Bessel potential space on the Laguerre hypergroup

In this article, we define the fractional differentiation D_δ of order δ, δ > 0, induced by the Laguerre operator L and associated with respect to the Haar measure dm_α. We obtain a characterization of the Bessel potential space using D_δ and different equivalent norms.

During the second half of the twentieth century (until the 1990s), the Continuous Time Random Walk (CTRW) method was practically the only tool available to describe subdiffusive and/or superdiffusive phenomena associated with complex systems for many groups of research. The main reason behind the usefulness of fractional derivatives have been until this moment the close link that exists between fractional models and the so called Jump stochastic models, such as the CTRW or those of the multiple trapping type.

Note that fractional operators also provide a method for reflecting the memory properties and non-locality of many anomalous processes. In any case, at the moment it is not clear what is the best fractional time derivative or the spatial fractional derivative to be used in the different models.

Fractional calculus deals with the study of so-called fractional order integral and derivative operators over real or complex domains and their applications.

Since 1990, there has been a spectacular increase in the use of fractional models to simulate the dynamics of many different anomalous processes, especially those involving ultraslow diffusion. We hereby propose a few examples of fields where the fractional models have been used: materials theory, transport theory, fluid of contaminant flow phenomena through heterogeneous porous media, physics theory, electromagnetic theory, thermodynamics or mechanics, signal theory, chaos theory and/or fractals, geology and astrophysics, biology and other life sciences, economics or chemistry, etc.

As one would expect, since a fractional derivative is a generalization of an ordinary derivative, it is going to lose many of its basic properties. For example, it loses its geometric or physical interpretation but the index law is only valid when working on very specific function spaces and the derivative of the product of two functions is difficult to obtain and the chain rule is not straightforward to apply.

It is natural to ask then, what properties fractional derivatives have that make them so suitable for modeling certain complex systems. The answer lies in the property exhibited by many of the aforementioned systems of non-local dynamics, that is, the processes dynamics have a certain degree of memory. While fractional operators naturally incorporate the interesting property of no locality. They do lose some of the typical, basic properties of ordinary differential operators. The ordinary derivative is clearly, by definition, local [1].

According to the ideas presented by Stein [2], the fundamental operators of the harmonic analysis (fractional integrals, Riesz transformation, g-functions, ...) can be considered in the context of the Laguerre operator L.

It is important to mention that this way of describing harmonic operators in the Laguerre context was initiated by Muckenhoupt [3].

The organization of the article is as follows. Section 2 contains some basic facts needed in the sequel about the Laguerre hypergroup. Section 3 is devoted to some generation and representation for the semigroups also we define the fractional power, the heat-diffusion and the Poisson-Laguerre semigroups based on a Laguerre operator. Finally, Sect. 4 is devoted to proving the main result of this article (Theorem 1) where we establish that ||D_δf||_p and ||f||_δ,p are equivalent when the fractional differentiation D_δ is defined for δ > 0.

In this section we set some notations and we recall some basic results in harmonic analysis related to Laguerre hypergroups (see [4-6]).

First we begin with some notation.

• We denote by equipped with the weighted Lebesgue measure m_α on given by

For every 1 ≤ p ≤ ∞,we denote by the spaces of complex-valued functions f, measurable on such that:

and

• the subspace of of functions ψ satisfying the following:

(i) There exists m₀ ∈ ℕ satisfying ψ(λ, m) = 0, for all such that m > m₀.

(ii) for all m ≤ m₀, the function λ ↦ ψ(λ,m) is on ℝ with compact support and vanishes in a neighborhood of zero.

• the topological dual space of .

• the dual space of .

• the spaces of complex-valued functions f, measurable on such that:

and

where dγ_α(λ, m) being the positive measure defined on by:

For (x, t) ]0, ∞[×ℝ and α ∈ [0, ∞[, we consider the following partial differential operator, named the Laguerre operator:

Remark 1. For α = n - 1, n ∈ ℕ*, the operator L is the radial part of the sublaplacian on the Heisenberg group ℍⁿ.

For and , we put , where is the Laguerre function defined on [0, ∞] by and is the Laguerre polynomial of degree m and order α.

Proposition 1. For , the function φ_λ,m, is the unique solution of the following problem:

We denote by: .

Definition 1. (i) The generalized Fourier transform F is defined on by:

(ii) We have also the inverse formula of the generalized Fourier transform F^-1 on by:

For , we denote by: P_(λ,m)f = F(f)(λ, m)φ_λ,m.

3.1 The heat-diffusion semigroup

The heat-diffusion semigroup {T_t}_t≥0, associated to (-L), is then defined by

where

is the heat kernel of the integral representation T_tf.

Proposition 2. This semigroup {T_t}_t≥0 is a strongly continuous semigroup on with infinitesimal generator L(see [7]).

Proof. Let then

By the definition of the heat-diffusion semigroup {T_t}_t≥0, we establish the following result.

Corollary 1. For , we have

Proof. we have

3.2 The fractional power

For δ > 0, the negative power L^-δ of L with respect to the measure dm_α is defined, as in [8], by

It is not hard to prove that L^-δ can be expressed, for , by means of the following integral

L^-δ is also called δth fractional integral associated with L. This kind of fractional integrals has been investigated by several authors ([9-12]).

Corollary 2. If f(y, s) = φ_λ,m(y, s), we have:

Proof. The proof is trivial by using and the change of variable .

3.3 The Poisson-Laguerre semigroup

The Poisson-Laguerre semigroup {P_t}_t≥0, associated to (-L), is given by

where L^1/2 is defined by using the spectral theorem.

Now, by using the Bochner subordination formula

After the change of variable , we obtain:

Proposition 3. This semigroup {P_t}_t≥0 is also a strongly continuous semigroup on ,with infinitesimal generator L^1/2.

Proof. We use the fact that is strongly continuous.

By the definition of the Poisson-Laguerre semigroup {P_t}_t≥0 ,we establish also the following result

Corollary 3. For , we have

Proof. We replace c_μ,η by in the proof of Corollary 1, then the result is immediate.

3.4 The Riesz potential

For δ > 0, the Riesz potential of order δ, I_δ, with respect to the measure dm_α is defined, as in the classical case [13], by

Proposition 4. The Riesz potential can be also writed as

Proof. By using (-L)^-δ, we have

After to replace P_tf(y, s) with his expression, the change of variable and the property of the function Gamma, we obtain:

Corollary 4. If f(y, s) = φ_λ,m(y, s), we have

Proof. The proof is trivial by using and the change of variable .

4.1 The fractional differentiation

Following the classical case, the fractional differentiation D_δ of order δ > 0 on the Laguerrre hypergroup is defined formally by

Corollary 5. In the case of 0 <δ < 1, we have

Proof. In the case of 0 <δ < 1, we can write using [13] that

where

By a change of variable and the definition of c_δ, we have again:

Remark 2. Observe that:

As an application of the operator fractional derivative D_δ, we will give a characterization of the potential spaces , which is simpler and more powerful, valid for any 1 <p < ∞ and δ ≥ 0.

4.2 Bessel potential space on

We mention that the Laguerre potential spaces is defined as

equipped with the norm

Let us define the Laguerre Bessel operator as

where c_λ,m is the homogenous norm of

Proposition 5. If 0 ≤ δ₁ <δ₂ then for each 1 <p < ∞

Proof. We have

Now, let us establish a relation among different norms of potential spaces.

Proposition 6. Given 1 <p < ∞ and δ ≥ 1, if then

(i) .

(ii) .

Moreover,

Proof. (i) is immediate, since such that δ₁ <δ₂.

(ii) We use the fact that L is symmetric, F(Lf) = -c_λ,mF(f) = -c_λ,mF(f) and , then:

Then, we get

Next we show that if is equivalent to . The main tool is Meyer's multiplier theorem and let us underline that the definition of D_δ on all the spaces , 1 <p < ∞, is also based on an application of Meyer's theorem [13].

Theorem 1. Let δ ≥ 0 and 1 <p < ∞, we have:

if and only if Moreover, there exist a constant B_p,δ and A_p,δ such that:

To prove this result we need the following lemma.

Lemma 1. Let and ψ = (I - L)^δ/2f, for δ ≥ 0 and 1 <p < ∞, then:

(i) .

(ii) P_λ,mψ = (1 + c_λ,m)^-δ/2 P_λ,mf.

Proof.

(i) We have

Then

(ii) We know that

then

Using the definition of P_λ,m, we obtain

Now let to prove the Theorem 1

Proof. Let and ψ = (I - L)^δ/2f, then:

Since ||f||_p,δ = ||ψ||_p, by Meyer's multipliers theorem and using the multipliers h(z) = (1 + z)^-δ/2, we obtain that:

To prove the converse, suppose and consider

so by Meyer's multipliers theorem, using the multiplier h(z) = (z + 1)^δ/2, we have:

Finally, we can write that

The author declares that they have no competing interests.

1. Trujillo, JJ: On best fractional derivative to be applied in fractionel modeling. 3rd IFAC Workshop 2008 Fractional Differentiation and its Applications, Ankara, Turkey (2008)

2. Stein, EM: Topics in Harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematical Studies, Princenton University Press, Princenton. 63, (1970)

3. Muckenhoupt, B: Poisson integrals for Hermite and Laguerre expansions. Trans Am Math Soc. 139, 231–242 (1969). Publisher Full Text

4. Assal, M, Nessibi, MM: Soblev type spaces on the dual of the Laguerre hypergroup. Potential Anal. 20, 85–103 (2004). Publisher Full Text

5. Kortas, H, Sifi, M: Lévy-Khintchine formula and dual convolution semigroups associated with Laguerre and Bessel functions. Potential Anal. 15, 43–58 (2001). Publisher Full Text

6. Nessibi, MM, Trimeche, K: Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets. J Math Anal Appl. 208, 337–363 (1997). Publisher Full Text

7. Pazy, A: Semigroups of Linear Operators and Applications to Partial Diffrential Equations. Springer-Verlag, New York (1983)

8. Stempak, K, Torrea, JL: Poisson integrals and Riesz transforms for Hermite function expansions with weights. J Funct Anal. 202, 443–472 (2003). Publisher Full Text

9. Gasper, G, Stempak, K, Trebels, W: Fractional integration for Laguerre expansions. Methods Appl Anal. 2, 67–75 (1995)

10. Graczyk, P, Loeb, J-L, Lopez, IA, Nowak, A, Urbina, W: Higher order Riesz transforms, fractional derivatives and Sobolev spaces for Laguerre expansions. J Math Pure et Appl. 84(3), 375–405 (2005). Publisher Full Text

11. Kanjin, Y, Sato, E: The Hardy-Littlewood theorem on fractional integration for Laguerre series. Proc Am Math Soc. 123, 2165–2171 (1995). Publisher Full Text

12. Stempak, K: Heat-diffusion and Poisson integrals for Laguerre expansions. Tohoku Math J. 46(1), 83–104 (1994). Publisher Full Text

13. Lopez, IA, Urbina, WO: Fractional differentiation for the Gaussian measure and applications. Bulletin des sciences mathematiques. 128, 587–603 (2004). Publisher Full Text