Strichartz Inequalities for the Wave Equation with the Full Laplacian on H-Type Groups
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.
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 , 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 , 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 , 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 Schrödinger equation: We shall address this problem in a forthcoming paper .
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 .
Remark 4. It is well know that H-type algebras are closely related to Clifford modules (see ). 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 ).
Remark 5. We identify with . We shall denote the topological dimension of by . Following Folland and Stein (see ), 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
Korányi, 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 ) and By  (see Proposition 6), there exists such that By standard arguments (see , 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 spaceis a Banach space with the norm;(ii)the definition ofdoes not depend on the choice of the functionin the Littlewood-Paley decomposition;(iii)forthe dual space ofis;(iv)forwe have the continuous inclusion(v)for allwe have the continuous inclusion;(vi);(vii)forwe 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 .
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 .
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
and our assertion simply read
Putting and , we first integrate on , and then
Performing the change of variable , we obtain
For , we have
Because of (55), it is implied that
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
By a direct induction, for , we have
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
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
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