L e t M be a Coo manifold and L a real Coo vector field on M. We shall study the existence of solutions of the equation
L u = / (6.4.I)
when u , / 6 C ~ ( M ) . I f K is a compact subset we denote as before b y C~(K) the quotient of G~(M) b y the subspace consisting of elements vanishing of infinite order on K. The dual space is then 8 ' ( K ) .
TH]~OR~M 6.4.1. Let K be a compact subset o/ M. Then the /ollowing conditions are equivalent:
(a) LCoo(K)=Coo(K).
(b) (L +a) C~(K) = Coo(K)/or every a E C~(K).
(c) There exists a /unction q~EC ~~ such that L 2 ~ > 0 on K.
(d) /Vo complete integral curve o/ L is contained in K.
Proo]. (a) ~ (b) for if L v = a and Lw=eV/ then (L+a)(we-V)=/. T h a t (b) ~ (a) is evident. Using (a) twice we find a function ~ 6 Coo(X) with L ~ ~0 = 1 which proves (c). F r o m (c) we obtain (d) b y noting t h a t if an integral curve F is contained in K and the m a x i m u m of ~ in F is attained at y, then the integral curve through y is contained in F and Lop(y) = O, L2qg(y)>0. This is a contradiction. Finally, to prove t h a t (d) ~ (a) we first note t h a t (d) implies
(d') No integral curve of L is contained in K for all positive or all negative values of the parameter.
I n fact, the solution curve starting at a limit point of a half integral curve with this pro- p e r t y would be entirely contained in K in contradiction with (d). (This a r g u m e n t already occurred in the proof of Theorem 6.3.3.) I n view of (d') every point y e K lies on an interval of an integral curve with end points outside K so if /EC~176 has support sufficiently close to y it is clear t h a t the equation L u = ] can be solved in a neighborhood of K.
B y a partition of unity we conclude t h a t this is also true for an a r b i t r a r y / E Coo(M). The proof is complete.
Theorem 6.4.1 is of course analogous to Theorem 6.3.1 which contains a less elementary proof t h a t (d) ~ (a). We now give an analogue of Theorem 6.3.3.
THEOREM 6.4.2. The /ollowing conditions (a)-(f) on L are equivalent:
(a) LCoo(i) = C~176
(b) (L+a)C~176 = C~176 /or every a6Coo(M).
F O U R I E R I N T E G R A L O P E R A T O R S . I I 213 (c) There exists a / u n c t i o n q~ E C~176 such that L~q~ > 0 and
{ y E M ; q~(y) <~c}
is compact/or every c.
(d) (1) No complete integral curve o~ L is contained in a compact subset o/ M,
(2) /or every compact subset K o/ M there exists a compact subset K' o/ M such that every compact interval on an integral curve with end points in K is contained in K'.
(e) There are no periodic integral curves and the relation R = {(yl,y2)EM • M; Yl and y~ are on the same integral curve o~ L} is a closed Coo submani/old o / M • M.
(f) There exists a mani/old M0, an open neighborhood M 1 o/ M 0 • {0} in M o • It which is :convex in the It direction, and a di//eomorphism M ~ M 1 which carries L into the operator ~/~t i/ points in M o • It are denoted by (Y0, t).
Proo/: (a) ~ (b) is obvious as in t h e proof of T h e o r e m 6.4.1. T h a t result also shows t h a t (a) ~ (dl). To prove t h a t (a) ~ (d2) we assume t h a t (d2) is n o t true. T h e n there exist intervals [yj, y~'] on solution curves a n d Ys [YJ, YJ'] such t h a t y~, yr E K b u t yj is n o t contained in a n y fixed c o m p a c t set for all j. T a k i n g a subsequence we m a y assume t h a t
I i /p it
y j ~ y , yj->y a n d t h a t a n y c o m p a c t set contains only finitely m a n y Yr W e can n o w t a k e a Coo n o n - n e g a t i v e function / on M which is so large near t h e points yj t h a t u(y'j) -u(y~), b e i n g t h e integral of / over t h e integral curve, tends to + with ] if L u = / . T h u s (a) c a n n o t be valid. B y T h e o r e m 6.4.1 we also have (c) ~ (dl), a n d t h a t (c) ~ (d2) is obvious.
(d) ~ (e). D e n o t e t h e L-flow b y ~ so t h a t t+q~(y, t) is t h e solution of t h e e q u a t i o n dz/dt = L(z) with z(0) = y defined on a m a x i m a l open interval c It. I f D e is t h e d o m a i n of F, t h e n
R = {(~(x, t), x); (x, t) e~)e}.
T h e m a p (x, t)->(q)(x, t), x) is injective since there are no closed bicharacteristic strips, a n d it is clear t h a t t h e differential is also injective. To prove t h a t R is a closed Coo sub- manifold it suffices therefore t o show t h a t t h e m a p is proper. L e t (x~, t~)E D e a n d assume t h a t x~-~x, cp(x~, t~)-+y in M. W e h a v e to show t h a t (xi, t~) has a limit point in D e. I n doing so we m a y assume t h a t t~-> T E [ - ~ , c~]. B y (d2) there is a c o m p a c t set K ' such t h a t q~(xi, t) EK' w h e n tE [0, t~]. I f T = _ ~ it follows t h a t ~0(x, s) E K ' for s ~>0 or for s < 0 . I n view of t h e equivalence of (d) a n d (d') in T h e o r e m 6.4.1 this contradicts (dl). H e n c e T is finite a n d (x~, t~)-+ (x, T) E D e.
(e) ~ (f). I t follows f r o m (e) t h a t the quotient space M o = M / R is a H a u s d o r f f space, a n d identifying a n e i g h b o r h o o d of t h e equivalence class of x with a manifold transversal t o L at x we o b t a i n a structure of Coo manifold in M 0. T h e m a p M - ~ M o has a C ~ cross
214 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
section Mo--~M. This is obvious locally a n d using a p a r t i t i o n of u n i t y in M 0 we can piece local sections t o g e t h e r to a global one, for o n l y a n affine s t r u c t u r e is required to f o r m averages. W e can n o w t a k e M 1 = {(x, t);x E Mo, (x, t)E D~} a n d t h e m a p M I - ~ M given b y ~0.
(See also S t e e n r o d [50], sections 12.2 a n d 6.7.)
(f) ~ (a) for t h e e q u a t i o n ~u/~t=/ECoo(Mx) has a unique solution uEC~176 w i t h u = O for t = 0 .
(f) ~ (e). I f ~0 a n d ~1 are positive C :r functions in M 0 a n d in M1, t h e n
~(Yo, t) = %(Yo) + ( t - s ) q ~ ( y o, s)ds
is in C~(M1) a n d L ~ 0 > 0 . I f q % ~ a t co in M 0 and ~ 0 1 ~ sufficiently r a p i d l y a t ~ in M1, it follows t h a t <p--, ~ a t oo in M 1.
Remark. T h e equivalence of (a), (b) a n d (d) is of course essentially c o n t a i n e d in Malgrange [48]. According to W h i t n e y [52] t h e L-flow is called parallelizable w h e n (f) is fulfilled. F o r conditions e q u i v a l e n t to (f) see also Birkhoff [30, Chap. V I I / a n d in t h e topological case Dugundji-Ant~siewicz [36].
I n t h e applications of T h e o r e m 5.3.2 in section 6.5 we m u s t solve equations of t h e f o r m (6.4.1) w h e n M is a cone m a n i f o l d (section 1.1, p. 87) a n d u, / are s y m b o l s on M . W e a s s u m e t h a t t h e v e c t o r field L c o m m u t e s with multiplication b y positive scalars as is t h e ease for t h e H a m i l t o n field of a function which is h o m o g e n e o u s of degree I. T h u s / ~ is h o m o g e n e o u s of degree m if u is, a n d LuES~+I-~(M) if uES~. I n particular, if M~ is the q u o t i e n t of M b y t h e action of It+, L induces a v e c t o r field L~ on Ms since t h e C ~~ functions on M s are precisely t h e C ~ functions o n M which are h o m o g e n e o u s o~ degree 0. W e write zr f o r t h e projection M ~ M~.
T H ~ O R ~ 6.4.3. When M is a cone mani/old and L a C ~~ vector field commuting with multiplication by positive scalars in M , the /ollowing conditions are equivalent:
(i) For e v e r y / E S ~ ( M ) , m E R , 8 9 the equation L u = / has a solution u E S ~ ( M ) . (ii) The vector field L~ on M s satis/ies one o/ the equivalent conditions in Theorem 6.4.2.
(iii) The vector field L on M satis/ies one o/ the equivalent conditions in Theorem 6.4.2, wad i~ • is a positive C OO /unction on M which is homogeneous o/ degree 1, t h e n / o r any compact set K c M ~
~ ( y ) <~ CKN(z) (6.4.2)
i/ ~y, ~z e K and y, z are on the same orbit o / L .
(iv) There exists a C ~ maui/old Mo, an open neighborhood M ' o/ M o • 0 in M o • R which is convex in the direction o / R , and a di//eomorphism M--> M" • R+ commuting with
F O U R I E R I N T E G R A L O P E R A T O R S . 215 multiplication by posifive scalars (de]ined as identity in M' and standard multiplication in R+) such that L is mapped to the vector/ield 8/~t if the variables in M 0 • R • It+ are denoted by (Yo, t, r).
Proo/. (i) ~ (ii). L e t /E C~176 a n d consider ] as a homogeneous f u n c t i o n of degree 0 on M. Choose a solution uES~(M) of t h e e q u a t i o n JLu=/. I f K is a c o m p a c t subset of M a n d if K~ = {ty;t>~ 1, y E g } , K s = (ty; t > 0 , y E K ) , t h e n there is a c o n s t a n t C such t h a t
lu(y)I < c , y E K 1, a n d we claim t h a t
[u(y)-u(z)[ < 2 C , if y, z e K 2 are on t h e same orbit. (6.4.3) I n fact, if t E R + a n d we write t*u(y)=u(ty), y E M , t > 0 , t h e n
L ( t * u - u ) = t*Lu - L u = t * / - / = 0
so t * u - u is c o n s t a n t on t h e orbits of L. H e n c e u(tz)-u(z) = u(ty)-u(y), so l u(y) -u(z) ] = ] n(ty) -u(tz) I .
F o r large t we h a v e ty, t z E K 1 a n d (6.4.3) follows. Since 7eK2=~K m a y be a n y c o m p a c t set in M s conditions (d l) a n d (d2) of T h e o r e m 6.4.2 for Ls follow immediately, already if we t a k e / = 1 in t h e case of (dl).
(ii) ~ (iv). First n o t e t h a t there is a cross section Ms-~M. I n fact, we can c o n s t r u c t a positive C ~~ function N on M which is homogeneous of degree 1 b y means of a p a r t i t i o n of u n i t y on M s. T h e section M s ~ M is t h e n u n i q u e l y d e t e r m i n e d if we require t h a t N(m) = 1 in t h e range. F r o m condition (f) in T h e o r e m 6.4.2 applied to t h e v e c t o r field Ls on Ms we n o w obtain a diffeomorphism M-+M' x R+ with M ' as in condition (iv), which t r a n s f o r m s L t o a v e c t o r field of t h e f o r m
L 1 =~/~t § t)r~/Sr where a E C~(M'). N o w solve t h e e q u a t i o n
~b(y0, t)/~t § a(yo, t) = 0
with b E C~176 ') a n d i n t r o d u c e t h e function Nl(y0, t, r) = r exp b(yo, t) which is h o m o g e n e o u s of degree 1 a n d satisfies t h e e q u a t i o n L 1 N I = 0 . T h e composition of t h e m a p M--~M'• R+
a n d t h e m a p
M ' • R+~ (Yo, t, r)-~ (Yo, t, 2Vl(y0, t, r ) ) E M ' • R+
will t h e n h a v e t h e properties required in (iv).
T h e implications (iv) ~ (i) a n d (iv) ~ (iii) are trivial so it only remains to show t h a t (iii) ~ (ii). I t follows f r o m (6.4.2) t h a t e v e r y integral curve o f L s c o n t a i n e d in a c o m p a c t
216 J . J . I ) U I S T E R M A A T AI~D L. HOR/YIANDER
subset of Ms c a n be lifted to an integral curve of L contained in a compact subset of M.
Hence (dl) of Theorem 6A.2 m u s t be valid for Ls. To prove (d2) we let K be a compact set in Ms and set
K 1 = {y e M; z~(m) e K, C~ 1 < iV(y) < UK}.
I f we lift an integral curve of Ls with end points in K to M so t h a t it starts at a point with /Y(y) = 1, then the other end point will belong to K 1 too. Hence the whole integral curve belongs to a fixed compact set K~ in M which completes the proof of (if) and of the theorem.
Remark 1. Under the hypotheses of Theorem 6.4.2 (or 6.4.3) the vector field (L, O) on M • M defines a vector field g on the relation manifold R which satisfies the same conditions. I n fact, this is obvious from conditions (f) and (iv} respectively.
Remark 2. The obvious proof of the equivalence (a) ~ (b) in Theorem 6.4.2 also gives t h a t (i) in Theorem 6.4.3 is equivalent to
(i') For e v e r y / e S ' ~ ( M ) , m E R , 89 ~ ~<1, and ceS~ the equation ( L + c ) u = [ has a solution uES~(M).
Moreover, the solution can be prescribed arbitrarily within the class S~(M o • on M 0 • 0 • R+ ~-, M, the injection being as in condition (iv) of Theorem 6.4.3.
Remark 3: The situation changes drastically if for example periodic integral curves occur. Assume t h a t dy(t)/dt =L(y(t)) and t h a t y(t) is periodic with period T. The equation L u + c u = / r e d u c e s to du/dt+c(y(t))u=/(y(t)) on this curve. Because t-->u(y(t)) has to be periodic, this leads if dh(t)/dt=c(y(t)), h(0)=0, to
u(y(O)) exp h(T) = u(y(O)) + ](y(t)) exp h(t) dr.
I t follows t h a t the equation L u + c u = / i s solvable if and only if exp h(T) 4=1, t h a t is,
f:e(y(t)) g~ 2 ke Z. (6.4.4)
Secondly, if this condition is satisfied t h e n the solution u is uniquely determined b y ].
Now suppose t h a t all integral curves of L are periodic with a positive minimal period depending continuously on the initial point. This means t h a t the relation R intro- duced in Theorem 6.4.2 is a closed submanifold of M • M, t h a t the orbit space M / R is a manifold and t h a t M - + M / R is a fibration with fibers diffeomorphie to the circle. T h e n the equation Lu 4-cu = / i s globally solvable for every i t if a n d only if (6,4.4) is valid for each
F O U R I E R I N T E G R A L O P E R A T O R S . I I 217 integral curve y(t) of L, with T denoting the period of the curve. This will be called the non-resonance case. Moreover, in this case the solution u is uniquely determined b y / . Analogous statements are of course valid in the case of cone manifolds.