# Generalized Lorentz Spaces and Applications

Authors: Hatem Mejjaoli
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We define and study the Lorentz spaces associated with the Dunkl operators onℝd. 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.


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## Body

## 1. Introduction

Dunkl operatorsTj(j=1,…,d) introduced by Dunkl in [1] are parameterized differential-difference operators on ℝd 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(1)i∂tu+△ku=f(t,x),(t,x)∈(0,∞)×ℝdu|t=0=g,
where △k=∑j=1d‍Tj2 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 Section2, 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

Letℝd be the Euclidean space equipped with a scalar product 〈,〉, and let ||x||=〈x,x〉. For α in ℝd∖{0}, σα denotes the reflection in the hyperplane Hα⊂ℝd perpendicular to α, that is, for x∈ℝd, σα(x)=x-2∥α∥-2〈α,x〉α. A finite set R⊂ℝd∖{0} is called a root system if R∩ℝα={±α} and σαR=R for all α∈R. We normalize each α∈R as 〈α,α〉=2. We fix a β∈ℝd∖∪α∈RHα and define a positive root system R+ of R as R+={α∈R∣〈α,β〉>0}. The reflections σα, 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
(2)γ=γ(k)=∑α∈R+‍k(α).
Throughout this paper, we will assume that k(α)≥0 for all α∈R. We denote by ωk the weight function on ℝd given by
(3)ωk(x)=∏α∈R+|〈α,x〉|2k(α),
which is invariant and homogeneous of degree 2γ. In the case that the reflection group G is the group ℤ2d of sign changes, the weight function ωk is a product function of the form ∏j=1d‍|xj|kj, kj≥0. We denote by ck the Mehta-type constant defined by
(4)ck=∫ℝd‍e-||x||2/2ωk(x)dx.
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(ℝd):
the space of continuous functions onℝd.Cp(ℝd):
the space of functions of classCp on ℝd.Cbp(ℝd):
the space of bounded functions of classCp.ℰ(ℝd):
the space ofC∞ functions on ℝd.𝒮(ℝd):
the Schwartz space of rapidly decreasing functions onℝd.D(ℝd):
the space ofC∞ functions on ℝd which are of compact support.𝒮′(ℝd):
the space of temperate distributions onℝd.


### 2.2. The Dunkl Operators

Letk:R→ℂ be a multiplicity function on R and R+ a fixed positive root system of R. Then, the Dunkl operators Tj, 1≤j≤d, are defined on C1(ℝd) by
(5)Tjf(x)=∂∂xjf(x)+∑α∈R+‍k(α)αjf(x)-f(σα(x))〈α,x〉,
where α=(α1,α2,…,αd). Similarly, as ordinary derivatives, each Tj satisfies for all f, g in C1(ℝd) and at least one of them is G-invariant,
(6)Tj(fg)=(Tjf)g+f(Tjg),
and for all f in Cb1(ℝd) and g in 𝒮(ℝd),
(7)∫ℝd‍Tjf(x)g(x)ωk(x)dx=-∫ℝd‍f(x)Tjg(x)ωk(x)dx.
Furthermore, according to [1, 14], the Dunkl operators Tj, 1≤j≤d commute, and there exists the so-called Dunkl's intertwining operator Vk such that TjVk=Vk(∂/∂xj) for 1≤j≤d and Vk(1)=1. We define the Dunkl-Laplace operator △k on ℝd by
(8)△kf(x):=∑j=1d‍Tj2f(x)Δkf(x):=△f(x)+2∑α∈R+‍k(α)(〈∇f(x),α〉〈α,x〉=Δf(x)+2∑α∈R+‍k(α)-f(x)-f(σα(x))〈α,x〉2),
where △ and ∇ are the usual Euclidean Laplacian and nabla operators on ℝd, respectively. Since the Dunkl operators commute, their joint eigenvalue problem is significant, and for each y∈ℝd, the system
(9)Tju(x,y)=yju(x,y),j=1,…,d,u(0,y)=1
admits a unique analytic solution K(x,y), x∈ℝd, called the Dunkl kernel, which has a holomorphic extension to ℂd×ℂd. For x, y∈ℂd, 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 functionsf on ℝd, we define Lp-norms of f with respect to ωk(x)dx as
(10)∥f∥Lkp(ℝd)=(∫ℝd‍|f(x)|pωk(x)dx)1/p,
if 1≤p<∞ and ∥f∥Lk∞(ℝd)=esssupx∈ℝd|f(x)|. We denote by Lkp(ℝd) the space of all measurable functions f on ℝd with finite Lkp-norm.The Dunkl transformℱD on Lk1(ℝd) is given by
(11)ℱD(f)(y)=1ck∫ℝd‍f(x)K(x,-iy)ωk(x)dx.
Some basic properties are the following (cf. [5, 6]).For allf∈Lk1(ℝd),(a)
∥ℱD(f)∥Lk∞(ℝd)≤ck-1∥f∥Lk1(ℝd),(b)
ℱD(f(·/λ))(y)=λ2γ+dℱD(f)(λy) for λ>0,(c)
ifℱD(f) belongs to Lk1(ℝd), then
(12)f(y)=1ck∫ℝd‍ℱD(f)(x)K(ix,y)ωk(x)dx,a.e.,
and moreover, for all f∈𝒮(ℝd),(d)
ℱD(Tjf)(y)=iyjℱD(f)(y),(e)
if we defineℱD¯(f)(y)=ℱD(f)(-y), then
(13)ℱDℱD¯=ℱD¯ℱD=Id.Proposition 1.
The Dunkl transformℱD is a topological isomorphism from 𝒮(ℝd) onto itself, and for all f in 𝒮(ℝd),
(14)∫ℝd‍|f(x)|2ωk(x)dx=∫ℝd‍|ℱD(f)(ξ)|2ωk(ξ)dξ.
In particular, the Dunkl transform f→ℱD(f) can be uniquely extended to an isometric isomorphism on Lk2(ℝd).We define the tempered distribution𝒯f associated with f∈Lkp(ℝd) by
(15)〈𝒯f,ϕ〉=∫ℝd‍f(x)ϕ(x)ωk(x)dx,
for ϕ∈𝒮(ℝd) and denote by 〈f,ϕ〉k the integral in the right hand side.Definition 2.
The Dunkl transformℱD(τ) of a distribution τ∈𝒮'(ℝd) is defined by
(16)〈ℱD(τ),ϕ〉=〈τ,ℱD(ϕ)〉,
for ϕ∈𝒮(ℝd).In particular, forf∈Lkp(ℝd), it follows that for ϕ∈𝒮(ℝd),
(17)〈ℱD(f),ϕ〉=〈ℱD(𝒯f),ϕ〉=〈𝒯f,ℱD(ϕ)〉〈ℱD(f),ϕ〉=〈f,ℱD(ϕ)〉k.Proposition 3.
The Dunkl transformℱD is a topological isomorphism from 𝒮'(ℝd) onto itself.


### 2.4. The Dunkl Convolution

By using the Dunkl kernel in Section2.2, we introduce a generalized translation and a convolution structure in our Dunkl setting. For a function f∈𝒮(ℝd) and y∈ℝd, the Dunkl translation τyf is defined by
(18)τyf(x)=1ck∫ℝd‍ℱD(f)(z)K(ix,z)K(iy,z)ωk(z)dz.
Clearly, τyf(x)=τxf(y), and by using the Dunkl's intertwining operator Vk, τyf is related to the usual translation as τyf(x)=(Vk)x(Vk)y((Vk)-1(f)(x+y)) (cf. [11, 17]). Hence, τy can also be defined for f∈ℰ(ℝd). We define the Dunkl convolution product f*Dg of functions f, g∈𝒮(ℝd) as follows:
(19)f*Dg(x)=∫ℝd‍τxf(-y)g(y)ωk(y)dy.
This convolution is commutative and associative (cf. [17]).SinceℱD(τyf)(x)=K(ix,y)ℱD(f)(x) by the previous definition of τyf, it follows that(a)
for allf, g∈D(ℝd) (resp., 𝒮(ℝd)), f*Dg belongs to D(ℝd) (resp. 𝒮(ℝd)) and
(20)ℱD(f*Dg)(y)=ℱD(f)(y)ℱD(g)(y). Moreover, as pointed in [16] and Sections 4 and 7, the operator f→f*Dg is bounded on Lkp(ℝd), 1≤p≤∞, provided that g is a radial function in Lk1(ℝd) or an arbitrary function in Lk1(ℝd) for G=ℤ2d. Hence, the standard argument yields the following Young's inequality.(b)
Let1≤p,q,r≤∞ such that 1/p+1/q-1/r=1. Assume that f∈Lkp(ℝd) and g∈Lkq(ℝd). If ∥τxg∥Lkq(ℝd)≤C∥g∥Lkq(ℝd) for all x∈ℝd, then f*Dg∈Lkr(ℝd) and
(21)∥f*Dg∥Lkr(ℝd)≤C∥f∥Lkp(ℝd)∥g∥Lkq(ℝd).Definition 4.
The Dunkl convolution product of a distributionS in 𝒮'(ℝd) and a function ϕ in 𝒮(ℝd) is the function S*Dϕ defined by
(22)S*Dϕ(x)=〈Sy,τ-yϕ(x)〉.Proposition 5.
Letf be in Lkp(ℝd), 1≤p≤∞, and ϕ in 𝒮(ℝd). Then, the distribution 𝒯f*Dϕ is given by the function f*Dϕ. If one assumes that ϕ is arbitrary for d=1 and radial for d≥2, then 𝒯f*Dϕ belongs to Lkp(ℝd). Moreover, for all ψ∈𝒮(ℝd),
(23)〈𝒯f*Dϕ,ψ〉=〈f,ˇϕ*Dψˇ〉k,
where ψˇ(x)=ψ(-x), and
(24)ℱD(𝒯f*Dϕ)=ℱD(𝒯f)ℱD(ϕ).For eachu∈𝒮'(ℝd), we define the distributions Tju,1≤j≤d, by
(25)〈Tju,ψ〉=-〈u,Tjψ〉,
for all ψ∈𝒮(ℝd). Then, 〈△ku,ψ〉=〈u,△kψ〉, and these distributions satisfy the following properties (see Section 2.3 (d)):
(26)ℱD(Tju)=iyjℱD(u),ℱD(△ku)=-∥y∥2ℱD(u).
In the following, we denote 𝒯f given by (15) by f for simplicity.


## 2.1. Root System, Reflection Group and Multiplicity Function

Letℝd be the Euclidean space equipped with a scalar product 〈,〉, and let ||x||=〈x,x〉. For α in ℝd∖{0}, σα denotes the reflection in the hyperplane Hα⊂ℝd perpendicular to α, that is, for x∈ℝd, σα(x)=x-2∥α∥-2〈α,x〉α. A finite set R⊂ℝd∖{0} is called a root system if R∩ℝα={±α} and σαR=R for all α∈R. We normalize each α∈R as 〈α,α〉=2. We fix a β∈ℝd∖∪α∈RHα and define a positive root system R+ of R as R+={α∈R∣〈α,β〉>0}. The reflections σα, 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
(2)γ=γ(k)=∑α∈R+‍k(α).
Throughout this paper, we will assume that k(α)≥0 for all α∈R. We denote by ωk the weight function on ℝd given by
(3)ωk(x)=∏α∈R+|〈α,x〉|2k(α),
which is invariant and homogeneous of degree 2γ. In the case that the reflection group G is the group ℤ2d of sign changes, the weight function ωk is a product function of the form ∏j=1d‍|xj|kj, kj≥0. We denote by ck the Mehta-type constant defined by
(4)ck=∫ℝd‍e-||x||2/2ωk(x)dx.
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(ℝd):
the space of continuous functions onℝd.Cp(ℝd):
the space of functions of classCp on ℝd.Cbp(ℝd):
the space of bounded functions of classCp.ℰ(ℝd):
the space ofC∞ functions on ℝd.𝒮(ℝd):
the Schwartz space of rapidly decreasing functions onℝd.D(ℝd):
the space ofC∞ functions on ℝd which are of compact support.𝒮′(ℝd):
the space of temperate distributions onℝd.


## 2.2. The Dunkl Operators

Letk:R→ℂ be a multiplicity function on R and R+ a fixed positive root system of R. Then, the Dunkl operators Tj, 1≤j≤d, are defined on C1(ℝd) by
(5)Tjf(x)=∂∂xjf(x)+∑α∈R+‍k(α)αjf(x)-f(σα(x))〈α,x〉,
where α=(α1,α2,…,αd). Similarly, as ordinary derivatives, each Tj satisfies for all f, g in C1(ℝd) and at least one of them is G-invariant,
(6)Tj(fg)=(Tjf)g+f(Tjg),
and for all f in Cb1(ℝd) and g in 𝒮(ℝd),
(7)∫ℝd‍Tjf(x)g(x)ωk(x)dx=-∫ℝd‍f(x)Tjg(x)ωk(x)dx.
Furthermore, according to [1, 14], the Dunkl operators Tj, 1≤j≤d commute, and there exists the so-called Dunkl's intertwining operator Vk such that TjVk=Vk(∂/∂xj) for 1≤j≤d and Vk(1)=1. We define the Dunkl-Laplace operator △k on ℝd by
(8)△kf(x):=∑j=1d‍Tj2f(x)Δkf(x):=△f(x)+2∑α∈R+‍k(α)(〈∇f(x),α〉〈α,x〉=Δf(x)+2∑α∈R+‍k(α)-f(x)-f(σα(x))〈α,x〉2),
where △ and ∇ are the usual Euclidean Laplacian and nabla operators on ℝd, respectively. Since the Dunkl operators commute, their joint eigenvalue problem is significant, and for each y∈ℝd, the system
(9)Tju(x,y)=yju(x,y),j=1,…,d,u(0,y)=1
admits a unique analytic solution K(x,y), x∈ℝd, called the Dunkl kernel, which has a holomorphic extension to ℂd×ℂd. For x, y∈ℂd, 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 functionsf on ℝd, we define Lp-norms of f with respect to ωk(x)dx as
(10)∥f∥Lkp(ℝd)=(∫ℝd‍|f(x)|pωk(x)dx)1/p,
if 1≤p<∞ and ∥f∥Lk∞(ℝd)=esssupx∈ℝd|f(x)|. We denote by Lkp(ℝd) the space of all measurable functions f on ℝd with finite Lkp-norm.The Dunkl transformℱD on Lk1(ℝd) is given by
(11)ℱD(f)(y)=1ck∫ℝd‍f(x)K(x,-iy)ωk(x)dx.
Some basic properties are the following (cf. [5, 6]).For allf∈Lk1(ℝd),(a)
∥ℱD(f)∥Lk∞(ℝd)≤ck-1∥f∥Lk1(ℝd),(b)
ℱD(f(·/λ))(y)=λ2γ+dℱD(f)(λy) for λ>0,(c)
ifℱD(f) belongs to Lk1(ℝd), then
(12)f(y)=1ck∫ℝd‍ℱD(f)(x)K(ix,y)ωk(x)dx,a.e.,
and moreover, for all f∈𝒮(ℝd),(d)
ℱD(Tjf)(y)=iyjℱD(f)(y),(e)
if we defineℱD¯(f)(y)=ℱD(f)(-y), then
(13)ℱDℱD¯=ℱD¯ℱD=Id.Proposition 1.
The Dunkl transformℱD is a topological isomorphism from 𝒮(ℝd) onto itself, and for all f in 𝒮(ℝd),
(14)∫ℝd‍|f(x)|2ωk(x)dx=∫ℝd‍|ℱD(f)(ξ)|2ωk(ξ)dξ.
In particular, the Dunkl transform f→ℱD(f) can be uniquely extended to an isometric isomorphism on Lk2(ℝd).We define the tempered distribution𝒯f associated with f∈Lkp(ℝd) by
(15)〈𝒯f,ϕ〉=∫ℝd‍f(x)ϕ(x)ωk(x)dx,
for ϕ∈𝒮(ℝd) and denote by 〈f,ϕ〉k the integral in the right hand side.Definition 2.
The Dunkl transformℱD(τ) of a distribution τ∈𝒮'(ℝd) is defined by
(16)〈ℱD(τ),ϕ〉=〈τ,ℱD(ϕ)〉,
for ϕ∈𝒮(ℝd).In particular, forf∈Lkp(ℝd), it follows that for ϕ∈𝒮(ℝd),
(17)〈ℱD(f),ϕ〉=〈ℱD(𝒯f),ϕ〉=〈𝒯f,ℱD(ϕ)〉〈ℱD(f),ϕ〉=〈f,ℱD(ϕ)〉k.Proposition 3.
The Dunkl transformℱD is a topological isomorphism from 𝒮'(ℝd) onto itself.


## 2.4. The Dunkl Convolution

By using the Dunkl kernel in Section2.2, we introduce a generalized translation and a convolution structure in our Dunkl setting. For a function f∈𝒮(ℝd) and y∈ℝd, the Dunkl translation τyf is defined by
(18)τyf(x)=1ck∫ℝd‍ℱD(f)(z)K(ix,z)K(iy,z)ωk(z)dz.
Clearly, τyf(x)=τxf(y), and by using the Dunkl's intertwining operator Vk, τyf is related to the usual translation as τyf(x)=(Vk)x(Vk)y((Vk)-1(f)(x+y)) (cf. [11, 17]). Hence, τy can also be defined for f∈ℰ(ℝd). We define the Dunkl convolution product f*Dg of functions f, g∈𝒮(ℝd) as follows:
(19)f*Dg(x)=∫ℝd‍τxf(-y)g(y)ωk(y)dy.
This convolution is commutative and associative (cf. [17]).SinceℱD(τyf)(x)=K(ix,y)ℱD(f)(x) by the previous definition of τyf, it follows that(a)
for allf, g∈D(ℝd) (resp., 𝒮(ℝd)), f*Dg belongs to D(ℝd) (resp. 𝒮(ℝd)) and
(20)ℱD(f*Dg)(y)=ℱD(f)(y)ℱD(g)(y). Moreover, as pointed in [16] and Sections 4 and 7, the operator f→f*Dg is bounded on Lkp(ℝd), 1≤p≤∞, provided that g is a radial function in Lk1(ℝd) or an arbitrary function in Lk1(ℝd) for G=ℤ2d. Hence, the standard argument yields the following Young's inequality.(b)
Let1≤p,q,r≤∞ such that 1/p+1/q-1/r=1. Assume that f∈Lkp(ℝd) and g∈Lkq(ℝd). If ∥τxg∥Lkq(ℝd)≤C∥g∥Lkq(ℝd) for all x∈ℝd, then f*Dg∈Lkr(ℝd) and
(21)∥f*Dg∥Lkr(ℝd)≤C∥f∥Lkp(ℝd)∥g∥Lkq(ℝd).Definition 4.
The Dunkl convolution product of a distributionS in 𝒮'(ℝd) and a function ϕ in 𝒮(ℝd) is the function S*Dϕ defined by
(22)S*Dϕ(x)=〈Sy,τ-yϕ(x)〉.Proposition 5.
Letf be in Lkp(ℝd), 1≤p≤∞, and ϕ in 𝒮(ℝd). Then, the distribution 𝒯f*Dϕ is given by the function f*Dϕ. If one assumes that ϕ is arbitrary for d=1 and radial for d≥2, then 𝒯f*Dϕ belongs to Lkp(ℝd). Moreover, for all ψ∈𝒮(ℝd),
(23)〈𝒯f*Dϕ,ψ〉=〈f,ˇϕ*Dψˇ〉k,
where ψˇ(x)=ψ(-x), and
(24)ℱD(𝒯f*Dϕ)=ℱD(𝒯f)ℱD(ϕ).For eachu∈𝒮'(ℝd), we define the distributions Tju,1≤j≤d, by
(25)〈Tju,ψ〉=-〈u,Tjψ〉,
for all ψ∈𝒮(ℝd). Then, 〈△ku,ψ〉=〈u,△kψ〉, and these distributions satisfy the following properties (see Section 2.3 (d)):
(26)ℱD(Tju)=iyjℱD(u),ℱD(△ku)=-∥y∥2ℱD(u).
In the following, we denote 𝒯f given by (15) by f for simplicity.


## 3.ℬ˙p,qs,k, ℱ˙p,qs,k(ℝd), and ℋ˙p,ks 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 by𝒞 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(ℝd) such that
(27)suppψ⊂B(0,1),suppφ⊂𝒞∀ξ∈ℝd,ψ(ξ)+∑j≥0‍φ(2-jξ)=1∀ξ∈𝒞,∑j∈ℤ‍φ(2-jξ)=1|n-m|≥2⟹suppφ(2-n·)∩suppφ(2-m·)=∅j≥1⟹suppψ∩suppφ(2-j·)=∅.Notations.
We denote by(28)Δjf=ℱD-1(φ(ξ2j)ℱD(f)),Sjf=∑n≤j-1‍Δnf,∀j∈ℤ.
The distribution Δjf is called the jth dyadic block of the homogeneous Littlewood-Paley decomposition of f associated with the Dunkl operators.Throughout this paper, we defineϕ and χ by ϕ=ℱD-1(φ) and χ=ℱD-1(ψ).When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functionsψ~ and φ~ belonging to D(ℝd) such that ψ~≡1 on suppψ and φ~≡1 on suppφ.Remark 7.
We remark that(29)ℱD(Sjf)(ξ)=ψ~(ξ2j)ℱD(Sjf)(ξ),ℱD(Δjf)(ξ)=φ~(ξ2j)ℱD(Δjf)(ξ).We put(30)ϕ~=ℱD-1(φ~),χ~=ℱD-1(ψ~).Definition 8.
Let one denote by𝒮'h,k(ℝd) the space of tempered distribution such that
(31)limj→-∞Sju=0in𝒮′(ℝd).On the follow, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Dunkl operators onℝd and obtain their basic properties.From now, we make the convention that for all nonnegative sequence{aq}q∈ℤ, the notation (∑q‍aqr)1/r stands for supqaq in the case r=∞.Definition 9.
Lets∈ℝ and p,q∈[1,∞]. The homogeneous Dunkl-Besov spaces ℬ˙p,qs,k(ℝd) are the space of distribution in 𝒮'h,k(ℝd) such that
(32)∥f∥ℬ˙p,qs,k(ℝd):=(∑j∈ℤ‍(2sj∥Δjf∥Lkp(ℝd))q)1/q<∞.Definition 10.
Lets∈ℝ and 1≤p,q≤∞, the homogeneous Dunkl-Triebel-Lizorkin space ℱ˙p,qs,k(ℝd) is the space of distribution in 𝒮'h,k(ℝd) such that
(33)∥f∥ℱ˙p,qs,k(ℝd):=∥(∑j∈ℤ‍2sjq|Δjf|q)1/q∥Lkp(ℝd)<∞.Let us recall that the operators(-△k)s/2 and (I-△k)s/2 have been defined, respectively, by (cf. [19])
(34)(-△k)s/2f=ℱD-1(∥·∥sℱDf),(I-△k)s/2f=ℱD-1((1+∥·∥2)s/2ℱDf).
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
(35)(I-△k)-s/2f=f*DBk,s,
where
(36)Bk,s(y)=1Γ(s/2)∫0∞‍e-te-∥y∥2/4tt(s-d-2γ)/2dtt.
We note that Bk,s(y)≥0 for all y∈ℝd, Bk,s∈Lk1(ℝd), and
(37)Bk,s(y)≤C∥y∥s-d-2γe-∥y∥2/2,∥y∥>0.Definition 11.
Fors∈ℝ and 1≤p≤∞, the Dunkl-Bessel potential space Hp,ks(ℝd) is defined as the space (I-△k)s/2(Lkp(ℝd)), equipped with the norm ∥f∥Hp,ks(ℝd)=∥(I-△k)s/2f∥Lkp(ℝd).Furthermore,p=2, H2,ks(ℝd):=Hks(ℝd).Definition 12.
The operators(-△k)k-s/2, 0<s<d+2γ, are called Dunkl-Riesz potentials operators, and one has
(38)(-△k)-s/2f=Rk,s*Df,
where Rk,s is the Dunkl-Riesz potential given by
(39)Rk,s(y):=C(k,s,d)∥y∥s-d-2γ,whereC(k,s,d)=Γ((d+2γ-s)/2)2(d+2γ-s)/2Γ(s/2).Definition 13.
Fors∈ℝ and 1≤p≤∞, the homogeneous Dunkl-Riesz potential space ℋ˙p,ks(ℝd) is defined as the space (-△k)s/2(Lkp(ℝd)), equipped with the norm ∥f∥ℋ˙p,ks(ℝd)=∥(-△k)s/2f∥Lkp(ℝd).Proposition 14.
Letq∈(1,∞), and let s∈ℝ such that 0<s<(d+2γ)/q, then one has
(40)ℬ˙q,qs,k(ℝd)=ℱ˙q,qs,k(ℝd)↪ℱ˙q,∞s,k(ℝd)↪ℱ˙∞,∞s-(d+2γ)/q,k(ℝd),(41)ℋ˙q,ks(ℝd)=ℱ˙q,2s,k(ℝd)↪ℱ˙q,∞s,k(ℝd)↪ℱ˙∞,∞s-(d+2γ)/q,k(ℝd).Proof.
We obtain these results by similar ideas used in the nonhomogeneous case. (cf. [12]).Theorem 15.
Leta,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∈ℱ˙q1,∞a,k(ℝd)∩ℱ˙q2,∞-b,k(ℝd), then one has
(42)|f(x)|≤C(supj∈ℤ2aj|Δjf(x)|)1-θ(supj∈ℤ2-bj|Δjf(x)|)θ.
In particular, one gets
(43)∥f∥Lkp(ℝd)≤C∥f∥ℱ˙q1,∞a,k(ℝd)1-θ∥f∥ℱ˙q2,∞-b,k(ℝd)θ.Proof.
Letf be a Schwartz class, we have
(44)|f(x)|≤∑j∈ℤ‍|Δjf(x)||f(x)|≤∑j∈ℤ‍min(2-ajsupj∈ℤ(2aj|Δjf(x)|),2jbsupj∈ℤ(2-jb|Δjf(x)|)).
We define N(x) as the largest index such that
(45)2jbsupj∈ℤ(2-jb|Δjf(x)|)≤2-ajsupj∈ℤ(2aj|Δjf(x)|),
and we write
(46)|f(x)|≤∑j≤N(x)‍2jbsupj∈ℤ(2-jb|Δjf(x)|)+∑j>N(x)‍2-ajsupj∈ℤ(2aj|Δjf(x)|)≤C(supj∈ℤ2aj|Δjf(x)|)b/(a+b)(supj∈ℤ2-bj|Δjf(x)|)a/(a+b).
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.
Letq∈(1,∞), and let s∈ℝ such that 0<s<(d+2γ)/q, then one has
(47)∥f∥Lkp(ℝd)≤C∥f∥ℬ˙∞,∞-((2γ+d)/q-s),k(ℝd)1-q/p∥f∥ℬ˙q,qs,k(ℝd)q/p,(48)∥f∥Lkp(ℝd)≤C∥f∥ℬ˙∞,∞-((2γ+d)/q-s),k,k(ℝd)1-q/p∥f∥ℋ˙q,ks(ℝd)q/p,
where p=q(2γ+d)/(2γ+d-qs).Proof.
We takea=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, ℬ˙p,rs,k(ℝd)∩Lk∞(ℝd) is an algebra, and there exists a positive constant C such that
(49)∥uv∥ℬ˙p,rs,k(ℝd)≤C[∥u∥Lk∞(ℝd)∥v∥ℬ˙p,rs,k(ℝd)+∥v∥Lk∞(ℝd)∥u∥ℬ˙p,rs,k(ℝd)].
(
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
(50)∥uv∥ℬ˙p2,r2s,k(ℝd)≤C[∥u∥ℬ˙p1,∞s1,k(ℝd)∥v∥ℬ˙p2,r2s2,k(ℝd)+∥u∥ℬ˙p2,r2s2,k(ℝd)∥v∥ℬ˙p1,∞s1,k(ℝd)],
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
(51)∥uv∥ℬ˙p,∞s,k(ℝd)≤C[∥u∥ℬ˙p1,r1s1,k(ℝd)∥v∥ℬ˙p2,r2s2,k(ℝd)+∥u∥ℬ˙p2,r2s2,k(ℝd)∥v∥ℬ˙p1,r1s1,k(ℝd)].
(
4
) Moreover, for any (s1,s2), any (p1,p2,p) and any (r1,r2) such that sj<(d+2γ)/pj, s1+s2>(d+2γ)(1/p1+1/p2-1/p) and p≥max(p1,p2), one has
(52)∥uv∥ℬ˙p,rs1,2,k(ℝd)≤C∥u∥ℬ˙p1,r1s1,k(ℝd)∥v∥ℬ˙p2,r2s2,k(ℝd).
with s1,2=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 bylq(ℤ) the set of sequence (aj)j∈ℤ such that
(53)(∑j∈ℤ‍|aj|q)1/q<∞
stands for supj|aj| 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 discreteJ-method and the K-method which are the simplest ones.We consider two Banach spacesA0 and A1 which are continuously imbedded into a common topological vector space V and t>0.TheJ-method and the K-method consist to consider the J-functional and the K-functional defined on A0⋂‍A1 by
(54)J(t,a;A0,A1)=max(∥a∥A0,t∥a∥A1),K(t,a;A0,A1)=min(∥a0∥A0+t∥a1∥A1:a=a0+a1).Definition 18 (J-method of interpolation).
For0<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∈ℤ‍aj, where the series converge in A0+A1, each aj belongs to A0⋂‍A1 and (2-jθJ(2j,aj;A0,A1))j∈ℤ∈lq(ℤ).
The norm of[A0,A1]θ,q,J is defined by
(55)∥a∥[A0,A1]θ,q,J=infa=∑j∈ℤaj(∑j∈ℤ2-jθq∥aj∥A0q‍)1/q+(∑j∈ℤ‍2j(1-θ)q∥aj∥A1q)1/q.Definition 19 (K-method of interpolation).
For0<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(2j,a;A0,A1))j∈ℤ∈lq(ℤ).
The norm of[A0,A1]θ,q,K is defined as follows:
(56)∥a∥[A0,A1]θ,q,K=(∑j∈ℤ‍2-jθqK(2j,a;A0,A1)q)1/q.Proposition 20 (Equivalence theorem).
For0<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.
Fora=∑j∈ℤ‍aj and ϱ>0, with ϱ≠1, one has
(57)∥a∥[A0,A1]θ,q≤C(q,θ,ϱ)(∑j∈ℤ‍ϱ-jθq∥aj∥A0q)(1-θ)/q∥a∥[A0,A1]θ,q≤×(∑j∈ℤ‍ϱj(1-θ)q∥aj∥A1q)θ/q.Proposition 23.
(i) Forθ0≠θ1, one has
(58)[[A0,A1]θ0,q0,[A0,A1]θ1,q1]θ,q=[A0,A1](1-θ)θ0+θθ1,q.(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 spacesA0′,A1′ 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′=[A0′,A1′]θ,q′, where q′ is the conjugate component of q.For any measurable functionf on ℝd, we define its distribution and rearrangement functions
(59)df,k(λ):=∫{x∈ℝd:|f(x)|≥λ}‍ωk(x)dx,fk*(s):=inf{λ:df,k(λ)≤s}.For1≤p≤∞ and 1≤q≤∞, define
(60)∥f∥Lkp,q(ℝd)={(∫0∞(s1/pfk*(s))qdss‍)1/qif1≤p,q<∞sups>0s1/pfk*(s)if1≤p≤∞,q=∞.
The generalized Lorentz spaces Lkp,q(ℝd) is defined as the set of all measurable functions f such that ||f||Lkp,q(ℝd)<∞.Proposition 25.
(i) For1<p<∞, 1≤q≤∞(61)Lkp,q(ℝd)=[Lk1(ℝd),Lk∞(ℝd)]θ,q,
with 1/p=1-θ.
(ii) Forp0≠p1, one has
(62)[Lkp0,q0(ℝd),Lkp1,q1(ℝd)]θ,q=[Lkp0(ℝd),Lkp1(ℝd)]θ,q=Lkp,q(ℝd),
with 1/p=(1-θ)/p0+θ/p1.
(iii) In the casep0=p1=p, one has
(63)[Lkp,q0(ℝd),Lkp,q1(ℝd)]θ,q=Lkp,q(ℝd),
with 1/q=(1-θ)/q0+θ/q1.
(iv) If1≤p≤∞ and 1≤q1<q2≤∞, then
(64)Lkp,q1(ℝd)↪Lkp,q2(ℝd).Proof.
We obtain these results by similar ideas used in the Euclidean case.Proposition 26.
(i) Let1<p<∞, 1≤q≤∞. Then, there exists a constant C such that every f∈Lkp,q(ℝd) can be decomposed as f=∑j∈ℤ‍fj, where
(65)∥(2-j(p-1)/p∥fj∥Lk1(ℝd))∥lr+∥(2j/p∥fj∥Lk∞(ℝd))∥lr≤C∥f∥Lkp,q(ℝd),
the fj have disjoint supports if j≠n, fjfn=0.
(ii) Let1<p<∞, 1≤q≤∞. Then, there exists a constant C such that every f∈Lkp,q(ℝd) and every g∈Lkp/(p-1),q/(q-1)(ℝd), one has fg∈Lk1(ℝd) and
(66)|∫ℝd‍f(x)g(x)ωk(x)dx|≤C∥f∥Lkp,q(ℝd)∥g∥Lkp/(p-1),q/(q-1)(ℝd).Proof.
We obtain these results by similar ideas used in the Euclidean case.


## 5. Inequalities for the Fractional Dunkl-Laplace Operators

Lemma 27.
Lets be a real number such that 0<s<d+2γ, and let 1<p<q<∞ satisfy
(67)1p-1q=s2γ+d.
For f∈Lkp(ℝd), one has
(68)∥(I-△k)-s/2f∥Lkq(ℝd)≤C∥f∥Lkp(ℝd).Proof.
We obtain this result by similar ideas used for the Dunkl-Riesz potential (cf. [21]).Proposition 28.
Lets<(d+2γ)/2 and q=(2d+4γ)/(d+2γ-2s). Then,
(69)∥f∥Lkq(ℝd)≤C∥(I-△k)s/2f∥Lk2(ℝd),f∈Hks(ℝd).Proof.
Let us first observe that sinceD(ℝd) is dense in Hks(ℝd), it is enough to prove (69) for f∈D(ℝd). Let f,g∈D(ℝd). Then, we have
(70)〈f,g〉Lk2(ℝd)=〈ℱD(f),ℱD(g)〉Lk2(ℝd)=∫ℝd‍(1+∥ξ∥2)s/2ℱD(f)(ξ)ℱD(g)(ξ)¯(1+∥ξ∥2)-s/2==ii×ωk(ξ)dξ=〈(I-△k)s/2f,(I-△k)-s/2g〉Lk2(ℝd).
Hence,
(71)|〈f,g〉Lk2(ℝd)|≤∥(I-△k)s/2f∥Lk2(ℝd)∥(I-△k)-s/2g∥Lk2(ℝd).
Now, by the previous lemma we obtain
(72)|〈f,g〉Lk2(ℝd)|≤C∥(I-△k)s/2f∥Lk2(ℝd)∥g∥Lkp(ℝd),
where p=(2d+4γ)/(d+2γ+2s). Now, let us take g=fq-1, with 1/p+1/q=1, that is, q=(2d+4γ)/(d+2γ-2s). Then, the relation (72) gives that
(73)∥f∥Lkq(ℝd)q≤C∥(I-△k)s/2f∥Lk2(ℝd)∥f∥Lkq(ℝd)q-1.
Thus, we obtain (69).Proposition 29.
Let1≤p,p2<∞, 0<θ<p<∞, 0<s<d+2γ, and 1<p1<(d+2γ)/s. Then, one has the inequality
(74)∥f∥Lkp(ℝd)≤C∥(I-△k)s/2f∥Lkp1(ℝd)θ/p∥f∥Lkp2(ℝd)(p-θ)/p,
with
(75)θ(1p1-sd+2γ)+p-θp2=1.Proof.
Hölder's inequality yields(76)∥f∥Lkp(ℝd)p≤C∥f∥Lkp0(ℝd)θ∥f∥Lkp2(ℝd)p-θ,
where
(77)1p0=1θ(1-1p),1p2=1(p-θ)p.
Applying Lemma 27, with p1=((d+2γ)p0)/(d+2γ+sp0), we obtain the result.Theorem 30.
Let1<q<∞, 0<s<d+2γ and 1<p1<(d+2γ)/s. Then, the inequality(78)exp((|f(x)|q∥f∥Lkq(ℝd)q)(1q+sd+2γ-1p1)×∫ℝd‍|f(x)|q∥f∥Lkq(ℝd)qln(|f(x)|q∥f∥Lkq(ℝd)q)ωk(x)dx)≤C∥(I-△k)s/2f∥Lkp1(ℝd)∥f∥Lkq(ℝd)
holds for
(79)1q+sd+2γ-1p1>0.Proof.
Using the convexity of the functiong(h)=hln(∫ℝd‍|f(x)|1/hωk(x)dx), and the logarithmic Hölder's inequality proved by Merker [22], we obtain
(80)∫ℝd‍|f(x)|q∥f∥Lkq(ℝd)qln(|f(x)|q∥f∥Lkq(ℝd)q)ωk(x)dx≤pp-qln(∥f∥Lkq(ℝd)q∥f∥Lkq(ℝd)q),
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
(81)∫ℝd‍|f(x)|q∥f∥Lkq(ℝd)qln(|f(x)|q∥f∥Lkq(ℝd)q)ωk(x)dx≤pp-qln((C∥(I-△k)s/2f∥Lkp1(ℝd)θ/p∥f∥Lkp2(ℝd)(p-θ)/p)q∥f∥Lkq(ℝd)q)≤qθp-qln(C∥(I-△k)s/2f∥Lkp1(ℝd)∥f∥Lkq(ℝd)q).
By a simple calculation, we obtain the result.Corollary 31.
Let0<s<d+2γ and 1<q<(d+2γ)/s, f∈Hq,ks(ℝd) such that ∥f∥Lkq(ℝd)=1, one has
(82)exp(sd+2γ∫ℝd‍|f(x)|qln(|f(x)|q)ωk(x)dx)≤C∥(I-△k)s/2f∥Lkq(ℝd).Proof.
It suffices to apply the previous theorem forq=p1.Lemma 32 (see [23]).
One assumes thatG=ℤ2d. If f∈Lkp1,q1(ℝd), g∈Lkp2,q2(ℝd), and 1/p1+1/p2>1, then f*Dg∈Lkp3,q3(ℝd) where 1/p3=1/p1+1/p2-1 and q3≥1 is any number such that 1/q3≤1/q1+1/q2. Moreover,
(83)∥f*Dg∥Lkp3,q3(ℝd)≤C∥f∥Lkp1,q1(ℝd)∥g∥Lkp2,q2(ℝd).Remark 33.
The analogues of this lemma for the general reflection groupG, together with other additional results, will appear in a forthcoming paper.Theorem 34.
One assumes thatG=ℤ2d. Let 1≤p<∞, 1≤p2,q,q1,q2<∞, 0<θ<q, 0<s<d+2γ, and 1<p1<(d+2γ)/s. Then, the inequality
(84)∥f∥Lkp,q(ℝd)≤C∥(I-△k)s/2f∥Lkp1,q1(ℝd)θ/q∥f∥Lkp2,q2(ℝd)(q-θ)/q
holds for
(85)θq1+q-θq2=1,θ(1p1-sd+2γ)+q-θp2=qp.Proof.
Applying the Hölder inequality and simple computation yields(86)∥f∥Lkp,q(ℝd)q≤C∥f∥Lkp3,q1(ℝd)θ∥f∥Lkp2,q2(ℝd)q-θ,
where
(87)1p3=1p-1q+1q1,1p2=1p-1q+1q2.
Note that
(88)f(x)=(I-△k)s/2f*DBk,s(x),
where Bk,s is the Dunkl-Bessel kernel defined by relation (36). From the relation (37), we see that Bk,s∈Lk(d+2γ)/(d+2γ-s),∞(ℝd). Using now Lemma 32, we deduce that
(89)∥f∥Lkp3,q1(ℝd)≤C∥(I-△k)s/2f∥Lkp1,q1(ℝd),
for
(90)1p3=1p1-sd+2γ,0<s<d+2γp1.
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.
Lets<(d+2γ)/2 and q=(2d+4γ)/(d+2γ-2s). Then,
(91)∥f∥Lkq(ℝd)≤C∥(-△k)s/2f∥Lk2(ℝd),f∈Hks(ℝd).Proposition 36.
Let1≤p,p2<∞, 0<θ<p<∞, 0<s<d+2γ, and 1<p1<(d+2γ)/s. Then, the inequality
(92)∥f∥Lkp(ℝd)≤C∥(-△k)s/2f∥Lkp1(ℝd)θ/p∥f∥Lkp2(ℝd)(p-θ)/p
with
(93)θ(1p1-sd+2γ)+p-θp2=1.Theorem 37.
Let1<q<∞, 0<s<d+2γ, and 1<p1<(d+2γ)/s. Then, the inequality
(94)exp((|f(x)|q∥f∥Lkq(ℝd)q)(1q+sd+2γ-1p1)×∫ℝd‍|f(x)|q∥f∥Lkq(ℝd)qln(|f(x)|q∥f∥Lkq(ℝd)q)ωk(x)dx)≤C∥(-△k)s/2f∥Lkp1(ℝd)∥f∥Lkq(ℝd)
holds for
(95)1q+sd+2γ-1p1>0.Corollary 38.
Let0<s<d+2γ and 1<q<(d+2γ)/s, f∈ℋ˙q,ks(ℝd) such that ∥f∥Lkq(ℝd)=1, one has
(96)exp(sd+2γ∫ℝd‍|f(x)|qln(|f(x)|q)ωk(x)dx)≤C∥(-△k)s/2f∥Lkq(ℝd).Theorem 39.
One assumes thatG=ℤ2d. Let 1≤p<∞, 1≤p2,q,q1,q2<∞, 0<θ<q, 0<s<d+2γ, and 1<p1<(d+2γ)/s. Then, the inequality
(97)∥f∥Lkp,q(ℝd)≤C∥(-△k)s/2f∥Lkp1,q1(ℝd)θ/q∥f∥Lkp2,q2(ℝd)(q-θ)/q
holds for
(98)θq1+q-θq2=1,θ(1p1-sd+2γ)+q-θp2=qp.Remark 40.
(i) We assume thatG=ℤ2d. It follows from the special case p1=q1 and p2=q2 of (97) that the inequality
(99)∥f∥Lkp,q(ℝd)≤C∥(-△k)s/2f∥Lkp1(ℝd)θ/q∥f∥Lkp2(ℝd)(q-θ)/q
with q=p(1-θs/(d+2γ)). Equation (99) can be thought of a refinement of (92) from (64).
(ii) We assume thatG=ℤ2d. It follows from the special case p1=q=θ that (99) becomes
(100)∥f∥Lkq(d+2γ)/(d+2γ-qs),q(ℝd)≤C∥(-△k)s/2f∥Lkq(ℝd),
which can also be thought of as a refinement of the Hardy-Littlewood-Sobolev fractional integration theorem in Dunkl setting (cf. [21]):
(101)∥(-△k)-s/2f∥Lkq(ℝd)≤C∥f∥Lkp(ℝd).
(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 thatG=ℤ2d. Let 1<p<∞, 0<s<(d+2γ)/p, and 1≤q≤∞. There exists a positive constant C such that one has
(102)∥f(x)∥x∥s∥Lkp,q(ℝd)≤C∥(-△k)s/2f∥Lkp,q(ℝd).For proof of this result, we need the following lemma which we prove as the Euclidean case.Lemma 42.
Let1≤p1,p2,q1,q2≤∞. If f∈Lkp1,q1(ℝd) and g∈Lkp2,q2(ℝd), then
(103)∥fg∥Lkp,q(ℝd)≤C∥f∥Lkp1,q1(ℝd)∥g∥Lkp2,q2(ℝd),
where 1/p=1/p1+1/p2 and 1/q=1/q1+1/q2.Proof of Theorem41.
Let1<p<∞ and s∈(0,(d+2γ)/p). We take g(x)=1/∥x∥s and apply (103) in the specific form
(104)∥fg∥Lkp,q(ℝd)≤C∥f∥Lkp1,q(ℝd)∥g∥Lkr,∞(ℝd),
where r=(d+2γ)/s and p1=(q(d+2γ))/(d+2γ-qs). As g∈Lkr,∞(ℝd), we have
(105)∥f(x)∥x∥s∥Lkp,q(ℝd)≤C∥f∥Lk((d+2γ)p)/(d+2γ-ps),q(ℝd),
with 1≤q≤∞. On the other hand, from [23], Theorem 1.2, we have
(106)∥f∥Lk((d+2γ)p)/(d+2γ-ps),q(ℝd)≤C∥(-△k)s/sf∥Lkp,q(ℝd),
for any f∈Lkp,q(ℝd) with 1≤q≤∞, 1<p<∞, and 0<s<(d+2γ)/p. Thus, we obtain (102).


## 6. Dispersion Phenomena

Notations.
We denote byℐk(t) the Dunkl-Schrödinger semigroup on Lk2(ℝd) defined by
(107)ℐk(t)v:=1ck|t|γ+d/2e-i(d+2γ)(π/4)sgntei(∥·∥2/4t)ℐk(t)v:=×[ℱD(ei(∥·∥2/4t)v)](·2t).Wk1,r(ℝd) (1≤r≤∞) Banach space of (classes of) measurable functions u:ℝd→ℂ such that Tμu∈Lkr(Rd) in the sense of distributions, for every multi-index μ with |μ|≤1. Wk1,r(ℝd) is equipped with the norm
(108)∥u∥Wk1,r(ℝd)=∑|μ|≤1‍∥Tμu∥Lkr(ℝd).Wk,G1,r(ℝd) (1≤r≤∞) the subspace of Wk1,r(ℝd), 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
(109)1q+d+2γ2r≤d+2γ4.
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
(110)P=(2,2d+4γd+2γ-2)
is sharp (d+2γ)/2-admissible.Lemma 44 (see [25]).
LetE and F be Banach spaces, and let ℒ: Lp,r(0,∞;E)→Lq,s(0,∞;F) be an integral operator for some p,r,q,s, with a kernel k(t,τ) such that
(111)ℒf(t)=∫0∞‍k(t,τ)f(τ)dτ.
If 1≤p≤r<s≤q<∞, then one has
(112)∥ℒ~f∥Lq,s(0,∞;F)≤C∥f∥Lp,r(0,∞;E),
where ℒ~ is the low diagonal operator defined by
(113)ℒ~f(t)=∫0t‍k(t,τ)f(τ)dτ.Lemma 45.
For any(d+2γ)/2-admissible pair (q,r) with q>2(114)∥ℐk(t)f∥Lq,2(0,∞;Lkr(ℝd))≤C∥f∥Lk2(ℝd),(115)∥∫0t‍ℐk(t-τ)g(τ)dτ∥Lq,2(0,∞;Lkr(ℝd))∩L∞(0,∞;Lk2(ℝd))≤C∥g∥Lq′,2(0,∞;Lkr′(ℝd)).Proof.
From the dispersion ofℐk(t) such that
(116)∥ℐk(t)g∥Lkr(ℝd)≤Ct-(d+2γ)(1/2-1/r)∥g∥Lkr′(ℝd),
for any r∈[2,∞], (cf. [8]), and the fact that
(117)t-(d+2γ)(1/2-1/r)∈L2r/(d+2γ)(r-2),∞,foranyr∈[2,2(d+2γ)d+2γ-2],
one can easily prove the result.Theorem 46.
Suppose thatd≥1, (q,r) and (q1,r1) are (d+2γ)/2-admissible pairs and 2<a≤q. If u is a solution to the problem
(118)i∂tu(t,x)+△ku(t,x)=f(t,x),(t,x)∈ℝ×ℝd,u|t=0=u0,
for some data, u0, f, then
(119)∥u∥Lq,a(ℝ;Lkr(ℝd))+∥u∥C(ℝ;Lk2(ℝd))≤C(∥f∥Lq1′,2(ℝ;Lkr1′(ℝd))⋂‍L2(ℝ;Lk(2d+4γ)/(d+2γ+2),2(ℝd))∥u0∥Lk2(ℝd)+∥f∥Lq1′,2(ℝ;Lkr1′(ℝd))⋂‍L2(ℝ;Lk(2d+4γ)/(d+2γ+2),2(ℝd))).Proof.
Letu be a solution of (118). We write u as
(120)u(t,x)=ℐk(t)u0(x)+∫0t‍ℐk(t-τ)f(τ,x)dτ,(t,x)∈ℝ×ℝd.
Let k(t,τ)=ℐk(t-τ), E=Lkr1′(ℝd) or Lk(2d+4γ)/(d+2γ+2),2(ℝd), F=Lkr(ℝd), and ℒf(t)=∫0∞‍k(t,τ)f(τ)dτ. Then, since q1′≤2<s≤q, in view of Lemma 44, we only have to show that
(121)∥∫0∞‍k(t,τ)f(τ)dτ∥Lq,s(0,∞;Lkr(ℝd))≤C∥f∥Lq1′,2(0,∞;Lkr1′(ℝd))∩L2(0,∞;Lk(2d+4γ)/(d+2γ+2),2(ℝd)).
To show this, observe from (114) and Lq,s⊂Lq,2 for all s≥2 that
(122)∥∫0∞k(t,τ)f(τ)dτ‍∥Lq,s(0,∞;Lkr(ℝd))2≤C∫∫0∞‍〈ℐk(-τ)f(τ),ℐk(-y)f(y)〉dτdy.
Then, from the endpoint result of Keel and Tao [26], the right-hand side of (122) is bounded by ∥f∥L2(0,∞;Lk(2d+4γ)/(d+2γ+2),2(ℝd))2. The remaining part of theorem can be obtained by the duality of Lorentz space (Lq,s)′=Lq′,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:(123)i∂tu(t,x)+△ku(t,x)=-|u(t,x)|4/(d+2γ-2)u(t,x),(t,x)∈ℝ×ℝd,u|t=0=u0∈Hk1(ℝd)inℝd.Theorem 47.
If the initial data is sufficiently small andG-invariant, then there exists a unique solution u∈Lq,s(0,∞; Wk,G1,r(ℝd))∩L2(0,∞; Wk,G1,(2d+4γ)/(d+2γ-2)(ℝd))∩C([0,∞); Hk,G1(ℝd) for every sharp (d+2γ)/2-admissible pair (q,r) with q>2 and 2<s≤q.Proof.
The existence of a uniqueHk,G1(ℝd)-solution is proved in [9], it suffices to prove that u∈Lq,s(0,∞;Wk,G1,r(ℝd)). From Duhamel's principle, we deduce that
(124)u(t,x)=ℐk(t)u0(x)u(t,x)=+∫0t‍ℐk(t-τ)(|u(τ,x)|4/(d+2γ-2)u(τ,x))dτ.
Using (114) and (119), we have
(125)∥Tu∥Lq,s(ℝ;Lkr(ℝd))≤C(∥Tu0∥Lk2(ℝd)+∥T(|u|4/(d+2γ-2)u)∥Lq1′,2(ℝ;Lkr′1(ℝd))).
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
(126)1q1=4(d+2γ-2)q0+1q0,1r1=4(d+2γ-2)r1+1r0,1s1=4(d+2γ-2)s0+1s0,
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
(127)∥Tu∥Lq0,s0(ℝ;Lkr0(ℝd))≤C(∥Tu0∥Lk2(ℝd)+∥Tu∥Lq0,s0(ℝ;Lkr0(ℝd))(d+2γ+2)/(d+2γ-2)).
Since ||u0||Hk1(ℝd) is small, we have
(128)∥Tu∥Lq0,s0(ℝ;Lkr0(ℝd))≤C∥Tu0∥Lk2(ℝd).
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
(129)∥Tu∥Lq,s(ℝ;Lkr(ℝd))≤C(∥Tu0∥Lk2(ℝd)+∥Tu∥Lq0,s0(ℝ;Lkr0(ℝd))(d+2γ+2)/(d+2γ-2))≤C(∥Tu0∥Lk2(ℝd)+∥Tu0∥Lk2(ℝd)(d+2γ+2)/(d+2γ-2)).
In a similar way, we can also derive from the smallness of ||u0||Hk1(ℝd),
(130)∥u∥Lq,s(ℝ;Lkr(ℝd))≤C∥u0∥Lk2(ℝd).


## 7. Embedding Sobolev Theorems and Applications

Theorem 48.
Lets,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∈ℬ˙q1,rs,k(ℝd)∩ℬ˙q2,r-t,k(ℝd), then f∈Lkp,r(ℝd), and one has
(131)∥f∥Lkp,r(ℝd)≤C∥f∥ℬ˙q1,rs,k(ℝd)1-θ∥f∥ℬ˙q2,r-t,k(ℝd)θ.Proof.
We start pickingp1,p2 such that 1≤q1<p1<p<p2<q2≤∞ with 2/p=1/p1+1/p2. We have then 1/pi=(1-ai)/q1+ai/q2 with ai∈(0,1) and i=1,2. We write
(132)∥Δjf∥Lkpi(ℝd)≤∥Δjf∥Lkq1(ℝd)1-ai∥Δjf∥Lkq2(ℝd)ai.
Using Hölder's inequality and by simple calculations, we obtain
(133)∑j∈ℤ‍ϱ-jr/2∥Δjf∥Lkp1(ℝd)r≤∥f∥ℬ˙q1,rs,k(ℝd)(1-a1)r∥f∥ℬ˙q2,r-t,k(ℝd)ra1,∑j∈ℤ‍ϱjr/2∥Δjf∥Lkp2(ℝd)r≤∥f∥ℬ˙q1,rs,k(ℝd)(1-a2)r∥f∥ℬ˙q2,r-t,k(ℝd)ra2,
where ϱ=2-2(s(1-a1)-ta1)>0. From this, and applying Proposition 25, we deduce that if f∈ℬ˙q1,rs,k(ℝd)∩ℬ˙q2,r-t,k(ℝd), then f∈[Lkp1(ℝd),Lkp2(ℝd)]1/2,r=Lkp,r(ℝd). Furthermore, using (57), we finally have:
(134)∥f∥Lkp,r(ℝd)≤C∥f∥ℬ˙q1,rs,k(ℝd)1-θ∥f∥ℬ˙q2,r-t,k(ℝd)θ.Corollary 49.
Lets 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∈ℬ˙q,qs,k(ℝd), the following inequality holds:
(135)(∫ℝd‍|f(x)|q∥x∥sqωk(x)dx)1/q≤C∥f∥ℬ˙q,qs,k(ℝd)θ∥f∥ℬ˙∞,qs-(d+2γ)/q,k(ℝd)1-θ,
where θ=1-qs/(d+2γ).Proof.
Letp∈(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
(136)∥fg∥Lkq,q(ℝd)≤C∥f∥Lkp,q(ℝd)∥g∥Lkr,∞(ℝd),
where r=(d+2γ)/s and p=(q(d+2γ))/(d+2γ-qs). As g∈Lkr,∞(ℝd), we have
(137)(∫ℝd‍|f(x)|q∥x∥sqωk(x)dx)1/q≤C∥f∥Lkp,q(ℝd).
Combining this with (131), we obtain (135).Theorem 50.
Let0<s<(d+2γ)/2 be given. There exists a positive constant C such that for all function u∈ℋ˙2,ks(ℝd), one has
(138)∫ℝd‍|u(x)|2∥x∥2sωk(x)dx≤C∥u∥ℋ˙2,ks(ℝd)2.For proof of this theorem, we need the following lemma, which we obtain by simple calculations.Lemma 51.
Lets be a real number in the interval (0,γ+d/2). Then, the function x↦||x||-2s belongs to the Dunkl-Besov space ℬ˙1,∞d+2γ-2s,k(ℝd).Proof of Theorem50.
Let us define(139)Is,k(u):=∫ℝd‍|u(x)|2∥x∥2sωk(x)dx=〈∥·∥-2s,u2〉.
Using homogeneous Littlewood-Paley decomposition and the fact that u2 belongs to 𝒮'h,k(ℝd), we can write
(140)Is,k(u)=∑|n-m|≤2‍〈Δn(∥·∥-2s),Δm(u2)〉Is,k(u)≤C∑|n-m|≤2‍〈2n((d+2γ)/2-2s)Is,k(u)=Is,k(u)=×Δn(∥·∥-2s),2-m((d+2γ)/2-2s)Δm(u2)〉.
Lemma 51 claims that ∥·∥-2s belongs to ℬ˙2,∞(d+2γ)/2-2s,k(ℝd). Theorem 17 yields
(141)∥u2∥ℬ˙2,12s-(d+2γ)/2,k(ℝd)≤C∥u∥ℋ˙2,ks(ℝd)2.
Thus,(142)Is,k(u)≤C∥u∥ℋ˙2,ks(ℝd)2.The following results of this section are in sprit of the classical case (cf. [27]).Theorem 52.
Lets,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 everyf∈ℬ˙q1,r1s,k(ℝd)∩ℬ˙q2,r2-t,k(ℝd), and if r>r0, one has f∈Lkp,r(ℝd) and
(143)∥f∥Lkp,r(ℝd)≤C∥f∥ℬ˙q1,r1s,k(ℝd)1-θ∥f∥ℬ˙q2,r2-t,k(ℝd)θ.(ii)
Moreover, this inequality is valid forr=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 conditionr≥r0 is sharp.Proof.
(i) Caser>r0. With no loss of generality, we may assume that q1<q2, and we fix ε>0 such that
(144)1q2<1p-ε(1q1-1q2)=1p2<1p+ε(1q1-1q2)1q2<1p-ε(1q1-1q2)=1p1<1q1.
The proof follows essentially the same ideas used in the previous theorem. Indeed, we have for Mj=2js∥Δjf∥Lkq1(ℝd) and Nj=2-jt∥Δjf∥Lkq2(ℝd) and for ε0=1 and ε1=-1,
(145)∥Δjf∥Lkpi(ℝd)≤∥Δjf∥Lkq1(ℝd)1-θ+εεi∥Δjf∥Lkq2(ℝd)θ-εεi∥Δjf∥Lkpi(ℝd)=Mj1-θ+εεiNjθ-εεi2-jεεi(s+t).
As r1≠r2, we can only say that (Mj1-θ+εεiNjθ-εεi)j∈ℤ∈lϱi, where 1/ϱi=(1-θ+εεi)/r1+(θ-εεi)/r2. We may use (57), but we get only that f∈Lkp,ϱ(ℝd)=[Lkp1(ℝd),Lkp2(ℝd)]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) Caser=r0.
(a) Ifr=r1=r2: this case was treated in Theorem 48.
(b) Ifr1=q1 and r2=q2: this is a direct consequence of (43), since we have
(146)∥f∥ℬ˙qi,qis,k(ℝd)=∥f∥ℱ˙qi,qis,k(ℝd)≤C∥f∥ℱ˙qi,∞s,k(ℝd),∥f∥ℬ˙qi,qi-t,k(ℝd)=∥f∥ℱ˙qi,qi-t,k(ℝd)≤C∥f∥ℱ˙qi,∞-t,k(ℝd),
we obtain(147)∥f∥Lkp(ℝd)≤C∥f∥ℬ˙q1,q1s,k(ℝd)1-θ∥f∥ℬ˙q2,q2-t,k(ℝd)θ.
(c) Case1<p≤2andr0=p.
We just write(148)∥Δjf∥Lkp(ℝd)≤∥Δjf∥Lkq1(ℝd)1-θ∥Δjf∥Lkq2(ℝd)θ∥Δjf∥Lkp(ℝd)=(2js∥Δjf∥Lkq1(ℝd))1-θ(2-jt∥Δjf∥Lkq2(ℝd))θ
and get by Hölder's inequality:
(149)∥f∥ℬ˙p,p0,k(ℝd)≤C∥f∥ℬ˙q1,r1s,k(ℝd)1-θ∥f∥ℬ˙q2,r2-t,k(ℝd)θ.
We then use the embedding ℬ˙p,p0,k(ℝd)⊂Lkp(ℝd)=Lkp,p(ℝd) which is valid for p≤2.Theorem 53.
Lets,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)
Ifq1≤r1≤q2 and let 1/r=(1-θ)/r1+θ/q2. Then, one has
(150)∥f∥Lkp,r(ℝd)≤C∥f∥ℬ˙q1,r1s,k(ℝd)1-θ∥f∥ℬ˙q2,q2-t,k(ℝd)θ.(ii)
Ifq1≤r2≤q2, and let 1/r=(1-θ)/q1+θ/r2. Then, one has
(151)∥f∥Lkp,r(ℝd)≤C∥f∥ℬ˙q1,q1s,k(ℝd)1-θ∥f∥ℬ˙q2,r2-t,k(ℝd)θ.Proof.
We only prove the first inequality, as the proof for the second one is similar. Sincef∈ℬ˙q1,r1s,k(ℝd), noting that λj=2sj∥Δjf∥Lkq1(ℝd), we have (λj)j∈ℤ∈lr1. Thus, using Proposition 26 (i) for the interpolation
(152)lr1=[lq1,lq2]a,r,
with 1/r1=(1-a)/q1+a/q2, we see that we have a partition ℤ=∑j∈ℤ‍Zj such that
(153)∥2-aj(∑n∈Zj‍λnq1)1/q1∥lr1+∥2(1-a)j(∑n∈Zj‍λnq2)1/q2∥lr1≤C∥λj∥lr1.
Moreover, since f∈ℬ˙q2,q2-t,k(ℝd), we have
(154)((∑j∈Zn‍2-jq2t∥Δjf∥Lkq2(ℝd)q2)1/q2)n∈ℤ∈lq2.
Let us note that Mn=(∑j∈Zn‍2-jq2t∥Δjf∥Lkq2(ℝd)q2)1/q2, Nn=2-an(∑j∈Zn‍λjq1)1/q1, Ln=2(1-a)n(∑j∈Zn‍λjq2)1/q2, and fn=∑j∈Zn‍Δjf. We apply now (147) and Theorem 48 to obtain
(155)∥fn∥Lkp(ℝd)≤C∥fn∥ℬ˙q1,q1s,k(ℝd)1-θ∥fn∥ℬ˙q2,q2-t,k(ℝd)θ∥fn∥Lkp(ℝd)≤CNn1-θMnθ2na(1-θ),∥fn∥Lkp,q2(ℝd)≤C∥fn∥ℬ˙q1,q2s,k(ℝd)1-θ∥fn∥ℬ˙q2,q2-t,k(ℝd)θ∥fn∥Lkp,q2(ℝd)≤CLn1-θMnθ2-n(1-a)(1-θ).
Since we have f=∑n∈ℤ‍fn, with these two inequalities at hand, and using (57), we find that f∈[Lkp(ℝd),Lkp,q2(ℝd)]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 [Lkp(ℝd),Lkp,q2(ℝd)]a,r=Lkr(ℝd) with 1/r=(1-θ)/r1+θ/q2.Theorem 54.
Lets,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
(156)∥f∥Lkp,r(ℝd)≤C∥f∥ℬ˙q1,r1s,k(ℝd)1-θ∥f∥ℬ˙q2,r2-t,k(ℝd)θ.Proof.
Once the previous theorem is proved, it is enough to reapply similar arguments to obtain Theorem54. As q1<r1<r2<q2, we start using
(157)lr1=[lq1,lr2]a,r1
instead of (152), and we obtain a partition ℤ=∑j∈ℤ‍Zj such that
(158)∥2-aj(∑n∈Zj‍λnq1)1/q1∥lr1+∥2(1-a)j(∑n∈Zj‍λnr2)1/r2∥lr1≤C∥λj∥lr1,
with 1/r1=(1-a)/q1+a/r2 and where λj=2sj∥Δjf∥Lkq1(ℝd) belongs to lr1, since f∈ℬ˙q1,r1s,k(ℝd). Moreover, since f∈ℬ˙q2,r2-t,k(ℝd), we have
(159)((∑j∈Zn‍2-jq2t∥Δjf∥Lkq2(ℝd)q2)1/q2)n∈ℤ∈lq2.
Let us note that Mn=(∑j∈Zn‍2-jq2t∥Δjf∥Lkq2(ℝd)q2)1/q2, Nn=2-an(∑j∈Zn‍λjq1)1/q1, Ln=2(1-a)n(∑j∈Zn‍λjq2)1/q2, and fn=∑j∈Zn‍Δjf. We apply now (151) and Theorem 48 instead of (155) to obtain
(160)∥fn∥Lkp,b(ℝd)≤C∥fn∥ℬ˙q1,q1s,k(ℝd)1-θ∥fn∥ℬ˙q2,r2-t,k(ℝd)θ∥fn∥Lkp,b(ℝd)≤CNn1-θMnθ2na(1-θ),
where 1/b=(1-θ)/q1+θ/r2 and
(161)∥fn∥Lkp,r2(ℝd)≤C∥fn∥ℬ˙q1,r2s,k(ℝd)1-θ∥fn∥ℬ˙q2,r2-t,k(ℝd)θ∥fn∥Lkp,r2(ℝd)≤CNn1-θMnθ2-n(1-a)(1-θ).
Finally, we have via (57) that f∈[Lkp,b(ℝd),Lkp,r2(ℝd)]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∈Lkp,r(ℝd) with 1/r=(1-θ)/r1+θ/r2.Conjecture 55.
Theorems34, 39, and 41 are true for the general reflection group G.



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*Source: 302941-2013-09-08.xml*