classical microlocal analysis in the space of hyperfunctions

Seiichiro Wakabayashi Classical Microlocal Analysis in the Space of Hyperfunctions Springer... Institute of Mathematics University o f Tsukuba Tsukuba-shi, Ibaraki 305-8571, Japan E-

Lecture Notes in Mathematics

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Seiichiro Wakabayashi

Classical Microlocal Analysis in the Space

of Hyperfunctions

Springer

Institute of Mathematics

University o f Tsukuba

Tsukuba-shi, Ibaraki 305-8571, Japan

E-mail: wkbysh @ math.tsukuba.ac.jp

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Classical microloca analysis in the space of hyperfunctions /

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P r e f a c e

Many author have studied the theory of hyperfunctions from the view- point of "Algebraic Analysis," which is not necessarily accessible to us, studying partial differential equations (P.D.E.) in the framework of distri- butions The t r e a t m e n t there is considably different from ours Although

we think t h a t it is natural to work in the space of hyperfunctions for the purpose of studying P.D.E with analytic coefficients, we do not think that "Algebraic Analysis" is indispensable for this purpose We want to apply various m e t h o d s in the framework of distributions to the studies

on P.D.E with analytic coefficients In so doing the major difficulty is not to be able to use the "cut-off" technique For there is obviously no non-trivial real analytic function with compact support We shall use here "cut-off" operators ( pseudodifferential operators) instead of "cut- off" functions, which m a p real analytic functions and hyperfunctions to real analytic functions and hyperfunctions, respectively

In this lecture notes we a t t e m p t to establish "Classical Microlocal Analysis" in the space of hyperfunctions ( or in a rather wider class of functions) which makes it possible to apply the methods in the C °O_ distribution category to the studies on P.D.E in the hyperfunction cat- egory Here "Classical Microlocal Analysis" means that it does not use

"Algebraic Analysis" and that it is very similar to microlocal analysis

in the Coo-distribution category Our main tool is, in some sense, inte- gration by parts, which is equivalent to the fundamental theorem of the infinitesimal calculus In our direction there are two books One of them

is HSrmander's book [Hr5] which gives a short introduction to the theory

of hyperfunctions The other is Treves' book [Tr2] Treves developed in [Tr2] the theory of analytic pseudodifferential operators in the framework

of distributions, which had been studied by Boutet de Monvel and Kree [BK] On the basis of the m e t h o d s in these two books, we shall establish

"Classical Microlocal Analysis" in the space of hyperfunctions

Some parts of this lecture notes are simple generalizations of the re-

suits obtained in joint work with Prof Kajitani, and I would like to t h a n k him for many useful discussions

C o n t e n t s

1 H y p e r f u n c t i o n s

3

4

5

1.1 F u n c t i o n s p a c e s 5

1.2 S u p p o r t s 13

1.3 L o c a l i z a t i o n 23

1.4 H y p e r f u n c t i o n s 28

1.5 F u r t h e r a p p l i c a t i o n s of t h e R u n g e a p p r o x i m a t i o n t h e o r e m • 34 B a s i c c a l c u l u s o f F o u r i e r i n t e g r a l o p e r a t o r s a n d p s e u d o - d i f f e r e n t i a l o p e r a t o r s 41 2.1 P r e l i m i n a r y l e m m a s 41

2.2 S y m b o l classes 52

2.3 D e f i n i t i o n o f F o u r i e r i n t e g r a l o p e r a t o r s 57

2.4 P r o d u c t f o r m u l a o f 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 65

2.5 P r o d u c t f o r m u l a o f 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 87

2.6 P s e u d o l o c a l p r o p e r t i e s 93

2.7 P s e u d o d i f f e r e n t i a l o p e r a t o r s in B 107

2.8 P a r a m e t r i c e s of e l l i p t i c o p e r a t o r s 112

A n a l y t i c w a v e f r o n t s e t s a n d m i c r o f u n c t i o n s 3.1 3.2 3.3 3.4 3.5 3.6 3.7 1 1 5 A n a l y t i c w a v e f r o n t s e t s 115

A c t i o n of F o u r i e r i n t e g r a l o p e r a t o r s on w a v e f r o n t s e t s • • • 130 T h e b o u n d a r y v a l u e s o f a n a l y t i c f u n c t i o n s 155

O p e r a t i o n s on h y p e r f u n c t i o n s 165

H y p e r f u n c t i o n s s u p p o r t e d b y a h a l f - s p a c e 183

M i c r o f u n c t i o n s 192

F o r m a l a n a l y t i c s y m b o l s 201

M i c r o l o c a l u n i q u e n e s s 205 4.1 P r e l i m i n a r y l e m m a s 205

4.2 G e n e r a l r e s u l t s 222

A

B

4.3 M i c r o h y p e r b o l i c o p e r a t o r s 231

4.4 C a n o n i c a l t r a n s f o r m a t i o n 239

4.5 H y p o e l l i p t i c i t y 244

Local s o l v a b i l i t y 2 5 9 5.1 P r e l i m i n a r i e s 259

5.2 N e c e s s a r y c o n d i t i o n s on locM s o l v a b i l i t y a n d h y p o e l l i p t i c i t y 268 5.3 Sufficient c o n d i t i o n s on local s o l v a b i l i t y 272

5.4 S o m e e x a m p l e s 285

P r o o f s of p r o d u c t f o r m u l a e A.1 P r o o f o f T h e o r e m 2 4 4 A 2 P r o o f o f C o r o l l a r y 2 4 5 A 3 P r o o f o f T h e o r e m 2 4 6 A 4 P r o o f o f C o r o l l a r y 2 4 7 A 5 P r o o f o f T h e o r e m 2 5 3 2 9 5 295

323

328

336

338

A p r i o r i e s t i m a t e s 3 5 1 B.1 Gru~in o p e r a t o r s 351

B 2 A class o f o p e r a t o r s w i t h d o u b l e c h a r a c t e r i s t i c s 355

I n t r o d u c t i o n

Let 4 be the space of entire analytic functions on C ~ An analytic functional is a continuous linear functional on ,4 with usual topology We say t h a t an analytic functional u is carried by a c o m p a c t subset K of C ~,

i.e., u E 4t(K), if for any neighborhood w of K in C n there is C~ :> 0 such t h a t

lu(@l < C~osupb,(z)l for ~ e A

u E A r we can define supp u by s u p p u = N { K ; u E A t ( K ) } , which is called the s u p p o r t of u T h e concept of "support" is relating to restriction mappings and s h e a v e s 4 t defines the sheaf B of hyperfunctions while C r does the sheaf l) r of distributions In order to s t u d y partial differential equations ( P D E ) in the space of hyperfunctions it is usually sufficient

to consider problems in A t ( or T0 defined below) For u E A t we can define the Fourier transform fi(~) of u by

- 7 [ u ] ( , , ) : =

Therefore, we can formally define pseudodifferential operators p(x, D)

with appropriate symbols as

n)u := (2~r) - ~ f ei*4p(x,()~(~) d~ (0.1)

p(x,

However, p(x, D)u does not always belong to ¢4 r if p(x,~) is not a polyno- mial of ~ In microlocal analysis pseudodifferential operators play essen- tial roles So we need the corresponding spaces to the Schwartz spaces S

a n d s ~ F o r e E R w e p u t

We introduce the topology to S~ in a s t a n d a r d waỵ T h e n the dual space

ge I of S~ is given by

g , ' = {~(~) c v'; e-~<~)~(~) e s'}

If ¢ _> 0, then S+ is a dense subset of S and we can define S, := T-I[$+]

( C S) By duality we can define t h e transposed operators tic, tic-1 S~' ~ S~' for ¢ _> 0 We rewrite t j - = ~- since t U = 5 on S' Noting t h a t

~ ' e D S _ e f o r ¢ _>0, we define

S_~ := : 1[~_~] for e > 0

Moreover, S J E is defined as the dual space of S_~ when ¢ _> 0 We define ~'o := N~>o S_~ and 9% := N~>o S / From estimates of the Fourier transforms of analytic functoinals we see t h a t

£' c Á c £o c Fọ

Let V be an open conic subset of R n x ( R n \ {0}) We say t h a t p(x,~) E

C°°(F) is an analytic symbol in F if p(x,~) satisfies the estimates

Õ D~p(x,~) I <_ CAJ"'+l~llo, l!lfll!<~> '~-Ĩi for (~,~) ~ r with Ĩ1 > R and ~,/~ e Z ~ Ifp(~,~) is an analytic symbol

in R n x (R~ \{0}), p(x, D) defined by (0.1) maps C0 and ~'0 to E0 and To, respectivelỵ However, we can not define p(x, D) as an operator on £0 and 9% by (0.1) i f p ( z , ~ ) is an analytic symbol in F and F does not coincide with R n x ( R ~ \ { 0 } ) We do not want to a b a n d o n (0.1) as the definition of

p(x, D) So we introduce some symbol classes which contain symbols with

c o m p a c t supports We say t h a t a symbol ẵ, y, 71) E C°~(R ~ x R n x R n)

belongs to S m~'m2'~'~2 (R, A) if ẵ, y, T/) satisfies

I N T R O D UCTION 3

for u E 8oo A~ 8~ Then we have

82~++~2 -+ 8~-~1, 8-~+~ 2 + 8-2~+-~i,

a(D~:, y, Dy) : 8~2~, > 82'~++82,

8' -2~+-~1 ~ -~+~2 8'

if It >_ 2enA and ¢ <_ 1/R ( see T h e o r e m 2.3.3 below) In particular,

a(D,, y, Dy) maps continuously Jr0 to ~-0 if 61 = 62 = 0 and tt >_ 2enA

Therefore, we get "cut-off" operators, although R must be chosen to be large at each step of the calculation We do not fix one "cut-off" symbol and consider a family of "cut-off" symbols depending on R This is a disadvantage in comparison with usual calculus in the ditribution cate- gory However, we can overcome this disadvantage in most cases Using

"cut-off" operators we can define pseudodifferential operators and Fourier integral operators acting on the spaces ( or the sheaves) of hyperfunctions and microfunctions Since we must deal with operators with non-analytic symbols, the proof of the product formulas of pseudodifferential operators ( and Fourier integral operators) becomes longer t h a n usual one This is

a n o t h e r disadvantage of our methods However, as a consequence, we obtain the same symbol calculus as usual one

For u E ~-o the analytic wave front set WFA(u) ( C T * R '~ \ 0) of u is defined as follows: (x°,~ ¢°) E T * R n \ O does not belong to WFA(u) if there are a conic neighborhood F of ~0, R0 > 0 and {gn(~)}R>R0 C C ° ° ( R '~)

such t h a t gR(~) = 1 in F M {(~) >_ R},

Iog+ R( )l <

if (~> >_ RI(~I, and gR(D)u is analytic at x ° for R _> R0 The precise definition that gR(D)u is analytic at x ° will be given in Definition 1.2.8 Our definition of WFA(u), of course, coincides with usual definitions Our definition of WFA(U) is very similar to the definition of the wave front set of distributions Therefore, we can study P.D.E in the hyperfunction category in the almost same way as in the distribution category Our aim here is to provide microlocal analysis in the space of hyperfunctions in the same way as for distributions As applications we shall consider microlo- cal uniqueness and local solvability in the last two chapters These are still basic problems in the theory of linear partial differential operators

It is well-known in the framework of C °O and distributions that Carleman type estimates play an essential role in microlocal versions of the Holm- gren uniqueness theorem This is also true in the framework of analytic

functions and hyperfunctions General criteria on microlocal uniqueness will be given in C h a p t e r 4 Microlocal uniqueness yields results not mere-

ly on propagation of analytic singularities but on analytic hypoellipticity

We can also apply the same arguments to the studies on local solvability

in the framework of hyperfunctions as in the framework of distributions

We shall prove in C h a p t e r 5 that tp(x, D) ( resp p(x, D)) satisfies energy estimates if p(x, D) is locally solvable ( resp analytic hypoelliptic) We shall also show t h a t a little strengthen estimates guarantee local solvabil- ity So the problems on microlocal uniqueness, analytic hypoellipticity and local solvability will be reduced to the problems to derive energy es- timates ( or a priori estimates), which was carried out in the framework

of C °o and distributions by us

We should remark t h a t SjSstrand studied P.D.E in the framework of analytic functions and distributions in [Sj], using the FBI transformation

It may be possible to deal with hyperfunctions by his methods Using a

priori estimates he got many remarkable results However, we think t h a t his theory is different from usual microlocal analysis in the distribution category, although it is new and powerful So we will establish microlocal analysis in the space of hyperfunctions which is very similar to microlocal analysis in the framework of C °O and distributions

Chapter 1

Hyperfunctions

In this chapter we shall introduce the function spaces S~ and S~' corre- sponding to the Schwartz spaces 8 and 8 ~, respectively These spaces play a key role in our calculus The spaces S_~ and S~ ~ ( ~ > 0) include the space A ' of analytic functionals We shall define the supports and the restrictions of functions belonging to these spaces Hyperfunctions ( in a bounded open subset of R n) will be defined as residue classes of analytic functionals after the m a n n e r of H S r m a n d e r ' s book [Hr5] in Section 1.4

We shall prove t h a t the presheaf B of hyperfunctions is a flabby sheaf

We shall also prove flabbiness of the quotient sheaf B/`4 of B by the sheaf 4 of real analytic functions in Section 1.5

Since D ( = C ~ ( R n) ) is dense in L , t h e dual space & of L can be identified with {d({)v(~) E 79'; v E S'} For e >_ 0 we define

& ; jr l[ge] ( ,~'[L] = {'U E S; ee{{}'/~({) E S}),

where j r and j r - 1 d e n o t e the Fourier t r a n s f o r m a t i o n and the inverse Fourier t r a n s f o r m a t i o n on S ( or S' ), respectively, and ~(~) = jr[u](~)

We introduce the topology in S~ so t h a t j r : ,~¢ ) S¢ is h o m e o m o r p h i c Denote by S~ the dual space of S~ for e _> 0 Since & is dense in S for _> 0, we can regard S ' as a subspace of S~ T h e n we can define the

t r a n s p o s e d operators t~- and t j r - 1 of j r and j r - l , which m a p SJ and , ~ onto S~ and S~, respectively Since ,~_~ C S~ ( C l ) ' ) for ~ >_ 0, we can define S_¢ := tjr-l[~_~] for e > 0, and introduce the topology so t h a t

t j r - 1 : ~_~ _~ S - e is h o m e o m o r p h i c S_~ d e n o t e s the dual space of S_¢ for E _> 0 T h e n we have S_~ = Sr[~_~] C S ' C S~ for c _> 0 and j r = t j r

on S ( So we write t j r as jr Note t h a t & is a Fr~chet space with the topology d e t e r m i n e d by the s e m i n o r m s ]ul,s,,t := ]jr[u]l~,,t ( g E Z + )

We d e n o t e by A t h e space of entire analytic functions in C TM Let K

be a c o m p a c t subset of C ~, and denote by A ' ( K ) the space of analytic functionals carried by K , i.e., u E A'(K) if and only if

(i) u : A ~ qa ~ u(~) E C is a linear functional, and

(ii) for any neighborhood w of K ( in C '~ ) there is C~ _> 0 such t h a t

I~(~)1 ~ C~supl~,(z)l for 9o E A

Let ~2 ( C C '~ ) be a domain of holomorphy We call f~ a R u n g e domain

if every function in A(12) can be a p p r o x i m a t e d locally uniformly in fl by polynomials, where A(l)) d e n o t e s the space of analytic functions in ~2

It is known t h a t fl is a R u n g e d o m a i n if and only if K c - n f~ (E fl for any K (Z fl, and t h a t K has a f u n d a m e n t a l system of neighborho_oods consisting of R u n g e d o m a i n s if K is polynomially convex, i.e., K = K c ,

( see, e.g., [Hr8]) Here A ( Z B means t h a t t h e closure ~ of A is c o m p a c t

and a C int(B), where A , B C R '~ ( or C ~) and int(B) denotes the

interior of B If K is polynomially convex, u E A ' ( K ) and qa is analytic

in a n e i g h b o r h o o d of K , t h e n we can define u(qa), a p p r o x i m a t i n g qa by entire functions

1.1 F U N C T I O N SPACES 7

L e m m a 1 1 1 Let K be a compact subset of R n Then, for E > 0 the set

A

K~ := { z ~ C~; [Re z - xI + IIm zl <_ ~ for some x ~ K ) (1.1)

is polynomially convex In particular, u(~) can be defined for u E A'(K)

if ~ is analytic in a neighborhood of K

R e m a r k A direct proof of the second part of t h e l e m m a is given in Proposition 9.1.2 of [Hr5]

P r o o f Let z ° ~ K , This implies t h a t IRe z ° - x I + IIm z°l > ¢ for any

x E K First assume t h a t Re z ° E K Then, p u t t i n g f ( z ) = e x p [ - ( z -

Re z°)2], we ihave

sup If(z)l < exp[e 2] < If(z°)l, (1.2)

A

zEK~

where z 2 ( = z z) = E j ~ l z2 for z = (Z1,''',Zn) e C n Since f ( z )

is entire analytic in C n and, therefore, can be approximated uniformly

in any comp~u=t subsets of C n by polynomials, (1.2) gives z ° • ( K ~ ) ~ , Next assume t h a t R e z ° ¢ ( K Choose x ° E K so t h a t I R e z ° - x ° l = dis({Re z°}, K ) ( _ - infzeK IRe z ° - x I ), and p u t

A - IRe z ° - x°l + IIm z°l > IRe z I - xll, we have

subset of C ~ and u E .A'(K) We can define the Fourier transform fi(~)

Fourier transformation ~" is injective on A'(C~) So We can regard A ' ( K )

as a subspace of E~ ( C ~ ) if g C {z E C~; IIm zl < ~}

L e m m a 1 1 2 Let ~ >_ O, and let K be a compact subset of C ~ such

t h a t g C { z E C~; I l m z l _ < ~ ) / f u E A ' ( K ) , 5 >E andqDES~, then

R e m a r k If ~ E S~, then ~ can be continued analytically to an analytic function in {z E Ca; IIm zl < ~ ( see Lemma 1.1.3 below) Moreover, the polynomially convex hull K c - of K is included in the convex hull ch[K] ( C {z E C~; IIm zl < ~} ) So we can define u@) for u E A ' ( K )

where ~R = {~ E Rn; I jl < R ( 1 < j _ n)} Therefore, we have

lim u~(ff~ e-iZ'~'-l[~](~)d~)-~-u(~)

where r = (x 2 + x~+l) 112 and K.x(r) is a modified Bessel function of the

second kind It is known that

z < 0} ( see, e.g., [Ol]) Moreover, we have

• Tx[Po](~,Xn+l) = (sgn x~+l)exp[-Ix,~+ll(~)]/2 if xn+, ¢ 0,

7"l(u)(x,x,+l) = eo(x,x,~+l) * u ( = (u(y),Po(x - y,x,~+l))u)

i f • _> 0, Ix~+ll > e and u • S], since P o ( x - y, xn+l) • S~(R~) for 5 _> 0 and (x,x~+l) • R T M with Ix~+ll > &

L e m m a 1 1 3 Let u e 8:, and put V(x,Xn+l) : n(u)(X, Xn+l) Then

~e ha.e the following: (i) V(x,x.+i)lx.+,>o e C~([O,~);S:), ( 1 -

v(x, x.+l)= -v(~,-x~+s) i n S " f o r ~ + ~ ¢ o, ~here A~,~.+ 1 = - ~ j = l

D~ and 0 + 1 = -iO/OX.+l (ii) U(x, X~+l) can be regarded as a function

in C ° ° ( R n × ( R \ [-¢+,~+])), where ~+ = max{e,O} Moreover, there is

(1.4) is obvious If ~ < O, then we have

u(x) = (21r)-n(exp[-e(~r)]fi(~¢), exp[ix- ~ + ~(()])(

R e m a r k It follows from L e m m a 1.1.3 t h a t 7-l(u)(z,t) is real analytic

in x for t > 0 if u E ~-0 Thus,

(u, q 0 ) = 2 lim f 7-l(u)(x,t)qo(x)dx f o r u E g - - 0 a n d q 0 E S o o

t ~ +O J R n

P r o o f By L e m m a 1.1.3 (1.5) is obvious Assume t h a t K is a c o m p a c t subset o f R ~, V is a n e i g h b o r h o o d of K in R n and u E A ' ( K ) T h e n

U(x, x~+1) ( = ( u ( y ) , Po(x - y, x,~+x))u ) can be continued analytically to

R T M \ K × {0} and, therefore, U(x, 0) = 0 for x ~ K Let w be a complex neighborhood of K such t h a t dis(w, R n \ V) > 0 There are C~ > 0 and

L e m m a 1 1 5 Multiplication by a(x) is well-defined and continuous on S~, i.e., the mapping S~ 9 u ~-+ au E Se is continuous, if I~l < (v~A) -1

P r o o f Let u E `9~ T h e n au E ,S is well-defined and

]D~(m'x)-2M I < CM(A/2)HIa[!(A'x) -2M-I~I

( see L e m m a 2.1.1 below) Therefore, we have

[D '~ ( (A' x)-2M a(x) )l < 2CCM A['~I I~]!(A'x)-2M<x)k

P u t i ]~1-2 Z ~ = I ~jD~j for ~c E R " \ {0} T h e n we have

I.~[(Atx)2M a(x)](5)[ < f [Lg{(A'x)-2Ma(x)}] dx < Ck,A(V/-~A/ISI)Q!

i f " ~ E R " \ { 0 } and g E Z + " or "~ = 0 and g = 0." It follows from

L e m m a 2.1.1 t h a t

_ ~ t /c\1/2 exp[_(V/-~A)-l(~)]

I.T[(A'x)2Ma(x)] (~)l < "~k,A V,/

for ~ E R '~ This yields

= (27r)-" (~)/[gr[(A 'x) -2Ma(x)] * (D~ (AtD~) 2Mfi)(~) ]

<_ Ck,A,t,],~l,~ exp[ e(~)]luis, ,2M+}al+g+n+l

and, therefore,

[aui,~,,t < Ck,A,I,~iUIS,,2M+t+n+I,

if le] < (v/-~a) -1 Since Soo is dense in `9~, this proves the lemma []

If a E C°°(R ") satisfies t h e e s t i m a t e (1.7) and lel < ( v ~ A ) -1, then

we can define au for u E SJ by

(au, ~) : = (u, a~) for ~v E S~

We note t h a t au E J4'(K) can be defined by (au)(~) = u(a~) ( ~ E A),

if u E A~(K) and a is analytic in a n e i g h b o r h o o d of K c - , where K is a

c o m p a c t subset of C "

1.2 S U P P O R T S 13

D e f i n i t i o n 1 2 1 Let e >_ 0 For u • S [ we define

supp u : N { K ; K is a closed subset of R "~ and there exists a real

analytic function V(x, Xn+i) in R n + a \ K × [ - e , e ]

such that V(x, Xn.4_l) = •(U)(X, Xn+I) f o r IXn+l] > 6")

R e m a r k (i) The definition of supp u does not d e p e n d on the choice of

e satisfying u E S~' (ii) For u, v • S [ we have supp (u ± v) C supp u U supp v, and supp u O X = supp v M X if supp (u - v) N X = 0, where

X C R n (iii) I f s u p p u C K~ ( A • A), then s u p p u C N ~ e h K ~ (iv) For u • ~-~ there is a real analytic function U ( x , x ~ + l ) in R n+l \

supp u × [-e,¢] such t h a t U ( x , x ~ + l ) = 7-l(u)(x, xn+l) for ]Xn+l] > e (v) Let u • U0 and x ° • R ~ T h e n 7t(u)(x, xn+l) can be extended to a C2-function near (x °, 0) • R T M if and only if x ° ~ supp u

L e m m a 1 2 2 (i) Let X be an open subset of R n, and assume that

W ( x , X.+l) e C°~(X × ( R \ {0})) satisfies ( 1 - A~,~.+,)W(x, XnTi) = O,

W ( x , x + l ) + 0 in D ' ( X ) as X.+l + O, and W ( x , - x + l ) = - W ( x , xn+l) for x E X and Xn+l > O Then W ( x , xn+l) can be extended to a real analytic function in X × R (ii) I f u E ,9 ~, supp u coincides with the distribution support of u, which is the support of u as a distribution

P r o o f (i) By assumptions we can regard W ( x , Xn+l) a.s a function in

C ( R ; D ' ( X ) ) ( C D ( X x R ) ) Since (O2/Ox~+,)W(x,x,+l) (1- A~)W(x, Xn+l) for x E X and xn+l • 0, it follows t h a t (02/Ox~+l)W(x,

in D ' ( X ) for ~ > 0 This yields (OW/Oxn+l)(x,=l=xn+l) e C([+0, cx~);

D'(X)) On the other hand, we have (OW/OXn+l)(X, +0) = (OW/Ox,~+l)

( x , - 0 ) Therefore, the mean value t h e o r e m implies t h a t W ( x , x,~+l) E

e l ( R ; D ' ( X ) ) Similarly, we have W e C 2 ( R ; D ' ( X ) ) This gives ( 1 - Ax,x,+I)W = 0 in D ' ( X x R) T h a t (1 - A~,x,+~) is analytic hypoelliptic also follows from the fact t h a t the f u n d a m e n t a l solution

E0(x, Xn-bl ) of ( 1 - A~,x.+, ) is real analytic for ( x , x + l ) # (0,0), al- though it is well-known So W ( x , X.+l) is real analytic in X × R We remark t h a t we shall prove analytic hypoellipticity of general elliptic op- erators in T h e o r e m 2.8.1 (ii) Let (z °, 0) E R n+l, and assume t h a t there are a closed subset K of R = and a real analytic function U(x, x~+l) in

R n+l \ It" × {0} satisfying x ° ~ K and ~t~(u)(X, Xn÷l) = U(x, Xn÷l) for

xn+i 5£ 0 U(x, Xn+l) is an odd function with respect to x5+1 Therefore,

U ( x , 0 ) = 0 for x ~ K Let qv E C ~ ( R '~ \ K) Then ~2(x)U(x, xn+l) -+ 0

in S ~ as X=+l -+ +0 On the other hand, it follows from L e m m a 1.1

3 t h a t ~(x)7-l(u)(x,x=+l)(= ~(x)~f(X, XnTI) ) + ~ ( x ) u ( x ) / 2 in S ' a s

Xn+l -+ +0 So we have qa(x)u(x) = 0 in S t, which implies x ° does not belong to the distribution support of u Next assume t h a t an open sub- set X of R = does not meet the distribution support of u, If qo E C ~ ( X ) ,

then, by L e m m a 1 1 3 , qa(x)7t(u)(x,x~+l) ~ 0 in S' as xn+l -+ +O, i.e., 7"l(u)(x, xn+i) -+ 0 in 79'(X) as Xn+l r +0 From L e m m a 1.1.3 and the assertion (i) we can see t h a t 7-l(u)(x, Xn+l) can be extended to a real analytic function in X × R and t h a t supp u M X = q} This completes

L e m m a 1 2 3 For any ~v E A there exists a unique 42 E C°~(R TM)

such that

(1 - A~,~.+1)42 = 0, 42l~.+1=o = 0, (010x,~+1)421~.+,=o = ~ (1.8)

Moreover, 42 can be continued analytically to C n+l and satisfies the fol- lowing estimates; for any R > 1, ~ > O, ~ E Z ~ and j E Z+ there is CR,~,~,j > 0 satisfying

1.2 SUPPORTS 15

-(02/Oy])~(z,t) and u = (I)(z, t) satisfies

( 0 2 / O t 2 ) u ( x , y , t ) - E ' ~ = ~ ( O ~ / O y ] ) u ( x , y , t ) - u ( z , y , t ) = O,

(1.9)

Conversely, if u satisfies (1.9), then (O/Oxj + iO/Oyj)u(x,y,t) = 0 ( 1

_~ j _< n) and u(x,y,t) = (~(z,t) For a more general treatment we re- fer to [Wk4] Regarding x as a parameters, (1.9) is a simple hyperbolic Cauchy problem with propagation speed 1 Let r > 0 and u0 E C °O ( R 2~) satisfying supp uo C {(x,y) E R2n; lY] ~- r) Using the Fourier transfor- mation, we can show that

satisfies (1.9) with ~o(x + iy) replaced by uo(x, 9), where fLo(x,,~) = Ty[uo (x, y)](~) and x / - a = iv/-a for a > 0 Moreover, we have

j

]DxDtu(x,y,t)[ <_ ( 2 1 r ) - ~ f I~12- 1 (j-1)/2 exp[it i ~ f ~ - 1]

+( 1) j-1 e x p [ - i t ~ ] IDaho(X, ~)I/2 dE _~ Cjr~(1 + ]ti)e Itl sup ]D~D~uo(x, w)l (1.10)

weR",l~l<n+j

for a E Z ~ and j E Z + Let R > 1 and T > 0, and choose + XR(s) E

C ~ ( R ) so that XR(s) = 1 ( Isl _< 1 ) and supp Xn C {s E R~; Isl < R)

If u0(x, y) = xn(lYl/T)~(x + iy), then, by finite propagation property, we have u(x, y, t) = ~(x + iy, t) for y E R '~ with lYl < Z - Itl This, together with (1.10), gives

L e m m a 1 2 4 Let X be an open subset of R ~ and R > O (i) Assume that U(x, x=+i) is a smooth function defined in a neighborhood of X x {0)

in n '~+' and satisfies (1 - A~,~.+,)U(x, Xn+i) = 0 there Moreover, if V(x, O) and (OU/()Xn+I)(X , O) van be continued analytically to {z E C'~;

Re z • X and IIm z I < R } , then U(x, Xn+l) can be continued analytically

to { ( z , x ~ + l ) • C '~ × R; a e z • X , IIm zl + Ix,~+ll < R ) (ii) / f u •

I)'(XR) satisfies (1 - A ) u = 0 ( in X R ) , then u van be regarded as an analytic function in {z • Cn; IRe z - x I + IIm z I < R for some x • X } ,

~here x R = {x • n~; Ix - vl < n for some v • x )

P r o o f (i) From analytic hypoellipticity it follows t h a t U(x, X n + l ) is real analytic in a neighborhood of X × {0) in R n+l By a s s u m p t i o n s we can regard U(x,O) and (OU/Ox=+l)(x,O) as analytic functions in {z E C'~;

Re z E X and ]Im z[ < R} Let us consider the Cauchy problem

(02/Ot - Au - 1 ) V ( x , y , t ) = O, V(x,y,O) = U(x + iv, O),

(OUlOt)(x, v, o) = ( o u I o x ~ + , ) ( x + iv, o)

(1.11)

This is a simple hyperbolic Cauchy problem with propagation speed 1, regarding x as a p a r a m e t e r So there is a unique solution V ( x , y , t ) of (1.11) which is real analytic in {(x, y, t) E X × R n × R; ]yi+it] < R ) Note

t h a t U(x, t) is analytic and satisfies ((02/Ot 2) - Ay - 1)U(x + iy, t) = 0

if (x + iy, t) belongs to a neighborhood of X x {0} in C ~ x R Therefore,

V ( x , y , t ) = U(x + iy, t) in a n e i g h b o r h o o d of X x {0) in C = × R , which proves t h e assertion (i) (ii) By a s s u m p t i o n u is real analytic in XR Let

us consider the Cauchy problem

{ (a21ot ~ - E~=:O:lOx~ + 1 ) v ( x , t ) = o ,

T h e n we have a unique solution v(z, t) of (1.12) which is real analytic in {(x,t) E X n × R; [x - y[ + It[ < R f o r some y E X } Moreover, we have

v ( x , t ) u ( x + i t e i ) if (x,t) belongs to a n e i g h b o r h o o d of X R x {0}, where

el ( 1 , 0 , , 0 ) E R n A is rotation invariant and, therefore, we can

c o n s t r u c t real analytic function v(x, y), defined in {(x, y) E X R × Rn;

[x - ~[ + [y[ < R for some & E X } , such t h a t VT(X,t) v ( T x , t T e l )

satisfies

1.2 S U P P O R T S 17

in { ( T - I x , t) E R n × R; Ix - ~21 + Ill < R for some ~ E X ) for each orthoganal m a t r i x T of order n, where el = t(1, 0 , - , 0) E R n and x is regarded as a column vector In fact, if T and S are orthogonal matrices

of order n and satisfy T e l = S e l , then we have VT(x,t) = v s ( S - t T z , t )

It is easily seen t h a t v ( z , y ) = u(z) in a complex neighborhood of XR, where z x + iy This proves the assertion (ii) []

We need the following l e m m a (see, e.g., T h e o r e m 7.3.2 and L e m m a 7.3.7 in [Hr5])

L e m m a 1.2 5 Let P ( D ) be a differential operator with constant coeffi- cients, and let v E C ' ( R n) satisfy the following: @, f ( x ) e ix'C) = 0 if f ( x )

is a polynomial and P ( - D ) ( / ( x ) d ~~) = o, where ~ E C n Then there is

a unique solution u E E ' ( R n) of P ( D ) u = v Here £ ' ( R n) denotes the space of distributions in R n with compact supports

P r o p o s i t i o n 1 2 6 Let K be a compact subset of R n and e >_ O We put K ~ := {z E C~; Re z E K and IIm zl < e) T h e n we have the following: (i) I f u E A'(Kt), then u E £~ and supp u C K~ := {x E R n ;

Ix - yl < e for some y E K } Moreover,

X 2 ~I12]

IF(x,xn+l)l _ C(Ixl = + x~+l)-("+2)/4exp[-(Ixl 2 + n+ls J

if Izl + Iz~+~l >> 1, where F;(x, X~+l) is a real analytic function in R n + l \

K~ × [ - e , e ] satisfying U(x, Xn+l) = 7t(u)(x, Xn+l) for Ix~+tl > e (ii) I f

u E A'(C n) N Y t and supp u C K, then u E A'(K ~) and

u(O<~lOx,,+,l~,,+,=o) = f fs(x, x,,+,)(i - A~,x.+,)(x<I>) dxd:r,,+l

for ¢~ E C ° ° ( R n+l) with (1 - Az,zz+,)~b = O, where X E C ~ ( R n+l) satisfies X = 1 near K × [ - e , e] and U(x, z.+~) is a real analytic function

in R n + X \ K × [-e, e] satisfying U(x, x~+l) = 7t(u)(x, xn+l) for Ix~+xl > E

R e m a r k (i) We can prove t h a t u = 0 if u E A ' ( C '~) and supp u = 0 (ii)

If ¢ E C ~ ° ( R TM) satisfies (1 - A~,x,+~)~ = 0 in R TM, then, by L e m m a

1 2 4 , ~ ( x , x~+l) can be regarded as an entire analytic function in C ~+1

P r o o f Following [Hr5], we shall prove the proposition (i) Assume t h a t

u E A ' ( K ~ ) T h e n

U(x,x,~+l) =- 7-l(u)(x, xn+x) = uv(Po(x - y , x ~ + l ) ) for Ixn+ll > e Since Re ( x - y ) 2 > 0 for x E R n \ g e and y E K , z, P ( x - y , xn+l) is analytic in a complex neighborhood of ( R n \ K~,) x K~ x R with respect

A

to ( x , y , x ~ + l ) for 6' > 6, where K~ is defined as in (1.1) Therefore,

U(x, x,~+x) can be e x t e n d e d to a real analytic function defined in ( R n \

K~) × R and

~](x, xn+l) = uy(Po(x - y, xn+x)) for (x,x,,+:) E ( R n \ Ke) x R

It follows t h a t there are R > 0 and a c o m p a c t complex neighborhood w

of K~ such t h a t Po(x - y, xn+x) is analytic in a neighborhood of w with respect to y if Ix] + Ix,~+al _> R This yields, with some c o n s t a n t C,

I~](x,x.+l)l << C s u p l P o ( x - y,x.+l)] i f l x l + l x + l ] _ > R

y e w

On the other hand, we have

2 /1/21

IP(x - y,x,~+,)l < C~,,R(lx - yl 2 + x~+,)-(n+2)/4exp[-(Ixl 2 + ~,,+aJ J

if y E w and Ixl + Ix,+1] >_ R This proves the assertion (i) (ii) Let

X E C ~ ( R ~+1) be a function satisfying X = 1 near K × [ - e , e ] , and let

E C ~ ( R TM) satisfy (1 - A~,x,+l)¢ = 0 T h e n the integral

u,(q0) = f U(1 - A~,~,+,)(x~)dxdxn+l

for qo E A, where @ is a unique solution of (1.8) T h e n It follows from

L e m m a 1.2.3 t h a t

lul(v)l < cR, sup I:(x + iu)l

xEK~,Iy[<R(~+~)

1.2 S U P P O R T S 19

for any R > 1, where ~ > 0 This yields U 1 E A'(K~) For a fixed y E R n

Po(x - y, X~+l) is a distribution of (x, x=+:) and we have

to a sufficiently small complex neighborhood of K~ By L e m m a 1.2.4 we

m a y assume t h a t q~ is entire analytic Therefore, it follows from L e m m a 1.1.1 t h a t Ul o p e r a t e s on (6Q(I)/0XnT1) (x, 0) and

U 1 ((O¢~/COX,,+I)(X, 0))

= uly ( f Po(x - y,x,~+i)(1- ,~,x,+i)(X'~) dxdz,~+l)

A p p r o x i m a t i n g the above integral by its Riemann sum, we can show that

Ul((O~/OXn+I) (X, 0)) : / Vl (x, Xn+l)(1 Az,x=+, )(X ~ ) dxdxn+l,

where UI(;T,Xn+I) = ~(?/1)(~g, Xn+l) for I~+11 > and Ul(X, Xn+l) is

a real analytic function in R n+l \ K , × [ E,¢] Here we have used the assertion (i) We may assume t h a t X is an even function with respect to x~+l Then we have

H ( 1 - A z , ~ + I ) ( X ¢ ) d x d x n + l = O,

where H -= U1 - U In fact, p u t t i n g q): (x, xn+l) = ( ¢ ( x , xn+l) - ¢ ( x , - Xn+l))/2, we have

(1 - Az,x,+l)@l = 0,

• l(X,0) =0, ( a , r l l O x n + l ) ( x , o ) =

ul ((O¢:/Oxn+l)(x, 0)) : [ U(1 - A~,~,+ 1 )(X¢1) dxdxn+l,

J since U is an odd function with respect to xn+: Choose X1 E C~°(R n+l)

so t h a t X = 1 in supp Xl and Xl = 1 near K~ x [-¢,¢], and p u t H I =

( 1 - x i ) H E C m ( R n + ' ) Then we have v G ( l - A x , x n + , ) H 1 E CF(R"+')

and

/ ( ( I - Az,z,+,)Hi)@ dxdxn+l = 0-

From Lemma 1.2.5 and hypoellipticity of ( 1 - A,,,,,+,) i t follows that

there is f E C,OO(Rn+l) satisfying ( 1 - f = v Since (1 -

Az,xn+l)(H1 - f) = 0 , H1 - f is analytic and H = H1 - f outside

implies that H1 - f G 0 , i.e., Ul = X ( u ) for Ixn+ll > E and ul = u , which proves the assertion (ii)

Lemma 1.2 7 Let X be a n open subset of Rn and E 2 0 Assume that u E S: and v E S t , and represent % ( U ) ( X , X ~ + ~ ) or its analytic continuation by U ( x , x,+~) (i) supp ( u - v) n X = 0 if and only if

U ( x , x,+l) can be continued analytically from Rn x (R\[-E, E ] ) t o X x (R\

( 0 ) ) and U ( X , X , + ~ ) + v ( x ) / 2 i n V t ( X ) as xn+l 4 0 (ii) If v E C m ( X ) and supp ( u - v ) n X = 0 , then U ( x , xn+1) can be regarded as a function

i n C m ( X x [O,oo)) and i n C m ( X x (-m,O]), and U ( x , f O ) = f v ( x ) / 2 for x E X If v is real analytic i n X , then U ( x , xn+1) can be continued analytically t o a neighborhood of X x [0, co) and one of X x (-oo,O] (iii)

Assume that X is bounded and that supp u n X 6 = 0, where b > 0 and

Xs denotes the &-neighborhood of X Then, U ( x , x,+~) can be continued analytically t o the set { ( z , zn+1) E [Re z-xl+l(Im z , Im ~ ~ + ~ ) l < 6

for some x E X ) I n particular, for any p > 0 there is C > 0 such that

if x E X and -p < xn+1 < p

Proof V ( x , x,+l) X ( v ) ( x , z,+l) can be regarded as a real analytic function in Rn x ( R \ ( 0 ) ) (i) First assume that supp (u - v ) n X = 0

Then U ( x , xn+1) - V ( x , xn+1) can be extended to a real analytic function

in X x R, which is an odd function with repect t o xn+l So U ( X , X , + ~ )

is real analytic in X x ( R \ ( 0 ) ) and U ( x , xn+1) + v ( x ) / 2 in V t ( X ) as

xn+l 4 0 , since V ( x , x,+~) + v ( x ) / 2 in S t as xn+l 4 0 Next assume that

U ( x , xn+l) is real analytic in X x (R\{O)) and that U ( x , xn+1) + v ( x ) / 2

in V t ( X ) as xn+l 4 0 P u t W ( x , x , + ~ ) = U ( x , x,+~) - V ( x , x,+~) Then

we have W ( x , x,+l) + 0 in D t ( X ) as x,+l 4 0 Moreover, we have ( 1 -

Az,xn+1)W(xixn+1) = 0 in X x ( R \ {O)), W ( x 1 -xn+l) = - W ( X , xn+l)

for x E X and xn+l > 0 By Lemma 1.2.2 W ( x , xn+1) can be extended

t o a real analytic in X x Rn, which gives supp ( u - v ) n X = 0 (ii)

NNe-N({) -N for x + l > 0 and N E N , taking g = max{u, I~1 + I~1 +

j + n + 1}, we have

[(y)~'n~F(x, xn+l,y)[ <_ C,~,j,,~,~(~v, ~b)

for x,y E R n and Xn+~ > O, i.e., {F(x, xn+i,y)}xen",z,,+~>_o is a bounded subset of S(R~) We can also see that { (O/Oxk)F(x, x~+i, y)}~eR~,~.+,_>0 ( 1 < k < n + l ) a r e bounded subsets of S(R~) Therefore, V2~,j(X, Xn+I)E

C ° & n x [0, oc)), which gives V(x, xn+l) E C°°(X x [0, oc)) By as- sumption we have U(x, xn+,) E C°°(X × [0, oc)) Similarly, we have

U(x,x,~+a) E C°°(X × ( - o c , 0 ] ) It is obvious that U(x, +O) = +v(x)/2

for x E X Moreover, V(x, x~+l) satisfies

( 1 - Az,~.+,)V(x, xn+l) = 0 in X x [0, oc),

v(~, +0) = v(x)/2, (OVlO~,~+~)(~, +o) = -(D>v(x)/2 in X

(1.15)

It is well-known t h a t (D)v is analytic in X if v ( E S') is analytic in

X We shall prove this fact ( analytic pseudolocality) for general ana- lytic pseudodifferential operators in T h e o r e m 2 6 5 Now assume t h a t

v is analytic in X Apply the Cauchy-Kowalevsky theorem to (1.15),

we can see t h a t V(x,x,~+l) can be extended to a real analytic func- tion in a neighborhood of X × [0, oc) This proves the assertion (ii) (iii) By assumption U(x, Xn+l) is real analytic in X~ x R and satis- fies ( 1 - A,,,,,+l)U(x,x,,+l ) = 0 So, from L e m m a 1.2.4 it follows

t h a t U(x,x,~+l) can be continued analytically to {(z,z~+l) E C ~ + l ; IRe z - x I + I(Im z, im Zn+i)l < (f for some x E X} (1.14) easily fol-

D e f i n i t i o n 1 2 8 (i) Let ¢ ~_ 0, u E ~'~ and x ° E R ~ We say t h a t

u is analytic at x ° if H(u)(x,x,~+~) can be continued analytically from

R ~ x (E, oo) to a neighborhood of {x °} x [0, ¢] in R ~+1 (ii) For u E To

we define

sing supp u : - {x E R n ; u is not analytic at x}

We give some remarks on supp u One can not expect t h a t supp u C

K if UN -+ u in S [ and supp UN C K In fact, let u E A~(Cn), and put

as = u(z '~) for a E Z n + T h e n there are C > 0 and A > 0 such t h a t

la,~ ] ~_ C A Ic'l for a E Z~_ Let ~ > v/-~A P u t t i n g

since I~1 + ' ' " + I~,~1 < ',~1~1 and I~lJe-~(~) _< (jlC~e)) j for j e g Here

( E l)'(Rn)) denotes Dirac's delta function on R " Therefore, {UN} is convergent in S[ For f E Soo C 4, ~_,,~ zC'(O~ f)(O)/a! converges locally uniformly to f ( z ) in C ~ So we have

u ( f ) ( = ( u , f } ) = ~_,(a,~(-O:~)'~/a!,f}

o~

for f E Soo This yields u g + u in S [ as N + oo It is obvious t h a t supp u y C {0}

We shall also need an existence t h e o r e m for elliptic differential oper- ators with constant coefficients ( see, e.g., T h e o r e m 4.4.6 in [Hr5])

P r o p o s i t i o n 1 3 2 Let P ( D ) be an elliptic differential operator with constant coeJ~cients, and let X be an open subset of R n Then, for any

f E I)'(X) there exists u E Z)~(X) such that P ( n ) u = f in X

T h e o r e m 1 3 3 Let K be a compact subset of R n, ~ >_ 0 and u E

~ Then there is v E zU(K ~) such that supp v C K and supp (u -

v) C R'~ \ K , where K ~ is defined as in Proposition 1 2 6 Moreover,

if vl E A'(K ~) satisfies supp V 1 C K and supp (u - vl) C R " \ K , then

supp ( v - Vl) (~ OK

R e m a r k By T h e o r e m 1.3.3 one can define t h e restriction m a p from T,

to A ' ( K * ) / { u E .A'(K*); supp u C OK}

P r o o f Following [Hr5], we shall prove the theorem From L e m m a 1.3.1

it follows t h a t there is ¢ E C °O ( R T M \ O K x [ - 6 , 6]) satisfying ¢(x, x~+l) =

P u t U(x, x , + l ) = H(u)(x, x,~+l) for Ix~+ll > 6 T h e n CU can be regarded

as a function in Coo(R T M \ K x [-6,6]) Moreover, (1 - A , , , + I ) ( ¢ U ) can be regarded as a function in Coo(R n+I \ OK x [ - ¢ , ¢ ] ) and satisfies

(1 - Ax,~.+,)(¢U) : 0 near ( R '~ \ OK) x [-6,6],

noting t h a t (1 - Az,x,+,)U = 0 for [Xn+l[ > 6 Since 1 - Ax,~.+, is elliptic, it follows from Proposition 1.3.2 t h a t there is f E Coo(R n+l \

OK x [ - e , e ] ) such t h a t f ( x , xn+l) : - f ( x , - x n + i ) and

(1 - A = , z , + , ) f = (1 - A z , x , + l ) ( ¢ U ) in R n+l \ OK x [-~,6]

P u t C / = CU - f E C ° ° ( R n+l \ K x [-~,E]) T h e n we have

(1 - AX,Xn.J,.1)~(/r = 0 in R ~+1 \ K x [ - e , ~ ] Regarding (1 - ¢)V as a function in C ° ° ( R ~+1 \ ( R ~ \ K ) x [-~,~]), we

have V - Y • C ° ° ( R n+l \ ( R n \ K ) x I •,El) and

( 1 - A ~ , ~ , + I ) ( U - ~ " ) - - 0 i n R n + t \ ( R ' ~ \ g ) x [ - g , ~ ]

Define v : A ~ ~ ~-+ v(~) • C by

where X is a function in C~ ° ( R n+l) satisfying X = 1 near K x [-E, El and (I)

is a unique solution of (1.8) T h e n we can prove v • ~4'(g ~) by t h e same

a r g u m e n t as in the proof of Proposition 1 2 6 P u t V 7/(v)(x, Xn+l) for ]Xn+l] > c and H V - V Applying the same a r g u m e n t as in the proof

of Proposition 1 2 6 , we can regard H as a function in C°°(R'~+I), which satisfies ( 1 - A ~ , ~ , + I ) H = 0 Since U - V = U - V 4 - H for Ix,~+l] > ~, U - V

can be continued analytically to a function defined in R n+l \ ( ( R n \ K ) x [-E, El) This gives supp (u - v) C R ~ \ g Y = V - H can be continued analytically to R ~+1 \ K x [ - ~ , c] and, therefore, supp v C K The second part of t h e t h e o r e m is obvious, since Y - 7/(Vl) = ( U - 7/(Vl)) - ( U - V)

[]

We denote A ' ( K ) : {u • A ' ( C ~) (3 J~; supp u C K } for a closed subset K of R ~ and E • R

T h e o r e m 1 3 4 Let K1 and K2 be compact subsets of R "~ and ~ > O I f

u • T~ and supp u C K1 U K2, then there are ul • A ' ( K 1 ) and u2 • T~ such that supp u2 C K2 and u - ul 4- u2

P r o o f Assume t h a t u • ~'~ and supp u C K I U K 2 , and put U(x,x,~+l) = 7/(u)(x,x,~+l) for IX~+ll > ~ T h e n V can be continued analytically to

R n+l \ ( g l U g 2 ) x I - v , v] and satisfies ( 1 - Ax,x,+~)V(x,x,~+t) = 0 there

1.3 L O C A L I Z A T I O N 25

and (1 - A,,~.+,)((1 - ¢)U) can be regarded as functions in C ° ° ( R '~+1 \

(K1 71/£2) × [-6, 6]) and satisfies

(1 - A ~ , ~ + I ) ( ¢ U ) = 0, (1 - A~,~.+I)((1 - ¢ ) U ) = 0

near { (K1 k / K 2 ) \ ( g l N g 2 ) } x [-e,¢] From Proposition 1.3.2 it follows

t h a t there is v E C°~(R n+l \ (K1 ¢3 K2) x [-e,e]) such that v(x,x,~+l) =

V(X, Xn+l) and

( 1 - A~,~.+~)v = ( 1 - A~,~.+~)(¢U) in R T M \ (KI N K2) × [-~,~]

P u t

81 = C V - v, 82 = (1 - ¢ ) U + v, where we regard Uj as a function in C ° ° ( R n+l \ Kj x [-e,e]) ( j = 1,2)

Then,

(1 - A ~ , ~ + , ) S j = 0 in R T M \ Kj x [ - e , e ] ( j = 1,2)

Define Ul : A 9 ~ ~-~ Ul(~0) E C by

Ul(~O) = f 8 1 ( 1 - Ax,xn+l)()(.~) dxdXn+l,

where X is a function in C ~ ( R TM) satisfying X = 1 near K1 × [ - e , e ] and

is a unique solution of (1.8) By the same argument as in the proof

of Proposition 1 2 6 , we have Ul E .A'(K[) Moreover, applying the

same argument as in the proof of Proposition 1 2 6 , we can regard H _= 81 7-~(Ul) as a function in C°~(R'~+I), which satisfies ( 1 - A ~ , ~ , + , ) H = 0 This implies t h a t s u p p u l C K1 and supp ( u - ul) C K2, since U -

H ( u l ) = U - 8 1 + H = 8 2 + H f o r l x ~ + l l > e a n d 8 2 + H • C ~ ( R ~ + 1 \

C o r o l l a r y 1 3 5 Let X1, X2 and X be bounded open subsets of R n such that X = X1 U X2, and let e > O Assume that uj • Y~ ( j = 1, 2) satisfy

supp uj C X -j ( j = 1, 2) and supp (ul - u2) 71 (XI N X2) = 0 Then there

is u • A'(-X) such that supp (u - uj) 71Xj = 0 ( j = 1, 2)

T h e o r e m 1 3 6 Let K1, K2 and K be compact subsets of R '~ such that K1 U K2 C K Then, for any u E To with sing s u p p u C K1 U K2 there are ul E A~(K) and us E To such that u =- Ul + u2 and sing supp uj C It~ u i ) g ( j = 1,2) Moreover, 7i(ul) can be continued analytically from

R ~ x (0, oo) to a neighborhood of (K2 \ g l ) x R \ OK x {0}

R e m a r k In the above t h e o r e m one can replace K1 with K1 \ int(K2)

T h e n one can improve the result on analytic continuation of 7/(Ul)

P r o o f Assume t h a t u E 5ro and s i n g s u p p u C K1 U K 2 , and put

U(x,x,~+l) = 7t(u)(x, xn+l) for X~+l • 0 T h e n there is an open neigh-

b o r h o o d f~0 of R n x [0, oe) \ (K1U g 2 ) x {0} such t h a t U(x, xn+l) can be

continued analytically to 12o We m a y assume t h a t 12oN (K1MK2) x {0} q} By L e m m a 1.3.1 with F0 = K~ x R and F1 = K1 x {0}, we can choose

n e a r (/~'1 \/4"2) X {0}

T h e n ¢1U ( resp (1-(~I)U) c a n be regarded as a function in C°° (~'~oU~~l) ( resp C°°(12o U f~2)), where 121 = R TM \ ( ( g l r3 K2) x {0) u s u p p ¢1) and ~2 = R TM \ ( ( g l M K2) x {0) U s u p p (1 - ¢1)) Moreover, (1 - Ax,x,+~)(¢lU) can be regarded as a function in C°°(120 U ~1 U ~2) and satisfies

- V l ( x , - X n + l ) ( Xn+l < 0)

T h e n ¢2V can be regarded as a function in C ° ° ( R T M \ K × {0}) More- over, (1 - A~,x.+,)(¢2Y ) can be regarded as a function in C ° ° ( R ~+1 \

OK × {0}) Similarly, there is v2 E C ° ° ( R ~+1 \ OK × {0)) such t h a t

v2(x,x~+l) = - v 2 ( x , - x n + l ) and

(1 - A~,~.+,)v2 (1 - A~,~.+I)(¢2V ) in R ~+1 \ OK x {0} Applying the same a r g u m e n t as in the proof of Proposition 1 2 6 , we can define Ul E A I ( K ) by

Ul(~) = f ~rl (1 - Az,zn+I)(X~) dxdxn+l for ~ E A,

where U1 = ¢ 2 V - v 2 E C °° ( R n+l \ K x {0}), X is a function in C ~ ( R TM) satisfying X = 1 near K x {0} and (I) is a unique solution of (1.8) More- over, H U, - 7/(Ul) can be regarded as a function in C ° ~ ( R ~+1) and satisfies (1 - A~,~.+,)H 0 in R ~+1 Since 71(ul) = ¢2V - v2 - H

¢2V1 - v2 - H for Xn+l > 0, H(Ul) can be continued analytically from

R ~ x (0, oc) to a neighborhood of ( R ~ × [0, oc) \ (OK U g l ) x {0}) U ((K2 \

g l ) x R \ O K x {0}), which gives sing supp Ul C O K U K 1 On the other hand,

U - ~ ( U l ) = U - ¢ 2 V I + v 2 + H = ( 1 - ¢ 2 ) U + ¢ 2 V 2 + v 2 + H for Zn+ 1 > 0

and U - "]-[(Ul) can be continued analytically from R n × (0, o¢) to a neighborhood of R ~ × [0, co) \ (OK U g 2 ) × {0}, since 1 - ¢2 = 0 in a neighborhood of ( g \ OK) × {0} in R TM \ OK × {0} This implies t h a t

C o r o l l a r y 1 3 7 Let X1, X2 and X be bounded open subsets of R ~ such that X j C X ( j = 1, 2 ) Assume that uj E -~o and sing supp uj C X j

( j = 1,2 ), andsing supp ( u I - u 2 ) N ( X 1 N X 2 ) = 0, Then there is u E ~'o such that sing supp (u - uj) N X j = 0 ( j = 1,2 ) and sing supp u C

X1 (J X2 U OX Moreover, ?{(u - ul) can be continued analytically from

R n x (0, oc) to a neighborhood o f { X 1 \(X2u(OX1NOX2))} × R \ O X x {0)

P r o o f It is obvious t h a t

sing supp (ul - u2) C (X2 \ X1) U (X1 \ X2)

From T h e o r e m 1.3.6 there are Vl C A t ( F ) and v2 • 3r0 such t h a t Ul -

u 2 = V 1 132, sing s u p p 131 C (-X2\X1)UOX and sing supp v2 C ( X I \ X 2 ) U

OX Moreover, 7/(Vl) can be continued analytically from R " x (0, o~) to

a neighborhood of {X1 \ (X2 U (OXl M OX2))} x R \ OX × {0} Therefore,

p u t t i n g u = Ul - Vl ( = u2 - v2), we have u • -To and sing supp u C

X 1 U X 2 U O X Moreover, we have sing supp ( u - u j ) M X j = 0 ( j = 1,2),

We also write B(X) = Bo(X), which is called the space of hyperfunctions

in X (ii) For an open s u b s e t Y of X and u ° E B~(X) the restriction u°iY E B~(Y) of u ° to Y is defined by the residue class [v] of v E ~4~(Y) which satisfies supp ( u - v ) C X \ Y, where the residue class of u E A~(X)

is u ° in Be(X) ( see T h e o r e m 1 3 3 ) (iii) For u ° e Be(X) we define supp u ° := supp u M X , where the residue class of u E ~4~(X) is u ° in

Be(X) (iv) For u ° E 13(X) we define sing supp u ° : sing s u p p u M X , where the residue class of u E A ' ( X ) is u ° in B(X) (v) It follows from the remark of T h e o r e m 1.3.3 t h a t each u in 9r~ uniquely determines

v E B~(X) such t h a t s u p p (u - vl) N X = 0 if the residue class in B(X)

of vl E A~(X) is v We also call v the restriction of u to X , and denote

U I x = V

R e m a r k (i) It is obvious t h a t supp u ° and sing supp u ° are well-defined (ii) If u E Be(X) and Y C X is open, then s u p p ulY = supp u M Y Moreover, sing s u p p uiY = sing supp u M Y if u e B(X) (iii) B~(X) can

be also defined by .A~(R'*)/A~e(R '~ \ X) (iv) For u • A~e(X) uix ( • I3~(X)) is the residue class of u

We shall define the sheaf B~ and prove t h a t the sheaf Be is a flabby sheaf ( see Definition 1.4 6 and T h e o r e m 1.4.8 below) In doing so, we need the following propositions

1.4 H Y P E R F U N C T I O N S 29

P r o p o s i t i o n 1 4 2 Let P(D) be an elliptic differential operator with constant coe]flcients, and let X and Y be open subsets of R n such that

Y C X , and K =O if X \ Y = F U K, F N K =O, F is closed in X and

K is compact I f u E C°°(Y) satisfies P ( D ) u = 0 in Y , then there is a sequence {u3) C C ~ ( X ) such that P ( D ) u j = 0 in X and ujly -+ u in C¢¢(Y), i.e., D"(uilY ) ~ u uniformly in every compact subset of Y for any t~ E Z ~ , where ujly denotes the restriction of uj to Y

The above proposition is an extension of the Runge approximation theorem ( see, e.g., Theorem 4.4.5 in [Hrh])

P r o p o s i t i o n 1 4 3 Let Ko and K be compact subsets of R n with Ko C

K, and let e >_ 0 and {ej} be a sequence in R such that ej $ e Assume that (i) uj E Y ~ ( j E N ), (ii) for any neighborhood ll of K there

is jo E N such that s u p p u j C Ll for j >_ jo, and that (iii) for any neighborhood llo of Ko there is Jl E N such that supp (uj - uk) C blo for

j, k > jx Then there is u E A ' ( K ) ( C Y~) such that for any neighborhood ldo of Ko there is j2 E N satisfying supp ( u - uj) C l, to for j >_ j2 Moreover, if v has the same properties as u, then supp (u - v) C K0

P r o o f Following [Hrh], we shall prove the proposition We may assume without loss of generality t h a t ( i f uj E ~-~ and supp uj C K~j_~ ( j =

1, 2,-.-), where g ~ = {x E R~; Ix - Yl -< 5 for some y E g ) , and t h a t (ii)' supp (uj - uk) C (Ko)Ej-~ if k _> j , modifying {ej} and omitting the first

several terms from {uj) if necessary Let Uj(x, xn+l) be an real analytic

continuation of 7t(uj)(x, xn+l) ( j E N ) Then, by the assumption (i)',

we have Uj E C ~ ( R '~+1 \ KE~_~ × [-ej,ej]) and

(1 - A ~ , x + , ) U 5 = 0 in R T M \ K~j_E x [ - e j , ej]

By the assumption (ii)' U j - Uk can be regarded as a function in C ~ ( R n+x

\(Ko),~-E x [ - e j , e / ] ) and satisfies (1 - A~:,~:,,+,)(Uj - Uk) = 0 there if

k > j Now we can apply Proposition 1.4.2 with P(D) = (1 - A~,x,+,),

X = R n+l \ K0 × [ - e , e ] and Y = R TM \ (K0)Ej-~ × [ - e j , e j ] In fact, assume that there are a nonvoid colsed subset F1 of R n+l \ K0 x [ - e , e ]

and a nonvoid compact subset F2 of R n+l satisfying F1 I"1 F2 = 0 and F1 t.J F2 = (K0)~,-~ × [-gj,g'j] \ Ko x [ - e , e ] Then F1 IJ Ko x [ - e , e ]

is colsed in R n+t and, therefore, compact By assumptions there are

(x o, o x~+l) E F2 and (y0, 0 Y~+I) E K0 × [ - e , e] such t h a t I x° y° I < e j - and IXn+l 0 Yn+ll o <_ ~j C P u t Q = {(z, Xn+l) E Rn+l; I x - Y°l < e j - ¢ and IXn+l -Y=+I[ -< e / - e } ( C (Ko)~¢-~ × [ - ¢ j , e i ] ) Then Q is connected o

and Q = {Q o (F1 U K0 × [ - e , e ] ) } U (Q f3 F2) This is a contradiction,

since (y°,y°+l) E Qn(FiUKo × [-e,E]) and (x ,Xn+l) E Q (1F2 P u t , 0 0

f o r j E N ,

M j = { ( z , x , + l ) E R " + I ; I ( x , x , + l ) l < j and I x - y i + l x , + l - y - + l I

_> 3(ej - e ) for any (Y,Y~+I) E K0 × [-~,~]},

which is a c o m p a c t subset of R n+l \ (Ko)~-~ > [-ej,ej] Moreover,

M j t ( R~+I \ K0 x [ - e , e]) as j + oc From Proposition 1.4.2 there

is a sequence {Ivy} C C ° ° ( R n+l \ K0 > [ - e , e ] ) such t h a t I/~(X, Xn+I) =

- ~ ( x , - X n + l ) , (1 - A~,~,,+I)V j = 0 in R n+l \ go X [ ~,~] and IUj+I -

and U E C ~ ( R '~+~ \ K × [-¢,~]) A p p l y i n g the same a r g u m e n t as in the

p r o o f of Proposition 1 2 6 , we can define u E A ' ( K ~) by

u(~) = f U(1 - A~,~,+~)(X(I )) dxdx,~+l for ~ E A,

where X is a ruction in C ~ ( R n+l) satisfying X = 1 near K × [-~,¢] and (I) is a unique solution of (1.8) Moreover, U - ?/(u) can be continued analytically to R T M and we have s u p p u C K Since t h e infinite series

P r o p o s i t i o n 1.4 4 Let X be a bounded open subset of R '~, and let

{X.~}~eA be a family of open sets in X such that X = U;~eAX;~ A s s u m e that ~ > 0 and u~ E B~(X~) ( A E A) satisfy

u~]xxnx, = u~,lxxnx, for every A,# E A (1.16)

Then there is a unique u E Be(X) such that uIx x = u~ for every A E A

1.4 H Y P E R F U N C T I O N S 31

P r o o f Uniqueness of u E Be(X) is obvious In order to prove the propo- sition it suffices to show the following: If v~ E ~-e, supp v~ C X~ and supp (v~ - v~) M (X~ n X , ) = q} for any A,# E A, then there is v • }'e satisfying

supp v C X and supp (v - v~) O X~ 0 for any A E A (1.17)

If {X~}~eA = {X1, X2}, then the above assertion follows from Corollary

1 3 5 Next assume t h a t A = N Then there are wj • Te ( j • N ) such

t h a t

J supp wj C U x k and supp (wj - Vk) M Xk = q} for 1 _< k _< j

for j > k _> 1 If/4o is an open neighborhood of OX, then X \ L/0 is a

c o m p a c t subset of X and there is k E N such t h a t Y \ ( U k = l Xe) C /a(O •

Applying Proposition 1.4.3 with K = X and K0 = OX, we can show

t h a t there is v E ice such t h a t supp v C X , and for any neighborhood I;0 of OX there is jo E N satisfying supp (v - wj) C ]30 for j _~ jo Let

k E N and x ° E Xk T h e n there is j0 E N such t h a t x ° ~ supp (v - wj)

for j ~_ j0 On t h e o t h e r hand, supp ( w j - v k ) M X k = 0 for j _> k This shows t h a t supp (V Vk) n X k = 0 Now we assume t h a t A is uncountable Since X has the Linderhf property, there is {Aj}je y C A

such t h a t X = Uj~ I x ~ j So there is v E }'e satisfying supp v C X and supp (v - v ~ ) M X~j = 0 ( j E N ) Then, we have (1.17) In fact, by t h e same a r g u m e n t as the above, for a fixed A E A there is 9 E ~'e satisfying supp 9 C X , supp ( 9 - v ~ j ) M X ~ = 0 ( j E N ) and supp ( b - v ~ ) M X ~ = q}

It is obvious t h a t supp (v - ~,) C O X This gives (1.17) O

D e f i n i t i o n 1.4 5 Let X be an open subset of R ~ and E _~ 0, and let (X~}~e^ be a family of bounded open subsets of X such t h a t X = U;~eAX ~ We define Be(X,{X~}~eA ) as the collection of {u~}~eA sat- isfying u~ E 13e(X:~) and (1.16) We identify an element {u~}~e^ E

Be(X, {XA}AeA) with {vu}ue M • Be(X, {'XU}#EM) if

U~[x~n~ ' = Vu]x~n.~ ' for any A • A and # • M

Then, with this identification, we define Be(X) as Be(X, {X~}:~eh ) We also write B ( X ) = Bo(X)

R e m a r k (i) If X is a b o u n d e d open subset of R n, then it follows from Proposition 1.4.4 t h a t B~(X) can be identified with .A~(X)/.A~(OX) So,

in the above definition each element {vu)ueM E B~(X, { ) f u } u e / ) deter- mines uniquely an element {u~}~en E B~(X, {X~}~eA)- Moreover, each element of ~'~ determines uniquely an element of /~e(Rn) (ii) Opera- tions on B~(X) can be naturally defined by those on B~(X) O In par- ticular, for an open subset Y of X and u {u~} E B~(X) we define U[y = {'ttAlx~nV } E B~(Y) and supp u = U~ s u p p u~

Let us define several operations in A'(R'~) We have already defined differentiation and multiplication:

(Dju)(~) = -u(Dj~), (au)(~) = u(aT)

if K is a c o m p a c t subset of R n, u E A'(K), ~ E ,4 and a is analytic near

K It is obvious t h a t s u p p Dju C supp u and supp au C supp u The tensor p r o d u c t u ® v of u E A'(R n) and v E A'(R m) is defined by

for every polynomial ~ in C '~+m Then u®v can be regarded as an element

of .A'(Rn+m) Moreover, we have J'[u @ v] = fi(~)fi(rl) and supp u ® v = supp u × s u p p v Let K be a c o m p a c t subset of R ~, and let f be a real analytic mapping of an open subset w of R n on an open neighborhood /4 of K We assume t h a t f is a diffeomorphism Then we can define the pull-back f*u E A ' ( f - l ( g ) ) of u E A ' ( g ) by

(f*u)(~2) = u ( ( ~ o h ) l d e t h ' D for ~o E ,4, where h = f - 1 :/4 ~ W and h' denotes the differential of h ( the Jacobian matrix of h) T h e above operations on A ' ( R ~) can be easily extended to those on B(X) This enables us to define B(X) for a real analytic manifold

X in the s t a n d a r d way Let X be a real analytic manifold, and let/C be an atlas for X, i.e., let ]C be a family of h o m e o m o r p h i s m s a of open subsets

X s of X onto open subsets )(~ of R = such t h a t X = U~eK:X~ and the mapping

/%'o/%-1 : /%(Xa N X n , ) +/%'(Xt~NXt¢,)

is real analytic for every a, a' E K; Then we define B ( X ) as the collection

of {u~}~e~: satisfying u~ E B()(~) and

- - 1 *

for ~, ~' E/C