Low frequency dispersive estimates for the Schrödinger group in
higher dimensions
Simon Moulin and Georgi Vodev
Abstract
For a large class of real-valued potentials, V (x), x ∈ Rn, n ≥ 4 , we prove dispersive
estimates for the low frequency part of eit(−∆+V )Pac, provided the zero is neither an eigen-
value nor a resonance of −∆+ V , where Pac is the spectral projection onto the absolutely
continuous spectrum of −∆ + V . This class includes potentials V ∈ L∞(Rn) satisfying
V (x) = O
〈x〉−(n+2)/2−ǫ
, ǫ > 0. As a consequence, we extend the results in [4] to a larger
class of potentials.
1 Introduction and statement of results
Let V ∈ L∞(Rn), n ≥ 4, be a real-valued function satisfying
|V (x)| ≤ C〈x〉−δ , ∀x ∈ Rn, (1.1)
with constants C > 0, δ > (n + 2)/2. Denote by G0 and G the self-adjoint realizations of
the operators −∆ and −∆+V on L2(Rn), respectively. It is well known that the absolutely
continuous spectrums of the operators G0 and G coincide with the interval [0,+∞), and that
G has no embedded strictly positive eigenvalues nor strictly positive resonances. However,
G may have in general a finite number of non-positive eigenvalues and that the zero may be
a resonance. We will say that the zero is a regular point for G if it is neither an eigenvalue
nor a resonance in the sense that the operator 1− V∆−1 is invertible on L1 with a bounded
inverse. Let Pac denote the spectral projection onto the absolutely continuous spectrum of
G. When n ≥ 3, Journé, Sofer and Sogge [4] proved the following dispersive estimate
∥∥eitGPac
L1→L∞
≤ C|t|−n/2, t 6= 0, (1.2)
provided the zero is neither an eigenvalue nor a resonance, for potentials satisfying (1.1) with
δ > n+ 4 as well as the condition
V̂ ∈ L1. (1.3)
This was later improved by Yajima [9] for potentials satisfying (1.1) with δ > n+ 2. When
n = 3, the estimate (1.2) in fact holds without (1.3). In this case, it was proved in [2] for
potentials satisfying (1.1) with δ > 3 and was later improved in [6] and [10] for potentials
satisfying (1.1) with δ > 5/2. Goldberg [1] has recently showed that (1.2) holds for potentials
V ∈ L3/2−ǫ∩L3/2+ǫ, 0 < ǫ≪ 1, which includes potentials satisfying (1.1) with δ > 2. When
n = 2, (1.2) is proved by Schlag [5] for potentials satisfying (1.1) with δ > 3.
Given any a > 0, set χa(σ) = χ1(σ/a), where χ1 ∈ C∞(R), χ1(σ) = 0 for σ ≤ 1,
χ1(σ) = 1 for σ ≥ 2. Set ηa = χ(1 − χa), where χ denotes the characteristic function of the
interval [0,+∞). Clearly, ηa(G) + χa(G) = Pac. When n ≥ 4, dispersive estimates with loss
of (n − 3)/2 derivatives for the operator eitGχa(G), ∀a > 0, have been recently proved in
http://arxiv.org/abs/0704.1200v2
[7] under the assumption (1.1), only. The loss of derivatives in this case is a high frequency
phenomenon and cannot be avoided unless one imposes some regularity condition on the
potential (see [3]). The condition (1.3) in [4] plays this role but it seems too strong. The
natural conjecture would be that we have dispersive estimates for eitGχa(G) with loss of ν
derivatives, 0 ≤ ν ≤ (n − 3)/2, provided V ∈ C(n−3)/2−ν(Rn) (with a suitable decay at
infinity). It turns out that no regularity on the potential is needed in order to get dispersive
estimates for the low frequency part eitGηa(G), a > 0 small. One just needs some decay
at infinity. In fact, the low frequency analysis turns out to be easier in dimensions n ≥ 4
compared with the cases of n = 2 and n = 3, and can be carried out for a larger class of
potentials satisfying (with some 0 < ǫ≪ 1)
|x− y|−n+2 + |x− y|−(n−2)/2+ǫ
|V (x)|dx ≤ C < +∞. (1.4)
Clearly, (1.4) is fulfilled for potentials satisfying (1.1). Our main result is the following
Theorem 1.1 Let n ≥ 4, let V satisfy (1.4) and assume that the zero is a regular point for
G. Then, there exists a constant a0 > 0 so that for 0 < a ≤ a0 we have the estimate
∥∥eitGηa(G)
L1→L∞
≤ C|t|−n/2, t 6= 0. (1.5)
Remark 1. We expect that (1.5) holds true for the larger class of potentials satisfying
|x− y|−n+2 + |x− y|−(n−1)/2
|V (x)|dx ≤ C < +∞, (1.6)
but the proof in this case would require a different approach.
Combining (1.5) with the estimates of [7], we obtain the following
Theorem 1.2 Let n ≥ 4, let V satisfy (1.1) and assume that the zero is a regular point for
G. Then, we have the estimates, ∀t 6= 0, 0 < ǫ≪ 1,
∥∥eitGPacf
≤ C|t|−n/2
∥∥∥〈G〉(n−3)/4f
, (1.7)
∥∥eitGPacf
≤ Cǫ|t|−n/2
∥∥∥〈x〉n/2+ǫf
. (1.8)
Remark 2. The proof in [7] is based on uniform estimates for the operator ψ(h2G), 0 <
h ≤ 1, ψ ∈ C∞0 ((0,+∞)) (see Lemma 2.2 of [7] or Lemma 2.3 of [8]). In the proof of this
lemma (which is given in [8]), however, there is a mistake. That is why, we will give a new
proof in Appendix 1 of the present paper.
Remark 3. We conjecture that the estimates (1.7) and (1.8) hold true for potentials satis-
fying (1.1) with δ > (n+ 1)/2.
Theorem 1.1 also allows to extend the results in [4] to a larger class of potentials. More
precisely, we have the following
Theorem 1.3 Let n ≥ 4, let V satisfy (1.1) with δ > n − 1 as well as (1.3), and assume
that the zero is a regular point for G. Then, the estimate (1.2) holds true.
Theorem 1.3 follows from (1.5) and the dispersive estimate for eitGχa(G) proved in Ap-
pendix 2.
To prove (1.5) we adapt the semi-classical approach of [7] based on the semi-classical
version of Duhamel’s formula (which in our case is of the form (3.4) or (3.5)). While in [7]
the estimates had to be uniform with respect to the semi-classical parameter 0 < h ≤ 1,
in the case of low frequency we need to make them uniform for h ≫ 1 (see (3.1)). This,
however, turns out to be easier (when n ≥ 4) as we can absorb the remaining terms taking
h big enough (see Section 3). That is why, we do not need any more to work on weighted
L2 spaces (as in [7]), which in turn allows to cover a much larger class of potentials. As
mentioned in Remark 1, the natural class of potentials for which the low frequency analysis
works out (for n ≥ 4) is given by (1.6), and the fact that the crucial Proposition 2.1 below
holds true under (1.6) is a strong indication for that. In fact, (1.4) is used in the proof of
Proposition 2.3, only.
2 Preliminary estimates
Let ψ ∈ C∞0 ((0,+∞)). We will first prove the following
Proposition 2.1 Let n ≥ 4, let V satisfy (1.6) and assume that the zero is a regular point
for G. Then, there exist positive constants C, β and h0 so that the following estimates hold
∥∥ψ(h2G0)
L1→L1
≤ C, h > 0, (2.1)
∥∥ψ(h2G)
L1→L1
≤ C, h ≥ h0, (2.2)
∥∥ψ(h2G)− ψ(h2G0)T
L1→L1
≤ Ch−β, h ≥ h0, (2.3)
where the operator
1− V∆−1
: L1 → L1 (2.4)
is bounded by assumption.
Proof. Set ϕ(λ) = ψ(λ2). We are going to take advantage of the formula
ψ(h2G) =
(z)(h2G− z2)−1zL(dz), (2.5)
where L(dz) denotes the Lebesgue measure on C, ϕ̃ ∈ C∞0 (C) is an almost analytic contin-
uation of ϕ supported in a small complex neighbourhood of suppϕ and satisfying
∣∣∣∣ ≤ CN |Im z|
N , ∀N ≥ 1.
For ±Im z > 0, denote
R±0,h(z) = (h
2G0 − z2)−1, R±h (z) = (h
2G− z2)−1.
The kernel of the operator R±0,h(z) is of the form R
h (|x− y|, z), where
R±h (σ, z) = ±h
−2 iσ
4(2π)ν
H±ν (σz/h) = h−nR±1 (σh−1, z),
where ν = (n − 2)/2, H±ν (λ) = λνH±ν (λ), H±ν being the outgoing and incoming Henkel
functions of order ν. It is well known that these functions satisfy the bound
∣∣H±ν (λ)
∣∣ ≤ C〈λ〉(n−3)/2e−|Imλ|, ∀λ, ±Imλ ≥ 0, (2.6)
while near λ = 0 they are of the form
H±ν (λ) = a±ν,1(λ) + λn−2 logλa
ν,2(λ), (2.7)
where a±ν,j are analytic functions, a
ν,2 ≡ 0 if n is odd. By (2.6) and (2.7), we have
∣∣H±ν (λ)−H±ν (0)
∣∣ ≤ C|λ|1/2〈λ〉(n−4)/2, ∀λ, ±Imλ ≥ 0. (2.8)
Hence, the functions R±h satisfy the bounds (for z ∈ C±ϕ := {z ∈ supp ϕ̃, ±Im z ≥ 0}, σ > 0,
h ≥ 1) ∣∣R±h (σ, z)
∣∣ ≤ Ch−2
σ−n+2 + σ−(n−1)/2
, (2.9)
∣∣R±h (σ, z)−R
h (σ, 0)
∣∣ ≤ Ch−5/2
σ−n+5/2 + σ−(n−1)/2
. (2.10)
Using the above bounds we will prove the following
Lemma 2.2 For z ∈ C±ϕ , we have
‖VR0,h(z)‖L1→L1 ≤ Ch
−2, h ≥ 1, (2.11)
‖VR0,h(z)− VR0,h(0)‖L1→L1 ≤ Ch
−5/2, h ≥ 1, (2.12)
‖VRh(z)‖L1→L1 ≤ Ch
−2, h ≥ h0, (2.13)
∥∥∥R±0,h(z)
L1→L1
≤ C|Im z|−q, h > 0, Im z 6= 0, (2.14)
∥∥R±h (z)
L1→L1
≤ C|Im z|−q, h ≥ h0, Im z 6= 0, (2.15)
with constants C, q, h0 > 0 independent of z and h.
Proof. In view of (2.9), the norm in the LHS of (2.11) is upper bounded by
|V (x)|
∣∣R±h (|x − y|, z)
≤ Ch−2 sup
|V (x)|
|x− y|−n+2 + |x− y|−(n−1)/2
dx ≤ Ch−2.
The estimate (2.12) follows in the same way using (2.10). To prove (2.14), we use (2.6) to
get (for z ∈ C±ϕ , Im z 6= 0)
∣∣R±1 (σ, z)
∣∣ ≤ Cσ−2ν〈σ〉(n−3)/2e−σ|Im z| ≤ Cσ−2ν〈σ〉−5/2|Im z|−(n+2)/2. (2.16)
By (2.16), the norm in the LHS of (2.14) is upper bounded by
∣∣R±h (σ, z)
∣∣ dσ = C
∣∣R±1 (σ, z)
∣∣ dσ
≤ C|Im z|−(n+2)/2
〈σ〉−3/2dσ ≤ C|Im z|−(n+2)/2.
To prove (2.13) and (2.15), we will use the identity
R±h (z)
1 + h2VR±0,h(z)
= R±0,h(z), ±Im z > 0. (2.17)
Observe that 1 + h2VR±0,h(0) = 1 − V∆−1, which is supposed to be invertible on L1 with
a bounded inverse denoted by T . Thus, it follows from (2.12) that there exists a constant
h0 > 0 so that for h ≥ h0 the operator 1 + h2VR±0,h(z) is invertible on L1 with an inverse
satisfying ∥∥∥∥
1 + h2VR±0,h(z)
)−1∥∥∥∥
L1→L1
≤ C, z ∈ C±ϕ , (2.18)
with a constant C > 0 independent of z and h. Hence, we can write
R±h (z) = R
0,h(z)
1 + h2VR±0,h(z)
. (2.19)
Now (2.13) follows from (2.11), (2.18) and (2.19), while (2.15) follows from (2.14), (2.18) and
(2.19). ✷
Clearly, (2.1) and (2.2) follow from (2.5) and (2.14), (2.15), respectively. To prove (2.3)
we rewrite the identity (2.19) in the form
R±h (z)−R
0,h(z)T
= R±0,h(z)T
h2VR±0,h(z)− h
2VR±0,h(0)
h2VR±0,h(z)− h
2VR±0,h(0)
(2.20)
By Lemma 2.2, (2.18) and (2.20) we conclude
∥∥∥R±h (z)−R
0,h(z)T
L1→L1
≤ Ch−β |Im z|−q, h ≥ h0, z ∈ C±ϕ , Im z 6= 0, (2.21)
with constants C, q, β > 0 independent of z and h. Now (2.3) follows from (2.5) and (2.21).
Let ψ1 ∈ C∞0 ((0,+∞)), ψ1 = 1 on suppψ.
Proposition 2.3 Under the assumptions of Theorem 1.1, there exist positive constants h0
and β so that we have the estimates
∥∥V eitG0ψ(h2G0)
L1→L1
dt ≤ Ch−β, h ≥ 1, (2.22)
∥∥V ψ(h2G)eitG0ψ1(h2G0)
L1→L1
dt ≤ Ch−β , h ≥ h0. (2.23)
Proof. It is shown in [7] (Section 2) that the kernel of the operator eitG0ψ(h2G0) is of
the form Kh(|x− y|, t) with a function Kh satisfying
Kh(σ, t) = h
−nK1(σh
−1, th−2),
|K1(σ, t)| ≤ C|t|−s−1/2σs−(n−1)/2, 0 ≤ s ≤ (n− 1)/2, σ > 0, t 6= 0.
Hence, for all 0 ≤ s ≤ (n− 1)/2, σ > 0, t 6= 0, h > 0, we have
|Kh(σ, t)| ≤ Chs−(n−1)/2|t|−s−1/2σs−(n−1)/2,
which together with (1.4) imply
∥∥V eitG0ψ(h2G0)
L1→L1
≤ Chs−(n−1)/2|t|−s−1/2, 1/2− ǫ ≤ s ≤ 1/2 + ǫ, (2.24)
where 0 < ǫ≪ 1. Clearly, (2.22) follows from (2.24). Furthermore, using (2.5), (2.13), (2.14)
and (2.24), we get
ψ(h2G)− ψ(h2G0)
eitG0ψ1(h
L1→L1
≤ Ch2
∥∥∥VR±h (z)V e
itG0ψ1(h
2G0)R±0,h(z)
L1→L1
L(dz)
≤ Ch2
∥∥VR±h (z)
L1→L1
∥∥V eitG0ψ1(h2G0)
L1→L1
∥∥∥R±0,h(z)
L1→L1
L(dz)
≤ Chs−(n−1)/2|t|−s−1/2
∣∣∣∣ |Im z|
−qL(dz)
≤ Chs−(n−1)/2|t|−s−1/2, 1/2− ǫ ≤ s ≤ 1/2 + ǫ, (2.25)
which clearly implies (2.23). ✷
3 Proof of Theorem 1.1
Denote
Ψ(t, h) = eitGψ(h2G)− T ∗eitG0ψ(h2G0)T,
T being given by (2.4). We will first show that (1.5) follows from the following
Proposition 3.1 Under the assumptions of Theorem 1.1, there exist positive constants C,
h0 and β so that for h ≥ h0 we have
‖Ψ(t, h)‖L1→L∞ ≤ Ch
−β |t|−n/2, t 6= 0. (3.1)
Recall that χa(σ) = χ1(σ/a), a > 0 small. Then we can write the function ηa as follows
ηa(σ) =
ψ(σθ)
, σ > 0,
where ψ(σ) = σχ′1(σ) ∈ C∞0 ((0,+∞)). Thus, we obtain from (3.1),
∥∥eitGηa(G) − T ∗eitG0ηa(G0)T
L1→L∞
∥∥∥Ψ(t,
L1→L∞
≤ C|t|−n/2
θ−1−β/2dθ ≤ C|t|−n/2, (3.2)
provided a is taken small enough. Clearly, (1.5) follows from (3.2).
Proof of Proposition 3.1. We will first prove the following
Proposition 3.2 Under the assumptions of Theorem 1.1, there exist positive constants C,
h0 and β so that for h ≥ h0 we have
∥∥V eitGψ(h2G)
L1→L1
dt ≤ Ch−β. (3.3)
Proof. Using Duhamel’s formula
eitG = eitG0 + i
ei(t−τ)GV eiτG0dτ,
we get the identity
eitGψ(h2G) = ψ(h2G)eitG0ψ1(h
2G0)T + e
itGψ(h2G)
2G)− ψ1(h2G0)T
ψ(h2G)ei(t−τ)GV eiτG0ψ1(h
2G0)Tdτ. (3.4)
Using Propositions 2.1 and 2.3, (3.4) together with Young’s inequality we obtain
∥∥V eitGψ(h2G)
L1→L1
dt ≤ Ch−β + Ch−β
∥∥V eitGψ(h2G)
L1→L1
∥∥∥V ψ(h2G)ei(t−τ)G
L1→L1
∥∥V eiτG0ψ1(h2G0)
L1→L1
≤ Ch−β + Ch−β
∥∥V eitGψ(h2G)
L1→L1
which clearly implies (3.3) if we take h large enough. ✷
Using Duhamel’s formula
eitG = eitG0 + i
ei(t−τ)G0V eiτGdτ,
we get the identity
Ψ(t;h) =
Ψj(t;h), (3.5)
where
Ψ1(t;h) = T
∗ψ1(h
2G0)e
ψ(h2G)− ψ(h2G0)T
2G)− T ∗ψ1(h2G0)
eitGψ(h2G),
Ψ2(t;h) = i
T ∗ψ1(h
2G0)e
i(t−τ)G0V eiτGψ(h2G)dτ.
By (2.1) and (2.3) together with the well known estimate
∥∥eitG0
L1→L∞
≤ C|t|−n/2,
we get
‖Ψ1(t;h)f‖L∞ ≤ Ch
−β |t|−n/2‖f‖L1 + Ch−β ‖Ψ(t;h)f‖L∞ , t 6= 0. (3.6)
By Propositions 2.3 and 3.2, ∀f ∈ L1, t > 0, we have
tn/2 ‖Ψ2(t;h)f‖L∞
∫ t/2
(t− τ)n/2
∥∥∥ψ1(h2G0)ei(t−τ)G0
L1→L∞
∥∥V eiτGψ(h2G)f
∥∥∥ψ1(h2G0)ei(t−τ)G0V
L∞→L∞
∥∥eiτGψ(h2G)f
∥∥V eiτGψ(h2G)f
+C sup
t/2≤τ≤t
∥∥eiτGψ(h2G)f
∥∥V eiτG0ψ1(h2G0)
L1→L1
≤ Ch−β‖f‖L1 + Ch−β sup
t/2≤τ≤t
∥∥eiτGψ(h2G)f
. (3.7)
Combining (3.5), (3.6) and (3.7), we conclude, ∀f ∈ L1, t > 0,
tn/2 ‖Ψ(t;h)f‖L∞ ≤ Ch
−β‖f‖L1 + Ch−βtn/2 ‖Ψ(t;h)f‖L∞
+Ch−β sup
t/2≤τ≤t
τn/2 ‖Ψ(τ ;h)f‖L∞ . (3.8)
Taking h big enough we can absorb the second and the third terms in the RHS of (3.8), thus
obtaining (3.1). Clearly, the case of t < 0 can be treated in the same way. ✷
A Appendix 1
We will prove the following
Lemma A.1 Let ψ ∈ C∞0 ((0,+∞)). Then, for all h > 0, s ≥ 0, we have the estimates
∥∥ψ(h2G0)
L1→L1
≤ C, (A.1)
∥∥〈x〉sψ(h2G0)〈x〉−s
L1→L2
≤ Ch−n/2〈h〉s, (A.2)
where the constant C is of the form
C = C′ sup
0≤k≤k0
|∂kλψ(λ)|, (A.3)
with some integer k0 independent of ψ and a constant C
′ > 0 depending on the support of ψ,
only. Furthermore, if V satisfies (1.1) with δ > n/2, we have the estimates (for 0 < h ≤ 1)
∥∥ψ(h2G)− ψ(h2G0)
L1→L1
≤ Ch2, (A.4)
∥∥〈x〉δ
ψ(h2G)− ψ(h2G0)
L1→L2
≤ Ch2−n/2. (A.5)
Proof. The estimate (A.1) is proved in Section 2 using the formula (2.5) and (2.14). It
can be also seen by using the fact that the kernel of the operator ψ(h2G0) is of the form
kh(|x− y|) with a function kh satisfying
kh(σ) = h
−nk1(σ/h), (A.6)
|k1(σ)| ≤ Cm〈σ〉−m, ∀σ > 0, (A.7)
for all integers m ≥ 0, with a constant Cm of the form
Cm = C
m sup
0≤j≤jm
|∂jλψ(λ)|, (A.8)
where jm is some integer independent of ψ, while C
m > 0 depends on the support of ψ. By
Young’s inequality, the norm in the LHS of (A.1) is upper bounded by
|kh(|ξ|)|dξ =
|k1(|ξ|)|dξ ≤ Cn+1.
The norm in the LHS of (A.2) is upper bounded by
〈x〉2s〈y〉−2s|kh(|x− y|)|2dx
〈x− y〉2s|kh(|x − y|)|2dx
≤ C〈h〉s
〈ξ/h〉2s|kh(|ξ|)|2dξ
= C〈h〉sh−n/2
〈ξ〉2s|k1(|ξ|)|2dξ
≤ Csn〈h〉sh−n/2,
where sn is some integer depending on n and s. To prove (A.4) observe that by (2.5) we
ψ(h2G)− ψ(h2G0) =
(z)(h2G0 − z2)−1V (h2G− z2)−1zL(dz). (A.9)
Clearly, (A.4) would follow from (A.9), (2.14) and the estimate (for z ∈ supp ϕ̃)
∥∥(h2G− z2)−1
L1→L1
≤ C|Im z|−q, 0 < h ≤ 1, Im z 6= 0. (A.10)
Let φ ∈ C∞0 ([1, 2]) be such that
φ(θ2)θ−1dθ = 1. Given a parameter 0 < ε ≪ 1, we
decompose the free resolvent as follows
(h2G0 − z2)−1 = Aε(z;h) + Bε(z;h), (A.11)
where
Aε(z;h) =
(εθh)2G0; (εθ)
Bε(z;h) =
(εθh)2G0; (εθ)
where
f(λ;µ; z) =
λµ−1 − z2 .
It is easy to see that there exist constants 0 < µ1 < µ2 so that the function f satisfies the
following bounds ∣∣∣∂jλf(λ;µ; z)
∣∣∣ ≤ Cjµ, 0 < µ ≤ µ1, (A.12)
∣∣∣∂jλf(λ;µ; z)
∣∣∣ ≤ C′j |Im z|−j−1, µ1 ≤ µ ≤ µ2, (A.13)
∣∣∣∂jλf(λ;µ; z)
∣∣∣ ≤ C′′j , µ ≥ µ2, (A.14)
for every integer j ≥ 0. By (A.1), (A.3) and (A.12), we have
(εθh)2G0; (εθ)
L1→L1
≤ C(εθ)2, 0 < θ ≤ 1, (A.15)
provided ε > 0 is taken small enough. We deduce from (A.15),
‖Aε(z;h)‖L1→L1 ≤ Cε
2, z ∈ supp ϕ̃, (A.16)
with a constant C > 0 independent of z, h and ε. By (A.2), (A.3), (A.12)-(A.14), we have
∥∥〈x〉sf
(εθh)2G0; (εθ)
〈x〉−s
L1→L2
≤ C(εθh)−n/2〈εθh〉s|Im z|−q, (A.17)
with constants C, q > 0 independent of z, θ, h and ε. We deduce from (A.17),
‖Bε(z;h)‖L1→L2 ≤ C
−n/2|Im z|−q
θ−1−n/2dθ
≤ Cεh−n/2|Im z|−q, z ∈ supp ϕ̃, (A.18)
with a constant Cε > 0 independent of z and h. It follows from (A.16) that the operator
1+ h2VAε(z;h) is invertible on L1, provided ε > 0 is taken small enough, independent of h.
Therefore, we can write the identity
h2G− z2
h2G0 − z2
h2G− z2
h2G0 − z2
, (A.19)
in the form (
h2G− z2
= M(z;h) + h2
h2G− z2
)−1 N (z;h), (A.20)
where the operators
M(z;h) =
h2G0 − z2
)−1 (
1 + h2VAε(z;h)
N (z;h) = V Bε(z;h)
1 + h2VAε(z;h)
satisfy the estimates
‖M(z;h)‖L1→L1 + ‖N (z;h)‖L1→L1 ≤ C|Im z|
−q, (A.21)
‖N (z;h)‖L1→L2 ≤ Ch
−n/2|Im z|−q. (A.22)
By (A.20) we have
h2G− z2
M(z;h)N (z;h)j + h2J
h2G− z2
)−1 N (z;h)J
M(z;h)N (z;h)j + h2J
h2G0 − z2
)−1 N (z;h)J
+h2J+2
h2G0 − z2
h2G− z2
)−1 N (z;h)J , (A.23)
for every integer J ≥ 1. By (A.22) and (2.14), we obtain
h2G0 − z2
h2G− z2
)−1 N (z;h)
L1→L1
≤ ‖V ‖L2
h2G0 − z2
)−1∥∥∥
L1→L1
h2G− z2
)−1∥∥∥
L2→L2
‖N (z;h)‖L1→L2
≤ Ch−n/2|Im z|−q2 . (A.24)
Now, (A.10) follows from (A.21), (A.23) and (A.24).
To prove (A.5) we rewrite (A.20) in the form
h2G− z2
)−1 −
h2G0 − z2
Fj(z;h), (A.25)
where
F1(z;h) = h2Aε(z;h)VAε(z;h)
1 + h2VAε(z;h)
F2(z;h) = h2Bε(z;h)VAε(z;h)
1 + h2VAε(z;h)
F3(z;h) = h2
h2G− z2
V Bε(z;h)
1 + h2VAε(z;h)
It is easy to see that we have the estimate
∥∥∥〈x〉s
h2G− z2
)−1 〈x〉−s
L2→L2
≤ C|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1, (A.26)
for every s ≥ 0 with constants C, q > 0 depending on s but independent of z and h. By
(A.18) and (A.26),
∥∥〈x〉δF3(z;h)
L1→L2
≤ Ch2−n/2|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1. (A.27)
Observe now that we can write the operator Aε(z;h) in the form
Aε(z;h) = χ(3)ε (h2G0)
h2G0 − z2
where
χ(3)ε (σ) =
∫ εσ1/2
φ(θ2)
Similarly, we can decompose the operator Bε(z;h) as B(1)ε + B(2)ε , where
B(j)ε (z;h) = χ(j)ε (h2G0)
h2G0 − z2
, j = 1, 2,
χ(1)ε (σ) =
∫ ε−1σ1/2
εσ1/2
φ(θ2)
, χ(2)ε (σ) =
ε−1σ1/2
φ(θ2)
Taking ε > 0 small enough we can arrange that suppχ
ε ∩ suppϕ = ∅, j = 2, 3, so the
operator-valued functions Aε(z;h) and B(2)ε (z;h) are analytic on supp ϕ̃. Therefore, we can
write (A.9) in the form
ψ(h2G)− ψ(h2G0) =
(z)Fj(z;h)zL(dz), (A.28)
where
F4(z;h) = h2B(1)ε (z;h)VAε(z;h)
1 + h2VAε(z;h)
By (A.17) (with s = δ), we have
∥∥〈x〉δF4(z;h)
L1→L2
≤ Ch2
∥∥∥〈x〉δB(1)ε (z;h)〈x〉−δ
L1→L2
≤ Ch2−n/2|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1. (A.29)
Now (A.5) follows from (A.27)-(A.29). ✷
B Appendix 2
Combining some ideas from [6],[7] and [4] we will prove the following
Theorem B.1 Let n ≥ 4, let V satisfy (1.1) with δ > n − 1 as well as (1.3). Then, for
every a > 0 we have the estimate
∥∥eitGχa(G)
L1→L∞
≤ C|t|−n/2, t 6= 0. (B.1)
Remark. Note that (B.1) is proved in [4] for potentials satisfying (1.1) with δ > n, the
condition (1.3) as well as an extra technical assumption. Here we eliminate this extra as-
sumption.
Proof. The key point in the proof in [4] is the bound
∥∥e−itG0V eitG0
L1→L1
≤ ‖V̂ ‖L1 , ∀t. (B.2)
Combining (B.2) with Duhamel’s formula one easily gets
∥∥e−itG0V eitG
L1→L1
≤ C, |t| ≤ 1, (B.3)
with a constant C > 0 independent of t. In what follows we will derive (B.1) from (B.2) and
(B.3). To this end, given a function ψ ∈ C∞0 ((0,+∞)) and a parameter 0 < h ≤ 1, as in [6],
[7], denote
Ψ(t, h) = eitGψ(h2G)− eitG0ψ(h2G0),
F (t) = i
ei(t−τ)G0V eiτG0dτ.
As in these papers, it is easy to see that (B.1) follows from the following
Theorem B.2 Under the assumptions of Theorem B.1, there exist constants C, β > 0 so
that we have the estimates (for 0 < h ≤ 1, t 6= 0)
‖F (t)‖L1→L∞ ≤ C|t|
−n/2, (B.4)
∥∥Ψ(t, h)− F (t)ψ(h2G0)
L1→L∞
≤ Chβ |t|−n/2. (B.5)
Proof. Clearly, (B.4) follows from (B.2) for |t| ≤ 2. Let |t| ≥ 2. Without loss of generality
we may suppose t ≥ 2. Write F = F1 + F2, where
F1(t) = i
∫ t−1
ei(t−τ)G0V eiτG0dτ,
F2(t) = i
ei(t−τ)G0V eiτG0dτ.
It follows from (B.2) that F2(t) satisfies (B.4). To deal with the operator F1(t), observe that
its kernel is of the form
U(|x− ξ|2/4, |y − ξ|2/4, t)V (ξ)dξ,
where cn is a constant and
U(σ1, σ2, t) =
∫ t−1
eiσ1/(t−τ)+iσ2/τ (t− τ)−n/2τ−n/2dτ.
To prove that F1(t) satisfies (B.4), it suffices to show that
|U(σ1, σ2, t)| ≤ Ct−n/2
1 + σ
, ∀σ1, σ2 > 0, t ≥ 2. (B.6)
To do so, observe that
U(σ1, σ2, t) = t
u(σ1t
−1, σ2t
−1, t−1) + u(σ2t
−1, σ1t
−1, t−1)
, (B.7)
where
u(σ′1, σ
2, κ) =
∫ 1/2
/(1−τ ′)+iσ′
/τ ′(1− τ ′)−n/2(τ ′)−n/2dτ ′.
It is easy to see that (B.6) follows from (B.7) and the bound
|u(σ′1, σ′2, κ)| ≤ Cκ−(n−3)/2(σ′2)−1/2, ∀σ′1, σ′2 > 0, 0 < κ ≤ 1/2. (B.8)
To prove (B.8), we make a change of variables µ = 1/τ ′ and write the function u in the form
u(σ′1, σ
2, κ) =
∫ κ−1
eiϕ(µ,σ
µn/2−2dµ,
where
ϕ(µ, σ′1, σ
2) = µσ
µ− 1σ
We have
|u(σ′1, σ′2, κ)| ≤ C
∫ κ−1
µn/2−2dµ ≤ Cκ−(n−2)/2. (B.9)
Furthermore, observe that
ϕ′(µ) =
= σ′2 −
(µ− 1)2 ,
so ϕ′ vanishes at µ0 = 1 + (σ
1/2. We will consider now two cases.
Case 1. µ0 6∈ [3/2, 3κ−1/2]. Then, we have
|ϕ′(µ)| ≥ σ′2
|µ− µ0|
µ− 1 ≥
, µ ∈ [2, κ−1].
Therefore, integrating by parts, we obtain
u(σ′1, σ
2, κ) =
∫ κ−1
(iϕ′)−1
µn/2−2deiϕ
= eiϕ(iϕ′)−1
µn/2−2
∣∣∣∣∣
∫ κ−1
eiϕf(µ)dµ, (B.10)
where
f(µ) =
(iϕ′)−1
µn/2−2
= (iϕ′)−1
µn/2−2
µn/2−2.
Since ∣∣∣∣
∣∣∣∣ ≤
2σ′1(µ− 1)−2
(µ− 1) |σ′2 − σ′1(µ− 1)−2|
µ− 1 ,
we have (for µ ≥ 2)
|f(µ)| ≤ C(σ′2)−1µn/2−3. (B.11)
By (B.10) and (B.11),
|u(σ′1, σ′2, κ)| ≤ C(σ′2)−1κ−(n−4)/2. (B.12)
Clearly, in this case (B.8) follows from (B.9) and (B.12).
Case 2. µ0 ∈ [3/2, 3κ−1/2]. Denote I(µ0) = [9µ0/10, 11µ0/10] ∩ [2, κ−1]. We write the
function u as u1 + u2, where
I(µ0)
µn/2−2dµ = µ
n/2−1
Ĩ(µ0)
eiλφ(z)g(z)dz, (B.13)
where we have made a change of variables µ = µ0(1 + z), Ĩ(µ0) ⊂ [−1/10, 1/10], λ = µ0σ′2,
g(z) =
1 + z
1 + z − µ−10
(1 + z)n/2−2,
φ(z) = (1 + z)
(µ0 − 1)2
µ0(1 + z)− 1
= µ0 +
µ0 − 1
z2 +O(z3), |z| ≪ 1,
uniformly in µ0. It is easy to see that we have the estimate
eiλφ(z)g(z)dz
∣∣∣∣ ≤ Cλ
−1/2, |a| ≤ 1/10. (B.14)
Indeed, the functions g(z) and φ(z) are analytic in |z| ≤ 1/10 with |g(z)| bounded there
uniformly in µ0. Therefore, we can change the contour of integration to obtain (with some
0 < γ ≪ 1)
eiλφ(z)g(z)dz
∣∣∣∣ ≤
eiλφ(e
iγy)g(eiγy)dy
∣∣∣∣+
∣∣∣∣a
eiλφ(e
iθa)g(eiθa)dθ
e−Cλy
dy + C′1
′λθdθ = O(λ−1/2),
with some constants C,C′, C1, C
1 > 0. By (B.13) and (B.14) we conclude
|u1| ≤ C(σ′2)−1/2µ
(n−3)/2
0 ≤ C̃(σ′2)−1/2κ−(n−3)/2. (B.15)
On the other hand, if µ ∈ [2, κ−1] \ I(µ0), then
|µ− µ0|
µ− 1 ≥ C > 0,
so we can bound from below |ϕ′(µ)|. Therefore, the function u2 can be treated in the same
way as does u in Case 1. Thus, u2 satisfies (B.8) and hence, in view of (B.15), so does u.
This completes the proof of (B.4).
It suffices to prove (B.5) for 0 < h ≤ h0 with some constant 0 < h0 ≤ 1, since for
h0 ≤ h ≤ 1 it follows from (B.4) and the estimate of the L1 → L∞ norm of Ψ(t, h) proved in
[7] for the larger class of potentials satisfying (1.1) with δ > (n+2)/2 (without using (1.3)).
Without loss of generality we may suppose t > 0. Now, using Duhamel’s formula as in [6],
[7] we get the identity
Ψ(t;h)− F (t)ψ(h2G0) =
Ψj(t;h), (B.16)
where
Ψ1(t;h) = ψ1(h
2G0)e
ψ(h2G)− ψ(h2G0)
2G)− ψ1(h2G0)
eitG0ψ(h2G0) +
2G)− ψ1(h2G0)
Ψ(t;h),
Ψ2(t;h) = i
2G0)e
i(t−τ)G0V eiτGψ(h2G)dτ,
Ψ3(t;h) = −i
ei(t−τ)G0V eiτG0ψ(h2G0)dτ,
Ψ4(t;h) = i
∫ t−γ
2G0)e
i(t−τ)G0VΨ(τ ;h)dτ,
Ψ5(t;h) = −i
∫ t−γ
(1− ψ1)(h2G0)ei(t−τ)G0V eiτG0ψ(h2G0)dτ,
where ψ1 ∈ C∞0 ((0,+∞)), ψ1 = 1 on suppψ, and 0 < γ ≪ 1 is a parameter to be fixed later
on, depending on h. In view of (A.4), we have
‖Ψ1(t;h)f‖L∞ ≤ Ch
2t−n/2‖f‖L1 + Ch2 ‖Ψ(t;h)f‖L∞ , ∀f ∈ L
1. (B.17)
By (B.2) and (B.3),
‖Ψj(t;h)f‖L∞ ≤ Cγt
−n/2‖f‖L1 + Cγ ‖Ψ(t;h)f‖L∞ , ∀f ∈ L
1, j = 2, 3, (B.18)
with a constant C > 0 independent of t, h and γ.
Proposition B.3 Let V satisfy (1.1) with δ > n− 1. Then, there exist constants C, β1 > 0
so that for 0 < h ≤ 1, t ≥ 2γ, we have the estimate
‖Ψ4(t, h)‖L1→L∞ ≤ Ch
β1γ−(n−3)/2t−n/2. (B.19)
Proof. We will make use of the following estimates proved in [7].
Proposition B.4 Let V satisfy (1.1) with δ > (n + 2)/2. Then, for every 0 < ǫ ≪ 1,
1/2− ǫ/4 ≤ s ≤ (n− 1)/2, 0 < h ≤ 1, t 6= 0, we have the estimates
∥∥∥ψ(h2G0)eitG0〈x〉−s−1/2−ǫ
L2→L∞
≤ Chs−(n−1)/2|t|−s−1/2, (B.20)
∥∥∥Ψ(t, h)〈x〉−s−1/2−ǫ
L2→L∞
≤ Chs−(n−3)/2−ǫ/4|t|−s−1/2. (B.21)
By (B.20) and (B.21), we get (with some 0 < ε0 ≪ 1)
‖Ψ4(t, h)‖L1→L∞
∫ t/2
∥∥∥ψ1(h2G0)ei(t−τ)G0〈x〉−n/2−ε0
L2→L∞
∥∥∥〈x〉−(n−2)/2−ε0Ψ(τ, h)
L1→L2
∫ t−γ
∥∥∥ψ1(h2G0)ei(t−τ)G0〈x〉−(n−2)/2−ε0
L2→L∞
∥∥∥〈x〉−n/2−ε0Ψ(τ, h)
L1→L2
≤ Chε0/4t−n/2
∫ t/2
τ−(n−2)/2dτ + Chε0/4t−n/2
∫ t−γ
(t− τ)−(n−2)/2dτ
≤ Chε0/4γ−(n−3)/2t−n/2.
Proposition B.5 Let V satisfy (1.1) with δ > n−1. Then, for every 0 < ǫ≪ 1, 0 < h ≤ 1,
t ≥ 2γ, we have the estimate
‖Ψ5(t, h)‖L1→L∞ ≤ Cǫh
ǫγ−(n−3)/2−ǫt−n/2. (B.22)
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) =
(2π)ν+1
Jν(σλ)ψ(h2λ2)λdλ = h−nK1(σh−1, th−2), (B.23)
where Jν(z) = zνJν(z), Jν(z) = (H+ν (z) +H−ν (z)) /2 is the Bessel function of order ν =
(n− 2)/2. So, the kernel of the operator Ψ5 is of the form
Wh(|x− ξ|, |y − ξ|, t, γ)V (ξ)dξ,
where
Wh(σ1, σ2, t, γ) = −i
∫ t−γ
K̃h(σ1, t− τ)Kh(σ2, τ)dτ
= h−2n+2W1(σ1h
−1, σ2h
−1, th−2, γh−2), (B.24)
where K̃h is defined by replacing in the definition of Kh the function ψ by 1−ψ1. It is easy
to see that (B.22) follows from the bound (for all σ1, σ2, γ > 0, 0 < ǫ≪ 1, t ≥ 2γ)
|Wh(σ1, σ2, t, γ)| ≤ Cǫhǫγ−(n−3)/2−ǫt−n/2
σ−n+21 + σ
1 + σ
2 + σ
. (B.25)
In view of (B.24), it suffices to prove (B.25) with h = 1. Now, observe thatW1 =W
where
1 (σ1, σ2, t, γ) =
(σ1σ2)
4(2π)2ν+2
ei(t−γ)λ
+iγλ2
2ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)dλ21dλ22,
1 (σ1, σ2, t, γ) =
(σ1σ2)
4(2π)2ν+2
ei(t−γ)λ
+iγλ2
1ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)dλ21dλ22,
where the function
ρ(λ21, λ
(1− ψ1)(λ21)ψ(λ22)
λ22 − λ21
= (1 − ψ1)(λ21)ψ1(λ22)
ψ(λ22)− ψ(λ21)
λ22 − λ21
satisfies the bound
∣∣∣∂α1
ρ(λ21, λ
∣∣∣ ≤ Cα1,α2〈λ21〉−1−α1 , ∀(λ1, λ2). (B.26)
Given any integers 0 ≤ k,m < n/2, since Jν(z) = O(zn−2) as z → 0, we can integrate by
parts to get
1 (σ1, σ2, t, γ) = i
−m−k(t− γ)−kγ−m (σ1σ2)
4(2π)2ν+2
ei(t−γ)λ
+iγλ2
×∂kλ2
ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)
dλ21dλ
1 (σ1, σ2, t, γ) = i
−m−k(t− γ)−kγ−m (σ1σ2)
4(2π)2ν+2
ei(t−γ)λ
+iγλ2
×∂mλ2
ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)
dλ21dλ
Using the inequality ∣∣∣∣
ϕ(λ)dλ
∣∣∣∣ ≤ C|t|
−1/2 ‖ϕ̂‖L1 , (B.27)
we obtain (for t ≥ 2γ)
∣∣∣W (1)1 (σ1, σ2, t, γ)
∣∣∣ ≤ Ct−k−1/2γ−m(σ1σ2)−2ν
eiτλ1+iγλ
×λ1∂kλ2
ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)
dλ1dλ
∣∣∣ dτ, (B.28)
∣∣∣W (2)1 (σ1, σ2, t, γ)
∣∣∣ ≤ Ct−k−1/2γ−m(σ1σ2)−2ν
eiτλ2+iγλ
×λ2∂mλ2
ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)
dλ2dλ
∣∣∣ dτ. (B.29)
Recall now that the function Jν is of the form Jν(z) = eizb+ν (z) + e−izb−ν (z), where b±ν (z)
are symbols of order (n− 3)/2 for z ≥ 1, while near z = 0 the function Jν(z) is equal to z2ν
times an analytic function. Therefore, it satisfies the bounds
∣∣∂jzJν(z)
∣∣ ≤ Czn−2−j〈z〉j−(n−1)/2, ∀z > 0, 0 ≤ j ≤ n− 2, (B.30)
∣∣∂jzJν(z)
∣∣ ≤ Cj〈z〉(n−3)/2, ∀z > 0, j ≥ 0. (B.31)
Moreover, the functions b±ν (z) are of the form (near z = 0)
b±ν (z) = b
ν,1(z) + z
n−2 log z b±ν,2(z),
where b±ν,j(z) are analytic functions, b
ν,2(z) ≡ 0 if n is odd. Therefore, we have
∣∣∂jzb±ν (z)
∣∣ ≤ C, 0 < z ≤ 1, 0 ≤ j ≤ n− 3,
∣∣∂jzb±ν (z)
∣∣ ≤ Cǫz−ǫ, 0 < z ≤ 1, j = n− 2,
∣∣∂jzb±ν (z)
∣∣ ≤ Cjzn−2−j, 0 < z ≤ 1, j ≥ n− 1,
which imply ∣∣∂jzb±ν (z)
∣∣ ≤ C〈z〉(n−3)/2−j , ∀z > 0, 0 ≤ j ≤ n− 3, (B.32)
∣∣∂jzb±ν (z)
∣∣ ≤ Cǫz−ǫ〈z〉−(n−1)/2+ǫ, ∀z > 0, j = n− 2, (B.33)
∣∣∂jzb±ν (z)
∣∣ ≤ Cjzn−2−j〈z〉−(n−1)/2, ∀z > 0, j ≥ n− 1. (B.34)
± (λ1, λ2, σ1, σ2) = λ1e
∓iσ1λ1∂kλ2
ρ(λ21, λ
±iσ1λ1b±ν (σ1λ1)Jν(σ2λ2)
± (λ1, λ2, σ1, σ2) = λ2e
∓iσ2λ2∂mλ2
ρ(λ21, λ
2)Jν(σ1λ1)e±iσ2λ2b±ν (σ2λ2)
By (B.26), (B.30)-(B.34), we have (with ℓ = 0, 1)
∣∣∣∂ℓλ1A
± (λ1, λ2, σ1, σ2)
≤ C〈σ1〉k+(n−3)/2σn−22 〈σ2〉m−(n−1)/2〈λ1〉(n−3)/2−k−1, ∀(λ1, λ2), (B.35)
∣∣∣∂ℓλ2A
± (λ1, λ2, σ1, σ2)
≤ Cσn−21 〈σ1〉m−(n−1)/2〈σ2〉k+(n−3)/2〈λ1〉(n−3)/2−m−2, ∀(λ1, λ2). (B.36)
Using the inequality
‖ϕ̂(τ)‖L1 ≤ C‖〈τ〉ϕ̂(τ)‖L2 ≤ C
∥∥∂ℓλϕ(λ)
∣∣∂ℓλϕ(λ)
we obtain from (B.28) and (B.35) (if k > (n− 3)/2)
∣∣∣W (1)1 (σ1, σ2, t, γ)
∣∣∣ ≤
Ct−k−1/2γ−m(σ1σ2)
eiτλ1+iγλ
± (λ1, λ2, σ1, σ2)dλ1dλ
∣∣∣∣ dτ
Ct−k−1/2γ−m(σ1σ2)
−2ν sup
λ1,λ2
∣∣∣∂ℓλ1A
± (λ1, λ2, σ1, σ2)
≤ Ct−k−1/2γ−mσ−(n−2)/21 〈σ1〉k+(n−3)/2〈σ2〉m−(n−1)/2, (B.37)
where we have made a change of variables τ → τ ± σ1. Similarly, by (B.29) and (B.36), we
get (if m > (n− 3)/2)
∣∣∣W (2)1 (σ1, σ2, t, γ)
∣∣∣ ≤
Ct−k−1/2γ−m(σ1σ2)
eiτλ2+iγλ
± (λ1, λ2, σ1, σ2)dλ2dλ
∣∣∣∣ dτ
Ct−k−1/2γ−m(σ1σ2)
−2ν sup
∣∣∣∂ℓλ2A
± (λ1, λ2, σ1, σ2)
∣∣∣ dλ21
≤ Ct−k−1/2γ−mσ−(n−2)/22 〈σ2〉k+(n−3)/2〈σ1〉m−(n−1)/2. (B.38)
We would like to apply (B.37) and (B.38) with k = (n−1)/2,m = (n−3)/2+ǫ, 0 < ǫ≪ 1. To
this end, we need to show that these estimates are valid for all real (n−3)/2 < m ≤ (n−2)/2,
(n−2)/2 ≤ k < n/2 if n is even, and for k = (n−1)/2 and all real (n−3)/2 < m ≤ (n−1)/2
if n is odd. This can be done by interpolation as follows. Let φ ∈ C∞0 (R), φ(λ) = 1
for |λ| ≤ 1, φ(λ) = 0 for |λ| ≥ 2. Decompose W (j)1 as X(j) + Y (j), j = 1, 2, where X(j)
and Y (j) are defined by replacing in the definition of W
1 the function ρ by φ(λj)ρ and
(1−φ)(λj)ρ, respectively. Clearly, the functions X(j) satisfy (B.37) and (B.38), respectively,
for all integers 0 ≤ k,m < n/2, while the functions Y (j) satisfy (B.37) and (B.38) for all
integers k > (n − 3)/2, (n − 3)/2 < m < n/2, respectively. When n is odd, this is fulfilled
with k = (n− 1)/2. To show this in the case of even n, we write the function φ as
φ(λ) =
with some function φ1 ∈ C∞0 (R), φ1(λ) = 0 in a neighbourhood of λ = 0. Thus,
X(j) =
X(j)p ,
whereX
p is defined by replacing in the definition ofX
(j) the function φ(λj) by φ1(2
pλj). As
above, one can see that the functions X
p , j = 1, 2, satisfy (B.37) and (B.38), respectively,
with an extra factor in the RHS of the form 2p(k−n/2) for all integers k ≥ (n − 2)/2, and
hence, by interpolation, for all real k ≥ (n−2)/2. Therefore, summing up these estimates we
conclude that X(j), j = 1, 2, satisfy (B.37) and (B.38), respectively, for all real (n− 2)/2 ≤
k < n/2, and in particular for k = (n− 1)/2. Hence, so do the functions W (j)1 . Furthermore,
1 satisfies (B.37) for all integers 0 ≤ m < n/2, and hence, by interpolation, for all real
0 ≤ m ≤ (n− 1)/2 if n is odd, and for all real 0 ≤ m ≤ (n− 2)/2 if n is even. In particular,
this is valid with m = (n − 3)/2 + ǫ. To show that the function W (2)1 satisfies (B.38) with
m = (n − 3)/2 + ǫ, we decompose it as Z +N , where Z and N are defined by replacing in
the definition of W
1 the function ρ by φ(λ1)ρ and (1 − φ)(λ1)ρ, respectively. Clearly, the
function Z satisfies (B.37) for all integers 0 ≤ m < n/2, and hence, by interpolation, for all
real 0 ≤ m ≤ (n − 1)/2 if n is odd, and for all real 0 ≤ m ≤ (n− 2)/2 if n is even. To deal
with the function N , we write the function 1− φ as
(1 − φ)(λ) =
−pλ),
with some function φ2 ∈ C∞0 (R), φ2(λ) = 0 in a neighbourhood of λ = 0. Thus,
where Np is defined by replacing in the definition of N the function (1−φ)(λ1) by φ2(2−pλ1).
Now, the functionsNp satisfy (B.38) with an extra factor in the RHS of the form 2
−p(m−(n−3)/2)
for all integers 0 ≤ m < n/2, and hence, by interpolation, for all real 0 ≤ m ≤ (n − 1)/2
if n is odd, and for all real 0 ≤ m ≤ (n − 2)/2 if n is even. Therefore, summing up these
estimates we conclude that N satisfies (B.38) for all real (n− 3)/2 < m ≤ (n − 1)/2 if n is
odd, and for all real (n− 3)/2 < m ≤ (n− 2)/2 if n is even. In particular, this is valid with
m = (n− 3)/2 + ǫ.
By (B.37) and (B.38) with k = (n− 1)/2, m = (n− 3)/2 + ǫ, we obtain
|W1(σ1, σ2, t, γ)|
≤ Ct−n/2γ−(n−3)/2−ǫ
σ−n+21 〈σ1〉n−2〈σ2〉−1+ǫ + σ
2 〈σ2〉n−2〈σ1〉−1+ǫ
≤ Ct−n/2γ−(n−3)/2−ǫ
σ−n+21 + σ
2 + 〈σ1〉−1+ǫ + 〈σ2〉−1+ǫ
≤ Ct−n/2γ−(n−3)/2−ǫ
σ−n+21 + σ
2 + σ
1 + σ
which is the desired bound. ✷
Taking γ = hβ
with a suitably chosen constant β′ > 0, we deduce from (B.4), (B.16)-
(B.19) and (B.22), ∥∥Ψ(t;h)f − F (t)ψ(h2G0)f
≤ Chβt−n/2‖f‖L1 + Chβ
∥∥Ψ(t;h)f − F (t)ψ(h2G0)f
, ∀f ∈ L1, (B.39)
with some constant β > 0. Taking h small enough, we can absorb the second term in the
RHS of (B.39), thus obtaining (B.5). ✷
References
[1] M. Goldberg, Dispersive bounds for the three dimensional Schrödinger equation with
almost critical potentials, GAFA 16 (2006), 517-536.
[2] M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in
dimensions one and three, Commun. Math. Phys. 251 (2004), 157-178.
[3] M. Goldberg and M. Visan, A conterexample to dispersive estimates for Schrödinger
operators in higher dimensions, Commun. Math. Phys. 266 (2006), 211-238.
[4] J.-L. Journé, A. Sofer and C. Sogge, Decay estimates for Schrödinger operators,
Commun. Pure Appl. Math. 44 (1991), 573-604.
[5] W. Schlag, Dispersive estimates for Schrödinger operators in two dimensions, Com-
mun. Math. Phys. 257 (2005), 87-117.
[6] G. Vodev, Dispersive estimates of solutions to the Schrödinger equation, Ann. H.
Poincaré 6 (2005), 1179-1196.
[7] G. Vodev, Dispersive estimates of solutions to the Schrödinger equation in dimensions
n ≥ 4, Asymptot. Anal. 49 (2006), 61-86.
[8] G. Vodev, Dispersive estimates of solutions to the wave equation with a potential in
dimensions n ≥ 4, Commun. Partial Diff. Equations 31 (2006), 1709-1733.
[9] K. Yajima, The W k,p continuity of wave operators for Schrödinger operators, J. Math.
Soc. Japan 47 (1995), 551-581.
[10] K. Yajima, Dispersive estimates for Schrödinger equations with threshold resonance
and eigenvalue, Commun. Math. Phys. 259 (2005), 475-509.
Université de Nantes, Département de Mathématiques, UMR 6629 du CNRS, 2, rue de
la Houssinière, BP 92208, 44332 Nantes Cedex 03, France
e-mail: simon.moulin@math.univ-nantes.fr
e-mail: georgi.vodev@math.univ-nantes.fr
	Introduction and statement of results
	Preliminary estimates
	Proof of Theorem 1.1
	Appendix 1
	Appendix 2

  We prove dispersive estimates for the low frequency part of the Schrodinger
group for a large class of potentials in dimensions greater or equal to four.
As a consequence, we extend the result of Journe, Sofer and Sogge to a larger
class of potentials. In this revised version a mistake in the proof of the
estimate (B.4) is removed.

Introduction and statement of results
Let V ∈ L∞(Rn), n ≥ 4, be a real-valued function satisfying
|V (x)| ≤ C〈x〉−δ , ∀x ∈ Rn, (1.1)
with constants C > 0, δ > (n + 2)/2. Denote by G0 and G the self-adjoint realizations of
the operators −∆ and −∆+V on L2(Rn), respectively. It is well known that the absolutely
continuous spectrums of the operators G0 and G coincide with the interval [0,+∞), and that
G has no embedded strictly positive eigenvalues nor strictly positive resonances. However,
G may have in general a finite number of non-positive eigenvalues and that the zero may be
a resonance. We will say that the zero is a regular point for G if it is neither an eigenvalue
nor a resonance in the sense that the operator 1− V∆−1 is invertible on L1 with a bounded
inverse. Let Pac denote the spectral projection onto the absolutely continuous spectrum of
G. When n ≥ 3, Journé, Sofer and Sogge [4] proved the following dispersive estimate
∥∥eitGPac
L1→L∞
≤ C|t|−n/2, t 6= 0, (1.2)
provided the zero is neither an eigenvalue nor a resonance, for potentials satisfying (1.1) with
δ > n+ 4 as well as the condition
V̂ ∈ L1. (1.3)
This was later improved by Yajima [9] for potentials satisfying (1.1) with δ > n+ 2. When
n = 3, the estimate (1.2) in fact holds without (1.3). In this case, it was proved in [2] for
potentials satisfying (1.1) with δ > 3 and was later improved in [6] and [10] for potentials
satisfying (1.1) with δ > 5/2. Goldberg [1] has recently showed that (1.2) holds for potentials
V ∈ L3/2−ǫ∩L3/2+ǫ, 0 < ǫ≪ 1, which includes potentials satisfying (1.1) with δ > 2. When
n = 2, (1.2) is proved by Schlag [5] for potentials satisfying (1.1) with δ > 3.
Given any a > 0, set χa(σ) = χ1(σ/a), where χ1 ∈ C∞(R), χ1(σ) = 0 for σ ≤ 1,
χ1(σ) = 1 for σ ≥ 2. Set ηa = χ(1 − χa), where χ denotes the characteristic function of the
interval [0,+∞). Clearly, ηa(G) + χa(G) = Pac. When n ≥ 4, dispersive estimates with loss
of (n − 3)/2 derivatives for the operator eitGχa(G), ∀a > 0, have been recently proved in
http://arxiv.org/abs/0704.1200v2
[7] under the assumption (1.1), only. The loss of derivatives in this case is a high frequency
phenomenon and cannot be avoided unless one imposes some regularity condition on the
potential (see [3]). The condition (1.3) in [4] plays this role but it seems too strong. The
natural conjecture would be that we have dispersive estimates for eitGχa(G) with loss of ν
derivatives, 0 ≤ ν ≤ (n − 3)/2, provided V ∈ C(n−3)/2−ν(Rn) (with a suitable decay at
infinity). It turns out that no regularity on the potential is needed in order to get dispersive
estimates for the low frequency part eitGηa(G), a > 0 small. One just needs some decay
at infinity. In fact, the low frequency analysis turns out to be easier in dimensions n ≥ 4
compared with the cases of n = 2 and n = 3, and can be carried out for a larger class of
potentials satisfying (with some 0 < ǫ≪ 1)
|x− y|−n+2 + |x− y|−(n−2)/2+ǫ
|V (x)|dx ≤ C < +∞. (1.4)
Clearly, (1.4) is fulfilled for potentials satisfying (1.1). Our main result is the following
Theorem 1.1 Let n ≥ 4, let V satisfy (1.4) and assume that the zero is a regular point for
G. Then, there exists a constant a0 > 0 so that for 0 < a ≤ a0 we have the estimate
∥∥eitGηa(G)
L1→L∞
≤ C|t|−n/2, t 6= 0. (1.5)
Remark 1. We expect that (1.5) holds true for the larger class of potentials satisfying
|x− y|−n+2 + |x− y|−(n−1)/2
|V (x)|dx ≤ C < +∞, (1.6)
but the proof in this case would require a different approach.
Combining (1.5) with the estimates of [7], we obtain the following
Theorem 1.2 Let n ≥ 4, let V satisfy (1.1) and assume that the zero is a regular point for
G. Then, we have the estimates, ∀t 6= 0, 0 < ǫ≪ 1,
∥∥eitGPacf
≤ C|t|−n/2
∥∥∥〈G〉(n−3)/4f
, (1.7)
∥∥eitGPacf
≤ Cǫ|t|−n/2
∥∥∥〈x〉n/2+ǫf
. (1.8)
Remark 2. The proof in [7] is based on uniform estimates for the operator ψ(h2G), 0 <
h ≤ 1, ψ ∈ C∞0 ((0,+∞)) (see Lemma 2.2 of [7] or Lemma 2.3 of [8]). In the proof of this
lemma (which is given in [8]), however, there is a mistake. That is why, we will give a new
proof in Appendix 1 of the present paper.
Remark 3. We conjecture that the estimates (1.7) and (1.8) hold true for potentials satis-
fying (1.1) with δ > (n+ 1)/2.
Theorem 1.1 also allows to extend the results in [4] to a larger class of potentials. More
precisely, we have the following
Theorem 1.3 Let n ≥ 4, let V satisfy (1.1) with δ > n − 1 as well as (1.3), and assume
that the zero is a regular point for G. Then, the estimate (1.2) holds true.
Theorem 1.3 follows from (1.5) and the dispersive estimate for eitGχa(G) proved in Ap-
pendix 2.
To prove (1.5) we adapt the semi-classical approach of [7] based on the semi-classical
version of Duhamel’s formula (which in our case is of the form (3.4) or (3.5)). While in [7]
the estimates had to be uniform with respect to the semi-classical parameter 0 < h ≤ 1,
in the case of low frequency we need to make them uniform for h ≫ 1 (see (3.1)). This,
however, turns out to be easier (when n ≥ 4) as we can absorb the remaining terms taking
h big enough (see Section 3). That is why, we do not need any more to work on weighted
L2 spaces (as in [7]), which in turn allows to cover a much larger class of potentials. As
mentioned in Remark 1, the natural class of potentials for which the low frequency analysis
works out (for n ≥ 4) is given by (1.6), and the fact that the crucial Proposition 2.1 below
holds true under (1.6) is a strong indication for that. In fact, (1.4) is used in the proof of
Proposition 2.3, only.
2 Preliminary estimates
Let ψ ∈ C∞0 ((0,+∞)). We will first prove the following
Proposition 2.1 Let n ≥ 4, let V satisfy (1.6) and assume that the zero is a regular point
for G. Then, there exist positive constants C, β and h0 so that the following estimates hold
∥∥ψ(h2G0)
L1→L1
≤ C, h > 0, (2.1)
∥∥ψ(h2G)
L1→L1
≤ C, h ≥ h0, (2.2)
∥∥ψ(h2G)− ψ(h2G0)T
L1→L1
≤ Ch−β, h ≥ h0, (2.3)
where the operator
1− V∆−1
: L1 → L1 (2.4)
is bounded by assumption.
Proof. Set ϕ(λ) = ψ(λ2). We are going to take advantage of the formula
ψ(h2G) =
(z)(h2G− z2)−1zL(dz), (2.5)
where L(dz) denotes the Lebesgue measure on C, ϕ̃ ∈ C∞0 (C) is an almost analytic contin-
uation of ϕ supported in a small complex neighbourhood of suppϕ and satisfying
∣∣∣∣ ≤ CN |Im z|
N , ∀N ≥ 1.
For ±Im z > 0, denote
R±0,h(z) = (h
2G0 − z2)−1, R±h (z) = (h
2G− z2)−1.
The kernel of the operator R±0,h(z) is of the form R
h (|x− y|, z), where
R±h (σ, z) = ±h
−2 iσ
4(2π)ν
H±ν (σz/h) = h−nR±1 (σh−1, z),
where ν = (n − 2)/2, H±ν (λ) = λνH±ν (λ), H±ν being the outgoing and incoming Henkel
functions of order ν. It is well known that these functions satisfy the bound
∣∣H±ν (λ)
∣∣ ≤ C〈λ〉(n−3)/2e−|Imλ|, ∀λ, ±Imλ ≥ 0, (2.6)
while near λ = 0 they are of the form
H±ν (λ) = a±ν,1(λ) + λn−2 logλa
ν,2(λ), (2.7)
where a±ν,j are analytic functions, a
ν,2 ≡ 0 if n is odd. By (2.6) and (2.7), we have
∣∣H±ν (λ)−H±ν (0)
∣∣ ≤ C|λ|1/2〈λ〉(n−4)/2, ∀λ, ±Imλ ≥ 0. (2.8)
Hence, the functions R±h satisfy the bounds (for z ∈ C±ϕ := {z ∈ supp ϕ̃, ±Im z ≥ 0}, σ > 0,
h ≥ 1) ∣∣R±h (σ, z)
∣∣ ≤ Ch−2
σ−n+2 + σ−(n−1)/2
, (2.9)
∣∣R±h (σ, z)−R
h (σ, 0)
∣∣ ≤ Ch−5/2
σ−n+5/2 + σ−(n−1)/2
. (2.10)
Using the above bounds we will prove the following
Lemma 2.2 For z ∈ C±ϕ , we have
‖VR0,h(z)‖L1→L1 ≤ Ch
−2, h ≥ 1, (2.11)
‖VR0,h(z)− VR0,h(0)‖L1→L1 ≤ Ch
−5/2, h ≥ 1, (2.12)
‖VRh(z)‖L1→L1 ≤ Ch
−2, h ≥ h0, (2.13)
∥∥∥R±0,h(z)
L1→L1
≤ C|Im z|−q, h > 0, Im z 6= 0, (2.14)
∥∥R±h (z)
L1→L1
≤ C|Im z|−q, h ≥ h0, Im z 6= 0, (2.15)
with constants C, q, h0 > 0 independent of z and h.
Proof. In view of (2.9), the norm in the LHS of (2.11) is upper bounded by
|V (x)|
∣∣R±h (|x − y|, z)
≤ Ch−2 sup
|V (x)|
|x− y|−n+2 + |x− y|−(n−1)/2
dx ≤ Ch−2.
The estimate (2.12) follows in the same way using (2.10). To prove (2.14), we use (2.6) to
get (for z ∈ C±ϕ , Im z 6= 0)
∣∣R±1 (σ, z)
∣∣ ≤ Cσ−2ν〈σ〉(n−3)/2e−σ|Im z| ≤ Cσ−2ν〈σ〉−5/2|Im z|−(n+2)/2. (2.16)
By (2.16), the norm in the LHS of (2.14) is upper bounded by
∣∣R±h (σ, z)
∣∣ dσ = C
∣∣R±1 (σ, z)
∣∣ dσ
≤ C|Im z|−(n+2)/2
〈σ〉−3/2dσ ≤ C|Im z|−(n+2)/2.
To prove (2.13) and (2.15), we will use the identity
R±h (z)
1 + h2VR±0,h(z)
= R±0,h(z), ±Im z > 0. (2.17)
Observe that 1 + h2VR±0,h(0) = 1 − V∆−1, which is supposed to be invertible on L1 with
a bounded inverse denoted by T . Thus, it follows from (2.12) that there exists a constant
h0 > 0 so that for h ≥ h0 the operator 1 + h2VR±0,h(z) is invertible on L1 with an inverse
satisfying ∥∥∥∥
1 + h2VR±0,h(z)
)−1∥∥∥∥
L1→L1
≤ C, z ∈ C±ϕ , (2.18)
with a constant C > 0 independent of z and h. Hence, we can write
R±h (z) = R
0,h(z)
1 + h2VR±0,h(z)
. (2.19)
Now (2.13) follows from (2.11), (2.18) and (2.19), while (2.15) follows from (2.14), (2.18) and
(2.19). ✷
Clearly, (2.1) and (2.2) follow from (2.5) and (2.14), (2.15), respectively. To prove (2.3)
we rewrite the identity (2.19) in the form
R±h (z)−R
0,h(z)T
= R±0,h(z)T
h2VR±0,h(z)− h
2VR±0,h(0)
h2VR±0,h(z)− h
2VR±0,h(0)
(2.20)
By Lemma 2.2, (2.18) and (2.20) we conclude
∥∥∥R±h (z)−R
0,h(z)T
L1→L1
≤ Ch−β |Im z|−q, h ≥ h0, z ∈ C±ϕ , Im z 6= 0, (2.21)
with constants C, q, β > 0 independent of z and h. Now (2.3) follows from (2.5) and (2.21).
Let ψ1 ∈ C∞0 ((0,+∞)), ψ1 = 1 on suppψ.
Proposition 2.3 Under the assumptions of Theorem 1.1, there exist positive constants h0
and β so that we have the estimates
∥∥V eitG0ψ(h2G0)
L1→L1
dt ≤ Ch−β, h ≥ 1, (2.22)
∥∥V ψ(h2G)eitG0ψ1(h2G0)
L1→L1
dt ≤ Ch−β , h ≥ h0. (2.23)
Proof. It is shown in [7] (Section 2) that the kernel of the operator eitG0ψ(h2G0) is of
the form Kh(|x− y|, t) with a function Kh satisfying
Kh(σ, t) = h
−nK1(σh
−1, th−2),
|K1(σ, t)| ≤ C|t|−s−1/2σs−(n−1)/2, 0 ≤ s ≤ (n− 1)/2, σ > 0, t 6= 0.
Hence, for all 0 ≤ s ≤ (n− 1)/2, σ > 0, t 6= 0, h > 0, we have
|Kh(σ, t)| ≤ Chs−(n−1)/2|t|−s−1/2σs−(n−1)/2,
which together with (1.4) imply
∥∥V eitG0ψ(h2G0)
L1→L1
≤ Chs−(n−1)/2|t|−s−1/2, 1/2− ǫ ≤ s ≤ 1/2 + ǫ, (2.24)
where 0 < ǫ≪ 1. Clearly, (2.22) follows from (2.24). Furthermore, using (2.5), (2.13), (2.14)
and (2.24), we get
ψ(h2G)− ψ(h2G0)
eitG0ψ1(h
L1→L1
≤ Ch2
∥∥∥VR±h (z)V e
itG0ψ1(h
2G0)R±0,h(z)
L1→L1
L(dz)
≤ Ch2
∥∥VR±h (z)
L1→L1
∥∥V eitG0ψ1(h2G0)
L1→L1
∥∥∥R±0,h(z)
L1→L1
L(dz)
≤ Chs−(n−1)/2|t|−s−1/2
∣∣∣∣ |Im z|
−qL(dz)
≤ Chs−(n−1)/2|t|−s−1/2, 1/2− ǫ ≤ s ≤ 1/2 + ǫ, (2.25)
which clearly implies (2.23). ✷
3 Proof of Theorem 1.1
Denote
Ψ(t, h) = eitGψ(h2G)− T ∗eitG0ψ(h2G0)T,
T being given by (2.4). We will first show that (1.5) follows from the following
Proposition 3.1 Under the assumptions of Theorem 1.1, there exist positive constants C,
h0 and β so that for h ≥ h0 we have
‖Ψ(t, h)‖L1→L∞ ≤ Ch
−β |t|−n/2, t 6= 0. (3.1)
Recall that χa(σ) = χ1(σ/a), a > 0 small. Then we can write the function ηa as follows
ηa(σ) =
ψ(σθ)
, σ > 0,
where ψ(σ) = σχ′1(σ) ∈ C∞0 ((0,+∞)). Thus, we obtain from (3.1),
∥∥eitGηa(G) − T ∗eitG0ηa(G0)T
L1→L∞
∥∥∥Ψ(t,
L1→L∞
≤ C|t|−n/2
θ−1−β/2dθ ≤ C|t|−n/2, (3.2)
provided a is taken small enough. Clearly, (1.5) follows from (3.2).
Proof of Proposition 3.1. We will first prove the following
Proposition 3.2 Under the assumptions of Theorem 1.1, there exist positive constants C,
h0 and β so that for h ≥ h0 we have
∥∥V eitGψ(h2G)
L1→L1
dt ≤ Ch−β. (3.3)
Proof. Using Duhamel’s formula
eitG = eitG0 + i
ei(t−τ)GV eiτG0dτ,
we get the identity
eitGψ(h2G) = ψ(h2G)eitG0ψ1(h
2G0)T + e
itGψ(h2G)
2G)− ψ1(h2G0)T
ψ(h2G)ei(t−τ)GV eiτG0ψ1(h
2G0)Tdτ. (3.4)
Using Propositions 2.1 and 2.3, (3.4) together with Young’s inequality we obtain
∥∥V eitGψ(h2G)
L1→L1
dt ≤ Ch−β + Ch−β
∥∥V eitGψ(h2G)
L1→L1
∥∥∥V ψ(h2G)ei(t−τ)G
L1→L1
∥∥V eiτG0ψ1(h2G0)
L1→L1
≤ Ch−β + Ch−β
∥∥V eitGψ(h2G)
L1→L1
which clearly implies (3.3) if we take h large enough. ✷
Using Duhamel’s formula
eitG = eitG0 + i
ei(t−τ)G0V eiτGdτ,
we get the identity
Ψ(t;h) =
Ψj(t;h), (3.5)
where
Ψ1(t;h) = T
∗ψ1(h
2G0)e
ψ(h2G)− ψ(h2G0)T
2G)− T ∗ψ1(h2G0)
eitGψ(h2G),
Ψ2(t;h) = i
T ∗ψ1(h
2G0)e
i(t−τ)G0V eiτGψ(h2G)dτ.
By (2.1) and (2.3) together with the well known estimate
∥∥eitG0
L1→L∞
≤ C|t|−n/2,
we get
‖Ψ1(t;h)f‖L∞ ≤ Ch
−β |t|−n/2‖f‖L1 + Ch−β ‖Ψ(t;h)f‖L∞ , t 6= 0. (3.6)
By Propositions 2.3 and 3.2, ∀f ∈ L1, t > 0, we have
tn/2 ‖Ψ2(t;h)f‖L∞
∫ t/2
(t− τ)n/2
∥∥∥ψ1(h2G0)ei(t−τ)G0
L1→L∞
∥∥V eiτGψ(h2G)f
∥∥∥ψ1(h2G0)ei(t−τ)G0V
L∞→L∞
∥∥eiτGψ(h2G)f
∥∥V eiτGψ(h2G)f
+C sup
t/2≤τ≤t
∥∥eiτGψ(h2G)f
∥∥V eiτG0ψ1(h2G0)
L1→L1
≤ Ch−β‖f‖L1 + Ch−β sup
t/2≤τ≤t
∥∥eiτGψ(h2G)f
. (3.7)
Combining (3.5), (3.6) and (3.7), we conclude, ∀f ∈ L1, t > 0,
tn/2 ‖Ψ(t;h)f‖L∞ ≤ Ch
−β‖f‖L1 + Ch−βtn/2 ‖Ψ(t;h)f‖L∞
+Ch−β sup
t/2≤τ≤t
τn/2 ‖Ψ(τ ;h)f‖L∞ . (3.8)
Taking h big enough we can absorb the second and the third terms in the RHS of (3.8), thus
obtaining (3.1). Clearly, the case of t < 0 can be treated in the same way. ✷
A Appendix 1
We will prove the following
Lemma A.1 Let ψ ∈ C∞0 ((0,+∞)). Then, for all h > 0, s ≥ 0, we have the estimates
∥∥ψ(h2G0)
L1→L1
≤ C, (A.1)
∥∥〈x〉sψ(h2G0)〈x〉−s
L1→L2
≤ Ch−n/2〈h〉s, (A.2)
where the constant C is of the form
C = C′ sup
0≤k≤k0
|∂kλψ(λ)|, (A.3)
with some integer k0 independent of ψ and a constant C
′ > 0 depending on the support of ψ,
only. Furthermore, if V satisfies (1.1) with δ > n/2, we have the estimates (for 0 < h ≤ 1)
∥∥ψ(h2G)− ψ(h2G0)
L1→L1
≤ Ch2, (A.4)
∥∥〈x〉δ
ψ(h2G)− ψ(h2G0)
L1→L2
≤ Ch2−n/2. (A.5)
Proof. The estimate (A.1) is proved in Section 2 using the formula (2.5) and (2.14). It
can be also seen by using the fact that the kernel of the operator ψ(h2G0) is of the form
kh(|x− y|) with a function kh satisfying
kh(σ) = h
−nk1(σ/h), (A.6)
|k1(σ)| ≤ Cm〈σ〉−m, ∀σ > 0, (A.7)
for all integers m ≥ 0, with a constant Cm of the form
Cm = C
m sup
0≤j≤jm
|∂jλψ(λ)|, (A.8)
where jm is some integer independent of ψ, while C
m > 0 depends on the support of ψ. By
Young’s inequality, the norm in the LHS of (A.1) is upper bounded by
|kh(|ξ|)|dξ =
|k1(|ξ|)|dξ ≤ Cn+1.
The norm in the LHS of (A.2) is upper bounded by
〈x〉2s〈y〉−2s|kh(|x− y|)|2dx
〈x− y〉2s|kh(|x − y|)|2dx
≤ C〈h〉s
〈ξ/h〉2s|kh(|ξ|)|2dξ
= C〈h〉sh−n/2
〈ξ〉2s|k1(|ξ|)|2dξ
≤ Csn〈h〉sh−n/2,
where sn is some integer depending on n and s. To prove (A.4) observe that by (2.5) we
ψ(h2G)− ψ(h2G0) =
(z)(h2G0 − z2)−1V (h2G− z2)−1zL(dz). (A.9)
Clearly, (A.4) would follow from (A.9), (2.14) and the estimate (for z ∈ supp ϕ̃)
∥∥(h2G− z2)−1
L1→L1
≤ C|Im z|−q, 0 < h ≤ 1, Im z 6= 0. (A.10)
Let φ ∈ C∞0 ([1, 2]) be such that
φ(θ2)θ−1dθ = 1. Given a parameter 0 < ε ≪ 1, we
decompose the free resolvent as follows
(h2G0 − z2)−1 = Aε(z;h) + Bε(z;h), (A.11)
where
Aε(z;h) =
(εθh)2G0; (εθ)
Bε(z;h) =
(εθh)2G0; (εθ)
where
f(λ;µ; z) =
λµ−1 − z2 .
It is easy to see that there exist constants 0 < µ1 < µ2 so that the function f satisfies the
following bounds ∣∣∣∂jλf(λ;µ; z)
∣∣∣ ≤ Cjµ, 0 < µ ≤ µ1, (A.12)
∣∣∣∂jλf(λ;µ; z)
∣∣∣ ≤ C′j |Im z|−j−1, µ1 ≤ µ ≤ µ2, (A.13)
∣∣∣∂jλf(λ;µ; z)
∣∣∣ ≤ C′′j , µ ≥ µ2, (A.14)
for every integer j ≥ 0. By (A.1), (A.3) and (A.12), we have
(εθh)2G0; (εθ)
L1→L1
≤ C(εθ)2, 0 < θ ≤ 1, (A.15)
provided ε > 0 is taken small enough. We deduce from (A.15),
‖Aε(z;h)‖L1→L1 ≤ Cε
2, z ∈ supp ϕ̃, (A.16)
with a constant C > 0 independent of z, h and ε. By (A.2), (A.3), (A.12)-(A.14), we have
∥∥〈x〉sf
(εθh)2G0; (εθ)
〈x〉−s
L1→L2
≤ C(εθh)−n/2〈εθh〉s|Im z|−q, (A.17)
with constants C, q > 0 independent of z, θ, h and ε. We deduce from (A.17),
‖Bε(z;h)‖L1→L2 ≤ C
−n/2|Im z|−q
θ−1−n/2dθ
≤ Cεh−n/2|Im z|−q, z ∈ supp ϕ̃, (A.18)
with a constant Cε > 0 independent of z and h. It follows from (A.16) that the operator
1+ h2VAε(z;h) is invertible on L1, provided ε > 0 is taken small enough, independent of h.
Therefore, we can write the identity
h2G− z2
h2G0 − z2
h2G− z2
h2G0 − z2
, (A.19)
in the form (
h2G− z2
= M(z;h) + h2
h2G− z2
)−1 N (z;h), (A.20)
where the operators
M(z;h) =
h2G0 − z2
)−1 (
1 + h2VAε(z;h)
N (z;h) = V Bε(z;h)
1 + h2VAε(z;h)
satisfy the estimates
‖M(z;h)‖L1→L1 + ‖N (z;h)‖L1→L1 ≤ C|Im z|
−q, (A.21)
‖N (z;h)‖L1→L2 ≤ Ch
−n/2|Im z|−q. (A.22)
By (A.20) we have
h2G− z2
M(z;h)N (z;h)j + h2J
h2G− z2
)−1 N (z;h)J
M(z;h)N (z;h)j + h2J
h2G0 − z2
)−1 N (z;h)J
+h2J+2
h2G0 − z2
h2G− z2
)−1 N (z;h)J , (A.23)
for every integer J ≥ 1. By (A.22) and (2.14), we obtain
h2G0 − z2
h2G− z2
)−1 N (z;h)
L1→L1
≤ ‖V ‖L2
h2G0 − z2
)−1∥∥∥
L1→L1
h2G− z2
)−1∥∥∥
L2→L2
‖N (z;h)‖L1→L2
≤ Ch−n/2|Im z|−q2 . (A.24)
Now, (A.10) follows from (A.21), (A.23) and (A.24).
To prove (A.5) we rewrite (A.20) in the form
h2G− z2
)−1 −
h2G0 − z2
Fj(z;h), (A.25)
where
F1(z;h) = h2Aε(z;h)VAε(z;h)
1 + h2VAε(z;h)
F2(z;h) = h2Bε(z;h)VAε(z;h)
1 + h2VAε(z;h)
F3(z;h) = h2
h2G− z2
V Bε(z;h)
1 + h2VAε(z;h)
It is easy to see that we have the estimate
∥∥∥〈x〉s
h2G− z2
)−1 〈x〉−s
L2→L2
≤ C|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1, (A.26)
for every s ≥ 0 with constants C, q > 0 depending on s but independent of z and h. By
(A.18) and (A.26),
∥∥〈x〉δF3(z;h)
L1→L2
≤ Ch2−n/2|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1. (A.27)
Observe now that we can write the operator Aε(z;h) in the form
Aε(z;h) = χ(3)ε (h2G0)
h2G0 − z2
where
χ(3)ε (σ) =
∫ εσ1/2
φ(θ2)
Similarly, we can decompose the operator Bε(z;h) as B(1)ε + B(2)ε , where
B(j)ε (z;h) = χ(j)ε (h2G0)
h2G0 − z2
, j = 1, 2,
χ(1)ε (σ) =
∫ ε−1σ1/2
εσ1/2
φ(θ2)
, χ(2)ε (σ) =
ε−1σ1/2
φ(θ2)
Taking ε > 0 small enough we can arrange that suppχ
ε ∩ suppϕ = ∅, j = 2, 3, so the
operator-valued functions Aε(z;h) and B(2)ε (z;h) are analytic on supp ϕ̃. Therefore, we can
write (A.9) in the form
ψ(h2G)− ψ(h2G0) =
(z)Fj(z;h)zL(dz), (A.28)
where
F4(z;h) = h2B(1)ε (z;h)VAε(z;h)
1 + h2VAε(z;h)
By (A.17) (with s = δ), we have
∥∥〈x〉δF4(z;h)
L1→L2
≤ Ch2
∥∥∥〈x〉δB(1)ε (z;h)〈x〉−δ
L1→L2
≤ Ch2−n/2|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1. (A.29)
Now (A.5) follows from (A.27)-(A.29). ✷
B Appendix 2
Combining some ideas from [6],[7] and [4] we will prove the following
Theorem B.1 Let n ≥ 4, let V satisfy (1.1) with δ > n − 1 as well as (1.3). Then, for
every a > 0 we have the estimate
∥∥eitGχa(G)
L1→L∞
≤ C|t|−n/2, t 6= 0. (B.1)
Remark. Note that (B.1) is proved in [4] for potentials satisfying (1.1) with δ > n, the
condition (1.3) as well as an extra technical assumption. Here we eliminate this extra as-
sumption.
Proof. The key point in the proof in [4] is the bound
∥∥e−itG0V eitG0
L1→L1
≤ ‖V̂ ‖L1 , ∀t. (B.2)
Combining (B.2) with Duhamel’s formula one easily gets
∥∥e−itG0V eitG
L1→L1
≤ C, |t| ≤ 1, (B.3)
with a constant C > 0 independent of t. In what follows we will derive (B.1) from (B.2) and
(B.3). To this end, given a function ψ ∈ C∞0 ((0,+∞)) and a parameter 0 < h ≤ 1, as in [6],
[7], denote
Ψ(t, h) = eitGψ(h2G)− eitG0ψ(h2G0),
F (t) = i
ei(t−τ)G0V eiτG0dτ.
As in these papers, it is easy to see that (B.1) follows from the following
Theorem B.2 Under the assumptions of Theorem B.1, there exist constants C, β > 0 so
that we have the estimates (for 0 < h ≤ 1, t 6= 0)
‖F (t)‖L1→L∞ ≤ C|t|
−n/2, (B.4)
∥∥Ψ(t, h)− F (t)ψ(h2G0)
L1→L∞
≤ Chβ |t|−n/2. (B.5)
Proof. Clearly, (B.4) follows from (B.2) for |t| ≤ 2. Let |t| ≥ 2. Without loss of generality
we may suppose t ≥ 2. Write F = F1 + F2, where
F1(t) = i
∫ t−1
ei(t−τ)G0V eiτG0dτ,
F2(t) = i
ei(t−τ)G0V eiτG0dτ.
It follows from (B.2) that F2(t) satisfies (B.4). To deal with the operator F1(t), observe that
its kernel is of the form
U(|x− ξ|2/4, |y − ξ|2/4, t)V (ξ)dξ,
where cn is a constant and
U(σ1, σ2, t) =
∫ t−1
eiσ1/(t−τ)+iσ2/τ (t− τ)−n/2τ−n/2dτ.
To prove that F1(t) satisfies (B.4), it suffices to show that
|U(σ1, σ2, t)| ≤ Ct−n/2
1 + σ
, ∀σ1, σ2 > 0, t ≥ 2. (B.6)
To do so, observe that
U(σ1, σ2, t) = t
u(σ1t
−1, σ2t
−1, t−1) + u(σ2t
−1, σ1t
−1, t−1)
, (B.7)
where
u(σ′1, σ
2, κ) =
∫ 1/2
/(1−τ ′)+iσ′
/τ ′(1− τ ′)−n/2(τ ′)−n/2dτ ′.
It is easy to see that (B.6) follows from (B.7) and the bound
|u(σ′1, σ′2, κ)| ≤ Cκ−(n−3)/2(σ′2)−1/2, ∀σ′1, σ′2 > 0, 0 < κ ≤ 1/2. (B.8)
To prove (B.8), we make a change of variables µ = 1/τ ′ and write the function u in the form
u(σ′1, σ
2, κ) =
∫ κ−1
eiϕ(µ,σ
µn/2−2dµ,
where
ϕ(µ, σ′1, σ
2) = µσ
µ− 1σ
We have
|u(σ′1, σ′2, κ)| ≤ C
∫ κ−1
µn/2−2dµ ≤ Cκ−(n−2)/2. (B.9)
Furthermore, observe that
ϕ′(µ) =
= σ′2 −
(µ− 1)2 ,
so ϕ′ vanishes at µ0 = 1 + (σ
1/2. We will consider now two cases.
Case 1. µ0 6∈ [3/2, 3κ−1/2]. Then, we have
|ϕ′(µ)| ≥ σ′2
|µ− µ0|
µ− 1 ≥
, µ ∈ [2, κ−1].
Therefore, integrating by parts, we obtain
u(σ′1, σ
2, κ) =
∫ κ−1
(iϕ′)−1
µn/2−2deiϕ
= eiϕ(iϕ′)−1
µn/2−2
∣∣∣∣∣
∫ κ−1
eiϕf(µ)dµ, (B.10)
where
f(µ) =
(iϕ′)−1
µn/2−2
= (iϕ′)−1
µn/2−2
µn/2−2.
Since ∣∣∣∣
∣∣∣∣ ≤
2σ′1(µ− 1)−2
(µ− 1) |σ′2 − σ′1(µ− 1)−2|
µ− 1 ,
we have (for µ ≥ 2)
|f(µ)| ≤ C(σ′2)−1µn/2−3. (B.11)
By (B.10) and (B.11),
|u(σ′1, σ′2, κ)| ≤ C(σ′2)−1κ−(n−4)/2. (B.12)
Clearly, in this case (B.8) follows from (B.9) and (B.12).
Case 2. µ0 ∈ [3/2, 3κ−1/2]. Denote I(µ0) = [9µ0/10, 11µ0/10] ∩ [2, κ−1]. We write the
function u as u1 + u2, where
I(µ0)
µn/2−2dµ = µ
n/2−1
Ĩ(µ0)
eiλφ(z)g(z)dz, (B.13)
where we have made a change of variables µ = µ0(1 + z), Ĩ(µ0) ⊂ [−1/10, 1/10], λ = µ0σ′2,
g(z) =
1 + z
1 + z − µ−10
(1 + z)n/2−2,
φ(z) = (1 + z)
(µ0 − 1)2
µ0(1 + z)− 1
= µ0 +
µ0 − 1
z2 +O(z3), |z| ≪ 1,
uniformly in µ0. It is easy to see that we have the estimate
eiλφ(z)g(z)dz
∣∣∣∣ ≤ Cλ
−1/2, |a| ≤ 1/10. (B.14)
Indeed, the functions g(z) and φ(z) are analytic in |z| ≤ 1/10 with |g(z)| bounded there
uniformly in µ0. Therefore, we can change the contour of integration to obtain (with some
0 < γ ≪ 1)
eiλφ(z)g(z)dz
∣∣∣∣ ≤
eiλφ(e
iγy)g(eiγy)dy
∣∣∣∣+
∣∣∣∣a
eiλφ(e
iθa)g(eiθa)dθ
e−Cλy
dy + C′1
′λθdθ = O(λ−1/2),
with some constants C,C′, C1, C
1 > 0. By (B.13) and (B.14) we conclude
|u1| ≤ C(σ′2)−1/2µ
(n−3)/2
0 ≤ C̃(σ′2)−1/2κ−(n−3)/2. (B.15)
On the other hand, if µ ∈ [2, κ−1] \ I(µ0), then
|µ− µ0|
µ− 1 ≥ C > 0,
so we can bound from below |ϕ′(µ)|. Therefore, the function u2 can be treated in the same
way as does u in Case 1. Thus, u2 satisfies (B.8) and hence, in view of (B.15), so does u.
This completes the proof of (B.4).
It suffices to prove (B.5) for 0 < h ≤ h0 with some constant 0 < h0 ≤ 1, since for
h0 ≤ h ≤ 1 it follows from (B.4) and the estimate of the L1 → L∞ norm of Ψ(t, h) proved in
[7] for the larger class of potentials satisfying (1.1) with δ > (n+2)/2 (without using (1.3)).
Without loss of generality we may suppose t > 0. Now, using Duhamel’s formula as in [6],
[7] we get the identity
Ψ(t;h)− F (t)ψ(h2G0) =
Ψj(t;h), (B.16)
where
Ψ1(t;h) = ψ1(h
2G0)e
ψ(h2G)− ψ(h2G0)
2G)− ψ1(h2G0)
eitG0ψ(h2G0) +
2G)− ψ1(h2G0)
Ψ(t;h),
Ψ2(t;h) = i
2G0)e
i(t−τ)G0V eiτGψ(h2G)dτ,
Ψ3(t;h) = −i
ei(t−τ)G0V eiτG0ψ(h2G0)dτ,
Ψ4(t;h) = i
∫ t−γ
2G0)e
i(t−τ)G0VΨ(τ ;h)dτ,
Ψ5(t;h) = −i
∫ t−γ
(1− ψ1)(h2G0)ei(t−τ)G0V eiτG0ψ(h2G0)dτ,
where ψ1 ∈ C∞0 ((0,+∞)), ψ1 = 1 on suppψ, and 0 < γ ≪ 1 is a parameter to be fixed later
on, depending on h. In view of (A.4), we have
‖Ψ1(t;h)f‖L∞ ≤ Ch
2t−n/2‖f‖L1 + Ch2 ‖Ψ(t;h)f‖L∞ , ∀f ∈ L
1. (B.17)
By (B.2) and (B.3),
‖Ψj(t;h)f‖L∞ ≤ Cγt
−n/2‖f‖L1 + Cγ ‖Ψ(t;h)f‖L∞ , ∀f ∈ L
1, j = 2, 3, (B.18)
with a constant C > 0 independent of t, h and γ.
Proposition B.3 Let V satisfy (1.1) with δ > n− 1. Then, there exist constants C, β1 > 0
so that for 0 < h ≤ 1, t ≥ 2γ, we have the estimate
‖Ψ4(t, h)‖L1→L∞ ≤ Ch
β1γ−(n−3)/2t−n/2. (B.19)
Proof. We will make use of the following estimates proved in [7].
Proposition B.4 Let V satisfy (1.1) with δ > (n + 2)/2. Then, for every 0 < ǫ ≪ 1,
1/2− ǫ/4 ≤ s ≤ (n− 1)/2, 0 < h ≤ 1, t 6= 0, we have the estimates
∥∥∥ψ(h2G0)eitG0〈x〉−s−1/2−ǫ
L2→L∞
≤ Chs−(n−1)/2|t|−s−1/2, (B.20)
∥∥∥Ψ(t, h)〈x〉−s−1/2−ǫ
L2→L∞
≤ Chs−(n−3)/2−ǫ/4|t|−s−1/2. (B.21)
By (B.20) and (B.21), we get (with some 0 < ε0 ≪ 1)
‖Ψ4(t, h)‖L1→L∞
∫ t/2
∥∥∥ψ1(h2G0)ei(t−τ)G0〈x〉−n/2−ε0
L2→L∞
∥∥∥〈x〉−(n−2)/2−ε0Ψ(τ, h)
L1→L2
∫ t−γ
∥∥∥ψ1(h2G0)ei(t−τ)G0〈x〉−(n−2)/2−ε0
L2→L∞
∥∥∥〈x〉−n/2−ε0Ψ(τ, h)
L1→L2
≤ Chε0/4t−n/2
∫ t/2
τ−(n−2)/2dτ + Chε0/4t−n/2
∫ t−γ
(t− τ)−(n−2)/2dτ
≤ Chε0/4γ−(n−3)/2t−n/2.
Proposition B.5 Let V satisfy (1.1) with δ > n−1. Then, for every 0 < ǫ≪ 1, 0 < h ≤ 1,
t ≥ 2γ, we have the estimate
‖Ψ5(t, h)‖L1→L∞ ≤ Cǫh
ǫγ−(n−3)/2−ǫt−n/2. (B.22)
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) =
(2π)ν+1
Jν(σλ)ψ(h2λ2)λdλ = h−nK1(σh−1, th−2), (B.23)
where Jν(z) = zνJν(z), Jν(z) = (H+ν (z) +H−ν (z)) /2 is the Bessel function of order ν =
(n− 2)/2. So, the kernel of the operator Ψ5 is of the form
Wh(|x− ξ|, |y − ξ|, t, γ)V (ξ)dξ,
where
Wh(σ1, σ2, t, γ) = −i
∫ t−γ
K̃h(σ1, t− τ)Kh(σ2, τ)dτ
= h−2n+2W1(σ1h
−1, σ2h
−1, th−2, γh−2), (B.24)
where K̃h is defined by replacing in the definition of Kh the function ψ by 1−ψ1. It is easy
to see that (B.22) follows from the bound (for all σ1, σ2, γ > 0, 0 < ǫ≪ 1, t ≥ 2γ)
|Wh(σ1, σ2, t, γ)| ≤ Cǫhǫγ−(n−3)/2−ǫt−n/2
σ−n+21 + σ
1 + σ
2 + σ
. (B.25)
In view of (B.24), it suffices to prove (B.25) with h = 1. Now, observe thatW1 =W
where
1 (σ1, σ2, t, γ) =
(σ1σ2)
4(2π)2ν+2
ei(t−γ)λ
+iγλ2
2ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)dλ21dλ22,
1 (σ1, σ2, t, γ) =
(σ1σ2)
4(2π)2ν+2
ei(t−γ)λ
+iγλ2
1ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)dλ21dλ22,
where the function
ρ(λ21, λ
(1− ψ1)(λ21)ψ(λ22)
λ22 − λ21
= (1 − ψ1)(λ21)ψ1(λ22)
ψ(λ22)− ψ(λ21)
λ22 − λ21
satisfies the bound
∣∣∣∂α1
ρ(λ21, λ
∣∣∣ ≤ Cα1,α2〈λ21〉−1−α1 , ∀(λ1, λ2). (B.26)
Given any integers 0 ≤ k,m < n/2, since Jν(z) = O(zn−2) as z → 0, we can integrate by
parts to get
1 (σ1, σ2, t, γ) = i
−m−k(t− γ)−kγ−m (σ1σ2)
4(2π)2ν+2
ei(t−γ)λ
+iγλ2
×∂kλ2
ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)
dλ21dλ
1 (σ1, σ2, t, γ) = i
−m−k(t− γ)−kγ−m (σ1σ2)
4(2π)2ν+2
ei(t−γ)λ
+iγλ2
×∂mλ2
ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)
dλ21dλ
Using the inequality ∣∣∣∣
ϕ(λ)dλ
∣∣∣∣ ≤ C|t|
−1/2 ‖ϕ̂‖L1 , (B.27)
we obtain (for t ≥ 2γ)
∣∣∣W (1)1 (σ1, σ2, t, γ)
∣∣∣ ≤ Ct−k−1/2γ−m(σ1σ2)−2ν
eiτλ1+iγλ
×λ1∂kλ2
ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)
dλ1dλ
∣∣∣ dτ, (B.28)
∣∣∣W (2)1 (σ1, σ2, t, γ)
∣∣∣ ≤ Ct−k−1/2γ−m(σ1σ2)−2ν
eiτλ2+iγλ
×λ2∂mλ2
ρ(λ21, λ
2)Jν(σ1λ1)Jν(σ2λ2)
dλ2dλ
∣∣∣ dτ. (B.29)
Recall now that the function Jν is of the form Jν(z) = eizb+ν (z) + e−izb−ν (z), where b±ν (z)
are symbols of order (n− 3)/2 for z ≥ 1, while near z = 0 the function Jν(z) is equal to z2ν
times an analytic function. Therefore, it satisfies the bounds
∣∣∂jzJν(z)
∣∣ ≤ Czn−2−j〈z〉j−(n−1)/2, ∀z > 0, 0 ≤ j ≤ n− 2, (B.30)
∣∣∂jzJν(z)
∣∣ ≤ Cj〈z〉(n−3)/2, ∀z > 0, j ≥ 0. (B.31)
Moreover, the functions b±ν (z) are of the form (near z = 0)
b±ν (z) = b
ν,1(z) + z
n−2 log z b±ν,2(z),
where b±ν,j(z) are analytic functions, b
ν,2(z) ≡ 0 if n is odd. Therefore, we have
∣∣∂jzb±ν (z)
∣∣ ≤ C, 0 < z ≤ 1, 0 ≤ j ≤ n− 3,
∣∣∂jzb±ν (z)
∣∣ ≤ Cǫz−ǫ, 0 < z ≤ 1, j = n− 2,
∣∣∂jzb±ν (z)
∣∣ ≤ Cjzn−2−j, 0 < z ≤ 1, j ≥ n− 1,
which imply ∣∣∂jzb±ν (z)
∣∣ ≤ C〈z〉(n−3)/2−j , ∀z > 0, 0 ≤ j ≤ n− 3, (B.32)
∣∣∂jzb±ν (z)
∣∣ ≤ Cǫz−ǫ〈z〉−(n−1)/2+ǫ, ∀z > 0, j = n− 2, (B.33)
∣∣∂jzb±ν (z)
∣∣ ≤ Cjzn−2−j〈z〉−(n−1)/2, ∀z > 0, j ≥ n− 1. (B.34)
± (λ1, λ2, σ1, σ2) = λ1e
∓iσ1λ1∂kλ2
ρ(λ21, λ
±iσ1λ1b±ν (σ1λ1)Jν(σ2λ2)
± (λ1, λ2, σ1, σ2) = λ2e
∓iσ2λ2∂mλ2
ρ(λ21, λ
2)Jν(σ1λ1)e±iσ2λ2b±ν (σ2λ2)
By (B.26), (B.30)-(B.34), we have (with ℓ = 0, 1)
∣∣∣∂ℓλ1A
± (λ1, λ2, σ1, σ2)
≤ C〈σ1〉k+(n−3)/2σn−22 〈σ2〉m−(n−1)/2〈λ1〉(n−3)/2−k−1, ∀(λ1, λ2), (B.35)
∣∣∣∂ℓλ2A
± (λ1, λ2, σ1, σ2)
≤ Cσn−21 〈σ1〉m−(n−1)/2〈σ2〉k+(n−3)/2〈λ1〉(n−3)/2−m−2, ∀(λ1, λ2). (B.36)
Using the inequality
‖ϕ̂(τ)‖L1 ≤ C‖〈τ〉ϕ̂(τ)‖L2 ≤ C
∥∥∂ℓλϕ(λ)
∣∣∂ℓλϕ(λ)
we obtain from (B.28) and (B.35) (if k > (n− 3)/2)
∣∣∣W (1)1 (σ1, σ2, t, γ)
∣∣∣ ≤
Ct−k−1/2γ−m(σ1σ2)
eiτλ1+iγλ
± (λ1, λ2, σ1, σ2)dλ1dλ
∣∣∣∣ dτ
Ct−k−1/2γ−m(σ1σ2)
−2ν sup
λ1,λ2
∣∣∣∂ℓλ1A
± (λ1, λ2, σ1, σ2)
≤ Ct−k−1/2γ−mσ−(n−2)/21 〈σ1〉k+(n−3)/2〈σ2〉m−(n−1)/2, (B.37)
where we have made a change of variables τ → τ ± σ1. Similarly, by (B.29) and (B.36), we
get (if m > (n− 3)/2)
∣∣∣W (2)1 (σ1, σ2, t, γ)
∣∣∣ ≤
Ct−k−1/2γ−m(σ1σ2)
eiτλ2+iγλ
± (λ1, λ2, σ1, σ2)dλ2dλ
∣∣∣∣ dτ
Ct−k−1/2γ−m(σ1σ2)
−2ν sup
∣∣∣∂ℓλ2A
± (λ1, λ2, σ1, σ2)
∣∣∣ dλ21
≤ Ct−k−1/2γ−mσ−(n−2)/22 〈σ2〉k+(n−3)/2〈σ1〉m−(n−1)/2. (B.38)
We would like to apply (B.37) and (B.38) with k = (n−1)/2,m = (n−3)/2+ǫ, 0 < ǫ≪ 1. To
this end, we need to show that these estimates are valid for all real (n−3)/2 < m ≤ (n−2)/2,
(n−2)/2 ≤ k < n/2 if n is even, and for k = (n−1)/2 and all real (n−3)/2 < m ≤ (n−1)/2
if n is odd. This can be done by interpolation as follows. Let φ ∈ C∞0 (R), φ(λ) = 1
for |λ| ≤ 1, φ(λ) = 0 for |λ| ≥ 2. Decompose W (j)1 as X(j) + Y (j), j = 1, 2, where X(j)
and Y (j) are defined by replacing in the definition of W
1 the function ρ by φ(λj)ρ and
(1−φ)(λj)ρ, respectively. Clearly, the functions X(j) satisfy (B.37) and (B.38), respectively,
for all integers 0 ≤ k,m < n/2, while the functions Y (j) satisfy (B.37) and (B.38) for all
integers k > (n − 3)/2, (n − 3)/2 < m < n/2, respectively. When n is odd, this is fulfilled
with k = (n− 1)/2. To show this in the case of even n, we write the function φ as
φ(λ) =
with some function φ1 ∈ C∞0 (R), φ1(λ) = 0 in a neighbourhood of λ = 0. Thus,
X(j) =
X(j)p ,
whereX
p is defined by replacing in the definition ofX
(j) the function φ(λj) by φ1(2
pλj). As
above, one can see that the functions X
p , j = 1, 2, satisfy (B.37) and (B.38), respectively,
with an extra factor in the RHS of the form 2p(k−n/2) for all integers k ≥ (n − 2)/2, and
hence, by interpolation, for all real k ≥ (n−2)/2. Therefore, summing up these estimates we
conclude that X(j), j = 1, 2, satisfy (B.37) and (B.38), respectively, for all real (n− 2)/2 ≤
k < n/2, and in particular for k = (n− 1)/2. Hence, so do the functions W (j)1 . Furthermore,
1 satisfies (B.37) for all integers 0 ≤ m < n/2, and hence, by interpolation, for all real
0 ≤ m ≤ (n− 1)/2 if n is odd, and for all real 0 ≤ m ≤ (n− 2)/2 if n is even. In particular,
this is valid with m = (n − 3)/2 + ǫ. To show that the function W (2)1 satisfies (B.38) with
m = (n − 3)/2 + ǫ, we decompose it as Z +N , where Z and N are defined by replacing in
the definition of W
1 the function ρ by φ(λ1)ρ and (1 − φ)(λ1)ρ, respectively. Clearly, the
function Z satisfies (B.37) for all integers 0 ≤ m < n/2, and hence, by interpolation, for all
real 0 ≤ m ≤ (n − 1)/2 if n is odd, and for all real 0 ≤ m ≤ (n− 2)/2 if n is even. To deal
with the function N , we write the function 1− φ as
(1 − φ)(λ) =
−pλ),
with some function φ2 ∈ C∞0 (R), φ2(λ) = 0 in a neighbourhood of λ = 0. Thus,
where Np is defined by replacing in the definition of N the function (1−φ)(λ1) by φ2(2−pλ1).
Now, the functionsNp satisfy (B.38) with an extra factor in the RHS of the form 2
−p(m−(n−3)/2)
for all integers 0 ≤ m < n/2, and hence, by interpolation, for all real 0 ≤ m ≤ (n − 1)/2
if n is odd, and for all real 0 ≤ m ≤ (n − 2)/2 if n is even. Therefore, summing up these
estimates we conclude that N satisfies (B.38) for all real (n− 3)/2 < m ≤ (n − 1)/2 if n is
odd, and for all real (n− 3)/2 < m ≤ (n− 2)/2 if n is even. In particular, this is valid with
m = (n− 3)/2 + ǫ.
By (B.37) and (B.38) with k = (n− 1)/2, m = (n− 3)/2 + ǫ, we obtain
|W1(σ1, σ2, t, γ)|
≤ Ct−n/2γ−(n−3)/2−ǫ
σ−n+21 〈σ1〉n−2〈σ2〉−1+ǫ + σ
2 〈σ2〉n−2〈σ1〉−1+ǫ
≤ Ct−n/2γ−(n−3)/2−ǫ
σ−n+21 + σ
2 + 〈σ1〉−1+ǫ + 〈σ2〉−1+ǫ
≤ Ct−n/2γ−(n−3)/2−ǫ
σ−n+21 + σ
2 + σ
1 + σ
which is the desired bound. ✷
Taking γ = hβ
with a suitably chosen constant β′ > 0, we deduce from (B.4), (B.16)-
(B.19) and (B.22), ∥∥Ψ(t;h)f − F (t)ψ(h2G0)f
≤ Chβt−n/2‖f‖L1 + Chβ
∥∥Ψ(t;h)f − F (t)ψ(h2G0)f
, ∀f ∈ L1, (B.39)
with some constant β > 0. Taking h small enough, we can absorb the second term in the
RHS of (B.39), thus obtaining (B.5). ✷
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Université de Nantes, Département de Mathématiques, UMR 6629 du CNRS, 2, rue de
la Houssinière, BP 92208, 44332 Nantes Cedex 03, France
e-mail: simon.moulin@math.univ-nantes.fr
e-mail: georgi.vodev@math.univ-nantes.fr
	Introduction and statement of results
	Preliminary estimates
	Proof of Theorem 1.1
	Appendix 1
	Appendix 2
