Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 302941, 14 pages

http://dx.doi.org/10.1155/2013/302941

Research Article

Generalized Lorentz Spaces and Applications

Hatem Mejjaoli

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.

Abstract.

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”

1. Introduction

Dunkl operators introduced by Dunkl in [1] 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. Preliminaries

In order to confirm the basic and standard notations, we briefly overview the theory of Dunkl operators and related harmonic analysis. Main references are [1, 5, 6, 11, 14–17].

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. [18]) 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.

The Dunkl transform on is given by

Some basic properties are the following (cf. [5, 6]).

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. [17]).

Since by the previous definition of , it follows that

(a)for all , (resp., ), belongs to (resp. ) and

Moreover, as pointed in [16] 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

We put

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. [19])

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. [12]).

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 .

Proof. We take , , , and , and we deduce the inequality (47) from the relations (43) and (40). In the same way, we deduce (48) from the relations (43) and (41).

Theorem 17 (see [13]). 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. [20]).

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 .

Proposition 25. (i) For ,

with .

(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.

Proposition 26. (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. [21]).

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 [22], 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 [23]). 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. [21]):

(iii) We note that the results of Dunkl-Riesz potential of this section are in sprit of the classical case (cf. [24]).

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 [23], 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

() the subspace of , which these elements are -invariant.

Definition 43. One says that the exponent pair is -admissible if , , and

If equality holds in (109) one says that is sharp -admissible, otherwise, one says that is nonsharp -admissible. Note in particular that when , the endpoint

is sharp -admissible.

Lemma 44 (see [25]). Let and be Banach spaces, and let : be an integral operator for some , with a kernel such that

If , then one has

where is the low diagonal operator defined by

Lemma 45. For any -admissible pair with

Proof. From the dispersion of such that

for any , (cf. [8]), and the fact that

one can easily prove the result.

Theorem 46. Suppose that , and are -admissible pairs and . If is a solution to the problem

for some data, , , then

Proof. Let be a solution of (118). We write as

Let , or , , and . Then, since , in view of Lemma 44, we only have to show that

To show this, observe from (114) and for all that

Then, from the endpoint result of Keel and Tao [26], the right-hand side of (122) is bounded by . The remaining part of theorem can be obtained by the duality of Lorentz space and the second part of (115).

As an application of the previous theorem, we can derive Strichartz estimates of the solution to the following nonlinear problem:

Theorem 47. If the initial data is sufficiently small and -invariant, then there exists a unique solution ; ; ; for every sharp -admissible pair with and .

Proof. The existence of a unique -solution is proved in [9], it suffices to prove that . From Duhamel's principle, we deduce that

Using (114) and (119), we have

We can always find an admissible pair with and and and such that

where . Thus, from the Leibnitz rule, Hölder's inequality on Lorentz space, and Sobolev embedding, we deduce that

Since is small, we have

Finally, since we can choose arbitrarily to be -admissible, for any -admissible pair and with and , we have

In a similar way, we can also derive from the smallness of ,

7. Embedding Sobolev Theorems and Applications

Theorem 48. Let , with . Let , , and . If , then , and one has

Proof. We start picking such that with . We have then with and . We write

Using Hölder's inequality and by simple calculations, we obtain

where . From this, and applying Proposition 25, we deduce that if , then . Furthermore, using (57), we finally have:

Corollary 49. Let be a real number in the interval , and let be a real number in . There is a constant such that, for any function , the following inequality holds:

where .

Proof. Let and with . We take and apply (103) in the specific form

where and . As , we have

Combining this with (131), we obtain (135).

Theorem 50. Let be given. There exists a positive constant such that for all function , one has

For proof of this theorem, we need the following lemma, which we obtain by simple calculations.

Lemma 51. Let be a real number in the interval . Then, the function belongs to the Dunkl-Besov space .

Proof of Theorem 50. Let us define

Using homogeneous Littlewood-Paley decomposition and the fact that belongs to , we can write

Lemma 51 claims that belongs to . Theorem 17 yields

Thus,

The following results of this section are in sprit of the classical case (cf. [27]).

Theorem 52. Let , and let , with , .

(i)For every , and if , one has and

(ii)Moreover, this inequality is valid for in the following cases:

(a),(b) and ,(c) and .

(iii)Finally, the condition is sharp.

Proof. (i) Case . With no loss of generality, we may assume that , and we fix such that

The proof follows essentially the same ideas used in the previous theorem. Indeed, we have for and and for and ,

As , we can only say that , where . We may use (57), but we get only that with and that satisfies (143) with . However, we may choose as small as we want and thus, as close to as we want; thus satisfies (143) for every .

(ii) Case .

(a) If : this case was treated in Theorem 48.

(b) If and : this is a direct consequence of (43), since we have

we obtain

(c) Case and .

We just write

and get by Hölder's inequality:

We then use the embedding which is valid for .

Theorem 53. Let , let with . Let , and let .

(i)If and let . Then, one has

(ii)If , and let . Then, one has

Proof. We only prove the first inequality, as the proof for the second one is similar. Since , noting that , we have . Thus, using Proposition 26 (i) for the interpolation

with , we see that we have a partition such that

Moreover, since , we have

Let us note that , , , and . We apply now (147) and Theorem 48 to obtain

Since we have , with these two inequalities at hand, and using (57), we find that , with , but, since and , we obtain with .

Theorem 54. Let , and let with . Let , and let . Let , and let . Then, one has

Proof. Once the previous theorem is proved, it is enough to reapply similar arguments to obtain Theorem 54. As , we start using

instead of (152), and we obtain a partition such that

with and where belongs to , since . Moreover, since , we have

Let us note that , , , and . We apply now (151) and Theorem 48 instead of (155) to obtain

where and

Finally, we have via (57) that , with . To conclude, we use the fact that and in order to obtain that with .

Conjecture 55. Theorems 34, 39, and 41 are true for the general reflection group .

Acknowledgments

The author gratefully acknowledges the Deanship of Scientific Research at the University of Taibah University on material and moral support in the financing of this research Project No. 4001. The author is deeply indebted to the referees for providing constructive comments and for helping in improving the contents of this paper.

References

C. F. Dunkl, “Differential-difference operators associated to reflection groups,” Transactions of the American Mathematical Society, vol. 311, no. 1, pp. 167–183, 1989.

T. H. Baker and P. J. Forrester, “Non symmetric Jack polynomials and integral kernels,” Duke Mathematical Journal, vol. 95, no. 1, pp. 1–50, 1998.

J. F. van Diejen and L. Vinet, Calogero-Sutherland-Moser Models, CRM Series in Mathematical Physics, Springer, New York, NY, USA, 2000.

K. Hikami, “Dunkl operator formalism for quantum many-body problems associated with classical root systems,” Journal of the Physical Society of Japan, vol. 65, no. 2, pp. 394–401, 1996.

M. F. E. de Jeu, “The dunkl transform,” Inventiones Mathematicae, vol. 113, no. 1, pp. 147–162, 1993.

C. F. Dunkl, “Hankel transforms associated to finite reflection groups,” Contemporary Mathematics, vol. 138, pp. 123–138, 1992.

H. Mejjaoli, “Strichartz estimates for the Dunkl wave equation and application,” Journal of Mathematical Analysis and Applications, vol. 346, no. 1, pp. 41–54, 2008.

H. Mejjaoli, “Dispersion phenomena in Dunkl-Schrodinger equation and applications,” Serdica Mathematical Journal, vol. 35, pp. 25–60, 2009.

H. Mejjaoli, “Global well-posedness and scattering for a class of nonlinear Dunkl-Schrödinger equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 72, no. 3-4, pp. 1121–1139, 2010.

H. Mejjaoli, “Dunkl-heat semigroup and applications,” Applicable Analysis, 2012.

M. Rösler, “Generalized Hermite polynomials and the heat equation for Dunkl operators,” Communications in Mathematical Physics, vol. 192, no. 3, pp. 519–542, 1998.

T. Kawazoe and H. Mejjaoli, “Generalized Besov spaces and their applications,” Tokyo Journal of Mathematics, vol. 35, no. 2, pp. 297–320, 2012.

H. Mejjaoli, “Generalized homogeneous Besov spaces and their applications,” Serdica Mathematical Journal, vol. 38, no. 4, pp. 575–614, 2012.

C. F. Dunkl, “Integral kernels with re ection group invariant,” Canadian Journal of Mathematics, vol. 43, pp. 1213–1227, 1991.

M. Rösler, “A positive radial product formula for the Dunkl kernel,” Transactions of the American Mathematical Society, vol. 355, no. 6, pp. 2413–2438, 2003.

S. Thangavelu and Y. Xu, “Convolution operator and maximal function for the Dunkl transform,” Journal d'Analyse Mathematique, vol. 97, pp. 25–55, 2005.

K. Trimèche, “Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators,” Integral Transforms and Special Functions, vol. 13, no. 1, pp. 17–38, 2002.

P. Etingof, “A uniform proof of the macdonald-Mehta-Opdam identity for finite coxeter groups,” Mathematical Research Letters, vol. 17, no. 2, pp. 277–282, 2010.

S. Thangavelyu and Y. Xu, “Riesz transform and Riesz potentials for Dunkl transform,” Journal of Computational and Applied Mathematics, vol. 199, no. 1, pp. 181–195, 2007.

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, New York, NY, USA, 1976.

S. Hassani, S. Mustapha, and M. Sifi, “Riesz potentials and fractional maximal function for the dunkl transform,” Journal of Lie Theory, vol. 19, no. 4, pp. 725–734, 2009.

J. Merker, “Regularity of solutions to doubly nonlinear diffusion equations,” Electronic Journal of Differential Equations, vol. 17, pp. 185–195, 2009.

M. G. Hajibayov, “Boundedness of the Dunkl convolution operators,” in Analele Universitatii de Vest, vol. 49 of Timisoara Seria Matematica Informatica, pp. 49–67, 2011.

H. Hajaiej, X. Yu, and Z. Zhai, “Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms,” Journal of Mathematical Analysis and Applications, vol. 396, no. 2, pp. 569–577, 2012.

C. Ahn and Y. Cho, “Lorentz space extension of Strichartz estimates,” Proceedings of the American Mathematical Society, vol. 133, no. 12, pp. 3497–3503, 2005.

M. Keel and T. Tao, “Endpoint Strichartz estimates,” American Journal of Mathematics, vol. 120, no. 5, pp. 955–980, 1998.

D. Chamorro and P. G. Lemarié-Rieusset, “Real Interpolation method, Lorentz spaces and refined Sobolev inequalities,” http://arxiv.org/abs/1211.3320