Some estimates of intrinsic square functions on weighted Herz-type Hardy spaces

In this paper, by using the atomic decomposition theory of weighted Herz-type Hardy spaces, we will obtain some strong type and weak type estimates for intrinsic square functions including the Lusin area function, Littlewood-Paley $\mathcal G$-function and $\mathcal G^*_\lambda$-function on these spaces.


Introduction and preliminaries
First, let's recall some standard definitions and notations. The classical A p weight theory was first introduced by Muckenhoupt in the study of weighted L p boundedness of Hardy-Littlewood maximal functions in [15]. A weight w is a locally integrable function on R n which takes values in (0, ∞) almost everywhere, B = B(x 0 , R) denotes the ball with the center x 0 and radius R. We say that w ∈ A p , 1 < p < ∞, if where C is a positive constant which is independent of B. For the case p = 1, w ∈ A 1 , if A weight function w is said to belong to the reverse Hölder class RH r if there exist two constants r > 1 and C > 0 such that the following reverse Hölder inequality holds It is well known that if w ∈ A p with 1 < p < ∞, then w ∈ A r for all r > p, and w ∈ A q for some 1 < q < p. If w ∈ A p with 1 ≤ p < ∞, then there exists r > 1 such that w ∈ RH r .
Given a ball B and λ > 0, λB denotes the ball with the same center as B whose radius is λ times that of B. For a given weight function w, we denote the Lebesgue measure of B by |B| and the weighted measure of B by w(B), where w(B) = B w(x) dx.
We shall need the following lemmas.
Lemma A ( [4]). Let w ∈ A p , p ≥ 1. Then, for any ball B, there exists an absolute constant C such that In general, for any λ > 1, we have where C does not depend on B nor on λ.
Lemma B ( [4,5]). Let w ∈ A p ∩ RH r , p ≥ 1 and r > 1. Then there exist constants C 1 , C 2 > 0 such that for any measurable subset E of a ball B.
Next we shall give the definitions of the weighted Herz space, weak weighted Herz space and weighted Herz-type Hardy space. In 1964, Beurling [2] first introduced some fundamental form of Herz spaces to study convolution algebras. Later Herz [6] gave versions of the spaces defined below in a slightly different setting. Since then, the theory of Herz spaces has been significantly developed, and these spaces have turned out to be quite useful in harmonic analysis. For instance, they were used by Baernstein and Sawyer [1] to characterize the multipliers on the classical Hardy spaces, and used by Lu and Yang [12] in the study of partial differential equations.
On the other hand, a theory of Hardy spaces associated with Herz spaces has been developed in [3,10]. These new Hardy spaces can be regarded as the local version at the origin of the classical Hardy spaces H p (R n ) and are good substitutes for H p (R n ) when we study the boundedness of nontranslation invariant operators(see [11]). For the weighted case, in 1995, Lu and Yang introduced the following weighted Herz-type Hardy spaces HK α,p q (w 1 , w 2 ) (HK α,p q (w 1 , w 2 )) and established their atomic decompositions. In 2006, Lee gave the molecular characterizations of these spaces, he also obtained the boundedness of the Hilbert transform and the Riesz transforms on HK n(1/p−1/q),p q (w, w) and HK n(1/p−1/q),p q (w, w) for 0 < p ≤ 1. For the results mentioned above, we refer the readers to the book [14] and the papers [7,8,9,13] for further details. Let Given a weight function w on R n , for 1 ≤ p < ∞, we denote by L p w (R n ) the space of all functions satisfying Definition 1. Let α ∈ R, 0 < p, q < ∞ and w 1 , w 2 be two weight functions on R n . (i) The homogeneous weighted Herz spaceK α,p q (w 1 , w 2 ) is defined bẏ Definition 2. Let α ∈ R, 0 < p, q < ∞ and w 1 , w 2 be two weight functions on R n .
Let S (R n ) be the class of Schwartz functions and let S ′ (R n ) be its dual space. For f ∈ S ′ (R n ), the grand maximal function of f is defined by Definition 3. Let 0 < α < ∞, 0 < p < ∞, 1 < q < ∞ and w 1 , w 2 be two weight functions on R n . (i) The homogeneous weighted Herz-type Hardy space HK α,p q (w 1 , w 2 ) associated with the spaceK α,p q (w 1 , w 2 ) is defined by

The atomic decomposition
In this article, we will use the atomic decomposition theory for weighted Herz-type Hardy spaces in [8,9]. We characterize these spaces in terms of atoms in the following way.
where the infimum is taken over all the above decompositions of f .
where the infimum is taken over all the above decompositions of f .

The intrinsic square functions and our main results
The intrinsic square functions were first defined by Wilson in [17] and [18]. For 0 < β ≤ 1, let C β be the family of functions ϕ defined on R n , such that ϕ has support containing in {x : |x| ≤ 1}, R n ϕ(x) dx = 0 and for all For (y, t) ∈ R n+1 Then we define the intrinsic square function of f (of order β) by the formula where Γ(x) denotes the usual cone of aperture one: We can also define varying-aperture version of S β (f ) by the formula where Γ γ (x) is the usual cone of aperture γ > 0: The intrinsic Littlewood-Paley g-function(could be viewed as "zero-aperture" version of S β (f )) and the intrinsic g * λ -function(could be viewed as "infinite aperture"version of S β (f )) will be defined respectively by In [18], Wilson showed the following weighted L p boundedness of the intrinsic square functions.
Theorem D. Let w ∈ A p , 1 < p < ∞ and 0 < β ≤ 1. Then there exists a constant C > 0 such that In [16], the author obtained some boundedness properties of intrinsic square functions on the homogeneous and non-homogeneous weighted Herztype Hardy spaces. As a continuation of [16], the aim of this paper is to discuss their weak type estimates. Our main results are stated as follows.

Proofs of Theorems 1 and 2
Proof of Theorem 1. First we note that our assumption α = n(1 − 1/q) + β implies that s = [α + n(1/q − 1)] = [β] = 0. For every f ∈ HK α,p q (w 1 , w 2 ), then by Theorem C, we have the decomposition f = j∈Z λ j a j , where j∈Z |λ j | p < ∞ and each a j is a central (α, q, 0; w 1 , w 2 )-atom. Without loss of generality, we may assume that supp a j ⊆ B(0, R j ) and R j = 2 j . For any given σ > 0, we write Since w 2 ∈ A 1 , then w 2 ∈ A q for any 1 < q < ∞. Note that 0 < p ≤ 1. Applying Chebyshev's inequality and Theorem D, we have Changing the order of summation yields Since w 1 ∈ A 1 , then we know w ∈ RH r for some r > 1. When k ≤ j + 1, then B k ⊆ B j+1 . It follows from Lemma B that where δ = (r − 1)/r > 0. By using Lemma A and the inequality (1), we can get where the last series is convergent. Furthermore, it is bounded by a constant which is independent of j ∈ Z. Hence We now turn to estimate I 2 . We first claim that for any (x, t) ∈ R n+1 + and j ∈ Z, the following inequality holds Actually, this result was already given in [16] in a general form. Here, we give its proof for completeness. For any ϕ ∈ C β , by the vanishing moment condition of central atom a j , we can get Denote the conjugate exponent of q > 1 by q ′ = q/(q−1). Hölder's inequality and the A q condition imply Substituting the above inequality (5) into (4) and taking the supremum over all functions ϕ ∈ C β , we obtain the estimate (3). Note that when j ≤ k − 2, then for any y ∈ B j and x ∈ C k = B k \B k−1 , we have |x| ≥ 2|y|. We also note that supp ϕ ⊆ {x ∈ R n : |x| ≤ 1}, then we can get t ≥ |x − y| ≥ 1 2 |x|. Hence by the inequality (3), we deduce Since B j ⊆ B k−2 , then by using Lemma B, we obtain From our assumption α = n(1 − 1/q) + β and (6), it follows immediately that

then the inequality
holds trivially.
If {x ∈ C k : k−2 j=−∞ |λ j ||g β (a j )(x)| > σ/2} = Ø, then by the inequality (7), we have . It is easy to verify that lim k→∞ A k = 0. Then for any fixed σ > 0, we are able to find a maximal positive integer k σ such that σ < C · A kσ f HK α,p q (w 1 ,w 2 ) . From the above discussion, we have that B k−2 ⊆ B kσ−2 , then by using Lemma B again, we obtain Furthermore, it follows from Lemma A that Therefore Combining the above estimates (2) and (8) and taking the supremum over all σ > 0, we complete the proof of Theorem 1.
Proof of Theorem 2. The proof is similar. We only point out the main differences. Write Using the same arguments as in the proof of Theorem 1, we can prove To estimate J 2 , we note that if j ≤ k−2, then for any x ∈ C k = B k \B k−1 and z ∈ B j , we have |z| ≤ 1 2 |x|. Furthermore, when |x−y| < t and |y−z| < t, then we deduce 2t > |x − z| ≥ |x| − |z| ≥ 1 2 |x|.
By using the inequality (3), we thus obtain which is equivalent to The rest of the proof is exactly the same as that of Theorem 1, we can get This completes the proof of Theorem 2.

Proof of Theorems 3
In [16], the author have already established the following propositions.
Changing the order of summation gives Following the same lines as that of Theorem 1, we can show . We now turn to deal with K 2 . In the proof of Theorem 2, for any fixed j with j ≤ k − 2, we have already showed S β (a j )(x) 2 ≤ C · 2 j(n+β) w 1 (B j ) −α/n w 2 (B j ) −1/q 2 |x| −2n−2β .
Again, the rest of the proof is exactly the same as that of Theorem 1, we can obtain K 2 ≤ C f p HK α,p q (w 1 ,w 2 ) . Therefore, we conclude the proof of Theorem 3.
Remark. The corresponding results for non-homogeneous weighted Herztype Hardy spaces can also be proved by atomic decomposition theory. The arguments are similar, so the details are omitted here.