Weighted Central BMO Spaces and Their Applications

In this paper, the central BMO spaces with Muckenhoupt 
 
 
 
 A
 
 
 p
 
 
 
 weight is introduced. As an application, we characterize these spaces by the boundedness of commutators of Hardy operator and its dual operator on weighted Lebesgue spaces. The boundedness of vector-valued commutators on weighted Herz spaces is also considered.


Introduction
For 1 < p < ∞ and a nonnegative locally integrable function ω on ℝ n , it is said that ω is in the Muckenhoupt A p class if it satisfies the condition A weight function ω belongs to the class A 1 if It is well-known that A ∞ = S 1≤p<∞ A p : Let ω ∈ A ∞ and p ∈ ð0,∞Þ; we denote L p ðωÞ as the space of all measurable functions f such that The definition of A p weight was introduced by Muckenhoupt [1]. Weighted inequalities arise naturally in Fourier analysis, but their use is best justified by the variety of applications in which they appear. For example, the theory of weights plays an important role in the study of boundary value problems for the Laplace equation on Lipschitz domains. Other applications of weighted inequalities include vector-valued inequalities, extrapolation of operators, and applications to certain classes of integral equations and nonlinear partial differential equations. There are a number of classical results which demonstrate that the Muckenhoupt A p classes are the right collections of weights to do harmonic analysis on weighted spaces. The main results along these lines are the equivalence between the ω ∈ A p condition and the L p ðωÞ boundedness (or weak boundedness) of maximal operator and singular integral operators.
A well-known result of Muckenhoupt [1] showed that the Hardy-Littlewood maximal operator M, that is is (weak) bounded on weighted Lebesgue spaces L p ðωÞ if and only if ω ∈ A p for 1 < p < ∞ (for the case n = 1). Hunt et al. [2] proved that the A p condition also characterizes the L p ðω Þ boundedness of the Hilbert transform H, where Later, Coifman and Fefferman [3] extended the A p theory to the case n ≥ 1 and the general Calderón-Zygmund operators; they also proved that A p weights satisfy the crucial reverse Hölder condition.
On the other hand, it is well-known that BMOðℝ n Þ is just the dual space of Hardy space H 1 ðℝ n Þ. Like this, the dual space of Herz-type Hardy space is the so-called central BMO space which is defined by where The space CBMO p ðℝ n Þ can be regarded as a local version of BMOðℝ n Þ at the origin, that is, BMOðℝ n Þ ⊊ CBMO p ðℝ n Þ for 1 ≤ p < ∞ (see [4]). However, they have quite different properties. For example, there is no analysis of the famous John-Nirenberg inequality of BMOðℝ n Þ for CBMO p ðℝ n Þ. See also [5][6][7][8][9] and [10] for more details. In 2007, Fu et al. [11] characterized CBMO p ðℝ n Þ space in terms of the boundedness of commutators of the Hardy operator.
In this paper, we will introduce the space of central BMO with Muckenhoupt A p weight and characterize these spaces by the boundedness of commutator of the Hardy operator and its dual operator on weighted Lebesgue spaces. The boundedness of vector-valued commutators on weighted Herz spaces is also considered.
Throughout this paper, the letter C denotes constants which are independent of the main variables and may change from one occurrence to another. Denote B k = fx ∈ ℝ n : jxj ≤ 2 k g and C k = B k \ B k−1 , and χ k is the characteristic function for k ∈ ℤ.

Weighted Central BMO Spaces
In this section, we will introduce the definition of weighted central BMO spaces and give some properties of CBMO p ðω Þ.
Let 1 ≤ p < ∞, and ω is a nonnegative locally integrable function. A function f ∈ L p loc ðℝ n Þ is said to belong to the When ω ≡ 1 is a constant, CBMO p ðωÞ is just CBMO p ð ℝ n Þ.
We recall some properties of the weighted Lebesgue spaces. Let Y n denote the set of all families of disjoint and open cubes in ℝ n . In [12], Diening et al. obtained the following lemma in the general case on Musielak-Orilicz spaces. But we only describe the special case on the weighted Lebesgue spaces now.

Lemma 1.
If ω ∈ A ∞ , then there exist 0 < δ < 1 and C > 0 which only depend on the A ∞ -constant of ω such that for all Q ∈ Y n ; all ft Q g Q∈Q , t Q ≥ 0; and all f ∈ L 1 loc ðℝ n Þ with f Q ≠ 0, Q ∈ Q.
Lemma 2 (see [1]). Let ω ∈ A p , 1 ≤ p < ∞; then, there exist constants C 1 , C 2 , δ > 0 such that for all balls B in ℝ n and all measurable subsets E ⊂ B, In fact, the first inequality of Lemma 2 can be improved as follows.

Lemma 3.
Let ω ∈ A p , 1 < p < ∞. Then, there exist p 0 with 1 < p 0 < p and C > 0 such that for all balls B in ℝ n and all measurable subsets E ⊂ B, Proof. By the fact that A p = ∪ q<p A q and ω ∈ A p (see [13]), there exist 1 < p 0 < p such that ω ∈ A p/p 0 . Applying Lemma 2, there exists a constant C > 0 such that for any ball B and any measurable set E ⊂ B,

Journal of Function Spaces
That is, Therefore we have proved Lemma 3. ☐ Now, we show the relationship between CBMO p ðωÞ and central BMO spaces.
Proof. Let f ∈ CBMO q ðωÞ. For any B ≔ Bð0, rÞ, by Hölder's inequality, we have On the other hand, Let f ∈ CBMO p ðωÞ, For any B ≔ Bð0 , rÞ, by Hölder's inequality and the condition ω ∈ A p , we have Therefore, we only need to prove that there exists a function f such that f ∈ CBMOðℝ n Þ \ CBMO p ðωÞ. Without loss of generality, we may assume that n = 1. Let sgn ðxÞ, then for any B ≔ Bð0, rÞ, When r ≤ 1, we have f ðxÞ ≡ 0 and When r > 1, there exists k 0 ∈ ℤ + such that 2 k 0 < r ≤ 2 k 0 +1 ; then, From (19) and (20), it follows that f ∈ CBMOðℝ n Þ.
which implies that Since By Lemma 3, there exists a constant 1 < p 0 < p such that This gives us Hence, the proof of Proposition 5 is completed. ☐ Proof. We set c Bð0,rÞ = f Bð0,rÞ for all balls Bð0, rÞ; the necessity of the condition in Proposition 6 holds. Let us check the sufficiency of Proposition 6. A similar argument as Proposition 4, we have, for any B ≔ Bð0, rÞ, Thus, Therefore, f ∈ CBMO p ðωÞ; the proof of Proposition 6 is completed. ☐

Proposition 7.
If ω ∈ A p and 1 < p < ∞, then f ∈ CBMO p ðωÞ if and only if Proof. The proof of Proposition 7 is similar as that of Proposition 6; we omit the details. ☐

Characterization of CBMO p ðωÞ Spaces via Commutators
We first review the definitions of the n-dimensional Hardy operator and its dual operator. For a locally integrable function f in ℝ n , the n-dimensional Hardy operator H is defined by The dual Hardy operator H * is defined by Let b be a locally integrable function on ℝ n . The commutators of H and H * are defined by The study of the Hardy operator has a very long history, and a number of papers involved its generalizations, variants, and applications. For the earlier development of this kind of integrals and many important applications, we refer the interested reader to the masterpiece [14]. We are interested in the characterization of commutator of the Hardy operator. Now, we give a remarkable result about the commutator of the Hardy operator; that is, Fu et al. [11] showed the following. The following consequence improves Theorem 8.

Theorem 9.
If ω ∈ A p , 1 < p < ∞ and μ = ω 1−p′ . Then, the following statements are equivalent: It is easy to see that Journal of Function Spaces By Hölder's inequality, we get In [11], Fu et al. showed that for b ∈ CBMOðℝ n Þ and j, By Proposition 4, for b ∈ CBMO p ðωÞ ⊂ CBMOðℝ n Þ and This gives us ð Combing (38) and (41), we get From the condition ω ∈ A p and Lemma 2, it follows that for k ≥ j, there exists a constant δ ∈ ð0, 1Þ such that Therefore, generalized Minkowski's inequality implies ðiiÞ ⇒ ðiÞ. The condition b ∈ CBMO p ðωÞ ∩ CBMO p ′ ðμÞ turns out to be necessary for the conclusion that both ½b, H and ½b, H * are bounded on L p ðωÞ.
For any ball B ≔ Bð0, rÞ and x ∈ B, we have where f 0 = jxj n jBj −1 χ B ðxÞ. From ½b, H and ½b, H * that are bounded on L p ðωÞ, it follows that Therefore, we obtain that b belongs to CBMO p ðωÞ. Note that ðL p ðωÞÞ ′ = L p ′ ðω 1−p ′ Þ (see [15]). We know that ½b, H and ½b, H * are bounded on L p ′ ðμÞ. Therefore, we obtain that b ∈ CBMO p ′ ðμÞ.
This completes the proof of Theorem 9. ☐

Vector-Valued Inequality
In this section, we give the definition of weighted Herz spaces ( [16]). Let α ∈ ℝ, 0 < p, q < ∞, and ω be weight functions on ℝ n . The homogeneous weighted Herz space _ K for all sequences of functions f f j g ∞ j=1 satisfying In order to prove Theorem 10, we additionally introduce the next lemma well-known as the generalized Minkowski inequality.
Lemma 11. If 1 < r < ∞, then there exists a constant C > 0 such that for all sequences of functions f f j g ∞ j=1 satisfying Proof of Theorem 10. We focus on the proof of the boundedness of ½b, H, since the arguments of ½b, H * are similar with necessary modifications. For every f f j g ∞ For convenience, below, we denote F ≔ ∥f f j g j ∥ ℓ r . For x ∈ C k , generalized Hölder's inequality and generalized Minkowski's inequality (51) imply from the fact that which gives us By Lemma 3 and α/n < 1/p′, there exists a constant 1 < p 0 < p such that α/n < 1/p ′ 0 and 6 Journal of Function Spaces Then, This implies that If 0 < q ≤ 1, then we obtain If 1 < q < ∞, then we use Hölder's inequality and obtain b, This completes the proof of Theorem 10.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.