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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
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
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 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
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
>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
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'),
0ii:
(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) 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 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 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
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 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
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
>
£)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 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),
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) 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 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
r 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 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 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
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 £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
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
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- (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
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 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
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 (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 —
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
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, then (2.23)
xeLt.
2. SOME NATURAL FIRST-ORDER OPERATORS
151
If the metric tensor belongs to Lip'(M), and 1
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 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
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
0.
Taking p
TO PDE
3.
164
This implies that, if
E
C3(ôIfl, then
Va' 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
(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,
(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 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 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)
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 —
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 + ... + 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
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 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 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
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, 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')
(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 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 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.