Generalized Lorentz Spaces and Applications

Abstract

We define and study the Lorentz spaces associated with the Dunkl operators on ℝ^𝑑. 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 𝑇_𝑗(𝑗=1,…,𝑑) introduced by Dunkl in [1] are parameterized differential-difference operators on ℝ^𝑑 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 𝑖𝜕_𝑡𝑢+△_𝑘𝑢=𝑓(𝑡,𝑥),(𝑡,𝑥)∈(0,∞)×ℝ^𝑑𝑢_|𝑡=0=𝑔, (1) where △_𝑘=∑^𝑑_𝑗=1𝑇²_𝑗 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 ℝ^𝑑 be the Euclidean space equipped with a scalar product ⟨,⟩, and let ||𝑥||=⟨𝑥,𝑥⟩. For 𝛼 in ℝ^𝑑⧵{0}, 𝜎_𝛼 denotes the reflection in the hyperplane 𝐻_𝛼⊂ℝ^𝑑 perpendicular to 𝛼, that is, for 𝑥∈ℝ^𝑑, 𝜎_𝛼(𝑥)=𝑥−2‖𝛼‖⁻²⟨𝛼,𝑥⟩𝛼. A finite set 𝑅⊂ℝ^𝑑⧵{0} is called a root system if 𝑅∩ℝ𝛼={±𝛼} and 𝜎_𝛼𝑅=𝑅 for all 𝛼∈𝑅. We normalize each 𝛼∈𝑅 as ⟨𝛼,𝛼⟩=2. We fix a 𝛽∈ℝ^𝑑⧵∪_𝛼∈𝑅𝐻_𝛼 and define a positive root system 𝑅₊ of 𝑅 as 𝑅₊={𝛼∈𝑅∣⟨𝛼,𝛽⟩>0}. The reflections 𝜎_𝛼, 𝛼∈𝑅, generate a finite group 𝐺⊂𝑂(𝑑), called the reflection group. A function 𝑘∶𝑅→ℂ on 𝑅 is called a multiplicity function if it is invariant under the action of 𝐺. We introduce the index 𝛾 as 𝛾=𝛾(𝑘)=_𝛼∈𝑅+𝑘(𝛼). (2) Throughout this paper, we will assume that 𝑘(𝛼)≥0 for all 𝛼∈𝑅. We denote by 𝜔_𝑘 the weight function on ℝ^𝑑 given by 𝜔_𝑘(𝑥)=_𝛼∈𝑅+|⟨𝛼,𝑥⟩|^2𝑘(𝛼), (3) which is invariant and homogeneous of degree 2𝛾. In the case that the reflection group 𝐺 is the group ℤ^𝑑₂ of sign changes, the weight function 𝜔_𝑘 is a product function of the form ∏^𝑑_𝑗=1|𝑥_𝑗|^𝑘𝑗, 𝑘_𝑗≥0. We denote by 𝑐_𝑘 the Mehta-type constant defined by 𝑐_𝑘=_ℝ𝑑𝑒^{−||𝑥||2/2}𝜔_𝑘(𝑥)𝑑𝑥. (4) 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 by𝐶(ℝ^𝑑): the space of continuous functions on ℝ^𝑑.𝐶^𝑝(ℝ^𝑑): the space of functions of class 𝐶^𝑝 on ℝ^𝑑.𝐶^𝑝_𝑏(ℝ^𝑑): the space of bounded functions of class 𝐶^𝑝.ℰ(ℝ^𝑑): the space of 𝐶^∞ functions on ℝ^𝑑.𝒮(ℝ^𝑑): the Schwartz space of rapidly decreasing functions on ℝ^𝑑.𝐷(ℝ^𝑑): the space of 𝐶^∞ functions on ℝ^𝑑 which are of compact support.𝒮^′(ℝ^𝑑): the space of temperate distributions on ℝ^𝑑.

2.2. The Dunkl Operators

Let 𝑘∶𝑅→ℂ be a multiplicity function on 𝑅 and 𝑅₊ a fixed positive root system of 𝑅. Then, the Dunkl operators 𝑇_𝑗, 1≤𝑗≤𝑑, are defined on 𝐶¹(ℝ^𝑑) by 𝑇_𝑗𝑓(𝑥)=𝜕𝜕𝑥_𝑗𝑓(𝑥)+_𝛼∈𝑅+𝑘(𝛼)𝛼_𝑗𝑓(𝑥)−𝑓𝜎_𝛼(𝑥)⟨⟩,𝛼,𝑥 (5) where 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑑). Similarly, as ordinary derivatives, each 𝑇_𝑗 satisfies for all 𝑓, 𝑔 in 𝐶¹(ℝ^𝑑) and at least one of them is 𝐺-invariant, 𝑇_𝑗𝑓𝑔=𝑇_𝑗𝑓𝑔+𝑓𝑇_𝑗𝑔, (6) and for all 𝑓 in 𝐶¹_𝑏(ℝ^𝑑) and 𝑔 in 𝒮(ℝ^𝑑), _ℝ𝑑𝑇_𝑗𝑓(𝑥)𝑔(𝑥)𝜔_𝑘(𝑥)𝑑𝑥=−_ℝ𝑑𝑓(𝑥)𝑇_𝑗𝑔(𝑥)𝜔_𝑘(𝑥)𝑑𝑥. (7) Furthermore, according to [1, 14], the Dunkl operators 𝑇_𝑗, 1≤𝑗≤𝑑 commute, and there exists the so-called Dunkl's intertwining operator 𝑉_𝑘 such that 𝑇_𝑗𝑉_𝑘=𝑉_𝑘(𝜕/𝜕𝑥_𝑗) for 1≤𝑗≤𝑑 and 𝑉_𝑘(1)=1. We define the Dunkl-Laplace operator △_𝑘 on ℝ^𝑑 by △_𝑘𝑓(𝑥)∶=^𝑑_𝑗=1𝑇²_𝑗𝑓(𝑥)=△𝑓(𝑥)+2_𝛼∈𝑅+𝑘(𝛼)∇𝑓(𝑥),𝛼⟨⟩−𝑓(𝑥)𝜎𝛼,𝑥−𝑓_𝛼(𝑥)⟨𝛼,𝑥⟩², (8) where △ and ∇ are the usual Euclidean Laplacian and nabla operators on ℝ^𝑑, respectively. Since the Dunkl operators commute, their joint eigenvalue problem is significant, and for each 𝑦∈ℝ^𝑑, the system 𝑇_𝑗𝑢𝑥,𝑦=𝑦_𝑗𝑢𝑥,𝑦,𝑗=1,…,𝑑,𝑢0,𝑦=1 (9) admits a unique analytic solution 𝐾(𝑥,𝑦), 𝑥∈ℝ^𝑑, called the Dunkl kernel, which has a holomorphic extension to ℂ^𝑑×ℂ^𝑑. For 𝑥, 𝑦∈ℂ^𝑑, the kernel satisfies(a)𝐾(𝑥,𝑦)=𝐾(𝑦,𝑥),(b)𝐾(𝜆𝑥,𝑦)=𝐾(𝑥,𝜆𝑦) for 𝜆∈ℂ,(c)𝐾(𝑔𝑥,𝑔𝑦)=𝐾(𝑥,𝑦) for 𝑔∈𝐺.

2.3. The Dunkl Transform

For functions 𝑓 on ℝ^𝑑, we define 𝐿^𝑝-norms of 𝑓 with respect to 𝜔_𝑘(𝑥)𝑑𝑥 as 𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}=_ℝ𝑑𝑓(𝑥)^𝑝𝜔_𝑘(𝑥)𝑑𝑥^1/𝑝, (10) if 1≤𝑝<∞ and ‖𝑓‖_{𝐿∞𝑘(ℝ𝑑)}=esssup_{𝑥∈ℝ𝑑}|𝑓(𝑥)|. We denote by 𝐿^𝑝_𝑘(ℝ^𝑑) the space of all measurable functions 𝑓 on ℝ^𝑑 with finite 𝐿^𝑝_𝑘-norm.

The Dunkl transform ℱ_𝐷 on 𝐿¹_𝑘(ℝ^𝑑) is given by ℱ_𝐷𝑓𝑦=1𝑐_𝑘ℝ^𝑑𝑓(𝑥)𝐾𝑥,−𝑖𝑦𝜔_𝑘(𝑥)𝑑𝑥. (11) Some basic properties are the following (cf. [5, 6]).

For all 𝑓∈𝐿¹_𝑘(ℝ^𝑑),(a)‖ℱ_𝐷(𝑓)‖_𝐿^∞_𝑘(ℝ^𝑑₎≤𝑐_𝑘⁻¹‖𝑓‖_𝐿¹_𝑘(ℝ^𝑑₎,(b)ℱ_𝐷(𝑓(⋅/𝜆))(𝑦)=𝜆^2𝛾+𝑑ℱ_𝐷(𝑓)(𝜆𝑦) for 𝜆>0,(c)if ℱ_𝐷(𝑓) belongs to 𝐿¹_𝑘(ℝ^𝑑), then 𝑓𝑦=1𝑐_𝑘ℝ^𝑑ℱ_𝐷𝑓(𝑥)𝐾𝑖𝑥,𝑦𝜔_𝑘(𝑥)𝑑𝑥,a.e., (12) and moreover, for all 𝑓∈𝒮(ℝ^𝑑),(d)ℱ_𝐷(𝑇_𝑗𝑓)(𝑦)=𝑖𝑦_𝑗ℱ_𝐷(𝑓)(𝑦),(e)if we define ℱ_𝐷(𝑓)(𝑦)=ℱ_𝐷(𝑓)(−𝑦), then ℱ_𝐷ℱ_𝐷=ℱ_𝐷ℱ_𝐷=𝐼𝑑. (13)

Proposition 1. The Dunkl transform ℱ_𝐷 is a topological isomorphism from 𝒮(ℝ^𝑑) onto itself, and for all f in 𝒮(ℝ^𝑑), _ℝ𝑑𝑓(𝑥)²𝜔_𝑘(𝑥)𝑑𝑥=_ℝ𝑑ℱ_𝐷𝑓(𝜉)²𝜔_𝑘(𝜉)𝑑𝜉. (14) In particular, the Dunkl transform 𝑓→ℱ_𝐷(𝑓) can be uniquely extended to an isometric isomorphism on 𝐿²_𝑘(ℝ^𝑑).

We define the tempered distribution 𝒯_𝑓 associated with 𝑓∈𝐿^𝑝_𝑘(ℝ^𝑑) by 𝒯_𝑓,𝜙=_ℝ𝑑𝑓(𝑥)𝜙(𝑥)𝜔_𝑘(𝑥)𝑑𝑥, (15) for 𝜙∈𝒮(ℝ^𝑑) and denote by ⟨𝑓,𝜙⟩_𝑘 the integral in the right hand side.

Definition 2. The Dunkl transform ℱ_𝐷(𝜏) of a distribution 𝜏∈𝒮^′(ℝ^𝑑) is defined by ℱ_𝐷(𝜏),𝜙=𝜏,ℱ_𝐷𝜙, (16) for 𝜙∈𝒮(ℝ^𝑑).

In particular, for 𝑓∈𝐿^𝑝_𝑘(ℝ^𝑑), it follows that for 𝜙∈𝒮(ℝ^𝑑), ℱ_𝐷𝑓,𝜙=ℱ_𝐷𝒯_𝑓,𝜙=𝒯_𝑓,ℱ_𝐷𝜙=𝑓,ℱ_𝐷(𝜙)_𝑘. (17)

Proposition 3. The Dunkl transform ℱ_𝐷 is a topological isomorphism from 𝒮^′(ℝ^𝑑) 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 𝑓∈𝒮(ℝ^𝑑) and 𝑦∈ℝ^𝑑, the Dunkl translation 𝜏_𝑦𝑓 is defined by 𝜏_𝑦𝑓(𝑥)=1𝑐_𝑘ℝ^𝑑ℱ_𝐷𝑓(𝑧)𝐾(𝑖𝑥,𝑧)𝐾𝑖𝑦,𝑧𝜔_𝑘(𝑧)𝑑𝑧. (18) Clearly, 𝜏_𝑦𝑓(𝑥)=𝜏_𝑥𝑓(𝑦), and by using the Dunkl's intertwining operator 𝑉_𝑘, 𝜏_𝑦𝑓 is related to the usual translation as 𝜏_𝑦𝑓(𝑥)=(𝑉_𝑘)_𝑥(𝑉_𝑘)_𝑦((𝑉_𝑘)⁻¹(𝑓)(𝑥+𝑦)) (cf. [11, 17]). Hence, 𝜏_𝑦 can also be defined for 𝑓∈ℰ(ℝ^𝑑). We define the Dunkl convolution product 𝑓∗_𝐷𝑔 of functions 𝑓, 𝑔∈𝒮(ℝ^𝑑) as follows: 𝑓∗_𝐷𝑔(𝑥)=_ℝ𝑑𝜏_𝑥𝑓−𝑦𝑔𝑦𝜔_𝑘𝑦𝑑𝑦. (19) This convolution is commutative and associative (cf. [17]).

Since ℱ_𝐷(𝜏_𝑦𝑓)(𝑥)=𝐾(𝑖𝑥,𝑦)ℱ_𝐷(𝑓)(𝑥) by the previous definition of 𝜏_𝑦𝑓, it follows that(a)for all 𝑓, 𝑔∈𝐷(ℝ^𝑑) (resp., 𝒮(ℝ^𝑑)), 𝑓∗_𝐷𝑔 belongs to 𝐷(ℝ^𝑑) (resp. 𝒮(ℝ^𝑑)) and ℱ_𝐷𝑓∗_𝐷𝑔𝑦=ℱ_𝐷𝑓𝑦ℱ_𝐷𝑔𝑦. (20) Moreover, as pointed in [16] and Sections 4 and 7, the operator 𝑓→𝑓∗_𝐷𝑔 is bounded on 𝐿^𝑝_𝑘(ℝ^𝑑), 1≤𝑝≤∞, provided that 𝑔 is a radial function in 𝐿¹_𝑘(ℝ^𝑑) or an arbitrary function in 𝐿¹_𝑘(ℝ^𝑑) for 𝐺=ℤ^𝑑₂. Hence, the standard argument yields the following Young's inequality.(b)Let 1≤𝑝,𝑞,𝑟≤∞ such that 1/𝑝+1/𝑞−1/𝑟=1. Assume that 𝑓∈𝐿^𝑝_𝑘(ℝ^𝑑) and 𝑔∈𝐿^𝑞_𝑘(ℝ^𝑑). If ‖𝜏_𝑥𝑔‖_𝐿^𝑞_{𝑘(ℝ𝑑)}≤𝐶‖𝑔‖_𝐿^𝑞_{𝑘(ℝ𝑑)} for all 𝑥∈ℝ^𝑑, then 𝑓∗_𝐷𝑔∈𝐿^𝑟_𝑘(ℝ^𝑑) and 𝑓∗_𝐷𝑔_{𝐿𝑟𝑘(ℝ𝑑)}≤𝐶𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}𝑔_𝐿^𝑞_{𝑘(ℝ𝑑)}. (21)

Definition 4. The Dunkl convolution product of a distribution 𝑆 in 𝒮^′(ℝ^𝑑) and a function 𝜙 in 𝒮(ℝ^𝑑) is the function 𝑆∗_𝐷𝜙 defined by 𝑆∗_𝐷𝜙(𝑥)=𝑆_𝑦,𝜏_−𝑦𝜙(𝑥). (22)

Proposition 5. Let 𝑓 be in 𝐿^𝑝_𝑘(ℝ^𝑑), 1≤𝑝≤∞, and 𝜙 in 𝒮(ℝ^𝑑). Then, the distribution 𝒯_𝑓∗_𝐷𝜙 is given by the function 𝑓∗_𝐷𝜙. If one assumes that 𝜙 is arbitrary for 𝑑=1 and radial for 𝑑≥2, then 𝒯_𝑓∗_𝐷𝜙 belongs to 𝐿^𝑝_𝑘(ℝ^𝑑). Moreover, for all 𝜓∈𝒮(ℝ^𝑑), 𝒯_𝑓∗_𝐷𝜙,𝜓=𝑓,̌𝜙∗_𝐷𝜓̌_𝑘, (23) where 𝜓̌(𝑥)=𝜓(−𝑥), and ℱ_𝐷𝒯_𝑓∗_𝐷𝜙=ℱ_𝐷𝒯_𝑓ℱ_𝐷𝜙. (24)

For each 𝑢∈𝒮^′(ℝ^𝑑), we define the distributions 𝑇_𝑗𝑢,1≤𝑗≤𝑑, by 𝑇_𝑗𝑢,𝜓=−𝑢,𝑇_𝑗𝜓, (25) for all 𝜓∈𝒮(ℝ^𝑑). Then, ⟨△_𝑘𝑢,𝜓⟩=⟨𝑢,△_𝑘𝜓⟩, and these distributions satisfy the following properties (see Section 2.3 (d)): ℱ_𝐷𝑇_𝑗𝑢=𝑖𝑦_𝑗ℱ_𝐷(𝑢),ℱ_𝐷△_𝑘𝑢=−𝑦²ℱ_𝐷(𝑢). (26) In the following, we denote 𝒯_𝑓 given by (15) by 𝑓 for simplicity.

3. ℬ̇^𝑠,𝑘_𝑝,𝑞, ℱ̇^𝑠,𝑘_𝑝,𝑞(ℝ^𝑑), and ℋ̇^𝑠_𝑝,𝑘 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 𝒞 the ring of center 0, of small radius 1/2, and great radius 2. There exist two radial functions 𝜓 and 𝜑 the values of which are in the interval [0,1] belonging to 𝐷(ℝ^𝑑) such that supp𝜓⊂𝐵(0,1),supp𝜑⊂𝒞∀𝜉∈ℝ^𝑑,𝜓(𝜉)+_𝑗≥0𝜑2^−𝑗𝜉=1∀𝜉∈𝒞,_𝑗∈ℤ𝜑2^−𝑗𝜉||2=1𝑛−𝑚≥2⟹supp𝜑^−𝑛⋅∩supp𝜑2^−𝑚⋅2=∅𝑗≥1⟹supp𝜓∩supp𝜑^−𝑗⋅=∅. (27)

Notations. We denote by Δ_𝑗𝑓=ℱ_𝐷⁻¹𝜑𝜉2^𝑗ℱ_𝐷𝑓,𝑆_𝑗𝑓=_{𝑛≤𝑗−1}Δ_𝑛𝑓,∀𝑗∈ℤ. (28) The distribution Δ_𝑗𝑓 is called the 𝑗th dyadic block of the homogeneous Littlewood-Paley decomposition of 𝑓 associated with the Dunkl operators.

Throughout this paper, we define 𝜙 and 𝜒 by 𝜙=ℱ_𝐷⁻¹(𝜑) and 𝜒=ℱ_𝐷⁻¹(𝜓).

When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functions 𝜓 and 𝜑 belonging to 𝐷(ℝ^𝑑) such that 𝜓≡1 on supp𝜓 and 𝜑≡1 on supp𝜑.

Remark 7. We remark that ℱ_𝐷𝑆_𝑗𝑓(𝜉)=𝜓𝜉2^𝑗ℱ_𝐷𝑆_𝑗𝑓(𝜉),ℱ_𝐷Δ_𝑗𝑓(𝜉)=𝜑𝜉2^𝑗ℱ_𝐷Δ_𝑗𝑓(𝜉). (29)

We put 𝜙=ℱ_𝐷⁻¹𝜑,𝜒=ℱ_𝐷⁻¹𝜓. (30)

Definition 8. Let one denote by 𝒮^′_ℎ,𝑘(ℝ^𝑑) the space of tempered distribution such that lim_{𝑗→−∞}𝑆_𝑗𝑢=0in𝒮^′ℝ^𝑑. (31)

On the follow, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Dunkl operators on ℝ^𝑑 and obtain their basic properties.

From now, we make the convention that for all nonnegative sequence {𝑎_𝑞}_𝑞∈ℤ, the notation (∑_𝑞𝑎^𝑟_𝑞)^1/𝑟 stands for sup_𝑞𝑎_𝑞 in the case 𝑟=∞.

Definition 9. Let 𝑠∈ℝ and 𝑝,𝑞∈[1,∞]. The homogeneous Dunkl-Besov spaces ℬ̇^𝑠,𝑘_𝑝,𝑞(ℝ^𝑑) are the space of distribution in 𝒮^′_ℎ,𝑘(ℝ^𝑑) such that 𝑓_{̇ℬ𝑠,𝑘𝑝,𝑞(ℝ𝑑)}∶=_𝑗∈ℤ2^𝑠𝑗Δ_𝑗𝑓_{𝐿𝑝𝑘(ℝ𝑑)}^𝑞1/𝑞<∞. (32)

Definition 10. Let 𝑠∈ℝ and 1≤𝑝,𝑞≤∞, the homogeneous Dunkl-Triebel-Lizorkin space ℱ̇^𝑠,𝑘_𝑝,𝑞(ℝ^𝑑) is the space of distribution in 𝒮^′_ℎ,𝑘(ℝ^𝑑) such that 𝑓_{̇ℱ𝑠,𝑘𝑝,𝑞(ℝ𝑑)}∶=_𝑗∈ℤ2^𝑠𝑗𝑞Δ_𝑗𝑓^𝑞1/𝑞_{𝐿𝑝𝑘(ℝ𝑑)}<∞. (33)

Let us recall that the operators (−△_𝑘)^𝑠/2 and (𝐼−△_𝑘)^𝑠/2 have been defined, respectively, by (cf. [19]) −△_𝑘^𝑠/2𝑓=ℱ_𝐷⁻¹‖⋅‖^𝑠ℱ_𝐷𝑓,𝐼−△_𝑘^𝑠/2𝑓=ℱ_𝐷⁻¹1+‖⋅‖^2𝑠/2ℱ_𝐷𝑓. (34) The operators (𝐼−△_𝑘)^−𝑠/2, for 𝑠>0, are called Dunkl-Bessel potential operators, and they are given by Dunkl convolution with the Dunkl-Bessel potential 𝐼−△_𝑘^−𝑠/2𝑓=𝑓∗_𝐷𝐵_𝑘,𝑠, (35) where 𝐵_𝑘,𝑠𝑦=1Γ(𝑠/2)^∞₀𝑒^−𝑡𝑒^{−‖𝑦‖2/4𝑡}𝑡^{(𝑠−𝑑−2𝛾)/2}𝑑𝑡𝑡. (36) We note that 𝐵_𝑘,𝑠(𝑦)≥0 for all 𝑦∈ℝ^𝑑, 𝐵_𝑘,𝑠∈𝐿¹_𝑘(ℝ^𝑑), and 𝐵_𝑘,𝑠𝑦≤𝐶𝑦^{𝑠−𝑑−2𝛾}𝑒^{−‖𝑦‖2/2},𝑦>0. (37)

Definition 11. For 𝑠∈ℝ and 1≤𝑝≤∞, the Dunkl-Bessel potential space 𝐻^𝑠_𝑝,𝑘(ℝ^𝑑) is defined as the space (𝐼−△_𝑘)^𝑠/2(𝐿^𝑝_𝑘(ℝ^𝑑)), equipped with the norm ‖𝑓‖_{𝐻𝑠𝑝,𝑘(ℝ𝑑)}=‖(𝐼−△_𝑘)^𝑠/2𝑓‖_𝐿^𝑝_{𝑘(ℝ𝑑)}.

Furthermore, 𝑝=2, 𝐻^𝑠_2,𝑘(ℝ^𝑑)∶=𝐻^𝑠_𝑘(ℝ^𝑑).

Definition 12. The operators (−△_𝑘)_𝑘^−𝑠/2, 0<𝑠<𝑑+2𝛾, are called Dunkl-Riesz potentials operators, and one has −△_𝑘^−𝑠/2𝑓=𝑅_𝑘,𝑠∗_𝐷𝑓, (38) where 𝑅_𝑘,𝑠 is the Dunkl-Riesz potential given by 𝑅_𝑘,𝑠𝑦∶=𝐶(𝑘,𝑠,𝑑)𝑦^{𝑠−𝑑−2𝛾},()=Γwhere𝐶𝑘,𝑠,𝑑𝑑+2𝛾−𝑠/22^{(𝑑+2𝛾−𝑠)/2}Γ().𝑠/2 (39)

Definition 13. For 𝑠∈ℝ and 1≤𝑝≤∞, the homogeneous Dunkl-Riesz potential space ℋ̇^𝑠_𝑝,𝑘(ℝ^𝑑) is defined as the space (−△_𝑘)^𝑠/2(𝐿^𝑝_𝑘(ℝ^𝑑)), equipped with the norm ‖𝑓‖_̇ℋ^𝑠_{𝑝,𝑘(ℝ𝑑)}=‖(−△_𝑘)^𝑠/2𝑓‖_𝐿^𝑝_{𝑘(ℝ𝑑)}.

Proposition 14. Let 𝑞∈(1,∞), and let 𝑠∈ℝ such that 0<𝑠<(𝑑+2𝛾)/𝑞, then one has ℬ̇^𝑠,𝑘_𝑞,𝑞ℝ^𝑑=ℱ̇^𝑠,𝑘_𝑞,𝑞ℝ^𝑑↪ℱ̇^𝑠,𝑘_𝑞,∞ℝ^𝑑↪ℱ̇^{𝑠−(𝑑+2𝛾)/𝑞,𝑘}_∞,∞ℝ^𝑑,ℋ̇^𝑠_𝑞,𝑘ℝ^𝑑=ℱ̇^𝑠,𝑘_𝑞,2ℝ^𝑑↪ℱ̇^𝑠,𝑘_𝑞,∞ℝ^𝑑↪ℱ̇^{𝑠−(𝑑+2𝛾)/𝑞,𝑘}_∞,∞ℝ^𝑑. (40)(41)

Proof. We obtain these results by similar ideas used in the nonhomogeneous case. (cf. [12]).

Theorem 15. Let 𝑎,𝑏>0, and let 𝑞₁,𝑞₂∈[1,∞]. Let 𝜃=𝑎/(𝑎+𝑏)∈(0,1), and let 1/𝑝=(1−𝜃)/𝑞₁+𝜃/𝑞₂. Then, there exists a constant 𝐶 such that for every 𝑓∈ℱ̇_𝑞^𝑎,𝑘_1,∞(ℝ^𝑑)∩ℱ̇_𝑞^−𝑏,𝑘_2,∞(ℝ^𝑑), then one has 𝑓(𝑥)≤𝐶sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥)^1−𝜃sup_𝑗∈ℤ2^−𝑏𝑗Δ_𝑗𝑓(𝑥)^𝜃. (42) In particular, one gets 𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}≤𝐶𝑓_̇ℱ^1−𝜃_{𝑞1𝑎,𝑘,∞ℝ𝑑}𝑓^𝜃_{̇ℱ𝑞2−𝑏,𝑘,∞ℝ𝑑}. (43)

Proof. Let 𝑓 be a Schwartz class, we have 𝑓(𝑥)≤_𝑗∈ℤΔ_𝑗𝑓(𝑥≤_𝑗∈ℤ)min2^−𝑎𝑗sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥,2_𝑗𝑏)sup_𝑗∈ℤ2^−𝑗𝑏Δ_𝑗𝑓(𝑥). (44) We define 𝑁(𝑥) as the largest index such that 2^𝑗𝑏sup_𝑗∈ℤ2^−𝑗𝑏Δ_𝑗𝑓(𝑥)≤2^−𝑎𝑗sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥), (45) and we write 𝑓(𝑥)≤_{(𝑥)𝑗≤𝑁}2^𝑗𝑏sup_𝑗∈ℤ2^−𝑗𝑏Δ_𝑗𝑓(𝑥)+_{(𝑥)𝑗>𝑁}2^−𝑎𝑗sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥)≤𝐶sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥)^{𝑏/(𝑎+𝑏)}sup_𝑗∈ℤ2^−𝑏𝑗Δ_𝑗𝑓(𝑥)^{𝑎/(𝑎+𝑏)}. (46) Thus, (42) is proved. In order to obtain (43), it is enough to apply Hölder’s inequality in the expression previous since we have 𝜃=𝑎/(𝑎+𝑏)∈(0,1) and 1/𝑝=(1−𝜃)/𝑞₁+𝜃/𝑞₂.

Corollary 16. Let 𝑞∈(1,∞), and let 𝑠∈ℝ such that 0<𝑠<(𝑑+2𝛾)/𝑞, then one has 𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}≤𝐶𝑓_̇ℬ^{1−𝑞/𝑝}_{−((2𝛾+𝑑)/𝑞−𝑠),𝑘∞,∞ℝ𝑑}𝑓_̇ℬ^𝑞/𝑝_{𝑠,𝑘𝑞,𝑞ℝ𝑑},𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}≤𝐶𝑓_̇ℬ^{1−𝑞/𝑝}_{−((2𝛾+𝑑)/𝑞−𝑠),𝑘,𝑘∞,∞ℝ𝑑}𝑓_̇ℋ^𝑞/𝑝_{𝑠𝑞,𝑘ℝ𝑑}, (47)(48) where 𝑝=𝑞(2𝛾+𝑑)/(2𝛾+𝑑−𝑞𝑠).

Proof. We take 𝑎=𝑠>0, −𝑏=𝑠−(𝑑+2𝛾)/𝑞<0, 𝑞₁=𝑞, and 𝑞₂=∞, 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]). (1) Let 𝑠>0 and 𝑝,𝑟∈[1,∞]. Then, ℬ̇^𝑠,𝑘_𝑝,𝑟(ℝ^𝑑)∩𝐿^∞_𝑘(ℝ^𝑑) is an algebra, and there exists a positive constant 𝐶 such that ‖𝑢𝑣‖_̇ℬ^𝑠,𝑘_{𝑝,𝑟(ℝ𝑑)}≤𝐶‖𝑢‖_𝐿^∞_𝑘(ℝ^𝑑₎‖𝑣‖_̇ℬ^𝑠,𝑘_{𝑝,𝑟(ℝ𝑑)}+‖𝑣‖_𝐿^∞_𝑘(ℝ^𝑑₎‖𝑢‖_̇ℬ^𝑠,𝑘_{𝑝,𝑟(ℝ𝑑)}. (49)
(2) Moreover, for any (𝑠₁,𝑠₂), any 𝑝₂ and any 𝑟₂ such that 𝑠₁+𝑠₂>(𝑑+2𝛾)/𝑝₁ and 𝑠₁<(𝑑+2𝛾)/𝑝₁, one has ‖𝑢𝑣‖_̇ℬ𝑝22^𝑠,𝑘_{,𝑟(ℝ𝑑)}≤𝐶‖𝑢‖_̇ℬ^𝑠_1𝑝1^,𝑘_{,∞(ℝ𝑑)}‖𝑣‖_̇ℬ^𝑠_2𝑝22^,𝑘_{,𝑟(ℝ𝑑)}+‖𝑢‖_̇ℬ^𝑠_2𝑝22^,𝑘_{,𝑟(ℝ𝑑)}‖𝑣‖_̇ℬ^𝑠_1𝑝1^,𝑘_{,∞(ℝ𝑑)}, (50) where 𝑠=𝑠₁+𝑠₂−(𝑑+2𝛾)/𝑝₁.
(3) Moreover, for any (𝑠₁,𝑠₂), any 𝑝₂ and any (𝑟₁,𝑟₂) such that 𝑠₁+𝑠₂>(𝑑+2𝛾)/𝑝₁, 𝑠₁<(𝑑+2𝛾)/𝑝₁, 1/𝑟₁+1/𝑟₂=1, one has ‖𝑢𝑣‖_̇ℬ^𝑠,𝑘_{𝑝,∞(ℝ𝑑)}≤𝐶‖𝑢‖_̇ℬ^𝑠_1𝑝11^,𝑘_{,𝑟(ℝ𝑑)}‖𝑣‖_̇ℬ^𝑠_2𝑝22^,𝑘_{,𝑟(ℝ𝑑)}+‖𝑢‖_̇ℬ^𝑠_2𝑝22^,𝑘_{,𝑟(ℝ𝑑)}‖𝑣‖_̇ℬ^𝑠_1𝑝11^,𝑘_{,𝑟(ℝ𝑑)}. (51)
(4) Moreover, for any (𝑠₁,𝑠₂), any (𝑝₁,𝑝₂,𝑝) and any (𝑟₁,𝑟₂) such that 𝑠_𝑗<(𝑑+2𝛾)/𝑝_𝑗, 𝑠₁+𝑠₂>(𝑑+2𝛾)(1/𝑝₁+1/𝑝₂−1/𝑝) and 𝑝≥max(𝑝₁,𝑝₂), one has ‖𝑢𝑣‖_̇ℬ^𝑠_1,2^,𝑘_{𝑝,𝑟(ℝ𝑑)}≤𝐶‖𝑢‖_̇ℬ^𝑠_1𝑝11^,𝑘_{,𝑟(ℝ𝑑)}‖𝑣‖_̇ℬ^𝑠_2𝑝22^,𝑘_{,𝑟(ℝ𝑑)}. (52) with 𝑠_1,2=𝑠₁+𝑠₂−(𝑑+2𝛾)(1/𝑝₁+1/𝑝₂−1/𝑝) and 𝑟=max(𝑟₁,𝑟₂).

4. A Primer to Real Interpolation Theory and Generalized Lorentz Spaces

From now, we denote by 𝑙^𝑞(ℤ) the set of sequence (𝑎_𝑗)_𝑗∈ℤ such that _𝑗∈ℤ𝑎_𝑗^𝑞1/𝑞<∞ (53) stands for sup_𝑗|𝑎_𝑗| in the case 𝑞=∞.

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 𝐴₀ and 𝐴₁ which are continuously imbedded into a common topological vector space 𝑉 and 𝑡>0.

The 𝐽-method and the 𝐾-method consist to consider the 𝐽-functional and the 𝐾-functional defined on 𝐴₀⋂𝐴₁ by 𝐽𝑡,𝑎;𝐴₀,𝐴₁=max‖𝑎‖_𝐴0,𝑡‖𝑎‖_𝐴1,𝐾𝑡,𝑎;𝐴₀,𝐴₁=min𝑎_0𝐴0+𝑡𝑎_1𝐴1∶𝑎=𝑎₀+𝑎₁. (54)

Definition 18 (𝐽-method of interpolation). For 0<𝜃<1 and 1≤𝑞≤∞, the interpolation space [𝐴₀,𝐴₁]_{𝜃,𝑞,𝐽} is defined as follows: 𝑎∈[𝐴₀,𝐴₁]_{𝜃,𝑞,𝐽} if and only if 𝑎 can be written as a sum 𝑎=∑_𝑗∈ℤ𝑎_𝑗, where the series converge in 𝐴₀+𝐴₁, each 𝑎_𝑗 belongs to 𝐴₀⋂𝐴₁ and (2^−𝑗𝜃𝐽(2^𝑗,𝑎_𝑗;𝐴₀,𝐴₁))_𝑗∈ℤ∈𝑙^𝑞(ℤ).
The norm of [𝐴₀,𝐴₁]_{𝜃,𝑞,𝐽} is defined by ‖𝑎‖_{[𝐴0,𝐴1]𝜃,𝑞,𝐽}=inf_{𝑎=∑𝑗∈ℤ𝑎𝑗𝑗∈ℤ}2^{−𝑗𝜃𝑞}𝑎_𝑗^𝑞_𝐴0^1/𝑞+_𝑗∈ℤ2^{𝑗(1−𝜃)𝑞}𝑎_𝑗^𝑞_𝐴1^1/𝑞. (55)

Definition 19 (K-method of interpolation). For 0<𝜃<1 and 1≤𝑞≤∞, the space [𝐴₀,𝐴₁]_{𝜃,𝑞,𝐾} is defined by 𝑎∈[𝐴₀,𝐴₁]_{𝜃,𝑞,𝐾} if and only if 𝑎∈𝐴₀+𝐴₁, and (2^−𝑗𝜃𝐾(2^𝑗,𝑎;𝐴₀,𝐴₁))_𝑗∈ℤ∈𝑙^𝑞(ℤ).
The norm of [𝐴₀,𝐴₁]_{𝜃,𝑞,𝐾} is defined as follows: ‖𝑎‖_{[𝐴0,𝐴1]𝜃,𝑞,𝐾}=_𝑗∈ℤ2^{−𝑗𝜃𝑞}𝐾2^𝑗,𝑎;𝐴₀,𝐴₁^𝑞1/𝑞. (56)

Proposition 20 (Equivalence theorem). For 0<𝜃<1 and 1≤𝑞≤∞, one has [𝐴₀,𝐴₁]_{𝜃,𝑞,𝐾}=[𝐴₀,𝐴₁]_{𝜃,𝑞,𝐽}.

Remark 21. In the following, we will denote this space by [𝐴₀,𝐴₁]_𝜃,𝑞.

Lemma 22. For 𝑎=∑_𝑗∈ℤ𝑎_𝑗 and 𝜚>0, with 𝜚≠1, one has ‖𝑎‖_{[𝐴0,𝐴1]𝜃,𝑞}≤𝐶𝑞,𝜃,𝜚_𝑗∈ℤ𝜚^{−𝑗𝜃𝑞}𝑎_𝑗^𝑞_𝐴0^{(1−𝜃)/𝑞}×_𝑗∈ℤ𝜚^{𝑗(1−𝜃)𝑞}𝑎_𝑗^𝑞_𝐴1^𝜃/𝑞. (57)

Proposition 23. (i) For 𝜃₀≠𝜃₁, one has 𝐴₀,𝐴_1𝜃0,𝑞0,𝐴₀,𝐴_{1𝜃1,𝑞1𝜃,𝑞}=𝐴₀,𝐴_{1(1−𝜃)𝜃0+𝜃𝜃1,𝑞}. (58)(ii) For 𝜃₀=𝜃₁, (58) is still valid if 1/𝑞=(1−𝜃)/𝑞₀+𝜃/𝑞₁.

Proposition 24 (Duality theorem for the real method). One considers the dual spaces 𝐴^′₀,𝐴^′₁ and [𝐴₀,𝐴₁]^′_𝜃,𝑞 for 0<𝜃<1 and 1≤𝑞<∞ of the spaces 𝐴₀,𝐴₁ and [𝐴₀,𝐴₁]_𝜃,𝑞. If 𝐴₀⋂𝐴₁ is dense in 𝐴₀ and in 𝐴₁, one has [𝐴₀,𝐴₁]^′_𝜃,𝑞=[𝐴^′₀,𝐴^′₁]_𝜃,𝑞′, where 𝑞^′ is the conjugate component of 𝑞.

For any measurable function 𝑓 on ℝ^𝑑, we define its distribution and rearrangement functions 𝑑_𝑓,𝑘(𝜆)∶=_{{𝑥∈ℝ𝑑∶|𝑓(𝑥)|≥𝜆}}𝜔_𝑘(𝑥)𝑓𝑑𝑥,_∗𝑘(𝑠)∶=inf𝜆∶𝑑_𝑓,𝑘(𝜆)≤𝑠. (59)

For 1≤𝑝≤∞ and 1≤𝑞≤∞, define 𝑓_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)=∫^∞₀𝑠^1/𝑝𝑓^∗_𝑘(𝑠)^𝑞𝑑𝑠𝑠^1/𝑞if1≤𝑝,𝑞<∞sup_𝑠>0𝑠^1/𝑝𝑓^∗_𝑘(𝑠)if1≤𝑝≤∞,𝑞=∞. (60) The generalized Lorentz spaces 𝐿_𝑘^𝑝,𝑞(ℝ^𝑑) is defined as the set of all measurable functions 𝑓 such that ||𝑓||_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)<∞.

Proposition 25. (i) For 1<𝑝<∞, 1≤𝑞≤∞𝐿_𝑘^𝑝,𝑞ℝ^𝑑=𝐿¹_𝑘ℝ^𝑑,𝐿^∞_𝑘ℝ^𝑑_𝜃,𝑞, (61) with 1/𝑝=1−𝜃.
(ii) For 𝑝₀≠𝑝₁, one has 𝐿^𝑝0,𝑞0_𝑘ℝ^𝑑,𝐿^𝑝1,𝑞1_𝑘ℝ^𝑑_𝜃,𝑞=𝐿^𝑝0_𝑘ℝ^𝑑,𝐿^𝑝1_𝑘ℝ^𝑑_𝜃,𝑞=𝐿_𝑘^𝑝,𝑞ℝ^𝑑, (62) with 1/𝑝=(1−𝜃)/𝑝₀+𝜃/𝑝₁.
(iii) In the case 𝑝₀=𝑝₁=𝑝, one has 𝐿^𝑝,𝑞0_𝑘ℝ^𝑑,𝐿^𝑝,𝑞1_𝑘ℝ^𝑑_𝜃,𝑞=𝐿_𝑘^𝑝,𝑞ℝ^𝑑, (63) with 1/𝑞=(1−𝜃)/𝑞₀+𝜃/𝑞₁.
(iv) If 1≤𝑝≤∞ and 1≤𝑞₁<𝑞₂≤∞, then 𝐿^𝑝,𝑞1_𝑘ℝ^𝑑↪𝐿^𝑝,𝑞2_𝑘ℝ^𝑑. (64)

Proof. We obtain these results by similar ideas used in the Euclidean case.

Proposition 26. (i) Let 1<𝑝<∞, 1≤𝑞≤∞. Then, there exists a constant 𝐶 such that every 𝑓∈𝐿_𝑘^𝑝,𝑞(ℝ^𝑑) can be decomposed as 𝑓=∑_𝑗∈ℤ𝑓_𝑗, where 2^{−𝑗(𝑝−1)/𝑝}𝑓_{𝑗𝐿1𝑘(ℝ𝑑)𝑙𝑟}+2^𝑗/𝑝𝑓_{𝑗𝐿∞𝑘(ℝ𝑑)𝑙𝑟}≤𝐶𝑓_𝐿𝑘^𝑝,𝑞_(ℝ𝑑), (65) the 𝑓_𝑗 have disjoint supports if 𝑗≠𝑛, 𝑓_𝑗𝑓_𝑛=0.
(ii) Let 1<𝑝<∞, 1≤𝑞≤∞. Then, there exists a constant 𝐶 such that every 𝑓∈𝐿_𝑘^𝑝,𝑞(ℝ^𝑑) and every 𝑔∈𝐿_𝑘^{𝑝/(𝑝−1),𝑞/(𝑞−1)}(ℝ^𝑑), one has 𝑓𝑔∈𝐿¹_𝑘(ℝ^𝑑) and _ℝ𝑑𝑓(𝑥)𝑔(𝑥)𝜔_𝑘(𝑥)𝑑𝑥≤𝐶𝑓_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)𝑔_𝐿𝑘^{𝑝/(𝑝−1),𝑞/(𝑞−1)}_(ℝ𝑑). (66)

Proof. We obtain these results by similar ideas used in the Euclidean case.

5. Inequalities for the Fractional Dunkl-Laplace Operators

Lemma 27. Let 𝑠 be a real number such that 0<𝑠<𝑑+2𝛾, and let 1<𝑝<𝑞<∞ satisfy 1𝑝−1𝑞=𝑠.2𝛾+𝑑 (67) For 𝑓∈𝐿^𝑝_𝑘(ℝ^𝑑), one has 𝐼−△_𝑘^−𝑠/2𝑓_{𝐿𝑞𝑘(ℝ𝑑)}≤𝐶𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}. (68)

Proof. We obtain this result by similar ideas used for the Dunkl-Riesz potential (cf. [21]).

Proposition 28. Let 𝑠<(𝑑+2𝛾)/2 and 𝑞=(2𝑑+4𝛾)/(𝑑+2𝛾−2𝑠). Then, 𝑓_𝐿^𝑞_{𝑘(ℝ𝑑)}≤𝐶𝐼−△_𝑘^𝑠/2𝑓_{𝐿2𝑘(ℝ𝑑)},𝑓∈𝐻^𝑠_𝑘ℝ^𝑑. (69)

Proof. Let us first observe that since 𝐷(ℝ^𝑑) is dense in 𝐻^𝑠_𝑘(ℝ^𝑑), it is enough to prove (69) for 𝑓∈𝐷(ℝ^𝑑). Let 𝑓,𝑔∈𝐷(ℝ^𝑑). Then, we have 𝑓,𝑔_{𝐿2𝑘(ℝ𝑑)}=ℱ_𝐷(𝑓),ℱ_𝐷𝑔_{𝐿2𝑘(ℝ𝑑)}=_ℝ𝑑1+𝜉^2𝑠/2ℱ_𝐷𝑓(𝜉)ℱ_𝐷𝑔(𝜉)1+𝜉^2−𝑠/2×𝜔_𝑘(𝜉)=𝑑𝜉𝐼−△_𝑘^𝑠/2𝑓,𝐼−△_𝑘^−𝑠/2𝑔_{𝐿2𝑘(ℝ𝑑)}. (70) Hence, 𝑓,𝑔_{𝐿2𝑘(ℝ𝑑)}≤𝐼−△_𝑘^𝑠/2𝑓_{𝐿2𝑘(ℝ𝑑)}𝐼−△_𝑘^−𝑠/2𝑔_{𝐿2𝑘(ℝ𝑑)}. (71) Now, by the previous lemma we obtain 𝑓,𝑔_{𝐿2𝑘(ℝ𝑑)}≤𝐶𝐼−△_𝑘^𝑠/2𝑓_{𝐿2𝑘(ℝ𝑑)}𝑔_𝐿^𝑝_{𝑘(ℝ𝑑)}, (72) where 𝑝=(2𝑑+4𝛾)/(𝑑+2𝛾+2𝑠). Now, let us take 𝑔=𝑓^𝑞−1, with 1/𝑝+1/𝑞=1, that is, 𝑞=(2𝑑+4𝛾)/(𝑑+2𝛾−2𝑠). Then, the relation (72) gives that 𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}≤𝐶𝐼−△_𝑘^𝑠/2𝑓_{𝐿2𝑘(ℝ𝑑)}𝑓_𝐿^𝑞−1_{𝑞𝑘ℝ𝑑}. (73) Thus, we obtain (69).

Proposition 29. Let 1≤𝑝,𝑝₂<∞, 0<𝜃<𝑝<∞, 0<𝑠<𝑑+2𝛾, and 1<𝑝₁<(𝑑+2𝛾)/𝑠. Then, one has the inequality 𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}≤𝐶𝐼−△_𝑘^𝑠/2𝑓_𝐿^𝜃/𝑝_{𝑝1𝑘ℝ𝑑}𝑓_𝐿^{(𝑝−𝜃)/𝑝}_{𝑝2𝑘ℝ𝑑}, (74) with 𝜃1𝑝₁−𝑠+𝑑+2𝛾𝑝−𝜃𝑝₂=1. (75)

Proof. Hölder's inequality yields 𝑓^𝑝_{𝐿𝑝𝑘ℝ𝑑}≤𝐶𝑓^𝜃_𝐿^𝑝_0𝑘ℝ𝑑𝑓_𝐿^𝑝−𝜃_{𝑝2𝑘ℝ𝑑}, (76) where 1𝑝₀=1𝜃11−𝑝,1𝑝₂=1𝑝.𝑝−𝜃 (77) Applying Lemma 27, with 𝑝₁=((𝑑+2𝛾)𝑝₀)/(𝑑+2𝛾+𝑠𝑝₀), we obtain the result.

Theorem 30. Let 1<𝑞<∞, 0<𝑠<𝑑+2𝛾 and 1<𝑝₁<(𝑑+2𝛾)/𝑠. Then, the inequality1exp𝑞+𝑠−1𝑑+2𝛾𝑝₁×_ℝ𝑑𝑓(𝑥)^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}𝑓(𝑥ln)^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}𝜔_𝑘(𝑥)𝑑𝑥≤𝐶𝐼−△_𝑘^𝑠/2𝑓_{𝐿𝑝1𝑘(ℝ𝑑)}𝑓_𝐿^𝑞_{𝑘(ℝ𝑑)} (78) holds for 1𝑞+𝑠−1𝑑+2𝛾𝑝₁>0. (79)

Proof. Using the convexity of the function 𝑔(ℎ)=ℎln(∫_ℝ𝑑|𝑓(𝑥)|^1/ℎ𝜔_𝑘(𝑥)𝑑𝑥), and the logarithmic Hölder's inequality proved by Merker [22], we obtain _ℝ𝑑𝑓(𝑥)^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}𝑓(𝑥ln)^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}𝜔_𝑘(𝑥)≤𝑝𝑑𝑥𝑓𝑝−𝑞ln^{𝑞𝐿𝑞𝑘ℝ𝑑}𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}, (80) for 0<𝑞<𝑝≤∞. We can choose 𝑝=((𝑑+2𝛾)𝑞)/(𝑑+2𝛾−𝑞𝑠)∈(𝑞,∞) for 𝑝₂=𝑞 and 𝜃 satisfying the condition of Proposition 29, and we get _ℝ𝑑𝑓(𝑥)^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}𝑓(𝑥ln)^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}𝜔_𝑘(𝑥)≤𝑝𝑑𝑥𝐶𝑝−𝑞ln𝐼−△_𝑘^𝑠/2𝑓_𝐿^𝜃/𝑝_{𝑝1𝑘ℝ𝑑}𝑓_𝐿^{(𝑝−𝜃)/𝑝}_{𝑝2𝑘ℝ𝑑}^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}≤𝑞𝜃𝐶𝑝−𝑞ln𝐼−△_𝑘^𝑠/2𝑓_{𝐿𝑝1𝑘(ℝ𝑑)}𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}. (81) By a simple calculation, we obtain the result.

Corollary 31. Let 0<𝑠<𝑑+2𝛾 and 1<𝑞<(𝑑+2𝛾)/𝑠, 𝑓∈𝐻^𝑠_𝑞,𝑘(ℝ^𝑑) such that ‖𝑓‖_𝐿^𝑞_{𝑘(ℝ𝑑)}=1, one has 𝑠exp𝑑+2𝛾_ℝ^𝑑𝑓(𝑥)^𝑞ln𝑓(𝑥)^𝑞𝜔_𝑘(𝑥)𝑑𝑥≤𝐶𝐼−△_𝑘^𝑠/2𝑓_{𝐿𝑞𝑘(ℝ𝑑)}. (82)

Proof. It suffices to apply the previous theorem for 𝑞=𝑝₁.

Lemma 32 (see [23]). One assumes that 𝐺=ℤ^𝑑₂. If 𝑓∈𝐿^𝑝1,𝑞1_𝑘(ℝ^𝑑), 𝑔∈𝐿^𝑝2,𝑞2_𝑘(ℝ^𝑑), and 1/𝑝₁+1/𝑝₂>1, then 𝑓∗_𝐷𝑔∈𝐿^𝑝3,𝑞3_𝑘(ℝ^𝑑) where 1/𝑝₃=1/𝑝₁+1/𝑝₂−1 and 𝑞₃≥1 is any number such that 1/𝑞₃≤1/𝑞₁+1/𝑞₂. Moreover, 𝑓∗_𝐷𝑔_𝐿^𝑝_33𝑘^,𝑞_(ℝ𝑑)≤𝐶𝑓_𝐿^𝑝_11𝑘^,𝑞_(ℝ𝑑)𝑔_𝐿^𝑝_22𝑘^,𝑞_(ℝ𝑑). (83)

Remark 33. The analogues of this lemma for the general reflection group 𝐺, together with other additional results, will appear in a forthcoming paper.

Theorem 34. One assumes that 𝐺=ℤ^𝑑₂. Let 1≤𝑝<∞, 1≤𝑝₂,𝑞,𝑞₁,𝑞₂<∞, 0<𝜃<𝑞, 0<𝑠<𝑑+2𝛾, and 1<𝑝₁<(𝑑+2𝛾)/𝑠. Then, the inequality 𝑓_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)≤𝐶𝐼−△_𝑘^𝑠/2𝑓_𝐿^𝜃/𝑞_{𝑝11𝑘,𝑞ℝ𝑑}𝑓_𝐿^{(𝑞−𝜃)/𝑞}_{𝑝22𝑘,𝑞ℝ𝑑} (84) holds for 𝜃𝑞₁+𝑞−𝜃𝑞₂𝜃1=1,𝑝₁−𝑠+𝑑+2𝛾𝑞−𝜃𝑝₂=𝑞𝑝. (85)

Proof. Applying the Hölder inequality and simple computation yields 𝑓^𝑞_{𝐿𝑘𝑝,𝑞ℝ𝑑}≤𝐶𝑓^𝜃_𝐿^𝑝_31𝑘^,𝑞_ℝ𝑑𝑓_𝐿^𝑞−𝜃_{𝑝22𝑘,𝑞ℝ𝑑}, (86) where 1𝑝₃=1𝑝−1𝑞+1𝑞₁,1𝑝₂=1𝑝−1𝑞+1𝑞₂. (87) Note that 𝑓(𝑥)=𝐼−△_𝑘^𝑠/2𝑓∗_𝐷𝐵_𝑘,𝑠(𝑥), (88) where 𝐵_𝑘,𝑠 is the Dunkl-Bessel kernel defined by relation (36). From the relation (37), we see that 𝐵_𝑘,𝑠∈𝐿_𝑘^{(𝑑+2𝛾)/(𝑑+2𝛾−𝑠),∞}(ℝ^𝑑). Using now Lemma 32, we deduce that 𝑓_𝐿^𝑝_31𝑘^,𝑞_(ℝ𝑑)≤𝐶𝐼−△_𝑘^𝑠/2𝑓_{𝐿𝑝11𝑘,𝑞(ℝ𝑑)}, (89) for 1𝑝₃=1𝑝₁−𝑠𝑑+2𝛾,0<𝑠<𝑑+2𝛾𝑝₁. (90) 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 𝑠<(𝑑+2𝛾)/2 and 𝑞=(2𝑑+4𝛾)/(𝑑+2𝛾−2𝑠). Then, 𝑓_𝐿^𝑞_{𝑘(ℝ𝑑)}≤𝐶−△_𝑘^𝑠/2𝑓_{𝐿2𝑘(ℝ𝑑)},𝑓∈𝐻^𝑠_𝑘ℝ^𝑑. (91)

Proposition 36. Let 1≤𝑝,𝑝₂<∞, 0<𝜃<𝑝<∞, 0<𝑠<𝑑+2𝛾, and 1<𝑝₁<(𝑑+2𝛾)/𝑠. Then, the inequality 𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}≤𝐶−△_𝑘^𝑠/2𝑓_𝐿^𝜃/𝑝_{𝑝1𝑘ℝ𝑑}𝑓_𝐿^{(𝑝−𝜃)/𝑝}_{𝑝2𝑘ℝ𝑑} (92) with 𝜃1𝑝₁−𝑠+𝑑+2𝛾𝑝−𝜃𝑝₂=1. (93)

Theorem 37. Let 1<𝑞<∞, 0<𝑠<𝑑+2𝛾, and 1<𝑝₁<(𝑑+2𝛾)/𝑠. Then, the inequality 1exp𝑞+𝑠−1𝑑+2𝛾𝑝₁×_ℝ𝑑𝑓(𝑥)^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}𝑓(𝑥ln)^𝑞𝑓^𝑞_{𝐿𝑞𝑘ℝ𝑑}𝜔_𝑘(𝑥)𝑑𝑥≤𝐶−△_𝑘^𝑠/2𝑓_{𝐿𝑝1𝑘(ℝ𝑑)}𝑓_𝐿^𝑞_{𝑘(ℝ𝑑)} (94) holds for 1𝑞+𝑠−1𝑑+2𝛾𝑝₁>0. (95)

Corollary 38. Let 0<𝑠<𝑑+2𝛾 and 1<𝑞<(𝑑+2𝛾)/𝑠, 𝑓∈ℋ̇^𝑠_𝑞,𝑘(ℝ^𝑑) such that ‖𝑓‖_𝐿^𝑞_{𝑘(ℝ𝑑)}=1, one has 𝑠exp𝑑+2𝛾_ℝ^𝑑𝑓(𝑥)^𝑞ln𝑓(𝑥)^𝑞𝜔_𝑘(𝑥)𝑑𝑥≤𝐶−△_𝑘^𝑠/2𝑓_{𝐿𝑞𝑘(ℝ𝑑)}. (96)

Theorem 39. One assumes that 𝐺=ℤ^𝑑₂. Let 1≤𝑝<∞, 1≤𝑝₂,𝑞,𝑞₁,𝑞₂<∞, 0<𝜃<𝑞, 0<𝑠<𝑑+2𝛾, and 1<𝑝₁<(𝑑+2𝛾)/𝑠. Then, the inequality 𝑓_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)≤𝐶−△_𝑘^𝑠/2𝑓_𝐿^𝜃/𝑞_{𝑝11𝑘,𝑞ℝ𝑑}𝑓_𝐿^{(𝑞−𝜃)/𝑞}_{𝑝22𝑘,𝑞ℝ𝑑} (97) holds for 𝜃𝑞₁+𝑞−𝜃𝑞₂𝜃1=1,𝑝₁−𝑠+𝑑+2𝛾𝑞−𝜃𝑝₂=𝑞𝑝. (98)

Remark 40. (i) We assume that G=ℤ^𝑑₂. It follows from the special case 𝑝₁=𝑞₁ and 𝑝₂=𝑞₂ of (97) that the inequality 𝑓_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)≤𝐶−△_𝑘^𝑠/2𝑓_𝐿^𝜃/𝑞_{𝑝1𝑘ℝ𝑑}𝑓_𝐿^{(𝑞−𝜃)/𝑞}_{𝑝2𝑘ℝ𝑑} (99) with 𝑞=𝑝(1−𝜃𝑠/(𝑑+2𝛾)). Equation (99) can be thought of a refinement of (92) from (64).
(ii) We assume that 𝐺=ℤ^𝑑₂. It follows from the special case 𝑝₁=𝑞=𝜃 that (99) becomes 𝑓_𝐿𝑘^{𝑞(𝑑+2𝛾)/(𝑑+2𝛾−𝑞𝑠),𝑞}_(ℝ𝑑)≤𝐶−△_𝑘^𝑠/2𝑓_{𝐿𝑞𝑘(ℝ𝑑)}, (100) which can also be thought of as a refinement of the Hardy-Littlewood-Sobolev fractional integration theorem in Dunkl setting (cf. [21]): −△_𝑘^−𝑠/2𝑓_{𝐿𝑞𝑘(ℝ𝑑)}≤𝐶𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}. (101)
(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 𝐺=ℤ^𝑑₂. Let 1<𝑝<∞, 0<𝑠<(𝑑+2𝛾)/𝑝, and 1≤𝑞≤∞. There exists a positive constant 𝐶 such that one has 𝑓(𝑥)‖𝑥‖^𝑠_𝐿𝑘^𝑝,𝑞_(ℝ^𝑑₎≤𝐶−△_𝑘^𝑠/2𝑓_{𝐿𝑘𝑝,𝑞(ℝ𝑑)}. (102)

For proof of this result, we need the following lemma which we prove as the Euclidean case.

Lemma 42. Let 1≤𝑝₁,𝑝₂,𝑞₁,𝑞₂≤∞. If 𝑓∈𝐿^𝑝1,𝑞1_𝑘(ℝ^𝑑) and 𝑔∈𝐿^𝑝2,𝑞2_𝑘(ℝ^𝑑), then 𝑓𝑔_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)≤𝐶𝑓_𝐿^𝑝_11𝑘^,𝑞_(ℝ𝑑)𝑔_𝐿^𝑝_22𝑘^,𝑞_(ℝ𝑑), (103) where 1/𝑝=1/𝑝₁+1/𝑝₂ and 1/𝑞=1/𝑞₁+1/𝑞₂.

Proof of Theorem 41. Let 1<𝑝<∞ and 𝑠∈(0,(𝑑+2𝛾)/𝑝). We take 𝑔(𝑥)=1/‖𝑥‖^𝑠 and apply (103) in the specific form 𝑓𝑔_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)≤𝐶𝑓_𝐿^𝑝_1𝑘^,𝑞_(ℝ𝑑)𝑔_𝐿𝑘^𝑟,∞_(ℝ𝑑), (104) where 𝑟=(𝑑+2𝛾)/𝑠 and 𝑝₁=(𝑞(𝑑+2𝛾))/(𝑑+2𝛾−𝑞𝑠). As 𝑔∈𝐿_𝑘^𝑟,∞(ℝ^𝑑), we have 𝑓(𝑥)‖𝑥‖^𝑠_𝐿𝑘^𝑝,𝑞_(ℝ^𝑑₎≤𝐶𝑓_𝐿𝑘^{((𝑑+2𝛾)𝑝)/(𝑑+2𝛾−𝑝𝑠),𝑞}_(ℝ𝑑), (105) with 1≤𝑞≤∞. On the other hand, from [23], Theorem 1.2, we have 𝑓_𝐿𝑘^{((𝑑+2𝛾)𝑝)/(𝑑+2𝛾−𝑝𝑠),𝑞}_(ℝ𝑑)≤𝐶−△_𝑘^𝑠/𝑠𝑓_{𝐿𝑘𝑝,𝑞(ℝ𝑑)}, (106) for any 𝑓∈𝐿_𝑘^𝑝,𝑞(ℝ^𝑑) with 1≤𝑞≤∞, 1<𝑝<∞, and 0<𝑠<(𝑑+2𝛾)/𝑝. Thus, we obtain (102).

6. Dispersion Phenomena

Notations. We denote by ℐ_𝑘(𝑡) the Dunkl-Schrödinger semigroup on 𝐿²_𝑘(ℝ^𝑑) defined by ℐ_𝑘(𝑡)1𝑣∶=𝑐_𝑘|𝑡|^𝛾+𝑑/2𝑒^{−𝑖(𝑑+2𝛾)(𝜋/4)sgn𝑡}𝑒^{𝑖(‖⋅‖2/4𝑡)}×ℱ_𝐷𝑒^{𝑖(‖⋅‖2/4𝑡)}𝑣⋅.2𝑡 (107)𝑊_𝑘^1,𝑟(ℝ^𝑑) (1≤𝑟≤∞) Banach space of (classes of) measurable functions 𝑢∶ℝ^𝑑→ℂ such that 𝑇^𝜇𝑢∈𝐿^𝑟_𝑘(𝑅^𝑑) in the sense of distributions, for every multi-index 𝜇 with |𝜇|≤1. 𝑊_𝑘^1,𝑟(ℝ^𝑑) is equipped with the norm ‖𝑢‖_𝑊𝑘^1,𝑟_(ℝ^𝑑₎=_|𝜇|≤1𝑇^𝜇𝑢_{𝐿𝑟𝑘(ℝ𝑑)}. (108)𝑊^1,𝑟_𝑘,𝐺(ℝ^𝑑) (1≤𝑟≤∞) the subspace of 𝑊_𝑘^1,𝑟(ℝ^𝑑), which these elements are 𝐺-invariant.

Definition 43. One says that the exponent pair (𝑞,𝑟) is (𝑑+2𝛾)/2-admissible if 𝑞,𝑟≥2, (𝑞,𝑟,(𝑑+2𝛾)/2)≠(2,∞,1), and 1𝑞+𝑑+2𝛾≤2𝑟𝑑+2𝛾4. (109) If equality holds in (109) one says that (𝑞,𝑟) is sharp (𝑑+2𝛾)/2-admissible, otherwise, one says that (𝑞,𝑟) is nonsharp (𝑑+2𝛾)/2-admissible. Note in particular that when 𝑑+2𝛾>2, the endpoint 𝑃=2,2𝑑+4𝛾𝑑+2𝛾−2 (110) is sharp (𝑑+2𝛾)/2-admissible.

Lemma 44 (see [25]). Let 𝐸 and 𝐹 be Banach spaces, and let ℒ: 𝐿^𝑝,𝑟(0,∞;𝐸)→𝐿^𝑞,𝑠(0,∞;𝐹) be an integral operator for some 𝑝,𝑟,𝑞,𝑠, with a kernel 𝑘(𝑡,𝜏) such that ℒ𝑓(𝑡)=^∞₀𝑘(𝑡,𝜏)𝑓(𝜏)𝑑𝜏. (111) If 1≤𝑝≤𝑟<𝑠≤𝑞<∞, then one has ℒ𝑓_{𝐿𝑞,𝑠(0,∞;𝐹)}≤𝐶𝑓_{𝐿𝑝,𝑟(0,∞;𝐸)}, (112) where ℒ is the low diagonal operator defined by ℒ𝑓(𝑡)=^𝑡₀𝑘(𝑡,𝜏)𝑓(𝜏)𝑑𝜏. (113)

Lemma 45. For any (𝑑+2𝛾)/2-admissible pair (𝑞,𝑟) with 𝑞>2ℐ_𝑘(𝑡)𝑓_{𝐿𝑞,2(0,∞;𝐿𝑟𝑘(ℝ𝑑))}≤𝐶𝑓_{𝐿2𝑘(ℝ𝑑)},_𝑡0ℐ_𝑘(𝑡−𝜏)𝑔(𝜏)𝑑𝜏_{𝐿𝑞,2(0,∞;𝐿𝑟𝑘(ℝ𝑑))∩𝐿∞(0,∞;𝐿2𝑘(ℝ𝑑))}≤𝐶𝑔_{𝐿𝑞′,2(0,∞;𝐿𝑟′𝑘(ℝ𝑑))}. (114)(115)

Proof. From the dispersion of ℐ_𝑘(𝑡) such that ℐ_𝑘(𝑡)𝑔_{𝐿𝑟𝑘(ℝ𝑑)}≤𝐶𝑡^{−(𝑑+2𝛾)(1/2−1/𝑟)}𝑔_{𝐿𝑟′𝑘(ℝ𝑑)}, (116) for any 𝑟∈[2,∞], (cf. [8]), and the fact that 𝑡^{−(𝑑+2𝛾)(1/2−1/𝑟)}∈𝐿^{2𝑟/(𝑑+2𝛾)(𝑟−2),∞},2forany𝑟∈2,𝑑+2𝛾,𝑑+2𝛾−2 (117) one can easily prove the result.

Theorem 46. Suppose that 𝑑≥1, (𝑞,𝑟) and (𝑞₁,𝑟₁) are (𝑑+2𝛾)/2-admissible pairs and 2<𝑎≤𝑞. If 𝑢 is a solution to the problem 𝑖𝜕_𝑡𝑢(𝑡,𝑥)+△_𝑘𝑢(𝑡,𝑥)=𝑓(𝑡,𝑥),(𝑡,𝑥)∈ℝ×ℝ^𝑑,𝑢_|𝑡=0=𝑢₀, (118) for some data, 𝑢₀, 𝑓, then ‖𝑢‖_𝐿^𝑞,𝑎_(ℝ;𝐿^𝑟_𝑘(ℝ^𝑑₎₎+‖𝑢‖_{𝐶(ℝ;𝐿}²_𝑘(ℝ^𝑑₎₎≤𝐶𝑢_{0𝐿2𝑘(ℝ𝑑)}+𝑓_{𝐿𝑞′1,2(ℝ;𝐿𝑟′1𝑘(ℝ𝑑⋂𝐿))2(ℝ;𝐿𝑘(2𝑑+4𝛾)/(𝑑+2𝛾+2),2(ℝ𝑑))}. (119)

Proof. Let 𝑢 be a solution of (118). We write 𝑢 as 𝑢(𝑡,𝑥)=ℐ_𝑘(𝑡)𝑢₀(𝑥)+^𝑡₀ℐ_𝑘()𝑓()()𝑡−𝜏𝜏,𝑥𝑑𝜏,𝑡,𝑥∈ℝ×ℝ^𝑑. (120) Let 𝑘(𝑡,𝜏)=ℐ_𝑘(𝑡−𝜏), 𝐸=𝐿^𝑟′1_𝑘(ℝ^𝑑) or 𝐿_𝑘^{(2𝑑+4𝛾)/(𝑑+2𝛾+2),2}(ℝ^𝑑), 𝐹=𝐿^𝑟_𝑘(ℝ^𝑑), and ℒ𝑓(𝑡)=∫^∞₀𝑘(𝑡,𝜏)𝑓(𝜏)𝑑𝜏. Then, since 𝑞^′₁≤2<𝑠≤𝑞, in view of Lemma 44, we only have to show that ^∞₀𝑘(𝑡,𝜏)𝑓(𝜏)𝑑𝜏_{𝐿𝑞,𝑠(0,∞;𝐿𝑟𝑘(ℝ𝑑))}≤𝐶𝑓_{𝐿𝑞′1,2(0,∞;𝐿𝑟′1𝑘(ℝ𝑑))∩𝐿2(0,∞;𝐿𝑘(2𝑑+4𝛾)/(𝑑+2𝛾+2),2(ℝ𝑑))}. (121) To show this, observe from (114) and 𝐿^𝑞,𝑠⊂𝐿^𝑞,2 for all 𝑠≥2 that ^∞₀𝑘(𝑡,𝜏)𝑓(𝜏)𝑑𝜏²_{𝐿𝑞,𝑠0,∞;𝐿𝑟𝑘ℝ𝑑}≤𝐶^∞₀ℐ_𝑘(−𝜏)𝑓(𝜏),ℐ_𝑘−𝑦𝑓𝑦𝑑𝜏𝑑𝑦. (122) Then, from the endpoint result of Keel and Tao [26], the right-hand side of (122) is bounded by ‖𝑓‖²_{𝐿2(0,∞;𝐿𝑘(2𝑑+4𝛾)/(𝑑+2𝛾+2),2(ℝ𝑑))}. The remaining part of theorem can be obtained by the duality of Lorentz space (𝐿^𝑞,𝑠)^′=𝐿^{𝑞′,𝑠′} 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: 𝑖𝜕_𝑡𝑢(𝑡,𝑥)+△_𝑘𝑢()|𝑢(𝑡,𝑥=−𝑡,𝑥)|^{4/(𝑑+2𝛾−2)}𝑢(𝑡,𝑥),(𝑡,𝑥)∈ℝ×ℝ^𝑑,𝑢_|𝑡=0=𝑢₀∈𝐻¹_𝑘ℝ^𝑑inℝ^𝑑. (123)

Theorem 47. If the initial data is sufficiently small and 𝐺-invariant, then there exists a unique solution 𝑢∈𝐿^𝑞,𝑠(0,∞; 𝑊^1,𝑟_𝑘,𝐺(ℝ^𝑑))∩𝐿²(0,∞; 𝑊^{1,(2𝑑+4𝛾)/(𝑑+2𝛾−2)}_𝑘,𝐺(ℝ^𝑑))∩𝐶([0,∞); 𝐻¹_𝑘,𝐺(ℝ^𝑑) for every sharp (𝑑+2𝛾)/2-admissible pair (𝑞,𝑟) with 𝑞>2 and 2<𝑠≤𝑞.

Proof. The existence of a unique 𝐻¹_𝑘,𝐺(ℝ^𝑑)-solution is proved in [9], it suffices to prove that 𝑢∈𝐿^𝑞,𝑠(0,∞;𝑊^1,𝑟_𝑘,𝐺(ℝ^𝑑)). From Duhamel's principle, we deduce that 𝑢(𝑡,𝑥)=ℐ_𝑘(𝑡)𝑢₀(𝑥)+^𝑡₀ℐ_𝑘(𝑡−𝜏)|𝑢(𝜏,𝑥)|^{4/(𝑑+2𝛾−2)}𝑢(𝜏,𝑥)𝑑𝜏. (124) Using (114) and (119), we have ‖𝑇𝑢‖_𝐿^𝑞,𝑠_(ℝ;𝐿^𝑟_𝑘(ℝ^𝑑₎₎≤𝐶𝑇𝑢_{0𝐿2𝑘(ℝ𝑑)}+𝑇|𝑢|^{4/(𝑑+2𝛾−2)}𝑢_{𝐿𝑞′1,2(ℝ;𝐿𝑟′1𝑘(ℝ𝑑))}. (125) We can always find an admissible pair (𝑞₀,𝑟₀) with 𝑟₀<𝑑+2𝛾 and 2<𝑠₀<𝑞₀ and (𝑞₁,𝑟₁) and 1<𝑠₁<2 such that 1𝑞₁=4𝑑+2𝛾−2𝑞₀+1𝑞₀,1𝑟₁=4𝑑+2𝛾−2𝑟₁+1𝑟₀,1𝑠₁=4𝑑+2𝛾−2𝑠₀+1𝑠₀, (126) where 𝑟^∗=((𝑑+2𝛾)𝑟₀)/(𝑑+2𝛾−𝑟₀). Thus, from the Leibnitz rule, Hölder's inequality on Lorentz space, and Sobolev embedding, we deduce that ‖𝑇𝑢‖_𝐿^𝑞₀₀^,𝑠_(ℝ;𝐿^𝑟0_𝑘(ℝ^𝑑₎₎≤𝐶𝑇𝑢_{0𝐿2𝑘(ℝ𝑑)}+‖𝑇𝑢‖_𝐿^{(𝑑+2𝛾+2)/(𝑑+2𝛾−2)}_{𝑞00,𝑠ℝ;𝐿𝑟0𝑘ℝ𝑑}. (127) Since ||𝑢₀||_𝐻¹_𝑘(ℝ^𝑑₎ is small, we have ‖𝑇𝑢‖_𝐿^𝑞₀₀^,𝑠_(ℝ;𝐿^𝑟0_𝑘(ℝ^𝑑₎₎≤𝐶𝑇𝑢_{0𝐿2𝑘(ℝ𝑑)}. (128) Finally, since we can choose (𝑞₁,𝑟₁) arbitrarily to be (𝑑+2𝛾)/2-admissible, for any (𝑑+2𝛾)/2-admissible pair (𝑞,𝑟) and 𝑠 with 𝑞>2 and 2<𝑠≤𝑞, we have ‖𝑇𝑢‖_𝐿^𝑞,𝑠_(ℝ;𝐿^𝑟_𝑘(ℝ^𝑑₎₎≤𝐶𝑇𝑢_{0𝐿2𝑘(ℝ𝑑)}+‖𝑇𝑢‖_𝐿^{(𝑑+2𝛾+2)/(𝑑+2𝛾−2)}_{𝑞00,𝑠ℝ;𝐿𝑟0𝑘ℝ𝑑}≤𝐶𝑇𝑢_{0𝐿2𝑘(ℝ𝑑)}+𝑇𝑢_0𝐿^{(𝑑+2𝛾+2)/(𝑑+2𝛾−2)}_2𝑘ℝ𝑑. (129) In a similar way, we can also derive from the smallness of ||𝑢₀||_𝐻¹_𝑘(ℝ^𝑑₎, ‖𝑢‖_𝐿^𝑞,𝑠_(ℝ;𝐿^𝑟_𝑘(ℝ^𝑑₎₎≤𝐶𝑢_{0𝐿2𝑘(ℝ𝑑)}. (130)

7. Embedding Sobolev Theorems and Applications

Theorem 48. Let 𝑠,𝑡>0, 𝑞₁,𝑞₂∈[1,∞] with 𝑞₁≠𝑞₂. Let 𝜃=𝑠/(𝑠+𝑡)∈(0,1), 1/𝑝=(1−𝜃)/𝑞₁+𝜃/𝑞₂, and 𝑟∈[1,∞]. If 𝑓∈ℬ̇_𝑞^𝑠,𝑘_1,𝑟(ℝ^𝑑)∩ℬ̇_𝑞^−𝑡,𝑘_2,𝑟(ℝ^𝑑), then 𝑓∈𝐿_𝑘^𝑝,𝑟(ℝ^𝑑), and one has 𝑓_𝐿𝑘^𝑝,𝑟_(ℝ𝑑)≤𝐶𝑓_̇ℬ^1−𝜃_{𝑞1𝑠,𝑘,𝑟ℝ𝑑}𝑓^𝜃_{̇ℬ𝑞2−𝑡,𝑘,𝑟(ℝ𝑑)}. (131)

Proof. We start picking 𝑝₁,𝑝₂ such that 1≤𝑞₁<𝑝₁<𝑝<𝑝₂<𝑞₂≤∞ with 2/𝑝=1/𝑝₁+1/𝑝₂. We have then 1/𝑝_𝑖=(1−𝑎_𝑖)/𝑞₁+𝑎_𝑖/𝑞₂ with 𝑎_𝑖∈(0,1) and 𝑖=1,2. We write Δ_𝑗𝑓_{𝐿𝑝𝑖𝑘(ℝ𝑑)}≤Δ_𝑗𝑓^1−𝑎𝑖_{𝐿𝑞1𝑘ℝ𝑑}Δ_𝑗𝑓^𝑎𝑖_{𝐿𝑞2𝑘(ℝ𝑑)}. (132) Using Hölder's inequality and by simple calculations, we obtain _𝑗∈ℤ𝜚^{−𝑗𝑟/2}Δ_𝑗𝑓^𝑟_{𝐿𝑝1𝑘(ℝ𝑑)}≤𝑓^(1−𝑎1_̇ℬ^)𝑟_{𝑞1𝑠,𝑘,𝑟ℝ𝑑}𝑓^𝑟𝑎1_{̇ℬ𝑞2−𝑡,𝑘,𝑟ℝ𝑑},_𝑗∈ℤ𝜚^𝑗𝑟/2Δ_𝑗𝑓^𝑟_{𝐿𝑝2𝑘(ℝ𝑑)}≤𝑓^(1−𝑎2_̇ℬ^)𝑟_{𝑞1𝑠,𝑘,𝑟ℝ𝑑}𝑓^𝑟𝑎2_{̇ℬ𝑞2−𝑡,𝑘,𝑟(ℝ𝑑)}, (133) where 𝜚=2^{−2(𝑠(1−𝑎1)−𝑡𝑎1)}>0. From this, and applying Proposition 25, we deduce that if 𝑓∈ℬ̇_𝑞^𝑠,𝑘_1,𝑟(ℝ^𝑑)∩ℬ̇_𝑞^−𝑡,𝑘_2,𝑟(ℝ^𝑑), then 𝑓∈[𝐿^𝑝1_𝑘(ℝ^𝑑),𝐿^𝑝2_𝑘(ℝ^𝑑)]_1/2,𝑟=𝐿_𝑘^𝑝,𝑟(ℝ^𝑑). Furthermore, using (57), we finally have: 𝑓_𝐿𝑘^𝑝,𝑟_(ℝ𝑑)≤𝐶𝑓_̇ℬ^1−𝜃_{𝑞1𝑠,𝑘,𝑟ℝ𝑑}𝑓^𝜃_{̇ℬ𝑞2−𝑡,𝑘,𝑟(ℝ𝑑)}. (134)

Corollary 49. Let 𝑠 be a real number in the interval (0,(𝑑+2𝛾)/𝑞), and let 𝑞 be a real number in [1,∞]. There is a constant 𝐶 such that, for any function 𝑓∈ℬ̇^𝑠,𝑘_𝑞,𝑞(ℝ^𝑑), the following inequality holds: _ℝ𝑑𝑓(𝑥)^𝑞‖𝑥‖^𝑠𝑞𝜔_𝑘(𝑥)𝑑𝑥^1/𝑞≤𝐶𝑓^𝜃_{̇ℬ𝑠,𝑘𝑞,𝑞ℝ𝑑}𝑓_̇ℬ^1−𝜃_{𝑠−(𝑑+2𝛾)/𝑞,𝑘∞,𝑞ℝ𝑑}, (135) where 𝜃=1−𝑞𝑠/(𝑑+2𝛾).

Proof. Let 𝑝∈(1,∞) and 𝑠∈(0,(𝑑+2𝛾)/𝑞) with 1/𝑝=1/𝑞−𝑠/(𝑑+2𝛾). We take 𝑔(𝑥)=1/||𝑥||^𝑠 and apply (103) in the specific form 𝑓𝑔_𝐿𝑘^𝑞,𝑞_(ℝ𝑑)≤𝐶𝑓_𝐿𝑘^𝑝,𝑞_(ℝ𝑑)𝑔_𝐿𝑘^𝑟,∞_(ℝ𝑑), (136) where 𝑟=(𝑑+2𝛾)/𝑠 and 𝑝=(𝑞(𝑑+2𝛾))/(𝑑+2𝛾−𝑞𝑠). As 𝑔∈𝐿_𝑘^𝑟,∞(ℝ^𝑑), we have _ℝ𝑑𝑓(𝑥)^𝑞‖𝑥‖^𝑠𝑞𝜔_𝑘(𝑥)𝑑𝑥^1/𝑞≤𝐶𝑓_𝐿𝑘^𝑝,𝑞_(ℝ𝑑). (137) Combining this with (131), we obtain (135).

Theorem 50. Let 0<𝑠<(𝑑+2𝛾)/2 be given. There exists a positive constant 𝐶 such that for all function 𝑢∈ℋ̇^𝑠_2,𝑘(ℝ^𝑑), one has _ℝ𝑑|𝑢(𝑥)|²‖𝑥‖^2𝑠𝜔_𝑘(𝑥)𝑑𝑥≤𝐶‖𝑢‖²_{̇ℋ𝑠2,𝑘ℝ𝑑}. (138)

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

Lemma 51. Let 𝑠 be a real number in the interval (0,𝛾+𝑑/2). Then, the function 𝑥↦||𝑥||^−2𝑠 belongs to the Dunkl-Besov space ℬ̇^{𝑑+2𝛾−2𝑠,𝑘}_1,∞(ℝ^𝑑).

Proof of Theorem 50. Let us define 𝐼_𝑠,𝑘(𝑢)∶=_ℝ𝑑|𝑢(𝑥)|²‖𝑥‖^2𝑠𝜔_𝑘(𝑥)𝑑𝑥=‖⋅‖^−2𝑠,𝑢². (139) Using homogeneous Littlewood-Paley decomposition and the fact that 𝑢² belongs to 𝒮^′_ℎ,𝑘(ℝ^𝑑), we can write 𝐼_𝑠,𝑘(𝑢)=_{||𝑛−𝑚≤2}Δ_𝑛‖⋅‖^−2𝑠,Δ_𝑚𝑢²≤𝐶_{||𝑛−𝑚≤2}2^{𝑛((𝑑+2𝛾)/2−2𝑠)}×Δ_𝑛‖⋅‖^−2𝑠,2^{−𝑚((𝑑+2𝛾)/2−2𝑠)}Δ_𝑚𝑢². (140) Lemma 51 claims that ‖⋅‖^−2𝑠 belongs to ℬ̇^{(𝑑+2𝛾)/2−2𝑠,𝑘}_2,∞(ℝ^𝑑). Theorem 17 yields 𝑢²_{̇ℬ2𝑠−(𝑑+2𝛾)/2,𝑘2,1(ℝ𝑑)}≤𝐶‖𝑢‖²_{̇ℋ𝑠2,𝑘(ℝ𝑑)}. (141)
Thus, 𝐼_𝑠,𝑘(𝑢)≤𝐶‖𝑢‖²_{̇ℋ𝑠2,𝑘(ℝ𝑑)}. (142)

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

Theorem 52. Let 𝑠,𝑡>0, 𝜃=𝑠/(𝑠+𝑡) and let 𝑞₁,𝑞₂,𝑟₁,𝑟₂∈[1,∞], 𝑝,𝑟₀∈[1,∞) with 1/𝑝=(1−𝜃)/𝑞₁+𝜃/𝑞₂, 1/𝑟₀=(1−𝜃)/𝑟₁+𝜃/𝑟₂.(i)For every 𝑓∈ℬ̇_𝑞^𝑠,𝑘_1,𝑟1(ℝ^𝑑)∩ℬ̇_𝑞^−𝑡,𝑘_2,𝑟2(ℝ^𝑑), and if 𝑟>𝑟₀, one has 𝑓∈𝐿_𝑘^𝑝,𝑟(ℝ^𝑑) and 𝑓_𝐿𝑘^𝑝,𝑟_(ℝ𝑑)≤𝐶𝑓_̇ℬ^1−𝜃_{𝑞11𝑠,𝑘,𝑟ℝ𝑑}𝑓^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑟ℝ𝑑}. (143)(ii)Moreover, this inequality is valid for 𝑟=𝑟₀ in the following cases:(a)𝑟=𝑟₁=𝑟₂,(b)𝑟₁=𝑞₁ and 𝑟₂=𝑞₂,(c)1<𝑝≤2 and 𝑟₀=𝑝.(iii)Finally, the condition 𝑟≥𝑟₀ is sharp.

Proof. (i) Case 𝑟>𝑟₀. With no loss of generality, we may assume that 𝑞₁<𝑞₂, and we fix 𝜀>0 such that 1𝑞₂<1𝑝1−𝜀𝑞₁−1𝑞₂=1𝑝₂<1𝑝1+𝜀𝑞₁−1𝑞₂=1𝑝₁<1𝑞₁. (144) The proof follows essentially the same ideas used in the previous theorem. Indeed, we have for 𝑀_𝑗=2^𝑗𝑠‖Δ_𝑗𝑓‖_{𝐿𝑞1𝑘(ℝ𝑑)} and 𝑁_𝑗=2^−𝑗𝑡‖Δ_𝑗𝑓‖_{𝐿𝑞2𝑘(ℝ𝑑)} and for 𝜀₀=1 and 𝜀₁=−1, Δ_𝑗𝑓_{𝐿𝑝𝑖𝑘(ℝ𝑑)}≤Δ_𝑗𝑓^{1−𝜃+𝜀𝜀𝑖}_{𝐿𝑞1𝑘ℝ𝑑}Δ_𝑗𝑓^{𝜃−𝜀𝜀𝑖}_{𝐿𝑞2𝑘ℝ𝑑}=𝑀^{1−𝜃+𝜀𝜀𝑖}_𝑗𝑁^{𝜃−𝜀𝜀𝑖}_𝑗2^{−𝑗𝜀𝜀𝑖(𝑠+𝑡)}. (145) As 𝑟₁≠𝑟₂, we can only say that (𝑀^{1−𝜃+𝜀𝜀𝑖}_𝑗𝑁^{𝜃−𝜀𝜀𝑖}_𝑗)_𝑗∈ℤ∈𝑙^𝜚𝑖, where 1/𝜚_𝑖=(1−𝜃+𝜀𝜀_𝑖)/𝑟₁+(𝜃−𝜀𝜀_𝑖)/𝑟₂. We may use (57), but we get only that 𝑓∈𝐿_𝑘^𝑝,𝜚(ℝ^𝑑)=[𝐿^𝑝1_𝑘(ℝ^𝑑),𝐿^𝑝2_𝑘(ℝ^𝑑)]_1/2,𝜚 with 𝜚=max(𝜚₁,𝜚₂) and that satisfies (143) with 𝑟=𝜚. However, we may choose 𝜀 as small as we want and thus, 𝜚 as close to 𝑟₀ as we want; thus 𝑓 satisfies (143) for every 𝑟>𝑟₀.
(ii) Case 𝑟=𝑟₀.
(a) If 𝑟=𝑟₁=𝑟₂: this case was treated in Theorem 48.
(b) If 𝑟₁=𝑞₁ and 𝑟₂=𝑞₂: this is a direct consequence of (43), since we have 𝑓_{̇ℬ𝑞𝑖𝑖𝑠,𝑘,𝑞(ℝ𝑑)}=𝑓_{̇ℱ𝑞𝑖𝑖𝑠,𝑘,𝑞(ℝ𝑑)}≤𝐶𝑓_{̇ℱ𝑞𝑖𝑠,𝑘,∞(ℝ𝑑)},𝑓_{̇ℬ𝑞𝑖𝑖−𝑡,𝑘,𝑞(ℝ𝑑)}=𝑓_{̇ℱ𝑞𝑖𝑖−𝑡,𝑘,𝑞(ℝ𝑑)}≤𝐶𝑓_{̇ℱ𝑞𝑖−𝑡,𝑘,∞(ℝ𝑑)}, (146)
we obtain 𝑓_𝐿^𝑝_{𝑘(ℝ𝑑)}≤𝐶𝑓_̇ℬ^1−𝜃_{𝑞11𝑠,𝑘,𝑞ℝ𝑑}𝑓^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑞ℝ𝑑}. (147)
(c) Case 1<𝑝≤2 and 𝑟₀=𝑝.
We just write Δ_𝑗𝑓_{𝐿𝑝𝑘(ℝ𝑑)}≤Δ_𝑗𝑓_𝐿^1−𝜃_{𝑞1𝑘ℝ𝑑}Δ_𝑗𝑓^𝜃_{𝐿𝑞2𝑘ℝ𝑑}=2^𝑗𝑠Δ_𝑗𝑓_{𝐿𝑞1𝑘(ℝ𝑑)}^1−𝜃2^−𝑗𝑡Δ_𝑗𝑓_{𝐿𝑞2𝑘(ℝ𝑑)}^𝜃 (148) and get by Hölder's inequality: 𝑓_{̇ℬ0,𝑘𝑝,𝑝(ℝ𝑑)}≤𝐶𝑓_̇ℬ^1−𝜃_{𝑞11𝑠,𝑘,𝑟ℝ𝑑}𝑓^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑟ℝ𝑑}. (149) We then use the embedding ℬ̇^0,𝑘_𝑝,𝑝(ℝ^𝑑)⊂𝐿^𝑝_𝑘(ℝ^𝑑)=𝐿_𝑘^𝑝,𝑝(ℝ^𝑑) which is valid for 𝑝≤2.

Theorem 53. Let 𝑠,𝑡>0, let 𝑞₁,𝑞₂∈[1,∞] with 𝑞₁<𝑞₂. Let 𝜃=𝑠/(𝑠+𝑡)∈(0,1), and let 1/𝑝=(1−𝜃)/𝑞₁+𝜃/𝑞₂.(i)If 𝑞₁≤𝑟₁≤𝑞₂ and let 1/𝑟=(1−𝜃)/𝑟₁+𝜃/𝑞₂. Then, one has 𝑓_𝐿𝑘^𝑝,𝑟_(ℝ𝑑)≤𝐶𝑓_̇ℬ^1−𝜃_{𝑞11𝑠,𝑘,𝑟ℝ𝑑}𝑓^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑞ℝ𝑑}. (150)(ii)If 𝑞₁≤𝑟₂≤𝑞₂, and let 1/𝑟=(1−𝜃)/𝑞₁+𝜃/𝑟₂. Then, one has 𝑓_𝐿𝑘^𝑝,𝑟_(ℝ𝑑)≤𝐶𝑓_̇ℬ^1−𝜃_{𝑞11𝑠,𝑘,𝑞ℝ𝑑}𝑓^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑟ℝ𝑑}. (151)

Proof. We only prove the first inequality, as the proof for the second one is similar. Since 𝑓∈ℬ̇_𝑞^𝑠,𝑘_1,𝑟1(ℝ^𝑑), noting that 𝜆_𝑗=2^𝑠𝑗‖Δ_𝑗𝑓‖_{𝐿𝑞1𝑘(ℝ𝑑)}, we have (𝜆_𝑗)_𝑗∈ℤ∈𝑙^𝑟1. Thus, using Proposition 26 (i) for the interpolation 𝑙^𝑟1=𝑙^𝑞1,𝑙^𝑞2_𝑎,𝑟, (152) with 1/𝑟₁=(1−𝑎)/𝑞₁+𝑎/𝑞₂, we see that we have a partition ℤ=∑_𝑗∈ℤ𝑍_𝑗 such that 2^−𝑎𝑗_{𝑛∈𝑍𝑗}𝜆^𝑞1_𝑛^1/𝑞1_𝑙𝑟1+2^{(1−𝑎)𝑗}_{𝑛∈𝑍𝑗}𝜆^𝑞2_𝑛^1/𝑞2_𝑙𝑟1≤𝐶𝜆_{𝑗𝑙𝑟1}. (153) Moreover, since 𝑓∈ℬ̇_𝑞^−𝑡,𝑘_2,𝑞2(ℝ^𝑑), we have _{𝑗∈𝑍𝑛}2^{−𝑗𝑞2𝑡}Δ_𝑗𝑓^𝑞2_{𝐿𝑞2𝑘ℝ𝑑}^1/𝑞2_𝑛∈ℤ∈𝑙^𝑞2. (154) Let us note that 𝑀_𝑛=(∑_{𝑗∈𝑍𝑛}2^{−𝑗𝑞2𝑡}‖Δ_𝑗𝑓‖^𝑞2_{𝐿𝑞2𝑘(ℝ𝑑)})^1/𝑞2, 𝑁_𝑛=2^−𝑎𝑛(∑_{𝑗∈𝑍𝑛}𝜆^𝑞1_𝑗)^1/𝑞1, 𝐿_𝑛=2^{(1−𝑎)𝑛}(∑_{𝑗∈𝑍𝑛}𝜆^𝑞2_𝑗)^1/𝑞2, and 𝑓_𝑛=∑_{𝑗∈𝑍𝑛}Δ_𝑗𝑓. We apply now (147) and Theorem 48 to obtain 𝑓_𝑛𝐿^𝑝_{𝑘(ℝ𝑑)}≤𝐶𝑓_𝑛̇ℬ^1−𝜃_{𝑞11𝑠,𝑘,𝑞ℝ𝑑}𝑓_𝑛^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑞ℝ𝑑}≤𝐶𝑁_𝑛^1−𝜃𝑀^𝜃_𝑛2^{𝑛𝑎(1−𝜃)},𝑓_{𝑛𝐿2𝑘}^𝑝,𝑞_(ℝ𝑑)≤𝐶𝑓_𝑛̇ℬ^1−𝜃_{𝑞12𝑠,𝑘,𝑞ℝ𝑑}𝑓_𝑛^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑞ℝ𝑑}≤𝐶𝐿_𝑛^1−𝜃𝑀^𝜃_𝑛2^{−𝑛(1−𝑎)(1−𝜃)}. (155) Since we have 𝑓=∑_𝑛∈ℤ𝑓_𝑛, with these two inequalities at hand, and using (57), we find that 𝑓∈[𝐿^𝑝_𝑘(ℝ^𝑑),𝐿^𝑝,𝑞2_𝑘(ℝ^𝑑)]_𝑎,𝑟, with 1/𝑟=(1−𝑎)/𝑝+𝑎/𝑞₂, but, since 1/𝑟₁=(1−𝑎)/𝑞₁+𝑎/𝑞₂ and 1/𝑝=(1−𝜃)/𝑞₁+𝜃/𝑞₂, we obtain [𝐿^𝑝_𝑘(ℝ^𝑑),𝐿^𝑝,𝑞2_𝑘(ℝ^𝑑)]_𝑎,𝑟=𝐿^𝑟_𝑘(ℝ^𝑑) with 1/𝑟=(1−𝜃)/𝑟₁+𝜃/𝑞₂.

Theorem 54. Let 𝑠,𝑡>0, and let 𝑞₁,𝑞₂∈[1,∞] with 𝑞₁<𝑞₂. Let 𝜃=𝑠/(𝑠+𝑡)∈(0,1), and let 1/𝑝=(1−𝜃)/𝑞₁+𝜃/𝑞₂. Let 𝑞₁≤𝑟₁≤𝑟₂≤𝑞₂, and let 1/𝑟=(1−𝜃)/𝑟₁+𝜃/𝑟₂. Then, one has 𝑓_𝐿𝑘^𝑝,𝑟_(ℝ𝑑)≤𝐶𝑓_̇ℬ^1−𝜃_{𝑞11𝑠,𝑘,𝑟ℝ𝑑}𝑓^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑟ℝ𝑑}. (156)

Proof. Once the previous theorem is proved, it is enough to reapply similar arguments to obtain Theorem 54. As 𝑞₁<𝑟₁<𝑟₂<𝑞₂, we start using 𝑙^𝑟1=𝑙^𝑞1,𝑙^𝑟2_𝑎,𝑟1 (157) instead of (152), and we obtain a partition ℤ=∑_𝑗∈ℤ𝑍_𝑗 such that 2^−𝑎𝑗_{𝑛∈𝑍𝑗}𝜆^𝑞1_𝑛^1/𝑞1_𝑙𝑟1+2^{(1−𝑎)𝑗}_{𝑛∈𝑍𝑗}𝜆^𝑟2_𝑛^1/𝑟2_𝑙𝑟1≤𝐶𝜆_{𝑗𝑙𝑟1}, (158) with 1/𝑟₁=(1−𝑎)/𝑞₁+𝑎/𝑟₂ and where 𝜆_𝑗=2^𝑠𝑗‖Δ_𝑗𝑓‖_{𝐿𝑞1𝑘(ℝ𝑑)} belongs to 𝑙^𝑟1, since 𝑓∈ℬ̇_𝑞^𝑠,𝑘_1,𝑟1(ℝ^𝑑). Moreover, since 𝑓∈ℬ̇_𝑞^−𝑡,𝑘_2,𝑟2(ℝ^𝑑), we have _{𝑗∈𝑍𝑛}2^{−𝑗𝑞2𝑡}Δ_𝑗𝑓^𝑞2_{𝐿𝑞2𝑘ℝ𝑑}^1/𝑞2_𝑛∈ℤ∈𝑙^𝑞2. (159) Let us note that 𝑀_𝑛=(∑_{𝑗∈𝑍𝑛}2^{−𝑗𝑞2𝑡}‖Δ_𝑗𝑓‖^𝑞2_{𝐿𝑞2𝑘(ℝ𝑑)})^1/𝑞2, 𝑁_𝑛=2^−𝑎𝑛(∑_{𝑗∈𝑍𝑛}𝜆^𝑞1_𝑗)^1/𝑞1, 𝐿_𝑛=2^{(1−𝑎)𝑛}(∑_{𝑗∈𝑍𝑛}𝜆^𝑞2_𝑗)^1/𝑞2, and 𝑓_𝑛=∑_{𝑗∈𝑍𝑛}Δ_𝑗𝑓. We apply now (151) and Theorem 48 instead of (155) to obtain 𝑓_𝑛𝐿𝑘^𝑝,𝑏_(ℝ𝑑)≤𝐶𝑓_𝑛̇ℬ^1−𝜃_{𝑞11𝑠,𝑘,𝑞ℝ𝑑}𝑓_𝑛^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑟ℝ𝑑}≤𝐶𝑁_𝑛^1−𝜃𝑀^𝜃_𝑛2^{𝑛𝑎(1−𝜃)}, (160) where 1/𝑏=(1−𝜃)/𝑞₁+𝜃/𝑟₂ and 𝑓_{𝑛𝐿2𝑘}^𝑝,𝑟_(ℝ𝑑)≤𝐶𝑓_𝑛̇ℬ^1−𝜃_{𝑞12𝑠,𝑘,𝑟ℝ𝑑}𝑓_𝑛^𝜃_{̇ℬ𝑞22−𝑡,𝑘,𝑟ℝ𝑑}≤𝐶𝑁_𝑛^1−𝜃𝑀^𝜃_𝑛2^{−𝑛(1−𝑎)(1−𝜃)}. (161) Finally, we have via (57) that 𝑓∈[𝐿_𝑘^𝑝,𝑏(ℝ^𝑑),𝐿^𝑝,𝑟2_𝑘(ℝ^𝑑)]_𝑎,𝑟, with 1/𝑟=(1−𝑎)/𝑏+𝑎/𝑟₂. To conclude, we use the fact that 1/𝑏=(1−𝜃)/𝑞₁+𝜃/𝑟₂ and 1/𝑟₁=(1−𝑎)/𝑞₁+𝑎/𝑟₂ in order to obtain that 𝑓∈𝐿_𝑘^𝑝,𝑟(ℝ^𝑑) with 1/𝑟=(1−𝜃)/𝑟₁+𝜃/𝑟₂.

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

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.