Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 118, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
STRICHARTZ ESTIMATES ON α-MODULATION SPACES
WEICHAO GUO, JIECHENG CHEN
Abstract. In this article, we consider some dispersive equations, including
Schrödinger equations, nonelliptic Schrödinger equations, and wave equations.
We develop some Strichartz estimates in the frame of α-modulation spaces.
1. Introduction
We study the Cauchy problem for the Schrödinger type equation
iut + (−∆)β/2 u = F
u(0, x) = u0 ,
(1.1)
the Cauchy problem for nonelliptic Schrödinger equation
ivt + ψ(D)v = F
v(0, x) = v0
,
(1.2)
where ψ(ξ) = Σnl=1 ± |ξl |β , and the Cauchy problem for wave equation
wtt − ∆w = F
w(0, x) = w0 , wt (0, x) = w1 .
(1.3)
0,α
The initial data belongs to the α-modulation space M2,1
, and we use F to denote
some nonlinear terms.
We recall Duhamel’s formula for above three dispersive equations. The solution
to (1.1) is
Z t
β/2
β/2
u(t, x) = eit(−∆) u0 − i
ei(t−s)(−∆) F (s)ds.
(1.4)
0
The solution to (1.2) is
v(t, x) = eitψ(D) v0 − i
Z
t
ei(t−s)ψ(D) F (s)ds.
(1.5)
0
The solution to (1.3) is
√
√
Z t
sin(t −∆)
sin((t − s) −∆)
√
√
w(t, x) = cos(t −∆)w0 +
w1 +
F (s)ds.
−∆
−∆
0
√
2000 Mathematics Subject Classification. 42B37, 46E35, 35L05.
Key words and phrases. Strichartz estimate; Schrödinger equation; wave equation;
α-modulation space.
c
2013
Texas State University - San Marcos.
Submitted March 12, 2013. Published May 10, 2013.
1
(1.6)
2
W. GUO, J. CHEN
EJDE-2013/118
There are many publications about the theoretical and applied aspects of the
Schrödinger equation and the wave equation; see for example Tao [14] and Sogge
[11] for a nice introduction. We refer the reader to [1, 2, 3, 4, 10, 16, 17, 18] for the
study of modulation space and dispersive equation.
In this paper, we are concerned mainly with the Strichartz estimates for the
solutions of the above three equations. The original estimates are due to Strichartz
[13], and they became fundamental and important tools in the study of dispersive
equations. The theory of Strichartz estimates has also been studied by many authors. One is referred to [6], [7] and [9] for classical Strichartz estimates. We also
refer the readers to [5] and [20] for the Strichartz estimates in the frame of Wiener
amalgam spaces and modulation spaces. The following lemma is a basic Strchartz
estimate, proved by Keel-Tao [9], that we will use it frequently in our proofs.
Definition 1.1. An exponent pair (r, p) is called σ-admissible if r, p ≥ 2, (r, p, n) 6=
(2, ∞, 2) and
σ
1 σ
+ ≤ .
(1.7)
r
p
2
If the equality holds, we say that (r, p) is sharp σ-admissible, otherwise we say that
(r, p) is nonsharp σ-admissible. If σ > 1 we say the sharp σ-admissible pair
(2,
2σ
)
σ−1
(1.8)
is an endpoint.
Next we have the Strichartz estimates.
Lemma 1.2 ([9]). Let {U (t)}t∈R be a semigroup of operators that obey energy
estimate
kU (t)f kL2x . kf kL2x
(1.9)
and dispersive estimate
kU (t)(U (s))∗ gkL∞
. |t − s|−σ kgkL1x .
x
(1.10)
kU (t)f kLrt Lpx . kf kL2x ,
(1.11)
Then the estimates
Z
k
R
Z
k
s<t
(U (s))∗ F (s)dskL2x . kF kLr0 Lp0 ,
t
(1.12)
x
U (t)(U (s))∗ F (s)dskLrt Lpx . kF kLr̃0 Lp̃0
t
x
(1.13)
hold for all sharp σ-admissible pairs (r, p) and (r̃, p̃). If U (t) satisfies stronger
condition
kU (t)(U (s))∗ gkL∞
. (1 + |t − s|)−σ kgkL1x ,
(1.14)
x
then the above estimates hold for all σ-admissible pairs.
In 2012, Zhang[20] established some Strichartz estimates in the frame of modulation spaces, here we will study the estimates in the frame of α-modulation spaces,
our theorems will cover the estimates in [20]. First, we recall the definition of
α-modulation space.
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STRICHARTZ ESTIMATES
3
Definition 1.3. Let ρ(ξ) be a smooth radial bump supported in the ball |ξ| < 2,
satisfying ρ(ξ) = 1 as |ξ| ≤ 1. For any k ∈ Zn , we set
ξ − hkiα/(1−α) k ρα
,
(1.15)
k (ξ) = ρ
Chkiα/(1−α)
and denote
X
−1
α
ηkα (ξ) = ρα
(ξ)
ρ
(ξ)
.
(1.16)
k
l
l∈Zn
For any k ∈ Zn , we define
−1 α
α
ηk F .
(1.17)
k =F
When α ∈ [0, 1), the α-modulation space associated with the above decomposition
is defined by
X
1/q
s,α
q
s,α
<∞
Mp,q
(Rn ) = f ∈ S 0 (Rn ) : kf kMp,q
hkisq/(1−α) kα
f
k
(Rn ) =
k
p
k∈Zn
with the usual modifications when q = ∞.
The α-modulation space was introduced by Gröbner [8], and it is an intermediate
space between modulation space and Besov space, when α = 0 it is the usual
modulation space, and the Besov space can be regarded as the case α → 1. A
comprehensive study of α-modulation space has been done in [19]. We also need
some frequency decomposition spaces which are similar with the spaces defined in
[16].
Definition 1.4. If X = Lrt Lpx (R × Rn ) (1 ≤ r, p, q < ∞), we denote
X
1/q
s,q
q
l
u ∈ S 0 (Rn+1 ) : kukls,qα (X) =
hkisq/(1−α) kα
uk
<∞
α (X) =
k
X
k∈Zn
with the usual modifications when q = ∞.
Using Minkowski’s inequality, we can verify that for r ≥ q,
α,s ≤ kf k s,q
kf kLrt (R,Mp,q
.
)
l α (Lrt Lp
x)
(1.18)
In Section 2, we will give some basic definitions and properties associated with
α-modulation spaces, and recall some basic estimates of oscillatory integral which
are useful in our proof. We will give the proof of main theorems in Section 3.
Standard techniques involving T T ∗ method and duality argument will be used to
establish some Strichartz estimates. In the case when (r, p) and (r̃, p̃) are sharp, we
will use a dilation argument, based on some basic Strichartz estimates. Now, we
present our main results. First the Strichartz estimates for Schrödinger equation:
Theorem 1.5. Suppose s ∈ R, q ≥ 1, α ∈ [0, 1), β ∈ (0, 2] and β 6= 1, (p, r) and
(p̃, r̃) are both n2 -admissible pairs, then the solution of (1.1) satisfies
ku(t, x)kls,qα (Lrt Lpx ) . ku0 kM s+δ(r,p),α + kF kls+δ(r,p)+δ(r̃,p̃),q (Lr̃0 Lp̃0 )
2,q
where δ(r, p) = α( n2 −
2
r
β/2
k
s<t
β/2
ei(t−s)(−∆)
t
x
(1.19)
− np ) + (2 − β) 1r . More precisely, we have
keit(−∆)
Z
α
u0 kls,qα (Lrt Lpx ) . ku0 kM s+δ(r,p),α ,
(1.20)
2,q
F (s)dskls,qα (Lrt Lpx ) . kF kls+δ(r,p)+δ(r̃,p̃),q (Lr̃0 Lp̃0 ) .
α
t
x
(1.21)
4
W. GUO, J. CHEN
EJDE-2013/118
Next we have the Strichartz estimates for nonelliptic Schrödinger equation:
Theorem 1.6. Suppose s ∈ R, q ≥ 1, α ∈ [0, 1), β ∈ (0, 2] and β 6= 1, (p, r) and
(p̃, r̃) are both n2 -admissible pairs, then the solution of (1.2) satisfies
kv(t, x)kls,qα (Lrt Lpx ) . kv0 kM s+δ(r,p),α + kF kls+δ(r,p)+δ(r̃,p̃),q (Lr̃0 Lp̃0 )
where δ(r, p) = α( n2 −
α
2,q
2
r
t
x
− np ) + (2 − β) 1r . More precisely, we have
keitψ(D) v0 kls,qα (Lrt Lpx ) . kv0 kM s+δ(r,p),α ,
Z
k
s<t
(1.22)
(1.23)
2,q
ei(t−s)ψ(D) F (s)dskls,qα (Lrt Lpx ) . kF kls+δ(r,p)+δ(r̃,p̃),q (Lr̃0 Lp̃0 ) .
α
t
x
(1.24)
Next we have the Strichartz estimates for wave equation:
Theorem 1.7. Suppose s ∈ R, q ≥ 1, α ∈ [0, 1), (p, r) and (p̃, r̃) are both n−1
2 −
admissible pairs, if n2 − np − 1r − 1 > 0 and n − 1 − np − 1r − np̃ − 1r̃ > 0, then the
solution of (1.3) satisfies
kwkls,qα (Lrt Lpx ) . kw0 kM s+θ(r,p),α + kw1 kM s+θ(r,p)−1,α + kF kls+θ(r,p)+θ(r̃,p̃)−1,q (Lr̃0 Lp̃0 )
2,q
k
s<t
α
2,q
2
r
n−1
p )
t
x
n+1 1
n−1 r .
− −
+
More precisely, we have
√
k cos(t −∆)w0 kls,qα (Lrt Lpx ) . kw0 kM s+θ(r,p),α ,
2,q
√
sin(t −∆)
w1 kls,qα (Lrt Lpx ) . kw1 kM s+θ(r,p)−1,α ,
k √
2,q
−∆
√
sin((t − s) −∆)
√
F (s)dskls,qα (Lrt Lpx ) . kF kls+θ(r,p)+θ(r̃,p̃)−1,q (Lr̃0 Lp̃0 ) .
x
t
α
−∆
where θ(r, p) =
Z
n
α n−1
( n−1
2
(1.25)
(1.26)
(1.27)
We must point out that our results can’t be deduced directly by a simple interpolation between the modulation space(α = 0) and the Besov space, since in [19],
the authors have pointed out that α−modulation space can’t be reformulated by
interpolations between modulation and Besov spaces at least for some special cases.
2. Preliminaries
We will often use the notation X . Y whenever there exists some constant C
so that X ≤ CY , C can depend on n, p, r, p̃, r̃, α, β. For k = (k1 , k2 , . . . , kn ) ∈
1
Zn , hki := (1 + |k|2 ) 2 . We denote by S := S(Rn ) the Schwartz space and S 0 :=
S 0 (Rn ) the tempered distribution space. We use Lrt (R) to denote the Banach space
of functions f : R → C whose norm
Z
1/r
kf kLrt (R) :=
|f (t)|r dt
(2.1)
R
is finite, with the usual modifications when r = ∞. We use Lrt Lpx (R×Rn ) to denote
the spacetime norm
Z
1/r
p
kF kLrt Lx (R×Rn ) =
kF krLpx dt
(2.2)
R
with the usual modifications when p, q or r is infinity.
In our proofs, the almost orthogonality property will be used frequently, this
property is independent of α. Here we give a proof which is different from [19] and
seems more regular. Firstly, we establish following position lemma.
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STRICHARTZ ESTIMATES
5
Lemma 2.1. For every x, y ∈ Rn , we have the following
min(hxiα/(1−α) , hyiα/(1−α) )|x − y| . |hxiα/(1−α) x − hyiα/(1−α) y|
. max(hxiα/(1−α) , hyiα/(1−α) )|x − y|.
Particularly, if we choose y = k, |x − y| = r, then
min(hxiα/(1−α) , hkiα/(1−α) )r . |hxiα/(1−α) x − hkiα/(1−α) k|
. max(hxiα/(1−α) , hkiα/(1−α) )r,
so we have
|hxiα/(1−α) x − hkiα/(1−α) k| ∼ hkiα/(1−α)
if |k| −→ ∞ and |x − k| = r.
(2.3)
Proof. By the symmetry of x and y, we only prove the case |x| ≤ |y|. Let
h(t) = |x − ty|2 ,
(2.4)
and derivative it, we have
h0 (t) =
n
X
2tyi2 −
i=1
n
X
2xi yi .
(2.5)
i=1
Using the fact |x| ≤ |y| and Cauchy-Schwartz inequality, we verify that
h0 (t) ≥ 0
(2.6)
when t ≥ 1. Using this inequality, we conclude
|hxiα/(1−α) x − hyiα/(1−α) y| = hxiα/(1−α) |x −
hyiα/(1−α)
y|
hxiα/(1−α)
(2.7)
≥ hxiα/(1−α) |x − y|
= min(hxiα/(1−α) , hyiα/(1−α) )|x − y|.
On the other hand,
|hxiα/(1−α) x − hyiα/(1−α) y| ≤ |hxiα/(1−α) − hyiα/(1−α) ||x| + |hyiα/(1−α) ||x − y|.
By mean-valued theorem and the fact |x| ≤ |y|, one can verify that
α
|hxiα/(1−α) − hyiα/(1−α) | . |x − y|hyi 1−α −1 .
(2.8)
Since |x| ≤ |y| ≤ hyi, so we have
|hxiα/(1−α) − hyiα/(1−α) ||x| . |x − y|hyiα/(1−α) ,
(2.9)
so
|hxiα/(1−α) x − hyiα/(1−α) y| . |hxiα/(1−α) − hyiα/(1−α) ||x| + |hyiα/(1−α) ||x − y|
. |hyiα/(1−α) ||x − y|
= max(hxiα/(1−α) , hyiα/(1−α) )|x − y|.
By the position lemma, we can conclude that there exists a positive constant c1
independent of x and k, such that
c1 hkiα/(1−α) |x − k| ≤ |hxiα/(1−α) x − hkiα/(1−α) k|.
(2.10)
6
W. GUO, J. CHEN
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In fact, if |k| > 2|x|, we have
|x − k| ≤ |x| + |k| ≤
3
|k|,
2
(2.11)
then
|hxiα/(1−α) x − hkiα/(1−α) k| ≥ |hkiα/(1−α) k| − |hxiα/(1−α) x|
1
1
≥ |hkiα/(1−α) k| − |hkiα/(1−α) k| = |hkiα/(1−α) k|
2
2
1 2
1
& · |hkiα/(1−α) ||x − k| = |hkiα/(1−α) ||x − k|.
2 3
3
If |k| ≤ 2|x|, then
|hxiα/(1−α) x − hkiα/(1−α) k| ≥ min(hxiα/(1−α) , hkiα/(1−α) )|x − k|
≥ min(cα hkiα/(1−α) , hkiα/(1−α) )|x − k|
& hki
α/(1−α)
(2.12)
|x − k|.
Similarly, for a fixed constant G > 0, we can find a constant c2 only depend on G,
such that if |x − k| < G,
|hxiα/(1−α) x − hkiα/(1−α) k| ≤ c2 hkiα/(1−α) |x − k|.
(2.13)
This can be easily concluded by position lemma.
We set a map Jα from Rn to Rn
Jα (x) = hxiα/(1−α) x.
(2.14)
Using inequality (2.10), we can take R sufficiently large such that Rc1 > 2C, thus
supp ηkα ⊂ B(hkiα/(1−α) k, c1 hkiα/(1−α) R) ⊂ Jα (B(k, R)).
(2.15)
Similarly, we can choose r small enough such that rc2 < C, thus
Jα (B(k, r)) ⊂ B(hkiα/(1−α) k, c2 hkiα/(1−α) r) ⊂ supp ηkα .
(2.16)
Jα (B(k, r)) ⊂ supp ηkα ⊂ Jα (B(k, R)),
(2.17)
So we have
and
{(i, j) : supp ηiα ∩ supp ηjα 6= ∅} ⊂ {(i, j) : Jα (B(i, R)) ∩ Jα (B(j, R)) 6= ∅}
= {(i, j) : (B(i, R) ∩ B(j, R)) 6= ∅}.
(2.18)
So, in some sense, the α-modulation space is as regular as modulation space up
to a transform Jα . We recall some estimates of oscillatory integrals, which can be
deduced by principle of stationary phase. One can find the methods in [12] and
[11]. We use ϕ(ξ) to denote the symbol of the Littlewood-Paley operator ∆0 .
Lemma 2.2. If 0 < β 6= 1, then
Z
β
eit|ξ| ϕ(ξ)eix·ξ dξ . |t|−n/2 .
(2.19)
Rn
If β = 1, then
Z
Rn
eit|ξ| ϕ(ξ)eix·ξ dξ . |t|(n−1)/2 .
(2.20)
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STRICHARTZ ESTIMATES
7
We omit the proof of the above lemma, and refer the reader to [20]. This lemma
can be also concluded by a lemma by Littman [17]. Then we will show another
inequality which will be used in the proof of Theorem 1.2, one can find the proof
in [20] for the case that α = 0.
Lemma 2.3. If 0 < β ≤ 2 and β 6= 1, then
Z
Pn
2−β n
β
eit l=1 ±|ξl | ηkα (ξ)eix·ξ dξ . hki 1−α 2 |t|−n/2 .
(2.21)
Rn
Proof. We only prove the case β < 2. One can easily find a one dimension smooth
α/(1−α)
Qn
kl
bump function φ(ξ) such that l=1 φ( ξl −hki
)ηkα = ηkα for every k ∈ Zn .
hkiα/(1−α)
Then we have
n
Z
Y
P
ξl − hkiα/(1−α) kl ix·ξ it n
±|ξl |β
l=1
e
)e dξl φ(
hkiα/(1−α)
Rn
l=1
(2.22)
n Z
Y
ξl − hkiα/(1−α) kl
i(xl ·ξl ±t|ξl |β )
)dξl =
φ(
e
hkiα/(1−α)
R
l=1
and
Z
β
ξl − hkiα/(1−α) kl
)dξl ei(xl ·ξl ±t|ξl | ) φ(
α/(1−α)
hki
R
Z
Z
ξl − hkiα/(1−α) kl
−ξl − hkiα/(1−α) kl
iP+ (ξl )
iP− (ξl )
e
φ(
e
φ(
.
)dξ
+
)dξ
.
l
l
α/(1−α)
α/(1−α)
hki
hki
+
+
R
R
β−2
When kl 6= 0 (kl is large), we have |P±00 (ξl )| & t hkiα/(1−α) kl
, we use Van de
Corputs lemma to deduce
Z
β
ξl − hkiα/(1−α) kl
)dξ
ei(xl ·ξl ±t|ξl | ) φ(
l
α/(1−α)
hki
R
ξ − hkiα/(1−α) k 0
(2−β)/2
l
l
α/(1−α)
(2.23)
1
. φ
k
hki
k
|t|−1/2
l
L
hkiα/(1−α)
(2−β)/2
(2−β)/2
1
|t|−1/2 .
. hkiα/(1−α) kl
|t|−1/2 . hki 1−α
When kl = 0 (kl small), we have
Z
β
ξl
ei(xl ·ξl ±t|ξl | ) φ( α/(1−α) )dξl hki
R
Z
αβ
α/(1−α)
xl ·ξl ±hki 1−α t|ξl |β )
φ(ξl )dξl .
= hkiα/(1−α) ei(hki
R
j
Let ϕj (ξ) = ϕ(ξ/2 ) be the symbol of one dimension Littlewood-Paley operator
∆j , we use Lemma 2.2(one dimension version) and dilation to deduce that
Z
αβ
β
α/(1−α)
α 2−β
xl ·ξl ±hki 1−α t|ξl |β )
ϕj (ξl )dξl . 2j(1− 2 ) hki 1−α 2 |t|−1/2 ,
hkiα/(1−α) ei(hki
R
then
Z
αβ
α/(1−α)
xl ·ξl ±hki 1−α t|ξl |β )
φ(ξl )dξl ei(hki
α/(1−α) hki
R
8
W. GUO, J. CHEN
.
0
X
β
α
2j(1− 2 ) hki 1−α
2−β
2
EJDE-2013/118
|t|−1/2
j=−∞
α
2−β
. hki 1−α 2 |t|−1/2
(2−β)/2
1
|t|−1/2 .
. hki 1−α
Using above estimates and the fact that kF −1 ηkα kL1 . 1, we complete the proof.
3. Proof of main results
Proof of Theorem 1.5. We prove only the case that β < 2. Let ϕj = ϕ( 2ξj ) be
the symbol of the littlewood-paley operator ∆j . Using dilation and Lemma 2.2, we
can deduce that
Z
β
β
ξ
(3.1)
eit|ξ| ϕ( j )eix·ξ dξ . 2jn(1− 2 ) |t|−n/2 ,
2
Rn
so we have
X
it(−∆)β/2
it(−∆)β/2
kα
f kL∞
.
k∆j α
f kL∞
0e
0e
x
x
j≤c
.
X
β
2jn(1− 2 ) |t|−n/2 kα
0 f kL1x
(3.2)
j≤c
−n/2
. |t|−n/2 |α
kf kL1x .
0 f kL1x . |t|
For k 6= 0, we have
β/2
it(−∆)
kα
ke
X
f kL∞
.
x
β/2
it(−∆)
k∆j α
ke
f kL∞
x
α 6=∅
supp ϕj ∩supp ηk
X
.
β
2jn(1− 2 ) |t|−n/2 kα
k f kL1x .
(3.3)
α 6=∅
supp ϕj ∩supp ηk
1
Using the almost orthogonality property and 2j ∼ hki 1−α , we can deduce that
β−2 −n/2
it(−∆)β/2
kα
kf kL1x .
(3.4)
f kL∞
. hki 1−α |t|
ke
x
On the other hand, we have
Z
−n/2
β
α
α
.
eit|ξ| ηkα (ξ)eix·ξ dξ . kηkα kL1 . hki 1−α n = hki 1−α (−2)
(3.5)
The above two estimates imply that
−n/2
β−2
α
it(−∆)β/2
1−α (−2) + hki 1−α |t|
∞ .
kf kL1x .
kα
e
f
k
hki
L
k
x
(3.6)
Rn
When k = 0, we have
β/2
it(−∆)
kα
0e
f kL∞
. (1 + |t|)
x
−n/2
kf kL1x
(3.7)
and the energy estimate
β/2
it(−∆)
kα
0e
f kL2x . kf kL2x ,
we can use Lemma 1.2 to deduce that
it(−∆)
kα
0e
β/2
f kLrt Lpx . kf kL2x ,
(3.8)
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STRICHARTZ ESTIMATES
Z
kα
0
β/2
ei(t−s)(−∆)
9
F (s)dskLrt Lpx . kF kLr̃0 Lp̃0 .
x
t
s<t
Using Lemma 2.2, one can easily verify that
β/2
k∆0 eit(−∆)
f kL∞
. (1 + |t|)−n/2
x
(3.9)
and deduce that
β/2
k∆0 eit(−∆)
Z
β/2
ei(t−s)(−∆)
k∆0
f kLrt Lpx . kf kL2x ,
F (s)dskLrt Lpx . kF kLr̃0 Lp̃0 .
x
t
s<t
Using dilation, we obtain
2−β
β/2
n
n
2
k∆j eit(−∆) f kLrt Lpx . 2j (( 2 − p − r )+ r ) kf kL2x ,
Z
β/2
k∆j
ei(t−s)(−∆) F (s)dskLrt Lpx
(3.10)
s<t
2
n
n
2
n
1
1
j (( n
2 − r − p )+( 2 − r̃ − p̃ )+(2−β)( r + r̃ ))
.2
When (r, p) is sharp
β/2
it(−∆)
kα
ke
x
t
n
2 -admissible,
f kLrt Lpx .
we have
X
(3.11)
kF kLr̃0 Lp̃0 .
β/2
k∆j eit(−∆)
f kLrt Lpx
α 6=∅
supp ϕj ∩supp ηk
.
X
2j
2−β
r
(3.12)
kf kL2x .
α 6=∅
supp ϕj ∩supp ηk
1
Using the almost orthogonality property and 2j ∼ hki 1−α , we can deduce that
β/2
it(−∆)
kα
ke
1
f kLrt Lpx . hki 1−α (
2−β
r )
kf kL2x .
(3.13)
Similarly, we can deduce that
Z
β/2
1
1
1
kα
ei(t−s)(−∆) F (s)dskLrt Lpx . hki 1−α (2−β)( r + r̃ ) kF kLr̃0 Lp̃0
k
x
t
s<t
(3.14)
for all sharp n2 -admissible pairs (r, p). When (r, p) is nonsharp n2 -admissible; that
is, 2r + np < n2 . Combining with 3.6 and energy estimate, we have
β/2
it(−∆)
kα
ke
− n2 (1− p2 )
β−2
α
f kLpx . hki 1−α (−2) + hki 1−α |t|
kf kLp0 .
x
(3.15)
So
kα
k
Z
β/2
ei(t−s)(−∆)
F (s)dskLrt Lpx
R
.
Z − n2 (1− p2 )
β−2
α
kF (s)kLp0 dsLr
hki 1−α (−2) + hki 1−α |t − s|
x
R
(3.16)
t
− n2 (1− p2 )
β−2
α
. k hki 1−α (−2) + hki 1−α |t|
kLr/2 kF (s)kLr0 Lp0 .
t
t
x
One can check that
− n2 (1− p2 ) 2δ(r,p)
β−2
α
(−2)
hki 1−α
r/2 . hki 1−α ,
+ hki 1−α |t|
L
t
(3.17)
10
W. GUO, J. CHEN
EJDE-2013/118
then deduce that
Z
2δ(r,p)
β/2
kα
ei(t−s)(−∆) F (s)dskLrt Lpx . hki 1−α kF (s)kLr0 Lp0 .
k
x
t
R
(3.18)
Then homogeneous estimate(1.20) follows by using T T ∗ method, standard duality
argument and the almost orthogonality property of α-modulation space. For the
inhomogeneous part, for every n2 -admissible pairs (r, p) and (r̃, p̃), there exist two
constant p1 and p̃1 such that (r, p1 ) and (r̃, p˜1 ) are sharp. Combining the following
inequalities
1
α
1
1−α n( p1 − p )
p
p
kα
kα
k f kLx1 ,
k f kLx . hki
α
1
1
1−α n( p̃1 − p̃ )
kα
kα
k f kLp˜1 0 . hki
k f kLp̃0
x
x
with (3.14), we have
Z
β/2
kα
ei(t−s)(−∆) F (s)dskLrt Lpx
k
s<t
1
α
1
. hki 1−α n( p1 − p ) kα
k
Z
t
β/2
ei(t−s)(−∆)
0
. hki
. hki
Recall that
kα
k
n
p1
1
α
1−α n( p1
1
)
−p
1
α
1−α n( p1
1
−p
)
hki
hki
1
1
1
1−α (2−β)( r + r̃ )
1
1
1
1−α (2−β)( r + r̃ )
F (s)dskLrt Lpx1
(3.19)
kF kLr̃0 Lp˜1 0
hki
x
t
α
1
1−α n( p̃1
1
− p̃
)
kF kLr̃0 Lp̃0 .
x
t
=
Z
n
2
−
2
r
and
n
p˜1
β/2
ei(t−s)(−∆)
s<t
=
n
2
−
2
r̃ ,
so we have
F (s)dskLrt Lpx . hki
δ(r,p)+δ(r̃,p̃)
1−α
kF kLr̃0 Lp̃0 .
t
x
(3.20)
Then the inhomogeneous estimate (1.21) follows by the definition and almost orthogonality property of α-modulation space.
Remark 3.1. In the case (r, p) is nonsharp n2 -admissible, one can also deduce the
R
it(−∆)β/2
i(t−s)(−∆)β/2
estimates of kα
f kLrt Lpx and kα
F (s)dskLrt Lpx by using
ke
k s<t e
(3.10) and (3.11) respectively, but it will lose more regularity.
Proof of Theorem 1.6. We prove only the case β < 2. Using Lemma 2.3 and
the fact that
Z
−n/2
Pn
β
α
α
, (3.21)
eit l=1 ±|ξl | ηkα (ξ)eix·ξ dξ . kηkα kL1 . hki 1−α n = hki 1−α (−2)
Rn
we have
−n/2
β−2
α
itψ(D)
kα
f kL∞
. hki 1−α (−2) + hki 1−α |t|
kf kL1x .
ke
x
(3.22)
When k = 0 we can deduce the estimates as the proof of Theorem 1.5. When k 6= 0,
we can also obtain the homogeneous estimates (1.23) for nonsharp pair (r, p) by
T T ∗ method and standard duality argument.
For the case that (r, p) and (r̃, p̃) are sharp, we use dilation
argument,Pbut it’s
P
a little different from the proof Theorem 1.5. Let Sj = l≤j ∆l , ϕ0 = j≤0 ϕj ,
using Lemma 2.3 with k = 0, we can deduce
Z
Pn
β
eit l=1 ±|ξl | ϕ0 (ξ)eix·ξ dξ . |t|−n/2 ,
(3.23)
Rn
EJDE-2013/118
STRICHARTZ ESTIMATES
11
so
kS0 eitψ(D) f kL∞
. (1 + |t|)−n/2 kf kL1x .
x
Using Lemma 1.2, we can obtain the following estimates:
(3.24)
kS0 eitψ(D) f kLrt Lpx . kf kL2x ,
Z
kS0
s<t
ei(t−s)ψ(D) F (s)dskLrt Lpx . kF kLr̃0 Lp̃0 .
x
t
A dilation argument then yields
kSj eitψ(D) f kLrt Lpx . 2j
Z
2−β
r
1
kSj
s<t
to deduce
1
s<t
x
t
α
k
itψ(D)
kα
f kLrt Lpx . hki 1−α (
ke
kα
k
1
ei(t−s)ψ(D) F (s)dskLrt Lpx . 2j(2−β)( r + r̃ ) kF kLr̃0 Lp̃0 .
Then we use Sj to cover
Z
kf kL2x ,
ei(t−s)ψ(D) F (s)dskLrt Lpx . hki
2−β
r )
kf kL2x ,
1
1
1
1−α (2−β)( r + r̃ )
kF kLr̃0 Lp̃0
t
x
for any sharp n2 -admissible pairs (r, p) and (r̃, p̃).
The remain case is that inhomogeneous estimate (1.24) for nonsharp admissible
pairs, it can be deduced like the proof of Lemma 1.1, we omit the details.
Proof of Theorem 1.3. We need to prove only the following estimates:
√
it
kα
ke
−∆
√
it −∆
e
kα
k √
√
−∆
w0 kLrt Lpx . hki
w1 kLrt Lpx . hki
θ(r,p)
1−α
kw0 kL2x ,
θ(r,p)−1
1−α
(3.25)
kw1 kL2x ,
(3.26)
θ(r,p)+θ(r̃,p̃)−1
ei(t−s) −∆
1−α
√
F (s)dskLrt Lpx . hki
kF kLr̃0 Lp̃0 .
x
t
−∆
s<t
Using the same techniques as before, we can deduce (3.25) and
Z
√
θ(r,p)+θ(r̃,p̃)
1−α
kα
ei(t−s) −∆ F (s)dskLrt Lpx . hki
kF kLr̃0 Lp̃0 .
k
kα
k
Z
t
s<t
(3.27)
x
If k 6= 0, then (3.26) and (3.27) will then follow by
1
−1/2
kα
f kL2x . hki− 1−α kf kL2x .
k (−∆)
it
√
−∆
If k = 0, we do not have energy estimate for the operator e√−∆ , so we can not use
the T T ∗ argument. One can deduce
Z
eit|ξ| |ξ|−1 ϕ(ξ)eix·ξ dξ . |t|(n−1)/2 ,
Rn
Z
eit|ξ| |ξ|−1/2 ϕ(ξ)eix·ξ dξ . |t|(n−1)/2
Rn
by principle of stationary phase as in Lemma 2.2, then Lemma 1.2 will yield
√
−∆
eit
k∆0 √
−∆
√
Z
k∆0
s<t
f kLrt Lpx . kf kL2x ,
ei(t−s) −∆
√
F dskLrt Lpx . kF kLr̃0 Lp̃0 .
x
t
−∆
12
W. GUO, J. CHEN
EJDE-2013/118
Using a dilation argument, we obtain
√
eit
k∆j √
Z
k∆j
s<t
e
−∆
−∆
√
i(t−s) −∆
√
−∆
n
n
1
f kLrt Lpx . 2j( 2 − p − r −1) kf kL2x ,
n
1
n
1
F dskLrt Lpx . 2j(n−1− p − r − p̃ − r̃ ) kF kLr̃0 Lp̃0 .
t
x
If n2 − np − 1r − 1 > 0 and n − 1 − np − 1r − np̃ − 1r̃ > 0, we can use ∆j to cover α
0
and get the estimates (3.26) and (3.27) for k = 0.
Remark 3.2. If we take α = 0 in Theorem 1.1 − 1.3, we obtain the Strichartz estimates in the frame of modulation spaces. The Strichartz estimates of Schrödinger
equation in the frame of α modulation spaces is the case that β = 2.
Acknowledgments. This work was supported by grant Y604563 from the NSFZJ
of China, and grant 11271330 from the NSF of China.
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EJDE-2013/118
STRICHARTZ ESTIMATES
Weichao Guo
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
E-mail address: maodunguo@163.com
Jiecheng Chen
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
E-mail address: jcchen@zjnu.edu.cn
13