Taylor M.E. Partial Differential Equations III: Nonlinear Equations

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58 13. Function Space and Operator Theory for Nonlinear Analysis

That p

.x; / satisﬁes an estimate of the form (9.2), with m replaced by mrı,

follows from (9.32), with t D 0. That it satisﬁes (9.1), with m replaced by m rı,

is a consequence of the estimate (9.33).

It will also be useful to smooth out a symbol p.x; / 2 C



1;ı

,forı 2 .0; 1/.

Pick  2 .ı; 1/, and apply (9.28), with "

D 2

j.ı/

, obtaining p

.x; / and

hence a decomposition of the form (9.27). In this case, we obtain

(9.37) p.x; / 2 C



1;ı

H) p

.x; / 2 S

1;

.x; / 2 C



m.ı/r

1;

We use the symbol decomposition (9.27) to establish the following variant of

Theorem 9.1, which will be most useful in Chap. 14.

Proposition 9.10. If ı 2 Œ0; 1/ and p.x; / 2 C



1;ı

,then

p.x; D/ W H

sCm;p

! H

s;p

;

p.x; D/ W C

sCm



! C



;

(9.38)

provided p 2 .1; 1/ and

(9.39) .1  ı/r < s < r:

Proof. The result follows directly from Theorem 9.1 if 0<s<r, so it remains

to consider s 2



.1 ı/r; 0



. Use the decomposition (9.27), p D p

, with

(9.37) holding. Thus p

.x; D/ has the mapping property (9.38)foralls 2 R.

Applying Theorem 9.1 to p

.x; D/ yields mapping properties such as

.x; D/ W H

Cm.ı/r;p

! H

;p

;>0;

or, setting s D   .  ı/r,

.x; D/ W H

sCm;p

! H

sC.ı/r;p

 H

s;p

;s>.  ı/r;

and similar results on C

sCm



. Then letting  % 1 completes the proof of (9.38).

Recall that, for r 2 .0; 1/,wehavedeﬁnedp.x;/ to belong to the space



1;ı

/ provided the estimates (9.1)and(9.2) hold. If r 2 Œ0; 1/, we will

say that p.x; / 2 C

1;ı

/ provided that (9.1)–(9.2) hold and, additionally,

(9.40) kD



p.;/k

 C

hi

mj˛jCjı

;0 j  r; j 2 Z:

In particular, we make a semantic distinction between C



1;ı

and C

1;ı

even

when r … Z

, in which cases C



and C

coincide. The differences between the

two symbol classes are minor, especially when r … Z, but natural examples of

symbols often do have this additional property, and we sometimes use the symbol

classes just deﬁned to record this fact.

Exercises 59

Exercises

1. Young’s inequality implies

kf  gk

q kf k

kgk

q ;

where f D .f

/; g D .g

/,and.f  g/

j k

. Show how this applies

(with q D 2) to the estimate of (9.17).

2. Supplement Lemma 9.8 with the estimates

f k

 C kf k

; jˇjs;

.jˇjs/

kf k



; jˇj >s;(9.41)

given s>0.

3. Show that if p.x; / 2 C



1;0

has the decomposition (9.27), then

.x; / 2 S

1;ı

; for jˇj <r;

mCı.jˇjr/

1;ı

; for jˇj >r:(9.42)

4. Strengthen part of Proposition 9.10 to obtain, for ı 2 Œ0; 1/; r > 0,

(9.43) p.x;/ 2 C



1;ı

H) p.x; D/ W C

sCm



! C



; for  .1  ı/r < s  r:

(Hint: Apply Proposition 9.7 to p

.x; D/,arisingin(9.37).)

5. Given s 2 R;1 p; q 1,wesayf 2 S

/ belongs to the Triebel space

p;q

/ provided

(9.44) kf k

p;q



.D/f g



< 1;

where f

g is the partition of unity (9.9). Note that F

p;2

D H

s;p

if 1<p<1,

by Lemma 9.4. Also, we say that f 2 S

/ belongs to the Besov space B

p;q

provided

(9.45) kf k

p;q



.D/f g



< 1:

Note that B

1;1

D C



.Also,B

2;2

D H

,since`

// D L

Extend Theorem 9.1 to results of the form

p.x; D/ W F

sCm

p;q

! F

p;q

;p.x;D/W B

sCm

p;q

! B

p;q

(See [Ma1].)

6. We deﬁne the symbol class C



to consist of p.x; / 2 C



1;0

such that

(9.46) p.x;/ 

j 0

.x; /

where p

.x; / 2 C



1;0

is homogeneous of degree m  j in ,forjj1,and

(9.46) means that the difference between the left side and the sum over 0  j<N

belongs to C



mN

1;0

.Ifr 2 R

n Z

, we also denote the symbol class by C

Show that estimates of the form (9.3)and(9.4) have simpler proofs in this case, derived

60 13. Function Space and Operator Theory for Nonlinear Analysis

from expansions of the form

(9.47) p

.x; / D



j

.x/jj

mj



jj

1





;

for jj1,wheref!



g is an orthonormal basis of L

n1

/ consisting of eigenfunc-

tions of the Laplace operator.

10. Paradifferential operators

Here we develop the paradifferential operator calculus, introduced by J.-M. Bony

in [Bon]. We begin with Y. Meyer’s ingenious formula for F.u/ as M.x;D/uCR

where F is smooth in its argument(s), u belongs to a H¨older or Sobolev space,

M.x;D/ is a pseudodifferential operator of type .1; 1/,andR is smooth. From

there, one applies symbol smoothing to M.x;/ and makes use of results estab-

lished in  9.

Following [Mey], we discuss the connection between F.u/, for smooth non-

linear F , and the action on u of certain pseudodifferential operators of type .1; 1/.

Let

./ D '

./

be the Littlewood–Paley partition of unity (5.37), and set

‰

./ D

j k

./.Givenu (e.g., in C

/), set

(10.1) u

D ‰

.D/u;

and write

(10.2) F.u/ D F.u

/ C ŒF .u

/  F.u

/ CCŒF .u

kC1

/  F.u

/ C:

Then write

F.u

kC1

/  F.u

/ D F



kC1

.D/u



 F.u

D m

.x/

kC1

.D/u;(10.3)

where

(10.4) m

.x/ D



‰

.D/u Ct

kC1

.D/u



dt:

Consequently, we have

F.u/ D F.u

/ C

kD0

.x/

kC1

.D/u

D M.x;D/u CF.u

/;(10.5)

10. Paradifferential operators 61

where

(10.6) M.x;/ D

kD0

.x/

kC1

./ D M

.uIx; /:

We claim

(10.7) M.x;/ 2 S

1;1

;

provided u is continuous. To estimate M.x;/, note ﬁrst that by (10.4)

(10.8) km

 sup jF

./j:

To estimate higher derivatives, we use the elementary estimate

(10.9) kD

g.h/k

 C

CC`



`

1

kD



to obtain

(10.10) kD

 C

`1

hkuk

`1

 2

;

granted the following estimates, which hold for all u 2 L

(10.11) k‰

.D/u Ct

kC1

.D/uk

 C kuk

and

(10.12) kD

Œ‰

.D/u Ct

kC1

.D/uk

 C

kuk

for t 2 Œ0; 1. Consequently, (10.6) yields

(10.13) jD



M.x;/jC

sup



./jhi

j˛j

and, for jˇj1,

(10.14) jD



M.x;/jC

˛ˇ

jˇj1

hkuk

jˇj1

hi

jˇjj˛j

We give a formal statement of the result just established.

Proposition 10.1. If F is C

and u 2 C

with r  0,then

(10.15) F.u/ D M

.uIx; D/u CR.u/;

62 13. Function Space and Operator Theory for Nonlinear Analysis

where

R.u/ D F.

.D/u/ 2 C

and

(10.16) M

.uIx; / D M.x;/ 2 S

1;1

Following [Bon]and[Mey], we call M

.uIx; D/ a paradifferential operator.

Applying Theorem 9.1,wehave

(10.17) kM.x;D/f k

s;p

 Kkf k

s;p

;

for p 2 .1; 1/; s > 0, with

(10.18) K D K

.F; u/ D C kF

Œ1 Ckuk

;

provided 0<s<N, and similarly

(10.19) kM.x;D/f k



 Kkf k



Using f D u, we have the following important Moser-type estimates, extending

Proposition 3.9:

Proposition 10.2. If F is smooth with kF

.R/

< 1, and 0<s<N,then

(10.20) kF.u/k

s;p

 K

.F; u/kuk

s;p

CkR.u/k

s;p

and

(10.21) kF.u/k



 K

.F; u/kuk



CkR.u/k



;

given 1<p<1, with K

.F; u/ as in (10.18).

This expression for K

.F; u/ involves the L

-norm of u, and one can use

.I /

,whereI contains the range of u. Note that if F.u/ D u

,then

.u/ D 2u, and higher powers of kuk

do not arise; hence we obtain the

estimate

(10.22) ku

s;p

 C

kuk

kuk

s;p

;s>0;

and a similar estimate on ku



It will be useful to have further estimates on the symbol M.x;/ D

.uIx; / when u 2 C

with r>0. The estimate (10.12) extends to

(10.23)





‰

.D/f C t

kC1

.D/f





 C

kf k

;` r;

k.`r/

kf k

;`>r;

10. Paradifferential operators 63

so we have, when u 2 C

(10.24)



M.x;/j K

˛ˇ

hi

j˛j

; jˇjr;

˛ˇ

hi

j˛jCjˇjr

; jˇj >r;

with

(10.25) K

˛ˇ

D K

˛ˇ

.F; u/ D C

˛ˇ

jˇj

Œ1 Ckuk

jˇj

:

Also, since ‰

.D/ C t

kC1

.D/ is uniformly bounded on C

,fort 2 Œ0; 1 and

k  0,wehave

(10.26) kD



M.;/k

 K

˛r

hi

j˛j

;

where K

˛r

is as in (10.25), with jˇjDŒr C 1. This last estimate shows that

(10.27) u 2 C

H) M

.uIx; / 2 C

1;0

This is useful additional information; for example, (10.17)and(10.19) hold for

s>r, and of course we can apply the symbol smoothing of  9.

It will be useful to have terminology expressing the structure of the symbols

we produce. Given r  0,wesay

(10.28)

p.x; / 2 A

1;ı

”kD



p.;/k

 C

hi

mj˛j

and



p.x; /jC

˛ˇ

hi

mj˛jCı.jˇjr/

; jˇj >r:

Thus (10.24)–(10.26) yield

(10.29) M.x;/ 2 A

1;1

for the M.x;/ of Proposition 10.1.Ifr 2 R

n Z

, the class A

1;1

coincides

with the symbol class denoted by A

by Meyer [Mey]. Clearly, A

1;ı

D S

1;ı

and

1;ı

 C

1;0

\ S

1;ı

Also, from the deﬁnition we see that

(10.30)

p.x; / 2 A

1;ı

H) D

p.x; / 2 S

1;ı

; for jˇjr;

mCı.jˇjr/

1;ı

; for jˇjr:

It is also natural to consider a slightly smaller symbol class:

(10.31) p.x; / 2 A

1;ı

”kD



p.;/k

rCs

 C

˛s

hi

mj˛jCıs

;s 0:

64 13. Function Space and Operator Theory for Nonlinear Analysis

Considering the cases s D 0 and s Djˇjr, we see that

1;ı

 A

1;ı

We also say

(10.32) p.x;/ 2

1;ı

” the right side of (10.30) holds;

1;ı



1;ı

The following result reﬁnes (10.29).

Proposition 10.3. For the symbol M.x;/ D M

.uIx; / of Proposition 10.1,

we have

(10.33) M.x;/ 2 A

1;1

;

provided u 2 C

;r 0.

Proof. For this, we need

(10.34) km

rCs

 C  2

Now, extending (10.9), we have

(10.35) kg.h/k

rCs

 C kgk

Œ1 Ckhk

.khk

rCs

C 1/;

with N D Œr Cs C 1, as a consequence of (10.21)whenr C s is not an integer,

and by (10.9) when it is. This gives, via (10.4),

(10.36) km

rCs

 C.kuk

/ sup

t2I

k.‰

C t

kC1

/uk

rCs

;

where I D Œ0; 1.However,

(10.37) k.‰

C t

kC1

/uk

rCs

 C  2

kuk

For r C s 2 Z

, this follows from (9.41); for r C s … Z

, it follows as in the

proof of Lemma 9.8,since

(10.38) 2

ks

.‰

C t

kC1

/ is bounded in OP S

1;0

This establishes (10.34), and hence (10.33) is proved.

10. Paradifferential operators 65

Returning to symbol smoothing, if we use the method of  9 to write

(10.39) M.x;/ D M

.x; / C M

.x; /;

then (10.27) implies

(10.40) M

.x; / 2 S

1;ı

.x; / 2 C

mrı

1;ı

We now reﬁne these results; for M

we have a general result:

Proposition 10.4. For the symbol decomposition deﬁned by the formulas (9.27)–

(9.30),

(10.41) p.x; / 2 C

1;0

H) p

.x; / 2 A

1;ı

Proof. This is a simple modiﬁcation of (9.42) which essentially says that

.x; / 2 A

1;ı

; we simply supplement (9.41) with

(10.42) kJ

f k

rCs



 C"

s

kf k



;s 0;

which is basically the same as (10.37).

To treat M

.x; /,wehave,forı  ,

(10.43) p.x; / 2 A

1;

H) p

.x; / 2 C

mır

1;ı

\ A

1;

 S

mır

1;

;

where containment in C

mır

1;ı

follows from (9.35). To see the last inclusion,

note that for p

.x; / to belong to the intersection above implies

(10.44)



.;/k

 C hi

mj˛jırCıs

; for 0  s  r;

C hi

mj˛jC.sr/

; for s  r:

In particular, these estimates imply p

.x; / 2 S

mrı

1;

. This proves the following:

Proposition 10.5. For the symbol M.x;/ D M

.uIx; / with decomposition

(10.39),

(10.45) u 2 C

H) M

.x; / 2 S

rı

1;1

Results discussed above extend easily to the case of a function F of several

variables, say u D .u

;:::;u

/. Directly extending (10.2)–(10.6), we have

(10.46) F.u/ D

j D1

.x; D/u

C F.‰

.D/u/;

66 13. Function Space and Operator Theory for Nonlinear Analysis

with

(10.47) M

.x; / D

.x/

kC1

./;

where

(10.48) m

.x/ D



‰

.D/u Ct

kC1

.D/u



dt:

Clearly, the results established above apply to the M

.x; / here; for example,

(10.49) u 2 C

H) M

.x; / 2 A

1;1

In the particular case F.u;v/D uv, we obtain

(10.50) uv D A.uIx; D/v C A.vIx; D/u C ‰

.D/u ‰

.D/v;

where

(10.51) A.uIx; / D

kD1

‰

.D/u C

kC1

.D/u

kC1

./:

Since this symbol belongs to S

1;1

for u 2 L

, we obtain the following extension

of (10.22), which generalizes the Moser estimate (3.21):

Corollary 10.6. For s>0;1<p<1, we have

(10.52) kuvk

s;p

 C



kuk

kvk

s;p

Ckuk

s;p

kvk



We now analyze a nonlinear differential operator in terms of a paradifferential

operator. If F is smooth in its arguments, in analogy with (10.46)–(10.48)wehave

(10.53) F.x;D

u/ D

j˛jm

.x; D/D

u C F.x;D

‰

.D/u/;

where F.x;D

‰

.D/u/ 2 C

and

(10.54) M

.x; / D

.x/

kC1

./;

with

(10.55) m

.x/ D

.@F=@



‰

.D/D

u C t

kC1

.D/D



dt:

10. Paradifferential operators 67

As in Propositions 10.1 and 10.3,wehave,forr  0,

(10.56) u 2 C

mCr

H) M

.x; / 2 A

1;1

 S

1;1

\ C

1;0

In other words, if we set

(10.57) M.uIx; D/ D

j˛jm

.x; D/D

;

we obtain

Proposition 10.7. If u 2 C

mCr

;r 0,then

(10.58) F.x;D

u/ D M.uIx; D/u C R;

with R 2 C

and

(10.59) M.uIx; / 2 A

1;1

 S

1;1

\ C

1;0

As in Propositions 10.4 and 10.5, in this case symbol smoothing yields

(10.60) M.uIx; / D M

.x; / C M

.x; /;

with

(10.61) M

.x; / 2 A

1;ı

.x; / 2 S

mrı

1;1

A speciﬁc choice for symbol smoothing which leads to paradifferential opera-

tors of [Bon]and[Mey] is the following operation on M.x;/:

(10.62) M

.x; / D

‰

k5

M.x;/

./;

where, as in (9.28), ‰

k5

acts on M.x;/ as a function of x.Weuse‰

k5

‰

k5

.D/, with ‰

./ D

j `

./.Wehave

(10.63) M.x;/ 2 L

1;0

H) M

.x; / 2 B



1;1

;

with  D 1=16, where we deﬁne B



1;1

for <1to be

(10.64) B



1;1

Dfb.x; / 2 S

1;1

b.;/ supported in jjjjg;

and where

b.;/ D

b.x; /e

ix

dx.SetBS

1;1

<1



1;1