R E S E A R C H
Open Access
Decay estimates for fractional wave
equations on
H
-type groups
Manli Song
**Correspondence:
[email protected] School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, China
Abstract
The aim of this paper is to establish the decay estimate for the fractional wave equation semigroup onH-type groups given byeitα
, 0 <
α
< 1. Combining the dispersive estimate and a standard duality argument, we also derive the corresponding Strichartz inequalities.MSC: 22E25; 33C45; 35H20; 35B40
Keywords: fractional wave equation; decay estimate;H-type groups
1 Introduction
In this paper, we study the decay estimate for a class of dispersive equations:
i∂tu+αu=f, u() =u, ()
whereis the sub-Laplacian onH-type groupsG,α> .
The partial differential equation in () is significantly interesting in mathematics. When
α= , it is reduced to the wave equation; whenα= , it is reduced to the Schrödinger
equation. The two equations are most important fundamental types of partial differential equations.
In , Bahouriet al.[] derived the Strichartz inequalities for the wave equation on
the Heisenberg group via a sharp dispersive estimate and a standard duality argument (see [] and []). The dispersive estimate
eitαϕ
L∞≤C|t|
–θ
()
plays a crucial role, whereϕ is the kernel function on the Heisenberg group related to
a Littlewood-Paley decomposition introduced in Section andθ > . Such an estimate
does not exist for the Schrödinger equation (see []). The sharp dispersive estimate is also
generalized toH-type groups for the wave equation and the Schrödinger equation (see [–
]). Motivated by Guoet al.[] on the Euclidean space, we consider the fractional wave
equation () onH-type groups and will prove a sharp dispersive estimate.
Theorem . Let N be the homogeneous dimension of the H-type group G,and p the di-mension of its center.For <α< ,we have
eitαu
∞≤Cα|t|–p/uB˙N,–p/,
and the result is sharp in time.Here,the constant Cα> does not depend on u,t,andB˙ ρ
q,r
is the homogeneous Besov space associated to the sublaplacianintroduced in the next section.
Following the work by Keel and Tao [] or by Ginibre and Velo [], we also get a useful estimate on the solution of the fractional wave equation.
Corollary . If <α< and u is the solution of the fractional wave equation(),then for q∈[(N–p)/p, +∞)and r such that
/q+N/r=N/ – ,
we have the estimate
uLq((,T),Lr)≤CαuH˙+fL((,T),H˙)
,
where the constant Cα> does not depend on u,f or T.
Remark . In this article, we assume <α< . Forα= , the decay estimate has been proved (see []). For other cases, we could investigate the problem in a similar way to <α≤.
2 Preliminaries
2.1 H-Type groups
Letgbe a two step nilpotent Lie algebra endowed with an inner product·,·. Its center is denoted byz.gis said to be ofH-type if [z⊥,z⊥] =z, and, for everys∈z, the mapJs:z⊥→z⊥
defined by
Jsu,w:=
s, [u,w], ∀u,w∈z⊥,
is an orthogonal map whenever|s|= .
AnH-type group is a connected and simply connected Lie groupGwhose Lie algebra
is ofH-type.
Given =a∈z∗, the dual ofz, we can define a skew-symmetric mappingB(a) onz⊥by
B(a)u,w=a[u,w], ∀u,w∈z⊥.
We denote byzathe element ofzdetermined by
SinceB(a) is skew symmetric and non-degenerate, the dimension ofz⊥is even,i.e.dimz⊥= d.
We can choose an orthonormal basis
E(a),E(a), . . . ,Ed(a),E(a),E(a), . . . ,Ed(a)
ofz⊥such that
B(a)Ei(a) =|za|J|za|za Ei(a) =|a|Ei(a)
and
B(a)Ei(a) = –|a|Ei(a).
We setp=dimz. We can choose an orthonormal basis{,, . . . ,p}ofzsuch thata() =
|a|,a(j) = ,j= , , . . . ,p. Then we can denote the element ofgby
(z,t) = (x,y,t) =
d
i=
(xiEi+yiEi) + p
j=
sjj.
We identifyGwith its Lie algebragby the exponential map. The group law onH-type
groupGhas the form
(z,s)z,s=
z+z,s+s+
z,z, ()
where [z,z]j=z,Ujzfor a suitable skew-symmetric matrixUj,j= , , . . . ,p.
Theorem . G is an H-type group with underlying manifoldRd+p,with the group law
()and the matrix Uj,j= , , . . . ,p,satisfies the following conditions: (i) Ujis ad×dskew-symmetric and orthogonal matrix,j= , , . . . ,p. (ii) UiUj+UjUi= ,i,j= , , . . . ,pwithi=j.
Proof See [].
Remark . It is well known thatH-type algebras are closely related to Clifford modules
(see []).H-type algebras can be classified by the standard theory of Clifford algebras.
Specially, onH-type groupG, there is a relation between the dimension of the center and
its orthogonal complement space. That isp+ ≤d(see []).
Remark . We identifyGwithRd×Rpand denote byn= d+pits topological dimen-sion. Following Folland and Stein (see []), we will exploit the canonical homogeneous structure, given by the family of dilations{δr}r>,
δr(z,s) =
rz,rs.
The left invariant vector fields which agree, respectively, with ∂∂xj,∂∂yj at the origin are given by
Xj=
∂ ∂xj
+
p
k=
d
l=
zlUlk,j
∂ ∂sk
,
Yj=
∂ ∂yj
+
p
k=
d
l=
zlUlk,j+d
∂ ∂sk
,
wherezl=xl,zl+d=yl,l= , , . . . ,d. In terms of these vector fields we introduce the
sub-laplacianby
= –
d
j=
Xj+Yj.
2.2 Spherical Fourier transform
Korányi [], Damek, and Ricci [] have computed the spherical functions associated to the Gelfand pair (G,O(d)) (we identifyO(d) with O(d)⊗Idp). They involve, as on the
Heisenberg group, the Laguerre functions
L(mγ)(τ) =L(mγ)(τ)e–τ/, τ∈R,m,γ ∈N,
whereL(mγ)is the Laguerre polynomial of typeγ and degreem.
We say a functionf onGis radial if the value off(z,s) depends only on|z|ands. We denote, respectively, bySrad(G) andLqrad(G), ≤q≤ ∞, the spaces of radial functions inS(G) andLp(G). In particular, the set ofL
rad(G) endowed with the convolution
prod-uct
f∗f(g) =
G
f
gg–f
gdg, g∈G,
is a commutative algebra. Letf ∈L
rad(G). We define the spherical Fourier transform,m∈N,λ∈Rp,
ˆ
f(λ,m) =
m+d–
m
–
Rd+pe
iλsf(z,s)L(d–)
m |
λ| |z|
dz ds.
By a direct computation, we have f∗f=fˆ· ˆf. Thanks to a partial integration on the
sphereSp–, we deduce from the Plancherel theorem on the Heisenberg group its analog
for theH-type groups.
Proposition . For all f∈Srad(G)such that
m∈N
m+d–
m Rp
we have
f(z,s) =
π
d+p
m∈N
Rpe
–iλ·sˆ
f(λ,m)L(md–)
|λ| |z|
|λ|ddλ, ()
the sum being convergent in L∞norm.
Moreover, iff ∈Srad(G), the functionsf is also inSrad(G) and its spherical Fourier transform is given by
f(λ,m) = (m+d)|λ|ˆf(λ,m).
The sublaplacianis a positive self-adjoint operator densely defined onL(G). So by the
spectral theorem, for any bounded Borel functionhonR, we have
h()f(λ,m) =h(m+d)|λ|fˆ(λ,m). 2.3 Homogeneous Besov spaces
We shall recall the homogeneous Besov spaces given in []. LetRbe a non-negative, even
function inCc∞(R) such thatsuppR⊆ {τ∈R:≤ |τ| ≤}and
j∈Z
R–jτ= , ∀τ= .
Forj∈Z, we denote by ϕ and ϕj, respectively, the kernel of the operator R() and
R(–j). AsR∈C∞
c (R), Hulanicki [] proved thatϕ∈Srad(G) and obviouslyϕj(z,s) =
Njϕ(δ
j(z,s)). For anyf ∈S(G), we setjf =f∗ϕj.
By the spectral theorem, for anyf ∈L(G), the following homogeneous Littlewood-Paley
decomposition holds:
f =
j∈Z
jf inL(G).
So
fL∞(G)≤ j∈Z
jfL∞(G), f∈L(G), ()
where both sides of () are allowed to be infinite.
Let ≤q,r≤ ∞,ρ<N/q, we define the homogeneous Besov spaceB˙ρq,r as the set of
distributionsf ∈S(G) such that
fB˙ρq,r= j∈Z
jρr jfrq
r
<∞,
andf=j∈Zjf inS(G).
Letρ<N/q. The homogeneous Sobolev spaceH˙ρis
˙
which is equivalent to
u∈ ˙Hρ ⇔ ρ/u∈L.
Analogous to Proposition of [] on the Heisenberg group, we list some properties of the spacesB˙ρq,rin the following proposition.
Proposition . Let q,r∈[,∞]andρ<N/q.
(i) The spaceB˙ρq,ris a Banach space with the norm · B˙ρq,r;
(ii) the definition ofB˙ρq,rdoes not depend on the choice of the functionRin the Littlewood-Paley decomposition;
(iii) for–Nq <ρ<Nq the dual space ofB˙ρq,risB˙–qρ,r;
(iv) for anyu∈S(G)andσ> ,thenu∈ ˙Bρq,rif and only ifLσ/u∈ ˙Bρq,–rσ; (v) for anyq,q∈[,∞],the continuous inclusion holds:
˙
Bρ q,r⊆ ˙B
ρ q,r,
q –ρ
N =
q –ρ
N, ρ≥ρ;
(vi) for allq∈[,∞]we have the continuous inclusionB˙
q,⊆Lq;
(vii) B˙,=L.
3 Technical lemmas
By the inversion Fourier formula (), we may writeeitαϕexplicitly into a sum of a list of
oscillatory integrals. In order to estimate the oscillatory integrals, we recall the stationary phase lemma.
Lemma .(see []) Let g∈C∞([a,b])be real-valued such that
g(x)≥δ
for any x∈[a,b]withδ> .Then for any function h∈C∞([a,b]),there exists a constant C which does not depend onδ,a,b,g or h,such that
abeig(x)h(x)dx≤Cδ–/h∞+h.
In order to prove the sharpness of the time decay in Theorem ., we describe the asymp-totic expansion of oscillating integrals.
Lemma .(see []) Supposeφ is a smooth function onRpand has a non-degenerate
critical point atλ.¯ Ifψ is supported in a sufficiently small neighborhood ofλ,¯ then
Rpe
itφ(λ)ψ(λ)dλ∼ |t|–p/, as t→ ∞.
Lemma .(see [])
τ d
dτ
γ
L(md–)(τ)≤Cγ,d(m+d)d–/
for all≤γ ≤d.
Finally, we introduce the following properties of the Fourier transform of surface-carried measures.
Theorem .(see []) Let S be a smooth hypersurface inRpwith non-vanishing Gaussian
curvature and dμa C∞measure on S.Suppose that⊂Rp\{}is the cone consisting of all
ξwhich are normal of some point x∈S belonging to a fixed relatively compact neighborhood N ofsuppdμ.Then
∂ ∂ξ
ν
dμ(ξ) =O +|ξ|–M, ∀M∈N, ifξ∈/,
dμ(ξ) = e–i(xj,ξ)a
j(ξ), ifξ∈,
where the(finite)sum is taken over all points x∈N havingξas a normal and
∂
∂ξ
ν
aj(ξ) ≤Cν
+|ξ|–(p–)/–|ν|.
Here, we need the following properties of the Fourier transform of the measuredσ on
the sphereSp–. Obviously,dσis radial. By Theorem ., we have the radical decay prop-erties of the Fourier transform of the spherical measure.
Lemma . For anyξ∈Rp,the estimate holds
dσ(ξ) =ei|ξ|φ+
|ξ|+e–i|ξ|φ–
|ξ|,
where
φ±(k)(r)≤ck( +r)–(p–)/–k, for all r> ,k∈N.
4 Dispersive estimates
Lemma . Let <α< .The kernel of ϕ of R()introduced in Sectionsatisfies the estimate
sup
z
eitαϕ(z,s)≤C
α|t|–/|s|(–p)/.
Proof By the inversion Fourier formula () and polar coordinate changes, we have
eitαϕ(z,s) =
π
d+p
m∈N
Rpe
–iλ·s+it(m+d)α|λ|α
×R(m+d)|λ|L(md–) |
λ| |z|
=
π
d+p
m∈N
Sp– +∞
e–iλε·s+it(m+d)αλα
×R(m+d)λL(md–)
λ |z|
λd+p–dλdσ(ε). ()
The expression after theSp–integral sign in () is very similar to an integral computed in [] or [] (see the proof of Lemma .). Integrating the result overSp–gives us
sup
z
eitαϕ(z,s)≤Cαmin
,|t|–/, ()
and Lemma . will come out only if we prove the case forp≥ and|s|> . By switching the order of the integration in (), it follows from Lemma . that
eitαϕ(z,s) =
π
d+p
m∈N
+∞
dσ(λs)eit(m+d)αλαR(m+d)λ
×L(md–)
λ |z|
λd+p–dλ
=
π
d+p
m∈N
+∞
eiλ|s|φ+
λ|s|+e–iλ|s|φ–
λ|s|eit(m+d)αλα
×R(m+d)λL(md–)
λ |z|
λd+p–dλ
:=
π
d+p
m∈N
Im++Im–.
Then it suffices to study
Im±=
+∞
ei(±λ|s|+t(m+d)αλα)φ±λ|s|R(m+d)λL(md–)
λ |z|
λd+p–dλ.
Performing the change of variables,μ= (m+d)λ, recall thatR∈C∞c (R),
Im±=
/
eitgm±,s,t(μ)h
m,s,z(μ)dλ,
where
gm±,s,t(μ) =± μ|s| (m+d)t+μ
α,
hm,s,z(μ) =φ±
μ|s|
m+d
R(μ)L(md–)
μ|z| (m+d)
μd+p– (m+d)d+p.
By Lemma . and Lemma ., we get
hm,s,z∞+hm,s,z≤C(m+d)
–(p+)/|s|–(p–)/. Since|(gm±,s,t)| ≥α|α– |–α–, applying Lemma . onI±
mgives us
To conclude it suffices to sum these estimates since
m∈N
(m+d)–(p+)/< +∞.
The decay estimate of time is sharp in the joint space-time cone
(s,t)∈Rp×R:s=Ct.
We will prove the sharp dispersive estimate.
Lemma . Let <α< .The kernel ofϕ of R()introduced in Section satisfies the estimate
sup
z,s
eitαϕ(z,s)≤Cα|t|–p/.
Proof From (), it suffices to show the inequality|t|> . Recall from () that
eitαϕ(z,s) =
π
d+p
m∈N
Sp–Im,εd
σ(ε),
where
Im,ε=
+∞
e–iλε·s+it(m+d)αλα
R(m+d)λL(md–)
λ |z|
λd+p–dλ
=
/
eitGm,ε,s,t(μ)H
m,z(μ)dμ
with
Gm,ε,s,t(μ) =μα–
μ (m+d)tε·s, Hm,z(μ) =R(μ)L(md–)
μ|z|
(m+d)
μd+p– (m+d)d+p.
We will try to applyQtimes a non-critical phase estimate to the oscillatory integralIm,ε.
Case :|s| ≥α–α–(m+d)|t|. By (),
Sp–Im,εd
σ(ε)=Im++Im–≤Cα(m+d)–p–/|t|–p/.
Case :|s| ≤α–α–(m+d)|t|. We get
Gm,ε,s,t(μ) =αμ
α–– ε·s
(m+d)t ≥α
–α–– |s|
(m+d)|t|≥α
–α–.
Here the phase functionGm,ε,s,t has no critical point on [/, ]. ByQ-fold (≤Q≤d)
integration by parts, we have
Im,ε= (it)–Q
/
eitGm,ε,s,t(μ)DQH m,z(μ)
where the differential operatorDis defined by
DHm,z=
d dμ
Hm,z(μ)
Gm,ε,s,t(μ)
.
By a direct induction,
DQHm,z=
Q
k=Qβ=k
C(β,k,Q)H (β)
m,z(Gm,ε,s,t)β· · ·(G
(Q+)
m,ε,s,t)βQ+
(Gm,ε,s,t)k
,
whereβ= (β, . . . ,βQ+)∈ {, . . . ,Q} ×NQandβ= Q+
j= jβj.
A direct calculation shows that
G(ml),ε,s,t(μ)=α l–
j=
(j–α)μ–l+α≤l+αα
l–
j=
(j–α)≤C(α,Q), l≥.
Using Lemma .,
H(β)
m,z(μ)≤C(β)(m+d)–p–/.
Hence, we have
|Im,ε| ≤C(α,Q)|t|–Q sup
≤β≤Q H(β)
m,z∞≤C(α,Q)|t|–Q(m+d)–p–/.
TakingQ=d, since|t|> andp≤d– , which impliesp/ <d, it follows that
|Im,ε| ≤Cα|t|–p/(m+d)–p–/.
It immediately leads to
Sp–Im,εdσ(ε)
≤Cα|t|–p/(m+d)–p–/.
Combining the two cases, by a straightforward summation
eitαϕ(z,s)≤Cα|t|–p/
m∈N
(m+d)–p–/≤Cα|t|–p/.
The lemma is proved.
Proof of Theorem. The dispersive inequality in Theorem . is a direct consequence of Lemma . (see []). It suffices to show the sharpness of the estimate. LetQ∈Cc∞([/, ]) withQ() = . Chooseusuch that
ˆ
u(λ,m) =
⎧ ⎨ ⎩
Q(|λ|), m= ,
By the inversion Fourier formula (), then we have
eitαu(z,s) =
π
d+p
Rpe
–iλ·s+itdα|λ|α
Q|λ|e–|λ||z|/|λ|ddλ.
Considereitαu(,ts¯) for a fixed¯s=αdα(, . . . , , ). The above oscillatory integral has a
phase
(λ) = –λ· ¯s+dα|λ|α
with a unique non-degenerate critical pointλ¯=α–d–α¯s= (, . . . , , ). Indeed, the Hessian
is equal to
H(λ) =¯ αdα|¯λ|α–(α– )λ¯kλ¯l+|¯λ|δk,l
≤k,l≤p=αd
α
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
. ..
α–
⎞ ⎟ ⎟ ⎟ ⎟ ⎠.
So by Lemma ., it yields
eitαu(,ts¯)∼C|t|–p/.
5 Strichartz inequalities
In this section, we shall prove the Strichartz inequalities by the decay estimate in Lemma .. We obtain the intermediate results as follows. We omit the proof and refer to [, ].
Theorem . Let <α< .For i= , ,let qi,ri∈[,∞]andρi∈Rsuch that
() /qi=p(/ – /ri);
() ρi= –(N–p/)(/ – /ri),
except for(qi,ri,p) = (,∞, ).Let qi,ridenote the conjugate exponent of qi,rifor i= , .
Then the following estimates are satisfied:
eitαuLq(R,B˙ρ
r,)≤CuL ,
tei(t–τ)αf(τ)dτ
Lq((,T),B˙ρ r,)
≤Cf
Lq((,T),B˙–ρ r,)
,
where the constant C> does not depend on u,f or T.
Consider the non-homogeneous fractional wave equation (). The general solution is given by
u(t) =eitαu–i
t
Theorem . Under the same hypotheses as in Theorem.,the solution of the fractional wave equation()satisfies the following estimate:
uLq((,T),B˙ρr ,)≤
CuL+f
Lq((,T),B˙–ρ r,)
,
where the constant C> does not depend on u,f or T.
Applying Proposition ., by direct Besov space injections, we immediately obtain the Strichartz inequalities on Lebesgue spaces in Corollary ..
Competing interests
The author declares to have no competing interests.
Acknowledgements
The work is supported by the National Natural Science Foundation of China (Grant No. 11371036) and the Fundamental Research Funds for the Central Universities (Grant No. 3102015ZY068).
Received: 1 September 2016 Accepted: 26 September 2016
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