# Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals

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

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## Abstract

Letℒ=−Δ+V be a Schrödinger operator on ℝd, d≥3, where Δ is the Laplacian operator on ℝd, and the nonnegative potential V belongs to the reverse Hölder class RHs with s≥d/2. For given 0<α<d, the fractional integrals associated with the Schrödinger operator ℒ is defined by ℐα=ℒ−α/2. Suppose that b is a locally integrable function on ℝd and the commutator generated by b and ℐα is defined by b.ℐαfx=bx⋅ℐαfx−ℐαbfx. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHs with s≥d/2. Then, we will establish the boundedness properties of the fractional integrals ℐα on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator b,ℐα in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than Ap,q, BMOℝd, and Lp,κμ,ν corresponding to the classical case (that is V≡0).


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## Body

## 1. Introduction

### 1.1. Critical Radius Functionρx

Letd≥3 be a positive integer and ℝd be the d-dimensional Euclidean space, and let V:ℝd⟶ℝ, d≥3, be a nonnegative locally integrable function that belongs to the reverse Hölder class RHs for some exponent 1<s<∞; i.e., there exists a positive constant C>0 such that the following reverse Hölder inequality(1)1B∫BVysdy1/s≤C1B∫BVydy,holds for every ball B in ℝd. For given V∈RHs with s≥d/2, we introduce the critical radius function ρx=ρx;V which is given by(2)ρx≔supr>0:1rd−2∫Bx,rVydy≤1,x∈ℝd,where Bx,r denotes the open ball centered at x and with radius r. It is well known that this auxiliary function satisfies 0<ρx<∞ for any x∈ℝd under the above condition on V (see [1]). We need the following known result concerning the critical radius function (2).Lemma 1 (see [1]).
IfV∈RHs with s≥d/2, then there exist two constants C0≥1 and N0>0 such that, for all x and y in ℝd,(3)1C01+x−yρx−N0≤ρyρx≤C01+x−yρxN0/N0+1.As a straightforward consequence of (3), we can see that, for each integer k≥1, the following estimate(4)1+2krρy≥1C01+rρx−N0/N0+11+2krρx,is valid for any y∈Bx,r with x∈ℝd and r>0, and C0 is defined in (3).


### 1.2. Fractional Integrals Associated with Schrödinger Operators

LetV∈RHs with s≥d/2. For such a potential V, we consider the Schrödinger differential operator on ℝd, d≥3,(5)ℒ≔−Δ+V,and its associated semigroup(6)Ttfx≔e−tℒfx=∫ℝdptx,yfydy,t>0,where ptx,y denotes the kernel of the operator e−tℒ,t>0. From the Feynman–Kac formula, it is well known that(7)ptx,y≤htx−y,t>0,where ht is the classical heat kernel; i.e.,(8)htx−y≔4πt−d/2exp−x−y24t.Moreover, this estimate (7) can be improved when V belongs to the reverse Hölder class RHs for some s≥d/2 (see [2,3], for instance). The auxiliary function ρx arises naturally in this context.Proposition 1.
LetV∈RHs with s≥d/2. For every positive integer N≥1, there exists a positive constant CN>0 such that, for all x and y in ℝd,(9)ptx,y≤CN⋅t−d/2exp−x−y25t1+tρx+tρy−N,t>0.
For given0<α<d, the ℒ-Riesz potential or ℒ-fractional integral operator is defined by(10)ℐαfx≔ℒ−α/2fx=∫0∞e−tℒfxtα/2−1dt.
In this work, we shall be interested in the behavior of theℒ-fractional integral operator ℐα=ℒ−α/2.


### 1.3.Apρ,∞ and Ap,qρ,∞ Weights

A weight will always mean a nonnegative function which is locally integrable onℝd. Given a Lebesgue measurable set E and a weight w, E will denote the Lebesgue measure of E and(11)wE=∫Ewxdx.GivenB=Bx0,r and t>0, we will write tB for the t-dilate ball, which is the ball with the same center x0 and radius tr. As in [2] (see also [4, 5]), we say that a weight w belongs to the class Apρ,θ for 1<p<∞ and 0<θ<∞, if there is a positive constant C>0 such that, for all balls B=Bx0,r⊂ℝd with x0∈ℝd and r>0,(12)1B∫Bwxdx1/p1B∫Bwx−p′/pdx1/p′≤C⋅1+rρx0θ,where p′ is the dual exponent of p such that 1/p+1/p′=1. For p=1 and 0<θ<∞, we also say that a weight w belongs to the class A1ρ,θ, if there is a positive constant C>0 such that, for all balls B=Bx0,r⊂ℝd with x0∈ℝd and r>0,(13)1B∫Bwxdx≤C⋅1+rρx0θessinfx∈Bwx.For given1≤p<∞, we define(14)Apρ,∞≔∪θ>0Apρ,θ.For any givenθ>0, let us introduce the maximal operator which is given in terms of the critical radius function (2):(15)Mρ,θfx≔supr>01+rρx−θ1Bx,r∫Bx,rfydy,x∈ℝd.Observe that a weightw belongs to the class A1ρ,∞ if and only if there exists a positive number θ>0 such that Mρ,θwx≤Cwx, for a.e. x∈ℝd, where the constant C>0 is independent of w. Since(16)1≤1+rρx0θ1≤1+rρx0θ2,for 0<θ1<θ2<∞, then for given p with 1≤p<∞, one has(17)Ap⊂Apρ,θ1⊂Apρ,θ2,where Ap denotes the classical Muckenhoupt’s class (see [6], Chapter 7) and hence Ap⊂Apρ,∞. In addition, for some fixed θ>0 (see [7]),(18)A1ρ,θ⊂Ap1ρ,θ⊂Ap2ρ,θ,whenever 1≤p1<p2<∞. Obviously, for any fixed θ>0,(19)1≤1+2rρx0θ≤2θ1+rρx0θ.To establish weighted norm inequalities for fractional integrals, we need to introduce another weight classAp,qρ,∞. As in [7], we say that a weight w satisfies the condition Ap,qρ,θ for 1<p<q<∞ and 0<θ<∞, if there exists a positive constant C>0 such that, for any ball B=Bx0,r⊂ℝd with x0∈ℝd and r>0,(20)1B∫Bwxqdx1/q1B∫Bwx−p′dx1/p′≤C⋅1+rρx0θ,where p′=p/p−1. We also say that a weight w satisfies the condition A1,qρ,θ for 1<q<∞ and 0<θ<∞, if there exists a positive constant C>0 such that, for any ball B=Bx0,r⊂ℝd with x0∈ℝd and r>0,(21)1B∫Bwxqdx1/q≤C⋅1+rρx0θessinfx∈Bwx.Similarly, for givenp,q with 1≤p<q<∞, by (16), one has(22)Ap,q⊂Ap,qρ,θ1⊂Ap,qρ,θ2,whenever 0<θ1<θ2<∞. Here, Ap,q denotes the classical Muckenhoupt–Wheeden's class (see [8]). We also define(23)Ap,qρ,∞≔∪θ>0Ap,qρ,θ,1≤p<q<∞.So, we haveAp,q⊂Ap,qρ,∞. The following results (Lemmas 2–5) are extensions of well-known properties of Ap and Ap,q weights. We first present an important property of the classes of weights in Apρ,θ with 1≤p<∞, which was given by Bongioanni, Harboure, and Salinas in Lemma 5 of [2].Lemma 2 (see [2]).
Ifw∈Apρ,θ with 0<θ<∞ and 1≤p<∞, then there exist positive constants ϵ,η>0, and C>0 such that(24)1B∫Bwx1+εdx1/1+ε≤C1B∫Bwxdx1+rρx0η,for every ball B=Bx0,r in ℝd.
As a direct consequence of Lemma2, we have the following result.Lemma 3.
Ifw∈Apρ,θ with 0<θ<∞ and 1≤p<∞, then there exist two positive numbers δ>0 and η>0 such that(25)wEwB≤CEBδ1+rρx0η,for any measurable subset E of a ball B=Bx0,r, where C>0 is a constant which does not depend on E and B.Proof.
For any given ballB=Bx0,r with x0∈ℝd and r>0, suppose that E⊂B, then by Hölder’s inequality with exponent 1+ϵ and (24), we can deduce that(26)wE=∫BχEx⋅wxdx≤∫Bwx1+εdx1/1+ε∫BχEx1+ε/εdxε/1+ε≤CB1/1+ε1B∫Bwxdx1+rρx0ηEε/1+ε=CEBε/1+ε1+rρx0ηwB.
This gives (25) with δ=ϵ/1+ϵ. Here and in the sequel, the characteristic function of E is denoted by χE.
In view of Lemma2, we now define the reverse Hölder-type class RHqρ,θ that is given in terms of the critical radius function (2). We say that w∈RHqρ,θ for some 1<q<∞ and 0<θ<∞, if there exists a positive constant C>0 such that the following reverse Hölder-type inequality(27)1B∫Bwxqdx1/q≤C1B∫Bwxdx1+rρx0θholds for every ball B=Bx0,r in ℝd. The class RHqρ,∞ is defined as(28)RHqρ,∞≔∪θ>0RHqρ,θ,1<q<∞.
Clearly, one hasRHq⊂RHqρ,∞.
Let1<q<∞ and A1q=w:wq∈A1. It is known that, for the classical case (see [9]),(29)A1q=A1∩RHq.Lemma 4.
If1<q<∞ and 0<θ<∞, then wq∈A1ρ,θ implies that w∈A1ρ,θ/q∩RHqρ,θ/q.Proof.
The conclusionw∈A1ρ,θ/q follows easily by Hölder’s inequality and the definition of A1ρ,θ. Indeed, for any given ball B=Bx0,r with x0∈ℝd and r>0,(30)1B∫Bwxdx≤1B∫Bwqxdx1/q≤C⋅1+rρx0θ/qessinfx∈Bwqx1/q=C⋅1+rρx0θ/qessinfx∈Bwx.
On the other hand, fix a ballB=Bx0,r⊂ℝd and take y∈B. Let E be a ball centered at y0 and with radius h which contains y. By picking h small enough so that E⊂B, then by the condition wq∈A1ρ,θ, we can deduce that(31)1B∫BχExdx≤CwqB⋅essinfx∈Bwqx∫BχExdx1+rρx0θ≤CwqB∫BχEx⋅wqxdx1+rρx0θ,which is equivalent to(32)wqBB≤C⋅wqEE1+rρx0θ.
Since this holds for allE⊂B and y∈E, then by taking h⟶0+ and using Lebesgue differentiation theorem,(33)wqBB≤C⋅wqy1+rρx0θ.
Thus, by raising both sides to the power1/q and integrating over B, we get(34)1B∫Bwyqdy1/q≤C1B∫Bwydy1+rρx0θ/q.
This amounts tow∈RHqρ,θ/q.
A subtle interplay between these two classes of weights,Apρ,∞ and Ap,qρ,∞, is expressed by the following lemma:Lemma 5.
Suppose that1≤p<q<∞. Then, the following statements are true:(i) Ifp>1 and 0<θ<∞, then w∈Ap,qρ,θ implies that wq∈Atρ,θ⋅1/1/q+1/p′ with t=1+q/p′ and w−p′∈At′ρ,θ⋅1/1/q+1/p′ with t′=1+p′/q;(ii) Ifp=1 and 0<θ<∞, then w∈A1,qρ,θ implies that wq∈A1ρ,θ⋅q.Proof.
In fact, whent=1+q/p′, then a simple computation shows that(35)1t=1q⋅11/q+1/p′,1t′=t−1t=1p′⋅11/q+1/p′,q⋅−t′t=−q⋅1t−1=−p′.
Ifw∈Ap,qρ,θ with 1<p<q<∞ and 0<θ<∞, then we have(36)1B∫Bwqxdx1/t1B∫Bwqx−t′/tdx1/t′=1B∫Bwxqdx1/q1B∫Bwx−p′dx1/p′1/1/q+1/p′≤C⋅1+rρx0θ⋅1/1/q+1/p′,which means that wq∈Atρ,θ⋅1/1/q+1/p′ with t=1+q/p′. Here and in the sequel, for any positive number γ>0, we denote wγx=wxγ by convention. Analogously, it can be easily shown that w−p′∈At′ρ,θ⋅1/1/q+1/p′ with t′=1+p′/q. On the other hand, if w∈A1,qρ,θ with 1<q<∞ and 0<θ<∞, then we have(37)1B∫Bwqxdx≤C⋅1+rρx0θ⋅qessinfx∈Bwxq=C⋅1+rρx0θ⋅qessinfx∈Bwqx,which means that wq∈A1ρ,θ⋅q.
Given a weightw on ℝd, as usual, the weighted Lebesgue space Lpw for 1≤p<∞ is defined to be the set of all functions f such that(38)fLpw≔∫ℝdfxpwxdx1/p<∞.
We also denote byWLpw the weighted weak Lebesgue space consisting of all measurable functions f for which(39)fWLpw≔supλ>0λ⋅wx∈ℝd:fx>λ1/p<∞.
Note that ifℒ=−Δ is the Laplacian on ℝd, then ℒ−α/2 is the classical fractional integral operator Iα=−Δ−α/2 of order α. Let 0<α<d and 1≤p<d/α. Define 1<q<∞ by the relation 1/q=1/p−α/d. It is well known that when p>1 and w∈Ap,q, Iα is bounded from Lpwp into Lqwq. When p=1 and w∈A1,q, Iα is bounded from L1w into WLqwq (see [8]). Recently, Bongioanni et al. ([2], Theorem 4) established the weighted boundedness for fractional integral operators ℐα associated with Schrödinger operators defined in (10). They showed that the same estimates also hold for weights in the class Ap,qρ,∞, which is larger than Muckenhoupt–Wheeden’s class (another proof was later given by Tang in Theorem 3.8 of [7]). Their results can be summarized as follows:Theorem 1 (see [2]).
Let0<α<d, 1<p<d/α, 1/q=1/p−α/d, and w∈Ap,qρ,∞. If V∈RHs with s≥d/2, then the ℒ-fractional integral operator ℐα is bounded from Lpwp into Lqwq.Theorem 2 (see [2]).
Let0<α<d, p=1, q=d/d−α, and w∈A1,qρ,∞. If V∈RHs with s≥d/2, then the ℒ-fractional integral operator ℐα is bounded from L1w into WLqwq.


### 1.4. TheBMOρ,∞ℝd Space

For a locally integrable functionb on ℝd (usually called the symbol), we will also consider the commutator operator(40)b,ℐαfx≔bx⋅ℐαfx−ℐαbfx,x∈ℝd.Recently, Bongioanni et al. [5] introduced a kind of new spaces BMOρ,∞ℝd defined by(41)BMOρ,∞ℝd≔∪θ>0BMOρ,θℝd,where for 0<θ<∞, the space BMOρ,θℝd is defined to be the set of all locally integrable functions b satisfying(42)1Bx0,r∫Bx0,rbx−bBx0,rdx≤C⋅1+rρx0θ,for all balls Bx0,r with x0∈ℝd and r>0 and bBx0,r denotes the mean value of b on Bx0,r; that is,(43)bBx0,r≔1Bx0,r∫Bx0,rbydy.A norm forb∈BMOρ,θℝd, denoted by bBMOρ,θ, is given by the infimum of the constants satisfying (42), or equivalently(44)bBMOρ,θ≔supBx0,r1+rρx0−θ1Bx0,r∫Bx0,rbx−bBx0,rdx,where the supremum is taken over all balls Bx0,r with x0∈ℝd and r>0. With the above definition in mind, one has(45)BMOℝd⊂BMOρ,θ1ℝd⊂BMOρ,θ2ℝd,for 0<θ1<θ2<∞ by virtue of (16), and hence, BMOℝd⊂BMOρ,∞ℝd. Moreover, the classical BMO space [10] is properly contained in BMOρ,∞ℝd (see [4, 5] for more examples). We need the following key result for the space BMOρ,θℝd, which was proved by Tang in Proposition 4.2 of [7].Proposition 2 (see [7]).
Letb∈BMOρ,θℝd with 0<θ<∞. Then, there exist two positive constants C1 and C2 such that, for any given ball Bx0,r in ℝd and for any λ>0, we have(46)x∈Bx0,r:bx−bBx0,r>λ≤C1Bx0,rexp−1+rρx0−θ∗C2λbBMOρ,θ,where θ∗=N0+1θ and N0 is the constant appearing in Lemma 1.
As a consequence of Proposition2 and Lemma 3, we have the following result:Proposition 3.
Letb∈BMOρ,θℝd with 0<θ<∞ and w∈Apρ,∞ with 1≤p<∞. Then, there exist positive constants C1,C2 and η>0 such that, for any given ball Bx0,r in ℝd and for any λ>0, we have(47)wx∈Bx0,r:bx−bBx0,r>λ≤C1wBx0,rexp−1+rρx0−θ∗C2λbBMOρ,θ1+rρx0η,where θ∗=N0+1θ and N0 is the constant appearing in Lemma 1.
Notice that ifℒ=−Δ is the Laplacian operator on ℝd, then b,ℐα is just the commutator b,Iα of the classical fractional integrals. It is well known that when b∈BMOℝd, the commutator b,Iα is bounded from Lpwp (1<p<d/α) into Lqwq whenever 1/q=1/p−α/d and w∈Ap,q (see [12]). This corresponds to the norm inequalities satisfied by Iα. In Theorem 4.4 of [7], Tang obtained weighted strong-type estimate for the commutator b,ℐα of fractional integrals associated with Schrödinger operators, when b in a larger space than BMOℝd, that is the space BMOρ,∞ℝd. More precisely, he gave the following weighted result (see also [11], Theorem 3.5).Theorem 3 (see [7]).
Let0<α<d, 1<p<d/α, 1/q=1/p−α/d, and w∈Ap,qρ,∞. If V∈RHs with s≥d/2, then the commutator operator b,ℐα is bounded from Lpwp into Lqwq, whenever b belongs to BMOρ,∞ℝd.
In this paper, firstly, we will define several classes of weighted Morrey spaces related to certain nonnegative potentials satisfying appropriate reverse Hölder inequality. Secondly, we establish weighted estimates of fractional integralsℐα associated with ℒ on these new spaces. Finally, we also study the boundedness property for the commutators b,ℐα of fractional integrals with the new BMO functions defined above.
Throughout this paper,C denotes a positive constant not necessarily the same at each occurrence, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. 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 aandb such that C1a≤b≤C2a.


## 1.1. Critical Radius Functionρx

Letd≥3 be a positive integer and ℝd be the d-dimensional Euclidean space, and let V:ℝd⟶ℝ, d≥3, be a nonnegative locally integrable function that belongs to the reverse Hölder class RHs for some exponent 1<s<∞; i.e., there exists a positive constant C>0 such that the following reverse Hölder inequality(1)1B∫BVysdy1/s≤C1B∫BVydy,holds for every ball B in ℝd. For given V∈RHs with s≥d/2, we introduce the critical radius function ρx=ρx;V which is given by(2)ρx≔supr>0:1rd−2∫Bx,rVydy≤1,x∈ℝd,where Bx,r denotes the open ball centered at x and with radius r. It is well known that this auxiliary function satisfies 0<ρx<∞ for any x∈ℝd under the above condition on V (see [1]). We need the following known result concerning the critical radius function (2).Lemma 1 (see [1]).
IfV∈RHs with s≥d/2, then there exist two constants C0≥1 and N0>0 such that, for all x and y in ℝd,(3)1C01+x−yρx−N0≤ρyρx≤C01+x−yρxN0/N0+1.As a straightforward consequence of (3), we can see that, for each integer k≥1, the following estimate(4)1+2krρy≥1C01+rρx−N0/N0+11+2krρx,is valid for any y∈Bx,r with x∈ℝd and r>0, and C0 is defined in (3).


## 1.2. Fractional Integrals Associated with Schrödinger Operators

LetV∈RHs with s≥d/2. For such a potential V, we consider the Schrödinger differential operator on ℝd, d≥3,(5)ℒ≔−Δ+V,and its associated semigroup(6)Ttfx≔e−tℒfx=∫ℝdptx,yfydy,t>0,where ptx,y denotes the kernel of the operator e−tℒ,t>0. From the Feynman–Kac formula, it is well known that(7)ptx,y≤htx−y,t>0,where ht is the classical heat kernel; i.e.,(8)htx−y≔4πt−d/2exp−x−y24t.Moreover, this estimate (7) can be improved when V belongs to the reverse Hölder class RHs for some s≥d/2 (see [2,3], for instance). The auxiliary function ρx arises naturally in this context.Proposition 1.
LetV∈RHs with s≥d/2. For every positive integer N≥1, there exists a positive constant CN>0 such that, for all x and y in ℝd,(9)ptx,y≤CN⋅t−d/2exp−x−y25t1+tρx+tρy−N,t>0.
For given0<α<d, the ℒ-Riesz potential or ℒ-fractional integral operator is defined by(10)ℐαfx≔ℒ−α/2fx=∫0∞e−tℒfxtα/2−1dt.
In this work, we shall be interested in the behavior of theℒ-fractional integral operator ℐα=ℒ−α/2.


## 1.3.Apρ,∞ and Ap,qρ,∞ Weights

A weight will always mean a nonnegative function which is locally integrable onℝd. Given a Lebesgue measurable set E and a weight w, E will denote the Lebesgue measure of E and(11)wE=∫Ewxdx.GivenB=Bx0,r and t>0, we will write tB for the t-dilate ball, which is the ball with the same center x0 and radius tr. As in [2] (see also [4, 5]), we say that a weight w belongs to the class Apρ,θ for 1<p<∞ and 0<θ<∞, if there is a positive constant C>0 such that, for all balls B=Bx0,r⊂ℝd with x0∈ℝd and r>0,(12)1B∫Bwxdx1/p1B∫Bwx−p′/pdx1/p′≤C⋅1+rρx0θ,where p′ is the dual exponent of p such that 1/p+1/p′=1. For p=1 and 0<θ<∞, we also say that a weight w belongs to the class A1ρ,θ, if there is a positive constant C>0 such that, for all balls B=Bx0,r⊂ℝd with x0∈ℝd and r>0,(13)1B∫Bwxdx≤C⋅1+rρx0θessinfx∈Bwx.For given1≤p<∞, we define(14)Apρ,∞≔∪θ>0Apρ,θ.For any givenθ>0, let us introduce the maximal operator which is given in terms of the critical radius function (2):(15)Mρ,θfx≔supr>01+rρx−θ1Bx,r∫Bx,rfydy,x∈ℝd.Observe that a weightw belongs to the class A1ρ,∞ if and only if there exists a positive number θ>0 such that Mρ,θwx≤Cwx, for a.e. x∈ℝd, where the constant C>0 is independent of w. Since(16)1≤1+rρx0θ1≤1+rρx0θ2,for 0<θ1<θ2<∞, then for given p with 1≤p<∞, one has(17)Ap⊂Apρ,θ1⊂Apρ,θ2,where Ap denotes the classical Muckenhoupt’s class (see [6], Chapter 7) and hence Ap⊂Apρ,∞. In addition, for some fixed θ>0 (see [7]),(18)A1ρ,θ⊂Ap1ρ,θ⊂Ap2ρ,θ,whenever 1≤p1<p2<∞. Obviously, for any fixed θ>0,(19)1≤1+2rρx0θ≤2θ1+rρx0θ.To establish weighted norm inequalities for fractional integrals, we need to introduce another weight classAp,qρ,∞. As in [7], we say that a weight w satisfies the condition Ap,qρ,θ for 1<p<q<∞ and 0<θ<∞, if there exists a positive constant C>0 such that, for any ball B=Bx0,r⊂ℝd with x0∈ℝd and r>0,(20)1B∫Bwxqdx1/q1B∫Bwx−p′dx1/p′≤C⋅1+rρx0θ,where p′=p/p−1. We also say that a weight w satisfies the condition A1,qρ,θ for 1<q<∞ and 0<θ<∞, if there exists a positive constant C>0 such that, for any ball B=Bx0,r⊂ℝd with x0∈ℝd and r>0,(21)1B∫Bwxqdx1/q≤C⋅1+rρx0θessinfx∈Bwx.Similarly, for givenp,q with 1≤p<q<∞, by (16), one has(22)Ap,q⊂Ap,qρ,θ1⊂Ap,qρ,θ2,whenever 0<θ1<θ2<∞. Here, Ap,q denotes the classical Muckenhoupt–Wheeden's class (see [8]). We also define(23)Ap,qρ,∞≔∪θ>0Ap,qρ,θ,1≤p<q<∞.So, we haveAp,q⊂Ap,qρ,∞. The following results (Lemmas 2–5) are extensions of well-known properties of Ap and Ap,q weights. We first present an important property of the classes of weights in Apρ,θ with 1≤p<∞, which was given by Bongioanni, Harboure, and Salinas in Lemma 5 of [2].Lemma 2 (see [2]).
Ifw∈Apρ,θ with 0<θ<∞ and 1≤p<∞, then there exist positive constants ϵ,η>0, and C>0 such that(24)1B∫Bwx1+εdx1/1+ε≤C1B∫Bwxdx1+rρx0η,for every ball B=Bx0,r in ℝd.
As a direct consequence of Lemma2, we have the following result.Lemma 3.
Ifw∈Apρ,θ with 0<θ<∞ and 1≤p<∞, then there exist two positive numbers δ>0 and η>0 such that(25)wEwB≤CEBδ1+rρx0η,for any measurable subset E of a ball B=Bx0,r, where C>0 is a constant which does not depend on E and B.Proof.
For any given ballB=Bx0,r with x0∈ℝd and r>0, suppose that E⊂B, then by Hölder’s inequality with exponent 1+ϵ and (24), we can deduce that(26)wE=∫BχEx⋅wxdx≤∫Bwx1+εdx1/1+ε∫BχEx1+ε/εdxε/1+ε≤CB1/1+ε1B∫Bwxdx1+rρx0ηEε/1+ε=CEBε/1+ε1+rρx0ηwB.
This gives (25) with δ=ϵ/1+ϵ. Here and in the sequel, the characteristic function of E is denoted by χE.
In view of Lemma2, we now define the reverse Hölder-type class RHqρ,θ that is given in terms of the critical radius function (2). We say that w∈RHqρ,θ for some 1<q<∞ and 0<θ<∞, if there exists a positive constant C>0 such that the following reverse Hölder-type inequality(27)1B∫Bwxqdx1/q≤C1B∫Bwxdx1+rρx0θholds for every ball B=Bx0,r in ℝd. The class RHqρ,∞ is defined as(28)RHqρ,∞≔∪θ>0RHqρ,θ,1<q<∞.
Clearly, one hasRHq⊂RHqρ,∞.
Let1<q<∞ and A1q=w:wq∈A1. It is known that, for the classical case (see [9]),(29)A1q=A1∩RHq.Lemma 4.
If1<q<∞ and 0<θ<∞, then wq∈A1ρ,θ implies that w∈A1ρ,θ/q∩RHqρ,θ/q.Proof.
The conclusionw∈A1ρ,θ/q follows easily by Hölder’s inequality and the definition of A1ρ,θ. Indeed, for any given ball B=Bx0,r with x0∈ℝd and r>0,(30)1B∫Bwxdx≤1B∫Bwqxdx1/q≤C⋅1+rρx0θ/qessinfx∈Bwqx1/q=C⋅1+rρx0θ/qessinfx∈Bwx.
On the other hand, fix a ballB=Bx0,r⊂ℝd and take y∈B. Let E be a ball centered at y0 and with radius h which contains y. By picking h small enough so that E⊂B, then by the condition wq∈A1ρ,θ, we can deduce that(31)1B∫BχExdx≤CwqB⋅essinfx∈Bwqx∫BχExdx1+rρx0θ≤CwqB∫BχEx⋅wqxdx1+rρx0θ,which is equivalent to(32)wqBB≤C⋅wqEE1+rρx0θ.
Since this holds for allE⊂B and y∈E, then by taking h⟶0+ and using Lebesgue differentiation theorem,(33)wqBB≤C⋅wqy1+rρx0θ.
Thus, by raising both sides to the power1/q and integrating over B, we get(34)1B∫Bwyqdy1/q≤C1B∫Bwydy1+rρx0θ/q.
This amounts tow∈RHqρ,θ/q.
A subtle interplay between these two classes of weights,Apρ,∞ and Ap,qρ,∞, is expressed by the following lemma:Lemma 5.
Suppose that1≤p<q<∞. Then, the following statements are true:(i) Ifp>1 and 0<θ<∞, then w∈Ap,qρ,θ implies that wq∈Atρ,θ⋅1/1/q+1/p′ with t=1+q/p′ and w−p′∈At′ρ,θ⋅1/1/q+1/p′ with t′=1+p′/q;(ii) Ifp=1 and 0<θ<∞, then w∈A1,qρ,θ implies that wq∈A1ρ,θ⋅q.Proof.
In fact, whent=1+q/p′, then a simple computation shows that(35)1t=1q⋅11/q+1/p′,1t′=t−1t=1p′⋅11/q+1/p′,q⋅−t′t=−q⋅1t−1=−p′.
Ifw∈Ap,qρ,θ with 1<p<q<∞ and 0<θ<∞, then we have(36)1B∫Bwqxdx1/t1B∫Bwqx−t′/tdx1/t′=1B∫Bwxqdx1/q1B∫Bwx−p′dx1/p′1/1/q+1/p′≤C⋅1+rρx0θ⋅1/1/q+1/p′,which means that wq∈Atρ,θ⋅1/1/q+1/p′ with t=1+q/p′. Here and in the sequel, for any positive number γ>0, we denote wγx=wxγ by convention. Analogously, it can be easily shown that w−p′∈At′ρ,θ⋅1/1/q+1/p′ with t′=1+p′/q. On the other hand, if w∈A1,qρ,θ with 1<q<∞ and 0<θ<∞, then we have(37)1B∫Bwqxdx≤C⋅1+rρx0θ⋅qessinfx∈Bwxq=C⋅1+rρx0θ⋅qessinfx∈Bwqx,which means that wq∈A1ρ,θ⋅q.
Given a weightw on ℝd, as usual, the weighted Lebesgue space Lpw for 1≤p<∞ is defined to be the set of all functions f such that(38)fLpw≔∫ℝdfxpwxdx1/p<∞.
We also denote byWLpw the weighted weak Lebesgue space consisting of all measurable functions f for which(39)fWLpw≔supλ>0λ⋅wx∈ℝd:fx>λ1/p<∞.
Note that ifℒ=−Δ is the Laplacian on ℝd, then ℒ−α/2 is the classical fractional integral operator Iα=−Δ−α/2 of order α. Let 0<α<d and 1≤p<d/α. Define 1<q<∞ by the relation 1/q=1/p−α/d. It is well known that when p>1 and w∈Ap,q, Iα is bounded from Lpwp into Lqwq. When p=1 and w∈A1,q, Iα is bounded from L1w into WLqwq (see [8]). Recently, Bongioanni et al. ([2], Theorem 4) established the weighted boundedness for fractional integral operators ℐα associated with Schrödinger operators defined in (10). They showed that the same estimates also hold for weights in the class Ap,qρ,∞, which is larger than Muckenhoupt–Wheeden’s class (another proof was later given by Tang in Theorem 3.8 of [7]). Their results can be summarized as follows:Theorem 1 (see [2]).
Let0<α<d, 1<p<d/α, 1/q=1/p−α/d, and w∈Ap,qρ,∞. If V∈RHs with s≥d/2, then the ℒ-fractional integral operator ℐα is bounded from Lpwp into Lqwq.Theorem 2 (see [2]).
Let0<α<d, p=1, q=d/d−α, and w∈A1,qρ,∞. If V∈RHs with s≥d/2, then the ℒ-fractional integral operator ℐα is bounded from L1w into WLqwq.


## 1.4. TheBMOρ,∞ℝd Space

For a locally integrable functionb on ℝd (usually called the symbol), we will also consider the commutator operator(40)b,ℐαfx≔bx⋅ℐαfx−ℐαbfx,x∈ℝd.Recently, Bongioanni et al. [5] introduced a kind of new spaces BMOρ,∞ℝd defined by(41)BMOρ,∞ℝd≔∪θ>0BMOρ,θℝd,where for 0<θ<∞, the space BMOρ,θℝd is defined to be the set of all locally integrable functions b satisfying(42)1Bx0,r∫Bx0,rbx−bBx0,rdx≤C⋅1+rρx0θ,for all balls Bx0,r with x0∈ℝd and r>0 and bBx0,r denotes the mean value of b on Bx0,r; that is,(43)bBx0,r≔1Bx0,r∫Bx0,rbydy.A norm forb∈BMOρ,θℝd, denoted by bBMOρ,θ, is given by the infimum of the constants satisfying (42), or equivalently(44)bBMOρ,θ≔supBx0,r1+rρx0−θ1Bx0,r∫Bx0,rbx−bBx0,rdx,where the supremum is taken over all balls Bx0,r with x0∈ℝd and r>0. With the above definition in mind, one has(45)BMOℝd⊂BMOρ,θ1ℝd⊂BMOρ,θ2ℝd,for 0<θ1<θ2<∞ by virtue of (16), and hence, BMOℝd⊂BMOρ,∞ℝd. Moreover, the classical BMO space [10] is properly contained in BMOρ,∞ℝd (see [4, 5] for more examples). We need the following key result for the space BMOρ,θℝd, which was proved by Tang in Proposition 4.2 of [7].Proposition 2 (see [7]).
Letb∈BMOρ,θℝd with 0<θ<∞. Then, there exist two positive constants C1 and C2 such that, for any given ball Bx0,r in ℝd and for any λ>0, we have(46)x∈Bx0,r:bx−bBx0,r>λ≤C1Bx0,rexp−1+rρx0−θ∗C2λbBMOρ,θ,where θ∗=N0+1θ and N0 is the constant appearing in Lemma 1.
As a consequence of Proposition2 and Lemma 3, we have the following result:Proposition 3.
Letb∈BMOρ,θℝd with 0<θ<∞ and w∈Apρ,∞ with 1≤p<∞. Then, there exist positive constants C1,C2 and η>0 such that, for any given ball Bx0,r in ℝd and for any λ>0, we have(47)wx∈Bx0,r:bx−bBx0,r>λ≤C1wBx0,rexp−1+rρx0−θ∗C2λbBMOρ,θ1+rρx0η,where θ∗=N0+1θ and N0 is the constant appearing in Lemma 1.
Notice that ifℒ=−Δ is the Laplacian operator on ℝd, then b,ℐα is just the commutator b,Iα of the classical fractional integrals. It is well known that when b∈BMOℝd, the commutator b,Iα is bounded from Lpwp (1<p<d/α) into Lqwq whenever 1/q=1/p−α/d and w∈Ap,q (see [12]). This corresponds to the norm inequalities satisfied by Iα. In Theorem 4.4 of [7], Tang obtained weighted strong-type estimate for the commutator b,ℐα of fractional integrals associated with Schrödinger operators, when b in a larger space than BMOℝd, that is the space BMOρ,∞ℝd. More precisely, he gave the following weighted result (see also [11], Theorem 3.5).Theorem 3 (see [7]).
Let0<α<d, 1<p<d/α, 1/q=1/p−α/d, and w∈Ap,qρ,∞. If V∈RHs with s≥d/2, then the commutator operator b,ℐα is bounded from Lpwp into Lqwq, whenever b belongs to BMOρ,∞ℝd.
In this paper, firstly, we will define several classes of weighted Morrey spaces related to certain nonnegative potentials satisfying appropriate reverse Hölder inequality. Secondly, we establish weighted estimates of fractional integralsℐα associated with ℒ on these new spaces. Finally, we also study the boundedness property for the commutators b,ℐα of fractional integrals with the new BMO functions defined above.
Throughout this paper,C denotes a positive constant not necessarily the same at each occurrence, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. 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 aandb such that C1a≤b≤C2a.


## 2. Main Results

In this section, we introduce some types of weighted Morrey spaces related to the nonnegative potentialV and then give our main results.Definition 1.
Let1≤p<∞, 0≤κ<1, and μ,ν be two weights on ℝd. For given 0<θ<∞, the weighted Morrey space Lρ,θp,κμ,ν is defined to be the set of all p-locally integrable functions f on ℝd such that(48)1νBκ∫Bfxpμxdx1/p≤C⋅1+rρx0θ,for every ball B=Bx0,r in ℝd. A norm for f∈Lρ,θp,κμ,ν, denoted by fLρ,θp,κμ,ν, is given by the infimum of the constants in (48), or equivalently,(49)fLρ,θp,κμ,ν≔supB1+rρx0−θ1νBκ∫Bfxpμxdx1/p<∞,where the supremum is taken over all balls B in ℝd and x0 and r denote the center and radius of B, respectively. Define(50)Lρ,∞p,κμ,ν≔∪θ>0Lρ,θp,κμ,ν.
For any givenf∈Lρ,∞p,κμ,ν, let(51)θ∗≔infθ>0:f∈Lρ,θp,κμ,ν.
Now, define(52)f⋆=fLρ,∞p,κμ,ν≔fLρ,θ∗p,κμ,ν.
It is easy to see that⋅⋆ satisfies the axioms of a norm; i.e., for f,g∈Lρ,∞p,κμ,ν and λ∈ℝ, we have(i)f⋆≥0(ii)f⋆=0⟺f=0(iii)λf⋆=λf⋆(iv)f+g⋆≤f⋆+g⋆
Ifμ=ν=w, then we denote Lρ,θp,κw and Lρ,∞p,κw for simplicity.
Note that this definition does not coincide with the one given in [13] (see also [14] for the unweighted case), but in view of the space BMOρ,∞ℝd and the class Ap,qρ,∞ defined above, it is more natural in our setting. Obviously, if we take θ=0 or V≡0, then this new space is just the familiar weighted Morrey space Lp,κμ,ν (or Lp,κw), which was first introduced by Komori and Shirai in [15] (see also [16]).Definition 2.
Let1≤p<∞, 0≤κ<1, and w be a weight on ℝd. For given 0<θ<∞, the weighted weak Morrey space WLρ,θp,κw is defined to be the set of all measurable functions f on ℝd such that(53)1wBκ/psupλ>0λ⋅wx∈B:fx>λ1/p≤C⋅1+rρx0θfor every ball B=Bx0,r in ℝd, or equivalently,(54)fWLρ,θp,κw≔supB1+rρx0−θ1wBκ/psupλ>0λ⋅wx∈B:fx>λ1/p<∞.
Correspondingly, we define(55)WLρ,∞p,κw≔∪θ>0WLρ,θp,κw.
For any givenf∈WLρ,∞p,κw, let(56)θ∗∗≔infθ>0:f∈WLρ,θp,κw.
Similarly, we define(57)f⋆⋆=fWLρ,∞p,κw≔fWLρ,θ∗∗p,κw.
By the definition, we can easily show that⋅⋆⋆ satisfies the axioms of a (quasi) norm, and WLρ,∞p,κw is a (quasi) normed linear space. Clearly, if we take θ=0 or V≡0, then this space is just the weighted weak Morrey space WLp,κw (see [17]). According to the above definitions, one has(58)Lp,κμ,ν⊂Lρ,θ1p,κμ,ν⊂Lρ,θ2p,κμ,ν,WLp,κw⊂WLρ,θ1p,κw⊂WLρ,θ2p,κw,for 0<θ1<θ2<∞. Hence, Lp,κμ,ν⊂Lρ,∞p,κμ,ν (in particular Lp,κw⊂Lρ,∞p,κw) and WLp,κw⊂WLρ,∞p,κw for p,κ∈1,∞×0,1.
The spaceLρ,θp,κw (or WLρ,θp,κw) could be viewed as an extension of weighted (or weak) Lebesgue space (when κ=θ=0). Naturally, one may ask the question whether the above conclusions (i.e., Theorems 1 and 2 as well as Theorem 3) still hold if replacing the weighted Lebesgue spaces by the weighted Morrey spaces. In this work, we will give a positive answer to this question. We now state our main results as follows.
For weighted strong-type and weak-type estimates of fractional integrals associated withℒ on Lρ,∞p,κwp,wq, where w belongs to the new classes of weights, we have the following.Theorem 4.
Let0<α<d, 1<p<d/α, 1/q=1/p−α/d, and w∈Ap,qρ,∞. If V∈RHs with s≥d/2 and 0<κ<p/q, then the ℒ-fractional integral operator ℐα is bounded from Lρ,∞p,κwp,wq into Lρ,∞q,κq/pwq.Theorem 5.
Let0<α<d, p=1, q=d/d−α, and w∈A1,qρ,∞. If V∈RHs with s≥d/2 and 0<κ<1/q, then the ℒ-fractional integral operator ℐα is bounded from Lρ,∞1,κw,wq into WLρ,∞q,κqwq.Concerning the boundedness property of the commutatorsb,ℐα in the setting of weighted Morrey space, where b is in the new BMO-type space, we shall prove the following.Theorem 6.
Let0<α<d, 1<p<d/α, 1/q=1/p−α/d, and w∈Ap,qρ,∞. If V∈RHs with s≥d/2 and 0<κ<p/q, then the commutator operator b,ℐα is bounded from Lρ,∞p,κwp,wq into Lρ,∞q,κq/pwq, whenever b∈BMOρ,∞ℝd.Moreover, for the extreme caseκ=p/q of Theorem 4, we will show that the fractional integrals associated with ℒ maps Lρ,∞p,κwp,wq continuously into the new space BMOρ,∞ℝd. This result may be regarded as a supplement of Theorem 4.Theorem 7.
Let0<α<d, 1<p<d/α, 1/q=1/p−α/d, and w∈Ap,qρ,∞. If V∈RHs with s≥d/2 and κ=p/q, then the ℒ-fractional integral operator ℐα is bounded from Lρ,∞p,κwp,wq into BMOρ,∞ℝd.Remark 1.
It is worth pointing out that, in the classical case whenV≡0, Theorems 4, 5, and 6 have been proved by Komori and Shirai in [15], while Theorem 7 has been shown by the author in [18].


## 3. Proofs of Theorems4 and 5

In this section, we will prove the conclusions of Theorems4 and 5. Let us remind that the ℒ-fractional integral operator of order α∈0,d can be written as(59)ℐαfx=ℒ−α/2fx=∫0∞e−tℒfxtα/2−1dt=∫ℝd∫0∞ptx,ytα/2−1dtfydy.The kernel of the fractional integral operatorℐα will be denoted by Kαx,y. Then, (see [19, 20])(60)Kαx,y=∫0∞ptx,ytα/2−1dt,ℐαfx=∫ℝdKαx,yfydy.The following lemma plays a key role in the proof of our main theorems, which can be found in Proposition 8 of [20] (see also [7], Lemma 3.7).Lemma 6 (see [20]).
LetV∈RHs with s≥d/2 and 0<α<d. For every positive integer N≥1, there exists a positive constant CN>0 such that, for all x and y in ℝd,(61)Kαx,y≤CN1+x−yρx−N1x−yd−α.Proof of Theorem 4.
By definition, we only have to show that for any given ballB=Bx0,r of ℝd, there is some ϑ>0 such that(62)1wqBκq/p∫Bℐαfxqwqxdx1/q≤C⋅1+rρx0ϑ,holds for any f∈Lρ,∞p,κwp,wq with 1<p<q<∞ and 0<κ<p/q. Suppose that f∈Lρ,θp,κwp,wq for some θ>0 and w∈Ap,qρ,θ′ for some θ′>0. We decompose the function f as(63)f=f1+f2∈Lρ,θp,κwp,wq;f1=f⋅χ2B;f2=f⋅χ2Bc,where 2B is the ball centered at x0 of radius 2r>0 and χ2B is the characteristic function of 2B. Then, by the linearity of ℐα, we write(64)1wqBκq/p∫Bℐαfxqwqxdx1/q≤1wqBκq/p∫Bℐαf1xqwqxdx1/q+1wqBκq/p∫Bℐαf2xqwqxdx1/q≔I1+I2.
We now analyze each term separately. By Theorem1, we get(65)I1=1wqBκq/p∫Bℐαf1xqwqxdx1/q≤C⋅1wqBκ/p∫ℝdf1xpwpxdx1/p=C⋅1wqBκ/p∫2Bfxpwpxdx1/p≤CfLρ,θp,κwp,wq⋅wq2Bκ/pwqBκ/p⋅1+2rρx0θ.
Sincew∈Ap,qρ,θ′ with 1<p<q<∞ and 0<θ′<∞, then we know that wq∈Atρ,θ′⋅1/1/q+1/p′ with t=1+q/p′ according to Lemma 5. Now we claim that, for every weight v∈Atρ,τ and every ball B in ℝd, there exists a dimensional constant C>0 independent of v and B such that(66)v2Bx0,r≤C⋅1+2rρx0tτvBx0,r.
In fact, for1<t<∞, by Hölder’s inequality and the definition of Atρ,τ, we have(67)12B∫2Bℏxdx=12B∫2Bℏxvx1/tvx−1/tdx≤12B∫2Bℏxtvxdx1/t∫2Bvx−t′/tdx1/t′≤Cvv2B1/t∫2Bℏxtvxdx1/t1+2rρx0τ.
If we takeℏx=χBx, then the above expression becomes(68)B2B≤Cv⋅vB1/tv2B1/t1+2rρx0τ,which in turn implies (66). Also observe that(69)tθ′⋅11/q+1/p′=1+qp′θ′⋅11/q+1/p′=qθ′.
Therefore, in view of (66) and (19),(70)I1≤CwfLρ,θp,κwp,wq⋅1+2rρx0qθ′⋅κ/p⋅1+2rρx0θ=CwfLρ,θp,κwp,wq⋅1+2rρx0ϑ′≤C⋅1+rρx0ϑ′,where ϑ′≔qθ′κ/p+θ. For the other term I2, notice that, for any x∈B and y∈2Bc, one has x−y≈x0−y. It then follows from Lemma 6 that, for any x∈Bx0,r and any positive integer N,(71)ℐαf2x≤∫2BcKαx,y⋅fydy≤CN∫2Bc1+x−yρx−N1x−yd−α⋅fydy≤CN,d∫2Bc1+x0−yρx−N1x0−yd−α⋅fydy=CN,d∑k=1∞∫2k+1B\2kB1+x0−yρx−N1x0−yd−α⋅fydy≤CN,d∑k=1∞12k+1B1−α/d∫2k+1B\2kB1+2krρx−Nfydy.
In view of (4) in Lemma 1 and (19), we further obtain(72)ℐαf2x≤C∑k=1∞12k+1B1−α/d∫2k+1B1+rρx0N⋅N0/N0+11+2krρx0−Nfydy≤C∑k=1∞12k+1B1−α/d∫2k+1B1+rρx0N⋅N0/N0+11+2k+1rρx0−Nfydy.
We consider each term in the sum of (72) separately. By using Hölder’s inequality and Ap,qρ,θ′ condition on w, we obtain that, for each k≥1,(73)12k+1B1−α/d∫2k+1Bfydy≤12k+1B1−α/d∫2k+1Bfypwpydy1/p∫2k+1Bwy−p′dy1/p′≤CfLρ,θp,κwp,wq⋅wq2k+1Bκ/pwq2k+1B1/q1+2k+1rρx0θ1+2k+1rρx0θ′.
Hence,(74)I2≤CfLρ,θp,κwp,wq⋅wqB1/qwqBκ/p∑k=1∞wq2k+1Bκ/pwq2k+1B1/q1+rρx0N⋅N0/N0+11+2k+1rρx0−N+θ+θ′=CfLρ,θp,κwp,wq1+rρx0N⋅N0/N0+1∑k=1∞wqB1/q−κ/pwq2k+1B1/q−κ/p1+2k+1rρx0−N+θ+θ′.
Recall thatwq∈Atρ,θ′⋅1/1/q+1/p′ with t=1+q/p′ and 0<θ′<∞, then there exist two positive numbers δ,η>0 such that (25) holds. This allows us to obtain(75)I2≤CfLρ,θp,κwp,wq1+rρx0N⋅N0/N0+1∑k=1∞B2k+1Bδ1/q−κ/p×1+2k+1rρx0η1/q−κ/p1+2k+1rρx0−N+θ+θ′=CfLρ,θp,κwp,wq1+rρx0N⋅N0/N0+1×∑k=1∞B2k+1Bδ1/q−κ/p1+2k+1rρx0−N+θ+θ′+η1/q−κ/p.
Thus, by choosingN large enough so that N>θ+θ′+η1/q−κ/p, and the last series is convergent, we then have(76)I2≤CfLρ,θp,κwp,wq1+rρx0N⋅N0/N0+1∑k=1∞B2k+1Bδ1/q−κ/p≤C1+rρx0N⋅N0/N0+1,where the last inequality follows from the fact that 1/q−κ/p>0. Summing up the above estimates for I1 and I2 and letting ϑ=maxϑ′,N⋅N0/N0+1, we obtain our desired inequality (62). This completes the proof of Theorem 4.Proof of Theorem 5.
To prove Theorem5, by definition, it suffices to prove that, for each given ball B=Bx0,r of ℝd, there is some ϑ>0 such that(77)1wqBκsupλ>0λ⋅wqx∈B:ℐαfx>λ1/q≤C⋅1+rρx0ϑ,holds for any f∈Lρ,∞1,κw,wq with 1<q<∞ and 0<κ<1/q. Now suppose that f∈Lρ,θ1,κw,wq for some θ>0 and w∈A1,qρ,θ′ for some θ′>0. We decompose the function f as(78)f=f1+f2∈Lρ,θ1,κw,wq,f1=f⋅χ2B,f2=f⋅χ2Bc.
Then, for any givenλ>0, by the linearity of ℐα, we can write(79)1wqBκλ⋅wqx∈B:ℐαfx>λ1/q≤1wqBκλ⋅wqx∈B:ℐαf1x>λ21/q+1wqBκλ⋅wqx∈B:ℐαf2x>λ21/q≔I1′+I2′.
We first give the estimate for the termI1′. By Theorem 2, we get(80)I1′=1wqBκλ⋅wqx∈B:ℐαf1x>λ21/q≤C⋅1wqBκ∫ℝdf1xwxdx=C⋅1wqBκ∫2Bfxwxdx≤CfLρ,θ1,κw,wq⋅wq2BκwqBκ⋅1+2rρx0θ.
Sincew∈A1,qρ,θ′ with 1<q<∞ and 0<θ′<∞, then we know that wq∈A1ρ,θ′⋅q according to Lemma 5. We now claim that there exists a dimensional constant C>0 independent of v and B such that, for every weight v∈A1ρ,τ and every ball B in ℝd,(81)v2Bx0,r≤C⋅1+2rρx0τvBx0,r.
Similar to the proof of (66), by the definition of A1ρ,τ, we can deduce that(82)12B∫2Bℏxdx≤Cvv2B⋅essinfx∈2Bvx∫2Bℏxdx1+2rρx0τ≤Cvv2B∫2Bℏxvxdx1+2rρx0τ.
If we chooseℏx=χBx, then the above expression becomes(83)B2B≤Cv⋅vBv2B1+2rρx0τ,which in turn implies (81). Therefore, in view of (81) and (19),(84)I1′≤C⋅1+2rρx0θ′q⋅κ⋅1+2rρx0θ=C⋅1+2rρx0ϑ′≤C⋅1+rρx0ϑ′,where ϑ′≔θ′qκ+θ. As for the second term I2′, by using the pointwise inequality (72) and Chebyshev’s inequality, we deduce that(85)I2′=1wqBκλ⋅wqx∈B:ℐαf2x>λ21/q≤2wqBκ∫Bℐαf2xqwqxdx1/q≤C⋅wqB1/qwqBκ∑k=1∞12k+1B1−α/d∫2k+1B1+rρx0N⋅N0/N0+11+2k+1rρx0−Nfydy.
We consider each term in the sum of (85) separately. By Lemma 4, we know that w∈A1ρ,θ′∩RHqρ,θ′ because wq is in A1ρ,θ′⋅q. Thus, for each k≥1, we compute(86)12k+1B1−α/d∫2k+1Bfydy≤Cw⋅2k+1Bα/dw2k+1B⋅essinfy∈2k+1Bwy∫2k+1Bfydy1+2k+1rρx0θ′≤Cw⋅2k+1Bα/dw2k+1B∫2k+1Bfywydy1+2k+1rρx0θ′≤CfLρ,θ1,κw,wq⋅2k+1Bα/dw2k+1Bwq2k+1Bκ1+2k+1rρx0θ1+2k+1rρx0θ′.
In addition, by reverse Hölder-type inequality (w∈RHqρ,θ′), we can see that(87)wq2k+1B1/q=∫2k+1Bwqydy1/q≤C2k+1B1/q12k+1B∫2k+1Bwydy1+2k+1rρx0θ′.
This indicates that(88)2k+1Bα/dw2k+1B≤C⋅1wq2k+1B1/q1+2k+1rρx0θ′.
Consequently,(89)I2′≤CfLρ,θ1,κw,wq⋅wqB1/qwqBκ∑k=1∞wq2k+1Bκwq2k+1B1/q1+rρx0N⋅N0/N0+11+2k+1rρx0−N+θ+2θ′=CfLρ,θ1,κw,wq1+rρx0N⋅N0/N0+1∑k=1∞wqB1/q−κwq2k+1B1/q−κ1+2k+1rρx0−N+θ+2θ′.
Recall thatwq∈A1ρ,θ′⋅q with 0<θ′<∞ and 1<q<∞, then there exist two positive numbers δ′,η′>0 such that (25) holds. Therefore,(90)I2′≤CfLρ,θ1,κw,wq1+rρx0N⋅N0/N0+1∑k=1∞B2k+1Bδ′1/q−κ×1+2k+1rρx0η′1/q−κ1+2k+1rρx0−N+θ+2θ′=CfLρ,θ1,κw,wq1+rρx0N⋅N0/N0+1×∑k=1∞B2k+1Bδ′1/q−κ1+2k+1rρx0−N+θ+2θ′+η′1/q−κ.
By selectingN large enough so that N>θ+2θ′+η′1/q−κ, we thus have(91)I2′≤C1+rρx0N⋅N0/N0+1∑k=1∞B2k+1Bδ′1/q−κ≤C1+rρx0N⋅N0/N0+1,where the last inequality is due to 0<κ<1/q. Let ϑ=maxϑ′,N⋅N0/N0+1. Here N is an appropriate constant. Summing up the above estimates for I1′ and I2′, and then taking the supremum over all λ>0, we obtain the desired inequality (77). This finishes the proof of Theorem 5.


## 4. Proof of Theorem6

For the result involving commutators of fractional integrals associated with Schrödinger operators, we need the following properties ofBMOρ,∞ℝd functions, which are extensions of well-known properties of BMOℝd functions.Lemma 7.
Ifb∈BMOρ,∞ℝd and w∈Apρ,∞ with 1≤p<∞, then there exist positive constants C>0 and μ>0 such that for every ball B=Bx0,r in ℝd, we have(92)∫Bbx−bBpwxdx1/p≤C⋅wB1/p1+rρx0μ,where bB=1/B∫Bbydy.Proof.
We may assume thatb∈BMOρ,θℝd with 0<θ<∞. According to Proposition 3, we can deduce that(93)∫Bbx−bBpwxdx1/p=∫0∞pλp−1wx∈B:bx−bB>λdλ1/p≤C11/p⋅wB1/p∫0∞pλp−1exp−1+rρx0−θ∗C2λbBMOρ,θ1+rρx0ηdλ1/p=C11/p⋅wB1/p∫0∞pλp−1exp−1+rρx0−θ∗C2λbBMOρ,θdλ1/p1+rρx0η/p.
By a change of variables, we thus have(94)∫Bbx−bBpwxdx1/p≤C11/p⋅wB1/p∫0∞psp−1e−sds1/pbBMOρ,θC21+rρx0θ∗1+rρx0η/p=C1pΓp1/pbBMOρ,θC2⋅wB1/p1+rρx0θ∗+η/p,which yields the desired inequality (92), if we choose C=C1pΓp1/pC2−1bBMOρ,θ and μ=θ∗+η/p.Lemma 8.
Ifb∈BMOρ,θℝd with 0<θ<∞, then for any positive integer k, there exists a positive constant C>0 such that, for every ball B=Bx0,r in ℝd, we have(95)b2k+1B−bB≤C⋅k+11+2k+1rρx0θ.Proof.
For any positive integerk, we have(96)b2k+1B−bB≤∑j=1k+1b2jB−b2j−1B=∑j=1k+112j−1B∫2j−1Bb2jB−bydy≤∑j=1k+12d2jB∫2jBby−b2jBdy≤Cb,dbBMOρ,θ∑j=1k+11+2jrρx0θ.
Since for any1≤j≤k+1, the following estimate(97)1+2jrρx0θ≤1+2k+1rρx0θ,holds trivially and hence(98)b2k+1B−bB≤C∑j=1k+11+2k+1rρx0θ=C⋅k+11+2k+1rρx0θ.
This is just our desired conclusion.
Now, we are ready to prove the main result of this section.Proof of Theorem6.
By definition, we only need to prove that, for an arbitrary ballB=Bx0,r of ℝd and 0<α<d, there is some ϑ>0 such that(99)1wqBκq/p∫Bb,ℐαfxqwqxdx1/q≤C⋅1+rρx0ϑ,holds for any f∈Lρ,∞p,κwp,wq with 1<p<q<∞ and 0<κ<p/q, whenever b belongs to BMOρ,∞ℝd. Suppose that f∈Lρ,θp,κwp,wq for some θ>0, w∈Ap,qρ,θ′ for some θ′>0 as well as b∈BMOρ,θ″ℝd for some θ″>0. As before, we decompose the function f as(100)f=f1+f2∈Lρ,θp,κwp,wq;f1=f⋅χ2B;f2=f⋅χ2Bc.
Then by the linearity ofb,ℐα, we write(101)1wqBκq/p∫Bb,ℐαfxqwqxdx1/q≤1wqBκq/p∫Bb,ℐαf1xqwqxdx1/q+1wqBκq/p∫Bb,ℐαf2xqwqxdx1/q≔J1+J2.
Now, we give the estimates forJ1 and J2, respectively. According to Theorem 3, we have(102)J1≤C⋅1wqBκ/p∫ℝdf1xpwpxdx1/p=C⋅1wqBκ/p∫2Bfxpwpxdx1/p≤CfLρ,θp,κwp,wq⋅wq2Bκ/pwqBκ/p⋅1+2rρx0θ.
Moreover, notice thatwq∈Atρ,θ′⋅1/1/q+1/p′ with t=1+q/p′ by Lemma 5, and that(103)tθ′⋅11/q+1/p′=1+qp′θ′⋅11/q+1/p′=qθ′.
Thus, in view of the inequalities (66) and (19), we get(104)J1≤CfLρ,θp,κwp,wq⋅1+2rρx0qθ′⋅κ/p⋅1+2rρx0θ=CfLρ,θp,κwp,wq1+2rρx0ϑ′≤C⋅1+rρx0ϑ′,where ϑ′≔qθ′κ/p+θ. On the other hand, by definition (40), we can see that, for any x∈Bx0,r,(105)b,ℐαf2x≤∫ℝdbx−byKαx,yf2ydy.
Adding and subtractingbB inside the integral, we write(106)b,ℐαf2x≤bx−bB∫ℝdKαx,yf2ydy+∫ℝdby−bBKαx,yf2ydy≔ξx+ζx.
So, we can divideJ2 into two parts:(107)J2≤1wqBκq/p∫Bξxqwqxdx1/q+1wqBκq/p∫Bζxqwqxdx1/q=J3+J4.
To deal with the termJ3, since t=1+q/p′<q, one has wq∈Atρ,∞⊂Aqρ,∞. From the pointwise estimate (72) and (92) in Lemma 7, it then follows that(108)J3≤C⋅1wqBκ/p∫Bbx−bBqwqxdx1/q×∑k=1∞12k+1B1−α/d∫2k+1B1+rρx0N⋅N0/N0+11+2k+1rρx0−Nfydy≤Cb⋅wqB1/qwqBκ/p1+rρx0μ×∑k=1∞12k+1B1−α/d∫2k+1B1+rρx0N⋅N0/N0+11+2k+1rρx0−Nfydy.
Following along the same lines as that of Theorem4, we are able to show that(109)J3≤CfLρ,θp,κwp,wq1+rρx0μ1+rρx0N⋅N0/N0+1×∑k=1∞wqB1/q−κ/pwq2k+1B1/q−κ/p1+2k+1rρx0−N+θ+θ′≤CfLρ,θp,κwp,wq1+rρx0μ+N⋅N0/N0+1×∑k=1∞B2k+1Bδ1/q−κ/p1+2k+1rρx0−N+θ+θ′+η1/q−κ/p.
The last inequality is obtained by using (25). Next, we estimate ζx for any x∈Bx0,r. For any positive integer N, similar to the proof of (71) and (72), we can also deduce that(110)ζx=∫2Bcby−bBKαx,yfydy≤CN∫2Bcby−bB1+x−yρx−N1x−yd−α⋅fydy≤CN,d∑k=1∞∫2k+1B\2kBby−bB1+x0−yρx−N1x0−yd−α⋅fydy≤CN,d∑k=1∞12k+1B1−α/d∫2k+1B\2kBby−bB1+2krρx−Nfydy≤C∑k=1∞12k+1B1−α/d∫2k+1Bby−bB1+rρx0N⋅N0/N0+11+2k+1rρx0−Nfydy,where in the last inequality we have used (4) in Lemma 1. Hence, by the above pointwise estimate for ζx,(111)J4≤C⋅wqB1/q−κ/p∑k=1∞1+rρx0N⋅N0/N0+11+2k+1rρx0−N×12k+1B1−α/d∫2k+1Bby−bBfydy.
Let us consider each term in the sum of (111) separately. For each integer k≥1,(112)12k+1B1−α/d∫2k+1Bby−bBfydy≤12k+1B1−α/d∫2k+1Bby−b2k+1Bfydy+12k+1B1−α/d∫2k+1Bb2k+1B−bBfydy.
By using Hölder’s inequality, the first term of the expression (112) is bounded by(113)12k+1B1−α/d∫2k+1Bfypwpydy1/p∫2k+1Bby−b2k+1Bp′wy−p′dy1/p′≤CfLρ,θp,κwp,wq⋅wq2k+1Bκ/p2k+1B1−α/d1+2k+1rρx0θ∫2k+1Bby−b2k+1Bp′wy−p′dy1/p′.
Sincew∈Ap,qρ,θ′ with 0<θ′<∞ and 1<p<q<∞, then by Lemma 5, we know that w−p′∈At′ρ,θ′⋅1/1/q+1/p′ with t′=1+p′/q. If we denote v=w−p′, then v∈At′ρ,θ′⋅1/1/q+1/p′⊂Ap′ρ,∞ because t′=1+p′/q<p′. This fact together with Lemma 7 implies(114)∫2k+1Bby−b2k+1Bp′vydy1/p′≤Cb⋅v2k+1B1/p′1+2k+1rρx0μ=Cb⋅∫2k+1Bwy−p′dy1/p′1+2k+1rρx0μ≤Cb,w⋅2k+1B1−α/dwq2k+1B1/q1+2k+1rρx0θ′1+2k+1rρx0μ.
Therefore, the first term of the expression (112) can be bounded by a constant times(115)wq2k+1Bκ/pwq2k+1B1/q1+2k+1rρx0θ+θ′+μ.
Sinceb∈BMOρ,θ″ℝd with 0<θ″<∞, then by Lemma 8, Hölder’s inequality, and the Ap,qρ,θ′ condition on w, the latter term of the expression (112) can be estimated by(116)Cbk+11+2k+1rρx0θ″12k+1B1−α/d∫2k+1Bfydy≤Cbk+11+2k+1rρx0θ″12k+1B1−α/d∫2k+1Bfypwpydy1/p∫2k+1Bwy−p′dy1/p′≤CfLρ,θp,κwp,wq⋅k+1wq2k+1Bκ/pwq2k+1B1/q1+2k+1rρx0θ+θ′+θ″.
Consequently,(117)12k+1B1−α/d∫2k+1Bby−bBfydy≤CfLρ,θp,κwp,wq⋅k+1wq2k+1Bκ/pwq2k+1B1/q1+2k+1rρx0θ+θ′+θ″+μ.
Substituting the above inequality (117) into (111), we thus obtain(118)J4≤CfLρ,θp,κwp,wq⋅wqB1/q−κ/p∑k=1∞k+11+rρx0N⋅N0/N0+11+2k+1rρx0−N×1wq2k+1B1/q−κ/p1+2k+1rρx0θ+θ′+θ″+μ=CfLρ,θp,κwp,wq1+rρx0N⋅N0/N0+1×∑k=1∞k+1wqB1/q−κ/pwq2k+1B1/q−κ/p1+2k+1rρx0−N+θ+θ′+θ″+μ⋅θ″.
Note thatwq∈Atρ,θ′⋅1/1/q+1/p′ with t=1+q/p′. A further application of (25) yields(119)J4≤CfLρ,θp,κwp,wq1+rρx0N⋅N0/N0+1∑k=1∞k+1B2k+1Bδ1/q−κ/p×1+2k+1rρx0η1/q−κ/p1+2k+1rρx0−N+θ+θ′+θ″+μ=CfLρ,θp,κwp,wq1+rρx0N⋅N0/N0+1∑k=1∞k+1B2k+1Bδ1/q−κ/p×1+2k+1rρx0−N+θ+θ′+θ″+μ+η1/q−κ/p⋅θ″.
Hence, combining the above estimates forJ3 and J4, we conclude that(120)J2≤J3+J4≤CfLρ,θp,κwp,wq1+rρx0μ+N⋅N0/N0+1×∑k=1∞k+1B2k+1Bδ1/q−κ/p1+2k+1rρx0−N+θ+θ′+θ″+μ+η1/q−κ/p⋅θ″.
By choosingN large enough so that N>θ+θ′+θ″+μ+η1/q−κ/p, we thus have(121)J2≤C1+rρx0μ+N⋅N0/N0+1∑k=1∞k+1B2k+1Bδ1/q−κ/p≤C1+rρx0μ+N⋅N0/N0+1,where the last series is convergent since 0<κ<p/q. Finally, collecting the above estimates for J1andJ2, and letting ϑ=maxϑ′,μ+N⋅N0/N0+1, we obtain the desired inequality (99). The proof of Theorem 6 is finished.
The higher-order commutators generated byBMOρ,∞ℝd functions b and the fractional integrals ℐα are usually defined by(122)b,ℐαmfx≔∫ℝnbx−bymKαx,yfydy,x∈ℝd;0<α<d,m=1,2,3,….
Obviously,b,ℐα1=b,ℐα which is just the linear commutator (40) and(123)b,ℐαm=b,b,ℐαm−1,m=2,3,….
By induction onm, we are able to show that the conclusion of Theorem 6 also holds for the higher-order commutators b,ℐαm with m≥2. The details are omitted here.Theorem 8.
Let0<α<d, 1<p<d/α, 1/q=1/p−α/d, and w∈Ap,qρ,∞. If V∈RHs with s≥d/2 and 0<κ<p/q, then for any positive integer m≥2, the higher order commutators b,ℐαm are bounded from Lρ,∞p,κwp,wq into Lq,κq/pwq, whenever b∈BMOρ,∞ℝd.


## 5. Proof of Theorem7

The following lemma plays a key role in the proof of our main theorem, which can be found in Proposition 8 of [20] (see also [7], Lemma 3.7).Lemma 9 (see [20]).
LetV∈RHs with s≥d/2 and 0<α<d. For every positive integer N≥1, there exists a positive constant CN>0 such that, for all x and y in ℝd and for some fixed 0<ε≤1,(124)Kαx,z−Kαy,z≤CN1+x−zρx−Nx−yεx−zd−α+ε,whenever x−y≤x−z/2.Proof of Theorem 7.
For an arbitrary ballB=Bx0,r in ℝd and 0<α<d, it suffices to prove that the following inequality(125)1B∫Bℐαfx−ℐαfBdx≤C⋅1+rρx0ϑ,holds for any f∈Lρ,∞p,κwp,wq with 1<p<q<∞ and κ=p/q, where ℐαfB denotes the average of ℐαf over B. Suppose that f∈Lρ,θp,κwp,wq for some θ>0 and w∈Ap,qρ,θ′ for some θ′>0. Decompose the function f as f=f1+f2, where f1=f⋅χ4B, f2=f⋅χ4Bc, and 4B=Bx0,4r. By the linearity of the ℒ-fractional integral operator ℐα, the left-hand side of (125) can be divided into two parts. That is,(126)1B∫Bℐαfx−ℐαfBdx≤1B∫Bℐαf1x−ℐαf1Bdx+1B∫Bℐαf2x−ℐαf2Bdx≔K1+K2.
First, let us consider the termK1. Applying the weighted Lp,Lq-boundedness of ℐα (see Theorem 1) and Hölder’s inequality, we obtain(127)K1≤2B∫Bℐαf1xdx≤2B∫Bℐαf1xqwqxdx1/q∫Bwx−q′dx1/q′≤CB∫4Bfxpwpxdx1/p∫Bwx−q′dx1/q′≤CfLρ,θp,κwp,wq⋅wq4Bκ/pB∫Bwx−q′dx1/q′1+4rρx0θ.
Sinceq′<p′, by Hölder’s inequality, it is easy to check that(128)1B∫Bwx−q′dx1/q′≤1B∫Bwx−p′dx1/p′.
Moreover, sincew is a weight in the class Ap,qρ,θ′, one has(129)1B∫Bwxqdx1/q1B∫Bwx−q′dx1/q′≤1B∫Bwxqdx1/q1B∫Bwx−p′dx1/p′≤C⋅1+rρx0θ′,which implies(130)∫Bwx−q′dx1/q′≤C⋅1+rρx0θ′BwqB1/q.
Also observe thatwq∈Atρ,θ′⋅1/1/q+1/p′ with t=1+q/p′. Using the inequalities (66) and (130) and noting the fact that κ=p/q, we have(131)K1≤CfLρ,θp,κwp,wq⋅wq4B1/qwqB1/q1+4rρx0θ1+rρx0θ′≤CfLρ,θp,κwp,wq1+4rρx02θ′1+4rρx0θ1+rρx0θ′≤C⋅1+rρx0ϑ′,where ϑ′≔3θ′+θ. Now we turn to estimate K2. For any x∈Bx0,r,(132)ℐαf2x−ℐαf2B=1B∫Bℐαf2x−ℐαf2ydy=1B∫B∫4BcKαx,z−Kαy,zfzdzdy≤1B∫B∫4BcKαx,z−Kαy,z⋅fzdzdy.
Next note that, by a purely geometric argument, one has(133)x−y≤x−z2,x−z≈x0−z,whenever x,y∈B and z∈4Bc. This fact along with Lemma 9 yields(134)ℐαf2x−ℐαf2B≤CNB∫B∫4Bc1+x−zρx−Nx−yεx−zd−α+ε⋅fzdzdy≤CN,d∫4Bc1+x0−zρx−Nrεx0−zd−α+ε⋅fzdz=CN,d∑k=2∞∫2k+1B\2kB1+x0−zρx−Nrεx0−zd−α+ε⋅fzdz≤CN,d∑k=2∞12kε⋅12k+1B1−α/d∫2k+1B\2kB1+2krρx−Nfzdz.
Furthermore, by using Hölder’s inequality, (4) and Ap,qρ,θ′ condition on w, we get that, for any x∈Bx0,r,(135)ℐαf2x−ℐαf2B≤C∑k=2∞12kε⋅12k+1B1−α/d1+rρx0N⋅N0/N0+11+2k+1rρx0−N×∫2k+1Bfzpwpzdz1/p∫2k+1Bwz−p′dz1/p′≤CfLρ,θp,κwp,wq∑k=2∞12kε⋅1+rρx0N⋅N0/N0+11+2k+1rρx0−N×wq2k+1Bκ/pwq2k+1B1/q1+2k+1rρx0θ1+2k+1rρx0θ′=CfLρ,θp,κwp,wq∑k=2∞12kε⋅1+rρx0N⋅N0/N0+11+2k+1rρx0−N+θ+θ′,where the last equality is due to the assumption κ=p/q. From the pointwise estimate (135), it readily follows that(136)K2=1B∫Bℐαf2x−ℐαf2Bdx≤CfLρ,θp,κwp,wq∑k=2∞12kε⋅1+rρx0N⋅N0/N0+11+2k+1rρx0−N+θ+θ′.
NowN can be chosen sufficiently large so that N>θ+θ′, and hence, the above series is convergent. Therefore,(137)K2≤C1+rρx0N⋅N0/N0+1∑k=2∞12kε≤C1+rρx0N⋅N0/N0+1.
Fix thisN and set ϑ=maxϑ′,N⋅N0/N0+1. Finally, combining the above estimates for K1 and K2, the inequality (125) is proved, and then the proof of Theorem 7 is finished.



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*Source: 6907170-2020-03-16.xml*