Anisotropic variable Hardy–Lorentz spaces and their real interpolation

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Anisotropic variable Hardy–Lorentz spaces and their real interpolation ✩ Jun Liu, Dachun Yang ∗ , Wen Yuan School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 26 April 2017 Available online xxxx Submitted by A. Cianchi Keywords: Variable exponent (Hardy–)Lorentz space Expansive matrix Atom Lusin area function Real interpolation

a b s t r a c t Let p(·) : Rn → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous condition, q ∈ (0, ∞] and A be a general expansive matrix on Rn . In this article, the authors first introduce the anisotropic variable Hardy– p(·),q Lorentz space HA (Rn ) associated with A, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. p(·),q Moreover, the authors also obtain characterizations of HA (Rn ), respectively, in terms of the atom and the Lusin area function. As an application, the authors p(·),q prove that the anisotropic variable Hardy–Lorentz space HA (Rn ) serves as the p(·) intermediate space between the anisotropic variable Hardy space HA (Rn ) and the space L∞ (Rn ) via the real interpolation. This, together with a special case of the real interpolation theorem of H. Kempka and J. Vybíral on the variable Lorentz p(·),q space, further implies the coincidence between HA (Rn ) and the variable Lorentz p(·),q n n space L (R ) when ess inf x∈R p(x) ∈ (1, ∞). © 2017 Elsevier Inc. All rights reserved.

1. Introduction As a generalization of the classical Lebesgue spaces Lp (Rn ), the variable Lebesgue spaces Lp(·) (Rn ), in which the constant exponent p is replaced by an exponent function p(·) : Rn → (0, ∞], were studied by Musielak [54] and Nakano [56,57], which can be traced back to Orlicz [60,61]. But the modern theory of function spaces with variable exponents was started with the articles [45] of Kováčik and Rákosník and [32] of Fan and Zhao as well as [21] of Cruz-Uribe and [24] of Diening, and nowadays has been widely used in harmonic analysis (see, for example, [22,25,80]). In addition, the theory of variable function spaces also has interesting applications in fluid dynamics [2], image processing [17], partial differential equations and variational calculus [3,30,40,66]. ✩

This project is supported by the National Natural Science Foundation of China (Grant Nos. 11571039, 11671185 and 11471042).

* Corresponding author. E-mail addresses: [email protected] (J. Liu), [email protected] (D. Yang), [email protected] (W. Yuan). http://dx.doi.org/10.1016/j.jmaa.2017.07.003 0022-247X/© 2017 Elsevier Inc. All rights reserved.

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Recently, Nakai and Sawano [55] and, independently, Cruz-Uribe and Wang [23] with some weaker assumptions on p(·) than those used in [55], extended the theory of variable Lebesgue spaces via investigating the variable Hardy spaces on Rn . Later, Sawano [69], Zhuo et al. [86] and Yang et al. [84] further completed the theory of these variable Hardy spaces. For more developments of function spaces with variable exponents, we refer the reader to [6,26,44,58,59,76–79,83] and their references. In particular, Kempka and Vybíral [44] introduced the variable Lorentz spaces which are a generalization of both the variable Lebesgue spaces and the classical Lorentz spaces and obtained some basic properties of these spaces including several embedding conclusions. The real interpolation result that the variable Lorentz space serves as the intermediate space between the variable Lebesgue space Lp(·) (Rn ) and the space L∞ (Rn ) was also presented in [44]. Very recently, Yan et al. [79] first introduced the variable weak Hardy spaces on Rn and established their various real-variable characterizations; as an application, the boundedness of some Calderón–Zygmund operators on these spaces in the critical case was also presented. Based on these results, via establishing a very interesting decomposition for any distribution of the variable weak Hardy space, Zhuo et al. [87] proved the following real interpolation theorem between the variable Hardy space H p(·)(Rn ) and the space L∞ (Rn ): (H p(·) (Rn ), L∞ (Rn ))θ,∞ = WH p(·)/(1−θ) (Rn ),

θ ∈ (0, 1),

(1.1)

where WH p(·)/(1−θ) (Rn ) denotes the variable weak Hardy space and (·, ·)θ,∞ the real interpolation. As was well known, Fefferman et al. [34] showed that the Hardy–Lorentz space H p,q (Rn ) was actually the intermediate space between the classical Hardy space H p (Rn ) and the space L∞ (Rn ) under the real interpolation, which is the main motivation to develop the real-variable theory of H p,q (Rn ). Thus, it is natural and interesting to ask whether or not the variable Hardy–Lorentz space also serves as the intermediate space between the variable Hardy space H p(·) (Rn ) and the space L∞ (Rn ) via the real interpolation, namely, if (·, ·)θ,∞ in (1.1) is replaced by (·, ·)θ,q with q ∈ (0, ∞], what happens? On the other hand, as the series of works (see, for example, [1,5,7,34,35,49,62]) reveal, the Hardy–Lorentz spaces (as well the weak Hardy spaces) serve as a more subtle research object than the usual Hardy spaces when studying the boundedness of singular integrals, especially, in some critical cases, due to the fact that these function spaces own finer structures. Moreover, after the celebrated articles [14–16] of Calderón and Torchinsky on parabolic Hardy spaces, there has been an enormous interest in extending classical function spaces arising in harmonic analysis from Euclidean spaces to some more general underlying spaces; see, for example, [28,36,38,39,70,67,68,72,73,81]. The function spaces in the anisotropic setting have proved of wide generality (see, for example, [10–12]), which include the classical isotropic spaces and the parabolic spaces as special cases. For more progresses about this theory, we refer the reader to [46,47,51–53,31,74,75] and their p,q references. In particular, the authors recently introduced the anisotropic Hardy–Lorentz spaces HA (Rn ), associated with some dilation A, and obtained their various real-variable characterizations (see [51,52]). Also, very recently, Zhuo et al. [85] developed the real-variable theory of the variable Hardy space H p(·) (X ) on an RD-space X . Recall that a metric measure space of homogeneous type X is called an RD-space if it is a metric measure space of homogeneous type in the sense of Coifman and Weiss [19,20] and satisfies some reverse doubling property, which was originally introduced by Han et al. [39] (see also [82] for some equivalent characterizations). To further study the intermediate space between the variable Hardy space H p(·) (Rn ) and the space ∞ L (Rn ) via the real interpolation and also to give a complete theory of variable Hardy–Lorentz spaces in anisotropic setting, in this article, we first introduce the anisotropic variable Hardy–Lorentz space, via the radial grand maximal function, and then establish its several real-variable characterizations, respectively, in terms of the atom, the radial or the non-tangential maximal functions, and the Lusin area function. p(·),q As an application, we prove that the anisotropic variable Hardy–Lorentz space HA (Rn ) serves as the p(·) intermediate space between the anisotropic variable Hardy space HA (Rn ) and the space L∞ (Rn ) via the

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real interpolation. This, together with a special case of the real interpolation theorem of Kempka and Vybíral p(·),q in [44] on the variable Lorentz space, further implies the coincidence between HA (Rn ) and the variable Lorentz space Lp(·),q (Rn ) when ess inf x∈Rn p(x) ∈ (1, ∞). To be precise, this article is organized as follows. In Section 2, we first recall some notation and notions on Euclidean spaces, with anisotropic dilations, and variable Lebesgue spaces as well as some basic properties of these spaces to be used later in this article. p(·),q Then we introduce the anisotropic variable Hardy–Lorentz space HA (Rn ) via the radial grand maximal function. p(·),q Section 3 is aimed to characterize HA (Rn ) by means of the radial or the non-tangential maximal functions (see Theorem 3.8 below). To this end, via the Aoki–Rolewicz theorem (see [8,64]), we first prove N (K,L) that the Lp(·),q (Rn ) quasi-norm of the tangential maximal function Tφ (f ) can be controlled by that of the non-tangential maximal function Mφ (f ) for any f ∈ S  (Rn ) (see Lemma 3.5 below), where K  is the truncation level, L the decay level and S (Rn ) denotes the set of all tempered distributions on Rn . Then, by the boundedness of the Hardy–Littlewood maximal function as in (3.1) below on Lp(·) (Rn ) (see Lemma 3.3 below) with p(·) satisfying the so-called globally log-Hölder continuous condition (see (2.6) and (2.7) below) and 1 < p− ≤ p+ < ∞, where p− and p+ are as in (2.4) below, we obtain the boundedness of the Hardy–Littlewood maximal function on Lp(·),q (Rn ) (see Lemma 3.4 below) with p(·) satisfying the same condition as that in Lemma 3.3 and q ∈ (0, ∞]. We point out that the monotone convergence property for increasing sequences on Lp(·),q (Rn ) (see Proposition 2.8 below) as well as Lemmas 3.3 and 3.4 play a key role in proving Theorem 3.8. In Section 4, via borrowing some ideas from [51, Theorem 3.6] and [79, Theorem 4.4], we establish the p(·),q atomic characterization of HA (Rn ). Indeed, we first introduce the anisotropic variable atomic Hardy– p(·),r,s,q Lorentz space HA (Rn ) in Definition 4.2 below and then prove (K,L)

p(·),q

HA

p(·),r,s,q

(Rn ) = HA

(Rn ) p(·),r,s,q

with equivalent quasi-norms (see Theorem 4.8 below). To prove that HA (Rn ) is continuously embedded p(·),q n into HA (R ), motivated by [69, Lemma 4.1], we first conclude that some estimates related to Lp(·) (Rn ) norms for some series of functions can be reduced into dealing with the Lr (Rn ) norms of the corresponding functions (see Lemma 4.5 below), which actually is an anisotropic version of [69, Lemma 4.1]. Then, by using this key lemma and the Fefferman–Stein vector-valued inequality of the Hardy–Littlewood maximal p(·),r,s,q p(·),q operator MHL on Lp(·) (Rn ) (see Lemma 4.3 below), we prove that HA (Rn ) ⊂ HA (Rn ) and the inclusion is continuous. The method used in the proof for the converse embedding is different from that used in the proof for the corresponding embedding of variable Hardy spaces H p(·)(Rn ) (or, resp., anisotropic p p Hardy spaces HA (Rn )). Recall that L1loc (Rn ) ∩H p(·) (Rn ) (or, resp., L1loc (Rn ) ∩HA (Rn )) is dense in H p(·) (Rn ) p p n p(·) (or, resp., HA (R )), which plays a key role in the atomic decomposition of H (Rn ) (or, resp., HA (Rn )). p(·),∞ However, this standard procedure is invalid for the space HA (Rn ), due to its lack of a dense function subspace. To overcome this difficulty, we borrow some ideas from [51, Theorem 3.6] (see also [27]), in which p(·),∞ the authors directly obtained an atomic decomposition for convolutions of distributions in HA (Rn ) and Schwartz functions instead of some dense function subspace. p(·),q As an application of the atomic characterization of HA (Rn ) obtained in Theorem 4.8, in Section 5, we p(·),q establish the Lusin area function characterization of HA (Rn ) (see Theorem 5.2 below). In the proof of Theorem 5.2, the anisotropic Calderón reproducing formula and the method used in the proof of the atomic p(·),q characterization of HA (Rn ) play a key role. However, when we decompose a distribution into a sum of atoms, the dual method used in estimating the norm of each atom in the classic case does not work anymore in the present setting. Instead, a strategy, used in [52], originated from Fefferman [33], that obtains a subtle estimate (see, for example, [52, (3.23)]) plays a key role here; see the estimate (5.16) below.

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p(·),q

In Section 6, as another application of the atomic characterization of HA (Rn ), we prove the following p(·) real interpolation result between the anisotropic variable Hardy space HA (Rn ) and the space L∞ (Rn ): (HA (Rn ), L∞ (Rn ))θ,q = HA p(·)

p(·)/(1−θ),q

(Rn ),

θ ∈ (0, 1), q ∈ (0, ∞]

(1.2)

(see Theorem 6.2 below). To prove this result, via borrowing some ideas from [87], we first obtain a decomposition for any distribution of the anisotropic variable Hardy–Lorentz space into “good” and “bad” parts (see Lemma 6.5 below), which is of independent interest. We point out that the atomic characp(·) terization of HA (Rn ) (see [50, Theorem 4.8]) and the Fefferman–Stein vector-valued inequality of the Hardy–Littlewood maximal function on the variable Lebesgue space (see Lemma 4.3 below), play a key role in the proof of Lemma 6.5. Applying (1.2), together with [85, Corollary 4.20] on the coincidence bep(·) tween HA (Rn ) and Lp(·) (Rn ) as well as [44, Remark 4.2(ii)], we further obtain the coincidence between p(·),q HA (Rn ) and Lp(·),q (Rn ) when ess inf x∈Rn p(x) ∈ (1, ∞); see Corollary 6.3 below. We should point out that, if A := d In×n for some d ∈ R with |d| ∈ (1, ∞), here and hereafter, In×n p(·),q denotes the n × n unit matrix and | · | denotes the Euclidean norm in Rn , then the space HA (Rn ) becomes the classical isotropic variable Hardy–Lorentz space. In this case, the results in this article are also independently obtained by Jiao et al. [43] via some slightly different methods. Finally, we make some conventions on notation. Throughout this article, we always let N := {1, 2, . . .} and Z+ := {0} ∪ N. For any multi-index β := (β1 , . . . , βn ) ∈ (Z+ )n =: Zn+ , let |β| := β1 + · · · + βn . We denote by C a positive constant which is independent of the main parameters, but its value may change from line to line. Moreover, we use f  g to denote f ≤ Cg and, if f  g  f , we then write f ∼ g. For any q ∈ [1, ∞], we denote by q  its conjugate index, namely, 1/q + 1/q  = 1. In addition, for any set E ⊂ Rn , we denote by E  the set Rn \ E, by χE its characteristic function and by E the cardinality of E. The symbol

s, for any s ∈ R, denotes the largest integer not greater than s. 2. Preliminaries In this section, we introduce the anisotropic variable Hardy–Lorentz space via the radial grand maximal function. To this end, we first recall some notation and notions on spaces of homogeneous type associated with dilations and variable Lebesgue spaces as well as some basic conclusions of these spaces to be used later in this article. For an exposition of these concepts, we refer the reader to the monographs [10,22,25]. We begin with recalling the notion of expansive matrices in [10]. Definition 2.1. A real n × n matrix A is called an expansive matrix (shortly, a dilation) if min |λ| > 1,

λ∈σ(A)

here and hereafter, σ(A) denotes the collection of all eigenvalues of A. Throughout this article, A always denotes a fixed dilation and b := | det A|. Then we easily find that b ∈ (1, ∞) by [10, p. 6, (2.7)]. Let λ− and λ+ be two positive numbers satisfying that 1 < λ− < min{|λ| : λ ∈ σ(A)} ≤ max{|λ| : λ ∈ σ(A)} < λ+ . In the case when A is diagonalizable over C, we can even take λ− := min{|λ| : λ ∈ σ(A)} and λ+ := max{|λ| : λ ∈ σ(A)}. Otherwise, we need to choose them sufficiently close to these equalities according to what we need in the arguments below. It was proved in [10, p. 5, Lemma 2.2] that, for a given dilation A, there exist an open ellipsoid Δ and r ∈ (1, ∞) such that Δ ⊂ rΔ ⊂ AΔ, and one may additionally assume that |Δ| = 1, where |Δ| denotes

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the n-dimensional Lebesgue measure of the set Δ. For any k ∈ Z, let Bk := Ak Δ. Obviously, Bk is open, Bk ⊂ rBk ⊂ Bk+1 and |Bk | = bk . An ellipsoid x + Bk for some x ∈ Rn and k ∈ Z is called a dilated ball. Denote by B the set of all such dilated balls, namely, B := {x + Bk : x ∈ Rn , k ∈ Z} .

(2.1)

Throughout this article, let τ be the minimal integer such that rτ ≥ 2. Then, for any k ∈ Z, it holds true that Bk + Bk ⊂ Bk+τ

(2.2)

Bk + (Bk+τ ) ⊂ (Bk ) ,

(2.3)

and

where E + F denotes the algebraic sum {x + y : x ∈ E, y ∈ F } of sets E, F ⊂ Rn . The notion of the homogeneous quasi-norm induced by A was introduced in [10, p. 6, Definition 2.3] as follows. Definition 2.2. A measurable mapping ρ : Rn → [0, ∞) is called a homogeneous quasi-norm, associated with a dilation A, if (i) x = 0n implies that ρ(x) ∈ (0, ∞), here and hereafter, 0n denotes the origin of Rn ; (ii) ρ(Ax) = bρ(x) for any x ∈ Rn ; (iii) ρ(x + y) ≤ H[ρ(x) + ρ(y)] for any x, y ∈ Rn , where H ∈ [1, ∞) is a constant independent of x and y. In the standard dyadic case A := 2In×n , ρ(x) := |x|n for any x ∈ Rn is an example of the homogeneous quasi-norm associated with A. In [10, p. 6, Lemma 2.4], it was proved that all homogeneous quasi-norms associated with A are equivalent. Therefore, for a given dilation A, in what follows, we always use the step homogeneous quasi-norm ρ defined by setting, for any x ∈ Rn , ρ(x) :=



bj χBj+1 \Bj (x) when x = 0n ,

or else ρ( 0n ) := 0

j∈Z

for convenience. Obviously, for any k ∈ Z, Bk = {x ∈ Rn : ρ(x) < bk }. Observe that (Rn , ρ, dx) is a space of homogeneous type in the sense of Coifman and Weiss [19,20], here and hereafter, dx denotes the n-dimensional Lebesgue measure, and, moreover, (Rn , ρ, dx) is actually an RD-space (see [39,82]). Recall that a measurable function p(·) : Rn → (0, ∞] is called a variable exponent. For any variable exponent p(·), let p− := ess inf p(x) and p+ := ess sup p(x). n x∈R

(2.4)

x∈Rn

Denote by P(Rn ) the set of all variable exponents p(·) satisfying 0 < p− ≤ p+ < ∞. Let f be a measurable function on Rn and p(·) ∈ P(Rn ). Then the modular functional (or, for simplicity, the modular) p(·) , associated with p(·), is defined by setting  |f (x)|p(x) dx

p(·) (f ) := Rn

and the Luxemburg (also called Luxemburg–Nakano) quasi-norm f Lp(·) (Rn ) by

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  f Lp(·) (Rn ) := inf λ ∈ (0, ∞) : p(·) (f /λ) ≤ 1 . Moreover, the variable Lebesgue space Lp(·) (Rn ) is defined to be the set of all measurable functions f satisfying that p(·) (f ) < ∞, equipped with the quasi-norm f Lp(·) (Rn ) . Remark 2.3. Let p(·) ∈ P(Rn ). (i) Obviously, for any r ∈ (0, ∞) and f ∈ Lp(·) (Rn ), |f |r Lp(·) (Rn ) = f rLrp(·) (Rn ) . Moreover, for any μ ∈ C and f, g ∈ Lp(·) (Rn ), μf Lp(·) (Rn ) = |μ|f Lp(·) (Rn ) and p

p

p

f + gLp(·) (Rn ) ≤ f Lp(·) (Rn ) + gLp(·) (Rn ) , here and hereafter, p := min{p− , 1}

(2.5)

with p− as in (2.4). In particular, when p− ∈ [1, ∞], Lp(·) (Rn ) is a Banach space (see [25, Theorem 3.2.7]). (ii) It was proved, in [22, Proposition 2.21], that, for any function f ∈ Lp(·) (Rn ) with f Lp(·) (Rn ) > 0, p(·) (f /f Lp(·) (Rn ) ) = 1 and, in [22, Corollary 2.22], that, if f Lp(·) (Rn ) ≤ 1, then p(·) (f ) ≤ f Lp(·) (Rn ) . A function p(·) ∈ P(Rn ) is said to satisfy the globally log-Hölder continuous condition, denoted by p(·) ∈ C log (Rn ), if there exist two positive constants Clog (p) and C∞ , and p∞ ∈ R such that, for any x, y ∈ Rn , |p(x) − p(y)| ≤

Clog (p) log(e + 1/ρ(x − y))

(2.6)

C∞ . log(e + ρ(x))

(2.7)

and |p(x) − p∞ | ≤

The following variable Lorentz space Lp(·),q (Rn ) is known as a special case of the variable Lorentz space Lp(·),q(·) (Rn ) investigated by Kempka and Vybíral in [44]. Definition 2.4. Let p(·) ∈ P(Rn ). The variable Lorentz space Lp(·),q (Rn ) is defined to be the set of all measurable functions f such that

f Lp(·),q (Rn ) :=

⎧⎡ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎣ λq χ{x∈Rn : ⎨

0 ⎪  ⎪ ⎪ ⎪ ⎪ ⎩ sup λ χ{x∈Rn : λ∈(0,∞)

is finite.

q p(·) |f (x)|>λ} L



(Rn )

⎤ q1 dλ ⎦ λ 

|f (x)|>λ} Lp(·) (Rn )

when q ∈ (0, ∞), when q = ∞

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From [44, Lemma 2.4 and Theorem 3.1], we deduce the following Lemmas 2.5 and 2.6, respectively. Lemma 2.5. Let p(·) ∈ P(Rn ) and q ∈ (0, ∞]. Then, for any measurable function f ,  f Lp(·),q (Rn ) ∼



2

kq

χ{x∈Rn :

q p(·) |f (x)|>2k } L

1/q (Rn )

k∈Z

with the usual interpretation for q = ∞, where the equivalent positive constants are independent of f . Lemma 2.6. Let p(·) ∈ P(Rn ) and q ∈ (0, ∞]. Then  · Lp(·),q (Rn ) defines a quasi-norm on Lp(·),q (Rn ). It is easy to obtain the following result, the details being omitted. Lemma 2.7. Let p(·) ∈ P(Rn ) and q ∈ (0, ∞]. Then, for any r ∈ (0, ∞) and f ∈ Lp(·),q (Rn ), |f |r Lp(·),q (Rn ) = f rLrp(·),rq (Rn ) . From the monotone convergence theorem of Lp(·) (Rn ) (see [22, Corollary 2.64]), we easily deduce the following monotone convergence property of Lp(·),q (Rn ), the details being omitted. Proposition 2.8. Let p(·) ∈ P(Rn ) and {fk }k∈N ⊂ Lp(·),q (Rn ) be any sequence of non-negative functions satisfying that fk , as k → ∞, increases pointwisely almost everywhere to f ∈ Lp(·),q (Rn ) in Rn . Then f − fk Lp(·),q (Rn ) → 0

as k → ∞.

Throughout this article, denote by S(Rn ) the space of all Schwartz functions, namely, the set of all C (Rn ) functions ϕ satisfying that, for every integer ∈ Z+ and multi-index α ∈ (Z+ )n , ∞

ϕα, := sup [ρ(x)] |∂ α ϕ(x)| < ∞. x∈Rn

These quasi-norms { · α, }α∈(Z+ )n ,∈Z+ also determine the topology of S(Rn ). We use S  (Rn ) to denote the dual space of S(Rn ), namely, the space of all tempered distributions on Rn equipped with the weak-∗ topology. For any N ∈ Z+ , let SN (Rn ) := {ϕ ∈ S(Rn ) : ϕα, ≤ 1, |α| ≤ N, ≤ N }; equivalently,    N ϕ ∈ SN (Rn ) ⇐⇒ ϕSN (Rn ) := sup sup |∂ α ϕ(x)| max 1, [ρ(x)] ≤ 1. |α|≤N x∈Rn

In what follows, for any ϕ ∈ S(Rn ) and k ∈ Z, let ϕk (·) := b−k ϕ(A−k ·). Definition 2.9. Let ϕ ∈ S(Rn ) and f ∈ S  (Rn ). The non-tangential maximal function Mϕ (f ) and the radial maximal function Mϕ0 (f ) of f with respect to ϕ are defined, respectively, by setting, for any x ∈ Rn , Mϕ (f )(x) :=

sup y∈x+Bk ,k∈Z

|f ∗ ϕk (y)|

(2.8)

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and Mϕ0 (f )(x) := sup |f ∗ ϕk (x)|.

(2.9)

k∈Z

For any given N ∈ N, the non-tangential grand maximal function MN (f ) and the radial grand maximal 0 function MN (f ) of f ∈ S  (Rn ) are defined, respectively, by setting, for any x ∈ Rn , MN (f )(x) :=

sup

Mϕ (f )(x)

(2.10)

ϕ∈SN (Rn )

and 0 MN (f )(x) :=

sup ϕ∈SN (Rn )

Mϕ0 (f )(x).

We now introduce anisotropic variable Hardy–Lorentz spaces as follows. Definition 2.10. Let p(·) ∈ C log (Rn ), q ∈ (0, ∞] and N ∈ N ∩ [ ( p1 − 1) lnlnλb−  + 2, ∞), where p is as in (2.5). p(·),q

The anisotropic variable Hardy–Lorentz space, denoted by HA p(·),q

HA p(·),q

and, for any f ∈ HA

(Rn ), is defined by setting

  0 (Rn ) := f ∈ S  (Rn ) : MN (f ) ∈ Lp(·),q (Rn )

0 (Rn ), let f H p(·),q (Rn ) := MN (f )Lp(·),q (Rn ) . A

Remark 2.11. p(·),q

(i) Even though the quasi-norm of HA (Rn ) in Definition 2.10 depends on N , it follows from Theorem 3.8 p(·),q below that the space HA (Rn ) is independent of the choice of N as long as     1 ln b + 2, ∞ ∩ N. N∈ −1 p ln λ− p(·),q

p,q If p(·) ≡ p ∈ (0, ∞), then the space HA (Rn ) is just the anisotropic Hardy–Lorentz space HA (Rn ) investigated by Liu et al. in [51] and, if A := d In×n for some d ∈ R with |d| ∈ (1, ∞) and q = ∞, then p(·),∞ the space HA (Rn ) becomes the variable weak Hardy space introduced by Yan et al. in [79]. (ii) Very recently, via the variable Lorentz spaces Lp(·),q(·) (Rn ) in [29], where

p(·), q(·) : (0, ∞) → (0, ∞) are bounded measurable functions, Almeida et al. [4] investigated the anisotropic variable Hardy– Lorentz spaces H p(·),q(·) (Rn , A) on Rn . As was mentioned in [44, Remark 2.6], the space Lp(·),q(·) (Rn ) in [29] never goes back to the space Lp(·) (Rn ), since the variable exponent p(·) in Lp(·),q(·) (Rn ) is only p(·),q defined on (0, ∞) while not on Rn . On the other hand, the space HA (Rn ), in this article, is defined p(·),q(·) n via the variable Lorentz space L (R ) (with q(·) ≡ a constant ∈ (0, ∞]) from [44], which is not p(·),q(·) n covered by the space H (R , A) in [4]. Moreover, as was pointed out in [4, p. 6], the key tool of [4] is the fact that the set L1loc (Rn ) ∩ H p(·),q(·) (Rn , A) is dense in H p(·),q(·) (Rn , A). Therefore, the method p(·),q used in [4] does not work for HA (Rn ) in the present article, due to the lack of a dense function p(·),∞ subspace of HA (Rn ) even when p(·) ≡ a constant ∈ (0, ∞) and A := d In×n for some d ∈ R with |d| ∈ (1, ∞).

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p(·),q

3. Maximal function characterizations of HA

9

Rn ) (R

p(·),q

In this section, we characterize HA (Rn ) in terms of the radial maximal function Mϕ0 (see (2.9)) or the non-tangential maximal function Mϕ (see (2.8)). We begin with the following Definitions 3.1 and 3.2 from [10]. Definition 3.1. For any function F : Rn × Z → [0, ∞), K ∈ Z ∪ {∞} and ∈ Z, the maximal function F∗K of F with aperture is defined by setting, for any x ∈ Rn , F∗K (x) :=

sup

sup

F (y, k).

k∈Z, k≤K y∈x+Bk+ 0(K,L)

Definition 3.2. Let K ∈ Z, L ∈ [0, ∞) and N ∈ N. For any φ ∈ S, the radial maximal function Mφ (K,L) Mφ (f )

the non-tangential maximal function and the tangential maximal function S  (Rn ) are, respectively, defined by setting, for any x ∈ Rn , 0(K,L)

Mφ

(K,L)

Mφ

(f )(x) :=

N (K,L) Tφ (f )

(f ),

of f ∈

   −L  −L |(f ∗ φk )(x)| max 1, ρ A−K x , 1 + b−k−K

sup k∈Z, k≤K

(f )(x) :=

   −L  −L sup |(f ∗ φk )(y)| max 1, ρ A−K y 1 + b−k−K

sup

k∈Z, k≤K y∈x+Bk

and N (K,L)

Tφ

(f )(x) :=

sup

sup

k∈Z, k≤K y∈Rn

|(f ∗ φk )(y)| (1 + b−k−K )−L . −k N [max {1, ρ (A (x − y))}] [max {1, ρ (A−K y)}]L 0(K,L)

Furthermore, the radial grand maximal function MN (f ) and the non-tangential grand maximal function (K,L) MN (f ) of f ∈ S  (Rn ) are, respectively, defined by setting, for any x ∈ Rn , 0(K,L)

MN

(f )(x) :=

sup φ∈SN

(Rn )

0(K,L)

Mφ

(f )(x)

and (K,L)

MN

(f )(x) :=

sup φ∈SN

(Rn )

(K,L)

Mφ

(f )(x).

For any r ∈ (0, ∞), denote by Lrloc (Rn ) the set of all locally r-integrable functions on Rn and, for any measurable set E ⊂ Rn , by Lr (E) the set of all measurable functions f such that

f Lr (E)

⎡ ⎤1/r  := ⎣ |f (x)|r dx⎦ < ∞. E

Recall that the Hardy–Littlewood maximal operator MHL (f ) is defined by setting, for any f ∈ L1loc (Rn ) and x ∈ Rn ,   1 1 MHL (f )(x) := sup sup |f (z)| dz = sup |f (z)| dz, (3.1) x∈B∈B |B| k∈Z y∈x+Bk |Bk | y+Bk

where B is as in (2.1).

B

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Observe that (Rn , ρ, dx) is a space of homogeneous type in the sense of Coifman and Weiss [19,20]. From this and [41, Theorems 5.2 and 4.3], we deduce the following lemma, which is an anisotropic version of [22, Theorem 3.16], the details being omitted. Lemma 3.3. Let p(·) ∈ C log (Rn ). (i) If 1 ≤ p− ≤ p+ < ∞, then, for any s ∈ [1, ∞), sp(·) ∈ C log (Rn ) and, for any f ∈ Lsp(·) (Rn ), sup λχ{x∈Rn :

λ∈(0,∞)



MHL (f )(x)>λ} Lsp(·) (Rn )

≤ Cf Lsp(·) (Rn ) ,

where C is a positive constant independent of f . (ii) If 1 < p− ≤ p+ < ∞, then, for any s ∈ [1, ∞), sp(·) ∈ C log (Rn ) and, for any f ∈ Lsp(·) (Rn ),  Lsp(·) (Rn ) , MHL (f )Lsp(·) (Rn ) ≤ Cf  is a positive constant independent of f . where C Moreover, as a simple consequence of [44, Theorem 4.1], [79, Theorem 3.1] and Lemma 3.3(ii), we immediately obtain the following boundedness of MHL on Lp(·),q (Rn ), which is of independent interest, the details being omitted. Lemma 3.4. Let p(·) ∈ C log (Rn ) satisfy 1 < p− ≤ p+ < ∞, where p− and p+ are as in (2.4), and q ∈ (0, ∞]. Then the Hardy–Littlewood maximal operator MHL is bounded on Lp(·),q (Rn ). Lemma 3.5. Let p(·) ∈ C log (Rn ), N ∈ (1/p− , ∞) ∩ N and φ ∈ S(Rn ). Then there exists a positive constant C such that, for any K ∈ Z, L ∈ [0, ∞) and f ∈ S  (Rn ), N (K,L) (K,L) (f ) p(·),q ≤ C Mφ (f ) p(·),q . (3.2) Tφ L

(Rn )

L

(Rn )

Proof. We first prove that, for any p(·) ∈ C log (Rn ), K ∈ Z and ∈ [  , ∞) ∩ Z, ∗K F p(·),q n  b(− )/p− F∗K p(·),q n ,  L (R ) L (R )

(3.3)

where F∗K is as in Definition 3.1 and, for any φ ∈ S(Rn ), f ∈ S  (Rn ), k ∈ Z and y ∈ Rn , −L    −L  1 + b−k−K F (y, k) := |(f ∗ φk )(y)| max 1, ρ A−K y . Indeed, by a proof similar to that of [10, p. 42, Lemma 7.2], we easily find that, for any ,  ∈ Z with ≥  and λ ∈ (0, ∞),     x ∈ Rn : F∗K (x) > λ ⊂ x ∈ Rn : MHL (χΩλ )(x) ≥ b −−τ ,



(3.4)

where, for any λ ∈ (0, ∞), Ωλ := {x ∈ Rn : F∗K (x) > λ}. Then, by (3.4), Lemma 3.3(i) and Remark 2.3(i),  we know that χ{x∈Rn : F∗K (x)>λ} p(·) n ≤ χ{x∈Rn : MHL (χΩ )(x)≥b −−τ } p(·) n L

(R )

λ

= χ{x∈Rn :

MHL (χΩλ

L

(R )

1/p−  −−τ p(·)  )(x)≥b } p− L

(Rn )

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∼b b

− p−

− p−

  −−τ χ{x∈Rn : b

MHL (χΩλ

1/p−

− p−

χΩλ 

∼b

p(·)

)(x)≥b −−τ }

11

1/p− p(·) p− L

(Rn )

χΩλ Lp(·) (Rn ) ,

L p− (Rn )

(3.5)

which, together with the definition of Ωλ and Definition 2.4, further implies (3.3). On the other hand, by [51, (4.7)], for any N ∈ N, K ∈ Z, L ∈ [0, ∞), f ∈ S  (Rn ) and x ∈ Rn , we have N (K,L)

Tφ

(f )(x) ≤

∞ 

∗K Fj+1 (x)b−jN .

(3.6)

j=0

Now we show (3.2). By (3.6), the Aoki–Rolewicz theorem (see [8,64]), (3.5) and the fact that N ∈ (1/p− , ∞) ∩ N, it is easy to see that there exists υ ∈ (0, 1] such that N (K,L) υ (f ) Tφ p(·),q L

(Rn )

≤

∞ 

∗K υ p(·),q n b−jN υ Fj+1 (R ) L

j=0



∞ 

υ (K,L) υ b−jN υ b(j+1)υ/p− F0∗K Lp(·),q (Rn )  Mφ (f ) p(·),q L

j=0

(Rn )

,

2

which implies (3.2) and hence completes the proof of Lemma 3.5.

The following Lemmas 3.6 and 3.7 are just [10, p. 45, Lemma 7.5 and p. 46, Lemma 7.6], respectively.  Lemma 3.6. Let φ ∈ S(Rn ) and Rn φ(x) dx = 0. Then, for any given N ∈ N and L ∈ [0, ∞), there exist an I ∈ N and a positive constant C(N,L) , depending on N and L, such that, for any K ∈ Z+ , f ∈ S  (Rn ) and x ∈ Rn , 0(K,L)

MI

N (K,L)

(f )(x) ≤ C(N,L) Tφ

(f )(x).

 Lemma 3.7. Let φ ∈ S(Rn ) and Rn φ(x) dx = 0. Then, for any given M ∈ (0, ∞) and K ∈ Z+ , there exist L ∈ (0, ∞) and a positive constant C(K, M ) , depending on K and M , such that, for any f ∈ S  (Rn ) and x ∈ Rn , (K,L)

Mφ

−M

(f )(x) ≤ C(K, M ) [max {1, ρ(x)}]

.

(3.7)

Now we state the main result of this section as follows. Theorem 3.8. Suppose that p(·) ∈ C log (Rn ) and φ ∈ S(Rn ) with the following statements are mutually equivalent:

 Rn

φ(x) dx = 0. Then, for any f ∈ S  (Rn ),

p(·),q

(i) f ∈ HA (Rn ); (ii) Mφ (f ) ∈ Lp(·),q (Rn ); (iii) Mφ0 (f ) ∈ Lp(·),q (Rn ). In this case, it holds true that f H p(·),q (Rn ) ≤ C1 Mφ0 (f ) Lp(·),q (Rn ) ≤ C1 Mφ (f )Lp(·),q (Rn ) ≤ C2 f H p(·),q (Rn ) , A

where C1 and C2 are two positive constants independent of f .

A

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12

Proof. Obviously, (i) implies (ii) and (ii) implies (iii). Thus, to prove Theorem 3.8, it suffices to show that (ii) implies (i) and that (iii) implies (ii). We first prove that (ii) implies (i). To this end, notice that, by Lemma 3.6 with N ∈ (1/p− , ∞) ∩ N 0(K,0) N (K,0) and L = 0, we find that there exists an I ∈ N such that MI (f )(x)  Tφ (f )(x) for any x ∈ Rn ,  n f ∈ S (R ) and K ∈ Z+ . From this and Lemma 3.5, we further deduce that, for any K ∈ Z+ and f ∈ S  (Rn ), 0(K,0) (f ) MI

Lp(·),q (Rn )

(K,0)  Mφ (f )

Lp(·),q (Rn )

.

(3.8)

Letting K → ∞ in (3.8), by Proposition 2.8, we know that 0 MI (f )

Lp(·),q (Rn )

 Mφ (f )Lp(·),q (Rn ) ,

which shows that (ii) implies (i). Now we show that (iii) implies (ii). Assume now that Mφ0 (f ) ∈ Lp(·),q (Rn ). By Lemma 3.7 via choosing M ∈ (1/p− , ∞), we find that there exists L ∈ (0, ∞) such that (3.7) holds true, which further implies that (K,L) Mφ (f ) ∈ Lp(·),q (Rn ) for any K ∈ Z+ . Indeed, when q ∈ (0, ∞) and M ∈ (1/p− , ∞), we have (K,L) q (f ) p(·),q Mφ L

∞ (Rn )

=

λq−1 χ{x∈Rn :

(K,L) Mφ (f )(x)>λ}

λq−1 χ{x∈Rn :

(K,L) Mφ (f )(x)>λ}

q

dλ

q

dλ

Lp(·) (Rn )

0

1 

Lp(·) (B1 )

0 −kM

+

∞ b  k=1



∞ 

λq−1 χ{x∈Rn :

(K,L)

Mφ

q

(f )(x)>λ}

0

b−kM q b

(k+1)q p−

Lp(·) (Bk+1 \Bk )

dλ

∼ 1.

(3.9)

k=0 (K,L)

Clearly, when q = ∞, (3.9) still holds true. Thus, Mφ (f ) ∈ Lp(·),q (Rn ). On the other hand, by Lemmas 3.6 and 3.5, we know that, for any L ∈ (0, ∞), there exists some I ∈ N such that, for any K ∈ Z+ and f ∈ S  (Rn ), 0(K,L) (f ) MI

Lp(·),q (Rn )

(K,L) ≤ C3 Mφ (f )

Lp(·),q (Rn )

,

where C3 is a positive constant independent of K and f . For any fixed K ∈ Z+ , let   0(K,L) (K,L) ΩK := x ∈ Rn : MI (f )(x) ≤ C4 Mφ (f )(x) with C4 := 2C3 . Then (K,L) (f ) Mφ

Lp(·),q (Rn )

(K,L)  Mφ (f )

Lp(·),q (ΩK )

,

(3.10)

because (K,L) (f ) Mφ

Lp(·),q (Ω K)

0(K,L) ≤ C4−1 MI (f )

Lp(·),q (Ω K)

(K,L) ≤ C3 /C4 Mφ (f )

Lp(·),q (Rn )

.

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For any given L ∈ [0, ∞), by an argument similar to that used in the proof of [51, (4.17)], we conclude that, for any t ∈ (0, p− ), K ∈ Z+ , f ∈ S  (Rn ) and x ∈ ΩK , (K,L) Mφ (f )(x)



 !1/t t  0(K,L) (x) MHL Mφ (f ) .

(3.11)

Then, by (3.10), Lemma 2.7, (3.11) and Lemma 3.4, we further find that, for any K ∈ Z+ and f ∈ S  (Rn ),  t (K,L) t (K,L) t (K,L) M (f ) p(·),q n  Mφ (f ) p(·),q ∼ (f ) Mφ φ p(·) , q (R ) (ΩK ) L L L t t (ΩK )    t 0(K,L)  (f ) MHL Mφ p(·) , q L t t (ΩK )  t 0(K,L) t 0(K,L) M  (f ) ∼ (f ) , M φ φ p(·) q L

(K,L)

t

,

t

Lp(·),q (Rn )

(Rn )

0(K,L)

which, together with the fact that Mφ (f ) and Mφ (f ) converge pointwisely and monotonically, respectively, to Mφ (f ) and Mφ0 (f ) as K → ∞ and Proposition 2.8, implies that Mφ (f )Lp(·),q (Rn )  Mφ0 (f ) Lp(·),q (Rn ) . This shows that (iii) implies (ii) and hence finishes the proof of Theorem 3.8. p(·),q

4. Atomic characterization of HA

2

Rn ) (R p(·),q

In this section, we establish the atomic characterization of HA notion of anisotropic (p(·), r, s)-atoms.

(Rn ). We begin with the following

Definition 4.1. Let p(·) ∈ P(Rn ), r ∈ (1, ∞] and     1 ln b , ∞ ∩ Z+ . s∈ −1 p− ln λ−

(4.1)

A measurable function a on Rn is called an anisotropic (p(·), r, s)-atom if (i) supp a ⊂ B, where B ∈ B and B is as in (2.1); (ii) aLr (Rn ) ≤ χB|B| Lp(·) (Rn ) ;  α (iii) Rn a(x)x dx = 0 for any α ∈ Zn+ with |α| ≤ s. 1/r

For the presentation simplicity, throughout this article, we call an anisotropic (p(·), r, s)-atom simply by a (p(·), r, s)-atom. Now, via (p(·), r, s)-atoms, we introduce the notion of the anisotropic variable atomic p(·),r,s,q Hardy–Lorentz space HA (Rn ) as follows. Definition 4.2. Let p(·) ∈ C log (Rn ), q ∈ (0, ∞], r ∈ (1, ∞], s be as in (4.1) and A a dilation. The anisotropic p(·),r,s,q variable atomic Hardy–Lorentz space HA (Rn ) is defined to be the set of all f ∈ S  (Rn ) satisfying that k there exist a sequence of (p(·), r, s)-atoms, {ai }i∈N, k∈Z , supported, respectively, on {Bik }i∈N, k∈Z ⊂ B and n  such that "  a positive constant C i∈N χAj0 Bik (x) ≤ C for any x ∈ R and k ∈ Z, with some j0 ∈ Z \ N, and f=

 k∈Z i∈N

λki aki

in

S  (Rn ),

(4.2)

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where λki ∼ 2k χBik Lp(·) (Rn ) for any k ∈ Z and i ∈ N with the equivalent positive constants independent of k and i. p(·),r,s,q Moreover, for any f ∈ HA (Rn ), define ⎤ q1

⎡

f H p(·),r,s,q (Rn ) A

$ p %1/p  q λki χBik ⎢ := inf ⎣ χ  k p(·) (Rn ) L B p(·) i i∈N k∈Z L

⎥ ⎦ (Rn )

with the usual interpretation for q = ∞, where the infimum is taken over all decompositions of f as above. p(·),q

In order to establish the atomic characterization of HA (Rn ), we need several technical lemmas as follows. First, observe that (Rn , ρ, dx) is an RD-space (see [39,82]). From this and [85, Theorem 2.7], we deduce the following Fefferman–Stein vector-valued inequality of the maximal operator MHL on the variable Lebesgue space Lp(·) (Rn ), the details being omitted. Lemma 4.3. Let r ∈ (1, ∞]. Assume that p(·) ∈ C log (Rn ) satisfies 1 < p− ≤ p+ < ∞. Then there exists a positive constant C such that, for any sequence {fk }k∈N of measurable functions, $ %1/r  r [M (f )] HL k k∈N

Lp(·) (Rn )

' (1/r  r ≤C |f | k k∈N

Lp(·) (Rn )

with the usual modification made when r = ∞, where MHL denotes the Hardy–Littlewood maximal operator as in (3.1). Remark 4.4. (i) Let p(·) ∈ C log (Rn ) and i ∈ Z+ . Then, by Lemma 4.3 and the fact that, for any dilated ball B ∈ B i 1 and r ∈ (0, p), χAi B ≤ b r [MHL (χB )] r , we conclude that there exists a positive constant C such that, for any sequence {B (k) }k∈N ⊂ B,  χAi B (k) k∈N

p(·),r,s,q

 ≤ Cb χB (k) i r

Lp(·) (Rn )

k∈N

. Lp(·) (Rn )

p(·),r,s,q

(ii) Let f ∈ HA (Rn ). By the definition of HA (Rn ), we know that there exists a sequence k {ai }i∈N,k∈Z of (p(·), r, s)-atoms supported, respectively, on {Bik }i∈N, k∈Z ⊂ B, satisfying that, for any "   k ∈ Z and x ∈ Rn , i∈N χAj0 Bik (x) ≤ C with some j0 ∈ Z \ N and C being a positive constant independent of k and x such that f admits a decomposition as in (4.2) with λki ∼ 2k χBik Lp(·) (Rn ) for any k ∈ Z and i ∈ N, where the equivalent positive constants are independent of k and i, and ⎤ q1

⎡

f H p(·),r,s,q (Rn ) A

$ p %1/p  q k  λ χ k B i ⎢ i ∼⎣ χBik Lp(·) (Rn ) p(·) k∈Z i∈N L

with the equivalent positive constants independent of f . Moreover, by the definition of λki and (i) of this remark, we further conclude that

" i∈N

⎥ ⎦ (Rn )

 for any k ∈ Z, χAj0 Bik ≤ C

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⎤ q1

⎡

f H p(·),r,s,q (Rn ) A

' (1/p  q  ⎢ ∼⎣ 2kq χBik p(·) i∈N k∈Z L

⎡ ∼⎣



k∈Z

⎤ q1

⎡

' (1/p  q  ⎥ ⎢ 2kq χAj0 Bik ⎦ ∼⎣ p(·) i∈N k∈Z L

⎡

⎤ q1

q  kq 2 χAj0 Bik i∈N

(Rn )

15

⎦ ∼⎣



k∈Z

Lp(·) (Rn )

(Rn )

⎤ q1

q  kq 2 χBik i∈N

⎥ ⎦

⎦ ,

Lp(·) (Rn )

where the equivalent positive constants are independent of f . We also need the following useful technical lemma, whose proof is similar to that of [69, Lemma 4.1], the details being omitted. Lemma 4.5. Let p(·) ∈ C log (Rn ), t ∈ (0, p] and r ∈ [1, ∞] ∩ (p+ , ∞], where p and p+ are, respectively, as in (2.5) and (2.4). Then there exists a positive constant C such that, for any sequence {B (k) }k∈N ⊂ B of dilated balls, numbers {λk }k∈N ⊂ C and measurable functions {ak }k∈N satisfying that, for each k ∈ N, supp ak ⊂ B (k) and ak Lr (Rn ) ≤ |B (k) |1/r , it holds true that ' ( 1t  t |λk ak | k∈N

Lp(·) (Rn )

' ( 1t  t ≤C |λk χB (k) | k∈N

.

Lp(·) (Rn )

The following Proposition 4.6 and Lemma 4.7 are just [10, p. 17, Proposition 3.10 and p. 9, Lemma 2.7], respectively. Proposition 4.6. For any given N ∈ N, there exists a positive constant C(N ) , depending only on N , such that, for any f ∈ S  (Rn ) and x ∈ Rn , 0 0 MN (f )(x) ≤ MN (f )(x) ≤ C(N ) MN (f )(x), 0 where MN (f ) and MN (f ) are as in Definition 2.9.

Lemma 4.7. Let Ω ⊂ Rn be an open set with |Ω| < ∞. Then, for any m ∈ Z+ , there exist a sequence of points, {xk }k∈N ⊂ Ω, and a sequence of integers, { k }k∈N ⊂ Z, such that (i) (ii) (iii) (iv) (v)

) Ω = k∈N (xk + Bk ); {xk + Bk −τ }k∈N are pairwise disjoint, where τ is as in (2.2) and (2.3); for each k ∈ N, (xk + Bk +m ) ∩ Ω = ∅, but (xk + Bk +m+1 ) ∩ Ω = ∅; if (xi + Bi +m−2τ ) ∩ (xj + Bj +m−2τ ) = ∅, then | i − j | ≤ τ ; for any i ∈ N, {j ∈ N : (xi + Bi +m−2τ ) ∩ (xj + Bj +m−2τ ) = ∅} ≤ R, where R is a positive constant independent of Ω and i.

Now, it is a position to state the main result of this section. Theorem 4.8. Let p(·) ∈ C log (Rn ), q ∈ (0, ∞], r ∈ (max{p+ , 1}, ∞] with p+ as in (2.4) and s be as in (4.1). p(·),q p(·),r,s,q Then HA (Rn ) = HA (Rn ) with equivalent quasi-norms. Proof. First, we show that p(·),r,s,q

HA

p(·),q

(Rn ) ⊂ HA

(Rn ).

(4.3)

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16

p(·),r,s,q

To this end, for any f ∈ HA (Rn ), by Remark 4.4(ii), we find that there exists a sequence of k (p(·), r, s)-atoms, {ai }i∈N, k∈Z , supported, respectively, on {Bik }i∈N, k∈Z ⊂ B such that f=



in S  (Rn ),

λki aki

k∈Z i∈N

where λki ∼ 2k χBik Lp(·) (Rn ) for any k ∈ Z and i ∈ N, x ∈ Rn and k ∈ Z, and ⎡ f H p(·),r,s,q (Rn ) ∼ ⎣ A

 k∈Z

" i∈N

χAj0 Bik (x)  1 with some j0 ∈ Z \ N for any ⎤ q1

q  2kq χBik i∈N

⎦ .

(4.4)

Lp(·) (Rn )

0 For the notational simplicity, in the remainder of this proof, for any f ∈ S  (Rn ), we denote MN (f ) simply p(·),q 1 ln b n by M (f ) when N ∈ N ∩ [ ( p − 1) ln λ−  + 2, ∞). To prove f ∈ HA (R ), by Definition 2.10 and Lemma 2.5, it suffices to show that





2

kq

χ{x∈Rn :

q p(·) M (f )(x)>2k } L

 q1 (Rn )

 f H p(·),r,s,q (Rn ) . A

k∈Z

For any fixed k0 ∈ Z, we write

f=

k 0 −1



λki aki +

k=−∞ i∈N

∞  

· · · =: f1 + f2 .

k=k0 i∈N

Then, by Remark 2.3(i), we know that χ{x∈Rn :



M (f )(x)>2k0 } Lp(·) (Rn )

 χ{x∈Rn :



M (f1 )(x)>2k0 −1 } Lp(·) (Rn )

+ χ{x∈(Ek



+ χ{x∈Ek0 :



M (f2 )(x)>2k0 −1 } p(·) n L (R )

k0 −1 }  0 ) : M (f2 )(x)>2 Lp(·) (Rn )

=: J1 + J2 + J3 ,

(4.5)

)∞ ) where Ek0 := k=k0 i∈N Aτ Bik . For J1 , clearly, we have " " χ J1  k −1 0 k M (ak )(x)χ k0 −2 } {x∈Rn : k=−∞ λ (x)>2 i i∈N i Aτ Bik Lp(·) (Rn ) " " + χ{x∈Rn : k0 −1 k k k −2 0 λ M (a )(x)χ (x)>2 } k=−∞

i∈N

=: J1,1 + J1,2 .

i

i

(Aτ Bik )

Lp(·) (Rn )

(4.6)

To deal with J1,1 , by the Hölder inequality, we find that, for any given t ∈ (0, min{p, q}), q ∈ r (1, min{ max{p , 1 }), δ ∈ (0, 1 − 1q ) and any x ∈ Rn , + ,1} t k 0 −1



k=−∞ i∈N

λki M (aki )(x)χAτ Bik (x)

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' ≤

k 0 −1



2kδq

(1/q ⎧ k −1 0 ⎨  ⎩

k=−∞

=

q⎫1/q ⎬

 2−kδq



λki M (aki )(x)χAτ Bik (x)

⎭

i∈N

k=−∞

⎧ 0 −1 ⎨ k

k0 δ

17

2 2−kδq (2δq − 1)1/q ⎩k=−∞

q⎫1/q ⎬





λki M (aki )(x)χAτ Bik (x)

,

⎭

i∈N

where q denotes the conjugate index of q, namely, 1q + q1 = 1. By this, the facts that qt < 1 and M (f )(x)  MHL (f )(x) for any x ∈ Rn , Remark 2.3(i) and [22, Theorem 2.61], we further conclude that "k0 −1 " J1,1  χ{x∈Rn : 2k0 δ { 2−kδq [ λk M (ak )(x)χ (x)]q }1/q >2k0 −2 }   (2δ q −1)1/q

k=−∞

i∈N

i

i

 q k 0 −1  −k0 q(1−δ) −kδ q k k 2 2 λi M (ai )χAτ Bik k=−∞ i∈N

Aτ Bik

Lp(·) (Rn )

Lp(·) (Rn )

1 k −1  -qt 0  t   2−k0 q(1−δ) 2(1−δ)kqt χBik p(·) n MHL (aki )χAτ Bik p(·) L (R )

i∈N

k=−∞

L

t

(Rn )

⎡

$   -qt % 1t t ⎢  2−k0 q(1−δ) ⎣ 2(1−δ)kqt χBik p(·) n MHL (aki )χAτ Bik L (R ) i∈N p(·) k=−∞

⎤ 1t

k 0 −1

L

⎥ ⎦ . (Rn )

On the other hand, since r > 1, then, from the boundedness of MHL on Lr (Rn ), we deduce that, for any k ∈ Z and i ∈ N,  -q q q . . q k  χBik p(·) n MHL (aki )χAτ Bik r n  .Bik . r , χBik p(·) n MHL (ai )χAτ Bik r/q n L (R ) (R ) L L (R ) L

(R )

which, combined with Lemma 4.5, Remark 4.4(i), the fact that t < q and the Hölder inequality, implies that ⎡

J1,1

' ( 1t  t ⎢  2−k0 q(1−δ) ⎣ 2(1−δ)kqt χAτ Bik p(·) i∈N k=−∞

⎤ 1t

k 0 −1

L

⎥ ⎦ (Rn )

⎧ ⎪ 0 −1 ⎨ k

' ( 1t  t  2−k0 q(1−δ) 2[(1−δ)q−ε]kt 2εkt χAj0 Bik ⎪ i∈N p(·) ⎩k=−∞ L

 2−k0 q(1−δ)

⎧ 0 −1 ⎨ k ⎩

k=−∞

$

k 0 −1

 2−k0 q(1−δ)

2

q [(1−δ) q −ε]kt q−t

k=−∞

⎡ ∼ 2−εk0 ⎣

k 0 −1

k=−∞

Lp(·) (Rn )

i∈N

⎪ ⎭

⎭

⎡ q % q−t qt k 0 −1  εkq ⎣ 2 χBik k=−∞

q  2εkq χBik

(Rn )

⎫ 1t ⎬

t  [(1−δ) q −ε]kt εkt 2 2 χAj0 Bik i∈N

⎫ 1t ⎪ ⎬

Lp(·) (Rn )

⎤ q1 ⎦ ,

i∈N

Lp(·) (Rn )

⎤ q1 ⎦

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18

where ε ∈ (1, (1 − δ) q ). From this and (4.4), we deduce that 

⎡

 q1



⎣

2k0 q (J1,1 )q

k0 ∈Z



2k0 q 2−εk0 q

k0 ∈Z

⎡ ∼⎣

k=−∞

∞   k∈Z k0 =k+1

⎡ ∼⎣



k 0 −1

2

kq

i∈N

⎦

Lp(·) (Rn )

⎤ q1

q  2(1−ε)k0 q 2εkq χBik i∈N

⎦

Lp(·) (Rn )

⎤ q1

q  χBik i∈N

k∈Z

⎤ q1

q  εkq 2 χBik

⎦ ∼ f  p(·),r,s,q n . H (R )

(4.7)

A

Lp(·) (Rn )

To estimate J1,2 , for any i ∈ N, k ∈ Z and x ∈ (Aτ Bik ) , by an argument similar to that used in the proof of [51, (3.27)], we have −1 M (aki )(x)  χBik p(·) L

(Rn )

−1 β  |Bik |β  (χ , M k k )(x) χ HL B B i i Lp(·) (Rn ) [ρ(x − xki )]β

(4.8)

where, for any i ∈ N, k ∈ Z, xki denotes the center of the dilated ball Bik and  β :=

ln b +s+1 ln λ−



1 ln λ− > . ln b p

(4.9)

By this, Remark 2.3(i), Lemma 4.3, Remark 4.4(i) and the Hölder inequality, we find that, for any t ∈ 1 1 , t ) and δ ∈ (0, 1 − q11 ), (0, min{q, 1/β}), q1 ∈ ( βt J1,2

 χ{x∈Rn :

2k0 δ  (2δq1 −1)1/q1

"k0 −1 " k M (ak )(x)χ −kδq1 [ q1 }1/q1 >2k0 −2 } { 2 λ (x)]  i k=−∞ i∈N i (Aτ B k )

k −1  q 1 0   −k0 q1 (1−δ) −kδq1 k k 2 2 λi M (ai )χ(Aτ Bik ) i∈N

k=−∞

 2−k0 q1 (1−δ)

⎧ 0 −1 ⎨ k ⎩

k=−∞

⎡

k 0 −1

 2−k0 q1 (1−δ) ⎣

k=−∞

 2−k0 q1 (1−δ)

⎩

k=−∞

$  2−k0 q1 (1−δ)

k 0 −1

i∈N

∼ 2−εk0 ⎣

k 0 −1 k=−∞

p(·) t

(Rn )

i∈N

⎫ 1t ⎬

Lp(·) (Rn )

⎡ % q−t qt ⎣

q [(1−δ)q1 −ε]kt q−t

k 0 −1

k=−∞

q  2εkq χBik i∈N

Lp(·) (Rn )

⎭

⎦ p(·) L t

t  2[(1−δ)q1 −ε]kt 2εkt χAj0 Bik

2

(Rn )

⎤ 1t

k=−∞

⎡

⎫ 1t ⎬

L

 (1−δ)kq1 t 2 χBik

⎧ 0 −1 ⎨ k

Lp(·) (Rn )

  / 0βtq1 (1−δ)kq1 t 2 MHL χBik i∈N

Lp(·) (Rn )

i

⎤ q1 ⎦ ,

⎭

q  εkq 2 χBik i∈N

Lp(·) (Rn )

⎤ q1 ⎦

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19

where ε ∈ (1, (1 − δ)q1 ). This, together with a calculation similar to that of (4.7), further implies that 

 q1



2k0 q (J1,2 )q

⎡ ⎣

k0 ∈Z



k∈Z

⎤ q1

q  2kq χBik i∈N

⎦ ∼ f  p(·),r,s,q n . H (R )

(4.10)

A

Lp(·) (Rn )

By this, (4.6) and (4.7), we have 



 q1 2k0 q (J1 )q

⎡ ⎣

k0 ∈Z



k∈Z

⎤ q1

q  kq 2 χBik i∈N

⎦ ∼ f  p(·),r,s,q n . H (R )

(4.11)

A

Lp(·) (Rn )

For J2 , by Remark 4.4(i), the Hölder inequality, we know that, for any t ∈ (0, min{p, q}) and δ ∈ (0, 1), J2  χEk0 Lp(·) (Rn )

∞    χAτ Bik k=k0 i∈N

∞    χBik k=k0 i∈N

' 

∞ 

⎡

Lp(·) (Rn )

( q−t qt 2

Lp(·) (Rn )

q −kδt q−t

⎡ ⎣

k=k0

t ∞   ⎣ χBik

⎤1/t

q  kδq 2 χBik

⎤1/q

i∈N

k=k0

∞ 

k=k0

⎦

Lp(·) (Rn )

i∈N

⎦

⎡

∞ 

∼ 2−k0 δ ⎣

k=k0

Lp(·) (Rn )

⎤1/q

q  kδq 2 χBik i∈N

⎦

.

Lp(·) (Rn )

From this, similarly to (4.7), we deduce that 



 q1 2k0 q (J2 )q

⎡ ⎣

k0 ∈Z

For J3 , since β >

1 p,

k∈Z

0

) :

⎤ q1

q  2kq χBik i∈N

⎦ ∼ f  p(·),r,s,q n . H (R )

(4.12)

A

Lp(·) (Rn )

it follows that there exist 0 < t < δ < 1 and L ∈ (1, ∞) such that βt >

qβtL > 1. By this, the value of J3  χ{x∈(Ek



"∞

λki ,

"

k=k0

(4.8), Lemma 4.3 and the Hölder inequality, we find that

k k k0 −1 } i∈N λi M (ai )(x)>2

∞    −tk0 k k t 2 λi M (ai ) χ(Ek ) 0 k=k0 i∈N



Lp(·) (Rn )

Lp(·) (Rn )

⎡

$ 1 % βt   k  k t λi M (ai ) χ(Ek ) 0 i∈N k=k0

⎤βt

∞  −tk0 ⎢ 2 ⎣

⎥ ⎦ Lβtp(·) (Rn )

⎡

⎤βt

$ %1  ∞ / 0βt βt  ⎢  2−tk0 ⎣ 2k/β MHL χBik i∈N k=k0

⎥ ⎦

Lβtp(·) (Rn )

⎡  2−tk0 ⎣

∞ 

k=k0

1 βt  k/β 2 χBik

i∈N

Lp(·) (Rn )

⎤βt ⎦

⎡  2−tk0 ⎣

∞  k=k0

k

δ

kδ

2 βL (1− t ) 2 βtL

1 βtL  χBik

i∈N

Lp(·) (Rn )

⎤βtL ⎦

1 p

and

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20



⎡  qβtL q

∞ 

 2−tk0

2

⎣

qβtL k δ βL (1− t ) qβtL−1

k=k0

⎡ ∼ 2−δk0 ⎣

∞ 

k=k0

∞  k=k0

i∈N

i∈N

⎦

⎦

Lp(·) (Rn )

⎤1/q

q  kδq 2 χBik

⎤1/q

q  kδq 2 χBik

.

Lp(·) (Rn )

From this and an argument similar to that used in the proof of (4.7), we deduce that 

 q1



2k0 q (J3 )q

⎡ ⎣

k0 ∈Z



k∈Z

⎤ q1

q  2kq χBik i∈N

⎦ ∼ f  p(·),r,s,q n , H (R )

(4.13)

A

Lp(·) (Rn )

which, combined with (4.5), (4.11) and (4.12), implies that  f H p(·),q (Rn ) ∼ A

 



2k0 q χ{x∈Rn :

k0 ∈Z



q p(·) M (f )(x)>2k0 } 

 q1 2k0 q (J1 )q

L

+

k0 ∈Z



 q1 (Rn )



 q1 2k0 q (J2 )q

+

k0 ∈Z



 q1 2k0 q (J3 )q

k0 ∈Z

 f H p(·),r,s,q (Rn ) . A

p(·),q

This further implies that f ∈ HA (Rn ) and hence finishes the proof of (4.3). p(·),q p(·),r,s,q (Rn ). To this end, it suffices to show that We now prove that HA (Rn ) ⊂ HA p(·),q

HA

p(·),∞,s,q

(Rn ) ⊂ HA

(Rn ),

(4.14)

due to the fact that each (p(·), ∞, s)-atom is also a (p(·), r, s)-atom and hence p(·),∞,s,q

HA

p(·),r,s,q

(Rn ) ⊂ HA

(Rn ).

Now we prove (4.14) by three steps.  p(·),q Step 1. To show (4.14), for any f ∈ HA (Rn ), φ ∈ S(Rn ) with Rn φ(x) dx = 1 and ν ∈ N, let f (ν) := f ∗ φ−ν . By this and [10, p. 15, Lemma 3.8], we know that f (ν) → f in S  (Rn ) as ν → ∞. Moreover, by [10, p. 39, Lemma 6.6], we conclude that there exists a positive constant C(N,φ) , depending on N and φ, but independent of f , such that, for any ν ∈ N and x ∈ Rn , / 0 MN +2 f (ν) (x) ≤ C(N,φ) MN (f )(x). p(·),q

Thus, by Definition 2.10 and Proposition 4.6, we find that f (ν) ∈ HA (ν) f

p(·),q

HA

(Rn )

(Rn ) and

 f H p(·),q (Rn ) .

The aim of this step is to prove that, for any ν ∈ N,   ν,k f (ν) = hi

A

in S  (Rn ),

(4.15)

k∈Z i∈N

where, for each ν, i ∈ N and k ∈ Z, hν,k is a (p(·), ∞, s)-atom multiplied by a constant depending on k i and i, but independent of f and ν.

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To show (4.15), we borrow some ideas from the proofs of [10, p. 38, Theorem 6.4] and [51, Theorem 3.6]. For any k ∈ Z and N ∈ N ∩ [ ( p1 − 1) lnlnλb−  + 2, ∞), let   Ωk := x ∈ Rn : MN (f )(x) > 2k . Clearly, Ωk is open. From this and Lemma 4.7 with m = 6τ , we deduce that there exist a sequence {xki }i∈N ⊂ Ωk and a sequence { ki }i∈N ⊂ Z such that Ωk =

1

(xki + Bki );

(4.16)

i∈N

(xki + Bki −τ ) ∩ (xkj + Bkj −τ ) = ∅ for any i, j ∈ N with i = j;

(4.17)

(xki + Bki +6τ ) ∩ Ωk = ∅, (xki + Bki +6τ +1 ) ∩ Ωk = ∅ for any i ∈ N; if (xki + Bki +4τ ) ∩ (xkj + Bkj +4τ ) = ∅, then | ki − kj | ≤ τ ;    j ∈ N : (xki + Bki +4τ ) ∩ (xkj + Bkj +4τ ) = ∅ ≤ R for any i ∈ N,

(4.18)

where τ and R are same as in Lemma 4.7. Fix η ∈ S(Rn ) satisfying supp η ⊂ Bτ , 0 ≤ η ≤ 1 and η ≡ 1 on B0 . For any i ∈ N, k ∈ Z and x ∈ Rn , let k k ηi (x) := η(A−i (x − xki )) and ηik (x) . Σj∈N ηjk (x)

θik (x) :=

" Then θik ∈ S(Rn ), supp θik ⊂ xki + Bki +τ , 0 ≤ θik ≤ 1, θik ≡ 1 on xki + Bki −τ by (4.17), and i∈N θik = χΩk . From this, it follows that {θik }i∈N forms a smooth partition of unity of Ωk . For any m ∈ Z+ , define Pm (Rn ) to be the linear space of all polynomials on Rn with degree not greater than m. Moreover, for each i and P ∈ Pm (Rn ), let ⎡ 1 k (x) dx θ Rn i

P i,k := ⎣ 



⎤1/2 |P (x)|2 θik (x) dx⎦

.

(4.19)

Rn

Then (Pm (Rn ),  · i,k ) is a finite dimensional Hilbert space. For each i, since f (ν) induces a linear functional on Pm (Rn ) via 2 3 1 (ν) k f , Qθ , Q ∈ Pm (Rn ), i θk (x) dx Rn i

Q → 

by the Riesz lemma, it follows that there exists a unique polynomial Piν,k ∈ Pm (Rn ) such that, for any Q ∈ Pm (Rn ), 2 2 3 3 1 1  f (ν) , Qθik =  Piν,k , Qθik k k θ (x) dx θ (x) dx Rn i Rn i  1 = Piν,k (x)Q(x)θik (x) dx. k (x) dx θ n i R Rn

For each i ∈ N, k ∈ Z and ν ∈ N, let bν,k := [f (ν) − Piν,k ]θik and i g ν,k := f (ν) −

 i∈N

bν,k = f (ν) − i



  ν,k Pi θik . f (ν) − Piν,k θik = f (ν) χΩ + k

i∈N

i∈N

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Then, by an argument similar to that used in [51, p. 1679], we conclude that, for any ν ∈ N, g ν,k L∞ (Rn )  2k and g ν,k L∞ (Rn ) → 0 as k → −∞. For any k ∈ Z, let    k := x ∈ Rn : MN (f (ν) )(x) > 2k . Ω p(·),q

Then, for any  ∈ (0, 1), by f (ν) ∈ HA (Rn ), we know that there exists an integer k() such that, for any  k | < . Since, for any k ∈ [k() , ∞) ∩ Z and α ∈ (0, ∞), k ∈ [k() , ∞) ∩ Z, |Ω . . . p−  . . . . . −p− .Ωk ∩ x ∈ Rn : MN (f (ν) )(x) > α . ≤ min .Ω MN (f (ν) ) p(·),∞ k. , α L

. . p− . . −p− ≤ min .Ω MN (f (ν) ) p(·),∞ k. , α L

!  k) (Ω

! ,

(Rn )

it follows that, for any p0 ∈ (0, p− ) satisfying (1/p0 − 1) ln b/ ln λ−  ≤ s,   p0 dx MN (f (ν) )(x) k Ω

∞ =

. . .  k : MN (f (ν) )(x) > α .. dα p0 αp0 −1 . x ∈ Ω

0

γ ≤

∞ . . p− .  . p0 −1 p0 .Ωk . α dα + p0 αp0 −1−p− MN (f (ν) ) p(·),∞ L

p− p− − p0

. .1− pp0 p0 . . − .Ωk . MN (f (ν) ) p(·),∞ L

− p1

k| where γ := MN (f (ν) )Lp(·),∞ (Rn ) |Ω and p0 as in (4.20),  

−



(Rn )

( bν,k i

p0 dx 

(x)

i∈N

Rn

L

(Rn )

,

(4.20)

 

MN (f (ν) )(x)

p0

dx,

k Ω

which, together with (4.20), further implies that ' (   ν,k ν,k bi := MN bi p i∈N

HA0 (Rn )

. .1− pp0 p0 . . −  .Ω MN (f (ν) ) p(·),q k.

. On the other hand, by [51, (3.9)], we know that, for any k ∈ Z

' MN

i∈N

dα

γ

0

=

(Rn )

. . p1 − p1 . . 0 −  .Ω MN (f (ν) ) k. Lp0 (Rn )

Lp(·),q (Rn )

→0

p as k → ∞. Here and hereafter, for any p ∈ (0, ∞), HA (Rn ) denotes the anisotropic Hardy space introduced by Bownik in [10]. By this and the fact that, for any ν ∈ N, g ν,k L∞ (Rn ) → 0 as k → −∞, we have

N   ν,k+1  (ν) g − g ν,k f − k=−N

 ν,N +1  bi i∈N

p

HA0 (Rn )+L∞ (Rn )

+ g ν,−N L∞ (Rn ) → 0 as N → ∞, p

HA0 (Rn )

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23

p0 where, for any f ∈ HA (Rn ) + L∞ (Rn ),

 f HAp0 (Rn )+L∞ (Rn ) := inf f1 HAp0 (Rn ) + f2 L∞ (Rn ) :



p0 (Rn ), f2 ∈ L∞ (Rn ) f = f1 + f2 , f1 ∈ HA

with the infimum being taken over all decompositions of f as above. From this and a proof similar to the proofs of [51, (3.12), (3.13), (3.14) and (3.16)], we deduce that, for any ν ∈ N, f (ν) =



g ν,k+1 − g ν,k

k∈Z

=



⎡ ⎣bν,k − i

k∈Z i∈N



/

⎤ 0   ν,k ν,k+1 k+1 ⎦ θik − Pi,j θj hi in S  (Rn ), bν,k+1 =: j

j∈N

k∈Z i∈Z

ν,k+1 where, for any ν, i, j ∈ N and k ∈ Z, Pi,j is the orthogonal projection of [f (ν) − Pjν,k+1 ]θik on Pm (Rn ) with respect to the norm defined by (4.19) and hν,k is a multiple of a (p(·), ∞, s)-atom satisfying i

 hν,k i (x)Q(x) dx = 0

for any Q ∈ Pm (Rn ),

(4.21)

Rn

⊂ (xki + Bki +4τ ) supp hν,k i

(4.22)

and ν,k hi

L∞ (Rn )

 2k .

(4.23)

This finishes the proof of (4.15). Step 2. From (4.23) and the Alaoglu theorem (see, for example, [65, Theorem 3.17]), it follows that there νι ,k exists a subsequence {νι }∞ → hki as ι → ∞ in the weak-∗ ι=1 ⊂ N such that, for each i ∈ N and k ∈ Z, hi  topology of L∞ (Rn ). Moreover, supp hki ⊂ (xki + Bki +4τ ), hki L∞ (Rn )  2k and Rn hki (x)Q(x)dx = 0 for any Q ∈ Pm (Rn ). Therefore, hki is a multiple of a (p(·), ∞, s)-atom aki . Let hki := λki aki , where λki ∼ 2k χxki +B k Lp(·) (Rn ) . Then, by (4.16), (4.18) and Lemma 2.5, we have i +4τ

⎧ ⎡ ⎤p ⎫1/p q k ⎨ χ λ k ⎬   xi +Bk +4τ i ⎢ i ⎣ ⎦ ⎣ ⎩ χxki +B k Lp(·) (Rn ) ⎭ k∈Z i∈N i +4τ p(·) ⎡

L

L

' ∼



k∈Z

⎥ ⎦ (Rn )

⎤ q1

⎡

' ( p1  q ⎢ kq ∼⎣ 2 χxki +B k i +4τ p(·) i∈N k∈Z

⎤ q1

⎥ ⎦ (Rn )

( q1 2

kq

q χΩk Lp(·) (Rn )

∼ MN (f )Lp(·),q (Rn ) ∼ f H p(·),q (Rn ) A

with the usual modification made when q = ∞ and p as in (2.5). " " k Step 3. By Step 2, we conclude that, to prove (4.14), it suffices to show that f = k∈Z i∈N hi in S  (Rn ).

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24

To this end, for any k ∈ Z, let fk :=



hki

i∈N

and, for any ν ∈ N and k ∈ Z, fk := g ν,k+1 − g ν,k . Then fk ι → fk in S  (Rn ) as ι → ∞. Indeed, by (4.18) n and the support conditions of hki and hν,k i , we find that, for any ϕ ∈ S(R ), (ν)

2

(ν )

4

3

(ν ) fk ι , ϕ

=

5



hνi ι ,k , ϕ

=

i∈N

→

 2 ν ,k 3 hi ι , ϕ i∈N

6

7 hki , ϕ =

4 

i∈N

5

hki , ϕ

= fk , ϕ as ι → ∞.

i∈N

Next we aim to prove that, for any ν ∈ N, 

(ν)

fk

→ 0 in S  (Rn ) as K1 → ∞.

(4.24)

|k|≥K1

Indeed, for any p0 ∈ (0, p− ) with (1/p0 − 1) ln b/ ln λ−  ≤ s, by (4.21), (4.22) and (4.23), we know that (2k |Bki +4τ |1/p0 )−1 hν,k is a (p0 , ∞, s)-atom multiplied by a constant. From this, [10, p. 19, Theorem 4.2], i (4.18), (4.16) and Lemma 2.5, we deduce that, as K2 → ∞, p0  (ν) fk k≥K2 p0

≤

  ν,k p0 hi p0

HA (Rn )

k≥K2 i∈N

HA (Rn )









. . . . 2kp0 .Bki +4τ .

  k≥K2 i∈N

p

p

2kp0 χΩk L−p(·) (Rn ) + χΩk L+p(·) (Rn )



k≥K2

  p 2k(p0 −p− ) f  −p(·),∞  HA

k≥K2



  p 2k(p0 −p− ) f  −p(·),q HA

k≥K2

(Rn )

(Rn )

+2

k(p0 −p+ )

f 

+ 2k(p0 −p+ ) f 

p+ p(·),∞

HA

(Rn )

p+ p(·),q

HA

(Rn )

→ 0.

(4.25)

Let E1 := {x ∈ Rn : p(x) ∈ (0, 1)}, E2 := {x ∈ Rn : p(x) ∈ [1, ∞)} and, for any j ∈ {1, 2}, p− (Ej ) := ess inf p(x) and p+ (Ej ) := ess sup p(x). x∈Ej

x∈Ej

Then, by (4.21), (4.22) and (4.23) again, we find that (2k |Bki +4τ |)−1 hν,k is a (1, ∞, s)-atom multiplied by i a constant. Therefore, by [10, p. 19, Theorem 4.2], (4.18), (4.16) and Lemma 2.5 again, we conclude that     ν,k (ν) f ≤ h χ  2k |Ωk ∩ E1 | E 1 i k k≤K3 1 k≤K3 i∈N k≤K3 H 1 (Rn ) HA (E1 )







A

p (E )

p (E )

1 + 1 2k χΩk L−p(·) (R n ) + χΩk Lp(·) (Rn )



k≤K3

  p (E1 )  2k(1−p− (E1 )) f  −p(·),∞ k≤K3

HA

(Rn )

+2

k(1−p+ (E1 ))

f 

p+ (E1 ) p(·),∞

HA

(Rn )

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25

2

k(1−p− (E1 ))

f 

k≤K3

p− (E1 ) p(·),q

HA

(Rn )

+2

k(1−p+ (E1 ))

f 

p+ (E1 ) p(·),q

HA

→0

(Rn )

(4.26)

as K3 → −∞. Similarly, for any p+ ∈ (p+ (E2 ), ∞), we have   ν,k   1 (ν) fk ≤ hi χE2  2k |Ωk ∩ E2 | p+ p k≤K4 p+ k≤K4 i∈N k≤K4 L + (Rn ) L (E2 )   p− (E2 ) p+ (E2 )  p  p   2k χΩk Lp(·)+(Rn ) + χΩk Lp(·)+(Rn ) k≤K4



 k≤K4





 2

k(1−

 2

k(1−

p− (E2 ) ) p +

p− (E2 ) ) p +

f  f 

k≤K4

p− (E2 ) p + p(·),∞

HA

(Rn )

p− (E2 ) p + p(·),q

HA

(Rn )

+2

+2

k(1−

k(1−

p+ (E2 ) ) p +

p+ (E2 ) ) p +

f 

f 



p+ (E2 ) p + p(·),∞

HA

(Rn )

p+ (E2 ) p + p(·),q

HA

(Rn )

 →0

as K4 → −∞. This, combined with (4.25) and (4.26), implies that (4.24) holds true. " By an argument similar to that used in the proof of (4.24), we know that |k|≥K1 fk → 0 in S  (Rn ) as K1 → ∞. From this, (4.24) and a proof similar to that used in [51, pp. 1682–1683], we further deduce that f=



fk =



hki

in

S  (Rn ),

k∈Z i∈N

k∈Z

p(·),q

which completes the proof of (4.14). This shows HA of Theorem 4.8. 2 p(·),q

5. Lusin area function characterizations of HA

p(·),r,s,q

(Rn ) ⊂ HA

(Rn ) and hence finishes the proof

Rn ) (R p(·),q

In this section, using the atomic characterization of HA (Rn ) obtained in Theorem 4.8, we establish p(·),q the Lusin area function characterization of HA (Rn ) in Theorem 5.2. To this end, we first recall the notion of the anisotropic Lusin area function (see [48,51]).  Definition 5.1. Let ψ ∈ S(Rn ) satisfy Rn ψ(x)xγ dx = 0 for any multi-index γ ∈ (Z+ )n with |γ| ≤ s, where s ∈ N ∩ [ ( p1− − 1) ln b/ ln λ− , ∞) and p− is as in (2.4). Then, for any f ∈ S  (Rn ), the anisotropic Lusin area function S(f ) is defined by setting, for any x ∈ Rn , ⎡ S(f )(x) := ⎣

 k∈Z

b−k



⎤1/2 |f ∗ ψk (y)| dy ⎦ 2

,

x+Bk

where ψk (·) := b−k ψ(A−k ·). Recall also that a distribution f ∈ S  (Rn ) is said to vanish weakly at infinity if, for each ψ ∈ S(Rn ), f ∗ ψk → 0 in S  (Rn ) as k → ∞. Denote by S0 (Rn ) the set of all f ∈ S  (Rn ) vanishing weakly at infinity. The main result of this section is the following Theorem 5.2. Theorem 5.2. Let p(·) ∈ C log (Rn ) and q ∈ (0, ∞]. Then f ∈ HA (Rn ) if and only if f ∈ S0 (Rn ) and p(·),q S(f ) ∈ Lp(·),q (Rn ). Moreover, there exists a positive constant C such that, for any f ∈ HA (Rn ), p(·),q

C −1 S(f )Lp(·),q (Rn ) ≤ f H p(·),q (Rn ) ≤ CS(f )Lp(·),q (Rn ) . A

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26

To prove Theorem 5.2, we need several technical lemmas. First, by an argument similar to that used in the proof of [79, Lemma 6.5], it is easy to see that the following lemma holds true, the details being omitted. p(·),q

Lemma 5.3. Let p(·) ∈ C log (Rn ) and q ∈ (0, ∞]. Then HA

(Rn ) ⊂ S0 (Rn ).

Via borrowing some ideas from [86, Lemma 2.6] and [55, Lemma 2.2], we obtain the following result, which is an anisotropic version of [86, Lemma 2.6]. Lemma 5.4. Let p(·) ∈ C log (Rn ) and p− ∈ (1, ∞). Then there exists a positive constant C such that, for all subsets E1 , E2 of Rn with E1 ⊂ E2 , C −1



|E1 | |E2 |

 p1

−

≤



χE1 Lp(·) (Rn ) ≤C χE2 Lp(·) (Rn )

|E1 | |E2 |

 p1

+

.

Proof. In view of similarity, we only show that χE1 Lp(·) (Rn )  χE2 Lp(·) (Rn )



|E1 | |E2 |

 p1

+

.

(5.1)

To this end, for any i ∈ {1, 2}, let p− (Ei ) := ess inf p(x) and p+ (Ei ) := ess sup p(x). x∈Ei

x∈Ei

If |E2 | ≤ 1, then, by (2.6) and p− ∈ (1, ∞), we know that, for any i ∈ {1, 2} and x ∈ Ei , 1

1

1

|Ei | p(x) ∼ |Ei | p− (Ei ) ∼ |Ei | p+ (Ei ) , which, combined with 1

1

|Ei | p− (Ei ) ≤ χEi Lp(·) (Rn ) ≤ |Ei | p+ (Ei ) , implies that 1

1

1

χEi Lp(·) (Rn ) ∼ |Ei | p(x) ∼ |Ei | p− (Ei ) ∼ |Ei | p+ (Ei ) .

(5.2)

By this, we conclude that, for any x ∈ E1 , χE1 Lp(·) (Rn ) ∼ χE2 Lp(·) (Rn )



|E1 | |E2 |

1  p(x)

 

|E1 | |E2 |

 p1

+

.

(5.3)

If |E1 | ≥ 1, let {Qj }j∈N be a partition of Rn such that, for any i, j ∈ N, |Qi | = |Qj | = 1 and, when dist( 0n , Qi ) > dist( 0n , Qj ), i > j, where, for any i ∈ N, dist( 0n , Qi ) := inf{|x| : x ∈ Qi }. Then, by [42, Theorem 2.4] and (5.2), we find that, for any i ∈ {1, 2},   1 χEi Lp(·) (Rn ) ∼ χEi Lp(·) (Qj ) ∼ |Ei | p∞ , (5.4) j∈N p∞

where p∞ is as in (2.7). From this, it follows that χE1 Lp(·) (Rn ) ∼ χE2 Lp(·) (Rn )



|E1 | |E2 |

 p1

∞

 

|E1 | |E2 |

 p1

+

.

(5.5)

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27

If |E1 | < 1 < |E2 |, then, from (5.2) and (5.4), we deduce that 1

χE1 Lp(·) (Rn ) |E1 | p+ (E1 ) ∼  1 χE2 Lp(·) (Rn ) |E2 | p∞



|E1 | |E2 |

 p1

+

.

(5.6)

Combining (5.3), (5.5) and (5.6), we obtain (5.1). This finishes the proof of Lemma 5.4. 2 The following lemma is just [13, Lemma 2.3], which is a slight modification of [18, Theorem 11]. Lemma 5.5. Let A be a dilation. Then there exists a collection   Q := Qkα ⊂ Rn : k ∈ Z, α ∈ Ik of open subsets, where Ik is certain index set, such that (i) (ii) (iii) (iv)

. . ) for any k ∈ Z, .Rn \ α Qkα . = 0 and, when α = β, Qkα ∩ Qkβ = ∅; for any α, β, k, with ≥ k, either Qkα ∩ Qβ = ∅ or Qα ⊂ Qkβ ; for each ( , β) and each k < , there exists a unique α such that Qβ ⊂ Qkα ; there exist some negative integer v and positive integer u such that, for any Qkα with k ∈ Z and α ∈ Ik , there exists xQkα ∈ Qkα satisfying that, for any x ∈ Qkα , xQkα + Bvk−u ⊂ Qkα ⊂ x + Bvk+u .

In what follows, we call Q := {Qkα }k∈Z, α∈Ik from Lemma 5.5 dyadic cubes and k the level, denoted by

(Qkα ), of the dyadic cube Qkα with k ∈ Z and α ∈ Ik . Remark 5.6. In the definition of (p(·), r, s)-atoms (see Definition 4.1), if we replace dilated balls B by dyadic cubes, then, from Lemma 5.5, we deduce that the corresponding anisotropic variable atomic Hardy–Lorentz space coincides with the original one (see Definition 4.2) in the sense of equivalent quasi-norms. The following Calderón reproducing formula is just [13, Proposition 2.14]. Recall that, for any ψ ∈  √ 8 L1 (Rn ) and ξ ∈ Rn , ψ(ξ) := (2π)−n/2 Rn ψ(x)e−ıx·ξ dx, where ı := −1 and, for any x := (x1 , . . . , xn ), "n ξ := (ξ1 , . . . , ξn ) ∈ Rn , x · ξ := k=1 xk ξk . Lemma 5.7. Let s ∈ Z+ and A be a dilation. Then there exist ϕ, ψ ∈ S(Rn ) such that (i) supp ϕ ⊂ B0 ,

 xγ ϕ(x) dx = 0 Rn

for any γ ∈ (Z+ ) with |γ| ≤ s, ϕ(ξ) 8 ≥ C for any ξ ∈ {x ∈ Rn : a ≤ ρ(x) ≤ b}, where 0 < a < b < 1 and C are positive constants; (ii) supp ψ8 is compact and bounded away from the origin; " 8 ∗ j 8 ∗ )j ξ) = 1 for any ξ ∈ Rn \ { 0n }, where A∗ denotes the adjoint matrix of A. (iii) j∈Z ψ((A ) ξ)ϕ((A n

Moreover, for any f ∈ S0 (Rn ), f = Now we prove Theorem 5.2.

" j∈Z

f ∗ ψj ∗ ϕj in S  (Rn ).

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J. Liu et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

28

p(·),q

Proof of Theorem 5.2. We first show the necessity of this theorem. Let f ∈ HA (Rn ). It follows from Lemma 5.3 that f ∈ S0 (Rn ). On the other hand, for any k0 ∈ Z, due to Theorem 4.8 and Remark 5.6, we can decompose f as follows k 0 −1

f=



λki aki +

k=−∞ i∈N

∞  

· · · =: f1 + f2 ,

k=k0 i∈N

where {λki }i∈N,k∈Z and {aki }i∈N,k∈Z are as in Theorem 4.8 satisfying (4.2). Let v, u be as in Lemma 5.5 and w := u − v + 2τ . Then we have χ{x∈Rn :

S(f )(x)>2k0 }



Lp(·) (Rn )

 χ{x∈Rn :

S(f1

)(x)>2k0 −1 }

+ χ{x∈(Ek ) : 0



+ χ{x∈Ek0 : Lp(·) (Rn )

S(f2

)(x)>2k0 −1 }

k −1 0 S(f2 )(x)>2 }



Lp(·) (Rn )

Lp(·) (Rn )

=: I1 + I2 + I3 , where Ek0 := Obviously,

)∞

)

k=k0

i∈N

(5.7)

Aw Qki and {Qki }i∈N,k∈Z ⊂ Q are as in Lemma 5.5.

I1  χ{x∈Rn :

k S(ak )(x)χ k0 −2 } λ (x)>2 k w i i∈N i A Q

"k0 −1 "

+ χ{x∈Rn :

k=−∞

i

"k0 −1 " k=−∞

Lp(·) (Rn )

k S(ak )(x)χ k0 −2 } λ (x)>2 i i∈N i (Aw Qk )

Lp(·) (Rn )

i

=: I1,1 + I1,2 .

(5.8)

For I1,1 , by [13, Theorem 3.2], Lemma 4.5, Remark 4.4(i) and a proof similar to that of (4.7), we conclude that 

 k0 ∈Z

 q1 2k0 q (I1,1 )q

⎡ ⎣

 k∈Z

q  kq 2 χQki i∈N

⎤ q1 ⎦ ∼ f  p(·),r,s,q n . H (R )

(5.9)

A

Lp(·) (Rn )

To deal with I1,2 , assume that a is a (p(·), r, s)-atom supported on a dyadic cube Q. For any j ∈ N, let   Uj := xQ + Bv[(Q)−j−1]+u+2τ \ Bv[(Q)−j]+u+2τ . Then, by Lemma 5.5(iv), we know that, for any x ∈ (Aw Q) , there exists some j0 ∈ N such that x ∈ Uj0 . For this j0 , choose N ∈ N lager enough such that  (N − β)vj0 + (1 − v)

 1 − β (Q) < 0 r

with β as in (4.9). By this and an argument similar to that used in the proof [52, (3.3)], we find that S(a)(x)  bN vj0 b−

v(Q) r

aLr (Q) .

From this and the size condition of a, we deduce that, for any x ∈ (Aw Q) ,

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|Q|β

−1

S(a)(x)  b(N −β)vj0 +(1−v)( r −β)(Q) χQ Lp(·) (Rn ) 1



−1

 χQ Lp(·) (Rn )

|Q| ρ(x − xQ )

-β

29

b[(Q)−j0 ]vβ

−1

β

 χQ Lp(·) (Rn ) [MHL (χQ )(x)] .

(5.10)

By (5.10), similarly to (4.10), we conclude that 



 q1 2k0 q (I1,2 )q

⎡ ⎣

k0 ∈Z



k∈Z

⎤ q1

q  2kq χQki i∈N

⎦ ∼ f  p(·),r,s,q n , H (R ) A

Lp(·) (Rn )

which, together with (5.8) and (5.9), further implies that 



 q1 2k0 q (I1 )q

k0 ∈Z

 f H p(·),r,s,q (Rn ) .

(5.11)

A

For I2 and I3 , from (5.10) and a proof similar to those of (4.12) and (4.13), it follows that 





 q1 2

k0 q

q

(I2 )

k0 ∈Z

 f H p(·),r,s,q (Rn ) and A



 q1 2

k0 q

 f H p(·),r,s,q (Rn ) .

q

(I3 )

(5.12)

A

k0 ∈Z

Combining (5.7), (5.11) and (5.12), we obtain  S(f )Lp(·),q (Rn ) ∼



2

k0 q

2

k0 q

χ{x∈Rn :

k0 ∈Z

 



 q1 q

(I1 )

k0 ∈Z

q p(·) |S(f )(x)|>2k0 } L

 +



 q1 (Rn )

 q1 2

k0 q

q

(I2 )

k0 ∈Z

 +



 q1 2

k0 q

q

(I3 )

k0 ∈Z

 f H p(·),r,s,q (Rn ) ∼ f H p(·),q (Rn ) A

A

with the usual modification made when q = ∞, which implies that S(f ) ∈ Lp(·),q (Rn ) and S(f )Lp(·),q (Rn )  f H p(·),q (Rn ) . A

This shows the necessity of Theorem 5.2. Next we prove the sufficiency of Theorem 5.2. Let f ∈ S0 (Rn ) and S(f ) ∈ Lp(·),q (Rn ). Then we need to p(·),q prove that f ∈ HA (Rn ) and f H p(·),q (Rn )  S(f )Lp(·),q (Rn ) .

(5.13)

A

To this end, for any k ∈ Z, let Ωk := {x ∈ Rn : S(f )(x) > 2k } and Qk :=

Q ∈ Q : |Q ∩ Ωk | >

|Q| |Q| and |Q ∩ Ωk+1 | ≤ 2 2

! .

Clearly, for each Q ∈ Q, there exists a unique k ∈ Z such that Q ∈ Qk . Let {Qki }i be the set of all maximal dyadic cubes in Qk , namely, there exists no Q ∈ Qk such that Qki  Q for any i. Let u, v be as in Lemma 5.5 and, for any Q ∈ Q,

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30

8 := {(y, t) ∈ Rn × R : y ∈ Q, v (Q) + u + τ ≤ t < v[ (Q) − 1] + u + τ } . Q 8 Q∈Q are mutually disjoint and Then {Q} 11

Rn × R =

1

k∈Z i

11

8 =: Q

Bk, i .

(5.14)

k∈Z i

Q⊂Qk i , Q∈Qk

Obviously, {Bk,i }k∈Z, i are mutually disjoint by Lemma 5.5(ii). Let ψ and ϕ be as in Lemma 5.7. Then, by Lemma 5.7, the properties of the tempered distributions (see [37, Theorem 2.3.20] or [71, Theorem 3.13]) and (5.14), we find that, for any f ∈ S0 (Rn ) with S(f ) ∈ Lp(·),q (Rn ) and x ∈ Rn ,   f (x) = f ∗ ψk ∗ ϕk (x) = f ∗ ψt (y) ∗ ϕt (x − y) dydm(t) k∈Z

=

 

k∈Z

Rn ×R

f ∗ ψt (y) ∗ ϕt (x − y) dydm(t) =:

i B k, i

 k∈Z

hki (x)

i

in S  (Rn ), where m(t) is the counting measure on R, namely, for any set E ⊂ R, m(E) := E if E has only finite elements, or else m(E) := ∞. Moreover, by an argument similar to that used in the proofs of [52, (3.24), (3.29) and (3.30)], it is easy to see that there exists some C0 ∈ (0, ∞) such that, for any r ∈ (max{p+ , 1}, ∞), k ∈ Z, i and α ∈ (Z+ )n as in Definition 4.1, supp hki ⊂ xQki + Bv[(Qki )−1]+u+3τ =: Bik , . . k hi r n ≤ C0 2k .Bik .1/r L (R ) and

(5.15) (5.16)

 hki (x)xα dx = 0,

(5.17)

Rn

namely, hki is a (p(·), r, s)-atom multiplied by a constant. For any k ∈ Z and i, let λki := C0 2k χBik Lp(·) (Rn ) and aki := (λki )−1 hki , where C0 is a positive constant as in (5.16). Then we obtain f=

 k∈Z

hki =

i

 k∈Z

λki aki

in

S  (Rn ).

i

By (5.15) and (5.17), we know that supp aki ⊂ Bik and aki also has the vanishing moments up to s. From (5.16) and Lemma 5.5(iv), it follows that aki Lr (Rn ) ≤ χBik −1 |Bik |1/r . Therefore, aki is a (p(·), r, s)-atom Lp(·) (Rn ) for any k ∈ Z and i. Moreover, by Theorem 4.8, the mutual disjointness of {Qki }k∈Z, i , Lemma 5.5(iv) again, |Qki ∩ Ωk | ≥

|Qk i| 2 ,

Lemmas 5.4 and 2.5, we conclude that ⎤ q1

⎡

f H p(·),q (Rn ) A

$ p %1/p  q k  λ χ k B i ⎢ i ∼⎣ χBik Lp(·) (Rn ) p(·) i∈N k∈Z L

⎤ q1

⎡

' ( p1  q ⎢ kq ∼⎣ 2 χBik p(·) i∈N k∈Z L

(Rn )

⎥ ⎦ (Rn )

⎤ q1

⎡

' ( p1  q ⎥ ⎢ kq 2 χQki ⎦ ∼⎣ p(·) i∈N k∈Z L

⎥ ⎦ (Rn )

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⎤ q1

⎡

2q ' (1/2  p  ⎢ ∼⎣ 2kq χQki 2p(·) i∈N p k∈Z

L

' ∼



(Rn )

31

⎤ q1

⎡

2q ' (1/2  p  ⎥ ⎢ 2kq χQki ∩Ωk ⎦ ⎣ 2p(·) i∈N p k∈Z

L

⎥ ⎦ (Rn )

( q1 q

2kq χΩk Lp(·) (Rn )

∼ S(f )Lp(·),q (Rn )

k∈Z p(·),q

with the usual modification made when q = ∞ and p as in (2.5), which implies that f ∈ HA

(Rn ) and

f H p(·),q (Rn )  S(f )Lp(·),q (Rn ) . A

This finishes the proof of (5.13) and hence Theorem 5.2.

2

6. Real interpolation p(·),q

As another application of the atomic characterization of HA (Rn ), in this section, we obtain a real p(·) interpolation result between HA (Rn ) and L∞ (Rn ). Moreover, using this result, together with [85, Corollary p(·),q 4.20] and [44, Remark 4.2(ii)], we then show that the anisotropic variable Hardy–Lorentz space HA (Rn ) with p− ∈ (1, ∞) coincides with the variable Lorentz space Lp(·),q (Rn ). To state the main result of this section, we first recall some basic notions about the real interpolation (see [9]). Assume that (X1 , X2 ) is a compatible couple of quasi-normed spaces, namely, X1 and X2 are two quasi-normed linear spaces which are continuously embedded in some larger topological vector space. Let X1 + X2 := {f1 + f2 : f1 ∈ X1 , f2 ∈ X2 }. For any t ∈ (0, ∞], the Peetre K-functional on X1 + X2 is defined by setting, for any f ∈ X1 + X2 , K(t, f ; X1 , X2 ) := inf {f1 X1 + tf2 X2 : f = f1 + f2 , f1 ∈ X1 and f2 ∈ X2 } . Moreover, for any θ ∈ (0, 1) and q ∈ (0, ∞], the real interpolation space (X1 , X2 )θ,q is defined as ⎧ ⎫ ⎡∞ ⎤1/q ⎪ ⎪  ⎨ ⎬  −θ q dt ⎣ ⎦ t K(t, f ; X1 , X2 ) (X1 , X2 )θ,q := f ∈ X1 + X2 : f θ,q := <∞ . ⎪ ⎪ t ⎩ ⎭ 0

Definition 6.1. Let p(·) ∈ C log (Rn ) and N ∈ N ∩[ ( p1 −1) lnlnλb−  +2, ∞), where p is as in (2.5). The anisotropic p(·)

variable Hardy space, denoted by HA (Rn ), is defined by setting   p(·) 0 HA (Rn ) := f ∈ S  (Rn ) : MN (f ) ∈ Lp(·) (Rn ) p(·)

0 and, for any f ∈ HA (Rn ), let f H p(·) (Rn ) := MN (f )Lp(·) (Rn ) . A

The main result of this section is stated as follows. Theorem 6.2. Let p(·) ∈ C log (Rn ), q ∈ (0, ∞] and θ ∈ (0, 1). Then it holds true that p(·),q 

(HA (Rn ), L∞ (Rn ))θ,q = HA p(·)

where

1 p(·) 

=

1−θ p(·) .

(Rn ),

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32

As a consequence of Theorem 6.2, [85, Corollary 4.20] and [44, Remark 4.2(ii)], we immediately obtain the following conclusion, the details being omitted. p(·),q

Corollary 6.3. Let p(·) ∈ C log (Rn ). If p− ∈ (1, ∞) and q ∈ (0, ∞], then HA equivalent quasi-norm.

(Rn ) = Lp(·),q (Rn ) with

Remark 6.4. (i) When p(·) ≡ p ∈ (0, 1], Theorem 6.2 goes back to [51, Lemma 6.3], which states that, for any θ ∈ (0, 1) and q ∈ (0, ∞], p (HA (Rn ), L∞ (Rn ))θ,q = HA

p/(1−θ),q

(Rn ).

(ii) When p(·) ≡ p ∈ (1, ∞), Theorem 6.2 coincides with [51, Remark 6.7] (see also [63, Theorem 7]), namely, for any θ ∈ (0, 1) and q ∈ (0, ∞], (Lp (Rn ), L∞ (Rn ))θ,q = Lp/(1−θ),q (Rn ). p/(1−θ),q

p (iii) Let A := d In×n for some d ∈ R with |d| ∈ (1, ∞). Then HA (Rn ) and HA (Rn ) in (i) of this remark become the classical isotropic Hardy and Hardy–Lorentz spaces, respectively. In this case, the result in p(·) p(·),q  (i) of this remark is just [34, Theorem 1]. In addition, HA (Rn ) and HA (Rn ) in Theorem 6.2 become the classical isotropic variable Hardy and Hardy–Lorentz spaces, respectively. In this case, Theorem 6.2 includes the result in [87, Theorem 1.5] as a special case, and Theorem 6.2 with p− ∈ (1, ∞) coincides with [44, Remark 4.2(ii)].

To prove Theorem 6.2, we need the following technical lemma, which decomposes any distribution f from p(·),q  the anisotropic variable Hardy–Lorentz space HA (Rn ) into “good” and “bad” parts and plays a key role in the proof of Theorem 6.2. p(·),q 

Lemma 6.5. Let θ ∈ (0, 1), q ∈ (0, ∞], p(·) and p(·) be as in Theorem 6.2. Then, for any f ∈ HA (Rn ) and k ∈ Z, there exist gk ∈ L∞ (Rn ) and bk ∈ S  (Rn ) such that f = gk + bk in S  (Rn ), gk L∞ (Rn ) ≤ C2k and  MN (f )χ{x∈Rn : bk H p(·) (Rn ) ≤ C A



MN (f )(x)>2k } Lp(·) (Rn )

< ∞,

(6.1)

 are two where, for any N ∈ N ∩ [ ( p1 − 1) lnlnλb−  + 2, ∞) with p as in (2.5), MN (f ) is as in (2.10), C and C positive constants independent of f and k. p(·),q 

Proof. Let all the notation be the same as those used in the proof of Theorem 4.8. For any f ∈ HA k ∈ Z and N ∈ N ∩ [ ( p1 − 1) lnlnλb−  + 2, ∞), let   Ωk := x ∈ Rn : MN (f )(x) > 2k . Then, for any k0 ∈ Z, by an argument similar to that used in the proof of (4.14), we have f=

 k∈Z i∈N

hki =

k0   k=−∞ i∈N

hki +

∞  

· · · =: gk0 + bk0

in S  (Rn ),

k=k0 +1 i∈N

where, for any k ∈ Z and i ∈ N, hki is a (p(·), ∞, s)-atom multiplied by a constant and satisfies that

(Rn ),

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33

supp hki ⊂ (xki + Bki +4τ ), k hi ∞ n  2k L (R )

(6.2) (6.3)

and  hki (x)Q(x)dx = 0

for any Q ∈ Pm (Rn ).

(6.4)

Rn

By the finite intersection property of {xki + Bki +4τ }i∈N for each k ∈ Z (see (4.18)), (6.2) and (6.3), we conclude that, for any k0 ∈ Z, gk0 L∞ (Rn ) 

k0 

2k ∼ 2k0 .

k=−∞

On the other hand, for any k ∈ Z and i ∈ N, let aki := (λki )−1 hki , where λki ∼ 2k χxki +B k

i +4τ

. Lp(·) (Rn )

Then, by this, (6.2), (6.3) and (6.4), we know that, for any k ∈ Z and i ∈ N, aki is a (p(·), ∞, s)-atom. Therefore, by the finite intersection property of {xki + Bki +4τ }i∈N for each k ∈ Z again and the fact that ) Ωk = i∈N (xki + Bki ) (see (4.16)), we find that ⎧ ⎡ ⎤p ⎫ p1 k ∞ χ λ k ⎨  ⎬  i xi +Bk +4τ i ⎣ ⎦ ⎩ χxki +B k Lp(·) (Rn ) ⎭ i +4τ k=k0 +1 i∈N ' ( p1 ∞   kp  2 χ k xi +Bk +4τ i k=k0 +1 i∈N  1 ∞   k p p ∼ 2 χΩk \Ωk+1 k=k0 +1 ∼ MN (f )χΩ

k0 +1



Lp(·) (Rn )

Lp(·) (Rn )

Lp(·) (Rn )

Lp(·) (Rn )

' ( p1 ∞  kp  2 χ Ωk k=k0 +1

 MN (f )χ{x∈Rn :

MN

(f )(x)>2k0 }

Lp(·) (Rn )

Moreover, from Remark 2.3(i) and Lemma 2.5, it follows that MN (f )χ{x∈Rn :



MN (f )(x)>2k0 } Lp(·) (Rn )

⎧ ⎨ [MN (f )]p χ{x∈Rn : ≤ ⎩

2j+k0
(f )(x)≤2j+k0 +1 }

j∈Z+

⎡ ⎣



j∈Z+

(2j+k0 )p χ{x∈Rn :

MN

Lp(·) (Rn )

' ∞ ( p1   MN (f ) χΩk \Ωk+1 k=k0 +1

p p(·) (f )(x)>2j+k0 } L

⎫ p1 ⎬ L

⎤ p1 (Rn )

⎦

p(·) p

(Rn ) ⎭

.

Lp(·) (Rn )

(6.5)

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34

∼

⎧ ⎨ ⎩

(2j+k0 )−θp/(1−θ)



2j+k0 χ{x∈Rn :

MN

(f )(x)>2j+k0 }

⎫1 p/(1−θ) ⎬ p ⎭

 Lp(·) (Rn )

j∈Z+

⎡  MN (f )

1 1−θ p(·),∞ L (Rn )

⎣

 

2j+k0

−θp/(1−θ)

⎤ p1 ⎦

j∈Z+ 1  −θ/(1−θ) ∼ 2k0 f  1−θ p(·),∞ 

HA

(Rn )

1  −θ/(1−θ)  2k0 f  1−θ p(·),q 

HA

(Rn )

< ∞.

Then, from [50, Theorem 4.8], (6.5) and (6.6), we further deduce that ⎧ ⎡ ⎤p ⎫ p1 ∞ λki χxki +B k ⎨  ⎬  i +4τ ⎣ ⎦ bk0 H p(·) (Rn )  A ⎩ χxki +B k Lp(·) (Rn ) ⎭ i +4τ k=k0 +1 i∈N  MN (f )χ{x∈Rn : This finishes the proof of Lemma 6.5.

MN

(f )(x)>2k0 }

Lp(·) (Rn )

(6.6)

Lp(·) (Rn )

.

2

Now we prove Theorem 6.2. Proof of Theorem 6.2. We first prove that p(·),q 

HA

(Rn ) ⊂ (HA (Rn ), L∞ (Rn ))θ,q . p(·)

(6.7)

p(·)q 

To this end, let f ∈ HA (Rn ). Then, by Lemma 6.5, we know that, for any k ∈ Z, there exist gk ∈ p(·) L (Rn ) and bk ∈ HA (Rn ) such that f = gk + bk in S  (Rn ), gk L∞ (Rn )  2k and bk satisfies (6.1). By this and a proof similar to that of [87, (3.3)], we find that, for any t ∈ (0, ∞), ∞

K(t, f ; HA (Rn ), L∞ (Rn ))  2k(t) t, p(·)

(6.8)

where, for any t ∈ (0, ∞), ⎧ ⎪ ⎨ k(t) := inf

⎪ ⎩

⎡

∈Z: ⎣

 

p

2j h(2j+ )

⎤ p1 ⎦ ≤t

j∈Z+

⎫ ⎪ ⎬ ⎪ ⎭

and, for any λ ∈ (0, ∞), h(λ) := χ{x∈Rn :



MN (f )(x)>λ} Lp(·) (Rn )

.

On the other hand, by (6.8), it is easy to see that, for any θ ∈ (0, 1) and q ∈ (0, ∞], ⎧∞ ⎨ ⎩

t−θq

⎫1/q q dt ⎬ p(·) K(t, f ; HA (Rn ), L∞ (Rn )) t⎭



0

⎡ ⎢ q ⎣ 2 ∈Z

⎤1/q

 t(1−θ)q {t∈(0,∞): 2 <2k(t) ≤2+1 }

dt ⎥ ⎦ t

⎡ ⎢ ⎢ ⎢ 2q ⎣ ∈Z

{

"

⎤1/q

1 j j+ p p )] } j∈Z+ [2 h(2



t(1−θ)q 0

dt ⎥ ⎥ ⎥ t⎦

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⎤1/q ⎡ ⎧ ⎫ (1−θ)q p ⎨  ⎬   p ⎥ ⎢ ∼⎣ 2q 2j h(2j+ ) =: CJ, ⎦ ⎩ ⎭

(6.9)

j∈Z+

∈Z

where C is a positive constant independent of f . Next we estimate J by considering two cases. Case 1. (1−θ)q ∈ (0, 1]. In this case, by the well-known inequality that, for any d ∈ (0, 1] and {ai }i∈N ⊂ C, p ' 

(d |ai |

≤

i∈N



|ai |d ,

i∈N

we obtain Jq ≤



2q

j∈Z+

∈Z

=



 (1−θ)q   (−j)q j(1−θ)q   (1−θ)q h(2 ) 2j(1−θ)q h(2j+ ) = 2 2



2

 q

(1−θ)q 

h(2 )

j∈Z+

∈Z (1−θ)q ∈ (1, ∞]. p θ ∈ (0, 1−θ ),

Case 2. for any δ

J≤



∼

⎧ ⎪ ⎨ ⎪ ⎩ ∈Z ⎧ ⎨ ⎩

⎩

2q ⎝

⎞ η



2q



2



2−δpjη ⎠

 (1+δ)(1−θ)jq

2

j+

h(2

 (1−θ)q 2q h(2 ) .

(1−θ)q . p

⎫1/q ⎪ pη ⎬

 2(1+δ)pjη h(2j+ )

⎪ ⎭

⎫1/q ⎬  (1−θ)q h(2j+ ) ⎭

(1−θ)q 

)

2

[δ(1−θ)−θ]jq

j∈Z+

∈Z

(6.10)

Then, from the Hölder inequality, it follows that,

j∈Z+

j∈Z+

 q



η

j∈Z+

∈Z j∈Z+

∈Z

For this case, let η :=

⎛

∈Z

⎧ ⎨

2−jθq ∼



⎫1/q ⎬ ⎭

$ ∼



 (1−θ)q 2 h(2 )

%1/q

q

.

(6.11)

∈Z

Combining (6.9), (6.10) and (6.11), we conclude that ⎧∞ ⎨ ⎩

0



t−θq 





⎫1/q q dt ⎬ p(·) K(t, f ; HA (Rn ), L∞ (Rn )) t⎭

2 χ{x∈Rn : q

MN

(1−θ)q p(·) n (f )(x)>2 } L

1/q

(R )

∈Z

 ∼



2 χ{x∈Rn : q

MN

q p(·) (f )(x)>2 }  L

1/q (Rn )

∈Z

∼ MN (f )Lp(·),q   (Rn ) ∼ f H p(·),q (Rn ) A

with the usual modification made when q = ∞, which implies that f ∈ (HA (Rn ), L∞ (Rn ))θ,q p(·)

and hence completes the proof of (6.7). Conversely, we need to show that p(·),q 

(HA (Rn ), L∞ (Rn ))θ,q ⊂ HA p(·)

(Rn ).

(6.12)

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To this end, we claim that MN is bounded from the space (HA (Rn ), L∞ (Rn ))θ,q to the space p(·) (Lp(·) (Rn ), L∞ (Rn ))θ,q . Indeed, let f ∈ (HA (Rn ), L∞ (Rn ))θ,q . Then, by definition, we know that there p(·) n ∞ n exist f1 ∈ HA (R ) and f2 ∈ L (R ) such that p(·)

⎧∞ ⎨ ⎩

t−θq

0

⎫1/q q dt ⎬  f (H p(·) (Rn ),L∞ (Rn )) . f1 H p(·) (Rn ) + tf2 L∞ (Rn ) θ,q A A t⎭



(6.13)

On the other hand, since MN is bounded from HA (Rn ) to Lp(·) (Rn ) and also from L∞ (Rn ) to L∞ (Rn ), it follows that MN (f1 ) ∈ Lp(·) (Rn ) and MN (f2 ) ∈ L∞ (Rn ). For any i ∈ {1, 2}, let p(·)

Ei :=

! 1 x ∈ R : MN (f )(x) ≤ MN (fi )(x) . 2 n

Then Rn = E1 ∪ E2 . Thus, we have MN (f ) = MN (f )χE1 + MN (f )χE2 \E1 ∈ Lp(·) (Rn ) + L∞ (Rn ), which, combined with (6.13), further implies that MN (f )(Lp(·) (Rn ),L∞ (Rn ))θ,q ⎫1/q ⎧∞ ⎨  q dt ⎬ ≤ t−θq MN (f )χE1 Lp(·) (Rn ) + t MN (f )χE2 \E1 L∞ (Rn ) ⎩ t⎭ 0





⎧∞ ⎨ ⎩

⎫1/q q dt ⎬ MN (f1 )Lp(·) (Rn ) + t MN (f2 )L∞ (Rn ) t⎭



⎫1/q q dt ⎬  f (H p(·) (Rn ),L∞ (Rn )) f1 H p(·) (Rn ) + tf2 L∞ (Rn ) θ,q A A t⎭

0

⎧∞ ⎨ ⎩

t−θq



0

t−θq

with the usual modification made when q = ∞. Therefore, the above claim holds true. p(·) By this claim and [44, Remark 4.2(ii)], we find that, if f ∈ (HA (Rn ), L∞ (Rn ))θ,q , then MN (f ) belongs p(·),q   to Lp(·),q (Rn ), namely, f ∈ HA (Rn ). Thus, (6.12) holds true. This finishes the proof of Theorem 6.2. 2 Acknowledgments The authors would like to thank the referee for her/his carefully reading and useful comments which improve the presentation of this article. References [1] W. Abu-Shammala, A. Torchinsky, The Hardy–Lorentz spaces H p,q (Rn ), Studia Math. 182 (2007) 283–294. [2] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002) 213–259. [3] E. Acerbi, G. Mingione, Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math. 584 (2005) 117–148. [4] V. Almeida, J.J. Betancor, L. Rodríguez, Anisotropic Hardy–Lorentz spaces with variable exponents, arXiv:1601.04487. [5] A. Almeida, A.M. Caetano, Generalized Hardy spaces, Acta Math. Sin. (Engl. Ser.) 26 (2010) 1673–1692. [6] A. Almeida, P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010) 1628–1655. [7] J. Álvarez, H p and weak H p continuity of Calderón–Zygmund type operators, in: Fourier Analysis, in: Lecture Notes in Pure and Appl. Math., vol. 157, Dekker, New York, 1994, pp. 17–34.

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