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

Подождите немного. Документ загружается.

18 13. Function Space and Operator Theory for Nonlinear Analysis

and

(5.6) jr

x;y

K.x; y/jC jx  yj

n1

Furthermore, at least when ı<1,wehaveanL

-bound:

(5.7) kP uk

 Kkuk

;

and smoothings of such an operator have smooth Schwartz kernels satisfying

(5.5)–(5.7)forﬁxedC; K. (Results in  9 of this chapter will contain another

proof of this L

-estimate. Note that when p.x; / D p./ the estimate (5.7)fol-

lows from the Plancherel theorem.) Our main goal here is to give a proof of the

following fundamental result of A. P. Calderon and A. Zygmund:

Theorem 5.2. Suppose P W L

/ ! L

/ is a weak limit of operators with

smooth Schwartz kernels satisfying (5.5)–(5.7) uniformly. Then

(5.8) P W L

/ ! L

/; 1 < p < 1:

In particular, this holds when P 2 OP S

1;ı

/; ı 2 Œ0; 1/.

The hypotheses do not imply boundedness on L

/ or on L

/.They

will imply that P is of weak type .1; 1/. By deﬁnition, an operator P is of weak

type .q; q/ provided that, for any >0,

(5.9) meas fx WjP u.x/j >gC

q

kuk

Any bounded operator on L

is a fortiori of weak type .q; q/, in view of the simple

inequality

(5.10) meas fx Wju.x/j >g

1

kuk

A key ingredient in proving Theorem 5.2 is the following result:

Proposition 5.3. Under the hypotheses of Theorem 5.2, P is of weak type .1; 1/.

Once this is established, Theorem 5.2 will then follow from the next result,

known as the Marcinkiewicz interpolation theorem.

Proposition 5.4. If r<p<qand if T is both of weak type .r; r/ and of weak

type .q; q/,thenT W L

! L

Proof. Write u D u

C u

, with u

.x/ D u.x/ for ju.x/j >and u

.x/ D u.x/

for ju.x/j. With the notation

(5.11) 

./ D meas fx Wjf.x/jg;

5. Singular integral operators on L

we have



T u

.2/  

T u

./ C 

T u

./

 C



r

C C



q

:(5.12)

Also, there is the formula

jf.x/j

dx D p



./

p1

d:

Hence

jT u.x/j

dx D p



T u

./

p1

d

 C



p1r



juj>

ju.x/j



d(5.13)

C C



p1q



juj

ju.x/j



d:

Now

(5.14)



p1r



juj>

ju.x/j



d D

p  r

ju.x/j

and, similarly,

(5.15)



p1q



juj

ju.x/j



d D

q  p

ju.x/j

dx:

Combining these gives the desired estimate on kT uk

We will apply Proposition 5.4 in conjunction with the following covering

lemma of Calderon and Zygmund:

Lemma 5.5. Let u 2 L

/ and >0be given. Then there exist v; w

/ and disjoint cubes Q

;1 k<1, with centers x

, such that

u D v C

; kvk

 3kuk

;(5.16)

jv.x/j2

;(5.17)

.x/ dx D 0 and supp w

 Q

;(5.18)

20 13. Function Space and Operator Theory for Nonlinear Analysis

meas.Q

/  

1

kuk

:(5.19)

Proof. Tile R

with cubes of volume greater than 

1

kuk

. The mean value of

ju.x/j over each such cube is <. Divide each of these cubes into 2

equal cubes,

and let I

;::: be those so obtained over which the mean value of ju.x/j

is  . Note that

(5.20)  meas.I

/ 

ju.x/j dx  2

 meas.I

Now set

(5.21) v.x/ D

meas.I

u.y/ dy; for x 2 I

;

and

(5.22)

.x/ D u.x/  v.x/; for x 2 I

;

0; for x … I

Next take all the cubes that are not among the I

, subdivide each into 2

equal

parts, select those new cubes I

;:::, over which the mean value of ju.x/j is

 , and extend the deﬁnitions (5.21)–(5.22) to these cubes, in the natural fash-

ion. Continue in this way, obtaining disjoint cubes I

and functions w

.Then

reorder these cubes and functions as Q

;:::,andw

;:::. Complete the

deﬁnition of v by setting v.x/ D u.x/,forx …[Q

. Then we have the ﬁrst part

of (5.16). Since

(5.23)



jv.x/jCjw

.x/j



dx  3

ju.x/j dx;

and since the cubes are disjoint, w

is supported in Q

,andv D u on R

n[Q

we obtain the rest of (5.16).

Next, (5.17) follows from (5.20)ifx 2[Q

.Butifx …[Q

,thereare

arbitrarily small cubes containing x over which the mean value of ju.x/j is <,

so (5.17) holds almost everywhere on R

n[Q

as well. The assertion (5.18)is

obvious from the construction, and (5.19) follows by summing (5.20). The lemma

is proved.

One thinks of v as the “good” piece and w D

as the “bad” piece. What

is “good” about v is that kvk

 2

kuk

,so

(5.24) kPvk

 K

kvk

 4

kuk

5. Singular integral operators on L

Hence

(5.25)





meas

x WjPv.x/j >



 Ckuk

To treat the action of P on the “bad” term w, we make use of the following es-

sentially elementary estimate on the Schwartz kernel K. The proof is an exercise.

Lemma 5.6. There is a C

< 1 such that, for any t>0,ifjyjt; x

2 R

(5.26)

jxx

j2t

K.x; x

C y/  K.x; x

dx  C

To estimate Pw,wehave

.x/ D

K.x; y/w

.y/ dy



K.x; y/  K.x; x



.y/ dy:(5.27)

Before we make further use of this, a little notation: Let Q



be the cube concentric

with Q

, enlarged by a linear factor of 2n

1=2

, so meas Q



D .4n/

n=2

meas Q

For some t

>0, we can arrange that

fx Wjx  x

jt

D R

n Q



fx Wjx  x

j >2t

(5.28)

Furthermore, set O D[Q



, and note that

(5.29) meas O  L

1

kuk

;

with L D .4n/

n=2

. Now, from (5.27), we have

jPw

.x/j dx



jyjt

;

jxj2t

K.x C x

C y/  K.x C x

(5.30)

jw

.y C x

/j dx dy

 C

;

the last estimate using Lemma 5.6. Thus

(5.31)

jPw.x/j dx  3C

kuk

22 13. Function Space and Operator Theory for Nonlinear Analysis

Together with (5.29), this gives

(5.32) meas

x WjPw.x/j >



kuk

;

and this estimate together with (5.25) yields the desired weak (1,1)-estimate:

(5.33) meas

x WjP u.x/j >





kuk

This proves Proposition 5.3.

To complete the proof of Theorem 5.2, we apply Marcinkiewicz interpolation

to obtain (5.8)forp 2 .1; 2. Note that the Schwartz kernel of P



also satisﬁes

the hypotheses of Theorem 5.2,sowehaveP



W L

! L

,for1<p 2. Thus

the result (5.8)forp 2 Œ2; 1/ follows by duality.

We remark that if (5.6) is weakened to jr

K.x; y/jC jx yj

n1

, while the

hypotheses (5.5)and(5.7) are retained, then Lemma 5.6 still holds, and hence so

does Proposition 5.3. Thus, we still have P W L

/ ! L

/ for 1<p 2,

but the duality argument gives only P



W L

/ ! L

/ for 2  p<1.

We next describe an important generalization to operators acting on Hilbert

space-valued functions. Let H

and H

be Hilbert spaces and suppose

(5.34) P W L

; H

/ ! L

; H

Then P has an L.H

; H

/-operator-valued Schwartz kernel K. Let us impose on

K the hypotheses of Theorem 5.2, where now jK.x; y/jstands for the L.H

; H

norm of K.x; y/. Then all the steps in the proof of Theorem 5.2 extend to this

case. Rather than formally state this general result, we will concentrate on an

important special case.

Proposition 5.7. Let P./ 2 C

; L.H

; H

// satisfy

(5.35) kD



P./k

L.H

 C

hi

j˛j

;

for all ˛  0.Then

(5.36) P.D/ W L

; H

/ ! L

; H

/; for 1<p<1:

This leads to an important circle of results known as Littlewood–Paley theory.

To obtain this, start with a partition of unity

(5.37) 1 D

j D0

./

;

5. Singular integral operators on L

where '

2 C

./ is supported on jj1; '

./ is supported on 1=2 

jj2,and'

./ D '

1j

/ for j  2.WetakeH

D C; H

D `

,and

look at

(5.38) ˆ W L

/ ! L

given by

(5.39) ˆ.f / D .'

.D/f; '

.D/f; : : : /:

This is clearly an isometry, though of course it is not surjective. The adjoint

(5.40) ˆ



W L

/ ! L

given by

(5.41) ˆ



;:::/ D

.D/g

;

satisﬁes

(5.42) ˆ



ˆ D I

on L

/. Note that ˆ D ˆ.D/,where

(5.43) ˆ./ D .'

./; '

./; : : : /:

It is easy to see that the hypothesis (5.35) is satisﬁed by both ˆ./ and ˆ



./.

Hence, for 1<p<1,

ˆ W L

/ ! L



W L

/ ! L

/:(5.44)

In particular, ˆ maps L

/ isomorphically onto a closed subspace of

/, and we have compatibility of norms:

(5.45) kuk

kˆuk

In other words,

(5.46) C

kuk





j D0

.D/uj

1=2



 C

kuk

;

for 1<p<1.

24 13. Function Space and Operator Theory for Nonlinear Analysis

Exercises

1. Estimate the family of symbols a

./ Dhi

;y2 R. Show that if ƒ

D a

.D/,

then

(5.47) kƒ

 C

hyi

nC1

kuk

This estimate will be useful for the development of the Sobolev spaces H

s;p

in the next

section.

2. Let

./ be supported on 1=4 jj4;

./ D 1 for 1=2 jj2,and

./ D

1j

/ for j  2.Lets 2 R. Show that

A.D/; B.D/ W L

/ ! L

/; 1 < p < 1;

for

./ D 2

hi

s

./ı

;

./ D 2

ks

hi

./ı

;

by applying Proposition 5.7.

3. Give a proof that

(5.48)

jf.x/j

dx D p



./ 

p1

d;

used in (5.13). Also, demonstrate (5.14)and(5.15). (Hint: After doing (5.48), get an

analogous identity for the integral of jf.x/j

over the set fx Wjf.x/j >g,resp.,

 :)

4. Give a detailed proof of Lemma 5.6.

5. Let A 2 OP S

1;0

/, and suppose A.x; / D 0 for x

D 0.DeﬁneTf D Af



where R

Dfx 2 R

W˙x

 0g. Show that, for 1  p 1,

(5.49) f 2 L

/; supp f  R

H) Tf 2 L



(Hint: Apply Proposition 5.1 of Appendix A. Compare with Exercise 3 in  5 of Ap-

pendix A.)

6. The spaces H

s;p

Here we deﬁne and study H

s;p

for any s 2 R;p2 .1; 1/. In analogy with the

characterization of H

/ D H

s;2

/ given in  1 of Chap.4, we set

(6.1) H

s;p

/ D ƒ

s

Given the results of  5, we can establish the following.

Proposition 6.1. When s D k is a positive integer, p 2 .1; 1/, the spaces

k;p

/ of  1 coincide with (6.1).

6. The spaces H

s;p

Proof. For j˛jk; 

hi

k

belongs to S

/. Thus, by Theorem 5.1, D

k

maps L

/ to itself. Thus any u 2 ƒ

k

/ satisﬁes the deﬁnition of

k;p

/ given in  1. For the converse, note that one can write

(6.2) hi

j˛jk

./

;

with coefﬁcients q

2 S

/. Thus if D

u 2 L

/ for all j˛jk, it follows

that ƒ

u 2 L

We next prove an interpolation theorem generalizing the identity



/; H





D H

s

/; for  2 Œ0; 1;

proven in  2 of Chap. 4.

Proposition 6.2. Fo r s 2 R;2 .0; 1/, and p 2 .1; 1/,

(6.3)



/; H

s;p





D H

s;p

Proof. The proof is parallel to that of Proposition 2.2 of Chap. 4, except

that we use the estimate (5.47) of the last section in place of the obvious

identity kA

kD1 for a unitary operator A

on a Hilbert space. Thus, if

v 2 H

s;p

/,let

(6.4) u.z/ D e

.z/s

Then u./ D e



v; u.iy/ D e

y

iys



s



is bounded in L

/,by

(5.47), and also u.1 C iy/ D e

.yi/

s

iys



s



is bounded in the space

s;p

/. Therefore, such a function v belongs to the left side of (6.3). The

reverse containment is similarly established as in the proof of Proposition 2.2 of

Chap. 4.

This sort of argument yields more generally that, for ; s 2 R;2 .0; 1/,and

p 2 .1; 1/,

(6.5) ŒH

;p

/; H

s;p

/



D H

sC.1/;p

With Proposition 6.2 established, we can deﬁne and analyze spaces H

s;p

compact manifolds in the same way as we did for p D 2 in Chap. 4. If M is a com-

pact manifold without boundary, one deﬁnes H

s;p

.M / in analogy with H

.M /,

via coordinate charts, and proves

(6.6) ŒH

;p

.M /; H

s;p

.M /



D H

sC.1/;p

.M /;

for p 2 .1; 1/;  2 .0; 1/.If is a compact subdomain of M with smooth

boundary, we deﬁne H

k;p

./ as in  1, and recall the extension operator E W

k;p

./ ! H

k;p

.M /.IfwedeﬁneH

s;p

./ for s>0by

26 13. Function Space and Operator Theory for Nonlinear Analysis

(6.7) H

s;p

./ D ŒL

./; H

k;p

./



;2 .0; 1/; s D k;

it follows that E W H

s;p

./ ! H

s;p

.M / and hence

(6.8) H

s;p

./  H

s;p

.M /=fu W u D 0 on g:

Also, of course, H

s;p

./ agrees with the characterization of  1 when s D k is a

positive integer. Generalizing the theorem of Rellich, Proposition 4.4 of Chap. 4,

one has, for s  0; 1 < p < 1,

(6.9)  W H

sC;p

./ ,! H

s;p

./ is compact for >0:

By the arguments used in Chap. 4, we easily reduce this to showing that, for >

0; 1 < p < 1,

(6.10) ƒ



W L

/ ! L

/ is compact.

Indeed, the operator (6.10) is of the form ƒ



u D k



 u, with k



2 L

for any >0. Thus k



is an L

-norm limit of k

;j

2 C

/,soƒ



is an

operator norm limit of convolution maps L

/ ! C

/, which are clearly

compact on L

We now extend some of the Sobolev imbedding theorems of  2. Once they

are obtained on R

, they easily yield similar results for functions on compact

manifolds, perhaps with boundary.

Proposition 6.3. If s>n=p,thenH

s;p

/  C.R

/ \ L

Proof. ƒ

s

u D J

 u,where

./ Dhi

s

. It sufﬁces to show that

(6.11) J

2 L

/; for s>

;

D 1:

Indeed, estimates established in  8 of Chap.3 imply that J

.x/ is smooth on

n 0, rapidly decreasing as jxj!1,and

(6.12) jJ

.x/jC jxj

nCs

; jxj1; s < n;

which is sufﬁcient. Compare estimates for s D n=2 in (4.4)–(4.9).

Next we generalize (2.9).

Proposition 6.4. Fo r sp < n; p 2 .1; 1/, we have

(6.13) H

s;p

/  L

np=.nsp/

Proof. Suppose s D k C ; k 2 Z

;2 Œ0; 1/.Thenu 2 H

s;p

) ƒ



u 2

k;p

, and by (2.9)thisgivesƒ



u 2 L

/, with q D np = .n  kp/. Note that

6. The spaces H

s;p

q 2 .1; 1/ and np =.n  sp/ D nq=.n  q/,soalsoq < n. Hence it sufﬁces

to show that

(6.14) ƒ



W L

/ ! L

nq=.nq/

when  2 .0; 1/; q 2 .1; 1/,andq < n. We divide the analysis into cases.

Case I: 1<q<n. In this case, we have, by (2.2),

(6.15) H

1;q

/  L

nq=.nq/

Fixing v 2 L

/, consider ƒ

z

v for z 2  Dfz 2 C W 0  Re z  1g.Note

that Proposition 5.7 implies

(6.16) kƒ

 Ae

Bjyj

kvk

;

for y 2 R. Making use also of (6.15), we have

(6.17) kƒ

.1Ciy/

nq=.nq/

 Ae

Bjyj

kvk

From here a complex interpolation argument gives (6.14) in this case.

Case II: 2  n  q<1. In this case, set r D nq=.n  q/. Note that

(6.18)





and



;

where r

is the dual exponent to r.Wehaver>q n  2,sor

<2 n,and

Case I gives

(6.19) ƒ



W L

/ ! L

Then (6.14) follows by duality.

Case III: n D 1. Here one needs a different approach. Since this case is not so

crucial for PDE, we omit it. Various proofs that include this case can be found in

[S1], [S3], and [BL].

The following result is an immediate consequence of the deﬁnition (6.1),

the pseudodifferential operator calculus, and the L

-boundedness result of

Theorem 5.2.

Proposition 6.5. If P 2 OP S

1;ı

/; 0  ı<1, and 1<p<1,then

(6.20) P W H

s;p

/ ! H

sm;p