Strichartz Inequalities for the Wave Equation with the Full Laplacian on H-Type Groups Heping

and Applied Analysis 3 Remark 5. We identify G with R2d × R. We shall denote the topological dimension of G by n = 2d + p. Following Folland and Stein (see [8]), we will exploit the canonical homogeneous structure, given by the family of dilations {δ r } r>0 , δ r (z, s) = (rz, r 2 s) . (16) We then define the homogeneous dimension of G by N = 2d + 2p. The left invariant vector fields which agree, respectively, with ∂/∂x j , ∂/∂y j at the origin are given by X j = ∂ ∂x j + 1 2 p


Introduction
The aim of this paper is to study Strichartz inequalities for the solution for the following Cauchy problem of the wave equation related to the full Laplacian on H-type groups  with topological dimension  and homogeneous dimension : where L is the full Laplacian on  and the Besov spaces Ḃ  , (L) (written by Ḃ  , for short) are defined by a Littlewood-Paley decomposition related to the full Laplacian.In [1], Bahouri et al. found sharp dispersive estimates and Strichartz inequalities for the Cauchy problem for the wave equation related to the Kohn-Laplacian Δ on the Heisenberg group, using the Besov spaces Ḃ  , (Δ).In [2], Furioli et al. studied the corresponding Cauchy problem for the wave equation with the full Laplacian on the Heisenberg group, using the Besov spaces Ḃ  , .They also proved that there was no hope to obtain a dispersive inequality as in Theorem 1 with the space Ḃ  , (Δ).Later, in [3], Del Hierro generalized the dispersive and Strichartz estimates for the wave equation on H-type groups, using the Besov spaces Ḃ  , (Δ).In this paper, we will show that the wave equation related to the full Laplacian on H-type groups is also dispersive, using the Besov space Ḃ  , .To deal with the problem, we have to pay attention to two points compared with [2,3].On the one hand, the full Laplacian does not have the homogeneous properties.On the other hand, the dimension of the center of H-type groups is in general bigger than 1 (actually, in the H-type groups, only the Heisenberg groups have a one dimensional centre).
It is well known that the general solution (1) can be written as  = V +  where V is a solution of (1) with  = 0 and  is the solution of (1) with  0 =  1 = 0.They are classically given by We can now state the main results of the paper.As always when dealing with Strichartz inequalities, we prove first the following dispersive inequality on V. ) ,  ∈ R * .(3) The Strichartz inequalities we have obtained are listed as follows.
Thus, it is natural to wonder whether such a generalization for Strichartz inequalities, obtained for the wave equation on H-type groups (with full Laplacian), remains true also for the corresponding Schrödinger equation: We shall address this problem in a forthcoming paper [4].

H-Type Groups and Spherical Fourier Transform
2.1.H-Type Groups.Let g be a two-step nilpotent Lie algebra endowed with an inner product ⟨⋅, ⋅⟩.Its center is denoted by z. g is said to be of H-type if [z ⊥ , z ⊥ ] = z and for every  ∈ z, the map   : z ⊥ → z ⊥ defined by is an orthogonal map whenever || = 1.
An H-type group is a connected and simply connected Lie group  whose Lie algebra is of H-type.
Remark 4. It is well know that H-type algebras are closely related to Clifford modules (see [6]).H-type algebras can be classified by the standard theory of Clifford algebras.Specially, on H-type group , there is a relation between the dimension of the center and its orthogonal complement space.That is  + 1 ≤ 2 (see [7]).
Remark 5. We identify  with R 2 × R  .We shall denote the topological dimension of  by  = 2 + .Following Folland and Stein (see [8]), we will exploit the canonical homogeneous structure, given by the family of dilations We then define the homogeneous dimension of  by  = 2 + 2.
We say a function  on  is radial if the value of (, ) depends only on || and .We denote by S rad () and   rad (), 1 ≤  ≤ ∞ the spaces of radial functions in S() and   (), respectively.In particular, the set of  1 rad () endowed with the convolution product is a commutative algebra.
Let  ∈  1 rad ().We define the spherical Fourier transform By a direct computation, we have Thanks to a partial integration on the sphere  −1 we deduce from the Plancherel theorem on the Heisenberg group its analogue for the H-type groups.
Proposition 6.For all  ∈ S rad () such that we have the sum being convergent in  ∞ norm.
Moreover, if  ∈ S rad (), the functions L are also in S rad () and its spherical Fourier transform is given by The full Laplacian L is a positive self-adjoint operator densely defined on  2 ().So by the spectral theorem, for any bounded Borel function ℎ on R, we have

Littlewood-Paley Decomposition
In this paper we use the Besov spaces defined by a Littlewood-Paley decomposition related to the spectral of the full Laplacian L. Let  be a nonnegative, even function in For  ∈ Z, we denote by   the kernel of the operator (2 −2 L) and we set Δ   =  *   .As  ∈  ∞ 0 (R), Hulanicki proved that   ∈ S rad () (see [11]) and By [12] (see Proposition 6), there exists  > 0 such that By standard arguments (see [12], Proposition 9), we can deduce from (29) that where both sides of (30) are allowed to be infinite.By the spectral theorem, for any  ∈  2 (), the following homogeneous Littlewood-Paley decomposition holds: So where both sides of (32) are allowed to be infinite.Let 1 ≤ ,  ≤ ∞,  < /.We define the homogeneous Besov space Ḃ  , as the set of distributions  ∈ S  () such that and  = ∑ ∈Z Δ   in S  ().
We collect in the following proposition all the properties we need about the spaces Ḃ  , .

Dispersive Estimates
It is a very classical way to get a dispersive estimate if we want to reach Strichartz inequalities.Hence, first what we want to do is to get a dispersive estimate ‖ −√L   ‖  ∞ () .
Our main tool is to apply oscillating integral estimates to the wave equation.First of all, we recall the stationary phase lemma (see [13,  Next, we will need some estimates of the Laguerre functions.
Proof.We refer the reader to the proof of Lemma 3.2 in [3].
Furthermore, we will exploit the following estimates, which can be easily proved by comparing the sums with the corresponding integrals.
Lemma 11.Fix  ∈ R.There exists   > 0 such that for  > 0 and  ∈ Z + , and we have Finally, we introduce the following properties of the Bessel functions.Let   be the Bessel function of order  > −1/2, By -fold integration by parts we obtain the following.
Lemma 12.For any  ∈ N, where  ±  are complex coefficients.
Lemma 13.For any  ∈ N, Proof.See the proof of Lemma 3.4 in [3].
We can now prove the following.
(61) Applying the stationary phase Lemma 8, we obtain a consistent estimate Hence, we have where Case 1 ( is odd).Using Lemma 12, we put where Analogous to what we have done in Lemma 14, we obtain Case 2 ( is even).Using Lemma 13, we put where and the estimate holds To improve the time decay, we will try to apply  times a noncritical phase estimate.First, we need to give an estimate of the derivatives of the phase function  ,,, .Lemma 15.For any  ∈ [ , ,  , ],  ≥ 2, we obtain Proof.According to (58), we have where By a direct induction, for  ≥ 2, we have Because of for any  ∈ [ , ,  , ].By (57), when  ≥ 2  , we have  ∼ 1.Hence, (77) yields Then, according to (75), ( 76), (78), and (79), we have By (57), when  ≤ 2  , we have  ∼ 2 − .Hence, (77) yields Similarly, we prove that Furthermore, we will exploit the following estimates for the derivatives of ℎ ,, .Lemma 16.For any  ∈ [ , ,  , ], 0 ≤  ≤ , we have where Proof.Recall that By an induction we get where Applying Lemma 9 and (57), Lemma 16 comes out easily.
We can now prove the following.
We can obtain Corollary 18 by the same proof as in [14, Corollary 10].
In the end of the section, let us show as in [3] the sharpness of the time decay in Corollary 18.First we recall the asymptotic expansion of oscillating integrals.

Strichartz Estimates
We are now to prove our Strichartz estimates.