CUBO A Mathematical Journal

Vol.10, N
o
¯ 02, (01–14). July 2008

Semi-Classical Dispersive Estimates for the

Wave and Schrödinger Equations with a

Potential in Dimensions n ≥ 4

F. Cardoso

Universidade Federal de Pernambuco, Departamento de Matemática,

Av. Prof. Luiz Freire, S/N, Cid. Universitária,

CEP. 50.540-740 – Recife-Pe, Brazil

email: fernando@dmat.ufpe.br

and

G. Vodev

Université de Nantes, Département de Mathématiques,

UMR 6629 du CNRS, 2, rue de la Houssinière, BP 92208,

44332 Nantes Cedex 03, France

email: georgi.vodev@math.univ-nantes.fr

ABSTRACT

We expand the operators |t|(n−1)/2eit
√

−∆+V ϕ(h
√
−∆ + V ) and |t|n/2eit(−∆+V )

ψ(h2(−∆ + V )), 0 < h ≪ 1, modulo operators whose L1 → L∞ norm is ON (h
N

),

∀N ≥ 1, where ϕ,ψ ∈ C∞0 ((0, +∞)) and V ∈ L
∞

(R
n
), n ≥ 4, is a real-valued po-

tential satisfying V (x) = O
(
〈x〉−δ

)
, δ > (n + 1)/2 in the case of the wave equation

and δ > (n + 2)/2 in the case of the Schrödinger equation. As a consequence, we give

sufficent conditions in order that the wave and the Schrödinger groups satisfy dispersive

estimates with a loss of ν derivatives, 0 ≤ ν ≤ (n − 3)/2. Roughly speaking, we reduce

this problem to estimating the L1 → L∞ norms of a finite number of operators with



2 F. Cardoso and G. Vodev CUBO
10, 2 (2008)

almost explicit kernels. These kernels are completely explicit when 4 ≤ n ≤ 7 in the

case of the wave group, and when n = 4, 5 in the case of the Schrödinger group.

RESUMEN

En este trabajo son expandidos los operadores |t|(n−1)/2eit
√

−∆+V ϕ(h
√
−∆ + V ) y

|t|n/2eit(−∆+V )ψ(h2(−∆+V )), 0 < h ≪ 1, modulo operadores cuja L1 → L∞ norma es

ON (h
N

), ∀N ≥ 1, donde ϕ,ψ ∈ C∞0 ((0, +∞)) y V ∈ L
∞

(Rn), n ≥ 4, es un potencial

real satisfaziendo V (x) = O
(
〈x〉−δ

)
, δ > (n + 1)/2 en el caso de la ecuación de la

onda y δ > (n + 2)/2 en el caso de la ecuación de Schrödinger. Como consequencia

presentamos condiciones suficientes a fin de que los grupos de la onda y Schrödinger

cumplan estimativas dispersivas con una perdida de ν derivadas 0 ≤ ν ≤ (n − 3)/2.

Rigurosamente hablando, reduzimos este problema a estimar las normas L1 → L∞ de

un número finito de operadores con nucleos casi explicitos. Estos nucleos son comple-

tamente explicitos cuando 4 ≤ n ≤ 7 en el caso del grupo de la onda y cuando n = 4, 5

en el caso del grupo de Schrödinger.

Key words and phrases: Potential, dispersive estimates.

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

1 Introduction and statement of results

Denote by G the self-adjoint realization of the operator −∆ + 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 + 1)/2. It is well known that G has no strictly positive eigenvalues

and resonances. We will also denote by G0 the self-adjoint realization of the operator −∆ on

L2(Rn). It is well known that the free wave group satisfies the following semi-classical dispersive

estimate ∥∥∥eit
√

G0ϕ(h
√
G0)

∥∥∥
L1→L∞

≤ Ch−(n+1)/2|t|−(n−1)/2, ∀t 6= 0, h > 0, (1.2)

where ϕ ∈ C∞0 ((0, +∞)). The natural question is to find the bigest possible class of potentials for

which we have an analogue of (1.2) for the perturbed wave group. It is proved in [16] that under

the assumption (1.1) only, we have such an estimate but with a significant loss in h for 0 < h ≪ 1,

namely ∥∥∥eit
√

Gϕ(h
√
G)

∥∥∥
L

1
→L

∞

≤ Ch−n+1|t|−(n−1)/2, ∀t 6= 0, 0 < h ≤ 1, (1.3)



CUBO
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Semi-classical Dispersive Estimates for the wave ... 3

and this seems hard to improve without extra assumptions on the potential. This estimate is then

used in [16] to obtain dispersive estimates with a loss of (n − 3)/2 derivatives for eit
√

Gχa(
√
G),

∀a > 0, where χa ∈ C
∞

((−∞, +∞)), χa(λ) = 0 for λ ≤ a, χa(λ) = 1 for λ ≥ 2a.

In the present work we will expand eit
√

Gϕ(h
√
G) modulo remainders whose L1 → L∞ norm

is upper bounded by Cmh
m−n+1|t|−(n−1)/2, 0 < h ≤ h0 ≪ 1, for every integer m ≥ 0. In order to

state the precise result we need to introduce some notations. Let ϕ1 ∈ C
∞

0 ((0, +∞)) be such that

ϕ1 = 1 on supp ϕ, and set ϕ̃(λ) = λϕ(λ), ϕ̃1(λ) = λ
−1ϕ1(λ). Under (1.1) there exists a constant

h0 > 0 so that for 0 < h ≤ h0, the operator

T (h) :=
(
Id + ϕ1(h

√
G0) − ϕ1(h

√
G)

)
−1

= Id + O(h2)

is uniformely bounded on Lp, 1 ≤ p ≤ +∞, as well as on weighted L2 spaces (see Lemma 2.3 of

[16] and Lemma A.1 of [11]). Set

U0(t,h) = ϕ̃1(h
√
G0) sin(t

√
G0), E

0
0 (t,h) = e

it
√

G0ϕ(h
√
G0),

E0(t,h) = ϕ1(h
√
G0) cos(t

√
G0)ϕ(h

√
G) + iϕ̃1(h

√
G0) sin(t

√
G0)ϕ̃(h

√
G).

Furthermore, given any integer j ≥ 1, define the operators

Ej (t,h) = −h

∫
t

0

U0(t − τ,h)V T (h)Ej−1(τ,h)dτ,

E0
j
(t,h) = −h

∫
t

0

U0(t − τ,h)V E
0
j−1(τ,h)dτ.

Theorem 1.1 Let V satisfy (1.1). Then, there exists a constant h0 > 0 so that for all 0 < h ≤ h0,

t 6= 0, we have the estimate

∥∥∥∥∥∥
eit

√

Gϕ(h
√
G) − T (h)

m∑

j=0

Ej (t,h)

∥∥∥∥∥∥
L

1
→L

∞

≤ Cmh
m−n+1

|t|−(n−1)/2, (1.4)

for every integer m ≥ 0 with a constant Cm > 0 independent of t and h. Moreover, the operators

Ej satisfy the estimates

‖E0(t,h)‖L1→L∞ ≤ Ch
−(n+1)/2

|t|−(n−1)/2, (1.5)

‖Ej (t,h)‖L1→L∞ ≤ Cjh
j−n

|t|−(n−1)/2, j ≥ 1, (1.6)

∥∥Ej (t,h) − E0j (t,h)
∥∥

L1→L∞
≤ Cjh

j+2−n
|t|−(n−1)/2, j ≥ 1. (1.7)

It follows from this theorem that to improve the estimate (1.3) in h, it suffices to improve the

estimate (1.6). We also have the following



4 F. Cardoso and G. Vodev CUBO
10, 2 (2008)

Corollary 1.2 Let V satisfy (1.1) and suppose in addition that there exists 0 ≤ k ≤ (n − 3)/2

such that the operators Ej satisfy the estimate

‖Ej (t,h)‖L1→L∞ ≤ Ch
k−n+1

|t|−(n−1)/2, (1.8)

for all integers 1 ≤ j < k + 1. Then, for every a > 0, 0 < ǫ ≪ 1, we have the estimate
∥∥∥eit

√

G
(
√
G)k−n+1−ǫχa(

√
G)

∥∥∥
L1→L∞

≤ Cǫ|t|
−(n−1)/2, ∀t 6= 0, (1.9)

while for every 0 ≤ q ≤ (n − 3)/2 − k, 2 ≤ p <
2(n−1−2q−2k)

n−3−2q−2k
, we have

∥∥∥eit
√

G
(
√
G)

−α((n+1)/2+q)
χa(

√
G)

∥∥∥
L

p
′
→L

p

≤ C|t|
−α(n−1)/2

, ∀t 6= 0, (1.10)

where 1/p + 1/p′ = 1, α = 1 − 2/p. Moreover, when 4 ≤ n ≤ 7 the estimates (1.9) and (1.10) hold

true if we suppose (1.8) fulfilled with Ej replaced by E
0
j
.

The estimate (1.8) with k > 0 seems hard to establish (even if we replace Ej by E
0
j
) and the

proof would probably require some regularity condition on the potential. Note that when n = 2

and n = 3 the estimates (1.9) and (1.10) (with k = (n − 3)/2, q = 0) are proved in [2] under (1.1)

only. In the case of n = 2 these estimates are proved (for a large enough) in [10] for a much larger

class of potentials satisfying

sup
y∈R2

∫

R2

|V (x)|dx

|x − y|1/2
≤ C < +∞. (1.11)

When n = 3 these estimates are proved in [4] for a quite large subclass of potentials satisfying

sup
y∈R3

∫

R3

|V (x)|dx

|x − y|
≤ C < +∞. (1.12)

When n ≥ 4 optimal dispersive estimates (that is, without loss of derivatives) are proved in [1] for

potentials belonging to the Schwartz class. When n ≥ 4, as mentioned above, the estimates (1.9)

and (1.10) with k = 0 are proved in [16] under (1.1) only. The proof of Theorem 1.1 and Corollary

1.2, which will be given in Section 2, is based very much on the analysis developed in [16].

A similar analysis as above can be carried out for the Schrödinger group as well. The free one

satisfies the following dispersive estimate

∥∥eitG0ψ(h2G0)
∥∥

L
1
→L

∞
≤ C|t|

−n/2
, ∀t 6= 0, h > 0, (1.13)

where ψ ∈ C∞0 ((0, +∞)). On the other hand, it is proved in [15] that under the assumption (1.1)

with δ > (n + 2)/2 only, the perturbed Schrödinger group satisfies

∥∥eitGψ(h2G)
∥∥

L1→L∞
≤ Ch−(n−3)/2|t|−n/2, ∀t 6= 0, 0 < h ≤ 1. (1.14)

This estimate is used in [15] to obtain dispersive estimates with a loss of (n − 3)/2 derivatives for

eitGχa(G), ∀a > 0.



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Semi-classical Dispersive Estimates for the wave ... 5

In this work we will also expand eitGψ(h2G) modulo remainders whose L1 → L∞ norm is

upper bounded by Cmh
m−(n−2)/2−ǫ

|t|−n/2, 0 < h ≤ h0 ≪ 1, for every integer m ≥ 0, similarly

to the wave group above. To this end, choose a function ψ1 ∈ C
∞

0 ((0, +∞)) such that ψ1 = 1 on

supp ψ, and set

T (h) :=
(
Id + ψ1(h

2G0) − ψ1(h
2G)

)
−1

= Id + O(h2),

F 00 (t,h) = e
itG0ψ(h2G0), F0(t,h) = ψ1(h

2G0)e
itG0ψ(h2G), W0(t,h) = e

itG0ψ1(h
2G0),

Fj (t,h) = i

∫
t

0

W0(t − τ,h)V T (h)Fj−1(τ,h)dτ, j ≥ 1,

F 0
j
(t,h) = i

∫
t

0

W0(t − τ,h)V F
0
j−1(τ,h)dτ, j ≥ 1.

Theorem 1.3 Let V satisfy (1.1) with δ > (n + 2)/2. Then, there exists a constant h0 > 0 so

that for all 0 < h ≤ h0, t 6= 0, 0 < ǫ ≪ 1, we have the estimate
∥∥∥∥∥∥
eitGψ(h2G) − T (h)

m∑

j=0

Fj (t,h)

∥∥∥∥∥∥
L

1
→L

∞

≤ Cmh
m−(n−2)/2−ǫ

|t|−n/2, (1.15)

for every integer m ≥ 0 with a constant Cm > 0 independent of t and h. Moreover, the operators

Fj satisfy the estimates

‖F0(t,h)‖L1→L∞ ≤ C|t|
−n/2, (1.16)

‖Fj (t,h)‖L1→L∞ ≤ Cjh
j−n/2−ǫ

|t|−n/2, j ≥ 1, (1.17)
∥∥Fj (t,h) − F 0j (t,h)

∥∥
L1→L∞

≤ Cjh
j+2−n/2−ǫ

|t|−n/2, j ≥ 1. (1.18)

Thus, to improve the estimate (1.14) in h, it suffices to improve the estimate (1.17). We also

have the following

Corollary 1.4 Let V satisfy (1.1) with δ > (n + 2)/2 and suppose in addition that there exists

0 ≤ k ≤ (n − 3)/2 such that the operators Fj satisfy the estimate

‖Fj (t,h)‖L1→L∞ ≤ Ch
k−(n−3)/2

|t|−n/2, (1.19)

for all integers 1 ≤ j ≤ k + 3/2. Then, for every a > 0, 0 < ǫ ≪ 1, we have the estimate
∥∥∥eitGGk/2−(n−3)/4−ǫχa(G)

∥∥∥
L1→L∞

≤ Cǫ|t|
−n/2, ∀t 6= 0, (1.20)

while for every 0 ≤ q ≤ (n − 3)/2 − k, 2 ≤ p <
2(n−1−2q−2k)

n−3−2q−2k
, we have

∥∥∥eitGG−αq/2χa(G)
∥∥∥

L
p
′
→L

p

≤ C|t|−αn/2, ∀t 6= 0, (1.21)

where 1/p + 1/p′ = 1, α = 1 − 2/p. Moreover, if there exists an operator Fk(t), independent of h,

such that the following estimates hold
∥∥∥Fk(t)Gk/2−(n−3)/40

∥∥∥
L1→L∞

≤ C|t|−n/2, (1.22)



6 F. Cardoso and G. Vodev CUBO
10, 2 (2008)

∥∥F1(t,h) − Fk(t)ψ(h2G0)
∥∥

L
1
→L

∞
≤ Chk−(n−3)/2+ε|t|−n/2, (1.23)

‖Fj (t,h)‖L1→L∞ ≤ Ch
k−(n−3)/2+ε

|t|
−n/2

, (1.24)

for 2 ≤ j ≤ k + 3/2 with some ε > 0, then we have
∥∥∥eitGGk/2−(n−3)/4χa(G)

∥∥∥
L

1
→L

∞

≤ C|t|−n/2, ∀t 6= 0. (1.25)

Furthermore, when n = 4, 5 the estimates (1.20), (1.21) and (1.25) hold true if we suppose (1.19),

(1.23) and (1.24) fulfilled with Fj replaced by F
0
j
.

As in the case of the wave group above, the estimates (1.19), (1.22), (1.23) and (1.24) with

k > 0 seem hard to establish (even if we replace Fj by F
0
j
) and the proof would certainly require

some regularity condition on the potential. Indeed, it follows from the results in [5] that there

exist compactly supported potentials V ∈ Cν (Rn), ∀ν < (n− 3)/2, for which these estimates with

k = (n − 3)/2 fail to hold. Therefore, it is naural to expect that they hold true for potentials

V ∈ C(n−3)/2−k(Rn). We also conjecture that the statements of Theorem 1.3 and Corollary 1.4

hold true for potentials satisfying (1.1) with δ > (n + 1)/2 as for the wave group above. Note

that when n = 2 the estimate (1.25) without loss of derivatives (that is, with k = (n − 3)/2) is

proved in [12] under (1.1) with δ > 2. In this case this estimate is proved (for a large enough)

in [10] for potentials satisfying (1.11). When n = 3 this estimate is proved in [6] for potentials

V ∈ L3/2−ǫ ∩L3/2+ǫ, 0 < ǫ ≪ 1, and in particular for potentials satisfying (1.1) with δ > 2. In this

case it is also proved in [13] for potentials satisfying (1.12) with C < 4π. When n ≥ 4 the optimal

dispersive estimate (that is, without loss of derivatives) is proved in [9] for potentials satisfying

(1.1) with δ > n as well as V̂ ∈ L1. This result has been recently extended in [11] to potentials

satisfying (1.1) with δ > n − 1 as well as V̂ ∈ L1. When n ≥ 4, as mentioned above, the estimates

(1.21) and (1.25) with k = 0 are proved in [15] under (1.1) with δ > (n + 2)/2 only. The proof of

Theorem 1.3 and Corollary 1.4, which will be given in Section 3, relies very much on the analysis

developed in [15].

Acknowledgements. A part of this work was carried out while F. C. was visiting the

University of Nantes in May 2007 with the support of the agreement Brazil-France in Mathematics

- Proc. 69.0014/01-5. The first author has also been partially supported by the CNPq-Brazil.

2 Semi-classical expansion of eit
√
Gϕ(h

√
G)

We keep the same notations as in the introduction. Our starting point is the following identity

which can be derived easily from Duhamel’s formula (see [16])

(
Id + ϕ1(h

√
G0) − ϕ1(h

√
G)

)
eit

√

Gϕ(h
√
G)

= E0(t,h) − h

∫
t

0

U0(t − τ,h)V e
iτ

√

G
ϕ(h

√
G)dτ. (2.1)



CUBO
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Semi-classical Dispersive Estimates for the wave ... 7

We rewrite (2.1) as follows

eit
√

Gϕ(h
√
G) = Ẽ0(t,h) +

∫
t

0

Ũ0(t − τ,h)V e
iτ

√

Gϕ(h
√
G)dτ, (2.2)

where

Ẽ0(t,h) = T (h)E0(t,h), Ũ0(t,h) = −hT (h)U0(t,h).

Iterating (2.2) m times leads to the identity

e
it

√

G
ϕ(h

√
G) =

m∑

j=0

Ẽj (t,h) + Rm+1(t,h), (2.3)

where the operators Ẽj , j ≥ 1, are defined by

Ẽj (t,h) =

∫
t

0

Ũ0(t − τ,h)V Ẽj−1(τ,h)dτ,

while the operators Rm, m ≥ 0, are defined as follows

R0(t,h) = e
it

√

Gϕ(h
√
G),

Rm+1(t,h) =

∫
t

0

Ũ0(t − τ,h)V Rm(τ,h)dτ.

It is clear from (2.3) that the estimate (1.4) follows from the following

Proposition 2.1 Under the assumptions of Theorem 1.1, for all 0 < h ≤ h0, t 6= 0, 1/2 − ǫ/4 ≤

s ≤ (n − 1)/2, 0 < ǫ ≪ 1, we have the estimates

‖Rm+1(t,h)‖L1→L∞ ≤ Cmh
m−n+1

|t|
−(n−1)/2

, (2.4)

∥∥〈x〉−s−ǫRm+1(t,h)
∥∥

L1→L2
≤ Cmh

m−n/2+1
|t|−s, (2.5)

for every integer m ≥ 0.

Proof. For m = 0 the estimate (2.4) is proved in [16] (see (4.10)). We will now derive (2.4)

for m ≥ 1 from (2.5) and the following estimate proved in [16] (see (2.4)):

∫
∞

−∞

|t|2s
∥∥∥〈x〉−1/2−s−ǫeit

√

G0ϕ(h
√
G0)f

∥∥∥
2

L
2

dt ≤ Ch−n ‖f‖
2
L

1 , ∀f ∈ L
1, (2.6)

for 0 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. By (2.5) and (2.6), we have

|t|(n−1)/2 |〈Rm+1(t,h)f,g〉|

≤ C

∫
t

t/2

|τ|(n−1)/2
∥∥∥〈x〉−1−ǫŨ0(t − τ,h)∗g

∥∥∥
L2

∥∥∥〈x〉−(n−1)/2−ǫRm(τ,h)f
∥∥∥

L2
dτ



8 F. Cardoso and G. Vodev CUBO
10, 2 (2008)

+C

∫
t

t/2

|τ|
(n−1)/2

∥∥∥〈x〉−n/2−ǫŨ0(τ,h)∗g
∥∥∥

L
2

∥∥∥〈x〉−1/2−ǫRm(t − τ,h)f
∥∥∥

L
2

dτ

≤ Ch
m−n/2

‖f‖L1

(∫
∞

−∞

〈τ
′

〉
1+ǫ/2

∥∥∥〈x〉−1−ǫŨ0(τ′,h)∗g
∥∥∥

2

L
2

dτ
′

)1/2

+C

(∫
∞

−∞

|τ|n−1
∥∥∥〈x〉−n/2−ǫŨ0(τ,h)∗g

∥∥∥
2

L2
dτ

)1/2 (∫ ∞

−∞

∥∥∥〈x〉−1/2−ǫRm(τ′,h)f
∥∥∥

2

L2
dτ′

)1/2

≤ Chm+1−n‖f‖L1‖g‖L1.

We will now prove (2.5) by induction in m. For m = 0 it is proved in [16] (see (4.6)) with

s = (n − 1)/2 but the proof for general s is the same. We will show that (2.5) for Rm+1 follows

from (2.5) for Rm and the following estimate proved in [16] (see (2.1)):

∥∥∥〈x〉−seit
√

G0ϕ(h
√
G0)〈x〉

−s

∥∥∥
L2→L2

≤ C〈t〉−s, ∀t, 0 < h ≤ 1. (2.7)

Consider first the case 1 ≤ s ≤ (n − 1)/2. We have

|t|
s
∥∥〈x〉−s−ǫRm+1(t,h)

∥∥
L

1
→L

2

≤ C

∫
t

t/2

|τ|s
∥∥∥〈x〉−s−ǫŨ0(t − τ,h)〈x〉−1−ǫ

∥∥∥
L

2
→L

2

∥∥〈x〉−s−ǫRm(τ,h)
∥∥

L
1
→L

2
dτ

+C

∫
t

t/2

|τ|s
∥∥∥〈x〉−s−ǫŨ0(τ,h)〈x〉−s−ǫ

∥∥∥
L2→L2

∥∥〈x〉−1−ǫRm(t − τ,h)
∥∥

L1→L2
dτ

≤ Chm+1−n/2
∫

∞

−∞

〈τ′〉−1−ǫdτ′ + Ch

∫
∞

−∞

∥∥〈x〉−1−ǫRm(τ′,h)
∥∥

L
1
→L

2
dτ′ ≤ Chm+1−n/2.

Let now 1/2 − ǫ/4 ≤ s ≤ 1. We have

|t|
s
∥∥〈x〉−s−ǫRm+1(t,h)

∥∥
L

1
→L

2

≤ C

∫
t

t/2

|τ|s
∥∥∥〈x〉−s−ǫŨ0(t − τ,h)〈x〉−1/2−ǫ

∥∥∥
L2→L2

∥∥∥〈x〉−s−1/2−ǫRm(τ,h)
∥∥∥

L1→L2
dτ

+C

∫
t

t/2

|τ|s
∥∥∥〈x〉−s−ǫŨ0(τ,h)〈x〉−s−ǫ

∥∥∥
L2→L2

∥∥〈x〉−1−ǫRm(t − τ,h)
∥∥

L1→L2
dτ

≤ Ch

(∫
∞

−∞

〈τ′〉−1−ǫdτ′
)1/2 (∫ ∞

−∞

|τ|2s
∥∥∥〈x〉−s−1/2−ǫRm(τ,h)

∥∥∥
2

L1→L2
dτ

)1/2

+Ch

∫
∞

−∞

∥∥〈x〉−1−ǫRm(τ′,h)
∥∥

L
1
→L

2
dτ

′

≤ Ch
m+1−n/2

.

2

To prove (1.6) observe first that in the same way as in the proof of (2.5) one can show that

the operator Ej satisfies the estimate

∥∥〈x〉−s−ǫEj (t,h)
∥∥

L
1
→L

2
≤ Cjh

j−n/2
|t|

−s
, j ≥ 1, (2.8)



CUBO
10, 2 (2008)

Semi-classical Dispersive Estimates for the wave ... 9

for 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. On the other hand, proceeding as in the proof of (2.4),

one can easily see that (2.8) implies (1.6). To prove (1.7) we decompose Ej − E
0
j

as follows

Ej (t,h) − E
0
j
(t,h) = −h

∫
t

0

U0(t − τ,h)V (T (h) − Id)Ej−1(τ,h)dτ

+h

∫
t

0

U0(t − τ,h)V (Ej−1(τ,h) − E
0
j−1(τ,h))dτ := E

1
j
(t,h) + E2

j
(t,h). (2.9)

Using (2.8), in the same way as in the proof of (1.6), one gets

∥∥E1
j
(t,h)

∥∥
L1→L∞

≤ Cjh
j+2−n

|t|−(n−1)/2. (2.10)

On the other hand, it is easy to see from (2.9) by induction in j that we have the estimate

∥∥〈x〉−s−ǫ(Ej (t,h) − E0j (t,h))
∥∥

L
1
→L

2
≤ Cjh

j+2−n/2
|t|−s, j ≥ 0, (2.11)

for 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. It follows from (2.11) that the operator E2
j

satisfies the

estimate ∥∥E2
j
(t,h)

∥∥
L

1
→L

∞
≤ Cjh

j+2−n
|t|−(n−1)/2. (2.12)

Now (1.7) follows from (2.9), (2.10) and (2.12).

Proof of Corollary 1.2. Following [16] we set

Φ(t,h) = eit
√

Gϕ(h
√
G) − eit

√

G0ϕ(h
√
G0).

It follows from (1.4) and (1.8) that the operator Φ satisfies the estimate

‖Φ(t,h)‖
L

1
→L

∞ ≤ Ch
k−n+1

|t|
−(n−1)/2

. (2.13)

On the other hand, we have (see Theorem 3.1 of [16])

‖Φ(t,h)‖
L2→L2

≤ Ch, ∀t. (2.14)

By interpolation between (2.13) and (2.14) we conclude

‖Φ(t,h)‖
L

p
′
→L

p ≤ Ch
1−α(n−k)

|t|
−α(n−1)/2

, (2.15)

for every 2 ≤ p ≤ +∞, where 1/p + 1/p′ = 1, α = 1 − 2/p. Now we will make use of the identity

σ−α((n+1)/2+q)χa(σ) =

∫ 1

0

ϕ(θσ)θα((n+1)/2+q)−1dθ,

where ϕ(σ) = σ1−α((n+1)/2+q)χ′
a
(σ) ∈ C∞0 ((0, +∞)). By (2.15) we get

∥∥∥eit
√

G
(
√
G)−α((n+1)/2+q)χa(

√
G) − eit

√

G0 (
√
G0)

−α((n+1)/2+q)χa(
√
G0)

∥∥∥
Lp

′
→Lp



10 F. Cardoso and G. Vodev CUBO
10, 2 (2008)

≤

∫ 1

0

‖Φ(t,θ)‖
Lp

′
→Lp

θα((n+1)/2+q)−1dθ

≤ C|t|−α(n−1)/2
∫ 1

0

θ−α((n−1)/2−k−q)dθ ≤ C|t|−α(n−1)/2, (2.16)

provided α((n − 1)/2 − k − q) < 1, that is, for 2 ≤ p <
2(n−1−2q−2k)

n−3−2q−2k
. Now (1.10) follows from

(2.16) and the fact that it holds for the free operator. Similarly, by (2.13) we get

∥∥∥eit
√

G
(
√
G)

k−n+1−ǫ
χa(

√
G) − e

it
√

G0 (

√
G0)

k−n+1−ǫ
χa(

√
G0)

∥∥∥
L

1
→L

∞

≤

∫ 1

0

‖Φ(t,θ)‖
L1→L∞

θn−k−2+ǫdθ

≤ C|t|−(n−1)/2
∫ 1

0

θ−1+ǫdθ ≤ Cǫ|t|
−(n−1)/2. (2.17)

Now (1.9) follows from (2.17) and the fact that it holds for the free operator. 2

3 Semi-classical expansion of eitGψ(h2G)

We keep the same notations as in the introduction. We will make use of the following identity

which can be derived easily from Duhamel’s formula (see [15])

(
Id + ψ1(h

2G0) − ψ1(h
2G)

)
eitGψ(h2G)

= F0(t,h) + i

∫
t

0

W0(t − τ,h)V e
iτ Gψ(h2G)dτ. (3.1)

We rewrite (3.1) as follows

e
itG
ψ(h

2
G) = F̃0(t,h) +

∫
t

0

W̃0(t − τ,h)V e
iτ G

ψ(h
2
G)dτ, (3.2)

where

F̃0(t,h) = T (h)F0(t,h), W̃0(t,h) = iT (h)W0(t,h).

Iterating (3.2) m times leads to the identity

eitGψ(h2G) =

m∑

j=0

F̃j (t,h) + Rm+1(t,h), (3.3)

where the operators F̃j , j ≥ 1, are defined by

F̃j (t,h) =

∫
t

0

W̃0(t − τ,h)V F̃j−1(τ,h)dτ,

while the operators Rm, m ≥ 0, are defined as follows

R0(t,h) = e
itG
ψ(h

2
G),



CUBO
10, 2 (2008)

Semi-classical Dispersive Estimates for the wave ... 11

Rm+1(t,h) =

∫
t

0

W̃0(t − τ,h)V Rm(τ,h)dτ.

It is clear from (3.3) that the estimate (1.15) follows from the following

Proposition 3.1 Under the assumptions of Theorem 1.2, for all 0 < h ≤ h0, t 6= 0, 1/2 − ǫ/4 ≤

s ≤ (n − 1)/2, 0 < ǫ ≪ 1, we have the estimates

‖Rm+1(t,h)‖L1→L∞ ≤ Cmh
m−(n−2)/2−ǫ

|t|
−n/2

, (3.4)

∥∥∥〈x〉−1/2−s−ǫRm+1(t,h)
∥∥∥

L1→L2
≤ Cmh

m+s−(n−3)/2−ǫ/6
|t|−s−1/2, (3.5)

for every integer m ≥ 0.

Proof. For m = 0 these estimates are proved in Section 4 of [15]. We will now derive (3.4) for

m ≥ 1 from (3.5) and the following estimate proved in [15] (see (2.1)):

∥∥∥eitG0ψ(h2G0)〈x〉−1/2−s−ǫ
∥∥∥

L2→L∞
≤ Chs−(n−1)/2|t|−s−1/2, t 6= 0, 0 < h ≤ 1, (3.6)

for 0 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. We have

|t|n/2 ‖Rm+1(t,h)‖L1→L∞

≤ C

∫
t

t/2

|τ|n/2
∥∥∥W̃0(t − τ,h)〈x〉−1−ǫ

∥∥∥
L2→L∞

∥∥∥〈x〉−n/2−ǫRm(τ,h)
∥∥∥

L1→L2
dτ

+C

∫
t

t/2

|τ|
n/2

∥∥∥W̃0(τ,h)〈x〉−n/2−ǫ
∥∥∥

L
2
→L

∞

∥∥〈x〉−1−ǫRm(t − τ,h)
∥∥

L
1
→L

2
dτ

≤ Chm−ǫ/6
∫

∞

−∞

∥∥∥W̃0(τ′,h)〈x〉−1−ǫ
∥∥∥

L
2
→L

∞

dτ′

+C

∫
∞

−∞

∥∥〈x〉−1−ǫRm(τ′,h)
∥∥

L
1
→L

2
dτ

′

≤ Ch
m−(n−2)/2−ǫ/3

.

We will now show that (3.5) for Rm+1 follows from (3.5) for Rm and the following estimate proved

in [15] (see (2.2)):

∥∥〈x〉−seitG0ψ(h2G0)〈x〉−s
∥∥

L
2
→L

2
≤ C〈t/h〉−s, ∀t, 0 < h ≤ 1. (3.7)

We have

|t|s+1/2
∥∥∥〈x〉−1/2−s−ǫRm+1(t,h)

∥∥∥
L1→L2

≤ C

∫
t

t/2

|τ|
s+1/2

∥∥∥〈x〉−1/2−s−ǫW̃0(t − τ,h)〈x〉−1−ǫ
∥∥∥

L
2
→L

2

∥∥∥〈x〉−1/2−s−ǫRm(τ,h)
∥∥∥

L
1
→L

2

dτ

+C

∫
t

t/2

|τ|s+1/2
∥∥∥〈x〉−1/2−s−ǫW̃0(τ,h)〈x〉−1/2−s−ǫ

∥∥∥
L2→L2

∥∥〈x〉−1−ǫRm(t − τ,h)
∥∥

L1→L2
dτ



12 F. Cardoso and G. Vodev CUBO
10, 2 (2008)

≤ Chm+s−(n−1)/2−ǫ/6
∫

∞

−∞

〈τ′/h〉−1−ǫ/2dτ′

+Chs+1/2
∫

∞

−∞

∥∥〈x〉−1−ǫRm(τ′,h)
∥∥

L1→L2
dτ′ ≤ Chm+s−(n−3)/2−ǫ/6.

2

To prove (1.17) observe that in the same way as in the proof of (3.5) one can show that the

operator Fj satisfies the estimate

∥∥∥〈x〉−1/2−s−ǫFj (t,h)
∥∥∥

L
1
→L

2

≤ Cjh
j+s−(n−1)/2−ǫ/6

|t|
−s−1/2

, j ≥ 1, (3.8)

for 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. On the other hand, proceeding as in the proof of (3.4),

one can easily see that (3.8) implies (1.17). To prove (1.18) we decompose Fj − F
0
j

as follows

Fj (t,h) − F
0
j
(t,h) = i

∫
t

0

W0(t − τ,h)V (T (h) − Id)Fj−1(τ,h)dτ

−i

∫
t

0

W0(t − τ,h)V (Fj−1(τ,h) − F
0
j−1(τ,h))dτ := N

1
j
(t,h) + N2

j
(t,h). (3.9)

Using (3.8), in the same way as in the proof of (1.17), one gets

∥∥N1
j
(t,h)

∥∥
L1→L∞

≤ Cjh
j+2−n/2−ǫ

|t|−n/2. (3.10)

On the other hand, it is easy to see from (3.9) by induction in j that we have the estimate

∥∥∥〈x〉−1/2−s−ǫ(Fj (t,h) − F 0j (t,h))
∥∥∥

L
1
→L

2

≤ Cjh
j+2+s−(n−1)/2−ǫ/6

|t|
−s−1/2

, j ≥ 0, (3.11)

for 1/2 − ǫ/4 ≤ s ≤ (n − 1)/2, 0 < ǫ ≪ 1. It follows from (3.11) that the operator N2
j

satisfies the

estimate ∥∥N2
j
(t,h)

∥∥
L1→L∞

≤ Cjh
j+2−n/2−ǫ

|t|−n/2. (3.12)

Now (1.18) follows from (3.9), (3.10) and (3.12).

Proof of Corollary 1.4. Following [15] we set

Ψ(t,h) = eitGψ(h2G) − eitG0ϕ(h2G0).

It follows from (1.15) and (1.19) that the operator Ψ satisfies the estimate

‖Ψ(t,h)‖
L

1
→L

∞ ≤ Ch
k−(n−3)/2

|t|−n/2. (3.13)

On the other hand, we have (see Theorem 3.1 of [15])

‖Ψ(t,h)‖
L

2
→L

2 ≤ Ch, ∀t. (3.14)



CUBO
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Semi-classical Dispersive Estimates for the wave ... 13

By interpolation between (3.13) and (3.14) we conclude

‖Ψ(t,h)‖
L

p
′
→L

p ≤ Ch
1−α((n−1)/2−k)

|t|
−αn/2

, (3.15)

for every 2 ≤ p ≤ +∞, where 1/p + 1/p′ = 1, α = 1 − 2/p. Now we will make use of the identity

σ−αq/2χa(σ) =

∫ 1

0

ψ(θσ)θαq/2−1dθ,

where ψ(σ) = σ1−αq/2χ′
a
(σ) ∈ C∞0 ((0, +∞)). By (3.15) we get

∥∥∥eitGG−αq/2χa(G) − eitG0G−αq/20 χa(G0)
∥∥∥

L
p
′
→L

p

≤

∫ 1

0

∥∥∥Ψ(t,
√
θ)

∥∥∥
Lp

′
→Lp

θαq/2−1dθ

≤ C|t|
−αn/2

∫ 1

0

θ
−1/2−α((n−1)/2−k−q)/2

dθ ≤ C|t|
−αn/2

, (3.16)

provided 1/2 + α((n − 1)/2 − k − q)/2 < 1, that is, for 2 ≤ p <
2(n−1−2q−2k)

n−3−2q−2k
. Now (1.21) follows

from (3.16) and the fact that it holds for the free operator. Similarly, by (3.13) we get

∥∥∥eitGGk/2−(n−3)/4−ǫχa(G) − eitG0Gk/2−(n−3)/4−ǫ0 χa(G0)
∥∥∥

L1→L∞

≤

∫ 1

0

∥∥∥Ψ(t,
√
θ)

∥∥∥
L

1
→L

∞

θ−k/2+(n−3)/4−1+ǫdθ

≤ C|t|−n/2
∫ 1

0

θ−1+ǫdθ ≤ Cǫ|t|
−n/2. (3.17)

Now (1.20) follows from (3.17) and the fact that it holds for the free operator. By (1.15), (1.19),

(1.23) and (1.24), we have

∥∥Ψ(t,h) − Fk(t)ψ(h2G0)
∥∥

L
1
→L

∞
≤ Chk−(n−3)/2+ε|t|−n/2. (3.18)

Proceeding as above with a suitably chosen function ψ, we obtain from (3.18)

∥∥∥eitGGk/2−(n−3)/4χa(G) − eitG0Gk/2−(n−3)/40 χa(G0) − Fk(t)G
k/2−(n−3)/4
0

∥∥∥
L

1
→L

∞

≤

∫ 1

0

∥∥∥Ψ(t,
√
θ) − Fk(t)ψ(θG0)

∥∥∥
L

1
→L

∞

θ−k/2+(n−3)/4−1dθ

≤ C|t|
−n/2

∫ 1

0

θ
−1+ε/2

dθ ≤ C|t|
−n/2

. (3.19)

Now (1.25) follows from (3.19), (1.22) and the fact that it holds for the free operator. 2

Received: October 2007. Revised: December 2007.



14 F. Cardoso and G. Vodev CUBO
10, 2 (2008)

References

[1] M. Beals, Optimal L∞ decay estimates for solutions to the wave equation with a potential,

Commun. Partial Diff. Equations 19 (1994), 1319–1369.

[2] F. Cardoso, C. Cuevas and G. Vodev, Dispersive estimates of solutions to the wave

equation with a potential in dimensions two and three, Serdica Math. J. 31 (2005), 263–278.

[3] F. Cardoso, C. Cuevas and G. Vodev, Weighted dispersive estimates for solutions of the

Schrödinger equation, Serdica Math. J., 34 (2008), 39–54.

[4] P. D’ancona and V. Pierfelice, On the wave equation with a large rough potential, J.

Funct. Analysis 227 (2005), 30–77.

[5] V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential,

Commun. Partial Diff. Equations 28 (2003), 1325–1369.

[6] M. Goldberg, Dispersive bounds for the three dimensional Schrödinger equation with almost

critical potentials, GAFA 16 (2006), 517–536.

[7] M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions

one and three, Commun. Math. Phys. 251 (2004), 157–178.

[8] M. Goldberg and M. Visan, A conterexample to dispersive estimates for Schrödinger op-

erators in higher dimensions, Commun. Math. Phys. 266 (2006), 211–238.

[9] J.-L. Journé, A. Soffer and C. Sogge, Decay estimates for Schrödinger operators, Com-

mun. Pure Appl. Math. 44 (1991), 573–604.

[10] S. Moulin, High frequency dispersive estimates in dimension two, submitted.

[11] S. Moulin and G. Vodev, Low frequency dispersive estimates for the Schrödinger group in

higher dimensions, Asymptot. Anal., 55 (2007), 49–71.

[12] W. Schlag, Dispersive estimates for Schrödinger operators in two dimensions, Commun.

Math. Phys. 257 (2005), 87–117.

[13] I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with

rough and time-dependent potentials, Invent. Math. 155 (2004), 451–513.

[14] G. Vodev, Dispersive estimates of solutions to the Schrödinger equation, Ann. H. Poincaré

6 (2005), 1179–1196.

[15] G. Vodev, Dispersive estimates of solutions to the Schrödinger equation in dimensions n ≥ 4,

Asymptot. Anal. 49 (2006), 61–86.

[16] G. Vodev, Dispersive estimates of solutions to the wave equation with a potential in dimen-

sions n ≥ 4, Commun. Partial Diff. Equations 31 (2006), 1709–1733.


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