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

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Pseudo-diﬀerential calculus with a parameter 455

Thus the result will be proved if we show that

Op(a + a

∗

)u, u≥

9α

u

− C u

−1

In some sense this reduces the problem to Hermitian symbols.

• By assumption, the Hermitian symbol a := a + a

∗

− α



, with α



3α/4, is positive-deﬁnite. By Lemma C.1, there exists b ∈ S

such that b

∗

b = a.

Denoting B =Op(b)and



A =Op(a + a

∗

− α



), we know from Theorems C.1

and C.3 i) that

∗

B −



A ∈ OPS

−1

Consequently, there exists c>0sothat



Au, u≥B

∗

Bu, u−cu

−1

u

≥Bu

−



u

−

c



u

−1

which implies that

Op(a + a

∗

)u, u≥

3α



u

−

c



u

−1

• Finally, we have the inequality in (C.1.5) with C =(4c

+3c

)/(2α). 

We complete this section by stating without proof the sharp G˚arding inequal-

ity, which amounts to allowing α = 0 in the standard one. In other words, it

shows that non-negative symbols imply a gain of derivatives: an operator of order

s with non-negative symbol satisﬁes a lower bound as though it were of order

s − 1. The sharp G˚arding inequality was originally proved by H¨ormander [86] for

scalar operators and by Lax and Nirenberg [112] for matrix-valued symbols. The

proof was later simpliﬁed by several authors; it can be found in [88, 205, 210].

Theorem C.5 (Sharp G˚arding inequality) If A is a pseudo-diﬀerential opera-

tor of symbol a ∈ S

,orA is associated with a ∈

by a low frequency cut-oﬀ,

such that for some positive α

a(x, ξ)+a(x, ξ)

∗

≥ 0

(in the sense of Hermitian matrices) for all x ∈ R

and ξ large, then there

exists C so that

Re Au , u ≥−C u

(m−1)/2

(C.1.6)

for all u ∈ H

m/2

C.2 Pseudo-diﬀerential calculus with a parameter

The introduction of a parameter γ is intended to deal with weighted-in-time

estimates, typically in L

(R, e

−γt

dt).

We shall consider symbols that depend uniformly on a parameter γ ∈ [1, +∞).

To avoid overcomplicated notations, we shall use, as far as possible, the same

456 Pseudo-/para-diﬀerential calculus

notations as in standard pseudo-diﬀerential calculus. The main diﬀerence is that

λ now stands for the rescaled weight

(ξ,γ):=(γ

+ ξ

)

s/2

Alternatively, we shall sometimes denote λ

s,γ

= λ

(·,γ). Associated with λ

s,γ

is an equivalent norm on H

,namelyλ

s,γ

u 

. To avoid confusion with the

standard norm, we shall denote

u 

= λ

s,γ

u 

One may also observe that

u 

≤u 

≤ γ

u 

for s>0 (and the converse for s<0). Another useful remark about these

weighted norms is that

u 

≤ γ

s−m

u 

for s ≤ m.

Remark C.1 For m ∈ N, there exist c

> 0andC

> 0sothat



|α|≤m

2(m−|α|)

∂

u

≤u

≤ C



|α|≤m

2(m−|α|)

∂

u

for all γ ≥ 1andu ∈ H

Deﬁnition C.3 For any real number m, we deﬁne the set S

of functions

a ∈ C

∞

× R

× [1, +∞[; C

N×N

) such that for all d-uples α and β there exists

α,β

> 0 so that

∂

∂

a(x, ξ, γ) ≤C

α,β

m−|β|

(ξ,γ) . (C.2.7)

(One may relax the smoothness with respect to γ assumption, which is of no use

actually.)

Basic examples. Of course the ﬁrst example is given by functions λ

them-

selves. Clearly, λ

belongs to S

. More generally, if (ξ,γ) → a(ξ,γ)isC

∞

on (R

× R

)\{(0, 0)} and homogeneous degree m, then its restriction to

× [1, +∞) belongs to S

. (Compared to the usual symbols, the singularity

at the origin is eliminated by taking γ ≥ 1.)

Needless to say, Lemma C.1 extends to this setting in a straightforward way.

Also, Proposition C.1 and Theorem C.1 enable us to deﬁne Op

(a) in such a way

that

(Op

(a)u)(x)=

(2π)



ix·ξ

a(x, ξ, γ) u(ξ)dξ (C.2.8)

Pseudo-diﬀerential calculus with a parameter 457

for all u ∈ S . And Theorem C.2 applies to the operator Op

for all γ ≥ 1.

However, one may wish to keep track of the dependence on γ in the estimates.

This leads to the following.

Deﬁnition C.4 A family of pseudo-diﬀerential operators {P

}

γ≥1

is said to

be of order m if P

belongs to OPS

for all γ ≥ 1 and for all s there exists a

constant C, independent of γ, so that

P

u

s−m

≤ C u

The basic example of a family of order m is precisely

{Λ

m,γ

}

γ≥1

where Λ

m,γ

is by deﬁnition the pseudo-diﬀerential operator of symbol λ

m,γ

.As

a matter of fact, the little calculation made in Section C.1.3 just becomes

Λ

m,γ

u

s−m

= Λ

s−m,γ

m,γ

u

= Λ

s,γ

u

= u

More generally, it can be shown that {Op

(a)}

γ≥1

is a family of order m for

any a ∈ S

. We actually have a collection of results of this kind – extending

Theorems C.1 and C.3 – that we summarize in the following.

Theorem C.6 If a and b belong to S

and S

, respectively, then

i) {Op

(a)}

γ≥1

is a family of order m,

ii) {(Op

(a))

∗

− (Op

∗

)) }

γ≥1

is a family of order m − 1,

iii) {Op

(a) ◦ Op

(b) − Op

(ab)}

γ≥1

is a family of order m + n − 1,

iv) {[Op

(a) , Op

(b)] − Op

([a, b]) }

γ≥1

is a family of order m + n − 1.

The proof of course relies on the fact that the estimates in (C.2.7) are

independent of γ. Let us sketch the proof of i). We need a bound for Op

(a)

in B(H

; H

s−m

) that is uniform in γ. So we go back to the usual estimation

of Op

(a), and pay attention to the dependence on γ.Thecasem = s =0 is

rather easy, because the norm of Op

(a) as an operator on L

only depends on

bounds on derivatives of a (even though this fact is not so easy to prove, it is

well-known), which are independent of γ by assumption. For arbitrary m and s,

the proof amounts to playing with commutators involving Λ

s,γ

and Λ

−m,γ

,thus

reducing the problem to the case m = s = 0. So it is closely related to the proof

of iii). The details are left to the reader.

Remark C.2 Since

u 

≤ γ

s−p

u 

for s ≤ p, a family {P

}

γ≥1

of order m satisﬁes

P

u

s−m

≤ Cγ

s−p

u

458 Pseudo-/para-diﬀerential calculus

for s ≤ p. In particular, for a family of negative order, we can take p = s − m

and obtain

P

u

≤ Cγ

u

We complete this section by parameter versions of G˚arding’s inequality.

Theorem C.7 (G˚arding inequality with parameter) If a ∈ S

is such that for

some positive α,

a(x, ξ, γ)+a(x, ξ, γ)

∗

≥ αλ

(ξ,γ) I

(in the sense of Hermitian matrices) for all (x, ξ, γ) ∈ R

× R

× [1, +∞), then

there exists γ

≥ 1 so that for all γ ≥ γ

and u ∈ H

m/2

Re Op

(a)u, u≥

u

m/2

. (C.2.9)

Compated to the standard G˚arding’s inequality in (C.1.5), the inequality (C.2.9)

does not contain any remainder term. This is made possible by absorbing the

errors in the main term for γ large enough.

Proof The proof parallels that of Theorem C.4.

• By Theorem C.6 ii) and the Cauchy–Schwarz inequality, there exists c>0

so that

(Op

(a)+Op

(a)

∗

)u, u≥Op

(a + a

∗

)u, u−c u

2m−1−m

u

• The symbol a := a + a

∗

− α



, with α



<α, is positive-deﬁnite.

By Lemma C.1, there exists b ∈ S

such that b

∗

b = a. Then, denoting



(a + a

∗

− α



), by Theorem C.6 iii) ii) and the Cauchy–Schwarz

inequality there exists c>0sothat



u, u≥Op

(b)

∗

(b)u, u−c u

m−1

u

which implies that

Op

(a + a

∗

)u, u≥α



u

− c u

m−1

u

• Therefore, by Young’s inequality,

(Op

(a)+Op

(a)

∗

)u, u≥



u

−



(c + c)

u

m−1

The conclusion just follows from the fact that



2m,γ

−



(c + c)

2(m−1),γ

≥



2m,γ

for γ large enough. 

A sharpened version of Theorem C.7 is the following.

Littlewood–Paley decomposition 459

Theorem C.8 (Sharp G˚arding inequality with parameter) If a ∈ S

satisﬁes

a(x, ξ, γ)+a(x, ξ, γ)

∗

≥ 0

(in the sense of Hermitian matrices) for all (x, ξ, γ) ∈ R

× R

× [1, +∞), then

there exist γ

≥ 1 and C>0 so that for all γ ≥ γ

and all u ∈ H

m/2

Re Op

(a)u, u≥−C u

(m−1)/2

. (C.2.10)

The proof is sketched in [125] (pp. 82–84, in the case m = 1), by adapting

the approach of Nagase [146].

C.3 Littlewood–Paley decomposition

C.3.1 Introduction

Littlewood–Paley decomposition is a well-known tool in modern analysis, of

which various versions are available [57,211,212]. Here we adopt the presentation

of Meyer [137], and also G´erard and Rauch [68]. For a recent, PDE oriented

presentation, see also [33].

We consider a reference cut-oﬀ function ψ ∈ D (R

), monotonically decaying

along rays and so that

ψ(ξ)=1 if ξ≤1/2 ,

0 ≤ ψ(ξ) ≤ 1if 1/2 ≤ξ≤1 ,

ψ(ξ)=0 if ξ≥1 .

(C.3.11)

This function is associated with the other cut-oﬀ function φ ∈ D(R

) deﬁned by

φ(ξ):=ψ(ξ/2) − ψ(ξ) .

The monotonicity property of ψ implies that φ(ξ) is everywhere non-negative.

The other main feature of φ is that it is supported by the set {1/2 ≤ξ≤2 }.

Now we consider the functions φ

deﬁned by

(ξ):=φ(2

−q

ξ )

for all q ∈ N. By construction we have

Supp φ

⊂{2

q−1

≤ξ≤2

q+1

Then it is elementary to show the following.

Proposition C.2 For the functions ψ and φ

deﬁned as above, we have

≡ 0 if |p − q|≥2 , (C.3.12)

ψ +



q≥0

≡ 1 , (C.3.13)

460 Pseudo-/para-diﬀerential calculus

≤ ψ



q≥0

≤ 1 . (C.3.14)

Observe that because of (C.3.12) the sums in (C.3.13) and (C.3.14) are locally

ﬁnite. Indeed for all ξ ∈ R

there are at most two indices q such that φ

(ξ) =0.

For convenience we also denote φ

−1

:= ψ.

All functions φ

for q ≥−1 can be viewed as constant-coeﬃcient symbols

in S

−∞

and thus associated with pseudo-diﬀerential operators, denoted by ∆

Equivalently, ∆

is deﬁned on S



∆

:= F

−1

F , (C.3.15)

where F denotes the Fourier transform on S



The interest of these operators is that, for all u ∈ S



,wehavebecauseof

(C.3.13)

u =



q≥−1

∆

where the convergence of the series holds true in S



, and the terms ∆

u are C

∞

functions (since the φ

are compactly supported).

We introduce the notation for partial sums

q−1



p=−1

∆

By convention, we put ∆

= 0 for p ≤−2andS

= 0 for q ≤−1.

By deﬁnition we have for all u ∈ S



F (∆

u)=φ

u and F (S

u)=ψ

u (C.3.16)

with the rescaled functions ψ

being deﬁned similarly as the φ

(ξ):=ψ(2

−q

ξ )

for q ≥ 0.

A ﬁrst interesting property of the operators ∆

is that the L

∞

norms of ∆

u and their derivatives are all controlled by the L

∞

norm of u. The cost of

one derivative is found to be 2

Proposition C.3 (Bernstein) For al l m ∈ N, there exists C

> 0 so that for

all u ∈ L

∞

, for all d-uple α, |α|≤m, for all q ≥−1,

∂

(∆

u) 

∞

≤ C

q|α|

u 

∞

and ∂

u) 

∞

≤ C

q|α|

u 

∞

(C.3.17)

Proof By (C.3.16) we have

∆

u = F

−1

∗ u and S

u = F

−1

∗ u.

Littlewood–Paley decomposition 461

All functions F

−1

are integrable because of the regularity of φ

, with the

additional invariance property

F

−1



= F

−1

φ 

for all q ≥ 0. We also easily compute that

∂

−1

) 

q|α|

∂

−1

φ) 

The same is of course true for ψ

. Then a basic convolution inequality yields the

conclusion with

C =max

|α|≤m

(∂

−1

ψ) 

, ∂

−1

φ) 

) .



C.3.2 Basic estimates concerning Sobolev spaces

All results displayed in this section but the very last are concerned with the most

classical Sobolev spaces H

on the whole space R

First, we note that if u belongs to H

the equality u =



∆

u holds true

not only in S



but also in H

. As a matter of fact, we have

F (S

u − u)

q→∞

−−−→ 0and|λ

(ξ)F (S

u − u)(ξ)|

≤ (1 + ψ

∞

) |λ

(ξ)u(ξ)|

Thus by Lebesgue’s theorem we have

lim

q→∞

S

u − u

=0.

Furthermore, the operators ∆

appear to give rise to equivalent norms on the

Sobolev spaces.

Proposition C.4 For al l s ∈ R, there exist C

> 1 such that for all u ∈ H



q≥−1

2qs

∆

u

≤u

≤ C



q≥−1

2qs

∆

u

. (C.3.18)

Proof We begin with the case s = 0. We claim that the estimate in (C.3.18)

works with C

= 2. One may remark that the equality, that is (C.3.18) with

= 1, could be true if the ∆

were pairwise orthogonal. But we only have, in

view of (C.3.12),

∆

u, ∆

u  = 0 provided that |p − q|≥2 . (C.3.19)

The inequalities in (C.3.18) can be viewed as measuring the default of orthogo-

nality. Their proof is almost straightforward. As a matter of fact, the inequalities

in (C.3.14) imply that



q≥−1

|φ

(ξ) u(ξ) |

≤|u(ξ) |

≤ 2



q≥−1

|φ

(ξ) u(ξ) |

462 Pseudo-/para-diﬀerential calculus

for all u ∈ L

and almost all ξ ∈ R

. Integrating in ξ we get, in view of the

deﬁnition (C.3.15) of ∆



q≥−1





∆

u 

≤u 

≤ 2



q≥−1





∆

u 

and we just conclude by Plancherel’s theorem.

The general case is not much more diﬃcult. The inequalities in (C.3.14) imply

that



q≥−1

|λ

(ξ) φ

(ξ) u(ξ) |

≤|λ

(ξ) u(ξ) |

≤ 2



q≥−1

|λ

(ξ) φ

(ξ) u(ξ) |

Assume, for instance, that s is positive. Then for q ≥ 0 and for

ξ ∈ Supp φ

⊂{2

q−1

≤ξ≤2

q+1

}

we have

−2s

2qs

≤ λ

(ξ)=(1+ξ

)

≤ 2

2qs

while for

ξ ∈ Supp φ

−1

⊂{ξ≤1 }

we have

−2s

=1≤ λ

(ξ) ≤ 2

−2s

Therefore, the inequalities in (C.3.14) holds true with C

3s+1

.Whens is

negative the estimates on λ

are reversed and thus (C.3.14) holds true with

−3s+1



In particular, this proposition shows that for all u ∈ H

and all q ≥−1

∆

u 

≤



−qs

u 

. (C.3.20)

Of course the constant

√

becomes 1 if we replace the usual H

norm by

the equivalent norm

u







q≥−1

2qs

∆

u





1/2

. (C.3.21)

As regards the operation of ∆

on L

, we actually have much more infor-

mation than that obtained by setting s = 0 in the inequality (C.3.20). We have

similar estimates as in Proposition C.3.

Littlewood–Paley decomposition 463

Proposition C.5 For al l m ∈ N, there exists C

> 0 so that for all u ∈ L

for all d-uple α, |α|≤m, for all q ≥−1,

∂

(∆

u) 

≤ C

q|α|

u 

and ∂

u) 

≤ C

q|α|

u 

(C.3.22)

In other words, we have a kind of ‘symmetric’ counterpart of (C.3.20). For

all positive integers s, there exists C>0 so that for all q ≥−1andu ∈ L

∆

u

≤ C 2

u

and S

u

≤ C 2

u

. (C.3.23)

(Note that the constant C here depends on s. It is strictly increasing with s.)

The proof of Proposition C.5 is exactly the same as that of Proposition C.3,

replacing the L

− L

∞

convolution estimates by L

− L

convolution estimates.

Another noteworthy remark is that the L

∞

norms of ∆

u and S

u can be

controlled even for unbounded u (to which Proposition C.3 does not apply),

provided that u belongs to some H

(which is not embedded in L

∞

for s ≤ d/2!),

as shown in the following.

Proposition C.6 For al l s ∈ R, there exists C>0 so that for all u ∈ H

)

and all q ≥−1,

∆

u 

∞

≤ C 2

−q(s−d/2)

u 

and S

u

≤ C 2

−q(s−d/2)

u 

(C.3.24)

Proof The proof is somewhat analogous to that of Proposition C.3. Since both



∆

u and



u are supported by the ball {ξ≤2

q+1

} we have, for instance,



∆

u = ψ

q+2



∆

and similarly for



u. Therefore,

∆

u = F

−1

q+2

∗ ∆

Now, to get the correct estimate we just have to pay attention to the fact that

the L

norm is not invariant by the rescaling. We have indeed

ψ



qd/2

ψ

and thus Plancherel’s theorem and a basic convolution inequality yield

∆

u 

∞

≤ 2

(q+2)d/2

ψ

∆

u 

Together with (C.3.20) this gives (C.3.24) for ∆

u with C =2

ψ

√

.The

same computation shows the inequality for S

u. 

A straightforward consequence of this proposition is, of course, the well-known

Sobolev embedding H

) → L

∞

for s>d/2. For, the inequality in (C.3.24)

shows the series



∆

u is normally convergent in L

∞

if u belongs to H

)

and s>d/2, and its sum must be u (by uniqueness of limits in the space of

distributions).

464 Pseudo-/para-diﬀerential calculus

Remark C.3 By a similar calculation as in the proof of Proposition C.6, we

have L

estimates of ∆

u for u ∈ L

. Namely, there exists C>0sothat

∆

u 

≤ C 2

qd/2

u 

for all u ∈ L

)andq ≥−1. Indeed, by deﬁnition of ∆

u, Plancherel’s theorem

shows that

∆

u 

= φ

u 

≤φ



u 

∞

for q ≥ 0(forq = −1 just replace φ

by ψ)and

φ



qd/2

φ

while, of course, u 

∞

≤u 

. As a consequence of these estimates and

Proposition C.4 we ﬁnd the embedding L

) → H

−s

)fors>

To complete this section, we prove an additional result in the same spirit as

Proposition C.6, which gives an estimate of ∆

u 

∞

in terms of ∆

u 

m,∞

(instead of ∆

u 

in the proof of Proposition C.6).

Proposition C.7 For al l m ∈ N, there exists C

> 0 so that for all u ∈ L

∞

and all q ≥ 0,

∆

u 

∞

≤ C

−qm



|α|=m

∂

(∆

u) 

∞

. (C.3.25)

Proof There is nothing to prove for m = 0. Let us assume m ≥ 1. We consider

some function χ ∈ D (R

) vanishing near 0 and being equal to 1 on the support

of φ (for instance take χ(ξ)=ψ(ξ/4) − ψ(2ξ)), so that φ = χφ. With obvious

notations we also have

= χ

for all q ≥ 0. Since χ vanishes near 0 we can deﬁne for all d-uples α of length m

a function χ

∈ D (R

)by

(ξ)=

(iξ)



|α|=m

(iξ

)

χ(ξ) .

By construction we have

χ(ξ)=



|α|=m

(iξ)

(ξ) ,

and

(ξ)=2

−qm



|α|=m

(iξ)

(ξ) ,