Variable Anisotropic Hardy Spaces with Variable Exponents

: Let p (·) : R n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of R n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces H p (·) ( Θ ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces H p ( Θ ) on R n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from H p (·) ( Θ ) to L p (·) ( R n ) in general and from H p (·) ( Θ ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on H p ( Θ ).


Introduction
The main purpose of this article is to introduce and to investigate the variable anisotropic Hardy spaces H p(·) (Θ) with variable exponents. Due to the celebrated work [19] of Fe erman and Stein on classical isotropic Hardy spaces, there has been an increasing interest in extending classical Hardy spaces. In 2002, Bownik [3] investigated a special form of Hardy spaces H p (R n ), i.e., anisotropic Hardy spaces H p A (R n ) de ned over R n , where the Euclidian balls are replaced by images of the unit ball by powers of a xed expansion matrix A. In 2011, Dekel et al. [16] introduced a more general Hardy space H p (Θ) de ned over R n , where the Euclidian balls are replaced by continuous ellipsoid cover Θ including the anisotropic Hardy space of Bownik [3]. Different from Bownik's spaces, the anisotropy in H p (Θ) can change rapidly from point to point in R n and from level to level in depth.
As we all know, the variable function spaces have found their applications in uid dynamics [1,2], image processing [9,22], partial di erential equations, variational calculus [8,18] and harmonic analysis [10]. In 2012, Nakai and Sawano [27] introduced the variable Hardy space H p(·) (R n ), via the radial grand maximal function, and then obtained some real-variable characterizations of the space, such as the characterizations in terms of the atomic and the molecular decompositions. Then, in 2018, Liu et al. [24] introduced the variable anisotropic Hardy space H p(·) A (R n ) de ned over R n , where A is a general expansive matrix on R n , it extends the theory of Hardy spaces on R n of Bownik [3].
Inspired by Dekel et al. [16] and Liu et al. [24], the rst goal is to further introduce the variable anisotropic Hardy space H p(·) (Θ) with variable exponent de ned via the radial grand maximal function and then obtain its atomic decomposition.
The theory of singular integral operators plays an important role in harmonic analysis and partial differential equations; see, for example, [20,21,26,30]. In the classical isotropic setting of R n we consider Calderón-Zygmund operators T with regularity s of the form x ∉ supp f , f ∈ C ∞ c (R n ), whose kernel K(x, y) satis es the bound It is well-known that operators T are bounded on isotropic Hardy spaces H p (R n ) provided that s > n( /p − ) and T preserves vanishing moments T * (x α ) = for |α| < s, see [25,Proposition 7.4.4], [26,Theorem III.4]. Bownik [3] introduced anisotropic Calderón-Zygmund operators associated with expansive dilations and has shown their boundedness on anisotropic Hardy spaces, where the anisotropy is xed and global on R n . An extension of these results to anisotropic Hardy spaces with variable exponents was done in [29].
Recently, Bownik et al. [6] further introduced the following class of singular integral operators adapted to variable anisotropy depending on a point x ∈ R n and a scale t ∈ R. Precisely, suppose that Θ is a continuous ellipsoid cover of Dekel et al. [12] consisting of ellipsoids θ x,t with center x ∈ R n and scale t ∈ R of the form θ x,t = M x,t (B n ) + x, where M x,t is a nonsingular matrix and B n is the unit ball in R n , see De nition 2.1. This ellipsoid cover Θ de nes a space of homogeneous type [12] with quasi-distance ρ Θ de ned as in mum of ellipsoid volumes for all x ≠ y and multi-indices |α| ≤ s, (1.2) where m = − log ρ Θ (x, y). Then, they obtain the boundedness of the extended variable anisotropic singular integral operator T from H p (Θ) to L p (R n ) in general and from H p (Θ) to itself under the moment condition. Inspired by this, and also as an application of the atomic decomposition of H p(·) (Θ), the second goal is to extend these boundedness results to the variable exponents setting.
To be precise, this article is organized as follows.
In Section 2, we rst recall notation, de nitions and properties of continuous ellipsoid cover Θ, and quasidistance ρ Θ that are used throughout the paper. We also introduce the variable Lebesgue spaces L p(·) (R n ) and some related properties. In Section 3, we de ne the variable Hardy space H p(·) (Θ) by means of the radial grand maximal function and show some relate properties.
In Section 4 via borrowing some ideas from those used in the proofs of [ ∞, l (Θ) and hence also into H p(·) q, l (Θ) due to the fact that each (p(·), ∞, l)-atom is also a (p(·), q, l)-atom for any q ∈ ( , ∞).
Section 5 is devoted to showing the boundedness of variable anisotropic singular integral operators T from H p(·) (Θ) to L p(·) (R n ) in general (see Theorem 5.7 below) or from H p(·) (Θ) to itself under the moment condition (see Theorem 5.8 below). Worthwhile to point out that the crucial pointwise estimate of M • Ta(x) simpli es the corresponding pointwise estimate of M • Tã(x) in [6] even when (p(·), ∞)-atom a reduced to (p, ∞)-atom a. Precisely, we draw more on the estimate of M • a(x) in the proof of Lemma 4.4.
Finally, we make some conventions on notation. Let N := { , , . . .} and N := { } ∪ N. For any α := (α , . . . , αn) ∈ N n , |α| := α + · · · + αn, and ∂ α := ( ∂ ∂x ) α · · · ( ∂ ∂xn ) αn . Throughout the whole paper, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. The symbol D F means that D ≤ CF. If D F and F D, we then write D ∼ F. For any sets E ⊂ R n , we use E to denote the set R n \ E. Let S(R n ) be the space of Schwartz functions, S (R n ) the space of tempered distributions, and C N (R n ) the space of continuously di erentiable functions of order N.

Preliminaries
In this section we recall some notion, notations and basic properties on continuous ellipsoid covers (see, for example, [12,16]) and variable Lebesgue spaces (see, for example, [10,11]), respectively. An ellipsoid θ in R n is an image of the Euclidean unit ball B n := {x ∈ R n : |x| < } under an a ne transform, i.e., where M θ is a nonsingular matrix and c θ is the center.
Let us begin with the de nition of continuous ellipsoid covers, which is from [12, De nition 2.4].
De nition 2.1. We say that is a continuous ellipsoid cover of R n , or shortly a cover, if there exist constants p(Θ) := {a , . . . , a } such that: (i) For every x ∈ R n and t ∈ R, there exists an ellipsoid θ x, t := M x, t (B n ) + x, where M x, t is a nonsingular matrix and x is the center, satisfying (ii) Intersecting ellipsoids from Θ satisfy a "shape condition", i.e., for any x, y ∈ R n , t ∈ R and s ≥ , if θ x, t ∩ θ y, t+s ≠ ∅, then Here, · is the matrix norm of M given by M := max |x|= |Mx|.
There are many examples and results for ellipsoid cover in [7,[12][13][14]. Let us show one example from [12] to explain exactly why we need to range x as well in ellipsoid cover. Via suitable ellipsoid cover needing range x, Dahmen, Dekel and Petrushev showed a higher adaptive anisotropic Besov spaces (B-spaces) smoothness than their regular Besov space smoothness. To better illustrate this, in [12, Section 7.1], they showed that, for a suitable ellipsoid cover Θ, the B-space smoothness of the characteristic function of the unit ball B( , ) ⊂ R inḂ α ττ (Θ) is essentially /p, while in the corresponding (classical isotropic) Besov spaces it is /p. More strikingly, in the adaptive B-space scalesḂ α ττ (Θ), the smoothness of the characteristic function of any square Q ⊂ R is arbitrarily high, i.e., can be any α > , while in the corresponding isotropic Besov spaces it is essentially /p (see [12,Section 7.2] ). However, it is important to note that the cover Θ needed to describe that level of smoothness depends on α.
Next we collect results about ellipsoid covers from [12,16] which will be used through the whole paper.
Proposition 2.2. Let Θ be a continuous ellipsoid cover.

De nition 2.3.
A quasi-distance on a set X is a mapping ρ : X × X → [ , ∞) that satis es the following conditions for all x, y, z ∈ X: Dekel, Han, and Petrushev have shown that an ellipsoid cover Θ induces a quasi-distance ρ Θ on R n , see [12,Proposition 2.7]. Moreover, R n equipped with the quasi-distance ρ Θ and the Lebesgue measure is a space of homogeneous type, [ where the constants of equivalence depend only on p(Θ). De ne P(R n ) the set of all measurable functions p(·) satisfying < p− ≤ p+ < ∞. For any measurable function p(·) ∈ P(R n ), the variable Lebesgue space L p(·) (R n ) denotes the set of measurable functions f on R n such that, for some λ > , This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm Let C log (R n ) be the set of all functions p(·) ∈ P(R n ) satisfying the globally log-Hölder continuous condition, namely, there exist C log (p), C∞ ∈ ( , ∞) and p∞ ∈ R such that, for any x, y ∈ R n , and .
The following Lemma 2.6 shows Hölder's inequality adapted to the variable Lebesgue space L p(·) (R n ).
where p is as in (2.4).
We nish the proof of Proposition 2.8.

Variable Anisotropic Hardy Spaces with Variable Exponents
In this section we recall the de nitions and properties of Hardy spaces with variable anisotropy which were originally introduced by Dekel, Petrushev and Weissblat [16]. Let Θ be a continuous ellipsoid cover. For any locally integrable function f on R n , the Hardy-Littlewood maximal operators M Bρ Θ and M Θ are de ned, respectively, to be and where Bρ Θ (y, r) is as in (2.3). By [16,Lemma 3.2], these two maximal functions are pointwise equivalent with the usual modi cation made when r = ∞, where M Θ denotes the Hardy-Littlewood maximal operator as in (3.2).

De nition 3.2. Let
By a proof similar to that of [4, Proposition 2.11(i)], we obtain the following lemma.

Lemma 3.5. [16, Theorem 3.8] For any cover Θ and f
, there exist constants c , c > depending on the parameters of the cover such that Let Θ be a continuous ellipsoid cover of R n with parameters p(Θ) = {a , . . . , a }, p(·) ∈ P(R n ) and let p be as in (2.4). We de ne N p(·) := N p(·) (Θ) as the minimal integer satisfying 5) and then N p(·) := N p(·) (Θ) as the minimal integer satisfying De nition 3.6. Let Θ be a continuous ellipsoid cover, p(·) ∈ C log (R n ) and . The variable anisotropic Hardy space with variable exponent is de ned as

Lemma 3.7. Let Θ be a continuous ellipsoid cover and p(·)
which implies that the inclusion H p(·) (Θ) → S (R n ) is continuous.
(ii) By (i) and repeating the proof of [11, Proposition 4.1], we can obtain Lemma 3.7(ii) holds true.

Atomic Characterization of H p(·) (Θ)
In this section, we establish the atomic characterization of variable (iii) R n a(y)y α dy = for all α ∈ N n such that |α| ≤ l.

De nition 4.2.
Let Θ be a continuous ellipsoid cover, p(·) ∈ C log (R n ) and (p(·), q, l) an admissible triple as in De nition 4.1. The variable anisotropic atomic Hardy space H p(·) q, l (Θ) associated with Θ is de ned to be the set of all tempered distribution f ∈ S (R n ) of the form f = ∞ i= λ i a i , where the series converges in S (R n ), where the in mum is taken over all admissible decompositions of f as above.
Now we state the main result of this section as follows. To establish the atomic characterization of H p(·) (Θ), we need several technical lemmas as follows.  .4). Then there exists a positive constant C, such that, for any Proof. We show this lemma by borrowing some ideas from the proofs of [16,Theorem 4.3] and [23, pp. 1686-1687]. Let θ z, t be the ellipsoid associated with an atom a, where z ∈ R n and t ∈ R. We estimate the integral of the function M • a(x) on x ∈ (θ z, t−J ) , where J is from Proposition 2.2(i). By De nition 3.3 and Lemma 3.5, we estimate R n a(y)φx, s(y) dy , where φ ∈ S N p(·) , N p(·) (R n ) with support in B n , s ∈ R and x ∈ θ z, t−kJ \θ z, t−kJ+J , k ≥ . It is easy to see that if θ z, t ∩ θx, s = ∅ then R n a(y)φx, s(y)dy = . Thus, we may assume Suppose P is a polynomial of degree N p(·) − , by repeating the proof of [16, pp. 1077-1079], we obtain where /q + /q = and Moreover, by [16,Formula (4.6)], we also know that Repeating the estimate of [16, pp. 1078-1079], we conclude that From this, (4.2), (3.5) and (4.4), we conclude that Since q > , q > and p ≤ , we have q + a n q − max{ , a }n + p < .
Therefore, the (4.5) over s ≥ t has the largest value when s = t and hence max{ ,a }n+ p (−kJ) .
Since t − s ≥ and − q − a n q < . Therefore, the (4.9) over s ≤ t has the largest value when s = t and hence  From (4.6), (4.10), (3.2) and x ∈ θ z, t−kJ−γ , we deduce that where p is as in (2.4). Therefore, we nish the proof of Lemma 4.4.
The following Lemma 4.5 is essentially from [28, Theorem 1.1], which plays an important role in this section and is also of independent interest. Since its proof remains the same with that of [28, Theorem 1.1], the details being omitted. Lemma 4.5. Let r(·) ∈ C log (R n ) and q ∈ [ , ∞] ∩ (r+, ∞] with r+ as in (2.4).
where C is a positive constant independent of λ i , a i and θ x i , t i .
We say that an ellipsoid cover Θ is pointwise continuous if for every t ∈ R, the matrix valued function The condition (4.11) is implicitly used in [16] to guarantee that the superlevel set Ω corresponding to the grand maximal function, which is given by (4.13), is open. In this paper, since it is always possible to construct an equivalent ellipsoid cover such that Ξ is pointwise continuous and Ξ is equivalent to Θ (see [6,Theorem 2.2]). We say that two ellipsoid covers Θ and Ξ are equivalent if there exists a constant C > such that for any x ∈ R n and t ∈ R, we have Therefore, in this paper, we always assume that the ellipsoid cover is pointwise continue. Now, let us recall the Calderón-Zygmund decomposition established in [16]. Throughout this section for a given continuous ellipsoid cover Θ, we consider f ∈ L q (R n ), q ≥ , for every λ > , let (4.13) We shall assume that Θ is pointwise continuous, that (4.11) holds. By [6, Lemma 3.7], the set Ω is open. Since M • is bounded from L (R n ) to weak L (R n ) and bounded on L r (R n ), r > (see [ where J and γ are as in Proposition 2.2. Moreover, there exists a constant L > such that where E denotes the cardinality of a set E. Fix ϕ ∈ C ∞ (R n ) such that supp ϕ ⊂ B n , ≤ ϕ ≤ and ϕ ≡ on B n . For every i ∈ N , de ne Observe that ϕ i is well de ned since by (4.14) and (4.18), ≤ i ϕ i (x) ≤ L for every x ∈ Ω. Also ϕ i ∈ C ∞ (R n ) and supp ϕ i ⊂ θ x i , t i −J . By (4.14) and (4.19), we have i ϕ i (x) = χ Ω (x), which implies that the family {ϕ i } i∈N forms a smooth partition of unitary subordinate to the cover of Ω by the ellipsoids {θ x i , t i −J } i∈N . Let P l (R n ) denote the space of polynomials of n variables with degree ≤ l, where N p(·) ≤ l, see (3.5). For each i ∈ N we introduce an Hilbert space structure on the space P l (R n ) by setting P, Q i := ϕ i R n P(x)Q(x)ϕ i (x) dx for any P, Q ∈ P l (R n ). (4.20) The distribution f ∈ S (R n ) induces a linear functional on P l (R n ) given by By Riesz's lemma it is represented by a unique polynomial P i ∈ P l (R n ) such that f , Q i = P i , Q i for any Q ∈ P l (R n ).
De nition 4.6. For every i ∈ N , de ne the locally "bad part" b i := (f − P i )ϕ i and the "good part" g := f − i b i . The representation f = g+ i b i , where g and b i are as above, is a Calderón-Zygmund decomposition of degree l and height λ associated with M • .
To prove Proposition 4.7, we will use the following three results which are from [ where ν := a JN and where v := a JN .
Proof of Proposition 4.7. From Lemma 4.8, we know that there exists a positive constant C, independent of f ∈ S (R n ) and λ > , such that For any where M Θ is as in (3.2). Then, we have Following [16,Section 4.3], for each k ∈ Z, we consider the Calderón-Zygmund decomposition of f of degree l ≥ N p(·) at height k associated with M • , where Here, sequences {x k i } i∈N ⊂ Ω k and {t k i } i∈N ⊂ R satisfy (4.14)- (4.18) for Ω k , functions {ϕ k i } i∈N are de ned as in (4.19), and polynomials {P k i } i∈N are projections of f onto P l (R n ) with respect to the inner product given by (4.19).
Let l ∈ N with l ≥ N p(·) . For each i ∈ N and P, Q ∈ P l (R n ), de ne which induces a nite dimensional Hilbert space (P l (R n ), ·, · i, k ). The distribution f ∈ S (R n ) induces a linear functional on P l (R n ) by which by Riesz's lemma is represented by a unique polynomial P k i ∈ P l (R n ) such that Obviously P k i is the orthogonal projection of f onto P l (R n ) with respect to the inner product induced by (4.23). That is, P k+ ij is the unique polynomial in P l (R n ) such that, for all Q ∈ P l (R n ), For each k ∈ Z, de ne the index set The following Lemmas 4.11, 4.13 and 4.14 show some properties of the smooth partition of unity ϕ k i . Lemma 4.12 gives some results for these ellipsoids from the Whitney covering lemma. These lemmas play an important role in the proof of H p(·) (Θ) ⊂ H p(·) q, l (Θ).
(i) For any (i, j) ∈ I k , we have θ x k+ Let f ∈ H p(·) q, l (Θ). By De nition 4.2, we know that there exist {λ i } i∈N ⊂ C and a sequence of (p(·), q, l)-atoms and . (4.28) Fix an x ∈ R n for the time being. By (4.27) and Lemma 4.4, we have By Lemma 2.7(i), Lemma 3.1 and the fact that, for any {θ x i , t i } i∈N ⊂ Θ and r ∈ ( , p), .
Combining this and f ∈ H p(·) q, l (Θ), we have Thus, by Lemma 4.5 with (4.30) and (4.33), and (4.28), we obtain To deal with I , by Lemma 2.7(i), Lemma 3.1 and (4.29), we have which together with De nition 3.6 and (4.34) implies that Thus, (4.26) holds true. We now prove that H p(·) (Θ) ⊂ H p(·) q, l (Θ). To this end, it su ces to show that due to the fact that each (p(·), ∞, l)-atom is also a (p(·), q, l)-atom and hence Next we prove (4.36) by two steps.
Step 1. In this step, we show that, for any holds true.
To prove (4.37), we borrow some ideas from those used in the proofs of [24,Theorem 4.8] and [16,Theorem 4.19]. Let f ∈ H p(·) (Θ) ∩ L q (R n ). For each k ∈ Z, we consider the Calderón-Zygmund decomposition of f of degree l ≥ N p(·) at height k associated with M • , f = g k + i∈N b k i . From this, the de nition of b k i and i∈N ϕ k i = χ Ω k , it follows that By this, (3.4), Lemma 4.11 and (4.18) we have Notice that f ∈ L q (R n ) = H q (Θ) with q ∈ (max{p+, }, ∞) (see [16, p. 1075]). Then, repeating the proof of [16,Lemma 4.11] with some slight modi cations, we nd that, for any k ∈ Z, and hence converges in S (R n ). By this, Lemma 4.9, Lemma 4.10, (4.14), (4.16), (4.18) and (3.2), we conclude that, for any k ∈ Z, x ∈ R n and r ∈ ( , p) close to p, where the last inequality holds due to a N p(·) − /r > . From this, the fact that r ∈ ( , p), Lemma 2.7(i), Lemma 3.1, (4.14), (4.18) and the de nition of Ω k , it follows that, for any k ∈ Z, This together with (4.22) and De nition 3.6 further implies that, From this, the fact that g k L ∞ (R n ) → as k → −∞ (see (4.38)), and Lemma 3.7, we deduce that On the other hand, by an argument same as that used in [16,Theorem 4.19], we obtain and where C is a positive constant independent of k and i. Now, for any k ∈ Z and i ∈ N, let and where C is as in (4.42). Then, by (4.40), (4.41) and (4.42), we easily know that, for any k ∈ Z and i ∈ N, a k i is a (p(·), ∞, l)-atom. Moreover, we have In addition, from (4.43), (4.14), (4.18), the de nition of Ω k and De nition 3.6, we further deduce that This implies that (4.37) holds true.
Step 2. In this step, we prove that (4.37) also holds true for any f ∈ H p(·) (Θ).
To this end, let f ∈ H p(·) (Θ). Then, by Proposition 4.7, we know that there exists a sequence {f j } j∈N ⊂ H p(·) (Θ) ∩ L q (R n ) with q ∈ (max{p+, }, ∞) such that f = j∈N f j in H p(·) (Θ) and for any j ∈ N, Notice that, for any j ∈ N, by the conclusion obtained in Step 1, we nd that f j has an atomic decomposition, namely, where {λ j, k i } k∈N, i∈Z and {a j, k i } k∈N,i∈Z are constructed as in (4.43). Thus, {a j, k i } k∈N, i∈Z are (p(·), ∞, l)atoms. By this, (4.44) and Proposition 2.8, we have which implies that (4.37) holds true for any f ∈ H p(·) (Θ) and hence completes the proof of Theorem 4.3.

Variable Anisotropic Singular Integral Operators
In this section, we introduce the notion of variable anisotropic singular integral operators associated with a continuous ellipsoid cover Θ and show that such operators are bounded from H p(·) (Θ) to L p(·) (R n ) in general and from H p(·) (Θ) to itself under the moment condition. |K(x, y) − K(x , y)|dy ≤ C, (5.2) where r > , y ∈ Bρ Θ (y, r), y ∈ R n and Bρ Θ (y, r) is as in (2.3). We say that T is a variable anisotropic singular integral operator (VASIO) of order if T : L (R n ) → L (R n ) is a bounded linear operator if there exists a kernel K satisfying (5.1) and (5.2) such that x ∉ supp (f ).

Example 5.2. (i) Let K be a locally integrable function on
and there exist positive constants δ and C such that for all x ≠ y ∈ R n we have , where κ ≥ is the triangle inequality constant of ρ Θ . This kernel K satis es (5.1) and (5.2) by referring to [6,Proposition 5.7]. And hence a bounded linear operator T : L (R n ) → L (R n ) associated with the above kernel K is a singular integral operator as in De nition 5.
Since we are interested in the boundedness of singular integral operators on the Hardy spaces H p(·) (Θ), < p(·) ≤ , we need to impose smoothness hypothesis on the kernel K, which is much stricter than that given by De nition 5.1. To this end we shall extend the de nition of Calderón-Zygmund operators in anisotropic setting which was given in [6, De nition 5.4].
De nition 5.4. Let s ∈ N and let T be a VASIO as in De nition 5.1 with kernel K(x, y) in the class C s (R n ) as a function of y. Then we say that T is a VASIO of order s if there exists a constant C > such that for any (x, y) ∈ Ω and for any multi-index |α| ≤ s we have Our ultimate goal is to show that anisotropic Calderón-Zygmund operators T are bounded on H p(·) (Θ). Generally, we can not expect this unless we also assume that T preserves vanishing moments. Hence, we adopt the following de nition motivated by [6, De nition 5.9].
De nition 5.5. Let s ∈ N and < q < ∞. We say that a VASIO T of order s satis es where l < a s/a , if for any f ∈ L q (R n ) with compact support with vanishing moments R n f (x)x α dx = for all |α| < s, we have The actual value of q is not relevant in De nition 5.5 as we merely need that T : L q (R n ) → L q (R n ) is bounded (see [6, Theorem 5.2 and Remark 5.3]). However, the requirement that l < a s/a is essential to guarantee that the integrals R n Tf (x)x α dx are well de ned for all |α| ≤ l. This is a consequence of the following lemma.
Lemma 5.6. [6,Lemma 5.10] Let l, s ∈ N, < q < ∞. Let T be a VASIO of order s. Suppose that f ∈ L q (R n ) satis es supp f ⊂ θ z,t for some z ∈ R n , t ∈ R, and R n f (x)x α dx = for all |α| < s. Then, for some C > depending only on T (s) and p(Θ), In particular, if l < a s/a , then We are now ready to state the main results of the paper, Theorems 5.7 and 5.8. There are generalizations of [6, Theorems 5.12 and 5.11] from H p(·) (Θ) to L p(·) (R n ) and from H p(·) (Θ) to itself. where p is as in (2.4). Then, T extends to a bounded linear operator from H p(·) (Θ) to L p(·) (R n ). T * (x α ) = for all α ∈ N n , |α| ≤ N p(·) . (5.8) Then, T extends to a bounded linear operator from H p(·) (Θ) to itself.
To prove Theorems 5.7 and 5.8, we need the following de nition and technical lemmas.
Lemma 5.10. Let p(·) ∈ C log (R n ), < q < ∞, and l ≥ N p(·) . Then, for any f ∈ L q (R n ) ∩ H p(·) (Θ), there exist a sequence of (p(·), ∞, l)-atoms {a k i } k∈Z,i∈N , a sequence {λ k i } k∈Z,i∈N ⊂ C, and a positive constant C independent of f such that ≤ C f H p(·) (Θ) (5.9) and f = k∈Z i∈N λ k i a k i converges in L q (R n ) and almost everywhere, (5.10) where p is as in (2.4).
Proof. By checking the proof of Theorem 4.3, we obtain (5.9) and (5.10) holds almost everywhere. By referring to the proof of [6,Theorem 4.10] with the existing estimates in the proof of Theorem 4.10, we conclude that (5.10) also holds in L q (R n ).
Proof of Theorem 5.7. First, we show that T(f ) L p(·) (R n ) f H p(·) (Θ) holds true for any f ∈ H p(·) (Θ) ∩ L q (R n ) with q ∈ ( , ∞)∩(p+, ∞). For f ∈ H p(·) (Θ)∩L q (R n ), by Lemma 5.10, we know that there exist {λ k i } k∈Z,i∈N ⊂ C and a sequence of (p(·), ∞, s − )-atoms, {a k i } k∈Z,i∈N , supported, respectively, on {θ x k i , t k i } k∈Z,i∈N ⊂ Θ such that where the last inequality holds due to (5.16). For K , let φ ∈ S N p(·) , N p(·) (R n ) with support in B n , and s ∈ R. Suppose P is a polynomial of degree N p(·) − .
For any (p(·), ∞, s − )-atom a k i , shortly wrote as a supported on some θ z, t := θ x k i , t k i , by Lemma 5.11, we