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Strichartz estimates in the frame of modulation spaces
Nonlinear Analysis 78 (2013) 156–167
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Strichartz estimates in the frame of modulation spaces Chunjie Zhang ∗ School of Science, Hangzhou Dianzi University, Hangzhou, 310016, PR China
article
abstract
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Article history: Received 28 December 2011 Accepted 3 October 2012 Communicated by Enzo Mitidieri
In this paper, we study the Strichartz estimates for three types of differential equations in the framework of modulation spaces. The equations considered here include the wave equation, the Schrödinger equation and the nonelliptic Schrödinger equation. Inhomogeneous equations of the above three types are also studied. © 2012 Elsevier Ltd. All rights reserved.
MSC: 35L05 46E35 42B37 Keywords: Modulation space Strichartz estimate Schrödinger equation Wave equation Inhomogeneous equation
1. Introduction The Schrödinger type dispersive equation
i∂t u + (∆)α/2 u = 0 , u(0, x) = f (x)
(1.1)
and the wave equation
∂tt v − ∆v = 0 v(0, x) = f (x),
(1.2)
vt (0, x) = g (x)
have been vastly studied by numerous mathematicians due to both their strong physical background and theoretical importance. Their solutions are formally written as uα (t , x) = e
α it ∆ 2
f ( x) =
α
Rn
eit |ξ | fˆ (ξ )ei⟨x,ξ ⟩ dξ
and
v(t , x) =
Rn
cos(t |ξ |)fˆ (ξ )ei⟨x,ξ ⟩ dξ +
sin(t |ξ |)
Rn
|ξ |
gˆ (ξ )ei⟨x,ξ ⟩ dξ .
One can easily find thousands of theorems on them, among which we are especially interested in the Strichartz estimates. Such estimate also have a long history and are often found to be useful in the study of nonlinear equations. One is referred to [1–8] for some historical work on this subject. Following is an abstract Strichartz estimate proved by Keel and Tao in [9].
∗
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0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.10.002
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
157
Definition. We call a pair (p, q) is δ -admissible, if p, q ≥ 2, (p, q, δ) ̸= (∞, 2, 1) and 1 q
+
δ p
δ
≤
2
.
(1.3)
If the equality in (1.3) holds, we call that (p, q) is sharp δ -admissible, otherwise, (p, q) is non-sharp δ -admissible. Theorem A. Let {U (t )} be a semigroup of operators satisfying ∥U (t )f ∥L2 ≤ ∥f ∥L2 and x
x
∥U (s)U (t )g ∥L∞ ≼ |t − s| ∥g ∥L1x . x
(1.4)
∥U (t )f (x)∥Lq Lpx ≼ ∥f ∥L2x , t ∗ U (t )U (s)F (s)ds q
(1.5)
∗
−δ
Then
s
p
≼ ∥F ∥Ll′ Lr ′
Lt Lx
(1.6)
t x
hold for all sharp δ -admissible pairs (p, q) and (l, r ). If (1.4) is replaced by
∥U (s)U ∗ (t )g ∥L∞ ≼ (1 + |t − s|)−δ ∥g ∥L1x , x
(1.7)
then (1.5) and (1.6) hold for all δ -admissible pairs (p, q) and (l, r ). Using the above theorem, Keel and Tao proved that, the solution to (1.1) (with α = 2) and (1.2) satisfy
∥u(t , x)∥Lq Lpx ≼ ∥f ∥L2x , t
n
∀ -admissible (p, q), 2
and
∥v(t , x)∥Lq Lpx ≼ ∥f ∥H˙ β + ∥g ∥H˙ β−1 ,
∀
t
Here, β = n
1 2
−
1 p
−
1 q
n−1 2
-admissible (p, q).
˙ β denotes time homogeneous L2 -Sobolev space. and H
In this paper we will study the Strichartz estimates on the modulation spaces, which were first introduced by Feichtinger [10] and soon found to be useful in the study of pseudo-differential operators; see [11–14]. Several equivalent characterizations of modulation spaces have been found in [15–17] among which we only present the definition using frequency uniform decomposition below. Let σ be a non-negative real function in C0∞
− 45 , 54
n
satisfying σ (ξ ) ≡ 1 for ξ ∈
2 2 n − 5 , 5 . Denoting Zn the all
integers in Rn and σk (ξ ) = σ (ξ − k), we further assume
σk (ξ ) ≡ 1,
∀ξ ∈ Rn .
k∈Zn
Now for s ∈ R, 1 ≤ d ≤ ∞ and a Banach function space X , we define the modulation space as M s (X , l d ) =
1/d
f ∈ S ′ (Rn ) : ∥f ∥M s (X ,ld )
= ⟨z ⟩sd ∥πk f ∥dX k∈Z
< +∞
where ⟨k⟩ = 1 + |k| and πk f = F −1 (σk fˆ ). When X = Lp , we denote M s (X , ld ) by Mps,d , which is the modulation space mostly
referred. Note that Mps,d is different from Lp . In fact, one can easily show Mp0,1 ⊂ Lp . For more relationship among Mps,d , Lp and Besov spaces, one is referred to [18,13,14,19]. Estimates on Mps,d for pseudo-differential operators have attracted many researchers in recent years. Here we list for example the following result from [20,21],
∥uα (t , x)∥Mps,d ≤ Ct ∥f ∥Mps,d .
(1.8)
It is interesting to see that the norm Mps,d is preserved by the time evolution of Schrödinger equation (1.1) while there is a different case for the Lp estimate; see [20]. The sharp decay of Ct in (1.8) is also studied in [22]. Our purpose of this paper is to obtain some Strichartz estimates on modulation spaces. Below are two of our main results.
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C. Zhang / Nonlinear Analysis 78 (2013) 156–167
Theorem 1. Let s ∈ R, 0 < α ≤ 2, α ̸= 1 and 1 ≤ d ≤ ∞. If (p, q) is 2n -admissible, then the solution to (1.1) satisfies
∥uα (t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+β , t
2,d
where β = 2−α . In particular, when q ≥ d we have q ∞
∥uα (t , x)∥qM s dt
1/q
p,d
0
≼ ∥f ∥M s+β . 2,d
Theorem 2. Let n ≥ 2, s ∈ R and 1 ≤ d ≤ ∞. If n
1 2
−
1 p
≥ 1 + 1q , then the solution to (1.2) satisfies
∥v(t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+γ + ∥g ∥M s+γ −1 , 2,d
t
2,d
where γ = q(nn+−11) . In particular, when q ≥ d we have ∞
∥v(t , x)∥qM s dt
1/q
p,d
0
≼ ∥f ∥M s+γ + ∥g ∥M s+γ −1 . 2,d
2,d
To the best of our knowledge, Strichartz estimate on modulation space for wave equation has not been reported. For the standard Schrödinger equation, Wang and Hudzik [19] obtained
∥eit ∆ f (x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M2s ,d , t
which is the α = 2 case of Theorem 1. In fact, their method can be modified to adapt our case uα (t , x) and prove the sharp admissible part of Theorem 1. We will use a different way to prove our theorem so that less regularity is required on initial data when (p, q) is non-sharp admissible. We will also study the dispersive equation
i∂t w + ψ(D)w = 0 w(0, x) = f (x),
(1.9)
±|ξl |α . Its formal solution is wα (t , x) = eit ψ(D) f (x) = eit ψ(ξ ) fˆ (ξ )ei⟨x,ξ ⟩ dξ .
where ψ(ξ ) =
n
l =1
Rn
When α = 2 and the signs in the sum are not all same, (1.9) is called the non-elliptic Schrödinger equation. One is referred to [23–25] for its physical background. The well posedness for such equations can be found for example, in [26–28]. There is also an interesting pointwise convergence theorem for this equation [29]. Here we will prove the following. Theorem 3. Let s ∈ R, 0 < α ≤ 2, α ̸= 1 and 1 ≤ d ≤ ∞. If (p, q) is 2n -admissible, then the solution to (1.9) satisfies
∥wα (t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+β , t
2,d
where β = 2−α . In particular, when q ≥ d we have q ∞
0
∥wα (t , x)∥qM s dt p,d
1/q
≼ ∥f ∥M s+β . 2,d
We point that when α = 2, the above theorem is proved in [28]. In fact in [19,28], Strichartz estimates on modulation spaces for inhomogeneous equations are also partially studied and they use those estimates to study the existence of nonlinear solutions on modulation spaces. Here in Section 4, we will present the more complete Strichartz estimates for inhomogeneous equations of all three types. These estimates are proved in a standard and similar way. So we will only sketch their proofs and point some differences.
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
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2. Some preliminary lemmas By the frequency uniform decomposition we used in the definition of modulation spaces, we can write
uα (t , x) =
Vk (t )f (x) =
k∈Zn
α
eit |ξ | σk (ξ )fˆ (ξ )ei⟨x,ξ ⟩ dξ .
Rn
k∈Zn
We will also use the Littlewood–Paley decomposition uα (t , x) =
Uj (t )f (x) =
j∈Z
α
Rn
j∈Z
eit |ξ | Φ (2−j ξ )fˆ (ξ )ei⟨x,ξ ⟩ dξ ,
where Φ (ξ ) is a standard cutoff function supported in {ξ , 1 < |ξ | < 3}. The C0∞ functions σ (ξ ) and Φ (ξ ) may vary in different occasions during the discussion, but we will always use the notations Vk (t ) and Uj (t ). Lemma 2.1. If 0 < α ̸= 1, then
|U0 (t )(x)| =
e
it |ξ |α
Rn
Φ (ξ )e
i⟨x,ξ ⟩
n dξ ≼ (1 + |t |)− 2 .
If α = 1, we have
|U0 (t )(x)| =
eit |ξ | Φ (ξ )ei⟨x,ξ ⟩ dξ ≼ (1 + |t |)−
Rn
n−1 2
.
Proof. The two inequalities in fact follows from some standard stationary phase argument. One may find the second inequality in many papers ([30] for example). For clarity, we still present the proof of the first one here. It is enough to consider t > 0. If t ≤ 1, then since |U0 (t )(x)| ≼ C , the lemma follows. So we assume t > 1. Note by polar decomposition, U0 (t )(x) =
∞
itr α
Φ (r )r
e
n −1
0
S n−1
′ ei⟨rx,ξ ⟩ dσ (ξ ′ )dr .
When |x| > 1/4, using the asymptotic of Fourier transform on S n−1 , we have U0 (t )(x) ≃ |x|
1−n 2
∞
n−1 α ei(tr ±|x|r ) Φ (r )r 2 dr
0
Denoting ψ(r ) = tr α ± |x|r, we easily find some Cα > 1 such that
|ψ ′ (r )| = |α tr α−1 ± |x ∥≥ |x|/2 whenever |x| > Cα t and
|ψ ′ (r )| = |α tr α−1 ± |x ∥≥ t /2 whenever |x| < Cα−1 t, since 1 < r < 3 by the support of Φ . In either case, we can use integrating by parts and obtain
|U0 (t )(x)| ≤ t −N ,
∀N > 0 .
When |x| ≃ t, it is easy to see that
ψ ′′ (r ) = |α(α − 1)r α−2 t | ≥ cα t . By Van de’Coupt’s lemma, we have
|U0 (t )(x)| ≼ |x|
1−n 2
1 n t− 2 ≼ t− 2 .
When |x| ≤ 1/4, we have r |x| ≤ 1 so
S n−1
′ ei⟨rx,ξ ⟩ dσ (ξ ′ ) = O(1)
and consequently U0 (t )(x) ≃
∞
α eitr Φ (r )r n−1 dr ≼ t −N
0
by integrating by parts. Now we have finished the proof of Lemma 2.1.
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C. Zhang / Nonlinear Analysis 78 (2013) 156–167
Lemma 2.2. Let k ∈ Zn . If 0 < α ≤ 2 and α ̸= 1, then n
∥Vk (t )g ∥L∞ ≼ (1 + t |k|α−2 )− 2 ∥g ∥L1x . x If α = 1, then
n−1 +1 − 2 − nn− 1 ∥Vk (t )g ∥L∞ ≼ 1 + t | k | ∥g ∥L1x . x ⃗ then the lemma reduces to Proof. We only prove the case α ̸= 1. The other one follows a similar way. If k = 0, n
∥V0 (t )g ∥L∞ ≼ (1 + t )− 2 ∥g ∥L1x . x
(2.1)
2 If α = 2, we note that the kernel of the Schrödinger operator is t −n/2 ei|x| /t . By Young’s inequality, n
n
∥V0 (t )g ∥L∞ = ∥eit ∆ (π0 g )∥L∞ ≤ t − 2 ∥π0 g ∥L1x ≼ t − 2 ∥g ∥L1x . x x On the other hand, by the Hausdorff–Young inequality and Hölder’s inequality, α
α
∥Vk (t )g ∥L∞ = ∥(eit |ξ | σk gˆ )∨ ∥L∞ ≼ ∥eit |ξ | σk gˆ ∥L1 ≼ ∥ˆg (ξ )∥L∞ ≼ ∥g ∥L1x . x x ξ ξ
(2.2)
In particular
∥V0 (t )g ∥L∞ ≼ ∥g ∥L1x , x and (2.1) is proved. If α < 2, then by Lemma 2.1 and Young’s inequality, n
∥U0 (t )g (x)∥L∞ ≼ (1 + t )− 2 ∥g ∥L1x . x A simple dilation argument then yields n
∥Uj (t )g ∥L∞ ≼ 2jn (1 + 2jα t )− 2 ∥g ∥L1x . x
(2.3)
Therefore,
∥V0 (t )g ∥L∞ = x
0
∥Uj (t )(π0 g )∥L∞ ≼ x
0
n
2jn (1 + 2jα t )− 2 ∥π0 g ∥L1 x
j=−∞
j=−∞
≤
0
n
2jn (2jα + 2jα t )− 2 ∥g ∥L1 x
j=−∞
≤
0
n
2jn(2−α)/2 (1 + t )− 2 ∥g ∥L1 . x
j=−∞
Note the above sum is finite and (2.1) follows. ⃗ Note the support of σk is contained in finite number of annulus with radius r ≃ 2j ≃ |k|, Now we consider the case k ̸= 0. and the number of such annulus is independent of k. Therefore, we have Vk (t )g (x) =
Uj (t )(πk g )(x)
j≃log |x|
and consequently, by (2.2)
∥Vk (t )g ∥L∞ x
n ≼ Uj (t )(πk g ) ≼ 2jn (1 + 2jα t )− 2 ∥πk g ∥L1 x j ∞ j Lx n n ≤ 2jn (2jα t )− 2 ∥πk g ∥L1 ≤ (2j(α−2) t )− 2 ∥g ∥L1x x j
j
(α−2)
≃ (|k|
− 2n
t)
∥g ∥L1x .
Combining (2.3), we have proved the lemma.
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
161
To study the nonelliptic Schrödinger equation, we need the following decomposition
wα (t , x) =
Wk (t )f (x) =
k∈Zn
k∈Zn
eit
±|ξl |α
Rn
σk (ξ )fˆ (ξ )ei⟨x,ξ ⟩ dξ .
Lemma 2.3. Let k ∈ Zn , 0 < α ≤ 2 and α ̸= 1. We have n
∥Wk (t )g ∥L∞ ≼ (1 + t |k|α−2 )− 2 ∥g ∥L1x . x Proof. Similar to (2.3), we easily have
∥Wk (t )g (x)∥L∞ ≼ ∥g ∥L1x . x So by Young’s inequality, we are left to show n
∥Wk (t )(x)∥L∞ ≼ (|k|α−2 |t |)− 2 . (2.4) x 9 9 which satisfies φ(t ) ≡ 1 when t ∈ − 45 , 45 . Then for any fixed k = (k1 , k2 , . . . , kn ) ∈ Zn , Let us take φ(t ) ∈ C0∞ − 10 , 10
we have
σk (ξ ) = σk (ξ )
n
φ(ξl − kl ).
l=1
Therefore, we only have to prove (2.4) for
k (t )(x) = W
eit
±|ξl |α
Rn
n
φ(ξl − kl )ei⟨x,ξ ⟩ dξ
l =1
instead of Wk (t ). And this will be done by splitting the variables,
k (t )(x) = W
n
n
α ei(t |ξl | +xl ξl ) φ(ξl − kl )dξl =
R
l =1
kl (t )(xl ). W
l =1
By the change of variables and the Van de Corputs Lemma,
kl (t )(xl )| = |W
∞
α ei(xl ξl ±t ξl ) φ(ξl − kl )dξl +
0
∞
0 1
α ei(−xl ξl ±t ξl ) φ(−ξl − kl )dξl
1
≼ (|kl |α−2 |t |)− 2 ≼ (|k|α−2 |t |)− 2 . Therefore we have
k (t )(x)| = |W
n
n
kl (t )(xl )| ≼ (|k|α−2 |t |)− 2 . |W
l =1
3. Proof of the theorems Proof of Theorem 1. By the definition of modulation space, we only have to show
∥Vk (t )f ∥Lq Lpx ≼ ⟨k⟩
2−α q
t
∥πk f ∥L2
(3.1)
⃗ this reduces to for all 2n -admissible pairs (p, q). When k = 0, ∥V0 (t )f ∥Lq Lpx ≼ ∥π0 f ∥L2 .
(3.2)
t
By Plancherel’s theorem, V0 (t ) satisfies the energy estimate
∥V0 (t )f ∥L2x ≤ ∥f ∥L2x , and by (2.1) we also have n
∥V0 (t )V0∗ (s)g ∥L∞ = ∥V0 (t − s)g ∥L∞ ≼ (1 + |t − s|)− 2 ∥g ∥L1x . x x (3.2) then follows by applying Theorem A.
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C. Zhang / Nonlinear Analysis 78 (2013) 156–167
⃗ By Lemma 2.1, (1.7) holds with δ = n/2. So applying Theorem A we have, for any Next, we prove (3.1) for k ̸= 0. (p, q),
n -admissible 2
∥U0 (t )f (x)∥Lq Lpx ≼ ∥f ∥L2x , t
which further implies j 2n − np − αq
∥Uj (t )f (x)∥Lq Lpx ≼ 2
t
∥f ∥L2x
(3.3)
by simple dilation argument. Note the support of σk is contained in finite number of annulus with radius r ≃ 2j ≃ |k|. Therefore, by (3.3) we obtain
∥Vk (t )f ∥
= Uj (t )(πk f ) j q
q p Lt Lx
≼
p
2
∥πk f ∥L2x
j
Lt Lx
α
n
n
j 2n − np − αq
≃ ⟨k⟩ 2 − p − q ∥πk f ∥L2x . If (p, q) is sharp 2n -admissible, then n 2
n
−
−
p
α
=
q
2−α q
2 q
=
n 2
−
n p
and
,
from which (3.1) follows. If (p, q) is non-sharp 2n -admissible, we cannot use Theorem A directly but we can modify its proof. Unlike the Littlewood–Paley decomposition, we are not able to obtain (3.1) by a dilation method in the frequency uniform decomposition case. So below we will prove (3.1) for each k ∈ Zn using the bilinear method of [9], but track the power of ⟨k⟩ in each step. Since the dual of (3.1) is
∞
Vk∗ (t )F (t , ·)(x)dt
0
≼ ⟨k⟩
2−α q
L2x
∥F (t , x)∥Lq′ Lp′ , t
x
we further write it into the bilinear form ∞
I =
∞
0
2(2−α) ⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩dtds ≼ ⟨k⟩ q ∥F ∥Lq′ Lp′ ∥G∥Lq′ Lp′ . t
0
x
t
(3.4)
x
By Plancherel’s theorem α
∥Vk (t )f ∥L2x = ∥eit |ξ | σk fˆ ∥L2x ≼ ∥f ∥L2x . We take its bilinear form
|⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩| ≼ ∥F (s)∥L2x ∥G(t )∥L2x .
(3.5)
On the other hand, by Lemma 2.2, n
∥Vk (t )Vk∗ (s)g ∥L∞ = ∥Vk (t − s)g ∥L∞ ≼ (1 + |k|α−2 |t − s|)− 2 ∥g ∥L1x . x x Again we take its bilinear form n
|⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩| ≼ (1 + |k|α−2 |t − s|)− 2 ∥F (s)∥L1x ∥G(t )∥L1x .
(3.6)
Interpolating between (3.5) and (3.6) yields nθ
|⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩| ≼ (1 + |k|α−2 |t − s|)− 2 ∥F (s)∥Lp′ ∥G(t )∥Lp′ x
x
where θ = 1 − 2p . Therefore, by Hölder’s inequality and Young’s inequality, ∞
∥F (s)∥Lp′
∞
x ∥G(t )∥Lp′ dsdt nθ x (1 + |k|α−2 |t − s|) 2 ∞ ∥F (s)∥Lp′ x ≼ ∥G(t )∥Lq′ Lp′ ds nθ α− 2 t x 0 (1 + |k| |t − s|) 2 Lq t α−2 − n2θ ≼ ∥G(t )∥Lq′ Lp′ ∥F (t )∥Lq′ Lp′ (1 + |k| |t |) r
I ≼
0
0
t
x
t
x
Lt
(3.7)
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
where r =
q . 2
Since (p, q) is now assumed to be non-sharp 2n -admissible, it is easy to see check that
163 nr θ 2
> 1, thus
nθ
(1 + |k|α−2 |t |)− 2 ∈ Lrt . By easy calculation, the Lrt norm is bounded by |k|
2(2−α) q
. Now we have proved (3.4), hence the whole theorem.
Remark. In the above proof, we applied Lemma 2.1 and Theorem A to get the sharp admissible part of Theorem 1, while for the non-sharp admissible part we only used Lemma 2.2. In fact, the sharp admissible part can also be obtained using Lemma 2.2 only. To see this, we note by Lemma 2.2, n
∥Vk (t )g (x)∥L∞ ≼ (|k|α−2 t )− 2 ∥g ∥L1x . x Invoking the argument in obtaining (3.7), we similarly have nθ
|⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩| ≼ (|k|α−2 |t − s|)− 2 ∥F (s)∥Lp′ ∥G(t )∥Lp′ . x
x
The term
I =
∞ 0
∞
⟨Vk (s)F (s), Vk (t )G(t )⟩dtds ∗
0
∗
is then estimated using the Hardy–Sobolev inequality; see page 960 of [9]. But using the Hardy–Sobolev inequality, one still misses the sharp 2n -admissible endpoint (p, q) = n2n , 2 . Keel and Tao spent much effort to fix this point, and a −2 single ‘‘by similar argument’’ is not always convincing, especially when the ‘‘argument’’ is long and complicated. So in the proof of Theorem 1, we chose to treat the sharp admissible line (which contains the endpoint) differently. Nevertheless, their argument does adapt to the endpoint here. We may carefully follow their proof, with some difference of tracking the constants in each step; see also the explanation in [19]. Proof of Theorem 2. By the expression of the wave solution v(t , x), we consider two operators
v1 (t )f (x) =
v2 (t )g (x) =
Rn
Rn
eit |ξ | fˆ (ξ )ei⟨x,ξ ⟩ dξ ,
|ξ |−1 eit |ξ | gˆ (ξ )ei⟨x,ξ ⟩ dξ .
Theorem 2 will be proved if we obtain
∥v1 (t )f ∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+γ , 2,d
t
∥v2 (t )f ∥M s (Lq Lpx ,ld ) ≼ ∥g ∥M s+γ −1 . t
2,d
Thus by the definition of modulation space, we only have to show n+1
∥πk v1 (t )f ∥Lq Lpx ≼ ⟨k⟩ q(n−1) ∥πk f ∥L2x
(3.8)
t
∥πk v2 (t )g ∥Lq Lpx ≼ ⟨k⟩
n+1 −1 q(n−1)
t
∥πk g ∥L2x
(3.9)
for any k ∈ Zn . 1 ⃗ -admissible requirement on (p, q). For (3.9), if k ̸= 0, The proof of (3.8) is almost the same as (3.1), but with a different n− 2 then
∥πk v2 (t )g ∥Lq Lpx = ∥πk v1 (t )(|ξ |−1 gˆ (ξ ))∨ ∥Lq Lpx t
t
≼ ⟨k⟩γ ∥πk ((|ξ |−1 gˆ (ξ ))∨ )∥L2x ≼ ⟨k⟩γ −1 ∥πk g ∥L2x , ⃗ we define where we have applied (3.8) in the second inequality. When k = 0, Uj (t )f (x) =
Rn
|ξ |−1 eit |ξ | Φ (2−j ξ )fˆ (ξ )ei⟨x,ξ ⟩ dξ .
Similar to (3.3), we obtain j ∥ Uj (t )f ∥Lq Lp ≼ 2 x t
n − np − 1q −1 2
∥f ∥L2x
164
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
by applying Theorem A and the α = 1 part of Lemma 2.1. Using this inequality and the Littlewood–Paley theory, also in view of the assumption n 12 − 1p ≥ 1 + 1q , we have
0 = Uj (t )(π0 g ) j=−∞
∥π0 v2 (t )g ∥Lq Lpx t
Noting that n
1 2
−
1 p
≥1+
≼ ∥π0 g ∥L2x . q p
Łt Lx
1 implies (p, q) n− -admissible, we have proved Theorem 2. 2
1 q
The proof of Theorem 3 follows from the Remark and Lemma 2.3. 4. Estimates for inhomogeneous equations The solution to inhomogeneous Schrödinger type equation
i∂t u + (∆)α/2 u = F (t , x) u(0, x) = f (x)
(4.1)
is written as α
u(t , x) = eit ∆ f (x) − i 2
α
t
ei(t −s)∆ F (s, ·)(x)ds. 2
0
For the nonelliptic equation
i∂t u + ψ(D)u = 0 , u(0, x) = f (x)
(4.2)
the formal solution is u(t , x) = eit ψ(D) f (x) − i
t
ei(t −s)ψ(D) F (s, ·)(x)ds.
0
Theorem 4. Let s ∈ R, 1 ≤ d ≤ ∞ and 0 < α ≤ 2 with α ̸= 1. If (p, q) and (r , l) are both solutions to (4.1) and (4.2) satisfy
n -admissible 2
pairs, then the
∥u(t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+β + ∥F (t , x)∥M s+β˜ (Ll′ Lr ′ ,ld ) , t
2,d
where β = 2−α and β˜ = (2 − α) q ∞
∥ u∥ 0
q dt Mps ,d
t x
1 l
+
1 q
1/q
. In particular, if l′ ≤ d ≤ q, then
≼ ∥f ∥M s+β +
1/l′
∞
∥F ∥
2,d
0
l′
dt s+β˜ M ′ r ,d
.
Proof. We only prove the theorem for the solution of (4.1). By linearity and Theorem 1, we only have to show
t α ei(t −s)∆ 2 F (s)ds 0
q p
M s (Lt Lx ,ld )
≼ ∥F (t , x)∥M s+β˜ (Ll′ Lr ′ ,ld ) . t x
And this will be done if we prove
t α i(t −s)∆ 2 πk F ( s ) ds e q 0
≼ ⟨k⟩β ∥πk F (t )∥Ll′ Lr ′ . ˜
p Lt Lx
t x
Using the notation of Section 2, we rewrite the above inequality as
t ∗ V ( t ) V ( s ) F ( s ) ds k k q 0
≼ ⟨k⟩β ∥πk F (t )∥Ll′ Lr ′ . ˜
p
Lt Lx
t x
(4.3)
Note (4.3) is formally similar to (1.6). We can prove it by a similar argument as the proof of (1.6) in [9], since we still have the energy estimate
∥Vk (t )f ∥L2x ≼ ∥f ∥L2x
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
165
and the replacement of (1.7), n
∥Vk (t )g ∥L∞ ≼ (1 + t |k|α−2 )− 2 ∥g ∥L1x , x by Lemma 2.2 (or Lemma 2.3). As we previously remarked, the difference to Keel and Tao’s proof is still tracking the constants. Here we only sketch briefly how we reach the constant
β˜ = (2 − α)
1 l
+
1
q
.
First, by the compact support of πk F and Vk (t )Vk∗ (s)F , we may assume both (p, q) and (l, r ) are sharp (4.3). By duality, (3.4) implies
t ∗ V ( t ) V ( s ) F ( s ) ds k k q 0
≼ ⟨k⟩
2(2−α) q
p
∥πk F (t )∥Lq′ Lp′ t
Lt Lx
n -admissible 2
in
(4.4)
x
for all admissible pair (p, q). Following the argument in [9], one can also prove
t ∗ Vk (t )Vk (s)F (s)ds ∞
≼ ∥πk F (t )∥L∞ L2x
t ∗ Vk (t )Vk (s)F (s)ds ∞
≼ ⟨k⟩
(4.5)
t
Lt L2x
0
and 2−α q
0
∥πk F (t )∥L2 LMx
(4.6)
t
Lt L2x
where (M , 2) = n2n , 2 is the endpoint of sharp admissible line. Now (4.3) and the index β˜ comes from the interpolations −2 among (4.4)–(4.6).
Finally we turn to the wave equation
∂tt v − ∆v = F (t , x) v(0, x) = f (x), vt (0, x) = g (x).
Theorem 5. Let n ≥ 2 and 1 ≤ d ≤ ∞. If n n r′
+
1 l′
≥2+
n p
(4.7)
1 2
−
1 p
≥ 1 + 1q , (r , l) is
n −1 -admissible 2
and they satisfy the gap condition
1
+ , q
then the solution to (4.7) satisfies
∥v(t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+γ + ∥g ∥M s+γ −1 + ∥F (t , x)∥M s+γ (Ll′ Lr ′ ,ld ) , 2,d
t
+1 where γ = q(nn+−11) and γ = nn− 1
1 l
2,d
+
1 q
t x
− 1.
Proof. By Duhamel’s principle, we write the solution of (4.7) as
√ √ t sin t −∆ sin (t − s) −∆ v(t , x) = cos t −∆ f (x) + g (x) + F (s)ds. √ √ −∆ −∆ 0 √
In view of Theorem 2, we only have to show
t sin (t − s)√−∆ F (s)ds √ 0 −∆
q p
M s (Lt Lx ,ld )
≼ ∥F (t , x)∥M s+γ (Ll′ Lr ′ ,ld ) . t x
And this will be accomplished if we prove
√ t i(t −s) −∆ e F (s)ds ≼ ⟨k⟩γ ∥πk F (t )∥Ll′ Lr ′ . Jk = πk √ t x qp − ∆ 0 L L t x
Recall the notation Uj (t ) in Section 3. By Lemma 2.1, n−1 ∥ U0 (t )g ∥L∞ ≼ (1 + |t |)− 2 ∥g ∥L1x x
(4.8)
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C. Zhang / Nonlinear Analysis 78 (2013) 156–167
So we apply Theorem A and obtain
t ∗ U0 (t )U0 (s)F (s)ds q 0
p
≼ ∥F ∥Ll′ Lr ′ t x
Lt Lx
1 for all n− -admissible (p, q) and (l, r ). A simple dilation argument then yields 2
t ∗ U ( t ) U ( s ) F ( s ) ds j j q 0
j n′ + 1′ −2− np − 1q r l
≼2
∥F (t , x)∥Ll′ Lr ′ . t x
p
Lt Lx
(4.9)
Therefore, by the Littlewood–Paley theory and the gap condition, we get
t 0 ∗ J0 = Uj (t )Uj (s)F (s)ds j=−∞ 0 q
≼ ∥F (t , x)∥Ll′ Lr ′ , t x
p
Lt Lx
⃗ part of (4.8). which is just the k = 0 ⃗ similar to (4.3), we obtain When k ̸= 0, t √ i(t −s) −∆ πk F ( s ) ds e q 0
n+1
≼ ⟨k⟩ (n−1)
1 + 1q l
∥πk F (t )∥Ll′ Lr ′ . t x
p
Lt Lx
Consequently,
Jk = πk
≼ ⟨k⟩
t
0
√
1 ei(t −s) −∆ (−∆) 2 F (s) ds
n+1 (n−1)
n+1
1 + 1q l
1 + 1q l
q p
Lt Lx
1 πk (−∆) 2 F (t ) l′
′ Lt Lrx
≼ ⟨k⟩ (n−1)
−1
∥πk F (t )∥Ll′ Lr ′ . t x
Now we have finished the whole proof.
One may compare Theorems 4 and 5 to the Strichartz estimate on Sobolev space (see [9] for example). Roughly speaking, ˙ s by the modulation norm M2s ,d (see the ‘‘in particular’’ part of Theorems 1–4). we have substituted the Sobolev norm H For the Standard Schrödinger equation, the admissible area of index is the same. For the wave equation, we imposed the 1 assumption a bit stronger that n− -admissible 2
n
1 2
−
1
p
≥1+
1 q
⃗ part of the wave operators. This in fact reflexes that the modulation space to estimate the low frequency ((3.9) when k = 0) we considered here is some sort of inhomogeneous function space. Note also the ‘‘gap’’ condition in Theorem 5 is an inequality rather than an equality. In fact we can only assume the equality, then Theorem 5 still holds with γ = 0, which makes the theorem more close to previous Strichartz estimates. Note also that γ =
n+1 n−1
1 l
+
1 q
−1≤
n r′
+
1 l′
−2−
n p
−
1 q
1 with the equality holds only when both (p, q) and (r , l) are sharp n− -admissible. 2
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
J. Ginibre, G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 123 (1989) 535–573. J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995) 50–68. L. Kapitanski, Some generalizations of the Strichartz–Brenner inequality, Leningrad Math. J. 1 (1990) 693–726. H. Lindblad, C.D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995) 357–426. G. Mockenhaupt, A. Seeger, C.D. Sogge, Local smoothing of Fourier integrals and Carleson–Sjölin estimates, J. Amer. Math. Soc. 6 (1993) 65–130. I.E. Segal, Space–time decay for solutions of wave equations, Adv. Math. 22 (1976) 304–311. R.S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977) 705–714. K. Yajima, Existence of solutions for Schrodinger evolution equations, Comm. Math. Phys. 110 (1987) 415–426. M. Keel, T. Tao, End point Strichartz estimates, Amer. J. Math. 120 (1998) 955–980. H.G. Feichtinger, Modulation spaces on locally compact abelian groups. Technical Report, University of Vienna, 1983. K. Gröchening, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, MA, 2001. J. Sjöstrand, An algebra of pseudo-differential operators, Math. Res. Lett. 1 (1994) 185–192.
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J. Toft, Continuity properties for modulation spaces with applications to pseudo-differential calculus, I. J. Funct. Anal. 207 (2004) 399–429. J. Toft, Continuity properties for modulation spaces with applications to pseudo-differential calculus, II, Ann Global Anal Geom. 26 (2004) 73–106. H.G. Feichtinger, Modulation spaces: looking back and ahead, Sampl. Theory Signal Image Process. 5 (2006) 109–140. M. Kobayashi, Modulation spaces M p,q for 0 < p, q ≤ ∞, Funct. Spaces Appl. 4 (2006) 329–341. M. Kobayachi, Dual of modulation spaces, Funct. Spaces Appl. 5 (2007) 1–8. K.A. Okoudjou, Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc. 132 (2004) 1639–1647. B.X. Wang, H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations 231 (2007) 36–73. A. Bényi, K. Gröchenig, K.A. Okoudjou, L.G. Rogers, Unimodular Fourier multiplier for modulation spaces, J. Funct. Anal. 246 (2007) 366–384. B.X. Wang, L.F. Zhao, B.L. Guo, Isometric decomposition operators, function spaces Epλ,q and their applications to nonlinear evolution equations, J. Funct. Anal. 233 (2006) 1–39. J. Chen, D. Fan, Estimates for wave and Klein–Gordon equations on modulation spaces, Sci. China Math. 55 (2012) 2109–2123. A. Davey, K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. Lond. Set. A 338 (1974) 101–110. Y. Ishimori, Multi vortex solutions of a two dimensinal nonlinear wave equation, Progr. Theoret. Phys. 72 (1984) 33–37. C. Sulem, P-L. Sulem, The Nonlinear Schrödinger Equation, in: Appl. Math. Sci., vol. 139, Springer-Verlag, New York, 1999. C.E. Kenig, G. Ponce, L. Vega, Small solutions to nonlinear Schrodinger equation, Ann. Inst. H. Poincaré Sect. C 10 (1993) 255–288. C.E. Kenig, G. Ponce, L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math. 134 (1998) 489–545. B.X. Wang, L.J. Han, C.Y. Huang, Global well-Posedness and scattering for the derivative nonlinear Schröodinger equation with small rough data, Ann. Inst. Henri Poincaré Anal Non Linéaire 26 (2009) 2253–2281. K. Rogers, A. Vargas, L. Vega, Pointwise convergence of solutions to the nonelliptic Schrödinger equation, Indiana Univ. Math. J. 55 (2006) 1893–1906. C.D. Sogge, Lectures on Nonlinear Wave Equations, International Press, Cambridge, MA, 1995.
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Strichartz estimates in the frame of modulation spaces Chunjie Zhang ∗ School of Science, Hangzhou Dianzi University, Hangzhou, 310016, PR China
article
abstract
info
Article history: Received 28 December 2011 Accepted 3 October 2012 Communicated by Enzo Mitidieri
In this paper, we study the Strichartz estimates for three types of differential equations in the framework of modulation spaces. The equations considered here include the wave equation, the Schrödinger equation and the nonelliptic Schrödinger equation. Inhomogeneous equations of the above three types are also studied. © 2012 Elsevier Ltd. All rights reserved.
MSC: 35L05 46E35 42B37 Keywords: Modulation space Strichartz estimate Schrödinger equation Wave equation Inhomogeneous equation
1. Introduction The Schrödinger type dispersive equation
i∂t u + (∆)α/2 u = 0 , u(0, x) = f (x)
(1.1)
and the wave equation
∂tt v − ∆v = 0 v(0, x) = f (x),
(1.2)
vt (0, x) = g (x)
have been vastly studied by numerous mathematicians due to both their strong physical background and theoretical importance. Their solutions are formally written as uα (t , x) = e
α it ∆ 2
f ( x) =
α
Rn
eit |ξ | fˆ (ξ )ei⟨x,ξ ⟩ dξ
and
v(t , x) =
Rn
cos(t |ξ |)fˆ (ξ )ei⟨x,ξ ⟩ dξ +
sin(t |ξ |)
Rn
|ξ |
gˆ (ξ )ei⟨x,ξ ⟩ dξ .
One can easily find thousands of theorems on them, among which we are especially interested in the Strichartz estimates. Such estimate also have a long history and are often found to be useful in the study of nonlinear equations. One is referred to [1–8] for some historical work on this subject. Following is an abstract Strichartz estimate proved by Keel and Tao in [9].
∗
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0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.10.002
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
157
Definition. We call a pair (p, q) is δ -admissible, if p, q ≥ 2, (p, q, δ) ̸= (∞, 2, 1) and 1 q
+
δ p
δ
≤
2
.
(1.3)
If the equality in (1.3) holds, we call that (p, q) is sharp δ -admissible, otherwise, (p, q) is non-sharp δ -admissible. Theorem A. Let {U (t )} be a semigroup of operators satisfying ∥U (t )f ∥L2 ≤ ∥f ∥L2 and x
x
∥U (s)U (t )g ∥L∞ ≼ |t − s| ∥g ∥L1x . x
(1.4)
∥U (t )f (x)∥Lq Lpx ≼ ∥f ∥L2x , t ∗ U (t )U (s)F (s)ds q
(1.5)
∗
−δ
Then
s
p
≼ ∥F ∥Ll′ Lr ′
Lt Lx
(1.6)
t x
hold for all sharp δ -admissible pairs (p, q) and (l, r ). If (1.4) is replaced by
∥U (s)U ∗ (t )g ∥L∞ ≼ (1 + |t − s|)−δ ∥g ∥L1x , x
(1.7)
then (1.5) and (1.6) hold for all δ -admissible pairs (p, q) and (l, r ). Using the above theorem, Keel and Tao proved that, the solution to (1.1) (with α = 2) and (1.2) satisfy
∥u(t , x)∥Lq Lpx ≼ ∥f ∥L2x , t
n
∀ -admissible (p, q), 2
and
∥v(t , x)∥Lq Lpx ≼ ∥f ∥H˙ β + ∥g ∥H˙ β−1 ,
∀
t
Here, β = n
1 2
−
1 p
−
1 q
n−1 2
-admissible (p, q).
˙ β denotes time homogeneous L2 -Sobolev space. and H
In this paper we will study the Strichartz estimates on the modulation spaces, which were first introduced by Feichtinger [10] and soon found to be useful in the study of pseudo-differential operators; see [11–14]. Several equivalent characterizations of modulation spaces have been found in [15–17] among which we only present the definition using frequency uniform decomposition below. Let σ be a non-negative real function in C0∞
− 45 , 54
n
satisfying σ (ξ ) ≡ 1 for ξ ∈
2 2 n − 5 , 5 . Denoting Zn the all
integers in Rn and σk (ξ ) = σ (ξ − k), we further assume
σk (ξ ) ≡ 1,
∀ξ ∈ Rn .
k∈Zn
Now for s ∈ R, 1 ≤ d ≤ ∞ and a Banach function space X , we define the modulation space as M s (X , l d ) =
1/d
f ∈ S ′ (Rn ) : ∥f ∥M s (X ,ld )
= ⟨z ⟩sd ∥πk f ∥dX k∈Z
< +∞
where ⟨k⟩ = 1 + |k| and πk f = F −1 (σk fˆ ). When X = Lp , we denote M s (X , ld ) by Mps,d , which is the modulation space mostly
referred. Note that Mps,d is different from Lp . In fact, one can easily show Mp0,1 ⊂ Lp . For more relationship among Mps,d , Lp and Besov spaces, one is referred to [18,13,14,19]. Estimates on Mps,d for pseudo-differential operators have attracted many researchers in recent years. Here we list for example the following result from [20,21],
∥uα (t , x)∥Mps,d ≤ Ct ∥f ∥Mps,d .
(1.8)
It is interesting to see that the norm Mps,d is preserved by the time evolution of Schrödinger equation (1.1) while there is a different case for the Lp estimate; see [20]. The sharp decay of Ct in (1.8) is also studied in [22]. Our purpose of this paper is to obtain some Strichartz estimates on modulation spaces. Below are two of our main results.
158
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
Theorem 1. Let s ∈ R, 0 < α ≤ 2, α ̸= 1 and 1 ≤ d ≤ ∞. If (p, q) is 2n -admissible, then the solution to (1.1) satisfies
∥uα (t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+β , t
2,d
where β = 2−α . In particular, when q ≥ d we have q ∞
∥uα (t , x)∥qM s dt
1/q
p,d
0
≼ ∥f ∥M s+β . 2,d
Theorem 2. Let n ≥ 2, s ∈ R and 1 ≤ d ≤ ∞. If n
1 2
−
1 p
≥ 1 + 1q , then the solution to (1.2) satisfies
∥v(t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+γ + ∥g ∥M s+γ −1 , 2,d
t
2,d
where γ = q(nn+−11) . In particular, when q ≥ d we have ∞
∥v(t , x)∥qM s dt
1/q
p,d
0
≼ ∥f ∥M s+γ + ∥g ∥M s+γ −1 . 2,d
2,d
To the best of our knowledge, Strichartz estimate on modulation space for wave equation has not been reported. For the standard Schrödinger equation, Wang and Hudzik [19] obtained
∥eit ∆ f (x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M2s ,d , t
which is the α = 2 case of Theorem 1. In fact, their method can be modified to adapt our case uα (t , x) and prove the sharp admissible part of Theorem 1. We will use a different way to prove our theorem so that less regularity is required on initial data when (p, q) is non-sharp admissible. We will also study the dispersive equation
i∂t w + ψ(D)w = 0 w(0, x) = f (x),
(1.9)
±|ξl |α . Its formal solution is wα (t , x) = eit ψ(D) f (x) = eit ψ(ξ ) fˆ (ξ )ei⟨x,ξ ⟩ dξ .
where ψ(ξ ) =
n
l =1
Rn
When α = 2 and the signs in the sum are not all same, (1.9) is called the non-elliptic Schrödinger equation. One is referred to [23–25] for its physical background. The well posedness for such equations can be found for example, in [26–28]. There is also an interesting pointwise convergence theorem for this equation [29]. Here we will prove the following. Theorem 3. Let s ∈ R, 0 < α ≤ 2, α ̸= 1 and 1 ≤ d ≤ ∞. If (p, q) is 2n -admissible, then the solution to (1.9) satisfies
∥wα (t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+β , t
2,d
where β = 2−α . In particular, when q ≥ d we have q ∞
0
∥wα (t , x)∥qM s dt p,d
1/q
≼ ∥f ∥M s+β . 2,d
We point that when α = 2, the above theorem is proved in [28]. In fact in [19,28], Strichartz estimates on modulation spaces for inhomogeneous equations are also partially studied and they use those estimates to study the existence of nonlinear solutions on modulation spaces. Here in Section 4, we will present the more complete Strichartz estimates for inhomogeneous equations of all three types. These estimates are proved in a standard and similar way. So we will only sketch their proofs and point some differences.
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
159
2. Some preliminary lemmas By the frequency uniform decomposition we used in the definition of modulation spaces, we can write
uα (t , x) =
Vk (t )f (x) =
k∈Zn
α
eit |ξ | σk (ξ )fˆ (ξ )ei⟨x,ξ ⟩ dξ .
Rn
k∈Zn
We will also use the Littlewood–Paley decomposition uα (t , x) =
Uj (t )f (x) =
j∈Z
α
Rn
j∈Z
eit |ξ | Φ (2−j ξ )fˆ (ξ )ei⟨x,ξ ⟩ dξ ,
where Φ (ξ ) is a standard cutoff function supported in {ξ , 1 < |ξ | < 3}. The C0∞ functions σ (ξ ) and Φ (ξ ) may vary in different occasions during the discussion, but we will always use the notations Vk (t ) and Uj (t ). Lemma 2.1. If 0 < α ̸= 1, then
|U0 (t )(x)| =
e
it |ξ |α
Rn
Φ (ξ )e
i⟨x,ξ ⟩
n dξ ≼ (1 + |t |)− 2 .
If α = 1, we have
|U0 (t )(x)| =
eit |ξ | Φ (ξ )ei⟨x,ξ ⟩ dξ ≼ (1 + |t |)−
Rn
n−1 2
.
Proof. The two inequalities in fact follows from some standard stationary phase argument. One may find the second inequality in many papers ([30] for example). For clarity, we still present the proof of the first one here. It is enough to consider t > 0. If t ≤ 1, then since |U0 (t )(x)| ≼ C , the lemma follows. So we assume t > 1. Note by polar decomposition, U0 (t )(x) =
∞
itr α
Φ (r )r
e
n −1
0
S n−1
′ ei⟨rx,ξ ⟩ dσ (ξ ′ )dr .
When |x| > 1/4, using the asymptotic of Fourier transform on S n−1 , we have U0 (t )(x) ≃ |x|
1−n 2
∞
n−1 α ei(tr ±|x|r ) Φ (r )r 2 dr
0
Denoting ψ(r ) = tr α ± |x|r, we easily find some Cα > 1 such that
|ψ ′ (r )| = |α tr α−1 ± |x ∥≥ |x|/2 whenever |x| > Cα t and
|ψ ′ (r )| = |α tr α−1 ± |x ∥≥ t /2 whenever |x| < Cα−1 t, since 1 < r < 3 by the support of Φ . In either case, we can use integrating by parts and obtain
|U0 (t )(x)| ≤ t −N ,
∀N > 0 .
When |x| ≃ t, it is easy to see that
ψ ′′ (r ) = |α(α − 1)r α−2 t | ≥ cα t . By Van de’Coupt’s lemma, we have
|U0 (t )(x)| ≼ |x|
1−n 2
1 n t− 2 ≼ t− 2 .
When |x| ≤ 1/4, we have r |x| ≤ 1 so
S n−1
′ ei⟨rx,ξ ⟩ dσ (ξ ′ ) = O(1)
and consequently U0 (t )(x) ≃
∞
α eitr Φ (r )r n−1 dr ≼ t −N
0
by integrating by parts. Now we have finished the proof of Lemma 2.1.
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C. Zhang / Nonlinear Analysis 78 (2013) 156–167
Lemma 2.2. Let k ∈ Zn . If 0 < α ≤ 2 and α ̸= 1, then n
∥Vk (t )g ∥L∞ ≼ (1 + t |k|α−2 )− 2 ∥g ∥L1x . x If α = 1, then
n−1 +1 − 2 − nn− 1 ∥Vk (t )g ∥L∞ ≼ 1 + t | k | ∥g ∥L1x . x ⃗ then the lemma reduces to Proof. We only prove the case α ̸= 1. The other one follows a similar way. If k = 0, n
∥V0 (t )g ∥L∞ ≼ (1 + t )− 2 ∥g ∥L1x . x
(2.1)
2 If α = 2, we note that the kernel of the Schrödinger operator is t −n/2 ei|x| /t . By Young’s inequality, n
n
∥V0 (t )g ∥L∞ = ∥eit ∆ (π0 g )∥L∞ ≤ t − 2 ∥π0 g ∥L1x ≼ t − 2 ∥g ∥L1x . x x On the other hand, by the Hausdorff–Young inequality and Hölder’s inequality, α
α
∥Vk (t )g ∥L∞ = ∥(eit |ξ | σk gˆ )∨ ∥L∞ ≼ ∥eit |ξ | σk gˆ ∥L1 ≼ ∥ˆg (ξ )∥L∞ ≼ ∥g ∥L1x . x x ξ ξ
(2.2)
In particular
∥V0 (t )g ∥L∞ ≼ ∥g ∥L1x , x and (2.1) is proved. If α < 2, then by Lemma 2.1 and Young’s inequality, n
∥U0 (t )g (x)∥L∞ ≼ (1 + t )− 2 ∥g ∥L1x . x A simple dilation argument then yields n
∥Uj (t )g ∥L∞ ≼ 2jn (1 + 2jα t )− 2 ∥g ∥L1x . x
(2.3)
Therefore,
∥V0 (t )g ∥L∞ = x
0
∥Uj (t )(π0 g )∥L∞ ≼ x
0
n
2jn (1 + 2jα t )− 2 ∥π0 g ∥L1 x
j=−∞
j=−∞
≤
0
n
2jn (2jα + 2jα t )− 2 ∥g ∥L1 x
j=−∞
≤
0
n
2jn(2−α)/2 (1 + t )− 2 ∥g ∥L1 . x
j=−∞
Note the above sum is finite and (2.1) follows. ⃗ Note the support of σk is contained in finite number of annulus with radius r ≃ 2j ≃ |k|, Now we consider the case k ̸= 0. and the number of such annulus is independent of k. Therefore, we have Vk (t )g (x) =
Uj (t )(πk g )(x)
j≃log |x|
and consequently, by (2.2)
∥Vk (t )g ∥L∞ x
n ≼ Uj (t )(πk g ) ≼ 2jn (1 + 2jα t )− 2 ∥πk g ∥L1 x j ∞ j Lx n n ≤ 2jn (2jα t )− 2 ∥πk g ∥L1 ≤ (2j(α−2) t )− 2 ∥g ∥L1x x j
j
(α−2)
≃ (|k|
− 2n
t)
∥g ∥L1x .
Combining (2.3), we have proved the lemma.
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
161
To study the nonelliptic Schrödinger equation, we need the following decomposition
wα (t , x) =
Wk (t )f (x) =
k∈Zn
k∈Zn
eit
±|ξl |α
Rn
σk (ξ )fˆ (ξ )ei⟨x,ξ ⟩ dξ .
Lemma 2.3. Let k ∈ Zn , 0 < α ≤ 2 and α ̸= 1. We have n
∥Wk (t )g ∥L∞ ≼ (1 + t |k|α−2 )− 2 ∥g ∥L1x . x Proof. Similar to (2.3), we easily have
∥Wk (t )g (x)∥L∞ ≼ ∥g ∥L1x . x So by Young’s inequality, we are left to show n
∥Wk (t )(x)∥L∞ ≼ (|k|α−2 |t |)− 2 . (2.4) x 9 9 which satisfies φ(t ) ≡ 1 when t ∈ − 45 , 45 . Then for any fixed k = (k1 , k2 , . . . , kn ) ∈ Zn , Let us take φ(t ) ∈ C0∞ − 10 , 10
we have
σk (ξ ) = σk (ξ )
n
φ(ξl − kl ).
l=1
Therefore, we only have to prove (2.4) for
k (t )(x) = W
eit
±|ξl |α
Rn
n
φ(ξl − kl )ei⟨x,ξ ⟩ dξ
l =1
instead of Wk (t ). And this will be done by splitting the variables,
k (t )(x) = W
n
n
α ei(t |ξl | +xl ξl ) φ(ξl − kl )dξl =
R
l =1
kl (t )(xl ). W
l =1
By the change of variables and the Van de Corputs Lemma,
kl (t )(xl )| = |W
∞
α ei(xl ξl ±t ξl ) φ(ξl − kl )dξl +
0
∞
0 1
α ei(−xl ξl ±t ξl ) φ(−ξl − kl )dξl
1
≼ (|kl |α−2 |t |)− 2 ≼ (|k|α−2 |t |)− 2 . Therefore we have
k (t )(x)| = |W
n
n
kl (t )(xl )| ≼ (|k|α−2 |t |)− 2 . |W
l =1
3. Proof of the theorems Proof of Theorem 1. By the definition of modulation space, we only have to show
∥Vk (t )f ∥Lq Lpx ≼ ⟨k⟩
2−α q
t
∥πk f ∥L2
(3.1)
⃗ this reduces to for all 2n -admissible pairs (p, q). When k = 0, ∥V0 (t )f ∥Lq Lpx ≼ ∥π0 f ∥L2 .
(3.2)
t
By Plancherel’s theorem, V0 (t ) satisfies the energy estimate
∥V0 (t )f ∥L2x ≤ ∥f ∥L2x , and by (2.1) we also have n
∥V0 (t )V0∗ (s)g ∥L∞ = ∥V0 (t − s)g ∥L∞ ≼ (1 + |t − s|)− 2 ∥g ∥L1x . x x (3.2) then follows by applying Theorem A.
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C. Zhang / Nonlinear Analysis 78 (2013) 156–167
⃗ By Lemma 2.1, (1.7) holds with δ = n/2. So applying Theorem A we have, for any Next, we prove (3.1) for k ̸= 0. (p, q),
n -admissible 2
∥U0 (t )f (x)∥Lq Lpx ≼ ∥f ∥L2x , t
which further implies j 2n − np − αq
∥Uj (t )f (x)∥Lq Lpx ≼ 2
t
∥f ∥L2x
(3.3)
by simple dilation argument. Note the support of σk is contained in finite number of annulus with radius r ≃ 2j ≃ |k|. Therefore, by (3.3) we obtain
∥Vk (t )f ∥
= Uj (t )(πk f ) j q
q p Lt Lx
≼
p
2
∥πk f ∥L2x
j
Lt Lx
α
n
n
j 2n − np − αq
≃ ⟨k⟩ 2 − p − q ∥πk f ∥L2x . If (p, q) is sharp 2n -admissible, then n 2
n
−
−
p
α
=
q
2−α q
2 q
=
n 2
−
n p
and
,
from which (3.1) follows. If (p, q) is non-sharp 2n -admissible, we cannot use Theorem A directly but we can modify its proof. Unlike the Littlewood–Paley decomposition, we are not able to obtain (3.1) by a dilation method in the frequency uniform decomposition case. So below we will prove (3.1) for each k ∈ Zn using the bilinear method of [9], but track the power of ⟨k⟩ in each step. Since the dual of (3.1) is
∞
Vk∗ (t )F (t , ·)(x)dt
0
≼ ⟨k⟩
2−α q
L2x
∥F (t , x)∥Lq′ Lp′ , t
x
we further write it into the bilinear form ∞
I =
∞
0
2(2−α) ⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩dtds ≼ ⟨k⟩ q ∥F ∥Lq′ Lp′ ∥G∥Lq′ Lp′ . t
0
x
t
(3.4)
x
By Plancherel’s theorem α
∥Vk (t )f ∥L2x = ∥eit |ξ | σk fˆ ∥L2x ≼ ∥f ∥L2x . We take its bilinear form
|⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩| ≼ ∥F (s)∥L2x ∥G(t )∥L2x .
(3.5)
On the other hand, by Lemma 2.2, n
∥Vk (t )Vk∗ (s)g ∥L∞ = ∥Vk (t − s)g ∥L∞ ≼ (1 + |k|α−2 |t − s|)− 2 ∥g ∥L1x . x x Again we take its bilinear form n
|⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩| ≼ (1 + |k|α−2 |t − s|)− 2 ∥F (s)∥L1x ∥G(t )∥L1x .
(3.6)
Interpolating between (3.5) and (3.6) yields nθ
|⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩| ≼ (1 + |k|α−2 |t − s|)− 2 ∥F (s)∥Lp′ ∥G(t )∥Lp′ x
x
where θ = 1 − 2p . Therefore, by Hölder’s inequality and Young’s inequality, ∞
∥F (s)∥Lp′
∞
x ∥G(t )∥Lp′ dsdt nθ x (1 + |k|α−2 |t − s|) 2 ∞ ∥F (s)∥Lp′ x ≼ ∥G(t )∥Lq′ Lp′ ds nθ α− 2 t x 0 (1 + |k| |t − s|) 2 Lq t α−2 − n2θ ≼ ∥G(t )∥Lq′ Lp′ ∥F (t )∥Lq′ Lp′ (1 + |k| |t |) r
I ≼
0
0
t
x
t
x
Lt
(3.7)
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
where r =
q . 2
Since (p, q) is now assumed to be non-sharp 2n -admissible, it is easy to see check that
163 nr θ 2
> 1, thus
nθ
(1 + |k|α−2 |t |)− 2 ∈ Lrt . By easy calculation, the Lrt norm is bounded by |k|
2(2−α) q
. Now we have proved (3.4), hence the whole theorem.
Remark. In the above proof, we applied Lemma 2.1 and Theorem A to get the sharp admissible part of Theorem 1, while for the non-sharp admissible part we only used Lemma 2.2. In fact, the sharp admissible part can also be obtained using Lemma 2.2 only. To see this, we note by Lemma 2.2, n
∥Vk (t )g (x)∥L∞ ≼ (|k|α−2 t )− 2 ∥g ∥L1x . x Invoking the argument in obtaining (3.7), we similarly have nθ
|⟨Vk∗ (s)F (s), Vk∗ (t )G(t )⟩| ≼ (|k|α−2 |t − s|)− 2 ∥F (s)∥Lp′ ∥G(t )∥Lp′ . x
x
The term
I =
∞ 0
∞
⟨Vk (s)F (s), Vk (t )G(t )⟩dtds ∗
0
∗
is then estimated using the Hardy–Sobolev inequality; see page 960 of [9]. But using the Hardy–Sobolev inequality, one still misses the sharp 2n -admissible endpoint (p, q) = n2n , 2 . Keel and Tao spent much effort to fix this point, and a −2 single ‘‘by similar argument’’ is not always convincing, especially when the ‘‘argument’’ is long and complicated. So in the proof of Theorem 1, we chose to treat the sharp admissible line (which contains the endpoint) differently. Nevertheless, their argument does adapt to the endpoint here. We may carefully follow their proof, with some difference of tracking the constants in each step; see also the explanation in [19]. Proof of Theorem 2. By the expression of the wave solution v(t , x), we consider two operators
v1 (t )f (x) =
v2 (t )g (x) =
Rn
Rn
eit |ξ | fˆ (ξ )ei⟨x,ξ ⟩ dξ ,
|ξ |−1 eit |ξ | gˆ (ξ )ei⟨x,ξ ⟩ dξ .
Theorem 2 will be proved if we obtain
∥v1 (t )f ∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+γ , 2,d
t
∥v2 (t )f ∥M s (Lq Lpx ,ld ) ≼ ∥g ∥M s+γ −1 . t
2,d
Thus by the definition of modulation space, we only have to show n+1
∥πk v1 (t )f ∥Lq Lpx ≼ ⟨k⟩ q(n−1) ∥πk f ∥L2x
(3.8)
t
∥πk v2 (t )g ∥Lq Lpx ≼ ⟨k⟩
n+1 −1 q(n−1)
t
∥πk g ∥L2x
(3.9)
for any k ∈ Zn . 1 ⃗ -admissible requirement on (p, q). For (3.9), if k ̸= 0, The proof of (3.8) is almost the same as (3.1), but with a different n− 2 then
∥πk v2 (t )g ∥Lq Lpx = ∥πk v1 (t )(|ξ |−1 gˆ (ξ ))∨ ∥Lq Lpx t
t
≼ ⟨k⟩γ ∥πk ((|ξ |−1 gˆ (ξ ))∨ )∥L2x ≼ ⟨k⟩γ −1 ∥πk g ∥L2x , ⃗ we define where we have applied (3.8) in the second inequality. When k = 0, Uj (t )f (x) =
Rn
|ξ |−1 eit |ξ | Φ (2−j ξ )fˆ (ξ )ei⟨x,ξ ⟩ dξ .
Similar to (3.3), we obtain j ∥ Uj (t )f ∥Lq Lp ≼ 2 x t
n − np − 1q −1 2
∥f ∥L2x
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C. Zhang / Nonlinear Analysis 78 (2013) 156–167
by applying Theorem A and the α = 1 part of Lemma 2.1. Using this inequality and the Littlewood–Paley theory, also in view of the assumption n 12 − 1p ≥ 1 + 1q , we have
0 = Uj (t )(π0 g ) j=−∞
∥π0 v2 (t )g ∥Lq Lpx t
Noting that n
1 2
−
1 p
≥1+
≼ ∥π0 g ∥L2x . q p
Łt Lx
1 implies (p, q) n− -admissible, we have proved Theorem 2. 2
1 q
The proof of Theorem 3 follows from the Remark and Lemma 2.3. 4. Estimates for inhomogeneous equations The solution to inhomogeneous Schrödinger type equation
i∂t u + (∆)α/2 u = F (t , x) u(0, x) = f (x)
(4.1)
is written as α
u(t , x) = eit ∆ f (x) − i 2
α
t
ei(t −s)∆ F (s, ·)(x)ds. 2
0
For the nonelliptic equation
i∂t u + ψ(D)u = 0 , u(0, x) = f (x)
(4.2)
the formal solution is u(t , x) = eit ψ(D) f (x) − i
t
ei(t −s)ψ(D) F (s, ·)(x)ds.
0
Theorem 4. Let s ∈ R, 1 ≤ d ≤ ∞ and 0 < α ≤ 2 with α ̸= 1. If (p, q) and (r , l) are both solutions to (4.1) and (4.2) satisfy
n -admissible 2
pairs, then the
∥u(t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+β + ∥F (t , x)∥M s+β˜ (Ll′ Lr ′ ,ld ) , t
2,d
where β = 2−α and β˜ = (2 − α) q ∞
∥ u∥ 0
q dt Mps ,d
t x
1 l
+
1 q
1/q
. In particular, if l′ ≤ d ≤ q, then
≼ ∥f ∥M s+β +
1/l′
∞
∥F ∥
2,d
0
l′
dt s+β˜ M ′ r ,d
.
Proof. We only prove the theorem for the solution of (4.1). By linearity and Theorem 1, we only have to show
t α ei(t −s)∆ 2 F (s)ds 0
q p
M s (Lt Lx ,ld )
≼ ∥F (t , x)∥M s+β˜ (Ll′ Lr ′ ,ld ) . t x
And this will be done if we prove
t α i(t −s)∆ 2 πk F ( s ) ds e q 0
≼ ⟨k⟩β ∥πk F (t )∥Ll′ Lr ′ . ˜
p Lt Lx
t x
Using the notation of Section 2, we rewrite the above inequality as
t ∗ V ( t ) V ( s ) F ( s ) ds k k q 0
≼ ⟨k⟩β ∥πk F (t )∥Ll′ Lr ′ . ˜
p
Lt Lx
t x
(4.3)
Note (4.3) is formally similar to (1.6). We can prove it by a similar argument as the proof of (1.6) in [9], since we still have the energy estimate
∥Vk (t )f ∥L2x ≼ ∥f ∥L2x
C. Zhang / Nonlinear Analysis 78 (2013) 156–167
165
and the replacement of (1.7), n
∥Vk (t )g ∥L∞ ≼ (1 + t |k|α−2 )− 2 ∥g ∥L1x , x by Lemma 2.2 (or Lemma 2.3). As we previously remarked, the difference to Keel and Tao’s proof is still tracking the constants. Here we only sketch briefly how we reach the constant
β˜ = (2 − α)
1 l
+
1
q
.
First, by the compact support of πk F and Vk (t )Vk∗ (s)F , we may assume both (p, q) and (l, r ) are sharp (4.3). By duality, (3.4) implies
t ∗ V ( t ) V ( s ) F ( s ) ds k k q 0
≼ ⟨k⟩
2(2−α) q
p
∥πk F (t )∥Lq′ Lp′ t
Lt Lx
n -admissible 2
in
(4.4)
x
for all admissible pair (p, q). Following the argument in [9], one can also prove
t ∗ Vk (t )Vk (s)F (s)ds ∞
≼ ∥πk F (t )∥L∞ L2x
t ∗ Vk (t )Vk (s)F (s)ds ∞
≼ ⟨k⟩
(4.5)
t
Lt L2x
0
and 2−α q
0
∥πk F (t )∥L2 LMx
(4.6)
t
Lt L2x
where (M , 2) = n2n , 2 is the endpoint of sharp admissible line. Now (4.3) and the index β˜ comes from the interpolations −2 among (4.4)–(4.6).
Finally we turn to the wave equation
∂tt v − ∆v = F (t , x) v(0, x) = f (x), vt (0, x) = g (x).
Theorem 5. Let n ≥ 2 and 1 ≤ d ≤ ∞. If n n r′
+
1 l′
≥2+
n p
(4.7)
1 2
−
1 p
≥ 1 + 1q , (r , l) is
n −1 -admissible 2
and they satisfy the gap condition
1
+ , q
then the solution to (4.7) satisfies
∥v(t , x)∥M s (Lq Lpx ,ld ) ≼ ∥f ∥M s+γ + ∥g ∥M s+γ −1 + ∥F (t , x)∥M s+γ (Ll′ Lr ′ ,ld ) , 2,d
t
+1 where γ = q(nn+−11) and γ = nn− 1
1 l
2,d
+
1 q
t x
− 1.
Proof. By Duhamel’s principle, we write the solution of (4.7) as
√ √ t sin t −∆ sin (t − s) −∆ v(t , x) = cos t −∆ f (x) + g (x) + F (s)ds. √ √ −∆ −∆ 0 √
In view of Theorem 2, we only have to show
t sin (t − s)√−∆ F (s)ds √ 0 −∆
q p
M s (Lt Lx ,ld )
≼ ∥F (t , x)∥M s+γ (Ll′ Lr ′ ,ld ) . t x
And this will be accomplished if we prove
√ t i(t −s) −∆ e F (s)ds ≼ ⟨k⟩γ ∥πk F (t )∥Ll′ Lr ′ . Jk = πk √ t x qp − ∆ 0 L L t x
Recall the notation Uj (t ) in Section 3. By Lemma 2.1, n−1 ∥ U0 (t )g ∥L∞ ≼ (1 + |t |)− 2 ∥g ∥L1x x
(4.8)
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C. Zhang / Nonlinear Analysis 78 (2013) 156–167
So we apply Theorem A and obtain
t ∗ U0 (t )U0 (s)F (s)ds q 0
p
≼ ∥F ∥Ll′ Lr ′ t x
Lt Lx
1 for all n− -admissible (p, q) and (l, r ). A simple dilation argument then yields 2
t ∗ U ( t ) U ( s ) F ( s ) ds j j q 0
j n′ + 1′ −2− np − 1q r l
≼2
∥F (t , x)∥Ll′ Lr ′ . t x
p
Lt Lx
(4.9)
Therefore, by the Littlewood–Paley theory and the gap condition, we get
t 0 ∗ J0 = Uj (t )Uj (s)F (s)ds j=−∞ 0 q
≼ ∥F (t , x)∥Ll′ Lr ′ , t x
p
Lt Lx
⃗ part of (4.8). which is just the k = 0 ⃗ similar to (4.3), we obtain When k ̸= 0, t √ i(t −s) −∆ πk F ( s ) ds e q 0
n+1
≼ ⟨k⟩ (n−1)
1 + 1q l
∥πk F (t )∥Ll′ Lr ′ . t x
p
Lt Lx
Consequently,
Jk = πk
≼ ⟨k⟩
t
0
√
1 ei(t −s) −∆ (−∆) 2 F (s) ds
n+1 (n−1)
n+1
1 + 1q l
1 + 1q l
q p
Lt Lx
1 πk (−∆) 2 F (t ) l′
′ Lt Lrx
≼ ⟨k⟩ (n−1)
−1
∥πk F (t )∥Ll′ Lr ′ . t x
Now we have finished the whole proof.
One may compare Theorems 4 and 5 to the Strichartz estimate on Sobolev space (see [9] for example). Roughly speaking, ˙ s by the modulation norm M2s ,d (see the ‘‘in particular’’ part of Theorems 1–4). we have substituted the Sobolev norm H For the Standard Schrödinger equation, the admissible area of index is the same. For the wave equation, we imposed the 1 assumption a bit stronger that n− -admissible 2
n
1 2
−
1
p
≥1+
1 q
⃗ part of the wave operators. This in fact reflexes that the modulation space to estimate the low frequency ((3.9) when k = 0) we considered here is some sort of inhomogeneous function space. Note also the ‘‘gap’’ condition in Theorem 5 is an inequality rather than an equality. In fact we can only assume the equality, then Theorem 5 still holds with γ = 0, which makes the theorem more close to previous Strichartz estimates. Note also that γ =
n+1 n−1
1 l
+
1 q
−1≤
n r′
+
1 l′
−2−
n p
−
1 q
1 with the equality holds only when both (p, q) and (r , l) are sharp n− -admissible. 2
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