Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups

Xueting Han; Yanping Chen

doi:10.1515/anona-2025-0092

Article Open Access

Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups

Xueting Han and Yanping Chen

Published/Copyright: July 4, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

Let G be a stratified Lie group, and let { X j } 1 ≤ j ≤ n 1 be a basis of the left-invariant vector fields of degree one on G and Δ = − ∑ j = 1 n 1 X j 2 be the sub-Laplacian of G . Given that 0 ≤ α < Q , this article studies the commutators [ b , X j Δ − 1 + α 2 ] m , j = 1 , … , n 1 of order m ∈ N and establishes their uniform two-weight boundedness from L p ( μ p ) to L q ( w q ) for any 0 < α < Q and 1 q = 1 p − α Q via BMO ν 1 ⁄ m ( G ) space, assuming that μ , w ∈ A p , q and ν = μ w . Based on this, we also give the characterization of VMO ( G ) space with respect to the uniform weighted compactness of [ b , X j Δ − 1 + α 2 ] for 0 ≤ α < Q . As a consequence of our results, the corresponding boundedness and compactness for commutators of Riesz transforms on G can be recovered as α → 0 .

Keywords: stratified Lie groups; Riesz transform; BMO space; VMO space

MSC 2010: 42B20; 42B35; 43A15; 22E30

1 Introduction

Δ R n = − ∑ j = 1 n ∂ j 2 is the Laplace operator on Euclidean space R n . For 0 ≤ α < n and x ∈ R n , for j = 1 , … , n , a kind of singular integral is defined as follows:

(1) ∂ j Δ R n − 1 + α 2 f ( x ) = 2 − α π − n 2 Γ ( n + 1 − α 2 ) Γ α + 1 2 ∫ R n y j ∣ y ∣ n + 1 − α f ( x − y ) d y ,

where Γ denotes the gamma function. In the particular case, when α = 0 , the operator in (1) reduces to the well-known Riesz transform

R j f ( x ) = π − n + 1 2 Γ n + 1 2 ∫ R n y j ∣ y ∣ n + 1 f ( x − y ) d y , j = 1 , … , n .

Moreover, the singular integral (1) also shows in the generalized surface quasi-geostrophic (SQG) equation whose form is

ω t + u ⋅ ∇ ω = 0 , ( x , t ) ∈ R 2 × R + u = ∇ ⊥ Δ R 2 − 1 + β ω , ω ( x , 0 ) = ω 0 ,

where 0 ≤ β ≤ 1 2 and ∇ ⊥ = ( − ∂ 2 , ∂ 1 ) (please refer [6–8,30–32] for further details).

In 2019, Yu and Jiu [30] established a uniform bound for ∂ j Δ R n − 1 + α 2 , j = 1 , … , n on R n . Since then, significant progress has been made in this area, including the uniform boundedness for the Calderón-Zygmund-type singular integral with a smooth kernel (refer to [31]) and in the case without any smoothness assumed on the kernel [6–8]. Furthermore, the uniform weighted bounds for the commutator of singular integral with rough kernel were also obtained in [8]. Here, the commutator generated by a linear operator T and a locally integrable function b is defined as

[ b , T ] f = b T ( f ) − T ( b f )

and the iterated commutator of order m ∈ N is defined inductively as

[ b , T ] m f = [ b , [ b , T ] m − 1 ] f .

Particularly, [ b , T ] 0 f = T f .

Based on these references, we apply the results in [8] to the operator ∂ j Δ R n − 1 + α 2 :

Theorem A

[8, Theorems 1.7 and 1.8] Let α ∈ [ 0 , 1 ⁄ 4 ] , 1 < p < n α , and 1 q = 1 p − α n . Let b ∈ BMO ( R n ) . Then, for j = 1 , … , n and i = 0 , 1 . If w ∈ A p , q , we have that

[ b , ∂ j Δ R n − 1 + α 2 ] i f L q ( w q ) ≤ C ( n ) ( p ′ + q ) [ w ] A p , q i + max 1 , p ′ q [ w ] A p , q min max 1 , 1 − α n p ′ q , max 1 − α n , p ′ q ‖ f ‖ L p ( w p ) ,

where C ( n ) depends only on n .

Consequently, for each j = 1 , … , n , the operator ∂ j Δ R n − 1 + α 2 and the commutator [ b , ∂ j Δ R n − 1 + α 2 ] can be considered as extensions of the Riesz transform R j and its commutator, respectively, on R n [8].

This article is devoted to studying the extension of the commutators of Riesz transforms in a more general setting of the stratified Lie groups, which plays an important role in quantum physics and a variety of fields in mathematics. It is known that the famous Heisenberg group H n is a classic example of the stratified Lie group [2,12,29].

For the sake of the completeness, here we briefly recall [2,4,10,12,26,29] that a connected and simply connected nilpotent Lie group G is called a stratified Lie group, if its left-invariant Lie algebra g , which is real and finite dimensional, is endowed with a direct sum decomposition:

(2) g = V 1 ⊕ … ⊕ V k , [ V 1 , V i ] = V i + 1 , for 1 ≤ i ≤ k − 1 and [ V 1 , V k ] = 0 .

Here, the Lie algebra g is generated by V 1 . Let dim V 1 = n 1 . We establish a basis of degree one { X 1 , … , X n 1 } for V 1 . Therefore, the sub-Laplacian on G can be defined as

Δ = − ∑ j = 1 n 1 X j 2 .

Assume G is a stratified Lie group with the homogeneous dimension Q and given 0 ≤ α < Q , we will consider the operator

X j Δ − 1 + α 2 , j = 1 , … , n 1 .

It is evident that X j Δ − 1 + α 2 on G stands for the operator ∂ j Δ R n − 1 + α 2 in (1) on Euclidean spaces and that X j Δ − 1 + α 2 corresponds to the Riesz transform X j Δ − 1 2 on G when α = 0 .

The study of commutators on Lie groups attracted much attention over the past decades [3,15,16,20,25,27]. It is noteworthy to highlight that, on stratified Lie groups G , Duong et al. [10] gave a characterization of the boundedness of the commutator of Riesz transforms [ b , X j Δ − 1 2 ] , j = 1 , … , n 1 using the bounded mean oscillation function space BMO ( G ) . Recently, the main result established in [9] contains the case that the weighted BMO ν 1 ⁄ m ( G ) space with the weight ν 1 ⁄ m ( m ∈ N ) can be characterized by the two-weight boundedness property for m -order commutators [ b , X j Δ − 1 2 ] m .

At this point, it is natural to consider the sufficient and necessary conditions of the two-weight boundedness for m -order commutators [ b , X j Δ − 1 + α 2 ] m with m ∈ N and j = 1 , … , n 1 . Moreover, this boundedness is uniform in the choice of 0 < α < Q on G .

Hereby, we state our first result as follows.

Theorem 1.1

Suppose that G is a stratified Lie group. Given 0 < α < Q and 1 < p < Q α , let 1 p + 1 p ′ = 1 and 1 q = 1 p − α Q . For μ , w ∈ A p , q , ν = μ w and j = 1 , 2 , … , n 1 .

If b ∈ BMO ν 1 m ( G ) , m ∈ N , the commutator [ b , X j Δ − 1 + α 2 ] m satisfies
[ b , X j Δ − 1 + α 2 ] m f L w q q ( G ) ≤ ∑ h = 0 m C m h C Q , p , q , m , h C w , μ , m , h C α , Q , q ‖ b ‖ BMO ν 1 m ( G ) m ‖ f ‖ L μ p p ( G ) ,
where
C α , Q , q = Γ Q − α 2 Γ α + 1 2 + Γ q Q − α 2 Γ q α + 2 2 ,

C w , μ , m , h = [ w q ] A q m + ( h + 1 ) 2 [ μ q ] A q m − ( h + 1 ) 2 m − h m max 1 , q ′ q [ w p ] A p h − 1 2 [ μ p ] A p m − h − 1 2 h m max 1 , p ′ p × [ w ] A p , q h m [ μ ] A p , q m − h m max p ′ q , 1 q + 1 p ′ .
C Q , p , q , m , h is always bounded and independent of α , and C α , Q , q is uniformly bounded as α → 0 .
For every b ∈ L loc 1 ( G ) , if [ b , X j Δ − 1 + α 2 ] m with m ∈ N is bounded from L μ p p ( G ) to L w q q ( G ) , then there exists a constant C < ∞ such that b ∈ BMO ν 1 m ( G ) with
‖ b ‖ BMO ν 1 m ( G ) ≤ C b , X j Δ − 1 + α 2 m L μ p p ( G ) → L w q q ( G ) 1 m [ μ p ] A p 1 p m [ w q ] A q 1 q m .

Remark 1.2

It should be noted that as α → 0 , the expression C α , Q , q → π − 1 2 Γ Q 2 + Γ p Q 2 is bounded. Hence, for j = 1 , 2 , … , n 1 and m ∈ N , the commutator [ b , X j Δ − 1 + α 2 ] m can be regarded as an extension of commutators of Riesz transforms [ b , X j Δ − 1 2 ] m on G .

Remark 1.3

According to Remark 1.2, when m = 0 , the operator X j Δ − 1 + α 2 can be viewed as an extension of the Riesz transform X j Δ − 1 2 on G . Additionally, if m = 1 and μ = w = 1 , Theorem 1.1 can recover the estimates for the commutator [ b , X j Δ − 1 2 ] , j = 1 , … , n 1 , as presented in [10].

Considering the aforementioned remarks, if we restrict Theorem 1.1 to the case μ = w and m = 1 , the BMO ( G ) space can be characterized by the uniform weighted boundedness of the commutator [ b , X j Δ − 1 + α 2 ] , j = 1 , 2 , … , n 1 for 0 ≤ α < Q on stratified Lie groups G . It is known that the space VMO ( G ) denotes the closure of all the C 0 ∞ functions on G under the norm of the BMO ( G ) space, and it is characterized by the weighted compactness of the commutator of the Riesz transform [ b , X j Δ − 1 2 ] on G [4].

Motivated by these results, in this article, we are going to establish a characterization of the VMO ( G ) space in terms of the uniform weighted compactness of the commutator [ b , X j Δ − 1 + α 2 ] , j = 1 , … , n 1 for 0 ≤ α < Q on G , which shall recover the conclusion in [4].

The relevant theorem is presented as follows:

Theorem 1.4

Suppose that G is a stratified Lie group. Given 0 ≤ α < Q and 1 < p < Q α , let 1 q = 1 p − α Q . Let b ∈ L loc 1 ( G ) , w ∈ A p , q .

If b ∈ VMO ( G ) , then for all j ∈ { 1 , … , n 1 } , the commutator [ b , X j Δ − 1 + α 2 ] is compact from L w p p ( G ) to L w q q ( G ) uniformly for 0 ≤ α < Q ;
If for some j ∈ { 1 , … , n 1 } , the commutator [ b , X j Δ − 1 + α 2 ] is compact from L w p p ( G ) to L w q q ( G ) , then b ∈ VMO ( G ) .

Remark 1.5

Theorem 1.4 is still new on Euclidean space.

Several noteworthy features distinguish this article:

To prove Theorem 1.1, we mainly adopt the method in [1]. Moreover, we also establish the corresponding weak-type boundedness directly and precisely for X j Δ − 1 + α 2 , where j = 1 , … , n 1 , making calculations slightly more complicated.
If we simply follow the previous strategy to prove compactness, we will meet the unbounded terms. We overcome this difficulty by employing the sparse domination [9] to obtain the uniform compactness.

Throughout this article, “ C ( Q ) ” denotes a positive constant, which is independent of the essential variables, and may change from line to line. “ C Q ” denotes a fixed positive constant.

This article is organized as follows. First, we shall list a collection of useful definitions and lemmas in Section 2. Next, Section 3 is devoted to proving Theorem 1.1. Finally, the proof of Theorem 1.4 is presented in Section 4.

2 Some definitions and key results

2.1 Stratified Lie group and heat kernel

Assumed that Lie algebra g is real and finite-dimensional, and is equipped with the natural dilations δ r ( r > 0 ) :

δ r ( ∑ i = 1 k v i ) = ∑ i = 1 k r i v i , with v i ∈ V i .

Therefore, this allows a dilation on G , which we still denote by δ r .

Lie algebra g has decomposition (2). If n i ≔ dim ( V i ) , i = 1 , … , k , the number

Q = ∑ i = 1 k i n i

will be the homogeneous dimension of G . Moreover, G is of topological dimension n = ∑ i = 1 k n i (see, for example, [4]).

The exponential map from g to G is a diffeomorphism, and d g be a bi-invariant Haar measure on G , which is the lift of Lebesgue measure on g via the exponential map.

We recall from [12] that there always exists at least one homogeneous norm ρ on G , which is defined by a continuous function g → ρ ( g ) from G to [ 0 , ∞ ) , which is C ∞ on G \ { 0 } and satisfies

ρ ( g − 1 ) = ρ ( g ) ;
ρ ( δ r ( g ) ) = r ρ ( g ) , for any g ∈ G and r > 0 ;
ρ ( g ) = 0 , if and only if g = 0 .

Moreover, the homogeneous norms on G are equivalent to each other [2,29]. Indeed, ρ is a quasi-metric, i.e., there exists a constant A 0 ≥ 1 , such that for ∀ g 1 , g 2 , g ∈ G ,

ρ ( g 1 , g 2 ) ≤ A 0 ( ρ ( g 1 , g ) + ρ ( g , g 2 ) ) .

Recall that the Carnot-Carathéodory metric d associated with { X j } 1 ≤ j ≤ n 1 which is suitable for the study of Δ [12,26]. For convenience, we set

d ( g , g ′ ) = d ( g ′ − 1 g ) = d ( g − 1 g ′ ) , ∀ g , g ′ ∈ G .

Indeed, d satisfies triangle inequality and is equivalent to ρ , i.e., for every g , g ′ ∈ G , there exist constants C d 1 > 0 and C d 2 ≥ 1 such that

(3) C d 1 ρ ( g ′ , g ) ≤ d ( g ′ , g ) ≤ C d 2 ρ ( g ′ , g ) .

Moreover, the constants C d 1 and C d 2 are independent of the essential variables and α , p and q in 1 q = 1 p − α Q , and so on. For the sake of brevity, we omit their notation in the constants C ( Q ) throughout the subsequent calculations.

The ball with radius r that is centered at g is defined by B ( g , r ) = { g ′ ∈ G : d ( g , g ′ ) < r } . Note that the Haar measure of the ball B ( g , r ) is ∣ B ( g , r ) ∣ = r Q . We also denote by t B with t > 0 the ball B ( g , t r ) .

Let p t ( t > 0 ) be the heat kernel [12,26], the convolution kernel of e − t Δ on G ,

(4) p t ( g ) = t − Q 2 p ( δ 1 t ( g ) ) , ∀ t > 0 , g ∈ G .

For 0 ≤ α < Q , the kernel of the operator X j Δ − 1 + α 2 , j = 1 , … , n 1 can be written as K j , α ( g , g ′ ) = K j , α ( g ′ − 1 g ) and

(5) K j , α ( g ) = 1 Γ α + 1 2 ∫ 0 + ∞ t α − 1 2 X j p t ( g ) d t = 1 Γ α + 1 2 ∫ 0 + ∞ t − Q − α 2 − 1 ( X j p ) δ 1 t ( g ) d t .

Now, combining with (3), we recall some estimates from [26,29].

Lemma 2.1

[26,29] Let G be a stratified Lie group and p t be the associated heat kernel. Then, for all g ∈ G , t > 0 , i , j = 1 , 2 , … , n 1 , there exist C and c such that

∣ X j p t ( g ) ∣ ≤ C t − Q + 1 2 e − d ( g ) 2 c t and ∣ X i X j p t ( g ) ∣ ≤ C t − Q + 2 2 e − d ( g ) 2 c t .

From Lemma 2.1, (4) and (5), it is not hard to check that [4]

(6) K j , α ∈ C ∞ ( G \ { 0 } ) , K j , α ( δ r ( g ) ) = r − Q + α K j , α ( g ) , ∀ g ≠ 0 , r > 0 , 1 ≤ j ≤ n .

Now, given 0 ≤ α < Q , it is easy to provide two estimates for K j , α , j = 1 , … , n 1 .

From (5) and Lemma 2.1, we can obtain

(7) ∣ K j , α ( g ) ∣ ≤ C ( Q ) 1 Γ ( α + 1 2 ) ∫ 0 ∞ t α − 1 2 t − Q + 1 2 e − d 2 ( g ) c t d t ≤ C ( Q ) Γ Q − α 2 Γ ( α + 1 2 ) d ( g ) α − Q ≕ C ( Q ) τ α , Q d ( g ) α − Q ,

where τ α , Q = Γ Q − α 2 ⁄ Γ ( α + 1 2 ) . For i , j ∈ { 1 , … , n 1 } , using Lemma 2.1 again, we also obtain

∣ X i K j , α ( g ) ∣ = 1 Γ ( α + 1 2 ) ∫ 0 ∞ t α − 1 2 X i X j p t ( g ) d t ≤ C ( Q ) Γ Q − α + 1 2 Γ α + 1 2 d ( g ) α − Q − 1 .

Since 0 ≤ α < Q , we have α + 1 2 ≥ 1 2 and 1 2 < Q − α + 1 2 ≤ Q + 1 2 . By the properties of gamma function, we have Γ α + 1 2 > 4 5 and Γ Q − α + 1 2 ≤ max Γ 1 2 , Γ Q + 1 2 . Now, we conclude that there exists a bounded constant C ( Q ) , which depends only on Q such that

Γ Q − α + 1 2 Γ α + 1 2 < 5 4 max Γ 1 2 , Γ Q + 1 2 ≤ C ( Q ) .

Thus, it follows that

(8) ∣ X i K j , α ( g ) ∣ ≤ C ( Q ) d ( g ) α − Q − 1 .

For 0 ≤ α < Q , applying (3) and (6), the same arguments as in the proof in [4] show the following result.

Lemma 2.2

Suppose that G is a stratified Lie group with homogeneous dimension Q and that j ∈ { 1 , 2 , … , n 1 } . There exists a large positive constant r 0 such that for every g ∈ G , there exists a set G g ⊂ G such that inf g ′ ∈ G g d ( g , g ′ ) = r 0 R 0 and that for every g 1 ∈ B ( g , R 0 ) and g 2 ∈ G g , we have

∣ K j , α ( g 1 , g 2 ) ∣ ≥ C K j , α d ( g 1 , g 2 ) − Q + α , ∣ K j , α ( g 2 , g 1 ) ∣ ≥ C K j , α d ( g 1 , g 2 ) − Q + α .

Here, C K j , α ≠ 0 for any 0 ≤ α < Q . All K j , α ( g 1 , g 2 ) as well as all K j , α ( g 2 , g 1 ) have the same sign. Moreover, this “twisted truncated sector” G g is regular, in the sense that ∣ G g ∣ = ∞ and that for any R 2 > R 1 > 2 r 0 R 0 , ∣ ( B ( g , R 2 ) \ B ( g , R 1 ) ) ∩ G g ∣ is equivalent to ∣ B ( g , R 2 ) \ B ( g , R 1 ) ∣ , where the implicit constants are independent of g and R 1 and R 2 .

2.2 Some function spaces

For any ball B in G , we recall from [18] that a nonnegative locally integrable function w ∈ A p , 1 < p < ∞ if

[ w ] A p = sup B ⊂ G 1 ∣ B ∣ ∫ B w 1 ∣ B ∣ ∫ B w − 1 p − 1 p − 1 < ∞

and w ∈ A p , q , 1 < p < q < ∞ , 1 p + 1 p ′ = 1 , if

[ w ] A p , q = sup B ⊂ G 1 ∣ B ∣ ∫ B w q 1 ∣ B ∣ ∫ B w − p ′ q p ′ < ∞ .

Then, we recall a useful inequality obtained in [4]. We denote by w ( B ) ≔ ∫ B w .

Lemma 2.3

[4] Let w ∈ A p ( G ) , p ≥ 1 . Then, there exist constants r > 1 , C l , C r > 0 such that

C l ∣ E ∣ ∣ B ∣ p ≤ w ( E ) w ( B ) ≤ C r ∣ E ∣ ∣ B ∣ ( r − 1 ) ⁄ r ,

for any measurable subset E of a ball B.

Next, we recall the weighted BMO space on G from [9].

Given a weight w ∈ A p and any ball B ∈ G , the weighted bounded mean oscillation space BMO w ( G ) is the space of all locally integrable functions b on G such that

‖ b ‖ BMO w ( G ) ≔ sup B ⊂ G 1 w ( B ) ∫ B ∣ b ( g ) − b B ∣ d g < ∞ ,

where b B = 1 ∣ B ∣ ∫ B b .

Definition 2.4

[28] A median value of b over a ball B will be any real number m b ( B ) that satisfies simultaneously

∣ { x ∈ B : b ( g ) > m b ( B ) } ∣ ≤ 1 2 ∣ B ∣ and ∣ { x ∈ B : b ( g ) < m b ( B ) } ∣ ≤ 1 2 ∣ B ∣ .

We also define that

M ( b , B ) ≔ 1 ∣ B ∣ ∫ B ∣ b ( g ) − b B ∣ d g .

Then, it is easy to obtain the inequality

(9) M ( b , B ) ≤ 2 1 ∣ B ∣ ∫ B ∣ b ( g ) − m b ( B ) ∣ d g .

We now recall the definition of Hardy spaces [12].

Definition 2.5

[10,12] The space H 1 ( G ) is the set of functions of the form f = ∑ j = 1 ∞ λ j a j with { λ j } ∈ ℓ 1 and a j an L ∞ -atom, meaning that it is supported on a ball B ⊂ G , has mean value zero ∫ B a ( g ) d g = 0 , and has a size condition ‖ a ‖ L ∞ ( G ) ≤ ∣ B ∣ − 1 . One norms this space of functions by

‖ f ‖ H 1 ( G ) ≔ inf ∑ j = 1 ∞ ∣ λ j ∣ : { λ j } ∈ ℓ 1 , f = ∑ j = 1 ∞ λ j a j , a j an atom ,

with the infimum taken over all possible representations of f via atomic decompositions.

2.3 Dyadic cubes

Lemma 2.6

[9,17] (a system of dyadic cubes) On stratified Lie groups G , a countable family D ≔ ∪ k ∈ Z D k , D k ≔ { Q t k : t ∈ A k } , of Borel sets Q t k ⊆ G is called a system of dyadic cubes with parameters ζ ∈ ( 0 , 1 ) and 0 < a 1 ≤ A 1 < ∞ if it has the following properties:

G = ⋃ α ∈ A k Q α k (disjoint union) for all k ∈ Z ;
if ℓ ≥ k , then either Q s ℓ ⊆ Q t k or Q t k ∩ Q s ℓ = ∅ ;
for each ( k , t ) and each ℓ ≤ k , there exists a unique s such that Q t k ⊆ Q s ℓ ;
for each ( k , t ) there exist at most M (a fixed geometric constant) s such that Q s k + 1 ⊆ Q t k , and Q t k = ⋃ Q ∈ D k + 1 , Q ⊆ Q t k Q ;
B ( x t k , a 1 ζ k ) ⊆ Q t k ⊆ B ( x t k , A 1 ζ k ) ≕ B ( Q t k ) ;
if ℓ ≥ k and Q s ℓ ⊆ Q t k , then B ( Q s ℓ ) ⊆ B ( Q t k ) ;

The set Q t k is called a dyadic cube of generation k with center point x t k ∈ Q t k and side length ζ k .

Lemma 2.7

[9,17] (adjacent systems of dyadic cubes) On stratified Lie groups G , a finite collection { D t : t = 1 , 2 , … , ℐ } of the dyadic families is called a collection of adjacent systems of dyadic cubes with parameters ζ ∈ ( 0 , 1 ) , 0 < a 1 ≤ A 1 < ∞ and 1 ≤ C adj < ∞ if it has the following properties: individually, each D t is a system of dyadic cubes with parameters t ∈ ( 0 , 1 ) and 0 < a 1 ≤ A 1 < ∞ ; collectively, for each ball B ( x , r ) ⊆ G with ζ k + 3 < r ≤ ζ k + 2 , k ∈ Z , there exist t ∈ { 1 , 2 , … , ℐ } and Q ∈ D t of generation k and with center point x s k , t such that d ( x , x s k , t ) < 2 ζ k and

(10) B ( x , r ) ⊆ Q ⊆ B ( x , C adj r ) ,

where C adj depends only on ζ .

We refer to [17] for the constructions of the system of dyadic cubes and the adjacent systems of dyadic cubes in Definitions 2.6 and 2.7. There are more properties about dyadic cubes in [19] and [9].

From the properties of the dyadic system earlier, we can deduce that there exists a constant C Q depending only on Q , such that for any Q t k and Q s k + 1 with Q s k + 1 ⊂ Q t k ,

(11) ∣ Q s k + 1 ∣ ≤ ∣ Q t k ∣ ≤ C Q ∣ Q s k + 1 ∣ .

Definition 2.8

[9,17] Given 0 < δ < 1 , a collection S ⊂ D of dyadic cubes is said to be δ -sparse provided that for every Q ∈ S , there is a measurable subset E Q ⊂ Q such that ∣ E Q ∣ ≥ δ ∣ Q ∣ and the sets { E Q } Q ∈ S have only finite overlap, i.e., there exists a constant C S ≥ 1 such that ∑ Q χ E Q ( x ) ≤ C S for all x ∈ G .

2.4 Some key lemmas

We also require some basic results. First, we will use the L p boundedness of the Riesz transform on stratified Lie groups established by Folland in [11].

Lemma 2.9

[11] Suppose that G is a stratified Lie group. Let 1 < p < ∞ . Then, the Riesz transform X j Δ − 1 2 with j = 1 , 2 , … , n 1 satisfies

X j Δ − 1 2 f L p ( G ) ≤ C ( Q , p ) ‖ f ‖ L p ( G ) .

There are two crucial results from [4], which we will use to prove Theorem 1.4.

Lemma 2.10

[4] Let 1 < p < ∞ , g 0 ∈ G . Then, the subset ℰ of L w p p ( G ) is relatively compact in L w p p ( G ) if and only if the following conditions are satisfied:

ℰ is bounded.
lim R → ∞ ∫ G \ B ( g 0 , R ) ∣ f ( g ) ∣ p w p ( g ) d g = 0 ,
uniformly for f ∈ ℰ .
lim t → 0 ∫ G ∣ f ( g ) − f B ( g , t ) ∣ p w p ( g ) d g = 0 ,
uniformly for f ∈ ℰ .

Lemma 2.11

[4] Let b ∈ BMO ( G ) . Then, b ∈ VMO ( G ) if and only if b satisfies the following three conditions:

lim a → 0 sup r B = a M ( b , B ) = 0 .
lim a → ∞ sup r B = a M ( b , B ) = 0 .
lim r → ∞ sup B ⊂ G \ B ( 0 , r ) M ( b , B ) = 0 .

3 Proof of Theorem 1.1

3.1 b ∈ BMO ν 1 ⁄ m ( G ) ⇒ [ b , T j , α ] m is bounded

To simplify the notation, we denote by

(12) T j , α ≔ X j Δ − α + 1 2 , j = 1 , … , n 1 ,

on G . For the ball B 0 ∈ G and g ∈ B 0 , we can define the local grand maximal truncated operator by

(13) ℳ T j , α , B 0 f ( g ) ≔ sup B ∋ g , B ⊂ B 0 esssup ξ ∈ B ∣ T j , α ( f χ C s ˜ B 0 \ C s ˜ B ) ( ξ ) ∣ .

s ˜ is the smallest integer such that 2 s ˜ > max { 3 , 2 C adj } and

(14) C s ˜ ≔ 2 s ˜ + 2 .

It is easy to see that C s ˜ depends only on ζ .

In the present and the following sections, note that C s ˜ , C adj , and A 1 depend on ζ appeared in Definition 2.6. Moreover, the constants ζ and ℐ in Definition 2.7 are independent of the essential variables and α , p , and q in 1 q = 1 p − α Q . Therefore, we omit their notations in the constants C ( Q ) from line to line in the following proof.

We sketch now the method used to prove (i) in Theorem 1.1, which can be divided into four steps. In Step 1, we estimate the weak-type ( 1 , Q ⁄ ( Q − α ) ) boundedness for the operator T j , α , j = 1 , 2 , … , n 1 for 0 < α < Q , which yields two useful properties for the operator ℳ T j , α , B 0 in Step 2. In Step 3, we will establish the sparse domination for the commutator [ b , T j , α ] m with m ∈ N , which plays a vital role in the proof. Then, with the help of the sparse domination and the weighted norm inequalities of sparse operators, we obtain the uniform two-weight boundedness for [ b , T j , α ] m in Step 4.

3.1.1 Step 1. Weak-type boundedness

We are going to give the weak-type boundedness for operators X j Δ − 1 + α 2 , j = 1 , … , n 1 through the following lemma.

Lemma 3.1

Suppose that G is a stratified Lie group. Given 0 < α < Q and 1 < p < Q α , let 1 q = 1 p − α Q , for any λ > 0 , the operator X j Δ − 1 + α 2 , j = 1 , 2 , … , n 1 satisfies

∣ { g ∈ G : ∣ X j Δ − 1 + α 2 f ( g ) ∣ > λ } ∣ ≤ C T j , α ‖ f ‖ L 1 ( G ) λ Q Q − α ,

where

C T j , α ≔ κ Q , p , q Q − α Q Γ q Q − α 2 Γ q ( α + 2 2 ) Q − α Q ,

where, the constant κ Q , p , q depends on Q , p , and q and is bounded as α → 0 .

Proof

First, we start the proof with estimating X j Δ − 1 + α 2 f L p ( G ) → L q ( G ) .

For 1 < q < ∞ , using Lemma 2.9, it is easy to obtain

(15) X j Δ − 1 + α 2 f L q ( G ) = X j Δ − 1 2 Δ − α 2 f L q ( G ) ≤ C ( Q , q ) Δ − α 2 f L q ( G ) .

Let ω Q − 1 denote the surface area of the unit sphere { g ∈ G : d ( g ) = 1 } on stratified Lie groups. Then, we observe that

(16) ∣ Δ − α 2 f ( g ) ∣ = 1 Γ ( α 2 ) ∫ 0 + ∞ t α 2 − 1 f ∗ p t ( g ) d t ≤ C ( Q ) Γ Q − α 2 Γ α 2 ∫ d ( g ′ ) < R + ∫ d ( g ′ ) ≥ R ∣ f ( g g ′ − 1 ) ∣ d ( g ′ ) Q − α d g ′ ≕ C ( Q ) Γ Q − α 2 Γ α 2 ( I 1 + I 2 ) .

It is easy to verify that

I 1 ≤ C ( Q ) R α α M ( f ) ( g ) ,

where M is the Hardy-Littlewood maximal operator [12] defined by

M f ( g ) = sup ∣ B ∣ − 1 ∫ B ∣ f ( g ′ ) ∣ d g ′ : B is a ball containing g .

Next, applying Hölder’s inequality, we have

I 2 ≤ ∫ d ( g ′ ) ≥ R ( d ( g ′ ) − Q + α ) p ′ d g ′ 1 p ′ ‖ f ‖ L p ( G ) = q ω Q − 1 p ′ Q 1 p ′ R − Q q ‖ f ‖ L p ( G ) .

Taking R = α p Q q ω Q − 1 p ′ Q p Q p ′ ‖ f ‖ L p ( G ) p Q ( M ( f ) ( g ) ) − p Q , it follows that

I 1 + I 2 ≤ C ( Q ) 1 α α p α Q q ω Q − 1 p ′ Q p α Q p ′ ( M ( f ) ( g ) ) p q ‖ f ‖ L p ( G ) p α Q ≤ C ( Q ) 1 α q ω Q − 1 p ′ Q p α Q p ′ ( M ( f ) ( g ) ) p q ‖ f ‖ L p ( G ) p α Q .

Together with (16), we have

Δ − α 2 f ( g ) ≤ C ( Q ) Γ Q − α 2 Γ α 2 1 α q ω Q − 1 p ′ Q p α Q p ′ ( M ( f ) ( g ) ) p q ‖ f ‖ L p ( G ) p α Q .

Corollary 2.5 in [12] provides the fact that ‖ M ‖ L 1 ( G ) → L 1 , ∞ ( G ) depends only on Q and the fact that

‖ M ‖ L p ( G ) → L p ( G ) ≤ C ( Q ) p ′ , 1 < p < ∞ .

Thus, by preceding estimates, we obtain for 0 < α < Q , 1 < p < Q α , and 1 q = 1 p − α Q ,

(17) Δ − α 2 f L q ( G ) ≤ C ( Q ) 1 α Γ Q − α 2 Γ ( α 2 ) q ω Q − 1 p ′ Q p α Q p ′ ‖ ( M ( f ) ) p q ‖ L q ( G ) ‖ f ‖ L p ( G ) p α Q ≤ C ( Q ) p ′ p q q ω Q − 1 p ′ Q p α Q p ′ Γ Q − α 2 Γ α + 2 2 ‖ f ‖ L p ( G ) .

Combining (15) with (17), we obtain the desired result,

(18) X j Δ − 1 + α 2 f L q ( G ) ≤ C ( Q , q ) p ′ p q q ω Q − 1 p ′ Q p α Q p ′ Γ Q − α 2 Γ α + 2 2 ‖ f ‖ L p ( G ) .

Second, we will estimate X j Δ − 1 + α 2 f L 1 ( G ) → L Q ⁄ ( Q − α ) , ∞ ( G ) .

Taking advantage of the ideas in the proof of [13, Theorem 4.3.3], fix λ > 0 , by Definition 2.6, decompose G into a collection of disjoint dyadic cubes such that ∫ Q ∣ f ( g ) ∣ d g < λ ∣ Q ∣ holds for every cube Q in this collection. Then, applying Calderón-Zygmund decomposition [10] to f on each Q at height h = λ Q Q − α ‖ f ‖ L 1 ( G ) α α − Q , we obtain a disjoint family of dyadic cubes { Q k } k ∈ N with total measure ∑ k ∣ Q k ∣ ≤ h − 1 ‖ f ‖ L 1 ( G ) and f = f 1 + f 2 satisfies

(19) ‖ f 1 ‖ L ∞ ≤ C Q h , ‖ f 1 ‖ L 1 ( G ) ≤ ‖ f ‖ L 1 ( G ) ;

(20) f 2 = ∑ k f 2 , k , supp ( f 2 , k ) ⊆ Q k , ∫ Q k f 2 , k = 0 ;

(21) ∑ k ‖ f 2 , k ‖ L 1 ( G ) ≤ 2 C Q h ∑ k ∣ Q k ∣ ≤ 2 C Q ‖ f ‖ L 1 ( G ) ,

where C Q is defined as in (11). Thus, by Chebychev’s inequality, (15), and (19), we have

(22) ∣ { g : ∣ ( X j Δ − 1 + α 2 ) f 1 ( g ) ∣ > λ } ∣ ≤ λ − q X j Δ − 1 + α 2 L p ( G ) → L q ( G ) q ‖ f 1 ‖ L p ( G ) q ≤ λ − q X j Δ − 1 + α 2 L p ( G ) → L q ( G ) q ‖ f 1 ‖ L 1 ( G ) q p ‖ f 1 ‖ L ∞ ( G ) ( p − 1 ) q p ≤ C ( Q ) X j Δ − 1 + α 2 L p ( G ) → L q ( G ) q ‖ f ‖ L 1 ( G ) λ Q Q − α .

Next, for each cube Q k with k ∈ N , from properties of dyadic cubes, we can find the ball B ( Q k ) = B ( g B ( Q k ) , r B ( Q k ) ) covers Q k . Hence, applying (8) and (20), we obtain

g ∉ ∪ t 2 B ( Q t ) : ∣ X j Δ − 1 + α 2 f 2 ( g ) ∣ > λ Q − α Q ≤ λ − 1 ∑ k ∫ g ∉ U t 2 B ( Q t ) ∣ X j Δ − 1 + α 2 f 2 , k ( g ) ∣ Q Q − α d g Q − α Q = λ − 1 ∑ k ∫ g ∉ U t 2 B ( Q t ) ∫ Q k ( K j , α ( g , g ′ ) − K j , α ( g , g B ( Q k ) ) ) f 2 , k ( g ′ ) d g ′ Q Q − α d g Q − α Q ≤ C ( Q ) λ ∑ k ∫ B ( Q k ) ∣ f 2 , k ( g ′ ) ∣ ∑ i = 1 ∞ ∫ 2 i + 1 B ( Q k ) \ 2 i B ( Q k ) d ( g ′ , g B ( Q k ) ) d ( g , g B ( Q k ) ) Q − α + 1 Q Q − α d g Q − α Q d g ′ ≤ C ( Q ) λ ∑ k ∫ B ( Q k ) ∣ f 2 , k ( g ′ ) ∣ ∑ i = 1 ∞ 2 − i ∣ 2 i + 1 B ( Q k ) ∣ ( 2 i r B ( Q k ) ) Q Q − α Q d g ′ ≤ C ( Q ) ‖ f ‖ L 1 ( G ) λ .

This along with the previous estimate (22) implies that

(23) ∣ { g : ∣ X j Δ − 1 + α 2 f ( g ) ∣ > λ } ∣ ≤ g : ∣ X j Δ − 1 + α 2 f 1 ( g ) ∣ > λ 2 + g ∉ ∪ t 2 B ( Q t ) : ∣ X j Δ − 1 + α 2 f 2 ( g ) ∣ > λ 2 + ∣ ∪ t 2 B ( Q t ) ∣ ≤ C ( Q , q ) p ′ p q ω Q − 1 p ′ Q q − p p ′ Γ q Q − α 2 Γ q α + 2 2 + C ( Q ) Γ q α + 2 2 Γ q Q − α 2 Γ q Q − α 2 Γ q α + 2 2 ‖ f ‖ L 1 ( G ) λ Q Q − α ≤ C T j , α ‖ f ‖ L 1 ( G ) λ Q Q − α .

Here,

(24) C T j , α ≔ κ Q , p , q Q − α Q Γ q Q − α 2 Γ q ( α + 2 2 ) Q − α Q

and

(25) κ Q , p , q ≔ C ( Q , q ) p ′ p q ω Q − 1 p ′ Q q − p p ′ + C ( Q , q ) .

C T j , α is still bounded in the case α → 0 . Moreover, it is direct to observe that C T j , α > 1 and κ Q , p , q > 1 . Then, by the arbitrariness of λ , the proof of this lemma is complete.□

3.1.2 Step 2. Properties for the local grand maximal truncated operator

Consider the expressions (12) and (13) defining T j , α and ℳ T j , α , B 0 . We will give two important properties for ℳ T j , α , B 0 via the following lemma.

Lemma 3.2

For a.e. g ∈ B 0 , a ball in G , we have

(26) ∣ T j , α ( f χ C s ˜ B 0 ) ( g ) ∣ ≤ C ( Q ) C T j , α ∣ C s ˜ B 0 ∣ α Q ∣ f ( g ) ∣ + ℳ T j , α , B 0 f ( g )

and

(27) ℳ T j , α , B 0 f ( g ) ≤ C ( Q ) ( τ α , Q + C T j , α ) ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) + 4 M ( T j , α ( f χ C s ˜ B 0 ) ) 1 2 ( g ) 2 ,

where C s ˜ and C T j , α are defined as in (14) and (24), respectively; τ α , Q = Γ Q − α 2 ⁄ Γ ( α + 1 2 ) ,

Proof

First, we give the proof of (26), which is based on the ideas in [22]. Let g belong to the inside of the ball B 0 . Then, for any ε > 0 , we define the set

E s ( g ) ≔ { g ′ ∈ B ( g , s ) : ∣ T j , α ( f χ C s ˜ B 0 ) ( g ′ ) − T j , α ( f χ C s ˜ B 0 ) ( g ) ∣ < ε } ,

and lim s → 0 ∣ E s ( g ) ∣ ∣ B ( g , s ) ∣ = 1 . We can choose s small enough such that B ( g , s ) ⊂ B 0 . Then, for a.e. g ′ ∈ E s ( g ) , we obtain

(28) ∣ T j , α ( f χ C s ˜ B 0 ) ( g ) ∣ ≤ ∣ T j , α ( f χ C s ˜ B 0 ) ( g ) − T j , α ( f χ C s ˜ B 0 ) ( g ′ ) ∣ + ∣ T j , α ( f χ C s ˜ B 0 ) ( g ′ ) ∣ ≤ ε + ∣ T j , α ( f χ C s ˜ B 0 \ C s ˜ B ( g , s ) ) ( g ′ ) ∣ + ∣ T j , α ( f χ C s ˜ B ( g , s ) ) ( g ′ ) ∣ .

Then, Lemma 3.1 yields

∣ E s ( g ) ∣ ≤ ∣ { g ′ : ∣ T j , α ( f χ C s ˜ B ( g , s ) ) ( g ′ ) ∣ ≤ essinf g ′ ∈ E s ( g ) ∣ T j , α ( f χ C s ˜ B ( g , s ) ) ( g ′ ) ∣ } ∣ ≤ C T j , α ‖ f χ C s ˜ B ( g , s ) ‖ L 1 ( G ) essinf g ′ ∈ E s ( g ) ∣ T j , α ( f χ C s ˜ B ( g , s ) ) ( g ′ ) ∣ Q Q − α .

Together with (28), this implies that

∣ T j , α ( f χ C s ˜ B 0 ) ( g ) ∣ ≤ ε + ℳ T j , α , B 0 f ( g ) + essinf g ′ ∈ E s ( g ) ∣ T j , α ( f χ C s ˜ B ( g , s ) ) ( g ′ ) ∣ ≤ ε + ℳ T j , α , B 0 f ( g ) + ∣ E s ( g ) ∣ − Q − α Q C T j , α ‖ f χ C s ˜ B ( g , s ) ‖ L 1 ( G ) ≤ ε + ℳ T j , α , B 0 f ( g ) + ∣ C s ˜ B 0 ∣ α Q C T j , α C s ˜ Q ∣ B ( g , s ) ∣ ∣ E s ( g ) ∣ 1 ∣ C s ˜ B ( g , s ) ∣ ∫ C s ˜ B ( g , s ) f ( g ′ ) d g ′ .

Let s → 0 and ε → 0 in the aforementioned inequality. We obtain the desired result (26).

Next, we continue to prove (27). For any g ∈ B ⊂ B 0 and B is a ball with radius r , denote by B g ≔ B ( g , 4 ( 1 + C s ˜ ) r ) , then C s ˜ B ⊂ B g . For any ξ ∈ B ⊂ B 0 , we have

∣ T j , α ( f χ C s ˜ B 0 \ C s ˜ B ) ( ξ ) ∣ ≤ ∣ T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( ξ ) − T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( g ) ∣ + ∣ T j , α ( f χ C s ˜ B 0 χ B g \ C s ˜ B ) ( ξ ) ∣ + ∣ T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( g ) ∣ ≕ I I 1 + I I 2 + I I 3 .

First, using (8), for ξ , g ∈ B and g ′ ∈ B g = B ( g , 4 ( 1 + C s ˜ ) r ) , we obtain

(29) I I 1 ≤ C ( Q ) ∫ G \ B g d ( g , ξ ) d ( g , g ′ ) Q − α + 1 ∣ f ( g ′ ) ∣ χ C s ˜ B 0 ( g ′ ) d g ′ ≤ C ( Q ) 2 r ∣ C s ˜ B 0 ∣ α Q ∑ i = 0 ∞ ∫ 2 i + 1 B g \ 2 i B g ∣ f ( g ′ ) ∣ χ C s ˜ B 0 ( g ′ ) d ( g , g ′ ) Q + 1 d g ′ ≤ C ( Q ) ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) .

Then, it follows from (7) that

(30) I I 2 ≤ C ( Q ) τ α , Q ∫ B g \ C s ˜ B ∣ f ( g ′ ) ∣ χ C s ˜ B 0 ( g ′ ) d ( ξ , g ′ ) Q − α d g ′ ≤ C ( Q ) τ α , Q ∣ C s ˜ B 0 ∣ α Q 1 ( ( C s ˜ − 1 ) r ) Q ∫ B g ∣ f ( g ′ ) ∣ χ C s ˜ B 0 ( g ′ ) d g ′ ≤ C ( Q ) τ α , Q ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) .

Next, let g ˜ ∈ 1 2 B g = B ( g , 2 ( 1 + C s ˜ ) r ) . Then, the last term is controlled by

I I 3 = 1 2 B g − 1 ∫ 1 2 B g ∣ T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( g ) ∣ 1 2 d g ˜ 2 ≤ 4 1 2 B g − 1 ∫ 1 2 B g ∣ T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( g ) − T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( g ˜ ) ∣ 1 2 d g ˜ 2 + 4 1 2 B g − 1 ∫ 1 2 B g ∣ T j , α ( f χ C s ˜ B 0 χ B g ) ( g ˜ ) ∣ 1 2 d g ˜ 2 + 4 1 2 B g − 1 ∫ 1 2 B g ∣ T j , α ( f χ C s ˜ B 0 ) ( g ˜ ) ∣ 1 2 d g ˜ 2 ≕ I I I 1 + I I I 2 + I I I 3 .

The similar calculations as (29) show that

I I I 1 ≤ C ( Q ) ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) .

It is not hard to check that

I I I 3 ≤ 4 M ( T j , α ( f χ C s ˜ B 0 ) ) 1 2 ( g ) 2 .

Now, it remains to estimate I I I 2 . Let s = 1 2 B g − Q − α Q C T j , α ‖ f χ C s ˜ B 0 χ B g ‖ L 1 ( G ) , where C T j , α is defined as in (24). We have

I I I 2 ≤ 4 1 2 1 2 B g − 1 ∫ 0 ∞ t − 1 2 g ˜ ∈ 1 2 B g : ∣ T j , α ( f χ C s ˜ B 0 χ B g ) ( g ˜ ) ∣ > t d t 2 ≤ 1 2 B g − 1 ∫ 0 s t − 1 2 d t 1 2 B g + ∫ s ∞ t − 1 2 g ˜ ∈ 1 2 B g : ∣ T j , α ( f χ C s ˜ B 0 χ B g ) ( g ˜ ) > t ∣ d t 2 ≤ 1 2 B g − Q − α Q C T j , α ‖ f χ C s ˜ B 0 χ B g ‖ L 1 ( G ) ≤ C ( Q ) C T j , α ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) .

Observing the fact from (23) that C T j , α > 1 , we can conclude that

(31) I I 3 ≤ C ( Q ) C T j , α ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) + 4 M ( T j , α ( f χ C s ˜ B 0 ) ) 1 2 ( g ) 2 .

By combining (29), (30), and (31), for any g , ξ ∈ B , we have

∣ T j , α ( f χ C s ˜ B 0 \ C s ˜ B ) ( ξ ) ∣ ≤ C ( Q ) ( τ α , Q + C T j , α ) ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) + 4 M ( T j , α ( f χ C s ˜ B 0 ) ) 1 2 ( g ) 2 .

We end the proof of (27), and therefore, this lemma is proved.□

3.1.3 Step 3. Sparse domination

To give the sparse domination for [ b , T j , α ] m , j = 1 , 2 , … , n 1 , m ∈ N , our proof will rely upon the ideas in [1,9,22]. We present the sparse domination in the following lemma.

Lemma 3.3

Let b ∈ L loc 1 ( X ) . For every f ∈ L ∞ ( G ) with a bounded support, there exist ℐ dyadic systems D t , t = 1 , 2 , … , ℐ and δ -sparse families S t ⊂ D t such that for almost every g ∈ G , m ∈ N and j = 1 , 2 , … , n 1 , we have for 0 < α < Q , 1 < p < Q α , and 1 q = 1 p − α Q ,

∣ [ b , T j , α ] m f ( g ) ∣ ≤ C ( Q , m ) κ Q , p , q C α , Q , q ∑ h = 0 m C m h ∑ t = 1 ℐ ∑ Q ∈ S t ∣ Q ∣ α Q ∣ b ( g ) − b Q ∣ m − h ∣ ∣ f ∣ ∣ b − b Q ∣ h ∣ Q χ Q ( g ) ,

where

C α , Q , q = Γ Q − α 2 Γ α + 1 2 + Γ q Q − α 2 Γ q α + 2 2

is bounded when α → 0 . κ Q , p , q is defined as in (25).

Proof

Assume that supp f ⊂ B 0 ≔ B ( x 0 , r 0 ) , a ball in G , then there exists a decomposition of G with respect to this ball B 0 as in [9]. By (10), we can find an integer t 0 ∈ { 1 , 2 , … , ℐ } and Q 0 ∈ D t 0 such that B 0 ⊆ Q 0 ⊆ C adj B 0 . In addition, for the ball Q 0 , it is easy to obtain that B ( Q 0 ) covers B 0 and ∣ B ( Q 0 ) ∣ ≤ C ( C adj , A 1 ) ∣ B 0 ∣ .

To prove Lemma 3.3, we need to prove a 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. g ∈ B 0 ,

(32) ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ ≤ C ( Q , m ) κ Q , p , q C α , Q , q ∑ h = 0 m C m h ∑ Q ∈ ℱ t 0 ∣ C s ˜ B ( Q ) ∣ α Q ∣ b ( g ) − b R Q ∣ m − h ∣ ∣ f ∣ ∣ b − b R Q ∣ h ∣ C s ˜ B ( Q ) χ Q ( g ) ,

where R Q is the dyadic cube in D t for some t ∈ { 1 , 2 , … , ℐ } such that C s ˜ B ( Q ) ⊂ R Q ⊂ C adj C s ˜ B ( Q ) .

To prove the claim, it is sufficient to prove the following estimate: there exist cubes P i ∈ D t 0 ( Q 0 ) that are pairwise disjoint such that ∑ i ∣ P i ∣ ≤ 1 2 ∣ Q 0 ∣ and

(33) ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ χ Q 0 ( g ) ≤ C ( Q , m ) κ Q , p , q C α , Q , q ∑ h = 0 m C m h ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ b ( g ) − b R Q 0 ∣ m − h ∣ ∣ f ∣ ∣ b − b R Q 0 ∣ h ∣ C s ˜ B ( Q 0 ) χ Q 0 ( g ) + ∑ i ∣ [ b , T j , α ] m ( f χ C s ˜ B ( P i ) ) ( g ) ∣ χ P i ( g ) ,

for almost every g ∈ B 0 . By iterating the aforementioned estimate, we can prove (32), where ℱ t 0 is the union of all the families { P i k } with { P i 0 } = { Q 0 } , { P i 1 } = { P i } as mentioned earlier, and { P i k } are the cubes obtained at the k th stage of the iterative process. It is also clear that ℱ t 0 is a 1 2 -sparse family.

We note the fact that for any family of disjoint cubes { P i } ⊂ D t 0 ( Q 0 ) ,

(34) ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ χ Q 0 ( g ) ≤ ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ χ Q 0 \ ∪ i P i ( g ) + ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ χ ∪ i P i ( g ) ≤ ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ χ Q 0 \ ∪ i P i ( g ) + ∑ i ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) \ C s ˜ B ( P i ) ) ( g ) ∣ χ P i ( g ) + ∑ i ∣ [ b , T j , α ] m ( f χ C s ˜ B ( P i ) ) ( g ) ∣ χ P i ( g )

and

(35) ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ χ Q 0 \ ∪ i P i ( g ) + ∑ i ∣ [ b , T j , α ] m ( f χ C s ˜ B ( Q 0 ) \ C s ˜ B ( P i ) ) ( g ) ∣ χ P i ( g ) ≤ ∑ h = 0 m C m h ∣ T j , α ( ( b − b R Q 0 ) h f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ ∣ b ( g ) − b R Q 0 ∣ m − h χ Q 0 \ ∪ i P i ( g ) + ∑ i ∑ h = 0 m C m h ∣ T j , α ( ( b − b R Q 0 ) h f χ C s ˜ B ( Q 0 ) \ C s ˜ B ( P i ) ) ( g ) ∣ ∣ b ( g ) − b R Q 0 ∣ m − h χ P i ( g ) .

Thus, to prove (33), by (34) and (35), we need to provide the following estimate:

(36) ∑ h = 0 m C m h ∣ T j , α ( ( b − b R Q 0 ) h f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ ∣ b ( g ) − b R Q 0 ∣ m − h χ Q 0 \ ∪ i P i ( g ) + ∑ i ∑ h = 0 m C m h ∣ T j , α ( ( b − b R Q 0 ) h f χ C s ˜ B ( Q 0 ) \ C s ˜ B ( P i ) ) ( g ) ∣ ∣ b ( g ) − b R Q 0 ∣ m − h χ P i ( g ) ≤ C ( Q , m ) κ Q , p , q C α , Q , q ∑ h = 0 m C m h ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ b ( g ) − b R Q 0 ∣ m − h ∣ ∣ f ∣ ∣ b − b R Q 0 ∣ h ∣ C s ˜ B ( Q 0 ) χ Q 0 ( g ) .

For a constant C E , which will be fixed later on, we define the set E h for any h = 0 , 1 , … , m by

E h ≔ { g ∈ B 0 : ∣ b ( g ) − b R Q 0 ∣ h ∣ f ( g ) ∣ > C E ∣ ∣ b − b R Q 0 ∣ h ∣ f ∣ ∣ C s ˜ B ( Q 0 ) } ∪ { g ∈ B 0 : ℳ T j , α , B 0 ( ( b − b R Q 0 ) h f ) ( g ) > C E ( τ α , Q + C T j , α ) ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ ∣ b − b R Q 0 ∣ h ∣ f ∣ ∣ C s ˜ B ( Q 0 ) } .

We recall that C T j , α defined as in (24). To simplify the notation in the following, we write

∣ b − b R Q 0 ∣ h ∣ f ∣ ≕ F , h = 0 , 1 , … , m .

By combining Lemma 3.2 with the fact that B 0 ⊂ B ( Q 0 ) , it follows that

(37) ∣ { g ∈ B 0 : ℳ T j , α , B 0 ( F ) ( g ) > C E ( τ α , Q + C T j , α ) ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ F ∣ C s ˜ B ( Q 0 ) } ∣ ≤ ∣ { g ∈ B 0 : C ( Q ) M ( F χ C s ˜ B 0 ) ( g ) > C E ∣ F ∣ C s ˜ B ( Q 0 ) } ∣ + ( C E ( τ α , Q + C T j , α ) ∣ C s ˜ B 0 ∣ α Q ∣ F ∣ C s ˜ B ( Q 0 ) ) − Q Q − α ‖ 8 ( M ( T j , α ( F χ C s ˜ B 0 ) ) 1 2 ( g ) ) 2 ‖ L Q ⁄ ( Q − α ) , ∞ ( G ) Q ⁄ ( Q − α ) .

Now, by Whitney lemma and the maximal Theorem 2.4 in [12], for any λ > 0 , we have

(38) ∣ { g : M f ( g ) > λ } ∣ ≤ C ( Q ) λ ∫ { g : M f ( g ) > λ } ∣ f ( g ) ∣ d g .

Using Exercise 1.1.11 in [13], there exists an estimate on stratified Lie groups for p > 1 ,

(39) ∫ { g : M f ( g ) > λ } ∣ f ( g ) ∣ d g ≤ p ′ ∣ { g : M f ( g ) > λ } ∣ p − 1 p ‖ f ‖ L p , ∞ ( G ) .

(38) and (39) yield for p > 1 ,

(40) ‖ M f ‖ L p , ∞ ( G ) ≤ C ( Q ) p ′ ‖ f ‖ L p , ∞ ( G ) .

Thus, we deduce from (37), (40), and Lemma 3.1 that

(41) ∣ { g ∈ B 0 : ℳ T j , α , B 0 ( F ) ( g ) > C E ( τ α , Q + C T j , α ) ∣ C s ˜ B 0 ∣ α Q ∣ F ∣ C s ˜ B ( Q 0 ) } ∣ ≤ C ( Q ) ‖ F χ C s ˜ B 0 ‖ L 1 ( G ) C E ∣ F ∣ C s ˜ B ( Q 0 ) + C ( Q ) ‖ T j , α ( F χ C s ˜ B 0 ) ‖ L Q ⁄ ( Q − α ) , ∞ ( G ) C E ( τ α , Q + C T j , α ) ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ F ∣ C s ˜ B ( Q 0 ) Q Q − α ≤ C ( Q ) ∣ C s ˜ B ( Q 0 ) ∣ C E + C T j , α ‖ F χ C s ˜ B 0 ‖ L 1 ( G ) C E ( τ α , Q + C T j , α ) ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ F ∣ C s ˜ B ( Q 0 ) Q Q − α ≤ C ( Q ) C E − 1 + C E − Q Q − α ∣ C s ˜ B ( Q 0 ) ∣ .

It is easy to see from B 0 ⊂ B ( Q 0 ) that

(42) ∣ { g ∈ B 0 : ∣ F ( g ) ∣ > C E ∣ F ∣ C s ˜ B ( Q 0 ) } ∣ ≤ ∫ B 0 ∣ F ( g ) ∣ d g C E ∣ F ∣ C s ˜ B ( Q 0 ) ≤ ∣ C s ˜ B ( Q 0 ) ∣ C E .

Then, we can conclude from (41) and (42) that

∣ E h ∣ ≤ C ( Q ) C E − 1 + C E − Q Q − α ∣ C s ˜ B ( Q 0 ) ∣ + C E − 1 ∣ C s ˜ B ( Q 0 ) ∣ .

Due to the fact that B 0 ⊂ B ( Q 0 ) and ∣ B ( Q 0 ) ∣ ≤ C ( C adj , A 1 ) ∣ B 0 ∣ , we now choose C E > 1 big enough in such a way that

∣ E ∣ ≔ ∣ ∪ h = 0 m E h ∣ = ∑ h = 0 m ∣ E h ∣ ≤ C ( Q , m ) C E − 1 + C E − Q Q − α ∣ C s ˜ B ( Q 0 ) ∣ + ( m + 1 ) C E − 1 ∣ C s ˜ B ( Q 0 ) ∣ ≤ C ( Q , m ) C E − 1 ∣ B 0 ∣ = ( 4 C Q ) − 1 ∣ B 0 ∣ ,

where

(43) C E = 4 C ( Q , m ) C Q = C ( Q , m ) ,

and C Q > 1 is a fixed constant defined as in (11). In fact, we can always set C E > 1 . Otherwise, in the case C E ≤ 1 ,

( m + 1 ) C E − 1 ∣ C s ˜ B ( Q 0 ) ∣ ≥ ( m + 1 ) ∣ C s ˜ B ( Q 0 ) ∣ > ( 4 C Q ) − 1 ∣ B 0 ∣ .

This leads to a contradiction.

After choosing the constant C E , we are now in the position to prove the lemma. We use the Calder ó n-Zygmund decomposition to the function χ E on B 0 at the height h ≔ 1 2 C Q in order to obtain pairwise disjoint cubes { P i } ⊂ D t 0 ( Q 0 ) such that ∣ P i ∩ E ∣ ≤ 1 2 ∣ P i ∣ and ∣ E \ ∪ i P i ∣ = 0 . This yields

∑ i ∣ P i ∣ ≤ 1 2 ∣ B 0 ∣ and P i ∩ E c ≠ ∅ .

Then, there exist some g ∈ P i , such that g ∈ ( ∪ h E h ) c . We have

(44) ℳ T j , α , B ( Q 0 ) ( ( b − b R Q 0 ) h f ) ( g ) ≤ C E ( τ α , Q + C T j , α ) ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ ∣ b − b R Q 0 ∣ h ∣ f ∣ ∣ C s ˜ B ( Q 0 ) .

Considering the definition of ℳ T j , α , B ( Q 0 ) , for any g ∈ P i , P i ⊂ B ( Q 0 ) , using (43) and (44), we have

(45) ∑ i ∑ h = 0 m C m h ∣ T j , α ( ( b − b R Q 0 ) h f χ C s ˜ B ( Q 0 ) \ C s ˜ B ( P i ) ) ( g ) ∣ ∣ b ( g ) − b R Q 0 ∣ m − h χ P i ( g ) ≤ C ( Q , m ) ( τ α , Q + C T j , α ) ∑ h = 0 m C m h ∣ b ( g ) − b R Q 0 ∣ m − h ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ ∣ b − b R Q 0 ∣ h ∣ f ∣ ∣ C s ˜ B ( Q 0 ) .

Taking into consideration (26) and (44), for every g ∈ Q 0 \ ∪ i P i , we have g ∈ Q 0 \ E and

∣ T j , α ( ( b − b R Q 0 ) h f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ ≤ C ( Q ) C T j , α ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ ( b − b R Q 0 ) h f ( g ) ∣ + ℳ T j , α , B ( Q 0 ) ( ( b − b R Q 0 ) h f ) ( g ) ≤ C ( Q ) C E ( C T j , α + τ α , Q + C T j , α ) ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ ∣ b − b R Q 0 ∣ h ∣ f ∣ ∣ C s ˜ B ( Q 0 ) .

Then, by the definition of C E , we have

(46) ∑ h = 0 m C m h ∣ T j , α ( ( b − b R Q 0 ) h f χ C s ˜ B ( Q 0 ) ) ( g ) ∣ ∣ b ( g ) − b R Q 0 ∣ m − h χ Q 0 \ ∪ i P i ( g ) ≤ C ( Q , m ) ( τ α , Q + C T j , α ) ∣ C s ˜ B ( Q 0 ) ∣ α Q ∑ h = 0 m C m h ∣ b ( g ) − b R Q 0 ∣ m − h ∣ ∣ b − b R Q 0 ∣ h ∣ f ∣ ∣ C s ˜ B ( Q 0 ) .

Applying (24) with 1 q = 1 p + α Q and κ Q , p , q , C T j , α > 1 , it is easy to obtain

τ α , Q + C T j , α ≤ κ Q , p , q Γ Q − α 2 Γ α + 1 2 + Γ q Q − α 2 Γ q ( α + 2 2 ) ≕ κ Q , p , q C α , Q , q ,

where

(47) C α , Q , q = Γ Q − α 2 Γ α + 1 2 + Γ q Q − α 2 Γ q α + 2 2

and is bounded when α → 0 . Now, (36) can be deduced from (45) and (46). Then, using the argument in [9], we can obtain (32).

The same arguments as in [9] show that there exists a 1 C ( s ˜ , C adj ) -sparse family S t such that

where κ Q , p , q and C α , Q , q are defined as in (25) and (47), respectively. Set δ = C ( s ˜ , C adj ) , Lemma 3.3 can be proved.□

3.1.4 Step 4. Uniform two-weight boundedness

In this section, for m ∈ N and j = 1 , 2 , … , n 1 , we shall give the uniform two-weight boundedness for [ b , T j , α ] m from L p ( μ p ) to L q ( w q ) for 0 < α < Q , 1 q = 1 p + α Q .

It is easy to check that when m = 0 , the boundedness for T j , α can be obtained from (18).

For m ∈ Z + , to obtain the boundedness for [ b , T j , α ] m , we will follow the ideas in [1,9,22]. First, we need to borrow a result from [9], which can be stated as the following lemma.

Lemma 3.4

[9] Let S be a sparse family contained in a dyadic lattice D , η a weight, b ∈ BMO η , and f ∈ C c ∞ ( G ) . There exists a possibly larger sparse family S ˜ ⊂ D containing S such that, for every positive integer h and every Q ∈ S ˜ ,

∫ Q ∣ b − b Q ∣ h ∣ f ∣ ≤ C ( Q ) ‖ b ‖ BMO η h ∫ Q A S ˜ , η h f ( g ) d g ,

where A S ˜ , η h f stands for the hth iteration of A S ˜ , η , which is defined by A S ˜ , η f ≔ A S ˜ ( f ) η , with A S ˜ being the sparse operator given by

A S ˜ f ( g ) ≔ ∑ Q ∈ S ˜ 1 ∣ Q ∣ ∫ Q ∣ f ∣ χ Q ( g ) .

We refer to the estimate in [24] that for any w p ∈ A p , 1 < p < ∞ ,

(48) ‖ A S ‖ L w p p ( G ) ≤ max 2 p p ′ , 2 p ′ p p ′ p [ w p ] A p max { p ′ p , 1 } .

Now, for 0 < α < Q , let I S , η α f ≔ I S α ( f ) η , where

I S α f ( g ) ≔ ∑ Q ∈ S 1 ∣ Q ∣ 1 − α Q ∫ Q ∣ f ∣ χ Q ( g ) .

For 1 < p < Q α , 1 q = 1 p − α Q and w ∈ A p , q , the following estimate is due to [24] that when p ′ q ≤ 1 − α Q ,

(49) ‖ I S α ‖ L w p p ( G ) → L w q q ( G ) ≤ p ′ 1 + q p ′ 1 − α Q 2 q p ′ ( 1 − α Q ) [ w ] A p , q 1 − α Q .

It remains to providing an estimate for the case p ′ q > 1 − α Q . We start with recalling an estimate from [21] that for w ∈ A p , q ,

∣ E ∣ ∣ Q ∣ ≤ w − p ′ ( E ) w − p ′ ( Q ) 1 p ′ [ w ] A p , q 1 q ,

where E ⊂ Q and ∣ Q ∣ ≤ 2 ∣ E ∣ . This estimate implies that

(50) w − p ′ ( Q ) 1 − q p ′ ( 1 − α Q ) ≤ 2 p ′ − q + α q Q [ w ] A p , q p ′ q − ( 1 − α Q ) w − p ′ ( E ) 1 − q p ′ ( 1 − α Q ) .

Taking into account the discussion in [24] for (49), it follows from (50) that for p ′ q > 1 − α Q with α < Q ,

(51) ‖ I S α ‖ L w p p ( G ) → L w q q ( G ) ≤ p ′ 1 + q p ′ 1 − α Q 2 q p ′ ( 1 − α Q ) + p ′ − q + α q Q [ w ] A p , q p ′ q .

Then, for 0 < α < Q , 1 q = 1 p − α Q , 1 < p , q < ∞ , and w ∈ A p , q . Since 0 < 1 − α Q < 1 , we conclude from (49) and (51) that

(52) ‖ I S α ‖ L w p p ( G ) → L w q q ( G ) ≤ 2 q p ′ 1 − α Q max { 1 , 2 p ′ − q + α q Q } p ′ 1 + q p ′ 1 − α Q [ w ] A p , q max { p ′ q , 1 − α Q } ≤ max { 2 q p ′ , 2 p ′ p } ( p ′ + q ) [ w ] A p , q max { p ′ q , 1 q + 1 p ′ } .

We observe that when α → 0 , (52) corresponds to (48). Thus, (52) holds for 0 ≤ α < ∞ .

Next, we define the sparse operators

A α , S m , h ( b , f ) ( g ) ≔ ∑ Q ∈ S ∣ b ( g ) − b Q ∣ m − h ∣ Q ∣ α Q ∣ ∣ f ∣ ∣ b − b Q ∣ h ∣ Q χ Q ( g ) .

Then, by combining with Lemma 3.3, we have

(53) ∣ [ b , T j , α ] m f ( g ) ∣ ≤ C ( Q , m ) κ Q , p , q C α , Q , q ∑ h = 0 m C m h ∑ t = 1 ℐ A α , S t m , h ( b , f ) ( g ) .

Equipped with these estimates, we will give the uniform two-weight boundedness for [ b , T j , α ] m .

Note that p ′ + p = p p ′ , making use of Lemma 3.4 and (48), we have

(54) ‖ A α , S m , h ( b , f ) ‖ L w q q ( G ) ≤ C ( Q ) ‖ b ‖ BMO η m ‖ A S ˜ A S ˜ , η m − h − 1 I S , η α A S ˜ , η h f ‖ L w q q ( G ) ≤ C ( Q ) ‖ b ‖ BMO η m max 2 q q ′ , 2 q ′ q q ′ q m − h ∏ l = 0 m − h − 1 [ w q η l q ] A q max 1 , q ′ q ‖ I S , η α A S ˜ , η h f ‖ L w q η q ( m − h − 1 ) q ( G ) .

Then, (52) implies that

(55) ‖ I S , η α A S ˜ , η h f ‖ L w q η q ( m − h − 1 ) q ( G ) ≤ max { 2 q p ′ , 2 p ′ p } ( p ′ + q ) [ w η m − h ] A p , q max { p ′ q , 1 q + 1 p ′ } ‖ A S ˜ , η h f ‖ L w p η p ( m − h ) p ( G ) .

Next, (48) yields that

(56) ‖ A S ˜ , η h f ‖ L w p η p ( m − h ) p ( G ) ≤ max 2 p p ′ , 2 p ′ p p ′ p h ∏ l = m − h + 1 m [ w p η l p ] A p max 1 , p ′ p ‖ f ‖ L w p η p m p ( G ) .

Let η = ν 1 ⁄ m . Then, there exist estimates from [1] that

(57) [ w ν m − h m ] A p , q ≤ [ w ] A p , q h m [ μ ] A p , q m − h m and [ w p ν p l m ] A p ≤ [ w p ] A p m − l m [ μ p ] A p l m .

Then, it follows from (57) that

∏ l = 0 m − h − 1 [ w q η l q ] A q max 1 , q ′ q ∏ l = m − h + 1 m [ w p η l p ] A p max 1 , p ′ p [ w η m − h ] A p , q max { p ′ q , 1 q + 1 p ′ } ≤ ( [ w q ] A q m + ( h + 1 ) 2 [ μ q ] A q m − ( h + 1 ) 2 ) m − h m max 1 , q ′ q ( [ w p ] A p h − 1 2 [ μ p ] A p m − h − 1 2 ) h m max 1 , p ′ p ( [ w ] A p , q h m [ μ ] A p , q m − h m ) max { p ′ q , 1 q + 1 p ′ } ≕ C w , μ , m , h .

Gathering all the estimates (53)–(56), we obtain the desired result

‖ [ b , T j , α ] m f ‖ L w q q ( G ) ≤ ∑ h = 0 m C m h C Q , p , q , m , h C w , μ , m , h C α , Q , q ‖ b ‖ BMO ν 1 ⁄ m ( G ) m ‖ f ‖ L μ p p ( G ) ,

where C α , Q , q is defined as in (47),

C Q , p , q , m , h = C ( Q , m ) κ Q , p , q max 2 q q ′ , 2 q ′ q q ′ q m − h max 2 p p ′ , 2 p ′ p p ′ p h max { 2 q p ′ , 2 p ′ p } ( p ′ + q )

and κ Q , p , q is defined as in (25). When α → 0 , we have q = p and

(58) C Q , p , q , m , h → C ( Q , m ) C ( Q , p ) p ′ p max 2 p p ′ , 2 p ′ p p ′ p m + 1

and

(59) C α , Q , q → π − 1 2 Γ Q 2 + Γ p Q 2

are bounded. Hence, the first result in Theorem 1.1 is proved.

3.2 [ b , T j , α ] m is bounded ⇒ b ∈ BMO v 1 ⁄ m ( G )

We recall the definition

M ( b , B ) = 1 ∣ B ∣ ∫ B ∣ b ( g ) − b B ∣ d g .

The proof in this section is based on the ideas in [9]. We know that for any ball B ≔ B ( g 0 , r ) ⊂ G , there exist the balls B ˜ ≔ B ( g ˜ 0 , r ) such that 3 A 0 r ≤ d ( g 0 , g ˜ 0 ) . For any ( g , g ˜ ) ∈ B × B ˜ , we can see from Lemma 2.2, for 0 < α < Q ,

∣ K j , α ( g , g ′ ) ∣ ≥ C K j , α d ( g , g ′ ) − Q + α , j = 1 , 2 , … , n 1 ,

and K j , α ( g , g ′ ) does not change sign on B × B ˜ .

Let us introduce the following lemma, which will be quite useful.

Lemma 3.5

[9] Let b be a real-valued measurable function. For any ball B, B ˜ is defined as earlier. Then, there exist measurable sets E 1 , E 2 ⊂ B , and F 1 , F 2 ⊂ B ˜ , such that

B = E 1 ∪ E 2 , B ˜ = F 1 ∪ F 2 and ∣ F i ∣ ≥ 1 2 ∣ B ˜ ∣ , i = 1 , 2 ;
b ( g ) − b ( g ˜ ) does not change sign for all ( g , g ˜ ) in E i × F i , i = 1 , 2 ;
∣ b ( g ) − m b ( B ˜ ) ∣ ≤ ∣ b ( g ) − b ( g ˜ ) ∣ for all ( g , g ˜ ) in E i × F i , i = 1 , 2 .

Now, we are in the position to prove the second part of Theorem 1.1. By combining Lemma 3.5 with (9), we have

(60) 1 ∣ B ∣ ∑ i = 1 2 ∫ B ∣ [ b , T j , α ] m χ F i ( g ) ∣ d g ≥ 1 ∣ B ∣ ∑ i = 1 2 ∫ E i ∫ F i ∣ b ( g ′ ) − b ( g ) ∣ m ∣ K j , α ( g , g ′ ) ∣ d g ′ d g ≥ C ( Q ) C K j , α ∣ B ∣ − 1 + α Q 1 ∣ B ∣ ∑ i = 1 2 ∫ E i ∫ F i ∣ b ( g ′ ) − m b ( B ˜ ) ∣ m d g ′ d g ≥ C ( Q ) C K j , α ∣ B ∣ α Q 2 − m M ( b , B ) m .

For μ p ∈ A p , it was shown in [14] that

μ p ( B ˜ ) ≤ C ( Q , p ) [ μ p ] A p μ p ( B ) .

In addition, note that ν = μ w and the following result [1, Page 1224],

1 ∣ B ∣ ∫ B μ p 1 p ≤ C ( Q ) 1 ∣ B ∣ ∫ B ν 1 m m 1 ∣ B ∣ ∫ B w p 1 p .

Then, using the aforementioned two inequalities and (60), we can obtain

(61) M ( b , B ) m ≤ 2 m C K j , α ∣ B ∣ − 1 − α Q ∑ i = 1 2 ∫ B ∣ [ b , T j , α ] m χ F i ∣ ≤ C ( Q , m ) C K j , α ∣ B ∣ − 1 − α Q ∑ i = 1 2 ∫ B ∣ [ b , T j , α ] m χ F i ∣ q w q 1 q ∫ B w − q ′ 1 q ′ ≤ C ( Q , m ) C K j , α ‖ [ b , T j , α ] m ‖ L μ p p ( G ) → L w q q ( G ) ∣ B ∣ − 1 − α Q ∑ i = 1 2 ∫ B ∣ χ F i ∣ p μ p 1 p ∫ B w − q ′ 1 q ′ ≤ C ( Q , m , p ) C K j , α ‖ [ b , T j , α ] m ‖ L μ p p ( G ) → L w q q ( G ) [ μ p ] A p 1 p 1 ∣ B ∣ μ p ( B ) 1 p 1 ∣ B ∣ ∫ B w − q ′ 1 q ′ ≤ C ( Q , m , p ) C K j , α ‖ [ b , T j , α ] m ‖ L μ p p ( G ) → L w q q ( G ) [ μ p ] A p 1 p [ w q ] A q 1 q ν 1 m ( B ) ∣ B ∣ m .

Then, we can conclude from (61) that

‖ b ‖ BMO ν 1 m ( G ) = sup B ⊂ G 1 ν 1 m ( B ) ∫ B ∣ b ( g ) − b B ∣ d g = sup B ⊂ G ∣ B ∣ ν 1 m ( B ) M ( b , B ) ≤ C ( Q , m , p ) C K j , α − 1 m ‖ [ b , T j , α ] m ‖ L μ p p ( G ) → L w q q ( G ) 1 m [ μ p ] A p 1 p m [ w q ] A q 1 q m < ∞ .

Thus, we have b ∈ BMO ν 1 ⁄ m ( G ) .

4 Proof of Theorem 1.4

4.1 b ∈ VMO ( G ) ⇒ [ b , T j , α ] is compact

Note that T j , α = X j Δ − α + 1 2 , j = 1 , … , n 1 on G . Inspired by the method in [4,5], it is sufficient to show that [ b , T j , α ] , 0 ≤ α < Q is compact if b ∈ C 0 ∞ ( G ) . Suppose b ∈ C 0 ∞ ( G ) , we need to prove that for every bounded subset ℰ ⊂ L w p p ( G ) , the set [ b , T j , α ] ℰ , j = 1 , … , n 1 is precompact. Thus, we only need to show that [ b , T j , α ] ℰ ⊂ L w q q ( G ) satisfies (i)–(iii) in Lemma 2.10.

For 0 ≤ α < Q , the fact that [ b , T j , α ] ℰ satisfies condition (i) can be proved by Theorem 1.1 when m = 1 and μ = w directly.

4.1.1 Proof of condition (ii)

Given that b ∈ C 0 ∞ ( G ) with supp b ⊂ B ( 0 , R ) . For any f ∈ ℰ , t > 2 and 0 ≤ α < Q . First, we observe that b ( g ) = 0 if g ∈ G \ B ( 0 , t R ) . Thus,

(62) ∫ G \ B ( 0 , t R ) ∣ [ b , T j , α ] f ( g ) ∣ q w q ( g ) d g 1 q ≤ ∫ G \ B ( 0 , t R ) ∣ T j , α ( b f ) ( g ) ∣ q w q ( g ) d g 1 q .

We estimate by using (7) and Hölder’s inequality that

∣ T j , α ( b f ) ( g ) ∣ ≤ C ( Q ) τ α , Q ‖ b ‖ L ∞ ( G ) 1 d ( g , 0 ) Q − α ‖ f ‖ L w p p ( G ) ∫ B ( 0 , R ) w − p ′ 1 p ′ .

Note that for any ball B ∈ G , there exists an estimate that

∫ B w − p ′ 1 p ′ = 1 ∣ B ∣ ∫ B w − p ′ 1 p ′ 1 ∣ B ∣ ∫ B w q 1 q ∣ B ∣ 1 p ′ w q ( B ) ∣ B ∣ − 1 q ≤ [ w ] A p , q 1 q ∣ B ∣ 1 p ′ + 1 q ( w q ( B ) ) 1 q .

Due to the fact that w ∈ A p , q then w q ∈ A q Q − α Q with q Q − α Q > 1 [23]. Moreover, there also exists θ > 0 such that q Q − α Q − θ > 1 and w q ∈ A q Q − α Q − θ [23]. Applying Lemma 2.3 and aforementioned two estimates, we obtain

∫ G \ B ( 0 , t R ) ∣ T j , α ( b f ) ( g ) ∣ q w q ( g ) d g 1 q ≤ C ( Q ) τ α , Q ‖ b ‖ L ∞ ( G ) [ w ] A p , q 1 q ‖ f ‖ L w p p ( G ) ∫ G \ B ( 0 , t R ) w q ( g ) d ( g , 0 ) q ( Q − α ) d g 1 q ∣ B ( 0 , R ) ∣ 1 p ′ + 1 q ( w q ( B ( 0 , R ) ) ) 1 q ≤ C ( Q , b ) τ α , Q [ w ] A p , q 1 q ‖ f ‖ L w p p ( G ) ∑ i = [ log 2 t ] ∞ 1 2 i q ( Q − α ) w q ( 2 i + 1 B ( 0 , R ) \ 2 i B ( 0 , R ) ) ( w q ( B ( 0 , R ) ) ) 1 q ≤ C ( Q , b ) τ α , Q [ w ] A p , q 1 q ‖ f ‖ L w p p ( G ) 2 Q ∑ i = [ log 2 t ] ∞ 1 2 i θ Q 1 q ≤ C ( Q , b ) τ α , Q [ w ] A p , q 1 q ‖ f ‖ L w p p ( G ) t − θ Q 2 θ Q 1 − 2 − θ Q 1 q .

Then, the aforementioned estimate and (62) prove the desired result, for 0 ≤ α < Q ,

lim t → ∞ ∫ G \ B ( 0 , t R ) ∣ [ b , T j , α ] f ( g ) ∣ q w q ( g ) d g 1 q = 0 .

4.1.2 Proof of condition (iii)

Now, we turn to condition (iii). For any t > 0 small enough,

(63) ∫ G ∣ [ b , T j , α ] f ( g ) − ( [ b , T j , α ] f ) B ( g , t ) ∣ q w q ( g ) d g 1 q = ∫ G 1 ∣ B ( 0 , t ) ∣ ∫ B ( 0 , t ) [ b , T j , α ] f ( g ) − [ b , T j , α ] f ( g ˜ g ) d g ˜ q w q ( g ) d g 1 q .

For j = 1 , 2 , … , n 1 and g ˜ ∈ B ( 0 , t ) with t > 0 . Let

T j , g ˜ f ( g ) ≔ [ b , T j , α ] f ( g ) − [ b , T j , α ] f ( g ˜ g ) .

Then, for the ball B 0 ∈ G and g ∈ B 0 , we define the local grand maximal truncated operator by

ℳ T j , g ˜ , B 0 f ( g ) ≔ sup B ∋ g , B ⊂ B 0 esssup ξ ∈ B ∣ T j , g ˜ ( f χ C s ˜ B 0 \ C s ˜ B ) ( ξ ) ∣ .

s ˜ is defined as in (14). By the method of Section 3.1, to obtain the sparse domination for T j , g ˜ f , we need to obtain the next lemma.

Lemma 4.1

For a.e. g ∈ B 0 ∈ G , g ˜ ∈ B ( 0 , t ) ∈ G in (63), we have

(64) ∣ T j , g ˜ ( f χ C s ˜ B 0 ) ( g ) ∣ ≤ ℳ T j , g ˜ , B 0 f ( g )

and

(65) ℳ T j , g ˜ , B 0 f ( g ) ≤ C ( Q , b ) ( τ α , Q + 1 + t 1 2 ( τ α , Q + 2 C T j , α ) ) t 1 2 ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) + 4 ‖ ∇ b ‖ L ∞ ( G ) t M ( T j , α ( f χ C s ˜ B 0 ) ) 1 2 ( g ) 2 ,

where C s ˜ and C T j , α are defined as in (14) and (24), respectively; τ α , Q = Γ Q − α 2 ⁄ Γ ( α + 1 2 ) .

Proof

First, we will prove (64). For any g ∈ B 0 , we can find a constant s < 1 small enough, such that B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) ⊂ B 0 since d ( g ˜ , 0 ) < ∞ . Then,

(66) ∣ T j , g ˜ ( f χ C s ˜ B 0 ) ( g ) ∣ ≤ ∣ T j , g ˜ ( f χ C s ˜ B 0 \ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) ) ( g ) ∣ + ∣ T j , g ˜ ( f χ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) ) ( g ) ∣ ≤ ℳ T j , g ˜ , B 0 f ( g ) + ∣ [ b , T j , α ] ( f χ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) ) ( g ) ∣ + ∣ [ b , T j , α ] ( f χ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) ) ( g ˜ g ) ∣ .

Applying (7), we have

(67) ∣ [ b , T j , α ] ( f χ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) ) ( g ) ∣ ≤ C ( Q ) τ α , Q ‖ ∇ b ‖ L ∞ ( G ) ∫ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) d ( g , g ′ ) d ( g , g ′ ) Q − α ∣ f ( g ′ ) ∣ d g ′ ≤ C ( Q , b ) τ α , Q ∣ C s ˜ B 0 ∣ α Q d ( g ˜ , 0 ) s M f ( g ) .

Since g ′ ∈ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) , we have d ( g , g ′ ) ≤ s d ( g ˜ , 0 ) and ( 1 − s ) d ( g ˜ , 0 ) ≤ d ( g g ˜ , g ′ ) ≤ ( 1 + s ) d ( g ˜ , 0 ) . Then, the third term in the last line of (66) is controlled by

(68) ∣ [ b , T j , α ] ( f χ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) ) ( g ˜ g ) ∣ ≤ C ( Q ) τ α , Q ‖ ∇ b ‖ L ∞ ( G ) ∫ C s ˜ B ( g , s C s ˜ − 1 d ( g ˜ , 0 ) ) d ( g ˜ g , g ′ ) d ( g ˜ g , g ′ ) Q − α ∣ f ( g ′ ) ∣ d g ′ ≤ C ( Q , b ) τ α , Q d ( g ˜ , 0 ) α + 1 ( 1 + s ) α + 1 ( 1 − s ) Q s Q M f ( g ) .

Let s → 0 in (67) and (68). Combining with (66), we can obtain the desired result (64).

As for (65), for any g ∈ B ⊂ B 0 and B is a ball with radius r , denote by B g = B ( g , 4 ( 1 + C s ˜ ) r ) , then C s ˜ B ⊂ B g . Since d ( g ˜ , 0 ) < t , which is small enough as in (63), and B is not empty, without loss of generality, we can assume t < r . Then, for any g ∈ G , we have

(69) ∣ T j , g ˜ ( f χ C s ˜ B 0 \ C s ˜ B ) ( ξ ) ∣ ≤ ∣ T j , g ˜ ( f χ C s ˜ B 0 χ B g \ C s ˜ B ) ( ξ ) ∣ + ∣ T j , g ˜ ( f χ C s ˜ B 0 χ G \ B g ) ( ξ ) ∣ .

Based on the calculation process in (30), since ξ ∈ B , g ′ ∈ B g \ C s ˜ B and C s ˜ > 4 , we have d ( g ˜ ξ , g ′ ) ≤ 2 d ( ξ , g ′ ) and the estimate

(70) ∣ T j , g ˜ ( f χ C s ˜ B 0 χ B g \ C s ˜ B ) ( ξ ) ∣ ≤ ∫ G ( K j , α ( ξ , g ′ ) − K j , α ( g ˜ ξ , g ′ ) ) ( b ( g ˜ ξ ) − b ( g ′ ) ) f ( g ′ ) χ C s ˜ B 0 ( g ′ ) χ B g \ C s ˜ B ( g ′ ) d g ′ + ∫ G K j , α ( ξ , g ′ ) ( b ( g ˜ ξ ) − b ( ξ ) ) f ( g ′ ) χ C s ˜ B 0 ( g ′ ) χ B g \ C s ˜ B ( g ′ ) d g ′ ≤ C ( Q , b ) ( 1 + τ α , Q ) ∣ C s ˜ B 0 ∣ α Q d ( g ˜ , 0 ) M ( f χ C s ˜ B 0 ) ( g ) .

In terms of the second term on the right-hand side of (69), we divide it into four parts

∣ T j , g ˜ ( f χ C s ˜ B 0 χ G \ B g ) ( ξ ) ∣ = ∫ G K j , α ( ξ , g ′ ) ( b ( ξ ) − b ( g ˜ ξ ) ) f ( g ′ ) χ C s ˜ B 0 ( g ′ ) χ G \ B g ( g ′ ) d g ′ + ∫ d ( g , g ′ ) > d ( g ˜ , 0 ) 1 2 ( K j , α ( ξ , g ′ ) − K j , α ( g ˜ ξ , g ′ ) ) ( b ( g ˜ ξ ) − b ( g ′ ) ) f ( g ′ ) χ C s ˜ B 0 ( g ′ ) χ G \ B g ( g ′ ) d g ′ + ∫ d ( g , g ′ ) ≤ d ( g ˜ , 0 ) 1 2 K j , α ( ξ , g ′ ) ( b ( g ˜ ξ ) − b ( g ′ ) ) f ( g ′ ) χ C s ˜ B 0 ( g ′ ) χ G \ B g ( g ′ ) d g ′ − ∫ d ( g , g ′ ) ≤ d ( g ˜ , 0 ) 1 2 K j , α ( g ˜ ξ , g ′ ) ( b ( g ˜ ξ ) − b ( g ′ ) ) f ( g ′ ) χ C s ˜ B 0 ( g ′ ) χ G \ B g ( g ′ ) d g ′ ≕ I V 1 + I V 2 + I V 3 − I V 4 .

By applying (29), (31), and the fact that C T j , α = ‖ T j , α ‖ L 1 ( G ) → L Q ⁄ ( Q − α ) , ∞ ( G ) > 1 , we have

∣ I V 1 ∣ ≤ ‖ ∇ b ‖ L ∞ ( G ) d ( g ˜ , 0 ) ∣ T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( ξ ) − T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( g ) ∣ + ‖ ∇ b ‖ L ∞ ( G ) d ( g ˜ , 0 ) ∣ T j , α ( f χ C s ˜ B 0 χ G \ B g ) ( g ) ∣ ≤ C ( Q ) ‖ ∇ b ‖ L ∞ ( G ) d ( g ˜ , 0 ) ∣ C s ˜ B 0 ∣ α Q C T j , α M ( f χ C s ˜ B 0 ) ( g ) + 4 ‖ ∇ b ‖ L ∞ ( G ) d ( g ˜ , 0 ) M ( T j , α ( f χ C s ˜ B 0 ) ) 1 2 ( g ) 2 .

Note that when g ′ ∈ G \ B g and g , ξ ∈ B , we have d ( ξ , g ′ ) ≥ 1 2 d ( g , g ′ ) . It follows from (29) that

∣ I V 2 ∣ ≤ C ( Q ) ‖ b ‖ L ∞ ( G ) ∫ d ( g , g ′ ) > d ( g ˜ , 0 ) 1 2 d ( g ˜ , 0 ) d ( g , g ′ ) Q − α + 1 f ( g ′ ) χ C s ˜ B 0 ( g ′ ) d g ′ ≤ C ( Q ) ‖ b ‖ L ∞ ( G ) ∣ C s ˜ B 0 ∣ α Q d ( g ˜ , 0 ) 1 2 M ( f χ C s ˜ B 0 ) ( g ) .

Assume that t < r and for any g ′ ∈ G \ B g and g , ξ ∈ B , then d ( ξ , g ′ ) ≥ 1 2 d ( g , g ′ ) and 7 4 d ( g , g ′ ) ≥ d ( g ˜ ξ , g ′ ) ≥ 1 4 d ( g , g ′ ) . Now, from (67), we can estimate

∣ I V 3 − I V 4 ∣ ≤ C ( Q ) τ α , Q ‖ ∇ b ‖ L ∞ ( G ) ∫ d ( g , g ′ ) ≤ d ( g ˜ , 0 ) 1 2 d ( g ˜ ξ , g ′ ) d ( g , g ′ ) Q − α ∣ f ( g ′ ) ∣ χ C s ˜ B 0 ( g ′ ) d g ′ ≤ C ( Q ) τ α , Q ‖ ∇ b ‖ L ∞ ( G ) ∣ C s ˜ B 0 ∣ α Q d ( g ˜ , 0 ) 1 2 M ( f χ C s ˜ B 0 ) ( g ) .

Since d ( g ˜ , 0 ) < t and C T j , α > 1 , combining the estimates for I V 1 , I V 2 , I V 3 , and I V 4 with (69) and (70), this leads to the estimate

∣ T j , g ˜ ( f χ C s ˜ B 0 \ C s ˜ B ) ( ξ ) ∣ ≤ C ( Q , b ) ( τ α , Q + 1 + t 1 2 ( τ α , Q + 2 C T j , α ) ) t 1 2 ∣ C s ˜ B 0 ∣ α Q M ( f χ C s ˜ B 0 ) ( g ) + 4 ‖ ∇ b ‖ L ∞ ( G ) t M ( T j , α ( f χ C s ˜ B 0 ) ) 1 2 ( g ) 2 .

Now, the proof of (65) is complete.□

By (65) and Lemma 3.1, repeating Step 3 in Section 3.1.3, we can choose a constant C E ˜ = C ( Q , b ) with ∣ E ˜ ∣ ≤ ( 4 C Q ) − 1 ∣ B 0 ∣ , where C Q is a constant defined as in (11) and C E ˜ is bounded when α = 0 , such that

E ˜ ≔ { g ∈ B 0 : ℳ T j , g ˜ , B 0 f ( g ) > C E ˜ ( τ α , Q + 1 + t 1 2 ( τ α , Q + 2 C T j , α ) ) t 1 2 ∣ C s ˜ B ( Q 0 ) ∣ α Q ∣ f ∣ C s ˜ B ( Q 0 ) } .

Relying upon the proof of Section 3.1.3, we have S i is a δ -sparse family, δ > 0 and

(71) ∣ T j , g ˜ f ( g ) ∣ ≤ C ( Q , b ) ( τ α , Q + 1 + t 1 2 ( τ α , Q + 2 C T j , α ) ) t 1 2 ∑ i = 1 ℐ ∑ Q ∈ S i ∣ Q ∣ α Q ∣ f ∣ Q χ Q ( g ) .

Now, it follows from (63) and (71) that

lim t → 0 ∫ G ∣ [ b , T j , α ] f ( g ) − ( [ b , T j , α ] f ) B ( g , t ) ∣ q w q ( g ) d g 1 q = lim t → 0 ∫ G 1 ∣ B ( 0 , t ) ∣ ∫ B ( 0 , t ) ∣ T j , g ˜ f ( g ) ∣ d g ˜ ∣ q w q ( g ) d g 1 q ≤ C ( Q , b ) lim t → 0 t 1 2 ( τ α , Q + 1 + t 1 2 ( τ α , Q + 2 C T j , α ) ) ∣ Q ∣ α Q ℐ ‖ I S α f ‖ L w q q ( G ) ≤ C ( Q , b ) lim t → 0 t 1 2 ( τ α , Q + 1 + t 1 2 ( τ α , Q + 2 C T j , α ) ) ∣ Q ∣ α Q ‖ I S α ‖ L w p p ( G ) → L w q q ( G ) ‖ f ‖ L w p p ( G ) = 0 .

Therefore, by Lemma 2.10, we have [ b , T j , α ] is compact from L w p p ( G ) to L w q q ( G ) , where 1 q = 1 p − α Q for 0 ≤ α < Q .

4.2 [ b , T j , α ] is compact ⇒ b ∈ VMO ( G )

Given 0 ≤ α < Q , 1 < p < Q α . Let 1 q = 1 p − α Q . For some j = 1 , … , n 1 , assume that [ b , T j , α ] is compact from L w p p ( G ) to L w q q ( G ) with w ∈ A p , q . By condition (i) in Lemma 2.10 and Theorem 1.1, it is direct to see that b ∈ BMO ( G ) . Without loss of generality, we may assume that ‖ b ‖ BMO ( G ) = 1 .

To obtain b ∈ VMO ( G ) , we need to check conditions (iv)–(vi) in Lemma 2.11 hold. Applying the ideas from [4], we will use the contradiction argument through Lemma 2.11. Assume that b ∉ VMO ( G ) , then we will check b does not satisfy at least one of the three conditions in Lemma 2.11. The argument is similar in all three conditions, and we just present the case that condition (iv) does not hold.

Suppose b does not satisfy (iv). Then, we can find ϑ > 0 and a sequence { B l } l = 1 ∞ ≔ { B ( g l , r l ) } l = 1 ∞ of balls in G with r l → 0 as l → ∞ and

M ( b , B l ) > ϑ .

For every l ∈ N , by Definition 2.4, we can choose disjoint subsets E l 1 , E l 2 ⊂ B l such that

E l 1 ⊂ { g ∈ B l : b ( g ) ≥ m b ( B l ) } , E l 2 ⊂ { g ∈ B l : b ( g ) ≤ m b ( B l ) } ,

and ∣ E l 1 ∣ = ∣ E l 2 ∣ = 1 2 ∣ B l ∣ . Now, we define f l ( g ) = ( w p ( B l ) ) − 1 p ( χ E l 1 ( g ) − χ E l 2 ( g ) ) . Then, for every g ∈ B l , f l satisfies supp f l ⊂ B l and

(72) ∣ f l ( g ) ∣ = ( w p ( B l ) ) − 1 p , f l ( g ) ( b ( g ) − m b ( B l ) ) ≥ 0 ,

(73) ∫ B l f l ( g ) d g = 0 , ‖ f l ‖ L w p p ( G ) = 1 .

Note the fact that r l → 0 as l → ∞ , there exists a subsequence { B l i } of { B l } such that

∣ B l i + 1 ∣ ∣ B l i ∣ ≤ γ 1 − Q .

Fixing i , m ∈ N . γ 2 is to be determined later such that γ 1 > γ 2 . Let

Ω ≔ γ 1 B l i \ γ 2 B l i , Ω 1 ≔ Ω \ γ 1 B l i + m , Ω 2 ≔ G \ γ 1 B l i + m .

It is not hard to check that

Ω 1 ⊂ γ 1 B l i ∩ Ω 2 , Ω 1 = Ω \ ( Ω \ Ω 2 ) .

Now, it follows that

(74) ‖ [ b , T j , α ] ( f l i ) − [ b , T j , α ] ( f l i + m ) ‖ L w q q ( G ) ≥ ∫ Ω 1 ∣ [ b , T j , α ] ( f l i ) ( g ) − [ b , T j , α ] ( f l i + m ) ( g ) ∣ q w q ( g ) d g 1 q ≥ ∫ Ω \ ( Ω \ Ω 2 ) ∣ [ b , T j , α ] ( f l i ) ( g ) ∣ q w q ( g ) d g 1 q − ∫ Ω 2 ∣ [ b , T j , α ] ( f l i + m ) ( g ) ∣ q w q ( g ) d g 1 q ≕ V 1 − V 2 .

Assume that Ω \ Ω 2 ≠ ∅ , then Ω \ Ω 2 ⊂ γ 1 B l i + m . Hence, for each k ≥ [ log 2 γ 2 ] ,

∣ 2 k + 1 B l i \ 2 k B l i ∣ > ∣ 2 k B l i ∣ > ∣ B l i ∣ ≥ γ 1 Q ∣ B l i + m ∣ = ∣ γ 1 B l i + m ∣ ≥ ∣ Ω \ Ω 2 ∣ .

Then, there exist at most two rings, 2 k 0 + 2 B l i \ 2 k 0 + 1 B l i and 2 k 0 + 1 B l i \ 2 k 0 B l i such that

Ω \ Ω 2 ⊂ ( 2 k 0 + 2 B l i \ 2 k 0 + 1 B l i ) ∪ ( 2 k 0 + 1 B l i \ 2 k 0 B l i ) .

This yields

(75) Ω \ ( Ω \ Ω 2 ) ⊃ Ω \ ( ( 2 k 0 + 2 B l i \ 2 k 0 + 1 B l i ) ∪ ( 2 k 0 + 1 B l i \ 2 k 0 B l i ) ) .

Then, for k large enough, we claim that

(76) ∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ∣ [ b , T j , α ] f l ( g ) ∣ q w q ( g ) d g ≥ C 1 ( Q ) C K j , α q ϑ q 2 − ( k + 1 ) q Q − q w p ( 2 k + 1 B l ) w p ( B l ) q p

and

(77) ∫ 2 k + 1 B l \ 2 k B l ∣ [ b , T j , α ] f l ( g ) ∣ q w q ( g ) d g ≤ C 2 ( Q ) τ α , Q q 2 − k q Q − q α w p ( 2 k + 1 B l ) w p ( B l ) q p ,

where C K j , α is the constant as in Lemma 2.2. We will check them later.

By Lemma 2.3 and w p ∈ A p , there exists a constant σ ∈ ( 0 , p ) such that w p ∈ A p − σ [14]. Then, there exists a constant r > 1 such that

(78) w p ( 2 k + 1 B l ) w p ( B l ) ≥ 1 C r ∣ 2 k + 1 B l ∣ ∣ B l ∣ r − 1 r and w p ( 2 k + 1 B l ) w p ( B l ) ≤ 1 C l ∣ 2 k + 1 B l ∣ ∣ B l ∣ p − σ .

where C l and C r depend only on p , Q , and [ w p ] A p .

Now, combining (75)–(78), we obtain

V 1 q ≥ ∑ k = [ log 2 γ 2 ] + 1 , k ≠ k 0 , k 0 + 1 [ log 2 γ 1 ] − 1 ∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ∣ [ b , T j , α ] f l ( g ) ∣ q w q ( g ) d g ≥ C 1 ( Q ) C r C K j , α q ϑ q ∑ k = [ log 2 γ 2 ] + 3 [ log 2 γ 1 ] − 1 2 − ( k + 1 ) q Q − q + ( k + 1 ) Q r − 1 r q p ≥ C 1 ( Q ) C r C K j , α q ϑ q γ 2 − q Q ( 1 − r − 1 r p ) 2 − 4 q Q ( 1 − r − 1 r p ) − q ≕ ( 2 C ˜ ) q .

We can choose γ 1 > γ 2 large enough, such that

V 2 q ≤ ∑ k = [ log 2 γ 1 ] ∞ ∫ 2 k + 1 B l i + m \ 2 k B l i + m ∣ [ b , T j , α ] f l ( g ) ∣ q w q ( g ) d g ≤ C 2 ( Q ) C l τ α , Q q ∑ k = [ log 2 γ 1 ] ∞ 2 − k q Q − q α + ( k + 1 ) Q ( p − σ ) q p ≤ C 2 ( Q ) C l τ α , Q q 2 − Q q σ p [ log 2 γ 1 ] 2 ( Q − α ) q − Q q σ p 1 − 2 − Q q σ p < C ˜ q .

Thus, it follows from C K j , α ≠ 0 and (74) that

‖ [ b , T j , α ] ( f l i ) − [ b , T j , α ] ( f l i + m ) ‖ L w q q ( G ) ≥ C ˜ .

This leads to a contradiction with the fact that { [ b , T j , α ] f l } l = 1 ∞ is relatively compact in L w q q ( G ) . Hence, b satisfies condition (iv).

If Ω \ Ω 2 = ∅ , from the aforementioned estimate, condition (iv) also holds.

Proofs of (76) and (77)

We now give the proof of (76) and (77). First, we will recall an important inequality from [4], for each k ∈ N and B ∈ G ,

(79) ∫ 2 k + 1 B ∣ b ( g ) − m b ( B ) ∣ p d g ≤ C ( Q ) k p ∣ 2 k + 1 B ∣ .

From [14], w ∈ A p , for any r > 1 , there exists a constant C depending only on w , p , and Q such that

(80) 1 ∣ Q ∣ ∫ Q w ≤ C 1 ∣ Q ∣ ∫ Q w 1 r r .

Then, since q > p , for g ∈ ( G \ 2 k B l ) ∩ G g l , by Hölder’s inequality, we have

∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l w p ( g ) d g ≤ ∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l w q ( g ) d g p q ∣ 2 k + 1 B l ∣ q − p q .

Combine the aforementioned estimate with Lemma 2.2 and (72). Let k be large enough. We obtain

(81) ∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ∣ T j , α ( ( b − m b ( B l ) ) f l ) ( g ) ∣ q w q ( g ) d g = ∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ∫ E l 1 ∪ E l 2 ∣ K j , α ( g , g ′ ) ∣ ( b ( g ′ ) − m b ( B l ) ) f l ( g ′ ) d g ′ q w q ( g ) d g ≥ C K j , α q ( w p ( B l ) ) − q p ∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ∫ B l 1 d ( g , g ′ ) Q − α ∣ b ( g ′ ) − m b ( B l ) ∣ d g ′ q w q ( g ) d g ≥ C ( Q ) C K j , α q ( w p ( B l ) ) − q p ϑ q ∣ B l ∣ q 2 − ( k + 1 ) q ( Q − α ) r l − q ( Q − α ) ∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l w q ( g ) d g ≥ C ( Q ) C K j , α q ϑ q 2 − ( k + 1 ) q ( Q − α ) r l q α ∣ 2 k + 1 B l ∣ p − q p w p ( ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ) w p ( 2 k + 1 B l ) q p w p ( 2 k + 1 B l ) w p ( B l ) q p ≥ C 3 ( Q ) C K j , α q ϑ q 2 − k q Q 2 − q Q − q + q α w p ( 2 k + 1 B l ) w p ( B l ) q p ,

where C K j , α ≠ 0 as in Lemma 2.2. We deduce the last inequality from w ∈ A p , q , then w p ∈ A P [23],

w p ( ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ) w p ( 2 k + 1 B l ) 1 p ≥ C l ∣ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ∣ ∣ 2 k + 1 B l ∣ ≥ C l ∣ 2 k + 1 B l \ 2 k B l ∣ ∣ 2 k + 1 B l ∣ ≥ C l 2 ,

where C l depends only on p , Q , and [ w p ] A p .

Next, it follows from (8), (72), (73), (79), and (80) that

(82) ∫ 2 k + 1 B l \ 2 k B l ∣ ( b ( g ) − m b ( B l ) ) T j , α ( f l ) ( g ) ∣ q w q ( g ) d g ≤ ∫ 2 k + 1 B l \ 2 k B l ∣ b ( g ) − m b ( B l ) ∣ q ∫ B l ( K j , α ( g , g ′ ) − K j , α ( g , g l ) ) f l ( g ′ ) d g ′ q w q ( g ) d g ≤ C ( Q ) ∫ 2 k + 1 B l \ 2 k B l ∣ b ( g ) − m b ( B l ) ∣ q ∫ B l d ( g ′ , g l ) d ( g , g l ) Q − α + 1 ∣ f l ( g ′ ) ∣ d g ′ q w q ( g ) d g ≤ C ( Q ) ( 2 k r l ) − q ( Q − α + 1 ) r l q ( w p ( B l ) ) − q p ∣ B l ∣ q ∫ 2 k + 1 B l \ 2 k B l ∣ b ( g ) − m b ( B l ) ∣ q w q ( g ) d g ≤ C ( Q ) ( w p ( B l ) ) − q p 2 − k q ( Q − α + 1 ) r l q α ∣ 2 k + 1 B l ∣ 1 p + 1 p ′ k q 1 ∣ 2 k + 1 B l ∣ ∫ 2 k + 1 B l w p ( g ) d g q p ≤ C 4 ( Q ) k q 2 − k Q q − k q − q α w p ( 2 k + 1 B l ) w p ( B l ) q p .

Thus, for any α ∈ [ 0 , Q ) , we can find k ≥ [ log 2 γ 2 ] + 3 large enough such that

C 4 ( Q ) 2 q 2 q ( Q − α ) k q C 3 ( Q ) C K j , α q 2 k q ϑ q < 1 2 .

Then, (81) and (82) yield

∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ∣ [ b , T j , α ] f l ( g ) ∣ q w q ( g ) d g ≥ ∫ ( 2 k + 1 B l \ 2 k B l ) ∩ G g l ∣ T j , α ( ( b − m b ( B l ) ) f l ) ( g ) ∣ q w q ( g ) d g − ∫ 2 k + 1 B l \ 2 k B l ∣ ( b ( g ) − m b ( B l ) ) T j , α ( f l ) ( g ) ∣ q w q ( g ) d g ≥ C 1 ( Q ) C K j , α q ϑ q 2 − ( k + 1 ) q Q − q w p ( 2 k + 1 B l ) w p ( B l ) q p .

The proof of (76) is complete.

As for (77), by (7), (79), and (80), we have

(83) ∫ 2 k + 1 B l \ 2 k B l ∣ T j , α ( ( b − m b ( B l ) ) f l ) ( g ) ∣ q w q ( g ) d g ≤ C ( Q ) τ α , Q q ( w p ( B l ) ) − q p ∫ 2 k + 1 B l \ 2 k B l ∫ B l 1 d ( g , g ′ ) Q − α ∣ b ( g ′ ) − m b ( B l ) ∣ d g ′ q w q ( g ) d g ≤ C ( Q ) τ α , Q q ( w p ( B l ) ) − q p ∣ B l ∣ q ∫ 2 k + 1 B l \ 2 k B l 1 d ( g , g l ) q ( Q − α ) w q ( g ) d g ≤ C ( Q ) τ α , Q q ( w p ( B l ) ) − q p 2 − k q ( Q − α ) r l q α ∣ 2 k + 1 B l ∣ 1 ∣ 2 k + 1 B l ∣ ∫ 2 k + 1 B l w p ( g ) d g q p ≤ C 5 ( Q ) τ α , Q q 2 − k q Q − q α w p ( 2 k + 1 B l ) w p ( B l ) q p .

Now, for any 0 ≤ α < Q , we can choose k ≥ [ log 2 γ 1 ] large enough such that

C 4 ( Q ) k q < C 5 ( Q ) τ α , Q q 2 k q .

Then, together with (82) and (83), it follows that

∫ 2 k + 1 B l \ 2 k B l ∣ [ b , T j , α ] f l ( g ) ∣ q w q ( g ) d g ≤ ∫ 2 k + 1 B l \ 2 k B l ∣ T j , α ( ( b − m b ( B l ) ) f l ) ( g ) ∣ q w q ( g ) d g + ∫ 2 k + 1 B l \ 2 k B l ∣ ( b ( g ) − m b ( B l ) ) T j , α ( f l ) ( g ) ∣ q w q ( g ) d g ≤ C 2 ( Q ) τ α , Q q 2 − k q Q − q α w p ( 2 k + 1 B l ) w p ( B l ) q p .

Therefore, the proof of (77) is complete.

Acknowledgments

The authors sincerely thank the editor and the reviewers for their careful reading and valuable suggestions, which have helped improve the quality of this article.

Funding information: Yanping Chen was supported by the National Natural Science Foundation of China (Grant Nos. [12371092], [12326366], and [12326371]).
Author contributions: All authors contributed to the study and have read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: This research did not involve the use of any data.

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Received: 2024-07-06

Revised: 2024-11-19

Accepted: 2025-05-06

Published Online: 2025-07-04

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0092

Keywords for this article

stratified Lie groups; Riesz transform; BMO space; VMO space

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