CUBO A Mathematical Journal
Vol.11, No¯ 05, (57–70). December 2009

Dispersive Estimates for the Schrödinger Equation with

Potentials of Critical Regularity

Fernando Cardoso, Claudio Cuevas

Universidade Federal de Pernambuco, Departamento de Matemática,

CEP. 50540-740 Recife-Pe, Brazil.

emails: fernando@dmat.ufpe.br,

cch@dmat.ufpe.br

and

Georgi Vodev

Département de Mathématiques, UMR 6629 du CNRS,

Université de Nantes, 2 rue de la Houssinière, BP 92208,

FR-44322 Nantes Cedex 03, France

email : vodev@math.univ-nantes.fr

ABSTRACT

We prove L
1
→ L

∞
dispersive estimates with a logarithmic loss of derivatives for the Schrödinger

group e
it(−∆+V )

for a class of real-valued potentials V ∈ C
(n−3)/2(Rn), V (x) = O(〈x〉−δ),

where n = 4, 5, δ > 3 if n = 4 and δ > 5 if n = 5.

RESUMEN

Probamos L
1
→ L

∞
estimativas dispersivas con una perdida logaritmica de derivadas para el

grupo de Schrödinger eit(−∆+V ) para una clase de potenciales a valores reales V ∈ C(n−3)/2(Rn),
V (x) = O(〈x〉−δ), donde n = 4, 5, δ > 3 si n = 4 y δ > 5 si n = 5.

Key words and phrases: Schrodinger equation, dispersive estimates.

Math. Subj. Class.: 35L15, 35B40, 47F05.



58 Fernando Cardoso, Claudio Cuevas and Georgi Vodev CUBO
11, 5 (2009)

1 Introduction and Statement of Results

The present paper is a continuation of our previous one [2] where L1 → L∞ dispersive estimates

without loss of derivatives for the Schrödinger group eit(−∆+V ) have been proved for potentials

V ∈ Ck(Rn), V (x) = O(〈x〉−δ ), where n = 4, 5, k > (n − 3)/2, δ > 3 if n = 4 and δ > 5 if n = 5.

To be more precise, denote by G0 and G the self-adjoint realizations of the operators −∆ and

−∆ + V on L2(Rn), n ≥ 4, where V ∈ L∞(Rn) is a real-valued potential satisfying

|V (x)| ≤ C〈x〉−δ, ∀x ∈ Rn, (1.1)

with constants C > 0, δ > (n + 2)/2. In fact, we are interested in finding the biggest possible class

of real-valued potentials for which the perturbed Schrödinger group satisfies the following analogue

of the well-known dispersive estimate for the free one:

∥∥eitGPac
∥∥

L1→L∞
≤ C|t|−n/2, t 6= 0, (1.2)

where Pac denotes the spectral projection onto the absolutely continuous spectrum of G. There

have been many works studying this problem. In general, the proof of (1.2) goes in studying

separately and in a different maner three regions of frequencies - (1) low ones belonging to an

interval [0,ε], 0 < ε ≪ 1, (assuming additionally that zero is neither an eigenvalue nor a resonance),

(2) intermediate ones in [ε,ε−1], and (3) high frequencies in [ε−1, +∞). It became clear that in

dimensions one, two and three no regularity of the potential is needed to prove (1.2). In higher

dimensions the same conclusion remains true as far as the frequencies from the first two regions

are concerned, but it is no longer true at high frequencies. In fact, from purely mathematical point

of view the problem of proving (1.2) turns out to be interesting and difficult at high frequencies,

only. That is why, in the present paper we will be only interested in the following high frequency

analogue of (1.2): ∥∥eitGχa(G)
∥∥

L1→L∞
≤ C|t|−n/2, t 6= 0, (1.3)

where χa ∈ C
∞

((−∞, +∞)), χa(λ) = 0 for λ ≤ a, χa(λ) = 1 for λ ≥ a + 1, a ≫ 1. Note that

when n = 1 the estimate (1.3) is proved by Goldberg and Shlag [4] for potentials V ∈ L1, while in

dimension n = 2 it is proved by Moulin [7] for potentials satisfying

sup
y∈R2

∫

R2

|V (x)|

|x − y|1/2
dx < +∞.

When n = 3 Rodnianski and Shlag [10] proved (1.3) for small potentials belonging to a subclass

of Kato class with Kato norm satisfying

sup
y∈R3

∫

R3

|V (x)|

|x − y|
dx < 4π,

while for large potentials Goldberg [3] proved (1.3) for V ∈ L3/2−ǫ ∩ L3/2+ǫ, 0 < ǫ ≪ 1. The

situation, however, changes drastically when n ≥ 4. Indeed, in this case Goldberg and Visan [5]



CUBO
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Dispersive Estimates for the Schrödinger Equation ... 59

showed the existence of potentials V ∈ Ck0 (R
n
), ∀k < (n − 3)/2, for which (1.3) fails to hold. On

the other hand, Journé, Soffer and Sogge [6] proved (1.3) for potentials satisfying (1.1) with δ > n

as well as the following regularity condition

V̂ ∈ L1. (1.4)

Note that (1.3) has been recently proved by Moulin and Vodev [8] for potentials satisfying (1.1)

with δ > n − 1 and (1.4). Without any regularity conditions on the potential Vodev [12] proved

dispersive estimates with a loss of (n − 3)/2 derivatives. More precisely, it was shown in [12] that

under the condition (1.1) only, we have the estimates

∥∥eitGχa(G)f
∥∥

L∞
≤ C|t|−n/2

∥∥∥〈G〉(n−3)/4 f
∥∥∥

L1
, (1.5)

∥∥eitGχa(G)f
∥∥

L∞
≤ C|t|−n/2

∥∥∥〈x〉n/2+ǫ f
∥∥∥

L2
, (1.6)

for every 0 < ǫ ≪ 1. So, the natural question which arises when n ≥ 4 is that one of finding the

smallest possible regularity of the potential in order to have (1.3). In other words, is it possible

to replace the condition (1.4) by another one requiring less regularity on the potential? In view of

the counterexample of [5] mentioned above, when n ≥ 4 it is quite natural to make the following

Conjecture 1. The dispersive estimate (1.3) holds true for all potentials V ∈ C
(n−3)/2
0 (R

n
).

To our best knowledge, this is still an open problem. The following weaker statement, how-

ever, is more likely to be valid.

Conjecture 2. The dispersive estimate (1.3) holds true for all potentials V ∈ Ck0 (R
n
), where

k > (n − 3)/2.

Indeed, when n = 4 or n = 5 Conjecture 2 follows from the recent results of [2]. However,

it is still open when n ≥ 6. In fact, in [2] more general potentials are treated not necessarily

compactly supported. To describe this in more detials, introduce the spaces Ckδ (R
n
) and Vkδ (R

n
)

of all functions V ∈ Ck(Rn) satisfying

‖V ‖
Ck

δ
:= sup

x∈Rn

∑

0≤|α|≤k0

〈x〉δ |∂αx V (x)|

+ν sup
x∈Rn

∑

|β|=k0

〈x〉δ sup
x′∈Rn:|x−x′|≤1

∣∣∂βx V (x) − ∂βx V (x′)
∣∣

|x − x′|ν
< +∞,

‖V ‖
Vk

δ
:= sup

x∈Rn

∑

0≤|α|≤k0

〈x〉δ+|α| |∂αx V (x)|



60 Fernando Cardoso, Claudio Cuevas and Georgi Vodev CUBO
11, 5 (2009)

+ν sup
x∈Rn

∑

|β|=k0

〈x〉δ+k0+ν sup
x′∈Rn:|x−x′|≤1

∣∣∂βx V (x) − ∂βx V (x′)
∣∣

|x − x′|ν
< +∞,

where k0 ≥ 0 is an integer and ν = k − k0 satisfies 0 ≤ ν < 1. In [2] we have proved the following

Theorem 1.1. Let n = 4 or n = 5 and let V ∈ Ckδ (R
n
) with k > (n − 3)/2, δ > 3 if n = 4 and

δ > 5 if n = 5. Then, the dispersive estimate (1.3) holds true.

It is not clear, however, if this still holds with k = (n− 3)/2. Using the results of [2] we prove

in the present paper the following

Theorem 1.2. Let n = 4 or n = 5 and let V ∈ C
(n−3)/2
δ (R

n
), δ > 3 if n = 4 and δ > 5 if n = 5.

Then, we have the dispersive estimate

∥∥eitGχa(G)f
∥∥

L∞
≤ Cǫ|t|

−n/2
∥∥∥
(
log

(
2 + G2

))2+ǫ
f
∥∥∥

L1
, (1.7)

for every 0 < ǫ ≪ 1. Moreover, for every 2 ≤ p < +∞ we have the optimal dispersive estimate

∥∥eitGχa(G)
∥∥

Lp
′
→Lp

≤ C|t|−n(1/2−1/p), (1.8)

where 1/p + 1/p′ = 1.

Note that it is not clear if (1.7) and (1.8) hold true when n ≥ 6. To prove the dispersive

estimates above one needs to bound the quantity

A(t,h) = |t|n/2
∥∥eitGψ(h2G)

∥∥
L1→L∞

uniformly in both t and h, where ψ ∈ C∞0 ((0, +∞)) and 0 < h ≪ 1 is a semi-classical parameter.

It was shown in [12] that, under the assumption (1.1) only, we have the bound

A(t,h) ≤ Ch−(n−3)/2 (1.9)

in all dimensions n ≥ 4, where C > 0 is a constant independent of t and h. On the other hand, if

we suppose (1.1) fulfilled with δ > n − 1 as well as (1.4), then we have the optimal bound (see [8])

A(t,h) ≤ C. (1.10)

Note that (1.10) still holds under the assumptions of Theorem 1.1 (see [2]). To prove the estimates

(1.7) and (1.8) we show in the present paper that, under the assumptions of Theorem 1.2, we have

the bound

A(t,h) ≤ C log
1

h
. (1.11)

It is an open problem, however, to show that (1.11) still holds for potentials V ∈ C
(n−3)/2
0 (R

n
)

when n ≥ 6. Indeed, the strategy of proving (1.11) proposed in [1] leads to the study of a finite

number (∼ n/2) of operators, Tj (t,h), t > 0, j = 0, 1, ..., with explicit kernels defined as follows

Tj (t,h) = i

∫ t

0

ei(t−τ )G0ψ1(h
2G0)V Tj−1(τ,h)dτ, T0(t,h) = e

itG0ψ(h2G0),



CUBO
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Dispersive Estimates for the Schrödinger Equation ... 61

where ψ1 ∈ C
∞
0 ((0, +∞)), ψ1 = 1 on supp ψ. Roughly speaking, one needs to show that if

V ∈ C
(n−3)/2
0 (R

n
), then each Tj satisfies the bound

‖Tj (t,h)‖L1→L∞ ≤ Cjt
−n/2

log
1

h
, j ≥ 1. (1.12)

In the present paper we prove (1.12) with j = 1 in all dimensions n ≥ 4 (actually for a larger

class of potentials - see Section 3). However, this is hard to show for j ≥ 2. In fact, under the

assumption (1.1) only, we have the bounds (see [1])

‖Tj (t,h)‖L1→L∞ ≤ Cjt
−n/2hj−n/2, j ≥ 1. (1.13)

On the other hand, without any regularity assumption on V , the kernel of Tj behaves like Cjt
−n/2h−j(n−3)/2.

These observations show that if one wants to prove (1.12) (for j ≥ 2) it suffices to do it for j < n/2

only, and secondly one should better avoid using the kernels of Tj for this purpose, unless one

menages to show that some regularity on the potential improves the behaviour in h of the kernels

(which is far from being clear when j ≥ 2). Note that (1.11) follows from (1.12) with j = 1 and

the following theorem proved in [2].

Theorem 1.3. If n = 4 we suppose V ∈ Cεδ (R
4
) with ε > 0, δ > 3, while if n = 5 we suppose

V ∈ C1δ (R
5
) with δ > 5. Then, there exist constants C,ε0 > 0 so that we have the estimate

∥∥∥∥∥∥
eitGψ(h2G) −

1∑

j=0

Tj (t,h)

∥∥∥∥∥∥
L1→L∞

≤ Chε0t−n/2. (1.14)

Note again that it is a difficult open problem to show that (1.14) holds true for potentials

V ∈ C
(n−3)/2
0 (R

n
) when n ≥ 6. However, (1.14) holds in all dimensions n ≥ 4 for potentials

satisfying (1.1) with δ > n− 1 as well as (1.4) (see Appendix B of [8]). It is shown in [12], [1] that,

under the assumption (1.1) only, we have the bound (1.14) with h−(n−4)/2 in place of hε0 in the

right-hand side.

2 Proof of Theorem 1.2

In this section we will show that Theorem 1.2 follows from the estimate (1.11). To this end we will

use the identity

χa(σ) =

∫ 1

0

ψ(θσ)
dθ

θ
,

where ψ(σ) = σχ′a(σ) ∈ C
∞
0 ((0, +∞)). So we can write

χa(σ)
(
log

(
2 + σ2

))−2−ǫ
=

∫ 1

0

ψθ(θσ)
dθ

θ (log (4θ−2))
2+ǫ

,



62 Fernando Cardoso, Claudio Cuevas and Georgi Vodev CUBO
11, 5 (2009)

where

ψθ(σ) = ψ(σ)

(
1 −

log(θ2/2 + σ2/4)

log(θ2/4)

)−2−ǫ

belongs to C∞0 ((0, +∞)) uniformly in θ and having a support independent of θ. Therefore, we

have

eitGχa(G)
(
log

(
2 + G2

))−2−ǫ
=

∫ 1

0

eitGψθ(θG)
dθ

θ (log (4θ−2))
2+ǫ

.

Hence, by (1.11) we get

∥∥∥eitGχa(G)
(
log

(
2 + G2

))−2−ǫ∥∥∥
L1→L∞

≤

∫ 1

0

∥∥eitGψθ(θG)
∥∥

L1→L∞

dθ

θ (log (4θ−2))
2+ǫ

≤ C|t|−n/2
∫ 1

0

dθ

θ (log (2θ−1))
1+ǫ

≤ C|t|−n/2
∫ 1

0

−d log
(
2θ−1

)

(log (2θ−1))
1+ǫ

≤ C|t|−n/2,

which proves (1.7). To prove (1.8) we will use the following estimate proved in [12] (see Theorem

3.1) ∥∥eitGψ(θG) − eitG0ψ(θG0)
∥∥

L2→L2
≤ Ch. (2.1)

On the other hand, by (1.11) we have

∥∥eitGψ(θG) − eitG0ψ(θG0)
∥∥

L1→L∞
≤ Cǫh

−ǫ|t|−n/2, (2.2)

for every 0 < ǫ ≪ 1. An interpolation between (2.1) and (2.2) leads to the estimate

∥∥eitGψ(θG) − eitG0ψ(θG0)
∥∥

Lp
′
→Lp

≤ Cǫh
1−(1+ǫ)(1−2/p)|t|−n(1/2−1/p), (2.3)

for every 2 ≤ p ≤ +∞, where 1/p + 1/p′ = 1. Let 2 ≤ p < +∞. By (2.3) we get

∥∥eitGχa(G) − eitG0χa(G0)
∥∥

Lp
′
→Lp

≤

∫ 1

0

∥∥eitGψ(θG) − eitG0ψ(θG0)
∥∥

Lp
′
→Lp

dθ

θ

≤ C|t|−n(1/2−1/p)
∫ 1

0

θ−1+1/p−ǫ(1/2−1/p)dθ

≤ C|t|−n(1/2−1/p)
∫ 1

0

θ−1+1/(2p)dθ ≤ C|t|−n(1/2−1/p), (2.4)

provided ǫ > 0 is taken small enough. Clearly, (1.8) follows from (2.4).

3 Study of the Operator T1 in all Dimensions n ≥ 4

Let γ = t/2 if 0 < t ≤ 2, γ = 1 if t ≥ 2, and decompose the operator T1 as follows

T1 =

(∫ γ

0

+

∫ t

t−γ

)
... +

∫ t−γ

γ

... := T
(1)
1 + T

(2)
1 .

In this section we will prove the following



CUBO
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Dispersive Estimates for the Schrödinger Equation ... 63

Proposition 3.1. Let n ≥ 4 and let V ∈ V
(n−3)/2−k
δ (R

n
), where 0 ≤ k < (n− 3)/2 and δ > 2 + k.

Then, we have the estimate

∥∥∥T (1)1 (t,h)
∥∥∥

L1→L∞
≤ Ct−n/2h−k log

1

h
. (3.1)

Moreover, if V ∈ Vmδ (R
n
) with an integer 0 ≤ m < (n − 3)/2, and if k ≥ 0 is such that n − 1 −

2m − δ < k < (n − 3)/2 − m, then we have
∥∥∥T (2)1 (t,h)

∥∥∥
L1→L∞

≤ Ct−n/2h−k, t ≥ 2. (3.2)

Remark. It is proved in [12] that if V satisfies (1.1) with δ > (n + 1)/2, then we have

‖T1(t,h)‖L1→L∞ ≤ Ct
−n/2h−(n−3)/2.

Note also that (3.2) with m = k = 0 is proved in [8] (see Appendix B), and this is enough for the

proof of Theorem 1.2.

Proof. We will make use of the fact that the kernel of the operator eitG0ψ(h2G0) is of the

form Kh(|x − y|, t), where

Kh(σ,t) =
σ−2ν

(2π)ν+1

∫ ∞

0

eitλ
2

ψ(h2λ2)Jν (σλ)λdλ = h
−nK1(σ/h,t/h

2
), (3.3)

where Jν (z) = z
νJν (z), Jν (z) = (H

+
ν (z) + H

−
ν (z)) /2 is the Bessel function of order ν = (n− 2)/2.

Thus, the kernel of T1 is of the form

T (x,y,t,h) =

∫ t

0

∫

Rn

K̃h(|x − ξ|, t − τ)Kh(|y − ξ|,τ)V (ξ)dξdτ,

where K̃h denotes the kernel of the operator e
itG0ψ1(h

2G0). It is shown in [12] (see Proposition

2.1) that the function Kh satisfies the bound

|Kh(σ,t)| ≤ C(hσ)
s−(n−1)/2t−s−1/2, ∀t,σ > 0, 0 < h ≤ 1, 0 ≤ s ≤ (n − 1)/2. (3.4)

Set

ah(σ,t) = t
n/2eiσ

2/4tKh(σ,t) = a1(σ/h,t/h
2
). (3.5)

Clearly, (3.4) can be rewritten as

|ah(σ,t)| ≤ C

(
t

hσ

)s
, ∀t,σ > 0, 0 < h ≤ 1, 0 ≤ s ≤ (n − 1)/2. (3.6)

Denote a′h = dah/dt. We need the following

Lemma 3.2. For every t,σ > 0, 0 < h ≤ 1, 0 ≤ s ≤ (n − 1)/2 and every integer k ≥ 0 such that

k + s ≤ n/2, we have the bound

∣∣∂kσah(σ,t)
∣∣ ≤ C

(
1

σ

)k (
t

hσ

)s
. (3.7)



64 Fernando Cardoso, Claudio Cuevas and Georgi Vodev CUBO
11, 5 (2009)

Moreover, if k + s ≤ (n − 2)/2, we have

∣∣∂kσa
′
h(σ,t)

∣∣ ≤ Ct−1
(

1

σ

)k (
t

hσ

)s
. (3.8)

Proof. In view of (3.5), it suffices to prove (3.7) and (3.8) with h = 1. Consider first the case

0 < σ ≤ 1. We will use that Jν (z) = z
2νgν(z) with a function gν(z) analytic at z = 0. Hence,

given any integer k ≥ 0 we have

∂kσK1(σ,t) =

∫
∞

0

eitλ
2

ψk(λ)g
(k)
ν (σλ)dλ,

where ψk ∈ C
∞
0 ((0, +∞)) and g

(k)
ν (z) = d

kgν (z)/dz
k. Let t ≥ 1. Then, in the same way as in the

proof of Proposition 2.1 of [12] (see (2.10)) we have

∣∣∂kσK1(σ,t)
∣∣ ≤ Ck,mt−m−1/2, (3.9)

for every integer m ≥ 0, and hence for all real m ≥ 0. Using (3.9) we get

∣∣∂kσa1(σ,t)
∣∣ ≤ Ctn/2

k∑

j=0

∣∣∣∂jσ
(
eiσ

2/4t
)∣∣∣

∣∣∂k−jσ K1(σ,t)
∣∣

≤ Ct(n−1)/2−m
k∑

j=0

∣∣∣∂jσ
(
eiσ

2/4t
)∣∣∣ ≤ Ct(n−1)/2−m,

which proves (3.7) (with h = 1) in this case. Let 0 < t ≤ 1. Then we have

∣∣∂kσK1(σ,t)
∣∣ ≤ Ck. (3.10)

Using (3.10) we get

∣∣∂kσa1(σ,t)
∣∣ ≤ Ctn/2

k∑

j=0

∣∣∣∂jσ
(
eiσ

2/4t
)∣∣∣

∣∣∂k−jσ K1(σ,t)
∣∣

≤ Ctn/2
k∑

j=0

∣∣∣∂jσ
(
eiσ

2/4t
)∣∣∣ ≤ Ctn/2−k ≤ Cts.

Consider now the case σ ≥ 1. We will use that Jν (z) = e
izb+ν (z) + e

−izb−ν (z) with functions b
±
ν

satisfying ∣∣∂jzb
±
ν (z)

∣∣ ≤ Cjz(n−3)/2−j, ∀z ≥ z0, (3.11)

for every integer j ≥ 0 and every z0 > with a constant Cj > 0 depending on j and z0. We can

write K1 = K
+
1 + K

−

1 with K
±

1 defined by replacing in the definition of K1 the function Jν (z) by

e±izb±ν (z). Then the functions

a±1 (σ,t) = t
n/2eiσ

2/4tK±1 (σ,t)



CUBO
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Dispersive Estimates for the Schrödinger Equation ... 65

can be written in the form

a±1 (σ,t) = t
n/2

∫
∞

0

eit(λ±σ/2t)
2

b̃±ν (σλ)ϕ(λ)dλ,

where ϕ(λ) = (2π)−ν−1λ1+2νψ(λ2), b̃±ν (z) = z
−2νb±ν (z). Hence

∂kσa
±

1 (σ,t) = t
n/2

∫
∞

0

k∑

j=0

cj∂
j
σ

(
eit(λ±σ/2t)

2
)
∂k−jσ b̃

±
ν (σλ)ϕ(λ)dλ.

Using the identity

∂jσ

(
eit(λ±σ/2t)

2
)

= (∓2t)−j∂
j
λ

(
eit(λ±σ/2t)

2
)

and integrating by parts, we get

∂kσa
±

1 (σ,t) =

k∑

j=0

tn/2−jeiσ
2/4t

∫ ∞

0

eitλ
2
±iσλϕ̃(λ)B±ν,j (λ,σ)dλ,

where ϕ̃ ∈ C∞0 ((0, +∞)), ϕ̃ = 1 on supp ϕ, and

B±ν,j (λ,σ) = cj (±2)
−j
∂

j
λ

(
∂k−jσ b̃

±
ν (σλ)ϕ(λ)

)
.

It is easy to deduce from (3.11) that the functions B±ν,j satisfy the bounds

∣∣∂ℓλB
±

ν,j (λ,σ)
∣∣ ≤ Cℓ,jσ−(n−1)/2−k+j, (3.12)

for all integers ℓ,j ≥ 0. Using (3.12), in the same way as in the proof of Proposition 2.1 of [12]

(see (2.13)), we get

∣∣∣∣
∫ ∞

0

eitλ
2
±iσλϕ̃(λ)B±ν,j (λ,σ)dλ

∣∣∣∣ ≤ Cm,jt
−m−1/2σ−(n−1)/2−k+j+m, (3.13)

for all integers m, and hence for all real m. By (3.13) with m = (n − 1)/2 − s − j we obtain

∣∣∂kσa1(σ,t)
∣∣ ≤ Cσ−k

(
t

σ

)s
,

which is the desired result in this case.

To prove (3.8) with h = 1, observe that

a′1(σ,t) = t
n/2eiσ

2/4t

(
K′1(σ,t) +

n

2t
K1(σ,t) −

iσ2

4t2
K1(σ,t)

)
. (3.14)

On the other hand, integrating by parts twice with respect to the variable λ2 and using that the

function gν (z) = z
−2νJν (z) = z

−νJν (z) satisfies the equation

g′′ν (z) + (n − 1)z
−1g′ν (z) + gν (z) = 0,



66 Fernando Cardoso, Claudio Cuevas and Georgi Vodev CUBO
11, 5 (2009)

we get

K′1(σ,t) +
n

2t
K1(σ,t) −

iσ2

4t2
K1(σ,t) = t

−1
(
K

(0)
1 (σ,t) + K

(1)
1 (σ,t)

)
, (3.15)

where

K
(j)
1 (σ,t) =

(σ
t

)j ∫ ∞

0

eitλ
2

ψ(j)(λ)g(j)ν (σλ)dλ, j = 0, 1,

where ψ(j) ∈ C∞0 ((0, +∞)), g
(0)
ν (z) = gν (z), g

(1)
ν (z) = dgν (z)/dz. By (3.14) and (3.15),

a′1(σ,t) = t
−1

(
a
(0)
1 (σ,t) + a

(1)
1 (σ,t)

)
, (3.16)

where

a
(j)
1 (σ,t) = t

n/2eiσ
2/4tK

(j)
1 (σ,t), j = 0, 1.

Now, in the same way as above one can see that the functions a
(j)
1 satisfy (3.7) with h = 1, provided

k + s ≤ (n − 2)/2. 2

The kernel of the operator T1 is of the form

T (x,y,t,h) =

∫ t

0

∫

Rn

e−iϕ(t − τ)−n/2τ−n/2ãh(|x − ξ|, t − τ)ah(|y − ξ|,τ)V (ξ)dξdτ,

where

ϕ =
|x − ξ|2

4(t − τ)
+

|y − ξ|2

4τ
,

and ãh is defined by replacing in the definition of ah the function Kh by K̃h. Observe that by

Lemma 3.2 we have the bounds

∣∣∂αξ ah(|x − ξ|, t)
∣∣ ≤ C(t/h)k+ǫ|x − ξ|−|α|−k−ǫ, (3.17)

∣∣∂αξ a
′
h(|x − ξ|, t)

∣∣ ≤ Ct−1(t/h)k+ǫ|x − ξ|−|α|−k−ǫ, (3.18)

for every 0 ≤ ǫ ≪ 1, 0 ≤ k < (n − 3)/2, and all multi-indices α such that |α| ≤ (n − 2)/2 − k − ǫ.

Define the functions F(1) and F(2) by replacing
∫ t
0

in the definition of the function T by
∫ γ
0

and
∫ t/2

γ
, respectively. Let φ ∈ C∞0 (R), φ(λ) = 1 for |λ| ≤ 1/2, φ(λ) = 0 for |λ| ≥ 1, and write

1 =

∞∑

q=0

φq(λ),

where φ0 = φ, φq(λ) = φ̃(2
−qλ), q ≥ 1, with a function φ̃ ∈ C∞0 (R), φ̃(λ) = 0 for |λ| ≤ 1/2 and

|λ| ≥ 1. We can write

F(1) =
∞∑

p=1

∞∑

q=0

F(1)p,q , F
(2)

=

∞∑

q=0

F(2)q ,

where

F(1)p,q =

∫ γ

0

∫

Rn

e−iϕ(t − τ)−n/2τ−n/2φp

(γ
τ

)
φq(|ξ|)ãh(|x − ξ|, t − τ)ah(|y − ξ|,τ)V (ξ)dξdτ



CUBO
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Dispersive Estimates for the Schrödinger Equation ... 67

= t1−n
∫ ∞

t/γ

∫

Rn

e−iϕ
(

µ

µ − 1

)n/2
µn/2−2φp (γµ/t) ãh(|x − ξ|, t(µ − 1)/µ)ah(|y − ξ|, t/µ)Vq(ξ)dξdµ,

F(2)q =

∫ t/2

1

∫

Rn

e−iϕ(t − τ)−n/2τ−n/2φq(|ξ|)ãh(|x − ξ|, t − τ)ah(|y − ξ|,τ)V (ξ)dξdτ

= t1−n
∫ t

2

∫

Rn

e−iϕ
(

µ

µ − 1

)n/2
µn/2−2ãh(|x − ξ|, t(µ − 1)/µ)ah(|y − ξ|, t/µ)Vq(ξ)dξdµ,

where we have made a change of variables µ = t/τ and set Vq(ξ) = φq(|ξ|)V (ξ). Clearly, it suffices

to prove the following

Proposition 3.3. Under the assumptions of Proposition 3.1, there exist constants C,ε′ > 0 such

that we have the bounds ∣∣∣F(1)p,q
∣∣∣ ≤ C2−εp−ε

′qt−n/2h−k−ε, (3.19)

for every 0 < ε ≪ 1, and ∣∣∣F(2)q
∣∣∣ ≤ C2−ε

′qt−n/2h−k, t ≥ 2. (3.20)

Indeed, by (3.19) we have

∣∣∣F(1)
∣∣∣ ≤ Ct−n/2h−k(εhε)−1 = C′t−n/2h−k log

1

h
,

if we take ε so that h−ε = 2, while (3.20) yields

∣∣∣F(2)
∣∣∣ ≤ Ct−n/2h−k, t ≥ 2.

Proof. Let ρ ∈ C∞0 (R
n
) be a real-valued function such that

∫
ρ(x)dx = 1, and set ρθ(x) =

θ−nρ(x/θ), 0 < θ ≤ 1, Vq,θ = ρθ ∗ Vq. Let k0 ≥ 0 be an integer such that (n − 3)/2 − k = k0 + ν

with 0 ≤ ν < 1. Since V ∈ Vk0+νδ (R
n
), we have

∣∣∂αξ Vq(ξ)
∣∣ ≤ C2−q(δ+|α|), 0 ≤ |α| ≤ k0, (3.21)

∣∣∂αξ Vq(ξ) − ∂
α
ξ Vq(ξ

′
)
∣∣ ≤ C2−q(δ+k0+ν)|ξ − ξ′|ν, |ξ − ξ′| ≤ 1, |α| = k0. (3.22)

It is easy to see that these bounds imply

∣∣∂αξ Vq,θ(ξ)
∣∣ ≤ C2−q(δ+|α|), 0 ≤ |α| ≤ k0, (3.23)

∣∣∂αξ Vq,θ (ξ)
∣∣ ≤ Cθ−1+ν 2−q(δ+k0+ν), |α| = k0 + 1, (3.24)

∣∣∂αξ Vq(ξ) − ∂
α
ξ Vq,θ(ξ)

∣∣ ≤ Cθ2−q(δ+|α|+1), 0 ≤ |α| ≤ k0 − 1, (3.25)
∣∣∂αξ Vq(ξ) − ∂

α
ξ Vq,θ(ξ)

∣∣ ≤ Cθν 2−q(δ+|α|+ν), |α| = k0. (3.26)

Integrating by parts with respect to the variable ξ as in Section 4 of [12] (see the proof of (4.15))

we obtain the following



68 Fernando Cardoso, Claudio Cuevas and Georgi Vodev CUBO
11, 5 (2009)

Lemma 3.4. Let 0 ≤ m < (n − 1)/2 be an integer and let W(µ, ·) ∈ Cm0 (R
n
). Then we have the

estimate ∣∣∣∣∣
t1−n

∫
∞

t/γ

∫

Rn

e−iϕ
(

µ

µ − 1

)n/2
µn/2−2φp (γµ/t) W(µ,ξ)dξdµ

∣∣∣∣∣

≤ Ct−n/2 (2p/γ)
(n−3)/2−m

∑

0≤|α|≤m

∫

Rn

|ξ − y|
−2m+|α|−1

∣∣∂αξ W(∞,ξ)
∣∣ dξ

+Ct−n/2 (2p/γ)
(n−3)/2−m

∑

0≤|α|≤m

∫

Rn

|y − ξ|−1
∣∣ξ − y − γt−1(x − y)

∣∣−2m+|α| ∣∣∂αξ W(t/γ,ξ)
∣∣ dξ

+Ct−n/2−1 (2p/γ)
(n−3)/2−m−1

∑

0≤|α|≤m

∫ 2pt/γ

2p−1t/γ

∫

Rn

(
|y − ξ|−1

∣∣ξ − y − µ−1(x − y)
∣∣−2m+|α|

+
∣∣ξ − y − µ−1(x − y)

∣∣−2m+|α|−1
) ∣∣∂αξ W(µ,ξ)

∣∣ dξdµ

+Ct−n/2 (2p/γ)
(n−3)/2−m

∑

0≤|α|≤m

∫ 2pt/γ

2p−1t/γ

∫

Rn

|y − ξ|−1
∣∣ξ − y − µ−1(x − y)

∣∣−2m+|α|

×
∣∣∂µ∂αξ W(µ,ξ)

∣∣ dξdµ. (3.27)

We would like to apply this lemma with a function W of the form

W(µ,ξ) = ãh(|x − ξ|, t(µ − 1)/µ)ah(|y − ξ|, t/µ)Q(ξ)

where Q ∈ Cm0 (R
n
) is independent of the variable µ. In view of (3.17) and (3.18) we have

∣∣∂αξ W(µ,ξ)
∣∣ ≤ C

∑

|α1|+|α2|+|α3|=|α|

∣∣∣∂α1ξ ãh(|x − ξ|, t(µ − 1)/µ)
∣∣∣
∣∣∣∂α2ξ ah(|y − ξ|, t/µ)

∣∣∣
∣∣∣∂α3ξ Q(ξ)

∣∣∣

≤ Ch−k−ε(t/µ)k+ε
∑

|α1|+|α2|+|α3|=|α|

|x − ξ|−|α1||y − ξ|−|α2|−k−ε
∣∣∣∂α3ξ Q(ξ)

∣∣∣ , (3.28)

∣∣∂µ∂αξ W(µ,ξ)
∣∣ ≤ tµ−2

∣∣∂αξ (ã
′
h(|x − ξ|, t(µ − 1)/µ)ah(|y − ξ|, t/µ)Q(ξ))

∣∣

+tµ−2
∣∣∂αξ (ãh(|x − ξ|, t(µ − 1)/µ)a

′
h(|y − ξ|, t/µ)Q(ξ))

∣∣

≤ Ctµ−2
∑

|α1|+|α2|+|α3|=|α|

∣∣∣∂α1ξ ã
′
h(|x − ξ|, t(µ − 1)/µ)

∣∣∣
∣∣∣∂α2ξ ah(|y − ξ|, t/µ)

∣∣∣
∣∣∣∂α3ξ Q(ξ)

∣∣∣

+Ctµ−2
∑

|α1|+|α2|+|α3|=|α|

∣∣∣∂α1ξ ãh(|x − ξ|, t(µ − 1)/µ)
∣∣∣
∣∣∣∂α2ξ a

′
h(|y − ξ|, t/µ)

∣∣∣
∣∣∣∂α3ξ Q(ξ)

∣∣∣

≤ Cµ−1h−k−ε(t/µ)k+ε
∑

|α1|+|α2|+|α3|=|α|

|x − ξ|−|α1||y − ξ|−|α2|−k−ε
∣∣∣∂α3ξ Q(ξ)

∣∣∣ . (3.29)

By (3.27), (3.28) and (3.29), one can easily get the estimate
∣∣∣∣∣
t1−n

∫
∞

t/γ

∫

Rn

e−iϕ
(

µ

µ − 1

)n/2
µn/2−2φp (γµ/t) W(µ,ξ)dξdµ

∣∣∣∣∣



CUBO
11, 5 (2009)

Dispersive Estimates for the Schrödinger Equation ... 69

≤ Ct−n/2h−k−ε
(
γ2−p

)m+k+ε−(n−3)/2
M(m,Q), (3.30)

where

M(m,Q) =
∑

0≤|α|≤m

sup
Rn

〈ξ〉n+2ε
′
−2m−1−k+|α|

∣∣∂αξ Q(ξ)
∣∣ ,

for every 0 < ε′ ≪ 1 and 0 < ε ≤ ε′. Consider first the case k0 < (n − 3)/2. Then, we are going to

apply (3.30) with m = k0, Q = Vq − Vq,θ, and m = k0 + 1, Q = Vq,θ, respectively. Choose ε
′ such

that δ ≥ k + 2 + 4ε′. In view of (3.25) and (3.26), we have

M(k0,Vq − Vq,θ) ≤ Cθ
ν

(2
q
)
ν−ε′

, (3.31)

while by (3.23) and (3.24), we have

M(k0 + 1,Vq,θ) ≤ Cθ
ν−1

(2
q
)
ν−1−ε′

. (3.32)

Combining (3.30), (3.31) and (3.32) we conclude

∣∣∣F(1)p,q
∣∣∣ ≤ Ct−n/2h−k−ε

(
γ2−p

)ε
2
−ε′q

((
θ2p+q/γ

)ν
+

(
θ2p+q/γ

)ν−1)
. (3.33)

Taking θ = γ2−p−q we deduce (3.19) from (3.33). Let now k0 = (n − 3)/2. This implies that n is

odd and ν = 0. Then we apply (3.30) with m = k0 and Q = Vq. In view of (3.21) we have

M(k0,Vq) ≤ C2
−ε′q, (3.34)

with some constant 0 < ε′ ≪ 1, so in this case (3.19) follows from (3.30) and (3.34).

Proceeding as in the proof of (3.30) we obtain in the same way the estimate

∣∣∣∣∣
t1−n

∫ t

2

∫

Rn

e−iϕ
(

µ

µ − 1

)n/2
µn/2−2W(µ,ξ)dξdµ

∣∣∣∣∣

≤ Ch−k
(
t−n/2 + tm+k−n+3/2

)
M(m,Q)

+Ch−ktm+k−n+3/2M(m,Q)

∫ t

2

µ(n−3)/2−m−k−1dµ ≤ C′h−kt−n/2M(m,Q), (3.35)

where we have used that m + k < (n − 3)/2. On the other hand, since V ∈ Vmδ (R
n
) with

δ > n − 1 − 2m − k, it is easy to check that we have the bound

M(m,Vq) ≤ C2
−ε′q, (3.36)

with some constant 0 < ε′ ≪ 1. Now, combining (3.35) and (3.36) leads to (3.20). 2



70 Fernando Cardoso, Claudio Cuevas and Georgi Vodev CUBO
11, 5 (2009)

Acknowledgements

This work was carried out while G.V. was visiting the Universidade Federal de Pernambuco, Recife,

in August - September 2008 with the support of the agreement Brazil-France in Mathematics -

Proc. 69.0014/01-5. The first two authors have also been partially supported by the CNPq-Brazil.

Received: November, 2008. Revised: April, 2009.

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