is Hhlder continuous on TI~I

[]

Proof.

For

v ETI~/I

let again Kv: M • M • 0 M - ~ (0, co) be the Martin kernel of Lv.

Then for every fixed v e T I M the assignment

w--* dK~ (Pw, P~tw, ~(w))/dtit=o

is Hhlder continuous (Lemma 3.2 of [H1]) and hence we only have to show t h a t for every

vETI~/I

the assignment

w 6 T ~ I - - + -~K~(Pv, P~ v, ~(w))

t=0 = d t is Hhlder continuous at v.

For this recall from Corollary B.5 and the estimates in the proof of Corollary B.3 that there is a number X>0 such t h a t for every

vETI~/I,

every

wCT],~I

w i t h / ( v , w ) < ~ and every y c - ~ which is contained in the ball

B(Pv,

1) of radius 1 about

Pv

in (2~, ( . , . }) we have

I L , ~ ( y ) l < /(v, w) x

and

I ~ , - ~ l ( y ) < A ( v , w ) •

Let

x = / ( v , w ) x

and recall that there is a number Co>0 not depending on v such that %1 ~<~,(y)~<c0 for all

yeB(Pv,

1).

Define ~ - - ( l + 2 c 0 x ) ~ - ~ . Then >c~<~<(1+2c02)x and ] L v ~ i < x on

B(Pv,

1) which

104 u. HAMENSTA.DT

means t h a t there is a continuous function p:

B(Pv,

1)---,[-1, 1] such t h a t ( L v + p ) @ - - 0 . By our assumption on the coefficients of L , we then necessarily have

for some cl > 0 not depending on v, w and hence

ClX(1+2co+2c2).

This shows the corollary. []

We conclude this a p p e n d i x with some r e m a r k s a b o u t the relation between the oper- ator L and the o p e r a t o r L* which is leafwise formally adjoint to L. For this recall t h a t K~ denotes the M a r t i n kernel of the o p e r a t o r L~ which is formally adjoint to Lv. To explain the relation between K~ and K v assume for the m o m e n t t h a t for every

vETI~/I

the vector field Y~--Yiw8

(v)

on

W s (v),,~ff/I

is the g-gradient of the logarithm of a function f~ on 2~ which we assume to be normalized in such a way t h a t

f~(Pv)=l.

T h e n we have

LEMMA B.8.

K*(Pv, y,~)=fv(y)K,(Pv, y,~) for all vETI~I, ~EO~I and yEM.

Proof.

For a s m o o t h function ~ on

W ~ ( v ) ~ I

we have L*(Q) = A , ( r

Now if ~ is any positive L~-harmonic function on

W ~ ( v ) ~ I

then

L*(~fv)

= f ~ A ~ ( ~ ) + 2 g ( V ~ , V f ~ ) + ~ A ~ ( f ~ ) - d i v ( ~ V f v ) + ~ X ~

= f~(A.(~)+Y.(~)+~X~) = 0

and hence the assignment ~--*~fv m a p s the space of positive L~-harmonic functions on 2~r to the space of positive L~-haxmonic functions. From this the l e m m a immediately

follows. []

Assume now again t h a t L is an a r b i t r a r y weakly coercive o p e r a t o r on

T1M

with HSlder-continuous coefficients. T h e n we have

LEMMA B.9.

There is a number

Co>0

such that

Co 1 <. g , ( P w , pCtw, u(w))K*(Pw, P(~tw, ~(-w)) <~ co for all v, wET1AI and all t>~O.

Proof

(compare L e m m a 3.10 and Corollary 3.11 of [H1]). For

vETI~/I

let G~:

M• co)

be the Green function of the o p e r a t o r

Lv.

For fixed x E M the func- tion

y--~G~(y, x)

is positive and Lv-harmonic on M - ( x } and its values on the distance

H A R M O N I C M E A S U R E S F O R C O M P A C T N E G A T I V E L Y C U R V E D M A N I F O L D S 105 sphere of radius 1 about x are bounded from above and below by a positive constant not depending on v and x. T h e Harnack inequality at infinity of Ancona ([An]) as quoted in the text preceding Corollary B.4 then shows that there is a number 5 > 0 such t h a t

5 -1 ~ K v (p~tw, Pw, 7r(w))/G~ (Pw, P~tw)<<. 5

for all v, w E T 1 / ~ and all t~> 1.

Now

G*(x, y)=G~(y, x)

is the Green function of the formal adjoint L* of L~. Hence another application of the Harnack inequality at infinity for positive L*-harmonic func- tions on _~r shows that

5 -1 <<.K~ (Pw, P ~ w , lr(-w))/G~(Pw, P~tw)<~5.

Together this

shows the lemma. []

Acknowledgement.

This paper is the result of an a t t e m p t to understand the work of F. Ledrappier. I am very indebted to him not only for inspiring me to this work, but also for many helpful discussions and for answering with great patience all my questions.

I t h a n k the referee of an earlier version of this paper for pointing out several errors and for making valuable suggestions for improvements of the presentation. Part of this work was done while I visited the IHES in Bures-sur-Yvette; I thank the institute for its hospitality and for the almost ideal working conditions which it provided.

106 U. HAMENST~,DT

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URSULA HAMENSTADT Mathematisches Institut Universit~t Bonn Beringstrai3e 1 D-53115 Bonn Germany

ursula@rheimiam.uni-bonn.de Received March 3, 1995

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