On weighted compactness of commutators of Schrödinger operators
aa r X i v : . [ m a t h . C A ] F e b ON WEIGHTED COMPACTNESS OF COMMUTATORS OFSCHR ¨ODINGER OPERATORS
QIANJUN HE AND PENGTAO LI
Abstract.
Let L = − ∆ + V ( x ) be a Schr¨odinger operator, where ∆ is theLaplacian operator on R d ( d ≥ V ( x ) be-longs to the reverse H¨older class B q , q > d/
2. In this paper, we study weightedcompactness of commutators of some Schr¨odinger operators, which includeRiesz transforms, standard Calder´on-Zygmund operatos and Littlewood-Paleyfunctions. These results generalize substantially some well-know results. Introduction
Let us consider the Schr¨odinger differential operators L = − ∆ + V ( x ) on R d with d ≥
3, where ∆ is the Laplace operator on R d , and the potential V ≥ B q )for some q > d/ (cid:18) | B | Z B V ( y ) q (cid:19) /q ≤ C | B | Z B V ( y ) dy (1.1)for any ball B ⊂ R d . The general theory of semigroup, in particlar Yosida’s gen-erating theorm [54], implies that L is the infinitesimal generator of semigroups,formally denoted by T t = e − t L , that solves the diffusion problem ∂∂t u ( x, t ) = L u ( x, t ) , ( x, t ) ∈ R d × [0 , ∞ ); u ( x,
0) = f ( x ) , x ∈ R d , by setting u ( x, t ) = e − t L f ( x ).In this paper, we study weighted compactness of commutators of some classicaloperators of harmonic analysis associated with Schr¨odinger operators. In functionalanalysis, an important branch is the theory of compact operators. Let L be alinear operator from a Banach space X to another Banach space Y . We call L acompact operator if the image under L of any bounded subset of X is a relativelycompact subset of Y . One of classical examples of compact operators is the compactimbedding of Sobolev spaces. By such imbedding, it can be converted an ellipticboundary value problem into a Fredholm integral equation. For further informationon compact operators, we refer the reader to Conway [18], Folland-Stein [24] andKutateladze [35]. Mathematics Subject Classification.
Primary: 42B20; Secondary: 47B47, 42B35.
Key words and phrases.
Commutator; compactness; Schr¨oding operator; weight function.
In 1978, Uchiyama [48] first studied the compactness of commutators of a singularintegral operator with the kernel Ω ∈ Lip ( S d − ) defined by T Ω f ( x ) = p.v. Z R d Ω( y/ | y | ) | y | d f ( x − y ) dy. He obtained that the commutator [ b, T Ω ] is compact on L p ( R d ) , < p < ∞ , ifand only if b ∈ CMO( R d ), where CMO( R d ) denotes the closure of C ∞ c ( R d ) in thetopology of BMO( R d ). In 1984, Janson and Peetre [32] established the theory ofparacommutators T b defined by \ ( T s,tb f )( ξ ) = 12 π Z R d ˆ b ( ξ − η ) A ( ξ, η ) | ξ | s | η | t ˆ f ( η ) dη. Under some assumptions of A ( · , · ), Janson and Peetre proved that if b ∈ CMO( R d ),then T , b is compact, see [32, Theorems 13.2 and 13.3]. The commutators andhigher commutators of convolution singular integrals are special cases of T , b . Thenthe result of [32] is a generalization of that in [48].Since then, the study on the compactness of commutators of different operatorshas attracted much more attention. For examples, the compactness of commutatorsof linear Fourier multipliers and pseudodifferential operators was considered byCordes [19]. Peng [40] gave the the compactness of paracommutators T b . Beatrousand Li [1] studied the boundedness and compactness of the commutators of Hankeltype operators. Krantz and Li [33, 34] applied the compactness characterization ofthe commutator [ b, T Ω ] to study Hankel type operators on Bergman spaces. Wang[49] showed that the commutators of fractional integral operators are compact from L p ( R d ) to L q ( R d ). In 2009, Chen and Ding [15] proved that the commutatorof singular integrals with variable kernels is compact on R d if and only if b ∈ CMO( R d ). In [15], the authors also established the compactness of Littlewood-Paley square functions in [16]. After that, Chen, Ding and Wang [17] obtained thecompactness of commutators for Marcinkiewicz integrals on Morrey spaces. Liuand Tang [38] studied the compactness for higher order commutators of oscillatorysingular integral operators. Li and Peng [36] investigated compact commutatorsof Riesz transforms associated to Schr¨odinger operators. Li, Mo and Zhang [37]established a compactness criterion with applications to the commutators associatedwith Schr¨odinger operators. For more information about the compactness problemsof commutators, see also [2–4, 9–14, 20, 21, 26, 29–31, 46, 47, 50–53] and thereferences therein.The study of Schr¨odinger operator L = − ∆ + V recently attracted much atten-tion, see [5–8, 22, 23, 27, 42, 55]. In particular, Shen [42] considered L p estimatesfor Schr¨odinger operators L with certain potentials which include Schr¨odinger Riesztransforms R L j = ∂∂x j L − / , j = 1 , . . . , n. Shen also proved that the Schr¨odinger type operators: ∇ ( − ∆ + V ) − ∇ , ∇ ( − ∆ + V ) − / , ( − ∆ + V ) − / ∇ with V ∈ B d , and ( − ∆ + V ) iγ with γ ∈ R and V ∈ B d/ ,are standard Calder´on-Zygmund operators.Recently, Bongioanni, Harboure and Salinas [5] proved L p ( R d ) (1 < p < ∞ )boundedness for commutators of Riesz transforms associated with Schr¨odinger op-erators with BMO( ρ ) functions which include the class of BMO functions. In [6], N WEIGHTED COMPACTNESS OF COMMUTATORS OF SCHR ¨ODINGER OPERATORS 3
Bongioanni et al. established the weighted boundedness for Riesz transforms, frac-tional integrals and Littlewood-Paley functions associated to Schr¨odinger opera-tors with weight A ρp class which includes the Muckenhoupt weight class. Tangand his collaborators [43–45] have established weighted norm inequalities for someSchr¨odinger type operators, which include commutators of Riesz transforms, frac-tional integrals, and Littlewood-Paley functions related to Schr¨odinger operators(see also [7, 8]).Naturally, it will be a very interesting problem to ask whether we can establishthe weighted compactness of commutators of some Schr¨odinger type operators withCMO( ρ ) functions and weight A ρp class. In this paper, we give a positive answer.To obtain the conclusion, we will utilize a new estimate of kernels and new weightedinequalities for new maximal operators. It is worth pointing out that our method isapplicable to more general Schr¨odinger type operators, and generalizes the resultsobtained in [36, 37]. The paper is organized as follows. In Section 2, we give somenotation and several basic results which will play a crucial role in the sequel. In Sec-tion 3, we establish the weighted compactness of commutators of Riesz transforms,standard Calder´on-Zygmund operators and Littlewood-Paley functions associatedwith Schr¨odinger operators.Throughout this paper, we let C denote constants that are independent of themain parameters involved but whose value may differ from line to line. By A ∼ B ,we mean that there exists a constant C > /C ≤ A/B ≤ C . By A . B ,we mean that there exists a constant C > A ≤ CB .2. Some notation and basic results
We first recall some notation. Given B = B ( x, r ) and λ >
0, we will write λB for the λ -dilate ball, which is the ball centered at x and with radius λr . Given aLebesgue measurable set E and a weight w , the symbol | E | denotes the Lebesguemeasure of E , and w ( E ) := Z E wdx. For 0 < p < ∞ , k f k L p ( w ) = (cid:18)Z R d | f ( y ) | p w ( y ) dy (cid:19) /p . The following auxiliary function m V ( x ) was first introduced by Shen [42] and iswidely used in the research of Schr¨odinger operators: ρ ( x ) = 1 m V ( x ) = sup r> ( r : 1 r d − Z B ( x,r ) V ( y ) dy ≤ ) . Obviously, 0 < m V ( x ) < ∞ if V = 0. In particular, m V ( x ) = 1 with V = 1 and m V ( x ) ∼ (1 + | x | ) with V = | x | .For different x and y , Shen [42] gave the following important inequality. Lemma 2.1. ([42, Lemmas 1.4 and 1.8])
Assume that V ∈ B q for q > d/ . (i) There exist constants k > , C > and C > such that C (1 + | x − y | m V ( x )) − k ≤ m V ( x ) m V ( y ) ≤ C (1 + | x − y | m V ( x )) k / ( k +1) . (2.1) In particular, m V ( x ) ∼ m V ( y ) if | x − y | ≤ C/m V ( x ) . QIANJUN HE AND PENGTAO LI (ii)
For < r < R < ∞ , r d − Z B ( x,r ) V ( y ) dy ≤ C ( R/r ) d/q − R d − Z B ( x,R ) V ( y ) dy. (2.2)By 0 < m V ( x ) < ∞ and (2.2), Guo et al. [27] got the following result. Lemma 2.2. ([27, Lemma 1])
Suppose that V ∈ B q for some q > d/ and let K > log C + 1 , where C is the constant in (2.1) . Then for any x ∈ R d and R > , we have m V ( x ) R ) K Z B ( x,R ) V ( y ) dy ≤ CR d − . (2.3)For a number θ > B = B ( x , r ) with center at x and radius r , wedenote Ψ θ ( B ) = (1 + r/ρ ( x )) θ .A weight will always mean a nonnegative locally integrable function. As in [6],we say that a weight w belongs to the class A ρ,θp , < p < ∞ , if there is a constant C such that for all balls B = B ( x, r ), (cid:18) θ ( B ) | B | Z B w ( y ) dy (cid:19) (cid:18) θ ( B ) | B | Z B w − / ( p − ( y ) dy (cid:19) p − ≤ C. We also say that a nonnegative function w satisfies the A ρ,θ condition if there existsa constant C such that for all balls B , M θ V ( w )( x ) ≤ Cw ( x ) a.e. x ∈ R d , where M θ V f ( x ) = sup B ∋ x θ | B | Z B | f ( y ) | dy. Since Ψ θ ( B ) ≥
1, obviously, A p ⊂ A ρ,θp for 1 < p < ∞ , where A p denote theclassical Muckenhoupt weights ([25, 39]).Since Ψ θ ( B ) ≤ Ψ θ (2 B ) ≤ θ Ψ θ ( B ) , we remark that balls can be replaced by cubes in the definitions of A ρ,θp for p ≥ M θ V , For convenience, in the rest of this paper, for fixed θ >
0, we use thenotation Ψ( B ) and A ρp instead of Ψ θ ( B ) and A ρ,θp , respectively.The next lemma follows from the definition of A ρp (1 ≤ p < ∞ ): Lemma 2.3. ([43])
Let < p < ∞ . Then the following assertions hold. (i) If < p < p < ∞ , then A ρp ⊂ A ρp . (ii) w ∈ A ρp if and only if w − / ( p − ∈ A ρp ′ , where /p + 1 /p ′ = 1 . (iii) If w ∈ A ρp for ≤ p < ∞ , then Q ) | Q | Z Q | f ( y ) | dy ≤ C (cid:18) w (5 Q ) Z Q | f | p w ( y ) dy (cid:19) /p , where w ( E ) = R E w ( x ) dx. In particular, letting f = χ E for any measurable set E ⊂ Q , we have | E | Ψ( Q ) | Q | ≤ C (cid:18) w ( E ) w (5 Q ) (cid:19) /p . (2.4) N WEIGHTED COMPACTNESS OF COMMUTATORS OF SCHR ¨ODINGER OPERATORS 5
In [5], Bongioanni et al. introduced a new space BMO( ρ ) defined by k f k BMO( ρ ) = sup B ⊂ R d B ) | B | Z B | f ( x ) − f B | dx < ∞ , where f B = | B | R B f ( y ) dy , Ψ( B ) = (1 + r/ρ ( x )) θ , B = B ( x , r ) and θ >
0. Wedenote by CMO( ρ ) the closure of C ∞ c in the topology of BMO( ρ ), where C ∞ c is theset of all smooth functions on R d with compact supports.To prove the weighted boundedness for the area functions related with Schr¨odingeroperators, Tang et al. [45] consider the following variant of maximal operator M V,η , < η < ∞ , defined as M V,η f ( x ) := sup B ∋ x B )) η | B | Z B | f ( y ) | dy. One of the main results obtained in [45] is the weighted L p -boundedness of M V,η , <η < ∞ . Precisely, Lemma 2.4.
Let < p < ∞ and p ′ = p/ ( p − . If w ∈ A ρp , then there exists aconstant C > such that k M V,p ′ k L p ( w ) ≤ C k f k L p ( w ) . Remark 2.5. If η = 1 , Lemma . holds for < p < p < ∞ . There exists many operators related with − ∆+ V are standard Calder´on-Zygmundoperators ([42]), for instance, ∇ ( − ∆ + V ) − ∇ , V ∈ B n , ∇ ( − ∆ + V ) − / , V ∈ B n , ( − ∆ + V ) − / ∇ , V ∈ B n , ( − ∆ + V ) iγ , γ ∈ R & V ∈ B n/ , and ∇ ( − ∆ + V ) − ∇ with V being a nonnegative polynomial. In particular, thekernels K of the operators mentioned above all satisfy the following conditions: forsome δ > l ∈ N = N S { } , there exists a constant C l such that | K ( x, y ) | ≤ C l (1 + | x − y | ( m V ( x ) + m V ( y ))) l | x − y | d (2.5)and | K ( x + h, y ) − K ( x, y ) | + | K ( x, y + h ) − K ( x, y ) |≤ C l (1 + | x − y | ( m V ( x ) + m V ( y ))) l | h | δ | x − y | d + δ (2.6)whenever x, y, h ∈ R d and | h | < | x − y | / Lemma 2.6.
Let < p < ∞ . If w ∈ A ρp , then there exists a constant C > suchthat k T ∗ f k L p ( w ) ≤ C k f k L p ( w ) , where the maximal operator T ∗ is defined by T ∗ f ( x ) := sup ǫ> | T ǫ f ( x ) | = sup ǫ> (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z | y − x | >ǫ K ( x, y ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . QIANJUN HE AND PENGTAO LI
We remark that the maximal operator can be controlled by M V ,η in L p ( w ), andit is proved in [43]. Thus, using Lemma 2 .
4, it implies that Lemma 2 . L p boundedness of commutator [ b, T ]whichcan be found in [43]. Lemma 2.7.
Let < p < ∞ , b ∈ BMO( ρ ) and w ∈ A ρp . Then there exists aconstant C p > such that k [ b, T ] k L p ( w ) ≤ C p k b k BMO( ρ ) k f k L p ( w ) . Next we consider another class V ∈ B q for q ≥ d/ T = ( − ∆ + V ) − V , T = ( − ∆ + V ) − / V / and T = ( − ∆ + V ) − / ∇ . Tang considered the weighted estimates for the operators T i , i = 1 , , , in [43]. Lemma 2.8.
Suppose that V ∈ B q and q ≥ d/ . Then the following three state-ments hold. (i) If q ′ ≤ p < ∞ and w ∈ A ρp/q ′ , then k T f k L p ( w ) ≤ C k f k L p ( w ) . (ii) If (2 q ) ′ ≤ p < ∞ and w ∈ A ρp/ (2 q ) ′ , then k T f k L p ( w ) ≤ C k f k L p ( w ) . (iii) If p ′ ≤ p < ∞ and w ∈ A ρp/p ′ , where /p = 1 /q − /d and d/ ≤ q < d , then k T f k L p ( w ) ≤ C k f k L p ( w ) . In [43], using Lemma 2 . T i , i = 1 , , L p boundedness of commutator [ b, T i ] , i = 1 , ,
3, with b ∈ BMO( ρ ). Lemma 2.9.
Suppose that V ∈ B q , q ≥ d/ . Let b ∈ BMO( ρ ) . Then the followingthree statements hold. (i) If q ′ ≤ p < ∞ and w ∈ A ρp/q ′ , then k [ b, T ] f k L p ( w ) ≤ C k b k BMO( ρ ) k f k L p ( w ) . (ii) If (2 q ) ′ ≤ p < ∞ and w ∈ A ρp/ (2 q ) ′ , then k [ b, T ] f k L p ( w ) ≤ C k b k BMO( ρ ) k f k L p ( w ) . (iii) If p ′ ≤ p < ∞ and w ∈ A ρp/p ′ , where /p = 1 /q − /d and d/ ≤ q < d , then k [ b, T ] f k L p ( w ) ≤ C k b k BMO( ρ ) k f k L p ( w ) . We list some estimates of the kernel K i of operator T i , i = 1 , ,
3, and refer thereader to Guo-Li-Peng [27] and Shen [42].
Lemma 2.10.
Suppose V ∈ B q for some q > n/ . Then there exist constants δ > and C l such that for < | h | < | x − y | / and l > , | K ( x, y ) | ≤ C l (1 + | x − y | m V ( x )) l | x − y | d − V ( y ) , (2.7) | K ( x + h, y ) − K ( x, y ) | ≤ C l (1 + | x − y | m V ( x )) l | h | δ | x − y | d − δ V ( y ) , (2.8) | K ( x, y ) | ≤ C l (1 + | x − y | m V ( x )) l | x − y | d − V / ( y ) (2.9) N WEIGHTED COMPACTNESS OF COMMUTATORS OF SCHR ¨ODINGER OPERATORS 7 and | K ( x + h, y ) − K ( x, y ) | ≤ C l (1 + | x − y | m V ( x )) l | h | δ | x − y | d − δ V / ( y ) . (2.10) In particular, for d/ < q < d , we also have | K ( x, y ) | (2.11) ≤ C l (1 + | x − y | m V ( x )) l | x − y | d − Z B ( y, | x − y | ) V ( ξ ) | y − ξ | d − dξ + 1 | x − y | ! and | K ( x + h, y ) − K ( x, y ) | (2.12) ≤ C l (1 + | x − y | m V ( x )) l | h | δ | x − y | d − δ Z B ( y, | x − y | ) V ( ξ ) | y − ξ | d − dξ + 1 | x − y | ! . The maximal operator T i, Max of T i , i = 1 , , , is defined by T i, Max f ( x ) := sup r> (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z | x − y | >r K i ( x, y ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , i = 1 , , . To prove our results, we need the following weighted boundedness of maximaloperator T i, Max . Theorem 2.11.
Suppose that V ∈ B q , q > d/ . Then the following three state-ments are hold. (i) If q ′ < p < ∞ and w ∈ A ρp/q ′ , then k T , Max f k L p ( w ) ≤ C k f k L p ( w ) . (ii) If (2 q ) ′ < p < ∞ and w ∈ A ρp/ (2 q ) ′ , then k T , Max f k L p ( w ) ≤ C k f k L p ( w ) . (iii) If p ′ < p < ∞ and w ∈ A ρp/p ′ , where /p = 1 /q − /d and d/ < q < d , then k T , Max f k L p ( w ) ≤ C k f k L p ( w ) . To prove Theorem 2 .
11, we need the following two lemmas. The first is a coveringlemma.
Lemma 2.12. ([43])
For any ball B = B ( x , r ) , if r ≥ /m V ( x ) , the ball B canbe decomposed into finite disjoint cubes { Q i } i =1 ,...,m such that B ⊂ m [ i =1 Q i ⊂ √ dB and r i / ≤ m V ( x ) ≤ √ dC r i for some x ∈ Q i = Q ( x i , r i ) , where C is the same as in (2.1) of Lemma . Lemma 2.13.
Suppose that < η < ∞ and V ∈ B q , q ≥ d/ . For any ball B = B ( x , r ) , we have for x ∈ B | B | Z B | f ( y ) | dy ≤ (2 √ d ) d M V ,η f ( x ) . QIANJUN HE AND PENGTAO LI
Proof.
It is sufficient to consider two cases.
Case 1: r < /m V ( x ). Since r < /m V ( x ) implies Ψ( B ) ∼
1, this case is easyto handle and we omit the details.
Case 2: r ≥ /m V ( x ). Using Lemma 2 .
13, there exist finite disjoint cubes Q i ( x i , r i ) , i = 1 , . . . , m, such that R B | f ( y ) | dy ≤ m P i =1 R Q i | f ( y ) | dy, | B | ≤ m P i =1 | Q i | ≤ (2 √ d ) d | B | ,r i / ≤ /m V ( x i ) ≤ √ dC r i . Note that r i < C/m V ( x i ) implies Ψ( Q i ) ∼
1. For x ∈ B , we then have Z B | f ( y ) | dy ≤ m X i =1 | Q i | M V ,η f ( x ) ≤ (2 √ d ) d | B | M V ,η f ( x ) . This finished the proof. (cid:3)
Proof of Theorem . . We first prove (i). Take T ,r f ( x ) = Z | x − y | >r K ( x, y ) f ( y ) dy. For B = B ( x, r/ f as f = f + f , where f := f χ B . It follows fromLemma 2 .
13 and Lemma 2 . w = 1 that | T ,r f ( x ) | = 1 | B | Z B | T ,r f ( x ) | dy ≤ | B | Z B | T f ( y ) | dy + 1 | B | Z B | T f ( y ) | dy + 1 | B | Z B | T f ( y ) − T ,r f ( x ) | dy ≤ M V ,η ( T f )( x ) + 1 | B | /q ′ k T f k L q ′ + 1 | B | Z B | T f ( y ) − T ,r f ( x ) | dy ≤ M V ,η ( T f )( x ) + C (cid:18) | B | Z B | f ( y ) | q ′ dy (cid:19) /q ′ + 1 | B | Z B | T f ( y ) − T ,r f ( x ) | dy ≤ M V ,η ( T f )( x ) + C (cid:16) M V ,η ( | f | q ′ ) (cid:17) /q ′ + 1 | B | Z B | T f ( y ) − T ,r f ( x ) | dy. For the third term in the last inequaliy, we have1 | B | Z B | T f ( y ) − T ,r f ( x ) | dy = 1 | B | Z B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z (16) c K ( y, ξ ) f ( ξ ) dξ − Z | x − ξ | >r K ( x, ξ ) f ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy ≤ | B | Z B Z | x − ξ | >r | K ( y, ξ ) − K ( x, ξ ) || f ( ξ ) | dξ ! dy =: 1 | B | Z B I ( y ) dy. N WEIGHTED COMPACTNESS OF COMMUTATORS OF SCHR ¨ODINGER OPERATORS 9
Now, set h = | y − x | . Since | y − x | < r/ < | x − ξ | /
16 for y ∈ B , by (2.8) inLemma 2 .
10, we can obtain that for l = θη/q ′ + K , I ( y ) ≤ ∞ X k =0 Z k r< | x − ξ |≤ k +1 r C l (1 + | x − ξ | m V ( x )) l | y − x | δ | x − ξ | d − δ V ( ξ ) | f ( ξ ) | dξ ≤ ∞ X k =0 C l r δ (2 k r ) d − δ (1 + m V ( x )2 k r ) l Z | x − ξ |≤ k +1 r V q ( ξ ) dξ ! /q Z | x − ξ |≤ k +1 r | f ( ξ ) | q ′ dξ ! /q ′ ≤ C ∞ X k =0 C l ( M V ,η ( | f | q ′ )( x )) /q ′ (1 + m V ( x )2 k r ) K r δ (2 k r ) d − δ Z B ( x, k ) V ( ξ ) dξ ! ≤ C ( M V ,η ( | f | q ′ )( x )) /q ′ ∞ X k =0 r δ (2 k r ) d − δ (2 k r ) d − ≤ C ( M V ,η ( | f | q ′ )( x )) /q ′ . Here we have used (2.2) for R = 2 k +1 r , and (2.3) in Lemma 2 . R = 2 k r . Theestimate for I ( y ) implies that1 | B | Z B I ( y ) dy ≤ C ( M V ,η ( | f | q ′ )( x )) /q ′ and T , Max f ( x ) ≤ M V ,η ( T f )( x ) + C (cid:16) M V ,η ( | f | q ′ ) (cid:17) /q ′ . Hence, using Lemma 2 . . . k T , Max k L p ( w ) ≤ k M V ,η ( T f ) k L p ( w ) + C k ( M V ,η ( | f | q ′ )( x )) /q ′ k L p ( w ) ≤ C k f k L p ( w ) for p > q ′ >
1. This finishes the proof of (i).Similar to (i), (ii) can be obtained easily. We omit the details.It remains to handle the maximal operator T , Max . For any x , let T ,r = Z | x − y | >r K ( x, y ) f ( y ) dy. For B = B ( x, r/ f = f + f , where f = f χ B . Similarly, we canobtain T ,r f ( x ) ≤ M V ,η ( T f )( x ) + C (cid:16) M V ,η ( | f | q ′ ) (cid:17) /q ′ + 1 | B | Z B I ( y ) dy, where I ( y ) denotes the following integral: I ( y ) = Z | x − ξ | >r | K ( y, ξ ) − K ( x, ξ ) || f ( ξ ) | dξ. Since y ∈ B and h = | y − x | < r/ < | x − ξ | , we deduce from (2.12) that I ( y ) . I , ( x ) + I , ( x ), where I , ( x ) := Z | x − ξ | >r | r | δ | x − ξ | d − δ (1 + | x − ξ | m V ( x )) l Z B ( ξ, | x − ξ | ) V ( u ) | ξ − u | d − du ! | f ( ξ ) | dξ ; I , ( x ) := Z | x − ξ | >r | x − ξ | m V ( x )) l | r | δ | x − ξ | d + δ | f ( ξ ) | dξ. For I , ( x ), we have I , ( x ) . r δ ∞ X k =0 Z k r< | x − ξ |≤ k +1 r m V ( x )2 k r ) ηθ/p ′ k r ) d + δ dξ . r δ ∞ X k =0 k r ) δ ( M V ,η ( | f | p ′ )( x )) /p ′ . ( M V ,η ( | f | p ′ )( x )) /p ′ . Since | u − ξ | < | x − ξ | yields | x − u | ≤ | x − ξ | + | ξ − u | < | x − ξ | , we can applyH¨older’s inequality and the Hardy-Littlewood-Sobolev inequality with p = q − d to obtain I , ( x ) ≤ ∞ X k =0 Z k r< | x − ξ | < k +1 r C l | r | δ (2 k r ) d − δ (1 + m V ( x )2 k r ) l Z B ( x, k +2 r ) V ( u ) | ξ − u | d − du ! | f ( ξ ) | dξ ≤ ∞ X k =0 C l | r | δ (2 k r ) d − δ (1 + m V ( x )2 k r ) l (cid:13)(cid:13)(cid:13)(cid:13)Z R d V ( u ) χ B ( x, k +2 r ) | · − u | d − du (cid:13)(cid:13)(cid:13)(cid:13) L p Z B ( x, k +1 r ) | f ( ξ ) | p ′ dξ ! /p ′ ≤ C ∞ X k =0 ( M V ,η ( | f | p ′ )( x )) /p ′ (1 + m V ( x )2 k r ) K | r | δ (2 k r ) d − δ Z B ( x, k +2 r ) V q ( ξ ) dξ ! /q (2 k +1 r ) d/p ′ ≤ C ∞ X k =0 ( M V ,η ( | f | p ′ )( x )) /p ′ (1 + m V ( x )2 k r ) K | r | δ (2 k +2 r ) d/q − n (2 k r ) d − δ (2 k +1 r ) d/p ′ Z B ( x, k +2 r ) V ( ξ ) dξ ! ≤ C ( M V ,η ( | f | p ′ )( x )) /p ′ ∞ X k =0 | r | δ (2 k r ) d − δ (2 k r ) d/q − d +( d/p ′ )+ d − ≤ C ( M V ,η ( | f | p ′ )( x )) /p ′ . Here we have used the fact that d/q − d +( d/p ′ )+ d − d − /p = 1 /q − /d .Thus, by a similar manner as the case (i), we obtain the desired result. Thiscompletes the proof of Lemma 2 . (cid:3) Finally, we continue to investigate the Littlewood-Paley functions related toSchr¨odinger operators. We first introduce some notations. For ( x, t ) ∈ R d +1+ = R d × (0 , ∞ ), let T s = e − s L and( Q t f )( x ) = t (cid:18) dT s ds (cid:12)(cid:12)(cid:12)(cid:12) s = t f (cid:19) ( x ) . The Littlewood-Paley g -function g Q and the area function S Q related to Schr¨odingeroperators (cf. [3, 22, 44, 45]) are defined by g Q ( f )( x ) := (cid:18)Z ∞ | Q t ( f )( x ) | dtt (cid:19) / (2.13)and S Q ( f )( x ) := Z ∞ Z | x − y | Lemma 2.14. Let < p < ∞ . If w ∈ A ρp , then there exists a constant C such that k g Q ( f ) k L p ( w ) ≤ C k f k L p ( w ) and k S Q ( f ) k L p ( w ) ≤ C | f k L p ( w ) . The commutators of g Q and S Q with b ∈ BMO( ρ ) are defined by g Q,b ( f )( x ) = (cid:18)Z ∞ | Q t (( b ( x ) − b ( · )) f )( x ) | dtt (cid:19) / and S Q,b ( f )( x ) = Z ∞ Z | x − y | Lemma 2.15. Let b ∈ BMO( ρ ) and < p < ∞ . If w ∈ A ρp , then there exists aconstant C such that k g Q,b ( f ) k L p ( w ) ≤ C k b k BMO( ρ ) k f k L p ( w ) and k S Q,b ( f ) k L p ( w ) ≤ C k b k BMO( ρ ) k f k L p ( w ) . In [22], the autors introduce some properties for the integral kernnel Q t ( x, y ) ofthe operators Q t in (2.13) and (2.14). Lemma 2.16. There exist positive constants c and δ ≤ such that for any l ≥ there is a constant C l so that the following inequalities hold: (i) | Q t ( x, y ) | ≤ C l t − n (cid:16) tρ ( x ) + tρ ( y ) (cid:17) − l exp (cid:0) − c | x − y | /t (cid:1) , (ii) | Q t ( x + h, y ) − Q t ( x, y ) | ≤ C l ( | h | / | t | ) δ t − n (cid:16) tρ ( x ) + tρ ( y ) (cid:17) − l exp (cid:0) − c | x − y | /t (cid:1) for all | h | ≤ t . We define the space B = L ( R d +1 , dydt/t d ) to be the set of all measurablefunctions a : R d +1+ → C endwoed the norm | a | B = Z R d +1+ | a ( y, t ) | dydtt d ! / < ∞ . Let ϕ be a nongegative infinitely differentiable function on R + such that ϕ ( s ) = 1for 0 < s < ϕ ( x ) = 0 for s ≥ 2. Then the function ϕ t ( x, y ) := t ϕ ( | x − y | t )satisfies | ϕ t ( x, y ) − ϕ ( x ′ ,y ) | ≤ | x − x ′ | t χ [0 , (cid:18) min {| x − y | , | x ′ − y |} t (cid:19) for | x − y | > | x − x ′ | .Denote by e K ( · , · ) the kernel defined as follows: e K ( x, z ) := n t / ϕ t ( x, y ) Q t ( y, z ) o ( y,z ) ∈ R d +1+ . It is easy to see that Z ∞ Z | x − y | Let δ as same as in Lemma . . Then for any l we have | e K ( x, z ) | B ≤ C l (1 + | x − z | ( m V ( x ) + m V ( y ))) l | x − z | d . We end this section by a general weighted version of the Frechet-Kolmogrovtheorem, which was proved by Xue, Yabuta and Yan [53]. Lemma 2.18. Let w be a weight on R d . Assume that w − / ( p − is also a weighton R d for some p > . Let < p < ∞ and F be a subset in L p ( w ) , then F issequentially compact in L p ( w ) if the following three conditions are satisfied: (i) F is bounded, i.e., sup f ∈F k f k L p ( w ) < ∞ ; (ii) F uniformly vanishes at infinitly, i.e., lim N →∞ sup f ∈F Z | x | >N | f ( x ) | p w ( x ) dx = 0;(iii) F is uniformly equicontinous, i.e., lim | h |→ sup f ∈F Z R d | f ( x + h ) − f ( x ) | p w ( x ) dx = 0 . Note that a operator T : V → Y is said to be a compact operator if T iscontinuous and maps bounded subsets into sequentially compact subsets.3. Compactness of commutators of Schr¨odinger type operators In this section, we will establish the weighted compactness of commutatorsof Riesz transforms, standard Calder´on-Zygmund operators and Littlewood-Paleyfunctions associated with Schr¨odinger operators.3.1. The weighted compactness of [ b, T ] . First of all, we consider the weighted compactness of [ b, T ]. Theorem 3.1. Let < p < ∞ , b ∈ CMO( ρ ) and w ∈ A ρp . Then [ b, T ] is a compactoperator from L p ( R d ) to itself.Proof. Since | [ b , T ] f ( x ) − [ b , T ] f ( x ) | ≤ | [ b − b , T ] f ( x ) | and b ∈ CMO( ρ ) ⊂ BMO( ρ ), then by Lemma 2 . 7, the commutator [ b, T ] is con-tinuous on L p ( w ). Hence, for any bounded set F ⊂ L p ( w ), where f ∈ F with k f k L p ( w ) . 1, it suffices to prove that F = { [ b, T ] f : f ∈ F, b ∈ CMO( ρ ) } is a sequentially compact subset. According to a density argument, if b ∈ CMO( ρ ),then there exists a sequence of functions b ǫ ∈ C ∞ c ( R d ) such that k b − b ǫ k BMO( ρ ) < ǫ. Thus, by Lemma 2 . 7, we show that k [ b, T ] − [ b ǫ,T ] k L p ( w ) → L p ( w ) ≤ k [ b − b ǫ , T ] k L p ( w ) → L p ( w ) . ǫ. N WEIGHTED COMPACTNESS OF COMMUTATORS OF SCHR ¨ODINGER OPERATORS 13 Therefore, it is enough to prove that F is sequentially compact. Without loss ofgeneralization, we will verify F satisfies the conditions (i) − (iii) of Lemma 2 . 18 for b ∈ C ∞ c ( R d ). The proof is divided into three steps. Step I: F satisfies the condition (i). First, by Lemma 2 . 7, we havesup f ∈ F k [ b, T ] f k L p ( w ) ≤ C k b k BMO( ρ ) k f k L p ( w ) < ∞ , which yields the fact that the set F is bounded. Step II: F satisfies the condition (ii). We adapt the method using in [36] to verifythe condition (ii) of Lemma 2 . 18. Assume that b ∈ C ∞ c ( R d ) and supp b ⊂ B (0 , R ),where B (0 , R ) is a ball of radius R and centered at origin in R d . For ν > 2, set B c = { x ∈ R d : | x | > νR } . Then we have Z | x | >νR | [ b, T ] f ( x ) | p w ( x ) dx ≤ Z | x | >νR Z | y | <ρ (0) | K ( x, y ) || b ( y ) || f ( y ) | dy ! p w ( x ) dx. It can be deduced from (2.1) and the scaling technique directly that for any x, y ∈ R d and c ∈ (0 , C (1 + | x − y | m V ( y )) k +1 . 11 + c | x − y | m V ( x ) (3.1) . C c (1 + | x − y | m V ( y )) / (1+ k ) , where the constants k and C is as same as in (2.1) of Lemma 2 . | x | > νR implies | x − y | > (1 − /ν ) | x | with ν > 2, applying (2.5) andH¨older’s inequality, we have Z | y | Case I : R > ρ (0). Since R > ρ (0) implies 1 / (1+(2 j νR ) /ρ ) < / (1+2 j ν ) ≤ / j ν ,if l > θ ( k + 1), it holds ∞ X j =0 (1 + 2 j νR/ρ (0)) θ (1 + (2 j νR ) /ρ (0)) l/ ( k +1) ≤ ∞ X j =0 j ν ) l/ ( k +1) − θ ≤ Cν l/ ( k +1) − θ . Case II : R ≤ ρ (0). Note that R and ρ are finite, there exists finite integer N ≥ [log ( ρ (0) /R )] + 1 such that 2 N R > ρ (0). Hence, this case goes back to CaseI and we get ∞ X j =0 (1 + 2 j νR/ρ (0)) θ (1 + (2 j νR ) /ρ (0)) l/ ( k +1) ≤ ∞ X j =0 j ν ) l/ ( k +1) − θ ≤ C N ( l/ ( k +1) − θ ) ν l/ ( k +1) − θ . By the above argument, we obtain Z | x | >νR | [ b, T ] f ( x ) | p w ( x ) dx ! /p ≤ C k b k L ∞ ( R d ) k f k L p ( w ) k w k /pA ρp (1 − /ν ) d + l/ ( k +1) ν l/ ( k +1) − θ max { N ( l/ ( k +1) − θ ) , } , N WEIGHTED COMPACTNESS OF COMMUTATORS OF SCHR ¨ODINGER OPERATORS 15 which implies that for any p > l > θ ( k + 1),lim ν →∞ Z | x | >νR | [ b, T ] f ( z ) | p w ( x ) dx = 0holds whenever f ∈ F . Step III: F satisfies condition (iii). It remains to show that the set F is uniformlyequicontinuous. It suffices to verify that for any ǫ > 0, if | h | is sufficiently smalland only depends on ǫ , then k [ b, T ] f ( h + · ) − [ b, T ] f ( · ) k L p ( w ) ≤ Cǫ (3.2)holds uniformly for f ∈ F .For any x ∈ R d , we divide [ b, T ] f ( x + h ) − [ b, T ] f ( x ) = P i =1 I i ( x ), where I ( x ) := Z | x − y | >a | h | K ( x, y )( b ( x + h ) − b ( x )) f ( y ) dy ; I ( x ) := Z | x − y | >a | h | ( K ( x + h, y ) − K ( x, y ))( b ( x + h ) − b ( y )) f ( y ) dy ; I ( x ) := Z | x − y |≤ a | h | K ( x, y )( b ( x ) − b ( y )) f ( y ) dy ; I ( x ) := Z | x − y |≤ a | h | K ( x + h, y )( b ( x + h ) − b ( y )) f ( y ) dy. Clearly, by the definition of T ∗ and b ∈ C ∞ c ( R d ), we have | I ( x ) | ≤ | h |k∇ b k L ∞ ( R d ) T ∗ f ( x ) , which, together with Lemma 2 . 6, indicates that k I k L p ( w ) ≤ | h |k T ∗ f k L p ( w ) ≤ C | h |k f k L p ( w ) . For I ( x ), take a > 2. Using (2.6) and k b k L ∞ ( R d ) ≤ C , we have | I ( x ) | . Z | x − y | >a | h | | x − y | ( m V ( x ) + m V ( y ))) l | h | δ | x − y | d + δ | f ( y ) | dy . ∞ X k =0 Z k a | h | < | x − y |≤ k +1 a | h | | x − y | m V ( x )) l | h | δ | x − y | d + δ | f ( y ) | dy . ∞ X k =0 Z k a | h | < | x − y |≤ k +1 a | h | k a | h | ) m V ( x )) l | h | δ (2 k a | h | ) d + δ | f ( y ) | dy . ∞ X k =0 k a | h | ) m V ( x )) l | h | δ (2 k a | h | ) d + δ Z B ( x, k +1 a | h | ) | f ( y ) | dy . M V ,η f ( x ) ∞ X k =0 | h | δ (2 k a | h | ) δ . a − δ M V ,η f ( x ) , where we have used the constant l = θη . Hence, it follows from Lemma 2 . η = p ′ , k I k L p ( w ) ≤ Ca − δ k M V ,η f k L p ( w ) ≤ Ca − δ k f k L p ( w ) . For I ( x ), applying (2.5) and b ∈ C ∞ c ( R d ), we obtain | I ( x ) | . k∇ b k L ∞ ( R d ) Z | x − y |≤ a | h | C l (1 + | x − y | m V ( x )) l | x − y | d − | f ( y ) | dy . X j = −∞ Z j − a | h | < | x − y |≤ j a | h | j − a | h | ) m V ( x )) l j − a | h | ) d − | f ( y ) | dy . X j = −∞ j − a | h | ) m V ( x )) l j − a | h | ) d − Z B ( x, j a | h | ) | f ( y ) | dy . M V ,η f ( x ) X j = −∞ j − a | h | . a | h | M V ,η f ( x ) , where we have used the constant l = θη . Hence, we use Lemma 2 . k I k L p ( w ) ≤ Ca | h |k M V ,η f k L p ( w ) ≤ Ca | h |k f k L p ( w ) for η = p ′ .The estimate of I ( x ) is similar to that of I ( x ). Since | x − y | ≤ a | h | , we have | x + h − y | ≤ ( a + 1) | h | . Using the kernel property of T in (2.5) and b ∈ C ∞ c ( R d ),we have | I ( x ) | . Z | x + h − y |≤ ( a +1) | h | C l (1 + | x + h − y | m V ( x + h )) l | x + h − y | d − | f ( y ) | dy . X j = −∞ j − ( a +1) | h | ) d − (1 + (2 j − ( a + 1) | h | ) m V ( x + h )) l Z B ( x + h, j ( a +1) | h | ) | f ( y ) | dy . M V ,η f ( x ) X j = −∞ j − ( a + 1) | h | . ( a + 1) | h | M V ,η f ( x ) , where we have used the constant l = θη . Thus, k I k L p ( w ) ≤ C ( a + 1) | h |k M V ,η f k L p ( w ) ≤ C ( a + 1) | h |k f k L p ( w ) . Combining with the estimations of I ( x ), I ( x ), I ( x ) and I ( x ), it implies that k [ b, T ] f ( · + h ) − [ b, T ] f ( · ) k L p ( w ) ≤ X i =1 k I i k L p ( w ) ≤ C ( | h | + a − δ + a | h | + ( a + 1) | h | ) k f k L p ( w ) . Consequently, for any ǫ > 0, we can choose a large enough such thatmax { /a , / ( a + 1) , /a δ } < ǫ, and set | h | being sufficiently small satisfying | h | < min { /a , / ( a + 1) } . Letting a → ∞ , we can see that F is uniformly equicontinous (condition (iii)). Thiscompletes the proof of Theorem 3 . (cid:3) N WEIGHTED COMPACTNESS OF COMMUTATORS OF SCHR ¨ODINGER OPERATORS 17 The weighted compactness of g b and S Q,b .Theorem 3.2. Let < p < ∞ , b ∈ CMO( ρ ) and w ∈ A ρp . Then g b and S Q,b arecompact operators from L p ( w ) to itself.Proof. We first prove that S Q,b is a compact operator from L p ( w ) to itself. Since | S Q,b f ( x ) − S Q,b f ( x ) | ≤ | S Q,b − b f ( x ) | , by the argument in the proof of Theore 3 . . 15, we only need to provethat for b ∈ C ∞ c ( R d ), S = { [ b, T ] f : f ∈ F, b ∈ BMO( ρ ) } satisfies the conditions(ii) − (iii) of Lemma 2 . 18. We divide the proof into two steps. Step I: S satisfies the condition (ii).Suppose supp b ⊂ { z : | z | < R } and choose ν > 2. For | x | > νR > R , we have b ( x ) = 0. Therefore, by the Minkowski inequality, we have | S Q,b f ( x ) | = Z ∞ Z | x − y | 17, following the argument in Step II of the proof of Theorem 3 . 1, weobtain Z | x | >νR | S Q,b f ( x ) | p w ( x ) dx ! p ≤ C k b k L ∞ ( R d ) k f k L p ( w ) k w k p A ρp (1 − /ν ) d + lk ν lk − θ max { N ( lk − θ ) , } . For f ∈ F , letting ν → ∞ reaches Z | x | >νR | S Q,b f ( x ) | p w ( x ) dx → , i.e., S satisfies the condition (ii) in Lemma 2 . Step II: S satisfies the condition (iii). It suffices to prove that for 1 < p < ∞ and w ∈ A ρp , lim | h |→ k S Q,b f ( · + h ) − S Q,b f ( · ) k L p ( w ) = 0 . Note that | S Q,b f ( x + h ) − S Q,b f ( x ) | ≤ Z ∞ Z | x − y | 0, and write D ( x, y, y ) = D + D + D + D , where D ( x ) := Z | x − z | >a | h | Q t ( y, z )( b ( x + h ) − b ( z )) f ( z ) dz ; D ( x ) := Z | x − z | >a | h | ( Q t ( y + h, z ) − Q t ( y, z ))( b ( x + h ) − b ( z )) f ( z ) dz ; D ( x ) := Z | x − z |≤ a | h | Q t ( y, z )( b ( x ) − b ( z )) f ( z ) dz ; D ( x ) := Z | x − z |≤ a | h | Q t ( y + h, z )( b ( x + h ) − b ( z )) f ( z ) dz. For D and D , it can be deduced from (2.15) and Minkowski’s inequality that Z ∞ Z | x − y | 1, we use Lemma 2 . 17 to obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ Z | x − y | 16. In fact, consider | x − z | > h and define E = { y : | y − x | ≥ | x − z | / } . Hence, we can apply (ii) of Lemma 2 . 16 to obtain | K Q ( x, z ) | ≤ Z R d Z | y − x | / | Q t ( y + h, z ) − Q t ( y, z ) | dtdyt d +1N WEIGHTED COMPACTNESS OF COMMUTATORS OF SCHR ¨ODINGER OPERATORS 19 ≤ C | h | δ Z R d Z | y − x | / ρ ( x ) l t d +1+2 δ +2 l t N ( t + | z − y | ) N dtdy ≤ C | h | δ Z E Z | y − x | / ρ ( x ) l t d +1+2 δ +2 l t N ( t + | z − y | ) N dtdy + C | h | δ Z E c Z | y − x | / ρ ( x ) l t d +1+2 δ +2 l t N ( t + | z − y | ) N dtdy =: III + III . For III , we then have III ≤ C | h | δ Z E ρ ( x ) l | y − x | d +2 δ +2 l dy ≤ C (cid:18) | x − z | ρ ( x ) (cid:19) − l | h | δ | x − z | d +2 δ . If y ∈ E c , then | y − x | < | x − z | / < | y − z | < | x − z | , and hence III ≤ | h | δ Z E c Z | x − z || y − x | / + Z ∞| x − z | ! ρ ( x ) l t d +1+2 δ +2 l t N ( t + | z − y | ) N dtdy =: III a + III b . For III a and III b , letting d + δ < N − l < (3 d + 2 δ ) / 2, we can get III a ≤ C | h | δ | x − z | N Z E c Z ∞| y − x | / ρ ( x ) l t d +1+2 δ − N +2 l dtdy ≤ C | h | δ | x − z | N Z | y − x | < | x − z | / ρ ( x ) l t d +2 δ − N +2 l dtdy ≤ C (cid:18) | x − z | ρ ( x ) (cid:19) − l | h | δ | x − z | d +2 δ , and III b ≤ C | h | δ Z E c Z ∞| x − z | ρ ( x ) l t d +1+2 δ +2 l dtdy ≤ C ρ ( x ) l | h | δ | x − z | d +2 δ +2 l Z | y − x | < | x − z | / dy ≤ C (cid:18) | x − z | ρ ( x ) (cid:19) − l | h | δ | x − z | d +2 δ . Combining the above inequalities, we obtain the desired inequality (3.5).Taking a > 2, similar to estimate I ( x ) in Step III of proof of Theorem 3 . 1, weobtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ Z | x − y | 13, we get S Q,a,h f ( x ) = 1 | B | Z B S Q,a,h f ( x ) dξ ≤ | B | Z B S Q f ( ξ ) dξ + 1 | B | Z B S Q f ( ξ ) dξ + 1 | B | Z B | S Q f ( ξ ) − S Q,a,h f ( x ) | dξ ≤ M V ,η ( S Q f )( x ) + L + L . By Lemma 2 . 13, Lemma 2 . 14 with w = 1 and H¨older’s inequality, for any 1 < q a | h | Q t ( y + ξ − x, z ) f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dydtt d +1 − Z ∞ Z | x − y | 14 and Lemma 2 . 4, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ Z | x − y | 0, we can choose a large enough such thatmax { /a , / ( a + 1) , /a δ } < ǫ, and set | h | being small enough such that | h | < min { /a , / ( a + 1) } . Letting a → ∞ , we have the uniformly equicontinous (condition (iii)) of S .Next we will prove that g b is a compact operator from L p ( w ) to itself. In orderto this result, we need the following claim: Claim 3.3. Let δ as same as in Lemma . . Then for any l we have (cid:18)Z ∞ | Q t ( x, y ) | dtt (cid:19) ≤ C l (1 + | x − y | ( m V ( x ) + m V ( y ))) l | x − y | d and (cid:18)Z ∞ | Q t ( x, y ) − Q t ( ξ, y ) | dtt (cid:19) ≤ C l (1 + | x − y | ( m V ( x ) + m V ( y ))) l | x − ξ | δ | x − y | d + δ for | x − y | > | x − ξ | . If Claim 3 . S Q,b , we can easily obtain g b is acompact operator on L p ( w ). Now we proceed to prove Claim 3 . 3. Using inequality(a) in Lemma 2 . 16, we obtain Z ∞ | Q t ( x, y ) | dtt ≤ Z ∞ ρ ( x ) l t d +1+2 l t N ( t + | x − y | ) N dt ≤ Z | x − y | + Z ∞| x − y | ! ρ ( x ) l t d +1+2 l t N ( t + | x − y | ) N dt =: W + W . For W and W , let d + l < N < (2 d + 2 l + 1) / 2. We have the following estimates: W ≤ ρ ( x ) l | x − y | N Z | x − y | t d +1+2 l − N dt ≤ C (cid:18) | x − y | ρ ( x ) (cid:19) − l | x − y | d and W ≤ Z ∞| x − y | ρ ( x ) l t d +1+2 l ≤ C (cid:18) | x − y | ρ ( x ) (cid:19) − l | x − y | d . Combining the above inequalities, we get the first inequality of Claim 3 . 3. Thesecond inequality of Claim 3 . . (cid:3) The weighted compactness of [ b, T i ] , i = 1 , , . Next, we discuss the weighted compactness of [ b, T i ] , i = 1 , , , on L p ( w ). Theorem 3.4. Suppose that V ∈ B q , q > d/ . Let b ∈ CMO( ρ ) . Then thefollowing three statements hold. (i) If q ′ ≤ p < ∞ and w ∈ A ρp/q ′ , [ b, T ] is a compact operator from L p ( w ) to itself. (ii) If (2 q ) ′ < p < ∞ and w ∈ A ρp/ (2 q ) ′ , [ b, T ] is a compact operator from L p ( w ) toitself. (iii) If p ′ < p < ∞ and w ∈ A ρp/p ′ , where /p = 1 /q − /d and d/ < q < d , [ b, T ] is a compact operator from L p ( w ) to itself.Proof. Similar to the proof of Theorem 3 . 1, and following the process of the proofsof Theorem 2.1, Theorem 2.5 and Theorem 2.7 in [36], we can easily obtain thedesired results. Hence, we omit the details. (cid:3) Let T ∗ = V ( − ∆ + V ) − , T ∗ = V / ( − ∆ + V ) / and T ∗ = ∇ ( − ∆ + V ) − / . By duality, the following weighted L p -compactness of T ∗ i , i = 1 , , . Corollary 3.5. Suppose that V ∈ B q and q > d/ . Let b ∈ CMO( ρ ) . 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