# Boundedness of a Class of Sublinear Operators and Their Commutators on Generalized Morrey Spaces

Authors: Vagif S. Guliyev; Seymur S. Aliyev; Turhan Karaman
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The authors study the boundedness for a large class of sublinear operatorT generated by Calderón-Zygmund operator on generalized Morrey spaces Mp,φ. As an application of this result, the boundedness of the commutator of sublinear operators Ta on generalized Morrey spaces is obtained. In the case a∈BMO(ℝn), 1<p<∞ and Ta is a sublinear operator, we find the sufficient conditions on the pair (φ1,φ2) which ensures the boundedness of the operator Ta from one generalized Morrey space Mp,φ1 to another Mp,φ2. In all cases, the conditions for the boundedness of Ta are given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity of φ1,φ2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.


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## Body

## 1. Introduction

Forx∈ℝn and r>0, we denote by B(x,r) the open ball centered at x of radius r, and by B(x,r)∁ denote its complement. Let |B(x,r)| be the Lebesgue measure of the ball B(x,r).Letf∈L1loc(ℝn). The Hardy-Littlewood maximal operator M is defined by (1.1)Mf(x)=sup⁡t>0|B(x,t)|-1∫B(x,t)|f(y)|dy.LetK be a Calderón-Zygmund singular integral operator, briefly a Calderón-Zygmund operator, that is, a linear operator bounded from L2(ℝn) to L2(ℝn) taking all infinitely continuously differentiable functions f with compact support to the functions f∈L1loc(ℝn) represented by (1.2)Kf(x)=∫Rnk(x,y)f(y)dyx∉supp⁡f.
Such operators were introduced in [1]. Here, k(x,y) is a continuous function away from the diagonal which satisfies the standard estimates: there exist c1>0 and 0<ε≤1 such that (1.3)|k(x,y)|≤c1|x-y|-n,
for all x,y∈ℝn, x≠y, and (1.4)|k(x,y)-k(x′,y)|+|k(y,x)-k(y,x′)|≤c1(|x-x′||x-y|)ε|x-y|-n,
whenever 2|x-x′|≤|x-y|.It is well known that maximal operator and Calderón-Zygmund operators play an important role in harmonic analysis (see [2–6]).Suppose thatT represents a linear or a sublinear operator, which satisfies that for any f∈L1(ℝn) with compact support and x∉suppf(1.5)|Tf(x)|≤c0∫Rn|f(y)||x-y|ndy,
where c0 is independent of f and x.For a functiona, suppose that the commutator operator Ta represents a linear or a sublinear operator, which satisfies that for any f∈L1(ℝn) with compact support and x∉suppf(1.6)|Taf(x)|≤c0∫Rn|a(x)-a(y)||x-y|-n|f(y)|dy,
where c0 is independent of f and x.We point out that the condition (1.5) was first introduced by Soria and Weiss in [7]. The condition (1.5) are satisfied by many interesting operators in harmonic analysis, such as the Calderón-Zygmund operators, Carleson's maximal operator, Hardy-Littlewood maximal operator, C. Fefferman's singular multipliers, R. Fefferman's singular integrals, Ricci-Stein's oscillatory singular integrals, and the Bochner-Riesz means (see [7, 8] for details).In this work, we prove the boundedness of the sublinear operatorT satisfies the condition (1.5) generated by Calderón-Zygmund operator from one generalized Morrey space Mp,φ1 to another Mp,φ2, 1<p<∞, and from M1,φ1 to the weak space WM1,φ2. In the case a∈BMO(ℝn), 1<p<∞ and the commutator operator Ta is a sublinear operator, we find the sufficient conditions on the pair (φ1,φ2) which ensures the boundedness of the operators Ta from Mp,φ1 to Mp,φ2. Finally, as applications, we apply this result to several particular operators such as the pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.ByA≲B, we mean that A≤CB with some positive constant C independent of appropriate quantities. If A≲B and B≲A, we write A≈B and say that A and B are equivalent.


## 2. Morrey Spaces

The classical Morrey spacesMp,λ were originally introduced by Morrey Jr. in [9] to study the local behavior of solutions to second-order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [9, 10].We denote byMp,λ≡Mp,λ(ℝn) the Morrey space, the space of all functions f∈Lploc(ℝn) with finite quasinorm (2.1)‖f‖Mp,λ≡‖f‖Mp,λ(Rn)=sup⁡x∈Rn,r>0r-λ/p‖f‖Lp(B(x,r)),
where 1≤p<∞ and 0≤λ≤n.Note thatMp,0=Lp(ℝn) and Mp,n=L∞(ℝn). If λ<0 or λ>n, then Mp,λ=Θ, where Θ is the set of all functions equivalent to 0 on ℝn.We also denote byWMp,λ≡WMp,λ(ℝn) the weak Morrey space of all functions f∈WLploc(ℝn) for which (2.2)‖f‖WMp,λ≡‖f‖WMp,λ(Rn)=sup⁡x∈Rn,r>0r-λ/p‖f‖WLp(B(x,r))<∞,
where WLp(B(x,r)) denotes the weak Lp-space of measurable functions f for which (2.3)‖f‖WLp(B(x,r))≡‖fχB(x,r)‖WLp(Rn)=sup⁡t>0t|{y∈B(x,r):|f(y)|>t}|1/p=sup⁡0<t≤|B(x,r)|t1/p(fχB(x,r))*(t)<∞.
Here, g* denotes the nonincreasing rearrangement of a function g.Chiarenza and Frasca [11] studied the boundedness of the maximal operator M in these spaces. Their result can be summarized as follows.Theorem 2.1.
Let1≤p<∞ and  0≤λ<n. Then, for  p>1 the operator  M is bounded on  Mp,λ and for  p=1M is bounded from  M1,λ to  WM1,λ.Di Fazio and Ragusa [12] studied the boundedness of the Calderón-Zygmund operators in Morrey spaces, and their results imply the following statement for Calderón-Zygmund operators K.Theorem 2.2.
Let1≤p<∞,  0<λ<n. Then, for  1<p<∞ Calderón-Zygmund operator  K is bounded on  Mp,λ and for  p=1K is bounded from  M1,λ to  WM1,λ.Note that Theorem2.2 was proved by Peetre [10] in the case of the classical Calderón-Zygmund singular integral operators.


## 3. Generalized Morrey Spaces

We find it convenient to define the generalized Morrey spaces in the form as follows.Definition 3.1.
Letφ(x,r) be a positive measurable function on ℝn×(0,∞) and 1≤p<∞. We denote by Mp,φ≡Mp,φ(ℝn) the generalized Morrey space, the space of all functions f∈Lploc(ℝn) with finite quasinorm
(3.1)‖f‖Mp,φ≡‖f‖Mp,φ(Rn)=sup⁡x∈Rn,r>0φ(x,r)-1|B(x,r)|-1/p‖f‖Lp(B(x,r)).
Also, by WMp,φ≡WMp,φ(ℝn) we denote the weak generalized Morrey space of all functions f∈WLploc(ℝn) for which
(3.2)‖f‖WMp,φ≡‖f‖WMp,φ(Rn)=sup⁡x∈Rn,r>0φ(x,r)-1|B(x,r)|-1/p‖f‖WLp(B(x,r))<∞.According to this definition, we recover the spacesMp,λ and WMp,λ under the choice φ(x,r)=r(λ-n)/p: (3.3)Mp,λ=Mp,φ|φ(x,r)=r(λ-n)/p,WMp,λ=WMp,φ|φ(x,r)=r(λ-n)/p.In [13–19], there were obtained sufficient conditions on φ1 and φ2 for the boundedness of the maximal operator M and Calderón-Zygmund operator K from Mp,φ1 to Mp,φ2, 1<p<∞ (see also [20–23]). In [19], the following condition was imposed on φ(x,r): (3.4)c-1φ(x,r)≤φ(x,t)≤cφ(x,r),
whenever r≤t≤2r, where c (≥1) does not depend on t, r and x∈ℝn, jointly with the condition (3.5)∫r∞φ(x,t)pdtt≤Cφ(x,r)p,
for the sublinear operator T satisfies the condition (1.5), where C (>0) does not depend on r and x∈ℝn.


## 4. Sublinear Operators Generated by Calderón-Zygmund Operators in the SpacesMp,φ

In [24] (see, also [25, 26]), the following statements was proved by sublinear operator T satisfies the condition (1.5), containing the result in [18, 19].Theorem 4.1.
Let1<p<∞ and  φ(x,r) satisfy conditions (3.4) and (3.5). Let  T be a sublinear operator satisfies the condition (1.5) and bounded on  Lp(ℝn). Then, the operator  T is bounded on  Mp,φ.The following statements, containing results obtained in [18, 19] was proved in [13] (see also [14, 15]).Theorem 4.2.
Let1≤p<∞, and let  (φ1,φ2) satisfy the condition
(4.1)∫t∞φ1(x,r)drr≤Cφ2(x,t),
where C does not depend on x and t. Then, the operators M and K are bounded from Mp,φ1 to Mp,φ2 for p>1 and from M1,φ1 to WM1,φ2.In this section, we are going to use the following statement on the boundedness of the Hardy operator:(4.2)(Hg)(t):=1t∫0tg(r)dr,0<t<∞.Theorem 4.3 (see [27]).
The inequality(4.3)esssup⁡t>0⁡w(t)Hg(t)≤cesssup⁡t>0⁡v(t)g(t)
holds for all nonnegative and nonincreasing g on (0,∞) if and only if
(4.4)A:=sup⁡t>0w(t)t∫0tdresssup⁡0<s<r⁡v(s)<∞,
and c≈A.Lemma 4.4.
Let1≤p<∞,  T be a sublinear operator which satisfies the condition (1.5) bounded on  Lp(ℝn) for  p>1 and bounded from  L1(ℝn) to  WL1(ℝn).
Then, for1<p<∞,
(4.5)‖Tf‖Lp(B)≲rn/p∫2r∞t-n/p-1‖f‖Lp(B(x0,t))dt
holds for any ball B=B(x0,r) and for all f∈Lploc(ℝn).
Moreover, forp=1,
(4.6)‖Tf‖WL1(B)≲rn∫2r∞t-n-1‖f‖L1(B(x0,t))dt
holds for any ball B=B(x0,r) and for all f∈L1loc(ℝn).Proof.
Letp∈(1,∞). For arbitrary x0∈ℝn, set B=B(x0,r) for the ball centered at x0 and of radius r, 2B=B(x0,2r). We represent f as
(4.7)f=f1+f2,f1(y)=f(y)χ2B(y),f2(y)=f(y)χ(2B)c(y),r>0,
and have
(4.8)‖Tf‖Lp(B)≤‖Tf1‖Lp(B)+‖Tf2‖Lp(B).
Sincef1∈Lp(ℝn), Tf1∈Lp(ℝn) and from the boundedness of T in Lp(ℝn), it follows that
(4.9)‖Tf1‖Lp(B)≤‖Tf1‖Lp(Rn)≤C‖f1‖Lp(Rn)=C‖f‖Lp(2B),
where constant C>0 is independent of f.
It is clear thatx∈B, y∈(2B)c implies (1/2)|x0-y|≤|x-y|≤(3/2)|x0-y|. We get
(4.10)|Tf2(x)|≤2nc0∫(2B)c|f(y)||x0-y|ndy.
By Fubini's theorem, we have
(4.11)∫(2B)c|f(y)||x0-y|ndy≈∫(2B)c|f(y)|∫|x0-y|∞dttn+1dy≈∫2r∞∫2r≤|x0-y|<t|f(y)|dydttn+1≲∫2r∞∫B(x0,t)|f(y)|dydttn+1.
Applying Hölder's inequality, we get
(4.12)∫(2B)c|f(y)||x0-y|ndy≲∫2r∞‖f‖Lp(B(x0,t))dttn/p+1.
Moreover, for allp∈[1,∞),
(4.13)‖Tf2‖Lp(B)≲rn/p∫2r∞‖f‖Lp(B(x0,t))dttn/p+1
is valid. Thus,
(4.14)‖Tf‖Lp(B)≲‖f‖Lp(2B)+rn/p∫2r∞‖f‖Lp(B(x0,t))dttn/p+1.
On the other hand,
(4.15)‖f‖Lp(2B)≈rn/p‖f‖Lp(2B)∫2r∞dttn/p+1≲rn/p∫2r∞‖f‖Lp(B(x0,t))dttn/p+1.
Thus,
(4.16)‖Tf‖Lp(B)≲rn/p∫2r∞‖f‖Lp(B(x0,t))dttn/p+1.
Letp=1. From the weak (1,1) boundedness of T and (4.15), it follows that
(4.17)‖Tf1‖WL1(B)≤‖Tf1‖WL1(Rn)≲‖f1‖L1(Rn)=‖f‖L1(2B)≲rn∫2r∞∫B(x0,t)|f(y)|dydttn+1.
Then, by (4.13) and (4.17), we get (4.6).Theorem 4.5.
Let1≤p<∞, and let  (φ1,φ2) satisfy the condition
(4.18)∫r∞essinf⁡t<s<∞⁡φ1(x,s)sn/ptn/p+1dt≤Cφ2(x,r),
where C does not depend on x and r. Let T be a sublinear operator which satisfies the condition (1.5) bounded on Lp(ℝn) for p>1 and bounded from L1(ℝn) to WL1(ℝn). Then, the operator T is bounded from Mp,φ1 to Mp,φ2 for p>1 and from M1,φ1 to WM1,φ2. Moreover, for p>1(4.19)‖Tf‖Mp,φ2≲‖f‖Mp,φ1,
and for p=1(4.20)‖Tf‖WM1,φ2≲‖f‖M1,φ1.Proof.
By Lemma4.4 and Theorem 4.3, we have for p>1(4.21)‖Tf‖Mp,φ2≲sup⁡x∈Rn,r>0φ2(x,r)-1∫r∞‖f‖Lp(B(x,t))dttn/p+1≈sup⁡x∈Rn,r>0φ2(x,r)-1∫0r-n/p‖f‖Lp(B(x,t-p/n))dt=sup⁡x∈Rn,r>0φ2(x,r-p/n)-1∫0r‖f‖Lp(B(x,t-p/n))dt≲sup⁡x∈Rn,r>0φ1(x,r-p/n)-1r‖f‖Lp(B(x,t))=‖f‖Mp,φ1,
and for p=1(4.22)‖Tf‖WM1,φ2≲sup⁡x∈Rn,r>0φ2(x,r)-1∫r∞‖f‖L1(B(x,t))dttn+1≈sup⁡x∈Rn,r>0φ2(x,r)-1∫0r-n‖f‖L1(B(x,t-1/n))dt=sup⁡x∈Rn,r>0φ2(x,r-1/n)-1∫0r‖f‖L1(B(x,t-1/n))dt≲sup⁡x∈Rn,r>0φ1(x,r-1/n)-1r‖f‖L1(B(x,r-1/n))=‖f‖M1,φ1.Corollary 4.6.
Let1≤p<∞, and  (φ1,φ2) satisfies the condition (4.18). Then, the operators  M and  K are bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Note that Corollary4.6 was proved in [28].


## 5. Commutators of Sublinear Operators Generated byCalderón-Zygmund Operators in the SpacesMp,φ

LetT be a Calderón-Zygmund singular integral operator and a∈BMO(ℝn). A well-known result of Coifman et al. [29] states that the commutator operator [a,T]f=T(af)-aTf is bounded on Lp(ℝn) for 1<p<∞. The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, e.g., [12, 30, 31]).First, we introduce the definition of the space ofBMO(ℝn).Definition 5.1.
Suppose thatf∈L1loc(ℝn), and let
(5.1)‖f‖*=sup⁡x∈Rn,r>01|B(x,r)|∫B(x,r)|f(y)-fB(x,r)|dy<∞,
where
(5.2)fB(x,r)=1|B(x,r)|∫B(x,r)f(y)dy.
Define
(5.3)BMO(Rn)={f∈L1loc⁡(Rn):‖f‖*<∞}.If one regards two functions whose difference is a constant as one, then spaceBMO(ℝn) is a Banach space with respect to norm ∥·∥*.Remark 5.2.
(1) The John-Nirenberg inequality: there are constantsC1,C2>0 such that for all f∈BMO(ℝn) and β>0,
(5.4)|{x∈B:|f(x)-fB|>β}|≤C1|B|e-C2β/‖f‖*,∀B⊂Rn.
(2) The John-Nirenberg inequality implies that(5.5)‖f‖*≈sup⁡x∈Rn,r>0(1|B(x,r)|∫B(x,r)|f(y)-fB(x,r)|pdy)1/p,
for 1<p<∞.
(3) Letf∈BMO(ℝn). Then, there is a constant C>0 such that
(5.6)|fB(x,r)-fB(x,t)|≤C‖f‖*ln⁡trfor0<2r<t,
where C is independent of f, x, r, and t.In [24], the following statement was proved for the commutators of sublinear operators, containing the result in [18, 19].Theorem 5.3.
Let1<p<∞,  φ(x,r) which satisfies the conditions (3.4) and (3.5) and  a∈BMO(ℝn). Suppose that  T is a linear operator and satisfies the condition (1.5). If the operator  [a,T] is bounded on  Lp(ℝn), then the operator  [a,T] is bounded on  Mp,φ.Remark 5.4.
Note that Theorem5.3 in the following form is also valid. Let 1<p<∞, φ(x,r) satisfy the conditions (3.4) and (3.5) and a∈BMO(ℝn). Suppose that Ta is a sublinear operator satisfies the condition (1.6) and bounded on Lp(ℝn), then the operator Ta is bounded on Mp,φ.Lemma 5.5.
Let1<p<∞,  a∈BMO(ℝn), and a sublinear operator  Ta satisfies the condition (1.6) and bounded on  Lp(ℝn).
Then,(5.7)‖Taf‖Lp(B)≲‖a‖*rn/p∫2r∞(1+ln⁡tr)t-(n/p)-1‖f‖Lp(B(x0,t))dt
holds for any ball B=B(x0,r) and for all f∈Lploc(ℝn).Proof.
1<p<∞, a∈BMO(ℝn), and a sublinear operator Ta satisfies the condition (1.6). For arbitrary x0∈ℝn, set B=B(x0,r) for the ball centered at x0 and of radius r. Write f=f1+f2 with f1=fχ2B and f2=fχ(2B)c. Hence,
(5.8)‖Taf‖Lp(B)≤‖Taf1‖Lp(B)+‖Taf2‖Lp(B).
From the boundedness of Ta in Lp(ℝn), it follows that
(5.9)‖Taf1‖Lp(B)≤‖Taf1‖Lp(Rn)≲‖a‖*‖f1‖Lp(Rn)=‖a‖*‖f‖Lp(2B).
Forx∈B, we have
(5.10)|Taf2(x)|≲∫Rn|a(y)-a(x)||x-y|n|f2(y)|dy≈∫(2B)c|a(y)-a(x)||x0-y|n|f(y)|dy.
Then,(5.11)‖Taf2‖Lp(B)≲(∫B(∫(2B)c|a(y)-a(x)||x0-y|n|f(y)|dy)pdx)1/p≲(∫B(∫(2B)c|a(y)-aB||x0-y|n|f(y)|dy)pdx)1/p+(∫B(∫(2B)c|a(x)-aB||x0-y|n|f(y)|dy)pdx)1/p=I1+I2.
Let us estimate I1(5.12)I1≈rn/p∫(2B)c|a(y)-aB||x0-y|n|f(y)|dy≈rn/p∫(2B)c|a(y)-aB||f(y)|∫|x0-y|∞dttn+1dy≈rn/p∫2r∞∫2r≤|x0-y|≤t|a(y)-aB||f(y)|dydttn+1≲rn/p∫2r∞∫B(x0,t)|a(y)-aB||f(y)|dydttn+1.
Applying Hölder's inequality and by (5.5), (5.6), we get
(5.13)I1≲rn/p∫2r∞∫B(x0,t)|a(y)-aB(x0,t)||f(y)|dydttn+1+rn/p∫2r∞|aB(x0,r)-aB(x0,t)|∫B(x0,t)|f(y)|dydttn+1≲rn/p∫2r∞(∫B(x0,t)|a(y)-aB(x0,t)|p′dy)1/p′‖f‖Lp(B(x0,t))dttn+1+rn/p∫2r∞|aB(x0,r)-aB(x0,t)|‖f‖Lp(B(x0,t))dttn/p+1≲‖a‖*rn/p∫2r∞(1+ln⁡tr)‖f‖Lp(B(x0,t))dttn/p+1.
In order to estimate I2, note that
(5.14)I2=(∫B|a(x)-aB|pdx)1/p∫(2B)c|f(y)||x0-y|ndy.
By (5.5), we get
(5.15)I2≲‖a‖*rn/p∫(2B)c|f(y)||x0-y|ndy.
Thus, by (4.12),
(5.16)I2≲‖a‖*rn/p∫2r∞‖f‖Lp(B(x0,t))dttn/p+1.
Summing up I1 and I2, for all p∈[1,∞), we get
(5.17)‖Taf2‖Lp(B)≲‖a‖*rn/p∫2r∞(1+ln⁡tr)‖f‖Lp(B(x0,t))dttn/p+1.
Finally,
(5.18)‖Taf‖Lp(B)≲‖a‖*‖f‖Lp(2B)+‖a‖*rn/p∫2r∞(1+ln⁡tr)‖f‖Lp(B(x0,t))dttn/p+1,
and statement of Lemma 5.5 follows by (4.15).The following theorem is true.Theorem 5.6.
Let1<p<∞,  a∈BMO(ℝn) and  (φ1,φ2) satisfy the condition
(5.19)∫r∞(1+ln⁡tr)essinf⁡t<s<∞⁡φ1(x,s)sn/ptn/p+1dt≤Cφ2(x,r),
where C does not depend on x and r. Suppose that Ta is a sublinear operator which satisfies the condition (1.6) and bounded on Lp(ℝn).
Then, the operatorTa is bounded from Mp,φ1 to Mp,φ2. Moreover,
(5.20)‖Taf‖Mp,φ2≲‖a‖*‖f‖Mp,φ1.Proof.
The statement of Theorem5.6 is followed by Lemma 5.5 and Theorem 4.3 in the same manner as in the proof of Theorem 4.5.For the sublinear commutator of the maximal operator(5.21)Ma(f)(x)=sup⁡t>0|B(x,t)|-1∫B(x,t)|a(x)-a(y)||f(y)|dy,
and for the linear commutator of the Calderón-Zygmund operator [a,K] from Theorem 5.6, we get the following new results.Corollary 5.7.
Let1<p<∞,  (φ1,φ2) satisfy the condition (5.19) and  a∈BMO(ℝn). Then, the sublinear commutator operator  Ma is bounded from  Mp,φ1 to  Mp,φ2.Corollary 5.8.
Let1<p<∞,  (φ1,φ2) satisfy the condition (5.19) and  a∈BMO(ℝn). Then, Calderón-Zygmund singular integral  Kf(x) exists for a.e.  x∈ℝn and the operator  [a,K] is bounded from  Mp,φ1 to  Mp,φ2.Note that when the conditions of Corollary5.8 are satisfied, the existence of Kf(x) for a.e. x∈ℝn was proved in [28].


## 6. Some Applications

In this section, we will apply Theorems4.5 and 5.6 to several particular operators such as the pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.


### 6.1. Pseudodifferential Operators

Pseudodifferential operators are generalizations of differential operators and singular integrals. Letm be real number, 0≤δ<1 and 0≤ρ<1. Following [32, 33], a symbol in Sρ,δm is a smooth function σ(x,ξ) defined on ℝn×ℝn such that for all multi-indices α and β the following estimate holds: (6.1)|DxαDξβσ(x,ξ)|≤Cαβ(1+|ξ|)m-ρ|β|+δ|α|,
where Cαβ>0 is independent of x and ξ. A symbol in Sρ,δ-∞ is one which satisfies the above estimates for each real number m.The operatorA given by (6.2)Af(x)=∫Rnσ(x,ξ)e2πixξf̂(ξ)dξ
is called a pseudodifferential operator with symbol σ(x,ξ)∈Sρ,δm, where f is a Schwartz function and f̂ denotes the Fourier transform of f. As usual, Lρ,δm will denote the class of pseudodifferential operators with symbols in Sρ,δm.Miller [34] showed the boundedness of L1,00 pseudodifferential operators on weighted Lp(1<p<∞) spaces whenever the weight function belongs to Muckenhoupt’s class Ap. In [1], it is shown that pseudodifferential operators in L1,00 are Calderón-Zygmund operators, then from Corollary 5.8, we get the following new results.Corollary 6.1.
Let1≤p<∞, and let  (φ1,φ2) satisfy the condition (4.18). If  A is a pseudodifferential operator of the Hörmander class  L1,00, then the operator  A is bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Corollary 6.2.
Let1<p<∞,  (φ1,φ2) satisfy the condition (5.19) and  a∈BMO(ℝn). Let also  A be a pseudodifferential operator of the Hörmander class  L1,00. Then, the commutator operator  [a,A] is bounded from  Mp,φ1 to  Mp,φ2.


### 6.2. Littlewood-Paley Operator

The Littlewood-Paley functions play an important role in classical harmonic analysis, for example, in the study of nontangential convergence of Fatou type and boundedness of Riesz transforms and multipliers [4–6, 35]. The Littlewood-Paley operator (see [6, 36]) is defined as follows.Definition 6.3.
Suppose thatψ∈L1(ℝn)satisfies(6.3)∫Rnψ(x)dx=0.
Then, the generalized Littlewood-Paley g function gψ is defined by
(6.4)gψ(f)(x)=(∫0∞|Ft(f)(x)|2dtt)1/2,
where ψt(x)=t-nψ(x/t) for t>0 and Ft(f)=ψt*f.
The sublinear commutator of the operatorgψ is defined by
(6.5)[a,gψ](f)(x)=(∫0∞|Fta(f)(x)|2dtt)1/2,
where
(6.6)Fta(f)(x)=∫Rn[a(y)a(x)-a(y)]ψt(x-y)f(y)dy.The following theorem is valid (see [3, Theorem  5.1.2]).Theorem 6.4.
Suppose thatψ∈L1(ℝn) satisfies (6.3) and the following properties:
(6.7)|ψ(x)|≤C(1+|x|)n+α,x∈Rn,∫Rn|ψ(x+h)-ψ(x)|dx≤C|h|α,h∈Rn,
where C and α>0 are both independent of x and h. Then, gψ is bounded on Lp(ℝn) for all 1<p<∞, and bounded from L1(ℝn) to WL1(ℝn).LetH be the space H={h:∥h∥=(∫0∞|h(t)|2dt/t)1/2<∞}, then for each fixed x∈ℝn, Ft(f)(x) may be viewed as a mapping from [0,∞) to H, and it is clear that gψ(f)(x)=∥Ft(f)(x)∥. In fact, by Minkowski inequality and the conditions on ψ, we get (6.8)gψ(f)(x)≤∫Rn|f(y)|(∫0∞|ψt(x-y)|2dtt)1/2dy≤C∫Rn|f(y)|(∫0∞t-2n(1+|x-y|/t)2(n+1)dtt)1/2dy=C∫Rn|f(y)||x-y|ndy.
Thus, we get the following.Corollary 6.5.
Let1≤p<∞,  (φ1,φ2) satisfies the condition (4.18) and  ψ∈L1(ℝn) satisfies (6.3) and (6.7). Then the operator  gψ is bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Corollary 6.6.
Let1<p<∞,  (φ1,φ2) satisfies the condition (5.19),  a∈BMO(ℝn) and  ψ∈L1(ℝn) satisfies (6.3) and (6.7). Then the operator  [a,gψ] is bounded from  Mp,φ1 to  Mp,φ2.


### 6.3. Marcinkiewicz Operator

LetSn-1={x∈ℝn:|x|=1} be the unit sphere in ℝn equipped with the Lebesgue measure dσ. Suppose that Ω satisfies the following conditions.(a)
Ω is the homogeneous function of degree zero on ℝn∖{0}; that is,
(6.9)Ω(tx)=Ω(x),foranyt>0,x∈Rn∖{0}.(b)
Ω has mean zero on Sn-1; that is,
(6.10)∫Sn-1Ω(x′)dσ(x′)=0.(c)
Ω∈Lipγ(Sn-1), 0<γ≤1, that is there exists a constant M>0 such that
(6.11)|Ω(x′)-Ω(y′)|≤M|x′-y′|γforanyx′,y′∈Sn-1.In 1958, Stein [35] defined the Marcinkiewicz integral of higher dimension μΩ as (6.12)μΩ(f)(x)=(∫0∞|FΩ,t(f)(x)|2dtt3)1/2,
where (6.13)FΩ,t(f)(x)=∫|x-y|≤tΩ(x-y)|x-y|n-1f(y)dy.Since Stein's work in 1958, the continuity of Marcinkiewicz integral has been extensively studied as a research topic and also provides useful tools in harmonic analysis [3–6].The sublinear commutator of the operatorμΩ is defined by (6.14)[a,μΩ](f)(x)=(∫0∞|FΩ,t,a(f)(x)|2dtt3)1/2,
where (6.15)FΩ,t,a(f)(x)=∫|x-y|≤tΩ(x-y)|x-y|n-1[a(x)-a(y)]f(y)dy.LetH be the space H={h:∥h∥=(∫0∞|h(t)|2dt/t3)1/2<∞}. Then, it is clear that μΩ(f)(x)=∥FΩ,t(f)(x)∥.By Minkowski inequality and the conditions onΩ, we get (6.16)μΩ(f)(x)≤∫Rn|Ω(x-y)||x-y|n-1|f(y)|(∫|x-y|∞dtt3)1/2dy≤C∫Rn|f(y)||x-y|ndy.
Thus, μΩ satisfies the condition (1.5). It is known that μΩ is bounded on Lp(ℝn) for p>1 and bounded from L1(ℝn) to WL1(ℝn) (see [37]), then from Theorems 4.5 and 5.6, we get the following collory.Corollary 6.7.
Let1≤p<∞ and  (φ1,φ2) satisfy the condition (4.18), and let  Ω satisfy the conditions (a)–(c). Then,  μΩ is bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Corollary 6.8.
Let1<p<∞,  (φ1,φ2) satisfy the condition (5.19),  a∈BMO(ℝn), and  Ω satisfy the conditions (a)–(c). Then,  [a,μΩ] is bounded from  Mp,φ1 to  Mp,φ2.


### 6.4. Bochner-Riesz Operator

Letδ>(n-1)/2, Btδ(f̂)(ξ)=(1-t2|ξ|2)+δf̂(ξ) and Btδ(x)=t-nBδ(x/t) for t>0. The maximal Bochner-Riesz operator is defined by (see [38, 39]) (6.17)B*δ(f)(x)=sup⁡t>0|Btδ(f)(x)|.LetH be the space H={h:∥h∥=supt>0|h(t)|<∞}, then it is clear that B*δ(f)(x)=∥Btδ(f)(x)∥.By the condition onBrδ (see [2]), we have (6.18)|Brδ(x-y)|≤Cr-n(1+|x-y|/r)-(δ+(n+1)/2)=C(rr+|x-y|)δ-(n-1)/21(r+|x-y|)n≤|x-y|-n,B*δ(f)(x)≤C∫Rn|f(y)||x-y|ndy.
Thus, B*δ satisfies the condition (1.5). It is known that B*δ is bounded on Lp(ℝn) for p>1, and bounded from L1(ℝn) to WL1(ℝn), then from Theorems 4.5 and 5.6, we get the following corollary.Corollary 6.9.
Let1≤p<∞,  (φ1,φ2) satisfy the condition (4.18) and  δ>(n-1)/2. Then, the operator  B*δ is bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Corollary 6.10.
Let1<p<∞,  (φ1,φ2) satisfy the condition (5.19),  δ>(n-1)/2 and  a∈BMO(ℝn). Then, the operator  [a,Btδ] is bounded from  Mp,φ1 to  Mp,φ2.Remark 6.11.
Recall that under the assumption thatφ(x,r) satisfies the conditions (3.4) and (3.5), the Corollaries 6.9 and 6.10 were proved in [38].


## 6.1. Pseudodifferential Operators

Pseudodifferential operators are generalizations of differential operators and singular integrals. Letm be real number, 0≤δ<1 and 0≤ρ<1. Following [32, 33], a symbol in Sρ,δm is a smooth function σ(x,ξ) defined on ℝn×ℝn such that for all multi-indices α and β the following estimate holds: (6.1)|DxαDξβσ(x,ξ)|≤Cαβ(1+|ξ|)m-ρ|β|+δ|α|,
where Cαβ>0 is independent of x and ξ. A symbol in Sρ,δ-∞ is one which satisfies the above estimates for each real number m.The operatorA given by (6.2)Af(x)=∫Rnσ(x,ξ)e2πixξf̂(ξ)dξ
is called a pseudodifferential operator with symbol σ(x,ξ)∈Sρ,δm, where f is a Schwartz function and f̂ denotes the Fourier transform of f. As usual, Lρ,δm will denote the class of pseudodifferential operators with symbols in Sρ,δm.Miller [34] showed the boundedness of L1,00 pseudodifferential operators on weighted Lp(1<p<∞) spaces whenever the weight function belongs to Muckenhoupt’s class Ap. In [1], it is shown that pseudodifferential operators in L1,00 are Calderón-Zygmund operators, then from Corollary 5.8, we get the following new results.Corollary 6.1.
Let1≤p<∞, and let  (φ1,φ2) satisfy the condition (4.18). If  A is a pseudodifferential operator of the Hörmander class  L1,00, then the operator  A is bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Corollary 6.2.
Let1<p<∞,  (φ1,φ2) satisfy the condition (5.19) and  a∈BMO(ℝn). Let also  A be a pseudodifferential operator of the Hörmander class  L1,00. Then, the commutator operator  [a,A] is bounded from  Mp,φ1 to  Mp,φ2.


## 6.2. Littlewood-Paley Operator

The Littlewood-Paley functions play an important role in classical harmonic analysis, for example, in the study of nontangential convergence of Fatou type and boundedness of Riesz transforms and multipliers [4–6, 35]. The Littlewood-Paley operator (see [6, 36]) is defined as follows.Definition 6.3.
Suppose thatψ∈L1(ℝn)satisfies(6.3)∫Rnψ(x)dx=0.
Then, the generalized Littlewood-Paley g function gψ is defined by
(6.4)gψ(f)(x)=(∫0∞|Ft(f)(x)|2dtt)1/2,
where ψt(x)=t-nψ(x/t) for t>0 and Ft(f)=ψt*f.
The sublinear commutator of the operatorgψ is defined by
(6.5)[a,gψ](f)(x)=(∫0∞|Fta(f)(x)|2dtt)1/2,
where
(6.6)Fta(f)(x)=∫Rn[a(y)a(x)-a(y)]ψt(x-y)f(y)dy.The following theorem is valid (see [3, Theorem  5.1.2]).Theorem 6.4.
Suppose thatψ∈L1(ℝn) satisfies (6.3) and the following properties:
(6.7)|ψ(x)|≤C(1+|x|)n+α,x∈Rn,∫Rn|ψ(x+h)-ψ(x)|dx≤C|h|α,h∈Rn,
where C and α>0 are both independent of x and h. Then, gψ is bounded on Lp(ℝn) for all 1<p<∞, and bounded from L1(ℝn) to WL1(ℝn).LetH be the space H={h:∥h∥=(∫0∞|h(t)|2dt/t)1/2<∞}, then for each fixed x∈ℝn, Ft(f)(x) may be viewed as a mapping from [0,∞) to H, and it is clear that gψ(f)(x)=∥Ft(f)(x)∥. In fact, by Minkowski inequality and the conditions on ψ, we get (6.8)gψ(f)(x)≤∫Rn|f(y)|(∫0∞|ψt(x-y)|2dtt)1/2dy≤C∫Rn|f(y)|(∫0∞t-2n(1+|x-y|/t)2(n+1)dtt)1/2dy=C∫Rn|f(y)||x-y|ndy.
Thus, we get the following.Corollary 6.5.
Let1≤p<∞,  (φ1,φ2) satisfies the condition (4.18) and  ψ∈L1(ℝn) satisfies (6.3) and (6.7). Then the operator  gψ is bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Corollary 6.6.
Let1<p<∞,  (φ1,φ2) satisfies the condition (5.19),  a∈BMO(ℝn) and  ψ∈L1(ℝn) satisfies (6.3) and (6.7). Then the operator  [a,gψ] is bounded from  Mp,φ1 to  Mp,φ2.


## 6.3. Marcinkiewicz Operator

LetSn-1={x∈ℝn:|x|=1} be the unit sphere in ℝn equipped with the Lebesgue measure dσ. Suppose that Ω satisfies the following conditions.(a)
Ω is the homogeneous function of degree zero on ℝn∖{0}; that is,
(6.9)Ω(tx)=Ω(x),foranyt>0,x∈Rn∖{0}.(b)
Ω has mean zero on Sn-1; that is,
(6.10)∫Sn-1Ω(x′)dσ(x′)=0.(c)
Ω∈Lipγ(Sn-1), 0<γ≤1, that is there exists a constant M>0 such that
(6.11)|Ω(x′)-Ω(y′)|≤M|x′-y′|γforanyx′,y′∈Sn-1.In 1958, Stein [35] defined the Marcinkiewicz integral of higher dimension μΩ as (6.12)μΩ(f)(x)=(∫0∞|FΩ,t(f)(x)|2dtt3)1/2,
where (6.13)FΩ,t(f)(x)=∫|x-y|≤tΩ(x-y)|x-y|n-1f(y)dy.Since Stein's work in 1958, the continuity of Marcinkiewicz integral has been extensively studied as a research topic and also provides useful tools in harmonic analysis [3–6].The sublinear commutator of the operatorμΩ is defined by (6.14)[a,μΩ](f)(x)=(∫0∞|FΩ,t,a(f)(x)|2dtt3)1/2,
where (6.15)FΩ,t,a(f)(x)=∫|x-y|≤tΩ(x-y)|x-y|n-1[a(x)-a(y)]f(y)dy.LetH be the space H={h:∥h∥=(∫0∞|h(t)|2dt/t3)1/2<∞}. Then, it is clear that μΩ(f)(x)=∥FΩ,t(f)(x)∥.By Minkowski inequality and the conditions onΩ, we get (6.16)μΩ(f)(x)≤∫Rn|Ω(x-y)||x-y|n-1|f(y)|(∫|x-y|∞dtt3)1/2dy≤C∫Rn|f(y)||x-y|ndy.
Thus, μΩ satisfies the condition (1.5). It is known that μΩ is bounded on Lp(ℝn) for p>1 and bounded from L1(ℝn) to WL1(ℝn) (see [37]), then from Theorems 4.5 and 5.6, we get the following collory.Corollary 6.7.
Let1≤p<∞ and  (φ1,φ2) satisfy the condition (4.18), and let  Ω satisfy the conditions (a)–(c). Then,  μΩ is bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Corollary 6.8.
Let1<p<∞,  (φ1,φ2) satisfy the condition (5.19),  a∈BMO(ℝn), and  Ω satisfy the conditions (a)–(c). Then,  [a,μΩ] is bounded from  Mp,φ1 to  Mp,φ2.


## 6.4. Bochner-Riesz Operator

Letδ>(n-1)/2, Btδ(f̂)(ξ)=(1-t2|ξ|2)+δf̂(ξ) and Btδ(x)=t-nBδ(x/t) for t>0. The maximal Bochner-Riesz operator is defined by (see [38, 39]) (6.17)B*δ(f)(x)=sup⁡t>0|Btδ(f)(x)|.LetH be the space H={h:∥h∥=supt>0|h(t)|<∞}, then it is clear that B*δ(f)(x)=∥Btδ(f)(x)∥.By the condition onBrδ (see [2]), we have (6.18)|Brδ(x-y)|≤Cr-n(1+|x-y|/r)-(δ+(n+1)/2)=C(rr+|x-y|)δ-(n-1)/21(r+|x-y|)n≤|x-y|-n,B*δ(f)(x)≤C∫Rn|f(y)||x-y|ndy.
Thus, B*δ satisfies the condition (1.5). It is known that B*δ is bounded on Lp(ℝn) for p>1, and bounded from L1(ℝn) to WL1(ℝn), then from Theorems 4.5 and 5.6, we get the following corollary.Corollary 6.9.
Let1≤p<∞,  (φ1,φ2) satisfy the condition (4.18) and  δ>(n-1)/2. Then, the operator  B*δ is bounded from  Mp,φ1 to  Mp,φ2 for  p>1 and bounded from  M1,φ1 to  WM1,φ2.Corollary 6.10.
Let1<p<∞,  (φ1,φ2) satisfy the condition (5.19),  δ>(n-1)/2 and  a∈BMO(ℝn). Then, the operator  [a,Btδ] is bounded from  Mp,φ1 to  Mp,φ2.Remark 6.11.
Recall that under the assumption thatφ(x,r) satisfies the conditions (3.4) and (3.5), the Corollaries 6.9 and 6.10 were proved in [38].



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*Source: 356041-2011-07-24.xml*