Heat Equations Associated with Weinstein Operator and Applications

Hatem Mejjaoli Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah Al Munawarah, Saudi Arabia Correspondence should be addressed to HatemMejjaoli; hatem.mejjaoli@ipest.rnu.tn Received 31 May 2013; Accepted 3 August 2013 Academic Editor: Dashan Fan Copyright © 2013 Hatem Mejjaoli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish a characterization for the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup. Next, we obtain the generalized Sobolev embedding theorems.


Introduction
We consider the Weinstein operator defined on R  × ]0, ∞[ by where Δ  is the Laplacian for the -first variables and L  is the Bessel operator for the last variable given by For  > 2, the operator Δ  is the Laplace-Beltrami operator on the Riemannian space R  × ]0, ∞[ equipped with the following metric: (cf. [1,2]).The Weinstein operator Δ  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 We study (4) to focus on the following problems.
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,

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].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.

Harmonic Analysis Associated with the Weinstein
In the following, We denote by  * (R +1 ) the space of continuous functions on R +1 , even with respect to the last variable;   * (R +1 ) the space of functions of class   on R +1 , even with respect to the last variable; E * (R +1 ) the space of  ∞ -functions on R +1 , even with respect to the last variable; S * (R +1 ) the Schwartz space of rapidly decreasing functions on R +1 , even with respect to the last variable;  * (R +1 ) the space of  ∞functions on R +1 which are of compact support, even with respect to the last variable; S  * (R +1 ) the space of temperate distributions on R +1 , even with respect to the last variable.It is the topological dual of S * (R +1 ).
We consider the Weinstein operator Δ  defined by where Δ   is the Laplace operator on R  and L , +1 is the Bessel operator on ]0, ∞[ given by The Weinstein kernel Λ is given by where   ( +1  +1 ) is the normalized Bessel function.The Weinstein kernel satisfies the following properties.
(i) For each  ∈ R +1 + , we have (ii) For all ,  ∈ C +1 , we have (iii) For all ] ∈ N +1 ,  ∈ R +1 , and  ∈ C +1 , we have where The Weinstein intertwining operator is the operator R  defined on  * (R +1 ) by R  is a topological isomorphism from E * (R +1 ) onto itself satisfying the following transmutation relation: where where   is the measure on R +1 + given by The Weinstein transform is given for  in  1  (R +1 + ) by Some basic properties of this transform are as follows.
In the Fourier analysis, the translation operator is given by   → (⋅ + ).
In harmonic analysis associated for the operator Δ  , the generalized translation operator   ,  ∈ R +1 + is defined by where  ∈  * (R +1 ).
By using the Weinstein kernel, we can also define a generalized translation.For functions  ∈ S * (R +1 ) and  ∈ R +1 + the generalized translation    is defined by the following relation: By using the generalized translation, we define the generalized convolution product  *   of functions ,  ∈  1  (R +1 + ) as follows: This convolution is commutative and associative, and it satisfies the following.

Ḃ𝑠,𝛽
Notations.We denote by The distribution Δ   is called the th dyadic block of the homogeneous Littlewood-Paley decomposition of  associated with the Weinstein operators.Throughout this paper, we define  and  by  = F −1  () and  = F −1  ().When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functions ψ and φ belonging to  * (R +1 ) such that ψ ≡ 1 on supp  and φ ≡ 1 on supp .Remark 7. We remark that We put Definition 8.One denotes by S  ℎ,, * (R +1 ) the space of tempered distribution such that lim Proposition 9 (Bernstein inequalities).For all  ∈ N +1 and  ∈ R, for all  ∈ Z, for all 1 ≤ ,  ≤ ∞, and for all  ∈ S  * (R +1 ), one has the following Proof.Using Remark 7, we deduce from Proposition 5 that Thus, from the relation (29), we prove (i), (ii), and (iii).

Definitions.
In the following, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Weinstein operators on R +1 + and obtain their basic properties.
From now, we make the convention that for all nonnegative sequence {  } ∈Z , the notation (∑     ) 1/ stands for sup    in the case  = ∞.
Proposition 11 (see [6]).Let  ∈ R and  and  two elements of [1, ∞]; the space where The nonhomogeneous Besov space  Proposition 13.Let  ∈ R and  and  two elements of [1, ∞]; The operator R −  is called Weinstein-Riesz potential space.Proof.We obtain these results by the similar ideas used in the nonhomogeneous case (cf.[6]).
(1) is obvious from the Hölder's inequality.As for (2), we write where  is chosen here after.By the definition of the homogeneous Weinstein-Besov norms, we see that and thus ‖‖ Ḃ+(1−), Hence, in order to complete the proof of (2), it suffices to choose  such that As for (3), it is easy to see that ‖‖ Ḃ, ,1 (R +1 + ) is dominated as Hence, letting we can obtain the desired estimate.

Generalized Heat Equation
We introduce the Weinstein heat semigroup   () for the Weinstein-Laplace operator where Γ  is the Weinstein heat kernel defined by Γ  (, , ) :=   ( where Thus In practice, we use the integral formulation of ( 76  is the Gauss kernel associated with Weinstein operators.This function satisfies for all  > 0. Proof.It follows from the relations (80) and ( 29) combined with scaling property of the kernel  ()  .In this section, we prove estimates for the Weinstein heat semigroup.These estimates are based on the following result.
Proof.We again consider a function Θ in (R +1 + \ {0}), the value of which is identically 1 in neighborhood of annulus C. We can also assume without loss of generality that  = 1.We then have where The lemma is proved provided that we can find positive real numbers  and  such that To begin, we perform integrations by parts in (87).We get Using formula, we obtain and ( 88) follows.
For any interval  of R (bounded or unbounded), we define the mixed space-time   (;    (R +1 + )) Banach space of (classes of) measurable functions  :  → Corollary 28.Let C be an annulus and  a positive real number.Let  0 (resp.,  = (, )) satisfy supp F  ( 0 ) ⊂ C (resp., supp F  ((, ⋅)) ⊂ C for all  in [0, ]).Consider  a solution of and V a solution of There exist positive constants  and , depending only on C, such that for any 1 ≤  ≤  ≤ ∞ and 1 ≤  ≤  ≤ ∞, we have          (R + ,/) To prove this result we need the following lemma.
Lemma 30.There exist two positive constants  and  depending only on  such that for all 1 ≤  ≤ ∞,  ≥ 0 and  ∈ Z, one has Proof.The result follows immediately by applying Lemma 27 and because Δ  (  ()) = (  ()Δ  ).
Proof of Theorem 29.Using Lemma 30 and considering the fact that the operator Δ  commutes with the operator   () and the definition of the homogeneous Weinstein-Besov (semi) norm we get where ( , ) ∈Z denotes, as in all this proof, a generic element of the unit sphere of   (Z).In the case when  = ∞, the required inequality comes immediately from the following easy result.For any positive , we have In the case  < ∞, using the Hölder inequality with the weight 2 2  −2 2 , (99), and the Fubini theorem, we obtain In order to prove the other inequality, let us observe that for any  greater than −1, we have Then, Lemma 30, Proposition 9, and the fact that the operator Δ  commutes with the operator   () lead to the following: In the case  = ∞, we simply write In the case  < ∞, Hölder's inequality with the weight  −2 2 gives Thanks to (99) and Fubini's theorem, we infer from (102) that The theorem is proved.
Second Proof of Theorem 29.We only consider the case 1 ≤  < ∞.The case  = ∞ can be shown similarly.We first prove that It is easy to see that where The result is immediately from (117) and (119).
Proof.By density, we can suppose that  belongs to S * (R +1 ).It is easy to see that and decompose the integral in two parts as follows: where  is a constant to be fixed later.
On the other hand, by Theorem 29, we obtain Therefore after integrating, we get On the other hand, denoting  = (−Δ  ) /2 , we have We proceed as in [8], we prove that where   () is a maximal function of  associated with the Weinstein operators (cf.[12]).
In conclusion, we get and the choice of  such that where  = /,  = Indeed, we use the following identity (which may be easily proven by taking the Weinstein transform in  of both sides) with  =  1 −  > 0.
We decompose the integral in two parts as follows: where  is a constant to be fixed later.We proceed as in [8], we obtain We fix now  = (          Ḃ−− For any interval  of R (bounded or unbounded) and a Banach space , we define the mixed space-time (; ) space of continuous functions  → .When  is bounded, (; ) is a Banach space with the norm of  ∞ (, ).
Finally, taking the   -norm with respect to  in (155) and (157) with the usual convention if  = ∞, we can deduce the desired estimate.