# Two-Weight, Weak-Type Norm Inequalities for Fractional Integral Operators and Commutators on Weighted Morrey and Amalgam Spaces

Authors: Hua Wang
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

Let0<γ<n and Iγ be the fractional integral operator of order γ, Iγfx=∫ℝnx−yγ−nfydy and let b,Iγ be the linear commutator generated by a symbol function b and Iγ, b,Iγfx=bx⋅Iγfx−Iγbfx. This paper is concerned with two-weight, weak-type norm estimates for such operators on the weighted Morrey and amalgam spaces. Based on weak-type norm inequalities on weighted Lebesgue spaces and certain Ap-type conditions on pairs of weights, we can establish the weak-type norm inequalities for fractional integral operator Iγ as well as the corresponding commutator in the framework of weighted Morrey and amalgam spaces. Furthermore, some estimates for the extreme case are also obtained on these weighted spaces.


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## Body

## 1. Introduction

### 1.1. Fractional Integral Operators

Letℝn be the n-dimensional Euclidean space equipped with the Euclidean norm ⋅ and the Lebesgue measure dx. For given γ, 0<γ<n, the fractional integral operator (or Riesz potential) Iγ with order γ is defined by (see [1] for the basic properties of Iγ)(1)Iγfx≔1ζγ∫ℝnfyx−yn−γdy,ζγ=πn/22γΓγ/2Γn−γ/2.Weighted norm inequalities for fractional integral operators arise naturally in harmonic analysis and have been extensively studied by several authors. The study of two-weight problem forIγ was initiated by Sawyer in his pioneer paper [2]. By a weight w, we mean that w is a nonnegative and locally integrable function. In [2], Sawyer concerned the following question. Suppose that 1<p≤q<∞. For which pairs of weights w,ν on ℝn is the fractional integral operator bounded from Lpν into weak-Lqw? A necessary and sufficient condition for the weak-type p,q inequality was given by Sawyer. More specifically, he showed the following.Theorem 1 (see [2]).
Let0<γ<n and 1<p≤q<∞. Given a pair of weights w,ν on ℝn, the weak-type inequality(2)σ⋅wx∈ℝn:Iγfx>σ1/q≤C∫ℝnfxpνxdx1/p,holds for any σ>0 if and only if(3)∫QIγχQwxp′νx1−p′dx1/p′≤C⋅wQ1/q′<∞,for all cubes Q in ℝn. Here, χQ denotes the characteristic function of the cube Q, p′=p/p−1 denotes the conjugate index of p, and C is a universal constant.Sawyer’s result is interesting and important, and it promotes a series of research works on this subject (see, e.g., [3, 4, 5–9]), but it has the defect that condition (3) involves the fractional integral operator Iγ itself. In [3], Cruz-Uribe and Pérez considered the case when q=p and found a sufficient Ap-type condition on a pair of weights w,ν which ensures the boundedness of the operator Iγ from Lpν into weak-Lpw, where 1<p<∞. The condition (4) given below is simpler than (3) in the sense that it does not involve the operator Iγ itself, and hence it can be more easily verified.Theorem 2 (see [3]).
Let0<γ<n and 1<p<∞. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(4)Qγ/n⋅1Q∫Qwxrdx1/rp1Q∫Qνx−p′/pdx1/p′≤C<∞.Then, the fractional integral operatorIγ satisfies the weak-type p,p inequality(5)σ⋅wx∈ℝn:Iγfx>σ1/p≤C∫ℝnfxpνxdx1/p,for anyσ>0,where C does not depend on f nor on σ>0.The proof ofTheorem 2 is quite complicated. It depends on an inequality relating the Hardy–Littlewood maximal function and the sharp maximal function which is strongly reminiscent of the good-λ inequality of Fefferman and Stein. For another, more elementary proof, see also [4]. This solves a problem posed by Sawyer and Wheeden in [8]. Moreover, in [5], Li improved this result by replacing the “power bump” in (4) by a smaller “Orlicz bump” (see also [10]). On the other hand, for given 0<γ<n, the linear commutator b,Iγ generated by a suitable function b and Iγ is defined by(6)b,Iγfx≔bx⋅Iγfx−Iγbfx=1ζγ∫ℝnbx−by⋅fyx−yn−γdy.This commutator was first introduced by Chanillo in [11]. In [6], Liu and Lu obtained a sufficient Ap-type condition for the commutator b,Iγ to satisfy a two-weight weak-type p,p inequality, where 1<p<∞. That condition is an Ap-type condition in the scale of Orlicz spaces (see (7) given below).Theorem 3 (see [6]).
Let0<γ<n, 1<p<∞ and b∈BMOℝn. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(7)Qγ/n⋅1Q∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′1+log+tp′ and log+t≔maxlogt,0, that is,(8)log+t=logt,ast>1;0,otherwise.Then, the linear commutatorb,Iγ satisfies the weak-type p,p inequality(9)σ⋅wx∈ℝn:b,Iγfx>σ1/p≤C∫ℝnfxpνxdx1/p,for anyσ>0,where C does not depend on f nor on σ>0.In [7], Martell considered the case when q>p and gave a verifiable condition which is sufficient for the two-weight, weak-type p,q inequality for fractional integral operator Iγ. The condition (10) given below (in the Euclidean setting of [7]) is also simpler than the one in Theorem 1.Theorem 4 (see [7]).
Let0<γ<n and 1<p<q<∞. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(10)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rq1Q∫Qνx−p′/pdx1/p′≤C<∞.Then, the fractional integral operatorIγ satisfies the weak-type p,q inequality(11)σ⋅wx∈ℝn:Iγfx>σ1/q≤C∫ℝnfxpνxdx1/p,for anyσ>0,where C does not depend on f nor on σ>0.Furthermore, in [9], Zhang sharpened Martell’s result by replacing the local Lr norm on the left-hand side of (10) by the smaller Orlicz space norm. On the other hand, by comparing Theorem 4 with Theorems 2 and 3, it is natural to conjecture that when q>p, there is a two-weight, weak-type p,q inequality for the commutator b,Iγ of fractional integral operator. By using the same method as in the proof of Theorem 1 in [12] and certain Orlicz norm, we are able to obtain the following sufficient condition on a pair of weights w,ν to ensure the Lpν⟶WLqw boundedness of b,Iγ, whenever b belongs to BMOℝn. More specifically, the following statement is true.Theorem 5.
Let0<γ<n, 1<p<q<∞ and b∈BMOℝn. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(12)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rqν−1/pA,Q≤C<∞,where At=tp′1+log+tp′. Then, the linear commutator b,Iγ satisfies the weak-type p,q inequality(13)σ⋅wx∈ℝn:b,Iγfx>σ1/q≤C∫ℝnfxpνxdx1/p,for anyσ>0,where C does not depend on f nor on σ>0.The details are omitted here. Note that condition (12) reduces to condition (7) provided that p=q.Question. In view of Theorems 2–5, it is a natural and interesting problem to find some sufficient conditions for which the two-weight, weak-type norm inequalities hold for the operators Iγ and b,Iγ, in the endpoint case p=1.In this paper, we are mainly interested in the weighted Morrey spaces and weighted amalgam spaces. Let us recall their definitions.


### 1.2. Weighted Morrey Spaces

The classical Morrey spaceLp,λℝn was introduced by Morrey [12] in connection with elliptic partial differential equations. Let 1≤p<∞ and 0≤λ≤n. We recall that a real-valued function f is said to belong to the space Lp,λℝn on the n-dimensional Euclidean space ℝn, if the following norm is finite:(14)fLp,λ≔supx,r∈ℝn×0,∞rλ−n∫Bx,rfypdy1/p,where Bx,r=y∈ℝn:x−y<r is the Euclidean ball with center x∈ℝn and radius r∈0,∞ as well as the Lebesgue measure Bx,r=vn⋅rn. Here, vn is the volume of the unit ball of ℝn. In particular, one has(15)Lp,0ℝn=L∞ℝn,Lp,nℝn=Lpℝn.In [13], Komori and Shirai considered the weighted case and introduced a version of weighted Morrey space, which is a natural generalization of weighted Lebesgue space.Definition 1.
Let1<p<∞ and 0≤κ<1. For two weights w and ν on ℝn, the weighted Morrey space Lp,κν,w is defined by(16)Lp,κν,w≔f∈Llocpν:fLp,κν,w<∞,where the norm is given by(17)fLp,κν,w≔supQ⊂ℝn1wQκ∫Qfxpνxdx1/p,and the supremum is taken over all cubes Q in ℝn.Definition 2.
Let1<p<∞, 0≤κ<1, and w be a weight on ℝn. We define the weighted weak Morrey space WLp,κw as the set of all measurable functions f satisfying(18)fWLp,κw≔supQ⊂ℝnsupσ>01wQκ/pσ⋅wx∈Q:fx>σ1/p<∞.
By definition, it is clear that(19)Lp,0ν,w=Lpν,WLp,0w=WLpw.


### 1.3. Weighted Amalgam Spaces

Let1≤p,s≤∞, a function f∈Llocpℝn is said to be in the Wiener amalgam space Lp,Lsℝn of Lpℝn and Lsℝn, if the function y⟼f⋅⋅χBy,1Lp belongs to Lsℝn, where By,1 is an open ball in ℝn centered at y with radius 1, χBy,1 is the characteristic function of the ball By,1, and ⋅Lp is the usual Lebesgue norm in Lpℝn. In [14], Fofana introduced a new class of function spaces Lp,Lsαℝn which turned out to be the subspaces of Lp,Lsℝn. More precisely, for 1≤p,s,α≤∞, we define the amalgam space Lp,Lsαℝn of Lpℝn and Lsℝn as the set of all measurable functions f satisfying f∈Llocpℝn and fLp,Lsαℝn<∞, where(20)fLp,Lsαℝn≔supr>0∫ℝnBy,r1/α−1/p−1/sf⋅χBy,rLpℝnsdy1/s,=supr>0By,r1/α−1/p−1/sf⋅χBy,rLpℝnLsℝn,with the usual modification when p=∞ or s=∞, and By,r is the Lebesgue measure of the ball By,r. As it was shown in [14] that the space Lp,Lsαℝn is nontrivial if and only if p≤α≤s, in the remaining of this paper, we will always assume that this condition p≤α≤s is satisfied. Let us consider the following two special cases:(1)
If we takep=s, then p=α=s. By Fubini’s theorem, it is easy to check that(21)By,r−1/pf⋅χBy,rLpℝnLpℝn=∫ℝnBy,r−1∫ℝnfxp⋅χBy,rdxdy1/p=∫ℝnBy,r−1∫Bx,rfxpdydx1/p=∫ℝnfxpdx1/p,where the last equality holds sinceBy,r−1⋅Bx,r=1. Hence, the amalgam space Lp,Lsαℝn is equal to the Lebesgue space Lpℝn with the same norms provided that p=α=s.(2)
Ifs=∞, then we can see that in such a situation, the amalgam space Lp,Lsαℝn is equal to the classical Morrey space Lp,λℝn with equivalent norms, where λ=pn/α.In this paper, we will consider the weighted version ofLp,Lsαℝn.Definition 3.
Let1≤p≤α≤s≤∞ and let ν,w,andμ be three weights on ℝn. We denote by Lp,Lsαν,w;μ the weighted amalgam space, the space of all locally integrable functions f, such that(22)fLp,Lsαν,w;μ≔supℓ>0∫ℝnwQy,ℓ1/α−1/p−1/sf⋅χQy,ℓLpνsμydy1/s=supℓ>0wQy,ℓ1/α−1/p−1/sf⋅χQy,ℓLpνLsμ<∞,with wQy,ℓ=∫Qy,ℓwxdx and the usual modification when s=∞.Definition 4.
Let1≤p≤α≤s≤∞, and let w,μ be two weights on ℝn. We denote by WLp,Lsαw;μ the weighted weak amalgam space consisting of all measurable functions f such that(23)fWLp,Lsαw;μ≔supℓ>0∫ℝnwQy,ℓ1/α−1/p−1/sf⋅χQy,ℓWLpwsμydy1/s=supℓ>0wQy,ℓ1/α−1/p−1/sf⋅χQy,ℓWLpwLsμ<∞,with wQy,ℓ=∫Qy,ℓwxdx and the usual modification when s=∞.
Note that in the particular case whenμ≡1, this kind of weighted (weak) amalgam space was introduced by Feuto in [15] (see also [16]). We remark that Feuto [15] considered ball B instead of cube Q in his definition, but these two definitions are evidently equivalent. Also, note that when 1≤p≤α and s=∞, then Lp,Lsαν,w;μ is just the weighted Morrey space Lp,κν,w with κ=1−p/α, and WLp,Lsαw;μ is just the weighted weak Morrey space WLp,κw with κ=1−p/α.
Recently, in [17–19], the author studied the two-weight, weak-type p,p inequalities for fractional integral operator, as well as its commutators on weighted Morrey and amalgam spaces, under some Ap-type conditions (4) and (7) on the pair w,ν. As a continuation of the works mentioned above, in this paper, we consider related problems about two-weight, weak-type p,q inequalities for Iγ and b,Iγ, under some other Ap-type conditions (10) and (12) on w,ν and 1<p<q.


## 1.1. Fractional Integral Operators

Letℝn be the n-dimensional Euclidean space equipped with the Euclidean norm ⋅ and the Lebesgue measure dx. For given γ, 0<γ<n, the fractional integral operator (or Riesz potential) Iγ with order γ is defined by (see [1] for the basic properties of Iγ)(1)Iγfx≔1ζγ∫ℝnfyx−yn−γdy,ζγ=πn/22γΓγ/2Γn−γ/2.Weighted norm inequalities for fractional integral operators arise naturally in harmonic analysis and have been extensively studied by several authors. The study of two-weight problem forIγ was initiated by Sawyer in his pioneer paper [2]. By a weight w, we mean that w is a nonnegative and locally integrable function. In [2], Sawyer concerned the following question. Suppose that 1<p≤q<∞. For which pairs of weights w,ν on ℝn is the fractional integral operator bounded from Lpν into weak-Lqw? A necessary and sufficient condition for the weak-type p,q inequality was given by Sawyer. More specifically, he showed the following.Theorem 1 (see [2]).
Let0<γ<n and 1<p≤q<∞. Given a pair of weights w,ν on ℝn, the weak-type inequality(2)σ⋅wx∈ℝn:Iγfx>σ1/q≤C∫ℝnfxpνxdx1/p,holds for any σ>0 if and only if(3)∫QIγχQwxp′νx1−p′dx1/p′≤C⋅wQ1/q′<∞,for all cubes Q in ℝn. Here, χQ denotes the characteristic function of the cube Q, p′=p/p−1 denotes the conjugate index of p, and C is a universal constant.Sawyer’s result is interesting and important, and it promotes a series of research works on this subject (see, e.g., [3, 4, 5–9]), but it has the defect that condition (3) involves the fractional integral operator Iγ itself. In [3], Cruz-Uribe and Pérez considered the case when q=p and found a sufficient Ap-type condition on a pair of weights w,ν which ensures the boundedness of the operator Iγ from Lpν into weak-Lpw, where 1<p<∞. The condition (4) given below is simpler than (3) in the sense that it does not involve the operator Iγ itself, and hence it can be more easily verified.Theorem 2 (see [3]).
Let0<γ<n and 1<p<∞. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(4)Qγ/n⋅1Q∫Qwxrdx1/rp1Q∫Qνx−p′/pdx1/p′≤C<∞.Then, the fractional integral operatorIγ satisfies the weak-type p,p inequality(5)σ⋅wx∈ℝn:Iγfx>σ1/p≤C∫ℝnfxpνxdx1/p,for anyσ>0,where C does not depend on f nor on σ>0.The proof ofTheorem 2 is quite complicated. It depends on an inequality relating the Hardy–Littlewood maximal function and the sharp maximal function which is strongly reminiscent of the good-λ inequality of Fefferman and Stein. For another, more elementary proof, see also [4]. This solves a problem posed by Sawyer and Wheeden in [8]. Moreover, in [5], Li improved this result by replacing the “power bump” in (4) by a smaller “Orlicz bump” (see also [10]). On the other hand, for given 0<γ<n, the linear commutator b,Iγ generated by a suitable function b and Iγ is defined by(6)b,Iγfx≔bx⋅Iγfx−Iγbfx=1ζγ∫ℝnbx−by⋅fyx−yn−γdy.This commutator was first introduced by Chanillo in [11]. In [6], Liu and Lu obtained a sufficient Ap-type condition for the commutator b,Iγ to satisfy a two-weight weak-type p,p inequality, where 1<p<∞. That condition is an Ap-type condition in the scale of Orlicz spaces (see (7) given below).Theorem 3 (see [6]).
Let0<γ<n, 1<p<∞ and b∈BMOℝn. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(7)Qγ/n⋅1Q∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′1+log+tp′ and log+t≔maxlogt,0, that is,(8)log+t=logt,ast>1;0,otherwise.Then, the linear commutatorb,Iγ satisfies the weak-type p,p inequality(9)σ⋅wx∈ℝn:b,Iγfx>σ1/p≤C∫ℝnfxpνxdx1/p,for anyσ>0,where C does not depend on f nor on σ>0.In [7], Martell considered the case when q>p and gave a verifiable condition which is sufficient for the two-weight, weak-type p,q inequality for fractional integral operator Iγ. The condition (10) given below (in the Euclidean setting of [7]) is also simpler than the one in Theorem 1.Theorem 4 (see [7]).
Let0<γ<n and 1<p<q<∞. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(10)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rq1Q∫Qνx−p′/pdx1/p′≤C<∞.Then, the fractional integral operatorIγ satisfies the weak-type p,q inequality(11)σ⋅wx∈ℝn:Iγfx>σ1/q≤C∫ℝnfxpνxdx1/p,for anyσ>0,where C does not depend on f nor on σ>0.Furthermore, in [9], Zhang sharpened Martell’s result by replacing the local Lr norm on the left-hand side of (10) by the smaller Orlicz space norm. On the other hand, by comparing Theorem 4 with Theorems 2 and 3, it is natural to conjecture that when q>p, there is a two-weight, weak-type p,q inequality for the commutator b,Iγ of fractional integral operator. By using the same method as in the proof of Theorem 1 in [12] and certain Orlicz norm, we are able to obtain the following sufficient condition on a pair of weights w,ν to ensure the Lpν⟶WLqw boundedness of b,Iγ, whenever b belongs to BMOℝn. More specifically, the following statement is true.Theorem 5.
Let0<γ<n, 1<p<q<∞ and b∈BMOℝn. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(12)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rqν−1/pA,Q≤C<∞,where At=tp′1+log+tp′. Then, the linear commutator b,Iγ satisfies the weak-type p,q inequality(13)σ⋅wx∈ℝn:b,Iγfx>σ1/q≤C∫ℝnfxpνxdx1/p,for anyσ>0,where C does not depend on f nor on σ>0.The details are omitted here. Note that condition (12) reduces to condition (7) provided that p=q.Question. In view of Theorems 2–5, it is a natural and interesting problem to find some sufficient conditions for which the two-weight, weak-type norm inequalities hold for the operators Iγ and b,Iγ, in the endpoint case p=1.In this paper, we are mainly interested in the weighted Morrey spaces and weighted amalgam spaces. Let us recall their definitions.


## 1.2. Weighted Morrey Spaces

The classical Morrey spaceLp,λℝn was introduced by Morrey [12] in connection with elliptic partial differential equations. Let 1≤p<∞ and 0≤λ≤n. We recall that a real-valued function f is said to belong to the space Lp,λℝn on the n-dimensional Euclidean space ℝn, if the following norm is finite:(14)fLp,λ≔supx,r∈ℝn×0,∞rλ−n∫Bx,rfypdy1/p,where Bx,r=y∈ℝn:x−y<r is the Euclidean ball with center x∈ℝn and radius r∈0,∞ as well as the Lebesgue measure Bx,r=vn⋅rn. Here, vn is the volume of the unit ball of ℝn. In particular, one has(15)Lp,0ℝn=L∞ℝn,Lp,nℝn=Lpℝn.In [13], Komori and Shirai considered the weighted case and introduced a version of weighted Morrey space, which is a natural generalization of weighted Lebesgue space.Definition 1.
Let1<p<∞ and 0≤κ<1. For two weights w and ν on ℝn, the weighted Morrey space Lp,κν,w is defined by(16)Lp,κν,w≔f∈Llocpν:fLp,κν,w<∞,where the norm is given by(17)fLp,κν,w≔supQ⊂ℝn1wQκ∫Qfxpνxdx1/p,and the supremum is taken over all cubes Q in ℝn.Definition 2.
Let1<p<∞, 0≤κ<1, and w be a weight on ℝn. We define the weighted weak Morrey space WLp,κw as the set of all measurable functions f satisfying(18)fWLp,κw≔supQ⊂ℝnsupσ>01wQκ/pσ⋅wx∈Q:fx>σ1/p<∞.
By definition, it is clear that(19)Lp,0ν,w=Lpν,WLp,0w=WLpw.


## 1.3. Weighted Amalgam Spaces

Let1≤p,s≤∞, a function f∈Llocpℝn is said to be in the Wiener amalgam space Lp,Lsℝn of Lpℝn and Lsℝn, if the function y⟼f⋅⋅χBy,1Lp belongs to Lsℝn, where By,1 is an open ball in ℝn centered at y with radius 1, χBy,1 is the characteristic function of the ball By,1, and ⋅Lp is the usual Lebesgue norm in Lpℝn. In [14], Fofana introduced a new class of function spaces Lp,Lsαℝn which turned out to be the subspaces of Lp,Lsℝn. More precisely, for 1≤p,s,α≤∞, we define the amalgam space Lp,Lsαℝn of Lpℝn and Lsℝn as the set of all measurable functions f satisfying f∈Llocpℝn and fLp,Lsαℝn<∞, where(20)fLp,Lsαℝn≔supr>0∫ℝnBy,r1/α−1/p−1/sf⋅χBy,rLpℝnsdy1/s,=supr>0By,r1/α−1/p−1/sf⋅χBy,rLpℝnLsℝn,with the usual modification when p=∞ or s=∞, and By,r is the Lebesgue measure of the ball By,r. As it was shown in [14] that the space Lp,Lsαℝn is nontrivial if and only if p≤α≤s, in the remaining of this paper, we will always assume that this condition p≤α≤s is satisfied. Let us consider the following two special cases:(1)
If we takep=s, then p=α=s. By Fubini’s theorem, it is easy to check that(21)By,r−1/pf⋅χBy,rLpℝnLpℝn=∫ℝnBy,r−1∫ℝnfxp⋅χBy,rdxdy1/p=∫ℝnBy,r−1∫Bx,rfxpdydx1/p=∫ℝnfxpdx1/p,where the last equality holds sinceBy,r−1⋅Bx,r=1. Hence, the amalgam space Lp,Lsαℝn is equal to the Lebesgue space Lpℝn with the same norms provided that p=α=s.(2)
Ifs=∞, then we can see that in such a situation, the amalgam space Lp,Lsαℝn is equal to the classical Morrey space Lp,λℝn with equivalent norms, where λ=pn/α.In this paper, we will consider the weighted version ofLp,Lsαℝn.Definition 3.
Let1≤p≤α≤s≤∞ and let ν,w,andμ be three weights on ℝn. We denote by Lp,Lsαν,w;μ the weighted amalgam space, the space of all locally integrable functions f, such that(22)fLp,Lsαν,w;μ≔supℓ>0∫ℝnwQy,ℓ1/α−1/p−1/sf⋅χQy,ℓLpνsμydy1/s=supℓ>0wQy,ℓ1/α−1/p−1/sf⋅χQy,ℓLpνLsμ<∞,with wQy,ℓ=∫Qy,ℓwxdx and the usual modification when s=∞.Definition 4.
Let1≤p≤α≤s≤∞, and let w,μ be two weights on ℝn. We denote by WLp,Lsαw;μ the weighted weak amalgam space consisting of all measurable functions f such that(23)fWLp,Lsαw;μ≔supℓ>0∫ℝnwQy,ℓ1/α−1/p−1/sf⋅χQy,ℓWLpwsμydy1/s=supℓ>0wQy,ℓ1/α−1/p−1/sf⋅χQy,ℓWLpwLsμ<∞,with wQy,ℓ=∫Qy,ℓwxdx and the usual modification when s=∞.
Note that in the particular case whenμ≡1, this kind of weighted (weak) amalgam space was introduced by Feuto in [15] (see also [16]). We remark that Feuto [15] considered ball B instead of cube Q in his definition, but these two definitions are evidently equivalent. Also, note that when 1≤p≤α and s=∞, then Lp,Lsαν,w;μ is just the weighted Morrey space Lp,κν,w with κ=1−p/α, and WLp,Lsαw;μ is just the weighted weak Morrey space WLp,κw with κ=1−p/α.
Recently, in [17–19], the author studied the two-weight, weak-type p,p inequalities for fractional integral operator, as well as its commutators on weighted Morrey and amalgam spaces, under some Ap-type conditions (4) and (7) on the pair w,ν. As a continuation of the works mentioned above, in this paper, we consider related problems about two-weight, weak-type p,q inequalities for Iγ and b,Iγ, under some other Ap-type conditions (10) and (12) on w,ν and 1<p<q.


## 2. Statement of Our Main Results

We are now in a position to state our main results. Letp′ be the conjugate index of p whenever p>1; that is, 1/p+1/p′=1. First, we give the two-weight, weak-type norm inequalities for the fractional integral operator in the setting of weighted Morrey and amalgam spaces.Theorem 6.
Let0<γ<n,1<p<q<∞and0<κ<p/q. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(24)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rq1Q∫Qνx−p′/pdx1/p′≤C<∞.Ifw∈Δ2, then the fractional integral operator Iγ is bounded from Lp,κν,w into WLq,κq/pw.Theorem 7.
Let0<γ<n, 1<p<q<∞ and μ∈Δ2. Given a pair of weights w,ν on ℝn, assume that for some r>1 and for all cubes Q in ℝn,(25)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rq1Q∫Qνx−p′/pdx1/p′≤C<∞.Ifp≤α<β<s≤∞ and w∈Δ2, then the fractional integral operator Iγ is bounded from Lp,Lsαν,w;μ into WLq,Lsβw;μ with 1/β=1/α−1/p−1/q.Next, we introduce the definition of the space ofBMOℝn (see [20]). Suppose that b∈Lloc1ℝn and let(26)b*≔supQ⊂ℝn1Q∫Qbx−bQdx<∞,where bQ denotes the mean value of b on Q, namely,(27)bQ≔1Q∫Qbydy,and the supremum is taken over all cubes Q in ℝn. Define(28)BMOℝn≔b∈Lloc1ℝn:b*<∞.If we regard two functions whose difference is a constant as one, then the spaceBMOℝn is a Banach space with respect to the norm ⋅*. Concerning the two-weight weak-type estimates for the linear commutator b,Iγ in the context of weighted Morrey and amalgam spaces, we have the following results.Theorem 8.
Let0<γ<n, 1<p<q<∞, 0<κ<p/q, and b∈BMOℝn. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(29)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rqν−1/pA,Q≤C<∞,where At=tp′1+log+tp′. If w∈A∞, then the linear commutator b,Iγ is bounded from Lp,κν,w into WLq,κq/pw.Theorem 9.
Let0<γ<n, 1<p<q<∞, μ∈Δ2, and b∈BMOℝn. Given a pair of weights w,ν on ℝn, assume that for some r>1 and for all cubes Q in ℝn,(30)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rqν−1/pA,Q≤C<∞,where At=tp′1+log+tp′. If p≤α<β<s≤∞ and w∈A∞, then the linear commutator b,Iγ is bounded from Lp,Lsαν,w;μ into WLq,Lsβw;μ with 1/β=1/α−1/p−1/q.Moreover, for the extreme caseκ=p/q of Theorem 6, we will prove the following theorem, which could be viewed as a supplement of Theorem 6.Theorem 10.
Let0<γ<n and 1<p<q<∞. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(31)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rq1Q∫Qνx−p′/pdx1/p′≤C<∞.Ifκ=p/q and w∈Δ2, then the fractional integral operator Iγ is bounded from Lp,κν,w into BMO.In addition, we will also discuss the extreme caseβ=s of Theorem 7. In order to do so, we need to introduce the following new BMO-type space.Definition 5.
Let1≤s≤∞ and μ∈Δ2. The space BMO,Lsμ is defined as the set of all locally integrable functions f satisfying f**<∞, where(32)f**≔supℓ>01Qy,ℓ∫Qy,ℓfx−fQy,ℓdxLsμ.
Here, theLsμ-norm is taken with respect to the variable y. We also use the notation fQy,ℓ to denote the mean value of f on Qy,ℓ.
Observe that ifs=∞, then BMO,Lsμ is just the classical BMO space given above.
Now, we can show thatIγ is bounded from Lp,Lsαν,w;μ into our new BMO-type space defined above. This new result may be viewed as a supplement of Theorem 7.Theorem 11.
Let0<γ<n, 1<p<q<∞, and μ∈Δ2. Given a pair of weights w,ν on ℝn, assume that for some r>1 and for all cubes Q in ℝn,(33)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rq1Q∫Qνx−p′/pdx1/p′≤C<∞.Ifp≤α<s≤∞, 1/s=1/α−1/p−1/q, and w∈Δ2, then the fractional integral operator Iγ is bounded from Lp,Lsαν,w;μ into BMO,Lsμ.


## 3. Notation and Definitions

In this section, we recall some standard definitions and notation.


### 3.1. Weights

For giveny∈ℝn and ℓ>0, we denote by Qy,ℓ the cube centered at y and has side length ℓ>0, and all cubes are assumed to have their sides parallel to the coordinate axes. Given a cube Qy,ℓ and λ>0, λQy,ℓ stands for the cube concentric with Q and having side length λn times as long, i.e., λQy,ℓ≔Qy,λnℓ. A nonnegative function w defined on ℝn will be called a weight if it is locally integrable. For any given weight w and any Lebesgue measurable set E of ℝn, we denote the characteristic function of E by χE, the Lebesgue measure of E by E, and the weighted measure of E by wE, where wE≔∫Ewxdx. We also denote Ec≔ℝn\E the complement of E. Given a weight w, we say that w satisfies the doubling condition, if there exists a finite constant C>0 such that for any cube Q in ℝn, we have(34)w2Q≤C⋅wQ.Whenw satisfies this condition (34), we denote w∈Δ2 for brevity. A weight w is said to belong to Muckenhoupt’s class Ap for 1<p<∞, if there exists a constant C>0 such that(35)1Q∫Qwxdx1/p1Q∫Qwx−p′/pdx1/p′≤C,holds for every cube Q in ℝn. The class A∞ is defined as the union of the Ap classes for 1<p<∞, i.e., A∞=∪1<p<∞Ap. If w is an A∞ weight, then we have w∈Δ2 (see [21]). Moreover, this class A∞ is characterized as the class of all weights satisfying the following property: there exists a number δ>0 and a finite constant C>0 such that (see [21])(36)wEwQ≤CEQδ,holds for every cube Q⊂ℝn and all measurable subsets E of Q. Given a weight w on ℝn and for 1≤p<∞, the weighted Lebesgue space Lpw is defined to be the collection of all measurable functions f satisfying(37)fLpw≔∫ℝnfxpwxdx1/p<∞.For a weightw and 1≤p<∞, define the distribution function of f with w by(38)dfλ=wx∈ℝn:fx>λ,where λ is a positive number. We say that f is in the weighted weak Lebesgue space WLpw, if there exists a constant C>0 such that(39)fWLpw≔supλ>0λ⋅dfλ1/p≤C<∞.


### 3.2. Orlicz Spaces

We next recall some basic facts from the theory of Orlicz spaces needed for the proofs of the main results. For more information about these spaces the reader may consult the book [22]. Let A:0,+∞⟶0,+∞ be a Young function. That is, a continuous, convex, and strictly increasing function satisfying A0=0 and At⟶+∞ as t⟶+∞. For a Young function A and a cube Q in ℝn, we will consider the A-average of a function f given by the following Luxemburg norm:(40)fA,Q≔infλ>0:1Q∫QAfxλdx≤1.In particular, whenAt=tp with 1<p<∞, it is easy to see that(41)fA,Q=1Q∫Qfxpdx1/p,that is, the Luxemburg norm in such a situation coincides with the normalized Lp norm. The main examples that we are going to consider are At=tp1+log+tp with 1<p<∞.Throughout the paper,C always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. We will use A≈B to denote the equivalence of A and B; that is, there exist two positive constants C1 and C2 independent of A and B such that C1A≤B≤C2A.


## 3.1. Weights

For giveny∈ℝn and ℓ>0, we denote by Qy,ℓ the cube centered at y and has side length ℓ>0, and all cubes are assumed to have their sides parallel to the coordinate axes. Given a cube Qy,ℓ and λ>0, λQy,ℓ stands for the cube concentric with Q and having side length λn times as long, i.e., λQy,ℓ≔Qy,λnℓ. A nonnegative function w defined on ℝn will be called a weight if it is locally integrable. For any given weight w and any Lebesgue measurable set E of ℝn, we denote the characteristic function of E by χE, the Lebesgue measure of E by E, and the weighted measure of E by wE, where wE≔∫Ewxdx. We also denote Ec≔ℝn\E the complement of E. Given a weight w, we say that w satisfies the doubling condition, if there exists a finite constant C>0 such that for any cube Q in ℝn, we have(34)w2Q≤C⋅wQ.Whenw satisfies this condition (34), we denote w∈Δ2 for brevity. A weight w is said to belong to Muckenhoupt’s class Ap for 1<p<∞, if there exists a constant C>0 such that(35)1Q∫Qwxdx1/p1Q∫Qwx−p′/pdx1/p′≤C,holds for every cube Q in ℝn. The class A∞ is defined as the union of the Ap classes for 1<p<∞, i.e., A∞=∪1<p<∞Ap. If w is an A∞ weight, then we have w∈Δ2 (see [21]). Moreover, this class A∞ is characterized as the class of all weights satisfying the following property: there exists a number δ>0 and a finite constant C>0 such that (see [21])(36)wEwQ≤CEQδ,holds for every cube Q⊂ℝn and all measurable subsets E of Q. Given a weight w on ℝn and for 1≤p<∞, the weighted Lebesgue space Lpw is defined to be the collection of all measurable functions f satisfying(37)fLpw≔∫ℝnfxpwxdx1/p<∞.For a weightw and 1≤p<∞, define the distribution function of f with w by(38)dfλ=wx∈ℝn:fx>λ,where λ is a positive number. We say that f is in the weighted weak Lebesgue space WLpw, if there exists a constant C>0 such that(39)fWLpw≔supλ>0λ⋅dfλ1/p≤C<∞.


## 3.2. Orlicz Spaces

We next recall some basic facts from the theory of Orlicz spaces needed for the proofs of the main results. For more information about these spaces the reader may consult the book [22]. Let A:0,+∞⟶0,+∞ be a Young function. That is, a continuous, convex, and strictly increasing function satisfying A0=0 and At⟶+∞ as t⟶+∞. For a Young function A and a cube Q in ℝn, we will consider the A-average of a function f given by the following Luxemburg norm:(40)fA,Q≔infλ>0:1Q∫QAfxλdx≤1.In particular, whenAt=tp with 1<p<∞, it is easy to see that(41)fA,Q=1Q∫Qfxpdx1/p,that is, the Luxemburg norm in such a situation coincides with the normalized Lp norm. The main examples that we are going to consider are At=tp1+log+tp with 1<p<∞.Throughout the paper,C always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. We will use A≈B to denote the equivalence of A and B; that is, there exist two positive constants C1 and C2 independent of A and B such that C1A≤B≤C2A.


## 4. Proofs ofTheorems 6 and 7

Proof of Theorem 6.
Letf∈Lp,κν,w with 1<p<q<∞ and 0<κ<p/q. For an arbitrary fixed cube Q=Qx0,ℓ in ℝn, we decompose f as(42)f=f1+f2∈Lp,κν,w;f1=f⋅χ2Q;f2=f⋅χ2Qc,where 2Q≔Qx0,2nℓ and χ2Q denotes the characteristic function of 2Q. For any given σ>0, we then write(43)1wQκq/p⋅1/qσ⋅wx∈Q:Iγfx>σ1/q≤1wQκ/pσ⋅wx∈Q:Iγf1x>σ21/q+1wQκ/pσ⋅wx∈Q:Iγf2x>σ21/q≔I1+I2.
Let us consider the first termI1. Using Theorem 4 and the condition w∈Δ2, we have(44)I1≤C⋅1wQκ/p∫ℝnf1xpνxdx1/p=C⋅1wQκ/p∫2Qfxpνxdx1/p≤CfLp,κν,w⋅w2Qκ/pwQκ/p≤CfLp,κν,w.
This is exactly what we want. We now deal with the second termI2. Note that x−y≈x0−y, whenever x,x0∈Q and y∈2Qc. For 0<γ<n and all x∈Q, using the standard technique, we can see that(45)Iγf2x≤C∫ℝnf2yx−yn−γdy≤C∫2Qcfyx0−yn−γdy=C∑j=1∞∫2j+1Q\2jQfyx0−yn−γdy≤C∑j=1∞12j+1Q1−γ/n∫2j+1Qfydy.
This pointwise estimate (45) together with Chebyshev’s inequality yields(46)I2≤2wQκ/p⋅∫QIγf2xqwxdx1/q≤C⋅wQ1/q−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qfydy.
By using Hölder’s inequality with exponentp>1, we can deduce that(47)I2≤C⋅wQ1/q−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qfypνydy1/p×∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w⋅wQ1/q−κ/p∑j=1∞w2j+1Qκ/p2j+1Q1−γ/n×∫2j+1Qνy−p′/pdy1/p′=CfLp,κν,w∑j=1∞wQ1/q−κ/pw2j+1Q1/q−κ/p⋅w2j+1Q1/q2j+1Q1−γ/n×∫2j+1Qνy−p′/pdy1/p′.
Moreover, we apply Hölder’s inequality again with exponentr>1 to get(48)w2j+1Q=∫2j+1Qwydy≤2j+1Q1/r′∫2j+1Qwyrdy1/r.
This indicates that(49)I2≤CfLp,κν,w∑j=1∞wQ1/q−κ/pw2j+1Q1/q−κ/p⋅2j+1Q1/r′q2j+1Q1−γ/n×∫2j+1Qwyrdy1/rq∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w×∑j=1∞wQ1/q−κ/pw2j+1Q1/q−κ/p.
The last inequality is obtained by theAp-type condition (10) assumed on w,ν. Furthermore, since w∈Δ2, we can easily check that there exists a reverse doubling constantD=Dw>1 independent of Q such that (see ([13], Lemma 4.1))(50)w2Q≥D⋅wQ,for any cubeQ⊂ℝn,which implies that for any positive integer j,(51)w2j+1Q≥Dj+1⋅wQ,by induction principle. Hence,(52)∑j=1∞wQ1/q−κ/pw2j+1Q1/q−κ/p≤∑j=1∞wQDj+1⋅wQ1/q−κ/p=∑j=1∞1Dj+11/q−κ/p≤C,where the last series is convergent since the reverse doubling constantD>1 and 1/q−κ/p>0. Therefore, in view of (52), we get(53)I2≤CfLp,κν,w,which is our desired inequality. Combining the above estimates for I1 and I2 and then taking the supremum over all cubes Q⊂ℝn and all σ>0, we complete the proof of Theorem 6.Proof of Theorem 7.
Let1<p≤α<s≤∞ and f∈Lp,Lsαν,w;μ with w∈Δ2 and μ∈Δ2. For an arbitrary point y∈ℝn, we set Q=Qy,ℓ for the cube centered at y and of side length ℓ. Decompose f as(54)f=f1+f2∈Lp,Lsαν,w;μ;f1=f⋅χ2Q;f2=f⋅χ2Qc,where 2Q=Qy,2nℓ. Then, for given y∈ℝn and ℓ>0, we write(55)wQy,ℓ1/β−1/q−1/sIγf⋅χQy,ℓWLqw≤2⋅wQy,ℓ1/β−1/q−1/sIγf1⋅χQy,ℓWLqw+2⋅wQy,ℓ1/β−1/q−1/sIγf2⋅χQy,ℓWLqw≔I1y,ℓ+I2y,ℓ.
Let us consider the first termI1y,ℓ. According to Theorem 4, we get(56)I1y,ℓ≤2⋅wQy,ℓ1/β−1/q−1/sIγf1WLqw≤C⋅wQy,ℓ1/β−1/q−1/s∫ℝnf1xpνxdx1/p=C⋅wQy,ℓ1/β−1/q−1/s∫Qy,2nℓfxpνxdx1/p.
Observe that(57)1β−1q−1s=1α−1p−1s,is valid by our assumption 1/β=1/α−1/p−1/q. This allows us to obtain(58)I1y,ℓ≤C⋅wQy,ℓ1/α−1/p−1/sf⋅χQy,2nℓLpν=C⋅wQy,2nℓ1/α−1/p−1/sf⋅χQy,2nℓLpν×wQy,ℓ1/α−1/p−1/swQy,2nℓ1/α−1/p−1/s.
Moreover, since1/α−1/p−1/s<0 and w∈Δ2, by doubling inequality (34), we get(59)wQy,ℓ1/α−1/p−1/swQy,2nℓ1/α−1/p−1/s≤C.
Substituting the above inequality (59) into (58), we thus obtain(60)I1y,ℓ≤C⋅wQy,2nℓ1/α−1/p−1/sf⋅χQy,2nℓLpν.
We now estimate the second termI2y,ℓ. Recall that by the definition of Iγ, the following estimate holds for any x∈Qy,ℓ:(61)Iγf2x≤C∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz.
This pointwise estimate (61) along with Chebyshev’s inequality implies that(62)I2y,ℓ≤2⋅wQy,ℓ1/β−1/q−1/s∫Qy,ℓIγf2xqwxdx1/q≤C⋅wQy,ℓ1/β−1/s∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz.
A further application of Hölder’s inequality yields(63)I2y,ℓ≤C⋅wQy,ℓ1/β−1/s∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓνz−p′/pdz1/p′=C∑j=1∞wQy,2j+1nℓ1/β−1/q−1/sf⋅χQy,2j+1nℓLpν×wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s⋅wQy,2j+1nℓ1/qQy,2j+1nℓ1−γ/n×∫Qy,2j+1nℓνz−p′/pdz1/p′.
In addition, we apply Hölder’s inequality again with exponentr>1 to get(64)wQy,2j+1nℓ=∫Qy,2j+1nℓwzdz≤Qy,2j+1nℓ1/r′∫Qy,2j+1nℓwzrdz1/r.
Hence, in view of (64) and (57), we have(65)I2y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpν⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s×Qy,2j+1nℓ1/r′qQy,2j+1nℓ1−γ/n∫Qy,2j+1nℓwzrdz1/rq∫Qy,2j+1nℓνz−p′/pdz1/p′≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpν⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s.
The last inequality is obtained by theAp-type condition assumed on w,ν. Furthermore, arguing as in the proof of Theorem 6, we know that for any positive integer j, there exists a reverse doubling constantD=Dw>1 independent of Qy,ℓ such that(66)wQy,2j+1nℓ≥Dj+1⋅wQy,ℓ.
Hence, we compute(67)∑j=1∞wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s≤∑j=1∞wQy,ℓDj+1⋅wQy,ℓ1/β−1/s=∑j=1∞1Dj+11/β−1/s≤C,where the last series is convergent since the reverse doubling constantD>1 and 1/β−1/s>0. Therefore, by taking the Lsμ-norm of both sides of (55) (with respect to the variable y) and then using Minkowski’s inequality, (60) and (65), we obtain(68)wQy,ℓ1/β−1/q−1/sIγf⋅χQy,ℓWLqwLsμ≤I1y,ℓLsμ+I2y,ℓLsμ≤CwQy,2nℓ1/α−1/p−1/sf⋅χQy,2nℓLpνLsμ+C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpνLsμ×wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s≤CfLp,Lsαν,w;μ+CfLp,Lsαν,w;μ×∑j=1∞wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s≤CfLp,Lsαν,w;μ,where the last inequality follows from (67). Thus, by taking the supremum over all ℓ>0, we finish the proof of Theorem 7.


## 5. Proofs of Theorems 8 and 9

For the results involving commutators, we need the following properties ofBMOℝn, which can be found in [23, 24].Lemma 1.
Let b be a function inBMOℝn.(i)
For every cubeQ in ℝn and for any positive integer j,(69)b2j+1Q−bQ≤C⋅j+1b*.(ii)
Let1<p<∞. For every cube Q in ℝn and for any w∈A∞,(70)∫Qbx−bQpwxdx1/p≤Cb*⋅wQ1/p.Before proving our main theorems, we will also need a generalization of Hölder’s inequality due to O’Neil [25].Lemma 2.
LetA, B, and C be Young functions such that for all t>0,(71)A−1t⋅B−1t≤C−1t,where A−1t is the inverse function of At. Then, for all functions f and g and all cubes Q in ℝn,(72)f⋅gC,Q≤2fA,QgB,Q.We are now ready to give the proofs of Theorems8 and 9.Proof of Theorem 8.
Letf∈Lp,κν,w with 1<p<q<∞ and 0<κ<p/q. For any given cube Q=Qx0,ℓ⊂ℝn, as before, we decompose f as(73)f=f1+f2∈Lp,κν,w;f1=f⋅χ2Q;f2=f⋅χ2Qc,where 2Q≔Qx0,2nℓ. Then, for any given σ>0, one writes(74)1wQκq/p⋅1/qσ⋅wx∈Q:b,Iγfx>σ1/q≤1wQκ/pσ⋅wx∈Q:b,Iγf1x>σ21/q+1wQκ/pσ⋅wx∈Q:b,Iγf2x>σ21/q≔J1+J2.
Applying Theorem5 and doubling inequality (34), we have(75)J1≤C⋅1wQκ/p∫ℝnf1xpνxdx1/p=C⋅1wQκ/p∫2Qfxpνxdx1/p≤CfLp,κν,w⋅w2Qκ/pwQκ/p≤CfLp,κν,w.
On the other hand, for anyx∈Q, from (6), it then follows that(76)b,Iγf2x≤C∫ℝnbx−by⋅f2yx−yn−γdy≤Cbx−bQ⋅∫ℝnf2yx−yn−γdy+C∫ℝnby−bQ⋅f2yx−yn−γdy≔ξx+ηx.
Thus, we can further splitJ2 into two parts as follows:(77)J2≤1wQκ/pσ⋅wx∈Q:ξx>σ41/q+1wQκ/pσ⋅wx∈Q:ηx>σ41/q≔J3+J4.
Using the pointwise estimate (45) and Chebyshev’s inequality, we obtain that(78)J3≤4wQκ/p⋅∫Qξxqwxdx1/q≤CwQκ/p⋅∫Qbx−bQqwxdx1/q∑j=1∞12j+1Q1−γ/n∫2j+1Qfydy≤Cb*⋅wQ1/q−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qfydy,where the last inequality is due to the assumption w∈A∞ and Lemma 1ii. By the same manner as in the proof of Theorem 6, we can also show that(79)J3≤CfLp,κν,w.
Similar to the proof of (45), for all x∈Q, we can show the following pointwise inequality as well:(80)ηx≤C∫2Qcby−bQ⋅fyx0−yn−γdy,≤C∑j=1∞12j+1Q1−γ/n∫2j+1Qby−bQ⋅fydy.
This, together with Chebyshev’s inequality, yields(81)J4≤4wQκ/p⋅∫Qηxqwxdx1/q≤C⋅wQ1/q−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qby−bQ⋅fydy≤C⋅wQ1/q−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qby−b2j+1Q⋅fydy+C⋅wQ1/q−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qb2j+1Q−bQ⋅fydy≔J5+J6.
An application of Hölder’s inequality leads to that(82)J5≤C⋅wQ1/q−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qfypνydy1/p×∫2j+1Qby−b2j+1Qp′νy−p′/pdy1/p′≤CfLp,κν,w⋅wQ1/q−κ/p∑j=1∞w2j+1Qκ/p2j+1Q1−γ/n×2j+1Q1/p′b−b2j+1Q⋅ν−1/pC,2j+1Q,where Ct=tp′ is a Young function by (41). For 1<p<∞, it is immediate that the inverse function of Ct is C−1t=t1/p′. Also, observe that the following identity is true:(83)C−1t=t1/p′=t1/p′1+log+t⋅1+log+t≔A−1t⋅B−1t.
Moreover, it is easy to see that(84)At≈tp′1+log+tp′,Bt≈expt−1.
LethexpL,Q denote the mean Luxemburg norm of h on cube Q with Young function Bt≈expt−1. According to Lemma 2, we have(85)b−b2j+1Q⋅ν−1/pC,2j+1Q≤Cb−b2j+1QexpL,2j+1Q⋅ν−1/pA,2j+1Q≤Cb*⋅ν−1/pA,2j+1Q,where in the last inequality, we have used the well-known fact that (see [23])(86)b−bQexpL,Q≤Cb*,for any cubeQ⊂ℝn.
Indeed, the above inequality (86) is equivalent to the following inequality:(87)1Q∫Qexpby−bQc0b*dy≤C,for any cubeQ⊂ℝn,which is an immediate consequence of the celebrated John–Nirenberg’s inequality (see [20]). Consequently, in view of (86) and (48), we can deduce that(88)J5≤CfLp,κν,w∑j=1∞wQ1/q−κ/pw2j+1Q1/q−κ/p⋅w2j+1Q1/q2j+1Q1/p−γ/n⋅b*ν−1/pA,2j+1Q≤Cb*fLp,κν,w∑j=1∞wQ1/q−κ/pw2j+1Q1/q−κ/p×2j+1Qγ/n+1/q−1/p⋅12j+1Q∫2j+1Qwxrdx1/rq⋅ν−1/pA,2j+1Q.
Sincew∈A∞, we know that w∈Δ2. Furthermore, by the Ap-type condition (12) on w,ν and the estimate (52), we obtain(89)J5≤CfLp,κν,w×∑j=1∞wQ1/q−κ/pw2j+1Q1/q−κ/p≤CfLp,κν,w.
It remains to estimate the last termJ6. Making use of the first part of Lemma 1 and Hölder’s inequality, we get(90)J6≤C⋅wQ1/q−κ/p∑j=1∞j+1b*2j+1Q1−γ/n∫2j+1Qfydy≤C⋅wQ1/q−κ/p∑j=1∞j+1b*2j+1Q1−γ/n∫2j+1Qfypνydy1/p∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w⋅wQ1/q−κ/p∑j=1∞j+1⋅w2j+1Qκ/p2j+1Q1−γ/n∫2j+1Qνy−p′/pdy1/p′=CfLp,κν,w∑j=1∞j+1⋅wQ1/q−κ/pw2j+1Q1/q−κ/p⋅w2j+1Q1/q2j+1Q1−γ/n∫2j+1Qνy−p′/pdy1/p′.
LetCt and At be the same as before. Clearly, Ct≤At for all t>0; then, for any cube Q in ℝn, one has fC,Q≤fA,Q by definition, which implies that condition (12) is stronger than condition (10). This fact together with (48) yields(91)J6≤CfLp,κν,w∑j=1∞j+1⋅wQ1/q−κ/pw2j+1Q1/q−κ/p⋅2j+1Q1/r′q2j+1Q1−γ/n×∫2j+1Qwyrdy1/rq∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w∑j=1∞j+1⋅wQ1/q−κ/pw2j+1Q1/q−κ/p.
Moreover, by our hypothesis onw:w∈A∞, there exists a number δ>0 such that the inequality (36) holds, and hence we compute(92)∑j=1∞j+1⋅wQ1/q−κ/pw2j+1Q1/q−κ/p≤C∑j=1∞j+1⋅Q2j+1Qδ1/q−κ/p=C∑j=1∞j+1⋅12j+1nδ1/q−κ/p≤C,where the last series is convergent since the exponent δ1/q−κ/p is positive. This implies our desired estimate(93)J6≤CfLp,κν,w.
Summarizing the estimates derived above and then taking the supremum over all cubesQ⊂ℝn and all σ>0, we conclude the proof of Theorem 8.Proof of Theorem 9.
Let1<p≤α<s≤∞ and f∈Lp,Lsαν,w;μ with w∈A∞ and μ∈Δ2. For any fixed cube Q=Qy,ℓ in ℝn, as usual, we decompose f as(94)f=f1+f2∈Lp,Lsαν,w;μ;f1=f⋅χ2Q;f2=f⋅χ2Qc,where 2Q=Qy,2nℓ. Then, for given y∈ℝn and ℓ>0, we write(95)wQy,ℓ1/β−1/q−1/sb,Iγf⋅χQy,ℓWLqw≤2⋅wQy,ℓ1/β−1/q−1/sb,Iγf1⋅χQy,ℓWLqw+2⋅wQy,ℓ1/β−1/q−1/sb,Iγf2⋅χQy,ℓWLqw≔J1y,ℓ+J2y,ℓ.
Next, we shall calculate the two terms, respectively. According toTheorem 5, we get(96)J1y,ℓ≤2⋅wQy,ℓ1/β−1/q−1/sb,Iγf1WLqw≤C⋅wQy,ℓ1/β−1/q−1/s∫Qy,2nℓfxpνxdx1/p=C⋅wQy,2nℓ1/α−1/p−1/sf⋅χQy,2nℓLpν×wQy,ℓ1/α−1/p−1/swQy,2nℓ1/α−1/p−1/s,where the last identity is due to (57). Moreover, since w∈A∞, we know that w∈Δ2, and hence by inequality (59),(97)J1y,ℓ≤C⋅wQy,2nℓ1/α−1/p−1/sf⋅χQy,2nℓLpν.
On the other hand, from (6), one can see that for any x∈Qy,ℓ,(98)b,Iγf2x≤bx−bQy,ℓ⋅Iγf2x+IγbQy,ℓ−bf2x≔ξ˜x+η˜x.
Consequently, we can further divideJ2y,ℓ into two parts:(99)J2y,ℓ≤4⋅wQy,ℓ1/β−1/q−1/sξ˜⋅⋅χQy,ℓWLqw+4⋅wQy,ℓ1/β−1/q−1/sη˜⋅⋅χQy,ℓWLqw≔J3y,ℓ+J4y,ℓ.
For the termJ3y,ℓ, it follows directly from Chebyshev’s inequality and estimate (61) that(100)J3y,ℓ≤4⋅wQy,ℓ1/β−1/q−1/s∫Qy,ℓξ˜xqwxdx1/q≤C⋅wQy,ℓ1/β−1/q−1/s∫Qy,ℓbx−bQy,ℓqwxdx1/q×∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz≤C⋅wQy,ℓ1/β−1/s∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz,where in the last inequality, we have used the fact that w∈A∞ and Lemma 1ii. Arguing as in the proof of Theorem 7, we can also obtain that(101)J3y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpν⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s.
Let us now estimate the other termJ4y,ℓ. As it was shown in Theorem 8 (see (81)), the pointwise estimate(102)η˜x=IγbQy,ℓ−bf2x≤C∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓbz−bQy,ℓ⋅fzdz,holds for any x∈Qy,ℓ by a routine argument. This, together with Chebyshev’s inequality, implies that(103)J4y,ℓ≤4⋅wQy,ℓ1/β−1/q−1/s∫Qy,ℓη˜xqwxdx1/q≤C⋅wQy,ℓ1/β−1/s∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓbz−bQy,ℓ⋅fzdz≤C⋅wQy,ℓ1/β−1/s∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓbz−bQy,2j+1nℓ⋅fzdz+C⋅wQy,ℓ1/β−1/s∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓbQy,2j+1nℓ−bQy,ℓ⋅fzdz≔J5y,ℓ+J6y,ℓ.
An application of Hölder’s inequality leads to that(104)J5y,ℓ≤C⋅wQy,ℓ1/β−1/s∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓbz−bQy,2j+1nℓp′νz−p′/pdz1/p′=C⋅wQy,ℓ1/β−1/s∑j=1∞1Qy,2j+1nℓ1−γ/nf⋅χQy,2j+1nℓLpν×Qy,2j+1nℓ1/p′b−bQy,2j+1nℓ⋅ν−1/pC,Qy,2j+1nℓ,where Ct=tp′ is a Young function. Recall that the inequalities(105)b−bQy,2j+1nℓ⋅ν−1/pC,Qy,2j+1nℓ≤Cb−bQy,2j+1nℓB,Qy,2j+1nℓ⋅ν−1/pA,Qy,2j+1nℓ≤Cb*⋅ν−1/pA,Qy,2j+1nℓ,hold by generalized Hölder’s inequality and the estimate (87), where(106)At≈tp′1+log+tp′,Bt≈expt−1.
Moreover, in view of (64) and (106), we can deduce that(107)J5y,ℓ≤Cb*⋅wQy,ℓ1/β−1/s∑j=1∞f⋅χQy,2j+1nℓLpνQy,2j+1nℓ1/p−γ/n⋅ν−1/pA,Qy,2j+1nℓ=Cb*∑j=1∞wQy,2j+1nℓ1/β−1/q−1/sf⋅χQy,2j+1nℓLpν⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s×wQy,2j+1nℓ1/qQy,2j+1nℓ1/p−γ/n⋅ν−1/pA,Qy,2j+1nℓ≤Cb*∑j=1∞wQy,2j+1nℓ1/β−1/q−1/sf⋅χQy,2j+1nℓLpν⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/sQy,2j+1nℓγ/n+1/q−1/p⋅1Qy,2j+1nℓ∫Qy,2j+1nℓwzrdz1/rqν−1/pA,Qy,2j+1nℓ≤Cb*∑j=1∞wQy,2j+1nℓ1/β−1/q−1/sf⋅χQy,2j+1nℓLpν⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s.
The last inequality is obtained by theAp-type condition (12) assumed on w,ν. We now turn our attention to the last term J6y,ℓ. Applying Lemma 1i and Hölder’s inequality, we get(108)J6y,ℓ≤C⋅wQy,ℓ1/β−1/s∑j=1∞j+1b*Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz≤C⋅wQy,ℓ1/β−1/s∑j=1∞j+1b*Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓνz−p′/pdz1/p′=Cb*∑j=1∞wQy,2j+1nℓ1/β−1/q−1/sf⋅χQy,2j+1nℓLpν×j+1⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s⋅wQy,2j+1nℓ1/qQy,2j+1nℓ1−γ/n∫Qy,2j+1nℓνz−p′/pdz1/p′.
Also, observe that condition (12) is stronger than condition (10). Using this fact along with (64), we have(109)J6y,ℓ≤Cb*∑j=1∞wQy,2j+1nℓ1/β−1/q−1/sf⋅χQy,2j+1nℓLpν×j+1⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s×Qy,2j+1nℓ1/r′qQy,2j+1nℓ1−γ/n∫Qy,2j+1nℓwzrdz1/rq∫Qy,2j+1nℓνz−p′/pdz1/p′≤Cb*∑j=1∞wQy,2j+1nℓ1/β−1/q−1/sf⋅χQy,2j+1nℓLpν×j+1⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s.
Summing up all the above estimates and taking into consideration (57), we conclude that(110)J2y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/β−1/q−1/sf⋅χQy,2j+1nℓLpν×j+1⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s=C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpν×j+1⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s.
Moreover, by our hypothesis onw:w∈A∞ and inequality (36) with exponent δ*>0, we compute(111)∑j=1∞j+1⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s≤C∑j=1∞j+1⋅Qy,ℓQy,2j+1nℓδ*1/β−1/s=C∑j=1∞j+1⋅12j+1nδ*1/β−1/s≤C.
Notice that the exponentδ*1/β−1/s is positive because β<s, which guarantees that the last series is convergent. Thus, by taking the Lsμ-norm of both sides of (96) (with respect to the variable y) and then using Minkowski’s inequality, (98) and (111), we finally obtain(112)wQy,ℓ1/β−1/q−1/sb,Iγf⋅χQy,ℓWLqwLsμ≤J1y,ℓLsμ+J2y,ℓLsμ≤CwQy,2nℓ1/α−1/p−1/sf⋅χQy,2nℓLpνLsμ+C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpνLsμ×j+1⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s≤CfLp,Lsαν,w;μ+CfLp,Lsαν,w;μ×∑j=1∞j+1⋅wQy,ℓ1/β−1/swQy,2j+1nℓ1/β−1/s≤CfLp,Lsαν,w;μ,where the last inequality is due to (112). We therefore conclude the proof of Theorem 9 by taking the supremum over all ℓ>0.
The higher order commutators formed byIγ and a symbol function b are usually defined by(113)b,Iγmfx≔1ζγ∫ℝnbx−bym⋅fyx−yn−γdy,m=1,2,3,….
Obviously,b,Iγ1=b,Iγ which is the linear commutator (6), and(114)b,Iγm=b,b,Iγm−1,m=2,3,….
By induction argument, we will then obtain the following conclusions.Theorem 12.
Let0<γ<n, 1<p<q<∞, 0<κ<p/q, and b∈BMOℝn. Given a pair of weights w,ν on ℝn, suppose that for some r>1 and for all cubes Q in ℝn,(115)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rqν−1/pAm,Q≤C<∞,where Amt=tp′1+log+tmp′, m=2,3,…. If w∈A∞, then the higher order commutators b,Iγm are bounded from Lp,κν,w into WLq,κq/pw.Theorem 13.
Let0<γ<n, 1<p<q<∞, μ∈Δ2, and b∈BMOℝn. Given a pair of weights w,ν on ℝn, assume that for some r>1 and for all cubes Q in ℝn,(116)Qγ/n+1/q−1/p⋅1Q∫Qwxrdx1/rqν−1/pAm,Q≤C<∞,where Amt=tp′1+log+tmp′, m=2,3,…. If p≤α<β<s≤∞ and w∈A∞, then the higher order commutators b,Iγm are bounded from Lp,Lsαν,w;μ into WLq,Lsβw;μ with 1/β=1/α−1/p−1/q.


## 6. Proofs of Theorems 10 and 11

In the last section, we will prove the conclusions ofTheorems 10 and 11.Proof of Theorem 10.
Letf∈Lp,κν,w with 1<p<q<∞ and κ=p/q. For any given cube Q=Qx0,ℓ in ℝn, it suffices to prove that the inequality(117)1Q∫QIγfx−IγfQdx≤CfLp,κν,w,holds. Decompose f as f=f1+f2, where f1=f⋅χ4Q, f2=f⋅χ4Qc, 4Q=Qx0,4nℓ. By the linearity of the fractional integral operator Iγ, the left-hand side of (117) can be divided into two parts. That is,(118)1Q∫QIγfx−IγfQdx≤1Q∫QIγf1x−Iγf1Qdx+1Q∫QIγf2x−Iγf2Qdx≔I+II.
For the first termI, it follows directly from Fubini’s theorem that(119)I≤2Q∫QIγf1xdx≤CQ∫Q∫4Q1x−yn−γfydydx=CQ∫4Q∫Q1x−yn−γdxfydy.
It is clear that(120)x−y≤x−x0+y−x0≤5n2ℓ,when x∈Q and y∈4Q. Using the transform x−y↦z and polar coordinates, one has(121)∫Q1x−yn−γdx≤∫z≤5n/2ℓ1zn−γdz=wn−1⋅∫05n/2ℓ1ϱn−γϱn−1dϱ=wn−1⋅1γ5n2ℓγ.
Here, we usewn−1 to denote the measure of the unit sphere in ℝn. This indicates that(122)I≤CQ1−γ/n∫4Qfydy.
Using Hölder’s inequality and noting the fact thatκ=p/q, we have(123)I≤CQ1−γ/n∫4Qfypνydy1/p∫4Qνy−p′/pdy1/p′≤CfLp,κν,w⋅w4Qκ/pQ1−γ/n∫4Qνy−p′/pdy1/p′=CfLp,κν,w⋅w4Q1/qQ1−γ/n∫4Qνy−p′/pdy1/p′.
Moreover, it follows from condition (10) on w,ν and (48) (consider 4Q instead of 2j+1Q) that(124)I≤CfLp,κν,w⋅4Q1/r′q4Q1−γ/n∫4Qwyrdy1/rq∫4Qνy−p′/pdy1/p′≤CfLp,κν,w.
For the second termII, by (1), we have that for any x∈Q,(125)Iγf2x−Iγf2Q=1Q∫QIγf2x−Iγf2ydy=CQ∫Q∫4Qc1x−zn−γ−1y−zn−γfzdzdy≤CQ∫Q∫4Qc1x−zn−γ−1y−zn−γ⋅fzdzdy.
Since bothx and y are in Q, z∈4Qc, by a routine geometric observation, we must have x−z≥2x−y and x−z≈z−x0. This fact along with the mean value theorem yields(126)Iγf2x−Iγf2Q≤CQ∫Q∫4Qcx−yx−zn−γ+1⋅fzdzdy≤CQ∫Q∫4Qcℓz−x0n−γ+1⋅fzdzdy≤C∫4Qcℓz−x0n−γ+1⋅fzdz≤C∑j=2∞12j⋅12j+1Q1−γ/n∫2j+1Qfzdz.
Another application of Hölder’s inequality gives that(127)Iγf2x−Iγf2Q≤C∑j=2∞12j⋅12j+1Q1−γ/n×∫2j+1Qfzpνzdz1/p∫2j+1Qνz−p′/pdz1/p′≤CfLp,κν,w∑j=2∞12j⋅w2j+1Qκ/p2j+1Q1−γ/n∫2j+1Qνz−p′/pdz1/p′=CfLp,κν,w∑j=2∞12j⋅w2j+1Q1/q2j+1Q1−γ/n∫2j+1Qνz−p′/pdz1/p′,where the last equality is due to the fact that κ=p/q. Moreover, we apply estimate (48) and condition (10) to get(128)Iγf2x−Iγf2Q≤CfLp,κν,w∑j=2∞12j⋅2j+1Q1/r′q2j+1Q1−γ/n×∫2j+1Qwzrdz1/rq∫2j+1Qνz−p′/pdz1/p′≤CfLp,κν,w×∑j=2∞12j≤CfLp,κν,w.
From the pointwise estimate (128), it readily follows that(129)II=1Q∫QIγf2x−Iγf2Qdx≤CfLp,κν,w.
By combining the above estimates forI and II, we are done.Proof of Theorem 11.
Let1<p≤α<s≤∞ and f∈Lp,Lsαν,w;μ with w∈Δ2 and μ∈Δ2. For any fixed cube Q=Qy,ℓ in ℝn, we are going to estimate the following expression:(130)1Qy,ℓ∫Qy,ℓIγfx−IγfQy,ℓdx.
As usual, we decomposef as f=f1+f2, where f1=f⋅χ4Q, f2=f⋅χ4Qc, 4Q=Qy,4nℓ. By the linearity of the fractional integral operator Iγ, the above expression (130) can be divided into two parts. That is,(131)1Qy,ℓ∫Qy,ℓIγfx−IγfQy,ℓdx≤1Qy,ℓ∫Qy,ℓIγf1x−Iγf1Qy,ℓdx+1Qy,ℓ∫Qy,ℓIγf2x−Iγf2Qy,ℓdx≔Iy,ℓ+IIy,ℓ.
Let us first deal with the termIy,ℓ. Fubini’s theorem allows us to obtain(132)Iy,ℓ≤2Qy,ℓ∫Qy,ℓIγf1xdx≤CQy,ℓ∫Qy,ℓ∫Qy,4nℓ1x−zn−γfzdzdx=CQy,ℓ∫Qy,4nℓ∫Qy,ℓ1x−zn−γdxfzdz≤CQy,ℓ1−γ/n∫Qy,4nℓfzdz,where we have invoked (121) in the last inequality. Moreover, by Hölder’s inequality, we can see that(133)Iy,ℓ≤CQy,ℓ1−γ/n∫Qy,4nℓfzpνzdz1/p∫Qy,4nℓνz−p′/pdz1/p′=C⋅wQy,4nℓ1/α−1/p−1/sf⋅χQy,4nℓLpν×wQy,4nℓ1/qQy,ℓ1−γ/n∫Qy,4nℓνz−p′/pdz1/p′,where in the last equality, we have used the hypothesis 1/s=1/α−1/p−1/q. Taking into consideration (64) and condition (10) on w,ν, we further obtain that(134)Iy,ℓ≤C⋅wQy,4nℓ1/α−1/p−1/sf⋅χQy,4nℓLpν×Qy,4nℓ1/r′qQy,4nℓ1−γ/n∫Qy,4nℓwzrdz1/rq∫Qy,4nℓνz−p′/pdz1/p′≤C⋅wQy,4nℓ1/α−1/p−1/sf⋅χQy,4nℓLpν.
We now turn to estimate the second termIIy,ℓ. From (1), it then follows that for any x∈Qy,ℓ,(135)Iγf2x−Iγf2Qy,ℓ=1Qy,ℓ∫Qy,ℓIγf2x−Iγf2zdz=CQy,ℓ∫Qy,ℓ∫Qy,4nℓc1x−ζn−γ−1z−ζn−γfζdζdz≤CQy,ℓ∫Qy,ℓ∫Qy,4nℓc1x−ζn−γ−1z−ζn−γ⋅fζdζdz.
By the same reason as in the proof ofTheorem 10 (see (126)), we can show that for any x∈Qy,ℓ,(136)Iγf2x−Iγf2Qy,ℓ≤C∑j=2∞12j⋅1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfζdζ.
Furthermore, by using Hölder’s inequality, the preceding expression in (136) can be estimated as follows:(137)1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfζdζ≤1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfζpνζdζ1/p∫Qy,2j+1nℓνζ−p′/pdζ1/p′=wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpν×wQy,2j+1nℓ1/qQy,2j+1nℓ1−γ/n∫Qy,2j+1nℓνζ−p′/pdζ1/p′,where the last equality is also due to the fact that 1/α−1/p−1/s=−1/q. It then follows from (137) and (64) that(138)Iγf2x−Iγf2Qy,ℓ≤C∑j=2∞12j⋅wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpν×Qy,2j+1nℓ1/r′qQy,2j+1nℓ1−γ/n∫Qy,2j+1nℓwζrdζ1/rq∫Qy,2j+1nℓνζ−p′/pdζ1/p′≤C∑j=2∞12j⋅wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpν.
Consequently,(139)IIy,ℓ=1Qy,ℓ∫Qy,ℓIγf2x−Iγf2Qy,ℓdx≤C∑j=2∞12j⋅wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpν.
Therefore, by taking theLsμ-norm of (131) (with respect to the variable y) and then using Minkowski’s inequality, (134) and (139), we get(140)1Qy,ℓ∫Qy,ℓIγfx−IγfQy,ℓdxLsμ≤Iy,ℓLsμ+IIy,ℓLsμ≤CwQy,4nℓ1/α−1/p−1/sf⋅χQy,4nℓLpνLsμ+C∑j=2∞12j⋅wQy,2j+1nℓ1/α−1/p−1/sf⋅χQy,2j+1nℓLpνLsμ≤CfLp,Lsαν,w;μ+CfLp,Lsαν,w;μ×∑j=2∞12j≤CfLp,Lsαν,w;μ.
We end the proof by taking the supremum over allℓ>0.



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*Source: 3235942-2020-06-02.xml*