On weighted Compactness of Commutator of semi-group maximal function associated to Schrödinger operators
aa r X i v : . [ m a t h . C A ] F e b ON WEIGHTED COMPACTNESS OF COMMUTATOR OF SEMI-GROUP MAXIMALFUNCTION ASSOCIATED TO SCHR ¨ODINGER OPERATORS
SHIFEN WANG AND QINGYING XUE ∗ A BSTRACT . Let T ∗ be the semi-group maximal function associated to the Schr¨odinger operator − ∆ + V ( x ) with V satisfying an appropriate reverse H¨older inequality. In this paper, we show thatthe commutator of T ∗ is a compact operator on L p ( w ) for 1 < p < ∞ if b ∈ CMO θ ( ρ )( R n ) and w ∈ A ρ , θ p ( R n ) . Here CMO θ ( ρ )( R n ) denotes the closure of C ∞ c ( R n ) in the BMO θ ( ρ )( R n ) (whichis larger than the classical BMO ( R n ) space) topology. The space where b belongs and the weighsclass w belongs are more larger than the usual CMO ( R n ) space and the Muckenhoupt A p weightsclass, respectively.
1. I
NTRODUCTION
Consider the Schr¨odinger operator L = − ∆ + V ( x ) in R n , n ≥
3. Where ∆ is the Laplacian operator on R n and the function V is a nonnegativepotential belonging to certain reverse H ¨older class RH q with an exponent q > n /
2, that is, thereexists a constant C > (cid:16) | B ( x , r ) | ˆ B ( x , r ) V q ( y ) dy (cid:17) q ≤ C (cid:16) | B ( x , r ) | ˆ B ( x , r ) V ( y ) dy (cid:17) , for every ball B ( x , r ) ⊂ R n . It is worth pointing out that if V ∈ RH q for some q >
1, then thereexists ε >
0, such that V ∈ RH q + ε (see [12]). On the other hand, the H ¨older inequality gives that RH q ⊂ RH q if q ≥ q >
1. Therefore, the assumption q > n / q ≥ n / V V ∈ RH q with q ≥ n / L with nonnegative potentials is very useful in the study of certainsubelliptic operators. For instance, by taking the partial Fourier transform in the t variable, theoperator − ∆ x − V ( x ) ∂ t is reduced to − ∆ x + V ( x ) ξ (See [19]). Some basic results on L , includingcertain estimates of the fundamental solutions of L and the boundedness on L p of Riesz transforms ∆ L − / were obtained by Fefferman [11], Shen [20] and Zhong [31].Attentions have also been paid to the study of function spaces associated to L . It was Dzi-uba´nski and Zienkiewicz [10] who characterized the Hardy space H L related to the Schr¨odingeroperator. Later on, for 0 < p ≤
1, the H pL space with potentials from reverse H ¨older classes werestudied in [8] and [9]. Subsequently, Yang et al. [30] characterize the localized Hardy spaces byestablishing the boundedness of Riesz transforms, maximal operators and endpoint estimates offractional integrals associated with L .For the classical Schr¨odinger operators L , there are many interesting results of its associatedRiesz transforms, which essentially and heavily depend on the properties of e − tL . The propertiesof semi-group e − tL such as the positivity, Gaussian estimates and off-diagonal estimates play afundamental role in the study of Riesz transform. The maximal function defined by the semigroup Date : December 3, 2020.
Key words and phrases.
Schr¨odinger operator, semi-group maximal operator, commutator, compactness.
Primary 42B25, Secondary 35J10.The second author was supported partly by NNSF of China (Nos. 11671039, 11871101) and NSFC-DFG (No.11761131002). ∗ Corresponding author, e-mail address: [email protected]. e − tL ( t > ) or the Riesz transforms ∆ L − / were further generalized by Lin, et. al. [16] to thesetting of Heisenberg groups.In order to introduce more results, we need to give some definitions. The semi-group maximalfunction associated to the Schr¨odinger operator L is defined by(1.1) T ∗ ( f )( x ) = sup t > | e − tL f ( x ) | = sup t > (cid:12)(cid:12)(cid:12) ˆ R n k t ( x , y ) f ( y ) dy (cid:12)(cid:12)(cid:12) , where k t is the kernel of the operator e − tL . As in [20], we will use the auxiliary function ρ definedfor R n as(1.2) ρ ( x ) = sup r > n r : 1 r n − ˆ B ( x , r ) V ( y ) dy ≤ o . Remark 1.1.
Under the above assumptions on V , it is easy to get that 0 < ρ ( x ) < ∞ . In particular, ρ ( x ) = V =
1. For more details concerning the function ρ ( x ) and its applications in studyingthe Schr¨odinger operator L , we refer the reader to [11, 20, 21].For 1 < p < ∞ , the A ρ , θ p weights class is defined as follows. Definition 1.2. ( A ρ , θ p weights class, [2]) . Let w be a nonnegative, locally integrable function on R n . For 1 < p < ∞ , we say that a weight w belongs to the class A ρ , θ p if there exists a positiveconstant C such that for all balls B = B ( x , r ) , it holds that(1.3) (cid:16) | B | ˆ B w ( y ) dy (cid:17)(cid:16) | B | ˆ B w ( y ) − / ( p − ) dy (cid:17) p − ≤ C (cid:16) + r ρ ( x ) (cid:17) θ p . w is said to satisfy the A ρ , θ condition if there exists a constant C such that for all balls BM θ V ( w )( x ) ≤ Cw ( x ) , a . e . x ∈ R n , where M θ V ( f )( x ) = sup x ∈ B (cid:16) + r ρ ( x ) (cid:17) θ | B | ˆ B | f ( y ) | dy . Remark 1.3.
Clearly, the classes A ρ , θ p are increasing with θ , and A p ⊂ A ρ , θ p for 1 ≤ p < ∞ . More-over, from the Remark 1.6 below, it is easy to see that A p ( A ρ , θ p . Definition 1.4. (
BMO θ ( ρ )( R n ) space, [1]) . For θ >
0, we defined the class BMO θ ( ρ )( R n ) oflocally integrable functions f such that(1.4) 1 | B ( x , r ) | ˆ B ( x , r ) | f ( y ) − f B | dy ≤ C (cid:16) + r ρ ( x ) (cid:17) θ , for all x ∈ R n and r >
0, where f B = | B | ´ B b . A norm for f ∈ BMO θ ( ρ )( R n ) , denoted by k f k BMO θ ( ρ ) , is given by the infimum of the constants satisfying (1.4), after identifying functionsthat differ upon a constant. Clearly BMO ( R n ) ⊂ BMO θ ( ρ )( R n ) ⊂ BMO θ ′ ( ρ )( R n ) for 0 < θ < θ ′ .The commutator of T ∗ with b ∈ BMO θ ( ρ )( R n ) is defined by(1.5) T ∗ b ( f )( x ) = sup t > (cid:12)(cid:12)(cid:12) ˆ R n k t ( x , y )( b ( x ) − b ( y )) f ( y ) dy (cid:12)(cid:12)(cid:12) . In 2011, Bongioanni, Harboure and Salinas [1] considered the L p ( R n )( < p < ∞ ) boundednessof the commutators of Riesz transforms related to L with BMO θ ( ρ )( R n ) functions. In anotherpaper, they [2] established the weighted boundedness for the semi-group maximal function, Riesztransforms, fractional integrals and Littlewood-Paley functions related to L with weights belong to A ρ , θ p (see definition 1.2) class which includes the Muckenhoupt weight class A p . Theorem A ( [2]).
For < p < ∞ , the operators T ∗ is bounded on L p ( w ) when w ∈ A ρ , θ p . N WEIGHTED COMPACTNESS OF COMMUTATORS 3
Recently, Tang [21] considered the weighted norm inequalities for T ∗ b . Theorem B ( [21]).
Let < p < ∞ , w ∈ A ρ , θ p and b ∈ BMO θ ( ρ )( R n ) , then there exists a constantC such that k T ∗ b ( f ) k L p ( w ) ≤ C k b k BMO θ ( ρ ) k f k L p ( w ) . This paper is devoted to studying the weighted compactness for commutators of semi-groupmaximal function related to Schr¨odinger operators. Before stating our main results, we recallsome background for the compactness of the commutators of some classical operators. Given alocally integrable function b , the commutator [ b , T ] is defined by [ b , T ]( f )( x ) = bT f ( x ) − T ( b f )( x ) . In 1978, Uchiyama [25] first studied the compactness of commutators and showed that the com-mutator [ b , T Ω ] is compact on L p ( R n ) for 1 < p < ∞ if and only if b ∈ CMO ( R n ) , where T Ω isa singular integral operator with rough kernel Ω ∈ Lip ( S n − ) and CMO ( R n ) is the closure of C ∞ c ( R n ) in the BMO ( R n ) topology.Since then, the study on the compactness of commutators of different operators has attractedmuch more attention. Krantz and Li applied the compactness characterization of the commutator [ b , T Ω ] to study Hankel type operators on Bergman space in [14] and [15]. Wang [26] showed thecompactness of the commutator of fractional integral operator form L p ( R n ) to L q ( R n ) . In 2009,Chen and Ding [5] proved the commutator of singular integrals with variable kernels is compacton L p ( R n ) if and only if b ∈ CMO ( R n ) and they also establised the compactness of Littlewood-Paley square functions in [6]. Later on, Chen, Ding and Wang [7] obtained the compactness ofcommutators for Marcinkiewicz Integral in Morrey Spaces. Recently, Liu, Wang and Xue [18]showed the compactness of the commutator of oscillatory singular integrals with H ¨older classkernels of non-convolutional type. We refer the reader to [3,4,17,23,24,27,29] for the compactnessof commutators of multilinear operators.The above compactness results are all concerned with the space CMO ( R n ) . However, The-orem B shows that the L p boundedness holds for more larger space BMO θ ( ρ )( R n ) , rather thanBMO ( R n ) and the weights class A ρ , θ p is more larger than A p weights class. Let CMO θ ( ρ )( R n ) bethe closure of C ∞ c ( R n ) in the BMO θ ( ρ )( R n ) topology. Then, it is quite natural to ask the followingquestion: Question 1.5.
Is the operator T ∗ b compact from L p ( w ) to L p ( w ) when w ∈ A ρ , θ p and b belongs tothe space CMO θ ( ρ )( R n ) ?The main purpose of this paper is to give a firm answer to the above question. Our result is asfollows: Theorem 1.1.
Let < p < ∞ , w ∈ A ρ , θ p and b ∈ CMO θ ( ρ )( R n ) . If w satisfies the following condi-tion (1.6) lim A → + ∞ A − np + n ˆ | x | > w ( Ax ) | x | np dx = , then the operator T ∗ b defined by (1.1) is a compact operator from L p ( w ) to L p ( w ) . Remark 1.6.
We give some comments about Theorem 1.1:(1) The weights class in Theorem 1.1 is more larger than the classical Muckenhoupt weightsclass A p . In fact, if w ∈ A p , the classical Muckenhoupt weights class, then the condition(1.6) holds. Let 0 < γ < θ and w ( x ) = ( + | x | ) − ( n + γ ) , it is easy to see that w satisfies(1.6) and w ( x ) / ∈ A p (1 ≤ p < ∞ ), but w ∈ A ρ , θ ⊂ A ρ , θ p (1 < p < ∞ ) provided that V = θ ( ρ )( R n ) where b belongs is more larger than CMO ( R n ) space. S. WANG AND Q. XUE
The paper is organized as follows. In section 2 we give some definitions and preliminary lem-mas, which are the main ingredients of our proofs. In section 3 we will give the proof of Theorem1.1 via smooth truncated techniques. The domain of integration will be divided into several cases.In actuality some cases are combinable, but various subcases also arise, which increases the diffi-culty we need to deal with.Throughout the paper, the letter C or c , sometimes with certain parameters, will stand for posi-tive constants not necessarily the same one at each occurrence, but are independent of the essentialvariables. A ∼ B means that there exists constants C > C > C B ≤ A ≤ C B .2. P RELIMINARIES
We first recall some notation. Given a Lebesgue measurable set E ⊂ R n , | E | will denote theLebesgue measure of E . If B = B ( x , r ) is a ball in R n and λ is a real number, then λ B shallstand for the ball with the same center as B and radiu λ times that of B . A weight w is a non-negative measurable function on R n . The measure associated with w is the set function given by w ( E ) = ´ E wdx . For 0 < p < ∞ we denote by L p ( w ) the space of all Lebesgue measurable function f ( x ) such that k f k L p ( w ) = (cid:16) ˆ R n | f ( x ) | p w ( x ) dx | (cid:17) / p . The auxiliary function ρ enjoys the following property. Lemma 2.1. ( [20]) . There exists k ≥ and C > such that for all x , y ∈ R n ,C − ρ ( x ) (cid:16) + | x − y | ρ ( x ) (cid:17) − k ≤ ρ ( y ) ≤ C ρ ( x )( + | x − y | ρ ( x ) (cid:17) k k + . In particular, ρ ( x ) ∼ ρ ( y ) if | x − y | < C ρ ( x ) .A ρ , θ p weights class has some properties analogy to A p weights class for 1 ≤ p < ∞ . Lemma 2.2. ( [2] [22]) . Let < p < ∞ and w ∈ A ρ , ∞ p = S θ ≥ A ρ , θ p . Then (i) If ≤ p < p < ∞ , then A ρ , θ p ⊂ A ρ , θ p . (ii) w ∈ A ρ , θ p if and only if w − p − ∈ A ρ , θ p ′ , where / p + / p ′ = . (iii) If w ∈ A ρ , ∞ p , < p < ∞ , then there exists ε > such that w ∈ A ρ , ∞ p − ε . It should be pointed out that (iii) of Lemma 2.2 was proved by Bongioanni, Harboure andSalinas in [2].For convenience, we write Ψ θ ( B ) = ( + r / ρ ( x )) θ , if B = B ( x , r ) . Then M θ V can be rewrittenas M θ V ( f )( x ) = sup x ∈ B Ψ θ ( B ) | B | ˆ B | f ( y ) | dy , and the following result holds: Lemma 2.3. ( [22]) . Let < p < ∞ and suppose that w ∈ A ρ , θ p . If p < p < ∞ , then ˆ R n | M θ V f ( x ) | p w ( x ) dx ≤ C p ˆ R n | f ( x ) | p w ( x ) dx . By the Lemma 2.3, M θ V may not be bounded on L p ( w ) for all w ∈ A ρ , θ p and 1 < p < ∞ . So weneed the variant maximal operator M V , η defined by M V , η f ( x ) = sup x ∈ B ( Ψ θ ( B )) η | B | ˆ B | f ( y ) | dy , < η < ∞ . We have the following Lemma.
N WEIGHTED COMPACTNESS OF COMMUTATORS 5
Lemma 2.4. ( [22]) . Let < p < ∞ , p ′ = p / ( p − ) and suppose that w ∈ A ρ , θ p . Then there existsa constant C > such that k M V , p ′ f k L p ( w ) ≤ C k f k L p ( w ) . We also need the following properties of the kernel k t . Lemma 2.5. ( [8], [13]) . For every N, there is a constant C N such that < k t ( x , y ) ≤ C N t − n e − | x − y | t (cid:16) + √ t ρ ( x ) + √ t ρ ( y ) (cid:17) − N . Lemma 2.6. ( [9]) . There exists < δ < and a constant c > such that for every N > there isa constant C N > so that, for all | h | ≤ √ t | k t ( x + h ) − k t ( x , y ) | ≤ C N (cid:16) | h |√ t (cid:17) δ t − n e − c | x − y | t (cid:16) + √ t ρ ( x ) + √ t ρ ( y ) (cid:17) − N . We end this section by introducing the general weighted version of Frechet-Kolmogorov theo-rems, which was proved by Xue, Yabuta and Yan in [29].
Lemma 2.7. ( [29]) . Let w be a weight on R n . Assume that w − / ( p − ) is also a weight on R n forsome p > . Let < p < ∞ and F be a subset in L p ( w ) , then F is sequentially 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 infinity, i.e., lim N → ∞ sup f ∈ F ˆ | x | > N | f ( x ) | p w ( x ) dx = F is uniformly equicontinuous, i.e., lim | h |→ sup f ∈ F ˆ R n | f ( · + h ) − f ( · ) | p w ( x ) dx = .
3. P
ROOF OF T HEOREM
Proof of Theorem 1.1.
We shall prove Theorem 1.1 via smooth truncated techniques. First, weintroduce the following smooth truncated function. Let ϕ ∈ C ∞ ([ , ∞ )) satisfy0 ≤ ϕ ≤ and ϕ ( x ) = ( , x ∈ [ , ] , , x ∈ [ , ∞ ) . (3.1)For any γ >
0, let(3.2) k t , γ ( x , y ) = k t ( x , y ) (cid:16) − ϕ ( γ − | x − y | ) (cid:17) . Define(3.3) T ∗ γ f ( x ) = sup t > (cid:12)(cid:12)(cid:12) ˆ R n k t , γ ( x , y ) f ( y ) dy (cid:12)(cid:12)(cid:12) . and(3.4) T ∗ b , γ f ( x ) = sup t > (cid:12)(cid:12)(cid:12) ˆ R n k t , γ ( x , y )( b ( x ) − b ( y )) f ( y ) dy (cid:12)(cid:12)(cid:12) . S. WANG AND Q. XUE
For any b ∈ C ∞ c ( R n ) and γ , θ , η >
0, by (3.2), (3.4) and lemma 2.5 with N = θη , one has | T ∗ b f ( x ) − T ∗ b , γ f ( x ) | ≤ C γ sup t > ˆ | x − y | < γ t − n / e − | x − y | t (cid:16) + √ t ρ ( x ) (cid:17) − θη | f ( y ) | dy ≤ C γ n sup √ t < γ ˆ | x − y | < √ t t − n / e − | x − y | t (cid:16) + √ t ρ ( x ) (cid:17) − θη | f ( y ) | dy + sup √ t < γ ˆ √ t ≤| x − y | < γ t − n / e − | x − y | t (cid:16) + √ t ρ ( x ) (cid:17) − θη | f ( y ) | dy + sup √ t ≥ γ ˆ | x − y | < γ t − n / e − | x − y | t (cid:16) + √ t ρ ( x ) (cid:17) − θη | f ( y ) | dy o = : C γ { J + J + J } . (3.5)One may obtain J ≤ sup √ t < γ t − n / (cid:16) + √ t ρ ( x ) (cid:17) − θη ˆ | x − y | < √ t | f ( y ) | dy ≤ CM V , η f ( x ) . (3.6)and J ≤ θη sup √ t ≥ γ γ − n (cid:16) + γρ ( x ) (cid:17) − θη ˆ | x − y | < γ | f ( y ) | dy ≤ CM V , η f ( x ) . (3.7)It remains o estimate J . Using the estimate e − s ≤ Cs M / with M > n + θη and splitting to annuli,it follows that J ≤ sup √ t < γ ∞ ∑ k = t M − n (cid:16) + √ t ρ ( x ) (cid:17) − θη ˆ | x − y |∼ k √ t | f ( y ) || x − y | M dy ≤ sup √ t < γ ∞ ∑ k = − k ( M − n − θη ) ( k √ t ) n (cid:16) + k √ t ρ ( x ) (cid:17) θη ˆ | x − y | < k √ t | f ( y ) | dy ≤ CM V , η f ( x ) . (3.8)Combing (3.8) with (3.5), (3.6) and (3.7) may lead to | T ∗ b f ( x ) − T ∗ b , γ f ( x ) | ≤ C γ M V , η f ( x ) . Then Lemma 2.4 with p ′ ≤ η < ∞ gives that k T ∗ b f − T ∗ b , γ f k L p ( w ) ≤ C γ k f k L p ( w ) , which implies that(3.9) lim γ → k T ∗ b f − T ∗ b , γ f k L p ( w ) = . On the other hand, if b ∈ CMO θ ( ρ )( R n ) , then for any ε >
0, there exists b ε ∈ C ∞ c ( R n ) such that k b − b ε k BMO θ ( ρ ) < ε , so that k T ∗ b f − T ∗ b ε f k L p ( w ) ≤ k T ∗ b − b ε f k L p ( w ) ≤ C k b − b ε k BMO θ ( ρ ) k f k L p ( w ) ≤ C ε . Thus, to prove T ∗ b is compact on L p ( w ) for any b ∈ CMO θ ( ρ ) , it suffices to prove that T ∗ b iscompact on L p ( w ) for any b ∈ C ∞ c ( R n ) . By (3.9) and [28], it suffices to show that T ∗ b , γ is compactfor any b ∈ C ∞ c ( R n ) when γ > F in L p ( w ) ,let F = { T ∗ b , γ f : f ∈ F } . Then, we need to show that for b ∈ C ∞ c ( R n ) , F satisfies the conditions ( i ) - ( iii ) of Lemma 2.7. N WEIGHTED COMPACTNESS OF COMMUTATORS 7
From the definition of k t , γ , we know that 0 < k t , γ ( x , y ) ≤ k t ( x , y ) , then T ∗ γ f ( x ) ≤ T ∗ ( | f | )( x ) and T ∗ b , γ f ( x ) ≤ T ∗ ( | f | )( x ) . Hence, the boundedness of T ∗ γ and T ∗ b , γ also holds. Thus, we havesup f ∈ F k T ∗ b , γ f k L p ( w ) ≤ C sup f ∈ F k f k L p ( w ) ≤ C , which yields the fact that the set F is bounded.Assume b ∈ C ∞ c ( R n ) and supp ( b ) ⊂ B ( , R ) , where B ( , R ) is the ball of radius R center at originin R n . For any | x | > A > R , w ∈ A ρ , θ p , 1 < p < ∞ and f ∈ F . By Lemma 2.5 and the estimate e − | x − y | t ≤ C t n | x − y | n , we have | T ∗ b , γ f ( x ) | ≤ sup t > ˆ | y | < R k t ( x , y ) | b ( y ) f ( y ) | dy ≤ C sup t > t − n / (cid:16) + √ t ρ ( x ) (cid:17) − N ˆ | y | < R e − | x − y | t | f ( y ) | dy ≤ C | x | − n ˆ | y | < R | f ( y ) | dy ≤ C | x | − n k f k L p ( w ) (cid:16) ˆ | y | < R w − p ′ / p ( y ) dy (cid:17) / p ′ . Therefore ˆ | x | > A | T ∗ b , γ f ( x ) | p w ( x ) dx ≤ C ˆ | x | > A w ( x ) | x | np dx = C ∞ ∑ j = ˆ j A < | x | < j + A w ( x ) | x | np dx = CA − np + n ˆ | x | > w ( Ax ) | x | np dx . This together with the condition (1.6) yields thatlim A → ∞ ˆ | x | > A | T ∗ b , γ f ( x ) | p w ( x ) dx = , whenever f ∈ F .It remains to show that the set F is uniformly equicontinuous. It suffices to verify that(3.10) lim | h |→ k T ∗ b , γ f ( h + · ) − T ∗ b , γ f ( · ) k L p ( w ) = , holds uniformly for f ∈ F .In what follows, we fix γ ∈ ( , ) and | h | < γ . Then | T ∗ b , γ f ( x + h ) − T ∗ b , γ f ( x ) | ≤ sup t > ˆ R n | k t , γ ( x + h , y ) − k t , γ ( x , y ) || b ( x + h ) − b ( y ) || f ( y ) | dy + sup t > ˆ R n k t , γ ( x , y ) | b ( x + h ) − b ( x ) || f ( y ) | dy = : I ( x ) + II ( x ) . (3.11)For II ( x ) , it holds that II ( x ) = | b ( x + h ) − b ( x ) | sup t > ˆ R n k t , γ ( x , y ) | f ( y ) | dy ≤ C | h | T ∗ γ ( | f | )( x ) . Then, by the L p ( w ) -bounds of T ∗ γ , we have(3.12) k II k L p ( w ) ≤ C | h |k f k L p ( w ) . S. WANG AND Q. XUE
For I ( x ) , we decompose it into two parts I ( x ) ≤ sup √ t ≥| h | ˆ R n | k t , γ ( x + h , y ) − k t , γ ( x , y ) || b ( x + h ) − b ( y ) || f ( y ) | dy + sup √ t < | h | ˆ R n | k t , γ ( x + h , y ) − k t , γ ( x , y ) || b ( x + h ) − b ( y ) || f ( y ) | dy = : I ( x ) + I ( x ) . (3.13) Contribution of I . For I ( x ) , if | h | ≤ √ t , then by lemma 2.5 and lemma 2.6, we have | k t , γ ( x + h , y ) − k t , γ ( x , y ) | ≤ | k t ( x + h ) − k t ( x , y ) | + | k t ( x + h ) − k t ( x , y ) | ϕ ( γ − | x + h − y | )+ k t ( x , y ) | ϕ ( γ − | x + h − y | ) − ϕ ( γ − | x − y | ) |≤ C (cid:16) | h |√ t (cid:17) δ t − n e − c | x − y | t (cid:16) + √ t ρ ( x ) + √ t ρ ( y ) (cid:17) − N + C | h | γ t − n e − c | x − y | t (cid:16) + √ t ρ ( x ) + √ t ρ ( y ) (cid:17) − N . (3.14)Therefore, we have I ( x ) ≤ C sup √ t ≥| h | ˆ R n (cid:16)(cid:16) | h |√ t (cid:17) δ + | h | γ (cid:17) t − n e − c | x − y | t (cid:16) + √ t ρ ( x ) (cid:17) − N × | b ( x + h ) − b ( y ) || f ( y ) | dy ≤ C sup √ t ≥ n ˆ | x − y | < √ t + ˆ | x − y |≥√ t o(cid:16)(cid:16) | h |√ t (cid:17) δ + | h | γ (cid:17) t − n e − c | x − y | t × (cid:16) + √ t ρ ( x ) (cid:17) − N | b ( x + h ) − b ( y ) || f ( y ) | dy + C sup | h |≤√ t < n ˆ | x − y | < √ t + ˆ | x − y |≥√ t o(cid:16)(cid:16) | h |√ t (cid:17) δ + | h | γ (cid:17) t − n e − c | x − y | t × (cid:16) + √ t ρ ( x ) (cid:17) − N | b ( x + h ) − b ( y ) || f ( y ) | dy = : I ( x ) + I ( x ) + I ( x ) + I ( x ) . (3.15)Now, we are in the position to estimate the above four terms.For I ( x ) , if √ t ≥
1, then t − δ / ≤
1. Taking N = θη for any θ , η >
0, then we have I ( x ) ≤ C γ − ( | h | δ + | h | ) sup √ t ≥ ˆ | x − y | < √ t t − n (cid:16) + √ t ρ ( x ) (cid:17) − θη | f ( y ) | dy ≤ C γ − ( | h | δ + | h | ) sup √ t ≥ ( √ t ) n ( + √ t ρ ( x ) ) θη ˆ | x − y | < √ t t − n | f ( y ) | dy ≤ C γ − ( | h | δ + | h | ) M V , η f ( x ) . (3.16)In order to estimate I ( x ) , we need the following ineqality: for any M >
0, there exists aconstant C >
0, such that(3.17) e − c | x − y | t ≤ C t M | x − y | M . N WEIGHTED COMPACTNESS OF COMMUTATORS 9
Using (3.17) with M > n + θη , splitting into annuli, we obtain I ( x ) ≤ C γ − ( | h | δ + | h | ) sup √ t ≥ t M − n (cid:16) + √ t ρ ( x ) (cid:17) − θη ˆ | x − y |≥√ t | f ( y ) || x − y | M dy ≤ C γ − ( | h | δ + | h | ) sup √ t ≥ ∞ ∑ k = − k ( M − n ) ( k √ t ) n ( + √ t ρ ( x ) ) θη ˆ | x − y | < k √ t | f ( y ) | dy ≤ C γ − ( | h | δ + | h | ) sup √ t ≥ ∞ ∑ k = − k ( M − n − θη ) ( k √ t ) n ( + k √ t ρ ( x ) ) θη ˆ | x − y | < k √ t | f ( y ) | dy ≤ C γ − ( | h | δ + | h | ) M V , η f ( x ) . (3.18)If √ t <
1, then t − δ / < t − / . For any θ , η >
0, taking N = θη . For I ( x ) , if | h | ≤ √ t , | x − y | < √ t and b ∈ C ∞ c ( R n ) , then | b ( x + h ) − b ( y ) | ≤ C | x + h − y | ≤ C ( | x − y | + | h | ) ≤ C √ t . Then, it follows that I ( x ) ≤ C | h | δ sup | h |≤√ t < t − n + (cid:16) + √ t ρ ( x ) (cid:17) − θη ˆ | x − y | < √ t | b ( x + h ) − b ( y ) || f ( y ) | dy + C γ − | h | sup | h |≤√ t < t − n (cid:16) + √ t ρ ( x ) (cid:17) − θη ˆ | x − y | < √ t | f ( y ) | dy ≤ C γ − ( | h | δ + | h | ) sup | h |≤√ t < t − n (cid:16) + √ t ρ ( x ) (cid:17) − θη ˆ | x − y | < √ t | f ( y ) | dy ≤ C γ − ( | h | δ + | h | ) M V , η f ( x ) . (3.19)For I ( x ) , if | x − y | < k √ t , k = , , · · · , and | h | ≤ √ t , b ∈ C ∞ c ( R n ) , then | b ( x + h ) − b ( y ) | ≤ C | x + h − y | ≤ C ( | x − y | + | h | ) ≤ C k √ t , which combining with (3.17) for M > n + + θη yields that I ( x ) ≤ C | h | δ sup | h |≤√ t < t M − n − (cid:16) + √ t ρ ( x ) (cid:17) − θη ˆ | x − y |≥√ t | f ( x ) || b ( x + h ) − b ( y ) || x − y | M dy + C γ − | h | sup | h |≤√ t < t M − n (cid:16) + √ t ρ ( x ) (cid:17) − θη ˆ | x − y |≥√ t | f ( x ) || x − y | M dy ≤ C | h | δ sup | h |≤√ t < t M − n − (cid:16) + √ t ρ ( x ) (cid:17) − θη ∞ ∑ k = k √ t ( k √ t ) M ˆ | x − y |∼ k √ t | f ( y ) | dy + C γ − | h | sup | h |≤√ t < t M − n (cid:16) + √ t ρ ( x ) (cid:17) − θη ∞ ∑ k = ( k √ t ) M ˆ | x − y |∼ k √ t | f ( y ) | dy ≤ C γ − ( | h | δ + | h | ) sup | h |≤√ t < ∞ ∑ k = − k ( M − n − − θη ) ( k √ t ) n ( + k √ t ρ ( x ) ) θη ˆ | x − y | < k √ t | f ( y ) | dy ≤ C γ − ( | h | δ + | h | ) M V , η f ( x ) . (3.20)Sum up (3.15), (3.16), (3.18), (3.19) and (3.20) in all, we get(3.21) I ( x ) ≤ C γ − ( | h | δ + | h | ) M V , η f ( x ) . Contribution of I . Next we will estimate I ( x ) . When | x − y | < γ and | h | < γ , then | x + h − y | < γ . Hence ϕ ( γ − | x + h − y | ) = = ϕ ( γ − | x − y | ) . This implies k t , γ ( x + h , y ) = = k t , γ ( x , y ) . For I ( x ) , we decompose it as follows: I ( x ) ≤ sup √ t < | h | < ρ ( x ) ˆ | x − y |≥ γ | k t , γ ( x + h , y ) − k t , γ ( x , y ) || b ( x + h ) − b ( y ) || f ( y ) | dy + sup √ t < | h | ρ ( x ) ≤| h | ˆ | x − y |≥ γ | k t , γ ( x + h , y ) − k t , γ ( x , y ) || b ( x + h ) − b ( y ) || f ( y ) | dy : = I ( x ) + I ( x ) . (3.22)For I ( x ) , since | x − y | > γ > | h | and √ t < | h | < ρ ( x ) , then | x + h − y | ∼ | x − y | , | h | / ρ ( x ) < ( | h | / √ t ) M > M >
0. Choosing M > n + + θη and using Lemma 2.5 and (3.17),we get I ( x ) = sup √ t < | h | < ρ ( x ) ˆ | x − y |≥ γ | k t , γ ( x + h , y ) − k t , γ ( x , y ) || b ( x + h ) − b ( y ) || f ( y ) | dy ≤ C sup √ t < | h | < ρ ( x ) ˆ | x − y |≥ | h | t − n (cid:16) | h |√ t (cid:17) M − n e − c | x − y | t | x − y || f ( y ) | dy ≤ C | h | M − n sup √ t < | h | < ρ ( x ) ∞ ∑ k = ˆ | x − y |≥ | h | | f ( y ) || x − y | M − dy ≤ C | h | M − n sup √ t < | h | < ρ ( x ) ∞ ∑ k = ( k | h | ) M − ˆ | x − y |∼ k | h | | f ( y ) | dy ≤ C | h | sup √ t < | h | < ρ ( x ) ∞ ∑ k = − k ( M − n − − θη ) ( k | h | ) n k θη ˆ | x − y | < k | h | | f ( y ) | dy ≤ C | h | M V , η f ( x ) . (3.23)Finally, it remains to consider I ( x ) . Since b ∈ C ∞ c ( R n ) , | x − y | ≥ | h | , √ t < | h | , then | h | / √ t > | b ( x + h ) − b ( x ) | ≤ C | x − y | . In addition, if | x − y | < l ρ ( x ) , l = , , · · · , then by lemma 2.1,we have ρ ( y ) ≤ C k k + l ρ ( x ) . Then, it follows that (cid:16) + √ t ρ ( x ) + √ t ρ ( y ) (cid:17) − N + (cid:16) + √ t ρ ( x + h ) + √ t ρ ( y ) (cid:17) − N ≤ (cid:16) + √ t ρ ( y ) (cid:17) − N ≤ C N (cid:16) + − k k + l √ t ρ ( x ) (cid:17) − N . Choosing M , N such that N > M > n + + ( k + ) θη , and applying lemma 2.5 and (3.17) again,we obtain I ( x ) ≤ C | h | sup √ t < | h | ρ ( x ) ≤| h | ˆ | x − y |≥ ρ ( x ) t − n + e − c | x − y | t (cid:16) + √ t ρ ( y ) (cid:17) − N | x − y || f ( y ) | dy ≤ C | h | sup √ t < | h | ρ ( x ) ≤| h | t M − n − ∞ ∑ l = (cid:16) + − k k + l √ t ρ ( x ) (cid:17) − N ˆ | x − y |∼ l ρ ( x ) | f ( y ) || x − y | M − dy ≤ C | h | sup √ t < | h | ρ ( x ) ≤| h | ∞ ∑ l = (cid:16) √ t ρ ( x ) (cid:17) M − n − (cid:16) + − k k + l √ t ρ ( x ) (cid:17) − N − l ( M − n − − θη ) ( l ρ ( x )) n l θη (3.24) N WEIGHTED COMPACTNESS OF COMMUTATORS 11 × ˆ | x − y | < l ρ ( x ) | f ( y ) | dy ≤ C | h | sup √ t < | h | ρ ( x ) ≤| h | ∞ ∑ l = − l ( M − n − k + − θη ) ( l ρ ( x )) n l θη ˆ | x − y | < l ρ ( x ) | f ( y ) | dy ≤ C | h | M V , η f ( x ) . Inequality (3.24) together with (3.22) and (3.23) gives that(3.25) I ( x ) ≤ C | h | M V , η f ( x ) . Therefore, by (3.13) and (3.21) we have I ( x ) ≤ C ( | h | + | h | δ ) M V , η f ( x ) . By Lemma 2.4 for any p ′ ≤ η < ∞ , it holds that(3.26) k I k L p ( w ) ≤ C ( | h | + | h | δ ) k M V , η f k L p ( w ) ≤ C ( | h | + | h | δ ) k f k L p ( w ) . From (3.11), (3.12) and (3.26), we get k T ∗ b , γ f ( h + · ) − T ∗ b , γ f ( · ) k L p ( w ) ≤ C ( | h | + | h | δ ) k f k L p ( w ) , which yields (3.10) and finishes the proof of Theorem 1.1. (cid:3) R EFERENCES [1] B. Bongioanni, E. Harboure and O. Salinas,
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