Generalized Lorentz Spaces and Applications

Dunkl operators Tj (j = 1, . . . , d) introduced by Dunkl in [1] are parameterized differential-difference operators on R 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-Sutherlandtype 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, andwe 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

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.
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.

The Dunkl
Some basic properties are the following (cf.[5,6]).For all F  () ()  (, )   () , a.e., (12) and moreover, for all  ∈ S(R  ), Proposition 1.The Dunkl transform F  is a topological isomorphism from S(R  ) onto itself, and for all f in S(R  ), In particular, the Dunkl transform  → F  () can be uniquely extended to an isometric isomorphism on  2  (R  ).
We define the tempered distribution T  associated with for  ∈ S(R  ) and denote by ⟨, ⟩  the integral in the right hand side.
Definition 2. The Dunkl transform F  () of a distribution  ∈ S  (R  ) is defined by for  ∈ S(R  ).

Ḃ𝑠,𝑘
Notations.We denote by The distribution Δ   is called the jth dyadic block of the homogeneous Littlewood-Paley decomposition of  associated with the Dunkl operators.
Throughout this paper, we define  and  by  = F −1  () and  = F −1  ().When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functions ψ and φ belonging to (R  ) such that ψ ≡ 1 on supp  and φ ≡ 1 on supp .Remark 7. We remark that We put Definition 8. Let one denote by S  ℎ, (R  ) the space of tempered distribution such that lim On the follow, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Dunkl operators on R  and obtain their basic properties.

A Primer to Real Interpolation Theory and Generalized Lorentz Spaces
From now, we denote by   (Z) the set of sequence (  ) ∈Z such that 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 J-method and the K-method which are the simplest ones.
We consider two Banach spaces  0 and  1 which are continuously imbedded into a common topological vector space  and  > 0.
For any measurable function  on R  , we define its distribution and rearrangement functions The generalized Lorentz spaces  ,  (R  ) is defined as the set of all measurable functions  such that ||||  ,  (R  ) < ∞.
(ii) For  0 ̸ =  1 , one has Proof .We obtain these results by similar ideas used in the Euclidean case.         /(−1),/(−1) Proof.We obtain these results by similar ideas used in the Euclidean case.

Inequalities for the Fractional Dunkl-Laplace Operators
Lemma 27.Let  be a real number such that 0 <  <  + 2, and let 1 <  <  < ∞ satisfy Proof.We obtain this result by similar ideas used for the Dunkl-Riesz potential (cf.[21]).
Lemma 32 (see [23]).One assumes that where Note that where  , is the Dunkl-Bessel kernel defined by relation (36).From the relation (37), we see that  , ∈  (+2)/(+2−),∞  (R  ).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.
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
For proof of this result, we need the following lemma which we prove as the Euclidean case. where Proof of Theorem 41.Let 1 <  < ∞ and  ∈ (0, ( + 2)/).

Dispersion Phenomena
Notations.We denote by I  () the Dunkl-Schrödinger semigroup on  2  (R  ) defined by ) . (107) Banach space of (classes of) measurable functions  : R  → C such that    ∈    (  ) in the sense of distributions, for every multi-index  with , which these elements are -invariant.
Transform.For functions  on R  , we define   -norms of  with respect to   () as ) = ess sup ∈R  |()|.We denote by    (R  ) the space of all measurable functions  on R  with finite    -norm.The Dunkl transform F  on  1  (R  ) is given by ).Then, the distribution T  *   is given by the function  *  .If one assumes that  is arbitrary for  = 1 and radial for  ≥ 2, then T  *   belongs to then  *   ∈    (R  ) and      *         (R  ) ≤             (R  )            (R  ) .(21) Definition 4. The Dunkl convolution product of a distribution  in S  (R  ) and a function  in S(R  ) is the function  *   defined by  *   () = ⟨  ,  −  ()⟩ .(22) Proposition 5. Let  be in    (R  ), 1 ≤  ≤ ∞, and  in S(R Let one define by C the ring of center 0, of small radius 1/2, and great radius 2. There exist two radial functions  and  the values of which are in the interval [0, 1] belonging to (R  ) such that 1/ stands for sup    in the case  = ∞.