Publicacions
Ma e ná iques,
Vol
35
(1991),
187-207
.
FACTORIZATION
OF
THE
GREEN'S
OPERATOR
AND
WEAK-TYPE
ESTIMATES
FORA
RANDOM
WALK
ON
ATREE
RICHARD
ROCHBERG
AND
MITCHELL
TAIBLESON
1
.
In oduc ion
.
Le
X
be
a
ee,
which
is
o
say,
a
connec ed
g aph
wi hou
loops
.
The
o de
o a
e ex
o
he
ee
is
he
numbe
o
edges
ha
mee
a
he
e ex
.
To
a oid
some
messy,
bu
essen ially
i ial,
complica ions
we
will
assume
h oughou
ha
he
o de
o
e e y
e ex
is
a
leas
h ee
.
We
say
ha
a
e ex
u
is
a
neighbo
o
he
e ex
i
u and
a e
connec ed
by
an
edge
.
When
u and
a e
neighbo s
we
w i e
u
-
.
The
se
o
ansi ion
p obabili ies
{p(u,
)}
is
said o
de e mine
a
nea es
neighbo
andom
walk
i
0
_<
p(u,
)
<
1
o
all
u,
E
X
and
p(u,
)
>
0
i
and
only
i
u
-
.
The
walk
is
said
o be
s ochas ic
i
j
:,,
p(u,
)
=
1
o
all
u in
he
ee
.
The
ansi ion
p obabili ies
de e mine
a
ansi ion
ope a o ,
P,
as ollows
:
Fo
a
unc ion
de ined
on
he
( e ices
o
he)
ee,
P
(u) =
1
:
p(u,
)
( )
.
VEX
We
o en iden i y
he
ansi ion
ope a o
wi h
i s
associa ed
se
o ansi ion
p obabili ies
.
We
assume
u he
ha
he
ansi ion
ope a o
P
is
egula
in
he
sense
ha
he e
is
a posi i e
numbe
6
such
ha
p(u,
)
>_
b
whene e
u
-
.
This
implies
ha
he
o de s
o he
e ices
a e
bounded
abo e
.
The
isi ing
p obabili y,
U(u,
),
o
u
and
in
he
ee
is
he
p obabili y
ha
a
walk
s a ing a u
will
isi
.
The e
a e
wo
common
con en ions
i
u
=
.
The
s anda d
con en ion
.i
s
ha
U(u,
u)
is
he
p obabili y
ha
he
walk
will
isi
u a
some
ime
in
he
u u e,
and
his
is
ou
de ini ion
.
The
o he
con en ion
is
ha
U(u,
u)
=
1
.
Wi h
his
in
mind
we
de ine
:
{
a(u,
)

U
(u,
),

i
u
7É
1,
i u=
.
P
is
ansien
i
U(u,
)
<
1
o
all
u and
in he
ee
.
We
will
equi e
ha
P
be
s ongly
ansien
in
he
sense
ha he e
is
a S
>
0 so
ha
U(u,
)
<
1
-
S
Suppo ed
in
pa
by
NSF
G an s
DMS-8701271
and
DMS-8701203
.
18
8

R
.
ROCHBERG,
M
.
TAIBLESON
whene e
u
-
.
In
he
Appendix
o
[KPTI
i is
shown
ha
i
he e
is
a
posi i e
numbe
y
such ha p(u,
)
<
1/2
-y
hen
P
is
s ongly ansien
.
Le
q
+
1
-
q(u)
+
1
be
he
o de
o
he
e ex
u
.
We
say
ha
he
ansi ion
ope a o
P
is
iso opic
i
p(u,
)
=
1
/(q
+
1)
whene e
u
-
;
we
say
ha
i
is
symme ic
i
p(u,
)
=
p( , u) o
all
u and
.
A
ee
is
homogeneous
i
all
e ices
a e
o he
same
o de
.
A
ansi ion
ope a o
is
homogeneous
i
i is
de ined
on a homogeneous
ee
and
he
se
o
ansi ion
p obabili ies
a
each
e ex
a e
he
same
.
I is
easy
o see
ha
bo h an
iso opic
ansi ion
ope a o
onan
o de
bounded
ee
and
a
homogeneous
symme ic
ansi ion
ope a o
a e
always
s ongly
ansien
.
In he sequel
we
assume
ha
P
is
s ongly
ansien
and
we
se
Thus,0<5<1
.
=
sup{a(u,
)
:
u
-
}
.
Th oughou
his
pape
unc ions
on
he
ee
X
a e
complex
alued,
X
is
supplied
wi h
he
disc e e
opology
and
he
a omic
measu e
which
assigns
mass
one
o
each
e ex
.
We
hink
o
he
ee
as
a
collec ion
o
e ices
and
o
edges
as
a
ela ionship
on
he
ee
.
F om
his
poin
o
iew
X
is
a
locally
compac
measu e
space
.
Fo
a
unc ion,
,
on
he
ee,
1/
Il 11P
=
(E
l (
u
)I
P
UEX
li 11
.
=
sup
i (u)1
.
UEX
The
Laplacian
ope a o ,
A,
is
de ined
by
A
(u)
=
E
p(u,
)
( )
-
(u)
.
o-u
Tha
is,
A
=
P
-
d
.
The
G een's
ope a o ,
G, speaking
loosely,
is
he
in e se
o
-A
.
I is
de ined
by
a
ke nel
G(u,
)
,
he
G een's
ke nel,
G
(u)
=
1
:
G(u,
V)
( )
.
VEX
0<p<oo
We
de e mine
G
as ollows
:
Fix
u
and
in
X
.
A
pa h
in
X
connec ing
u o
is
a
ini e
sequence
o
e ices,
w
=
{wo,wl,
.. . .
w
n
},
whe e
w
o
=
u,
w
n
=
and
o
each
k
=
1,
.
. .
,
n,
wk_
1
is
a
neighbo
o
wk
.
The
leng h
o
he
pa h
w
is
2(w)
=
n
.
The
weigh
o
he
pa h
w
is
n
W(w)
=11
p(wk-1,
wk)
.
k=1
A
i ial
pa h
w=
{u}
has leng h
ze o
and
weigh
one
.
FACTORIZATION
ON
THE
GREEN'S
OPERATOR

189
De ini ion
.
G(u,
)

W(w)
whe e
w
anges
o e
all
pa hs
ha
connec
u
o
.
Obse a ions
.
I
is
immedia e
ha
G(u,
)
=
E'
o
p(k)
(U,
)
and
so
G=

o
Pk
whe e
he
p(k)
a e ansi ion
p obabili ies
associa ed
wi h
he
ansi ion
ope a o
P
k
.
I is
no
di icul
o see ha
G(u,
)
is
he
expec ed
numbe
o
isi s
o o
a
andom
walk
ha
s a s
a
u
.
We
no e
u he ha
P
is
ansien
i
and
only
i
G(u,
)
<
oo
o
all
u
and
.
We
do
no use
any
o
hese
obse a ions
.
Fo
any wo
e ices,
u
and
,
in
X
he e
is
a
unique
pa h
o
sho es
leng h
ha
connec s
u
o
.
This
pa h
is
called
he
geodesic
ha
connec s
u
o
.
We
deno e
i s
leng h
as
d(u, ),
and
obse e
ha
d
is
a
me ic
on
X
.
Gene al
e e ences
o
ma e s
aised in
his
in oduc ion
a e
[C],
[KPT],
and
[G]
.
2
.
Disk
ealiza ion
o he
ee
.
The e
a e
wo
na u al
ways
o gi e
an
o ien a ion
o
he
edges
in
a
ee,
he
disk
ealiza ion
o
he
ee
and
he
hal -plane
ealiza ion,
which
we
desc ibe
in
he
nex
sec ion
.
Fo he
disk
pic u e,
an
a bi a y
e ex
is
selec ed
and
is
deno ed
o
.
I
is
iewed
as
he
ini ial
poin
o a
andom
walk
ha
is
go e ned
by
he ansi ion
p obabili ies
{p(u, )}
.
A
poin
on
he
bounda y
o
X,
OX,
is
a
semi-in ini e
geodesic, x
=
{xo,
x
l .
...
,
xk,
. .
.
},
whe e
x
o
=
o
.
No e
ha
d(o,
xk)
=
k
o
all
k
.
Suppose
u
and
a e
in
X
.
We
say
ha
w
is
be ween
u
and
i
w
is
on
he
geodesic ha
connec s
u o
.
I
x
E
áX
hen
w
is
be ween
xand
u
i
w
is
be ween
u and
xk
o
all
k
la ge
enough
.
Le
D
=
X
U
OX
.
I
is
be ween
o
and
u
we
say
ha
is
abo e
u, o
ha
u
is
below
.
I
x
=
{xo,
x,
....
,
xk,
. . .
}
we
say
ha
he
geodesic
{xo, xi
...
.
.
xk,
. . .
}
connec s
x
o
o
.
I
ollows
ha
o
is
abo e
e e y poin
in
D
and
i
xE
CM
hen
x
is
below
xk
o
all
k
.
Fo
each
e ex
u
he e
a e
q(u)
neighbo s
o
u,
{uj},
ha
a e
below
u,
excep
o
o
ha
has
q(o)
- -
1
lowe
neighbo s
.
Fo
each
e ex
u,
u
7É
o, he e
is
a
unique
e ex
u -
such
ha
u-
-
u
and
u -
is
abo e
u
.
We
now
de ine a
subbase
o
he
opology
o
D
.
I
consis s
o
all
se s
N(u),
u
E
X
whe e
N(u)
=
{
:
is
below
u}
.
Wi h
his
opology
D
is
compac ,
he
es ic ion
o
he
opology
o
X
is
disc e e,
and
i s
es ic ion
o
áX
is
compac
.
I
{xo, xl
...
.
.
xk,
...
}
E
áX
hen
N(xk)
l
8X
is
e e ed
o as
an
in e al
o
le el
k
.
We
say
ha
xk
is
a
e ex
o
le e]
k
.
We
w i e
Iu
=
N(u)
l
OX
.
We
now
de ine he hi ing
(ha monic)
measu e,
p,
on
8X
.
Deno e
by
F
n
he
( andom)
e ex
a s ep
n
o
he
andom
walk
de e mined
by
P
wi h
F
o
=
o
.
p(I«)
=
P (3k
o
:
Fk
E
N(u)

Vk
>
k
o
1
F
o
=
o)
p
ex ends
o
a
Bo el
measu e
on
9X
.
The
unc ion
U(u,
)
de ined
in
Sec ion
1,
called
he
isi ing
p obabili y,
is
o mally
de ined
by
:
U(u, )=P (3n>0
:F
n
= 1
F
o
=u)
.
190

R
.
ROGHBERG,
M
.
TAIBLESON
Conside
he
andom
walk
de e mined
by
P
s a ing
a
u
and
condi ioned
o
emain
in
N(u)
om
some
s ep
onwa d
.
De ine
a
hi ing
measu e,
,
o
his
walk
jus as
we
de ined
p
.
Then
(I
u
)
=
1
and
o
-
u,
5A
u-
,
de ine
he
ela i e
o wa d
p obabili y,
7 (u,
)
=
(I,)
.
Clea ly
We
say
ha
we
a e
mo ing
o wa d
i
we
mo e
o a
posi ion
below
.
By
P oposi ion
2 o
[KPT],
o
u
=~
o
Obse e
ha
his
o mula
shows
ha
he
p obabili y
o
being
in
N(u)
om
some
s ep
onwa d,
condi ioning
en
he
e en
o
being
in
N(u
-
)
om some
s ep
onwa d,
is
he
same
as ha o
he
e en
o
s a ing a
u-
,
condi ioning
on
he
e en
o
ne e
e u ning
o
u- and
being
in
N(u)
om
he
i s
s ep
onwa d
.
Le
A(u)
be
he
p obabili y
ha a
andom
walk
s a ing a u
mo es
o wa d
en
i s
i s
s ep
and
ne e
e u ns
o u
.
We
see
ha
A(u
-
)
is
he
denomina o
in
equa ion
(1),
so ha
No a ion
.
neighbo s
o u
7
i
=
l (u,uj
)
1
esul o
A(o)
ollows
om
he
de ini ions
.
Fo
u
7É
o
we
use
he
ela ionship
Lemma
1
.
P oo
.
The
which
ollows
ew i e
his
7 (u,
)
=
(Ij
=
1
.
:
-
=u
p(u
-
,
u)
(1
-
U(
u,
u-
))
UW
:W
-
=
u
-
P(u
,
w)
(1
-
U(w,
u-»
7 (u )
=
p(u,
)(Á(
u
)
( +u)~

i
u
=
-
.
Suppose
u
is
a
e ex
and
u

o
.

Le
u
j
,
j
=
1,
. .
.
,
q
be
he
ha
a e
below
u
.
We
se
u
-
=
uo, Pi
=
P(U)
u
j
),
j
=
0, 1,
. . . ,
q
;
aj
=
a(uj,
u),
j
=
1,
...
,
q
;
and
ao
=
ca(u,
uo
)
.
A(o)
=
1
-
U(o,
o)
.
I
u
:~
o
hen
A(u)
=
p(u,
u-
)(1
-
U(u,
u -
))/U(u,
u -
)
.
q
ao
=
Po
+
E
p
.i
al ao
,
j=i
om
he
Ma ko
p ope y
o
andom
walks
(see
[KPT])
.
We
q

po
q

1
~Pia
.i
=1-
a
o
=~P
.i+poC1
-
~
o
l
.
j=i

j=i
F om
his
we
ob ain
FACTORIZATION
ON
TIIE
GREEN'S
OPERATOR

191
Pj(
1
-
aj)
=
1
-
ao
-Po
.
ao
j=1
Using
he
no a ion
in oduced
be o e
he
s a emen
o
Lemma
1
we
no e
ha
A(u)
=
Po(1
-
«o)

pj(
1
-
aj)ao
ao

j-_
p
o
(1
-
«o)
when
u
7É
o
.
I is
easy
o
see
ha
i
{w
o
,
wl,
...
,
w
n} is
he
geodesic
connec ing
o
o
u
hen
n
h(I
.)
=
117 (wk-1,wk)
k=1
I
w
=
{w
o
,
wl,
...
,
w
n
} is
he
geodesic
connec ing
u o
we
de ine
(7)

V(u,
)
=
W(w)
and
obse e
ha
n
(8)

a(u, )
=
Il
a(wk-
l ,
wk)
.
k=1
Obse a ion
.
I is
clea
ha he
egula i y
and
s ong
ansi i i y
o
he
walk
imply
ha
he e
is
a S
>
0
such
ha
whene e
u and
a e
e ices
and
u
=
-
hen
7 (u,
)
>
8
.
and
A(u)
, . ..
1 o
all
e ices
u
.
Theo em
2
.
I
u
and
a e
e ices
andu
is
abo e
hen
c(I

)
-
V(u, )
A( )
p(I
.)

V( ,
u)
A(u)
U( ,
u)
.
P oo
. .
Use
equa ions
(2)-(8),
ga he
e ms
and
simpli y
.
3
.
Hal -plane
ealiza ion
o
he
ee
.
We
begin
wi h
he
disk
pic u e
.
Selec
a
poin
on
óX
and
deno e
i
by
oo
.
To
each
o he
poin ,
y
on
CM
we
associa e
he
unique
doubly
in ini e
geodesic
{
. .
.
,
y-2,y_1,
yo,
yl,
y2,
. .
.
}
such
ha {yo,
yl,
...
}
is
co inal
wi h
he
geodesic
in
D
ha
de ines
y
and
{yo,
y_1,
. . .
}
is
co inal
wi h
he
geodesic
de ining
oc
.
Fo
his
ealiza ion
we
call
he
ee
Y
and
he
ini e
pa o
he
bounda y
is
8Y,
H
=
Y
U
aY
.
Fo
each
e ex
u
he e
is
a
unique
geodesic
ha
connec s
u
o oo
;
which
is
o
say
a
hal -in ini e
geodesic
wi h
ini ial
poin
u
ha
is
co inal
wi h
he
geodesic
in

192

R
.
ROCHBERG,
M
.
TAIBLESON
X
ha
de ines oo
.
We
say
ha
each
e ex
on
ha
geodesic
is
abo e
u,
and
ha
is
below
w
whene e
w
is
abo e
.
The
poin
oc
is
abo e
e e y
o he
poin
in
H, and a
poin
on
áY
is
below
e e y
e ex
ha
lies
on
he
geodesic
ha
connec s
i
o oo
.
A
c ucial
(simpli ying) di e ence
wi h
he
disk
pic u e
is
ha
o
e e y
e ex,
u,
on
he
ee
he e
is
a
unique
e ex,
u-
,
ha
is
a
neighbo
o u
and
is
abo e
u
.
As
be o e
we
se
N(u)
=
{
E
H
:
is
below
u}
.
This
de ines
a
subbase
o
a
opology
on
H
ha
is
locally
compac
.
The
es ic ion
o
he
opology
o
Y
is
disc e e
and
i s
es ic ion
o
áY
is
locally
compac
.
Again
we
se
I
u
=
N(u)
n
DY
.
Jus
as
each
e ex
in
he
disk
has
a
le el
we
can
de ine a
le e]
o
each
e ex
in
he
hal -plane
.
Selec
a
e e ence
e ex
e,
and
conside
he
geodesic
{yo,
y_1,
y_2
. . . .
}
ha
connec s
e o
oo
.
Thenwe
se
he
le e]
o e
as 0
and
each
e ex,
u,
on
he
geodesic
has
le el
-d(e,
u)
.
Obse e
ha
e e y
e ex
on
he
ee
is
below
some
e ex
on
ha
hal -in ini e
geodesic
.
Using
he
ule
:
he
le e]
o u
is
he
le el
o u -
plus one,
a
le el
o
each
e ex
is
de ined
.
As
in
Sec ion
2
we
can
de ine
he
unc ions
:
A(u)
and
V(u, )
.
To
de ine
A(u) we no
longe
need
a
special case
.
Fo
all
e ices
u
A(u)
=
p(u, u
-
)
(1
-
a(u,
u
-
))/a(u,
u-
)
.
Fu he mo e,
we
may
de ine
he
ela i e
o l a d
p obabili ies
(10)

7
(u,
)
=
p(u,
)
(1
-
a( ,
u))/A(u)
whene e
u
=
-
.
Since
he
bounda y
is
no
compac
we
may
no
longe
de ine he
M(Iu)
as
p obabili ies,
bu
we
will
be
able o
use
condi ional
p obabili ies
.
We
p oceed
by
choosing
a
e e ence
e ex, o
.
Fo
example,
one
may
choose
o
=
e
.
Se
p(I o)
=
1
.
Ex end
u o he
ee
by
he ule
:
«u)
=
h(Iu-)7 (u,u-),
ex ends
o a
Bo el
measu e
on
áY
.
As
in
Sec ion
3
:
Theo em
3
.
I
he
e ex
is
below
¡he e ex
u
hen
h(I,)
-
V(u, )
A( )
/,(Iu)

V
( ,
u)
A(u)
U
( ,
u)
.
Obse a ion
.
Theo em
2and
Theo em
3
ha e
he
same
o o
bu he
alues
o
he
unc ion
A
a e
de ined
di e en ly
in
he
wo
ealiza ions
.
We
no e
ha
p,(IJ/p(I
u
)is
he
p obabili y
ha
a
andom
walk
ha
is
condi ioned
o
e en ually
s ay
in
N(u),
is
e en ually
in
N( )
.
Mo e
in o mally
:
i is
he_
p obabili y
ha
he
andom
walk
hi s
he
bounda y
in
I

gi en
ha
i
will
hi
he
bounda y
in
I
u
.
We
also
no e
ha
A(u)
-
1
.
FACTORIZATION
ON
TIIE
GREENS
OPERATOR

193
4
.
Fac o iza ion
o
he Laplacian
and
he
G een's ope a o
.
We
begin
wi h
he
hal -plane
;
he
ex ension
o
he
disk will
be
gi en
in
Sec ion
6
.
below
.
Recall
ha
e e y
e ex
is
bo h
abo e
and
below
i sel
.
I
we
need
o
ule
ou
his
possibili y
we
will
use he
locu ion
s ic 1y
below
o
s icly
aboye
.
We
de ine
he
ke nels o
ope a o s
S
and
T
.
(11)

KS(u, )
=
(lu)'
0,
o he wise
.
a(u,
)
-
a(u,
-
),

i
is
aboye
u
0,

o he wise
.
(12)

KT(u,
)
Obse e
ha
lis(u,
u)
=
1
and
IiT(u, u)
=
1
-
a(u,
u
-
)
o
all
e ices u
.
We
le ,
whene e
he
se ies
con e ges
absolu ely
.
Fo
he
nex
de ini ions
and
he
heo em
ha
ollows
we
use he
no a ion
in oduced
jus
be o e
Lemma
1
and
he
o mula
o
A(u)
in
equa ion
(5)
.
We
in oduce
wo
di e ence
ope a o s
.
Fo
g a
unc ion de ined
on
he ee
:
g(u)
-
1
-
a(u,
u-)
g(u)

1
á
(
a(u,
u
)
)
g(u

)
Theo em
4
.
Fo
¢ny
unc ion
g
de ned
on
he
ee
A
+
Ag(u)
=
A(u)
~9(u)
P oo
.~
D+O
9(u)
_
h
P(IV)

i
is
below
u
S
(u)
_
1
:
Iis(u,
)
( ),
T
(u)
_
1
:
IiT(u,
)
( ),
A9(uj)
-
0
9(u)
194

R
.
ROCIIBERG,
M
.
TAIBLESON
Sol ing
o
we
ob ain
9
ao

E
pjg(u
.i)
-
Po(1
-
%)
j-i
-
1 l
a
g(u)
+
1
a
a
g(uo)
o

o
9
Diana
-

ao
-
po

1

u
~Po(1
-
e¿,)
+
1
-
ceo)
g()
ao
9
Po(1
-
ao)
~
pjg(uj)
-
g(u)
A(
)ág(u)
.
u
9
po(1a
ó
ao)
z
pjg(uj)
+
Po(1aó
ao)pog(uo)
j-i
po(1
-
ao)g(u)
A
he
ou h
s ep
we
used equa ion
(4)
.
This
comple es
he
p oo
.
We
show
nex ha
A
-
is
he in e se
o
T
and
ha
-0+
is
he
in e se o
S on
app op ia e
domains
.
Suppose
is
in
he
domain
o
T
.
I is
easy
o see
ha
T
(u)
=
(1
-
ce(u,
u-
))
(u)
-}-
a(u,
u -
)
T
(u
-
)
.
(u)
=
1
-
a
(
uu-
)
T (u)
-
1
-(a(u
))T (u
)
=
A

T
(u)
.
Since
he
coe icien s
ha
de ine
T
(u)
a e
non-nega i o
and
sum
o
one,
20°
(and
so
2P,
0
<
p
-<
oo)
is
in
he
domain
o
T
.
Thus on
2P,
0
<
p
<
oo,
0
-
T=I
.
The e
is
an a
such ha
0
<
ca(u,
u
-
)<_
á
<
1
o
all
u
.
Consequen ly
i
is
abo e
u,
n(u,
)
<
ád(u, )
.
Conside
he
semi-in ini o
geodesic
{x
o
,
x
y
,
x2
...
.
},
FACTORIZATION
ON
THE
GREENS
OPERATOR

195
whe e
x
o
=
u,
and
xk+1
=
xk
o
k
=
1,2,
....
Then
0o
TA
-
(u)
=E(a(x0,xk)
-
a(x0,xk+1))o
(xk)
k=0
0o
a(x0,xk)(
1
-
a(xk,xk+1))x
k=0
N
-
limo
j
,
a(x0,xk
) (
x
k)
-
a(x0,
xk+l) (xk+l)
k=0
=
(x0)
-

lim
a(x0,
XN+1) (XN
+
1)
X
[
1-
a(
k,xk+1)
(xk)

1
a(a(xk,k
k+l) (xk+l)
N-oo
p o ided
a(x0,xk+1) (xk+1)
=
o(1) as
k
->
oo
.
This
ce ainly
holds
i
is
bounded
and
so
:
Theo em
5
.
0
-
is
he
in e se
o
T
on
~P,
0
<
p
<
oo
.
Lemma
6
.
T
is
bounded
on 2
°°
wi h
no m
1
.
T
and
0
-
map
~°°
on o
i sel
.
P oo .
I
is
easy
o see ha
T
is
bounded on
~°°
wi h
no m
1
and
i is
i ial
ha
0
-
is
bounded
on
2°°
wi h
no m
a
mos
(1
+
a)/(1
-
á)
.
I
ollows ha
T
and
0
-
map
P°°
on o
Qo°
.
We
now
conside
S and
0+
.
No e
ha
S
is
no
de ined
on
2°°
.
In
ac
i
(u)
-
1
hen
S
(u)
-
oo
.
To
see
his
le
{xkj},
J
=1,
. .
.,j(k)',

k=0,1,2,
. .
.,
be
he
j(k)
e ices
ha
a e
below
u a
dis ance
k
.
Then
oo
7(k)

(I
=k,
)

o0
S (u)=E~
i~

.1=1
:
1=00
.
k=0j=1

h(I
.)

k=0
Obse a ion
.
The e
is
a
S,
0
<
S
<
1
such
ha
7 (u
-
,
u)
_< 1
-
S
o
all
u
.
This
ollows
om
he
egula i y
and
s ong
ansience
o
P
.
Lemma
7
.
I
is
in
£P,
0
<
p
<
oo,
hen
is
in
¡he
domain
o
S
.
P oo
..
Suppose
i s
ha
1
<
p
<
oo
.
j(k)

j(k)

1/P
(13)

(xkj)
<
~~ (xkj)IP
j=1

j=1
202

R
.
ROCHBERG,
M
.
TAIBLESON
Suppose
is
o
le el
k
o
.
Then
KT(u>V)Y'P(U)
G
l
:
gkak«ep(ku+k)
u

k=0
=A
E,«,qOP( )
p o ided
e
>
1/p(log
q/
log(1/ix)
-
1)
.
An
elemen a y
compu a ion
shows
ha
a
choice
o
such
an
e is
possible
p o ided
=
IXkOEP
l
:
ú
k(1+EP+10g
11lOg
a)
k=0
log
q

_
p
>
1011/ix)
-
pl
.
Le
p
o
=
max[1,p
l
] .
Ou
a gumen
shows
ha
T
is
bounded on
2P
i
p
>
po
.
An
example
is
gi en
in
Sec ion
7 o
which
p
o
>
1
.
We
use
a
no ion
in oduced
by
Ge l
in [G]
.
De lni ion
.
A
nea es
neighbo
andom
walk
is
s ongly
e e sible
i
he e
is
a
unc ion
m(u)
such
ha
whene e
u
and
a e
neighbo s
m(u)p(u,
)
=
m( )p( ,
u)
and
i
he e
a e
posi i e
cons an s
m
and
M
such
ha
m(u)p(u,
)>_
m
>
0
whene e
u
and
a e
neighbo s
and m(u)
<
M
<
oo
o
all
e ices
u
.
Examples
.
I
P
is
symme ic
;
ha
is,
p(x,y)=p(y,x)
o
all
e ices
x
and
y,
he
walk
is
s ongly
e e sible
.
Jus
se
m(u)
-
1
and
use he
egula i y o
P
.
A
walk
is
iso opic
i
p(u,
)
=
1/(q(u)
+
1)
when
is
a
neighbo
o
u
.
Recall
ha
a
egula
walk
is
o de
bounded,
so
i
we
se
m(u)
=
1
+
q(u)
we
see
ha
iso opic
walks
a e
s ongly
e e sible
.
One
says ha
wo
unc ions
and
g a e
equi alen
and
w i e
-
g
i
whene e
ei he
is
non-ze o so
is
he
o he
and
he
a ios
o
he
absolu e
alues
a e
bounded
abo e
and
below
by
posi i e
cons an s
.
Lemma
20
.
I
he
walk
P
is
s ongly
e e sible
( egula
and
s ongly
an-
sien )
hen
K
T
.
-
Ks
ajad
K
S
.
-KT
.
P oo
.
Bo h
esul s
ollow
om
showing
ha
KS(u,
)
-
KT( ,
u)
.
Re e ing
o (11)
and
(12) in
Sec ion
4
we
see
ha
bo h
exp essions
a e
ze o
i
is
no
below
u
.
I
u
=
we
a e asking
i 1
-
a(u,
u-
)
-
1
which
ollows since
1
-« <
1
-
a(u,
u-
)
<
1
.
Thus we
may
assume
ha
is
below
u
and
ha
u
:~
_
Re e ing
again
o (11)
and
(12)
and
using
Theo em
3
we wan
o
compa e
(19)
V(u, )
A( )
a( ,u)
V( ,u)
A(u)

o
(20)

a( , u)
-
a( ,
u
-
)
=
ce( ,
u)
(1
-
a(u,
u
-
))
.
We
no ed
a
he
end
o
Sec ion
3 ha
A(u)
-
1
so
we
only
need
o
show
ha
V(u,
)
V( ,
u)

1
o
all
e ices
u and
.
Le
us
ecall
he
de ini ions
.
Le
w=
{w0,
w
1
,.
.
.
,
w
n
be
he
geodesic ha
connec s
u
=
w
o
o
=w
n
.
Then
(21)
FACTORIZATION
ON
TIIE
GREENS
OPERATOR

203
V( ,u)
11_1p(w
.i,w
.i_1)
.
Conside
wo
special
cases
.
I
he
walk
is
symme ic
he
exp ession
in
(21)
is
1
and we
a e
done
.
I
he
walk
is
iso opic
i is
easy
o
check
ha
he exp ession
in (21)
is
q(u)
+
1
q( )
+
1
and
since iso opic
egula
walks
a e
o de
bounded we
a e
done
.
Mo e
gene ally,
i
he
walk
is
e e sible
hen
P(
w
.i-1,w
.i)
_-
M(wi-1)p(w,i-I,wi)
m(w
.i)
_-
m(wj)
p(wi,wi-1)

m(wi)p(wj,w,i-1)
M(wi-1) M(wi-1)
and
so
he
p oduc
in
(21)
is
(22)

m(wi)
=
m(u)
. .,
1
M(-j_1( )
j-1
since
he
walk
is
s ongly
e e sible
.
This
comple es
he
p oo
.
Theo em
21
.
I
P
is
s ongly
e e sible
( egula
and
s ongly
i ansien )
he
he
G een's
ope a o
is
o
weak
ype
(1,1),
is
bounded
on
~P
,
1
<p
<
oo,
and
sa is ies
he
conclusion
o
Lemma
13
.
P oo
..
S
is
bounded on
P,
1 <_
p <
oo,
and
sa is ies
he conclusion
o
Lemma
13
.
S*
is
o
weak
ype
(1,1)
and
is
bounded
on
QP,
1
<
p
<
oo
.
Since
KT
-
Ks
.,
T
is
o
weak
ype
(1,1),
and
is
bounded
on
2P,
1
<
p
<
oo
.
G
=
TSR
.
T,
S,
and
R
a e
bounded on QP, 1
<
p
<
oo so
G
is
bounded
on
QP
.
Fixs>0andan
E
P
.
I
{x
:
IG (x)I
>
s I
<
21ISR 1I11
<
Cll 111
s
Now
ake
a
ini e se
o
e ices,
E,
and
le
F
be
he
subse
desc ibed
in
Lemma
13
.
Then
IFI
>
z
IEl
and
IIGXFII
.
-< IIRII
.IIT(SXF)II
.<
IIRII
.IISXFII
.
-<

¡IR¡¡
.
.
20
4

R
.
ROCIIBERG,
M
.
TAIBLESON
Theo em
22
.
G=
-0
-1
on U,
0
< p
<
oo
.
P oo
..
Immedia e om
Theo em
4,
Theo em
5,
Theo em
8,
and
Co ol-
la y
12
.
Co olla y
.
23
.
Unde
he
condi ions
o
Theo em
20,
i 1
<p<
oo
G
maps
£P
on o
i sel
.
6
.
Fac o iza ion
o he Laplacian
and
G een's
ope a o -The
disk
ealiza ion
.
We
p esen
he e
he
modi ica ions
needed
o
ca y
o e
he
esul s
o
Sec ions
4and
5
.
In
his
se ing
he
pa hs
s a
om a
ini e
posi ion,
o
ins ead
o
"coming
om
in ini y"
.
In
his
Sec ion
we
use he
ollowing
no a ion
:
Fo
xE
9X,
x
=
{o
=
xo,
x1, xz,
. .
.,
xk,
.
. .
}
.
Thus,
xk
is
always
a
e ex
a
dis ance
k om
o
and
i k
>_
1
hen
(xk-
1
)
_
(xk)
-
.
Fo
k
>_
1
he
q(xk)
-
q
lowe
neighbo s
o
xk
a e
deno ed
xk
+1
j,
j

=
1,
2,
. .
.
,
q
.

Fo
x
o
we
ha e
q(o)
-}-
1
-
q
-F
1
lowe
neighbo s
x1
j,
j
=
1, 2,
. .
.
,
q
-I-
1
.
We
se
andi n
:~
0
T (xn)
=
Fo he
"backwa d
di e ence"
we
se
andi n ~
0
k=0
0
(xn)
=

1

(xn)
-

a(xn>xn-1)

(xn-1)
.
1
-
«(xn,xn-1)

1
-
a(xn,x"_1)
I is
easy
o
check
ha
T
and
A-
a e in e se o
each
o he
on
he
class
o
all
unc ions
on
he
ee
.
We
see
ha
T
is
de ined
by
he
ke nel
KT
whe e
a(u,
),

i
=
o
KT(u,
)
=

a(u,
)
-
a(u,
-
),

i
u
is
below
and
7~
o
.
0,

o he wise
S
is
de ined
by
he
same
o mula
as
in
Sec ion
4
;
howe e ,
he
alues
u(I
u
)
a e
de ined
as
in
Sec ion
3
.
Simila ly,
T (xo)
=
(xo)
(a(xn,
xn-k)
-
CY(xn,
X
.-k-1))
(
xn
-
k)
+
a(xn, xo)
(
x
o)
.
A
(xo)
=
(xo)
A+
(xn
)
-

7
(xn,xn+1,7) (xn+1,7)
-
(xn
7
Wi h
A
de ined
as
in
Sec ion
3
we
ind
ha
A+¿~,

(x)
=
R(x)O
(x)
.
We
ind
again
ha
G=
TSR
and
he
esul s
o
Sec ions
4
and
5
ca y
o e
almos
wi hou
change
.
FACTORIZATION
ON
TIIE
GREENS
OPERATOR

20
5
7
.
Examples
.
The
simples
example
is
an
iso opic
walk on
a
homogeneous
ee
o
deg ee
q
+
1,
q
>
2, in
he
hal -plane
ealiza ion
.
By
symme y
we
see
ha
7 (u
-
,
u)
-
7
=
1
/q
o
all
e ices
u
.
Simila ly,
a
-
a(u,
u -
)
o
all
u
.
Using
equa ion
(4)
we
ha e
1
a
=

+

q

a
2
q+1 q+1
om
which
i
ollows ha
a=
1/q
.
F om
equa ion
(5)
we
see
ha
A(u)
=
(q
-1)/(q
+
1)
o
all
e ices
u
.
We
also see
ha
excep
o
a
ac o o
(q
-
1)lq
he ope a o s
S and
T
a e
ádjoin s
.
The
G een's
ke nel
is
gi en
by
q1
d(u, )
G(u,
)
=

_l
q+
1
()
which
ollows
by a
i al
applica ion
o
Theo em
11
.
Since
(GX{xo})(x)
_
G(u,
x
o
)
we
see
ha
he
G een's
ope a o
is
no
bounded
on
Ql
.
An
example
ha
a ises
om
a
g oup
is
a
homogeneous
ee
wi h
a
symme ic
aniso opic
walk
.
Fo
a
speci ic
example
ake
a
g oup
wi h
iden i y
e
and
h ee
gene a o s
a,
b,
and
c
;
wi h
he
ela ions
a
2
=b2=
c
2
=
e
.
Le
Y
be
he
Cayley
g aph
o
he
g oup
.
I is
a
ee
ha
is
homogeneous
o
o de
h ee
.
Each
e ex
co esponds
o
a
educed
wo d
and
he
h ee
edges
a
each
e ex
co espond
o
he
igh
mul iplica ion
by
one
o
he
h ee
gene a o s
.
Assign
o
each
edge
p(g),
g
=
a,
b,
o
c,
as
he
case
may
be,
whe e
p(a)
-}-
p(b)
+
p(c)
=
1,
and
each
o he
p's
is
posi i e
.
Fo
any
such assignmen
o
p obabili ies
he
associa ed
walk
is
egula ,
s ongly
ansien ,
and
s ongly
e e sible
.
Le
us
now
cons uc
a
simple
example
o
which
he
G een's
ope a o
ails
o
be
bounded on
£P,
1
<
p
<
oc
.
Take
a
ee ha
is
homogeneous
o
o de
h ee
in
i s
hal -plane
ealiza ion
.
A
each
e ex
he e
is
one
neighbo
di ec ly
abo e
i
and
wo
di ec ly
below
.
To
he
edge
going
up
assign
he
p obabili y
.4
and
o
he
wo
edges going
down
spli
he
di e ence
and
gi e
each
he
p obabili y
.3
.
Wha
makes
his
example
wo k
is
ha
1/3
<
0
.4
<
1/2
.
I
he
"up"
p obabili y
is
g ea e
han
o
equal
o
1/2
he
walk
ails
o
be
ansien
and
i i
is
less
han
1/3 he
G een's
ope a o
is
bounded on h
.
The
case
whe e
i is
equal
o
1/3
gi es he
iso opic
walk
.
Take
he
p obabili ies
as
we
ga e
hem
.
F om
equa ion
(4)
we
ha e
he
equa ion
a=
.4+
.6a
2
whe e
a
is
he
isi ing
p obabili y
associa ed
wi h an
upwa d
ansi ion
.
F om
his
equa ion
we
see
ha
a
is
equal
o
2/3
.
The
a gumen
o
Theo em
19
shows
ha
T,
and
so also
G,
is
bounded on
QP
i
p >
log
3/
log(3/2)
=
p
o ,
which
is
abou
2
.7
.
Tes ing
T
agains
he
cha ac e is ic
unc ion
o
a
e ex
we
ind
ha he
esul
is
sha p
and
ha
T
ails
o
be bounded on
QP
i
p
<
p
o
.
A
mo e
ca e ul
analysis
shows
ha
T
is
o
es ic ed
weak
ype
(p
o
,
p
o )
.
(See
[SW
;
p
.197]
o
de ini ons
.)
20
6

R
.
ROGIIBERG,
M
.
TAIBLESON
8
.
Commen s
and
ques ions
.
The
p incipal
boundedness
esul s
in
his
pape
ollow
om
an
isope ime ic
inequali y
exp essed
in
Lemma
13,
which
hen
leads o
he
weak- ype
es ima e
s a ed
in
Theo em
14
.
This
ci cle
o ideas
is
closely
ela ed
o
he
esul s
ob ained
by
Ge l
in
[G],
whe e
he
ob ains
U
boundedness
esul s
s a ing
om
ano he
isope ime ic
inequali y
.
While
he
lowe
bounda y
desc ibed
in
he
p oo
o
Lemma
13
is
no he
same
as
he
bounda y
used by Ge l (and
o he s
in
combina o ic
g aph
heo y)
he e
mus
be
Glose
connec ions
and
hose
connec ions
should be
s udied
.
The
no ion
o
a
lowe
bounda y was
in oduced
in
[RT]
in a
di e en
se ing
.
The
possibili y
ha
i
could
lead
o
boundedness
esul s
o
he
G een's
ope a o
ca ne
o
mind
whenwe
lea ned
o Ge l's
wo k
in [G]
.
In
[RT]
Lemma
13
in
he
se ing
o he
dyadic ma ingale
was
used
o
com-
pu e
he
K- unc ional
o
he spaces
CMd,
he
space
o
disc e e
Ca leson
Mea-
su es,
and
Po,
he
space
o
unc ions
wi h
ini e
suppo
.
In
his
special
case
he
dyadic ma ingale
is
he
bounda y
ma ingale
o
an
iso opic
andom
walk on
a
ee
ha
is
homogeneous
o
deg ee
h ee
;
he
nodes
o
he
g aph
can
be
iewed
as
he
index
se o
he dyadic
in e als
o
R
.
Lemma
13
allows
he ex ension
o
he
in e pola ion
esul s
o
he
bounda y
ma ingale
o
any
s ongly
an-
sien
andom
walk
on
a
non-homogeneous
ee
.
In
his
mo e
gene al
se ing
he
nodes
o
he
ee
ep esen
in e als
in
a
nes ed
sys em
o
in e als
mo e
gene al
han
he
dyadic
sys em
.
In
Theo em
19
we
ind
a p
o
such
ha
he
G een's
ope a o
is
bounded on
QP
i
p >
po
.

This
p
o
depends
on q
+
1,
he
maximum
o de
o a e ex,
and
á,
he
"maximum
isi ing
p obabili y",
á
=
sup
a(u,
u-
)
.

Fo
iso opic
andom
walks
on a
homogeneous
ee
o
o de
q
+
1,
qand a
a e
cons an s,
a
=
1lq,
and
his
leads o
p
o
=
1
.
Fo
iso opic
andom
walks
on
an
o de
bounded
ee
weknow
ha
p
o
=
1
bu
he
a gumen
o
Theo em
19
leads o
a
g óss
o e es ima e
o
po
.
In
[L]
Lyons
uses
he no ion
o
an
a e age
b anching
numbe
o
a
andom
walk
on a
ee
whe e
he
b anching
numbe
a
a
e ex
u
is
q(u)
in
ou
no a ion
.
Lyons
shows
in a a ie y o
p oblems
ha
his
a e age
b anching
numbe
beha es
like
q
when
q
is
a
cons an
.
Lyons'
esul s
sugges
ha
he e
migh
be an
"a e age
is ing
p obabili y"
as well as
an
a e age
b anching
numbe
and
ha
he
in imum
o
hese
alues
o
p
o
which
he
G een's
ope a o
is
bounded
on
U
could
be
compu ed
om
hese
a e ages
.
Conside
now
he
si ua ion
when
he
andom
walk
is
s ongly
ansien
and
s ongly
e e sible
.
In
his
si ua ion
he
G een's
ope a o
maps
QP
on o
£P
i
1
<
p
<
oo
and
i is
ne e
bounded
on
£
1
.
Fo
i i
we e
bounded on
P
hen
T
would
be
bounded
on
Q1,
which
implies
ha
S
is
bounded
on
£°°
.
Bu S
is
ne e
bounded on
Q°°
as
we saw
in
ou
ema ks
ollowing
he
p oo
o
Lemma
6
.
This
aises
he
p oblem
o
desc ibing
he
class
o in eg able unc ions,
,
on
he
ee
such
ha
G
is
also
in eg able
.
We
can
de ine
a
kind
o
Ha dy
space,
H
=
{
:
E
£
1
,
G
E
Q
1
},
li
IIH
=
ji 111
- -
IiG 111
.
I is
easy
o
see
ha
his
no m
is
equi alen
o
an
"a omic
no m"
in
he
sense
ha
he e
a e
unc ions
called
a o as
and
E
H
i
and
only
i
=
1
:
Akak(x)
whe e
he
ak
a e
a oms
FACTORIZATION
ON
THE
GREENS
OPERATOR

20
7
and
1
:
JAki
<
oo
whe e
ji
IIH
-
in
1
:
¡Akl
o e
all
such
ep esen a ions
.
This
is
no
e y
sa is ac o y since
he
de ini ion
o
"a om"
is
es ic i e
(an
a om
being
he
Laplacian
o a
poin
mass)
.
One
would
wan
a
less
es ic i e
de ini ion
o
an
a om
as well as a
maximal
unc ion
cha ac e iza ion be o e
one
would
ha e
a
sa is ying
heo y
.
This
line
o
hough
also
sugges s
ha
he e
should be
a
BMO
heo y
.
We
only
ema k
ha
he
ee
as
a
measu e
space
wi h
disc e e
measu e
and
he
na u al
me ic
induced
by
geodesic dis ance
is
no
a
space
o
homogeneous
ype, in
he
sense
o
Coi man
and
Weiss
[CW]
.
Fo
spaces
o
homogeneous
ype
Coi man and
Weiss
cons uc ed
a
Ha dy
space
heo y
.
Re e ences
[C]
P
.
CARTIER,
Fonc ions
ha monique
su
un
a b e,
Symp
.
Ma h
.
9
(1972),
203-270
.
[CW]
R
.R
.
COIFMAN
AND
G
.
WEISS,
Ex ensions
o
Ha dy
spaces
and
hei
use
in
Analysis,
Bull
.
Ame
.
Ma h
.
Soc
.
83
(1977),
569-645
.
[G]
P
.
GERL,
Random
walks
on
g aphs wi h
a
s ong
isope ime ic
inequali y,
J
.
Theo e ical
P ob
.
1
(1988),
171-188
.
[KPT]
A
.
KORÁNYI,
M
.
PICARDELLO,
AND
M
.
TAIBLESON,
Ha dy
spaces
on
non-homogeneous
ees,
Symp
.
Ma h
.
29
(1988),
205-265
.
[L]
R
.
LYONS,
Random
walks
and
pe cola ion
on
ees,
Ann
.
P ob
.
( o ap-
pea )
.
[RT]
R
.
ROCHBERG
AND
M
.
TAIBLESON,
An
a e aging
ope a o
on
a
ee,
in
"Ha monic
Analysis
and
Pa ial
Di e en ial
Equa ions,"
Lec u es
No es
in
Ma h
.
1384,
Sp inge
Ve lag,
1989,
pp
.
207-213
.
[RW]
R
.
RoCHBERG
AND
G
.
WEISS,
De i a i es
o
analy ic
amilies
o
Ba-
nach
spaces,
Annals
Ma h
.
118
(1983),
315-347
.
[SW]
E
.M
.
STEIN
AND
G
.
WEISS,
"In oduc ion
o
Fou ie
Analysis
on
Euclidean
Spaces,"
P ince on
Uni
.
P ess,
1971
.
Depa men
o
Ma hema ics
Campus
Box
1146
Washing on
Uni e si y
Sain
Louis,
Missou i
63130
U
.S
.A
.
@da id
.wus l .ed
u