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

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Littlewood–Paley decomposition 465

with still the obvious notation χ

(ξ)=χ

−q

ξ). This easily implies that

∆

u =2

−qm



|α|=m

−1

∗ ∂

(∆

u) .

The result follows again from a convolution inequality and the identities

F

−1



= F

−1



We ﬁnd that

∆

u 

∞

≤ 2

−qm

max

|α|=m

F

−1





|α|=m

∂

(∆

u) 

∞



The proof here above would obviously fail for q = −1, because ∆

−1

u does

involve small frequencies. However, by Proposition C.3,

∆

−1

u

∞

≤ C

u

∞

≤ C

u

∞

for all k ∈ N. Therefore, using the commutation property ∂

∆

=∆

∂

consequence of Proposition C.7 is the following.

Corollary C.1 For al l k ∈ N, there exists C

> 0 so that for all u ∈ W

k,∞

= {u ; ∂

u 

∞

< ∞∀α ∈ N

; |α|≤k },

∀q ≥−1 , ∆

u 

∞

≤ C

−qk

u 

k,∞

. (C.3.26)

C.3.3 Para-products

The operators ∆

are also convenient tools to deﬁne para-products. The para-

product by (a not necessarily smooth) function u is intended to operate on

Sobolev spaces (and on H¨older spaces) when the standard product does not. Para-

products were originally introduced by Bony [20]. He actually deﬁned two kinds

of para-products, one based on the dyadic Littlewood–Paley decomposition, as

presented below, and one based on a continuous spectral decomposition, and

showed that they essentially enjoy the same properties.

To motivate the deﬁnition, let us consider two tempered distributions u and

v,andformally write

uv =



p,q≥−1

∆

u ∆



p ≥ 2

∆

p−3



q=−1

∆

v +



q ≥ 2

q−3



p=−1

∆

u ∆

v +



|p−q|≤2

∆

u ∆

466 Pseudo-/para-diﬀerential calculus

There is some arbitrariness in this decomposition. The idea is to separate

terms involving frequencies of the same order (last sum) from terms where the

frequencies of u dominate those of v (ﬁrst sum) or vice versa (middle sum).

Denoting as before by S

the truncated sums

q−1



p=−1

∆

for q ≥ 0 and setting by convention S

= 0 for q ≤−1, we introduce the para-

product of v by u as

v :=



q ≥−1

q−2

u ∆

v =



q ≥ 2

q−2

u ∆

v. (C.3.27)

This deﬁnition is still formal for arbitrary distributions. However, observing that

Supp F (S

q−2

u ∆

v) ⊂ Supp

q−3



p=−1

∗ φ

⊂{ξ≤2

q−2

} + {2

q−1

≤ξ≤2

q+1

}

and hence

Supp F ( S

q−2

u ∆

v ) ⊂



≤ξ≤



, (C.3.28)

it will be easy to give sense to (C.3.27) for a wide range of u and v.

Remark C.4 In the special, apparently trivial case when u is constant, the

para-product T

v is not exactly the usual product uv, but diﬀers from it by a

∞

function. Indeed, u = uδ and thus ∆

−1

u = u while ∆

u = 0 for q ≥ 0.

Therefore, T

v = u



q ≥ 2

∆

v and

uv − T

v = u



|q|≤1

∆

This means that the operator (u − T

) is inﬁnitely smoothing when u is constant.

We shall see in Theorem C.13 that for any Lipschitz function u the operator

(u − T

) is still smoothing, to a limited extent though.

In general, we formally have the symmetric decomposition

uv = T

u + T

v + R(u, v), (C.3.29)

where the remainder term is

R(u, v):=



|p−q|≤2

∆

u ∆

v, (C.3.30)

and will appear to be the smoothest term when it is well-deﬁned.

As regards the para-product T

, it does operate on H

for all s provided that

u belongs to L

∞

, as shown in Proposition C.8 below. The para-product by u is

Littlewood–Paley decomposition 467

a typical example of a para-diﬀerential operator (of order 0). See [20] for more

details, or Section C.4 for a sketch.

Proposition C.8 For al l s there exists C>0 so that for all u ∈ L

∞

and all

v ∈ H

T

v

≤ C u

∞

v

. (C.3.31)

Proof We begin with a remark on the meaning of the deﬁnition in (C.3.27)

for u ∈ L

∞

and v ∈ H

. By (C.3.17) and (C.3.20) we have

S

q−2

u ∆

v

≤ C u

∞

−qs

v

Thus if s>0 the series in (C.3.27) is normally convergent in L

. However, the

following computations justify a posteriori the deﬁnition of T

v for all s,since

they show that the series in (C.3.27) is convergent (though not normally, in

general) in H

Using the equivalent norm (C.3.21) from Proposition C.4, the estimate

(C.3.31) equivalently reads



p≥−1

2ps

∆

v

≤ C

u

∞



p≥−1

2ps

∆

v

To prove this, it is to be noted that

∆

( S

q−2

u ∆

v ) ≡ 0for|p − q|≥4 . (C.3.32)

Equation (C.3.32) is easy to check, since by (C.3.28) the support of

F ( S

q−2

u ∆

v ) is clearly disjoint from {2

p−1

≤ξ≤2

p+1

} for |p − q|≥4.

Therefore, and this is the crucial point in the proof, only a ﬁnite number of

terms from the sum in (C.3.27) are to persist under the operation of ∆

.We

have

∆

v =

p+3



q = p−3

∆

( S

q−2

u ∆

v ) ,

which implies by the Cauchy–Schwarz inequality that

2ps

∆

v

≤ 7 × 2

6|s|

p+3



q = p−3

2qs

∆

( S

q−2

u ∆

v )

Now all terms in this sum are easily estimated. By (C.3.22) we have

∆

( S

q−2

u ∆

v )

≤ C S

q−2

u ∆

v

and by (C.3.17)

S

q−2

u ∆

v

≤ C u

∞

∆

v

468 Pseudo-/para-diﬀerential calculus

Collecting and summing these successive inequalities, we arrive at the aimed

result



p≥−1

2ps

∆

v

≤ 7

× 2

6|s|

u

∞



q≥−1

2qs

∆

v



Actually, this proof can be reﬁned and extended to a more general framework.

Let r be a rational integer. If r ≥ 2 then we can ﬁnd k ∈ N so that we have

similarly as in (C.3.32)

∆

( S

q−r

u ∆

v ) ≡ 0for|p − q|≥k +1.

This is elementary by looking at

Supp F ( S

q−r

u ∆

v ) ⊂{ξ≤2

q−r

} + {2

q−1

≤ξ≤2

q+1

But we point out that for r ≤ 1, the support of F ( S

q−r

u ∆

v ) is not bounded

away from 0 and thus we only have ∆

( S

q−r

u ∆

v ) ≡ 0 for large p. More

precisely, there exists k ∈ N so that

∆

( S

q−r

u ∆

v ) ≡ 0forp − q ≥ k +1. (C.3.33)

However, these properties are suﬃcient to prove the analogous Proposition C.8,

under some restriction on s though.

Proposition C.9 For al l r ∈ Z and s>0 there exists C>0 so that for u ∈ L

∞

and v ∈ H





q ≥−1

q−r

u ∆



≤ C u

∞

v

. (C.3.34)

Proof The proof is very similar to that of Proposition C.8, except that we

must make use of a ﬁner Cauchy–Schwarz inequality (in 

instead of R

!) and

thus require s>0. To facilitate the reading, we denote

S(u, v)=



q ≥−1

q−r

u ∆

Because of (C.3.33) we have

∆

S(u, v)=

+∞



q = p−k

∆

( S

q−r

u ∆

v ) ,

which implies by the Cauchy–Schwarz inequality that

∆

S(u, v)

≤





+∞



q = p−k

−qs









+∞



q = p−k

∆

( S

q−r

u ∆

v )





Littlewood–Paley decomposition 469

The ﬁrst factor is clearly bounded by c

s,k

−ps

with c

s,k



l≥−k

−ls

for

positive s. And we know from (C.3.22) and (C.3.17) that

∆

( S

q−r

u ∆

v )

≤ C

u

∞

∆

v

Therefore, we ﬁnd that



p≥−1

2ps

∆

S(u, v)

≤ c

s,k

u

∞



p≥−1



q≥p−k

(p+q)s

∆

v

The conclusion then follows from the observation that



p≥−1



q≥p−k

(p+q)s

∆

v







l≥−k

−ls







q≥−1

2qs

∆

v

(The constant C in (C.3.34) is thus c

s,k

with our present notations.) 

As a consequence of this proposition, we get in particular an error estimate

for T

u, provided that both u and v belong to L

∞

∩ H

, s>0.

Proposition C.10 For al l s>0, there exists C>0 such that for all u and v

in L

∞

∩ H

, we have

uv − T

u 

≤ C u

∞

v

Proof The assumption s>0 ensures that for u, v ∈ H

the series



∆

u and



∆

v are normally convergent in L

(because of (C.3.20)). This justiﬁes the

formula

uv =



p,q≥−1

∆

u ∆

and thus by deﬁnition of T

uv − T

u =



q≥−1

q+3

u ∆

Consequently, Proposition C.9 applied to r = −3 yields the result. 

This in turn leads to a very simple proof of the following.

Proposition C.11 For al l s>0 there exists C>0 such that for all u and v

in L

∞

∩ H

, the product uv also belongs to H

and

uv

≤ C ( u

∞

v

+ v

∞

u

) .

Proof Just sum the estimate of T

u obtained in Proposition C.8 with the error

estimate in Proposition C.10. 

An alternative form of Proposition C.11 is the following.

470 Pseudo-/para-diﬀerential calculus

Proposition C.12 For each integer s>0 there exists C>0 such that for all

u and v in L

∞

∩ H

and all d-uples α, β with |α| + |β| = s we have

(∂

u)(∂

v) 

≤ C ( u

∞

v

+ v

∞

u

) . (C.3.35)

This proposition can actually be proved in a more classical way, by using the

H¨older inequality together with the Gagliardo–Nirenberg inequality [64,148]. The

latter holds indeed for each positive integer s, and gives a constant C>0sothat

for u ∈ L

∞

∩ H

and |α|≤s

∂

u 

2s/|α|

≤ C u

1−|α|/s

∞

u

|α|/s

. (C.3.36)

An easy consequence of Proposition C.12 is the following commutator esti-

mate.

Proposition C.13 If s>1 and α is a d-uple of length |α|≤s, there exists

C>0 such that for all u and a in H

with ∇u and ∇a in L

∞

 [ ∂

,a∇] u 

≤ C ( ∇a

∞

u

+ ∇u

∞

a

) .

Proof We ﬁrst note that the assumptions on a and u imply by Proposition

C.11 that a∇u belongs to H

s−1

and so

[ ∂

,a∇] u = ∂

(a∇u) − a ∇(∂

u) ∈ H

−1

for |α|≤s. Our purpose is to show that in fact [ ∂

,a∇] u belongs to L

(which

is in general the best one can expect since for a ∈ C

∞

the commutator [ ∂

,a∇]

is a diﬀerential operator of order |α|).

We consider an index j ∈{1, ···,d} and want to estimate the L

norm of

[ ∂

,a∂

] u. By the Leibniz rule there exist coeﬃcients c

with c

= 1 so that

∂

( a∂

u)=



|β|≤|α|

(∂

a)(∂

α−β

∂

u) .

Consequently, we have

[ ∂

,a∂

] u =



1≤|β|≤|α|

(∂

a)(∂

α−β

∂

u) .

In the latter sum, all terms are of the form

(∂

∂

a)(∂

α−β

∂

for some index k ∈{1, ···,d} and d-uple β

of length |β

| = |β|−1. Both ∂

and ∂

u belong to L

∞

∩ H

|α|−1

and thus meet the assumptions of Proposition

C.12 with s = |α|−1. This yields the estimate

(∂

∂

a)(∂

α−β

∂

u)

≤ C (∂

a

∞

∂

u

|α|−1

+ ∂

u

∞

∂

a

|α|−1

) ,

where the right-hand side is clearly bounded by ∇a

∞

u

+ ∇u

∞

a

for |α|≤s. Hence, by summing on β and k we get the desired estimate. 

Littlewood–Paley decomposition 471

To illustrate the power of para-products, let us just show the following result

on the remainder R, where we see that the regularity of R(u, v) is ‘almost’ the

one of u plus the one of v.

Theorem C.9 For al l s and t with s + t>0, there exists C>0 so that for all

u ∈ H

and all v ∈ H

, R(u, v) is well-deﬁned by (C.3.30) and meets the estimate

R(u, v)

s+t−d/2

≤ C u

v

. (C.3.37)

Proof • At ﬁrst, we check that the assumption s + t>0 ensures that R is

well-deﬁned and

R(u, v)=



q≥−1

(u, v) with R

(u, v):=

q+2



r=q−2

∆

u ∆

As a matter of fact, we have

R

(u, v)

≤

q+2



r=q−2

∆

u

∆

v

≤ C

q+2



r=q−2

−rs

u

−qt

v

by (C.3.20), hence

R

(u, v)

≤ 5 × 2

2|s|

C u

v

−q(s+t)

which shows that the series



(u, v) is normally convergent in L

• To prove the estimate in (C.3.37) we must evaluate the L

norm of

∆

R(u, v). Similarly as in the proof of Proposition C.9, we note that ∆

R(u, v)

only involves some terms ∆

(u, v). This is due to the fact (already used in

(C.3.33)) that there is an integer k such that

∆

(∆

u ∆

v ) ≡ 0forp − q ≥ k +1 and|r − q|≤2 . (C.3.38)

Therefore, we have

∆

R(u, v)=



q≥p−k

∆

(u, v) .

For all p ≥−1wehave

∆

(u, v)=

q+2



r=q−2

−1

∗ (∆

u ∆

v )

and thus a standard convolution inequality and Plancherel’s theorem show that

∆

(u, v) 

≤φ



q+2



r=q−2

∆

u ∆

v 

Since for p ≥ 0wehave

φ



pd/2

φ

472 Pseudo-/para-diﬀerential calculus

the latter inequality implies that for all p ≥−1

∆

(u, v) 

≤ c 2

pd/2

q+2



r=q−2

∆

u

∆

v 

with c := max(φ

, 2

d/2

ψ

Like in Proposition C.9 we can apply the Cauchy–Schwarz inequality to

obtain

∆

R(u, v)

≤ c

t+s,k

−p(t+s)



q≥p−k

q(t+s)

∆

(u, v)

with c

t+s,k



+∞

l=−k

−l(t+s)

. Consequently, we have

∆

R(u, v)

≤ C



−p(t+s−d)



q≥p−k

q(t+s)

∆

v 

q+2



r=q−2

∆

u

with C



:= 5 × c

t+s,k

and thus

2p(s+t−d/2)

∆

R(u, v)

≤ 5 × 2

2|s|





q≥p−k

(p−q)(t+s)

2qt

∆

v

q+2



r=q−2

2rs

∆

u

So we have by (C.3.18)



p≥−1

2p(s+t−d/2)

∆

R(u, v)

≤ C



v

u

with C



:= 5 × 2

2|s|



t+s,k

. This ﬁnally proves the estimate in (C.3.37)

with C =





t+s−d/2

. 

Remark C.5 This result on the remainder R(u, v) gives a slightly bigger index

than in the classical result recalled below for the full product uv.

Theorem C.10 For al l s and t with s + t>0,ifu ∈ H

and v ∈ H

then the

product belongs to H

for all r ≤ min(s, t) such that r<s+ t − d/2. Further-

more, there exists C (depending only on r, s, t and d) such that

uv

≤ C u

v

In the case r = s = t, this result may be viewed as a consequence of Proposi-

tion C.11; an alternative, elementary proof, which also gives the result in Sobolev

spaces on any smooth domain Ω, may be found in [1], p. 115–117. For more

general values of r, s and t (but Ω = R

), Theorem C.10 can be deduced from

Theorem C.9 and the following additional result on para-products.

Littlewood–Paley decomposition 473

Proposition C.14 For al l s and t,ifu ∈ H

and v ∈ H

then the para-product

v is well-deﬁned and belongs to H

for all r<s+ t − d/2. Furthermore, there

exists C>0 independent of u and v so that

T

v

≤ C u

v

. (C.3.39)

Proof It is very similar to the proof of Proposition C.8, replacing the use of the

estimate of S

u

∞

in (C.3.17) by the estimate in (C.3.24): the condition r<

s + t − d/2 is here to ensure the convergence of the series



−2p(s+t−d/2−r)

hence of



2ps

∆

v

. Details are left to the reader. 

An easy consequence of Theorem C.10 is the following commutator estimate.

Corollary C.2 If m is an integer greater than d/2+1 and α is a d-uple of

length |α|∈[1,m], there exists C>0 such that for all a in H

and all u ∈

|α|−1

 [ ∂

,a] u 

≤ C a

u

|α|−1

C.3.4 Para-linearization

Proposition C.8 and Theorem C.9 show in particular that for all s>0, if

u ∈ H

∩ L

∞

then

=2T

u + R(u, u)=T

u + R(u, u),

with the uniform estimates

T

u

≤ C u

∞

u

, R(u, u)

2s−d/2

≤ C u

A very strong result from para-diﬀerential calculus says that this decompo-

sition of F (u)=u

can be generalized to any C

∞

function F vanishing at 0,

under the only constraint that s>d/2.

Theorem C.11 (Bony–Meyer) If F ∈ C

∞

(R), F (0) = 0,ifs>d/2 then for

all u ∈ H

) we have

F (u)=T



(u)

u + R(u), (C.3.40)

with R(u) ∈ H

2s−d/2

(Note that the assumption s>d/2 automatically implies u ∈ L

∞

if u ∈

).)

Equation (C.3.40) is often referred to as the para-linearization formula

of Bony. Historically, Bony proved that the remainder term R(u) belongs to

2s−d/2−ε

for ε>0 [20], and Meyer proved the actual result with ε = 0 [138].

In particular, (C.3.40) shows that F (u) belongs to H

. We do not intend to

give the extensive proof of Theorem C.11. We ‘directly’ show that F (u) enjoys

the same estimate as its para-linearized counterpart T



(u)

474 Pseudo-/para-diﬀerential calculus

Theorem C.12 If F ∈ C

∞

(R), F (0) = 0,ifs>d/2 then there exists a con-

tinuous function C :[0, +∞) → [0, +∞) such that for all u ∈ H

)

F (u) 

≤ C (u

∞

) u

Proof The assumption s>d/2 obviously implies that each u ∈ H

) nec-

essarily belongs to L

∞

∩ L

• We begin by showing the estimate for F (S

u) instead of F (u). To do so,

it is suﬃcient to bound ∂

F (S

u)

for all d-uples α of length |α|≤m with

m − 1 ≤ s<m.For|α| = 0 this is almost trivial. By Propositions C.3 and C.5

we have

S

u

∞

≤ C u

∞

and S

u

≤ C u

and thus the mean value theorem applied to F in the ball of radius R = u

∞

implies that

F (S

u)

≤ C

u

for some C

> 0 depending continuously on R.For|α|≥1, the chain rule shows

there exist coeﬃcients c

, with b = {β

, ···,β

} being a family of d-uples of

positive length and of sum β

+ ···+ β

= α,sothat

∂

( F (S

u)) =



1 ≤ n ≤|α|

+ ···+ β

= α

|β

|≥1

(n)

u) ∂

u) ···∂

u) .

By Propositions C.3 and C.5 we have

∂

u)

∞

≤ C u

∞

and ∂

u)

≤ C u

Therefore, using L

∞

bounds for the successive derivatives F

(n)

, n ≤ m,onthe

ball of radius R we obtain a uniform estimate

∂

( F (S

u))

≤ C

u

for all α with |α|≤m. In particular, up to modifying C

,wehave

F (S

u) 

≤ C

u

for all s ≤ m.

• The other main part of the proof consists in bounding the ‘error’ F (u) −

F (S

u). Since S

u is known to tend to u in H

,weformally have

F (u) − F (S

u)=

∞



p=0

( F (S

p+1

u) − F (S

u)).