classical microlocal analysis in the space of hyperfunctions

Lecture Notes in Mathematics
Editors:
A. Dold, Heidelberg
F. Takens, Groningen
B. Teissier, Paris
1737
Springer
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan
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Seiichiro Wakabayashi
Classical Microlocal
Analysis in the Space
of Hyperfunctions
Springer
Author
Seiichiro Wakabayashi
Institute of Mathematics
University of Tsukuba
Tsukuba-shi, Ibaraki 305-8571, Japan
E-mail: wkbysh @ math.tsukuba.ac.jp
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Die Deutsche Bibliothek - CIP-Einlaeitsaufnahme
Wakabayashi, Seiichiro:

Classical microloca analysis in the space of hyperfunctions /
Seiichiro Wakabayashi. - Berlin ; Heidelberg ; New York ; Barcelona ;
Hong Kong ; London ; Milan ; Pads ; Singapore ; Tokyo : Springer,
2000
(Lecture notes in mathematics ; 1737)
ISBN 3-540-67603- l
Mathematics Subject Classification (2000): 35-02, 35S05, 35S30, 35A27, 35A20,
35A07, 35HI0, 35A21
ISSN 0075- 8434
ISBN 3-540-67603-1 Springer-Verlag Berlin Heidelberg New York
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Preface
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 treatment there is considably different from ours. Although
we think that 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 methods 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 map real analytic functions and hyperfunctions to
real analytic functions and hyperfunctions, respectively.
In this lecture notes we attempt 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 methods 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-
vi
suits obtained in joint work with Prof. Kajitani, and I would like to thank

him for many useful discussions.
Contents
1 Hyperfunctions
3
4
5
1.1 Function spaces 5
1.2 Supports 13
1.3 Localization 23
1.4 Hyperfunctions 28
1.5 Further applications of the Runge approximation theorem • 34
Basic
calculus of Fourier integral
operators and pseudo-
differential operators 41
2.1 Preliminary lemmas 41
2.2 Symbol classes 52
2.3 Definition of Fourier integral operators 57
2.4 Product formula of Fourier integral operators I 65
2.5 Product formula of Fourier integral operators II 87
2.6 Pseudolocal properties 93
2.7 Pseudodifferential operators in B 107
2.8 Parametrices of elliptic operators 112
Analytic wave front
sets and microfunctions
3.1
3.2
3.3
3.4
3.5

3.6
3.7
115
Analytic wave front sets 115
Action of Fourier integral operators on wave front sets • • • 130
The boundary values of analytic functions 155
Operations on hyperfunctions 165
Hyperfunctions supported by a half-space 183
Microfunctions 192
Formal analytic symbols 201
Microlocal uniqueness 205
4.1 Preliminary lemmas 205
4.2 General results 222
viii
CONTENTS
A
B
4.3 Microhyperbolic operators 231
4.4 Canonical transformation 239
4.5 Hypoellipticity 244
Local
solvability 259
5.1 Preliminaries 259
5.2 Necessary conditions on locM solvability and hypoellipticity 268
5.3 Sufficient conditions on local solvability 272
5.4 Some examples 285
Proofs
of product
formulae
A.1 Proof of Theorem 2.4.4

A.2 Proof of Corollary 2.4.5
A.3 Proof of Theorem 2.4.6
A.4 Proof of Corollary 2.4.7
A.5 Proof of Theorem 2.5.3
295
295
323
328
336
338
A priori estimates 351
B.1 Gru~in operators 351
B.2 A class of operators with double characteristics 355
Introduction
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 that an analytic functional u is carried by a compact 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 that
lu(@l < C~osupb,(z)l for ~ e A.
zEw
We denote
.4' := U .4'(K).
K(::I:R '~
The space A' of analytic functionals carried by R n is very similar to the
space :D t of distributions, particularly to ~F. One can identify ~-t with a
subspace of A r, and anlytic functionals have compact "supports." For

u E A r we can define supp u by supp u = N{K; u E At(K)}, which is
called the support of u. The concept of "support" is relating to restriction
mappings and sheaves 4 t defines the sheaf B of hyperfunctions while C r
does the sheaf l) r of distributions. In order to study 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
2 INTROD UCTION
ands ~. ForeERweput
We introduce the topology to S~ in a standard way. Then 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 the transposed operators tic, tic-1 .
S~' ~ S~' for ¢ _> 0. We rewrite t j- = ~- since tU = 5 on S'. Noting that

~' S_efor _>0, we
e D ¢ define
S_~ := : 1[~_~] for e > 0.
Moreover, SJ 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 that
£'
c
A'
c
£o
c
Fo.
Let V be an open conic subset of
R n x (R n \
{0}). We say that
p(x,~) E
C°°(F) is an analytic symbol in F if
p(x,~)
satisfies the estimates
O~ D~p(x,~) I <_
CAJ"'+l~llo, l!lfll!<~> '~-I~i
for (~,~) ~ r with I~1 > 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,
respectively. However, we can not define
p(x, D) as
an operator on £0 and
9% by (0.1) ifp(z,~) is an analytic symbol in F and F does not coincide

with R n x (R~\{0}). We do not want to abandon (0.1) as the definition of
p(x, D).
So we introduce some symbol classes which contain symbols with
compact supports. We say that a symbol
a(~, y,
71) E
C°~(R ~
x
R n
x
R n)
belongs to
S m~'m2'~'~2 (R, A)
if a(~, y, T/) satisfies
"-"u '-'n ~',, - Clal+l~l+l.~l
x (A/R)I,~1+
I~' I+W I+M (,~)~, -I,~ I+1~' I (r/>,,,2-1"~1+1~2 I
x
exp[(~l(~>
-4-
(~2(~>]
if <¢> _> R(lal +
I/~11)
and (r/> _>
R(lfl21
+ b'l). For
a(,~,y, rl) ~
S ~,'~2'~''~2
(R, A) we define the pseudodifferential operator
a(D~, y, D~)

by
INTROD 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' 8'
-2~+-~1 ~ -~+~2
if
It >_ 2enA
and
¢ <_ 1/R (
see Theorem 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 than usual one. This is
another 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*Rn\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 that 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
INTROD UCTION
functions and hyperfunctions. General criteria on microlocal uniqueness
will be given in Chapter 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 Chapter 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 that 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 that 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 that
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 manner of HSrmander's book [Hr5] in Section 1.4.
We shall prove that 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.
1.1 Function spaces
Let ~ E R, and denote (~) = (1 +
1 12)1/ ,
where (~1,""
,~n) E R n
and
]~[ = ()-~jn__ 1
I~jl2) 1/2.
We define
Here S denotes the Schwartz space. We introduce a family of seminorms
on S~ as follows:
(= sup
)

( l= 0,1, 2, - - -),
where a = (al,-", an) e Z~ ( = (N U {0}) n ),
lal
= Ej~_-I
aj, D~ =
(D6, ,D~,) = -iO~ = -i(O/O~l, ,i)/O~n)
and D~ = D~ ~ D~".
6 CHAPTER 1. HYPERFUNCTIONS
Since D ( =
C~(R n) )
is dense in L, the 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 jr and jr-1 denote the Fourier transformation and the inverse
Fourier transformation on S ( or
S'
), respectively, and ~(~) = jr[u](~).
We introduce the topology in S~ so that jr : ,~¢ ) S¢ is homeomorphic.
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~. Then we can define the
transposed operators t~- and t jr-1 of jr and jr-l, which map 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 that
t jr-1 : ~_~ _~ S-e is homeomorphic. S_~ denotes the dual space of S_¢
for E _> 0. Then we have S_~ = Sr[~_~] C S' C S~ for c _> 0 and jr = t jr
on S( So we write t jr as jr. Note that & is a Fr~chet space with the
topology determined by the seminorms
]ul,s,,t
:= ]jr[u]l~,,t ( g E Z+).
We denote by A the space of entire analytic functions in C

TM.
Let K
be a compact 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 that
I~(~)1
~
C~supl~,(z)l for 9o
E A.
zEw
Put
A
Kc- := {z E C~; IP(z)l _ sup IPI for any polynomial P}.
K
Let ~2 ( C C '~ ) be a domain of holomorphy. We call f~ a Runge domain
if every function in A(12) can be approximated locally uniformly in fl by
polynomials, where A(l)) denotes the space of analytic functions in ~2.
It is known that fl is a Runge domain if and only if Kc- n f~ (E fl for
any K (Z fl, and that K has a fundamental system of neighborho_oods
consisting of Runge domains if K is polynomially convex,
i.e., K = Kc,
( see,
e.g.,
[Hr8]). Here A (Z B means that the closure ~ of A is compact
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 neighborhood of K, then we can define u(qa), approximating qa by
entire functions.
1.1. FUNCTION SPACES 7
Lemma 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.
Remark A direct proof of the second part of the lemma is given in
Proposition 9.1.2 of [Hr5].
Proof Let z ° ~ K,. This implies that IRe z ° - x I + IIm z°l > ¢ for any
x E K. First assume that Re z ° E K. Then, putting
f(z)
= exp[-(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) = Ej~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 that Rez° ¢(K. Choose x ° E K so that IRez °-x°l =
dis({Re z°}, K) ( _ infzeK IRe z ° - x I ), and put
f(z)
= exp[-(z- Re z °- Jim z°lu°)2],
u ° = [Re z °- x°[ -1(Re z °- x°).
Then we have
If(z°)l = 1 and If(z)l
=
exp[h(z)],
(1.3)
where
h(z) =:
]Imz] 2-1Rez-Rez °-lImz°lu°] 2. Let z 1 e K~, and
choose x 1 E K so that IRe z 1 -xl[ dis({Re
zl},g) ( < ¢).
Noting that
A - IRe z ° - x°l + IIm z°l > IRe z I - xll, we have
h(z 1) < (E - IRe z 1 - xll) 2 - (A
-
[Re z I - xl[) 2
= (E- A)(~ + A- 21Re
z 1- xll).
Since A > ¢, we have
h(z 1)
< 0. This, with (1.3), proves the lemma. []
Denote
.4'( = .A'(R n)
) :=

I,.JKG:?.R,~ A'(K), A'(C n) := UKG=C"
A'(K), Soo
:=: N~en S~, • := Ns>~ 8-5 and ~'~ : N~>~ S•. We introduce
the projective., topologies to Soo, & and .7 We note that
.T-I[C~(Rn)] C
Soo and that Soo is dense in S~ and S[ for e E R. Let K be a compact
8 CHAPTER 1. HYPERFUNCTIONS
subset of C ~ and u E
.A'(K).
We can define the Fourier transform fi(~)
of u by
fi(~) (= :Cu](~)) = u~(e-~'~),
where
z. ~ = ~-~d~___l zj~j
for z = (zl, ', z~) E C '~ and ~ = (~1,'",~) E
R ~. It is obvious that fi(~) can be continued analytically to an entire
analytic function in C ~, and that ~2(~) E S-8 ifK C {z E C~; lIm zl < e}
and ~ > e. Since
u(P)
= (P(-D~)fi)(0) for every polynomial P, the
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 < ~}.
Lemma 1.1.2
Let ~ >_ O, and let K be a compact subset of C ~ such
thatg
C {zE C~; Ilmzl_<~). /fu EA'(K),
5 >E andqDES~, then
(u, @ = u@), wher
(., .) denotes
the duality of

S[
and
&.
Remark 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 Kc- 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)
and qo E S~ with 5 > e.
Proof Leth>~,uEA~(K) and qaES~. SinceuES_~cS~/,wehave
(U, ~) : (U z (e-iZ'~),
.T -1 [~](~'))
=
fR n
Uz(e-iZ'~).~-l[~](~)d~ :
fR n
Uz(e-iz'~.~-l[~t9](~) )
d~.
Let w be a neighborhood of K such that w ~ {z • ca; IIm zl < (~ +
5)/2}. Then it is easily seen that
supl f
e-iZ".T-l[qv](,) d~ ,0 as R +
(x),
zeta IJRn\gl n
where
~R = {~ E Rn;
I jl
< R
(
1 < j _

n)}.
Therefore, we have
lim
u~(ff~
e-iZ'~'-l[~](~)d~)-~-u(~).
R ~oo R
Since the Riemann sum of the integral far e-i~'~'-~[V](~)d~ uniformly
converges to the integral in w, we have
fR- =
This proves the lemma. []
1.1. FUNCTION SPACES 9
Eo(x, x,~+l)
Po(~, ~+l)
Then we have
Let e • R and u • S]. We can define
D~u(= D~ ~ D~"u) :=
9r-x[~fi(~)](x)• S[,
where a = (al, ,an) • Z~( = (N U {0})"), x = (xi, ,x,) E R",
D = (D1,'",
Dn) = -iO = -i(OlOxl,"', OlOxn).
Following [Hr5], we
define
7-l(u)(x,x,~+l)
:=
(sgnx,~+l)exp[-[x,~+li(D)]u(x)/2
(= (sgn
xn+l)Yr(l[exp[-Ix,~+ll(~)lC~(~)](x)l 2) • S/
for
Xn+l • R \ {0}.
Put

:: .r~.+,)[(1 + I,fl 2 +,¢~+,)-'](~,~.+1),
:=
-
(O/Ox.+x)Eo(x, Xn+l).
Eo(x, zn+~) = (27r)-('~+~)12r-('~-l)12K(,,_,)12(r),
where
r = (x 2 + x~+l) 112
and K.x(r) is a modified Bessel function of the
second kind. It is known that
g~(z) = ~ie~"~/~H(~l)(ei~/~z)
I5
= e-ZC°Shtcosh(At) dt
(largzl < rr/2),
K~(z) = r()~)(z12)-~12 +
o(Izl -~) as Izl -+ 0 for ~ > 0,
go(z)
=
- log z + o(log lzl)
as
Izl-~ 0,
e-z - ~)k(+~)k(2z) -k
g~(z) =
\-2"zz)
k=o k!
(7
+o(1~1-~)]
asiz I +ccif~>Oand largz I <31r/2-~,
where (A)k F(A + k)/r(,~), and that
K~(z)
is analytic in C \ {z • R;

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)
if• _> 0, Ix~+ll > e and u • S],
since
Po(x- y, xn+l) • S~(R~)
for 5 _> 0 and (x,x~+l) • R
TM
with
Ix~+ll > &
10 CHAPTER 1. HYPERFUNCTIONS
Lemma 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-
Ax,xn+t )
V(x, XnTl) : 0 for Xn+ 1 ¢ O, V(x,-~-O) : u(x)/2 in S: and
v(x, x.+l)=
-v(~,-x~+s)
inS" for~+~ ¢ 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
g 6 Z+ such that
Ol
]D~:D~+IU(x,x.+t)I <_ C~j,S(1 + Ixl + Ix~+ll) / exp[-]xn+lI]

if5 > 0 and [X.+l[ > e+ + 5, and
(1-A~,~.+,)U(x,x,+l)=O inR nx (R\[-¢+,e+]). (1.4)
(iii) Ire < O, then u(x) can be continued analytically to {z E C"; Jim z] <
Proof Since
.~x[U(x, x,+l)](~) = (sgn Xn+l) exp[-Ix.+11(~)]~(~)/2,
the assertion (i) is obvious. Note that e-*(~)h(~) E S'. Therefore, there
are g E Z+ and C > 0 such that
for ~ E S. This gives
O/
ID~D3~+IU(X,X.+I)I
= [(exp[-e(~)]fi(~),~DJ+x exp[-(IX,+ll - ¢)(~)]ei~'~/(2(2~r)"))~l
< c.,j,s,~(l +
Izl
+
IZ.+ll)eexp[-lx.+,l]
if
IX~+ll
> ~+ + ~
( > ~÷).
In fact, by Fubini's theorem we can prove
that if 5 > 0 and exp[~(~)]O(~) E S', v(x) is a function and
v(x) (21r)-n(exp[6(~)]~(~¢), exp[ix • ~ - 8(~[)])~.
(1.4) is obvious. If ~ < O, then we have
u(x) = (21r)-n(exp[-e(~r)]fi(~¢), exp[ix- ~ + ~(()])(.
This proves the assertion (iii). []
1.1. FUNCTION SPACES
11
Corollary 1.1.4
Let u E S/,
and put U(x, Xn+l)

:
"]~(U)(X,Xn+I).
Then
(u, V) = 2 lim
(U(x, t), ~o(x))~ for ~ E S~.
(1.5)
t-+TO
Moreover, ilK is a compact subset of R n and u E .A'(K), then
u(~p)( (u,~)) =2 lim
f U(x,t)~(x)dx for~ESoo,
(1.6)
t r+o Jv
where V is a neighborhood of K in R n.
Remark It follows from Lemma 1.1.3 that
7-l(u)(z,t)
is real analytic
in x for t > 0 if u E ~-0. Thus,
(u, q0)=2 lim
f 7-l(u)(x,t)qo(x)dx
foruEg 0andq0ESoo.
t ~ +O J Rn
Proof
By Lemma 1.1.3 (1.5) is obvious. Assume that K is a compact
subset ofR ~, V is aneighborhood of K in R n and u E A'(K). Then
U(x,
x~+1) (=(u(y),
Po(x - y,
x,~+x))u ) can be continued analytically to
R
TM

\K × {0} and, therefore,
U(x,
0) = 0 for x ~ K. Let w be a complex
neighborhood of K such that dis(w, R n \ V) > 0. There are C~ > 0 and
C~,v
> 0 such that
IU(x,t)l < C~suplPo(x - z,t)l < C~,ve -Ixl
zEw
if x E R ~ \ V and 0 < t <_ 1. By Lebesgue's convergence theorem we have
lim
[ U(x,t)q~(x)dx=O
for~ESoo,
t~+o
JRn\V
which gives (1.6). []
Let a(x) E
C~(R ~)
satisfy
[D%(x)l <_ CAl"ll~l!(x)k
for x E R ~, (1.7)
where C > 0, A > 0 and k E R. Then we have the following
Lemma 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.
12
CHAPTER 1. HYPERFUNCTIONS
Proof Let u E `9~. Then
au E ,S
is well-defined and

/ e-iZ'(~-'7)(A'x) -2Ma(x)(A'D,1)2Mct(y) d~Tdx
3C[au](~)
f "T[(a'x) -2Ma(x)](~ - rl)(atDn)2M~(~7)d~l, (2,~)-"
where
(A'Dn) 2 = 1 - Y]j~=t A'2(O/OyJ) ~,
M = max{[(n + k)/2] + 1, 0},
A' = A/(4(1 + v#2)) and [c] denotes the largest integer <_ c for c E R.
Note that
]D~(m'x)-2M I < CM(A/2)HIa[!(A'x) -2M-I~I
( see Lemma 2.1.1 below). Therefore, we have
[D '~ ( (A' x)-2M a(x) )l < 2CCM A['~I
I~]!(A'x)-2M<x)k.
Put i ]~1-2 Z~=I
~jD~j
for ~c E R" \ {0}. Then we have
I.~[(Atx)2M a(x)](5)[ < f [Lg{(A'x)-2Ma(x)}] dx < Ck,A(V/-~A/ISI)Q!
if"~ E R"\{0} and g E Z+" or "~ = 0 and g = 0." It follows from
Lemma 2.1.1 that
_ ~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 the estimate (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 that
au
E J4'(K) can be
defined
by
(au)(~) = u(a~) ( ~ E .A),
if u E A~(K) and a is analytic in a neighborhood of Kc-, where K is a
compact subset of C".
1.2. SUPPORTS 13
1.2 Supports
Definition 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
Rn+a\ K × [-e,e]
such

that
V(x, Xn.4_l) =
•(U)(X, Xn+I) for
IXn+l] > 6").
Remark (i) The definition of supp u does not depend 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) Ifsuppu C K~ ( A • A), then suppu C N~ehK~.
(iv) For u • ~-~ there is a real analytic function U(x,x~+l) in R n+l \
supp u × [-e,¢] such that U(x,x~+l) = 7-l(u)(x, xn+l) for ]Xn+l] > e.
(v) Let u • U0 and x ° • R ~. Then 7t(u)(x, xn+l) can be extended to a
C2-function near (x °, 0) • R
TM
if and only if x ° ~ supp u.
Lemma 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) If u E ,9 ~, supp u coincides with the
distribution support of u, which is the support of u as a distribution.
Proof (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 that (02/Ox~+l)W(x,
±xn+x) e C([+O,~);D'(X)), (O2W/Ox2+l)(x,+O) = 0 in D'(X) and
that

(oW/Ox.+,)(x,
/±,-i
= (OW/OXn+l)(X,±l) - (02W/Ox2n+l)(x, Xn+l)dXn+l
6
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 theorem implies that W(x, x,~+l) E
el(R; D'(X)). Similarly, we have W e C2(R;D'(X)). This gives
(1- Ax,x,+I)W = 0 in D'(X x R). That (1 - A~,x,+~) is analytic
hypoelliptic also follows from the fact that the fundamental solution
14
CHAPTER 1. HYPERFUNCTIONS
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 that we shall prove analytic hypoellipticity of general elliptic op-
erators in Theorem 2.8.1 (ii) Let (z °, 0) E R n+l, and assume that 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 Lemma 1.1.
3 that
~(x)7-l(u)(x,x=+l)(=
~(x)~f(X, XnTI) ) +
~(x)u(x)/2
in
S'as
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 that an open sub-
set X of R = does not meet the distribution support of u, If qo E
C~(X),
then, by Lemma 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 Lemma 1.1.3 and
the assertion (i) we can see that
7-l(u)(x, Xn+l)
can be extended to a real
analytic function in X × R. and that supp u M X = q}. This completes
the proof. [3
Lemma 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
IDgDJ.+142(x,x~+l)l
< CR,s,~,j(t
TM
A- t-J)e Iz"+ll
sup I~(x
+ w + iy)l
lyl<Rt+S, lwl<5
if
(x, Xn+l) E
R n+l and
IXn+lI _< t.
Proof If 42 E C°¢(R n+l) satisfies (1.8), then it follows from analytic
hypoellipticity of (1 - A,,,,+, ) that 42 is real analytic in
R n+l.
On the
other hand, by the Cauchy-Kowalevsky theorem (1.8) has a unique entire
analytic solution which is an analytic continuation of 42. In particular,
(1.8) has a unique solution 42 E C°~(Rn+l). Write z =
x+iy
for x, y E R n,
and let
42(z, xn+1)

be the analytic continuation. Since
(O/O-;~j)42(z,t)
( = (O/Oxj + iO/Oyj)42(z,t)/2) = 0 ( 1 <_ j <_ n), (02/Oz~)42(z,t) =
1.2. SUPPORTS
15
-(02/Oy])~(z,t)
and u = (I)(z, t) satisfies
(02/Ot2)u(x,y,t) - E'~=~(O~/Oy])u(x,y,t) - u(z,y,t) = O,
(1.9)
u(~,u,o) = o, (Ou/Ot)(~,u,o) = ~(~ + iu).
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)[ <_
(21r)-~f I~12- 1
(j-1)/2
exp[it i~f~ - 1]
+( 1) j-1
exp[-it~] 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
]D~D~(~(x,t)]
<
CR,jTn(1 + ]tl)eltl(1 + T-l) ~+j
× sup

]n~n~(x + iy)]
lYI<RT, I~Ign+j
for It I _< T. Applying Cauchy's estimate ( or Cauchy's integral formula),
we have
ID;DJ(~(x,t)I <_ CR,~,~,j(T
TM
+ T-J)e Itl
if(i>O, xER '~,tERandjtl_<T.
sup I~(x + w + iy)l
[]
We shall also need the following lemma concerning analytic continu-
ation of solutions of (1 - Ax,x,+l)u = 0.
16 CHAPTER 1. HYPERFUNCTIONS
Lemma 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; aez • X, IIm zl + Ix,~+ll < R). (ii) /fu •
I)'(XR) satisfies (1 - A)u = 0 ( in XR), 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 xR = {x • n~; Ix - vl < n for some v • x).
Proof (i) From analytic hypoellipticity it follows that U(x,
Xn+l)
is real
analytic in a neighborhood of X × {0) in
R n+l.

By assumptions 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) = (ouIox~+,)(x + iv, o).
(1.11)
This is a simple hyperbolic Cauchy problem with propagation speed 1,
regarding x as a parameter. 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
that 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 neighborhood of X x {0) in C = × R, which
proves the assertion (i). (ii) By assumption u is real analytic in XR. Let
us consider the Cauchy problem
{ (a21ot ~
-
E~=:O:lOx~ + 1)v(x,t)=o,
v(x, o) -~- u(x), (()v/Ot)(x, o) : i(()u/oXl)(X ).
(1.12)
Then we have a unique solution v(z, t) of (1.12) which is real analytic in
{(x,t) E Xn × R; [x - y[ + It[ < Rfor some y E X}. Moreover, we have
v(x,t) u(x+itei) if (x,t) belongs to a neighborhood of XR x {0}, where
el (1,0, ,0) E R n. A is rotation invariant and, therefore, we can
construct real analytic function v(x, y), defined in {(x, y) E XR × Rn;
[x - ~[ + [y[ < R for some & E X}, such that VT(X,t) v(Tx,tTel)
satisfies
{
(o~/ot ~-
Ej\~ o~/o~ + 1)vr(~,t)=o,

VT(X,O) = u(Tx), (OVT/Ot)(x,O) = i(O/OXl)U(Tx)
1.2. SUPPORTS
17
in
{(T-Ix, t) E R n × R; Ix - ~21 + Ill < R
for some ~ E X) for each
orthoganal matrix 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 Tel =
Sel,
then we have
VT(x,t) = vs(S-tTz,t).
It is easily seen that
v(z,y) = u(z)
in a complex neighborhood of XR,
where
z x + iy.
This proves the assertion (ii). []
We need the following lemma (see,
e.g.,
Theorem 7.3.2 and Lemma
7.3.7 in [Hr5]).
Lemma 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.

Proposition 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).
Then
we have the
following:
(i)
If u E .A'(Kt), then u E £~ and
supp u C K~ := {x ERn;
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 Rn+l\
K~ × [-e,e]
satisfying U(x, Xn+l) = 7t(u)(x, Xn+l) for
Ix~+tl > e. (ii)
If
u E .A'(C n) N Yt 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 Rn+X\K × [-e, e] satisfying
U(x, x~+l) =
7t(u)(x, xn+l) for
Ix~+xl > E.
Remark (i) We can prove that 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 Lemma
1.2.4, ~(x, x~+l) can be regarded as an entire analytic function in C ~+1.
Proof Following [Hr5], we shall prove the proposition. (i) Assume that
u E A'(K~).
Then
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\ge
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