is natural and interesting to ask whether or not we can give a unified dual space of ${H^{p(\cdot ),q}({\mathbb {R}^n})}$ for the full range $p_+\in (0, \infty )$and $q\in (0,\infty )$?

Inspired by this question and the recent articles [17, 50], via introducing some new anisotropic variable $\eta$-type Campanato spaces, in this article, we give a complete dual characterization of anisotropic variable Hardy-Lorentz spaces $H^{p(\cdot ),q}_A({\mathbb {R}^n})$ for full range $p_+\in (0, \infty )$ and $q\in (0,\infty )$, which includes the variable Hardy-Lorentz space $H^{p(\cdot ),q}({\mathbb {R}^n})$ studied in [21]. Therefore, the obtained result in this article unifies the known duality theorems obtained in [21, 43] and also affirms the above question. Furthermore, as applications, we deduce several equivalent characterizations of these $\eta$-type Campanato spaces, and also introduce some anisotropic variable tent-Lorentz spaces. Then, via establishing atomic decomposition of the tent-Lorentz spaces and using the obtained duality theorem of $H^{p(\cdot ),q}_A({\mathbb {R}^n})$, we deduce the Carleson measure characterizations of these anisotropic variable $\eta$-type Campanato spaces. Note that, when $A:=d\text {I}_{n\times n}$ and $p(\cdot )\equiv p\in (0,\infty )$, where $d\in {\mathbb R}$ with $|d|\in (1,\infty )$ and $\text {I}_{n\times n}$ denotes the $n\times n$ unit matrix, the space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ comes back to the isotropic Hardy–Lorentz space $H^{p,q}({\mathbb {R}^n})$ of Abu-Shammala and Torchinsky [1]. Even in this case, all these results obtained in the present article are also new.

To be precise, this article is organized as follows. In Sect. 2, we first recall some notions of expansive matrices, variable Lebesgue spaces ${L^{p(\cdot )}({\mathbb {R}^n})}$ and variable Lorentz spaces ${L^{p(\cdot ),q}({\mathbb {R}^n})}$ as well as anisotropic variable Hardy–Lorentz spaces ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$. Then we introduce the anisotropic variable $\eta$-type Campanato space and show some basic properties. Finally, we state the main results.

Section 3 is aimed to prove Theorem 1. To this end, we recall the atomic and finite atomic characterizations of the anisotropic variable Hardy–Lorentz space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ established in [25, 26]. Combining these, the special structure of the anisotropic variable $\eta$-type Campanato space and some basic tools from functional analysis, we show Theorem 1. We point out that, as a special case, Theorem 1 with $p_+,\,q\in (0,1]$ is just the dual result obtained in [43, Theorem 4.6]; see Remark 5(i) below. At the meantime, Theorem 1 also gives a unified dual space of ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ [particularly, of ${H^{p(\cdot ),q}({\mathbb {R}^n})}$] for the full range $q\in (0,\infty )$; see Remark 5(ii) below.

In Sect. 4, we prove Theorem 2. For this purpose, we first show several equivalent characterizations of the anisotropic variable $\eta$-type Campanato space (see Proposition 1 and Corollary 3 below). Then we introduce a kind of anisotropic variable tent-Lorentz spaces (see Definition 10 below) and give their atomic decompositions (see Lemma 6 below), which plays a key role in the proof of Theorem 2. Finally, applying this atomic decomposition, the obtained equivalent characterizations, the Lusin area function characterization of ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ from [26, Theorem 5.2] and Theorem 1, we further establish the Carleson measure characterization of the anisotropic variable $\eta$-type Campanato space $\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})$ (see Theorem 2 below).

Finally, we make some conventions on notation. Let ${\mathbb N}:=\{1,2,\ldots \}$, ${\mathbb Z}_+:=\{0\}\cup {\mathbb N}$, and ${\mathbb Z}_+^n:=({\mathbb Z}_+)^n$. Denote by $\textbf{0}$ the origin of ${\mathbb {R}^n}$. For each multi-index $\beta :=(\beta _1,\ldots ,\beta _n)\in (\mathbb {Z}_+)^n=:\mathbb {Z}_+^n,$ let $|\beta |:=\beta _1+\cdots +\beta _n$ and $\partial ^{\beta }:=(\frac{\partial }{\partial x_1})^{\beta _1} \cdots (\frac{\partial }{\partial x_n})^{\beta _n}.$ We always use C to denote a positive constant which is independent of the main parameters, but may vary from line to line. The notation $f\lesssim g$ means $f\le Cg$ and, if $f\lesssim g\lesssim f$, then we write $f\sim g$. We also use the following convention: If $f\le Cg$ and $g=h$ or $g\le h$, we then write $f\lesssim g\sim h$ or $f\lesssim g\lesssim h$, rather than $f\lesssim g=h$ or $f\lesssim g\le h$. For any $q\in [1,\infty ]$, we denote by $q'$ its conjugate exponent, namely, $1/q+1/q'=1$. The symbol $\lfloor t\rfloor$ for any $t\in \mathbb {R}$ denotes the largest integer not greater than t. For any $\Omega \subset {\mathbb {R}^n}$, we denote the set ${\mathbb {R}^n}\setminus \Omega$ by $\Omega ^\complement$, its characteristic function by $\textbf{1}_\Omega$, and its n-dimensional Lebesgue measure by $|\Omega |$.

2 Preliminaries and main results

In this section, we first recall the definitions of expansive matrices and variable Lorentz spaces ${L^{p(\cdot ),q}({\mathbb {R}^n})}$. Then the anisotropic variable $\eta$-type Campanato spaces $\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})$ are introduced. Finally, we state the main theorems, namely, the duality theorem and the Carleson measure characterizations of the anisotropic variable $\eta$-type Campanato space.

We begin with recalling the notion of expansive matrices from [3, p. 5, Definition 2.1].

Definition 1

A real $n\times n$ matrix A is called an $expansive \ matrix$ (or dilation) if $\min _{\lambda \in \sigma (A)}|\lambda |>1,$ where $\sigma (A)$ denotes the ${collection \ of \ all \ eigenvalues \ of}$ A.

In this article, we always let A be a fixed dilation and $b:=|\det A|\in (1,\infty )$. For a given dilation A, we find that there exist an open ellipsoid $\Delta$, with $|\Delta |=1$, and $r\in (1,\infty )$ satisfying that $\Delta \subset r\Delta \subset A\Delta$ (see [3, p. 5, Lemma 2.2]), where $|\Delta |$ denotes the n-dimensional Lebesgue measure of the set $\Delta$. For any $k\in {\mathbb Z}$, let $B_k:=A^k\Delta$. Then $B_k\subset rB_k\subset B_{k+1}$ and $|B_k|=b^k$. For any $x\in {\mathbb {R}^n}$ and $k\in \mathbb {Z}$, $x+B_k$ is called a dilated ball. Here and thereafter, we always denote by $\mathfrak {B}$ the set of all such dilated balls, namely,

$$\begin{aligned} \mathfrak {B}:=\left\{ x+B_k:\ x\in {\mathbb {R}^n}\ \textrm{and}\ k\in \mathbb {Z}\right\} \end{aligned}$$

(2.1)

and

$$\begin{aligned} \omega :=\inf \left\{ \ell \in {\mathbb Z}:\ r^\ell \ge 2\right\} . \end{aligned}$$

(2.2)

The following notion of the homogeneous quasi-norm induced by A is just [3, p. 6, Definition 2.3].

Definition 2

A homogeneous quasi-norm, associated with a dilation A, is a measurable mapping $\rho :\ {\mathbb {R}^n}\rightarrow [0,\infty )$ satisfying that

• (i) if $x\ne \textbf{0}$, then $\rho (x)\in (0,\infty )$;

• (ii) $\rho (Ax)=b\rho (x)$ for any $x\in {\mathbb {R}^n}$;

• (iii) there exists a $C\in [1,\infty )$ such that, for any $x,\,y\in {\mathbb {R}^n}$, $\rho (x+y)\le C[\rho (x)+\rho (y)].$

Observe that [3, p. 6, Lemma 2.4] implies that all homogeneous quasi-norms associated with a given A are equivalent. Thus, for any given dilation A, we always use the step homogeneous quasi-norm $\rho$ defined by setting, for any $x\in {\mathbb {R}^n}$,

$$\begin{aligned} \rho (x):=\sum _{k\in \mathbb {Z}} b^k\textbf{1}_{B_{k+1}\setminus B_k}(x)\, \textrm{when} x\ne \textbf{0},\, \mathrm{or\ else} \,\rho (\textbf{0}):=0 \end{aligned}$$

for both simplicity and convenience.

Recall also that a measurable function $p(\cdot ):\ {\mathbb {R}^n}\rightarrow (0,\infty ]$ is called a variable exponent if it satisfies

(2.3)

Moreover, let

$$\begin{aligned} \underline{p}:=\min \{p_-,1\} \end{aligned}$$

(2.4)

with $p_-$ as in (2.3). The collection of all variable exponents $p(\cdot )$ is denoted by ${\mathcal P}({\mathbb {R}^n})$. For any $p(\cdot )\in {\mathcal P}({\mathbb {R}^n})$ and a measurable function f, the ${modular \ functional}$ $\varrho _{p(\cdot )}(f)$ is given by setting

$$\begin{aligned} \varrho _{p(\cdot )}(f):=\int _{\mathbb {R}^n}|f(x)|^{p(x)}\,dx \end{aligned}$$

and the Luxemburg quasi-norm $\Vert f\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}$ of f is defined by

$$\begin{aligned} \Vert f\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}:=\inf \left\{ \lambda \in (0,\infty ):\ \varrho _{p(\cdot )}(f/\lambda )\le 1\right\} . \end{aligned}$$

Furthermore, the ${variable \ Lebesgue \ space}$ ${L^{p(\cdot )}({\mathbb {R}^n})}$ is defined to be the set of all measurable functions f such that $\varrho _{p(\cdot )}(f)<\infty$, equipped with the quasi-norm $\Vert f\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}$.

We denote by $C^{\log }({\mathbb {R}^n})$ the set of all functions $p(\cdot )\in {\mathcal P}({\mathbb {R}^n})$ satisfying the so-called globally log-Hölder continuous condition, namely, there exist two positive constants $C_{\log }(p)$, $C_\infty$, and $p_\infty \in {\mathbb R}$ such that, for any $x,\ y\in {\mathbb {R}^n}$,

$$\begin{aligned} |p(x)-p(y)|\le \frac{C_{\log }(p)}{\log (e+1/\rho (x-y))} \end{aligned}$$

and

$$\begin{aligned} |p(x)-p_\infty |\le \frac{C_\infty }{\log (e+\rho (x))}. \end{aligned}$$

The following variable Lorentz space ${L^{p(\cdot ),q}({\mathbb {R}^n})}$ is known as a special case of the variable Lorentz space $L^{p(\cdot ),q(\cdot )}({\mathbb {R}^n})$ investigated by Kempka and Vybíral in [22].

Definition 3

Let $p(\cdot )\in {\mathcal P}({\mathbb {R}^n})$ and $q\in (0,\infty ]$. The ${variable \ Lorentz \ space}$ ${L^{p(\cdot ),q}({\mathbb {R}^n})}$ is defined to be the set of all measurable functions f such that

$$\begin{aligned} \Vert f\Vert _{{L^{p(\cdot ),q}({\mathbb {R}^n})}}:=\left\{ \begin{array}{ll} \left[ \displaystyle \int _0^\infty \lambda ^q\left\| \textbf{1}_{\{x\in {\mathbb {R}^n}:\ |f(x)|>\lambda \}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\,\frac{d\lambda }{\lambda }\right] ^{{1}/{q}} &{}\textrm{when} q\in (0,\infty ),\\ \displaystyle \sup _{\lambda \in (0,\infty )}\left[ \lambda \left\| \textbf{1}_{\{x\in {\mathbb {R}^n}:\ |f(x)|>\lambda \}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}\right] &{}\textrm{when} q=\infty \end{array}\right. \end{aligned}$$

is finite.

Remark 1

Let $p(\cdot )\in {\mathcal P}({\mathbb {R}^n})$ and $q\in (0,\infty ]$. Then, for any measurable function f,

$$\begin{aligned} \Vert f\Vert _{{L^{p(\cdot ),q}({\mathbb {R}^n})}}\sim \left[ \sum _{k\in {\mathbb Z}}2^{kq} \left\| \textbf{1}_{\{x\in {\mathbb {R}^n}:\ |f(x)|>2^k\}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right] ^{1/q} \end{aligned}$$

with the usual interpretation for $q=\infty$, where the positive equivalence constants are independent of f.

For any $s\in {\mathbb Z}_+$, the set of all polynomials on ${\mathbb {R}^n}$ with degree not greater than s is denoted by $\mathbb {P}_s({\mathbb {R}^n})$; for any locally integrable function g on ${\mathbb {R}^n}$ and any ball $B\in \mathfrak {B}$ with $\mathfrak {B}$ as in (2.1), the ${minimizing \ polynomial}$ of g with degree not greater than s is denoted by $P^s_Bg$, which means that $P^s_Bg$ is the unique polynomial $Q\in \mathbb {P}_s({\mathbb {R}^n})$ such that, for any $R\in \mathbb {P}_s({\mathbb {R}^n})$, $\int _{B}[g(x)-Q(x)]R(x)\,dx=0.$

We now introduce the following anisotropic variable $\eta$-type Campanato space. For any given $r\in [1,{\infty })$, the set of all r-order locally integrable functions on ${\mathbb {R}^n}$ is denoted by $L_{{\mathop {\mathrm {\,loc\,}}}}^r({\mathbb {R}^n})$.

Definition 4

Let $p(\cdot )\in \mathcal {P}({\mathbb {R}^n})$, $r\in [1,{\infty })$, $\eta \in (0,\infty )$, $q\in (0,\infty )$, and $s\in {\mathbb Z}_+$. Then the anisotropic variable $\eta$-type Campanato space $\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})$ is defined to be the set of all $f\in L^r_\textrm{loc}({\mathbb {R}^n})$ such that

$$\begin{aligned} \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})}&:=\,\sup \left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{B_i^k}}{\Vert \textbf{1}_{B_i^k}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\eta }\right\} ^{1/\eta }\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\\&\quad \times \left\{ \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|B_l^j|}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \left[ \frac{1}{|B_l^j|}\int _{B_l^j}\left| f(x)-P^s_{B_l^j}f(x)\right| ^r \,dx\right] ^{1/r}\right\} \end{aligned}$$

is finite, where the supremum is taken over all $m\in {\mathbb N}$, $v\in {\mathbb N}$, $\{B_l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B}$, and $\{\lambda _l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset [0,\infty )$ with $\sum _{j=1}^v\sum _{l=1}^m\lambda _l^j\ne 0$.

Remark 2

• (i) Since $\mathbb {P}_s({\mathbb {R}^n})\subset \mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})$, we always identify $f\in \mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})$ with $\{f+P:\ P\in \mathbb {P}_s({\mathbb {R}^n})\}$ in what follows.

• (ii) For any $f\in L^r_\textrm{loc}({\mathbb {R}^n})$, let

  $$\begin{aligned} \Vert |f\Vert |_{\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})}&:=\,\sup \inf \left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{B_i^k}}{\Vert \textbf{1}_{B_i^k}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\eta }\right\} ^{1/\eta }\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\\&\quad \times \left\{ \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|B_l^j|}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \left[ \frac{1}{|B_l^j|}\int _{B_l^j}\left| f(x)-P(x)\right| ^r \,dx\right] ^{1/r}\right\} , \end{aligned}$$

  where the infimum is taken over all $P\in \mathbb {P}_s({\mathbb {R}^n})$ and the supremum is the same as in Definition 4. Then it is easy to check that $\Vert |\cdot \Vert |_{\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})}$ is an equivalent quasi-norm of the $\eta$-type Campanato space ${\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})}$ and the details are omitted.

The following definition of the anisotropic variable Campanato spaces $\mathcal {L}_{p(\cdot ),r,s}^A({\mathbb {R}^n})$ is just [41, Definition 4.1].

Definition 5

Let $p(\cdot )\in \mathcal {P}({\mathbb {R}^n})$, $r\in [1,\infty )$, and $s\in {\mathbb Z}_+$. The anisotropic variable Campanato space $\mathcal {L}_{p(\cdot ),r,s}^A({\mathbb {R}^n})$ is defined to be the set of all $f\in L^r_\textrm{loc}({{{\mathbb R}}^n})$ such that

$$\begin{aligned} \Vert f\Vert _{\mathcal {L}_{p(\cdot ),r,s}^A({\mathbb {R}^n})}:=\sup _{B\in \mathfrak {B}}\inf _{P\in \mathbb {P}_s({\mathbb {R}^n})} \frac{|B|}{\Vert \textbf{1}_B\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\left[ \frac{1}{|B|} \int _B\left| f(x)-P(x)\right| ^r\,dx\right] ^{{1/r}}<\infty . \end{aligned}$$

Remark 3

• (i) From [43, Lemma 4.4], we infer that, for any $f\in L^r_\textrm{loc}({{{\mathbb R}}^n})$,

  $$\begin{aligned} \Vert f\Vert _{\mathcal {L}_{p(\cdot ),r,s}^A({\mathbb {R}^n})}\sim \Vert |f\Vert |_{\mathcal {L}_{p(\cdot ),r,s}^A({\mathbb {R}^n})}:=\sup _{B\in \mathfrak {B}}\frac{|B|}{\Vert \textbf{1}_{B}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\left[ \frac{1}{|B|}\int _{B}\left| f(x)-P_B^sf(x)\right| ^r \,dx\right] ^{1/r}. \end{aligned}$$

• (ii) By Remark 2(ii) and Definition 5, it is easy to verify that

  $$\begin{aligned} \mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})\subset \mathcal {L}_{p(\cdot ),r,s}^A({\mathbb {R}^n}) \end{aligned}$$

  with continuous inclusion for all indices as in Definition 4.

Throughout this article, the symbol $C^{\infty }({\mathbb {R}^n})$ denotes the set of all infinitely differentiable functions on ${\mathbb {R}^n}$. Recall that a $C^\infty ({\mathbb {R}^n})$ function $\varphi$ is called a Schwartz function if, for any multi-index $\gamma \in {\mathbb Z}_+^n$ and $t\in {\mathbb Z}_+$, $\Vert \varphi \Vert _{\gamma ,t}:= \sup _{x\in {\mathbb {R}^n}}[\rho (x)]^t |\partial ^\gamma \varphi (x)|<\infty .$ Denote by ${\mathcal S}({\mathbb {R}^n})$ the set of all Schwartz functions, equipped with the topology determined by the family $\{\Vert \cdot \Vert _{\gamma ,t}\}_{\gamma \in {\mathbb Z}_+^n,t\in {\mathbb Z}_+}$, and ${\mathcal S}'({\mathbb {R}^n})$ the dual space of ${\mathcal S}({\mathbb {R}^n})$, namely, the space of all tempered distributions on ${\mathbb {R}^n}$, equipped with the weak-$*$ topology. For any $N\in \mathbb {Z}_+$, let ${\mathcal S}_N({\mathbb {R}^n}):=\left\{ \varphi \in {\mathcal S}({\mathbb {R}^n}):\ \Vert \varphi \Vert _{\gamma ,t}\le 1,\ |\gamma |\le N,\ t\le N\right\} ;$ equivalently,

$$\begin{aligned} \varphi \in {\mathcal S}_N({\mathbb {R}^n})\Longleftrightarrow \Vert \varphi \Vert _{{\mathcal S}_N({\mathbb {R}^n})}:=\sup _{|\gamma |\le N} \sup _{x\in {\mathbb {R}^n}}\left[ \left| \partial ^\gamma \varphi (x)\right| \max \left\{ 1,\left[ \rho (x)\right] ^N\right\} \right] \le 1. \end{aligned}$$

Let $\lambda _-$, $\lambda _+\in (1,\infty )$ be two numbers such that

$$\begin{aligned} \lambda _-\le \min \{|\lambda |:\ \lambda \in \sigma (A)\} \le \max \{|\lambda |:\ \lambda \in \sigma (A)\}\le \lambda _+. \end{aligned}$$

(2.5)

We should point out that, if A is diagonalizable over $\mathbb {C}$, then we may let $\lambda _-:=\min \{|\lambda |:\ \lambda \in \sigma (A)\}$ and $\lambda _+:=\max \{|\lambda |:\ \lambda \in \sigma (A)\}$. Otherwise, we may choose them sufficiently close to these equalities in accordance with what we need in our arguments.

Recall that the anisotropic variable Hardy–Lorentz space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ is defined as follows; see [26, Definition 2.10].

Definition 6

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$, $q\in (0,\infty ]$ and $N\in \mathbb {N}\cap [\lfloor (\frac{1}{\underline{p}}-1)\frac{\ln b}{\ln \lambda _-}\rfloor +2,\infty )$. The anisotropic variable Hardy–Lorentz space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ is defined to be the set of all $f\in {\mathcal S}'({\mathbb {R}^n})$ such that

$$\begin{aligned} \Vert f\Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}:=\left\| M_N^0(f)\right\| _{{L^{p(\cdot ),q}({\mathbb {R}^n})}}<\infty , \end{aligned}$$

where $M_N^0(f)$ denotes the radial grand maximal function of f defined by setting, for any $x\in {\mathbb {R}^n}$,

$$\begin{aligned} M_N^0(f)(x):= \sup _{\varphi \in {\mathcal S}_N}\sup _{k\in {\mathbb Z}}\left| f*\varphi _k(x)\right| , \end{aligned}$$

here and thereafter, for any $\varphi \in {\mathcal S}({\mathbb {R}^n})$ and $k\in \mathbb {Z}$, let $\varphi _k(\cdot ):=b^k\varphi (A^k\cdot )$.

The next notions of anisotropic variable $(p(\cdot ),r,s)$-atoms and anisotropic variable finite atomic Hardy–Lorentz spaces are just [25, Definitions 2.12 and 2.13].

Definition 7

Let $p(\cdot )\in {\mathcal P}({\mathbb {R}^n})$, $r\in (1,\infty ]$ and

$$\begin{aligned} s\in \left[ \left\lfloor \left( \dfrac{1}{p_-}-1\right) \dfrac{\ln b}{\ln \lambda _-}\right\rfloor ,\infty \right) \cap {\mathbb Z}_+, \end{aligned}$$

(2.6)

where $\lambda _-$ and $p_-$ are as in (2.5) and (2.3), respectively. An anisotropic $(p(\cdot ),r,s)$-atom is a measurable function a on ${\mathbb {R}^n}$ satisfying

• (i) $\mathop {\mathrm {\,supp\,}}a \subset B$, where $B\in \mathfrak {B}$ and $\mathfrak {B}$ is as in (2.1);

• (ii) $\Vert a\Vert _{L^r({\mathbb {R}^n})}\le \frac{|B|^{1/r}}{\Vert \textbf{1}_B\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}$;

• (iii) for any $\alpha \in {\mathbb Z}_+^n$ with $|\alpha |\le s$, $\int _{\mathbb R^n}a(x)x^\alpha \,dx=0$.

Definition 8

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$, $q\in (0,\infty ]$, $r\in (1,\infty ]$, s be as in (2.6) and A a dilation. The anisotropic variable finite atomic Hardy–Lorentz space ${H_{A,\textrm{fin}}^{p(\cdot ),r,s,q}({\mathbb {R}^n})}$ is defined to be the set of all $f\in {\mathcal S}'({\mathbb {R}^n})$ satisfying that there exist K, $I\in {\mathbb N}$, a finite sequence of $(p(\cdot ),r,s)$-atoms, $\{a_i^k\}_{i\in [1,I]\cap {\mathbb N},\,k\in [1,K]\cap {\mathbb N}}$, supported, respectively, in $\{B_i^k\}_{i\in [1,I]\cap {\mathbb N},\,k\in [1,K]\cap {\mathbb N}}\subset \mathfrak {B}$ and a positive constant $\widetilde{C}$, independent of I and K, such that $\sum _{i=1}^{I}\textbf{1}_{A^{j_0}B_i^k}(x)\le \widetilde{C}$ for any $x\in {\mathbb {R}^n}$ and $k\in [1,K]\cap {\mathbb N}$, with some $j_0\in {\mathbb Z}\setminus {\mathbb N}$, and $f=\sum _{k=1}^{K}\sum _{i=1}^I\lambda _i^ka_i^k$ in ${\mathcal S}'({\mathbb {R}^n}),$ where, for any $k\in [1,K]\cap {\mathbb N}$ and $i\in [1,I]\cap {\mathbb N}$, $\lambda _i^k\sim 2^k\Vert \textbf{1}_{B_i^k}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}$ with the positive equivalence constant independent of k, K, i and I. Moreover, for any $f\in {H_{A,\textrm{fin}}^{p(\cdot ),r,s,q}({\mathbb {R}^n})}$, define

$$\begin{aligned} \Vert f\Vert _{{H_{A,\textrm{fin}}^{p(\cdot ),r,s,q}({\mathbb {R}^n})}}:= \inf \left[ \sum _{k=1}^K\left\| \left\{ \sum _{i=1}^I \left[ \frac{\lambda _i^k\textbf{1}_{B_i^k}}{\Vert \textbf{1}_{B_i^k}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right] ^{1/q} \end{aligned}$$

with the usual modification made when $q=\infty$, where $\underline{p}$ is as in (2.4) and the infimum is taken over all decompositions of f as above.

We now state the first main result as follows, which establishes the complete duality theorem between the anisotropic variable $\eta$-type Campanato space and the anisotropic variable Hardy–Lorentz space for all $p(\cdot )\in C^{\log }({\mathbb {R}^n})$ and $q\in (0,\infty )$. This extends the known dual result of [21, 43].

Theorem 1

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$, $q\in (0,\infty )$, $r\in (\max \{1,p_+\},{\infty }]$ with $p_+$ as in (2.3), and s is as in (2.6). Then the dual space of ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, denoted by $({H_A^{p(\cdot ),q}({\mathbb {R}^n})})^*$, is $\mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})$ in the following sense:

• (i) Let $g\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})$. Then the linear functional

  $$\begin{aligned} G_g:\ f\rightarrow G_g(f):=\int _{{\mathbb {R}^n}}f(x)g(x)\,dx, \end{aligned}$$

  (2.7)

  initially defined for any $f\in H_{A,\textrm{fin}}^{p(\cdot ),r,s,q}({\mathbb {R}^n})$, has a bounded extension to ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$.

• (ii) Conversely, any continuous linear functional on ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ arises as in (2.7) with a unique $g\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})$.

Moreover, $\Vert g\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})}\sim \Vert G_g\Vert _{({H_A^{p(\cdot ),q}({\mathbb {R}^n})})^*}$, where the positive equivalence constants are independent of g.

As an application, we also establish the Carleson measure characterizations of $\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})$. We begin with introducing the $[p(\cdot ),q]$-Carleson measure as follows.

Definition 9

Let $p(\cdot )\in \mathcal {P}({\mathbb {R}^n})$, $q\in (0,\infty )$. A Borel measure $d\mu$ on ${\mathbb {R}^n}\times {\mathbb Z}$ is called a $[p(\cdot ),q]$-Carleson measure if

$$\begin{aligned} \left\| d\mu \right\| _{\mathcal {C}_{p(\cdot ),q,A}}:=\sup&\left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{B_i^k}}{\Vert \textbf{1}_{B_i^k}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\eta }\right\} ^{1/\eta }\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\\&\quad \times \left\{ \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|B_l^j|^{1/2}}{\Vert \textbf{1}_{B_l^j} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \left[ \int _{\widehat{B_l^j}}\,|d\mu (x,k)|\right] ^{1/2}\right\} \end{aligned}$$

is finite, where $\eta \in (0,\infty )$ and the supremum is taken over all $m\in {\mathbb N}$, $v\in {\mathbb N}$, $\{B_l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B}$ and $\{\lambda _l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset [0,\infty )$ with $\sum _{j=1}^v\sum _{l=1}^m\lambda _l^j\ne 0$, and, for any $l\in [1,m]\cap {\mathbb N}$, $j\in [1,v]\cap {\mathbb N}$, the tent over ${B_l^j}$, denoted by $\widehat{B_l^j}$, is defined by setting $\widehat{B_l^j}:=\{(y,k)\in {\mathbb {R}^n}\times {\mathbb Z}:\ y+B_k\subset B_l^j\}.$

Remark 4

Let $p(\cdot )$, $\eta$, q and $d\mu$ be as in Definition 9 and

where the supremum is taken over all $\{B_l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset \mathfrak {B}$ and $\{\lambda _l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset [0,\infty )$ satisfying

Then .

To state the Carleson measure characterizations, we need some notions. For any given $k\in {\mathbb Z}$, let $\delta _k(j)=1$ when $j=k$, otherwise, $\delta _k(j)=0$. The set of all infinitely differentiable functions with compact support on ${\mathbb {R}^n}$ is denoted by $C_\textrm{c}^{\infty }({\mathbb {R}^n})$ and, for any $\varphi \in {\mathcal S}({\mathbb {R}^n})$, its Fourier transform is denoted by $\widehat{\varphi }$, namely, for any $\xi \in {\mathbb {R}^n}$, ${\widehat{\varphi }}(\xi ):= \int _{{\mathbb {R}^n}} \varphi (x) e^{-2\pi \imath x \cdot \xi } \, dx,$ where $\imath :=\sqrt{-1}$ and $x\cdot \xi :=\sum _{i=1}^{n}x_i \xi _i$ for any $x:=(x_1,\ldots ,x_n)$, $\xi :=(\xi _1,\ldots ,\xi _n) \in {\mathbb {R}^n}$. Let $s\in {\mathbb Z}_+$ and $\phi \in C_\textrm{c}^{\infty }({\mathbb {R}^n})$ satisfy

$$\begin{aligned} \mathop {\mathrm {\,supp\,}}\phi \subset B_0,~~~ \int _{{\mathbb {R}^n}}x^\alpha \phi (x)\,dx=0,~~~\forall ~\alpha \in {\mathbb Z}_+^n~~ \textrm{with}~~|\alpha |\le s, \end{aligned}$$

(2.8)

and there exists a positive constant C such that, for any $\xi \in \{x\in {\mathbb {R}^n}:\ (2\Vert A\Vert )^{-1}\le \rho (x)\le 1\}$,

$$\begin{aligned} \left| \widehat{\phi }(\xi )\right| \ge C \end{aligned}$$

(2.9)

where, for any dilation $A:=(a_{i,j})_{1\le i,j\le n}$, $\Vert A\Vert :=(\sum _{i,j=1}^n|a_{i,j}|^2)^{1/2}$.

Then we state the second main result as follows.

Theorem 2

Let $p(\cdot )$, q, s, and $\underline{p}$ be as in Theorem 1, $p_+\in (0,2)$, and $\phi \in {\mathcal S}({\mathbb {R}^n})$ be a radial real-valued function satisfying (2.8) and (2.9).

• (i) If $h\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})$, then, for any $(x,k)\in {\mathbb {R}^n}\times {\mathbb Z}$, $d\mu (x,k):=\sum _{\ell \in {\mathbb Z}} |\phi _{-\ell }*h(x)|^2\,dx\,\delta _{\ell }(k)$ is a $[p(\cdot ),q]$-Carleson measure on ${\mathbb {R}^n}\times {\mathbb Z}$; moreover, there exists a positive constant C, independent of h, such that

  $$\begin{aligned} \Vert d\mu \Vert _{\mathcal {C}_{p(\cdot ),q,A}} \le C\Vert h\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})}. \end{aligned}$$

• (ii) If $h\in L^2_\textrm{loc}({\mathbb {R}^n})$ and, for any $(x,k)\in {\mathbb {R}^n}\times {\mathbb Z}$, $d\mu (x,k):=\sum _{\ell \in {\mathbb Z}} |\phi _{-\ell }*h(x)|^2\,dx\,\delta _{\ell }(k)$ is a $[p(\cdot ),q]$-Carleson measure on ${\mathbb {R}^n}\times {\mathbb Z}$, then $h\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})$ and, there exists a positive constant C, independent of h, such that

  $$\begin{aligned} \Vert h\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})} \le C\Vert d\mu \Vert _{\mathcal {C}_{p(\cdot ),q,A}}. \end{aligned}$$

3 Proof of Theorem 1

In this section, we give the proof of Theorem 1. To begin with, we present the atomic and finite atomic characterizations of ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, which are, respectively, established in [26, Theorem 4.8] and [25, Theorem 2.14].

Lemma 1

Let $p(\cdot )$, q, r, and s be as in Theorem 1, $\{a_l^j\}_{l\in {\mathbb N},\,j\in {\mathbb Z}}$ a sequence of $(p(\cdot ),r,s)$-atoms supported, respectively, in $\{B_l^j\}_{l\in {\mathbb N},\,j\in {\mathbb Z}}\subset \mathfrak {B}$ such that

$$\begin{aligned} \left( \sum _{j\in {\mathbb Z}}\left\| \left\{ \sum _{l\in {\mathbb N}} \left[ \frac{\lambda _l^j\textbf{1}_{B_l^j}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/{\underline{p}}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q}<\infty , \end{aligned}$$

where $\lambda _l^j\sim 2^j\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}$ for any $j\in \mathbb {Z}$ and $l\in \mathbb {N}$, $\sum _{l\in \mathbb {N}}\textbf{1}_{A^{j_0}B_l^j}(x)\lesssim 1$ for any $x\in {\mathbb {R}^n}$ and $j\in \mathbb {Z}$ with some $j_0\in {\mathbb Z}\setminus {\mathbb N}$. Then the series $f=\sum _{j\in {\mathbb Z}}\sum _{l\in {\mathbb N}}\lambda _l^ja_l^j$ converges in ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, $f\in {H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, and there exists a positive constant C, independent of f, such that

$$\begin{aligned} \left\| f\right\| _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}\le C\left( \sum _{j\in {\mathbb Z}}\left\| \left\{ \sum _{l\in {\mathbb N}} \left[ \frac{\lambda _l^j\textbf{1}_{B_l^j}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/{\underline{p}}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q}. \end{aligned}$$

Lemma 2

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$, $q\in (0,\infty ]$, $r\in (\max \{p_+,1\},\infty ]$ and s be as in (2.6).

• (i) If $r\in (\max \{p_+,1\},\infty )$, then $\Vert \cdot \Vert _{{H_{A,\textrm{fin}}^{p(\cdot ),r,s,q}({\mathbb {R}^n})}}$ and $\Vert \cdot \Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}$ are equivalent quasi-norms on ${H_{A,\textrm{fin}}^{p(\cdot ),r,s,q}({\mathbb {R}^n})}$;

• (ii) $\Vert \cdot \Vert _{{H_{A,\textrm{fin}}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})}}$ and $\Vert \cdot \Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}$ are equivalent quasi-norms on ${H_{A,\textrm{fin}}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})}\cap C({\mathbb {R}^n})$, here and thereafter, $C({\mathbb {R}^n})$ denotes the set of all continuous functions on ${\mathbb {R}^n}$.

The next lemma is also needed to prove Theorem 1, whose proof is similar to that of [50, Proposition 3.13]; the details are omitted.

Lemma 3

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$, $q\in (0,\infty )$ and s be as in (2.6). Then ${H_{A,\textrm{fin}}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})}\cap C({\mathbb {R}^n})$ is dense in ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$.

We now prove Theorem 1.

Proof

Let all the notation be as in Theorem 1. We first prove (i) by considering two cases: $r\in (\max \{1,p_+\},\infty )$ and $r=\infty$.

To deal with the case $r\in (\max \{1,p_+\},\infty )$, let $g\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})$. Then, for any $f\in H_{A,\mathrm fin}^{p(\cdot ),r,s,q}({\mathbb {R}^n})$, from Definition 8, we infer that there exist v, $m\in {\mathbb N}$ and a finite sequence of $(p(\cdot ),r,s)$-atoms, $\{a_l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}$, supported, respectively, in $\{B_l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B}$ such that

$$\begin{aligned} f=\sum _{j=1}^v\sum _{l=1}^m\lambda _l^ja_l^j \end{aligned}$$

in ${\mathcal S}'({\mathbb {R}^n}),$ and

$$\begin{aligned} \left( \sum _{j=1}^v\left\| \left\{ \sum _{l=1}^m \left[ \frac{\lambda _l^j\textbf{1}_{B_l^j}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q} \sim \Vert f\Vert _{H_{A,\mathrm fin}^{p(\cdot ),r,s,q}({\mathbb {R}^n})}, \end{aligned}$$

where, for any $j\in [1,v]\cap {\mathbb N}$ and $l\in [1,m]\cap {\mathbb N}$, $\lambda _l^j\sim 2^j\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}$ with the positive equivalence constant independent of j, v, l and m. From this, Definition 7, the Hölder inequality, Remark 2(ii), and Lemma 2(i), we deduce that

$$\begin{aligned} \left| G_g(f)\right|&\le \sum _{j=1}^{v}\sum _{l=1}^{m}\lambda _l^j\left| \int _{{\mathbb {R}^n}}a_l^j(x)g(x)\,dx\right| \\&=\sum _{j=1}^{v}\sum _{l=1}^{m}\lambda _l^j\inf _{P\in \mathbb {P}_s({\mathbb {R}^n})}\left| \int _{B_l^j}a_l^j(x)\left[ g(x)-P(x)\right] \,dx\right| \\&\le \sum _{j=1}^{v}\sum _{l=1}^{m}\frac{\lambda _l^j|B_l^j|}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \inf _{P\in \mathbb {P}_s({\mathbb {R}^n})}\left[ \frac{1}{|B_l^j|} \int _{B_l^j}|g(x)-P(x)|^{r'}\,dx\right] ^{{1}/{r'}}\\&\lesssim \Vert g\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})} \left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{B_i^k}}{\Vert \textbf{1}_{B_i^k}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q}\\&\sim \Vert g\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})} \Vert f\Vert _{H_{A,\mathrm fin}^{p(\cdot ),r,s,q}({\mathbb {R}^n})} \sim \Vert g\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})} \Vert f\Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}. \end{aligned}$$

This, combined with the fact that $H_{A,\mathrm fin}^{p(\cdot ),r,s,q}({\mathbb {R}^n})$ is dense in ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, further implies that (i) holds true in this case.

In this case $r=\infty$, by Lemma 3 and repeating the above proof with some slight modifications, we conclude that each $g\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({\mathbb {R}^n})$ induces a bounded linear functional on ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, which is initially defined on $H_{A,\mathrm fin}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})\cap C{({\mathbb {R}^n})}$ by setting, for any $f\in H_{A,\mathrm fin}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})\cap C{({\mathbb {R}^n})}$,

$$\begin{aligned} G_g:\ f\mapsto \ G_g(f):=\int _{{\mathbb {R}^n}}f(x)g(x)\,dx, \end{aligned}$$

(3.1)

and also has a bounded extension to ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$. Thus, to prove (i) in this case, we only need to show that, for any $f\in H_{A,\mathrm fin}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})$,

$$\begin{aligned} G_g(f)=\int _{{\mathbb {R}^n}}f(x)g(x)\,dx. \end{aligned}$$

(3.2)

To this end, suppose that $f\in H_{A,\mathrm fin}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})$ and $\mathop {\mathrm {\,supp\,}}f\subset B(\textbf{0},D)$ with some $D\in (0,\infty )$. Let $\varphi \in \mathcal {S}({\mathbb {R}^n})$ satisfy $\mathop {\mathrm {\,supp\,}}\varphi \subset B(\textbf{0},1)$ and $\int _{{\mathbb {R}^n}}\varphi (x)\,dx=1$. Therefore, for any $t\in (0,1)$, $\varphi _t*f\in H_{A,\mathrm fin}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})\cap C{({\mathbb {R}^n})}$, where, for any $t\in (0,\infty )$, $\varphi _{t}(\cdot ):=t^{-n}\varphi (t^{-1}\cdot )$. Let $d\in (\max \{1,p_+\},\infty )$. Then $f\in L^{d}({\mathbb {R}^n})$ and hence, by [7, Theorem 2.1], we find that $\lim _{t\in (0,1),t\rightarrow 0} \left\| f-\varphi _t*f\right\| _{L^{d}({\mathbb {R}^n})}=0.$ This, together with the Riesz lemma, implies that there exists a sequence $\{t_k\}_{k\in {\mathbb N}}\subset (0,1)$ such that $\lim _{k\rightarrow \infty }t_k=0$ and $\lim _{k\rightarrow \infty }\varphi _{t_k}*f(x)=f(x)$ holds true for almost every $x\in {\mathbb {R}^n}$. Thus,

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert f-\varphi _{t_k}*f\Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}=0. \end{aligned}$$

(3.3)

Indeed, it suffices to show that, for any given $(p(\cdot ),\infty ,s)$-atom, (3.3) is valid. To do this, let a be a $(p(\cdot ),\infty ,s)$-atom supported in the dilated ball $x+B_i$ with some $x\in {\mathbb {R}^n}$ and $i\in {\mathbb Z}$. Then, from [7, Theorem 2.1], it follows that

$$\begin{aligned} \lim _{t\in (0,1),t\rightarrow 0}\left\| a-\varphi _{t}*a\right\| _{L^{d}({\mathbb {R}^n})}=0. \end{aligned}$$

(3.4)

Observe that, for any $t\in (0,1)$, $\frac{|x+B_{i+2\omega }|^{1/d}(a-\varphi _{t}*a)}{\Vert \textbf{1}_{x+B_{i+2\omega }}\Vert _{L^{p(\cdot )}({\mathbb {R}^n})}\Vert a-\varphi _t*a\Vert _{L^{d}({\mathbb {R}^n})}}$ is a $(p(\cdot ),d,s)$-atom supported in the dilated ball $x+B_{i+2\omega }$, where $\omega$ is as in (2.2). From this, Lemma 1 and (3.4), we further deduce that

$$\begin{aligned} \left\| a-\varphi _t*a\right\| _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}} \lesssim \frac{\Vert \textbf{1}_{x+B_{i+2\omega }}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}\Vert a-\varphi _t*a\Vert _{L^{d}({\mathbb {R}^n})}}{|x+B_{i+2\omega }|^{1/d}} \lesssim \Vert a-\varphi _t*a\Vert _{L^{d}({\mathbb {R}^n})}\rightarrow 0 \end{aligned}$$

as $t\rightarrow 0$. This proves (3.3).

Moreover, by (3.3), (3.1), the fact that $|\left( \varphi _{t_k}*f\right) g|\le \Vert f\Vert _{L^\infty ({\mathbb {R}^n})}\textbf{1}_{B(\textbf{0},D+1)}|g|\in L^1({\mathbb {R}^n})$ with $D\in (0,\infty )$ and the Lebesgue dominated convergence theorem, we conclude that

$$\begin{aligned} G_g(f)=\lim _{k\rightarrow \infty }G_g(\varphi _{t_k}*f) =\lim _{k\rightarrow \infty }\int _{{\mathbb {R}^n}}\varphi _{t_k}*f(x)g(x)\,dx=\int _{{\mathbb {R}^n}}f(x)g(x)\,dx. \end{aligned}$$

This finishes the proof of (3.2) and hence of (i).

We now show (ii). For any $G\in ({H_A^{p(\cdot ),q}({\mathbb {R}^n})})^*$, by an argument similar to that used in the proof of [21, Theorem 7.2(ii)], we know that there exists a unique $g\in \mathcal {L}_{p(\cdot ),r',s}^A({\mathbb {R}^n})$ such that, for any $f\in H_{A,\mathrm fin}^{p(\cdot ),r,s,q}({\mathbb {R}^n})$, $G(f) =\int _{{\mathbb {R}^n}}f(x)g(x)\,dx.$ To prove (ii), it remains to show that $g\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})$. For this purpose, for any $m\in {\mathbb N}$, $v\in {\mathbb N}$, $\{B_l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B}$, and $\{\lambda _l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset [0,\infty )$ satisfying $\lambda _l^j\sim 2^j\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}$ with the positive equivalence constant independent of j, v, l and m, let $\eta _l^j\in L^r(B_l^j)$ with $\Vert \eta _l^j\Vert _{L^r(B_l^j)}=1$ satisfy that

$$\begin{aligned} \left[ \int _{B_l^j}\left| g(x)-P^s_{B_l^j}g(x)\right| ^{r'} \,dx\right] ^{1/{r'}} =\int _{B_l^j}\left[ g(x)-P^s_{B_l^j}g(x)\right] \eta _l^j(x)\,dx. \end{aligned}$$

(3.5)

Moreover, for any $x\in {\mathbb {R}^n}$, set

$$\begin{aligned} a_l^j(x):=\frac{|B_l^j|^{1/r} [\eta _l^j(x)-P^s_{B_l^j}\eta _l^j(x)]\textbf{1}_{B_l^j}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}} \Vert \eta _l^j-P^s_{B_l^j}\eta _l^j\Vert _{L^r(B_l^j)}}. \end{aligned}$$

Then, for any $l\in [1,m]\cap {\mathbb N}$, $j\in [1,v]\cap {\mathbb N}$, it is easy to check that $a_l^j$ is a $(p(\cdot ),r,s)$-atom. By this and Lemma 1, we find that $\sum _{j=1}^v\sum _{l=1}^m\lambda _l^j a_l^j\in {H_A^{p(\cdot ),q}({\mathbb {R}^n})}$. Combining this, (3.5) and the facts that $G \in ({H_A^{p(\cdot ),q}({\mathbb {R}^n})})^*$ and $\Vert \eta _l^j-P^s_{B_l^j}\eta _l^j\Vert _{L^r(B_l^j)}\lesssim 1,$ we obtain

$$\begin{aligned}&\sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|B_l^j|}{\Vert \textbf{1}_{B_l^j} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\left[ \frac{1}{|B_l^j|}\int _{B_l^j}\left| g(x)-P^s_{B_l^j}g(x)\right| ^{r'} \,dx\right] ^{1/r'}\\&\quad =\sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|B_l^j|^{1/r}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\int _{B_l^j}\left[ g(x)-P^s_{B_l^j}g(x)\right] \eta _l^j(x)\,dx\\&\quad =\sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|B_l^j|^{1/r}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \int _{B_l^j}\left[ \eta _l^j(x)-P^s_{B_l^j}\eta _l^j(x)\right] g(x){\textbf{1}}_{B_l^j}(x)\,dx\\&\quad \lesssim \sum _{j=1}^v\sum _{l=1}^m{\lambda }_l^j \int _{B_l^j}a_l^j(x)g(x)\,dx \sim \sum _{j=1}^v\sum _{l=1}^m{\lambda }_l^j G(a_l^j) \sim G\left( \sum _{j=1}^v\sum _{l=1}^m{\lambda }_l^j a_l^j\right) \\&\quad \lesssim \left\| \sum _{j=1}^v\sum _{l=1}^m{\lambda }_l^j a_l^j\right\| _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}} \lesssim \left( \sum _{j=1}^v\left\| \left\{ \sum _{l=1}^m \left[ \frac{\lambda _l^j\textbf{1}_{B_l^j}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q}. \end{aligned}$$

By this and Definition 4, we further conclude that $g\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},r'}^A({\mathbb {R}^n})$, which completes the proof of (ii) and hence of Theorem 1.

The next lemma shows that the new introduced space ${\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})}$ includes the Campanato-type space $\mathcal {L}_{p(\cdot ),r,s}^A({\mathbb {R}^n})$ in some cases, whose proof is omitted.

Lemma 4

Let $p(\cdot )\in \mathcal {P}({\mathbb {R}^n})$, $p_+\in (0,1]$ with $p_+$ as in (2.3), $r\in [1,\infty )$, $\eta \in (0,1]$, $q\in (0,1]$ and $s\in {\mathbb Z}_+$. Then $\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})=\mathcal {L}_{p(\cdot ),r,s}^A({\mathbb {R}^n})$ with equivalent quasi-norms.

Remark 5

• (i) Note that W. Wang and A. Wang [43, Theorem 4.6] studied the dual of the anisotropic variable Hardy–Lorentz space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ as $p(\cdot )\in C^{\log }({\mathbb {R}^n})$ with $p_+\in (0,1]$ and $q\in (0,1]$. However, using Lemma 4, it is easy to see that [43, Theorem 4.6] is a special case of Theorem 1. Indeed, in the case when $p_+\in (1,\infty )$ or $q\in (1,\infty )$, the dual space of ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ can not be obtained from [43, Theorem 4.6] anymore. But, Theorem 1 establishes the duality theorem of ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ for the full range $p_+,\,q\in (0,\infty )$. This is the main contribution of Theorem 1.

• (ii) Let $\text {I}_{n\times n}$ denote the $n\times n$ unit matrix. If $A:=d\text {I}_{n\times n}$ for some $d\in {\mathbb R}$ with $|d|\in (1,\infty )$, then the space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ becomes the classical variable Hardy–Lorentz space $H^{p(\cdot ),q}$ from [21]. In this case, Theorem 1 shows the duality theorem of $H^{p(\cdot ),q}$ with $p_+,\,q\in (0,\infty )$. Recall that Jiao et al. [21] investigated the dual space of ${H^{p(\cdot ),q}({\mathbb {R}^n})}$ with $p_+\in (0,1]$ and $q\in (0,\infty )$. Thus, Theorem 1 completes the duality theorem of ${H^{p(\cdot ),q}({\mathbb {R}^n})}$ for the case when $p_+\in (1,\infty )$ and $q\in (0,\infty )$. Observe also that, for the case $q\in (0,1]$ and $q\in (1,\infty )$, the definitions of dual space of ${H^{p(\cdot ),q}({\mathbb {R}^n})}$ obtained in [21] takes two completely different forms. However, Theorem 1 gives a unified dual space of ${H^{p(\cdot ),q}({\mathbb {R}^n})}$ for the full range $q\in (0,\infty )$. This is another contribution of Theorem 1.

• (iii) When $A:=d\text {I}_{n\times n}$ for some $d\in {\mathbb R}$ with $|d|\in (1,\infty )$, and $p(\cdot )\equiv p\in (0,\infty )$, the space ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ comes back to the isotropic Hardy–Lorentz space $H^{p,q}({\mathbb {R}^n})$ of Abu-Shammala and Torchinsky [1]. Even in this case, Theorem 1 is also new.

Using Theorem 1, we easily obtain the following equivalence of the $\eta$-type Campanato space; the details are omitted.

Corollary 1

Let $p(\cdot )$, q, s and $\underline{p}$ be as in Theorem 1, $r\in [1,\infty )$ when $p_+\in (0,1)$, or $r\in [1,p_+')$ when $p_+\in [1,\infty )$. Then

$$\begin{aligned} \mathcal {L}_{p(\cdot ),q,s,\underline{p},r}^A({\mathbb {R}^n}) =\mathcal {L}_{p(\cdot ),q,s_0,\underline{p},1}^A({\mathbb {R}^n}) \end{aligned}$$

with equivalent quasi-norms, where $s_0:=\lfloor (\frac{1}{p_-}-1)\frac{\ln b}{\ln \lambda _-}\rfloor$ with $p_-$ as in (2.3), and $\underline{p}$ as in (2.4).

From [26, Corollary 6.3] and Corollary 1, we deduce the next conclusion.

Corollary 2

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$ with $p_-\in (1,\infty )$, and q, $s_0$ and $\underline{p}$ be as in Corollary 1. Then

$$\begin{aligned} (L^{{p(\cdot )},q}({\mathbb {R}^n}))^*=\mathcal {L}_{p(\cdot ),q,s_0,\underline{p},1}^A({\mathbb {R}^n}) \end{aligned}$$

with equivalent quasi-norms, where $(L^{{p(\cdot )},q}({\mathbb {R}^n}))^*$ denotes the dual space of $L^{{p(\cdot )},q}({\mathbb {R}^n})$.

4 Proof of Theorem 2

This section is devoted to giving the proof of Theorem 2. To this end, we first deduce several equivalent characterizations of the anisotropic variable $\eta$-type Campanato spaces. Then we introduce the anisotropic variable tent-Lorentz spaces and establish their atomic decomposition. Combining these and Theorem 1, we finally characterize these anisotropic variable $\eta$-type Campanato spaces in terms of Carleson measure.

First, we show the following equivalent characterizations of $\mathcal {L}_{p(\cdot ),q,s,\underline{p},r}^A({\mathbb {R}^n})$.

Proposition 1

Assume that $p(\cdot ),\ q,\ s$, $\underline{p}$ and r are the same as in Corollary 1. Let $\varepsilon \in ([2/c-1]\ln b/\ln \lambda _-,\infty )$ for some $c\in (0,\underline{p})$. Then the following statements are mutually equivalent:

• (i) $f\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},r}^A({\mathbb {R}^n})$;

• (ii) $f\in L^r_\textrm{loc}({\mathbb {R}^n})$ and

  $$\begin{aligned} \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({\mathbb {R}^n})}&:=\,\sup \left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\\&\quad \times \left\{ \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \int _{x_l^j+B_{\epsilon _l^j}}\left| f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)\right| \,dx\right\} \end{aligned}$$

  is finite, where the supremum is taken over all $m\in {\mathbb N}$, $v\in {\mathbb N}$,

  $$\begin{aligned} \left\{ x_l^j+B_{\epsilon _l^j}\right\} _{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B} \end{aligned}$$

  with $\{x_l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset {\mathbb {R}^n}$ and $\{{\epsilon _l^j}\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset {\mathbb Z}$, and

  $$\begin{aligned} \left\{ \lambda _l^j\right\} _{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset [0,\infty )\ with\ \sum _{j=1}^v\sum _{l=1}^m\lambda _l^j\ne 0; \end{aligned}$$

• (iii) $f\in L^r_\textrm{loc}({\mathbb {R}^n})$ and

  $$\begin{aligned} \Vert |f\Vert |_{\mathcal {L}_{p(\cdot ),q,s,\underline{p},r}^A({\mathbb {R}^n})}&:=\,\sup \inf \left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\\&\quad \times \sum _{j=1}^v \sum _{l=1}^m\frac{{\lambda }_l^j|x_l^j+B_{\epsilon _l^j}|}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \left[ \frac{1}{|x_l^j+B_{\epsilon _l^j}|} \int _{x_l^j+B_{\epsilon _l^j}}\left| f(x)-P(x)\right| ^r \,dx\right] ^{1/r} \end{aligned}$$

  is finite, where the infimum is taken over all $P\in \mathbb {P}_s({\mathbb {R}^n})$ and the supremum is the same as in (ii);

• (iv) $f\in L^r_\textrm{loc}({\mathbb {R}^n})$ and

  $$\begin{aligned} \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^{A,\varepsilon }({\mathbb {R}^n})}&:=\,\sup \left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\\&\quad \times \sum _{j=1}^v \sum _{l=1}^m\frac{{\lambda }_l^j|x_l^j+B_{\epsilon _l^j}|}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \int _{{\mathbb {R}^n}}\frac{b^{\varepsilon \epsilon _l^j\frac{\ln \lambda _-}{\ln b}}|f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)|}{b^{\epsilon _l^j(1+\varepsilon \frac{\ln \lambda _-}{\ln b})}+[\rho (x-x_l^j)]^{1+\varepsilon \frac{\ln \lambda _-}{\ln b}}}\,dx \end{aligned}$$

  is finite, where the supremum is the same as in (ii).

Moreover, for any $f\in L^r_\textrm{loc}({\mathbb {R}^n})$,

$$\begin{aligned} \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},r}^A({\mathbb {R}^n})}\sim \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({\mathbb {R}^n})}\sim \Vert |f\Vert |_{\mathcal {L}_{p(\cdot ),q,s,\underline{p},r}^A({\mathbb {R}^n})}\sim \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^{A,\varepsilon }({\mathbb {R}^n})} \end{aligned}$$

with the positive equivalence constants independent of f.

To show Proposition 1, we need the following inequality, which is just [26, Remark 4.4(i)].

Lemma 5

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$ and c be as in Proposition 1. Then there exists a positive constant C such that, for any $\{x_i+B_i\}_{i\in {\mathbb N}}\subset \mathfrak {B}$ with $\{x_i\}_{i\in {\mathbb N}}\subset {\mathbb {R}^n}$, and $k\in {\mathbb N}$,

$$\begin{aligned} \left\| \sum _{i\in {\mathbb N}}\textbf{1}_{x_i+B_{i+k}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}} \le Cb^{k/c}\left\| \sum _{i\in {\mathbb N}}\textbf{1}_{x_i+B_i}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}. \end{aligned}$$

Now we prove Proposition 1.

Proof

The equivalence of (i) and (ii) can be easily verified by Corollary 1. In addition, (i) obviously implies (iii) and conversely, by a proof similar to that of [27, Proposition 4.1], we find that (iii) implies (i). Thus, to show Proposition 1, we only need to prove that (ii) is equivalent to (iv). Indeed, (iv) obviously implies (ii). Next, we prove that (ii) implies (iv).

To do this, assume that (ii) holds true. Observe that, for any $m\in {\mathbb N}$, $v\in {\mathbb N}$, $\{x_l^j+B_{\epsilon _l^j}\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B}$ with $\{x_l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset {\mathbb {R}^n}$ and $\{{\epsilon _l^j}\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset {\mathbb Z}$, and $\{\lambda _l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset [0,\infty )$ with $\sum _{j=1}^v\sum _{l=1}^m\lambda _l^j\ne 0$, we have

$$\begin{aligned}&\sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|x_l^j+B_{\epsilon _l^j}|}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\int _{{\mathbb {R}^n}}\frac{b^{\varepsilon \epsilon _l^j\frac{\ln \lambda _-}{\ln b}}|f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)|}{b^{\epsilon _l^j(1+\varepsilon \frac{\ln \lambda _-}{\ln b})}+[\rho (x-x_l^j)]^{1+\varepsilon \frac{\ln \lambda _-}{\ln b}}}\,dx\\&=\sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|x_l^j+B_{\epsilon _l^j}|}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\\&\times \left[ \int _{x_l^j+B_{\epsilon _l^j}}+\sum _{u=0}^{\infty } \int _{b^{\epsilon _l^j+u}\le \rho (x-x_l^j)<b^{\epsilon _l^j+u+1}}\right] \frac{b^{\varepsilon \epsilon _l^j\frac{\ln \lambda _-}{\ln b}}|f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)|}{b^{\epsilon _l^j(1+\varepsilon \frac{\ln \lambda _-}{\ln b})}+[\rho (x-x_l^j)]^{1+\varepsilon \frac{\ln \lambda _-}{\ln b}}}\,dx\\&\lesssim \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\int _{x_l^j+B_{\epsilon _l^j}}\left| f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)\right| \,dx+\sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\\&\times \sum _{u=0}^{\infty }b^{-u({1+\varepsilon \frac{\ln \lambda _-}{\ln b}})}\int _{b^{\epsilon _l^j+u}\le \rho (x-x_l^j)<b^{\epsilon _l^j+u+1}}\left| f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)\right| \,dx. \end{aligned}$$

Thus,

(4.1)

where

$$\begin{aligned} \Xi :=&\left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\left\{ \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right. \\&\times \left. \sum _{u=0}^{\infty }b^{-u({1+\varepsilon \frac{\ln \lambda _-}{\ln b}})} \int _{b^{\epsilon _l^j+u}\le \rho (x-x_l^j)<b^{\epsilon _l^j+u+1}} \left| f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)\right| \,dx\right\} \\ \lesssim&\left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\\&\times \left\{ \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\sum _{u\in {\mathbb N}}^{\infty }b^{-u({1+\varepsilon \frac{\ln \lambda _-}{\ln b}})}\int _{x_l^j+B_{\epsilon _l^j+u}}\left| f(x)-P^s_{x_l^j+B_{\epsilon _l^j+u}}f(x)\right| \,dx\right\} \\&+\left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q} \left\{ \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right. \\&\times \left. \sum _{u\in {\mathbb N}}b^{-u({1+\varepsilon \frac{\ln \lambda _-}{\ln b}})}\int _{x_l^j+B_{\epsilon _l^j+u}}\left| P^s_{x_l^j+B_{\epsilon _l^j+u}}f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)\right| \,dx\right\} . \end{aligned}$$

From this, the Tonelli theorem, $c\in (0,\underline{p})$, and Lemma 5, it follows that

$$\begin{aligned} \Xi&\lesssim \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({\mathbb {R}^n})} \sum _{u\in {\mathbb N}}b^{-u({1-\frac{2}{c}+\varepsilon \frac{\ln \lambda _-}{\ln b}})} +\left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\nonumber \\&\times \left\{ \sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}} \Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right. \nonumber \\&\qquad \quad \times \left. \sum _{u\in {\mathbb N}}b^{-u({1+\varepsilon \frac{\ln \lambda _-}{\ln b}})}\int _{x_l^j+B_{\epsilon _l^j+u}}\left| P^s_{x_l^j+B_{\epsilon _l^j+u}}f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)\right| \,dx\right\} . \end{aligned}$$

(4.2)

On another hand, by [3, (8.9)], we conclude that, for any $x\in x_l^j+B_{\epsilon _l^j+u}$,

$$\begin{aligned}&\left| P^s_{x_l^j+B_{\epsilon _l^j+u}}f(x)-P^s_{x_l^j+B_{\epsilon _l^j}}f(x)\right| \\&\quad \lesssim \sum _{\tau =1}^u\frac{1}{|x_l^j+B_{\epsilon _l^j+\tau -1}|} \int _{x_l^j+B_{\epsilon _l^j+\tau }}\left| f(y)-P^s_{x_l^j+B_{\epsilon _l^j+\tau }}f(y)\right| \,dy. \end{aligned}$$

Combining this, (4.2), the Tonelli theorem, Lemma 5 again, and the assumption $\varepsilon \in ([2/c-1]\ln b/\ln \lambda _-,\infty )$, we obtain

$$\begin{aligned} \Xi&\lesssim \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({\mathbb {R}^n})} \sum _{u\in {\mathbb N}}b^{-u({1-\frac{2}{c}+\varepsilon \frac{\ln \lambda _-}{\ln b}})} +\Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({\mathbb {R}^n})} \sum _{u\in {\mathbb N}}b^{-u\varepsilon \frac{\ln \lambda _-}{\ln b}}\sum _{\tau =1}^u b^{\tau (2/c-1)}\\&\lesssim \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({\mathbb {R}^n})} \sum _{u\in {\mathbb N}}b^{-u({1-\frac{2}{c}+\varepsilon \frac{\ln \lambda _-}{\ln b}})} \lesssim \Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({\mathbb {R}^n})}. \end{aligned}$$

From this and (4.1), we further deduce that (iv) holds true and hence complete the proof of Proposition 1. $\square$

The next proposition presents a equivalent quasi-norm characterization of the anisotropic variable $\eta$-type Campanato space $\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})$, whose proof is similar to that of [18, Proposition 1] and hence we omit the details here.

Proposition 2

Let $p(\cdot )\in \mathcal {P}({\mathbb {R}^n})$, $r\in [1,{\infty })$, $\eta \in (0,\infty )$, $q\in (0,{\infty })$, and $s\in {\mathbb Z}_+$. For any $f\in L^r_\textrm{loc}({\mathbb {R}^n})$, let

$$\begin{aligned} \widetilde{\Vert f\Vert }_{\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})}&:=\,\sup \left( \sum _{k\in {\mathbb N}}\left\| \left\{ \sum _{i\in {\mathbb N}} \left[ \frac{\lambda _i^k\textbf{1}_{B_i^k}}{\Vert \textbf{1}_{B_i^k}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\eta }\right\} ^{1/\eta }\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q}\\&\quad \times \left\{ \sum _{j\in {\mathbb N}}\sum _{l\in {\mathbb N}}\frac{{\lambda }_l^j|B_l^j|}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \left[ \frac{1}{|B_l^j|}\int _{B_l^j}\left| f(x)-P^s_{B_l^j}f(x)\right| ^r \,dx\right] ^{1/r}\right\} , \end{aligned}$$

where the supremum is taken over all $\{B_l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset \mathfrak {B}$ and $\{\lambda _l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset [0,\infty )$ satisfying

$$\begin{aligned} \sum _{k\in {\mathbb N}}\left\| \left\{ \sum _{i\in {\mathbb N}} \left[ \frac{\lambda _i^k\textbf{1}_{B_i^k}}{\Vert \textbf{1}_{B_i^k}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\eta }\right\} ^{1/\eta }\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\in (0,\infty ). \end{aligned}$$

Then, for any $f\in L^r_\textrm{loc}({\mathbb {R}^n})$, $\widetilde{\Vert f\Vert }_{\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})} =\Vert f\Vert _{\mathcal {L}_{p(\cdot ),q,s,\eta ,r}^A({\mathbb {R}^n})}.$

Applying Propositions 1 and 2, it is easy to check the following equivalent characterizations of $\mathcal {L}_{p(\cdot ),q,s,\underline{p},r}^A({\mathbb {R}^n})$; we omit the details.

Corollary 3

Let $p(\cdot ),\ q,\ s$, $\underline{p}$, r and $\varepsilon$ be as in Proposition 1. Then all the conclusions of Proposition 1 still hold true with m and v replaced by $\infty$, and the supremum therein taken over all $\{x_l^j+B_{\epsilon _l^j}\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset \mathfrak {B}$ with $\{x_l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset {\mathbb {R}^n}$ and $\{{\epsilon _l^j}\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset {\mathbb Z}$, and $\{\lambda _l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset [0,\infty )$ satisfying

$$\begin{aligned} \sum _{j\in {\mathbb N}}\left\| \left\{ \sum _{l\in {\mathbb N}} \left[ \frac{\lambda _l^j\textbf{1}_{x_l^j+B_{\epsilon _l^j}}}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/{\underline{p}}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\in (0,\infty ). \end{aligned}$$

To show the Carleson measure characterizations, we need the anisotropic variable tent-Lorentz space and its atomic decomposition. Recall that for any measurable function H on ${\mathbb {R}^n}\times {\mathbb Z}$, the anisotropic discrete Lusin area function ${\mathcal A}(H)$ is defined by setting, for any $x\in {\mathbb {R}^n}$,

$$\begin{aligned} {\mathcal A}(H)(x):=\left[ \sum _{\ell \in {\mathbb Z}}b^{-\ell } \int _{\{y\in {\mathbb {R}^n}:\ (y,\ell )\in \Gamma (x)\}}|H(y,\ell )|^2\, dy\right] ^{1/2}, \end{aligned}$$

where $\Gamma (x):=\{(y,k)\in {\mathbb {R}^n}\times {\mathbb Z}:\ y\in x+B_k\}$ denotes the cone of aperture 1 with vertex $x\in {\mathbb {R}^n}$.

Via this anisotropic discrete Lusin area function, we next introduce the anisotropic variable tent-Lorentz space.

Definition 10

Let $p(\cdot )\in \mathcal {P}({\mathbb {R}^n})$ and $q\in (0,\infty )$. The anisotropic variable tent-Lorentz space $T^{p(\cdot ),q}_A({\mathbb {R}^n}\times {\mathbb Z})$ is defined to be the set of all measurable functions H on ${\mathbb {R}^n}\times {\mathbb Z}$ satisfying ${\mathcal A}(H)\in {L^{p(\cdot ),q}({\mathbb {R}^n})}$. Moreover, for any $H\in T^{p(\cdot ),q}_A({\mathbb {R}^n}\times {\mathbb Z})$, define $\Vert H\Vert _{T^{p(\cdot ),q}_A({\mathbb {R}^n}\times {\mathbb Z})}:=\Vert {\mathcal A}(H)\Vert _{{L^{p(\cdot ),q}({\mathbb {R}^n})}}.$

We next give the notion of anisotropic $(p(\cdot ),\infty )$-atoms.

Definition 11

Let $p(\cdot )\in \mathcal {P}({\mathbb {R}^n})$ and $r\in (1,\infty )$. A measurable function a on ${\mathbb {R}^n}\times {\mathbb Z}$ is said to be an anisotropic $(p(\cdot ),r)$-atom if there exists a dilated ball $B\in {\mathfrak B}$ such that

• (i) $\mathop {\mathrm {\,supp\,}}a:=\{(x,k)\in {\mathbb {R}^n}\times {\mathbb Z}:\ a(x,k)\ne 0\}\subset \widehat{B}$;

• (ii) $\Vert a\Vert _{T_2^{r}({\mathbb {R}^n}\times {\mathbb Z})}:=\Vert {\mathcal A}(a)\Vert _{L^r({\mathbb {R}^n})}\le \frac{|B|^{1/r}}{\Vert \textbf{1}_B\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}$.

Furthermore, if a is an anisotropic $(p(\cdot ),r)$-atom for any $r\in (1,\infty )$, then a is said to be an anisotropic $(p(\cdot ),\infty )$-atom.

The following atomic decomposition of anisotropic variable tent-Lorentz spaces $T^{p(\cdot ),q}_A({\mathbb {R}^n}\times {\mathbb Z})$ plays a key role in the proof of Theorem 2 and also is of independent interest.

Lemma 6

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$ and $q\in (0,\infty )$. Then, for any $H\in T^{p(\cdot ),q}_A({\mathbb {R}^n}\times {\mathbb Z})$, there exist $\{B_l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset \mathfrak {B}$, $\{\lambda _l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset [0,\infty )$, and a sequence $\{G_l^j\}_{l\in {\mathbb N},j\in {\mathbb N}}$ of anisotropic $(p(\cdot ),\infty )$-atoms supported, respectively, in $\{\widehat{B_l^j}\}_{l\in {\mathbb N},j\in {\mathbb N}}$ such that, for almost every $(x,k)\in {\mathbb {R}^n}\times {\mathbb Z}$,

$$\begin{aligned} H(x,k)=\sum _{j\in {\mathbb N}}\sum _{l\in {\mathbb N}}\lambda _l^jG_l^j(x,k)\quad \text {and}\quad |H(x,k)| =\sum _{j\in {\mathbb N}}\sum _{l\in {\mathbb N}}\lambda _l^j|G_l^j(x,k)| \end{aligned}$$

pointwisely, and

$$\begin{aligned} \left( \sum _{j\in {\mathbb N}}\left\| \left\{ \sum _{l\in {\mathbb N}} \left[ \frac{\lambda _l^j\textbf{1}_{B_l^j}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/{\underline{p}}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q} \lesssim \Vert H\Vert _{T^{p(\cdot ),q}_A({\mathbb {R}^n}\times {\mathbb Z})}, \end{aligned}$$

(4.3)

where the implicit positive constant is independent of H.

Proof

Let $p(\cdot )\in C^{\log }({\mathbb {R}^n})$, $q\in (0,\infty )$ and $H\in T^{p(\cdot ),q }_A({\mathbb {R}^n}\times {\mathbb Z})$. For each $j\in {\mathbb Z}$, define

$$\begin{aligned} O_j:=\left\{ x\in {\mathbb {R}^n}:\ {\mathcal A}(H)(x)>2^j\right\} ,\quad H_j:=(O_j)^{\complement }, \end{aligned}$$

and, for each given $\varepsilon \in (0,1)$, let $(O_j)^*_{\varepsilon }:=\left\{ x\in {\mathbb {R}^n}:\ M_{\textrm{HL}}(\textbf{1}_{O_j})(x)>1-\varepsilon \right\} ,$ where, for any $f\in L^1_{\textrm{loc}}({\mathbb {R}^n})$ and $x\in {\mathbb {R}^n}$, $M_{\textrm{HL}}(f)(x):=\sup _{k\in \mathbb {Z}} \sup _{y\in x+B_k}\frac{1}{|B_k|} \int _{y+B_k}|f(z)|\,dz.$ Therefore, from the proof of [8, (1.14)], it follows that

$$\begin{aligned} \mathop {\mathrm {\,supp\,}}H\subset \left[ \bigcup _{j\in {\mathbb Z}}\widehat{(O_j)^*_{\varepsilon }}\cup E\right] , \end{aligned}$$

(4.4)

where $E\subset {\mathbb {R}^n}\times {\mathbb Z}$ such that $\sum _{\ell \in {\mathbb Z}}\int _{\{y\in {\mathbb {R}^n}:\ (y,\ell )\in E\}}\,dy=0$. Furthermore, by [8, (1.15)], we find that, for any $j\in {\mathbb Z}$, there exist an integer $R_j\in {\mathbb N}\cup \{\infty \}$, $\{x_k^j\}_{k=1}^{R_j}\subset (O_j)^*_{\varepsilon }$, and $\{l_k\}_{k=1}^{R_j}\subset {\mathbb Z}$ satisfying that $\{x_k^j+B_{l_k}^j\}_{k=1}^{R_j}$ has finite intersection property and

$$\begin{aligned} (O_j)^*_{\varepsilon }&=\bigcup _{k=1}^{R_j} \left( x_k^j+B_{l_k}^j\right) \nonumber \\&=\left( x_1^j+B_{l_1}^j\right) \cup \left\{ \left( x_2^j+B_{l_2}^j\right) \setminus \left( x_1^j+B_{l_1}^j\right) \right\} \cup \cdots \cup \left\{ \left( x_{R_j}^j+B_{l_{R_j}}^j\right) \setminus \bigcup _{i=1}^{R_j-1}\left( x_i^j+B_{l_i}^j\right) \right\} \nonumber \\&=:\bigcup _{k=1}^{R_j} B_{j,k}. \end{aligned}$$

(4.5)

Observe that, for each given $j\in {\mathbb Z}$, $\{B_{j,k}\}_{k=1}^{R_j}$ are mutually disjoint. Then $\widehat{(O_j)^*_{\varepsilon }}=\bigcup _{k=1}^{R_j} \widehat{B_{j,k}}.$ For any $j\in {\mathbb Z}$ and $k\in \{1,\ldots ,R_j\}$, let $\Omega _{j,k}:=\widehat{B_{j,k}}\cap \left[ \widehat{(O_j)^*_{\varepsilon }} {\setminus } \widehat{(O_{j+1})^*_{\varepsilon }}\right] ,$

$$\begin{aligned} G_{j,k}:=2^{-j}\left\| \textbf{1}_{x_k^j+B_{l_k}^j}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^{-1}H\textbf{1}_{\Omega _{j,k}}, \end{aligned}$$

(4.6)

and $\lambda _{j,k}:=2^j\Vert \textbf{1}_{x_k^j+B_{l_k}^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}$. Thus, by (4.4), we conclude that

$$\begin{aligned} H=\sum _{j\in {\mathbb N}}\sum _{k=1}^{R_j} \lambda _{j,k}G_{j,k}~~~~\textrm{and}~~~~|H|=\sum _{j\in {\mathbb N}}\sum _{k=1}^{R_j} \lambda _{j,k}|G_{j,k}| \end{aligned}$$

almost everywhere on ${\mathbb {R}^n}\times {\mathbb Z}$. We next prove that, for any $j\in {\mathbb Z}$ and $k\in \{1,\ldots ,R_j\}$, $G_{j,k}$ is an anisotropic $(p(\cdot ),\infty )$-atom supported in $\widehat{x_k^j+B_{l_k}^j}$. Clearly,

$$\begin{aligned} \mathop {\mathrm {\,supp\,}}G_{j,k}\subset \Omega _{j,k}\subset \widehat{B_{j,k}}\subset \widehat{x_k^j+B_{l_k}^j}. \end{aligned}$$

Moreover, let $r\in (1,\infty )$ and $g\in T_2^{r'}({\mathbb {R}^n}\times {\mathbb Z})$ satisfy $\Vert g\Vert _{T_2^{r'}({\mathbb {R}^n}\times {\mathbb Z})}\le 1$. Observe that $\Omega _{j,k}\subset \widehat{(O_{j+1})^*_{\varepsilon }}^{\complement } =\bigcup _{x\in (H_{j+1})^*_{\varepsilon }}\Gamma (x).$ From this, [8, Lemma 1.3], the Hölder inequality, and (4.6), we deduce that

$$\begin{aligned} \left| \left\langle G_{j,k},g\right\rangle \right|&\lesssim \int _{H_{j+1}}\sum _{\ell \in {\mathbb Z}}\int _{\{y\in {\mathbb {R}^n}:\ (y,\ell )\in \Gamma (x)\}}b^{-\ell }\left| G_{j,k}(y,\ell )g(y,\ell )\right| \,dy\,dx\\&\lesssim \int _{(O_{j+1})^{\complement }}{\mathcal A}(G_{j,k})(x){\mathcal A}(g)(x)\,dx\\&\lesssim \left\{ \int _{(O_{j+1})^{\complement }}\left[ {\mathcal A}(G_{j,k})(x)\right] ^r\,dx\right\} ^{1/r} \left\{ \int _{(O_{j+1})^{\complement }}\left[ {\mathcal A}(g)(x)\right] ^{r'}\,dx\right\} ^{1/{r'}}\\&\lesssim 2^{-j}\left\| \textbf{1}_{x_k^j+B_{l_k}^j}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^{-1} \left\{ \int _{(x_k^j+B_{l_k}^j)\cap (O_{j+1})^{\complement }}\left[ {\mathcal A}(H)(x)\right] ^r\,dx\right\} ^{1/r} \Vert g\Vert _{T_2^{r'}({\mathbb {R}^n}\times {\mathbb Z})}\\&\lesssim \frac{|x_k^j+B_{l_k}^j|^{1/r}}{\Vert \textbf{1}_{x_k^j+B_{l_k}^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}. \end{aligned}$$

Combining this and the fact that $(T_2^{r}({\mathbb {R}^n}\times {\mathbb Z}))^*=T_2^{r'}({\mathbb {R}^n}\times {\mathbb Z})$ (see [5, 8]), we obtain

$$\begin{aligned} \Vert G_{j,k}\Vert _{T_2^{r}({\mathbb {R}^n}\times {\mathbb Z})} \lesssim \frac{|x_k^j+B_{l_k}^j|^{1/r}}{\Vert \textbf{1}_{x_k^j+B_{l_k}^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}. \end{aligned}$$

Thus, for any $j\in {\mathbb Z}$ and $k\in \{1,\ldots ,R_j\}$, $G_{j,k}$ is an anisotropic $(p(\cdot ),r)$-atom up to a harmless constant multiple for all $r\in (1,\infty )$. Therefore, for any $j\in {\mathbb Z}$ and $k\in \{1,\ldots ,R_j\}$, $G_{j,k}$ is an anisotropic $(p(\cdot ),\infty )$-atom up to a harmless constant multiple.

We now show (4.3). Indeed, by (4.5), the finite intersection property of $\{x_k^j+B_{l_k}^j\}_{k=1}^{R_j}$, the fact that $\textbf{1}_{(O_j)^*_{\varepsilon }}\lesssim [M_{\textrm{HL}}(\textbf{1}_{O_j})]^{1/u}$ with $u\in (0,\underline{p})$, [24, Lemma 3.3(ii)], and Remark 1, we conclude that

$$\begin{aligned}&\left( \sum _{j\in {\mathbb Z}}\left\| \left\{ \sum _{k=1}^{R_j} \left[ \frac{\lambda _{j,k}\textbf{1}_{x_k^j+B_{l_k}^j}}{\Vert \textbf{1}_{x_k^j+B_{l_k}^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^{\underline{p}} \right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q} \\&=\left( \sum _{j\in {\mathbb Z}}\left\| \left\{ \sum _{k=1}^{R_j}\left( 2^j \textbf{1}_{x_k^j+B_{l_k}^j}\right) ^{\underline{p}} \right\} ^{1/\underline{p}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q} \lesssim \left( \sum _{j\in {\mathbb Z}}2^{jq}\left\| \textbf{1}_{(O_j)^*_{\varepsilon }}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q}\\&\lesssim \left( \sum _{j\in {\mathbb Z}}2^{jq}\left\| \textbf{1}_{O_j}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q} \sim \left\| {\mathcal A}(H)\right\| _{{L^{p(\cdot ),q}({\mathbb {R}^n})}} \sim \Vert H\Vert _{T^{p(\cdot ),q}_A({\mathbb {R}^n}\times {\mathbb Z})}. \end{aligned}$$

This implies that (4.3) is valid and hence finishes the proof of Lemma 6. $\square$

Applying Theorem 1, Proposition 1 and Lemma 6, we now show Theorem 2.

Proof

Let $p(\cdot )$, s, $\underline{p}$, q and $\phi$ be as in Theorem 2. We first prove (i). For this purpose, let $h\in \mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})$ and $\{x_l^j+B_{\epsilon _l^j}\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B}$, where $m\in {\mathbb N}$, $v\in {\mathbb N}$, $\{x_l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset {\mathbb {R}^n}$, and $\{{\epsilon _l^j}\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset {\mathbb Z}$. Then, for any $l\in [1,m]\cap {\mathbb N}$, $j\in [1,v]\cap {\mathbb N}$, it holds true that

$$\begin{aligned} h&=P^s_{x_l^j+B_{\epsilon _l^j}}h+ \left( h-P^s_{x_l^j+B_{\epsilon _l^j}}h\right) \textbf{1}_{x_l^j+B_{\epsilon _l^j+\omega }}+ \left( h-P^s_{x_l^j+B_{\epsilon _l^j}}h\right) \textbf{1}_{(x_l^j+B_{\epsilon _l^j+\omega })^{\complement }}\nonumber \\&=:h_{l,j}^{(1)}+h_{l,j}^{(2)}+h_{l,j}^{(3)}, \end{aligned}$$

(4.7)

where $\omega$ is as in (2.2). Observe that, for any $\gamma \in {\mathbb Z}_+^n$ with $|\gamma |\le s$, $\int _{{\mathbb {R}^n}}\phi (x)x^{\gamma }\,dx=0$. Thus, for any $k\in {\mathbb Z}$, $l\in [1,m]\cap {\mathbb N}$ and $j\in [1,v]\cap {\mathbb N}$, $\phi _k*h_{l,j}^{(1)}\equiv 0$. This implies that, for any $l\in [1,m]\cap {\mathbb N}$, $j\in [1,v]\cap {\mathbb N}$,

$$\begin{aligned} \sum _{\ell \in {\mathbb Z}}\int _{\{x\in {\mathbb {R}^n}:\ (x,\ell )\in \widehat{x_l^j+B_{\epsilon _l^j}}\}}\left| \phi _{-\ell }*h_{l,j}^{(1)}(x)\right| ^2\,dx=0. \end{aligned}$$

(4.8)

On another hand, by the Tonelli theorem and the boundedness of the g-function (see [25, Theorem 2.9]), we conclude that, for any $l\in [1,m]\cap {\mathbb N}$, $j\in [1,v]\cap {\mathbb N}$,

$$\begin{aligned}&\sum _{\ell \in {\mathbb Z}}\int _{\{x\in {\mathbb {R}^n}:\ (x,\ell )\in \widehat{x_l^j+B_{\epsilon _l^j}}\}}\left| \phi _{-\ell }*h_{l,j}^{(2)}(x)\right| ^2\,dx\nonumber \\&\le \int _{{\mathbb {R}^n}}\sum _{\ell \in {\mathbb Z}}\left| \phi _{-\ell }*h_{l,j}^{(2)}(x)\right| ^2\,dx \lesssim \left\| h_{l,j}^{(2)}\right\| _{L^2({\mathbb {R}^n})}^2 \sim \int _{x_l^j+B_{\epsilon _l^j+\omega }}\left| h(x)-P^s_{x_l^j+B_{\epsilon _l^j}}h(x)\right| ^2\,dx\nonumber \\&\lesssim \int _{x_l^j+B_{\epsilon _l^j+\omega }}\left| h(x)-P^s_{x_l^j+B_{\epsilon _l^j+\omega }}h(x)\right| ^2\,dx\nonumber \\&\qquad +\int _{x_l^j+B_{\epsilon _l^j+\omega }}\left| P^s_{x_l^j+B_{\epsilon _l^j+\omega }} h(x)-P^s_{x_l^j+B_{\epsilon _l^j}}h(x)\right| ^2\,dx. \end{aligned}$$

(4.9)

In addition, from [3, (8.9)], we deduce that, for any $x\in x_l^j+B_{\epsilon _l^j+\omega }$,

$$\begin{aligned} \left| P^s_{x_l^j+B_{\epsilon _l^j+\omega }}h(x)-P^s_{x_l^j+B_{\epsilon _l^j}}h(x)\right| \lesssim \frac{1}{|x_l^j+B_{\epsilon _l^j}|}\int _{x_l^j+B_{\epsilon _l^j+\omega }} \left| h(y)-P^s_{x_l^j+B_{\epsilon _l^j+\omega }}h(y)\right| \,dy. \end{aligned}$$

This, combined with (4.9) and Lemma 5, further implies that, for any $v\in {\mathbb N}$, $m\in {\mathbb N}$, $\{x_l^j+B_{\epsilon _l^j}\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B}$, and $\{\lambda _l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset [0,\infty )$ with $\sum _{j=1}^v\sum _{l=1}^m\lambda _l^j\ne 0$,

By this, the assumption $p_+\in (0,2)$ and Corollary 1, we find that

$$\begin{aligned}&\left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/{\underline{p}}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{-1/q} \sum _{j=1}^v\sum _{l=1}^m \frac{{\lambda }_l^j|x_l^j+B_{\epsilon _l^j}|^{1/2}}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\nonumber \\&\times \left[ \sum _{\ell \in {\mathbb Z}}\int _{\{x\in {\mathbb {R}^n}:\ (x,\ell )\in \widehat{x_l^j+B_{\epsilon _l^j}}\}}\left| \phi _{-\ell }*h_{l,j}^{(2)}(x)\right| ^2\,dx\right] ^{1/2}\nonumber \\&\lesssim \Vert h\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})}. \end{aligned}$$

(4.10)

To deal with $h_{l,j}^{(3)}$, let $c\in (0,\underline{p})$ and $\varepsilon \in ([2/c-1]\ln b/\ln \lambda _-,\infty )$. Then, for any $l\in [1,m]\cap {\mathbb N}$, $j\in [1,v]\cap {\mathbb N}$ and $(x,\ell )\in \widehat{x_l^j+B_{\epsilon _l^j}}$, it holds true that

$$\begin{aligned} \left| \phi _{-\ell }*h_{l,j}^{(3)}(x)\right|&\lesssim \int _{(x_l^j+B_{\epsilon _l^j+\omega })^{\complement }}\frac{b^{\varepsilon \ell \frac{\ln \lambda _-}{\ln b}}}{[b^{\ell }+\rho (x-y)]^{1+\varepsilon \frac{\ln \lambda _-}{\ln b}}} \left| h(y)-P^s_{x_l^j+B_{\epsilon _l^j}}t(y)\right| \,dy\\&\lesssim \frac{b^{\varepsilon \ell \frac{\ln \lambda _-}{\ln b}}}{b^{\varepsilon \epsilon _l^j\frac{\ln \lambda _-}{\ln b}}}\int _{(x_l^j+B_{\epsilon _l^j+\omega })^{\complement }} \frac{b^{\varepsilon \epsilon _l^j\frac{\ln \lambda _-}{\ln b}}|h(y)-P^s_{x_l^j+B_{\epsilon _l^j}}t(y)|}{b^{\epsilon _l^j(1+\varepsilon \frac{\ln \lambda _-}{\ln b})} +[\rho (x_l^j-y)]^{1+\varepsilon \frac{\ln \lambda _-}{\ln b}}}\,dy.\nonumber \end{aligned}$$

From this, Proposition 1, and the Tonelli theorem, it follows that, for any $v\in {\mathbb N}$, $m\in {\mathbb N}$, $\{x_l^j+B_{\epsilon _l^j}\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset \mathfrak {B}$, and $\{\lambda _l^j\}_{l\in [1,m]\cap {\mathbb N},\,j\in [1,v]\cap {\mathbb N}}\subset [0,\infty )$ with $\sum _{j=1}^v\sum _{l=1}^m\lambda _l^j\ne 0$,

$$\begin{aligned}&\left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^{\underline{p}}\right\} ^{1/{\underline{p}}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{- 1/q}\sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|x_l^j+B_{\epsilon _l^j}|^{1/2}}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\\&\times \left[ \sum _{\ell \in {\mathbb Z}}\int _{\{x\in {\mathbb {R}^n}:\ (x,\ell )\in \widehat{x_l^j+B_{\epsilon _l^j}}\}}\left| \phi _{-\ell }*h_{l,j}^{(3)}(x)\right| ^2\,dx\right] ^{1/2}\\&\lesssim \Vert h\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^{A,\varepsilon }({\mathbb {R}^n})} \sim \Vert h\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^{A}({\mathbb {R}^n})}. \end{aligned}$$

By this, (4.7), (4.8), and (4.10), we conclude that

$$\begin{aligned}&\left( \sum _{k=1}^v\left\| \left\{ \sum _{i=1}^m \left[ \frac{\lambda _i^k\textbf{1}_{x_i^k+B_{\epsilon _i^k}}}{\Vert \textbf{1}_{x_i^k+B_{\epsilon _i^k}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^{\underline{p}}\right\} ^{1/{\underline{p}}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{- 1/q}\sum _{j=1}^v\sum _{l=1}^m\frac{{\lambda }_l^j|x_l^j+B_{\epsilon _l^j}|^{1/2}}{\Vert \textbf{1}_{x_l^j+B_{\epsilon _l^j}}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\\&\times \left[ \sum _{\ell \in {\mathbb Z}}\int _{\{x\in {\mathbb {R}^n}:\ (x,\ell )\in \widehat{x_l^j+B_{\epsilon _l^j}}\}}\left| \phi _{-\ell }*h(x)\right| ^2\,dx\right] ^{1/2}\\&\lesssim \Vert h\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^{A}({\mathbb {R}^n})} \end{aligned}$$

Therefore, for any $(x,k)\in {\mathbb {R}^n}\times {\mathbb Z}$, $d\mu (x,k):=\sum _{\ell \in {\mathbb Z}}|\phi _{-\ell }*h(x)|^2\,dx\,\delta _{\ell }(k)$ is a $[p(\cdot ),q]$-Carleson measure on ${\mathbb {R}^n}\times {\mathbb Z}$ and $\Vert d\mu \Vert _{\mathcal {C}_{p(\cdot ),q,A}} \lesssim \Vert h\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})}$, which completes the proof of (i).

We now prove (ii). To do this, let $f\in H_{A,\textrm{fin}}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})$ with $\Vert f\Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}\ne 0$. Then $f\in L^{\infty }({\mathbb {R}^n})$ with compact support. Thus, by the fact that $h\in L^2_\textrm{loc}({\mathbb {R}^n})$ and [8, (2.10)], we find that

$$\begin{aligned} \int _{{\mathbb {R}^n}}f(x)\overline{h(x)}\,dx \sim \sum _{\ell \in {\mathbb Z}}\int _{{\mathbb {R}^n}}\varphi _{-\ell }*f(x)\overline{\phi _{-\ell }*h(x)}\,dx, \end{aligned}$$

(4.11)

where $\varphi \in {\mathcal S}({\mathbb {R}^n})$ satisfies that $\mathop {\mathrm {\,supp\,}}\widehat{\varphi }$ is compact and away from $\textbf{0}$ and, for any $\xi \in {\mathbb {R}^n}\setminus \{\textbf{0}\}$, $\sum _{i\in {\mathbb Z}} \widehat{\varphi }((A^*)^i\xi )\widehat{\phi }((A^*)^i\xi )=1$ with $A^*$ as the adjoint matrix of A. Moreover, by the Lusin area function characterization of ${H_A^{p(\cdot ),q}({\mathbb {R}^n})}$ (see [26, Theorem 5.2]) and the fact that $f\in {H_A^{p(\cdot ),q}({\mathbb {R}^n})}$, we know that $\left\| \varphi _{-\ell }*f\right\| _{T_A^{p(\cdot ),q}({\mathbb {R}^n}\times {\mathbb Z})}\sim \Vert f\Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}<\infty .$ Combining this and Lemma 6, we conclude that there exist $\{\lambda _l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset [0,\infty )$ and a sequence $\{G_l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}$ of anisotropic $(p(\cdot ),\infty )$-atoms supported, respectively, in $\{\widehat{B_l^j}\}_{l\in {\mathbb N},\,j\in {\mathbb N}}$ with $\{B_l^j\}_{l\in {\mathbb N},\,j\in {\mathbb N}}\subset {\mathfrak B}$ such that, for almost every $(x,\ell )\in {\mathbb {R}^n}\times {\mathbb Z}$, $\varphi _{-\ell }*f(x)=\sum _{j\in {\mathbb N}}\sum _{l\in {\mathbb N}}\lambda _l^jG_l^j(x,\ell )$ and

$$\begin{aligned} 0<\left( \sum _{j\in {\mathbb N}}\left\| \left\{ \sum _{l\in {\mathbb N}} \left[ \frac{\lambda _l^j\textbf{1}_{B_l^j}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}}\right] ^ {\underline{p}}\right\} ^{1/{\underline{p}}}\right\| _{{L^{p(\cdot )}({\mathbb {R}^n})}}^q\right) ^{ 1/q}\lesssim \Vert f\Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}. \end{aligned}$$

This, together with (4.11), the Hölder inequality, the Tonelli theorem, and Definition 11(ii), further implies that, for any $f\in H_{A,\textrm{fin}}^{p(\cdot ),\infty ,s,q}({\mathbb {R}^n})$,

$$\begin{aligned} \left| \int _{{\mathbb {R}^n}}f(x)\overline{h(x)}\,dx\right|&\lesssim \sum _{j\in {\mathbb N}}\sum _{l\in {\mathbb N}}\lambda _l^j\left[ \sum _{\ell \in {\mathbb Z}}\int _{\{x\in {\mathbb {R}^n}:\ (x,\ell )\in \widehat{B_l^j}\}}\left| G_l^j(x,\ell )\right| ^2 \,dx\right] ^{1/2}\\&\quad \times \left[ \sum _{\ell \in {\mathbb Z}}\int _{\{x\in {\mathbb {R}^n}:\ (x,\ell )\in \widehat{B_l^j}\}} \left| \phi _{-\ell }*h(x)\right| ^2 \,dx\right] ^{1/2}\\&\lesssim \sum _{j\in {\mathbb N}}\sum _{l\in {\mathbb N}}\frac{\lambda _l^j|B_l^j|^{1/2}}{\Vert \textbf{1}_{B_l^j}\Vert _{{L^{p(\cdot )}({\mathbb {R}^n})}}} \left[ \sum _{\ell \in {\mathbb Z}}\int _{\{x\in {\mathbb {R}^n}:\ (x,\ell )\in \widehat{B_l^j}\}}\left| \phi _{-\ell }*h(x)\right| ^2 \,dx\right] ^{1/2}\\&\lesssim \Vert f\Vert _{{H_A^{p(\cdot ),q}({\mathbb {R}^n})}}\widetilde{\Vert d\mu \Vert }_{\mathcal {C}_{p(\cdot ),q,A}}. \end{aligned}$$

From this, Theorem 1, the fact that $p_+\in (0,2)$, Corollary 1, and Remark 4, we further deduce that $\Vert h\Vert _{\mathcal {L}_{p(\cdot ),q,s,\underline{p},1}^A({{\mathbb {R}^n}})} \lesssim \Vert d\mu \Vert _{\mathcal {C}_{p(\cdot ),q,A}}.$ This finishes the proof of (ii) and hence of Theorem 2. $\square$