Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

Li, Baode; Yang, Dachun; Yuan, Wen

doi:https://doi.org/10.1155/2014/306214

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Research Article | Open Access

Volume 2014 | Article ID 306214 | https://doi.org/10.1155/2014/306214

Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

Baode Li,¹Dachun Yang,²and Wen Yuan²

Academic Editor: A. Fiorenza, A. Agouzal, A. Ibeas

Received17 Nov 2013

Accepted15 Jan 2014

Published16 Mar 2014

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., [1–5]). 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 Calderón and Torchinsky [7], and weighted Hardy spaces of García-Cuerva [8] as well as Strömberg 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., [13–15]). Moreover, as a development of the theory of Orlicz spaces, Orlicz-Hardy spaces and their dual spaces were studied by Strömberg [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., [20–23]). 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 Strömberg [16] and Janson [17] and the weighted Hardy space with studied by García-Cuerva [8] and Strömberg 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, 23–27]). 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, 30–32]).

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 Calderón-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 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 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 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 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 , 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 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 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 García-Cuerva [8] as well as Strömberg 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, Lemmas 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. Calderón-Zygmund Decompositions

In this section, by using the Calderón-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 Calderón-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 ,

Lemma 21. There exists a positive constant , independent of and , such that, for all ,

Lemma 22. There exists a positive constant , independent of and , such that, for all and , .

Lemma 23. If , then there exists a positive constant , independent of and , such that, for all , , and , .

Lemma 24. If , then there exists a positive constant such that, for all , , and , Moreover, the series converges in and where is as in (69).

Proof. By Lemma 22, we know that Notice that implies that for sufficient small and sufficient large . Using Lemma 10 with , Lemma 23, and the fact that for all , we have which gives (75).
By (75) and (69), we see that which, together with the completeness of (see Proposition 7), implies that converges in . So, by Proposition 6, we know that the series converges in and therefore . From this and Lemma 13, we deduce (76). This finishes the proof of Lemma 24.

Let . We denote by the usually anisotropic weighted Lebesgue space with the anisotropic Muckenhoupt weight . Then we have the following technical lemma (see [6, Lemma 4.8]), the details being omitted.

Lemma 25. If and , then the series converges in , and there exists a positive constant , independent of and , such that .

The following conclusion is essentially [9, Lemma 4.9], the details being omitted.

Lemma 26. If and converges in , then there exists a positive constant , independent of and , such that, for all , where

Lemma 27. Let and .(i)If and , then , and there exists a positive constant , independent of and , such that (ii)If and , then , and there exists a positive constant , independent of and , such that .

Proof. Since , by Lemma 24, we know that converges in and therefore in by Proposition 6. Then, by Lemma 26, we have where is as in Lemma 26. Observe that implies that . Moreover, for any fixed with , we find that From this, the -boundedness of the vector-valued maximal function (see [42, Theorem 2.5]), (65), and (69), it follows that and hence
Noticing that on , then, for some , we find that On the other hand, since on , for any , using we see that Combining the above two estimates with (86), we obtain the desired conclusion of Lemma 27(i).
Moreover, notice that, if , then and are functions. By Lemma 25, converges in and hence in due to the fact that is continuous embedding (see [6, Lemma 2.8]). Write By Lemma 21 and (69), we have for all , and for almost every , which leads to and hence (ii) holds true. This finishes the proof of Lemma 27.

Corollary 28. For any and , the subset is dense in .

Proof. Let . For any , let be the Calderón-Zygmund decomposition of of degree with and height associated with as in Definition 19. Here, we rewrite and in Definition 19 into and , respectively. By (76) of Lemma 24, we know that and therefore in as . Moreover, by Lemma 27(i), we see that , which, together with Lemma 17, implies that . This finishes the proof of Corollary 28.

5. Atomic Characterizations of

In this section, we establish the equivalence between and anisotropic atomic Hardy spaces of Musielak-Orlicz type (see Theorem 40 below).

Let be the collection of all dilated balls.

Definition 29. For any and , let be the set of all measurable functions , supported in , such that

It is easy to show that is a Banach space. Next we introduce anisotropic atomic Hardy spaces of Musielak-Orlicz type.

Definition 30. We have the following definitions.(i)An anisotropic triplet is said to be admissible, if and such that with as in (14).(ii)For an admissible anisotropic triplet , a measurable function is called an anisotropic -atom if(a) for some ;(b);(c) for any .(iii)For an admissible anisotropic triplet , the anisotropic atomic Hardy space of Musielak-Orlicz type, , is defined to be the set of all distributions which can be represented as a sum of multiples of anisotropic -atoms, that is, in , where for is a multiple of an anisotropic -atom supported in the dilated ball , with the property Define where the infimum is taken over all admissible decompositions of as above.

Remark 31. (i) In Definition 30, if we assume that can be represented as in , where are -atoms supported in dilated balls , and where the infimum is taken over all admissible decompositions of as above with then the induced space and the space coincide with equivalent (quasi)norms.
Indeed, if in for some -atoms, , and such that . Write . It is easy to see that .
Conversely, if in with , by defining we see that and . Thus, the above claim holds true.
(ii) If is as in (15) with an anisotropic Muckenhoupt weight and for all with , then the atomic space is just the weighted anisotropic atomic Hardy space introduced in [6].

The following lemma shows that anisotropic -atoms of Musielak-Orlicz type are in .

Lemma 32. Let be an anisotropic admissible triplet and let . Then there exists a positive constant such that, for any anisotropic -atom associated with some , and hence .

Proof. The case is easy. We just consider . Now let us write By using Lemma 10, the proof of is similar to that of [20, Lemma 5.1], the details being omitted.
To estimate , we claim that, for all and , where . If this claim is true, choosing and such that , then, by and Lemma 10, we have
Combining the estimates for and , we obtain (98).
To prove the estimate (100), we borrow some techniques from the proof of Theorem 4.2 in [9]. By Hölder’s inequality, , and we obtain Let , , and . For and , we have . Observe that . By this, (103), , and , we conclude that For , let be the Taylor expansion of at the point of order . Thus, by the Taylor remainder theorem and for all (see [9, Section 2]), we see that where, in the last step, we used (8) and the fact that since . By this, (103), , and the fact that has vanishing moments up to order , we find that Observe that, when , by , we know that Finally, when , from (107), we immediately deduce (108). This shows that (108) holds for all . Combining this with (104), and taking supremum over , we see that From this estimate and (see [9, Propostion 3.10]), we further deduce (100) and hence complete the proof of Lemma 37.

Then, by using Lemma 32, together with an argument similar to that used in the proof of [20, Theorem 5.1], we obtain the following theorem, the details being omitted.

Theorem 33. Let be an admissible triplet and let . Then and the inclusion is continuous.

To obtain the conclusion , we use the Calderón-Zygmund decomposition obtained in Section 4. Let be an anisotropic growth function, let , and let . For each , as in Definition 19, has a Calderón-Zygmund decomposition of degree and height associated with as follows: where Recall that, for fixed , is a sequence in and is a sequence of integers such that (65) through (69) hold for , are given by (70), and are projections of onto with respect to the norms given by (71). Moreover, for each and , let be the orthogonal projection of onto with respect to the norm associated with given by (71), namely, the unique element of such that, for all , For convenience, let .

Lemmas 34 through 36 are just [9, Lemmas 5.1 through 5.3], respectively.

Lemma 34. The following hold true.(i)If , then and .(ii)For any , , where is as in (69).

Lemma 35. There exists a positive constant , independent of , such that, for all , , and , where .

Lemma 36. For every , , where the series converges pointwise and also in .

The proof of the following lemma is similar to that of [20, Lemma 5.4], the details being omitted.

Lemma 37. Let and let . Then, for any , there exists a positive constant , independent of and , such that

The following lemma establishes the atomic decompositions for a dense subspace of .

Lemma 38. Let and let . Then, for any , there exists a sequence of multiples of -atoms such that converges almost everywhere and also in , and Moreover, there exists a positive constant , independent of , such that, for all and , and, for any ,

Proof. Let . For each , has a Calderón-Zygmund decomposition of degree and height associated with , as above. The conclusions (117) and (118) follow immediately from (65) and (66). By (76) of Lemma 24 and Proposition 6, we know that in both and as . It follows, from Lemma 27(ii), that as , which further implies that almost everywhere as , and, moreover, by the fact that is continuously embedding into (see [6, Lemma 2.8]), we conclude that in as . Therefore, we obtain in .
Since and as , then almost everywhere as . Thus, (121) also holds almost everywhere. By Lemma 36 and for all , we see that where all the series converge in and almost everywhere. Furthermore, By definitions of and , for all , we have
Moreover, since , we rewrite (123) into By Lemma 17, we know that for almost every , and, by Lemmas 21, 34(ii), and 35, we find that
Recall that implies , and hence, by Lemma 34(i), we see that . Therefore, by applying (123), we further conclude that
Obviously, (126) and (127) imply (119) and (116), respectively. Moreover, by (124), (126), and (127), we know that is a multiple of a -atom. By Lemma 10, (118), (126), uniformly upper type 1 property of , and Lemma 37, for any , we have which gives (120). This finishes the proof of Lemma 38.

The following Lemma 39 is just [20, Lemma 4.3(ii)].

Lemma 39. Let be an anisotropic growth function. For any given positive constant , there exists a positive constant such that, for some , the inequality implies that

Theorem 40. Let be as in (13). If and , then with equivalent (quasi)norms.

Proof. Observe that, by (103), Definition 30, and Theorem 33, it holds true that where , and all the inclusions are continuous. Thus, to finish the proof of Theorem 40, it suffices to prove that, for all with , , which implies that .
To this end, let ; by Lemma 38, we obtain Consequently, by Lemma 39, we see that
Let . By Corollary 28, there exists a sequence of functions in such that and in . By Lemma 38, for each , has an atomic decomposition in , where are multiples of -atoms with . Since then, by Lemma 39, we further see that and which completes the proof of Theorem 40.

For simplicity, from now on, we denote simply by the anisotropic Hardy space of Musielak-Orlicz type with .

6. Finite Atomic Decompositions and Their Applications

The goal of this section is to obtain the finite atomic decomposition characterization of , and, as an application, a bounded criterion on of quasi-Banach space-valued sublinear operators is also obtained.

6.1. Finite Atomic Decompositions

In this subsection, we prove that, for any given finite linear combination of atoms when (or continuous atoms when ), its norm in can be achieved via all its finite atomic decompositions. This extends the conclusion [39, Theorem 3.1] by Meda et al. to the setting of anisotropic Hardy spaces of Musielak-Orlicz type.

Definition 41. Let be an admissible triplet. Denote by the set of all finite linear combinations of multiples of -atoms, and the norm of in is defined by

Obviously, for any admissible triplet , the set is dense in with respect to the quasinorm .

In order to obtain the finite atomic decomposition, we need the notion of the uniformly locally dominated convergence condition from [20]. An anisotropic growth function is said to satisfy the uniformly locally dominated convergence condition if the following holds: for any compact set in and any sequence of measurable functions such that tends to for almost every , if there exists a nonnegative measurable function such that for almost every and then We remark that the anisotropic growth functions for all and with and as in (15) satisfy the uniformly locally dominated convergence condition; see [20, page 12].

Theorem 42. Let be an anisotropic growth function satisfying the uniformly locally dominated convergence condition, as in (13), and an admissible triplet.(i)If , then and are equivalent quasinorms on .(ii) and are equivalent quasinorms on .

Proof. Obviously, by Theorem 40, and, for all , Thus, we only need to prove that, for all when and for all when , .
Now we prove this by three steps.
Step 1 (a new decomposition of ). Assume that . Without loss of generality, we may assume that and . Notice that has compact support. Suppose that for some , where is as in Section 2. For each , let We use the same notation as in Lemma 38. Since , where if and if , by Lemma 38, there exists a sequence of multiples of -atoms such that holds almost everywhere and in . Moreover, by and Theorem 40, we know that
On the other hand, by Step 2 of the proof of [6, Theorem 6.2], we know that there exists a positive constant , depending only on , such that for all . Hence, for all , we have
We now denote by the largest integer such that . Then
Let and let , where the series converge almost everywhere and in . Clearly, and , which, together with , further yields .
Step 2 (prove to be a multiple of a -atom). Notice that, for any and , by Hölder’s inequality and , we have Observing that and has vanishing moments up to order , we know that is a multiple of a -atom and therefore . Then, by (142), (116), (117), and (119) of Lemmas 38 and 34(ii), for any , we conclude that (Notice that in (119) is replaced by here.) This, together with the vanishing moments of , implies that has vanishing moments up to order and, hence, so does by . Using Lemma 34(ii), (119) of Lemma 38, and the facts that and , we obtain Thus, there exists a positive constant , independent of , such that is a -atom, and, by Definition 30, it is also a -atom for any admissible triplet .
Step 3 (prove (i)). Let . We first show . For any , since , there exists such that . Since for , applying Lemma 34(ii) and (119) of Lemma 38, we conclude that, for all , By , we further have . Since satisfies the uniformly locally dominated convergence condition, it follows that converges to in .
Now, for any positive integer , let and let . Since converges in , for any , if is large enough, we have that is a -atom. Thus, is a finite linear combination of atoms. By (120) of Lemma 38 and Step 2, we further find that which completes the proof of (i).
To prove (ii), assume that is a continuous function in ; then is also continuous by examining its definition (see (123)). Since for any , where the positive constant only depends on and , it follows that the level set is empty for all satisfying that . We denote by the largest integer for which the above inequality does not hold. Then the index in the sum defining will run only over .
Let . Since is uniformly continuous, it follows that there exists a such that if , then . Write with and , where and .
On the other hand, for any fixed integer , by (118) of Lemma 38 and , we see that is a finite set and hence is continuous. Furthermore, from Step 5 of the proof of [6, Theorem 6.2], it follows that . Since is arbitrary, we can hence split into a continuous part and a part that is uniformly arbitrarily small. This fact implies that is continuous. Thus, is a multiple of a continuous -atom by Step 2.
Now we can give a finite atomic decomposition of . Let us use again the splitting . By (140), the part is a finite sum of multiples of -atoms and Notice that are continuous and have vanishing moments up to order and, hence, so does . Moreover, and . Thus, we can choose small enough such that becomes a sufficient small multiple of a continuous -atom; that is, Therefore, is a finite linear combination of continuous atoms. Then, by (150) and the fact that is a -atom, we have This finishes the proof of (ii) and hence Theorem 42.

6.2. Applications

As an application of the finite atomic decompositions obtained in Theorem 42, we establish the boundedness on of quasi-Banach-valued sublinear operators.

Recall that a quasi-Banach space is a vector space endowed with a quasinorm which is nonnegative, nondegenerate (i.e., if and only if ), and homogeneous and obeys the quasitriangle inequality; that is, there exists a constant such that, for all , .

Definition 43. Let . A quasi-Banach space with the quasinorm is a -quasi-Banach space if for all .

Notice that any Banach space is a -quasi-Banach space, and the quasi-Banach spaces , , and with are typical -quasi-Banach spaces. Also, when is of uniformly lower type , the space is a -quasi-Banach space. Moreover, according to the Aoki-Rolewicz theorem (see [47] or [48]), any quasi-Banach space is, in essential, a -quasi-Banach space, where .

For any given -quasi-Banach space with and a sublinear space , an operator from to is said to be -sublinear if, for any and complex numbers , , it holds true that and .

We remark that, if is linear, then is -sublinear. Moreover, if with , is a Muckenhoupt weight, and is sublinear in the classical sense, then is also -sublinear.

Theorem 44. Let be an admissible triplet. Assume that is an anisotropic growth function satisfying the uniformly locally dominated convergence condition and being of uniformly upper type and a quasi-Banach space. If one of the following holds true,(i), and is a -sublinear operator such that (ii) is a -sublinear operator defined on continuous -atoms such that then has a unique bounded -sublinear operator extension from to .

Proof. Suppose that the assumption (i) holds true. For any , by Theorem 42(i), there exist complex numbers and -atoms supported in balls such that pointwise. By Remark 31(i), we know that Since is of uniformly upper type , it follows that there exists a positive constant such that, for all , and , If there exists such that , then Otherwise, it follows, from (157), that which implies that Therefore, by the assumption (i), we obtain Since is dense in , a density argument then gives the desired conclusion.
Suppose now that the assumption (ii) holds true. Similar to the proof of (i), by Theorem 42(ii), we also conclude that, for all , . To extend to the whole , we only need to prove that is dense in . Since is dense in , it suffices to prove is dense in in the quasinorm . Actually, we only need to show that is dense in due to Theorem 42.
To see this, let . Since is a finite linear combination of functions with bounded supports, it follows that there exists such that . Take such that and . By (7), it is easy to show that for any , and has vanishing moments up to order , where for all . Hence, .
Likewise, for any , and has vanishing moments up to order . Take any . By [6, Proposition 2.9 (ii)] and the fact that satisfies the uniformly locally dominated convergence condition, we know that and hence for some -atom , where is a constant depending on and as . Thus, we obtain as . This finishes the proof of Theorem 44.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project is partially supported by the National Natural Science Foundation of China (Grants nos. 11001234, 11161044, 11171027, 11361020, and 11101038), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20120003110003), and the Fundamental Research Funds for Central Universities of China (Grant no. 2012LYB26).

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Copyright © 2014 Baode Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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