Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces
aa r X i v : . [ m a t h . C A ] J a n Two-weight Norm Inequalities for Local Fractional Integrals onGaussian Measure Spaces
Boning Di, Qianjun He ∗ , Dunyan Yan Abstract
In this paper, the authors establish the two-weight boundedness of the local fractional maximaloperators and local fractional integrals on Gaussian measure spaces associated with the localweights. More precisely, the authors first obtain the two-weight weak-type estimate for the local- a fractional maximal operators of order α from L p ( v ) to L q, ∞ ( u ) with 1 ≤ p ≤ q < ∞ undera condition of ( u, v ) ∈ S b ′ >a A b ′ p,q,α , and then obtain the two-weight weak-type estimate for thelocal fractional integrals. In addition, the authors obtain the two-weight strong-type boundednessof the local fractional maximal operators under a condition of ( u, v ) ∈ M a +9 √ da p,q,α and the two-weight strong-type boundedness of the local fractional integrals. These estimates are establishedby the radialization method and dyadic approach. The Gaussian measure space, denoted by (cid:0) R d , | · | , γ (cid:1) , is the Euclidean space R d endowed with theEuclidean distance | · | and the Gaussian probability measure γ , whered γ ( x ) := π − d/ e −| x | d x. The Gaussian harmonic analysis extends the classical results obtained in harmonic analysis of trigono-metric expansions to orthogonal polynomial expansions [13]. On the other hand, Gaussian harmonicanalysis is widely used in the second quantization [14], Malliavin calculus [15], hypercontractivity [2],and geometric applications[3], etc.On Gaussian measure spaces, due to the special locally doubling and reverse doubling propertywhich will be shown in Section 2, we concentrate on the local Hardy-Littlewood maximal operator M a defined by M a ( f )( x ) := sup Q ∈ Q a ( x ) γ ( Q ) Z Q | f ( y ) | d γ ( y ) , where Q a ( x ) = { Q ∈ Q a : Q ∋ x } , Q a = { Q ⊂ R d : ℓ ( Q ) ≤ am ( c Q ) } , m ( x ) = min { , / | x |} with c Q denoting the center of the cube Q and ℓ ( Q ) denoting the side length of the cube Q . Fur-thermore, the local fractional Hardy-Littlewood maximal operator M aα is defined by M aα ( f )( x ) := sup Q ∈ Q a ( x ) γ ( Q )] − ˙ α Z Q | f ( y ) | d γ ( y ) Mathematics Subject Classification : Primary 42B35; Secondary 42B20, 42B25.
Key words and phrases : Local fractional integral, local fractional maximal operator, two-weight inequality, Gaussianmeasure space. ∗ Correspoding author α = α/d . Here all the sides of the cubes are parallel to the coordinate axes.Recently the study of Gaussian harmonic analysis has aroused extensive attention. In 2007,Mauceri and Meda [12] introduced the local maximal operator and used it to develop the singularintegral operator theory on the Gaussian measure spaces; in the same year, Aimar et al. [1] obtainedthe weak type (1 ,
1) inequalities for the Gaussian Riesz transform and a general maximal operatorwhich dominates the Ornstein-Uhlenbeck maximal operator; then in 2010, Liu and Yang[11] obtainedthe boundedness of the local fractional integral operator defined by I aα ( f )( x ) := Z B ( x,am ( x )) f ( y )[ γ ( B ( x, | x − y | ))] − ˙ α d γ ( y )and the local fractional maximal operator M aα on the Gaussian Lebesgue spaces; in 2014, Liu et al. [10]obtained the boundedness of the local fractional integral operator and the local maximal operatoron the Gaussian Morrey-type spaces; later in 2016, Wang et al. [19] characterized the one-weightboundedness of M a on Gaussian Lebesgue spaces by the local A ap weights defined as[ ω ] A ap = sup B ∈ B a (cid:18) γ ( B ) Z B ω ( x )d γ ( x ) (cid:19) (cid:18) γ ( B ) Z B ω ( x ) − p ′ d γ ( x ) (cid:19) p − < ∞ , and obtained the one-weight boundedness of M aα on the Gaussian Lebesgue spaces by the local A ap,q weights defined as[ w ] A ap,q = sup B ∈ B a (cid:18) γ ( B ) Z B ω ( x ) q d γ ( x ) (cid:19) /q (cid:18) γ ( B ) Z B ω ( x ) − p ′ d γ ( x ) (cid:19) /p ′ < ∞ ;more recently in 2020, Lin and Mao [8] established the one-weight norm inequalities associated withthe local A ap,q weights for the fractional operators I aα and M aα on the Gaussian measure spaces.Based on the results above, this article aims to establish the two-weight norm inequalities forthe local fractional maximal and integral operators with respect to the local Muckenhoupt typeweights and the local Sawyer type weights. To state our main results, we introduce the followinglocal Muckenhoupt type weights. Definition 1.1.
Given 0 < a < ∞ , 0 ≤ ˙ α < α = ˙ αd and 1 < p, q < ∞ , we say that a pair ofweights ( u, v ) ∈ A ap,q,α if[ u, v ] A ap,q,α := sup Q ∈ Q a γ ( Q ) ( ˙ α +1 /q − /p ) (cid:18) γ ( Q ) Z Q u ( x )d γ ( x ) (cid:19) /q (cid:18) γ ( Q ) Z Q v ( x ) − p ′ d γ ( x ) (cid:19) /p ′ < ∞ . For p = 1, we say ( u, v ) ∈ A a ,q,α if[ u, v ] A a ,q,α := sup Q ∈ Q a ess sup x ∈ Q γ ( Q ) ( ˙ α +1 /q − (cid:18) γ ( Q ) Z Q u ( x )d γ ( x ) (cid:19) /q v ( x ) − < ∞ , where the essential supremum is associated with the measure γ . Remark 1.2.
Note that if α + d/q − d/p = 0 and u ( x ) = ω ( x ) q , v ( x ) = ω ( x ) p , then the two-weight condition A ap,q,α goes back to the one-weight condition A ap,q first introduced byWang et al [19]; if α + d/q − d/p <
0, by letting suitable γ ( Q ) →
0, we conclude the fact A ap,q,α = ∅ ;finally if α + d/q − d/p >
0, it is obvious that u ( x ) = v ( x ) = 1 satisfy the A ap,q,α condition, i.e., A ap,q,α = ∅ . 2ne of our main results in this paper is the following two-weight weak-type estimate for M aα . Theorem 1.3.
Given a ∈ (0 , ∞ ) , ≤ p ≤ q < ∞ , ≤ α < d and a pair of weights ( u, v ) , if ( u, v ) ∈ [ b ′ >a A b ′ p,q,α , then M aα is bounded from L p (cid:0) R d , v, γ (cid:1) to L q, ∞ (cid:0) R d , u, γ (cid:1) , that is, there exists a constant C > suchthat Z { x ∈ R d : M aα ( f )( x ) >λ } u ( x )d γ ( x ) ≤ Cλ q (cid:18)Z R d | f ( x ) | p v ( x )d γ ( x ) (cid:19) q/p holds for all λ > . In addition, we get the following two-weight weak-type estimate for the local fractional integralopeartor ˜ I aα with cubes on Gaussian measure spaces defined by˜ I aα ( f )( x ) := Z Q ( x,am ( x )) f ( y )[ γ ( Q ( x, | x − y | ))] − ˙ α d γ ( y ) , where Q ( x, am ( x )) denotes the cube with c Q = x and ℓ ( Q ) = am ( x ). Theorem 1.4.
Given a ∈ (0 , ∞ ) , ≤ p ≤ q < ∞ , < α < d and a pair of weights ( u, v ) , if ( u, v ) ∈ [ a suchthat Z { x ∈ R d : ˜ I aα ( f )( x ) >λ } u ( x )d γ ( x ) ≤ Cλ q (cid:18)Z R d | f ( x ) | p v ( x )d γ ( x ) (cid:19) q/p holds for all λ > . Based on the radialization method in [5] and inspired by [6], we prove Theorem 1.3 by introducinga radial version of the local fractional maximal operator M aα and the A ap,q,α weights. So that we can usethe dyadic analysis on Gaussian measure spaces to prove the desired conclusion. Then Theorem 1.4comes from an extended Welland type inequality on Gaussian measure spaces (see Lemma 3.5 below)and the boundedness of M aα . Furthermore using a similar method, we also establish the two-weightstrong-type estimates for M aα and ˜ I aα under the following local Sawyer type condition. Definition 1.5.
Let a ∈ (0 , ∞ ), α ∈ [0 , d ) and 1 < p ≤ q < ∞ . We say that a pair of weights ( u, v )satisfies the local- a testing condition if[ u, v ] M ap,q,α := sup Q ∈ Q a (cid:20)Z Q (cid:16) M aα ( v − p ′ χ Q )( x ) (cid:17) q u ( x )d γ ( x ) (cid:21) /q (cid:18)Z Q v ( x ) − p ′ d γ ( x ) (cid:19) − /p < ∞ . In this situation, we write ( u, v ) ∈ M ap,q,α . Remark 1.6.
We can rewrite the definition of [ u, v ] M ap,q,α assup Q ∈ Q a "Z Q sup Q ′ ∈ Q a ( x ) γ ( Q ′ ) − ˙ α Z Q ′ ∩ Q v ( y ) − p ′ d γ ′ ( y ) ! q u ( x )d γ ( x ) q (cid:18)Z Q v ( x ) − p ′ d γ ( x ) (cid:19) − p . Thereby when d/q − d/p + α <
0, similar to Remark 1.2, we conclude M ap,q,α = ∅ by letting suitable Q = Q ′ and γ ( Q ′ ) = γ ( Q ) →
0. 3hen we obtain the two-weight strong-type boundedness of M aα and ˜ I aα as follows. Theorem 1.7.
Given a ∈ (0 , ∞ ) , < p ≤ q < ∞ , ≤ α < d and a pair of weights ( u, v ) , if ( u, v ) ∈ M a +9 √ da p,q,α , then M aα is bounded from L p ( R d , v, γ ) to L q ( R d , u, γ ) , that is, there exists a constant C > such that (cid:18)Z R d (cid:16) M aα ( f )( x ) (cid:17) q u ( x )d γ ( x ) (cid:19) /q ≤ C (cid:18)Z R d | f ( x ) | p v ( x )d γ ( x ) (cid:19) /p . Theorem 1.8.
Given a ∈ (0 , ∞ ) , < p ≤ q < ∞ , < α < d and a pair of weights ( u, v ) , if ( u, v ) ∈ [ a +9 √ da such that (cid:18)Z R d (cid:16) ˜ I aα ( f )( x ) (cid:17) q u ( x )d γ ( x ) (cid:19) /q ≤ C (cid:18)Z R d | f ( x ) | p v ( x )d γ ( x ) (cid:19) /p . The highlights of the paper are as follows. In Section 2, we give some basic facts used in theproofs of the desired main results, as well as the radial versions of M aα and A ap,q,α . In Section 3,we investigate some properties for the local Muckenhoupt type weights and prove the two-weightweak-type boundedness of M aα and ˜ I aα . Meanwhile a natural question (see Question 1 below) ariseshere. In Section 4, we study the similar properties for the local Sawyer type weights and prove thetwo-weight strong-type boundedness of M aα and ˜ I aα . Furthermore, a similar question (see Question 2below) is stated at the end of Section 4.We end this section with some notions and notations. Hereafter, we will be working in R d and d will always denote the dimension. We will denote by C or its variants a positive constant independentof the main involved parameters, and use f . g to denote f ≤ Cg ; particularly, if f . g . f , thenwe will write f ∼ g . If necessary, we will denote the dependence of the constants parenthetically,e.g., C = C ( a, d ) or C = C a,d . Similarly, f . a,d g will denote f ≤ C a,d g and f ∼ a,d g will denote C a,d f ≤ g ≤ C a,d f . By a weight we will always mean a locally integrable function and non-negativealmost everywhere on R d (with respect to the associated measure γ ). For a given weight ω , theweighted Gaussian Lebesgue norms on R d will be denoted by k f k L p ( R d ,ω,γ ) := (cid:18)Z R d | f ( x ) | p ω ( x )d γ ( x ) (cid:19) p , and k f k L p, ∞ ( R d ,ω,γ ) := sup λ> λ Z { x ∈ R d : f ( x ) >λ } ω ( x )d γ ( x ) ! p . It is easy to see that the probability measure γ is highly concentrated around the origin with ex-ponential decay at infinity. Thereby it is not a doubling measure, i.e., there is no constant C > x ∈ R d and r >
0, such that γ ( B ( x, r )) ≤ Cγ ( B ( x, x ))4olds for all x ∈ R d and r >
0. Here γ ( B ) := R B d γ ( x ). See [18, Appendix 10.3] for more details.Hence we know that the Gaussian measure space is not a homogeneous type space in the sense ofCoifman and Weiss [4]. However, as we have mentioned in Section 1, if we define the family ofadmissible cubes Q a , then [12, Proposition 2.1] points out that e | c Q | ∼ a e | x | (1)holds for all Q ∈ Q a and all x ∈ Q . From this estimate we conclude that γ ( Q ) = Z Q d γ ( x ) ∼ a,d e −| c Q | Z Q d x = e −| c Q | ℓ ( Q ) d (2)holds for all Q ∈ Q a . Furthermore the Gaussian measure is doubling and reverse doubling if werestrict it to Q a . In other words, there exist constants C = C a,d ≥ C = C ′ a,d > Q ∈ Q a we have γ (2 Q ) ≤ C γ ( Q ) , γ (2 Q ) ≥ C γ ( Q ) . Hence we say γ satisfies the locally doubling condition and locally reverse doubling condition on Q a .On the other hand, the Gaussian measure is trivially a d -dimensional measure in R d , i.e., γ ( Q ) ≤ ℓ ( Q ) d holds for all cubes Q in R d . Therefore some results on the d -dimensional measure, such as [17] and [6],may be useful in Gaussian harmonic analysis.By using the estimates (1) and (2), we conclude that1[ γ ( Q )] − ˙ α Z Q | f ( y ) | d γ ( y ) ∼ a,d e −| c Q | γ ( Q )] − ˙ α Z Q | f ( y ) | d y ∼ a,d e − ˙ α | c Q | ℓ ( Q ) d − α Z Q | f ( y ) | d y ∼ a,d ℓ ( Q ) d − α Z Q | f ( y ) | e − ˙ α | y | d y holds for all Q ∈ Q a . Now we can give the pointwise equivalent radial version of M aα as follows. Definition 2.1.
Let a ∈ (0 , ∞ ), ˙ α ∈ [0 , α = ˙ αd and f ∈ L loc ( γ ). We define the local fractionalmaximal operator M aα on Gaussian measure spaces by setting M aα ( f )( x ) := sup Q ∈ Q a ( x ) γ ( Q )] − ˙ α Z Q | f ( y ) | d γ ( y ): ∼ a,d sup Q ∈ Q a ( x ) ℓ ( Q ) d − α Z Q | f ( y ) | d γ ′ ( y ) , where d γ ′ ( y ) = e − ˙ α | y | d y = e − α | y | /d d y .To use the dyadic analysis on Gaussian measure spaces more conveniently and adapt the A ap,q,α condition to our radialization method, we introduce the following A ap,q,α condition. Definition 2.2.
Given 0 < a < ∞ , 0 ≤ ˙ α < α = ˙ αd and 1 < p ≤ q < ∞ , we say that a pair ofweights ( u, v ) ∈ A ap,q,α if[ u, v ] A ap,q,α = sup Q ∈ Q a ℓ ( Q ) d − α (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) /q (cid:18)Z Q v ( x ) − p ′ d γ ′ ( x ) (cid:19) /p ′ < ∞ .
5n the case p = 1, we say ( u, v ) ∈ A a ,q,α if there exists a constant C such that for every cube Q ∈ Q a the inequality 1 ℓ ( Q ) d − α (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) /q ≤ Cv ( x )holds for γ ′ -a.e. x ∈ Q .We can show the following close relation between the A ap,q,α condition and the A ap,q,α condition. Propsition 2.3.
Let u ′ ( x ) := u ( x ) e − (1 − ˙ α ) | x | , v ′ ( x ) := v ( x ) e − (1 − ˙ α ) | x | . Then ( u, v ) ∈ A ap,q,α ⇔ ( u ′ , v ′ ) ∈ A ap,q,α . Proof.
We focus on proving the case p > p = 1 is essentially the same. For everyfixed Q ∈ Q a , we have known that e −| x | ∼ a e | y | (3)holds for any x ∈ Q and any y ∈ Q . Hence we conclude γ ( Q ) ( ˙ α +1 /q − /p ) (cid:18) γ ( Q ) Z Q u ( x )d γ ( x ) (cid:19) /q (cid:18) γ ( Q ) Z Q v ( x ) − p ′ d γ ( x ) (cid:19) /p ′ ∼ a,d ℓ ( Q ) α + d/q − d/p e −| c Q | ( ˙ α +1 /q − /p ) e −| c Q | ℓ ( Q ) d e −| c Q | Z Q u ( x )d x ! /q e −| c Q | ℓ ( Q ) d e −| c Q | Z Q v ( x ) − p ′ d x ! /p ′ = ℓ ( Q ) α + d/q − d/p e −| c Q | ( ˙ α +1 /q − /p ) (cid:18) ℓ ( Q ) d Z Q u ( x )d x (cid:19) /q (cid:18) ℓ ( Q ) d Z Q v ( x ) − p ′ d x (cid:19) /p ′ . On the other hand, using the estimate (3) again we have1 ℓ ( Q ) d − α (cid:18)Z Q u ( x ) e − (1 − ˙ α ) | x | d γ ′ ( x ) (cid:19) /q (cid:18)Z Q v ( x ) − p ′ e − (1 − ˙ α ) | x | (1 − p ′ ) d γ ′ ( x ) (cid:19) /p ′ = 1 ℓ ( Q ) d − α (cid:18)Z Q u ( x )d x (cid:19) /q e −| c Q | /q (cid:18)Z Q v ( x ) − p ′ d x (cid:19) /p ′ e −| c Q | ( ˙ α − /p ) = ℓ ( Q ) α + d/q − d/p e −| c Q | ( ˙ α +1 /q − /p ) (cid:18) ℓ ( Q ) d Z Q u ( x )d x (cid:19) /q (cid:18) ℓ ( Q ) d Z Q v ( x ) − p ′ d x (cid:19) /p ′ . These two facts yield the desired result.
Due to the Proposition 2.3 above, we investigate some properties of the A ap,q,α condition. In theone-weight case , Wang et al. [19] point out that the local Moukenhoupt weights A ap on Gaussianmeasure spaces have the property A ap = A bp and then A ap,q = A bp,q ; but in the two-weight case, thesimilar result A ap,q,α = A bp,q,α is not always true. To see this fact, we need the following example first. Example 3.1.
Let 0 < a < b < ∞ , n ∈ Z + , u ( x ) and v ( x ) be even functions on R . When x ∈ R + , u ( x ) = , x ∈ (0 , ,n q , x ∈ (cid:0) n − a + ( n − b, n − a + ( n − b + b − a (cid:1) , /n q , x ∈ (cid:0) n − a + ( n − b + b − a , na + nb (cid:1) , n ≥ v ( x ) = , x ∈ (0 , b ) ,n p , x ∈ (cid:0) n − a + ( n − b, n − a + nb − b − a (cid:1) , /n p , x ∈ (cid:0) n − a + nb − b − a , n − a + nb (cid:1) , with n ≥
2. Define[ u, v ] A ap,q,α = sup Q ∈ Q ′ a ℓ ( Q ) − α (cid:18)Z Q u ( x )d x (cid:19) /q (cid:18)Z Q v ( x ) − p ′ d x (cid:19) /p ′ , where Q ′ a = { Q ⊂ R d : ℓ ( Q ) ≤ a } . If 1 /q − /p + α ≥
0, then[ u, v ] A ap,q,α < ∞ , [ u, v ] A bp,q,α = ∞ . Proof.
We only need to consider all the cubes Q ∈ Q ′ a on R + due to the symmetry. Setting σ ( x ) = v ( x ) − p ′ , we can draw the function graphs of u ( x ) /q and σ ( x ) /p on R + .10 . O B B ′ B B ′ B B ′ A A ′ A A ′ A A ′ Figure 1:
The fuctions u ( x ) /q and σ ( x ) /p on R + As shown in Figure 1, the green full line segments are the function graph of σ ( x ) /p and thepurple dotted line segments are the function graph of u ( x ) /q . The lengths of these intervals satisfythe following( B i , B ′ i ) = b, ( A i , A ′ i ) = a, ( B i , A i ) = ( A ′ i , B ′ i ) = b − a , ( B ′ i , B i +1 ) = a for every i ∈ Z + . Based on this Figure 1, we shall prove the desired results more intuitively. Set Q n = ( B n , A n ) = (cid:18) n − a + ( n − b, n − a + ( n − b + b − a (cid:19) ,Q n = ( A ′ n , B ′ n ) = (cid:18) n − a + nb − b − a , n − a + nb (cid:19) and Q n = ( B n , B ′ n ) = (cid:16) n − a + ( n − b, n − a + nb (cid:17) . For fixed Q ∈ Q ′ a , since ( A i , A ′ i ) = ( B ′ i , B i +1 ) = a and ℓ ( Q ) ≤ a , the open interval Q cannot intersectwith more than one of these intervals Q n and Q n . Therefore, when Q ∩ Q n = ∅ we obtain1 ℓ ( Q ) − α (cid:18)Z Q u ( x )d x (cid:19) /q (cid:18)Z Q v ( x ) − p ′ d x (cid:19) /p ′ ≤ ℓ ( Q ) − α (cid:18)Z Q n q d x (cid:19) /q (cid:18)Z Q n p ′ d x (cid:19) /p ′ = 1 ℓ ( Q ) − α ℓ ( Q ) /q +1 /p ′ = ℓ ( Q ) /q − /p + α ≤ a /q − /p + α ;7nd when Q ∩ Q n = ∅ we also obtain1 ℓ ( Q ) − α (cid:18)Z Q u ( x )d x (cid:19) /q (cid:18)Z Q v ( x ) − p ′ d x (cid:19) /p ′ ≤ ℓ ( Q ) − α (cid:18)Z Q n q d x (cid:19) /q (cid:18)Z Q n p ′ d x (cid:19) /p ′ = 1 ℓ ( Q ) − α ℓ ( Q ) /q +1 /p ′ = ℓ ( Q ) /q − /p + α ≤ a /q − /p + α , where we have used the assumption that 1 /q − /p + α ≥
0. Hence we have proved the result[ u, v ] A ap,q,α ≤ a /q − /p + α < ∞ . On the other hand, by chosing Q = Q n ∈ Q ′ b , we conclude that[ u, v ] A bp,q,α = sup Q ∈ Q ′ b ℓ ( Q ) d − α (cid:18)Z Q u ( x )d x (cid:19) /q (cid:18)Z Q v ( x ) − p ′ d x (cid:19) /p ′ ≥ ℓ ( Q n ) − α Z Q n u ( x )d x ! /q Z Q n v ( x ) − p ′ d x ! /p ′ = 1 ℓ ( Q n ) − α · n · ℓ ( Q n ) d/q · ℓ ( Q n ) d/p ′ = n (cid:0) b − a (cid:1) /q +1 /p ′ b d − α . Letting n → ∞ , we get the desired result [ u, v ] A bp,q,α = ∞ . This completes the proof. Propsition 3.2.
Let < a < b < ∞ , ≤ α < d and < p ≤ q < ∞ . Then A bp,q,α $ A ap,q,α . Proof.
It’s trivial that A bp,q,α ⊂ A ap,q,α and by Remark 1.2 together with Propsition 2.3 we only needto concentrate on the situation 1 /q − /p + α ≥
0. In other words, it is enough to find ( u, v ) belongsto A ap,q,α but not in A bp,q,α under the assumption 1 /q − /p + α ≥
0. Inspired by Example 3.1, on R + , we set u ( x ) = e ˙ α | x | , x ∈ (0 , ,n kq e ˙ α | x | , x ∈ (cid:16) n − a + P n − i =1 b i , n − a + P n − i =1 b i + b n − a n (cid:17) ,n − kq e ˙ α | x | , x ∈ (cid:16) n − a + P n − i =1 b i + b n − a n , na + P ni =1 b i (cid:17) , with n ≥ v ( x ) = e ˙ α | x | − p ′ , x ∈ (0 , ,n kp e ˙ α | x | − p ′ , x ∈ (cid:16) n − a + P n − i =1 b i , n − a + P ni =1 b i − b n − a n (cid:17) ,n − kp e ˙ α | x | − p ′ , x ∈ (cid:0) n − a + P ni =1 b i − b n − a n , n − a + P ni =1 b i (cid:1) , with n ≥ a n , b n satisfy the condition x − b x = 1 b n = bx n >
01 + ( n − a + P n − i =1 b i = x n − b x n a n = ax n , (4)8 O B B ′ B B ′ A A ′ x A A ′ x B B ′ A A ′ x Figure 2:
The admissible partition of R + and the parameter k will be chosen later. The exponent terms in u ( x ), v ( x ) are used to offset themeasure d γ ′ ( x ) to Lebesgue measure d x and the condition (4) can be shown in Figure 2.As you can see in Figure 2, the red intervals ( B n , B ′ n ) and the black intervals ( A n , A ′ n ) have thesame centers x n . Furthermore, the lengths of these intervals satisfy the following B ′ n − B n = b/x n = b n , A ′ n − A n = a/x n = a n , B n +1 − B ′ n = a so that ( B n , B ′ n ) ∈ Q b , ( A n , A ′ n ) ∈ Q a and these intervals ( B n , B ′ n ) are “far enough to each other”.By even extension, we get u ( x ) and v ( x ) on R . Similarly set Q n = ( B n , A n ) = n − a + n − X i =1 b i , n − a + n − X i =1 b i + b n − a n ! ,Q n = ( A ′ n , B ′ n ) = n − a + n X i =1 b i − b n − a n , n − a + n X i =1 b i ! and Q n = ( B n , B ′ n ) = n − a + n − X i =1 b i , n − a + n X i =1 b i ! . For fixed Q ∈ Q a , we should avoid the situation that Q ∩ Q n = ∅ and Q ∩ Q n = ∅ hold simultaneously.Actually, this situation may appear if n = 1. However, we claim that the open interval Q ∈ Q a cannot intersect with more than one of these intervals Q n and Q n if n ≥ /q − /p + α ≥
0, we can follow the scheme of the proofin Example 3.1. For every fixed Q ∈ Q a , if Q ∩ Q n = ∅ with n ≥
2, we conclude that1 ℓ ( Q ) − α (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) /q (cid:18)Z Q v ( x ) − p ′ d γ ′ ( x ) (cid:19) /p ′ ≤ ℓ ( Q ) − α · n k · n − k · | Q | /q · | Q | /p ′ = ℓ ( Q ) /q − /p + α ≤ a /q − /p + α , where | Q | is the Lebesgue measure of Q ; if Q ∩ Q = ∅ , we also conclude that1 ℓ ( Q ) − α (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) /q (cid:18)Z Q v ( x ) − p ′ d γ ′ ( x ) (cid:19) /p ′ ≤ ℓ ( Q ) − α · | Q | /q · | Q | /p ′ = ℓ ( Q ) /q − /p + α ≤ a /q − /p + α . Hence we can obtain the result [ u, v ] A ap,q,α ≤ a /q − /p + α < ∞ . Q = Q n ∈ Q ′ b , we can similarly get[ u, v ] A bp,q,α ≥ ℓ ( Q n ) − α Z Q n u ( x )d γ ′ ( x ) ! /q Z Q n v ( x ) − p ′ d γ ′ ( x ) ! /p ′ = n k · ℓ ( Q n ) − α · | Q n | /q · | Q n | /p ′ = n k · b − αn · (cid:18) b n − a n (cid:19) /q +1 /p ′ = (cid:0) b − a (cid:1) /q +1 /p ′ b − α n k · | x n | − α − /q +1 /p ≥ (cid:0) b − a (cid:1) /q +1 /p ′ b − α n k · (1 + na + nb ) − α − /q +1 /p , where we have used the facts that α + 1 /q − /p ≥ | x n | ≤ na + n X i =1 b i ≤ na + nb. Finally by taking k = α + 1 /q − /p + 1 >
0, we get the desired weights u ( x ) and v ( x ).It remains to prove the claim above. It’s easy to see that if n ≥
2, then x n > a > p a/ x > p a/
2, the function f ( x ) = x + a x is increasing. Hence if n ≥
2, then for all0 < c Q ≤
1, we have x n + a x n > a + 1 ≥ c Q + a ;and for all 1 < c Q < x n , we have x n + a x n > c Q + a c Q . On the other hand, it is obvious that if c Q ≥ x n , then c Q − a c Q ≥ x n − a x n . These comments deduce that there is no interval Q ∈ Q a satisfies the following two conditionssimultaneously ( c Q + am ( c Q )2 > A ′ n = x n + a x n c Q − am ( c Q )2 < A n = x n − a x n when n ≥
2. This finishes the proof of the claim.Now we turn to the proof of Theorem 1.3. Firstly, based on the radial versions of the localoperators and weights constructed above, we can prove the following two lemmas by imitating theproofs in [6] and using some basic facts on the Gaussian measure spaces.
Lemma 3.3.
Let < a < ∞ and ≤ q ≤ dd − α . Then the pair of weights ( u, v ) ∈ A a ,q,α if and onlyif (cid:0) M aβ u ( x ) (cid:1) /q ≤ Cv ( x ) (5) with β = d − ( d − α ) q for γ ′ -almost everywhere x ∈ R d . Indeed, both conditions hold with the sameconstant. roof. Set e Q = n Q ⊂ R d : c Q ∈ Q d and ℓ ( Q ) ∈ Q + o , e Q a = e Q ∩ Q a . By a continuity argument, it is enough to consider the cubes just in e Q a in the definition of M aα .Suppose that ( u, v ) ∈ A a ,q,α with constant C . For every Q ∈ e Q a , define N ( Q ) = ( x ∈ Q : 1 ℓ ( Q ) d − α (cid:18)Z Q u ( y )d γ ′ ( y ) (cid:19) /q > C v ( x ) ) , N = [ Q ∈ e Q a N ( Q ) . Then by the definition of A a ,q,α we have γ ′ ( N ( Q )) = 0 and, since e Q a is countable, γ ′ ( N ) = 0. Set F = n y ∈ R d : (cid:0) M aβ ( u )( y ) (cid:1) /q > C v ( y ) o , where d − βq = d − α . Hence for every y ∈ F , there exists a cube Q ∈ e Q a ( y ) such that1 ℓ ( Q ) d − α (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) /q > C v ( y ) . This yields that F ⊂ N and γ ′ ( F ) = 0. Equivalently, we have proved (cid:0) M aβ u ( x ) (cid:1) /q ≤ C v ( x ) γ ′ − a.e. x ∈ R d . On the other hand if (5) holds with constant C , then for any fixed Q ∈ e Q a and γ ′ -a.e. y ∈ Q we directly conclude1 ℓ ( Q ) d − α (cid:18)Z Q u ( y )d γ ′ ( y ) (cid:19) /q ≤ sup Q ∈ Q a ( y ) ℓ ( Q ) ( d − α ) q Z Q u ( x )d γ ′ ( x ) ! /q = (cid:0) M aβ ( u )( y ) (cid:1) /q ≤ C v ( y ) . Lemma 3.4.
Given a ∈ (0 , ∞ ) , ≤ p ≤ q < ∞ , ≤ α < d and a pair of weights ( u, v ) , then thefollowing two statements are equivalent:(i) ( u, v ) ∈ A ap,q,α .(ii) For every f ≥ and every cube Q ∈ Q a , (cid:18) ℓ ( Q ) d − α Z Q f ( x )d γ ′ ( x ) (cid:19) q (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) ≤ C (cid:18)Z Q f ( x ) p v ( x )d γ ′ ( x ) (cid:19) q/p . Proof.
We adapt some ideas from [6]. Firstly, let us prove the conclusion (ii) with the assumptionthat ( u, v ) ∈ A ap,q,α . In the case p = 1, for every f ≥ Q ∈ Q a we have (cid:18) ℓ ( Q ) d − α Z Q f ( x )d γ ′ ( x ) (cid:19) q (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) = "Z Q f ( x ) 1 ℓ ( Q ) d − α (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) /q d γ ′ ( x ) q ≤ C (cid:18)Z Q f ( x ) v ( x )d γ ′ ( x ) (cid:19) , u, v ) ∈ A a ,q,α . In the case 1 < p < ∞ , by H¨older’s inequality (cid:18) ℓ ( Q ) d − α Z Q f ( x )d γ ′ ( x ) (cid:19) q ≤ ℓ ( Q ) ( d − α ) q (cid:18)Z Q f ( x ) p v ( x )d γ ′ ( x ) (cid:19) q/p (cid:18)Z Q v ( x ) − p ′ d γ ′ ( x ) (cid:19) q/p ′ . Thus the condition ( u, v ) ∈ A ap,q,α yields that (cid:18) ℓ ( Q ) d − α Z Q f ( x )d γ ′ ( x ) (cid:19) q (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) ≤ ℓ ( Q ) ( d − α ) q (cid:18)Z Q f ( x ) p v ( x )d γ ′ ( x ) (cid:19) q/p (cid:18)Z Q v ( x ) − p ′ d γ ′ ( x ) (cid:19) q/p ′ (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) ≤ [ u, v ] q A ap,q,α (cid:18)Z Q f ( x ) p v ( x )d γ ′ ( x ) (cid:19) q/p . On the other hand, suppose the statement (ii) is right. Take f ≥
0. For any S ⊂ Q ∈ Q a , using(ii) with f χ S ( x ) we get the following (cid:18) ℓ ( Q ) d − α Z S f ( x )d γ ′ ( x ) (cid:19) q (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) ≤ C (cid:18)Z S f ( x ) p v ( x )d γ ′ ( x ) (cid:19) q/p . (6)Then by taking f ≡ (cid:18) γ ′ ( S ) ℓ ( Q ) d − α (cid:19) q (cid:2) u ( Q, γ ′ ) (cid:3) ≤ C (cid:2) v ( S, γ ′ ) (cid:3) q/p , (7)where γ ′ ( S ) = Z S d γ ′ ( x ) , u ( Q, γ ′ ) = Z Q u ( x )d γ ′ ( x ) , v ( S, γ ′ ) = Z S v ( x )d γ ′ ( x ) . For any Q ∈ Q a , from this inequality (7) we claim that v ( x ) > γ ′ − a.e. x ∈ Q unless u ( x ) = 0 γ ′ − a.e. x ∈ Q ; u ( Q, γ ′ ) < ∞ unless v ( x ) = ∞ γ ′ − a.e. x ∈ Q. Indeed, if v ( x ) = 0 on some S ⊂ Q with γ ′ ( S ) >
0, from the inequality (7) we have u ( Q, γ ′ ) = 0;if u ( Q, γ ′ ) = ∞ , using this estimate (7) again, we conclude that v ( S, γ ′ ) = ∞ for all S ⊂ Q with γ ′ ( S ) >
0. Hence we finish the proof of the claim.Once we have done these observations shown above, we can prove this lemma by following thescheme of the proof in [6, Theorem 4.3]. For the case 1 < p < ∞ , we can obtain the desired conclusionby setting f ( x ) = v ( x ) − p ′ , S j = (cid:26) x ∈ Q : v ( x ) > j (cid:27) where Q ∈ Q a and using some locally integrable arguments together with the fact v ( x ) > γ ′ -a.e. x ∈ Q ; for the case p = 1, we can get the desired result by rewriting the estimate (7) as1 ℓ ( Q ) d − α (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) /q ≤ C v ( S, γ ′ ) γ ′ ( S ) ∀ S ⊂ Q ∈ Q a with γ ′ ( S ) > , considering a > ess inf Q := inf (cid:8) t > γ ′ { x ∈ Q : v ( x ) < t } > (cid:9) , S a = { x ∈ Q : v ( x ) < a } ⊂ Q ∈ Q a and using a continuity argument. This part is a little long but essentially the same as [6, Theorem4.3] since the restriction Q ∈ Q a does not metter. Thereby we omit the detailed proof here.12ext we can use the dyadic analysis and covering theorem together with some techniques onGaussian measure spaces to establish the two-weight weak-type boundedness of M aα with respect tothe A ap,q,α condition on Gaussian Lebesgue spaces. Proof of Theorem 1.3 . By using Proposition 2.3, we will prove that if ( u, v ) ∈ A bp,q,α for some b > a , then Z { x ∈ R d : M aα ( f )( x ) >λ } u ( x )d γ ′ ( x ) . λ q (cid:18)Z R d | f ( x ) | p v ( x )d γ ′ ( x ) (cid:19) q/p (8)holds for every λ > f ∈ L p ( R d , v, γ ′ ). By Lemma 3.4, weknow that the following estimate (cid:18) ℓ ( Q ) d − α Z Q f ( x )d γ ′ ( x ) (cid:19) q (cid:18)Z Q u ( x )d γ ′ ( x ) (cid:19) . (cid:18)Z Q f ( x ) p v ( x )d γ ′ ( x ) (cid:19) q/p (9)holds for every f ≥ Q ∈ Q b . When f ∈ L p ( R d , v, γ ′ ) and Q ∈ Q b with u ( Q, γ ′ ) >
0, theinequality (9) yields that Z Q f ( x )d γ ′ ( x ) < ∞ . It follows that if R Q f ( x )d γ ′ ( x ) = ∞ for some cube Q , then u ( Q , γ ′ ) = 0 and there is a improperpoint x of the function f ( x ) contained in the closure of the cube Q . For this x , using the estimate(9) again we conclude that u ( Q ′ , γ ′ ) = 0 for all Q ′ ∈ Q b ( x ). On the other hand, by the definition of M aα we konw that M aα ( f )( x ) = ∞ for all x ∈ Q ′ ∈ Q a ( x ). Thus we observe that u ( x ) ≡ , M aα ( f )( x ) ≡ ∞ for γ ′ -a.e. x ∈ S Q ′ ∈ Q a ( x ) Q ′ . Notice that the desired result (8) also holds in this case. Thereby weonly need to focus on the situation x / ∈ S Q ′ ∈ Q b ( x ) Q ′ where M aα ( f )( x ) < ∞ . In this way, withoutloss of generality, we can assume that Z Q f ( x )d γ ′ ( x ) < ∞ for all Q ∈ Q a . By the Heine-Borel theorem, we can assume f ∈ L ( R d , γ ′ ). Moreover, by defining f k = f χ Q (0 ,k ) , if the estimate (8) holds for each f k ∈ L ( R d , γ ′ ) with the constant independentof k , then using Fatou’s lemma we can obtain (8) for all f ∈ L ( R d , γ ′ ). Hence we can assume f ∈ L ( R d , γ ′ ). Take all these remarks into account, we only need to prove the desired result for f ≥ , f ∈ L p ( R d , v, γ ′ ) ∩ L ( R d , γ ′ ) . Define E λ := { x ∈ R d : M aα ( f )( x ) > λ } . Then for every x ∈ E λ , there exists a cube Q x ∈ Q a ( x ) such that1 ℓ ( Q x ) d − α Z Q x | f ( y ) | d γ ′ ( y ) > λ. Note that ℓ ( Q x ) ≤ a . By [16, Theorem 1.5], there exists a constant N = N ( d, k ) depending only onthe dimension d and the ratio k = b/a such that E λ ⊂ [ x ∈ E λ Q x ⊂ N [ j =1 [ x ρ ∈ E λ,j kQ x ρ , { Q x ρ } x ρ ∈ E λ,j are disjiont for fixed j . Recall that (9) leads to (6).By taking S = Q x ρ and Q = kQ x ρ ∈ Q b in this estimate (6) we can obtain Z E λ u ( x )d γ ′ ( x ) ≤ N X j =1 X x ρ ∈ E λ,j Z kQ xρ u ( x )d γ ′ ( x ) . N X j =1 X x ρ ∈ E λ,j ℓ ( Q x ρ ) d − α Z Q xρ f ( x )d γ ′ ( x ) ! − q Z Q xρ f ( x ) p v ( x )d γ ′ ( x ) ! q/p ≤ N X j =1 X x ρ ∈ E λ,j λ q Z Q xρ f ( x ) p v ( x )d γ ′ ( x ) ! q/p ≤ N X j =1 λ q X x ρ ∈ E λ,j Z Q xρ f ( x ) p v ( x )d γ ′ ( x ) q/p ≤ Nλ q (cid:18)Z R d f ( x ) p v ( x )d γ ′ ( x ) (cid:19) q/p , where we have used the fact p ≤ q and the disjointness of { Q x ρ } E λ,j for fixed j .Now we turn to the proof of the two-weight weak-type estimate for the local fractional integraloperator ˜ I aα with cubes on Gaussian measure spaces. We need to introduce the following lemma first. Lemma 3.5.
Let a ∈ (0 , ∞ ) , ≤ α < α < α ≤ d and f ∈ L ( γ ) . If ˜ b > a , then there exists aconstant C independent of the function f such that | ˜ I aα ( f )( x ) | ≤ C (cid:0) M aα ( f )( x ) (cid:1) α − αα − α (cid:16) M ˜ bα ( f )( x ) (cid:17) α − α α − α holds for all x ∈ R d .Proof. We adapt some ideas from [8, Lemma 3.4] and [9, Lemma 4.1]. Set ˙ α = α /d and ˙ α = α /d .If x = ( x , . . . , x d ) ∈ R d , we define the norm k · k on R d by k x k = max {| x | , . . . , | x d |} . Then for any x ∈ R d , we divide the proof into two cases. Case 1: γ [ Q ( x, am ( x ))] ≤ (cid:18) M ˜ bα ( f )( x ) M aα ( f )( x ) (cid:19) α − ˙ α .By using the locally reverse doubling property of γ on the admissiable cubes Q a , we obtain aconstant R = R d,a ∈ (1 , ∞ ) such that | ˜ I aα ( f )( x ) | ≤ ∞ X j =0 Z − j − am ( x ) ≤ k x − y k < − j am ( x ) | f ( y ) | [ γ ( Q ( x, | x − y | ))] − ˙ α d γ ( y ) . ∞ X j =0 [ γ ( Q ( x, − j am ( x )))] ˙ α − ˙ α [ γ ( Q ( x, − j am ( x )))] − ˙ α Z Q ( x, − j am ( x )) | f ( y ) | d γ ( y ) ≤ [ γ ( Q ( x, am ( x )))] ˙ α − ˙ α ∞ X j =0 R − j ( ˙ α − ˙ α ) M aα ( f )( x ) . [ γ ( Q ( x, am ( x )))] ˙ α − ˙ α M aα ( f )( x ) . | ˜ I aα ( f )( x ) | . (cid:0) M aα ( f )( x ) (cid:1) ˙ α − ˙ α ˙ α − ˙ α (cid:16) M ˜ bα ( f )( x ) (cid:17) ˙ α − ˙ α α − ˙ α = (cid:0) M aα ( f )( x ) (cid:1) α − αα − α (cid:16) M ˜ bα ( f )( x ) (cid:17) α − α α − α . Case 2: γ [ Q ( x, am ( x ))] > (cid:18) M ˜ bα ( f )( x ) M aα ( f )( x ) (cid:19) α − ˙ α .From [10, Lemma 2.11] we know there exists an r ∈ (0 , a ) such that12 M ˜ bα ( f )( x ) M aα ( f )( x ) ! α − ˙ α < γ [ Q ( x, rm ( x ))] < M ˜ bα ( f )( x ) M aα ( f )( x ) ! α − ˙ α . (10)Then we can write the following | ˜ I aα ( f )( x ) | ≤ Z k x − y k Proof of Theorem 1.4 . Assume that ( u, v ) ∈ A bp,q,α − ε for some b > a and ε > 0. First, the fact γ ( Q ) < A bp,q,α ⊂ A bp,q,α if 0 < α < α < d . Hence it is sufficient to consider the case 0 < ε < min { α, d − α } . Then usingLemma 3.5 with ˜ b = a + b , α − α = α − α = ε and using the H¨older’s inequality for weak spaces (see [7, Exercise 1.1.15]) with q = q = 2 q weconclude that (cid:13)(cid:13)(cid:13) ˜ I aα ( f ) (cid:13)(cid:13)(cid:13) L q, ∞ ( R d ,u,γ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) M a + b α + ε ( f )( · ) (cid:19) / (cid:0) M aα − ε ( f )( · ) (cid:1) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q, ∞ ( R d ,u,γ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) M a + b α + ε ( f )( · ) (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q , ∞ ( R d ,u,γ ) (cid:13)(cid:13)(cid:13)(cid:0) M aα − ε ( f )( · ) (cid:1) / (cid:13)(cid:13)(cid:13) L q , ∞ ( R d ,u,γ ) = (cid:13)(cid:13)(cid:13)(cid:13) M a + b α + ε ( f ) (cid:13)(cid:13)(cid:13)(cid:13) / L q, ∞ ( R d ,u,γ ) (cid:13)(cid:13) M aα − ε ( f ) (cid:13)(cid:13) / L q, ∞ ( R d ,u,γ ) . k f k L p ( R d ,u,γ ) , where the last inequality comes from Theorem 1.3.Based on the Theorem 1.3 proved above and the one-weight results in [19], it is natural to askthe following question. Question 1. Is ( u, v ) ∈ A ap,q,α rather than ( u, v ) ∈ S b ′ >a A b ′ p,q,α sufficient for the two-weight weak-type boundedness of M aα ?It is not hard to see that ( u, v ) ∈ A ap,q,α is necessary for the two-weight weak-type boundednessof M aα . This problem (the equivalence) may be solved by means of using other covering theoreminstead of [16, Theorem 1.5] or other more delicate approach, but we have no progress yet. On theother hand, we have the following example to yield that the relation A ap,q,α = [ b ′ >a A b ′ p,q,α is not always true. Here we only give the example for the case a = 1, the method is also valid forgeneral a > Example 3.6. Let u ( x ) and v ( x ) be even functions. When x ∈ R + , v ( x ) = x p ′− e ˙ αx − p ′ , < x < ,e ˙ αx − p ′ , otherwise, u ( x ) = (cid:26) , < x < ,e ˙ αx , otherwise. Then for any b ′ > u, v ] A p,q,α < ∞ , [ u, v ] A b ′ p,q,α = ∞ . Proof. Let’s first consider the following[ u, v ] A b ′ p,q,α = sup ( c,c ) ∈ Q b ′ | c − c | − α (cid:18)Z c c u ( x )d γ ′ ( x ) (cid:19) /q (cid:18)Z c c v ( x ) − p ′ d γ ′ ( x ) (cid:19) /p ′ = sup ( c,c ) ∈ Q b ′ | c − c | − α (cid:18)Z c c u ′ ( x )d x (cid:19) /q (cid:18)Z c c σ ′ ( x )d x (cid:19) /p ′ where b ′ > R + σ ′ ( x ) = (cid:26) /x, < x < , , otherwise, u ′ ( x ) = (cid:26) , < x < , , otherwise. We take the interval ( c, c ) := (0 , c ′ ) where 1 < c ′ < min { , b ′ } . It’s easy to see that (0 , c ′ ) ∈ Q b ′ .Hence we conclude [ u, v ] A b ′ p,q,α ≥ c ′ − α Z c ′ d x ! /q Z c ′ x d x ! /p ′ = 1 c ′ − α (cid:0) c ′ − (cid:1) /q Z c ′ x d x ! /p ′ = ∞ . On the other hand, since[ u, v ] A p,q,α = sup ( c,c ) ∈ Q | c − c | − α (cid:18)Z c c u ( x )d γ ′ ( x ) (cid:19) /q (cid:18)Z c c v ( x ) − p ′ d γ ′ ( x ) (cid:19) /p ′ = sup ( c,c ) ∈ Q | c − c | − α (cid:18)Z c c u ′ ( x )d x (cid:19) /q (cid:18)Z c c σ ′ ( x )d x (cid:19) /p ′ , we divide the proof into three cases as follows. Case 1: If the interval [ c, c ] ⊂ [ − , u, v ] A p,q,α = 0 by the definition of u ( x ); Case 2: If c > c < − 1, we deduce that[ u, v ] A p,q,α ≤ ( c − c ) /q +1 /p ′ | c − c | − α ≤ ( c − c ) α +1 /q − /p ≤ Case 3: If 1 ∈ ( c, c ) or − ∈ ( c, c ), we only need to prove the result under the situation1 ∈ ( c, c ) by the symmetry. Moreover, by a continuity argument it is enough to consider thebehavior of c → c − c = 1. In this case, set y = c − 1. Then it follows that1 | c − c | − α (cid:18)Z c c u ( x )d γ ′ ( x ) (cid:19) /q (cid:18)Z c c v ( x ) − p ′ d γ ′ ( x ) (cid:19) /p ′ = 1 | c − c | − α (cid:18)Z y d x (cid:19) /q (cid:18)Z y x d x + Z y d x (cid:19) /p ′ = y /q ( − ln y + y ) /p ′ → y → 0. Combining all these three cases, we get the desired result [ u, v ] A p,q,α < ∞ .17 The local Sawyer type weights In this section, we begin with a result about the strict inclusion relation on the local- a testingcondition. The similar result on the A ap,q,α condition has been proved in Proposition 3.2. Propsition 4.1. Let < a < b < ∞ , ≤ α < d and < p ≤ q < ∞ . Then M bp,q,α $ M ap,q,α . Proof. It’s trivial that M bp,q,α ⊂ M ap,q,α and it is enough to focus on the situation 1 /q − /p + α ≥ M ap,q,α = ∅ when 1 /q − /p + α < 0. The desired weights are analogous to that inProposition 3.2. On R + , we set u ( x ) = e ( ˙ α − q | x | + | x | , x ∈ (0 , ,n − kq ( p − e ( ˙ α − q | x | + | x | , x ∈ (cid:16) n − a + P n − i =1 b i , n − a + P n − i =1 b i + b n − a n (cid:17) ,n − kq ( p +1) e ( ˙ α − q | x | + | x | , x ∈ (cid:16) n − a + P n − i =1 b i + b n − a n , na + P ni =1 b i (cid:17) , with n ≥ n ≥ v ( x ) = e | x | − p ′ , x ∈ (0 , b ) ,n − kp ( p − e | x | − p ′ , x ∈ (cid:16) n − a + P n − i =1 b i , n − a + P ni =1 b i − b n − a n (cid:17) ,n kp ( p − e | x | − p ′ , x ∈ (cid:0) n − a + P ni =1 b i − b n − a n , n − a + P ni =1 b i (cid:1) , where a n , b n satisfy the same condition (4). Then by even extension, we can get u ( x ) and v ( x ) on R . For similar reasons we consider the following expressionsup Q ∈ Q a "Z Q sup Q ′ ∈ Q a ( x ) ℓ ( Q ′ ) − α Z Q ′ ∩ Q v ( y ) − p ′ d γ ′ ( y ) ! q u ( x )d γ ( x ) q (cid:18)Z Q v ( x ) − p ′ d γ ( x ) (cid:19) − p (11)only on R + . Using the same notations as in Proposition 3.2, we have u ( x ) ≤ e ( ˙ α − q | x | + | x | , v ( x ) − p ′ ≤ e | x | (12)on (0 , b + a ) and on R \ {∪ Q n ∪ Q n } =: S . Hence if Q ⊂ S or if Q ∩ Q = ∅ which means Q ⊂ (0 , b + a ), we conclude that "Z Q sup Q ′ ∈ Q a ( x ) ℓ ( Q ′ ) − α Z Q ′ ∩ Q v ( y ) − p ′ d γ ′ ( y ) ! q u ( x )d γ ( x ) q (cid:18)Z Q v ( x ) − p ′ d γ ( x ) (cid:19) − p . (cid:20)(cid:18) ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α (cid:19) q ℓ ( Q ) (cid:21) /q ℓ ( Q ) − /p ≤ ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α ℓ ( Q ) /q − /p ≤ ℓ ( Q ′ ∩ Q ) α ℓ ( Q ) /q − /p ≤ ℓ ( Q ) α +1 /q − /p ≤ a α +1 /q − /p by the estimates (12). Thereby for the expression (11), we only need to focus on the case Q ∩ Q n = ∅ with n ≥ Q ∩ Q n = ∅ or Q ∩ Q n = ∅ where n ≥ Q ∩ Q n = ∅ with n ≥ 2, then Q ′ ∈ Q a yields that Q ′ ∩ Q n = ∅ and (cid:18) ℓ ( Q ′ ) − α Z Q ′ ∩ Q v ( y ) − p ′ d γ ′ ( y ) (cid:19) q ∼ a (cid:18) n kp e (1 − ˙ α ) | x | ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α (cid:19) q . (13)18f Q ∩ Q n = ∅ with n ≥ 2, from n − kp ( p − ≤ n kp ( p − and 1 − p ′ ≤ (cid:18) ℓ ( Q ′ ) − α Z Q ′ ∩ Q v ( y ) − p ′ d γ ′ ( y ) (cid:19) q . a (cid:18) n kp e (1 − ˙ α ) | x | ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α (cid:19) q . (14)Based on these two estimates above, when n ≥ 2, for every Q ∈ Q a we divide the proof into twocases. Firstly, if Q ∩ Q n = ∅ , by the estimate (13) we have "Z Q sup Q ′ ∈ Q a ( x ) ℓ ( Q ′ ) − α Z Q ′ ∩ Q v ( y ) − p ′ d γ ′ ( y ) ! q u ( x )d γ ( x ) /q . a "Z Q ∩ Q n (cid:18) n kp e (1 − ˙ α ) | x | ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α (cid:19) q e ( ˙ α − q | x | + | x | n − kq ( p − e −| x | d x /q = n k ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α | Q ∩ Q n | /q ≤ n k ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α ℓ ( Q ) /q . Then from the assumption 1 /q − /p + α ≥ u, v ] M ap,q,α . a n k ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α ℓ ( Q ) /q · (cid:18)Z Q v ( x ) − p ′ d γ ( x ) (cid:19) − /p ∼ n k ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α ℓ ( Q ) /q · n − k | Q | − /p = ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α ℓ ( Q ) /q − /p ≤ a α +1 /q − /p . Secondly, if Q ∩ Q n = ∅ , then the inequality (14) holds. Noticing that u ( x ) = n − kq ( p +1) e ( ˙ α − q | x | + | x | on Q n , we can use an analogous method as in the first case to obtain[ u, v ] M ap,q,α . a n − k ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α ℓ ( Q ) /q · (cid:18)Z Q v ( x ) − p ′ d γ ( x ) (cid:19) − /p . a n − k ℓ ( Q ′ ∩ Q ) ℓ ( Q ′ ) − α ℓ ( Q ) /q · n k | Q | − /p ≤ a α +1 /q − /p . Combining these two cases together with the former cases Q ∩ Q = ∅ and Q ⊂ S , we have provedthe result [ u, v ] M ap,q,α . a α +1 /q − /p < ∞ . Next we consider the term [ u, v ] M bp,q,α . By taking Q = Q n ∈ Q b and Q ′ = Q n ∈ Q b ( y ) for y ∈ Q n ,similar to the proof in Proposition 3.2, we deduce the following[ u, v ] bp,q,α ≥ "Z Q n ℓ ( Q n ) d − α Z Q n v ( y ) − p ′ d γ ′ ( y ) ! q u ( x )d γ ( x ) /q · Z Q n v ( x ) − p ′ d γ ( x ) ! − /p ∼ b Z Q n n kpq e (1 − ˙ α ) q | x | ℓ ( Q n ) αq · n − kq ( p − e ( ˙ α − q | x | + | x | e −| x | d x ! /q · n k | Q n | − /p = n k ℓ ( Q n ) α · | Q n | /q | Q n | − /p ≥ (cid:18) b − a (cid:19) α +1 /q − /p n k (1 + na + nb ) − α − /q +1 /p , α + 1 /q − /p ≥ | x n | ≤ na + n X i =1 b i ≤ na + nb. Finally taking k = α + 1 /q − /p + 1 > 0, we get the desired weights.Then we consider the two-weight strong-type estimate for M aα with respect to the local- a testingcondition on Gaussian Lebesgue spaces. To use the dyadic analysis in this situation, we need thefollowing lemma first. Lemma 4.2. Consider a ∈ (0 , ∞ ) , α ∈ [0 , d ) , f ∈ L ( R d , γ ′ ) and f ≥ . If for some cube Q ∈ Q a and for some t > we have ℓ ( Q ) d − α Z Q f ( y )d γ ′ ( y ) > t, then there exists a dyadic cube P ∈ Q a +3 √ da such that Q ⊂ P and ℓ ( P ) d − α Z P f ( y )d γ ′ ( y ) > α − d t. Proof. Take k ∈ Z such that 2 k − ≤ ℓ ( Q ) < k . Then there exist at most 2 d dyadic cubes { P i } Ni =1 which have the following two properties: ℓ ( P i ) = 2 k and Q ∩ P i = ∅ . From ℓ ( Q ) < ℓ ( P i ) ≤ ℓ ( Q ) and Q ∩ P i = ∅ we conclude that Q ⊂ P i . Moreover, it is easy to seethat at least one of these dyadic cubes, say P , satisfies the condition R P f ( y )d γ ′ ( y ) ℓ ( Q ) d − α > t d . As a consequence, for this dyadic cube P we have1 ℓ ( P ) d − α Z P f ( y )d γ ′ ( x ) > ℓ ( Q ) d − α tℓ ( P ) d − α d ≥ α − d t. Then it remains to prove that P ∈ Q a +3 √ da . We divide the proof into four cases as follows. Case 1: | c Q | ≤ | c P | ≤ 1, then ℓ ( Q ) ≤ a and ℓ ( P ) ≤ ℓ ( Q ) ≤ a ; Case 2: | c Q | ≤ | c P | > 1, then ℓ ( Q ) ≤ a and ℓ ( P ) · | c P | ≤ ℓ ( Q ) · | c Q | + 3 √ dℓ ( Q )2 ! = 2 ℓ ( Q ) | c Q | + 2 ℓ ( Q ) 3 √ dℓ ( Q )2 ≤ a + 3 √ da ; Case 3: | c Q | > | c P | ≤ 1, then ℓ ( Q ) ≤ a/ | c Q | ≤ a and ℓ ( P ) ≤ ℓ ( Q ) ≤ a ; Case 4: | c Q | > | c P | > 1, then ℓ ( Q ) ≤ a/ | c Q | ≤ a and ℓ ( P ) · | c P | ≤ ℓ ( Q ) · | c Q | + 3 √ dℓ ( Q )2 ! = 2 ℓ ( Q ) | c Q | + 2 ℓ ( Q ) 3 √ dℓ ( Q )2 ≤ a + 3 √ da . Combining all these four cases, we obtain the desired conclusion P ∈ Q a +3 √ da .On the basis of this Lemma 4.2, by following the scheme of the proof in [6, Theorem 3.1] and usingthe radialization method mentioned before, we prove the two-weighted strong-type boundedness ofthe local fractional maximal operator M aα on Gaussian Lebesgue spaces.20 roof of Theorem 1.7 . Without loss of generality, we can assume that f ∈ L p ( R d , v, γ ) is a non-negative bounded function with compact support. Thereby we conclude that M aα ( f )( x ) is finite γ -almost everywhere. Decompose R d by R d = [ k ∈ Z Ω k , Ω k = { x ∈ R d : 2 k < M aα ( f )( x ) ≤ k +1 } . Then for every k ∈ Z and every x ∈ Ω k , there exists a cube Q kx ∈ Q a satisfying the condition1 ℓ ( Q kx ) d − α Z Q kx f ( y )d γ ′ ( y ) > k . By Lemma 4.2 we get a dyadic cube P kx ∈ Q a +3 √ da such that Q kx ⊂ P kx and1 ℓ ( P kx ) d − α Z P kx f ( y )d γ ′ ( y ) > α − d k . (15)Notice that ℓ ( P kx ) ≤ a + 3 √ da . Hence for every fixed k , there is a subcollection of maximal disjointdyadic cubes { P kj } j in the sense that for any Q kx there exists a P kj satisfying Q kx ⊂ P kj . Thisconstruction yields that Ω k ⊂ S j P kj . Thus we can decompose Ω k by defining E k = 3 P k \ Ω k , E k = (cid:16) P k \ P k (cid:17) \ Ω k , . . . , E kj = P kj \ j − [ r =1 P kr ! \ Ω k , . . . . In conclusion we obtain R d = [ k ∈ Z Ω k = [ j,k E kj where the sets E kj are disjoint and Ω k are disjoint. For a fixed integer K > 0, defineΛ K = { ( j, k ) ∈ N × Z : | k | ≤ K } . By using the fact E kj ⊂ Ω k and the condition (15) we deduce I K := Z S Kk = − K Ω k ( M aα ( f )( x )) q u ( x )d γ ( x ) = X ( j,k ) ∈ Λ K Z E kj ( M aα ( f )( x )) q u ( x )d γ ( x ) ≤ X ( j,k ) ∈ Λ K Z E kj u ( x )d γ ( x )2 ( k +1) q ≤ (2 d − α +1) q X ( j,k ) ∈ Λ K Z E kj u ( x )d γ ( x ) ℓ ( P kj ) d − α Z P kj f ( y )d γ ′ ( y ) ! q = 2 (2 d − α +1) q ( d − α ) q X ( j,k ) ∈ Λ K Z E kj u ( x )d γ ( x ) ℓ (3 P kj ) d − α Z P kj σ ( y )d γ ′ ( y ) ! q R P kj ( f σ − )( y ) σ ( y )d γ ′ ( y ) R P kj σ ( y )d γ ′ ( y ) q = 2 (2 d − α +1) q ( d − α ) q Z Y T K ( f σ − ) q d ν, where Y = N × Z , σ ( x ) = v ( x ) − p ′ . Furthermore, the measure ν in Y is given by ν ( j, k ) := Z E kj u ( x )d γ ( x ) ℓ (3 P kj ) d − α Z P kj σ ( y )d γ ′ ( y ) ! q T K for measurable function h is defined by the following three equivalent forms T K ( h )( j, k ) := R P kj h ( y ) σ ( y )d γ ′ ( y ) R P kj σ ( y )d γ ′ ( y ) χ Λ K ( j, k ): ∼ a,d R P kj h ( y ) σ ( y )d y R P kj σ ( y )d y χ Λ K ( j, k ): ∼ a,d R P kj h ( y ) σ ( y )d γ ( y ) R P kj σ ( y )d γ ( y ) χ Λ K ( j, k )due to the fact that P kj ∈ Q a +3 √ da . In this way, if we can prove that the operator T K is boundedfrom L p ( R d , σ, γ ) to L q ( Y , ν ) independently of K , we can obtain I K . Z Y T K ( f σ − ) q d ν . (cid:18)Z R d ( f σ − ) p σ d γ (cid:19) q/p = (cid:18)Z R d f ( x ) p v ( x )d γ ( x ) (cid:19) q/p . Based on the uniformity of K and the monotone convergence theorem, we shall prove the desiredresult by letting K → ∞ .Consequently, it remains to prove the uniform boundedness of T K . It is easy to see that T K : L ∞ ( R d , σ, γ ) → L ∞ ( Y , ν )with norm equal or less than 1. Marcinkiewicz interpolation theorem yields that it is sufficient toshow T K : L ( R d , σ, γ ) → L q/p, ∞ ( Y , ν )with norm independent of K . In other words, it is sufficient to prove that ν { ( j, k ) ∈ Y : T K ( h )( j, k ) > λ } . (cid:18) λ Z R d | h ( x ) | σ d γ ( x ) (cid:19) q/p holds for all λ > 0. To establish this estimate, for the fixed nonnegative bounded function h ( x ) withcompact support, set F λ = { ( j, k ) ∈ Y : T K ( h )( j, k ) > λ } = { ( j, k ) ∈ Λ K : T K ( h )( j, k ) > λ } . Noticing the fact E kj ⊂ P kj ∈ Q a +9 √ da , we conclude ν ( F λ ) = X ( j,k ) ∈ F λ Z E kj u ( x )d γ ( x ) ℓ (3 P kj ) d − α Z P kj σ ( y )d γ ′ ( y ) ! q = X ( j,k ) ∈ F λ Z E kj ℓ (3 P kj ) d − α Z P kj σ ( y )d γ ′ ( y ) ! q u ( x )d γ ( x ) ≤ X ( j,k ) ∈ F λ Z E kj (cid:16) M a +9 √ da α (cid:16) σχ P kj (cid:17) ( x ) (cid:17) q u ( x )d γ ( x ) . Taking ℓ ( P kj ) ≤ a + 3 √ da into account, we can extract a maximal disjoint subcollection { P i } i fromthe collection { P kj : ( j, k ) ∈ F λ } in the sense that for every ( j, k ) ∈ F λ there exists an i such that22 kj ⊂ P i . By the disjointness of E kj , the construction of P i and the fact ( u, v ) ∈ M a +9 √ da p,q,α we canobtain ν ( F λ ) ≤ X i X P kj ⊂ P i Z E kj (cid:16) M a +9 √ da α (cid:16) σχ P kj (cid:17) ( x ) (cid:17) q u ( x )d γ ( x ) ≤ X i Z P i (cid:16) M a +9 √ da α ( σχ P i ) ( x ) (cid:17) q u ( x )d γ ( x ) . X i (cid:18)Z P i σ ( x )d γ ( x ) (cid:19) q/p . Recall that these cubes P i were extracted from n P kj : ( j, k ) ∈ F λ o where T K ( h )( j, k ) > λ . This yields Z P i σ ( x )d γ ( x ) < λ Z P i h ( x ) σ ( x )d γ ( x ) . Then by using p ≤ q and the disjointness of P i we shall conclude ν ( F λ ) . X i (cid:18) λ Z P i h ( x ) σ ( x )d γ ( x ) (cid:19) q/p ≤ X i λ Z P i h ( x ) σ ( x )d γ ( x ) ! q/p ≤ (cid:18) λ Z R d h ( x ) σ ( x )d γ ( x ) (cid:19) q/p . All these estimates above are independent of K , in other words, we have completed the proof. Proof of Theorem 1.8 . Assume that ( u, v ) ∈ M bp,q,α − ε for some b > a + 9 √ da and ε > 0. First,the fact γ ( Q ) < M bp,q,α ⊂ M bp,q,α if 0 < α < α < d . Hence it is sufficient to consider the case 0 < ε < min { α, d − α } . Since b > a + 9 √ da , we can choose ˜ a = ˜ a ( a, b ) > a such that 6˜ a + 9 √ d ˜ a < b . For example, we can take˜ a = ab a +9 √ da + a . Then using Lemma 3.5 with ˜ b = ˜ a, α − α = α − α = ε and using the H¨older’s inequality with q = q = 2 q we conclude that (cid:13)(cid:13)(cid:13) ˜ I aα ( f ) (cid:13)(cid:13)(cid:13) L q ( R d ,u,γ ) . (cid:13)(cid:13)(cid:13)(cid:0) M ˜ aα + ε ( f )( · ) (cid:1) / (cid:0) M aα − ε ( f )( · ) (cid:1) / (cid:13)(cid:13)(cid:13) L q ( R d ,u,γ ) ≤ (cid:13)(cid:13)(cid:13)(cid:0) M ˜ aα + ε ( f )( · ) (cid:1) / (cid:13)(cid:13)(cid:13) L q ( R d ,u,γ ) (cid:13)(cid:13)(cid:13)(cid:0) M aα − ε ( f )( · ) (cid:1) / (cid:13)(cid:13)(cid:13) L q ( R d ,u,γ ) = (cid:13)(cid:13) M ˜ aα + ε ( f ) (cid:13)(cid:13) / L q ( R d ,u,γ ) (cid:13)(cid:13) M aα − ε ( f ) (cid:13)(cid:13) / L q ( R d ,u,γ ) . k f k L p ( R d ,u,γ ) , where the last inequality comes from Theorem 1.7.23s we have discussed in Question 1, there is also a similar problem for the local Sawyer typeweights. We state the question as follows. Question 2. Is ( u, v ) ∈ M ap,q,α rather than ( u, v ) ∈ M a +9 √ da p,q,α sufficient for the two-weight strong-type boundedness of M aα ?It is easy to see that ( u, v ) ∈ M ap,q,α is necessary for the two-weight strong-type boundedness of M aα . The parameter 6 a + 9 √ da in our theorem may be smaller by finer estimates. We expect it canbe a but have no idea for this yet. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos. 11871452 and12071473), Beijing Information Science and Technology University Foundation (Grant Nos. 2025031).