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

Let {φn}n∈N be the Laguerre functions of Hermite type with index α. These are eigenfunctions of the Laguerre differential operator Lα 1/2 −d2/dy2 y2 ( 1/y2 ) α2 − 1/4 . In this paper, we investigate the boundedness of the Hardy-Littlewood maximal function, the heat maximal function, and the Littlewood-Paley g-function associated with Lα in the localized BMO space BMOLα , which is the dual space of the Hardy space H 1 Lα .


Introduction
Let n ∈ N, α > −1.The Laguerre function of Hermite type ϕ α on 0, ∞ is defined as where L α n x denotes the Laguerre polynomial of degree n and order α, see 1 .It is well known that for every α > −1 the system {ϕ α n } ∞ n 0 forms an orthonormal basis of L 2 0, ∞ .Moreover, these functions are eigenfunctions of the Laguerre differential operator satisfying L α ϕ α n 2n α 1 ϕ α n .The operator L α can be extended to a positive self-adjoint operator on L 2 0, ∞ by giving a suitable domain of definition, see 2 ; we also denote 2 Journal of Function Spaces and Applications the extension by L α .Let {T α t } t≥0 be the heat-diffusion semigroup generated by −L α .More precisely, for f ∈ L 2 0, ∞ , we define where 1 − e −2t x 2 y 2 . 1.4 I α is the modified Bessel function of the first kind and order α.
In 3 , we introduced and developed a localized BMO space BMO L α associated with the operator L α , which is the dual space of the Hardy space H 1 L α introduced by Dziuba ński 4 .More precisely, let Definition 1.1.Let α > −1/2, B s y be any ball in 0, ∞ with the center y and the radius s and f a locally integrable function on 0, ∞ .We say f ∈ BMO L α if there exists a constant C ≥ 0 independent of s and y such that

1.6
Here, f B s y 1/|B s y | B s y fdx.We let f BMO Lα denote the smallest C in the two inequalities above.
It is readily seen that BMO L α is a Banach space with norm • BMO Lα .In this paper, we obtain the boundedness on BMO L α of several operators including the Hardy-Littlewood maximal operator defined on 0, ∞ , the heat maximal function, and the Littlewood-Paley g-function associated with T α t .These results were investigated by Dziuba ński et al. in 5 for Schr ödinger operators on R d with d ≥ 3 and with potentials satisfying a reverse H ölder's inequality.Recently, a theory of localized BMO spaces on RD-spaces associated with an admissible function ρ was investigated in 6 ; the authors also established the similar results above for their BMO spaces.The admissible function ρ in 6 is required to satisfy Obviously, our ρ L α in 1.5 does not satisfy this condition.Indeed, let x tend to zero and y 1; then the left side becomes greater than the right.It is notable the generalized square functions associated to Schr ödinger 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 g-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 g-functions.In Section 3, we establish the boundedness of two maximal operators the Hardy-Littlewood maximal operator and the heat maximal function from BMO L α to BMO L α .In Section 4, we obtain the boundedness on BMO L α of the Littlewood-Paley gfunction associated with the heat semigroup for L α .We make some conventions.Throughout this paper by C we always denote a positive constant that may vary at each occurrence; B r y 0 stands for {y > 0, |y − y 0 | ≤ r}; A ∼ B means 1/C A ≤ B ≤ CA, and the notation X Y is used to indicate that X ≤ CY with an independent positive constant C.

Preliminaries
Now we give the following covering lemma for 0, ∞ which will be used frequently below.The proof is trivial and left to the reader.Lemma 2.1.Let x 0 1, x j x j−1 ρ L α x j−1 for j ≥ 1, and x j x j 1 − ρ L α x j 1 for j < 0. One defines the family of "critical balls" of c for any y 0 ∈ 0, ∞ , at most three balls in B have nonempty intersection with B y 0 , ρ L α y 0 .

Corollary 2.2.
There exists a constant C > 0 such that for every B R x ⊆ 0, ∞ with R > ρ L α x , one has Corollary 2.3.There exists a constant C such that, for f ∈ BMO L α , one has where, for any ball B, the norm • BMO B is given by

2.3
Corollary 2.4 see 3, Corollary 3 .Let B B r y 0 ⊂ 0, ∞ .There exists a constant C > 0 such that, for all f ∈ BMO L α , one has 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 h ∈ BMO B * k and g 1 and g 2 be functions in L ∞ 0, ∞ .If f is any measurable function satisfying 2.4 Lemma 2.6.For all f ∈ BMO L α and B B r y 0 with r < ρ L α y 0 .There exists a constant C > 0 such that Let H be the Hermite operator

2.6
One considers the heat diffusion semigroup {W t } t>0 associated with H and defined by, for every f ∈ L 2 R , where for each x, y ∈ R and t > 0, 2.8 see 10 .
Proposition 2.7.Let α > −1/2, W t x, y be in 2.8 .There exists C > 0 such that, for t > 0, Parts a , b , and c are the contents of Lemma 2.11 in 8 .Part d is from 2.6 in 4 .
Now we consider the estimates of the integral kernel for the g-function, which will be defined in Section 4:

2.10
Proposition 2.9.One has, Parts a and b are contained in 9, 3.4 and 3.6 .
Proposition 2.10.For every N ≥ 1, there is a constant C N such that Proof.By using 2.8 we can write, for every x, y ∈ R and s > 0,

6 Journal of Function Spaces and Applications
By the simple fact which implies a .
To prove b , we also directly compute the x partial derivative:

2.16
By an elementary manipulation and 2.14 , we have

2.17
This together with the mean value theorem and the condition |h| ≤ t leads to b .Let φ n y φ y/n ; φ y is a smooth function satisfying φ y 1 for |y| ≤ 1, φ y 0 for |y| ≥ 2 and Δφ y ≤ 1 for y ∈ R. From the above, for fixed s and x, a straightforward manipulation shows that ∞ −∞ ∂W s x, y ∂s dy < ∞.

2.18
Hence, we have W s x, y y 2 dy.
Lemma 2.11 see 3, Theorem 2 .For all f ∈ BMO L α and B B r y 0 ⊆ 0, ∞ , there exists a constant C > 0 such that 2.21

Maximal Operators
First of all, we define the following notions: In this section, we will show I * α and M are bounded on BMO L α .

Theorem 3.1.
There exists a constant C > 0 such that, for all f ∈ BMO L α , M f < ∞, for a.e.
x ∈ 0, ∞ , and Proof.First of all, we show that for a.e.x ∈ 0, ∞ , M f < ∞.To do this, we only need to show that for, at almost We turn to the boundedness in BMO L α .Let M denote the Hardy-Littlewood function on R; it is well known in 11 that M is bounded on BMO R .Let f 0 be a function defined on R which is f on 0, ∞ and 0 on −∞, 0 .Notice that M f Mf 0 , for x ∈ 0, ∞ , so Now, we need to show that

3.5
On the other hand, we again split where in the last inequality we have used Corollary 2.4.
Theorem 3.2.Let α > −1/2.There exists a constant C > 0 such that Proof.By the definition of BMO L α and Corollary 2.3, it suffices to prove the following: for every fixed "critical ball" B k ∈ B see Lemma 2.1 we have It remains to show 2 .By Lemma 2.5, we split I * α f x into several parts.First, we shall show sup From d of Proposition 2.7, we have

3.10
Notice that, for j ≥ 0 and t > ρ 2 L α x k , we have Therefore, sup

3.13
By Proposition 2.7, it easily follows that sup

3.14
Indeed, since x ∼ x k , for x ∈ B * k , by a of Proposition 2.7 and Remark 2.8, we have

3.16
Now, we come to treat f 1 .We make further decompositions.Split where 1 − e −2t x − y 2 g y dy.

3.18
For the first term, by c of Proposition 2.7, we have

3.21
Finally, by Lemma 2.5 again, we need to show that sup 0<t≤ρ 2 Consider B B r x 0 ⊂ B * k and write

3.22
By Corollary 2.3, we choose a constant

3.23
For the first integral, by Corollary 2.4 it easily follows that

3.24
For the second integral,

3.25
By the mean value theorem and the elementary inequality we have

3.27
On the other hand, by the fact |f B | ≤ C 1 log ρ L α x 0 /r f BMO Lα in Lemma 2.6, we obtain

3.28
Therefore, we obtain which establishes the proof.

g-Function
For all f ∈ L 1 loc 0, ∞ and x ∈ 0, ∞ , define the Littlewood-Paley g-function by where, {Q t } t>0 is a family of operators with the integral kernels Proof.By Proposition 2.9 and a of Proposition 2.10, we have For f ∈ BMO L α , because of this and the integrability of 1 |x| −2 f x see 12, page 141 , is well defined absolutely convergent integral for all x, t ∈ 0, ∞ × 0, ∞ .Similar to the proof of Theorem 3.2, we will try to show that, for B k ⊂ B in Lemma 2.1, We split

4.5
By Lemma 2.11 and H ölder inequality, assertion 1 holds for g 1 f x .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: f y P t x, y dy

4.9
For I t 1 x and x ∈ B * k , we shall first show the inequality Using a of Proposition 2.9, if

4.11
If The previous two inequalities above imply

4.13
For I t 2 x and x ∈ B * k , we shall also prove the inequality

4.14
We split this integral as

4.15
To deal with J 3 x , we discuss two cases.In the first case of x k ≤ 1, notice that y > x, when According to b of Proposition 2.9,

4.16
The last inequality is from the same proof of J 1 x .In the second case of x k > 1, using b of Proposition 2.9 again, for t > 20ρ L α x k we obtain

4.17
The last inequality is also from the same proof of J 1 x .Inserting this into J 3 x leads to Using b of Proposition 2.9, by the standard argument it easily follows that

4.18
To complete the proof of 4.6 , we need to show that We also consider two cases of x k ≤ 1 and x k > 1.When x k ≤ 1, repeating the above argument for J 3 x and using a of Proposition 2.10, we have J 5 ≤ c f 2 BMO Lα .When x k > 1, using a of Proposition 2.10 again, for t ≥ 20ρ L α x k , we obtain Therefore, by Lemma 2.5 and Corollary 2.3, it suffices to prove

4.23
To prove 4.23 , we first claim that, for all f ∈ BMO L α , x ∈ B * k , and t ≤ 20ρ L α x k , We shall split into three different estimates: f y P t x, y dy ≤ C f BMO Lα .

4.27
Let us first treat 4.25 .Since y ≤ c ρ L α x k 2 /x k , when y ∈ X t 1 x , notice that x ∼ x k when x ∈ B * k , using a of Proposition 2.9, and recalling the definition of ρ L α x , we have

4.28
For 4.26 , using b of Proposition 2.9, for x ∈ B * k , the left side of 4.26 is controlled by

4.29
The third inequality is from the same argument for dealing with 3.21 in the proof of Theorem 3.

4.31
The third and fourth inequalities come from the fact 1 − e −2t
since supp f 2 is in the complement of B * k , we have 1/|B| B |f y |dy 0. Otherwise, by the definition of BMO L α , 1/|B| B |f y |dy ≤ 4/|B 4r x k | B 4r x k |f y |dy ≤ c f BMO Lα .
Obviously, since f is locally integrable, we have M f 1 < ∞ for a.e.x ∈ 0, ∞ .For f 2 , if Journal of Function Spaces and Applications x ∈ B and B∩B * k ∅, 2 e t 2 /xt x/t and x ∼ x k , when t ≤ 20ρ L α x k and x ∈ B * k .It is notable that, for t ≤ 20ρ L α x k and x ∈ B * k , the inequality above also implies On the other hand, by a of Proposition 2.10 and the simple fact that |f 2 j B t x − f B t x | ≤ Cj f BMO Lα , we obtain BMO Lα .4.37 It is easy to check 4.37 .Indeed, without loss of generality, we assume x < x 0 .According to a of Proposition 2.10, Inserting this into 4.37 gives the desired result.Moreover, from the above we also have proved