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Para-diﬀerential calculus 485

and we want to show that

∂

(a − R

(a))

∞

≤



m−1−|β|

(ξ) .

For convenience, we shall denote b = ∂

(a − R

(a)). There is nothing to do

for ξ≤1sinceλ

is bounded on the unit ball! It is more delicate to obtain a

bound of b(·,ξ)

∞

for ξ > 1. Littewood–Paley decomposition will be of help

again. Indeed, by deﬁnition of R

we have

F (b(·,ξ)) = ∂



(1 − χ(·,ξ))F (a(·,ξ))



which vanishes identically on B(0; ε

ξ)forξ≥1. Therefore, recalling that

∆

= F

−1

F with Supp φ

⊂ B(0; 2

q+1

), ∆

(b(·,ξ)) ≡ 0forq and ξ such

that

ξ > 2

q+1

and ξ≥1 .

Consequently, when ξ≥1 the Littlewood–Paley decomposition of b(·,ξ)reads

b(·,ξ)=



q ; ε

ξ≤2

q+1

∆

(b(·,ξ))

and this sum is locally ﬁnite. Furthermore, by Corollary C.1,

∆

(b(·,ξ))

∞

≤ 2

−q

b(·,ξ) 

1,∞

and for 1 ≤ξ≤2

q+1

we have

−q

≤

ξ

≤

(ξ)

This implies

b(·,ξ) 

∞

≤ C

(ξ)

b(·,ξ) 

1,∞

≤ 4 CC

m−|β|−1

(ξ) .



A straightforward consequence of Proposition C.16 is the following.

Corollary C.5 If χ

and χ

are two admissible cut-oﬀ functions, for all a ∈

, R

(a) − R

(a) belongs to Γ

m−1

Deﬁnition C.7 Let χ be an admissible frequency cut-oﬀ according to Deﬁnition

C.6. To any symbol a ∈ Γ

we associate the so-called para-diﬀerential operator,

said to be of order m,

:= Op(R

(a)) .

In particular, Corollary C.5 shows that for a ∈ Γ

, T

are unique modulo

operators of order m − 1.

Another interesting point is the following.

486 Pseudo-/para-diﬀerential calculus

Remark C.8 If χ is constructed through Littlewood–Paley decomposition as

explained above, and k ≥ 1, for any function of x only, a ∈ W

k,∞

viewed as a

symbol in Γ

, the operators T

coincide with the para-product operator T

to an inﬁnitely smoothing operator.Indeed,if

χ(η, ξ)=



p ≥ 0

ψ(2

2−p

η) φ(2

−p

ξ)=



p ≥ 0

p−2

(η) φ

(ξ) ,

then

F (R

(a)(·,ξ)) = (



|p|≤1

p−2

(·) φ

(ξ)) F (a)+



p≥2

F (S

p−2

(a)) φ

(ξ) ,

or equivalently

(a)(x, ξ)=



|p|≤1

−1

( ψ

p−2

F (a))(x) φ

(ξ)+



p≥2

p−2

(a)(x) φ

(ξ) .

In terms of operators, this means that for all u ∈ F

−1



u =Op(b) u + T

where the last term is the usual para-product, while

b(x, ξ)=



|p|≤1

−1

( ψ

p−2

F (a))(x) φ

(ξ)

satisﬁes (C.4.46) with ε =1/2 and is compactly supported in ξ.

C.4.2 Basic results

We omit below the superscript χ, all results being valid for any admissible

frequency cut-oﬀ χ.

Theorem C.16 For al l a ∈ Γ

, the adjoint operator (T

)

∗

is of order m and

)

∗

− T

∗

is of order m − 1.

Theorem C.17 For al l a ∈ Γ

and b ∈ Γ

, the product ab belongs to Γ

m+n

and

◦ T

− T

is a para-diﬀerential operator of order m + n − 1, associated with

asymbolinΓ

m+n−1

. In particular, if the symbols a and b commute – for example,

if at least one of the operators is scalar-valued – the commutator [ T

] is of

order m + n − 1.

Proposition C.17 If a ∈ Γ

, there exists C>0 such that for all u ∈ H

|Re T

u, u| ≤ C u

Proof We have

|Re T

u, u| = |Re λ

−m



u, λ

u|≤(Λ

−m

◦ T

)(u) 

Λ

u 

Para-diﬀerential calculus with a parameter 487

Since Λ

u 

= u

, the ﬁnal estimate follows from the fact that Λ

−m

◦ T

is an operator of order −m +2m = m, which is a consequence of Theorem C.17

and the fact that T

− Λ

is inﬁnitely smoothing. 

Theorem C.18 (G˚arding inequality) If a ∈ Γ

is such that for some positive

α,

a(x, ξ)+a(x, ξ)

∗

≥ αλ

(ξ) I

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

× R

, then there exists

C>0 so that for all u ∈ H

Re T

u, u≥

u

− C u

m−1/2

. (C.4.49)

One may also state a sharpened version of G˚arding’s inequality in this context,

but for smoother symbols (at least C

in x).

C.5 Para-diﬀerential calculus with a parameter

The ﬁnal reﬁnement in this overview of modern analysis tools concerns families of

para-diﬀerential operators depending on one parameter, as extensions of pseudo-

diﬀerential operators with parameter.

As in Section C.2, we denote

s,γ

(ξ)=(γ

+ ξ

)

s/2

and deﬁne parameter-dependent symbols of limited regularity as follows.

Deﬁnition C.8 For any real number m and any natural integer k, the set

consists of functions, a : R

× R

× [1, +∞) → C

N×N

that are C

∞

in ξ

and such that for all d-uple β, there exists C

> 0 so that for all (ξ,γ) ∈ R

[1, +∞),

∂

a(·,ξ,γ) 

k,∞

≤ C

m−|β|,γ

(ξ) . (C.5.50)

The subset Σ

is made of symbols a ∈ Γ

satisfying the spectral requirement

Supp ( F (a(·,ξ,γ)) ) ⊂ B(0; ελ

1,γ

(ξ)) (C.5.51)

for some ε ∈ (0, 1) independent of (ξ,γ).

The analogue of Theorem C.15 is the following fundamental result.

Theorem C.19 Any symbol a ∈ Σ

can be associated with a family of operators

denoted by {Op

(a)}

γ ≥ 1

, deﬁned on temperate distributions with a compact

spectrum by

(a): F

−1



) −→ C

∞

u → Op

(a) u; (Op

(a) u)(x)=

(2π)

e

ix·

a(x, ·,γ), u

∞



)

This deﬁnition of Op

(a) coincides with (C.2.8) if a belongs to S

. Furthermore,

for all s ∈ R and γ ≥ 1, Op

(a) extends in a unique way into a bounded operator

488 Pseudo-/para-diﬀerential calculus

from H

to H

s−m

, and there exists C

> 0 independent of γ and u so that

Op

(a) u 

s−m

≤ C

u 

The proof, which we omit here, makes use of a parameter version of

Littlewood–Paley decomposition, based on cut-oﬀ functions in the (ξ,γ)-space.

Namely, taking ψ ∈ D (R

× R) with ψ(ξ, γ) = Ψ((γ

+ ξ

)

1/2

) and Ψ monot-

ically decaying such that

Ψ(r)=1if r ≤ 1/2 , Ψ(r)=0if r ≥ 1 ,

and denoting

(ξ)=ψ(2

−q

ξ,2

−q

γ) ,φ(ξ, γ):=ψ(ξ/2,γ/2) − ψ(ξ,γ),

(ξ)=φ(2

−q

ξ,2

−q

γ) ,

we may deﬁne operators S

and ∆

of symbols, respectively, ψ

and φ

Observing that ∆

=0 for γ ≥ 2

q+1

, and in particular ∆

−1

=0 for γ ≥ 1, we

easily check that



p≥0

∆

=id

in S



. Furthermore, the analogue of Proposition C.4 for the standard H

norm

is the following for the H

norm.

Proposition C.18 For al l s ∈ R, u ∈ H

(R) if and only if



p≥0

2ps

∆

u

< ∞

for all γ ≥ 1. In addition, there exists C

> 1 so that



p≥0

2ps

∆

u

≤u

≤ C



p≥0

2ps

∆

u

for all γ ≥ 1.

Knowing Theorem C.19, it is then possible to deﬁne a family of operators

associated with all symbols a in Γ

. The procedure is the same as in standard

(that is, without parameter) para-diﬀerential calculus. The basic tool is a so-

called admissible cut-oﬀ function.

Deﬁnition C.9 A C

∞

function χ :(η, ξ, γ) ∈ R

× R

× [1, +∞) →

χ(η, ξ, γ) ∈ R

is termed an admissible frequency cut-oﬀ if there exist ε

1,2

with 0 <ε

<ε

< 1 so that



χ(η, ξ, γ)=1, if η≤ε

(ξ,γ) ,

χ(η, ξ, γ)=0, if η≥ε

(ξ,γ) ,

Para-diﬀerential calculus with a parameter 489

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

α,β

> 0 so that

|∂

∂

χ(η, ξ, γ) |≤C

α,β

−|α|−|β|

(ξ,γ) . (C.5.52)

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

parameter described above, the function χ deﬁned by

χ(η, ξ, γ)=



p ≥ 0

ψ(2

2−p

η, 0) φ(2

−p

ξ,2

−p

γ)

is an admissible frequency cut-oﬀ with ε

=1/16 and ε

=1/2. The veriﬁcation

is left to the reader.

We now continue the machinery of Section C.4.

Proposition C.19 Let χ be an admissible frequency cut-oﬀ according to Deﬁ-

nition C.9 and consider the operator

: a ∈ Γ

→ σ ∈ C

∞

; σ(·,ξ,γ)=K

(·,ξ,γ) ∗

a(·,ξ,γ) ,

where the kernel K

is deﬁned by

(·,ξ,γ)=F

−1

(χ(·,ξ,γ)) .

Then R

maps into

= {a ∈ Γ

; Supp ( F (a(·,ξ,γ)) ) ⊂ B(0; ε

1,γ

(ξ)) }.

Furthermore, if k ≥ 1, for all a ∈ Γ

, a − R

(a) belongs to Γ

m−1

k−1

Note: Since χ(0,ξ,γ) ≡ 1, R

(a)=a for all symbols a depending only on ξ.

Deﬁnition C.10 If χ is an admissible frequency cut-oﬀ, to any symbol a ∈ Γ

we associate the family of para-diﬀerential operators {T

χ,γ

}

γ≥1

deﬁned by

χ,γ

:= Op

(a)) .

Remark C.9 If the symbol a is a function of x only, a ∈ W

k,∞

,itcanbe

viewed as a symbol in Γ

and T

χ,γ

u is a parameter version of the para-product

of a and u. More precisely, if the cut-oﬀ function χ is based on the Littlewood–

Paley decomposition with parameter in the way explained above, we have

χ,γ

u =



p≥0

p−2

a ∆

where S

:= F

−1

F with ψ

(ξ):=ψ(2

−p

ξ,0) = Ψ(2

−p

ξ) (as in the

standard Littlewood–Paley decomposition

For simplicity, we shall now omit the dependence on χ and just denote T

Remark C.10 For a symbol of the form

a(x, ξ)=p(ξ) b(x, γ) ,

Except for the deﬁnitions of S

−2

, S

−1

, which were taken to be zero in Section C.3.3

490 Pseudo-/para-diﬀerential calculus

the regularized symbol is given by R

(a)(x, ξ, γ)=p(ξ) R

(b)(x, ξ, γ). Applying

this in particular to polynomials p, we see that for any d-uple α,

∂

u = T

|α|

It is important for the applications to be able to estimate the error done when

replacing products by para-products. Such estimates are given in Theorem C.13

(and its corollary) for standard para-products. We have similar results for T

which show that a − T

is of order −1assoonasa is Lipschitz.

Theorem C.20 There exists C>0 so that, for all a ∈ W

1,∞

and u ∈ L

for all γ ≥ 1,

γ au − T

u 

≤ C a

1,∞

u

a∂

u − T

∂

u 

= a∂

u − T

iξ

u 

≤ C a

1,∞

u

au − T

u 

≤ C a

1,∞

u

Proof The ﬁrst inequality is easy to show. The factor γ comes from the

fact that ∆

u = 0 for γ ≥ 2

q+1

. Indeed, the fact that a is Lipschitz implies,

by Corollary C.1 for the standard Littlewood–Paley decomposition, that

∆

a

∞

 2

−q

a

1,∞

and therefore that the series



∆

a is normally convergent in L

∞

.Takeu ∈ S .

Then the series



∆

u is normally convergent in L

and u =



∆

u. Therefore,

au − T

u =



q≥−1



p≥0

∆

a ∆

u −



p≥0

p−2

a ∆

u =



q+3

≥γ

∆

q+3

(Recall that S

= 0 for γ ≥ 2

.) Hence

au − T

u



q+3

≥γ

−q

a

1,∞

u



a

1,∞

u

Here, we have used the fact that

S

u   u

which comes from the deﬁnition of S

,aL

− L

convolution estimate and a

uniform bound for F

−1

(ψ

)

. The derivation of the latter bound comes from

the observation that

sup

1≤ γ≤2

F

−1

(ψ

)

)

≤F

−1

(γ,ξ)

(ψ

)

(R×R

)

= F

−1

(γ,ξ)

(ψ)

(R×R

)

We omit the proof of the second inequality, which is much trickier (see [38,

136]).

Para-diﬀerential calculus with a parameter 491

The third inequality is an easy consequence of the ﬁrst two. Indeed, by

deﬁnition,

f

≤ γ

f

+ ∇f

for all f ∈ H

,and

∂

( au − T

u )

≤ 3 a∂

u − T

∂

u

+3(∂

a) u

+3T

∂

u

≤ 3 C

a

1,∞

u

+3(1+C

) ∂

a

∞

u

where C

comes from the basic estimate

T

u

≤ C

b

∞

u

Therefore,

au − T

u 

≤ γ

au − T

u 



∂

( au − T

u )

≤ ((1+3d) C

+3d(1 + C

)) a

1,∞

u



Other basic results, similar to those in pseudo-diﬀerential calculus with

parameter, are the following.

Theorem C.21 For al l a ∈ Γ

, the family of adjoint operators {(T

)

∗

}

γ≥1

of order m and the family {(T

)

∗

− T

∗

}

γ≥1

is of order (less than or equal to)

m − 1.

Theorem C.22 For al l a ∈ Γ

and b ∈ Γ

, the product ab belongs to Γ

m+n

and the family {T

◦ T

− T

}

γ≥1

is of order (less than or equal to) m + n − 1.

Theorem C.23 (G˚arding inequality) If a ∈ Γ

is such that for some positive

α,

a(x, ξ, γ)+a(x, ξ, γ)

∗

≥ αλ

2m,γ

(ξ) 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 all u ∈ H

Re T

u, u≥

u

. (C.5.53)

Again, we omit the sharp form of G˚arding’s inequality, which is valid only

for smoother symbols.

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