# Heat Equations Associated with Weinstein Operator and Applications

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

---

## Abstract

We establish a characterization for the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup. Next, we obtain the generalized Sobolev embedding theorems.


---

## Body

## 1. Introduction

We consider the Weinstein operator defined onℝd  ×  ]0,∞[ by
(1)Δβ:=∑i=1d+1‍∂2∂xi2+2β+1xd+1∂∂xd+1,β>-12=Δd+ℒβ,
where Δd is the Laplacian for the d-first variables and ℒβ is the Bessel operator for the last variable given by
(2)ℒβ=∂2∂xd+12+2β+1xd+1∂∂xd+1,β>-12.
For d>2, the operator Δβ is the Laplace-Beltrami operator on the Riemannian space ℝd×]0,∞[ equipped with the following metric:
(3)ds2=xd+12(2β+1)/(d-1)∑i=1d+1‍dxi2
(cf. [1, 2]). The Weinstein operator Δβ has several applications in pure and applied mathematics especially in the fluid mechanics (cf. [3]).The harmonic analysis associated with the Weinstein operator is studied by Ben Nahia and Ben Salem (cf. [1, 2]). In particular, the authors have introduced and studied the generalized Fourier transform associated with the Weinstein operator. This transform is called the Weinstein transform. We note that, for this transform, we have studied the uncertainty principle (cf. [4]) and the Gabor transform (cf. [5]).In the present paper, we intend to continue our study of generalized spaces of type Sobolev associated with the Weinstein operator that started in [6].In this paper we consider the Weinstein heat equation(4)∂tu-Δβu=f(t,x),(t,x)∈(0,∞)×ℝ+d+1,u|t=0=g.
We study (4) to focus on the following problems.(1)
Characterize the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup.(2)
Prove the imbedding Sobolev theorems.I have studied the generalized Sobolev spaces in the context of differential-differences operators (cf. [7–10]).The remaining part of the paper is organized as follows. Section2 is a summary of the main results in the harmonic analysis associated with the Weinstein operators. In Section 3, we introduce and study the homogeneous Weinstein-Besov spaces, the homogeneous Weinstein-Triebel-Lizorkin spaces and the homogeneous Weinstein-Riesz potential spaces. In Section 4, we characterize the homogeneous Weinstein Besov spaces via the Weinstein heat semigroup. Next, we prove the Sobolev embedding theorems.


## 2. Preliminaries

In order to confirm the basic and standard notations, we briefly overview the Weinstein operator and the related harmonic analysis. Main references are [1, 2].


### 2.1. Harmonic Analysis Associated with the Weinstein Operator

In this subsection, we collect some notations and results on the Weinstein kernel, the Weinstein intertwining operator and its dual, the Weinstein transform, and the Weinstein convolution.In the following,(5)ℝ+d+1=ℝd×[0,∞[,x=(x1,…,xd,xd+1)=(x′,xd+1)∈ℝ+d+1,∥x∥=x12+⋯+xd+12.
We denote by C*(ℝd+1) the space of continuous functions on ℝd+1, even with respect to the last variable; C*p(ℝd+1) the space of functions of class Cp on ℝd+1, even with respect to the last variable; ℰ*(ℝd+1) the space of C∞-functions on ℝd+1, even with respect to the last variable; 𝒮*(ℝd+1) the Schwartz space of rapidly decreasing functions on ℝd+1, even with respect to the last variable; D*(ℝd+1) the space of C∞-functions on ℝd+1 which are of compact support, even with respect to the last variable; 𝒮*′(ℝd+1) the space of temperate distributions on ℝd+1, even with respect to the last variable. It is the topological dual of 𝒮*(ℝd+1).We consider the Weinstein operatorΔβ defined by
(6)++++++++∀x=(x′,xd+1)∈ℝd×]0,∞[,Δβf(x)=Δx′f(x′,xd+1)+ℒβ,xd+1f(x′,xd+1),++++++++++++++++f∈C*2(ℝd+1),
where Δx′ is the Laplace operator on ℝd and ℒβ,xd+1 is the Bessel operator on ]0,∞[ given by
(7)ℒβ,xd+1:=d2dxd+12+2β+1xd+1ddxd+1,β>-12.The Weinstein kernelΛ is given by
(8)Λ(x,z):=ei〈x′,z′〉jβ(xd+1zd+1),++++ii∀(x,z)∈ℝd+1×ℂd+1,
where jβ(xd+1zd+1) is the normalized Bessel function. The Weinstein kernel satisfies the following properties.(i)
For eachz∈ℝ+d+1, we have
(9)ΔβΛ(x,z)=-∥z∥2Λ(x,z)∀x∈ℝ+d+1.(ii)
For allz,t∈ℂd+1, we have
(10)Λ(z,t)=Λ(t,z),Λ(z,0)=1,Λ(λz,t)=Λ(z,λt),∀λ∈ℂ.(iii)
For allν∈ℕd+1,  x∈ℝd+1, and z∈ℂd+1, we have
(11)|DzνΛ(x,z)|≤∥x∥|ν|exp(∥x∥∥Imz∥),where Dzν=∂ν/(∂z1ν1⋯∂zd+1νd+1) and |ν|=ν1+⋯+νd+1. In particular,
(12)|Λ(x,y)|≤1,∀x,y∈ℝd+1.The Weinstein intertwining operator is the operatorℛβ defined on C*(ℝd+1) by
(13)ℛβf(x′,xd+1)={2Γ(β+1)πΓ(β+1/2)xd+1-2β×∫0xd+1(xd+12-t2)β-1/2f(x′,t)dt,xd+1>0,f(x′,0),xd+1=0.ℛ
β is a topological isomorphism from ℰ*(ℝd+1) onto itself satisfying the following transmutation relation:
(14)Δβ(ℛβf)=ℛβ(Δd+1f),∀f∈ℰ*(ℝd+1),
where Δd+1=∑j=1d+1‍∂j2 is the Laplacian on ℝd+1.We denote byLβp(ℝ+d+1) the space of measurable functions on ℝ+d+1 such that
(15)∥f∥Lβp(ℝ+d+1)=(∫ℝ+d+1‍|f(x)|pdμβ(x))1/p<∞,++++++++++++++++if1≤p<∞,∥f∥Lβ∞(ℝ+d+1)=esssupx∈ℝ+d+1|f(x)|<∞,
where dμβ is the measure on ℝ+d+1 given by
(16)dμβ(x′,xd+1):=xd+12β+1dx′dxd+1.The Weinstein transform is given forf in Lβ1(ℝ+d+1) by
(17)ℱW(f)(y)=∫ℝ+d+1‍f(x)Λ(-x,y)dμβ(x),∀y∈ℝ+d+1.Some basic properties of this transform are as follows.(i)
Forf in Lβ1(ℝ+d+1),
(18)∥ℱW(f)∥Lβ∞(ℝ+d+1)≤∥f∥Lβ1(ℝ+d+1).(ii)
Forf in 𝒮*(ℝd+1), we have
(19)ℱW(Δβf)(y)=-∥y∥2ℱW(f)(y),∀y∈ℝ+d+1.(iii)
For allf in Lβ1(ℝ+d+1), if ℱW(f) belongs to Lβ1(ℝ+d+1), then
(20)f(y)=C(β)∫ℝ+d+1‍ℱW(f)(x)Λ(x,y)dμβ(x),a.e.,
where
(21)C(β):=1πd4β+d/2(Γ(β+1))2.(iv)
Forf∈𝒮*(ℝd+1), if we define
(22)ℱW¯(f)(y)=ℱW(f)(-y),
then
(23)(ℱW)-1=1C(β)ℱW¯.Proposition 1 (see [2]).
(i)  The Weinstein transformℱW is a topological isomorphism from 𝒮*(ℝd+1) onto itself, and for all f in 𝒮*(ℝd+1),
(24)∫ℝ+d+1‍|f(x)|2dμβ(x)=C(β)∫ℝ+d+1‍|ℱW(f)(ξ)|2dμβ(ξ).
(ii)  In particular, the renormalized Weinstein transformf→C(β)1/2ℱW(f) can be uniquely extended to an isometric isomorphism from Lβ2(ℝ+d+1) onto itself.In the Fourier analysis, the translation operator is given byf↦f(·+x).In harmonic analysis associated for the operatorΔβ, the generalized translation operator τx,x∈ℝ+d+1 is defined by
(25)∀y∈ℝ+d+1,τxf(y)=Γ(β+1)πΓ(β+1/2)+++==×∫0π‍f(x′+y′,xd+12+yd+12+2xd+1yd+1cosθ)+++++++×(sinθ)2βdθ,
where f∈C*(ℝd+1).By using the Weinstein kernel, we can also define a generalized translation. For functionsf∈𝒮*(ℝd+1) and y∈ℝ+d+1 the generalized translation τyf is defined by the following relation:
(26)ℱW(τyf)(x)=Λ(x,y)ℱW(f)(x).
By using the generalized translation, we define the generalized convolution product f*Wg of functions f,g∈Lβ1(ℝ+d+1) as follows:
(27)f*Wg(x)=∫ℝ+d+1‍τxf(-y′,yd+1)g(y)dμβ(y).
This convolution is commutative and associative, and it satisfies the following.(i)
For allf,g∈Lβ1(ℝ+d+1), f*Wg belongs to Lβ1(ℝ+d+1) and
(28)ℱW(f*Wg)=ℱW(f)ℱW(g).(ii)
Let1≤p,q,r≤∞ such that 1/p+1/q-1/r=1. If f∈Lβp(ℝ+d+1) and g∈Lβq(ℝ+d+1), then f*Wg∈Lβr(ℝ+d+1) and
(29)∥f*Wg∥Lβr(ℝ+d+1)≤∥f∥Lβp(ℝ+d+1)∥g∥Lβq(ℝ+d+1).We define the tempered distribution𝒯f associated with f∈Lβp(ℝ+d+1) by
(30)〈𝒯f,ϕ〉=∫ℝ+d+1‍f(x)ϕ(x)dμβ(x),
for ϕ∈𝒮*(ℝd+1) and denote by 〈f,ϕ〉β the integral in the righthand side.Definition 2.
The Weinstein transformℱW(τ) of a distribution τ∈𝒮*′(ℝd+1) is defined by
(31)〈ℱW(τ),ϕ〉=〈τ,ℱW(ϕ)〉,
for ϕ∈𝒮*(ℝd+1).In particular, forf∈Lβp(ℝ+d+1), it follows that for ϕ∈𝒮*(ℝd+1),
(32)〈ℱW(f),ϕ〉=〈ℱW(𝒯f),ϕ〉=〈𝒯f,ℱW(ϕ)〉=〈f,ℱW(ϕ)〉β.Proposition 3.
The Weinstein transformℱW is a topological isomorphism from 𝒮*′(ℝd+1) onto itself.Definition 4.
The generalized convolution product of a distributionS in 𝒮*′(ℝd+1) and a function ϕ in 𝒮*(ℝd+1) is the function S*Wϕ defined by
(33)S*Wϕ(x)=〈Sy,τ-yϕ(x)〉.Proposition 5.
Letf be in Lβp(ℝ+d+1), 1≤p≤∞, and ϕ in 𝒮*(ℝd+1). Then, the distribution 𝒯f*Wϕ is given by the function f*Wϕ and 𝒯f*Wϕ belongs to Lβp(ℝ+d+1). Moreover, for all ψ∈𝒮*(ℝd+1),
(34)〈𝒯f*Wϕ,ψ〉=〈fˇ,ϕ*Wψˇ〉β,
where ψˇ(x)=ψ(-x) and
(35)ℱW(𝒯f*Wϕ)=ℱW(𝒯f)ℱW(ϕ).For eachu∈𝒮*′(ℝd+1), we define the distribution Δβu by 〈Δβu,ψ〉=〈u,Δβψ〉, and this distribution satisfies the following property:
(36)ℱW(Δβu)=-∥y∥2ℱW(u).
In the following, we denote 𝒯f given by (30) by f for simplicity.


## 2.1. Harmonic Analysis Associated with the Weinstein Operator

In this subsection, we collect some notations and results on the Weinstein kernel, the Weinstein intertwining operator and its dual, the Weinstein transform, and the Weinstein convolution.In the following,(5)ℝ+d+1=ℝd×[0,∞[,x=(x1,…,xd,xd+1)=(x′,xd+1)∈ℝ+d+1,∥x∥=x12+⋯+xd+12.
We denote by C*(ℝd+1) the space of continuous functions on ℝd+1, even with respect to the last variable; C*p(ℝd+1) the space of functions of class Cp on ℝd+1, even with respect to the last variable; ℰ*(ℝd+1) the space of C∞-functions on ℝd+1, even with respect to the last variable; 𝒮*(ℝd+1) the Schwartz space of rapidly decreasing functions on ℝd+1, even with respect to the last variable; D*(ℝd+1) the space of C∞-functions on ℝd+1 which are of compact support, even with respect to the last variable; 𝒮*′(ℝd+1) the space of temperate distributions on ℝd+1, even with respect to the last variable. It is the topological dual of 𝒮*(ℝd+1).We consider the Weinstein operatorΔβ defined by
(6)++++++++∀x=(x′,xd+1)∈ℝd×]0,∞[,Δβf(x)=Δx′f(x′,xd+1)+ℒβ,xd+1f(x′,xd+1),++++++++++++++++f∈C*2(ℝd+1),
where Δx′ is the Laplace operator on ℝd and ℒβ,xd+1 is the Bessel operator on ]0,∞[ given by
(7)ℒβ,xd+1:=d2dxd+12+2β+1xd+1ddxd+1,β>-12.The Weinstein kernelΛ is given by
(8)Λ(x,z):=ei〈x′,z′〉jβ(xd+1zd+1),++++ii∀(x,z)∈ℝd+1×ℂd+1,
where jβ(xd+1zd+1) is the normalized Bessel function. The Weinstein kernel satisfies the following properties.(i)
For eachz∈ℝ+d+1, we have
(9)ΔβΛ(x,z)=-∥z∥2Λ(x,z)∀x∈ℝ+d+1.(ii)
For allz,t∈ℂd+1, we have
(10)Λ(z,t)=Λ(t,z),Λ(z,0)=1,Λ(λz,t)=Λ(z,λt),∀λ∈ℂ.(iii)
For allν∈ℕd+1,  x∈ℝd+1, and z∈ℂd+1, we have
(11)|DzνΛ(x,z)|≤∥x∥|ν|exp(∥x∥∥Imz∥),where Dzν=∂ν/(∂z1ν1⋯∂zd+1νd+1) and |ν|=ν1+⋯+νd+1. In particular,
(12)|Λ(x,y)|≤1,∀x,y∈ℝd+1.The Weinstein intertwining operator is the operatorℛβ defined on C*(ℝd+1) by
(13)ℛβf(x′,xd+1)={2Γ(β+1)πΓ(β+1/2)xd+1-2β×∫0xd+1(xd+12-t2)β-1/2f(x′,t)dt,xd+1>0,f(x′,0),xd+1=0.ℛ
β is a topological isomorphism from ℰ*(ℝd+1) onto itself satisfying the following transmutation relation:
(14)Δβ(ℛβf)=ℛβ(Δd+1f),∀f∈ℰ*(ℝd+1),
where Δd+1=∑j=1d+1‍∂j2 is the Laplacian on ℝd+1.We denote byLβp(ℝ+d+1) the space of measurable functions on ℝ+d+1 such that
(15)∥f∥Lβp(ℝ+d+1)=(∫ℝ+d+1‍|f(x)|pdμβ(x))1/p<∞,++++++++++++++++if1≤p<∞,∥f∥Lβ∞(ℝ+d+1)=esssupx∈ℝ+d+1|f(x)|<∞,
where dμβ is the measure on ℝ+d+1 given by
(16)dμβ(x′,xd+1):=xd+12β+1dx′dxd+1.The Weinstein transform is given forf in Lβ1(ℝ+d+1) by
(17)ℱW(f)(y)=∫ℝ+d+1‍f(x)Λ(-x,y)dμβ(x),∀y∈ℝ+d+1.Some basic properties of this transform are as follows.(i)
Forf in Lβ1(ℝ+d+1),
(18)∥ℱW(f)∥Lβ∞(ℝ+d+1)≤∥f∥Lβ1(ℝ+d+1).(ii)
Forf in 𝒮*(ℝd+1), we have
(19)ℱW(Δβf)(y)=-∥y∥2ℱW(f)(y),∀y∈ℝ+d+1.(iii)
For allf in Lβ1(ℝ+d+1), if ℱW(f) belongs to Lβ1(ℝ+d+1), then
(20)f(y)=C(β)∫ℝ+d+1‍ℱW(f)(x)Λ(x,y)dμβ(x),a.e.,
where
(21)C(β):=1πd4β+d/2(Γ(β+1))2.(iv)
Forf∈𝒮*(ℝd+1), if we define
(22)ℱW¯(f)(y)=ℱW(f)(-y),
then
(23)(ℱW)-1=1C(β)ℱW¯.Proposition 1 (see [2]).
(i)  The Weinstein transformℱW is a topological isomorphism from 𝒮*(ℝd+1) onto itself, and for all f in 𝒮*(ℝd+1),
(24)∫ℝ+d+1‍|f(x)|2dμβ(x)=C(β)∫ℝ+d+1‍|ℱW(f)(ξ)|2dμβ(ξ).
(ii)  In particular, the renormalized Weinstein transformf→C(β)1/2ℱW(f) can be uniquely extended to an isometric isomorphism from Lβ2(ℝ+d+1) onto itself.In the Fourier analysis, the translation operator is given byf↦f(·+x).In harmonic analysis associated for the operatorΔβ, the generalized translation operator τx,x∈ℝ+d+1 is defined by
(25)∀y∈ℝ+d+1,τxf(y)=Γ(β+1)πΓ(β+1/2)+++==×∫0π‍f(x′+y′,xd+12+yd+12+2xd+1yd+1cosθ)+++++++×(sinθ)2βdθ,
where f∈C*(ℝd+1).By using the Weinstein kernel, we can also define a generalized translation. For functionsf∈𝒮*(ℝd+1) and y∈ℝ+d+1 the generalized translation τyf is defined by the following relation:
(26)ℱW(τyf)(x)=Λ(x,y)ℱW(f)(x).
By using the generalized translation, we define the generalized convolution product f*Wg of functions f,g∈Lβ1(ℝ+d+1) as follows:
(27)f*Wg(x)=∫ℝ+d+1‍τxf(-y′,yd+1)g(y)dμβ(y).
This convolution is commutative and associative, and it satisfies the following.(i)
For allf,g∈Lβ1(ℝ+d+1), f*Wg belongs to Lβ1(ℝ+d+1) and
(28)ℱW(f*Wg)=ℱW(f)ℱW(g).(ii)
Let1≤p,q,r≤∞ such that 1/p+1/q-1/r=1. If f∈Lβp(ℝ+d+1) and g∈Lβq(ℝ+d+1), then f*Wg∈Lβr(ℝ+d+1) and
(29)∥f*Wg∥Lβr(ℝ+d+1)≤∥f∥Lβp(ℝ+d+1)∥g∥Lβq(ℝ+d+1).We define the tempered distribution𝒯f associated with f∈Lβp(ℝ+d+1) by
(30)〈𝒯f,ϕ〉=∫ℝ+d+1‍f(x)ϕ(x)dμβ(x),
for ϕ∈𝒮*(ℝd+1) and denote by 〈f,ϕ〉β the integral in the righthand side.Definition 2.
The Weinstein transformℱW(τ) of a distribution τ∈𝒮*′(ℝd+1) is defined by
(31)〈ℱW(τ),ϕ〉=〈τ,ℱW(ϕ)〉,
for ϕ∈𝒮*(ℝd+1).In particular, forf∈Lβp(ℝ+d+1), it follows that for ϕ∈𝒮*(ℝd+1),
(32)〈ℱW(f),ϕ〉=〈ℱW(𝒯f),ϕ〉=〈𝒯f,ℱW(ϕ)〉=〈f,ℱW(ϕ)〉β.Proposition 3.
The Weinstein transformℱW is a topological isomorphism from 𝒮*′(ℝd+1) onto itself.Definition 4.
The generalized convolution product of a distributionS in 𝒮*′(ℝd+1) and a function ϕ in 𝒮*(ℝd+1) is the function S*Wϕ defined by
(33)S*Wϕ(x)=〈Sy,τ-yϕ(x)〉.Proposition 5.
Letf be in Lβp(ℝ+d+1), 1≤p≤∞, and ϕ in 𝒮*(ℝd+1). Then, the distribution 𝒯f*Wϕ is given by the function f*Wϕ and 𝒯f*Wϕ belongs to Lβp(ℝ+d+1). Moreover, for all ψ∈𝒮*(ℝd+1),
(34)〈𝒯f*Wϕ,ψ〉=〈fˇ,ϕ*Wψˇ〉β,
where ψˇ(x)=ψ(-x) and
(35)ℱW(𝒯f*Wϕ)=ℱW(𝒯f)ℱW(ϕ).For eachu∈𝒮*′(ℝd+1), we define the distribution Δβu by 〈Δβu,ψ〉=〈u,Δβψ〉, and this distribution satisfies the following property:
(36)ℱW(Δβu)=-∥y∥2ℱW(u).
In the following, we denote 𝒯f given by (30) by f for simplicity.


## 3.ℬ˙p,qs,β, ℱ˙p,qs,β(ℝ+d+1), ℋ˙p,βs Spaces and Basic Properties

### 3.1. Homogeneous Weinstein-Littlewood-Paley Decomposition

One of the main tools in this paper is the homogeneous Littlewood-Paley decomposition of distribution associated with the Weinstein operators into dyadic blocs of frequencies.Lemma 6.
One defines by𝒞 the ring of center 0, of small radius 1/2 and great radius 2. There exists two radial functions ψ and φ the values of which are in the interval [0,1] belonging to D*(ℝd+1) such that
(37)suppψ⊂B(0,1),suppφ⊂𝒞,∀ξ∈ℝd+1,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
(38)Δjf=ℱW-1(φ(ξ2j)ℱW(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 Weinstein operators.Throughout this paper, we defineϕ and χ by ϕ=ℱW-1(φ) and χ=ℱW-1(ψ).When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functionsψ~ and φ~ belonging to D*(ℝd+1) such that ψ~≡1 on suppψ and φ~≡1 on suppφ.Remark 7.
We remark that(39)ℱW(Sjf)(ξ)=ψ~(ξ2j)ℱW(Sjf)(ξ),ℱW(Δjf)(ξ)=φ~(ξ2j)ℱW(Δjf)(ξ).We put(40)ϕ~=ℱW-1(φ~),χ~=ℱW-1(ψ~).Definition 8.
One denotes by𝒮h,β,*′(ℝd+1) the space of tempered distribution such that
(41)limj→-∞Sju=0in𝒮*′(ℝd+1).Proposition 9 (Bernstein inequalities).
For allμ∈ℕd+1 and σ∈ℝ, for all j∈ℤ, for all 1≤p, q≤∞, and for all f∈𝒮*′(ℝd+1), one has the following
(42)(i)∥Δjf∥Lβq(ℝ+d+1)≤∥ϕ~∥Lβr(ℝ+d+1)∥Δjf∥Lβp(ℝ+d+1)2j(d+2β+2)(1/p-1/q),++++++++++++with1q=1p+1r-1.(ii)∥Sjf∥Lβq(ℝ+d+1)≤∥χ~∥Lβr(ℝ+d+1)∥Sjf∥Lβp(ℝ+d+1)2j(d+2β+2)(1/p-1/q),++++++++++++with1q=1p+1r-1.(iii)∥(-Δβ)σΔjf∥Lβp(ℝ+d+1)≤∥ℱW-1(∥ξ∥σφ~)∥Lβ1(ℝ+d+1)++++++++++++++×∥Δjf∥Lβp(ℝ+d+1)2jσ.Proof.
Using Remark7, we deduce from Proposition 5 that
(43)Sjf=2j(d+2β+2)χ~(2j·)*WSjf,Δjf=2j(d+2β+2)ϕ~(2j·)*WΔjf.
Thus, from the relation (29), we prove (i), (ii), and (iii).


### 3.2. Definitions

In the following, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Weinstein operators onℝ+d+1 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 10.
Lets∈ℝ and p,q∈[1,∞]. The homogeneous Weinstein-Besov spaces ℬ˙p,qs,β(ℝ+d+1) are the spaces of distribution in 𝒮h,β,*′(ℝd+1) such that
(44)∥f∥ℬ˙p,qs,β(ℝ+d+1):=(∑j∈ℤ‍(2sj∥Δjf∥Lβp(ℝ+d+1))q)1/q<∞.Proposition 11 (see [6]).
Lets∈ℝ and p and q two elements of [1,∞]; the space ℬ˙p,qs,β(ℝ+d+1) is the set of f∈𝒮h,β,*′(ℝd+1) verifying
(45)∥f˙∥ℬ˙p,qs,β(ℝ+d+1)=(∫0∞‍(t-s∥f*Wϕt∥Lβp(ℝ+d+1))qdtt)1/q<∞,
where ϕt(x)=(1/t2β+2+d)ϕ(x/t), for all t∈(0,∞) and x∈ℝ+d+1.Definition 12.
Fors∈ℝ and p,q∈[1,∞], one writes
(46)∥f∥Bp,qs,β(ℝ+d+1)=∥S0f∥Lβp(ℝ+d+1)+(∑j≥1‍(2sj∥Δjf∥Lβp(ℝ+d+1))q)1/q.
The nonhomogeneous Besov spaceBp,qs,β(ℝ+d+1) associated with the Weinstein operators is defined by
(47)Bp,qs,β(ℝ+d+1)={f∈𝒮′(ℝd):∥f∥Bp,qs,β(ℝ+d+1)<∞}.We give now another definition equivalent to the nonhomogeneous Besov spaceBp,qs,β(ℝ+d+1).Proposition 13.
Lets∈ℝ and p and q two elements of [1,∞]; the space Bp,qs,β(ℝ+d+1) is the set of f∈𝒮′(ℝd) verifying
(48)∥f˙∥Bp,qs,β(ℝ+d+1)=∥f*Wψ∥Lβp(ℝ+d+1)+(∫01‍(t-s∥f*Wϕt∥Lβp(ℝ+d+1))qdtt)1/q<∞.Definition 14.
Lets∈ℝ and 1≤p, q≤∞, the homogeneous Weinstein-Triebel-Lizorkin space ℱ˙p,qs,β(ℝ+d+1) is the space of distribution in 𝒮h,β,*′(ℝd+1) such that
(49)∥f∥ℱ˙p,qs,β(ℝ+d+1):=∥(∑j∈ℤ‍2sjq|Δjf|q)1/q∥Lβp(ℝ+d+1)<∞.Definition 15.
Fors∈ℝ, the operator ℛβs from 𝒮h,β,*′(ℝd+1) to 𝒮h,β,*′(ℝd+1) is defined by
(50)ℛβs(f)=ℱW-1(∥·∥sℱWf).
The operator ℛβ-s is called Weinstein-Riesz potential space.Definition 16.
Fors∈ℝ and 1≤p≤∞, the homogeneous Weinstein-Riesz potential space ℋ˙p,βs(ℝ+d+1) is defined as the space ℛβ-s(Lβp(ℝ+d+1)), equipped with the norm ∥f∥ℋ˙p,βs(ℝ+d+1)=∥ℛβs(f)∥Lβp(ℝ+d+1).Proposition 17.
Lets∈ℝ and 1≤p,q≤∞.
The operatorΔβ is a linear continuous operator from ℬ˙p,qs,β(ℝ+d+1) into ℬ˙p,qs-2,β(ℝ+d+1) and from ℋ˙p,βs(ℝ+d+1) into ℋ˙p,βs-2(ℝ+d+1).Proof.
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).Proposition 18.
Lets,t∈ℝ and 1≤p,q≤∞. The operator ℛβt is a linear continuous injective operator from ℬ˙p,qs,β(ℝ+d+1) onto ℬ˙p,qs-t,β(ℝ+d+1) and from ℋ˙p,βs(ℝ+d+1) onto ℋ˙p,βs-t(ℝ+d+1).Proof.
We obtain these results by the similar ideas used in the nonhomogeneous case  (cf. [6]).


### 3.3. Embeddings

As in the Euclidean case (cf. [11]) the monotone character of lq-spaces and the Minkowskis inequality yield the following.Proposition 19.
If1≤q1<q2≤∞ one has
(51)ℬ˙p,q1s,β(ℝ+d+1)↪ℬ˙p,q2s,β(ℝ+d+1),(1≤p≤∞,s∈ℝ).
Moreover,
(52)ℬ˙p,1s,β(ℝ+d+1)↪ℋ˙p,βs(ℝ+d+1)↪ℬ˙p,∞s,β(ℝ+d+1),++++++++++++(1≤p≤∞,s∈ℝ).
If s0≠s1 one also has
(53)(ℋ˙p,βs0(ℝ+d+1),ℋ˙p,βs1(ℝ+d+1))θ,q=ℬ˙p,qs,β(ℝ+d+1),++++++++++(1≤p,q≤∞,θ∈(0,1)),
where s=(1-θ)s0+θs1.Proposition 20.
One assumes thats-(d+2β+2)/p=s1-(d+2β+2)/p1. Then the following inclusion holds
(54)+++++sis++ℬ˙p,qs,β(ℝ+d+1)↪ℬ˙p1,q1s1,β(ℝ+d+1),(1≤p≤p1≤∞,1≤q≤q1≤∞,s,s1∈ℝ).Proof.
In order to prove the inclusion, we use the estimate(55)Δjf=2j(d+2β+2)ϕ~(2j·)*WΔjf.
Proposition 9(i) gives that
(56)∥Δjf∥Lβp1(ℝ+d+1)=∥2j(d+2β+2)ϕ~(2j·)*WΔjf∥Lβp1(ℝ+d+1)≤C2j(d+2β+2)(1/p-1/p1)∥Δjf∥Lβp(ℝ+d+1).
By definition of the homogeneous Weinstein-Besov spaces, we therefore infer
(57)∥f∥ℬ˙p1,q1s1,β(ℝ+d+1)=(∑j=-∞∞(2js1∥Δjf∥Lβp1(ℝ+d+1))q1‍)1/q1≤C(∑j∈ℤ‍(2js12j(d+2β+2)(1/p-1/p1)∥Δjf∥Lβp(ℝ+d+1))q1)1/q1≤C(∑j∈ℤ(2js∥Δjf∥Lβp(ℝ+d+1))q1‍)1/q1≤C∥f∥ℬ˙p,qs,β(ℝ+d+1),
since q≤q1. This gives the inclusion.Proposition 21.
(1)  Ifu belongs to ℬ˙p,qs,β(ℝ+d+1)∩ℬ˙p,qt,β(ℝ+d+1), then u belongs to ℬ˙p,qθs+(1-θ)t,β(ℝ+d+1) for all θ∈[0,1] and
(58)∥u∥ℬ˙p,qθs+(1-θ)t,β(ℝ+d+1)≤∥u∥ℬ˙p,qs,β(ℝ+d+1)θ∥u∥ℬ˙p,qt,β(ℝ+d+1)1-θ.
(2)  Ifu belongs to ℬ˙p,∞s,β(ℝ+d+1)∩ℬ˙p,∞t,β(ℝ+d+1) and s<t, then u belongs to ℬ˙p,1θs+(1-θ)t,β(ℝ+d+1) for all θ∈(0,1) and there exists a positive constant C(t,s) such that
(59)∥u∥ℬ˙p,1θs+(1-θ)t,β(ℝ+d+1)≤C(t,s)∥u∥ℬ˙p,∞s,β(ℝ+d+1)θ∥u∥ℬ˙p,∞t,β(ℝ+d+1)1-θ.
(3)  Ifu belongs to ℬ˙p,∞s,β(ℝ+d+1)∩ℬ˙p,∞s+ε,β(ℝ+d+1) and ε>0, then u belongs to ℬ˙p,1s,β(ℝ+d+1) and there exists a positive constant C such that
(60)∥u∥ℬ˙p,1s,β(ℝ+d+1)≤Cε∥u∥ℬ˙p,∞s,β(ℝ+d+1)log2(e+∥u∥ℬ˙p,∞s+ε,β(ℝ+d+1)∥u∥ℬ˙p,∞s,β(ℝ+d+1)).Proof.
(1)  is obvious from the Hölder’s inequality. As for(2), we write ∥u∥ℬ˙p,1θs+(1-θ)t,β(ℝ+d+1) as
(61)∑j≤N‍2j(θs+(1-θ)t)∥Δju∥Lβp(ℝ+d+1)+∑j>N‍2j(θs+(1-θ)t)∥Δju∥Lβp(ℝ+d+1),
where N is chosen here after. By the definition of the homogeneous Weinstein-Besov norms, we see that
(62)2j(θs+(1-θ)t)∥Δju∥Lβp(ℝ+d+1)≤2j(1-θ)(t-s)∥u∥ℬ˙p,∞s,β(ℝ+d+1),2j(θs+(1-θ)t)∥Δju∥Lβp(ℝ+d+1)≤2-jθ(t-s)∥u∥ℬ˙p,∞t,β(ℝ+d+1),
and thus ∥u∥ℬ˙p,1θs+(1-θ)t,β(ℝ+d+1) is dominated by
(63)∥u∥ℬ˙p,∞s,β(ℝ+d+1)∑j≤N‍2j(1-θ)(t-s)+∥u∥ℬ˙p,∞t,β(ℝ+d+1)∑j>N‍2-jθ(t-s)≤C∥u∥ℬ˙p,∞s,β(ℝ+d+1)2(N+1)(1-θ)(t-s)2(1-θ)(t-s)-1+∥u∥ℬ˙p,∞t,β(ℝ+d+1)2-Nθ(t-s)1-2-θ(t-s).
Hence, in order to complete the proof of (2), it suffices to choose N such that
(64)∥u∥ℬ˙p,∞t,β(ℝd)∥u∥ℬ˙p,∞s,β(ℝd)≤2N(t-s)<2∥u∥ℬ˙p,∞t,β(ℝd)∥u∥ℬ˙p,∞s,β(ℝd).
As for (3), it is easy to see that ∥u∥ℬ˙p,1s,β(ℝ+d+1) is dominated as
(65)∑j≤N-1‍2js∥Δju∥Lβp(ℝ+d+1)+∑j≥N‍2js∥Δju∥Lβp(ℝ+d+1)≤(N+1)∥u∥ℬ˙p,∞s,β(ℝ+d+1)+2-(N-1)ε2ε-1∥u∥ℬ˙p,∞s+ε,β(ℝ+d+1).
Hence, letting
(66)N=1+[1εlog2∥u∥ℬ˙p,∞s+ε,β(ℝ+d+1)∥u∥ℬ˙p,∞s,β(ℝ+d+1)],
we can obtain the desired estimate.Proposition 22.
Letq∈(1,∞) and let s∈ℝ such that 0<s<(d+2β+2)/q, then one has
(67)ℬ˙q,qs,β(ℝ+d+1)=ℱ˙q,qs,β(ℝ+d+1)↪ℱ˙q,∞s,β(ℝ+d+1)↪ℱ˙∞,∞s-((d+2β+2)/q),β(ℝ+d+1),(68)ℋ˙q,βs(ℝ+d+1)=ℱ˙q,2s,β(ℝ+d+1)↪ℱ˙q,∞s,β(ℝ+d+1)↪ℱ˙∞,∞s-((d+2β+2)/q),β(ℝ+d+1).Proof.
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).Theorem 23.
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,β(ℝ+d+1)∩ℱ˙q2,∞-b,β(ℝ+d+1); then one has
(69)|f(x)|≤C(supj∈ℤ2aj|Δjf(x)|)1-θ(supj∈ℤ2-bj|Δjf(x)|)θ.
In particular, one gets
(70)∥f∥Lβp(ℝ+d+1)≤C∥f∥ℱ˙q1,∞a,β(ℝ+d+1)1-θ∥f∥ℱ˙q2,∞-b,β(ℝ+d+1)θ.Proof.
Letf be a Schwartz class; we have
(71)|f(x)|≤∑j∈ℤ‍|Δjf(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
(72)2jbsupj∈ℤ(2-jb|Δjf(x)|)≤2-ajsupj∈ℤ(2aj|Δjf(x)|)
and we write
(73)|f(x)|≤∑j≤N(x)‍2jbsupj∈ℤ(2-jb|Δjf(x)|)+∑j>N(x)‍2-ajsupj(2aj|Δjf(x)|)≤C(supj∈ℤ2aj|Δjf(x)|)b/(a+b)×(supj∈ℤ2-bj|Δjf(x)|)a/(a+b).
Thus, (69) is proved. In order to obtain (70), it is enough to apply the Hölder inequality in the expression above since we have θ=a/(a+b)∈(0,1) and let 1/p=(1-θ)/q1+θ/q2.Corollary 24.
Letq∈(1,∞) and let s∈ℝ such that 0<s<(d+2β+2)/q, then one has
(74)∥f∥Lβp(ℝ+d+1)≤C∥f∥ℬ˙∞,∞-((2β+2+d)/q-s),β(ℝ+d+1)1-(q/p)∥f∥ℬ˙q,qs,β(ℝ+d+1)q/p,(75)∥f∥Lβp(ℝ+d+1)≤C∥f∥ℬ˙∞,∞-((2β+2+d)/q-s),β(ℝ+d+1)1-(q/p)∥f∥ℋ˙q,βs(ℝ+d+1)q/p,
where p=q(2β+2+d)/(2β+2+d-qs).Proof.
By choosinga=s>0, -b=s-(d+2β+2)/q<0, q1=q, and q2=∞, we deduce (74) from the relations (70) and (67). In the same way, we deduce (75) from the relations (70) and (68).


## 3.1. Homogeneous Weinstein-Littlewood-Paley Decomposition

One of the main tools in this paper is the homogeneous Littlewood-Paley decomposition of distribution associated with the Weinstein operators into dyadic blocs of frequencies.Lemma 6.
One defines by𝒞 the ring of center 0, of small radius 1/2 and great radius 2. There exists two radial functions ψ and φ the values of which are in the interval [0,1] belonging to D*(ℝd+1) such that
(37)suppψ⊂B(0,1),suppφ⊂𝒞,∀ξ∈ℝd+1,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
(38)Δjf=ℱW-1(φ(ξ2j)ℱW(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 Weinstein operators.Throughout this paper, we defineϕ and χ by ϕ=ℱW-1(φ) and χ=ℱW-1(ψ).When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functionsψ~ and φ~ belonging to D*(ℝd+1) such that ψ~≡1 on suppψ and φ~≡1 on suppφ.Remark 7.
We remark that(39)ℱW(Sjf)(ξ)=ψ~(ξ2j)ℱW(Sjf)(ξ),ℱW(Δjf)(ξ)=φ~(ξ2j)ℱW(Δjf)(ξ).We put(40)ϕ~=ℱW-1(φ~),χ~=ℱW-1(ψ~).Definition 8.
One denotes by𝒮h,β,*′(ℝd+1) the space of tempered distribution such that
(41)limj→-∞Sju=0in𝒮*′(ℝd+1).Proposition 9 (Bernstein inequalities).
For allμ∈ℕd+1 and σ∈ℝ, for all j∈ℤ, for all 1≤p, q≤∞, and for all f∈𝒮*′(ℝd+1), one has the following
(42)(i)∥Δjf∥Lβq(ℝ+d+1)≤∥ϕ~∥Lβr(ℝ+d+1)∥Δjf∥Lβp(ℝ+d+1)2j(d+2β+2)(1/p-1/q),++++++++++++with1q=1p+1r-1.(ii)∥Sjf∥Lβq(ℝ+d+1)≤∥χ~∥Lβr(ℝ+d+1)∥Sjf∥Lβp(ℝ+d+1)2j(d+2β+2)(1/p-1/q),++++++++++++with1q=1p+1r-1.(iii)∥(-Δβ)σΔjf∥Lβp(ℝ+d+1)≤∥ℱW-1(∥ξ∥σφ~)∥Lβ1(ℝ+d+1)++++++++++++++×∥Δjf∥Lβp(ℝ+d+1)2jσ.Proof.
Using Remark7, we deduce from Proposition 5 that
(43)Sjf=2j(d+2β+2)χ~(2j·)*WSjf,Δjf=2j(d+2β+2)ϕ~(2j·)*WΔjf.
Thus, from the relation (29), we prove (i), (ii), and (iii).


## 3.2. Definitions

In the following, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Weinstein operators onℝ+d+1 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 10.
Lets∈ℝ and p,q∈[1,∞]. The homogeneous Weinstein-Besov spaces ℬ˙p,qs,β(ℝ+d+1) are the spaces of distribution in 𝒮h,β,*′(ℝd+1) such that
(44)∥f∥ℬ˙p,qs,β(ℝ+d+1):=(∑j∈ℤ‍(2sj∥Δjf∥Lβp(ℝ+d+1))q)1/q<∞.Proposition 11 (see [6]).
Lets∈ℝ and p and q two elements of [1,∞]; the space ℬ˙p,qs,β(ℝ+d+1) is the set of f∈𝒮h,β,*′(ℝd+1) verifying
(45)∥f˙∥ℬ˙p,qs,β(ℝ+d+1)=(∫0∞‍(t-s∥f*Wϕt∥Lβp(ℝ+d+1))qdtt)1/q<∞,
where ϕt(x)=(1/t2β+2+d)ϕ(x/t), for all t∈(0,∞) and x∈ℝ+d+1.Definition 12.
Fors∈ℝ and p,q∈[1,∞], one writes
(46)∥f∥Bp,qs,β(ℝ+d+1)=∥S0f∥Lβp(ℝ+d+1)+(∑j≥1‍(2sj∥Δjf∥Lβp(ℝ+d+1))q)1/q.
The nonhomogeneous Besov spaceBp,qs,β(ℝ+d+1) associated with the Weinstein operators is defined by
(47)Bp,qs,β(ℝ+d+1)={f∈𝒮′(ℝd):∥f∥Bp,qs,β(ℝ+d+1)<∞}.We give now another definition equivalent to the nonhomogeneous Besov spaceBp,qs,β(ℝ+d+1).Proposition 13.
Lets∈ℝ and p and q two elements of [1,∞]; the space Bp,qs,β(ℝ+d+1) is the set of f∈𝒮′(ℝd) verifying
(48)∥f˙∥Bp,qs,β(ℝ+d+1)=∥f*Wψ∥Lβp(ℝ+d+1)+(∫01‍(t-s∥f*Wϕt∥Lβp(ℝ+d+1))qdtt)1/q<∞.Definition 14.
Lets∈ℝ and 1≤p, q≤∞, the homogeneous Weinstein-Triebel-Lizorkin space ℱ˙p,qs,β(ℝ+d+1) is the space of distribution in 𝒮h,β,*′(ℝd+1) such that
(49)∥f∥ℱ˙p,qs,β(ℝ+d+1):=∥(∑j∈ℤ‍2sjq|Δjf|q)1/q∥Lβp(ℝ+d+1)<∞.Definition 15.
Fors∈ℝ, the operator ℛβs from 𝒮h,β,*′(ℝd+1) to 𝒮h,β,*′(ℝd+1) is defined by
(50)ℛβs(f)=ℱW-1(∥·∥sℱWf).
The operator ℛβ-s is called Weinstein-Riesz potential space.Definition 16.
Fors∈ℝ and 1≤p≤∞, the homogeneous Weinstein-Riesz potential space ℋ˙p,βs(ℝ+d+1) is defined as the space ℛβ-s(Lβp(ℝ+d+1)), equipped with the norm ∥f∥ℋ˙p,βs(ℝ+d+1)=∥ℛβs(f)∥Lβp(ℝ+d+1).Proposition 17.
Lets∈ℝ and 1≤p,q≤∞.
The operatorΔβ is a linear continuous operator from ℬ˙p,qs,β(ℝ+d+1) into ℬ˙p,qs-2,β(ℝ+d+1) and from ℋ˙p,βs(ℝ+d+1) into ℋ˙p,βs-2(ℝ+d+1).Proof.
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).Proposition 18.
Lets,t∈ℝ and 1≤p,q≤∞. The operator ℛβt is a linear continuous injective operator from ℬ˙p,qs,β(ℝ+d+1) onto ℬ˙p,qs-t,β(ℝ+d+1) and from ℋ˙p,βs(ℝ+d+1) onto ℋ˙p,βs-t(ℝ+d+1).Proof.
We obtain these results by the similar ideas used in the nonhomogeneous case  (cf. [6]).


## 3.3. Embeddings

As in the Euclidean case (cf. [11]) the monotone character of lq-spaces and the Minkowskis inequality yield the following.Proposition 19.
If1≤q1<q2≤∞ one has
(51)ℬ˙p,q1s,β(ℝ+d+1)↪ℬ˙p,q2s,β(ℝ+d+1),(1≤p≤∞,s∈ℝ).
Moreover,
(52)ℬ˙p,1s,β(ℝ+d+1)↪ℋ˙p,βs(ℝ+d+1)↪ℬ˙p,∞s,β(ℝ+d+1),++++++++++++(1≤p≤∞,s∈ℝ).
If s0≠s1 one also has
(53)(ℋ˙p,βs0(ℝ+d+1),ℋ˙p,βs1(ℝ+d+1))θ,q=ℬ˙p,qs,β(ℝ+d+1),++++++++++(1≤p,q≤∞,θ∈(0,1)),
where s=(1-θ)s0+θs1.Proposition 20.
One assumes thats-(d+2β+2)/p=s1-(d+2β+2)/p1. Then the following inclusion holds
(54)+++++sis++ℬ˙p,qs,β(ℝ+d+1)↪ℬ˙p1,q1s1,β(ℝ+d+1),(1≤p≤p1≤∞,1≤q≤q1≤∞,s,s1∈ℝ).Proof.
In order to prove the inclusion, we use the estimate(55)Δjf=2j(d+2β+2)ϕ~(2j·)*WΔjf.
Proposition 9(i) gives that
(56)∥Δjf∥Lβp1(ℝ+d+1)=∥2j(d+2β+2)ϕ~(2j·)*WΔjf∥Lβp1(ℝ+d+1)≤C2j(d+2β+2)(1/p-1/p1)∥Δjf∥Lβp(ℝ+d+1).
By definition of the homogeneous Weinstein-Besov spaces, we therefore infer
(57)∥f∥ℬ˙p1,q1s1,β(ℝ+d+1)=(∑j=-∞∞(2js1∥Δjf∥Lβp1(ℝ+d+1))q1‍)1/q1≤C(∑j∈ℤ‍(2js12j(d+2β+2)(1/p-1/p1)∥Δjf∥Lβp(ℝ+d+1))q1)1/q1≤C(∑j∈ℤ(2js∥Δjf∥Lβp(ℝ+d+1))q1‍)1/q1≤C∥f∥ℬ˙p,qs,β(ℝ+d+1),
since q≤q1. This gives the inclusion.Proposition 21.
(1)  Ifu belongs to ℬ˙p,qs,β(ℝ+d+1)∩ℬ˙p,qt,β(ℝ+d+1), then u belongs to ℬ˙p,qθs+(1-θ)t,β(ℝ+d+1) for all θ∈[0,1] and
(58)∥u∥ℬ˙p,qθs+(1-θ)t,β(ℝ+d+1)≤∥u∥ℬ˙p,qs,β(ℝ+d+1)θ∥u∥ℬ˙p,qt,β(ℝ+d+1)1-θ.
(2)  Ifu belongs to ℬ˙p,∞s,β(ℝ+d+1)∩ℬ˙p,∞t,β(ℝ+d+1) and s<t, then u belongs to ℬ˙p,1θs+(1-θ)t,β(ℝ+d+1) for all θ∈(0,1) and there exists a positive constant C(t,s) such that
(59)∥u∥ℬ˙p,1θs+(1-θ)t,β(ℝ+d+1)≤C(t,s)∥u∥ℬ˙p,∞s,β(ℝ+d+1)θ∥u∥ℬ˙p,∞t,β(ℝ+d+1)1-θ.
(3)  Ifu belongs to ℬ˙p,∞s,β(ℝ+d+1)∩ℬ˙p,∞s+ε,β(ℝ+d+1) and ε>0, then u belongs to ℬ˙p,1s,β(ℝ+d+1) and there exists a positive constant C such that
(60)∥u∥ℬ˙p,1s,β(ℝ+d+1)≤Cε∥u∥ℬ˙p,∞s,β(ℝ+d+1)log2(e+∥u∥ℬ˙p,∞s+ε,β(ℝ+d+1)∥u∥ℬ˙p,∞s,β(ℝ+d+1)).Proof.
(1)  is obvious from the Hölder’s inequality. As for(2), we write ∥u∥ℬ˙p,1θs+(1-θ)t,β(ℝ+d+1) as
(61)∑j≤N‍2j(θs+(1-θ)t)∥Δju∥Lβp(ℝ+d+1)+∑j>N‍2j(θs+(1-θ)t)∥Δju∥Lβp(ℝ+d+1),
where N is chosen here after. By the definition of the homogeneous Weinstein-Besov norms, we see that
(62)2j(θs+(1-θ)t)∥Δju∥Lβp(ℝ+d+1)≤2j(1-θ)(t-s)∥u∥ℬ˙p,∞s,β(ℝ+d+1),2j(θs+(1-θ)t)∥Δju∥Lβp(ℝ+d+1)≤2-jθ(t-s)∥u∥ℬ˙p,∞t,β(ℝ+d+1),
and thus ∥u∥ℬ˙p,1θs+(1-θ)t,β(ℝ+d+1) is dominated by
(63)∥u∥ℬ˙p,∞s,β(ℝ+d+1)∑j≤N‍2j(1-θ)(t-s)+∥u∥ℬ˙p,∞t,β(ℝ+d+1)∑j>N‍2-jθ(t-s)≤C∥u∥ℬ˙p,∞s,β(ℝ+d+1)2(N+1)(1-θ)(t-s)2(1-θ)(t-s)-1+∥u∥ℬ˙p,∞t,β(ℝ+d+1)2-Nθ(t-s)1-2-θ(t-s).
Hence, in order to complete the proof of (2), it suffices to choose N such that
(64)∥u∥ℬ˙p,∞t,β(ℝd)∥u∥ℬ˙p,∞s,β(ℝd)≤2N(t-s)<2∥u∥ℬ˙p,∞t,β(ℝd)∥u∥ℬ˙p,∞s,β(ℝd).
As for (3), it is easy to see that ∥u∥ℬ˙p,1s,β(ℝ+d+1) is dominated as
(65)∑j≤N-1‍2js∥Δju∥Lβp(ℝ+d+1)+∑j≥N‍2js∥Δju∥Lβp(ℝ+d+1)≤(N+1)∥u∥ℬ˙p,∞s,β(ℝ+d+1)+2-(N-1)ε2ε-1∥u∥ℬ˙p,∞s+ε,β(ℝ+d+1).
Hence, letting
(66)N=1+[1εlog2∥u∥ℬ˙p,∞s+ε,β(ℝ+d+1)∥u∥ℬ˙p,∞s,β(ℝ+d+1)],
we can obtain the desired estimate.Proposition 22.
Letq∈(1,∞) and let s∈ℝ such that 0<s<(d+2β+2)/q, then one has
(67)ℬ˙q,qs,β(ℝ+d+1)=ℱ˙q,qs,β(ℝ+d+1)↪ℱ˙q,∞s,β(ℝ+d+1)↪ℱ˙∞,∞s-((d+2β+2)/q),β(ℝ+d+1),(68)ℋ˙q,βs(ℝ+d+1)=ℱ˙q,2s,β(ℝ+d+1)↪ℱ˙q,∞s,β(ℝ+d+1)↪ℱ˙∞,∞s-((d+2β+2)/q),β(ℝ+d+1).Proof.
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).Theorem 23.
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,β(ℝ+d+1)∩ℱ˙q2,∞-b,β(ℝ+d+1); then one has
(69)|f(x)|≤C(supj∈ℤ2aj|Δjf(x)|)1-θ(supj∈ℤ2-bj|Δjf(x)|)θ.
In particular, one gets
(70)∥f∥Lβp(ℝ+d+1)≤C∥f∥ℱ˙q1,∞a,β(ℝ+d+1)1-θ∥f∥ℱ˙q2,∞-b,β(ℝ+d+1)θ.Proof.
Letf be a Schwartz class; we have
(71)|f(x)|≤∑j∈ℤ‍|Δjf(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
(72)2jbsupj∈ℤ(2-jb|Δjf(x)|)≤2-ajsupj∈ℤ(2aj|Δjf(x)|)
and we write
(73)|f(x)|≤∑j≤N(x)‍2jbsupj∈ℤ(2-jb|Δjf(x)|)+∑j>N(x)‍2-ajsupj(2aj|Δjf(x)|)≤C(supj∈ℤ2aj|Δjf(x)|)b/(a+b)×(supj∈ℤ2-bj|Δjf(x)|)a/(a+b).
Thus, (69) is proved. In order to obtain (70), it is enough to apply the Hölder inequality in the expression above since we have θ=a/(a+b)∈(0,1) and let 1/p=(1-θ)/q1+θ/q2.Corollary 24.
Letq∈(1,∞) and let s∈ℝ such that 0<s<(d+2β+2)/q, then one has
(74)∥f∥Lβp(ℝ+d+1)≤C∥f∥ℬ˙∞,∞-((2β+2+d)/q-s),β(ℝ+d+1)1-(q/p)∥f∥ℬ˙q,qs,β(ℝ+d+1)q/p,(75)∥f∥Lβp(ℝ+d+1)≤C∥f∥ℬ˙∞,∞-((2β+2+d)/q-s),β(ℝ+d+1)1-(q/p)∥f∥ℋ˙q,βs(ℝ+d+1)q/p,
where p=q(2β+2+d)/(2β+2+d-qs).Proof.
By choosinga=s>0, -b=s-(d+2β+2)/q<0, q1=q, and q2=∞, we deduce (74) from the relations (70) and (67). In the same way, we deduce (75) from the relations (70) and (68).


## 4. Generalized Heat Equation

### 4.1. Characterization for the Weinstein-Besov Spaces

The Weinstein heat equation reads(76)∂tu(t,x)-Δβu(t,x)=f(t,x),(t,x)∈[0,∞)×ℝ+d+1,u|t=0=g.
We introduce the Weinstein heat semigroup Hβ(t) for the Weinstein-Laplace operator
(77)Hβ(t)g(x)≔{∫ℝ+d+1‍Γβ(t,x,y)g(y)dμβ(y)ift>0g(x)ift=0,
where Γβ is the Weinstein heat kernel defined by
(78)Γβ(t,x,y):=τx(Et(β))(y),
where
(79)Et(β)(y)=2πd/2Γ(β+1)(4t)β+1+d/2e-∥y∥2/4t.
Thus
(80)Hβ(t)g(x)=g*WEt(β)(x).
In practice, we use the integral formulation of (76)
(81)u(t,x)=Hβ(t)g(x)+Gβ(f)(t,x)u(t,x)∶=Hβ(t)g(x)+∫0t‍Hβ(t-s)f(s,x)ds.Remark 25.
The functionEt(β) is the Gauss kernel associated with Weinstein operators. This function satisfies
(82)∀ξ∈ℝ+d+1,ℱW(Et(β))(ξ)=e-t∥ξ∥2.Proposition 26.
Let1≤p≤r≤∞ and let f∈Lβp(ℝ+d+1). Then the operator Hβ(t) maps Lβp(ℝ+d+1) continuously to Lβr(ℝ+d+1) and
(83)∥Hβ(t)f∥Lβr(ℝd)≤Ct-((d+2β+2)/2)(1/p-1/r)∥f∥Lβp(ℝd).
Moreover,
(84)∥(-Δβ)δ/2Hβ(t)f∥Lβr(ℝd)≤Ct-δ/2-((d+2β+2)/2)(1/p-1/r)∥f∥Lβp(ℝd)
for all δ>0.Proof.
It follows from the relations (80) and (29) combined with scaling property of the kernel Et(β).In this section, we prove estimates for the Weinstein heat semigroup. These estimates are based on the following result.Lemma 27.
Let𝒞 be an annulus. Positive constants c and C exist such that for any p in [1,∞] and any couple (t,λ) of positive real numbers, one has
(85)suppℱW(u)⊂λ𝒞⟹∥Hβ(t)u∥Lβp(ℝ+d+1)≤Ce-ctλ2∥u∥Lβp(ℝ+d+1).Proof.
We again consider a functionΘ in D(ℝ+d+1∖{0}), the value of which is identically 1 in neighborhood of annulus 𝒞. We can also assume without loss of generality that λ=1. We then have
(86)Hβ(t)u=g(t,·)*Wu,
where
(87)g(t,·)=ℱW-1(Θ(ξ)e-t∥ξ∥2).
The lemma is proved provided that we can find positive real numbers c and C such that
(88)∀t>0,∥g(t,·)∥Lβ1(ℝ+d+1)≤Ce-ct.
To begin, we perform integrations by parts in (87). We get
(89)|g(t,x)|=1cβ(1+∥x∥2)-(d+[2β]+3)×∫ℝ+d+1Λ(x,ξ)(Id-Δβ)d+[2β]+3‍×(Θ(ξ)e-t∥ξ∥2)dμβ(ξ).
Using Leibniz's formula, we obtain
(90)|g(t,x)|≤C(1+∥x∥2)-(d+[2β]+3)e-ct,
and (88) follows.For any intervalI of ℝ (bounded or unbounded), we define the mixed space-time Lp(I;Lβq(ℝ+d+1)) Banach space of (classes of) measurable functions u:I→Lβq(ℝ+d+1) such that ∥u∥Lp(I;Lβq(ℝ+d+1))<∞, with
(91)∥u∥Lp(I;Lβq(ℝ+d+1))=(∫I‍∥u(t,·)∥Lβq(ℝ+d+1)pdt)1/p,+++++++++++ui++if1≤p,q<∞,∥u∥L∞(I;Lβq(ℝ+d+1))=esssupt∈I∥u(t,·)∥Lβq(ℝ+d+1),if1≤q<∞.Corollary 28.
Let𝒞 be an annulus and λ a positive real number. Let u0 (resp., f=f(t,x)) satisfy suppℱW(u0)⊂λ𝒞 (resp., suppℱW(f(t,·))⊂λ𝒞 for all t in [0,T]). Consider u a solution of
(92)∂tu-Δβu=0,u|t=0=u0,
and v a solution of
(93)∂tv-Δβv=f(t,·),v|t=0=0.
There exist positive constants c and C, depending only on 𝒞, such that for any 1≤a≤b≤∞ and 1≤p≤q≤∞, we have
(94)∥u∥Lq([0,T],Lβb(ℝ+d+1))≤Cλ(d+2β+2)(1/a-1/b)λ-2/q∥u0∥Lβa(ℝ+d+1),∥v∥Lq([0,T],Lβb(ℝ+d+1))≤Cλ-2(1+1/q-1/p)λ(d+2β+2)(1/a-1/b)×∥f∥Lp([0,T],Lβa(ℝ+d+1)).Proof.
It suffices to use the fact that(95)u(t,·)=Hβ(t)u0,v(t,·)=∫0t‍Hβ(t-s)f(s,·)ds.
Combining Lemma 27 and Young's inequality (29) with scaling property of the kernel Et(β) now yields the result.Theorem 29.
Lets be a positive real number and (p,r)∈[1,∞]2. A constant C exists which satisfies the following property. For u∈ℬ˙p,r-2s,β(ℝ+d+1), one has
(96)C-1∥u∥ℬ˙p,r-2s,β(ℝ+d+1)≤∥∥tsHβ(t)u∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t)≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1).To prove this result we need the following lemma.Lemma 30.
There exist two positive constantsκ and C depending only on φ such that for all 1≤p≤∞, τ≥0 and j∈ℤ, one has
(97)∥Δj(Hβ(τ)u)∥Lβp(ℝ+d+1)≤Ce-κ22jτ∥Δju∥Lβp(ℝ+d+1).Proof.
The result follows immediately by applying Lemma27 and because Δj(Hβ(τ))u=(Hβ(τ)Δj)u.Proof of Theorem29.
Using Lemma30 and considering the fact that the operator Δj commutes with the operator Hβ(t) and the definition of the homogeneous Weinstein-Besov (semi) norm we get
(98)∥tsHβ(t)u∥Lβp(ℝ+d+1)≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1)∑j∈ℤ‍ts22jse-κt22jcr,j,
where (cr,j)j∈ℤ denotes, as in all this proof, a generic element of the unit sphere of lr(ℤ). In the case when r=∞, the required inequality comes immediately from the following easy result. For any positive s, we have
(99)supt>0∑j∈ℤ‍ts22jse-κt22j<∞.
In the case r<∞, using the Hölder inequality with the weight 22jse-κt22j, (99), and the Fubini theorem, we obtain
(100)∫0∞‍trs∥Hβ(t)u∥Lβp(ℝ+d+1)rdtt≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1)r∫0∞‍(∑j∈ℤ‍ts22jse-κt22j)r-1×(∑j∈ℤ‍ts22jse-κt22jcr,jr)dtt≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1)r∫0∞‍(∑j∈ℤ‍ts22jse-κt22jcr,jr)dtt≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1)r∑j∈ℤ‍cr,jr∫0∞‍(ts22jse-κt22j)dtt≤CΓ(s)∥u∥ℬ˙p,r-2s,β(ℝ+d+1)r.
In order to prove the other inequality, let us observe that for anys greater than -1, we have
(101)Δju=1Γ(s+1)∫0∞‍ts(-Δβ)s+1Hβ(t)Δjudt.
Then, Lemma 30, Proposition 9, and the fact that the operator Δj commutes with the operator Hβ(t) lead to the following:
(102)∥Δju∥Lβp(ℝ+d+1)≤C∫0∞‍ts22j(s+1)e-κt22j∥Hβ(t)u∥Lβp(ℝ+d+1)dt.
In the case r=∞, we simply write
(103)∥Δju∥Lβp(ℝ+d+1)≤C(supt>0ts∥Hβ(t)u∥Lβp(ℝ+d+1))×∫0∞‍22j(s+1)e-κt22jdt≤C22js(supt>0ts∥Hβ(t)u∥Lβp(ℝ+d+1)).
In the case r<∞, Hölder’s inequality with the weight e-κt22j gives
(104)(∫0∞‍tse-κt22j∥Hβ(t)u∥Lβp(ℝ+d+1)dt)r≤C2-2j(r-1)∫0∞‍trse-κt22j∥Hβ(t)u∥Lβp(ℝ+d+1)rdt.
Thanks to (99) and Fubini’s theorem, we infer from (102) that
(105)∑j∈ℤ‍2-2jrs∥Δju∥Lβp(ℝ+d+1)r≤C∫0∞‍trs∥Hβ(t)u∥Lβp(ℝ+d+1)rdtt.
The theorem is proved.Second Proof of Theorem29. We only consider the case 1≤r<∞. The case r=∞ can be shown similarly. We first prove that
(106)C-1∥u∥ℬ˙p,r-2s,β(ℝ+d+1)≤∥ts∥Hβ(t)u∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t).
It is easy to see that
(107)Δju=χj*WE2-2j(β)*Wu,
where χj=ℱW-1(φ(2-jξ)e2-2j∥ξ∥2) and E2-2j(β) is the Gauss kernel associated with Weinstein operators. By relation (29), we get
(108)∥Δju∥Lβp(ℝ+d+1)≤∥χj∥Lβ1(ℝ+d+1)∥E2-2j(β)*Wu∥Lβp(ℝ+d+1).
As
(109)∥χj∥Lβ1(ℝ+d+1)=∫ℝ+d+1‍|ℱW-1(φ(ξ)e∥ξ∥2)|dμβ(ξ)<∞,
we obtain
(110)∥Δju∥Lβp(ℝ+d+1)≤C∥E2-2j(β)*Wu∥Lβp(ℝ+d+1).
Moreover simple calculations give that
(111)E2-2j(β)*Wu=Hβ(2-4j-t2)(Et2(β)*Wu).
Thus, from Proposition 26, it follows that
(112)∥E2-2j(β)*Wu∥Lβp(ℝ+d+1)≤C∥Et2(β)*Wu∥Lβp(ℝ+d+1),
for any t∈[2-j-1,2-j], which implies that
(113)∑j∈ℤ‍2-2sjr∥Δju∥Lβp(ℝ+d+1)r≤C∑j∈ℤ‍∫2-j-12-j‍(t2s∥E2-2j(β)*Wu∥Lβp(ℝ+d+1))rdtt≤C∫0∞‍(t2s∥Et2(β)*Wu∥Lβp(ℝ+d+1))rdtt≤C∫0∞‍(ts∥Hβ(t)u∥Lβp(ℝ+d+1))rdtt,
where we have used the fact that Et2(β)*Wu=Hβ(t2)u.We now prove that(114)∥ts∥Hβ(t)u∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t)≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1).
Indeed, one has
(115)E2-2j(β)*Wu=∑n∈ℤ‍E2-2j(β)*WΔn+ju.
Arguing as above, we have
(116)∥Et2(β)*Wu∥Lβp(ℝ+d+1)≤C∥E2-2j(β)*Wu∥Lβp(ℝ+d+1),
for any t∈[2-j,21-j]. Thus
(117)∥ts∥Hβ(t)u∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t)=2∥∥t2sEt2(β)*Wu∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t)≤C∑j∈ℤ‍∫2-j21-j‍(2-2js∥E2-2j(β)*Wu∥Lβp(ℝ+d+1))rdtt≤C∑j∈ℤ‍(2-2js∑n∈ℤ‍∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1))r.
On the other hand, it is easy to see that
(118)∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1)≤C2-2ns∥Δn+ju∥Lβp(ℝ+d+1),
for any s>0. For 0<s1<s<s2 and by using the Minkowski inequality, we have
(119)∑j∈ℤ‍(2-2js∑n∈ℤ‍∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1))r≤∑j∈ℤ‍(2-2js∑-∞0‍2-2ns1∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1))r+∑j∈ℤ‍(2-2js∑n∈ℕ‍2-2ns2∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1))r≤C(∑-∞0‍2-2n(s1-s)(∑j∈ℤ‍(2-2(n+j)rs∥Δn+ju∥Lβp(ℝ+d+1))r)1/r)r+C(∑ℕ‍2-2n(s2-s)(∑j∈ℤ‍(2-2(n+j)rs∥Δn+ju∥Lβp(ℝ+d+1))r)1/r)r≤C∑j∈ℤ‍2-2jrs∥Δju∥Lβp(ℝ+d+1)r.
The result is immediately from (117) and (119).


### 4.2. Embedding Sobolev Theorems

Theorem 31.
Let1<p<∞ and let 0<s<(d+2β+2)/p. There exists a positive constant C such that for all function f∈ℋ˙p,βs(ℝ+d+1) one has
(120)∥f∥Lβq(ℝ+d+1)≤C∥f∥ℋ˙p,βs(ℝ+d+1)1-θ∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1)θ,
where θ=sp/(d+2β+2) and q=p(d+2β+2)/(d+2β+2-ps).Proof.
By density, we can suppose thatf belongs to 𝒮*(ℝd+1). It is easy to see that
(121)f=∫0∞‍Hβ(t)Δβfdt
and decompose the integral in two parts as follows:
(122)f=∫0A‍Hβ(t)Δβfdt+∫A∞‍Hβ(t)Δβfdt,
where A is a constant to be fixed later.
On the other hand, by Theorem29, we obtain(123)∥Hβ(t)Δβf∥Lβ∞(ℝ+d+1)≤Ct1-(1/2)(s-(d+2β+2)/p)∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1).
Therefore after integrating, we get
(124)∫A∞‍∥Hβ(t)Δβf∥Lβ∞(ℝ+d+1)dt≤A(1/2)(s-(d+2β+2)/p)∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1).
On the other hand, denoting g=(-Δβ)s/2f, we have
(125)Hβ(t)Δβf=1(-t)1-s/2Hβ(t)(-tΔβ)1-s/2g.
We proceed as in [8], we prove that
(126)|Hβ(t)(-tΔβ)1-s/2g(x)|≤C(s)Mβ(g)(x),
where Mβ(g) is a maximal function of g associated with the Weinstein operators (cf. [12]).
This leads to(127)|∫0A‍Hβ(t)Δβf(x)dt|≤CAs/2Mβ(g)(x).
In conclusion, we get
(128)|∫0∞‍Hβ(t)Δβf(x)dt|≤C(+A(1/2)(s-((d+2β+2)/p))∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1)As/2Mβ(g)(x)+++++A(1/2)(s-(d+2β+2)/p)∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1)),
and the choice of A such that
(129)A(d+2β+2)/2pMβ(g)(x)=∥f∥ℬ˙∞,∞s-(d+2β+2)/p,β(ℝ+d+1)
ensures that
(130)|∫0∞‍Hβ(t)Δβf(x)dt|≤C(Mβ(g)(x))1-(ps/(d+2β+2))∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1)ps/(d+2β+2).
Finally, taking the Lβq norm with q=p(d+2β+2)/(d+2β+2-ps) ends the proof thanks to the fact the maximal function Mβ is bounded of Lβq(ℝ+d+1) into itself for q>1.Theorem 32.
Let1<p<q<∞. For all function f such that f∈ℋ˙p,βs1(ℝ+d+1)⋂‍ℬ˙∞,∞-β,β(ℝ+d+1), one has
(131)∥f∥ℋ˙p,βs(ℝ+d+1)≤C∥f∥ℋ˙p,βs1(ℝ+d+1)θ∥f∥ℬ˙∞,∞-β,β(ℝ+d+1)1-θ,
where θ=p/q, s=θs1-(1-θ)β with β>0, -β<s<s1.Proof.
It suffices to prove that(132)∥(-Δβ)(s-s1)/2f∥Lβq(ℝ+d+1)≤C∥f∥Lβp(ℝ+d+1)θ∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1)1-θ.
Indeed, we use the following identity (which may be easily proven by taking the Weinstein transform in x of both sides)
(133)(-Δβ)-δ/2f(x)=1Γ(δ/2)∫0∞‍tδ/2-1Hβ(t)f(x)dt,
with δ=s1-s>0.
We decompose the integral in two parts as follows:(134)(-Δβ)-δ/2f(x)=1Γ(δ/2)∫0T‍tδ/2-1Hβ(t)f(x)dt+1Γ(δ/2)∫T∞‍tδ/2-1Hβ(t)f(x)dt,
where T is a constant to be fixed later.
We proceed as in [8], we obtain
(135)|Hβ(t)f(x)|≤CMβ(f)(x).
On the other hand, we use Theorem 29 and the fact that f belongs to ℬ˙∞,∞-β-s1,β(ℝ+d+1) to deduce that
(136)|Hβ(t)f(x)|≤Ct(-β-s1)/2∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1).
Thus, by applying the preceding estimates on the right part of (134) we obtain
(137)|(-Δβ)-δ/2f(x)|≤C1Γ(δ/2)Tδ/2Mβ(f)(x)+C2Γ(δ/2)T(δ-β-s1)/2∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1).
We fix now
(138)T=(∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1)Mβ(f)(x))2/(β+s1).
We obtain
(139)|(-Δβ)-δ/2f(x)|≤C1+C2Γ(δ/2)(Mβ(f)(x))θ∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1)1-θ.
Thus we deduce that
(140)∥(-Δβ)-δ/2f∥Lβq(ℝ+d+1)≤C1+C2Γ(δ/2)∥Mβ(f)∥Lβp(ℝ+d+1)θ∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1)1-θ.To conclude, we used the fact that the maximal function Mβ is bounded of Lβq(ℝ+d+1) into itself for q>1.


### 4.3. Estimates in Generalized Besov Spaces

For any intervalI of ℝ (bounded or unbounded) and a normed space F(ℝ+d+1), we define the mixed space-time Lp(I;F(ℝ+d+1)) space of (classes of) measurable functions u:I→F(ℝ+d+1) such that ||u||Lp(I;F(ℝ+d+1))<∞, with
(141)∥u∥Lp(I;F(ℝ+d+1))=(∫I‍∥u(t,·)∥F(ℝ+d+1)pdt)1/p,if1≤p<∞,∥u∥L∞(I;F(ℝ+d+1))=esssupt∈I∥u(t,·)∥F(ℝ+d+1).For any intervalI of ℝ (bounded or unbounded) and a Banach space X, we define the mixed space-time C(I¯;X) space of continuous functions I¯→X. When I is bounded, C(I¯;X) is a Banach space with the norm of L∞(I,X).Theorem 33.
Lets∈ℝ and 1≤p,q,r≤∞. Let T>0, g∈ℬ˙p,rs,β(ℝ+d+1), and f in Lq((0,T),ℬ˙p,rs-2+(2/q),β(ℝ+d+1)). Then (76) has a unique solution:
(142)u∈Lq((0,T),ℬ˙p,rs+(2/q),β(ℝ+d+1))⋂‍L∞((0,T),ℬ˙p,rs,β(ℝ+d+1)),
and there exists a constant C such that for all q1∈[q,∞], one has
(143)∥u∥Lq1((0,T),ℬ˙p,rs+(2/q1),β(ℝ+d+1))≤C(∥g∥ℬ˙p,rs,β(ℝ+d+1)+∥f∥Lq((0,T),ℬ˙p,rs-2+(2/q),β(ℝ+d+1))).
If in addition r<∞, then u∈C([0,T],ℬ˙p,rs,β(ℝ+d+1)).Proof.
Sinceg and f are temperate distributions, (76) has a unique solution u in 𝒮′((0,T)×ℝ+d+1), which satisfies
(144)ℱW(u)(t,ξ)=e-t∥ξ∥2ℱW(g)(ξ)+∫0t‍e(τ-t)∥ξ∥2ℱW(f)(τ,ξ)dτ.
Next, we notice that applying Δj to (76) and using formula (81) yield
(145)Δju(t,·)=Hβ(t)Δjg+∫0t‍Hβ(t-τ)Δjf(τ,·)dτ.
Therefore,
(146)∥Δju(t,·)∥Lβp(ℝ+d+1)≤∥Hβ(t)Δjg∥Lβp(ℝ+d+1)+∫0t‍∥Hβ(t-τ)Δjf(τ,·)∥Lβp(ℝ+d+1)dτ.
By virtue of Lemma 30, we thus have, for some κ>0,
(147)∥Δju(t,·)∥Lβp(ℝ+d+1)≤C[e-κ22jt∥Δjg∥Lβp(ℝ+d+1)+∫0t‍e-κ22j(t-τ)∥Δjf(τ,·)∥Lβp(ℝ+d+1)dτ].
Applying convolution inequalities, we get
(148)∥Δju∥Lq1((0,T),Lβp(ℝ+d+1))≤C[((1-e-κTq122jκq122j)1/q1)∥Δjg∥Lβp(ℝ+d+1)+((1-e-κTq222jκq222j)1/q2)∥Δjf∥Lq((0,T),Lβp(ℝ+d+1))],
with 1/q2=1+1/q1-1/q. Finally, taking the lr(ℤ) norm, we conclude that (with the usual convention if r=∞)
(149)∥u∥Lq1((0,T),ℬ˙p,rs+(2/q1),β(ℝ+d+1))≤C[∑j∈ℤ‍((1-e-κTq122jκq122j)r/q1)(2js∥Δjg∥Lβp(ℝ+d+1))r]1/r++C[∑j∈ℤ‍((1-e-κTq222jκq222j)r/q2)++++×(2j(s-2+2/q)∥Δjf∥Lq(0,T;Lβp(ℝ+d+1))r∑j∈ℤ‍((1-e-κTq222jκq222j)r/q2)]1/r,
which insures that u  ∈ Lq((0,T),ℬ˙p,rs+(2/q),β(ℝ+d+1))  ⋂ L∞((0,T),ℬ˙p,rs,β(ℝ+d+1)) and yields the desired inequality. Since u belongs to C([0,T],ℬ˙p,rs,β(ℝ+d+1)) in the case where r is finite may be easily deduced from the density of 𝒮*(ℝd+1)⋂‍ℬ˙p,rs,β(ℝ+d+1) in ℬ˙p,rs,β(ℝ).Theorem 34.
Lets∈ℝ, T>0, and 1≤p,q,r≤∞. One supposes that g∈Bp,rs,β(ℝ+d+1) and f∈Lq((0,T),Bp,rs-2+(2/q),β(ℝ+d+1)). Then (76) has a unique solution u belonging to
(150)Lq((0,T),Bp,rs+(2/q),β(ℝ+d+1))⋂‍L∞((0,T),Bp,rs,β(ℝ+d+1)),
and there exists a constant C such that, for all q≤q1≤∞,
(151)∥u∥Lq1((0,T),Bp,rs+(2/q1),β(ℝ+d+1))≤C[(1+T1+(1/q1)-(1/q))∥f∥Lq((0,T),Bp,rs-2+(2/q),β(ℝ+d+1))(1+T1/q1)∥g∥Bp,rs,β(ℝ+d+1)+(1+T1+1/q1-1/q)∥f∥Lq((0,T),Bp,rs-2+(2/q),β(ℝ+d+1))(1+T1/q1)∥g∥Bp,rs,β(ℝ+d+1)].
If in addition r<∞, then u∈C([0,T],Bp,rs,β(ℝ+d+1)).Proof.
Sinceg,f are tempered, (76) has a unique solution u in 𝒮′((0,T)×ℝ+d+1) satisfying
(152)ℱW(u)(t,ξ)=e-t∥ξ∥2ℱW(g)(ξ)+∫0t‍e(τ-t)∥ξ∥2ℱW(f)(τ,ξ)dτ.
Hence, applying Δj, j≥0, to (81), we see that
(153)Δju(t,·)=Hβ(t)Δjg+∫0t‍Hβ(t-τ)Δjf(τ,·)dτ,
and thus, by Lemma 30, we can deduce that
(154)∥Δju(t,·)∥Lβp(ℝ+d+1)≤∥Hβ(t)Δjg∥Lβp(ℝ+d+1)+∫0t‍∥Hβ(t-τ)Δjf(τ,·)∥Lβp(ℝ+d+1)dτ≤C[∫0t‍e-κ22j(t-τ)e-κ22jt∥Δjg∥Lβp(ℝ+d+1)++++∫0t‍e-κ22j(t-τ)∥Δjf(τ,·)∥Lβp(ℝ+d+1)dτ∥Δjg∥Lβp(ℝ+d+1)].
Then it follows from convolution inequalities that ∥Δju∥Lq1((0,T),Lβp(ℝ+d+1)) is dominated by
(155)(1-e-κTq122jκq122j)1/q1∥Δjg∥Bp,rs,β(ℝ+d+1)+(1-e-κTq222jκq222j)1/q2∥Δjf∥Lq((0,T),Lβp(ℝ+d+1)),
with 1/q2=1+1/q1-1/q. Moreover, similarly as above, we can obtain that
(156)∥Δ-1u(t,·)∥Lβp(ℝ+d+1)≤∥Δ-1g∥Lβp(ℝ+d+1)+∫0t‍∥Δ-1f(τ,·)∥Lβp(ℝ+d+1)dτ,
and thus if 1≤q≤q1≤∞,
(157)∥Δ-1u∥Lq1((0,T),Lβp(ℝ+d+1))≤C(T1/q1∥Δ-1g∥Lβp(ℝ+d+1)+T1/q2∥Δ-1f∥Lq((0,T),Lβp(ℝ+d+1))).
Finally, taking the lr-norm with respect to j in (155) and (157) with the usual convention if r=∞, we can deduce the desired estimate.


## 4.1. Characterization for the Weinstein-Besov Spaces

The Weinstein heat equation reads(76)∂tu(t,x)-Δβu(t,x)=f(t,x),(t,x)∈[0,∞)×ℝ+d+1,u|t=0=g.
We introduce the Weinstein heat semigroup Hβ(t) for the Weinstein-Laplace operator
(77)Hβ(t)g(x)≔{∫ℝ+d+1‍Γβ(t,x,y)g(y)dμβ(y)ift>0g(x)ift=0,
where Γβ is the Weinstein heat kernel defined by
(78)Γβ(t,x,y):=τx(Et(β))(y),
where
(79)Et(β)(y)=2πd/2Γ(β+1)(4t)β+1+d/2e-∥y∥2/4t.
Thus
(80)Hβ(t)g(x)=g*WEt(β)(x).
In practice, we use the integral formulation of (76)
(81)u(t,x)=Hβ(t)g(x)+Gβ(f)(t,x)u(t,x)∶=Hβ(t)g(x)+∫0t‍Hβ(t-s)f(s,x)ds.Remark 25.
The functionEt(β) is the Gauss kernel associated with Weinstein operators. This function satisfies
(82)∀ξ∈ℝ+d+1,ℱW(Et(β))(ξ)=e-t∥ξ∥2.Proposition 26.
Let1≤p≤r≤∞ and let f∈Lβp(ℝ+d+1). Then the operator Hβ(t) maps Lβp(ℝ+d+1) continuously to Lβr(ℝ+d+1) and
(83)∥Hβ(t)f∥Lβr(ℝd)≤Ct-((d+2β+2)/2)(1/p-1/r)∥f∥Lβp(ℝd).
Moreover,
(84)∥(-Δβ)δ/2Hβ(t)f∥Lβr(ℝd)≤Ct-δ/2-((d+2β+2)/2)(1/p-1/r)∥f∥Lβp(ℝd)
for all δ>0.Proof.
It follows from the relations (80) and (29) combined with scaling property of the kernel Et(β).In this section, we prove estimates for the Weinstein heat semigroup. These estimates are based on the following result.Lemma 27.
Let𝒞 be an annulus. Positive constants c and C exist such that for any p in [1,∞] and any couple (t,λ) of positive real numbers, one has
(85)suppℱW(u)⊂λ𝒞⟹∥Hβ(t)u∥Lβp(ℝ+d+1)≤Ce-ctλ2∥u∥Lβp(ℝ+d+1).Proof.
We again consider a functionΘ in D(ℝ+d+1∖{0}), the value of which is identically 1 in neighborhood of annulus 𝒞. We can also assume without loss of generality that λ=1. We then have
(86)Hβ(t)u=g(t,·)*Wu,
where
(87)g(t,·)=ℱW-1(Θ(ξ)e-t∥ξ∥2).
The lemma is proved provided that we can find positive real numbers c and C such that
(88)∀t>0,∥g(t,·)∥Lβ1(ℝ+d+1)≤Ce-ct.
To begin, we perform integrations by parts in (87). We get
(89)|g(t,x)|=1cβ(1+∥x∥2)-(d+[2β]+3)×∫ℝ+d+1Λ(x,ξ)(Id-Δβ)d+[2β]+3‍×(Θ(ξ)e-t∥ξ∥2)dμβ(ξ).
Using Leibniz's formula, we obtain
(90)|g(t,x)|≤C(1+∥x∥2)-(d+[2β]+3)e-ct,
and (88) follows.For any intervalI of ℝ (bounded or unbounded), we define the mixed space-time Lp(I;Lβq(ℝ+d+1)) Banach space of (classes of) measurable functions u:I→Lβq(ℝ+d+1) such that ∥u∥Lp(I;Lβq(ℝ+d+1))<∞, with
(91)∥u∥Lp(I;Lβq(ℝ+d+1))=(∫I‍∥u(t,·)∥Lβq(ℝ+d+1)pdt)1/p,+++++++++++ui++if1≤p,q<∞,∥u∥L∞(I;Lβq(ℝ+d+1))=esssupt∈I∥u(t,·)∥Lβq(ℝ+d+1),if1≤q<∞.Corollary 28.
Let𝒞 be an annulus and λ a positive real number. Let u0 (resp., f=f(t,x)) satisfy suppℱW(u0)⊂λ𝒞 (resp., suppℱW(f(t,·))⊂λ𝒞 for all t in [0,T]). Consider u a solution of
(92)∂tu-Δβu=0,u|t=0=u0,
and v a solution of
(93)∂tv-Δβv=f(t,·),v|t=0=0.
There exist positive constants c and C, depending only on 𝒞, such that for any 1≤a≤b≤∞ and 1≤p≤q≤∞, we have
(94)∥u∥Lq([0,T],Lβb(ℝ+d+1))≤Cλ(d+2β+2)(1/a-1/b)λ-2/q∥u0∥Lβa(ℝ+d+1),∥v∥Lq([0,T],Lβb(ℝ+d+1))≤Cλ-2(1+1/q-1/p)λ(d+2β+2)(1/a-1/b)×∥f∥Lp([0,T],Lβa(ℝ+d+1)).Proof.
It suffices to use the fact that(95)u(t,·)=Hβ(t)u0,v(t,·)=∫0t‍Hβ(t-s)f(s,·)ds.
Combining Lemma 27 and Young's inequality (29) with scaling property of the kernel Et(β) now yields the result.Theorem 29.
Lets be a positive real number and (p,r)∈[1,∞]2. A constant C exists which satisfies the following property. For u∈ℬ˙p,r-2s,β(ℝ+d+1), one has
(96)C-1∥u∥ℬ˙p,r-2s,β(ℝ+d+1)≤∥∥tsHβ(t)u∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t)≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1).To prove this result we need the following lemma.Lemma 30.
There exist two positive constantsκ and C depending only on φ such that for all 1≤p≤∞, τ≥0 and j∈ℤ, one has
(97)∥Δj(Hβ(τ)u)∥Lβp(ℝ+d+1)≤Ce-κ22jτ∥Δju∥Lβp(ℝ+d+1).Proof.
The result follows immediately by applying Lemma27 and because Δj(Hβ(τ))u=(Hβ(τ)Δj)u.Proof of Theorem29.
Using Lemma30 and considering the fact that the operator Δj commutes with the operator Hβ(t) and the definition of the homogeneous Weinstein-Besov (semi) norm we get
(98)∥tsHβ(t)u∥Lβp(ℝ+d+1)≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1)∑j∈ℤ‍ts22jse-κt22jcr,j,
where (cr,j)j∈ℤ denotes, as in all this proof, a generic element of the unit sphere of lr(ℤ). In the case when r=∞, the required inequality comes immediately from the following easy result. For any positive s, we have
(99)supt>0∑j∈ℤ‍ts22jse-κt22j<∞.
In the case r<∞, using the Hölder inequality with the weight 22jse-κt22j, (99), and the Fubini theorem, we obtain
(100)∫0∞‍trs∥Hβ(t)u∥Lβp(ℝ+d+1)rdtt≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1)r∫0∞‍(∑j∈ℤ‍ts22jse-κt22j)r-1×(∑j∈ℤ‍ts22jse-κt22jcr,jr)dtt≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1)r∫0∞‍(∑j∈ℤ‍ts22jse-κt22jcr,jr)dtt≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1)r∑j∈ℤ‍cr,jr∫0∞‍(ts22jse-κt22j)dtt≤CΓ(s)∥u∥ℬ˙p,r-2s,β(ℝ+d+1)r.
In order to prove the other inequality, let us observe that for anys greater than -1, we have
(101)Δju=1Γ(s+1)∫0∞‍ts(-Δβ)s+1Hβ(t)Δjudt.
Then, Lemma 30, Proposition 9, and the fact that the operator Δj commutes with the operator Hβ(t) lead to the following:
(102)∥Δju∥Lβp(ℝ+d+1)≤C∫0∞‍ts22j(s+1)e-κt22j∥Hβ(t)u∥Lβp(ℝ+d+1)dt.
In the case r=∞, we simply write
(103)∥Δju∥Lβp(ℝ+d+1)≤C(supt>0ts∥Hβ(t)u∥Lβp(ℝ+d+1))×∫0∞‍22j(s+1)e-κt22jdt≤C22js(supt>0ts∥Hβ(t)u∥Lβp(ℝ+d+1)).
In the case r<∞, Hölder’s inequality with the weight e-κt22j gives
(104)(∫0∞‍tse-κt22j∥Hβ(t)u∥Lβp(ℝ+d+1)dt)r≤C2-2j(r-1)∫0∞‍trse-κt22j∥Hβ(t)u∥Lβp(ℝ+d+1)rdt.
Thanks to (99) and Fubini’s theorem, we infer from (102) that
(105)∑j∈ℤ‍2-2jrs∥Δju∥Lβp(ℝ+d+1)r≤C∫0∞‍trs∥Hβ(t)u∥Lβp(ℝ+d+1)rdtt.
The theorem is proved.Second Proof of Theorem29. We only consider the case 1≤r<∞. The case r=∞ can be shown similarly. We first prove that
(106)C-1∥u∥ℬ˙p,r-2s,β(ℝ+d+1)≤∥ts∥Hβ(t)u∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t).
It is easy to see that
(107)Δju=χj*WE2-2j(β)*Wu,
where χj=ℱW-1(φ(2-jξ)e2-2j∥ξ∥2) and E2-2j(β) is the Gauss kernel associated with Weinstein operators. By relation (29), we get
(108)∥Δju∥Lβp(ℝ+d+1)≤∥χj∥Lβ1(ℝ+d+1)∥E2-2j(β)*Wu∥Lβp(ℝ+d+1).
As
(109)∥χj∥Lβ1(ℝ+d+1)=∫ℝ+d+1‍|ℱW-1(φ(ξ)e∥ξ∥2)|dμβ(ξ)<∞,
we obtain
(110)∥Δju∥Lβp(ℝ+d+1)≤C∥E2-2j(β)*Wu∥Lβp(ℝ+d+1).
Moreover simple calculations give that
(111)E2-2j(β)*Wu=Hβ(2-4j-t2)(Et2(β)*Wu).
Thus, from Proposition 26, it follows that
(112)∥E2-2j(β)*Wu∥Lβp(ℝ+d+1)≤C∥Et2(β)*Wu∥Lβp(ℝ+d+1),
for any t∈[2-j-1,2-j], which implies that
(113)∑j∈ℤ‍2-2sjr∥Δju∥Lβp(ℝ+d+1)r≤C∑j∈ℤ‍∫2-j-12-j‍(t2s∥E2-2j(β)*Wu∥Lβp(ℝ+d+1))rdtt≤C∫0∞‍(t2s∥Et2(β)*Wu∥Lβp(ℝ+d+1))rdtt≤C∫0∞‍(ts∥Hβ(t)u∥Lβp(ℝ+d+1))rdtt,
where we have used the fact that Et2(β)*Wu=Hβ(t2)u.We now prove that(114)∥ts∥Hβ(t)u∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t)≤C∥u∥ℬ˙p,r-2s,β(ℝ+d+1).
Indeed, one has
(115)E2-2j(β)*Wu=∑n∈ℤ‍E2-2j(β)*WΔn+ju.
Arguing as above, we have
(116)∥Et2(β)*Wu∥Lβp(ℝ+d+1)≤C∥E2-2j(β)*Wu∥Lβp(ℝ+d+1),
for any t∈[2-j,21-j]. Thus
(117)∥ts∥Hβ(t)u∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t)=2∥∥t2sEt2(β)*Wu∥Lβp(ℝ+d+1)∥Lr(ℝ+,dt/t)≤C∑j∈ℤ‍∫2-j21-j‍(2-2js∥E2-2j(β)*Wu∥Lβp(ℝ+d+1))rdtt≤C∑j∈ℤ‍(2-2js∑n∈ℤ‍∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1))r.
On the other hand, it is easy to see that
(118)∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1)≤C2-2ns∥Δn+ju∥Lβp(ℝ+d+1),
for any s>0. For 0<s1<s<s2 and by using the Minkowski inequality, we have
(119)∑j∈ℤ‍(2-2js∑n∈ℤ‍∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1))r≤∑j∈ℤ‍(2-2js∑-∞0‍2-2ns1∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1))r+∑j∈ℤ‍(2-2js∑n∈ℕ‍2-2ns2∥E2-2j(β)*WΔn+ju∥Lβp(ℝ+d+1))r≤C(∑-∞0‍2-2n(s1-s)(∑j∈ℤ‍(2-2(n+j)rs∥Δn+ju∥Lβp(ℝ+d+1))r)1/r)r+C(∑ℕ‍2-2n(s2-s)(∑j∈ℤ‍(2-2(n+j)rs∥Δn+ju∥Lβp(ℝ+d+1))r)1/r)r≤C∑j∈ℤ‍2-2jrs∥Δju∥Lβp(ℝ+d+1)r.
The result is immediately from (117) and (119).


## 4.2. Embedding Sobolev Theorems

Theorem 31.
Let1<p<∞ and let 0<s<(d+2β+2)/p. There exists a positive constant C such that for all function f∈ℋ˙p,βs(ℝ+d+1) one has
(120)∥f∥Lβq(ℝ+d+1)≤C∥f∥ℋ˙p,βs(ℝ+d+1)1-θ∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1)θ,
where θ=sp/(d+2β+2) and q=p(d+2β+2)/(d+2β+2-ps).Proof.
By density, we can suppose thatf belongs to 𝒮*(ℝd+1). It is easy to see that
(121)f=∫0∞‍Hβ(t)Δβfdt
and decompose the integral in two parts as follows:
(122)f=∫0A‍Hβ(t)Δβfdt+∫A∞‍Hβ(t)Δβfdt,
where A is a constant to be fixed later.
On the other hand, by Theorem29, we obtain(123)∥Hβ(t)Δβf∥Lβ∞(ℝ+d+1)≤Ct1-(1/2)(s-(d+2β+2)/p)∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1).
Therefore after integrating, we get
(124)∫A∞‍∥Hβ(t)Δβf∥Lβ∞(ℝ+d+1)dt≤A(1/2)(s-(d+2β+2)/p)∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1).
On the other hand, denoting g=(-Δβ)s/2f, we have
(125)Hβ(t)Δβf=1(-t)1-s/2Hβ(t)(-tΔβ)1-s/2g.
We proceed as in [8], we prove that
(126)|Hβ(t)(-tΔβ)1-s/2g(x)|≤C(s)Mβ(g)(x),
where Mβ(g) is a maximal function of g associated with the Weinstein operators (cf. [12]).
This leads to(127)|∫0A‍Hβ(t)Δβf(x)dt|≤CAs/2Mβ(g)(x).
In conclusion, we get
(128)|∫0∞‍Hβ(t)Δβf(x)dt|≤C(+A(1/2)(s-((d+2β+2)/p))∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1)As/2Mβ(g)(x)+++++A(1/2)(s-(d+2β+2)/p)∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1)),
and the choice of A such that
(129)A(d+2β+2)/2pMβ(g)(x)=∥f∥ℬ˙∞,∞s-(d+2β+2)/p,β(ℝ+d+1)
ensures that
(130)|∫0∞‍Hβ(t)Δβf(x)dt|≤C(Mβ(g)(x))1-(ps/(d+2β+2))∥f∥ℬ˙∞,∞s-((d+2β+2)/p),β(ℝ+d+1)ps/(d+2β+2).
Finally, taking the Lβq norm with q=p(d+2β+2)/(d+2β+2-ps) ends the proof thanks to the fact the maximal function Mβ is bounded of Lβq(ℝ+d+1) into itself for q>1.Theorem 32.
Let1<p<q<∞. For all function f such that f∈ℋ˙p,βs1(ℝ+d+1)⋂‍ℬ˙∞,∞-β,β(ℝ+d+1), one has
(131)∥f∥ℋ˙p,βs(ℝ+d+1)≤C∥f∥ℋ˙p,βs1(ℝ+d+1)θ∥f∥ℬ˙∞,∞-β,β(ℝ+d+1)1-θ,
where θ=p/q, s=θs1-(1-θ)β with β>0, -β<s<s1.Proof.
It suffices to prove that(132)∥(-Δβ)(s-s1)/2f∥Lβq(ℝ+d+1)≤C∥f∥Lβp(ℝ+d+1)θ∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1)1-θ.
Indeed, we use the following identity (which may be easily proven by taking the Weinstein transform in x of both sides)
(133)(-Δβ)-δ/2f(x)=1Γ(δ/2)∫0∞‍tδ/2-1Hβ(t)f(x)dt,
with δ=s1-s>0.
We decompose the integral in two parts as follows:(134)(-Δβ)-δ/2f(x)=1Γ(δ/2)∫0T‍tδ/2-1Hβ(t)f(x)dt+1Γ(δ/2)∫T∞‍tδ/2-1Hβ(t)f(x)dt,
where T is a constant to be fixed later.
We proceed as in [8], we obtain
(135)|Hβ(t)f(x)|≤CMβ(f)(x).
On the other hand, we use Theorem 29 and the fact that f belongs to ℬ˙∞,∞-β-s1,β(ℝ+d+1) to deduce that
(136)|Hβ(t)f(x)|≤Ct(-β-s1)/2∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1).
Thus, by applying the preceding estimates on the right part of (134) we obtain
(137)|(-Δβ)-δ/2f(x)|≤C1Γ(δ/2)Tδ/2Mβ(f)(x)+C2Γ(δ/2)T(δ-β-s1)/2∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1).
We fix now
(138)T=(∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1)Mβ(f)(x))2/(β+s1).
We obtain
(139)|(-Δβ)-δ/2f(x)|≤C1+C2Γ(δ/2)(Mβ(f)(x))θ∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1)1-θ.
Thus we deduce that
(140)∥(-Δβ)-δ/2f∥Lβq(ℝ+d+1)≤C1+C2Γ(δ/2)∥Mβ(f)∥Lβp(ℝ+d+1)θ∥f∥ℬ˙∞,∞-β-s1,β(ℝ+d+1)1-θ.To conclude, we used the fact that the maximal function Mβ is bounded of Lβq(ℝ+d+1) into itself for q>1.


## 4.3. Estimates in Generalized Besov Spaces

For any intervalI of ℝ (bounded or unbounded) and a normed space F(ℝ+d+1), we define the mixed space-time Lp(I;F(ℝ+d+1)) space of (classes of) measurable functions u:I→F(ℝ+d+1) such that ||u||Lp(I;F(ℝ+d+1))<∞, with
(141)∥u∥Lp(I;F(ℝ+d+1))=(∫I‍∥u(t,·)∥F(ℝ+d+1)pdt)1/p,if1≤p<∞,∥u∥L∞(I;F(ℝ+d+1))=esssupt∈I∥u(t,·)∥F(ℝ+d+1).For any intervalI of ℝ (bounded or unbounded) and a Banach space X, we define the mixed space-time C(I¯;X) space of continuous functions I¯→X. When I is bounded, C(I¯;X) is a Banach space with the norm of L∞(I,X).Theorem 33.
Lets∈ℝ and 1≤p,q,r≤∞. Let T>0, g∈ℬ˙p,rs,β(ℝ+d+1), and f in Lq((0,T),ℬ˙p,rs-2+(2/q),β(ℝ+d+1)). Then (76) has a unique solution:
(142)u∈Lq((0,T),ℬ˙p,rs+(2/q),β(ℝ+d+1))⋂‍L∞((0,T),ℬ˙p,rs,β(ℝ+d+1)),
and there exists a constant C such that for all q1∈[q,∞], one has
(143)∥u∥Lq1((0,T),ℬ˙p,rs+(2/q1),β(ℝ+d+1))≤C(∥g∥ℬ˙p,rs,β(ℝ+d+1)+∥f∥Lq((0,T),ℬ˙p,rs-2+(2/q),β(ℝ+d+1))).
If in addition r<∞, then u∈C([0,T],ℬ˙p,rs,β(ℝ+d+1)).Proof.
Sinceg and f are temperate distributions, (76) has a unique solution u in 𝒮′((0,T)×ℝ+d+1), which satisfies
(144)ℱW(u)(t,ξ)=e-t∥ξ∥2ℱW(g)(ξ)+∫0t‍e(τ-t)∥ξ∥2ℱW(f)(τ,ξ)dτ.
Next, we notice that applying Δj to (76) and using formula (81) yield
(145)Δju(t,·)=Hβ(t)Δjg+∫0t‍Hβ(t-τ)Δjf(τ,·)dτ.
Therefore,
(146)∥Δju(t,·)∥Lβp(ℝ+d+1)≤∥Hβ(t)Δjg∥Lβp(ℝ+d+1)+∫0t‍∥Hβ(t-τ)Δjf(τ,·)∥Lβp(ℝ+d+1)dτ.
By virtue of Lemma 30, we thus have, for some κ>0,
(147)∥Δju(t,·)∥Lβp(ℝ+d+1)≤C[e-κ22jt∥Δjg∥Lβp(ℝ+d+1)+∫0t‍e-κ22j(t-τ)∥Δjf(τ,·)∥Lβp(ℝ+d+1)dτ].
Applying convolution inequalities, we get
(148)∥Δju∥Lq1((0,T),Lβp(ℝ+d+1))≤C[((1-e-κTq122jκq122j)1/q1)∥Δjg∥Lβp(ℝ+d+1)+((1-e-κTq222jκq222j)1/q2)∥Δjf∥Lq((0,T),Lβp(ℝ+d+1))],
with 1/q2=1+1/q1-1/q. Finally, taking the lr(ℤ) norm, we conclude that (with the usual convention if r=∞)
(149)∥u∥Lq1((0,T),ℬ˙p,rs+(2/q1),β(ℝ+d+1))≤C[∑j∈ℤ‍((1-e-κTq122jκq122j)r/q1)(2js∥Δjg∥Lβp(ℝ+d+1))r]1/r++C[∑j∈ℤ‍((1-e-κTq222jκq222j)r/q2)++++×(2j(s-2+2/q)∥Δjf∥Lq(0,T;Lβp(ℝ+d+1))r∑j∈ℤ‍((1-e-κTq222jκq222j)r/q2)]1/r,
which insures that u  ∈ Lq((0,T),ℬ˙p,rs+(2/q),β(ℝ+d+1))  ⋂ L∞((0,T),ℬ˙p,rs,β(ℝ+d+1)) and yields the desired inequality. Since u belongs to C([0,T],ℬ˙p,rs,β(ℝ+d+1)) in the case where r is finite may be easily deduced from the density of 𝒮*(ℝd+1)⋂‍ℬ˙p,rs,β(ℝ+d+1) in ℬ˙p,rs,β(ℝ).Theorem 34.
Lets∈ℝ, T>0, and 1≤p,q,r≤∞. One supposes that g∈Bp,rs,β(ℝ+d+1) and f∈Lq((0,T),Bp,rs-2+(2/q),β(ℝ+d+1)). Then (76) has a unique solution u belonging to
(150)Lq((0,T),Bp,rs+(2/q),β(ℝ+d+1))⋂‍L∞((0,T),Bp,rs,β(ℝ+d+1)),
and there exists a constant C such that, for all q≤q1≤∞,
(151)∥u∥Lq1((0,T),Bp,rs+(2/q1),β(ℝ+d+1))≤C[(1+T1+(1/q1)-(1/q))∥f∥Lq((0,T),Bp,rs-2+(2/q),β(ℝ+d+1))(1+T1/q1)∥g∥Bp,rs,β(ℝ+d+1)+(1+T1+1/q1-1/q)∥f∥Lq((0,T),Bp,rs-2+(2/q),β(ℝ+d+1))(1+T1/q1)∥g∥Bp,rs,β(ℝ+d+1)].
If in addition r<∞, then u∈C([0,T],Bp,rs,β(ℝ+d+1)).Proof.
Sinceg,f are tempered, (76) has a unique solution u in 𝒮′((0,T)×ℝ+d+1) satisfying
(152)ℱW(u)(t,ξ)=e-t∥ξ∥2ℱW(g)(ξ)+∫0t‍e(τ-t)∥ξ∥2ℱW(f)(τ,ξ)dτ.
Hence, applying Δj, j≥0, to (81), we see that
(153)Δju(t,·)=Hβ(t)Δjg+∫0t‍Hβ(t-τ)Δjf(τ,·)dτ,
and thus, by Lemma 30, we can deduce that
(154)∥Δju(t,·)∥Lβp(ℝ+d+1)≤∥Hβ(t)Δjg∥Lβp(ℝ+d+1)+∫0t‍∥Hβ(t-τ)Δjf(τ,·)∥Lβp(ℝ+d+1)dτ≤C[∫0t‍e-κ22j(t-τ)e-κ22jt∥Δjg∥Lβp(ℝ+d+1)++++∫0t‍e-κ22j(t-τ)∥Δjf(τ,·)∥Lβp(ℝ+d+1)dτ∥Δjg∥Lβp(ℝ+d+1)].
Then it follows from convolution inequalities that ∥Δju∥Lq1((0,T),Lβp(ℝ+d+1)) is dominated by
(155)(1-e-κTq122jκq122j)1/q1∥Δjg∥Bp,rs,β(ℝ+d+1)+(1-e-κTq222jκq222j)1/q2∥Δjf∥Lq((0,T),Lβp(ℝ+d+1)),
with 1/q2=1+1/q1-1/q. Moreover, similarly as above, we can obtain that
(156)∥Δ-1u(t,·)∥Lβp(ℝ+d+1)≤∥Δ-1g∥Lβp(ℝ+d+1)+∫0t‍∥Δ-1f(τ,·)∥Lβp(ℝ+d+1)dτ,
and thus if 1≤q≤q1≤∞,
(157)∥Δ-1u∥Lq1((0,T),Lβp(ℝ+d+1))≤C(T1/q1∥Δ-1g∥Lβp(ℝ+d+1)+T1/q2∥Δ-1f∥Lq((0,T),Lβp(ℝ+d+1))).
Finally, taking the lr-norm with respect to j in (155) and (157) with the usual convention if r=∞, we can deduce the desired estimate.



---

*Source: 723976-2013-10-07.xml*