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

Authors: Heping Liu; Manli Song
Journal: Abstract and Applied Analysis
 (2014)

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## 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 isp and we assume that p>1. A key point consists in estimating the decay in time of the L∞ norm of the free solution. This requires a careful analysis due also to the nonhomogeneous nature of the full Laplacian.


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## Body

## 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 groupsG with topological dimension n and homogeneous dimension N:
(1)∂ttu+ℒu=f∈L1((0,T),L2),u|t=0=u0∈B˙2,21,∂tu|t=0=u1∈L2,
where ℒ is the full Laplacian on G and the Besov spaces B˙q,rρ(ℒ) (written by B˙q,rρ 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 B˙q,rρ(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 B˙q,rρ. They also proved that there was no hope to obtain a dispersive inequality as in Theorem 1 with the space B˙q,rρ(Δ). Later, in [3], Del Hierro generalized the dispersive and Strichartz estimates for the wave equation on H-type groups, using the Besov spaces B˙q,rρ(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 spaceB˙q,rρ. 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 u=v+w where v is a solution of (1) with f=0 and w is the solution of (1) with u0=u1=0. They are classically given by
(2)v(t)=cos⁡(tℒ)u0+sin(tℒ)ℒu1,w(t)=∫0t‍sin((t-τ)ℒ)ℒf(τ)dτ.
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.Theorem 1.
Letρ∈[n-1/2,n+1/2] and u0∈B˙1,1ρ, u1∈B˙1,1ρ-1. Then there exists a constant C>0, which does not depend on u0, u1, such that
(3)∥v(t)∥L∞(G)≤C|t|-p/2(∥u0∥B˙1,1ρ+∥u1∥B˙1,1ρ-1),t∈ℝ*.The Strichartz inequalities we have obtained are listed as follows.Theorem 2.
Letq1,q2,r1,r2∈[2,∞] and ρ1,ρ2∈ℝ such that (a)
(4)2qi=p(12-1ri);i=1,2,(b)
(5)-(n+12)(12-1r1)+1≤ρ1≤-(n-12)(12-1r1)+1,(c)
(6)-(n+12)(12-1r1)≤ρ2≤-(n-12)(12-1r1),except for (qi,ri,p)=(2,∞,2). Let qi′, ri′ denote the conjugate exponent of qi and ri. Then the following estimates are satisfied:
(7)∥v∥Lq1(ℝ,B˙r1,2ρ1)+∥∂tv∥Lq1(ℝ,B˙r1,2ρ1-1)≤C(∥u0∥B˙2,21+∥u1∥L2),∥w∥Lq1((0,T),B˙r1,2ρ1)+∥∂tw∥Lq1((0,T),B˙r1,2ρ1-1)≤C∥f∥Lq2′((0,T),B˙r2′,2-ρ2),
where the constant C>0 does not depend on u0, u1, f or T.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:(8)∂tu-iℒu=f∈L1((0,T),L2),u|t=0=u0∈B˙2,21.
We shall address this problem in a forthcoming paper [4].


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

### 2.1. H-Type Groups

Let𝔤 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 Js:𝔷⊥→𝔷⊥ defined by
(9)〈Jsu,w〉:=〈s,[u,w]〉,∀u,w∈𝔷⊥
is an orthogonal map whenever |s|=1.An H-type group is a connected and simply connected Lie groupG whose Lie algebra is of H-type.For a given0≠a∈𝔷*, the dual of 𝔷, we can define a skew-symmetric mapping B(a) on 𝔷⊥ by
(10)〈B(a)u,w〉=a([u,w]),∀u,w∈𝔷⊥.
We denote by za the element of 𝔷 determined by
(11)〈B(a)u,w〉=a([u,w])=〈Jzau,w〉.
Since B(a) is skew symmetric and nondegenerate, the dimension of 𝔷⊥ is even; that is, dim⁡𝔷⊥=2d.For a given0≠a∈𝔷*, we can choose an orthonormal basis
(12){E1(a),E2(a),…,Ed(a),E¯1(a),E¯2(a),…,E¯d(a)}
of 𝔷⊥ such that
(13)B(a)Ei(a)=|za|Jza/|za|Ei(a)=|a|E¯i(a),B(a)E¯i(a)=-|a|Ei(a).
We set p=dim⁡𝔷. Throughout this paper we assume that p>1. We can choose an orthonormal basis {ϵ1,ϵ2,…,ϵp} of 𝔷 such that a(ϵ1)=|a|,a(ϵj)=0,j=2,3,…,p. Then we can denote the element of 𝔤 by
(14)(z,t)=(x,y,t)=∑i=1d(xiEi+yiE¯i)+∑j=1psjϵj.
We identifyG with its Lie algebra 𝔤 by exponential map. The group law on H-type group G has the form
(15)(z,s)(z′,s′)=(z+z′,s+s′+12[z,z′]),
where [z,z′]j=〈z,Ujz′〉 for a suitable skew-symmetric matrix Uj,j=1,2,…,p.Theorem 3.
G is an H-type group with underlying manifoldℝ2d+p, with the group law (15), and the matrix Uj,j=1,2,…,psatisfies the following conditions.(i)
Uj is a 2d×2d skew-symmetric and orthogonal matrix, j=1,2,…,p.(ii)
UiUj+UjUi=0,i,j=1,2,…,p with i≠j.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 G, there is a relation between the dimension of the center and its orthogonal complement space. That is p+1≤2d (see [7]).Remark 5.
We identifyG with ℝ2d×ℝp. 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 {δr}r>0,
(16)δr(z,s)=(rz,r2s).
We then define the homogeneous dimension of G by N=2d+2p.The left invariant vector fields which agree, respectively, with∂/∂xj,∂/∂yj at the origin are given by
(17)Xj=∂∂xj+12∑k=1p(∑l=12dzlUl,jk)∂∂sk,Yj=∂∂yj+12∑k=1p(∑l=12dzlUl,j+dk)∂∂sk,
where zl=xl,zl+d=yl,l=1,2,…,d.The vector fieldsSk=∂/∂sk,k=1,2,…,p correspond to the center of G. In terms of these vector fields we introduce the sub-Laplacian Δ and full Laplacian ℒ, respectively,
(18)Δ=-∑j=1n(Xj2+Yj2)=-Δz+14|z|2𝒮-∑k=1p〈z,Uk∇z〉Skℒ=Δ+𝒮,
where
(19)Δz=∑j=12d∂2∂zj2,𝒮=-∑k=1p∂2∂sk2,∇z=(∂∂z1,∂∂z2,…,∂∂z2d)t.


### 2.2. Spherical Fourier Transform

Korányi, Damek, and Ricci (see [9, 10]) have computed the spherical functions associated to the Gelfand pair (G,O(2d)) (we identify O(2d) with O(2d)⊗Idp). They involve, as on the Heisenberg group, the Laguerre functions
(20)𝔏m(α)(τ)=Lm(α)(τ)e-τ/2,τ∈ℝ,m,α∈ℕ,
where Lm(α) is the Laguerre polynomial of type α and degree m.We say a functionf on G is radial if the value of f(z,s) depends only on |z| and s. We denote by 𝒮rad(G) and Lradq(G), 1≤q≤∞ the spaces of radial functions in 𝒮(G) and Lp(G), respectively. In particular, the set of Lrad1(G) endowed with the convolution product
(21)f1*f2(g)=∫G‍f1(gg′-1)f2(g′)dg′,g∈G
is a commutative algebra.Letf∈Lrad1(G). We define the spherical Fourier transform
(22)𝔉(f)(λ,m)=f^(λ,m)=(m+d-1m)-1×∫ℝ2d+p‍eiλsf(z,s)𝔏m(d-1)(|λ|2|z|2)dzds,111111111111111111111111m∈ℕ,λ∈ℝp.
By a direct computation, we have 𝔉(f1*f2)=𝔉(f1)·𝔉(f2). Thanks to a partial integration on the sphere Sp-1 we deduce from the Plancherel theorem on the Heisenberg group its analogue for the H-type groups.Proposition 6.
For allf∈𝒮rad(G) such that
(23)∑‍m∈ℕ(m+d-1m)∫ℝp‍|f^(λ,m)||λ|ddλ<∞
we have
(24)f(z,s)=(12π)d+p∑‍m∈ℕ∫ℝp‍e-iλsf^(λ,m)𝔏m(d-1)×(|λ|2|z|2)|λ|ddλ
the sum being convergent in L∞ norm.Moreover, iff∈𝒮rad(G), the functions ℒf are also in 𝒮rad(G) and its spherical Fourier transform is given by
(25)ℒf^(λ,m)=((2m+d)|λ|+|λ|2)f^(λ,m).
The full Laplacian ℒ is a positive self-adjoint operator densely defined on L2(G). So by the spectral theorem, for any bounded Borel function h on ℝ, we have
(26)h(ℒ)f^(λ,m)=h((2m+d)|λ|+|λ|2)f^(λ,m).


## 2.1. H-Type Groups

Let𝔤 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 Js:𝔷⊥→𝔷⊥ defined by
(9)〈Jsu,w〉:=〈s,[u,w]〉,∀u,w∈𝔷⊥
is an orthogonal map whenever |s|=1.An H-type group is a connected and simply connected Lie groupG whose Lie algebra is of H-type.For a given0≠a∈𝔷*, the dual of 𝔷, we can define a skew-symmetric mapping B(a) on 𝔷⊥ by
(10)〈B(a)u,w〉=a([u,w]),∀u,w∈𝔷⊥.
We denote by za the element of 𝔷 determined by
(11)〈B(a)u,w〉=a([u,w])=〈Jzau,w〉.
Since B(a) is skew symmetric and nondegenerate, the dimension of 𝔷⊥ is even; that is, dim⁡𝔷⊥=2d.For a given0≠a∈𝔷*, we can choose an orthonormal basis
(12){E1(a),E2(a),…,Ed(a),E¯1(a),E¯2(a),…,E¯d(a)}
of 𝔷⊥ such that
(13)B(a)Ei(a)=|za|Jza/|za|Ei(a)=|a|E¯i(a),B(a)E¯i(a)=-|a|Ei(a).
We set p=dim⁡𝔷. Throughout this paper we assume that p>1. We can choose an orthonormal basis {ϵ1,ϵ2,…,ϵp} of 𝔷 such that a(ϵ1)=|a|,a(ϵj)=0,j=2,3,…,p. Then we can denote the element of 𝔤 by
(14)(z,t)=(x,y,t)=∑i=1d(xiEi+yiE¯i)+∑j=1psjϵj.
We identifyG with its Lie algebra 𝔤 by exponential map. The group law on H-type group G has the form
(15)(z,s)(z′,s′)=(z+z′,s+s′+12[z,z′]),
where [z,z′]j=〈z,Ujz′〉 for a suitable skew-symmetric matrix Uj,j=1,2,…,p.Theorem 3.
G is an H-type group with underlying manifoldℝ2d+p, with the group law (15), and the matrix Uj,j=1,2,…,psatisfies the following conditions.(i)
Uj is a 2d×2d skew-symmetric and orthogonal matrix, j=1,2,…,p.(ii)
UiUj+UjUi=0,i,j=1,2,…,p with i≠j.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 G, there is a relation between the dimension of the center and its orthogonal complement space. That is p+1≤2d (see [7]).Remark 5.
We identifyG with ℝ2d×ℝp. 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 {δr}r>0,
(16)δr(z,s)=(rz,r2s).
We then define the homogeneous dimension of G by N=2d+2p.The left invariant vector fields which agree, respectively, with∂/∂xj,∂/∂yj at the origin are given by
(17)Xj=∂∂xj+12∑k=1p(∑l=12dzlUl,jk)∂∂sk,Yj=∂∂yj+12∑k=1p(∑l=12dzlUl,j+dk)∂∂sk,
where zl=xl,zl+d=yl,l=1,2,…,d.The vector fieldsSk=∂/∂sk,k=1,2,…,p correspond to the center of G. In terms of these vector fields we introduce the sub-Laplacian Δ and full Laplacian ℒ, respectively,
(18)Δ=-∑j=1n(Xj2+Yj2)=-Δz+14|z|2𝒮-∑k=1p〈z,Uk∇z〉Skℒ=Δ+𝒮,
where
(19)Δz=∑j=12d∂2∂zj2,𝒮=-∑k=1p∂2∂sk2,∇z=(∂∂z1,∂∂z2,…,∂∂z2d)t.


## 2.2. Spherical Fourier Transform

Korányi, Damek, and Ricci (see [9, 10]) have computed the spherical functions associated to the Gelfand pair (G,O(2d)) (we identify O(2d) with O(2d)⊗Idp). They involve, as on the Heisenberg group, the Laguerre functions
(20)𝔏m(α)(τ)=Lm(α)(τ)e-τ/2,τ∈ℝ,m,α∈ℕ,
where Lm(α) is the Laguerre polynomial of type α and degree m.We say a functionf on G is radial if the value of f(z,s) depends only on |z| and s. We denote by 𝒮rad(G) and Lradq(G), 1≤q≤∞ the spaces of radial functions in 𝒮(G) and Lp(G), respectively. In particular, the set of Lrad1(G) endowed with the convolution product
(21)f1*f2(g)=∫G‍f1(gg′-1)f2(g′)dg′,g∈G
is a commutative algebra.Letf∈Lrad1(G). We define the spherical Fourier transform
(22)𝔉(f)(λ,m)=f^(λ,m)=(m+d-1m)-1×∫ℝ2d+p‍eiλsf(z,s)𝔏m(d-1)(|λ|2|z|2)dzds,111111111111111111111111m∈ℕ,λ∈ℝp.
By a direct computation, we have 𝔉(f1*f2)=𝔉(f1)·𝔉(f2). Thanks to a partial integration on the sphere Sp-1 we deduce from the Plancherel theorem on the Heisenberg group its analogue for the H-type groups.Proposition 6.
For allf∈𝒮rad(G) such that
(23)∑‍m∈ℕ(m+d-1m)∫ℝp‍|f^(λ,m)||λ|ddλ<∞
we have
(24)f(z,s)=(12π)d+p∑‍m∈ℕ∫ℝp‍e-iλsf^(λ,m)𝔏m(d-1)×(|λ|2|z|2)|λ|ddλ
the sum being convergent in L∞ norm.Moreover, iff∈𝒮rad(G), the functions ℒf are also in 𝒮rad(G) and its spherical Fourier transform is given by
(25)ℒf^(λ,m)=((2m+d)|λ|+|λ|2)f^(λ,m).
The full Laplacian ℒ is a positive self-adjoint operator densely defined on L2(G). So by the spectral theorem, for any bounded Borel function h on ℝ, we have
(26)h(ℒ)f^(λ,m)=h((2m+d)|λ|+|λ|2)f^(λ,m).


## 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 Laplacianℒ. Let R be a nonnegative, even function in C0∞(ℝ) such that suppR⊆{τ∈ℝ:1/2≤|τ|≤4} and
(27)∑‍j∈ℤR(2-2jτ)=1,∀τ≠0.
For j∈ℤ, we denote by ψj the kernel of the operator R(2-2jℒ) and we set Δjf=f*ψj. As R∈C0∞(ℝ), Hulanicki proved that ψj∈𝒮rad(G) (see [11]) and
(28)ψ^j(λ,m)=R(2-2j((2m+d)|λ|+|λ|2)).
By [12] (see Proposition 6), there exists C>0 such that
(29)∥ψj∥L1(G)≤C,∀j∈ℤ.
By standard arguments (see [12], Proposition 9), we can deduce from (29) that
(30)∥ℒσ/2Δjf∥Lq(G)≤C2jσ∥Δjf∥Lq(G),σ∈ℝ,j∈ℤ,1≤q≤∞,f∈𝒮′(G),
where both sides of (30) are allowed to be infinite.By the spectral theorem, for anyf∈L2(G), the following homogeneous Littlewood-Paley decomposition holds:
(31)f=∑j∈ℤ‍ΔjfinL2(G).
So
(32)∥f∥L∞(G)≤∑j∈ℤ‍∥Δjf∥L∞(G),f∈L2(G),
where both sides of (32) are allowed to be infinite.Let1≤q,r≤∞, ρ<N/q. We define the homogeneous Besov space B˙q,rρ as the set of distributions f∈𝒮′(G) such that
(33)∥f∥B˙q,rρ=(∑‍j∈ℤ2jρr∥Δjf∥qr)1/r<∞
and f=∑j∈ℤΔjf in 𝒮′(G).We collect in the following proposition all the properties we need about the spacesB˙q,rρ.Proposition 7.
Letq,r∈[1,∞] and ρ<N/q.(i)
The spaceB˙q,rρis a Banach space with the norm∥·∥B˙q,rρ;(ii)
the definition ofB˙q,rρdoes not depend on the choice of the functionRin the Littlewood-Paley decomposition;(iii)
for-N/q′<ρ<N/qthe dual space ofB˙q,rρisB˙q′,r′-ρ;(iv)
forα∈[n,N]we have the continuous inclusion(34)B˙q1,rρ1⊂B˙q2,rρ2,1q1-ρ1α=1q2-ρ2α,ρ1≥ρ2;(v)
for allq∈[2,∞]we have the continuous inclusionB˙q,20⊂Lq;(vi)
B˙2,20=L2;(vii)
forθ∈[0,1]we have(35)[B˙q1,r1ρ1,B˙q2,r2ρ2]θ=B˙q,rρ with ρ=(1-θ)ρ1+θρ2, 1/q=(1-θ)/q1+θ/q2, and 1/r=(1-θ)/r1+θ/r2.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∥e-itℒψj∥L∞(G).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).
Letg∈C∞([a,b]) be real valued such that
(36)|g′′(x)|≥δ
for any x∈[a,b] with δ>0. Then for any function h∈C∞([a,b]), there exists a constant C which does not depend on δ,a,b,g or h, such that
(37)|∫ab‍eig(x)h(x)dx|≤Cδ-1/2[∥h∥∞+∫ab‍|h′(x)|dx].Next, we will need some estimates of the Laguerre functions.Lemma 9.
Consider the following:(38)|(τddτ)α𝔏m(d-1)(τ)|≤Cα,d(2m+d)d-1/4
for all 0≤α≤d.Proof.
We refer the reader to the proof of Lemma 3.2 in [3].Remark 10.
In fact, for0≤α≤d-1, we have a better estimate
(39)|(τddτ)α𝔏m(d-1)(τ)|≤Cα,d(2m+d)d-1.Furthermore, we will exploit the following estimates, which can be easily proved by comparing the sums with the corresponding integrals.Lemma 11.
Fixβ∈ℝ. There exists Cβ>0 such that for A>0 and d∈ℤ+, and we have
(40)∑‍m∈ℕ2m+d≥A(2m+d)β≤CβAβ+1,β<-1,(41)∑‍m∈ℕ2m+d≤A(2m+d)β≤CβAβ+1,β>-1.Finally, we introduce the following properties of the Bessel functions. LetJμ be the Bessel function of order μ>-1/2,
(42)Jμ(r)=(r/2)μΓ(μ+1/2)π1/2∫-11‍eirt(1-t2)μ-1/2dt.
By m-fold integration by parts we obtain the following.Lemma 12.
For anym∈ℕ,
(43)Jm+1/2=r-1/2∑k=0m(ak+eir+ak-e-ir)r-k,
where ak± are complex coefficients.Lemma 13.
For anym∈ℕ,
(44)Jm(r)=eir[a+r1/2+ϕ+(r)]+e-ir[a-r1/2+ϕ-(r)],
where ϕ±∈𝒮(ℝ+) are such that
(45)∀r>0,|ϕ±(r)|≤r-1/2,|ϕ±′(r)|≤r-3/2.Proof.
See the proof of Lemma 3.4 in [3].We can now prove the following.Lemma 14.
There exists aC>0, which depends only on d and p, such that for any ρ∈[n-1/2,n+1/2], j∈ℤ, and t∈ℝ*=ℝ∖{0} we have
(46)∥e-itℒψj∥L∞(G)≤C|t|-1/22jρ.Proof.
Fixingt∈ℝ*, j∈ℤ, and (z,s)∈G and by the inversion Fourier formula, we have
(47)e-itℒψj(z,s)=(12π)d+p∑‍m∈ℕ∫ℝp‍e-iλse-it(2m+d)|λ|+|λ|2×R(2-2j((2m+d)|λ|+|λ|2))×𝔏m(d-1)(|λ|2|z|2)|λ|ddλ=(12π)d+p∑‍m∈ℕIm,
where
(48)Im=∫ℝp‍e-iλse-it(2m+d)|λ|+|λ|2R(2-2j((2m+d)|λ|+|λ|2))×𝔏m(d-1)(|λ|2|z|2)|λ|ddλ
and our assertion simply read
(49)∑‍m∈ℕ|Im|≲{|t|-1/22j(2d+p-1/2),j>0,|t|-1/22j(2d+p+1/2),j≤0.
Putting σ=s/t and M=2m+d, we first integrate on ℝ+, and then
(50)Im=∫ℝp‍e-it(σ·λ+M|λ|+|λ|2)R(2-2j(M|λ|+|λ|2))1111111×𝔏m(d-1)(|λ|2|z|2)|λ|ddλ=∫Sp-1‍Iϵ,mdσ(ϵ),
where
(51)Iϵ,m=∫0+∞‍e-it(λσ·ϵ+Mλ+λ2)R(2-2j(Mλ+λ2))×𝔏m(d-1)(λ2|z|2)λd+p-1dλ.
Performing the change of variable x=2-2jMλ, we obtain
(52)Iϵ,m=2j(2d+2p)Kϵ,m,
where
(53)Kϵ,m=∫0+∞‍e-it2jGj,σ,ϵ,m(x)hj,z,m(x)dx.
Here,
(54)Gj,σ,ϵ,m(x)=2jM(xσ·ϵ+2-2jM2x+x2),hj,z,m(x)=R(x+22jM2x2)𝔏m(d-1)(22j-1x|z|2M)xd+p-1Md+p.
So
(55)supp⁡hj,z,m⊆{x∈ℝ+:12≤x+22jM2x2≤4}=[aj,m,bj,m],
where
(56)aj,m=11+1+22j+1M-2,bj,m=81+1+22j+4M-2.
Note that
(57)aj,m,bj,m~min⁡⁡(1,2-jM).
For x∈[aj,m,bj,m], we have
(58)Gj,σ,ϵ,m′′(x)=-2-3j-2M3(2-2jM2x+x2)3/2.
Because of (55), it is implied that
(59)2-2j-1M2≤2-2jM2x+x2≤2-2j+2M2,x∈[aj,m,bj,m].
Therefore,
(60)2-5≤|Gj,σ,ϵ,m′′(x)|≤2-1/2,x∈[aj,m,bj,m]
follows immediately from (58) and (59).
Moreover, by Lemma9 and (57), one can easily verify that
(61)∥hj,z,m∥L∞[aj,m,bj,m]+∥hj,z,m′∥L1[aj,m,bj,m]≲{M-(p+1),M≥2j,2-j(d+p-1)Md-2,M<2j.
Applying the stationary phase Lemma 8, we obtain a consistent estimate
(62)|Kϵ,m|≲{|t|-1/22-j/2M-(p+1),M≥2j,|t|-1/22-j(d+p-1/2)Md-2,M<2j.
Hence, we have
(63)|Im|≲{|t|-1/22j(2d+2p-1/2)M-(p+1),M≥2j,|t|-1/22j(d+p+1/2)Md-2,M<2j.
For j≤0, ∑m∈ℕ|Im|≲|t|-1/22j(2d+2p-1/2)≲|t|-1/22j(2d+p+1/2). For j>0, ∑m∈ℕ|Im|≲|t|-1/22j(2d+p-1/2) follows from (63) by applying Lemma 11 separately to the sums ∑M≥2j‍|Im| and ∑M<2j‍|Im|.
Next, we integrate first overSp-1 to estimate Im,
(64)Im=∫0+∞‍dσ^(λs)e-itMλ+λ2×R(2-2j(Mλ+λ2))𝔏m(d-1)(λ2|z|2)λd+p-1dλ,
where
(65)dσ^(ξ)=∫Sp-1‍e-ix·ξdσ(x)=2π(|ξ|2π)(2-p)/2J(p-2)/2(|ξ|).
Case
1 (p is odd). Using Lemma 12, we put
(66)Im=(2π)p/2∑±∑k=0(p-3)/2ak±Im,k±,
where
(67)Im,k±=|s|(1-p)/2-k∫0+∞‍e±iλ|s|-itMλ+λ2×R(2-2j(Mλ+λ2))𝔏m(d-1)(λ2|z|2)λd+(p-1)/2-kdλ.
Analogous to what we have done in Lemma 14, we obtain
(68)|Im,k±|≲{|t|-1/2|s|(1-p)/2-k2j(2d+p+1/2-2k)M-((p+3)/2-k),M≥2j,|t|-1/2|s|(1-p)/2-k2j(d+p/2+1-k)Md-2,M<2j.
Case
2 (p is even). Using Lemma 13, we put
(69)Im=(2π)p/2∑‍±a±(Im,0±+Υm±),
where
(70)Υm±=|s|(2-p)/2∫0+∞‍e±iλ|s|-itMλ+λ2ϕ±(λ|s|)×R(2-2j(Mλ+λ2))𝔏m(d-1)(λ2|z|2)λd+p/2dλ
and the estimate holds
(71)|Υm±|≲{|t|-1/2|s|(1-p)/22j(2d+p+1/2)M-(p+3)/2,M≥2j,|t|-1/2|s|(1-p)/22j(d+p/2+1)Md-2,M<2j.To improve the time decay, we will try to applyp times a noncritical phase estimate. First, we need to give an estimate of the derivatives of the phase function Gj,σ,ϵ,m.Lemma 15.
For anyx∈[aj,m,bj,m], l≥2, we obtain
(72)|Gj,σ,ϵ,m(l)(x)|≲{1,M≥2j,(2jM-1)l-2,M<2j.Proof.
According to (58), we have
(73)Gj,σ,ϵ,m′′(x)=-2-3j-2M3(φ(x))3/2,
where
(74)φ(x)=2-2jM2x+x2.
By a direct induction, for l≥2, we have
(75)Gj,σ,ϵ,m(l)(x)=(Gj,σ,ϵ,m′′)(l-2)(x)=-2-3j-2M3×∑l1+2l2=l-2‍C(l,l1,l2)(φ′(x))l1(φ′′(x))l2(φ(x))3/2+l-2-l2.
Because of
(76)φ(x)~2-2jM2,(77)φ′(x)=2-2jM2+2x,(78)φ′′(x)=2,
for any x∈[aj,m,bj,m].
By (57), when M≥2j, we have x~1. Hence, (77) yields
(79)φ′(x)~2-2jM2.
Then, according to (75), (76), (78), and (79), we have
(80)|Gj,σ,ϵ,m(l)(x)|≲2-3j-2M3∑l1+2l2=l-2‍(2-2jM2)-(3/2+l-2-l2-l1)≲2-3j-2M3∑0≤l2≤[(l-2)/2]‍(2-2jM2)-(3/2+l2)≲2-3j-2M3(2-2jM2)-3/2≲1.
By (57), when M≤2j, we have x~2-jM. Hence, (77) yields
(81)φ′(x)~2-jM.
Similarly, we prove that
(82)|Gj,σ,ϵ,m(l)(x)|≲(2jM-1)l-2.Furthermore, we will exploit the following estimates for the derivatives ofhj,z,m.Lemma 16.
For anyx∈[aj,m,bj,m], 0≤l≤d, we have
(83)|hj,z,m(l)(x)|≲{M-(p+θl),M≥2j,2-j(d+p-l-1)Md-l-θl-1,M<2j,
where
(84)θl={1,0≤l≤d-1,14,l=d.Proof.
Recall that(85)hj,z,m(x)=R(x+22jM2x2)𝔏m(d-1)(22j-1x|z|2M2)xd+p-1Md+p.
By an induction we get
(86)hj,z,m(l)(x)=∑‍α∈ℱA(l,α)R(α1)(x+22jM2x2)×(1+22j+1M2x)α2(22j+1M2)α3×[(xddx)α4𝔏m(d-1)](22j-1x|z|2M2)xd+p-α5-1Md+p,
where ℱ={α=(α1,…,α5)∈ℕ5:α1=α2+α3,α1+α3+α5=l,α4≤α5}.
Applying Lemma9 and (57), Lemma 16 comes out easily.We can now prove the following.Lemma 17.
There exists aC>0, which depends only on d and p, such that for any ρ∈[n-1/2,n+1/2], j∈ℤ, and t∈ℝ* we have
(87)∥e-itℒψj∥L∞(G)≤C|t|-p/22jρ.Proof.
From Lemma14, it suffices to prove the case |t|>1. In the following, we only give a detailed proof about the case when p is odd. For the case p is even, the proof is similar.
Recall that(88)Kϵ,m=∫0+∞‍e-it2jGj,σ,ϵ,m(x)hj,z,m(x)dx,
where
(89)Gj,σ,ϵ,m′(x)=2jM(σ·ϵ+1+2-4j-2M42-2jM2x+x2).
For j>0, we divide ℕ into three (possible empty) disjoint subsets:
(90)A1={m∈ℕ:M≥2j,|σ|≲2-jM},A2={m∈ℕ:M≥2j,|σ|≳2-jM},A3={m∈ℕ:M<2j}.
Then our assertion reads
(91)∑‍m∈Ar|Im|≲|t|-p/22j(2d+p-1/2),r=1,2,3.
For r=1, by (89), we obtain
(92)|Gj,σ,ϵ,m′(x)|≳1,foranyx∈[aj,m,bj,m].
The phase function Gj,σ,ϵ,m′(x) for Kϵ,m has no critical points on [aj,m,bj,m]. By Q-fold integration by parts, we get
(93)Kϵ,m=(it2j)-Q∫0+∞‍e-it2jGj,σ,ϵ,m(x)DQhj,z,m(x)dx,
where the differential operator D is defined by
(94)Dhj,z,m(x)=ddx(hj,z,m(x)Gj,σ,ϵ,m′(x)).
By a direct induction, we have
(95)DQhj,z,m=∑k=Q2Q∑∑l=1Q+1lαl=kC(α,k,Q)×hj,z,m(α1)(Gj,σ,ϵ,m′′)α2⋯(Gj,σ,ϵ,m(Q+1))αQ+1(Gj,σ,ϵ,m′)k
with α=(α1,α2,…,αQ+1)∈{0,1,…,Q}×ℕQ.
For anyl≥2, Lemma 15 implies
(96)|Gj,σ,ϵ,m(l)(x)|≲1,foranyx∈[aj,m,bj,m].
The estimates (92) and (96) yield
(97)∥DQhj,z,m∥∞≲sup⁡0≤α1≤Q∥hj,z,m(α1)∥∞.
Applying Lemma 16, we obtain
(98)sup⁡0≤α1≤Q∥hj,z,m(α1)∥∞≲M-(p+1/4).
By (57),
(99)aj,m,bj,m~1.
So
(100)|Kϵ,m|≲|t|-Q2-jQM-(p+1/4).
It follows from (40) that
(101)∑‍A1|Im|≲|t|-Q2j(2d+2p-Q)×∑‍M≥2jM-(p+1/4)≲|t|-Q2j(2d+p+3/4-Q).
Let Q=d. Since p≤2d-1 and p>1, we have d>p/2 and d≥2. Hence,
(102)∑‍A1|Im|≲|t|-d2j(d+p+3/4)≤|t|-p/22j(2d+p-1/2).
For r=2, the estimate (68) yields
(103)|Im,k±|≲|t|-p/2-k2j(2d+3p/2-k)M-(p+1)≲|t|-p/22j(2d+3p/2)M-(p+1).
Then it follows from (40) that
(104)∑‍m∈A2|Im|≲|t|-p/22j(2d+3p/2)×∑‍M≥2jM-(p+1)≲|t|-p/22j(2d+p/2)≲|t|-p/22j(2d+p-1/2).
For r=3, when |σ|≳1, the estimate (68) yields
(105)|Im,k±|≲|t|-p/2-k2j(d+p/2+1-k)Md-2≲|t|-p/22j(d+p/2+1)Md-2.
Thanks to (41), we have
(106)∑‍m∈A3|Im|≲|t|-p/22j(d+p/2+1)×∑‍M<2jMd-2≲|t|-p/22j(2d+p/2)≲|t|-p/22j(2d+p-1/2).
When |σ|≲1, similar to r=1, the estimates
(107)|Gj,σ,ϵ,m′(x)|≳2jM-1,|Gj,σ,ϵ,m(l)(x)|≲(2jM-1)l-2,l≥2
hold for any x∈[aj,m,bj,m]. Therefore,
(108)∥DQhj,z,m∥∞≲sup⁡0≤α1≤Q∥hj,z,m(α1)∥∞×sup⁡Q≤k≤2Qsup⁡∑l=1Q+1lαl=k(2jM-1)∑l=2Q+1(l-2)αl-k.
Because of
(109)∑l=2Q+1(l-2)αl-k=-∑l=2Q+12αl-α1≤-2(Q+1)∑l=1Q+1lαl=-2k(Q+1)≤-2Q(Q+1)
and according to Lemma 16(110)sup⁡0≤α1≤Q∥hj,z,m(α1)∥∞≲2-j(p+d-Q-1)Md-Q-5/4,
it follows that
(111)∥DQhj,z,m∥∞≲2-j(p+d+2Q/(Q+1)-Q-1)Md+2Q/(Q+1)-Q-5/4.
Moreover, by (57),
(112)aj,m,bj,m~2-jM.
Therefore, we obtain
(113)|Kϵ,m|≲|t|-Q2-jQ∥DQhj,z,m∥∞2-jM=|t|-Q2-j(p+d+2Q/(Q+1))Md+2Q/(Q+1)-Q-1/4.
Let Q=d, and then
(114)|Kϵ,m|≲|t|-d2-j(d+p+2d/(d+1))M2d/(d+1)-1/4.
Because of (41) and d>p/2,
(115)∑A3|Kϵ,m|≲|t|-p/22-j(d+p+2d/(d+1))×∑‍M<2jM2d/(d+1)-1/4≲|t|-p/22-j(d+p-3/4).
Noticing that d≥2, we have
(116)∑A3|Im|≲2j(2d+2p)∑A3|Kϵ,m|≲|t|-p/22j(d+p+3/4)≤|t|-p/22j(2d+p-1/2).
For j≤0, we divide ℕ into two (possible empty) disjoint subsets
(117)B1={m∈ℕ:|σ|≲2-jM},B2={m∈ℕ:|σ|≳2-jM}.
Then our assertion reads
(118)∑‍m∈Br|Im|≲|t|-p/22j(2d+p+1/2),r=1,2.
For B1, analogous to the case A1 for j>0, we get
(119)|Kϵ,m|≲|t|-Q2-jQM-(p+1/4).
So
(120)∑‍m∈B1|Im|≲|t|-Q2j(2d+2p-Q)×∑‍m∈ℕM-(p+1/4)≲|t|-Q2j(2d+2p-Q).
Let Q=(p+1)/2≤d. Because of p>1, it is implied that
(121)∑‍m∈B1|Im|≲|t|-p/22j(2d+3p/2-1/2)≲|t|-p/22j(2d+p+1/2).
For B2, the estimate (68) yields
(122)|Im,k±|≲|t|-p/2-k2j(2d+3p/2-k)M-(p+1)≲|t|-p/22j(2d+p+3/2)M-(p+1).
It follows that
(123)∑‍m∈B2|Im|≲|t|-p/22j(2d+p+3/2)∑‍m∈ℕM-(p+1)≲|t|-p/22j(2d+p+3/2)≲|t|-p/22j(2d+p+1/2).From Lemma17, it is easy to obtain our sharp dispersive inequality.Corollary 18.
There existsC>0, which depends only on d and p, such that for any ρ∈[n-1/2,n+1/2], t∈ℝ* and f∈𝒮(G) we have
(124)∥e-itℒf∥L∞(G)≤C|t|-p/2∥f∥B˙1,1ρ,(125)∥e-itℒf∥B˙∞,1-1≤C|t|-p/2∥f∥B˙1,1ρ-1.We can obtain Corollary18 by the same proof as in [14, Corollary 10].The dispersive inequality in Theorem1 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.
Supposeϕ is a smooth function on ℝp and has a nondegenerate critical point at x0. If ψ is supported in a sufficiently small neighborhood of x0, then
(126)|∫ℝp‍eitϕ(x)ψ(x)dx|~|t|-p/2,ast⟶∞.A proof can be found in [13, Proposition 6, page344].LetQ∈C0∞(D0) with Q(d)=1, where D0 is a small neighborhood of d such that 0∉D0. Then
(127)u^0(λ,m)=Q(|λ|)δm,0
and u1:=0 determines a solution of the Cauchy problem (1) with f=0:
(128)u((z,s),t)=cos⁡(tℒ)u0=C∫ℝp‍e-iλ·s-|λ||z|2/4cos⁡(td|λ|+|λ|2)×Q(|λ|)|λ|ddλ.
Consider u((0,ts0),t) for a fixed s0 such that |s0|=(3/22). This oscillating integral has a phase ϕ±(λ):=-λ·s0±d|λ|+|λ|2 with a unique critical point λ0±=∓(22d/3)s0 which is not degenerate. Indeed, the Hessian is equal to
(129)H(λ)=∓{4|λ|2+6d|λ|+3d24|λ|2(d|λ|+|λ|2)3/2λkλl-δk,ld+2|λ|2|λ|(d|λ|+|λ|2)1/2}1≤k,l≤p.
Let s0=(3/22)(0,…,0,1), so λ0±=∓(22d/3)s0=∓(0,…,0,d). The Hessian at λ0± is
(130)H(λ0±)=±182d{12⋱12-1}.
Applying asymptotic expansion of oscillating integrals, we get
(131)u((0,ts0),t)~|t|-p/2.


## 5. Strichartz Estimates

We are now to prove our Strichartz estimates.Proposition 20.
Fori=1,2, let qi,ri∈[2,∞] and ρi∈ℝ such that (a)
(132)2qi=p(12-1ri),(b)
(133)-(n+12)(12-1ri)≤ρi≤-(n-12)(12-1ri),except for (qi,ri,p)=(2,∞,2). Then the following estimates are satisfied:
(134)∥e-itℒu0∥Lq1(ℝ,B˙r1,2ρ1)≤C∥u0∥L2,∥∫0te-i(t-τ)ℒf(τ)dτ∥Lq1((0,T),B˙r1,2ρ1)≤C∥f∥Lq2′((0,T),B˙r2′,2-ρ2),
where the constant C>0 does not depend on u0, f, or T.Once we have obtained the estimate in Lemma17, the proof is classical and a good reference is, for example, the papers by Ginibre and Velo [15] or by Keel and Tao [16]. A detailed presentation in this framework is also given by [14] in the proof of Theorem 11.Theorem2 follows easily from the above proposition by the same proof that in [2].In particular, by Besov interpolation we get the Strichartz estimates on Lebesgue spaces.Theorem 21.
Letu be the solution of the Cauchy problem (1). If q and r satisfy 0≤2/q≤p(1/2-1/r) and p[n(1/2-1/r)-1]≤1/q≤(p/(2p-1))[N(1/2-1/r)-1], then there exists a constant C>0, which does not depend on u0, u1, f, or T, such that the following estimate is satisfied:
(135)∥u∥Lq((0,T),Lr)≤C(∥u0∥B˙2,21+∥u1∥L2+∥f∥L1((0,T),L2)).



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