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

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88 13. Function Space and Operator Theory for Nonlinear Analysis

We now turn to the proof of Lemma 12.2. We can pick a compact K  U

such that m.K/ > m

. Then the covering C yields a ﬁnite covering of K,

say A

;:::;A

.LetB

be the ball A

of the largest radius. Throw out all A

that meet B

,andletB

be the remaining ball of largest radius. Continue until

;:::;A

g is exhausted. One gets disjoint balls B

;:::;B

in C. Now each

meets some B

, having the property that the radius of B

is  the radius of

.Thus,if

is the ball concentric with B

, with three times the radius, we

have

[

j D1



[

`D1

 K:

This yields (12.9).

Note that clearly

(12.11) f 2 L

/ H) kM.f /k

kf k

Now the method of proof of the Marcinkiewicz interpolation theorem, Proposition

5.4, yields the following.

Corollary 12.3. If 1<p<1,then

(12.12) kM.f /k

 C

kf k

Our ﬁrst result on Hardy spaces is the following, relating h

/ to the smaller

space H

Proposition 12.4. If u 2 h

/ has compact support and

u.x/ dx D 0,then

u 2 H

Proof. It sufﬁces to show that

(12.13) v.x/ D supfj'

 u.x/jW' 2 F;t  1g

belongs to L

/. Clearly, v is bounded. Also, if supp u fjxjRg,thenwe

can write u D

; u

2 L

/.Then

(12.14)

 u.x/ D

1

 u

.x/;

.x/ D t

n

1

x/;

.x/ D @

'.x/:

If jxjDR C 1 C ,then

 u

.x/ D 0 for t<,so

(12.15) v.x/  C

1

M.u

/.x/:

The weak (1,1) bound (12.7)onM now readily yields an L

-bound on v.x/.

12. Hardy spaces 89

One advantage of h

/ is its localizability. We have the following useful

result:

Proposition 12.5. If r>0and g 2 C

/ has compact support, then

(12.16) u 2 h

/ H) gu 2 h

Proof. If g 2 C

and 0<r 1,wehave,forall' 2 F,

(12.17)

 .gu/.x/  g.x/'

 u.x/

 Ct

rn

.x/

ju.y/j dy:

Hence it sufﬁces to show that

(12.18) v.x/ D sup

0<t1

rn

.x/

ju.y/j dy

belongs to L

/.Since

(12.19) v.x/ 

.x  y/

jx yj

nr

ju.y/j dy;

where .x/ is the characteristic function of fjxj1g,thisisclear.

Given   R

open, u 2 L

loc

./,wesay

(12.20) u 2 H

loc

./ ” gu 2 h

/; 8 g 2 C

./:

This is equivalent to the statement that, for any compact K  ,thereisav 2

/ such that u D v on a neighborhood of K. To see this, note that if u 2

loc

./ and g 2 C

./; g D 1 on a neighborhood of K,thengu 2 h

Now take v D gu Ch,whereh 2 C

/ has support disjoint from supp g,and

h.x/ dx D

g.x/u.x/ dx. By Proposition 12.4, v 2 H

/. The converse

is established similarly.

Not every compactly supported element of L

/ belongs to h

/,butwe

do have the following.

Proposition 12.6. If p>1and u 2 L

/ has compact support, then u 2

Proof. We have

(12.21) .G

f /.x/  .Gf /.x/  C Mf.x/:

90 13. Function Space and Operator Theory for Nonlinear Analysis

Hence, given p>1;u 2 L

/ ) G

u 2 L

/.Also,G

u has support in

jxjR C 1 if supp u fjxjRg,soG

u 2 L

The spaces H

/ and h

/ are Banach spaces, with norms

(12.22) kuk

DkGuk

; kuk

DkG

It is useful to have the following approximation result.

Proposition 12.7. Fix 2 C

/ such that

.x/ dx D 1.Ifu2 H

then

(12.23) k

 u uk

! 0; as " ! 0:

Proof. One easily veriﬁes from the deﬁnition that, for some C<1; G.

 u/

.x/  C Gu.x/; 8 x; 8 " 2 .0; 1. Hence, by the dominated convergence theorem,

it sufﬁces to show that

(12.24) G.

 u u/.x/ ! 0; a.e. x; as " ! 0I

that is,

sup

t>0;'2F



 u '

 u/.x/

! 0; a.e. x; as " ! 0:

To prove this, it sufﬁces to show that

(12.25) lim

";ı !0

sup

0<tı

sup

'2F



 u '

 u/.x/

D 0; a.e. x;

and that, for each ı>0,

(12.26) lim

"!0

sup

tı

sup

'2F



 '

/  u.x/

D 0:

In fact, (12.25) holds whenever x is a Lebesgue point for u (see the exercises for

more on this), and (12.26) holds for all x 2 R

,sinceu 2 L

/ and, for all

' 2 F ,wehavek'



 '

 C"t

n1

Corollary 12.8. Let T

u.x/ D u.x C y/. Then, for u 2 H

(12.27) kT

u  uk

! 0; as jyj!0:

Proof. Since kT k

L.H

D 1 for all y, it sufﬁces to show that (12.27) holds for

u in a dense subspace of H

/. Thus it sufﬁces to show that, for each ">0;

u 2 H

(12.28) lim

jyj!0

 u/ 

 uk

D 0:

12. Hardy spaces 91

But T

 u/ 

 u D .



/  u,where

(12.29)

.x/ 

.x/ D "

n



1

.x C y//  ."

1



Thus

 u/ 

 uk

D sup

t>0;'2F



/  '

 uk

k



kuk

(12.30)

 C jyj"

n1

kuk

;

which ﬁnishes the proof.

It is clear that we can replace H

by h

in Proposition 12.7 and Corollary 12.8,

obtaining, for u 2 h

(12.31) k

 u uk

! 0;

as " ! 0,and

(12.32) kT

u  uk

! 0;

as jyj!0.

We can also approximate by cut-offs:

Proposition 12.9. Fix  2 C

/, so that .x/ D 1 for jxj1; 0 for jxj2,

and 0    1.Set

.x/ D .x=R/. Then, given u 2 h

/, we have

(12.33) lim

R!1

ku  

D 0:

Proof. Clearly, G

.u  

u/.x/ D 0,forjxjR  1,so

lim

R!1

.u  

u/.x/ D 0; 8 x 2 R

To get (12.33), we would like to appeal to the dominated convergence theorem. In

fact, the estimates (12.17)–(12.19) (with g D 1  

)give

(12.34) G

.u  

u/.x/  G

u.x/ C Av.x/; 8 R  1;

where A Dkrk

,andv.x/ isgivenby(12.19), with r D 1,sov 2 L

Thus dominated convergence does give

(12.35) lim

R!1

.u  

u/k

D 0;

and the proof is done.

92 13. Function Space and Operator Theory for Nonlinear Analysis

Together with (12.31), this gives

Corollary 12.10. The space C

/ is dense in h

A slightly more elaborate argument shows that

(12.36) D

u 2 C

/ W

u.x/ dx D 0

is dense in H

/;see[Sem].

One signiﬁcant measure of how much smaller H

/ is than L

/ is the

following identiﬁcation of an element of the dual of H

/ that does not belong

to L

Proposition 12.11. We have

(12.37)

f.x/ log jxj dx

 C kf k

Proof. Let .x/ 2 C

/ satisfy .x/ D 1 for jxj1; .x/ D 0 for jxj2.

Set

(12.38) `.x/ D

j D1

.2

x/ C

j D0



1  .2

j



It is easy to check that

(12.39) log jxj.log 2/`.x/ 2 L

Thus it sufﬁces to estimate

f.x/`.x/ dx.Wehave

(12.40)

f.x/`.x/ dx



j D1

f.x/.2

x/ dx

We claim that, for each j 2 Z,

(12.41)

f.x/.2

x/ dx

 C2

jn

inf

j

.0/

Gf:

In fact, given j 2 Z; z 2 B

j

.0/, we can write

(12.42)

f.x/2

.2

x/ dx D K'

 f.z/;

12. Hardy spaces 93

with r D 2

2j

;KD K.; n/,forsome' 2 F ;say'.x/ is a multiple of a

translate of .4x/. Consequently, with S

D B

j

.0/,wehave

(12.43)

f.x/`.x/ dx

 C

j D1

j C1

Gf D C kf k

By Corollary 12.8, we have the following:

Corollary 12.12. Given f 2 H

(12.44) logf 2 C.R

The result (12.37) is a very special case of the fact that the dual of H

/ is

naturally isomorphic to a space of functions called BMO.R

/. This was estab-

lished in [FS]. The special case given above is the only case we will use in this

book. More about this duality and its implications for analysis can be found in

the treatise [S3]. Also, [S3] has other important information about Hardy spaces,

including a study of singular integral operators on these spaces.

The next result is a variant of the div-curl lemma (discussed in Exercises for

 6), due to [CLMS]. It states that a certain function that obviously belongs to

/ actually belongs to H

/. Together with Corollary 12.12, this produces

a useful tool for PDE. An application will be given in  12B of Chap.14. The proof

below follows one of L. Evans and S. Muller, given in [Ev2].

Proposition 12.13. If u 2 L

; R

/; v 2 H

/, and div u D 0,thenu

rv 2 H

Proof. Clearly, u rv 2 L

/. Now, with ' 2 C

/, supported in the unit

ball, set '

.y/ D r

n



1

.x  y/



.Wehave

(12.45)

.u rv/'

dy D

.x/

.v  v

x;r

/u r'

dy;

since div u D 0.Thus,withC

Dkr'k

(12.46)

.u rv/'



.x/

ju v

x;r

jjuj dy:

Take

(12.47) p 2



n  2



;qD

p  1

2 .1; 2/:

94 13. Function Space and Operator Theory for Nonlinear Analysis

Then

.u rv/'



.x/

jv  v

x;r

1=p

.x/

juj

1=q



.x/

jrvj



1=

.x/

juj

1=q

;(12.48)

where  D pn=.p C n/ < 2 and a D n C1. Consequently,

.u rv/'

 C



jrvj





1=



juj



1=q

 C



jrvj





2=

C M



juj



2=q



:(12.49)

By Corollary 12.3,wehave





jrvj







2=

 C



jrvj





2=

,andso



jrvj





2=

dx  C

jrvj

dx:

Similarly,



juj



2=q

dx  C

juj

dx:

Hence

(12.50) ku rvk

D sup

'2F ;r>0



.u rv/'



 C



krvk

Ckuk



We next establish a localized version of Proposition 12.13.

Proposition 12.14. Let   R

be open. If u 2 L

.; R

/; div u D 0, and

v 2 H

./,thenurv 2 H

loc

./.

Proof. We may as well suppose n>1.Takeany

O  , diffeomorphic to a

ball. It sufﬁces to show that u rv is equal on

O to an element of H

/.Say

O  U  , with U also diffeomorphic to a ball. Pick  2 C

.U /;  D 1

Let Qu 2 L

.; ƒ

n1

/ correspond to u via the volume element on .Then

d Qu D 0. We use the Hodge decomposition of L

.U; ƒ

n1

/, with absolute bound-

ary condition:

(12.51) Qu D dıG

Qu C ıdG

Qu C P

Qu on u:

Exercises 95

Since d Qu D 0, we have by (9.48) of Chap. 5 that ıdG

Qu D 0. Also, given n>1,

n1

.U/ D 0,soP

Qu D 0, too, and so

(12.52) Qu D d Qw; Qw 2 H

.U; ƒ

n2

Having this, we deﬁne a vector ﬁeld u

on R

so that Qu

D d. Qw/,andweset

D v. It follows that u

satisfy the hypotheses of Proposition 12.13,so

rv

2 H

/.Butu

rv

D u rv on O, so the proof is done.

Let us ﬁnally mention that while we have only brieﬂy alluded to the space

BMO, it has also proven to be of central importance, especially since the work

of [FS]. More about the role of BMO in paradifferential operator calculus can

be found in [T2]. Also, Proposition 12.13 can be deduced from a commutator

estimate involving BMO, as explained in [CLMS]; see also [AT].

Exercises

We say x 2 R

is a Lebesgue point for f 2 L

/ provided

lim

r!0

vol.B

.x/

jf.y/ f.x/j dy D 0:

In Exercises 1 and 2, we establish that, given f 2 L

/,a.e.x 2 R

is a Lebesgue

point of f .

1. Set

M.f /.x/ D sup

r>0

vol.B

.x/

jf.y/ f.x/j dy:

Show that, for all x 2 R

M.f /.x/  M.f /.x/ Cjf.x/j:

2. Given >0,let



x 2 R

W lim sup

r!0

vol.B

.x/

jf.y/ f.x/j dy > 

Take ">0,andtakeg 2 C

/ so that kf gk

<". Show that E



is unchanged

if f is replaced by f g. Deduce that







x W M.f g/.x/ >





[



x Wjf.x/ g.x/j >





;

and hence, via Proposition 12.1,

meas.E



/ 



kf  gk



Deduce that meas.E



/ D 0; 8 >0, and hence a.e. x 2 R

is a Lebesgue point for f .

96 13. Function Space and Operator Theory for Nonlinear Analysis

3. Now verify that (12.25) holds whenever x is a Lebesgue point of u.

4. If u W R

! R

, show that

u 2 H

/ H) det Du 2 H

(Hint: Compute div w,whenw D .@

; @

/:)

5. If u W R

! R

, show that

u 2 H

/ H) u

 u

2 H

(Hint. Show that the ﬁrst argument of u

 u

is det Dv,wherev D .u

; u

/:)

A. Variations on complex interpolation

Let X and Y be Banach spaces, assumed to be linear subspaces of a Hausdorff

locally convex space V (with continuous inclusions). We say .X;Y;V/is a com-

patible triple. For  2 .0; 1/, the classical complex interpolation space ŒX; Y 



introduced in Chap. 4 and much used in this chapter, is deﬁned as follows. First,

Z D X C Y gets a natural norm; for v 2 X C Y ,

(A.1) kvk

D inf fkv

Ckv

W v D v

C v

2 X; v

2 Y g:

One has X C Y  X ˚ Y=L,whereL Df.v; v/ W x 2 X \ Y g is a closed

linear subspace, so X C Y is a Banach space. Let  Dfz 2 C W 0<Re z <1g,

with closure

.DeﬁneH



.X; Y / to be the space of functions f W  ! Z D

X C Y , continuous on

, holomorphic on  (with values in X C Y ), satisfying

f WfIm z D 0g!X continuous, f WfIm z D 1g!Y continuous, and

(A.2) ku.z/k

 C; ku.iy/k

 C; ku.1 C iy/k

 C;

for some C<1, independent of z 2

 and y 2 R. Then, for  2 .0; 1/,

(A.3) ŒX; Y 



Dfu. / W u 2 H



.X; Y /g:

One has

(A.4) ŒX; Y 



 H



.X; Y /=fu 2 H



.X; Y / W u./ D 0g;

giving ŒX; Y 



the sructure of a Banach space. Here

(A.5) kuk



.X;Y /

D sup

z2

ku.z/k

C sup

ku.iy/k

C sup

ku.1 C iy/k

If I is an interval in R, we say a family of Banach spaces X

;s2 I (subspaces

of V ) forms a complex interpolation scale provided that for s; t 2 I;  2 .0; 1/,

(A.6) ŒX





D X

.1/sCt

A. Variations on complex interpolation 97

Examples of such scales include L

-Sobolev spaces X

D H

s;p

.M /; s 2 R,

provided p 2 .1; 1/,asshownin 6 of this chapter, the case p D 2 having been

done in Chap. 4. It turns out that (A.6) fails for Zygmund spaces X

D C



.M /,

but an analogous identity holds for some closely related interpolation functors,

which we proceed to introduce.

If .X;Y;V/is a compatible triple, as deﬁned in above, we deﬁne H



.X;Y;V/

to be the space of functions u W

 ! X CY D Z such that

u W  ! Z is holomorphic,(A.7)

ku.z/k

 C; ku.iy/k

 C; ku.1 C iy/k

 C;(A.8)

and

(A.9) u W

 ! V is continuous:

For such u, we again use the norm (A.5). Note that the only difference with



.X; Y / is that we are relaxing the continuity hypothesis for u on .



.X;Y;V/is also a Banach space, and we have a natural isometric inclusion

(A.10) H



.X; Y / ,! H



.X;Y;V/:

Now for  2 .0; 1/ we set

(A.11) ŒX; Y 

IV

Dfu./ W u 2 H



.X;Y;V/g:

Again this space gets a Banach space structure, via

(A.12) ŒX; Y 

IV

 H



.X;Y;V/=fu 2 H



.X;Y;V/ W u./ D 0g;

and there is a natural continuous injection

(A.13) ŒX; Y 



,! ŒX; Y 

IV

Sometimes this is an isomorphism. In fact, sometimes ŒX; Y 



D ŒX; Y 

IV

for

practically all reasonable choices of V . For example, one can verify this for X D

/; Y D H

s;p

/,theL

-Sobolev space, with p 2 .1; 1/; s 2 .0; 1/.

On the other hand, there are cases where equality in (A.10) does not hold, and

where ŒX; Y 

IV

is of greater interest than ŒX; Y 



We next deﬁne ŒX; Y 



.InthiscaseweassumeX and Y are Banach spaces and

Y  X (continuously). We take  as above, and set

 Dfz 2 C W 0<Re z  1g,

i.e., we throw in the right boundary but not the left boundary. We then deﬁne



.X; Y / to be the space of functions u W

 ! X such that

u W  ! X is holomorphic,

ku.z/k

 C; ku.1 C iy/k

 C;(A.14)

u W

 ! X is continuous: