# Boundedness ofθ-Type Calderón–Zygmund Operators and Commutators in the Generalized Weighted Morrey Spaces

Authors: Hua Wang
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space as special cases. Then, we discuss the strong-type and weak-type estimates for a class of Calderón–Zygmund type operatorsTθ in these new Morrey type spaces. Furthermore, the strong-type estimate and endpoint estimate of commutators [b,Tθ] formed by b and Tθ are established. Also, we study related problems about two-weight, weak-type inequalities for Tθ and [b,Tθ] in the Morrey type spaces and give partial results.


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## Body

## 1. Introduction

Calderón–Zygmund singular integral operators and their generalizations on the Euclidean spaceRn have been extensively studied (see [1–5], for instance). In particular, Yabuta [5] introduced certain θ-type Calderón–Zygmund operators to facilitate his study of certain classes of pseudodifferential operators. Following the terminology of Yabuta [5], we introduce the so-called θ-type Calderón–Zygmund operators.Definition 1.
Letθ be a nonnegative, nondecreasing function on R+=(0,+∞) with(1)∫01θttdt<∞.A measurable function K(·,·) on Rn×Rn∖{(x,x):x∈Rn} is said to be a θ-type kernel if it satisfies(2)iKx,y≤Cx-yn,foranyx≠y;(3)iiKx,y-Kz,y+Ky,x-Ky,z≤Cx-yn·θx-zx-y,forx-z<x-y2.Definition 2.
LetTθ be a linear operator from S(Rn) into its dual S′(Rn). One can say that Tθ is a θ-type Calderón–Zygmund operator if(1)
Tθ can be extended to be a bounded linear operator on L2(Rn);(2)
there is aθ-type kernel K(x,y) such that(4)Tθfx≔∫RnKx,yfydyfor all f∈C0∞(Rn) and for all x∉supp⁡f, where C0∞(Rn) is the space consisting of all infinitely differentiable functions on Rn with compact supports.Note that the classical Calderón–Zygmund operator with standard kernel (see [1, 2]) is a special case of θ-type operator Tθ when θ(t)=tδ with 0<δ≤1.Definition 3.
Given a locally integrable functionb defined on Rn and given a θ-type Calderón–Zygmund operator Tθ, the linear commutator [b,Tθ] is defined for smooth, compactly supported functions f as(5)b,Tθfx≔bx·Tθfx-Tθb·fx=∫Rnbx-byKx,yfydy.Throughout the paper, let us suppose thatθ is a nonnegative, nondecreasing function on R+=(0,+∞) satisfying condition (1). Let us give the following weighted result of Tθ obtained by Quek and Yang in [6].Theorem 4 (see [6]).
Let1≤p<∞ and w∈Ap. Then, the θ-type Calderón–Zygmund operator Tθ is bounded on Lwp(Rn) for p>1 and bounded from Lw1(Rn) into WLw1(Rn) for p=1.Since linear commutator has a greater degree of singularity than the correspondingθ-type Calderón–Zygmund operator, we need a slightly stronger version of condition (8) given below. The following weighted endpoint estimate for commutator [b,Tθ] of the θ-type Calderón–Zygmund operator was established in [7] under a stronger version of condition (8) assumed on θ, if b∈BMO(Rn) (for the unweighted case, see [8]).Let us now recall the definition of the space ofBMO(Rn) (see [9]). BMO(Rn) is the Banach function space modulo constants with the norm ·∗ defined by(6)b∗≔supB1B∫Bbx-bBdx<∞,where the supremum is taken over all balls B in Rn and bB stands for the mean value of b over B; that is,(7)bB≔1B∫Bbydy.Theorem 5 (see [7]).
Letθ be a nonnegative, nondecreasing function on R+=(0,+∞) with(8)∫01θt·log⁡ttdt<∞,and let w∈A1 and b∈BMO(Rn). Then, for all σ>0, there is a constant C>0 independent of f and σ>0 such that(9)wx∈Rn:b,Tθfx>σ≤C∫RnΦfxσ·wxdx,where Φ(t)=t·(1+log+t) and log+t=max⁡{log⁡t,0}.On the other hand, the classical Morrey space was originally introduced by Morrey in [10] to study the local behavior of solutions to second-order elliptic partial differential equations. Since then, this space played an important role in studying the regularity of solutions to partial differential equations. In [11], Mizuhara introduced the generalized Morrey space Lp,Ψ which was later extended and studied by many authors. In [12], Komori and Shirai defined a version of the weighted Morrey space Lp,κ(w) which is a natural generalization of the weighted Lebesgue space. Let Tθ be the θ-type Calderón–Zygmund operator, and let [b,Tθ] be its linear commutator. The main purpose of this paper is twofold. We first define a new kind of Morrey type spaces Mp,ψ(w) containing generalized Morrey space Lp,Ψ and weighted Morrey space Lp,κ(w) as special cases, and then we will establish the weighted strong type and endpoint estimates for Tθ and [b,Tθ] in these Morrey type spaces Mp,ψ(w) for all 1≤p<∞ and w∈Ap. In addition, we will discuss two-weight, weak-type norm inequalities for Tθ and [b,Tθ] in Mp,ψ(w) and give some partial results.Throughout this paper,C will denote a positive constant whose value may change at each appearance. 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. Statements of the Main Results

### 2.1. Notation and Preliminaries

LetRn be the n-dimensional Euclidean space of points x=(x1,x2,…,xn) with norm |x|=(∑i=1nxi2)1/2. For x0∈Rn and r>0, let B(x0,r)={x∈Rn:|x-x0|<r} denote the open ball centered at x0 of radius r, B(x0,r)c denote its complement, and |B(x0,r)| be the Lebesgue measure of the ball B(x0,r). A weight w is a nonnegative locally integrable function on Rn that takes values in (0,+∞) almost everywhere. A weight w is said to belong to Muckenhoupt’s class Ap for 1<p<∞, if there exists a constant C>0 such that(10)1B∫Bwxdx1/p1B∫Bwx-p′/pdx1/p′≤Cfor every ball B⊂Rn, where p′ is the dual of p such that 1/p+1/p′=1. The class A1 is defined replacing the inequality above by(11)1B∫Bwxdx≤C·essinfx∈Bwxfor every ball B⊂Rn. We also define A∞=⋃1≤p<∞Ap. Given a ball B and λ>0, λB will denote the ball with the same center as B whose radius is λ times that of B. Given a Lebesgue measurable set E and a weight function w, we denote the characteristic function of E by χE, the Lebesgue measure of E by |E|, and the weighted measure of E by w(E), where w(E)=∫Ew(x)dx. It is well known that if w∈Ap with 1≤p<∞ (or w∈A∞), then w satisfies the doubling condition; that is, for any ball B, there exists an absolute constant C>0 such that (see [2])(12)w2B≤CwB.Moreover, if w∈A∞, then for any ball B and any measurable subset E of a ball B, there exists a number δ>0 independent of E and B such that (see [2])(13)wEwB≤CEBδ.Given a weight functionw on Rn, as usual, the weighted Lebesgue space Lwp(Rn) for 1≤p<∞ is defined as the set of all functions f such that(14)fLwp≔∫Rnfxpwxdx1/p<∞.We also denote by WLwp(Rn) (1≤p<∞) the weighted weak Lebesgue space consisting of all measurable functions f such that(15)fWLwp≔supλ>0⁡λ·wx∈Rn:fx>λ1/p<∞.We next recall some basic definitions and facts about Orlicz spaces needed for the proof of the main results. For further information on the subject, one can see [13]. A function A is called a Young function if it is continuous, nonnegative, convex, and strictly increasing on [0,+∞) with A(0)=0 and A(t)→+∞ as t→+∞. An important example of Young function is A(t)=tp(1+log+t)p with some 1≤p<∞. Given a Young function A, we define the A-average of a function f over a ball B by means of the following Luxemburg norm:(16)fA,B≔inf⁡λ>0:1B∫BAfxλdx≤1.When A(t)=tp, 1≤p<∞, it is easy to see that(17)fA,B=1B∫Bfxpdx1/p;that is, the Luxemburg norm coincides with the normalized Lp norm. Given a Young function A, we use A¯ to denote the complementary Young function associated with A. Then, the following generalized Hölder’s inequality holds for any given ball B:(18)1B∫Bfx·gxdx≤2fA,BgA¯,B.In particular, when A(t)=t·(1+log+t), we know that its complementary Young function is A¯(t)≈exp⁡(t)-1. In this situation, we denote(19)fLlog⁡L,B=fA,B,gexp⁡L,B=gA¯,B.So we have(20)1B∫Bfx·gxdx≤2fLlog⁡L,Bgexp⁡L,B.


### 2.2. Morrey Type Spaces

Let us begin with the definitions of the weighted Morrey space and generalized Morrey space.Definition 6 (see [12]).
Let1≤p<∞, 0<κ<1, and w be a weight function on Rn. Then, the weighted Morrey space Lp,κ(w) is defined by(21)Lp,κw≔f∈Llocpw:fLp,κw=supB1wBκ∫Bfxpwxdx1/p<∞,where the supremum is taken over all balls B in Rn. We also denote by WL1,κ(w) the weighted weak Morrey space of all measurable functions f such that(22)supB⁡supλ>0⁡1wBκλ·wx∈B:fx>λ≤C<∞.LetΨ=Ψ(r), r>0, be a growth function, that is, a positive increasing function in (0,+∞), and satisfy the following doubling condition:(23)Ψ2r≤D·Ψr,∀r>0,where D=D(Ψ)≥1 is a doubling constant independent of r.Definition 7 (see [11]).
Let1≤p<∞ and Ψ be a growth function in (0,+∞). Then, the generalized Morrey space Lp,Ψ(Rn) is defined by(24)Lp,ΨRn≔f∈LlocpRn:fLp,Ψ=supr>0;Bx0,r1Ψr∫Bx0,rfxpdx1/p<∞,where the supremum is taken over all balls B(x0,r) in Rn with x0∈Rn. One can also denote by WL1,Ψ(Rn) the generalized weak Morrey space of all measurable functions f for which(25)supBx0,rsupλ>01Ψrλ·x∈Bx0,r:fx>λ≤C<∞.In order to unify these two definitions, we now introduce Morrey type spaces associated withψ as follows. Let 0≤κ<1. Assume that ψ(·) is a positive increasing function defined in (0,+∞) and satisfies the following Dκ condition:(26)ψξξκ≤C·ψξ′ξ′κ,forany0<ξ′<ξ<+∞,where C>0 is a constant independent of ξ and ξ′.Definition 8.
Let1≤p<∞, 0≤κ<1, ψ satisfy the Dκ condition (26), and w be a weight function on Rn. We denote by Mp,ψ(w) the generalized weighted Morrey space, the space of all locally integrable functions f with finite norm:(27)fMp,ψw≔supB1ψwB∫Bfxpwxdx1/p≤C<∞.Then, we know that Mp,ψ(w) becomes a Banach function space with respect to the norm ·Mp,ψ(w). Furthermore, we denote by WMp,ψ(w) the generalized weighted weak Morrey space of all measurable functions f for which(28)fWMp,ψw≔supBsupσ>0⁡1ψwB1/pσ·wx∈B:fx>σ1/p≤C<∞.Definition 9.
In the unweighted case (whenw equals a constant function), one can denote the generalized unweighted Morrey space by Mp,ψ(Rn) and weak Morrey space by WMp,ψ(Rn). That is, let 1≤p<∞ and ψ satisfy the Dκ condition (26) with 0≤κ<1; one can define(29)Mp,ψRn≔f∈LlocpRn:fMp,ψ=supB1ψB∫Bfxpdx1/p<∞,WMp,ψRn≔f:fWMp,ψ=supB⁡supσ>0⁡1ψB1/pσ·x∈B:fx>σ1/p<∞.Note the following:(i)
Ifψ(x)≡1, then Mp,ψ(w)=Lwp(Rn) and WMp,ψ(w)=WLwp(Rn). Thus, our (weak) Morrey type space is an extension of the weighted (weak) Lebesgue space.(ii)
Ifψ(x)=xκ with 0<κ<1, then Mp,ψ(w) is just the weighted Morrey space Lp,κ(w), and WM1,ψ(w) is just the weighted weak Morrey space WL1,κ(w).(iii)
Ifw(x)≡1, below we will show that Mp,ψ(Rn) reduces to the generalized Morrey space Lp,Ψ(Rn), and WM1,ψ(Rn) reduces to the generalized weak Morrey space WL1,Ψ(Rn).Our main results on the boundedness ofTθ in the Morrey type spaces Mp,ψ(w) can be formulated as follows.Theorem 10.
Let1<p<∞ and w∈Ap. Assume that ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the θ-type Calderón–Zygmund operator Tθ is bounded on Mp,ψ(w).Theorem 11.
Letp=1 and w∈A1. Assume that ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the θ-type Calderón–Zygmund operator Tθ is bounded from M1,ψ(w) into WM1,ψ(w).Letθ be a nonnegative, nondecreasing function on R+=(0,+∞) satisfying condition (8), and let [b,Tθ] be the commutator formed by Tθ and BMO function b. For the strong-type estimate of the linear commutator [b,Tθ] in Mp,ψ(w) with 1<p<∞, we will prove the following.Theorem 12.
Let1<p<∞, w∈Ap, and b∈BMO(Rn). Assume that θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the commutator operator [b,Tθ] is bounded on Mp,ψ(w).To obtain endpoint estimate for the linear commutator[b,Tθ] in M1,ψ(w), we first need to define the weighted A-average of a function f over a ball B by means of the weighted Luxemburg norm; that is, given a Young function A and w∈A∞, we define (see [13, 14])(30)fAw,B≔infσ>0:1wB∫BAfxσ·wxdx≤1.When A(t)=t, this norm is denoted by ·L(w),B; when A(t)=t·(1+log+t), this norm is also denoted by ·LlogL(w),B. The complementary Young function of t·(1+log+t) is exp⁡t-1 with mean Luxemburg norm denoted by ·exp⁡L(w),B. For w∈A∞ and for every ball B in Rn, we can also show the weighted version of (20). Namely, the following generalized Hölder’s inequality in the weighted setting(31)1wB∫Bfx·gxwxdx≤CfLlog⁡Lw,Bgexp⁡Lw,Bis valid (see [14], for instance). Now we introduce new Morrey type spaces of Llog⁡L type associated with ψ as follows.Definition 13.
Letp=1, 0≤κ<1, ψ satisfy the Dκ condition (26), and w be a weight function on Rn. One can denote by MLlog⁡L1,ψ(w) the generalized weighted Morrey space of LlogL type, the space of all locally integrable functions f defined on Rn with finite norm fMLlog⁡L1,ψ(w).(32)MLlog⁡L1,ψw≔f∈Lloc1w:fMLlog⁡L1,ψw<∞,where(33)fMLlog⁡L1,ψw≔supBwBψwB·fLlog⁡Lw,B.Note thatt≤t·(1+log+t) for all t>0; then, for any ball B⊂Rn and w∈A∞, we have fLw,B≤fLlog⁡L(w),B by definition; that is, the inequality(34)fLw,B=1wB∫Bfx·wxdx≤fLlog⁡Lw,Bholds for any ball B⊂Rn. From this, we can further see that when ψ satisfies the Dκ condition (26) with 0≤κ<1,(35)1ψwB∫Bfx·wxdx=wBψwB·1wB∫Bfx·wxdx=wBψwB·fLw,B≤wBψwB·fLlogLw,B.Hence, we have MLlogL1,ψ(w)⊂M1,ψ(w) by definition.Definition 14.
In the unweighted case (whenw equals a constant function), one can denote by MLlog⁡L1,ψ(Rn) the generalized unweighted Morrey space of Llog⁡L type. That is, let p=1 and ψ satisfy the Dκ condition (26) with 0≤κ<1; one can define(36)MLlog⁡L1,ψRn≔f∈Lloc1Rn:fMLlog⁡L1,ψRn<∞,where(37)fMLlog⁡L1,ψRn≔supB⁡BψB·fLlog⁡L,B.We also consider the special case whenψ is taken to be ψ(x)=xκ with 0<κ<1 and denote the corresponding space by LLlog⁡L1,κ(w).Definition 15.
Letp=1, 0<κ<1, and w be a weight function on Rn. One can denote by LLlog⁡L1,κ(w) the weighted Morrey space of Llog⁡L type, the space of all locally integrable functions f defined on Rn with finite norm fLLlog⁡L1,κ(w).(38)LLlog⁡L1,κw≔f∈Lloc1w:fLLlog⁡L1,κw<∞,where(39)fLLlogL1,κw≔supBwB1-κ·fLlog⁡Lw,B.In this situation, we have LLlog⁡L1,κ(w)⊂L1,κ(w).For the endpoint case, we will also prove the following weak-typeLlog⁡L estimate of the linear commutator [b,Tθ] in the Morrey type space associated with ψ.Theorem 16.
Letp=1, w∈A1, and b∈BMO(Rn). Assume that θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1; then, for any given σ>0 and any ball B⊂Rn, there exists a constant C>0 independent of f, B, and σ>0 such that(40)1ψwB·wx∈B:b,Tθfx>σ≤C·ΦfσMLlog⁡L1,ψw,where Φ(t)=t·(1+log+t). From the definitions, we can roughly say that the commutator operator [b,Tθ] is bounded from MLlog⁡L1,ψ(w) into WM1,ψ(w).In particular, if we takeψ(x)=xκ with 0<κ<1, then we immediately get the following strong-type estimate and endpoint estimate of Tθ and [b,Tθ] in the weighted Morrey spaces Lp,κ(w) for all 0<κ<1 and 1≤p<∞.Corollary 17.
Let1<p<∞, 0<κ<1, and w∈Ap. Then, the θ-type Calderón–Zygmund operator Tθ is bounded on Lp,κ(w).Corollary 18.
Letp=1, 0<κ<1, and w∈A1. Then, the θ-type Calderón–Zygmund operator Tθ is bounded from L1,κ(w) into WL1,κ(w).Corollary 19.
Let1<p<∞, 0<κ<1, w∈Ap, and b∈BMO(Rn). Assume that θ satisfies (8); then, the commutator operator [b,Tθ] is bounded on Lp,κ(w).Corollary 20.
Letp=1, 0<κ<1, w∈A1, and b∈BMO(Rn). Assume that θ satisfies (8); then, for any given σ>0 and any ball B⊂Rn, there exists a constant C>0 independent of f, B, and σ>0 such that(41)1wBκ·wx∈B:b,Tθfx>σ≤C·ΦfσLLlog⁡L1,κw,where Φ(t)=t·(1+log+t).Naturally, whenw(x)≡1, we have the following unweighted results.Corollary 21.
Let1<p<∞. Assume that ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the θ-type Calderón–Zygmund operator Tθ is bounded on Mp,ψ(Rn).Corollary 22.
Letp=1. Assume that ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the θ-type Calderón–Zygmund operator Tθ is bounded from M1,ψ(Rn) into WM1,ψ(Rn).Corollary 23.
Let1<p<∞ and b∈BMO(Rn). Assume that θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the commutator operator [b,Tθ] is bounded on Mp,ψ(Rn).Corollary 24.
Letp=1 and b∈BMO(Rn). Assume that θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1; then, for any given σ>0 and any ball B⊂Rn, there exists a constant C>0 independent of f, B, and σ>0 such that(42)1ψB·x∈B:b,Tθfx>σ≤C·ΦfσMLlog⁡L1,ψRn,where Φ(t)=t·(1+log+t).LetΨ=Ψ(r), r>0, be a growth function with doubling constant D(Ψ):1≤D(Ψ)<2n. If, for any fixed x0∈Rn and r>0, we set ψ(|B(x0,r)|)=Ψ(r), then(43)ψ2nBx0,r=ψBx0,2r=Ψ2r.For the doubling constant D(Ψ) satisfying 1≤D(Ψ)<2n, which means that D(Ψ)=2κ·n for some 0≤κ<1, then we are able to verify that ψ is an increasing function and satisfies the Dκ condition (26) with some 0≤κ<1.Definition 25.
Letp=1 and Ψ be a growth function in (0,+∞). One can denote by LLlog⁡L1,Ψ(Rn) the generalized Morrey space of Llog⁡L type, which is defined by(44)LLlog⁡L1,ΨRn≔f∈Lloc1Rn:fLLlog⁡L1,ΨRn<∞,where(45)fLLlog⁡L1,ΨRn≔supr>0;Bx0,rBx0,rΨr·fLlog⁡L,Bx0,r.In this situation, we also have LLlogL1,Ψ(Rn)⊂L1,Ψ(Rn).From the definitions given above, we getMp,ψ(Rn)=Lp,Ψ(Rn), WM1,ψ(Rn)=WL1,Ψ(Rn), and MLlog⁡L1,ψ(Rn)=LLlog⁡L1,Ψ(Rn) by the choice of Ψ. Thus, by the above unweighted results (Corollaries 21–24), we can also obtain strong-type estimate and endpoint estimate of Tθ and [b,Tθ] in the generalized Morrey spaces Lp,Ψ(Rn) when 1≤p<∞ and Ψ satisfies the doubling condition (23).Corollary 26.
Let1<p<∞. Suppose that Ψ satisfies the doubling condition (23) and 1≤D(Ψ)<2n; then, the θ-type Calderón–Zygmund operator Tθ is bounded on Lp,Ψ(Rn).Corollary 27.
Letp=1. Suppose that Ψ satisfies the doubling condition (23) and 1≤D(Ψ)<2n; then, the θ-type Calderón–Zygmund operator Tθ is bounded from L1,Ψ(Rn) into WL1,Ψ(Rn).Corollary 28.
Let1<p<∞ and b∈BMO(Rn). Suppose that θ satisfies (8) and Ψ satisfies the doubling condition (23) with 1≤D(Ψ)<2n; then, the commutator operator [b,Tθ] is bounded on Lp,Ψ(Rn).Corollary 29.
Letp=1 and b∈BMO(Rn). Suppose that θ satisfies (8) and Ψ satisfies the doubling condition (23) with 1≤D(Ψ)<2n; then, for any given σ>0 and any ball B(x0,r)⊂Rn, there exists a constant C>0 independent of f, B(x0,r), and σ>0 such that(46)1Ψr·x∈Bx0,r:b,Tθfx>σ≤C·ΦfσLLlog⁡L1,ΨRn,where Φ(t)=t·(1+log+t).


## 2.1. Notation and Preliminaries

LetRn be the n-dimensional Euclidean space of points x=(x1,x2,…,xn) with norm |x|=(∑i=1nxi2)1/2. For x0∈Rn and r>0, let B(x0,r)={x∈Rn:|x-x0|<r} denote the open ball centered at x0 of radius r, B(x0,r)c denote its complement, and |B(x0,r)| be the Lebesgue measure of the ball B(x0,r). A weight w is a nonnegative locally integrable function on Rn that takes values in (0,+∞) almost everywhere. A weight w is said to belong to Muckenhoupt’s class Ap for 1<p<∞, if there exists a constant C>0 such that(10)1B∫Bwxdx1/p1B∫Bwx-p′/pdx1/p′≤Cfor every ball B⊂Rn, where p′ is the dual of p such that 1/p+1/p′=1. The class A1 is defined replacing the inequality above by(11)1B∫Bwxdx≤C·essinfx∈Bwxfor every ball B⊂Rn. We also define A∞=⋃1≤p<∞Ap. Given a ball B and λ>0, λB will denote the ball with the same center as B whose radius is λ times that of B. Given a Lebesgue measurable set E and a weight function w, we denote the characteristic function of E by χE, the Lebesgue measure of E by |E|, and the weighted measure of E by w(E), where w(E)=∫Ew(x)dx. It is well known that if w∈Ap with 1≤p<∞ (or w∈A∞), then w satisfies the doubling condition; that is, for any ball B, there exists an absolute constant C>0 such that (see [2])(12)w2B≤CwB.Moreover, if w∈A∞, then for any ball B and any measurable subset E of a ball B, there exists a number δ>0 independent of E and B such that (see [2])(13)wEwB≤CEBδ.Given a weight functionw on Rn, as usual, the weighted Lebesgue space Lwp(Rn) for 1≤p<∞ is defined as the set of all functions f such that(14)fLwp≔∫Rnfxpwxdx1/p<∞.We also denote by WLwp(Rn) (1≤p<∞) the weighted weak Lebesgue space consisting of all measurable functions f such that(15)fWLwp≔supλ>0⁡λ·wx∈Rn:fx>λ1/p<∞.We next recall some basic definitions and facts about Orlicz spaces needed for the proof of the main results. For further information on the subject, one can see [13]. A function A is called a Young function if it is continuous, nonnegative, convex, and strictly increasing on [0,+∞) with A(0)=0 and A(t)→+∞ as t→+∞. An important example of Young function is A(t)=tp(1+log+t)p with some 1≤p<∞. Given a Young function A, we define the A-average of a function f over a ball B by means of the following Luxemburg norm:(16)fA,B≔inf⁡λ>0:1B∫BAfxλdx≤1.When A(t)=tp, 1≤p<∞, it is easy to see that(17)fA,B=1B∫Bfxpdx1/p;that is, the Luxemburg norm coincides with the normalized Lp norm. Given a Young function A, we use A¯ to denote the complementary Young function associated with A. Then, the following generalized Hölder’s inequality holds for any given ball B:(18)1B∫Bfx·gxdx≤2fA,BgA¯,B.In particular, when A(t)=t·(1+log+t), we know that its complementary Young function is A¯(t)≈exp⁡(t)-1. In this situation, we denote(19)fLlog⁡L,B=fA,B,gexp⁡L,B=gA¯,B.So we have(20)1B∫Bfx·gxdx≤2fLlog⁡L,Bgexp⁡L,B.


## 2.2. Morrey Type Spaces

Let us begin with the definitions of the weighted Morrey space and generalized Morrey space.Definition 6 (see [12]).
Let1≤p<∞, 0<κ<1, and w be a weight function on Rn. Then, the weighted Morrey space Lp,κ(w) is defined by(21)Lp,κw≔f∈Llocpw:fLp,κw=supB1wBκ∫Bfxpwxdx1/p<∞,where the supremum is taken over all balls B in Rn. We also denote by WL1,κ(w) the weighted weak Morrey space of all measurable functions f such that(22)supB⁡supλ>0⁡1wBκλ·wx∈B:fx>λ≤C<∞.LetΨ=Ψ(r), r>0, be a growth function, that is, a positive increasing function in (0,+∞), and satisfy the following doubling condition:(23)Ψ2r≤D·Ψr,∀r>0,where D=D(Ψ)≥1 is a doubling constant independent of r.Definition 7 (see [11]).
Let1≤p<∞ and Ψ be a growth function in (0,+∞). Then, the generalized Morrey space Lp,Ψ(Rn) is defined by(24)Lp,ΨRn≔f∈LlocpRn:fLp,Ψ=supr>0;Bx0,r1Ψr∫Bx0,rfxpdx1/p<∞,where the supremum is taken over all balls B(x0,r) in Rn with x0∈Rn. One can also denote by WL1,Ψ(Rn) the generalized weak Morrey space of all measurable functions f for which(25)supBx0,rsupλ>01Ψrλ·x∈Bx0,r:fx>λ≤C<∞.In order to unify these two definitions, we now introduce Morrey type spaces associated withψ as follows. Let 0≤κ<1. Assume that ψ(·) is a positive increasing function defined in (0,+∞) and satisfies the following Dκ condition:(26)ψξξκ≤C·ψξ′ξ′κ,forany0<ξ′<ξ<+∞,where C>0 is a constant independent of ξ and ξ′.Definition 8.
Let1≤p<∞, 0≤κ<1, ψ satisfy the Dκ condition (26), and w be a weight function on Rn. We denote by Mp,ψ(w) the generalized weighted Morrey space, the space of all locally integrable functions f with finite norm:(27)fMp,ψw≔supB1ψwB∫Bfxpwxdx1/p≤C<∞.Then, we know that Mp,ψ(w) becomes a Banach function space with respect to the norm ·Mp,ψ(w). Furthermore, we denote by WMp,ψ(w) the generalized weighted weak Morrey space of all measurable functions f for which(28)fWMp,ψw≔supBsupσ>0⁡1ψwB1/pσ·wx∈B:fx>σ1/p≤C<∞.Definition 9.
In the unweighted case (whenw equals a constant function), one can denote the generalized unweighted Morrey space by Mp,ψ(Rn) and weak Morrey space by WMp,ψ(Rn). That is, let 1≤p<∞ and ψ satisfy the Dκ condition (26) with 0≤κ<1; one can define(29)Mp,ψRn≔f∈LlocpRn:fMp,ψ=supB1ψB∫Bfxpdx1/p<∞,WMp,ψRn≔f:fWMp,ψ=supB⁡supσ>0⁡1ψB1/pσ·x∈B:fx>σ1/p<∞.Note the following:(i)
Ifψ(x)≡1, then Mp,ψ(w)=Lwp(Rn) and WMp,ψ(w)=WLwp(Rn). Thus, our (weak) Morrey type space is an extension of the weighted (weak) Lebesgue space.(ii)
Ifψ(x)=xκ with 0<κ<1, then Mp,ψ(w) is just the weighted Morrey space Lp,κ(w), and WM1,ψ(w) is just the weighted weak Morrey space WL1,κ(w).(iii)
Ifw(x)≡1, below we will show that Mp,ψ(Rn) reduces to the generalized Morrey space Lp,Ψ(Rn), and WM1,ψ(Rn) reduces to the generalized weak Morrey space WL1,Ψ(Rn).Our main results on the boundedness ofTθ in the Morrey type spaces Mp,ψ(w) can be formulated as follows.Theorem 10.
Let1<p<∞ and w∈Ap. Assume that ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the θ-type Calderón–Zygmund operator Tθ is bounded on Mp,ψ(w).Theorem 11.
Letp=1 and w∈A1. Assume that ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the θ-type Calderón–Zygmund operator Tθ is bounded from M1,ψ(w) into WM1,ψ(w).Letθ be a nonnegative, nondecreasing function on R+=(0,+∞) satisfying condition (8), and let [b,Tθ] be the commutator formed by Tθ and BMO function b. For the strong-type estimate of the linear commutator [b,Tθ] in Mp,ψ(w) with 1<p<∞, we will prove the following.Theorem 12.
Let1<p<∞, w∈Ap, and b∈BMO(Rn). Assume that θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the commutator operator [b,Tθ] is bounded on Mp,ψ(w).To obtain endpoint estimate for the linear commutator[b,Tθ] in M1,ψ(w), we first need to define the weighted A-average of a function f over a ball B by means of the weighted Luxemburg norm; that is, given a Young function A and w∈A∞, we define (see [13, 14])(30)fAw,B≔infσ>0:1wB∫BAfxσ·wxdx≤1.When A(t)=t, this norm is denoted by ·L(w),B; when A(t)=t·(1+log+t), this norm is also denoted by ·LlogL(w),B. The complementary Young function of t·(1+log+t) is exp⁡t-1 with mean Luxemburg norm denoted by ·exp⁡L(w),B. For w∈A∞ and for every ball B in Rn, we can also show the weighted version of (20). Namely, the following generalized Hölder’s inequality in the weighted setting(31)1wB∫Bfx·gxwxdx≤CfLlog⁡Lw,Bgexp⁡Lw,Bis valid (see [14], for instance). Now we introduce new Morrey type spaces of Llog⁡L type associated with ψ as follows.Definition 13.
Letp=1, 0≤κ<1, ψ satisfy the Dκ condition (26), and w be a weight function on Rn. One can denote by MLlog⁡L1,ψ(w) the generalized weighted Morrey space of LlogL type, the space of all locally integrable functions f defined on Rn with finite norm fMLlog⁡L1,ψ(w).(32)MLlog⁡L1,ψw≔f∈Lloc1w:fMLlog⁡L1,ψw<∞,where(33)fMLlog⁡L1,ψw≔supBwBψwB·fLlog⁡Lw,B.Note thatt≤t·(1+log+t) for all t>0; then, for any ball B⊂Rn and w∈A∞, we have fLw,B≤fLlog⁡L(w),B by definition; that is, the inequality(34)fLw,B=1wB∫Bfx·wxdx≤fLlog⁡Lw,Bholds for any ball B⊂Rn. From this, we can further see that when ψ satisfies the Dκ condition (26) with 0≤κ<1,(35)1ψwB∫Bfx·wxdx=wBψwB·1wB∫Bfx·wxdx=wBψwB·fLw,B≤wBψwB·fLlogLw,B.Hence, we have MLlogL1,ψ(w)⊂M1,ψ(w) by definition.Definition 14.
In the unweighted case (whenw equals a constant function), one can denote by MLlog⁡L1,ψ(Rn) the generalized unweighted Morrey space of Llog⁡L type. That is, let p=1 and ψ satisfy the Dκ condition (26) with 0≤κ<1; one can define(36)MLlog⁡L1,ψRn≔f∈Lloc1Rn:fMLlog⁡L1,ψRn<∞,where(37)fMLlog⁡L1,ψRn≔supB⁡BψB·fLlog⁡L,B.We also consider the special case whenψ is taken to be ψ(x)=xκ with 0<κ<1 and denote the corresponding space by LLlog⁡L1,κ(w).Definition 15.
Letp=1, 0<κ<1, and w be a weight function on Rn. One can denote by LLlog⁡L1,κ(w) the weighted Morrey space of Llog⁡L type, the space of all locally integrable functions f defined on Rn with finite norm fLLlog⁡L1,κ(w).(38)LLlog⁡L1,κw≔f∈Lloc1w:fLLlog⁡L1,κw<∞,where(39)fLLlogL1,κw≔supBwB1-κ·fLlog⁡Lw,B.In this situation, we have LLlog⁡L1,κ(w)⊂L1,κ(w).For the endpoint case, we will also prove the following weak-typeLlog⁡L estimate of the linear commutator [b,Tθ] in the Morrey type space associated with ψ.Theorem 16.
Letp=1, w∈A1, and b∈BMO(Rn). Assume that θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1; then, for any given σ>0 and any ball B⊂Rn, there exists a constant C>0 independent of f, B, and σ>0 such that(40)1ψwB·wx∈B:b,Tθfx>σ≤C·ΦfσMLlog⁡L1,ψw,where Φ(t)=t·(1+log+t). From the definitions, we can roughly say that the commutator operator [b,Tθ] is bounded from MLlog⁡L1,ψ(w) into WM1,ψ(w).In particular, if we takeψ(x)=xκ with 0<κ<1, then we immediately get the following strong-type estimate and endpoint estimate of Tθ and [b,Tθ] in the weighted Morrey spaces Lp,κ(w) for all 0<κ<1 and 1≤p<∞.Corollary 17.
Let1<p<∞, 0<κ<1, and w∈Ap. Then, the θ-type Calderón–Zygmund operator Tθ is bounded on Lp,κ(w).Corollary 18.
Letp=1, 0<κ<1, and w∈A1. Then, the θ-type Calderón–Zygmund operator Tθ is bounded from L1,κ(w) into WL1,κ(w).Corollary 19.
Let1<p<∞, 0<κ<1, w∈Ap, and b∈BMO(Rn). Assume that θ satisfies (8); then, the commutator operator [b,Tθ] is bounded on Lp,κ(w).Corollary 20.
Letp=1, 0<κ<1, w∈A1, and b∈BMO(Rn). Assume that θ satisfies (8); then, for any given σ>0 and any ball B⊂Rn, there exists a constant C>0 independent of f, B, and σ>0 such that(41)1wBκ·wx∈B:b,Tθfx>σ≤C·ΦfσLLlog⁡L1,κw,where Φ(t)=t·(1+log+t).Naturally, whenw(x)≡1, we have the following unweighted results.Corollary 21.
Let1<p<∞. Assume that ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the θ-type Calderón–Zygmund operator Tθ is bounded on Mp,ψ(Rn).Corollary 22.
Letp=1. Assume that ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the θ-type Calderón–Zygmund operator Tθ is bounded from M1,ψ(Rn) into WM1,ψ(Rn).Corollary 23.
Let1<p<∞ and b∈BMO(Rn). Assume that θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1; then, the commutator operator [b,Tθ] is bounded on Mp,ψ(Rn).Corollary 24.
Letp=1 and b∈BMO(Rn). Assume that θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1; then, for any given σ>0 and any ball B⊂Rn, there exists a constant C>0 independent of f, B, and σ>0 such that(42)1ψB·x∈B:b,Tθfx>σ≤C·ΦfσMLlog⁡L1,ψRn,where Φ(t)=t·(1+log+t).LetΨ=Ψ(r), r>0, be a growth function with doubling constant D(Ψ):1≤D(Ψ)<2n. If, for any fixed x0∈Rn and r>0, we set ψ(|B(x0,r)|)=Ψ(r), then(43)ψ2nBx0,r=ψBx0,2r=Ψ2r.For the doubling constant D(Ψ) satisfying 1≤D(Ψ)<2n, which means that D(Ψ)=2κ·n for some 0≤κ<1, then we are able to verify that ψ is an increasing function and satisfies the Dκ condition (26) with some 0≤κ<1.Definition 25.
Letp=1 and Ψ be a growth function in (0,+∞). One can denote by LLlog⁡L1,Ψ(Rn) the generalized Morrey space of Llog⁡L type, which is defined by(44)LLlog⁡L1,ΨRn≔f∈Lloc1Rn:fLLlog⁡L1,ΨRn<∞,where(45)fLLlog⁡L1,ΨRn≔supr>0;Bx0,rBx0,rΨr·fLlog⁡L,Bx0,r.In this situation, we also have LLlogL1,Ψ(Rn)⊂L1,Ψ(Rn).From the definitions given above, we getMp,ψ(Rn)=Lp,Ψ(Rn), WM1,ψ(Rn)=WL1,Ψ(Rn), and MLlog⁡L1,ψ(Rn)=LLlog⁡L1,Ψ(Rn) by the choice of Ψ. Thus, by the above unweighted results (Corollaries 21–24), we can also obtain strong-type estimate and endpoint estimate of Tθ and [b,Tθ] in the generalized Morrey spaces Lp,Ψ(Rn) when 1≤p<∞ and Ψ satisfies the doubling condition (23).Corollary 26.
Let1<p<∞. Suppose that Ψ satisfies the doubling condition (23) and 1≤D(Ψ)<2n; then, the θ-type Calderón–Zygmund operator Tθ is bounded on Lp,Ψ(Rn).Corollary 27.
Letp=1. Suppose that Ψ satisfies the doubling condition (23) and 1≤D(Ψ)<2n; then, the θ-type Calderón–Zygmund operator Tθ is bounded from L1,Ψ(Rn) into WL1,Ψ(Rn).Corollary 28.
Let1<p<∞ and b∈BMO(Rn). Suppose that θ satisfies (8) and Ψ satisfies the doubling condition (23) with 1≤D(Ψ)<2n; then, the commutator operator [b,Tθ] is bounded on Lp,Ψ(Rn).Corollary 29.
Letp=1 and b∈BMO(Rn). Suppose that θ satisfies (8) and Ψ satisfies the doubling condition (23) with 1≤D(Ψ)<2n; then, for any given σ>0 and any ball B(x0,r)⊂Rn, there exists a constant C>0 independent of f, B(x0,r), and σ>0 such that(46)1Ψr·x∈Bx0,r:b,Tθfx>σ≤C·ΦfσLLlog⁡L1,ΨRn,where Φ(t)=t·(1+log+t).


## 3. Proof of Theorems10 and 11

Proof of Theorem10.
Letf∈Mp,ψ(w) with 1<p<∞ and w∈Ap. For an arbitrary point x0∈Rn, set B=B(x0,rB) for the ball centered at x0 and of radius rB, 2B=B(x0,2rB). We represent f as(47)f=f·χ2B+f·χ2Bc≔f1+f2;by the linearity of the θ-type Calderón–Zygmund operator Tθ, we write(48)1ψwB1/p∫BTθfxpwxdx1/p≤1ψwB1/p∫BTθf1xpwxdx1/p+1ψwB1/p∫BTθf2xpwxdx1/p≔I1+I2.Below, we will give the estimates of I1 and I2, respectively. By the weighted Lp boundedness of Tθ (see Theorem 4), we have(49)I1≤1ψwB1/pTθf1Lwp≤C·1ψwB1/p∫2Bfxpwxdx1/p≤CfMp,ψw·ψw2B1/pψwB1/p.Moreover, since 0<w(B)<w(2B)<+∞ when w∈Ap with 1<p<∞, then by the Dκ condition (26) of ψ and inequality (12), we obtain(50)I1≤CfMp,ψw·w2Bκ/pwBκ/p≤CfMp,ψw.As for the term I2, it is clear that when x∈B and y∈(2B)c, we get |x-y|≈|x0-y|. We then decompose Rn into a geometrically increasing sequence of concentric balls and obtain the following pointwise estimate:(51)Tθf2x≤∫Rnf2yx-yndy≤C∫2Bcfyx0-yndy≤C∑j=1∞12j+1B∫2j+1Bfydy.From this, it follows that(52)I2≤C·wB1/pψwB1/p∑j=1∞12j+1B∫2j+1Bfydy.By using Hölder’s inequality and Ap condition on w, we get(53)12j+1B∫2j+1Bfydy≤12j+1B∫2j+1Bfypwydy1/p∫2j+1Bwy-p′/pdy1/p′≤CfMp,ψw·ψw2j+1B1/pw2j+1B1/p.Hence,(54)I2≤CfMp,ψw∑j=1∞ψw2j+1B1/pψwB1/p·wB1/pw2j+1B1/p.Notice that w∈Ap⊂A∞ for 1<p<∞; then, by using the Dκ condition (26) of ψ again, inequality (13), and the fact that 0≤κ<1, we find that(55)∑j=1∞ψw2j+1B1/pψwB1/p·wB1/pw2j+1B1/p≤C∑j=1∞wB1-κ/pw2j+1B1-κ/p≤C∑j=1∞B2j+1Bδ1-κ/p≤C∑j=1∞12j+1nδ1-κ/p≤C,which gives our desired estimate I2≤CfMp,ψ(w). Combining the estimates above for I1 and I2 and then taking the supremum over all balls B⊂Rn, we complete the proof of Theorem 10.Proof of Theorem11.
Letf∈M1,ψ(w) with w∈A1. For an arbitrary ball B=B(x0,rB)⊂Rn, we represent f as(56)f=f·χ2B+f·χ2Bc≔f1+f2;then, for any given σ>0, by the linearity of the θ-type Calderón–Zygmund operator Tθ, one can write(57)1ψwBσ·wx∈B:Tθfx>σ≤1ψwBσ·wx∈B:Tθf1x>σ2+1ψwBσ·wx∈B:Tθf2x>σ2≔I1′+I2′.We first consider the term I1′. By the weighted weak (1, 1) boundedness of Tθ (see Theorem 4), we have(58)I1′≤C·1ψwBf1Lw1=C·1ψwB∫2Bfxwxdx≤CfM1,ψw·ψw2BψwB.Moreover, since 0<w(B)<w(2B)<+∞ when w∈A1, then we apply the Dκ condition (26) of ψ and inequality (12) to obtain that(59)I1′≤CfM1,ψw·w2BκwBκ≤CfM1,ψw.As for the term I2′, it follows directly from Chebyshev’s inequality and the pointwise estimate (51) that(60)I2′≤1ψwBσ·2σ∫BTθf2xwxdx≤C·wBψwB∑j=1∞12j+1B∫2j+1Bfydy.Another application of A1 condition on w gives that(61)12j+1B∫2j+1Bfydy≤C1w2j+1B·essinfy∈2j+1Bwy∫2j+1Bfydy≤C1w2j+1B∫2j+1Bfywydy≤CfM1,ψw·ψw2j+1Bw2j+1B.Consequently,(62)I2′≤CfM1,ψw∑j=1∞ψw2j+1BψwB·wBw2j+1B.Recall that w∈A1⊂A∞; therefore, by using the Dκ condition (26) of ψ again, inequality (13), and the fact that 0≤κ<1, we get(63)∑j=1∞ψw2j+1BψwB·wBw2j+1B≤C∑j=1∞wB1-κw2j+1B1-κ≤C∑j=1∞B2j+1Bδ∗1-κ≤C∑j=1∞12j+1nδ∗1-κ≤C,which implies our desired estimate I2′≤CfM1,ψ(w). Summing up the estimates above for I1′ and I2′ and then taking the supremum over all balls B⊂Rn and all σ>0, we finish the proof of Theorem 11.


## 4. Proof of Theorems12 and 16

To prove our main theorems in this section, we need the following lemma aboutBMO functions.Lemma 30.
Letb be a function in BMO(Rn). Then,(i)
for every ballB in Rn and for all j∈Z+,(64)b2j+1B-bB≤C·j+1b∗;(ii)
for every ballB in Rn and for all w∈Ap with 1≤p<∞,(65)∫Bbx-bBpwxdx1/p≤Cb∗·wB1/p.Proof.
For the proof of(i), we refer the reader to [3]. For the proof of (ii), we refer the reader to [15].Proof of Theorem12.
Letf∈Mp,ψ(w) with 1<p<∞ and w∈Ap. For each fixed ball B=B(x0,rB)⊂Rn, as before, we represent f as f=f1+f2, where f1=f·χ2B and 2B=B(x0,2rB)⊂Rn. By the linearity of the commutator operator [b,Tθ], we write(66)1ψwB1/p∫Bb,Tθfxpwxdx1/p≤1ψwB1/p∫Bb,Tθf1xpwxdx1/p+1ψwB1/p∫Bb,Tθf2xpwxdx1/p≔J1+J2.Since Tθ is bounded on Lwp(Rn) for 1<p<∞ and w∈Ap, then by the well-known boundedness criterion for the commutators of linear operators, which was obtained by Álvarez et al. in [16], we know that [b,Tθ] is also bounded on Lwp(Rn) for all 1<p<∞ and w∈Ap, whenever b∈BMO(Rn). This fact together with the Dκ condition (26) of ψ and inequality (12) implies(67)J1≤1ψwB1/pb,Tθf1Lwp≤C·1ψwB1/p∫2Bfxpwxdx1/p≤CfMp,ψw·ψw2B1/pψwB1/p≤CfMp,ψw·w2Bκ/pwBκ/p≤CfMp,ψw.Let us now turn to the estimate of J2. By definition, for any x∈B, we have(68)b,Tθf2x≤bx-bB·Tθf2x+TθbB-bf2x.In the proof of Theorem 10, we have already shown that (see (51))(69)Tθf2x≤C∑j=1∞12j+1B∫2j+1Bfydy.Following the same arguments as in (51), we can also prove that(70)TθbB-bf2x≤∫RnbB-byf2yx-yndy≤C∫2BcbB-byfyx0-yndy≤C∑j=1∞12j+1B∫2j+1Bby-bB·fydy.Hence, from the pointwise estimates above for |Tθ(f2)(x)| and |Tθ([bB-b]f2)(x)|, it follows that(71)J2≤CψwB1/p∫Bbx-bBpwxdx1/p∑j=1∞12j+1B∫2j+1Bfydy+C·wB1/pψwB1/p∑j=1∞12j+1B∫2j+1Bb2j+1B-bB·fydy+C·wB1/pψwB1/p∑j=1∞12j+1B∫2j+1Bby-b2j+1B·fydy≔J3+J4+J5.Below, we will give the estimates of J3, J4, and J5, respectively. Using (ii) of Lemma 30, Hölder’s inequality, and the Ap condition, we obtain(72)J3≤Cb∗·wB1/pψwB1/p∑j=1∞12j+1B∫2j+1Bfydy≤Cb∗·wB1/pψwB1/p∑j=1∞12j+1B∫2j+1Bfypwydy1/p∫2j+1Bwy-p′/pdy1/p′≤CfMp,ψw∑j=1∞ψw2j+1B1/pψwB1/p·wB1/pw2j+1B1/p≤CfMp,ψw,where in the last inequality we have used the estimate (55). Applying (i) of Lemma 30, Hölder’s inequality, and the Ap condition, we can deduce that(73)J4≤Cb∗·wB1/pψwB1/p∑j=1∞j+12j+1B∫2j+1Bfydy≤Cb∗·wB1/pψwB1/p∑j=1∞j+12j+1B∫2j+1Bfypwydy1/p∫2j+1Bwy-p′/pdy1/p′≤CfMp,ψw∑j=1∞j+1·ψw2j+1B1/pψwB1/p·wB1/pw2j+1B1/p.For any j∈Z+, since 0<w(B)<w(2j+1B)<+∞ when w∈Ap with 1<p<∞, then by using the Dκ condition (26) of ψ and inequality (13) together with the fact that 0≤κ<1, we thus obtain(74)∑j=1∞j+1·ψw2j+1B1/pψwB1/p·wB1/pw2j+1B1/p≤C∑j=1∞j+1·wB1-κ/pw2j+1B1-κ/p≤C∑j=1∞j+1·B2j+1Bδ1-κ/p≤C∑j=1∞j+1·12j+1nδ1-κ/p≤C,where the last series is convergent since the exponent δ(1-κ)/p is positive. This implies our desired estimate J4≤CfMp,ψ(w). It remains to estimate the last term J5. An application of Hölder’s inequality gives us that(75)J5≤C·wB1/pψwB1/p∑j=1∞12j+1B∫2j+1Bfypwydy1/p∫2j+1Bby-b2j+1Bp′wy-p′/pdy1/p′.If we set μ(y)=w(y)-p′/p, then we have μ∈Ap′ because w∈Ap (see [1, 2]). Thus, it follows from (ii) of Lemma 30 and the Ap condition that(76)∫2j+1Bby-b2j+1Bp′μydy1/p′≤Cb∗·μ2j+1B1/p′=Cb∗·∫2j+1Bwy-p′/pdy1/p′≤Cb∗·2j+1Bw2j+1B1/p.Therefore, in view of estimate (55), we conclude that(77)J5≤Cb∗·wB1/pψwB1/p∑j=1∞1w2j+1B1/p∫2j+1Bfypwydy1/p≤CfMp,ψw∑j=1∞ψw2j+1B1/pψwB1/p·wB1/pw2j+1B1/p≤CfMp,ψw.Summarizing the estimates derived above and then taking the supremum over all balls B⊂Rn, we complete the proof of Theorem 12.Proof of Theorem16.
For any fixed ballB=B(x0,rB) in Rn, as before, we represent f as f=f1+f2, where f1=f·χ2B and 2B=B(x0,2rB)⊂Rn. Then, for any given σ>0, by the linearity of the commutator operator [b,Tθ], one can write(78)1ψwB·wx∈B:b,Tθfx>σ≤1ψwB·wx∈B:b,Tθf1x>σ2+1ψwB·wx∈B:b,Tθf2x>σ2≔J1′+J2′.By using Theorem 5 and the previous estimate (35), we get(79)J1′≤C·1ψwB∫RnΦf1xσ·wxdx=C·1ψwB∫2BΦfxσ·wxdx=C·ψw2BψwB·1ψw2B∫2BΦfxσ·wxdx≤C·ψw2BψwB·w2Bψw2B·ΦfσLlog⁡Lw,2B.Moreover, since 0<w(B)<w(2B)<+∞ when w∈A1, then by the Dκ condition (26) of ψ and inequality (12), we have(80)J1′≤C·w2BκwBκ·w2Bψw2B·ΦfσLlogLw,2B≤C·ΦfσMLlogL1,ψw,which is our desired estimate. We now turn to deal with the term J2′. Recall that the following inequality(81)b,Tθf2x≤bx-bB·Tθf2x+TθbB-bf2xis valid. So we can further decompose J2′ as(82)J2′≤1ψwB·wx∈B:bx-bB·Tθf2x>σ4+1ψwB·wx∈B:TθbB-bf2x>σ4≔J3′+J4′.By using the previous pointwise estimate (51) and Chebyshev’s inequality together with (ii) of Lemma 30, we deduce that(83)J3′≤1ψwB·4σ∫Bbx-bB·Tθf2xwxdx≤C∑j=1∞12j+1B∫2j+1Bfyσdy1ψwB·∫Bbx-bBwxdx≤Cb∗∑j=1∞12j+1B∫2j+1BfyσdywBψwB.Furthermore, note that t≤Φ(t)=t·(1+log+t) for any t>0. It then follows from the A1 condition and the previous estimate (34) that(84)J3′≤C∑j=1∞1w2j+1B∫2j+1Bfyσ·wydywBψwB≤C∑j=1∞1w2j+1B∫2j+1BΦfyσ·wydywBψwB≤C∑j=1∞ΦfσLlogLw,2j+1BwBψwB=C∑j=1∞w2j+1Bψw2j+1B·ΦfσLlogLw,2j+1Bψw2j+1BψwB·wBw2j+1B≤C·ΦfσMLlogL1,ψw∑j=1∞ψw2j+1BψwB·wBw2j+1B≤C·ΦfσMLlogL1,ψw,where in the last inequality we have used estimate (63). On the other hand, applying the pointwise estimate (70) and Chebyshev’s inequality, we have(85)J4′≤1ψwB·4σ∫BTθbB-bf2xwxdx≤wBψwB·Cσ∑j=1∞12j+1B∫2j+1Bby-bB·fydy≤wBψwB·Cσ∑j=1∞12j+1B∫2j+1Bby-b2j+1B·fydy+wBψwB·Cσ∑j=1∞12j+1B∫2j+1Bb2j+1B-bB·fydy≔J5′+J6′.For the term J5′, since w∈A1, by the A1 condition and the fact that t≤Φ(t),(86)J5′≤Cσ·wBψwB∑j=1∞1w2j+1B∫2j+1Bby-b2j+1B·fywydy≤C·wBψwB∑j=1∞1w2j+1B∫2j+1Bby-b2j+1B·Φfyσwydy.Furthermore, we use the generalized Hölder’s inequality with weight (31) to obtain(87)J5′≤C·wBψwB∑j=1∞b-b2j+1Bexp⁡Lw,2j+1BΦfσLlogLw,2j+1B≤Cb∗·wBψwB∑j=1∞ΦfσLlogLw,2j+1B.In the last inequality, we have used the well-known fact that (see [14])(88)b-bBexp⁡Lw,B≤Cb∗,foranyballB⊂Rn.It is equivalent to the inequality(89)1wB∫Bexp⁡by-bBc0b∗wydy≤C,which is just a corollary of the well-known John–Nirenberg’s inequality (see [9]) and the comparison property of A1 weights. Hence, by estimate (63),(90)J5′≤Cb∗∑j=1∞w2j+1Bψw2j+1B·ΦfσLlogLw,2j+1Bψw2j+1BψwB·wBw2j+1B≤C·ΦfσMLlogL1,ψw∑j=1∞ψw2j+1BψwB·wBw2j+1B≤C·ΦfσMLlogL1,ψw.For the last term J6′, we proceed as follows. Using (i) of Lemma 30 together with the facts w∈A1 and t≤Φ(t)=t·(1+log+t), we deduce that(91)J6′≤C·wBψwB∑j=1∞j+1b∗·12j+1B∫2j+1Bfyσdy≤C·wBψwB∑j=1∞j+1b∗·1w2j+1B∫2j+1Bfyσ·wydy≤Cb∗·wBψwB∑j=1∞j+1w2j+1B∫2j+1BΦfyσ·wydy=Cb∗∑j=1∞w2j+1Bψw2j+1B·ΦfσLlogLw,2j+1Bj+1·ψw2j+1BψwB·wBw2j+1B≤C·ΦfσMLlogL1,ψw∑j=1∞j+1·ψw2j+1BψwB·wBw2j+1B.Recall that w∈A1⊂A∞. We can now argue exactly as we did in the estimation of (74) to get(92)∑j=1∞j+1·ψw2j+1BψwB·wBw2j+1B≤C∑j=1∞j+1·wB1-κw2j+1B1-κ≤C∑j=1∞j+1·B2j+1Bδ∗1-κ≤C∑j=1∞j+1·12j+1nδ∗1-κ≤C.Let us now substitute this estimate (92) into the term J6′; we get the desired inequality(93)J6′≤C·ΦfσMLlogL1,ψw.This completes the proof of Theorem 16.


## 5. Partial Results on Two-Weight Problems

In the last section, we consider related problems about two-weight, weak-type(p,p) inequalities with 1<p<∞. Let T be the classical Calderón–Zygmund operator with standard kernel; that is, T=Tθ when θ(t)=tδ with 0<δ≤1. It is well known that T is a bounded operator on Lwp(Rn) for all 1<p<∞ and w∈Ap, and, of course, T is a bounded operator from Lwp(Rn) into WLwp(Rn). In the two-weight context, however, the Ap condition is “not” sufficient for the weak-type (p,p) inequality for T. More precisely, given a pair of weights (u,v) and p, 1<p<∞, the weak-type inequality(94)ux∈Rn:Tfx>σ≤Cσp∫Rnfxpvxdxdoes not hold if (u,v)∈Ap: there exists a positive constant C such that, for every cube Q⊂Rn,(95)1Q∫Quxdx1/p1Q∫Qvx-p′/pdx1/p′≤C<∞;one can see [17, 18] for some counterexamples. Here, all cubes are assumed to have their sides parallel to the coordinate axes; Q(x0,l) will denote the cube centered at x0 and has side length l. In [17, 19], Cruz-Uribe and Pérez considered the problem of finding sufficient conditions on a pair of weights (u,v) such that T satisfies the weak-type (p,p) inequality (94) (1<p<∞). They showed in [19] that if we strengthened the Ap condition (95) by adding a “power bump” to the left-hand term, then inequality (94) holds for all f∈Lvp(Rn). More specifically, if there exists a number r>1 such that, for every cube Q in Rn,(96)1Q∫Quxrdx1/rp1Q∫Qvx-p′/pdx1/p′≤C<∞,then the classical Calderón–Zygmund operator T is bounded from Lvp(Rn) into WLup(Rn). Moreover, in [17], the authors improved this result by replacing the “power bump” in (96) by a smaller “Orlicz bump.” To be more precise, they introduced the following Ap-type condition in the scale of Orlicz spaces:(97)uLlogLp-1+δ,Q1/p1Q∫Qvx-p′/pdx1/p′≤C<∞,δ>0,where uL(logL)p-1+δ,Q is the mean Luxemburg norm of u on cube Q with Young function A(t)=t·(1+log+t)p-1+δ. It was shown that inequality (94) still holds under the Ap-type condition on (u,v), and this result is sharp since it does not hold in general when δ=0.On the other hand, the following Sharp function estimate forTθ was established in [8]: there exists some δ, 0<δ<1, and a positive constant C=Cδ such that, for any f∈C0∞(Rn) and x∈Rn,(98)M♯Tθfδx1/δ≤CMfx,where M is the standard Hardy–Littlewood maximal operator and M♯ is the well-known Sharp maximal operator defined as(99)M♯fx≔supx∈Q1Q∫Qfy-fQdy.Here, the supremum is taken over all the cubes containing x and fQ denotes the mean value of f over Q; namely, fQ=(1/|Q|)∫Qf(x)dx. It was pointed out in [19] (Remark 1.3) that, by using this Sharp function estimate (98), we can also show inequality (94) is true for more general operator Tθ, under condition (96) on (u,v). Then, we obtain a sufficient condition for Tθ to be weak (p,p) with 1<p<∞.Theorem 31.
Let1<p<∞. Given a pair of weights (u,v), suppose that, for some r>1 and for all cubes Q,(100)1Q∫Quxrdx1/rp1Q∫Qvx-p′/pdx1/p′≤C<∞.Then, the θ-type Calderón–Zygmund operator Tθ satisfies the weak-type (p,p) inequality:(101)ux∈Rn:Tθfx>σ≤Cσp∫Rnfxpvxdx,where C does not depend on f and σ>0.We want to extend Theorem31 to the Morrey type spaces. In order to do so, we need to define Morrey type spaces associated with ψ with two weights.Definition 32.
Let1≤p<∞, 0≤κ<1, and ψ satisfy the Dκ condition (26). For two weights u and v, one can denote by Mp,ψ(v,u) the generalized weighted Morrey space, the space of all locally integrable functions f with finite norm.(102)Mp,ψv,u≔f∈Llocpv:fMp,ψv,u<∞,where the norm is given by(103)fMp,ψv,u≔supQ1ψuQ∫Qfxpvxdx1/p.Note that(i)
ifu=v=w, then Mp,ψ(v,u) is the space Mp,ψ(w) in Definition 8;(ii)
ifψ(x)=xκ with 0<κ<1, then Mp,ψ(v,u) is just the weighted Morrey space with two weights Lp,κ(v,u), which was introduced by Komori and Shirai in [12].We are now ready to prove the following result.Theorem 33.
Let1<p<∞ and u∈A∞. Given a pair of weights (u,v), suppose that, for some r>1 and for all cubes Q,(104)1Q∫Quxrdx1/rp1Q∫Qvx-p′/pdx1/p′≤C<∞.If ψ satisfies the Dκ condition (26) with 0≤κ<1, then the θ-type Calderón–Zygmund operator Tθ is bounded from Mp,ψ(v,u) into WMp,ψ(u).Proof of Theorem33.
Letf∈Mp,ψ(v,u) with 1<p<∞. For any cube Q=Q(x0,l)⊂Rn and λ>0, we will denote by λQ the cube concentric with Q whose each edge is λ times as long; that is, λQ=Q(x0,λl). Let(105)f=f·χ2Q+f·χ2Qc≔f1+f2,where χ2Q denotes the characteristic function of 2Q=Q(x0,2l). Then, for any given σ>0, we write(106)1ψuQ1/pσ·ux∈Q:Tθfx>σ1/p≤1ψuQ1/pσ·ux∈Q:Tθf1x>σ21/p+1ψuQ1/pσ·ux∈Q:Tθf2x>σ21/p≔K1+K2.Using Theorem 31, the Dκ condition (26) of ψ, and inequality (12) (consider cube Q instead of ball B), we get(107)K1≤C·1ψuQ1/p∫Rnf1xpvxdx1/p=C·1ψuQ1/p∫2Qfxpvxdx1/p≤CfMp,ψv,u·ψu2Q1/pψuQ1/p≤CfMp,ψv,u·u2Qκ/puQκ/p≤CfMp,ψv,u.As for the term K2, using the same methods and steps as those we dealt with I2 in Theorem 10, we can also obtain that, for any x∈Q,(108)Tθf2x≤C∑j=1∞12j+1Q∫2j+1Qfydy.This pointwise estimate together with Chebyshev’s inequality implies(109)K2≤2ψuQ1/p·∫QTθf2xpuxdx1/p≤C·uQ1/pψuQ1/p∑j=1∞12j+1Q∫2j+1Qfydy.Moreover, an application of Hölder’s inequality gives that(110)K2≤C·uQ1/pψuQ1/p∑j=1∞12j+1Q∫2j+1Qfypvydy1/p∫2j+1Qvy-p′/pdy1/p′≤CfMp,ψv,u·uQ1/pψuQ1/p∑j=1∞ψu2j+1Q1/p2j+1Q∫2j+1Qvy-p′/pdy1/p′.For any j∈Z+, since 0<u(Q)<u(2j+1Q)<+∞ when u is a weight function, then by the Dκ condition (26) of ψ with 0≤κ<1, we can see that(111)ψu2j+1Q1/pψuQ1/p≤u2j+1Qκ/puQκ/p.In addition, we apply Hölder’s inequality with exponent r to get(112)u2j+1Q=∫2j+1Quydy≤2j+1Q1/r′∫2j+1Quyrdy1/r.Hence, in view of (111) and (112) derived above, we have(113)K2≤CfMp,ψv,u∑j=1∞uQ1-κ/pu2j+1Q1-κ/p·u2j+1Q1/p2j+1Q∫2j+1Qvy-p′/pdy1/p′≤CfMp,ψv,u∑j=1∞uQ1-κ/pu2j+1Q1-κ/p·2j+1Q1/r′p2j+1Q∫2j+1Quyrdy1/rp∫2j+1Qvy-p′/pdy1/p′≤CfMp,ψv,u∑j=1∞uQ1-κ/pu2j+1Q1-κ/p.The last inequality is obtained by condition (96) on (u,v). Furthermore, by our additional hypothesis on u:u∈A∞ and inequality (13) (consider cube Q instead of ball B), we get(114)∑j=1∞uQ1-κ/pu2j+1Q1-κ/p≤C∑j=1∞Q2j+1Qδ1-κ/p≤C∑j=1∞12j+1nδ1-κ/p≤C,which implies our desired estimate K2≤CfMp,ψ(v,u). Summing up the estimates above for K1 and K2 and then taking the supremum over all cubes Q⊂Rn and all σ>0, we finish the proof of Theorem 33.LetM denote the Hardy–Littlewood maximal operator and M♯ denote the Sharp maximal operator. For δ>0, we define(115)Mδf≔Mfδ1/δ,Mδ♯f≔M♯fδ1/δ.The maximal function associated with A(t)=t(1+log+t) is defined as(116)MLlogLfx≔supx∈QfLlogL,Q,where the supremum is taken over all the cubes containing x. Let b∈BMO(Rn) and [b,Tθ] be the commutator of the θ-type Calderón–Zygmund operator. In [8], it was proved that if θ satisfies condition (8), then, for 0<δ<ε<1, there exists a positive constant C=Cδ,ε such that, for any f∈C0∞(Rn) and x∈Rn,(117)Mδ♯b,Tθfx≤Cb∗MεTθfx+MLlogLfx.Using this Sharp function estimate (117) and following the idea of the proof in [19], we can also establish the two-weight, weak-type norm inequality for [b,Tθ].Theorem 34.
Let1<p<∞ and b∈BMO(Rn). Given a pair of weights (u,v), suppose that, for some r>1 and for all cubes Q,(118)1Q∫Quxrdx1/rpv-1/pA,Q≤C<∞,where A(t)=tp′(1+log+t)p′ is a Young function. If θ satisfies (8), then the commutator operator [b,Tθ] satisfies the weak-type (p,p) inequality:(119)ux∈Rn:b,Tθfx>σ≤Cσp∫Rnfxpvxdx,where C>0 does not depend on f and σ>0.We will extend Theorem34 to the Morrey type spaces. In order to do so, we need the following key lemma.Lemma 35.
Given three Young functionsA, B, and C such that, for all t>0,(120)A-1t·B-1t≤C-1t,where A-1(t) is the inverse function of A(t), then one has the following generalized Hölder’s inequality due to O’Neil [20]: for any cube Q⊂Rn and all functions f and g,(121)f·gC,Q≤2fA,QgB,Q.Theorem 36.
Let1<p<∞, u∈A∞, and b∈BMO(Rn). Given a pair of weights (u,v), suppose that, for some r>1 and for all cubes Q,(122)1Q∫Quxrdx1/rpv-1/pA,Q≤C<∞,where A(t)=tp′(1+log+t)p′. If θ satisfies (8) and ψ satisfies the Dκ condition (26) with 0≤κ<1, then the commutator operator [b,Tθ] is bounded from Mp,ψ(v,u) into WMp,ψ(u).Proof of Theorem36.
Letf∈Mp,ψ(v,u) with 1<p<∞. For an arbitrary cube Q=Q(x0,l) in Rn, as before, we set(123)f=f1+f2,f1=f·χ2Q,f2=f·χ2Qc.Then, for any given σ>0, we write(124)1ψuQ1/pσ·ux∈Q:b,Tθfx>σ1/p≤1ψuQ1/pσ·ux∈Q:b,Tθf1x>σ21/p+1ψuQ1/pσ·ux∈Q:b,Tθf2x>σ21/p≔K1′+K2′.Using Theorem 34, the Dκ condition (26) of ψ, and inequality (12) (consider cube Q instead of ball B), we get(125)K1′≤C·1ψuQ1/p∫Rnf1xpvxdx1/p=C·1ψuQ1/p∫2Qfxpvxdx1/p≤CfMp,ψv,u·ψu2Q1/pψuQ1/p≤CfMp,ψv,u·u2Qκ/puQκ/p≤CfMp,ψv,u.Next we estimate K2′. For any x∈Q, from the definition of [b,Tθ], we can see that(126)b,Tθf2x≤bx-bQ·Tθf2x+TθbQ-bf2x≔ξx+ηx.Thus, we have(127)K2′≤1ψuQ1/pσ·ux∈Q:ξx>σ41/p+1ψuQ1/pσ·ux∈Q:ηx>σ41/p≔K3′+K4′.For the term K3′, it follows from the pointwise estimate (108) mentioned above and Chebyshev’s inequality that(128)K3′≤4ψuQ1/p·∫Qξxpuxdx1/p≤CψuQ1/p·∫Qbx-bQpuxdx1/p∑j=1∞12j+1Q∫2j+1Qfydy≤C·uQ1/pψuQ1/p∑j=1∞12j+1Q∫2j+1Qfydy,where in the last inequality we have used the fact that Lemma 30(ii) still holds when u is an A∞ weight with B replaced by Q. Repeating the arguments in the proof of Theorem 33, we can show that K3′≤CfMp,ψ(v,u). As for the term K4′, using the same methods and steps as those we dealt with J2 in Theorem 12, we can show the following pointwise estimate as well:(129)ηx=TθbQ-bf2x≤C∑j=1∞12j+1Q∫2j+1Qby-bQ·fydy.This together with Chebyshev’s inequality yields(130)K4′≤4ψuQ1/p·∫Qηxpuxdx1/p≤C·uQ1/pψuQ1/p·∑j=1∞12j+1Q∫2j+1Qby-bQ·fydy≤C·uQ1/pψuQ1/p·∑j=1∞12j+1Q∫2j+1Qby-b2j+1Q·fydy+C·uQ1/pψuQ1/p·∑j=1∞12j+1Q∫2j+1Qb2j+1Q-bQ·fydy≔K5′+K6′.An application of Hölder’s inequality yields that(131)K5′≤C·uQ1/pψuQ1/p·∑j=1∞12j+1Q∫2j+1Qfypvydy1/p∫2j+1Qby-b2j+1Qp′vy-p′/pdy1/p′≤CfMp,ψv,u·uQ1/pψuQ1/p·∑j=1∞ψu2j+1Q1/p2j+1Q2j+1Q1/p′b-b2j+1Q·v-1/pC,2j+1Q,where C(t)=tp′ is a Young function. For 1<p<∞, we know the inverse function of C(t) is C-1(t)=t1/p′. Observe that(132)C-1t=t1/p′=t1/p′1+log+t1+log+t=A-1t·B-1t,where(133)At≈tp′1+log+tp′,Bt≈exp⁡t-1. Thus, by Lemma 35 and estimate (88) (when w≡1), we have(134)b-b2j+1Q·v-1/pC,2j+1Q≤Cb-b2j+1QB,2j+1Q·v-1/pA,2j+1Q≤Cb∗·v-1/pA,2j+1Q.Moreover, in view of (111) and (112), we can deduce that(135)K5′≤Cb∗fMp,ψv,u∑j=1∞u2j+1Qκ/puQκ/p·uQ1/p2j+1Q1/p·v-1/pA,2j+1Q≤Cb∗fMp,ψv,u∑j=1∞uQ1-κ/pu2j+1Q1-κ/p12j+1Q∫2j+1Quxrdx1/rp·v-1/pA,2j+1Q≤CfMp,ψv,u∑j=1∞uQ1-κ/pu2j+1Q1-κ/p≤CfMp,ψv,u.The last inequality is obtained by condition (122) on (u,v) and estimate (114). It remains to estimate the last term K6′. Applying Lemma 30(i) (use Q instead of B) and Hölder’s inequality, we get(136)K6′≤C·uQ1/pψuQ1/p∑j=1∞j+1b∗2j+1Q∫2j+1Qfydy≤C·uQ1/pψuQ1/p∑j=1∞j+1b∗2j+1Q∫2j+1Qfypvydy1/p∫2j+1Qvy-p′/pdy1/p′≤CfMp,ψv,u·uQ1/pψuQ1/p∑j=1∞j+1·ψu2j+1Q1/p2j+1Q∫2j+1Qvy-p′/pdy1/p′.Let C(t) and A(t) be the same as before. Obviously, C(t)≤A(t) for all t>0; then, for any cube Q⊂Rn, we have fC,Q≤fA,Q by definition, which implies that condition (122) is stronger than condition (96). This fact together with (111) and (112) yields(137)K6′≤CfMp,ψv,u∑j=1∞j+1·uQ1-κ/pu2j+1Q1-κ/p·u2j+1Q1/p2j+1Q∫2j+1Qvy-p′/pdy1/p′≤CfMp,ψv,u∑j=1∞j+1·uQ1-κ/pu2j+1Q1-κ/p·2j+1Q1/r′p2j+1Q∫2j+1Quyrdy1/rp∫2j+1Qvy-p′/pdy1/p′≤CfMp,ψv,u∑j=1∞j+1·uQ1-κ/pu2j+1Q1-κ/p.Moreover, by our additional hypothesis on u:u∈A∞ and inequality (13) (use Q instead of B), we finally obtain(138)∑j=1∞j+1·uQ1-κ/pu2j+1Q1-κ/p≤C∑j=1∞j+1·Q2j+1Qδ1-κ/p≤C∑j=1∞j+1·12j+1nδ1-κ/p≤C,which in turn gives that K6′≤CfMp,ψ(v,u). Summing up all the estimates above and then taking the supremum over all cubes Q⊂Rn and all σ>0, we therefore conclude the proof of Theorem 36.In particular, if we takeψ(x)=xκ with 0<κ<1, then we immediately get the following two-weight, weak-type (p,p) inequalities for Tθ and [b,Tθ] in the weighted Morrey spaces.Corollary 37.
Let1<p<∞, 0<κ<1, and u∈A∞. Given a pair of weights (u,v), suppose that, for some r>1 and for all cubes Q,(139)1Q∫Quxrdx1/rp1Q∫Qvx-p′/pdx1/p′≤C<∞.Then, the θ-type Calderón–Zygmund operator Tθ is bounded from Lp,κ(v,u) into WLp,κ(u).Corollary 38.
Let1<p<∞, 0<κ<1, u∈A∞, and b∈BMO(Rn). Given a pair of weights (u,v), suppose that, for some r>1 and for all cubes Q,(140)1Q∫Quxrdx1/rpv-1/pA,Q≤C<∞,where A(t)=tp′(1+log+t)p′. If θ satisfies (8), then the commutator operator [b,Tθ] is bounded from Lp,κ(v,u) into WLp,κ(u).



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*Source: 1309348-2016-10-31.xml*