Maximal Function Characterizations of Hardy Spaces on R n with Pointwise Variable Anisotropy

: In 2011, Dekel et al. developed highly geometric Hardy spaces H p ( Θ ) , for the full range 0 < p ≤ 1, which were constructed by continuous multi-level ellipsoid covers Θ of R n with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in Θ rapidly change shape from level to level, the authors further obtain some real-variable characterizations of H p ( Θ ) in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.


Introduction
As a generalization of the classical isotropic Hardy spaces H p (R n ) [1], anisotropic Hardy spaces H p A (R n ) were introduced and investigated by Bownik [2] in 2003. These spaces were defined on R n , associated with a fixed expansive matrix, which acts on an ellipsoid instead of Euclidean balls. In [3][4][5][6][7][8], many authors also studied Bownik's anisotropic Hardy spaces. In 2011, Dekel et al. [9] further generalized Bownik's spaces by constructing Hardy spaces with pointwise variable anisotropy H p (Θ), 0 < p ≤ 1, associated with an ellipsoid cover Θ. The anisotropy in Bownik's Hardy spaces is the same one at each point in R n , while the anisotropy in H p (Θ) can change rapidly from point to point and from level to level. Moreover, the ellipsoid cover Θ is a very general setting that includes the classical isotropic setting, non-isotropic setting of Calderón and Torchinsky [10], and the anisotropic setting of Bownik [2] as special cases; see more details in ( [2], pp. 2-3) and ( [11], p. 157).
On the other hand, maximal function characterizations are very fundamental characterizations of Hardy spaces, and they are crucial to conveniently apply the real-variable theory of Hardy spaces H p (R n ) with p ∈ (0, 1]. Maximal function characterizations were first shown for the classical isotropic Hardy spaces H p (R n ) by Fefferman and Stein in their fundamental work [1], ([12], Chapter III). Analogous results were shown by Calderón and Torchinsky [10,13] for parabolic H p spaces and Uchiyama [14] for H p on a homogeneoustype space. In 2003, Bownik ([2], p. 42) obtained the maximal function characterizations of the anisotropic Hardy space H p A (R n ). This was further extended to anisotropic Hardy spaces of the Musielak-Orlicz type in [15], to anisotropic Hardy-Lorentz spaces in [16], to variable anisotropic Hardy spaces in [17], and to anisotropic mixed-norm Hardy spaces in [18].
Motivated by the abovementioned facts, a natural question arises: Do the maximal function characterizations still hold for Hardy spaces H p (Θ) with variable anisotropy? In this article, we answer this question affirmatively in the sense that the ellipsoids in Θ can change shape rapidly from level to level, which is a variable anisotropic extension of Bownik's [2].
This article is organized as follows.
In Section 2, we recall some notation and definitions concerning anisotropic continuous ellipsoid cover Θ, several maximal functions, and anisotropic Hardy spaces H p (Θ) defined via the grand radial maximal function. We also give some propositions about H p (Θ), several classes of variable anisotropic maximal functions, and Schwartz functions since they provide tools for further work. In Section 3, we first state the main result: if the ellipsoids in Θ can rapidly change shape from level to level (see Definition 1), denoted as Θ t , we may obtain some real-variable characterizations of H p (Θ t ) in terms of the radial, the non-tangential, and the tangential maximal functions (see Theorem 1). Then, we present several lemmas that are isotropic extensions in the setting of variable anisotropy, and finally, we show the proof for the main result.
In the process of proving the main result, we used the methods from Stein [1] and Bownik [2]. However, it is worth pointing out that these ellipsoids of Bownik were images of the unit ball by powers of a fixed expansive matrix, whereas in our case, the ellipsoids of Dekel are images of the unit ball by powers of a group of matrices satisfying some "shape condition". This makes the proof complicated and needs many subtle estimates such as Propositions 5 and 6, and Lemma 1.
However, this article left an open question: if the maximal function characterizations of H p (Θ) still hold true in the sense that the ellipsoids of Θ change rapidly from level to level and from point to point?
Finally, we note some conventions on notation. Let N 0 := {0, 1, 2, . . .} and t be the smallest integer no less than t. For any α := (α 1 , . . . , α n ) ∈ N n 0 , |α| := α 1 + · · · + α n and ∂ α := ( ∂ ∂x 1 ) α 1 · · · ( ∂ ∂x n ) α n . Throughout the whole paper, we denote by C a positive constant that is independent on the main parameters but may vary from line to line. For any sets E, F ⊂ R n , we use E to denote the set R n \ E. If there are no special instructions, any space X (R n ) is denoted simply by X . Denote by S the space of all Schwartz functions and S the space of all tempered distributions.

Preliminary and Some Basic Propositions
In this section, we first recall the notion of continuous ellipsoid covers Θ and we introduce the pointwise continuity for Θ. An ellipsoid ξ in R n is an image of the Euclidean unit ball B n := {x ∈ R n : |x| < 1} under an affine transform, i.e., where M ξ is a non-singular matrix and c ξ ∈ R n is the center.
Let us begin with the definition of continuous ellipsoid covers, which was introduced in ([11], Definition 2.4).

Definition 1. We say that
is a continuous ellipsoid cover of R n or, in short, an ellipsoid cover if there exist positive constants p(Θ) := {a 1 , . . . , a 6 } such that (i) For every x ∈ R n and t ∈ R, there exists an ellipsoid θ(x, t) := M x, t (B n ) + x satisfying (ii) Intersecting ellipsoids from Θ satisfy a "shape condition", i.e., for any x, y ∈ R n , t ∈ R and s ≥ 0, if θ(x, t) ∩ θ(y, t + s) = ∅, then where · is the matrix norm given by M := max |x|=1 |Mx| for an n × n real matrix M. Particularly, for any θ(x, t) ∈ Θ, when the related matrix function M x,t of x ∈ R n and t ∈ R is reduced to the matrix function M t of t ∈ R, we call a cover Θ a t-continuous ellipsoid cover, denoted as Θ t .
The word continuous refers to the fact that ellipsoids θx, t are defined for all values of x ∈ R n and t ∈ R, and we say that a continuous ellipsoid cover Θ is pointwise continuous if, for every t ∈ R, the matrix valued function x → M x,t is continuous: Remark 1. By ( [19], Theorem 2.2), we know that the pointwise continuous assumption is not necessary since it is always possible to construct an equivalent ellipsoid cover such that Ξ is pointwise continuous and Ξ is equivalent to Θ. Here, we say that two ellipsoid covers Θ and Ξ are equivalent if there exists a constant C > 0 such that, for any x ∈ R n and t ∈ R, we have Taking M y, t+s = M x, t in (2), we have a 3 ≤ 1 and a 5 ≥ 1.
For any ϕ ∈ S, x ∈ R n , t ∈ R and θ( Particularly, when the matrix M x, t is reduced to M t , ϕ x, t (y) is simply denoted as ϕ t (y). Now, we give the notions of anisotropic variants of the non-tangential, the grand non-tangential, the radial, the grand radial, and the tangential maximal functions.
Definition 2. Let f ∈ S , ϕ ∈ S and N, N ∈ N 0 with N ≤ N. We define the non-tangential, the grand non-tangential, the radial, the rand radial, and the tangential maximal functions, respectively as Here and hereafter, the symbol " * " always represents a convolution.

Remark 2.
We immediately have the following pointwise estimate among the radial, the nontangential, and the tangential maximal functions: Next, we recall the definition of Hardy spaces with pointwise variable anisotropy ( [9], Definition 3.6) via the grand radial maximal function.
Let Θ be an ellipsoid cover of R n with parameters p(Θ) = {a 1 , · · · , a 6 } and 0 < p ≤ 1. We define N p (Θ) as the minimal integer satisfying and then N p (Θ) as the minimal integer satisfying Definition 3. Let Θ be a continuous ellipsoid cover and 0 < p ≤ 1.
Proof. This proposition is a corollary of ( [9], Theorems 4.4 and 4.19). Indeed, by Definition 3, we obtain that, for any N ≥ N p and N ≥ (a 4 N + 1)/a 6 , ([9], Theorem 4.4)), we obtain By checking the definition of anisotropic (p, q, l)-atom (see ([9], Definition 4.1)), we know that every (p, ∞, l)-atom is also a (p, q, l)-atom and hence Let l ≥ max(l, N). By a similar argument to the proof of ( [9], Theorem 4.19), we obtain where N ≥ N p and N ≥ (a 4 N + 1)/a 6 . Thus, Combining (7) and (8), we conclude that with equivalent (quasi-)norms. Let Θ be an ellipsoid cover. Then, there exists a constant J := J(p(Θ)) ≥ 1 such that, for any x ∈ R n and t ∈ R, Here and hereafter, let J always be as in Proposition 2.

Definition 4 ([9]
, Definition 3.1). Let Θ be an ellipsoid cover. For any locally integrable function f , the maximal function of the Hardy-Littlewood type of f is defined by

Proposition 3 ([9], Theorem 3.3).
Let Θ be an ellipsoid cover. Then, (i) There exists a constant C depending only on p(Θ) and n such that for all f ∈ L 1 and α > 0, (ii) For 1 < p < ∞, there exists a constant C p depending only on C and p such that, for all f ∈ L p , We give some useful results about variable anisotropic maximal functions with different apertures. They also play important roles in obtaining the maximal function characterizations of H p (Θ). For any given x ∈ R n , suppose that F : R n × R → (0, ∞) is a Lebesgue measurable function. Let Θ be an ellipsoid cover. For fixed l ∈ Z and t 0 < 0, define the maximal function of F with aperture l as Proposition 4. For any l ∈ Z and t 0 < 0, let F * t 0 l be as in (11). If the ellipsoid cover Θ is pointwise continuous, then F * t 0 l : R n → (0, ∞] is lower semi-continuous, i.e., {x ∈ R n : F * t 0 l (x) > λ} is open for any λ > 0.
Proof. If F * t 0 l (x) > λ for some x ∈ R n , then there exist t ≥ t 0 and y ∈ θ(x, t − l J) such that F(y, t) > λ. Since θ(x, t) is continuous for variable x (see Remark 1), there exists δ 1 > 0 such that, for any x ∈ U(x, δ) := {z ∈ R n : |z − x| < δ}, y ∈ θ(x , t − l J) and hence By Proposition 4, we obtain that {x ∈ R n : F * t 0 l (x) > λ} is Lebesgue measurable. Based on this and inspired by ([2], Lemma 7.2), the following Proposition 5 shows some estimates for maximal function F * t 0 l .

Proposition 5.
Let Θ be an ellipsoid cover, F * t 0 l and F * t 0 l as in (11) with integers l > l and t 0 < 0. Then, there exists a constant C > 0 that depends on parameters p(Θ) such that, for any functions F * t 0 l , F * t 0 l and λ > 0, we have and Proof.
The following result enables us to pass from one function in S to the sum of dilates of another function in S with nonzero mean, which is a variable anisotropic extension of ( [12], p. 93, Lemma 2) of Stein and ([2], Lemma 7.3) of Bownik.

Proposition 6.
Let Θ be an ellipsoid cover of R n and ϕ ∈ S, with R n ϕ(x) dx = 0. Then, for any ψ ∈ S, x ∈ R n , and t ∈ R, there exists a sequence {η k } ∞ k=0 and η k ∈ S, such that where J > 0 is as in Proposition 2. Furthermore, for any positive integers N, N and L, there exists a constant C > 0 depending on ϕ, L, N, N, and p(Θ) but not ψ, such that Proof. The following simplified proof is accomplished by Dekel. By scaling ϕ, we can assume that R n ϕ(x)dx = 1 and | ϕ(ξ)| ≥ 1/2, for |ξ| ≤ 2. This assumption only impacts the constant in (19). Let ζ ∈ S such that 0 ≤ ζ ≤ 1 on B n and supp (ζ) ⊂ 2B n . We fix x ∈ R n and t ∈ R, denote M k := M x, t+kJ , and define the sequence of functions {ζ k } ∞ k=0 , where ζ 0 := ζ, and where M T denotes the transpose of a matrix M. We claim that Indeed, by the properties of ζ, Proposition 2 and (2), In the other direction, Proposition 2 and the properties of ζ yield Applying (2), we have This proves (20). Additionally, by (2), for any ξ ∈ R n , From this, we deduce that, for any ξ ∈ R n , for a large enough k, (M −1 x, t M k ) T ξ ∈ B n . This implies that Thus, formally, a Fourier transform of (18) is given by Observe that η k is well defined and in S. Indeed, η k is well defined with 0/0 := 0, since by our assumption on ϕ, From this, it is obvious that η k ∈ S, and therefore, η k ∈ S. We now proceed to prove (19). First, observe that, for any η ∈ S, N, N ∈ N, Next, we claim that, for any K ∈ N, Indeed, on its support, any partial derivative of ζ k / ϕ((M −1 x, t M k ) T ·) has a denominator with its absolute value bounded from below and a numerator that is a superposition of compositions of partial derivatives of η and ϕ with contractive matrices of the type (M −1 x, t M k ) T . Using (20)-(22), we obtain

Maximal Function Characterizations of H p (Θ t )
In this section, we show the maximal function characterizations of H p (Θ t ) using the radial, the non-tangential, and the tangential maximal functions of a single test function ϕ ∈ S. Theorem 1. Let Θ t be a t-continuous ellipsoid cover, 0 < p ≤ 1, and ϕ ∈ S satisfy R n ϕ(x) dx = 0. Then, for any f ∈ S , the following are mutually equivalent: M 0 ϕ f ∈ L p ; (25) In this case, where the positive constants C 1 , C 2 , C 3 and C 4 are independent of f . , p. 47), we now start with maximal functions obtained from truncation with an additional extra decay term. Namely, for t 0 < 0 representing the truncation level and real number L ≥ 0 representing the decay level, we define the radial, the non-tangential, the tangential, the grand radial, and the grand non-tangential maximal functions, respectively, as The following Lemma 1 guarantees control of the tangential by the non-tangential maximal function in L p (R n ) independent of t 0 and L. Lemma 1. Let Θ t be a t-continuous ellipsoid cover. Suppose p > 0, N > 1/(a 6 p), and ϕ ∈ S. Then, there exists a positive constant C such that, for any t 0 < 0, L ≥ 0 and f ∈ S , Proof. Consider the function F : R n × R −→ [0, ∞) given by Let F * t 0 l be as in (11) with l = 0. When y ∈ θ(x, t), we have M −1 t (x − y) ∈ B n and hence By (2), we obtain M −1 t−(k−1)J M t ≤ a 5 2 −a 6 (k−1)J and hence, M −1 t−(k−1)J M t (B n ) ⊆ a 5 2 −a 6 (k−1)J B n , which implies (2 a 6 (k−1)J /a 5 )B n ⊆ M −1 t M t−(k−1)J (B n ). (27), it follows that |M −1 t (x − y)| ≥ 2 a 6 (k−1)J /a 5 . Thus, for any t ≥ t 0 , we have

From this and
By taking the supremum over all y ∈ R n and t ≥ t 0 , we know that Therefore, using this and Proposition 5, we obtain The following Lemma 2 gives the pointwise majorization of the grand radial maximal function by the tangential one, which is a variable anisotropic extension of ([2], Lemma 7.5).
The following Lemma 3 shows that the radial and the grand non-tangential maximal functions are pointwise equivalent, which is a variable anisotropic extension of ([2], Proposition 3.10).

Lemma 3 ([19]
, Theorem 3.4). For any N, N ∈ N with N ≤ N, there exists a positive constant C := C( N) such that, for any f ∈ S , The following Lemma 4 is a variable anisotropic extension of ( [2], p. 46, Lemma 7.6).

Lemma 4.
Let Θ t be a t-continuous ellipsoid cover, ϕ ∈ S, and f ∈ S . Then, for every M > 0 and t 0 < 0, there exist L > 0 and N > 0 large enough such that where C is a positive constant dependent on p(Θ), N , f , and ϕ.
Proof. For any ϕ ∈ S, there exist an integer N > 0 and positive constant C := C(ϕ) such that, for any N ≥ N and y ∈ R n , Therefore, for any t 0 < 0, t ≥ t 0 and x ∈ R n , by (34), we have Moreover, notice that, for any x ∈ M t +l J (B n ) + y , there exists z ∈ B n such that