Abstract

The authors prove that Marcinkiewicz integral operator is not only are bounded on , for , but also a bounded map from to weak . Meanwhile, the -boundedness and -boundedness are also obtained. Finally, the -boundedness and -boundedness for the commutator of Marcinkiewicz integral of schrödinger type are established.

1. Introduction and Notation

Let us consider the Schrödinger operator in . The function is nonnegative, , and belongs to a reverse Hölder class , for some exponent ; that is, there exists a constant such that for every ball .

We introduce the definition of the reverse Hölder index of as . It is known that implies , for some . Therefore, under the assumption , we may conclude .

The classical Marcinkiewicz integral operator is defined by

The above operator was introduced by Stein in [1] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimension. Meanwhile, Stein [1] showed that if , for some , then is a bounded operator on , for , and is a bounded map from to weak . Benedek et al. [2] showed that if is continuously differentiable in , then is a bounded operator on , for . Ding et al. [3] proved that the Marcinkiewicz function is bounded from to with satisfying cancelation condition on and -Dini condition.

Similar to the classical Marcinkiewicz function , we define the Marcinkiewicz integral associated with the Schrödinger operator by where and is the kernel of . In particular, when and is the kernel of .

We also give the definition of the commutator generalized by and by In this paper, we write and

For a given potential , with , we introduce the auxiliary function The above assumptions are finite, for all .

Proposition 1 (see [4]). There exist and such that for all .

In particular, , if . A ball is called critical.

Proposition 2 (see [5]). There exist a sequence of points in , so that the family , satisfies the following:(1);(2)there exists such that, for every , .

Function is an element of , if such that, for every ball , if , and if , where . Let be the smallest in the inequalities above. It is easy to verify that , for all .

Lemma 3 (see [6]). For any , there exists such that

Tang and Dong [6] have shown that Marcinkiewicz integral is bounded on , for , and are bounded from to weak . Meanwhile, they also proved that are bounded on and are also mapped from to under the assumption that satisfy the condition in Lemma 3.

Now, we introduce a new -type space of schrödinger operator. Let ; we define the class of locally integrable functions such that for all and , where . A norm for , denoted by , is given by the infimum of the constants in the inequalities above. Notice that if we let , we obtain the John-Nirenberg space .

With the above definition, we define . Clearly, , for , and hence . Moreover, it is in general a larger class.

For , we denote by the set of functions such that for all and . A norm for , denoted by , is given by the infimum of the constants in the inequalities above. Correspondingly, we define .

Next, we give some information on the Hardy space associated with Schrödinger operator . We say that a function is said to belong to , if the semigroup maximal operator is bounded on . The - norm of is given by , where is a semigroup generated by the Schrödinger operator (see [7]).

Shen [4] gave the following kernel estimate that we needed.

Lemma 4. If , then,(i)for every , there exists a constant such that (ii)for every and , there exists a constant such that where ,(iii)if denotes the vector valued kernel of the classical Riesz operator, for every , we have where .

Inspired by [6], we consider the same boundedness of Marcinkiewicz integral whose kernel satisfies Lemma 4 and -boundedness of its commutator for and -boundedness.

Theorem 5. The operators are bounded on , for , and are bounded from to weak . On the other hand, are bounded on and are bounded from to .

In the note, we devote ourselves to establish the following boundedness of the commutators of Marcinkiewicz integral of schrodinger operator type.

Theorem 6. Let , , and such that = , where is the reverse Hölder index of ; then, for all , we have where .

Theorem 7. Let and ; if satisfies the stronger condition , then are bounded from to .

Throughout this paper, denotes the constants that are independent of the main parameters involved but whose value may differ from line to line. By , we mean that there exists a constant such that . We use the symbol to denote that there exists a positive constant such that .

2. Notation and Preliminaries

Denote the following maximal functions for , , and : where and .

Lemma 8 (Fefferman-Stein type inequality [8]). For , there exist and such that if is a sequence of balls as that in Proposition , then for all .

Lemma 9 (see [9]). Let and . If , then for all , with and , where and is the constant appearing in Proposition 1.

Lemma 10 (see [9]). Let , , and ; then for all , with as in Lemma 9.

Next, we introduce our important Lemmas.

We denoted by the Hardy-Littlewood maximal function and, for , by , the operator is defined as .

Lemma 11. Let , and . Then, for any , there exists a constant such that for all and every ball .

Proof. Let and . We first observe and so we have to deal with the average on of each term.
Thanks to Hölder's inequality with and Lemma 10, one has
Let with ; then -boundedness of , for , says that Now, for , and using Lemma 4 and Minkowski's inequality, we get
For the second term, we split again . By choosing and denoting , by the boundedness of on and Hölder's inequality, we obtain where, in the last inequality, we have used Lemma 9.
For the remaining term, let , using Lemma 9 , Lemma 10, and Minkowski's inequality, we arrive to Since can be chosen large enough, the last series converges. Thus, we finished the proof.