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; 1) be a variable exponent function satisfying the globally log-H¨older 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 Calder´on-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 2 (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; 1) be a variable exponent function satisfying the glob- ally log-H¨oldercontinuous 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 bound- edness of δ-type Calder´on-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 Calder´on-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 expan- sive 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 Calder´on-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 ) estab- lished 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 Calder´on-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 := f1; 2;:::g and n n Z+ := f0g[N. For any α := (α1; : : : ; αn) 2 Z+ := (Z+) , let jαj := 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 2 [1; 1], we denote by q0 its conjugate index, namely, 1=q+1=q0 = 1. For any a 2 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 minλ2σ(A) jλj > 1, where σ(A) denotes the set of all eigenvalues of A. Let λ− and λ+ be two positive numbers such that 1 < λ− < minfjλj : λ 2 σ(A)g ≤ maxfjλj : λ 2 σ(A)g < λ+: In the case when A is diagonalizable over C, we can even take λ− := minfjλj : λ 2 σ(A)g and λ+ := maxfjλj : λ 2 σ(A)g. 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 2 (1; 1) and a set ∆ := fx 2 Rn : jP xj < 1g, where P is some non- degenerate n × n matrix, such that ∆ ⊂ r∆ ⊂ A∆; and one can additionally assume that j∆j = 1, where j∆j denotes the n-dimensional Lebesgue measure k of the set ∆. Let Bk := A ∆ for k 2 Z: Then Bk is open, Bk ⊂ rBk ⊂ Bk+1 k and jBkj = b , here and hereafter, b := j det Aj. An ellipsoid x + Bk for some x 2 Rn and k 2 Z is called a dilated ball. Denote by B the set of all such dilated balls, namely, n (2.1) B := fx + Bk : x 2 R ; k 2 Zg: 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 n E. Then, for all k; j 2 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 fx+y : x 2 E; y 2 F g 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; 1), for simplicity, denoted by ρ, satisfying n ~ ~ (i) ρ(x) > 0 for all x 2 R nf0ng, here and hereafter, 0n denotes the origin of Rn; (ii) ρ(Ax) = bρ(x) for all x 2 Rn, where, as above, b := j det Aj; (iii) ρ(x + y) ≤ H [ρ(x) + ρ(y)] for all x; y 2 Rn, where H 2 [1; 1) is a constant independent of x and y. 368 W. WANG n n In the standard dyadic case A := 2In×n, ρ(x) := jxj for all x 2 R is an example of homogeneous quasi-norms associated with A, here and hereafter, In×n denotes the n × n unit matrix, j · j 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 2 Rn, X k ~ ~ ρ(x) := b χBk+1nBk (x) if x 6= 0n; or else ρ(0n) := 0: k2Z By (2.2), we know that, for all x; y 2 Rn, ρ(x + y) ≤ bσ (max fρ(x); ρ(y)g) ≤ 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; 1) is called a variable exponent. For any variable exponent p(·), let (2.4) p− := ess inf p(x) and p+ := ess sup p(x): x2 n n R x2R Denote by P the set of all variable exponents p(·) satisfying 0 < p− ≤ p+ < 1.