Publicacions
Ma emá iques,
Vol
36
(1992),
421-426
.
A
UNIQUENESS
THEOREM
FOR
INVARIANTLY
HARMONIC
FUNCTIONS
IN
THE
UNIT
BALL
OF
en
Abs ac
JOAQUIM
BRUNA
Dedica
a
la
memó ia
den
Pe e
Menal
We
p o e
a
bounda y
uniqueness
heo em
o
ha monic
unc ions
wi h
espec
o
Be gman
me ic
in
he
uni
ball
o
Cn
and
gi e
an
applica ion
o
a
Runge
ype
app oxima ion
heo em
o
such
unc ions
.
Le
B
be
he
uni
ball in
en
and
S
i s
bounda y
.
The
in a ian
laplacian
0
in
B
is
he Laplacian
associa ed
o
he
Be gman
me ic
and
i is
gi en
in
coo dina es
by
Pa ially
suppo ed
by
DGICYT
g an
PB89-0311
.
(6i7
-
zizj)DiDj
.
The
e m
in a ian
comes
om
he
ac
ha
i
commu es
wi h
all
au-
omo phisms
kP
o
B
:
0(u
oT)
=
Du
o
T
.
Co espondingly,
hose
unc ions
u E
C
2
(B)
annihila ed
by
O
a e
called
in a ian ly
ha monic
o
M-ha monic
(see
[4,
chap e
4]
o
he
mo e
ele an
p ope ies
o
hese
unc ions)
.
The
aim
o his
no e
is
o
gi e
a
bounda y
uniqueness
heo em
o
M-
ha monic
unc ions
and an
applica ion
o
a
Runge
ype
app oxima ion
p oblem
.
1
.
The
uniqueness
heo em
essen ially
s a es
ha
S,
hough
A
com-
ple ely
degene a es
he e,
is
non-cha ac e is ic
o
a
ce ain
Cauchy
p ob-
lem
:
42
2

J
.
BRUNA
Theo em
.
Le
U
be a
ball
cen e ed
a ~
E
S,
and
le
uEC°°
(U
1
B)
sa is y
Du=
0
in
U
.
Then om
n
u=
a n
=0
onuns
i
ollows ha
all
de i a i es
o
u
a e
ze o
on
U
l
S
(hence
u
-
0
i
i is
eal-analy ic
acc oss
S)
.
The
p oo
will
show
in
ac
ha
u
and
onñ
de e mine
all
he o he s
de i a i es
o
u on
S
.
The
s a emen
sugges s
ha
he
ollowing
Cauchy-
Kowale ski
ype
heo em
is
p obably
ue
:
i
,
g
a e
eal-analy ic
unc-
ions
de ined
on
U
n
S,
he e
is
ano he
ball
V
C
U
con aining
S
and a
eal-analy ic
unc ion
u
in
V
such
ha
Du=
0 in
V
n
B
and
u
=
,
án ñ
=
g on
V
1
S
.
P oo
.
Le
Ao
=
Ei
j-1(Si7
-ziz
j
)DiD
j
and
le
R=
~
iz
j
D
jbe
he
adial
(holomo phic)
de i a i e
;
w i e
R=
N
+
iT,
hen
N
=
-2-
and
T
is
a
eal
angen
ield
o
S
.
We
will
show
ha
u
alone
de e mines
N3
u, j
<n-
1
and
ha
u,
N'
2
u
de e mine
all
de i a i es
a
poin s
o
S
.
A
compu a ion
shows
ha
AoN
-
NAo=
Do
+N
2
+T
2
D
O
T
-
TDo
=
0
.
TN-NT=0
.
F om
his
i
easily
ollows
by
induc ion
on
k
ha
DoN
k
=
Pk(N)Do
+
kNk+1
+
Rk(N,T)
whe e
Pk(x)
is
a
monic
polynomial
o
deg ee
k
and
Rk(x,
y)
is
o
deg ee
<
k
+
1 in
x, y,
bu
o
deg ee
<
k
in
x
.
Nex ,
he
ollowing
no mal- angen ial
decomposi ion
in
[21
is
needed
Do
=
I
i
2
{(1
-
Iz1
2
)RR
+
A
+
(n
-
1)N}
z
He e
A
is
he
box-laplacian
on
S
;
i s
pa icula
exp ession
will
no
be
needed,
only
he
ac
ha
i
is
a
angen ial
ope a o
.
I
ollows
ha
a
poin s
w
E
S
(4)

(Do )(w)
=
(A )(w)
+
(n
-
1)N (w)
.
UNIQUENESS
THEOREM
FOR
HARMONIC
FUNCTIONS

423
Applying
his
o
u
we
see
ha
Nu
=
0
on
S
.
Assume
by
induc ion
we
ha e
p o ed
N(
k
)u
=
0
on
S
whene e
u E C°°(U n
B)
is
M-ha monic
and
ze o
on
U
n
S,
k
<
n
-
1
.
Then
by
(3)
and
(4)
on
U
n
S
.
By
(2),
(
n-
1)N
(k+l)
u
=
D
o
N(k)
U
=
kNk+1U
+R
k
(N
,
T)
u
R
k
(N,T)u
=
i<k
i+j<k+1
N
Z
Tju
and by
(1),
Tju
is
also
M-ha monic
and
ob iously
ze o
on
S
.
By
he
induc ion
hypo hesis,
Rk(N,T)u
=
0 on
U
n
S
.
Then
we
conclude
ha
N(k+1)
u
=
0
i
k
<
n-1
.
I
k
=
n-1
we
canno
conclude
N(n)u
=
0
bu
i is
clea
ha
i
his
is
known
o
hold,
hen
he
induc ion
can
con inue
and
so
NW
u
=
0
o
all
j
on
U
n
S,
which
p o es
he
heo em
.
The e
is
some
connec ion
o his esul
wi h
a esul
om
Folland
[1]
acco ding
o
which
an
M-ha monic
unc ion
u
in
he
whole
ball
o
class
Cn
up
o
he
bounda y
mus
be
in
ac
plu iha monic
.
2
.
As an
applica ion
o
he
heo em
we
p o e
:
Theo em
.
Le
K
C
B
be
a
compac
se
such
ha
B K
is
connec ed
.
Then,
e e y
sa is ying
O
=
0
in a
neighbou hood
o
K
is
he
uni o m
limi
on
K
o
a
sequence
o
M-ha monic
unc ions
u
n
in
B,
con inuous
on
B
.
I
mus
be
poin ed
ou
ha
his esul
can be
p o ed
as
well
by com-
bining
a
gene al
esul o
[3]
on
analy ic-hypoellip ic
ope a o s
and
[4,
5
.5
.4]
.
Ou
p oo p oceeds
by
duali y
and
elies
on
some
well-known
ac s
ha
we
p oceed
o
ecall
.
The e
is
a
decomposi ion
o mula,
alid a
leas
o
u E
C
2
(B),
U(Z)
=
I
P(~,
z)u(~)
do,«)
+
I
Au(S)G(~,
z)
d>,«)
s

s
ha
co esponds
o
he
Poisson-G een
o mula
in
Euclidean
space
.
He e
do,
is
he
no malized
Lebesgue
measu e on
S,
d>,«)
=
(1-1
~
j2)-n-1dV«)
is
he
in a ian
measu e,
P«,
z)
is
he
in a ian
Poisson
(o
Poisson-
Szegó)
ke nel
_

2
n
P«,
z)
=
Í
1-
(Z
I
2
,
(ES,
zEB,
42
4

J
.
BRUNA
and
G«,
z)
is
he
G een
unc ion
wi h
pole
a
z,
(1
_
)n-1
G«,
z)
=
G(wz«),
0
)
=
cn

,~,

d ,
~~Pz(S)j2
c,
z
being
a
cons an
and
~o
z
he
au omo phism
o
B,
unique
up
o
uni a y
ans o ma ions,
ha
sends
z o 0
and
0 o
z
.
Mo eo e ,
1
-
I(Pz«)I
2
=
(1
-
IZI
2
)(1
-
I(12)
11
-
zl2
so
ha
G
is
in
ac
symme ic
.
Oneway
o
ob aining
(5)
is
o
w i e
p ecisely
he
Poisson-c een
o mula
o
u
o
co
z
a 0
and
change
a iables
in
he
esul ing
in eg als
.
A
second
(and
be e )
way
o
looking
a
(5)
is
h ough
he
c een
iden i y
in
he
Be gman
me ic
o
AC
B
(u0
-
0u)
dA
_
a
(U
á
-
8
)
a

a

dS
.
He e
is
he
ou wa d
uni a y
no mal
(by
he
Be gmann)
me ic
o
áA
and
dS
is
he
induced
measu e on
8A
.
Fo mally,
one
ob ains
(5)
by
specializing
o
A
=
B,
(~)
=
G,«)=G«,
z)
and
checking
ha
OCG«,
z)
dA(~)
=
Sz,

z
E
B
G«,
z)
=
0

E
S
(6)

wC
G«,
z)
dS
=
P(~,
z)
da

E
S,
z
E
B
(in
a
igo ous
way
one
should
choose
as
A
he
ball
o
adious
<
1
and
hen
make
-->
1)
.
Fo mula
(5)
implies
he
ollowing
ac s
:
(a)
The
gene al
o m
o
an
M-ha monic
unc ion
u
in
B,
con inuous
on
B
is
u(z)
=
P[ ](z)
=
P«,
z) «)
do
,
s
wi h
E
C(S)
;
equi alen ly,
P[ ]
is
he unique
solu ion
o
he
Di ichle
p oblem
Du=
0 in
B,
u
=
on
S
.
(b)
Fo
u
E
CZ(B)
o
compac
suppo
u
(z)
=
B
Du(S)G«,
z)
dN(~)
Le
.
u
coincides
wi h
he
G een
po en ial
o
i s
Laplacian
.
The
simme y
o
G
hen
implies
ha
o
a measu e
wi h
compac
suppo
in
B,
he
G een
po en ial
Gp
a
n
(e)

P(~,
z)
=
cn
9 n
G( ('
z)
=
1
'

E
S, z
E
B
.
This
is
because
wha
(6)
eally
means
is,
as said be o e,
UNIQUENESS
THEOREM
FOR
HARMONIC
FUNCTIONS

425
Gj¿(z)
=
I
G(~,
z)
dp(~)
s
sa is ies
(AGí )
dA
=
dM
in
he
weak
sense,
and
in
pa icula
OGil
=
0
in
he
usual sense
o
he
suppo
o
p,
.
Finally,
we
will
need
a
e o mula ion
o
(6)
in
e ms
o
he
Euclidean
no mal,
which
is
P«,
z)
=
lim
aG( (,
z)
do,
.
Since
=
c,,
(1
-
)
n
á
and
dS
=
c,
(1
-
)
1-2
'
do
,
(we
deno e
by c
',
all
cons an s
depending
on
n),
(c)
ollows
by
L'Hopi al's
ule
.
Al e na i ely,
(c)
can
be
p o ed
o
cou se
by
di ec
compu a ion
.
No e
ha
~G«,
z)j
=
0(1-
I(j2)n
o
a ixed z
.
Hence
á
;
G( (,
z)
=
0
o (
E
S, j
=0,
. . .
.
n-
1
.
P oo o
he
heo em
:
Le
p be a measu e on
K
which
is
o hogonal
o
all
M-ha monic
unc ions
in
B,
con inuous
on
B
.
By
(a)
abo e
his
is
equi alen
o
I
P«,
z)
dp(z)
=
0
o (
E
S
.
x
We
conside
he
G een
po en ial
o
p
G(p)(w)
=
I
G(z,
w)
dM(z)
x
so
ha
OG(p,)
=
0
o
K
.
Mo eo e G(p)
is s ill
de ined
and
is
eal
analy ic
in
a
neighbou hood
o
B,
o
K,
because
so
is
each
cp,
z
o
z
E
K
.
No e
ha
all
such
po en ials
sa is y
7
á ~
G(p)«)
=
0,
j
=
0,
. .
.,n
-
1,

E
S
.

426

J
.
BRUNA
By
(c),
(7)
says ha
also
(z)
=
B
O «)G«,
z)
d, «)
=
L
x
O «)G(~,
z)
dA«)
.
Hence
by
Fubini's
heo em
n
Ó n
G(P)(~)
=
0,

ES
.
The e o e,
by
he
heo em
in
Sec ion
1,
all
de i a i es
o
G(p)
anish
a
S
and,
since
B K
is
assumed
o
be
connec ed,
we
conclude
ha
G(M)
is
iden ically
ze o
in
B K
.
Le
now
as in
he
s a emen
;
mul iplying
by
a es
unc ion,
we
can
assume
ha
is
compac ly suppo ed
in
B
.
By
(b),
L
(z)
dp(z)
=

B~ x
O (~)
{Ix
G«,
z)
dp(z)
}
dA«)
=
x
-

A «)Gli(~)
dA(~)
=
0
~B~x
which
inishes
he
p oo ,
by Hahn-Banach's heo em
.
Re e en es
1
.

G
.
FOLLAND,
The
sphe ical
ha monic
expansion
o
he
Poisson-
Szegd
ke nel
o
he
ball,
PAMS
47
(1975),
401-408
.
2
.

D
.
GELLER,
Some
esul s
on
HP
heo y
o
he
Heisenbe g
g oup,
Duke
Ma h
.
J
.
47,
no
.
2
(1980),
365-390-
3
.

B
.
MALGRANGE,
Exis en e
e
app oxima ion
des
solu ions
des
equa-
ions
aux
de i ees
pa ielles
e
des
equa ions
de
con olu ion,
Ann
.
Ins
.
Fou ie
6
(1956),
271-355
.
4
.

W
.
RUDIN,
"Func ion
heo y
in
he
uni
ball
o en,"
G undleh en
241,
Sp inge -Ve lag
.
Depa amen
de
Ma emá iques
Uni e si a
Au ónoma
de
Ba celona
08193
Bella e a
(Ba celona)
SPAIN
Rebu
el
24 de
No emb e
de
1992