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

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48 13. Function Space and Operator Theory for Nonlinear Analysis

providing a right inverse for the surjective restriction operator

(8.39)  W C

.M / ! C

./:

From Proposition 8.6, we can deduce that whenever r>0and s

and  are

as above, C



.M / D



.M /; C

.M /





. Thus, by interpolation, we have, for

r>0,

(8.40) E W C



./ ! C



.M /;  W C



.M / ! C



./;

and E D I on C



./. Hence

(8.41) C



./  C



.M /

fu 2 C



.M / W u



D 0g:

This characterization is manifestly independent of the choices made in (8.37).

Note that the right side of (8.41) is meaningful even for r  0.

By Propositions 8.1 and 8.2, we know that C



.M / D C

.M /,forr 2 R

,so

(8.42) C



./ D C

./; for r 2 R

n Z

Using the spaces C



./, we can ﬁll in the gaps (at r 2 Z

) in the estimates of

Theorem 8.9.

Proposition 8.10. If .P; B

;1  j  `/ is a regular elliptic boundary problem

as in Theorem 8.9 and u solves (8.34), then, for all r 2 .0; 1/,

(8.43) g

2 C

rm



.@/ H) u 2 C



./:

Proof. For r 2 R

, this is equivalent to (8.35). Since the solution u is given,

mod C

./, by the operator (8.36), the rest follows by interpolation.

In a sense, the C



-norm is only a tad weaker than the C

-norm. The following

is a quantitative version of this statement, which will prove very useful for the

study of nonlinear evolution equations, particularly in Chap.17.

Proposition 8.11. If s>n=2C ı, then there is C<1 such that, for all " 2

.0; 1,

(8.44) kuk

 C"

kuk

C C



log



kuk



Proof. By (8.6), kuk



D sup k

.D/uk

. Now, with ‰

`j

,make

the decomposition u D ‰

.D/u C



1  ‰

.D/



u;let" D 2

j

. Clearly,

(8.45) k‰

.D/uk

 j kuk



8. H¨older spaces and Zygmund spaces 49

Meanwhile, using the Sobolev imbedding theorem, since n=2 < s  ı,

k.1  ‰

.D//uk

 C k.1  ‰

.D//uk

sı

 C2

jı

k.1  ‰

.D//uk

;(8.46)

the last identity holding since f2

jı

hi

ı



1  ‰

./



W j 2 Z

g is uniformly

bounded. This proves (8.44).

Suppose the norms satisfy kuk



 C kuk

. If we substitute "

1

kuk



=kuk

into (8.44), we obtain the estimate (for a new C D C.ı/)

(8.47) kuk

 C kuk



1 C log

kuk



We note that a number of variants of (8.44)and(8.47) hold. For some of them,

it is useful to strengthen the last observation in the proof above to

(8.48)

jı

hi

ı



1  ‰

./



W j 2 Z



is bounded in S

An argument parallel to the proof of Proposition 8.11 gives estimates

(8.49) kuk

.M /

 C"

kuk

.M /

C C



log



kuk



.M /

;

given k 2 Z

;s>n=2C k C ı, and consequently

(8.50) kuk

.M /

 C kuk



.M /

1 C log

kuk



when M is a compact manifold without boundary.

We can also establish such an estimate for the C

./-norm when  is a com-

pact manifold with boundary. If

  M as above, this follows easily from (8.50),

via:

Lemma 8.12. For any r 2 .0; N /,

(8.51) kuk



./

kEuk



.M /

Proof. If Eu

! v in C



.M /,thenE u

! v in C



./,thatis,u

! v in



./,sinceEu

D u

. Thus v D Ev, in this case. This proves the lemma,

which is also equivalent to the statement that E in (8.40) has closed range.

We also have such a result for Sobolev spaces:

(8.52) kuk

r;p

./

kEuk

r;p

.M /

;1<p<1:

50 13. Function Space and Operator Theory for Nonlinear Analysis

Thus (8.50) yields

(8.53) kuk

./

 C kuk



./

1 C log

kuk

./

kuk



./

;

provided s>n=2C k.

Exercises

1. Extend the estimates of Theorem 8.9 and Proposition 8.10 to solutions of

(8.54) P u D f on ; B

u D g

on @:

Show that, for r 2 .; 1/;  D max.m

(8.55) f 2 C

rm



./; g

2 C

rm



.@/ H) u 2 C



./:

Note that we allow r  m<0, in which case C

rm



./ is deﬁned by the right side of

(8.41) (with r replaced by r m).

2. Establish the following result, similar to (8.44):

(8.56) kuk

 C"

kuk

s;p

C C



log



11=q

kuk

n=q;q

;

where s>n=pC ı; q 2 Œ2; 1/, and a similar estimate for q 2 .1; 2,using



log 1="



1=q

.(See[BrG]and[BrW].)

3. From (8.15) it follows that H

1;p

/  C

/ if p>n;rD 1n=p. Demonstrate

the following more precise result:

(8.57) ju.x/  u.y/jC jx  yj

1n=p

kruk

;p>n;

where B

D B

jxyj

.x/ \ B

jxyj

.y/.

(Hint: Apply scaling to (2.16) to obtain

jv.re

/  v.0/jCr

pn

.0/

jrv.x/j

dx:

To pass from B

jxyj

.x/ to B

in (8.57), note what the support of ' is in Exercise 5 of

 2.) There is a stronger estimate, known as Morrey’s inequality. See Chap. 14 for more

on this.

9. Pseudodifferential operators with nonregular symbols

We establish here some results on H¨older and Sobolev space continuity for pseu-

dodifferential operators p.x; D/ with symbols p.x; / which are somewhat more

ill behaved than those for which we had L

-Sobolev estimates in Chap. 7 or

-Sobolev estimates and H¨older estimates in  5 and 8 of this chapter. These

results will be very useful in the analysis of nonlinear, elliptic PDE in Chap. 14

and will also be used in Chaps. 15 and 16.

9. Pseudodifferential operators with nonregular symbols 51

Let r 2 .0; 1/.Wesayp.x; / 2 C



1;ı

/ provided

(9.1) jD



p.x; /jC

hi

mj˛j

and

(9.2) kD



p.;/k



 C

hi

mj˛jCır

Here ı 2 Œ0; 1. The following rather strong result is due to G. Bourdaud [Bou],

following work of E. Stein [S2].

Theorem 9.1. If r>0and p 2 .1; 1/, then, for p.x; / 2 C



1;1

(9.3) p.x; D/ W H

sCm;p

! H

s;p

;

provided 0<s<r. Furthermore, under these hypotheses,

(9.4) p.x; D/ W C

sCm



! C



Before giving the proof of this result, we record some implications. Note

that any p.x; / 2 S

1;1

satisﬁes the hypotheses for all r>0. Since operators

in OP S

1;ı

possess good multiplicative properties for ı 2 Œ0; 1/,wehavethe

following:

Corollary 9.2. If p.x; / 2 S

1;ı

;0 ı<1, we have the mapping properties

(9.3) and (9.4) for all s 2 R.

It is known that elements of OPS

1;1

need not be bounded on L

,evenfor

p D 2, but by duality and interpolation we have the following:

Corollary 9.3. If p.x; D/ and p.x;D/



belong to OP S

1;1

,then(9.3) holds for

all s 2 R.

We prepare to prove Theorem 9.1. It sufﬁces to treat the case m D 0. Following

[Bou]andalso[Ma2], we make use of the following results from Littlewood–

Paley theory. These results follow from (6.23)and(6.25), respectively.

Lemma 9.4. Let f

2 S

/ be such that, for some A>0,

(9.5) supp

f W A  2

k1

jjA  2

kC1

g;k 1:

Say

has compact support. Then, for p 2 .1; 1/; s 2 R, we have

(9.6)



kD0



s;p

 C



kD0

1=2



52 13. Function Space and Operator Theory for Nonlinear Analysis

If f

D '

.D/f with '

supported in the shell deﬁned in (9.5) and bounded in

1;0

, then the converse of the estimate (9.6) also holds.

Lemma 9.5. Let f

2 S

/ be such that

(9.7) supp

f WjjA  2

kC1

g;k 0:

Then, for p 2 .1; 1/; s > 0, we have

(9.8)



kD0



s;p

 C



kD0

1=2



The next ingredient is a symbol decomposition. We begin with the Littlewood–

Paley partition of unity (5.37),

(9.9) 1 D

./

./;

and with

(9.10) p.x; / D

j D0

p.x; /

./ D

j D0

.x; /:

Now, let us take a basis of L



j

j <



of the form

./ D e

i˛

;

and write (for j  1)

(9.11) p

.x; / D

j˛

.x/e

j

/

./;

where

./ has support on 1=2 < jj <2and is 1 on supp

;

./ D

j C1

/, with an analogous decomposition for p

./. Inserting these decom-

positions into (9.10) and summing over j , we obtain p.x; / as a sum of a rapidly

decreasing sequence of elementary symbols.

By deﬁnition, an elementary symbol in C



1;ı

is of the form

(9.12) q.x; / D

kD0

.x/'

./;

where '

is supported on hi2

and bounded in S

—in fact, '

./ D

kC1

/,fork  2—and Q

.x/ satisﬁes

9. Pseudodifferential operators with nonregular symbols 53

(9.13) jQ

.x/jC; kQ



 C  2

krı

For the purpose of proving Theorem 9.1,wetakeı D 1. It sufﬁces to estimate the

r;p

-operator norm of q.x; D/ when q.x; / is such an elementary symbol.

Set Q

.x/ D

.D/Q

.x/, with f

g the partition of unity described in

(9.9). Set

q.x; / D

k4

j D0

.x/ C

kC3

j Dk3

.x/ C

j DkC4

.x/

;

./

D q

.x; / C q

.x; /:(9.14)

We will perform separate estimates of these three pieces. Set f

D '

.D/f .

First we estimate q

.x; D/f . By Lemma 9.4,sincehi2

on the spectrum

of Q

.x; D/f k

s;p

 C



kD4

k4

j D0

;

1=2



 C



(

kD4

)

1=2



(9.15)

 C kf k

s;p

;

for all s 2 R.

To estimate q

.x; D/f , note that kQ

 C 2

jrCkr

. Then Lemma 9.5

implies

(9.16) kq

.x; D/f k

s;p

 C



(

kD0

)

1=2



 C kf k

s;p

;

for s>0.

To estimate q

.x; D/f , we apply Lemma 9.4 to h

j 4

kD0

, to obtain

.x; D/f k

s;p

 C



j D4

j 4

kD0

;

1=2



 C



j D4

j.sr/

j 4

kD0

;

1=2



:(9.17)

Now, if we set g

j 4

kD0

.kj/r

j and then set G

D 2

and F

j, we see that

54 13. Function Space and Operator Theory for Nonlinear Analysis

j 4

kD0

.kj /.rs/

As long as r>s, Young’s inequality (see Exercise 1 at the end of this section)

yields k.G

 C k.F

, so the last line in (9.17) is bounded by



j D0

;

1=2



 C kf k

s;p

This proves (9.3).

The proof of (9.4) is similar. We replace (9.6)by

(9.18) kf k



 sup

k0

.D/f k

;r>0:

We also need an analogue of Lemma 9.5:

Lemma 9.6. If f

2 S

/ and supp

f WjjA 2

kC1

g, then, for r>0,

(9.19)



kD0





 C sup

k0

Proof. For some ﬁnite N ,wehave

.D/

k0

.D/

kj N

Suppose sup

D S.Then



.D/

k0



 CS

kj N

kr

 C

jr

This proves (9.19).

Now, to prove (9.4), as before it sufﬁces to consider elementary symbols, of the

form (9.12)–(9.13), and we use again the decomposition q.x; / D q

C q

of (9.14). Thus it remains to obtain analogues of the estimates (9.15)–(9.17).

Parallel to (9.15), using the fact that

k4

j D0

.x/f

has spectrum in the shell

hi2

,andkQ

 C , we obtain

.x; D/f k



 C sup

k0



k4

j D0



 C sup

k0

(9.20)

 C kf k



;

9. Pseudodifferential operators with nonregular symbols 55

for all s 2 R. Parallel to (9.16), using kQ

 C  2

jrCkr

and Lemma 9.6,

we have

.x; D/f k







kD0





 C sup

k0

(9.21)

 C sup

k0

 C kf k



;

for all s>0, where the sum of seven terms

kC3

j Dk3

.x/f

has spectrum contained in jjC  2

,andkg

 C kf

Finally, parallel to (9.17), since

j 4

kD0

has spectrum in the shell hi

,wehave

.x; D/f k



 C sup

j 0



j 4

kD0



 C sup

j 0

j.sr/

j 4

kD0

:(9.22)

If we bound this last sum by

(9.23)

j 4

kD0

k.rs/

sup

;

then

(9.24) kq

.x; D/f k



 C

sup

j 0

j.sr/

j 4

kD0

k.rs/

kf k



;

and the factor in brackets is ﬁnite as long as s<r. The proof of Theorem 9.1 is

complete.

Things barely blow up in (9.24)whens D r. We will establish the following

result here. A sharper result (for p.x; / 2 C



1;ı

with ı<1)isgivenin(9.43).

56 13. Function Space and Operator Theory for Nonlinear Analysis

Proposition 9.7. If p.x; / 2 C



1;1

,then

(9.25) p.x; D/ W C

mCrC"



! C



; for all ">0:

Proof. It sufﬁces to treat the case m D 0. We follow the proof of (9.4). The

estimates (9.20)and(9.21) continue to work; (9.22) yields

.x; D/f k



 C sup

j 0

j 4

kD0

D C

kD0

 C

kD0

 2

krk"

kf k

rC"



;

(9.26)

which proves (9.25).

The way symbols in C



1;ı

most frequently arise is the following. One has in

hand a symbol p.x; / 2 C



1;0

, such as the symbol of a differential operator,

with H¨older-continuous coefﬁcients. One is then motivated to decompose p.x; /

as a sum

(9.27) p.x; / D p

.x; / C p

.x; /;

where p

.x; / 2 S

1;ı

,forsomeı 2 .0; 1/, and there is a good operator calculus

for p

.x; D/, while p

.x; / 2 C





1;ı

(for some <m) is treated as a remain-

der term, to be estimated. We will refer to this construction as symbol smoothing.

The symbol decomposition (9.27) is constructed as follows. Use the partition

of unity

./ of (9.9). Given p.x; / 2 C



1;0

, choose ı 2 .0; 1 and set

(9.28) p

.x; / D

j D0

p.x; /

./;

where J

is a smoothing operator on functions of x, namely

(9.29) J

f.x/ D ."D/f .x/;

with  2 C

/; ./ D 1 for jj1 (e.g.,  D

), and we take

(9.30) "

D 2

jı

We then deﬁne p

.x; / to be p.x;/  p

.x; /, yielding (9.27).

To analyze these terms, we use the following simple result.

9. Pseudodifferential operators with nonregular symbols 57

Lemma 9.8. Fo r " 2 .0; 1,

(9.31) kD

f k



 C

jˇj

kf k



and

(9.32) kf  J

f k

st



 C"

kf k



; for t  0:

Furthermore, if s>0,

(9.33) kf  J

f k

 C

kf k



Proof. Theestimate(9.31) follows from the fact that, for each ˇ  0,

jˇj

."D/ is bounded in OPS

1;0

;

and the estimate (9.32) follows from the fact that, with ƒ D .1  /

1=2

W C



! C

st



isomorphically,

plus the fact that

t

.1  ."D// is bounded in OPS

1;0

;

for 0<" 1.Asfor(9.33), if "  2

j

,wehave



1  ."D/



f k



`j

.D/f k

 C

`j

`s

kf k



;

and since

`j

`s

 C

js

for s>0,(9.33) follows.

Using this, we easily derive the following conclusion:

Proposition 9.9. If p.x; / 2 C



1;0

, then, in the decomposition (9.27),

(9.34) p

.x; / 2 S

1;ı

and

(9.35) p

.x; / 2 C



mrı

1;ı

Proof. Theestimate(9.31) yields

(9.36) kD



.;/k



 C

˛ˇ

hi

mj˛jCıjˇ j

;

which implies (9.34).