Decay estimates for fractional wave equations on H-type groups

The aim of this paper is to establish the decay estimate for the fractional wave equation semigroup on H-type groups given by $e^{it\Delta^\alpha}$, $0<\alpha<1$. Combing the dispersive estimate and a standard duality argument, we also derive the corresponding Strichartz inequalties.


Introduction
In this paper, we stduy the decay estimate for a class of dispersive equations: where ∆ is the sub-Laplacian on H-type groups G, α > 0.
The partial differential equation in (1) is significantly interesting in mathematics. When α = 1 2 , it is reduced to the wave equation; when α = 1, it is reduced to the Schrödinger equation. The two equations are most important fundamental types of partial differential equations.
In 2000, Bahouri et al. [1] derived the Strichartz inequalities for the wave equation on the Heisenberg group via a sharp dispersive estimate and a standard duality argument (see [8] and [14]). The dispersive estimate plays a crucial role, where ϕ is the kernel function on the Heisenberg group related to a Littlewood Paley decomposition introduced in Section 2 and θ > 0. Such estimate does not exist for the Schrödinger equation (see [1]). The sharp dispersive estimate is also generalized to H-type groups for the wave equation and Schrödinger equation (see [10] and [20]). Motivated by the work by Guo et al. [9] on the Euclidean space, we consider the fractional wave equation (1) on H-type groups and will prove a sharp dispersive estimate.
Theorem 1.1. Let N be the homogeneous dimension of the H-type group G, and p the dimension of its center. For 0 < α < 1, we have ||e it∆ α u 0 || ∞ C α |t| −p/2 ||u 0 ||ḂN−p/2 1,1 , and the result is sharp in time. Here, the constant C α > 0 does not depend on u 0 , t, andḂ ρ q,r is the homogeneous Besov space associated to the sublaplacian ∆ introduced in the next section.
Following the work by Keel and Tao [14] or by Ginibre and Velo [8], we also get a useful estimate on the solution of the fractional wave equation.
Corollary 1.1. If 0 < α < 1 and u is the solution of the fractional wave equation (1), then for q ∈ [(2N − p)/p, +∞) and r such that we have the estimate where the constant C α > 0 does not depend on u 0 , f or T . Remark 1.1. In this article, we assume 0 < α < 1. For α = 1, the decay estimate has been proved (see [10]). For other cases, we could investigate the problem in the similar way to 0 < α 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 s ∈ z, the map J s : z ⊥ → z ⊥ defined by is an orthogonal map whenever |s| = 1.
An H-type group is a connected and simply connected Lie group G whose Lie algebra is of H-type. For a given 0 = a ∈ z * , the dual of z, we can define a skew-symmetric mapping B(a) on z ⊥ by We denote by z a be the element of z determined by B(a)u, w = a([u, w]) = J za u, w .
Since B(a) is skew symmetric and non-degenerate, the dimension of z ⊥ is even, i.e. dimz ⊥ = 2d. For a given 0 = a ∈ z * , we can choose an orthonormal basis We set p = dimz. We can choose an orthonormal basis {ǫ 1 , ǫ 2 , · · · , ǫ p } of z such that a(ǫ 1 ) = |a|, a(ǫ j ) = 0, j = 2, 3, · · · , p. Then we can denote the element of g by We identify G with its Lie algebra g by exponential map. The group law on H-type group G has the form where [z, z ′ ] j = z, U j z ′ for a suitable skew symmetric matrix U j , j = 1, 2, · · · , p.
Remark 2.1. It is well know that H-type algebras are closely related to Clifford modules (see [18]). H-type algebras can be classified by the standard theory of Clifford algebras. Specially, on H-type group G, there is a relation between the dimension of the center and its orthogonal complement space. That is p+1 2d (see [13]).

Remark 2.2.
We identify G with R 2d × R p . We shall denote the topological dimension of G by n = 2d + p. Following Folland and Stein (see [4]), 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).
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 where z l = x l , z l+d = y l , l = 1, 2, · · · , d. In terms of these vector fields we introduce the sublaplacian ∆ by

Spherical Fourier transform
Korányi [15], Damek and Ricci [3] have computed the spherical functions associated to the Gelfand pair (G, . They involve, as on the Heisenberg group, the Laguerre functions m is the Laguerre polynomial of type γ and degree m. We say a function f on G is radial if the value of f (z, s) depends only on |z| and s. We denote respectively by S rad (G) and L q rad (G),1 q ∞, the spaces of radial functions in S (G) and L p (G), respectively. In particular, the set of L 1 rad (G) endowed with the convolution product By a direct computation, we have f 1 * f 2 =f 1 ·f 2 . Thanks to a partial integration on the sphere S p−1 , we deduce from the Plancherel theorem on the Heisenberg group its analogue for the H-type groups.
the sum being convergent in L ∞ norm.
Moreover, if f ∈ S rad (G), the functions ∆f is also in S rad (G) and its spherical Fourier transform is given by The sublaplacian ∆ is a positive self-adjoint operator densely defined on L 2 (G). So by the spectral theorem, for any bounded Borel function h on R, we have
By the spectral theorem, for any f ∈ L 2 (G), the following homogeneous Littlewood-Paley decomposition holds: where both sides of (5) are allowed to be infinite. Let 1 q, r ∞, ρ < N/q, we define the homogeneous Besov spaceḂ ρ q,r as the set of distributions f ∈ S ′ (G) such that Let ρ < N/q. The homogeneous Sobolev spaceḢ ρ iṡ Analogous to Proposition 6 of [6] on the Heisenberg group, we list some properties of the spacesḂ ρ q,r in the following proposition.

Technical Lemmas
By the inversion Fourier formula (4), we may write e it∆ α ϕ explicitly into a sum of a list of oscillatory integrals. In order to estimate the oscillatory integrals, we recall the stationary phase lemma. In order to prove the sharpness of the time decay in Theorem 1.1. We describe the asymptotic expansion of oscillating integrals.
Besides, it will involve the Laguerre functions when we estimate the osciallatory integrals. We need the following estimates.
for all 0 γ d.
Finally, we introduce the following properties of the Fourier transform of surface-carried measures.
Theorem 3.1. (see [19]) Let S be a smooth hypersurface in R p with nonvanishing Gaussian curvature and dµ a C ∞ 0 measure on S. Suppose that Γ ⊂ R p \ {0} is the cone consisting of all ξ which are normal of some point x ∈ S belonging to a fixed relatively compact neighborhood N of supp dµ. Then, where the (finite) sum is taken over all points x ∈ N having ξ as a normal and Here, we need the following properties of the Fourier transform of the measure dσ on the sphere S p−1 . Obviously, dσ is radial. By Theorem 3.1, we have the radical decay properties of the Fourier transform of the spherical measure.
Lemma 3.4. For any ξ ∈ R p , the estimate holds Proof. By Fourier inversion (4) and polar coordinate changes, we have

Dispersive Estimates
The expression after the S p−1 integral sign in (6) is very similar to an integral computed in [1] or [10]( see the proof Lemma 4.1). Integrating the result over S p−1 gives us and Lemma 4.1 will come out only if we prove the case for p 2 and |s| > 1.
By switching the order of the integration in (6), it follows from Lemma 3.4 Then it suffices to study Performing the change of variables, µ = (2m + d)λ, recalling that R ∈ C ∞ c (R),
Since | g ± m,s,t ′′ | α|α − 1|2 −α−4 , applying Lemma 3.1 on I ± m gives us To conclude it suffices to sum these estimates since The decay estimate of time is sharp in the joint space-time cone We will prove the sharp dispersive estimate. Proof. From (7), it suffices to show the inequality |t| > 1. Recall from (6) that where We will try to apply Q times a non-critical phase estimate to the oscillatory integral I m,ε .

Hence, we have
Taking Q = d, since |t| > 1 and p 2d − 1 which implies p/2 < d, it follows

It immediately leads to
Combining the two cases, by a straightforward summation The lemma is proved.

Strichartz Inequalities
In this section, we shall prove the Strichartz inequalities by the decay estimate in Lemma 4.2. We obtain the intermediate results as follows. We omit the proof and refer to [8], [14].
Consider the non-homogeneous fractional wave equation (1). The general solution is given by where the constant C > 0 does not depend on u 0 , f or T .
Applying Propostion 2.2, by direct Besov space injections, we immediately obtain the Strichartz inequalities on Lebesgue spaces in Corollary 1.1.