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 .
Let , . The Laguerre function of Hermite type on is defined as
where denotes the Laguerre polynomial of degree and order , see . 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 ; we also denote the extension by . Let be the heat-diffusion semigroup generated by . More precisely, for , we define
is the modified Bessel function of the first kind and order .
In , we introduced and developed a localized BMO space associated with the operator , which is the dual space of the Hardy space introduced by Dziubański . 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  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 ; the authors also established the similar results above for their spaces. The admissible function in  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 . The authors of  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 .
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 .
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 ,
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 . Part (d) is from (2.6) in .
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  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
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
Now, we come to treat . We make further decompositions. Split
For the first term, by (c) of Proposition 2.7, we have
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
On the other hand, by the fact in Lemma 2.6, we obtain
Therefore, we obtain
which establishes the proof.
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