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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 302941, 14 pages
Generalized Lorentz Spaces and Applications
Department of Mathematics, College of Sciences, Taibah University, P.O. B 30002, Al Madinah Al Munawarah, Saudi Arabia
Received 21 March 2013; Accepted 6 June 2013
Academic Editor: Jose Luis Sanchez
Copyright © 2013 Hatem Mejjaoli. 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.
We define and study the Lorentz spaces associated with the Dunkl operators on . Furthermore, we obtain the Strichartz estimates for the Dunkl-Schrödinger equations under the generalized Lorentz norms. The Sobolev inequalities between the homogeneous Dunkl-Besov spaces and generalized Lorentz spaces are also considered.
“Dedicated to Khalifa Trimèche”
Dunkl operators introduced by Dunkl in  are parameterized differential-difference operators on that are related to finite reflection groups. Over the last years, much attention has been paid to these operators in various mathematical (and even physical) directions. In this prospect, Dunkl operators are naturally connected with certain Schrödinger operators for Calogero-Sutherland-type quantum many-body systems [2–4]. Moreover, Dunkl operators allow generalizations of several analytic structures, such as Laplace operator, Fourier transform, heat semigroup, wave equations, and Schrödinger equations [5–11].
In the present paper, we intend to continue our study of generalized spaces of type Sobolev associated with Dunkl operators started in [12, 13]. In this paper, we study the generalized Lorentz spaces, and we establish Sobolev inequalities between the homogeneous Dunkl-Besov spaces and many spaces as the homogeneous Dunkl-Riesz spaces and generalized Lorentz spaces.
As an application, we consider the Dunkl-Schrödinger equation where is the Dunkl Laplace operator. We study the previous equation focusing on the following problems:(1)Establish the Strichartz estimate under the generalized Lorentz norms.(2)Illustrate applications to well posedness.
The contents of the paper are as follows. In Section 2, we recall some basic results about the harmonic analysis associated with the Dunkl operators. In Section 3, we introduce the homogeneous Dunkl-Besov spaces, the homogeneous Dunkl-Triebel-Lizorkin spaces, and the homogeneous Dunkl-Riesz potential spaces and we prove new embedding Sobolev theorem. In Section 4, we recall some facts about a real interpolation method. Next, we define the generalized Lorentz spaces and will pay special attention to the interpolation definition of these spaces. Section 5 is devoted to give a complete picture of the Sobolev type inequalities for the fractional Dunkl-Laplace operators. In Section 6, Strichartz estimates for the solution of the Dunkl-Schrödinger evolution equation are considered on a mixed normed space with generalized Lorentz norm with respect to the time variable. Finally, we establish Sobolev inequalities between the homogeneous Dunkl-Besov spaces and generalized Lorentz spaces, and we give many applications.
2.1. Root System, Reflection Group and Multiplicity Function
Let be the Euclidean space equipped with a scalar product , and let . For in , denotes the reflection in the hyperplane perpendicular to , that is, for , . A finite set is called a root system if and for all . We normalize each as . We fix a and define a positive root system of as . The reflections , , generate a finite group , called the reflection group. A function on is called a multiplicity function if it is invariant under the action of . We introduce the index as Throughout this paper, we will assume that for all . We denote by the weight function on given by which is invariant and homogeneous of degree . In the case that the reflection group is the group of sign changes, the weight function is a product function of the form , . We denote by the Mehta-type constant defined by We note that Etingof (cf. ) has given a derivation of the Mehta-type constant valid for all finite reflection groups.
In the following, we denote by: the space of continuous functions on .: the space of functions of class on .: the space of bounded functions of class .: the space of functions on .: the Schwartz space of rapidly decreasing functions on .: the space of functions on which are of compact support.: the space of temperate distributions on .
2.2. The Dunkl Operators
Let be a multiplicity function on and a fixed positive root system of . Then, the Dunkl operators , , are defined on by where . Similarly, as ordinary derivatives, each satisfies for all , in and at least one of them is -invariant, and for all in and in , Furthermore, according to [1, 14], the Dunkl operators , commute, and there exists the so-called Dunkl's intertwining operator such that for and . We define the Dunkl-Laplace operator on by where and are the usual Euclidean Laplacian and nabla operators on , respectively. Since the Dunkl operators commute, their joint eigenvalue problem is significant, and for each , the system admits a unique analytic solution , , called the Dunkl kernel, which has a holomorphic extension to . For , , the kernel satisfies(a),(b) for ,(c) for .
2.3. The Dunkl Transform
For functions on , we define -norms of with respect to as if and . We denote by the space of all measurable functions on with finite -norm.
For all ,(a),(b) for ,(c)if belongs to , then and moreover, for all ,(d),(e)if we define , then
Proposition 1. The Dunkl transform is a topological isomorphism from onto itself, and for all f in , In particular, the Dunkl transform can be uniquely extended to an isometric isomorphism on .
We define the tempered distribution associated with by for and denote by the integral in the right hand side.
Definition 2. The Dunkl transform of a distribution is defined by for .
In particular, for , it follows that for ,
Proposition 3. The Dunkl transform is a topological isomorphism from onto itself.
2.4. The Dunkl Convolution
By using the Dunkl kernel in Section 2.2, we introduce a generalized translation and a convolution structure in our Dunkl setting. For a function and , the Dunkl translation is defined by Clearly, , and by using the Dunkl's intertwining operator , is related to the usual translation as (cf. [11, 17]). Hence, can also be defined for . We define the Dunkl convolution product of functions , as follows: This convolution is commutative and associative (cf. ).
Since by the previous definition of , it follows that(a)for all , (resp., ), belongs to (resp. ) and Moreover, as pointed in  and Sections 4 and 7, the operator is bounded on , , provided that is a radial function in or an arbitrary function in for . Hence, the standard argument yields the following Young's inequality.(b)Let such that . Assume that and . If for all , then and
Definition 4. The Dunkl convolution product of a distribution in and a function in is the function defined by
Proposition 5. Let be in , , and in . Then, the distribution is given by the function . If one assumes that is arbitrary for and radial for , then belongs to . Moreover, for all , where , and
For each , we define the distributions , by for all . Then, , and these distributions satisfy the following properties (see Section 2.3 (d)): In the following, we denote given by (15) by for simplicity.
3. , , and Spaces and Basic Properties
One of the main tools in this paper is the homogeneous Littlewood-Paley decompositions of distributions associated with the Dunkl operators into dyadic blocs of frequencies.
Lemma 6. Let one define by the ring of center , of small radius , and great radius . There exist two radial functions and the values of which are in the interval belonging to such that
Notations. We denote by The distribution is called the th dyadic block of the homogeneous Littlewood-Paley decomposition of associated with the Dunkl operators.
Throughout this paper, we define and by and .
When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functions and belonging to such that on and on .
Remark 7. We remark that
Definition 8. Let one denote by the space of tempered distribution such that
On the follow, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Dunkl operators on and obtain their basic properties.
From now, we make the convention that for all nonnegative sequence , the notation stands for in the case .
Definition 9. Let and . The homogeneous Dunkl-Besov spaces are the space of distribution in such that
Definition 10. Let and , the homogeneous Dunkl-Triebel-Lizorkin space is the space of distribution in such that
Let us recall that the operators and have been defined, respectively, by (cf. ) The operators , for , are called Dunkl-Bessel potential operators, and they are given by Dunkl convolution with the Dunkl-Bessel potential where We note that for all , , and
Definition 11. For and , the Dunkl-Bessel potential space is defined as the space , equipped with the norm .
Furthermore, , .
Definition 12. The operators , , are called Dunkl-Riesz potentials operators, and one has where is the Dunkl-Riesz potential given by
Definition 13. For and , the homogeneous Dunkl-Riesz potential space is defined as the space , equipped with the norm .
Proposition 14. Let , and let such that , then one has
Proof. We obtain these results by similar ideas used in the nonhomogeneous case. (cf. ).
Theorem 15. Let , and let . Let , and let . Then, there exists a constant such that for every , then one has In particular, one gets
Proof. Let be a Schwartz class, we have We define as the largest index such that and we write Thus, (42) is proved. In order to obtain (43), it is enough to apply Hölder’s inequality in the expression previous since we have and .
Corollary 16. Let , and let such that , then one has where .
Theorem 17 (see ). Let and . Then, is an algebra, and there exists a positive constant such that
Moreover, for any (), any and any such that and , one has where .
Moreover, for any (), any and any such that , , , one has
Moreover, for any (), any and any such that , and , one has with and .
4. A Primer to Real Interpolation Theory and Generalized Lorentz Spaces
From now, we denote by the set of sequence such that stands for in the case .
The theory of interpolation spaces was introduced in the early sixties by J. Lions and J. Peetre for the real method and by Caldéron for the complex method (cf. ).
In this section, we present the real method. There are many equivalent ways to define the method; we will present the discrete -method and the -method which are the simplest ones.
We consider two Banach spaces and which are continuously imbedded into a common topological vector space and .
The -method and the -method consist to consider the -functional and the -functional defined on by
Definition 18 (-method of interpolation). For and , the interpolation space is defined as follows: if and only if can be written as a sum , where the series converge in , each belongs to and .
The norm of is defined by
Definition 19 (K-method of interpolation). For and , the space is defined by if and only if , and .
The norm of is defined as follows:
Proposition 20 (Equivalence theorem). For and , one has .
Remark 21. In the following, we will denote this space by .
Lemma 22. For and , with , one has
Proposition 23. (i) For , one has (ii) For , (58) is still valid if .
Proposition 24 (Duality theorem for the real method). One considers the dual spaces and for and of the spaces and . If is dense in and in , one has , where is the conjugate component of .
For any measurable function on , we define its distribution and rearrangement functions
For and , define The generalized Lorentz spaces is defined as the set of all measurable functions such that .
(i) For ,
(ii) For , one has with .
(iii) In the case , one has with .
(iv) If and , then
Proof. We obtain these results by similar ideas used in the Euclidean case.
(i) Let , . Then, there exists a constant such that every can be decomposed as , where
the have disjoint supports if , .
(ii) Let , . Then, there exists a constant such that every and every , one has and
Proof. We obtain these results by similar ideas used in the Euclidean case.
5. Inequalities for the Fractional Dunkl-Laplace Operators
Lemma 27. Let be a real number such that , and let satisfy For , one has
Proof. We obtain this result by similar ideas used for the Dunkl-Riesz potential (cf. ).
Proposition 28. Let and . Then,
Proof. Let us first observe that since is dense in , it is enough to prove (69) for . Let . Then, we have Hence, Now, by the previous lemma we obtain where . Now, let us take , with , that is, . Then, the relation (72) gives that Thus, we obtain (69).
Proposition 29. Let , , , and . Then, one has the inequality with
Proof. Hölder's inequality yields where Applying Lemma 27, with , we obtain the result.
Theorem 30. Let , and . Then, the inequality holds for
Proof. Using the convexity of the function , and the logarithmic Hölder's inequality proved by Merker , we obtain for . We can choose for and satisfying the condition of Proposition 29, and we get By a simple calculation, we obtain the result.
Corollary 31. Let and , such that , one has
Proof. It suffices to apply the previous theorem for .
Lemma 32 (see ). One assumes that . If , , and , then where and is any number such that . Moreover,
Remark 33. The analogues of this lemma for the general reflection group , together with other additional results, will appear in a forthcoming paper.
Theorem 34. One assumes that . Let , , , , and . Then, the inequality holds for
Proof. Applying the Hölder inequality and simple computation yields where Note that where is the Dunkl-Bessel kernel defined by relation (36). From the relation (37), we see that . Using now Lemma 32, we deduce that for The result then follows.
Now, we state the results for the Dunkl-Riesz potential operators. The proofs are essentially as for the Dunkl-Bessel potential operators. We will not repeat them.
Proposition 35. Let and . Then,
Proposition 36. Let , , , and . Then, the inequality with
Theorem 37. Let , , and . Then, the inequality holds for
Corollary 38. Let and , such that , one has
Theorem 39. One assumes that . Let , , , , and . Then, the inequality holds for
Remark 40. (i) We assume that . It follows from the special case and of (97) that the inequality
with . Equation (99) can be thought of a refinement of (92) from (64).
(ii) We assume that . It follows from the special case that (99) becomes which can also be thought of as a refinement of the Hardy-Littlewood-Sobolev fractional integration theorem in Dunkl setting (cf. ):
(iii) We note that the results of Dunkl-Riesz potential of this section are in sprit of the classical case (cf. ).
Theorem 41. One assumes that . Let , , and . There exists a positive constant such that one has
For proof of this result, we need the following lemma which we prove as the Euclidean case.
Lemma 42. Let . If and , then where and .
Proof of Theorem 41. Let and . We take and apply (103) in the specific form where and . As , we have with . On the other hand, from , Theorem 1.2, we have for any with , , and . Thus, we obtain (102).
6. Dispersion Phenomena
Notations. We denote by the Dunkl-Schrödinger semigroup on defined by () Banach space of (classes of) measurable functions such that in the sense of distributions, for every multi-index with . is equipped with the norm