Heat Equations Associated with Weinstein Operator and Applications

Abstract

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

Dedicated to Khalifa Trimèche

1. Introduction

We consider the Weinstein operator defined on ℝ^𝑑  ×  ]0,∞[ by Δ_𝛽∶=^𝑑+1_𝑖=1𝜕²𝜕𝑥²_𝑖+2𝛽+1𝑥_𝑑+1𝜕𝜕𝑥_𝑑+11,𝛽>−2=Δ_𝑑+ℒ_𝛽, (1) where Δ_𝑑 is the Laplacian for the 𝑑-first variables and ℒ_𝛽 is the Bessel operator for the last variable given by ℒ_𝛽=𝜕²𝜕𝑥²_𝑑+1+2𝛽+1𝑥_𝑑+1𝜕𝜕𝑥_𝑑+11,𝛽>−2. (2) For 𝑑>2, the operator Δ_𝛽 is the Laplace-Beltrami operator on the Riemannian space ℝ^𝑑×]0,∞[ equipped with the following metric: 𝑑𝑠²=𝑥^{2(2𝛽+1)/(𝑑−1)}_𝑑+1^𝑑+1_𝑖=1𝑑𝑥²_𝑖 (3) (cf. [1, 2]). The Weinstein operator Δ_𝛽 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 𝜕_𝑡𝑢−Δ_𝛽𝑢=𝑓(𝑡,𝑥),(𝑡,𝑥)∈(0,∞)×ℝ₊^𝑑+1,𝑢_|𝑡=0=𝑔. (4) 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. Section 2 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, ℝ₊^𝑑+1=ℝ^𝑑×[[,𝑥0,∞𝑥=₁,…,𝑥_𝑑,𝑥_𝑑+1=𝑥^′,𝑥_𝑑+1∈ℝ₊^𝑑+1,‖𝑥‖=𝑥²₁+⋯+𝑥²_𝑑+1. (5) We denote by 𝐶_∗(ℝ^𝑑+1) the space of continuous functions on ℝ^𝑑+1, even with respect to the last variable; 𝐶^𝑝_∗(ℝ^𝑑+1) the space of functions of class 𝐶^𝑝 on ℝ^𝑑+1, even with respect to the last variable; ℰ_∗(ℝ^𝑑+1) the space of 𝐶^∞-functions on ℝ^𝑑+1, even with respect to the last variable; 𝒮_∗(ℝ^𝑑+1) the Schwartz space of rapidly decreasing functions on ℝ^𝑑+1, even with respect to the last variable; 𝐷_∗(ℝ^𝑑+1) the space of 𝐶^∞-functions on ℝ^𝑑+1 which are of compact support, even with respect to the last variable; 𝒮^′_∗(ℝ^𝑑+1) the space of temperate distributions on ℝ^𝑑+1, even with respect to the last variable. It is the topological dual of 𝒮_∗(ℝ^𝑑+1).

We consider the Weinstein operator Δ_𝛽 defined by ∀𝑥=𝑥^′,𝑥_𝑑+1∈ℝ^𝑑×][,Δ0,∞_𝛽𝑓(𝑥)=Δ_𝑥′𝑓𝑥^′,𝑥_𝑑+1+ℒ_{𝛽,𝑥𝑑+1}𝑓𝑥^′,𝑥_𝑑+1,𝑓∈𝐶²_∗ℝ^𝑑+1, (6) where Δ_𝑥′ is the Laplace operator on ℝ^𝑑 and ℒ_{𝛽,𝑥𝑑+1} is the Bessel operator on ]0,∞[ given by ℒ_{𝛽,𝑥𝑑+1}𝑑∶=²𝑑𝑥²_𝑑+1+2𝛽+1𝑥_𝑑+1𝑑𝑑𝑥_𝑑+11,𝛽>−2. (7)

The Weinstein kernel Λ is given by Λ(𝑥,𝑧)∶=𝑒^{𝑖⟨𝑥′,𝑧′⟩}𝑗_𝛽𝑥_𝑑+1𝑧_𝑑+1,∀(𝑥,𝑧)∈ℝ^𝑑+1×ℂ^𝑑+1, (8) where 𝑗_𝛽(𝑥_𝑑+1𝑧_𝑑+1) is the normalized Bessel function. The Weinstein kernel satisfies the following properties.(i)For each 𝑧∈ℝ₊^𝑑+1, we have Δ_𝛽Λ(𝑥,𝑧)=−‖𝑧‖²Λ(𝑥,𝑧)∀𝑥∈ℝ₊^𝑑+1. (9)(ii)For all 𝑧,𝑡∈ℂ^𝑑+1, we have Λ()()()Λ()()𝑧,𝑡=Λ𝑡,𝑧,Λ𝑧,0=1,𝜆𝑧,𝑡=Λ𝑧,𝜆𝑡,∀𝜆∈ℂ. (10)(iii)For all 𝜈∈ℕ^𝑑+1,  𝑥∈ℝ^𝑑+1, and 𝑧∈ℂ^𝑑+1, we have 𝐷^𝜈_𝑧Λ(𝑥,𝑧)≤‖𝑥‖^|𝜈|exp(‖𝑥‖‖Im𝑧‖), (11)where 𝐷^𝜈_𝑧=𝜕^𝜈/(𝜕𝑧^𝜈1₁⋯𝜕𝑧^𝜈𝑑+1_𝑑+1) and |𝜈|=𝜈₁+⋯+𝜈_𝑑+1. In particular, Λ𝑥,𝑦≤1,∀𝑥,𝑦∈ℝ^𝑑+1. (12)

The Weinstein intertwining operator is the operator ℛ_𝛽 defined on 𝐶_∗(ℝ^𝑑+1) by ℛ_𝛽𝑓𝑥^′,𝑥_𝑑+1=2Γ𝛽+1√𝑥𝜋Γ𝛽+1/2^{−2𝛽𝑑+1}×^𝑥𝑑+1₀𝑥²_𝑑+1−𝑡^2𝛽−1/2𝑓𝑥^′,𝑡𝑑𝑡,𝑥_𝑑+1𝑓𝑥>0,_′,0,𝑥_𝑑+1=0. (13)

ℛ_𝛽 is a topological isomorphism from ℰ_∗(ℝ^𝑑+1) onto itself satisfying the following transmutation relation: Δ_𝛽ℛ_𝛽𝑓=ℛ_𝛽Δ_𝑑+1𝑓,∀𝑓∈ℰ_∗ℝ^𝑑+1, (14) where Δ_𝑑+1=∑^𝑑+1_𝑗=1𝜕²_𝑗 is the Laplacian on ℝ^𝑑+1.

We denote by 𝐿^𝑝_𝛽(ℝ₊^𝑑+1) the space of measurable functions on ℝ₊^𝑑+1 such that 𝑓_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}=_ℝ+𝑑+1𝑓(𝑥)^𝑝𝑑𝜇_𝛽(𝑥)^1/𝑝𝑓<∞,if1≤𝑝<∞,_{𝐿∞𝛽(ℝ+𝑑+1)}=esssup_{𝑥∈ℝ+𝑑+1}𝑓(𝑥)<∞, (15) where 𝑑𝜇_𝛽 is the measure on ℝ₊^𝑑+1 given by 𝑑𝜇_𝛽𝑥^′,𝑥_𝑑+1∶=𝑥^2𝛽+1_𝑑+1𝑑𝑥^′𝑑𝑥_𝑑+1. (16)

The Weinstein transform is given for 𝑓 in 𝐿¹_𝛽(ℝ₊^𝑑+1) by ℱ_𝑊𝑓𝑦=_ℝ+𝑑+1𝑓(𝑥)Λ−𝑥,𝑦𝑑𝜇_𝛽(𝑥),∀𝑦∈ℝ₊^𝑑+1. (17)Some basic properties of this transform are as follows.(i)For 𝑓 in 𝐿¹_𝛽(ℝ₊^𝑑+1), ℱ_𝑊𝑓_{𝐿∞𝛽(ℝ+𝑑+1)}≤𝑓_{𝐿1𝛽(ℝ+𝑑+1)}. (18)(ii)For 𝑓 in 𝒮_∗(ℝ^𝑑+1), we have ℱ_𝑊Δ_𝛽𝑓𝑦=−𝑦²ℱ_𝑊𝑓𝑦,∀𝑦∈ℝ₊^𝑑+1. (19)(iii)For all 𝑓 in 𝐿¹_𝛽(ℝ₊^𝑑+1), if ℱ_𝑊(𝑓) belongs to 𝐿¹_𝛽(ℝ₊^𝑑+1), then 𝑓𝑦=𝐶𝛽_ℝ+𝑑+1ℱ_𝑊𝑓(𝑥)Λ𝑥,𝑦𝑑𝜇_𝛽(𝑥),a.e., (20) where 𝐶𝛽1∶=𝜋^𝑑4^𝛽+𝑑/2Γ𝛽+1². (21)(iv)For 𝑓∈𝒮_∗(ℝ^𝑑+1), if we define ℱ_𝑊𝑓𝑦=ℱ_𝑊𝑓−𝑦, (22) then ℱ_𝑊⁻¹=1𝐶𝛽ℱ_𝑊. (23)

Proposition 1 (see [2]). (i)  The Weinstein transform ℱ_𝑊 is a topological isomorphism from 𝒮_∗(ℝ^𝑑+1) onto itself, and for all f in 𝒮_∗(ℝ^𝑑+1), _ℝ+𝑑+1𝑓(𝑥)²𝑑𝜇_𝛽(𝑥)=𝐶𝛽_ℝ+𝑑+1ℱ_𝑊𝑓(𝜉)²𝑑𝜇_𝛽(𝜉). (24)
(ii)  In particular, the renormalized Weinstein transform 𝑓→𝐶(𝛽)^1/2ℱ_𝑊(𝑓) can be uniquely extended to an isometric isomorphism from 𝐿²_𝛽(ℝ₊^𝑑+1) onto itself.

In the Fourier analysis, the translation operator is given by 𝑓↦𝑓(⋅+𝑥).

In harmonic analysis associated for the operator Δ_𝛽, the generalized translation operator 𝜏_𝑥,𝑥∈ℝ₊^𝑑+1 is defined by ∀𝑦∈ℝ₊^𝑑+1,𝜏_𝑥𝑓𝑦=Γ𝛽+1√×𝜋Γ𝛽+1/2_𝜋0𝑓𝑥^′+𝑦^′,𝑥²_𝑑+1+𝑦²_𝑑+1+2𝑥_𝑑+1𝑦_𝑑+1×()cos𝜃sin𝜃^2𝛽𝑑𝜃, (25) where 𝑓∈𝐶_∗(ℝ^𝑑+1).

By using the Weinstein kernel, we can also define a generalized translation. For functions 𝑓∈𝒮_∗(ℝ^𝑑+1) and 𝑦∈ℝ₊^𝑑+1 the generalized translation 𝜏_𝑦𝑓 is defined by the following relation: ℱ_𝑊𝜏_𝑦𝑓(𝑥)=Λ𝑥,𝑦ℱ_𝑊𝑓(𝑥). (26) By using the generalized translation, we define the generalized convolution product 𝑓∗_𝑊𝑔 of functions 𝑓,𝑔∈𝐿¹_𝛽(ℝ₊^𝑑+1) as follows: 𝑓∗_𝑊𝑔(𝑥)=_ℝ+𝑑+1𝜏_𝑥𝑓−𝑦^′,𝑦_𝑑+1𝑔𝑦𝑑𝜇_𝛽𝑦. (27) This convolution is commutative and associative, and it satisfies the following.(i)For all 𝑓,𝑔∈𝐿¹_𝛽(ℝ₊^𝑑+1), 𝑓∗_𝑊𝑔 belongs to 𝐿¹_𝛽(ℝ₊^𝑑+1) and ℱ_𝑊𝑓∗_𝑊𝑔=ℱ_𝑊𝑓ℱ_𝑊𝑔. (28)(ii)Let 1≤𝑝,𝑞,𝑟≤∞ such that 1/𝑝+1/𝑞−1/𝑟=1. If 𝑓∈𝐿^𝑝_𝛽(ℝ₊^𝑑+1) and 𝑔∈𝐿^𝑞_𝛽(ℝ₊^𝑑+1), then 𝑓∗_𝑊𝑔∈𝐿^𝑟_𝛽(ℝ₊^𝑑+1) and 𝑓∗_𝑊𝑔_{𝐿𝑟𝛽(ℝ+𝑑+1)}≤𝑓_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}𝑔_𝐿^𝑞_{𝛽(ℝ+𝑑+1)}. (29)

We define the tempered distribution 𝒯_𝑓 associated with 𝑓∈𝐿^𝑝_𝛽(ℝ₊^𝑑+1) by 𝒯_𝑓,𝜙=_ℝ+𝑑+1𝑓(𝑥)𝜙(𝑥)𝑑𝜇_𝛽(𝑥), (30) for 𝜙∈𝒮_∗(ℝ^𝑑+1) and denote by ⟨𝑓,𝜙⟩_𝛽 the integral in the righthand side.

Definition 2. The Weinstein transform ℱ_𝑊(𝜏) of a distribution 𝜏∈𝒮^′_∗(ℝ^𝑑+1) is defined by ℱ_𝑊(𝜏),𝜙=𝜏,ℱ_𝑊𝜙, (31) for 𝜙∈𝒮_∗(ℝ^𝑑+1).

In particular, for 𝑓∈𝐿^𝑝_𝛽(ℝ₊^𝑑+1), it follows that for 𝜙∈𝒮_∗(ℝ^𝑑+1), ℱ_𝑊𝑓,𝜙=ℱ_𝑊𝒯_𝑓,𝜙=𝒯_𝑓,ℱ_𝑊𝜙=𝑓,ℱ_𝑊𝜙_𝛽. (32)

Proposition 3. The Weinstein transform ℱ_𝑊 is a topological isomorphism from 𝒮^′_∗(ℝ^𝑑+1) onto itself.

Definition 4. The generalized convolution product of a distribution 𝑆 in 𝒮^′_∗(ℝ^𝑑+1) and a function 𝜙 in 𝒮_∗(ℝ^𝑑+1) is the function 𝑆∗_𝑊𝜙 defined by 𝑆∗_𝑊𝜙(𝑥)=𝑆_𝑦,𝜏_−𝑦𝜙(𝑥). (33)

Proposition 5. Let 𝑓 be in 𝐿^𝑝_𝛽(ℝ₊^𝑑+1), 1≤𝑝≤∞, and 𝜙 in 𝒮_∗(ℝ^𝑑+1). Then, the distribution 𝒯_𝑓∗_𝑊𝜙 is given by the function 𝑓∗_𝑊𝜙 and 𝒯_𝑓∗_𝑊𝜙 belongs to 𝐿^𝑝_𝛽(ℝ₊^𝑑+1). Moreover, for all 𝜓∈𝒮_∗(ℝ^𝑑+1), 𝒯_𝑓∗_𝑊𝜙,𝜓=𝑓,̌𝜙∗_𝑊𝜓̌_𝛽, (34) where 𝜓̌(𝑥)=𝜓(−𝑥) and ℱ_𝑊𝒯_𝑓∗_𝑊𝜙=ℱ_𝑊𝒯_𝑓ℱ_𝑊𝜙. (35)

For each 𝑢∈𝒮^′_∗(ℝ^𝑑+1), we define the distribution Δ_𝛽𝑢 by ⟨Δ_𝛽𝑢,𝜓⟩=⟨𝑢,Δ_𝛽𝜓⟩, and this distribution satisfies the following property: ℱ_𝑊Δ_𝛽𝑢=−𝑦²ℱ_𝑊(𝑢). (36) In the following, we denote 𝒯_𝑓 given by (30) by 𝑓 for simplicity.

3. ℬ̇^𝑠,𝛽_𝑝,𝑞, ℱ̇^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1), ℋ̇^𝑠_𝑝,𝛽 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 by 𝒞 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_∗(ℝ^𝑑+1) such that supp𝜓⊂𝐵(0,1),supp𝜑⊂𝒞,∀𝜉∈ℝ^𝑑+1,𝜓(𝜉)+_𝑗≥0𝜑2^−𝑗𝜉=1,∀𝜉∈𝒞,_𝑗∈ℤ𝜑2^−𝑗𝜉||2=1,𝑛−𝑚≥2⟹supp𝜑^−𝑛⋅∩supp𝜑2^−𝑚⋅2=∅,𝑗≥1⟹supp𝜓∩supp𝜑^−𝑗⋅=∅. (37)

Notations. We denote by Δ_𝑗𝑓=ℱ_𝑊⁻¹𝜑𝜉2^𝑗ℱ_𝑊𝑓,𝑆_𝑗𝑓=_{𝑛≤𝑗−1}Δ_𝑛𝑓,∀𝑗∈ℤ. (38) The distribution Δ_𝑗𝑓 is called the 𝑗th dyadic block of the homogeneous Littlewood-Paley decomposition of 𝑓 associated with the Weinstein operators.

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

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

Remark 7. We remark that ℱ_𝑊𝑆_𝑗𝑓(𝜉)=𝜓𝜉2^𝑗ℱ_𝑊𝑆_𝑗𝑓(𝜉),ℱ_𝑊Δ_𝑗𝑓(𝜉)=𝜑𝜉2^𝑗ℱ_𝑊Δ_𝑗𝑓(𝜉). (39)

We put 𝜙=ℱ_𝑊⁻¹𝜑,𝜒=ℱ_𝑊⁻¹𝜓. (40)

Definition 8. One denotes by 𝒮^′_ℎ,𝛽,∗(ℝ^𝑑+1) the space of tempered distribution such that lim_{𝑗→−∞}𝑆_𝑗𝑢=0in𝒮^′_∗ℝ^𝑑+1. (41)

Proposition 9 (Bernstein inequalities). For all 𝜇∈ℕ^𝑑+1 and 𝜎∈ℝ, for all 𝑗∈ℤ, for all 1≤𝑝, 𝑞≤∞, and for all 𝑓∈𝒮^′_∗(ℝ^𝑑+1), one has the following (𝑖)Δ_𝑗𝑓_{𝐿𝑞𝛽(ℝ+𝑑+1)}≤𝜙_{𝐿𝑟𝛽(ℝ+𝑑+1)}Δ_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}2^{𝑗(𝑑+2𝛽+2)(1/𝑝−1/𝑞)},1𝑤𝑖𝑡ℎ𝑞=1𝑝+1𝑟(𝑆−1.𝑖𝑖)_𝑗𝑓_{𝐿𝑞𝛽(ℝ+𝑑+1)}≤𝜒_{𝐿𝑟𝛽(ℝ+𝑑+1)}𝑆_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}2^{𝑗(𝑑+2𝛽+2)(1/𝑝−1/𝑞)},1𝑤𝑖𝑡ℎ𝑞=1𝑝+1𝑟(−1.𝑖𝑖𝑖)−Δ_𝛽^𝜎Δ_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤ℱ_𝑊⁻¹𝜉^𝜎𝜑_{𝐿1𝛽(ℝ+𝑑+1)}×Δ_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}2^𝑗𝜎. (42)

Proof. Using Remark 7, we deduce from Proposition 5 that 𝑆_𝑗𝑓=2^{𝑗(𝑑+2𝛽+2)}𝜒2^𝑗⋅∗_𝑊𝑆_𝑗Δ𝑓,_𝑗𝑓=2^{𝑗(𝑑+2𝛽+2)}𝜙2^𝑗⋅∗_𝑊Δ_𝑗𝑓. (43) 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 on ℝ₊^𝑑+1 and obtain their basic properties.

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

Definition 10. Let 𝑠∈ℝ and 𝑝,𝑞∈[1,∞]. The homogeneous Weinstein-Besov spaces ℬ̇^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1) are the spaces of distribution in 𝒮^′_ℎ,𝛽,∗(ℝ^𝑑+1) such that 𝑓_̇ℬ^𝑠,𝛽_{𝑝,𝑞(ℝ+𝑑+1)}∶=_𝑗∈ℤ2^𝑠𝑗Δ_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑞1/𝑞<∞. (44)

Proposition 11 (see [6]). Let 𝑠∈ℝ and 𝑝 and 𝑞 two elements of [1,∞]; the space ℬ̇^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1) is the set of 𝑓∈𝒮^′_ℎ,𝛽,∗(ℝ^𝑑+1) verifying 𝑓̇_{̇ℬ𝑠,𝛽𝑝,𝑞(ℝ+𝑑+1)}=^∞₀𝑡^−𝑠𝑓∗_𝑊𝜙_𝑡𝐿^𝑝_{𝛽(ℝ+𝑑+1)}^𝑞𝑑𝑡𝑡^1/𝑞<∞, (45) where 𝜙_𝑡(𝑥)=(1/𝑡^2𝛽+2+𝑑)𝜙(𝑥/𝑡), for all 𝑡∈(0,∞) and 𝑥∈ℝ₊^𝑑+1.

Definition 12. For 𝑠∈ℝ and 𝑝,𝑞∈[1,∞], one writes 𝑓_𝐵^𝑠,𝛽_{𝑝,𝑞(ℝ+𝑑+1)}=𝑆₀𝑓_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}+_𝑗≥12^𝑠𝑗Δ_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑞1/𝑞. (46)
The nonhomogeneous Besov space 𝐵^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1) associated with the Weinstein operators is defined by 𝐵^𝑠,𝛽_𝑝,𝑞ℝ₊^𝑑+1=𝑓∈𝒮^′ℝ^𝑑∶𝑓_𝐵^𝑠,𝛽_{𝑝,𝑞(ℝ+𝑑+1)}<∞. (47)

We give now another definition equivalent to the nonhomogeneous Besov space 𝐵^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1).

Proposition 13. Let 𝑠∈ℝ and 𝑝 and 𝑞 two elements of [1,∞]; the space 𝐵^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1) is the set of 𝑓∈𝒮^′(ℝ^𝑑) verifying 𝑓̇_{𝐵𝑠,𝛽𝑝,𝑞(ℝ+𝑑+1)}=𝑓∗_𝑊𝜓_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}+¹₀𝑡^−𝑠𝑓∗_𝑊𝜙_𝑡𝐿^𝑝_{𝛽(ℝ+𝑑+1)}^𝑞𝑑𝑡𝑡^1/𝑞<∞. (48)

Definition 14. Let 𝑠∈ℝ and 1≤𝑝, 𝑞≤∞, the homogeneous Weinstein-Triebel-Lizorkin space ℱ̇^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1) is the space of distribution in 𝒮^′_ℎ,𝛽,∗(ℝ^𝑑+1) such that 𝑓_̇ℱ^𝑠,𝛽_{𝑝,𝑞(ℝ+𝑑+1)}∶=_𝑗∈ℤ2^𝑠𝑗𝑞Δ_𝑗𝑓^𝑞1/𝑞_{𝐿𝑝𝛽(ℝ+𝑑+1)}<∞. (49)

Definition 15. For 𝑠∈ℝ, the operator ℛ^𝑠_𝛽 from 𝒮^′_ℎ,𝛽,∗(ℝ^𝑑+1) to 𝒮^′_ℎ,𝛽,∗(ℝ^𝑑+1) is defined by ℛ^𝑠_𝛽𝑓=ℱ_𝑊⁻¹‖⋅‖^𝑠ℱ_𝑊𝑓. (50) The operator ℛ_𝛽^−𝑠 is called Weinstein-Riesz potential space.

Definition 16. For 𝑠∈ℝ and 1≤𝑝≤∞, the homogeneous Weinstein-Riesz potential space ℋ̇^𝑠_𝑝,𝛽(ℝ₊^𝑑+1) is defined as the space ℛ_𝛽^−𝑠(𝐿^𝑝_𝛽(ℝ₊^𝑑+1)), equipped with the norm ‖𝑓‖_̇ℋ^𝑠_{𝑝,𝛽(ℝ+𝑑+1)}=‖ℛ^𝑠_𝛽(𝑓)‖_{𝐿𝑝𝛽(ℝ+𝑑+1)}.

Proposition 17. Let 𝑠∈ℝ and 1≤𝑝,𝑞≤∞.
The operator Δ_𝛽 is a linear continuous operator from ℬ̇^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1) into ℬ̇^{𝑠−2,𝛽}_𝑝,𝑞(ℝ₊^𝑑+1) and from ℋ̇^𝑠_𝑝,𝛽(ℝ₊^𝑑+1) into ℋ̇^𝑠−2_𝑝,𝛽(ℝ₊^𝑑+1).

Proof. We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).

Proposition 18. Let 𝑠,𝑡∈ℝ and 1≤𝑝,𝑞≤∞. The operator ℛ^𝑡_𝛽 is a linear continuous injective operator from ℬ̇^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1) onto ℬ̇^{𝑠−𝑡,𝛽}_𝑝,𝑞(ℝ₊^𝑑+1) and from ℋ̇^𝑠_𝑝,𝛽(ℝ₊^𝑑+1) onto ℋ̇^𝑠−𝑡_𝑝,𝛽(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 𝑙_𝑞-spaces and the Minkowskis inequality yield the following.

Proposition 19. If 1≤𝑞₁<𝑞₂≤∞ one has ℬ̇^𝑠,𝛽_𝑝,𝑞1ℝ₊^𝑑+1↪ℬ̇^𝑠,𝛽_𝑝,𝑞2ℝ₊^𝑑+1,1≤𝑝≤∞,𝑠∈ℝ. (51) Moreover, ℬ̇^𝑠,𝛽_𝑝,1ℝ₊^𝑑+1↪ℋ̇^𝑠_𝑝,𝛽ℝ₊^𝑑+1↪ℬ̇^𝑠,𝛽_𝑝,∞ℝ₊^𝑑+1,1≤𝑝≤∞,𝑠∈ℝ. (52) If 𝑠₀≠𝑠₁ one also has ℋ̇^𝑠0_𝑝,𝛽ℝ₊^𝑑+1,ℋ̇^𝑠1_𝑝,𝛽ℝ₊^𝑑+1_𝜃,𝑞=ℬ̇^𝑠,𝛽_𝑝,𝑞ℝ₊^𝑑+1,1≤𝑝,𝑞≤∞,𝜃∈(0,1), (53) where 𝑠=(1−𝜃)𝑠₀+𝜃𝑠₁.

Proposition 20. One assumes that 𝑠−(𝑑+2𝛽+2)/𝑝=𝑠₁−(𝑑+2𝛽+2)/𝑝₁. Then the following inclusion holds ℬ̇^𝑠,𝛽_𝑝,𝑞ℝ₊^𝑑+1↪ℬ̇^𝑠1_𝑝^,𝛽_1,𝑞1ℝ₊^𝑑+1,1≤𝑝≤𝑝₁≤∞,1≤𝑞≤𝑞₁≤∞,𝑠,𝑠₁∈ℝ. (54)

Proof. In order to prove the inclusion, we use the estimate Δ_𝑗𝑓=2^{𝑗(𝑑+2𝛽+2)}𝜙2^𝑗⋅∗_𝑊Δ_𝑗𝑓. (55) Proposition 9(i) gives that Δ_𝑗𝑓_{𝐿𝑝1𝛽(ℝ+𝑑+1)}=2^{𝑗(𝑑+2𝛽+2)}𝜙2^𝑗⋅∗_𝑊Δ_𝑗𝑓_{𝐿𝑝1𝛽(ℝ+𝑑+1)}≤𝐶2^{𝑗(𝑑+2𝛽+2)(1/𝑝−1/𝑝1)}Δ_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}. (56) By definition of the homogeneous Weinstein-Besov spaces, we therefore infer 𝑓_̇ℬ^𝑠_1𝑝11^,𝛽_{,𝑞(ℝ+𝑑+1)}=^∞_𝑗=−∞2^𝑗𝑠1Δ_𝑗𝑓_{𝐿𝑝1𝛽(ℝ+𝑑+1)}^𝑞11/𝑞1≤𝐶_𝑗∈ℤ2^𝑗𝑠12^{𝑗(𝑑+2𝛽+2)(1/𝑝−1/𝑝1)}Δ_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑞11/𝑞1≤𝐶_𝑗∈ℤ2^𝑗𝑠Δ_𝑗𝑓_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑞11/𝑞1≤𝐶𝑓_̇ℬ^𝑠,𝛽_{𝑝,𝑞(ℝ+𝑑+1)}, (57) since 𝑞≤𝑞₁. This gives the inclusion.

Proposition 21. (1)  If 𝑢 belongs to ℬ̇^𝑠,𝛽_𝑝,𝑞(ℝ₊^𝑑+1)∩ℬ̇^𝑡,𝛽_𝑝,𝑞(ℝ₊^𝑑+1), then 𝑢 belongs to ℬ̇^{𝜃𝑠+(1−𝜃)𝑡,𝛽}_𝑝,𝑞(ℝ₊^𝑑+1) for all 𝜃∈[0,1] and ‖𝑢‖_̇ℬ^{𝜃𝑠+(1−𝜃)𝑡,𝛽}_{𝑝,𝑞(ℝ+𝑑+1)}≤‖𝑢‖^𝜃_{̇ℬ𝑠,𝛽𝑝,𝑞(ℝ+𝑑+1)}‖𝑢‖_̇ℬ^1−𝜃_{𝑡,𝛽𝑝,𝑞(ℝ+𝑑+1)}. (58)
(2)  If 𝑢 belongs to ℬ̇^𝑠,𝛽_𝑝,∞(ℝ₊^𝑑+1)∩ℬ̇^𝑡,𝛽_𝑝,∞(ℝ₊^𝑑+1) and 𝑠<𝑡, then 𝑢 belongs to ℬ̇^{𝜃𝑠+(1−𝜃)𝑡,𝛽}_𝑝,1(ℝ₊^𝑑+1) for all 𝜃∈(0,1) and there exists a positive constant 𝐶(𝑡,𝑠) such that ‖𝑢‖_̇ℬ^{𝜃𝑠+(1−𝜃)𝑡,𝛽}_{𝑝,1(ℝ+𝑑+1)}≤𝐶(𝑡,𝑠)‖𝑢‖^𝜃_{̇ℬ𝑠,𝛽𝑝,∞(ℝ+𝑑+1)}‖𝑢‖_̇ℬ^1−𝜃_{𝑡,𝛽𝑝,∞(ℝ+𝑑+1)}. (59)
(3)  If 𝑢 belongs to ℬ̇^𝑠,𝛽_𝑝,∞(ℝ₊^𝑑+1)∩ℬ̇^{𝑠+𝜀,𝛽}_𝑝,∞(ℝ₊^𝑑+1) and 𝜀>0, then 𝑢 belongs to ℬ̇^𝑠,𝛽_𝑝,1(ℝ₊^𝑑+1) and there exists a positive constant 𝐶 such that ‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,1(ℝ+𝑑+1)}≤𝐶𝜀‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ+𝑑+1)}log₂‖𝑢‖𝑒+^{̇ℬ𝑠+𝜀,𝛽𝑝,∞(ℝ+𝑑+1)}‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ+𝑑+1)}. (60)

Proof. (1)  is obvious from the Hölder’s inequality. As for (2), we write ‖𝑢‖_̇ℬ^{𝜃𝑠+(1−𝜃)𝑡,𝛽}_{𝑝,1(ℝ+𝑑+1)} as _𝑗≤𝑁2^{𝑗(𝜃𝑠+(1−𝜃)𝑡)}Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}+_𝑗>𝑁2^{𝑗(𝜃𝑠+(1−𝜃)𝑡)}Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}, (61) where 𝑁 is chosen here after. By the definition of the homogeneous Weinstein-Besov norms, we see that 2^{𝑗(𝜃𝑠+(1−𝜃)𝑡)}Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤2^{𝑗(1−𝜃)(𝑡−𝑠)}‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ+𝑑+1)},2^{𝑗(𝜃𝑠+(1−𝜃)𝑡)}Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤2^{−𝑗𝜃(𝑡−𝑠)}‖𝑢‖_̇ℬ^𝑡,𝛽_{𝑝,∞(ℝ+𝑑+1)}, (62) and thus ‖𝑢‖_̇ℬ^{𝜃𝑠+(1−𝜃)𝑡,𝛽}_{𝑝,1(ℝ+𝑑+1)} is dominated by ‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ+𝑑+1)𝑗≤𝑁}2^{𝑗(1−𝜃)(𝑡−𝑠)}+‖𝑢‖_̇ℬ^𝑡,𝛽_{𝑝,∞(ℝ+𝑑+1)𝑗>𝑁}2^{−𝑗𝜃(𝑡−𝑠)}≤𝐶‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ+𝑑+1)}2^{(𝑁+1)(1−𝜃)(𝑡−𝑠)}2^{(1−𝜃)(𝑡−𝑠)}+‖𝑢‖−1_{̇ℬ𝑡,𝛽𝑝,∞(ℝ+𝑑+1)}2^{−𝑁𝜃(𝑡−𝑠)}1−2^{−𝜃(𝑡−𝑠)}. (63) Hence, in order to complete the proof of (2), it suffices to choose 𝑁 such that ‖𝑢‖_̇ℬ^𝑡,𝛽_{𝑝,∞(ℝ𝑑)}‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ𝑑)}≤2^{𝑁(𝑡−𝑠)}‖𝑢‖<2^{̇ℬ𝑡,𝛽𝑝,∞(ℝ𝑑)}‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ𝑑)}. (64) As for (3), it is easy to see that ‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,1(ℝ+𝑑+1)} is dominated as _{𝑗≤𝑁−1}2^𝑗𝑠Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}+_𝑗≥𝑁2^𝑗𝑠Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤(𝑁+1)‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ+𝑑+1)}+2^{−(𝑁−1)𝜀}2^𝜀‖𝑢‖−1^{̇ℬ𝑠+𝜀,𝛽𝑝,∞(ℝ+𝑑+1)}. (65) Hence, letting 1𝑁=1+𝜀log₂‖𝑢‖_̇ℬ^{𝑠+𝜀,𝛽}_{𝑝,∞(ℝ+𝑑+1)}‖𝑢‖_̇ℬ^𝑠,𝛽_{𝑝,∞(ℝ+𝑑+1)}, (66) we can obtain the desired estimate.

Proposition 22. Let 𝑞∈(1,∞) and let 𝑠∈ℝ such that 0<𝑠<(𝑑+2𝛽+2)/𝑞, then one has ℬ̇^𝑠,𝛽_𝑞,𝑞ℝ₊^𝑑+1=ℱ̇^𝑠,𝛽_𝑞,𝑞ℝ₊^𝑑+1↪ℱ̇^𝑠,𝛽_𝑞,∞ℝ₊^𝑑+1↪ℱ̇^{𝑠−((𝑑+2𝛽+2)/𝑞),𝛽}_∞,∞ℝ₊^𝑑+1,ℋ̇^𝑠_𝑞,𝛽ℝ₊^𝑑+1=ℱ̇^𝑠,𝛽_𝑞,2ℝ₊^𝑑+1↪ℱ̇^𝑠,𝛽_𝑞,∞ℝ₊^𝑑+1↪ℱ̇^{𝑠−((𝑑+2𝛽+2)/𝑞),𝛽}_∞,∞ℝ₊^𝑑+1. (67)(68)

Proof. We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).

Theorem 23. Let 𝑎,𝑏>0 and let 𝑞₁,𝑞₂∈[1,∞]. Let, 𝜃=𝑎/(𝑎+𝑏)∈(0,1) and let 1/𝑝=(1−𝜃)/𝑞₁+𝜃/𝑞₂. Then, there exists a constant 𝐶 such that for every 𝑓∈ℱ̇_𝑞^𝑎,𝛽_1,∞(ℝ₊^𝑑+1)∩ℱ̇_𝑞^−𝑏,𝛽_2,∞(ℝ₊^𝑑+1); then one has 𝑓(𝑥)≤𝐶sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥)^1−𝜃sup_𝑗∈ℤ2^−𝑏𝑗Δ_𝑗𝑓(𝑥)^𝜃. (69) In particular, one gets 𝑓_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}≤𝐶𝑓_̇ℱ^1−𝜃_{𝑞1𝑎,𝛽,∞(ℝ+𝑑+1)}𝑓^𝜃_{̇ℱ𝑞2−𝑏,𝛽,∞(ℝ+𝑑+1)}. (70)

Proof. Let 𝑓 be a Schwartz class; we have 𝑓(𝑥)≤_𝑗∈ℤΔ_𝑗𝑓(𝑥≤_𝑗∈ℤ)min2^−𝑎𝑗sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥,2_𝑗𝑏)sup_𝑗∈ℤ2^−𝑗𝑏Δ_𝑗𝑓(𝑥). (71) We define 𝑁(𝑥) as the largest index such that 2^𝑗𝑏sup_𝑗∈ℤ2^−𝑗𝑏Δ_𝑗𝑓(𝑥)≤2^−𝑎𝑗sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥) (72) and we write 𝑓(𝑥)≤_{𝑗≤𝑁(𝑥)}2^𝑗𝑏sup_𝑗∈ℤ2^−𝑗𝑏Δ_𝑗𝑓(𝑥)+_{𝑗>𝑁(𝑥)}2^−𝑎𝑗sup_𝑗2^𝑎𝑗Δ_𝑗𝑓(𝑥)≤𝐶sup_𝑗∈ℤ2^𝑎𝑗Δ_𝑗𝑓(𝑥)^{𝑏/(𝑎+𝑏)}×sup_𝑗∈ℤ2^−𝑏𝑗Δ_𝑗𝑓(𝑥)^{𝑎/(𝑎+𝑏)}. (73) 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. Let 𝑞∈(1,∞) and let 𝑠∈ℝ such that 0<𝑠<(𝑑+2𝛽+2)/𝑞, then one has 𝑓_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}≤𝐶𝑓_̇ℬ^{1−(𝑞/𝑝)}_{−((2𝛽+2+𝑑)/𝑞−𝑠),𝛽∞,∞ℝ+𝑑+1}𝑓_̇ℬ^𝑞/𝑝_{𝑠,𝛽𝑞,𝑞ℝ+𝑑+1},𝑓_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}≤𝐶𝑓_̇ℬ^{1−(𝑞/𝑝)}_{−((2𝛽+2+𝑑)/𝑞−𝑠),𝛽∞,∞(ℝ+𝑑+1)}𝑓_̇ℋ^𝑞/𝑝_{𝑠𝑞,𝛽(ℝ+𝑑+1)}, (74)(75) where 𝑝=𝑞(2𝛽+2+𝑑)/(2𝛽+2+𝑑−𝑞𝑠).

Proof. By choosing 𝑎=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 𝜕_𝑡𝑢(𝑡,𝑥)−Δ_𝛽𝑢(𝑡,𝑥)=𝑓(𝑡,𝑥),(𝑡,𝑥)∈[0,∞)×ℝ₊^𝑑+1,𝑢_|𝑡=0=𝑔. (76) We introduce the Weinstein heat semigroup 𝐻_𝛽(𝑡) for the Weinstein-Laplace operator 𝐻_𝛽(𝑡)𝑔(𝑥)∶=_ℝ+^𝑑+1Γ_𝛽𝑡,𝑥,𝑦𝑔𝑦𝑑𝜇_𝛽𝑦𝑔(𝑥)if𝑡>0if𝑡=0, (77) where Γ_𝛽 is the Weinstein heat kernel defined by Γ_𝛽𝑡,𝑥,𝑦∶=𝜏_𝑥𝐸_𝑡^(𝛽)𝑦, (78) where 𝐸_𝑡^(𝛽)𝑦=2𝜋^𝑑/2Γ𝛽+1(4𝑡)^{𝛽+1+𝑑/2}𝑒^{−‖𝑦‖2/4𝑡}. (79) Thus 𝐻_𝛽(𝑡)𝑔(𝑥)=𝑔∗_𝑊𝐸_𝑡^(𝛽)(𝑥). (80) In practice, we use the integral formulation of (76) 𝑢(𝑡,𝑥)=𝐻_𝛽(𝑡)𝑔(𝑥)+𝐺_𝛽𝑓(𝑡,𝑥)∶=𝐻_𝛽(𝑡)𝑔(𝑥)+^𝑡₀𝐻_𝛽(𝑡−𝑠)𝑓(𝑠,𝑥)𝑑𝑠. (81)

Remark 25. The function 𝐸_𝑡^(𝛽) is the Gauss kernel associated with Weinstein operators. This function satisfies ∀𝜉∈ℝ₊^𝑑+1,ℱ_𝑊𝐸_𝑡^(𝛽)(𝜉)=𝑒^{−𝑡‖𝜉‖2}. (82)

Proposition 26. Let 1≤𝑝≤𝑟≤∞ and let 𝑓∈𝐿^𝑝_𝛽(ℝ₊^𝑑+1). Then the operator 𝐻_𝛽(𝑡) maps 𝐿^𝑝_𝛽(ℝ₊^𝑑+1) continuously to 𝐿^𝑟_𝛽(ℝ₊^𝑑+1) and 𝐻_𝛽(𝑡)𝑓_{𝐿𝑟𝛽(ℝ𝑑)}≤𝐶𝑡^{−((𝑑+2𝛽+2)/2)(1/𝑝−1/𝑟)}𝑓_𝐿^𝑝_{𝛽(ℝ𝑑)}. (83) Moreover, −Δ_𝛽^𝛿/2𝐻_𝛽(𝑡)𝑓_{𝐿𝑟𝛽(ℝ𝑑)}≤𝐶𝑡^{−𝛿/2−((𝑑+2𝛽+2)/2)(1/𝑝−1/𝑟)}𝑓_𝐿^𝑝_{𝛽(ℝ𝑑)} (84) for all 𝛿>0.

Proof. It follows from the relations (80) and (29) combined with scaling property of the kernel 𝐸_𝑡^(𝛽).

In this section, we prove estimates for the Weinstein heat semigroup. These estimates are based on the following result.

Lemma 27. Let 𝒞 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 suppℱ_𝑊(𝑢)⊂𝜆𝒞⟹𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶𝑒^{−𝑐𝑡𝜆2}‖𝑢‖_𝐿^𝑝_𝛽(ℝ+^𝑑+1₎. (85)

Proof. We again consider a function Θ 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 𝐻_𝛽(𝑡)𝑢=𝑔(𝑡,⋅)∗_𝑊𝑢, (86) where 𝑔(𝑡,⋅)=ℱ_𝑊⁻¹Θ(𝜉)𝑒^{−𝑡‖𝜉‖2}. (87) The lemma is proved provided that we can find positive real numbers 𝑐 and 𝐶 such that ∀𝑡>0,𝑔(𝑡,⋅)_{𝐿1𝛽(ℝ+𝑑+1)}≤𝐶𝑒^−𝑐𝑡. (88) To begin, we perform integrations by parts in (87). We get 𝑔(=1𝑡,𝑥)𝑐_𝛽1+‖𝑥‖^{2−(𝑑+[2𝛽]+3)}×_ℝ+𝑑+1Λ(𝑥,𝜉)𝐼𝑑−Δ_𝛽^{𝑑+[2𝛽]+3}×Θ(𝜉)𝑒^{−𝑡‖𝜉‖2}𝑑𝜇_𝛽(𝜉). (89) Using Leibniz's formula, we obtain 𝑔(𝑡,𝑥)≤𝐶1+‖𝑥‖^{2−(𝑑+[2𝛽]+3)}𝑒^−𝑐𝑡, (90) and (88) follows.

For any interval 𝐼 of ℝ (bounded or unbounded), we define the mixed space-time 𝐿^𝑝(𝐼;𝐿^𝑞_𝛽(ℝ₊^𝑑+1)) Banach space of (classes of) measurable functions 𝑢∶𝐼→𝐿^𝑞_𝛽(ℝ₊^𝑑+1) such that ‖𝑢‖_𝐿^𝑝_(𝐼;𝐿^𝑞_𝛽(ℝ+^𝑑+1₎₎<∞, with ‖𝑢‖_𝐿^𝑝_(𝐼;𝐿^𝑞_𝛽(ℝ+^𝑑+1₎₎=_𝐼‖𝑢(𝑡,⋅)‖^𝑝_{𝐿𝑞𝛽(ℝ+𝑑+1)}𝑑𝑡^1/𝑝,‖𝑢‖if1≤𝑝,𝑞<∞,_{𝐿∞(𝐼;𝐿𝑞𝛽(ℝ+𝑑+1))}=esssup_𝑡∈𝐼‖𝑢(𝑡,⋅)‖_𝐿^𝑞_𝛽(ℝ+^𝑑+1₎,if1≤𝑞<∞. (91)

Corollary 28. Let 𝒞 be an annulus and 𝜆 a positive real number. Let 𝑢₀ (resp., 𝑓=𝑓(𝑡,𝑥)) satisfy suppℱ_𝑊(𝑢₀)⊂𝜆𝒞 (resp., suppℱ_𝑊(𝑓(𝑡,⋅))⊂𝜆𝒞 for all 𝑡 in [0,𝑇]). Consider 𝑢 a solution of 𝜕_𝑡𝑢−Δ_𝛽𝑢=0,𝑢_|𝑡=0=𝑢₀, (92) and 𝑣 a solution of 𝜕_𝑡𝑣−Δ_𝛽𝑣=𝑓(𝑡,⋅),𝑣_|𝑡=0=0. (93) There exist positive constants 𝑐 and 𝐶, depending only on 𝒞, such that for any 1≤𝑎≤𝑏≤∞ and 1≤𝑝≤𝑞≤∞, we have ‖𝑢‖_𝐿^𝑞_{([0,𝑇],𝐿}^𝑏_𝛽(ℝ+^𝑑+1₎₎≤𝐶𝜆^{(𝑑+2𝛽+2)(1/𝑎−1/𝑏)}𝜆^−2/𝑞𝑢_{0𝐿𝑎𝛽(ℝ+𝑑+1)},‖𝑣‖_𝐿^𝑞_{([0,𝑇],𝐿}^𝑏_𝛽(ℝ+^𝑑+1₎₎≤𝐶𝜆^{−2(1+1/𝑞−1/𝑝)}𝜆^{(𝑑+2𝛽+2)(1/𝑎−1/𝑏)}×𝑓_{𝐿𝑝([0,𝑇],𝐿𝑎𝛽(ℝ+𝑑+1))}. (94)

Proof. It suffices to use the fact that 𝑢(𝑡,⋅)=𝐻_𝛽(𝑡)𝑢₀,𝑣(𝑡,⋅)=^𝑡₀𝐻_𝛽(𝑡−𝑠)𝑓(𝑠,⋅)𝑑𝑠. (95) Combining Lemma 27 and Young's inequality (29) with scaling property of the kernel 𝐸_𝑡^(𝛽) now yields the result.

Theorem 29. Let 𝑠 be a positive real number and (𝑝,𝑟)∈[1,∞]². A constant 𝐶 exists which satisfies the following property. For 𝑢∈ℬ̇^{−2𝑠,𝛽}_𝑝,𝑟(ℝ₊^𝑑+1), one has 𝐶⁻¹‖𝑢‖_̇ℬ^{−2𝑠,𝛽}_{𝑝,𝑟(ℝ+𝑑+1)}≤𝑡^𝑠𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)𝐿𝑟(ℝ+,𝑑𝑡/𝑡)}≤𝐶‖𝑢‖_̇ℬ^{−2𝑠,𝛽}_{𝑝,𝑟(ℝ+𝑑+1)}. (96)

To prove this result we need the following lemma.

Lemma 30. There exist two positive constants 𝜅 and 𝐶 depending only on 𝜑 such that for all 1≤𝑝≤∞, 𝜏≥0 and 𝑗∈ℤ, one has Δ_𝑗𝐻_𝛽(𝜏)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶𝑒^{−𝜅22𝑗𝜏}Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}. (97)

Proof. The result follows immediately by applying Lemma 27 and because Δ_𝑗(𝐻_𝛽(𝜏))𝑢=(𝐻_𝛽(𝜏)Δ_𝑗)𝑢.

Proof of Theorem 29. Using Lemma 30 and considering the fact that the operator Δ_𝑗 commutes with the operator 𝐻_𝛽(𝑡) and the definition of the homogeneous Weinstein-Besov (semi) norm we get 𝑡^𝑠𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶‖𝑢‖_̇ℬ^{−2𝑠,𝛽}_{𝑝,𝑟(ℝ+𝑑+1)𝑗∈ℤ}𝑡^𝑠2^2𝑗𝑠𝑒^{−𝜅𝑡22𝑗}𝑐_𝑟,𝑗, (98) where (𝑐_𝑟,𝑗)_𝑗∈ℤ denotes, as in all this proof, a generic element of the unit sphere of 𝑙^𝑟(ℤ). In the case when 𝑟=∞, the required inequality comes immediately from the following easy result. For any positive 𝑠, we have sup_{𝑡>0𝑗∈ℤ}𝑡^𝑠2^2𝑗𝑠𝑒^{−𝜅𝑡22𝑗}<∞. (99) In the case 𝑟<∞, using the Hölder inequality with the weight 2^2𝑗𝑠𝑒^{−𝜅𝑡22𝑗}, (99), and the Fubini theorem, we obtain ^∞₀𝑡^𝑟𝑠𝐻_𝛽(𝑡)𝑢^𝑟_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑑𝑡𝑡≤𝐶‖𝑢‖^𝑟_{̇ℬ−2𝑠,𝛽𝑝,𝑟(ℝ+𝑑+1)}^∞_0𝑗∈ℤ𝑡^𝑠2^2𝑗𝑠𝑒^{−𝜅𝑡22𝑗𝑟−1}×_𝑗∈ℤ𝑡^𝑠2^2𝑗𝑠𝑒^{−𝜅𝑡22𝑗}𝑐^𝑟_𝑟,𝑗𝑑𝑡𝑡≤𝐶‖𝑢‖^𝑟_{̇ℬ−2𝑠,𝛽𝑝,𝑟(ℝ+𝑑+1)}^∞_0𝑗∈ℤ𝑡^𝑠2^2𝑗𝑠𝑒^{−𝜅𝑡22𝑗}𝑐^𝑟_𝑟,𝑗𝑑𝑡𝑡≤𝐶‖𝑢‖^𝑟_{̇ℬ−2𝑠,𝛽𝑝,𝑟(ℝ+𝑑+1)𝑗∈ℤ}𝑐^𝑟_𝑟,𝑗^∞₀𝑡^𝑠2^2𝑗𝑠𝑒^{−𝜅𝑡22𝑗}𝑑𝑡𝑡≤𝐶Γ(𝑠)‖𝑢‖^𝑟_{̇ℬ−2𝑠,𝛽𝑝,𝑟(ℝ+𝑑+1)}. (100)
In order to prove the other inequality, let us observe that for any 𝑠 greater than −1, we have Δ_𝑗1𝑢=Γ(𝑠+1)^∞₀𝑡^𝑠−Δ_𝛽^𝑠+1𝐻_𝛽(𝑡)Δ_𝑗𝑢𝑑𝑡. (101) Then, Lemma 30, Proposition 9, and the fact that the operator Δ_𝑗 commutes with the operator 𝐻_𝛽(𝑡) lead to the following: Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶^∞₀𝑡^𝑠2^{2𝑗(𝑠+1)}𝑒^{−𝜅𝑡22𝑗}𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑑𝑡. (102) In the case 𝑟=∞, we simply write Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶sup_𝑡>0𝑡^𝑠𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}×^∞₀2^{2𝑗(𝑠+1)}𝑒^{−𝜅𝑡22𝑗}𝑑𝑡≤𝐶2^2𝑗𝑠sup_𝑡>0𝑡^𝑠𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}. (103) In the case 𝑟<∞, Hölder’s inequality with the weight 𝑒^{−𝜅𝑡22𝑗} gives ^∞₀𝑡^𝑠𝑒^{−𝜅𝑡22𝑗}𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑑𝑡^𝑟≤𝐶2^{−2𝑗(𝑟−1)∞}₀𝑡^𝑟𝑠𝑒^{−𝜅𝑡22𝑗}𝐻_𝛽(𝑡)𝑢^𝑟_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑑𝑡. (104) Thanks to (99) and Fubini’s theorem, we infer from (102) that _𝑗∈ℤ2^{−2𝑗𝑟𝑠}Δ_𝑗𝑢^𝑟_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶^∞₀𝑡^𝑟𝑠𝐻_𝛽(𝑡)𝑢^𝑟_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑑𝑡𝑡. (105) The theorem is proved.

Second Proof of Theorem 29. We only consider the case 1≤𝑟<∞. The case 𝑟=∞ can be shown similarly. We first prove that 𝐶⁻¹‖𝑢‖_{̇ℬ−2𝑠,𝛽𝑝,𝑟(ℝ+𝑑+1)}≤𝑡^𝑠𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)𝐿𝑟(ℝ+,𝑑𝑡/𝑡)}. (106) It is easy to see that Δ_𝑗𝑢=𝜒_𝑗∗_𝑊𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢, (107) where 𝜒_𝑗=ℱ_𝑊⁻¹(𝜑(2^−𝑗𝜉)𝑒^{2−2𝑗‖𝜉‖2}) and 𝐸₂^(𝛽)_−2𝑗 is the Gauss kernel associated with Weinstein operators. By relation (29), we get Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝜒_{𝑗𝐿1𝛽(ℝ+𝑑+1)}𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}. (108) As 𝜒_{𝑗𝐿1𝛽(ℝ+𝑑+1)}=_ℝ+𝑑+1ℱ_𝑊⁻¹𝜑(𝜉)𝑒^‖𝜉‖2𝑑𝜇_𝛽(𝜉)<∞, (109) we obtain Δ_𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}. (110) Moreover simple calculations give that 𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢=𝐻_𝛽2^−4𝑗−𝑡²𝐸_𝑡^(𝛽)₂∗_𝑊𝑢. (111) Thus, from Proposition 26, it follows that 𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶𝐸_𝑡^(𝛽)₂∗_𝑊𝑢_{𝐿𝑝𝛽ℝ+𝑑+1}, (112) for any 𝑡∈[2^−𝑗−1,2^−𝑗], which implies that _𝑗∈ℤ2^{−2𝑠𝑗𝑟}Δ_𝑗𝑢^𝑟_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶_𝑗∈ℤ^2−𝑗_2−𝑗−1𝑡^2𝑠𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟𝑑𝑡𝑡≤𝐶^∞₀𝑡^2𝑠𝐸_𝑡^(𝛽)₂∗_𝑊𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟𝑑𝑡𝑡≤𝐶^∞₀𝑡^𝑠𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟𝑑𝑡𝑡, (113) where we have used the fact that 𝐸_𝑡^(𝛽)₂∗_𝑊𝑢=𝐻_𝛽(𝑡²)𝑢.

We now prove that 𝑡^𝑠𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)𝐿𝑟(ℝ+,𝑑𝑡/𝑡)}≤𝐶‖𝑢‖_{̇ℬ−2𝑠,𝛽𝑝,𝑟(ℝ+𝑑+1)}. (114) Indeed, one has 𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢=_𝑛∈ℤ𝐸₂^(𝛽)_−2𝑗∗_𝑊Δ_𝑛+𝑗𝑢. (115) Arguing as above, we have 𝐸_𝑡^(𝛽)₂∗_𝑊𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢_{𝐿𝑝𝛽ℝ+𝑑+1}, (116) for any 𝑡∈[2^−𝑗,2^1−𝑗]. Thus 𝑡^𝑠𝐻_𝛽(𝑡)𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)𝐿𝑟(ℝ+,𝑑𝑡/𝑡)}=2𝑡^2𝑠𝐸_𝑡^(𝛽)₂∗_𝑊𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)𝐿𝑟(ℝ+,𝑑𝑡/𝑡)}≤𝐶_𝑗∈ℤ^21−𝑗_2−𝑗2^−2𝑗𝑠𝐸₂^(𝛽)_−2𝑗∗_𝑊𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟𝑑𝑡𝑡≤𝐶_𝑗∈ℤ2^−2𝑗𝑠_𝑛∈ℤ𝐸₂^(𝛽)_−2𝑗∗_𝑊Δ_𝑛+𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟. (117) On the other hand, it is easy to see that 𝐸₂^(𝛽)_−2𝑗∗_𝑊Δ_𝑛+𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶2^−2𝑛𝑠Δ_𝑛+𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}, (118) for any 𝑠>0. For 0<𝑠₁<𝑠<𝑠₂ and by using the Minkowski inequality, we have _𝑗∈ℤ2^−2𝑗𝑠_𝑛∈ℤ𝐸₂^(𝛽)_−2𝑗∗_𝑊Δ_𝑛+𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟≤_𝑗∈ℤ2^{0−2𝑗𝑠}_−∞2^{−2𝑛𝑠1}𝐸₂^(𝛽)_−2𝑗∗_𝑊Δ_𝑛+𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟+_𝑗∈ℤ2^−2𝑗𝑠_𝑛∈ℕ2^{−2𝑛𝑠2}𝐸₂^(𝛽)_−2𝑗∗_𝑊Δ_𝑛+𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟≤𝐶⁰_−∞2^{−2𝑛(𝑠1−𝑠)}_𝑗∈ℤ2^{−2(𝑛+𝑗)𝑟𝑠}Δ_𝑛+𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^{𝑟1/𝑟𝑟}+𝐶_ℕ2^{−2𝑛(𝑠2−𝑠)}_𝑗∈ℤ2^{−2(𝑛+𝑗)𝑟𝑠}Δ_𝑛+𝑗𝑢_{𝐿𝑝𝛽(ℝ+𝑑+1)}^{𝑟1/𝑟𝑟}≤𝐶_𝑗∈ℤ2^{−2𝑗𝑟𝑠}Δ_𝑗𝑢^𝑟_{𝐿𝑝𝛽(ℝ+𝑑+1)}. (119) The result is immediately from (117) and (119).

4.2. Embedding Sobolev Theorems

Theorem 31. Let 1<𝑝<∞ and let 0<𝑠<(𝑑+2𝛽+2)/𝑝. There exists a positive constant 𝐶 such that for all function 𝑓∈ℋ̇^𝑠_𝑝,𝛽(ℝ₊^𝑑+1) one has 𝑓_𝐿^𝑞_{𝛽(ℝ+𝑑+1)}≤𝐶𝑓_̇ℋ^1−𝜃_{𝑠𝑝,𝛽(ℝ+𝑑+1)}𝑓^𝜃_{̇ℬ𝑠−((𝑑+2𝛽+2)/𝑝),𝛽∞,∞(ℝ+𝑑+1)}, (120) where 𝜃=𝑠𝑝/(𝑑+2𝛽+2) and 𝑞=𝑝(𝑑+2𝛽+2)/(𝑑+2𝛽+2−𝑝𝑠).

Proof. By density, we can suppose that 𝑓 belongs to 𝒮_∗(ℝ^𝑑+1). It is easy to see that 𝑓=^∞₀𝐻_𝛽(𝑡)Δ_𝛽𝑓𝑑𝑡 (121) and decompose the integral in two parts as follows: 𝑓=^𝐴₀𝐻_𝛽(𝑡)Δ_𝛽𝑓𝑑𝑡+^∞_𝐴𝐻_𝛽(𝑡)Δ_𝛽𝑓𝑑𝑡, (122) where 𝐴 is a constant to be fixed later.
On the other hand, by Theorem 29, we obtain𝐻_𝛽(𝑡)Δ_𝛽𝑓_{𝐿∞𝛽(ℝ+𝑑+1)}≤𝐶𝑡^{1−(1/2)(𝑠−(𝑑+2𝛽+2)/𝑝)}𝑓_{̇ℬ𝑠−((𝑑+2𝛽+2)/𝑝),𝛽∞,∞(ℝ+𝑑+1)}. (123) Therefore after integrating, we get ^∞_𝐴𝐻_𝛽(𝑡)Δ_𝛽𝑓_{𝐿∞𝛽(ℝ+𝑑+1)}𝑑𝑡≤𝐴^{(1/2)(𝑠−(𝑑+2𝛽+2)/𝑝)}𝑓_{̇ℬ𝑠−((𝑑+2𝛽+2)/𝑝),𝛽∞,∞(ℝ+𝑑+1)}. (124) On the other hand, denoting 𝑔=(−Δ_𝛽)^𝑠/2𝑓, we have 𝐻_𝛽(𝑡)Δ_𝛽1𝑓=(−𝑡)^1−𝑠/2𝐻_𝛽(𝑡)−𝑡Δ_𝛽^1−𝑠/2𝑔. (125) We proceed as in [8], we prove that 𝐻_𝛽(𝑡)−𝑡Δ_𝛽^1−𝑠/2𝑔(𝑥)≤𝐶(𝑠)𝑀_𝛽𝑔(𝑥), (126) where 𝑀_𝛽(𝑔) is a maximal function of 𝑔 associated with the Weinstein operators (cf. [12]).
This leads to ^𝐴₀𝐻_𝛽(𝑡)Δ_𝛽𝑓(𝑥)𝑑𝑡≤𝐶𝐴^𝑠/2𝑀_𝛽𝑔(𝑥). (127) In conclusion, we get ^∞₀𝐻_𝛽(𝑡)Δ_𝛽𝑓(𝑥)𝐴𝑑𝑡≤𝐶^𝑠/2𝑀_𝛽𝑔(𝑥)+𝐴^{(1/2)(𝑠−(𝑑+2𝛽+2)/𝑝)}𝑓_{̇ℬ𝑠−((𝑑+2𝛽+2)/𝑝),𝛽∞,∞(ℝ+𝑑+1)}, (128) and the choice of 𝐴 such that 𝐴^{(𝑑+2𝛽+2)/2𝑝}𝑀_𝛽𝑔(𝑥)=𝑓_{̇ℬ𝑠−(𝑑+2𝛽+2)/𝑝,𝛽∞,∞(ℝ+𝑑+1)} (129) ensures that ^∞₀𝐻_𝛽(𝑡)Δ_𝛽𝑓(𝑥)𝑀𝑑𝑡≤𝐶_𝛽(𝑔)(𝑥)^{1−(𝑝𝑠/(𝑑+2𝛽+2))}𝑓_̇ℬ^{𝑝𝑠/(𝑑+2𝛽+2)}_{𝑠−((𝑑+2𝛽+2)/𝑝),𝛽∞,∞(ℝ+𝑑+1)}. (130) Finally, taking the 𝐿^𝑞_𝛽 norm with 𝑞=𝑝(𝑑+2𝛽+2)/(𝑑+2𝛽+2−𝑝𝑠) ends the proof thanks to the fact the maximal function 𝑀_𝛽 is bounded of 𝐿^𝑞_𝛽(ℝ₊^𝑑+1) into itself for 𝑞>1.

Theorem 32. Let 1<𝑝<𝑞<∞. For all function 𝑓 such that 𝑓∈ℋ̇^𝑠1_𝑝,𝛽(ℝ₊^𝑑+1)⋂ℬ̇^−𝛽,𝛽_∞,∞(ℝ₊^𝑑+1), one has 𝑓_{̇ℋ𝑠𝑝,𝛽(ℝ+𝑑+1)}≤𝐶𝑓^𝜃_̇ℋ^𝑠_{1𝑝,𝛽(ℝ+𝑑+1)}𝑓_̇ℬ^1−𝜃_{−𝛽,𝛽∞,∞(ℝ+𝑑+1)}, (131) where 𝜃=𝑝/𝑞, 𝑠=𝜃𝑠₁−(1−𝜃)𝛽 with 𝛽>0, −𝛽<𝑠<𝑠₁.

Proof. It suffices to prove that −Δ_𝛽^{(𝑠−𝑠1)/2}𝑓_{𝐿𝑞𝛽(ℝ+𝑑+1)}≤𝐶𝑓^𝜃_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}𝑓_̇ℬ^1−𝜃_{1−𝛽−𝑠,𝛽∞,∞(ℝ+𝑑+1)}. (132) Indeed, we use the following identity (which may be easily proven by taking the Weinstein transform in 𝑥 of both sides) −Δ_𝛽^−𝛿/2𝑓(𝑥)=1Γ(𝛿/2)^∞₀𝑡^𝛿/2−1𝐻_𝛽(𝑡)𝑓(𝑥)𝑑𝑡, (133) with 𝛿=𝑠₁−𝑠>0.
We decompose the integral in two parts as follows: −Δ_𝛽^−𝛿/2𝑓(𝑥)=1Γ(𝛿/2)^𝑇₀𝑡^𝛿/2−1𝐻_𝛽(𝑡)𝑓(𝑥)+1𝑑𝑡Γ(𝛿/2)^∞_𝑇𝑡^𝛿/2−1𝐻_𝛽(𝑡)𝑓(𝑥)𝑑𝑡, (134) where 𝑇 is a constant to be fixed later.
We proceed as in [8], we obtain 𝐻_𝛽(𝑡)𝑓(𝑥)≤𝐶𝑀_𝛽𝑓(𝑥). (135) On the other hand, we use Theorem 29 and the fact that 𝑓 belongs to ℬ̇^{−𝛽−𝑠1,𝛽}_∞,∞(ℝ₊^𝑑+1) to deduce that 𝐻_𝛽(𝑡)𝑓(𝑥)≤𝐶𝑡^{(−𝛽−𝑠1)/2}𝑓_{̇ℬ1−𝛽−𝑠,𝛽∞,∞(ℝ+𝑑+1)}. (136) Thus, by applying the preceding estimates on the right part of (134) we obtain −Δ_𝛽^−𝛿/2𝑓(𝑥≤𝐶)¹Γ()𝑇𝛿/2^𝛿/2𝑀_𝛽𝑓(𝑥)+𝐶₂Γ()𝑇𝛿/2^{(𝛿−𝛽−𝑠1)/2}𝑓_{̇ℬ1−𝛽−𝑠,𝛽∞,∞(ℝ+𝑑+1)}. (137) We fix now 𝑓𝑇=^{̇ℬ1−𝛽−𝑠,𝛽∞,∞(ℝ+𝑑+1)}𝑀_𝛽𝑓(𝑥)^{2/(𝛽+𝑠1)}. (138) We obtain −Δ_𝛽^−𝛿/2𝑓(𝑥≤𝐶)¹+𝐶₂Γ()𝑀𝛿/2^𝛽(𝑓)(𝑥)^𝜃𝑓_̇ℬ^1−𝜃_{1−𝛽−𝑠,𝛽∞,∞(ℝ+𝑑+1)}. (139) Thus we deduce that −Δ_𝛽^−𝛿/2𝑓_{𝐿𝑞𝛽(ℝ+𝑑+1)}≤𝐶₁+𝐶₂Γ()𝑀𝛿/2^𝛽𝑓^𝜃_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑓_̇ℬ^1−𝜃_{1−𝛽−𝑠,𝛽∞,∞(ℝ+𝑑+1)}. (140)To conclude, we used the fact that the maximal function 𝑀_𝛽 is bounded of 𝐿^𝑞_𝛽(ℝ₊^𝑑+1) into itself for 𝑞>1.

4.3. Estimates in Generalized Besov Spaces

For any interval 𝐼 of ℝ (bounded or unbounded) and a normed space 𝐹(ℝ₊^𝑑+1), we define the mixed space-time 𝐿^𝑝(𝐼;𝐹(ℝ₊^𝑑+1)) space of (classes of) measurable functions 𝑢∶𝐼→𝐹(ℝ₊^𝑑+1) such that ||𝑢||_𝐿^𝑝_{(𝐼;𝐹(ℝ+}^𝑑+1₎₎<∞, with ‖𝑢‖_{𝐿𝑝(𝐼;𝐹(ℝ+𝑑+1))}=_𝐼‖𝑢(𝑡,⋅)‖^𝑝_{𝐹(ℝ+𝑑+1)}𝑑𝑡^1/𝑝‖𝑢‖,if1≤𝑝<∞,_{𝐿∞(𝐼;𝐹(ℝ+𝑑+1))}=esssup_𝑡∈𝐼‖𝑢(𝑡,⋅)‖_{𝐹(ℝ+𝑑+1)}. (141)

For any interval 𝐼 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. Let 𝑠∈ℝ and 1≤𝑝,𝑞,𝑟≤∞. Let 𝑇>0, 𝑔∈ℬ̇^𝑠,𝛽_𝑝,𝑟(ℝ₊^𝑑+1), and 𝑓 in 𝐿^𝑞((0,𝑇),ℬ̇^{𝑠−2+(2/𝑞),𝛽}_𝑝,𝑟(ℝ₊^𝑑+1)). Then (76) has a unique solution: 𝑢∈𝐿^𝑞(0,𝑇),ℬ̇^{𝑠+(2/𝑞),𝛽}_𝑝,𝑟ℝ₊^𝑑+1𝐿^∞(0,𝑇),ℬ̇^𝑠,𝛽_𝑝,𝑟ℝ₊^𝑑+1, (142) and there exists a constant 𝐶 such that for all 𝑞₁∈[𝑞,∞], one has ‖𝑢‖_{𝐿𝑞1̇ℬ((0,𝑇),1𝑠+(2/𝑞),𝛽𝑝,𝑟(ℝ+𝑑+1))}≤𝐶𝑔_{̇ℬ𝑠,𝛽𝑝,𝑟(ℝ+𝑑+1)}+𝑓_{𝐿𝑞((),̇ℬ0,𝑇𝑠−2+(2/𝑞),𝛽𝑝,𝑟(ℝ+𝑑+1))}. (143) If in addition 𝑟<∞, then 𝑢∈𝐶([0,𝑇],ℬ̇^𝑠,𝛽_𝑝,𝑟(ℝ₊^𝑑+1)).

Proof. Since 𝑔 and 𝑓 are temperate distributions, (76) has a unique solution 𝑢 in 𝒮^′((0,𝑇)×ℝ₊^𝑑+1), which satisfies ℱ_𝑊(𝑢)(𝑡,𝜉)=𝑒^{−𝑡‖𝜉‖2}ℱ_𝑊𝑔(𝜉)+^𝑡₀𝑒^{(𝜏−𝑡)‖𝜉‖2}ℱ_𝑊𝑓(𝜏,𝜉)𝑑𝜏. (144) Next, we notice that applying Δ_𝑗 to (76) and using formula (81) yield Δ_𝑗𝑢(𝑡,⋅)=𝐻_𝛽(𝑡)Δ_𝑗𝑔+^𝑡₀𝐻_𝛽(𝑡−𝜏)Δ_𝑗𝑓(𝜏,⋅)𝑑𝜏. (145) Therefore, Δ_𝑗𝑢(𝑡,⋅)_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐻_𝛽(𝑡)Δ_𝑗𝑔_{𝐿𝑝𝛽(ℝ+𝑑+1)}+^𝑡₀𝐻_𝛽(𝑡−𝜏)Δ_𝑗𝑓(𝜏,⋅)_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑑𝜏. (146) By virtue of Lemma 30, we thus have, for some 𝜅>0, Δ_𝑗𝑢(𝑡,⋅)_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐶𝑒^{−𝜅22𝑗𝑡}Δ_𝑗𝑔_{𝐿𝑝𝛽(ℝ+𝑑+1)}+^𝑡₀𝑒^{−𝜅22𝑗(𝑡−𝜏)}Δ_𝑗𝑓(𝜏,⋅)_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑑𝜏. (147) Applying convolution inequalities, we get Δ_𝑗𝑢_{𝐿𝑞1((0,𝑇),𝐿𝑝𝛽(ℝ+𝑑+1))}≤𝐶1−𝑒^{−𝜅𝑇𝑞122𝑗}𝜅𝑞₁2^2𝑗1/𝑞1Δ_𝑗𝑔_{𝐿𝑝𝛽(ℝ+𝑑+1)}+1−𝑒^{−𝜅𝑇𝑞222𝑗}𝜅𝑞₂2^2𝑗1/𝑞2Δ_𝑗𝑓_{𝐿𝑞((0,𝑇),𝐿𝑝𝛽(ℝ+𝑑+1))}, (148) with 1/𝑞₂=1+1/𝑞₁−1/𝑞. Finally, taking the 𝑙^𝑟(ℤ) norm, we conclude that (with the usual convention if 𝑟=∞) ‖𝑢‖_{𝐿𝑞1̇ℬ((0,𝑇),1}^{𝑠+(2/𝑞),𝛽}_{𝑝,𝑟(ℝ+𝑑+1))}≤𝐶_𝑗∈ℤ1−𝑒^{−𝜅𝑇𝑞122𝑗}𝜅𝑞₁2^{2𝑗𝑟/𝑞1}2^𝑗𝑠Δ_𝑗𝑔_{𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟1/𝑟+𝐶_𝑗∈ℤ1−𝑒^{−𝜅𝑇𝑞222𝑗}𝜅𝑞₂2^{2𝑗𝑟/𝑞2}×2^{𝑗(𝑠−2+2/𝑞)}Δ_𝑗𝑓_{𝐿𝑞(0,𝑇;𝐿𝑝𝛽(ℝ+𝑑+1)}^𝑟1/𝑟, (149) which insures that 𝑢  ∈ 𝐿^𝑞((0,𝑇),ℬ̇^{𝑠+(2/𝑞),𝛽}_𝑝,𝑟(ℝ₊^𝑑+1))  ⋂ 𝐿^∞((0,𝑇),ℬ̇^𝑠,𝛽_𝑝,𝑟(ℝ₊^𝑑+1)) and yields the desired inequality. Since 𝑢 belongs to 𝐶([0,𝑇],ℬ̇^𝑠,𝛽_𝑝,𝑟(ℝ₊^𝑑+1)) in the case where 𝑟 is finite may be easily deduced from the density of 𝒮_∗(ℝ^𝑑+1)⋂ℬ̇^𝑠,𝛽_𝑝,𝑟(ℝ₊^𝑑+1) in ℬ̇^𝑠,𝛽_𝑝,𝑟(ℝ).

Theorem 34. Let 𝑠∈ℝ, 𝑇>0, and 1≤𝑝,𝑞,𝑟≤∞. One supposes that 𝑔∈𝐵^𝑠,𝛽_𝑝,𝑟(ℝ₊^𝑑+1) and 𝑓∈𝐿^𝑞((0,𝑇),𝐵^{𝑠−2+(2/𝑞),𝛽}_𝑝,𝑟(ℝ₊^𝑑+1)). Then (76) has a unique solution 𝑢 belonging to 𝐿^𝑞(0,𝑇),𝐵^{𝑠+(2/𝑞),𝛽}_𝑝,𝑟ℝ₊^𝑑+1𝐿^∞(0,𝑇),𝐵^𝑠,𝛽_𝑝,𝑟ℝ₊^𝑑+1, (150) and there exists a constant 𝐶 such that, for all 𝑞≤𝑞₁≤∞, ‖𝑢‖_{𝐿𝑞1((0,𝑇),𝐵1}^{𝑠+(2/𝑞),𝛽}_{𝑝,𝑟(ℝ+𝑑+1))}≤𝐶1+𝑇^1/𝑞1𝑔_𝐵^𝑠,𝛽_{𝑝,𝑟(ℝ+𝑑+1)}+1+𝑇^{1+1/𝑞1−1/𝑞}𝑓_{𝐿𝑞((0,𝑇),𝐵}^{𝑠−2+(2/𝑞),𝛽}_{𝑝,𝑟(ℝ+𝑑+1))}. (151) If in addition 𝑟<∞, then 𝑢∈𝐶([0,𝑇],𝐵^𝑠,𝛽_𝑝,𝑟(ℝ₊^𝑑+1)).

Proof. Since 𝑔,𝑓 are tempered, (76) has a unique solution 𝑢 in 𝒮^′((0,𝑇)×ℝ₊^𝑑+1) satisfying ℱ_𝑊(𝑢)(𝑡,𝜉)=𝑒^{−𝑡‖𝜉‖2}ℱ_𝑊𝑔(𝜉)+^𝑡₀𝑒^{(𝜏−𝑡)‖𝜉‖2}ℱ_𝑊𝑓(𝜏,𝜉)𝑑𝜏. (152) Hence, applying Δ_𝑗, 𝑗≥0, to (81), we see that Δ_𝑗𝑢(𝑡,⋅)=𝐻_𝛽(𝑡)Δ_𝑗𝑔+^𝑡₀𝐻_𝛽(𝑡−𝜏)Δ_𝑗𝑓(𝜏,⋅)𝑑𝜏, (153) and thus, by Lemma 30, we can deduce that Δ_𝑗𝑢(𝑡,⋅)_{𝐿𝑝𝛽(ℝ+𝑑+1)}≤𝐻_𝛽(𝑡)Δ_𝑗𝑔_{𝐿𝑝𝛽(ℝ+𝑑+1)}+^𝑡₀𝐻_𝛽(𝑡−𝜏)Δ_𝑗𝑓(𝜏,⋅)_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑒𝑑𝜏≤𝐶^{−𝜅22𝑗𝑡}Δ_𝑗𝑔_{𝐿𝑝𝛽(ℝ+𝑑+1)}+^𝑡₀𝑒^{−𝜅22𝑗(𝑡−𝜏)}Δ_𝑗𝑓(𝜏,⋅)_{𝐿𝑝𝛽(ℝ+𝑑+1)}𝑑𝜏. (154) Then it follows from convolution inequalities that ‖Δ_𝑗𝑢‖_{𝐿𝑞1((0,𝑇),𝐿𝑝𝛽(ℝ+𝑑+1))} is dominated by 1−𝑒^{−𝜅𝑇𝑞122𝑗}𝜅𝑞₁2^2𝑗1/𝑞1Δ_𝑗𝑔_{𝐵𝑠,𝛽𝑝,𝑟(ℝ+𝑑+1)}+1−𝑒^{−𝜅𝑇𝑞222𝑗}𝜅𝑞₂2^2𝑗1/𝑞2Δ_𝑗𝑓_{𝐿𝑞((0,𝑇),𝐿𝑝𝛽(ℝ+𝑑+1))}, (155) with 1/𝑞₂=1+1/𝑞₁−1/𝑞. Moreover, similarly as above, we can obtain that Δ₋₁𝑢(𝑡,⋅)_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}≤Δ₋₁𝑔_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}+^𝑡₀Δ₋₁𝑓(𝜏,⋅)_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}𝑑𝜏, (156) and thus if 1≤𝑞≤𝑞₁≤∞, Δ₋₁𝑢_{𝐿𝑞1((0,𝑇),𝐿}^𝑝_{𝛽(ℝ+𝑑+1))}≤𝐶𝑇^1/𝑞1Δ₋₁𝑔_𝐿^𝑝_{𝛽(ℝ+𝑑+1)}+𝑇^1/𝑞2Δ₋₁𝑓_{𝐿𝑞((0,𝑇),𝐿}^𝑝_{𝛽(ℝ+𝑑+1))}. (157) Finally, taking the 𝑙^𝑟-norm with respect to 𝑗 in (155) and (157) with the usual convention if 𝑟=∞, we can deduce the desired estimate.

Acknowledgment

The author gratefully acknowledges the Deanship of Scientific Research at the University of Taibah. The author is deeply indebted to the referee for providing constructive comments and help in improving the contents of this paper.