A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
A DAM K ORÁNYI
M ASSIMO A. P ICARDELLO
Boundary behaviour of eigenfunctions of the Laplace operator on trees
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4
esérie, tome 13, n
o3 (1986), p. 389-399
<http://www.numdam.org/item?id=ASNSP_1986_4_13_3_389_0>
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Boundary Eigenfunctions
of the Laplace Operator
onTrees.
ADAM
KORÁNYI (*) -
MASSIMO A. PICARDELLO(**)
1. - Introduction.
Let T be any
tree,
andlet s,
t be vertices of T. If s and t areneighbours,
,the ordered
pair (s, t)
is called an(oriented) edge issuing
from s. A nearest-neighbour
transition matrix is determinedby assigning
apositive
numberp(s, t)
to eachedge (s, t).
This matrixgives
rise to a transitionoperator
Pon functions F on the vertices of T:
where the sum is taken over all
edges issuing
from s. TheLaplace operator
associated with P is definedby
The
Laplace operator
has been studied in detail in the case where the tree ishomogeneous
ofdegree q +1,
and the transition matrixgives
rise to a(nearest-neighbour) isotropic
random walk:p(s, t) =1/(q -E- 1)
for everyedge (s, t) [5, 2].
In this case,eigenfunctions
of theLaplace operator (or, equi- valently,
of the transitionoperator P)
can berepresented
as Poisson trans-forms of
generalized
functions on the4 boundary)> 92
of the tree[5],
thatis,
its set of u ends >>. On asymmetric
space, where a similarrepresentation holds,
theboundary
behaviour ofeigenfuctions
of theLaplace-Beltrami operator
can be describedexplicitly [6]. Indeed,
let F, be the Poisson(*) Supported by
NSFgrant
no. MCS 8201815.(**) Supported by
MPI -Gruppo
Nazionale « Analisi Funzionale».Pervenuto alla Redazione il 30 Gennaio 1985 ed in forma definitiva il 15 Mar-
zo 1985.
transform of an
integrable
function defined on the Poissonboundary
of asymmetric
space.Then,
if F isappropriately normalized,
itsasymptotic
values
along
certainsets,
introduced in[3]
and called « admissibledomains »,
coincide
almosteverywhere
with the values off.
In the case when the sym- metric space is thehyperbolic
disc and the function F isharmonic,
thisresult coincides with the classical Fatou theorem on
nontangential
con-vergence.
The purpose of this paper is to introduce the
analogues
of admissible domains fortrees,
and to use them to describe theboundary
behaviour ofeigenfunctions
of A. Our results bear a close resemblance to those of[6]
for
symmetric
spaces.In section 2 we restrict attention to the
homogeneous
treeTq
withbranching degree q + l,
and theisotropic
transition matrix. Fix a reference vertex o inTq,
and consider the randomwalk, starting
at o, inducedby
the transition matrix. For each
complex
number y, thereexists,
up tonormalization, only
oney-eigenfunction
of the transitionoperator
which isradial, i.e.,
whichdepends only
on the distance from o : these functions arecalled
« spherical
functions ». If p is acomplex
measure on theboundary
S2of
T,,,
denoteby ,ur
theregular part of a
withrespect
to the Poisson meas- ure v on Q. For eachcomplex
number z, consider the associated Poisson transformF z
of 1", and thecorresponding spherical
function qz.Let e
bethe
spectral
radius of the transitionoperator
P onl2(Tq).
Then the main result of section 2 statesthat,
for everycomplex eigenvalue
of P outsidethe interval
(-
o,p),
the functionF,199,
convergesasymptotically
almosteverywhere, along
admissiblesets,
to theRadon-Nikodym
derivativedyldv.
Furthermore,
for theeigenvalues ±9,
we show that convergence toboundary
values holds in astronger
sense than the usual admissible con-vergence. This is the
analogue
of a recent result ofSjogren
for thehyper-
bolic disc
[7].
The results of section 2 follow from
sufficiently precise
estimates for the Poisson kernel and thespherical functions, along
the lines of[2].
Thecrucial
part
of theargument
shows that the normalized Poisson transform of anintegrable
functionf
on theboundary
Q is bounded on an admissibledomain
by
theHardy-Littlewood
maximal function at the vertex.On the other
hand,
fornon-homogeneous trees, sharp
estimates of thistype
are not available.Moreover,
a Poissonrepresentation
theorem is knownonly
for not too small realeigenvalues
and forpositive eigenfunctions
ofthe
Laplace operator [1,
thm.2.1].
For theseeigenfunctions
ageneraliza-
tion of Fatou’s convergence theorem was
proved by
Cartier[1].
Cartier’stheorem concerns radial >> convergence to the
boundary Q,
thatis,
con-vergence
along geodesics
in the tree. In section 3 weslightly improve
thisresult to
give
admissible convergence instead of radial convergence. Ourargument
isindependent
fromCartier’s,
and is based upon thegeneral
Fatou-Naim-Doob theorem
(that is,
thetheory
of fine convergence: see, forinstance, [9]).
A radial Fatou theorem for harmonic functions withrespect
to theisotropic Laplace operator
on ahomogeneous
tree had beenpreviously
obtained in[8],
in the framework of local fields.2. - The
isotropic Laplace operator
on ahomogeneous
tree.In this
section,
we restrict attention to thehomogeneous
treeTq
withbranching degree q + 1, q>2.
Most of thepreliminaries
are as in[2],
andnotations will
be,
as much aspossible,
the same as in that reference withonly
minor modifications. Most of these notations also make sense(and
will be
tacitly adopted)
fornon-homogeneous
trees. Inparticular,
Q denotesthe
boundary
of thetree,
thatis,
the set of infinitegeodesics, starting
at thereference vertex o, and
N(w, w’)
denotes the number ofedges
in commonbetween co, m’e S2. We can
identify
each vertex x ofTq
with the finitesimple path connecting
o with x ; thenN(x, m)
denotes the number ofedges
in common between the finite
geodesic
x and the infinitegeodesic
m. Thedistance
d(o, x)
is denotedby lxl,
and the vertex withlength n ixl
in thegeodesic connecting
o and x is denotedby
x...Similarly,
mn denotes the vertex withlength n
in thegeodesic
wE S2.Finally,
we endow Q with acompact topology
as follows.As in
[2],
we introduce the sets.Ex = {w: N(x, m)
=x }.
Then an open basis at w E Q consists of all the setsB.., n
e N. The Poisson measure v onS2,
thatis,
thehitting
distribution on 92 of the randomwalk, starting
at o, defined
by
the transitionoperator,
isgiven by
and the
corresponding
Poisson kernel is[2].
Inparticular
We are now
ready
to define admissible sets in the tree(this
definitionis also valid for
non-homogeneous trees).
Admissible sets in a tree are theanalogue
of euclidean cones(that is, hyperbolic cylinders)
in the upperhalf-spaces,
with vertex at theboundary.
DEFINITION. For every
integer 0153>O,
and co E S2: for some
We say that a function .F’ on the tree converges
admissibly
to l at wif,
forevery
x>0y
limF(.r)
= l.REMARK. The notion of admissible convergence is
independent
of thechoice of reference vertex o.
Indeed,
denoteby T( (m)
the admissible domains withrespect
to a different reference vertex o’.Then,
since there is aunique simple path connecting
o ando’,
it followsthat,
ifIx I
islarge enough, x E ra(cv)
if andonly
ifz e T( (m) .
DGiven two
complex-valued
functionsf, g
we writef
sw gif ( ffg
is boundedabove and below.
LEMMA 1..Let
co° E S2
andx E Fa(wO) for
somex>0y Ix
= n. ThenK(r, m) sw K(wt, co)y
with boundsdepending only
on a.. Inparticular
PROOF. It is
enough
to show thatIN(x, ro) - N(a)o, m) I a.
Forthis, let k = N(x, coO) > it - cx. Then,
if rok=Xk, bothN(x, co)
andN(co.0, co)
arenot less
than k,
and notlarger than n,
and theinequality
follows. On the otherhand,
ifWk =1= Xk,
I thenIt has been
proved
in[5]
that alleigenfunctions
of P can berepresented
as Poisson
transf orms,
thatis, integrals
of the formXz f = g(x, w)’f(w) dv(w),
where
f
is afinitely
additiz e measure(that is,
amartingale) on Q,
andz E C. The
corresponding eigenvalue
isBy
thesymmetry properties
of(1),
we can restrict attention tocomplex
numbers z such that Re z
> I
and -7t/log q
Imz7t/log
q, or Re z--. 2
and
0 Im z a/log
q. The latter set ofparameters corresponds
toeigen-
values
y(z)
in the12-spectrum
of P.Explicit expressions
are known for the Poisson transforms of the constant function1
onS2. Indeed,
there exists a constant az,depending
on z, such that ifIx ---
n, the «spherical
function » fp.. =Xz 1
has theproperties [2r
ch.
3] :
if if if
We are interested in the
asymptotic
behaviour of Poisson transforms nor--malized
by
thecorresponding spherical
functions.However, (2)
showsthat,,
for
z = 2 I + it 7 0 t n/log q,
thespherical
function qzoscillates, and
there-fore
dividing by
it cannot beexpected
to lead to results aboutasymptotic
convergence. For the other values of z we introduce the normalized Poisson kernel
gx(x, m)
=.gx(x, co)lgg,.(x).
Lemma 1 and the estimates
(2) yield
LEMMA 2. Let
Then, ’
with bounds
depending only
on z and a.Denote
by X
the group of all isometries ofTo
which fix o.COROLLARY 1..Assume Re z
> 2
or z= 1 + imnjlog
q, m =0, 1.
Then.i) for
every k in X andfor
every x, co,ii) f or
every x,iii)
there exists a constantM,
suchthat, for
every x,iv) f or
everyinteger j
andf or
everywo,
PROOF.
(i)
isobvious, because N(x, cv)
is invariant underX,
and(ii)
isnothing
else but the definition of pz .(iii)
follows from Lemma 2 and the .fact thatv(Ey) q-lvl
with boundsindependent
of y.Finally, (iv)
is animmediate consequence of Lemma 2. CI
By
standardarguments (see [10, chapter 17,
thms.1.20, 1.23])
the esti-mates in
corollary
1yield:
PROPOSITION 1. Assume Re
z > 2
or z== t + imnjlog
q, m =0,
1. Letf
be
defined
onQ,
and denoteby
F its normalized Poissontransform :
Then lim
F(a).,,)
=f(w)
i) uniformly i f f
iscontinuous;
ii)
inL1J(Q) if f
EL1J(Q) (1
p00);
iii)
in theweak*-topology of LOO(Q) if f
ELOO(Q);
iv)
in theweak*-topology of M(Q) if f
is aregular signed
measure.For the main result of this
section,
we need a final tool: theHardy-
Littlewood maximal theorem. If
f
isintegrable
function onQ,
itsHardy-
_Litttewood maximal
f unction
is defined as’The
operator f
-->MI
is called the maximaloperator,.
LEMMA 3. The maximal
operator
is weaktype (1,1 )
andstrong type (p, p) for
1 p oo.PROOF. Denote
by A
the stabilizer of coO in thegroup X
of all iso- metries ofT,, fixing
o. Then we introduce a « gauge » on S2by
the ruleIt is
readily
seen that this is a gauge for(X, A)
in thesense of
[4,
definition1.1]. Indeed,
weonly
need to check condition(iii)
of
[4,
definition1.1]:
this followsby observing that,
ifro1, ro2 satisfy N(ojO, oil) N(roo, ro2),
thenN(roo, ro1)
==.N(ao, ro2).
Then the lemma follows from[4, Corollary,
pg.580].
CJWe can now prove the
nontangential
convergence theorem.THEOREM 1. Let z be as in
Proposition 1,
and let u be a measure on Q.Denote
by F(x) =fK,(x, ro) du(w)
=K.,p/gg,
the normalized Poissontransform.
of
p, andby p,
theregular part of
p withrespect
to the Poisson measure v.Then F has admissible limits a. e. on
Q, equal
todPr/dv(ro).
PROOF. Standard methods
[10, chapter 17]
reduce theproof
to the caseof
absolutely
continuous P, thatis,
to Poisson transforms of .L1-functions.Since every .L1-function can be
approximated
in L1by
continuousfunctions, by Proposition
l.i. weonly
need to show thatv(m:
supIF(x) >
A forXEra{ro)} IlflBl/Â.
Because of Lemma
3,
this follows if we provethat,
forf
inL1(Q),
We
distinguish
two cases.Case 1.
Re z > 1.
For x inF,,((oO), Ixl
= n, Lemmas 1 and 2yield:
We have used the estimate
Case 2.
ikn/log q, k
=0,
1.Again by
Lemma2,
asabove,
In both cases,
(2)
isproved,
and the theorem follows. 0We recall that the case z
= 2 + ika/log q corresponds
to the twoeigen-
values
+e, where g
is thespectral
radius of P onl2(Tq).
For these criticaleigenvalues,
we can prove astronger
convergencetheorem, along
the linesof
[7].
Let us considerenlarged
admissible domainsand the
corresponding c strengthened
admissible convergence ».THEOREM
2. If
z= 1 -J- i7cnflog
q, ly =0, 1, then,
with notations as in Theorem1,
F converges a.e. todyldv
in thestrengthened
admissible sense.PROOF. Let x E
ha (00°), Ix ( =
n, and m = n -[log,, n] -
a. We can as-sume m > 0. Observe that
N(r, m°) >7n. Moreover,
IXEr£¥+logCln(ooO):
thusLemma 2
yields (write
E =E0153:n):
again
asv(E) ,. q-m.
On the other
hand,
ifoi c- Q - B
then mm#co’ ,
henceN(m), w)
mN(x, WO).
ThusKz(x, w) == Kz(wO, w),
and theargument
of Theorem 1applies again
togive
By combining
theseestimates,
y we obtain a constantC(a, z)
such thatIF(x) I C(a, z) Mf(m°)
for every x E(coO),
and the theorem follows. C1 REMARK. The end of theproof
of Theorem 2exploits
thegeometry
of the tree in the same way as the
argument
of Lemma 1. Thisargument
is
equivalent
toobserving
that the metric on Q inducedby
the gauge,d(ro, ro’)
=q-N(W,(O,) ,
satisfies the ultrametricinequality d(w, cv’)
max(d(m,
"CV"), d(GVn, cof)}.
.3. -
Non-homogeneous
trees.In this
section,
T is anon-homogeneous
tree.Following [1 ],
we considera transition
operator
.P suchthat,
for everyvertex 8,
the cc transition coef- ficients»p (s, t)
arepositive
andnearest-neighbour,
but do notsatisfy,
y ingeneral,
the « markoviancondition » z p(s, t)
= 1.i
Denote
by A
thecorresponding Laplace operator,
andby A.
the mar-kovian
Laplace operator
on ahomogeneous
tree considered in theprevious
section. Note that the
eigenfunctions g
ofAm corresponding
to agiven -eigenvaliie y > - 1 satisfy Ag
= 0 for anappropriate
non-markovianLaplace operator
A :nevertheless,
y in this section we follow theterminology
of[1],
and refer to the solutions of
Ag
= 0(or Ag > 0)
as « harmonic »(respectively,
,superharmonic)
functions. Apath
c in T is a finite or infinite collection ofadjoining edges (ci, ci+,)
in T: we write c =(co,
cl, ..., cn,...).
In the markovian case, the transition
operator
Pgives
rise to a randomwalk
X,,
on the vertices of T. In thegeneral
case, for every infinitepath
c,we write
Xn(c)
= cn[1,
section3.1].
nFor every finite
path
c =(co, ..., cn),
definep(c)=
and’i=1
consider the Green kernel
G(x, y) == Ip(c),
where the sum is taken overall
paths joining
the vertices x and y.Following [1],
we assume thatG(x, y)
is
finite
for every x, y(in
the markovian case, thisimplies
that the random walkX n
istransient).
Let
wE Q,
and letK(X, w)
=lim G(X, wn)/G(Q, (On):
the functionsK (0153)
= K(x, co)
are the minimal harmonic functions onT,
normalizedby KQ)(o)
=1.Every positive
harmonic function h has aunique
Poissonrepresentation
h
== f Kro dyltg
where lzh is apositive
Bore] measure on Q[1,
thm.2.1].
Infact,
Qgives
rise to the Martincompactification
of Z’(a
basis for thetopology
of T u Q isgiven by
the setsB,, U fy: EtJ C -Ex}).
Denote
by
W be the set of infinitepaths
andby
W’ the subset ofpaths
cwhich have a limit
(i.e.,
for which there exists m in Q such that cn - m in the abovetopology). By [1, Corollary 3.1 ],
for every x in T and (o in Q there is aprobability measure v’
onW,
withsupport
in the set ofpaths starting
at x andtending
to w, whichhas the following property:
for every finitepath
cjoining x and y,
,where
We
is the set of infinitepaths
whosestarting segment
is c.We are now
ready-
to state thenontangential
convergence theorem for thenon-homogeneous
case. Our theorem is an extension of[1,
thm.3.3].
The
argument
relies upon thefollowing fact,
which is a consequence of the Fatou-Naim-Doob theorem[9] (but
can also beproved easily,
yby using Corollary 3.1 b,
Theorem 3.1 and Theorem 3.2of [1 ]) : for
everypositive
harmonic h and
positive superharmonic
g, the limitlim
ngfh(Xn)
existsvx-atmost
surely tor Iz,,-almost
every w.’n
In order to connect the next statement with Theorem
1,
suppose that the tree ishomogeneous
and the transitionoperator
is as in section2,
and observethat,
if h =-1,
then itsrepresenting
measure a,, is v.THEOREM 3. Assume that there exists 6 > 0 such that
p(x, y) >
6for
allneighbours
x, y inT,
and that the Green kernel isf inite..Let
h be apositive
harmonic
function
withrepresenting
measure fth, and let g be apositive
super- harmonicfunction.
Theng/h
has admissible limitsfth-almost everywhere
on Q.PROOF.
By
the remarkpreceding
thestatement,
it suffices to show thatg/h
has admissible limit at m for all m such thatlim
4glh(X..)
existsvl-almost surely. By contradiction,
let l= lim g/h(Xn)
and suppose that there exists e >0,
aninteger
a and a sequence{Xk}
such that XkEFa(w),
lXk I --io-
00 andIg/h(Xk)
-I > B
for all k. Then we claim that there exists>
0 suchthat,
for every x EFa(w), v’[X,,,
= x for some71] >q.
Once theclaim is
proved,
the theorem follows :indeed,
the processXn
meetsinfinitely
many of the
xk’s
withv-probability >,q,
which contradicts theassumption
lim
g/h(Xn)
= 1.To prove this
claim, let .F’w(o, x)
=V’[Xk - X
for somek]. As
in[1, chap-
ter
2]
let where the sum is taken over allpaths
from oto x
reaching
xonly
at the end(in
the markovian case,F(o, x)
is theproba- bility
ofhitting x starting
ato). By
definition ofv#/, FW(o, x)
=F(o, x) K(x, a)).
We have to prove
-E’ (o, x) > 77
> 0 for all x in-P,, (w). By
definition ofK(x, co),
this amounts to
showing that,
forlarge n,
By [1,
prop.2.5],
the left hand sideequals F(o, x)F(x, cvn)/.F’(o, con).
AsxEFl¥(CO),
there exists aninteger j
such thatd,(x, (oj) a.
Without loss ofgenerality,
we can assume 6 1.Then, by [1,
coroll.2.3],
the above ex-pression equals F(o, a)j)F(coj, x)F(x, wj)F(wj, con)/F(o, coj)F(a)j, con) = Ti(o)j,
amd the claim is
proved.
C7REMARK. A more
general
but lessprecise
statement holds without theassumption
that the transition coefficients be bounded below.Indeed,
thetheorem holds with the same
proof
if we define1 a(cv)
==fx: F(o, x).K(x, b)
oc-ll.
On the otherhand,
a lessgeneral
but moreexplicit
statement is obtainedby assuming 0 6 p(x, y) 77 -1
for all x, y : this conditionautomatically guarantees
the finiteness(even
the uniformboundedness)
ofthe Green kernel.
In the case of a
homogeneous
tree withisotropic
transitionprobabilities,
,it is
interesting
to consider theoverlapping
between Theorems 1 and 3.As noticed
above, multiplication
of the transition matrixby
apositive
constant
gives
rise to a dilation of itseigenvalues. By
thistoken, Theorem
3handles all
positive eigenfunctions
of theLaplace operator
whoseeigen-
values
satisfy y > -1.
On the otherhand,
it followsimmediately by [2,
ch.
3]
that the existence of apositive eigenfunction
isequivalent
toy > O -1, where,
asbefore,
p denotes thespectral
radius of P in 12. In otherwords,
Theorem 3 can be used to deal with a proper subset of the set of
eigenvalues
considered in Theorem 1.
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harmoniques
sur un arbre,Sympos.
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Boundary
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onsymmetric
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