L e t X be an n-dimensional manifold and P EL~(X) a properly supported pseudo- differential operator with homogeneous principal part p. Let Nc be the set of all (x, ~)E T * ( X ) ~ O such t h a t p(x, ~)=0 and H~r v, Hr~v and the direction of the cone axis are hnearly independent. This is a C ~ manifold of codimension 2. The interior relative to 2V c of the set in ~V~ where the Poisson bracket {p, 73} vanishes we shall denote b y h r. As pointed out in the introduction the vector fields HR+v, Hn=v are tangent to h r and the tangent system on N spanned by them is integrable. The corresponding foliation will be called the
F O U R I E R I N T E G R A L O P E R A T O R S . H 249 bicharacteristie foliation and the leaves bicharacteristic strips. The projection of a bieharac- teristic strip in X will be called a bicharacteristic. Note t h a t the projection is non-singular only in the open subset of N where d~ Re p and d~ I m p are linearly independent.
If u E ~ ' ( X ) and (x, ~ ) E T * ( X ) ~ O we shall write s* (x, ~ ) = sup {t; u e//(~) at (x, ~)}.
That u EH(~ at (x, ~) means t h a t u = u I + u~ with u~ EH(t) and (x, ~) (~ WF(u~). The function s*(x, $) is lower semi-continuous and homogeneous of degree 0, and the proof of Theorem 2.5.3 shows that
s~ (x) = inf s* (x, ~) where s u is the function occurring in Lemma 7.1.2.
THEOREM 7.2.1. Let u E~'(X), P u =], and let s be a positively homogeneous ]unction in an open conic subset ~ o / N with s <~s r. Then it ]ollows that min (s*, s § - 1) is superharmonic in ~ i / s is superharraonic in ~, and that rain ( s * - s , m - l ) is superharmonic in ~ i] s is subharmonic in ~ (with respect to H~). I n particular, s* is superharmonic in ~ i/~2 N W F(]) = ~ . COROLLARY 7.2.2. I] U E ~ ' ( X ) and P u = / , then (N;) W F ( u ) ) ~ W F ( ] ) is invariant under the bicharacteristie /oliation in N ~ WF(]).
The corollary follows as Proposition 7.1.3 b y taking s = § co and ~ = N ~ W F ( ] ) in Theorem 7.2.1. The proof of Theorem 7.2.1 is parallel to that of Theorem 6.1.1 so we shall first consider the operator 8/85 in R n, z=x~_~§ and then pass to the general case b y means of Fourier integral operators.
First we shall prove t h a t Lemma 7.1.2 remains valid if s~, sf are replaced b y 8u, s~ $
and s b y a function of x and ~ which is homogeneous of degree 0. I n doing so we m a y assume t h a t uEe'(R~). If a(~) is a homogeneous function of degree 0, then 8u/OS=f implies
~(a(D) u)/~5 =a(D)]. We shall apply L e m m a 7.1.2 to such equations noting t h a t for every
~ 0 e t t ~ \ 0 one can choose aj with %(~0)=1 vanishing outside closed cones Fj which
__> $
decrease to the r a y through ~0 as i-~ c~. If uj = %(D)u we have suj(x) s~(x, ~0)so the superharmonicity of
min (suj, sj), where sj (x) = inf s(x, ~),
SeF~
gives in the limit t h a t min (s*, s) is superharmonic. This proves the first statement in Theorem 7.2.1 for the operator 8 / ~ . To prove the second we first note t h a t it is equiva- lent to the first if s is harmonic. Hence the second part follows if s is the supremum of
250 J . J . D U I S T E R M A A T A N D L . H O R M A N D E R
a f a m i l y of h a r m o n i c functions. If s E C 2 is strictly subharmonic, t h e n s(x', z, ~) >~ h(x', z, ~) in a n e i g h b o r h o o d of (x', w, ~) with e q u a l i t y at (x', w, ~) when h is t h e h a r m o n i c func- t i o n
h(x', z, ~) = s(x', w, ~) + R e ( 2 ( z - w)~s(x', w, ~)/~w + (z - w) 2 a2s(x ', w, ~)/~w2).
I n view of t h e local c h a r a c t e r of s u p e r h a r m o n i c i t y this proves t h e second s t a t e m e n t in T h e o r e m 7.2.1 w h e n s is strictly s u b h a r m o n i c a n d t h e general case follows b y a p p r o x i m a t i o n of s with such functions.
I n order t o pass f r o m ~]8~ t o P we shall use Fourier integral operators corresponding to a canonical t r a n s f o r m a t i o n o b t a i n e d f r o m Proposition 6.1.3 with Pl = Re p a n d P2 = ] m p . However, this requires t h a t {p, 15} = 2 i {Re p, I m p} vanishes n o t o n l y o n ~V b u t in a n e i g h b o r h o o d in T * ( X ) ~ O . T h e following l e m m a gives a r e d u c t i o n to t h a t case.
L]~MMA 7.2.3. I] (Xo, ~o) e N one can find a homogeneous C ~ ]unction a o/degree 1 - m with a(x o, ~o) :#0 such that {q, ~} = 0 in a neighborhood o] (x o, ~o) i/ q =ap.
Proo]. W e m a y assume w i t h o u t restriction t h a t m = 1. H~ can t h e n be regarded as a complex v e c t o r field on t h e cosphere bundle so there exists a fixed conic n e i g h b o r h o o d V of (x 0, ~0) such t h a t t h e e q u a t i o n
(H~ + a ) w = /
has a C ~~ homogeneous solution of degree 0 in N N V for all such a a n d / . T h e same is t r u e if w a n d ] are t a k e n h o m o g e n e o u s of a n y degree/~, for we can introduce w =bwl, /=b/1 with b h o m o g e n e o u s of degree # a n d different f r o m 0 a n d o b t a i n an equation of t h e same f o r m for w 1 a n d /1. T h e proof n o w proceeds in three steps.
1) Assume t h a t {p, 75} vanishes of order k>~l in V w h e n p = O . B y T a y l o r ' s f o r m u l a we can t h e n write
k
where we can choose aj so t h a t 5~ = - a k _ s since (p, 15} is p u r e l y imaginary. We claim t h a t it is possible to choose w vanishing of order ]r - 1 w h e n p = 0 so t h a t if q = p exp ~ t h e Poisson b r a c k e t {q, ~} vanishes of order k + 1 in V w h e n i~ = 0 . To prove this we note t h a t
{q, e = {p, + p{p, w} + + pp{ , w}.
T h e last t e r m vanishes of order 2 + m a x (2 k - 3, 0) >/k + 1 so it can be ignored. W r i t e
k - I
w = ~ w j p ~ ~-l-j.
0
FOURIER INTEGRAL OPERATORS. II k-1
T h e n (p, W} = ~ {p, Wt} p t ~ k - l - t _ ~ O(pr) 0
where r = m a x (k - 2 + k, 1) ~> k. T h u s we m u s t m a k e sure t h a t
k-1 k - l _ _ k
(H, wj)pJ~ k - ~ - ~ H~,wlp~-J~ + E a t p ~ - ' = O(p~+~),
0 0 0
251
t h a t i s , H ~ , w ~ - H ~ w k _ j + a j = O i n / V N V for j = 0 . . . k.
Here w k should be read as 0. Since 5j = - a k _ ~ these equations follow if in N N V H ~ w j + a j = 0 when
i<k/2, w~=0
when ~>k/2, Hpwt+aj/2=O if ~=k/2.9 This proves the assertion.
2) W e have n o w p r o v e d t h a t there exist functions wo, w 1 .... h o m o g e n e o u s of degree 0 a n d defined in a fixed n e i g h b o r h o o d V of (x0, ~0) such t h a t if q~ = p exp (w0 +---+u~z) t h e n {qk, qk} vanishes of order k + 2 in N N V. Moreover, w, vanishes of order k in N N V.
B y a classical t h e o r e m of E. Borel we can choose a homogeneous C ~ function w such t h a t w - w o - . . . - w ~ vanishes of order k + l in N ; / V for e v e r y k. I f q = p e x p ~ it is clear t h a t {q, ~} vanishes of infinite order w h e n q = 0 .
3) W e have n o w reduced t h e proof to t h e ease where {Re p, I m p} vanishes of infinite order when 19=0. W r i t e p = p l + i p 2 with real Pl, P~ a n d set {px, p~}=,~lPl§ where 2~ E C ~ vanishes of infinite order in N N V. Mow choose a f u n c t i o n / 1 such t h a t in a conic n e i g h b o r h o o d of (x0, ~0)
{ e f l , ~ 9 "1, P~} = e1'22P~, i.e., {/1, P2} -~ ~1 = 0.
There is a unique solution with initial d a t a / 1 = 0 on a conic hypersurface transversal to H~2 at (x0, ~0) a n d it is homogeneous of degree 0. Since N is i n v a r i a n t u n d e r t h e vector field H~: it is clear t h a t /1 vanishes of infinite order w h e n p = 0 . ~ e x t note t h a t
{e1'pl, e1~pe} = 0 if {e~'pl, ]3} + efl2, = O.
This e q u a t i o n also h a s a s o l u t i o n / ~ vanishing of infinite order when p = 0 . T h u s a ~- (eI'pl + ieS*p~)/(p 1 + ip2 ) = 1 + ((e I' - 1) Pl + i( e~* - 1) P2)/(Pl + ip2)
is infinitely differentiable a n d a - 1 vanishes of infinite order w h e n p = 0. Since q = ap = erlpl +ier~p2 t h e l e m m a is proved.
252 J . J . D U I S T E R M A A T A N D L , H O R M A N D E R
End o] p r o o / o / Theorem 7.2.1. L e t (xo, ~0) E ~ . Using L e m m a 7.2.3 we can choose an elliptic operator E of order 1 - m such t h a t for the principal symbol q of Q = E P we h a v e {l~e q, I m q } = 0 in a neighborhood of (x 0, ~0)- Since Q u = E ] and s ~ f = s ~ + m - 1 b y the regularity theory of elliptic operators it suffices to prove the theorem for the operator Q instead of P. B y Proposition 6.1.3 we can find a homogeneous canonical transformation Z from a conic neighborhood of (x 0, ~0) to a conic neighborhood of (X0,
F.0)eT*(R~)\0
such t h a t q is the pullback of Z~-I + i Z ~ b y the m a p Z. As in Proposition 6.1.4 we can t h e n find a corresponding Fourier integral operator with the properties stated there except t h a t D~ is replaced b y ~/~5 in (iii), z=X~_~+iX~. I n fact, the only change is t h a t to satisfy (6.1.12) with D~ replaced b y ~/a~ we have to solve a C a u c h y - R i e m a n n equation in each step. This can be done b y Cauchy's integral formula. The proof of Theorem 6.1.1 now serves again to deduce Theorem 7.2.1 for the operator Q from the special case of the operator
~/02 already established, and we do not repeat the details.
We shall now derive existence theorems from Theorem 7.2.1. I n doing so we assume for simplicity t h a t the set N there is equal to the characteristic set p-~ (0) although it would be easy to consider a m i x t u r e of this case and the one discussed in Chapter VI.
T ~ O ~ E ~ 7.2.4. Assume that P E L ~ ( X ) has a homogeneous principal symbol p, that {p, ~} =0 and that HRep, Hm~ and the cone axis are linearly independent when p =0. Let K be a compact subset o/ X such that no bicharacteristic strip o] P stays over K. I] s is upper semi- continuous in T * ( X ) ~ O and subharmonic in p-l(0), it ]ollows that
The space
u E ~ ( K ) , P u = / , s~ >~s ~ s u > ~ s + m - l : t
N(K) = {v E,C(K); tpv =
0}
(7.2.1) (7.2.2)
is a /inite dimensional subspace o/ C~(K). I / S is a lower semi-continuous /unction in T * ( X ) ~ O which is superharmonic in p-l(0), i/ /E@'(X), s~>~S, and / is orthogonal to N(K), then one can lind u E ~ ' ( X ) with s*~>~S+m-1 so that P u = / in a neighborhood o~ K.
s *u >~ s + m
Proo/. L e t n be as in (7.2.1). B y the elliptic theory in the complement of p-l(0), and b y Theorem 7.2.1 we know t h a t ~o = r a i n ( s * ~ - s - r e + l , 0) is superharmonic in p-l(0). W e can now argue as in the proof of (e) ~ (a) in Theorem 7.1.5: L e t the m i n i m u m of ~ in p-l(0) be t a k e n at (x, ~). Then ~ is constant in the bicharacteristic strip through (x, ~) which b y hypothesis contains points over CK. There s* = + ~ so ~ ~ 0 . Hence ~ ~>0 everywhere and (7.2.1) is proved.
The hypotheses on P are also fulfilled b y tp. Replacing P b y tp in (7.2.1) we conclude t h a t N ( K ) c C~ and therefore b y Fredholm theory t h a t dim N ( K ) < ~ .
F O U R I E R I N T E G R A L O P E R A T O R S . H 253
The existence theorem is now obtained b y standard functional analysis as in the proof of Theorem 6.3.1: L e t F = { u E ~ ' ( X ) , s*>~S) which is a Frdchet space with the topology defined b y the seminorms u--> []AuIIL, where A ELf(X) has a kernel of compact support and t < S in W F ( A ) . The dual space consists of all v E t ' ( X ) with s * > - S . Since
*> - S - m + l =~ s*> - S , vES'(K), tPv=g, sg
the functional analytic arguments at the end of the proof of Theorem 6.3.1 can be applied with C ~176 replaced b y F. The details are left for the reader.
T ~ O R W M 7.2.5. Assume that P E L T ( X ) has a homogeneous principal symbol p, that (p, ~} = 0 and that HRe~, Him p and the cone axis are linearly independent when p = O . Assume that no bicharacteristic strip stays over a compact subset o/ X . Then we have (a) ~ (b) ~ (c) where
(a) For every compact set K c X there is another compact set K ' c X such that/or each bicharacteristic strip B and each component C o ] B N Cz-I(K) which has relatively compact projection in X we have CcT~-I(K').
(b) For every compact set K ~ X there is another compact set K ' ~ X such that v E S ' ( X ) , sing supp t P v c K ~ sing supp v c K ' .
(c) P de/ines a 8urjective map ]rom ~ ' ( X ) to ~ ' ( X ) / C~(X).
Proo/. (a) ~ (b) follows from Corollary 7.2.2, for the projection in X of the wave front set is equal to the singular support. (b) ~ (e) follows from [46, Theorem 1.2.4], for we can take K ' = O when K = O because no bicharacteristic strip stays over a compact subset of X.
After constructing solutions with wave front set in a bicharacteristic strip we shall see in section 7.4 under additional hypotheses concerning P t h a t (c) implies (a). (See Theo- rem 7.4.2.)
I t is easily verified t h a t no bicharacteristic strip stays over a compact subset of K if and only if no leaf of the bicharacteristic foliation in the cosphere bundle stays in a com- pact set. Secondly condition (a) in Theorem 7.2.5 is equivalent to condition ( a ) o f Theorem 7,1.6 for the bicharacteristic foliation in the eosphere bundle. F o r this reason the manifold X will be called pseudo.convex with respect to P if condition (a) of Theorem 7.2.5 is fulfilled and P is of principal t y p e in the sense of Definition 6,3.2 which applies with no change in the present situation.
254 J . J . DUISTERMAAT AND L. H()RMANDER