Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 24, pp. 1–10.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ALMOST OPTIMAL LOCAL WELL-POSEDNESS FOR MODIFIED

BOUSSINESQ EQUATIONS

DAN-ANDREI GEBA, BAI LIN

Abstract. In this article, we investigate a class of modified Boussinesq equa-

tions, for which we provide first an alternate proof of local well-posedness in
the space (Hs ∩ L∞) × (Hs ∩ L∞)(R) (s ≥ 0) to the one obtained by Con-

stantin and Molinet [7]. Secondly, we show that the associated flow map is

not smooth when considered from Hs × Hs(R) into Hs(R) for s < 0, thus
providing a threshold for the regularity needed to perform a Picard iteration

for these equations.

1. Introduction

1.1. Background. Our goal is to study the initial value problem (IVP)

utt − uxx − uxxtt = (f(u))xx, u = u(t, x) ∈ R+ × R→ R,
u(0, x) = u0(x), ut(0, x) = u1(x),

(1.1)

for which the differential equation is known in the literature as a modified Boussi-
nesq (imBq) equation. Initially, Makhankov [10] derived the equation with f(u) =
u2 in the context of ion-sound wave propagation and mentioned the one with
f(u) = u3 as modeling nonlinear Alfvén waves. Later, Clarkson, LeVeque, and
Saxton [6] discovered that the equations with either f(u) = u3/3 or f(u) = u5/5
describe the propagation of longitudinal deformation waves in an elastic rod. For
related parabolic models, please see work by Chen and Liu [4].

The imBq equation is also known as an improved frequency dispersion version
of the classical Boussinesq equation

utt − uxx − uxxxx = (u2)xx,

derived in relation to shallow water waves. The latter has the dispersive relation

ω2 = k2 − k4,

which leads to a nonphysical instability when k > 1. This is not the case for the
imBq equation, whose dispersive relation is given by

ω2 =
k2

1 + k2
.

2010 Mathematics Subject Classification. 35B30, 35Q55.

Key words and phrases. Modified Boussinesq equation; well-posedness; ill-posedness.
c©2020 Texas State University.

Submitted June 12, 2019. Published March 19, 2020.

1



2 D.-A. GEBA, B. LIN EJDE-2020/24

In the same context, another well-known improved frequency dispersion version of
the classical Boussinesq equation is the “good” or “well-posed” Boussinesq equation

utt − uxx + uxxxx = (f(u))xx, (1.2)

which was found to describe electromagnetic waves in nonlinear dielectrics, magne-
toelastic waves in antiferromagnets, and shape-memory alloys.

Past investigations concerning the IVP (1.1) mainly focused on two directions.
The first one concentrated on the existence and uniqueness of various types of local
solution (e.g., strong, classical), as well as on sufficient conditions for the existence
or the blow-up in finite time of such global solutions. We mention here work by
Constantin and Molinet [7], who looked at the equation

utt − uxxtt = (F (u))xx, F ∈ C∞(R), F (0) = 0, (1.3)

and showed that the associated IVP is locally well-posed (LWP) for (u(0), ut(0)) ∈
(Hs∩L∞)(R)×(Hs∩L∞)(R), with s ≥ 0 being arbitrary. Moreover, the same paper
contains both continuation criterions for local-in-time solutions to be extended into
global ones and conditions on F which guarantee either global solutions or blow-up
in finite time for certain data profiles. Similar results were obtained by Wang and
Chen [14] for the multidimensional problem (i.e, x ∈ Rn, n ≥ 2, and every ∂2x is
replaced by ∆).

The other type of question that was studied in connection to the IVP (1.1) is
the existence and scattering of global small amplitude solutions. A very formal
description of this question is as follows: what are the values of p > 1 for which
global, small Hs solutions to (1.1), with |f(u)| ' |u|p, scatter? Cho and Ozawa
[5] gave an almost optimal answer to this question both for (1.1) and the IVP for
the “good” Boussinesq equation (1.2). We refer the interested reader to this article
and references therein for a comprehensive discussion of this issue.

1.2. Description and statement of main results. One topic which is usually
studied in relation to evolution equations, especially dispersive ones, is the ill-
posedness (IP) of the associated IVP. To our best knowledge, such an inquiry has
not been conducted yet for (1.1). The goal of this article is to do just that, in the
case when f(u) = ±up and p > 1 is an integer. Our results are in the same spirit
with the ones originally obtained by Bourgain [3] and Tzvetkov [13] for the KdV
equation and then also derived for other dispersive equations (e.g., Molinet and
Ribaud [11], Bona and Tzvetkov [2], Geba, Himonas, and Karapetyan [8]).

They establish loss of smoothness for the flow map, which is defined for a fixed
time t as

(u0, u1) 7→ S(t)(u0, u1) := u(t).

The loss of regularity occurs when the domain of the flow map is chosen to be
Hs × Hs(R), with s < 0 being arbitrary. To argue for the optimality claimed in
the title, we show that (1.1) is LWP in (Hs ∩L∞)× (Hs ∩L∞)(R) when s ≥ 0, by
running a contraction argument for one of its integral formulations. In particular,
this implies that the flow map is smooth as a map from (Hs∩L∞)× (Hs∩L∞)(R)
to (Hs ∩ L∞)(R) for all times in the interval of existence. It is not clear that
this conclusion can be drawn, at least easily, from the analysis done by Constantin
and Molinet in [7]. There, equation (1.3) is recast in the form of an ODE system
in Banach spaces and the LWP is obtained by classical Picard iteration. Another



EJDE-2020/24 MODIFIED BOUSSINESQ EQUATIONS 3

reason for the inclusion of our LWP argument is that it contains some new harmonic
analysis facts that may be of independent interest.

Following this, we derive the integral version of the IVP (1.1), which is the
central object of study from this point onward. This is obtained by first rewriting
the imBq equation as

utt + P (D)u = −P (D)f(u), P (D) := F−1ξ
ξ2

1 + ξ2
Fx, (1.4)

and then applying Duhamel’s principle to infer

u(t) = L(u0, u1)(t)−
∫ t

0

L (0, P (D)(f(u(τ)))) (t− τ) dτ, (1.5)

where

̂L(v0, v1)(t)(ξ) := cos(tλ(ξ)) v̂0(ξ) +
sin(tλ(ξ))

λ(ξ)
v̂1(ξ),

λ(ξ) :=
|ξ|
〈ξ〉

=
|ξ|

(1 + ξ2)1/2
.

(1.6)

We can now state our main results.

Theorem 1.1. Consider the integral equation (1.5) with f(u) = ±up and p > 1
being an arbitrary integer.

(i) (LWP) If s ≥ 0 and (u0, u1) ∈ (Hs ∩ L∞)× (Hs ∩ L∞)(R), then there exist

T = T (‖(u0, u1)‖(Hs∩L∞)×(Hs∩L∞)(R)) > 0

and a unique solution u satisfying

u ∈ C([0, T ], (Hs ∩ L∞)(R)).

Moreover,

S(t) : (Hs ∩ L∞)× (Hs ∩ L∞)(R)→ (Hs ∩ L∞)(R), S(t)(u0, u1) := u(t),

is smooth for all t ∈ [0, T ].
(ii) (IP) If s < 0, then there exists T > 0 such that S(t) : Hs×Hs(R)→ Hs(R)

does not admit a p-th order Fréchet derivative at zero for all 0 < t < T .

The LWP part of this theorem will be addressed in the next section, whereas
the argument for IP will occupy the final section.

2. LWP argument

In proving the LWP claim, we rely on the classical approach of verifying that
the right-hand side of (1.5), when seen as a functional in u (with the data u0 and
u1 being fixed), is a contraction on a suitably chosen closed ball of a Banach space.

For this purpose, we are first concerned with the mapping properties of the
multiplier operators P (D) (defined in (1.4)) and

Qt(D) := F−1ξ cos(tλ(ξ))Fx, Rt(D) := F−1ξ
sin(tλ(ξ))

λ(ξ)
Fx, (2.1)

where t ∈ R is arbitrary, yet fixed. Given the trivial bounds

0 ≤ λ2(ξ) =
ξ2

1 + ξ2
< 1, | cos(tλ(ξ))| ≤ 1,

∣∣ sin(tλ(ξ))

λ(ξ)

∣∣ ≤ |t|,



4 D.-A. GEBA, B. LIN EJDE-2020/24

Plancherel’s formula implies

‖P (D)v‖Hs ≤ ‖v‖Hs , ‖Qt(D)v‖Hs ≤ ‖v‖Hs , ‖Rt(D)v‖Hs ≤ |t|‖v‖Hs . (2.2)

From here on, for a functional space Y , we write Y = Y (R) as the majority of such
norms refers to this particular situation.

Next, which is one of the novelties in our paper, we show that the symbols of
these operators are also Fourier multipliers on L∞ in the sense of Bergh-Löfström
[1, Def. 6.1.1]. By comparison, Constantin and Molinet [7] proved that P (D) maps
Hs ∩ L∞ into itself for all s ≥ 0. In arguing for this claim, we rely on a number
of facts, some of which are contained in the book by Bergh and Löfström. One of
them is [1, Exercise 16 on page 164] which states that the homogeneous Besov space

Ḃ
n/2
2,1 (Rn) is a subspace of the normed space of Fourier multipliers on L∞(Rn). A

second fact (Inferred from [1, Theorem 6.3.1] and from Shatah-Struwe [12, Section

3.2].) is the equivalence between the original seminorm for Ḃ
n/2
2,1 (Rn) and

‖w‖∗
Ḃ

n/2
2,1 (Rn)

:=

∫
Rn

‖w(·+ h)− w(·)‖L2(Rn)

|h|n+ 1
2

dh.

We will also use the following integration result, which is a special case of Ginibre-
Tsutsumi-Velo [9, Lemma 4.2],∫

R

1

〈z − a〉2〈z − b〉4
dz .

1

〈a− b〉2
, (∀) a, b ∈ R. (2.3)

Lemma 2.1. The symbols

m1(ξ) = λ2(ξ), m2(ξ) = e±itλ(ξ), m3(ξ) = sin(tλ(ξ))/λ(ξ)

are all Fourier multipliers on L∞ and

‖P (D)v‖L∞ . ‖v‖L∞ , (2.4)

‖Qt(D)v‖L∞ . |t|‖v‖L∞ , ‖Rt(D)v‖L∞ . max{|t|, |t|3}‖v‖L∞ . (2.5)

Proof. Based on the facts listed above, it is clear that the lemma is proved if we
show that

‖m1‖∗Ḃ1/2
2,1

. 1, ‖m2‖∗Ḃ1/2
2,1

. |t|, ‖m3‖∗Ḃ1/2
2,1

. max{|t|, |t|3}.

A direct application of the Cauchy-Schwarz inequality yields

‖w(·+ h)− w(·)‖L2 ≤ |h|‖w′‖L2

and straightforward computations provide us with the bounds

|m′1(ξ)| . 1

〈ξ〉3
, |m′2(ξ)| . |t|

〈ξ〉3
, |m′3(ξ)| . |t|

3

〈ξ〉3
.

Then we infer that ∫
|h|≤2

‖m1(·+ h)−m1(·)‖L2

|h|3/2
dh . 1,∫

|h|≤2

‖m2(·+ h)−m2(·)‖L2

|h|3/2
dh . |t|,∫

|h|≤2

‖m3(·+ h)−m3(·)‖L2

|h|3/2
dh . |t|3.



EJDE-2020/24 MODIFIED BOUSSINESQ EQUATIONS 5

All what is left to discuss is the scenario when |h| ≥ 2. In this case, since

|ξ|+ |ξ + h| ≥ |h|,

it follows that

max{|ξ|, |ξ + h|} ' max{〈ξ〉, 〈ξ + h〉}. (2.6)

Coupled with

λ(ξ + h)− λ(ξ) =
h(2ξ + h)

〈ξ + h〉〈ξ〉(|ξ + h|〈ξ〉+ |ξ|〈ξ + h〉)
, (2.7)

this implies

|λ(ξ + h)− λ(ξ)| . |h|max
{ 1

〈ξ + h〉〈ξ〉2
,

1

〈ξ + h〉2〈ξ〉
}
.

As a consequence of (2.3), we deduce∫
|h|≥2

‖λ(·+ h)− λ(·)‖L2

|h|3/2
dh . 1. (2.8)

On the other hand, from 0 ≤ λ(ξ) < 1, we have

|m1(ξ + h)−m1(ξ)| ≤ 2|λ(ξ + h)− λ(ξ)|. (2.9)

We also have

|m2(ξ + h)−m2(ξ)| = 2 |sin(t(λ(ξ + h)− λ(ξ))/2)| ≤ |t||λ(ξ + h)− λ(ξ)|. (2.10)

For m3, we can write

m3(ξ + h)−m3(ξ) =
sin(tλ(ξ + h))− sin(tλ(ξ))

λ(ξ + h)
+ sin(tλ(ξ))

( 1

λ(ξ + h)
− 1

λ(ξ)

)
which leads to

|m3(ξ + h)−m3(ξ)| ≤ 2|t||λ(ξ + h)− λ(ξ)|
λ(ξ + h)

.

By symmetry, we obtain

|m3(ξ + h)−m3(ξ)| ≤ 2|t||λ(ξ + h)− λ(ξ)|
max{λ(ξ + h), λ(ξ)}

.

From (2.6), we infer max{λ(ξ + h), λ(ξ)} ' 1, and, subsequently,

|m3(ξ + h)−m3(ξ)| . |t||λ(ξ + h)− λ(ξ)|.

Estimates (2.8)-(2.10) imply∫
|h|≥2

‖m1(·+ h)−m1(·)‖L2

|h|3/2
dh . 1,∫

|h|≥2

‖m2(·+ h)−m2(·)‖L2

|h|3/2
dh . |t|,∫

|h|≥2

‖m3(·+ h)−m3(·)‖L2

|h|3/2
dh . |t|,

and the proof of the lemma is complete. �



6 D.-A. GEBA, B. LIN EJDE-2020/24

Now, we can start in earnest the LWP argument. For fixed u0 and u1, we denote
the right-hand side of (1.5) by Zu0,u1(u) and, using (1.4) and (2.1), we infer

Zu0,u1
(u) = Qt(D)u0 +Rt(D)u1 −

∫ t

0

Rt−τ (D)(P (D)(f(u(τ)))) dτ.

The well-known Moser-type estimate

‖vw‖Hs . ‖v‖Hs‖w‖L∞ + ‖v‖L∞‖w‖Hs ,

which is valid for all s ≥ 0, implies that Hs ∩ L∞ is an algebra in this case. If we
assume 0 ≤ t ≤ T and use (2.2), (2.4), and (2.5), we deduce

‖Zu0,u1
(u)(t)‖Hs∩L∞ . max{1, t}‖u0‖Hs∩L∞ + max{t, t3}‖u1‖Hs∩L∞

+

∫ t

0

max{t− τ, (t− τ)3}‖f(u(τ))‖Hs∩L∞ dτ.

Choosing now f(u) as in Theorem 1.1, it follows that

‖Zu0,u1(u)‖C([0,T ];Hs∩L∞) . max{1, T}‖u0‖Hs∩L∞ + max{T, T 3}‖u1‖Hs∩L∞

+ max{T 2, T 4}‖u‖pC([0,T ];Hs∩L∞).

Therefore, by working with T < 1, and with the ball of radius R centered at the
origin in the Banach space C([0, T ];Hs ∩ L∞), B(0, R) ⊂ C([0, T ];Hs ∩ L∞) we
obtain that

u ∈ B(0, R) 7→ Zu0,u1
(u) ∈ B(0, R)

if

R ' ‖u0‖Hs∩L∞ + ‖u1‖Hs∩L∞ and T . R−
p−1
2 .

Furthermore, using a similar argument, we derive

‖Zu0,u1
(u)− Zu0,u1

(ũ)‖C([0,T ];Hs∩L∞)

. T 2‖u− ũ‖C([0,T ];Hs∩L∞)

(
‖u‖p−1C([0,T ];Hs∩L∞) + ‖ũ‖p−1C([0,T ];Hs∩L∞)

)
. T 2Rp−1‖u− ũ‖C([0,T ];Hs∩L∞).

Thus, with an eventual additional adjustment on the size of T , we conclude that u 7→
Zu0,u1

(u) is a contraction on the ball B(0, R) and, consequently, a unique solution
to the integral equation (1.5) exists on the time interval [0, T ]. The smoothness of
the flow map follows then by an application of the analytic version of the implicit
function theorem. With this, the LWP part of Theorem 1.1 has been proved.

3. IP argument

As explained in the introduction, the scheme for proving IP consists in showing
that a putative flow map, when acting from Hs ×Hs into Hs, fails to be smooth
for any s < 0. Our approach is very similar to the one used in [8], from which it
borrows the main framework.

From (1.5), we see that if f(u) = ±up then

S(t)(u0, u1) = L(u0, u1)(t)∓
∫ t

0

L
(

0, P (D)
(

(S(τ)(u0, u1))p
))

(t− τ) dτ,

for all t ∈ [0, T ], where T > 0 is such that the flow map makes sense on [0, T ] near
the origin in Hs ×Hs. When the flow map is sufficiently regular, we can use this



EJDE-2020/24 MODIFIED BOUSSINESQ EQUATIONS 7

equation to compute explicitly (by relying on implicit differentiation) its Fréchet
derivatives at the origin. Precisely, we have

DS(t)(v0,v1)(u0, u1)

= L(v0, v1)(t)

∓ p
∫ t

0

L
(

0, P (D)
(
DS(τ)(v0,v1)(u0, u1)

(
S(τ)(u0, u1)

)p−1))
(t− τ) dτ,

where DS(t)(v0,v1)(u0, u1) stands for the first order Fréchet derivative of the flow
map at (u0, u1), evaluated for (v0, v1). Given that LWP ensures S(t)(0, 0) = 0, we
deduce

DS(t)(v0,v1)(0, 0) = L(v0, v1)(t).

Arguing along the same lines, we derive

DkS(t)(v10 ,v11),...,(vk0 ,vk1 )(0, 0) = 0, for 1 < k < p,

and, eventually,

DpS(t)(v10 ,v11),...,(v
p
0 ,v

p
1 )

(0, 0)

= ∓p!
∫ t

0

L
(
0, P (D)

(
L(v10 , v

1
1)(τ) · · ·L(vp0 , v

p
1)(τ)

))
(t− τ) dτ,

(3.1)

Hence, if the flow map had Cp regularity at the origin, the estimate

‖DpS(t)(v10 ,v11),...,(v
p
0 ,v

p
1 )

(0, 0)‖Hs .
p∏
j=1

‖(vj0, v
j
1)‖Hs×Hs

would hold uniformly for t ∈ [0, T ]. However, when s < 0, we show that this bound
fails by constructing a sequence

(
uN0 , u

N
1

)
N
⊂ Hs ×Hs satisfying

lim
N→∞

‖DpS(t)(uN
0 ,u

N
1 ),...,(uN

0 ,u
N
1 )(0, 0)‖Hs(

‖uN0 ‖Hs + ‖uN1 ‖Hs

)p =∞, for 0 < t < T. (3.2)

For ease of notation, we use onward the abbreviation

Ap(u0, u1)(t) := DpS(t)(u0,u1),...,(u0,u1)(0, 0).

We work with the data

ûN0 (ξ) = ϕBN
(ξ) + ϕ−BN

(ξ), ûN1 (ξ) = −iλ(ξ) (ϕBN
(ξ)− ϕ−BN

(ξ)) , (3.3)

where (BN )N≥1 is a sequence of subsets of R and ϕA is the characteristic function
of the set A. It is easy to check that

ûN0 (ξ) = ûN0 (−ξ), ûN1 (ξ) = ûN1 (−ξ),

and, thus, our data are real-valued. By using (1.6) and (3.1), we infer that

̂L(uN0 , u
N
1 )(t)(ξ) = e−itλ(ξ)ϕBN

(ξ) + eitλ(ξ)ϕ−BN
(ξ)

and, subsequently,

̂Ap(uN0 , u
N
1 )(t)(ξ)

= ∓p!λ(ξ)

∫ t

0

sin((t− τ)λ(ξ))
{∫

η1+···+ηp=ξ

p∏
j=1

ϕ±BN
(ηj) · e∓iτλ(ηj)

}
dτ,

(3.4)



8 D.-A. GEBA, B. LIN EJDE-2020/24

where the inner integral is∫
η1+···+ηp=ξ

f :=

∫
Rp−1

f(η1, . . . , ηp−1, ξ − η1 − . . .− ηp−1) dη1 . . . dηp−1

and, with its integrand, we assumed an Einstein summation convention for the
symbol ±; i.e., if η ∈ ±BN , then the corresponding exponent is ∓itλ(η).

Note the generic term in the time integral which yields ̂Ap(uN0 , u
N
1 )(t)(ξ) is of

the type ∫ t

0

sin(α(t− τ)) eiβτ dτ,

where

α = λ(ξ), β = −ε1λ(a1)− ε2λ(a2)− . . .− εpλ(ap),

ξ = ε1a1 + ε2a2 + . . .+ εpap, εj = ±1, aj ∈ BN , for 1 ≤ j ≤ p. (3.5)

Moreover, a direct computation reveals that for real parameters α and β we have

Re
{∫ t

0

sin(α(t−τ)) eiβτ dτ
}

=

{
α

α2−β2 (cos(βt)− cos(αt)), for |α| 6= |β|,
1
2 t sin(αt), for |α| = |β|.

(3.6)

There are two key facts which allow us to argue for (3.2). The first one is the
localization in frequency of our data, which is enforced by choosing

BN = [N,N + 1], ∀N ≥ 1.

Coupled with (3.3), this localization easily implies

‖uN0 ‖Hs + ‖uN1 ‖Hs ' Ns. (3.7)

The second important point is that we are interested only in the output of the
function Ap(u

N
0 , u

N
1 )(t) at preferred frequencies, depending on the parity of p. This

enables us to have control on the relative size of the parameter β in (3.5), which in
turn reduces the argument to obtaining good asymptotics for the generic term.

3.1. Argument for p even. In this case, we restrict our attention to the behavior

of ̂Ap(uN0 , u
N
1 )(t) on the interval [1/4, 1/2] and first deduce that

‖Ap
(
uN0 , u

N
1

)
(t)‖Hs ≥ ‖Ap

(
uN0 , u

N
1

)
(t)‖Hs(ξ∈[ 14 ,

1
2 ])

' ‖Ap
(
uN0 , u

N
1

)
(t)‖L2(ξ∈[ 14 ,

1
2 ])
.

(3.8)

Next, from (3.5), we obtain that for N sufficiently large (depending on p) we must
have an equal number of +1s and −1s in (3.5) for ξ ∈ [1/4, 1/2] to be true. Thus,
eventually relabelling the indices, we can write

ξ = a1 − a2 + . . .+ ap−1 − ap, β = λ(a1)− λ(a2) + . . .+ λ(ap−1)− λ(ap).

Following this, we use (2.7) to infer that

|λ(a)− λ(b)| . 1

N3
, ∀a, b ∈ BN , (3.9)

which leads to |β| . 1/N3. We also notice that α = λ(ξ) ' 1 if ξ ∈ [1/4, 1/2]. On
the basis of (3.6), we derive that for such values of α and β,

Re
{∫ t

0

sin(α(t− τ))eiβτ dτ
}
' sin2(αt) + O

( 1

N6

)
holds if 0 < t < 1 and N is large enough.



EJDE-2020/24 MODIFIED BOUSSINESQ EQUATIONS 9

These facts tell us that, for fixed 0 < t < 1, the real part of ̂Ap(uN0 , u
N
1 )(t) is

correctly described by the real part of the generic term in (3.4). As a consequence,
we obtain

lim inf
N→∞

‖Ap
(
uN0 , u

N
1

)
(t)‖L2(ξ∈[ 14 ,

1
2 ])
&
(∫ 1/2

1/4

sin4(λ(ξ)t) dξ
)1/2

and, factoring in (3.7) and (3.8), we conclude that

lim
N→∞

‖Ap(uN0 , uN1 )(t)‖Hs

(‖uN0 ‖Hs + ‖uN1 ‖Hs)p
=∞, for 0 < t < 1.

This proves (3.2) in the case when p is even.

3.2. Argument for p odd. For this scenario, we focus on how ̂Ap(uN0 , u
N
1 )(t)

evolves on the interval [N,N + 1] and, accordingly, proceed with

‖Ap
(
uN0 , u

N
1

)
(t)‖Hs ≥ ‖Ap

(
uN0 , u

N
1

)
(t)‖Hs(ξ∈[N,N+1])

' Ns‖Ap
(
uN0 , u

N
1

)
(t)‖L2(ξ∈[N,N+1]).

(3.10)

Arguing as in the even case, we deduce that the representation of ξ ∈ [N,N + 1] in
(3.5) requires precisely one more +1 than −1s. Hence, we obtain

ξ = a1 − a2 + . . .− ap−1 + ap, β = λ(a1)− λ(a2) + . . .− λ(ap−1) + λ(ap),

following a possible relabeling of the indices.
Next, it is straightforward to derive

|λ(a)− 1| ' 1

N2
, ∀a ∈ BN .

This estimate and (3.9) imply

α = 1 + O
( 1

N2

)
, −β = 1 + O

( 1

N2

)
.

Invoking (3.6) again, we infer that

Re
{∫ t

0

sin(α(t− τ))eiβτ dτ
}
' t sin(t) + O

( 1

N2

)
is valid for 0 < t < 1 and N big enough.

As in the case when p is even, the real part of the generic term in (3.4) describes

accurately the real part of ̂Ap(uN0 , u
N
1 )(t) and, thus,

‖Ap
(
uN0 , u

N
1

)
(t)‖L2(ξ∈[N,N+1]) ' t sin(t) + O

( 1

N2

)
.

Now, we can use (3.7) and (3.10) to conclude that

‖Ap(uN0 , uN1 )(t)‖Hs

(‖uN0 ‖Hs + ‖uN1 ‖Hs)p
' t sin(t) + O(1/N2)

Ns(p−1) , for 0 < t < 1,

which yields (3.2) also in the odd case.

Acknowledgements. The first author was supported in part by a grant from the
Simons Foundation #359727. Both authors are grateful to Allan Greenleaf and
Alex Iosevich for helpful discussions during the preparation of this manuscript.
Both authors kindly thank the careful referee for comments and suggestions which
improved the quality of this article.



10 D.-A. GEBA, B. LIN EJDE-2020/24

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Dan-Andrei Geba
Department of Mathematics, University of Rochester, Rochester, NY 14627, USA

Email address: dangeba@math.rochester.edu

Bai Lin

Department of Mathematics, University of Rochester, Rochester, NY 14627, USA

Email address: blin13@ur.rochester.edu


	1. Introduction
	1.1. Background
	1.2. Description and statement of main results

	2. LWP argument
	3. IP argument
	3.1. Argument for p even
	3.2. Argument for p odd
	Acknowledgements

	References