#### Abstract

We generalize the dispersive estimates and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on H-type groups, by means of Besov spaces defined by a Littlewood-Paley decomposition related to the spectral of the full Laplacian. The dimension of the center on those groups is *p* and we assume that . A key point consists in estimating the decay in time of the norm of the free solution. This requires a careful analysis due also to the nonhomogeneous nature of the full Laplacian.

#### 1. 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 is the full Laplacian on and the Besov spaces (written by 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 on the Heisenberg group, using the Besov spaces . 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 . They also proved that there was no hope to obtain a dispersive inequality as in Theorem 1 with the space . Later, in [3], Del Hierro generalized the dispersive and Strichartz estimates for the wave equation on H-type groups, using the Besov spaces .

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 . 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 where is a solution of (1) with and is the solution of (1) with . 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 .

Theorem 1. *Let and , . Then there exists a constant , which does not depend on , , such that
*

The Strichartz inequalities we have obtained are listed as follows.

Theorem 2. *Let and such that *(a)*(b)**(c)**except for . Let , denote the conjugate exponent of and . Then the following estimates are satisfied:
**
where the constant does not depend on , , or .*

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 SchrÃ¶dinger equation: We shall address this problem in a forthcoming paper [4].

#### 2. H-Type Groups and Spherical Fourier Transform

##### 2.1. H-Type Groups

Let be a two-step nilpotent Lie algebra endowed with an inner product . Its center is denoted by . is said to be of H-type if and for every , the map defined by is an orthogonal map whenever .

An H-type group is a connected and simply connected Lie group whose Lie algebra is of H-type.

For a given , the dual of , we can define a skew-symmetric mapping on by We denote by the element of determined by Since is skew symmetric and nondegenerate, the dimension of is even; that is, .

For a given , we can choose an orthonormal basis
of such that
We set . Throughout this paper we assume that . We can choose an orthonormal basis of such that ,,. Then we can denote the element of by
We identify* G* with its Lie algebra by exponential map. The group law on H-type group has the form
where for a suitable skew-symmetric matrix ,.

Theorem 3. *G is an H-type group with underlying manifold , with the group law (15), and the matrix ,satisfies the following conditions.*(i)* is a skew-symmetric and orthogonal matrix, .*(ii)*, with .*

*Proof. *See [5].

*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 (see [7]).

*Remark 5. *We identify with . We shall denote the topological dimension of by . 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 .

The left invariant vector fields which agree, respectively, with , at the origin are given by where ,,.

The vector fields , correspond to the center of . In terms of these vector fields we introduce the sub-Laplacian and full Laplacian , respectively, where

##### 2.2. Spherical Fourier Transform

KorÃ¡nyi, Damek, and Ricci (see [9, 10]) have computed the spherical functions associated to the Gelfand pair (we identify with ). They involve, as on the Heisenberg group, the Laguerre functions where is the Laguerre polynomial of type and degree .

We say a function on is radial if the value of depends only on and . We denote by and , the spaces of radial functions in and , respectively. In particular, the set of endowed with the convolution product is a commutative algebra.

Let . We define the spherical Fourier transform By a direct computation, we have . Thanks to a partial integration on the sphere we deduce from the Plancherel theorem on the Heisenberg group its analogue for the H-type groups.

Proposition 6. *For all such that
**
we have
**
the sum being convergent in norm.*

Moreover, if , the functions are also in and its spherical Fourier transform is given by The full Laplacian is a positive self-adjoint operator densely defined on . So by the spectral theorem, for any bounded Borel function on , we have

#### 3. 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 . Let be a nonnegative, even function in such that supp and For , we denote by the kernel of the operator and we set . As , Hulanicki proved that (see [11]) and By [12] (see Proposition 6), there exists 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 , the following homogeneous Littlewood-Paley decomposition holds: So where both sides of (32) are allowed to be infinite.

Let ,, . We define the homogeneous Besov space as the set of distributions such that and in .

We collect in the following proposition all the properties we need about the spaces .

Proposition 7. *Let and .*(i)*The space**is a Banach space with the norm**;*(ii)*the definition of**does not depend on the choice of the function**in the Littlewood-Paley decomposition;*(iii)*for**the dual space of**is**;*(iv)*for**we have the continuous inclusion*(v)*for all**we have the continuous inclusion**;*(vi)*;*(vii)*for**we have** with , , and .*

We omit the proof of the proposition which is analogous to (see [2, Proposition 3.3]).

#### 4. 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 .

Our main tool is to apply oscillating integral estimates to the wave equation. First of all, we recall the stationary phase lemma (see [13, Chapter VIII]).

Lemma 8 (stationary phase estimate). *Let be real valued such that
**
for any with . Then for any function , there exists a constant which does not depend on ,,, or , such that
*

Next, we will need some estimates of the Laguerre functions.

Lemma 9. *
Consider the following:**
for all .*

*Proof. *We refer the reader to the proof of Lemma 3.2 in [3].

*Remark 10. *In fact, for , we have a better estimate

Furthermore, we will exploit the following estimates, which can be easily proved by comparing the sums with the corresponding integrals.

Lemma 11. *Fix . There exists such that for and , and we have
*

Finally, we introduce the following properties of the Bessel functions. Let be the Bessel function of order , By -fold integration by parts we obtain the following.

Lemma 12. *For any ,
**
where are complex coefficients.*

Lemma 13. *For any ,
**
where are such that
*

*Proof. *See the proof of Lemma 3.4 in [3].

We can now prove the following.

Lemma 14. *There exists a , which depends only on and , such that for any , , and we have
*

*Proof. *Fixing , , and and by the inversion Fourier formula, we have
where
and our assertion simply read
Putting and , we first integrate on , and then
where
Performing the change of variable , we obtain
where
Here,
So
where
Note that
For , we have
Because of (55), it is implied that
Therefore,
follows immediately from (58) and (59).

Moreover, by Lemma 9 and (57), one can easily verify that
Applying the stationary phase Lemma 8, we obtain a consistent estimate
Hence, we have
For , . For , follows from (63) by applying Lemma 11 separately to the sums and .

Next, we integrate first over to estimate ,
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 , , we obtain
*

*Proof. *According to (58), we have
where
By a direct induction, for , we have
Because of
for any .

By (57), when , we have . Hence, (77) yields
Then, according to (75), (76), (78), and (79), we have
By (57), when , we have . Hence, (77) yields
Similarly, we prove that

Furthermore, we will exploit the following estimates for the derivatives of .

Lemma 16. *For any , , 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.

Lemma 17. *There exists a , which depends only on and , such that for any , , and we have
*

*Proof. *From Lemma 14, it suffices to prove the case . In the following, we only give a detailed proof about the case when is odd. For the case is even, the proof is similar.

Recall that
where
For , we divide into three (possible empty) disjoint subsets:
Then our assertion reads
For , by (89), we obtain
The phase function for has no critical points on . By -fold integration by parts, we get
where the differential operator is defined by
By a direct induction, we have
with .

For any , Lemma 15 implies
The estimates (92) and (96) yield
Applying Lemma 16, we obtain
By (57),
So

It follows from (40) that
Let . Since and , we have and . Hence,
For , the estimate (68) yields
Then it follows from (40) that
For , when , the estimate (68) yields
Thanks to (41), we have
When , similar to , the estimates
hold for any . Therefore,
Because of
and according to Lemma 16
it follows that
Moreover, by (57),
Therefore, we obtain
Let , and then
Because of (41) and ,
Noticing that , we have
For , we divide into two (possible empty) disjoint subsets
Then our assertion reads
For , analogous to the case for , we get
So
Let . Because of , it is implied that
For , the estimate (68) yields
It follows that

From Lemma 17, it is easy to obtain our sharp dispersive inequality.

Corollary 18. *There exists , which depends only on and , such that for any , and we have
*

We can obtain Corollary 18 by the same proof as in [14, Corollary 10].

The dispersive inequality in Theorem 1 is straightforward (see [2, Proposition 1.1]).

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.

Proposition 19. *Suppose is a smooth function on and has a nondegenerate critical point at . If is supported in a sufficiently small neighborhood of , then
*