# Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

Authors: Hua Wang
Journal: Journal of Function Spaces
 (2019)

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## Abstract

LetL=-Δ+V be a Schrödinger operator, where Δ is the Laplacian on Rd and the nonnegative potential V belongs to the reverse Hölder class RHq for q≥d. The Riesz transform associated with the operator L=-Δ+V is denoted by R=∇(-Δ+V)-1/2 and the dual Riesz transform is denoted by R⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHq for q≥d. Then we will establish the mapping properties of the operator R and its adjoint R⁎ on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators [b,R] and [b,R⁎] are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger than Ap, BMO(Rd), and Lp,κ(w) corresponding to the classical Riesz transforms (V≡0).


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## Body

## 1. Introduction

### 1.1. The Critical Radius Functionρ(x)

Letd≥3 be a positive integer and let Rd be the d-dimensional Euclidean space. A nonnegative locally integrable function V(x) on Rd is said to belong to the reverse Hölder class RHq for some exponent 1<q<∞, if there exists a positive constant C>0 such that the reverse Hölder inequality (1)1B∫BVyqdy1/q≤C1B∫BVydy holds for every ball B in Rd. For given V∈RHq with q≥d, we introduce thecritical radius functionρ(x)=ρ(x,V) which is given by(2)ρx≔sup⁡r>0:1rd-2∫Bx,rVydy≤1,x∈Rd,where B(x,r) denotes the open ball centered at x and with radius r. It is well known that 0<ρ(x)<∞ for any x∈Rd under our assumption (see [1]). We need the following known result concerning the critical radius function.Lemma 1 (from [1]).
IfV∈RHq with q≥d, then there exist two constants C>0 and N0≥1 such that(3)1C1+x-yρx-N0≤ρyρx≤C1+x-yρxN0/N0+1,for all x,y∈Rd. As a straightforward consequence of (3), we have that, for all k=1,2,3,…, the estimate(4)1+2krρy≥C1+rρx-N0/N0+11+2krρxis valid for any y∈B(x,r) with x∈Rd and r>0.


### 1.2. Schrödinger Operators

OnRd, d≥3, we consider theSchrödinger operator(5)L≔-Δ+V, where V∈RHq for q≥d. TheRiesz transform associated with the Schrödinger operator L is defined by(6)R≔∇L-1/2,and the associateddual Riesz transform is defined by(7)R∗≔L-1/2∇.Boundedness properties of R and its adjoint R∗ have been obtained by Shen in [1], where he showed that they are all bounded on Lp(Rd) for any 1<p<∞ when V∈RHq with q≥d. Actually, R and its adjoint R∗ are standard Calderón-Zygmund operators in such a situation. The operators R and R∗ have singular kernels that will be denoted by K(x,y) and K∗(x,y), respectively. For such kernels, we have the following key estimates, which can be found in [1–3].Lemma 2.
LetV∈RHq with q≥d. For any positive integer N, there exists a positive constant CN>0 such that (8)Kx,y≤CN1+x-yρx-N1x-yd;K∗x,y≤CN1+x-yρx-N1x-yd.


### 1.3.Apρ,∞ 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 (9)wE=∫Ewxdx. Given B=B(x0,r) and t>0, we will write tB for the t-dilate ball, which is the ball with the same center x0 and with radius tr. In [4] (see also [2, 3]), Bongioanni, Harboure, and Salinas introduced the following classes of weights that are given in terms of critical radius function (2). Following the terminology of [4], for given 1<p<∞, we define (10)Apρ,∞≔⋃θ>0Apρ,θ, where Apρ,θ is the set of all weights w such that (11)1B∫Bwxdx1/p1B∫Bwx-p′/pdx1/p′≤C·1+rρx0θ holds for every ball B=B(x0,r)⊂Rd with x0∈Rd and r>0, where p′ is the dual exponent of p such that 1/p+1/p′=1. For p=1 we define (12)A1ρ,∞≔⋃θ>0A1ρ,θ, where A1ρ,θ is the set of all weights w such that (13)1B∫Bwxdx≤C·1+rρx0θessinfx∈Bwx holds for every ball B=B(x0,r) in Rd. For θ>0, let us introduce the maximal operator that is given in terms of critical radius function (2). (14)Mρ,θfx≔supr>01+rρx-θ1Bx,r∫Bx,rfydy,x∈Rd. Observe that a weight w belongs to the class A1ρ,∞ if and only if there exists a positive number θ>0 such that Mρ,θw≤Cw, where the constant C>0 is independent of w. Since (15)1≤1+rρx0θ1≤1+rρx0θ2 for 0<θ1<θ2<∞, then, for given p with 1≤p<∞, one has (16)Ap⊂Apρ,θ1⊂Apρ,θ2, where Ap denotes the classical Muckenhoupt class (see [5, Chapter 7]), and hence Ap⊂Apρ,∞. In addition, for some fixed θ>0, (17)A1ρ,θ⊂Ap1ρ,θ⊂Ap2ρ,θ whenever 1≤p1<p2<∞. Now, we present an important property of the classes of weights in Apρ,θ with 1≤p<∞, which was given by Bongioanni et al. in [4, Lemma 5].Lemma 3 (from [4]).
Ifw∈Apρ,θ with 0<θ<∞ and 1≤p<∞, then there exist positive constants ϵ, η>0, and C>0 such that(18)1B∫Bwx1+ϵdx1/1+ϵ≤C1B∫Bwxdx1+rρx0ηfor every ball B=B(x0,r) in Rd.As a direct consequence of Lemma3, we have the following result.Lemma 4.
Ifw∈Apρ,θ with 0<θ<∞ and 1≤p<∞, then there exist two positive numbers δ>0 and η>0 such that(19)wEwB≤CEBδ1+rρx0ηfor any measurable subset E of a ball B=B(x0,r), where C>0 is a constant which does not depend on E and B.For any given ballB=B(x0,r) with x0∈Rd and r>0, suppose that E⊂B; then by Hölder’s inequality with exponent 1+ϵ and (18), we can deduce that (20)wE=∫BχEx·wxdx≤∫Bwx1+ϵdx1/1+ϵ∫BχEx1+ϵ/ϵdxϵ/1+ϵ≤CB1/1+ϵ1B∫Bwxdx1+rρx0ηEϵ/1+ϵ=CEBϵ/1+ϵ1+rρx0η. This gives (19) with δ=ϵ/(1+ϵ).Given a weightw on Rd, as usual, the weighted Lebesgue space Lp(w) for 1≤p<∞ is defined to be the set of all functions f such that (21)fLpw≔∫Rdfxpwxdx1/p<∞. We also denote by WL1(w) the weighted weak Lebesgue space consisting of all measurable functions f for which (22)fWL1w≔supλ>0⁡λ·wx∈Rd:fx>λ<∞.Recently, Bongioanni et al. [4] obtained weighted strong-type and weak-type estimates for the operators R and R∗ defined in (6) and (7). Their results can be summarized as follows.Theorem 5 (from [4]).
Let1<p<∞ and w∈Apρ,∞. If V∈RHq with q≥d, then the operators R and R∗ are all bounded on Lp(w).Theorem 6 (from [4]).
Letp=1 and w∈A1ρ,∞. If V∈RHq with q≥d, then the operators R and R∗ are all bounded from L1(w) into WL1(w).


### 1.4. The SpaceBMOρ,∞(Rd)

We denote byT either R or R∗. For a locally integrable function b on Rd (usually called thesymbol), we will also consider the commutator operator(23)b,Tfx≔bx·Tfx-Tbfx,x∈Rd.Recently, Bongioanni et al. [3] introduced a new space BMOρ,∞(Rd) defined by (24)BMOρ,∞Rd≔⋃θ>0BMOρ,θRd,where for 0<θ<∞ the space BMOρ,θ(Rd) is defined to be the set of all locally integrable functions b satisfying(25)1Bx0,r∫Bx0,rbx-bBx0,rdx≤C·1+rρx0θ,for all x0∈Rd and r>0, and bB(x0,r) denotes the mean value of b on B(x0,r); that is, (26)bBx0,r≔1Bx0,r∫Bx0,rbydy. A norm for b∈BMOρ,θ(Rd), denoted by bBMOρ,θ, is given by the infimum of the constants satisfying (25), or, equivalently, (27)bBMOρ,θ≔supBx0,r1+rρx0-θ1Bx0,r∫Bx0,rbx-bBx0,rdx, where the supremum is taken over all balls B(x0,r) with x0∈Rd and r>0. With the above definition in mind, one has (28)BMORd⊂BMOρ,θ1Rd⊂BMOρ,θ2Rd for 0<θ1<θ2<∞, and hence BMO(Rd)⊂BMOρ,∞(Rd). Moreover, the classical BMO space [6] is properly contained in BMOρ,∞(Rd) (see [2, 3] for some examples). We need the following key result for the space BMOρ,θ(Rd), which was proved by Tang in [7].Proposition 7 (from [7]).
Letb∈BMOρ,θ(Rd) with 0<θ<∞. Then there exist two positive constants C1 and C2 such that, for any given ball B(x0,r) in Rd and for any λ>0, we have(29)x∈Bx0,r:bx-bBx0,r>λ≤C1Bx0,rexp⁡-1+rρx0-θ∗C2λbBMOρ,θ,where θ∗=(N0+1)θ and N0 is the constant appearing in Lemma 1.As a consequence of Proposition7 and Lemma 4, we have the following result.Proposition 8.
Letb∈BMOρ,θ(Rd) 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 B(x0,r) in Rd and for any λ>0, we have(30)wx∈Bx0,r:bx-bBx0,r>λ≤C1wBx0,rexp⁡-1+rρx0-θ∗C2λbBMOρ,θ1+rρx0η,where θ∗=(N0+1)θ and N0 is the constant appearing in Lemma 1.


### 1.5. Orlicz Spaces

In this subsection, let us give the definition of and some basic facts about Orlicz spaces. For more information on this subject, the reader may consult book [8]. Recall that a function A:[0,∞)→[0,∞) is called a Young function if it is continuous, convex, and strictly increasing with (31)A0=0,limt→∞At→∞. An important example of Young functions is A(t)=t·(1+log+t)m with some 1≤m<∞. Given a Young function A and a function f defined on a ball B, we consider the A-average of a function f given by the following Luxemburg norm: (32)fA,B≔inf⁡λ>0:1B∫BAfxλdx≤1. Associated with each Young function A, one can define its complementary function A¯ as follows: (33)A¯s≔supt>0⁡st-At,0≤s<∞.Such a function A¯ is also a Young function. It is well known that the followinggeneralized Hölder inequality in Orlicz spaces holds for any given ball B⊂Rd: (34)1B∫Bfx·gxdx≤2fA,BgA-,B. In particular, for the Young function A(t)=t·(1+log+t), the Luxemburg norm will be denoted by ·Llog⁡L,B=·A,B. A simple computation shows that the complementary Young function of A(t)=t·(1+log+t) is A¯(t)≈et-1 (see [9, 10] for instance). The corresponding Luxemburg norm will be denoted by ·exp⁡L,B=·A-,B. In this situation, we have(35)1B∫Bfx·gxdx≤2fLlog⁡L,Bgexp⁡L,B.We next define the weighted A-average of a function f over a ball B. Given a Young function A and a weight function w, let (see [8] for instance) (36)fAw,B≔inf⁡λ>0:1wB∫BAfxλ·wxdx≤1. When A(t)=t, we denote ·L(w),B=·A(w),B, and when Φ(t)=t·(1+log+t), we denote ·Llog⁡L(w),B=·Φ(w),B. Also, the complementary Young function of Φ is given by Φ¯(t)≈et-1 with the corresponding Luxemburg norm denoted by ·exp⁡L(w),B. Given a weight w on Rd, we can also show the weighted version of (35). That is, the generalized Hölder inequality in the weighted setting (see [11] for instance)(37)1wB∫Bfx·gxwxdx≤CfLlog⁡Lw,Bgexp⁡Lw,Bholds for every ball B in Rd. It is a simple but important observation that, for any ball B in Rd, (38)fLw,B≤fLlog⁡Lw,B. This is because t≤t·(1+log+t) for all t>0. So we have(39)fLw,B=1wB∫Bfx·wxdx≤fLlog⁡Lw,B.In [2], Bongioanni et al. obtained weighted strong (p,p), 1<p<∞, and weak Llog⁡L estimates for the commutators of the Riesz transform and its adjoint associated with the Schrödinger operator L=-Δ+V, where V satisfies some reverse Hölder inequality. Their results can be summarized as follows.Theorem 9 (from [2]).
Let1<p<∞ and w∈Apρ,∞. If V∈RHq with q≥d, then the commutator operators [b,R] and [b,R∗] are all bounded on Lp(w), whenever b belongs to BMOρ,∞(Rd).Theorem 10 (from [2]).
Letp=1 and w∈A1ρ,∞. If V∈RHq with q≥d and b∈BMOρ,∞(Rd), then, for any given λ>0, there exists a positive constant C>0 such that, for those functions f such that Φ(|f|)∈L1(w), (40)wx∈Rn:b,Rfx>λ≤C∫RdΦfxλ·wxdx,wx∈Rn:b,R∗fx>λ≤C∫RdΦfxλ·wxdx,where Φ(t)=t·(1+log+t) and log+t:=max⁡{log⁡t,0}; that is, (41)log+t=log⁡t,ast>1;0,otherwise.In this paper, firstly, we will define some kinds of weighted Morrey spaces related to certain nonnegative potentials. Secondly, we prove that the Riesz transformR and its adjoint R∗ are both bounded operators on these new spaces. Finally, we also obtain the weighted estimates for the commutators [b,R] and [b,R∗] defined in (23).Throughout this paperC 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 a,b such that C1a≤b≤C2a.


## 1.1. The Critical Radius Functionρ(x)

Letd≥3 be a positive integer and let Rd be the d-dimensional Euclidean space. A nonnegative locally integrable function V(x) on Rd is said to belong to the reverse Hölder class RHq for some exponent 1<q<∞, if there exists a positive constant C>0 such that the reverse Hölder inequality (1)1B∫BVyqdy1/q≤C1B∫BVydy holds for every ball B in Rd. For given V∈RHq with q≥d, we introduce thecritical radius functionρ(x)=ρ(x,V) which is given by(2)ρx≔sup⁡r>0:1rd-2∫Bx,rVydy≤1,x∈Rd,where B(x,r) denotes the open ball centered at x and with radius r. It is well known that 0<ρ(x)<∞ for any x∈Rd under our assumption (see [1]). We need the following known result concerning the critical radius function.Lemma 1 (from [1]).
IfV∈RHq with q≥d, then there exist two constants C>0 and N0≥1 such that(3)1C1+x-yρx-N0≤ρyρx≤C1+x-yρxN0/N0+1,for all x,y∈Rd. As a straightforward consequence of (3), we have that, for all k=1,2,3,…, the estimate(4)1+2krρy≥C1+rρx-N0/N0+11+2krρxis valid for any y∈B(x,r) with x∈Rd and r>0.


## 1.2. Schrödinger Operators

OnRd, d≥3, we consider theSchrödinger operator(5)L≔-Δ+V, where V∈RHq for q≥d. TheRiesz transform associated with the Schrödinger operator L is defined by(6)R≔∇L-1/2,and the associateddual Riesz transform is defined by(7)R∗≔L-1/2∇.Boundedness properties of R and its adjoint R∗ have been obtained by Shen in [1], where he showed that they are all bounded on Lp(Rd) for any 1<p<∞ when V∈RHq with q≥d. Actually, R and its adjoint R∗ are standard Calderón-Zygmund operators in such a situation. The operators R and R∗ have singular kernels that will be denoted by K(x,y) and K∗(x,y), respectively. For such kernels, we have the following key estimates, which can be found in [1–3].Lemma 2.
LetV∈RHq with q≥d. For any positive integer N, there exists a positive constant CN>0 such that (8)Kx,y≤CN1+x-yρx-N1x-yd;K∗x,y≤CN1+x-yρx-N1x-yd.


## 1.3.Apρ,∞ 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 (9)wE=∫Ewxdx. Given B=B(x0,r) and t>0, we will write tB for the t-dilate ball, which is the ball with the same center x0 and with radius tr. In [4] (see also [2, 3]), Bongioanni, Harboure, and Salinas introduced the following classes of weights that are given in terms of critical radius function (2). Following the terminology of [4], for given 1<p<∞, we define (10)Apρ,∞≔⋃θ>0Apρ,θ, where Apρ,θ is the set of all weights w such that (11)1B∫Bwxdx1/p1B∫Bwx-p′/pdx1/p′≤C·1+rρx0θ holds for every ball B=B(x0,r)⊂Rd with x0∈Rd and r>0, where p′ is the dual exponent of p such that 1/p+1/p′=1. For p=1 we define (12)A1ρ,∞≔⋃θ>0A1ρ,θ, where A1ρ,θ is the set of all weights w such that (13)1B∫Bwxdx≤C·1+rρx0θessinfx∈Bwx holds for every ball B=B(x0,r) in Rd. For θ>0, let us introduce the maximal operator that is given in terms of critical radius function (2). (14)Mρ,θfx≔supr>01+rρx-θ1Bx,r∫Bx,rfydy,x∈Rd. Observe that a weight w belongs to the class A1ρ,∞ if and only if there exists a positive number θ>0 such that Mρ,θw≤Cw, where the constant C>0 is independent of w. Since (15)1≤1+rρx0θ1≤1+rρx0θ2 for 0<θ1<θ2<∞, then, for given p with 1≤p<∞, one has (16)Ap⊂Apρ,θ1⊂Apρ,θ2, where Ap denotes the classical Muckenhoupt class (see [5, Chapter 7]), and hence Ap⊂Apρ,∞. In addition, for some fixed θ>0, (17)A1ρ,θ⊂Ap1ρ,θ⊂Ap2ρ,θ whenever 1≤p1<p2<∞. Now, we present an important property of the classes of weights in Apρ,θ with 1≤p<∞, which was given by Bongioanni et al. in [4, Lemma 5].Lemma 3 (from [4]).
Ifw∈Apρ,θ with 0<θ<∞ and 1≤p<∞, then there exist positive constants ϵ, η>0, and C>0 such that(18)1B∫Bwx1+ϵdx1/1+ϵ≤C1B∫Bwxdx1+rρx0ηfor every ball B=B(x0,r) in Rd.As a direct consequence of Lemma3, we have the following result.Lemma 4.
Ifw∈Apρ,θ with 0<θ<∞ and 1≤p<∞, then there exist two positive numbers δ>0 and η>0 such that(19)wEwB≤CEBδ1+rρx0ηfor any measurable subset E of a ball B=B(x0,r), where C>0 is a constant which does not depend on E and B.For any given ballB=B(x0,r) with x0∈Rd and r>0, suppose that E⊂B; then by Hölder’s inequality with exponent 1+ϵ and (18), we can deduce that (20)wE=∫BχEx·wxdx≤∫Bwx1+ϵdx1/1+ϵ∫BχEx1+ϵ/ϵdxϵ/1+ϵ≤CB1/1+ϵ1B∫Bwxdx1+rρx0ηEϵ/1+ϵ=CEBϵ/1+ϵ1+rρx0η. This gives (19) with δ=ϵ/(1+ϵ).Given a weightw on Rd, as usual, the weighted Lebesgue space Lp(w) for 1≤p<∞ is defined to be the set of all functions f such that (21)fLpw≔∫Rdfxpwxdx1/p<∞. We also denote by WL1(w) the weighted weak Lebesgue space consisting of all measurable functions f for which (22)fWL1w≔supλ>0⁡λ·wx∈Rd:fx>λ<∞.Recently, Bongioanni et al. [4] obtained weighted strong-type and weak-type estimates for the operators R and R∗ defined in (6) and (7). Their results can be summarized as follows.Theorem 5 (from [4]).
Let1<p<∞ and w∈Apρ,∞. If V∈RHq with q≥d, then the operators R and R∗ are all bounded on Lp(w).Theorem 6 (from [4]).
Letp=1 and w∈A1ρ,∞. If V∈RHq with q≥d, then the operators R and R∗ are all bounded from L1(w) into WL1(w).


## 1.4. The SpaceBMOρ,∞(Rd)

We denote byT either R or R∗. For a locally integrable function b on Rd (usually called thesymbol), we will also consider the commutator operator(23)b,Tfx≔bx·Tfx-Tbfx,x∈Rd.Recently, Bongioanni et al. [3] introduced a new space BMOρ,∞(Rd) defined by (24)BMOρ,∞Rd≔⋃θ>0BMOρ,θRd,where for 0<θ<∞ the space BMOρ,θ(Rd) is defined to be the set of all locally integrable functions b satisfying(25)1Bx0,r∫Bx0,rbx-bBx0,rdx≤C·1+rρx0θ,for all x0∈Rd and r>0, and bB(x0,r) denotes the mean value of b on B(x0,r); that is, (26)bBx0,r≔1Bx0,r∫Bx0,rbydy. A norm for b∈BMOρ,θ(Rd), denoted by bBMOρ,θ, is given by the infimum of the constants satisfying (25), or, equivalently, (27)bBMOρ,θ≔supBx0,r1+rρx0-θ1Bx0,r∫Bx0,rbx-bBx0,rdx, where the supremum is taken over all balls B(x0,r) with x0∈Rd and r>0. With the above definition in mind, one has (28)BMORd⊂BMOρ,θ1Rd⊂BMOρ,θ2Rd for 0<θ1<θ2<∞, and hence BMO(Rd)⊂BMOρ,∞(Rd). Moreover, the classical BMO space [6] is properly contained in BMOρ,∞(Rd) (see [2, 3] for some examples). We need the following key result for the space BMOρ,θ(Rd), which was proved by Tang in [7].Proposition 7 (from [7]).
Letb∈BMOρ,θ(Rd) with 0<θ<∞. Then there exist two positive constants C1 and C2 such that, for any given ball B(x0,r) in Rd and for any λ>0, we have(29)x∈Bx0,r:bx-bBx0,r>λ≤C1Bx0,rexp⁡-1+rρx0-θ∗C2λbBMOρ,θ,where θ∗=(N0+1)θ and N0 is the constant appearing in Lemma 1.As a consequence of Proposition7 and Lemma 4, we have the following result.Proposition 8.
Letb∈BMOρ,θ(Rd) 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 B(x0,r) in Rd and for any λ>0, we have(30)wx∈Bx0,r:bx-bBx0,r>λ≤C1wBx0,rexp⁡-1+rρx0-θ∗C2λbBMOρ,θ1+rρx0η,where θ∗=(N0+1)θ and N0 is the constant appearing in Lemma 1.


## 1.5. Orlicz Spaces

In this subsection, let us give the definition of and some basic facts about Orlicz spaces. For more information on this subject, the reader may consult book [8]. Recall that a function A:[0,∞)→[0,∞) is called a Young function if it is continuous, convex, and strictly increasing with (31)A0=0,limt→∞At→∞. An important example of Young functions is A(t)=t·(1+log+t)m with some 1≤m<∞. Given a Young function A and a function f defined on a ball B, we consider the A-average of a function f given by the following Luxemburg norm: (32)fA,B≔inf⁡λ>0:1B∫BAfxλdx≤1. Associated with each Young function A, one can define its complementary function A¯ as follows: (33)A¯s≔supt>0⁡st-At,0≤s<∞.Such a function A¯ is also a Young function. It is well known that the followinggeneralized Hölder inequality in Orlicz spaces holds for any given ball B⊂Rd: (34)1B∫Bfx·gxdx≤2fA,BgA-,B. In particular, for the Young function A(t)=t·(1+log+t), the Luxemburg norm will be denoted by ·Llog⁡L,B=·A,B. A simple computation shows that the complementary Young function of A(t)=t·(1+log+t) is A¯(t)≈et-1 (see [9, 10] for instance). The corresponding Luxemburg norm will be denoted by ·exp⁡L,B=·A-,B. In this situation, we have(35)1B∫Bfx·gxdx≤2fLlog⁡L,Bgexp⁡L,B.We next define the weighted A-average of a function f over a ball B. Given a Young function A and a weight function w, let (see [8] for instance) (36)fAw,B≔inf⁡λ>0:1wB∫BAfxλ·wxdx≤1. When A(t)=t, we denote ·L(w),B=·A(w),B, and when Φ(t)=t·(1+log+t), we denote ·Llog⁡L(w),B=·Φ(w),B. Also, the complementary Young function of Φ is given by Φ¯(t)≈et-1 with the corresponding Luxemburg norm denoted by ·exp⁡L(w),B. Given a weight w on Rd, we can also show the weighted version of (35). That is, the generalized Hölder inequality in the weighted setting (see [11] for instance)(37)1wB∫Bfx·gxwxdx≤CfLlog⁡Lw,Bgexp⁡Lw,Bholds for every ball B in Rd. It is a simple but important observation that, for any ball B in Rd, (38)fLw,B≤fLlog⁡Lw,B. This is because t≤t·(1+log+t) for all t>0. So we have(39)fLw,B=1wB∫Bfx·wxdx≤fLlog⁡Lw,B.In [2], Bongioanni et al. obtained weighted strong (p,p), 1<p<∞, and weak Llog⁡L estimates for the commutators of the Riesz transform and its adjoint associated with the Schrödinger operator L=-Δ+V, where V satisfies some reverse Hölder inequality. Their results can be summarized as follows.Theorem 9 (from [2]).
Let1<p<∞ and w∈Apρ,∞. If V∈RHq with q≥d, then the commutator operators [b,R] and [b,R∗] are all bounded on Lp(w), whenever b belongs to BMOρ,∞(Rd).Theorem 10 (from [2]).
Letp=1 and w∈A1ρ,∞. If V∈RHq with q≥d and b∈BMOρ,∞(Rd), then, for any given λ>0, there exists a positive constant C>0 such that, for those functions f such that Φ(|f|)∈L1(w), (40)wx∈Rn:b,Rfx>λ≤C∫RdΦfxλ·wxdx,wx∈Rn:b,R∗fx>λ≤C∫RdΦfxλ·wxdx,where Φ(t)=t·(1+log+t) and log+t:=max⁡{log⁡t,0}; that is, (41)log+t=log⁡t,ast>1;0,otherwise.In this paper, firstly, we will define some kinds of weighted Morrey spaces related to certain nonnegative potentials. Secondly, we prove that the Riesz transformR and its adjoint R∗ are both bounded operators on these new spaces. Finally, we also obtain the weighted estimates for the commutators [b,R] and [b,R∗] defined in (23).Throughout this paperC 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 a,b such that C1a≤b≤C2a.


## 2. Our Main Results

In this section, we introduce some types of weighted Morrey spaces related to the potentialV and then give our main results.Definition 11.
Let1≤p<∞, let 0≤κ<1, and let w be a weight. For given 0<θ<∞, the weighted Morrey space Lρ,θp,κ(w) is defined to be the set of all Lp locally integrable functions f on Rd for which(42)1wBκ∫Bfxpwxdx1/p≤C·1+rρx0θfor every ball B=B(x0,r) in Rd. A norm for f∈Lρ,θp,κ(w), denoted by fLρ,θp,κ(w), is given by the infimum of the constants in (42), or, equivalently, (43)fLρ,θp,κw≔supB⁡1+rρx0-θ1wBκ∫Bfxpwxdx1/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 (44)Lρ,∞p,κw≔⋃θ>0Lρ,θp,κw.Note that this definition does not coincide with the one given in [12] (see also [13] for the unweighted case), but in view of the space BMOρ,∞(Rd) 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 weighted Morrey space Lp,κ(w), which was first defined by Komori and Shirai in [14] (see also [15]).Definition 12.
Letp=1, let 0≤κ<1, and let w be a weight. For given 0<θ<∞, the weighted weak Morrey space WLρ,θ1,κ(w) is defined to be the set of all measurable functions f on Rd for which (45)1wBκsupλ>0⁡λ·wx∈B:fx>λ≤C·1+rρx0θ for every ball B=B(x0,r) in Rd, or, equivalently, (46)fWLρ,θ1,κw≔supB1+rρx0-θ1wBκsupλ>0λ·wx∈B:fx>λ<∞. Correspondingly, we define (47)WLρ,∞1,κw≔⋃θ>0WLρ,θ1,κw.Clearly, if we takeθ=0 or V≡0, then this space is just the weighted weak Morrey space WL1,κ(w) (see [16]). According to the above definitions, one has (48)Lp,κw⊂Lρ,θ1p,κw⊂Lρ,θ2p,κw;WL1,κw⊂WLρ,θ11,κw⊂WLρ,θ21,κw, for 0<θ1<θ2<∞. Hence Lp,κ(w)⊂Lρ,∞p,κ(w) for (p,κ)∈[1,∞)×[0,1) and WL1,κ(w)⊂WLρ,∞1,κ(w) for 0≤κ<1.The spaceLρ,θp,κ(w) (or WLρ,θ1,κ(w)) could be viewed as an extension of the weighted (or weak) Lebesgue space (when κ=θ=0). Naturally, one may ask the question whether the above conclusions (i.e., Theorems 5 and 6 as well as Theorems 9 and 10) still hold if we replace the weighted Lebesgue spaces by the weighted Morrey spaces. In this work, we give a positive answer to this question. Our main results in this work are presented as follows.Theorem 13.
Let1<p<∞, 0<κ<1, and w∈Apρ,∞. If V∈RHq with q≥d, then the operators R and R∗ map Lρ,∞p,κ(w) into itself.Theorem 14.
Letp=1, 0<κ<1, and w∈A1ρ,∞. If V∈RHq with q≥d, then the operators R and R∗ map Lρ,∞1,κ(w) into WLρ,∞1,κ(w).Theorem 15.
Let1<p<∞, 0<κ<1, and w∈Apρ,∞. If V∈RHq with q≥d, then the commutator operators [b,R] and [b,R∗] map Lρ,∞p,κ(w) into itself, whenever b∈BMOρ,∞(Rd).To deal with the commutators in the endpoint case, we need to consider a new kind of weighted Morrey spaces of theLlog⁡L type.Definition 16.
Letp=1, let 0≤κ<1, and let w be a weight. For given 0<θ<∞, the weighted Morrey space (Llog⁡L)ρ,θ1,κ(w) is defined to be the set of all locally integrable functions f on Rd for which (49)wB1-κfLlog⁡Lw,B≤C·1+rρx0θ for every ball B=B(x0,r) in Rd, or, equivalently, (50)fLlog⁡Lρ,θ1,κw≔supB1+rρx0-θwB1-κfLlog⁡Lw,B<∞.Concerning the mapping properties of[b,R] and [b,R∗] in the weighted Morrey spaces of the Llog⁡L type, we have the following.Theorem 17.
Letp=1, 0<κ<1, and w∈A1ρ,∞. If V∈RHq with q≥d and b∈BMOρ,∞(Rd), then, for any given λ>0 and any given ball B=B(x0,r) of Rd, there exist some constants C>0 and ϑ>0 such that the inequalities(51)1wBκ·wx∈B:b,Rfx>λ≤C1+rρx0ϑΦfλLlog⁡Lρ,θ1,κw1wBκ·wx∈B:b,R∗fx>λ≤C1+rρx0ϑΦfλLlog⁡Lρ,θ1,κw, hold for those functions f such that Φ(|f|)∈(Llog⁡L)ρ,θ1,κ(w) with some fixed θ>0, where Φ(t)=t·(1+log+t).If we denote(52)Llog⁡Lρ,∞1,κw≔⋃θ>0Llog⁡Lρ,θ1,κw, then Theorem 17 now tells us that the commutators [b,R] and [b,R∗] map (Llog⁡L)ρ,∞1,κ(w) into WLρ,∞1,κ(w), when b is in BMOρ,∞(Rd).


## 3. Proofs of Theorems13 and 14

In this section, we will prove the conclusions of Theorems13 and 14.Proof of Theorem13.
We denote byT either R or R∗. By definition, we only have to show that, for any given ball B=B(x0,r) of Rd, there is some ϑ>0 such that(53)1wBκ∫BTfxpwxdx1/p≤C·1+rρx0ϑholds for any f∈Lρ,∞p,κ(w) with 1<p<∞ and 0<κ<1. Suppose that f∈Lρ,θp,κ(w) for some θ>0 and w∈Apρ,θ′ for some θ′>0. We decompose f, in the classical way, as (54)f=f1+f2∈Lρ,θp,κw;f1=f·χ2B;f2=f·χ2Bc,where 2B is the ball centered at x0 and radius 2r>0 and χ2B is the characteristic function of 2B. Then by the linearity of T, we write (55)1wBκ∫BTfxpwxdx1/p≤1wBκ∫BTf1xpwxdx1/p+1wBκ∫BTf2xpwxdx1/p≔I1+I2. We now analyze each term separately. By Theorem 5, we get(56)I1=1wBκ∫BTf1xpwxdx1/p≤C·1wBκ/p∫Rdf1xpwxdx1/p=C·1wBκ/p∫2Bfxpwxdx1/p≤CfLρ,θp,κw·w2Bκ/pwBκ/p·1+2rρx0θ. Since w∈Apρ,θ′ with 1<p<∞ and 0<θ′<∞, then we know that the inequality(57)w2Bx0,r≤C·1+2rρx0pθ′wBx0,ris valid. In fact, for 1<p<∞, by Hölder’s inequality and the definition of Apρ,θ′, we have (58)12B∫2Bħxdx=12B∫2Bħxwx1/pwx-1/pdx≤12B∫2Bħxpwxdx1/p∫2Bwx-p′/pdx1/p′≤Cww2B1/p∫2Bħxpwxdx1/p1+2rρx0θ′.If we take ħ(x)=χB(x), then the above expression becomes(59)B2B≤Cw·wB1/pw2B1/p1+2rρx0θ′,which in turn implies (57). Therefore, (60)I1≤CfLρ,θp,κw·1+2rρx0pθ′·κ/p·1+2rρx0θ=CfLρ,θp,κw·1+2rρx0ϑ′≤C·1+rρx0ϑ′,where ϑ′≔κ·θ′+θ. For the other term I2, notice that, for any x∈B and y∈(2B)c, one has x-y≈x0-y. It then follows from Lemma 2 that, for any x∈B(x0,r) and any positive integer N,(61)Tf2x≤∫2BcKx,y·fydyorK∗x,y≤CN∫2Bc1+x-yρx-N1x-yd·fydy≤CN,d∫2Bc1+x0-yρx-N1x0-yd·fydy=CN,d∑k=1∞∫2k+1B\2kB1+x0-yρx-N1x0-yd·fydy≤C∑k=1∞12k+1B∫2k+1B\2kB1+2krρx-Nfydy.In view of (4) in Lemma 1, we further obtain(62)Tf2x≤C∑k=1∞12k+1B∫2k+1B1+rρx0N·N0/N0+11+2krρx0-Nfydy≤C∑k=1∞12k+1B∫2k+1B1+rρx0N·N0/N0+11+2k+1rρx0-Nfydy.Moreover, by using Hölder’s inequality and the Apρ,θ′ condition on w, we get (63)12k+1B∫2k+1Bfydy≤12k+1B∫2k+1Bfypwydy1/p∫2k+1Bwy-p′/pdy1/p′≤CfLρ,θp,κw·w2k+1Bκ/pw2k+1B1/p1+2k+1rρx0θ1+2k+1rρx0θ′.Hence, (64)I2≤CfLρ,θp,κw·wB1/pwBκ/p∑k=1∞w2k+1Bκ/pw2k+1B1/p1+rρx0N·N0/N0+11+2k+1rρx0-N+θ+θ′=CfLρ,θp,κw1+rρx0N·N0/N0+1∑k=1∞wB1-κ/pw2k+1B1-κ/p1+2k+1rρx0-N+θ+θ′.Recall that w∈Apρ,θ′ with 0<θ′<∞ and 1<p<∞; then there exist two positive numbers δ,η>0 such that (19) holds. This allows us to obtain (65)I2≤CfLρ,θp,κw1+rρx0N·N0/N0+1∑k=1∞B2k+1Bδ1-κ/p×1+2k+1rρx0η1-κ/p1+2k+1rρx0-N+θ+θ′=CfLρ,θp,κw1+rρx0N·N0/N0+1∑k=1∞B2k+1Bδ1-κ/p1+2k+1rρx0-N+θ+θ′+η1-κ/p. Thus, by choosing N large enough so that N>θ+θ′+η(1-κ)/p, we then have (66)I2≤CfLρ,θp,κw1+rρx0N·N0/N0+1∑k=1∞B2k+1Bδ1-κ/p≤C1+rρx0N·N0/N0+1. Summing up the above estimates for I1 and I2 and letting ϑ=max⁡{ϑ′,N·N0/N0+1}, we obtain our desired inequality (53). This completes the proof of Theorem 13.Proof of Theorem14.
We denote byT either R or R∗. To prove Theorem 14, by definition, it is sufficient to prove that, for any given ball B=B(x0,r) of Rd, there is some ϑ>0 such that(67)1wBκsupλ>0λ·wx∈B:Tfx>λ≤C·1+rρx0ϑholds for any f∈Lρ,∞1,κ(w) with 0<κ<1. Now suppose that f∈Lρ,θ1,κ(w) for some θ>0 and w∈A1ρ,θ′ for some θ′>0. We decompose f, in the classical way, as (68)f=f1+f2∈Lρ,θ1,κw;f1=f·χ2B;f2=f·χ2Bc. Then for any given λ>0, by the linearity of T, we can write(69)1wBκλ·wx∈B:Tfx>λ≤1wBκλ·wx∈B:Tf1x>λ2+1wBκλ·wx∈B:Tf2x>λ2≔I1′+I2′. We first give the estimate for the term I1′. By Theorem 6, we get (70)I1′=1wBκλ·wx∈B:Tf1x>λ2≤C·1wBκ∫Rdf1xwxdx=C·1wBκ∫2Bfxwxdx≤CfLρ,θ1,κw·w2BκwBκ·1+2rρx0θ. Since w∈A1ρ,θ′ with 0<θ′<∞, similar to the proof of (57), we can also show the following estimate as well:(71)w2Bx0,r≤C·1+2rρx0θ′wBx0,r.In fact, by the definition of A1ρ,θ′, we can deduce that (72)12B∫2Bħxdx≤Cww2B·essinfx∈2Bwx∫2Bħxdx1+2rρx0θ′≤Cww2B∫2Bħxwxdx1+2rρx0θ′. If we choose ħ(x)=χB(x), then the above expression becomes(73)B2B≤Cw·wBw2B1+2rρx0θ′,which in turn implies (71). Therefore, (74)I1′≤C·1+2rρx0θ′·κ·1+2rρx0θ=C·1+2rρx0ϑ′≤C·1+rρx0ϑ′,where ϑ′≔θ′·κ+θ. As for the other term I2′, by using pointwise inequality (62) and Chebyshev’s inequality, we deduce that (75)I2′=1wBκλ·wx∈B:Tf2x>λ2≤2wBκ∫BTf2xwxdx≤C·wBwBκ∑k=1∞12k+1B∫2k+1B1+rρx0N·N0/N0+11+2k+1rρx0-Nfydy. Moreover, by the A1ρ,θ′ condition on w, we compute (76)12k+1B∫2k+1Bfydy≤Cww2k+1B·essinfy∈2k+1Bwy∫2k+1Bfydy1+2k+1rρx0θ′≤Cww2k+1B∫2k+1Bfywydy1+2k+1rρx0θ′≤CfLρ,θ1,κw·w2k+1Bκw2k+1B1+2k+1rρx0θ1+2k+1rρx0θ′. Consequently, (77)I2′≤CfLρ,θ1,κw·wBwBκ∑k=1∞w2k+1Bκw2k+1B1+rρx0N·N0/N0+11+2k+1rρx0-N+θ+θ′=CfLρ,θ1,κw1+rρx0N·N0/N0+1∑k=1∞wB1-κw2k+1B1-κ1+2k+1rρx0-N+θ+θ′. Recall that w∈A1ρ,θ′ with 0<θ′<∞; then there exist two positive numbers δ′,η′>0 such that (19) holds. Therefore, (78)I2′≤CfLρ,θ1,κw1+rρx0N·N0/N0+1∑k=1∞B2k+1Bδ′1-κ×1+2k+1rρx0η′1-κ1+2k+1rρx0-N+θ+θ′=CfLρ,θ1,κw1+rρx0N·N0/N0+1×∑k=1∞B2k+1Bδ′1-κ1+2k+1rρx0-N+θ+θ′+η′1-κ. By selecting N large enough so that N>θ+θ′+η′(1-κ), we thus have (79)I2′≤C1+rρx0N·N0/N0+1∑k=1∞B2k+1Bδ′1-κ≤C1+rρx0N·N0/N0+1. Let ϑ=max⁡{ϑ′,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 our desired inequality (67). This finishes the proof of Theorem 14.


## 4. Proofs of Theorems15 and 17

For the results involving commutators, we need the following properties ofBMOρ,∞(Rd) functions, which are extensions of well-known properties of BMO(Rd) functions.Lemma 18.
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=B(x0,r) in Rd, we have(80)∫Bbx-bBpwxdx1/p≤C·wB1/p1+rρx0μ,where bB=1/|B|∫Bb(y)dy.Proof.
We may assume thatb∈BMOρ,θ(Rd) with 0<θ<∞. According to Proposition 8, we can deduce that (81)∫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. Making change of variables, then we get (82)∫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 if we choose C=[C1pΓ(p)]1/pbBMOρ,θ/C2 and μ=θ∗+η/p.Lemma 19.
Ifb∈BMOρ,θ(Rd) with 0<θ<∞ and w∈A1ρ,∞, then there exist positive constants C,γ>0 and η>0 such that, for every ball B=B(x0,r) in Rd, we have(83)∫Bexp⁡1+rρx0-θ∗γbBMOρ,θbx-bB-1wxdx≤C·wB1+rρx0η,where bB=1/|B|∫Bb(y)dy and θ∗=(N0+1)θ and N0 is the constant appearing in Lemma 1.Proof.
Recall the following identity (see Proposition 1.1.4 in [5]): (84)∫Bexp⁡fx-1wxdx=∫0∞eλwx∈B:fx>λdλ. Using this identity and Proposition 8, we obtain(85)∫Bexp⁡1+rρx0-θ∗γbBMOρ,θbx-bB-1wxdx=∫0∞eλwx∈B:bx-bB>λ∗dλ≤C1·wB∫0∞eλexp⁡-1+rρx0-θ∗C2λ∗bBMOρ,θdλ1+rρx0η=C1·wB∫0∞eλ·e-C2λ/γdλ1+rρx0η, where λ∗ is given by (86)λ∗=λbBMOρ,θγ1+rρx0θ∗. If we take γ small enough so that 0<γ<C2, then the conclusion follows immediately.Lemma 20.
Ifb∈BMOρ,θ(Rd) with 0<θ<∞, then, for any positive integer k, there exists a positive constant C>0 such that, for every ball B=B(x0,r) in Rd, we have (87)b2k+1B-bB≤C·k+11+2k+1rρx0θ.Proof.
For any positive integerk, we have (88)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 any 1≤j≤k+1, the estimate (89)1+2jrρx0θ≤1+2k+1rρx0θ holds trivially, then (90)b2k+1B-bB≤C∑j=1k+11+2k+1rρx0θ=C·k+11+2k+1rρx0θ. We obtain the desired result. This completes the proof.Now, we are in a position to prove our main results in this section.Proof of Theorem15.
We denote by[b,T] either [b,R] or [b,R∗]. By definition, we only need to show that, for any given ball B=B(x0,r) of Rd, there is some ϑ>0 such that(91)1wBκ∫Bb,Tfxpwxdx1/p≤C·1+rρx0ϑholds for any f∈Lρ,∞p,κ(w) with 1<p<∞ and 0<κ<1, whenever b belongs to BMOρ,∞(Rd). Suppose that f∈Lρ,θp,κ(w) for some θ>0, w∈Apρ,θ′ for some θ′>0, and b∈BMOρ,θ′′(Rd) for some θ′′>0. We decompose f as (92)f=f1+f2∈Lρ,θp,κw;f1=f·χ2B;f2=f·χ2Bc. Then, by the linearity of [b,T], we write (93)1wBκ∫Bb,Tfxpwxdx1/p≤1wBκ∫Bb,Tf1xpwxdx1/p+1wBκ∫Bb,Tf2xpwxdx1/p≔J1+J2. Now we give the estimates for J1 and J2, respectively. According to Theorem 9, we have (94)J1≤C·1wBκ/p∫Rdf1xpwxdx1/p=C·1wBκ/p∫2Bfxpwxdx1/p≤CfLρ,θp,κw·w2Bκ/pwBκ/p·1+2rρx0θ. Moreover, in view of inequality (57), we get (95)J1≤CfLρ,θp,κw·1+2rρx0pθ′·κ/p·1+2rρx0θ≤C·1+rρx0ϑ′, where ϑ′≔θ′·κ+θ. On the other hand, by definition (23), we can see that, for any x∈B(x0,r),(96)b,Tf2x≤∫Rdbx-byKx,yf2ydyorK∗x,y≤bx-bB∫RdKx,yf2ydy+∫Rdby-bBKx,yf2ydy≔ξx+ζx.So we can divide J2 into two parts: (97)J2≤1wBκ∫Bξxpwxdx1/p+1wBκ∫Bζxpwxdx1/p≔J3+J4. From pointwise estimate (62) and (80) in Lemma 18, it then follows that (98)J3≤C·1wBκ/p∫Bbx-bBpwxdx1/p×∑k=1∞12k+1B∫2k+1B1+rρx0N·N0/N0+11+2k+1rρx0-Nfydy≤Cb·wB1/pwBκ/p1+rρx0μ×∑k=1∞12k+1B∫2k+1B1+rρx0N·N0/N0+11+2k+1rρx0-Nfydy. Following along the same lines as that of Theorem 13, we are able to show that (99)J3≤CfLρ,θp,κw1+rρx0μ1+rρx0N·N0/N0+1×∑k=1∞wB1-κ/pw2k+1B1-κ/p1+2k+1rρx0-N+θ+θ′≤CfLρ,θp,κw1+rρx0μ+N·N0/N0+1×∑k=1∞B2k+1Bδ1-κ/p1+2k+1rρx0-N+θ+θ′+η1-κ/p. The last inequality is obtained by using (19). For any x∈B(x0,r) and any positive integer N, similar to the proof of (61) and (62), we can also deduce that(100)ζx=∫2Bcby-bBKx,yfydy≤CN∫2Bcby-bB1+x-yρx-N1x-yd·fydy≤CN,d∑k=1∞∫2k+1B\2kBby-bB1+x0-yρx-N1x0-yd·fydy≤CN,d∑k=1∞12k+1B∫2k+1B\2kBby-bB1+2krρx-Nfydy≤C∑k=1∞12k+1B∫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 ζ(x), (101)J4≤C·wB1-κ/p∑k=1∞1+rρx0N·N0/N0+11+2k+1rρx0-N×12k+1B∫2k+1Bby-bBfydy. Moreover, for each integer k≥1,(102)12k+1B∫2k+1Bby-bBfydy≤12k+1B∫2k+1Bby-b2k+1Bfydy+12k+1B∫2k+1Bb2k+1B-bBfydy.By using Hölder’s inequality, the first term of expression (102) is bounded by (103)12k+1B∫2k+1Bfypwydy1/p∫2k+1Bby-b2k+1Bp′wy-p′/pdy1/p′≤CfLρ,θp,κw·w2k+1Bκ/p2k+1B1+2k+1rρx0θ∫2k+1Bby-b2k+1Bp′wy-p′/pdy1/p′. Since w∈Apρ,θ′ with 0<θ′<∞ and 1<p<∞, then, by the definition of Apρ,θ′, it can be easily shown that w∈Apρ,θ′ if and only if w-p′/p∈Ap′ρ,θ′, where 1/p+1/p′=1 (see [7, 17, 18] and the references therein). If we denote v=w-p′/p, then v∈Ap′ρ,θ′. This fact together with Lemma 18 implies (104)∫2k+1Bby-b2k+1Bp′vydy1/p′≤Cb·v2k+1B1/p′1+2k+1rρx0μ=Cb·∫2k+1Bwy-p′/pdy1/p′1+2k+1rρx0μ≤Cb,w·2k+1Bw2k+1B1/p1+2k+1rρx0θ′1+2k+1rρx0μ. Therefore, the first term of expression (102) can be bounded by a constant times (105)w2k+1Bκ/pw2k+1B1/p1+2k+1rρx0θ+θ′+μ. Since b∈BMOρ,θ′′(Rd) with 0<θ′′<∞, then, by Lemma 20, Hölder’s inequality, and the Apρ,θ′ condition on w, the latter term of expression (102) can be estimated by (106)Cbk+11+2k+1rρx0θ′′12k+1B∫2k+1Bfydy≤Cbk+11+2k+1rρx0θ′′12k+1B∫2k+1Bfypwydy1/p∫2k+1Bwy-p′/pdy1/p′≤CfLρ,θp,κw·k+1w2k+1Bκ/pw2k+1B1/p1+2k+1rρx0θ+θ′+θ′′. Consequently,(107)12k+1B∫2k+1Bby-bBfydy≤CfLρ,θp,κw·k+1w2k+1Bκ/pw2k+1B1/p1+2k+1rρx0θ+θ′+θ′′+μ.Thus, in view of (107),(108)J4≤CfLρ,θp,κw·wB1-κ/p∑k=1∞k+11+rρx0N·N0/N0+11+2k+1rρx0-N×1w2k+1B1-κ/p1+2k+1rρx0θ+θ′+θ′′+μ=CfLρ,θp,κw1+rρx0N·N0/N0+1∑k=1∞k+1wB1-κ/pw2k+1B1-κ/p1+2k+1rρx0-N+θ+θ′+θ′′+μ. Notice that w∈Apρ,θ′ with 0<θ′<∞. A further application of (19) yields (109)J4≤CfLρ,θp,κw1+rρx0N·N0/N0+1∑k=1∞k+1B2k+1Bδ1-κ/p×1+2k+1rρx0η1-κ/p1+2k+1rρx0-N+θ+θ′+θ′′+μ=CfLρ,θp,κw1+rρx0N·N0/N0+1∑k=1∞k+1B2k+1Bδ1-κ/p×1+2k+1rρx0-N+θ+θ′+θ′′+μ+η1-κ/p. Combining the above estimates for J3 and J4, we get (110)J2≤J3+J4≤CfLρ,θp,κw1+rρx0μ+N·N0/N0+1×∑k=1∞k+1B2k+1Bδ1-κ/p1+2k+1rρx0-N+θ+θ′+θ′′+μ+η1-κ/p. By choosing N large enough so that N>θ+θ′+θ′′+μ+η(1-κ)/p, we thus have (111)J2≤C1+rρx0μ+N·N0/N0+1∑k=1∞k+1B2k+1Bδ1-κ/p≤C1+rρx0μ+N·N0/N0+1. Finally, collecting the above estimates for J1 and J2 and letting ϑ=max⁡ϑ′,μ+N·N0/N0+1, we obtain the desired result (91). The proof of Theorem 15 is finished.Proof of Theorem17.
We denote by[b,T] either [b,R] or [b,R∗]. We are going to prove that, for any given λ>0 and any given ball B=B(x0,r) of Rd, there is some ϑ>0 such that the inequality(112)1wBκ·wx∈B:b,Tfx>λ≤C1+rρx0ϑΦfλLlog⁡Lρ,θ1,κwholds for those functions f such that Φ(f)∈(Llog⁡L)ρ,θ1,κ(w) with some fixed θ>0. Now assume that w∈A1ρ,θ′ for some θ′>0 and b∈BMOρ,θ′′(Rd) for some θ′′>0. As before, we decompose f as (113)f=f1+f2;f1=f·χ2B,f2=f·χ2Bc.Then for any given λ>0, by the linearity of [b,T], we can write (114)1wBκ·wx∈B:b,Tfx>λ≤1wBκ·wx∈B:b,Tf1x>λ2+1wBκ·wx∈B:b,Tf2x>λ2≔J1′+J2′. Let us first estimate the term J1′. By using Theorem 10, we get (115)J1′=1wBκ·wx∈B:b,Tf1x>λ2≤C·1wBκ∫RdΦf1xλ·wxdx=C·1wBκ∫2BΦfxλ·wxdx. A further application of (39) yields(116)J1′≤C·w2BwBκΦfλLlog⁡Lw,2B≤C·w2BκwBκ·1+2rρx0θΦfλLlog⁡Lρ,θ1,κw≤C·1+2rρx0κ·θ′·1+2rρx0θΦfλLlog⁡Lρ,θ1,κw, where the last inequality is due to (71). If we denote ϑ′=κ·θ′+θ, then(117)J1′≤C·1+2rρx0ϑ′ΦfλLlog⁡Lρ,θ1,κw≤C·1+rρx0ϑ′ΦfλLlog⁡Lρ,θ1,κwas desired. Next let us deal with the term J2′. Taking into account (96), we can divide it into two parts, namely, (118)J2′=1wBκ·wx∈B:b,Tf2x>λ2≤1wBκ·wx∈B:ξx>λ4+1wBκ·wx∈B:ζx>λ4≔J3′+J4′, where (119)ξx=bx-bB∫RdKx,yf2ydy,ζx=∫Rdby-bBKx,yf2ydy. Since b∈BMOρ,θ′′(Rd) for some θ′′>0, from pointwise inequality (62) and Chebyshev’s inequality, we then have (120)J3′≤1wBκ·4λ∫Bξxwxdx≤C·1wBκ∫Bbx-bBwxdx×∑k=1∞12k+1B∫2k+1B1+rρx0N·N0/N0+11+2k+1rρx0-Nfyλdy≤Cb·wBwBκ1+rρx0μ×∑k=1∞12k+1B∫2k+1B1+rρx0N·N0/N0+11+2k+1rρx0-Nfyλdy,where in the last inequality we have used (80) in Lemma 18. Moreover, it follows directly from the condition A1ρ,θ′ that, for each integer k≥1, (121)12k+1B∫2k+1Bfyλdy≤Cww2k+1B·essinfy∈2k+1Bwy∫2k+1Bfyλdy1+2k+1rρx0θ′≤Cww2k+1B∫2k+1Bfyλ·wydy1+2k+1rρx0θ′.Notice also that trivially(122)t≤t·1+log+t=Φt,foranyt>0.This fact along with (39) implies that, for each integer k≥1, (123)12k+1B∫2k+1Bfyλdy≤Cww2k+1B∫2k+1BΦfyλ·wydy1+2k+1rρx0θ′≤C·1+2k+1rρx0θ′ΦfλLlog⁡Lw,2k+1B≤C·1w2k+1B1-κ1+2k+1rρx0θ1+2k+1rρx0θ′ΦfλLlog⁡Lρ,θ1,κw. Consequently,(124)J3′≤C1+rρx0μ1+rρx0N·N0/N0+1×∑k=1∞wB1-κw2k+1B1-κ1+2k+1rρx0-N+θ+θ′ΦfλLlog⁡Lρ,θ1,κw. Since w∈A1ρ,θ′ with 0<θ′<∞, then there exist two positive numbers δ′ and η′>0 such that (19) holds. Therefore, (125)J3′≤C1+rρx0μ+N·N0/N0+1×∑k=1∞B2k+1Bδ′1-κ1+2k+1rρx0η′1-κ1+2k+1rρx0-N+θ+θ′ΦfλLlog⁡Lρ,θ1,κw=C1+rρx0μ+N·N0/N0+1×∑k=1∞B2k+1Bδ′1-κ1+2k+1rρx0-N+θ+θ′+η′1-κΦfλLlog⁡Lρ,θ1,κw. On the other hand, it follows from pointwise inequality (100) and Chebyshev’s inequality that(126)J4′≤1wBκ·4λ∫Bζxwxdx≤C·wBwBκ∑k=1∞1+rρx0N·N0/N0+11+2k+1rρx0-N×12k+1B∫2k+1Bby-bBfyλdy≤C·wBwBκ∑k=1∞1+rρx0N·N0/N0+11+2k+1rρx0-N×12k+1B∫2k+1Bby-bBΦfyλdy,where the last inequality follows from (122). Furthermore, by the definition of A1ρ,θ′, we compute(127)12k+1B∫2k+1Bby-bBΦfyλdy≤Cww2k+1B·essinfy∈2k+1Bwy∫2k+1Bby-bBΦfyλdy1+2k+1rρx0θ′≤Cww2k+1B∫2k+1Bby-b2k+1BΦfyλwydy1+2k+1rρx0θ′+Cww2k+1B∫2k+1Bb2k+1B-bBΦfyλwydy1+2k+1rρx0θ′.By using generalized Hölder inequality (37), the first term of expression (127) is bounded by (128)Cb-b2k+1Bexp⁡Lw,2k+1BΦfλLlog⁡Lw,2k+1B1+2k+1rρx0θ′≤CΦfλLlog⁡Lw,2k+1B1+2k+1rρx0η′1+2k+1rρx0θ′, where in the last inequality we have used the fact that (129)b-bBexp⁡Lw,B≤C1+rρx0η′,foranyballB=Bx0,r⊂Rn,which is equivalent to inequality (83) in Lemma 19. By Lemma 20 and (39), the latter term of expression (127) can be estimated by (130)Cbk+1w2k+1B∫2k+1BΦfyλwydy1+2k+1rρx0θ′′1+2k+1rρx0θ′≤Ck+1ΦfλLlog⁡Lw,2k+1B1+2k+1rρx0θ′1+2k+1rρx0θ′′. Consequently,(131)J4′≤C·wBwBκ∑k=1∞1+rρx0N·N0/N0+11+2k+1rρx0-N×k+1ΦfλLlog⁡Lw,2k+1B1+2k+1rρx0η′+θ′+θ′′≤C1+rρx0N·N0/N0+1∑k=1∞k+1wB1-κw2k+1B1-κΦfλLlog⁡Lρ,θ1,κw×1+2k+1rρx0-N+η′+θ+θ′+θ′′≤C1+rρx0N·N0/N0+1∑k=1∞k+1B2k+1Bδ′1-κ×1+2k+1rρx0-N+η′+θ+θ′+θ′′+η′1-κΦfλLlog⁡Lρ,θ1,κw. Hence, combining the above estimates for J3′ and J4′, we have (132)J2′≤J3′+J4′≤C1+rρx0μ+N·N0/N0+1∑k=1∞k+1B2k+1Bδ′1-κ×1+2k+1rρx0-N+η′+θ+θ′+θ′′+η′1-κΦfλLlog⁡Lρ,θ1,κw. Now N can be chosen sufficiently large such that N>η′+θ+θ′+θ′′+η′(1-κ), and hence the above series is convergent. Finally, (133)J2′≤C1+rρx0μ+N·N0/N0+1∑k=1∞k+1B2k+1Bδ′1-κΦfλLlog⁡Lρ,θ1,κw≤C1+rρx0μ+N·N0/N0+1ΦfλLlog⁡Lρ,θ1,κw.Fix this N and set ϑ=max⁡{ϑ′,μ+N·N0/N0+1}. Thus, combining the above estimates for J1′ and J2′, inequality (112) is proved and then the proof of Theorem 17 is finished.The higher order commutators formed by aBMOρ,∞(Rd) function b and the operator R and its adjoint R∗ are usually defined by(134)b,Rmfx≔∫Rnbx-bymKx,yfydy,x∈Rd;b,R∗mfx≔∫Rnbx-bymK∗x,yfydy,x∈Rd;m=1,2,3,….Let T denote R or R∗. Obviously, [b,T]1=[b,T] which is just the linear commutator (23), and (135)b,Tm=b,b,Tm-1,m=2,3,….By induction on m, we are able to show that the conclusions of Theorems 15 and 17 also hold for the higher order commutators [b,T]m with m≥2. The details are omitted here.Theorem 21.
Let1<p<∞, 0<κ<1, and w∈Apρ,∞. If V∈RHq with q≥d, then, for any positive integer m≥2, the higher order commutators [b,R]m and [b,R∗]m map Lρ,∞p,κ(w) into itself, whenever b∈BMOρ,∞(Rd).Theorem 22.
Letp=1, 0<κ<1, and w∈A1ρ,∞. If V∈RHq with q≥d and b∈BMOρ,∞(Rd), then, for any given λ>0 and any given ball B=B(x0,r) of Rd, there exist some constants C>0 and ϑ>0 such that the inequalities (136)1wBκ·wx∈B:b,Rmfx>λ≤C1+rρx0ϑΦmfλLlog⁡Lρ,θ1,κw,(137)1wBκ·wx∈B:b,R∗mfx>λ≤C1+rρx0ϑΦmfλLlog⁡Lρ,θ1,κwhold for those functions f such that Φm(f)∈(Llog⁡L)ρ,θ1,κ(w) with some fixed θ>0, where Φm(t)=t·(1+log+t)m, m=2,3,….



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