Sobolev space, Besov space and Triebel-Lizorkin space on the Laguerre hypergroup
© Huang; licensee Springer. 2012
Received: 9 February 2012
Accepted: 17 August 2012
Published: 31 August 2012
In this paper, we will investigate function spaces, including a Sobolev space, a Besov space and a Triebel-Lizorkin space, on the Laguerre hypergroup.
MSC:42B20, 42B25, 42C05.
1 Introduction and preliminaries
In  and , the authors investigated a Sobolev space on the dual of the Laguerre hypergroup and a generalized Besov space on the Laguerre hypergroup. In this paper, we define a Sobolev space on the Laguerre hypergroup by the Bessel potential. Then, we define a Besov space by the real interpolation of a Sobolev space and prove that our definition is a generalization of that given in . For the completeness, we also study a Triebel-Lizorkin space on the Laguerre hypergroup.
The following lemma follows from (1).
is a hypergroup in the sense of Jewett (cf. [4, 9]), where i denotes the involution defined by . If is a nonnegative integer, then the Laguerre hypergroup K can be identified with the hypergroup of radial functions on the Heisenberg group .
We also introduce a homogeneous norm defined by (cf. ). Then we can define the ball centered at of radius r, i.e., the set .
L is positive and symmetric in , and is homogeneous of degree 2 with respect to the dilations defined above. When , L is the radial part of the sublaplacian on the Heisenberg group . We call L the generalized sublaplacian.
The following proposition summarizes some basic properties of functions .
where A is a constant.
- (i)for all , the function
- (ii)for all , we have
there exists such that , for all such that .
for all , the function is on R, with compact support and vanishes out of a neighborhood of zero.
In the following, we introduce some basic notation about the real and complex interpolation, more about these can be found in .
where . The K-method of real interpolation consists in taking to be the set of all u in such that .
The norm of u is .
For , one defines , with the norm .
The paper is organized as follows. In Section 2, we will investigate Sobolev spaces on K. A Besov space and a Triebel-Lizorkin space will be studied in Section 3 and Section 4 respectively.
Throughout the paper, we will use C to denote the positive constant, which is not necessarily the same at each occurrence.
2 Sobolev spaces on K
In this section, we will study a Bessel potential space on the Laguerre hypergroup K.
It is easy to prove that the Bessel potentials satisfy the following semigroup property: and , where , and .
The Bessel potentials also satisfy the following property.
Proposition 2 The Bessel potentialis bounded, whereand.
is bounded for and follows from (6). This gives the proof of Proposition 2. □
Now, we define the Bessel potential space on K.
Definition 1 For , , we define the Bessel potential space as follows:
If , then is the collection of all functions such that for some with the norm ;
If , then is the collection of all distributions such that for some , where with , and ;
If , then .
When and , we call the Sobolev space on K.
It is easy to know that the definition of the space with is independent of m.
In the following, we prove that the spaces are complete.
Proposition 3 The Bessel potential spaces, whereand, are complete.
By Proposition 2, . This proves that is complete with and .
Therefore, is complete with .
If , the result is obvious. This completes the proof of Proposition 3. □
The Bessel potential space satisfies:
If , then ;
is an isomorphism;
, where .
- (1)Let . Then there exists such that
- (2)For , there exists such that . Therefore,
- (3)For and , there exist , such that , . Since , where , we have
Note , , we can get .
Therefore, . We complete the proof of Proposition 4. □
3 Besov space on K
In this section, we will define a Besov space on K by the real interpolation of the Bessel potential spaces.
where . The linear space of all such ρ is denoted by , the norm on is .
We have the following property about the Fourier multiplier on .
Proposition 5 If, thenand, whereand.
where . This proves Proposition 5. □
, for , ;
, for , .
We define functions and ψ on K by , . Then, we have:
Since , we get .
Since , we get . Therefore, and Lemma 2 is proved. □
Remark 2 By Theorem 3.4.2 in , we know is complete with , and .
In the following, we prove that our definition coincides with Definition 4.1 in  for , and .
where, , and.
This gives the proof of Theorem 1. □
We have the following version of the Calderón reproducing formula on K, the proof is standard (cf. ).
Remark 3 When for , it is easy to prove that f satisfies the condition of Lemma 3.
whereis the generalized homogeneous Besov-Laguerre type space defined in .
This completes the proof of Theorem 2. □
By Theorem 1 and Theorem 2, we know our definition coincides with the Definition 4.1 in  for , and .
By the properties of the Bessel potential space and the real interpolation, we can get the following properties about the Besov space, which are similar to those of the classical Besov space.
If , then , , , .
If , then , where , .
, , .
, , , .
is a linear bounded one-to-one operator.
, , , , .
4 Triebel-Lizorkin space on K
In this section, we will define a Triebel-Lizorkin space on K by the complex interpolation of the Bessel potential space and the Besov space. Then, we study some basic properties about the Triebel-Lizorkin space on K.
In order to give an equivalent norm for , we need the following Lemma (cf. ).
for, , and T be an operator defined by. Then T is bounded on, where.
The proof of the following lemma can be found in .
Then, Lemma 5 follows from Lemma 4. □
By Lemma 5, we can prove
Then Theorem 3 follows from (8) and (9). □
The following lemma has been proved in .
- (1)If is an interpolation couple, then
- (2)If , are Banach spaces and
Now we can prove the main result of this section.
where, , , , , and, .
Let , , where is the set of complex numbers. Then, by Lemma 6, we can get our theorem (cf. ). □
By the properties of the Sobolev space and the Besov space, we can get the following properties of the Triebel-Lizorkin space on K.
- (1)Let , , . Then
Let , , . Then .
- (3)Let , , . Then
, , .
, where , , and , .
Supported by National Natural Science Foundation of China (11001002), the Beijing Foundation Program (2010D005002000002).
- Assal M, Abdallah HB: Generalized Besov type spaces on the Laguerre hypergroup. Ann. Math. Blaise Pascal 2005, 12: 117–145. 10.5802/ambp.198MathSciNetView ArticleMATHGoogle Scholar
- Assal M, Nessibi MM: Sobolev spaces on the dual of the Laguerre hypergroup. Potential Anal. 2004, 20: 85–103. 10.1023/A:1025542009772MathSciNetView ArticleMATHGoogle Scholar
- Bergh J, Löfström J: Interpolation Spaces: An Introduction. Springer, Berlin; 1976.View ArticleMATHGoogle Scholar
- Bloom WR, Heyer H de Gruyter Studies in Mathematics 20. In Harmonic Analysis of Probability Measures on Hypergroups. de Gruyter, Berlin; 1994.Google Scholar
- Cruz-Báez DI, Rodríguez J: Spaces of distributions of Besov and Triebel-Lizorkin type for the Fourier-Bessel transform. J. Math. Anal. Appl. 2001, 259: 51–63. 10.1006/jmaa.2000.7380MathSciNetView ArticleMATHGoogle Scholar
- Folland G, Stein EM: Hardy Spaces on Homogeneous Groups. Princeton University Press and University of Tokyo Press, Princeton; 1982.MATHGoogle Scholar
- Huang JZ, Liu HP:Weak type estimates of the maximal functions on the Laguerre hypergroup. Can. Math. Bull. 2010, 53: 491–502. 10.4153/CMB-2010-058-6View ArticleMathSciNetMATHGoogle Scholar
- Huang JZ: Littlewood-Paley g -functions and multipliers for the Laguerre hypergroup. J. Inequal. Appl. 2011., 2011: Article ID 741095Google Scholar
- Jewett RI: Spaces with an abstract convolution of measures. Adv. Math. 1975, 18: 1–101. 10.1016/0001-8708(75)90002-XMathSciNetView ArticleMATHGoogle Scholar
- Stein EM: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton; 1970.MATHGoogle Scholar
- Stempak K: Mean summability methods for Laguerre series. Trans. Am. Math. Soc. 1990, 322: 671–690. 10.1090/S0002-9947-1990-0974528-8MathSciNetView ArticleMATHGoogle Scholar
- Triebel H: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam; 1978.MATHGoogle Scholar
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