A FATO
A FATOU-TYPE THEOREM FOR HARMONIC FUNCTIONS
ON SYMMETRIC SPACES1
BY S. HELGASON
AND A. KORANYI
Communicated by G. D. Mostow, August 31, 1967
1. Introduction. The result to be proved in this article is that if u
is a bounded harmonic function on a symmetric space X and x0 any
point in X then u has a limit along almost every geodesic in X starting
at x0 (Theorem 2.3). In the case when X is the unit disk with the nonEuclidean metric this result reduces to the classical Fatou theorem
(for radial limits). When specialized to this case our proof is quite
different from the usual one; in fact it corresponds to transforming the
Poisson integral of the unit disk to that of the upper half-plane and
using only a homogeneity property of the Poisson kernel. The kernel
itself never enters into the proof.
nition of boundary
[4]). In particular
The two actions
(gEG, bEB, xEX
on B normalized b)
mapping f--u whei
u
(1)
is a bijection of the
onto the set of bou
on u in (:
let k(g)E
bserve th
(i)
.
], the m;
2. Harmonic functions on symmetric spaces. Let G be a semisimple
those cor
nction or
connected Lie group with finite center, K a maximal compact subgroup of G and g and their respective Lie algebras. Let B denote the
Killing form of g and the corresponding orthogonal complement of
f in g. Let Ad denote the adjoint representation
for
of G. As usual we
fB
view as the tangent space to the symmetric space X = G/K at the
origin o= {K} and accordingly give X the G-invariant Riemannian
structure induced by the restriction of B to X, . Let A denote the
corresponding Laplace-Beltrami operator.
Fix a maximal abelian subspace aCp and let M denote the
for a suitably norn
centralizer of a in K. If X is a linear function on a and X0O let gx
= {XEGj [H, X] =X(H)X for all HEa}; X is called a restricted root
so the action of a
-- a on N.
if gx #0. Let a' denote the open subset of a where all restricted roots
are #0. Fix a Weyl chamber a+ in a, i.e. a connected component of
a'. A restricted root a is called positive (denoted a > 0) if its values on
a+ are positive. Let the linear function p on a be determined by
2p= >0o(dim g,)a and denote the subalgebras Ea>oga and
Ec>0 g- of g by n and nfrespectively. Let N and N denote the corresponding analytic subgroups of G.
By a Weyl chamber in p we understand a Weyl chamber in some
maximal abelian subspace of p. The boundary of X is defined as the
set B of all Weyl chambers in the tangent space to X at o; since this
boundary is via the map kM--Ad(k)a+ identified with K/M, which
by the Iwasawa decomposition G=KAN equals G/MAN, this defi1 This
work was supported by the National Science Foundation, GP 7477 and
GP 6155.
258
ak(i) MAN=k( a)
(3)
Let El, ** , E,
g_,a say g_.. Sinc
each HE a+, consi
= exp( l aEi)
Since
(4)
eXP
we have
For r>0 let BH,r C
note the volume of
A FATOU-TYPE THEOREM ON SYMMETRIC SPACES
FUNCTIONS
259
nition of boundary is equivalent to Furstenberg's [2] (see also [6] and
[4]). In particular the group G acts transitively on B as well as on X.
The two actions will be denoted (g, b)--g(b) and (g, x)-->g.x
(gEG, bB,
xEX). Let db denote the unique K-invariant measure
)67
on B normalized by fB db = 1. Then according to Furstenberg [2], the
-ticle is that if u
mapping f--u where
ce X and x0o any
!sic in X starting
sk with the non-
(1)
Fatou theorem
u(g.o) = jf(g(b))db
is a bijection of the set L-(B) of bounded measurable functions on B
onto the set of bounded solutions of Laplace's equation Au = 0 on X.
ir proof is quite
The function u in (1) is called the Poissonintegraloff.
If gEG let k(g) EK, H(g)Ea be determined by g= k(g) exp H(g)n
:ransforming the
r half-plane and
rnel. The kernel
(nEN). Observe that if ghdenotes hgh -l for hEG then k(nm) =k()",
H(iit)=H()
for nEN, mEM. According to Harish-Chandra [3,
Lemma 44], the mapping fi--k(R)M is a bijection of N onto a subset
of K/M whose complement is of lower dimension and if f is a continuous function on B, then
be a semisimple
al compact sub-
,et B denote the
I complement of
G. As usual we
(2)
f(b)db
B
X= G/K at the
a restricted root
restricted roots
-d component of
I)if its values on
n=exp(~ aEi) (aiER) we put
determined by
~E,>o ga and
f
;denote the cor-
ith K/M, which
MAN, this defiLtion, GP 7477 and
1V"=
so the action of a on the boundary corresponds to the conjugation
i--na on N.
Let E1, · ·. , E, be a basis of Ii such that each E lies in some
g-,, say g-,,. Since the map exp: i-*N is a bijection we can, for
each HEa+, consider the function -+l n[ H defined as follows: If
and X~0O let gx
o; since this
f f(k(i)M) exp(-2p(H(n)))da
a(k(i)M) = k(a3)M
(3)
M denote the
Y[ at
=
for a suitably normalized Haar measure dii on N. If aEA we have
ak (fi) MA N = k ( a) MA N whence
ant Riemannian
,et A denote the
hamber in some
s defined as the
(g E G),
a= Max (I a Ic%(H))
Since
I
noi m = exp (
(4)
I
I
Ea
exp(-ai()t)E)
I
i
we have
Il jH
forniEN, tER, H E a+.
For r>0 let BH,, denote the set {fiEN] InI<r} and let VH,,de(5)
I
eLs H
lH
= -
note the volume of BH,, (with respect to the Haar measure on T).
_ __
d
260
S. HELGASON AND A. KORANYI
[March
VB
,
A FATO
19681
I
and u the Poisson integral (1) of f. Put
LEMMA2.1. Let fEL-(B)
F(n)=f(k()M) for s ETW.Fix ioEN and HEa+ and assume
1r
(6)
E, ,J
for r-O. Then
(8)
H has
n--*nexP
By the Iwasawa decomposition we can write i-o=k(to0)
·(alnl)-l (alEA, nlEN) so
uz(k(oo)exp tH-o) = u(Soalnl exp tH o) = u(io exp tHalnxP((-tK)o).
But G=ANK so n'P(-tff)=a(t)ti(t)k(t),
t--+ 0o. If HEa is determined by
exp tl
0J
Since p(H(n))
lim u(k(So) exp tH(.o)) = f(k(io)M).
PROOF.
S
·i
IF(*oti) - F(iio) d -O 0
Jaco
I
(9)
Now iENc eP tat
each factor tending to e as
= e
and tHt-tH
is boi
= exp tHala(t)
we have
° P t exp tH,.o).
u(k(fo) exp tH.o) = u(fio(t)e
d=d(H,
Sio, c) bei
measure-preservinl
The function f(b)=f(fofi(t)eP tat(b)) has Poisson integral u'(x)
HSr.x); using (1) on u' and
=u(tOii(t)eIP
f' with g=exp tHi we get
from (2) and (3)
V
where dl=d(H,
ic
u(k(io) exp tH o) - f(k(o)M)
-
N
(F(IoI(t)eP s'Ht=IP
') - F(io))exp(--2p(H()))d
where d2 (H, o) is
majorized for all t
exp 2p(tHg)j
SO
Iu(k(0o)
(7)
exp tH o) -f(k(
<
0)M)
I F( 0ofieptla)
-
I
(10)
F(io) exp(- 2p(H(n(t)-i)))di.
where d and da are
On the other h;
second term in (8)
Now if c> 0 let N, denote the "square"
N = {exp
aiE)I
aI_
c,,
I
The integral on the right in (7) equals the sum
<i r}
21IF1. i:N(11)
[March
s9681
egral (1) of f. Put
nd assume
261
A FATOU-TYPE THEOREM ON SYMMETRIC SPACES
f
I F(ffiexP H,)
(8)
+
f
-
exp(-2p(H(i(t)%-x)))df
F(o)
I F(ioxP
Fifi)
I exp(-2p(H((t)-
1Ht) -
1i)))di.
Since p(H(g)) _>Ofor all gEN ([3, p. 287]) and since the mapping
.jn=xp H has Jacobian exp(-2p(H))
(cf. (4)) we see that
f
a write go0 =k(go)
) talinlerP(-)
F(ogOXntt)
(9)
.o).
-
1
F(go) exp(-2p(H(i(t)-
_ exp(2p(tHN))
)))df
F(Lo)
F(ao) d.
-
Now E N, -P t t if and only if
)r tending to e as
= exp(,
and tHt-tH
aie-1 ('H')E) where I a
c
is bounded (for fixed go and H). It follows that
N,xpM
BEd-'
Bdi-t
for all t 2 0,
d=d(H, o, c) being a constant. But since the map exp: ft--R is
measure-preserving it is clear that
=exp tHt we get
where dl=dl(H,ao, c) is another constant. Also
exp(2p(tHt))
exp(2p(tH))d2
where d2 (H, lo) is a constant. Thus the right hand side of (9) can be
majorized for all t >_0:
'xp(- 2p(H()))df
exp 2p(tHe)
t
2p(Hfi()-A)))d-fi
t _0
VH,d,- = exp(-2p(H)t)di
on integral u'(x)
(10)
e p
EI
F(ioi) - F(go) do
,ds
d
VH,d.-
B
B ,de-t
IF( 0 ,)
-
I
F(io)
0 dn
where d and da are constants depending on H, go and c.
On the other hand, if II jII denotes the uniform norm on N the
second term in (8) is majorized by
21[II=
(11)
r_
exp (-2p(fH((t)-ld)))d)
= 2I. . -f-.,,
A(t)No
ep
(-2p(H(n)))d)
262
S. HELGASON AND A. KORANYI
i968]
[Mauch
follows immediately
Now given > 0 we first choose c so large that
convergence theore
u has the same limi
Weyl chamber in p
(ii) In the case M
exp(-2p(H(i)))df) < / 2 ;
21IF11 (1 -
since A(t)--e for t-- + oo we can choose t such that A(t)NWioND for
t Ž ti. Then the expression in (11) is < /2 for t > ti; by our assumption
has proved (13), ev
(6) we can choose t2 such that the right hand side of (10) is <e/2
for t>t 2. In view of (7) and (8) this proves the lemma.
The next lemma shows that, for a fixed H, the assumption of
Lemma 2.1 actually holds for almost all oE-N.
0
lim
r
--+O
(12)
VH .
1. R. E. Edwards
operators,Acta Math. 1
2. H. Furstenberg, i
(1963), 335-386.
3. Harish-Chandra,
Math. 80 (1958), 241-3
4. F. I. KarpeleviE,
and fix HEa + . Then
LEMMA 2.2. Let FEL(i)
Br r
IF(i 0tio)-
F(so) I df = 0
operatoron symmetricsj
5. A. W. Knapp, un
6. C. Moore, Comp
201-218.
for almostall aoE N.
The proof of this result is essentially in the literature: In 11]
Edwards and Hewitt give all the necessary arguments for the case of
a discrete sequence tending to 0 and everything they do remains
trivially valid in the case r--0. The result in the exact form required
here was also proved by E. M. Stein independently of [1] (cf. his
expository article [6]).
7. E. M. Stein, Ma:
on bounded homogeneo
MASSACHUSETTS IN
YESHIVA UNIVE!
THEOREM 2.3. Let u be a bounded solution of Laplace's equation
Au=0 on the symmetric space X. Then for almost all geodesics (t)
starting at o
lim u( 7 (t))
exists.
,--.
(13)
PROOF.Let S+= {IHEa+
B(H, H)=11. Then the mapping
(kM,H)--Ad(k)H is a bijection of (K/M)XS + onto a subset of the
unit sphere S in p whose complement has lower dimension. Since
dim(K/M-k(N)/M) <dim K/M the mapping(, H)--Ad(k())H
is a bijection of N XS+ onto a subset of S whose complement in S has
lower dimension. If NH denotes the set of hnofor which (12) holds
(with F(t)=f(k(h)M))
and if S=UHEs+ Ad(k(NH))H it follows
from the Fubini theorem that S-So is a null set. This concludes the
proof.
REMARKS. (i) If f is continuous the limit relation
I
I
lim u(k exp tH. o) = f(kM)
I-0+e0
(H C a, W E KM)
A E'ATO
I968]
[March
Lat A(t)NDN./2
A FATOU-TYPE THEOREM ON SYMMETRIC
SPACES
263
follows immediately from (1), (2) and (3), by use of the dominated
convergence theorem. (See also [4, Theorem 18.3.2.1) In particular,
u has the same limit along all geodesics from o which lie in the same
Weyl chamber in p.
(ii) In the case when X has rank one (dim a= 1) A. W. Knapp [5]
has proved (13), even under the weaker assumption that fEL 1(B).
for
; by our assumption
.de of (10) is < e/2
REFERENCES
,mma.
the assumption of
1. R. E. Edwards and E. Hewitt, Pointwise limits for sequencesof convolution
operators, Acta Math. 113 (1965), 181-218.
2. H. Furstenberg, A Poissonformula for semisimple Lie groups, Ann. of Math. 77
(1963), 335-386.
3. Harish-Chandra, Sphericalfunctions on a semisimple Lie group. I, Amer. J.
Math. 80 (1958), 241-310.
-0
4. F. I. Karpelevi6, Geometryof geodesicsand eigenfunctionsof the Laplace-Beltrami
operatoron symmetric spaces,Trudy Moscov. Mat. Ob. 14 (1965), 48-185.
5. A. W. Knapp, unpublished manuscript.
6. C. Moore, Compactifications of symmetric spaces, Amer. J. Math. 86 (1964),
201-218.
literature: In [1]
ients for the case of
g they do remains
exact form required
7. E. M. Stein, Maximal functions and Fatou's theorem,C.I.M.E. summer course
on bounded homogeneous domains, Cremonese, 1967.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY AND
YESHIVA UNIVESITY
,ntly of [1] (cf. his
Laplace's equation
st all geodesics y(t)
II
'hen the mapping
nto a subset of the
:r dimension. Since
(n, H)--Ad(k(nfi))H
omplement in S has
,r which (12) holds
'(NH))H it follows
This concludesthe
i
I
I
I
I11
M E KIM)