Benzoni-Gavage S., Serre D. Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications

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Littlewood–Paley decomposition 475

To justify this decomposition we must show that the series involved is convergent

in H

. At ﬁrst, we note that

F (S

p+1

u) − F (S

u)=G(S

u, ∆

u)∆

where

G(v, w)=





(v + tw)dt

is a C

∞

function of both its arguments. By Proposition C.3 and a piece of

calculus we can bound G(S

u, ∆

u)inL

∞

in exactly the same way we bounded

F (S

u)inL

. Thus we ﬁnd another constant depending continuously on R, still

denoted by C

,sothat

∂

( G(S

u, ∆

u))

∞

≤ C

p|α|

for all α with |α|≤m. Then a ﬁne result, postponed to Lemma C.2 below, enables

us to conclude. As a matter of fact, Lemma C.2 applies to M

= G(S

u, ∆

and shows that



∞



p=0

( F (S

p+1

u) − F (S

u))





p≥0

G(S

u, ∆

u)∆



≤ cC

u

• We have

F (u)=F (S

u)+

∞



p=0

( F (S

p+1

u) − F (S

u)).

Collecting and summing the estimates of F (S

u) and the series



( F (S

p+1

u) −

F (S

u) ) we ﬁnd that

F (u) 

≤ C (u

∞

) u

with C (u

∞

) = (1 + c) C

. (We recall that R ≥u

∞



Lemma C.2 (Meyer) Let {M

}

p≥0

be a sequence of C

∞

functions enjoying the

uniform estimates

∂



∞

≤ c

p|α|

(C.3.41)

for all α with |α|≤m.Thenforall0 <s<m, there exists c so that for all

u ∈ H

the series



∆

u is convergent in H

and





p≥0

∆



≤ cc

u

. (C.3.42)

Proof The proof resembles the one of Proposition C.9, in that the sequence

} satisﬁes by assumption the same estimates as S

u (derived in Proposition

476 Pseudo-/para-diﬀerential calculus

C.3). However, there is an additional diﬃculty due to the fact that, unlike S

is not supposed to be compactly supported. This is why we ﬁrst perform a

frequency decomposition of M

. For this we use the dilated functions φ

−p−3

·)

and deﬁne

p,q

:= F

−1

(φ

−p−3

·)

)

for all q ≥−1. Observe that, for q ≥ 0, we merely have

p,q

:= ∆

q+p+3

of which the spectrum is included in

{ξ ;2

p+q+2

≤ξ≤2

p+q+4

and that the ﬁrst term of the expansion,

p,−1

= F

−1

(ψ(2

−p−3

·)

) ,

has a spectrum included in

{ξ ; ξ≤2

p+3

Because of (C.3.13) (evaluated at 2

−p−3

ξ), we have



q≥−1

p,q

in the sense of S



. In fact, this series is normally convergent in L

∞

,sinceby

Proposition C.7,

M

p,q



∞

≤ C



|α|=m

∂

p,q



∞

−(p+q+3)m

for q ≥ 0, and by Proposition C.3 applied to ∂

∂

p,q



∞

≤ C

∂



∞

(we have used here the fact that [∂

, ∆

] = 0 for all r) and so the assumption

(C.3.41) implies that

M

p,q



∞

≤



−qm

• Let us now look at the two parameters family {M

p,q

∆

p≥0,q≥−1

.By

(C.3.20) and the previous inequality we have



p≥0



q≥−1

M

p,q

∆

u

≤





p≥0



q≥−1

−qm

−ps

u

< ∞,

which justiﬁes the interchanging formula



p≥0



q≥−1

p,q

∆

u =



q≥−1



p≥0

p,q

∆

Littlewood–Paley decomposition 477

This equivalently reads



p≥0

∆

u =



q≥−1

with



p≥0

p,q

∆

• We can now estimate Σ

in H

We begin with the special case q = −1. We have

Supp F ( M

p,−1

∆

u ) ⊂{ξ≤2

p+4

}

and thus

∆

( M

p,−1

∆

u ) ≡ 0forr ≥ p +5.

Therefore,

∆

−1

+∞



p=r−4

∆

( M

p,−1

∆

u )

for all r ≥−1. By exactly the same procedure as in the proof of Proposition C.9

and the uniform estimate

M

p,−1



∞

≤ C M



∞

≤ CC

we show that



r≥−1

2rs

∆

−1



≤ c

s,4



p≥−1

2ps

∆

u

with c

s,4



l≥−4

−ls

. Using the equivalent norm in (C.3.21) this precisely

means that

Σ

−1



≤ c

s,4

u

The general case q ≥ 0 is no more diﬃcult. We have

Supp F ( M

p,q

∆

u ) ⊂{2

p+q+1

≤ξ≤2

p+q+5

}

and thus

∆

( M

p,q

∆

u ) ≡ 0forr ≥ p + q +6 or r ≤ p + q − 1 .

Consequently,

∆

r−q



p=r−q−5

∆

( M

p,q

∆

u )

478 Pseudo-/para-diﬀerential calculus

and

∆

( M

p,q

∆

u ) 

≤



−qm

∆

u 

By the Cauchy–Schwarz inequality, we obtain



r≥−1

2rs

∆

( M

p,q

∆

u ) 

≤ 6

× 2



−2q(m−s)



p≥−1

2ps

∆

u 

which means that

Σ



≤ 6 × 2



u

−2q(m−s)

• The conclusion then follows from the summation on q of the estimates for

Σ



. 

Let us mention an easy consequence of Theorem C.12 and Proposition C.11.

Corollary C.3 If F ∈ C

∞

(R) and s>d/2, then there exists a continuous

function C :(0, +∞) → (0, +∞) such that for all u and v in H

F (u) − F (v) 

≤ C(max(u

, v

)) u − v

Proof Without loss of generality, we may assume F



(0) equals 0. By Taylor’s

formula and Proposition C.11, we have

F (u) − F (v)

≤



F



(v + θ(u − v)) (u − v)

dθ

≤ C



max

|w|≤max(u

∞

,v

∞

)



(w)|u−v

+max

θ∈[0,1]

F



(v+θ(u−v))

u−v

∞



The ﬁrst term inside parentheses is already of the wanted form, by the Sobolev

embedding H

→ L

∞

. And in the second one we have

F



(v + θ(u−v))

u−v

∞

≤ C

(v+θ(u−v)

∞

)v+θ(u−v)

u−v

∞

by Theorem C.12, which yields the wanted inequality using again the Sobolev

embedding H

→ L

∞



C.3.5 Further estimates

A useful result that was not pointed out yet is the smoothing eﬀect of the operator

(a − T

)whena is at least Lipschitz.

Theorem C.13 For al l k ∈ N, there exists C>0 such that for all a ∈ W

k,∞

and all u in L

au − T

u 

≤ C a

k,∞

u

Littlewood–Paley decomposition 479

The case k = 0 (with no smoothing eﬀect!) is a trivial consequence of Propo-

sition C.8 and the triangular inequality. The diﬃcult case is of course for k ≥ 1.

A detailed proof can be found in [136] or [38] – theses references deal in fact

with the operator with parameter T

but the method works for T

A straightforward consequence of Theorem C.13 is the following.

Corollary C.4 There exists C>0 such that for all a ∈ W

1,∞

and u in L

for j ∈{1,...,d},

a∂

u − T

∂

u 

≤ C a

1,∞

u

Proof Observe that

a∂

u − T

∂

u = ∂

( au − T

u ) − ((∂

a) u − T

∂

u ) .

The second term is obviously bounded by ∂

a

∞

u

(according to Propo-

sition C.8 and the triangular inequality, or Theorem C.13 with k =0!)

And the ﬁrst one is bounded by a

k,∞

u

according to Theorem C.13

with k =1. 

A further useful result in this direction is the following.

Theorem C.14 If R

is a smoothing operator deﬁned as the convolution

operator by a kernel ρ

satisfying the standard properties of molliﬁers, namely

(x)=ε

−d

ρ(x/ε) with ρ ∈ D (R

; R

) and



ρ =1, then for all a ∈ W

1,∞

and u in L

,forj ∈{1,...,d},

R

(a∂

u) − a∂

u) 

≤ C a

1,∞

u

(C.3.43)

and

lim

ε→0

R

(a∂

u) − a∂

u) 

=0. (C.3.44)

Proof We can prove separately the results for the commutators [ R

∂

]

and [ R

, (a − T

)∂

]. The estimate for the latter comes directly from Corollary

C.4 and the boundedness of R

on L

. The estimate of [ R

∂

] is, in fact, a

consequence of para-diﬀerential calculus (see Section C.4 below, Theorem C.17),

once we observe that (R

)

ε∈(0,1)

is a family of pseudo-diﬀerential operators of

order 0. Indeed, each operator R

is inﬁnitely smoothing and R

goes to identity

when ε → 0. To be more precise, the symbol of R

, ρ

= ρ(ε·), belongs to S

and thus to S

−∞

. However, in the estimate

∂

ρ

(ξ) ≤C

m,β

(ε)(1 +ξ)

m−|β|

∀ξ ∈ R

the constant C

m,β

(ε) is uniformly bounded for ε ∈ (0, 1) only if m ≥ 0. So

Theorem C.17 shows that [ R

∂

] is a family of para-diﬀerential operators

of order 0 + 1 − 1 = 0. Regarded as operators on L

, their norms are uniformy

controlled by a

1,∞

. This shows the estimate for [ R

∂

]. Summing with

the estimate for [ R

, (a − T

)∂

] we get (C.3.43).

480 Pseudo-/para-diﬀerential calculus

The proof of the limit in (C.3.44) then proceeds in a classical way, approximat-

ing any function u of L

by a sequence of smoother functions u

is suﬃcient)

for which we know the limit holds true and applying (C.3.43) to u − u

. 

Remark C.6 A slightly more general statement of (C.3.44) is that for all a ∈

1,∞

and all u ∈ H

−1

lim

ε→0

[ a, R

] u 

=0.

Another smoothing result, which admits a proof much more elementary than

Theorems C.13 and C.14, holds true in Sobolev spaces of large negative index.

Proposition C.15 If s>

+1, there exists C>0 so that for all a ∈ H

)

and all ϕ ∈ D (R

aϕ − T

ϕ 

−s+1

≤ C a

ϕ

−s

Proof In fact, we shall prove that aϕ − T

ϕ belongs to L

– and use Remark

C.3 to conclude. The assumptions show that both series



∆

a and



∆

are normally convergent in L

, which allows us to write

aϕ =



p,q≥−1

∆

ϕ ∆

and thus by deﬁnition of T

ϕ:

au − T

ϕ =



q≥−1

q+2



p=−1

∆

ϕ ∆

By Proposition C.4, ∆

a 

− qs

and ∆

ϕ 

with



≤ C

ϕ

−s

. Therefore, by the Cauchy–Schwarz inequality in L



q≥−1

q+2



p=−1

∆

ϕ ∆

a

≤



q≥−1

q+2



p=−1

∆

ϕ

∆

a



q≥−1

q+2



p=−1

(p−q) s



k≥−2

− ks



p≥−1

p+k

≤



k≥−2

− ks

β



α



≤



a

ϕ

−s

since s>0. This proves that au − T

ϕ belongs to L

and

aϕ − T

ϕ 

≤



a

ϕ

−s

Para-diﬀerential calculus 481

Finally, since s>

+ 1 we have the embedding L

) → H

−s+1

) (see

Remark C.3). This completes the proof. 

C.4 Para-diﬀerential calculus

The tools introduced in the previous section provide a basis for what is called

para-diﬀerential calculus, involving operators whose ‘symbol’ has a limited reg-

ularity in x. In particular, the operators T

encountered in para-products are

special cases of para-diﬀerential operators.

The purpose of this section is not to develop the whole theory but only

some major aspects that are used elsewhere in the book. We shall use again the

notation

(ξ):=(1+ξ

)

s/2

for all s ∈ R.

C.4.1 Construction of para-diﬀerential operators

Deﬁnition C.5 For any real number m and any natural integer k, we deﬁne

the set Γ

of functions, also called symbols, a : R

× R

→ C

N×N

such that

for almost all x ∈ R

, the mapping ξ ∈ R

→ a(x, ξ) is C

∞

for all d-uple β and all ξ ∈ R

, the mapping x ∈ R

→ ∂

a(x, ξ) belongs

to W

k,∞

and there exists C

> 0 so that for all ξ ∈ R

∂

a(·,ξ) 

k,∞

≤ C

m−|β|

(ξ) . (C.4.45)

Of course, by Deﬁnition C.1 we have S

⊂ Γ

for all k. The novelty is

that functions with rather poor regularity in x are allowed. In particular, W

k,∞

functions of x only may be viewed as symbols in Γ

Symbols belonging to Γ

are said to be of order m and regularity k. Unlike

inﬁnitely smooth symbols in S

, functions in Γ

are not naturally associated

with bounded operators H

→ H

s−m

. But this will be the case for the subclass

of symbols in Γ

satisfying the additional, spectral property:

Supp ( F (a(·,ξ)) ) ⊂ B(0; ελ

(ξ)) (C.4.46)

for some ε ∈ (0, 1) independent of ξ, see Theorem C.15 below. One may argue that

since their Fourier transform is compactly supported such symbols are necessarily

∞

in x. And we want to handle non-smooth symbols. So where is the trick? In

fact, it relies on a special smoothing procedure, associating any symbol a ∈ Γ

with a symbol σ ∈ Σ

. We shall give more details below. Let us start with the

study of operators associated with symbols in Σ

, k ≥ 0.

Theorem C.15 For al l a ∈ Γ

satisfying (C.4.46), consider

Op(a): F

−1



) −→ C

∞

u → Op(a) u ;(Op(a) u)(x)=

(2π)

e

ix·

a(x, ·) , u 

∞



)

482 Pseudo-/para-diﬀerential calculus

where E



denotes the space of temperate distributions having compact support

and the unusual ordering (C

∞

, E



) is just meant to account for matrix-valued a.

This deﬁnition of Op(a) coincides with (C.1.2) if a belongs to S

. Furthermore,

for all s ∈ R, Op(a) extends in a unique way into a bounded operator from H

to H

s−m

This is a fundamental result, which we admit here. Its proof shows in

particular that F (Op(a) u) is compactly supported.

Remark C.7 The set Σ

is strictly bigger than S

, as those functions a ∈ Σ

only satisfy a weakened version of (C.1.1), namely

|∂

∂

a(x, ξ) |≤C

α,β

m−|β|+|α|

(ξ) . (C.4.47)

This estimate is a consequence of what is known as Berstein’s Lemma, in fact

the special, easy case stated below.

Lemma C.3 If u ∈ L

∞

and Supp u ⊂ B(0; ελ) with ε>0 and λ>0, then for

all d-uple α, there exists C

ε,α

> 0 (independent of λ) so that

∂

u

∞

≤ C

ε,α

|α|

u

∞

Proof Just write

u(ξ)=ϕ

(ξ/λ) u(ξ),

with ϕ

a smooth compactly supported function such that ϕ

≡ 1onB(0; ε),

and proceed with a basic convolution estimate like in the proof of Proposition

C.3. 

Let us now describe the smoothing procedure for symbols in Γ

,which

amounts to a frequency cut-oﬀ depending on the ξ-variable.

Deﬁnition C.6 A C

∞

function χ :(η,ξ) → χ(η, ξ) ∈ R

is called an admis-

sible frequency cut-oﬀ if there exist ε

1,2

with 0 <ε

<ε

< 1 so that



χ(η, ξ)=1, if η≤ε

ξ and ξ≥1 ,

χ(η, ξ)=0, if η≥ε

(ξ) or ξ≤ε

and if for all d-uples α and β there exists C

α,β

> 0 so that

|∂

∂

χ(η, ξ) |≤C

α,β

−|α|−|β|

(ξ) . (C.4.48)

Example. If ψ and φ are as in the Littlewood–Paley decomposition of Section

C.3, the function χ deﬁned by

χ(η, ξ)=



p ≥ 0

ψ(2

2−p

η) φ(2

−p

ξ)=



p ≥ 0

p−2

(η) φ

(ξ)

The dual space of C

∞

Para-diﬀerential calculus 483

is an admissible frequency cut-oﬀ. Indeed, recall ﬁrst that there are at most two

indices p for which φ

(ξ) is non-zero. So the sum is locally ﬁnite. Furthermore,

recalling that

Supp ψ

p−2

⊂{η ; η≤2

p−2

} and Supp φ

⊂{ξ ;2

p−1

≤ξ≤2

p+1

it is easy to check that χ vanishes as requested with ε

=1/2. Indeed, for

ξ≤1/2, φ

(ξ) = 0 for all p ≥ 0, which implies χ(η, ξ)=0whereverη is.

Additionally, for all (η, ξ)wehave

χ(η, ξ)=



p ; η≤2

p−2

(η) φ

(ξ) ,

and therefore χ(η, ξ)=0assoonasξ≤2 η. A fortiori, this means that

χ(η, ξ)=0forη≥

(ξ). On the ‘contrary’, since ψ ≡ 1 on the sphere of

radius 1/2, if ξ≥16 η then ψ

p−2

(η) = 1 for all p ≥ 0 such that φ

(ξ) =0.

And if ξ≥1, ψ(ξ) = 0 and thus



p≥0

(ξ) = 1. This shows that χ(η, ξ)=

1ifξ≥16 η and ξ≥1. So the ﬁrst requirement on χ holds true with

=1/16. The inequalities in (C.4.48)

are trivially satisﬁed for ξ < 1/2.

Otherwise, for ξ≥1/2, let us rewrite

χ(η, ξ)=



p ; ξ≤2

p+2

p−2

(η) φ

(ξ) ,

hence

∂

χ(η, ξ)=



p ;ξ≤2

p+2

(2−p)|α|−p|β|

∂

ψ(2

2−p

η) ∂

φ(2

−p

ξ) .

For 1 ≤ 2ξ≤2

p+2

we have

−p

≤

ξ

≤

(ξ)

so, recalling that the sum is locally ﬁnite we ﬁnd that

|∂

∂

χ(η, ξ) |≤C 2

2|α|



(ξ)



−|α|−|β|

∂

ψ∂

φ

∞

Proposition C.16 Let χ be an admissible frequency cut-oﬀ according to Deﬁ-

nition C.6 and consider the operator

: a ∈ Γ

→ σ ∈ C

∞

; σ(·,ξ)=K

(·,ξ) ∗

a(·,ξ) ,

where the kernel K

is deﬁned by

(·,ξ)=F

−1

(χ(·,ξ)) .

Observe that they mean χ belongs to S

as a function of 2d variables.

484 Pseudo-/para-diﬀerential calculus

Then R

maps into

= {a ∈ Γ

; Supp ( F (a(·,ξ)) ) ⊂ B(0; ε

(ξ)) }.

Furthermore, if k ≥ 1, for all a ∈ Γ

, a − R

(a) belongs to Γ

m−1

k−1

In other words, the symbol σ = R

(a) is related to a in Fourier space by

F (σ(·,ξ)) = χ(·,ξ) F (a(·,ξ))

for all ξ ∈ R

. In particular, if a is independent of x, F (a(·,ξ)) = a(ξ) δ hence

F (σ(·,ξ)) = χ(0,ξ) a(ξ) δ. So we see that if χ(0,ξ) were equal to 1 for all ξ

we would have σ = a. This is not exactly the case

, but σ and a diﬀer by a

compactly supported function of ξ. In terms of operators, this means that Op(σ)

diﬀers from the Fourier multiplier associated with a by an inﬁnitely smoothing

operator, which is harmless in terms of para-diﬀerential calculus.

Proof Take a ∈ Γ

and consider σ = R

(a). Since Supp χ(·,ξ) ⊂

B(0; ε

(ξ)), by construction

Supp ( F (σ(·,ξ)) ) ⊂ B(0; ε

(ξ)) .

The fact that σ belongs to Γ

requires an L

estimate of the kernel K

,namely

∂

(·,ξ) 

)

≤ C

−|β|

(ξ) ,

which is left to the reader. Once we know this, we immediately obtain

σ(·,ξ)

∞

≤ C

a(·,ξ)

∞

≤



(ξ),

since a belongs to Γ

. The estimates of partial derivatives ∂

σ then follow from

the observation that

∂

(a)=R

(∂

a) .

Finally, the estimates of crossed partial derivatives ∂

∂

σ use the bilinearity of

∗ and the Leibniz formula.

One may observe that for σ ∈ Σ

with the number ε in (C.4.46) less than

/2, R

(σ) is ‘almost’ equal to σ. Indeed, since χ(η, ξ)=1forη≤ε

ξ

and ξ≥1, (C.4.46) with ε ≤ ε

/2 implies

F (R

(σ)(·,ξ)) = F (σ(·,ξ))

for ξ≥1. So in terms of operators it means that Op(R

(σ) − σ) is inﬁnitely

smoothing.

For a ∈ Γ

, let us show now that a − R

(a) belongs to Γ

m−1

. By triangular

inequality, we already know that

∂

(a − R

(a))

1,∞

≤ C

m−|β|

(ξ)

This problem is, in fact, overcome when using para-diﬀerential calculus with a parameter, see

Section C.5.