A note on commutators on weighted Morrey spaces on spaces of homogeneous type
aa r X i v : . [ m a t h . C A ] S e p A NOTE ON COMMUTATORS ON WEIGHTED MORREY SPACESON SPACES OF HOMOGENEOUS TYPE
Ruming Gong, Ji Li, Elodie Pozzi and Manasa N. Vempati
Abstract:
In this paper we study the boundedness and compactness characterizationsof the commutator of Calder´on–Zygmund operators T on spaces of homogeneous type( X, d, µ ) in the sense of Coifman and Weiss. More precisely, We show that the commuta-tor [ b, T ] is bounded on weighted Morrey space L p,κω ( X ) ( κ ∈ (0 , , ω ∈ A p ( X ) , < p < ∞ ) if and only if b is in the BMO space. Moreover, the commutator [ b, T ] is compact onweighted Morrey space L p,κω ( X ) ( κ ∈ (0 , , ω ∈ A p ( X ) , < p < ∞ ) if and only if b is inthe VMO space. Keywords : commutator, compact operator, BMO space, VMO space, weighted Morreyspace, space of homogeneous typeMathematics Subject Classification 2010: 42B20, 43A80
It is well-known that the boundedness and compactness of Calder´on–Zygmund operator com-mutators on certain function spaces and their characterizations play an important role in var-ious area, such as harmonic analysis, complex analysis, (nonlinear) PDE, etc. See for example[10, 9, 3, 19, 20, 13, 22, 18, 24, 25, 34] and the references therein. Recently, equivalent charac-terizations of the boundedness and the compactness of commutators were further extended toMorrey spaces over the Euclidean space by Di Fazio and Ragusa [16] and Chen et al. [5], and toweighted Morrey spaces by Komori and Shirai [27] for Calder´on–Zygmund operator commuta-tors and by Tao, Da. Yang and Do. Yang [31, 32] for the Cauchy integral and Buerling-Ahlforstransformation commutator, respectively. For more results on the boundedness of operatorson Morrey spaces in different settings, we refer the reader to other studies [1, 15, 30, 17].Thus, along this literature, it is natural to study the boundedness and compactness ofCalder´on–Zygmund operator commutators on weighted Morrey spaces in a more general setting:spaces of homogeneous type in the sense of Coifman and Weiss [8], as Yves Meyer remarked inhis preface to [11], “One is amazed by the dramatic changes that occurred in analysis duringthe twentieth century. In the 1930s complex methods and Fourier series played a seminalrole. After many improvements, mostly achieved by the Calder´on–Zygmund school, the actiontakes place today on spaces of homogeneous type. No group structure is available, the Fouriertransform is missing, but a version of harmonic analysis is still present. Indeed the geometryis conducting the analysis.”We say that (
X, d, µ ) is a space of homogeneous type in the sense of Coifman and Weissif d is a quasi-metric on X and µ is a nonzero measure satisfying the doubling condition. A quasi-metric d on a set X is a function d : X × X −→ [0 , ∞ ) satisfying (i) d ( x, y ) = d ( y, x ) ≥ x , y ∈ X ; (ii) d ( x, y ) = 0 if and only if x = y ; and (iii) the quasi-triangle inequality :there is a constant A ∈ [1 , ∞ ) such that for all x , y , z ∈ X , d ( x, y ) ≤ A [ d ( x, z ) + d ( z, y )] . (1.1)1 R, Gong, J. Li, E. Pozzi, M. N. Vempati
We say that a nonzero measure µ satisfies the doubling condition if there is a constant C µ suchthat for all x ∈ X and r > µ ( B ( x, r )) ≤ C µ µ ( B ( x, r )) < ∞ , (1.2)where B ( x, r ) is the quasi-metric ball by B ( x, r ) := { y ∈ X : d ( x, y ) < r } for x ∈ X and r > n (the upper dimension of µ ) such that for all x ∈ X , λ ≥ r > µ ( B ( x, λr )) ≤ C µ λ n µ ( B ( x, r )) . (1.3)Throughout this paper we assume that µ ( X ) = ∞ and that µ ( { x } ) = 0 for every x ∈ X .We now recall the definition of Calder´on–Zygmund operators on spaces of homogeneoustype. Definition 1.1.
We say that T is a Calder´on–Zygmund operator on ( X, d, µ ) if T is boundedon L ( X ) and has an associated kernel K ( x, y ) such that T ( f )( x ) = R X K ( x, y ) f ( y ) dµ ( y ) forany x supp f , and K ( x, y ) satisfies the following estimates: for all x = y , | K ( x, y ) | ≤ CV ( x, y ) , (1.4) and for d ( x, x ′ ) ≤ (2 A ) − d ( x, y ) , | K ( x, y ) − K ( x ′ , y ) | + | K ( y, x ) − K ( y, x ′ ) | ≤ CV ( x, y ) β (cid:18) d ( x, x ′ ) d ( x, y ) (cid:19) , (1.5) where V ( x, y ) = µ ( B ( x, d ( x, y ))) , β : [0 , → [0 , ∞ ) is continuous, increasing, subadditive, and ω (0) = 0 . Throughout this paper we assume that β ( t ) = t σ , for some σ > . Note that by the doubling condition we have that V ( x, y ) ≈ V ( y, x ). From [12] we assume forany Calder´on–Zygmund operator T as in Definition 1.1 with β ( t ) → t →
0, the following“non-degenerate” condition holds:There exists positive constant c o and ¯ A such that for every x ∈ X and r >
0, there exists y ∈ B ( x, ¯ Ar ) \ B ( x, r ), satisfying | K ( x, y ) | ≥ c µ ( B ( x, r )) . (1.6)This condition gives a lower bound on the kernel and in R n this “non degenerate” conditionwas first proposed in [22]. On stratified Lie groups, a similar condition of the Riesz transformkernel lower bound was verified in [13].Let T be a Calder´on–Zygmund operator on X . Suppose b ∈ L ( X ) and f ∈ L p ( X ). Let[ b, T ] be the commutator defined by[ b, T ] f ( x ) := b ( x ) T ( f )( x ) − T ( bf )( x ) . Let p ∈ (1 , ∞ ) , κ ∈ (0 ,
1) and ω ∈ A p ( X ). The weighted Morrey space L p,κω ( X ) is defined by L p,κω ( X ) := { f ∈ L ploc ( X ) : k f k L p,κω ( X ) < ∞} Here k f k L p,κω ( X ) := sup B (cid:26) ω ( B ) κ Z B | f ( x ) | p ω ( x ) dµ ( x ) (cid:27) p . Our main results are the following theorems. oundedness and compactness of commutators Theorem 1.2.
Let p ∈ (1 , ∞ ) , κ ∈ (0 , and ω ∈ A p ( X ) . Suppose b ∈ L ( X ) and that T isa Calder´on–Zygmund operator as in Definition 1.1 and satisfies the non-degenerate condition (1.6) . Then the commutator [ b, T ] has the following boundedness characterization: (i) If b ∈ BM O ( X ) , then [ b, T ] is bounded on L p,κω ( X ) . (ii) If b is real valued and [ b, T ] is bounded on L p,κω ( X ) , then b ∈ BM O ( X ) . Theorem 1.3.
Let p ∈ (1 , ∞ ) , κ ∈ (0 , and ω ∈ A p ( X ) . Suppose b ∈ L ( X ) and that T isa Calder´on–Zygmund operator as in Definition 1.1 and satisfies the non-degenerate condition (1.6) . Then the commutator [ b, T ] has the following compactness characterization: (i) If b ∈ V M O ( X ) , then [ b, T ] is compact on L p,κω ( X ) . (ii) If b is real valued and [ b, T ] is compact on L p,κω ( X ) , then b ∈ V M O ( X ) . Throughout the paper, we denote by C and e C positive constants which are independent ofthe main parameters, but they may vary from line to line. For every p ∈ (1 , ∞ ), we denote by p ′ the conjugate of p , i.e., p ′ + p = 1. If f ≤ Cg or f ≥ Cg , we then write f . g or f & g ; andif f . g . f , we write f ≈ g. Let (
X, d, µ ) be a space of homogeneous type as mentioned in Section 1. We now recall theBMO and VMO space.
Definition 2.1.
A function b ∈ L ( X ) belongs to the BMO space BM O ( X ) if k b k BMO ( X ) := sup B M ( b, B ) := sup B µ ( B ) Z B | b ( x ) − b B | dµ ( x ) < ∞ , where the sup is taken over all quasi-metric balls B ⊂ X and b B = 1 µ ( B ) Z B b ( y ) dµ ( y ) . The following John-Nirenberg inequalities on spaces of homogeneous type comes from [26].
Lemma 2.2 ([26]) . If f ∈ BMO( X ) , then there exist positive constants C and C such thatfor every ball B ⊂ X and every α > , we have µ ( { x ∈ B : | f ( x ) − f B | > α } ) ≤ C λ ( B ) exp n − C k f k BMO( X ) α o . We recall the median value α B ( f ) ([4]). For any real valued function f ∈ L ( X ) and B ⊂ X , let α B ( f ) be a real number such thatinf c ∈ R µ ( B ) Z B | f ( x ) − c | dµ ( x )is attained. Moreover, it is known that α B ( f ) satisfies that µ ( { x ∈ B : f ( x ) > α B ( f ) } ) ≤ µ ( B )2 (2.1) R, Gong, J. Li, E. Pozzi, M. N. Vempati and µ ( { x ∈ B : f ( x ) < α B ( f ) } ) ≤ µ ( B )2 . (2.2)And it is easy to see that for any ball B ⊂ X , M ( b, B ) ≈ µ ( B ) Z B | b ( x ) − α B ( b ) | dµ ( x ) , (2.3)where the implicit constants are independent of the function b and the ball B .By Lip( β ), 0 < β < ∞ , we denote the set of all functions φ ( x ) defined on X such that thereexists a finite constant C satisfying | φ ( x ) − φ ( y ) | ≤ Cd ( x, y ) β for every x and y in X . k φ k β will stand for the least constant C satisfying the condition above.By Lip c ( β ), we denote the set of all Lip( β ) functions with compact support on X . Definition 2.3.
We define
VMO( X ) as the closure of the Lip c ( β ) functions X under the normof the BMO space. We also need to establish the characterisation of VMO( X ). We will give its proof in Ap-pendix. For the Euclidean and the stratified Lie groups case one can refer to [33] and [4]. Lemma 2.4.
Let f ∈ BMO ( X ) . Then f ∈ VMO ( X ) if and only if f satisfies the followingthree conditions: (i) lim a → sup r B = a M ( f, B ) = 0;(ii) lim a →∞ sup r B = a M ( f, B ) = 0;(iii) lim r →∞ sup B ⊂ X \ B ( x ,r ) M ( f, B ) = 0 , where r B is the radius of the ball B and x is a fixed point in X . To this end, we recall the definition of A p weights. Definition 2.5.
Let ω ( x ) be a nonnegative locally integrable function on X . For < p < ∞ ,we say ω is an A p weight , written ω ∈ A p , if [ ω ] A p := sup B (cid:18) − Z B ω (cid:19) − Z B (cid:18) ω (cid:19) / ( p − ! p − < ∞ . Here the suprema are taken over all balls B ⊂ X . The quantity [ ω ] A p is called the A p constantof ω . For p = 1 , we say ω is an A weight , written ω ∈ A , if M ( ω )( x ) ≤ ω ( x ) for µ -almostevery x ∈ X , and let A ∞ := ∪ ≤ p< ∞ A p and we have [ ω ] A ∞ := sup B (cid:0) − R B ω (cid:1) exp (cid:0) − R B log (cid:0) ω (cid:1)(cid:1) < ∞ . Next we note that for ω ∈ A p the measure ω ( x ) dµ ( x ) is a doubling measure on X . To bemore precise, we have that for all λ > B ⊂ X , ω ( λB ) ≤ λ np [ ω ] A p ω ( B ) , (2.4) oundedness and compactness of commutators n is the upper dimension of the measure µ , as in (1.3).We also point out that for ω ∈ A ∞ , there exists γ > B , µ (cid:16)n x ∈ B : ω ( x ) ≥ γ − Z B ω o(cid:17) ≥ µ ( B ) . And this implies that for every ball B and for all δ ∈ (0 , − Z B ω ≤ C (cid:18) − Z B ω δ (cid:19) /δ ; (2.5)see also [25].By the definition of A p weight and H¨older’s inequality, we can easily obtain the followingstandard properties. Lemma 2.6.
Let ω ∈ A p ( X ) , p ≥ . Then there exists constants ˆ C , ˆ C > and σ ∈ (0 , such that the following holds ˆ C (cid:18) µ ( E ) µ ( B ) (cid:19) p ≤ ω ( E ) ω ( B ) ≤ ˆ C (cid:18) µ ( E ) µ ( B ) (cid:19) σ for any measurable set E of a quasi metric ball B . According to [2, Theorem 5.5], we have the following result for BMO functions on X . Lemma 2.7.
Let < p < ∞ , v ∈ A ∞ ( X ) , f ∈ BMO( X ) . Then k f k BMO( X ) ≈ sup B ⊂ X (cid:26) v ( B ) Z B (cid:12)(cid:12) f ( x ) − f B,v (cid:12)(cid:12) p v ( x ) dµ ( x ) (cid:27) p , where f B,v = v ( B ) R B f ( y ) v ( y ) dµ ( y ) . In this section, we will give the proof of Theorem 1.2.
In order to prove Theorem 1.2(i), we need the following lemma.
Lemma 3.1 ([12]) . Let b ∈ BM O ( X ) and T be Calder´on–Zygmund operator on ( X, d, µ ) aSpace of homogeneous type. If κ ∈ (0 , , < p < ∞ and ω ∈ A p ( X ) , then [ b, T ] is bounded on L p,κω ( X ) .Proof of Theorem 1.2(i). Let 1 < p < ∞ . Then it suffices to show that (cid:18) ω ( B )] κ Z B | [ b, T ]( x ) | p ω ( x ) dµ ( x ) (cid:19) p . k b k BMO( X ) k f k L p,κω ( X ) , holds for any ball B .Now we will fix a ball B = B ( x , r ) and then decompose f = f χ B + f χ X \ B =: f + f .Then we have1 ω ( B ) κ Z B | [ b, T ] f ( x ) | p ω ( x ) dµ ( x ) R, Gong, J. Li, E. Pozzi, M. N. Vempati . (cid:18) ω ( B ) κ Z B | [ b, T ] f ( x ) | p ω ( x ) dµ ( x ) + 1 ω ( B ) κ Z B | [ b, T ] f ( x ) | p ω ( x ) dµ ( x ) (cid:19) =: I + II.
For the first term I here, we use Lemma 3.1 and we obtain1 ω ( B ) κ Z B | [ b, T ] f ( x ) | p ω ( x ) dµ ( x ) ≤ ω ( B ) κ Z X | [ b, T ] f ( x ) | p ω ( x ) dµ ( x ) . k b k p BMO( X ) ω ( B ) κ Z B | f ( x ) | p ω ( x ) dµ ( x ) . k b k p BMO( X ) k f k pL p,κω ( X ) . So we have k [ b, T ] f k L p,κω ( X ) . k b k p BMO( X ) k f k pL p,κω ( X ) . Now for the second term II , observe that for x ∈ B , by (1.4), we have | [ b, T ] f ( x ) | p ≤ (cid:18)Z X | b ( x ) − b ( y ) || K ( x, y ) || f ( y ) | dµ ( y ) (cid:19) p . Z X \ B | b ( x ) − b ( y ) | V ( x, y ) | f ( y ) | dµ ( y ) ! p . Z X \ B | f ( y ) | V ( x , y ) {| b ( x ) − b B,ω | + | b B,ω − b ( y ) |} dµ ( y ) ! p . Z X \ B | f ( y ) | V ( x , y ) dµ ( y ) ! p | b ( x ) − b B,ω | p + Z X \ B | f ( y ) | V ( x , y ) | b B,ω − b ( y ) | dµ ( y ) ! p , where b B,ω = ω ( B ) R B b ( y ) ω ( y ) dµ ( y ). Hence we have the following1 ω ( B ) κ Z B | [ b, T ] f ( x ) | p ω ( x ) dµ ( x ) . ω ( B ) κ Z X \ B | f ( y ) | V ( x , y ) dµ ( y ) ! p Z B | b ( x ) − b B,ω | p ω ( x ) dµ ( x )+ Z X \ B | f ( y ) | V ( x , y ) | b B,ω − b ( y ) | dµ ( y ) ! p ω ( B ) − κ =: III + IV.
Note that lim k →∞ µ (2 k B ) = ∞ . Then there exist j k ∈ N such that µ (2 j B ) ≥ µ ( B ) and µ (2 j k +1 B ) ≥ µ (2 j k B ) . For
III , using the H¨older inequality, and using Lemma 2.6 and Lemma 2.7, we get
III . k f k pL p,κω ( X ) ω ( B ) κ ∞ X k =0 Z jk +1 B \ jk B | f ( y ) | V ( x , y ) dµ ( y ) ! p Z B | b ( x ) − b B,ω | p ω ( x ) dµ ( x ) . k f k pL p,κω ( X ) ω ( B ) κ ∞ X k =0 ω (2 j k +1 B ) − κp ! p Z B | b ( x ) − b B,ω | p ω ( x ) dµ ( x ) oundedness and compactness of commutators . k f k pL p,κω ( X ) k b k p BMO(X) ∞ X k =0 (cid:18) ω ( B ) ω (2 j k +1 B ) (cid:19) − κp ! p . k f k pL p,κω ( X ) k b k p BMO(X) ∞ X k =0 − kσ − κp ! p . k f k pL p,κω ( X ) k b k p BMO(X) . Using H¨older’s inequality for the term IV , we get IV . ∞ X k =0 µ (2 j k B ) Z jk +1 B | f ( y ) || b B,ω − b ( y ) | dµ ( y ) ! p ω ( B ) − κ . (cid:18) ∞ X k =0 µ (2 j k B ) (cid:18)Z jk +1 B | f ( y ) | p ω ( y ) dµ ( y ) (cid:19) p × (cid:18)Z jk +1 B | b B,ω − b ( y ) | p ′ ω ( y ) − p ′ dµ ( y ) (cid:19) p ′ (cid:19) p ω ( B ) − κ . k f k pL p,κω ( X ) ( ∞ X k =0 ω (2 j k +1 B ) κp µ (2 j k B ) (cid:18)Z jk +1 B | b B,ω − b ( y ) | p ′ ω ( y ) − p ′ dµ ( y ) (cid:19) p ′ ) p ω ( B ) − κ . Now observe that (cid:18)Z jk +1 B | b B,ω − b ( y ) | p ′ ω ( y ) − p ′ dµ ( y ) (cid:19) p ′ ≤ (cid:18)Z jk +1 B (cid:16) | b ( y ) − b jk +1 B,ω − p ′ | + | b jk +1 B,ω − p ′ − b B,ω | (cid:17) p ′ ω ( y ) − p ′ dµ ( y ) (cid:19) p ′ ≤ (cid:18)Z jk +1 B (cid:16) | b ( y ) − b jk +1 B,ω − p ′ | (cid:17) p ′ ω ( y ) − p ′ dµ ( y ) (cid:19) p ′ + (cid:18)Z jk +1 B (cid:16) | b jk +1 B,ω − p ′ − b B,ω | (cid:17) p ′ ω ( y ) − p ′ dµ ( y ) (cid:19) p ′ =: V + V I.
We have ω − p ′ ∈ A p ′ ( X ) since ω ∈ A p ( X ). So we obtain V . k b k BMO(X) ω − p ′ (2 j k +1 B ) p ′ . For
V I , we have (cid:12)(cid:12)(cid:12) b jk +1 B,ω − p ′ − b B,ω (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) b jk +1 B,ω − p ′ − b jk +1 B (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) b jk +1 B − b B (cid:12)(cid:12) + | b B − b B,ω | . ω − p ′ (2 jk +1 B ) Z jk +1 B (cid:12)(cid:12) b ( y ) − b jk +1 B (cid:12)(cid:12) ω ( y ) − p ′ dµ ( y )+ ( k + 1) k b k BMO(X) + 1 ω ( B ) Z B | b ( y ) − b B | ω ( y ) dµ ( y ) . As we have b ∈ BMO( X ), by Lemma 2.2, there exists some constants C > C > α > µ ( { x ∈ B : | b ( x ) − b B | > α } ) ≤ C µ ( B ) e − C α k b k BMO(X) . R, Gong, J. Li, E. Pozzi, M. N. Vempati
Then using Lemma 2.6, we get ω ( { x ∈ B : | b ( x ) − b B | > α } ) ≤ C ω ( B ) e − C ασ k b k BMO(X) for some σ ∈ (0 , Z B | b ( y ) − b B | ω ( y ) dµ ( y ) = Z ∞ ω ( { y ∈ B : | b ( y ) − b B | > α } ) dα . ω ( B ) Z ∞ e − ¯ C ασ k b k BMO(X) dα . ω ( B ) k b k BMO(X) . Similarly, we also get (cid:16) Z jk +1 B (cid:12)(cid:12) b ( y ) − b jk +1 B (cid:12)(cid:12) ω ( y ) − p ′ dµ ( y ) (cid:17) p ′ . ( k + 1) k b k BMO(X) ω − p ′ (2 j k +1 B ) /p ′ . Together with Lemma 2.6, we have the following IV . k f k pL p,κω ( X ) k b k p BMO(X) " ∞ X k =0 ω (2 j k +1 B ) κp µ (2 j k B ) ( k + 1) ω − p ′ (2 j k +1 B ) /p ′ p ω ( B ) − κ . k f k pL p,κω ( X ) k b k p BMO(X) " ∞ X k =0 ( k + 1) ω ( B ) − κp ω (2 j k +1 B ) − κp p . k f k pL p,κω ( X ) k b k p BMO(X) " ∞ X k =0 ( k + 1)2 − ( k +1)(1 − κ ) σp p . k f k pL p,κω ( X ) k b k p BMO(X) . Therefore we have k [ b, T ] f k L p,κω ( X ) . k f k L p,κω ( X ) k b k BMO(X) . This completes the proof.
We first recall another version of the homogeneous condition (formulated in [12]): there existpositive constants 3 ≤ A ≤ A such that for any ball B := B ( x , r ) ⊂ X , there exist balls e B := B ( y , r ) such that A r ≤ d ( x , y ) ≤ A r , and for all ( x, y ) ∈ ( B × e B ), K ( x, y ) does notchange sign and | K ( x, y ) | & µ ( B ) . (3.1)If the kernel K ( x, y ) := K ( x, y ) + iK ( x, y ) is complex-valued, where i = −
1, then at leastone of K i satisfies (3.1).Then we first point out that the homogeneous condition (1.6) implies (3.1). Lemma 3.2 ([12]) . Let T be the Calder´on–Zygmund operator as in Definition 1.1 and satisfythe homogeneous condition as in (1.6) . Then T satisfies (3.1) . oundedness and compactness of commutators Proof of Theorem 1.2(ii).
To prove b ∈ BM O ( X ), it is sufficient to show for any ball B ⊂ X ,we have M ( b, B ) .
1. Let B = B ( x , r ) be a quasi metric ball in X . Also let e B := B ( y , r ) ⊂ X be the measurable set in (3.1). Following [12], we take E := { x ∈ B : b ( x ) ≥ α ˜ B ( b ) } E := { x ∈ B : b ( x ) < α ˜ B ( b ) } ; F ⊂ { y ∈ ˜ B : b ( y ) ≤ α ˜ B ( b ) } F ⊂ { y ∈ ˜ B : b ( y ) ≥ α ˜ B ( b ) } , with α ˜ B ( b ) the median value of b over ˜ B , such that µ ( F ) = µ ( F ) = µ ( ˜ B ) and F ∩ F = ∅ .For any ( x, y ) ∈ E j × F j , j ∈ { , } , we have | b ( x ) − b ( y ) | = | b ( x ) − α ˜ B ( b ) | + | α ˜ B ( b ) − b ( y ) | ≥ | b ( x ) − α ˜ B ( b ) | . As b is a real valued, using Lemma 2.6, H¨older’s inequality, and using the boundedness of[ b, T ] on L p,κω ( X ) and (3.1), we get that M ( b, B ) . µ ( B ) Z B (cid:12)(cid:12) b ( x ) − α ˜ B ( b ) (cid:12)(cid:12) dµ ( x ) ≈ X j =1 µ ( B ) Z E j | b ( x ) − α ˜ B ( b ) | dµ ( x ) . X j =1 µ ( B ) Z E j Z F j (cid:12)(cid:12) b ( x ) − α ˜ B ( b ) (cid:12)(cid:12) µ ( B ) dµ ( y ) dµ ( x ) ≈ X j =1 µ ( B ) Z E j Z F j (cid:12)(cid:12) b ( x ) − α ˜ B ( b ) (cid:12)(cid:12) V ( x, y ) dµ ( y ) dµ ( x ) . X j =1 µ ( B ) Z E j Z F j | b ( x ) − b ( y ) | V ( x, y ) dµ ( y ) dµ ( x ) . X j =1 µ ( B ) Z E j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z F j | b ( x ) − b ( y ) | K ( x, y ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( x ) ∼ X j =1 µ ( B ) Z E j (cid:12)(cid:12)(cid:12) [ b, T ] χ F j ( x ) (cid:12)(cid:12)(cid:12) dµ ( x ) . X j =1 µ ( B ) Z E j (cid:12)(cid:12)(cid:12) [ b, T ] χ F j ( x ) (cid:12)(cid:12)(cid:12) dµ ( x ) . X j =1 µ ( B ) Z E j (cid:13)(cid:13) [ b, T ] χ F j (cid:13)(cid:13) L p,κω ( X ) [ ω ( B )] κ − p µ ( B ) . X j =1 k [ b, T ] k L p,κω ( X ) → L p,κω ( X ) (cid:13)(cid:13) χ F j (cid:13)(cid:13) L p,κω ( X ) [ ω ( B )] κ − p . k [ b, T ] k L p,κω ( X ) → L p,κω ( X ) [ ω ( ˜ B )] − κp [ ω ( B )] κ − p . k [ b, T ] k L p,κω ( X ) → L p,κω ( X ) . This completes the proof of Theorem 1.2(ii).
Now we will prove Theorem 1.3.0
R, Gong, J. Li, E. Pozzi, M. N. Vempati
Now we will give sufficient conditions for the subsets of weighted Morrey spaces to be relativelycompact. We define a subset F of L p,κω ( X ) to be totally bounded if the L p,κω ( X ) closure of F is compact. Lemma 4.1.
Let p ∈ (1 , ∞ ) , κ ∈ (0 , and ω ∈ A p ( X ) , then a subset F of L p,κω ( X ) is totallybounded if the set F satisfies the following three conditions:(i) F is bounded, namely, sup f ∈F k f k L p,κω ( X ) < ∞ ; (ii) F vanishes uniformly at infinity, namely, for any ǫ ∈ (0 , ∞ ) , there exists some positiveconstant M such that, for any f ∈ F , k f χ { x ∈ X : d ( x ,x ) >M } k L p,κω ( X ) < ǫ, where x is a fixed point in X ;(iii) F is uniformly equicontinuous, namely, lim r → k f ( x ) − f B ( x,r ) k L p,κω ( X ) = 0 uniformly for f ∈ F . The proof for the lemma above, follows from [29] using a small modification from Euclideansetting to space of homogeneous type, this only requires following properties of the underlyingspace: metric on space and doubling measure.We will now show the boundedness of the maximal operator T ∗ of a family of smoothtruncated operators { T η } η ∈ (0 , ∞ ) as follows. For η ∈ (0 , ∞ ), we take T η f ( x ) := Z X K η ( x, y ) f ( y ) dµ ( y ) , where the kernel K η := K ( x, y ) ϕ ( d ( x,y ) η ) with ϕ ∈ C ∞ ( R ) and ϕ satisfies the following ϕ ( t ) = ϕ ( t ) ≡ t ∈ (cid:0) −∞ , (cid:1) ϕ ( t ) ∈ [0 , , if t ∈ (cid:2) , (cid:3) ϕ ( t ) ≡ , if t ∈ (1 , ∞ ) . Let [ b, T η ] f ( x ) := Z X [ b ( x ) − b ( y )] K η ( x, y ) f ( y ) dµ ( y ) . The maximal operator T ∗ is defined as below T ∗ f ( x ) := sup η ∈ (0 , ∞ ) (cid:12)(cid:12)(cid:12)(cid:12)Z X K η ( x, y ) f ( y ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . Observe that the
Hardy-Littlewood maximal Operator M is given as M f ( x ) := sup B ∋ x µ ( B ) Z B | f ( y ) | dµ ( y )for any f ∈ L ( X ) and x ∈ X , here we take the supremum over all quasi-metric balls B of X that contain x .Then we have the following lemmas. oundedness and compactness of commutators Lemma 4.2.
There exists a positive constant C such that we have, for any b ∈ Lip ( β ) , <β < ∞ , f ∈ L ( X ) and x ∈ X | [ b, T η ] f ( x ) − [ b, T ] f ( x ) | ≤ Cη β M f ( x ) . Proof.
Let f ∈ L ( X ). Now for any x ∈ X , we get | [ b, T η ] f ( x ) − [ b, T ] f ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z η/ Let p ∈ (1 , ∞ ) , κ ∈ (0 , and ω ∈ A p ( X ) . Then there exists a positive constant C such that, for any f ∈ L p,κω ( X ) , k T ∗ k L p,κω ( X ) + kM f k L p,κω ( X ) ≤ C k f k L p,κω ( X ) . Proof. To show the boundedness of M on L p,κω ( X ) one can refer to [2]. We will now onlyconsider the boundedness of T ∗ . For any fixed quasi-metric ball B ⊂ X and f ∈ L p,κω ( X ) , wewrite the following f := f + f := f χ B + f χ X \ B . Note that lim k →∞ µ (2 k B ) = ∞ . Then there exist j k ∈ N such that µ (2 j B ) ≥ µ ( B ) and µ (2 j k +1 B ) ≥ µ (2 j k B ) . Observe that f ∈ L pω ( X ) . Then using the boundedness of T ∗ on L pω ( X ) (see, for example, [23,Theorem 1.1] ) and from the H¨older inequality, also using size and smoothness of Kernel, wehave that (cid:20)Z B | T ∗ f ( x ) | p ω ( x ) dµ ( x ) (cid:21) p . (cid:20)Z B | T ∗ f ( x ) | p ω ( x ) dµ ( x ) (cid:21) p + ∞ X k =0 (Z B "Z jk +1 B \ jk B | f ( y ) | V ( x, y ) dµ ( y ) p ω ( x ) dµ ( x ) ) p . (cid:20)Z B | f ( x ) | p ω ( x ) dµ ( x ) (cid:21) p + ∞ X k =0 (cid:20) ω ( B ) µ (2 j k B ) p (cid:26)Z jk +1 B | f ( y ) | [ ω ( y )] p [ ω ( y )] − p dµ ( y ) (cid:27) p (cid:21) p R, Gong, J. Li, E. Pozzi, M. N. Vempati . k f k L p,κω ( X ) [ ω ( B )] κp + ∞ X k =0 n ω ( B ) (cid:2) ω (cid:0) j k B (cid:1)(cid:3) κ − k f k pL p,κω ( X ) o p . k f k L p,κω ( X ) ω ( B ) κp + ∞ X k =0 n [ ω ( B )] κ − kσ (1 − κ ) k f k pL p,κω ( X ) o p . k f k L p,κω ( X ) [ ω ( B )] κp , in the fourth inequality above, we have used Lemma 2.6 for some σ ∈ (0 , Proof of Theorem . i ) . When b ∈ VMO ( X ) , then for any ε ∈ (0 , ∞ ) , there exists b ( ε ) ∈ Lip c ( β ) , < β < ∞ such that we have (cid:13)(cid:13) b − b ( ε ) (cid:13)(cid:13) BMO( X ) < ε . Then, using the boundedness ofthe commutator [ b, T ] on L p,κω ( X ), we obtain (cid:13)(cid:13)(cid:13) [ b, T ] f − [ b ( ε ) , T ] f (cid:13)(cid:13)(cid:13) L p,κω ( X ) = (cid:13)(cid:13)(cid:13) [ b − b ( ε ) , T ] f (cid:13)(cid:13)(cid:13) L p,κω ( X ) . (cid:13)(cid:13)(cid:13) b − b ( ε ) (cid:13)(cid:13)(cid:13) BMO( X ) k f k L p,κω ( X ) . ε k f k L p,κω ( X ) . Also using Lemmas 4.2 and 4.3, we have the followinglim η → k [ b, T η ] − [ b, T ] k L p,κω ( X ) → L p,κω ( X ) = 0 . It sufficient to show that, for any b ∈ Lip c ( β ) , < β < ∞ and η ∈ (0 , ∞ ) small enough,[ b, T η ] is a compact operator on L p,κω ( X ) , this is equivalent to showing that, for any boundedsubset F ⊂ L p,κω ( X ) , [ b, T η ] F is relatively compact. Which means, we need to show that [ b, T η ]satisfies the conditions (i) through (iii) of Lemma 4.1.Observe by [27, Theorem 3.4] and using the fact that b ∈ BMO ( X ), we have that [ b, T η ]is bounded on L p,κω ( X ) for the given p ∈ (1 , ∞ ) , κ ∈ (0 , 1) and ω ∈ A p ( X ), this shows that[ b, T η ] F satisfies condition (i) of Lemma 4.1.Now, let x be a fixed point in X . Since b ∈ Lip c ( β ) , we can further assume that k b k L ∞ =1. Recall that there exists a positive constant R such that supp ( b ) ⊂ B ( x , R ). Let M ∈ (10 R , ∞ ) . Thus, for any y ∈ B ( x , R ) and x ∈ X with d ( x , x ) > M, d ( x, y ) ∼ d ( x , x ). Then, for x ∈ X with d ( x , x ) > M, by H¨older inequality and using that V ( x, y ) ∼ µ ( B ( x, d ( x, y ))) we deduce that | [ b, T η ] f ( x ) | ≤ Z X | b ( x ) − b ( y ) | | K η ( x, y ) | | f ( y ) | dµ ( y ) ≤ Z X | b ( y ) | | K ( x, y ) | | f ( y ) | dµ ( y ) . Z B ( x ,R ) | f ( y ) | V ( x, y ) dµ ( y ) . Z B ( x ,R ) | f ( y ) | µ ( B ( x , d ( x, x )) dµ ( y ) . µ ( B ( x , d ( x, x )) "Z B ( x ,R ) | f ( y ) | p ω ( y ) dµ ( y ) p (Z B ( x ,R ) [ ω ( y )] − p ′ p dµ ( y ) ) p ′ oundedness and compactness of commutators . µ ( B ( x , R )) µ ( B ( x , d ( x, x )) [ ω ( B ( x , R ))] κ − p k f k L p,κω ( X ) . From lim k →∞ µ ( B ( x , kM )) = ∞ , we have that there exist j k ∈ N such that µ ( B ( x , j M )) ≥ µ ( B ( x , M )) and µ ( B ( x , j k +1 M )) ≥ µ ( B ( x , j k M )) . Hence, for any fixed ball B := B ( e x, e r ) ⊂ X , by Lemma 2.6 , we get that1[ ω ( B )] κ Z B ∩{ x ∈ X : d ( x ,x ) >M } | [ b, T η ] f ( x ) | p ω ( x ) dµ ( x ) . µ ( B ( x , R )) p [ ω ( B ( x , R ))] ( κ − [ ω ( B )] κ k f k L p,κω ( X ) ∞ X k =0 ω (cid:0) B ∩ (cid:8) x ∈ X : 2 j k M < d ( x , x ) ≤ j k +1 M (cid:9)(cid:1) µ ( B ( x , j k M )) p . k f k L p,κω ( X ) ∞ X k =0 ω (cid:0) B ( x , j k +1 M ) (cid:1) − κ ω ( B ( x , R )) − κ µ ( B ( x , R )) p µ ( B ( x , j k M )) p . k f k L p,κω ∞ X k =0 µ ( B ( x , R )) pκ µ ( B ( x , j k M )) pκ . k f k L p,κω ∞ X k =0 − k µ ( B ( x , R )) pκ µ ( B ( x , M )) pκ . µ ( B ( x , R )) pκ µ ( B ( x , M )) pκ k f k pL p,κω ( X ) . Therefore the condition ( ii ) of Lemma 4.1 holds for [ b, T η ] F with large M .Now we will prove [ b, T η ] F also satisfies ( iii ) of Lemma 4.1. Let η be a fixed positive constantsmall enough and r < η A . Now, for any x ∈ X , we have[ b, T η ] f ( x ) − ([ b, T η ] f ) B ( x,r ) = 1 µ ( B ( x, r )) Z B ( x,r ) [ b, T η ] f ( x ) − [ b, T η ] f ( y ) dµ ( y ) . Note that[ b, T η ] f ( x ) − [ b, T η ] f ( y )= [ b ( x ) − b ( y ] Z X K η ( x, z ) f ( z ) dµ ( z ) + Z X [ K η ( x, z ) − K η ( y, z )] [ b ( y ) − b ( z )] f ( z ) dµ ( z )=: L ( x, y ) + L ( x, y ) . As b ∈ Lip c ( β ) , it follows that, for any y ∈ B ( x, r ) | L ( x, y ) | = | b ( x ) − b ( y ) | (cid:12)(cid:12)(cid:12)(cid:12)Z X K η ( x, z ) f ( z ) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . r β T ∗ ( f )( x ) . To estimate L ( x, y ) , we first recall that K η ( x, z ) = 0 , K η ( y, z ) = 0 for any y ∈ B ( x, r ), d ( x, z ) ≤ η A and r < η A . Using the definition of K η we have that, for any y ∈ B ( x, r ), d ( x, z ) > η A and r < η A , | K η ( x, z ) − K η ( y, z ) | . V ( x, z ) d ( x, y ) σ d ( x, z ) σ . R, Gong, J. Li, E. Pozzi, M. N. Vempati Hence this implies, for any y ∈ B ( x, r ) | L ( x, y ) | . Z d ( x,z ) > η A | f ( z ) | V ( x, z ) d ( x, y ) σ d ( x, z ) σ dµ ( z ) . ∞ X k =0 r σ (2 k η ) σ µ ( B ( x, k η A )) Z kη A Let p ∈ (1 , ∞ ) , κ ∈ (0 , and ω ∈ A p ( X ) . Suppose that b ∈ BMO ( X ) is a real-valued function with k b k BMO( X ) = 1 and there exists γ ∈ (0 , ∞ ) and a sequence { B j } j ∈ N := { B ( x j , r j ) } j ∈ N of balls in X , with { x j } j ∈ N ⊂ X and { r j } j ∈ N ⊂ (0 , ∞ ) such that, for any j ∈ N M ( b, B j ) > γ. (4.1) Then there exist real-valued functions { f j } j ∈ N ⊂ L p,κω ( X ) , positive constants K large enough, e C , e C and e C such that, for any j ∈ N and integer k ≥ K , k f j k L p,κω ( X ) ≤ ¯ C , Z B kj | [ b, T ] f j ( x ) | p ω ( x ) dµ ( x ) ≥ ˜ C γ p µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) , (4.2) where B kj := ^ A k − B j is the ball associates with A k − B j in (3.1) and Z A k +12 B j \ A k B j | [ b, T ] f j ( x ) | p ω ( x ) dµ ( x ) ≤ ˜ C µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) . (4.3) Proof. For each j ∈ N , we define function f j as follows: f (1) j := χ B j, − χ B j, := χ n x ∈ B j ; b ( x ) >α Bj ( b ) o − χ n x ∈ B j ; b ( x ) <α Bj ( b ) o , f (2) j := a j χ B j and f j := [ ω ( B j )] κ − p (cid:16) f (1) j − f (2) j (cid:17) , oundedness and compactness of commutators B j is as in the assumption of Lemma 4.4 and a j ∈ R is a constant such that Z X f j ( x ) dµ ( x ) = 0 . (4.4)Then, using the definition of a j , (2.1) and (2.2) we have | a j | ≤ / , supp ( f j ) ⊂ B j and, for any x ∈ B j , f j ( x ) (cid:0) b ( x ) − α B j ( b ) (cid:1) ≥ . (4.5)Also, since | a j | ≤ / , we obtain that, for any x ∈ ( B j, ∪ B j, ), | f j ( x ) | ∼ [ ω ( B j )] κ − p (4.6)and therefore k f j k L p,κw ( X ) . sup B ⊂ X (cid:26) ω ( B ∩ B j )[ ω ( B )] κ (cid:27) p [ ω ( B j )] k − p . sup B ⊂ X [ ω ( B ∩ B j )] − κp [ ω ( B j )] k − p . . Observe that, for any k ∈ N , we get A k − B j ⊂ ( A + 1) B kj ⊂ A k +12 B j (4.7)hence we have ω (cid:16) B kj (cid:17) ∼ ω (cid:16) A k B j (cid:17) (4.8)Observe that [ b, T ]( f ) = [ b − α B ( b )] T ( f ) − T ([ b − α B j ( b )] f ) . (4.9)Using Kernel estimates, (4.4), (4.6) and the fact that d ( x, x j ) ∼ d ( x, ξ ) for any x ∈ B kj withinteger k ≥ ξ ∈ B j , we have, for any x ∈ B kj , | (cid:2) b ( x ) − α B j ( b ) (cid:3) T ( f j ) ( x ) | = (cid:12)(cid:12) b ( x ) − α B j ( b ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Z B j [ K ( x, ξ ) − K ( x, x j )] f j ( ξ ) dµ ( ξ ) (cid:12)(cid:12)(cid:12) (4.10) ≤ (cid:12)(cid:12) b ( x ) − α B j ( b ) (cid:12)(cid:12) Z B j | K ( x, ξ ) − K ( x, x j ) | | f j ( ξ ) | dµ ( ξ ) . [ ω ( B j )] κ − p (cid:12)(cid:12) b ( x ) − α B j ( b ) (cid:12)(cid:12) Z B j V ( x, x j ) (cid:18) d ( ξ, x j ) d ( x, x j ) (cid:19) σ dµ ( ξ ) . [ ω ( B j )] κ − p A kσ µ ( B j ) µ ( A k B j ) (cid:12)(cid:12) b ( x ) − α B j ( b ) (cid:12)(cid:12) . As k b k BMO( X ) = 1 by John-Nirenberg inequality(c.f.[6]), for each k ∈ N and ball B ⊂ X , wehave Z A k +12 B | b ( x ) − α B ( b ) | p dµ ( x ) . Z A k +12 B (cid:12)(cid:12)(cid:12) b ( x ) − α A k +12 B ( b ) (cid:12)(cid:12)(cid:12) p dµ ( x ) + µ ( A k +12 B ) (cid:12)(cid:12)(cid:12) α A k +12 B ( b ) − α B ( b ) (cid:12)(cid:12)(cid:12) p . k p µ ( A k B ) , (4.11)6 R, Gong, J. Li, E. Pozzi, M. N. Vempati where the last inequality follows from the fact that (cid:12)(cid:12)(cid:12) α A k +12 B ( b ) − α B ( b ) (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) α A k +12 B ( b ) − b A k +12 B (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) b A k +12 B − b B (cid:12)(cid:12)(cid:12) + | b B − α B ( b ) | . k. As ω ∈ A p ( X ) , we observe that there exists ǫ ∈ (0 , ∞ ) such that the reverse H¨older inequality (cid:20) µ ( B ) Z B ω ( x ) ǫ dµ ( x ) (cid:21) ǫ . µ ( B ) Z B ω ( x ) dµ ( x )holds for any ball B ⊂ X . Then using the H¨older inequality, (4.11), (4.7) and (4.10) we canobtain a positive constant e C such that, for any k ∈ N Z B kj (cid:12)(cid:12)(cid:2) b ( x ) − α B j ( b ) (cid:3) T ( f j ) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) (4.12) . [ ω ( B j )] κ − A kσ p µ ( B j ) p µ ( A k B j ) p Z A k +12 B j (cid:12)(cid:12) b ( x ) − α B j ( b ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) . [ ω ( B j )] κ − A kσ p µ ( B j ) p µ ( A k B j ) p − " µ ( A k +12 B j ) Z A k +12 B j | b ( x ) − α B j ( b ) | p (1+ ǫ ) ′ dµ ( x ) ǫ ) ′ × " µ ( A k +12 B j ) Z A k +12 B j ω ( x ) ǫ dµ ( x ) ǫ ≤ e C k p A kσ p µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) . Using Lemma 4.1, (4.5), (4.6), (2.3), (4.1) and (1.6) for any x ∈ B kj , we get that (cid:12)(cid:12) T (cid:0)(cid:2) b − α B j ( b ) (cid:3) f j (cid:1) ( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B j, ∪ B j, K ( x, ξ ) (cid:2) b ( ξ ) − α B j ( b ) (cid:3) f j ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & Z B j, ∪ B j, (cid:12)(cid:12)(cid:2) b ( ξ ) − α B j ( b ) (cid:3) f j ( ξ ) (cid:12)(cid:12) µ ( B ( x, d ( x, ξ ))) dµ ( ξ ) & µ ( A k B j ) [ ω ( B j )] κ − p Z B j (cid:12)(cid:12) b ( ξ ) − α B j ( b ) (cid:12)(cid:12) dµ ( ξ ) & γµ ( B j ) µ ( A k B j ) [ ω ( B j )] κ − p . Along with (4.8) we deduce that there exists a positive constant e C such that Z B kj (cid:12)(cid:12) T (cid:0)(cid:2) b − α B j ( b ) (cid:3) f j (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) ≥ γ p µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) B kj (cid:17) (4.13) ≥ e C γ p µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) . Now let us take K ∈ (0 , ∞ ) large enough such that, for any integer k ≥ K ˜ C γ p p − − e C k p A kσ p ≥ e C γ p p . oundedness and compactness of commutators Z B kj | [ b, T ] f j ( x ) | p ω ( x ) dµ ( x ) ≥ p − Z B kj (cid:12)(cid:12) T (cid:0)(cid:2) b − α B j ( b ) (cid:3) f j (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) − Z B kj (cid:12)(cid:12)(cid:2) b ( x ) − α B j ( b ) (cid:3) T ( f j ) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) ≥ ˜ C γ p p − − e C k p A kσ p ! µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) ≥ e C γ p p µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) . This implies (4.2).Also, since supp ( f j ) ⊂ B j , by (4.6) and (2.3) and k b k BMO( X ) = 1, we deduve that, for any x ∈ A k +12 B j \ A k B j (cid:12)(cid:12) T (cid:0)(cid:2) b − α B j ( b ) (cid:3) f j (cid:1) ( x ) (cid:12)(cid:12) . [ ω ( B j )] κ − p Z B j (cid:12)(cid:12) b ( ξ ) − α B j ( b ) (cid:12)(cid:12) V ( x, ξ ) dµ ( ξ ) . [ ω ( B j )] κ − p µ ( B j ) µ ( A k B j ) . Therefore, by (4.12) with B kj replaced by A k +12 B j \ A k B j , we can deduce that, for any integer k ≥ K Z A k +12 B j \ A k B j | [ b, T ] f j ( x ) | p ω ( x ) dµ ( x ) . Z A k +12 B j \ A k B j (cid:12)(cid:12) T (cid:0)(cid:2) b − α B j ( b ) (cid:3) f j (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x )+ Z A k +12 B j \ A k B j (cid:12)(cid:12)(cid:2) b ( x ) − α B j ( b ) (cid:3) T ( f j ) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) . µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) + k p A kσ p µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) . µ ( B j ) p µ ( A k B j ) p [ ω ( B j )] κ − ω (cid:16) A k B j (cid:17) . This completes the proof of Lemma 4.4.The following technical result are needed to handle the weighted estimate for showing thenecessity of the compactness of the commutators. Lemma 4.5. Let < p < ∞ , < κ < , ω ∈ A p ( X ) , b ∈ BMO ( X ) , γ, K > , { f j } j ∈ N and { B j } j ∈ N be as given in Lemma 4.4. Now assume that { B j } j ∈ N := { B ( x j , r j ) } j ∈ N also satisfiesthe following two conditions:(i) ∀ ℓ, m ∈ N and ℓ = m A C B ℓ \ A C B m = ∅ , (4.14) where C := A K > C := A K for some K ∈ N large enough. R, Gong, J. Li, E. Pozzi, M. N. Vempati (ii) { r j } j ∈ N is either non-increasing or non-decreasing in j , or there exist positive constants C min and C max such that, for any j ∈ N C min ≤ r j ≤ C max . Then there exists a positive constant C such that, for any j, m ∈ N k [ b, T ] f j − [ b, T ] f j + m k L p,κω ( X ) ≥ C. Proof. Without loss of generality, we assume that k b k BMO( X ) = 1 and { r j } j ∈ N is non-increasing.Let { f j } j ∈ N , e C , e C be as in Lemma 4.4 associated with { B j } j ∈ N .By (4.2), (4.8), Lemma 2.6 with ω ∈ A p ( X ), we observe that, for any j ∈ N , "Z A K B j | [ b, T ] f j ( x ) | p ω ( x ) dµ ( x ) /p h ω (cid:16) A K B j (cid:17)i − κ/p (4.15) ≥ h ω (cid:16) A K B j (cid:17)i − κ/p (Z B K − j | [ b, T ] f j ( x ) | p ω ( x ) dµ ( x ) ) /p ≥ h ω (cid:16) A K B j (cid:17)i − κ/p ( e C γ p µ ( B j ) p µ ( A K − B j ) p [ ω ( B j )] κ − ω (cid:16) A K − B j (cid:17)) /p & h ω (cid:16) A K B j (cid:17)i − κ/p ( γ p [ ω ( B j )] κ A np ( K − ) /p ≥ C γA − n ( κK + K − [ ω ( B j )] − κ/p [ ω ( B j )] κ/p = C γA − n ( κK + K − holds for a positive constant C independent of γ and A . We also show that, for any j, m ∈ N , "Z A K B j | [ b, T ] f j + m ( x ) | p ω ( x ) dµ ( x ) /p h ω (cid:16) A K B j (cid:17)i − κ/p ≤ C γA − n ( κK + K − . (4.16)As supp ( f j + m ) ⊂ B j + m , from (2.3), (4.6), (4.14) and k b k BMO( X ) = 1 , it follows that, for any x ∈ A K B j (cid:12)(cid:12) T (cid:0)(cid:2) b − α B j + m ( b ) (cid:3) f j + m (cid:1) ( x ) (cid:12)(cid:12) . [ ω ( B j + m )] κ − p R B j + m | K ( x, ξ ) | (cid:12)(cid:12) b ( x ) − α B j + m ( b ) (cid:12)(cid:12) dµ ( ξ ) . [ ω ( B j + m )] κ − p µ ( B j + m ) V ( x j ,x j + m ) . So we have (Z A K B j (cid:12)(cid:12) T (cid:0)(cid:2) b − α B j + m ( b ) (cid:3) f j + m (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) ) p h ω (cid:16) A K B j (cid:17)i − κp (4.17) . [ ω ( B j + m )] κ − p µ ( B j + m ) V ( x j , x j + m ) h ω (cid:16) A K B j (cid:17)i − κp . oundedness and compactness of commutators x ∈ A K B j | T ( f j + m ) ( x ) | ≤ Z B j + m | K ( x, ξ ) − K ( x, x j + m ) | | f j + m ( ξ ) | dµ ( ξ ) . [ ω ( B j + m )] κ − p µ ( B j + m ) V ( x j , x j + m ) r σ j + m d ( x j , x j + m ) σ . (4.18)Hence, using (4.18) and the fact { r j } j ∈ N is non-increasing in j and from H¨olders and reverseH¨older inequalities we get that (Z A K B j (cid:12)(cid:12)(cid:2) b ( x ) − α B j + m ( b ) (cid:3) T ( f j + m ) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) ) /p h ω (cid:16) A K B j (cid:17)i − κ/p . [ ω ( B j + m )] κ − p µ ( B j + m ) V ( x j , x j + m ) r σ j + m d ( x j , x j + m ) σ h ω (cid:16) A K B j (cid:17)i − κ/p × "Z A K B j (cid:12)(cid:12) b ( x ) − α B j + m ( b ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) /p . [ ω ( B j + m )] κ − p µ ( B j + m ) V ( x j , x j + m ) r σ j + m d ( x j , x j + m ) σ h ω (cid:16) A K B j (cid:17)i − κp × (cid:18) log d ( x j , x j + m ) r j + m + log d ( x j , x j + m ) r j (cid:19) . [ ω ( B j + m )] κ − p µ ( B j + m ) V ( x j , x j + m ) h ω (cid:16) A K B j (cid:17)i − κp r σ j + m d ( x j , x j + m ) σ log d ( x j , x j + m ) r j + m . Observe that, for C large enough, using (4.14) we know that d ( x j , x j + m ) is also large enoughand so we have (cid:18) d ( x j , x j + m ) r j + m (cid:19) − σ log d ( x j , x j + m ) r j + m . . (4.19)Using (4.17), (4.18) and (4.19), we obtain that (Z A K B j | [ b, T ] ( f j + m ) ( x ) | p ω ( x ) dµ ( x ) ) /p h ω (cid:16) A K B j (cid:17)i − κ/p ≤ (Z A K B j (cid:12)(cid:12) T (cid:0)(cid:2) b − α B j + m ( b ) (cid:3) f j + m (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) ) /p h ω (cid:16) A K B j (cid:17)i − κ/p + (Z A K B j (cid:12)(cid:12)(cid:2) b ( x ) − α B j + m ( b ) (cid:3) T ( f j + m ) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) ) /p h ω (cid:16) A K B j (cid:17)i − κ/p . [ ω ( B j + m )] κ − p µ ( B j + m ) V ( x j , x j + m ) h ω (cid:16) A K B j (cid:17)i − κp . µ ( B j + m ) V ( x j , x j + m ) (cid:20) ω ( B ( x j , d ( x j , x j + m ))) ω ( B j + m ) (cid:21) − κp ≤ C ′ (cid:20) µ ( B j + m ) V ( x j , x j + m ) (cid:21) κ . R, Gong, J. Li, E. Pozzi, M. N. Vempati Note that lim k →∞ µ ( A k B j + m ) = ∞ . Then for C large enough, we have µ ( C B j + m ) ≥ (cid:16) C ′ C γA − n ( κK + K − (cid:17) κ µ ( B j + m ) . This implies that C ′ h µ ( B j + m ) V ( x j ,x j + m ) i κ ≤ C ′ h µ ( B j + m ) µ ( C B j + m ) i κ ≤ C γA − n ( κK + K − . This gives theproof of (4.16). Using (1.6) and (4.16) we know that, for any j, m ∈ N and C large enough (Z A K B j | [ b, T ] ( f j ) ( x ) − [ b, T ] ( f j + m ) ( x ) | p ω ( x ) dµ ( x ) ) /p h ω (cid:16) A K B j (cid:17)i − κ/p ≥ (Z A K B j | [ b, T ] ( f j ) ( x ) | p ω ( x ) dµ ( x ) ) /p h ω (cid:16) A K B j (cid:17)i − κ/p − (Z A K B j | [ b, T ] ( f j + m ) ( x ) | p ω ( x ) dµ ( g ) ) /p h ω (cid:16) A K B j (cid:17)i − κ/p ≥ C γA − n ( κK + K − . This completes the proof of Lemma 4.7. Proof of Theorem 1.3(ii). Without loss of generality, we assume that k b k BMO( X ) = 1. To prove b ∈ VMO ( X ), observe that b ∈ BMO ( X ) is a real-valued function, we will use a contradictionargument via Lemmas 2.4, 4.4 and 4.5. Now note that, if b / ∈ VMO ( X ) , then b does not satisfyat least one of (i) through (iii) of Lemma 2.4. We show that [ b, T ] is not compact on L p,κω ( X )in any of the following three cases. Case (i) b does not satisfy condition (i) Lemma 2.4. Hence there exist γ ∈ (0 , ∞ ) and asequence n B (1) j o j ∈ N := n B ( x (1) j , r (1) j ) o j ∈ N of balls in X satisfying (4.1) and that r (1) j → j → ∞ . Let x be a fixed point in X . Wenow consider the following two subcases. Subcase (i) There exists a positive constant M such that 0 ≤ d ( x , x (1) j ) < M for all x (1) j , j ∈ N . That is, x (1) j ∈ B := B ( x , M ) , ∀ j ∈ N . Let { f j } j ∈ N be associated with thesequence { B j } j ∈ N , ˜ C ˜ C , K and C be as in Lemmas 4.4 and 4.5. Let p ∈ (1 , p ) be such that ω ∈ A p ( X ) and C := A K > C = A K for K ∈ N large enough so that C := e C ˆ C γ p C µ A nK ( σ − p ) > e C ˆ C A K ( p − p )2 − A K ( p − p )2 , (4.20)where ˆ C and ˆ C are as in Lemma 2.6. As we know (cid:12)(cid:12)(cid:12) r (1) j (cid:12)(cid:12)(cid:12) → j → ∞ and n x (1) j o j ∈ N ⊂ B ,we choose a subsequence n B (1) j ℓ o ℓ ∈ N of n B (1) j o j ∈ N so that, for any j ∈ N , µ (cid:16) B (1) j ℓ +1 (cid:17) µ (cid:16) B (1) j ℓ (cid:17) < C n and ω (cid:16) B (1) j ℓ +1 (cid:17) ≤ ω (cid:16) B (1) j ℓ (cid:17) . (4.21) oundedness and compactness of commutators ℓ, m ∈ N J := C B (1) j ℓ \ C B (1) j ℓ , J := J \ C B (1) j ℓ + m and J := X \ C B (1) j ℓ + m . Observe that J ⊂ h(cid:16) C B (1) jℓ (cid:17) ∩ J i and J = J ∩ J . Hence we have (Z C B (1) jℓ (cid:12)(cid:12) [ b, T ] ( f j ℓ ) ( x ) − [ b, T ] (cid:0) f j i + m (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) ) /p (4.22) ≥ (cid:26)Z J (cid:12)(cid:12) [ b, T ] ( f j ℓ ) ( x ) − [ b, T ] (cid:0) f j ℓ + m (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) (cid:27) /p ≥ (cid:26)Z J | [ b, T ] ( f j ℓ ) ( x ) | p ω ( x ) dµ ( x ) (cid:27) /p − (cid:26)Z J (cid:12)(cid:12) [ b, T ] (cid:0) f j ℓ + m (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) (cid:27) /p = (cid:26)Z J ∩J | [ b, T ] ( f j ℓ ) ( x ) | p ω ( x ) dµ ( x ) (cid:27) /p − (cid:26)Z J (cid:12)(cid:12) [ b, T ] (cid:0) f j ℓ + m (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) (cid:27) /p =: F − F . First we consider the term F and assume that E j ℓ := J \J = ∅ . Then E j ℓ ⊂ C B (1) j ℓ + m by(4.21) we have µ ( E j ℓ ) ≤ C n µ (cid:16) B (1) j ℓ + m (cid:17) < µ (cid:16) B (1) j ℓ (cid:17) . (4.23)Now take B (1) j ℓ,k := ^ A k − B (1) j ℓ , to be the ball associates with A k − B (1) j ℓ in (3.1). Now using (4.23), we get µ (cid:16) B (1) j ℓ,k (cid:17) = µ (cid:16) A k − B (1) j ℓ (cid:17) > µ ( E j ℓ ) . Using this, we further know that there exist finite n B (1) j ℓ,k o K − k = K intersecting E j ℓ . Then, from(4.2) and Lemma 2.6, we deduce thatF p ≥ K − X k = K ,B (1) jℓ,k ∩ E jℓ = ∅ Z B (1) jℓ,k | [ b, T ] ( f j ℓ ) ( x ) | p ω ( x ) dµ ( x ) (4.24) ≥ e C γ p K − X k = K ,B (1) jℓ,k ∩ E jℓ = ∅ µ (cid:16) B (1) j ℓ (cid:17) p µ (cid:16) A k B (1) j ℓ (cid:17) p ω (cid:16) B (1) j ℓ (cid:17) κ − ω (cid:16) A k B (1) j ℓ (cid:17) ≥ K − X k = K ,B (1) jℓ,k ∩ E jℓ = ∅ e C ˆ C γ p µ (cid:16) B (1) j ℓ (cid:17) p µ (cid:16) A k B (1) j ℓ (cid:17) p A nkσ ω (cid:16) B (1) j ℓ (cid:17) κ R, Gong, J. Li, E. Pozzi, M. N. Vempati ≥ e C ˆ C γ p C µ A nK ( σ − p ) ω (cid:16) B (1) j ℓ (cid:17) κ = C ω (cid:16) B (1) j ℓ (cid:17) κ . If E j l := J \ J = ∅ , the inequality is still holds true.Observe that lim k →∞ µ ( A k B (1) j l + m ) = ∞ . Then there exist j k ∈ N such that µ ( A j B (1) j l + m ) ≥ A K µ ( A K B (1) j l + m ) and µ ( A j k +1 B (1) j l + m ) ≥ A K µ ( A j k B (1) j l + m ) . Also, using the proof of (4.3), Lemma 4.4, (4.20) and (4.21), we obtain thatF p ≤ ∞ X k =0 Z A jk +12 B (1) jl + m \ A jk B (1) jl + m | [ b, T ]( f j l + m )( x ) | p ω ( x ) dµ ( x ) (4.25) ≤ e C ∞ X k =0 µ (cid:16) B (1) j ℓ + m (cid:17) p µ (cid:16) A j k B (1) j ℓ + m (cid:17) p h ω (cid:16) B (1) j ℓ + m (cid:17)i κ − ω (cid:16) A j k B (1) j ℓ + m (cid:17) ≤ e C ∞ X k =0 µ (cid:16) B (1) j ℓ + m (cid:17) p A ( k +1) K p µ (cid:16) A K B (1) j ℓ + m (cid:17) p h ω (cid:16) B (1) j ℓ + m (cid:17)i κ − C A ( k +1) K p µ (cid:16) A K B (1) j ℓ + m (cid:17) p µ (cid:16) B (1) j ℓ + m (cid:17) p ω (cid:16) B (1) j ℓ + m (cid:17) ≤ e C ˆ C ∞ X k =0 A ( k +1) K ( p − p )2 µ (cid:16) B (1) j ℓ + m (cid:17) p − p µ (cid:16) A K B (1) j ℓ + m (cid:17) p − p h ω (cid:16) B (1) j ℓ + m (cid:17)i κ ≤ e C ˆ C A K ( p − p )2 − A K ( p − p )2 h ω (cid:16) B (1) j ℓ + m (cid:17)i κ . By (4.21), (4.22),(4.24) and (4.25) we deduce (cid:26)R C B (1) jℓ (cid:12)(cid:12) [ b, T ] ( f j ℓ ) ( x ) − [ b, T ] (cid:0) f j ℓ + m (cid:1) ( x ) (cid:12)(cid:12) p ω ( x ) dµ ( x ) (cid:27) /p ≥ C /p h ω (cid:16) B (1) j ℓ (cid:17)i κ/p − (cid:0) C (cid:1) /p h ω (cid:16) B (1) j ℓ (cid:17)i κ/p & h ω (cid:16) B (1) j ℓ (cid:17)i κ/p . Hence we get, { [ b, T ] f j } j ∈ N is not relatively compact in L p,κω ( X ) , which implies that [ b, T ] isnot compact on L p,κω ( X ). So, b satisfies condition (i) of Lemma 2.4. Subcase (ii) There exists a subsequence n B (1) j e o ℓ ∈ N := n B (cid:16) x (1) j ℓ , r (1) j ℓ (cid:17)o ℓ ∈ N of n B (1) j o j ∈ N such that d ( x , x (1) j ℓ ) → ∞ as ℓ → ∞ . In this subcase, by µ (cid:16) B (1) jℓ (cid:17) → ℓ → ∞ , we take amutually disjoint subsequence of n B (1) j ℓ o ℓ ∈ N , and denote by n B (1) j i o ℓ ∈ N , satisfying (4.14) as well.This, via Lemma 4.5 implies that [ b, T ] is not compact on L p,κω ( X ) , which is a contradictionto our assumption. Hence, b satisfies condition (i) of Lemma 2.4. Case (ii) If b does not satisfy condition (ii) of Lemma 2.4. In this case, there exist γ ∈ (0 , ∞ ) and a sequence n B (2) j o j ∈ N of balls in X satisfying (4.1) and that | r B (2) j | → ∞ as j → ∞ .We also consider the following two subcases as well. oundedness and compactness of commutators Subcase (i) There exists an infinite subsequence n B (2) jℓ o ℓ ∈ N of n B (2) j o j ∈ N and a point x ∈ X such that, for any ℓ ∈ N , x ∈ A C B (2) j ℓ . As | r B (2) jℓ | → ∞ as ℓ → ∞ , it follows thatthere exists a subsequence, denoted as earlier by n B (2) jℓ o ℓ ∈ N , such that, for any ℓ ∈ N µ (cid:16) B (2) j e (cid:17) µ (cid:16) B (2) j ℓ +1 (cid:17) < C n . (4.26)Note that 2 A C B (2) j ℓ ⊂ A C B (2) j ℓ +1 for any j ℓ ∈ N and hence ω (cid:16) A C B (2) j ℓ +1 (cid:17) ≥ ω (cid:16) A C B (2) j ℓ (cid:17) , M ( b, A C B j ℓ ) > γ A C . (4.27)Using similar method as that used in Subcase (i) of Case (i) and we redefine our sets in areversed order. That is, for any fixed ℓ, k ∈ N , we let e J := 2 A C C B (2) ℓ + k \ A C C B (2) ℓ + k , e J := e J \ A C C B (2) j l , e J := X \ A C C B (2) j ℓ . As in Case (i), by Lemma 4.4, (4.26) and (4.27), we deduce that the commutator [ b, T ] is notcompact on L p,κω ( X ) . This contradiction gives that b satisfies condition (ii) of Lemma 4.4. Subcase (ii) For any z ∈ X the number of n A C B (2) j o j ∈ N containing z is finite. Inthis subcase, for each square B (2) j ∈ n B (2) j o j ∈ N , the number of n A C B (2) j o j ∈ N intersecting A C B (2) j is finite. Then we take a mutually disjoint subsequence n B (2) j ℓ o ℓ ∈ N satisfying (4.1)and (4.14). From Lemma 4.5, we can deduce that [ b, T ] is not compact on L p,κω ( X ). Thus, b satisfies condition (ii) of Lemma 2.4. Case (iii) Condition (iii) of Lemma 2.4 does not hold for b . Then there exists γ > r > 0, there exists B ⊂ X \ B ( x , r ) with M ( b, B ) > γ . As in [4] for the γ above,there exists a sequence n B (3) j o j of balls such that for any j , M (cid:16) b, B (3) j (cid:17) > γ, (4.28)and for any i = m , γ B (3) i ∩ γ B (3) m = ∅ , (4.29)for sufficiently large γ since, by Case (i) and (ii), n B (3) j o j ∈ N satisfies the conditions (i) and(ii) of Lemma 2.4, it follows that there exist positive constants C min and C max such that C min ≤ r j ≤ C max , ∀ j ∈ N . Using this and Lemma 4.5 we deduce that, if [ b, T ] is compact on L p,κω ( X ) , then b also satisfiescondition (iii) of Lemma 2.4. This completes the proof of Theorem 1.3(ii) and hence of Theorem1.3.4 R, Gong, J. Li, E. Pozzi, M. N. Vempati VMO( X ) In this section, we provide the characterisation of VMO space on X by giving the proof ofLemma 2.4. Proof of Lemma 2.4. In the following, for any integer m , we use B m to denote the ball B ( x , m ),where x is a fixed point in X . Necessary condition: Assume that f ∈ VMO( X ). If f ∈ Lip c ( β ), then (i)-(iii) hold. Infact, by the uniform continuity, f satisfies (i). Since f ∈ L ( X ), f satisfies (ii). By the factthat f is compactly supported, f satisfies (iii). If f ∈ VMO( X ) \ Lip c ( β ), by definition, for anygiven ε > 0, there exists f ε ∈ Lip c ( β ) such that k f − f ε k BMO( X ) < ε . Since f ε satisfies (i)-(iii),by the triangle inequality of BMO( X ) norm, we can see (i)-(iii) hold for f . Sufficient condition: In this proof for j = 1 , , · · · , 8, the value α j is a positive constantdepending only on n and α i for 1 ≤ i < j . Assume that f ∈ BMO( X ) and satisfies (i)-(iii).To prove that f ∈ VMO( X ), it suffices to show that there exist positive constants α , α suchthat, for any ε > 0, there exists φ ε ∈ BMO( X ) satisfyinginf h ∈ Lip c ( β ) k φ ε − h k BMO( X ) < α ε, (5.1)and k φ ε − f k BMO( X ) < α ε. (5.2)By (i), there exist i ε ∈ N such thatsup (cid:8) M ( f, B ) : r B ≤ − i ε +4 (cid:9) < ε. (5.3)By (iii), there exists j ε ∈ N such thatsup (cid:8) M ( f, B ) : B ∩ B j ε = ∅ (cid:9) < ε. (5.4)We first establish a cover of X . Observe that B j ε = B − i ε [ jε + iε − [ ν =1 B (cid:0) x , ( ν + 1)2 − i ε (cid:1) \ B (cid:0) x , ν − i ε (cid:1) =: jε + iε − [ ν =0 R j ε ν, − i ε For m > j ε , B m \ B m − = jε + iε − − [ ν =0 B (cid:0) x , m − + ( ν + 1)2 m − j ε − i ε (cid:1) \ B (cid:0) x , m − + ν m − j ε − i ε (cid:1) =: jε + iε − − [ ν =0 R mν,m − j ε − i ε . For each R j ε ν, − i ε , ν = 1 , , · · · , j ε + i ε − 1, let ˜ B j ε ν, − i ε be an open cover of R j ε ν, − i ε consistingof open balls with radius 2 − i ε and center on the sphere S ( x , ( ν + 2 − )2 − i ε ). Let B j ε , − i ε = { B ( x , − i ε ) } and B j ε ν, − i ε be the finite subcover of ˜ B j ε ν, − i ε . Similarly, for each m > j ε and ν = 0 , , · · · , j ε + i ε − − 1, let B mν,m − j ε − i ε be the finite cover of R mν,m − j ε − i ε consisting of openballs with radius 2 m − j ε − i ε and center on the sphere S ( x , (2 m − + ( ν + 2 − )2 m − j ε − i ε ). oundedness and compactness of commutators B x as follows. If x ∈ B j ε , then there is ν ∈ { , , · · · , j ε + i ε − } such that x ∈ R j ε ν, − i ε , let B x be a ball in B j ε ν, − i ε that contains x . If x ∈ B m \ B m − , m > j ε , then thereis ν ∈ { , , · · · , j ε + i ε − − } such that x ∈ R mν,m − j ε − i ε , let B x be a ball in B mν,m − j ε − i ε thatcontains x . We can see that if B x ∩ B x ′ = ∅ , theneither r B x ≤ r B x ′ or r B x ′ ≤ r B x . (5.5)In fact, if r B x > r B x ′ , then there is m ∈ N such that x ∈ B m +2 \ B m +1 and x ′ ∈ B m , thus d ( x, x ′ ) ≥ d ( x , x ) − d ( x , x ′ ) ≥ m +1 − m > m +2 − j ε − i ε + 2 m − j ε − i ε = r B x + r B x ′ , which is contradict to the fact that B x ∩ B x ′ = ∅ (Without loss of generality, here we assumethat A = 1 in the quasi-triangle inequality. Otherwise, we just need to take r B m = ([2 A ]+1) m and make some modifications).Now we define φ ε . By (ii), there exists m ε > j ε large enough such that when r B > m ε − i ε − j ε ,we have M ( f, B ) < n ( − i ε − j ε − − ε. (5.6)Define φ ε ( g ) = ( f B x , if x ∈ B m ε ,f B mε \ B mε − , if x ∈ X \ B m ε . We claim that there exists a positive constant α , α such that if B x ∩ B x ′ = ∅ or x, x ′ ∈ X \ B m ε − , then (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) < α ε. (5.7)And if 2 B x ∩ B x ′ = ∅ , then for any x ∈ B x , x ∈ B x ′ , we have | φ ε ( x ) − φ ε ( x ) | < α ε. (5.8)Assume (5.7) and (5.8) at the moment, we now continue to prove the sufficiency of Lemma 2.4.Now we show (5.1). Let ˜ h ε ( x ) := φ ε ( x ) − f B mε \ B mε − . By definition of φ ε , we can see that˜ h ε ( x ) = 0 for x ∈ X \ B m ε and k ˜ h ε − φ ε k BMO( X ) = 0 . Observe that supp (˜ h ε ) ⊂ B m ε and there exists a function h ε ∈ C c ( X ) such that for any x ∈ X , | ˜ h ε ( x ) − h ε ( x ) | < ε. Let η ( s ) be an infinitely differentiable function defined on [0 , ∞ )such that 0 ≤ η ( s ) ≤ , η ( s ) = 1 for 0 ≤ s ≤ η ( s ) = 0 for s ≥ 2. And let ρ ( x, y, t ) = (cid:16) Z X η ( d ( x, z ) /t ) dµ ( z ) (cid:17) − η ( d ( x, z ) /t )and h tε ( x ) = Z X ρ ( x, y, t ) h ε ( y ) dµ ( y ) . Then by [28, Lemmas 3.15 and 3.23], h tε ( x ) approaches to h ε ( x ) uniformly for x ∈ X as t goesto 0 and h tε ∈ Lip c ( β ) for β > 0. Since k h tε − φ ε k BMO( X ) ≤ k h tε − h ε k BMO( X ) + k h ε − ˜ h ε k BMO( X ) + k ˜ h ε − φ ε k BMO( X ) ≤ k h tε − h ε k BMO( X ) + 2 ε, R, Gong, J. Li, E. Pozzi, M. N. Vempati we can obtain (5.1) by letting t go to 0 and by taking α = 2.Now we show (5.2). To this end, we only need to prove that for any ball B ⊂ X , M ( f − φ ε , B ) < α ε. We first prove that for every B x with x ∈ B m ε , Z B x (cid:12)(cid:12) f ( x ′ ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) ≤ α εµ ( B x ) . (5.9)In fact, Z B x (cid:12)(cid:12) f ( x ′ ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) = Z B x ∩ B mǫ | f ( x ′ ) − f B x ′ | dµ ( x ′ ) + Z B x ∩ ( X \ B mǫ ) | f ( x ′ ) − f B mε \ B mε − | dµ ( x ′ ) . When x ∈ B ( x , m ε − m ε − i ε − j ε ), then B x ⊂ B m ǫ , thus Z B x (cid:12)(cid:12) f ( x ′ ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) = Z B x | f ( x ′ ) − f B x ′ | dµ ( x ′ ) ≤ Z B x | f ( x ′ ) − f B x | dµ ( x ′ ) + Z B x | f B x − f B x ′ | dµ ( x ′ )= µ ( B x ) M ( f, B x ) + Z B x | f B x − f B x ′ | dµ ( x ′ ) . Note that if x ′ ∈ B x , then B x ∩ B x ′ = ∅ . Therefore, If B x ∩ B j ε = ∅ , by (5.4) and (5.7), wehave Z B x (cid:12)(cid:12) f ( x ′ ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) < ( ε + α ε ) µ ( B x ) . If B x ∩ B j ε = ∅ , then r B x ≤ − i ε +1 , then by (5.3) and (5.7), Z B x (cid:12)(cid:12) f ( x ′ ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) < ( ε + α ε ) µ ( B x ) . When x ∈ B m ε \ B ( x , m ε − m ε − j ε − i ε ), it is clear that B x ∩ B j ε = ∅ , then by (5.4), (5.6)and (5.7), we have Z B x (cid:12)(cid:12) f ( x ′ ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) ≤ Z B x ∩ B mǫ | f ( x ′ ) − f B x | dµ ( x ′ ) + Z B x ∩ B mǫ | f B x − f B x ′ | dµ ( x ′ )+ Z B x ∩ ( X \ B mǫ ) | f ( x ′ ) − f B mε +1 | dµ ( x ′ ) + Z B x ∩ ( X \ B mǫ ) | f B mε +1 − f B mε \ B mε − | dµ ( x ′ ) ≤ µ ( B x ) M ( f, B x ) + α εµ ( B x ) + µ ( B m ε +1 ) M ( f, B m ε +1 ) + µ ( B m ε +1 ) µ ( B x ) µ ( B m ε \ B m ε − ) M ( f, B m ε +1 ) < ( C ε + α ε ) µ ( B x ) . Then (5.9) holds by taking α = ( C + α ).Let B be an arbitrary ball in X , then M ( f − φ ε , B ) ≤ M ( f, B ) + M ( φ ε , B ) . If B ⊂ B m ε and max { r B x : B x ∩ B = ∅} > r B , thenmin { r B x : B x ∩ B = ∅} > r B . (5.10) oundedness and compactness of commutators r B b x = max { r B x : B x ∩ B = ∅} and b x ∈ B l \ B l − for some l ∈ Z .Then B ⊂ B l ∩ B b x . If l ≤ j ε , then (5.10) holds. If l > j ε , then r B b x = 2 l − j ε − i ε , and r B < r B b x = 2 l − j ε − i ε − . Since for any x ′ ∈ B b x , d ( x , x ′ ) ≥ d ( x , b x ) − d ( b x, x ′ ) ≥ l − − 32 2 l − j ε − i ε > l − − l − j ε − i ε +1 , we have dist( x , B b x ) := inf x ′ ∈ B b x d ( x , x ′ ) > l − − l − j ε − i ε +1 . Thus B ⊂ B l \ B l − . Therefore, if B x ∩ B = ∅ , then x ∈ B l \ B l − , which implies that r B x ≥ l − − j ε − i ε > r B . From (5.10) we can see that if B x i ∩ B = ∅ and B x j ∩ B = ∅ , then 2 B x i ∩ B x j = ∅ . Thenby (5.8), we can get M ( φ ε , B ) ≤ µ ( B ) Z B µ ( B ) Z B (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) dµ ( x )= 1 µ ( B ) X i : B xi ∩ B = ∅ Z B xi ∩ B X j : B xj ∩ B = ∅ Z B xj ∩ B (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) dµ ( x ) < α ε µ ( B ) X i : B xi ∩ B = ∅ µ ( B x i ∩ B ) X i : B xj ∩ B = ∅ µ ( B x j ∩ B ) < α α ε. Moreover, if B ∩ B j ǫ = ∅ , then by (5.10), r B < − i ε , thus by (5.3), we have M ( f, B ) < ε. If B ∩ B j ǫ = ∅ , then by (5.4), M ( f, B ) < ε. Consequently, M ( f − φ ε , B ) ≤ M ( f, B ) + M ( φ ε , B ) < (cid:0) α α (cid:1) ε. If B ⊂ B m ε and max { r B x : B x ∩ B = ∅} ≤ r B , since the number of B x with x ∈ B m ε thatcovers B is bounded by α , by (5.9), we have M ( f − φ ε , B ) ≤ µ ( B ) Z B | f ( x ) − φ ε ( x ) | dµ ( x ) ≤ µ ( B ) X i : B xi ∩ B = ∅ Z B xi | f ( x ) − φ ε ( x ) | dµ ( x ) ≤ µ ( B ) α ε X i : B xi ∩ B = ∅ µ ( B x i ) ≤ µ ( B ) α α εµ (8 B ) ≤ C α α ε. If B ⊂ X \ B m ε − , then B ∩ B j ε = ∅ , from (5.4) we can see M ( f, B ) < ε . By (5.7), M ( φ ε , B ) ≤ µ ( B ) Z B µ ( B ) Z B (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) dµ ( x ′ ) dµ ( x ) < α ε. Therefore, M ( f − φ ε , B ) ≤ M ( f, B ) + M ( φ ε , B ) < (1 + α ) ε. R, Gong, J. Li, E. Pozzi, M. N. Vempati If B ∩ ( X \ B m ε ) = ∅ and B ∩ B m ε − = ∅ . Let p B be the smallest integer such that B ⊂ B p B ,then p B > m ε . If p B = m ε + 1, then r B > (2 m ε − m ε − ) = 2 m ε − . If p B > m ε + 1, then r B > (2 p B − − m ε − ). Thus µ ( B p B ) µ ( B ) ≤ C . Therefore, M ( f − φ ε , B ) ≤ µ ( B ) Z B | f ( x ) − φ ε ( x ) − ( f − φ ε ) B pB | dµ ( x ) + | ( f − φ ε ) B pB − ( f − φ ε ) B |≤ µ ( B p B ) µ ( B ) 1 µ ( B p B ) Z B pB | f ( x ) − φ ε ( x ) − ( f − φ ε ) B pB | dµ ( x ) ≤ C ( M ( f, B p B ) + M ( φ ε , B p B )) ≤ C ( ε + M ( φ ε , B p B )) , where the last inequality comes from (5.6). By definition, M ( φ ε , B p B ) ≤ µ ( B p B ) Z B pB (cid:12)(cid:12) φ ε ( x ) − ( φ ε ) B pB \ B mε (cid:12)(cid:12) dµ ( x ) + (cid:12)(cid:12) ( φ ε ) B pB \ B mε − ( φ ε ) B pB (cid:12)(cid:12) ≤ µ ( B p B ) Z B pB (cid:12)(cid:12) φ ε ( x ) − ( φ ε ) B pB \ B mε (cid:12)(cid:12) dµ ( x ) . By (5.4), (5.9) and the fact that φ ε ( x ) = f B mε \ B mε − if x ∈ X \ B m ε , we have Z B pB (cid:12)(cid:12) φ ε ( x ) − ( φ ε ) B pB \ B mε (cid:12)(cid:12) dµ ( x ) ≤ Z B pB µ ( B p B \ B m ε ) Z B pB \ B mε | φ ε ( x ) − φ ε ( x ′ ) | dµ ( x ′ ) dµ ( x )= Z B mε | φ ε ( x ) − f B mε \ B mε − | dµ ( x ) ≤ Z B mε | φ ε ( x ) − f ( x ) | dµ ( x ) + Z B mε | f ( x ) − f B mε | dx + µ ( B m ε ) | f B mε − f B mε \ B mε − |≤ X i : B xi ∩ B mε = ∅ ,x i ∈ B mε Z B xi | φ ε ( x ) − f ( x ) | dµ ( x ) + (cid:18) µ ( B m ε ) + µ ( B m ε ) µ ( B m ε \ B m ε − ) (cid:19) M ( f, B m ε ) < α ε X i : B xi ∩ B mε = ∅ ,x i ∈ B mε µ ( B x i ) + 3 εµ ( B m ε ) < ( α α + 3) εµ ( B m ε ) . Therefore, M ( f − φ ε , B ) ≤ C ( ε + M ( φ ε , B p B )) ≤ C (cid:18) ε + 2 µ ( B m ε ) µ ( B p B ) ( α α + 3) ε (cid:19) < C ( α α + 3) ε. Then (5.2) holds by taking α = max { α α , α , C ( α α + 3) } . This finishes the proofof Lemma 2.4. Proof of (5.7) : We first claim that sup (cid:8)(cid:12)(cid:12) f B x − f B x ′ (cid:12)(cid:12) : x, x ′ ∈ B m ε \ B m ε − (cid:9) < C ε. (5.11)By (5.6), for any x ∈ B m ε \ B m ε − , we have | f B x − f B mε +1 | ≤ µ ( B m ε +1 ) µ ( B x ) 1 µ ( B m ε +1 ) Z B mε +1 (cid:12)(cid:12) f ( x ′ ) − f B mε +1 (cid:12)(cid:12) dµ ( x ′ ) oundedness and compactness of commutators µ ( B m ε +1 ) µ ( B x ) M ( f, B m ε +1 ) < C ε. Similarly, for any x ′ ∈ B m ε \ B m ε − , | f B x ′ − f B mε +1 | < C ε. Consequently, (5.11) holds.For the case x, x ′ ∈ X \ B m ε − , firstly, if x, x ′ ∈ X \ B m ε , then by definition (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) = 0 . Secondly, if x, x ′ ∈ B m ε \ B m ε − , then by (5.11), we have (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) < C ε. Thirdly, without loss of generality, we may assume that x ∈ B m ε \ B m ε − and x ′ ∈ X \ B m ε ,then by (5.6), we have (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) = (cid:12)(cid:12) f B x − f B mε \ B mε − (cid:12)(cid:12) ≤ | f B x − f B mε +1 | + (cid:12)(cid:12) f B mε +1 − f B mε \ B mε − (cid:12)(cid:12) ≤ µ ( B m ε +1 ) µ ( B x ) M ( f, B m ε +1 ) + µ ( B m ε +1 ) µ ( B m ε \ B m ε − ) M ( f, B m ε +1 ) ≤ (cid:18) µ ( B m ε +1 ) µ ( B x ) + µ ( B m ε +1 ) µ ( B m ε \ B m ε − ) (cid:19) M ( f, B m ε +1 ) < C ε. For the case B x ∩ B x ′ = ∅ and x, x ′ ∈ B m ε − , we may assume B x = B x ′ and r B x ≤ r B x ′ . By(5.5), B x ′ ⊂ B x ⊂ B x ′ . If x ′ ∈ B j ε +1 , then by (5.3), we have (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) = (cid:12)(cid:12) f B x − f B x ′ (cid:12)(cid:12) ≤ (cid:12)(cid:12) f B x − f B x ′ (cid:12)(cid:12) + (cid:12)(cid:12) f B x ′ − f B x ′ (cid:12)(cid:12) ≤ (cid:18) µ (3 B x ′ ) µ ( B x ) + µ (3 B x ′ ) µ ( B x ′ ) (cid:19) M ( f, B x ′ ) ≤ C ε. If x ′ / ∈ B j ε +1 , then 3 B x ′ ∩ B j ε = ∅ , by (5.4), we have (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) ≤ C M ( f, B x ′ ) ≤ C ε. Therefore, (5.7) holds by taking α = max { C , C , C } . Proof of (5.8) : Since x ∈ B x , x ∈ B x ′ , we have B x ∩ B x = ∅ and B x ∩ B x ′ = ∅ , by (5.7), | φ ε ( x ) − φ ε ( x ) | ≤ | φ ε ( x ) − φ ε ( x ) | + (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) + (cid:12)(cid:12) φ ε ( x ′ ) − φ ε ( x ) (cid:12)(cid:12) ≤ α ε + (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) . We may assume B x = B x ′ and r B x ≤ r B x ′ . If x, x ′ ∈ X \ B m ε − , then (5.8) follows from (5.7). If x, x ′ ∈ B m ε − , when x ′ ∈ B j ε +1 , then 2 − i ε ≤ r B x ≤ r B x ′ ≤ − i ε +1 , thus B x ′ ⊂ B x ⊂ B x ′ ,by (5.3), we have (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) f B x − f B x ′ (cid:12)(cid:12) + (cid:12)(cid:12) f B x ′ − f B x ′ (cid:12)(cid:12) = (cid:18) µ (6 B x ′ ) µ ( B x ) + µ (6 B x ′ ) µ ( B x ′ ) (cid:19) M ( f, B x ′ ) ≤ C ε. R, Gong, J. Li, E. Pozzi, M. N. Vempati When x ′ / ∈ B j ε +1 , then there exist ˜ m ∈ N and ˜ m ≥ j ε + 2 such that x ′ ∈ B ˜ m \ B ˜ m − . Since2 B x ∩ B x ′ = ∅ , we have B x ⊂ B x ′ . Note that 6 B x ′ ∩ B ˜ m − = ∅ , (in fact, for any ˜ x ∈ B x ′ , d ( x , ˜ x ) ≥ d ( x , x ′ ) − d ( x ′ , ˜ x ) ≥ ˜ m − − · ˜ m − j ε − i ε > ˜ m − ), thus B x ∩ B ˜ m − = ∅ and then r B x ′ = 2 ˜ m − − j ε − i ε ≤ r B x ≤ ˜ m − j ε − i ε = r B x ′ . Therefore, B x ′ ⊂ B x . Then by (5.4), wehave (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) ≤ C M ( f, B x ′ ) < C ε. If x ∈ B m ε − and x ′ ∈ X \ B m ε − , since 2 B x ∩ B x ′ = ∅ , by the construction of B x we can seethat x ∈ B m ε − \ B m ε − and x ′ ∈ B m ε \ B m ε − . Thus, B x ′ ⊂ B x ⊂ B x ′ . Then by (5.6),we have (cid:12)(cid:12) φ ε ( x ) − φ ε ( x ′ ) (cid:12)(cid:12) < C M ( f, B x ′ ) < C ε. Taking α = C + C + 2 α , then (5.8) holds. Acknowledgement: R.M. Gong is supported by the State Scholarship Fund of China (No.201908440061). J. Li is supported by ARC DP 170101060. References [1] D. R. Adams and J. Xiao. Morrey spaces in harmonic analysis, Ark Mat., Math. Nachr. , 185 (1997), 5–20.[3] S. Bloom, A commutator theorem and weighted BMO, Trans. Amer. Math. Soc., J. Funct. Anal. , 277 (6) (2019), 1639–1676.[5] Y. Chen, Y. Ding and X. Wang, Compactness of commutators for singular integrals onMorrey spaces, Canad. J. Math. , 64 (2012), 257–281.[6] R. E. Castillo, J. C. Ramos Fern andez, and E. Trousselot, Functions of bounded ( φ, p )mean oscillation, Proyecciones , 27 (2008), 163–177.[7] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains es-paces homog`enes. ´Etude de certaines int´egrales singuli`eres , Lecture Notes in Math. 242,Springer-Verlag, Berlin, 1971.[8] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull.Amer. Math. Soc. , 83 (1977), 569–645.[9] R. Coifman, P.L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardysapces, J. Math. Pures Appl. , 72 (1993), 247–286.[10] R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces inseveral variables, Ann. of Math., (2) 103 (1976), 611–635.[11] D. G. Deng and Y. S. Han, Harmonic analysis on spaces of homogeneous type, with apreface by Yves Meyer, Lecture Notes in Math. 1966, Springer-Verlag, Berlin, 2009. oundedness and compactness of commutators J.Geom. Anal., arXiv:1809.07942v1.[13] X. T. Duong, H.-Q. Li, J. Li and B. D. Wick, Lower bound for Riesz transform kernels andcommutator theorems on stratified nilpotent Lie groups, J. Math. Pures Appl.(9) , 124(2019), 273–299.[14] X. T. Duong and L. Yan, New function spaces of BMO type, the John-Nirenberg inequal-ity, interpolation, and applications, Communications on Pure and Applied Mathematics ,58(2005), 1375–1420.[15] D. S. Fan, S. Z. Lu and D. C. Yang. Boundedness of operators in Morrey spaces onhomogeneous spaces and its applications, Acta Math Sinica (N.S.), Boll. Un. Mat. Ital. A , 5 (7) (1991), 323–332.[17] Z. Fu, R. Gong, E. Pozzi and Q. Wu, Cauchy–Szeg¨o Commutators on Weighted MorreySpaces, arXiv:2006.10546.[18] W. Guo, J. Lian and H. Wu, The unified theory for the necessity of bounded commutatorsand applications, to appear in J. Geom. Anal., arXiv:1709.008279v1.[19] I. Holmes, M. Lacey and B. D. Wick, Commutators in the two-weight setting, Math. Ann. ,367 (2017), 51–80.[20] I. Holmes, S. Petermichl and B. D. Wick, Weighted little bmo and two-weight inequalitiesfor Journ´e commutators, Anal & PDE , 11 (2018), 1693–1740.[21] T. Hyt¨onen, The sharp weighted bound for general Calder´on-Zygmund operators, Ann. ofMath., (2) 175 (2012), 1473–1506.[22] T. Hyt¨onen, The L p → L q boundedness of commutators with applications to the Jacobianoperator, arXiv:1804.11167.[23] G. Hu, X. Shi and Q. Zhang, Weighted norm inequalities for the maximal singular integraloperators on spaces of homogeneous type, J. Math. Anal. Appl. , 336 (1) (2007), 1–17.[24] A.K. Lerner, S. Ombrosi and I.P. Rivera-R´ıos, On pointwise and weighted estimates forcommutators of Calder´on-Zygmund operators, Adv. Math., 319 (2017), 153–181.[25] A.K. Lerner, S. Ombrosi and I.P. Rivera-R´ıos, Commutators of singular integrals revisited, Bull. London Math. Soc. , 51 (2019), 107–119.[26] M. Kronz, Some function spaces on spaces of homogeneous type, Manuscripta Math. , 106(2) (2001), 219–248.[27] Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math.Nachr. , 282 (2009), 219–231.[28] R. A. Mac´ıas and C. Segovia, A decomposition into atoms of distributions on spaces ofhomogeneous type, Adv. Math. , 33 (1979), 271–309.2 R, Gong, J. Li, E. Pozzi, M. N. Vempati [29] S. Mao, L. Sun and H. Wu, Boundedness and compactness for commutators of bilinearFourier multipliers, Acta Math. Sinica (Chin. Ser.) , 59 (2016), 317–334.[30] S. Mao, H. Wu and D. Y. Yang. Boundedness and compactness characterizations of Riesztransform commutators on Morrey spaces in the Bessel setting, Anal Appl., Math Meth Appl Sci., Potential Anal , (2020), doi:10.1007/s11118-019-09814-7.[33] A. Uchiyama, On the compactness of operators of Hankel type, Tˆohoku Math. J. , 30(1978), 163–171.[34] T. C. Anderson and W. Damin, Calder´on-Zygmund operators and commutators inspaces of homogeneous type: weighted inequalities, to appear in J. of Math Inequalities ,arXiv:1401.2061v1.Ruming Gong, School of Mathematical Sciences, Guangzhou University, Guangzhou, China. E-mail : [email protected] Ji Li, Department of Mathematics, Macquarie University, NSW, 2109, Australia. E-mail : [email protected] Elodie Pozzi, Department of Mathematics and Statistics, Saint Louis University, 220 N.Grand Blvd, 63103 St Louis MO, USA. E-mail : [email protected] Manasa N. Vempati, Department of Mathematics, Washington University – St. Louis, OneBrookings Drive, St. Louis, MO USA 63130-4899 E-mail ::