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

and Applied Analysis 3 particular operators such as the pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator. By A 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 by Mp,λ ≡ Mp,λ R the Morrey space, the space of all functions f ∈ Lloc p R n with finite quasinorm ∥f∥Mp,λ ≡ ∥f∥Mp,λ Rn sup x∈Rn, r>0 r−λ/p ∥f∥Lp B x,r , 2.1 where 1 ≤ p <∞ and 0 ≤ λ ≤ n. Note thatMp,0 Lp R andMp,n L∞ R . If λ < 0 or λ > n, thenMp,λ Θ, where Θ is the set of all functions equivalent to 0 on R. We also denote by WMp,λ ≡ WMp,λ R the weak Morrey space of all functions f ∈ WLloc p R n for which ∥f∥WMp,λ ≡ ∥f∥WMp,λ Rn sup x∈Rn, r>0 r−λ/p ∥f∥WLp B x,r <∞, 2.2 whereWLp B x, r denotes the weak Lp-space of measurable functions f for which ∥f∥WLp B x,r ≡ ∥∥fχB x,r ∥WLp Rn sup t>0 t ∣∣{y ∈ B x, r : ∣∣f(y)∣∣ > t}∣∣1/p sup 0<t≤|B x,r | t1/p ( fχB x,r )∗ t <∞. 2.3 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. Let 1 ≤ p < ∞ and 0 ≤ λ < n. Then, for p > 1 the operator M is bounded on Mp,λ and for p 1M is bounded fromM1,λ toWM1,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 CalderónZygmund operators K. Theorem 2.2. Let 1 ≤ p < ∞, 0 < λ < n. Then, for 1 < p < ∞ Calderón-Zygmund operator K is bounded onMp,λ and for p 1K is bounded fromM1,λ toWM1,λ. 4 Abstract and Applied Analysis Note that Theorem 2.2 was proved by Peetre 10 in the case of the classical CalderónZygmund 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. Let φ x, r be a positive measurable function on R × 0,∞ and 1 ≤ p <∞. We denote by Mp,φ ≡ Mp,φ R the generalized Morrey space, the space of all functions f ∈ Lloc p R n with finite quasinorm ∥f∥Mp,φ ≡ ∥f∥Mp,φ Rn sup x∈Rn, r>0 φ x, r −1 |B x, r |−1/p∥f∥Lp B x,r . 3.1 Also, byWMp,φ ≡WMp,φ R we denote the weak generalized Morrey space of all functions f ∈WLloc p R for which ∥f∥WMp,φ ≡ ∥f∥WMp,φ Rn sup x∈Rn, r>0 φ x, r −1 |B x, r |−1/p∥f∥WLp B x,r <∞. 3.2 According to this definition, we recover the spacesMp,λ andWMp,λ under the choice φ x, r r λ−n : Mp,λ Mp,φ ∣∣ φ x,r r λ−n /p , WMp,λ WMp,φ ∣∣ φ x,r r λ−n /p . 3.3 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 : c−1φ x, r ≤ φ x, t ≤ c φ x, r , 3.4 whenever r ≤ t ≤ 2r, where c ≥1 does not depend on t, r and x ∈ R, jointly with the condition ∫∞ r φ x, t p dt t ≤ Cφ x, r , 3.5 for the sublinear operator T satisfies the condition 1.5 , where C >0 does not depend on r and x ∈ R. Abstract and Applied Analysis 5 4. Sublinear Operators Generated by Calderón-Zygmund Operators in the Spaces Mp,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. Let 1 < 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 R . Then, the operator T is bounded onMp,φ. The following statements, containing results obtained in 18, 19 was proved in 13 see also 14, 15 . Theorem 4.2. Let 1 ≤ p <∞, and let φ1, φ2 satisfy the condition ∫∞and Applied Analysis 5 4. Sublinear Operators Generated by Calderón-Zygmund Operators in the Spaces Mp,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. Let 1 < 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 R . Then, the operator T is bounded onMp,φ. The following statements, containing results obtained in 18, 19 was proved in 13 see also 14, 15 . Theorem 4.2. Let 1 ≤ p <∞, and let φ1, φ2 satisfy the condition ∫∞ t φ1 x, r dr r ≤ Cφ2 x, t , 4.1 where C does not depend on x and t. Then, the operatorsM and K are bounded fromMp,φ1 toMp,φ2 for p > 1 and fromM1,φ1 toWM1,φ2 . In this section, we are going to use the following statement on the boundedness of the Hardy operator: ( Hg ) t : 1 t ∫ t 0 g r dr, 0 < t <∞. 4.2 Theorem 4.3 see 27 . The inequality ess sup t>0 w t Hg t ≤ c ess sup t>0 v t g t 4.3 holds for all nonnegative and nonincreasing g on 0,∞ if and only if A : sup t>0 w t t ∫ t 0 dr ess sup0<s<rv s <∞, 4.4 and c ≈ A. Lemma 4.4. Let 1 ≤ p < ∞, T be a sublinear operator which satisfies the condition 1.5 bounded on Lp R for p > 1 and bounded from L1 R toWL1 R . Then, for 1 < p <∞, ∥Tf∥Lp B r ∫∞ 2r t−n/p−1 ∥f∥Lp B x0,t dt 4.5 holds for any ball B B x0, r and for all f ∈ Lloc p R . 6 Abstract and Applied Analysis Moreover, for p 1, ∥Tf∥WL1 B r ∫∞ 2r t−n−1 ∥f∥L1 B x0,t dt 4.6 holds for any ball B B x0, r and for all f ∈ Lloc 1 R . Proof. Let p ∈ 1,∞ . For arbitrary x0 ∈ R, set B B x0, r for the ball centered at x0 and of radius r, 2B B x0, 2r . We represent f as f f1 f2, f1 ( y ) f ( y ) χ2B ( y ) , f2 ( y ) f ( y ) χ c 2B ( y ) , r > 0, 4.7 and have ∥Tf∥Lp B ≤ ∥Tf1∥Lp B ∥Tf2∥Lp B . 4.8 Since f1 ∈ Lp R , Tf1 ∈ Lp R and from the boundedness of T in Lp R , it follows that ∥Tf1∥Lp B ≤ ∥Tf1∥Lp Rn ≤ C ∥f1∥Lp Rn C ∥f∥Lp 2B , 4.9 where constant C > 0 is independent of f . It is clear that x ∈ B, y ∈ c 2B implies 1/2 |x0 − y| ≤ |x − y| ≤ 3/2 |x0 − y|. We get ∣∣Tf2 x ∣∣ ≤ 2c0 ∫ c 2B ∣∣f(y)∣∣ ∣∣x0 − y∣∣n dy. 4.10 By Fubini’s theorem, we have ∫ c 2B ∣∣f(y)∣∣ ∣∣x0 − y∣∣n dy ≈ ∫ c 2B ∣∣f(y)∣∣ ∫∞ |x0−y| dt tn 1 dy ≈ ∫∞ 2r ∫ 2r≤|x0−y|<t ∣∣f(y)∣∣dy dt tn 1 ∫∞ 2r ∫ B x0,t ∣∣f(y)∣∣dy dt tn 1 . 4.11 Applying Hölder’s inequality, we get ∫ c 2B ∣∣f(y)∣∣ ∣∣x0 − y∣∣n dy ∫∞ 2r ∥f∥Lp B x0,t dt tn/p 1 . 4.12 Abstract and Applied Analysis 7 Moreover, for all p ∈ 1,∞ , ∥Tf2∥Lp B r ∫∞ 2r ∥f∥Lp B x0,t dt tn/p 1 4.13and Applied Analysis 7 Moreover, for all p ∈ 1,∞ , ∥Tf2∥Lp B r ∫∞ 2r ∥f∥Lp B x0,t dt tn/p 1 4.13 is valid. Thus, ∥Tf∥Lp B ∥f∥Lp 2B r ∫∞ 2r ∥f∥Lp B x0,t dt tn/p 1 . 4.14 On the other hand, ∥f∥Lp 2B ≈ r ∥f∥Lp 2B ∫∞ 2r dt tn/p 1 r ∫∞ 2r ∥f∥Lp B x0,t dt tn/p 1 . 4.15 Thus, ∥Tf∥Lp B r ∫∞ 2r ∥f∥Lp B x0,t dt tn/p 1 . 4.16 Let p 1. From the weak 1,1 boundedness of T and 4.15 , it follows that ∥Tf1∥WL1 B ≤ ∥Tf1∥WL1 Rn ∥f1∥L1 Rn ∥f∥L1 2B r ∫∞ 2r ∫ B x0,t ∣∣f(y)∣∣dy dt tn 1 . 4.17 Then, by 4.13 and 4.17 , we get 4.6 . Theorem 4.5. Let 1 ≤ p <∞, and let φ1, φ2 satisfy the condition ∫∞ r ess inft<s<∞φ1 x, s s tn/p 1 dt ≤ Cφ2 x, r , 4.18 where C does not depend on x and r. Let T be a sublinear operator which satisfies the condition 1.5 bounded on Lp R for p > 1 and bounded from L1 R toWL1 R . Then, the operator T is bounded fromMp,φ1 toMp,φ2 for p > 1 and fromM1,φ1 toWM1,φ2 . Moreover, for p > 1 ∥Tf∥Mp,φ2 ∥f∥Mp,φ1 , 4.19 and for p 1 ∥Tf∥WM1,φ2 ∥f∥M1,φ1 . 4.20 8 Abstract and Applied Analysis Proof. By Lemma 4.4 and Theorem 4.3, we have for p > 1 ∥Tf∥Mp,φ2 sup x∈Rn, r>0 x, r −1 ∫∞ r ∥f∥Lp B x,t dt tn/p 1 ≈ sup x∈Rn, r>0 φ2 x, r −1 ∫ r−n/p 0 ∥f∥Lp B x,t−p/n dt sup x∈Rn, r>0 φ2 ( x, r−p/n )−1 ∫ r 0 ∥f∥Lp B x,t−p/n dt sup x∈Rn, r>0 φ1 ( x, r−p/n )−1 r ∥f∥Lp B x,t ∥f∥Mp,φ1 , 4.21


Introduction
For x ∈ R n 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 .
Let f ∈ L loc 1 R n .The Hardy-Littlewood maximal operator M is defined by Let K be a Calder ón-Zygmund singular integral operator, briefly a Calder ón-Zygmund operator, that is, a linear operator bounded from L 2 R n to L 2 R n taking all infinitely 2 Abstract and Applied Analysis continuously differentiable functions f with compact support to the functions f ∈ L loc 1 R n represented by Kf x R n k x, y f y dy x / ∈ supp f. 1.2 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 c 1 > 0 and 0 < ε ≤ 1 such that for all x, y ∈ R n , x / y, and It is well known that maximal operator and Calder ón-Zygmund operators play an important role in harmonic analysis see 2-6 .
Suppose that T represents a linear or a sublinear operator, which satisfies that for any f ∈ L 1 R n with compact support and x / ∈ supp f Tf x ≤ c 0 R n f y x − y n dy, 1.5 where c 0 is independent of f and x.
For a function a, suppose that the commutator operator T a represents a linear or a sublinear operator, which satisfies that for any f ∈ L 1 R n with compact support and x / ∈ supp f where c 0 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 Calder ón-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 operator T satisfies the condition 1.5 generated by Calder ón-Zygmund operator from one generalized Morrey space M p,ϕ 1 to another M p,ϕ 2 , 1 < p < ∞, and from M 1,ϕ 1 to the weak space WM 1,ϕ 2 .In the case a ∈ BMO R n , 1 < p < ∞ and the commutator operator T a is a sublinear operator, we find the sufficient conditions on the pair ϕ 1 , ϕ 2 which ensures the boundedness of the operators T a from M p,ϕ 1 to M p,ϕ 2 .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.
By A 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.

Morrey Spaces
The classical Morrey spaces M p,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 by M p,λ ≡ M p,λ R n the Morrey space, the space of all functions f ∈ L loc p R n with finite quasinorm , where Θ is the set of all functions equivalent to 0 on R n .
We also denote by WM p,λ ≡ WM p,λ R n the weak Morrey space of all functions f ∈ WL loc p R n for which where WL p B x, r denotes the weak L p -space of measurable functions f for which

2.3
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.
Di Fazio and Ragusa 12 studied the boundedness of the Calder ón-Zygmund operators in Morrey spaces, and their results imply the following statement for Calder ón-Zygmund operators K. Theorem 2.2.Let 1 ≤ p < ∞, 0 < λ < n.Then, for 1 < p < ∞ Calderón-Zygmund operator K is bounded on M p,λ and for p 1K is bounded from M 1,λ to WM 1,λ .
Note that Theorem 2.2 was proved by Peetre 10 in the case of the classical Calder ón-Zygmund singular integral operators.

Generalized Morrey Spaces
We find it convenient to define the generalized Morrey spaces in the form as follows.
Definition 3.1.Let ϕ x, r be a positive measurable function on R n × 0, ∞ and 1 ≤ p < ∞.We denote by M p,ϕ ≡ M p,ϕ R n the generalized Morrey space, the space of all functions f ∈ L loc p R n with finite quasinorm Also, by WM p,ϕ ≡ WM p,ϕ R n we denote the weak generalized Morrey space of all functions f ∈ WL loc p R n for which According to this definition, we recover the spaces M p,λ and WM p,λ under the choice ϕ x, r r λ−n /p : M p,λ M p,ϕ ϕ x,r r λ−n /p , WM p,λ WM p,ϕ ϕ x,r r λ−n /p .

3.3
In 13-19 , there were obtained sufficient conditions on ϕ 1 and ϕ 2 for the boundedness of the maximal operator M and Calder ón-Zygmund operator K from M p,ϕ 1 to M p,ϕ 2 , 1 < p < ∞ see also 20-23 .In 19 , the following condition was imposed on ϕ x, r : whenever r ≤ t ≤ 2r, where c ≥1 does not depend on t, r and x ∈ R n , jointly with the condition for the sublinear operator T satisfies the condition 1.5 , where C >0 does not depend on r and x ∈ R n .

Sublinear Operators Generated by Calder ón-Zygmund Operators in the Spaces M p,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.Let 1 < 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 L p R n .Then, the operator T is bounded on M p,ϕ .
The following statements, containing results obtained in 18, 19 was proved in 13 see also 14 holds for any ball B B x 0 , r and for all f ∈ L loc p R n .
Moreover, for p 1, holds for any ball B B x 0 , r and for all f ∈ L loc 1 R n .
Proof.Let p ∈ 1, ∞ .For arbitrary x 0 ∈ R n , set B B x 0 , r for the ball centered at x 0 and of radius r, 2B B x 0 , 2r .We represent f as and have

4.11
Applying H ölder's inequality, we get c 2B f y

4.12
Moreover, for all p ∈ 1, ∞ , is valid.Thus, 4.14 On the other hand,

4.16
Let p 1. From the weak 1,1 boundedness of T and 4.15 , it follows that
and for p 1
Note that Corollary 4.6 was proved in 28 .

Commutators of Sublinear Operators Generated by Calder ón-Zygmund Operators in the Spaces M p,ϕ
Let T be a Calder ón-Zygmund singular integral operator and a ∈ BMO R n .A well-known result of Coifman et al. 29 states that the commutator operator a, T f T af − a Tf is bounded on L p R n for 1 < p < ∞.The commutator of Calder ón-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 of BMO R n .Definition 5.1.Suppose that f ∈ L loc 1 R n , and let where If one regards two functions whose difference is a constant as one, then space BMO R n is a Banach space with respect to norm • * .
Remark 5.2. 1 The John-Nirenberg inequality: there are constants C 1 , C 2 > 0 such that for all f ∈ BMO R n and β > 0, 5.4 2 The John-Nirenberg inequality implies that 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.Let 1 < p < ∞, ϕ x, r which satisfies the conditions 3.4 and 3.5 and a ∈ BMO R n .Suppose that T is a linear operator and satisfies the condition 1.5 .If the operator a, T is bounded on L p R n , then the operator a, T is bounded on M p,ϕ .Remark 5.4.Note that Theorem 5.3 in the following form is also valid.Let 1 < p < ∞, ϕ x, r satisfy the conditions 3.4 and 3.5 and a ∈ BMO R n .Suppose that T a is a sublinear operator satisfies the condition 1.6 and bounded on L p R n , then the operator T a is bounded on M p,ϕ .Lemma 5.5.Let 1 < p < ∞, a ∈ BMO R n , and a sublinear operator T a satisfies the condition 1.6 and bounded on L p R n . Then, holds for any ball B B x 0 , r and for all f ∈ L loc p R n .
Proof. 1 < p < ∞, a ∈ BMO R n , and a sublinear operator T a satisfies the condition 1.6 .For arbitrary x 0 ∈ R n , set B B x 0 , r for the ball centered at x 0 and of radius r.Write f f 1 f 2 with f 1 fχ 2B and f 2 fχ c 2B .Hence,

5.8
From the boundedness of T a in L p R n , it follows that

5.9
For x ∈ B, we have x 0 − y n f y dy.

5.11
Let us estimate I 1

5.13
In order to estimate I 2 , note that By 5.5 , we get Thus, by 4.12 ,

5.16
Summing up I 1 and I 2 , for all p ∈ 1, ∞ , we get 5.17 Finally, and statement of Lemma 5.5 follows by 4.15 .
The following theorem is true.
where C does not depend on x and r.Suppose that T a is a sublinear operator which satisfies the condition 1.6 and bounded on L p R n .Then, the operator T a is bounded from M p,ϕ 1 to M p,ϕ 2 .Moreover, Proof.The statement of Theorem 5.6 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 Note that when the conditions of Corollary 5.8 are satisfied, the existence of Kf x for a.e.x ∈ R n was proved in 28 .

Some Applications
In this section, we will apply Theorems 4.5 and 5.6 to several particular operators such as the pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

Pseudodifferential Operators
Pseudodifferential operators are generalizations of differential operators and singular integrals.Let m be real number, 0 ≤ δ < 1 and 0 ≤ ρ < 1.Following 32, 33 , a symbol in S m ρ,δ is a smooth function σ x, ξ defined on R n × R n such that for all multi-indices α and β the following estimate holds: where C αβ > 0 is independent of x and ξ.A symbol in S −∞ ρ,δ is one which satisfies the above estimates for each real number m.
The operator A given by Af x is called a pseudodifferential operator with symbol σ x, ξ ∈ S m ρ,δ , where f is a Schwartz function and f denotes the Fourier transform of f.As usual, L m ρ,δ will denote the class of pseudodifferential operators with symbols in S m ρ,δ .Miller 34 showed the boundedness of L 0 1,0 pseudodifferential operators on weighted L p 1 < p < ∞ spaces whenever the weight function belongs to Muckenhoupt's class A p .In 1 , it is shown that pseudodifferential operators in L 0 1,0 are Calder ón-Zygmund operators, then from Corollary 5.8, we get the following new results.

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.Then, the generalized Littlewood-Paley g function g ψ is defined by where ψ t x t −n ψ x/t for t > 0 and F t f ψ t * f.The sublinear commutator of the operator g ψ is defined by where The following theorem is valid see 3, Theorem 5.1.2 .
Theorem 6.4.Suppose that ψ ∈ L 1 R n satisfies 6.3 and the following properties: where C and α > 0 are both independent of x and h.Then, g ψ is bounded on L p R n for all 1 < p < ∞, and bounded from L 1 R n to WL 1 R n .
Let H be the space H {h : h ∞ 0 |h t | 2 dt/t 1/2 < ∞}, then for each fixed x ∈ R n , F t f x may be viewed as a mapping from 0, ∞ to H, and it is clear that g ψ f x F t f x .In fact, by Minkowski inequality and the conditions on ψ, we get

Marcinkiewicz Operator
Let S n−1 {x ∈ R n : |x| 1} be the unit sphere in R n equipped with the Lebesgue measure dσ.Suppose that Ω satisfies the following conditions.
a Ω is the homogeneous function of degree zero on R n \ {0}; that is, Ω tx Ω x , for any t > 0, x ∈ R n \ {0}.6.9 b Ω has mean zero on S n−1 ; that is, The sublinear commutator of the operator μ Ω is defined by

14 Abstract and Applied Analysis Definition 6 . 3 .
Suppose that ψ ∈ L 1 R n satisfies

S n− 1 Ω x dσ x 0. 6 . 10 c 0 F
Ω ∈ Lip γ S n−1 , 0 < γ ≤ 1, that is there exists a constant M > 0 such thatΩ x − Ω y ≤ M x − y γ for any x , y ∈ S n−1 .6.11In 1958, Stein 35 defined the Marcinkiewicz integral of higher dimension μ Ω as μ Ω f x ∞ 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 .
, 15 .Let 1 ≤ p < ∞, and let ϕ 1 , ϕ 2 satisfy the condition ∞ t ϕ 1 x, r dr r ≤ Cϕ 2 x, t , 4.1 where C does not depend on x and t.Then, the operators M and K are bounded from M p,ϕ 1 to M p,ϕ 2 for p > 1 and from M 1,ϕ 1 to WM 1,ϕ 2 .In this section, we are going to use the following statement on the boundedness of the Hardy operator: Hg t : 1 t t 0 g r dr, 0 < t < ∞. 4.2 Theorem 4.3 see 27 .The inequality ess sup t>0 w t Hg t ≤ c ess sup t>0 v t g t 4.3 holds for all nonnegative and nonincreasing g on 0, ∞ if and only if Lemma 4.4.Let 1 ≤ p < ∞, T be a sublinear operator which satisfies the condition 1.5 bounded on L p R n for p > 1 and bounded from Let T be a sublinear operator which satisfies the condition 1.5 bounded on L p R n for p > 1 and bounded from L 1 R n to WL 1 R n .Then, the operator T is bounded from M p,ϕ 1 to M p,ϕ 2 for p > 1 and from M 1,ϕ 1 to WM 1,ϕ 2 .Moreover, for p > 1 Proof.By Lemma 4.4 and Theorem 4.3, we have for p > 1 Theorem 4.5.Let 1 ≤ p < ∞, and let ϕ 1 , ϕ 2 satisfy the condition ∞ r ess inf t<s<∞ ϕ 1 x, s s n/p t n/p 1 dt ≤ Cϕ 2 x, r , 4.18where C does not depend on x and r.
Let 1 < p < ∞, ϕ 1 , ϕ 2 satisfy the condition 5.19 and a ∈ BMO R n .Then, the sublinear commutator operator M a is bounded from M p,ϕ 1 to M p,ϕ 2 .Let 1 < p < ∞, ϕ 1 , ϕ 2 satisfy the condition 5.19 and a ∈ BMO R n .Then, Calderón-Zygmund singular integral Kf x exists for a.e.x ∈ R n and the operator a, K is bounded from M p,ϕ 1 to M p,ϕ 2 .

Corollary 6.1. Let
1 ≤ p < ∞, and let ϕ 1 , ϕ 2 satisfy the condition 4.18 .If A is a pseudodifferential operator of the Hörmander class L 0 1,0 , then the operator A is bounded from M p,ϕ 1 to M p,ϕ 2 for p > 1 and bounded from M 1,ϕ 1 to WM 1,ϕ 2 .Let 1 < p < ∞, ϕ 1 , ϕ 2 satisfy the condition 5.19 and a ∈ BMO R n .Let also A be a pseudodifferential operator of the Hörmander class L 0 1,0 .Then, the commutator operator a, A is bounded from M p,ϕ 1 to M p,ϕ 2 .