Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 923874, 21 pages

doi:10.1155/2012/923874

Research Article

BMO-Boundedness of Maximal Operators and g-Functions Associated with Laguerre Expansions

Li Cha and Heping Liu

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

Received 29 February 2012; Revised 21 May 2012; Accepted 21 June 2012

Academic Editor: Aurelian Gheondea

Copyright © 2012 Li Cha and Heping Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be the Laguerre functions of Hermite type with index . These are eigenfunctions of the Laguerre differential operator . In this paper, we investigate the boundedness of the Hardy-Littlewood maximal function, the heat maximal function, and the Littlewood-Paley -function associated with in the localized BMO space , which is the dual space of the Hardy space .

1. Introduction

Let , . The Laguerre function of Hermite type on is defined as

where denotes the Laguerre polynomial of degree and order , see [1]. It is well known that for every the system forms an orthonormal basis of . Moreover, these functions are eigenfunctions of the Laguerre differential operator

satisfying . The operator can be extended to a positive self-adjoint operator on by giving a suitable domain of definition, see [2]; we also denote the extension by . Let be the heat-diffusion semigroup generated by . More precisely, for , we define

where

is the modified Bessel function of the first kind and order .

In [3], we introduced and developed a localized BMO space associated with the operator , which is the dual space of the Hardy space introduced by Dziubański [4]. More precisely, let

Definition 1.1. Let , be any ball in with the center and the radius and a locally integrable function on . We say if there exists a constant independent of and such that

Here, . We let denote the smallest in the two inequalities above.

It is readily seen that is a Banach space with norm .

In this paper, we obtain the boundedness on of several operators including the Hardy-Littlewood maximal operator defined on , the heat maximal function, and the Littlewood-Paley -function associated with .

These results were investigated by Dziubański et al. in [5] for Schrödinger operators on with and with potentials satisfying a reverse Hölder's inequality. Recently, a theory of localized BMO spaces on RD-spaces associated with an admissible function was investigated in [6]; the authors also established the similar results above for their spaces. The admissible function in [6] is required to satisfy

Obviously, our in (1.5) does not satisfy this condition. Indeed, let tend to zero and ; then the left side becomes greater than the right.

It is notable the generalized square functions associated to Schrödinger operators are studied in [7]. The authors of [7] gave several of equivalent conditions for BMO-boundedness of square functions.

In this paper, in order to obtain some key estimates, we will employ the differences in integral kernels (the heat kernel, the -function kernel) associated with the Hermite operator and the Laguerre operator, respectively (see [8, 9]).

The paper is organized as follows. In the next section we present some preliminary lemmas and collect some useful estimates of the kernels associated with the heat semigroups and the -functions. In Section 3, we establish the boundedness of two maximal operators (the Hardy-Littlewood maximal operator and the heat maximal function) from to . In Section 4, we obtain the boundedness on of the Littlewood-Paley -function associated with the heat semigroup for . We make some conventions. Throughout this paper by we always denote a positive constant that may vary at each occurrence; stands for ; means , and the notation is used to indicate that with an independent positive constant .

2. Preliminaries

Now we give the following covering lemma for which will be used frequently below. The proof is trivial and left to the reader.

Lemma 2.1. Let ,   for , and for . One defines the family of “critical balls” of , where . Then one has

(a), (b)for every , provided that , (c)for any , at most three balls in have nonempty intersection with .

Corollary 2.2. There exists a constant such that for every with , one has

Corollary 2.3. There exists a constant such that, for , one has

where, for any ball , the norm is given by

Corollary 2.4 (see [3, Corollary  3]). Let . There exists a constant such that, for all , one has

(1)if , then , (2)if , then .

We give two elementary lemmas, which will be used frequently in next section. The proofs are trivial, and the reader also refer to Lemmas 9 and 2 in [5].

Lemma 2.5. Let and and be functions in . If is any measurable function satisfying

then and .

Lemma 2.6. For all and with . There exists a constant such that

Let be the Hermite operator

One considers the heat diffusion semigroup associated with and defined by, for every ,

where for each and ,

(see [10]).

Proposition 2.7. Let , be in (2.8). There exists such that, for ,

(a), , (b), , (c), , (d).

Parts (a), (b), and (c) are the contents of Lemma 2.11 in [8]. Part (d) is from (2.6) in [4].

Remark 2.8. The ranges and are not critical; Proposition 2.7 also holds when and , where .

Now we consider the estimates of the integral kernel for the -function, which will be defined in Section 4:

Proposition 2.9. One has,

(a)for every such that ,

(b)for every such that ,

Parts (a) and (b) are contained in [9, (3.4) and (3.6)].

Proposition 2.10. For every , there is a constant such that

(a)if , ; (b)for , , is independent of ; (c), is independent of and .

Proof. By using (2.8) we can write, for every and ,

By the simple fact

a straightforward manipulation leads to

which implies (a).

To prove (b), we also directly compute the partial derivative:

By an elementary manipulation and (2.14), we have

This together with the mean value theorem and the condition leads to (b).

Let ; is a smooth function satisfying for , for and for . From the above, for fixed and , a straightforward manipulation shows that

Hence, we have

Using (2.8) again,

which implies (c).

Lemma 2.11 (see [3, Theorem 2]). For all and , there exists a constant such that

3. Maximal Operators

First of all, we define the following notions:

In this section, we will show and are bounded on .

Theorem 3.1. There exists a constant such that, for all , , for a.e. , and

Proof. First of all, we show that for a.e. , . To do this, we only need to show that for, at almost in Lemma 2.1, . Let us split with . Obviously, since is locally integrable, we have for a.e. . For , if and , since supp is in the complement of , we have . Otherwise, by the definition of , .

We turn to the boundedness in . Let denote the Hardy-Littlewood function on ; it is well known in [11] that is bounded on . Let be a function defined on which is on and on . Notice that , for , so

Now, we need to show that . Indeed, if , it is obvious that . If , then . If and , let , here and , then

On the other hand, we again split with , from the argument above, , for a.e . So

where in the last inequality we have used Corollary 2.4.

Theorem 3.2. Let . There exists a constant such that

Proof. By the definition of and Corollary 2.3, it suffices to prove the following: for every fixed “critical ball” (see Lemma 2.1) we have

(1), (2).

Let us start to prove (1). It is immediate from Theorem 3.1 and (d) of Proposition 2.7; since , for , therefore,

It remains to show (2). By Lemma 2.5, we split into several parts. First, we shall show

From (d) of Proposition 2.7, we have

Notice that, for and , we have , for . Thus

Therefore,

By Lemma 2.5, it suffices to show that satisfies (2). Write

By Proposition 2.7, it easily follows that

Indeed, since , for , by (a) of Proposition 2.7 and Remark 2.8, we have

Similarly,

Now, we come to treat . We make further decompositions. Split

where

For the first term, by (c) of Proposition 2.7, we have

By (2.8),

Notice that , when , and . Therefore, for and , we obtain

Finally, by Lemma 2.5 again, we need to show that satisfies (2). Consider and write

By Corollary 2.3, we choose a constant ,

For the first integral, by Corollary 2.4 it easily follows that

For the second integral,

By the mean value theorem and the elementary inequality

we have

On the other hand, by the fact in Lemma 2.6, we obtain

Therefore, we obtain

which establishes the proof.

4. -Function

For all and , define the Littlewood-Paley -function by

where, is a family of operators with the integral kernels

Theorem 4.1. Let . There exists a constant such that, for all , and .

Proof. By Proposition 2.9 and (a) of Proposition 2.10, we have

For , because of this and the integrability of (see [12, page 141]),

is well defined absolutely convergent integral for all . Similar to the proof of Theorem 3.2, we will try to show that, for in Lemma 2.1,

(1), (2).

We split

By Lemma 2.11 and Hölder inequality, assertion (1) holds for . To finish the proof of (1), it suffices to show that

In the next proof, for the sake of brevity we introduce the additional notations:

By and , we split as

For and , we shall first show the inequality

Using (a) of Proposition 2.9, if , , we get

If , , we have

The previous two inequalities above imply

For and , we shall also prove the inequality

We split this integral as

To deal with , we discuss two cases. In the first case of , notice that , when , and . According to (b) of Proposition 2.9,

The last inequality is from the same proof of . In the second case of , using (b) of Proposition 2.9 again, for we obtain

The last inequality is also from the same proof of . Inserting this into leads to . Now, it remains to show . Using (b) of Proposition 2.9, by the standard argument it easily follows that

To complete the proof of (4.6), we need to show that

We also consider two cases of and . When , repeating the above argument for and using (a) of Proposition 2.10, we have . When , using (a) of Proposition 2.10 again, for , we obtain

which shows that (4.19) holds.

Next, we come to prove assertion (2). By (4.6) and Lemma 2.5, we only need to show

Consider any ball . By Lemma 2.11, we have

Therefore, by Lemma 2.5 and Corollary 2.3, it suffices to prove

To prove (4.23), we first claim that, for all , , and ,

We shall split into three different estimates:

Let us first treat (4.25). Since , when , notice that when , using (a) of Proposition 2.9, and recalling the definition of , we have

For (4.26), using (b) of Proposition 2.9, for , the left side of (4.26) is controlled by

The third inequality is from the same argument for dealing with (3.21) in the proof of Theorem 3.2.

For (4.27), we write

By (a), (c) of Proposition 2.10 and the fact that , we have

The third and fourth inequalities come from the fact and , when and . It is notable that, for and , the inequality above also implies

On the other hand, by (a) of Proposition 2.10 and the simple fact that , we obtain

From (4.24), we deduce that

By (4.28) and (4.29), for , we have

To finish the proof of (4.23), it suffices to show

It is easy to check (4.37). Indeed, without loss of generality, we assume . According to (a) of Proposition 2.10,

Inserting this into (4.37) gives the desired result. Moreover, from the above we also have proved

For , we write

By (b) of Proposition 2.10, we obtain

From the inequality above, we deduce that

On the other hand, by using (b) of Proposition 2.10 again, we also have

This together with (4.32), (4.39), and gives that

Thus we finish the proof of (4.36) and, hence, finish the proof of Theorem 4.1.

Acknowledgments

The authors would like to express their gratitude to the referee for many valuable suggestions and revisions which improve the presentation of this paper. Supported by National Natural Science Foundation of China under Grant no. 10990012.

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