Open Access

Anisotropic interpolation theorems of Musielak-Orlicz type

Journal of Inequalities and Applications20162016:243

https://doi.org/10.1186/s13660-016-1184-z

Received: 18 June 2016

Accepted: 21 September 2016

Published: 3 October 2016

Abstract

Anisotropy is a common attribute of Nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations \(\{A^{k}: k\in\mathbb{Z}\}\), where A is a real \(n\times n\) matrix with all its eigenvalues λ satisfy \(|\lambda|>1\). Let \(\varphi: \mathbb{R}^{n}\times[0, \infty)\to[0,\infty)\) be an anisotropic Musielak-Orlicz function such that \(\varphi(x,\cdot)\) is an Orlicz function and \(\varphi(\cdot,t)\) is a Muckenhoupt \(\mathbb {A}_{\infty}(A)\) weight. The aim of this article is to obtain two anisotropic interpolation theorems of Musielak-Orlicz type, which are weighted anisotropic extension of Marcinkiewicz interpolation theorems. The above results are new even for the isotropic weighted settings.

Keywords

anisotropic expansive dilation Muckenhoupt weight Musielak-Orlicz function weighted Hardy space interpolation

MSC

42B35 46E30 42B25 42B30

1 Introduction

The aim of this article is to obtain two anisotropic interpolation theorems of Musielak-Orlicz type. Anisotropy is a common attribute of Nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. For example, the elastic modulus, hardness or fracture strength of a crystal is different in different directions, which shows the anisotropic property of the crystal. The anisotropic property, in mathematics, can be expressed by a general discrete group of dilations \(\{A^{k}: k\in\mathbb{Z}\}\), where A is a real \(n\times n\) matrix with all its eigenvalues λ satisfy \(|\lambda|>1\).

As a natural generalization of \(L^{p}({{\mathbb{R}}^{n}})\), the Orlicz space was introduced by Birnbaum-Orlicz [1] and Orlicz [2], which is defined via an Orlicz function. Recall that a Musielak-Orlicz space is defined via a Musielak-Orlicz function (see, for example, [3]) and a Musielak-Orlicz function is a natural generalization of an Orlicz function. Observe that, different from Orlicz functions, Musielak-Orlicz functions may also vary in the spatial variables. Musielak-Orlicz spaces include many function spaces far beyond \(L^{p}({{\mathbb{R}}^{n}})\), and the motivation to study function spaces of Musielak-Orlicz type comes from various applications in mathematics and physics (see, for example, [49] and the references therein).

On the other hand, there were several efforts of extending classical function spaces arising in harmonic analysis from Euclidean spaces to other domains and non-isotropic settings; see [1018]. Calderón and Torchinsky initiated the study of Hardy spaces on \(\mathbb{R}^{n}\) with anisotropic dilations [1113]. The theory of Hardy spaces associated to expansive dilations was recently developed in [10, 19]. Another research direction of extending classical function spaces is the study of weighted function spaces associated with general Muckenhoupt weights; see [2025]. García-Cuerva [25] and Strömberg-Torchinsky [26] established a theory of weighted Hardy spaces on \(\mathbb{R}^{n}\).

Moreover, many problems in Fourier analysis concern the boundedness of operators on Lebesgue spaces, and interpolation provides a framework that often simplifies this study. For instance, in order to show that a linear operator maps \(L^{p}\) to itself for all \(1< p<\infty \), it is sufficient to show that it maps the (smaller) Lorentz space \(L^{p, 1}\) into the (larger) Lorentz space \(L^{p, \infty }\) for the same range of p. Moreover, some further reductions can be made in terms of the Lorentz space \(L^{p, 1}\). This and other considerations indicate that interpolation is a powerful tool in the study of boundedness of operators. Therefore, many kinds of interpolation theorems were established. For example, Ding and Lan [27], Theorems 3.2 and 3.4, obtained two interpolation theorems associated with anisotropic (weak) Hardy spaces and Lebesgue spaces. Liang et al. [28], Theorem 2.7, obtained an interpolation theorem associated with weighted (weak) Lebesgue spaces. Cao et al. [29], Proposition 2.21, also obtained an interpolation theorem associated with weighted Hardy spaces and weighted (weak) Lebesgue spaces.

Let \(\mathbb{A}_{q}(A)\) with \(q\in[1,\infty]\) denote the class of anisotropic Muckenhoupt weights (see, for example, [21, 30] for their definitions and properties) and \(\varphi: \mathbb{R}^{n}\times [0,\infty)\to[0,\infty)\) be an anisotropic Musielak-Orlicz function such that \(\varphi(x,\cdot)\) is an Orlicz function and \(\varphi (\cdot,t)\in\mathbb{A}_{\infty}(A)\). The main goal of this article is to give two anisotropic interpolation theorems of Musielak-Orlicz type. Precisely, by the boundedness of sublinear operator T on weighted anisotropic Hardy spaces and weighted (weak) Lebesgue spaces, we can further obtain the boundedness of T on anisotropic Musielak-Orlicz function spaces (see Theorems 2.5 and 2.9 below). The interpolation theorems mentioned above are weighted anisotropic extension of classical Marcinkiewicz interpolation theorems, see, for example, [31], and they are also a complement of Liang et al. [28], Theorem 2.7. And it is worth mentioning that the classical cubes are not suitable for the anisotropic settings, so we introduce a class of general dyadic cubes of Christ [32] (see Lemma 3.11 below), which plays an important role in the proof of Theorem 2.9 mentioned above.

This article is organized as follows.

In Section 2, we first recall some notation and definitions concerning expansive dilations, anisotropic Muckenhoupt weights and anisotropic Musielak-Orlicz functions. Then we give two anisotropic interpolation theorems of Musielak-Orlicz type, the proofs of which are given in Section 3.

Finally, we make some conventions on notation. Let \(\mathbb{Z}_{+}:=\{1, 2, \ldots\}\) and \(\mathbb{N}:=\{0\}\cup \mathbb{Z}_{+}\). For any \(\alpha :=(\alpha _{1},\ldots,\alpha _{n})\in\mathbb{N}^{n}\), \(|\alpha |:=\alpha _{1}+\cdots+\alpha _{n}\), and \(\partial^{\alpha}:= (\frac{\partial}{\partial x_{1}})^{\alpha _{1}}\cdots (\frac{\partial}{\partial x_{n}})^{\alpha _{n}}\). Throughout the whole paper, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. The symbol \(D\lesssim F\) means that \(D\le CF\). If \(D\lesssim F\) and \(F\lesssim D\), we then write \(D\sim F\). If E is a subset of \({{\mathbb{R}}^{n}}\), we denote by \(\chi_{E}\) its characteristic function. For any \(a\in\mathbb{R}\), \(\lfloor a\rfloor\) denotes the maximal integer not larger than a. If there are no special instructions, any space \(\mathcal{X}({{\mathbb {R}}^{n}})\) is denoted simply by \(\mathcal{X}\) and any space \(\mathcal{X}({{\mathbb {R}}^{n}}; A)\) is denoted simply by \(\mathcal{X}(A)\). Denote by \(\mathcal{S}\) the space of all Schwartz functions, \(\mathcal{S}'\) the space of all tempered distributions. For any set \(E\subset{{\mathbb{R}}^{n}}\) and \(t\in(0, \infty)\), let \(\varphi (E, t):=\int_{E}\varphi (x, t) \,dx\) and, for any measurable function f and \(t\in(0, \infty)\), let \(\{|f|>t\} :=\{x\in{{\mathbb{R}}^{n}}: |f(x)|>t\}\).

2 Preliminaries and main results

In this section, let us first recall the notion of expansive dilations on \({{\mathbb{R}}^{n}}\); see [10], p.5. A real \(n\times n\) matrix A is called an expansive dilation, shortly a dilation, if \(\min_{\lambda \in\sigma (A)}|\lambda |>1\), where \(\sigma (A)\) denotes the set of all eigenvalues of A. Let \(\lambda _{-}\) and \(\lambda _{+}\) be two positive numbers such that
$$1< \lambda _{-}< \min\bigl\{ |\lambda |: \lambda \in\sigma (A)\bigr\} \le\max\bigl\{ | \lambda |: \lambda \in\sigma (A)\bigr\} < \lambda _{+}. $$
In the case when A is diagonalizable over \(\mathbb{C}\), we can even take \(\lambda _{-}:=\min\{|\lambda |: \lambda \in\sigma (A)\} \) and \(\lambda _{+}:=\max\{|\lambda |: \lambda \in\sigma (A)\}\). Otherwise, we need to choose them sufficiently close to these equalities according to what we need in our arguments.
It was proved in [10], p.5, Lemma 2.2, that, for a given dilation A, there exist a number \(r\in(1, \infty )\) and a set \(\Delta:=\{x\in {{\mathbb{R}}^{n}} : |Px|<1\}\), where P is some non-degenerate \(n\times n\) matrix, such that \(\Delta\subset r\Delta\subset A\Delta\) and, by a scaling, one can additionally assume that \(|\Delta|=1\), where \(|\Delta|\) denotes the n-dimensional Lebesgue measure of the set Δ. Let \(B_{k}:=A^{k}\Delta\) for \(k\in\mathbb{Z}\). Then \(B_{k}\) is open, \(B_{k}\subset rB_{k}\subset B_{k+1}\), and \(|B_{k}|=b^{k}\), here and hereafter \(b:=|\det A|\). Throughout the whole paper, let σ be the minimum positive integer such that
$$r^{\sigma}\ge2 $$
and, for any subset E of \({{\mathbb{R}}^{n}}\), let \(E^{\complement}:={{\mathbb {R}}^{n}}\setminus E\). Then, for all \(k, j\in\mathbb{Z}\) with \(k\le j\), it holds true that
$$\begin{aligned}& B_{k}+B_{j}\subset B_{j+\sigma }, \end{aligned}$$
(2.1)
$$\begin{aligned}& B_{k}+(B_{j+\sigma })^{\complement}\subset (B_{j})^{\complement}, \end{aligned}$$
(2.2)
where \(E+F\) denotes the algebraic sum \(\{x+y: x\in E, y\in F\}\) of sets \(E, F\subset{{\mathbb{R}}^{n}}\).

Definition 2.1

A quasi-norm, associated with an expansive matrix A, is a Borel measurable mapping \(\rho_{A}: \mathbb{R}^{n}\to[0,\infty)\), for simplicity, denoted by ρ, satisfying
  1. (i)

    \(\rho(x)>0\) for all \(x \in{{\mathbb{R}}^{n}}\setminus\{ \vec{0}_{n}\}\), here and hereafter, \(\vec{0}_{n}:=(\overbrace{0, \ldots, 0}^{n})\);

     
  2. (ii)

    \(\rho(Ax)= b\rho(x)\) for all \(x\in\mathbb{R}^{n}\), where, as above, \(b:=|\det A|\);

     
  3. (iii)

    \(\rho(x+y)\le H [\rho(x)+\rho(y) ]\) for all \(x, y\in\mathbb{R}^{n}\), where \(H\in[1, \infty )\) is a constant independent of x and y.

     

In the standard dyadic case \(A:=2I_{n\times n}\), \(\rho(x):=|x|^{n}\) for all \(x\in{{\mathbb{R}}^{n}}\) is an example of quasi-norms associated with A, here and hereafter, \(|\cdot|\) always denotes the Euclidean norm in \({{\mathbb{R}}^{n}}\).

It was proved, in [10], p.6, Lemma 2.4, that all quasi-norms associated with a given dilation A are equivalent. Therefore, for a given expansive dilation A, in the following, for simplicity, we always use the step quasi-norm ρ defined by setting, for all \(x\in{{\mathbb{R}}^{n}}\),
$$ \rho(x):=\sum_{k\in\mathbb{Z}}b^{k} \chi_{B_{k+1}\setminus B_{k}}(x) \quad \mbox{if } x\ne\vec{0}_{n},\quad \mbox{or else} \quad \rho(\vec{0}_{n}):=0. $$
By (2.1) and (2.2), we know that, for all \(x, y\in{{\mathbb{R}}^{n}}\),
$$\rho(x+y)\le b^{\sigma}\bigl(\max \bigl\{ \rho(x), \rho(y) \bigr\} \bigr) \le b^{\sigma}\bigl[\rho (x)+\rho(y)\bigr]. $$
Moreover, \(({{\mathbb{R}}^{n}}, \rho, dx)\) is a space of homogeneous type in the sense of Coifman and Weiss [33, 34], where dx denotes the n-dimensional Lebesgue measure.

Definition 2.2

Let \(q\in[1, \infty )\). A function \(w: {{\mathbb{R}}^{n}}\to [0,\infty )\) is said to satisfy the anisotropic Muckenhoupt condition \(\mathbb{A}_{q}(A)\), denoted by \(w\in\mathbb{A}_{q}(A)\), if there exists a positive constant C such that, when \(q\in(1,\infty )\),
$$\sup_{x\in{{\mathbb{R}}^{n}}}\sup_{k\in\mathbb{Z}} \biggl\{ b^{-k} \int_{x+B_{k}}w(y)\,dy \biggr\} \biggl\{ b^{-k} \int_{x+B_{k}}\bigl[w(y)\bigr]^{-\frac{1}{q-1}} \,dy \biggr\} ^{q-1}\le [w]_{\mathbb{A}_{q}(A)}< \infty , $$
and, when \(q=1\),
$$\sup_{x\in{{\mathbb{R}}^{n}}}\sup_{k\in\mathbb{Z}} \biggl\{ b^{-k} \int_{x+B_{k}} w(y)\,dy \biggr\} \bigl\{ \mathop {\operatorname{ess sup}}_{y\in x+B_{k}} \bigl[w(y)\bigr]^{-1} \bigr\} \le[w]_{\mathbb{A}_{1}(A)}< \infty . $$
Define \(\mathbb{A}_{\infty}(A):=\bigcup_{1\le q<\infty } \mathbb {A}_{q}(A)\) and, for any \(w\in\mathbb{A}_{\infty}(A)\), let
$$ q(w):=\inf\bigl\{ q\in [1, \infty ): w\in\mathbb{A}_{q}(A) \bigr\} . $$
(2.3)

Obviously, \(q(w)\in[1,\infty )\). Moreover, it is known (see [35]) that, if \(q(w)\in(1,\infty )\), then \(w\notin \mathbb{A}_{q(w)}(A)\) and there exists a \(w\in(\bigcap_{q>1}\mathbb{A}_{q}(A))\setminus\mathbb {A}_{1}(A)\) such that \(q(w)=1\).

Let φ be a nonnegative function on \({{\mathbb{R}}^{n}}\times [0,\infty )\). The function φ is called a Musielak-Orlicz function if, for any \(x\in{{\mathbb{R}}^{n}}\), \(\varphi(x, \cdot)\) is an Orlicz function on \([0,\infty )\) and, for any \(t\in[0,\infty ),\varphi(\cdot, t)\) is measurable on \({{\mathbb{R}}^{n}}\). Here a function \(\Phi: [0,\infty )\to[0,\infty )\) is called an Orlicz function if it is nondecreasing, \(\Phi (0)=0\), \(\Phi(t)>0\) for \(t\in[0,\infty )\) and \(\lim_{t\rightarrow\infty }\Phi (t)=\infty \) (see, for example, [3]). Remark that, unlike the usual case, such a Φ may not be convex.

For an Orlicz function Φ, the most useful tool to study its growth property may be the upper and the lower types of Φ. More precisely, for \(p\in(-\infty ,\infty )\), a function Φ is said to be of upper (resp. lower) type p, if there exists a positive constant C such that, for all \(s\in[1,\infty )\) (resp. \(s\in[0,1]\)) and \(t\in (0,\infty )\),
$$ \Phi(st)\leq Cs^{p}\Phi(t). $$
(2.4)
Let φ be a Musielak-Orlicz function. The Musielak-Orlicz space \(L^{\varphi}\), which was first introduced by Musielak [3], is defined to be the set of all measurable functions f such that \(\int_{\mathbb{R}^{n}}\varphi (x, |f(x)|)\,dx<\infty \) with the Luxembourg-Nakano (quasi-)norm:
$$\|f\|_{L^{\varphi}}:=\inf \biggl\{ \lambda \in(0, \infty ): \int _{\mathbb {R}^{n}}\varphi \biggl(x, \frac{|f(x)|}{\lambda} \biggr)\,dx\le 1 \biggr\} . $$

Remark 2.3

Let \(x\in{{\mathbb{R}}^{n}}\), \(t\in[0,\infty )\), w be a classical or an anisotropic Muckenhoupt \(\mathbb{A}_{\infty}\) weight with \(q(w)\) being as in (2.3), and \(\Phi(x)\) an Orlicz function on \({{\mathbb{R}}^{n}}\). If \(\varphi (x, t):=t^{p}w(x)\), then \(L^{\varphi}=L^{p}_{w}\); if \(\varphi (x, t):=\Phi(t)\), then \(L^{\varphi}=L^{\Phi}\); if \(\varphi (x, t):=\Phi(t)w(x)\), then \(L^{\varphi}=L^{\Phi}_{w}\).

Moreover, throughout the whole article, we always assume that the Musielak-Orlicz functions satisfy the following growth assumptions.

Assumption (φ)

Let \(\varphi: {{\mathbb {R}}^{n}}\times[0,\infty )\to[0,\infty )\) be a Musielak-Orlicz function satisfying the following two conditions:
  1. (i)

    for any \(t\in(0,\infty)\), \(\varphi(\cdot, t)\in\mathbb{A} _{\infty}(A)\);

     
  2. (ii)

    there exists \(p^{-}_{\varphi},p^{+}_{\varphi}\in(0,\infty )\) such that, for every \(x\in{{\mathbb{R}}^{n}}\), \(\varphi(x, \cdot)\) is of uniformly upper type \(p^{+}_{\varphi}\) and of uniformly lower type \(p^{-}_{\varphi}\).

     

When \(p^{-}_{\varphi}=p\in(0,1]\) and \(p^{+}_{\varphi}=1\), Assumption (φ) of φ coincides with that of [35], Definition 3.

For a Musielak-Orlicz function φ satisfying Assumption (φ), the following critical indices are useful. Let
$$\begin{aligned} i(\varphi ) :=&\sup \bigl\{ p \in(-\infty ,\infty ) : \mbox{for any } x\in{{\mathbb{R}}^{n}} , \varphi (x, \cdot) \mbox{ is of uniformly} \\ &\mbox{lower type } p \mbox{ with }C \mbox{ being as in (2.4) independent of } x \bigr\} \end{aligned}$$
(2.5)
and
$$\begin{aligned} q(\varphi ) :=&\inf \bigl\{ q\in[1, \infty ): \mbox{for any } t\in (0,\infty), \varphi (\cdot, t)\in\mathbb{A}_{q}(A) \\ &\mbox{with } \bigl[\varphi (\cdot, t)\bigr]_{\mathbb{A}_{q}(A)} \mbox{ independent of } t \bigr\} . \end{aligned}$$
(2.6)
Observe that \(i(\varphi )\) may be not attainable, namely, φ may be not of uniformly lower type \(i(\varphi )\); see, for example, [28] for some examples. Clearly,
$$ \varphi (x, t):= w(x)\Phi(t)\quad \mbox{for all } x\in{{\mathbb {R}}^{n}} \mbox{ and } t\in[0, \infty ) $$
satisfies Assumption (φ) if w is a classical or an anisotropic \(\mathcal{A}_{\infty}(A)\) Muckenhoupt weight (see, for example, [21]) and Φ is of lower type \(p^{-}_{\varphi}\) for some \(p^{-}_{\varphi}\in(0, \infty )\) and of upper type \(p^{+}_{\varphi}\) with \(p^{+}_{\varphi}\in (0, \infty )\). More examples of growth functions can be found in [9, 28, 36, 37].
Now, let us recall the notion of a weighted Lebesgue space. For any nonnegative locally integrable function w on \(\mathbb{R}^{n}\) and \(p\in(0,\infty ]\), the space \(L^{p}_{w}\) is defined to be the space of all measurable functions f such that, when \(p\in(0, \infty )\),
$$\|f\|_{L^{p}_{w}}:= \biggl\{ \int_{\mathbb{R}^{n}} \bigl\vert f(x)\bigr\vert ^{p} w(x) \,dx \biggr\} ^{\frac {1}{p}}< \infty $$
and, when \(p=\infty \),
$$\|f\|_{L^{\infty}_{w}}:=\inf_{w(E)=0}\sup_{x\in{{\mathbb {R}}^{n}}\setminus E} \bigl\vert f(x)\bigr\vert < \infty . $$
The space weak \(L^{p}_{w}\) denotes the set of all measurable functions f such that
$$\|f\|_{\text{weak } L^{p}_{w}}:=\sup_{\lambda >0} \lambda \bigl[w\bigl(\bigl\{ \vert f\vert >\lambda \bigr\} \bigr)\bigr]^{\frac{1}{p}}< \infty . $$
Next, let us introduce the notion of a weighted anisotropic Hardy space. For \(m\in\mathbb{N}\), let
$$ \mathcal{S}_{m}:= \Bigl\{ \phi\in\mathcal{S}: \sup _{x\in{{\mathbb {R}}^{n}}}\sup_{\substack{\alpha \in\mathbb{N} ^{n} \\ |\alpha |\le m+1}}\bigl[1+\rho(x) \bigr]^{m+2} \bigl\vert \partial^{\alpha}\phi(x)\bigr\vert \le1 \Bigr\} $$
and, for \(\phi\in\mathcal{S}\), \(k\in\mathbb{Z}\), and \(x\in{{\mathbb{R}}^{n}}\), let \(\phi_{k}(x):= b^{k}\phi(A^{k}x)\).
For \(f\in\mathcal{S}'\), the non-tangential grand maximal function \(f^{\ast}_{m}\) of f is defined by setting, for all \(x\in{{\mathbb{R}}^{n}}\),
$$ f^{\ast}_{m}(x) := \sup_{\phi\in\mathcal{S}_{m}} \sup _{k\in\mathbb {Z}, y\in x+B_{k}} \bigl\vert f\ast\phi_{k}(y)\bigr\vert . $$
For \(p\in(0,1]\), \(w\in\mathbb{A}_{\infty}(A)\), and \(q(w)\) being as in (2.3), let
$$ m_{p, w}:= \biggl\lfloor \biggl(\frac{q(w)}{p}-1 \biggr)\frac{\ln b}{\ln (\lambda_{-})} \biggr\rfloor . $$
(2.7)

The definition of the following weighted anisotropic Hardy space comes from [30], Definition 3.1.

Definition 2.4

For any \(p\in(0,1]\), \(w\in\mathbb{A}_{\infty}(A)\), \(m_{p, w}\) being as in (2.7) and \(m\in\mathbb{Z}_{+}\) satisfying \(m\geq m_{p, w}\), the weighted anisotropic Hardy space \(H^{p}_{w, m}(A)\) is defined as the set of all \(f\in\mathcal{S}'\) such that \(f^{\ast}_{m}\in L^{p}_{w}\) with the (quasi-)norm \(\|f\|_{H^{p}_{w, m}(A)}:=\|f^{\ast}_{m}\|_{L^{p}_{w}}\).

For any \(m_{1}, m_{2}\ge m_{p, w}\), since \(H^{p}_{w, m_{1}}(A)=H^{p}_{w, m_{2}}(A)\) with equivalent norms (see [30], Theorem 5.5), then from now on, we denote simply by \(H^{p}_{w}(A)\) the weighted anisotropic Hardy space \(H^{p}_{w, m}(A)\) with \(m\ge m_{p, w}\).

Finally, we need to make several further explanations for the \(\operatorname{BMO}(A)\) functions. For any \(f\in L^{1}_{\mathrm{loc}}\) and \(E\subset{{\mathbb{R}}^{n}}\), set \(f_{E}:=\frac{1}{|E|}\int_{E}f(x)\,dx\), and define the sharp maximal function associated with dilation A by setting, for all \(x\in{{\mathbb{R}}^{n}}\),
$$\mathcal{M}^{\sharp}f(x):=\sup_{x\in B\in\mathcal{B}}\frac {1}{\vert B\vert } \int_{B}\bigl\vert f(y)-f_{B}\bigr\vert \,dy, $$
here and hereafter, let \(\mathcal{B}:=\{B:=x+B_{k}: x\in{{\mathbb{R}}^{n}}, k\in\mathbb{Z}\} \) be the collection of all dilated balls.
Moreover, we also define anisotropic BMO space by setting
$$\operatorname{BMO}(A):= \bigl\{ f\in L^{1}_{\mathrm{loc}}: \mathcal{M}^{\sharp}f\in L^{\infty}\bigr\} $$
and
$$\Vert f\Vert _{\operatorname{BMO}(A)}:=\bigl\Vert \mathcal{M}^{\sharp}f \bigr\Vert _{\infty}. $$

Now, we introduce the interpolation theorem associated with a Musielak-Orlicz function, which may have independent interest.

Theorem 2.5

Let \(p_{1}\in(0,1]\), φ be a Musielak-Orlicz function satisfying Assumption  (φ), \(q(\varphi )\) as in (2.6) with \([q(\varphi )]^{2}< p^{-}_{\varphi}\leq p^{+}_{\varphi}<\infty \) and \(p_{2}\in(p^{+}_{\varphi}, \infty ]\). Assume that T is a sublinear operator defined on \(H^{p_{1}}_{\varphi (\cdot, t)}(A)+L^{p_{2}}_{\varphi (\cdot, t)}\) satisfying the requirement that there exist positive constants \(C_{1}\) and \(C_{2}\) such that, for all \(\alpha\in(0,\infty )\) and \(t\in (0,\infty )\),
$$ \varphi \bigl({\bigl\{ \vert Tf\vert >\alpha\bigr\} }, t\bigr) \leq C_{1}\alpha^{-{p_{1}}}{\|f\| ^{p_{1}}_{{H^{p_{1}}_{\varphi (\cdot, t)}(A)}}} $$
(2.8)
and
$$ \varphi \bigl({\bigl\{ \vert Tf\vert >\alpha\bigr\} }, t\bigr) \leq C_{2}\alpha^{-{p_{2}}} \|f\| ^{p_{2}}_{L^{p_{2}}_{\varphi (\cdot, t)}} $$
(2.9)
with the usual modification when \(p_{2}=\infty \). Then T is bounded on \(L^{\varphi}\) and, moreover, there exists a positive constant C such that, for any \(f\in L^{\varphi}\),
$$\int_{\mathbb{R}^{n}}\varphi \bigl(x,\bigl\vert Tf(x)\bigr\vert \bigr) \,dx\leq C \int_{\mathbb {R}^{n}}\varphi \bigl(x,\bigl\vert f(x)\bigr\vert \bigr) \,dx. $$

By Remark 2.3 and Theorem 2.5, we have the following three corollaries.

Corollary 2.6

Let w be a classical or an anisotropic Muckenhoupt \(\mathbb {A}_{\infty}\) weight with \(q(w)\) being as in (2.3) and \(p_{1}, p, p_{2}\in\mathbb{R}\) with \(0< p_{1}\le1\le[q(w)]^{2}< p< p_{2}\le\infty \). If a sublinear operator T is bounded from \(H^{p_{1}}_{w}\) to weak \(L^{p_{1}}_{w}\) and from \(L^{p_{2}}_{w}\) to weak \(L^{p_{2}}_{w}\), then T is bounded on \(L^{p}_{w}\).

Particularly, when \(w(x)=1\), the above consequence coincides with a generalization of Marcinkiewicz interpolation theorem.

Corollary 2.7

Let \(p_{1}, p_{2}\in\mathbb{R}\) and Φ be an Orlicz function with uniformly lower type \(p^{-}_{\Phi}\) and uniformly upper type \(p^{+}_{\Phi}\), where \(0< p_{1}\le1< p^{-}_{\Phi}\le p^{+}_{\Phi}< p_{2}\le\infty \). If a sublinear operator T is bounded from \(H^{p_{1}}\) to weak \(L^{p_{1}}\) and from \(L^{p_{2}}\) to weak \(L^{p_{2}}\), then T is bounded on \(L^{\Phi}\).

Corollary 2.8

Let w be a classical or an anisotropic Muckenhoupt \(\mathbb {A}_{\infty}\) weight with \(q(w)\) being as in (2.3), Φ an Orlicz function with uniformly lower type \(p^{-}_{\Phi}\) and uniformly upper type \(p^{+}_{\Phi}\) with \([q(w)]^{2}< p^{-}_{\Phi}\), and \(p_{1}, p_{2}\in\mathbb{R}\) with \(0< p_{1}\le1\) and \(p^{+}_{\Phi}< p_{2}\le\infty \). If a sublinear operator T is bounded from \(H^{p_{1}}_{w}\) to weak \(L^{p_{1}}_{w}\) and from \(L^{p_{2}}_{w}\) to weak \(L^{p_{2}}_{w}\), then T is bounded on \(L^{\Phi}_{w}\).

It should be pointed that when \(p_{1}=1\) and \(p_{2}=\infty \), there also exists an interpolation theorem which differs from Theorem 2.5. It can be stated as follows.

Theorem 2.9

Suppose φ is a Musielak-Orlicz function satisfying \(\varphi (\cdot, t)\in\mathbb{A}_{1}(A)\) for any \(t\in(0,\infty )\) and Assumption  (φ)(ii) with \(p^{-}_{\varphi}\in (1,\infty )\). Assume that T is a sublinear operator defined on
$$H^{1}_{\varphi (\cdot, t)}(A)+\operatorname{BMO}(A) $$
satisfying that there exist positive constants \(C_{1}\) and \(C_{2}\) such that, for all \(\alpha\in(0,\infty )\) and \(t\in(0,\infty )\),
$$ \|Tf\|_{L^{1}_{\varphi (\cdot, t)}}\leq C_{1}\alpha^{-1}{ \|f\| ^{1}_{{H^{1}_{\varphi (\cdot, t)}(A)}}} $$
(2.10)
and
$$ \|Tf\|_{\operatorname{BMO}(A)}\leq C_{2}\|f\|_{\infty}. $$
(2.11)
Then T is bounded on \(L^{\varphi}\) and, moreover, there exists a positive constant C such that, for any \(f\in L^{\varphi}\),
$$\int_{\mathbb{R}^{n}}\varphi \bigl(x,\bigl\vert Tf(x)\bigr\vert \bigr) \,dx\leq C \int_{\mathbb {R}^{n}}\varphi \bigl(x,\bigl\vert f(x)\bigr\vert \bigr) \,dx. $$

Remark 2.10

For any \(x\in{{\mathbb{R}}^{n}}\) and \(t\in(0, \infty )\), if we let \(\varphi (x, t):=t^{p}w(x)\), \(\varphi (x, t):=\Phi(t)\) or \(\varphi (x, t):=\Phi(t)w(x)\), respectively, then, by Theorem 2.9, we may also obtain three corollaries similar to Corollaries 2.6, 2.7, and 2.8, respectively.

3 Proofs of Theorems 2.5 and 2.9

In order to prove Theorem 2.5, we begin with several lemmas. Lemma 3.1 is from [26], pp.7-8.

Lemma 3.1

Let \(q\in[1, \infty )\) and \(w\in\mathbb{A}_{q}(A)\). Then, for any measurable set \(E\subset B\in\mathcal{B}\), there exist two positive constants \(C_{1}\) and \(C_{2}\) such that
$$C_{1} \biggl(\frac{|B|}{|E|} \biggr)^{1/q}\le \frac{w(B)}{w(E)}\le C_{2} \biggl(\frac{|B|}{|E|} \biggr)^{q}. $$

The following lemma is similar to [9], Lemma 4.1(ii).

Lemma 3.2

Let φ be a Musielak-Orlicz function with uniformly lower type \(p^{-}_{\varphi}\) and uniformly upper type \(p^{+}_{\varphi}\), where \(0< p^{-}_{\varphi}\le p^{+}_{\varphi}<\infty \), and, for all \((x, t)\in{{\mathbb{R}}^{n}}\times[0, \infty )\),
$$\tilde{\varphi} (x, t):= \int^{t}_{0}\frac{\varphi (x, s)}{s} \,ds. $$
Then φ̃ is a Musielak-Orlicz function, which is equivalent to φ, moreover, \(\tilde{\varphi} (x,\cdot)\) is continuous and strictly increasing for all \(x\in{{\mathbb{R}}^{n}}\).

Proof

Since φ is a Musielak-Orlicz function, it is easy to see that, for all \(x\in{{\mathbb{R}}^{n}}\), the function \(\tilde{\varphi} (x, \cdot)\) is continuous and strictly increasing. Moreover, φ is of uniformly lower type \(p^{-}_{\varphi}\) with \(p^{-}_{\varphi}\in(0, \infty )\), then we have, for any \(x\in{{\mathbb{R}}^{n}}\),
$$\tilde{\varphi} (x, t)= \int^{t}_{0}\frac{\varphi (x, s)}{s} \,ds\le C \frac{\varphi (x,t)}{t^{p^{-}_{\varphi}}} \int^{t}_{0}\frac{1}{s^{1-p^{-}_{\varphi}}} \,ds\leq C\varphi (x,t), $$
where C is a positive constant as in (2.4).
On the other hand, since φ is of uniformly upper type \(p^{+}_{\varphi}\in (0, \infty )\), we get, for any \(x\in{{\mathbb{R}}^{n}}\),
$$\tilde{\varphi} (x, t)= \int^{t}_{0}\frac{\varphi (x, s)}{s} \,ds\geq C \frac{\varphi (x,t)}{t^{p^{+}_{\varphi}}} \int^{t}_{0}\frac{1}{s^{1-p^{+}_{\varphi}}} \,ds\geq C\varphi (x,t), $$
where C is a positive constant as in (2.4). This finishes the proof of Lemma 3.2. □

Remark 3.3

By Lemma 3.2, in the future, we always consider a Musielak-Orlicz function φ of uniformly lower type \(p^{-}_{\varphi}\) and of uniformly upper type \(p^{+}_{\varphi}\), and \(\varphi (x, \cdot)\) is continuous and strictly increasing for all \(x\in{{\mathbb{R}}^{n}}\).

By using Remark 3.3 and repeating the proofs of [9], Lemma 4.2(i), and [9], Lemma 4.3(i), we obtain the following lemma.

Lemma 3.4

Let φ be a Musielak-Orlicz function with uniformly lower type \(p^{-}_{\varphi}\) and uniformly upper type \(p^{+}_{\varphi}\), where \(0< p^{-}_{\varphi}\le p^{+}_{\varphi}<\infty \).
  1. (i)

    \(\int_{{\mathbb{R}}^{n}}\varphi (x,\frac {|f(x)|}{\|f\|_{L^{\varphi}}} )\,dx=1\) for all \(f\in L^{\varphi}\setminus\{\mathbf{0}\}\).

     
  2. (ii)
    Given c is a positive constant. Then there exists a positive constant C such that the inequality, for any \(t\in(0, \infty )\),
    $$\int_{{\mathbb{R}}^{n}}\varphi \biggl(x, \frac{|f(x)|}{t} \biggr)\,dx\leq c $$
    holds true implies that \(\|f\|_{L^{\varphi}}\leq Ct\).
     

Lemma 3.5

Let T be a sublinear operator. If there exists a positive constant C such that, for any \(f\in L^{\varphi}\),
$$ \int_{\mathbb{R}^{n}}\varphi \bigl(x,\bigl\vert Tf(x)\bigr\vert \bigr) \,dx\leq C \int_{\mathbb {R}^{n}}\varphi \bigl(x,\bigl\vert f(x)\bigr\vert \bigr) \,dx, $$
(3.1)
then T is bounded on \(L^{\varphi}\).

Proof

If \(f\in L^{\varphi}\setminus\{\mathbf{0}\}\), then we have \(f\|f\| ^{-1}_{L^{\varphi}}\in L^{\varphi}\). From this, the homogeneity property of sublinear operator T, (3.1) and Lemma 3.4(i), it follows that
$$\int_{{\mathbb{R}}^{n}}\varphi \biggl(x,\frac{|Tf(x)|}{\|f\| _{L^{\varphi}}} \biggr)\,dx= \int_{{\mathbb{R}}^{n}} \varphi \biggl(x,\biggl\vert T \biggl( \frac{f(x)}{\|f\|_{L^{\varphi}}} \biggr)\biggr\vert \biggr)\,dx\lesssim \int_{{\mathbb{R}}^{n}}\varphi \biggl(x,\frac{|f(x)|}{\|f\| _{L^{\varphi}}} \biggr)\,dx\sim1, $$
which, together with Lemma 3.4(ii), implies that
$$\|Tf\|_{L^{\varphi}}\lesssim\|f\|_{L^{\varphi}}. $$
This finishes the proof of Lemma 3.5. □

Lemma 3.6

Let φ be a Musielak-Orlicz function satisfying Assumption  (φ) and let \(q(\varphi )\) be as in (2.6) with \(q(\varphi )< p^{-}_{\varphi}<\infty \). If \(f\in L^{\varphi}\), then \(f\in L^{1}_{\mathrm{loc}}\).

Proof

For any \(f\in L^{\varphi}\), we only need to prove, for any \(B\in \mathcal{B}\),
$$\mathrm{I}:=\frac{1}{\vert B\vert } \int_{B} \bigl\vert f(x)\bigr\vert \,dx< \infty . $$
Let \(q(\varphi )\) be as in (2.6) and \(p^{-}_{\varphi}\in(q(\varphi ), \infty )\). For any \(t\in(0, \infty )\), since \(\varphi (\cdot, t)\in\mathbb{A}_{p^{-}_{\varphi}}(A)\), the Hölder inequality yields
$$\begin{aligned} \mathrm{I}&\lesssim \biggl(\frac{1}{\vert B\vert } \int_{B} \bigl\vert f(x)\bigr\vert ^{p^{-}_{\varphi}}\varphi (x,t)\,dx \biggr)^{\frac{1}{p^{-}_{\varphi}}} \biggl(\frac{1}{\vert B\vert } \int_{B} \bigl[\varphi(x,t)\bigr]^{-\frac{1}{p^{-}_{\varphi}-1}} \,dx \biggr)^{p^{-}_{\varphi}-1} \\ &\lesssim \biggl(\frac{1}{\varphi(B,t)} \int_{B} \bigl\vert f(x)\bigr\vert ^{p^{-}_{\varphi }} \varphi(x,t)\,dx \biggr)^{\frac{1}{p^{-}_{\varphi}}} \end{aligned}$$
and hence, it suffices to prove \(\int_{B} |f(x)|^{p^{-}_{\varphi }}\varphi (x,t)\,dx<\infty \). Notice that
$$\begin{aligned} \begin{aligned} \int_{B} \bigl\vert f(x)\bigr\vert ^{p^{-}_{\varphi }} \varphi(x,t)\,dx &= \int_{\{x\in B: \vert f(x)\vert >t\}} \bigl\vert f(x)\bigr\vert ^{p^{-}_{\varphi }}\varphi (x,t)\,dx+ \int _{\{x\in B: \vert f(x)\vert \leq t\}} \bigl\vert f(x)\bigr\vert ^{p^{-}_{\varphi }} \varphi(x,t)\,dx \\ &=: \mathrm{I}_{1}+\mathrm{I}_{2}. \end{aligned} \end{aligned}$$
For \(\mathrm{I}_{1}\), by \(t/|f(x)|< 1\), the uniformly lower type \(p^{-}_{\varphi}\) property of φ and \(f\in L^{\varphi}\), we obtain, for any \(t\in(0, \infty )\),
$$\begin{aligned} \mathrm{I}_{1}&\lesssim \int_{\{x\in B: \vert f(x)\vert >t\}} \bigl\vert f(x)\bigr\vert ^{p^{-}_{\varphi}} \biggl(\frac{t}{\vert f(x)\vert } \biggr)^{p^{-}_{\varphi}}\varphi\bigl(x,\bigl\vert f(x) \bigr\vert \bigr)\,dx \\ &\lesssim t^{p^{-}_{\varphi}} \int_{{\mathbb{R}}^{n}}\varphi \bigl(x,\bigl\vert f(x)\bigr\vert \bigr) \,dx< \infty . \end{aligned}$$

For \(\mathrm{I}_{2}\), by \(|f(x)|\le t\) and \(\varphi (\cdot, t)\in L^{1}_{\mathrm{loc}}\), we have \(\mathrm{I}_{2}\lesssim t^{p^{-}_{\varphi}}\varphi(B, t)<\infty \), which, together with \(\mathrm{I}_{1}<\infty \), we finally obtain \(\mathrm{I}<\infty \) and finish the proof of Lemma 3.6. □

For any locally integrable function f, the anisotropic Hardy-Littlewood maximal function \(\mathcal{M}_{A} f\) is defined by
$$\mathcal{M}_{A} f(x):= \sup_{x\in B\in\mathcal{B}} \frac{1}{\vert B\vert } \int _{B} \bigl\vert f(y)\bigr\vert \,dy, \quad x\in \mathbb{R}^{n}. $$
Let \(w\in\mathbb{A}_{\infty}(A)\). For any locally integrable function f with respect to measure \(w(x) \,dx\), the anisotropic weighted Hardy-Littlewood maximal function \(\mathcal{M}_{w} f\) is defined by
$$\mathcal{M}_{w} f(x):= \sup_{x\in B\in\mathcal{B}} \frac {1}{w(B)} \int_{B} \bigl\vert f(y)\bigr\vert w(y) \,dy,\quad x\in \mathbb{R}^{n}. $$
For any \(q\in(q(w),\infty )\) with \(q(w)\) as in (2.3), by [19], Theorem 2.8, we have the boundedness of \(\mathcal{M}_{w}\) from \(L^{q}_{w}\) to \(L^{q}_{w}\). From this and Lemma 3.2, with an argument similar to that of [28], Theorem 2.7 and Corollary 2.8, we deduce the following lemma, the details being omitted.

Lemma 3.7

Let φ be a Musielak-Orlicz function satisfying Assumption  (φ) with \({q(\varphi )}<{p^{-}_{\varphi}}\leq{p^{+}_{\varphi}}<{\infty }\), where \(q(\varphi )\) is as in (2.6). Then, for any \(t\in(0, \infty )\), the weighted Hardy-Littlewood maximal operator \(\mathcal{M}_{\varphi (\cdot, t)}\) is bounded on \(L^{\varphi}\) and, moreover, there exists a positive constant C such that, for all \(f\in L^{\varphi}\),
$$\int_{\mathbb{R}^{n}}\varphi \bigl(x,\mathcal{M}_{\varphi (\cdot , t)} f(x) \bigr)\,dx\leq C \int_{\mathbb{R}^{n}}\varphi \bigl(x,\bigl\vert f(x)\bigr\vert \bigr) \,dx. $$

The definition of weighted atomic anisotropic Hardy spaces is from [30], Definition 3.2.

Definition 3.8

Let \(w\in\mathbb{A}_{\infty}(A)\) and \(q(w)\) be as in (2.3). A triplet \((p,q,s)_{w}\) is called admissible, if \(p\in(0,1]\), \(q\in(q(w),\infty ]\), and \(s\in\mathbb{N}\) with \(s\geq\lfloor(q(w)/p-1)\ln b/\ln {(\lambda _{-})}\rfloor\). A function a on \({{\mathbb{R}}^{n}}\) is said to be a \((p,q,s)_{w}\)-atom if it satisfies the following three conditions:
  1. (i)

    \(\operatorname{supp}a\subset x_{0}+B_{j}\) for some \(j\in \mathbb{Z}\) and \(x_{0}\in{{\mathbb{R}}^{n}}\);

     
  2. (ii)

    \(\|a\|_{L^{q}_{w}}\leq[w(x_{0}+B_{j})]^{\frac{1}{q}-\frac{1}{p}}\);

     
  3. (iii)

    \(\int_{\mathbb{R}^{n}}a(x)x^{\alpha}\,dx=0\) for any multi-index α satisfying \(|\alpha|\leq s\).

     

Definition 3.9

Let \(w\in\mathbb{A}_{\infty}(A)\) and \((p, q, s)_{w}\) be an admissible triplet as in Definition 3.8. The weighted atomic anisotropic Hardy space \(H^{p,q,s}_{w}(A)\) is defined to be the set of all \(f\in\mathcal{S}'\) satisfying that \(f=\sum_{i}\lambda_{i}a_{i}\) in \(\mathcal{S}'\), where \(\{\lambda_{i}\}_{i}\subset \mathbb{C}\), \(\sum_{i}|\lambda_{i}|^{p}<\infty \), and \(\{a_{i}\}_{i}\) are \((p, q, s)_{w}\)-atoms. Moreover, the (quasi-)norm of \(f\in H^{p,q,s}_{w}(A)\) is defined by
$$\|f\|_{H^{p,q,s}_{w}(A)}:=\inf \biggl\{ \biggl[\sum_{i}| \lambda_{i}|^{p} \biggr]^{1/p} \biggr\} , $$
where the infimum is taken over all admissible decompositions of f as above.

Proof of Theorem 2.5

Let φ be a Musielak-Orlicz function, \(q(\varphi )\) as in (2.6), \(p_{2}\in(p^{+}_{\varphi},\infty )\), \(p^{-}_{\varphi}\in ([q(\varphi )]^{2},\infty )\), and \(q\in(q(\varphi ),\infty )\) close to \(q(\varphi )\) such that \(q(\varphi )< p^{-}_{\varphi}/q\). Suppose \(f\in L^{\varphi}\), for any \(t\in (0,\infty )\), define
$$ \mathcal{M}^{q}_{\varphi (\cdot, t)} f(x):=\sup _{x\in B\in\mathcal {B}} \biggl(\frac {1}{\varphi (B,t)} \int_{B} \bigl\vert f(y)\bigr\vert ^{q}\varphi (y,t)\,dy \biggr)^{\frac {1}{q}}, \quad x\in{{\mathbb{R}}^{n}}. $$
(3.2)
For any \(\alpha\in(0,\infty )\), let
$$\Omega_{\alpha}:= \bigl\{ \mathcal{M}^{q}_{\varphi (\cdot, t)} f> \alpha \bigr\} . $$
By an anisotropic variant of Whitney covering lemma in [30], Lemma 2.3, we know that there exist a positive constant L depending only on σ, a sequence \(\{ x_{i}\}_{i}\subset\Omega_{\alpha}\), and a sequence \(\{\ell_{i}\}_{i}\) of integers such that
$$\begin{aligned}& \Omega_{\alpha}= \bigcup_{i}(x_{i}+B_{\ell_{i}}) , \end{aligned}$$
(3.3)
$$\begin{aligned}& (x_{i}+B_{\ell_{i}-2\sigma })\cap(x_{j}+B_{\ell_{j}-2\sigma }) = \emptyset \quad \mbox{for all } i, j \mbox{ with } i\ne j , \end{aligned}$$
(3.4)
$$\begin{aligned}& \sharp\bigl\{ j: (x_{i}+B_{\ell_{i}+2\sigma })\cap(x_{j}+ B_{\ell_{j}+2\sigma })\ne\emptyset\bigr\} \leq L \quad \mbox{for all } i, \end{aligned}$$
(3.5)
where we denote by ♯E the cardinality of the set E and, for any i,
$$ (x_{i}+B_{\ell_{i}+4\sigma})\cap\Omega^{\complement}_{\alpha}= \emptyset \quad \mbox{and} \quad (x_{i}+B_{\ell_{i}+4\sigma+1})\cap\Omega ^{\complement}_{\alpha}\neq\emptyset. $$
(3.6)
For any i and any \(x\in{{\mathbb{R}}^{n}}\), let \(\chi_{i}(x):=\chi _{x_{i}+{B_{\ell _{i}}}}(x)\) and
$$\eta_{i}(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{\chi_{i}(x)}{\sum_{j}\chi_{j}(x)}, & \mbox{for } x \in\Omega _{\alpha}, \\ 0,& \mbox{for } x\in\Omega^{\complement}_{\alpha}. \end{array}\displaystyle \right . $$
Let \(s\in\mathbb{N}\) with \(s\ge m_{p_{1}, \varphi (\cdot, t)}\), where \(p_{1}\in (0,1]\) and \(m_{p_{1}, \varphi (\cdot, t)}\) is as in (2.7), and \(\mathcal{P}_{s}\) denote the linear space of polynomials of degrees not more than s. For any \(B\in\mathcal{B}\), let \(\pi_{B}: L^{1}(B)\to\mathcal{P}_{s}\) be the natural projection defined, via the Riesz lemma, by setting, for all \(\tilde{f}\in L^{1}(B)\) and \(Q\in\mathcal{P}_{s}\),
$$\int_{B} \pi_{B}\tilde{f}(x)Q(x) \,dx= \int_{B} \tilde{f}(x)Q(x) \,dx. $$
Then, by [10], (8.9), there exists a positive constant C, depending only on s, such that, for all \(B\in\mathcal {B}\) and \(\tilde{f}\in L^{1}(B)\),
$$ \sup_{x\in B}\bigl\vert \pi_{B} \tilde{f}(x)\bigr\vert \le C \frac{1}{\vert B\vert } \int_{B} \bigl\vert \tilde{f}(x)\bigr\vert \,dx. $$
(3.7)
Setting
$$g(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} f(x), & \mbox{for } x \in\Omega^{\complement}_{\alpha}, \\ \sum_{i}(\pi_{x_{i}+{B_{\ell_{i}}}}f\eta_{i})(x)\chi_{i}(x), & \mbox{for } x\in\Omega_{\alpha}, \end{array}\displaystyle \right . $$
and, for any \(x\in{{\mathbb{R}}^{n}}\),
$$h(x):=\sum_{i} h_{i}(x), \quad \mbox{where } h_{i}(x):=f(x)\chi_{i}(x)-(\pi _{x_{i}+{B_{\ell_{i}}}}f\eta_{i}) (x)\chi_{i}(x). $$
Obviously, for any \(x\in{{\mathbb{R}}^{n}}\), we have \(f(x)=g(x)+h(x)\).
By Hölder’s inequality and the definition of \(\mathbb{A}_{q}(A)\) with \(\varphi(\cdot, t)\in\mathbb{A}_{q}(A)\), we have, for any \(B\in\mathcal{B}\) and \(t\in(0, \infty )\),
$$ \frac{1}{\vert B\vert } \int_{B} \bigl\vert f(x)\bigr\vert \,dx\leq \biggl( \frac{1}{\varphi(B,t)} \int_{B} \bigl\vert f(x)\bigr\vert ^{q} \varphi(x,t)\,dx \biggr)^{\frac{1}{q}}. $$
(3.8)
Moreover, notice that \((x_{i}+B_{\ell_{i}+4\sigma+1})\cap\Omega ^{\complement}_{\alpha}\neq\emptyset\) (see (3.6)), then there exists some
$$ y_{0}\in(x_{i}+B_{\ell_{i}+4\sigma+1})\cap \Omega^{\complement}_{\alpha}. $$
(3.9)
Hence, for any \(x\in{{\mathbb{R}}^{n}}\) and \(t\in(0,\infty )\), by (3.7) with \(f\in L^{1}(B)\) (see Lemma 3.6 with \(f\in L^{\varphi}\)), (3.8), Lemma 3.1 with \(\varphi (\cdot, t)\in\mathbb{A}_{q}(A)\) and (3.9), we obtain
$$\begin{aligned} \bigl\vert h_{i}(x)\bigr\vert &\le\bigl\vert f(x)\bigr\vert +C \biggl(\frac{1}{\varphi (x_{i}+{B_{\ell_{i}},t})} \int _{x_{i}+{B_{\ell_{i}}}} \bigl\vert f(y)\bigr\vert ^{q}\varphi (y,t)\,dy \biggr)^{\frac{1}{q}}\chi _{i}(x) \\ &\le\bigl\vert f(x)\bigr\vert +C \biggl(\frac{1}{\varphi (x_{i}+B_{\ell_{i}+4\sigma +1},t)} \int _{x_{i}+B_{\ell_{i}+4\sigma+1}} \bigl\vert f(y)\bigr\vert ^{q}\varphi (y,t)\,dx \biggr)^{\frac {1}{q}}\chi _{i}(x) \\ &\le\bigl\vert f(x)\bigr\vert +C \mathcal{M}^{q}_{\varphi (\cdot, t)} f(y_{0})\chi_{i}(x) \\ &\le\bigl\vert f(x)\bigr\vert +C\alpha\chi_{i}(x). \end{aligned}$$
Therefore, from the above estimate, Minkowski’s inequality, Lemma 3.1 with \(\varphi (\cdot, t)\in\mathbb{A}_{q}(A)\), and (3.9) again, it follows that
$$\begin{aligned}& \biggl(\frac{1}{\varphi (x_{i}+B_{\ell_{i}},t)} \int_{x_{i}+B_{\ell_{i}}} \bigl\vert h_{i}(x)\bigr\vert ^{q}\varphi (x,t)\,dx \biggr)^{\frac{1}{q}} \\& \quad \leq{ \biggl(\frac{1}{\varphi (x_{i}+B_{\ell_{i}},t)} \int _{x_{i}+B_{\ell _{i}}} \bigl\vert f(x)\bigr\vert ^{q}\varphi (x,t)\,dx \biggr)^{\frac{1}{q}}}+C\alpha \\& \quad \le C \biggl(\frac{1}{\varphi (x_{i}+B_{\ell_{i}+4\sigma+1},t)} \int _{x_{i}+B_{\ell_{i}+4\sigma+1}} \bigl\vert f(x)\bigr\vert ^{q}\varphi (x,t)\,dx \biggr)^{\frac {1}{q}}+C\alpha \\& \quad \le C\alpha. \end{aligned}$$
(3.10)
For any \(p_{1}\in(0,1]\), \(x\in{{\mathbb{R}}^{n}}\), and \(t\in(0, \infty )\), set
$$a_{i}(x):=\frac{h_{i}(x)}{C\alpha[\varphi (x_{i}+B_{\ell_{i}}, t)]^{1/p_{1}}}, $$
where C is a constant as in (3.10), then \(a_{i}\) is a \((p_{1},q,s)_{\varphi {(\cdot,t)}}\)-atom with integer \(s\ge m_{p_{1}, \varphi (\cdot, t)}\), where \(m_{p_{1}, \varphi (\cdot, t)}\) is as in (2.7). Thus
$$h(x)=C\alpha\sum_{i}\bigl[\varphi (x_{i}+B_{\ell_{i}},t)\bigr]^{\frac {1}{p_{1}}}a_{i}(x)\in H^{p_{1},q,s}_{\varphi (\cdot, t)}(A) $$
and, by Lemma 3.1 with \(\varphi (\cdot, t)\in\mathbb {A}_{q}(A)\), (3.3) and (3.4), we obtain
$$ \|h\|_{H^{p_{1},q,s}_{\varphi (\cdot, t)}(A)}\leq C\alpha \biggl[\sum _{i}\varphi (x_{i}+B_{\ell_{i}-2\sigma },t) \biggr]^{\frac{1}{p_{1}}}\leq C\alpha \bigl[\varphi (\Omega _{\alpha},t)\bigr]^{\frac{1}{p_{1}}}. $$
(3.11)
Also, when \(x\in\Omega_{\alpha}\), for any \(t\in(0,\infty )\), by (3.5), (3.7), (3.8), Lemma 3.1 with \(\varphi(\cdot, t)\in\mathbb{A}_{q}(A)\), and (3.9) again, we have
$$ \bigl\vert g(x)\bigr\vert \leq C\alpha. $$
(3.12)
Moreover, for a.e. \(x\in\Omega^{\complement}_{\alpha}\),
$$\bigl\vert g(x)\bigr\vert =\bigl\vert f(x)\bigr\vert \leq \mathcal{M}^{q}_{\varphi (\cdot, t)} f(x)\leq \alpha. $$
Thus, we obtain, for a.e. \(x\in{{\mathbb{R}}^{n}}\),
$$ \bigl\vert g(x)\bigr\vert \leq C\alpha. $$
(3.13)

We prove the theorem by two cases.

Case (i): \(p_{2}\in(p^{+}_{\varphi}, \infty )\). In this case, by Lemma 3.2, Fubini’s theorem, and the fact that T is a sublinear operator, we further have
$$\begin{aligned} \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert Tf(x)\bigr\vert \bigr) \,dx \sim& \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\vert Tf\vert >\alpha\}}\varphi (x, \alpha ) \,dx\,d\alpha \\ \lesssim& \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\vert Th\vert >\frac{\alpha }{2}\}}\varphi (x, \alpha) \,dx\,d\alpha \\ &{} + \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\vert Tg\vert >\frac{\alpha}{2}\} }\varphi (x, \alpha) \,dx\,d\alpha \\ =:& \mathrm{I}+\mathrm{II}. \end{aligned}$$
By (2.8), \(H^{p_{1}}_{\varphi (\cdot, \alpha )}(A)=H^{p_{1},q,s}_{\varphi (\cdot, \alpha )}(A)\) with equivalent norms (see [30], Theorem 5.5), (3.11) and Lemma 3.2, we conclude that
$$\begin{aligned} \mathrm{I}&\lesssim \int_{0}^{\infty}\frac{1}{\alpha} \biggl( \frac{\|h\| _{{H^{p_{1},q,s}_{\varphi (\cdot,\alpha )}}(A)}}{\alpha} \biggr)^{p_{1}}\,d\alpha\lesssim \int _{0}^{\infty}\frac{1}{\alpha} \varphi ( \Omega_{\alpha},\alpha)\,d\alpha \\ &\sim \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\mathcal {M}^{q}_{\varphi (\cdot, t)}f>\alpha\}}\varphi (x, \alpha) \,dx\,d\alpha\sim \int_{{\mathbb {R}}^{n}}\varphi \bigl(x, \mathcal{M}^{q}_{\varphi (\cdot, t)} f(x) \bigr)\,dx. \end{aligned}$$
Let \(\ddot{\varphi} (x,t):=\varphi (x,t^{1/q})\). Obviously, \(\ddot{\varphi} (x,t)\) is uniformly lower type \(p^{-}_{\varphi}/q\). From this, the definitions of \(\mathcal{M}^{q}_{\varphi (\cdot, t)}f(x)\) and \(\mathcal{M}_{\varphi (\cdot, t)}(|f|^{q})(x)\), \(q(\varphi )< p^{-}_{\varphi}/q\) with \(q(\varphi )\) being as in (2.6) and Lemma 3.7 with \(q(\ddot{\varphi} )=q(\varphi )< p^{-}_{\varphi}/q\), it follows that
$$\begin{aligned} \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl(\mathcal{M}^{q}_{\varphi (\cdot, t)} f\bigr) (x) \bigr) \,dx &= \int_{{\mathbb{R}}^{n}}\ddot{\varphi} \bigl(x, \bigl[\mathcal {M}^{q}_{\varphi (\cdot, t)}f(x) \bigr]^{q} \bigr) \,dx \\ &= \int_{{\mathbb{R}}^{n}}\ddot{\varphi} \bigl(x, \mathcal {M}_{\varphi (\cdot, t)} \bigl(\vert f\vert ^{q}\bigr) (x) \bigr) \,dx \\ &\lesssim \int_{{\mathbb{R}}^{n}}\ddot{\varphi} \bigl(x, \bigl\vert f(x)\bigr\vert ^{q} \bigr) \,dx \\ &\lesssim \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert f(x)\bigr\vert \bigr) \,dx. \end{aligned}$$
(3.14)
By (2.9), we obtain
$$\begin{aligned} \mathrm{II} &\lesssim \int_{0}^{\infty}\frac{1}{\alpha^{1+p_{2}}} \int_{{\mathbb {R}}^{n}}\bigl\vert g(x)\bigr\vert ^{p_{2}}\varphi (x, \alpha)\,dx\,d\alpha \\ &\sim \int_{0}^{\infty}\frac{1}{\alpha^{1+p_{2}}} \int_{\Omega_{\alpha}}\bigl\vert g(x)\bigr\vert ^{p_{2}}\varphi (x, \alpha) \,dx\,d\alpha+ \int_{0}^{\infty}\frac {1}{\alpha^{1+p_{2}}} \int_{\Omega^{\complement}_{\alpha}}\bigl\vert g(x)\bigr\vert ^{p_{2}}\varphi (x, \alpha) \,dx\,d\alpha \\ &=: \mathrm{II}_{1}+\mathrm{II}_{2}. \end{aligned}$$
By (3.13), Lemma 3.2, and (3.14), we conclude that
$$\begin{aligned} \mathrm{II}_{1}&\lesssim \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\mathcal {M}^{q}_{\varphi (\cdot, t)}f>\alpha\}}\varphi (x, \alpha) \,dx\,d\alpha \\ &\sim \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \mathcal {M}^{q}_{\varphi (\cdot, t)}f(x) \bigr)\,dx\lesssim \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert f(x)\bigr\vert \bigr) \,dx. \end{aligned}$$
Moreover, by Fubini’s theorem, \(|g(x)|=|f(x)|\leq \mathcal {M}^{q}_{\varphi (\cdot , t)}f(x)\) for \(x\in\Omega^{\complement}_{\alpha}\), and the uniformly upper type \(p^{+}_{\varphi}\) property of φ with \(p^{+}_{\varphi}< p_{2}\), we have
$$\begin{aligned} \mathrm{II}_{2}&\sim \int_{0}^{\infty}\frac{1}{\alpha^{1+p_{2}}} \int_{\{ \mathcal{M} ^{q}_{\varphi (\cdot, t)}f\leq\alpha\}} \bigl\vert f(x)\bigr\vert ^{p_{2}}\varphi (x, \alpha)\,dx \,d\alpha \\ &\sim \int_{{\mathbb{R}}^{n}}\bigl\vert f(x)\bigr\vert ^{p_{2}} \int_{\mathcal {M}^{q}_{\varphi (\cdot, t)} f(x)}^{\infty}\frac{1}{\alpha^{1+p_{2}}}\varphi (x, \alpha) \,d\alpha \,dx \\ &\lesssim \int_{{\mathbb{R}}^{n}}\bigl\vert f(x)\bigr\vert ^{p_{2}} \int_{\vert f(x)\vert }^{\infty}\frac{1}{\alpha ^{1+p_{2}}}\varphi (x, \alpha) \,d\alpha \,dx \\ &\lesssim \int_{{\mathbb{R}}^{n}}\bigl\vert f(x)\bigr\vert ^{p_{2}} \int_{\vert f(x)\vert }^{\infty}\frac{1}{\alpha ^{1+p_{2}}} \biggl( \frac{\alpha}{ \vert f(x)\vert } \biggr)^{p^{+}_{\varphi}}\varphi \bigl(x, \bigl\vert f(x)\bigr\vert \bigr)\,d\alpha \,dx \\ &\lesssim \int_{{\mathbb{R}}^{n}}\bigl\vert f(x)\bigr\vert ^{p_{2}-p^{+}_{\varphi}} \varphi \bigl(x, \bigl\vert f(x)\bigr\vert \bigr) \int _{\vert f(x)\vert }^{\infty}\alpha^{p^{+}_{\varphi}-{p_{2}}-1}\,d\alpha \,dx \\ &\sim \int_{\mathbb{R}^{n}}\varphi \bigl(x,\bigl\vert f(x)\bigr\vert \bigr) \,dx, \end{aligned}$$
as desired.
Case (ii): \(p_{2}=\infty \). In this case, obviously we have \(\|Tg\| _{\infty}\le C\|g\|_{\infty}\) (see (2.9) with the usual modification). From this and (3.13), it follows that
$$ \|Tg\|_{\infty}\le C\|g\|_{\infty}\le C\alpha . $$
(3.15)
By Lemma 3.2, Fubini’s theorem, Assumption (φ)(ii), and the fact that T is a sublinear operator, we further have
$$\begin{aligned} \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert Tf(x)\bigr\vert \bigr) \,dx \sim& \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\vert Tf\vert >\alpha\} }\varphi (x, \alpha) \,dx\,d\alpha \\ \lesssim& \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\vert Tf\vert >2C\alpha\} }\varphi (x, \alpha) \,dx\,d\alpha \\ \lesssim& \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\vert Th\vert >C\alpha\} }\varphi (x, \alpha) \,dx\,d\alpha \\ &{} + \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\vert Tg\vert >C\alpha\}}\varphi (x, \alpha) \,dx\,d\alpha \\ =:& \mathrm{I}+\mathrm{II}. \end{aligned}$$

For I, similar to the estimate of I of Case (i), we obtain \(\mathrm{I}\lesssim \int_{{\mathbb{R}}^{n}}\varphi (x, |f(x)|) \,dx\).

For II, by (3.15), we see that
$$\mathrm{II}\lesssim \int_{0}^{\infty}\frac{1}{\alpha} \int_{\{\|Tg\| _{\infty}>C\alpha \}}\varphi (x, \alpha) \,dx\,d\alpha=0\le \int_{{\mathbb {R}}^{n}}\varphi \bigl(x, \bigl\vert f(x)\bigr\vert \bigr) \,dx, $$
as desired.
Using the above estimates of Case (i) and Case (ii), we have, for any \(f\in L^{\varphi}\),
$$\int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert Tf(x)\bigr\vert \bigr) \,dx\lesssim \int _{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert f(x)\bigr\vert \bigr) \,dx, $$
which, together with Lemma 3.5, implies that T is bounded on \(L^{\varphi}\). This finishes the proof of Theorem 2.5. □

In order to prove Theorem 2.9, we need some lemmas.

Let
$$L^{\infty}_{0}:= \biggl\{ f\in L^{\infty}_{c}: \int_{{\mathbb {R}}^{n}}f(x)\,dx=0 \biggr\} , $$
where \(L^{\infty}_{c}\) is the space of bounded measurable functions with compact supports.

Lemma 3.10

Let φ be a Musielak-Orlicz function satisfying Assumption  (φ) with \(q(\varphi )<{p^{-}_{\varphi}}\leq{p^{+}_{\varphi}}<{\infty }\), where \(q(\varphi )\) is as in (2.6). Then \(L^{\infty}_{0}\) is dense in \(L^{\varphi}\).

Proof

Let φ be a Musielak-Orlicz function satisfying Assumption (φ) with \(q(\varphi )<{p^{-}_{\varphi}}\leq{p^{+}_{\varphi}}<{\infty }\), where \(q(\varphi )\) is as in (2.6). First, we prove that \(L^{\infty}_{c}\) is dense in \(L^{\varphi}\). For any \(f\in L^{\varphi}\), \(j\in\mathbb{Z}_{+}\) and \(x\in{{\mathbb {R}}^{n}}\), let \(f_{j}(x):=f(x)\chi_{B_{j}}(x)\chi_{\{|f|< j\}}(x)\). Obviously, we have \(|f_{j}(x)-f(x)|\rightarrow0\) (\(j\rightarrow\infty \)) for a.e. \(x\in{{\mathbb{R}}^{n}}\). From this, \(\varphi (x, \cdot)\) is continuous and strictly increasing for all \(x\in{{\mathbb{R}}^{n}}\) (see Remark 3.3), \(\varphi (x, |f_{j}(x)-f(x)|)\) decreasingly converges to 0 as \(j\rightarrow\infty \) for a.e. \(x\in {{\mathbb{R}}^{n}}\) and Levi’s theorem, it follows that
$$\int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert f_{j}(x)-f(x) \bigr\vert \bigr)\,dx\rightarrow0\quad (j\rightarrow\infty ), $$
which implies that, \(L^{\infty}_{c}\) is dense in \(L^{\varphi}\).
It remains to prove that \(L^{\infty}_{0}\) is dense in \(L^{\infty}_{c}\) with respect to the norm \(\|\cdot\|_{L^{\varphi}}\). For any \(f\in L^{\infty}_{c}\) with \(\operatorname{supp} f\subset B_{N}\), where N is some integer, integer \(j\geq N\) and \(x\in {{\mathbb{R}}^{n}}\), let
$$f_{j}(x):= \biggl(f(x)-\frac{1}{|B_{j}|} \int_{B_{j}}f(y)\,dy \biggr)\chi_{B_{j}}(x). $$
Obviously, \(\{f_{j}\}^{\infty }_{j=N}\subset L^{\infty}_{0}\). Thus, for any \(x\in{{\mathbb{R}}^{n}}\), we only need to prove
$$\int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert f_{j}(x)-f(x) \bigr\vert \bigr)\,dx\rightarrow0\quad (j\rightarrow\infty ). $$
By \(\operatorname{supp}f\subset B_{N}\subset B_{j}\), \(1/|B_{j}|=1/b^{j}<1\) for sufficient large j (since \(1/b^{j}\rightarrow0\) as \(j\rightarrow\infty \)), the uniformly lower type \(p^{-}_{\varphi}\) property of φ with \(q(\varphi )< p^{-}_{\varphi}\), Lemma 3.1, and \(\varphi (\cdot, t)\in \mathbb{A}_{q}(A)\) with some \(q\in(q(\varphi ), p^{-}_{\varphi})\) for any \(t\in(0, \infty )\), it follows that
$$\begin{aligned}& \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert f(x)-f_{j}(x) \bigr\vert \bigr)\,dx \\& \quad = \int _{{\mathbb{R}}^{n}}\varphi \biggl(x, \biggl\vert f(x) \chi_{B_{j}}(x)- \biggl(f(x)\chi_{B_{j}}(x)-\frac {1}{|B_{j}|} \int _{B_{j}}f(y)\,dy\chi_{B_{j}}(x) \biggr)\biggr\vert \biggr)\,dx \\& \quad \le \int_{B_{j}}\varphi \biggl(x, \frac{1}{|B_{j}|} \int _{B_{N}}\bigl\vert f(y)\bigr\vert \,dy \biggr)\,dx \lesssim \frac{1}{|B_{j}|^{p^{-}_{\varphi}}}\varphi \biggl(B_{j}, \int_{B_{N}}\bigl\vert f(y)\bigr\vert \,dy \biggr) \\& \quad \lesssim b^{-j(p^{-}_{\varphi}-q)}\varphi \bigl(B_{0}, \Vert f\Vert _{\infty}|B_{N}|\bigr)\rightarrow0\quad (j\rightarrow\infty ). \end{aligned}$$
This finishes the proof of Lemma 3.10. □

Since \(({{\mathbb{R}}^{n}},\rho,dx)\) is a space of homogeneous type in the sense of Coifman and Weiss [33, 34]. On such homogeneous spaces, the following lemma provides an analog of the grid of Euclidean dyadic cubes, which comes from [38], Lemma 2.3; see also [32].

Lemma 3.11

Let A be a dilation. There exists a collection \(\mathcal{Q}:=\{ Q^{k}_{\alpha}\subset{{\mathbb{R}}^{n}}: k\in\mathbb{Z}, \alpha\in I_{k}\}\) of open subsets, where \(I_{k}\) is certain index set, such that
  1. (i)

    \(|{{\mathbb{R}}^{n}}\setminus\bigcup_{\alpha}Q^{k}_{\alpha}|=0\) for each fixed k and \(Q^{k}_{\alpha}\cap Q^{k}_{\beta}=\emptyset\) if \(\alpha\neq \beta\);

     
  2. (ii)

    for any α, β, k, with \(\ell\ge k\), either \(Q^{k}_{\alpha}\cap Q^{\ell}_{\beta}=\emptyset\) or \(Q^{\ell}_{\alpha}\subset Q^{k}_{\beta}\);

     
  3. (iii)

    for each \((\ell,\beta)\) and each \(k<\ell\) there exists a unique α such that \(Q^{\ell}_{\beta}\subset Q^{k}_{\alpha}\);

     
  4. (iv)

    there exist a certain negative integer v and a positive integer u such that, for all \(Q^{k}_{\alpha}\) with \(k\in \mathbb{Z}\) and \(\alpha\in I_{k}\), there exists \(x_{Q^{k}_{\alpha}}\in Q^{k}_{\alpha}\) satisfying that, for any \(x\in Q^{k}_{\alpha}\), \(x_{Q^{k}_{\alpha}}+B_{vk-u}\subset Q^{k}_{\alpha}\subset x+B_{vk+u}\).

     

In the following, for convenience, we call \(\{Q^{k}_{\alpha}\}_{k\in \mathbb{Z} , \alpha\in I_{k}}\) in Lemma 3.11 dyadic cubes, k the level of the dyadic cube \(Q^{k}_{\alpha}\) with \(k\in\mathbb{Z}\) and \(\alpha \in I_{k}\) and we denote it by \(\ell(Q^{k}_{\alpha})\).

Now we recall the definition of the dyadic maximal function. For any given measurable function \(f\in L^{1}_{\mathrm{loc}}\) and \(x\in{{\mathbb {R}}^{n}}\), we define the dyadic maximal function by \(\mathcal{M}_{d} f(x):=\sup_{k\in \mathbb{Z}}E_{k} f(x)\), where
$$E_{k}f(x):=\sum_{Q\in\mathcal{Q}_{k}} \biggl( \frac{1}{\vert Q\vert } \int _{Q}\bigl\vert f(y)\bigr\vert \,dy \biggr) \chi_{Q}(x) $$
and \(\mathcal{Q}_{k}:=\{Q^{k}_{\alpha}: \alpha\in I_{k}\}\) denotes the set of dyadic cubes as in Lemma 3.11. The following lemma provides the Calderón-Zygmund decomposition in our setting with a non-typical assumption on f instead of the usual \(f\in L^{1}\), which is from [38], Proposition A.5.

Lemma 3.12

Given a measurable function \(f\in L^{p}_{w}\) for certain \(p\in[1,\infty )\) and \(w\in\mathbb{A}_{p}(A)\), and a positive number λ, then exists a sequence \(\{Q_{j}\}_{j}\subset\mathcal{Q}\) of disjoint dyadic cubes such that
  1. (i)

    \(\bigcup_{j}Q_{j}=\{\mathcal{M}_{d}f>\lambda\}\);

     
  2. (ii)

    \(|f(x)|<\lambda\) for almost every \(x\notin\bigcup_{j}Q_{j}\);

     
  3. (iii)

    \(\lambda<\frac{1}{|Q_{j}|}\int_{Q_{j}}|f(x)|\,dx\le C\lambda\), where \(C>1\) is a constant independent of f and λ.

     

Let \(\mu_{f}(\lambda):=\sum_{j} w(Q^{\lambda}_{j})\), where \(Q^{\lambda}_{j}:=Q_{j}\) is as in Lemma 3.12. We also need the following two lemmas.

Lemma 3.13

If \(f\in L^{1}_{w}\) with \(w\in\mathbb{A}_{1}(A)\). Then, for any \(\lambda \in (0,\infty )\), there exists a positive constant C such that
$$w\bigl(\bigl\{ \mathcal{M}f>\bigl(b^{3\sigma}C_{1}+1\bigr)\lambda \bigr\} \bigr)\le C\mu_{f}(\lambda), $$
where \(C_{1}\) is a positive constant as in Lemma  3.12(iii).

Proof

Let us first prove the inequality
$$ \mathcal{M}f(x)\le\bigl(b^{3\sigma}C_{1}+1\bigr) \lambda\quad \mbox{if } x\in\biggl(\bigcup_{j} x_{Q_{j}}+B_{\ell(Q_{j})v+u+2\sigma}\biggr)^{\complement}, $$
(3.16)
where \(x_{Q_{j}}+B_{\ell(Q_{j})v+u+2\sigma}\) is the dilated ball containing \(Q_{j}\) as in Lemma 3.11(iv). Pick any dilated ball \(y+B_{l}\) such that \(x\in y+B_{l}\). When \(y+B_{l}\subseteq(\bigcup_{j} Q_{j})^{\complement}\), by Lemma 3.12(ii), we have
$$ \frac{1}{|y+B_{l}|} \int_{y+B_{l}}\bigl\vert f(z)\bigr\vert \,dz\le\lambda . $$
(3.17)
In the following, we assume that \(y+B_{l}\nsubseteq(\bigcup_{j} Q_{j})^{\complement}\), and therefore, \((y+B_{l})\cap(\bigcup_{j} Q_{j})\neq \emptyset\). Thus, there exists at least one j such that \((y+B_{l})\cap (x_{Q_{j}}+B_{\ell(Q_{j})v+u})\neq\emptyset\). So there exists some \(\tilde{x}\in(y+B_{l})\cap(x_{Q_{j}}+B_{\ell(Q_{j})v+u})\). By \(\tilde{x}\in y+B_{l}\), we have \(y\in\tilde{x}+B_{l}\). From this, \(\tilde{x}\in x_{Q_{j}}+B_{\ell(Q_{j})v+u}\), and (2.1), it follows that
$$ x\in y+B_{l}\subset\tilde{x}+B_{l}+B_{l} \subset x_{Q_{j}}+B_{\ell (Q_{j})v+u}+B_{l+\sigma}. $$
(3.18)
If \(l+\sigma\le\ell(Q_{j})v+u\), we obtain \(x\in x_{Q_{j}}+B_{\ell (Q_{j})v+u+\sigma}\), which contradicts with \(x\notin\bigcup_{j} x_{Q_{j}}+B_{\ell(Q_{j})v+u+2\sigma}\). Therefore, we have \(l+\sigma>\ell (Q_{j})v+u\). By this and (3.18), we obtain
$$ x_{Q_{j}}+B_{\ell(Q_{j})v+u}\subset x_{Q_{j}}+B_{l+\sigma} $$
(3.19)
and \(x\in x_{Q_{j}}+B_{l+2\sigma}\). From this and (2.1), we conclude that \(x_{Q_{j}}\in x+B_{l+2\sigma}\) and
$$ x_{Q_{j}}+B_{l+\sigma}\subset x+B_{l+2\sigma}+B_{l+\sigma} \subset x+B_{l+3\sigma}. $$
(3.20)
By \(Q_{j}\subset x_{Q_{j}}+B_{\ell(Q_{j})v+u}\) (see Lemma 3.11(iv)), (3.19), and (3.20), we have
$$\bigcup_{j: {Q_{j}\cap(y+B_{l})\neq\emptyset}} Q_{j}\subset x+B_{l+3\sigma}. $$
Using this, (ii) and (iii) of Lemma 3.12, and the disjoint property of \(\{Q_{j}\}_{j}\), we find that
$$\begin{aligned} \int_{y+B_{l}}\bigl\vert f(z)\bigr\vert \,dz&= \int_{(y+B_{l})\cap(\bigcup_{j} Q_{j})^{\complement}}\bigl\vert f(z)\bigr\vert \,dz+ \int_{(y+B_{l})\cap(\bigcup_{j} Q_{j})}\bigl\vert f(z)\bigr\vert \,dz \\ &\le\lambda b^{l}+\sum_{j: Q_{j}\cap( y+B_{l})\neq\emptyset }C_{1} \lambda \vert Q_{j}\vert \\ &=\lambda b^{l}+C_{1}\lambda\biggl\vert \bigcup _{j: {Q_{j}\cap(y+B_{l})\neq \emptyset}}Q_{j}\biggr\vert \\ &\le\lambda b^{l}+C_{1}\lambda b^{l+3\sigma} = \bigl(b^{3\sigma}C_{1}+1\bigr)\lambda b^{l}, \end{aligned}$$
where \(C_{1}\) is a positive constant as in Lemma 3.12(iii). That is,
$$ \frac{1}{|y+B_{l}|} \int_{y+B_{l}}f(z)\,dz\le\bigl(b^{3\sigma}C_{1}+1 \bigr)\lambda. $$
(3.21)
Combining the estimates (3.17) and (3.21), we conclude that (3.16) holds true. Then, by (3.16), Lemma 3.11(iv), and Lemma 3.1 with \(w\in\mathbb{A}_{1}(A)\), we obtain
$$\begin{aligned}& w\bigl(\bigl\{ \mathcal{M}f>\bigl(b^{3\sigma}C_{1}+1\bigr)\lambda \bigr\} \bigr) \\& \quad =w\biggl(\biggl\{ x\in\biggl(\bigcup_{j} x_{Q_{j}}+B_{\ell(Q_{j})v+u+2\sigma }\biggr)^{\complement}: \mathcal{M}f(x)> \bigl(b^{3\sigma}C_{1}+1\bigr)\lambda\biggr\} \biggr) \\& \qquad {}+w\biggl(\biggl\{ x\in\biggl(\bigcup_{j} x_{Q_{j}}+B_{\ell(Q_{j})v+u+2\sigma}\biggr): \mathcal{M} f(x)> \bigl(b^{3\sigma}C_{1}+1\bigr)\lambda\biggr\} \biggr) \\& \quad =w\biggl(\biggl\{ x\in\biggl(\bigcup_{j} x_{Q_{j}}+B_{\ell(Q_{j})v+u+2\sigma}\biggr): \mathcal{M} f(x)> \bigl(b^{3\sigma}C_{1}+1\bigr)\lambda\biggr\} \biggr) \\& \quad \le w\biggl(\bigcup_{j} x_{Q_{j}}+B_{\ell(Q_{j})v+u+2\sigma} \biggr)\le\sum_{j} w(x_{Q_{j}}+B_{\ell(Q_{j})v+u+2\sigma}) \\& \quad \lesssim b^{2\sigma+2u}\sum_{j} w(x_{Q_{j}}+B_{\ell (Q_{j})v-u})\lesssim \sum_{j} w(Q_{j})=\mu_{f}(\lambda). \end{aligned}$$
This finishes the proof of Lemma 3.13. □

Lemma 3.14

Suppose that \(f\in L^{1}_{w}\) with \(w\in\mathbb{A}_{1}(A)\). Then, for any \(\alpha \in(0,\infty )\) and \(\beta\in(0, \infty )\) satisfying \(\beta b^{2u}<1/2\), there exists a positive constant C such that
$$\mu_{f}(\alpha )\le w\bigl(\bigl\{ \mathcal{M}^{\sharp}f>\alpha \beta/2\bigr\} \bigr)+\frac {\beta b^{2u}}{1-\beta b^{2u}}C\mu_{f}\bigl(2^{-1}C^{-1}_{1} \alpha\bigr), $$
where \(C_{1}\) is a positive constant as in Lemma  3.12(iii).

Proof

Let \(\nu:=2^{-1}C^{-1}_{1}\alpha\), and make the Calderón-Zygmund decompositions of f with the heights ν and α, respectively (see Lemma 3.12). Since \(\nu<\alpha\), from the proof of Lemma 3.12, we see that each cube \(Q_{k}^{\alpha}\) in Calderón-Zygmund decomposition with the height α must be included in some cube \(Q_{j}^{\nu}\) in Calderón-Zygmund decomposition with the height ν. Thus, by letting
$$\mathrm{I}:=\bigl\{ Q_{j}^{\nu}: j\in\mathrm{J}\bigr\} ,\quad \mbox{where } \mathrm {J}:=\bigl\{ j: Q_{j}^{\nu}\subset\bigl\{ \mathcal{M}^{\sharp}f>\alpha\beta/2\bigr\} \bigr\} $$
and
$$\mathrm{II}:=\bigl\{ Q_{j}^{\nu}: j\notin\mathrm{J}\bigr\} , $$
we have
$$\mu_{f}(\alpha)=\sum_{k} w \bigl(Q_{k}^{\alpha}\bigr)=\sum_{Q\in\mathrm{I}} \sum_{Q_{k}^{\alpha}\subset Q}w\bigl(Q_{k}^{\alpha}\bigr)+\sum_{Q\in\mathrm{II}}\sum_{Q_{k}^{\alpha}\subset Q}w \bigl(Q_{k}^{\alpha}\bigr). $$
For simplicity, for any dyadic cube Q, let \(B^{Q}:=x_{Q}+B_{\ell(Q)v+u}\) and \(B_{Q}:=x_{Q}+B_{\ell(Q)v-u}\).
On the one hand, when \(Q\in\mathrm{II}\), there exists some \(x\in Q\) such that
$$\mathcal{M}^{\sharp}f(x)\le\alpha\beta/2. $$
From this and \(Q\subset B^{Q}\) (see Lemma 3.11(iv)), it follows that
$$ \frac{1}{\vert B^{Q}\vert } \int_{B^{Q}}\bigl\vert f(y)-f_{B^{Q}}\bigr\vert \,dy \le\alpha\beta/2. $$
(3.22)
Moreover, by Lemma 3.12(iii), we conclude that
$$ |f_{Q}|\le C_{1}\nu=\alpha/2. $$
(3.23)
Thus, by Lemma 3.12(iii), the disjoint property of \(\{Q^{\alpha}_{k}\}_{k}\), (3.23), Lemma 3.11(iv), and (3.22), we have
$$\begin{aligned} \sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert < &\frac{1}{\alpha}\sum_{Q_{k}^{\alpha}\subset Q} \int_{Q_{k}^{\alpha}}\bigl\vert f(y)\bigr\vert \,dy \\ \le&\frac{1}{\alpha}\sum_{Q_{k}^{\alpha}\subset Q} \int _{Q_{k}^{\alpha}}\bigl\vert f(y)-f_{B^{Q}}\bigr\vert \,dy \\ &{}+\frac{1}{\alpha}\sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert \vert f_{B^{Q}}-f_{Q} \vert +\frac{1}{\alpha} \vert f_{Q}\vert \sum _{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert \\ \le&\frac{1}{\alpha} \int_{\bigcup_{Q_{k}^{\alpha}\subset Q}Q_{k}^{\alpha}}\bigl\vert f(y)-f_{B^{Q}}\bigr\vert \,dy \\ &{}+ \frac{1}{\alpha}\sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert \frac{1}{\vert Q\vert } \int_{Q}\bigl\vert f(y)-f_{B^{Q}}\bigr\vert \,dy+ \frac {1}{2}\sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert \\ \le&\frac{1}{\alpha} \int_{B^{Q}}\bigl\vert f(y)-f_{B^{Q}}\bigr\vert \,dy \\ &{}+ \frac{1}{\alpha }\sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert \frac{\vert B^{Q}\vert }{\vert Q\vert }\frac {1}{\vert B^{Q}\vert } \int_{B^{Q}}\bigl\vert f(y)-f_{B^{Q}}\bigr\vert \,dy+ \frac{1}{2}\sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert \\ \le&\frac{\beta}{2}\bigl\vert B^{Q}\bigr\vert + \frac{\beta}{2}\sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert \frac{\vert B^{Q}\vert }{\vert B_{Q}\vert }+\frac{1}{2} \sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert . \end{aligned}$$
From this, it follows that
$$ \sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert < \beta\bigl\vert B^{Q} \bigr\vert +\beta b^{2u}\sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert . $$
(3.24)
Choose β small enough such that \(\beta b^{2u}<1/2\). By this, the disjoint property of \(\{Q^{\alpha}_{k}\}_{k}\), (3.24), and Lemma 3.11(iv), we further have
$$ \biggl\vert \bigcup_{Q^{\alpha}_{k}\subset Q}Q^{\alpha}_{k} \biggr\vert =\sum_{Q_{k}^{\alpha}\subset Q}\bigl\vert Q_{k}^{\alpha}\bigr\vert < \frac{\beta}{1-\beta b^{2u}}\bigl\vert B^{Q}\bigr\vert \le\frac {\beta b^{2u}}{1-\beta b^{2u}}\vert Q\vert . $$
(3.25)
Moreover, by \(Q\subset B^{Q}\) (see Lemma 3.11(iv)), we obtain \(\bigcup_{Q^{\alpha}_{k}\subset Q}Q^{\alpha}_{k}\subset B^{Q}\). Therefore, by Lemma 3.11(iv) and Lemma 3.1 with \(w\in \mathbb{A}_{1}(A)\), we obtain
$$\frac{w(\bigcup_{Q^{\alpha}_{k}\subset Q}Q^{\alpha}_{k})}{w(Q)}\lesssim \frac {w(\bigcup_{Q^{\alpha}_{k}\subset Q}Q^{\alpha}_{k})}{w(B_{Q})}\lesssim \frac {w(\bigcup_{Q^{\alpha}_{k}\subset Q}Q^{\alpha}_{k})}{b^{-2u}w(B^{Q})}\lesssim \frac {|\bigcup_{Q^{\alpha}_{k}\subset Q}Q^{\alpha}_{k}|}{|Q|}. $$
From this, the disjoint property of \(\{Q^{\alpha}_{k}\}_{k}\) again, and (3.25), we deduce that
$$\sum_{Q_{k}^{\alpha}\subset Q}w\bigl(Q_{k}^{\alpha}\bigr)=w \biggl(\bigcup_{Q^{\alpha}_{k}\subset Q}Q^{\alpha}_{k} \biggr)\lesssim \frac{|\bigcup_{Q_{k}^{\alpha}\subset Q}Q_{k}^{\alpha}|}{|Q|}w(Q)\lesssim \frac{\beta b^{2u}}{1-\beta b^{2u}}w(Q). $$
Therefore, we have
$$ \sum_{Q\in\mathrm{II}}\sum _{Q_{k}^{\alpha}\subset Q}w\bigl(Q_{k}^{\alpha}\bigr)\lesssim \frac{\beta b^{2u}}{1-\beta b^{2u}}\sum_{Q\in\mathrm {II}}w(Q)\lesssim \frac{\beta b^{2u}}{1-\beta b^{2u}}\mu_{f}(\nu). $$
(3.26)
On the other hand,
$$\sum_{Q\in\mathrm{I}}\sum_{Q_{k}^{\alpha}\subset Q}w \bigl(Q_{k}^{\alpha}\bigr)\le \sum_{Q\in\mathrm{I}}w(Q) \le w\bigl(\bigl\{ \mathcal{M}^{\sharp}f>\alpha\beta /2\bigr\} \bigr), $$
by which, together with (3.26), we finally finish the proof of Lemma 3.14. □

In the following, for any \(\lambda\in(0, \infty )\), let \(\mu _{f}(\lambda , \lambda):=\sum_{j} \varphi (Q^{\lambda}_{j}, \lambda)\), where \(Q^{\lambda}_{j}:=Q_{j}\) is as in Lemma 3.12.

Proof of Theorem 2.9

Let φ be a Musielak-Orlicz function satisfying \(\varphi (\cdot, t)\in\mathbb{A}_{1}(A)\) for any \(t\in(0,\infty )\) and Assumption (φ)(ii) with \(p^{-}_{\varphi}\in (1,\infty )\) and \(q\in (1,\infty )\) close to 1 such that \(1< p^{-}_{\varphi}/q\). For any \(f\in L^{\infty}_{0}\), f is a multiple of \((1,\infty ,0)_{\varphi (\cdot, t)}\)-atom. From the assumption of Theorem 2.9, it follows that \(Tf\in L^{1}_{\varphi (\cdot, t)}\). Therefore, for any \(\lambda\in(0,\infty )\), by \(|f(x)|\le\mathcal{M}_{A} f(x)\) for any a.e. \(x\in{{\mathbb{R}}^{n}}\), Lemma 3.2, Assumption (φ)(ii), and Lemma 3.13, we have
$$\begin{aligned}& \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert Tf(x)\bigr\vert \bigr)\,dx \\& \quad \lesssim \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \mathcal {M}_{A}(Tf) (x) \bigr)\,dx\sim \int^{\infty}_{0}\frac {1}{\lambda}\varphi \bigl(\bigl\{ \mathcal{M}_{A}(Tf)>\lambda\bigr\} , \lambda \bigr)\,d\lambda \\& \quad \lesssim \int^{\infty}_{0}\frac{1}{\lambda}\varphi \bigl(\bigl\{ \mathcal {M}_{A}(Tf)>\bigl(b^{3\sigma }C_{1}+1\bigr) \lambda\bigr\} , \lambda\bigr)\,d\lambda \lesssim \int^{\infty}_{0}\frac{1}{\lambda}\mu_{Tf}( \lambda,\lambda )\,d\lambda, \end{aligned}$$
where \(C_{1}\) is a positive constant as in Lemma 3.12(iii).
Moreover, by Lemma 3.14 and Assumption (φ)(ii) again, we further have
$$\begin{aligned}& \int^{\infty}_{0}\frac{1}{\lambda}\mu_{Tf}( \lambda,\lambda )\,d\lambda \\& \quad \lesssim \int^{\infty}_{0}\frac{1}{\lambda}\varphi \bigl(\bigl\{ \mathcal {M}^{\sharp}(Tf)>\lambda \beta/2\bigr\} , \lambda\bigr)\,d\lambda+ \frac{\beta b^{2u}}{1-\beta b^{2u}} \int^{\infty}_{0}\frac{1}{\lambda}\mu_{Tf}( \lambda,\lambda )\,d\lambda \\& \quad \lesssim \int^{\infty}_{0}\frac{1}{\lambda}\varphi \bigl(\bigl\{ \mathcal {M}^{\sharp}(Tf)>\lambda \bigr\} , \lambda\bigr)\,d\lambda+ \frac{\beta b^{2u}}{1-\beta b^{2u}} \int ^{\infty}_{0}\frac{1}{\lambda} \mu_{Tf}(\lambda,\lambda)\,d\lambda. \end{aligned}$$
Taking β small enough such that \(\beta b^{2u}/(1-\beta b^{2u})<1\), by Lemma 3.2, we obtain
$$\begin{aligned} \int^{\infty}_{0}\frac{1}{\lambda}\mu_{Tf}( \lambda,\lambda)\,d\lambda & \lesssim \int^{\infty}_{0}\frac{1}{\lambda}\varphi \bigl(\bigl\{ \mathcal {M}^{\sharp}(Tf)>\lambda \bigr\} , \lambda\bigr)\,d\lambda \\ & \sim \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \mathcal{M}^{\sharp}(Tf) (x) \bigr)\,dx. \end{aligned}$$
Therefore, we have
$$ \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert Tf(x)\bigr\vert \bigr)\,dx\lesssim \int _{{\mathbb{R}}^{n}}\varphi \bigl(x, \mathcal{M}^{\sharp}(Tf) (x) \bigr)\,dx. $$
(3.27)
For any \(\lambda\in(0,\infty )\), let
$$\Omega_{\lambda}:= \bigl\{ \mathcal{M}^{q}_{\varphi (\cdot, \lambda )} f> \lambda \bigr\} , $$
where \(\mathcal{M}^{q}_{\varphi (\cdot, \lambda)}f\) is as in (3.2).
By checking the proof of Theorem 2.5 and using the same notation as in the proof of Theorem 2.5 and an anisotropic variant of the Whitney covering lemma (see [30], Lemma 2.3) associated with \(\Omega_{\lambda}\), then we may define
$$g(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} f(x), & \mbox{for } x \in\Omega^{\complement}_{\lambda}, \\ \sum_{i}(\pi_{x_{i}+{B_{\ell_{i}}}}f\eta_{i})(x)\chi_{i}(x), & \mbox{for } x\in\Omega_{\lambda}, \end{array}\displaystyle \right . $$
and, for any \(x\in{{\mathbb{R}}^{n}}\),
$$h(x):=\sum_{i} h_{i}(x), \quad \mbox{where } h_{i}(x):=f(x)\chi_{i}(x)-(\pi _{x_{i}+{B_{\ell_{i}}}}f\eta_{i}) (x)\chi_{i}(x). $$
Obviously, for any \(x\in{{\mathbb{R}}^{n}}\), we have \(f(x)=g(x)+h(x)\).
Moreover, for any \(x\in{{\mathbb{R}}^{n}}\), set
$$a_{i}(x):=\frac{h_{i}(x)}{C\lambda\varphi (x_{i}+B_{\ell_{i}}, \lambda)}, $$
where C is a constant as in (3.11), then \(a_{i}\) is a \((1,q,0)_{\varphi {(\cdot,\lambda)}}\)-atom by checking the proof of Theorem 2.5 with \(\varphi (\cdot, \lambda)\in\mathbb {A}_{1}(A)\subset \mathbb{A}_{q}(A)\). Therefore, we have
$$h(x)=C\lambda\sum_{i}\varphi (x_{i}+B_{\ell_{i}}, \lambda)a_{i}(x)\in H^{1,q,0}_{\varphi (\cdot, \lambda)}(A) $$
and
$$ \|h\|_{H^{1,q,0}_{\varphi (\cdot, \lambda)}(A)}\leq C\lambda\sum _{i}\varphi (x_{i}+B_{\ell_{i}-2\sigma },\lambda)\leq C\lambda\varphi (\Omega _{\lambda},\lambda). $$
(3.28)
By \(|g(x)|\leq C\lambda\) for a.e. \(x\in{{\mathbb{R}}^{n}}\) (see (3.12)) and (2.11), we obtain
$$\|Tg\|_{\operatorname{BMO}(A)}\le C\lambda \quad \textit{i.e.}\quad \mathcal{M}^{\sharp}(Tg) (x)\le C\lambda\quad \mbox{for a.e. } x\in{{ \mathbb{R}}^{n}}. $$
From this and T being a sublinear operator, it follows that
$$\begin{aligned}& \varphi \bigl( \bigl\{ \mathcal{M}^{\sharp}(Tf)>( C+1)\lambda \bigr\} , \lambda \bigr) \\& \quad \le\varphi \bigl( \bigl\{ \mathcal{M}^{\sharp}(Tg)>C\lambda \bigr\} , \lambda \bigr) +\varphi \bigl( \bigl\{ \mathcal{M}^{\sharp}(Th)>\lambda \bigr\} , \lambda \bigr) \\& \quad =\varphi \bigl( \bigl\{ \mathcal{M}^{\sharp}(Th)>\lambda \bigr\} , \lambda \bigr). \end{aligned}$$
(3.29)
Thus, by Lemma 3.2, the uniformly upper type \(p^{+}_{\varphi}\) property of φ, and (3.29), we see that
$$\begin{aligned} \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \mathcal{M}^{\sharp}(Tf) (x) \bigr)\,dx&\sim \int^{\infty}_{0}\frac {1}{\lambda} \varphi \bigl( \bigl\{ \mathcal{M}^{\sharp}(Tf)>(C+1)\lambda \bigr\} , (C+1)\lambda \bigr)\,d \lambda \\ &\lesssim \int^{\infty}_{0}\frac{1}{\lambda}\varphi \bigl( \bigl\{ \mathcal{M}^{\sharp}(Th)>\lambda \bigr\} , \lambda \bigr)\,d\lambda. \end{aligned}$$
From this and (3.27), it follows that
$$ \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert Tf(x)\bigr\vert \bigr)\,dx\lesssim \int^{\infty}_{0}\frac{1}{\lambda}\varphi \bigl(\bigl\{ \mathcal{M}^{\sharp}(Th)>\lambda\bigr\} , \lambda\bigr)\,d\lambda. $$
(3.30)
Since \(f\in L^{\infty}_{0}\), we have \(f\in L^{1}_{\mathrm{loc}}\). Therefore, for any \(x\in{{\mathbb{R}}^{n}}\), by \(\mathcal{M}^{\sharp}f(x)\le2\mathcal {M}_{A} f(x)\), Assumption (φ)(ii), the boundedness of \(\mathcal{M}_{A}\) from \(L^{1}_{\varphi (\cdot , \lambda )}\) to weak \(L^{1}_{\varphi (\cdot, \lambda)}\) with \(\varphi (\cdot , \lambda )\in\mathbb{A}_{1}(A)\) (see [30], Proposition 2.6(ii)), (2.10), \(H^{1}_{\varphi (\cdot, \lambda)}(A)=H^{1,q,0}_{\varphi (\cdot, \lambda)}(A)\) with equivalent norms (see [30], Theorem 5.5), (3.28), Lemma 3.2, and (3.14) with \(1=q(\varphi )< p^{-}_{\varphi}/q\), we obtain
$$\begin{aligned}& \int^{\infty}_{0}\frac{1}{\lambda}\varphi \bigl( \bigl\{ \mathcal {M}^{\sharp}(Th)>\lambda \bigr\} , \lambda \bigr)\,d\lambda \\& \quad \lesssim \int^{\infty}_{0}\frac{1}{\lambda}\varphi \biggl( \biggl\{ \mathcal{M}_{A}(Th)>\frac {\lambda}{2} \biggr\} , \lambda \biggr)\,d\lambda \lesssim \int^{\infty}_{0}\frac{1}{\lambda} \biggl( \frac{\|Th\| _{L^{1}_{\varphi (\cdot , \lambda)}}}{\lambda} \biggr)\,d\lambda \\& \quad \lesssim \int^{\infty}_{0}\frac{1}{\lambda} \biggl( \frac{\|h\| _{H^{1,q,0}_{\varphi (\cdot, \lambda)}(A)}}{\lambda} \biggr)\,d\lambda\lesssim \int^{\infty}_{0}\frac{1}{\lambda}\varphi ( \Omega_{\lambda}, \lambda )\,d\lambda \\& \quad \sim \int_{0}^{\infty}\frac{1}{\lambda} \int_{\{\mathcal {M}^{q}_{\varphi (\cdot, \lambda)}f>\lambda\}}\varphi (x, \lambda) \, dx\, d\lambda\sim \int _{{\mathbb{R}}^{n}} \varphi \bigl(x, \mathcal{M}^{q}_{\varphi (\cdot, \lambda)} f(x) \bigr)\,dx \\& \quad \lesssim \int_{{\mathbb{R}}^{n}}\varphi \bigl(x, \bigl\vert f(x)\bigr\vert \bigr) \,dx, \end{aligned}$$
which, together with (3.30) and Lemma 3.10, completes the proof of Theorem 2.9. □

Declarations

Acknowledgements

This project is supported by the National Natural Science Foundation of China (Grant Nos. 11461065 and 11661075). The authors would like to thank the anonymous referees for their constructive comments.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics and System Science, Xinjiang University

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