# Estimates for Fractional Integral Operators and Linear Commutators on Certain Weighted Amalgam Spaces

Authors: Hua Wang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we first introduce some new classes of weighted amalgam spaces. Then, we give the weighted strong-type and weak-type estimates for fractional integral operatorsIγ on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators b,Iγ generated by b and Iγ are established as well. In addition, we are going to study related problems about two-weight, weak-type inequalities for Iγ and b,Iγ on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator Mγ and generalized fractional integrals ℒ−γ/2 in the context of weighted amalgam spaces, where 0<γ<n and L is the infinitesimal generator of an analytic semigroup on L2Rn with Gaussian kernel bounds.


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## Body

## 1. Introduction

One of the most significant operators in harmonic analysis is the fractional integral operator. Letn be a positive integer. The n-dimensional Euclidean space Rn is endowed with the Lebesgue measure dx and the Euclidean norm ·. For given γ, 0<γ<n, the fractional integral operator (or Riesz potential) Iγ of order γ is defined by
(1)Iγfx≔1ζγ∫Rnfyx−yn−γdy,ζγ=πn/22γΓγ/2Γn−γ/2.The boundedness properties ofIγ between various function spaces have been studied extensively. It is well-known that the Hardy-Littlewood-Sobolev theorem states that the fractional integral operator Iγ is bounded from LpRn to LqRn for 0<γ<n, 1<p<n/γ, and 1/q=1/p−γ/n. Also, we know that Iγ is bounded from L1Rn to WLqRn for 0<γ<n and q=n/n−γ (see [1]). In 1974, Muckenhoupt and Wheeden [2] studied the weighted boundedness of Iγ and obtained the following two results (for sharp weighted norm inequalities, see [3]).Theorem 1 (see [2]).
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Then, the fractional integral operator Iγ is bounded from Lpwp to Lqwq.Theorem 2 (see [2]).
Let0<γ<n, p=1, q=n/n−γ, and w∈A1,q. Then, the fractional integral operator Iγ is bounded from L1wtoWLqwq.For0<γ<n, the linear commutator b,Iγ generated by a suitable function b and Iγ is defined by
(2)b,Iγfx≔bx⋅Iγfx−Iγbfx=1ζγ∫Rnbx−by⋅fyx−yn−γdy.This commutator was first introduced by Chanillo in [4]. In 1991, Segovia and Torrea [5] showed that b,Iγ is bounded from Lpwp (1<p<n/γ) to Lqwq whenever b∈BMORn (see [6] for sharp weighted bounds; see also [4] for the unweighted case). This corresponds to the norm inequalities satisfied by Iγ. Let us recall the definition of the space of BMORn (see [7]). BMORn is the Banach function space modulo constants with the norm ⋅∗ defined by
(3)b∗≔supB:ball1B∫Bbx−bBdx<∞,where the supremum is taken over all balls B in Rn and bB stands for the mean value of b over B; that is, bB≔1/B∫Bbydy.Theorem 3 (see [5]).
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Suppose that b∈BMORn, then the linear commutator b,Iγ is bounded from Lpwp to Lqwq.In the endpoint casep=1 and q=n/n−γ, since linear commutator b,Iγ has a greater degree of singularity than Iγ itself, a straightforward computation shows that b,Iγ fails to be of weak type (1,n/n−γ) when b∈BMORn (see [8] for some counterexamples). However, if we restrict ourselves to a bounded domain Ω in Rn, then the following weighted endpoint estimate for commutator b,Iγ of the fractional integral operator is valid, which was established by Cruz-Uribe and Fiorenza [9] in 2007 (see also [8] for the unweighted case).Theorem 4 (see [9]).
Let0<γ<n, p=1, q=n/n−γ, and wq∈A1. Suppose that b∈BMORn, then for any given λ>0 and any bounded domain Ω in Rn, there exists a constant C>0, which does not depend on f, Ω, and λ>0, such that
(4)wqx∈Ω:b,Iγfx>λ1/q≤C∫ΩΦfxλ⋅wxdx,where Φt=t·1+log+t and log+t=maxlogt,0.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,1⋅LPRn belongs to LsRn, where By,r=x∈Rn:x−y<r is the open ball centered at y and with radius r, χBy,r is the characteristic function of the ball By,r, and ⋅Lp is the usual Lebesgue norm in LpRn. Define
(5)Lp,LsRn≔f:fLp,LsRn=∫Rnf⋅χBy,1LpRnsdy1/s<∞.Then, we know thatLp,LsRn becomes a Banach function space with respect to the norm ⋅Lp,LsRn. This amalgam space was first introduced by Wiener in the 1920’s, but its systematic study goes back to the works of Holland [10] and Fournier and Stewart [11]. Let 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
(6)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. This generalization of amalgam space was originally introduced by Fofana in [12]. As proved in [12], the space Lp,LsαRn is nontrivial if and only if p≤α≤s; thus, in the remaining of the paper, we will always assume that this condition p≤α≤s is fulfilled. Note that(i) for1≤p≤α≤s≤∞, one can easily see that Lp,LsαRn⊆Lp,LsRn, where Lp,LsRn is the Wiener amalgam space defined by (5)(ii)if1≤p<α and s=∞, then Lp,LsαRn is just the classical Morrey space ℒp,κRn defined by (with κ=1−p/α, see [13])
(7)ℒp,κRn≔f:fℒp,κRn=supy∈Rn,r>01By,rκ∫By,rfxpdx1/p<∞(iii) ifp=α and s=∞, then Lp,LsαRn reduces to the usual Lebesgue space LpRnIn [14] (see also [15, 16]), Feuto considered a weighted version of the amalgam space Lp,Lsαw. A nonnegative measurable function w defined on Rn is called a weight if it is locally integrable. Let 1≤p≤α≤s≤∞ and w be a weight on Rn. We denote by Lp,Lsαw the weighted amalgam space, the space of all locally integrable functions f satisfying fLp,Lsαw<∞, where
(8)fLp,Lsαw≔supr>0∫RnwBy,r1/α−1/p−1/sf⋅χBy,rLpwsdy1/s=supr>0wBy,r1/α−1/p−1/sf⋅χBy,rLpwLsRn,with the usual modification when s=∞ and wBy,r≔∫By,rwxdx is the weighted measure of By,r. Similarly, for 1≤p≤α≤s≤∞, we can see that Lp,Lsαw becomes a Banach function space with respect to the norm ⋅Lp,Lsαw. Furthermore, we denote by WLp,Lsαw the weighted weak amalgam space consisting of all measurable functions f such that (see [14])
(9)fWLp,Lsαw≔supr>0∫RnwBy,r1/α−1/p−1/sf⋅χBy,rWLpwsdy1/s=supr>0wBy,r1/α−1/p−1/sf⋅χBy,rWLpwLsRn<∞.Notice that(i) if1≤p<α and s=∞, then Lp,Lsαw is just the weighted Morrey space ℒp,κw defined by (with κ=1−p/α, see [17])
(10)ℒp,κw≔f:fℒp,κw=supy∈Rn,r>01wBy,rκ∫By,rfxpwxdx1/p<∞and WLp,Lsαw is just the weighted weak Morrey space Wℒp,κw defined by (with κ=1−p/α, see [18])
(11)Wℒp,κw≔f:fWℒp,κw=supy∈Rn,r>0supλ>01wBy,rκ/pλ⋅wx∈By,r:fx>λ1/p<∞(ii) ifp=α and s=∞, then Lp,Lsαw reduces to the weighted Lebesgue space Lpw and WLp,Lsαw reduces to the weighted weak Lebesgue space WLpwRecently, many works in classical harmonic analysis have been devoted to norm inequalities involving several integral operators in the setting of weighted amalgam spaces (see [14–16, 19] and [20]). These results obtained are extensions of well-known analogues in the weighted Lebesgue spaces.LetIγ be the fractional integral operator, and let b,Iγ be its linear commutator. The aim of this paper is twofold. We first define some new classes of weighted amalgam spaces. As the weighted amalgam space may be considered as an extension of the weighted Lebesgue space, it is natural and important to study the weighted boundedness of Iγ and b,Iγ in these new spaces. We will prove that Iγ as well as its commutator b,Iγ, which are known to be bounded on weighted Lebesgue spaces, are bounded on weighted amalgam spaces under appropriate conditions. In addition, we will discuss two-weight, weak-type norm inequalities for Iγ and b,Iγ in the context of weighted amalgam spaces and give some partial results. Using these results and pointwise domination, we will establish the corresponding strong-type and weak-type estimates for fractional maximal operator Mγ and generalized fractional integrals ℒ−γ/2, where 0<γ<n and L is the infinitesimal generator of an analytic semigroup on L2Rn with Gaussian kernel bounds.The present paper is organized as follows. In Section2, we first state some preliminary definitions and results about Ap weights, Orlicz spaces, and weighted amalgam spaces, and the main results of the present paper are also given in Section 2. The following Sections 3, 4, and 5 are devoted to their proofs. Finally, in Section 6, we discuss some related two-weight problems.


## 2. Statement of Our Main Results

### 2.1. Notations and Preliminaries

Let us first recall the definitions of two weight classes:Ap and Ap,q.Definition 5 (Ap weights [21]).
A weightw is said to belong to the class Ap for 1<p<∞, if there exists a positive constant C such that for any ball B in Rn,
(12)1B∫Bwxdx1/p1B∫Bwx−p′/pdx1/p′≤C<∞,where we denote the conjugate exponent of p>1 by p′=p/p−1. The class A1 is defined replacing the above inequality by
(13)1B∫Bwxdx≤C·essinfx∈Bwx,for any ball B in Rn. We also define A∞=∪1≤p<∞Ap.Definition 6 (Ap,q weights [2]).
A weightw is said to belong to the class Ap,q for 1<p,q<∞, if there exists a positive constant C such that for any ball B in Rn,
(14)1B∫Bwxqdx1/q1B∫Bwx−p′dx1/p′≤C<∞.
The classA1,q (1<q<∞) is defined replacing the above inequality by
(15)1B∫Bwxqdx1/qesssupx∈B1wx≤C<∞,for any ball B in Rn.
There is a close connection betweenAp weights and Ap,q weights (see [22]).Lemma 7.
Suppose that0<γ<n, 1≤p<n/γ, and 1/q=1/p−γ/n. Then, the following statements are true:
(i) Ifp>1, then w∈Ap,q implies wq∈Aq and w−p′∈Ap′
(ii) ifp=1, then w∈A1,q if and only if wq∈A1Given a ballB and λ>0, we write λB for the ball with the same center as B whose radius is λ times that of B. For any r>0 and y∈Rn, we denote by By,rc the complement of By,r in Rn; that is, By,rc≔Rn\By,r. Given a weight w, we say that w satisfies the doubling condition if there exists a universal constant C>0 such that for any ball B in Rn, we have
(16)w2B≤C·wB.Whenw satisfies this doubling condition (16), we denote w∈Δ2 for brevity. An important fact here is that if w is in A∞, then w∈Δ2 (see [23]). Moreover, if w∈A∞, then there exists a number δ>0 such that (see [23])
(17)wEwB≤CEBδholds for any measurable subset E of a ball B.Given a weightw on Rn, for 1≤p<∞, the weighted Lebesgue space Lpw is defined as the set of all functions f such that
(18)fLpw≔∫Rnfxpwxdx1/p<∞.We also denote byWLpw(1≤p<∞) the weighted weak Lebesgue space consisting of all measurable functions f such that
(19)fWLpw≔supλ>0λ·wx∈Rn:fx>λ1/p<∞.We next recall some definitions and basic facts about Orlicz spaces needed for the proofs of our main results. For further information on this subject, we refer to [24]. A function A:0,+∞⟶0,+∞ is said to be a Young function if it is continuous, convex, and strictly increasing satisfying A0=0 and At⟶+∞ as t⟶+∞. An important example of Young function is At=tp·1+log+tp with some 1≤p<∞. Given a Young function A, we define the A-average of a function f over a ball B by means of the following Luxemburg norm:
(20)fA,B≔infλ>0:1B∫BAfxλdx≤1.In particular, whenAt=tp, 1≤p<∞, it is easy to see that A is a Young function and
(21)fA,B=1B∫Bfxpdx1/p;that is, the Luxemburg norm coincides with the normalized Lp norm. Recall that the following generalization of Hölder’s inequality holds
(22)1B∫Bfx·gxdx≤2fA,BgA¯,B,where A¯ is the complementary Young function associated with A, which is given by A¯s≔sup0≤t<∞st−At, 0≤s<∞. Obviously, Φt=t·1+log+t is a Young function, and its complementary Young function is Φ¯t≈et−1. In the present situation, we denote fΦ,B and gΦ¯,B by fLlogL,B and gexpL,B, respectively. Now, the above generalized Hölder’s inequality reads
(23)1B∫Bfx·gxdx≤2fLlogL,BgexpL,B.There is a further generalization of Hölder’s inequality that turns out to be useful for our purpose (see [25]): Let A, B, and C be Young functions such that for all t>0,
(24)A−1t·B−1t≤C−1t,where A−1t is the inverse function of At. Then, for all functions f and g and for all balls B in Rn,
(25)f·gC,B≤2fA,BgB,B.


### 2.2. Weighted Amalgam Spaces

Let us begin with the definitions of the weighted amalgam spaces with Lebesgue measure in (8) and (9) replaced by weighted measure.Definition 8.
Let1≤p≤α≤s≤∞, and let ν,w,μ 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
(26)fLp,Lsαν,w;μ≔supr>0∫RnwBy,r1/α−1/p−1/sf·χBy,rLpνsμydy1/s=supr>0wBy,r1/α−1/p−1/sf·χBy,rLpνLsμ<∞.
Ifν=w, then we denote Lp,Lsαw;μ for brevity, i.e., Lp,Lsαw,w;μ≔Lp,Lsαw;μ. Furthermore, we denote by WLp,Lsαw;μ the weighted weak amalgam space consisting of all measurable functions f for which
(27)fWLp,Lsαw;μ≔supr>0∫RnwBy,r1/α−1/p−1/sf·χBy,rWLpwsμydy1/s=supr>0wBy,r1/α−1/p−1/sf·χBy,rWLpwLsμ<∞,with the usual modification when s=∞.
The aim of this paper is to extend Theorems1–4 to the corresponding weighted amalgam spaces. We are going to prove that the fractional integral operator Iγ, which is bounded on weighted Lebesgue spaces, is also bounded on our new weighted spaces under appropriate conditions. Our first two results in this paper are stated as follows.Theorem 9.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Assume that p≤α<β<s≤∞ and μ∈Δ2, then the fractional integral operator Iγ is bounded from Lp,Lsαwp,wq;μ to Lq,Lsβwq;μ with 1/β=1/α−γ/n.Theorem 10.
Let0<γ<n, p=1, q=n/n−γ, and w∈A1,q. Assume that 1≤α<β<s≤∞ and μ∈Δ2, then the fractional integral operator Iγ is bounded from L1,Lsαw,wq;μ to WLq,Lsβwq;μ with 1/β=1/α−γ/n.Letb,Iγ be the linear commutator generated by Iγ and BMO function b. For the strong-type estimate of b,Iγ on the weighted amalgam spaces, we have the following result.Theorem 11.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Assume that p≤α<β<s≤∞, μ∈Δ2, and b∈BMORn, then the linear commutator b,Iγis bounded from Lp,Lsαwp,wq;μ to Lq,Lsβwq;μ with 1/β=1/α−γ/n.To obtain endpoint estimate for the linear commutatorb,Iγ, we first need to define the weighted A-average of a function f over a ball B by means of the weighted Luxemburg norm; that is, given a Young function A and w∈A∞, we define (see [24], for instance)
(28)fAw,B≔infλ>0:1wB∫BAfxλ·wxdx≤1.WhenAt=t, this norm is denoted by ·Lw,B, and when Φt=t·1+log+t, this norm is also denoted by ·LlogLw,B. The complementary Young function is given by Φ¯t≈et−1 with corresponding mean Luxemburg norm denoted by ·expLw,B. For w∈A∞ and for every ball B in Rn, we can also show the weighted version of (23). Namely, the following generalized Hölder’s inequality in the weighted setting
(29)1wB∫Bfx·gxwxdx≤CfLlogLw,BgexpLw,Bis true (see [26], for instance). Now, we introduce the new amalgam spaces of LlogL type as follows.Definition 12.
Letp=1, 1≤α≤s≤∞, and let ν,w,μ be three weights on Rn. We denote by LlogL,Lsαν,w;μ the weighted amalgam space of LlogL type, the space of all locally integrable functions f defined on Rn with finite norm fLlogL,Lsαν,w;μ:
(30)LlogL,Lsαν,w;μ≔f:fLlogL,Lsαν,w;μ<∞,where
(31)fLlogL,Lsαν,w;μ≔supr>0∫RnwBy,r1/α−1−1/svBy,rfLlogLν,By,rsμydy1/s=supr>0wBy,r1/α−1−1/svBy,rfLlogLν,By,rLsμ.
Note thatt≤t·1+log+t for all t>0. Then, for any ball B in Rn and ν∈A∞, it is immediate that fLv,B≤fLlogLv,B by definition, i.e., the inequality
(32)fLν,B=1νB∫Bfx·νxdx≤fLlogLν,Bholds for any ball B in Rn. From this, we can further see the following inclusion:
(33)LlogL,Lsαν,w;μ⊆L1,Lsαν,w;μ,when 1≤α≤s≤∞ and w,μ are some other weights.
In the endpoint casep=1, we will prove the following weak-type LlogL estimate of linear commutator b,Iγ in the setting of weighted amalgam spaces.Theorem 13.
Let0<γ<n,p=1, q=n/n−γ, and w∈A1,q. Assume that 1≤α<β<s≤∞, μ∈Δ2, and b∈BMORn, then for any given λ>0 and any ball By,r in Rn, there exists a constant C>0 independent of f, By,r, and λ>0 such that
(34)wqBy,r1/β−1/q−1/s·wqx∈By,r:b,Iγfx>λ1/qLsμ≤C·ΦfλLlogL,Lsαw,wq;μ,where Φt=t·1+log+t and 1/β=1/α−γ/n. From the above definitions, we can roughly say that the linear commutator b,Iγ is bounded from LlogL,Lsαw,wq;μ to WLq,Lsβwq;μ.Moreover, we will discuss the extreme caseβ=s of Theorem 9. In order to do so, we need to introduce a new BMO-type space given below.Definition 14.
Let1≤s≤∞ and μ∈Δ2. We define the space BMO,Lsμ as the set of all locally integrable functions f satisfying f∗∗<∞, where
(35)f∗∗≔supr>01By,r∫By,rfx−fBy,rdxLsμ.
Here, theLsμ-norm is taken with respect to the variable y. We also use the notation fBy,r to denote the mean value of f over By,r.
Observe that ifs=∞, then BMO,Lsμ is the classical BMO space.
Now, we can show thatIγ is bounded from Lp,Lsαwp,wq;μ to our new BMO-type space defined above. This result can be regarded as a supplement of Theorem 9.Theorem 15.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and let w∈Ap,q and μ∈Δ2. If p≤α<s≤∞ and 1/s=1/α−γ/n, then the fractional integral operator Iγ is bounded from Lp,Lsαwp,wq;μ to BMO,Lsμ.Throughout this paper, the letterC always denotes a positive constant that is independent of the essential variables but whose value may vary at each occurrence. We also use A≈B to denote the equivalence of A and B; that is, there exist two positive constants C1 and C2 independent of quantities A and B such that C1A≤B≤C2A. Equivalently, we could define the above notions of this section with cubes in place of balls and we will use whichever is more appropriate, depending on the circumstances.


## 2.1. Notations and Preliminaries

Let us first recall the definitions of two weight classes:Ap and Ap,q.Definition 5 (Ap weights [21]).
A weightw is said to belong to the class Ap for 1<p<∞, if there exists a positive constant C such that for any ball B in Rn,
(12)1B∫Bwxdx1/p1B∫Bwx−p′/pdx1/p′≤C<∞,where we denote the conjugate exponent of p>1 by p′=p/p−1. The class A1 is defined replacing the above inequality by
(13)1B∫Bwxdx≤C·essinfx∈Bwx,for any ball B in Rn. We also define A∞=∪1≤p<∞Ap.Definition 6 (Ap,q weights [2]).
A weightw is said to belong to the class Ap,q for 1<p,q<∞, if there exists a positive constant C such that for any ball B in Rn,
(14)1B∫Bwxqdx1/q1B∫Bwx−p′dx1/p′≤C<∞.
The classA1,q (1<q<∞) is defined replacing the above inequality by
(15)1B∫Bwxqdx1/qesssupx∈B1wx≤C<∞,for any ball B in Rn.
There is a close connection betweenAp weights and Ap,q weights (see [22]).Lemma 7.
Suppose that0<γ<n, 1≤p<n/γ, and 1/q=1/p−γ/n. Then, the following statements are true:
(i) Ifp>1, then w∈Ap,q implies wq∈Aq and w−p′∈Ap′
(ii) ifp=1, then w∈A1,q if and only if wq∈A1Given a ballB and λ>0, we write λB for the ball with the same center as B whose radius is λ times that of B. For any r>0 and y∈Rn, we denote by By,rc the complement of By,r in Rn; that is, By,rc≔Rn\By,r. Given a weight w, we say that w satisfies the doubling condition if there exists a universal constant C>0 such that for any ball B in Rn, we have
(16)w2B≤C·wB.Whenw satisfies this doubling condition (16), we denote w∈Δ2 for brevity. An important fact here is that if w is in A∞, then w∈Δ2 (see [23]). Moreover, if w∈A∞, then there exists a number δ>0 such that (see [23])
(17)wEwB≤CEBδholds for any measurable subset E of a ball B.Given a weightw on Rn, for 1≤p<∞, the weighted Lebesgue space Lpw is defined as the set of all functions f such that
(18)fLpw≔∫Rnfxpwxdx1/p<∞.We also denote byWLpw(1≤p<∞) the weighted weak Lebesgue space consisting of all measurable functions f such that
(19)fWLpw≔supλ>0λ·wx∈Rn:fx>λ1/p<∞.We next recall some definitions and basic facts about Orlicz spaces needed for the proofs of our main results. For further information on this subject, we refer to [24]. A function A:0,+∞⟶0,+∞ is said to be a Young function if it is continuous, convex, and strictly increasing satisfying A0=0 and At⟶+∞ as t⟶+∞. An important example of Young function is At=tp·1+log+tp with some 1≤p<∞. Given a Young function A, we define the A-average of a function f over a ball B by means of the following Luxemburg norm:
(20)fA,B≔infλ>0:1B∫BAfxλdx≤1.In particular, whenAt=tp, 1≤p<∞, it is easy to see that A is a Young function and
(21)fA,B=1B∫Bfxpdx1/p;that is, the Luxemburg norm coincides with the normalized Lp norm. Recall that the following generalization of Hölder’s inequality holds
(22)1B∫Bfx·gxdx≤2fA,BgA¯,B,where A¯ is the complementary Young function associated with A, which is given by A¯s≔sup0≤t<∞st−At, 0≤s<∞. Obviously, Φt=t·1+log+t is a Young function, and its complementary Young function is Φ¯t≈et−1. In the present situation, we denote fΦ,B and gΦ¯,B by fLlogL,B and gexpL,B, respectively. Now, the above generalized Hölder’s inequality reads
(23)1B∫Bfx·gxdx≤2fLlogL,BgexpL,B.There is a further generalization of Hölder’s inequality that turns out to be useful for our purpose (see [25]): Let A, B, and C be Young functions such that for all t>0,
(24)A−1t·B−1t≤C−1t,where A−1t is the inverse function of At. Then, for all functions f and g and for all balls B in Rn,
(25)f·gC,B≤2fA,BgB,B.


## 2.2. Weighted Amalgam Spaces

Let us begin with the definitions of the weighted amalgam spaces with Lebesgue measure in (8) and (9) replaced by weighted measure.Definition 8.
Let1≤p≤α≤s≤∞, and let ν,w,μ 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
(26)fLp,Lsαν,w;μ≔supr>0∫RnwBy,r1/α−1/p−1/sf·χBy,rLpνsμydy1/s=supr>0wBy,r1/α−1/p−1/sf·χBy,rLpνLsμ<∞.
Ifν=w, then we denote Lp,Lsαw;μ for brevity, i.e., Lp,Lsαw,w;μ≔Lp,Lsαw;μ. Furthermore, we denote by WLp,Lsαw;μ the weighted weak amalgam space consisting of all measurable functions f for which
(27)fWLp,Lsαw;μ≔supr>0∫RnwBy,r1/α−1/p−1/sf·χBy,rWLpwsμydy1/s=supr>0wBy,r1/α−1/p−1/sf·χBy,rWLpwLsμ<∞,with the usual modification when s=∞.
The aim of this paper is to extend Theorems1–4 to the corresponding weighted amalgam spaces. We are going to prove that the fractional integral operator Iγ, which is bounded on weighted Lebesgue spaces, is also bounded on our new weighted spaces under appropriate conditions. Our first two results in this paper are stated as follows.Theorem 9.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Assume that p≤α<β<s≤∞ and μ∈Δ2, then the fractional integral operator Iγ is bounded from Lp,Lsαwp,wq;μ to Lq,Lsβwq;μ with 1/β=1/α−γ/n.Theorem 10.
Let0<γ<n, p=1, q=n/n−γ, and w∈A1,q. Assume that 1≤α<β<s≤∞ and μ∈Δ2, then the fractional integral operator Iγ is bounded from L1,Lsαw,wq;μ to WLq,Lsβwq;μ with 1/β=1/α−γ/n.Letb,Iγ be the linear commutator generated by Iγ and BMO function b. For the strong-type estimate of b,Iγ on the weighted amalgam spaces, we have the following result.Theorem 11.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Assume that p≤α<β<s≤∞, μ∈Δ2, and b∈BMORn, then the linear commutator b,Iγis bounded from Lp,Lsαwp,wq;μ to Lq,Lsβwq;μ with 1/β=1/α−γ/n.To obtain endpoint estimate for the linear commutatorb,Iγ, we first need to define the weighted A-average of a function f over a ball B by means of the weighted Luxemburg norm; that is, given a Young function A and w∈A∞, we define (see [24], for instance)
(28)fAw,B≔infλ>0:1wB∫BAfxλ·wxdx≤1.WhenAt=t, this norm is denoted by ·Lw,B, and when Φt=t·1+log+t, this norm is also denoted by ·LlogLw,B. The complementary Young function is given by Φ¯t≈et−1 with corresponding mean Luxemburg norm denoted by ·expLw,B. For w∈A∞ and for every ball B in Rn, we can also show the weighted version of (23). Namely, the following generalized Hölder’s inequality in the weighted setting
(29)1wB∫Bfx·gxwxdx≤CfLlogLw,BgexpLw,Bis true (see [26], for instance). Now, we introduce the new amalgam spaces of LlogL type as follows.Definition 12.
Letp=1, 1≤α≤s≤∞, and let ν,w,μ be three weights on Rn. We denote by LlogL,Lsαν,w;μ the weighted amalgam space of LlogL type, the space of all locally integrable functions f defined on Rn with finite norm fLlogL,Lsαν,w;μ:
(30)LlogL,Lsαν,w;μ≔f:fLlogL,Lsαν,w;μ<∞,where
(31)fLlogL,Lsαν,w;μ≔supr>0∫RnwBy,r1/α−1−1/svBy,rfLlogLν,By,rsμydy1/s=supr>0wBy,r1/α−1−1/svBy,rfLlogLν,By,rLsμ.
Note thatt≤t·1+log+t for all t>0. Then, for any ball B in Rn and ν∈A∞, it is immediate that fLv,B≤fLlogLv,B by definition, i.e., the inequality
(32)fLν,B=1νB∫Bfx·νxdx≤fLlogLν,Bholds for any ball B in Rn. From this, we can further see the following inclusion:
(33)LlogL,Lsαν,w;μ⊆L1,Lsαν,w;μ,when 1≤α≤s≤∞ and w,μ are some other weights.
In the endpoint casep=1, we will prove the following weak-type LlogL estimate of linear commutator b,Iγ in the setting of weighted amalgam spaces.Theorem 13.
Let0<γ<n,p=1, q=n/n−γ, and w∈A1,q. Assume that 1≤α<β<s≤∞, μ∈Δ2, and b∈BMORn, then for any given λ>0 and any ball By,r in Rn, there exists a constant C>0 independent of f, By,r, and λ>0 such that
(34)wqBy,r1/β−1/q−1/s·wqx∈By,r:b,Iγfx>λ1/qLsμ≤C·ΦfλLlogL,Lsαw,wq;μ,where Φt=t·1+log+t and 1/β=1/α−γ/n. From the above definitions, we can roughly say that the linear commutator b,Iγ is bounded from LlogL,Lsαw,wq;μ to WLq,Lsβwq;μ.Moreover, we will discuss the extreme caseβ=s of Theorem 9. In order to do so, we need to introduce a new BMO-type space given below.Definition 14.
Let1≤s≤∞ and μ∈Δ2. We define the space BMO,Lsμ as the set of all locally integrable functions f satisfying f∗∗<∞, where
(35)f∗∗≔supr>01By,r∫By,rfx−fBy,rdxLsμ.
Here, theLsμ-norm is taken with respect to the variable y. We also use the notation fBy,r to denote the mean value of f over By,r.
Observe that ifs=∞, then BMO,Lsμ is the classical BMO space.
Now, we can show thatIγ is bounded from Lp,Lsαwp,wq;μ to our new BMO-type space defined above. This result can be regarded as a supplement of Theorem 9.Theorem 15.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and let w∈Ap,q and μ∈Δ2. If p≤α<s≤∞ and 1/s=1/α−γ/n, then the fractional integral operator Iγ is bounded from Lp,Lsαwp,wq;μ to BMO,Lsμ.Throughout this paper, the letterC always denotes a positive constant that is independent of the essential variables but whose value may vary at each occurrence. We also use A≈B to denote the equivalence of A and B; that is, there exist two positive constants C1 and C2 independent of quantities A and B such that C1A≤B≤C2A. Equivalently, we could define the above notions of this section with cubes in place of balls and we will use whichever is more appropriate, depending on the circumstances.


## 3. Proofs of Theorems9 and 10

In this section, we will prove the conclusions of Theorems9 and 10.Proof of Theorem 9.
The proof is inspired by [14, 15]. Let 1<p≤α<s≤∞ and f∈Lp,Lsαwp,wq;μ with w∈Ap,q and μ∈Δ2. For an arbitrary point y∈Rn, set B=By,r for the ball centered at y and of radius r, 2B=By,2r. We represent f as
(36)f=f·χ2B+f·χ2Bc≔f1+f2,where χ2B is the characteristic function of 2B. By the linearity of the fractional integral operator Iγ, one can write
(37)wqBy,r1/β−1/q−1/sIγf·χBy,rLqwq=wqBy,r1/β−1/q−1/s∫By,rIγfxqwqxdx1/q≤wqBy,r1/β−1/q−1/s∫By,rIγf1xqwqxdx1/q+wqBy,r1/β−1/q−1/s∫By,rIγf2xqwqxdx1/q≔I1y,r+I2y,r.
Here and in what follows, for any positive numberτ>0, we use the convention fτx≔fxτ. Below, we will give the estimates of I1y,r and I2y,r, respectively. By the weighted Lp,Lq-boundedness of Iγ (see Theorem 1), we have
(38)I1y,r≤wqBy,r1/β−1/q−1/sIγf1Lqwq≤C·wqBy,r1/β−1/q−1/s∫By,2rfxpwpxdx1/p.
Observe that1/β−1/q−1/s=1/α−1/p−1/s when 1/β=1/α−γ/n. This fact implies that
(39)I1y,r≤C·wqBy,r1/α−1/p−1/sf·χBy,2rLpwp=C·wqBy,2r1/α−1/p−1/sf·χBy,2rLpwp×wqBy,r1/α−1/p−1/swqBy,2r1/α−1/p−1/s.
Sincew∈Ap,q, we get wq∈Aq⊂A∞ by Lemma 7(i). Moreover, since 1/α−1/p−1/s<0, then by doubling inequality (16), we obtain
(40)wqBy,r1/α−1/p−1/swqBy,2r1/α−1/p−1/s≤C.
Substituting the above inequality (40) into (39), we can see that
(41)I1y,r≤C·wqBy,2r1/α−1/p−1/sf·χBy,2rLpwp.
Let us now turn to the estimate ofI2y,r. First, it is clear that when x∈By,r and z∈By,2rc, we get x−z≈y−z. We then decompose Rn into a geometrically increasing sequence of concentric balls and obtain the following pointwise estimate:
(42)Iγf2x≤1ζγ∫Rnf2zx−zn−γdz≤C∫By,2rcfzy−zn−γdz=C∑j=1∞∫By,2j+1rBy,2jrfzy−zn−γdz≤C∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzdz.
From this estimate (42), it then follows that
(43)I2y,r≤C·wqBy,r1/β−1/s∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzdz.
By using Hölder’s inequality andAp,q condition on w, we get
(44)1By,2j+1r1−γ/n∫By,2j+1rfzdz≤1By,2j+1r1−γ/n∫By,2j+1rfzpwpzdz1/p∫By,2j+1rwz−p′dz1/p′≤C∫By,2j+1rfzpwpzdz1/p·wqBy,2j+1r−1/q.
Hence,(45)I2y,r≤C·wqBy,r1/β−1/s×∑j=1∞∫By,2j+1rfzpwpzdz1/p·wqBy,2j+1r−1/q=C∑j=1∞wqBy,2j+1r1/β−1/q−1/sf·χBy,2j+1rLpwp×wqBy,r1/β−1/swqBy,2j+1r1/β−1/s=C∑j=1∞wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwp×wqBy,r1/β−1/swqBy,2j+1r1/β−1/s,where in the last equality we have used the relation 1/β−1/q=1/α−1/p. Notice that wq∈Aq⊂A∞ for 1<q<∞; then, by using the inequality (17) with exponent δ>0 and our assumption β<s, we find that
(46)∑j=1∞wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤C∑j=1∞By,rBy,2j+1rδ1/β−1/s=C∑j=1∞12j+1nδ1/β−1/s≤C,where the last series is convergent since δ1/β−1/s>0. Therefore, by taking the Lsμ-norm of both sides of (37)(with respect to the variable y), and then using Minkowski’s inequality, (41), (45), and (46), we have
(47)wqBy,r1/β−1/q−1/sIγf·χBy,rLqwqLsμ≤I1y,rLsμ+I2y,rLsμ≤CwqBy,2r1/α−1/p−1/sf·χBy,2rLpwpLsμ+C∑j=1∞wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwpLsμ×wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤CfLp,Lsαwp,wq;μ+CfLp,Lsαwp,wq;μ×∑j=1∞wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤CfLp,Lsαwp,wq;μ.
Thus, by taking the supremum over allr>0, we complete the proof of Theorem 9.Proof of Theorem 10.
Letp=1, 1≤α<s≤∞, and f∈L1,Lsαw,wq;μ with w∈A1,q and μ∈Δ2. For an arbitrary ball B=By,r in Rn, we represent f as
(48)f=f·χ2B+f·χ2Bc≔f1+f2;then, by the linearity of the fractional integral operator Iγ, one can write
(49)wqBy,r1/β−1/q−1/sIγf·χBy,rWLqwq≤2·wqBy,r1/β−1/q−1/sIγf1·χBy,rWLqwq+2·wqBy,r1/β−1/q−1/sIγf2·χBy,rWLqwq≔I′1y,r+I′2y,r.
We first consider the termI′1y,r. By the weighted weak 1,q-boundedness of Iγ (see Theorem 2), we have
(50)I′1y,r≤2·wqBy,r1/β−1/q−1/sIγf1WLqwq≤C·wqBy,r1/β−1/q−1/s∫By,2rfxwxdx.
Observe that1/β−1/q−1/s=1/α−1−1/s when 1/β=1/α−γ/n and q=n/n−γ. Then, we have
(51)I′1y,r≤C·wqBy,r1/α−1−1/s∫By,2rfxwxdx=C·wqBy,2r1/α−1−1/sf·χBy,2rL1w×wqBy,r1/α−1−1/swqBy,2r1/α−1−1/s.
Sincew is in the class A1,q, we get wq∈A1⊂A∞ by Lemma 7(ii). Moreover, since 1/α−1−1/s<0, then we apply inequality (16) to obtain that
(52)wqBy,r1/α−1−1/swqBy,2r1/α−1−1/s≤C.
Substituting the above inequality (52) into (51), we thus obtain
(53)I′1y,r≤C·wqBy,2r1/α−1−1/sf·χBy,2rL1w.
As for the second termI′2y,r, it follows directly from Chebyshev’s inequality and the pointwise estimate (42) that
(54)I2′y,r≤2⋅wqBy,r1/β−1/q−1/s∫By,rIγf2xqwqxdx1/q≤C⋅wqBy,r1/β−1/s∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzdz.
Moreover, by applying Hölder’s inequality and then the reverse Hölder’s inequality in succession, we can show thatwq∈A1 if and only if w∈A1∩RHq (see [27]), where RHq denotes the reverse Hölder class (see [28] for further details). Another application of A1 condition on w gives that
(55)1By,2j+1r1−γ/n∫By,2j+1rfzdz≤C·By,2j+1rγ/nwBy,2j+1r·essinfz∈By,2j+1rwz∫By,2j+1rfzdz≤C·By,2j+1rγ/nwBy,2j+1r∫By,2j+1rfzwzdz.
In addition, note thatw∈RHq. We are able to verify that for any positive integer j∈ℤ+,
(56)wqBy,2j+1r1/q=∫By,2j+1rwqxdx1/q≤CBy,2j+1r1/q−1·wBy,2j+1r,which is equivalent to
(57)By,2j+1rγ/nwBy,2j+1r≤C·1wqBy,2j+1r1/q.
Consequently,(58)I′2y,r≤C·wqBy,r1/β−1/s×∑j=1∞∫By,2j+1rfzwzdz·wqBy,2j+1r−1/q=C∑j=1∞wqBy,2j+1r1/β−1/q−1/sf·χBy,2j+1rL1w×wqBy,r1/β−1/swqBy,2j+1r1/β−1/s=C∑j=1∞wqBy,2j+1r1/α−1−1/sf·χBy,2j+1rL1w×wqBy,r1/β−1/swqBy,2j+1r1/β−1/s,where in the last equality we have used the relation 1/β−1/q=1/α−1. Recall that wq∈A1⊂A∞, then by using the inequality (17) with exponent δ∗>0 and the assumption β<s, we find that
(59)∑j=1∞wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤C∑j=1∞By,rBy,2j+1rδ∗1/β−1/s=C∑j=1∞12j+1nδ∗1/β−1/s≤C,where the last series is convergent since δ∗1/β−1/s>0. Therefore, by taking the Lsμ-norm of both sides of (49)(with respect to the variable y) and then using Minkowski’s inequality, (53) and (58), we have
(60)wqBy,r1/β−1/q−1/sIγf·χBy,rWLqwqLsμ≤I′1y,rLsμ+I′2y,rLsμ≤CwqBy,2r1/α−1−1/sf·χBy,2rL1wLsμ+C∑j=1∞wqBy,2j+1r1/α−1−1/sf·χBy,2j+1rL1wLsμ×wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤CfL1,Lsαw,wq;μ+CfL1,Lsαw,wq;μ×∑j=1∞wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤CfL1,Lsαw,wq;μ,where the last inequality follows from (59). Thus, by taking the supremum over all r>0, we finish the proof of Theorem 10.Given0<γ<n, the related fractional maximal operator Mγ with order γ is given by
(61)Mγfx≔supB∋x1B1−γ/n∫Bfydy,where the supremum is taken over all balls B containing x. Let us point out that Mγfx can be controlled pointwise by Iγfx for any fx. In fact, fix r>0, then we have
(62)Iγfx≥∫y−x<rfyx−yn−γdy≥1rn−γ∫y−x<rfydy.Taking the supremum forr>0 on both sides of the above inequality, we get
(63)Iγfx≥supr>01rn−γ∫y−x<rfydy=C·Mγfx.This is just our desired conclusion. An immediate application of the above inequality (63) is the following weighted strong-type and weak-type estimates for the operator Mγ.Corollary 16.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Assume that p≤α<β<s≤∞ and μ∈Δ2, then the fractional maximal operator Mγ is bounded from Lp,Lsαwp,wq;μ to Lq,Lsβwq;μ with 1/β=1/α−γ/n.Corollary 17.
Let0<γ<n, p=1, q=n/n−γ, and w∈A1,q. Assume that 1≤α<β<s≤∞ and μ∈Δ2, then the fractional maximal operator Mγ is bounded from L1,Lsαw,wq;μ to WLq,Lsβwq;μ with 1/β=1/α−γ/n.Suppose thatL is a linear operator which generates an analytic semigroup e−tLt>0 on L2Rn with a kernel ptx,y satisfying Gaussian upper bound; that is, there exist two positive constants C and A such that for all x,y∈Rn and all t>0, we have
(64)ptx,y≤Ctn/2e−Ax−y2/t.For any0<γ<n, the generalized fractional integrals L−γ/2 associated with the operator L is defined by
(65)L−γ/2fx≔1Γγ/2∫0∞e−tLfxtγ/2−1dt.Note that ifL=−Δ is the Laplacian on Rn, then L−γ/2 is the classical fractional integral operator Iγ, which is given by (1). Since the semigroup e−tL has a kernel ptx,y which satisfies the Gaussian upper bound (64), it is easy to check that for all x∈Rn,
(66)L−γ/2fx≤C·Iγfx.In fact, if we denote the kernel ofL−γ/2 by Kγx,y, then it follows immediately from (65) that (see [29, 30])
(67)Kγx,y=1Γγ/2∫0∞ptx,ytγ/2−1dt,where ptx,y is the kernel of e−tL. Thus, by using the Gaussian upper bound (64) and the expression (67), we can deduce that (see [29, 30])
(68)Kγx,y≤1Γγ/2∫0∞ptx,ytγ/2−1dt≤C∫0∞e−Ax−y2/ttγ/2−n/2−1dt≤C⋅1x−yn−γ,which implies (66). Taking into account this pointwise inequality, as a consequence of Theorems 9 and 10, we have the following results.Corollary 18.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Assume that p≤α<β<s≤∞ and μ∈Δ2, then the generalized fractional integral L−γ/2 is bounded from Lp,Lsαwp,wq;μ to Lq,Lsβwq;μ with 1/β=1/α−γ/n.Corollary 19.
Let0<γ<n, p=1, q=n/n−γ, and w∈A1,q. Assume that 1≤α<β<s≤∞ and μ∈Δ2, then the generalized fractional integral L−γ/2 is bounded from L1,Lsαw,wq;μ to WLq,Lsβwq;μ with 1/β=1/α−γ/n.


## 4. Proofs of Theorems11 and 13

To prove our main theorems in this section, we need the following lemma aboutBMORn functions.Lemma 20.
Letb be a function in BMORn.
(i)For any ballB in Rn and for any positive integer j∈ℤ+, then
(69)b2j+1B−bB≤C·j+1b∗
(ii)Let1<q<∞. For any ball B in Rn and for any weight ν∈A∞, then
(70)∫Bbx−bBqνxdx1/q≤Cb∗·νB1/qProof.
For the proof of (i), we refer the reader to [31]. For the proof of (ii), we refer the reader to [32].Proof of Theorem 11.
Let1<p≤α<s≤∞ and f∈Lp,Lsαwp,wq;μ with w∈Ap,q and μ∈Δ2. For each fixed ball B=By,r in Rn, as before, we represent f as f=f1+f2, where f1=f·χ2B, 2B=By,2r⊂Rn. By the linearity of the commutator operator b,Iγ, we write
(71)wqBy,r1/β−1/q−1/sb,Iγf·χBy,rLqwq=wqBy,r1/β−1/q−1/s∫By,rb,Iγfxqwqxdx1/q≤wqBy,r1/β−1/q−1/s∫By,rb,Iγf1xqwqxdx1/q+wqBy,r1/β−1/q−1/s∫By,rb,Iγf2xqwqxdx1/q≔J1y,r+J2y,r.
Sincew is in the class Ap,q, we get wq∈Aq⊂A∞ by Lemma 7(i). Also, observe that 1/β−1/q=1/α−1/p by our assumption. By using Theorem 3, we obtain
(72)J1y,r≤wqBy,r1/β−1/q−1/sb,Iγf1Lqwq≤C·wqBy,r1/β−1/q−1/s∫By,2rfxpwpxdx1/p=C·wqBy,2r1/α−1/p−1/sf·χBy,2rLpwp×wqBy,r1/α−1/p−1/swqBy,2r1/α−1/p−1/s≤C·wqBy,2r1/α−1/p−1/sf·χBy,2rLpwp,where the last inequality is due to (16) and the fact that 1/α−1/p−1/s<0. Let us now turn to the estimate of J2y,r. By definition, for any x∈By,r, we have
(73)b,Iγf2x≤bx−bBy,r·Iγf2x+IγbBy,r−bf2x.
In the proof of Theorem9, we have already shown that (see (42))
(74)Iγf2x≤C∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzdz.
By the same manner as in the proof of (42), we can also show that
(75)IγbBy,r−bf2x≤1ζγ∫RnbBy,r−bzf2zx−zn−γdz≤C∫By,2rcbBy,r−bzfzy−zn−γdz=C∑j=1∞∫By,2j+1r\By,2jrbz−bBy,r·fzy−zn−γdz≤C∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rbz−bBy,r·fzdz.
Hence, from the above two pointwise estimates (74) and (75), it follows that
(76)J2y,r≤C·wqBy,r1/β−1/q−1/s∫By,rbx−bBy,rqwqxdx1/q×∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzdz+C·wqBy,r1/β−1/s∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rbBy,2j+1r−bBy,r·fzdz+C·wqBy,r1/β−1/s∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rbz−bBy,2j+1r·fzdz≔J3y,r+J4y,r+J5y,r.
Below, we will give the estimates ofJ3y,r, J4y,r, and J5y,r, respectively. To estimate J3y,r, note that wq∈Aq⊂A∞ with 1<q<∞. Using the second part of Lemma 20, Hölder’s inequality, and the Ap,q condition on w, we compute
(77)J3y,r≤Cb∗·wqBy,r1/β−1/s×∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzdz≤Cb∗·wqBy,r1/β−1/s∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzpwpzdz1/p×∫By,2j+1rwz−p′dz1/p′≤Cb∗·wqBy,r1/β−1/s×∑j=1∞∫By,2j+1rfzpwpzdz1/p·wqBy,2j+1r−1/q.
To estimateJ4y,r, applying the first part of Lemma 20, Hölder’s inequality, and the Ap,q condition on w, we can deduce that
(78)J4y,r≤Cb∗·wqBy,r1/β−1/s×∑j=1∞j+1By,2j+1r1−γ/n∫By,2j+1rfzdz≤Cb∗·wqBy,r1/β−1/s∑j=1∞j+1By,2j+1r1−γ/n∫By,2j+1rfzpwpzdz1/p×∫By,2j+1rwz−p′dz1/p′≤Cb∗·wqBy,r1/β−1/s×∑j=1∞j+1·∫By,2j+1rfzpwpzdz1/p·wqBy,2j+1r−1/q.
It remains to estimate the last termJ5y,r. An application of Hölder’s inequality gives us that
(79)J5y,r≤C·wqBy,r1/β−1/s∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzpwpzdz1/p×∫By,2j+1rbz−bBy,2j+1rp′wz−p′dz1/p′.
If we setνz=wz−p′, then we have ν∈Ap′⊂A∞ because w∈Ap,q by Lemma 7(i). Thus, it follows from the second part of Lemma 20 and the Ap,q condition on w that
(80)∫By,2j+1rbz−bBy,2j+1rp′νzdz1/p′≤Cb∗·νBy,2j+1r1/p′=Cb∗·∫By,2j+1rwz−p′dz1/p′≤Cb∗·By,2j+1r1−γ/nwqBy,2j+1r1/q.
Therefore, in view of the estimate (80), we get
(81)J5y,r≤Cb∗·wqBy,r1/β−1/s×∑j=1∞∫By,2j+1rfzpwpzdz1/p·wqBy,2j+1r−1/q.
Summarizing the estimates derived above, we conclude that(82)J2y,r≤Cb∗·wqBy,r1/β−1/s×∑j=1∞j+1·∫By,2j+1rfzpwpzdz1/p·wqBy,2j+1r−1/q=Cb∗∑j=1∞wqBy,2j+1r1/β−1/q−1/sf·χBy,2j+1rLpwp×j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s=Cb∗∑j=1∞wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwp×j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s,where in the last equality we have used the relation 1/β−1/q=1/α−1/p again. Since wq∈Aq with 1<q<∞, then by using the inequality (17) with exponent δ>0 together with the fact that β<s, we obtain
(83)∑j=1∞j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤C∑j=1∞j+1·By,rBy,2j+1rδ1/β−1/s=C∑j=1∞j+1·12j+1nδ1/β−1/s≤C,where the last series is convergent since the exponent δ1/β−1/s is positive. Therefore, by taking the Lsμ-norm of both sides of (71) (with respect to the variable y) and then using Minkowski’s inequality, (72) and (82), we can get
(84)wqBy,r1/β−1/q−1/sb,Iγf·χBy,rLqwqLsμ≤J1y,rLsμ+J2y,rLsμ≤CwqBy,2r1/α−1/p−1/sf·χBy,2rLpwpLsμ+C∑j=1∞wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwpLsμ×j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤CfLp,Lsαwp,wq;μ+CfLp,Lsαwp,wq;μ×∑j=1∞j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤CfLp,Lsαwp,wq;μ,where the last inequality follows from (83). Thus, by taking the supremum over all r>0, we complete the proof of Theorem 11.Proof of Theorem 13.
For any fixed ballB=By,r in Rn, as before, we represent f as f=f1+f2, where f1=f·χ2B and f2=f·χ2Bc. Then, for any given λ>0, by the linearity of the commutator operator b,Iγ, one can write
(85)wqBy,r1/β−1/q−1/s·wqx∈By,r:b,Iγfx>λ1/q≤wqBy,r1/β−1/q−1/s·wqx∈By,r:b,Iγf1x>λ/21/q+wqBy,r1/β−1/q−1/s·wqx∈By,r:b,Iγf2x>λ/21/q≔J′1y,r+J′2y,r.
We first consider the termJ′1y,r. By using Theorem 4, we get
(86)J′1y,r≤C·wqBy,r1/β−1/q−1/s∫By,2rΦfxλ·wxdx=C·wqBy,r1/α−1−1/s∫By,2rΦfxλ·wxdx,where in the last equality we have used our assumption 1/β=1/α−γ/n. Since w is a weight in the class A1,q, one has wq∈A1⊂A∞ by Lemma 7(ii). This fact, together with the inequalities (52) and (32), gives us that
(87)J′1y,r≤C·wqBy,2r1/α−1−1/swBy,2rwBy,2r∫By,2rΦfxλ·wxdx≤C·wqBy,2r1/α−1−1/swBy,2rΦfλLlogLw,By,2r.
We now turn to deal with the termJ′2y,r. Recall that the following inequality
(88)b,Iγf2x≤bx−bBy,r·Iγf2x+IγbBy,r−bf2xis valid. Thus, we can further decompose J′2y,r as
(89)J′2y,r≤wqBy,r1/β−1/q−1/s·wqx∈By,r:bx−bBy,r·Iγf2x>λ/41/q+wqBy,r1/β−1/q−1/s·wqx∈By,r:IγbBy,r−bf2x>λ/41/q≔J′3y,r+J′4y,r.
Applying the previous pointwise estimate (42), Chebyshev’s inequality together with Lemma 20(ii), we deduce that
(90)J′3y,r≤wqBy,r1/β−1/q−1/s·4λ∫By,rbx−bBy,rq·Iγf2xqwqxdx1/q≤C·wqBy,r1/β−1/s∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzλdz×1wqBy,r∫By,rbx−bBy,rqwqxdx1/q≤Cb∗∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rfzλdz×wqBy,r1/β−1/s.
Furthermore, note thatt≤Φt=t·1+log+t for any t>0. As we pointed out in Theorem 10 that wq∈A1 if and only if w∈A1∩RHq, it then follows from the A1 condition that
(91)J′3y,r≤Cb∗∑j=1∞By,2j+1rγ/nwBy,2j+1r∫By,2j+1rfzλ·wzdz×wqBy,r1/β−1/s≤Cb∗∑j=1∞By,2j+1rγ/nwBy,2j+1r∫By,2j+1rΦfzλ·wzdz×wqBy,r1/β−1/s≤Cb∗∑j=1∞ΦfλLlogLw,By,2j+1r×By,2j+1rγ/nwqBy,r1/β−1/s,where in the last inequality we have used the estimate (32). In view of (57) and our assumption 1/β=1/α−γ/n, we have
(92)J′3y,r≤Cb∗∑j=1∞ΦfλLlogLw,By,2j+1r×wqBy,r1/β−1/swqBy,2j+1r1/qwBy,2j+1r=Cb∗∑j=1∞ΦfλLlogLw,By,2j+1r×wqBy,2j+1r1/α−1−1/swBy,2j+1r·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s.
On the other hand, applying the pointwise estimate (75) and Chebyshev’s inequality, we get
(93)J′4y,r≤wqBy,r1/β−1/q−1/s·4λ∫By,rIγbBy,r−bf2xqwqxdx1/q≤wqBy,r1/β−1/s·Cλ∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rbz−bBy,r·fzdz≤wqBy,r1/β−1/s·Cλ∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rbz−bBy,2j+1r·fzdz+wqBy,r1/β−1/s·Cλ∑j=1∞1By,2j+1r1−γ/n∫By,2j+1rbBy,2j+1r−bBy,r·fzdz≔J′5y,r+J′6y,r.
For the termJ′5y,r, since w∈A1, it follows directly from the A1 condition and the inequality t≤Φt that
(94)J5′y,r≤wqBy,r1/β−1/s×Cλ∑j=1∞By,2j+1rγ/nwBy,2j+1r∫By,2j+1rbz−bBy,2j+1r⋅fzwzdz≤C⋅wqBy,r1/β−1/s×∑j=1∞By,2j+1rγ/nwBy,2j+1r∫B2j+1rbz−bBy,2j+1r⋅Φfzλwzdz.
Furthermore, we use the generalized Hölder’s inequality (29) to obtain
(95)J′5y,r≤C·wqBy,r1/β−1/s×∑j=1∞By,2j+1rγ/n·b−bBy,2j+1rexpLw,By,2j+1rΦfλLlogLw,By,2j+1r≤Cb∗∑j=1∞ΦfλLlogLw,By,2j+1r×By,2j+1rγ/nwqBy,r1/β−1/s.
In the last inequality, we have used the well-known fact that (see [26], for instance)
(96)b−bBexpLw,B≤Cb∗,foreveryballB⊂Rn.
It is equivalent to the inequality(97)1wB∫Bexpby−bBc0b∗wydy≤C,which is just a corollary of the well-known John-Nirenberg’s inequality (see [7]) and the comparison property of A1 weights. In addition, by the estimate (57)
(98)J′5y,r≤Cb∗∑j=1∞ΦfλLlogLw,By,2j+1r×wqBy,2j+1r1/α−1−1/swBy,2j+1r·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s.
For the last termJ′6y,r we proceed as follows. Using the first part of Lemma 20 together with the facts that w∈A1 and t≤Φt=t·1+log+t, we deduce that
(99)J′6y,r≤C·wqBy,r1/β−1/s∑j=1∞j+1b∗·1By,2j+1r1−γ/n∫By,2j+1rfzλdz≤C·wqBy,r1/β−1/s∑j=1∞j+1b∗·By,2j+1rγ/nwBy,2j+1r∫By,2j+1rfzλ·wzdz≤Cb∗·wqBy,r1/β−1/s∑j=1∞j+1·By,2j+1rγ/nwBy,2j+1r∫By,2j+1rΦfzλ·wzdz.
Making use of the inequalities (32) and (57), we further obtain
(100)J′6y,r≤Cb∗∑j=1∞j+1·ΦfλLlogLw,By,2j+1r×By,2j+1rγ/nwqBy,r1/β−1/s≤Cb∗∑j=1∞j+1·ΦfλLlogLw,By,2j+1r×wqBy,2j+1r1/α−1−1/swBy,2j+1r·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s.
Summarizing the above discussions, we conclude that(101)J′2y,r≤Cb∗∑j=1∞j+1·ΦfλLlogLw,By,2j+1r×wqBy,2j+1r1/α−1−1/swBy,2j+1r·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s=C∑j=1∞wqBy,2j+1r1/α−1−1/swBy,2j+1rΦfλLlogLw,By,2j+1r×j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s.
Recall thatwq∈A1⊂A∞ with 1<q<∞. We can now argue exactly as we did in the estimation of (83) to get (now, choose δ∗ in (17))
(102)∑j=1∞j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤C∑j=1∞j+1·By,rBy,2j+1rδ∗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 by our assumption, which guarantees that the last series is convergent. Therefore, by taking the Lsμ-norm of both sides of (85) (with respect to the variable y) and then using Minkowski’s inequality, (87) and (101), we have
(103)wqBy,r1/β−1/q−1/s·wqx∈By,r:b,Iγfx>λ1/qLsμ≤J′1y,rLsμ+J′2y,rLsμ≤CwqBy,2r1/α−1−1/swBy,2rΦfλLlogLw,By,2rLsμ+C∑j=1∞wqBy,2j+1r1/α−1−1/swBy,2j+1rΦfλLlogLw,By,2j+1rLsμ×j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤CΦfλLlogL,Lsαw,wq;μ+CΦfλLlogL,Lsαw,wq;μ×∑j=1∞j+1·wqBy,r1/β−1/swqBy,2j+1r1/β−1/s≤CΦfλLlogL,Lsαw,wq;μ,where the last inequality follows from (102). This completes the proof of Theorem 13.Letbx be a BMO function on Rn and 0<γ<n. The related commutator formed by fractional maximal operator Mγ and b is given by
(104)b,Mγfx≔supB∋x1B1−γ/n∫Bbx−by·fydy,where the supremum is taken over all balls B containing x. Obviously, b,Mγ is a sublinear operator. It should be pointed out that b,Mγf can be controlled pointwise by the expression given below. For any 0<γ<n, x∈Rn, and r>0, we have
(105)∫Rnbx−by·fyx−yn−γdy≥∫y−x<rbx−by·fyx−yn−γdy≥1rn−γ∫y−x<rbx−by·fydy.Taking the supremum for allr>0 on both sides of the above inequality, we get
(106)∫Rnbx−by·fyx−yn−γdy≥C·b,Mγfx,forallx∈Rn,which is our desired result. Moreover, on the commutator b,Mγ of the fractional maximal operator Mγ, we also have the following result.Theorem 21 (see [22]).
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Suppose that b∈BMORn, then the sublinear operator b,Mγ is bounded from Lpwp to Lqwq.Taking into account (106) and Theorem 21 and then using the same arguments as in the proof of Theorem 11, we know that the conclusion of Theorem 11 still holds for the sublinear operator b,Mγ.Corollary 22.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Assume that p≤α<β<s≤∞, μ∈Δ2, and b∈BMORn, then the sublinear operator b,Mγ is bounded from Lp,Lsαwp,wq;μ to Lq,Lsβwq;μ with 1/β=1/α−γ/n.LetL−γ/2 be the generalized fractional integrals of L for 0<γ<n, and let b be a locally integrable function on Rn. The generalized commutator generated by b and L−γ/2 is defined as follows:
(107)b,L−γ/2fx≔bxL−γ/2fx−L−γ/2bfx.By the kernel estimate (68),
(108)b,L−γ/2fx=∫Rnbx−byKγx,yfydy≤∫Rnbx−byKγx,yfydy≤C∫Rnbx−by·fyx−yn−γdy.In 2008, Auscher and Martell [33] considered the weighted estimate for b,L−γ/2 and obtained the following result (see also [34]).Theorem 23 (see [33]).
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Suppose that b∈BMORn, then the generalized commutator b,L−γ/2 is bounded from Lpwp to Lqwq.Hence, as a direct consequence of the above results, we can also obtain the following.Corollary 24.
Let0<γ<n, 1<p<n/γ, 1/q=1/p−γ/n, and w∈Ap,q. Assume that p≤α<β<s≤∞, μ∈Δ2, and b∈BMORn, then the generalized commutator b,L−γ/2 is bounded from Lp,Lsαwp,wq;μ to Lq,Lsβwq;μ with 1/β=1/α−γ/n.


## 5. Proof of Theorem 15

This section is concerned with the proof of Theorem15 and the corresponding result for generalized fractional integrals L−γ/2.Proof of Theorem 15.
Let1<p≤α<s≤∞ and f∈Lp,Lsαwp,wq;μ with w∈Ap,q and μ∈Δ2. For any fixed ball B=By,r in Rn, we are going to estimate the following expression:
(109)1By,r∫By,rIγfx−IγfBy,rdx.
Decomposef as f=f1+f2, where f1=f·χ4B, f2=f·χ4Bc, and 4B=By,4r. By the linearity of the fractional integral operator Iγ, the above expression (109) can be divided into two parts. That is,
(110)1By,r∫By,rIγfx−IγfBy,rdx≤1By,r∫By,rIγf1x−Iγf1By,rdx+1By,r∫By,rIγf2x−Iγf2By,rdx≔Iy,r+IIy,r.
Let us first consider the termIy,r. Applying the weighted Lp,Lq-boundedness of Iγ (see Theorem 1) and Hölder’s inequality, we obtain
(111)Iy,r≤2By,r∫By,rIγf1xdx≤2By,r∫By,rIγf1xqwqxdx1/q∫By,rwx−q′dx1/q′≤CBy,r∫By,4rfxpwpxdx1/p∫By,rwx−q′dx1/q′.
Sincew is a weight in the class Ap,q, one has wq∈Aq⊂A∞ by Lemma 7(i). By definition, it reads
(112)1By,r∫By,rwqxdx1/q1By,r∫By,rwqx−q′/qdx1/q′≤C,which implies
(113)∫By,rwx−q′dx1/q′≤C·By,rwqBy,r1/q.Since wq∈Aq⊂A∞, then wq∈Δ2. Using the inequalities (113) and (16), we have
(114)Iy,r≤C∫By,4rfxpwpxdx1/p·1wqBy,r1/q≤C∫By,4rfxpwpxdx1/p·1wqBy,4r1/q=C·wqBy,4r1/α−1/p−1/sf·χBy,4rLpwp,where in the last equality we have used the hypothesis 1/s=1/α−γ/n and 1/q=1/p−γ/n. We now turn to estimate the second term IIy,r. For any x∈By,r,
(115)Iγf2x−Iγf2By,r=1By,r∫By,rIγf2x−Iγf2zdz=1By,r∫By,r∫By,4rc1x−ζn−γ−1z−ζn−γfζdζdz≤1By,r∫By,r∫By,4rc1x−ζn−γ−1z−ζn−γ·fζdζdz.
Since bothx and z are in By,r, ζ∈By,4rc, by a purely geometric observation, we must have x−ζ≥2x−z. This fact along with the mean value theorem yields
(116)Iγf2x−Iγf2By,r≤CBy,r∫By,r∫By,4rcx−zx−ζn−γ+1·fζdζdz≤C∫By,4rcrζ−yn−γ+1·fζdζ=C∑j=2∞∫By,2j+1r\By,2jrrζ−yn−γ+1·fζdζ≤C∑j=2∞12j·1By,2j+1r1−γ/n∫By,2j+1rfζdζ.
Furthermore, by using Hölder’s inequality andAp,q condition on w, the last expression in (116) can be estimated as follows:
(117)1By,2j+1r1−γ/n∫By,2j+1rfζdζ≤1By,2j+1r1−γ/n∫By,2j+1rfζpwpζdζ1/p∫By,2j+1rwζ−p′dζ1/p′≤C∫By,2j+1rfζpwpζdζ1/p·1wqBy,2j+1r1/q=C·wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwp,where in the last equality we have used the fact that 1/α−1/p−1/s=−1/q again. From this, it readily follows that for any x∈By,r,
(118)Iγf2x−Iγf2By,r≤C∑j=2∞12j·wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwp.
Consequently,(119)IIy,r=1By,r∫By,rIγf2x−Iγf2By,rdx≤C∑j=2∞12j·wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwp.
Therefore, by taking theLsμ-norm of (109) (with respect to the variable y) and then using Minkowski’s inequality, (114) and (119), we get
(120)1By,r∫By,rIγfx−IγfBy,rdxLsμ≤Iy,rLsμ+IIy,rLsμ≤CwqBy,4r1/α−1/p−1/sf·χBy,4rLpwpLsμ+C∑j=2∞12j·wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwpLsμ≤CfLp,Lsαwp,wq;μ+CfLp,Lsαwp,wq;μ×∑j=2∞12j≤CfLp,Lsαwp,wq;μ.
By taking the supremum over allr>0, we are done.For anyf∈LpRn, 1≤p<∞, Martell [35] defined a kind of sharp maximal function ML#f associated with the semigroup e−tLt>0 by the following expression:
(121)ML#fx≔supx∈B1B∫Bfy−e−tBLfydy,where tB=rB2 and rB is the radius of the ball B. We say that f∈BMOL if the sharp maximal function ML#f∈L∞Rn, and we define fBMOL=ML#fL∞. Inspired by this notion and Theorem 15, a natural question for the generalized fractional integrals L−γ/2 is the following: can we get any result corresponding to Theorem 15 for the limiting case β=s? For this purpose, we need to introduce the following BMO-type space associated with the semigroup e−tLt>0.Definition 25.
Let1≤s≤∞ and μ∈Δ2. We define the space BMOL,Lsμ as the set of all locally integrable functions f satisfying f∗∗∗<∞, where
(122)f∗∗∗≔supr>01By,r∫By,rfx−e−r2LfxdxLsμ,and the Lsμ-norm is taken with respect to the variable y.
Based on the above notion, we can prove the following result.Theorem 26.
Let0<γ<n, 1<p<n/γ, and 1/q=1/p−γ/n, and let w∈Ap,q and μ∈Δ2. If p≤α<s≤∞ and 1/s=1/α−γ/n, then the generalized fractional integral L−γ/2 is bounded from Lp,Lsαwp,wq;μ to BMOL,Lsμ.Proof.
Let1<p≤α<s≤∞ and f∈Lp,Lsαwp,wq;μ with w∈Ap,q and μ∈Δ2. In this situation, for any given ball B=By,r in Rn, we need to consider the following expression:
(123)1By,r∫By,rL−γ/2fx−e−r2LL−γ/2fxdx.
Decomposef as f=f1+f2, where f1=f·χ4B, f2=f·χ4Bc, and 4B=By,4r. Similarly, the above expression (123) can be divided into three parts. That is,
(124)1By,r∫By,rL−γ/2fx−e−r2LL−γ/2fxdx≤1By,r∫By,rL−γ/2f1xdx+1By,r∫By,re−r2LL−γ/2f1xdx+1By,r∫By,rL−γ/2f2x−e−r2LL−γ/2f2xdx≔I′y,r+II′y,r+III′y,r.
First, let us consider the termI′y,r. By (66) and Theorem 1, we know that the generalized fractional integral L−γ/2 is also bounded from Lpwp to Lqwq whenever w∈Ap,q. This fact along with Hölder’s inequality implies
(125)I′y,r≤1By,r∫By,rL−γ/2f1xqwqxdx1/q∫By,rwx−q′dx1/q′≤CBy,r∫By,4rfxpwpxdx1/p∫By,rwx−q′dx1/q′.
We now proceed exactly as we did in the proof of Theorem15, then
(126)I′y,r≤C·wqBy,4r1/α−1/p−1/sf·χBy,4rLpwp.
For the termII′y,r, since the kernel of e−r2L is pr2x,z, then we may write
(127)II′y,r≤1By,r∫By,r∫Rnpr2x,z·L−γ2f1zdzdx=1By,r∫By,r∫By,4rpr2x,z·L−γ2f1zdzdx+∑j=2∞1By,r∫By,r∫By,2j+1r\By,2jrpr2x,z·L−γ2f1zdzdx=II′1y,r+II′2y,r.
For anyx∈By,r and z∈By,4r, by (64), we have pr2x,z≤C·r2−n/2. Thus,
(128)II′1y,r≤CBy,r∫By,r∫By,4r1r2n/2L−γ/2f1zdzdx≤CBy,4r∫By,4rL−γ/2f1zdz.
On the other hand, note that for anyx∈By,r, z∈By,4rc, then z−y≈z−x. In this case, by using (64) again, we get pr2x,z≤C·r2n/2/x−z2n. Hence,
(129)II′2y,r≤C∑j=2∞1By,r∫By,r∫By,2j+1r\By,2jrrnx−z2nL−γ/2f1zdzdx≤C∑j=2∞∫By,2j+1r\By,2jrrny−z2nL−γ/2f1zdz≤C∑j=2∞12jn1By,2j+1r∫By,2j+1rL−γ/2f1zdz.
Summing up the above estimates forII′1y,r and II′2y,r, we get
(130)II′y,r≤C∑j=1∞12jn1By,2j+1r∫By,2j+1rL−γ/2f1zdz.
Furthermore, using Hölder’s inequality and invoking the weighted boundedness ofL−γ/2 mentioned above, we can deduce that
(131)II′y,r≤C∑j=1∞12jn1By,2j+1r×∫By,2j+1rL−γ/2f1zqwqzdz1/q∫By,2j+1rwz−q′dz1/q′≤C∑j=1∞12jn1By,2j+1r×∫By,2j+1rfzpwpzdz1/p∫By,2j+1rwz−q′dz1/q′.
Again, arguing as in the proof of Theorem15, we can also show that
(132)II′y,r≤C∑j=1∞12jn·wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwp.
In order to estimate the last termIII′y,r, we need the following key lemma given in [36] (see also [29]).Lemma 27.
For0<γ<n, the difference operator I−e−tLL−γ/2 has an associated kernel K~γ,tx,z which satisfies the following estimate:
(133)K~γ,tx,z≤Cx−zn−γ·tx−z2.Let us return to the proof ofIII′y,r. By the above kernel estimate (133), we have
(134)III′y,r=1By,r∫By,rI−e−r2LL−γ/2f2xdx≤1By,r∫By,r∫By,4rcK~γ,r2x,zfzdzdx≤CBy,r∫By,r∫By,4rc1x−zn−γ·r2x−z2fzdzdx≤C∫By,4rc1y−zn−γ·r2y−z2fzdz,where the last inequality is due to x−z≈y−z when x∈By,r and z∈By,4rc. Hence,
(135)III′y,r≤C∑j=2∞12j2·1By,2j+1r1−γ/n∫By,2j+1rfzdz.Moreover, in view of the estimate (117), we obtain
(136)III′y,r≤C∑j=2∞12j2·wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwp.Therefore, by taking theLsμ-norm of (123) (with respect to the variable y) and then using Minkowski’s inequality, (126), (132), and (136), we get
(137)1By,r∫By,rL−γ/2fx−e−r2LL−γ/2fxdxLsμ≤I′y,rLsμ+II′y,rLsμ+III′y,rLsμ≤CwqBy,4r1/α−1/p−1/sf·χBy,4rLpwpLsμ+C∑j=1∞12jn·wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwpLsμ+C∑j=2∞12j2·wqBy,2j+1r1/α−1/p−1/sf·χBy,2j+1rLpwpLsμ≤CfLp,Lsαwp,wq;μ+CfLp,Lsαwp,wq;μ×∑j=1∞12jn+CfLp,Lsαwp,wq;μ×∑j=2∞12j2≤CfLp,Lsαwp,wq;μ.We end the proof by taking the supremum over allr>0.


## 6. Some Results on Two-Weight Problems

In the last section, we consider related problems about two-weight, weak-type norm inequalities forIγ and b,Iγ on weighted amalgam spaces. In [37], Cruz-Uribe and Perez considered the problem of finding sufficient conditions on a pair of weights w,ν which ensure the boundedness of the operator Iγ from Lpν to WLpw, where 1<p<∞. They gave a sufficient Ap-type condition (see (138) below) and proved a two-weight, weak-type p,p inequality for Iγ (see also [38] for another, more simpler proof), which solved a problem posed by Sawyer and Wheeden in [39].Theorem 28 (see [37, 38]).
Let0<γ<n and 1<p<∞. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes QinRn,
(138)Qγ/n·1Q∫Qwxrdx1/rp1Q∫Qνx−p′/pdx1/p′≤C<∞.
Then, the fractional integral operatorIγ satisfies the weak-type p,p inequality
(139)wx∈Rn:Iγfx>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0.Moreover, in [40], Li improved this result by replacing the “power bump” in (138) by a smaller “Orlicz bump.” On the other hand, in [41], Liu and Lu obtained a sufficient Ap-type condition for the commutator b,Iγ to satisfy the two-weight weak type p,p inequality, where 1<p<∞. That condition is an Ap-type condition in the scale of Orlicz spaces (see (140) below).Theorem 29 (see [41]).
Let0<γ<n, 1<p<∞, and b∈BMORn. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes QinRn,
(140)Qγ/n·1Q∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′·1+log+tp′. Then, the linear commutator b,Iγ satisfies the weak-type p,p inequality
(141)wx∈Rn:b,Iγfx>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0.Here and in what follows, all cubes are assumed to have their sides parallel to the coordinate axes andQy,ℓ will denote the cube centered at y and has side length ℓ. For any cube Qy,ℓ and any λ>0, λQ stands for the cube concentric with Q and having side length λ times as long, i.e., λQ≔Qy,λℓ. We now extend the results mentioned above to the weighted amalgam spaces.Theorem 30.
Let0<γ<n, 1<p≤α<s≤∞, and μ∈Δ2. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes QinRn, (138) holds. If w∈Δ2, then the fractional integral operator Iγ is bounded from Lp,Lsαν,w;μ to WLp,Lsαw;μ.Theorem 31.
Let0<γ<n, 1<p≤α<s≤∞, μ∈Δ2, and b∈BMORn. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn, (140) holds. If w∈A∞, then the linear commutator b,Iγ is bounded from Lp,Lsαν,w;μ to WLp,Lsαw;μ.Proof of Theorem 30.
Let1<p≤α<s≤∞ and f∈Lp,Lsαν,w;μ with w∈Δ2 and μ∈Δ2. For arbitrary y∈Rn, set Q=Qy,ℓ for the cube centered at y and of the side length ℓ. Let
(142)f=f·χ2Q+f·χ2Qc≔f1+f2,where χ2Q denotes the characteristic function of 2Q=Qy,2ℓ. Then, for given y∈Rn and ℓ>0, we write
(143)wQy,ℓ1/α−1/p−1/sIγf·χQy,ℓWLpw≤2·wQy,ℓ1/α−1/p−1/sIγf1·χQy,ℓWLpw+2·wQy,ℓ1/α−1/p−1/sIγf2·χQy,ℓWLpw≔K1y,ℓ+K2y,ℓ.
Using Theorem28, we get
(144)K1y,ℓ≤2·wQy,ℓ1/α−1/p−1/sIγf1WLpw≤C·wQy,ℓ1/α−1/p−1/s∫Qy,2ℓfxpνxdx1/p=C·wQy,2ℓ1/α−1/p−1/sf·χQy,2ℓLpν×wQy,ℓ1/α−1/p−1/swQy,2ℓ1/α−1/p−1/s.
Moreover, since1/α−1/p−1/s<0 and w∈Δ2, then by doubling inequality (16)(consider cube Q instead of ball B), we obtain
(145)wQy,ℓ1/α−1/p−1/swQy,2ℓ1/α−1/p−1/s≤C.
Substituting the above inequality (145) into (144), we thus obtain
(146)K1y,ℓ≤C·wQy,2ℓ1/α−1/p−1/sf·χQy,2ℓLpν.
We now estimate the second termK2y,ℓ. Using the same methods and steps as we deal with I2y,r in Theorem 9, we can also obtain that for any x∈Qy,ℓ,
(147)Iγf2x≤C∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓfzdz.
This pointwise estimate (147) together with Chebyshev’s inequality implies
(148)K2y,ℓ≤2·wQy,ℓ1/α−1/p−1/s∫Qy,ℓIγf2xpwxdx1/p≤C·wQy,ℓ1/α−1/s∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓfzdz.
A further application of Hölder’s inequality yields(149)K2y,ℓ≤C·wQy,ℓ1/α−1/s∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓfzpνzdz1/p×∫Qy,2j+1ℓνz−p′/pdz1/p′=C∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν×wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s·wQy,2j+1ℓ1/pQy,2j+1ℓ1−γ/n∫Qy,2j+1ℓνz−p′/pdz1/p′.
In addition, we apply Hölder’s inequality with exponentr to get
(150)wQy,2j+1ℓ=∫Qy,2j+1ℓwzdz≤Qy,2j+1ℓ1/r′∫Qy,2j+1ℓwzrdz1/r.
Hence, in view of (150), we have
(151)K2y,ℓ≤C∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s×Qy,2j+1ℓ1/r′pQy,2j+1ℓ1−γ/n∫Qy,2j+1ℓwzrdz1/rp∫Qy,2j+1ℓνz−p′/pdz1/p′≤C∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s.
The last inequality is obtained by theAp-type condition (138) on w,ν. Furthermore, since w∈Δ2, we can easily check that there exists a reverse doubling constant D=Dw>1 independent of Q such that (see Lemma 4.1 in [17])
(152)w2Q≥D·wQ,foranycubeQ⊂Rn,which implies that for any positive integer j∈ℤ+, w2j+1Q≥Dj+1·wQ by iteration. Hence,
(153)∑j=1∞wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s≤∑j=1∞wQy,ℓDj+1·wQy,ℓ1/α−1/s=∑j=1∞1Dj+11/α−1/s≤C,where the last series is convergent since the reverse doubling constant D>1 and 1/α−1/s>0. Therefore, by taking the Lsμ-norm of both sides of (143) (with respect to the variable y) and then using Minkowski’s inequality, (146), (151), and (153), we have
(154)wQy,ℓ1/α−1/p−1/sIγf·χQy,ℓWLpwLsμ≤K1y,ℓLsμ+K2y,ℓLsμ≤CwQy,2ℓ1/α−1/p−1/sf·χQy,2ℓLpνLsμ+C∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpνLsμ×wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s≤CfLp,Lsαν,w;μ+CfLp,Lsαν,w;μ×∑j=1∞wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s≤CfLp,Lsαν,w;μ.
Finally, by taking the supremum over allℓ>0, we finish the proof of Theorem 30.Proof of Theorem 31.
Let1<p≤α<s≤∞ and f∈Lp,Lsαν,w;μ with w∈A∞ and μ∈Δ2. For an arbitrary cube Q=Qy,ℓ in Rn, as before, we set
(155)f=f1+f2,f1=f·χ2Q,f2=f·χ2Qc.
Then, for any giveny∈Rn and ℓ>0, we write
(156)wQy,ℓ1/α−1/p−1/sb,Iγf·χQy,ℓWLpw≤2·wQy,ℓ1/α−1/p−1/sb,Iγf1·χQy,ℓWLpw+2·wQy,ℓ1/α−1/p−1/sb,Iγf2·χQy,ℓWLpw≔K′1y,ℓ+K′2y,ℓ.
Sincew∈A∞, we know that w∈Δ2. Applying Theorem 29 and inequality (145), we get
(157)K′1y,ℓ≤2·wQy,ℓ1/α−1/p−1/sb,Iγf1WLpw≤C·wQy,ℓ1/α−1/p−1/s∫Qy,2ℓfxpνxdx1/p=C·wQy,2ℓ1/α−1/p−1/sf·χQy,2ℓLpν×wQy,ℓ1/α−1/p−1/swQy,2ℓ1/α−1/p−1/s≤C·wQy,2ℓ1/α−1/p−1/sf·χQy,2ℓLpν.
Next, we estimate the other termK′2y,ℓ. For any x∈Qy,ℓ, from the definition of b,Iγ, one can see that
(158)b,Iγf2x≤bx−bQy,ℓ·Iγf2x+IγbQy,ℓ−bf2x≔ξx+ηx.
Consequently, we can further divideK′2y,ℓ into two parts:
(159)K′2y,ℓ≤4·wQy,ℓ1/α−1/p−1/sξ··χQy,ℓWLpw+4·wQy,ℓ1/α−1/p−1/sη··χQy,ℓWLpw≔K′3y,ℓ+K′4y,ℓ.
For the termK′3y,ℓ, it follows directly from Chebyshev’s inequality and estimate (147) that
(160)K′3y,ℓ≤4·wQy,ℓ1/α−1/p−1/s∫Qy,ℓξxpwxdx1/p≤C·wQy,ℓ1/α−1/p−1/s∫Qy,ℓbx−bQy,ℓpwxdx1/p×∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓfzdz≤C·wQy,ℓ1/α−1/s∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓfzdz,where in the last inequality, we have used the fact that Lemma 20(ii) still holds with ball B replaced by cube Q, when w is an A∞ weight. Arguing as in the proof of Theorem 30, we can also obtain that
(161)K′3y,ℓ≤C∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s.
Let us now estimate the termK′4y,ℓ. Using the same methods and steps as we deal with J2y,r in Theorem 11, we can show the following pointwise estimate as well:
(162)ηx=IγbQy,ℓ−bf2x≤C∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓbz−bQy,ℓ·fzdz.
This, together with Chebyshev’s inequality, implies(163)K′4y,ℓ≤4·wQy,ℓ1/α−1/p−1/s∫Qy,ℓηxpwxdx1/p≤C·wQy,ℓ1/α−1/s∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓbz−bQy,ℓ·fzdz≤C·wQy,ℓ1/α−1/s∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓbz−bQy,2j+1ℓ·fzdz+C·wQy,ℓ1/α−1/s∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓbQy,2j+1ℓ−bQy,ℓ·fzdz≔K′5y,ℓ+K′6y,ℓ.
An application of Hölder’s inequality leads to that(164)K′5y,ℓ≤C·wQy,ℓ1/α−1/s∑j=1∞1Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓfzpνzdz1/p×∫Qy,2j+1ℓbz−bQy,2j+1ℓp′νz−p′/pdz1/p′=C·wQy,ℓ1/α−1/s∑j=1∞f·χQy,2j+1ℓLpνQy,2j+1ℓ1−γ/n×Qy,2j+1ℓ1/p′b−bQy,2j+1ℓ·ν−1/pC,Qy,2j+1ℓ,where Ct=tp′ is a Young function. For 1<p<∞, it is easy to see that the inverse function of Ct is C−1t=t1/p′. Also, observe that the following equality holds
(165)C−1t=t1/p′=t1/p′1+log+t×1+log+t=A−1t·B−1t,where
(166)At≈tp′·1+log+tp′Bt≈expt−1.
Thus, by generalized Hölder’s inequality (25) and estimate (96) (consider cube Q instead of ball B when w≡1), we have
(167)b−bQy,2j+1ℓ·ν−1/pC,Qy,2j+1ℓ≤Cb−bQy,2j+1ℓB,Qy,2j+1ℓ·ν−1/pA,Qy,2j+1ℓ≤Cb∗·ν−1/pA,Qy,2j+1ℓ.
Moreover, in view of (150), we can deduce that
(168)K′5y,ℓ≤Cb∗·wQy,ℓ1/α−1/s∑j=1∞f·χQy,2j+1ℓLpνQy,2j+1ℓ1/p−γ/n·ν−1/pA,Qy,2j+1ℓ=Cb∗∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s×wQy,2j+1ℓ1/pQy,2j+1ℓ1/p−γ/n·ν−1/pA,Qy,2j+1ℓ≤Cb∗∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s×Qy,2j+1ℓ1/r′pQy,2j+1ℓ1/p−γ/n∫Qy,2j+1ℓwzrdz1/rp·ν−1/pA,Qy,2j+1ℓ≤Cb∗∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s.
The last inequality is obtained by theAp-type condition (140) on w,ν. It remains to estimate the last term K′6y,ℓ. Applying Lemma 20(i) (use Q instead of B) and Hölder’s inequality, we get
(169)K′6y,ℓ≤C·wQy,ℓ1/α−1/s∑j=1∞j+1b∗Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓfzdz≤C·wQy,ℓ1/α−1/s∑j=1∞j+1b∗Qy,2j+1ℓ1−γ/n∫Qy,2j+1ℓfzpνzdz1/p×∫Qy,2j+1ℓνz−p′/pdz1/p′=Cb∗∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν×j+1·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s·wQy,2j+1ℓ1/pQy,2j+1ℓ1−γ/n∫Qy,2j+1ℓνz−p′/pdz1/p′.
LetCt and At be the same as before. Obviously, Ct≤At for all t>0; then, it is not difficult to see that for any given cube Q in Rn, we have fC,Q≤fA,Q by definition, which implies that condition (140) is stronger than condition (138). This fact together with (150) yields
(170)K′6y,ℓ≤Cb∗∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν×j+1·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s×Qy,2j+1ℓ1/r′pQy,2j+1ℓ1−γ/n∫Qy,2j+1ℓwzrdz1/rp∫Qy,2j+1ℓνz−p′/pdz1/p′≤Cb∗∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν×j+1·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s.
Summing up all the above estimates, we conclude that(171)K′2y,ℓ≤C∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpν×j+1·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s.
Moreover, sincew is an A∞ weight, one has w∈Δ2. Then, there exists a reverse doubling constant D=Dw>1 such that for any positive integer j∈ℤ+, wQy,2j+1ℓ≥Dj+1·wQy,ℓ. This allows us to get the following:
(172)∑j=1∞j+1·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s≤∑j=1∞j+1·wQy,ℓDj+1·wQy,ℓ1/α−1/s=∑j=1∞j+1·1Dj+11/α−1/s≤C.
Notice that the exponent1/α−1/s is positive because α<s, which guarantees that the last series is convergent. Thus, by taking the Lsμ-norm of both sides of (156) (with respect to the variable y) and then using Minkowski’s inequality, (157), (171), and (172), we finally obtain
(173)wQy,ℓ1/α−1/p−1/sb,Iγf·χQy,ℓWLpwLsμ≤K′1y,ℓLsμ+K′2y,ℓLsμ≤CwQy,2ℓ1/α−1/p−1/sf·χQy,2ℓLpνLsμ+C∑j=1∞wQy,2j+1ℓ1/α−1/p−1/sf·χQy,2j+1ℓLpνLsμ×j+1·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s≤CfLp,Lsαν,w;μ+CfLp,Lsαν,w;μ×∑j=1∞j+1·wQy,ℓ1/α−1/swQy,2j+1ℓ1/α−1/s≤CfLp,Lsαν,w;μ.
We therefore conclude the proof of Theorem31 by taking the supremum over all ℓ>0.In view of (63) and (66), as an immediate consequence of Theorem 30, we have the following result.Corollary 32.
Let0<γ<n, 1<p≤α<s≤∞, and μ∈Δ2. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes QinRn, (138) holds. Ifw∈Δ2; then, both fractional maximal operator Mγ and generalized fractional integrals L−γ/2 are bounded from Lp,Lsαν,w;μ to WLp,Lsαw;μ.



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*Source: 2697104-2020-08-03.xml*