Weighted Estimates for Iterated Commutators of Riesz Potential on Homogeneous Groups

: In this paper, we study the two weight commutators theorem of Riesz potential on an arbitrary homogeneous group H of dimension N . Moreover, in accordance with the results in the Euclidean space, we acquire the quantitative weighted bound on homogeneous group.

Finally, the Heisenberg group on R 3 is an example of a homogeneous group. If we define the multiplication (x, y, u)(x , y , u ) = (x + x , y + y , u + u + (xy − yx )/2), (x, y, u)(x , y , u ) ∈ R 3 , the R 3 with this group law is the Heisenberg group H 1 ; a dilation is defined by t • (x, y, u) = (tx, ty, t 2 u), that is the parameters β 1 = 1, β 2 = 1, β 3 = 2. Definition 1. Let w(x) is a function on H, which is non-negative locally integrable. For 1 < p < ∞, we call that w is an A p weight, denoted by w ∈ A p , if [w] A p := sup The supremum here is taken over of all balls B ⊂ H. We call that the quantity [w] A p is the A p constant of w. For p = 1, if M(w)(x) ≤ cw(x) for a.e.x ∈ H, then we say that w is an A 1 weight, denoted by w ∈ A 1 , where M represents the Hardy-Littlewood maximal function. In addition, let A ∞ := ∪ 1≤p≤∞ A p , then we have Definition 2. Let x ∈ H, and w(x) be a non-negative locally integrable function. For 1 < p < q < ∞, w ∈ A p,q if where p is the conjugate exponent of p, that is 1 p + 1 p = 1. We now review the definition of Riesz potential on homogeneous group. For 0 < α <N, and the corresponding associated maximal function M α by The reason why we study the weighted estimates for these operators is because they have a wide range of applications in partial differential equations, Sobolev embeddings or quantum mechanics (see [3] or [4]).
Muckenhoupt and Wheeden [5] are the first scholars to study the Riesz potential. When H is an isotropic Euclidean space, Muckenhoupt and Wheeden [5] show that I α is bounded from L p (w p ) to L q (w q ) for 1 < p < n α , 1 q = 1 p − α n , w ∈ A p,q . Moreover, the sharp constant in this inequality was given in [6]: The iterative commutators (I α ) m b , m ∈ N, are defined naturally by In 2016, Holmes, Rahm and Spencer [7] prove that Later, the quantitative estimates for iterated commutators of fractional integrals was obtained by N. Accomazzo, J. C. Martínez-Perales and I. P. Rivera-Ríos [8].
In 2013, Sato [9] gave the estimates for singular integrals on homogeneous groups. In [10], X. T. Duong, H. Q. Li and J. Li established the Bloom-type two weight estimates for the commutator of Riesz transform on stratified Lie groups. Moreover, Z. Fan and J. Li [11] obtained the quantitative weighted estimates for rough singular integrals on homogeneous groups.
Motivated by the above estimates, we investigate the quantitative weighted estimation for the higher order commutators of fractional integral operators on homogeneous groups.
In this paper, our main result is the follow theorem.

A System of Dyadic Cubes
We define a left-unchanged analogous-distance d on H by d(x, y) = ρ(x −1 y), which signifies that there has a constant A 0 ≥ 1 such that for any x, y, z ∈ H, Next, let B(x, r) := {y ∈ H : d(x, y) < r} be the open ball which is centered on x ∈ H and r > 0 is the radius.
Let A k be k-th denumerable index set. A denumerable class D := ∪ k∈Z D k , D k := {Q k β : β ∈ A k }, of Borel sets Q k β ⊆ H is known as a set of dyadic cubes with arguments δ ∈ (0, 1) and 0 < a 1 ≤ A 1 < ∞ if it has the characteristics below: (1) H = ∪ β∈A k Q k β (disjoint union) for all k ∈ Z; For arbitrary (k, β) and for any ≤ k, there is a exclusive γ such that Q k β ⊆ Q γ ; (4) For arbitrary (k, β) there exists no more that M (a settled geometric constant) γ such that Q k+1 γ ⊆ Q k β , and Q k β = ∪ Q∈D k+1 ,Q⊆Q k β Q; . The set Q k β is called a dyadic cube of generation k with centre x k β ∈ Q k β and side length (Q k β ) = δ k . From the natures of the dyadic system above, for any Q k β , Q k+1 γ and Q k+1 γ ⊂ Q k β , we get that there is a constant A 0 > 0 such that:

Sparse Operators
We review the concept of sparse family given in [12] on ordinary spaces of homogeneous description in the sense of Coifman and Weiss [13], which is also suitable in the case of homogeneous groups.

Definition 5.
Let 0 < η < 1, for every Q ∈ S, we call that the collection S ⊂ D of dyadic cubes be a η-sparse, if there exists a measurable subset E Q ⊂ Q such that |E Q | ≥ η|Q| and the sets {E Q } Q∈S have only limited overlap. Definition 6. Given a sparse family, the sparse operator A S is defined by In this subfraction, the primary target is to reveal the following quantitative edition of Lacey's pointwise domination inequality.
. . , T and η-sparse families S t ⊂ D t such that for a.e.x ∈ H, where for a sparse family S, A m,k α,S (b, ·) is the sparse operator given by To show the Proposition 1, we need some auxiliary maximal operators. To begin with, let j 0 be the smallest integer such that and let C j 0 := 2 j 0 +2 A 0 . Next we define the grand maximal truncated operator M I α as follows: where the first supremum is taken over of all balls B ⊂ H satisfying x ∈ B. We can know that this operator is of vital importance in the following proof, Given a ball B 0 ⊂ H, for x ∈ B 0 we also define a local edition of M I α by Now, we claim that the following lemma is true.
The following pointwise estimates holds: 1.
For a.e.x ∈ B 0 , There exists a constant C N,α > 0 such that for a.e.x ∈ H, Using the results of Lemma 1, we then prove the Proposition 1.
Proof of Proposition 1. In order to proof the Proposition 1, we refer to the thinking in [8] for this domination, which is adapted to our situation of homogeneous groups. Firstly, we suppose that f is supported in a ball B 0 := B(x 0 , r) ⊂ H, next we disintegrate H which respect to this ball B 0 . We can do it as follows. We start define the annuli U j := 2 j+1 B 0 \ 2 j B 0 , j ≥ 0 and select the minimum integer j 0 such that j 0 > j 0 and 2 j 0 > 4A 0 (5) Next, for any U j , we select the balls centred in U j and with radius 2 j− j 0 r to cover U j . From the doubling property [13], we obtain where C A 0 , j 0 is an positive constant that only relates on A 0 and j 0 .
We now go over the characters of these B j, . Denote B j, := B(x j, , 2 j− j 0 r), where j 0 is defines as in (4). Then we have C adj B j, := B(x j, , C adj 2 j− j 0 r), which was shown in the proof of Theorem 3.7 in [12] that C adj B j, ∩ U j+j 0 = ∅, ∀j ≥ 0 and ∀ = 1, 2, . . . , L j ; and Now, because of the Equation (8) and (9), we see that each C adj B j, , at most overlap with 2j 0 + 1 annuli U j 's. Moreover, for every j and , C j 0 B j, covers B 0 .
Next by observing the (2), there is an integer t 0 ∈ {1, 2, . . . , T } and Q 0 ∈ D t 0 such that B 0 ⊆ Q 0 ⊆ C adj B 0 . Additionally, for this Q 0 , as in Section 2.1 the ball that includes Q 0 and has comparable measure to Q 0 is represented by B(Q 0 ). Consequently, B 0 is overwritten by B(Q 0 ) and |B(Q 0 )| |B 0 |, where the implicit constant relates only to C adj and A 1 . Now we claim that there exists a 1 2 -sparse family F t 0 ⊂ D t 0 (Q 0 ), the set of all dyadic cubes in t 0 -th dyadic system that are contained in Q 0 , such that for a.e. x ∈ B 0 , where is defined as in Section 2.1, j 0 defined as in (5) and j 0 defined as in (4).
Assume that we have already proven the assertion (10). Let us take a partition of H as follows: We next consider the annuli U j := 2 j+1 B 0 \ 2 j B 0 for j ≥ 0 and the covering { B j, } L j =1 of U j as in (6). We note that for each B j, , there exist t j, ∈ {1, 2, . . . , T } and Q j, ∈ D t j, such that B j, ⊆ Q j, ⊆ C adj B j, . Therefore, we acquire that for each such B j, , the enlargement Next, we utilize (10) to each B j, , then we acquire a 1 2 -sparse family F j, ⊂ D t j, ( Q j, ) such that (10) can be established for a.e. x ∈ B j, . Now, set F := ∪ j, F j, . Then we observe that the balls C adj B j, are overlapping not more than C A 0 , j 0 (2j 0 + 1) times, where C A 0 , j 0 is the constant in (7). Then, we can obtain that F is a 1 2C A 0 , j 0 (2j 0 +1) -sparse family and for a.e. c ∈ H, Since C j 0 B(Q) ⊂ R Q , and it is clear that |R Q | ≤ C|C j 0 B(Q)| (C depends only on C adj ), we obtain that f C j 0 B(Q) ≤ C f R Q . Now, we set S t := {R Q ∈ D t : Q ∈ F }, t ∈ {1, 2, . . . , T }, then since the fact that F is 1 2C A 0 , j 0 (2j 0 +1) -sparse, we can acquire that each family S t is where c is a constant relating only on C, C j 0 . Then it follows that (3) holds, which finishes the proof.
Proof of the Assertion (10). To demonstrate the assertion it suffice to attest the following recursive computation: there exist the cubes P j ∈ D t 0 (Q 0 ) that does not intersect each other such that ∑ j |P j | ≤ 1 2 |Q 0 | and for a.e. x ∈ B 0 , Iterating this estimate, we acquire (10) with F t 0 being the union of all the families {P k j }, where {P 0 j } = {Q 0 }, {P 1 j } = {P j } as mentioned above, and {P k j } are the cubes acquired at the k-th stage of the iterative approach. Clearly F t 0 is a 1 2 -sparse family, since let Now we prove the recursive estimate. For any countable family {P j } j of disjoint cubes P j ⊂ D t 0 (Q 0 ), we have that So we just have to reveal that we can opt for a family of pairwise disjoint cubes {P j } ⊂ D t 0 (Q 0 ) such that ∑ j |P j | ≤ 1 2 |Q 0 | and that for a.e. x ∈ B 0 , Using that (I α ) m b f = (I α ) m b−c f for any c ∈ R, and also that with C N,m,α being a positive number to be chosen. From [8], we can choose C N,m,α big enough (depending on C j 0 , C adj , and A 1 ) such that |E| ≤ 1 where A 0 is defined in Section 2.1. We now utilize the Calderón-Zygmund decomposition to the function χ E on B 0 at the height λ := 1 Fix some j. Since we have P j ∩ E c = ∅, we observe that which allows us to control the summation in W 2 by considering the cube P j . Now by (i) in Lemma 1, we know that Since |E \ ∪ j P j | = 0, we have that Consequently, These estimates allow us to control the remaining terms in W 1 , so we are done.
Proof of Lemma 1. Now we give the proof process of Lemma 1. The result in the Euclidean space case can be referred to as [8]. Now, we can adapt the proof in [8] to our setting of homogeneous groups.
(i) Let r is close enough to 0 such that B(x, r) ⊂ B 0 . Then, the estimate for the first term follows by standard computations involving a dyadic annulitype decomposition of the B(x, r). Then, the estimate in (i) is settled letting r → 0 in (11).
(ii) Let x, ξ ∈ B := B(x 0 , r). Let B x be the closed ball with radius 4(A 0 + C j 0 )r, which centered at x. Then C j 0 B ⊂ B x , and we acquire For the first term, since ρ is homogeneous of degree α − N, and by using the Proposition 1.7 in [1], we get Finally, we observe that which finishes the proof of (ii).
Next, we review that the dyadic weighted BMO space associated with the system D t is defined as Then according to the dyadic structure theorem studies in [14], one has Now, to verify a function b is in BMO η (H), it suffices to verify it belongs to each weighted dyadic BMO space BMO η,D t (H). Given a dyadic cube Q ∈ D t with t = 1, 2, . . . , T , and a measurable function f on H, we define the local mean oscillation of f on Q by With these notation and dyadic structure theorem above, following the same proof in [10], we also acquire that for any weight η ∈ A 2 , we have where C depends on η.

Proposition 2.
Suppose that H is a homogeneous group with dimension N, b ∈ L 1 loc (H). Then for any cube Q ⊂ H, there exist measurable set F i ⊂ Q with i = 1, 2, such that Proof. We take ideas from N. Accomazzo, J. C. Martínez-Perales and I. P. Rivera-Ríos [8].
In [8], for any cube Q ∈ D t with t = 1, 2, . . . , T , there exists a subset E ⊂ Q with |E| = 1 2 N+2 |Q| such that for every x ∈ E, where m b (Q) is a not necessarily unique number that satisfies Let E 1 ⊂ Q with |E| = 1 2 |Q| and such that b(x) ≥ m b (Q) for every x ∈ E 1 . Further let E 2 = Q \ E 1 , then |E 2 | = 1 2 |Q| and for every x ∈ E 2 , b(x) ≤ m b (Q). We obtain that at least half of the set E is contained either in E 1 or in E 2 since Q is the disjoint union of E 1 and E 2 . Without loss of generality, we assume that half of E is in E 1 , then we let Then if x ∈ F 1 and y ∈ F 2 , we have that which shows that Proposition 2 holds. Given a dyadic grid D, define the dyadic Riesz potential operator Proposition 3. Given 0 < α < N, then for any dyadic grid D, Proof. The result in the Euclidean setting is from the Proposition 2.1 in [15]. Here, we can adapt the proof in [15] to our setting of spaces of homogeneous type.

Proof of Theorem 1
To proof (i), we are following the ideas in [16] or [8]. Let D be a dyadic system in H and let S be a sparse family from D. We know by duality, we have that By Lemma 3.5 in [12], there exists a sparse family S ⊂ D such that S ⊂ S and for every cube Q ∈ S, for a.e. x ∈ Q, where Ω(b, P) = 1 |P| P |b(x) − b P |dx Assume that b ∈ BMO η (H) with η to be chosen, then we have for a.e. x ∈ Q, Then, we further have Next, note that for each ∈ N, from [12], for an arbitrary function h, we have  (13) and the boundedness of I α f , if p, q, α are as in the hypothesis of Theorem 1.1 and w ∈ A p,q , S ⊂ D, then Observe that A S is self-adjoint, then By Hölder inequality, we have that (see, e.g., [17] ), [λ q η iq ] A q max{1, 1 q−1 } I α S,η A k S,η (| f |) L q λ q η (m−k−1)q (H) .
Using (14), we have that  Combining all the preceding estimates obtains (i).
To proof (ii), we are going to follow ideas in [10]. Based on (12), it suffices to show that there exists a positive constant C such that for all dyadic cubes Q ∈ D t , Using Proposition 2 and Hölder inequality implies that where we used that 1 q + α N = 1 p . Further, this yields