ALINE BONAMI 1
, DARIUSZ BURACZEWSKI 1;2
, EWA DAMEK 1;2
, ANDRZEJ
HULANICKI 1;2
, RICHARD PENNEY, AND BARTOSZ TROJAN 1;2
Abstra t. We onsiderhereageneralizationoftheHuasystemwhi hwasproved
byJohnsonandKoranyito hara terizePoisson-SzegointegralsforSiegeldomains
oftubetype. Weshowthatthesituationis ompletelydi erentwhendealingwith
nontubetypesymmetri irredu iblesymmetri domains: thenallfun tionswhi h
are annihilatedbythis se ond ordersystemand satisfyanH 2
typeintegrability
onditionarepluriharmoni fun tions.
1. Introdu tion
Let D be a bounded symmetri domain in C m
, and let G be the group of all
biholomorphi transformationsof D. Theaimof thispaperis tostudy H-harmoni
fun tions, where H is a naturally de ned G-invariant real system of se ond order
di erentialoperatorsonDwhi hannihilatespluriharmoni fun tions. ThesystemH
isde ned intermsofthe Kahlerstru tureof Dand makessenseoneveryKahlerian
manifold.
To de ne the system H, we re all some basi fa ts about D. Let T 1;0
(D) be the
holomorphi tangent bundle of D. The Riemannian onne tion r indu ed by the
Bergman metri onDpreservesT 1;0
(D)andsodoesthe urvaturetensor. ForZ;W
two omplex ve tor elds we denote by R (Z;W) = r
Z r
W r
W r
Z r
[Z ;W℄
the
urvature tensor restri ted toT 1;0
(D). Letf be asmooth fun tion onD and let
(Z;W)f =(ZW r
Z
W)f =(WZ r
W Z)f: (1)
Then (Z;W) may be seen as a se ond order operator whi h annihilates both
holomorphi and antiholomorphi fun tions, and onsequently, the pluriharmoni
fun tions. Conversely, ifall(Z;W)annihilatef, thenf ispluriharmoni . Indeed,
wehave (
z
j
;
z
k )=
z
j
z
k .
Let(; )bethe anoni alHermitianprodu tinT 1;0
(D). Fixingasmoothfun tion
f, we use ( ; ) to de ne a smooth se tion
f
of the bundle of endomorphisms of
1
Thisworkwaspartlydonewithin theproje tTMR Network\Harmoni Analysis", ontra t
no. ERBFMRX-CT97-0159. Wethank the European Commissionand the mentionedNetwork
forthesupportprovided.
2
TheauthorswerepartlysupportedbyKBN grant2P03A04316,FoundationforPolishS ien es,
Subsidy 3/99.
T (D):
(f Z;W)=(W;Z)f; (2)
where Z;W are holomorphi ve tor elds. Then we de ne Hf as another smooth
se tion of the bundle of endomorphisms of T 1;0
(D) by
(Hf Z;W)=Tr (R (Z;W)
f
)=Tr(R (W;Z)
f ): (3)
To omputeexpli itlyHf,wemaytakeanorthonormalframeofse tionsofT 1;0
(D),
whi h we denoteE
1
;E
2
; ;E
m
. Then
Hf = X
j;k ((E
j
;E
k
)f)R (E
j
;E
k ): (4)
The system His,of ourse,a ontra tionofthe tensor eld
f
. It isinvariantwith
respe t tobiholomorphisms, whi h means that
H(f Æ )= 1
[(Hf)Æ ℄
(5)
for every biholomorphi transformation of D,
being itsdi erential.
By de nition, H-harmoni fun tions are fun tions whi h are annihilated by H.
We will onsider here symmetri Siegel domains, for whi h these notions are well
de ned sin e they are holomorphi ally equivalent to bounded domains. When D is
asymmetri Siegeldomainoftube type,(3)isequivalenttothe lassi alHuasystem.
Thissystemisknownto hara terizethePoisson-Szegointegrals(see[FK℄and[JK℄).
This means that a fun tion on D is H-harmoni if, and only if, it is the Poisson-
Szego integral of ahyperfun tiononthe Shilov boundary. Originally,the urvature
tensor was not expli it in the Hua system. For lassi al domains, the system has
been de ned by L. K. Hua as a \quantization"of the equation de ning the Shilov
boundary(see[Hu℄and[BV ℄). L.K.HuaprovedthatthesystemannihilatesPoisson-
Szego integrals. Thenthe system wasextended by K.Johnsonand A.Koranyi [JK℄
toallsymmetri tubetypedomainsandwaswrittendownintermsoftheenveloping
algebra of the semi-simple Liegroup of automorphisms of the domain. K. Johnson
and A. Koranyi proved not only that for all tube domains the system annihilates
the Poisson-Szego kernel, butalsothattheH-harmoni fun tionsare Poisson-Szego
integrals. Rewriting Johnson{Koranyi formula C(;
) in terms of the urvature
tensor, assuggested byNolanWalla h,one obtainsthe same system asabove. Itis
why we all Hthe Hua-Walla hsystem.
Noti e that (4) and (5) have a perfe t sense on any Kahlerian manifold, and,
for general Siegel domains, the system (4) has been already studied in [DHP℄. In
parti ular, for non tube symmetri Siegel both (4) and Johnson{Koranyi formula
C(;
) take the same form. In the work of N. Berline and M. Vergne [BV ℄ it is
observed thatC(;
)doesnotannihilatePoisson-Szegointegrals,andtheproblem
of des ribingC(;
){harmoni fun tionsisrisen. Hereweare goingtoanswertheir
Main Theorem. Let D be a symmetri irredu ible Siegel domain of type II, and
let F be an H-harmoni fun tion on D whi h satis es the growth ondition
sup
z2D Z
N()
jF(uz)j 2
du<1;
(H 2
)
whereN() isa nilpotentsubgroupof S whosea tionisparalleltothe Shilovbound-
ary. Then F ispluriharmoni .
This is in astriking ontrast to the ase when D isa symmetri tube domain. It
requires some omments.
The Poisson-Szego integrals on type II domains have been hara terized by N.
Berlineand M.Vergne[BV℄ aszerosof aG-invariantsystemwhi h,ingeneral,isof
the thirdorder. It is obtained by \quantization"of the Shilov boundary equations.
Theyalsoprovethatfordomainsoverthe oneofhermitianpositivede nitematri es
one anusease ondordersystem,
Z
,to hara terizePoisson-Szegointegrals. This
system appears already in the book by Hua [Hu℄. It is obtained from C(;
) by a
proje tion that eliminatesa part of the equations.
All this shows that the system H does not seem to be anoni al in any sense,
although it is de ned with the aid of the urvature tensor, ertainly an important
invariant, the geometri meaning of the system being still un lear. Our present
work suggests that it would be interesting to understand se ond order systems of
operatorson symmetri Siegel domains whi h are invariant under the full group of
biholomorphisms.
Inthe proofofthe maintheorem,weuseheavilythetheoryofharmoni fun tions
with respe t to subellipti operators on solvable Lie groups [R℄, [D℄, [DH℄, [DHP℄.
To do this, we identify the domain D with a solvable Lie group S G that a ts
simply transitively on D. We then use a spe ial orthonormal frame of S-invariant
ve tor elds, E
1
;E
2
;;E
m
, to ompute the operator H by the formula (4). In
fa t, we only onsider the left-invariantse ond order ellipti operators built out of
the diagonalof H,
H
j
f =(HfE
j
;E
j ):
(6)
Ellipti operatorswhi hare linear ombinationsofoperators H
j
playthe main role
in our argument, and in parti ular the Lapla e-Beltrami operator , whi h is the
tra e of H. We represent H-harmoni fun tions as various Poisson integrals, and
weuse properties of these representations.
Linear ombinationsoftheoperators(E
j
;E
k
)havealreadybeenusedto hara -
terizepluriharmoni fun tions(see[DHMP℄). Weshouldemphasizethatthesystems
under study here are di erent from those of [DHMP ℄, and the proofs require new
ideas. Sin e a part of the onstru tion is the same in the two papers, we try to
simplify the presentation for the reader's onvenien e.
Our growth assumption (H 2
) is made mainly for te hni al reasons, L 2
harmoni
analysis being the easiest. We hope to be able to obtain similar on lusions for
however, a somewhat more deli ate te hni . In fa t, it is not lear that the on-
lusion requires any growth ondition at the boundary, and one may onje ture
that onlygrowth onditions atin nity are ne essary to insure pluriharmoni ityfor
H{harmoni fun tions on symmetri irredu ible Siegel domains of type II. On the
other hand, for tube type domains, one may onje ture that growth onditions on
derivatives atthe boundaryinsure pluriharmoni ity for H-harmoni fun tions as it
is the ase inthe unit ball([BBG℄) for -harmoni fun tions.
Finally,letusremark that,if wedonot insist oninvarian epropertiesof thesys-
tems onsidered, then itis always possible to hara terize pluriharmoni fun tions,
amongthefun tionswhi hareharmoni withrespe ttotheLapla e-Beltramioper-
ator , asthosewhi hareannihilatedbyasinglese ondorder operatorL(without
any growth ondition). Indeed,a lassi altheoremofForelli(seeRudin'sbook[Ru ℄)
asserts that every smooth fun tion in the unit ball whi h is annihilated by the op-
erator P
z
j z
k
2
zjz
k
ispluriharmoni inthe ball. SoL an be taken as thisoperator
suitably translated,sothat afun tionwhi hisannihilatedby Lispluriharmoni in
the neihborhoodof apoint. Thenthe real-analyti ityof thefun tion, whi h follows
from the fa t that itis {harmoni ,insures its pluriharmoni ityeverywhere.
In view of Forelli's Theorem, it is not so mu h the small number of operators in
the system used to hara terizepluriharmoni fun tionsthan the stronginvarian e
properties ofthe systemitselfwhi hare relevant. Inthis ontext, thepresent paper
an be viewed as a omplementto [DHP℄ and[DHMP ℄.
2. Hua-Walla h systems
2.1. General Hua-Walla h systems. In this subse tion, D is a general domain
inC m
whi hisholomorphi allyequivalenttoabounded domain. Were allherethe
propertiesof theKahlerianstru turerelatedtotheBergmanmatri aswellassome
elementaryfa tsabouttheHua-Walla hsystem whi hwewilluse later. Thereader
may referto [He ℄ and [KN℄ formore details onthe prerequisites.
Let T be the tangent bundle for the omplex domain D, and let T C
be the
omplexi ed tangent bundle. The omplex stru ture J and the Bergman metri g
areextendedfromT toT C
by omplexlinearity. LetT 1;0
andT 0;1
betheeigenspa es
of J su hthat Jj
T
1;0 =iId ,Jj
T
0;1 = iId . Wehave
T C
=T 1;0
T 0;1
:
The onjugation operatorex hangesT 1;0
and T 0;1
.
Thespa esofsmoothse tionsofT,T C
,T 1;0
willbedenoted (T), (T C
), (T 1;0
),
respe tively. Smooth se tions of T 1;0
are alled holomorphi ve tor elds.
The Riemannian onne tion r isalso extended from (T) to (T C
) by omplex
linearity and, sin e r is de ned by a Kahlerian stru ture, it ommutes with J.
An immediate onsequen e is that r
Z
W belongs to (T 1;0
) (respe tively (T 0;1
))
1;0 0;1
(T ), we have that r
U
V =r
U V and
[U;V℄=r
U
V r
V U;: (7)
Therefore, for Z;W holomorphi ve tor elds,
r
W Z =
(1;0)
([W;Z℄);
(8)
where
(1;0)
denotes the proje tion from T C
onto T 1;0
. The urvature tensor
R (U;V)=r
U r
V r
V r
U r
[U;V℄
preserves also T 1;0
and R (U;V)Z = R (U;V)Z. The restri tion of R (U;V) to T 1;0
is also denoted by R (U;V). On T 1;0
the Hermitian s alar produ t arising from the
Bergman metri is denoted by
(Z;W)= 1
2
g(Z;W):
ForU,V,Z,W holomorphi ve tor elds, we have
(R (V;U)Z;W)=(R (W;Z)U;V)=(U;R (Z;W)V): (9)
In parti ular,
R (W;Z)=R (Z;W)
: (10)
Letusnowgoba k tothe de nitions given intheintrodu tion. The identityZW
r
Z
W = WZ r
W
Z is a dire t onsequen e of (7). The fa t that all (Z;W)
annihilate pluriharmoni fun tions follows from (8) as well as from the identity
(
z
j
;
z
k )=
z
j
z
k
. Moreover,
( Z; W)f = (Z;W)f:
whi h means that
f
is a tensor eld. The equality in (3) omes from (7), while
one proves (4)using (9) for (R (W;Z)E
k
;E
j ).
Let us now show invarian e of H
D
with respe t to biholomorphisms. Let be
a biholomorphismfrom D ontoD 0
, and
the holomorphi di erentialof whi h
maps T 1;0
D
ontoT 1;0
D 0
. Alltensor elds aretransportedby ,in luding, of ourse,the
Riemannian stru ture and the urvature tensor. Thus
R
D
(W;Z)= 1
R
D 0
(
W;
Z)
:
Moreover, for a smooth fun tion f on D 0
and g =f Æ , we have
g
= 1
f
.
So
(H
D
gZ;W)=Tr R
D 0
(
W;
Z)
f
whi h implies
H
D
g = 1
(H
D 0
f)
:
Finally, letus remark that, fromformulas (1), (3)and (10), it follows that
f
=
(
f )
, and Hf = (Hf)
. So, to study H-harmoni fun tions, it is suÆ ient to
Wewantnowto omputeexpli itlythe Hua-Walla hoperatorforsymmetri irre-
du ibleSiegeldomains. Todoit,wewilluseFormula(4)foraparti ularorthonormal
basis E
1
;:::;E
m .
2.2. Preliminaries on irredu ible symmetri ones. Let be an irredu ible
symmetri one inanEu lideanspa e. Ouraimistodes ribepre isely thesolvable
groupthata ts simplytransitivelyon. Thegroupwillbeusedinthe onstru tion
of the orthonormal basis. We doit allin terms of Jordan algebras, and we refer to
the book of Faraut and Koranyi [FK℄ for these prerequisites, introdu inghere only
the notations and prin ipalresults that willbeneeded later.
A nitedimensional algebraV witha s alarprodu th;i isanEu lideanJordan
algebra if for allelementsx;y and z inV
xy=yx x(x
2
y)=x 2
(xy) hxy;zi=hy;xzi:
We denoteby L(x) the self-adjointendomorphismof V given by the multipli ation
by x, i.e. L(x)y =xy:
For an irredu ible symmetri one ontained in a linear spa e V of same di-
mension,the spa eV anbemadeasimplerealEu lideanJordanalgebrawithunit
elemente, so that
= intfx 2
: x2Vg:
Let G be the onne ted omponent of the group of all transformations in GL(V)
whi h leave invariant, and let G be its Lie algebra. Then G is a subspa e of the
spa e of endomorphisms of V whi h ontains all L(x) for all x 2 V, as well as all
x2y forx;y 2V,where x2y=L(xy)+[L(x);L(y)℄(see [FK℄ for these properties).
We x aJordan framef
1
;:::;
r
ginV,that is,a omplete system oforthogonal
primitive idempotents:
2
i
=
i
;
i
j
=0 ifi6=j;
1
+:::+
r
=e
and none of the
1
;:::;
r
is a sum of two non-zero idempotents. Let us re all that
the length r is independent of the hoi e of the Jordanframe. It is alled the rank
of V. To have anexample inmind, one may thinkof the spa e V of the symmetri
rr matri esendowed withthe symmetrizedprodu tofmatri es 1
2
(xy+yx). Then
the orresponding one is the set of symmetri positive de niterr matri es, the
set ofdiagonalmatri eswith allentries equalto0ex ept forone equalto1 being a
Jordan frame.
The Peir e de omposition of V related to the Jordan frame f
1
;:::;
r
g ([FK℄,
Theorem IV.2.1)may be writtenas
V = M
1ijr V
ij : (11)
It is given by the ommon diagonalization of the self-adjoint endomorphims L(
j )
with respe t to their only eigenvalues 0, 1
2
, 1. In parti ular V
jj
= R
j
is the
eigenspa e of L( ) related to 1, and, for i < j, V is the interse tion of the
eigenspa es of L(
i
) and L(
j
) related to
2
. All V
ij
, for i < j, have the same
dimension d.
For ea h i < j, we x on e for all an orthonormal basis of V
ij
, whi h we note
fe
ij
g,with1 d. To simplifythe notation,wewrite e
ii
=
i
( takingonlythe
value 1). Then the system fe
ij
g, for i j and 1 dimV
ij
, is an orthonormal
basis of V.
Let usdenoteby A the abelian subalgebraof G onsisting of elementsH =L(a),
where
a = r
X
j=1 a
j
j 2
M
i V
ii :
We set
j
the linear form on A given by
j
(H) = a
j
. It is lear that the Peir e
de ompositiongivesalsoa simultaneous diagonalizationof all H2A, namely
Hx=L(a)x=
i
(H)+
j (H)
2
x x2V
ij : (12)
Let A = expA. Then A is an abelian group, and this is the Abelian group in the
Iwasawade ompositionofG. Wenowdes ribethenilpotentpartN
0
. ItsLiealgebra
N
0
is the spa e of elements X 2G su h that, for allij,
XV
ij
M
kl;(k;l )>(i;j) V
kl
;
where the pairs ordered lexi ographi ally. On e N
0
is de ned, we de ne S
0 as
the dire t sum N
0
A. The groups S
0
and N
0
are then obtained by taking the
exponentials. It follows from the de nition of N
0
that the matri es of elements of
S
0
and S
0
, in the orthonormalbasis fe
ij
g, are upper-triangular.
The solvable group S
0
a ts simply transitively on . This may be found in[FK℄
Chapter VI, aswellasthe pre isedes riptionofN
0
whi hwillbeneeded later. One
has
N
0
= M
i<jr N
ij
;
where
N
ij
=fz2
i
:z 2V
ij g:
This de omposition orresponds toadiagonalizationoftheadjointa tionofAsin e
[H;X℄=
j
(H)
i (H)
2
X; X2 N
ij : (13)
Finally, letV C
=V +iV be the omplexi ationof V. We extend the a tion of G
C
2.3. Preliminaries on irredu ible symmetri Siegel domains of type II.
We onsider the Siegel domain de ned by anirredu ible symmetri one and an
additional omplex ve tor spa e Z together with a Hermitian symmetri bilinear
mapping
: ZZ !V C
;
su h that
(;)2 ; 2Z;
(;)=0 implies =0:
The Siegeldomain asso iatedwith these data is de ned as
D=f(;z)2ZV C
: =z (;)2g:
(14)
It is alled of tube type if Z is redu ed to f0g. Otherwise, it is alled of type II.
There is arepresentation :S
0
3s 7!(s)2GL(Z) su h that
s(;w)=((s);(s)w); (15)
and su h that all automorphisms (s), for s 2 A, admit a joint diagonalization
(see [KW℄). To redu e notations, we shall as well denote by the orresponding
representation of the algebraS
0
. ForX 2S
0
, (15) impliesthat
X(;w)=((X);w)+(;(X)w): (16)
As an easy onsequen e, one an prove that the onlypossible eigenvalues for(H),
with H 2A, are
j
(H)=2, for j =1;:::;r. So wemay write
Z = r
M
j=1 Z
j
with the property that
(H)=
j (H)
2
; 2Z
j : (17)
Moreover, all the spa es Z
j
have the same dimension 1
. A proof of these two fa ts
may be found in [DHMP℄. We all the dimension of Z
j
for j = 1;:::;r. Let us
remark, using (16) and (17), that for ;w 2 Z
j
, we have L(
j
)(;w) = (;w).
Therefore, (;w) = Q
j
(;w)
j
, for ;w 2 Z
j
. Moreover, h
j
;(;)i > 0 for
2Z
j
and so the Hermitianform Q
j
is positive de niteonZ
j .
The representation allows to onsider S
0
as a group of holomorphi automor-
phisms of D. More generally, the elements 2 Z, x 2 V and s 2 S
0
a t on D in
1
Infa t,thepresentstudygeneralizestoallhomogeneousSiegel domainsrelatedtoirredu ible
the following way:
(w;z)=(+w;z+2i(w;)+i(;));
x(w;z)=(w;z+x); (18)
s(w;z)=((s)w;sz):
We allN()thegroup orrespondingtothe rsttwoa tions,thatisN() =ZV
with the produ t
(;x)(
0
;x 0
)=(+ 0
;x+x 0
+2=(; 0
)):
(19)
All three a tions generate a solvable Liegroup
S=N()S
0
=N()N
0
A =NA;
whi h identi es with a group of holomorphi automorphisms a ting simply transi-
tively on D. The group N(), that is two-step nilpotent, is a normal subgroup of
S. The Lie algebraS of S admitsthe de omposition
S =N()S
0
=
r
M
j=1 Z
j
M
ij V
ij
M
i<j N
ij
A:
(20)
Moreover, by (12), (13) and (17), one knows the adjoint a tionof elementsH 2A:
[H;X℄=
j (H)
2
X for X 2Z
j
;
[H;X℄=
i
(H)+
j (H)
2
X for X2V
ij
; (21)
[H;X℄=
j
(H)
i (H)
2
X for X 2N
ij :
Sin eS a tssimplytransitivelyonthedomainD,wemayidentifySandD. More
pre isely, we de ne
:S 3s7!(s)=se2D;
(22)
where e is the point (0;ie) in D. The Lie algebra S is then identi ed with the
tangent spa e of D at e using the di erential d
e
. We identify e with the unit
element of S. We then transport both the Bergman metri g and the omplex
stru ture J from D to S, where they be ome left-invariant tensor elds on S. We
still write J for the omplex stru ture on S. Moreover, the omplexi ed tangent
spa e T C
e
is identi edwith the omplexi ationof S, whi h we denoteby S C
. The
de ompositionT C
e
=T (1;0)
e
T (0;1)
e
istransported into
S C
=QP: (23)
Elements of S C
are identi ed with left invariant ve tor elds on S, and are alled
left invariant holomorphi ve tor elds when they belong to Q. The onjugation
operator ex hanges Q and P, while the transported operator J oin ides with iId
seen as a Hermitian form on Q, and orthonormality for left invariant holomorphi
ve tor elds means orthonormality for the orresponding elements inQ.
Now, let us onstru t an orthonormal basis of left invariant holomorphi ve tor
elds. We rst build a basis in S. To do this, we use the de omposition given in
(20) and give a basis forea h blo k.
Wehavealready xedanorthonormalbasisfe
jk
ginV orrespondingtothePeir e
de omposition hosen. Forj <k and 1 d, wede neX
jk 2V
jk andY
jk 2N
jk
astheleft-invariantve tor eldsonS orrespondingtoe
jk
and2e
jk 2
j
,respe tively.
Forea hj wede neX
j
andH
j
astheleft-invariantve tor eldsonS orresponding
to
j 2V
jj
andL(
j
)2A, respe tively. It remainsto hoose a basisof ea hZ
j . We
hoose fore
j
anorthonormalbasis of Z
j
relatedto4Q
j
, whereQ
j
is the quadrati
form de ned above. Forz
j
= x
j +iy
j
the orresponding oordinates, we de ne
X
j
;Y
j
as the left-invariant ve tor elds onS whi h oin idewith
xj
and
yj at
e.
Finally, wede ne
Z
j
=X
j iH
j
; Z
jk
=X
jk iY
jk
; Z
j
=X
j iY
j :
We an nowstate the following lemma.
Lemma 2.1. The left invariant ve tor elds Z
j
, for j = 1;;r, Z
jk
, for j <
k r and = 1; ;d, and Z
j
for j = 1; ;r and = 1; ; , onstitute an
orthonormal basis of holomorphi left invariantve tor elds.
Proof. This lemma is already ontained in [DHMP ℄, to whi h we refer for details.
Toprove thatZ
j ,Z
jk
,and Z
j
are holomorphi ve tor elds,itissuÆ ient toprove
that
J(X
j )=H
j
; J(X
jk )=Y
jk
; J(X
j )=Y
j :
To do this, we ompute the image of the ve tor elds X
j
;H
j
;X
jk
;Y
ij
;X
j
, and Y
j
by the di erential d
e
. We nd the following tangent ve tors at e:
x
jj ,
y
jj ,
x
jk ,
y
jk ,
x
j
, and
y
j
. Here the oordinates thatwehaveused inZV C
aregiven by
(;z)=
X
j;
(x
j +iy
j )e
j
; X
ij;
(x
ij +iy
ij )e
ij
:
The assertionfollows aton e, using the omplex stru ture in ZV C
.
To show orthonormality, it is possible to use Koszul's formula whi h allows to
get the Bergman metri from the adjointa tion. This is done in [DHMP℄, Lemma
(1.18). 2
2.4. Hua-Walla h systems for irredu ible symmetri Siegel domains. We
now ompute the operator H in the orthonormal basis that we have built in the
previous subse tion. Infa t,it isenoughto ompute thefollowingoperators, alled
strongly diagonal HW operators,and de ned by
H f =(Hf Z ;Z ) ; j =1;;r:
We have the following proposition.
Proposition 2.2. Thestrongly diagonal HW operators H
j are
H
j
= X
L
j +2
j +
X
k<j X
kj +
X
l >j X
jl
; (24)
where
j
=X 2
j +H
2
j H
j
L
j
=(X
j )
2
+(Y
j )
2
H
j (25)
ij
=(X
ij )
2
+(Y
ij )
2
H
j :
Proof. We rst ompute the urvature tensor R (Z;Z), with Z =Z
1
;;Z
r
. From
(8), weknowthat, for Z;W in Q,
r
Z
W =
Q
([Z;W℄)=
Q
([Z;(W +W)℄);
(26)
where
Q
denotes the proje tion from S C
onto Q. We laim that
Lemma 2.3. The followingidentities hold:
r
Z
j Z
k
=iÆ
jk Z
j
r
Z
j Z
kl
= i
2 (Æ
l j Z
kj +Æ
kj Z
jl
) if k<l
r
Z
j Z
k
= i
2 Æ
jk Z
j :
Proof. In the omputation, we have seen that we may repla e the three left hand
sides of the formulas above by 2
Q ([Z
j
;X
k
℄), 2
Q ([Z
j
;X
kl
℄) and 2
Q ([Z
j
;X
k
℄), re-
spe tively. Moreover, if we repla e Z
j
by iH
j
inthese three expressions, we obtain
the righthand sides, by virtue of(21). Thus thelemma follows, on e weprovethat
all bra kets [Z
j
;X
k
℄, [Z
j
;X
kl
℄, and [Z
j
;X
k
℄ vanish. This last fa t follows from a
standard argument. One proves that ea h of these ve tor elds is annihilated by
all endomorphisms adH (H)Id, with H 2 A, for a value (H) that is not an
eigenvalue of adH forsome H. Soit vanishes. 2
Let usgo onwith the proof of the proposition. It is easyto dedu e the a tion of
r
Z
onQfromtheone ofr
Z
. Indeed,sin e the a tionofS preservesthe Hermitian
s alar produ t, and sin e Z is left-invariant,
0=Z (U;V)=(r
Z
U;V)+(U;r
Z V)
forany oupleU,V ofleft-invariantholomorphi ve tor elds. Sotheendomorphism
of Q de ned by r
Z
is the opposite of the adjointendomorphismde ned by r
Z . It
follows fromthe matrix representation given inthe lemma that they are equal, and
they ommute. So, forU 2Q,
R (Z
j
;Z
j
)U = r
[Z ;Z ℄
U = 2ir
Z U
sin e [Z
j
;Z
j
℄ = 2iX
j
= i(Z
j +Z
j
). Using again Lemma 2.3 and the expression of
Hf given in(4), we see that
H
j
= X
(Z
j
;Z
j
)+2(Z
j
;Z
j )+
X
k<j X
(Z
kj
;Z
kj )+
X
l >j X
(Z
jl
;Z
jl ):
Wereferto[DHMP℄ forthe omputationof(Z
j
;Z
j
),(Z
j
;Z
j
),and(Z
kj
;Z
kj ).
2
We also refer to [DHMP ℄ for the omputation of the Lapla e-Beltrami operator
,
= X
j
j +
X
k<j X
kj +
X
j;
L
j : (27)
Itisprovedin[DHP℄thattheLapla e-Beltramioperatoristhetra eoftheoperator
H.
All results, up to now, are also valid for the tube domain T
= V +i, whi h
identi es with the subgroup VS
0
of the group S and appears as a parti ular ase.
Left invariant di erential operators a t from the right. Therefore, we an identify
left invariantdi erential operators onthe tube domain with left invariantdi eren-
tial operators on the domain D itself. We add a subs ript or supers ript for su h
operators omingfromthetubedomain,and de neH T
j
,j =1;;r,and
T
asthe
operators omingfromthestronglydiagonalHW operatorsforthetubedomainand
the Lapla e-Beltrami operator,respe tively. Then, we have the following orollary,
the proof of whi h isimmediate:
Corollary 2.4. Thefollowing identities hold:
H T
j
=2
j +
X
k<j X
kj +
X
l >j X
jl
; (28)
T
= r
X
j=1 H
j
: (29)
2.5. Indu tion pro edure. We olle t in this subse tion some information and
some notations whi hwillbeused inallproofswhi h arebased onindu tiononthe
rank of the one. So, here, we assume that r>1. We rst de ne
A =linfL(
1
);:::;L(
r 1
)g and A +
=linfL(
r )g;
and, inananalogous way,
N
0
= M
i<jr 1 N
ij
and N +
0
= r 1
M
j=1 N
jr :
N +
0
is an ideal of N
0
, while N
0
is a subalgebra. Clearly A = A A +
and
N
0
=N N +
.
Next, we de ne A ;A ;N
0
;N
0
as the exponentials of the orresponding Lie
subalgebras. ThenS
0
=N
0
A isthe solvablegroup orrespondingtothe one ,
determined by the frame
1
; ;
r 1
, whi h is of rank r 1 as we wanted. The
underlying spa e V for is the subspa e
V =
M
1ij<r V
ij :
We willmake anextensive use of the fa t that
A =A A +
and N
0
=N
0 N
+
0
in the sense that the mappings
A A +
3(a ;a +
)7!a a +
2A;
and
N
0
N +
0
3(y ;y +
)7!y y +
2A
are di eomorphisms.
Now, let usde ne
Z =
r 1
M
j=1 Z
j :
Then it is easily seen that Z Z is mapped by into the subspa e (V ) C
.
Moreover, (;)belongsto when 2Z . So, wemayde netheSiegeldomain
D as
D =f(;z)2Z (V ) C
: =z (;)2 g:
Let usde ne N() =Z V and
N() +
=Z
r
M
jr V
jr :
Then, again, N() is a subalgebra and N() +
is an ideal of N(). We de ne
N() and N() +
as their exponentials. ThenN() isa semi-dire t produ t
N()=N() N() +
:
Clearly N() is the nilpotentstep two\boundary"group orresponding toD .
Finally, wewant to de ompose the group N. Let N =N() (N
0
) , and N +
=
N() +
(N
0 )
+
. Then N is a semi-dire t produ t N = N N +
. Moreover, the whole
group S may be writtenas
S =N N +
A A +
=N A N +
A +
:
3. Poisson integrals
The aimofthis se tionistoprove thefollowingpartialresultinviewof themain
theorem.
Theorem 3.1. Let F be a bounded fun tion on S annihilated by and by H
j , for
j =1;:::;r. Then
H T
j
F =0 for j =1;:::;r;
and
L
j F =
X
L
j
F =0 for j =1;:::;r:
>From the formulas of the previous se tion, it is lear that a bounded fun tion
on the domain D whi h is H{harmoni satis es the assumptions. Moreover, the
rst statement implies the se ond one. To prove the rst one, we shall use the
hara terization of H{harmoni fun tions in terms of Poisson-Szego integrals on
tube domains. More pre isely,followingHua [Hu℄and [FK℄, itis suÆ ient toprove
that F, onsidered as a fun tion on the tube domain T
=V +i, is the Poisson-
Szego integral of some bounded fun tion on V. To do this, our main tool will be
the possibility to write F, in di erent ways, as a Poisson integral related to some
ellipti operators whi h annihilateF.
Letus rst givesomenotations. From the lastse tion,we knowthat every g 2S
may be written in a unique way as a produ t (;x)na, with (;x) 2 N() and
n 2 N
0
. We write for the proje tion on N(), given by (g)= (;x), and ~ for
the proje tion on N, given by (g)~ =(;x)n.
We rst re allpreviousresultsof twoof theauthors. Even if they arevalidinthe
more general ontextof asemi-dire t produ t,we givethem inthe present ontext.
We onsider ellipti operatorswhi h may bewritten as
L = r
X
j=1
j L
j +
r
X
j=1
j H
T
j (30)
with
j
and
j
positive onstants. Then L is a sum of square of ve tor elds plus
a rst order term Z = Z(L), whi h is alled the drift, and may be written as
Z = P
j H
j
, with
j
=
j
+(2+(j 1)d)
j +d
P
k<j
k
. It follows from[DH℄
and [R℄that themaximalboundaryofL anbeeasily omputed (itdepends onthe
signs of
j
(Z)
i
(Z)for i <j). In parti ular, it is equal toN() if the sequen e
j
is a non{in reasing sequen e, and to N if it is an in reasing sequen e. Let us
summarize the results that weshall use inthe next proposition.
Proposition 3.2. Let L be givenby (30), and
j
as above.
(i) If L is su h that
j
is a non{in reasing sequen e, there is a unique positive,
bounded, smooth fun tion P
L
on N() with R
P
L
(y)dy = 1 su h that bounded
L-harmoni fun tions on S are in one-one orresponden e with L (N()) via the
Poisson integral
F(s)=P
L
f(s)= Z
N( )
f((sw))P
L
(w) dw:
(31)
(ii)IfLissu hthat
j
isanin reasingsequen e, thereisauniquepositive,bounded,
smooth fun tion
~
P
L
onN with R
N
~
P
L
(y)dy=1su h thatboundedL-harmoni fun -
tions on S are in one-one orresponden e with L 1
(N) via the Poissonintegral
F(s)=
~
P
L
f(s)= Z
N
f(~(sy))
~
P
L
(y) dy:
(32)
Moreover, forea h given>0, wemay hoose the oeÆ ients
j
and
j
sothat (i)
holds, and that
Z
N()
(w)
P
L
(w) dw<1 (33)
where (w) is thedistan e of w from the unit elemente2N() with respe t to any
left{invariant Riemannian metri .
Aswesaid, (i)and(ii)maybefoundin[DH℄ and[R℄. Theintegrability ondition
may befound in[D℄, Theorem (3.10): asuÆ ient ondition for (33) is that
X
2
j
(H
j )
2
+(Z)<0;
for all linear forms on A of the form =
k +
p
2
;
k
2
. The fa t that this ondition
may besatis ed is elementary.
Wehave hosen toaddatilde every timethat weare on ernedwith anoperator
whose maximal boundary is the whole groupN. Wethen de ne P
L
asan integral,
P
L (w)=
Z
N0
~
P
L
(wy) dy:
(34)
Let usremark that, in this ase, the fun tions F whi h may be writtenas
F(s)=P
L
f(s)= Z
N( )
f((sw))P
L
(w)dw;
(35)
with f a bounded fun tion on N(), onstitute a proper subspa e of the spa e of
bounded fun tions whi h are annihilated by L. It is in parti ular the ase for the
Lapla e-Beltrami operator, whi h is obtained for the values = 2 = 1, and has
maximal boundaryN.
The mainstep inthisse tionisthe nextproposition. Ithasbeen proved in[DHP℄
for general homogeneousSiegel domains(non ne essarilysymmetri ), and formore
general operators. However, in the ase of symmetri Siegel domains, whi h is the
ase under onsideration, the proof may be simpli ed onsiderably. We in lude it
Proposition 3.3. Let F be a bounded fun tion on S annihilated by and by H
j ,
for j =1;:::;r. Then, there exists a bounded fun tion f on N() su h that F may
be written as
F(s)=P
f(s)= Z
N( )
f((sw))P
(w) dw:
Proof. Wealready knowthat there exists somebounded fun tion
~
f onN su h that
F maybewrittenas
~
P
~
f. Moreover, wemayassumethat
~
f isa ontinuousfun tion
and provethat, inthis ase,f istherestri tionof
~
f onN( ). Indeed, inthe general
ase, we onsider the sequen e of fun tionsF
m
de ned by
F
m (s)=
Z
N
m
(n)F(n 1
s) dn=
~
P
(
m
~
f)(s):
with
m
an approximate identity whi h is ompa tly supported and of lass C 1
.
Clearly F
m
tends to F pointwise. Let us assume that we have already proved the
propositionfor ontinuousfun tions. ThenF
m
=P
(f
m
). Allthefun tions(f
m )are
boundedbykfk
L
1,sowe anextra ta
{weak onvergentsequen e whi h onverges
to f. Then P
(f
m
) onverges to P
(f) pointwise. Hen e F =P
f.
So, let
~
f be a bounded ontinuous fun tion on N, and let F =
~
P
~
f. To prove
the proposition, we want to prove that, for ea h xed w 2 N(), the fun tion
y 7!
~
f(wy) is onstant on N
0
. Indeed, assume that it is the ase and denote by f
the restri tion of
~
f toN(). Then,fors=wya withw2N(), n2N
0
anda2A,
we an write
F(wya)=
~
P
~
f(wya)= Z
N()N0
~
f(wyavua 1
)
~
P
(vu)dvdu;
= Z
N()N
0
f(wyava 1
y 1
)
~
P
(vu) dvdu;
=P
f(wya):
Let us nallyremark that it issuÆ ient to prove that y 7!
~
f(y) is onstant onN
0 .
Indeed, on e we have proved this, for ea h w 2N() we have the same on lusion
with F repla ed by
w
F, with
w
F(g) =F(wg)=
~
P
(
w
~
f)(g). Again
w
~
f(y) =
~
f(wy)
is onstant, whi h we wanted to prove.
So, letus showthat y7!
~
f(y) is onstant onN
0
. Let usde ne
F
H
(wya)= Z
N0
~
f(yaua 1
)
Z
N()
~
P
(vu) dv
du:
(36)
We laim that
F
H
(g)= lim F((exptH)g);
(37)
where H is the ve tor eld H=
j=1 H
j
. Indeed, writing
F(g)= Z
N()N
0
~
f(~(gvu))
~
P
(vu) dvdu;
wehave
F((exptH)wya)=F(w
t y
t
aexptH)= Z
N()N
0
~
f(w
t y
t av
t u
t a
1
)
~
P
(vu)dvdu:
Foranelement g of N we have used the notation
g
t
=(exptH)g(exp( tH)):
It follows from (21) that u
t
= u for every u 2 N
0
, and that w
t
tends to the unit
element. This implies (37). We now laimthat
F
H
(wya)=F
H (ya) (38)
H
j F
H
=0 for j =1;:::;r and F
H
=0 (39)
HF
H
=0:
(40)
We have already proved (38). Then (39) follows from the fa t that left and right
translations ommute. So, for every t, F((exptH)g) is annihilated by the HW
operators and the Lapla ian. To see (40), we use again the fa t that u
t
= u for
every u2N
0
and the formula(36) toobtain that
F
H
(ya(exptH))=F
H (ya):
Then (40) follows aton e.
Finally, uniqueness in Proposition 33implies that y 7!
~
f(y) if and only if F
H is
onstant. Toprovethat F
H
is onstant,we onsider thefun tion Gde ned onN
0 A
by G(ya)=F
H
(ya). Then learly Gis annihilated by alloperators
D
j
= H
j
+2(H 2
j H
j )+
X
i<j X
((Y
ij )
2
H
j )+
X
j<kr X
((Y
jk )
2
H
k ) (41)
and by H. So, to omplete the proof,it is suÆ ientto prove the following lemma.
Lemma 3.4. Let G be a bounded fun tion on N
0
A whi h annihilated bythe opera-
tors H;D
1
;:::;D
r
. Then G is onstant.
Proof. There is nothing to prove when r = 1. For r = 2, let us remark that G,
whi h isannihilated by H
1 +H
2
, is alsoannihilated by H
1 H
2 sin e
(D
1 D
2
)G= ( +2)(H
1 H
2
)G=0:
Therefore, H
1
G=0 and H
2
G=0 and so, G is a bounded fun tion onthe Abelian
group N
0
=R d
annihilated by the Lapla eoperator. Hen e G is onstant.
Let us now onsider r >2. We assume that the lemma has been proved with r
repla ed by r 1. We write G asa Poisson integralwith respe t to the operator
D= X
D ;
for whi hthedriftZ(D)isequaltoZ =
j H
j
,with
j
=( +2+(j 1)d)
j +
d P
k<j
k
. We rst remark that we may hoose the oeÆ ients
j
so that the
j
de rease for j 2 (when +2 > d, one an even nd a sequen e
j
su h that
j
isde reasing,and on lude dire tly sin eeveryD{harmoni bounded fun tionis
onstant). Withthis hoi e,the maximal boundaryof D isthe groupN
1
=expN
1 ,
with
N 1
=
j>1 N
1j :
Wealsode neN 1
=expN 1
, withN 1
=
1<i<jr N
ij
. Every yinN
0
an bewritten
in a unique way as y
1 y
0
, with y
1 2 N
1
and y 0
2 N 1
. We de ne
1 by
1
(ya) = y
1 .
Then, (see [DH℄),there exists fun tions
D
and su hthat
G(ya)= Z
N1
(
1
(yau))
D
(u) du:
The fun tion is bounded, and we an assume as before that it is ontinuous.
Usingnotationsofthe subse tion2.5. onthe indu tionpro edure, we analsowrite
y 2 N
0 as y
+
y . When y is in N
1
, then y +
belongs to N
1r
= expN
1r
. We shall
prove that (y) depends only on y +
. Again, to prove this, it is suÆ ient to prove
that (y ) is onstant. Indeed, on e we have proved this, we may apply it to
y +
(with
y
+ (n)= (y +
n)), using the fun tion
y
+Gin pla e of G.
In order to prove that (y ) is onstant, let usde ne, asbefore,
G
#
(ya)= lim
t! 1
G((exptH
r )ya)
= lim
t! 1 Z
N1
(
1 (y (y
+
)
t au (u
+
)
t a
1
))
D (u
+
u ) du +
du
= Z
N
1
(
1
(y au a 1
))
D (u
+
u ) du +
du :
Here u
t
= (exptH
r
)u(exp( tH
r
)). We have used the fa t that (y )
t
= y , and
(y +
)
t
tends to the unit element. We have
G
#
(y y +
a a +
)=G
#
(y a )=G
#
(y +
y a a +
)
D
#
j G
#
=0; for j =1;:::;r 1;
H
#
G
#
=0;
where H
#
=H
1 +H
2
++H
r 1
, and, forj =1;:::;r 1,
D
#
j
=2H 2
j
( +2)H
j +
X
i<j ((Y
ij )
2
H
j )+
X
j<kr 1 X
((Y
jk )
2
H
k ):
From the indu tion hypothesis, we on lude that G
#
is onstant. So (y ) is also
onstant. Hen e (y) = (y +
), and, using obvious notations, we on lude that G
may infa t be writtenas
G(ya)= Z
N1r
(
1r
(yau))
D
(u)du:
Sin e exptH
j
ommutes with elementsof N
1r
for j =2;r 1, we on lude that
H
j
G=0. So(H
1 +H
r
)G=0. Moreover,
D
1 G=
2H 2
1
( +2)H
1 +
X
((Y
1r )
2
H
r )
G=0
D
r G=
2H 2
r
( +2+(r 2)d)H
r +
X
((Y
1r )
2
H
r )
G=0;
whi h, asin the ase r=2, impliesthat G is onstant. 2
On e we have on luded forthe lemma,we on lude for the proposition 3.3. 2
Our next step is the followingtheorem.
Theorem 3.5. Let f be a bounded fun tion on N() and let F = P
f. Assume
that
T
F =0:
Then
F((;x)ya)= Z
V f
(xyava 1
y 1
)p(v) dv;
(42)
wheref
(x)=f(;x)andpisthePoisson-SzegokernelforthetubedomainV+i.
Proof. Usingthe same kindof proofasin the lastproposition,wemay assumethat
f is ontinuous. The maximal boundary for
T
onsidered as anoperator onVS
0
is VN
0
. Letp~be the orresponding kernel on VN
0
. Thenthe fun tionF
,whi h is
de ned for 2Z xed by F((;x)s)=F
(xs), may be writtenas
F
(xya)= Z
VN
0 g
(xyavua 1
))p(vu)~ dvdu;
(43)
where v 2V, u2N
0 .
We have also
F
(xya)=P
f((;x)ya)= Z
N()
f((;x)ya(;v)a 1
y 1
)P
(;v)ddv:
Let a
t
=expt(
P
r
j=1 jH
j
). Then, onone hand,
lim
t! 1 F
(xya
t )=g
(xy)
in *-weaktopology onL 1
(VN
0
)and on the other,
lim
t! 1 F
(xya
t
)=f(;x);
pointwise. Hen e g
(xy)=f(;x)=f
(x). Therefore,
F
(xya)= Z
VN0 f
(xyava 1
aua 1
)~p(vu) dvdu;
= Z
V f
(xyava 1
y 1
)(
Z
N0
~
p(vu)du) dv:
It remains to prove that it is also equal to the right hand side of (42). But this
last expression is a
T
{harmoni fun tion (sin e the Poisson{Szgego kernel is
T {
harmoni ), with the sameboundary values onVN
0
. This proves (42). 2
Proof of Theorem 3.1. Again, using Proposition 3.3, we may assume that F =
P
f. Moreover, we may assume that f is ontinuous, using the same tri k as in
the proof of Proposition 3.3. So, it follows from Theorem 3.5 that F
is a Poisson-
Szegointegralonthetubedomain. Weknowfrom[Hu ℄and[JK℄thatPoisson-Szego
integrals on symmetri tube domains are annihilated by Hua operators, i.e. in our
situation F
is annihilatedby H T
j
. This nishes the proof. 2
4. The proof of pluriharmoni ity
In thisse tionweprovethefollowingstatement, whi himpliesthemaintheorem:
Theorem 4.1. Assume that
sup
s2S
0 Z
N()
jF (;x)s
j 2
ddx<1 (44)
and
F =H
1
F =:::=H
r
F =0:
(45)
Then F is pluriharmoni .
We rst laim that the results of the lastse tion on bounded fun tions apply to
(H 2
) growth onditions. Indeed, wehave thefollowinglemma. Here Lis anellipti
operatoras in(30),
L= r
X
j=1
j L
j +
r
X
j=1
j H
T
j
;
with oeÆ ients hosen so that it has maximal boundary N() and satis es the
integrability ondition (33) forsome tobe hosen later.
Lemma 4.2. A fun tion F whi h satis es(44) and (45) maybe written as a Pois-
son integral
F(g)= Z
N()
f((gw))P
L
(w) dw; f 2L 2
(N()); g 2S:
(46)
Proof. We redu e to bounded fun tions by onvolving F from the left. More pre-
isely,let
n 2C
1
(N()) be anapproximate identity, and let
F
n (g)=
Z
N()
n
(w)F(w 1
g) dw:
Then F
n
is bounded, and satis es (45). Soit follows fromthe lastse tion that
F
n (g)=
Z
f
n
((gw))P
L
(w)dw;
for anf
n
2 L (N()). Moreover, f
n
, whi h may be obtained as a {weak limit in
L 1
,whent! 1,ofF
n
(exptH)aswellasaweaklimitinL 2
(N()),isuniformly
in L 2
. Hen e,
kf
n k
L 2
(N())
sup
s Z
N()
jF(ws)j 2
dw
Wemay take forf 2L 2
(N()) the weak limitof asubsequen e, andget (46). This
on ludes the proof of the lemma. 2
To prove Theorem 4.1, we may assume that F = P
L
f as above. Moreover,
eventually onvolvingf inthegroupN()withaC 1
ompa tlysupportedfun tion
as in the lastse tion,wemay assume that
Assumption on f: itmay be writtenas
~
f where isa C 1
ompa tly supported
fun tion.
Atthispoint,ourmaintoolwillbeharmoni analysisofthenilpotentgroupN().
On e we haveproved that the Fourier transformof f vanishesoutside [ , one
on ludes easilylikein [DHMP ℄.
Letus rstre allsomebasi fa tsaboutFourieranalysisonN(),following[OV℄.
Let ( ; ) be the Hermitians alar produ t on Z for whi h the basis e
j
, whi h was
introdu edinsubse tion 2.3, isorthonormal. It oin ides with4Q
j
onea hZ
j ,and
thesesubspa esarepairwiseorthogonal. Forea h2V,letusde netheHermitian
transformation M
:Z !Z by
4h;(;!)i=(M
;!); ;! 2Z:
and onsider the set
=f2V :detM
6=0g
forwhi htheaboveHermitianformisnondegenerate. Remarkthatitisinparti ular
the ase for 2 sin e we assumed that (;) belongs to nf0g for all 6= 0.
The same is valid for 2 . So detM
, whi h is a polynomial of , does not
vanish identi ally, and is an open set of full measure. It arries the Plan herel
measure (see [OV℄), given by
()d =jdetM
jd:
Let us des ribe the Fo k representation asso iated to 2 . For every 2 we
de ne a omplex stru ture J
, whi h determines the representation spa e H
. Let
jM
jbethe positive Hermitiantransformation su h that jM
j
2
=M 2
. Then
J
=ijM
j
1
M
:
If 2thenJ
=iI =J oin ides withthe ordinary omplexstru ture inZ. For
general , the omplex stru ture J
has a ni e des ription in anappropriate basis.
Namely, there is a -measurable hoi e of an ( ; ) orthogonal basis e
1
;:::;e
m su h
that
H
(e
;e
)=
j Æ
jk
with
j
=1(dependingonand lo ally onstant). Inthe basise
1
;:::;e
m
;Je
1
;:::,
Je
m
of Z over R we have
J
(e
j )=
j (Je
j
) and J
(Je
j
)=
j e
j :
Let
B
==H
:
A dire t al ulation shows that
B
(J
e
j
;e
k )=Æ
jk
and so
B
(J
;)>0 if 6=0:
We de ne H
as the set of all C 1
fun tions F on Z whi h are holomorphi with
respe t tothe omplex stru ture J
and su h that
F()() 1
2
e
2 B
(J
;)
2L 2
(Z;dz):
Here dz is the Lebesgue measure related tothe s alar produ t (;)on Z.
The spa e H
isa Hilbert spa e for the s alar produ t
(F
1
;F
2 )
= Z
Z F
1 ()
F
2 ()e
B
(J
;)
()d:
The Fo k representation U
, whi h is a unitary and irredu ible representation on
H
,is given by
U
(;x)F(!)=e
2ih;xi
2 jj
2
+!
F(! ); (47)
with !
= B
(J
!;)+iB
(!;) and jj 2
=
. Then the Fourier transform of
f 2L 1
(N()), whi h we note U
f
, isde ned asthe operatoronH
given by
(U
f F;G)
= Z
N()
f(;x)(U
(;x) F;G)
dx:
If f 2L 1
(N())\L 2
(N()), then the Plan herel theorem says that
Z
V kU
f k
2
HS
()d=kfk 2
L 2
(N()) :
It followsthat, for f 2L 2
(N()),U
f
isde ned for almostevery andisa Hilbert-
S hmidt operator.
NowwewriteanorthonormalbasisofH
,whi h hangesmeasurably with. For
2Z, wenote
j;
its oordinates inthe basis e
j
,so that, inparti ular,
B
(J
;)= X
j j
j;
j 2
:
Given a multi-index =(
1
;:::;
m ), let
=
j j
2
p
! Y
j (1+
j )
2
j;
j (1
j )
2
j;
;
Then every
is holomorphi with respe t to the omplex stru ture J
and the
family f
g formsa (; )
- orthonormalbasis. Indeed, one may verify that
(
;
)
=
j j+j j
2
p
! ! Y
j Z
C u
j (1+
j )
2
u
j (1
j )
2
u j
(1+
j )
2
u j
(1
j )
2
e
juj 2
du:
We nallyde ne, for f 2L 2
(N() and almost every ,
^
f(; ; )=(U
f
;
): (48)
We may nowgivethe main step of the proof.
Lemma 4.3. Let F = P
L
f a fun tion whi h satis es the assumptions of Theorem
4.1, with L and f 2 L 2
(N()) hosen as above. Then, for almost every and for
all , , we have
^
f(; ; )=0 for 2=
[
: (49)
ProofofTheorem 4.1. Forthe moment,wetakethelemmaforgranted and nish
the proof of Theorem 4.1. Let us rst give some notations. For s 2S
0
, we note F
s
the fun tionde ned onN() by
F
s
(;x)=F((;x)s) (50)
and
^
F(; ; ;s)its Fourier transform. We laim that
^
F(; ; ;s)=e
2h;sei
(U
f
;
); for a.e 2
;
=e 2h;sei
(U
f
;
); fora.e 2
; (51)
=0; for a.e 2=
[
:
Indeed, we know from Theorem 3.5 that F may be written as a Poisson-Szego
integral,i.e.
F((;x)s)= Z
V f
(xsvs
1
)p(v) dv= Z
V
f(;x u)p
s
(u)du;
with p
s
de ned by
p
s
(u)=det(s 1
)p(s 1
u):
Here the element s 1
is onsidered as a ting on V. If f 2 L 1
(N())\L 2
(N()),
then
(U
Fs
;
)=
Z
N() Z
V
f(;x u)p
s (u)(U
(;x)
;
)duddx
= Z
V (U
f U
(0;u)
;
)p
s (u)du
=(U
f
;
)
Z
e
2ih;ui
p
s
(u) du:
These formulas are still valid for a general fun tion f 2 L (N()): only use an
approximation of f and the Plan herel theorem.
It remains to al ulate the Fourier transform of p
s
for 2
[
. We shall
do this for 2
. For 2
the proof is analogous. If 2
we onsider the
bounded holomorphi fun tionon V +igiven by
G(z)=e 2ih;zi
=e
2ih;x+isei
=e
2ih;xi 2h;sei
:
Then G is the Poisson integral of its boundary value, i.e.
G(z)= Z
V e
2ih;x ui
p
s
(u)du:
Therefore,
G(ise)=e
2h;sei
= Z
V e
2ih;ui
p
s
(u) du:
Finally, for 2
, we have
(U
Fs
;
)=e
2h;sei
(U
f
;
):
From (51), a dire t omputation (see [DHMP ℄ for the details) shows that
j F =0
for j = 1;:::;r. Moreover, we already know that L
j
F = 0. Then it follows from
Theorem 3.1 in[DHMP℄ thatF isthe real part of anH 2
holomorphi fun tion. 2
Proofof Lemma 4.3. It remainsto prove the lemma. Letus remark that thereis
nothing to prove for r =1. So the theorem is ompletely proved in this ase. For
r >1, we an make the assumption that the theorem is valid for r 1, and prove
the lemmawith this additionalindu tionhypothesis.
Weuseagainthenotationsofthesubse tion2.5. fortheindu tionpro edure. An
elementa 2A will be writtenas a=a 0
a +
, a 0
2A , a 0
2 A +
. We allS 0
0
the group
N
0
A , and S 0
the group NA . Fors2S
0
, we may write s=ya=ya 0
a +
=s 0
a +
.
Wede neanewfun tionF 0
onS 0
byalimitpro ess. Morepre isely,for(;x)s 0
2
S 0
, we de ne
F 0
((;x)s 0
)=F 0
s 0
(;x)= lim
t! 1
F((;x)s 0
exptH
r ):
(52)
Using the same arguments as before, as well as our assumptions on the boundary
value f of F, one an see that this limitexists and is given by
F 0
s 0
(;x)= Z
N()
f((;x)s 0
w (s 0
) 1
)P 0
L
(w )dw ;
where
P 0
L
(w )= Z
N() +
P
L (w w
+
) dw +
:
We are now able to give a sket h of the proof. The fun tion f may be seen as the
boundary value of F 0
. So, we will onsider the Fourier transform of F 0
s 0
. Using the
indu tion hypothesis forallfun tions
w +F
0
,de ned onS by
w +F
0
(s )=F(w +
s ),
wewillshowthat +F 0
arepluriharmoni . This impliesfortheir Fouriertransforms
to satisfy a di erential equation with initial data f(; ; ). Then smoothness of
the Fourier transformwillfor e this fun tion tobe zero for 2=
[
.
Our main work will be to show the smoothness of Fourier transforms, and will
ask for many te hni alities.
Step 1: F 0
isa smooth fun tion of arbitrary orderon S 0
.
Proof. First, let W be a right-invariant di erential operator on N(). We know
from the assumptions on f that Wf is well de ned, and bounded. Therefore, we
have
WF 0
s 0
(;x)= Z
N()
Wf((;x)s 0
w (s 0
) 1
)P 0
L
(w )dw :
Moreover, partial derivatives of f grow at most polynomially. The a tion of s 0
is
linear, hen e there are onstants C( ;K) and M( ) su h that
j
s
0f((;x)s 0
w (s 0
) 1
)jC( ;K)(1+ (w )) M( )
for (;x)s 0
belonging to a ompa t set K S 0
, with any left{invariant distan e
as in (33). Now we sele t su h that P
L
integrates the right hand side above, to
obtain
Z
N() j
s 0
f((;x)s 0
w (s 0
) 1
)jP 0
L
(w ) dw <1
whi h allows todi erentiate F 0
with respe t tos 0
. 2
Step 2: the fun tion
w +F
0
satis es the indu tion hypothesis onS .
Proof. We laim rst that the assumption (44), with S in pla e of S, is satis ed
for almost every w +
. Indeed, itis suÆ ient toprove that
sup
s 0
2S 0
0 kF
0
s 0
k
L 2
(N())
<1:
(53)
This follows from the fa t that, for every s 0
2S 0
0
, the fun tion F(s 0
exptH
r
) has a
weaklimitin L 2
(N()) when t tends to 1. Indeed, for 2L 2
(N()),
I = Z
N() (F(ws
0
exp (t
1 H
r
)) F(ws 0
exp (t
2 H
r
))) (w) dw
= Z
N() Z
N() (f(wv
1
) f(wv
2 )P
L
(v) (w) dv dw
with v
j
= s 0
exp (t
j H
r )v(s
0
exp (t
j H
r ))
1
for j = 1;2. Integrating with respe t to v
over a ompa t set K and over its omplement weget
I sup
v2K kf(v
1
) f(v
2 )k
L 2
(N()) k k
L 2
(N())
+2kfk
L 2
(N()) k k
L 2
(N()) Z
K
P
L
(v)dv;
whi h tends tozero when t
1
;t
2
! 1.
We now prove that the fun tions
w +F
0
satisfy the ondition (45), again with
+
element. Noti e that the operators L
j
;
j
and
kj
have a perfe t sense as left-
invariantoperators onS asfar asindi es are smaller than r . Let
(H 0
j )
T
=2
j +
X
k<j X
kj +
X
j<k<r X
jk :
Again(H 0
j )
T
may be onsidered asoperatorsboth onS and S . In the se ond ase
(H 0
1 )
T
;:::;(H 0
r 1 )
T
are HW operators for the tube V +i . We want to prove
that, for j =1; ;r 1,we have
L
j F
0
=(H 0
j )
T
F 0
=0:
Sin e for i<j <r, L
j ,
ij
, and
j
ommute withA +
, we have,for g 0
2S 0
,
lim
t! 1 L
j F(g
0
exptH
r )=L
j F
0
(g 0
)
lim
t! 1
j F(g
0
exptH
r )=
j F
0
(g 0
)
lim
t! 1
ij F(g
0
exptH
r )=
ij F
0
(g 0
);
By hypothesis, F satis es(45). Sowe on lude dire tly for L
j F
0
, j =1;;r 1.
For(H 0
j )
T
F 0
, we on lude alsoon e we knowthat
lim
t! 1
jr F(g
0
exptH
r )=0 (54)
Before doing it, we give a last de nition. We note
~
X
jk ,
~
Y
jk , and
~
X
j ,
~
Y
j
the left-
invariant ve tor elds on N whi h oin ide, at the unit element of N, with the
orrespondingelementsofthe basis ofN thatwe onstru ted insubse tion2.3. We
de ne as well
~
L
r
= P
(
~
X
r )
2
+(
~
Y
r )
2
.
In the next omputation, we identify an element a with a n-uple (a
1
;a
2
;a
r ),
with a
j
>0,insu h away that a is the exponentialof P
j (loga
j )H
j
. Inparti ular,
an element a +
2 A +
identi es with a s alar, whi h we note a
r
for omprehension.
With these notations, the previous limitsare obtained for a
r
tendingto 0.
Then, it follows from the fa t that L
r
F =0 and a dire t omputationthat
a
r F(g
0
a
r )=
~
L
r F(g
0
a
r ):
Moreover,
jr F(g
0
a
r )=a
r
X
j<r X
a
j (
~
X
jr )
2
+a 1
j (
~
Y
jr )
2 1
~
L
r
F(g 0
a
r )!0
when a
r
! 0. This nishes the proof of (54), as well as the laim of this step.
Indeed, for almost w +
, the fun tion
w +F
0
is pluriharmoni as a fun tion on S 0
. It
follows that
1 F
0
vanishes identi ally. This is the main point whi h will be used
later. 2
^
0 0 0
