Local well-posedness of nonlinear dispersive equations on modulation
  spaces
LOCAL WELL-POSEDNESS OF NONLINEAR DISPERSIVE
EQUATIONS ON MODULATION SPACES
ÁRPÁD BÉNYI AND KASSO A. OKOUDJOU
Abstract. By using tools of time-frequency analysis, we obtain some improved
local well-posedness results for the NLS, NLW and NLKG equations with Cauchy
data in modulation spaces Mp,1
1. Introduction and statement of results
The theory of nonlinear dispersive equations (local and global existence, regularity,
scattering theory) is vast and has been studied extensively by many authors. Almost
exclusively, the techniques developed so far restrict to Cauchy problems with initial
data in a Sobolev space, mainly because of the crucial role played by the Fourier
transform in the analysis of partial differential operators. For a sample of results and
a nice introduction to the field, we refer the reader to Tao’s monograph [12] and the
references therein.
In this note, we focus on the Cauchy problem for the nonlinear Schrödinger equa-
tion (NLS), the nonlinear wave equation (NLW), and the nonlinear Klein-Gordon
equation (NLKG) in the realm of modulation spaces. Generally speaking, a Cauchy
data in a modulation space is rougher than any given one in a fractional Bessel poten-
tial space and this low-regularity is desirable in many situations. Modulation spaces
were introduced by Feichtinger in the 80s [6] and have asserted themselves lately as
the “right” spaces in time-frequency analysis. Furthermore, they provide an excellent
substitute in estimates that are known to fail on Lebesgue spaces. This is not entirely
surprising, if we consider their analogy with Besov spaces, since modulation spaces
arise essentially replacing dilation by modulation.
The equations that we will investigate are:
(1) (NLS) i
+∆xu+ f(u) = 0, u(x, 0) = u0(x),
(2) (NLW )
−∆xu+ f(u) = 0, u(x, 0) = u0(x),
(x, 0) = u1(x),
(3) (NLKG)
+ (I −∆x)u+ f(u) = 0, u(x, 0) = u0(x),
(x, 0) = u1(x),
Date: October 30, 2018.
2000 Mathematics Subject Classification. Primary 35Q55; Secondary 35C15, 42B15, 42B35.
Key words and phrases. Fourier multiplier, weighted modulation space, short-time Fourier trans-
form, nonlinear Schrödinger equation, nonlinear wave equation, nonlinear Klein-Gordon equation,
conservation of energy.
http://arxiv.org/abs/0704.0833v1
2 Á. Bényi and K. A. Okoudjou
where u(x, t) is a complex valued function on Rd×R, f(u) (the nonlinearity) is some
scalar function of u, and u0, u1 are complex valued functions on R
The nonlinearities considered in this paper will be either power-like
(4) pk(u) = λ|u|2ku, k ∈ N, λ ∈ R,
or exponential-like
(5) eρ(u) = λ(e
ρ|u|2 − 1)u, λ, ρ ∈ R.
Both nonlinearities considered have the advantage of being smooth. The correspond-
ing equations having power-like nonlinearities pk are sometimes referred to as alge-
braic nonlinear (Schrödinger, wave, Klein-Gordon) equations. The sign of the coeffi-
cient λ determines the defocusing, absent, or focusing character of the nonlinearity,
but, as we shall see, this character will play no role in our analysis on modulation
spaces.
The classical definition of (weighted) modulation spaces that will be used through-
out this work is based on the notion of short-time Fourier transform (STFT). For
z = (x, ω) ∈ R2d, we let Mω and Tx denote the operators of modulation and transla-
tion, and π(z) = MωTx the general time-frequency shift. Then, the STFT of f with
respect to a window g is
Vgf(z) = 〈f, π(z)g〉.
Modulation spaces provide an effective way to measure the time-frequency concen-
tration of a distribution through size and integrability conditions on its STFT. For
s, t ∈ R and 1 ≤ p, q ≤ ∞, we define the weighted modulation space Mp,qt,s (Rd) to
be the Banach space of all tempered distributions f such that, for a nonzero smooth
rapidly decreasing function g ∈ S(Rd), we have
‖f‖Mp,qt,s =
|Vgf(x, ω)|p < x >tp dx
< ω >qs dω
Here, we use the notation
< x >= (1 + |x|2)1/2.
This definition is independent of the choice of the window, in the sense that different
window functions yield equivalent modulation-space norms. When both s = t = 0, we
will simply write Mp,q = Mp,q0,0. It is well-known that the dual of a modulation space
is also a modulation space, (Mp,qs,t )′ = M
p′,q′
−s,−t, where p
′, q′ denote the dual exponents
of p and q, respectively. The definition above can be appropriately extended to
exponents 0 < p, q ≤ ∞ as in the works of Kobayashi [9], [10]. More specifically, let
β > 0 and χ ∈ S such that suppχ̂ ⊂ {|ξ| ≤ 1} and
k∈Zd χ̂(ξ − βk) = 1, ∀ξ ∈ Rd.
For 0 < p, q ≤ ∞ and s > 0, the modulation space Mp,q0,s is the set of all tempered
distributions f such that
|f ∗ (Mβkχ)(x)|p dx
< βk >sq
NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 3
When, 1 ≤ p, q ≤ ∞ this is an equivalent norm on Mp,q0,s, but when 0 < p, q < 1
this is just a quasi-norm. We refer to [9] for more details. For another definition of
the modulation spaces for all 0 < p, q ≤ ∞ we refer to [5, 15]. For a discussion of
the cases when p and/or q = 0, see [4]. These extensions of modulation spaces have
recently been rediscovered and many of their known properties reproved via different
methods by Baoxiang et all [1], [2]. There exist several embedding results between
Lebesgue, Sobolev, or Besov spaces and modulation spaces, see for example [11], [13];
also [1], [2]. We note, in particular, that the Sobolev space H2s coincides with M
For further properties and uses of modulation spaces, the interested reader is referred
to Gröchenig’s book [8].
The goal of this note is two fold: to improve some recent results of Baoxiang,
Lifeng and Boling [1] on the local well-posedness of nonlinear equations stated above,
by allowing the Cauchy data to lie in any modulation space Mp,10,s, p > dd+1 , s ≥ 0,
and to simplify the methods of proof by employing well-established tools from time-
frequency analysis. Ideally, one would like to adapt these methods to deal with global
well-posedness as well. We plan to address these issues in a future work.
In what follows, we assume that d ≥ 1, k ∈ N, d
< p ≤ ∞, λ, ρ ∈ R and s ≥ 0
are given. With pk and eρ defined by (4) and (5) respectively, our main results are
the following.
Theorem 1. Assume that u0 ∈ Mp,10,s(Rd) and f ∈ {pk, eρ}. Then, there exists
T ∗ = T ∗(‖u0‖Mp,10,s) such that (1) has a unique solution u ∈ C([0, T
∗],Mp,10,s(Rd)).
Moreover, if T ∗ < ∞, then lim sup
t→T ∗
‖u(·, t)‖Mp,10,s = ∞.
Theorem 2. Assume that u0, u1 ∈ Mp,10,s(Rd) and f ∈ {pk, eρ}. Then, there exists
T ∗ = T ∗(‖u0‖Mp,10,s , ‖u1‖Mp,10,s) such that (2) has a unique solution u ∈ C([0, T
∗],Mp,10,s(Rd)).
Moreover, if T ∗ < ∞, then lim sup
t→T ∗
‖u(·, t)‖Mp,10,s = ∞.
Theorem 3. Assume that u0, u1 ∈ Mp,10,s(Rd) and f ∈ {pk, eρ}. Then, there exists
T ∗ = T ∗(‖u0‖Mp,10,s , ‖u1‖Mp,10,s) such that (3) has a unique solution u ∈ C([0, T
∗],Mp,10,s(Rd)).
Moreover, if T ∗ < ∞, then lim sup
t→T ∗
‖u(·, t)‖Mp,10,s = ∞.
Remark 1. In Theorem 1 we can replace the (NLS) equation with the following
more general (NLS) type equation:
(7) (NLS)α i
+∆α/2x u+ f(u) = 0, u(x, 0) = u0(x),
for any α ∈ [0, 2] and p ≥ 1. The operator ∆α/2x is interpreted as a Fourier multiplier
operator (with t fixed), ∆̂
x u(ξ, t) = |ξ|αû(ξ, t). This strengthening will become
evident from the preliminary Lemma 1 of the next section.
Remark 2. Theorems 1.1 and 1.2 of [1] are particular cases of Theorem 1 with p = 2
and s = 0.
4 Á. Bényi and K. A. Okoudjou
2. Fourier multipliers and multilinear estimates
The generic scheme in the local existence theory is to establish linear and nonlinear
estimates on appropriate spaces that contain the solution u. As indicated by the main
theorems above, the spaces we consider here areMp,10,s, and we present the appropriate
estimates in the lemmas below. In fact, we will need estimates on Fourier multipliers
on modulation spaces. As proved in [3] and [7], a function σ(ξ) is a symbol of a
bounded Fourier multiplier on Mp,q for 1 ≤ p, q ≤ ∞ if σ ∈ W (FL1, ℓ∞) (see the
proofs of the following two lemmas for a definition of this space). As we shall indicate
below, this condition can be naturally extended to give a sufficient criterion for the
boundedness of the Fourier multiplier operator on Mp,q0,s for 0 < p, q ≤ ∞ and s ≥ 0.
The notation A . B stands for A ≤ cB for some positive constant c independent of
A and B.
Lemma 1. Let σ be a function defined on Rd and consider the Fourier multiplier
operator Hσ defined by
Hσf(x) =
σ(ξ) f̂(ξ) e2πξ·x dξ.
Let χ ∈ S such that supp χ̂ ⊂ {|ξ| ≤ 1}. Let d ≥ 1, s ≥ 0, 0 < q ≤ ∞, and 0 < p < 1.
If σ ∈ W (FLp, ℓ∞)(Rd), i.e.,
‖σ‖W (FLp,ℓ∞) = sup
‖σ · Tβnχ‖FLp < ∞
for β > 0, then Hσ extends to a bounded operator on Mp,q0,s(Rd).
Proof. We use the definition of the modulation spaces given by (6) (see also [9]). In
particular, let χ ∈ S such that supp χ̂ ⊂ {|ξ| ≤ 1}, and define g ∈ S by ĝ = χ̂2.
Denote g̃(x) = g(−x). For f ∈ S, β > 0, k ∈ Zd and x ∈ Rd we have:
|Hσf ∗ (Mβkg̃)(x)| = |VgHσf(x, βk)|
= |〈σf̂,M−xTβkĝ〉|
= |〈σf̂,M−xTβkχ̂2〉|
≤ |F−1(σ · Tβkχ̂)| ∗ |F−1(f̂ · Tβkχ̂)|(x)
≤ |F−1(σ · Tβkχ̂)| ∗ |f ∗ (Mβkχ̃)|(x).
Now, observe that supp
σ · Tβkχ̂
⊂ Γk := βk + {|ξ| ≤ 1} and supp
f̂ · Tβkχ̂
⊂ Γk.
Moreover, by assumption we know that σ ∈ W (FLp, ℓ∞) and so F−1(σ · Tβkχ̂) ∈ Lp
and f ∗(Mβkχ̃) ∈ Lp. Consequently, by [9, Lemma 2.6] we have the following estimate
‖Hσf ∗ (Mβkg̃)‖Lp ≤ C ‖F−1(σ · Tβkχ̂)‖Lp‖f ∗ (Mβkχ̃)‖Lp,
where C is a positive constant that depends only on the diameter of Γk and p. Clearly,
the diameter of Γk is independent of k, and this makes C a constant depending only
on the dimension d and the exponent p. Therefore, for 0 < q ≤ ∞ we have
‖Hσf‖Mp,q0,s . sup
‖F−1(σ · Tβkχ̂)‖Lp ‖f‖Mp,q0,s = ‖σ‖W (FLp,ℓ∞) ‖f‖Mp,q0,s .
NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 5
The result then follows from the density of S in Mp,q0,s for p, q < ∞; see [9, Theorem
3.10]. �
We are now ready to state and prove the boundedness of Fourier multipliers that
will be needed in establishing our main results.
Lemma 2. Let d ≥ 1, s ≥ 0, and 0 < q ≤ ∞ be given. Define mα(ξ) = ei|ξ|
1 ≤ p ≤ ∞ and α ∈ [0, 2], then the Fourier multiplier operator Hmα extends to a
bounded operator on Mp,q0,s(Rd).
Moreover, If α ∈ {1, 2} and d
< p ≤ ∞, then the Fourier multiplier operator
Hmα extends to a bounded operator on M
0,s(R
Proof. First, we prove the result when 1 ≤ p ≤ ∞, and 0 < q ≤ ∞. Let g ∈ S(Rd)
and define χ ∈ S by χ̂ = g2. For f ∈ S, we have
|VχHmαf(x, ξ)|
mα(t) f̂(t) e
2πix·t χ̂(t− ξ) dt
mα(t) Tξg(t) < t >
s f̂(t) <ξ>
<t>s <t−ξ>N < t− ξ >
N g(t− ξ) e2πix·t dt
mα(t) Tξg(t)φN(ξ, t) ̂< D >s f(t) TξgN(t) dt
∣∣∣∣F
mα · Tξg φN(ξ, ·) ̂< D >s f TξgN
∣∣∣∣F(mα · Tξg) ∗ F2(φN(ξ, ·)) ∗ F( ̂< D >s f · TξgN)(−x)
∣∣∣∣,
where N > 0 is an integer to be chosen later, gN(t) =< t >
N g(t), φN(ξ, t) =
<t>s <t−ξ>N , and < D >
s is the Fourier multiplier defined by ̂< D >s f(ξ) =< ξ >s
f̂(ξ). We also denote by Φ2,N (ξ, ·) := F2(φN(ξ, ·)) the Fourier transform in the second
variable of φN(ξ, ·)
6 Á. Bényi and K. A. Okoudjou
We can therefore estimate the weighted modulation norm of Hmαf as follows:
‖Hmαf‖Mp,q0,s
|Vχf(x, ξ)|p dx
< ξ >qs
∣∣∣∣F(mα · Tξg) ∗ Φ2,N (ξ, ·) ∗ F( ̂< D >s f · TξgN)(−x)
‖F−1(mα · Tξg)‖qL1 ‖Φ2,N(ξ, ·)‖
‖F( ̂< D >s f · TξgN)‖qLp dξ
≤ sup
‖F−1(mα · Tξg)‖L1 sup
‖Φ2,N(ξ, ·)‖L1
‖F−1( ̂< D >s f · TξgN)‖qLp dξ
≤ sup
‖F(mα · Tξg)‖L1 sup
‖Φ2,N (ξ, ·)‖L1 ‖f‖Mp,q0,s .
Now, it follows from [3, Lemma 8] that, for α ∈ [0, 2],
‖F−1(mα · Tξg)‖L1 := ‖mα‖W (FL1,ℓ∞) < ∞.
Moreover (see, for example, [13, Lemma 3.1] or [14, Lemma 2.1]), we can select a
sufficiently large N > 0 such that
‖Φ2,N (ξ, ·)‖L1 ≤
|Φ2,N (ξ, x))|dx < ∞.
Hence, using (8), we get
‖Hmαf‖Mp,q0,s ≤ Cα‖f‖Mp,q0,s .
To prove the second part of the result we shall use Lemma 1. In particular, we
need to show that for α ∈ {1, 2} and d
< p < 1, mα ∈ W (FLp, ℓ∞). This, however,
follows by straightforward adaptations of the proofs of [3, Theorems 9 and 11], which
we leave to the interested reader. �
In analogy to the proof of the previous lemma, we can prove the following weighted
version of [3, Theorem 16].
Lemma 3. Let d ≥ 1, s ≥ 0, d
< p ≤ ∞ and 0 < q ≤ ∞ be given, and let
m(1)(ξ) =
sin(|ξ|)
|ξ| and m
(2)(ξ) = cos(|ξ|), for ξ ∈ Rd. Then, the Fourier multiplier
operators Hm(1) , Hm(2) can be extended as bounded operators on M
A “smooth” version of Lemma 3 is obtained by replacing |ξ| with < ξ >.
Lemma 4. Let d ≥ 1, s ≥ 0, d
< p ≤ ∞ and 0 < q ≤ ∞ be given, and let
m(ξ) = ei<ξ>, m(1)(ξ) =
sin(<ξ>)
and m(2)(ξ) = cos(< ξ >), for ξ ∈ Rd. Then, the
Fourier multiplier operators Hm, Hm(1), Hm(2) can be extended as bounded operators
on Mp,q0,s.
NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 7
Proof. It is clear that m,m(1), m(2) are C∞(Rd) functions and that all their derivatives
are bounded. Therefore, m,m(1), m(2) ∈ Cd+1(Rd) ⊂ M∞,1(Rd) ⊂ W (FL1, ℓ∞)(Rd)
[8, 11]. Thus, for 1 ≤ p ≤ ∞, and 0 < q ≤ ∞ the result follows from [3] and Lemma 2.
For d
< p < 1 and 0 < q ≤ ∞, it can be showed that m,m(1), m(2) ∈ Cd+1(Rd) ⊂
W (FLp, ℓ∞)(Rd). Indeed, this follows from obvious modifications to the proof of
the embedding Cd+1(Rd) ⊂ M∞,1(Rd) ⊂ W (FL1, ℓ∞)(Rd) [8, 11]. Furthermore, if
we modify, for example, the multiplier m to mt(ξ) = e
it<ξ>, t ∈ R, we have for
< p ≤ 1
(9) ‖mt‖W (FLp,ℓ∞) ≤ (1 + |t|)d+1,
and similar estimates hold for modified multipliers m
t and m
t . �
Finally, we state a crucial multilinear estimate that will be used in our proofs.
Although the estimate will be needed only in the particular case of a product of
functions (see Corollary 1), we present it here in its full generality that applies to
multilinear pseudodifferential operators.
An m-linear pseudodifferential operator is defined à priori through its (distribu-
tional) symbol σ to be the mapping Tσ from the m-fold product of Schwartz spaces
S × · · · × S into the space S ′ of tempered distributions given by the formula
Tσ(u1, . . . , um)(x)
σ(x, ξ1, . . . , ξm) û1(ξ1) · · · ûm(ξm) e2πix·(ξ1+···+ξm) dξ1 · · · dξm,(10)
for u1, . . . , um ∈ S. The pointwise product u1 · · ·um corresponds to the case σ = 1.
Lemma 5. If σ ∈ M∞,10,s (R(m+1)d), then the m-linear pseudodifferential operator Tσ
defined by (10) extends to a bounded operator from Mp1,q10,s ×· · ·×M
pm,qm
0,s into M
p0,q0
when 1
+ · · ·+ 1
+ · · ·+ 1
= m − 1 + 1
, and 0 < pi ≤ ∞, 1 ≤ qi ≤ ∞
for 0 ≤ i ≤ m.
This result is a slight modification of [4, Theorem 3.1]. Its proof proceeds along
the same lines, and therefore it is omitted here. Note that if σ ∈ M∞,10,s , and we pick
u1 = · · · = um = u (some of them could be equal to ū since the modulation norm is
preserved), p1 = · · · = pm = mp, 0 < p ≤ ∞, and q1 = · · · = qm = 1 we have
(11) ‖Tσ(u, . . . , u)‖Mp,10,s . ‖u‖
Mmp,10,s
. ‖u‖mMp,10,s ,
where we used the obvious embedding Mp,10,s ⊆ M
0,s . The notation A . B stands
for A ≤ cB for some positive constant c independent of A and B. In particular, if
we select σ = 1 (the constant function 1), then σ ∈ M∞,10,s ⊂ M∞,1, and we obtain
Corollary 1. Let 0 < p ≤ ∞. If u ∈ Mp,10,s, then um ∈ M
0,s. Furthermore,
‖um‖Mp,10,s . ‖u‖
Mp,10,s
8 Á. Bényi and K. A. Okoudjou
This is of course just a particular case of the more general multilinear estimate
Mp0,q00,s
‖ui‖Mpi,qi0,s ,
where the exponents satisfy the same relations as in Lemma 1. When we consider
the power nonlinearity f(u) = pk(u) = λ|u|2ku = λuk+1ūk, Corollary 1 becomes
Corollary 2. Let 0 < p ≤ ∞. If u ∈ Mp,10,s, then pk(u) ∈ M
0,s. Furthermore,
‖pk(u)‖Mp,10,s . ‖u‖
Mp,10,s
For a different proof of the estimate in Corollary 2, see [1, Corollary 4.2]. It is
important to note that the previous estimate allows us to control the exponential
nonlinearity eρ as well. Indeed, since
eρ(u) = λ(e
ρ|u|2 − 1)u =
pk(u),
if we now apply the modulation norm on both sides and use the triangle inequality,
we arrive at
Corollary 3. Let 0 < p ≤ ∞. If u ∈ Mp,10,s, then eρ(u) ∈ M
0,s. Furthermore,
‖eρ(u)‖Mp,10,s . ‖u‖Mp,10,s(e
|ρ|‖u‖2
0,s − 1).
3. Proofs of the main results
We are now ready to proceed with the proofs of our main theorems. We will only
prove our results for the power nonlinearities f = pk, by making use of Corollary 2.
The case of exponential nonlinearity f = eρ is treated similarly, by now employing
Corollary 3. In all that follows we assume that u : [0, T )×Rd → C where 0 < T ≤ ∞
and that f(u) = pk(u) = λ|u|2ku.
3.1. The nonlinear Schrödinger equation: Proof of Theorem 1. We start by
noting that (1) can be written in the equivalent form
(13) u(·, t) = S(t)u0 − iAf(u)
where
(14) S(t) = eit∆, A =
S(t− τ) · dτ.
Consider now the mapping
J u = S(t)u0 − i
S(t− τ)(pk(u))(τ) dτ.
It follows from Lemma 2 (see also [3, Corollary 18]) that
‖S(t)u0‖Mp,10,s ≤ C (t
2 + 4π2)d/4 ‖u0‖Mp,10,s ,
NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 9
where C is a universal constant depending only on d. Therefore,
(15) ‖S(t)u0‖Mp,10,s ≤ CT ‖u0‖Mp,10,s ,
where CT = sup
t∈[0,T )
C (t2 + 4π2)d/4. Moreover, we have
S(t− τ)(pk(u))(τ) dτ
Mp,10,s
‖S(t− τ)(pk(u))(τ)‖Mp,10,s dτ
≤ T CT sup
t∈[0,T ]
‖pk(u)(t)‖Mp,10,s .(16)
By using now Corollary 2, we can further estimate in (16) to get
S(t− τ)(pk(u))(τ) dτ
Mp,10,s
. CT T ‖u(t)‖2k+1Mp,10,s .
Consequently, using (15) and (17) we have
(18) ‖J u‖
C([0,T ],Mp,10,s)
≤ CT (‖u0‖Mp,10,s + cT ‖u‖
Mp,10,s
for some universal positive constant c. We are now in the position of using a stan-
dard contraction argument to arrive to our result. For completeness, we sketch it
here. Let BM denote the closed ball of radius M centered at the origin in the space
C([0, T ],Mp,10,s). We claim that
J : BM → BM ,
for a carefully chosen M . Indeed, if we let M = 2CT‖u0‖Mp,10,s and u ∈ BM , from (18)
we obtain
‖J u‖
C([0,T ],Mp,10,s)
+ cCTTM
2k+1.
Now let T be such that cCTTM
2k ≤ 1/2, that is, T ≤ T̃ (‖u0‖Mp,10,s ). We obtain
‖J u‖
C([0,T ],Mp,10,s)
that is J u ∈ BM . Furthermore, a similar argument gives
‖J u− J v‖
C([0,T ],Mp,10,s)
‖u− v‖
C([0,T ],Mp,10,s)
This last estimate follows in particular from the following fact:
pk(u)(τ)− pk(v)(τ) = λ(u− v)|u|2k(τ) + λv(|u|2k − |v|2k)(τ).
Therefore, using Banach’s contraction mapping principle, we conclude that J has a
fixed point in BM which is a solution of (13); this solution can be now extended up
to a maximal time T ∗(‖u0‖Mp,10,s ). The proof is complete.
10 Á. Bényi and K. A. Okoudjou
3.2. The nonlinear wave equation: Proof of Theorem 2. Equation (2) can be
written in the equivalent form
(19) u(·, t) = K̃(t)u0 +K(t)u1 − Bf(u)
where
(20) K(t) =
sin(t
−∆ , K̃(t) = cos(t
−∆), B =
K(t− τ) · dτ
Consider the mapping
J u = K̃(t)u0 +K(t)u1 − Bf(u).
Recall that f = pk. If we now use Lemma 3 (see also [3, Corollary 21]) for the first
two inequalities below and Corollary 2 for the last estimate, we can write
‖K̃(t)u0‖Mp,10,s ≤ CT‖u0‖Mp,10,s ,
‖K(t)u1‖Mp,10,s ≤ CT‖u1‖Mp,10,s ,
‖Bf(u)‖Mp,10,s ≤ cT CT‖u‖
Mp,10,s
where c is some universal positive constant. The constants T and CT have the
same meaning as before. The standard contraction mapping argument applied to J
completes the proof.
3.3. The nonlinear Klein-Gordon equation: Proof of Theorem 3. The equiv-
alent form of equation (3) is
(22) u(·, t) = K̃(t)u0 +K(t)u1 + Cf(u)
where now
(23) K(t) =
sin t(I−∆)1/2
(I−∆)1/2 , K̃(t) = cos t(I −∆)
1/2, C =
K(t− τ) · dτ.
Consider the mapping
J u = K̃(t)u0 +K(t)u1 + Cf(u).
Using Lemma 4 and the notations above, we can write
‖K̃(t)u0‖Mp,10,s ≤ CT‖u0‖Mp,10,s ,
‖K(t)u1‖Mp,10,s ≤ CT‖u1‖Mp,10,s ,
‖Cf(u)‖Mp,10,s ≤ cT CT‖u‖
Mp,10,s
The standard contraction mapping argument applied to J completes the proof.
NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 11
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Árpád Bényi, Department of Mathematics, 516 High Street, Western Washington
University, Bellingham, WA 98225, USA
E-mail address : arpad.benyi@wwu.edu
Kasso A. Okoudjou, Department of Mathematics, University of Maryland, Col-
lege Park, MD 20742, USA
E-mail address : kasso@math.umd.edu
	1. Introduction and statement of results
	2. Fourier multipliers and multilinear estimates
	3. Proofs of the main results
	3.1. The nonlinear Schrödinger equation: Proof of Theorem ??
	3.2. The nonlinear wave equation: Proof of Theorem ??
	3.3. The nonlinear Klein-Gordon equation: Proof of Theorem ??
	References
