#### Abstract

Let be a Musielak-Orlicz function and an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, , via the grand maximal function. The authors then obtain some real-variable characterizations of in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space with and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization of , and, as an application, the authors prove that, for a given admissible triplet , if is a sublinear operator and maps all -atoms with (or all continuous -atoms with ) into uniformly bounded elements of some quasi-Banach spaces , then uniquely extends to a bounded sublinear operator from to . These results are new even for anisotropic Orlicz-Hardy spaces on .

#### 1. Introduction

The theory of Hardy spaces on the Euclidean space plays an important role in various fields of analysis and partial differential equations (see, e.g., [1â€“5]). One of the most important applications of Hardy spaces is that they are good substitutes of Lebesgue spaces when . For example, when , it is well known that Riesz transforms are not bounded on ; however, they are bounded on Hardy spaces . Moreover, there were several efforts to extend classical Hardy spaces, some of which are weighted anisotropic Hardy spaces [6] associated with general expansive dilations and Muckenhoupt weights. These Hardy spaces include classical isotropic Hardy spaces of Fefferman and Stein [1], parabolic Hardy spaces of CalderÃ³n and Torchinsky [7], and weighted Hardy spaces of GarcÃa-Cuerva [8] as well as StrÃ¶mberg and Torchinsky [5] as special cases. Apart from their theoretical consideration, such anisotropic function spaces also play an important role in allowing even more general discrete dilation structures which have originated from the theory of wavelets; see, for example, [9, 10].

On the other hand, as a generalization of , the Orlicz space was introduced by Birnbaum and Orlicz in [11] and Orlicz in [12]. Since then, the theory of the Orlicz spaces themselves has been well developed and these spaces have been widely used in many branches of analysis (see, e.g., [13â€“15]). Moreover, as a development of the theory of Orlicz spaces, Orlicz-Hardy spaces and their dual spaces were studied by StrÃ¶mberg [16] and Janson [17] on and, quite recently, Orlicz-Hardy spaces associated with divergence form elliptic operators by Jiang and Yang [18].

Let with denote the class of* Muckenhoupt weights* (see, e.g., [19] for their definitions and properties) and let be a* growth function* (see [20]) which means that is a Musielak-Orlicz function such that is an Orlicz function and is a Muckenhoupt weight. It is known that Musielak-Orlicz functions are the natural generalization of Orlicz functions that may vary in the spatial variables (see, e.g., [20â€“23]). Recently, Ky [20] introduced a new* Musielak-Orlicz Hardy* space , via the grand maximal function, and established its atomic characterization. It is known that generalizes both the Orlicz-Hardy space of StrÃ¶mberg [16] and Janson [17] and the weighted Hardy space with studied by GarcÃa-Cuerva [8] and StrÃ¶mberg and Torchinsky [5]. Recall that the motivation to study function spaces of Musielak-Orlicz type comes from their applications to many branches of mathematics and physics (see, e.g., [20, 23â€“27]). In [20], Ky further introduced the -type space , which was proven to be the dual space of ; as an interesting application, Ky proved that the class of pointwise multipliers for , characterized by Nakai and Yabuta [28, 29], is the dual space of , where denotes the Musielak-Orlicz Hardy space related to the growth function
for all and . It is worth noticing that some special Musielak-Orlicz Hardy spaces appear naturally in the study of the products of functions in and (see [25, 26, 30]), the endpoint estimates for the div-curl lemma, and the commutators of singular integral operators (see [25, 30â€“32]).

Moreover, observe that a distribution in Hardy spaces can be represented as a (finite or infinite) linear combination of atoms (see [33, 34]). Then, the boundedness of linear operators in Hardy spaces can be deduced from their behavior on atoms in principle. However, Meyer et al. [35, page 513] gave an example of whose norm can not be achieved by its finite atomic decompositions via -atoms. Applying this, Bownik [36] showed that there exists a linear functional defined on a dense subspace of , which maps all -atoms into bounded scalars, but yet can not extend to a bounded linear functional on the whole . Let and let be a nonnegative integer not less than . This implies that the uniform boundedness in some quasi-Banach space of a linear operator on all -atoms does not generally guarantee the boundedness of from to . This phenomenon has also essentially already been observed by Meyer et al. in [37, page 19]. Motivated by [36], via using the Lusin function characterization of Hardy spaces , Yang and Zhou [38] proved that a -sublinear operator uniquely extends to a bounded -sublinear operator from with to some quasi-Banach space if and only if maps all -atoms into uniformly bounded elements of . Independently, Meda et al. [39] established another more general bounded criterion via using the grand maximal function characterization of ; precisely, they proved that if is a linear operator and maps all -atoms with or all continuous -atoms into uniformly bounded elements of a Banach space , then uniquely extends to a bounded linear operator from to . This result was further generalized to the weighted anisotropic Hardy spaces in [6], weighted anisotropic product Hardy spaces in [40], and, especially, Hardy spaces of Musielak-Orlicz type by Ky in [20].

There are three goals in this paper. First, we introduce anisotropic Hardy spaces of Musielak-Orlicz type, , via grand maximal functions and characterize these spaces via anisotropic atomic decompositions. These Hardy spaces include classical Hardy spaces of Fefferman and Stein [1], weighted anisotropic Hardy spaces of Bownik [6], and Hardy spaces of Musielak-Orlicz type of Ky [20].

The second goal is to obtain some new real-variable characterizations of in terms of the radial, the nontangential, and the tangential maximal functions via some bounded estimates of the truncated maximal function pointwise or in anisotropic Musielak-Orlicz spaces which are motivated by [9, Section 7]. These real-variable characterizations of coincide with the known best results, when is the anisotropic Hardy space , with (see [9, Theorem 7.1]), or new even in its weighted variant.

The third goal is to generalize the result of Meda et al. [39] to the present setting. More precisely, we prove the existence of finite atomic decompositions achieving the norm in dense subspaces of . As an application, we prove that, for a given admissible triplet (see Definition 30 below), if is a -sublinear operator and maps all -atoms with (or all continuous -atoms with ) into uniformly bounded elements of some quasi-Banach spaces , then uniquely extends to a bounded -sublinear operator from to . These results are new even for the anisotropic Hardy-Orlicz spaces on .

This paper is organized as follows. In Section 2, we first recall some notation and definitions concerning Musielak-Orlicz functions, expansive dilations, and Muckenhoupt weights. Then we introduce the anisotropic Hardy spaces of Musielak-Orlicz type, , via grand maximal functions, and some basic properties of these spaces are also presented. In Section 3, we obtain some new real-variable characterizations of via the radial, the nontangential, and the tangential maximal functions. Section 4 is devoted to generalizing the CalderÃ³n-Zygmund decomposition associated to weighted anisotropic Hardy spaces in [6] to the more general spaces . Applying this, in Section 5, we introduce the anisotropic atomic Hardy spaces of Musielak-Orlicz type, , for any admissible triplet , and further prove that, for any admissible triplet , with equivalent norms (see Theorem 40 below). Moreover, in Section 6.1, we prove that and are equivalent quasinorms on when and on when , where denotes the space of all finite linear combinations of multiples of -atoms. In Section 6.2, we obtain criteria for boundedness of sublinear operators in (see Theorem 44 below). The results in Section 6 are also new even for the anisotropic Hardy-Orlicz spaces on .

Finally, we make some conventions on notation. Let and let . Denote by the* space of all Schwartz functions* and the* space of all tempered distributions*. For any , and . Throughout the whole paper, we denote by a* positive constant* which is independent of the main parameters, but it may vary from line to line. The* symbol*â€‰â€‰ means that . If and , we then write . If is a subset of , we denote by its* characteristic function*. For any , denotes the* maximal integer* not larger than .

#### 2. Anisotropic Hardy Spaces of Musielak-Orlicz Type

In this section, we introduce anisotropic Hardy spaces of Musielak-Orlicz type via grand maximal functions and give out some basic properties.

First let us recall some notation for Orlicz functions; see, for example, [20]. A function is called an* Orlicz function* if it is nondecreasing and , if , and . Observe that, differently from the classical Orlicz functions being convex, the Orlicz functions in this paper may not be convex. An Orlicz function is said to be of* lower *(resp.*, upper*)* type * with , if there exists a positive constant such that, for all and (resp., ),

Given the function such that, for any , is an Orlicz function, is said to be of* uniformly lower* (resp.,* upper*)* type*â€‰â€‰ with , if there exists a positive constant such that, for all , , and (resp., ),
is said to be of* positive uniformly lower *(resp*.*,* upper*)* type* if it is of uniformly lower (resp., upper) type for some . Let
denote the* uniformly critical lower type* and the* critical upper type* of the function , respectively.

Now we recall the notion of expansive dilations on ; see [9]. A real matrix is called an* expansive dilation*, shortly a* dilation*, if , where denotes the set of all* eigenvalues* of . Let and be two* positive numbers* such that
In the case when is diagonalizable over , we can even take and . Otherwise, we need to choose them sufficiently close to these equalities according to what we need in our arguments.

It was proved in [9, Lemma 2.2] that, for a given dilation , there exist an open ellipsoid and such that , and one can additionally assume that , where denotes the -dimensional Lebesgue measure of the set . Let for . Then is open, , and . Throughout the whole paper, let be the* minimal integer* such that and, for any subset of , let . Then, for all with , it holds true that
where denotes the algebraic sums of sets .

*Definition 1. *A* quasinorm*, associated with an expansive matrix , is a Borel measurable mapping , for simplicity, denoted by , such that(i) for all ;(ii) for all , where ;(iii) for all , where is a constant.

In the standard dyadic case , for all is an example of homogeneous quasinorms associated with ; here and hereafter, always denotes the â€‰* unit matrix* and the Euclidean norm in .

It was proved in [9, Lemma 2.4] that all homogeneous quasinorms associated with a given dilation are equivalent. Therefore, for a given expansive dilation , in what follows, for convenience, we always use the* step homogeneous quasinorm*â€‰â€‰ defined by setting, for all ,
By (7) and (8), we know that, for all ,
see [9, page 8]. Moreover, is a space of homogeneous type in the sense of Coifman and Weiss [41], where denotes the -dimensional Lebesgue measure.

*Definition 2. *Let . A function is said to satisfy the* uniform anisotropic Muckenhoupt condition*â€‰â€‰, denoted by , if there exists a positive constant such that, for all , when ,
and, when ,
The minimal constant as above is denoted by .

Define and

If is independent of , then is just an anisotropic Muckenhoupt weight in [42]. Obviously, . If , by a discussion similar to [6, page 3072], it is easy to know . Moreover, there exists a such that ; see Johnson and Neugebauer [43, page 254, Remark].

Now, we introduce anisotropic growth functions.

*Definition 3. *A function is called an* anisotropic growth function* if(i)the function is an anisotropic Musielak-Orlicz function; that is,(a)the function is an Orlicz function for all ,(b)the function is a Lebesgue measurable function for all ;(ii)the function belongs to ;(iii)the function is of positive uniformly lower type for some and of uniformly upper type 1.

Given a growth function , let Clearly, is an anisotropic growth function if is a classical or an anisotropic Muckenhoupt weight (cf. [42]) and of positive lower type for some and of upper type 1. More examples of growth functions can be found in [20, 22, 30, 32].

*Remark 4. *By Lemma 11 below (see also [20, Lemma 4.1]), without loss of generality, we may always assume that an anisotropic growth function is of positive uniformly lower type for some and of uniformly upper type 1 such that is continuous and strictly increasing for all given .

Throughout the whole paper, we always assume that is an anisotropic growth function. Recall that the* Musielak-Orlicz-type space*â€‰â€‰ is defined to be the set of all measurable functions such that, for some ,
with the Luxembourg (or called the Luxembourg-Nakano) (quasi)norm

For , let In what follows, for , , and , let .

For , the* nontangential grand maximal function * of is defined by setting, for all ,
If , we then write instead of .

*Definition 5. *For any and anisotropic growth function , the* anisotropic Hardy space*â€‰â€‰ of Musielak-Orlicz type is defined to be the set of all such that with the (quasi)norm . When , is denoted simply by .

Observe that, when and is as in (15) with a Muckenhoupt weight and an Orlicz function , the above Hardy spaces are just weighted Hardy-Orlicz spaces which include classical Hardy-Orlicz spaces of Janson [44] ( in this context) and classical weighted Hardy spaces of GarcÃa-Cuerva [8] as well as StrÃ¶mberg and Torchinsky [5] ( for all in this context); see also [19, 45, 46]. When is as in (15) with for all , the above Hardy spaces become weighted anisotropic Hardy spaces (see [6]) and, more generally, when is an Orlicz function, these Hardy spaces are new.

Now let us give some basic properties of .

Proposition 6. *For , it holds true that and the inclusion is continuous.*

*Proof. *Let . For any and , we have , where for all .

By Definition 1, we see that
Therefore, it holds true that
This implies that and the inclusion is continuous, which completes the proof of Proposition 6.

Using Proposition 6, with an argument similar to that of [20, Proposition 5.2], we have the following conclusion, the details being omitted.

Proposition 7. *Let and let be an anisotropic growth function. Then is complete.*

#### 3. Characterizations of via Maximal Functions

The goal of this section is to establish some maximal function characterizations of . Let us begin with the notions of anisotropic variants of the radial, the nontangential, and the tangential maximal functions.

*Definition 8. *Let with . The anisotropic radial, the nontangential, and the tangential maximal functions of associated to are defined, respectively, by setting, for all ,

Theorem 9. *Let be an anisotropic growth function and with . Then, for any , the following are equivalent:
**
Moreover, for sufficiently large , there exist positive constants , , , and , independent of , such that
*

The approach we use to prove Theorem 9 is motivated by Bownik [9, Theorem 7.1]. First, we need the following two lemmas which come from [5, pages 7-8] and [20, Lemma 4.1(ii)].

In what follows, for any set and , let

Lemma 10. *Let and . Then there exists a positive constant such that, for all , , and ,
*

Lemma 11. *Let be an anisotropic growth function. For all , is also an anisotropic growth function which is equivalent to ; moreover, for any given is continuous and strictly increasing.*

We now recall some Peetre-type maximal functions from [9]. These maximal functions are obtained via the truncation with an additional extra decay term. Namely, for an integer representing the truncation level and a real nonnegative number representing the decay level, any and , we define and the following Peetre-type radial, the nontangential, the tangential, the radial grand, and the nontangential grand maximal functions: where is as in (18).

We need some technical lemmas. To begin with, let be an arbitrary Borel measurable function. For fixed and , the* maximal function* of with aperture is defined by setting, for all ,
It was shown in [9, page 42] that is lower semicontinuous; namely, is open for any .

We have the following Lemma 12 associated to which is a uniformly weighted analogue of [9, Lemma 7.2].

Lemma 12. *Let and . Then there exists a positive constant such that, for any and ,
*

*Proof. *For any , let . For any satisfying , there exist and such that . Clearly, . Moreover, by (7) and , we find that
From this and with Lemma 10, it follows that
Consequently, by this and , we have
which implies that
where denotes the* centered Hardy-Littlewood maximal function* associated to the measure ; namely, for all ,
Thus,
From this and the weak- boundedness of with , it is easy to deduce (33).

Next we prove (34). By Lemma 11, we know that
which, together with (33), further implies that
which is desired. This finishes the proof of Lemma 12.

The following Lemma 13 is just [20, Lemma 4.1(i)].

Lemma 13. *Let be an anisotropic growth function. Then there exists a positive constant such that, for all with ,
*

The following Lemma 14 extends [9, Lemma 7.5] to the setting of anisotropic Musielak-Orlicz function spaces.

Lemma 14. *Let , let be an anisotropic growth function, and let . Then there exists a positive constant such that, for all , and ,
*

*Proof. *For any , , , and , consider a function given by setting, for all ,
with being as in (30). Fix and . If and , then
where is as in (32). If and for some , then
where is as in (32). By taking supremum over all and , we obtain
Moreover, since , we choose large enough and small enough such that . Therefore, from this, (48), Lemma 13, the uniformly lower type of , and Lemma 12, it follows that
which implies (44). This finishes the proof of Lemma 14.

The following Lemmas 16 and 18 are just [9, Lemmasâ€‰â€‰7.5 and 7.6], respectively.

Lemma 15. *Suppose with . Then, for any given , there exist a positive integer and a positive constant such that, for all , integers and ,
*

Lemma 16. *Let with and . Then, for every , there exists such that, for all ,
**
where is a positive constant depending on , , and , but independent of and .*

The following Lemma 17 is just [9, Proposition 3.10] and [6, Proposition 2.11].

Lemma 17. *There exists a positive constant such that, for almost every , , and ,
**
where for all and denotes the anisotropic Hardy-Littlewood maximal operator defined by setting, for all ,
*

The following lemma comes from [22, Corollary 2.8] with a slight modification, the details being omitted.

Lemma 18. *Let be an anisotropic Musielak-Orlicz function with uniformly lower type and uniformly upper type satisfying , where is as in (13). Then the Hardy-Littlewood maximal operator is bounded on .*

*Proof of Theorem 9. *Obviously, (23) (25) (26). Let be an anisotropic growth function and let satisfy . By (50) of Lemma 15 with and , we know that there exists a positive integer such that, for all , , and integers ,
From this and Lemma 14, it follows that, for all and ,
As , by the monotone convergence theorem and the continuity of (see Lemma 11), we have
which, together with Lemma 17, implies that (25) (24) (23). It remains to prove (26) (23).

Suppose . By Lemma 16, we find some such that (51) holds true, which implies that for all . By Lemmas 14 and 15, we find such that
with a positive constant being independent of . For any given , let
where with . We claim that
Indeed, by (57), the uniformly lower type of and , we have
Moreover, for any and , we choose small enough such that , where is as in (13), and, by [9, page 48, (7.16)], we know that there exists a constant such that, for all integers and ,
Furthermore, from the fact that is of uniformly upper type 1 and positive lower type with , it follows that is of uniformly upper and lower type . Consequently, using (59), (61), and Lemma 18 with , we obtain
where depends on but is independent of . This inequality is crucial, since it gives a bound of the nontangential maximal function by the radial maximal function in .

Since converges pointwise and monotonically to for all as , it follows that by (62), the continuity of (see Lemma 11), and the monotone convergence theorem. Therefore, by choosing and using (62), the continuity of , and the monotone convergence theorem, we conclude that , where now the positive constant corresponds to and is independent of . Combining this, (56), and Lemma 17, we obtain the desired conclusion and hence complete the proof of Theorem 9.

#### 4. CalderÃ³n-Zygmund Decompositions

In this section, by using the CalderÃ³n-Zygmund decomposition associated with grand maximal functions on anisotropic established in [6], we obtain some bounded estimates on . We follow the constructions in [2, 6].

Throughout this section we consider a tempered distribution so that, for all , , where is some fixed integer. For a given , let By referring to [6, page 3081], we know that there exist a positive constant , independent of and , a sequence , and a sequence of integers, , such that Here and hereafter, for a set , denotes its cardinality.

Fix such that , , and on . For each and all , define . Clearly, and on . By (65) and (69), for any , we have . For every and all , define Then , , , on by (66), and . Therefore, the family forms a smooth partition of unity on .

Let be some fixed integer and let denote the linear space of polynomials of degrees not more than . For each and , let Then is a finite dimensional Hilbert space. Let . For each , since induces a linear functional on via , by the Riesz lemma, we know that there exists a unique polynomial such that, for all , For every , define a distribution .

We will show that, for suitable choices of and , the series converges in and, in this case, we define in .

*Definition 19. *The representation , where and are as above, is called a* CalderÃ³n-Zygmund decomposition* of degree and height associated with .

The remainder of this section consists of a series of lemmas. In Lemmas 20 and 21, we give some properties of the smooth partition of unity . In Lemmas 22 through 25, we derive some estimates for the bad parts . Lemmas 26 and 27 give some estimates over the good part . Finally, Corollary 28 shows the density of in , where .

Lemmas 20 through 23 are essentially Lemmas 4.3 through 4.6 of [9], the details being omitted.

Lemma 20. *There exists a positive constant , depending only on , such that, for all and ,
*