AN ABSTRACT OF THE THESIS OF
Kirk Eugene Lancaster for the degree of
in
Mathematics
Title:
presented on
Doctor of Philosophy
August 15, 1980
ON TEMPERATE FUNDAMENTAL SOLUTIONS WITH SUPPORT IN A
NONSALIENT CONE
A
Signature redacted for privacy.
Abstract approved:
Bent Petersen
A sufficient condition for the existence of temperate fundamental
solutions of a system of constant coefficient partial differential
operators with support in a cone of the form R x F, where
closed, convex, and salient, is proved.
F
is
The approach is to solve a
system of polynomial equations with weighted L2-estimates.
Ce°
dependence on the parameters is only obtained if the weight function
need not change as we take derivatives in the parameters.
On Temperate Fundamental Solutions with
Support in a Nonsalient Cone
by
Kirk Eugene Lancaster
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Commencement June 1981
APPROVED:
Signature redacted for privacy.
Professor
thematics
in charge of major
Signature redacted for privacy.
_
Chairman of Depariment of Mathefflatics
Signature redacted for privacy.
Dean of G aduate Schoo
Date thesis is presented
August 15, 1980
Typed by Suzanne Elcrat for
Kirk Eugene Lancaster
ACLNOWLEDGaENTS
I wish to express my gratitude to Professor Bent Petersen for his
assistance, patience, encouragement, and, above all, generosity during my
stay in Corvallis.
I have enjoyed knowing Professor Petersen and learning
from him.
I would like to thank all the people with whom I have conversed,
especially David Finch, Harold Parks, Kennan Smith, John Leahy, Alan
Coppola, and Don Solmon.
I also wish to express my gratitude to my excellent teachers, Stuart
Newberger, Joel Davis, Arnold Kas, J. W. Smith, and Dave Carlson.
I wish to thank my wife, Deborah, for her patience, and my family
and friends for their help and encouragement.
TABLE OF CONTENTS
Chapter
Page
INTRODUCTION
1
1
SPECIALIZATION OF HILBERT RESOLUTIONS
3
2
EXACT SEQUENCES
11
3
A PARTITION OF UNITY
18
4
THE WEIERSTRASS THEOREMS
24
5
LOCAL EXISTENCE THEOREMS
34
6
SOLUTIONS OF THE CAUCHY-RIEMANN AND
COBOUNDARY EQUATIONS
45
GLOBAL EXISTENCE AND FUNDAMENTAL SOLUTIONS
56
BIBLIOGRAPHY
63
ON TEMPERATE FUNDAMENTAL SOLUTIONS WITH SUPPORT
IN A NONSALIE,IT CONE
INTRODUCTION
p x q (p
In this paper we prove a sufficient condition for a
system of constant coefficient partial differential operators
have a temperate fundamental solution with support in
a closed, convex, salient cone.
qxp
a system isa
P(D)K =
matrix
K
R x F
< q)
to
P(D)
when
is
7
A temperate fundamental solution of such
of temperate distributions such that
This can be considered as a partial extension of the result
I.
of Enqvist.
Ehrenpreis (1954) and Malgrange (1955) independently proved that
every constant coefficient partial differential operator has a fundamental
Since then, conditions for existence of fundamental solutions
solution.
with special properties have been studied.
For an early survey, see
Hormander (1957).
We are interested in the location of the support of the fundamental
This question has been studied by several researchers, including
solution.
Smith, Petersen (1975) and Enqvist (1976), among others.
One application of our result concerns the following overdetermined
Cauchy problem:
Let
be in the interior of the dual cone
X
closed half-space determined by
Suppose
w c D'(R11+1)
convolution algebra.
and
P(D)tu
=
P(D)tw?
with
(0, X), i.e.
supp (P(D)tw) C H
Does there exist
u c V'
of
7
and let
H = R x fx
and
P(D)tw
(R1)
with
H
be the
R111<x,X» 01.
in a suitable
supD(u) C H
2
If
P(D)
the answer is yes.
x F,
R
has a temperate fundamental solution with support in
For details, see Lancaster-Petersen (1980).
This paper can be viewed as one long proof of the result initially
indicated.
The method of proof is a modification of proofs by HOrmander
(1969) and Petersen (1975, 1976).
The paper divides into four parts.
Chapters One through Four form the first part.
In Chapter One we
prove that the specialization of an exact sequence of matrices of polynomials is exact off a proper variety.
In Chapter Two, we show that an
exact sequence of matrices of polynomials is exact over the sheaf
germs of
Cw
functions holomorphic in some of the variables.
of
EO
We prove
the existence of partitions of unity with polynomial growth conditions in
Chapter Three.
Chapter Four is the Weierstrass Preparation and Division
Theorems with polynomial dependence on parameters and with estimates.
The second part, Chapter Five, consists of several lemmas in which
we construct local solutions of systems of polynomial equations with
smooth dependence on a parameter and with estimates.
These are similar to
Lemmas 1 and l' of Petersen (1975).
In the third part, Chapter Six, we solve the Cauchy-Riemann and co-
boundary equations with smooth dependence on parameters and
L2
estimates.
with
weighted
Solutions of the Cauchy-Riemann equations with smooth de-
pendence on parameters, but without estimates, were established by
B. Weinstock.
In the last part, we use the results of Chapters Five and Six and a
Hilbert resolution to find global solutions of a system of polynomial
equations with smooth dependence on a parameter and with estimates.
then find fundamental solutions as indicated earlier.
We
3
CHAPTER ONE
SPECIALIZATION OF HILBERT RESOLUTIONS
In this chapter, we prove a "specialization" result for exact
A weaker
C [C,z]-homomorphisms which will be used later.
sequences of
result would suffice for the purposes of this paper, but this result is
The proof is a modification of an argument from
aesthetically appealing.
unpublished lecture notes of Aldo Andreotti.
z = (z1,
Throughout this paper
= (z1,
Cn,
of
e u
C'
z
z, and
will denote the first
c
=
will denote an n-tuple in
zn)
will denote an m-tuple.
denotes the polynomial ring
the polynomial ring
C [z]
Cm
C.
C(, z)
E
,
pxq
isa
.
C[
z]
V
in
all
px
is the principal
P, then det
px
p
M
of rational
,
then there exists
F E
C [,z]cl
such
0
n.
Let
p
be the rank of
P
over
Interchanging rows and columns, we may assume that
the field C(, z).
if M
zn)
F(,,), ) = f.
and
The proof is by induction on
Proof:
zn],
so that if
Cm
f e C [11q, and P(0, .)f = 0, then there exists
PF = 0
Rm).
both over the complex field
S11,z1,
matrix over
a proper (complex) algebraic variety
that
Cm
Em' z' zn.
P
If
Lemma 1.1
zn]
denotes the field C(1,
functions in
or
Em, zl,
C
C [z1,
will be in
E
in this chapter (in later chapters it will be in either
C [E,z1
coordinates
n - 1
and det
minors of
P.
p
minor (i.e., upper left hand corner) of
M has largest degree in
z, say
11,
among
We may assume, by a unitary transformation in
4
the z-coordinates, that
det M(, z) = a()z11
where
det
and
a e
NI
HI
CIV
and
of degree
1.
u
Let P=
Let
case
0.
in
z
On
Rotate
coordinates also by
Then M= (Pi'
a
and pick
=
(0,
n
so
[
be the z-homogeneous term of
n 6 Cn
so that
...,
1)
0,
and P'=
P1 ...Pq]
H(.,
0
and denote the new
= [Pi ... p].
P:3).
D = det M = det(P1 '
n = 0, V
H
(Let
zn
z.)
=
...
+ lower order terms in
c
and
P')
p
Cn1D()
that all minors of order
V' =
f
e CnIa() = 01.
is the required variety.)
= 01
(In
Notice
in the matrix
p + 1
-R.1
Ri
have zero determinant, for
p, since
i = 1,
sider the minor given by the first
p
N= -R!1 P..
13
R1
Pij
R'
P
P
.
PL.
P
has rank
columns and the
jth
p.
column
Con-
5
where
R! = (Pjl,
Pjp).
Expanding by minors about the first row, we see
0 = det N = E( -1)k+1 Pik Lkj + (-1)P p. .D
where
with the
Lk. = det
kth
column omitted.
P
R'
P
Lkj = (-1)k+p+1 frkj
Setting
L.
R.
= [P
we see
P.
...
]
iq
=0
L.
L.
L.
P3
0
where
D
is the
Thus we have
jth
q - p
component, for
vectors in ker
p +
P.
1
_<j <_
q.
Collecting these into a matrix,
we get
which is a
q x
over, if
V'
has rank
q - p.
(q - p)
(or
D
0
0
D
matrix over
V
if
C
z]
with rank
n = 0), then D(, )
By the choice of
M, each
L..
q - p.
More-
0 and so C()Z,.)
has degree
<
in
z.
6
Of course, PC = 0.
(i)
Now assume
n = 0.
A. =f./D(
Let
j
j
)
0
Suppose
for
j
Cq, and P(0)f
V, f
0
= p + 1,
..., q
and
A =
= 0.
Ap+1
A
-
Then C(0)A =
D(C0) A0+1
D(y A,
f - C(0)A
so
=
_
= (1).
Let 0=
e
1
eP
0
0
P(0)4) = P(o)f - P(0) C(0)A = 0, so
o =
Mgo)
[ao
Since M(0)
Thus
is nonsingular and
f = C(Eo)A
.
Let
F() = C()A
)0 = 0, 0 = 0.
.
7
(ii)
Assume
and assume inductively that the lemma has been
n > 0
proved in dimension
P(E0, .)f = 0.
zn
to divide
by
D(E0, .).
Recall
D(E, z) =
c C[z]
and the degree of
in
6.
zn
J
J
(Note that
A.
If we set A=
C(E0,
C[z]q
and
+
j = p + 1, ..., q.
3
A., 6.
Eo 4 V', f
zn.
f. = D(0'
E
)A. + Q.
where
Suppose
We now use the division algorithm relative to
fj
lower order in
n - 1.
contains powers of
E)
a(0
is
< p.
in its denominator.)
we see
Ap+1
-)A =
D(E0, )Ap+1
D(0, .)A
0
Define
q
0 = f - C(o, *)A =
C
C[z]cl
.
p+1
6
Notice
q
P(E0, .)0 = P(E0, )f - P(0, )C(0,
i.e., 0 e
ker P(E0, ).
= 0
Considering just the first
p
rows of
8
P(0, -)0
we obtain
-(P'*)
p+1
M(0, *)
q
0
By Cramer's Rule, if
D(Eo, )ej = det (Pi(E0,
E
=
))
p+1
< p,
< j
1
0
P'(0, ))
v
-)
eiCii
.)
L=p+1
v
(with
where
isa
jth place)
in the
Cij = - det (Pi ... Pi ...
pxp
cofactor of
D(0, )ej (1 < j
has degree
<u
zn
We have shown that if
in
.th
place) which
and so has degree in z< u.
P
has degree
< p)
in
(Pi
in
j = 1,
for all
V'
< 2p
and
zn
and so
Thus
ej
..., q.
P(10, )f = 0, then
f = C(0, )A +
where
degree
A e C[z]q-P, 0
< p
in
zn.
E C[z]cl
.
+ a
p
(z')
0(z) ' z11-11
n
(1)
z' =
8
has
If we write
1)(, z) = z(71(' z')
where
and each component of
ai
is a
Z+1
z')
'
(z1)
pxq matrix over
C[E,z1],
and.
(1)3
E IC[Z]q, then the condition
0'
.)f = 0
becomes
and this is equivalent to
= 0
P(Eo,
P(
01(0, .)(1)]. =
12(0' .)(P1
.)(P2 =
etc.
If we define
tp =
al
0
a1
0
a
(p + 24p x pq
matrix over
CE, z'j
and apply the induc-
tion hypothesis, we can find a proper (complex) algebraic
variety W C Cm
V = V' LIW.
e CE,
V, then we can find
If
z']q(1
(7)'i
so that the lemma holds for
_<j <_
p)
so that, if we set
=
4)].
(1)u
then
4 = 0
and
74;(0, -)
=
=
tp.
We set
10
Define
H(C, z) =
z1V-,171(C,z')
F = CA + H
Theorem 1.1
If
C[C,
+
E CE, Z]q
c
PF = 0
,P
C[C, 11(1
jrz
+
exists a proper (complex) algebraic variety V
ZIP
,
in
Cm
is exact, there
so that if
Co 4 V, then
zi
r
Q(Cny )
)
P(C
C[ z] cl
CI z] P
is exact.
Proof:
V
Let
Et
C0
V'
exists
Clearly
P(C, ) Q(C, -) = 0
for all
C
Cm.
E
be the variety determined in the previous lemma.
f
E
C[z]q, and
F E C[C,
0
ness, there exists
g = G(Co,
P(C0'
so that
G e C[C,
.) e C[z]r.
Then
.)f = O.
PF = 0
z]r
Now assume
By the previous lemma, there
and
so that
F(CO3
)
QG = F.
Q(Co, )g = F(Co, -) =
and
F(0, ) = f.
and
C[
C[C,z]g
= f.
Now set
f.
By exact-
11
CHAPTER TWO
Exact Sequences
In this chapter we prove that if
C[E, z]r
is exact, where
consider
functions
C
Q
and
P
and
P
C[E, z]q
Q
P
C[E, ZIP
are matrices of polynomials, then, if we
as maps of q-tuples and r-tuples respectively of
in UxWCRMxCn with W convex (and
and holamorphic in
W, this new sequence is also exact.
U
W
and
open)
We also introduce
some notation that we will use throughout this paper.
We denote the coordinates in
z1
=
+ iy
x1
'
Rm
zn = xn + iyn
.
by
El,
..., Elm
C
functions in
Rm x Cn
EO = sheaf of germs of
Cm
functions in
e
sheaf of germs of
x Cn
forms of type (p,q) relative to
C
AO = sheaf of germs of real-analytic functions in
holomorphic in
z
is a nonnegative integer, we denote by
k
Rm x Cn
which are
zn
CO = sheaf of germs of holomorphic functions in
which are
which are holomorphic in
zn
zi,
k
by
zn
zi,
If
Cn
We will use the following notation:
EE = sheaf of germs of
EE(I)q)=
and in
Cm 4- n
EkE, Ek0, etc., the sheaves
times continuously differentiable in all variables and
(or holomorphic, etc.) in
zi,
zn
We will denote the stalk of a sheaf at
e.g., EED, and similarly for germs.
see Chapter 7 of Hormander (1966).
Cm
.
0
(for example) by subscript,
For a short discussion of sheaves,
1?
denotes the continuous sections of the sheaf
F (V, S)
F (J x W, EE) =
over
S
Thus
V.
x W).
Let
R
be a commutative ring with identity and
R-module.
B
is R-flat if and only if it satisfies one of the following
Def:
be a right
B
equivalent conditions:
The functor
B
is exact.
B
B is injective, for every finitely generated left
I
R-ideal I.
If
.) ap ER; bl,
al,
thereexist.ER
and
rlj
..., bp E B, and
i=1
a.b. = 0
11
,
then
f. E B (i = 1, ..., p,j = 1, ..., q) such
)
q
f.r.
thatZa.r.=0(j=1 ,...,q)andb.=E
i
.
li
j=1 i
(i) and (ii) are equivalent by Theorems 3.53 and 3.54 of Rottman.
(iii)
(ii)
a =
If
E b. 0 a. E Ker(B 0 I
33
b.a. = 0
i.e.
j
)
3
then a=Eb.0a. =ZZfr.
j
)
j zZjZ
(iii)
(i)
B)
)
a. =Ef (s)Ea.r.
z
)
Define a morphism
RP
by
R
(c1,
= 0
.
)2,
)
c.a.
c )
.
1=1
K
and let
K
B
0
exact.
B
RP
Notice B0 K
(b'c
Ea.bt..
i=1 1 1
be its kernel.
and
b'cp)
Then
R
B
B
BP
P
0
K
is exact; thus
is
B
defined by
R
RP
b'
is defined by
is exact.
B
0
(c1,
(b'
l'
BP
K
cp)
b')
By
(i),
B
is
13
Now
B1,
is in the kernel of this last map, so there exists
bp)
(lb1,
..., Bq E K
and
fq E B
fl,
so that
ct
bp).
j=1
Bj = (r1j,
If we write
B. E K
3
E
implies
rpj),
a.r.. = 0
EE
bi = .E r..f.
3=1 13 3
and
.
13
i=1
Lemma 2.1
then
is flat over
A00.
0'
Proof:
Suppose
We may extend
aP E A00, bl,
al,
aP
a1,
EN,
bP E
and
A00
O.
to holamorphic germs (by power series) and
apply Oka's theorem (Theorem 6.4.1 of Hormander (1966)).
exists irE
K
= albi
i=1
p, j=1,
i = 1,
q
Then there
P
such that
E a r
= 0
i=1
P
and if
u1
upEA00
,
and
If
v. E AO
j
0
such that
..., q
= 1,
we expand
P
E a u. = 0
i=1
then there exists
1
u. =
1
j=ir
v..
3
i i
E a b
in formal power series about the origin, then
i=1
i
P i
E a b = 0
i=1 e e
(in the formal power series ring) for
e
near the origin.
The formal power series ring is flat over the convergent power series ring,
so
bi = E rij sj
e
e
for some formal power series
sj,
j
= 1,
..., q
.
By
Malgrange's theorem (Theorem 1 of Malgrange (1960)),
q
b- =
E
forsomet.EEE,j = I,
Notice that this proves
..., q
.
3
j=1
FE
is flat over
AO, i.e., BE is flat over AO
14
0x
at each point of
Lemma 2.2
r
u E
r (u x w,
x w,
Let
C[xl, yl,
is flat over
and W open and convex in
km
and
EE(p,q+1))
EkE
f = 0
in
such that Tu = f
U x W,
in
EE(10q))
P
If
Cn
only.
C[x,y] =
such that the matrix of partial differential
operatorsobtainedbyreplacingby
and
v .
by
Dy
'3
which we denote
Cn.
U x W.
be a matrix of polynomials over
xn, yn]
AO.
then there exists
T denotes the Cauchy-Riemann operator in
Recall that
Proof:
be open in
Let U
f E
We do not know if
Cn.
in
P(D(x,y)), represents the map
T: Cm
Cm
(Cn)
(p,q+1)
(12,q)
where we identify (p,q)-forms with
(n)
(n)
(e).
vectors of
C
functions.
P
Let
Q
be the matrix of polynomials over
C[x,y]
such that
Q(D(x,y)) represents the map
C(p,c0.2)(Cn).
C(p,q+1)(Cn)
Then
C[x
Now consider
YJA
P
and
C[,x,y]A
Notice
Q
as matrices over
D.
independent of F, =
is exact.
C[x, y]C
C[x, Y]B
)
C[,x,y]
is exact.
(which are
Then
C[,x,y]B
Q(D(x,y))f = 0,
By the Malgrange-Ehrenpreis theorem
since
C[E,.,x,y]C
--9-f = 0.
(Malgrange (1963)), there exists
P,
15
if
3
3
g.Er(11.xW)Asuclithatp(0(x,y))g7...-f
inU.Thusin U.
g3
x W.
Let
is open and convex
3
be an open cover of
(u.1
3
U.
U
3
consistingofconvexsetsandx.a.partition of unity of U
7
subordinate to
Define
U..
u = E x.g.
E
C
3 3(P,q)
j
(U x W)
Then
u = f
in U x W.
Weinstock (1973) proves this lemma for
and
q = p = 0,
Corollary 2.1
i.e., for
U
Let
negative integer.
W
a domain of holomorphy
(0,1)- forms.
W
and
be as in Lemma 2.2 and let
There exists an integer
N
k
be a non-
(independent of k) such
that:
If
f E
r
x W, E
k+N
k+N
and
in
-5-f = 0
U x W, then there
E(p,c14.1))
exists
Proof:
u E
r(u x w,
EkEkPg))
such that
Tu = f
in
U x W.
The proof of Lemma 2.2 is still valid if we use the version of
the Malgrange-Ehrenpreis theorem in Hormander (1966).
Lemma 2.3
Cn.
Suppose
U
is open in
W
and
is a convex, open set in
Then
r(u x
Q
EO)r
W,
,
C[C, z]r
Proof:
Q0 = Q, ro = r,
Let
need not be finite);
z]P
is exact.
r (1.1 x iv, E0)q----E-----)- F (U x
Q
is exact if
C[C,
km
that is,
C[, z]P
C[C,
and let
N3}
C[, z]cl Q
,
...
By Lemma 7.6.3 of HOrilander (1966)
is exact.
>
00q
Since we can identify A00
POOP
and
is exact.
be a resolution of
r.
----)- 00 3
W, EO)P
000, we see
C[C,
z]cl
Q
(which
--11-)-
16
r.
AO 3
is exact.
AOP
A0c1
Since
EE
is flat over AO,
r.
is exact.
"<EEcl
'"<...
EE 3
EEP
P
By a partition of unity argument, we se,e. that
r.
r(u x w,
is exact.
,j
EE)
r(u x w, EE)cl
This is clearly true for
P
also.
EE(p,q)
Suppose
f E
r(u x w,
with
E0)
u E
r (u x
Pf = 0.
Then there exists
W, EE)r
such that
Qu = f
Then Q
u1
= Tf
u
and so Tu = Q u,
= 0
E F(U x W, BE
U x W.
in
1
u1 = Q2 u2
for some
u2
for some
u = 0,
U xW
in
E CCU
U x W
Now
(1))
Q1Tul= TQ1u1 =
so
in
x W,
r2
.
EE(0,2))
r.
Inductivelywefin.du.Er
with
Qiu. =
.
uj -1
When
By Lemma 2.2, there exists
such that
un
=
vn-1.
Now
x W, BE
j = n,
vn-1
E r
W)
3
un = 0.
x W, BE
rn
(0, n-1))
17
Qn
so
-a-
Thus
=
un
(un..1
and
un-1
Qn u
Qn vn_i)
Qnvn-1
0
E
for some
un-1 - Qnvn-1 =n-2
vl
u1 - Q2 v1 =
for some
vo
Now Q1u1 = Ql(ul - Q2v1)
So
and
Q1u1 =
-a-(u
-
vo
r(U x
e rcu x w,
(u1 - Q2 v1) = 0
and
vn_l
-
Continuing in this manner we find
So
.Qn
E
W,
EE(0, n-2)
,r 2
EE(0,1))
with
.
r(u x w,
EE)
rl
.
Qiv
=
.
Qlvd=
0
i.e., u - Q1v0 E
Q(u - Q1 v0) = Q u = f
Corollary 2.2
If
f E
g E
F(U x W, EO)r
Proof:
Let
u = f
W, EO)r.
.
r(u x
exists
r(u x
W, EE)r
such that
with
-5"
Qf = 0
,
then there
Qg = Qf.
in the previous proof.
Then
1
g = u - Q v0 = f - Q1v0.
13
CHAPTER THREE
A PARTITION OF UNITY
We will eventually need special partitions of unity with polynomial
growth and so we prove a suitable result in this chapter.
{Udj
Let
= 1, ..., Z}
be a finite open cover of
Rm x
C.
We
define
B(t) = {(C, Z.)
E RM x
1
1
1z1
< t,
< t},
suP1(dist ('z),z) c
z) E
D(t) = inf (d(, z)1('
B(t)
- B(t - 2)1, and
d(t) = min (1, D(t)}
for
E Rm, and
=
t e R,
z = (z1,
we mean the (topological) boundary of the set
V.
zn) e Cn.
By
nr
Notice that d(, z)
thedistancefromtotheboundaryofthesetU.which contains
z)
and whose boundary is furthest from
V C Rm x Cn, then
If
Lemma 3.1
denotes
V(z)
There exist constants
Ca > 0
(E z).
(E,
E Rm1(, z) E
V}.
for each m-multi-index
a
such
13
that,foreachzce,thereexistsapartitimofunitY{1).}0f
Rm
subordinate to
(U.(z)1j
Ipaqii(U1
if
(, Z) E
B(t)
= 1,
< Ca(1
- B(t - 2).
...,
Z}
so that
17,1)1a12(d(t))-1a1(21al+m)-1
is
19
Proof:
of
We assume
(and of
z
is fixed and we obtain constants
z c Cn
For
t).
K(j, t, r) = {E
1 <
j
Z, t > o, r
<
z) E
E
Ui
n
E
independent
R, define
and
B(t)
Ca
dist ((E, z),
> 2-1-1
L(j, t, r) = K(j, t, r) - K(3, t-2, r).
2-r < D(t), then
Notice that if
cover of
by
r(t)
2-r(t) = d(t)/2.
Notice
characteristic function of
L(j, t, r(t))
=
and
E
by
s > o
picked so that
C2m(r(t)+1)f(22(r(t)+1) lc12
supp (w(t,j)) C closed
and w(t,j)(E) = 1
If
C
r(R)
E
fm p()g = 1; p
is a
Define
mollifier.
Notice
with
f
be the
x(t, j)
s < o
0
p(E) = Cf(1E12 - 1)
Let
and define
lexp (1/s)
f(s) =
Z,
t, r(t))1j=1,
Rm with intersection
consisting of relatively compact sets.
< 2t
pt(E)
4.1.,(j,
is a locally finite open cover of
t = 1, 2, 3 ...1
Let
is an open
B(t)(z) - B(t - 2)(z) = (E c Rml(E, z) c B(t) - B(t - 2)1.
Define
number
{L(j, t, r)lj = 1, ..., Z}
if
1)
2-(r(t)+1)
E c L(j,t,r(t))
-
- nbhd
d(q)
21-r(q)
and so, for some
L(j,q,r(q)) - 2-(r(q)+1)nbhd
of
w(t,j) = x(t,j) * pt.
of
L(j,t,r(t) C U(z)
2-(r(t)+1)nbhd
E e Rm, then for some positive integer
d(E,z)
and
q, (E,z)
E w(q ,j).
3=1 q=1
n(j,q,r(q)); then
e E C(R)
E
n(j,t,r(t)).
B(q)
- B(q-2)
1 < j < Z,
co
Define @= .E
of
and e > 1, since
w(q,j)(E) = 1.
20
{supp (0/(q,j))1j = 1, ...,
Z, q = 1, 2, 3,
is locally finite, has
...}
intersection number < 32., and consists of compact sets.
Define
CO
j
= q1 w(ct'i)/e
= 1,
11)3
...,
= 1, ..., Z.
is a partition of unity subordinate to
2.}
{U.(z)}
liftp.(C)1.
and it remains to estimate
By Leibniz's formula,
CO
((a3) 1Da-k(c1,3)(1){0-1)()1
-< q1
I
where
is the sum over all m-multi-indices
such that
S
S < a.
13<c(
Note
D8(01)
where
(b =
= I(-1)aa:(D6)1/8°4-1
is the sum over all mb-multi-indices
lal)
with
yi
<
+
1b'
(D8)1 = D -8
)1.....
y.
and
1D514(t,j)()1
<
<
<
if
D
volume of
C2m(r(t)÷1)r(t)m(t+1) I
l,
(C,z)
'
E B(t)
Here
D
=
Do8 = 1 (0 = (0,
...,
E zm).
0)
= 1x(q,j) * Dapt(W
max 1D5p (C)1
L(j,t,r(t))
C'245 +2m
-e.
(3111) ijm
\a
Also
is the number of
o
=
yb)
111
1-t
yj, and
nonzero
yb
+
'l
y = (yi,
Lt
al
t
max
21131(2r(t)+3)
If(k) (s)I
__
o<k<lel
-1<s<0
1
LII
1
1
lzi
4-
1)1
d(t)
- B(t - 2), since then
-2
mllog2(d(t))
t < max
{W
2,
-
lzl
2Im
21.
21
Clearly
2
lAp.()1
So
t, j, and
is independent of
C'a
0
0.
E
E
E ID"w(q,j)(E,Wal
$<a j=1 q=1
< C"
3
1
<C (1
+
a
(since
on
2.,
z.
since 6 =
E
E w(q,j)>1
j=1 q=1
12
d(q) -(21al+m)laHl
1
1
has intersection number < 3t).
{supp (w(q,j))1
depends
Ca
n, m.
Rm x Cn
algebraic open cover of
is a semi-
2,1
Weldlibeinterestedinthecasewhere{U).0=1,
and we want a partition of unity with
polynomial growth.
Lerma 3.2.
Let.
{U. 1j=1,
..., 2}
be a (real) semi-algebraic open cover
3
of
Then there exist constants
Rm x Cn.
C' > 0
a
and an integer
such that, for each
z c Cn, there exists a partition of unity
R7
U.(z)1
5;111)m-climate to
Proof:
Let
{Ipil
lemma; we assume
+W+
U
a.
Cn
has been fixed.
= dist ((c,
z),
in
We consider
is (real) semi-algebraic (i.e., U = {(, x,
Q(E x, y) < 01
with
DU))
of
IzONIal2+N
be the partition of unity obtained
zE
{tpj}
such that
IDay01 <
for each m-multi-index
N > 0
P
and
Q
y) 6
polynomials) then
the previous
Cn
as
Rm+2n1P(E,-,
{(11,
z)
R2n.
If
x, y) = 0,
E
Rm+2N+11
is a (real) semi-algebraic set.
Since the supremum (and so infimum) of a semi-algebraic collection
of semi-algebraic sets is semi-algebraic, {(t, d(t))1t > 01
algebraic.
is semi-
Then, by the Seidenberg-Tarski theorem, the following
22
{(t,d(t), p)It > 0, 0 < pd(t) < 11
sets are semi-algebraic:
and
M = {(t, p) 1t > 0, 0 < pd(t) < 1}.
Mt = (111(t,
Let
p(t) = sup p =
sup
O<T<t
ptMt
lim
d()-1
11)
= (1110 < p < d(t)1}
E
d(T)-1.
and
lim D(t) = d(0,0) > O.
Notice
Thus
t4-0
= max(1,1/d(0,0)}.
is nondecreasing and finite.
11
By Lemma 2.1
T+0
(HEirmander, 1969, p. 276), there exists a rational number
constant A > 0
Cta
p(t) <
be an integer with
d(t)-1
< p(t) <
t < max
then
d(t)
-1
{10
< C(3
+ 2,
W
if
t
t > o.
N'
=
If
< 10 + 1z1
1z1 + 2}
so that
p
t > 0
> a, say
N'
C(l+t)N'
as
depending on
C > 0
Then there exists a constant
N'
Eaj + 1.
(,z)
Then
E B(t)
- B(t-2),
+ 2, so
1z1)N' <C'(1 4-1E,1+
17-1)N'
Using the estimates of the previous lemma, we see
Ipaqii()1 <
Ca(1+1E,1+1,z1)1a12(d(t))-1a1(21al+m)-1
< C'(1+10 +1z1)Nla
where
N = 1 + ZN' + mN'
and
-+N
C'
a,
t E
(max
and a
so that
p(t) = ATa(1 + o(1))
Let
a > 0
1z11, max
{10, 1z11
= C C'.
a
+ 2)
In the first line,
is arbitrary.-
23
Corollary 3.1.
Lemma 3.2 is true with the estimate
Ipc'y)1
Proof:
< 4C;(1 +
Computation using the inequality
+
(a + b)2 < 2(a2 + b2).
?4
CHAPTER FOUR
THE WEIERSTRASS THEOREMS
In this chapter we prove a version of the Weierstrass Preparation
and Division theorems with bounds and with parameters.
methods of proof of the Weierstrass theorems.
There are several
Here we modify the proof
based on Cauchy's theorem.
If
P E
CE, z], then we define
P'()(z:r) =
11)(,z414)1 =
sup
1141<r
An m-multi-index is an m-tuple
The length
means
< a
a = (al, ..., am)
of an m-multi-index
at
< a.
13.
for
sup 11)(,z+rw)1.
lwl<1
(al,
m
j=1,
of nonnegative integers.
and
a, +
is
..., am)
means
< a
We continue to use the notation of Chapters 1 and 2.
r(u x
denotes the set of
W, EO)
holomorphic in
W,
r(u x
W, AO)
U x W which are holomorphic on
of
Ck
Let
U
be a connected open subset of
P E CI, z], s c (0, 1), and
Let
z e Cn
r > 0.
Assume
P()(z:r) < C
and assume
< a and 13
a.
Thus
W which are
F(U x W, Ek0)
W, and
U x W which are holomorphic in
functions on
am.
denotes the set of real-analytic functions
on
Lemma 4.1
U x
functions on
Cc°
13
+
W.
and
Rm
P(, -)
inf
denotes the set
0
P(E-z1,T)I
C > 0.
if
Let
U.
for each
e U.
1T-Znf=rs
Leto c
T
U
and let
13(, Z1, T)
number
s', o < s'
N+
be the number of roots of the polynomial
which satisfy
IT - zn
< 1, depending only on
< rs.
a,
Then there exists a
n, and
C
where
a
is the
25
degree of
P
A = {w E
01114j1
ball
B =
A.
z, such that the polydisc
in
j=1,...,n-1,
< s'
< 1}
{14 e
IwnI
and so that the following properties hold:
There exist unique functions
U x (z
with P,
+ rA)
and
P+
and
U
E
=
A'
e
Cn-1
1'w.
31
-1-5'()(z:r)
sup
z+rA
f e
<
r(u x
for each
sup
z+rA
0
and
P-(C,-)
if wez+r-E
U, then
E
C'
> 0
111{)1 +
w'
is monic, has degree
N+
in
wn,
rA', all roots of the
E Z/
IT -
satisfy
ZnI < rs, where
j=1,...,n-11.
depending only on
C1 > 0
< C rN inf
z+rA
P(,)1
u,
n, and
and
Cir-N+is"(C)(z:r).
(z + rA), EO)
is a polynomial in
constant
such that:
wn, P4-(,-)
P-(,w)
< s'
There exists a constant
If
rA), AO)
in
and
I
so that
wn
P+g, w', T)
T
w' = (.41,...,wn_i)
h
and
as a polynomial in
polynomial
B.
x Cz
P = PP-
so that
P-
are polynomials in
P-
and for each
C
and
e U.
p+
(iii)
P
c F(U
are bounded in z+ rA
and
is contained in the unit
< s}
and
f(,-)
f = Pg + h
where
of degree
wn
depending only on
P'{)(z:r)
suplg(,)1
z+rA
is bounded in
g, h
< N+
E F(U x
z + rA
(z+rA),E0),
and there exists a
u, n, and
C
so that
<C' suPlf(,*)1.
z+rA
Moreover, there exist nonnegative integers
Nk
depending on
x,
26
n, and
the top order terms of
depending on
Kk
and constants
k
such that
P
p.,(-1\14-(k+1)
)(z:r)
< Kk(1
same inequality for
1Dap-
2
2 Nk
-N4-(k+1
+r )
P(&-)(z.. r)
E
r-N÷(10-1)
sup 1121f(,)1
a<a
z+rA
1Dag(,w)I
and the same inequality for
of length
a
for every m-multi-index
and the
c,w)rN+(k+1)
Kk(1+1z12 +izI
ID°1-1(&,w)1 <
X, n, m, k, C, and
lal
and for
= k
U,
E
W E Z + rA.
C..
If
E r(u x
f
for each
h
E
U, then f = Pg + h
wn
is a polynomial in
C, N, and
constants
f(,.)
and
(z+rA), Ek0)
where
of degree
with
U, w
g
Z
and
+ rA
in place of
De = (-2)
Proof:
Let
C
so that
lal
z+rA
< k.
The same estimate is true
()m
3rn
and
x
is the total degree
We can expand
Q
in powers of
3
P.
Q(E,w) = P(E, z + rw).
i.e., Q(E,w) =
P(,w) =
P, k, and
a
'"
3E1
of
and there exist
NI+
h.
al
Here
(z+rA),Ek0),
sup 1E0f(,.)1
a<a
E
c,r(u x
z + rA
24.1z124.r2)N 1-5()(z:r)-Mr-M
. E
for
g, h
<
M depending on
IDah(E,w)I < C(1+W
is bounded in
n-multi-indices).
a (E)WY(-y
Y
(w-z)/r) =
a
Yl<u
(&-)(w-z)-Yr-IYI
Then
w,
27
Since any two norms on the finite dimensional vector space
(polynomials in
= az1,
C[ z]
there exists a constant
C01 Q()(0:1)
E
<
IYHP
Notice that
-Q-()(0:1)
[-Co, Co]
depending only on
E
x D, then
A =
U, WI E A'
IQ(,
0,
A C B
and
j=1,
sl
-
U x
has exactly
...,
T,(1
Denote these roots by
and
e U
vary in the
By a compactness
n.
p, c
n-11, D
IwnI = s.)
for
w')
if
0
and
n, and
s'
= {T E CIITI <
s},
&()(0:1)/2C, if
(Note that the hypothesis becomes
ITI < s - 6
roots with
N+
and
p
is connected, we see the
A'
so that
of Q(,w)/&()(0:1)
IQ(,w)1 >
< 6.
if
n
so that, if
s
and
wn)I > Z(C)(0:1)/C
fact that
P(,.)
since
depending only on
s'
and
U
E
n, and
C,
and
i
< CoQ()(0:1).
depending only on
d > o
argument, we can find
are equivalent,
Y
a ()/()(0:1)
that the coefficients
compact set
if
0
(0
A
p)
<
depending only on
> 0
C0
of degree
zn]
P(11,n)
By Rouche's theorem and the
polynomial
if
E
j=1,
U
WE, WI, T)
T
and
wf
E pl.
N+.
N+
Define Q+(, w) =
II
(wn
j=1
- Tj(E, w'))
for
E U, WI
E LI
and
N+
P+(, w)
=
n
j=1
(w
-zn
n
- rr.(&, (w'
a4Q
Set
Since
a(E,
Q
oZn
1
w'J, = 2Tri
z')/r))
(,
A'
for
in
U x
Cm+11.
E,
E U, w'
E z' +
rA'.
T)
Q(, w', T)
does not vanish on
neighborhood of U x
-
j
A'
dT
x 3D, eZ
for
i = 0, 1, 2,
..
is holomorphic on an open
By Cauchy's integral formula,
?8
0(, w') = E
(T.C, w'))
and the coefficients of powers of
j=1
are elementary symmetric polynomials of
(T1(E, w'),
wn
in
Q+
w')).
T
Since the elementary symmetric polynomials are polynomials with rational
coefficients of the "power sums"
rk(x) =
E
x., we can find rational
j=1
coefficients
a
=e
b2,
ao
N
are
(a
a
,N"
)
+ 1-multi-indices) so that, with
-N
.0
N+
Q+(E, w) = E I bt
,
2, a
and so
a (e(E,
14'))awz'
n
for
is holomorphic in a neighborhood of U
Q+
is holomorphic in a neighborhood of U
P
0 < t < N+,
'
w) =EEb
e(E,
-
x (z'
z')/r)a(wn
lal
x A'
+ rA')
<
N+,
x C.
x C
Similarly,
and
+
- zn) r-NT -"
.
La Z'a
Define
w'
21
E
P- = P/P+
on
P-(E,
rA', T
U
x (z' +
14f, T)
rAl)
x C.
For
each
U
and
U
x (z
T.
is a polynomial in
By Cauchy's integral formula
1)(,
P_(E, w) =
if
wn
e
zn
Since
IT(
1Q+(, w)I
'
IP
>
(E,
14)1
P-
+ rD, so
w')1
-
d
- s)N
(114 -z 1-r(s-d))
n
n
P+(E, w',
T)(T -
dT
wn)
is holomorphic on a neighborhood of
< s
6
(114111
r3D
w', T)
(by the choice of
if
6), we see
c U, w' E A',
if E eU,w'
E z'
+ rA).
w
> s - 6, and
rA',1wn -zn 1>r(s-6).
'79
Then
IP-(E,
w)I
<
(r)N+
ifIwn
sup 11)(,.)1
z+rA
-
znI =
and by the
rs
maximum principle,
1P-(, w)1
Since the roots of
cients of
T
<
Q+(E, w', T)
1Q+(,
-4-
w', T)
roots, there is a constant
ITI
satisfy
<
e U.
rA,
s < 1
and the coeffi-
are elementary symmetric polynomials in the
C"
coefficients of powers of
stant C"
if w E Z +
(rd)-N 1-3()(z:r)
are bounded by
T
depending only on
u
and
n
so that the
So there exists a con-
C".
so that
n
and
u
depending only on
sup IQ+(E,W,T)I < C".
lil<1
P+(,w)
Since
=
rN Q+(,(w-z)/r)
sup
1121+(, w',
and
e U
if
T)I
<
w' c z' +
Cu'rN
1T-zni<r
i.e.,
1134-g,w)1
1P-g,w)1
w' e z'
< C"'rN4-
if
1P(,w)lr-N4-/C"
+ rA', and
Iwn -
E
>
zni
U
and
w e z + rA.
Thus
-F(E)(z:r) r/2CC" if
e
U,
= rs, and so, by the maximum principle for
1/P-, we see
113-(,w)I >
-1()(z:r)r-N /2CC"
This proves A if we set
If
s
-
6 < p < s,
C1
E U,
if
maximum of
W'
E
E
U, w e z + rA.
2CC"
and
and
wn
+ rA'
f(,14',T)(P+(E,14',7)
-
6-N
-
znI
<
rp, define
P+(,w',wn))
dT
11('w) - 271
IT-1=r10
Zn
P+(,w1,T)(T-wn)
30
E
f(,W,T)
1
g( E,w) -
and
2TriP(E,W,wn)
P+(.,-,w',T)(T-wn)
These integrals are independent of
in
IT -
zni > r(s -
f = Pg + h in
x (z
U
For each
+ rA).
P1-(E,W,T)
since
p
Clearly, g, h
(5).
dT
E r(u x (z
E U,
E
has no roots
rA), EO)
WI C ZI
and
rA',
P(,w',wn)
P+(,w',T)
T-W
nomial in
wn
wn) of degree
(and in
T
is a polynomial in
of degree
<
h
N+, so
is a poly-
<
N+
Q+(E,w1,T)
As before, if
I dn(E,W)Tz, then
=
Z=0
if
E
E
WI E AI
and
U
Q+(E,14',T)
T
for
and
lwn
p,
<
1T1
w' c A'
C
Idi(E,w')1 <C"
;
and so
N+
Q+C,wf,wn)
< C"
W
= p, and
p>S
- 6.
I
I
z=1 j=1
C pN -1
n
So if
2
P, Hr=
<
U, then
P+(E,z'+rw',zn+rT) - P+(,z'+rw1,zn+rwn)
+ rT (zn + rwn)
zn
r
N+
+
(Q
-
Q Ct:04/tywn))
_< C2(rp)
r(T - wn)
where
C2
depends on
and
C"
N-1
1
and so on
and
n.
P,
P > s-6,
31
By the definition
of
h, we see that
sup Ifg,')1C2(1-P)N+-1/(Iwn-zn1
In(,w)1
1
r
21T JIT-zni =rp
if
-
z+rA
s - 6 < p < s, w'
E z' +
rA',
-N4-
C2
1dT1
1wn
1h(E,w)1 <C2 sup 1f(E,.) lim sup
z+rA
r(s-6))N
- zn1
<
(rp)N+(
rp, and
wn-zn1
-
E E U.
Thus
r(s-6))-N+
sup
1f(,.)1
z+rA
= C3 sup 1f(,-)1
z+rA
-N+
where
depends on
C3 = C26
sup
z+rA
1P(E,')11g()1 <
Also, 1P(E,w)1 > ii'()(z:r)/2C
and
1114'11
- zn1
-
sup
z+rA
Let
rs1 <
1g(E,.)1
C.
Since
Pg =
f-
h, we have
(C3 + 1) sup
z+rA
(by choice of
6) when
c U, w' c z' + rA',
rd, so, by the maximum principle, we see
< 2C(C3+1)17)-(r,)(z:r)-1 sup 1f(,-)1.
z+rA
C' = C3 + 2C(C3 + 1).
Recall
and
p, n, and
131-(,w) = E
E
i=o lal<N
bz a(6(,(141-z1)/r)'(wn-zn)r
1Q(E,(wf-z')/r,T)1 > Z-(E)(0:1)/2C = -15()(z:r)/2C
WI E ZI
rA',
E
wn
zn+
rD, and
1T1
= S.
if
E E U,
32
ez,
From the definition of
Iii
= k, and
DY =
it is clear that if
(441
/
'1
16Yezg,w1)!
_<
(2C)1Y1+1-13(E)(z:r)-1Y1-1C4sZ+1
(1+1E12+Izt+rw'12
and
4
w'
exists a nonnegative integer
p+(E,w)1
if
depending on
N'
K,T;()(z:r)-N1-(k+1)(1+1
<
U, and
IY1 _< k,
w
lwn - zn
P- = P/P+.
E E
<
If
k
so that
12.4.1z12+r2)N'
K'
depends on
A, k,
P.
e U, w'
= k,
y
Kfl(rd)-N+(k+1)13g)(z:r)-
DYP-(E,*)
lpyp-(z,.)1
if
e U
E Z'
rA', and
= rs, then
ID1P-(,w)1
and since
A, n, and
z + rA, and where
n, C, and the top order part of
Recall
P, if
P()(z:r) < C(14-1E12+HI2+r2)A/2, we see that there
Since
E A'.
Izn12 + r2s2)(IY1+1)(X-1)/2
k, A, n, C, and the top order part of
depends on
C
+
IY1-1(1+1E,2+1z121.r2)(1Y1+1)(X-1)/2
< C06()(z:r))
where
is an m-multi-index,
then
)Ym
(33,_
y
U, w E Z
N+(k+1)(i+w2+ z124.1.2)N"
is holomorphic
/c,1")(z:r)-N+(k+l)r-N1-(k+1)0..i.w24.1z124.r2)N"
rA.
From the definition of
g
h
and
we see that there exist constants
K
and the estimates on
K2 > 0
P+
and
and nonnegative integers
P-,
33
so that
N1, N2
+
IDYh(E,w)
N,
r(EAz:r),-N (k+1)(14-1E12+r2÷1z12)
(k+1);-_,
-< K1kr-N
E
sup
-Lk
ID'f(,Z,-)1
B<y z+rA
if
E e U, w e z + rA, and
Now let
K", K1
N = maximum of
'k
,
K2
1
Notice
concerned.
y= k.
N', N", Ni
Similarly
for
a
with
K
,
etc.
-k
, N2
and let
Kk = maximum of
K',
.
k
C.
is essentially the same
as B.
as far as
the
proof is
34
CHAPTER FIVE
LOCAL EXISTENCE THEOREMS
In this chapter we prove a version of Lemma 1 of Petersen (1975)
depending on one parameter.
We would prefer to establish the follow-
ing lemmas for any number of parameters, but the method of proof breaks
down if m > 1.
We are only interested in taking derivatives in the
parameter
E, and so we introduce the notation
Lemma 5.1
Let
B
Mk
V E r(R x
Nk, for
and
k = 0, 1,
Ck, M, and
Proof:
(i)
+ rB), E0)
rtB), E0)cl
(7.
sup
IDkv(&,*)1
z+rtB
Here
m=
1, and let
a e CW
and
N
<
Ck > 0, and nonnegative
zE
k = 0, 1, 2, ..., so that if
x (z
u e r(R
0 < r < E, and
for every
for this chapter.
There exist a polynomial
E > 0, then there exist constants
integers
3
such that:
(0, 1)
If
-z
P bea pxq matrix over C{, w], let
be the open unit ball in C.
tE
D =
with
Ck(l+r
Cn,
then there exists
Pv = aPu
in
R
)(1+1z124-W2+r2)Nk
x (z
+ rtB)
and
sup 1DiPu(E:,*)1
Z=o z+rB
...
and
E R.
depend on
and
k
The proof is by induction on
n = 0, p = 1.
P.
n
and
P = (P1,
p.
and we may assume
P/ 1 0.
Pct)
Leta=PleCW
'
V=
(v1,
vq).
are clearly true.
v1
=Pu,v.=0, j
Then
= 2,
..., q
and
3
Pv = P/Pu = aPu
and the estimates
35
(ii)
n > 0
Let
p > 1
and
and assume inductively that we have
proved the lemma for all
P1
Consider
a polynomial
a'
n' < n
f
= -Pll'
E
C[V
negative integers
and
Pig).
< p.
p'
By hypothesis, there exists
and there exist constants
and
v' c r(R x(v-rt'B),Eo)q
N'
k'
t'
so that
E (0,1)
P1v'
=
Ci, non-
and
a'P1v
in
R x (z+rt'B)
and
IDkvi(c,-)1 < C/'((l+r
E
)(14-1
sup
12+1zI2+r2)Nk
ID'Q'Flu('')1
9..=o z+rB
for
E R.
cmzig
CI,zir
Let
C[Ez]
be exact.
By Lemma 2.3
F(R x (z+rt113),E0)q
F(R x (z+rt'B),E0)r
1
F(R x (z+rt1B),E0)
f E F(R x (z+rt'B),E0)r
so
Consider the equation
p a" E
1
equations.
g E
r(R x (z+rt't"B), EO)r
in
R x (z+rt't"B)
(iii)
Let
n > 1
and
and
Qf =
Let
- v'.
PQg = PQf, which is a system of
By the induction hypothesis, there exist
CfV,CNj, and
t = t't"
is exact.
t"
6 (0,1)
such that
with the estimates.
and there exists
PQg = a"PQf = a"P(a'u-v')
Set
v = Qg + a"v',
a = a'a".
p = 1
proved the lemma for all
and assume inductively that we have
n' < n
and
p' c m
36
Pq)
P = (P1,
of
in w
P
= bA , where
P
A (,.) g 0
for
under the evaluation map
where
P(p,n)
polynomials in
Let
b c CE
]
,
Aq E
w.
in
P
04] , and
C[
C.
E
K CB - (01 C Cn
Let
and the degree
u = maximum of the degrees of
is
We may write
Pq X 0
We may assume
be a finite set such that its image
is a basis for
P(u,n)'
B
P(p,n)',
is the finite dimensional C-vector space of
of degree
C[w]
< p
and
P(u,n)
is its dual.
be the dual basis of
{1,(e) le c K} C P(u,n)
P(p,n).
Then
A (,w)= IeEK
and
Aq('e)L(e)(w)
-A- (E)(z:r) < C maxIA (Z,z+re)I
eEK
where
u
depends only on
C
q
and
Cr)
V(e) = ((,w) c R x
Let
IA (Z,w + re)I
open cover of
n.
such that
> 1/2 max IA (,w + re')}.
e'EK q
R x Cn
and, if
A (&)(w:r) < 2CIA (, w + re)
(0,,,)
I.
(V(e)le E Kl
is an
c V(e), then
Define
q
fe(T) = A(, w +
has
a < p
re T)
roots, which we call
(1/2 - 1/8u, 1 + 1/8u)
si
into
T1,
2u + 1
2a < 2u
R, there exists
Isi
-
>
2.
i"
E
(0,
for each
...,
20, such that
j=1,
fe
equal intervals and let
of these intervals.
...,
C.
We divide
Ta.
=1/2+Tibetheirlilidpoints.(1TWj=1,
intersect at most
TC
c CET] , a polynomial in
.
,
.
a}
can
So for each
37
V(e,i) = {(,w)
Define
SE
{V(e,i)Ie
K, 2=0,
E
E
V(e)IA (Fw + rse) # 0
IsI
with
C
20
...,
inf
< 2C(Bu+1)
22.-1
L2
8u
(,w)
8u
2
R x Cn
E V(e,t),
(,w+rseT)I
IA
ITI=1
If
2t+11
1
'
is an open cover of
consisting of semi-algebraic sets.
()(w:r)
rl
for all
q
TEC
zE
Assume now that
Cn
is fixed and
(,z) e V(e,t).
We
make a unitary change of coordinates and so assume that
ATh(z:r) < 2C(8u+1)
where
=
s
inf
1A,(,z',T)I
IT-znI=rs
Define W = W(e,t) = V(e,i)(z).
Iskel.
We apply Lemma 4.1 to the polynomial
and we find
Aq
s' c (0,1),
A={wEcnIlw.l<
A+, Aand
g,h e
E r(w
r(w
x
s'
rA), E0)
u =Ag+h in Wx (z
the Lemma.
where
T
Also, h
< sl,
114nI
rA), AO),
(z
(z
n-1,
j=1,
+ rA)
such that A
=
A+A-
and
and we have the estimates from
is a polynomial in
wn of degree
+
N
is the number of roots of the polynomial
Aq(r,Iz'T)
in
IT-
znI
< rs.
By the same argument as on page 197 of Hermander (1966), we
see:P1f1+..+Pq_Ifq-1+Aqfq=h1dheref.EF(W x(z+rA),E0)
38
and
are polynomials in
fq_i
fl,
is a polynomial in
Thus
of degree < N+ < p.
(of degree
wn
Aq fq
wn
by Lemma 7.6.9 of HOrmander (1966),
and,
< p + N+)
is a polynomial in
Aq fq
(of degree
p).
<
By Lemma 3.2, there exist
(e'
wn
functions
C
such that
E K, 2=0, ..., 2p)
tp(e',
9.,1)
on
R
supp(1.1)(e',Z1)) C W(e',2.1)
2U
and
E
4)(e', Z') = 1
e'EK Z.=o
P f' +
11
and
.
+ P
.
and
f! = 4)(e, Z)A f.
+ A f' = h'
q q
f'
q-1 q-1
are polynomials in
f!, h
Set
R.
Then
Z)A-h.
h' = 11./(e,
on
of degree
wn
< p.
Also, the
support of these functions is contained in
W(e, Z) x (zT+rA')x C.
This equation is equivalent to a system of
2p
coefficients (of powers of
h'
of the
wn)
f!
equations in the
on the left and of
and zeros on the right and in the variables
(, w').
By the induction hypothesis applied to this system, we can
find a polynomial
a2 c CP.,1 ,
nonnegative integers
Z=0, ..., 2p,
M1
and
constants
and, for
N1
2p
PP
+
1 1
and, if
a
j=1,
Fjz E F(R x (zi+rs't, B), E0)2p
F.(,w) =
functions
E
Z=0
F.
3Z
(,14')w2,
+Pq-1 Fq-1
satisfy
+ AF
q q = a2h'
is a nonnegative integer <
and
C1 > 0
k,
t1
(0,1),
q,
such that the
39
2p
q
E E
-M
sup
lpcIF
j=1 Z=o
z'+rs't1
*)
< C1 (l+r
I
12+1z,r2+1.2)N
1)(1,1
'
B'
2p
E
sup
z'+rs'B'
z=o
where
*)
2p
Cl -1
B' is the unit ball in
ID 511i(E,
hi(E,w) = E hz(E,W)wn.
and
L=0
Now set
Set
j
vi(e,Z) = alFj/A
< q
and
vq = a2t1)(e,Z)g +Fq/A-.
Then
a(e,Z) = a1a2.
Pv(e,Z) = a1(P1F1 +
+ AqFq)/A- + ala211)(e,2)Aqg
= a(e,Z)1,b(e,Z)u.
We made a unitary change of coordinates in
Cn
earlier, and
now we make the inverse coordinate transformation; this does not
affect
a(e,Z)
or
11)(e,Z), which depend only on
E.
a =a(e,Z) and
Set
eEK, Z=o,..., 2p
2p
I av(e,Z)/a(e,Z).
v =
a E C[
and
1
v
E F(R x
(z+rtB),E0)q
eEK Z=o
t = min {se, s'(e,9)t1(e'
where
e
K, Z=o,
E
20.
Then
Pv = Ze,i aPv(e,Z)/a(e,Z) = Ee,2a4)(e,Z)u = au.
We will estimate
v
A (E,w) = E a ()wY =
Y
ct
Y
Y bY
now.
Now
(E)(w - z)Y.
(w) zw-Y aw().
b (E) =
Let
Here
a
and
w
are n-multi-indices
w>1
of length
n
There exists
< p.
Co > 0
depending only on
so that
o
Y Y ()1
1A
A ()(z:r) <
q
lb
r
21y1
< CA ()(z:r).
o q
p
and
40
0 < r < E, E2(11-171) > r2(11-171)
Since
r2Iy1
u
1)
E-2 (U-1Yr2
>
_>
0 < E<_ 1, r21Y1
b() =
Let
,w) =
c (
R
on
c(E ,w) = bvi(E)
lb ()12
2u
c(,w)
-1
and
Notice
C.,
D
.
so that
A
1
(F.,
R, w
E
,w)1
C1
C-1 r214(1
W(e
b()
-1
<
In the unit ball in
.
Thus, if H=rnax{ 0,4 X] }
2
2)L,
+
L
2
C., (1 +
1E1
+ 1z1-)
Thus
.
+ IV
) ,
+
2
a
I
z
2)-1'
,w) > C4r-
z + rtB, from Lemma 4.1 (since
11014 follows from the e
Cn.
2)H/2 < C3(1 +
So
depends on A
and w e Cn. (Notice
E
p. 276, there exists a constant
,
(,w)1A1
1
(w)1
< C3(1 +
Recall that on
E
C(E ,z) = b().
has a positive lower bound.
(z:r) >
w
These are real polynomials
2.
By a result of B. Petersen using
0.)
bw(E)
C-21 min (1,
L = 1 +[H/2]
where
I
for real
0
c(E ,w)
Thus I< C2 max (1,
R x Cn, c
and
lb (E)12
IY1<U
Y
and a rational number
>
and if
C1r
Lemma 2.1 of Hormander (1969)
Ic(E,w)1
,
so that
E
( )ww- Yaw()
and
> E-2 11/-2u
Thus there is a constant
R2n+1, respectively, and
A (E,' ) g 0
C2 > 0
_< u.
1y1
n, and
I
and
if
>
(z:r)
3
If B> 1, r2 1Y1
.
r2ij,
u,
depending on
C1
and so
.
N+
tB C A).
estimates of
(z :r)
for
The estimate of
g,F.
,
etc.
41
Lemma 5.2
open unit ball in
Nk
u c (F(R
x (z
such that
rtB), E0)
E
f-(/)z+rtBV
ID v('')I 2 e
J
CW and
Ck > 0
6 (0,1)
t
and nonnegative
+ rB), E0))q, then there
Pv = bPu
is any continuous real-valued function on
(I)
Nke d
r
b
be the
B
such that:
Cn, 0 < r < E, and
z
C[F,,w], m=1, and
There exist a polynomial
Cn.
v E r(R X (Z
exists
and if
and
Mk
If
matrix over
E > 0, then there exist constants
such that, if
integers
pxq
P bea
Let
+ 1E12)
.< Ckea(1
x (z
R
in
+ rtB)
z + /11, then
k
JrID2Pu(,)12e-(1)dV
Z=0
where
Proof:
ties.
a = sup
z+rB
icp(w')
and
(w")1
-
z+rB
.0(w) = log (1 +
11412).
The proof is in Petersen (1974) and uses one of Peetre's inequaliThe proof in Petersen (1974) does not involve parameters, but the
proof still works in this case.
t c (0,1)
Definition: If
is good for
t
Let
Lemma 5.3
s
E (0,1),
Pl,
Pq
P (,.) ,E 0
P.
U
be an open, connected subset of
C > 0, and
E
satisfies the conclusion cf this lemma, we say
C[Fw]
and
p
and
k
Rm,
c Cn, r > 0,
z
be nonnegative integers.
all have degrees in
w = (wl,
()(z:r) <C inf
Suppose
wn)
< p,
for
F,
p, n, s, and
C
z', T)1
6 U.
1T-zn1=rs
Then there exists
that
ker (P) C F(U
s'
x (z
(0,1)
depending only on
+ rp), Ek0)q
elements which are polynomials in
wn
is generated by those of its
of degree
< p.
such
42
A = {14 6
Here
ell
114.1
Cq) E r(u
j < q,
< SI
x (z
< s}
Iwnl
and
c.P.
rA), Eko)ci
j=1
Proof:
.
3
3
This is a straightforward application of Lemma 4.1 and Lemma 7.6.9
of HEirmander (1966).
Lemma 5.4
Let
be the unit ball in
B
cWwIr
be exact with
constant
(z
rB), Ek0)cl
E r(u x
(z
rtB), EkO)r
Proof:
C[
(i)
Let
,w113
b 6 C[]
and a
is a nonnegative integer and
with
Pt = 0, then there exists
Qg = bf
such that
X (z
U
in
+ rtB).
The proof is by an Oka induction and the second and third cases
are very similar to (ii) and (iii) of the proof of Lemma 5.1.
nomial
R.
such that:
(0,1)
r(u x
f e
P
,w]cl
Then there exists a polynomial
in = 1.
z e Cn, r > 0, k
If
g
t e
Cf
and U be open in
C
b
The poly-
comes only from the proof of the first case, which we present.
n= O.
Let
E bea
qxq
matrix over
C(c)
is the unique reduced row echelon matrix of
Qg' = f
iff
Sg' = Ef.
so that S= EQ
Notice that
Q.
be the rank of
Let
Then there exists a permutation
ki <
Q
(over
< k,:;1. <
C()).
< Zs
of(1,...,rlsothatS..=Oifj<k.and
S.
13
1
= 6...
13
ikj
Then define
g'
by
gi. =
1
E
Z=1
E.
4
f
and
0 =
E
=1
By Theorem 1.1, we can solve
Q()g'
SVg'
off a proper variety.
= E()fM, for
= f(E)
E.,f
if i >p.
14 1
and so
S(F,)gi ()
43
is a reduced row echelon matrix, so the last
q -
rows are
11
zero, and thus the second condition in the definition of
Now let
entries of
and
Lemma 5.5
is satisfied.
Let
6 CIV
a
and
B
6 (0,1)
and
E > 0
If
and
P
t
E
be the least common denominator of the
b
Now
g = bg'.
be as in Lemma 5.1.
There exist a polynomial
and
NI
v
C(l+r-m)(1+ W2+lz
<
2
(z+rtB), Ek0)q
1DiPu(E,.)!
j=o z+rB
R
E
Proof:
E F(R x
N
+r-)E
sup
z+rtB
for
z E Cn, 0 < r < E, and
so that if
N
C > 0
and
IDv(,) 1
sup
Qg = bf.
is a nonnegative integer, then there exist
k
U c F(R x (z + rB), Ek0)q, then there exists
Pv = aPu
Sg = bEf, so
such that:
and nonnegative integers
with
g'
and
Z=0,
k.
The proof is the same as that of Lemma 5.1 except that we use
Lemma 5.4 in (ii) rather than Lemma 2.3.
Lemma 5.6
P bea
Let
open unit ball in C.
such that, if
constants
matrix over
and M and
u
v
r(R x
such that
if
rM
rtB), Ek0)q
E
r
Jz+rtBIL)
2
e
b c CEFJ
B
and
be the
t e (0,1)
F(R x (z + rB), Ek0)q, then there exists
Pv = bPu
is any continuous real-valued function on
(I)
,m= 1, and
nonnegative integers such that:
N
z E Cn, 0 < r < E, and
(z
,14]
is a nonnegative integer, then there exist
k
If
E
C[
There exists a polynomial
and
E > 0
C > 0
pxq
dV < Ce
a
(1+W
2 N
)
r
in
R x (z + rtB)
and
z + r-B, then
Z
E
jz+rB
3=0
DJ Pu
)!
dV
44
e R
for
a =
and
2=0,
k
where
8(w) = log (1
+
11412)
and
- gw)1.
lq)(141)
sup
z+rB
The proof is the same as Lemma 5.2 and the proof in Petersen (1974)
Proof:
(without parameters) works in this case.
Definition
that
t
If
t c (0,1)
is good for
P.
satisfies the condition of Lemma 5.6, we say
For a given
two definitions of good for
P
agree.
P, t
is independent of
k
and the
45
CI-IAPTER SIX
SOLUTIONS OF THE CAUCHY-RIEMANN AND COBOUNDARY EQUATIONS
In this chapter we solve
estimates in the
-ii(E-) =
with weighted
-)
variables and smooth dependence on
z
L2-
The proof
of this result is a modification of some arguments of L. Hormander (1965).
We then solve the coboundary equation
5c' = c
with estimates and smooth
dependence on parameters.
We will use the notation of Hormander (1966, 1965) and some special
Suppose
notation.
and
is open in
Rm
are nonnegative integers and
q
2
then
to
U
L(p,q)(U, W.
2
L(p,q)(W, (0).
cp)
W
and
On.
is open in
If
p
is a measurable function of
cp
denotes the set of continuous functions from
L2
(P,q )
is the Hilbert space of
(p)
W,
U
(p, q)-forms
whose coefficients are square integrable with respect to e-c1V and has
the norm
fw lce-
c
1c12
where
la1=P
Let
b
be a nonnegative integer or
2
the subset of
W, ¢)
+co.
!2.
lc
L2
Pq)(U,
151N
W, ¢, b)
denotes
consisting of those functions which are
L(p,q)(U,
Cb
functions as mappings from U
2
to
¢)
L(p,q)(W,
represents the Cauchy-Riemann operator in
variables only).
and if
a
D.
On
(i.e., in the
represents the partial derivative with respect to
is an m-multi-index, a = (al, ..., am),
Da
=
.Dm.
Dal
1
Let
W be an open set in
Cn with
C2
boundary; that is, there
n
exists a real function
z
p E C-(C)
such that
W = tz
Cn p(7.)
< 0},
46
Cn1p(z)
= (z e
and
= 0}
dp
on
n
E
0
pseudo-convex (or Levi-convex) iff
j=1
zeM,E....1(z)hr.--.0 and
DW = the boundary of W. W
n
E
w.71, > 0 when
k=1 azju'k 3
w = (.41,
6
wn)
j=1 azj
in
C.
If
W
is
is open
is pseudo-convex iff there is a continuous plurisubharmonic
Cn, W
in W such that
u
function
= {z
C C: W
c
for all
c c R, where
C}.
WIU(7.) <
A function
W
defined in an open set
u
W C Cn
is plurisubharmonic
iff
is upper semicontinuous, and
u
for
and w e Cn, the function
z
in the part of
open set
W C Cn
j=1
z e W
with a
C2
n
U
2
TW)
function u
1(
be open in R.
W
Let
pseudo-convex boundary, let
ci5
W, and let
be a bounded open set in
c C-
value of the matrix
Dzj"lc
f e Lfp,q)(U, W,
eK c C(W)
be the lowest eigen-
)
1)
such that
q > 0,
) = 0
,2
fwlf(, )
fwiDif(E -)12 -4)dV < 00
for
-(15-K
e
j=1,
aV <
.7
m
Cn
be strictly plurisub-
( 32(I)
and
defined in an
O.
harmonic in a neighborhood of
Suppose
is subharmonic
u(z)
k=1 Dzj3Tk
and w e Cn, w
Let
Lemma 6.1
C2
U(7.
is strictly plurisubharmonic iff
n
for
A
where it is defined.
C
T
and
c U.
47
and
q
L(p,q_1)(U, W, 0)
such that
5u(,-,
= f(,
-)
-)12e*KdV
-)12e-g5dV <
for all
Proof:
2
u
Then there exists
U.
We will use the notation and results of Hdrmander (1965).
2
A be the operator on
L(p,q)(W, 0)
Let
defined by
Ah = AT eK/2 h c L2(13,q)(W, 4)).
A
is bounded and self-adjoint.
For
(1965, p. 99).
h
Let
dom (T*)
n
< IIT*h
lAh 112
S
and
T
be as defined in Hormander
dom(S),
II2c5
11ShIl
As in the proof of Theorem 1.1.4 of Hormander (1965), we see
l(f(, .)
for all
F()
E
c U
and
2
(1,(p,q)(W,
Ilh(, -)110
dom(T*), where
g
I))/
R(T*)
lower bound (Hormander, 1965).
fkz = T*a )
is closed since
Therefore, F(ç)
lh()11
A
has a positive
is well defined for
ile-K/2f11/
be an orthonormal basis of
R(T*)
v6i
in
L2(p
2
define u() E L'(p,q-1)
01
0)
by
u() = E F(F,)( kz) kz =
for
Define
and
11F()
Let
f(E -) = A*11(, .).
by FW(k) = (f(, -), g)0 where
k = T*g + k2, k2 c R(T*)I.
e U
IIT*gll
c U.
Notice
-), gzyz
01 , 0)
and
48
111-4)(')11(2,
i1F()( kz)
=
u
It remains to show that
since Tu(C, ) = f(C, )
Now the map
E
12 <
L2 pq-1)(U, W, ¢)
.)12edV/q
where
u(C, -) = u(C)(.),
by construction.
(f(C, ), g)
C
fwifCC,
Also,sinceSisclosed,
c, )
if
C1(U)
is
= 0
for
2
g
0).
L(p,q-1)
C e U, so by Theorem 2.2.1
of Hormander(1965),D.f(C,.)=Tw.(C)(.), where
3
2
w.(C) e L(p,q-1)CW, (P)
exists
rN(T)1
for each
depending only on T
C > 0
and
C c U
so that
1 <
11wilIcb
<
j
There
< m.
CI1Twillq)
(Theorem 1.1.1 (HOrmander, 1965)).
From Taylor's Theorem, we see
1(f(c,
f(°, ),
m
1
(D.f(tC + (1-t)e, )(C. - C?), g )
=
=
o j1
(f1
o
dt
1
3
3
3
m
2
1
<
gzY
(w (t+ (1-t)e), kz)cp12dt)1/2 IC J
j=1
2
Then
E I(f(C,
f(°, *),aoz
)
')
1
Jr
2
(1)
in
I2
(w.(tC + (1-t)°), kz)2
(151
dt1
0
j=1
1
m
o
j1
=
E
lea.(tC + (1-t)°), k )
111,1-(t
1
c I,
o
12 dt1C
cp
3
m
j
02
-
3
3
(1-t)e)
2
Icp dt1F,
-
e
2
02
where the next to last inequality is by Parsavel's relation and
C1
49
depends on
T, f, and
Ilu(C,
Let
Lemma 6.2
U, W,
and
(1),
is a nonnegative integer or
with -5f(C,
E L2 (p,c1_1)(U, W,
- )
<
a
for all m-multi-indices
Proof:
and
Jrw
with
L2
(p,q-1)(11
to
f c L2
1
-cp-K
dV <
....
)
T*gd
and
u(C, ) =
We may assume
b.
Then
l(f(C + he., ) - f(c, ) - hy(c, .), gz)qsi
1
Ii(hD.f(C + thej, .) - hDjf(C, .),
gzyt1
=
2
-
Th.12,
i
OD.f(E; + sthe., .)th, g ) dsdt1
1
< !hi
2
1111
1(w.
010
Jj(C
+ sthe.' ), k2, ) dstdt1
3
11
(I I w..(C
o
o
33
+ sthe.
),
and all
)
kZ
12dst2dt)1/2
b = 1
and
is independent of
E(f(c, -), gz)1(
2.
previous lemma.
C c U
(q > 0)
< b.
prove the lemma in this case, since the choice of u
{kz =
for
b+1)
b
2e - (P-1(dV
We will prove this by induction on
Let
(1),
Suppose
and
-Tu = f
Daf (c,
lal
W,
Pq)(U,
(i))-
Then there exists
< b + 1.
such that
I
), gz)(p12 <C1
be as in the last lemma.
K
+.
lal
b)
Dau(C,
q
U
Thus
say).
f(e,
IwIDaf(C, -)12e
a with
m-multi-indices
u
and
= 0
.)
< I
= E l(f(C,
*)112(1)
is continuous as a map from
u
implies
u(e,
*)
-I
(for
C°
be as in the
b.
SO
), w..(E,
Thus
f(E, ) - 11.D.f(, ), o)
) -
+ he
Zi(f(
N(T)i.
E
JJ
JJ
12
11
Esr Jr 1(w. .(E
<
o
o
.Q.
+ sthe., .), k )
3
JJ
12dst2dt 11114
2,
,1 r1
i
00
j
-- -
-)112t2dsdt 11114
,E, + sthe. ,
1
Iwii(
< c11114
J
by Parsavel's relation and Theorem 1.1.1 of Hormander (1965).
limilu(E
+ h)
-
-
wj(E, ) = /:(Djf(E, ),
where
.)11cl)/11-11
j=1
h-oo
u e L2 (p,q_1)(U, W, 0, 1),
-) = f(E, ).
-)
=o
4)).
c
gz)(1)kz
),
=
Thus
Then
g9.)q5kz
and
,
Similarly, q I-1:11Dju(E,.)12e-gcldV = q fw1wi(E,)12e-(1)dV
<f1)
1-f(,
-)12e-'15-KdV
.
3
Theorem 6.1
Let
W be a pseudoconvex open set in
plurisubharmonic function on
of
plurisubharmonicity
Let
cp.
nonnegative integers with
Suppose
f c
W, and
a
Rm, and
p
and
q > O.
2
W, O, co)
with
-9-f(,
and
.) = 0
L(Pc1)(U,
,
fwiDaf(E, )12edV <
for Ec U
and all m-multi-indices
2
u E L(p,q_1)(U, W,
q
for all
E e U
(1),
m)
such that
IwIDau(,-)12e-dV
<
a.
Then there exists
7U(E, )
= f(E,
.)
fwiDaf(E,.)12e-4-KdV
and all m-multi-indices
a.
C2
be a lower bound for the
eK E C(W)
be open in
U
Cn,
and
q
be
51
2
Proof: Notice L(p,q-1)(U,
W,
a e Cm(W)
WM =
be a strictly pseudoconvex function on
is relatively compact in
{z 6 W1a(z) < M}
and
q
W, (0,
co)
pseudoconvex boundary
Cm
such that - gz(E, -) = f(E, ) in
f 1Daa (E, -)12e-15dV_<
-)12e-q5-KdV,
Iw
w
m-multi-indices a, and 2=0, 1, 2, ..., where
is an increasing sequence of reals, and M
by zero outside
gz
(Theorem 2.6.11 of
W
Then, by Lemma 6.2, there exists
(Hormander, 1965).
c L2pq-1)(U,
..
as
co
{MZ=0, 1, ...}
,
We extend
2
fgZ1Z=0, 1, 2, '..1 C C (j, L(p,q-1)CW,
(p)).
so
Wz,
= W,
Wz
U x Wz
for all E c U,
Notice this sequence is bounded in C(U, Lfp,c1_1)(W, (0)).
be an orthonormal basis of
1}
Let
such that
W
Hormander (1966)) and for almost all M, Wm has
g
()).
= Cm(U, Lfp,q_1)(W,
co)
L2(1),c1_1)(W, (0).
Let
Notice that
((gzC-, -),tc).(-))b 1Z=0, 1, ...} is bounded in Cm(U), a Frechet-Montel
space, for each j=1, 2, ... . Then there exists a subsequence (gz(1)
g
of fg }
such that (a°(1),
Z
a subsequence
{g
(s)
1
)
4)
a1 E C(U).
Similarly, there exists
of ia(s-1)} such that
(s)
(g2.
' 4)S)cP--.'aS e Cm(U),
for s = 2, 3, ..
Let
and
u(E, -)
Set vt(E,
Z
-0-
u(E, ) =
j
J
u E nU,
We claim that
3
is in the weak closure of (gz(E, -)1
=
4)(E,-)
co, for j=1, 2,
...,
Now (vz(E, -),
and so if
itpdj=1, 2, ...1 then (vZ (E, -),
(,
qij)(1)
-4-
L2(p,c1 1)(W4))
for each E E U.
.(u(E, -), ¢j)(0 as
is the subspace generated by
S
-4-
(1(E, -), ¢)
(1)
c S. Now let A c L-pq-1) (W, ¢). Since
S
as
2.
-0- w
for all
is dense in 1,-(p,q-1)(W, ¢),
52
if
E > 0, there exists
so that
IP E S
1(vtg, *), 'P)(1)
where
depends on
2,
) - u(, *),
i(vzg,
)
- u(,
*), 4))1
C()iin - 4)14
e
- (u(, *), 11(t,1
1(v2,(E, '), A
C()11A
< E.
For some
2.
> 0,
< E,
Then
of course.
1(/2,g,
11A - 1014)
-
1PY + 101(,
),A -0951
1P11gb
(1 + 2C())c
where C()
= IIf(,
weakly (in
L(p,q_/)CW, (0))
in
-)110.K.
2
Now let
Then
e---0.
to u(, -), for
vz(E, -)
converges
c U. Thus -5.1.4,-) = f(,-)
U x W, by standard arguments.
Now Dagz(,-) =
(Dag(), 4)j)
=
j
relation,
.
()I2 < liDaf(,')1142)+K
Daa.,g)tpi.
Jx,
for all
E
By Parsavel's
U
and m-multi-
j
qEiDaaz
indices
Thus,
a.
zIpaa.()12 < sup q ZI1Pa2,g)12 < IlDafg,')Iic2p+K
j
3
so "Dau(,-)" defined by
T Daaj()4)j
Lfp,c1-1)(W4), for each
U
anda.ileclainthatp.u, -), defined as the limit of a difference
3
quotient, exists and is continuous.
u
...
C
2
C (U, L(p,q_1)(W, (1)))
This will show that
(i.e., the proof
will
be clear).
53
u(+te.,-) - u(,-)
I.
El.u(E-)11,
11
cP,
(a (+te )
aZ())
I
= 11E
(D.a
)
3
112
lp
Z
- D.a ())4) 11j
Z
Da))
-
Z
Z
2"
(by the Mean-Value Theorem, where
is between
and
+ te.)
3
ZD.a
=
Z
(where ei
Zj
2
2"
is between
<
2,
3 Z
2
Z
)()4)z11(I)
and
(e.j)121
e)
-12
3
< It 12 Z1D2a (e))12
j2.
Z
--C2,X1t12
where
max
sup
C2
4,A
c..)(
11Daf(V,
U
containing a neighborhood of
u
is differentiable.
and
12
co+K
lal<2
Thus
X
u E C(U, L2
A similar argument shows
and Du is differentiable.
is a compact subset of
Now11Dau(,.)
u
E
pq-1)(W4))
and
Ci(U, L2cp,q_1)(W,(0))
1,2
Elpaa.(012
11,4)
ji
j
11Daf(E;,-)1124)+K
for all
E
Theorem 6.2
U
and all
Let
W,
a.
eK, U, p, and
q
be as in Theorem 6.1.
be a nonnegative integer.
Suppose
f E L-
(,
Pq)(U,
W,
b+2)
with
.f(E.-.) = 0
and
Let
b
54
fw!Daf(,.)12e-(1)-1( dV <
U
for E e
and
such that
.1_1(E,-)
for
and
E e U
Proof:
Then there exists
< b + 2.
lal
= f(E,)
and
u
Lfp,q_1)(U,
E
W,cp, b)
qfwIDU(E,-)12e-cipdV < fw1Daf(E,-)12e-4)-KdV
< b.
lai
We modify the proof of Theorem 6.1 as follows:
We note that the inclusion of
the sequence
into
is compact and
Cb(U)
2
is bounded in
(gz}
Cb+1 (U)
Cb+1(U, L(p,(1_1)(W,q5))
The proof
then follows.
W
Suppose
cover of
is a domain of holomorphy in
W, and
is open in
U
If
Rm.
is either a nonnegative integer or
b
Cn, fW
> 1}
j
is an open
is a nonnegative integer and
s
+w, then
Cs(U,(W.),Z
(P,q),ct),b)
denotes the set of all alternating s-cochains
(s + 1)-multi-index) with
for
tor
((5)
Lemma 6.3
(W.Ij
> 1)
(where
a
is an
2
Ca
L-(pq)
CU,
,
n
"1
Here W
E E U.
c = (ca)
= W
cp,
Wa'
and
b)
-5Ca(E,-) = 0
We define the coboundary operaWas.
in the usual manner (Petersen, 1975; Hbrmander, 1965).
Let
W
be a domain of holomorphy in
Cn, U
be open in
Rm, and
be an open cover of W with the properties of Lemma 5 of
3
Petersen (1976) (and A, B
and
a
as in the lemma).
strictly plurisubharmonic function in
W
and let
lower bound for the plurisubharmonicity of
some real
If
with
s
L.
Let
b
for
be a
cl)
C2
be a continuous
K < L
on
W, for
be a nonnegative integer.
is a positive integer, then for each
6c(E,-) = 0
eK
with
4)
Let
c
CS(U, (W.),
E c U, there exists
c' c Cs-1(Uj)
Z(p,q),
b)
p,c1)4+K,b+2s)
55
Sc'(,)- = c(,.)
such that
and
C11Dac(,*)11q).4.1(
for
and
U
E
and
lal
Here
< b.
1,b(z)
is a constant depending on
C
Proof:
=-
log d(z) = - log(minfl,dist(z,3w)l)
A, B, a, L, and
b.
The proof is essentially the same as the proof of Theorem 6 of
Petersen (1976) with the lemmas here replacing those in Petersen (1976)
Note that the lemma is true with
as necessary.
replace
b =
co
if we
by Cs.
C
Lemma6.4LetW,U,(W.)be as in Lemma 6.3 and let
subharmonic function in
nonnegative integer, and
Suppose
s
be a
is a positive integer, b
C2
pluri-
is a
q > 0.
c c Cs(U, (Wi),
If
C E
W.
p
b + 2s)
such that
6c(,-) = 0
for
U, then there exists
c' E CS-1(U, (W.),
3
such that
dc'(,-) = c(,.)
Z(pq)
274)
+ 20, b)
and
liDac'(E,*)11cp+211)+28 < CI IDc(,) 14)
if
e(z) =
E
U
and
log(l
a
is an in-multi-index with
+1z12),
11)(z)
= -log d(z), and
1a1
C
< b.
Here
depends only on
A, B, a, m,
n, and
b.
Proof:
The proof is essentially that of Corollary 7 of Petersen (1976).
Note that Lemma 6.4 is true with
k =
if we replace
C
by C.
56
CHAPTER SEVEN
GLOBAL EXISIENCE AND FUNDAI\ENTAL SOLUTIONS
In the chapter we prove an existence theorem with estimates similar
to Theorem 1 of Petersen (1975).
Using the division theorem we prove,
under certain conditions, the existence of temperate fundamental solutions
R x F, where
with support in
A
is a closed, convex, salient cone.
F
simple counterexample of the converse is provided.
If
isa
P
pxq
matrix over
to be the set of alternating s-cocycles
U
n
n
x Wa = U x
(Wao
f
E F(U x
c = (ca)
with
on
f c L2
W, Eb0)
n
L2(U, Wa, ¢, b)
and V C C W, then
00)(U V Z(00),
W, then
4)
,
F(U x V, Eb0).
f
Pca = 0
in
and
b)
.
f c L00)(U, V, Z(0,0), ¢, b)
for any locally (lower) bounded measurable function
if
Cs(U,(Wi) ,R(P)
Was),
ca c F(U x Wa, Eb0)q
If
we define
,z]
C[
on
¢
W.
Conversely,
for some plurisubharmonic function
¢
(The proof of this last comment follows
from Theorem 2.2.3 of Hormander (1966) and the fact that
is upper
¢
semi-continuous).
Lemma 7.1
and
Let
m = 1, U
j>1
(1v.)
3
Z=0, ..., 3n + 3
be open in
R, W
plurisubharmonic function in
<
d(z).
Cn,
be a collection of open covers of W with the
properties of Lemma 2 of Petersen (1975).
lz - wl
a domain of holomorphy in
W with
Let
¢ (z)
-¢
and
L > 0
<
L
be a
¢
if
z
E
W
C-
and
57
Suppose
pxq
isa
P
a Hilbert resolution of
good for
(20,
matrix over
(as matrices over
P
so that
Q2.
and
Q
0
n
9'
X
t 6
C[ E, z]) with
is
-
(0, 1)
satisfies the condition of Lemma 5.4.
t
a 6 C[E]
There exists a polynomial
C[E, z]
(independent of
t) and a nonnegative
integer E such that:
s > 1, b
If
is a nonnegative integer, and
c
Cs(U, (W.), R(P),(1),b +
3
6c(E, ) = 0
with
6 U), then there exist nonnegative integers
(E
M
and
N, C > 0, and
CI E
Sc'(,E )
and
3n+3
Oa.
Cs-1(U,
=
), R(P),
(/)
a(E) c(E, )
lID2. c' (E,')11(12)+mp+Ne< C(1 + W2)M
j=o
and
P
2.
= 0,
and
...,
b.
Here M and
N
are as before;
ky(z) =
-
log (d(z))
IID)c(Z, ')112
for
E 6 U
(I)
are nonnegative integers depending on
is a constant depending on
b, C
with
NO, b)
+
and
P, b, and
e(z) =
L, and
4
and
6
log (1 +
The proof is essentially the same as the proof of Theorem 3 of
Proof:
Petersen (1975).
However, we take three times as many refinements as in
Petersen (1975).
One set is due to Lemma 5.4.
since
r(u
f e
L2(U, W2.+1
a
,
Theorem 7.1
R, and W
x'a
Eb0)
may not be in
L-(U,
The other is required
Wa'
b)
but is in
b).
Let m= 1,P bea pxq matrix over
a domain of holomorphy in
Cn.
C[E,
z] ,U open in
There exists a polynomial
b c C [E] such that the following holds:
Let
u c
r(u x
b
be a nonnegative integer, 4
W, Eb+E0)q
(where
2
as in Lemma 7.1, and
is from Lemma 7.1).
Then there exists
58
v e F(U x W, Eb0)q
Pv = bPu
such that
f 141D2V(, -)12e--M4)-NedV < C(1 + 1Ci2)N
C e U
and
depending only on
and
P
Here M
..., b.
2. = 0,
and
and
b
C
and
)12e-dV
`2"
j=o
for
W
U x
in
-
N
and
are nonnegative integers
is a constant depending on
P, b,
L.
Proof: The proof is essentially the same as that of Theorem 1 of Petersen
(1975).
r
Let
be a closed, convex, salient cone in
r+ = {x e
r
Rn1
<x, y> > 0
y e Fl
if
and let
be its dual cone.
Salient means that
contains no one-dimensional subspaces, which is equivalent to the
interior
ro
+
of
being nonempty.
F
is a q-tuple of polynomials in
u e
Rn
r(R x
on W
iRn), E0)cl
= r+
and
(z)
iRn
-
with
c(w)I <.
7.1, there exists
z)I
ITY417('
where
C[C, 11
and
cp
is a
C2
L
if
z e W
lz
and
plurisubharmonic function
co
- w1
<
v e F(R x (Fo + iRn), Ek0)q
< C()(1
lz12)N+n+1 d(z)
for
2.
for
k +
d(z), then by Theorem
such that
-M-2N
= 0,
Pv = bPu
e R
and
and
2 =
are from Theorem 7.1 and
C
2 N
+ 1W)
=-nnn22n+N+M eL C(1
P = (P1,. ..P)
and
If
.
f1411)21pu(, -)12e-g5dV <
b e C[C], M, N, and
C(C)
m = 1
Suppose
DZP-u(, *)1120-
i=o
Then
distribution
T2'.(C)
E
wedefineT.:R---0-vF
3
D2.v(
S'(Rn)
is the Laplace transform of a temperate
-)
with support in
by T() = To(C)
3
j
F, for
= 1,
2. = 0,
..., q.
3
transform is as in Petersen (1972); defined folmally
by
k, and
Here the Laplace
59
L(f)(z)
I
=
e
-<z X>
We let
f(x) dx.
denote the Fourier transform in
F
Rn
S'.
Then, if
x e
r+0,
-<x,>
4dr,
(E), F(f))
'*>
=
dE
( v.(E, x + i.), f(.))
j
= (Dv(, x + i), f(.))
( e-<x'.>T(E), F(f))
=
if
f " and() has compact support.
-
as the limit of a difference quotient,
on
e<x
-1.111s
.
>
DT() = e-<x
1;H(Rn)ifxer+0,andsoaT(E)=TI:(V
1
T.(E) e
.
1
>
T.(E)
as distributions.
Since
Then
S'.
S', DT. (E) e
We define
y
3
1:127.(E)
= T51:(E)
(m), f)
e
S'(Rn), for
e
Ck (R)
f c
if
2.
k, and the map
= 0,
S(Rn).
By the division theorem (Atiyah, 1970), there exists
with
GA = 1
in
trivial in the case of one variable.)
some-norm
11
11H,I
S(R), so
on
Banach space determined by
Theorem 7.2
Suppose
(a
if A e C[E]
S'(R)
m = 1
1
I
and
C
0).
G e
S'(R)
(Of course, this is
is continuous with respect to
G e .51 C S', where
is the
SH,,
IIH,/.
Q =
(IQ Q q)
is a q-tuple over
'
c[,
which is not identically zero.
convex, salient cone in
Let
F
be a closed
Rn.
IfQ.(z)=a()Pg,z)forj=1,...,q(aandP.polynomials) such that
then
Q(- iD, D)
P =
(P1,
Pq)
has no common zeros in
has a temperate fundamental solution
R x (F++iRn),
S = (S1,
S )
60
R x F. Here
with support in
is the partial differential
Qj(- iD, D)
a
operatorobtainedbyreplacingEby,-4andz.by
3
+
e F(R x (Fo + iRn ), EE)
ui,
Let
Proof:
we may find by a partition of unity argument.
exist
u
where
b
iRn), E0)
(r+
and define
H,
I
with
G c S'(R)
=
F0
o
-
is the partial Fourier transform in the
Crj(E), Fo(g)(,
))
S
e
G E
S1. Let
(T.(E), F (g)(E,.)))
(G,
3
where
so that
by
S(Rn+1)
g)
Let A = ab,
Pu = 1.
(nonnegative integers) so that
on
S.
By Corollary 2.2, there
P), let
is given in Theorem 7.1 (for
GA = 1, and pick
k = H
r(R x
e
.
Pu' = 1, which
such that
-
E-variable.
Here
(in fact, rapidly decreasing) and we
applyGtothismap.(FremtheestimtesonDv.we see that
is continuous with respect to a fixed norm on
and
Si
Thus
<
S.
S(Rn), for all
T.(E)
E e R,
j<q,
is well-defined and a short computation shows that
is continuous with respect to a norm on
normswithrespecttowhichG,T.,and
S(Rn+1)
(depending on the
are continuous) and so
F0
S.ES'(Rn+l)forj=1,...,c1.ThesupportofS.is
contained in
3
R x r,
since
is the Fourier transform in
Fo
TheLaplacetransformofT.(E)
is
E
alone and
supp(Tj(E))c F.
v.(E, .), so the Laplace trans-
./
form of
(EAT. (E)
P(E, D)T(E)
j=1
P(C, D)T(E) = b(E)d e Si(Rn).
D)S, g)
=
Suppose
E
j=1
j
J
P(E,.)v(E,.) = b(E).
J
g
S(R). Then
(Q.(-iD, D)S., g)
3
( S., Q.(-iD, D)*g)
=
is
3
Thus
61
E(G,(T.(), F-1
o
=
D)*g)(,-)))
3
(G, (T()
=
)
0
(aG (EP.(,
D)Tj(), F-01(g)(,)) )
)
=
( b()6, F-01g(,.)) )
=
( aG,
=
(GA,
=
(1, F-01(g)(.,0) )
-1
Fo
(g)(E,0))
= g(0, 0)
since if
Here
S(R), then
f
D)
placing
-1
(
6, f ) = ( F0(6), Fo (f)
a
9
by
-
9
and 7c. by
is the partial differential operator on
by
Qi(- iD, D)
Rn
obtained
Rn+1
and
obtained by re-
in P(, D).
3x.
9xj
in
-1
(1, Fo (f)).
=
)
is the partial differential operator on
Q).(- iD, D)
by replacing
f(0) =
q=1
Notice that the converse of Theorem 7.2 is not true, even if
(see Enqvist, 1976).
z) = z +
solution of
P =
side function.
Theorem 7.3
m = 1)
and
-
1, then
(P1, P2)
Notice
Suppose
F
If we set
F,
S = (H 0 6, 0)
with
and Pl(E z) =
pxq
matrix over
a c Cp:1
so that
is the Heavy-
H
is a common zero of
P.
(2[, zl,
is a closed, convex, salient cone in
there is a polynomial
and
is a temperate fundamental
supp(S) = R x0}, where
= 0, z = 1
isa
P
T = [0,+ 00)
an
with
P(, z) = a()R(F z)
-n]
(with
p < q.
and
If
R
+
has rank
p
in
R x
(F
+ iRn ), then
mental solution with support in
R
x r.
P(- ID, D)
has a temperate funda-
62
has a temperate fundamental solution means that there is
P(- iD, D)
aqxp matrix
E
over
S'(RI1+1)
so that
P(- iD, D)E = SI
where
I
is the
pxp
Proof:
The proof of the sufficiency of the Lemma on page 248 of Lancaster
identity matrix.
and Petersen (1980) is still valid if we replace
F
by
R x
F.
The
theorem then follows from Theorem 7.2.
Notice that we only require that there is a polynomial
det M(E z) = a()Q(, z)
that
of
and the
P
+
0
.
1
x(q)
system
n
Q.
E
(Q
CI, z]
for each
P)
px
p
a c CE
submatrix M
has no common zeros in
submatrix of Pand J
p-multi-index (see Lancaster and Petersen 1980)).
such
is a
63
BIBLIOGRAPHY
M. F. Atiyah, Resolution of Singularities and Division of Distributions,
Communications of Pure and Applied Mathematics, )III (1970),
pp 145-150
L. Ehrenpreis, Solution of Some Problems of Division, Amer. J. Math. 76
(1954), pp. 883-903.
Enqvist, On temperate fundamental solutions supported by a convex cone,
Ark. Mat. 14 (1976), pp. 35-41.
L. Hormander, Local and Global Properties of Fundamental Solutions, Math.
Scand. 5 (1957) pp. 27-39.
L. HOrmander, L2 Estimates and Existence Theorems for the
Acta Math. 113 (1965), pp. 89-152.
7
Operator,
L. HOrmander, An Introduction to Complex Analysis in Several Variables,
Van Nostrand, Princeton, N. J., 1966.
L. HOrmander, Linear Partial Differential Operators, Springer-Verlag, New
York, 1969.
K. E. Lancaster and B. E. Petersen, Systems of Partial Differential
Operators with Fundamental Solutions supported by a Cone, Proc. Amer.
Math. Soc. 78 (1980), pp. 247-252.
S. Lojasiewicz, Ensembles Semi-Analytiques, IHES, 1965.
Malgrange, Existence et Approximation des Solutions des Equations aux
Derivees Partielles et des Equations de Convolution, Ann. Inst.
Fourier 6 (1955), pp. 271-354.
B. Malgrange, Division Des Distributions IV, Seminaire Schwartz, 1960.
B. Malgrange, Sur les Systemes Differentiels a Coefficients Constants, Coll.
N. R. S., Paris, 1963, pp. 113-122.
B.E. Petersen, "On the Laplace Transform of a Temperate Distribution
Supported by a Cone", Proc. Amer. Math. Soc. 35 (1972), pp. 123-128.
B.E. Petersen, Holomorphic Functions with Bounds, unpublished lecture
notes, 1974.
B.E. Petersen, Holomorphic Functions with Growth Conditions, Trans. Amer.
Math. Soc. 206 (1975), pp. 395-406.
64
B.E. Petersen, "Holomorphic Functions with Bounds", Complex Analysis and
its Applications III, International Atomic Energy Agency, Vienna,
1976.
J. Rottman, An Introduction to Homological Algebra, Academic Press,
New York, 1979.
T. Smith, Formulas to Represent Functions by their Derivatives, Math.
Ann. (1970), pp. 53-77.
B. Weinstock, Inhomogeneous Cauchy-Riemann Systems with Smooth Dependence
on Parameters, Duke Math. J. 40 (1973), pp. 307-312.