# Two-Weight, Weak-Type Norm Inequalities for a Class of Sublinear Operators on Weighted Morrey and Amalgam Spaces

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

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## Abstract

LetTα0≤α<n be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let b,Tα0≤α<n be the commutators generated by BMORn functions and Tα. This paper is concerned with two-weight, weak-type norm estimates for these sublinear operators and their commutators on the weighted Morrey and amalgam spaces. Some boundedness criteria for such operators are given, under the assumptions that weak-type norm inequalities on weighted Lebesgue spaces are satisfied. As applications of our main results, we can obtain the weak-type norm inequalities for several integral operators as well as the corresponding commutators in the framework of weighted Morrey and amalgam spaces.


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## Body

## 1. Introduction and Main Results

### 1.1. Sublinear Operators

LetRn be the n-dimensional Euclidean space equipped with the Euclidean norm · and the Lebesgue measure dx. Suppose that T represents a linear or a sublinear operator, which satisfies that for any f∈L1Rn with compact support and x∉suppf,
(1)Tfx≤c1∫Rnfyx−yndy,where c1 is a universal constant independent of f and x∈Rn. The size condition (1) was first introduced by Soria and Weiss in [1]. It can be proved that (1) is satisfied by many integral operators in harmonic analysis, such as the Hardy–Littlewood maximal operator, Calderón–Zygmund singular integral operators, Ricci–Stein’s oscillatory singular integrals, and Bochner–Riesz operators at the critical index and so on.Similarly, for any given0<γ<n, we assume that Tγ represents a linear or a sublinear operator with order γ, which satisfies that for any f∈L1Rn with compact support and x∉suppf,
(2)Tγfx≤c2∫Rnfyx−yn−γdy,where c2 is also a universal constant independent of f and x∈Rn. It can be easily checked that (2) is satisfied by some important operators such as the fractional maximal operator, Riesz potential operators, and fractional oscillatory singular integrals.Letb be a locally integrable function on Rn; suppose that the commutator operator b,T stands for a linear or a sublinear operator, which satisfies that for any f∈L1Rn with compact support and x∉suppf,
(3)b,Tfx≤c3∫Rnbx−by·fyx−yndy,where c3 is an absolute constant independent of f and x∈Rn.Similarly, for any given0<γ<n, we assume that the commutator operator b,Tγ stands for a linear or a sublinear operator, which satisfies that for any f∈L1Rn with compact support and x∉suppf,
(4)b,Tγfx≤c4∫Rnbx−by·fyx−yn−γdy,where c4 is also an absolute constant independent of f and x∈Rn. Clearly, based on the above assumptions,
(5)T0=T,b,T0=b,T.


### 1.2. Weighted Morrey Spaces

The classical Morrey space was introduced by Morrey [2] in connection with elliptic partial differential equations. Let 1≤p<∞ and 0≤λ≤n. We recall that a real-valued function f is said to belong to the space Lp,λ on the n-dimensional Euclidean space Rn, if the following norm is finite:
(6)fLp,λ≔supx,r∈Rn×0,∞rλ−n∫Bx,rfypdy1/p,where Bx,r=y∈Rn:x−y<r is the Euclidean ball with center x∈Rn and radius r∈0,∞. In particular, one has
(7)Lp,0=L∞,Lp,n=Lp.In [3], Komori and Shirai considered the weighted case and gave the definitions of the weighted Morrey spaces as follows.Definition 1.
Let1<p<∞ and 0≤κ<1. For two weights w and ν on Rn, the weighted Morrey space Lp,κν,w is defined by
(8)Lp,κν,w≔f∈Llocpν:fLp,κν,w<∞,where the norm is given by
(9)fLp,κν,w≔supQ⊂Rn1wQκ∫Qfxpνxdx1/p,and the supremum is taken over all cubes Q in Rn.Definition 2.
Let1<p<∞, 0≤κ<1, and w be a weight on Rn. We define the weighted weak Morrey space WLp,κw as the set of all measurable functions f satisfying
(10)fWLp,κw≔supQ⊂Rnsupσ>01wQκ/pσ·wx∈Q:fx>σ1/p<∞.
By definition, it is clear that(11)Lp,0ν,w=Lpν,WLp,0w=WLpw.


### 1.3. Weighted Amalgam Spaces

Let1≤p,q≤∞; a function f∈LlocpRn is said to be in the Wiener amalgam space Lp,LqRn of LpRn and LqRn, if the function y↦f··χBy,1Lp belongs to LqRn, where By,1 is the open ball in Rn centered at y with radius 1, χBy,1 is the characteristic function of the ball By,1, and ·Lp is the usual Lebesgue norm in LpRn. In [4], Fofana introduced a new class of function spaces Lp,LqαRn which turn out to be the subspaces of Lp,LqRn. More precisely, for 1≤p,q,α≤∞, we define the amalgam space Lp,LqαRn of LpRn and LqRn as the set of all measurable functions f satisfying f∈LlocpRn and fLp,LqαRn<∞, where
(12)fLp,LqαRn≔supr>0∫RnBy,r1/α−1/p−1/qf·χBy,rLpRnqdy1/q=supr>0By,r1/α−1/p−1/qf·χBy,r∥LpRn∥LqRn,with the usual modification when p=∞ or q=∞, and By,r is the Lebesgue measure of the ball By,r. It was shown in [4] that the space Lp,LqαRn is nontrivial if and only if p≤α≤q. Throughout this paper, we will always assume that the condition p≤α≤q is satisfied. Let us consider the following special cases:
(1)
If we takep=q, then p=α=q. It is easy to check that(13)By,r−1/pf·χBy,r∥LpRn∥LpRn=∫RnBy,r−1∫Rnfxp·χBy,rdxdy1/p=∫RnBy,r−1∫Bx,rfxpdydx1/p=∫Rnfxpdx1/p.Hence, the amalgam spaceLp,LqαRn is equal to the Lebesgue space LpRn with the same norms provided that p=α=q(2)
Ifq=∞, then we can see that the amalgam space Lp,LqαRn is equal to the Morrey space Lp,λRn with equivalent norms, where λ=pn/αIn this paper, we will consider the weighted version ofLp,LqαRn.Definition 3.
Let1≤p≤α≤q≤∞, and let ν, w, and μ be three weights on Rn. We denote by Lp,Lqαν,w;μ the weighted amalgam space, the space of all locally integrable functions f such that
(14)∥f∥Lp,Lqαν,w;μ≔supℓ>0∫RnwQy,ℓ1/α−1/p−1/qf·χQy,ℓLpνqμydy1/q=supℓ>0wQy,ℓ1/α−1/p−1/qf·χQy,ℓ∥Lpν∥Lqμ<∞,with wQy,ℓ=∫Qy,ℓwxdx and the usual modification when q=∞.Definition 4.
Let1≤p≤α≤q≤∞, and let w and μ be two weights on Rn. We denote by WLp,Lqαw;μ the weighted weak amalgam space consisting of all measurable functions f such that
(15)∥f∥WLp,Lqαw;μ≔supℓ>0∫RnwQy,ℓ1/α−1/p−1/q∥f·χQy,ℓ∥WLpwqμydy1/q=supℓ>0∥wQy,ℓ1/α−1/p−1/q∥f·χQy,ℓ∥WLpw∥Lqμ<∞,with wQy,ℓ=∫Qy,ℓwxdx and the usual modification when q=∞.Note that whenμ≡1, this kind of weighted (weak) amalgam space was introduced by Feuto in [5] (see also [6]). We remark that Feuto considered ball B instead of cube Q in his definition, but these two definitions are apparently equivalent. Also, note that when 1≤p≤α and q=∞, then Lp,Lqαν,w;μ is just the weighted Morrey space Lp,κν,w with κ=1−p/α, and WLp,Lqαw;μ is just the weighted weak Morrey space WLp,κw with κ=1−p/α.The two-weight problem for classical integral operators has been extensively studied. In [7–9] and [10], the authors gave some Ap-type conditions which are sufficient for the two-weight, weak-type p,p inequalities for Calderón–Zygmund operators, fractional integral operators, and their commutators on the weighted Lebesgue spaces. In [11], the authors established the two-weight, weak-type p,p estimates for the maximal Bochner–Riesz operators and their commutators. Inspired by the above results, it is natural and interesting to study the weak-type estimates for sublinear operators (1) and (2), as well as the corresponding commutators (3) and (4).Letp′ be the conjugate index of p whenever p>1; that is, 1/p+1/p′=1. The main purpose of this paper is to investigate the two-weight, weak-type norm inequalities in the setting of weighted Morrey and amalgam spaces. Our main results can be stated as follows. On the boundedness properties of the sublinear operators and their commutators on weighted Morrey spaces, we will prove the following.Theorem 5.
Let1<p<∞, 0<κ<1, and T satisfy condition (1). Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn,
(16)1Q∫Qwxrdx1/rp1Q∫Qνx−p′/pdx1/p′≤C<∞.
Furthermore, we suppose thatT satisfies the weak-type p,p inequality
(17)wx∈Rn:Tfx>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0. If w∈Δ2, then the operator T is bounded from Lp,κν,w into WLp,κw.Theorem 6.
Let1<p<∞, 0<κ<1, and Tγ satisfy condition (2) with 0<γ<n. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn,
(18)Qγ/n·1∣Q∣∫Qwxrdx1/rp1∣Q∣∫Qνx−p′/pdx1/p′≤C<∞.
Furthermore, we suppose thatTγ satisfies the weak-type p,p inequality
(19)wx∈Rn:∣Tγfx∣>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0. If w∈Δ2, then the operator Tγ is bounded from Lp,κν,w into WLp,κw.Theorem 7.
Let1<p<∞, 0<κ<1, b∈BMORn, and b,T satisfy condition (3). Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn,
(20)1∣Q∣∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′loge+tp′. Furthermore, we suppose that b,T satisfies the weak-type p,p inequality
(21)wx∈Rn:b,Tfx>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0. If w∈A∞, then the commutator operator b,T is bounded from Lp,κν,w into WLp,κw.Theorem 8.
Let1<p<∞, 0<κ<1, b∈BMORn, and b,Tγ satisfy condition (4) with 0<γ<n. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn,
(22)Qγ/n·1Q∫Qwxrdx1/rp∥ν−1/p∥A,Q≤C<∞,where At=tp′loge+tp′. Furthermore, we suppose that b,Tγ satisfies the weak-type p,p inequality
(23)wx∈Rn:b,Tγfx>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0. If w∈A∞, then the commutator operator b,Tγ is bounded from Lp,κν,w into WLp,κw.Concerning the boundedness properties on weighted amalgam spaces for these operators, we have the following results.Theorem 9.
Let1<p≤α<q≤∞ and μ∈Δ2. Given a pair of weights w,ν, assume that for some r>1 and for all cubes Q in Rn,
(24)1Q∫Qwxrdx1/rp1Q∫Qνx−p′/pdx1/p′≤C<∞.
Furthermore, we assume thatT satisfies the weak-type p,p inequality (17). If w∈Δ2, then the operator T is bounded from Lp,Lqαν,w;μ into WLp,Lqαw;μ.Theorem 10.
Let0<γ<n, 1<p≤α<q≤∞, and μ∈Δ2. Given a pair of weights w,ν, assume that for some r>1 and for all cubes Q in Rn,
(25)Qγ/n·1∣Q∣∫Qwxrdx1/rp1∣Q∣∫Qνx−p′/pdx1/p′≤C<∞.
Furthermore, we assume thatTγ satisfies the weak-type p,p inequality (19). If w∈Δ2, then the operator Tγ is bounded from Lp,Lqαν,w;μ into WLp,Lqαw;μ.Theorem 11.
Let1<p≤α<q≤∞, μ∈Δ2 and b∈BMORn. Given a pair of weights w,ν, assume that for some r>1 and for all cubes Q in Rn,
(26)1∣Q∣∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′loge+tp′. Furthermore, we assume that b,T satisfies the weak-type p,p inequality (21). If w∈A∞, then the commutator operator b,T is bounded from Lp,Lqαν,w;μ into WLp,Lqαw;μ.Theorem 12.
Let0<γ<n, 1<p≤α<q≤∞, μ∈Δ2, and b∈BMORn. Given a pair of weights w,ν, assume that for some r>1 and for all cubes Q in Rn,
(27)Qγ/n·1Q∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′loge+tp′. Furthermore, we assume that b,Tγ satisfies the weak-type p,p inequality (23). If w∈A∞, then the commutator operator b,Tγ is bounded from Lp,Lqαν,w;μ into WLp,Lqαw;μ.Remark 13.
It should be pointed out that the conclusions of our main theorems are natural generalizations of the corresponding weak-type estimates on the weighted Lebesgue spaces. The operators satisfying the assumptions of the above theorems include Calderón–Zygmund operators, Bochner–Riesz operators, and fractional integral operators. Hence, we are able to apply our main theorems to these classical integral operators.


## 1.1. Sublinear Operators

LetRn be the n-dimensional Euclidean space equipped with the Euclidean norm · and the Lebesgue measure dx. Suppose that T represents a linear or a sublinear operator, which satisfies that for any f∈L1Rn with compact support and x∉suppf,
(1)Tfx≤c1∫Rnfyx−yndy,where c1 is a universal constant independent of f and x∈Rn. The size condition (1) was first introduced by Soria and Weiss in [1]. It can be proved that (1) is satisfied by many integral operators in harmonic analysis, such as the Hardy–Littlewood maximal operator, Calderón–Zygmund singular integral operators, Ricci–Stein’s oscillatory singular integrals, and Bochner–Riesz operators at the critical index and so on.Similarly, for any given0<γ<n, we assume that Tγ represents a linear or a sublinear operator with order γ, which satisfies that for any f∈L1Rn with compact support and x∉suppf,
(2)Tγfx≤c2∫Rnfyx−yn−γdy,where c2 is also a universal constant independent of f and x∈Rn. It can be easily checked that (2) is satisfied by some important operators such as the fractional maximal operator, Riesz potential operators, and fractional oscillatory singular integrals.Letb be a locally integrable function on Rn; suppose that the commutator operator b,T stands for a linear or a sublinear operator, which satisfies that for any f∈L1Rn with compact support and x∉suppf,
(3)b,Tfx≤c3∫Rnbx−by·fyx−yndy,where c3 is an absolute constant independent of f and x∈Rn.Similarly, for any given0<γ<n, we assume that the commutator operator b,Tγ stands for a linear or a sublinear operator, which satisfies that for any f∈L1Rn with compact support and x∉suppf,
(4)b,Tγfx≤c4∫Rnbx−by·fyx−yn−γdy,where c4 is also an absolute constant independent of f and x∈Rn. Clearly, based on the above assumptions,
(5)T0=T,b,T0=b,T.


## 1.2. Weighted Morrey Spaces

The classical Morrey space was introduced by Morrey [2] in connection with elliptic partial differential equations. Let 1≤p<∞ and 0≤λ≤n. We recall that a real-valued function f is said to belong to the space Lp,λ on the n-dimensional Euclidean space Rn, if the following norm is finite:
(6)fLp,λ≔supx,r∈Rn×0,∞rλ−n∫Bx,rfypdy1/p,where Bx,r=y∈Rn:x−y<r is the Euclidean ball with center x∈Rn and radius r∈0,∞. In particular, one has
(7)Lp,0=L∞,Lp,n=Lp.In [3], Komori and Shirai considered the weighted case and gave the definitions of the weighted Morrey spaces as follows.Definition 1.
Let1<p<∞ and 0≤κ<1. For two weights w and ν on Rn, the weighted Morrey space Lp,κν,w is defined by
(8)Lp,κν,w≔f∈Llocpν:fLp,κν,w<∞,where the norm is given by
(9)fLp,κν,w≔supQ⊂Rn1wQκ∫Qfxpνxdx1/p,and the supremum is taken over all cubes Q in Rn.Definition 2.
Let1<p<∞, 0≤κ<1, and w be a weight on Rn. We define the weighted weak Morrey space WLp,κw as the set of all measurable functions f satisfying
(10)fWLp,κw≔supQ⊂Rnsupσ>01wQκ/pσ·wx∈Q:fx>σ1/p<∞.
By definition, it is clear that(11)Lp,0ν,w=Lpν,WLp,0w=WLpw.


## 1.3. Weighted Amalgam Spaces

Let1≤p,q≤∞; a function f∈LlocpRn is said to be in the Wiener amalgam space Lp,LqRn of LpRn and LqRn, if the function y↦f··χBy,1Lp belongs to LqRn, where By,1 is the open ball in Rn centered at y with radius 1, χBy,1 is the characteristic function of the ball By,1, and ·Lp is the usual Lebesgue norm in LpRn. In [4], Fofana introduced a new class of function spaces Lp,LqαRn which turn out to be the subspaces of Lp,LqRn. More precisely, for 1≤p,q,α≤∞, we define the amalgam space Lp,LqαRn of LpRn and LqRn as the set of all measurable functions f satisfying f∈LlocpRn and fLp,LqαRn<∞, where
(12)fLp,LqαRn≔supr>0∫RnBy,r1/α−1/p−1/qf·χBy,rLpRnqdy1/q=supr>0By,r1/α−1/p−1/qf·χBy,r∥LpRn∥LqRn,with the usual modification when p=∞ or q=∞, and By,r is the Lebesgue measure of the ball By,r. It was shown in [4] that the space Lp,LqαRn is nontrivial if and only if p≤α≤q. Throughout this paper, we will always assume that the condition p≤α≤q is satisfied. Let us consider the following special cases:
(1)
If we takep=q, then p=α=q. It is easy to check that(13)By,r−1/pf·χBy,r∥LpRn∥LpRn=∫RnBy,r−1∫Rnfxp·χBy,rdxdy1/p=∫RnBy,r−1∫Bx,rfxpdydx1/p=∫Rnfxpdx1/p.Hence, the amalgam spaceLp,LqαRn is equal to the Lebesgue space LpRn with the same norms provided that p=α=q(2)
Ifq=∞, then we can see that the amalgam space Lp,LqαRn is equal to the Morrey space Lp,λRn with equivalent norms, where λ=pn/αIn this paper, we will consider the weighted version ofLp,LqαRn.Definition 3.
Let1≤p≤α≤q≤∞, and let ν, w, and μ be three weights on Rn. We denote by Lp,Lqαν,w;μ the weighted amalgam space, the space of all locally integrable functions f such that
(14)∥f∥Lp,Lqαν,w;μ≔supℓ>0∫RnwQy,ℓ1/α−1/p−1/qf·χQy,ℓLpνqμydy1/q=supℓ>0wQy,ℓ1/α−1/p−1/qf·χQy,ℓ∥Lpν∥Lqμ<∞,with wQy,ℓ=∫Qy,ℓwxdx and the usual modification when q=∞.Definition 4.
Let1≤p≤α≤q≤∞, and let w and μ be two weights on Rn. We denote by WLp,Lqαw;μ the weighted weak amalgam space consisting of all measurable functions f such that
(15)∥f∥WLp,Lqαw;μ≔supℓ>0∫RnwQy,ℓ1/α−1/p−1/q∥f·χQy,ℓ∥WLpwqμydy1/q=supℓ>0∥wQy,ℓ1/α−1/p−1/q∥f·χQy,ℓ∥WLpw∥Lqμ<∞,with wQy,ℓ=∫Qy,ℓwxdx and the usual modification when q=∞.Note that whenμ≡1, this kind of weighted (weak) amalgam space was introduced by Feuto in [5] (see also [6]). We remark that Feuto considered ball B instead of cube Q in his definition, but these two definitions are apparently equivalent. Also, note that when 1≤p≤α and q=∞, then Lp,Lqαν,w;μ is just the weighted Morrey space Lp,κν,w with κ=1−p/α, and WLp,Lqαw;μ is just the weighted weak Morrey space WLp,κw with κ=1−p/α.The two-weight problem for classical integral operators has been extensively studied. In [7–9] and [10], the authors gave some Ap-type conditions which are sufficient for the two-weight, weak-type p,p inequalities for Calderón–Zygmund operators, fractional integral operators, and their commutators on the weighted Lebesgue spaces. In [11], the authors established the two-weight, weak-type p,p estimates for the maximal Bochner–Riesz operators and their commutators. Inspired by the above results, it is natural and interesting to study the weak-type estimates for sublinear operators (1) and (2), as well as the corresponding commutators (3) and (4).Letp′ be the conjugate index of p whenever p>1; that is, 1/p+1/p′=1. The main purpose of this paper is to investigate the two-weight, weak-type norm inequalities in the setting of weighted Morrey and amalgam spaces. Our main results can be stated as follows. On the boundedness properties of the sublinear operators and their commutators on weighted Morrey spaces, we will prove the following.Theorem 5.
Let1<p<∞, 0<κ<1, and T satisfy condition (1). Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn,
(16)1Q∫Qwxrdx1/rp1Q∫Qνx−p′/pdx1/p′≤C<∞.
Furthermore, we suppose thatT satisfies the weak-type p,p inequality
(17)wx∈Rn:Tfx>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0. If w∈Δ2, then the operator T is bounded from Lp,κν,w into WLp,κw.Theorem 6.
Let1<p<∞, 0<κ<1, and Tγ satisfy condition (2) with 0<γ<n. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn,
(18)Qγ/n·1∣Q∣∫Qwxrdx1/rp1∣Q∣∫Qνx−p′/pdx1/p′≤C<∞.
Furthermore, we suppose thatTγ satisfies the weak-type p,p inequality
(19)wx∈Rn:∣Tγfx∣>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0. If w∈Δ2, then the operator Tγ is bounded from Lp,κν,w into WLp,κw.Theorem 7.
Let1<p<∞, 0<κ<1, b∈BMORn, and b,T satisfy condition (3). Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn,
(20)1∣Q∣∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′loge+tp′. Furthermore, we suppose that b,T satisfies the weak-type p,p inequality
(21)wx∈Rn:b,Tfx>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0. If w∈A∞, then the commutator operator b,T is bounded from Lp,κν,w into WLp,κw.Theorem 8.
Let1<p<∞, 0<κ<1, b∈BMORn, and b,Tγ satisfy condition (4) with 0<γ<n. Given a pair of weights w,ν, suppose that for some r>1 and for all cubes Q in Rn,
(22)Qγ/n·1Q∫Qwxrdx1/rp∥ν−1/p∥A,Q≤C<∞,where At=tp′loge+tp′. Furthermore, we suppose that b,Tγ satisfies the weak-type p,p inequality
(23)wx∈Rn:b,Tγfx>σ≤Cσp∫Rnfxpνxdx,foranyσ>0,where C does not depend on f and σ>0. If w∈A∞, then the commutator operator b,Tγ is bounded from Lp,κν,w into WLp,κw.Concerning the boundedness properties on weighted amalgam spaces for these operators, we have the following results.Theorem 9.
Let1<p≤α<q≤∞ and μ∈Δ2. Given a pair of weights w,ν, assume that for some r>1 and for all cubes Q in Rn,
(24)1Q∫Qwxrdx1/rp1Q∫Qνx−p′/pdx1/p′≤C<∞.
Furthermore, we assume thatT satisfies the weak-type p,p inequality (17). If w∈Δ2, then the operator T is bounded from Lp,Lqαν,w;μ into WLp,Lqαw;μ.Theorem 10.
Let0<γ<n, 1<p≤α<q≤∞, and μ∈Δ2. Given a pair of weights w,ν, assume that for some r>1 and for all cubes Q in Rn,
(25)Qγ/n·1∣Q∣∫Qwxrdx1/rp1∣Q∣∫Qνx−p′/pdx1/p′≤C<∞.
Furthermore, we assume thatTγ satisfies the weak-type p,p inequality (19). If w∈Δ2, then the operator Tγ is bounded from Lp,Lqαν,w;μ into WLp,Lqαw;μ.Theorem 11.
Let1<p≤α<q≤∞, μ∈Δ2 and b∈BMORn. Given a pair of weights w,ν, assume that for some r>1 and for all cubes Q in Rn,
(26)1∣Q∣∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′loge+tp′. Furthermore, we assume that b,T satisfies the weak-type p,p inequality (21). If w∈A∞, then the commutator operator b,T is bounded from Lp,Lqαν,w;μ into WLp,Lqαw;μ.Theorem 12.
Let0<γ<n, 1<p≤α<q≤∞, μ∈Δ2, and b∈BMORn. Given a pair of weights w,ν, assume that for some r>1 and for all cubes Q in Rn,
(27)Qγ/n·1Q∫Qwxrdx1/rpν−1/pA,Q≤C<∞,where At=tp′loge+tp′. Furthermore, we assume that b,Tγ satisfies the weak-type p,p inequality (23). If w∈A∞, then the commutator operator b,Tγ is bounded from Lp,Lqαν,w;μ into WLp,Lqαw;μ.Remark 13.
It should be pointed out that the conclusions of our main theorems are natural generalizations of the corresponding weak-type estimates on the weighted Lebesgue spaces. The operators satisfying the assumptions of the above theorems include Calderón–Zygmund operators, Bochner–Riesz operators, and fractional integral operators. Hence, we are able to apply our main theorems to these classical integral operators.


## 2. Notations and Definitions

### 2.1. Weights

A nonnegative functionw defined on Rn will be called a weight if it is locally integrable. Qx0,ℓ will denote the cube centered at x0 and has side length ℓ>0; all cubes are assumed to have their sides parallel to the coordinate axes. Given a cube Q=Qx0,ℓ and λ>0, λQ stands for the cube concentric with Q having side length λn times as long, i.e., λQx0,ℓ≔Qx0,λnℓ. For any given weight w and any Lebesgue measurable set E of Rn, we denote the characteristic function of E by χE, the Lebesgue measure of E by E, and the weighted measure of E by wE, where wE≔∫Ewxdx. We also denote Ec≔Rn\E the complement of E. Given a weight w, we say that w satisfies the doubling condition, if there exists a universal constant C>0 such that for any cube Q⊂Rn, we have
(28)w2Q≤C·wQ.Whenw satisfies condition (28), we denote w∈Δ2 for brevity. A weight w is said to belong to Muckenhoupt’s class Ap for 1<p<∞, if there exists a constant C>0 such that
(29)1Q∫Qwxdx1/p1Q∫Qwx−p′/pdx1/p′≤C,holds for every cube Q⊂Rn. The class A∞ is defined as the union of the Ap classes for 1<p<∞, i.e., A∞=⋃1<p<∞Ap. If w is an A∞ weight, then we have w∈Δ2 (see [12]). Moreover, this class A∞ is characterized as the class of all weights satisfying the following property: there exists a number δ>0 and a finite constant C>0 such that (see [12])
(30)wEwQ≤CEQδ,holds for every cube Q⊂Rn and all measurable subsets E of Q. Given a weight w on Rn and for 1≤p<∞, the weighted Lebesgue space Lpw is defined as the set of all measurable functions f such that
(31)∥f∥Lpw≔∫Rnfxpwxdx1/p<∞.We also define the weighted weak Lebesgue spaceWLpw (1≤p<∞) as the set of all measurable functions f satisfying
(32)∥f∥WLpw≔supσ>0σ·wx∈Rn:fx>σ1/p<∞.


### 2.2. Orlicz Spaces and BMO

We next recall some basic facts about Orlicz spaces needed for the proofs of the main results. For further details, we refer the reader to [13]. 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⟶+∞. Given a Young function A, we define the A-average of a function f over a cube Q by means of the following Luxemburg norm:
(33)fA,Q≔infλ>0:1∣Q∣∫QAfxλdx≤1.In particular, whenAt=tp, 1<p<∞, it is easy to check that
(34)fA,Q=1∣Q∣∫Qfxpdx1/p;that is, the Luxemburg norm coincides with the normalized Lp norm. The main examples that we are going to consider are At=tploge+tp with 1<p<∞.Let us now recall the definition of the space ofBMORn (see [14]). A locally integrable function b is said to be in BMORn, if
(35)b∗≔supQ⊂Rn1Q∫Qbx−bQdx<∞,where bQ denotes the mean value of b over Q, namely,
(36)bQ≔1Q∫Qbydy,and the supremum is taken over all cubes Q in Rn. After identifying functions that differ by a constant, the space BMORn becomes a Banach space.Throughout this paper,C always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. We will use c1,⋯,c4 appearing in the first section of this paper to denote certain constants. 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 and B such that C1A≤B≤C2A.


## 2.1. Weights

A nonnegative functionw defined on Rn will be called a weight if it is locally integrable. Qx0,ℓ will denote the cube centered at x0 and has side length ℓ>0; all cubes are assumed to have their sides parallel to the coordinate axes. Given a cube Q=Qx0,ℓ and λ>0, λQ stands for the cube concentric with Q having side length λn times as long, i.e., λQx0,ℓ≔Qx0,λnℓ. For any given weight w and any Lebesgue measurable set E of Rn, we denote the characteristic function of E by χE, the Lebesgue measure of E by E, and the weighted measure of E by wE, where wE≔∫Ewxdx. We also denote Ec≔Rn\E the complement of E. Given a weight w, we say that w satisfies the doubling condition, if there exists a universal constant C>0 such that for any cube Q⊂Rn, we have
(28)w2Q≤C·wQ.Whenw satisfies condition (28), we denote w∈Δ2 for brevity. A weight w is said to belong to Muckenhoupt’s class Ap for 1<p<∞, if there exists a constant C>0 such that
(29)1Q∫Qwxdx1/p1Q∫Qwx−p′/pdx1/p′≤C,holds for every cube Q⊂Rn. The class A∞ is defined as the union of the Ap classes for 1<p<∞, i.e., A∞=⋃1<p<∞Ap. If w is an A∞ weight, then we have w∈Δ2 (see [12]). Moreover, this class A∞ is characterized as the class of all weights satisfying the following property: there exists a number δ>0 and a finite constant C>0 such that (see [12])
(30)wEwQ≤CEQδ,holds for every cube Q⊂Rn and all measurable subsets E of Q. Given a weight w on Rn and for 1≤p<∞, the weighted Lebesgue space Lpw is defined as the set of all measurable functions f such that
(31)∥f∥Lpw≔∫Rnfxpwxdx1/p<∞.We also define the weighted weak Lebesgue spaceWLpw (1≤p<∞) as the set of all measurable functions f satisfying
(32)∥f∥WLpw≔supσ>0σ·wx∈Rn:fx>σ1/p<∞.


## 2.2. Orlicz Spaces and BMO

We next recall some basic facts about Orlicz spaces needed for the proofs of the main results. For further details, we refer the reader to [13]. 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⟶+∞. Given a Young function A, we define the A-average of a function f over a cube Q by means of the following Luxemburg norm:
(33)fA,Q≔infλ>0:1∣Q∣∫QAfxλdx≤1.In particular, whenAt=tp, 1<p<∞, it is easy to check that
(34)fA,Q=1∣Q∣∫Qfxpdx1/p;that is, the Luxemburg norm coincides with the normalized Lp norm. The main examples that we are going to consider are At=tploge+tp with 1<p<∞.Let us now recall the definition of the space ofBMORn (see [14]). A locally integrable function b is said to be in BMORn, if
(35)b∗≔supQ⊂Rn1Q∫Qbx−bQdx<∞,where bQ denotes the mean value of b over Q, namely,
(36)bQ≔1Q∫Qbydy,and the supremum is taken over all cubes Q in Rn. After identifying functions that differ by a constant, the space BMORn becomes a Banach space.Throughout this paper,C always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. We will use c1,⋯,c4 appearing in the first section of this paper to denote certain constants. 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 and B such that C1A≤B≤C2A.


## 3. Proofs of Theorems5 and 6

Proof of Theorem 5.
Letf∈Lp,κν,w with 1<p<∞ and 0<κ<1. For an arbitrary fixed cube Q=Qx0,ℓ⊂Rn, we set 2Q≔Qx0,2nℓ. Decompose f as
(37)f=f1+f2∈Lp,κν,w,f1=f·χ2Q,f2=f·χ2Qc,where χE denotes the characteristic function of the set E. Then, for any given σ>0, we write
(38)1wQκ/pσ·wx∈Q:Tfx>σ1/p≤1wQκ/pσ·wx∈Q:Tf1x>σ21/p+1wQκ/pσ·wx∈Q:Tf2x>σ21/p≔I1+I2.
We first consider the termI1. Using assumption (17) and the condition w∈Δ2, we get
(39)I1≤C·1wQκ/p∫Rnf1xpνxdx1/p=C·1wQκ/p∫2Qfxpνxdx1/p≤C∥f∥Lp,κν,w·w2Qκ/pwQκ/p≤C∥f∥Lp,κν,w.
This is just our desired estimate. Let us estimate the second termI2. To this end, we observe that when x∈Q and y∈2Qc, one has x−y≈x0−y. We then decompose Rn into a geometrically increasing sequence of concentric cubes and obtain the following pointwise estimate by condition (1).
(40)Tf2x≤c1∫Rnf2yx−yndy≤C∫2Qcfyx0−yndy=C∑j=1∞∫2j+1Q\2jQ∣fy∣x0−yndy≤C∑j=1∞12j+1Q∫2j+1Qfydy.
This pointwise estimate (40) together with Chebyshev’s inequality implies that
(41)I2≤2wQκ/p·∫QTf2xpwxdx1/p≤C·wQ1−κ/p∑j=1∞12j+1Q∫2j+1Qfydy.
It follows directly from Hölder’s inequality with exponentp>1 that
(42)I2≤C·wQ1−κ/p∑j=1∞12j+1Q∫2j+1Qfypνydy1/p×∫2j+1Qνy−p′/pdy1/p′≤C∥f∥Lp,κν,w·wQ1−κ/p∑j=1∞w2j+1Qκ/p2j+1Q×∫2j+1Qνy−p′/pdy1/p′=C∥f∥Lp,κν,w∑j=1∞wQ1−κ/pw2j+1Q1−κ/p·w2j+1Q1/p∣2j+1Q∣×∫2j+1Qνy−p′/pdy1/p′.
Moreover, for any positive integerj, we apply Hölder’s inequality again with exponent r>1 to get
(43)w2j+1Q=∫2j+1Qwydy≤2j+1Q1/r′∫2j+1Qwyrdy1/r.
Thus, in view of (43), we conclude that
(44)I2≤CfLp,κν,w∑j=1∞wQ1−κ/pw2j+1Q1−κ/p·2j+1Q1/r′p2j+1Q×∫2j+1Qwyrdy1/rp∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w×∑j=1∞wQ1−κ/pw2j+1Q1−κ/p.
The last inequality is obtained by theAp-type condition (16) on w,ν. Since w∈Δ2, we can easily check that there exists a reverse doubling constantD=Dw>1 independent of Q such that (see [3], Lemma 4.1)
(45)w2Q≥D·wQ,forallcubesQ⊂Rn,which implies that for any positive integer j,
(46)w2j+1Q≥Dj+1·wQ,by iteration. Hence,
(47)∑j=1∞wQ1−κ/pw2j+1Q1−κ/p≤∑j=1∞wQDj+1·wQ1−κ/p=∑j=1∞1Dj+11−κ/p≤C,where the last series is convergent since the reverse doubling constantD>1 and 0<κ<1. This yields our desired estimate
(48)I2≤CfLp,κν,w.
Summing up the above estimates forI1 and I2, and then taking the supremum over all cubes Q⊂Rn and all σ>0, we finish the proof of Theorem 5.Proof of Theorem 6.
Letf∈Lp,κν,w with 1<p<∞ and 0<κ<1. For an arbitrary fixed cube Q=Qx0,ℓ in Rn, we decompose f as
(49)f=f1+f2∈Lp,κν,w,f1=f·χ2Q,f2=f·χ2Qc,where 2Q≔Qx0,2nℓ. For any given σ>0, we then write
(50)1wQκ/pσ·wx∈Q:∣Tγfx∣>σ1/p≤1wQκ/pσ·wx∈Q:Tγf1x>σ21/p+1wQκ/pσ·wx∈Q:Tγf2x>σ21/p≔I′1+I′2.
Let us consider the first termI′1. Using assumption (19) and the condition w∈Δ2, we have
(51)I′1≤C·1wQκ/p∫Rnf1xpνxdx1/p=C·1wQκ/p∫2Qfxpνxdx1/p≤C∥f∥Lp,κν,w·w2Qκ/pwQκ/p≤C∥f∥Lp,κν,w.
This is exactly what we want. We now deal with the second termI′2. Note that x−y≈x0−y, whenever x,x0∈Q and y∈2Qc. For 0<γ<n and all x∈Q, using the standard technique and condition (2), we can see that
(52)Tγf2x≤c2∫Rnf2yx−yn−γdy≤C∫2Qcfyx0−yn−γdy=C∑j=1∞∫2j+1Q\2jQfyx0−yn−γdy≤C∑j=1∞12j+1Q1−γ/n∫2j+1Qfydy.
This pointwise estimate (52) together with Chebyshev’s inequality yields
(53)I′2≤2wQκ/p·∫QTγf2xpwxdx1/p≤C·wQ1−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qfydy.
By using Hölder’s inequality with exponentp>1, we can deduce that
(54)I′2≤C·wQ1−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qfypνydy1/p×∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w·wQ1−κ/p∑j=1∞w2j+1Qκ/p2j+1Q1−γ/n×∫2j+1Qνy−p′/pdy1/p′=CfLp,κν,w∑j=1∞wQ1−κ/pw2j+1Q1−κ/p·w2j+1Q1/p2j+1Q1−γ/n×∫2j+1Qνy−p′/pdy1/p′.
Moreover, we apply estimate (43) to get
(55)I′2≤CfLp,κν,w∑j=1∞wQ1−κ/pw2j+1Q1−κ/p·2j+1Q1/r′p2j+1Q×2j+1Qγ/n·∫2j+1Qwyrdy1/rp∫2j+1Qνy−p′/pdy1/p′≤C∥f∥Lp,κν,w×∑j=1∞wQ1−κ/pw2j+1Q1−κ/p.
The last inequality is obtained by theAp-type condition (18) on w,ν. Therefore, in view of (47), we find that
(56)I′2≤CfLp,κν,w.
Combining the above estimates forI′1 and I′2, and then taking the supremum over all cubes Q⊂Rn and all σ>0, we complete the proof of Theorem 6.


## 4. Proofs of Theorems7 and 8

For the results involving commutators, we need the following properties ofBMORn, which can be found in [15].Lemma 14.
Letb be a function in BMORn.
(i)
For every cubeQ in Rn and for any positive integer j, then(57)b2j+1Q−bQ≤C·j+1b∗.(ii)
Let1<p<∞. For every cube Q in Rn and for any w∈A∞, then(58)∫Qbx−bQpwxdx1/p≤Cb∗·wQ1/p.Before proving our main theorems, we will also need a generalization of Hölder’s inequality due to O’Neil [16].Lemma 15.
LetA, B, and C be Young functions such that for all t>0,
(59)A−1t·B−1t≤C−1t,where A−1t is the inverse function of At. Then, for all functions f and g and all cubes Q in Rn,
(60)f·gC,Q≤2fA,Q∥g∥B,Q.We are now ready to give the proofs of Theorems7 and 8.Proof of Theorem 7.
Letf∈Lp,κν,w with 1<p<∞ and 0<κ<1. For any given cube Q=Qx0,ℓ⊂Rn, we split f as usual by
(61)f=f·χ2Q+f·χ2Qc≔f1+f2,where 2Q≔Qx0,2nℓ. Then, for any given σ>0, one writes
(62)1wQκ/pσ·wx∈Q:b,Tfx>σ1/p≤1wQκ/pσ·wx∈Q:b,Tf1x>σ21/p+1wQκ/pσ·wx∈Q:b,Tf2x>σ21/p≔J1+J2.
Sincew is an A∞ weight, we know that w∈Δ2. By our assumption (21) and inequality (28), we have
(63)J1≤C·1wQκ/p∫Rnf1xpνxdx1/p=C·1wQκ/p∫2Qfxpνxdx1/p≤CfLp,κν,w·w2Qκ/pwQκ/p≤CfLp,κν,w,which is exactly what we want. For any x∈Q, from the size condition (3), we can easily see that
(64)b,Tf2x≤c3∫Rnbx−by·∣f2y∣x−yndy≤c3bx−bQ·∫Rnf2yx−yndy+c3∫Rnby−bQ·∣f2y∣x−yndy≔ξx+ηx.
Hence, we can further splitJ2 into two parts as follows:
(65)J2≤1wQκ/pσ·wx∈Q:ξx>σ41/p+1wQκ/pσ·wx∈Q:ηx>σ41/p≔J3+J4.
For the termJ3, it follows from the pointwise estimate (40) and Chebyshev’s inequality that
(66)J3≤4wQκ/p·∫Qξxpwxdx1/p≤CwQκ/p·∫Qbx−bQpwxdx1/p∑j=1∞12j+1Q∫2j+1Qfydy≤Cb∗·wQ1−κ/p∑j=1∞12j+1Q∫2j+1Qfydy,where in the last inequality, we have used the second part of Lemma 14 since w∈A∞. Repeating the arguments used in Theorem 5, we can also prove that
(67)J3≤CfLp,κν,w.
Let us consider the termJ4. Similar to the proof of (40), for any given x∈Q, we can obtain the following pointwise estimate as well.
(68)ηx=c3∫Rnby−bQ·f2yx−yndy≤C∫2Qcby−bQ·fyx0−yndy≤C∑j=1∞12j+1Q∫2j+1Qby−bQ·fydy.
This, together with Chebyshev’s inequality, yields(69)J4≤4wQκ/p·∫Qηxpwxdx1/p≤C·wQ1−κ/p·∑j=1∞12j+1Q∫2j+1Qby−bQ·fydy≤C·wQ1−κ/p·∑j=1∞12j+1Q∫2j+1Qby−b2j+1Q·fydy+C·wQ1−κ/p·∑j=1∞12j+1Q∫2j+1Qb2j+1Q−bQ·fydy≔J5+J6.
An application of Hölder’s inequality leads to(70)J5≤C·wQ1−κ/p·∑j=1∞12j+1Q∫2j+1Qfypνydy1/p×∫2j+1Qby−b2j+1Qp′νy−p′/pdy1/p′≤C∥f∥Lp,κν,w·wQ1−κ/p∑j=1∞w2j+1Qκ/p2j+1Q×2j+1Q1/p′b−b2j+1Q·ν−1/pC,2j+1Q,where Ct=tp′ is a Young function by (34). For 1<p<∞, it is immediate that the inverse function of Ct is C−1t=t1/p′. Also, observe that the following identity is true:
(71)C−1t=t1/p′=t1/p′loge+t×loge+t=A−1t·B−1t,where
(72)At≈tp′loge+tp′,Bt≈expt−1.
LethexpL,Q denote the mean Luxemburg norm of h on cube Q with Young function Bt≈expt−1. Thus, by Lemma 15, we have
(73)b−b2j+1Q·ν−1/pC,2j+1Q≤Cb−b2j+1QexpL,2j+1Q·ν−1/pA,2j+1Q≤Cb∗·ν−1/pA,2j+1Q,where in the last inequality, we have used the well-known fact that (see [15])
(74)b−bQexpL,Q≤Cb∗,foranycubeQ⊂Rn.
This is equivalent to the following inequality:(75)1Q∫Qexpby−bQc0b∗dy≤C,foranycubeQ⊂Rn,which is just a corollary of the celebrated John–Nirenberg’s inequality (see [14]). Consequently,
(76)J5≤Cb∗fLp,κν,w∑j=1∞wQ1−κ/pw2j+1Q1−κ/p·w2j+1Q1/p2j+1Q1/p·ν−1/pA,2j+1Q.
Moreover, in view of (43), we can deduce that
(77)J5≤CfLp,κν,w∑j=1∞wQ1−κ/pw2j+1Q1−κ/p×12j+1Q∫2j+1Qwxrdx1/rp·ν−1/pA,2j+1Q≤CfLp,κν,w×∑j=1∞wQ1−κ/pw2j+1Q1−κ/p≤CfLp,κν,w.
The last inequality is obtained by theAp-type condition (20) on w,ν and estimate (47). It remains to estimate the last term J6. Applying the first part of Lemma 14 and Hölder’s inequality, we get
(78)J6≤C·wQ1−κ/p·∑j=1∞j+1b∗2j+1Q∫2j+1Qfydy≤C·wQ1−κ/p·∑j=1∞j+1b∗2j+1Q∫2j+1Qfypνydy1/p×∫2j+1Qνy−p′/pdy1/p′≤C∥f∥Lp,κν,w·wQ1−κ/p∑j=1∞j+1·w2j+1Qκ/p2j+1Q×∫2j+1Qνy−p′/pdy1/p′=CfLp,κν,w∑j=1∞j+1·wQ1−κ/pw2j+1Q1−κ/p·w2j+1Q1/p2j+1Q×∫2j+1Qνy−p′/pdy1/p′.
LetCt and At be the same as before. Obviously, Ct≤At for all t>0, then for any cube Q in Rn, one has fC,Q≤fA,Q by definition, which implies that condition (20) is stronger than condition (16). This fact together with (43) yields
(79)J6≤CfLp,κν,w∑j=1∞j+1·wQ1−κ/pw2j+1Q1−κ/p·2j+1Q1/r′p2j+1Q×∫2j+1Qwyrdy1/rp∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w∑j=1∞j+1·wQ1−κ/pw2j+1Q1−κ/p.
Moreover, by our hypothesis onw:w∈A∞ and inequality (30), we compute
(80)∑j=1∞j+1·wQ1−κ/pw2j+1Q1−κ/p≤C∑j=1∞j+1·∣Q∣∣2j+1Q∣δ1−κ/p≤C∑j=1∞j+1·12j+1nδ1−κ/p≤C,where the last series is convergent since the exponent δ1−κ/p is positive. This implies our desired estimate
(81)J6≤CfLp,κν,w.
Summing up all the above estimates, and then taking the supremum over all cubesQ⊂Rn and all σ>0, we conclude the proof of Theorem 7.Proof of Theorem 8.
Letf∈Lp,κν,w with 1<p<∞ and 0<κ<1. For any given cube Q=Qx0,ℓ⊂Rn, as before, we set
(82)f=f1+f2,f1=f·χ2Q,f2=f·χ2Qc,where 2Q≔Qx0,2nℓ. Then, for any given σ>0, one writes
(83)1wQκ/pσ·wx∈Q:b,Tγfx>σ1/p≤1wQκ/pσ·wx∈Q:b,Tγf1x>σ21/p+1wQκ/pσ·wx∈Q:b,Tγf2x>σ21/p≔J′1+J′2.
Sincew is an A∞ weight, then we have w∈Δ2. From our assumption (23) and inequality (28), it follows that
(84)J′1≤C·1wQκ/p∫Rnf1xpνxdx1/p=C·1wQκ/p∫2Qfxpνxdx1/p≤CfLp,κν,w·w2Qκ/pwQκ/p≤CfLp,κν,w.
On the other hand, for anyx∈Q, from the size condition (4), it then follows that
(85)b,Tγf2x≤c4∫Rnbx−by·f2yx−yn−γdy≤c4bx−bQ·∫Rnf2yx−yn−γdy+c4∫Rnby−bQ·f2yx−yn−γdy≔ξ~x+η~x.
Thus, we can further splitJ′2 into two parts as follows:
(86)J′2≤1wQκ/pσ·wx∈Q:ξ~x>σ41/p+1wQκ/pσ·wx∈Q:η~x>σ41/p≔J′3+J′4.
Using the pointwise estimate (52) and Chebyshev’s inequality, we obtain that
(87)J3′≤4wQκ/p·∫Qξ~xpwxdx1/p≤CwQκ/p·∫Qbx−bQpwxdx1/p∑j=1∞12j+1Q1−γ/n∫2j+1Qfydy≤Cb∗·wQ1−κ/p∑j=1∞12j+1Q1−γ/n∫2j+1Qfydy,where the last inequality is due to w∈A∞ and Lemma 14(ii). By using the same arguments as that of Theorem 6, we can also show that
(88)J′3≤CfLp,κν,w.
Similar to the proof of (52), for all x∈Q, we can show the following pointwise inequality as well.
(89)η~x=c4∫Rnby−bQ·f2yx−yn−γdy≤C∫2Qcby−bQ·fyx0−yn−γdy≤C∑j=1∞12j+1Q1−γ/n∫2j+1Qby−bQ·fydy.
This, together with Chebyshev’s inequality, yields(90)J4′≤4wQκ/p·∫Qη~xpwxdx1/p≤C·wQ1−κ/p·∑j=1∞12j+1Q1−γ/n∫2j+1Qby−bQ·fydy≤C·wQ1−κ/p·∑j=1∞12j+1Q1−γ/n∫2j+1Qby−b2j+1Q·fydy+C·wQ1−κ/p·∑j=1∞12j+1Q1−γ/n∫2j+1Qb2j+1Q−bQ·fydy≔J′5+J′6.
An application of Hölder’s inequality leads to(91)J′5≤C·wQ1−κ/p·∑j=1∞12j+1Q∣1−γ/n∫2j+1Qfypνydy1/p×∫2j+1Qby−b2j+1Qp′νy−p′/pdy1/p′≤CfLp,κν,w·wQ1−κ/p∑j=1∞w2j+1Qκ/p2j+1Q1−γ/n×2j+1Q1/p′b−b2j+1Q·ν−1/pC,2j+1Q,where Ct=tp′ is a Young function. Let Bt and At be the same as in Theorem 7. In view of (73) and (43), we can deduce that
(92)J′5≤CfLp,κν,w∑j=1∞wQ1−κ/pw2j+1Q1−κ/p·w2j+1Q1/p2j+1Q1/p−γ/n·b∗ν−1/pA,2j+1Q≤Cb∗fLp,κν,w∑j=1∞wQ1−κ/pw2j+1Q1−κ/p×2j+1Qγ/n·12j+1Q∫2j+1Qwxrdx1/rp·ν−1/pA,2j+1Q.
Furthermore, by theAp-type condition (22) on w,ν and estimate (47), we obtain
(93)J′5≤CfLp,κν,w×∑j=1∞wQ1−κ/pw2j+1Q1−κ/p≤CfLp,κν,w.
It remains to estimate the last termJ′6. Making use of the first part of Lemma 14 and Hölder’s inequality, we get
(94)J′6≤C·wQ1−κ/p·∑j=1∞j+1b∗2j+1Q1−γ/n∫2j+1Qfydy≤C·wQ1−κ/p·∑j=1∞j+1b∗2j+1Q1−γ/n∫2j+1Qfypνydy1/p×∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w·wQ1−κ/p∑j=1∞j+1·w2j+1Qκ/p2j+1Q1−γ/n×∫2j+1Qνy−p′/pdy1/p′=CfLp,κν,w∑j=1∞j+1·wQ1−κ/pw2j+1Q1−κ/p·w2j+1Q1/p2j+1Q1−γ/n×∫2j+1Qνy−p′/pdy1/p′.
It was pointed out in Theorem7 that for any cube Q in Rn, one has fC,Q≤fA,Q, where Ct=tp′ and At≈tp′loge+tp′. This implies that condition (22) is stronger than condition (20). Using this fact along with (43), we can see that
(95)J′6≤CfLp,κν,w∑j=1∞j+1·wQ1−κ/pw2j+1Q1−κ/p·2j+1Q1/r′p2j+1Q1−γ/n×∫2j+1Qwyrdy1/rp∫2j+1Qνy−p′/pdy1/p′≤CfLp,κν,w∑j=1∞j+1·wQ1−κ/pw2j+1Q1−κ/p≤CfLp,κν,w,where the last inequality follows from estimate (80). Summarizing the estimates derived above, and then taking the supremum over all cubes Q⊂Rn and all σ>0, we therefore conclude the proof of Theorem 8.


## 5. Proofs of Theorems9 and 10

Proof of Theorem 9.
Let1<p≤α<q≤∞ and f∈Lp,Lqαν,w;μ with w∈Δ2 and μ∈Δ2. For any cube Q=Qy,ℓ⊂Rn with y∈Rn and ℓ>0, we denote by λQ the cube concentric with Q whose each edge is λn times as long; that is, λQ=Qy,λnℓ. Decompose f as
(96)f=f1+f2∈Lp,Lqαν,w;μ,f1=f·χ2Q,f2=f·χ2Qc,where χ2Q denotes the characteristic function of 2Q=Qy,2nℓ. Then, for any given y∈Rn and ℓ>0, we write
(97)wQy,ℓ1/α−1/p−1/qTf·χQy,ℓWLpw≤2·wQy,ℓ1/α−1/p−1/qTf1·χQy,ℓWLpw+2·wQy,ℓ1/α−1/p−1/qTf2·χQy,ℓWLpw≔K1y,ℓ+K2y,ℓ.
Let us consider the first termK1y,ℓ. In view of (17), we get
(98)K1y,ℓ≤2·wQy,ℓ1/α−1/p−1/qTf1WLpw≤C·wQy,ℓ1/α−1/p−1/q∫Qy,2nℓfxpνxdx1/p=C·wQy,2nℓ1/α−1/p−1/q∥f·χQy,2nℓ∥Lpν×wQy,ℓ1/α−1/p−1/qwQy,2nℓ1/α−1/p−1/q.
Moreover, since1/α−1/p−1/q<0 and w∈Δ2, then by doubling inequality (28), we obtain
(99)wQy,ℓ1/α−1/p−1/qwQy,2nℓ1/α−1/p−1/q≤C.
Substituting the above inequality (99) into (98), we thus obtain
(100)K1y,ℓ≤C·wQy,2nℓ1/α−1/p−1/qf·χQy,2nℓLpν.
As for the second termK2y,ℓ, recall that by the size condition (1), the following inequality holds for any x∈Qy,ℓ:
(101)Tf2x≤C∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓfzdz.
This pointwise estimate (101) together with Chebyshev’s inequality yields
(102)K2y,ℓ≤2·wQy,ℓ1/α−1/p−1/q∫Qy,ℓTf2xpwxdx1/p≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓfzdz.
Moreover, an application of Hölder’s inequality gives us that(103)K2y,ℓ≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓνz−p′/pdz1/p′=C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q·wQy,2j+1nℓ1/pQy,2j+1nℓ∫Qy,2j+1nℓνz−p′/pdz1/p′.
In addition, we apply Hölder’s inequality with exponentr>1 to get
(104)wQy,2j+1nℓ=∫Qy,2j+1nℓwzdz≤Qy,2j+1nℓ1/r′∫Qy,2j+1nℓwzrdz1/r.
Consequently,(105)K2y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q×Qy,2j+1nℓ1/r′pQy,2j+1nℓ∫Qy,2j+1nℓwzrdz1/rp∫Qy,2j+1nℓνz−p′/pdz1/p′≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
The last inequality is obtained by theAp-type condition (24) on w,ν. Furthermore, arguing as in the proof of Theorem 5, we know that for any positive integer j, there exists a reverse doubling constantD=Dw>1 independent of Qy,ℓ such that
(106)wQy,2j+1nℓ≥Dj+1·wQy,ℓ.
Hence,(107)∑j=1∞wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤∑j=1∞wQy,ℓDj+1·wQy,ℓ1/α−1/q=∑j=1∞1Dj+11/α−1/q≤C,where the last series is convergent since the reverse doubling constantD>1 and 1/α−1/q>0. Therefore, by taking the Lqμ-norm of both sides of (97) (with respect to the variable y), and then using Minkowski’s inequality, (100) and (105), we have
(108)wQy,ℓ1/α−1/p−1/qTf·χQy,ℓ∥WLpw∥Lqμ≤K1y,ℓLqμ+K2y,ℓLqμ≤CwQy,2nℓ1/α−1/p−1/qf·χQy,2nℓ∥Lpν∥Lqμ+C∑j=1∞∥wQy,2j+1nℓ1/α−1/p−1/q∥f·χQy,2j+1nℓ∥Lpν∥Lqμ×wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤CfLp,Lqαν,w;μ+C∥f∥Lp,Lqαν,w;μ×∑j=1∞wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤CfLp,Lqαν,w;μ,where the last inequality follows from (107). By taking the supremum over all ℓ>0, we finish the proof of Theorem 9.Proof of Theorem 10.
Let1<p≤α<q≤∞ and f∈Lp,Lqαν,w;μ with w∈Δ2 and μ∈Δ2. For an arbitrary point y∈Rn, we set Q=Qy,ℓ for the cube centered at y and of the side length ℓ. Decompose f as
(109)f=f1+f2∈Lp,Lqαν,w;μ,f1=f·χ2Q,f2=f·χ2Qc,where 2Q=Qy,2nℓ. Then, for any given y∈Rn and ℓ>0, we write
(110)wQy,ℓ1/α−1/p−1/qTγf·χQy,ℓWLpw≤2·wQy,ℓ1/α−1/p−1/qTγf1·χQy,ℓWLpw+2·wQy,ℓ1/α−1/p−1/qTγf2·χQy,ℓWLpw≔K′1y,ℓ+K′2y,ℓ.
Let us consider the first termK′1y,ℓ. Using assumption (19) and inequality (99), we get
(111)K′1y,ℓ≤2·wQy,ℓ1/α−1/p−1/qTγf1WLpw≤C·wQy,ℓ1/α−1/p−1/q∫Qy,2nℓfxpνxdx1/p=C·wQy,2nℓ1/α−1/p−1/qf·χQy,2nℓLpν×wQy,ℓ1/α−1/p−1/qwQy,2nℓ1/α−1/p−1/q≤C·wQy,2nℓ1/α−1/p−1/qf·χQy,2nℓLpν.
We now estimate the second termK′2y,ℓ. Recall that by the size condition (2), the following estimate holds for any x∈Qy,ℓ:
(112)Tγf2x≤C∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz.
This pointwise estimate (112) along with Chebyshev’s inequality implies
(113)K′2y,ℓ≤2·wQy,ℓ1/α−1/p−1/q∫Qy,ℓTγf2xpwxdx1/p≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz.
A further application of Hölder’s inequality yields(114)K′2y,ℓ≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓνz−p′/pdz1/p′=C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q·wQy,2j+1nℓ1/pQy,2j+1nℓ1−γ/n×∫Qy,2j+1nℓνz−p′/pdz1/p′.
Hence, in view of (104), we have
(115)K′2y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q×Qy,2j+1nℓ1/r′pQy,2j+1nℓ1−γ/n∫Qy,2j+1nℓwzrdz1/rp∫Qy,2j+1nℓνz−p′/pdz1/p′≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
The last inequality is obtained by theAp-type condition (25) on w,ν. Therefore, by taking the Lqμ-norm of both sides of (110) (with respect to the variable y), and then using Minkowski’s inequality, (111) and (115), we obtain
(116)wQy,ℓ1/α−1/p−1/qTγf·χQy,ℓ∥WLpw∥Lqμ≤K′1y,ℓLqμ+K′2y,ℓLqμ≤CwQy,2nℓ1/α−1/p−1/qf·χQy,2nℓ∥Lpν∥Lqμ+C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓ∥Lpν∥Lqμ×wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤CfLp,Lqαν,w;μ+CfLp,Lqαν,w;μ×∑j=1∞wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤CfLp,Lqαν,w;μ,where the last inequality follows from (107). Thus, by taking the supremum over all ℓ>0, we complete the proof of Theorem 10.


## 6. Proofs of Theorems11 and 12

Proof of Theorem 11.
Let1<p≤α<q≤∞ and f∈Lp,Lqαν,w;μ with w∈A∞ and μ∈Δ2. For any fixed cube Q=Qy,ℓ in Rn, as before, we decompose f as
(117)f=f1+f2,f1=f·χ2Q,f2=f·χ2Qc,where 2Q=Qy,2nℓ. Then, for any given y∈Rn and ℓ>0, we write
(118)wQy,ℓ1/α−1/p−1/qb,Tf·χQy,ℓWLpw≤2·wQy,ℓ1/α−1/p−1/qb,Tf1·χQy,ℓWLpw+2·wQy,ℓ1/α−1/p−1/qb,Tf2·χQy,ℓWLpw≔L1y,ℓ+L2y,ℓ.
Sincew∈A∞, we know that w∈Δ2. By assumption (21) and inequality (99), then we have
(119)L1y,ℓ≤2·wQy,ℓ1/α−1/p−1/qb,Tf1WLpw≤C·wQy,ℓ1/α−1/p−1/q∫Qy,2nℓfxpνxdx1/p=C·wQy,2nℓ1/α−1/p−1/qf·χQy,2nℓLpν×wQy,ℓ1/α−1/p−1/qwQy,2nℓ1/α−1/p−1/q≤C·wQy,2nℓ1/α−1/p−1/qf·χQy,2nℓLpν.
Next, we estimateL2y,ℓ. For any x∈Qy,ℓ, from condition (3), we can see that
(120)b,Tf2x≤bx−bQy,ℓ·Tf2x+TbQy,ℓ−bf2x≔ξ∗x+η∗x.
Then, we have(121)L2y,ℓ≤4·wQy,ℓ1/α−1/p−1/qξ∗··χQy,ℓWLpw+4·wQy,ℓ1/α−1/p−1/qη∗··χQy,ℓWLpw≔L3y,ℓ+L4y,ℓ.
For the termL3y,ℓ, it follows directly from Chebyshev’s inequality and estimate (101) that
(122)L3y,ℓ≤4·wQy,ℓ1/α−1/p−1/q∫Qy,ℓξ∗xpwxdx1/p≤C·wQy,ℓ1/α−1/p−1/q∫Qy,ℓbx−bQy,ℓpwxdx1/p×∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓfzdz≤Cb∗·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓfzdz,where in the last inequality, we have used the fact that w∈A∞ and Lemma 14(ii). We can now argue exactly as we did in the proof of Theorem 9 to get
(123)L3y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
For the termL4y,ℓ, as it was shown in Theorem 7, the following pointwise inequality holds by the size condition (1).
(124)η∗x=TbQy,ℓ−bf2x≤C∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓbz−bQy,ℓ·fzdz.
This, together with Chebyshev’s inequality, yields(125)L4y,ℓ≤4·wQy,ℓ1/α−1/p−1/q∫Qy,ℓη∗xpwxdx1/p≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓ∣bz−bQy,ℓ∣·∣fz∣dz≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓbz−bQy,2j+1nℓ·fzdz+C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓbQy,2j+1nℓ−bQy,ℓ·fzdz≔L5y,ℓ+L6y,ℓ.
An application of Hölder’s inequality leads to(126)L5y,ℓ≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓbz−bQy,2j+1nℓp′νz−p′/pdz1/p′≤C·wQy,ℓ1/α−1/q∑j=1∞f·χQy,2j+1nℓLpνQy,2j+1nℓ×Qy,2j+1nℓ1/p′b−bQy,2j+1nℓ·ν−1/pC,Qy,2j+1nℓ,where Ct=tp′ is a Young function. Recall that the following inequality holds by generalized Hölder’s inequality and estimate (74):
(127)b−bQy,2j+1nℓ·ν−1/pC,Qy,2j+1nℓ≤Cb−bQy,2j+1nℓB,Qy,2j+1nℓν−1/pA,Qy,2j+1nℓ≤Cb∗·ν−1/pA,Qy,2j+1nℓ,where
(128)At≈tp′loge+tp′,Bt≈expt−1.
Consequently,(129)L5y,ℓ≤Cb∗·wQy,ℓ1/α−1/q∑j=1∞f·χQy,2j+1nℓLpνQy,2j+1nℓ1/p·ν−1/pA,Qy,2j+1nℓ=Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q×wQy,2j+1nℓ1/pQy,2j+1nℓ1/p·ν−1/pA,Qy,2j+1nℓ.
Moreover, in view of (104), we can deduce that
(130)L5y,ℓ≤Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q×Qy,2j+1nℓ1/r′pQy,2j+1nℓ1/p∫Qy,2j+1nℓwzrdz1/rp·ν−1/pA,Qy,2j+1nℓ≤Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
The last inequality is obtained by theAp-type condition (26) on w,ν. Let us now estimate the last term L6y,ℓ. Applying Lemma 14(i) and Hölder’s inequality, we get
(131)L6y,ℓ≤C·wQy,ℓ1/α−1/q∑j=1∞j+1b∗Qy,2j+1nℓ∫Qy,2j+1nℓfzdz≤C·wQy,ℓ1/α−1/q∑j=1∞j+1b∗Qy,2j+1nℓ∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓνz−p′/pdz1/p′=Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q·wQy,2j+1nℓ1/pQy,2j+1nℓ×∫Qy,2j+1nℓνz−p′/pdz1/p′.
It was pointed out in Theorem7 that condition (26) is stronger than condition (24). Taking into account this fact and (104), we obtain
(132)L6y,ℓ≤Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q×Qy,2j+1nℓ1/r′pQy,2j+1nℓ∫Qy,2j+1nℓwzrdz1/rp∫Qy,2j+1nℓνz−p′/pdz1/p′≤Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
Summing up all the above estimates, we get(133)L2y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
Moreover, by our hypothesis onw:w∈A∞ and inequality (30) with exponent δ∗>0, we compute
(134)∑j=1∞j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤C∑j=1∞j+1·Qy,ℓQy,2j+1nℓδ∗1/α−1/q=C∑j=1∞j+1·12j+1nδ∗1/α−1/q≤C.
Notice that the exponentδ∗1/α−1/q is positive because α<q, which guarantees that the last series is convergent. Thus, by taking the Lqμ-norm of both sides of (118) (with respect to the variable y), and then using Minkowski’s inequality, (119) and (133), we obtain
(135)wQy,ℓ1/α−1/p−1/qb,Tf·χQy,ℓ∥WLpw∥Lqμ≤L1y,ℓLqμ+L2y,ℓLqμ≤CwQy,2nℓ1/α−1/p−1/qf·χQy,2nℓ∥Lpν∥Lqμ+C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓ∥Lpν∥Lqμ×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤CfLp,Lqαν,w;μ+CfLp,Lqαν,w;μ×∑j=1∞j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤CfLp,Lqαν,w;μ,where the last inequality is due to (134). We conclude the proof of Theorem 11 by taking the supremum over all ℓ>0.Proof of Theorem 12.
Let1<p≤α<q≤∞ and f∈Lp,Lqαν,w;μ with w∈A∞ and μ∈Δ2. For any fixed cube Q=Qy,ℓ in Rn, as usual, we decompose f as
(136)f=f1+f2,f1=f·χ2Q,f2=f·χ2Qc,where 2Q=Qy,2nℓ. Then, for any given y∈Rn and ℓ>0, we write
(137)wQy,ℓ1/α−1/p−1/qb,Tγf·χQy,ℓWLpw≤2·wQy,ℓ1/α−1/p−1/qb,Tγf1·χQy,ℓWLpw+2·wQy,ℓ1/α−1/p−1/qb,Tγf2·χQy,ℓWLpw≔L′1y,ℓ+L′2y,ℓ.
Sincew∈A∞, we know that w∈Δ2. By assumption (23) and inequality (99), we get
(138)L′1y,ℓ≤2·wQy,ℓ1/α−1/p−1/qb,Tγf1WLpw≤C·wQy,ℓ1/α−1/p−1/q∫Qy,2nℓfxpνxdx1/p=C·wQy,2nℓ1/α−1/p−1/qf·χQy,2nℓLpν×wQy,ℓ1/α−1/p−1/qwQy,2nℓ1/α−1/p−1/q≤C·wQy,2nℓ1/α−1/p−1/qf·χQy,2nℓLpν.
Next, we estimate the other termL′2y,ℓ. For any x∈Qy,ℓ, from condition (4), one can see that
(139)b,Tγf2x≤bx−bQy,ℓ·Tγf2x+TγbQy,ℓ−bf2x≔ξ~∗x+η~∗x.
Consequently, we can further divideL′2y,ℓ into two parts:
(140)L′2y,ℓ≤4·wQy,ℓ1/α−1/p−1/qξ~∗··χQy,ℓWLpw+4·wQy,ℓ1/α−1/p−1/qη~∗··χQy,ℓWLpw≔L′3y,ℓ+L′4y,ℓ.
For the termL′3y,ℓ, it follows directly from Chebyshev’s inequality and estimate (112) that
(141)L′3y,ℓ≤4·wQy,ℓ1/α−1/p−1/q∫Qy,ℓξ~∗xpwxdx1/p≤C·wQy,ℓ1/α−1/p−1/q∫Qy,ℓbx−bQy,ℓpwxdx1/p×∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz,where in the last inequality, we have used the fact that w∈A∞ and Lemma 14(ii). Arguing as in the proof of Theorem 10, we can also obtain that
(142)L′3y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
Let us now estimate the termL′4y,ℓ. As it was proved in Theorem 8, the following pointwise estimate holds by the size condition (2).
(143)η~∗x=TγbQy,ℓ−bf2x≤C∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓbz−bQy,ℓ·fzdz.
This, together with Chebyshev’s inequality implies that(144)L′4y,ℓ≤4·wQy,ℓ1/α−1/p−1/q∫Qy,ℓη~∗xpwxdx1/p≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓbz−bQy,ℓ·fzdz≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓbz−bQy,2j+1nℓ·fzdz+C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓbQy,2j+1nℓ−bQy,ℓ·fzdz≔L′5y,ℓ+L′6y,ℓ.
An application of Hölder’s inequality leads to(145)L′5y,ℓ≤C·wQy,ℓ1/α−1/q∑j=1∞1Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓbz−bQy,2j+1nℓp′νz−p′/pdz1/p′=C·wQy,ℓ1/α−1/q∑j=1∞f·χQy,2j+1nℓLpνQy,2j+1nℓ1−γ/n×Qy,2j+1nℓ1/p′b−bQy,2j+1nℓ·ν−1/pC,Qy,2j+1nℓ,where Ct=tp′ is a Young function. Moreover, in view of (104) and (127), we can deduce that
(146)L′5y,ℓ≤Cb∗·wQy,ℓ1/α−1/q∑j=1∞f·χQy,2j+1nℓLpνQy,2j+1nℓ1/p−γ/n·ν−1/pA,Qy,2j+1nℓ=Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q×wQy,2j+1nℓ1/pQy,2j+1nℓ1/p−γ/n·ν−1/pA,Qy,2j+1nℓ≤Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q×Qy,2j+1nℓ1/r′pQy,2j+1nℓ1/p−γ/n∫Qy,2j+1nℓwzrdz1/rp·ν−1/pA,Qy,2j+1nℓ≤Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
The last inequality is obtained by theAp-type condition (27) on w,ν. Let us estimate the last term L′6y,ℓ. Applying Lemma 14(i) and Hölder’s inequality, we get
(147)L′6y,ℓ≤C·wQy,ℓ1/α−1/q∑j=1∞j+1b∗Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzdz≤C·wQy,ℓ1/α−1/q∑j=1∞j+1b∗Qy,2j+1nℓ1−γ/n∫Qy,2j+1nℓfzpνzdz1/p×∫Qy,2j+1nℓνz−p′/pdz1/p′=Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q·wQy,2j+1nℓ1/pQy,2j+1nℓ1−γ/n∫Qy,2j+1nℓνz−p′/pdz1/p′.
Also, observe that condition (27) is stronger than condition (25). Using this fact along with (104), we get
(148)L′6y,ℓ≤Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q×Qy,2j+1nℓ1/r′pQy,2j+1nℓ1−γ/n∫Qy,2j+1nℓwzrdz1/rp∫Qy,2j+1nℓνz−p′/pdz1/p′≤Cb∗∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
Summing up all the above estimates, we conclude that(149)L′2y,ℓ≤C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓLpν×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q.
Thus, by taking theLqμ-norm of both sides of (137) (with respect to the variable y), and then using Minkowski’s inequality, (138) and (149), we finally obtain
(150)wQy,ℓ1/α−1/p−1/qb,Tγf·χQy,ℓ∥WLpw∥Lqμ≤L′1y,ℓLqμ+L′2y,ℓLqμ≤CwQy,2nℓ1/α−1/p−1/qf·χQy,2nℓ∥Lpν∥Lqμ+C∑j=1∞wQy,2j+1nℓ1/α−1/p−1/qf·χQy,2j+1nℓ∥Lpν∥Lqμ×j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤CfLp,Lqαν,w;μ+CfLp,Lqαν,w;μ×∑j=1∞j+1·wQy,ℓ1/α−1/qwQy,2j+1nℓ1/α−1/q≤CfLp,Lqαν,w;μ,where the last inequality is due to (134). We therefore conclude the proof of Theorem 12 by taking the supremum over all ℓ>0.



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*Source: 3673921-2020-07-20.xml*