(1.22)

For another class of examples, consider (1.23)

w(h) = (log

o< hC

This yields (1.24) in (1.5) (up to a constant factor). Proposition 1.1 applies here for

any s> 0. To

apply Proposition 1.2, we need .s > 1. In such a case, (1.15) holds for any fi > 0, and we get h

a(h)=f (log7) —edt 1

(1.25)

1

Note that a(h) >> w(h) as h \ 0 in this case, again illustrating that Proposition 1.2 is not exactly a converse of Proposition 1.1. As another family of examples, let us take a > 0 and consider

(1.26)

This yields

A(j) = 5S2-j

(1.27)

Propositions 1.1 and 1.2 apply, for any a ? 0. We compute a(h) using (1.14). We have (1.28) Also,

(1.29)

f

= fh(10g 1)S

dt

0

dt =

f

For h small, i.e., for A =log(1/h) large, we have / j h

Hence, with w and A given

r8eT dr.

log(1/h)

o

\ on jA,cx), so this is /

JIog(1/h)

1\st1 Ft

by (1.26)—(1.27), we have

C" C C(A),

C

a(h) = h(log

1

st-i

'

UNUUUS

This extends (1.22). We mention that a closer analysis of (1.29) shows that it is

h(log(1/h))8 as h \ 0, so for this class of examples the second term on the right side of (1.14) is dominant. We will return to this point shortly; see (1.54)—(1.57). for It is desirable to estimate IP(2 in a class somewhat larger than that figuring in the definition (1.4). We will take (1.31)

b(Rn)

E

P(0) = 0,

where membership in the symbol class STb(PJ2) means (1.32)

We assume Li 0. To say P(0) = with

E

we have

means is a sum of terms of the form We may as well suppose Now = 0

P(23D)u= 2

=

(1.33)

+ If we assume u E 0(A), then (1.34)

CA(t),

V jre

(1.35)

V£

0,

j.

Hence

(1.36)

2

+C

Let us set A(j) = w(23), h = Then the two terms on the right side of (1.36) w(t)t(l_b) dl, respectively. We have the are equivalent to h w(t)C2 dt and to following.

PROPosITIoN 1.3. Assume and A(j) = then

e

Li 0, and P(0) = .

Cb(h)hf w(t)dtfhw(t)dl If Li >0, we can take (1.39)

ah(h)=w(h)+hf

0.

If it e C(A)

If b> 0

\ for some fi e (0,1), we can take

and

(1.40)

Cb(h) = w(h).

Note that, if b = 0, we require EA(j) C for the right side of (1.38) to be finite. If b> 0, this restriction is not required. Also note that all the estimates above work if we consider instead (141)

==

A(e)B(Iej),

Ae

si_b_l_m(R)

B

P(0) =

0.

This will be useful later on.

Note that the same argument as used in the proof of Proposition 1.1 shows that, if P E and P(0) = 0, then, when w is a modulus of continuity,

ue

.

w(h).

We will review that argument in more detail to treat hypotheses closer to those of Proposition 1.3. If P 6 and j' P(x) dx 0, then

=

- y) - u(x)]

h

(1.42)

Cw(h) + P(y)j

if

dt,

F(t). If we assume that P(y)j P(hD)u(x))
(1.43)

then dt.

Estimating the last term by such a device as used in (1.16), we have the following. PROPoSITION 1.4. Assume that P(0) =

0

and

(1.44)

If a > 0, then the estimate (1.43) holds. If also then

(1.45)

If

the a

=

u

\ for some fl

(0.a),

w(h).

then P satisfies

(1.44),

other hand, if satisfies (1.41), with b 1. In such a case, the estimate (1.43) on

for all a> 0, > 0. then it

as long as b

> 0. On

satisfies (1.44), with

has the same strength the estimate (1.39), which is valid under the weaker hypothesis that n 6 We next give a relationship between the modulus of continuity and an estimate, which is slightly closer to an equivalence than that given in Propositions as

1.1—1.2.

9

1. SPACES OF CONIINUOUS

PROPOSITION 1.5. Let

4'0(e)

be as in (1.2,). Then, for a modulus of continuity

uE

(1.46)

On the other hand, given u e L9W'), : u E Cc,

(1.47)

a(h)

i0h

=

PROOF. The result (1.46) follows from Proposition 1.4, applied to P(C) = To get (1.47), we argue as follows. Note that

Since

=

=

(1.48)

is supported on the shell 1

2, it follows that

(1.49)

Hence the hypothesis in (1.47) implies that (1.50)

Now, if x1 e IRTh and lxi

— x21

u(xi) — u(x2) (1.51)

=

h,

let us write

[u(xi) — 4'o(hI))u(xi)]

+ [Wo(hD)u(xi) — 'Fo(hD)u(x2)]

+

[Wo(hD)u(x2) —

and deduce that

(1.52)

which gives (1.47), upon absorbing w(h).

The conclusion in (1.47) is stronger than the conclusion of Proposition 1.2, when the second term on the right side of (1.14) is dominant. This happens when w(h) is given by (1.26), i.e., (1.53)

The analysis of the last term in (1.14), in this case, is done in (1.28), while the analysis of the other term was begun in (1.29). It is time to complete that analysis, i.e., to evaluate asymptotically (1.54)

frseTdr,

This is equal to

f t3e_At dt

(1.55)

dt +

tse_At dt.

The first integral on the right is

To

e

23f e_At dt =

(1.56)

estimate the last integral in (1.55), note that t8+2e_Jtt

enough,

\ on

[2,oc)

for A large

so this is

(1.57)

=

S

when A is

23e2A, for large A, so,

large enough. Hence (1.54) is 5

when w(h)

the form (1.53), we see that fr'w(t)dt S Cw(h). COROLLARY 1.6.

If

U6

(1.58)

w(h) has the form (153,), for some s

u

S

We mention that the s = Note

1

that

(1.59)

ju(x)

0, then, given

C4-J.

case of this played a role in [BC].

Wo(h'y){u(x) -n(x

- Wo(hD)u(x)I =

and this can be estimated in the same fashion reasoning leading to (1.45) applies, so we have (1.60)

has

as (1.42). Since 'I's 6

S(1R75, the

u

w(h). S (Compare [CM], Lemma 3, p. 43.) We can summarize some of the results above as follows. If we denote by the space of functions satisfying (1.61)

n

jJu —

S Cw(h),

then, with A(j) = w(23), we have (1.62)

When

c

c

c

a(h)

=

f

¶idt + hf w(t)d

a(h)

4/i), all these spaces are equal. The identity of 0° and c["'l when (0 <m < 1) is the content of classical resuits of Jackson and Bernstein. It is useful to separate some reverse inclusions; we have

4/i) =

c(A) C CIWJ

cC°,

n(h) =w(h)+hf w(t)dt

1. SPACES OF CONTINUOUS FUNCTIONS

11

The first inclusion in (1.63) follows by summing over k j, while the second follows from an examination of (1.12). Note that, in light of (1.15) (1.16), for some we have CH = CLQ whenever (0, 1). Another function space is suggested by the result of Proposition 1.5. We denote by c{w) the space of functions satisfying

\

e

(1.64)

c

c{w}

while,

by Proposition 1.3,

(1.66)

c

c

c(A)

= n

dt,

a(h) = w(h) + h

/1 w(t)

dt.

\

c{w) provided in particular, again by (1.15)—(1.16), C(A) for some (0,1). We collect the inclusions given above in the following diagram. Here, given we take A(j) = w(22) and a(h) are as in (1.62)—(1.63). cw

4CM

I

I

CLwl

c(A)

I

I

I I

follow by substituting one modulus for C C another in (1 .62)—(1 .63) and manipulating some double integrals.

We present the following examples in which the various function spaces examined above can be seen to coincide or differ, recalling several calculations done above:

w(h) = h(log We

s

0

=

=

make note of some desirable special properties one might require of a mod-

ulus of continuity. One is that w(h) be concave. All the examples given above have that property. Concavity implies slow variation; in fact, if u is concave, then w(2h) 2w(h). The following result is useful. PROPOSITiON 1.7. If w is a concave modulus of continuity, then (1.67)

Cu'.

1. OF'EI-LAIORS WITH MILDLY REGULAR SYMBOLS

PROOF.

and

_t21
ko(xi) — p(x2H = w(ti) — t21) <w(Ixi — the

first inequality by concavity.

To state the next requirement we might impose, set uio(h)

of order k if

= property.

forj = 1,2,3

=

ductively, set

w(h) and,

is concave and

for all j

= (log(1/h))_S(s

k.

Note that both

> 0, 0 < h 1/2) < r < 1) and w(h) The following is a useful complement to Proposition 1.7.

PRoPOSITION

1.8.

If w

in-

We will say that w is tame

have this

modulus of continuity which is tame of order 1,

is a

then 4'(x) =

(1.68)

V4

w(ixl)ixl

PROOF. We have

=&(lxl)Ho(x),

(1.69)

where &(h) = wi(h) +w(h) and Ho(x) =

x/ixi.

Our hypothesis implies that & is

concave and &(h)
— V'1(x2) = [&Uxii) —&Gx20]Ho(xi)

+&(1x21)[Ho(xi) — The first term on the right is estimated as in (1.67). It remains to show that

(1.71) To

&(lx2l) jH0(xi)

—

Ho(x2)l

— X2l).

see this, note that

C

Ho(x2)l

Ho(xi) —

x1 — Xii

so

it

+

1X21

suffices to show that

fi',ci\

-

XrX2

<

-

— X21).

Xi + X2

Say lxi — x21 = h, 1x21 = t. If t < h, this follows because the left side of (1.72) is &(t), which is then &(h). On the other hand, if t h, the estimate (1.72)

follows from the estimate (1.73)

&(t) <

which follows from the concavity of &.

for

This

t

proves (1.68).

is that, if iu(h) = o(h), then the space = w(23)) might not he trivial. For example, if w(h)

Note

h,

(with = r> 1, then CCX) = C:

trivial, but

with

i. SPACES OF CONTiNUOUS FUNCTIONS

hi

is the Zygmund space. Also, C1"'1 might not be trivial. We will encounter a number of occasions to use such spaces. It is desirable to record the simple fact that the spaces defined above are independent of the choice of the Littlewood-Paley partition of unity used in the definition. Thus, suppose {pj } is another such partititon of unity, and set 'I's = Then

-

there exist M, N such that, whenever j M,

(1.74)

£-j- N

if u

Hence,

as in the definition (1.4), we have j+N

x:

(1.75)

j

Note that

tF} is

space, including

a uniformly bounded family of operators on each L'

so

j+N

j

Similarly,

t}

is uniformly bounded on each

space, including

L°°(IR"), and hence (1.77)

IIV'1'a(D)uIIt=

Similarly, we have (1.78)

II('

We

—

—

and Ct"'1 under a can also establish the invariance of the spaces that coincides with a rotation outside some compact

diffeomorphism F 1EV' —p

set. Seeing this comes down to observing properties of the operators çoj (x, D) by the action of the and and obtained by conjugating diffeomorphism F. The symbols have the form

where

tity

+...,

+...,

(1.79)

the remainder terms have relatively milder behavior. To estimate the quanwe can write j+N

(1.80)

where, for j M, (1.81)

A(x,

is compactly supported in

and

we have, for the partial Fourier

transform, (1.82)

K,

with

independent of j.

see that similar

Since

j E

the integral kernel of

is uniformly bounded on

analysis of the remainder terms in (1.80), we obtain

D) is Ag(x, x

— y), we

Thus, making a


(1.83)

—

N

The invariance of under the action of such a diffeomorphism follows. In a similar fashion one establishes the invariance of and Ck}(IEr). It is then straightforward to define the spaces (M), CH (M), and (M), whenever M is a compact manifold. We need only remark that these spaces are easily shown to be invariant under multiplication by elements of (if course, is also well defined in such a case. We next establish the following characterization of the spaces C(A)(M), etc. PROPOSITION 1.9. Let M be a compact Riemannian manifold (with smooth metric tensor), with Laplace operator and let A = e OPS'(M). Let w and A be as before. Given n E D'(M), (1.84)

n


(1.85)

n

for every

smooth

S C(X) vector field X

on

M.

PROOF. These results follow from analyses of 1/5 (A) and W3 (A) as families of pseudodifferential operators. As shown in Chapter 12 of [Ti], we have, in local coordinates, (1.87)

where, with (1.88)

i/5(A) = wj(x, D),

= wj(x,e)

ri(x, E) (and subsequent terms in the

+23r1(x,2ie) +...

expansion) having compact support in

and vanishing for /5 near 0. Given U C0')(M), to verify the right side of (1.84), we use a partition of unity to write n as a sum of elements of (M)

patches. Thus it If we denote by some N.

with supports on coordinate

suffices to estimate

the principal

term

for cpj(x,/5) satisfying (1.88). on the right side of (1.88), we have, for

H-N

>

£=j -N

1. SPACES UY uurcllNtJoIj5 FtJNUTIONS

15

here, the same sort of arguments as in the proof of (1.83) (including an show that argument that {wij(x, D) j t} is uniformly bounded on :

(1.90) hence giving half of (1.84).

For one approach to the other half of (1.84), we could note that, for some N, the principal part of the symbol of is equal to 1 on a neighborhood N and that, by (1.87)—(1.88), of the support of j-f-N

(1.91)

>

with (1.92) S

< 1,

(1.93)

5;

estimates establish the other half of (1.84), at least when tends to 0 no faster than To treat more general A(j), one could take a closer look at the remainder terms in (1.88) and (1.91), and establish These

(1.94)

C

j-FN

An alternative, and perhaps more transparent approach can be given via an analysis

using techniques of 1ST] and and of families of operators such as [T6]. There one associates to a family p(h, x, D) of operators on M, an operator

on M x (1.95)

P#(u(x)euJCO)

p(k_l,x,D)u(x)etkO,

k

Z\0,

= ij51(hA), h == we associate settingit equalto 0 when k = 0. Thus, then, for N OPS°(M x S'). If the operator = equal to 1 on a conic has total symbol (mod large enough, B# = neighborhood in T*(M x S1) \ 0 of the essential support of the image of iJ'1(hD) under this transformation. This in turn implies the identity (1.91), with the better estimate (1.96)

Similar arguments apply to prove (1.85) and (1.86). We note parenthetically that this approach was used in §4 of [T6] specifically to study "excellent approximate identities," of which the family W3(A) is a paradigmatic example. As we have seen, elements of C(A) are continuous provided > AU)
note of the following, which is closely related to a differentiability result of [WZ], as improved by [JN]. PROPOSITION 1.10. We have

coo

(1.97)

c bmo.

and hiebel-Lizorkin spaces

We bring in Besov spaces with norms defined by

IifHk. = E (1.98)

I

where Øj is as in (1.2), and we allow any is supported in a shell ei in S?0(RTh) and

as

long as there are definite bounds

2. The

hypothesis EMi)2 Coo yields (1.99)

C

the latter inclusion by inspection of (1.98), noting that

lfI!h2 On the other hand, it is a consequence of work of [FS2] that

= bmo(IR");

(1.100)

see ITri2], p. 93 for a proof. This establishes (1.97). We will deal with bmo in more detail in §2. We now consider an estimate, complementary to Proposition 12, which will be of use in the study of operators on in §2.

PROPOSITION 1.11. If cr(h) is a modulus of continuity and A(j) \ 0, both slowly varying, and

c(h)ba(bh), then, with 1bk(e) as in (1.2), (1.102) with C =

111

independent of k.

b' AND DM0

2. OPERATOR ESTIMATES ON

PROOF. We (1.103)

iei

c02k, then

IlfkUGa S

(1.104)

We

has support in c1r2"

are stating that, if /k

17

-

S

2—k We

need only treat

Tyfk

(1.105)

set p

—

Ilk

= 2_kz

and write

fk = 42(D)fk

where

=

(1.106)

=

1)p(e),

and (1.107)

Then

1

'p

(1.104) holds if

we have

Izi Si.

S

(1.108)

On

the other hand it


is clear from (1.106) that

Ill

5

(1.109) Now

cc0.

for

Izi

51.

(1.109) implies (1.108) under the hypothesis (1.101).

2. Operator estimates on U,

and bmo

symbols p(x, e) which are well behaved in e but have limited regularity in x. Various specific symbol classes considered will be defined below.

We establish here some U bounds for operators p(x, D)

with

One tool in the analysis, used here as in other places, such as [CM], [Boar], [Mal]. and [T2], is the following result from Littlewood-Paley theory. Let {øk} be as in (1.2). LEMMA 2.1. Let

(2.2)

S'(R") be such that, for some

suppfkc{e:Bo.2k

(2.1) Say Jo

fk

has

S

compact support. Then, forp

s

iei


.2k+1},

we have

> 0,

k1.

SYMBOLS

If fk = lPk(D)f, then the converse inequality also holds.

The next ingredient (also used in {CM] and subsequent papers mentioned above) is a symbol decomposition. We begin with the Littlewood-Paley partition of unity (1.2): (2.3)

and with =

(2.4)

= Now,


let us take a basis of

of the form

= Then

we write (for j

1)

(2.5)

where

0

has support

<2

on 1/2 <

(2.6) where

for

k

=

and is 1 on supp

with an analogous decomposition for p0(x, positions into (2.4) and summing over j, we obtain p(x, decreasing sequence of "elementary symbols." By definition, an elementary symbol is of the form

Inserting these decomas

a sum of a rapidly

= pk is supported on

2k

and bounded in

S?,

in

fact

c°k(e)

= wi(2_k+lC,

2, and Qk(x) satisfies

(2.7)

IQk(x)I

A1,

5;

fi(k,i),

for j k —3. The array /3(k, j) will have properties which will be described below. In (2.7), as in (2.3), is the Littlewood-Paley partition of unity (1.2), so is supported on

'S-'

Our next task is to estimate the U-operator norm of q(x, D) when such an elementary symbol. Set = (D)Qk(x), and set (2.8)

q(x,C)=>{)Qkj(x)+ k

j=O

E

q(x,

is

>

We will perform separate estimates of these three pieces. In two of will apply Lemma 2.1, with fk = 50k(D)f.

these cases, we

23 on the spectrum of

First we estimate q1(x,D)f. By Lemma 2.1, since

00

k-4

k=-4 j—U 00

To estimate q2(x,D)f,

1/2

set k+3

k+3

8(k)= and

assume

(2.11)

E/3(k)


k>O

Then

xi: (2.12)

we again apply Lemma 2.1, with fk replaced by gj

To estimate

Qkjfk, to

obtain 00

j-4 k=O

(2.13)

212 /

cc

}

j4 k=O Now make the hypothesis that i —4

=

(2.14)

defines a bounded linear operator on £2: (2.15)

lThaU)12 < iO

We deduce from (2.13) that (2.16)

IfiI2}

1/2 1LP

S

sYMBOLS

yvi In W111.LJLT

Now

can

p(x,C)

be written as a sum

=

(2.17)

where (cu) is rapidly decreasing and

=

(2.18)

all satisfy (2.7), with fixed A1 and fi, and S 2} and Iccrllce polynomially bounded in ji, for each £, provided that,

where

c

in the sum (2.4), (2.19) for

C0fl(k,j)2

all a 0 and j k

—

3.

For this, it suffices to have

S

(2.20)

5 We have the following conclusion.

for all a 0 and 141

PRoposiTioN 2.2. Assume p(x, 4) satisfies (220), where fl(k,j) satisfies


(2.21)

—3 5i 53,

and (2.22)

T3a(j)

=

£2.

£2

T3

Then

p(x, D) L'(R't) —* L"(W),

(2.23)

In the particular case

1

that j3(log2(4),j) =

(2.24)

the hypothesis (2.20) is implied by (2.25)

IDp(x,4)l 5 A

and

tc do not vary too fast, the hypothesis (2.21) follows from

(2.26)

I


while (2.22) follows from (2.27)

= A(j)Ek(2k)a(k)

Tp

£2

£2.

2. OPERATOR LIbLIMAILS

UN L, V.

Note that the bound (2.22) on Tp holds provided

(2.28)

Efi(k,j)SAa, k=O

In particular, (2.27) holds provided (2.29)

k(2k)

A3A(j)',

On the other hand, assuming A(j)

A(j)

\

j=k+1 and

tc(j) /,

we have

\(j) >,c(2k)

(2.30)

and (2.31)

K(2k) >A(j) k-Fl

j=k+1

50 (2.29) follows from (2.26). More generally, suppose (2.32)

/3(k,j) is

\

in j, Vkand /

ink,Vj.

Then

(2.33)

5; Efl(k,k),

and

fl(j,j).

(2.34)

j=k+l

j=k+l

Thus we have the following.

PROPOSITION 2.3. Assume p(x,e) satisfies (2.20). If/3(k,j) satisfies (2.32) and

> fl(k+3,k) coo,

(2.35)

k

then (2.36)

p(x, D) :

17(R7t)

—* 17(R7t),

In particular, this holds if p(x,e) satisfies

\, tc(i) 7, and

(2.37)

provided k(2j)

Gtz(j).

1
C

(2.25), with

Coo,

Let us look at some examples. First, picking A(j) (2.38)

as

in (1.13), fix r > 0 and let

AU)

The hypothesis (2.25) is equivalent to

(2.39)

e

2_ir2fro .c cc, 0 and if 0 C S < 1. We recover the well known result that

in this case. Now (2.37) holds if and only if 6

and only

(2.40)

p(x,

for 1

E

c:s?6(Ir)

provided r > 0,

p(x, I))

i.e., if

—*

8 E [0, 1).

On the other hand, if we pick

(2.41)

then

A(j)

= 2-jr

=

0

the hypothesis (2.25) implies that p(x.

(2.42)

(log2 (c)) is just a little better behaved than

p(x,C) E

provided s > 0. In fact, to check (2.37), note that case; this

on

is finite if and

p(x, D) if p(x,

s>1.

satisfies

(2.25), with A, it given

Next) picking A(j) as in (1.15), fix

(2.43)

El5

only if s> 1. Consequently we

.\(j)

s,r> 0

=

in

this

have U' operator bounds by (2.41), provided r > 0 and

and let

=

> r + 1. The estimates involved in establishing Proposition 2.3 suggest that we pay special attention to certain spaces of symbols, of the following sort. Given a Banach space X of continuous functions on R", define X87'0 to consist of symbols p(x, such that Then the hypothesis (2.37) is equivalent to s

(2.44) Spaces X of particular interest include C ft > a symbol is decomposed into simple symbols, and written

as in

(2.6), we have

E

and

If such

such a simple symbol q(x, 0

HQkiJx S

is

Cr.

In particular, E C(A)s?O

we have Ø(k,i) = A(j) in (2.7). Hence, /3(k,j) A(j), we obtain the following. i.e.,

S CA(j), if

we apply Proposition 2.3 to the case

AND DM0

2. OPERATOR ESTIMATES ON

23

PROPOSITION 2.4. 17 Ni) \ and EA(j) <no, then (2.45)

CWS?o, then p(x,D)

In [CM] it is established that, if p(x, C)

< no,

long as

here.

where .\(j)

= w(22).

interest for the

2,5. Let )t(j) =

cHs10

p(x, C)

PROOF. The estimate q(x,

by

though

= cH in cases of greatest

have

;

<no. Then

and assume

p(x, D)

—*

argument here is essentially the same as in [CM]. It suffices to

D)f when q(x, C)

is an elementary symbol (2.6). We decompose q(x, C)

two pieces; instead of (2.8) we write q(x, C)

(2.47)

where

is

=

(x, C) + q2(x,

as in (2.8), and

=

(2.48)

The

as

application of this result.

PROPOSITION

(2.46)

L" —f

\Ve recall the proof of their result

To emphasize the mechanism of their proof we replace

by the remark made after (1.63) we actually

into

1

—p f7(W2),

p(x,C)

Bk(x) =

estimate (2.9) still applies to qi (x, D)f.

Now,

with fk

-

estimate

D)f,


q(x,C)

(2.49)

(I

note

that

CA(k). p/2,

= cPk(D)f, write .

(2.50)

where

and using

(>(Bk(x)12)(E(fk(x)(2) =

A(k)2

<no. Raising

both sides of (2.50) to the power

integrating

the converse inequality to (2.2), we obtain

(2.51)

1u12(x,D)f

CA!!f(Itv.

Comparing Propositions 2.4 and 2.5, we see that neither result contains the other. Some examples for which Proposition 2.4 has an advantage over Proposition 2.5 arise in §16. It is also useful to consider the following more general type of symbol classes. If X is a Banach space of continuous functions on 1W1, and ic is slowly increasing (so ic(2r) Ck(r)), we say

p(x,C) C XSrk) (252)

and

trreactrcrfVflflTrMrcDTYrTtFntJLAR SYMBOLS

Then the part of Proposition 2.3 to which (2.37) applies can be phrased as follows. PROPOSITION 2.6. We have p(x, D)

p(x,e) E

(2.53)

provided that A \, /

are

—'

1
L"(IW'),

slowly varying and

(2.54)

>A(j)Ic(23)

<00.

If we use X = C1"'1, we can establish the following extension of Proposition 2.5. A similar result, but with X = C"', was given in [Ma37. PROPOSITION 2.7. We have

1

p(x, D) L"(IIU') —' L"(IRT5,

p(x,E) E

(2.55)

provided


(2.56)

PaooF. Set A(j) = w(23). As in the proof of Proposition 2.5, we consider an make the decomposition (2.47)—(2.48), and note that elementary symbol q(x, the bound on qi(x, D) is already done. D), note that As for the bound on e

(2.57)

5;

5;

The rest of the proof follows

and then (2.50) holds, with A2 = as in the proof of Proposition 2.5.

We remark that you can also improve the estimate on q2(x,D)f in (2.12), by making an argument like that above, but there is no analogous improvement of the estimate on q3(x,D)f in (2.31)—(2.16), so one does not obtain by such reasoning an improvement of Proposition 2.2. we now consider Having analyzed a number of classes of operators on There are several approaches to the definition of their action on Hardy spaces. Hardy spaces. Here is one; set (2.58)

eF,t>O},

wherecotfr) = (2.59)

This

and

F = {'p e

: w(x) = 0 for xI

1,

IIVcQIILOC

5;

1}.

is called the grand maximal function of f. Then we define

(2.60)

551(r) = {f e L'(W') : çf e L2(llr)}.

A related but slightly larger space is I,1(r), defined as follows. Set (2.61)

(gbf)(x)

*-f(x)j WEF,0
2. OPERATOR ESTiMATES ON 1)', tj', AND BMO

25

and define (2.62)

b'(R") =

{f E

:

gbf

Ll(lRnl)}.

A great deal of information about hardy spaces is given in [S3J. Brief introductions can be found in [Sem] and in Chapter 13 of {T5J. Here we concentrate on which has the desirable localization property: (2.63)

aECC(Rfl), fE

E

To see this, note that for all

* (af)(x)

(2.64)

so it

suffices

to verify the

v(x)

(2.65)


provided f

e F,

-

*

f

f(x)I

membership in

a(t)

=

dy,

of

f f(y)I dy 5; f

- vi)

dy,

B,(x)

—

where x(x) is the characteristic function of condition on cr(h). (We may as well assume An alternative characterization of b'

5; 1}, which is clear given the Dm1

\.)

which will be of use here, is the

following:

Iflki

(2.66)

k=O

as in (1.2). Furthermore, in the setting of Lemma 2.1, with fk satisfying

with (2.1),

1/2

(2.67)

Li

5;

k=O

The can

results (2.66)—(2.67) are established in §2.5 of [Tri2]. Using these results, we

check that (2.9), (2.12), and (2.16) have analogues of the form <

(2.68)

Cjifiti,

D)fj under

b'

b'

the hypotheses (2.11) and (2.15). Consequently the propositions above ex-

tend, as follows:

PRoPosiTioN 2.8. Under the hypotheses of Propositions 2.2, 2.3, we also have p(x,D) : In particular —' (2.69)

p(x,e)

e

provided E A(j) <so,

and

(2.70)

E

—÷

:

more generally : bi(lftn)

L'(llr),

2.4,

or 2.6,

provided A \,

tc

/

vviiri IVILL4JL,T LtIiAIULAK 5YMUL)L5

slowly varying and

are

(2.71)

EA(j)K(23)
Furthermore,

(2.72)

EEBS?1

Here consists of symbols p(x, whose partial Fourier transforms ji(ij, supported in ijj p for further discussion of this symbol class (which will play an increasingly important role later in this work). The result (2.72) follows because when an element of is expanded into elementary symbols, only symbols of the type qi (x, arise. From (2.72) one obtains are

p(x,e) e

(2.73)

ä<

b'(r)

1

Examining the proofs of Propositions 2.5 and 2.7, we have the following, the first part of which was established in [CM] (with b' replaced by PROPOSITioN 2.9. If E

then E

general conditions under which P:

—*

This will involve a modification of (2.12). We have 1q2(x,

> k

(2.76)

where

/3#(k)

is the b'-multiplier norm of Qr:

(2.77)

.(

As shown in (2.63), we have

(2.78) given

fl#(k)

any u(h) satisfying the Dini condition. Taking jz(j)

(1.102), we have (assuming (1.101) holds) (2.79)

fl#(k)

u(2i)

and using

S

This gives the following result, proven by taking j4k) = A(k)1/2 and j4k) = 1/2 respectively:

2. OPERATOR ESTIMATES ON

b' AND DM0

27

PROPOSITION 2.10. We have

p(x,D) :

(2.80)

provided that

e

(2.81)

EAU)'/2 <

or more generally provided

p(x,e) e

(2.82)

Now we bring in the John-Nirenberg space BMO(IftTh), introduced in [JN], with semi-norm given by

(2.83)

MIMBMO

7

-

V(Q)

dx,

the sup taken over all n-dimensional cubes Q in r, where V(Q) is the volume of Q and fQ is the mean value of over Q. Actually, we concentrate on the space

bmo(r), with norm (2.84)

+

where 'Po E

satisfies

0.

The fundamental duality (2.85)

=

was established in [FS2J. The parallel result (2.86)

= bmo(RT')

was proven in [Gol}. Exploiting this duality, we can draw an immediate conclusion from (2.73):

P E OPS?5, a <

(2.87)

P : bmo(RTh)

bino(W1).

However most of the operator classes dealt with in Propositions 2.8—2.9 are not closed under the operation of passing to the adjoint. To get bounds on bmo for such operators we follow an idea used in [Mal}, in a related setting, and obtain estimates on the adjoints of these operators, acting on (Rn) To do this, we will use the p = 1, q = 2 case of the following vector Fourier multiplier result, established in §2.4.9 of ['1112]. LEMMA 2.11. Assume

I

= Mk(2C4fl, and assume

p cc

c'o,

1 q cc oo. Let Mk(e)

Then

CA provided (2.90)

SuPPIk C


be

given, set

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

28

We

= 1,2,3. To begin, we look at

look at q1(x,D)*f, j

qi(x,D)*f

(2.91)

= kO

Since

is

j-=O

c2k—3, we

supported in iCi

can

write k+5

(2.92)

fk k>O

>

We then have

(2.93)

C

the

first inequality by (2.65), the second by Lemma 2.11, the third under the hy-

pothesis (2.94)

as

in (2.7), and the fourth by (2.64).

Next we look at k+3

(2.95)

= >

kO

j=k—3

This time, since Qr(e) is supported in (2.96)

q2(x, D)*f =

c2k+4, we can write Fk = Wk+5(D)f.

>

k>O

We then have cc

(2.97)

k=o

2. OPERATOR ESTIMATES ON

the

i,1, AND BMO

29

first inequality by (2.65), the second by Lemma 2.11, and the third by the

definition of /3(k) in (2.10). Using

sup

(2.98)

we have FkI k

(2.99)

2

1

/ the

last inequality being essentially a consequence of (2.61)—(2.62), since Fk = * f with W0

8(Rtm).

We now look at

E

(2.100)

k=O j=k-t-4

Using the support properties of

and reversing the order of summation, we can

write cc j—4

fj= >

(2.101) j=4k=-O

A crude approach to (2.100) yields, with

E

ik+4 the estimates (2.102) and

(2.103) given

any cr(h)

From

satisfying the Dini condition.

(2.102) plus the estimates on qi(x,D)*f

PRoPosiTioN

2.12.

and q2(x,D)*f, we have:

Assume

(2.104)

Coo,

E

or more generally

(2105)

Coo.

E

Then (2.106)

p(x,D)* (j'(R't) —÷

p(x,D)

L°°(RTh) —÷ bmo(RTh).

CTWER7cFtJfCS WFFH MILDLY REGULAR SYMBOLS

PrtooF. Note that the hypothesis in (2.105) implies for q(x, D) (2.107)

-

1(1

Cw(21),c(2k)

so the sum in (2.102) is finite. F1'om (2.103) and the aforementioned estimates on we obtain:

(x, D)*f and q2(x, D)*f,

PRopOSiTION 2.13. Assnme

e

(2.108)

C[w],

or more generally (2.109)

< cc.

p(x,e) E

Then (2.110)

p(x, D)

p(x, D)

:

—÷

bmo(RTh).

Furthermore, :

p(x,

PROOF. To

\

use (2.103), we

whenever

D):

—

recall from the comment following (1.63) that

=

for some /9 e (0, 1), which we may as well assume here.

It is readily verified that, if &(h) is another modulus of continuity, (2.112)

RI

Ca(2_k)IIQI(craa].

—

Taking cv(h) = c(h) = w(h)V2 shows that, under hypothesis (2.108), the sum in

(2.103) is finite.

To treat the hypothesis (2.109), note that this implies hence, by (2.112), S CU(2_lt)k(2k)

(2.113)

Ck(2k), and

&(h)u(h) = w(h).

So take (2.114)

a(h) =

c(h) = [w(h)k(1/h)]"2.

Finally, the result (2.111) follows from the analysis of qi(x, D)f in (2.93).

AND SYMBOL SMOOTHING

3. sYMBOL

3. Symbol classes and symbol smoothing As in (2.44), if E A(j) < oo we define x such that p(x,e) on

to

5; A(Y(erHfl,

(3.1)

consist of functions

5;

is slowly increasing, we define More generally, as in (2.52), if to consist of functions p(x, such that

Tn

II Dp(.,

5;

(3.2)

5; C0ec(e)

if and only if the hypothesis (2.25) holds. particular, E into two pieces: We will find it convenient to split

+ji(x,e),

(3.3)

with

=

(3.4) Here,

is

\0

to 1 for 5; 1. We take (3.3) depend on the choice of is easy to verify the following.

LEMMA 3.1. Fore

E

— cc.

as j

as

is assumed to be equal

and 'P0 E

the partition of unity (1.2),

The properties of the decomposition

we will see below. Let us set

It

(0,1/2],

(3.5)

5;

Also, ifp(j)A(j), andA(j)/pt(j) \, then

hf

(3.6)

S Cy(e)h(fIhc(A,

—

with 7(E)

(3.7)

= A(log2 ji (log2

fl

satisfies (3.1), it follows that

Now, if

on supp

(3.8) If we pick 6 e

(0,1] and cE (0,oo)

and set

=

(3.9)

so

on supp

(3.10) Furthermore, if (3.11)

this implies i.e.,

5;

satisfies (3.1), 5;

then, by (3.6),

on supp

e

vviin

If

rvULVbY fl±AJIJLAIt SYMBOLS

is given by (3.9), then

ID;pb(.,e)1lC(u)

(3.12)

where A(8 (3.13)

7o(e)

,48

More generally, if p(x,C) satisfies (3.2) and 6j

S

(3.14)

If

=

then we have (3.10) for

we have

p#(x,e), and for

K(C)

< cx, then D?9(x,e)I is bounded by a constant times the left side of

(3.14). For a better estimate, note that Ill

ifs

S

>1

—

so, with 11(t) =

)t(j), as in (1.9), we have 5; 1l(log2

If —

or equivalently, if )t(j) = (3.15)

Ill

-

=

&(s)IIfIIc(A),

ft dt.

Hence 'c(C)

Dfr(x,C)I 5;

where (3.17)

A6(e)

=

These estimates suggest making a further generalization of the symbol classes defined by (3.1)—(3.2). Namely, if is a positive, slowly varying function, and X is one of our favorite function spaces, we say E

if and (3.19)

only if

5;

Estimates parallel to those done above prove the following.

a

3 SYMBOL CLASSES AND SYMBOL SMOOTHING

33

Then, in the decomposition

PROPOSITION 3.2. Assume p(x,e) e given by (3.9), we have (3.3) -(3.4), with p#(x,e) e Sf6, i.e.,

(3.20)

and

pb(x,e) e

(3.21)

where

we can take various functions

and

(3.22)

r(e), satisfying =

'r(e)

For example, we could take

= Ao(e)k(e) (3.23)

w((e)

—

—

and

with A(j) =

6)

given by (3.15). For this to be useful, we want

_

Note that, if we take p = A, so = 1, then r(e) = in (3.23). We will see below some examples for in (3.22), and r(e) = 0

—*

oc.

which it is not desirable to use (3.23). Let us consider some examples. First, as in (2.38), fix r > 0 and take

A(j) = 2-fl,

(3.24)

with 8 e [0, 1). Then (3.20) holds (with 8 replaced by

p(j)

(3.25)

Since 11(j)

in (3.4), then

If

=

i.e., p#(x,e) e 2—is,

<s

Meanwhile, let us take r.

when A(j) is given by (3.24), we have 1

=

(3.26)

=

(er.

r s Thus, pb(x,e) E provided 0 c s 0 and 0 < 8 ccy < 1, (327)

p(x,e)

=4.

e

e

pb(x,e) e

This is a known result; cf. (1.3.21) of [T2].

For the next example, as in (2.43), fix r, s > 0 and let (3.28)

A(j) =

=

0!

-.

Take if s

>

in (3.4), so again we have (3.20), i.e.,

1, Q(j)

so ci(o

A(j), we have

p(j)

E Sf6. Note that, Hence, if we pick

with

e

—

(3.29)

Thus, if 0cc 5

cc

1, and A(j),k(e)

are given by (3.28), then

==*

e

E 5r6(lw'),

(3.30)

E

where only 0

we also

and

r

cc

s

—

are given by (3.29). For this to be satisfactory. we need not

1, which implies

—p0 as

oo and EA(j),c(2i)

want


(3.31)

in order to apply Proposition 2.6, which states that q(x, D) is bounded on for 1 cc p cc Do, provided that e C(")S?(T) and (3.31) holds. As will be shown in Proposition 5.2, this condition also implies q(x, D) But if is given by (3.29), then A(j)'r(23) so the condition (3.31) fails. We can fix this by modifying and 'r(e) as follows. Take so that assuming as above that s —

1

—

0<

< s —

r

Then replace (3.29) by

> 0.

1

—

=

(3.32)

—

Now (3.30) holds also with these new

and this symbol decomposition can be useful. It is often useful to record more information on the term Ø(x, For example, as shown in Proposition 3.1.C of (P2], if = 2-M in (3.4), then, for r > 0, (3.33)

p#(x,e)

where we say (3.34) As

s

0.

noted in (3.1.30) of [T2], E

for

E

for

flj

r.

is significant for operator calculus. In examples considered above, we have taken S cc in (3.9). As first shown in [Bon], it can be very useful to take S = 1, provided we take c sufficiently large that This

1

is supported in


SMUUlnhtNt,

3. SYMBOL ULASSLS ANt)

for some p < 1. We will call this the paradifferential symbol smoothing. We define whose partial Fourier transforms satisfy such a to consist of symbols in support condition, and we set (3.37)

= U

7<1

Applying such a symbol smoothing we have, for r > 0, pb(x,e) c

p(x,e) cs;

(3.38)

where

=

(3.39)

fl

= U

zrsr,

One desirable property of these symbol classes is that operators with symbols in these classes operate on all the proper Sobolev spaces. As shown by [Meyl], if and 1

p(x, D) : H"(W1) —p H8"(Wt),

(3.40)

Vs

lEt.

A more general result will be established in Proposition 4.7. Also, the action on C(A) is considered in Proposition 5.7. There are also useful results on compositions of such operators, which will be discussed and extended in §6. The same analysis used to obtain Proposition 3.2 also yields: PROPOSITION 3.3. Suppose that A(j) satisfies paradifferential

(3.41)

C

p(x,e)

pb(x,e)

C(A)sr0

where, as in (3.22), we can take various Junctions i,b(e) and (3.42) with

and you use the

symbol smoothing. Then

satisfying

= and

)t(j) =

.D 9iven by (3.15).

Suppose (3.43)

\

23A(j)

so C(A) C

C'.

Then

(3.41), to get, with (3.44)

cc

0,


.

1 in

= pb(x,e)

p(x,e)

provided .X(j) satisfies (3.43). Note that, by Proposition 2.3 and (2.37), if (3.43) holds,

and

=

then q(x, D) :

(3.45)

—*

1

An important example of paradifferential symbol smoothing occurs in Bony's paraproduct (3.46)

T1u

= E@"k_5Df)

36

1.

OPERATORS WITH MILDLY REGULAR SYMBOLS

which arises in a decomposition of the product: (3.47)

fri

= Tf U +

+ R(f, ri),

with

R(f,u)=Rfu=

(3.48)

j-kI<5 Clearly

E 0P135?1.

1 E

(3.49)

Also, if we write Tf = F(x, D), then, for any modulus of continuity w, with

=

we have

1e

(3.50)

is useful in analyzing products of the form a(x, D)Tf, as will be in Proposition 6.1. As for estimating the remainder, we have easily from (3.48) the following:

This observation seen

(3.51)

1

for a pair

R1 C

C

'r(e) described as follows. You can choose a slowly varying

1

and set (3.52)

= B(IeDw((EY1),

r(e)

=

Mso we have

(3.53) We

1

e

E

=

briefly mention a variant of the symbol smoothing described by (3.4) and

(3.9). We take

=

(3.54)

instead of (3.9), where c(j) /

for example

c(j) =

(3.55)

Then (3.56)

oo;

(e)1

on supp

> so

0.

instead of (3.10) we have

p(x,e) C

particular, p#(x, e with ic(e) = and (2.37), whenever p#(r,e) has this property, with rn In

V'(W2)

—*

provided

By Proposition 2.3

0, we have Ø(x,D):

y> 1.

It is useful to consider smoothing of symbc1s in C[WISr() and in in the same fashion as (3.1) and (3.19), with replaced by the space which is defined by (1.61). This is particularly natural, since (3.15) sharpens

defined

to Ill

-

.wfr).

4. OPERATOR ESTIMATES ON SOBOLEV-LIKE SPACES

37

The same arguments used to establish Propositions 3.2—3.3 yield: PROPOSITION 3.4. If you use paradifferential symbol smoothing, then (3.58)

pb(xe)

=4.

p(x,e) e

1,

Here, you can choose a slowly varying (3.59)

and set

=

If you are able to choose a slowly varying (3.60)

1 50 that

B(Iej)ic(e)w((ey')—÷o

while (3.61)

then Proposition 2.7 applies to p"(x,

smoothing of the form (3.4), for

(3.9) with

8

e (0,

6 If you use symbol 1), then you have the result (3.58)

p"(x,e), with w((O—') replaced by w((e)—6) in (3.59), i.e.,

B(Iel)k(e)w((ey5),

(3.62)

T(e)

=

We illustrate Proposition 3.4 with a family of examples parallel to (3.28)—(3.32).

Given r, s > 0, take (for h e (3.63)

(0,

1/2])

w(h) =

k(e)

and (3.64)

=

r(e) =

and also

B(2k) =

(3.65)

Then we have desirable estimates (via Proposition 2.7) on pb(x, D) whenever s—r> = 1/2. whereas in the previous analysis we needed s — r> 1. In this case, by the remark following (1.63).

4. Operator estimates on Sobolev-like spaces Let Assume

with A(0) = 1.

1 be a slowly varying monotonic function of

(4.1)

ID?A(e)I

This implies that, for all s 6 IR, (4.2) We

D?A(e)81

set

(4.3)

= {f e

L"(TRTh)

: A(D)f €

=

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

38

Parallel to Lemma 2.1, we have the following result. LEMMA 4.1. Let fk e supp J'k C

(4.4)

Say

S'(R")

be such that. for some

{e:

> 0,

k

< ei

1.

has compact support. Then, forp E (1,x), we have

Jo

00

Ct

(4.5)

If .fk =

?I'k(D)f,

the converse inequality also holds.

We can use this to parallel the analysis of §2, to produce conditions under which —* As in §2, we can reduce the problem to examining p(x, D) elementary symbols, taking the form (4.6)

= and bounded in

where S0k is supported on

<Ø(k,j),

Qk(x)I C

(4.7)

where

and Qk(x) satisfies

again

is the partition of unity (1.2). We

make the decomposition

= as

in (2.8), and do separate estimates of qj(x,

D), 1

<j

To estimate q1(x, D)f, we apply Lemma 4.1 with fk =

on the spectrum

cok(D)f.

of Qk.', we have

1/2

(4.8)

CA1

IA(D)fIlLP.

We next tackle q2(x, D)f. As in §2, set

k+3

k+3

13(k)= E /3(k,j). j=k—3

We have (4.10)

1q2(',

< >

Since

2.'

4. OPERATOR ESTIMATES ON SOBOLEV-LJKE SPACES

Since

has spectrum in iei

we

can dominate this by

k

k

(4.11)


>13(k) A2
(4.12)

k>O J —4

To estiniate q3(x, D)f, we apply Lemma 4.1 to

= E Qkjfk, to get k=O

j—4

oo

k=O

(4.13)

j=4

This time, make the hypothesis that (4.14)

A(23)tØ(k,j)A(2t'b(k)

=

defines a bounded linear operator on (4.15)

>

>

jo It follows from (4.13) that


A(j)A(23) \,

can deduce the £2-operator bound (4.15) from (4.12), i.e., from

(4.17)

This proves the following.

EA(j)tc(23)
39

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

Assume

PROPOSITION 4-2.

(e)a(4)

1

as in Proposition 2.3. Let

are

A(j),

be a slowly increasing function of

=

and satisfy (4.1). If(4.16)-(4.17)

hold, then

p(x,D)

p(x,€)

(4.18)

for 1

:

cc.

More generally, (4.18) holds if we replace (4.16)—(4.17) by the hypothesis that

(4.19)

\0,

EA(j)k#(2i)
A(j)A(23)\.

,ç. for some To restate the sufficient condition for (4.18), suppose

i7(j)\Oasj—tcc,

(4.20)

= A(2k)fl(2k),

/ cc as k

cc.

k(2k) and

Then (4.18) holds provided

>1)(j)/3(23)
(4.21) COROLLARY

4.3.

The

= A(e).

(4.22)

PROOF. cc,

fl(2k)

It

implication (4.18) is valid if


ij(j) \ 0,

,\(j) =

is a simple exercise to pick / cc, slowly, such that E

<

and then we just set ,c#(e) = Taking COROLLARY

(4.23)

p(x,e)

(cr9,

4.4.

we have:

Given

s > 0,

4? p(x,D)

: H8"(JR")

1

cc,

provided (4.24)

'ç(e) = (eYe

A(j) =

2_S317(j),

with (4.25)

r1(j) \ 0,

>


The treatment of q2(x,D)f in (4.10) is cruder than the analogous treatment in [Bour} (and also in [Mal], [T2]) for operators on Sobolev spaces for s > 0. Such a treatment, needed for an analysis of operators with symbols in Sj'\, is based on the following sort of estimate, which, as we will see, does not extend far beyond its original setting.

4. OPERATOR ESTiMATES ON SOI3OLEV-LIKE SPACES

S'(R't) be such that, for some B > 0,

LEMMA 4.5. Let 1k (4.26)

supp

Ik

c

{C:

Ic

0.

= (e)a(t) 1 is a slowly varying monotonic function of

Then, if

(4.1),

41

satisfying

we have

(4.27)

provided that also, for some A1
>A(2k)_i

(4.28)

A4A(2')',

A4A(2c).

and f.Ck

PROOF. Set

Uk

= A(D)fk.

The left side

=

we

is

00

DC

Thus, with wk

of (4.27)

need to

show that

(4.29)

In

other words, we need

I'(D) : L1)(Rtt,t?2)

(4.30)

where IkE(e) (4.31)

Ic

= 0

k
Now (4.30) holds provided (4.32)

5

>

>11

and this follows from the hypothesis (4.28).

Note that, if we set a(t) =

(4.33)

A(2t),

f a(t)

Such an

estimate holds

= (log

ci)'

(for

oft)

for o(t) =

the standard Sobolev spaces,

then (4.28) is equivalent to

but

f n(t) dt

2st,5 > 0,

not

A4a(t).

which

i.e.,

for o(t) = t',

for

large).

The following is a little sharper than Corollary 4.4.

any r >

0,

i.e.,

defines

not for

PROPOSITION

=

p(x,

(4.34)

with s > 0 and E A(j) < cc. Set

4.6. Assume A(j) Then, for 1

E

:

provided A3

(4.35)

PRooF. As usual, it suffices to consider elementary symbols, of the form treated in the proof of Proposition 4.2. The estimate (4.8) of qi(x,D)f works = in this case. To estimate q2(x, D)f, instead of using (4.10), we use the case

of Lemma 4.5 to write

C{t22ksIfkj2}'/2h

(4.36)

Finally, to estimate qa(x, D)f, we need to verify the estimate (4.15), when Ts.Ab(j) =

(4.37)

The desired bound follows if (4.35) holds.

Note that (4.35) holds when

= j-r,

= kr_i, provided r> 1. Thus,

we can replace (4.24)—(4.25) by

=

(4.38)

Also,

+

(4.38) holds when zj(j)

(e)))r_l,

A(j)

2_si

.j-r,

= 2kr, provided

2

r

r> 1, s > 0. > 0.

Then we recover

the result of [Boul that (4.39)

assuming

p(x,D)

p(x,e) e

r,s >0,

—*

1

and more generally (ZSrS, are

defined as follows: (4.40)

p(x,e) e

ç Ca(e)m_Iod, ncs I

t\

As mentioned in §3, operators with symbols in Here

is a simple extension of (3.40). For any s

(4.41)

=

{f c

and

i-i

E R, we

have special properties. can set

: A(D)8f E U(IR't)},

A(5) =

There is an obvious extension of Lemma 4.1 and then an estimation parallel to that in (4.8) establishes the following.

4. OPERATOR ESTIMATES ON SOBOLEv-LIKE SPACES

(e)a(e)

PROPOSITION 4.7. if IR, pEE (1,oo), then

43

is as in Proposition 4.2 and s E

1

—* p(x,D) : 8S?J The following stands in a similar relation to Corollary 4.4 as Proposition 2.7

(4.42)

does to Proposition 2.6, except that, for simplicity, we restrict attention to integral orders of smoothness.

PRoPOSITION 4.8. Given s E Z+, 1

p(x,

HsP(r),

p(x. D)

E

provided (4.44)

=

=

and zj(j)\O,

(4.45)

>u](j)2
PROOF. We give details for s = 1; similar arguments work for other s In this case, we need to estimate the li-norm of D3p(x, D)u, given u follows from Proposition 2.7. Pushing the The analogous estimate on p(x, D3 past p(x, D), we are left with an Li-operator norm estimate on qj (x, D), where (4.46)

and this in turn also follows from Proposition 2.7.

Moving away from symbols of type (1, 1), we have the following simple but useful result. This result actually belongs as part of Proposition 2.1.E of [T2], but it escaped our attention at the time. PROPOSITION 4.9. Assume k

t, S

[0, 1), p

(1, x). Then, for

p(x,e)

(4.47)

we have (4.48)

p(x, D) :

—(1 — S)k

<

PROOF. The only part not contained in Proposition 2.1.E of [T2] is the endpoint result, s = k. We treat explicitly the case k = 1. It suffices to take m = —1; we need to show that (4.49)

p(x,e)

Li

D3(p(x, D)u)

Li.

That p(x, Li is clear from standard results. On the other hand, if D1 falls on the coefficients of p(x,D), we need only note that (4.50)

which follows from elementary integral kernel estimates.

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

The following limiting cases are also noteworthy. Set

=

(&51)

=

(1 —

(1—

is the localized John-

where bl(RhL) is the localized Hardy space and

Nirenberg space, introduced in §2. PROPOSITION 4.10. Given S E [0,

1), we

PROOF. It

suffices to take m = —1

and

show that

D3p(x, D) : if —* if, bmo

E

(4.53)

fact,

hiP.

p(x, D) :

(4.52)

In

have also for p = 1,00,

—*

bino.

given u E [ji or bmo, we have

D)u,

D)u = p3 (x, D)u +

(4.54)

where (4.55)

fi

: if

—* if, bmo —* bino,

by Propositions 2.10 and 2.13, and (4.56)

L1

by an elementary estimate.

5. Operator estimates on spaces —* Here we want to obtain conditions under which p(x, D) : Our analysis will be somewhat parallel to those in §2 and §4. As there, it suffices to consider elementary symbols, taking the form

(5.1)

= where

2k

is supported on

and

bounded in

S [3(k,j), where

is

=

(5.3)

as

the partition of unity (1.2). Again we make the decomposition +

+

in (2.8), and do separate estimates of qj(x, D),

1 5 j 5 3.

As in (2.8). we set

= In place

of Lemma 2.1 or 4.1. in

the following:

this case we have,

essentially from the definition,

5. OPERATOR ESTIMATES ON SPACES

LEMMA 5.1. Let 1k

8'(W') be as in Lemmas 2.1 and 4.1. Then

(5.4)

A(k)1

if 1k

and

To

=

l4'k(D)f, then the inequality is reversible.

estimate

(x, D)f,

on the spectrum of Qkj, we

since

have, by

Lemma 5.1,

sup

(5.5)


sup IIQkIILOC k>U

To tackle q2(x,D)f, we take start with

and

13(k) as

given in (2.10) and (4.9), and

sup A(E)1

(5.6)

has spectrum in iei
Now (5.7)

k?O

Thus the right side of (5.6) is


>

£

(5.8)

C sup

1

kfl—N

£

If

13(k)

1 and A(k) is slowly varying, we can write

(5.9)

k12(x,D)f(C(\)

:5;

provided that, with D2
Ve0.

(5.10) k>t

Note that, if (5.11) then

13(k,j) =

the condition (5.10) becomes

(5.12)

>,c(2c)A(k)2 D2A(ifl.

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

46

Finally, as

=

J—4

has Spectrum in

k=o

C

we have, by Lemma 5.1,

sup

(5.13)


If

Consequently,

(5.14)

D)fIIccA) S

provided that, with D3
Vj.

(5.15)

If fl(k,j) is given by (5.11), this condition becomes (5.16)

If, more generally, /13(k,j) is given by

/3(k,j) = tt(j)ac(2k),

(5.17)

1i(j) S

then the condition (5.15), controlling the estimate of q3 (x, D), becomes (5.18)

Vj.

5

that, if i9(k,j) is given by (5.17), then, for the condition (5.10), controlling the estimate of q2(x, D), we have Note

(5.19) in

5

place of (5.12). We therefore have the following result.

PR0P0S1TION

5.2. Assume

If j4j)

p(x, D)

(5.20)

provided

then

—+ C(A)(IPJ1),

that, for all j 0,

(5.21)

and k<j

j

conditions that hold whenever (5.22)

S A(j),


and

/.

5. OPERATOR ESTIMATES ON SPACES C

REMARK. We can take A(j)

here, in which case

1

the

47 conclusion of (5.20)

that —p

p(x, D)

Let us consider our favorite examples, starting with

(5.23)

in

j4j)

=

which case

as

SE

10,

ri',

defined

,q2k) = 2krS as in (4.40). Then (5.22) holds as long

1). Thus, for r > 0,

p(x,D)

E

(5.24)

if S

= A(j) =

—+

6 [0, 1), which is a well known result. Let us look at the case

(5.25) Note

JL(j)

that,

=

when (5.25) holds, even

k<j

though

2

AU)

(5.22) fails, we still have

Ek(2k)A(k),i(k) = k>j

kcZj

k?j

(5.21) holds. We have

(5.27)

a

k(2k) = 2kr,

=

(5.26)

50

2r,

6 C8?1(r)

C(W') —.C(I1U'),

0
result first established by E. Stein [S2].

Next, consider (5.28)

p(j) = j—r,

k(2k) =

A(k)

r> 1.

Note that (5.22) holds if and only if

s
(s

s+rS

T8)

(5.30)

logj

C while A(j)/p(j) =

j'3. Hence the

rS<s<1-+-rS,

if

0<s—rS<

if

s—rSl,

if

s—rS>l,

1,

first part of (5.21) holds provided that, either

and 1—s+rS
or else

s=1-4-rS,

and

r>s,

s>1+rS,

and

rs.

orelse

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

48

Meanwhile, r—s

(5.31)

k>j if s > 1 — (1 — S)r. Thus the second s—

1

+ r(1

(5.21)

of

— 8)

s, i.e.,

provided r

part of (5.21) holds provided s >

0 and

8). Hence, given (5.28), the condition

1/(1 —

Proposition 5.2 holds, provided 1

and

(5.32) and

either rSs0),

(5.33)

In summary, when /2,

are given by (5.28), and (5.32)—(5.33) hold, then

p(x,e) e

(5.34)

Recall

N, and A

==#'

p(x,D)

—+

from the discussion of (2.43) that this class of operators is bounded on

for 1

provided r> 1 + i'S,

If we take tc(j) = 1 and p(j) =

of s

1+rbccs
or

A(j)

i.e.,

provided (5.32) holds, with strict

conclusion (5.34) from (5.32)—(5.33) 5.3, specialized to A(k) = k_s.

r, the

Corollary

the following consequence T, r < 1, A(k) = k5, 0 <

in (5.20), we have

Proposition 5.2. Note however that in case p(j) =

j

(with S

0) is stronger then

COROLLARY 5.3. If A(k) \ 0 is slowly varying and EA(k)
p(x,D)

p(x,e)

PROOF. The two

—+ C(A)(RTh).

parts of (5.21) are

(5.36)

D2A(j),

>A(k) cc D3, k<j

k>j

in this case. The hypotheses of the corollary imply these results, since, under these hypotheses, A(k)2 A(j) A(k). COROLLARY 5.4. is

then

If A(k) \ 0 is slowly varying and EA(k) <00,

a Banach algebra, i.e.,

(5.37)

u,v E C(A)(W1)

If we set p(k) = PROPOSITION 5.5. (5.38)

p(x,

uv E

N(2k) = 2rk, and apply Proposition 5.2, we have:

<00 and

2—rk

0,

p(x, D)

then

—+

provided (5.39)

>A(k)2Tk < D3A(j)2T3, k<j

and

>A(k)

kj

D2A(j).

5. OPERATOR ESTIMATES ON SPACES c(A)

Suppose for example that

0 <s
A(k) =

(5.40)

Then the two conditions in (5.39) take the form <

(5.41)

< k?j

k<j

Note that the first inequality in (5.41) always holds when 7(k) while the second inequality in (5.41) always holds when 7(k) \ In particular. if there exist .

(0,r) such that

\ while 282kA(k) / (for large k) then both conditions in

(5.39) hold. For example, we have the following. PROPOSITION 5.6. If 0 (5.42)

< s
p(x,e) e

p(x, D) : C(A)(RTh)

A(k) =

(5.43)

any 7

ilk.

Propositions 5.5—5.6 do not apply when A(j) decreases to zero very slowly. It is useful to note the following result, which follows directly from the analysis in (5.5):

PROPOSITION 5.7. Whenever A(j) \ 0 is slowly varying, then p(x, D) C(A)(IRut)

(5.44)

—+

Hence the conclusion of (5.44) holds if S [0,1). For another important example, note that if Tf denotes the paraproduct (3.51),

(5.45)

Tf

fE

—*

We next study mapping properties of Rf, defined by (3.48), the remainder term in the expansion (3.47) of a product into paraproducts.

is slowly varying, p(k)

PRoPoSITIoN 5.8. Assume f e C(A). If p(k) A(k), and (5.46)

then (5.47)

PROOF.

(5.48)

Rf

=

Fly (3.51) we have, with

Rf

E

1(r)

with (5.49)

B(IeI)w((e)-'),

=

1

where

so that, with

OPERATORS WITH MILDLY REGULAR SYMBOLS

1 is a slowly varying function we are free to choose. We choose it =

5(2—k)

hence

Now the analysis of R1 is controlled see that (5.47) holds as long as

=

by the analysis of q2(x,

D)

in

(5.6)—(5.12). We

Er(2k)A(k)2

and since r(22) =

1i(j)/A(j),

this is equivalent to the hypothesis (5.46).

A more direct approach to the proof of Proposition

5.8 is to write

(5.50)

i

Ii—kI3

and, using the fact that the jth term in this sum for Rjg has Fourier transform in a ball CV, we have

supported

(5.51)

—

yielding the conclusion that (5.46)

(5.47).

We mention some cases in which (5.46) holds. Assuming as always that A(j) and ji(j) are decreasing, we have (5.46) whenever

>1a(k) ccoo.

(5.52) Also,

we have (5.46) whenever CA(fl.

(5.53) k>e

This includes the classical case, A(k) = holds for (5.54)

R(k) =

r > 0, 1.4k) = 1. In addition, (5.46)

A(k) = r3,

1.

Using the expansion (3.47), i.e., (5.55)

fg = Tfg + Tqf + R1g = T1g +

we deduce the following refinement of Corollary 5.4:

+ Rgf,

5. OPERATOR ESTIMATES ON SPACES C(A)

51

COROLLARY 5,9. When the hypotheses of Proposition 5.8 hold, S

(556)

+ C min{

G(A),

}.

It is useful to consider such spaces as Ck(A), for k e Z+, consisting of u e that for IcEI S k. More generally, we can define, for r E JR. E

such

CT(A)

(5.57)

= {(1

—

A)

ii e

It follows from Proposition 5.2 that this is equivalent to the characterization above when r = k E V, and also that

P:

P E OPS%

(5.58)

It also easily follows that, if r > 0, =

(5.59) Also, we

,4,j) = 2T3A(j).

note that, if k

c Cz,

(5.60)

A(j) c oo.

if

Results so far in this section have examined operators mapping a space C0') to From these results one can deduce further results, about operators between different spaces of the type or more generally We will not formuitself.

late general results here, but will record one case that will arise in the proof of Proposition 1.1 of Chapter III. PRoPosiTioN 5.10. Suppose p(x,e)

Assume that r(e) =

E

and that

(5.61)

A(j)r(22)

\ 0,

.coo,

Then

p(x,D) C

(5.62)

p(j) =

More generally,

(5.63)

p(x,D) :Cr(7)

PROOF. If A(D), with reversed)

= E C(A)S?(Y).

implies q(x,D)

v(j) =max(pQj)7(j),A(j)). then A(D) C —÷ and p(x,D) = q(x,D) Now, Proposition 5.2 (with the roles of p and A —÷

so we have (5.62). The property (5.63)

follows similarly.

The following is a useful compactness result.

1. OPERATORS WITH MILDLY IIEGULAu

PROPOSITION 5.11. Let A(j) = w(23), 1i(j) = Assume that w and c are tame of all orders (as defined before Proposition 1.8). Furthermore, assume

u(23).

that

(5.64)

Then,

for a compact manifold M, the inclusion

PROOF.

First, it suffices to consider M =

is

Next. it

TTh.

(5.65)

suffices

compact.

to show that, if

1,

=

then B(D)

(5.66)

and A(D)

(5.67)

is compact.

Of these, (5.66) follows from Proposition 5.10 and (5.67) follows from the fact that A(D) is a norm limit of finite rank operators, under the hypothesis (5.64). We have seen that uPS?1 maps

decreases to zero at only an algebraic a greater variety of cases here. PROPOSITION 5.12. Let Then

(5.68)

PROOF. Proposition

Pe

A(j) \ 0

into

itself sometimes,

rate. We record how

but not when AU) acts on C(A) in

be slowly varying and assume E

P c(A)

c(M),

A(j)
1i(j) =

It suffices to estimate for ji = 1,2,3, 5.2. An examination of (5.5) shows that

as in the proof of

(5.69) even without the restriction that

Proposition

E A(j) <

5.7.) Parallel to (5.6)—(5.8),

Csup <

(This has

we have

>

ri

k)'f—N 1

>

/ k>!-

N

already been noted in

5. OPERATOR ESTIMATES ON SPACES c(A)

53

< C. This result is consistent with (5.68). Finally,

since q(x,

parallel to (5.13), we have

j>O

(5.71)

IIIMC(A)

U

k—O

j4

sup since

(irs?,

(5.72)

Picking r sufficiently large that 2/t8A(k)

/ for some s E (0, r), we have C2irA(j),

(5.73)

and hence (5.74) again

without the restriction that

A(j) cccxi

It is natural to sharpen Proposition 5.12, making use of symbol classes defined by

= C(A)Sr(r).

(5.75)

= A(k)'.

with

We have the following.

PROPOSITION 5.13. Let u(j), AU) \

0,

u(j) A(j), and assume


Then (5.76)

P

P:

E

provided (5.77)

PROOF. We need only revisit the estimation of q3(x,D)f. In place of (5.71), we have (5.78)

which is

C as long as (5.77) holds.

Note that (5.77) holds when (5.79)

r4k) = A(k),

Cj.

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

54

(We continue to assume p(j) is given as in (5.76).) In particular, Proposition 5.13 applies to

v(j)=\(j)=j8,

(5.80)

s>

1.

6. Products Here we develop some operator calculus for the operators arising as A#(x, 13) of a symbol with limited in the symbol decomposition A(x,e)

regularity in x. We begin with a variant of a well known analysis of products and b(x,e) e BSfl, the symbol class a(x,D)b(x,D) =p(x,D) when a(x,e) E defined

in (3.36)—(3.37). We are particularly interested in estimating the remainder

arising

in

a(x,D)b(x,D)

(6.1)

where —IckI

(6.2)

>

IaIv Proposition 6.1 below is a variant of a number of results, including results in [Bon],

[Meyl], [H3}, and [AT]. To begin the analysis, we have the formula (6.3)

=

(2r)tL

+

-

Write

= j?O

with

=

f

(6.5)

=

where the terms

in these integrands are defined as

Littlewood-Paley

H

partition of unity

1, while for j

1,

(H

-

iaIv

= Note

that

=

(6.7)

Thus,

H

p3(ij) is supported in

=

with ,\(j)

11B(x4

follows. Pick t9>

O}, such that

j

66

—y) dy,

1 and take a

is supported in

Then

we set

6. PRODUCTs

Here we allow w p

=

1, in which case

55

Furthermore,

given that b(x,e) E

1,1'

C

(6.9)

= 0 for

',

We next estimate the L1-norm of Sobolev's

pICL

Now, by a standard proof of

imbedding theorem, given K> n/2,

we have

(6.10)

=

f(t93zj), so is supported in 0. Let us use the integral formula for the remainder term in the power series expansion to write where

(x,

O3zj)

(6.11)

Since

+

— s)

C 0

on the support of

if also t9'1 <

<

then

p E (0, 1), choose 0> 1 such that

p03 c 1. This implies (6.12)

+ sO-'zj), for all s e [0, 1]. We deduce that the hypothesis for

113?a(x,Ol

ii

+

1

implies

(613) Now,

5;

for

when (6.8) and (6.13) hold, we have

(6.14) and if also (6.9) applies, we have (6.15)

5;

provided w(h) has the following property:

(6.16)

fl <

>

To estimate derivatives of

we can write

(6.17)

= 1

C

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

56

is produced just like replaced by and with b(x,e) replaced by bol hypothesis (6.12) to

with the symbol a(x,e)

Now

Thus, ifwe strengthen the

v+

for

(6.18) we

—y) is produced just like the sym-

1,

have •)IIL'

(6.19)

for 19Y1 <

Furthermore, extending (6.8), we have (6.20)

(x,

D?2

) Ii L"

Cw(

lIc(A).

Now, if (6.16) holds, then

(6.21) so,

<

>

as long as (6.9) applies, (6.19)—(6.20) yield

(6.22)

I

C

>ii:

(e)P2(&1 1131— 'vt

'Yi

These estimates lead to the following result. PRoPoSITIoN 6.1. Assume (6.23)

a(x,e) E

b(x,e)

Then (6.24)

a(x, D)b(x, D) = p(x, D)

OPsrtm.

Assume furthermore that

for

(6.25)

al

u + 1,

with P2(E) p, and that, with A(j) = (6.26)

S

We assume that either w(h) is a modulus of continuity or w that w(h) has the property (6.27)

>

Then we have (6.1)- (6.2) with

(6.28)

1. Finally, assume

6. PRODUCTs

57

and (6.28). However, some examples arise typical case in which (6.26) arises is via symbol smoothing, of any of the sorts studied in when A(j)
Typically, rn2(e) =

in in (6.26)

where 1722(C) > in, but m2(C) — r(C) < Tfl. A

(6.29)

Another (related)

case is given by b(x,D) = T1, I see (3.50). Regarding the condition (6.27), note the following sufficient condition:

forsome s€(0,v+1)

(6.30)

(6.27) holds with

In the version of Proposition 6.1 given in [AT]. w(h) = (0, oc), As another example, given s b—s

/

w(h)=llog—1

hi

'.

(6.31)

(6.27)

is

The class

v+ 1 > r.

1

for

0
2

=

holds with

not quite

h'S. and one needs

an algebra, but the following result is a useful

substitute: PROPOSITION 6.2. If

p = Pi +P2 +PIP2 < 1, then

(6.32)

and

pi(x,D)p2(x,D)

(6.33)

The result (6.32) follows easily from the identity

= for A(x,D) =pi(x,D)p2(x,D). One easy consequence of

which

will

Propositions 6.1 —6.2 is the following result of [Meyl], is as in (3.39).

be useful in the next section. Here,

PROPOSITION 6.3. If

p

S 1/3, then

[pi(x,D),p2(x,D)]

(6.34)

Regarding the hypothesis (6.25), the following describes a class of operators for which this is relevant. It consists of operators of the form

which

636

fk(x) = 'I'k_o(Eflf(x),

T1u(x) =

(6.35) give

rise to paraproducts. Note that

f

F(x,D)

OPBSrI,

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

PRoPosiTioN

with A(j) \ 0, then

1

(6.37)

In

6.4. 1ff E

other words, (6.25)

where

holds with M2(e) =

= A(log (e)).

(6.38)

PROOF.

IfQte ={e:2e
ial 1

(6.39)

D?F(x,e)

on

J=4

Hence

J—ffGiILHe)

on

1. However,

for

life-i

-

.

=

This yields (6.37) for

When c(A)

0.

Estimates for general $ are similar.

C, such a result arose in [AT].

Proposition 6.4 is useful in the analysis of Tjh(x, D). As for an analysis of a(x, D)T1, we have noted that the following version of (6.26) holds:

f

(6.42) These

llD;F(.,e)liC(A)

two analyses will be put together for a commutator estimate, in §7.

7. Commutator estimates is less In many cases, the commutator of operators with symbols in We begin our explosingular than a general operator with symbol in ration of this phenomenon with an investigation of when (7.1)

B(x,e)

A(x,e)

implies

(7.2)

for I <

{A,B]

:

—* L'2(Ir),

< oo; for short, we set A(x, D) = A, etc. To tackle this, we use the paradifferential symbol decomposition defined in §3, of the form (3.9) with S = 1 p

and c large, giving rise to (3.44). Thus, assume that (7.3)

7. COMMUTATOR ESTIMATES

59

We have

A# +Ab,

A

A#(x,C) E

Ab(x,C) e

C(A)SIO(Y),

B=B#+Bb,

)

the classes are defined by (3.39). One can take p as small as desired; make sure p 1/3, so Proposition 6.2 applies. We expand the commutator [A, B]: with r(C) = (C). Here,

{A,B] =

(7.5)

[A#,B#]+R,

with

To

analyze [A#, B#], we can use Proposition 6.3 which, together with Proposition

4.7, implies

[A#,B#] :

(7.6)

Vs E

Ift,

p E (1,oo).

Thus, to check (7.2), it remains to investigate the U-boundedness of the six terms in F?.

By Proposition 2.6 we have Ab : U(IW2) —* U(IRTh),

(7.7)

for I

Since the crucial part of hypothesis (2.37) is the same as hypothesis

= = (C). As noted in Proposition 3.3, the inclusion B#(x,C) E (7.3), when for all s E so we have implies that B# is bounded on U. indeed on

desired bounds on the first two terms in F?. Also, Bb is clearly bounded on U, so the last two terms in R are under control. It remains to look at the terms A#Bb and BbA#. Now, by Proposition 3.2, the

VsE IR,

(7.8) A#(x,C)e131811

Hence the desired U bounds follow if we establish that, for 1

B"(x,C) E

Setting

the s = I

0

0 s

—*

obtain this in turn from the implication

p(x,

D) : H8'9(R'2) —+

0

s 5 1,

and given

=

case of Corollary 4.4,

PROPOSITION 7.1. If A(x,C) E

B(x,C)

e c(A)s100,

and (7.11)

then (7.12)

I.

A(j) satisfying (7.3). Now, (7.10) follows together with Proposition 2.3 (for the case in (7.10)), and interpolation. We have the following result.

given 1

s

= B5(x,C)(C), we p(x, C) E

(7.10)

from

:

[A,B] : U(RTh) —'

1

with A(j)23 \

0

I. OPERATORS WITH MILDLY REGULAR SYMBOLS

The following is the analogous result obtained by replacing pare [Ma2] for a similar result, in the framework of C"'. PROPOSITION 0 and

7.2. Assume A(x,e)

C"']s10,

Com-

by

with

E

\

3)]2 (7.13)

Then

(7.14)

—p L"(W'),

[A, B]:

1

PROOF. We have the symbol decompositions in (7.4), in

with the difference that,

this case,

(7.15)

e

E

= (c), by (3.47)- (3.48). The result (7.6) still applies in this case, as does (7.8). For the rest of the estimates of the terms in (7.5), we need only replace the application of Proposition 2.6 in (7.7) by an application of Proposition 2.7, and to replace the application of Corollary 4.4 in (7.10) by an application of Proposition 4.8. again with

e 8Sf'1 and f e We now analyze the commutator [T1,p(x,D)], given TI'1 where T1 is the paraproduct (3.46). We see by (6.36) that f E = and if > A(j)
T1p(x,D) =a(x,D),

(7.16) with

(7.17)

a(x,e) =

where

F(x,e)p(x,e)

+ro(x,e),

ro(x,e) E

Sf7",

is the analogue of (6.25) for F(x,e), i.e., by Proposition 6.4, =

(7.18)

On the other hand, as noted in (3.50) and again in (6.42), we have

(6.36) holds for F(x,e) with m2(e) = 0. If we assume w(t)rM \ 0 for some s e (0, 1), then (6.27) holds with ii = 0, and Proposition 6.1 also gives

i.e.,

(7.19)

p(x,

D)T1 = b(x, D),

with (7.20)

where (7.21)

b(x,e) r(e)

is

= p(x,e)F(x,e)

-i-fo(x,e),

fo(x,C) e

as in (6.27). Thus, by (6.30), w(t)t

\

for some s E (0,1)

—

r0(x,e)

5f7P,

8. OPERATORS WITH SOBOLEV COEFFICIENTS

given by (7.18). We have established the as in (7.17), i.e., following. (For a variant, see Proposition 10.2.) with the same

PROPOSITION 7.3.

Let

some sE (0,1). Set A(j)

\

w be a modulus of continuity such that w(t)t 8 for Let p(x,D) E OPf3l/2SV'I. and take f E

Then (7.22)

[T1,p(x,D)] E

with

=w((e)—').

(7.23)

A special case of Proposition 7.3 is that, for 0
fEC

(7.24)

[Tf,p(x,D)]

E

The following result treats the r / 1 limit of (7.24). The implication (7.25) was established in Proposition 4.2 of [AT]. An r \ 0 limit of (7.24) will be discussed in §10. PROPOSITION 7.4. Given (7.25)

f

p(x,e) e OPB1/25rl, [T1,p(x,D)]

E Lip'

E

and (7.26)

f

PROOF. We have (7.16) (7.27)

when (7.28)

a(x,C) =

f

e

as

=

{Tf,p(x, D)] E

e

with

F(x,e)p(x,e)

+ro(x,e),

ro(x,e) e

a special case of (7.17)—(7.18). This time we have (7.19) with

b(x,e)

= p(x,e)

—

i

.

+fi(z,e),

IcL=1

and (6.27) holds with v = 1, r = 1, yielding E The results (7.25)—(7.26) now follow from the observation that

(7.29)

fE Lip'

8.

f

provided

fE

E

Operators with Sobolev coefficients

Here we record some facts about a class of operators studied in [Mal] and to consist of r > 0, 6 E [0, 1), we define

[Ma5]. Given q E (1, functions p(x, satisfying (8 1)

1. t)t'bltAlUtW WIlt1 MILDLY REGULAI{ SYMBOLS We will require

(8.2) q

The following result is contained in Theorem 2.2 of [Mal]. PROPOSITION 8.1. Let

with 8 6 [0,1), and assume (8.2)

6

holds. Then

p(x,D) :

(8.3)

provided (8.4)

We refer to [Mal] for a proof, but mention that the proof involves treating the case of an elementary symbol. We also mention {BR] for the result in the case p = q = 2; their proof is also given in §2.1 of [T2. For use in Chapter III, we record a proof of the following result on symbol smoothing. PROPOSITION 8.2. Assume that p(x, e) e symbol smoothing operation given by

with with

r > n/q. Then the

as in (3.9,), yields

p#(x,e) +pb(x,e),

(8.5) with (8.6)

p#(x,e)

pb(x,e) e

e

r — —.

PRooF. Parallel to the proof of Proposition 3.2 (or the proof of Propositions

1.3.D—1.3.E in [T2]) we use the following elementary inequalities: (8.7) (8.8)

(8.9)

The estimate (8.7) implies (8.10) which gives p#(x,e) 6

since r > n/q.

The estimate (8.8) implies

=

(8.11)

and the estimate (8.9) implies (8.12)

IDpb(x,C)l

and together (8.11) (8.12) yield

e

cö

Will-I DOUBLE SYMBOLS

9.

63

We mention that Theorem 2.2 of [Mal] also establishes Proposition 8.1 in the cases q = 1, oc and p = 1, oc. In these cases we take

H8I(IRti) = (1— A)_s/2b1(Rnt),

(8.13)

=

(1

—

Here Ij' (RTh) is the localized Hardy space and bmo(Wi) is the localized JohnNirenberg space, introduced in §2. in particular, taking p = q = oc, one has, for 0 <8 C 1, (8.14)

One

p(x,e) E

—*

:

can compare the limiting case s

1, r \

1

—(1

with

the

—

8)r <8cc r.

following result, estab-

lished in Proposition 4.10: (8.15) given

Se

[0,1). 9.

Operators with double symbols

Here we consider symbols of the form a(x, y, prescription (9.1)

Au(x)

=

ff a(x, y,

e XSf% (and Ac (9.2) where

which yield operators via the

when ,

:s;;

X is this time a Banach space of functions on R' x RI'. Similarly, as in

(2.52), we say and

e

As in (3.18)—(3.i9), we could replace m e R by a function m(e)

with appropriate

behavior.

smooth and in one of Hormander's classes e (with 0 S cc p 1) then A can be rewritten as A = p(x,D), with This reduction does not work for symbols with limited smoothness, and these operators require a separate (though somewhat parallel) treatment. We begin with some estimates, which are variants of results in [Ma4]. As is well known, if a(x, y, C

PROPOSITION 9.1. If (9.4)

e

is

then

A: U(R')

—*

1
cc

cx.

PROOF. As in §2, it suffices to consider an "elementary" double symbol: (9.5) k-_u

with for k (9.6) with

bounded in S? and supported on shells 2' (say on 1), and with S C. Make the decomposition qk(x,y) =

qki(x,y) +

qkl(x,y)

as in §2 (but acting on functions of both x and y, so "k e

Thus

A(k)=w(2'5.

SCA(k),

(9.7)

Let Q be the operator associated to q(x, y, tion resulting from (9.6) by Q = Qi + Q2. To estimate Qi, write

Qiu(x) =

(9.8)

and denote the operator decomposi-

— y)u(y) dy

We have Now the restriction on the support of 2k1 implies

(9.9)

and

gk(X) =

also that

(9.10)

f

k+3

qkl(x,y)çok(x —

Uk

E

b= k —3

is supported on a shell cC

.

2". We have

* IUk((x) S CMUk(x),

where MU,, is the Hardy-Littlewood maximal function of property of we have

Uk. Due

to the support

s

(9.11)

We can estimate the right side of (9.11) using (9.10), together with the following vector maximal estimate, established in [FS1]; see also {RRT] or [S31 for a proof: (9.12)

for 1

In concert with the previous estimates, this yields

c

9. OPERATORS WITH l)OUHLE SYMBOLS

To estimate Q2, use (9.7) (9.14)

to

65

write * iuI(x).

A(k)

Q2n(x)i k=O

This implies the desired estimate (9.15) as long as EA(k)

Note bols of


that if a(x, y, 0

C[WIS?(,c) is decomposed into elementary double sym-

the form (9.5), we have the decomposition (9.6) with

CA(k)tc(2k).

(9.16)

Thus

the proof of Proposition 9.1 extends to yield:

< cc, then

PROPOSITION 9.2. If>w(2

A: L"(r) —÷

e

(9.17)

L71(r),

1

that the conditions on w(2 in Propositions 9.1—9.2 are more stringent those in Propositions 2.5 and 2.7. This is necessary in view of the following example, given in [Ma4]. Consider Note

than

=

(9.18)

k>O

where

One

e

is supported on 1/2 <

<2 and

=

=

Now,

with

6

sees that

iei < 3/2, then Qu = very weak boundedness of Q.

3/4 <

A(k).

Hence EA(k) < cc

1

for 3/4 < <3/2. is supported on

if ii

is necessary for even

We next establish some special properties of operators produced by double symbols that vanish at x = y. Let consist of C' functions whose first-order

derivatives belong to CEwI.


PROPOSITION 9.3. Assume (9.19)

A:

a(x,x,e) = 0

e

Then

1

—÷

PROOF. Write

=

(9.20)

Then writing (9.21)

Since

—

—

y4,

6

and

integrating by parts yields

Au(x) = (x, y, C) e C:w]s?0, the conclusion follows from Proposition 9.1.

1. UrrnttAlUttb VVI•i•I-1 M1L1)LY RISGULAK SYMbOLS

00

PRoPOSITIoN 9.4. Assume

Ew(2

k)


(9.22)

1
E

PROOF. We have

(9.23)

+ Proposition 9.3 the last term.

applies to the first

PROPOSITION

term on

9.5. Assume 0 C s

the right and

Proposition 9.1

applies to


(9.24)

e Moreover,

1
under the hypotheses in (9.24),

A:

(9.25) PROOF.

—*

With A

(1

—

1

—

(9.26) for z in

ft={zeC:0
=

(9.27)

Let

—

denote the associated family of operators. Now, given u E

sition 9.1 implies

E

for all z E

with

Propo-

an exponential bound as

—p oo. Also, Proposition 9.4 implies E whenever Re z = 1, again with an exponential bound. By complex interpolation, A3u E Since a(x, x, = 0 implies A3 = A, we have the desired result in (9.24). The result

lImzl

in (9.25) follows by duality and interpolation. The duality argument works since

A*u(x)

Using Proposition 9.5 we can obtain further results for symbols that do not vanish at x = y. PROPOSITIoN 9.6. Assume 0 <

(9.28)

a(x,y,e) E

r

<

1

and

—r < .s Cr. Then C p < 00.

9. OPERATORS WITH DOUBLE SYMBOLS

PROOF. If a(x, y,

satisfies

67

the hypothesis of (9.28), then we know by Corol-

lary 4.4 that for the = a(x,x,e) e C"S?0 satisfies p(x,D) : range of s and p considered. Meanwhile = — p(x,e) satisfies the hypotheses of (9.24). so this yields an operator satisfying (9.25). This proves Proposition 9.6.

PROPOSITION 9.7. Assume that 0 < s
(929)

A

1

—'

=

PROOF. We may as well treat E

with and integrating by parts, we have

=

Using

Au(x) =

ff ib(x, y,

—

—

(2n)1

Uju(y) dy

ff

y,

The result now follows from Proposition 9.6. If a(x, y, for 1

then

e

Proposition 9.7

applies to A and K, so we have,

A : H +s.P(w2)

6

—1 —

r


PROPOSITION 9.8. Assume 0 < s
A: Ha+sP(r)

6

1

PRooF. By duality, it suffices to show that, when a(x. y, pothesis in (9.31), then (9.32)

the hy-

1

—*

A:

satisfies

As in the proof of Proposition 9.5, set A =

(1

—

—

For each 6 > 0, set

= = {z e C : 0 < Rez 1}, and denote by operators. If Rez = (, we have in

the associated family of for each 6 > 0, and in uniformly in 6 (0, 11. Take ii 6 Then, for each 6 > 0, belongs to cc (as a for all z with an exponential bound as very special case of Proposition 9.6). Furthermore, we have (by (9.28) and (9.30)) on

H 8 E (0, 1], with an exponential bound as cc. From here a use of the Hadamard three lines lemma and complex interpolation theory yields H uniformly in 8 (0, 1], and taking 8 —' 0 gives (9.32).

1. urt,ttAluflb wiid MILDLY REGULAR SYMBOLS

We next derive some estimates on operators associated with double symbols of negative order. PROPOsiTION

Assume Ew(2_k),c(2k)
9.9.

E

(9.33)

E

then

1
(9.34)

PROOF. Look again at the formula (9.23) for

The two amplitudes on under the hypotheses (9.33), so the boundedness of follows from Proposition 9.2.

the right belong to on

Note that Proposition 9.9 applies when (9.35)

E

r

> 0,

where, as is natural, we say

e

and nct

/

By duality and interpolation we have, for r> (9.36)

=* A

e

provided —1 S s 5 0, 1

9.10.

PROOF.

5E

[0,

1),

—.

:

can apply interpolation in a different way, I

A:

Then

1< p

—*

As in the proof of Proposition 9.5, let A = (1

(9.38)

= {z

0,

C'

n

Assume0<s'cr< and0<8czl.

E

(9.37)

on

We

r\

—

—

Then

-(z—a)

y,

1}, and denote by the associated family of operators. we have with an exponential E for all z E Proposition 9.2. Also we have E H1'(r), for Rez = 1 (with an exponential bound) by Proposition 9.9. Interpolation gives Au = A3u E Given u E bound, by

E C : 0 5 Re z 5

LP(IR't),

We next study the action of operators of the form (9.1) on the spaces Again we can consider symbols of the form (9.5). We begin by splitting the associated operator Q into 9 pieces: Q by splitting qk(x,y) into 9 pieces: qk = qkafl. Take (9.39)

=

9. OPERATORS WITH DOUBLE SYMBOLS 4 for a 1, over k — 3 5; p 5; k + 3 for a = 2, and v varies over similar ranges for /3 = 1,2,3, respectively! We then need to estimate

where p varies over 0 and over p k + 4 for

(9.40)

sk—

a=

Q00u(x)

3,

— y)u(y) dy =

f

in these 9 cases, by examining supports of Fourier transforms of various functions. For example, for (a, /3) = (1, 1), we see that we can replace 11 by ILk (as in (9.9)), and that

2k• Hence

is supported on

5;

Thus, (9.41)

We have the same bounds on the supports of u by ük. Hence we have estimates

and 1k13, but we cannot replace

fl2,3.

(9.42)

on

For (a, /3) = (2, 1), we can again replace u by 12k; 2kt3, so 1/'L(D)Q21u = hence

(9.43)

this

time Ük21 is supported

>1 k>L—3

We have the same bounds on the supports of

and

but we cannot replace

u by 11krn Hence we have the estimates

c(

(9.44)

/3=2,3.

> k>L--3

we make a finer decomposition, To analyze ting qk3$ into pieces:

(9.45)

by

=

=

qk3$(x,y) =

vO

v?O

Then (9.46)

> k.vO

k,v>O

In

all cases,

(9.47)

has

support on

2v-i-k, and

= >

split-

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

If$ =

1,

<

we can replace it by Uk, SO

and

I

(9.48)

E

kC€ -3

Here,

=

For /3

2 or 3, we cannot replace it by iLk. We have

sc( >:i:

(9.49)

k
and

c( k
(9.50)

—3

Using the estimates (9.41)—(9.50), we obtain:

PRoPOSITIoN 9.11. Let w be a modulus of continuity, and A(k) = w(2_k). satisfies the following conditions: Assume E A(k) < c'o. Assume a(x, (9.51) (9.52)

C0A(k)

ID?(I -

(9.53) (9.54)

— wk(D.0))a(x,y,e)I

(9.55)

K(2k)fL(k)A(k) CA&),

k

CA(f). kCP+3

Then (9.56)

—p

A:

Note that (9.51)—(9.53) (supplemented by (9.55)) apply to (9.41) (9.43), respectively, that (9.54) applies to (9.44), (9.49), and (9.50), and that (9.55) applies to (9.48). PROOF.

Note that, when a(x, y, is independent of y, this result reduces to Proposition 5.2. As noted there, the condition (9.55) holds provided (9.57)

In particular, if (9-58)

Ek(2k)1t(k)

and

9. OPERATORS WiTH DOUBLE SYMBOLS

Regarding the conditions (9.51)- (9.54). note that (9.53) is actually a consequence of (9.54). Furthermore, when ic = 1 and Ep(k)
)t(k)/p(k) /,

and

that

satisfies

—

(9.59)

Then A

that (9.59) holds provided

(9.60) i.e.,

V k,f.

—÷

:

Note

C0A(k)1a(e)(C)H0I,

—

1°,

Wk(DX,Y))a(x,y,C)I

provided

(9.61)

a(2')

E

Considering the special case ji(k) = 2tk, A(k) (9.62)

6

= A(k)p(k). 2—sk

A : CQR'1) —+

we have, for r > 0,

G(lr), 0 s

r.

We will sharpen these results below. We next examine the behavior of operators of the form (9.1) on various Zygmund spaces, when a(x, x, C) = 0, paralleling Propositions 9.3—9.5.

PROPOSITION 9.13. Assume r> 0. Then, for 0 s

A :C(r) —÷c:(lr).

c

(9.63)

PROOF. We have (9.20)in (9.63) follows from (9.62).

PROPOSITION 9.14. If r> 0, then, (9.64)

for 0

: C(W')

a(x,x,C)

C

so the conclusion

6

(9.21) with

PROOF. As in the proof of Proposition 9.4. to analyze 151A we use (9.23). We

see that Proposition 9.13 applies to the first term on the right and (9.62) applies to the last term. PROPOSITION 9.15. IfO < a < 1, then, for ails C a, (9.65) PROOF.

C

0

A: C2(Rn1)

s+2r,

r> 0. Define

a(x,r,C)

Assumes C (0,a), and set a

of Proposition 9.5, via (9.26)—(9.27). Take it C for all z C and Proposition 9.14 implies

as in the proof

Then (9.62) implies

C

when ltez - 1 (with exponential bounds). By complex interpolation, Au = A5u C C(R11), as desired.

We can now improve (9.62), at least for r

C (0, 1):

c

_________

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

72

PROPOSITiON 9.16. If 0 C

r < 1, then

C:(r),

A: C:(lr)

E

(9.66)

PROOF. As in the proof of

C

p(x,

argument for s = = applies to known)

s< r.

Proposition 96, consider C

0, or

0

for

.s E [0, r), by

(5.24), pius a related (well

an appeal to (9.62). Furthermore, Proposition 9.15 — p(x,E).

This proves (9.66).

ilaving this improvement of (9.62), we can sharpen Propositions 9.13—9.14 by in (9.63) and (9.64), at least for r E (0, 1). We slightly, replacing omit details on sharpening these results for larger values of r. It is worth noting that we can use Proposition 9.5 to extend Proposition 9.16,

as follows. PROPOSITiON

9.17.

If 0

C r C 1,

e CS?0

(9.67)

then

A:

—r <8< T.

C:(ir)

the decomposition a(x, y, = p(x, + b(x, y, used above. C for —r <s Cr (cf. Proposition 2.1.E of [T2J). Meanwhile, Proposition 9.5 applies to the operator B associated with b(x, y, e); in particular (9.25) holds for B, and this in turn implies B : C: C s Cr. PROOF. Make

We

actually have p(x, D) : C:

In fact, it is useful to record the foiiowing improvement of Proposition 9.15, proof fo'lows from the argument just presented.

whose

PROPOSITION 9.18. If 0 Cc C 1 then, given r E (—c,0], s e (9.68)

on

A: C:(r) —÷

a(x,x,e) = 0

E

(0,cr),

We return to the level of generality of Proposition 9.11, and consider conditions C(\2). For this, the estimates (9.41)—(9.44) and that imply A: C(A)

(9.48)—(9.50) are still effective. This time we want to estimate Let us set

=

(9.69) We see that (9.70)

A(k) CA2(k)

:5;

while, for j3 = 2,3, 5;

Next,

(9.72)

5;

,\(k) 5; CA2(t) provided 5;

by

g. OPERATORS WITH DOUBLE SYMBOLS

and Ekt_37(2)k(2'il1@)#)t(k)

E

(9.73)

i.e.,

ic(2")li(k)A2(k)

CA2(t)

Furthermore, Ek>i3 —

(9.74)

2,3, Provided

for /3

y,

and (9.73) holds.

Examining provided

73

-

CA2(t)

(9.48)—(9.50), we see that

(9.75)

and

CA2(t), i.e.,

(9.76)

>

K(2k)A2(k)/i(t)

Finally, we have Ek
S

(9.76) holds. Note that (9.72), (9.74), (9.75), and (9.77) are implied by

-

(9.78)

If k =

K,

this also implies (9.71). We have the following result:

PROPOSITION 9.19. Given A and A2 (decreasing and slowly varying) we have provided a(x, e with

A: (9

provided

—

(9.77)

and

S CA2(t) and a similar estimate

replaced by I —

—

79)

/

—

cr(2_k)

(9.80)

= A(k)14k),

and

k?P

In

5

S

(9.81)

kcCE

particular (9.80,)-(9.81) hold if

(9.82)

a(2 k)

= A(k)A2(k), and

Ek(2k)A2(k)
As in §3, it is useful to apply symbol smoothing to double symbols with limited smoothness. Thus, if a(x, y, E XSr(k), we can write, parallel to (3.3)--(3.4), (9.83)

=

+

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

with

a#(x,

(9.84)

Pick

6 E

(0,1).

=>

E S.

(9.85)

parallel

We aLso have estimates on ab(x, y,

a(x,

(9.86)

generally, as

to those on pb(x,

derived

in §3. For

< 1,

example (cf. (3.27)) if 0
More

we have

As in (3.10), as long as X c

rb

ab(x,

C C'5f'0

in (3.58)—(3.59),

(9.87)

ab(x,y,e) C

C

with

=

B(lej)w((e)-6),

(9.88)

a variety of choices of 1. Let the operator decomposition associated with (9.83) be denoted A = A# + Ab. Note that, if a# satisfies (9.85), then and

A# = b(x,D) C OPSr6.

(9.89)

Consequently this piece is very well behaved. We can use Proposition 9.19 to estimate Ab. Consider the case = 0. Let A and A2 be decreasing and slowly varying; assume A2(k) A(k). Take = A(k)A2(k), and define as in (9.79). If (9.88) is to hold, this specifies ]3(2k\

p990"

A2(k)

1

'.

The necessity to have B(jCI) 1 puts a mild restriction on the relation between A and A2. If ab(x,y,C) C —* C(A2) provided : we see that = A(k)A2(k) and Er(2k)A2(k)
g

this

is equivalent to >2 A(k)

PRoPoSriloN

have Ab :

us summarize:

9.20. Assume that A2(k) A(k) and that (9.90) is provided

given by (9.79), r(e)

IJsing

<00. Let

=

with

e

w(2 -k) =

1.

We

A(k)A2(k),

and EA(k)
this we can establish that A

:

—*

c(A)

under a condition that is

(usually) weaker than (9.61). PROPOSITION 9.21. Assume that w(2 k) = p(k)A(k) with 14k) A(k) and 1 A(Sk) 1u(k)p(bk) —

10. THE CRW COMMUTATOR ESTIMATE

75

for some 6 E (0,1). Then

A:

e

(9.91)

PROOF. Replace A by p and A2 by A in the result of Proposition 9.20 about —* c(A). Hence Ab : c(A) —* CP'), if A(k) p(k). The Ab. We have A1' result for A# is clear. :

that (9.61) looks similar to the hypothesis above, but there we require

Note

p(k)

A(k).

We can also use Proposition 9.20 to estimate A when a(x,x,e) =

0.

PROPOSITION 9.22. Let w, A, and A2 be as in Proposition 9.20. Also assume 2)k for E (5.1). Then

A2(k)/A(k)

a(x, y, €)

(9.92)

a(x, x.

E

=0

A:

—*

PROOF. Use the symbol smoothing (9.84) and associated decomposition A = A#+Ab. We have and Proposition 9.20 implies Ab : c(A) E so we have Also a(x,x,C) 0 a#(x,x,e) =

(9.93)

jD?a#(x,x,e)j

in addition to a#(z, y,

E

a#(x, x,

In particular

satisfies

(9.94)

for all 'y E (& 1). Hence p#(x, D) _p#(x,e) = b#(x,y,e) belongs to

Interpolation with (9.93) gives p#(x, C(A2).

C(A)

also B#

:

Finally,

E

—*

10. The CRW commutator estimate In this section we establish the following estimate of [CRW]:

THEOREM 10.1. Given P E OPS?O, 1

Ilf(Pu)

—

P Since

0

< 5 .c

1,

and

also for P E

OPBS?1.

[CRW], there have been a number of proofs of this result. For example,

we mention one in [Jo]. The proof we give is taken from [AT]. It exploits the representation of a product in terms of the paraproduct, (10.2)

used

fu = Tju + T,,f + R(f.u),

here already in several places, such as (3.47) and (8.6).

We also make use of the following estimates: (10.3) These

IR(f,g)(ILP estimates are due to [CM]; proofs can also be found in [T2].

1. %JrrjItnitjrw VVLLfl

The space

was introduced in §2. We recall that it has norm Ifibmo

given by

= llfMursio +

where 'Po E satisfies 'P0(0) space, with semi-norm given by HflhjMo =

0, and BMO is the classical John-Nirenberg

11

7

V(Q) j 1

dx,

—

sup taken over all n-dimensional cubes Q in where V(Q) is the volume of Q and fq is the mean value of I over Q. Actually [CRW] established (10.1) with BMO in place of bnio, but the result as stated here is equally effective for local the

analysis.

Our first step in proving (10.1) is to write

fPu = T1Pu + Tpaf + R(f, Pu),

(104 As long as

P:

L'(IRTh) for 1

—÷

+

we have

+

+

Thus it remains to estimate [T1, P]. The desired estimate is a consequence of the following stronger estimate: (10.6)

lI[Ti,P]ulltp

In turn, this is a consequence of the following: PROPOSITION 10.2. If f

C2(llP) and P E

(with

p + 1/8 < 1),

then (10.7)

[Tf,P] E

The proof of Proposition 7.3 can be applied to proof

r

this case, but we will give the

from [AT], which is slightly different. To begin the proof, we note that, for

>0,

f e C;'(r)

(10.8)

Tf E

and hence (10.9)

fe

T1 E

Now, for any t9x,D) E

F(x, D)P(x, D)

P(x,D) e OP!38r1, and

belongs to

Qi(x.e) =

V E

E In


>0.

the product Qi(x,D) =

we have a symbol expansion .

+

10. THE CRW COMMUTATOR ESTIMATE

with estimates on T1 = F(x,e), we have

given by Proposition 6.1. Applying Proposition 6.4 to

(10.11)

for

5;

P E QPBS?1,

=

f E

1,

ctl

so we can use this in hypothesis (6.25) of Proposition 6.1., this Proposition (with P2 rrt2 = r 0), we deduce that (10.12)

77

with ji

= 0. Applying

V 6 > 0,

(21 E

and mod

(10.13)

Also by

Proposition 6.1,

(10.14)

Note

fE

PE

that, if

T1

=

V 6 > 0.

(22 E

= F(x, D), then F3(x, = UF(x,

is specified by

D) = T8,1,

(10.15) so, by (10.8),

(10.16)

fE

E

Consequently, again by Proposition 6.1, (10.14) holds with

(10.17)

mod

> IaI=1

the sum over 1 here also belongs to Together, (10.13) and (10.17) establish that [T1,P] E and invoking Proposition 6.2 we can replace by proving Proposition 10.2. As a consequence of Theorem 10.1, we now establish the following "supercommutator" estimate, which will be convenient for applications in Chapter III. Let f be an £-form, and set W1u = A u, and and

f

(10 18)

W1fl

W1]

if

£

if £

is even,

is odd,

where [A, B] = AB — BA and {A, B} = AB + BA. Here, d is the exterior derivative and A = (I — a)'/'2 There is the following estimate. PROPOSITiON 10.3. For 1

• is

L OPERAUO1-(S WItH MILDLY REGULAR SYMBOLS

PROOF. Write W1 =

where

are smooth £-forms and E Ilfililimo

If Ilhmcr Then 'd, W?]]/3 =

(10.20) Now

the estimate

symbol

of

A1d is

Mf.}W6./3 +

(10.1) applies

to the first sum on the

right. Since the principal

we have

wedge by

(10.21)

EOPSTd,

so the estimate on the second term on the right side of (10.20) is elementary.

11. Operators with vmo coefficients Consider a symbol

E bmoSr0,

(11.1) i.e.,

such that

(11.2) Assume

furthermore that

r1,

(11.3)

We say (11.4)

Related

p(x,e) E symbol classes include

(11.5) defined in the obvious way, where vmo denotes the closure in bmo of the space of

functions with bounded derivatives of all orders. We aim to study operators in such symbol classes, obtaining some results of [CFL] and some extensions, some of which have been discussed in [T4]. These results will be useful for some elliptic regularity results in Chapter III. We start with the following decomposition of a symbol p(x, C Let j 1 } be an orthonormal basis of L2 consisting of eigenfunctions of the Laplace operator As on Sn_i. We can write smooth

(11.6)

-

=po(x,e) +

where

I

Furthermore,

for

1/2, arid

is

CN(j). Now

write

=

-

=

supported on

1.

11. OPERATORS WITH VMO COEFFiCiENTS

so we have (11.9)

p(r,e) = i?0

The operator P0 (x, D) has a simple analysis. One can write (11.10)

po(x,e) =

with

(11.11) Thus

pg(x,D)u =

(11.12)

*

it,

where

(11.13)

x) = C,23(2z + £).

Hence

bmo(W),

po(x,D)

(11.14)

VpE [1,oo).

We use this decomposition to establish some commutator estimates.

suppose

First,

0ä<1.

(11.15)

Take in

(11.16)

=

0 above, and use

[B,p1(.x,D)]u

the

instead of

notation

=

Then

=

Since (11.17)

and

= [B,ag(D)] E

there are polynomial

bounds (in j)

on

the relevant seminorms of the symbols,

we have (11.18)

given

1

supp it

c K, compact.

Also, by the fundamental commutator estimate (10.1),

we have (11.19)

and

ClifjIIinrojlVjILP,

hence

(11.20) for j

1. Summing over j, we have:

az

WiTH MILDLY REGULAR SYMBOLS

1.

It is a general fact that, if 21 is a C' algebra and a closed *-subalgebra of 21, containing the identity element, and if f E then f is invertible in if and only if f is invertible in 21. To see this, consider h = 1*1 and expand (h + 1 — z = 0. Consequently we have (11.37)

This holds for matrix-valued a(x). Hence, as another consequence, if p(x, n

S C1 for

is elliptic, i.e.,

E (LD0

n

ei G2,

is an appropriate cut-off. This

where

e —

then

construction of Fredholm inverses of elliptic operators with coefficients in Applications are given in §1 of Chapter III.

allows the flvmo.

12. Estimates on a class of Besov spaces Assume s E IR, 1 p, q S oo. The Besov spaces Bq(W1)

E

f

(12.1)

are

defined by


3

with

?Pj as in (1.2). We aim to look at a subclass of this family of Besov spaces,

1. We begin with a result whose extension to general q is well known (cf.[RunJ), but for simplicity we stick to the case q = 1, with a view to some applications in Chapter III. with q

PRoposiTioN 12.1.

Given in, S E

P:

(12.2)

IR,

1 5; p 5; oc, we have

B;*m(Ir)

for all P E OPBSr1. Hence (12.2,) holds whenever Furthermore, if s > 0, then (12.2) holds for all P e

P e

0

5; 5 < 1.

with r> s.

PROOF. It is straightforward to reduce to the case in 0. As in 4, etc., it then suffices to consider the action of q(x, D) when q(x, is an elementary symbol, with decomposition as in (2.8). We first look at

(12.3)

j
k

Since QLWk(D)f has Fourier transform supported in (12.4) so

if

E 5;

we have

5; CA1

Hence 5;


qi(x,D)

we have

12. ESTIMATES ON A CLASS OF BESOV SPACES

This is enough to establish (12.2) for all P c any effort to general q E [1,ool.) Next we look at

83

(This part extends without

k+3 (12.7) k

j-k 3

In this case the Fourier transform of can write

has support in

C2k, and we

E

(12.8)

Hence

(12.9)

C>

8(k) =

kt—4 so

5; C>

>

£)Ok>e—4 k+4

=

(12.10)

k>O

k>O

the last inequality holding provided s > 0, and we have (12.11)

5; C ===t' q2(x,D)

—÷

Next we look at (12.12)

q3(x,D)f

>

k>Oj>k+4 In

this case Qkjcok(D)f has Fourier transform with support in

(12.13)

E

f+5

v5e(D)(Qkjçck(D)f),

k<E-j-5 j—1?—5

hence

(12.14)

5; C

>

We have

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

D)fIILP 1+10

C>11

(12.15)

>1

£O k=O

1k—1O

k-=O

Thus

we have

(12.16)

q3(x,D)

A2

>

—*

£ k —10

this completes the proof of Propo-

Since

sition

12.1.

If we look at the action of P on

we see that (12.10) needs to be modified

C> > (12.17)

C>(k k>O

and we have (12.18)

When

'.q2(x,D) s

= 0,

(12.16) continues to hold.

so

If q(x,€fl E

that

IIQkjIIL°' CA(j),c(2k),

(12.19)

the

condition (12.6) holds, and furthermore the conditions (12.11) and

(12.18)

hold

provided

A(j),c(2k)

A(k),c(2k)

(12.20)

A2
{j:jk} We

have the following result.

PRoposiTioN 12.2. Assume A \,tc then,

for 1

(12.21)

Note (12.22)

/ are slowly

varying. If (12.20) holds,

oc,

E

—*

that (12.20) is weaker than A(k)k(21')

\,

12. ESTIMATES ON A CLASS OF BESOV SPACES

defined by

The Besov space (12.23)

85

fE

is of particular relevance for one of the points of view we take in this chapter, namely to study spaces of functions very close to the space of functions, but on which pseudodifferential operators (with rough symbols) act. It is clear that (12.24)

C C(RTh),

C

We next show that certain spaces

provided >A(j)
(Rn) are algebras.

PROPOSITION 12.3. Given s > 0, p E [1,oo], the space an algebra under pointwise multiplication. We have (12.25)

when s> 0 and

c

(12.26)

then

is

+ If

5;

Consequently,

fl

(RTh) is an algebra under pointwise multiplication.

PROOF. We use the paraproduct expansion as in (3.46)—(3.47):

fg = Tjg + T9f + R(f,g),

(12.27)

with

Tjg = > Wk_4(D)f

(12.28)

and (12.29)

R(f,g)= >

It is readily verified that (12.30)

for s E R, p E [1, oc], by the analysis of (12.3)—(12.6). There is an obvious analogue

for T9f. Similarly the estimation of R(f, g) is a special case of the analysis of (12.7)—(12.11). In fact, with k+4

(12.31)

k)= >

Wk(D)fhI)k(D)f,

j=k—4

we have precisely

RHS of (12.10),

(12.32)

(12.33)

8>0

1. OPERATORS WiTH MILDLY REGULAR SYMBOLS

86

This finishes the proof of Proposition 12.3. REMARK. Proposition 12.3 is contained in results of [Run2]. The proof given above of this Proposition is essentially the same as the proof given by H. Triebel in [Tril], pp. 133—137. In fact, Thebel anticipates the use of paraproducts in establishing such results. The general result stated in Theorem 1 on p. 133 of [Till] would also imply that (It') is an algebra under pointwise multiplication. However, the proof given there does not work for this space. The problem arises in

the estimate (12.32) (or equivalently, in (9) on p. 136 of [Till]). This estimate

works for s > C) but fails for s

it

(Itt)

is shown that

= 0.

This correction has

is not an

been noted in [SIT], where

algebra.

As an example of spaces satisfying (12.26), we mention the well known result

that, for 1 p < no,

c

B7(Wt)

(12.34)

c

L°°(R't).

The first inclusion can be established by showing (12.35)

and,

C2't"1' IkI'dD)fHLP,

as indicated above, the second

inclusion in (12.34) is straightforward. Regard-

ing

fe

gE

In fact, the estimate (12.37) follows as in (12.17)—(12.18).

13. Operators with coefficients in a function algebra We recall that, whenever B C L°°(r) the symbol class (13.1)

is defined by

IID?p(.,CHIB S

p(x,tf)

If we assume furthermore that (13.2) we

p(x,re)

=

1,

1,

say

(13.3)

p(x,e) e

In this section we treat sonic results that are valid in great generality. At the end we indicate 5011W interest ing classes of examples to which these results apply.

13. OPERAToRS WITH COEFFiCIENTS IN A FUNCTION ALGEBRA

PROPoSITiON 13.1. Let B, B' be Banach spaces of functions on

87

with trans-

(R") is an algebra under pointwise lation-invariant norms. Assume that B c multiplication, and that B' is a B-module. Also assume (13.4)

P(D)

P(C) E

B' —*

B'.

Then

p(x,C) E

(13.5)

PROOF. 1)ecompose

B'

B'.

m=

0. An analysis

as in (11.6), with

p(x,

(11.10)—(11.13) applies topo(x,C), with lptIJn

behavior of B'

for j

1

to

as a B-module then gives p9(x,

Next,

parallel

in place of (11.11). The

we have, as

D) B'

>

B'.

in (11.8),

=

(13.6)

with S CN(J)N,

(13.7)

and,

and the operator bounds from

as a consequence of symbol estimates on a7 (C)

(13.4),

<

(13.8)

forsome

= M(B. B')
independent of j.

Using

again the fact that B'

is a

B-module, we have (13.9)

which

D)uIIp' 5

Cv(j)_NjIuIjiv,

proves (13.5).

in B.

Let us denote by B the closure of

The

following result lacks the

depth of Proposition 11.3, but it is still useful.

PlioPosiTloN

13.2.

In the setting of Proposition 13.1, assume that

E

(13.10)

Furthermore,

assume that E

(13 11)

with compact

x-support

r(x, D) B' —* B' is compact.

Then (13.12)

p(x, D)q(x. D) — a(x, D)

with (13.13)

a(x,C)

e

-I- K,

88

1. OPERATORS WITH MiLDLY REGULAR SYMBOLS

and

K: B' —÷ B' compact.

(13.14)

PROOF. We can construct with compact x-support, such that E —* p(x,e) in BS% with its natural topology. Hence p3(x,D) —* p(x,D) in £(B'), in operator norm. Similarly, qj(x, D) — q(x, D) and, with a(x,D). Now, for each j, a(x, D)—p(x,D)q(x,D) isa norm limit in £(B') of compact operators on B'. There are, of course, many examples of function spaces B and B' to which the results above apply. We mention

(13.15)

B

B' =

B1(IRTh),

,s>

0, .sp Ti,

as one interesting family of examples, studied in §12. Without going into details, we note that Propositions 13.1—13.2 have relatively straightforward analogues with replaced by TT' or some other compact manifold. Sometimes it is technically preferable to operate in this context. If one wants to apply Proposition 13.2 (or its analogue on a compact manifold M) to produce a Fredholm inverse of an operator q(x, D) which is elliptic, in the sense that (13.16)

E BSZ,

p(x,e) =

'(i

—

is a convenient cut-off and C = C(M) is the space of continuous func-

where

then it is desirable to know that (13.16) leads to p(x,e) e for function algebras B with the property that

tions on M,

works

This

(13J7) Generally, whenever all operators in OPS?1 are bounded on B, the paradifferential operator calculus gives (13.17); see the introduction to Chapter II to see how this works. This works for the spaces given in (13.15). A more elaborate argument, given in §9 of Chapter II, is needed when B = 14.

Some BKM-type estimates

As is well known, elements of OPS?U are not typically bounded on L°". It has proven useful to have estimates of the following form: (14.1)

Pe ops?,0

log

any r > 0. Such estimates are used in [BKM], which has stimulated much further work. In these estimates and some extensions were derived as a consequence of estimates of the form for

(14.2)

log log

i4. SOME BKM-TypE ESTIMATES

89

which in turn follow from (14.3)

+ c(log

CeT

we note several extensions of these estimates, involving other function spaces and rougher operators. We start with some results in which C is replaced by spaces of the form C[wI We retain our standard hypotheses on w(h) and on A(k) and With Here

as

in

(1.2), we write

u=

(14.4)

and

+ (I —

deduce that

E

(14.5)

+

1K'

-

VkE

Hence, by the definition (1.61), (14.6)

+

S

Vk

1,

V e

e (0,1].

or equivalently (14.7)

+

S Cw(e)

C(log

This extends (14.3), in light of the inclusions recorded in (1.62) (1.63), i.e., (14.8)

in

C Cc,

CW c

w(h) +!if

a(h)

To extend (14.1). we work with the spaces and apply estimates established We will also need the following inclusion, recorded in (1.63):

(14.9)

p(h) =

C(A) C

assuming w(h) satisfies the Dini condition:

1'

Jo

We

see from (14.7) that, if A(j)

(14.11)

f

dl < I

=

S Cp(e) IU11C(x) + (log

Replacing it by Pu we hence have

(14.12)

S

+ C(log

dt,

1. OF'EJUVIOI(S WITH MILDLY REGULAR SYMBOLS

Hence

P:

—+

P; c(A)

(14.13) 1

+ c(log

!IfhIC(A1

—)

Note that

=

(14.14)

and we can also state the conclusion of (14.13) as

(14.15)

C7(k)lIuhjcA) +

V k 1.

Regarding the hypothesis of (14.13), we can apply Proposition 5.2 and Corollary 5.3 to obtain: PROPOSITION 14.1. Assume A(k) \

0

is

slowly varying and

use (14.14) to define 7(k). Then (14.15) holds for all P PROOF. That P: That P :

—÷

A(k)
for such P is explicitly stated in Corollary 5.3.

follows from Proposition 5.2,

iii light

of (5.22).

A particular case to consider is (14.16)

If we

apply (14.15) and take k

(14.17)

for

s>0. Is),

P E oPc(A)sy0

we obtain

s

this class of spaces

As we have stated, the original point of the sort of estimates discussed above was

to obtain estimates. However, slight variants of these arguments are actually effective in obtaining estimates in a slightly stronger norm, namely the 1-norm. as we now demonstrate. PROPOSITION 14.2. In (14.18)

the setting of Proposition 14.1, we have C7(k)Ilu(ICCA) + CkIJuIIco,

V

k 1,

with 7(k) given by (14.14). Jlence, for P (14.19)

S C7(k);Iuljc(A\ + CkjIuI(co,

PROOF. We have — 'P k(D))uHBo

(14.20)

S C >1 f>k

>

2

V k

1.

14. sOME 13KM-TYPE ESTIMATES

91

and

C(k +

C >1

(14.21)

from which (14.18) follows. Then (14.19) follows from Proposi

In light of Proposition 14.2, we can replace the left side of (14.3) by and hence we can sharpen (14.2) to log IuHc? log IluiHo

(14.22)

),

r>a

Similarly, by the arguments leading to (14.17), we have (14.23)

I

lUIIuo

.s)

when A(k) is given by (14.16).

One application of BKM-type estimates made in [T2 was to the following result on persistence of solutions to higher-order hyperbolic equations, of the form —1

(14.24)

Uru= > :i

with (14.25)

=

where the coefficients t E Ift, z E Al, dim Al

> lnI<m j

depend smoothly on their arguments. We take = n. The following is Theorem 5.3.A of [T2].

THEOREM. Assume the equation (14.24) is strictly hyperbolic. Let initial data be given: (14.26) and

i,

assume s > n/2 + 1. Then there is a unique local solution C(J,H81Th_l(M))

(14.27) This

'

solution persists as long as

K
auxiliary space

C C' (Al) having the following two properties:

(14.30)

P:

PE

(14.29)

Ik4ci

*

(1 +

—*

C'(M),

log MuIIH.%)

log MuMc'

2

+ 1.

1. OPERATORS WITH MILDLY REGULAR SYMBOLS

92

At the time the author had not noticed that (14.22) is valid, and a somewhat ad hoc space (M) was constructed. At this point we can state the following:

PRoPosnioN 14.3. The space

can be used in place of

in the analysis

in §5.3 of {T2].

15. Variations on an estimate of Thmanov In the course of work on CR mappings, A. Tumanov [Th] established the following estimate. Define an operator P÷ on V'(S') by (15.1) k >0

PROPOSITION 15.1. If u E R(P+), r> 0, r

t, then

I(I—P+Xfu)Hcr

(15.2)

Note that (15.3)

[M1,P+]u,

uE

where M1tt = fu, so (15.2) is equivalent to the commutator estimate

Vu e

(15.4)

We can replace the Flölder spaces by Zygmund spaces C and then these estimates hold for all r > 0. Our first goal is to establish the following multi-dimensional version:

PROPOSITION 15.2. If 0 < r 1, P e (15.5)

then

+

The necessity to restrict r to (0, 1] when the dimension n is greater than 1 is due to well known results on symbol calculus. The special features that allow the estimates (15.2) and (15.4) to hold for r > 1 will be apparent when we rederive (15.4), at the end of this section. The proof of Proposition 15.2 will use techniques of paradifferential operator calculus. We begin with the following representation of the product:

fu =

+ R(f,u),

T1u

as in (3.47), where T1 is Bony's paraproduct. We have (15.7)

where

f

TI)

P(x, D) e 0P138?1,

denotes Y. Meyer's class of paradifferential operators, defined in (3.36)—

(3.37). When in = 0, these operators are bounded on all the standard Sobolev and Zygmund spaces. Thus we have

fPu = P(fu) =

T1Pu + + R(f, Pu), PT1u + PTUf + PR(f, u).

Directly using (15.7), we have (15.9)

+


+

15. VARIATIONS ON AN ESTiMATE OF TUMANOV

for all

= R(f,u),

r E R. Also, setting

E

93

we have

R11

Vr>

IIR(f,Pu)Hc: +

0.

hence, to prove (15.5), it suffices to have the following.

PROPOSITION 15.3. If 0 < r

then

1, P E

(15.11)

In fact,

for 0


1,

f C

(15.12)

e

This result is proved in [Bon] and also contained in results of [Mey2] and of [AT]. Furthermore, (15.12) is a special case of Proposition 7.3 for 0 < r < 1 and Proposition 7.4 for r 1. Next, we rederive the estimate (15.4). Note that, to prove (15.4), for all r > 0 (with C' replaced by C), since we have the estimates (15.9)—(15.10), it suffices to prove:

PROPOSITION (15.13)

15.4.

If r > 0, f e 1/981),

then

I{Ti,P+]uHc:

In fact, (15.14)

[T1,P+] E

In this case, we analyze = A(x, D) since P÷(e) is independent of x, we have (15.15) Next,

as

follows. First,

A(x,E) =

using Proposition 6.1, with v = 0,

(15.16)

= B(x. D)

and

we have

=P+(e)F(x,e)+n)(x,e),

the following analysis of the remainder. The symbol P+(e) has the following special property: with

(15.17) Thus (6.25) holds for any

al 1 e.g.,

a?P+(e) E =

—r. Since

fe

L°°

have (15.18)

n)(x,e)esll, Vr>0.

This proves (15.14), which in turn implies (15.13).

We next establish the following extension, to estimates in

E

we

YTtPLBAIIJTtS Will-I MILDLY REGULAIi

PROPOSiTION 15.5. Letw bearnodulus of continuity, A(k) =w(2_k). Assume

that

p(k) \,

A(k),

EA(k),t(k) CA(€). k>P

Then,

forf E

(15.20) If,

as in (15.1),

+ Iuk(M)).

+

I{Mi,

in addition, w(t)r' \ for soniC 5 E (0,1), then, given f e G(A)(lRTt), P e

uPS?11 (R"), (15.21)

+

1[MfPIuIHA C

+ Iuk(M)).

PROOF. This time, in the estimation of the terms in (15.8), we have +

+

(15.22)

by Proposition 5.7, and

+

(15.23)

by Proposition 5.8. It remains to estimate [T1, P]u. Proposition 15.4 is enough to yield (15.20). On the other hand, if w(t)t -s \ fcir some s e (0, 1), we have and this gives (mod from (7.22) that ITf,P] E when f (15.21).

Let us recall that examples when (15.19) holds are given in (5.52)- (5.54).

16. Estimates on Morrey-type spaces the Morrey space M"(lR") is defined by

For p E (1, (16.1) for all balls

fE

r"/lf(x)Idx

of radius r e (0, 1). PvIorrey's imbedding theorem is that

Vu E M"(r). p>

(16.2)

n

u E

We will consider the following Morrey-type spaces. Let w be a modulus of continuity. We say (16.3)

fE

r"/ f(x)j dx <

for all balls B, of radius r E (0, 1). We will demonstrate the following.

16. ESTIMATES ON MORREy-TyPE SPACES

95

THEOREM 16.1. We have

Vu e

(16.4)

Recall that

is

u

defined by (1.64) and satisfies the containment relation

(1.65). In particular, we have continuity of u in (16.4) as long as

w(t)r' dt <

We prove Theorem 16.1 by a method similar to that used to prove Morrey's theorem in [T21. Consider the family of operators pr(D) i.e., pr(e) = Nosvpr(D) is the operator of convolution with (4rrr2) the definition (16.3) we have (16.5)

1 e

Cr

v

so from

r

e (0,1).

On the other hand, the condition on the right side of (16.5) implies

which in turn implies <

(16.7)

Vu e condition that defines membership in In addition to the spaces it is useful to consider the following extension of the class of spaces Given 1 q p < oc, we say the

1/(;

(16.9)

fe

f(x)N dx)

S Cr

One all balls of radius r E (0, 1). Clearly Mff(r1) c = advantage of spaces of the form (16.9) is that large classes of pseudodifferential operators act on these spaces, provided 1 < q 5 p < oc, as we will see shortly. In with 4(r) = rlt(1 Q/P)w(r)O [P7 and [Sp] the space is denoted for

We also define homogeneous versions of the spaces considered above. Namely,

we say f e Mr(IW') if and only if the condition (16.9) holds for all r

E (0, oc). As a check on the sharpness of (16.4), consider the following family of examples:

= (log anY

(16.10) for

x E W, lxi 5

(16.11)

1/2,

cut off smoothly to be supported in {lxi < 1}. We have

= si.

log

xl S

and hence

e

(16.12)

with 1

(16.13)

w(h) = —i- ; r h Jo

dr —,

r

1. OPERATORS WITH MILDLY REGULAR SYMBOLS for 0< h < 1/2. If it = 1, one has w(h) e C"' in this case, if s <0. However,

it>

(16.14)

for h <

1/2,

if s

< 1.

1

co(h)

We see here that, while, by Theorem 16.1,

(16.15) as

i.e.,

E

long as s <1 (a fact that can also be verified directly from the definition (1.64)),

clearly C"'.

(16.16)

Note that, by Proposition 1.7, if s <0, then

h1/2,

(16.17)

in (1.65), for this family of moduli of

This optimizes the containment C{w) C continuity.

We record the result on modulus of continuity which follows from Theorem 16.1 combined with the containment (1.65). Such a result, with a different proof, is

contained in [Sp].

COROLLARY 16.2.

Provided ff3' w(t)r' dt, we have ph

Cc(wl),

VuE

(16.18)

cx(h)=J (1

M"

if Vu e MT1". Note that. if we apply the V-operator Let us say u E estimate Proposition 2.4 together with Theorem 16.1, we get

PROPOSITION 16.3.

co(i)t

As long as

1 dt cc

/c)

00, i.e., >w(2

< co, we

have

(16.19) Note

that (16.19) applies when

(16.20) On the other

(16.21)

I

p(x,

=

1.

hand, if co(t) satisfies (16.20), we have

M"""1(R")cC°(r),

a(2

sf1

If we were to apply Proposition 2.5 to V-estimates instead of Proposition 2.4, we would need cc co. i.e., s > 3/2. Thus, for such a family of examples, Proposition 2.4 is more effective than Proposition 2.5 (and also Theorem 16.1 is more effective than Corollary 16.2). Next we examine the action of various classes of pseudodifferential operators We begin with the following general result, extending a on the spaces result of [T4].

16. ESTIMATES ON MORREYTYPE SPACES

PROPOSITION 16.4. Assume the Schwartz kernel k(x,y) of T satisfies

k(x,y)I

(16.22)

for some M >

0. If 1


(16.23)

for

\,

some j3 E (0,n/p).

then

T:

(16.24)

PROOF.

T:

L"(W')

L9(IIV')

Let f E Mr(llVt). Pick z

(16.25)

ERTh,

r

E (0,1],

and write

f==f(J+ >

r <1

where fo =

and j

1

gj

Arj

x&3f,

= {x: Ix —

e

in the sum. We want to estimate Tf on Br(z). Clearly

(16.26)

S

S

and the estimate (16.22) for x — yj

a =

I implies

(16.27)

S

It remains to estimate > Tgj

on

lJr(z). To do this, write

(16.28)

=

where TI; has integral kernel

(16.29)

k3(x,y) =

Now, using (16.22) for Ix — yJ 5 1, we have

f

dx

dx

f

(16.30)

S

f

vol

Br(z)

dy= f Arj

(16.31)

S
vol

—

ChAPTER 2

Paradifferential Operators and Nonlinear Estimates Introduction This chapter presents a variety of estimates on compositions of functions F o typically where it = (iti,... Estimates on such compositions have long been important for nonlinear PDE. Various analytical tools have been brought to bear on such estimates, such as the Gagliardo-Nirenberg inequalities. Seminal work of J.-M. Bony [Bon] and Y. Meyer [Meyl] introduced the use of the paradifferential operator calculus. This has been quite effective, especially when F is smooth. S. Alinhac [Al] extended paradifferential techniques to cases where F has limited (though a fair amount of) regularity. Some more recent papers, including [CW], [KPV], and [Sta], have established useful results about F o u under weaker regularity hypotheses on F. We will recapitulate some of their results, and produce some more results along these lines, in ,

i.e., on where we establish estimates of [CW] on In §2 we establish estimates on [KPV] on As(fu) — fA8u. In both cases, we make use of Bony's paraproduct, in addition to techniques used in [CW] and [KPV], involving maximal functions. Using the decomposition (0.17) given below, we reduce the latter problem to an estimate on [A8, Tf]. Part of the analysis of this commutator involves estimates made in §3. Our choice of order of arguments was made iii order to motivate some of the rather technical arguments of §3; some of them are taken from [CW], and others are extensions, which will be useful in when F' is bounded and 0 < s < 1, obtaining §5. In §4 we estimate F o u in part of a result in [CW]. have been known before, our main goal in While many of the results of presenting our approach to their proofs is to lay the groundwork for the result of § 5, in which we estimate F o u under more general hypotheses on F'. The usefulness of such an estimate for problems in nonlinear PDE was perceived in [CW], and a similar result is stated there. However, the proof given there has a gap. This gap §1

A

—

was filled for an important class of functions F in [Sta], and the result there is useful for many problems in nonlinear PDE. however, in the course of work in [K], T. Nato formulated a more general estimate. The work presented in §5 was done by the current author in response to a question posed by Prof. Nato. In §6 we digress from applications of paradifferential operator calculus. We discuss some results on the continuity of u r—* 1(u) in H' due to [MM]. We include this partly to put in perspective some of the results of §5 and §7, and also to derive some implications for Sobolev spaces on irregular domains. This latter

1U4

rl t/kL)Ifl

K1-JNJIAI Ul-'EJ{ATOBs ANI) NONLINEAR ESTIMATES

category of results, in the case p = 2. played a role in some results of the author Ofl scattering of waves by very rough obstacles, given in Chapter 9 of [T3]. In §7 we return to paradifferential operator calculus and obtain estimates on F(u) -- F(v). These estimates generalize and sharpen some given in [Sog]. Also we correct a small error occurring there. In §8 we discuss a pseudodifferential operator estimate which to some degree combines methods of §1 with those of [Bourl, which have played an important role in paradifferential operator calculus. in §9 we analyze F(u) in terms of paradifferential operators, in yet a different setting. We assume F is smooth but ii belongs to a space introduced in Chapter T. We show that F(u) produces a paradifferential operator (by a process we recall below) that acts on given u E for slowly varying sequences A(j) O satisfying a condition just slightly stronger than the Dini condition .\(j) This is less straightforward for such cases as j .s > 1, than for the classical case A(j) = r > 0, because does not map to itself for such slowly decreasing sequences. We are able to succeed via a bootstrapping argument that treats progressively larger classes of sequences in Appendices A and B we discuss some results of [Al] on composition and One of our original goals was to link this material with the estimates treated in but in the end it seems that our task is to report on their essential differences, both in perspective and in the flavor of the results. We complement Alinhac's results with some new estimates, both in norms for p 2 and in Besov norms. Estimations done here involve As, with A = T. Kato has pointed out that one can make use of dilation arguments to obtain estimates involving instead D8, with D = A typical case of such an argument is described in [K2]. As preparation for material in the following sections, we briefly review the analysis of F(u), for smooth F, given in [Meyl]. bringing in the paradifferential operator calculus. More detailed accounts can be found in Chapter 3 of [T2] and in Chapter 13 of [T5]. Take W0 E 'I'o(e) = 1 for < 1. and set wk(e) = W0(2ke), Uk = Wk(D)u. Then

\

—

F(u) = M(x, D)u + F(u0)

(0.1)

where the formula (0.2)

M(x, D)u =

— F(uk)} k >0

yields (0.3)

(0.4)

M(x,e) =

mk(x)

Wk(C),

=

f

WkfrT(D) = 'Fk(D) +'ni5k±I(D).

INTRODUCTIoN

103

To estimate M(r,e), given u e L°°(W), we have, by the chain rule,

...

(0.5) 1
Also,

c

(0.6) Since

on the support of Ipk÷1(e). we see that uE

(0.7)

Now, as proved in [Si] (see also [Bour]), (0.8)

p(x,e)€

H3"(TRTt), p€ (1,cc),

as long as s> 0. Proofs of this are also given in [T2] and in Chapter 13 of [T5]. One immediate consequence of this is that, for s > 0, p (1, oc), < 1IM(x, D)uIIjjan +

(0 9)

+ 1),

In fact, the estimates establishing (0.8) show that, for 0 < s < N, (0.10)

<

+ 1),

with (0.11)

KN(F,u) = IF'IICN(o)(l +

The symbol class S?1 obtained in (0.7) lacks many desirable properties. In order to have a symbol calculus available, one splits such a symbol as M(x, into two pieces, as discussed in §3 of Chapter 1:

= M#(x,e) +

(0.12) where

M#(x,e)

(0.13)

=EJkmk(x) ?Pk-hl(O,

and Jk are smoothing operators (in the x variable), forming an approximate identity. Possible choices of Jk are (0.14) where 6

(0.15)

Jk

= Wo(2_"6D), or Jk

(0, 1). Given r > 0, we have u

cr

CV

n

M(x,€S

and

M#(x,e)

If we take 6 < I, then the standard symbol calculus applies. If instead we take Wk_5(D), then there is a replacement operator calculus, given by [Bon] and [Meyl]. We have M#(x, in the symbol class where (0.16)

Sh, and supp

C

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

104

for some p

and [HZ]. If

(0, 1). A more general operator calculus has been developed in [Bour2}

I3Sfl, then

(0.8) holds, for ails

lit.

There is the following representation of a product (cf. the discussion in §3 of Chapter I):

fg =

(0.17)

Tfg

+ T9f + R(f,g),

where Tf is Bony's "paraproduct," defined by

Tfg =

(0.18)

>i: 'Tik_5(D)f . k>5

This arises from the construction (0.1)—(0.14), applied to F(f,g)

4 given by the second formula in (0.14). Clearly (0.19)

Also,

fq. and with

OPBS?1.

1

if Rfg = R(f, g), a simple symbol estimate yields

f

(0.20)

OPS?,1.

Hence (0.8) applies to Tf and Rf. There are also the following important estimates (used already in §10 of Chapter I): (0.21)

(1, oo), which follow from work of [CM]; proofs are also given in [T2]. Prom (0.19)—(0.20) and the operator estimate (0.8) we have, for s > 0, 1

for p

CX), the Moser estimate

C

(0.22)

Compare

Hap +

(0.10). In subsequent sections we will see some variants of (0.22), such as

a result of [CW} in §1, and another variant in §6 of Chapter III.

Material just discussed will provide

following sections. Other tools maximal function (0.23)

some of the tools for the analysis in the

include known estimates on the

Mf(x)=sup r>0

I

vol Br(x)

Hardy-Littlewood

f

and Littlewood-Paley theory. We mention two results that will be used frequently. 2k, then, for p (1, First, if fk(e) are supported on shells s lit,

(0.24) k?O

k?O

If = ?I-'k(D)f, the converse estimate also holds. Second, if fk(e) are supported on balls iei C2k, then (0.24) holds, for p s > 0. Recall that these tools

have been used in Chapter I.

I. A PRODUCT ESTIMATE

1. A product estimate In this section we establish the following: PROPOSITION 1.1. We have, for s > 0, 1

+

ifgIIHaP <

IIfHJfst2

provided

(L2)

p

This

qi

result was

q2,r2E(1,00),ql,rlE(1,00].

r2

r1

q2

established in [CW]. Note

that the Moser estimate (0.22), i.e.,

IIYIIHsv +

!fgIIH•'P

is the special case = r1 = oc of (1.1). We give a proof that casts the analysis in terms of Bony's paraproduct. This will provide a warm-up for results in subsequent sections. As in the approach to (0.22) sketched in the introduction to this chapter, we begin by writing

fg =

(1.3)

Tfg + Tqf + R(f,g).

It suffices to show that, under the hypotheses of Proposition 1.1, (1.4) (1.5)

In

IR(f,g)IIH&P

fact, we have, for all s E IL 1/2

(1.6)

IgjIIvQ2.

Here, M is the Hardy-Littlewood maximal operator. This proves (1.4). Next, for

s >0,

cM( E (1.7)

C and,

Mf(E22k

as in (1.6), this last quantity is

2

IIgIIHaQ2, so we have (1.5).

am

z. PARADIFFERENTIAL OPERATORS AND NONLINEAR 2.

A commutator estimate

In this section we establish the following: PROPOSITION

2.1. Ifs E (0,1), p E (1. cc), then

IA8(fu) — fA8UIILP

(2.1)

result was given in [KPV}. As before, our main desire is to re-cast the in the language of Bony's paraproduct, and to motivate further arguments. As in treatments of commutator estimates in [AT} and [T2}, we write This

proof

A8(fu)

22

We will estimate four terms on the right side of (4.2) separately, and then estimate [A8,Ti]u. To begin, we have (2.3)

= R(f,u),

where

and hence

Vs

(24)

V

5

(Note

s>0.

that these results are special cases of (1.4)—(1.5).) Next,

(2.5)

u E

s > 0

TA3U E

Vs >0.

(2.6)

We next establish that (2.7) In

Vs >

IIR(f,A8u)IILP

0.

fact, by Proposition 3.5.B of [T2}, (keeping in mind that R(f,

v) =

have (2.8)

IIR(f,v)IlLr ç

as long as (2.9)

PE

P

:

-, BMO.

If X8 = then this criterion holds, and applies to give (2.7). It remains to estimate [A8,T1}u; for this we have the following. PRoPoSITIoN 2.2. Ifse(0,1), (2.10)

IASTIU

—

pE(1,cc), then

T1A3UHLP

CJIuIIcHIfIji,sp.

R(v, f)) we

2. A COMMUTATOR ESTIMATE

107

PROOF. We have [A8,T1]u =

E{A8((Wk_5f)tu)

—

= where

fk =

(2.12)

Wk_5f.

Due to the spectral properties of 1/.'ku and of 1k, we can write (modulo a negligible error) (2.13) where

[A8,Mfkjl/.iku =

4t(D) =

(2.14)

and 0(C) [A8, TjJu =

has

compact support. Thus we have [?/r , MfkJI/.Jku

and

(2.15)

({A8,Ti]uljtp

Now, with Vk = (2.16)

Ifk(x)

1k(Y)l

dy.

Furthermore, a result we will establish in the next section (Lemma 3.3) implies

(2.17)

fIfk(X) -fk(y)I

Plugging into (2.15), we get

(2.18)

LEMMA 2.3. Ifs < 1, then (2.19)

S

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

108

PROOF. Set

=

Then the left side

of (2.19)

is

equal to

>: >

k £
y k>niax(j.E)

The inner sum

is equal to C

2—2(1_a)

max(3O, provided

s

< 1,

so (2.20) is

equal to

(2.21)

provided

s < 1, where

=

(2.22)

It

=

2(i—s)[j

then elementary to dominate (2.21) by E

is

Returning to (2.18), we hence have, for s E

(2.23)

and this gives (2.19).

(0,

1), p E (1, co).

I[A8,TfIUIILP

Now there is the vector

maximal estimate S

(2.24)

of {FS1]; compare the use of such estimates in §9 of Chapter I. Thus we dominate

the

right side of (2.23) by

(2.25)

and

Proposition 2.2 is proved.

3. Some handy estimates involving maximal functions

= = e S(TRTh) and set some results of [CW], and some extensions. We begin

Here, take establish is

obvious. LEMMA 3.1. We have

f Next, write kPkv(x) — (3.2)

Hence:

=

v(x) -

-

dy

=

*

u. We

with a result that

3. SOME HANDY ESTiMATES INVOLVING MAXIMAL FUNCTIONS

109

LEMMA 3.2. We have

C.2kf

(3.3)

Mv(sx+(1—s)y)

Furthermore, (3.4)

for

Ix

5;

—

2-k =

Note that, v, obtaining

(3.5)

kPkv(x)

Ix — YI

2-k

C2kMv(x) Ix — yl.

—

=

NPkv(x)

1

on supp ?/)k, then we can substitute cokv


—

We next use (3.4) in an estimate of f 'e4'kv(x) — we have

-

f

—

—

dy.

First,

- y)I

.

(3.6) — yI .

I&(x —

dy

Next,

-

f

-

.

(3.7)

CN Using (1 + 231x —

f yIY' c

dy.

231x —

<

on

the region of integration, we

dominate this by (3.8)

Hence:

LEMMA 3.3. We have (3.9)

-

f

dy

This is useful when k <j. Otherwise, use (3.1), with v — In §5 we will need analogous estimates when a function 11(y) > 0 is thrown into the integrand. We begin with the following obvious extension of Lemma 3.1. LEMMA 3.4. We have (3.10)

f Iv(x) - v(y)I

-

dy < CIv(x)IMH(x) + CM(vH)(x).

110

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

Next,

extending (3.6), we have

-

f

- y)j H(y)

.

(3.11)


-

•

H(y) dy

MH(x),

and, extending (3.7)—(3.8), we have

tv(x) -

f

- y)i H(y)

Ix_yI2_k

(3.12)

f


-

+ .

dy

MH(x) +

Hence there is the following extension of Lemma 3.3: LEMMA 3.5. We have

f

(3.13)

.

4.

-

.

H(y) dy

MH(x) +

A composition estimate

Our goal here is to establish this result of [CW]: PROPOSITION 4.1. Assume F(O) = (1,cx), (4.1) In

0 and iF'i


Then, for s

CKiluiijp.p.

the proof, we will

want

to estimate

=

dy

(4.2)

= when j

1.

-

- y)

dy,

Using

(4.3) we have, for j

(4.4)

f[F(u(y))

F(u(y)) -

- u(y)j,

1, dy.

E (0,1), p E

4. A COMPOSITION ESTIMATE

111

Using Lemma 3.3 and the comment following it, we dominate the right side of (4.4) by dy

(4.o)

kj

k.cj

Now

(4.6)

(neglecting the term j =

and

0)

we have

o

> (4.7)

CK2 E2238(>2k_lMwku(x)) k<j j

(L Mcoku(x)).

+ CK2 > We use the following: LEMMA 4.2. Ifs < 1, then S

(4.8)

Ifs> 0, then (4.9)

The inequality (4.8) was established already in (2.19). To prove (4.9), so the left side of (4.9) is equal to we again set bk = PRooF.

(4.10)

with Hence

=

j

k>j £?j

= 2_sIk_tIC (4.10) is dominated by CE bId2, which gives (4.9).

Using Lemma 4.2 to dominate the right side of (4.7), we get (4.11)

oujdHs,p S IlWo(F

+

22k8IMwkuI2)

and, as in the estimation of (2.23), we can use (2.24) to dominate the last term by 1/2

(4.12)

CKh(E22181c01u12)

proving Proposition 4.1.

2. PARADIF'FERENTIAL OPERATORS AND NONLINEAR ESTIMATES

112

5. More general composition estimate in this

section, we establish the following result.

PROPOSITION

5.1. Assume F:

R' is

Tffl

F'(rv+ (1- r)w)I

(5.1)

a

C' map, satisfying F(O) =

0

and

p(r)[G(v) + G(w)J,

givenG>O, pEL1([O,1]). Then, forse(O,1), pe(1,cc),

IF o UIIHS.P

(5.2)

o

jkLIIua.Q2,

provided

(5.3)

q,

p

This was established The

in [Sta], in the case that (5.1) holds with C(v) =

IF'(v)I.

more general statement above is from {K]. In

(5.4)

the proof, we again want to estimate (4.2). This time, we replace (4.3) by

F(u(y)) — F(u(x)) C{H(x) + H(y)] Ju(x) — u(y)),

H(x) = C o

and hence we replace (4.4) by o

u)(x)I S CH(x)

(5.5)

f

u(x) -

+ cf u(x) -

dy

.

- y)j H(y) dy.

.

The first term on the right has an estimate of the form (4.5), while Lemmas 3.4-3.5

apply to the last term in (5.5).

Thus, as in (4.7) (again neglecting the term j

we have

oi4x)12 S C(MH(x))2

+ C(MH(x))2 E

2238

(E Mcoku(x))

3

5.6)

+ + Jsing

Lemma 4.2 to dominate the right side of (5.6), we have IF

5.7)

CE2238(EM(kbkupH)(x)).

S

22k8IMykuI2)

ou)IIHS.P + CM(MH) k

2)

0)

6. CONTINUITY OF U

f(u) ON

113

Using (2.24), we dominate the second term on the right by

(y

(5.8)

JJ8Q2.

Similarly, we estimate the last term in (5.7) by (5.9)

Lq'

LP

Thus Proposition 5.1 is proved.

6. Continuity of u

f(u) on

Throughout this section, we assume 1

except 1; we will find it convenient in (6.8)—(6. 11) to allow I to be complex valued.

then f(u) 6

PROPOSITION 6.1. If u 6 PROOF. Use

f(u(x+y))

(6.1)

M is the Lipschitz constant of f. In that way, one sees that (6.2) A simple localization

allows us to obtain the following result for arbitrary open

1? C 1R71, even with very rough boundary:

PRoPoSITIoN 6.2. If u

then f(u) 6

6

and (6.2) holds.

In [Zie] can be found the following result, for f Lipschitz, u 6 (6.3)

Vfou(x)=f'(u(x))Vu(x),

a.e.

particular, f'(u(x)) is well defined a.e. on {x 6 Il: Vu(x) implies (6.2). A particular case of this is that, if u 6

In

0}. Of course, (6.3) then

Vu(x) on {xeIl:t4x)O} on {xeII:u(x)<0},

—Vu(x)

and (6.5) We

Vu(x) =

0

a.e., on {x 5 Il u(x) = 0}.

mention that (6.3) is straightforward if also

f is

C'. The

results

(6.4)-(6.5)

can be obtained from this by a simple limiting argument; cf. [CT], pp. 151--152. However, (6.3) seems to be

We

more delicate for general Lipschitz f.

next establish a fairly simple, but weak, result on continuity of the map

u in

then f(uj) —* f(u) weakly in

2. PARADIFFERENTIAL

114

AND NONLINEAR ESTIMATES

PROOF. By Proposition 6.2, {f(u1)} is a bounded subset of given p E (1, cx)), the Banach space is reflexive. sequence which converges weakly, say to g E from (6.6)

S

—

that

1(u) in

L"ffl),

Now,

Hence, there is a sub-

However, it is clear

ul

—

so g = 1(u). The consequent uniqueness of the weak

limit proves the result.

The following gives a

significant improvement in the conclusion of Proposition

6.3. The proof differs only slightly from one in [MM]; that paper also treats the case p =1. PRoPoSITIoN 6.4.

If

u

in

then f(ug)

f(u) in

norm.

PROOF. It is clear that each 1

(1,.

—p

1(u) in Lu-norm. It remains to show that, for

.. ,

—p

(6.7)

Ojf(u) in

L"-norm.

We will first show that, if

=1

(6.8)

then,

as j

(6.9)

for

a.e.,

each £.

—p

Since

LT'ffl)

(u)IILP,

is uniformly convex for

each p E (1,

this

together with

in Proposition 6.3, will imply (6.7); cf. [Ko]. (This type of argument is rather familiar in the case p = 2, where L2(Q) is a Hilbert space.) In fact, using (6.3), we have = for a.e. x e fI, whenever (6.8) holds, and hence so (6.9) is clear, and hence we have the weak convergence established

(6.7), whenever (6.8) holds. We now establish (6.7) for all Lipschitz f. To do this, it suffices to assume the Lipschitz constant of f is S 1, so f'(s) S 1 a.e, It is then routine to write (6.10)

f'(s)

with 9k measurable,

(6.11) and

+ 92(5)],

=

Jgk(5)j

1

a.e.,

and, upon setting fk(s) = 509k(i7) da, we have f(s) =

+

the argument above applies to

for

each Ic. This finishes the proof.

Next, define functions 7b on JR by 0

(6.12)

=

if

s—} if s-i-1

if s<—1.

6. CONTINUITY OF u

In

other words, flj(8)

=

f(u) ON

115

where

$ —


if

S

(6.13)

if

if PROPOSITION

6.5. If it E

PROOF. It suffices to show that

U-norm.

0

in

Clearly

0

in

Meanwhile, by (6.3),

(6.14)

o it

V(3(it)=Vu

\Z=

(x

E 11:

it(x)

=

on

on

0

Now

it in

then

= O}, and, by the monotone convergence

dx NJ

=J

(6.16)

theorem,

dx =

the last identity by (6.5). This proves the proposition.

We apply some of the propositions above to deduce some results about the in H1P(Ifl. These results defined as the closure of structure of of [T51 for are of interest when OfI is perhaps very rough; cf. Chapter 9, applications in the case p = 2. LEMMA 6.6. Let R C fI (6.17)

be

open. Then

u=0

UE

PROOF. It suffices to assume it is real valued. Then the hypotheses apply to

and iC, so it Then

so

Now, with Tb

as in (6.12), set

=

(6.18)

Since min(u,w)

it in

it in

0 on 11. Take it,

suffices to assume it

= (it—w)+w, we

see that

e

and v,, —÷ it in

we have (6.17).

LEMMA 6.7. Let be open in RT1, component of Q1 fl 112. Then (6.19)

iti=it2=it on

Let (9 be a connected

116

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

PROOF. It suffices to assume are real valued. The hypotheses imply ut = uE on 0. Thus it suffices to assume in addition that and =

4 4

0 on R• Now we can find

Ui

H1"(111) and

such that V11 —4

WV

in

—+

Hence vj —+

u1

and 4 —j'

in

in

Now

has the properties Wv —i

Wv

u in

so (6.19) is proved.

7. Estimates on F(u)

We can estimate F(u)

(7.1)

—

—

F(v)

F(v), using

F(u)

F(v) = G(u, v)(u —

—

with (7.2)

f F'(ru + (1 — r)v)

G(u, v)

together with

estimates on G(u,

dr,

v) and product estimates. The simplest case is the

following.

PRoPosITION 7.1. Assume F

Then, wheneverp

(1,oc), s >0, 11cc

we have —

(7.3)

5;

—

(1+

+

PROOF. Thewell-known

special case qi = r1 = oc, = V2 p of Proposition applied to the right side of (7.1), dominates the left side of (7.3) by the first term on the right plus 1.1,

(7.4) Then,

CIlu the estimate (0.9) applied to G(u,v) yields

(7.3).

We remark that (7.3) is both sharper and more general than the estimate (3.8) in [Sog]. It also contains the estimate (3.8') of [Sog] when n is odd, but not when n is even. However, as we show below, (3.8') is false when it is even. The following provides an estimate on F(u) — F(v) when F is only C2. PR0P0SJTION 7.2. Assume F is C2. F'I < K1. and IF"I 5; Then, for

sE(0,1),p€(1,oo), IZccR",we/zave (7.5)

IF(u) 5; CKflJu —

+ CK2(1 +

+

—

7. ESTIMATES ON F(u) — F(v)

117

PROOF. As before, the product estimate dominates the left side of (7.5) by the

first term on the right plus (7.4). Applying Proposition 4.1 to the estimation of G(u,

v) in

this case gives the rest of (7.5).

Actually, to apply previous estimates to G(u,v), we need G(0,O) = 0, i.e., This can be arranged by replacing F(u) by F(u) — F'(O)u, a change that does not affect the validity of the estimates (7.3) and (7.5). F'(O) = 0.

Next, we give a counterexample to the following: ASSERTION. Let

tion. Then, for u, V

bounded subset

B be a

E

F a smooth func-

B,

C(F,

— F(v)II

(7.6)

—

VIIHI.

This is a special case (with n = 2) of the estimate (3.8') in [Sog], which is not correct. On the other hand, as mentioned above, the estimate (3.8) of [Sog] is correct, so none of the major results of that paper are affected. We take F(u) = u2. Then F(u) — F(v) = u2 — = (u + v)(u — v). Note that

consequently

a correct result, and a special case of Proposition 1.1, is (7.7)

Setting w = u — v and re-labeling u + v as F(u) = u2, then, given B C H'(T2) fl

u,w

(7.8)

Now,

we see that, if (7.6) is correct for

u,

bounded,

B

E

we can pick a,b

for arbitrary nonzero u,w c

= 1,

IIauIIHl +

IIbwIIui

such that

e

+

1.

Hence (7.8), applied to au, bw, yields

(7.9)

(ab)lIuwIIHl SCbUWIIHI.

Dividing

by b, we deduce that, for all u,v

(7.10)

IIuwllHl

Hence, this estimate must show this is not possible. Since

+

<

hold for all u

and IIUVWIIL2

E C9T2),

are

H'(T2) fl C(T2), w

H1(T2). We will

both dominated by the right side of (7.10), we

see that (7.10) holds if and only if (7.11)

ilwVuIIL2 < Q(j(Vu11L2 +

Let us set 1

for

xI

= 'Ø(x/e), 1/4. Let w(x)

supported in xI cc

given

=

so w

(Identify opposite sides of the unit square in ]R2 (7.12)

+

IIuelItno

1/2, equal to

H1(T2) but w

L9T2).

to get T2.) Then

C0,

of e (0, 1]. Thus the right side of (7.11) (with u = ue) is bounded for e (0,11. However, the left side of (7.11) blows up like (log fl"3. This shows that (7.11) is false; hence (7.10) is false, so (7.6) is false. independent

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

118

8. A pseudodiffereritial operator estimate Here we combine methods of [Bour] with methods of §1 to produce estimates on p(x, D)u. As with a number of estimates in Chapter I, we begin by considering an "elementary symbol:" (8.1)

= k>O

with cck(e) supported in 2k-1 < Let us write

< 2k+i (for k 1) and bounded in S?o(Rhl).

Qk(x) = Qkl(x) + Qk2(x),

(8.2) with

<2k+5

Qkl supported on

3"

Qk2 supported on iei

Thus, q(x, D)u = qi (x, D)u+q2(x, D)u, and we estimate these two terms separately. First, for all s > 0, p E (1, x), liqi (x, D)uII ff8P

lQkI

2

1wku12)

k

(8.4)

CQ1(>22Th0ku12)

12 /

where

Qi(x)=sup IQkl(x)I.

(8.5)

k

Hence (8.6)

a >0, p E (1,oo),

C

provided that (8.7)

= p

qi

+

q2

e

q2 E (1,oo).

Next, in view of the spectral properties of

we have

(8.8) k<j—4

Let us assume that, for some a > 0, T

0

(8.9)

Vj ? Iv +4.

We then dominate the right side of (8.8) by 1/2

2

(8.10)

j

)

8. A PSEUDODIFFERENTIAL OPERATOR ESTiMATE

119

Hence (by a variant of (4.8)), (8.11) provided

S


that

=

(8.12)

p

+

TI

r1 T2

For an elementary symbol Q1(x)

(8.13)

E (1,oo], r2

E (1,oo).

we have


sup

where M is the Hardy-Littlewood maximal operator and

q4(x)

(8.14)

Hence, for qi E (1,oo], (8.15)

:s;

On the other hand, we have sup

(8.16) Note that we can set

It follows that

=

(8.17)

Q2(x)

C

where

= sup

(8.18)

(e)—r

C

Hence, for r1

E (1, oo],

(8.19)

MQ2IIU'

an elementary symbol, 1

Consequently, when q(x, and

is

<

(8.20)

+

r)r2.

The dependence in (8.20) on e-derivatives is hidden, but implicit in the hypothesis that q(x, is an elementary symbol; the "constant" C in (8.20) depends on bounds on (e)knID?cck(e), for some finite range of a (as well as qj,rj, and other parameters). A general symbol can be written as a sum of elementary symbols (8.21)

p(x,e) =

= £

with (8.22)

k)O

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

and Qkl(x) obtained from a Fourier analysis of for some a C Details on this are recalled in §2 of Chapter 1. Thus, we have:

on

PROPoSITION 8.1. AssumeO<s
For sufficiently large N, take

= sup sup t

(8.24)

and

p(x) =

(8.25)

t

IaI
Then (8.26) 9.

Paradifferential operators on the spaces

function on Let F be a regularity, we will analyze F(u)

as

If u is a function (with values in R") with a paradifferential operator. As in (O.1)—(O.4),

we can write

F(u) = F(uo) + {F(ui) — F(uo)]

(9.1) with

uk = Wk(D)u,

(9.2)

and

+ ...

+ [F(uk÷1) — F(uk)]

then we write

F(uk+1) — F(uk) = mk(x)?i'k+l(D)u,

where (9.3)

mk(x)

=

f

dt

F'(Wk(D)u +

Hence (9.4)

F(u) = Mp(u; x,

D)u + R(u),

where

(9.5)

(9.6)

R(u)

= F(tbo(D)u)

MF(u;x,e) =

We have

(9.7)

MF(u;x,e)

u

E

As we noted in the introduction to this chapter, if r > 0, (9.8)

u E

c'

MF(u;x,e)

E

+"m

9. PABADIFFEB.ENTIAL OPERATORS ON THE SPACES

121

We want to draw an analogous conclusion when we assume u E To do this, we need to estimate Imk Ic(Ah which is essentially equivalent to an where estimation of

Vk4!k(D)U+flbk+1(D)U,

(9.9)

07-<1.

Note that, parallel to (9.4), we have

F'(v) = MF'(v;x,D)v+R(v),

(9.10)

and

Now

Mc"(v;x,E)

v E LOC

(9.11)

applying Proposition 5.12 of Chapter I, we have

Cp((IvIIJ]oo)(1 +

(9.12)

provided

A2(i) \ and 5 CA(j).

(9.13)

ej (In

fact, (9.12) can often be seen to hold via a modulus of continuity argument.)

Now

(9.14) so

IVkIlcxA2

we have

(9.15)

ImkHccA)

+

Hence, if (9.13) holds, and E

Coo, so

ue

E

(9.16) We

c

L°°, we have K(2k)

recall from Proposition 5.2 of Chapter I that

(9.17)

E

==*

p(x,D) c(A)

provided C 0°,

(9.18)

i.e., in the setting of (9.16), provided A(k)

E A2(k) C

(9.19)

00.

Thus, if (9.13) and (9.19) both hold, we have (9.20)

U E

Mp(u;x,D) : c(A)

so

(9.21)

+

=

2. PARADIFFERENTIA L OPERATORS AND NON LiNEAR ESTIMATES

122

For example, we can take

(922)

Then (9.13) holds for s > 0, and (9.19) holds for s > 2. The classical case, with c(A)

is

A2(j) =

A(j)

(9.23)

1,

r>

0.

Of course, in this case, we already have OPS?1 acting on C, so the exponent of 2 in (9.21) is not needed. We want to extend the collection of A(j) for which it can be shown that (9.20)

holds. As a first step, assume we have p(j) \ such that Ap satisfies (9.13) and (9.19), i.e., (9.24)

A(k)1i(k)

EA(t)p(t)A2(t)

Then, parallel to (9.21), we have (9.25)

+

Hence we can replace (9.12)—(9.15) by

+ (9.26)

2

1

Thus, when (9.24) holds,

(9.27)

Mp(u;x.e) e

u E

This time, Proposition 5.2 of (9.28)

provided

=

Chapter 1 gives

p(x,D)

e

0(A)


(9.29)

This

condition fits together with (9.24) best when p(j)

=

We thus see

that if (9.30)

EA(4\2&)3/2
< Ic

then

(9.20) holds, and we have

(9.31)

IF(u)IL'A

We see that (9.30) holds for

(1+ as in (9.22), with s >

3/2.

9. PARAD1FFERENT1AL OPERATORS ON THE SPACES

123

We can iterate this argument. Suppose Ap satisfies (9.30), i.e.1 A(l?)p(t)A2(4?)3"2

> (9.32)

CA(j)p(j)A2(j)'12,

1/2

> A(k)p(k)A2(k)

Then we have (9.25), with the exponent 2 replaced by 3, and similarly for (9.26). Thus, when (9.32) holds, (9.33)

Mp(u;x,e) e c(AP)S?(y),

u E

_

Another invocation of Proposition 5.2 of Chapter 1 gives (9.28) in this case, provided

<00.

(9.34)

Taking

p(k)

=

we

see that (9.32) and (9.34) both hold provided

<00.

EA(t)A2&)5/3 <

(9.35)

Then (9.20) holds and we have

+ IIuIIcA)).

(9.36)

We see that (9.35) holds for A, A2 as in (9.22), provided s Inductively,

PR0POsvJON 9.1. Let A(j) \ varying

A2(j) \

> 4/3.

we have the following. 0

be slowly varying. Suppose there exists slowly

that

such

< CA(j)A2(j)

(9.37)

EA(k)A2(k)

.c oo,

Ic

for somevEV. Then (9.38)

u

e

MF(u;x,D)

—>

and we have

+ IIlc(A)).

(9.39) We see that (9.37) holds for

(9.40)

A(y)=j

A2(j)=y

1

,

We now produce further information on the symbol M1.' (u; x, when u and A(j) satisfies the hypothesis (9.37). Then, parallel to (9.28), we have (9.41)

+

which we can use in place of (9.12) to obtain (9.42)

+ IkuIIC(A)),

e

2. PARADIPPERENTIAL OPERATORS AND NONLINEAR ESTIMATES

124

and hence, for )t(j) satisfying (9.37),

ue

(9.43)

Mp(u:x,e) E

substantial improvement over (9.16). We next obtain some more estimates on From the chain rule, we have a

and hence

(9.44)

the sum being over flt, > 0, if 0 > 0. We can deduce that, if )t(j) \ satisfies (9.37), then IIC(A)

(9.45)

.. .

>i:

This uses (9.39) plus the fact, established in Corollary 5.4 of Chapter I, that is a Banach algebra (as long as EA(j) coo), so that (9.46)

IIVWIIC(A)

Ii

Now, if v = vk has the form (9.9), we see that IlD0vkIIcA) = sup I?

C sup

(9.47)

I?

C SO

(9.45) yields

+

(9.48)

It follows that rnk(x) in (9.6) has such an estimate, and hence, for )t(j) satisfying (9.37), (9.49)

e

We are ready to record the following further properties of MF(u; x, PRoPOSITION

(9.50)

9.2. Under the hypotheses of Proposition 9.1, u e

MF(u;x,e)

E

Furthermore, we have the decomposition

(951)

MF(u;x,e) =M#(x,e)+Mb(x,e) = M#(x,e) + +

where the terms on the right have the following properties: (9.52)

M#(xC)

FD?M#(.,e)IICCA)

A. PARACOMPOSITIOM has support in

(9.53)

E

has

24JC(,

ku

125

support in IiuI

?

and

SjJ',

where (9.54)

with çh

= A(j),

(9.55)

&(h)

=

j

dt.

0

PROOF. We already have (9.50) and (9.52), from (9.42). To proceed, note that (9.48)

gives

+

(9.56) Hence (9.57)

k

5;

k

A

5;

=

These estimates yield (9.53)—(9.55).

We

note that the proof of Proposition 5.12 of Chapter I shows that

—s

(9.59)

P2(i) =

and (9.60)

We

P3(i) = i4i) >A(j).

M1(x,D)

have /22(j) 5; Cp3(j), and

typically /22 A. Paracomposition

F

In this appendix we discuss a construction of [Al], applied to a composition o u, and extend some of the estimates given there. The basic thrust of this

despite a similar appearance of material is somewhat different from that of the objects imder study. For one thing, it is necessary to assume here that u is a diffeomorphism. As we will see, that assumption will play a crucial role in Lemma

A.2.

Throughout this section, we make the standing hypothesis that all functions F have support in some fixed compact set. Then quantities like 'IuUca can be interpreted in that light. Also, subtracting off a smooth function (whose composition with u is estimated by previous techniques) we assume F(0) = 0.

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMA'I'F)S

126

=

To begin, set

and write

Fou >[13(Wku) - F,(Wklu)]. j,k

and

We decompose the double sum into (A.2)

Note

that. due

to cancellation,

= EFk(Wku).

—

j>k

kO

Meanwhile,

(A.3)

—

Fj(Wk_lu)J =

—

j
(Wk_1F)(4'k

k?1

hence °

F a u(x) = 4F(u; x, D)u +

(A.4)

(Wku)(x),

k >0

where

= k?0

(A.ö)

1

+ (1 — = f 0(WkF)'(rWk÷lu With 4'k+r = 7-Wk÷1 + (1 — r)Wk, we can estimate Ak(x)

a

(AM)

dr.

= a

>

In particular, for /9 > 0, (A.7)

C

IWk÷luhcølI

>:i: 1

131

Of course, (A.8)

One consequence of (Al) is the following, in case u is Lipschitz. Note that, for (AS)

s

02k(I/3,j

Hence (A.7) implies C

(A.l0)

0

=

C2k3 E LcZlvl
127

A.

Hence

F,u

(A.12) p

Lip'

(1, oc), x, D)UIIHS,P

(A.13)

:5;

c

Now a variant of Alinhac's "paracomposition" operator is given by (A.14)

wk(C)= k

N

Here,

is

which

a number, determined by

will

be specified in Appendix

B. If we set (A.15)

then,

Fk 0

(Wkn)(x) —utF(x),

from (A.4), we have

Fou(x)

(AiM)

+utF(x)+

=

In

fact, the operator defined as

(A.17)

u*F =

paracomposition in [Al]

is

ou).

We estimate the difference of (A.14) and (A.17), given by

= u*F

(A.18)

—

utF = >Wk(D)gk

with gk(x) = Fk(u) (AdO)

—

Fk(4'ku) = bk(x)(I —

bk(x)= / FLfrU+(1—r)Wku)dr. Jo

LEMMA A.1. Given s > 0, we have

(A.20)

C

PROOF. We have

(A.21)

—

:5;

Now (A.22)

s > 0

ks —

so (A.20) follows.

We next estimate if F. The

following is due to [Al].

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

128

LEMMA A.2. We have IIu*FIIc:

(A.23) PROOF. By (A.14),

i4'j(D)u*F=

(A.24)

j+N

C

(A.25)

CN2_Jr k=j-N

This gives (A.23).

We next estimate the "remainder" RFu in (A.16), folowing [Al]. LEMMA A.3. Assume u is a diffeomorphism of class CS, that u' is uniformly is uniformly bounded. Ifs> 1 and r + s > 0, then continuous, and that

(A.26)

IRp,uMc:+a

PROOF. Note

(A.27)

that

= Eu — Wk+N)(Fk °

+ EWk_N(Fk

We use the following two estimates of [Al], which will be discussed further in Appendix B: (A.28) for

2k(u-s+1)

°

j k + N, v > s —

1

0, and if also k K (depending on

and

IIDu1 (A.29)

2-k(s-1)

IWk.N(Fk o

will be seen in Appendix B, the hypothesis that u is a diffeomorphism is needed to establish (A.29). We can estimate in two parts. First, As

(A.30)

Eu - Wk+N)(Fk Ck<j—N E IkPj(Fk a 'Pku)lILoc Cu(MuIIcs) >

2jv2k(v—s-r)

k<j-N

by (A.28). Taking v> s + r, we dominate the last espression by (A.31)

r('+8)2.

A. PARACOMPOSITION

129

Next, we have, for k K, 0 WkuL)M

(A.32)

k>j-f-4 IIFIL;:+i,

by

(A.29). Combining this with (A.30)—(A.31), we have

2fr+sb,

(A.33)

which gives (A.26), upon taking note of the dependence on N in these estimates, and crudely estimating the last sum in (A.27) over k < K. We summarize what has been done above: is

PROPOSITION A.4. Assume it is a diffeornorphism of class C8, s> 1, that u' uniformly continuous, and (u')' is uniformly boundS. Assume F is Lipschitz.

Then (A.34)

Fou(x) = n*p(x) + 4F(u;x,D)u +

+

where the paracomposition u*F, given by (A.1 7), satisfies estimates (A.35) the second term

<

has

the property

F,u E Lip'

(A.36)

e OPS?1,

and the remainder terms satisfy the following estimates. (A.37)

and, for s 1. r (A.38)

s > 0, + 8

> 0,

IRF,tJIc:+s

[Al] there are estimates on L2-Sobolev norms of the various quantities in norm estimates. Such estimates on estimates x, D)u are already recorded in (A.13). We next give on the paracomposition. In

(A.34). Here, we produce some

PRoPosiTioN A.5. ff1

Iu*FJI1I3p <

ML')

gap.

2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

130

PROOF. In view of the location of the spectrum of the various terms in the sum (A.17) defining u*F, we have 1/2

(A.40) Now, since u is a diffeomorphism we have IIGOUIILP C(I1DU_hIlLoc), so the right

side of (A.40) is (A.41)

This gives (A.39).

C=

estimates on

We next obtain

PROPOSITION AM. If p E (1,cc) and 5> Ikt*F —

(A.42)

0,

we have

CMF'Ijc:

r,p

PROOF. Using (A.17) and the estimate (,Ok(D)gk(X)I CM(bk(x)(I — Wk)u(x)) — CIIFL

(A 43) we

have —

22k(s+r)

u*FIIH.H k

(A.44)

—

Wk)flI)M.

We dominate the last factor by (A.45)

provided s > 0, by (4.9). Making use of (2.24), we thus get (A.42). estimates on RF' seem less accessable from the results of Appendix B than Zygmund space estimates. However, Lemma B.2 does readily yield Besov space estimates. Given 1
HI11

>ii:

Then the estimates (B.15) (B.16) yield the following analogue of Lemma A.3:

A- PARACOMPOSITIOIc

131

PRoPosiTioN A.?. Under the hypotheses of Lemma A.3, if 1
IIRF,UIIB;tt

The boundedness of operators in 0 PS?I on Besov spaces (for s > 0) is well known; see, e.g., [Ma]. We recorded the proof for the case q = 1 in §12 of Chapter

1. We deduce from (A.12) that, for 1 0, (A.48)

We now obtain Besov space estimates on the other terms in (A.34). PROPOSITION A.8. Under the hypotheses of Proposition A.4, if 1
IJDu1

<

PROOF.

In view of the location of the spectrum of the various terms in (A.17),

we have C

12

ic

—

(A.50)

f
IHPj(D)(Fkou)Iliv

>i: k

which

1< N

-

gives (A.49).

PRoPosITIoN A.9. If 1 cp,q< oo ands >0, we have

C( Dulftoc,

(A.51) PROOF.

We have

<

HPjD)(bku —

> !k—jl
-

(A.52)

>

—

Ik—iI1V

To

obtain (A.51) from this, it suffices to establish

(A.53)

>

—


s

> 0,

which in turn follows from (A.54)

<

s>0.

This estimate is proved by the same argument as used for (4.9). Thus we have Besov space estimates for all the terms in (A.34). One has (A.55)

2. PARADEFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES

132

so the L2-Sobolev estimates of [Al] are contained in these. Whether the set of is a matter we will leave V'-Sobolev estimates can be completed (to include unsettled, recommending it as a problem for the reader to consider.

B. Alinhac's lemma Here we prove an estimate which was used in Appendix A in the estimation of the remainder term in the decomposition (A.16). As before, take Fk = lPk(D)F. LEMMA B. 1. There exists N sufficiently large, determined by that the following holds. For j k + N, ii s — 1 > 0, we have

(B.l)

II'cbj(D)(Fk

°

S Cv(IIuIlcs)

such

2—jv2k(v—s+1)

Furthermore, if u is a diffeomorphism, u' is uniformly continuous, and (u')' uniformly bounded, then there exists K, depending on and such

that, for k K,

s

°

(B.2)

2—k(s—1)

IIDu

In view of the localization properties of i,i51(D), it will suffice to establish estimates

such

when we throw in a factor A0(y) E 4/ku)),

etc.

i.e., consider estimates on Also, to simplify notation, set

(B.3)

Vk

=

Wku

The proof of Lemma 131, taken from [Al], makes use of some techniques one encounters in the theory of Fourier integral operators. To begin, write

ovk))(x) =

f

dy

(B.4)

=

ff

dii dy

Set

=

=

(B.5)

— y

.

1 / v,,(y)

"

Note that, if N is picked large enough, then (B.6)

provided j

—

k + N.

ei

CfliiI

+

Fj)

on supp

In fact, it suffices that

2" > sup

t4(y)j.

Furthermore,

= Thus, we can integrate by parts many

(B.8)

ff ei(+X

times, and write (13.4) as ii,

dii dy

B. ALINHAC'S LEMMA

133

with

= Note that support of

ii, C) is homogeneous of degree —'.' in (ij, and smooth on the Also (on the cone generated by this support) on

5

+

=

1,

with similar estimates on all C)-derivatives, assuming U E C9. Next, on a box containing supp for all j k, write

= > (cx.fl)EA

where A is an appropriate lattice. Note that (B.12)

akva/3(y)l < CN2

s+1)(1 +

+

We see that (B.8) (hence (B.4)) equals (B.13)

ff

2

= 2_3v

f

"i-vC

dy dC

Fk(vk(y) +

dy dC

= 23VEJFk(vk(Y)+

— y) +&) dy.

By (B.12), this is dominated by the right side of (B.1), so (B.1) is established. replaced by To establish (B.2), we analyze a quantity like (B.4), with Our hypotheses on u and imply that, for N large enough, and k large enough, (B.14) so

+ Cl) on supp

—

we have an argument parallel to (B.8)—(B.13), proving (B.2). Behind (B.14) is the fact

Then (B.13) also yields

that,

at least for large k,

estimates.

is uniformly bounded.

We record the variant of Lemma B.1 so

produced.

is uniLEMMA B.2. If u is a diffeomorphism of class Cs, s > 1, and and K, depending on forrnly bounded, then there exist N, depending on Ilullcs and

(B.15)

IlDu'lluo,

such

that,fork K, 5 k+N, ii s—i, and 1

OWku)llLP S

2—jv2k(v—s-I-1)

:5; 00, IIFkIILP,

and (B.16)

114'k_N(Fk ° Wku)lIf, S C(Ilullcs,

2—k(s—1)

IIFkI1LP.

CHAPTER 3

Applications to PDE Introduction In this chapter we apply some of the results of Chapters 1 arid TI to some problems in PDE. We provide a sampling of applications rather than any systematic development, as the main focus of this work is on the internal development of the theory of various classes of operators. Tn §1 we produce some results on regularity and Fredholm properties of elliptic differential operators with mildly smooth coefficients. An example is the Laplace operator on a Riemannian manifold with metric tensor of limited regularity. Among the various hypotheses on the mnetric tensor we consider, we mention particularly 92k

E cT, C(A),

or I7 fl vmno.

Tn §2 we study some natural first-order differential operators arising on a manifold with a Riemnannian metric that is Lipschitz in local coordinates, making contact with work in [Mor] on the Hodge decomnposition on such a class of T1.iemannian manifolds; we also produce some results when the metric tensor is in the Zygmund One particular operator we study here yields the trace-free part of the class deformation tensor associated to a vector field on a Riemannian manifold. Previous studies of this operator have played a role in works on quasiconformal mappings. Tn this section we also study the Heltrami operator

B= for

k<

1.

8

-

8

We establish regularity results in case A E

flvmo and in

case A e 2 are pursued further in [MT], which Some of the methods developed in considers the method of layer potentials on Lipschitz domains in Riemannian man-

ifolds with C' metric tensors, and also in [MMT1. This point will be discussed further in Chapter Tv. In §3 we produce some estimates on solutions to the Dirichlet problem, particularly with boundary data in C(k)(8fl), for various domains in R". We include some results on regions with nonsmnooth boundary, such as arbitrary convex domains in R't. Some of these results will be extended in §4. Others will be of further use in §5.

Tn §4 we consider the method of layer potentials for boundary problems on domains whose boundaries are C1w regular. We obtain results when w satisfies a Dliii condition. We also obtain mnore precise results when the boundary is C1+ regular, for some r E (0. 1). The study of boundary layer potentials on such regions

o

TO POE

is classical; cf. [Mik] and references therein. Some of the results of §4 reproduce results there, while others carry the study further. Results of this section can be compared and contrasted with results discussed in Chapter IV, for Lipschitz domains. In §5 we study trace asyinptotics for the semigroup where L = —LI + V and LI is the Laplace operator on a compact mainfold M, with a Riemannian metric for some k e r E (0, 1). We produce k+1 terms in the expansion. of class In §6 we study the Euler equations for ideal, incompressible fluid flow, on rough planar domains. We obtain global weak solutions on a class of domains that includes all bounded convex domains on the plane.

In §7 we produce a result on persistence of solutions to a class of semilinear wave equations, assuming an a priori estimate on the solutions weaker than required

by standard techniques. This complements results on persistence of solutions to quasilinear equations established by the author in [T2]. These conditions are of a different nature from results in [BB] and [KM]. In §8 we discuss div-curl estimates. The most basic div-curl lemmas provide v are vector fields on 1R3 for which there are knowu u estimates on (i) norm bounds on u and v, (ii) norm bounds on div u and curl v, estimates that are stronger than those obtainable from (i) alone. There are many ramifications of this. The sort we discuss here pertain to knowing that

pe(1,oc),

pp

and using bounds of the form (ii) to conclude that v belongs to the Hardy space rather than merely to L'. Such a result and many variants were presented in [CLMS]. We prove a number of these results here. We use two unifying principles, the "supercommutator estimate" produced in §10 of Chapter I, and an "abstract div-curl lemma," which will be stated and proved in §8 of this Chapter. In §9 we consider the problem of constructing harmonic coordinates on a neighborhood of a point in a Riemannian manifold with rough metric tensor, including (in two dimensions) the problem of constructing isothermal coordinates. We make constructions for metric tensors rougher than Holder. In §10 we consider Riemannian manifolds with bounded Ricci tensor. We begin with fairly weak assumptions on the metric tensor and conclude that there exists a coordinate system in which the metric tensor has two derivatives in bmo. This result builds on a number of previous works, beginning with [DeTK]. We also estimate the metric tensor under some other hypotheses on the Ricci tensor. In §11 we study propagation of singularities for solutions to linear PDE (or pseudodifferential equations) with coefficients of limited regularity. We obtain results on propagation of singularities along null bicharacteristics for operators whose coefficients are of class C'', so the associated Hamiltonian vector fields have Lipschitz coefficients. Then we extend these results to the case of operators whose coefficients have "log-Lipschitz" gradients. This class includes operators whose coWe have results both for non-divergence efficients belong to the Zygmund class form operators and for divergence form operators. For example, results here apply

1. INTERIOR ELLIPTIC REGULARITY

137

to solutions to the wave equation = 0,

—

is the Laplace operator on a Riemannian manifold with bounded Ricci

when tensor.

1. Interior elliptic regularity In this section we investigate solutions to various elliptic PDE with mildly smooth coefficients. We first take a look at a second order elliptic equation in divergence form:

= j..

(1.1) We assume A2k E

where w is a modulus of continuity and .X(j) =

The equation (1.1) might be an L x L system. We write it as

E

(1.2)

and then,

using the symbol decomposition of §3 in Chapter I, write

= A7(x,€) (1.3)

Ar(x,e) e S e (0, 1) and require (as a special case of (3.22) of Chapter I) (1.4)

r(e)

(CY

where

a(h)

=

f

dt.

We make the following hypotheses:

A(j)r(23) \ 0,

(1.6)

lim r(e)'

(1.7)


= 0,

We recall from Chapter I that these hypotheses imply operator bounds on various function spaces, for example :

One example to keep in mind is (the r = 0 case of) the example discussed in (3.28)—(3.32) of Chapter I. Assuming 0 <7 <s — 1, we take

(1.8)

A(j) =

=

=

and a(23) = Another, more classical example is the = in which case j41? case A(j)=2_Jr, r in(1.3). Then

The following is our first regularity result.

3. A['PLICATJONS TO POE

satisfies the elliptic system (1. 1), with

PROPOSiTION I . 1 Assume that it Aik E C(A). Assume that the conditions

hold. Let )(k)/.\(k) /, and

assume that (1.9)

f=

Assume that (with r(e)

E

in (1.4))

as

for some M.

(1.10)

Then Vu E

The operator P# = D) is an elliptic operator in OPSI be a parametrix for fl. Then (1.1) is equivalent to

PROOF. Let

E#

E

u = E#f

(1.11)

mod C°°,

—

using the summation convention, so (1.12)

Oku

dkPfiDjfj

Note that DkE#8j

mod (1.13)

E

E

D)u.

—

so,

by Proposition 5.7 of Chapter I,

OkE#Ojfj e

Now, by Proposition 5.10 of Chapter I, under the hypotheses (1.6)—(1.7),

(1.14) where

E

that Vu

/2 is as in (1.10). We deduce from

pi(j) = max(p(j), A(j)). (1.15) Iterating

Applying

E

A(j)=j"8,

where

E

Proposition 5.10 of Chapter 1 again, we have /223) =

rnax(/2(j)t1i(j),A(j)).

this procedure a finite number of times, we obtain Vu E

Note that, in case (1.8), p(j) (1.16)

..

Hence

as

desired.

Proposition 1.1 holds for

01.

We also record the more classical type of result to which Proposition 1.1 can be specialized, namely (1.17)

ifrs>0.

In Propositions 1.4 and 1.10 below, we will treat variants of the regularity result of Proposition 1.1, with other hypotheses on u. Next we tackle some existence results, starting with the following result. For simplicity of notation we work on the torus V1 to start with, extending the resuk to other manifolds shortly. The spaces are as in (5.57) of Chapter

1. INTFRJOII ELLIPTIC ftECULAIHTY

139

E c(A)(Tht). P =

PR0P0SITJON 1.2. If

is elliptic, and the

conditions (1.c)-(1.7, hold, then

c-I(A)(rL)

P:

(1.18)

is &edholrn.

The operator E# E OPSI maps

PROOF.

to

Note that

E#P=I+ we have

e

Since

compact.

:

Hence we have

E# is a 2-sided Fredhoim inverse of P. Suppose the operator P in Proposition 1.2 is strongly elliptic. Then P has index zero in (1.16). In fact, the family of operators (1.23) is

(1 —

t)P -F

0

UI

< t < 1,

a continuous family of Fredholm operators, of constant index, arid zX obviously has

index zero. In this case, ker P consists of constants. C C1 ci IP2, and hence so u E ker P E

ker P

Vu =

in fact, under our hypotheses,

0 = —(Pu,u) =

;

0. We are ready to prove the following existence theorem.

PROPOSJTION 1 .3.

Under the hypotheses of Proposition 1.2, suppose further the equation 0. In such a case,

that P is strongly elliptic, and scalar. Then given f e Pu = f has a solution u e if and only ifff(x) dx u is unique

up to an additive constant.

PRooF. From what we have done so far, we know that the range of P in

(1.18)

is a closed linear subspace of C of codimension 1. Now f f(x) dx = 0 is one linear condition satisfied by all elements of the range of P. Hence this is the only such linear relation. and the proposition is proved.

The standard Hilbert space theory implies that, whenever the operator P is strongly elliptic, P : iO2(r) consisting of constants. We have:

is Fredholm, of index zero, with kernel

3. APPLICATIONS TO PDE

140

PROPOSITION 1.4. Assume A3k E C(A)(Tm), P = elliptic, and the conditions 7) hold. Then, given f e

is strongly the equation Pu = has a solution u E if and only if f f(x) dx = 0. Such a solution is unique modulo an additive constant. Furthermore, if f then u E

f

H'2(T"),

PROOF. It remains only to justify the last assertion. In fact, if E f guaranteed by Proposition 1.3, then, since L A(j) < 00 implies C1(A) C Ifl2, we see that u1 — u is a constant.

the solution to Pui =

and Ea(k) Coo, so the hypotheses of PropoNote that, if Aic E sition 1.4 hold with A(k) = then f E C C2. We also mention that Proposition 1.4 will be localized below, in Proposition 1.10. Many natural PDEs arise in a form which is a variant of the divergence form (1.1). For example, suppose M is a compact Riemannian manifold. Then the Laplace operator A, acting on scalar functions, is given in local coordinates by so the equation Au f is equivalent, in local Au g"2 coordinates, to

EOjAJ"(x)Oku = f,

(1.24)

where

= gJk(x)g(x)l/2.

(1.25)

In (1.24), f is treated as a distribution. If f is a function (say in L2), then the right side should be replaced by This slightly subtle (though not difficult) point is often glossed over in the smooth case, when distributions and "generalized functions" are identified via the density dx. In the nonsmooth case, this identification breaks down for singular distributions. The following extension of Proposition 1.1 is established by the same sort of arguments as used above. PROPOSITION 1.5. Assume that M has a Riemannian metric of class (in some smooth local coordinate system), and associated Laplace-Beltrami operator A. Assume conditions (1.o,l—(1.7) hold, and that (1.10) holds, with A = A. Then (1.26)

uE

Au =

f

C"

u E

This leads to a Fredholm result for the Laplace operator, for a class of compact Riemannian manifolds with mildly smooth metrics.

PROPOSITION 1.6. Let M be a smooth, compact, connected manifold with a Riemannian metric which belongs to Assume A satisfies the hypotheses of Proposition 1.5. Then the Laplace operator A on scalar functions has the property that (1.27)

A:

—+

is Predholm, of index zero, and kernel consisting of constants. With f if and only if'(1,f) = 0. one can solve Au f foru e

C l(A)(pvi),

1. INTERiOR ELLIPTiC REGULARITY

141

PitooF. Using symbol smoothing (via local coordinates and a partition of = P# + pb with fl e elliptic, and show unity), we can write for that a parametrix E# e in (1.27), just is a Fredholm inverse of as in the proof of Proposition 1.2. If we consider a line connecting the given metric

on M and a smooth one, we obtain a continuous family of Fredholm operators connecting in (1.27) to the Laplace operator for a smooth metric on M; hence all these operators have index zero. The rest of the assertions above follow as in the proof of Proposition 1.3. Again, for examples of when Proposition 1.6 applies, we refer to (1.16)—(1.17). The formula (1.24) -(1.25) for the Laplace operator on scalar functions (0-forms) comes from —(LXU,v) = (dn,dv). For £-forms, with £ > 0, one has = (dii, dv) + (d*n, d*v), which gives in local coordinates an expression of the form

=

+

+ cn},

+

where b-' and c are matrix-valued functions, involving both the coefficients of the

metric tensor and their first-order derivatives. Thus 93k E C b-f, c E C'. In Chapter 7 of [Mor] there is a treatment of these operators, for manifolds with Riemannian metrics in Lip'(M), with data in Sobolev spaces H p 2. There is also a discussion of the related Hodge decomposition. We will not dwell on such problems further here. But see Proposition 2.5 in the next section for some results related to the lodge decomposition. We now produce some Sobolev space analogues of the results above. Our first result requires very little regularity for the metric tensor.

PRoPosITIoN 1.7. Let M be a compact, connected Riemannian manifold with metric tensor of class fl vmo. Then

—*

(1.28)

1

is Predholm, of index zero, with kernel consisting of constants.

when M = PROOF. We start by constructing a Fredholm inverse for SO (1.28) is given by defined and Z'1-periodic on with metric tensor = gJk(x)gl/'2(x),

= aj(Aik(x)Ukn),

(1.29)

keeping in mind the remark made below (1.25). Now, in terms of Fourier series on by define A V'(T") —÷ Au(x) = >

(1.30)

k

E Vt

and set

A=

(1.31)

A

=

E RJV(x)Rk,

=

OJA

142

3. APPLICATiONS TO PDE

Using the results of § 11 in Chapter I, we see that

A=a(x,D) mod

(1.33)

where

l
K,,, denotes the space of compact operators on

a(x.e)

(1.34)

and

e

Furthermore, if we set (1.35) where

b(xe) e

(W')

=

-

is equal to 1 on a neighborhood of 0,

then

b(x,e) E

(1.36)

and,

(e)2(1

by Proposition 1.11.3,

mod K1,,

(1.37)

hence, for M

= F', a

two-sided F'redholm inverse of A in (1.28) is

A'BA

(1.38)

l
—s

A standard argument using a coordinate chart and partition of unity

then yields a

inverse to A in (1.28), for any compact M. Given that A is Fredholm in (1.28), arguments as used above show that its index is zero. The description of ker A is clear for p E [2, oo). For p E (1, 2), we claim the annihilator of the range of A in (1.28) consists of constants; this follows by the last remark, applied to since = Ji'" (M). Then the fact that A has index zero and A annihilates constants implies ker A consists exactly of constants for p E (1,2). two-sided

The next result obtains somewhat stronger conclusions, granted more regularity for the metric tensor.

PaoPosJ'noN 1.8. Let M be a compact, connected Riemannian manifold, with metric tensor of class G0'), and associated Laplace operator A. Assume conditions (1.6,)—(J.7,) (1.39)

hold, with r(e)

given by (1.4)—(J.5). Then

A ii

1

30,

—1

s

1,

is Fredholm, of index zero, with kernel consisting of constants. PROOF. Bring in the sort of symbol smoothing used in the proof of Proposition 1.1, so A = + D), with fl Fredholm on all spaces of interest, X3 smooth vector fields on lvi, arid E We claim that, given (1.6)—(1.7).

D)

is compact.

INTERIOR ELLIPTIc REGULARITY

143

In fact, Proposition 4.2 of Chapter 1 gives

B(x,e) e

(1.42)

',

B(x,D)

under these circumstances. Also we have

(1.43)

so we can readily verify that Fourier multiplication by

is compact on LP(Tnl),

for 1

It is useful to extend these results to

L=A—V.

(1A4)

We will assume V E

(M) is real valued. Then the same argument as above gives L:

(1.45)

—*

1

Fredholm of index zero. If we assume V has the property that L is injective on II'2(M), then L in (1.45) is injective for 2 p < oo, hence an isomorphisrn for such p, hence, by duality, an isomorphism for all p e (1, oo). An example of this is when V 0 on M and V> 0 on a set of positive measure in M (which is assumed to be connected). For such L we record the following global regularity result. PROPOSITION 1.9. Let M be a compact, connected Riemannian manifold, with metric tensor in Assume L in (1.45,) is an isomorphism forp = 2 (hence

forallpe(1,oo)). Then,forl
uE

Lu= fE Jr'"(M)

Suppose that the metric tensor is of class C(A), and (1.6)- (1.7) hold. Then, given (1.47)

u c HLI)(M), Lu = f e H

uE

and (1.48)

uE

Lu=fe

be the unique PROOF. If the hypotheses of (1.46) hold, then let v E H1 solution to Lv = f. Hence ii — v E solves L(u — v) = 0, so u — v = 0, and we get the conclusion of (1.46). The demonstrations of (1.47) (1.48) are similar. Now we establish a local regularity result.

3. APPLICATIONS TO PDE

144

PRO POSITION 1.10. Let M be a compact, connected Riemannian manifold, with metric tensor in Loccflvmo, and let 0 C M be open. Then, for 1

u€

(1.49)

Lu=

Suppose that the metric tensor is of class C(A), and that (1.6)—(1.7) hold. Assume —1 s 1, A(2—) Cr63 for some 6 cc 1. Then, given 1

Lu =

uE

f

u6

and

u6

(1.51) PROoF.

Pick p e Cr(0). We L(çou) = y,f

(1.52)

f6

Lu

:

u

e

have

+ 8,

+

the hypotheses of (1.49), the right side of (1.52) belongs to H 1Pi (M), where = np/(n — p) if p cc n and q Pi, with obvious modifications in other cases. hence u 6 Then Proposition 1.9 gives 6 H1'P'(M), for all such A finite number of iterations of this argument gives the conclusion of (1.49). The

Under

demonstrations of (1.50)—(1.51) are similar, once we note that, by (1.49), if the for for all q cc cx, hence u 6 hypotheses of (1.51) hold, then u 6 all e> 0, so the right side of (1.52) is in

REMARK. We mention that the implication (1.49) refines the r \ 0 limit of Theorem 2.2.H of [T2]. A similar result is also established in [DIF]. A regularity result with a flavor similar to (1.49), but for operators in nondivergence form, was given in [CFL]. When C(A) = Cr, 0 cc r cc 1, (1.50) gives the following result. PROPOSITION 1.11. Assume the metric tensor on M is of class C',

uE

(1.53) This

u

< r cc 1.

Lu=f€

result is established in [MT4], with the hypothesis on u to

u 6

'r> 1 —

weakened from

r.

We now establish some estimates for metric tensors of class of use in §10. For simplicity of notation we use H'" for PRoPoSITIoN 1.12. Assume p>n. Then (L54)

0

—1
Then,

which

will be

etc.

the metric tensor on M is of class

with

1. INTERIOR ELLIPTIC REGULARITY

145

when p> n, we see from (1.49) that u e PROOF. Since L' C for afl r < 00. in particular we have u E Now we write Lu = f as

g"2f + g'12Vu,

(1.55)

and, as in the proof of Proposition 1.1, we proceed as follows. We have (1.56)

u = E#(g'/2f) + E#(gU2Vu)_

where, for some S

mod C°°,

(0, 1),

(1.57)

a

E

1—

p

Here we use the symbol class discussed in §8 of Chapter I. As noted there, in (8.2), we need to pick 8> 0 small enough that 1 — 8> n/p. The result on given in (1.57) follows from Proposition 8.2 of Chapter I. Now we have E#(g'/'2f) + Furthermore, by Proposition —* 8.1 of Chapter I, for —(1 —8) < 'r — 1 1, 50 :

(1.58)

—'

:

Applying this once to u we have u finite number of times, we obtain (L54).

8 < i- 2. H'4

Iterating this argument a

The following can be proven in a similar fashion.

PRoPosITIoN 1.13. Assume the metric tensor on M is of class H'P, r

1, rp>n. Then (1.59)

REMARK. An alternative approach to proving Propositions 1.12 -1.13, not using the results stated in §8 of Chapter I, can be given along the lines of Proposition 1.16 below; cf. also Proposition 9.4 in §9 of this Chapter. Next we look at a second order elliptic equation in nondivergence form: (1.60)

= f.

>

Let us denote the left side by Lu. We assume AJk

Performing a symbol

smoothing, we write (1.61)

L = L# +

We take and r as in (1.4)(1.5). Also, we assume that (1.6)—(1.7) hold. The ellipticity hypothesis on L implies L# OPS?6 is elliptic; let F# OPSI be a parametrix. Using this, we establish the following regularity result.

146

3.

TO PDE

PROPOSITION 1.14. Assume u Aik

satisfies th€ elliptic system (1.60), with and (1.10)

Let A(k)/A(k) /. Assume that the conditions

hold. Then

f

(1.62) If Lu

=

f,

then,

(1.63)

where

is

—

c(A)

as in (1.10).

mod

hirthermore,

Lbu

parallel to (1.14)-(1.15),

P#L6u

Now (1.63) (1.64) yield u

A(k)). Iterating this obtain

F#Lbu

C2(A).

u

(1.64)

parallel to (1.11), we have

=

u

As we know, f

u

argument, as in the proof

(1.62).

= with of Proposition 1.1, we

Next, we establish a Fredholm result. PROPOSITION 1.15. If AJk conditions (1.6)—(1.7,) hold, then L: C2(A)(TT?)

(1.65)

is

L = EAJk(x)OjOk is elliptic, and the

Fredholm.

PROOF. (1.66)

The operator fl in (1.63) maps to Note F#L = I + F#Lb, LF# = I -i- Lbn,

that

modulo operators that are infinitely smoothing, and hence clearly compact. Since we have (1.67)

Hence

Lb

compact.

:

we have

compact.

(1.68)

and

(1.69)

LbF#

Thus

is a 2-sided F'redholm inverse of L.

compact.

There are also analogues of Propositions 1.7—1.10, involving L etc. The resulting analogue of (1.49) was established in [CFL]; see of [T4] for further discussion. We next consider operators of the form (1.70)

_* also §5

Lu= 0 < 1

when the coefficients have simultaneously some Holder continuity and certain Soholev reguJarity.

I. INTERIOR ELLIPTIC REGULARITY

147

PROPOSITION 1.16. Assume L zn (1.70) is elliptic. Assume

n

(1.71)

for some r,s>

0 andp

Assume ii solves

Lu=f,

(1.72) with

f

u

U

PROOF. We by

this

with a = —E, for

have u

any e > 0. We will improve

stages. Write

(1.75)

Lu

=

L#u+

+

We have

L# =

(1.76)

L#(x,D)

OPsSr1,

Sir, with

We can take

=

for

ei large, and, by

C, K = E(x, D) satisfies

Proposition 6.1 of Chapter I, since

= I + F,

(1.77)

elliptic.

F

OPS;j,

of0
(1.78)

Ef —

—

Fit.

Now

Ef

f and

since we (1.81)

for

al

Fit

u

(1.80)

have

m+

+r

a

=t'

it

m, so

>0. To analyze

since

s

LOC

E4(a,u), note that OPS?1,

> 0,

(1.82)

Thus, under the hypotheses of our proposition, granted that in (1.83)

We

it

it

+ a + r > 0,

jpn+s.p + jjnikc{r,p

can iterate this until we get the desired conclusion (1.74).

In [Mila] there is a result similar to Proposition 1.16 (though with a different proof) in the special case that p 2, s Z and in = 2, and with the hypothesis that it In that paper boundary regularity is also established. Also [Mila]

TO PDF

3.

148

points out applications of this to regularity for some nonlinear elliptic PDE. Such nonlinear results also follow directly from the paradifferential operator calculus. See Theorem 3.3.C in [T2] and Theorem 4.8 in Chapter 14 of [T51 for interior regularity, and Chapter 14, §8 of [T5] for boundary estimates. We remark that there are analogues of Proposition 1.16 for elliptic equations in divergence form. For an example, see Proposition 9.4 later in this chapter.

2. Some natural first-order operators We now look at some first order systems. We emphasize slightly different phenomena than in § 1. As a guide, we focus on the trace-free deformation operator DYE, defined on a vector field X on a Riemannian manifold M by (2.1)

1

DYEX

—

2

1

(VX + VXt) —

—

—(div X)g,

n

where g is the metric tensor, V its Levi-Civita connection, and X the 1-form assois a section of ciated to X via "lowering indices." The tensor field the bundle of symmetric, trace-free, second order, covariant tensors on M. in index notation, (DYEX)Jk

+

2

(2.2)

2

— —

+

=

X1;Egjk.

where the connection coefficients are OkX'3 + expressed in terms of the metric tensor and its first derivatives. Thus, in local in local coordinates, X'3;k = coordinates, (2.3)

=

+

—

_gjkOeXe +

where Bike is expressed in terms of the metric tensor and its first derivatives. If the metric tensor is Lipschitz in local coordinates, then the coefficients of the first order derivatives in (2.3) are Lipschitz. while the zero order coefficients belong to The operator DYE is "overdetermined elliptic" i.e., its symbol is injective. If 2, then DYE is (determined) elliptic. The kernel of DTE consists of conformal Killing fields on M, i.e., vector fields whose flows consist of conformal (M) —* maps of M (granted sufficient regularity). It is of interest to study DYE C2(M). Estimates on such an operator can be established in the following more general context.

dim M =

PROPOSITION 2.1. Let M be a smooth compact manifold and (2.4)

—÷C2(M,F1),

a first order differential operator between sections of smooth vector bundles Fo and F1. Assume the principal symbol is injective. Assume that, in local coordinates, (2.5)

Dii = EAJ(x)Oju+B(x)u,

2. SOME NATURAL FIRST-ORDER OPERATORS

149

with

r>0, B€C2.

A3EC',

Then there exists a continuous linear map E

(2.7)

such

that

EVI+K, K compact on If

the principal symbol of V

is bijective (so V is elliptic), then V is Fredholm in

Using local coordinates, a partition of unity, and symbol smoothing,

PROOF.

we can write (2.9)

with (for some 5 < 1) (2.10)

V# E

V'

BE

If the symbol of V is injective, then V# has a left parametrix E —p Cs'. Hence, modulo a smoothing operator, E (2.11)

Thus

EV—I+EV1'+EB,

where (2.12)

EVb

:

—p

Thus EVb is compact on Also, clearly B: since it can be factored through the compact inclusion

—*

is compact,

(M) —*

so we

have (2.8).

to be a 2If V is (determined) elliptic, so is V#, so we can take E sided parametrix for V#. Then, in addition to (2.11), we have, modulo a smoothing operator, (2.13)

and we see that the right side of (2.13) is the identity plus an operator compact on C?, so the rest of the proposition is proved. Now VbE : C? —*

When (2.8) holds, we know that the kernel 11(V) of V in (2.4) is a finite dimensional subspace of (M, Fo) and its range 1Z.(V) is a closed linear subspace of C?(M, F1). Also 11(V) has a closed linear complement £ in (M, Fb), and V

maps £ bijectively onto lt(V). Thus there is a continuous inverse T : R(V) We have the following.

TO POE

3.

150

COROLLARY 2.2. Let V satisfy (2.4)

and have injective principal symbol.

F1) for

Then therr exists C
such

lu— ullci CIIfIlc2.

(2.14) It

P1(V) c

that

is useful to

complement these results

with a regularity result.

PROPOSITION 2.3. Let V be a first order differential If V has injective principal symbol, then

operator satisfying (2.5,)-

(2.6).

(215)

provided 0< s <

1 and r + s> 1. Ifr> 1, and B

E

in (2.6,),

then

(2.16)

If r > 0,

B E L0C in

(2.6),

and

I

then

(2.17)

PROOF. We have, modulo Ccc.

u=EVu—EVTu—EBu.

(2.18)

EVu E

Of course, Vu analyze

Since

Also,

uE

EBu E

It

remains to

we have

E

(2.19)

provided —(1—8)r
For a proof, see Proposition 2.1.E of [T2]. The condition on a in (2.19) is equivalent

Thus,ifr+s>1. se(0,1),andVuEC2,wehave =min(s+r&1). this argument a finite number of times, we have (2.15). The proof of (2.16) is similar, and (2.17) follows by an even easier application of (2.18).

Iterating

Applying the results of Propositions 2.1—2.3 to Vqp, we have the following:

PRoPoSITIoN 2.4. Let Al be a smooth compact manifold, with a metric tensor then the following results g. If the coefficients of g belong to the Zygmund space hold: (2.20)

VT!.':

has closed range and finite dimensional kerneL There exists C ct oo with the following property. If X e (Al, T) then there exists X e (Al, T) such that

(2.21)

There

VT!.'X =0,

-

is the regularity result:

s>0,

(2.22)

If the metric tensor belongs to C' (M) for some r > 1, then (2.23)

xeLt.

2. SOME NATURAL FIRST-ORDER OPERATORS

151

If the metric tensor belongs to Lip'(M), and 1

X E Cs, s > 0, VTFX E I?

XE

The interest in estimating X in the Zygmund class element of

is the following. An

has the modulus of continuity:

(2.25)

This well known result, which we have noted in (1.22) of Chapter 1. was one of the prime motivations for Proposition 1.5 in that chapter. In turn, the significance of (2.24) is that, by Osgood's Theorem, vector fields with such a modulus of continuity generate uniquely defined flows. A proof of this classical result can be found at the end of Chapter 1 of [T5]. As a consequence of these observations. Proposition 2.4 is related to studies of quasiconformal mappings. In fact, the special case of Proposition 2.4 in which Al = with its standard smooth metric, has played an important role in such studies; see [MeM], [Rei]. Proposition 2.4, specialized to M = contains Theorem A.1O of [MeM]. Another geometrical example of interest is d + 6, acting on differential forms, i.e., sections of A = BAIT*(M). here, d is the exterior derivative and 6 is its formal adjoint. with respect to a given Riemannian metric. In local coordinates, d + 6 has the form (2.5), where A1 is expressed in terms of the metric tensor Yjk and B in

terms of g3,, and its first derivatives. In this case, d + 6 is (determined) elliptic. We hence have the following, which to some extent complements results in Chapter 7 of [MorJ on the hlodge decomposition for a compact Riemannian manifold with Lipschitz metric tensor. PROPOSITION 2.5. Let Al be a smooth compact manifold with a Riemannian then metric tensor g. If g belongs to the Zygmund space (2.26)

is Fredholm. We have (2.27)

If the metric tensor belongs to Lip' (Al), then, for 1
ct Do,

(2.28)

In such a case, the null space of d + 6 in (2.26,) satisfies (2.29)

PROOF. Only (2.29) remains to be established. When the metric tensor is Tipschitz, it and its volume element can be used to implement the duality of II"2(M, A)

and H "2(M,A), and we have u 6 H"2, w C L2

(Su,w) = (u,dw).

Hence (2.30)

u,v C H"2(M,A)

(Su,dv) = (u,ddv) =

0.

3. APPLICATI0Isz5 TO PDE

Now, if n 6

N(d + S), we

know by (2.28) that n 6 H"2,

= IdnIII2 +

(2.31)

I1S?11ft2

so

+ (dn,Sn) + (Sn,dn)

+

and this implies (2.29).

The next first-order operator we consider is the Beltrami operator, given by B=

(2.32)

On

On

A(x,y)—.

—

Liz

Liz

We assume

k < 1.

(2.33)

This operator is important in the theory of quasiconformal mappings. The following result is fundamental. Assume that A satisfies (2.33) and has compact support in C = lR2. Then there exists a unique solution to Bn = 0 on C of the form

n(x,y) = z +v(x,y),

(2.34)

S

satisfying n. —

(2.35)

1

e

L1'(1R2),

for

some p> 2.

The map n is a quasiconformal homeomorphism of C onto itself. We refer to {Ahl] or [CG] for a proof of the result stated above. Here we establish further regularity results for solutions to fin = 0, given some mild regularity hypotheses on A. First, we establish some Fredholm properties. PROPOSITION

2.6.

Let A

fl vrno(T2) satisfy (2.33).

Then, for all p

(1,oo), —*

B

(2.36)

is Fredholrn, of index

zero, and Ker B consists of constants.

PROOF. Setting A = (—L\ + 1)1/2,

17(T2) —* L"(T2).

q(x, D) =

(2.37)

consider

and the hypothesis (2.33) This is an operator with symbol q(x, = fl vmo)Sg, where implies it is elliptic, i.e., — w(C) is an appropriate cut-off. It follows from Proposition 11.3 of Chapter I that D'(T2) is a two-sided Fredholm inverse of BA'. Thus B p(x, D) 17(T2) in (2.36) is F1edholm. To compute the index, consider the 1-parameter family of operators :

=

(2.38)

05r51.

We see that these operators are all Fredholm, with the same index. The index of B9 is clearly zero, and hence so is the index of B. Now suppose n H1"(T2) satisfies Bn = 0, i.e., (2.39)

On

On

2. SOME NATURAL FIRST-ORDER OPERATORS

153

If p 2 we can take the inner product of both sides with Du/82, obtaining 1

(2.40)

hence Vu

and

(2.41)

2

=

2

<

0. This plus the index calculation implies, for 1 < q

2, that

—* I-I-

B'

one-dimensional kernel, independent of q (1,21, from which we deduce that has one-dimensional kernel for all q (1, cx)). This implies that B in (2.36) has B' one-dimensional cokernel, independent of p (1, cx)), and the index computation implies that its kernel is one-dimensional. has

Here is another Fredholm result. PRoPOSiTIoN 2.7. Assume

A(j) \ 0

(2.42)

>A(j)a
Let A

satisfy (2.32). Then

(2.43)

B

is

slowly varying and satisfies

forsorne a<1.

is Predholrn, of index zero, and Ker B consists of constants. PROOF. If we take r(2') = NJY'1, then the hypothesis (2.42) implies that the (l.6)—(1.7) hold. Prom here, the proof is exactly parallel to the proofs of Propositions 1.2 1.3. conditions

Here is a third Fredholm result.

PRoPosiTioN 2.8. Assume A

B1(T2) with s >

0,

1
sp

2.

Then (2.44)

B

—*

B1(T2)

is Fredhoim, of index zero, and KerB consists of constants.

PROOF. Under our hypotheses, B1 (T2) is an algebra under pointwise multiplication, contained in C(T2). Furthermore, by Proposition 12.1 of Chapter I, OPS?I acts on B,1(T2), so paradifferential operators act on these spaces. Hence, under our hypotheses, whenever f B1 (T2) and C(T2), we can deduce

that

B1(T2). Now consider q(x,D) =

B1(T2)

B1(T2),

with (2.46)

q(x, D)

elliptic.

= — belongs to B1SC°I. Note also that is dense in B1 (T2). It follows from Proposition 13.2 of Chapter I that p(x, D) is a Fredholm inverse of q(x,D) in (2.45). Thus B in (2.44) is Fredholm. The Hence

same argument used in Proposition 2.6 shows B has index zero. Since B follows from Proposition 2.6.

3. APPLICATIONS TO POE

154

Propositions 2.6—2.8 lead to the following global regularity results.

PRoPosITIoN 2.9. Under the hypotheses of Proposition 2.6, if 1

UE

Ru E Ji'(T2)

uE

Under the hypotheses of Proposition 2.7, if 1

uE

Ru E

u E

Under the hypotheses of Proposition 2.8, zfs > 0, 1
cc, sp 2,

tiE H'"(T2), RUE

PROOF. The same sorts of arguments as used in the proofs of Proposition 1.4 and Proposition 1.9 are effective here. Now we have the following local regularity result. PROPOSiTiON 2.10. Let 0 C 'IF2 be open. Under the hypotheses of Proposition

2.6, if 1

(2.50)

RU E

U E

Under the hypotheses of Proposition 2.7, if 1

UE

Ru E

UE

Under the hypotheses of Proposition 2.8, if 0 < s < 1, 1
UE

Ru E

UE

PROOF. The cut-off argument used in the proof of Proposition 1.10 is also effective here.

Having this local regularity result, we can return to the set-up in which A has compact support in C, (2.32) holds, and one has the solution to RU = 0 satisfying (2.33)—(2.34). We can deduce that, if in addition A E U'° fl vmo(R2), with then Vu E for all q < cc. Furthermore, if in addition A E In such a case, the map u 15 a C' A(j) \ 0 satisfying (2.42), then VU E quasiconformal diffeomorphism of C onto itself. Actually, at this point we have that U is a C' quasiconformal homeomorphism. To deduce that U i5 a diffeomorphism, we need to know its Jacobian determinant is nowhere vanishing. One way to prove this is to show that each point p E C has a neighborhood 0 on which the I3eltrami equation (2.39) has a C1 solution ii that as in Proposition 2.7). Then is a diffeomorphism (under the hypothesis A E Uo is a C' homeomorphism that is conformal; it is well known that this must be a diffeomorphism. and this implies that U is a diffeomorphism. The existence of such ft can be deduced from Proposition 9.3 (later in this Chapter), given the relation between solutions to the Heltrami equation and the production of isothermal coordi,iates on a surface with a Riemannian metric, which we now recall.

3. ESTIMATES FOR THE DIRICIILET PROBLEM

For a complex-valued function u, write u = so 0 and A =

If

UT

a

=

v

155

+ iw, with v arid in real-valued.

.1 + A

—i1

—

computation gives

(2.53)

dv2

+ dw2

1—

is positive-definite if C 1. Conversely, if (x, y) preserving diffeomorphism and which

(2.54)

(v, w) is a C' orientation-

dv2 +dw2 =A(x,y)(Edx2 +2Fdxdy-}-Gdy2),

and we set u = given by J =

v

iUzl2

AI2dy2),

A >0,

+ iw, then the Jacobian determinant of this diffeomorphism is —

1u112, and

E—G+2iF

(255) Uz

Note

that the right side of (2.55) has absolute value cc 1, as a consequence of the

positive-definiteness of (2.54). We make some further comments about regularity results contained in Propo-

sition 2.10. That the solution ii is C' if A is Holder continuous is classical. The regularity result (2.52) contains results in Chapter 3 of [Bli] (with a somewhat different proof). There is a regularity result given on p. 235 of [LV], which can be compared with (2.51), though neither result contains the other. 3. Estimates for the Dirichlet problem

We begin with a study of the solution to the Dirichlet problem for the Laplace equation on a half space w1+l. Thus, we look for estimates on

u(y,x) =

(3.1) for

f

defined

on

PRoPosiTioN 3.1. Assume A(j) = U, defined (3.2)

where

(3.3)

e

for z =

(y,

x) e Rif'

and

by (8.1), satisfies

I'u(z,) — U(z2)l C Ca(Iz, —

AU) < oo. 1ff E

then

i56

3.

TO PDE

PROOF. From Proposition 1.3 of Chapter I, applied to ele—IeI, per the remark (1.41) given there, we have

=

and

to

(3.4) with

= w(h) +

(3.5)

and

hf

also

(3.6)

Then the standard method of setting h (3.7) u(z1) — u(z2) =

where

— z2j and writing

[u(yi,xi) — u(yi + h,xi)] + [n(yj + [u(yi +h,x2)—u(y2,x2)],

zj = (yj, xg),

+ h,xi) — u(y1

+ h,x2)]

yields

+

Iu(zi) — u(z2)I

(3.8)

cf

dy.

To deduce (3.2)—(3.3), note that

f

()

0

f

dy Y

w(t)

dy + f"

o

o

dl

y

It is of interest to dwell on some specific examples. For s > 0, consider the

family

of

moduli of continuity, given by

=

(3.10)

suitably

altered for h

EXAMPLE 1. Let (3.11)

PROOF. It (3.12)

0< h

1/2. e cg°(B112). Then, for s > 0,

f(x) = log jxIr can be shown that, if n

and

n 2,

u E 2,

fE

This can be verified directly from the definition (1.64) in Chapter I; alternatively, see the computations (A.1o) (A.15) of Chapter 1 (in which s is replaced by —s). Now (3.12) implies f E with = j(l+s) (cf. (1.65) of Chapter I), and then Proposition 3.1 gives the implication in (3.11).

3. ESTIMATES FOR THE DJRLCHLET PROBLEM

ExAMPLE 2.

f(x)

(3.13)

be as in Example 1. Then, for s > 0,

Let

PROOF. Note

=

1,

and n

1,

Vu E

$ w(x)

xI

157

that

Vf(x) =

(3.14)

+ R(x),

where R(x) is smoother than the principal term. Thus

Vf e

(3.15)

c

A9+i(j)

Thus Proposition 3.1 directly implies that tigate

1

It remains to inves-

E

gj(x) =

(3.16)

is the jth Riesz transform. Since is a classical pseudodifferential operator of order 0 and f has compact support, we have by Corollary 5.3 of Chapter I. Then Proposition 3.1 gives E

where

In view of Example 1, one might think you could replace —s—i by —sin (3.13), when ii 2, but in fact you cannot. The factor in (3.14) prevents such an improvement from working out. Proposition 3.1 contains the following result: CoROLLARY 3.2.

f E Cw(wl),

and

If

then

w is a modulus of continuity satisfying f w(t)r' dt < oo, u with a PIf, defined by (3.1), belongs to

given by (3.59. 3.1. will

As shown by Example 1 above, Corollary 3.2 is strictly weaker than Proposition All the same, it is interesting to consider a direct proof of Corollary 3.2, so we give an alternative proof of the estimates (3.4)—(36), given f E Cw(Rn). We

use the integral formula

f(x')dx'.

We

P1w E

use the relatively elementary fact that

x0 E

W', choose

ço(x) =

f(xo)r''o12.

the first-order derivatives of v =

PIg,

Given

Then it suffices to estimate, at (y,xo),

for g =

f—

using the fact that g(x)I

We have

Cyf

(y2 +r2)N*3)/2

rdr.

If we divide the range of integration into (O,y] and (y,oo), we readily obtain an estimate of the form (3.4). A similar attack works on xij)I. We next consider harmonic functions on a bounded convex domain in C With little effort we could extend these results to the case where Q satisfies a

3. AVPLICATIONS TO

uniform exterior sphere condition. The operator PT: C(812) —# C(12) produces the harmonic function in 12 with given boundary value. We want to provide a sufficient condition that P1 f e Let w(h) be a modulus of continuity. We will impose the following condition on w. Namely, let be a bounded continuous function on Rni such that (3.17)

=w(ix'j)ix'i

while ço(x') = that

on

I

for Ix'i

for

1. We assume that ço(x') is a monotone function of jx'j,

is smooth on ir '\ 0, and that the unique bounded harmonic function v equal to won R?t 1, has the property

(3.18)

where here PT is the solution operator (3.1), with n+1 replaced by n, and y replaced by Now let f be a function on R11 satisfying

fEC'(IW'),

(3.19)

VfECC&)(Rn),

and consider its restriction to 812, which we also denote f. The set of such restrictions to 812 is denoted Clw(812). We have the following. PROPOSITION 3.3. 1112 is a bounded convex domain in R" and w is a modulus of continuity for which the hypothesis (3. 18) is satisfied, then (3.20)

P1:

—> Lip(I2).

PRooF. Let f E and set ii = P11. By the maximum principle, it suffices to estimate Vu on 812. Take Po E 812. ]hnslating and rotating, we can assume Po = 0 and 12 is contained in > 0}. Let

v be given by (3.18).

The hypothesis there, plus Zaremba's principle,

implies

v(x)

(3.21)

for some C,C' follows that

Cw(ix'i)1x'I +

> 0, whenever x E

+Vf(0) on 812,

(3.22)

for

sufficiently large A.

C'w(ixj)ixj, lxi

diam(lfl. Given f E Clw(lRhl), it

<1(x) <1(0) +Vf(0) By

the maximum principle, we hence

+ Vf(0) x - Av(x) u(x) <1(0) + Vf(0) .

have

x + Av(x)

on Since v is Lipschitz, this is enough to estimate Vu(p3), and hence to prove (3.20).

s>

If w(h) has the form (3.10), then the Lipschitz hypothesis (3.18) holds whenever 1. as follows from the implication (3.13). Thus we have:

4. LAYER POTENTIALS ON C1"' SURFACES

159

COROLLARY 3.4. If Q is a bounded convex domain ink11, the mapping property (3.20) holds for moduli of continuity w(h) of the form (8.10,). as long as s > I.

A special case of Corollary 3.4 will be useful in a study of ideal fluid flow in convex planar regions, in §5. In fact, in §5 it will suffice to know that, for such a domain ft we have —* Lip(ftJ,

PT:

(3.23)

V r > 0.

The specific result we need for §5 is the following: PROPOsITION 3.5. Let 11 be a bounded convex domain in

=g

(3.24)

=

on ft

Assume it satis-

0.

Then

g e L'1(ftj, p > n

(3.25)

it E Lip(?'i).

v + w where (assuming p < cc)

PROOF. Write (3.26)

(3.27)

As long as p > n,

=g

on r,

=

on 11,

0

ci

v E

w=

—v

on Oft

for some r > 0, so (3.23) applies to w,

and we have (3.25).

4. Layer potentials on C'"' surfaces Here we study the method of layer potentials on domains whose boundaries are a bit smoother than C'. We begin with an analysis of a class of singular integral of the following sort. Assume operators on a-dimensional C1"'-surfaces on cc : W1 —+ ktm is C', that LAp E C"', and that Dco(x) is injective for each x E W1 (with Dw(x)*Dw(x) cI for some c> 0). Let k e \ 0) be homogeneous of odd, i.e., k(—z) = —k(z). We look at the singular integral operator degree —n, and with

kernel k(w(x) — ço(y)).

The fundamental theorem of calculus gives

(4.1)

k(p(x)—p(y)) =k(Dp(x)(x—y))+ki(x,y)=ko(x,y)+kJ(x,y),

where

(4.2)

k,(x,y)

f

+rW(x,y)](x— y))dr W(x,y)(x —

with (4.3)

W(x,y) =

+(1

—

t)y)

-

LAç(x)]dt.

3. APPLICATIoNs TO PDE

160

Thus we have, for the singular integral operator K0 given by (4.4) the

Kof(x) =P.V.fko(x,y)f(v)dv,

property

(4.5)

K0 6

Let us also assume

that, for

1W', re

—* R"2 of the form B = all linear maps B : [0,1], there isalower bound

BtB ci,

(4.6) for

OPCWSCO1.

some c>

0,

independent of x, y, 'r.

(4.7)

For

example. if p has the form

97(x) =

of class C1", then (4.6) holds with c = Now, under our current hypotheses, we have W(x,y)I

with

:

1ktm —*

(4.8) In

and hence

Iki(x,y)I

particular, if w

satisfies the Dini condition

A w(t)F'dt
(4.9)

then

1.

the operator K1

given by

(4.10)

compact from to for all p 6 [1, cc]. If we are interested in local analysis, we study the action of K K0 + K1 on functions supported on some fixed compact set C C 1W', and we consider Ku(x) for x 6 C. Thus, we can alter k(z) for zI large without affecting this action. We henceforth assume k(z) is so altered, to be rapidly decreasing as Izi — cc. In such a case, K1 can be expressed as a pseudodifferential operator with double symbol, such as considered in §9 of Chapter I: is

(4.11)

Kif(x) =

ff

The representation (4.2) implies that (when (4.6) holds) (4.12)

Note also that k1(x,x,e) = Chapter 1 to deduce:

tcj(x,y,e) e C"'S?0.

976 0.

Hence we can apply Propositions 9.5 and 9.18 of

SURFACES

4. LAYER POTENTIALS ON

161

0 < r < 1, and (4.6) holds, then, for all

PROPOSITION 4.1. If y., E

se[0,r), pe(1,oc), (4.13)

K1:

—+

K1 :

—p cY:(Irr).

Moreover, for a E (—r, 0], (4.14)

K1 : H°"(Wt) —f Hc+8P(Rni),

K1: C(W') —p

We can extend these results, replacing k(z) by k(w, z), smooth on x odd in z, and homogeneous of degree —n in z (for not large, and cut off for JzI large). We can consider operators of the form

(4.15)

-

Kf(x) = P.V.

dy,

where we have also thrown in a factor a. We assume ct E CW (1k11); for example ct(y) dy could be the area element of the n-surface

F = {p(x) : x e W'}

(4.16)

C ir.

Another important class of operators has the form (4.17)

Hf(x) =

f

-

with h(w, z) smooth on Izi

x and homogeneous of degree —(n—i) in z (for small), and even in z. In such a case, H e It is important to note

that

has better behavior than would 83T for a typical element T e

In fact,

DjHf(x) = P.V. f (4.18)

+

—

f

-

a(y) dy a(y) dy.

(w, z) is homogeneous of degree —n in z (for IzI small) and odd in z, while z) is homogeneous of degree —(n — 1) in z (for Izi small). Thus the first term on the right side of (4.17) defines an operator in 0PCWS?o and the last term defines an operator in Hence 83H E ØPCWS?o. We can decompose the first term on the right side of (4.17) in a fashion similar to (4.1), to get Here

(4.19)

where (4.20)

=

6

e 0PCWS?,o has double symbol that vanishes at x = y. Thus, if w and satisfies the Dini condition (4.9), then to is compact from if e C't'(W'), r 6 (0, 1), the estimates in Proposition 4.1 apply to B3. In the rest of this section we take in = n + 1, so F in (4.16) is a hypersurface. We can as well take F to be an n-dimensional C1 "j' hypersurface in an (it + 1)and

dimensional manifold M. The results derived above extend to the following setting.

3.

1b2

TO EWE

Let M be a compact (n + 1)-dimensional Riemannian manifold, and let fi be a connected open subset of M, with boundary F = 811, which we assume is a Let P E hypersurface of class (M) he an operator whose principal

symbol p_i(x, is odd in e. Let b E 'D'(M x M) denote the Schwartz kernel of P, so that, for a fuction u on

Pn(x)

(4.21)

where dv denotes the volume element on M arising from its metric tensor. Then (with da denoting surface area on F)

Bf(x)

(4.22)

P.V. f b(x, y)f(y) da(y) for 1

defines a bounded operator on write

B=B0+B1,

(4.23)

with

B0 e

(4.24)

and (4.25)

if w

L'(F) compact,

B1 :

1

satisfies the Dini condition (4.9). Furthermore, if F is of class

then

B1 satisfies

the estimates in (4.13)- (4.14), for all

r E

(0,

1),

E [0,r), p E (1,oo).

One has similar conclusions if a factor 13(x, y) is thrown into the integrand in (4.20), with j9 e CW(M x M).

There is a related extension of the analysis of (4.17). Let Q e OPSCI2(M) be an operator whose principal symbol is even in e V'(M x M) denote its Schwartz kernel, and consider (4.26)

Sf(x) =

zE F.

Then (4.27)

8:

—+

I

vector field on if w satisfies the Dini condition (4.9). Furthermore, if X is a LP(F) has properties similar to those listed in

tangent to F, then XS : LP(F) (4.23)—(4.25) for B.

Operators of this form arise when treating boundary problems for elliptic PDE

in 11, by the method of layer potentials. We illustrate this with the following be the Laplace operator on M (with its Riemannian matric), and example. Let (M) is > 0 everywhere and > 0 somewhere on each set L = — V where V component of A] \ = ft Then L is invertible, with inverse Q e .

4. LAYER POTENTIALS ON

SURFACES

163

Apply the construction in the previous paragraph to this operator Q. Note that

Sf =

where

Sf(x)

(4.28)

wE M.

is known as a single-layer potential. Clearly, given any decent 1, the function u = Sf is smooth on M \ /311 and satisfies the PDE Lu = 0 there. We produce some other important estimates, This

yielding behavior as one approaches /311. PROPOSITION 4.2. Assume 811

is of

class Cl+T,

f E C8(811)

(4.29)

0

< r < 1. Ifs

e

(0,r), then

E

1 <j < N} be a collection of smooth vector fields on M PROOF. Let for each x E M. We estimate on M \811 that they span the tangent space functions :

such the

(4.30)

Sf(x)

=

f

=

dci

E(x, y) in the i-variable. To do this, let V be a smooth vector where XiXk acts field on M, transverse to 811, and define for t E (ra, a) (for a small) the operators S3k(t) by

Sjk(t)f(z) =

(4.31)

dci

= x, 7!1(O) = V(x). We

is the constant-speed geodesic satisfying

where have

SJk(t) bounded in

(4.32) and

(4.33)

for t E

tSjk(t) bounded (—a,a).

in 0PCS?0(810,

By interpolation, we have for any 0 E

(4.34)

tl°Sjk(t)

(0,1),

bounded in

If we apply Proposition 9.8 of Chapter 1 to the family of operators in (4.34), we obtain,

for any a E (0, s), p E (1, oo),

bounded in ga

C8(811) C IF" (811)

(4.35)

f E

provided

a+0—

1

e (0. 1). Thus, take 0

large enough, we deduce that


(4.36)

In other words, (4.37)

X3XkSf(x)I

=

1

— a + F, for small

I

> 0.

Taking p

TO PDE

3.

164

This implies that, if

E

C3(ôIfl, then

Va'
(4.38)

(439)

E

This is just slightly

weaker than the

V

I Cs.

asserted result (4.29). We finish the proof as

follows.

Namely, we borrow an argument from the proof of Corollary 3.2. Fix e C81 (811) such that fi(xo) = and given. Pick fi be x0 E 811 f(xo) (s,r). Let satisfies Then the argument so far shows that IIfiIIc°i 5

XjXkSfl(x)I

(4.40)

On the other hand, the same sort of analysis as shows that (4.41) These

—

in the proof of Corollary 3.2

C

two estimates yield the optimal estimate

IXjXkSf(x)I

(4.42) and

used

the proof of Proposition 4.2 is complete. Using the argument in Corollary 3.2 once again, we can establish:

PROPOSITION 4.3. Let 811 be as in Proposition Assume w is a modulus of continuity satisfying (4.9) and also w(h) on (0, 1j, for some p (0, r). Then

f

(4.43) where

a(h) is given by

(4.44)

PROOF. Pick s (p,r). Given x0 811, pick fj f(xo). Proposition 4.2 implies that SfiL1 (4.40), and the previous argument applies to give

C'(811) such that fi(xo) = satisfies the estimate

(4.45)

with 7(h) as

in (3.5). The

conclusion in (4.43) follows from this, via calculations

parallel to (3.7)—(3.9).

For more general boundaries 812. of class

with w satisfying

(4.9), we will

an estimate on the non-tangential maximal function of VSf. Generally, for a function u defined on M \ aa the non-tangential maximal function u* is describe

4. LAYER POTENTIALS ON 01W SURFACES

defined on 811 as follows. Pick i3 E (0, 1). For x E

let

dist(y,311) t9dist(y,a)}. Then we set u*(x)

(4.46)

165

= {y

E

M \ 311

sup

The following is the basic estimate on (Vsf)*. It is a consequence of the results in

(4.21)—(4.25). Such arguments will arise in Chapter IV, in the context of rougher boundaries, so we will not present them here. PRoPosITION 4.4. Assume 311

is

of class

where w satisfies (4.9). Then,

forpE(1,Do),

I(VSf)*1ILP(aQ) C

(4.47)

x E F from either Sf(x) as z While Sf(z) = or (L, there is the following well known jump relation for the normal derivative:

&_sf) =

(4.48) where

K*f(x)

(4.49)

A derivation can be found in many places and will be omitted here. (See, e.g., [Mik]; compare also calculations in Chapter IV.) The integral in (4.49) is a weakly singular integral. To see this, note that since the leading singularity in E(x, y) is equal to CTfl dist(x, y)(fh_2) (for m = n+ 1 3, with a similar formula involving log for m = 2). one readily verifies that

Cwflx

(4.50)

—

(x

—

We deduce that

K4

(4.51) if

w satisfies

—+ 17(F)

:

is compact,

1

p

the Dini condition (4.9).

The general constructions described in the beginning of this section apply to

K*, to yield K4 =

+ Kt, with Kt E

and Kt E 0PCWS?o

For this operator there is having double symbol vanishing at x = y (mod a further special feature, following from the estimate (4.50). Of course, the Schwartz kernel of also satisfies such an estimate. Hence, so must the Schwartz kernel of This implies that the principal symbol of Kt is identically zero. Hence, if (1,oo), (4.52)

K4 : L"(F) —'

K* :

—+

Moreover, forcE (—r,0], (4.53)

K4 :

—*

K4 : C(F)

C(F).

thb

3.

TO PDE

In the study of layer potentials, one considers both the single-layer potentials (4.28) and the double-layer potentials

Vf(x)

(4.54)

=/

XE M \F.

En this case, the jump relations are

(Vf)±(X) =

(4.55)

+ K)f(X),

X E F,

where

Kf(X) =

(4.56)

P.V.

/ r(X,

y)f(y) da(y).

As the notation indicates, K and K* are adjoint operators. Parallel to (4.50), we have (4.57)

IX

—

—

and the properties (4.51)—(4.53) apply also to K, under appropriate hypotheses on

all.

There are also results parallel to Propositions 4.2—4.4. For example, if 0Q is of

re

class (4.58)

(0,1),

and ifs e (0,r), then f E C5(8ll)

Vf L

If w is as in Proposition 4.2, we have (4.59)

fe

E

with a as in (4.44). If Ull is of class

with w satisfying (4.9), then

(4.60) 1

We now establish solvability of some boundary integral equations that will apply to the study of the Dirichlet and Neumann problems. Some of these results will be extended to other spaces further on, but the following proposition will provide a useful start. PROPOSITION 4.5. If Oil is of class Ciw and w satisfies the Dini condition then we have isomorphisms

l
(4.63)

+ K,

r E (0,1), then we have isornorphisrns + KS

—*

+ K. Al + K' C(O!l) —k

—r < s < r, 1

—r < s
4. LAYER POTENTIALS UN

SURFACES

167

PROOF. Compactness results on K and K* established above imply that all these operators are Fredhoim of index zero. If we can show that I + is injective on each space listed above, it will follow that + K* is an isomorphism on all these spaces; hence, by duality the isomorphism property follows for I + K. in cases (4.61)—(4.62).

Thus we tackle injectivity of I + Kt. We start with the case where 31? is of for some r E (0,1). Suppose f IJsP(3Q) belongs to + Kt), for some S E (- r, r), p E (1, oo), so f — Repeated use of the mapping properties in (4.53) (plus the Sobolev imbedding theorem) show that, in this case class

Va
(4.64) Now given such f in the null space, set it According to (4.48), (3u/3v)_ = 0 on 30.

= Sf, so If f E

0 on Al \ with aq > n. then

— V)u

f c C8 (311) for some S > 0. One has for it = Sf that

E C'(II), by Proposition 4.2. Then Zaremba's principle implies that bit/dy is nonzero at a point E 30 where it is maximal, unless it is constant on each connected component of 0_. Since we are assuming that V = 0 somewhere on each such component, this requires it = o on 0-. Then Sf 0 on 311. Thus, by the maximum principle, it -= 0 on 0. Since,

by (4.48), f is equal to the jump of across 30, we have f = 0, so we have the isomorphism in (4.62) in the case that 30 is of class (0, 1). We now establish that the maps in (4.63) are isomorphisms. Again we need only prove injectivity, but this time we cannot appeal to duality, so we must treat both operators in (4.63). First, suppose f E Ker + K), f E C(30) for some s E (—r,r). This implies that f E for each a E (—r,s). p < oo. But the isomorphism just established in (4.62) then implies f = 0. SO I + K is injective in (4.63). Injectivity of + Kt on C(30) follows similarly. Now we tackle the case where 30 is of class Clw, w satisQving (4.9), and establish injectivity of + Kt on 11(30), for all p E 1, oo]. We first show that, for any f E

Vq
(4.65)

via an argument used in [FJR]. Pick x0 E 30. Civen 8 > 0, let {x C Al dist(r,rf)) < 8}. Pick C Cr(B) such that = 1 for dist(x,x0) < 8/3, 0

for dist(x,xo) 28/3, while 1 on suppp. Note that, if f = _2K*f, then = Kt]f — 2Kt(pf), so (4.66)

(I

= g,

g=

Examination of the integral kernel of [Mr, K*] shows that (4.67)

cf

Hence f C 11(30). i/p — 1/n = l/q

1

da(y).

g C 11(30), while 1/p — 1/n < 0 g C If S is small enough, tile operator norm of 2MeK*Mv, will for all q < he < 1 on V'(dII), and we can deduce from (4.66) that pf C A covering argument yields f C L"(31fl. Iterating this argument then gives (4.65).

TO PLIE

We complete the proof by establishing that I + K is injective on L2(311). Suppose! e L2(311) and = ft As above, set u = Sf,so (Lx— V)u = 0 on Al \ 811. The estimate (4.47) allows the use of Green's formula, to write f{Ivul2 + V1u12}dv(x) =

(4.68)

dcr(x),

and by (4.48) the right side of (4.68) vanishes when I Kt). Thus u is constant on each connected component of 11 and u 0 on supp V, so u 0 on IL. Thus Sf = 0 on 311 so, again by Green's formula (justified as before) we have (4.69)

+ V1u12} dv(x)

%

u(2)

dc(x) =0.

= 0 on 311. Since I is equal to the jump of Hence u constant on II, so across 311, we have f = 0, so the proof is complete. is

With Proposition 4.5 in hand, we can produce solutions to the Dirichlet problem Lu =

(4.70)

0

on 11,

UL,c1 = 1,

in the form (4.71)

u is given by (4.71), then (4.70) holds, follows from (4.55). We also have from (4.60)—(4.61) that, for 1

:s;

This holds for 311 of class

w satisfying (4.9). If 311 is of class Cl*r, r

(0, 1),

then, by (4.58) and (4.63), (4.73)

P1: C8 (311)

—*

C8(Ifl,

0<

s


Uniqueness of solutions satisfying (4.73) follows from the classical maximum principle. For solutions satisfying (4.72), uniqueness is a special case of results that we will establish in Chapter IV (cf. Proposition 5.5 there) when p 2. We omit a treatment of uniqueness for p (1, 2); cf. [FJR] for a uniqueness result of such a nature, when 311 is merely We next establish solvability results for Sf — g, which complement Proposition 4.5 as tooLs to treat the Dirichlet problem.

PRoPosiTioN 4.6.

If 311 is of class

and w satisfies the Dini condition

(4.9), then we have üomorphisms (4.74)

If

311

(4.75)

(4.76)

is of class

5

:

r

(0,

1), then

we have isomorphisms

Hi+8P(311)

S : C(311) —b C,!

—r < s

8(311)


—r <s


4. LAYER POTENTIALS ON C1"' SURFACES

169

PROOF. Our first order of business will be to show that S is Fredhoim in the cases listed in (4.74)—(4.76). To do this, we make some modifications. First, we can refine the differentiable structure on 811 to a smooth structure (via a process simpler for a hypersurface of a smooth manifold than for the general case), impose a smooth Riemannian metric on 81? (different from the metric we have been using) and consider D = d + d*, acting on sections of A*T*(Di1) = eL0 A3T* (811), the adjoint d* defined using this new metric. So far, S acts on sections of AOT* (81?). Using local trivializations and a partition of unity we can define operators we also denote S on sections of AJT* (DIfl, for 1 j it. Now consider

T = DS,

(4.77)

acting on sections of A*T*(811). By (4.18)--(4.20) we decomposition similar to (4.19), i.e.,

have

Te

OPCWS?O,

with a

(4.78)

with

T0 OPCWSZ, and with T1 ØPC"S?() having double symbol vanishing at x = y. Now the crucial thing to note is that

(4.79)

is elliptic.

T0

It follows that To is Fredholm on L"(DIfl, 1

W, r (0, 1). Also, T is Fredholm on these spaces. Since D = d + d* is a first-order, elliptic differential operator with smooth coefficients, the Fredholmness of the operators in (4.74)—(4.76) follows. Furthermore, since S has a positive, real principal symbol, all these operators have index zero. It remains to establish injectivity of S in (4.74)—(4.76). Again we first tackle the case when 81? is of class Cr, r (0, 1), so C" = cr in (4.74)- (4.76), and in particular T1 satisfies the estimates in (4.13)—(4.14). Then applying symbol smoothing to T0 to write T0 = T+TR with elliptic, TR we have taking a parametrix in for and using arguments as in that, whenever f belongs to one of the domain spaces in (4.75), then for —r <s
(4.80)

Sf =

0

Thus, considering u = Sf, we

f

Hsi'(811),

have

V

.s
p < oo.

C1*s(fl+)

Lu = 0 on 11+ and on 11_,

for all s < r, and uj00 = 0, so it = 0 on Al if Sf = 0. Then

=

SO f = 0. This establishes injectivity, hence isomorphism. in (4.75) and also in (4.76), when 811 is of class The injectivity property in (4.74) follows as in the proof of injectivity of on L2(81?) given in Proposition 4.5. This completes the proof of Proposition 4.6. 0,

We briefly describe how one can refine the differentiable structure of 81? as indicated in the proof above. Let V be a smooth vector field on Al, transversal to 81?. Define a function c1 on a neighborhood 0 of 811 in M by 1,

= 0.

¶WPDE

E C' Approximate by 'I' using a mollifier, and let E = (x E 0: W(x) = 0}. Then E is a hypersurface in M. Flowing along V Then

produces a Clw diffeomorphism between 811 and E, and this induces a refinement of the differentiable structure of 811, useful for the proof of Proposition 4.6. Using Proposition 4.6, we can obtain solutions to the Dirichlet problem (4.70) in the form

u=S(Th1f)

(4.81)

We can also denote this by u =

PT f,

in view of uniqueness results. We have

(4.82)

2

when 811 is of class

w satisfying (4.9), by (4.74) and (4.47). In fact, this can

be extended to p E (1, 2), but we will not carry this out here. If 811 is of class cI+r, r E (0,1), we do have the extension of (4.82) to all p E (1, DC), by (4.75). We also have (4.83)

0<s
P1:

by (4.76) and (4.29). Furthçrmore. we can interpolate between (4.83) and (4.43). to obtain

O<s<1+r,

(4.84)

We next obtain variants of the property (3.20) for

class

C' 3-r

domains with boundary of

PRoposiTioN 4.7. Assume 811 is of class C' r E (0, 1). Let w be a modulus of continuity satisfyinqw(t) t1', t e (0,1), for some p E (0,r). Then

P1:

(4.85)

with (4.86)

a(h)

PROOF. Set A(k) = (4.87)

(Here we refine the define C(A)(aQ) and

with

=

J

(log

dy.

We claim that PTf

f

S(S'f) has the

property

C

differentiable structure of 811

(811), as

to a smooth one, in order to

in Chapter I.) Granted (4.87), since

dt, by

Cr

Proposition 1.2 (together with the remarks (1.15) (1.16)) of Chapter I, we have PT f Sg, g C and the result (4.85) follows from Proposition 4.3 (plus the same

remarks from Chapter I), with cr(h) = y(t)t' dt. The relations just stated between w, 'y, and a then lead to (4.86). Thus Proposition 4.7 is proven once we establish the following. PROPOSITiON 4.8. Let Oil andw be as in Proposition 4.7, and A(k)

Then we have an isomorphisrrt

S:

C'(M (811).

w(2 k)

4. LAYER POTENTIALS ON

SURFACES

PROOF. As in the proof of Proposition 4.6. we can show that T: is Fredhoim, by combining the arguments used in that proof with results of §5 in Chapter 1, particularly Proposition 5.2 there. It follows that S in (4.88) is Fredhoim, of index zero. Injectivity is a consequence of results of Proposition 4.6, so the proof is complete.

An important operator for the study of boimdary problems for L is the Neumann operator N, defined by

NJ =

(4.89)

We have the following mapping properties.

PRoposiTioN 4.9.

If 811

is of class

and w satisfies

2
(4.90) 1f811

then

re(0,1),

is of class

N:

(4.91)

then

JrP(8cfl

0

.s


and

N:

(4.92)

Furthermore,

0< .s
—p

if w is as in Proposition 4.7,

(4.93)

N:

C (811),

with a given by (4.86,).

PRooF. The

result (4.90) follows from (4.82), (4.92) follows from (4.83),

follows from (4.85). Note that if we write P1 f we have (4.93)

N=

(4.94)

and

= S(S 1f) and apply (4.48),

+

on the various spaces listed in Proposition 4.9. This also establishes (4.91).

We can produce another formula relating N and 8, as follows. On M \ 811,

consider (4.95)

Vf(x)

- SNf(x)

=

f {f(y)r(x.

y)

- Nf(y) E(x, y)} da(y).

00

Applying Green's formula, we see that p496"I

xeQ, 0,

Approaching 811, we obtain (4.97)

Comparison with (4.94), which we can rewrite as (4.98)

x€Q.

TO POE

172

shows that

K=SKtS1.

(4.99)

In particular, we deduce that H lP(3Q)

K

(4.100)

is compact, for2 p < co if Ofl is of class w satisfying (4.9), and if 311 is of r E (0, 1), then we can combine (4.53) and Proposition 4.6 to deduce class further mapping properties of K. Since S is an isomorphism, we deduce from (4.94) that fit is Fredhoim of index zero, in (4.90)—(4.92). In fact, we have the following. PROPOSITION 4.10. Assume V = 0 on 11 (which, recall, is assumed to be considered in Proposition 4.9, KerN is the connected,). In the cases one-dimensional space of constant functions on 811. Furthermore, given f in any of the target spaces in fl.90,)—('4.92), (4.101)

PROOF. In case (4.92), the fact that Ker Jsf consists of constants follows from Zaremba's principle. In case (4.90) this fact follows from Green's formula (4.69),

justified by the estimate (4.47). In case (4.91), we have f 6 Kerflt if and only if + K*). The same regularity belongs to g = 51f (which is in for argument used in the proof of Proposition 4.5 then implies that g 6 all a C r, q c co, so f E if l+c,q(311), and the previous arguments apply. Knowing that fit has one-dimensional null space and is Fredhoim of index zero, we now know that R(flt) has codimension one in each case (4.90)—(4.92). Now the implication in (4.101) follows from Green's formula if f is sufficiently regular, which includes all cases in (4.90) and (4.92). For cases in (4.91) that lack sufficient regularity, the Green's formula argument still shows that the implication holds on a dense subspace, hence, by continuity, everywhere. This is enough to complete the proof.

Using these results, we can treat the Neumann boundary problem

Lu=0 on 11,

(4.102)

PROPOSITION 4.11. Assume V =

on 811. 0

811 is of class C1w, cv satisfying (4.9).

(4.102)

Then, given f

6

= 0. Assume 1 C p C co,

has a solution u satisfying

(4.103)

I(VuYHLJum

If 811 is of class cl+r. r 6 (0,]). then, a

on 11 and assume

solution u 6 ('I

unique

given f

c

up to an additive constant.

sE (0,r), (4.102) has

5. PARAMETBIX ESTIMATES AND TRACE ASYMPTOTICS

173

PROOF. The oniy part not a consequence of Propositions 4.9—4.10 is the exThis follows from p E (1,2), when Oil is of class istence for f E solving (4.104)

and writing it = 8g. In turn, solvability of (4.104) on IY(Oifl, for p E (1,cx),

fda

0, follows by arguments similar to those used to prove Propo+ K*), one shows that g E sition 4.5. In particular, if g E Ker for all q C cx). Then consider it = 8g. Formula (4.69) applies, so it is constant on 11; say it c0 there. Then, using (4.68), we have, granted

(4.105) 80

cz_

where the last identity uses the fact that g is equal to the jump of

across Oil.

Iff80gda=Oitfollowsthatu=Oonfl_. Henceit=Oonil,sog=0. This shows that — I + Kt is injective on = {g

(4.106) Thus its range on

has

L"(Oil)

fgda

o}.

codimension one, and the rest of the proof follows.

Note that, in the case (4.103), uniqueness of it (modulo an additive constant) follows from Green's formula, when p 2. Following our previous pattern, we omit a treatment of uniqueness for the case p E (1, 2). One can use the Neumann operator to investigate numerous other regular elliptic boundary problems. Compare treatments of the smooth case, discussed, e.g.,

in [Ti].

5. Parametrix estimates and trace asymptotics Let Al be a smooth, compact, n-dimensional manifold, equipped with a Riein local coordinates. We assume k Z+ and mannian metric tensor of class r E (0, 1). Let A be the associated Laplace operator, and consider L = —A + V, if k 2, or V C(M) if k 1. Assume L is positivewith V E definite (adding a constant to V if necessary). One of our goals here is to prove the following.

THEOREM 5.1. ff1 k C n, then there is an asymptotic erpanszon (5.1)

sTh(Ao + A1s

+ Aksk +

as s \ 0. Before proving this, we first demonstrate the following result, on the integral

kernel of L', given by (5.2)

3.

THEOREM 5.2. If

k

1

TO POE

< n — 2, then

there is an asymptotic expansion of

E(x,y), of the following sort, in local coordinates: (5.3)

where in z, and

z)

is continuous on z

0

and

homogeneous of degree —(n — 2

—

2-k)).

(5.4)

We can give more precise results on the smoothnessz),of t in (5.3), note that, in local coordinates, dv(y) = As we will see, Theorem 5.1 follows from a variant of Theorem 5.2, applied to + L, acting on functions on IR x M. Note that, if we omit the remainder on the right side of (5.3), we have the integral kernel of an operator F, also given by via (5.7) below. Regarding the appearance of

Fu(x) =

(5.5)

ff F(y,

i/)Cu(y) dy

where

(5.6)

with Fe+2(y,

+ ... +

+

=

homogeneous of degree —2— £ in e• We will construct these symbols

so that (5.7)

F F = F(x, This is technically convenient, to deal with the limited regularity of the symbols we work with. If L(x, D) is a differential (or pseudodifferential) operator with standard symbol representation, then (5.8)

L(x, D)Fu = (2n)

-fl

ff

L(x,

dy

This can be treated by methods used in §9 of Chapter I. Under our hypotheses on L = + V, we have, in a local coordinate system (5.9)

L(x,O =

with (x, a homogeneous polynomial of degree j in and coefficients that are 1+r, and to functions of x belonging respectively to to (In fact, L2(x,e) = and L0(x,e) = V(x).) We can then obtain satisfying (5.6)—(5.7), such that (5.10)

L(x, D)Fu = u(x) + f I'k(X, y)u(y) dy,

with —

We k.

give the details of this for the ease k =

1;

IC

r)

there is a routine extension to larger

5. PARAMETRIX ESTIMATES AND TRACE ASYMPTOTICS

In

this case, we

175

evaluate (5.8) by setting

(5.12)

=

(5.13)

We have (5.14)

e

where L2(x,C) E

(5.15)

—

Also we have

=

(5.16)

L1(x,e)

(5.17)

Note

1A10(x,y)I

—

that

(5.18) —

=

ff

Now, define

+ (x — y) .

y) .

.

aI—l

and

by 1,

(5.19)

=

- 4

.

=1

We

obtain

LFu(x) = u(x)

+

(5.20)

+

+ in turn yields (5.10)—(5.11), in the case k = 1, since the quantity in brackets in the integral on the right side of (5.20) is a sum of terms homogeneous in e of degrees —1. —2, and —3, and the terms homogeneous in e of degree —1 are O(Ix — This

To (5.21)

estimate Rk(x,y) in (5.3), given (5.10)—(5.11), note that

= —]Tk(X,y).

176

3.

TO PDE

Now (5.11) implies that, for given y in 0, a coordinate chart on M,

Vpc

(5.22)

n—k—r

Given our hypotheses on the coefficients of L, techniques used in §1 (or even simpler, more classical methods) give

VP<-.

(5.23)

Hence, by the Sobolev imbedding theorem,

Rk(.,y)eIfl(0),

(5.24)

Note that, under our hypotheses, n — k — r — 2 > 0. To obtain estimates on Rk(x, y) of the form (5.4), we do the following. Suppose = 2p. We want to estimate Rk(x,y) on {x Jxo — — xoI c p/2}. Shift coordinates so Xo = 0, and introduce the dilation operators (5.25)

u(x) = Rk(X,y) for

p, then ri,,

Ixl

(5.26)

— P2

satisfies the equation

VpUp =

where

L\,v =

(5.27) and

(5.28)

= V(px),

= I'k(PX,Y),

The equation (5.26) holds on B1 coefficients g3C are

> 0).

some

of

class

= {x

fr, and

jxl C

z\, is

gJk(x) 1}.

gik(px).

On this set, the family of

elliptic, uniformly in p E (0, P0] (for

Also, by (5.11),

(5.29)

Now the estimates (5.23)—(5.24) on Rk(',y)

imply

(5.30)

Vqc

for its dilates

ri-_k—r—2'

and

C(q1)p',

(5.31)

Note for

that q 5

1 — n/q1.

n/(n — k — r — 2)

n/q > n — k —r — 2, with a similar implication such bounds on ri,, in crude norms, and given the bound (5.29), satisfies the elliptic equation (5.26) and the regularity estimates on

C C(QE)P

(5.32)

(5.33)

#.

Given

the fact that this PDE give

Hence,

Vq1
for any E >

(n-k-r 2+-e)

V

q C oc,

>0.

5. PARAMETRIX ESTiMATES AND TRACE ASYMPTOT1CS

177

In other words, we have the following more precise version of (5.4): (5.34)

—

Rk(x,y)1 C

This completes the proof of Theorem 5.2.

The analysis just described has some points in common with estimates on E2(y,x — y) made in [MT], in the case where M has a C' metric tensor, though the analysis done there had a different purpose. To study the integral kernel of we use the following device. Given E L2(M), set

s>0,

u(s,x)=

r r (o.3o)

s <0.

We see that = Lu +

(5.36)

In

view of the decay of u(s,

x)

as

oo,

+

u =

(5.37)

so this gives, for s

we have

> 0,

(5.38)

(5.39)

p(s,x,y) =

—238E(s.x,0,y), There is an extra + in Theorem 5.2, but an entirely parallel

where E(s.x,t,y) is the integral kernel of

here, compared to the quantity analyzed

argument yields the following.

PROPOSITION 5.3. Assume 1 k C class

n,

and that M has a metric tensor of p(s,x,y) of

C1. If L is given as in Theorem 5.1, then the integral kernel has the following asymptotic behavior on the diagonal x =

p(s,x,x)

(5.40)

+B,(x)s+ ...

y, as s \ 0:

+o(sk)),

with

e

(5.41)

We

are now ready to prove Theorem 5.1. From (5.39) we easily have p(s, x, y)

on (0, oo) x M x M. Of course, an operator with a continuous integral kernel might not be of trace class. However, since M is compact, such continuity certainly implies that e s'/L is Hilbert-Schmidt, for each s > 0. Since = we deduce that is in fact of trace class. continuous

3. APPLICATIONS

178

TO PDE

A well known argument (cf. Appendix A of [T51) shows that, whenever a trace-class operator K on L2(M) has a continuous integral kernel k(x,y), then Tr K is equal to the integral of k(x,x), so we have (5.42) for

each s

Theorem

> 0.

Given this and the behavior (5.40) of p(s,x,x),

we

have (5.1), and

5.1 is proved.

6. Euler flows on rough planar domains Let 11 be a bounded open set in 1ft2.

The Euler equation for ideal incompressible

flow in Il is

divu=0. Here u is the fluid velocity and p the pressure. The boundary condition is that u

be tangent to the boundary. There is a good existence theory when Dcl is smooth, which can be found for example in Chapter 17 of [T5]. Here we want to allow 311 to be rough. have Our basic strategy will be to use an approximation procedure. Let smooth boundary and satisfy

fl1ccQ2cc...ccIl5/Q For convenience, we assume each domain has connected boundary. are vector fields, tangent to Dcli. satisfying div V2 = 0 and (6.2)

jVjIIrl(q)

K < 00,

V

Assume v5 e

E

We want to construct a weak solution to the Euler equation for incompressible flow x 114 on 11, with initial data v, as a limit of (a subsequence of) solutions uj e to (6.3)

+

=

div u3 =

We require to be vector fields on tangent to fluid pressure. We will obtain certain uniform bounds on the conservation law (6.4)

u5(0) =

0,

The functions P5 give the First, there is the well-known

C.

=

Next, we have a vorticity estimate. Recall that, if denotes the 1-form on corresponding to via the Riemannian metric, then the vorticity is given by = *dü5. As a consequence of the 2D vorticity equation, 3w5 —a;:-

=

+

0,

we have (6.5)

=

K.

6. EULER FLOWS ON ROUGH PLANAR DOMAINS

=

Now div

0.

*dü, =

is an elliptic system. In fact, we can write

=

(6.6) where

x Ilj) satisfies, for each t,

E

=wj,

(6.7)

Thus,

179

=0.

if 0 cc QN, we have

C(O),

(6.8)

N,

Vj

where we set

Hk00(0)

(6.9)

{u:

E BMO(O),

k}.

Hence

C(O),

(6.10)

V j N.

in the analysis of solutions to (6.3), it is convenient to rewrite the system as (6,11)

where 13 is the Helmholtz projection onto the space of divergence-free vector fields on tangent to the boundary. Also, it is useful to have the identity (6.12)

if

The pressure is eliminated from (6.11), but it can be recovered as the solution (well defined modulo a constant) to the boundary problem

= —Tr((Vuj)2)

(6.13)

on Ili,

= —ki

on

where Ni (x) is the curvature of 3cl3 at x. We now assume that 12 and all the are convex domains in R2. We can then use

Kadlec's formula,

E f ldkUflu(2 dx U

f

dx + (n — I)

f

H(x) dS(x),

803

U

(in it dimensions, where H(x) is the mean curvature of cf. [Gri], p. 139), to obtain a uniform bound IIfiUn2 S and hence, by (6.5),

< C2 < cxi

(6.14)

This in turn implies (6.15)

S C'

Furthermore, by Proposition 3.5, we have a uniform bound and hence

(6.16)

Ilfi(t)ULjI)(iiJ)

< øo,

which implies

(6.17)

5 C.

S

Ivc,

3. APPLICATIONS TO PDE

180

We have

C;

5;

(6.18)

hence

(6.19)

S

0 uj(t))

C,

a

(6.20) Thus we have 5; C.

(6.21)

Interpolation

with the bound (6.15) yields

VTcoc, s<1,

compactin

(6.22)

is a sequence of extension operators:

where (6.23)

—÷H'(Q),

E1

—÷L2(Q),

with uniform operator norm bounds. Also, interpolation of (6.21) and (6.8) yields, for any C cc

for

N}

:j

(6.24)

compact in C({—T,T],

any T C 00, S < 1, pc cxi Hence :j

(6.25)

Now,

compact

in C({—T,T] xb).

perhaps passing to a subsequence, we obtain

(6.26) with

N}

9).

—tuE

convergence in the weak* topology. Furthermore, (6.22) implies

(6.27)

—÷

for any given

(6.28) for any

T

c

s C 1.

®

Tc

(6.29)

it in norm, in C([—T,T], H8(Q)), Hence

—

it® u

L2(c1)),

weakK in

Boundedness results above then imply —+u®u weak* in

11),

and hence (6.30)

div(u®u)

weak*in

To see that ii solves the Euler equation, we rewrite (6.3) as (6.31)

div it3 =

+

where

(6.32)

Vp3

-(I— P4

6. EULER FLOWS ON ROUGH PLANAR DOMAINS

181

satisfies (6.33) as

C,

a consequence of (6.19). Normalizing Pi

that

dx =

0,

we have

bounded in

(6.34)

(perhaps passing to a subsequence), we have

so

—>p weak*

(6.35)

(6.36) We

Otu+

div(u®u)=—Vp,

divu=0.

next show that, for each t,

(6.37)

u(t)

E V°(Q) = L2-closure of {v 6

fact, for each j C cc,

In

in. L°°(R,H'(Ifl),

then results above yield

and

standard

belongs

T) : div v =

0}.

by

to the analogous space

theory for smooth boundaries. Hence, for each t, there exist C

such

the

(t) 6

that

div

It

so

follows from (6.27) that

=

0,

—

—÷

S

u(t) in norm in L2(Q)

as

j

—÷ cc.

so

(6.37) is

established.

Regarding other properties of the limit, we have from (6.25)

ii 6 C(TR x 11)

(6.38)

and, from (6.8), (6.39)

ueL°°(lR,H'°°(O)), VOccQ.

We derive some further local bounds, starting with bounds on the pressures Pi,

which solve (6.13). Given 0 CC 11N, the right side of the first equation of (6.13) is bounded in for j N, for any p < cc, as a consequence of (6.8). The global bound (6.33) plus local elliptic regularity yields IIVpjlIHlPco C,(O),

(6.40) for

j N.

Hence, if we use the equation

(6.41) plus

atuj +

=

the bound (6.10), we get

(6.42)

S C(O).

In the limit, these yield associated bounds on it.

We summarize what has been established:

TO PJJE

PRoPoSITION 6.1. Let 1? be a bounded Convex domain in JR2. Let v E

C'(Q) n

be given. Then there exists a solution u to

(6.43)

n V°(Q)

the Euler equation

divu=o in RxQ,

u(O)=v,

satisfying

uE (6.44)

Q)flLip(R%T°(fl))

E

Vp e

L2(Q)).

Furthermore, given any 0 cc ft we have UE

(6.45)

Ut e

Vpe

Vp< cc.

We can extend Proposition 6.1, obtaining some results of a flavor such as obtained for flows in all of IE&2 in [DiPM}. That is, we can obtain weak solutions to the Euler equations (6.43), with initial vorticity in for some range of p. Some of these results follow from the analysis done above. Indeed, given (6.46)

SC,

we have

5 C. As long as p 2, the bound (6.14) still holds. If p = 2, we do not quite get (6.16) -(6.17), but if p.> 2 we have so we do Lip S get (6.16)—(6.17) when (6.46) holds and p.> 2. The rest of the arguments involving (6.18)—(6.37) then go through (except we get weaker bounds on 0 thus Proposition 6.1 extends

(6.47)

to the case of initial data satisfying v e

cc ftv), and

V°(Q) n Hh2)(cl),

provided p> 2. We can go further, obtaining weak solutions for smaller p. To do this, we need a couple of more tools. One is the following result of V. Adolfsson [Adi, extending the consequence of Kadlec's formula described above. Namely, for our sequence of

convex domains estimate (6.48) This

and for

satisfying (6.7), there is for each p e (1, 2J a uniform

IIfj(t)11n2P(03) S

gives, in place of (6.15),

(6.49)

H1P(l,)

S C.

If p <2 here, we cannot expect the L°°-estimate (6.17). Sobolev's theorem yields S C,

ifpe

q=

(1,2). Hence, inplaceof (6.19). wehave

S

7. PERSISTENCE OF SOLUTIONS TO SEMJLINEAR WAVE EQUATiONS

183

with

=

E (1,21, provided p E

The second additional tool we need is an U-operator norm estimate on P3. In 3, with uniform bounds [FMMJ it is proven that P3 acts on U(Q3), for 3/2 r satisfy uniform Lipschitz estimates. That situation can be C',., as long as C assumed to hold for our family of convex domains, so we have, in place of (6.20), div (u3(t) ®

(6.53)

5; C,

provided (6.51)—(6.52) hold with r E (3/2,21. Note that

pE

(6.54)

As before, we can push through variants of (6.21) (6.37), and obtain the following. PROPOSITION 6.2. Let Q be a bounded convex domain in R2 Let (6.55)

v

V°(fl) n

be given. Assume p E (12/7, 2). Then there exists a solution u to the Euler equation (6.43), satisfying uE (6.56)

n

it,, E

VpE with r given by (6.52).

We end this section with a remark on planar Lipschitz domains. Namely, the (6.17) on the fluid velocity Lipschitz estimate (6.16) and consequent can definitely fail. To see this, start with f(x) = log IxI on the annulus Q = {x E 1/2 5; 5; 1}. It is harmonic, and ii = *df defines a vector field it on Ii which gives a steady, irrotational solution to the Euler equation. Now one can Il —' 0 of Il onto a planar region C with easily construct a conformal mapping is not Lipschitz, a corner whose internal angle lies in (ir, 2ir), such that g = f o = *dg is not hounded on 0, nor does it belong to H1(O), though it does and still generate a steady irrotational flow. Furthermore, one can arrange that v not for p in the range considered in Proposition 6.2. This illustrates that be in different methods will be required to prove existence theorems for Euler flows on such Lipschitz domains. 1

7. Persistence of solutions to semilinear wave equations Let IV! be a Lorentz manifold. For simplicity, assume M = R x X where X is a compact Riemannian manifold, of dimension n. and the metric on M is = —dt2 + gx. We consider hyperbolic systems of the form (7.1)

Du = B(x,u,Vu),

3.

184

TO EWE

a being

where flu = 8?u —

the Laplace operator on X. Also, Vu denotes smooth in (x,u) and a

We take u: M —÷ IRk. We assume B(x,u,p)

quadratic

form in p. As is well-known, given initial data

= 1€ H84l(X),

u(O)

(7.2)

there

=g

e

118(X),

is a local solution

uEC(I,H841(X))flCl(I,H8(X)), 1= (—Ti,?'2),

(7.3)

for some T1 > 0, provided s > n/2. Also, given a bound

IIu(t)IIcI(x) + IIôtu(t)IILx(x)

(7.4)

the

5;

K,

t e I,

solution extends to an open interval J D [—T1, T2]. A proof can be found in

We aim to

[T2].

are Zygmund

prove the following sharper result. As usual,

spaces.

PROPOSITION 7.1. The solution to (7. 1)—(7. 2) extends to an open neighborhood

of

[—T1 T2J provided ,

IIt4tHIcl(x) +

(7.5)

5; K,

t E I.

e.g., = be a Friedrichs mollifler commuting with Let E OPS'(M). Now, in general, for sufficiently smooth w(t, cc),

A=

= 2(wt,wtt

+

(7.6)

Applying

this to A8J5u,

- aw).

we have

+

jj

=

(7.7)

Set

£2) —

2(AsJeut,A8JeB(x,u,Vu)) 5;

.

1A8J6B(x,u,Vu)(1L2,

where the second identity in (7.7) uses — a)A8Je = Proposition 7.2 we will establish an estimate that implies: IIA3B(x,

u,

Vu)

(7.8)

+ i)}.

+

Thus, if (7.9) we

=

+

have, under the hypothesis (7.5),

(7.10)

and,

Ne(t)

C+Cf

+C)No(s) ds,

letting e \ 0, we have

(7.11)

—

N0(t) 5;

C+Cf

+C)No(s) ds.

Now, in

7. PERSISTENCE OF SOLUTIONS TO SEMtLINEAR WAVE EQUATIONS

185

Our next step is inspired by [BKM}. As in Proposition B.1.C of [T2], we have (7.12) given

[i

kIlc4

s > n/2. Also CIIOjujIcn [i + log

(7.13)

Hence, under the hypothesis (7.5), if .s > n/2 N0(t)

(7.14)


in

(C+log4

IIOtuiIHs] IOtuIi C9

(7.9),

No(s))No(s) ds.

Since

/

(7.15)

.h

rlogr

Gronwall's inequality applied to (7.14) yields a bound on N0(t) for all t e I, and the proof is complete.

We now establish an estimate that implies (7.8), upon setting v =

and

a=1.

PROPOSmON 7.2. If B(x, then, for s

it,

v)

is smooth in (x, it)

and

a quadratic form

in

v,

> 0, l
IIB(x,u,v)IIJ-rsr(x) (7.16)

+ i)}.

x {IIvIIHRP(x) +

PRooF. Write B(x,u,v) = B(x,u)v. v = Wv .

v,

where W

B(x,u) is a

matrix-valued function. By (0.22) of Chapter II. we have IB(x, u)v

.

VIIHa.P

jjsp +

(7.17)

It

remains to estimate the last factor in the last term, i.e.,

For this, we

use (7.18)

We estimate Twv by using the implication W E Tw E OPS?I, as stated in (0.19) of Chapter II. As for the other terms, we use the following implication, valid for a > 0: (7.19)

vE

2',,, Ii,, e

which follows from a straightforward symbol estimate; compare (3.5.7) and (3.5.11)

in [T2}. Hence, for s > 0,1

0, (7.20)

186

3. APPLICATIONS TO PDE

Applying the Moser-type estimate given in (0.9) of Chapter II to W = B(x, it), we obtain the desired estimate (7.16).

We mention another known improvement on the straightforward results described in (7.2)-(7.4). Namely, one can relax the requirement s > rr/2. For example, when is the standard Laplacian on W', it is shown in [BB] that (7.l)—(7.2) has a local solution of the form (7.3) as long as s> fri — l)/2 if ri = 3, and as long as s (ii — l)/2, if ri 4. If, in addition, B(x, it, Vu) belongs to a certain class of "null forms" that includes ones arising in "wave maps," then it is shown in [KS] that (7.1)—(7.2) has a local solution as long as s > fri — 2)/2. We refer the reader to [BB], [KM], [KS], and references therein for more on this. While those results do not imply Proposition 7.1, they do lead one to wonder whether this Proposition might be improved.

8. Div-curl estimates The most basic div-curl lemma takes the following form. Suppose it and v are vector fields on JR3 satisfying

pE(loo),

(8.1)

Then (8.2)

where 551 denotes the hardy space. Equivalently, in view of the duality result of [FS], the conclusion in (8.2) is that it v can be paired with an element of BMO. Such a result arid many variants were presented in [CLMS]. One of the analytical techniques used in [CLMS] was the commutator estimate (8.3)

P E 0PS10,

IfPu —

of [CRW], which was established in §10 of Chapter I. Using the identity

f Pu - P(fu)]v dx =

(8.4)

f

f[(Pu)v - u(P'v)] dx,

one obtains (8.5)

I(Pu)v —

<

1

which in turn was shown in [CLMS] to yield a number of estimates, including the div-curl estimate mentioned above. We recall a number of div-curl type results established in [CLMS], and present proofs. We use two general principles to derive these results. One is an div-curl lemma," generalizing (8.5). This result, given as Proposition 8.1, was formulated by the author arid P. Auscher. The other is a "super-commutator estimate," given in Propostion 10.3 of Chapter I, Rather than dealing specifically with

Hardy spaces, we show directly that the relevant quantities can be paired with elements of BMO. Due to the famous duality established in [FS], this is the same thing, but in fact it is t lie BMO-pairing that is most directly useful. Actually, we

8. f)IV-CUI-tL ES'I'IMATES

187

find it convenient to replace the BMO-seminorm by the slightly stronger bmo-norm, given by

+

(8.6)

1. The two types of estimates have the 'I'o(C) = for el same implications for local analysis. Most div-curl type estimates have been established in the context of constantcoefficient PDE. The end of this section deals with a variable-coefficient div-curl type estimate. We begin with the following div-curl lemma," whose statement in this form arose iii the course of correspondence of the author and P. Auscher: where '1'o E

1

PROPOSITION 8.1. Let u and v be defined on ir and take values in 1W and R', respectively. Let 2, Q E OPS?0 (or more generally in OPBS?1) be a k x N and an £ x N matrix of operators, and consider Qv

(8.7)

=

Assume that p E (1, oc) and

r>p.

(8.8)

(Note

that h takes values in 1W.) Then,

(8.9)

(f,Pu.

if supp f C K is compact in R't,

<

+

PRooF. We have (f,Pu

(8.10)

Qv) =ff(Pu).(Qv) dx=fvQt(fPu) dx =

(v, [Qt, Mf]Pu)

+ (fv, QtPu).

Now the hypothesis (8.8) implies (8.11)

(fv,h)I

while the basic commutator estimate (8.3) implies that (8.12)

l[Qt,A1fIwhLv

Hence we have (8.9).

We note that the div-curl lemma and a number of variants are special cases of

this result.

188

3. APPLICATIONS TO PDE 1.

The div-curl lemma.

Here, N =

n, £ =

n, k

Pu =

taking w =

2.

+ (Idiv

.

Estimate

of Tr((Vu)(Vv)).

Pu =

with ii =

Qv =

i3 =

=

(8.16)

so

uIIH 1.?)IJWIIH1,p',

implies the result stated in (8.1)-(8.2).

(8.15)

Then,

Then

we have

(f,u

(8.14)

which

We taken =

Pu.Qvz>ujdj(A_1v), Q'Pu=Jss' divu,

(8.13)

so,

1.

u, Qv

—831C2

divu.

the conclusion is

(8.17)

+

3. Estimate

of det Vu, n = 2.

v==A1u2, then det Vu(x)

2v) —

(82A1w)(8iiY'v).

Thus, herek=€=1, N=2, n=2.Wetake (8.18)

Pw =

Qv

Qv = det Vu,

so

QtPw

= (32iY1v. —OiiY1v).

—5i52A2w+DiO2iY2w =

0,

the conclusion is

(8.20)

(f,det

CpK hf

hhulIhHl,P

Hu21(111,P'.

There are a number of generalizations of #3 involving estimates of wedge prodtake a look at some of these. While the special case (j + 1) + (k + 1) = n of the next result can be easily deduced from Lemma 8.1, we find it more convenient to use Proposition 1.10.3 to establish the general result. ucts. We

8. DIV-CURL ESTIMATES

189

4. Estimate of duAdv.

Let it be a j-form and v a k-form on M, j + k C n

ff A du A dv =

(8.21)

—

2.

Let f be an L-form,

6A' *

= 0, the right side is equal to

Since

(8.22)

=

(A—1 dWfdA 4i,

([{A—' d,

*1)).

Applying Proposition 1.10.3, we deduce that (8.23)

5. Estimate of Assume

+ 1) = in

are Li-forms,

C n.

Let f be an (n — m)-form.

Then we claim that

ffAdui

(8.24) provided

p2 e (1,oc] and

(8.25)

PkE(1,00).

Pk

P1

To prove this, note that, since dii1 A to write (8.26)

du1 A

A duk_l is exact, we can use Hodge theory A duk_1 = dii,

with IIuUHhn C

(8.27)

Then, with v =

1

1

P1

Pk-1

P

1

jj'Pk-i,

PE(l,00),pk=p.

we have

(8.28)

The

IIH'fl

1

ffAduI...Aduk=ffAduAdv.

estimate (8.23) applies to the integral on the right side of (8.28). This proves

the desired estimate

estimate,

(8.24). The case k = n, = 0 yields a particularly significant role in

a Jacobian determinant

which played

6. Estimates on solutions to Maxwell's equations.

Let .1 be a 2-form on Minkowski space R3', representing an electromagnetic field. Part of Maxwell's equations is

=0.

3. APPLICATIONS TO PDE

If F

we can write F

we have

Thus by

ff

(8.30)

for

dA, A

f supported in K cc 0. If If and B are the electric and magnetic fields

encapsulated in F, (8.31)

i.e.,

(with t =

Xo)

F= EEjdxj Adt +Bidx2 A dx3 + B2dx3 A dx1 +B3dx1 Adx2,

then (8.30) is equivalent to an estimate on If B.

The rest of Maxwell's equations (in empty

space) is d*F =

0, where d* is

"adjoint" of d, defined by the (indefinite) Lorentz metric. Rather than directly augmenting (8.29) with this, we consider a 2-form Q, satisfying the

d*Q =

(8.32)

0.

This means d(*G) 0, where * is the Lorentz analogue of the Hodge star operator. Hence #4 provides an estimate on FA*Q = (F, Q)w, where w is the natural volume element

on R3'. Hence

ff(F,c)

(8.33)

If Q is

written in the form (8.31). with If3 replaced by D3 and B3 replaced by H3,

then (8.33) is equivalent to an estimate on If D — (29) in [CLMS]. We now

establish some results that include a

H. Compare

the analysis of

variable coefficient version of The-

orem VI.1 in [CLMS], and that complement estimates in [BHJ. (See also [LMZJ.) This work arose from conversations with P. Auscher, following up our work in EAT]. Let be matrices of operators, : C°°(RT', Rc) Wa). The principal symbol of takes values in Let Q OPS° be a k x k matrix of operators, with principal symbol q(x, taking values in End(Rk). Assume

that

(8.34)

where

q(x, (x,

=

+

are homogeneous of degree in takes values in £(Wa,Rk).

OPThm,. Here,

PROPOSITION 8.2. Assume that u

A1u€

A2v€

Then

€

principal We

symbols of B3

desire to estimate v Qu.

1
v

r

>p, p>p'.

< oc, and

8. DIV-CURL ESTIMA'I'ES

PRoOF. Given K C

191

compact, suppose supp f C K. and write

(8.37)

(f.

v Qu) = (fv, Qu).

Now (8.34) implies

EOPS

(8.38)

Then

(1 v,Qu) = (fv, B1A1u) + (fB2A2v,u) + ([132A2, M1}v,u) + (1 v,Ru).

(8.39)

Note that (8.36) implies (8.40)

B1A1u

L'S,

B2A2v

LI,

so we have (8.41)

(fv, B1A1u)I +

<

Similarly,

(fv, Rufl

(8.42)

Finally,

OPS°, the basic commutator estimate (8.3) gives

since B2A2

(8.43)

1RB2A2, A41]v

(8.44)

I(f,v

Such a result was established in Theorem VI.1 of [CLMS], under the following additional hypotheses. It was assumed that A1 = A2, a first order differential operator. with constant coefficients, Q was taken to be a constant symmetric k x k matrix, and it was assumed that u = v and p = p' = 2. In IBH], it was also assumed that A1 = A2 was a first order differential operator, but it could have variable coefficients, and {BH] took Qu = Q(x)u, with Q(x) a smooth k x k matrix-valued function; then [Bil] derived Besov space estimates for v Qu. Other extensions, again in the constant-coefficient case, are considered in {LMZ]. Note that, if (8.34) holds, then (8.45)

In

Ker ai(x,e) —÷

particular, if a1 = a2, then v

vanishes whenever v

Ker ai(x,C).

In Theorem VI.1 of [CLMS], this was part of the hypotheses, rather than (8.34);

W have constant rank on a1 (x,e) = a2(x,e) Rc proof of the following result, that (8.45) plus constant rank

another part was that

We record a simple

hypotheses

imply (8.34).

Let A1 OPS° have principal symbols Q Assume that, for each x,e, (3.12) holds. Also assume that a2(x,e) have constant rank on T*IRn \ 0. Then there exist with principal symboLs such that (8.34) holds.

PRoPosITIoN 8.3.

a

192

APPLICATIONS TO PDE

PROOF. The constant rank hypothesis on of 1R1 onto Ker a1 (x, is COC

a1 implies that the orthogonal pro-

jection ir1 (x,

degree 0 in e• is COC on

Similarly, \

0.

Now, for each (x,

(8.46)

E

it is also homogeneous of of Rt1 onto 1Z(ai(x,E)) we have an isomorphism \O;

the orthogonal projection

T*IRn \

0

(Ker ai(x,E))' —*

Denote this isomorphism by a(x,

and

1z(ai(x,e)).

set

bi(x,e) =

(8.47)

Thus for vi Kerai(x,e), A

)

(

o

for yE Kerai(x,e).

We will next define b2(x,e)t for (x,e) in a conic neighborhood F of any given 1 r = Dim Ker ai(x,e), T*IRIL \ 0. To begin, take smooth on F, forming for each (x, e F a basis of Ker a1 (x, Now belongs to the range of a2(x, by hypothesis (8.45). Since a2(x, has constant rank, we can find e IRt2, smooth on F, such that

i

(x04))

= a2(x,e)twv(x,e).

(8.49)

Then, set (8.50)

b2(x,e)t Ker

by linearity on Ker ai(x,e), and set b2(x,e)tv = 0

for v I

ai(x,e). then we have for

(851) 0

yE Kerai(x,e),

for vi Keraj(x,e),

given (x,e) e F.

Putting together

(8.48) and (8.51), we have (8.34), but so far only for (x,

e F.

However, we can take a locally finite covering of T*IRfl \ 0 by cones on which such a construction works, and use a partition of unity to obtain (8.34) globally. As an example of Proposition 8.2, when A1 the div-curl lemma, upon taking (8.52) acting

A1—d,

A2, we derive a result containing

A2=&

on differential forms. We take Q = I, and the relevant identity of the form

(8.38) is (8.53)

Q=- E(Sd+dS)+R=(E8)d+d(E8)+R,

where we take E = Laplacian. Then

(1

—

'

0P8 2, A

8.2 directly gives:

—(dS

+ Sd) being the Hodge

8. DIV-CURL ESTIMATES

193

COROLLARY 8.4. Assume it and v are j-forrns on a Riemannian manifold. TakepEs(1,oc). Then (8.54)

u v C

When j = 1, this is equivalent to the standard div-curl lemma. Also, via the lodge star operator, one deduces from Corollary 8.4 that (8.56) We

next apply Proposition 8.2 to an extension of (the N =

2

case of) an

estimate given on pp. 276—277 of [CLMS]. In this case, it and v are defined on JR2 and we take

A1=D1, A2=D2,

(8.57)

Again we take Q = I, and

=

where (8.58)

with E = (D? +

+ 1)-' =

(1

0P82. Then Proposition 8.2 gives:

—

COROLLARY 8.5. Assume that

1


(8.59)

v C

Then

r>p, p>p'

implies

uv

(8.60)

The following natural generalization of Corollary 8.5 is also a simple corollary of Proposition 8.2.

COROLLARY 8.6. Assume 1 C p C oc, it

C

vC

valued, and that

with

r>p, p>p',

Aiu€LT(IRT'),

(8.61)

C

0P50. Then Char

(8.62)

A1 fl

Char A2 = 0

uv

SjL(IRTh).

PROOF. The hypothesis of (8.62) implies that (8.34) holds with 1,

8°.

are real

3. APPLICATIONS TO POE

194

9. Harmonic coordinates The use of harmonic coordinates is an important tool in differential geometry. we produce harmonic coordinates when the metric tensor has limited regularity. We first consider the classical case of Holder continuous metric tensors. We then extend the results to a class of metric tensors with less regularity. To begin, let M be an n-dimensional C' manifold. with a continuous metric H110JM) —* (M) is well tensor. Then the Laplace-Beltrami operator defined, as is the notion of a harmonic function on an open subset of M. We now assume M has a finer structure. Namely, we assume there exist K0, K1 E (0, oc) Here

M a C'

C M (B1 centered at the origin), such that (0) = z and the denoting the unit ball in belongs to C8 (B1) and satisfies metric tensor pulled back to B1 via

and s e (0,

(9.1)

1) and, for each z e

fJjk(O) = 8jk,

0C

diffeomorphism Wz

:

B1

119jk11c8(B,)

(gjk(x))

K1.

We take up the task of constructing harmonic coordinates, centered at a given point

z e M. to translate the P1)E To begin, for 0 < p < 1, using the coordinate system where a neighborhood of the origin in solve on the ball + ... +
=

(9.2)

where aJk(x) =

Note

0<

(9.3)

cc

=

0,

XE,

<£

1

that

(aJk(x)) <

Here and below, will denote a constant, which might depend on n, s, and for < u, but not on other quantities. To analyze the solution to (9.2), it is convenient to dilate coordinates. Consider U(x) = p'i4(px), defined on B1. This solves (9.4)

= =

with

= XE,

Note that

0<

K.'I c

< K21,

K4.

We estimate how close are 11 and Xj. Note jk

i

Given

vE

(9.7)

that

jk

(x)OkxE =

p

= hE.

we have

=f[a3c(O)

—

and hence —

q

,

< oc.

Hence (9.9)

— XE) =

h,

u;

—

X438

9. HARMONIC COOJUJINATES

195

and (9.10)

K5,,4,

V q < oc.

An elementary consequence is that

—

(9.11)

< K6p8•

To derive further bounds on

note that the bound in (9.5) on

V a < 5,

(9.12)

implies

q<

Interpolation with (9.10) gives (9.13)

Then

V q < oc. interior regularity results established in §

(9.14)

1

yield, for all q < oc,

hÜ —

—

Also interior regularity results applied to (9.4) yield (9.15)

It

follows that there exists p =

p(n,Kio)

> 0 such that

_p (x) = (u1(x),.

—)

..,v4,(x))

is a diffeomorphism of B1,'2 onto a region containing B1,'3, such that

c B1,'3

(9.16)

C ((131/2),

having inverse map (" satisfying

(9.17)

BI/4 C ((B173) c

Bi/2

and (9.18)

This argument establishes the following.

PRoposmoN 9.1.

Assume the existence of an atlas

cM such

B1

po(n,s,Ko,K1) > 0 and such that (0) = z and such

that (9.1) holds. Then there exists Pu =

for each z E M

a C' diffeomorphism

that

1

C

—÷

via (9.19)

belongs to

0 <

—+

the inverse

is harmonic. Furthermore, the metric tensor pulled back to and satisfies

(93k(x))

K,3!,

KM.

in both overlaps the harmonic functions are of class (P systems, so we have a (P system. We can hence deduce that the old coordinate atlas and the z e M} must have been of class (P two coordinate atlases are Except perhaps for the bounds recorded above, particularly on po, Proposition 9.1 is classical, the n = 2 case going back to Korn and Lichtenstein; cf. [Mor], §9.3. See also [Che]. As shown in [11W], this cannot be extended to arbitrary continuous metric tensors. Here we obtain harmonic coordinates when belongs to a class of spaces Note that on

coordinate

flfll

a.

TO PDE

Let w be a slowly varying modulus of continuity and assume = w(23) satisfies the following condition, which is slightly stronger than the Dini condition:

EA(i)a
(9.20)

1.

For example, we could take A(i) = for any s > 1. (For such sequences, Proposition 1.10 can be applied.) Let us begin afresh, assuming we have a C1diffeomorphism : Bi —* C M, and that the metric tensor pulled back to B1 satisfies, in place of (9.1), the following: (9.21)

gjk(O) = 5jk,

0

<

5; (gjk(x))

5;

K01,

5;

where we fix 'F such that 41(x) = 1 for x <9/10. We solve the Dirithiet provided p 5; 1/4. This time, making use of Proposition 9.1 problem (9.2) for of Chapter II, we have

0<

(9.22)

5;

(aik(x)) 5; K21,

11Ws/lo&kIIcA

5; K3.

here and below we set 'Fb(x) = 4'(x/b). Again we set 14(x) = p'uç(px), which solves (9.4), this time with

where

(9.23) Again

0

we estimate i4 —

(9.24)

5; "2',

5;

using (9.6), 50 we need to estimate c(A) C

Using (9.7) we see that, for v

C, a(h) =

Now

iT

Cr(B1),

E

K5a(p)

(9.25)

This

5; K4.

implies

(9.26)

lhIIH_1P(B,) 5; K6,,yi(p).

5; K7a(p),

forp e (1,oo), and with Was in (9.21). We also have (9.27)

114'MIIc_i.x

From (9.26)—(9.27) we obtain, for each 0 e


(9.28)

where

is the usual Littlewood-Paley partition of unity, Pick 0 = with a as in (9.20), and set = A(j)'°. Then (9.29)

(1

—

a)/2

5; Kqu(p)°.

Parallel to (9.11), we have the elementary estimate 5; Kioa(p).

—

Then we can apply the regularity result (1.51) in Proposition 1.10 (with A replaced by to obtain —

S

.Kjio-(p)°.

9. HARMONIC COORDINATES

197

Proposition 1.10 also yields from (9.27) the estimate J1W3/414MC1(A) K12.

These estimates take the place of (9.14)-(9.15). It follows that there aists p =

p(n,Kii) >

0 such that

is a diffeomorphism of B112

(x) =

onto a region containing Bi/s, and

with inverse map (",

such

that

(9.16)—(9.17) hold,

IKI1C1c(B,,,) K13,

(9.33)

with a given by (9.24). Hence we have: : B1 —* U2 c Al such PROPOSITION 9.2. Assume the eristence of an atlas 0 a slowly varying sequence satisfying (9.20). Then there exists Pu = p0(n,A,K0,K1) > 0 and for each z E M a C1 diffeomorphisrn —+ B,,3 c RTh B,,3 —* = z and such that the inverse 4';1 : such that is harmonic. Furthermore, the metric tensor pulled back to B,,0 via belongs to C5 and satisfies

that (9.21) holds, with )t(j) \

0<

(9.34)

K15.

(gjk(x))

in these conclusions. would be of interest to see if we can replace C5 by In the original coordinate atlas, harmonic functions belong to c In view of (9.32)—(9.33), in the new coordinate atlas harmonic functions are still the harmonic functions are still of class Hence on overlaps U2', in in both coordinate systems, so we have a C1 5-coordinate system. We can hence and the two deduce that the old coordinate atlas must have been of class coordinate atlases are We note how the two-dimensional version of this analysis leads to the existence of isothermal coordinates. It

PROPOSITION 9.3. In the setting of Proposition 9.2, assume ii = 2. and assume (v1, v2) on B,,, satisfying

M is oriented. Then there exist p' > 0 and functions

on B,,,,

(9.35)

such that, for all x E B,,1, dv2(x) =

(9.36) (where

* is the Hodge star-operator given by the metric tensor (gjk)), and such that

e(x) = (vj(x),v2(x)) is a diffeornorphisrn of B,,112 onto a region containing B,,113. B,,,, 4 Hence there exists Pu > 0 and a conformal (0) = z, and such that the metric tensor pulled back to B,,0 via such that :

satisfies (9.37)

gjk(X) =

f(x)

K14,

If

198

3.

TO PEW

PROOF. Pick p = p(2, K11) as in the proof of Proposition 9.2 and let v1 = constructed there, so e C1(A). Set Pi H2. Perhaps decreasing Pi (by a controllable amount), we assume Mdvi(x) — and < — ) < 10 2• Then *dv1 is a 1-form of class C°, and it is closed, so we can as

define

v2(r)

(9.38)

f*d

the integral being independent of the path in from 0 to x. It is easily seen But also V2 is harmonic, i.e., it satisfies (9.35), so regularity gives that v2 E The rest of the proposition follows readily. v2 E C' REMARK. The construction of isothermal coordinates for a

tensor

(s > 0) was done by [Lic]. The theory of quasiconformal mappings provides a construction of a homeomorphism, of class H12 fl (for some a > 0) from c. 1k2 to C M, conformal almost everywhere, given a measurable metric tensor satisfying 0 < (gjk(x)) K0!, though such a map is not necessarily a diffeomorphism. For a global result along these lines, see [J]. One tool used in the theory of quasiconformal mappings is the Heltrami equa-

tion. We recall that results on the Beltrami equation produced in §2 tie in with Proposition 9.3. We record some results one gets on harmonic coordinates when the hypotheses on the regularity of the metric tensor are varied. As one example. consider the case when the hypothesis (9.1) is strengthened to (9.39)

0
K1,

where we are given p> ii. Then of course Proposition 9.1 applies. We have some additional estimates on the functions z4. Namely, by Proposition 1.12, (9.40)

t4 E

Then (9.39) can be seen to hold in the new coordinate atlas. Hence on overlaps fl the harmonic functions are of class in both coordinate systems, so we have an atlas. We can hence deduce that the old coordinate atlas must have been of class H2P and the two coordinate atlases are This result has also been established in [Sm2J. In view of some results to be discussed in §10, it is of interest to consider the following hypotheses:

(9.41)

K0 'I <

(gjk(x))

1(0!,

IIYjkIIcf'.P(21)

K2,

given 0 < s < 1 and 1 < p <

(not assuming p > n). Again Proposition 9.1

applies. Additional estimates cm a variant of Proposition 1.16.

are provided by the following regularity result,

199

9. HARMONIC COORDINATES

PROPOSITION 9.4. Suppose u e H'2(Q) solves the elliptic PDE

on 0.

(9.42)

Assume

O<s<1, 1
(9.43)

Then

uE

(9.44)

n

in place of PROOF. For simplicity of notation we use etc. We we use an argument similar to To prove u E already know that u E C' write (9.42) as that in our proof of Proposition 1.16. Setting L#u =

L#u =

(9.45)

+ F1(adk,Oku)].

L# E OPBS?, is elliptic and, since ajk E of Chapter I to obtain E E OPSI? such that Here

we can apply Proposition 6.1

EL#=I+F,

(9.46)

Then we have u = _Piaj[Tak.uaJk +

(9.47)

—

Eu.

Now

uE (9.48)

Taku, Rak.u E OPS?, T5kua3c + R(a3k,l3ku) e E

the second implication using the hypothesis that a3C E for all F > we have u E Knowing that u e Thus (9.47) yields Eu e (9.49)

< oo. Hence

Ve>O, q
This in turn implies more regularity on Eu, and iterating this a finite number of

times gives the desired conclusion.

Using this result, we now establish the following:

PROPOSITION 9.5. In the setting of Proposition 9.1, assume also that (9.41) holds. Then the harmonic coordinate system produced there has the property that via satisfies the metric tensor pulled back to (9.50)

K13.

3.

202

TO PDE

We need to show that the right side belongs to

(9.65)

E

it E

aik e

it E

aik

e C(A),

Indeed,

fl e

Oj

E

e

=:=*.

E

C 1? fl C

1(A)

so the proposition is proven. This result leads to an extension of Proposition 9.5, whose formulation we leave to the reader.

10. Riemannian manifolds with bounded Ricci tensor There has been a substantial amount of work in the area of deducing bounds on the metric tensor on a Riemannian manifold given bounds on the Ricci tensor (and other geometrically defined bounds). Since [DeTK], one important tool has been the use of harmonic coordinates. This is effective because, in harmonic coordinates. the metric tensor and the Ricci tensor are related by a quasilinear elliptic PDE, of the form

+Qpm(g,Vg) =Ricem,

(10.1)

where Vg) is a quadratic form in Vg, with coefficients that are smooth functions of the components of g (gik), as long as this is positive definite.

In effect, one is presented with two problems. One problem is to establish the existence of harmonic coordinate systems with sufficient regularity (and size). The second problem is to establish bounds on the metric tensor in such coordinate systems, given various bounds on the Ricci tensor. Regarding the first problem, some analytical conditions for existence of harmonic coordinates have been presented in §9. There is also work on geometrical conditions for existence of such harmonic coordinates, of a deeper nature. We mention particularly [An], [An2], and [AnC]. in which there are assumed a priori lower bounds on either the injectivity radius or on volumes of balls. Surveys of work done on these problems can be found in [G] and in Chapter 10 of [Pet]. These works have had important implications for various compactness and rigidity results. As mentioned above, work on the second problem began in [DeTK]. We state a result that extends that work. The following is established in Corollary 12B.5 in Chapter 13 of [T5]. PROPOsITiON 10.1. Assume there is a harmonic coordinate system ij': B1 satisfies gjk E C(B1)

C M and the metric tensor pulled back to B1, via such that H1(B1). Suppose given K0,K1 e U

(10.2)

Then, given p (10.3)

0 < K0 11

n/2.

q

Ric3k E

(gjk(x))

E (i,oc). and s L'>(B1) fl

IC. 0,

gjk E

fl

10. BIEMANNIAN MANIFOLDS WITH BOUNDED [IICCI TENSOR

wit/i

203

a naturally accompanying estimate (involving the modulus of continuity postu-

lated for fJjk

as well as the other quantities).

A consequence of this

result is that, under the hypotheses of Proposition 10.1,

Ric1k E L°°(B,)

(10.4) hence gjk E

V q < oo.

c for all r < 2. We establish a sharpening of this result.

PROPOSITION 10.2. In

the setting of Proposition 10.1,

(10.5)

assume

E bmo.

Then e bmo.

(10.6)

PROOF. By Proposition 10.1 we have

Vq

E

(10.7)

cc

hence we can deduce from (10.1) that EtijgYk(x)Ukgern

(10.8)

The following result is more than

=

Fern e

bmo.

adequate to finish off the proof of Proposition

10.2. PROPOSITION 10.3. Assume

u E H' solves the elliptic PDE

(10.9)

=

f E bmo,

with (10.10)

E Lip.

Then

V2u E bmo.

(10.11) PROOF.

To begin, we will establish the conclusion (10.11) under the stronger

hypothesis

aJlC E c,

(10.12)

r> 1,

which is already more than adequate for the proof of Proposition 10.2. As in the proof of Proposition 1.1, we can write (10.13)

where, (10.14)

mod for some ó E (0, 1),

OPSI,

3. APPLICATIONS TO PDE

204

1112 into the right side of (10.13) yields the improved regularity Plugging u H'4'62. A simple iteration gets us

Vs<2,q
(10.15)

Then we obtain

Vq
(10.16)

as long as 1 + cr6 < r. This implies (10.17)

11(2+rS)A(I+r—e)q

fl

c fl

qO

where a A b = min(a,b). Meanwhile, since V2E# (10.18)

V2E# : bmo

bmo,

so the conclusion (10.11) is established under the hypothesis (10.12). Under the weaker hypothesis (10.10), we have (10.13) with (10.19)

We still obtain (10.15). As a consequence of Proposition 4.9 of Chapter 1, we have

-(1 —6) <s < 1. q < oc,

(10.20)

while by Proposition 4.10 of Chapter 1 we have (10.21)

:

where, by definition, (10.22)

H3"° =

(1

—

In particular we have, for it satisfying the hypotheses of Proposition 10.3, (10.23)

hence (10.24)

and this gives (10.11). REMARK. We could carry out the proof replacing (10.21) by the weaker result H2_612°° —* Lip, which can be established by more elementary means.

We also note that, once we have the result (10.6) on the metric tensor, we can deduce from Proposition 1.13 that any harmonic function it on a coordinate neighborhood has the property

Vqccoc. —* actually provides a Hence the coordinate atlas system on Al, for all q < oc. We make a further remark on the significance of the conclusion (10.6). It implies that has a modulus of continuity w(h) = h log(1/h). Hence, by

ii. PROPAGATION OF SINGULARITIES

205

Osgood's theorem, the geodesic flow is uniquely defined. (Compare remarks made below (2.25).) However, we cannot say that the exponential map is differentiable, without stronger hypotheses.

We can show that the metric tensor is C2 (and hence the exponential map is C1) provided that in the coordinates (and hence in the components Ric3k have a modulus of continuity that satisfies a Dini condition. In fact, we can establish some slightly sharper results. Let w(h) be a slowly varying modulus of continuity for which for

\

some/3E (0,1). Take A(j)=w(2J).

PRoPoSITIoN 10.4. In the set-up of PrOposition JO.2, assume (10.26)

Assume w(h) satisfies the Dini condition. Then (10.27)

e

a(h)

C

=

dt.

If instead we make the still weaker hypothesis that (10.28)

E

then

e

(10.29)

C C.

gJk PROOF. We are already in the set-up of (10.9), with u = Yttrt and c' for all r < 2. Hence (10.13) holds, with E Under hypothesis (10.26), f e C(A) in (10.9). Meanwhile Proposition 5.7 of Chapter 1 gives

(10.30)

V2E#

C ca holding by Proposition 1.2 of Chapter 1. On the other hand, under hypothesis (10.28), the conclusion (10.29) follows because

(10.31) as

V2E#

—p

shown in Proposition 12.1 of Chapter I.

11. Propagation of singularities In 1981 [Bon] burst on the scene with an analysis of propagation of singularities for solutions to nonlinear PDE, introducing the theory of paradifferential operators as an essential tool. Other approaches to such results, given in {RR] and [BR], had an intermediate step of investigating solutions to linear PDE with rough coefficients. Further results on wave equations with rough coefficients include [Sm]. Here we take another look at propagation of singularities for solutions to linear PDE (or pseudodifferential equations) with rough coefficients. Assume (11.1)

e

3. APPLICATIONS TO PDE

206

has real principal symbol. In case r > 2, the analysis done in §6.1 of [T2] (though aimed primarily at the nonlinear case) essentially applies here, but we intend to take a closer look at the case C"1 and even rougher cases. We begin in a fairly general context. Assume (11.1) holds for some r> 1, and write pb e opcsrr6. p(x,D) p# + pb p# (11.2) Here

is as in (3.33)—(3.35) of Chapter I. We can also write

P#A+iB, A—At, B=Bt,

(11.3)

Let US assume that

p(x,D)u=f,

(11.4)

on a region 11 c

r, with

(11.5)

E

We will assume 8 e (0,1), rb> 1, and

—(1—8)r
(11.6)

is to say Here, F is an open conic subset of TtI1 \ 0, and to say f e is supported in F. Under these whenever E that ço(x,D)f E hypotheses, we have (11.7)

P#u=g,

and (11.8)

Pbu

E

We look for conditions under which we can deduce that (11.9)

E

for some open conic i' c c F, assuming that it conic set C c I'. (The closure is taken in TtQ \ 0.)

e

1(Q) for some open

The key estimates, following [H4], use the following basic commutator identity: (11.10)

Im(CP#u,Cu)=Re({_iC*[C,A]+C*BC+C*{B,C]}u,u),

with A and B as in (11.3) and C e oPSr0 to be specified below. Until further notice, we assume that m = 1, which involves no loss of generality. Thus B is bounded on L2, and we have the following basic commutator inequality: (11.11)

Re ({—iCt [C, A] —

MC*C}u,u)

where (11.12)

Al - IBM +

W = Re Ct [B, C],

+ (Wu,u)i,

ii. pROPAGATION OF SINGULARITIES

207

and to get from (11.10) to (11.11) we have used (Cflu,Cu)( IICuU2/4.

For C we will actually define a family of operators EL =

cjx,e) =

(11.13)

+

(T+T)/2, for an operator

In (11.12) we use the convention that ReT

(x, D),

with

÷e2g(x,e)2y'/2,

where A> 0 will be taken large (fixed) and e > 0 small (tending to 0). We assume (11.14)

d(x,E)

f(x,e) e

E

e

large. We will say more about /2 below, but we do take

all homogeneous for

We assume

0.

suppdcl, d0,

on

and, with Pi denoting the principal symbol of p(x, D), homogeneous in e of degree in = 1, we assume on f\O, where

on F\O,

C is some conic open subset of F. Furthermore, we assume

Hp,fl,

on F.

Later we will give geometrical conditions under which we can produce d, f, g satisfying (11.14) -(11.17); for now we take their existence as an hypothesis. Then, if is given by (11.13), we have

where 4

= d(Eg)'

— 2M),

and

=

(11.19)

Hence,

= 2dE(IIpld4c2M +

—

(11.18)

(eg)

'

E2(eg)'(HJ)1g)dj.

if (11.14)—(11.17) hold, we can pick A large enough that, for all e e (0, 1],

(11.20)

—

(CE{CE,pl} —

with w5(x,e) =

(11.21) and with

elliptic on F and supp E C

EE

0.

the real part by a(x, of ther complete symbol of A, given by (11.3), and we need to record the difference between symbols of products and commutators and the leading expansions of these symbols. A check of symbol expansions shows that, if Now to relate this to (11.11) we need to replace pi(x,

r=1+r0, 0
(11.22)

then (11.23)

bounded (11.24)

—ReiC[CE,AI(X,e) for 0 < e

e

l+6(1—ro)

1, and

—a(x,e) e

p= min(1,rb),

TO PDE

3.

208

so

(11.25)

j{c€,a}(x,e)

Thus,

- {cE.pl}(x,e)l

if we set

= (2E{CE,a}(x,e),

(11.26)

we have

E

(11.27)

as well as

—ReiC[C6, A]

(11.28)

with

natural bounds for 0 CE

D) E

—

< 1.

Furthermore,

—

(11.29)

—

+ E(x,C)2

so€(x,e)2

with p as in (11.24). This puts us in a position to exploit the following: SHARP GARDING INEQUALITY. Let q(x, —Co.

E

C'STh be scalar and satisfy q(x,

Then, for all u E

(11.30)

Re(q(x,D)u,u)

provided 2s s>0, rn<—.

(11.31)

2+s This result is Proposition 2.4.A of jT2]. The proof given there makes use of a symbol smoothing, q = q# + qb, and an application of the Fefferinan-Phong inequality

to q#(x,

We can apply

D).

sharp Gârding inequality when the left side of (11.29), provided

is given by

this

(1L32)

2+ro

and

,

2

2

where To is as in (11.22), and we can replace

since

by

(11.32) implies

-

(11.33)

-To).

When the corresponding consequence of the sharp Carding inequality is combined with (11.11), we obtain (11.34)

—

IIE(x,

—

<

K1

+ ((147ev, v)I.

is given by (11.12), with C = so in light of the behavior of B recorded (11.3) we see that WE is bounded in The inequality (11.33) hence implies an L2 -operator bound on W6, and we have Here in

(11.35)

cc

+

IE(x,

+

K11v1132.

U. PROPAGATION OF SINGULARITIES

We can carry out parallel estimates with J)# replaced

Pt

(11.36)

AS

200

by

=(1

In such a fashion we obtain estimates of the form (11.37)

c68 = A8CEA

with

be

+ HE5(x,

+

We are now ready to establish the following result.

etc.

PRoPOSITION 11.1. Assume with 0 < r0 E given by (11.2,). Suppose we have open conic sets 0, F,

1, and let J3# F, satisfying

(1L38)

Make the hypothesis that, for each j, there exist d = f satisfying (11.14)-(ll. 17), with F and F replaced by and

= fj,

and g =

Pick t E R

g1

and

assume ii E V (12) satisfies

ue

(11.39)

e

Then (11.40)

u E

PROOF. Shrinking 12, we can assume ii E H5(12) for some s E lR. If p> 0 is enough that (1l.32)—(1l.33) hold, and if s + p t, we can apply (11.37) and let s —* 0, to get u E Then we can construct fli(x,e) E supported while P#u in F1. equal to 1 on F2. and then it1 = fli(x,D)u belongs to coincides with P#u microlocally on F2. The proof is completed by a straightforward iteration. small

This gives the following result on p(x, D). PROPOSmON

Let

11.2.

satisfy the hypotheses of Proposition

e

11.1. (So 1
(11.41) with

(11.42) Assume

it

Se

E

f E

(0,1), rS> 1, and —(1 — S)r

(11.43)


Then (11.44)

it

E

it E

PROOF. By (11.7>—(11.8), we see that it satisfies

(11.38), with t =

a.

In case r 2, one can take p(x,e) E Furthermore, by Proposition 4.9 of Chapter I, we can allow the endpoint case of (11.6), a = r = 2, and still have given that it satisfies (11.5). This leads to the following: C

3. APPLICATIONS TO PDE

PROPOSITION 1L3. Let p(x,e) E

satisfy

f with

1. Assume p(x. D)u

11.

(1L45)

fc

u

Assume

the hypotheses of Proposition

8 E (1/2, 1) and

(11A6)

—2(1 —8)


2.

Then the implication (11.44) holds. in case

p(x,

D) has order

in,

by examining p(x, D)A1m

(11.6) hold and if pi(x,e)

we see that if (11.4)-

satisfies the hypotheses of Proposition

11.1, then, for a satisfying (11.43) we have

(11.47)

u E

u C HV—'(fl,

Let us also say something about

PDE

equations in divergence form. We consider

of order two:

p(x,D)u = OjAik(x)Oku.

(11.48)

In such a case, it is natural to replace (11.2)

(11.49)

p(x, D) =

by

=

UjA7 +

+

yb,

with (11.50)

AT

given A3k (11.51)

Hence,

c

C C. We see that, if —(1

u

if p(x,D)u

f

(11.52)

— S)r < a <

rS

r, then

C

e

with

ue

f

rS(Q),

we have

P#u

1(11.53)

C

From here, an argument parallel to that given above yields the following. PROPOSITION

11.4. Let p(x,D) be given by (17.48) with Aik f with u and f satisfying (11.52j Assume 0

IAssume p(x, D)u =

and

—(1

—

S)r < a < r.

C C, 1 < 8

<

1,



1,

FItrtherrriore, assume that

satisfies the hypotheses of Proposition 11.1. Then

a C

u C

Ask C C", this is valid for —2(1—8)


S 2.

The conclusions of Propositions 11.2—11.4 are primarily interesting when I' hugs characteristic set of p(x, D). In fact, if p(x, D) is elliptic on F then, in the setting f (11.1)-(11.6). we deduce from (11.7)—(11.8) that

russ)

uE

11. PROPAGATION OF SINGULARITIES

211

173 given in Proposition 11.1. without any need for hypotheses such as those on Of course this is stronger then (11.47). We now discuss conditions under which one can produce the symbols d, f, and p satisfying (11.14)—(11.17). If pi(x,e) were smooth, one could simply solve some = 1, but for Pi (x, E this would not have a smooth equations, such as However, we need only satisfy some differential inequalities in solution f(z, (11.16) (11.17), and we can exploit this flexibility. We work under the standing hypothesis that is nowhere radial on the characteristic set char p; in particular it is nowhere vanishing on char p. In cases where one has a uniqueness result for the orbits of HP! (e.g., when this vector field has Lipschitz coefficients) and one expects to prove that smoothness the constructions of d, and g can be localized. propagates along orbits of in fact, in the setting of (11.1)—(11.6), suppose 7 is a null bicharacteristic strip of for a conic neighborhood 0 of p(x,D), We want E 7, and it E microlocally on a conic neighborhoood of to show that it e Let I be and on which it is microlocally in an open interval in 7 containing It suffices to show that it is microlocally in H-' on a conic neighborhood of the endpoints of I. hence it sufFices to make constructions of d, f, and p on small conic has non-unique orbits, this localization neighborhoods of these endpoints. (if can fail, to be sure.) Let us hence take E (an endpoint of I), let V be a small conic neighborhood of (i, and consider constructions of d, f, and p on V. Here F will be a small conic neighborhood of intersected with V, F will be a small conic neighborhood of I, intersected with V. and 0 c V will be a small conic neighborhood of I. We assume maps Ci into I in small positive time. (If not, simply the flow generated by change (11.4) to —p(x,D)u= —f.) To take care of our homogeneity requirements on d, f, and p, it is convenient to take a "section" W of V, a smooth (2n — 1)-dimensional manifold containing Ci We write and transverse to the radial vector field R -= (11.56)

HP!

=

+Ø(x,e)R,

is tangent to W and its images under the flow generated by R, and Ø(x, where Note that our hypothesis that is real valued (and homogeneous of degree zero in

is nowhere vanishing. Different choices of is nowhere radial implies that sections yield different decompositions in (11.56), and it is convenient to note that we can construct a section W such that

(11.57)

fl(x,e) ? 0.

In these circumstances, our task of constructing d(x,

reduces to the following.

We require (11.58)

suppdcWfll', d0,

and we want dlpv to satisfy the differential inequalities (1159)

on Wn(F\0), on

212

3. APPLICATIONS TO PDE

Given that (11.57) has been arranged (and since p > 0),

on Wfl(1'\O),

(liMO)

it suffices

to have

on WflF\O.

satisfying (11.58) and (11.60), then the unique extension to V of degree p gives the desired symbol d(x, Let us assume that has unique orbits and generates a flow F consisting of a family of local homeomorphisms. We will assume there are smooth (2n — 2)transverse to dimensional surfaces E C and numbers and containing —a C —b < 0 Cc such that Once we have homogeneous

WflF= U F1E, WnF= U FL —a
(11.61)

—b
We also asume FE C W fl (9. Let us also take E#, with smooth boundary, such

that E CC E# CC

pick —a' E (—a, —b), and set

F#=

(11.62)

U

—a'
The following gives a relaxed set of conditions to verify in order to construct the symbol d.

LEMMA 11.5. Suppose there is a neighborhood U cc f# of wnr h

e

such

(1163)

and

on ?VnF, on hC2>0 on WnF, hC0 on F#\U. satisfying

Then there exists d

PROOF.

in f#

that

Take cC

Let d(() = a(h(()) for (

E

(11.58) and (11.60).

suchthat cr(t)=0 fort <0 and a'(t) >0 fort >0. and d(() = 0 for ( E W \ F#, and then alter d as

needed in W n (9 to ensure that d C We now show how Lemma 11.5 can be implemented when is a C1 vector field, which occurs when p(x,C) e Pick h0 & C9E) such that supp h0

and h0 > 0 on >#. Pick a

such that a(t)

0 for t C —a', a'(t) > 0 for

t> —a', and define h1 on W fl F by (11.64)

hi(Ft()=a(t)ho(cfl,

(eE.

We have h1 e C'(W n F), supported in r#. Inside F#, h1 > 0 and > 0. taking K = infp h1 > 0, set h2 = h1 — K/2. Approximating h2 in the CLtopology by elements of C'°(f'#) (and perhaps shrinking f# a bit), we obtain h C C00(F#) satisfying the conditions of Lemma 11.5. Thus we obtain d(x, C) with the desired properties, when (11.1) holds with r = 2. The constructions of f(x,C) and satisfying (l1.14)—(ll.17) are relatively simpler. Thus we can apply Propositions 11.2—11.4 to obtain propagation of singularities results along null bicharacteristics, for operators with C2 coefficients. Suppose now that is a Lipschitz vector field. Then the function h1 given by (11.64) belongs to Lip(I'VflF). and so does h2. Hence smoothings hE =

11. PROPAGATION OF SINGULARITIES

213

cannot be expected to converge in the C'-topolog-y. Nevertheless we can show that Xp1 converges locally imiformly to h2, as follows. Taking a smooth coordinate system on W in which (11.65)

X7,1

we have (11.66) Note

=

+ [Man

that = a'(t)ho((),

(11.67) so

under our current hypotheses X',, h2

E

Lip(W fl F). Thus

—*

(11.68)

( E E,

locally

uniformly,

E_mcs(x/s) (with in = dimW = mollifler. If support, an elementary estimate gives compact

is a Friedrichs

if

2n —

and

1)

has

(11.69) and

CE

,we(D)IuIILcc

Iki[

IIuIItc,

together with (11.68) this gives

(11.70)

—*

Xi,,

locally uniformly.

Thus the results on propagation of singularities along null bicharacteristics mentioned above work for operators with C11 coefficients as well as for operators with C2 coefficients. The scope of these last estimates can be extended, as follows. Suppose you have h2, as above, but with (11.71)

0cs
li2

We are hence dropping the hypothesis that a3(x) in (11.65) are

Lipschitz.

Let us

instead assume

&

(11.72)

Looking

1—sc s'

E

at (11.66), we see that (11.68) still holds. We need to estimate

of the commutators [Mai,

Cr'(WnF).

More generally, if a E Ce', P e OPS?0, and using the paraproduct expansions:

aPu = TaPu +

74" Now,

E Cr',

P(au)=PTau+PTua+PR(a.u). a E Cs'

(11.75)

(11.76)

u

+ R(a, Pu),

by (3.53) of Chapter I, for s' > 0,

Hence,

the action

on

(11.73)

by

ci.

as long as s —

1

+ s' >

Ra 6

0, we have

R(a,Pu), PR(a,u) 6

we examine [Ma, P]u

3. APPLICATioNS TO PDE

214

Furthermore,

given s

— 1

<0,

(11.77)

u E

e

so we have (11.78) Next,

E

by Proposition 7.3 of Chapter 1, for 0 < # < 1,

(11.79) a smoothing operator, so

[Ta,PIu

(11.80)

and hence, putting together the various ingredients in (11.74), we have (11.81)

[Ma,

H-s'

P]u e

We deduce that, under the hypotheses (11.71)—(11.72), for 0 <s S 1 we have

bounded in Cf.7(W nIT').

(11.82)

Hence this family of fimctions in precornpact in

fl IT). Since it

a3(x)ço6(D)(31h2) and

is clear that

—*

(11.83)

in for

each ö > 0, we deduce that [M03

0,

,

locally uniformly on 14) fl F,

so again (11.70) holds, under the more general hypotheses (11.71)—(11.72). Of course, the hypothesis (11.72) on the coefficients of the vector field X,1 is not

strong enough to imply that generates a uniquely defined flow F. Osgood's theorem (mentioned already in §2 and again in §10) guarantees that such a unique flow exists provided the coefficients have a modulus of continuity of the following sort: (11.84)

In

f

&ECW(Wflr),

such a case, if I&(xi) —

(11.85)

— x21) for all

5 —

w(s)

S

14' flY, we have

— x21,t),

where z9(a, t) is defined by

çt9(at) I ./a

—kt. c/s

w(s)

In particular this applies to the "log-Lipschitz" modulus of continuity given in (2.25):

11. PROPAGATiON OF SiNGULARiTiES

In

such a case, we have i9(a,t) =

If

215

akt).

(11.84) holds with w(s) given by (11.87), we say

e

iPxi

E LL. We have

lxi

We henceforth assume that is log-Lipschitz. Thus the flow F is Holder continuous for each 1, though the HOlder exponent decays exponentially. However, as we have noted, the task of establishing propagation of singularities along null bicharacteristics involves constructing the symbols d, f, and g only in a small conic neighborhood of a given point so in this construction we can keep t small and hence keep the HOlder exponent (call it s) as close to 1 as we like. It is then straightforward to show that () = F( produces a C8-homeomorphism of (—a, c) x onto a neighborhood of in ¾) (which we denote ¾) fl I', as in (11.61)), and furthermore c1 VVnI' —' (—a;c)xE is HOlder continuous of class C8. Hence (11.64) — K/2 satisfies (11.71). Of course, defines a function h1 e C8(Wn11), and h2 = if E LL. then (11.72) holds for all s' < 1. Hence we can apply Propositions 11.2—i 1.4 to obtain propagation of singularities results along null hicharacteristics. :

for operators with coefficients having one derivative in LL. As noted in (2.25), in particular it this happens if the coefficients belong to the Zygmund space happens if the coefficients have two derivatives in hmo. Proposition 11.4 can be applied to obtain propagation of singuFor larities results along null bicharacteristics for solutions to the wave equation (11.90)

—

L\u

=

0,

where is the Laplace-Beltrami operator on a Riemannian manifold with bounded Ricci tensor, as we see by writing (11.90) in local coordinates as (11.91)

O,g(x)'123,u

—

=

0,

and use Proposition 10.2 to see that the coefficients have one derivative in LL.

CHAPTER 4

Layer Potentials on Lipschitz Surfaces Introduction In this chapter we discuss results on layer potentials on Lipschitz surfaces and applications to the Dirichlet problem on Lipschitz domains. When a surface lacks moderate regularity beyond the class C', it becomes difficult to establish the basic operator norm estimates on single and double layer potentials. The first breakthrough on this was initiated by A. P. Calderón [Ca2], and completed by R. Coifman, A. McIntosh, and Y. Meyer [CMM], estimating the Cauchy kernel on Lipschitz curves. Estimates were also established for an appropriate class of potetitials on higher-dimensional Lipschitz surfaces in [CMM] and [CDM]. In we treat these estimates, in one and higher dimensions, respectively. Our treatment of the basic estimate of the Cauchy integral on Lipschitz curves follows a proof given in [CJS]. Other proofs have been produced; we mention particularly [GM] and [MeV]. These estimates on layer potentials allow one to apply Fredholm theory to the study of regular elliptic boundary problems in C' domains. This was carried out in [FJR]. However, for Lipschitz domains that are not C' one can lose such properties as compactness of double-layer potentials, and further effort is required. This was

accomplished in [ye]. for the Dirichlet and Neumann problems. Among other things, an identity of Rellich was brought to bear, to establish unique solvability of appropriate boundary integral equations. A number of other boundary problems on Lipschitz domains have subsequently been treated via layer potential techniques; we mention the works [DKVJ, [FKV], [EFV]. and [MMP]. All these works confine their attention to constant-coefficient Along with this restriction comes a topological equations on Lipschitz regions in restriction on the domain; only domains with connected boundary are treated. It was not expected that such a restriction should be necessary for the basic results to hold. Recently, [MiD] developed a technique to treat the Dirichlet problem for whose boundaries were not required to be the Laplace operator on domains in connected. In [MT], tools were developed to apply the method of layer potentials to equations with variable coefficients on Lipschitz domains. There the authors studied operators of the form L = — V where is the Laplace operator on a compact (M). The metric tensor was assumed to be Riemannian manifold M and V E

of class C' (an assumption that was relaxed to Lipschitz in [MT2] and relaxed further in [MT4]). The authors treated the Dirichlet and Neumann problems, and oblique derivative problems on Lipschitz domains in M. In [MMT] the scope of this work was extended to other boundary problems, including natural boundary

218

4. LAYER POTENTIALS ON LIPSCHITZ S[JRFACES

problems for the liodge Laplacian on Lipschitz domains in Riemannian manifolds. It is worth mentioning that, once one moves to the variable-coefficient setting, the need for topological restrictions evaporates; one can treat arbitrary compact Lipschitz domains in a smooth manifold. In

we present some of the material developed in [MT}, but here it is

specialized to the case of smooth metric tensors, for simplicity of exposition. Section 3 extends the layer potential estimates of §2 from potentials of "convolution type" to "variable-coefficient" generalizations. Section 4 investigates solvability of boundary integral equations arising in the layer potential approach to the Dirichlet problem.

Section 5 then appies these results to the Dirichlet problem. While we restrict attention to the case of smooth coefficients, we mention that some of the techniques used in this monograph, particularly in of Chapter III, were brought to bear in the more general cases treated in [MT], [MMT], and [MT2]—[MT4]. The key estimate on Cauchy integrals on Lipschitz curves in §1 makes use of the Koebe-Bieberbach distortion theorem. As this is outside the circle of results we have described as prerequisites, we present a proof of it in Appendix A, at the end of this chapter. Our treatment draws from those in [Porn] and [Mi!]. Taking a cue from [Mi!], we present an endgame to the proof that is somewhat more geometrical, and less computational, than usual.

1. Cauchy kerne!s on Lipschitz curves Let A: JR —+ JR be a Lipschitz function, with Lipschitz constant L, and consider the Lipschitz graph,

(1.1)

F={t+iA(t):teiR},

Denote by IL the region in C above F and by IL the region below F. We have the Cauchy integral (1.2)

ZEIL.

The main result of this section is the following result of [CMM]. following work of

[Ca2].

THEOREM 1.1. The limits (1.3)

= lim Cuf(z), +y\O

z

F,

exist and define operators

14 : L2(J')

L2(F),

sat £sfying

(Yo(1 + L)2,

for some absolute constant

it is technically convenient to treat first the case when A and f obtaining the estimate (1.5) purely in terms of the Lipschitz constant of

1. CAUCHY KERNELS ON UPSCHI'ry. CURVES

A. We discuss later iii this section how to pass from this case to the case of general Lipschitz A. The exponent 2 in (1.5) is better than obtained in [CMM]; the optimal expo-

nent 3/2 can be found in [Mur]. The proof we give here is taken from [CJS]. It exploits the behavior of

&f(z) =

(1.6)

on

Let

/

d(

The key analysis is contained in the three lemmas. denote the space of functions on !L satisfying

=f

(1.7)

where d(z) = dist(z. F). Let (

< oo,

denote

the inner product in

Define fl

similarly, using Il

LEMMA 1.2. Suppose F is holomorphic in (1.8)

+

IFIIL2(l') <

for some absolute constant

and vanishes at infinity. Then

There is an analogous estimate for F holomorphic

on

For the first step in the proof of Lemma 1.2, we let (1.9)

—÷

4?:

be a conformal mapping so that 4?(IR) = F and 4?(cx) = At this point, we invoke the Koebe-Bieberbach distortion theorem, which is the following: TIIE0REIvI. There are absolute constants (1. (0, oo) such that the following —p C is holomorphic and one-to-one, 'with image 11. then holds. If 4? :

(1.10)

In fact, the sharp values a = 1/2 and j3 2 are known. We give a proof of this result in Appendix A. at the end of this chapter. Using this distortion theorem, we see that Lemma 1.2 is equivalent to the estimate <

for holomorphic functions C decaying at for establishing this.

cf The following lemma will be useful

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

220

LEMMA 1.3. Let H and D be holornorphic on vanishes at infinity. Then IH(x)I2dx =

(1.12)

Assume ID(z)I

1

and H

4f

and

f

(1.13)

use

jH(x)j2dx.

PROOF. The identity (1.12) follows from Green's theorem. To obtain (1.13), to write

(1.12)

I:

H(x)D(x)I2dx =

H(x)l2dx

The estimate then follows from

the triangle inequality, (1.12), and the inequality

(H'j.

We turn to the proof of (1.11). Let A

denote

the left side of (1.11) and let B

denote the integral on the right side. Note that, since r — e

where F

= arccot(L) >

is Lipschitz, arg

0. Hence

dx c

I: so by Green's theorem

Ac =

+

CB +

Now set 1' = Hence (1.16)

e1",

so slY'

cf

= V'e" =

We can arrange that

=

<

<

1. CAUCIIY KERNELS ON LIPSCHITZ CURVES

where D =

221

Using this plus Cauchy's inequality, we have

f CC'W'fy dx

(f

(1.17)

R2

flu/29/2(f <

where the last inequality follows from (1.13) since Note that the factor C in (1.14)—(1.15) can be bounded by C1(1 + L). At this point we have an estimate of the form (1.18)

A

C(B +

+ CE +

which in turn implies A < CB (with a different C). The proof of Lemma 1.2 is complete.

LEMMA 1.4. Let f E 14 and define (1.19)

(El?.

Then (1.20)

for

5; C2(1

+

some absolute constant C2. PROOF. By Lemma 1.2,

(1.21) since

HTfIIL2(r) 5;

CII(Tf)'lIn_,

Tf is holomorphic in CL. Here C = C1(1 + L). Now I(Tf)'(w)l =

2f

(1.22) 5;

Hence

(1.23)

2f

dxdy.

Iz-wI

we will have (1.20) from (1.21) if we can show that

8:

—p

222

4. LAYER POTENTIALS ON LIPSCRITZ SURFACES

where S is given by

(1.24)

=

with k(z, w) = (1.25)

dCzi"2 dxdy

f

SF(w)

j

k(z,w)F(z)dxdy.

U

w E ft ,

—

we have

/ k(z, w) dA(w)

/ k(z, in) dx dy

L2 (IL), with

since d(z) Jz — wj. hence S is a bounded operator from norm 4ir, and Lemma 1.4 is proven.

We now prove Theorem 1.1 (under the restrictions mentioned after the statement of that result). Let f E L2(fl, and let g E fl+, and assume both are smooth and have compact support. Then

=

-

f(f

d()g(z)d(z)dxdy

r

(1.26)

= where T is given by (1.19). hence (1.27)

I

by Lemma 1.4. Thus

< CII!

(1.28)

On the other hand, Lemma 1.2 implies <

(1.29)

The estimate on KEf follows similarly. so we have the desired estimate on We can rewrite the Cauchy integral (1.2) as (1.30)

Krf(z)nf

I

iA(I)

with ((0 = + iA(I). If z = s + iA(s) + ic, and if we write g(I) = fK(t))('(I), then an essentially equivalent operator is (1.31)

/: s-t+i(A(s) -

A(i)) +ia

By Theorem 1.1, we have limiting operators as +a \ 0: (1.32)

L2(R) —s L2(R).

dl.

1. CAUCHY KERNELS ON LIPSCHLTZ CURVES

The operators the kernel (1.33)

are hence Calderón-Zygnumd operators, associated to

and kA(s

223

=

s-t+i(A(s) -A(t))

=

((s)-C(t)

which satisfies "standard estimates" (1.34)

S

V8.lkA(s,1)1

theory hence implies the houndedness properties

The general (1.35)

S C(p,L),

:

1

oo.

Also, these operators satisfy weak-type (1, 1) estimates. At this point we pass from the case A E to general Lipschitz A, with Lipschitz constant
Lipschitz constant
We want to estimate maximal functions associated to 1—1, O)u(O, 1],

Let

us set

(1.36)

= ((s)

Then

=

nY(s. 1) + (1.37)

- C(t)) a

a

- Na(s,1) =

C(1))

where

(1.38) Note

that, since A

W(z)

has ((s)

(1.39)

Note also that (1.40)

=

=

Lipschitz constant L,

_ ((0

as a ranges over

with a principal-value singular integral operator.

and to compare

E

r {z

E

we have

C: lmzl
224

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

so

C

s—t

U\ 0,

(1.41)

2

\1

+1)

it follows that

sup

(1.42)

CL Mg(s),

0<0<1

Mg is the Hardy-Littlewood maximal function associated with g. We next compare + to 2kA, and particularly to 2kg, where we set

where

k°(s,t)=kA(s,t)

'.

'143

if

a,

IC(s)

if

0

We have

C(s,t)

(1.44)

with

2

- 2kA(s,t) =

mccQ),

1(z) given by (1.38). Hence

+

(1.45)

=

—

with if

zz2+1

(L46)

—

if

z2 + 1 Thus,

if

—

denotes the integral operator with kernel V, i.e.,

f

(1.47)

{j((s)—((t) Io} we

have by arguments as in (1.39)—(1.42) that

(1.48)

sup

+

0
A kernel closely related to

is ka, given by

/f(s, t) =

p1.49)

CL Mg(s).

—

kA(s, t)

if

—

¼

if

0

Is

:1.50)

—

that

C(s)-C(t)

1.51)

denotes the operator with kernel

f

s-ti

otherwise.

0

L'hus, if

if

CO

i.e.,

C(s)-C(t)I,

1. CAUCHY KERNELS ON LIPSCHITZ CURVES

225

we have

sup

(1.52)

CL Mg(s).

—

O
is a Calderón-Zygmund operator with kernel as in (1.33), and Given that as in (1.51), it is a general that kC is defined by (1.49), yielding the operator result in Calderón-Zygmund theory that (1.53)

Mg(s).

sup

0<0<1

the

Cf. [Jo], pp. 56—57, or [S3J, pp. 34—35. Also, given g E (1.54)

=

Urn

Em

f

+

quantity

((s-t)-((s)] dt

clearly exists at each point s where A is differentiable. In view of the maximal estimates established above, it follows that there is a bounded operator

K,4:L"(R)—'L'2(R),

(1.55) such

VpE(1,oc),

that, for each g E

—>

(1.56) in V'-norm, as

a

0. One writes

KAg(s) = P.V.

(1.57)

(()((t)

From (1.50) we see that, if g E Cr(R), then,

a —' 0, —> 0

—

(1.58)

as

dt.

for each point s where A is differentiable. Hence the maximal function estimates established above imply that, for each g E L73(IR), p (1, oc), —*

(1.59) in

and pointwise a.e., as a 0. that (1.42), (1.48), (1.52), and (1.53) imply that Note

(1.60)

+CL Mg(s),

sup 0
on Also, we could replace and a similar estimate holds on the right side of (1.60) by M(KAg)(s). We are now ready for conclusions about various operators considered above. PROPOSITiON 1.5. Given g (1.61)

1

oc, we

=2KAg,

in L"-norrn. The convergence also holds pointwise a.e.

have

226

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

PROOF. In view of the estimates established above, it suffices to show that, if then

gE

(1.62)

bin

+

=

—

0,

for every point s where A is differentiable. This can be deduced from the formulas (1.45)—(1.46) and the following lemma.

for

LEMMA 1.6. Let g E a simple jump across {z: zj

Assume

0 is

continuous on C

(except

perhaps

1}), and

(1.63)

< Consider

(1.64)

Tag(s) =

Fix s e 1k. Assume

that A (hence () is differentiable at s, and that

$(('(s)t) dt

(1.65)

= b(s).

Then

lim Tag(s) = b(s)g(s).

(1.66)

a—?O

The proof is an exercise. (Clearly we have not picked a maximal set of functions for which the lemma holds.) This lemma applies to Proposition 1.5 with b(s) = 0. since Q(z) in (1.46) is odd in z (and can be altered off 0,, to satisfy all required conditions). Note furthermore that, if 1(z) is given by (1.38), then (1.67) ('(s)

as long

as

('(s) =

1

+

< L. This yields the following.

A'(s),

PRoposiTioN 1.7. Give

1

g e a

lim (YCA

c,—*0

in

and

—

)g =

have

27ri 4,

also pointwise a.e. Consequently lim

ra\O

(

and pointwise a.e.

While we have taken

ae

(0,

in the analysis of 'Cr,

tpplies if we allow a to vary over a cone 1.70)

a

similar argument

Co={aeC:0
)rovided S is sufficiently small that C, does not overlap with (9,,. 1)etails are an xercise.

1. CAUCHY KERNELS ON LLPSCHITZ CURVES

227

Reversing the passage from (1.30) to (1.31), we can phrase the last results in terms of the Cauchy integral (1.2). Given f E L2(F), z E F, set

(K1'f)t(z) r

(1.71)

sup {hCi'f(z a EC6

+ ia)I + lsCrf(z —

PROPOSiTION 1.8. ff1

(1.72)

Also,

we have

E

as

a —* 0 in C6,

Kr 1(z) + 7rif(z). — hi) —* Kr f(z) — nif(z). + ia)

)Crf(z

poiniwise

a.e. and in U-norm, where

Ki'f(z)=P.V.

(1.74)

,j

z—(

We next establish a couple of results from [CDM] that will play an important role in the extension to several dimensions in the next section. LEMMA 1.9.

Suppose A R —* R is

P. V. eA is the kernel of an operator EA

(1.76)

MEAl

Lipsehitz constant L. Set

— A(t) s—t

1

eA(s,t)=— s—t

(1.75) Then

Lipsehitz, with

on

L2(R), satisfying

<0(1+

PROOF. Let 9 be the region in C consisting of points of distance 1 from the interval [—2L. 2L] in the real axis, and denote its boundary by Now, for C E

(1.77)

Ac(t)

= (t — A(t),

kc(s,t)

= Ac(s)

—

Then Cauchy's formula gives

(1.78)

eA(s,t) =

To prove the lemma, it suffices to show that the operator 1
<0(1 +

V(

2L. we have two cases to consider. First, suppose —2L < Writing ( Then = +1 and Ac(t) = +i(e+iBc(t)), where Bc(t) = A(t)—et. Hence. up to a factor +i, t) is in this case precisely the Cauchy kernel (1.33) associated to the

Lipschitz graph of Bc, whose Lipschitz constant is < 3L, so the desired estimate Theorem 1.1.

follows from

on

Next we assume

Ac(t) = ((t

2L (the case

+ Cc(t)) with

IQ(t)

—2L being similar). Then we can write a.e. Again by Theorem 1.1, the kernel

1/2,

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

defines a bounded operator on L2(R), with uniformly (s — t — cc(s) — bounded norm. Hence IIKCIIL(L2) C/L for such (. PROPOSITION 1.10. Assume w: R —* N > n + 3, and consider the kernel

r(s,t)

(1.80) Then

P?t

is Lipschitz. Let F e

with

=

P. V.r(s,t) is the kernel of a bounded operator on L2(R).

has Lipschitz constant L, then the argument of F is contained in the ball {z E r_k: Izi
so

p

that 4)(z + 2irLsc) =

4)(z)

then we can expand 4) in a Fourier

for all ,c C

series 4)(z)

=

Hence

(1.82)

r(s,t)

with

gk(5,t) =

(1.83)

Now Lemma 1.9 implies that P.V. 9k(5, t)

the kernel of an operator Gk satisfying

< C(1

(1.84)

The

is

+ lCj)3.

estimate lahl

yields

2.


the result.

The method of rotations and extensions to higher dimensions

One can pass from the one-dimensional result of §1 to a useful multi-dimensional

result by the method of rotations. Our treatment of the next proposition follows [CDM] and [Day]. PROPOSITION 2.1. Let k e

\ 0)

be odd and homogeneous of degree —k.

Assume N > n — k + 3. Let I' be a k-dimensional Lipschitz graph, of the form

I'= where p

Rn—k is Lipschitz. Set

Kf(x)

- P.

xE = (x, ço(x)). Then -

dy

2. THE METHOD OF ROTATIONS AND EXTENSIONS TO HIGHER DIMENSIONS

229

is a well defined operator satisfying K : L2(Rk)

(2.3)

PROOF. Write (2.4)

Kf(x)=ck f

Sk1

where, for w E (2.5)

- Ø(x+ sw))f(x+

=

We estimate the operator norm of {x e Jftk : x w = 0}, and note that

ds.

on L2(Ilfl). To do this, let

=

(w)-'

=

=f

(2.6)

is the singular integral operator (acting on func-

where fe(t) = f(C + tui) and tions on IR) with kernel (2.7)

Thus our task is to estimate the operator norm of

on L2(R). Note that

(28)

s—t

s—t where

e

is given by

(2.9)

Now the function t F-+ c°(e + tw) is Lipschitz, uniformly in e and w. Hence the desired estimate on IITW.eIIr(L2) follows from Proposition 1.10, and the proof of Proposition 2.1 is complete.

As in §1, we see that the operator K in (2.2)—(2.3) is a Calderón-Zygmund —. LP(Rk): operator, and we have an estimate on K :

(2.10)

The operator operator:

1

IIKIIr(LP)

K

in


(2.2) is closely related to the principal-value singular integral

Krf(x)

= P.V.

(2.11)

=lim

J F

k(x - y)f(y) da(y)

f

{yEFIz—yI>e}

k(x-y)f(y)du(y),

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

230

where da is the area element of F, induced from the Euclidean structure of W1. This is also related to the following operator, defined for x E r \ F: (2.12)

= in a fashion parallel to analogues

/

k(x

- y)f(y) da(y).

in §1.

We now restrict attention to Lipschitz graphs of dimension n — the case Ic = n — 1 of Proposition 2.1.

1

in lftP, i.e., to

As in §1 we have estimates on nontangential maximal functions. If F is a Lipschitz graph, with Lipschitz constant L, and if i9> 1 is chosen, then for each 1}. For = C + x, where C = {x E RTh : x E F, consider the cone we define the nontangential maximal function: a function u defined on u*(x) = sup

(2.13)

x E F.

yEC1

From the analysis above and in §1 it 1

(2.14)

follows that,

with

I(CrfY((Lrcr S

C of the form (2.10). Extending jump relations given in (1.73), one can show that Krf has nontangential boundary values a.e. on F, which are related to Krf by:

(Crf)÷(x)

(2.15)

+ Krf(x).

Here, (Cçf) (x) is the limit from above F and (Krf)_(x) is the limit from below F (within the cone Cr), n(x) is the unit (downward-pointing) conormual to F at x (defined a.e. on F), and P_1 is the principal symbol (homogeneous of degree —1 in of the operator Pu(x) = k(x — y)u(y) dy. 3.

The variable-coefficient case

Our goal here is to extend results of §2 to the variable-coefficient case. As in

§2, let F be a Lipschitz graph in r, of the form x,T = W(xi,..

. ,

PRoposiTioN 3.1. There exists Al = M(n) such that the following holds. Let and homogeneous of deqree —(ii — 1) in z, and assume D?b(x, z) is continuous and bounded on x for al < M. Then b(x,x — y) is the kernel of an operator B, bounded on for 1

PROOF. The classical method of spherical harmonic decomposition due to Oalderón and Zygmund works in this case. Thus, we can write b(x.z)

lzV

(n1)

J> I

where )f the

: j 1} is an orthonormal basis of L2(S'hi I) consisting of eigenfunctions Furthermore, we can assume that

Laplace operator on the sphere 8rt

1

3. THE VARIABLE-COEFFICIENT CASE

231

is odd whenever 0. With N as in Proposition 2.1 and M than N, the regularity hypothesis implies

sufficiently

larger


(3.2) (n

Note that, if

= [1(F) with kernel k3(x

—

1) with

odd, then the operator

on

y) is estimable by (2.10), and, for f e [1(F). Bf(x)

(3.3)

=

Ebj(x)Kjf(x).

hence, (3.4)

S C(p,F) sup

1).

and the proof is done.

Proposition 3.1 applies to the Schwartz kernels of certain pseudodifferential have Schwartz kernels that differ from operators. In fact, operators in those treated in Proposition 3.1 by kernels with weak singularities, and with a different asymptotic behavior far from the diagonal. For our purposes it is sufficient to use the elementary consequence that the conclusions of these propositions hold,

provided one acts on functions with support on a given compact subset F0 of F, and estimates the norm of the resulting function over F0. In the rest of this section we will restrict attention to this case. We now state the consequence of Proposition 3.1 most directly relevant for the analysis in §4.

PRoposiTioN 3.2.

e C°Sj'

If

has a principal symbol that is odd in

then

the Schwartz kernel of p(x, D) is the kernel of an operator bounded on [1(F0),

for 1

The

operator B in Proposition 3.1

(3.5)

This

(3.6)

Bf(x)

is related to the

=

following

Bf(x)

is given by

fly.

- y)f(y)

da(y).

operator, defined for x e

IRT \

F:

- y)f(y)da(y).

=

There is an estimate on the nontangential maximal function for 13f. Under the hypotheses of Proposition 3.1, if 13 is as in (3.6) then, by (3.3), we have (3.7)

(j3f)*(7)

>

J?1

Thus, using estimates of the form (3.2) and (2.14), we have:

4. LAYER POTENTJALS ON LTPSCHJTZ SURFACES

232

then

PROPOSITiON 3.3. If E C°SJ1 has a principal symbol that is odd in its Schwartz kernel is the kernel of an operator B, satisfying

(3.8)

S

forlcpcoo Given

(andf supported on F0).

(2.15), the superposition arguments used above yield:

PROPOSITION 3.4. If p(x, P—i (a,

is

as in Proposition 3.3, with principal symbol

then, a.e. on F, we have nontangential limits

(Bf)±(x)

(3.9)

To consider an example for which the

results above apply, suppose

is given

a smooth Riemannian metric and Li is its Laplace-Beltrami operator, and E is a parametrix for L = Li — V, where V is smooth and 0. Assume for simplicity that the metric tensor is asymptotically Euclidean. Then Proposition and

3.2 applies to

so

the associated double layer potentials

are bounded

on 11(r0). It is standard to write the action of E = E(x, D) as (3.10)

where

dy

so g(y) = (3.11)

is

the volume element associated with the Riemannian metric Yjk' The single layer potential associated with this is then

Sf(x) =

dcr9(y)

/

E(x,y)f(y)p(y)da(y),

= dcr9(y) = p(y)dcr(y) is the area element on F inherited from the Riemannian metric gjk, which differs from that inherited from the Euclidean metric Sik by a factor p E 110(F), a formula for which we give below. In such a case, where

(3.12)

=

13, of the form treated in Proposition 3.3, corresponding to p5(x, principal symbol for

=

(3.13) where

is

G(x,e) =

the matrix inverse of (gjk). By

where (3.15)

Note

Iqf(x)

that

(3.16)

(3.9),

we have, for a.e. x

+ K,tf(x),

=

(3.14)

SJ1 with

=

F,

3. THE VAJUABLE-COEFFICJENT CASE

Now, the unit conormal to r with respect to the metric Yjk (3.17)

—_G(x,n(x))

—

233

is

given by

1/2

the unit outward normal to r with respect to this metric is given by

and

(3.18)

v3(x)

Thus we have (3.19)

+ zi3(x)K3tf(x),

with

gJk(x)nk(x)nj(x)c(x,n(x))_J/2

A(x) = p(x)

-1/2

=1,

last identity being a standard formula for the area element of a hypersurface in a Riemannian manifold (which works as well for Lipschitz hypersurfaces as for C' hypersurfaces). Hence we recover, in the context of a Lipschitz hypersurface in a smooth Riemanmian manifold, the standard formula the

(_sf)(x) =

(3.21)

+

with

K*f(x)

=

(3.22)

= There

P.V.

/

dag(y).

is a similar treatment of double

(3.23)

Vf(x)

=

f

layer potentials, defined by y)f(y) dcr9(y),

F

One

has

=

(3.24) nontangentially

(3.25)

a.e. on 1', with

Kf(x)

The operators K and K* are adjoints on the Hilbert space L2(I', da9). A variant of the case considered above arises on a smooth, compact, connected manifold M, endowed with a smooth Riemannian metric. If L = Es — V, with a smooth V 0 that is > 0 somewhere, then L is invertible, with inverse E E 0PS2(M). If 11 is a Lipschitz domain in M, then one can use local coordinates

4. LAYER POTENTiALS ON LIPSCHITZ SURFACES

234

and a partition of unity to construct operators to which Propositions 3.1—3.4 apply, and then obtain results parallel to (3.10)—(3.25). We collect some of the results discussed above:

LP(O1l), 1

PROPOSITION 3.5. Given f (3.26)

Sf-,-(x)

811,

= Sf_(x) = Sf(x),

and (3.27)

Vf÷(x)

where, for x (3.28)

Sf(x)

+ K)f(x),

=

811,

=

f

Kf(x) = P.V. f f(9)r(x,y) dcr(y),

f(y)E(x,y) dcr(y),

on

on

Furthermore,

+ K*)f(x).

(3.29)

In (3.30) for

addition, there are

estimates on nontangential maximal functions: S

IP(VSfYIIP(on) S

1

8: L2(811)

(3.31)

—*

H'(M)

In fact, there is the following simple general

is compact.

result.

PRoPoSITIoN 3.6. Let V be a Banach space. Assume that T: V a bounded linear map such that, for some s > 1, (3.32)

Tv

vEV

811), (VTv)'

H' (M) is

L2(811).

Then T is a compact operator from V to H'(M). PROOF. It

to

c V is bounded and Tv1 —> 0 weakly and let C be a collar neighborhood of 811, of thickness Sc. Since is bounded in H8(M\C), Rellich's theorem implies that —> 0. Meanwhile, the hypothesis (3.32) implies a bound j(VTvI)11L2(ocl) S C, and hence IIVTvI(112(c) S Cc, SO suffices

in H'(M), then Tv1

— 0

show that, if

in H'(M)-norm. Fix E > 0

limsup 2

This proves the proposition.

< Cc.

4. BOUNDARY INTEGRAL OPERATORS

235

4. Boundary integral operators The identities (3.27) -(3.29) suggest investigating the invertibility of various operators on L2(OIfl. We take this up here. Let M be a compact, connected, smooth manifold, with a smooth Riemannian metric tensor, II a domain in M with nonempty Lipschitz boundary. We will assume II is connected, but we do not assume its complement M \ 11 is connected. We take (4.1)

with smooth V ? 0. To study

0 on fl, we would take V 0 on but it is of interest to consider the more general case. It is technically important to assmrie that V is strictly positive somewhere on each connected component of 0 M \ We then define B, S, 23,5, and K as in §3. As before, v denotes the unit outward normal to 812, which is defined a.e. on OIL

We tackle the boundary integral equations that arise in the analysis of the Dirichlet problem for Lu = 0 on 12, when the method of layer potentials is used.

These equations were first treated in [ye] when L is the standard (constantcoefficient) Laplace operator on a Lipschitz domain in Euclidean space. The treatment here follows [MT], though as stated in the introduction we restrict attention here to operators with smooth coefficients, for simplicity. We mention that in IMT] the hypothesis that V ? 0 on 11 is also relaxed. One goal in this section is to prove that 1 + K is invertible on L2(8I1). We start with the following result.

PRoPoSITIoN 4.1. The map +

(4.2)

L2(8fl)

is injective.

= 0. Set it = Sf, so PROOF. Suppose f E L2(8I1) and on M \ 812. The estimate (3.30) allows the use of Green's formula, to write (4.3)

fIvul2 +V1u12}dv(x)

=

0

da(x).

= By (3.29), the right side of (4.3) vanishes when f E

+ K). Thus it

is

constant on each connected component of 0 and it = 0 on supp V. Hence it = 0 on 0. Hence, by (3.26), Sf = 0 a.e. on 812, so, again by (3.26) and Green's formula (justified as before) we have

f

+

dv(x) =

fit

du(x) =0.

Hence it is constant on 12, 50 = 0 a.e. on 811. Since, by (3.29), f is equal to the jump of 8,..u across 811, we have f = 0, so Proposition 4.1 is proven.

Our next step is to establish the following estimates.

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

236

PROPOSITION 4.2. For all f e L2(811), K*)IML2

(4.4)

+CUWSIIIL2M),

where W = (V2 + V)112. Also,

Sf

K*)fIFL2

(4.5)

+CIIvvSfJIL2(M).

Hence (4.6)

+

If

+

As in [ye], we use a Rellich-type identity, of the following sort. Suppose u e C2(Q), and LIu = h e L2(11). Let w be a smooth vector field on M. Then we have the identity

f (v, w)l Vu12 dc(x) =

2

f

811

(47)

+ whenever

dc(x) —2 f (Vwu)h

f

{(div

Vu)}

—

11 is smoothly bounded and cc

11.

dv(x),

To prove this identity, you just

compute div ((Vu, Vu)w) and 2 div Vu), and apply the divergence theorem to the difference. If, in addition, (Vu)* e L2(811), we can take ftj / 11 with bounded Lipschitz constants, and pass to the limit, replacing 11 by 11 in (4.7). In the last integral in (4.7), £wg denotes the Lie derivative with respect to w of the metric tensor g. Regarding the first integral on the right side of (4.7), note that (a.e. on 811)

= Tw is the component of w case of) identity (4.7) as where

+

tangent

to 811. ilence we can rewrite the (limiting

(4.8)

=

— 2f(Vwu)hdv(x)

+

/{(div w)IVuI2 +

Vu)} de(x),

VTU(x) denotes the tangential component of Vu(x), for a.e. x e 311. Now, pick w, smooth on M, such that

where

(w,u)a>O&e.on 1311.

4 BOUNDARY INTEGRAL OPERATORS

237

This can be done provided 811 is Lipschitz. We then deduce from (4.8) the inequality (4.10)

cf

f1vTu12&T(x)

+IVuI2}dv(x),

and also the inequality

cf

C

(4.11)

Furthermore,

+ IVuI2}dv(x).

if L'iu = h on 11, Green's formula gives

(4.12)

/

IVuI2 dv(x) =f

dc(x)

-

/

uh dv(x).

Also, there is the Poincaré estimate:

fIuI2dc

(4.13)

Hence

cf

(4.10) yields an estimate

S

Cf

+Cf{1h12 + IhuI}dv(x)

(4.14)

while

(4.11) yields

cf IVTuI2dcT(x)

+

(4.15)

We are now ready to prove Proposition 4.2. Given f e L2(8f1), let u = Sf; first restrict u to 11. Since L'iu = Vu on 11, we can apply (4J5) and use (3.29) to obtain (iT — L'*\t 2

" Ii

(4.16)

L2(t311)

CII VTSI

Next, we use (4.14) to estimate

+

cJ Sf

+ CII WSf

except we replace 11 by 0 = M\11. Then the first integral on the right side of (4.14) is equal to +

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

238

by (3.29). Hence, with u = Sf

on

0,

so

= Vu, we have

again

f IV'vu(2 (4.17)

312 2

+

Sf da(x)

+

+ CII W5f1112(o).

do

In concert with (4.16), this proves (4.4), and the proof of (4.5) is similar. Then (4.6) follows from (4.4) (4.5), together with the fact that the last two terms in each of these formulas are dominated by Vie mention another useful way of stating the Rellich estimate:

max{f

Duul2

da(x), f

(4.18)

C

dcr(x)}

(912

312

flVTuI2dU(X)}

min{f no

+

Cf

+ Vu12} dvol(x).

One easy consequence of Proposition 4.2 is the following closed range property PROPOSiTiON 4.3. The maps

+

have closed range on L2(DQ).

PROOF. As shown in (3.31), the map 5: L2(8O) H'(M) is compact. That (4.6) then implies the closed range property for + K* is a well known part of the Riesz theory of compact operators. COROLLARY 4.4. The map

+ K : L2(8fl)

(4.19) is

L2(UfI)

surjective.

PROOF. Proposition 4.1 implies that + K has dense range, and by Banach's closed range theorem, Proposition 4.3 implies that + K has closed range.

The last step in showing that + K in (4.19) is an isomorphism is to show that it is injective. Tn [ye] this was done by showing that 1+ K* has dense range. Here (as in [MT]), we will instead use the method of [EFV], and establish the following extension of Proposition 4.2. PROPoSiTiON 4.5. Given A E R. Al ? 1/2, there exists C = C(A, 11) (4.20)

If IL'win <

(Al + ItY)fIl L2(3u) + CIISIM Jjl (Al).

such that

4. BOUNDARY INTEGRAL OPERATORS

239

PROOF. Take the identity (4.7) and multiply by A — 1/2; take its counterpart with Q replaced by IL = (9 = M \ 0 and multiply by A + 1/2. Summing the two. we have (A

f

-

-

da(x)

DO.

-

(A

f

+

- (avu)2}

da(x)

arz_

(4.21)

=

(2A

—

1)

—

(2A

f

du(x)

+ 1) f

da(x)

+ R,

80-

where,

here and below, R denotes a quantity satisfying an estimate

RI

(4.22)

CII VuIft2(Iw) +

Recall that u = Sf and Ii =

-

f

= Vu. Now the left side of (4.21) is equal to

(v,w)IVyuI2da(x) -

(A

-

+ K)fl2dc(x) arz

(4.23)

+

+ (A+ 1*1

Furthermore, upon writing + last two terms in (4.23) sum to (4.24)

-(A2-

=

—

A)

+ (Al + K*), one sees that the

f(v,w)I(Al+K*)fI2da(x).

80

00

On the other hand, the right side of (4.23) is equal to R plus

+ K*)f)

(2A -1) Do

(4.25)

- (2A+ 1)

+ K*)f) da(x) 80

= _2f(V'111.Sf) . (Al + Kt)! da(x). Do

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

240

Thus, from (4.21)—(4.25) we have

(A2 (4.26)

1) f (w,w)jfj2da(x) + I

-

(Al + K*)f 2) da(x)

011

011

= 2f(V'rwSf) (AI+ K*)fdc(x)+R. 311

Now (4.20) follows directly from (4.26) if Al> 1/2. The cases A = ±1/2 follow from and using the estimates Proposition 4.2, upon writing = (4.4)—(4.5). Of course, the case A = 1/2 is already contained in Propositions 4.1 and 4.3, which imply the stronger estimate (4.27)

+ K*)flJL2(311).

If 1IL2(31l)

As noted in (3.31), S : L2(OQ) H1(M) is compact. Hence (4.20) implies 1/2, Al + K* : L2(OQ) —* L2(DQ) has closed range and finite that, for IAI dimensional kernel. Thus, for each such A, Al + K* is semi-Fredholm on L2(OQ), with a well defined index. Furthermore, the index is continuous in A, hence constant on (—cc, —1/2] and on [1/2, cc). Now, for Al > IIK1I, Al + K* is invertible, so we have:

PROPOSITION 4.6. If A E IR, Aj 1/2, the operator Al + K* is Fredholm on L2(UQ), of index zero; hence so is Al + K. In particular, the operators

+ K,

(4.28)

+ K* : L2(D11) —p L2(81l)

are Fredholm of index zero.

hom the injectivity (4.1) we deduce: COROLLARY 4.7. The operators (4.29)

+ K,

+ K* : L2(DQ) —.

are invertible.

We complement this with the following result on the subspace of L2(OQ) orthogonal to constants.

—

+ K*. Let

PROPOSITION 4.8. If V >0 somewhere on 11, then are invertible on L2(31l). If V = 0 on then (4.30)

denote and

K*

+ K* : Lg(811) —p L3(DQ)

is an isomorphism.

PROOF. We use reasoning parallel to the proof of Proposition 4.1. To begin, + K*)f = 0, and set u = Sf. Parallel to (4.3), we assume f L2(Ofl) and have (4.31)

f{IvuP2 + Vlu12}dv(x)

0,

5. THE DIRICELET PROBLEM ON LIPSCHITZ OOMAINS

241

so it is a constant (say co) on 11 (which we are assuming is connected). If V > C somewhere in 11, then c0 = 0; in any case, Sf = Co a.e. on 811. Hence (4.32)

f{Ivnl2

= _coffdcr(x),

+ V1u12}dv(x)

where the last identity uses the fact that f is equal to the jump of across Oil. If c0 = 0, then the right side of (4.32) vanishes, so it is constant on each connected component of 0 (and each such constant is 0). Thus the jump of across 811 is zero, i.e., f = 0, so + K* is injective on L2(DI1), if V > a somewhere on IL The invertibility on L2(DI1) then follows from Proposition 4.6.

On the other hand, if V = 0 on 11, then Green's formula implies + K*)f belongs to Lg(DI1) for all E L2(OI1), so (4.30) is well defined, and one deduces from Proposition 4.6 that this operator is also Predholm, of index zero. We show this operator is injective. Indeed, if f E belongs to its kernel, then the arguments involving (4.31) again hold, and again (4.32) vanishes, so again we have

1=0. 5. The Dirichiet problem on Lipschitz domains We now apply the invertibility results of §4 to the Dirichlet problem. As in §4, we assume M isa compact, connected, smooth manifold, with a smooth Riemannian metric tensor, 11 a domain in M with nonempty Lipschitz boundary, and L of thE form (4.1), with smooth V 0, and V > 0 somewhere on each connected component of C = M \ 11. We begin with the following existence result.

PRoPoSITIoN 5.1. Given f E L2(OIl), there exists it (5.1)

Lu=0 on 11, u*eL2(DIfl,

such that a.e.

PROOF. By Corollary 4.7, there exists a unique g E L2(OI1) such that + K)g = f. Then it = Pg satisfies (5.1), by (3.27) and (3.30). The interior regularity stated above is standard. Note that the solution to (5.1) constructed above is given by (5.2)

it =

K)1f).

We wish to establish uniqueness of it satisfying (5.1). For this, it will be usefu: to have sbme elementary results on solutions to the Dirichiet problem in spaces.

PROPOSITION 5.2. Given f E H1/2(D11), there exists a unique it satisfying (5.3)

Ln=0, neH'(Ifl,

4. LAYER POTENTiALS ON LIPSCIUTZ SURFACES

242

PROOF. Since the relevant Sobolev spaces are invariant under composition by bi-Lipschitz maps, one can locally flatten the boundary and produce E H'(Q) such that = f. If we write u = v + then (5.3) is equivalent to the statement — for all that v E and (Vv,V'i,b)L2 + (Ifl. The existence of a unique v with these properties is standard. E

We are not yet prepared to assert that the solution to (5.3) is given by (5.2); that will be done in Proposition 5.6. We will denote the solution operator to (5.3) by PT (without including in the notation the specific dependence on 11). Thus, for Lipschitz 11, we have PT: H'12(311) —+ H'(II).

(5.4)

The next two propositions deal with the case when 811 is smooth; these results will provide useful tools in the analysis of the Lipschitz case.

PRoPoSITIoN 5.3. If 311 is smooth, then 1/2(11)

P1: H9(311)

(5.5) PROOF.

This is

standard, as is

s

the fact that, when 311 is smooth,

Lv E H°(11)

vE

(5.6)

V

ve

Note that, as a byproduct of Propositions 5.2—5.3 and their proofs, we have the unique solvability of Lu = f, given f e L2(Q). for u E n H2 (Ifl, when 311 is smooth. This fact will play a role in the proof of Proposition 55.

smooth. Then there exists a constant C, depending only on the Lipschitz character of 11, such that the following holds: whenever v E and Lv h E L2(Q), we have E L2(311) and PROPOSITION 5.4.

Let 311 be

(5.7)

(811)

PROOF.

(5.6).

this

That

H'12 (311)

12(11).

c L2(311) under our hypotheses follows from

Also, the hypotheses imply

H2(12),

the

Lipschitz

Rellich type estimate

and

+ (Vv,v)L2(U)

plus Poincaré's inequality yields

Next, since v E

(4.11) implies that, with C depending

only on the

character of 11,

(5.8)

f

du(x) Cf {1Vv12 + 1h12} dv(x),

Ac?

since V7'v

0 on 311. This proves the desired estimate (5.7).

We now prove an estimate that implies uniqueness of solutions to (5.1). We return to the general case of Lipschitz 11.

5. THE DIRICIILET PROBLEM ON LIPSCH[TZ DOMAINS

243

PROPOSiTION 5.5. There is a constant C, depending only on the Lipschitz character of [I, such that, whenever

Lu=0, u*eL2(dIfl,

(5.9)

and u has a non-tangential boundary trace at almost every point in 011, we have

f

(5.10)

Cf

dv(x) C

du(x).

0

be a sequence of smooth domains, with bounded Lipschitz PROOF. Let constants, increasing to 11. Given f e L2(Ifl, define by Lv3=f on

(5.11)

e H2(ftj), and Proposition 5.4 applies to v3. Applying Green's formula Then and the estimate (5.7), we have

I ufdv(x)

j

(5.12)

=

I u—dcr(x)

j

dvi

603

C

HfUL2(cz3).

Given that ut E L2(0Q) and that we have non-tangential convergence to the limit on Oft we obtain (5.13)

f E L2(Ifl, which implies (5.10). Let us temporarily denote the solution operator to (5.1), produced by Proposition 5.1, by PT, so (5.14)

PT: L2(OQ) —p {u E Cocftl) : u* E L2(OIfl}.

We have the following compatibility result. PROPOSITION 5.6. We (5.15)

have

PIf = Hf.

f E H"2(O11)

V, the set of restrictions to 011 of elements of PROOF. First, consider f / 11, the maximum principle, C9M). Well known arguments involving smooth of [T5], yield P1: V —p and barrier functions, such as given in Chapter 5, Thus, for f V, PT f satisfies all the conditions in (5.11). Hence, by Proposition

5.5, fE

Now any f E H'/2(0Q) isalimit in

V. We have simultaneously PT in C'911), so we have (5.15).

sequence

f

Thus we drop the tilde from (5.14) and write (5.16)

PT: L2(0Q) —* {u

E

Ccc([I) : if E L2(OIfl}.

PT

f

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

244

Note also that, by reasoning similar to the proof of Proposition 5.6, when I E C(811), PIf coincides with the element u E C(fl) solving Lu = 0 provided by the Perron-Weiner-Brelot process:

P1: C(81l) —s

(5.17)

The following is a useful extension of (5.17).

the L2-solution of the Dirichiet prob-

PROPOSITION 5.7. For any I E tern

(5.18)

Lu =

0

in Il,

u* E L2(81Z),

=

f a.e. on 811

satisfies (5.19)

lJuIILoo(c2)

If IIL°(8c11).

(Oil) C L2(81l), we can construct a sequence PROOF. Given f } of continuous functions on 811 such that f3 —> f in L2(81l) as j cc, and = P1f5 in 12, we have that Then, if u uniIf If) formly on compact subsets of 11 and, by the maximum principle, Hence, passing to the limit, we have (5.19), if We can interpolate between the L2 and LOC results, to obtain:

PROPOSITION 5.8. For 2 p cc there exists a unique solution to the Dirichlet problem (5.20)

Lu=0 in

11, u*EL))(Offl,

This solution satisfies (5.21)

IIuJJLP(aII) 5; CIII IILP(difl.

PROOF. Consider the operator T : f F—* (PT f)* which is well defined and sublinear on L2(81l). Since T is a bounded mapping of L2(OIl) into itself as well as of L°°(O11) into itself, Marcinkiewicz's interpolation theorem implies that T is bounded on LF(Ofl) for 2 5; p 5; cc.

in view of (5.17), we know that evaluating P1 f at a point x e 12 produces the "harmonic measure" (5.22)

PIf(x)=

fe C(OIfl, xeIl.

We now have the following result, which was established for the Laplace operator on Lipschitz domains in flat Euclidean space by [Dahj.

PROPOSITION 5.9. For each x E fl, the measure on 812 are mutually absolutely continuous.

and the surface measure a

5. THE DIRICULET PROBLEM ON LIPSCHITZ DOMAINS

245

PRoOF. From (5.16) we have (5.23)

and (5.22) holds for all I L2(811). It remains to show that a <<wy, for each x e Il. Suppose E C 011 is a Borel set. Then (5.24)

Now u

=

= 0.

0

satisfies

Lu=0,

(5.25)

on Il.

and, for a.e. y E Oil,

bin u(z)

(5.26)

XFJ(Y),

z—+y

the limit taken nontangentially. if a(E)

0, then we deduce that u is not identically zero on Il. Then the Hopf maximum principle implies that u(x) > 0 for all x E 11. This shows that, for each x E 11, = 0 c(E) = 0, so the proof is complete.

We mention some further properties of the solution to (5.1). For example, one has (5.27)

1€ H1 (011)

(Vu)*

L2(011).

of IMTI, using techniques that also treat the Neumann boundThis is proven in ary problem. In the case of the Laplace operator on a Lipschitz domain in Euclidean space, (5.27) was established in [ye]. Also, in [MMT] it is shown that (528)

1

L2(311)

u

feH' (011)

In fMT2], extending work done in the constant coefficient case in [ye] and [DK], the authors show that, for some s = s(1l) > 0, (5.29)

f€H1'1(Olfl, I

As with other results treated here, these results were treated in [MT], and [MT2], for equations whose coefficients have minimal regularity. Here we have studied the homogeneous Dirichlet problem (5.1). There are further results on the inhomogeneous problem (5.30)

Lu=g on Il,

on Lipschitz domains, for the flat Laplacian on Euclidean space in [3K], and for Lipschitz domains in Riemannian manifolds with metric tensor in C' in [MT3]

b'*0

4.

LAYJIIIt FOTF)NTLALS

LIPSCHITZ SURFACES

A. The Koebe-Bieberbach distortion theorem The Koebe-Bieberbach distortion theorem, used in §1, is a result about univalent (i.e., one-to-one) holomorphic functions defined in the upper half-plane, or equivalently functions defined on the unit disk in C. It fits within a small collection of results about univalent functions, which we present here. Let S denote the set of univalent holomorphic functions, defined on the disk V = {z E C: zl c 1}, with the additional property that f(O) = 0 and f'(O) 1, so, for f C 5, 1(z) = z + a2i2 + a:1z3 +

(A.1)

The transformation f —g, given by g(() univalent holomorphic functions on W =

f(1/() takes S to E, consisting of C C: id> 1}, having the form

(A.2)

A calculation connects the coefficients in these expansions. In particular, one obtains (A.3)

b0 =

—a2.

There has been an intensive study of the coefficients ak in (A. 1), leading to the proof of the Bieberbach conjecture, that IakI Ic for all Ic. For the distortion theorem, the case Ic = 2 is relevant. The first tool for this is the following "area theorem," due to Gronwall: PRoPoSITION Al. For g C (A.4)

area of K =

the

A(K) =.

rr(l

C

\g(V*) is given by

—

PROOF. For any p > 1, the image of the circle zl =

p

under g is a simple

closed curve 7(p) in C, enclosing a region of area A(p). Green's theorem gives

A(p) =

f

xdy = —

7(p)

Setting z =

f

ydx =

7(p)

= Ek>l

ikO (with

b1 = 1),

kf

A(p) = (A.5)

= —'

Taking the limit

p\

1

yields (A.4).

As a corollary, we have

g C E —> b1( 1. Using this, we can prove E3ieberbach's theorem:

f

7(p)

we obtain

DISTORTIOr4 THEOREM

A. THE

PROPOSiTiON A.2. For

e

247

8, we have 1a21 2.

(A.7)

PROOF. Let g(V*). Now

g

f(1/()'. Note that 0

E be given as above, i.e., g(()

hK) = g((2)V2 =

(A.8)

is seen to belong to E. <2, and then (A.3) gives a2! 2. Applying Proposition A.1, we have bol C

The next result is the Koebe-Bieberbach quarter theorem. 11 is a biholomorphic map. Let w0 = f(O) THEOREM A.3. Suppose f : V and set To = dist(wo,811). Then lf'(O)I

To

(A.9)

and rescaling, we can assume that wo = 0 and f'(O) = 1, S. Then the claim is that 1/4 To 1. Let zo 811 have minimal distance

PROOF. Translating

so f

from the origin, so jzol =

To.

Consider

f(z)

(Ado)

2

+....

well as a2! 2. S. Hence Proposition A.2 gives a2 + 1/z01 2, as We have This gives 1/ro 4, i.e., To 1/4. 0 maps V into itself, with On the other hand, if To? 1, then = f is the identity map in and qV(O) = 1. The Schwartz lemma then implies that ?Ji

such

a case.

To

proceed, we bring in the notion of the Poincaré metric. On V. this metric

is given by ds

(A.11)

=

7(z) dzl is a metric on 11, the metric biholomorphic and if f*(da) induced on V is 7(f(z)) If'(z)I IdzI. Thus the Poincaré metric induced on 11 is of the form

1ff: V (A.12)

11

is

da

-

7(z) dz!,

fractional transforThere are biholomorphic maps of V to itself given by linear mations: (A.13)

1

+ baz

It is an exercise to show that each such map r (i

-

preserves

z12)V(z)! =1- r(z)!2,

z

the Poincaré metric, i.e., V.

4. LAYER POTENTIALS ON LIPSCHITZ SURFACES

it is also an exercise to show that the set of linear fractional transformations of the form (A.13) forms a group that acts transitively on V. We can now re-cast Theorem A.3 in terms of the Poincard metric induced on a simply connected domain in C. PRoPOSITiON A.4. Let Il be a proper, simply connected domain in C. be the Poirzcaré metric induced on 11, via a biholomorphic map I : V

'y(z) Idzl

Then, for z

E

Il,

1.dzst(z, Oil) S — 1

—

2

5;

Let

Il.

2 ds(z, Oil).

"y(z)

PROOF. Consider a point Zi e 1). Composing f with a linear fractional trans.. formation, we can assume without loss of generality that f(0) = Zi. In such a case, we have by (A.12) that

'y(zi) =

2

and the conclusion (A.14) follows directly from Theorem A.3.

NOTE. In

(A.14), dist(z,

Oil) of course

refers to the Euclidean distance from z

to Oil.

Applying (A.12) and (A.14) again, for general z e V, we have the distortion

theorem:

THEOREM (A.15)

1ff : V —. dist(f(z),Oil)

Il is a biholomorphic map, then, for all z (1

—

V,

S 4dist(f(z),Oil).

Via a linear fractional transformation taking the upper half plane U in C onto V, we readily translate this to the estimate (1.10), noting that the Poincaré metric induced on U is given by ds = REMARK. The

estimate

dist(f(z),OfZ) <(1-

zDjf'(z)I S 4dist(f(z),Oil)

follows directly from (A.9) and a scaling argument. This is only slightly weaker than (A.15), and it will suffice for application to (1.10), with nonoptimal a and 0, namely a = 1/4, fi = 4.

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List of Symbols Various function spaces and other objects occur in this monograph, labeled by the following special symbols. We give the chapter and formula number where each one is introduced.

BMO bmo

BSr1

1 (3.34) 1 (2.83)

I (2.84) I (2.72), (3.37)

B;,q c(A)

(0.14) 1 (12.1) 1(0.2), (1.4)

c{w}

I (2.45) i (5.75) I (0.1) 1(0.3), (1.61) I (0.4), (1.64)

1

1 (1.21) 1 (5.57) 1 (4.40)

DTF

III (2.1)

HsI

1 (4.3) 1 (4.51) 1 (4.51)

Si1 (RTh)

1(2.60) 1(2.62) 1(11.5)

MP(R'1)

1 (16.1)

R(f,u) Sr5

XSr(k)

I (16.3) I (9.2) 1(3.48) I (0.9) I (3.46); 11(0.18) I (0.8), (2.44) I (2.52) 1 (3.18)

Index Beltrami operator, 152 Besov

spaces, 16, 82

boundary integral equations, 235

(Jalderón-Zygmund

(Jauchy

operators, 223

integral, 218

commutator, 58 deformation operator, 148

metric tensor, 202 modulus of continuity, 3 Money space, 94 Morrey's imbedding theorem, 94 Neumann boundary problem, 172 Neumann operator, 171 nontangential maximal function, 230 null bicharacteristics, 215

Dini condition, 89, 160 Dirichiet problem, 155, 168, 217, 241

div-curl lemma, 186

layer potentials, 233 double symbols, 63 double

elementary symbol, 18 elliptic, 137 elliptic regularity, 137 Euler equation, 178 exterior derivative, 77, 151 Fourier multiplier, 1 Fredholm, 139, 240

paracoinposition, 127 paradifferential operator, 101 paradifferential symbol smoothing, 35 parametrix, 138 paraproduct, 35, 104 propagation of singularities, 205 pseudodifferential operators, 2 quasiconformal mappings, 152, 198 Rellich-type identity, 236 Ricci tensor, 202 Riernannian manifold, 140 Riesz transform, 157

global regularity, 143

Hardy spaces, 24 Hardy-Littlewood maximal function, 64, 104 harmonic coordinates, 194 harmonic measure, 244 Holder spaces, 5 index, 139, 240 isothermal coordinates,

Osgood's Theorem, 151

sharp Carding inequality, 208 single layer potential, 232 Sobolev spaces, 2 super-commutator, 77, 186 symbol smoothing, 31 symbols, 2

154

John-Nirenberg space 27

Triebel-Lizorkin

spaces, 16

univalent functions, 246

jump relations, 230

vorticity equation, 178 Kadlec's formula, 179

Koebe-Bieberbach distortion theorem, 219,

246 Laplace operator, 140 layer potentials, 159, 217

Lipschitz domains, 217

surfaces, 217 Littlewood-Paley partition of unity, 4 Littlewood-Paley theory, 17, 104 local regularity, 143 Lipschitz

Zygmund space, 1 Zyginund spaces, 5

This book develops three related tools that are useful in the analysis of partial differential equations (PDEs), arising from the classical study of singular integral operators: pseudodifferential operators. paradifferential operators, and layer potentials. A theme running throughout the work is the treatment of PDE in the presence of relatively

little regularity. The tirst chapter studies classes of pscudodifferentia) operators whose symbols have a limited degree of regularity: the second chapter shows how paradifferential opcrators yield sharp c'timates on the action ol various nonlinear operators on function this material to an assortment of results in PDE, includspaces. The third chapter ing regularity for elliptic PDE with rough planar fluid flows on rough

domains, estimates on Riemannian manifolds given weak bounds on Ricci tensor. div. curl e.tlmates, and results on propagation of singularities for wave equations with rough coefficients. The last chapter studies the method of layer potentials on Lipschiti domains. concentrating on applications to boundary problems for elliptic PDE with variable coefticienis.

U

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