Dualities of Variable Anisotropic Hardy Spaces and Boundedness of Singular Integral Operators

Bull. Korean Math. Soc. 58 (2021), No. 2, pp. 365–384 https://doi.org/10.4134/BKMS.b200301 pISSN: 1015-8634 / eISSN: 2234-3016

DUALITIES OF VARIABLE ANISOTROPIC HARDY SPACES AND BOUNDEDNESS OF SINGULAR INTEGRAL OPERATORS

Wenhua Wang

n n Abstract. Let A be an expansive dilation on R , and p(·): R → (0, ∞) be a variable exponent function satisfying the globally log-Hölder p(·) n continuous condition. Let HA (R ) be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the author obtains the boundedness of anisotropic convolutional p(·) n p(·) n δ-type Calderón-Zygmund operators from HA (R ) to L (R ) or from p(·) n HA (R ) to itself. In addition, the author also obtains the duality be- p(·) n tween HA (R ) and the anisotropic Campanato spaces with variable exponents.

1. Introduction

As a good substitute of the Lebesgue space Lp(Rn) when p ∈ (0, 1], Hardy space Hp(Rn) plays an important role in various fields of analysis and partial differential equations; see, for examples, [10,17–19,21,23]. On the other hand, variable exponent function spaces have their applications in fluid dynamics [1], image processing [3], partial differential equations and variational calculus [9, 20, 21]. Let p(·): Rn → (0, ∞) be a variable exponent function satisfying the globally log-Höldercontinuous condition (see Section2 below for its definition). Re- cently, Nakai and Sawano [15] introduced the variable Hardy space Hp(·)(Rn), 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. Moreover, they obtained the boundedness of δ-type Calderón-Zygmund operators from Hp(·)(Rn) to Lp(·)(Rn) or from Hp(·)(Rn) to itself. Then Sawano [16], Yang et al. [22] and Zhuo et al. [24] further contributed to the theory.

Received April 2, 2020; Revised August 9, 2020; Accepted September 15, 2020. 2010 Mathematics Subject Classiﬁcation. Primary 42B20; Secondary 42B30, 46E30. Key words and phrases. Anisotropy, Hardy space, atom, Calder´on-Zygmund operator, Campanato space.

c 2021 Korean Mathematical Society 365 366 W. WANG

Very recently, Liu et al. [12] introduced the variable anisotropic Hardy space p(·) n HA (R ) associated with a general expansive matrix A, via the non-tangential grand maximal function, and then established its various real-variable charac- p(·) n terizations of HA (R ), respectively, in terms of the atomic characterization and the Littlewood-Paley characterization. p(·) n To complete the theory of the variable anisotropic Hardy space HA (R ), in this article, as applications of the atomic characterization, we obtain the boundedness of anisotropic convolutional δ-type Calderón-Zygmund operators p(·) n p(·) n p(·) n from HA (R ) to L (R ) and from HA (R ) to itself. In addition, we also p(·) n obtain the dual space of HA (R ) is the anisotropic Campanato space with variable exponents. The rest of this paper is organized as follows. In Section2, we first recall some notation and definitions concerning expansive dilations, variable exponent, the variable Lebesgue space Lp(·)(Rn) and p(·) n the variable anisotropic Hardy space HA (R ), via the non-tangential grand maximal function. Section3 is devoted to getting the boundedness of anisotropic convolutional p(·) n p(·) n δ-type Calderón-Zygmund operators from HA (R ) to L (R ) and from p(·) n p(·) n HA (R ) to itself, by using the atomic characterization of HA (R ) established in [12, Theorem 4.8] (see also Lemma 3.3 below). It is worth pointing out that some of the proof methods of the boundedness of Calderón-Zygmund op- p n p, p n p(·) n erators T on HA(R ) = HA (R ) ([13, Theorem 3.9]) and H (R ) ([15, The- orem 5.2]) don’t work anymore in the present setting. For example, we search out some estimates related to Lp(·)(Rn) norms for some series of functions which can be reduced into dealing with the Lq(Rn) norms of the corresponding functions (see Lemma 3.6 below). p(·) n In Section4, we prove that the dual space of HA (R ) is the anisotropic Campanato space with variable exponents (see Theorem 4.4 below). For this purpose, we first introduce a new kind of anisotropic Campanato spaces with p(·), q, s n variable exponents LA (R ) in Definition 4.1 below, which includes the anisotropic Campanato space of Bownik (see [2, p. 50, Definition 8.1]) and the space BMO(Rn) of John and Nirenberg [11]. Finally, we make some conventions on notation. Let N := {1, 2,...} and n n Z+ := {0}∪N. For any α := (α1, . . . , αn) ∈ Z+ := (Z+) , let |α| := α1+···+αn and ∂ α1 ∂ αn ∂α := ··· . ∂x1 ∂xn 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. For any q ∈ [1, ∞], we denote by q0 its conjugate index, namely, 1/q+1/q0 = 1. For any a ∈ R, bac denotes the maximal integer not larger than a. The symbol D . F means that D ≤ CF . If D . F and F . D, we then write D ∼ F . If E is a VARIABLE ANISOTROPIC HARDY SPACES 367

n subset of R , we denote by χE its characteristic function. If there are no special instructions, any space X (Rn) is denoted simply by X . For instance, L2(Rn) is simply denoted by L2. Denote by S the space of all Schwartz functions and S0 its dual space (namely, the space of all tempered distributions).

p(·) 2. Variable anisotropic Hardy space HA In this section, we ﬁrst recall the notion of variable anisotropic Hardy space p(·) HA , via the non-tangential grand maximal function MN (f), and then given its molecular characterization. We begin with recalling the notion of expansive dilations on Rn; see [2, p. 5]. A real n × n matrix A is called an expansive dilation, shortly a dilation, if minλ∈σ(A) |λ| > 1, where σ(A) denotes the set of all eigenvalues of A. Let λ− and λ+ be two positive numbers such 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 our arguments. It was proved in [2, p. 5, Lemma 2.2] that, for a given dilation A, there exist a number r ∈ (1, ∞) and a set ∆ := {x ∈ Rn : |P x| < 1}, where P is some non- degenerate n × n matrix, such that ∆ ⊂ r∆ ⊂ A∆, and one can additionally assume that |∆| = 1, where |∆| denotes the n-dimensional Lebesgue measure k of the set ∆. Let Bk := A ∆ for k ∈ Z. Then Bk is open, Bk ⊂ rBk ⊂ Bk+1 k and |Bk| = b , here and hereafter, b := | det A|. 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, n (2.1) B := {x + Bk : x ∈ R , k ∈ Z}.

Throughout the whole paper, let σ be the smallest integer such that 2B0 ⊂ σ n n A B0 and, for any subset E of R , let E{ := R \ E. Then, for all k, j ∈ Z with k ≤ j, it holds true that

(2.2) Bk + Bj ⊂ Bj+σ,

(2.3) Bk + (Bk+σ){ ⊂ (Bk){, where E+F denotes the algebraic sum {x+y : x ∈ E, y ∈ F } of sets E,F ⊂ Rn. Deﬁnition 2.1. A quasi-norm, associated with dilation A, is a Borel measur- n able mapping ρA : R → [0, ∞), for simplicity, denoted by ρ, satisfying n ~ ~ (i) ρ(x) > 0 for all x ∈ R \{0n}, here and hereafter, 0n denotes the origin of Rn; (ii) ρ(Ax) = bρ(x) for all x ∈ Rn, where, as above, b := | det A|; (iii) ρ(x + y) ≤ H [ρ(x) + ρ(y)] for all x, y ∈ Rn, where H ∈ [1, ∞) is a constant independent of x and y. 368 W. WANG

n n In the standard dyadic case A := 2In×n, ρ(x) := |x| for all x ∈ R is an example of homogeneous quasi-norms associated with A, here and hereafter, In×n denotes the n × n unit matrix, | · | always denotes the Euclidean norm in Rn. It was proved, in [2, p. 6, Lemma 2.4], that all homogeneous quasi-norms associated with a given dilation A are equivalent. Therefore, for a given dilation A, in what follows, for simplicity, we always use the step homogeneous quasi- norm ρ deﬁned by setting, for all x ∈ Rn, X k ~ ~ ρ(x) := b χBk+1\Bk (x) if x 6= 0n, or else ρ(0n) := 0. k∈Z By (2.2), we know that, for all x, y ∈ Rn, ρ(x + y) ≤ bσ (max {ρ(x), ρ(y)}) ≤ bσ[ρ(x) + ρ(y)]; see [2, p. 8]. Moreover, (Rn, ρ, dx) is a space of homogeneous type in the sense of Coifman and Weiss [4,5], where dx denotes the n-dimensional Lebesgue measure. Now we recall that a measurable function p(·): Rn → (0, ∞) is called a variable exponent. For any variable exponent p(·), let

(2.4) p− := ess inf p(x) and p+ := ess sup p(x). x∈ n n R x∈R Denote by P the set of all variable exponents p(·) satisfying 0 < p− ≤ p+ < ∞. Let f be a measurable function on Rn and p(·) ∈ P. Then the modular function (or, for simplicity, the modular) %p(·), associated with p(·), is deﬁned by setting Z p(x) %p(·)(f) := |f(x)| dx n R and the Luxemburg (also called Luxemburg-Nakano) quasi-norm kfkLp(·) by kfkLp(·) := inf λ ∈ (0, ∞): %p(·)(f/λ) ≤ 1 . Moreover, the variable Lebesgue space Lp(·) is deﬁned to be the set of all measurable functions f satisfying that %p(·)(f) < ∞, equipped with the quasi-norm kfkLp(·) . The following remark comes from [14, Remark 2.3]. Remark 2.2. Let p(·) ∈ P. (i) Obviously, for any r ∈ (0, ∞) and f ∈ Lp(·), r r k|f| kLp(·) = kfkLrp(·) . p(·) Moreover, for any µ ∈ C and f, g ∈ L , kµfkLp(·) = |µ| kfkLp(·) and p p p kf + gkLp(·) ≤ kfkLp(·) + kgkLp(·) , here and hereafter,

(2.5) p := min{p−, 1} VARIABLE ANISOTROPIC HARDY SPACES 369

p(·) with p− as in (2.4). In particular, when p− ∈ [1, ∞], L is a Banach space (see [8, Theorem 3.2.7]). (ii) It was proved in [6, Proposition 2.21] that, for any function f ∈ Lp(·)

with kfkLp(·) > 0, %p(·)(f/kfkLp(·) ) = 1 and, in [6, Corollary 2.22] that, if kfkLp(·) ≤ 1, then %p(·)(f) ≤ kfkLp(·) . A function p(·) ∈ P is said to satisfy the globally log-H¨oldercontinuous log condition, denoted by p(·) ∈ C , if there exist two positive constants Clog(p) n and C∞, and p∞ ∈ R such that, for any x, y ∈ R , C (p) |p(x) − p(y)| ≤ log log(e + 1/ρ(x − y)) and C |p(x) − p | ≤ ∞ . ∞ log(e + ρ(x)) A C∞ function ϕ is said to belong to the Schwartz class S if, for every integer ` α ` ∈ Z+ and multi-index α, kϕkα,` := sup [ρ(x)] |∂ ϕ(x)| < ∞. The dual space n x∈R of S, namely, the space of all tempered distributions on Rn equipped with the 0 weak-∗ topology, is denoted by S . For any N ∈ Z+, let

SN := {ϕ ∈ S : kϕkα,` ≤ 1, |α| ≤ N, ` ≤ N} . n −k −k In what follows, for ϕ ∈ S, k ∈ Z and x ∈ R , let ϕk(x) := b ϕ A x . Deﬁnition 2.3. Let ϕ ∈ S and f ∈ S0. The non-tangential maximal function n Mϕ(f) with respect to ϕ is deﬁned by setting, for any x ∈ R ,

Mϕ(f)(x) := sup |f ∗ ϕk(y)|. y∈x+Bk,k∈Z Moreover, for any given N ∈ N, the non-tangential grand maximal function 0 n MN (f) of f ∈ S is deﬁned by setting, for any x ∈ R ,

MN (f)(x) := sup Mϕ(f)(x). ϕ∈SN

p(·) The following variable anisotropic Hardy space HA was introduced in [12, Deﬁnition 2.4].

log Deﬁnition 2.4. Let p(·) ∈ C , A be a dilation and N ∈ [b(1/p−1)/ ln λ−c+ p(·) 2, ∞), where p is as in (2.5). The variable anisotropic Hardy space HA is deﬁned as p(·) n 0 p(·)o HA := f ∈ S : MN (f) ∈ L

p(·) and, for any f ∈ HA , let kfk p(·) := kMN (f)kLp(·) . HA Remark 2.5. Let p(·) ∈ Clog. 370 W. WANG

p(·) (i) The quasi-norm of HA in Definition 2.4 depends on N, however, by p(·) [12, Theorem 3.10], we know that the HA is independent of the choice of N, as long as N ∈ [b(1/p − 1)/ ln λ−c + 2, ∞). p(·) (ii) When p(·) := p, where p ∈ (0, ∞), the space HA is reduced to the p anisotropic Hardy HA studied in [2, Definition 3.11]. p(·) (iii) When A := 2In×n, the space HA is reduced to the variable Hardy space Hp(·) studied in [15, p. 3674]. We begin with the following notion of anisotropic (p(·), q, s)-atoms introduced in [14, Definition 4.1].

Definition 2.6. Let p(·) ∈ P, q ∈ (1, ∞] and s ∈ [b(1/p− −1)ln b/ ln λ−c, ∞)∩ Z+ with p− as in (2.4). An anisotropic (p(·), q, s)-atom is a measurable function a on Rn satisfying (i) supp a ⊂ B, where B ∈ B and B is as in (2.1); |B|1/q (ii) kakLq ≤ kχ k ; B Lp(·) R α n (iii) n a(x)x dx = 0 for any α ∈ with |α| ≤ s. R Z+ Throughout this article, we call an anisotropic (p(·), q, s)-atom simply by a (p(·), q, s)-atom. The following variable anisotropic atomic Hardy space was introduced in [12, Definition 4.2]. log Definition 2.7. Let p(·) ∈ C , q ∈ (1, ∞], s ∈ [b(1/p− − 1)ln b/ ln λ−c, ∞) ∩ Z+ with p− as in (2.4), and A be a dilation. The variable anisotropic atomic p(·), q, s 0 Hardy space HA is defined to be the set of all distributions f ∈ S satisfying that there exist {λi}i∈N ⊂ C and a sequence of (p(·), q, s)-atoms, {ai}i∈N, (i) supported, respectively, on {B }i∈N ⊂ B such that X 0 f = λiai in S . i∈N p(·), q, s Moreover, for any f ∈ HA , let

( p)1/p X |λi|χB(i) kfkHp(·), q, s := inf , A kχ (i) k p(·) i∈ B L N Lp(·) where the inﬁmum is taken over all the decompositions of f as above.

p(·) 3. Calderón-Zygmund operators on HA In this section, we obtain the boundedness of anisotropic convolutional δ- p(·) p(·) p(·) type Calderón-Zygmund operators from HA to L or from HA to itself. Let us begin with the notion of anisotropic Calderón-Zygmund operators associated with dilation A. ln λ+ Let δ ∈ (0, ln b ). We call a linear operator T is an anisotropic convolutional δ-type Calderón-Zygmund operator, if T is bounded on L2 with kernel k ∈ S0 VARIABLE ANISOTROPIC HARDY SPACES 371 coinciding with a locally integrable function on Rn \{0}, and satisfying that there exists a positive constant C such that, for any x, y ∈ Rn with ρ(x) > b2σρ(y), [ρ(y)]δ |k(x − y) − k(x)| ≤ C . [ρ(x)]1+δ For any f ∈ L2, T (f) := p.v.k ∗ f(x).

log ln λ+ Theorem 3.1. Let p(·) ∈ C and δ ∈ (0, ln b ). Assume T is an anisotropic 1 convolutional δ-type Calder´on-Zygmundoperator. If p− ∈ ( 1+δ , 1) with p− as p(·) in (2.4), then T can be extended to a bounded linear operator from HA to p(·) p(·) p(·) L and from HA to HA . Moreover, there exists a positive constant C p(·) such that, for any HA ,

(i) kT (f)kLp(·) ≤ Ckfk p(·) ; HA (ii) kT (f)k p(·) ≤ Ckfk p(·) . HA HA

Remark 3.2. When A := 2In×n and T is a Calderon-Zygmund operator of convolution type, Theorem 3.1 coincides with [15, Proposition 5.3, Theorem 5.5] of Nakai and Sawano, respectively. To prove Theorem 3.1, we need some technical lemmas. The following lemma reveals the atomic characterization of the variable anisotropic Hardy space (see [12, Theorem 4.8]).

log Lemma 3.3. Let p(·) ∈ C , q ∈ (max{p+, 1}, ∞] with p+ as in (2.4), s ∈ [b(1/p− − 1)ln b/ ln λ−c, ∞) ∩ Z+ with p− as in (2.4) and N ∈ N ∩ [b(1/p − 1)ln b/ ln λ−c + 2, ∞). Then

p(·) p(·), q, s HA = HA with equivalent quasi-norms. By the proof of [12, Theorem 4.8], we obtain the following conclusion, which plays an important role in the section.

log Lemma 3.4. Let p(·) ∈ C , r ∈ (1, ∞] and s ∈ [b(1/p− −1)ln b/ ln λ−c, ∞)∩ p(·) r Z+ with p− as in (2.4). Then, for any f ∈ HA ∩ L , there exist {λi}i∈N ⊂ C, dilated balls {xi + B`i }i∈N ⊂ B and (p(·), ∞, s)-atoms {ai}i∈N such that X X k k r p(·) f = λi ai in L and HA , k∈Z i∈N where the series also converges almost everywhere.

p(·) r Proof. Let f ∈ HA ∩ L . For any k ∈ Z, by the proof of [12, Theorem k n k 4.8], we know that there exist {xi }i∈N ⊂ Ωk = {x ∈ R : MN f(x) > 2 }, 372 W. WANG

k k k {`i }i∈N ⊂ Z, a sequence of (p(·), ∞, s)-atoms, {ai }k∈Z,i∈N, supported on {xi + k B k }k∈ ,i∈ , respectively, and {λ }k∈ ,i∈ ⊂ , such that `i +4σ Z N i Z N C

X X k k X X k 0 (3.1) f = λi ai =: hi in S , k∈Z i∈N k∈Z i∈N k k and for any k ∈ and i ∈ , supp h ⊂ x + B k ⊂ Ωk, Z N i i `i +4σ

k k k k (3.2) kh kL∞ 2 and ]{j ∈ :(x + B k ) ∩ (x + B k ) 6= ∅} ≤ R, i . N i `i +4σ j `j +4σ

p(·) r where R is as in [12, Lemma 4.5]. Moreover, by f ∈ HA ∩ L , we have, for k(x) almost every x ∈ Ωk, there exists a k(x) ∈ Z such that 2 < MN f(x) ≤ k(x)+1 k n 2 . From this, supp hi ⊂ Ωk and (3.2), we deduce that, for a.e. x ∈ R ,

X X k X X k (3.3) |hi (x)| ∼ |hi (x)| k∈Z i∈N k∈(−∞,k(x)]∩Z i∈N X X k 2 χ k (x) . xi +B`k+4σ i k∈(−∞,k(x)]∩Z i∈N X k k(x) ∼ 2 ∼ 2 ∼ MN f(x). k∈(−∞,k(x)]∩Z This implies that there exists a subsequence of the series

( K ) X X k hi , k=−K i∈ Z K∈N denoted still by itself without loss of generality, which converges to some measurable function fe almost everywhere in Rn. On the other hand, from (3.3), it follows that, for any K ∈ N and a.e. x ∈ Rn,

K X k X X k fe(x) − hi (x) . |fe(x)| + |hi (x)| k=−K k∈(−∞,k(x)]∩Z i∈N

. |fe(x)| + MN f(x) . MN f(x).

r From this, the fact that MN (f) ∈ L with 1 < r ≤ ∞, and the Lebesgue dominated convergence theorem, we further deduce that f = P P hk in e k∈Z i∈N i Lr. By this and (3.3), we know f = fe ∈ Lr and hence

X X k r p(·) f = hi in L and HA , k∈Z i∈N and also almost everywhere. VARIABLE ANISOTROPIC HARDY SPACES 373

We recall the deﬁnition of anisotropic Hardy-Littlewood maximal function 1 n MHL(f). For any f ∈ Lloc and x ∈ R , 1 Z (3.4) MHL(f)(x) := sup |f(z)| dz, x∈B∈B |B| B where B is as in (2.1). The following lemma is just [14, Lemma 4.3]. log Lemma 3.5. Let q ∈ (1, ∞]. Assume that p(·) ∈ C satisﬁes 1 < p− ≤ p+ < ∞, where p− and p+ are as in (2.4). Then there exists a positive constant C such that, for any sequence {fk}k∈N of measurable functions,

( )1/q !1/q X q X q [MHL(fk)] ≤ C |fk|

k∈ k∈ N Lp(·) N Lp(·) with the usual modiﬁcation made when q = ∞, where MHL denotes the Hardy- Littlewood maximal operator as in (3.4). Let us recall some auxiliary estimates together with the completeness of function spaces. The following Lemma 3.6, Lemma 3.7 and Lemma 3.8, respectively, come from [12, Lemma 4.6, Lemma 4.7] and [6, Lemma 2.71]. log Lemma 3.6. Let p(·) ∈ C and q ∈ (1, ∞] ∩ (p+, ∞] with p+ as in (2.4). (i) q Assume that {λi}i∈N ⊂ C, {B }i∈N ⊂ B and {ai}i∈N ∈ L satisfy, for any (i) i ∈ N, supp ai ⊂ B , |B(i)|1/q kakLq ≤ kχB(i) kLp(·) and

( p)1/p X |λi|χB(i) < ∞. kχ (i) k p(·) i∈ B L N Lp(·) Then

" #1/p ( p)1/p X p X |λi|χB(i) |λiai| ≤ C , kχ (i) k p(·) i∈ i∈ B L N Lp(·) N Lp(·) where p is as in (2.5) and C is a positive constant independent of {λi}i∈N, (i) {B }i∈N and {ai}i∈N. log Lemma 3.7. Let p(·) ∈ C and r ∈ (1, ∞] ∩ (p+, ∞] with p+ as in (2.4). p(·) r p(·) Then HA ∩ L is dense in HA . Lemma 3.8. Given p(·) ∈ P, Lp(·) is complete: every Cauchy sequence in Lp(·) converges in norm.

log p(·) Lemma 3.9. Let p(·) ∈ C . Then HA is complete: every Cauchy sequence p(·) in HA converges in norm. 374 W. WANG

Proof. We show this lemma by borrowing some ideas from the proof of [7, p(·) Proposition 4.1]. To prove that HA is complete, it is suﬃcient to prove that ∞ p(·) if {fk}k=1 is a sequence in HA such that X p kfkk p(·) < ∞, HA k∈N P p(·) where p is as in (2.5), then the series fk in H converges in norm. For k∈N A Pj any j ∈ N, let Fj := k=1 fk. From Remark 2.2(i) and the fact that p is as in (2.5), we conclude that, for any m, n ∈ N with m > n, m p m p p X X kFm − Fnk p(·) = fk ≤ MN (fk) HA k=n+1 p(·) k=n+1 p(·) HA L " m #p m

X X p = MN (fk) ≤ [MN (fk)] k=n+1 Lp(·)/p k=n+1 Lp(·)/p m m X p X p ≤ k[MN (fk)] kLp(·)/p = kMN (fk)kLp(·) k=n+1 k=n+1 m X p = kfkk . Hp(·) k=n+1 A

p(·) p(·) By this, we know that {Fj}j∈N is a Cauchy sequence in HA . Since HA is 0 continuously contained in S (see [12, Lemma 4.3]), thus {Fj}j∈N is a Cauchy 0 sequence in S . Therefore we know that there exists a tempered distribution 0 0 f ∈ S such that Fj → f in S as j → ∞. p(·) Next we prove f ∈ HA . Since j X MN (f) ≤ lim MN (fk), j→∞ k=1 by Remark 2.2(i), we obtain

j p j p p X X kMN (f)k p(·) ≤ lim MN (fk) = lim MN (fk) L j→∞ j→∞ k=1 Lp(·) k=1 Lp(·) j ∞ X p X p ≤ lim [MN (fk)] = k[MN (fk)] k p(·)/p j→∞ L k=1 Lp(·)/p k=1 ∞ X p = kfkk < ∞, Hp(·) k=1 A p(·) which implies that f ∈ HA . VARIABLE ANISOTROPIC HARDY SPACES 375

Finally, by Remark 2.2(i), we ﬁnd that

p p p j s s X X X f − fk = lim fk ≤ lim fk s→∞ s→∞ k=1 Hp(·) k=j+1 p(·) k=j+1 p(·) A HA HA ∞ X p ≤ kfkk → 0 as j → ∞, Hp(·) k=j+1 A p(·) P which implies that {Fj}j∈ in H converges to f = fk in norm. This N A k∈N ﬁnishes the proof of Lemma 3.9.

Proof of Theorem 3.1. We only prove (i), by using Lemma 3.9, (ii) can be proved in the same way. p(·) r First, we show that (i) holds true for any f ∈ HA ∩ L with r ∈ (1, ∞] ∩ p(·) r (p+, ∞]. For any f ∈ HA ∩ L , from Lemma 3.4, we know that there exist numbers {λi}i∈N ⊂ C and a sequence of (p(·), ∞, s)-atom, {ai}i∈N, supported, respectively, on {xi + B`i }i∈N ⊂ B such that X r f = λiai in L i∈N and

  p1/p  |λ |χ  X i xi+B`i kfk p(·) .   . H  χ  A i∈N xi+B`i Lp(·) Lp(·) By the assumption that the operator T is bounded on Lr, we further have

X r T (f) = λi(T ai) in L , i∈N and hence in S0. Then, by Remark 2.2(i), we ﬁnd that, for any x ∈ Rn, ! p p X kT (f)k = T λ a Lp(·) i i i∈N Lp(·) p

X = λiT (ai) i∈N Lp(·) p p X X σ ≤ |λi|T (ai)χxi+A B` + |λi|T (ai)χ σ { i (xi+A B`i ) i∈N Lp(·) i∈N Lp(·) p ( )1/p X h ip σ . |λi|T (ai)χxi+A B` i i∈ N Lp(·) 376 W. WANG

X + |λi|T (ai)χ σ { =: L1 + L2. (xi+A Bì ) i∈N Lp(·) r For the term L1, by the boundedness of T on L with r ∈ (max{p+, 1}, ∞), and Lemma 3.6, we conclude that p   p1/p  |λ |χ  X i xi+Bì p L1 kfk . .   . Hp(·)  χ  A i∈N xi+Bì Lp(·) Lp(·)

For the term L2, assume ai(x) is a (p(·), ∞, s)-atom supported on xi + B`i . σ { From the size condition of ai(x), we conclude that, for any x ∈ (xi + A B`i ) ,

T ai(x) = k ∗ ai(x) Z ≤ |k(x − y) − k(x − xi)| |ai(y)| dy xi+Bì+σ Z δ ρ(y − xi) . 1+δ |ai(y)| dy ρ(x − xi) xi+Bì+σ δ 0 |xi + Bì | 1/r r . 1+δ kaikL |xi + Bì | ρ(x − xi) |x + B |δ 1 h i1+δ 1 i ì M (χ )(x) . . 1+δ . HL xi+Bì ρ(x − x ) kχ k p(·) kχ k p(·) i Bì L Bì L By this, Remark 2.2(i), Lemma 3.5 and the fact that β := 1 + δ > 1/p, we obtain p β X |λi| h i L2 MHL(χx +B ) . i li kχxi+B` kLp(·) i∈N i Lp(·) 1/β βp ( β) X |λi| h i M (χ ) ∼ HL xi+Bì kχx +B k p(·) i∈ i ì L N Lβp(·) ( )1/β βp p |λi|χx +B |λi|χx +B X i ì X i ì . ∼ kχxi+B` kLp(·) kχxi+B` kLp(·) i∈ i i∈ i p(·) N Lβp(·) N L p 1/p p ( " |λ |χ # ) X i xi+Bì p . ∼ kfk p(·), q, s . kχx +B k p(·) HA i∈ i ì L N Lp(·)

Combining the estimates of L1 and L2, we further conclude that, for any f ∈ p(·) r HA ∩ L , kT (f)kLp(·) . kfk p(·) . HA VARIABLE ANISOTROPIC HARDY SPACES 377

p(·) p(·) Next, we prove that (i) also holds true for any f ∈ HA . Let f ∈ HA , by p(·) r Lemma 3.7, we know that there exists a sequence {fj}j∈Z+ ⊂ HA ∩ L with p(·) r ∈ (1, ∞] ∩ (p+, ∞] such that fj → f as j → ∞ in HA . Therefore, {fj}j∈Z+ p(·) is a Cauchy sequence in HA . By this, we see that, for any j, k ∈ Z+,

kT (fj) − T (fk)kLp(·) = kT (fj − fk)kLp(·) . kfj − fkk p(·) . HA p(·) Notice that {T (fj)}j∈Z+ is also a Cauchy sequence in L . Applying Lemma p(·) 3.8, we conclude that there exist a g ∈ L such that T (fj) → g as j → ∞ in Lp(·). Let T (f) := g. We claim that T (f) is well deﬁned. Indeed, for any p(·) r p(·) other sequence {hj}j∈Z+ ⊂ HA ∩ L satisfying hj → f as j → ∞ in HA , by Remark 2.2(i), we have p p p kT (hj) − T (f)kLp(·) ≤ kT (hj) − T (fj)kLp(·) + kT (fj) − gkLp(·) . p p . khj − fjk p(·) + kT (fj) − gkLp(·) HA p p p . khj − fk p(·) + kf − fjk p(·) + kT (fj) − gkLp(·) → 0, HA HA as j → 0, which is wished. p(·) From this, we see that, for any f ∈ HA ,

kT (f)kLp(·) = kgkLp(·) = lim kT (fj)kLp(·) . lim kfjk p(·) ∼ kfk p(·) , j→∞ j→∞ HA HA p(·) which implies that (i) also holds true for any f ∈ HA and hence completes the proof of Theorem 3.1.

p(·) 4. Dual spaces of HA p(·) In this section, we give the dual space of HA . More precisely, as an ap- p(·) plication of the atomic characterizations of HA obtained in Lemma 3.3, we p(·) prove that the dual space of HA is the variable anisotropic Campanato space p(·), q, s LA . Now, we introduce the notion of the anisotropic Campanato space with p(·), q, s variable exponents LA . Deﬁnition 4.1. Let A be a given dilation, p(·) ∈ P, s be a nonnegative integer and r ∈ [1, ∞). Then the anisotropic Campanato space with variable exponent p(·), q, s q LA is deﬁned to be the set of all f ∈ Lloc such that Z 1/q |B| 1 q kfkLp(·), q, s := sup inf |f(x) − P (x)| dx < ∞ A B∈B P ∈Ps kχBkLp(·) |B| B and |B| ∞ kfkLp(·), ∞, s := sup inf kf(x) − P (x)kL < ∞, A B∈B P ∈Ps kχBkLp(·) 378 W. WANG where B is as in (2.1).

(i) Lemma 4.2. Let p(·) ∈ P. Then, for any {λi}i∈N ⊂ C and {B }i∈N ⊂ B,

( p)1/p X X |λi|χB(i) |λi| ≤ , kχ (i) k p(·) i∈ i∈ B L N N Lp(·) where p is as in (2.5).

Proof. Since p(·) ∈ P and the well-known inequality that, k · k`1 ≤ k · k`p with p ∈ (0, 1], then

X X χB(i) |λi| = |λi| kχB(i) kLp(·) Lp(·) i∈N i∈N

|λ |χ (i) X i B ≤ kχB(i) kLp(·) i∈N Lp(·)

( p)1/p X |λi|χB(i) ≤ . kχ (i) k p(·) i∈ B L N Lp(·) This ﬁnishes the proof of Lemma 4.2. Lemma 4.3. Let A be a given dilation, p(·) ∈ Clog, s be a nonnegative integer p(·) and q ∈ [1, ∞). Then, for any continuous linear functional ` on HA = p(·), q, s HA , n o k`k p(·), q, s ∗ := sup |`(f)| : kfk p(·), q, s ≤ 1 (HA ) HA = sup{|`(a)| : a is (p(·), q, s)-atom},

p(·), q, s ∗ p(·), q, s here and hereafter, (HA ) denotes the dual space of HA .

Proof. For any (p(·), q, s)-atom a, it is easy to show that kfk p(·), q, s ≤ 1. HA Therefore, n o sup{|`(a)| : a is (p(·), q, s)-atom} ≤ sup |`(f)| : kfk p(·), q, s ≤ 1 . HA p(·), q, s On the other hand, let f ∈ HA and kfk p(·), q, s ≤ 1. Then, for any HA ε > 0, we know that there exist {λi}i∈N ⊂ C and a sequence of (p(·), ∞, s)- atoms, {ai} , supported, respectively, on {Bi}i∈ ⊂ B such that i∈N N X 0 f = λiai in S and almost everywhere i∈N and

( p)1/p X |λi|χBi ≤ 1 + ε. kχB k p(·) i∈ i L N Lp(·) VARIABLE ANISOTROPIC HARDY SPACES 379

From this, the continuity of ` and Lemma 4.3, we further conclude that X X |`(f)| ≤ |λi||`(ai)| ≤ |λi| sup{|`(a)| : a is (p(·), q, s)-atom} i∈N i∈N ≤ (1 + ε) sup{|`(a)| : a is (p(·), q, s)-atom}. Combined with the arbitrariness of ε and hence ﬁnishes the proof of Lemma 4.3. q ∞ Let q ∈ [1, ∞] and s ∈ Z+. Denote by Lcomp the set of all function f ∈ L with compact and Z q, s q α Lcomp := {f ∈ Lcomp : f(x)x dx = 0, |α| ≤ s}. n R In this paper, for any r ∈ Z+, we use Pr to denote the set of polynomials on Rn with order not more than r. The main result of this section is the following a theorem. log Theorem 4.4. Let A be a given dilation, p(·) ∈ C , p+ ∈ (0, 1], q ∈ (p+, ∞) and s ∈ [b(1/p− − 1)ln b/ ln λ−c, ∞) ∩ Z+ with p− as in (2.4). Then the dual p(·) p(·), q, s p(·), q, s ∗ space of HA = HA , denoted by (HA ) , is the variable anisotropic p(·), q0, s p(·), q0, s Campanato space LA in the following sense: for any b ∈ LA , the linear functional Z (4.1) `b(g) := b(x)g(x) dx, n R q, s p(·), q, s p(·) initial deﬁned for all g ∈ Lcomp, has a bounded extension to HA = HA . p(·), q, s p(·) Conversely, if ` is a bounded linear functional on HA = HA , then ` p(·), q0, s has the form as in (4.1) with a unique b ∈ LA . Moreover,

0 kbk p(·), q , s ∼ k`bk p(·), q, s ∗ , LA (HA ) where the implicit positive constants are independent of b.

Remark 4.5. (i) When A := 2In×n, Theorem 4.4 goes back to [15, Theorem 7.5]. (ii) When p(·) := p, where p ∈ (0, 1], Theorem 4.4 is reduced to [2, Theorem 8.3]. Proof of Theorem 4.4. By Lemma 3.3, we only need to show p(·), q0, s p(·), q, s ∗ LA ⊂ (HA ) . p(·), q0, s Let b ∈ LA and a be a (p(·), q, s)-atom supported on B ∈ B. Then, by the vanishing moment condition of a, H¨older’sinequality and the size condition of a, we obtain Z Z

(4.2) b(x)a(x) dx = inf (b(x) − P (x))a(x) dx n P ∈P R s B 380 W. WANG

≤ kbk p(·), q0, s . LA

Therefore, for {λi}i∈N ⊂ C and a sequence {ai}i∈N of (p(·), q, s)-atoms sup- (i) ported, respectively, on {B }i∈N ⊂ B and X p(·), q, s g = λiai ∈ HA , i∈N by Lemma 4.2 and (4.2), we have Z Z X |`b(g)| = b(x)g(x) dx ≤ |λi| |b(x) − P (x)| |ai(x)| dx n B R i∈N X ≤ |λi|kbk p(·), q0, s ≤ kgk p(·), q, s kbk p(·), q0, s . LA HA LA i∈N

0 p(·), q , s p(·), q, s ∗ This implies that LA ⊂ (HA ) holds true. 0 p(·), q, s ∗ p(·), q , s Now we prove (HA ) ⊂ LA . For any B ∈ B, let 1 AB : L (B) → Ps

1 be the natural projection satisfying, for any g ∈ L and q ∈ Ps, Z Z AB(g)(x)q(x) dx = g(x)q(x) dx. B B From the similar proof of [2, (8.9)], we know that there exists a positive constant 1 Cs such that, for any B ∈ B and g ∈ L (B), R B |g(z)| dz sup |AB(g)(x)| ≤ Cs . x∈B |B| For any q ∈ (1, ∞] and B ∈ B, let Lq(B) := {f ∈ Lq : suppf ⊂ B}. Set

q q L0(B) := {g ∈ L (B): AB(g)(x) = 0 and g is not zero almost everywhere}. q For any g ∈ L0(B), set

( |B|1/q −1 kχ k kgkLq (B)g(x), x ∈ B; (4.3) a(x) := B Lp(·) 0, x∈ / B. VARIABLE ANISOTROPIC HARDY SPACES 381

p(·), q, s ∗ Then a(x) is a (p(·), q, s)-atom. By this, we obtain, for any ` ∈ (HA ) q and g ∈ L0(B),

kχBkLp(·) (4.4) |`(g)| ≤ k`k p(·), q, s kgk q . 1/q (H )∗ L (B) |B| A q This show that ` is a bounded linear function on L0(B). By the Hahn-Banach theorem, it can be extended to Lq(B) without increasing its norm. 0 When q ∈ (1, ∞), noticing the fact that the duality of Lq(B) is Lq (B), we q0 q know that there exists a ξ ∈ L (B) such that, for any f ∈ L0(B), Z `(f) = f(x)ξ(x) dx. B 0 When q = ∞, it’s easy to see that there exists a ξ ∈ Lq (B) such that, for any ∞ f ∈ L0 (B), Z `(f) = f(x)ξ(x) dx. B 0 Therefore, for any q ∈ (1, ∞], it’s easy to see that there exists a ξ ∈ Lq (B) q such that, for any f ∈ L0(B), Z `(f) = f(x)ξ(x) dx. B Let q ∈ (1, ∞]. Now we prove that, if there exists another function ξ0 ∈ q0 q L (B) such that, for any f ∈ L0(B) and Z `(f) = f(x)ξ(x) dx, B 0 0 1 then ξ − ξ ∈ Ps(B). To prove this, we only need to prove that, if ξ, ξ ∈ L (B) ∞ R 0 R 0 such that, for any f ∈ L0 (B), B f(x)ξ (x) dx = B f(x)ξ(x) dx, then ξ − ξ ∈ ∞ Ps(B). In fact, for any f ∈ L0 (B), we obtain Z 0 0 = [f(x) − AB(f)(x)][ξ (x) − ξ(x)] dx B Z Z 0 0 = f(x)[ξ (x) − ξ(x)] dx − f(x)AB[ξ (x) − ξ(x)] dx B B Z 0 0 = f(x)[ξ (x) − ξ(x) − AB(ξ − ξ)(x)] dx. B Therefore, for a.e. x ∈ B ⊂ B, we have 0 0 ξ (x) − ξ(x) = AB(ξ − ξ)(x). 0 q Hence ξ −ξ ∈ Ps(B). From this, we see that, for any q ∈ (1, ∞] and f ∈ L0(B), q0 there exists a unique ξ ∈ L (B)/Ps(B) such that Z `(f) = f(x)ξ(x) dx. B 382 W. WANG

q If q ∈ (1, ∞], for any j ∈ N and g ∈ L0(Bj) with q ∈ (1, ∞], let fj ∈ q0 L (B)/Ps(B) be a unique function such that Z `(g) = fj(x)g(x) dx. Bj

Then, we see that, for any i, j ∈ N with i < j, fj|Bi = fi. From this and the p(·), q, s ∗ q fact that, for any g ∈ (HA ) , there exists a j0 ∈ N such that g ∈ L0(Bj0 ). p(·), q, s ∗ Thus, for any g ∈ (HA ) , we have Z (4.5) `(g) = b(x)g(x) dx, B where b(x) := fj(x) with x ∈ Bj. p(·), q0, s Next we need to show that b ∈ LA . From (4.5) and (4.4), we conclude that, for any q ∈ (1, ∞], B ∈ B,

kχBkLp(·) (4.6) kbk q ∗ ≤ k`k p(·), q, s . (L0(B)) 1/q (H )∗ |B| A Moreover, by [2, (8.12)], we know that

kbk q ∗ = inf kb − P k q0 . (L0(B)) (L (B)) P ∈Ps From this and (4.6), we conclude that, for any q ∈ (1, ∞], |B|1/q kbk p(·), q0, s = sup inf kb − P k q0 L (L (B)) A B∈B kχBkLp(·) P ∈Ps |B|1/q = sup kbk q ∗ (L0(B)) B∈B kχBkLp(·)

≤ k`k p(·), q, s ∗ < ∞. (HA ) This ﬁnishes the proof of Theorem 4.4. Acknowledgements. The authors would like to express their deep thanks to the referees for their very careful reading and useful comments which do improve the presentation of this article.

References [1] E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological ﬂu- ids, Arch. Ration. Mech. Anal. 164 (2002), no. 3, 213–259. https://doi.org/10.1007/ s00205-002-0208-7 [2] M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003), no. 781, vi+122 pp. https://doi.org/10.1090/memo/0781 [3] Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406. https://doi.org/10. 1137/050624522 [4] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homog`enes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin, 1971. VARIABLE ANISOTROPIC HARDY SPACES 383

[5] , Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. https://doi.org/10.1090/S0002-9904-1977-14325-5 [6] D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer,Heidelberg, 2013. https://doi.org/10.1007/ 978-3-0348-0548-3 [7] D. Cruz-Uribe and L.-A. D. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447–493. https://doi.org/10.1512/iumj.2014.63.5232 [8] L. Diening, P. Harjulehto, P. Hästö,and M. Ruˇziˇcka,◦ Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8 [9] X. Fan, Global C1,α regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2007), no. 2, 397–417. https://doi.org/10.1016/ j.jde.2007.01.008 [10] C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. https://doi.org/10.1007/BF02392215 [11] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. https://doi.org/10.1002/cpa.3160140317 [12] J. Liu, F. Weisz, D. Yang, and W. Yuan, Variable anisotropic Hardy spaces and their applications, Taiwanese J. Math. 22 (2018), no. 5, 1173–1216. https://doi.org/10. 11650/tjm/171101 [13] J. Liu, D. Yang, and W. Yuan, Anisotropic Hardy-Lorentz spaces and their applications, Sci. China Math. 59 (2016), no. 9, 1669–1720. https://doi.org/10.1007/s11425-016- 5157-y [14] , Anisotropic variable Hardy-Lorentz spaces and their real interpolation, J. Math. Anal. Appl. 456 (2017), no. 1, 356–393. https://doi.org/10.1016/j.jmaa.2017.07.003 [15] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Cam- panato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748. https://doi.org/10.1016/ j.jfa.2012.01.004 [16] Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), no. 1, 123–148. https://doi.org/10.1007/s00020-013-2073-1 [17] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993. [18] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of Hp-spaces, Acta Math. 103 (1960), 25–62. https://doi.org/10.1007/ BF02546524 [19] J.-O. Strömberg and A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathe- matics, 1381, Springer-Verlag, Berlin, 1989. https://doi.org/10.1007/BFb0091154 [20] J. Tan, Atomic decompositions of localized Hardy spaces with variable exponents and applications, J. Geom. Anal. 29 (2019), no. 1, 799–827. https://doi.org/10.1007/ s12220-018-0019-1 [21] L. Tang, Lp(·),λ(·) regularity for fully nonlinear elliptic equations, Nonlinear Anal. 149 (2017), 117–129. https://doi.org/10.1016/j.na.2016.10.016 [22] D. Yang, C. Zhuo, and E. Nakai, Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Mat. Complut. 29 (2016), no. 2, 245–270. https://doi.org/ 10.1007/s13163-016-0188-z [23] H. Zhao and J. Zhou, Anisotropic Herz-type Hardy spaces with variable exponent and their applications, Acta Math. Hungar. 156 (2018), no. 2, 309–335. https://doi.org/ 10.1007/s10474-018-0851-6 384 W. WANG

[24] C. Zhuo, D. Yang, and Y. Liang, Intrinsic square function characterizations of Hardy spaces with variable exponents, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1541– 1577. https://doi.org/10.1007/s40840-015-0266-2

Wenhua Wang School of Mathematics and Statistics Wuhan University Wuhan 430072, P. R. China Email address: [email protected]