The boundedness of intrinsic square functions on the weighted Herz spaces

In this paper, we will obtain the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley $g$-function and $g^*_\lambda$-function on the weighted Herz spaces $\dot K^{\alpha,p}_q(w_1,w_2)$ ($K^{\alpha,p}_q(w_1,w_2)$) with general weights.

The Littlewood-Paley g-function (could be viewed as a "zero-aperture" version of S(f )) and the g * λ -function (could be viewed as an "infinite aperture" version of S(f )) are defined respectively by (see, for example, [13] and [14]) The modern (real-variable) variant of S γ (f ) can be defined in the following way (here we drop the subscript γ if γ = 1). Let ψ ∈ C ∞ (R n ) be real, radial, have support contained in {x : |x| ≤ 1}, and R n ψ(x) dx = 0. The continuous square function S ψ,γ (f ) is defined by (see, for example, [1] and [2]) In 2007, Wilson [22] introduced a new square function called intrinsic square function which is universal in a sense (see also [23]). This function is independent of any particular kernel ψ, and it dominates pointwise all the above-defined square functions. On the other hand, it is not essentially larger than any particular S ψ,γ (f ). 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 ∈ R n : |x| ≤ 1}, R n ϕ(x) dx = 0, and for all x, x ′ ∈ R n , |ϕ(x) − ϕ(x ′ )| ≤ |x − x ′ | β .
For (y, t) ∈ R n+1 + and f ∈ L 1 loc (R n ), we set Then we define the intrinsic square function of f (of order β) by the formula A β (f )(y, t) 2 dydt t n+1 We can also define varying-aperture versions of S β (f ) by the formula S β,γ (f )(x) =

Γγ (x)
A β (f )(y, t) 2 dydt t n+1 The intrinsic Littlewood-Paley G-function and the intrinsic G * λ -function will be given respectively by In [23], Wilson showed the following weighted L p boundedness of the intrinsic square functions.
Theorem A. Let 0 < β ≤ 1, 1 < p < ∞ and w ∈ A p (Muckenhoupt weight class). Then there exists a constant C > 0 independent of f such that [7], Lerner obtained sharp L p w norm inequalities for the intrinsic square functions in terms of the A p characteristic constant of w for all 1 < p < ∞. For further discussions about the boundedness of intrinsic square functions on various function spaces, we refer the readers to [5,18,19,20,21].
Before stating our main results, let us first recall some definitions about the weighted Herz and weak Herz spaces. For more information about these spaces, one can see [6,8,9,11,16] and the references therein. Let B k = B(0, 2 k ) = {x ∈ R n : |x| ≤ 2 k } and C k = B k \B k−1 for any k ∈ Z. Denote χ k = χ C k for k ∈ Z, χ k = χ k if k ∈ N and χ 0 = χ B 0 , where χ E is the characteristic function of the set E. For any given weight function w on R n and 0 < q < ∞, we denote by L q w (R n ) the space of all functions f satisfying . Let α ∈ R, 0 < p, q < ∞ and w 1 , w 2 be two weight functions on R n . (a) The homogeneous weighted Herz spaceK α,p q (w 1 , w 2 ) is defined bẏ The non-homogeneous weighted Herz space K α,p q (w 1 , w 2 ) is defined by For any k ∈ Z, λ > 0 and any measurable function f on R n , we set 11]). Let α ∈ R, 0 < p, q < ∞ and w 1 , w 2 be two weight functions on R n .
(c) A measurable function f (x) on R n is said to belong to the homogeneous weighted weak Herz space WK α,p (1.10) Obviously, if α = 0, thenK 0,q q (w 1 , w 2 ) = K 0,q q (w 1 , w 2 ) = L q w2 (R n ) for any 0 < q < ∞. We also have WK 0,q q (w 1 , w 2 ) = W K 0,q q (w 1 , w 2 ) = W L q w2 (R n ) when α = 0 and 0 < q < ∞, where Thus, weighted (weak) Herz spaces are generalizations of the weighted (weak) Lebesgue spaces. The main purpose of this paper is to consider the boundedness of intrinsic square functions on weighted Herz spaces with A p weights. At the extreme case, we will also prove that these operators are bounded from the weighted Herz spaces to the weighted weak Herz spaces. Our main results in the paper are formulated as follows.
In [22], Wilson also showed that for any 0 < β ≤ 1, the functions S β (f )(x) and G β (f )(x) are pointwise comparable, with comparability constants depending only on β and n. Thus, as a direct consequence of Theorems 1.1 and 1.2, we obtain the following: Then G β is bounded onK α,p q (w 1 , w 2 ) (K α,p q (w 1 , w 2 )) provided that w 1 and w 2 satisfy either of the following ).

A p weights
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 [12]. A weight w is a nonnegative, locally integrable function on R n , B = B(x 0 , r B ) denotes the ball with the center x 0 and radius r B . For any ball B and λ > 0, λB denotes the ball concentric with B whose radius is λ times as long. For a given weight function w and a measurable set E, we also denote the Lebesgue measure of E by |E| and set weighted measure where C is a positive constant which is independent of the choice of B. The smallest value of C such that the above inequalities hold is called the A p characteristic constant of w and denoted by [w] Ap . If there exist two constants r > 1 and C > 0 such that the following reverse Hölder inequality holds then we say that w satisfies the reverse Hölder condition of order r and write The following properties for A p weights will be repeatedly used in this paper.
Then, for any ball B, there exists an absolute constant C > 0 such that In general, for any λ > 1, we have where C does not depend on B nor on λ. 3,4]). 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.
Throughout this article, C always denotes a positive constant which is independent of the main parameters involved, but may vary from line to line.
Since S β (0 < β ≤ 1) is a sublinear operator, then we can write Since w 2 ∈ A q2 and 1 ≤ q 2 ≤ q, then w 2 ∈ A q . By Theorem A and Lemma 2.1, we have . For the term I 2 , we first use Minkowski's inequality to derive then by a direct computation, we can easily see that Thus, by using the above inequality (3.1) and Minkowski's inequality, we deduce Denote the conjugate exponent of q > 1 by q ′ = q/(q − 1). Applying Hölder's inequality and the A q condition, we can deduce that Substituting the above inequality (3.3) into (3.2), we thus obtain Here, we shall consider two cases. For the case of 0 < p ≤ 1, using the wellknown inequality ( ℓ |a ℓ |) p ≤ ℓ |a ℓ | p and changing the order of summation, we find that Moreover, it follows immediately from Lemma 2.1 that Since B k ⊇ B ℓ+2 when k ≥ ℓ + 2 and w i ∈ A qi for i = 1, 2. Then by Lemma 2.2, we can get Therefore where the last inequality holds since αq 1 < n(1 − q 2 /q). On the other hand, for the case of 1 < p < ∞, we will use Hölder's inequality to obtain Using the same arguments as above, we can also prove the following estimates under the assumption that Hence Summarizing the above estimates for the term I 2 , we obtain that for every 0 < p < ∞, Let us now turn to estimate the last term I 3 . In this case, for any x ∈ C k , (y, t) ∈ Γ(x) and z ∈ {2 ℓ−1 < |z| ≤ 2 ℓ } ∩ B(y, t) with ℓ ≥ k + 2, it is easy to check that Then it follows from the inequality (3.1) and Minkowski's inequality that This estimate together with (3.3) implies (3.8) Hence Now we will consider the following two cases again. For the case of 0 < p ≤ 1, by using the inequality ( ℓ |a ℓ |) p ≤ ℓ |a ℓ | p and changing the order of summation, we obtain Since w i ∈ A qi , then there exist r i > 1 such that w i ∈ RH ri for i = 1, 2. Thus by Lemma 2.2 again, we can get where δ i = (r i − 1)/r i > 0. Therefore, we have where in the last inequality we have used the fact that αδ 1 p/n+δ 2 p/q > 0 under our assumption (i) or (ii). On the other hand, for the case of 1 < p < ∞, an application of Hölder's inequality gives us that By using the same arguments as for I 3 , we are able to prove that the following two series is bounded by an absolute constant under the assumption (i) or (ii). Consequently From the above discussions for the term I 3 , we know that for any 0 < p < ∞, Summing up the above estimates for I 1 , I 2 and I 3 , we complete the proof of Theorem 1.1.
Proof of Theorem 1.2. Let f ∈K α,p q (w 1 , w 2 ). For any k ∈ Z, as in the proof of Theorem 1.1, we will split f (x) into three parts Then for any given λ > 0, we have Applying Chebyshev's inequality, Theorem A and Lemma 2.1, we obtain For any x ∈ C k , it follows from the inequalities (3.2) and (3.3) that By using Lemma 2.1, the inequality (3.4) and the fact that αq 1 = n(1 − q 2 /q), we deduce that Moreover, since 0 < p ≤ 1, then we have that for any x ∈ C k , holds trivially. Now we suppose that x ∈ C k : |S β (f 2 )(x)| > λ/3 = Ø. First it is easy to verify that lim k→∞ A k = 0. Then for any fixed λ > 0, we are able to find a maximal positive integer k λ such that Because B k ⊆ B k λ , then by Lemma 2.2 with the same notations δ i as in (3.9), we can get , for i = 1 and 2. Therefore On the other hand, it follows from the inequalities (3.3) and (3.7) that In the present situation, since B k ⊆ B ℓ−2 with ℓ ≥ k + 2, then it follows from the inequality (3.9) that Furthermore, recall that 0 < p ≤ 1, then for any x ∈ C k , we have Repeating the arguments used for the term I ′ 2 , we can also obtain Combining the above estimates for I ′ 1 , I ′ 2 and I ′ 3 , and then taking the supremum over all λ > 0, we finish the proof of Theorem 1.2.

Proofs of Theorems 1.3 and 1.4
In order to prove the main theorems of this section, let us first establish the following results.
Proposition 4.1. Let 0 < β ≤ 1, q = 2 and w ∈ A q2 with 1 ≤ q 2 ≤ q. Then for any j ∈ Z + , we have Proof. Since w ∈ A q2 , then by Lemma 2.1, we know that for any (y, t) ∈ R n+1 + , Taking square-roots on both sides of the above inequality, we are done.
Proposition 4.2. Let 0 < β ≤ 1, 2 < q < ∞ and w ∈ A q2 with 1 ≤ q 2 ≤ q. Then for any j ∈ Z + , we have Proof. For any j ∈ Z + and 0 < β ≤ 1, it is easy to see that Since q/2 > 1, then by duality, we have For w ∈ A q2 , we denote the weighted maximal operator by M w ; that is where the supremum is taken over all balls B which contain x. Then, by Lemma 2.1, we can get Substituting the above inequality (4.3) into (4.2) and then using Hölder's inequality together with the L (q/2) ′ w boundedness of M w , we thus obtain This estimate together with (4.1) implies the desired result.
Proposition 4.3. Let 0 < β ≤ 1, 1 < q < 2 and w ∈ A q2 with 1 ≤ q 2 ≤ q. Then for any j ∈ Z + , we have Proof. We will adopt the same method given in [17]. For any j ∈ Z + , set Observe that w Ω λ,j ≤ w Ω * λ + w Ω λ,j ∩ (R n \Ω * λ ) . Thus, for any j ∈ Z + , The weighted weak type estimate of M w yields To estimate II, we now claim that the following inequality holds.
The above inequality implies in particular that there is a point z ∈ B(y, t) ∩ (R n \Ω λ ) = ∅. In this case, we have (y, t) ∈ Γ(z) with z ∈ R n \Ω λ , which implies which is exactly what we want. This completes the proof of Proposition 4.3.
We are now in a position to give the proofs of the main theorems.