Publicacions
Ma emá iques,
Vol
34
(1990),
77-91
.
A
bs ac
HOLDER
AND
U'
ESTIMATES
FORTHE
SOLUTIONS
OF
THE
D-EQUATION
IN
NON-SMOOTH
STRICTLY
PSEUDOCONVEX
DOMAINS
J
.M
.
BURGUÉS
Le
D
a
bounded
s ic ly
pseudocon ex
non-smoo h
domain
in
C'
.
In
his
pape
we
p o e
ha
he
es ima es
in
Lp
and
Lipschi z
classes
o
he
solu ions
o
he
ó-equa ion
wi h
Lp-da a
in
egula
s ic ly
pseudocon ex
domains
(see[2])
a e
also
alid
o
D
.
We
also
gi e
es ima es
o
he
same
ype
o
he
86
in
he
egula
pa o
he
bounda y
o
hese
domains
.
0
.
In oduc ion
and
s a emen
o
esul s
This
pape
is
a
con inua ion
o
[1]
and
deals
wi h
he es ima es
o
he
D-
equa ion
on
s ic ly
pseudocon ex
non
smoo h
domains
.
By
his
we
mean
a
domain
D
de ined
by
he condi ion
D
=
{
G
0}
whe e
is
a
s ic ly
p
.s
.h
.
unc ion
o
class
C
2
de ined
in a
neighbo hood
o
bD
.
We
ecall
ha
i
is
no
assumed
ha
he
g adien
o
be
di e en
o 0 in
bD,
and
he
bounda y
ails
o
be
a
egula
submani old
o
Cn
jus
in
a
o ally
eal
se
.
Henkin and
Lei e e
p o ed
in
[3]
ha
he
equa ion
bu
=
has
a
bounded
solu ion
u o
any
(0,
q)- o m
,
wi h
bounded
coe icien s,
and
such
ha
O
=
0
.
In
[1]
i
was
p o ed
ha
he e
exis s
an
in eg al
ope a o
T
=
K«,
z)A (C)
D
mapping
Ll(°'
Q)
o
Llo
P_1)
such
ha
DT
=
i
O
=
0
and
sa is iying
he
es-
ima e
11T
11
LD(D)
<
cil
4y(D)
and
also
he
Lip
1/2
es ima e
11T
IILip(1/2,D)
:5
cil
11,,
in
he
case
is
o
class
C
.
He e
and
in
he
ollowing
he
Lp
spaces a e
wi h
espec
o
he
Lebesgue
measu e
dm,
and
Lip
(s,
D)
s ands
o
he
class
o
con inuous
unc ions
on
D
ha ing
modulus
o
con inui y
O(P)
.
The
Lp-
no ms
will
be
ab e ia ed
by
11
Ilp
.
Pa ially
suppo ed by
he
g an
PB85-0374
o
he
CYCIT
.
Minis e io
de
Educación
y
Cien-
cia,
Spain
.
78

J
.M
.
BURGUÉS
The
aim
o
his
pape
is
o
imp o e
hese
es ima es
ex ending
o
he
non-
smoo h
case
he
op imal
es ima es
ob ained
by
K an z
in
[2]
o
he
egula
case,
o
o ms
o
a bi a y
bedeg ee
.
The
pape
is
o ganized
as
ollows
.
In
sec ion
1
we
ecall
he
cons uc ion
o
he
ope a o
T
and
he es ima es
o
i s
ke nel
K
ha
we e
ob ained
in
[1]
.
In
sec ion
2
we
p o e
he
ollowing
h ee
heo ems
:
Theo em
1
.
The
ope a o
T
sa is ies
he
ollowing
LP-es ima es
:
(a)

IIT II9
<
eji
IIP,
1
<
p
<
2n
+
2,
9
=
p
-
2ñ+2
(b)

IIT
liq
c
QI
111,
`dq
<
2n
+
2
(c)
Fo
p
=
2n
+
2,
IT
I
sa is ies
¡he
es ima e
I
D
exp{cjT (z)I
sn+i
}dm«)
<
oo
o
some
cons an
c
depending
on
n
and
II
II2n+2
.
Theo em
2
.
(a) I
p
>
2n
-}-
2,
T
maps
Leo
1)
con inuously
in o
Lip
(á
-
~,
D)
(b)
I
ELP
«Q Ql
,
p
>
2n
+
2
and
l
>
1,
T
is
con inuous
on
D
.
(c)
In
case
is
o
class
C+7,
0
<
-y
<_
1/2,
hen o
p
>
2
,
T
maps
Le
a
Ql
in o
Lip
(min{-y,
2
-
np
},
D),
o
all
a,,3
.
No e
in
heo em
2
ha
i
is
jus
C,
Hdlde
es ima es
can
only
be ob ained
o
(0,1)- o ms
.
Fo
o ms
o
bedeg ee
(a,
/0),
,0
>
1,
one
needs
ex a
assump-
ions
on
o
ob ain Hdlde
es ima es
o
a
ce ain
ange
o
p's
.
In
case
is
o
class
C-4-1/2
,
hen
pa
(a)
holds
o
o ms
o
a bi a y
bedeg ee
.
In
o de
o s a e
ou
hi d
esul ,
which
gi es
an
imp o emen
o
he
es ima es
a
he
bounda y,
we
need
o
ecall
a de ini ion
om
[1]
and
[6]
:
Assuming
only
ha
is
de ined
in
an
neighbo hood
o
D,
we
pu
o
(,
z
E
D
p«,
z)
=
I
(a (z),
(
-
z)
j
+
I
(a «),
(
-
z)
j
+
II(
-
zjj2
This
is
a
pseudodis ance
in
he
sense
ha
iangle
inequali y
holds
wi h
some
cons an
C,
and
will
be
called
he
Ko anyi
pseudodis ance
.
We
w i e
Lip,(s,
D)
o
he
subspace
o
Lip
(s,
D)
such
ha
I
(w)
-
(z)
I
=
O(p(w,
z)
9 )
o
(,
z
E
bD
.
Theo em
3
.
(a)
Fo
2n
- -
2<p<
oo,
T
maps
LPo
1)
in o
Lip
p
(
2
-
nP
l
)
(b)
Fo
E
L-
1)
hen
IT (z)
-
T (w)I
<-eji II
.p(z,w)'
12
Ilogp(z,w)I
(c)
I
is
o
elass
C2+- ,
0
<
y
<-
1/2,
hen o
p
>
Z
1
n
27,
T
maps
L(a,Q)
in o
Lip,(min(y,
á
-
np
),
D)
Since
CHIC
-
x11
2
<
P(C, z)
<
c211
C
-
x11,
he
meaning
o
heo em
3
is
ha
he
solu ions
o
he
ó-equa ions
will
be,
o
he
ange
o
p
indica ed,
wice
as
egula
in
ce ain
di ec ions in
bD
.
This,
in
he
egula
case,
is
a
e o mula ion
o
he es ima es
in
he
non-iso opic
Lipschi z
spaces
I',,,2a
in oduced
by
S ein
.
Finally,
sec ion
3
con ains
he es ima es
o he in eg als in
he
p oo
o
he
heo ems
.
Ou
echnique
di e s
om
ha
in
[2]
in
wo
aspec s
:
i s
o
all,
o
cou se,
he
non-smoo hness
makes
mo e
in ol ed
he
es ima e
o
he
singula i y o
he
ke nels
and,
secondly,
we
use
di ec
me hods
ins ead
o
in e pola ion
esul s
in
ob aining
he Hdlde
es ima es
.
A
main
hechnical
di icul y
o
ha
is
ha
he
domain
being
non smoo h
we
do
no
ha e
a
ou
disposal
he
c i e ia
Du(z)
=
O(u(z)'
-1
)
o
u
o
be
in
Lip
(s,
D)
.
In
[1]
ke nels
a e
ob ained
o sol e
á
in
non egula
s ic ly
pseudocon ex
domains,
o
Henkin-Rami ez
ype wi h weigh
ac o s
.
Le
us
b ie ly
ecall
hei
cons uc ion
and
main
p ope ies
.
1 .1
.
Gene al
cons uc ion
.
Fo
U
a
C
1
bounded
domain
in
C
n,
le
s,
Q
:
U
x
U
->
Cn
whe e
s
is
a
sec ion
o
Bochne -Ma inelli
ype,
say
:
o
C,z
E
U,
and
ESTIMATES
FORTRE
Ó
EQUATION

79
whene e
(
E
U
and
z
E
L
compac
in
U,
and
Q
is
o
class
C
1
and
holomo phic
in
z
.
Le
also
G
be
a
holomo phic
unc ion
o
one
complex
a iable
de ined
in
a
neigbo hood
o
U
x
U
unde
he
map
(C,
z)
-
1
+
(Q(C,
z),
C
-
z)
and
wi h
G(1)
=
l
.
Finally
de ine
(1)

K«,
z)
=
en
~
(n
k!1)I
G(k)
(1
+
(Q,
C
-
z))
s
n
(dQ
)~
.
Az(di)k
-k-
1
k=0
1
.
The
ke nels
sol ing
á
IIS(C,
z)11=
o(II(
-
Z11)
I(S(C,
z),
(
-
z)1
?
CLII(
-
x112
whe e en
=
((27 i)n(n
-
1)!)
-1
,
s"
_
En
j=o
s j
d(Cj
-
zj
)
and
Q
=
~~
o
Qid(Cj
-
zj)
K
is
a2n
-
1
o m
in
d(,d(,dz
and
dz
oge he ,
and
o 0
<_
a,
a
<_
n,
le
K,,,p
he
componen
o
bedeg ee
(a,
P)
in z
and (n
-
a,
n-
/~
-
1) in
C
.
Then
:
80

J
.M
.
BURGUÉS
Theo em
1
.1
.

Whene e
Ñ
>
1, i
K<
,
Q((,
z)ISEbU
is
0 o
z
E
U,
he
ope a o
(2)

T
=
(_
1)a'+#
/
^
Ka,P-1
U
sa is ies
8T
=
i
EC
1
(a
a)(U)
.
1
.2
.
The
sec ion
and
weigh s
in
he
s ic 1y
pseudocon ex
case
.
I
D
is
a
s ic ly
pseudocon ex (non
egula
domain) and
is
a
C
de ining
unc ion
o
D,
le
U6
=
{ (z)
<
S}
and
V
=
{-S
<
(z)
<
b}
.
Then
Henkin
and
He e 's
lemmas
p o ides
us wi h
a
amily
o
unc ions
¿j,
j
=
1,
. .
.,
n,
and
cons an e
co,eo,
S
o
depending
only on
he
unc ion
(bu
no
on
i s
g adien ,
no
on
he
a iable
z),
such
ha
(Pj
EC
1
(V
6o
x
U6o)
and
a e
holomo phic
in z,
and
he
unc ion
4(C,
z)
=
E=
1
¿j«,
z)((i
-
zj) sa is ies
:
We
de ine
:
I<p((,z)I
?
co
i
IK-x11
>_
eo
2Ñ<P((,
z)
?
(C)
-
(z)
+
Cog
-
Z11
2
l
II(
-
zII
<
Eo
d«PIC-Z
=
dz,¿j,C
=
á (z)
~j«,
z)
_
(j
«),+
.
0(11(
-
zII)
when
I1(
-
zII
<
Eo,
I (z)I
<
6o
A(C,
z)
=
-
(z)
+
-¿((,
z)
and
i
x
EC
,
(C
n)
,
0
<
x
:5
1
and
x
-
1
on
V
!
,, de ine
z
and
Q=
(Q,,
. .
.,
`w
n
)
.
Qj((,z)
=
x
(C)
A«,
z)
W i e
now,
V
E,
6
=
{((
;
z)
:
1
7
-(01
<
b,
I (z)1
<

z
II
<
e}
and
de ine
Ib
;((,
z)
=
4>
.i(z,
(),

z)
_
~¿(z,
(),
. á((,
z)

z)+A(z,
()~¿j((,
z),
and
((,
z)
=Z
:'=1
.i(C,
z)(C
.i
-
zj)
.
Finally,
i
0
E
Cm(C"),
0
<
0
:5 1,
0
-
1
on
V~
s
,
de ine
_

a,s
S
.i(C
z)
=
«
C,
z)
.i((,
z)
+
(
1
-
0((,
z))(C
.i
-
zi)
s
=
(sl,
..
.sn)
and
H
=
(s,
(
-
z)
.
The
ollowing
es ima es
a e
c ucial
o
he
es ima es
o
he
ke nel's singula -
i y
:
2ReA
;z~
- (()
-
(z)
- -
coll(
-
x112
o
((,
z)
E
VEO,óo
n
(U x
(1)
.
This
implies
ha ,
in
e ms
o
he
pseudodis ance,
p
:
ESTIMATES
FORTHE
D
EQUATION

81
IHI~
:
c{( (()
-
(z))2
+
(
-
(()
-
(z))II(
-
zli
¡Al
^
- «)
-
(z)
+
p«,
z)
IHI
~~
e{(- «)
-
(z))II(
-
zjj2
+
P2((,
z)}
1
.3
.
The
esul ing
ke nels
and
hei
es ima es
.
Take
in
he
o mula
(1)
G(w)
=
w
n
,
and
he
sec ion
and
weigh
in oduced
be o e
.
De ine
also
:
n
w((,z)
_
1
:
4,i((,z)d((i
-
zi)
j=o
n
w
((,
z)
=
1
:4>
j
(z,
()d((i
-
zi)
j=1
A*((,
z)
=
A(z,
()
1
7((,
z)
=
(()Dz
w*
+
A*D(w
A e
a
combina o ic
compu a ion
one
ob ains
in
ha
case
:
n-1
_
1
(
3
)

K((,
z)
=
4J
A

Cn,k(

A~
)
)n-k
Hn
kAk
(D<w)k
A /n-k-1
k=0 n-1
+
[ (()&A
*
-
A*DS ) n
w
n
w*
n
E
C
n,
k(n
-
k
-
1)(
-
(()
)n-k
A
k=o
(á(w)k n
,n-k-2
n-1

(()
n-k
-
(()DSA
Aw
A
w*
A
E
cn,kk(

TA

)
k=1
+
11(
-
z11,
+
IMOIMO*},
Hn-kAk
1

(~
w
k-1
n
n-k-1
Hn-kAk+1
)
De ine
now
q((,
z)
=
Ild (z)jj
+
li(
-
zil,
and
deno e
jid (z)jj
=
A(z)
I is
clea
om
(3)
ha
K((,
z)
=
0 o ( E
bU
so
Theo em
1 .1
applies
and
he
ke nel
K
has
he
es ima e
(see
[1],
lemmas
2
.4
and
2
.5)
:
(
4
)

¡K((,
z)
1
=
G(I
~~nl
{
-
(()
+
q
2
((,
z)}11(
-
z11)}

82

J
.M
.
BURGUÉS
In
e ms
o
he
pseudodis ance,
we
ha e
(see[1],
lemma
6
.2)
(
5
)

IK((~
z)1
-
~(

1

+
(
-
(z)) n
_

11(
-
zilg2((,
z)
1l(
-
z
llzn

1

[- (z)11(
-
x11
2
+
P((,
z)
2
]n
+
11(-zll ?((,z)
)=de
O(N1)
P(('z)n+1
As showed
in
[1],
lemma
6
.3
he
ke nel
K((,
z)
is
in eg able
in
each
a iable,
uni o mely
in
he
o he
.
Using
a
s anda d
egula iza ion
p ocess,
one
can
hen
show
ha
i
E
h¡oc~a,a)
and
DT
=
0 in
he
weak
sense
hen
T
is
a
o m
in
Lioc(a,p-1) and
DT
=
in
he
weak
sense
.
No ice
ha
when
z
E
bU, he
es ima e
(5)
implies
(
6
)

Ih((,
z)
I
=
O(P((~
z)
z-n
+
A(z)
2
P((,
z)-n

1)
=de
O(N2)
and
also
he
wo se
bu
symme ic
es ima e
:
z

z
¡K«,
z)1
=
0(
ll(
-
z112n-1
+
11(
-
x11
2
.
3P2(('
z))
__de
D(N3)
1 .4
.
Es ima es
o he
di e ences
.
Ou me hod
in
p o ing
H51de
es ima es
in ol es
es ima es
o
he
di e ences
o
he
ke nels
¡K«,
z)
-
K((,
w)l
.
A
sui able
con ol
can
be
ob ained
in
e ms
o
he
g adien
o
K
wi h
espec
o
he
second
a iable
whene e
i
makes
sense,
ha
is
when
he
coe licien s
o
he
o m K((,
z) a e
C
1
in z
.
Fo mula
(3)
shows
ha
all
e ms
bu
hose
in ol ing
D
Z
w
*
a e
C
1
in
z
,
because
Dw*
=
E
D=ob~
n
d((j
-
zj)
and
D,ID~
((,
z)
=D
z
<Pj
(z,
()
is
only
con inuous,
since
i
in ol es
second
de i a i es
on
.
Obse e
also
ha
bad
e ms
may
appea
only
once
(because
Dz(*AD-,Y
_
0),
and
in
he
componen e
K«,0
hey
do
no appea
a
all
.
So
we
can
w i e
he
ke nel
K
as a
sum
K((,
z)
=
K,«,
z)
A
D=w*
+
K2((,
z)
whe e
K
1 ,
Kz
a e
C
1
in z
and
K,
,,,
o
=
(K
2
)
<
,,,
o
.
The
ke nels
K
l
and
IKz
sa is y
he
same
es ima es
as
K,
ha
is
(5)
and
(6)
.
The
g adien e
in z o
K
l
and
K
2
sa is y
he es ima es
con ained
in
lemma
6
.6
o
[1]
and
hence
i
ollows
ha
he e
exis s
c
such
ha
i
llz
-
wjl
is
small
enough and
ll(
-
zll
>
cllz
-
w11
1/2
,
(E
D,
hen,
o
j
=
1,
2
I
Kj((,
z)
-
Kj((,
w)I
=
0
(11z
-
w11M1((,
z))
whe e
(
8
')
__de

I (z)in¡¡(
-
zl1A(z)
2

11(
-
z11
.1(z)
2
Nh

(

+

+

)q(,
z)
11(
-
z112n+1
[I (z)I11(
-
x11
2
+
P(,
z)
2
]n+l

P(,
z)n+z
hus,
K«,o(C,
z)
-
Ka,o(C,
w)
will
sa is y
(8)
i
C,z,w
a e
as abo e,
wi hou
any
u he
equi emen
on
.
Fo he
componen s
K«,p
wi h
0
>
0
w i e
K(C,
z)
-
K«,
w)
=
{K(C,
z)
-
K
(C,
w)}
A
azw*
+
K2(C,
z)
-
K2(C,
w)
I
ollows
ha
i
d
2
sa is ies
a
Lipschi z
es ima e
o
o de
7
;
and
z,
w,
(
a e
as
abo e,
hen
(9)

IK«,p(C,z)-K,,p(C,w)I
=o(IIz-wJIMI(C,z)+IIz-wJI`NI(C,z))
The
es ima es
(8),
(8')
and
(9)
will
be
used
in
he
Euclidean Hólde
es ima es
.
Fo
he
non-iso opic
Hólde
es ima es
i
will
be
con enien
o
es ima e
he
di e ence
K«,p(C,
z)
-
K«,p(C,
w)
jus in
e ms
o p
.
In
a
simila
way
as be o e,
bu
using
now
lemma
6
.9
o
[1]
ins ead
o
lemma
6
.6,
we
see
ha
i
is
jus
C
2
and
p(C,
z)
>
cp(z,
w)
(,z,w
E
bU
hen
:
(10)

IK«,o(C, z)
-
K«,o(C,
w)
I
=0
(P(z,
w)'/
2
M2)
whe e
(
10
'
)

M2
=de
P(C,
z)
-n
+
(Z)
,P(C,
z)
-1-n
and
i
EC
2
+Y
hen
o
Q
>
0
(11)
whe e
(11')
2
.1
.
P oo
o
heo em
1
:
ESTIMATES
FORTHE
a
EQUATION

83
¡K
.,
#
«,
z)
-
K«,p(C,
w)I
=
O(P(z,
W)1/2M2
+
P(z,
w)
y/2
N2)
N2
=
P(C,
z)1/2-n
+
~X(z)
2
P(C,
z)
-n-1/2
2
.
P oo
o
heo ems
As
in
[2],
he
p oo
o
Theo em
1
is
based
on
he
ollowing
lemmas
:
Lemma
.
Le
K
:
R'
x
Rn
--
i
C
ha e
he
p ope y
ha
K(x,
.)
is
o
weak
ype s as
a unc ion
o
y,
uni o m1y
in x,
and
K(
.,
y)
is
o weak
ype
s as a
unc ion
o
x,
uni o m1y
in y
.
Then
he
linea
ans o ma ion
(x)
-
T (x)
=
J
n
K(x,y) (y)dy
R
+
K,(C,
z)
A
{0
.w*
-
aww*)
84

J
.M
.
BURGUÉS
de ined
o
:
R'
-->
C,
sa is ies
11T
II
Lq
<
Ap
li
II
Lp
whene e
s
>
1,
1
<
p
<
q
<
oo
and
q
=

+
9
-
1
.
Mo eo e ,
11T
L-<
A
E
jI
II
L
1
o
all
e
>
0
.
Lemma
.
Suppose
K,s,
as
aboye
and
E
L"
whe e
s
+
s,
=
1,
suppose
also
ha
m
=
n,
S2
CC
Rn,
and
suppK
C
S2
x
S2
.
Then
and
c,
M
do
no¡
depend
on
,
bu
on s
and
m(9)
Aco ding
o
he
lemma, we
ha e
o
p o e
ha
K«,
z)
is
o
2
2n~1-weak
ype
on
D
in
each
a iable
uni o mély
in
he
o he
.
Fo
his
we
use he
es ima e
(7)
and,
since
i is
symme ic,
i
will
be enough
o
p o e
he
ollowing
esul
:
(12)
Lemma
2
.1
.
m{z
:
(K«,
z)j
>
}
=
O(
-
~
),
uni o mly
in
(E
D
This
will
be done
in sec ion 3
.
2
.2
.
P oo
o
Theo em
2
:
Le
p
>
2n
+
2and
E
Le
o
1)
.
Fi s
we
p o e
ha
T
E
Lip(
1
-
-+',
D)
Le
z,
w
E
D,
b
=
~~z
-
w~~
.
We
de ine
i
=
61/2
and
es ima e
T
(z)
-T
(w),
using
(8),
by
In
case
mo e
e m
:
(z)

I (C)IN,(C,z)dm(C)+J
,n(z)nD
l «)¡NI«,w)dm«)
B
n

nD

B
+
6
1
l (o1Ml«,z)dm«)
B~
n(z)
EC
2
+
-
exp((c
IT (z)I
)')dm(z)
<
M
<
o0
~~
~~La'(sl)
and
E
LPa
Q),
,Q
>
1,
we
will
ha e,
aco ding o
(9),
one
ó7

1
(C)1
Ni«,
z)dm«)
Using
Hdlde
es ima es
we
a e
lead
o
he
ollowing
lemmms
which
will
be
p o ed
in
sec ion
3
:
Lemma
2
.2 .1
.
Fo
1
<
s
<
2n±2
we
ha e
-
2n+1
{

N
i
«,
z)
-
dm«)Í
1/e
=

~
O( ~
-2n
-
1
)
~B

(z)nD
Co olla y
.
Fo
1
<
s
<
2n+2
-
2n+1
Lemma
2
.2
.2
.
Fo
1
<
s
<
22n+
i
Using
lemma
2
.2 .1
he
i s
in eg al in
(12)
is
O(62
-
~)
.
The
second one
is
also
O(62
-
~),
because
B,,,(z)
C
B,,,,(w)
.
Finally
by
lemma
2
.2 .2,
he
las
e m
in
(12)
is
O(677i2jL-2
-2n-3
)
=
O(62-~)
This p o es
he
i s
pa
o
he
heo em
.
Unde
he condi ions
in (b)
we
ha e
one
mo e
e m which
is
O(1)
acco ding
o
he
co olla y o
lemma
2
.2 .1
.
Then
i
p
>
2n+2 we
ob ain
a
Lipschi z
condi ion
which
exponen
min

1

nn±l
)
(-Y,
2
-
which
gi es
pa
(b)
o
he
heo em
.
Finally,
o
pa
(c)
we
simply
w i e,
IT (z)
-
T (w)I
<_
II
IIp{
(IK«,
z)I
+
IK«,
w)I)
9
dm«)}
1/
s
B
whe e
ñ
+
9
=
1
ESTIMATES
FOR
TIiE
C9
EQUATION

$5
( D~B

N
I
«,
z)9dm«)}1/9
=
O(1)
n
(z)
{
1

Ml«,z)9dm«)}1/8
=
O(17~-2n-3)
B

(z)
+
II
IIp{
J

¡K«,
z)
-
K(C,
w)I9dm«)}lis
D~B
E
(z)
The
i s
e m
can
be
made
a bi a ily
small
by
choosing
e
small
and
he
second
oo
by
choosing
w
close
enough
o z
beacause
he
ke nels
a e
con inuous
o
he
diagonal
.
2
.3
.
P oo
o
he
Theo em
3
:
Fi s ,
le
A5(z)
=
{(
:
p«,
z)
<
6},
be
he
HS mande
ball
ela ed
o
p
.
Fo
p
>
2n
-}-
2
and
E
Le
o
1)
we
spli
now
he
in eg als
gi ing
T
(z)
-
T
(w)
o
z,w
E
bD
in
he
o m
:
(13)

JA
.6(Z)
I (C)IN2(C,
z)dm«)
+
JAa(Z)
I «)IN2(C,
w)dm«)
+
6112
JI (S)IM2«,z)dm«)
D~A~s(z)
He e
6
=
p(z,
w)
and we
ha e used
(10)
In
case
ECZ+
7and
E
Le
a
Q)
,
0
>
1,
we
will
ha e
by
(11)
ano he
e m
:
(
14
)
6y/2

I (QN2«,z)dm«)
D~A~a(z)
Using
HSlde 's
inequali y
as
be o e
we
a e
lead
now
o
he
ollowing
lemmas
: