1. Introduction

Let be the unit sphere in equipped with the Lebesgue measure . Suppose that satisfies the following conditions.

(a) is the homogeneous function of degree zero on , that is,

(1.1)

(b) has mean zero on , that is,

(1.2)

(c), that is,

(1.3)

In 1958, Stein [1] defined the Marcinkiewicz integral of higher dimension as

(1.4)

where

(1.5)

We refer to see [1, 2] for the properties of .

Let and The parameterized area integral and the parameterized Littlewood-Paley function are defined by

(1.6)

where and

(1.7)

respectively. and play very important roles in harmonic analysis and PDE (e.g., see [3–8]).

Before stating our result, let us recall some definitions. For the commutator formed by and the Marcinkiewicz integral are defined by

(1.8)

Let and The commutator of and the commutator of are defined, respectively, by

(1.9)
(1.10)

Let . It is said that if

(1.11)

where denotes the ball in centered at and with radius ,

(1.12)

and .

There are some results about the boundedness of the commutators formed by BMO functions with , , and (see [7, 9, 10]).

Many important operators gave a characterization of BMO space. In 1976, Coifman et al. [11] gave a characterization of BMO space by the commutator of Riesz transform; in 1982, Chanillo [12] studied the commutator formed by Riesz potential and BMO and gave another characterization of BMO space.

The purpose of this paper is to give a characterization of BMO space by the boundedness of the commutators of , , and on the generalized Morrey space .

Definition 1.1.

Let . Suppose that be such that is nonincreasing and is nondecreasing. The generalized Morrey space is defined by

(1.13)

where

(1.14)

We refer to see [13, 14] for the known results of the generalized Morrey space for some suitable . Noting that , we get the Lebesque space . For ,   coincides with the Morrey space .

The main result in this paper is as follows.

Theorem 1.2.

Assume that is nonincreasing and is nondecreasing. Suppose that is defined as (1.8), satisfies (1.1), (1.2), and

(1.15)

If is bounded on for some , then

Theorem 1.3.

Let and . Assume that is nonincreasing and is nondecreasing. Suppose that is defined as (1.9), satisfies (1.1), (1.2), and (1.15). If is a bounded operator on for some , then

Theorem 1.4.

Let , , and . Assume that is nonincreasing and is nondecreasing. Suppose that is defined as (1.10), satisfies (1.1), (1.2), and (1.15). If is on for some , then

Remark 1.5.

It is easy to check that (see, e.g., the proof of () in [15, page 89]), we therefore give only the proofs of Theorem 1.2 for and Theorem 1.3 for .

Remark 1.6.

It is easy to see that the condition (1.15) is weaker than for . In the proof of Theorems 1.2 and 1.3, we will use some ideas in [16]. However, because Marcinkiewicz integral and the parameterized Littlewood-Paley operators are neither the convolution operator nor the linear operators, hence, we need new ideas and nontrivial estimates in the proof.

2. Proof of Theorem 1.2

Let us begin with recalling some known conclusion.

Similar to the proof of [17], we can easily get the following.

Lemma 2.1.

If satisfies conditions (1.1), (1.2), and (1.15), let then for , we have

(2.1)

Now let us return to the proof of Theorem 1.2. Suppose that is a bounded operator on , we are going to prove that

We may assume that . We want to prove that, for any and , the inequality

(2.2)

holds, where Since we may assume that Let

(2.3)

where Since , we can easily get Then, has the following properties:

(2.4)
(2.5)
(2.6)
(2.7)
(2.8)

In this proof for , is a positive constant depending only on   , , , and . Since satisfies (1.2), then there exists an such that and

(2.9)

where is the measure on which is induced from the Lebesgue measure on . By the condition (1.15), it is easy to see that

(2.10)

is a closed set. We claim that

(2.11)

In fact, since note that we can get Taking , let

(2.12)

For , we have

(2.13)

For noting that if , then for . Thus, we have

(2.14)

Using (2.11), we get Noting that it follows from (2.5), (2.7), (2.8), and Hölder's inequality that

(2.15)

For , by , (2.4), (2.5), (2.6), the Minkowski inequality, and Lemma 2.1, we obtain

(2.16)

Let

(2.17)

Without loss of generality, we may assume that , otherwise, we get the desired result. Since is nonincreasing, it follows that . By (2.13), (2.15), and (2.16), we have

(2.18)

Thus,

(2.19)

Now, we claim that

(2.20)

where is independent of . In fact,

(2.21)

Now, we consider the norm of in the following two cases.

Case 1 ().

Since is nondecreasing in , then

(2.22)

Thus,

(2.23)

Case 2 ().

Since is nonincreasing in , then

(2.24)

Thus,

(2.25)

Now, (2.20) is established. Then, by (2.19) and (2.20), we get

(2.26)

If then Theorem 1.2 is proved. If then

(2.27)

Let . For , we have

(2.28)

Noting that if and , we get . Applying (2.11), we have . Since when and , it follows that

(2.29)

By ,   when and and the Minkowski inequality, we have

(2.30)

Thus, by (2.28), (2.29), and (2.30), we get, for ,

(2.31)

Similar to the proof of (2.20), we can easily get . Thus, by (2.31), , and when , we have

(2.32)

We first estimate Since for we have

(2.33)

Now, the estimate of is divided into two cases, namely, 1: ; 2: .

Case 1 ().

Since the function is decreasing for and for by (2.27), we get

(2.34)

Case 2 ().

Since the function is decreasing for and for , by (2.27), we have

(2.35)

From Cases 1 and 2, we know that there exists a constant such that

(2.36)

So by (2.32), (2.33), and (2.36), we get

(2.37)

Then, Theorem 1.2 is proved.

3. Proof of Theorem 1.3

Similar to the proof of Theorem 1.2, we only give the outline.

Suppose that is a bounded operator on , we are going to prove that

We may assume that . We want to prove that, for any and , the inequality

(3.1)

holds, where Since we may assume that Let be as (2.3), then (2.4)–(2.8) hold. In this proof for , is a positive constant depending only on , , , and . Since satisfies (1.2), then there exists a such that and

(3.2)

where is the measure on which is induced from the Lebesgue measure on . By the condition (1.15), it is easy to see that

(3.3)

is a closed set. As the proof of (2.11), we can get the following:

(3.4)

Taking , let

(3.5)

For , we have

(3.6)

For noting that if , , and then we get

(3.7)

Then by (3.4), we get . Since and we get and . Thus, by (2.5), (2.7), (2.8), and the Hölder inequality, we get

(3.8)

By (2.5) and (2.6), we have

(3.9)

In we have and . In we get and It is easy to see that Now, we estimate by , the Minkowski inequality, Lemma 2.1 for , and (2.4), we get

(3.10)

From (3.9) and (3.10), we get

(3.11)

Let

(3.12)

Without loss of generality, we may assume that , otherwise, we get the desired result. Since is nonincreasing, we have . Then by, (3.6), (3.8), and (3.11), we get

(3.13)

Thus,

(3.14)

Then, by (2.20) and (3.14), we get

(3.15)

If then Theorem 1.3 is proved. If then

(3.16)

Let . For , we have

(3.17)

For as above mentioned, we have Since and , it follows the Hölder inequality that

(3.18)

By , the Minkowski inequality, and for and , we get

(3.19)

Thus, by (3.17), (3.18), and (3.19), we get, for ,

(3.20)

Thus, by (3.20), , when and the Hölder inequality, we have

(3.21)

As the proof of (2.33) and (2.36), we can get that there exists a constant such that

(3.22)

So, by (3.21) and (3.22), we get

(3.23)

Then, Theorem 1.3 is proved.