Some applications of the trace condition for pluriharmonic functions in Cn

Publicacions Ma em`a iques, Vol. 44 (2000), 449–456
SOME APPLICATIONS OF THE TRACE CONDITION
FOR PLURIHARMONIC FUNCTIONS IN Cn
Alessand o Pe o i
Abs ac
In his pape we in es iga e some applica ions o he ace con-
di ion o plu iha monic unc ions on a smoo h, bounded domain
in Cn. This condi ion, ela ed o he no mal componen on ∂D
o he ∂-ope a o , pe mi s us o s udy he Neumann p oblem o
plu iha monic unc ions and he ∂-p oblem o (0,1)- o ms on D
wi h solu ions ha ing assigned eal pa on he bounda y.
1. In oduc ion
Le Dbe a smoo hly bounded domain in Cn. We a e in e es ed in
some esul s ha can be ob ained om he ace condi ion o plu iha -
monic unc ions in oduced by Fiche a in he pape s [F1], [F2], [F3] and
in es iga ed in [P]. This condi ion has a global cha ac e : a eal unc ion
on ∂D is he ace o a plu iha monic unc ion on Di i is o hogonal, in
he L2(∂D)-no m, wi h espec o a sui ably chosen space o unc ions.
This app oach is al e na i e o he local s udy o angen ial diffe en ial
condi ions (see o example [R,§18.3] and [F3] and he e e ences gi en
he e).
In his pape we fi s conside he Neumann p oblem o plu iha -
monic unc ions. Le λ>0. Gi en a eal unc ion φon he bounda y,
o class Cλ, we show (Theo em 1) ha he solu ions o he classical
Neumann p oblem
∂U
∂ν =φon ∂D
a e plu iha monic on Di and only i φis o hogonal in L2(∂D) o he
subspace o eal ha monic unc ions ha admi a decomposi ion H1+iH2
1991 Ma hema ics Subjec Classifica ion. P ima y 32F05; Seconda y 32F20, 35N15,
32D15.
Key wo ds. Plu iha monic unc ions, Neumann p oblem, ∂-p oblem.
Pa ially suppo ed by MURST (P ojec “P op ie `a geome iche delle a ie `a eali e
complesse”) and GNSAGA o CNR.
450 A. Pe o i
wi h H1,H
2∈Ha m1
0(D) (see §§2,3 o he p ecise defini ions). When
he domain is he uni ball o Cn, his esul can be exp essed in e ms
o sphe ical ha monics. We hus ob ain (Co olla y 1) ano he p oo o
a heo em gi en by Dzhu ae in [D2]: Uis plu iha monic i and only i
i sa isfies he Gauss compa ibili y condi ion Sφdσ = 0 and he ace
condi ion.
In §4 we in es iga e he ∂-p oblem o (0,1)- o ms on Dwi h a bound-
a y condi ion. Gi en a ∂-closed o m ∈C0
0,1(D) and a eal Cλ unc-
ion gon ∂D, we look o a solu ion u∈C1(D) o he p oblem
∂u = on D, Re u=gon ∂D.
We p o e (Theo em 2) ha a solu ion exis s i and only i ∂D g ∂nHdσ
equals he eal pa o D ∧∗∂H o e e y H∈Ha m1
0(D). Fo his
p oblem oo, we can ew i e he compa ibili y condi ion on he uni ball
in e ms o sphe ical ha monics (Co olla y 2). A simila app oach o his
p oblem on he ball o Cnappea s in [D1].
2. No a ions and p elimina ies
2.1. Le D={z∈Cn:ρ(z)<0}be a bounded domain in Cnwi h
bounda y o class Cm,m≥1. We assume ρ∈Cmon Cnand dρ =0
on ∂D.
Fo e e y α,0≤α≤m, we deno e by Phα(D) he space o eal plu i-
ha monic unc ions o class Cα(D) and simila ly o holomo phic unc-
ions Aα(D) and complex ha monic unc ions Ha mα(D). We deno e by
Phα
∂D(D) he space o es ic ions o ∂D o plu iha monic unc ions in
Phα(D) and by Re Aα(D) he space o eal pa s o elemen o Aα(D).
2.2. Fo e e y F∈C1(D), in a neighbou hood o ∂D we ha e he
decomposi ion o ∂F in he angen ial and he no mal pa s
∂F =∂bF+∂nF∂ρ
|∂ρ|
whe e ∂nF=k∂F
∂ζk
∂ρ
∂ζk
1
|∂ρ|.
I νdeno es he ou e uni no mal o ∂D and τ=iν, hen we can also
w i e ∂nF=1
2∂F
∂ν +i∂F
∂τ . The no mal pa o ∂F on ∂D can also be
exp essed by means o he Hodge ∗-ope a o and he Lebesgue su ace
measu e dσ:∂nFdσ=∗∂F|∂D (see [K,§§3.3 and 14.2]).
Some Applica ions o The T ace Condi ion 451
2.3. We shall deno e by Ha m1
0(D) he eal subspace o Ha m1(D)
Ha m1
0(D)={H∈Ha m1(D):∂nHis eal on ∂D}.
This space can be cha ac e ized in e ms o Bochne -Ma inelli ope -
a o M.In[P,§4] i was shown ha F∈Ha m1
0(D) i and only i
Im M(F)=ImFin D. We ecall he in eg al o hogonali y condi ion
ha cha ac e izes he aces on ∂D o plu iha monic unc ions on D
which was in oduced (in a diffe en o m) by Fiche a in [F1] (see also
[F2], [F3]) and ha was s udied in [P]:
∂D
U ∂nHdσ=0()
o e e y H∈Ha m1
0(D).
I was shown in [P] ha his condi ion is necessa y o plu iha monic-
i y when ∂D is o class C1and U∈Cα(D), α>0, o when ∂D is o
class C2and U∈C0(D). The ace condi ion is sufficien in he case
when U∈C1+λ,λ>0. I he bounda y alue uis only con inuous,
he same esul holds on s ongly pseudocon ex domains and on weakly
pseudocon ex domains wi h eal analy ic bounda y.
Rema k. When n= 1 and H1(D,R) = 0 condi ion () is oid, since H
belongs o Ha m1
0(D) i and only i His holomo phic on D(c . [P]).
In gene al (n≥1), he space o C1(D)-holomo phic unc ions on Dis
he maximal complex subspace in Ha m1(D) ha con ains Ha m1
0(D).
This ollows om a heo em o Ky mano and Aizenbe g [KA] (c . also
[K,§§14 and 15]): a C1(D)-ha monic unc ion Fis holomo phic on D
i and only i ∂nF=0on∂D.
2.4. Le Bbe he uni ball o Cnand le S=∂B. The space L2(S)
is he di ec sum o pai wise o hogonal spaces H(s, ), s≥0, ≥0,
whe e H(s, ) is he space o ha monic homogeneous polynomials o o al
deg ee sin zand o al deg ee in z(see [R,§12]). These spaces a e he
eigenspaces o he Bochne -Ma inelli ope a o .
In his case, he ace condi ion o plu iha monic unc ions educes
o he o hogonali y o he spaces H(s, ), s, > 0. This is he con en
o a heo em o Nagel and Rudin [NR] (see also [P,§5]).
3. The Neumann p oblem o plu iha monic unc ions
3.1. In his sec ion we s udy he Neumann p oblem o plu iha monic
unc ions:
452 A. Pe o i
Gi en φon he bounda y ∂D, find necessa y and sufficien condi ions
o he exis ence o a plu iha monic unc ion Uon Dsuch ha ∂U
∂ν =φ
on ∂D.
We s a om he ollowing esul announced by Fiche a in [F1], [F2],
[F3]. Le Dbe a simply connec ed domain, wi h bounda y o class C1+λ,
λ>0. Le A,B,Cbe eal unc ions o class C1(D), ha monic on D,
such ha ∂A
∂τ +∂B
∂ν =0,∂A
∂ν −∂C
∂τ =0
on ∂D. Gi en φ∈Cλ(∂D), he e exis s a plu iha monic unc ion U∈
C1(D) such ha ∂U
∂ν =φon ∂D i and only i
∂D
φ(B−C)dσ =0
o any iple A, B, C.
The p eceding condi ion is equi alen o he ollowing
∂D
φ(H1+iH2)dσ =0()
o e e y pai H1,H
2∈Ha m1
0(D) such ha H1+iH2is eal.
This can be seen se ing H1=−C+iA,H2=−A−iB. Then
H1,H
2∈Ha m1
0(D) and H1+iH2=B−C.
Theo em 1. Le H1(D, R)=0and ∂D o class C1+λ.Le φ∈Cλ(∂D)
be a eal unc ion, wi h λ>0. Then he e exis s U∈Ph1(D)such ha
∂U
∂ν =φon ∂D i and only i φsa isfies condi ion ().
P oo : We ela e he condi ion () o he ace condi ion (). Le H1,
H2be as be o e. F om he complex e sion o G een’s o mula (see o
example [K,§11.3]) we ge
∂D
U(∂nH1+i∂nH2)dσ =∂D
∂nU(H1+iH2)dσ
o e e y eal ha monic unc ion U∈C1(D). Taking eal pa s, we
ob ain ∂D
U∂nH1dσ =1
2∂D
∂U
∂ν (H1+iH2)dσ.
Now assume ha φsa isfies condi ion (). When H1=1,H2=0
() becomes he Gauss compa ibili y condi ion o he classical Neu-
mann p oblem o ha monic unc ions. Le λbe any posi i e numbe
smalle han λ. Le U∈Ha m1+λ
(D) be eal, such ha ∂U
∂ν =φon
∂D (de e mined up o an addi i e cons an ). We show ha Usa is-
fies he ace condi ion. I H∈Ha m1+λ
0(D), we can se H1=H,
Some Applica ions o The T ace Condi ion 453
H2=−Im H+iG, whe e G∈C1(D) is a eal ha monic solu ion o
∂G
∂ν =∂Im H
∂τ on ∂D. Since H1,H
2∈Ha m1
0and H1+iH2is eal, he in-
eg al ∂D φ(H1+iH2)dσ =2∂D U∂nHdσ anishes. F om Theo em 2
in [P] we ge ha Uis he eal pa o a holomo phic unc ion. No e
ha in he p oo o he ci ed heo em i is sufficien o conside unc ions
H∈Ha m1+λ
0(D).
Con e sely, i Uis plu iha monic on D, hen Theo em 1 in [P]says
ha Usa isfies condi ion () and hen ∂D φ(H1+iH2)dσ =0.
Rema ks. (i) In effec he p oo shows ha he condi ion () implies
ha U∈Re A1+λ(D) e en wi hou he opological assump ion
on D.
(ii) I n=1,H1,H
2belong o Ha m1
0(D) i and only i hey a e holo-
mo phic unc ions on D. Then H1+iH2is a eal alued holomo -
phic unc ion, ha is a cons an . As is expec ed, condi ion () e-
duces o he Gauss compa ibili y condi ion o he Neumann p ob-
lem.
3.2. I Bis he uni ball o Cnand S he uni sphe e, he p eceeding
esul can be ew i en in e ms o he spaces o ha monic homogeneous
polynomials H(s, ). Le N0be he eal linea p ojec ion in L2(S) in o-
duced in [P]. I is defined o Ps, ∈H(s, )by
N0(Ps, )=


s
s+ Ps, +
s+ Ps, , o >0
Ps, , o =0
.
The space Ha m1
0(B) coincides wi h Fix(N0)={F∈Ha m1(B):
N0(F)=F}. We show ha Theo em 1 in he case o he ball educes
o a esul gi en by Dzhu ae in [D2].
Co olla y 1. Le φ∈Cλ(S)be a eal unc ion, wi h λ>0. Then he e
exis s U∈C1(B), plu iha monic on Band such ha ∂U
∂ν =φon Si
and only i Sφdσ =0and φis o hogonal o he spaces H(s, )in L2(S)
o any s, > 0.
P oo : I s, > 0, we se H1=N0(Ps, ) and H2=N0(−Im H1). No e
ha o s, > 0weha eN0(Re F)=N0(F), N0(Im F)=N0(−iF ).
An easy compu a ion gi es H1+iH2=4s
(s+ )2Re Ps, . Replacing Ps,
wi h iPs, , we ge H1+iH2=4s
(s+ )2Im Ps, . Then he condi ion ()is
equi alen o he o hogonali y o φ o he spaces H(s, ) and Theo em 1
gi es he esul .

454 A. Pe o i
4. ∂-p oblem wi h assigned eal pa on he bounda y
4.1. In his sec ion we s udy he ∂-p oblem o (0,1)- o ms on Dwi h a
bounda y condi ion. Le ∈C0
0,1(D)bea∂-closed o m wi h con inuous
coefficien s on Dand le gbe a eal con inuous unc ion on ∂D.Welook
o a unc ion u∈C1(D) such ha
∂u = on D, Re u=gon ∂D.
The solu ion, i i exis s, is unique up o an imagina y cons an . This
p oblem was conside ed by Dzhu ae in [D1] and [D2] in he case o he
uni ball.
I H1(D,R) = 0 and he e exis s a solu ion wo ∂w = which is
con inuous on D, he p oblem can be educed o he ace condi ion
o plu iha monic unc ions, since hen i amoun s o finding a holomo -
phic unc ion hon Dsuch ha Re(h+w)=gon ∂D.I ∂D is o
class C2and Dis s ongly pseudocon ex, he e exis s a solu ion ope a-
o Sq:C0,q(D)→C0,q−1(D) such ha i ∂ = 0 and is o class Ck(D),
hen ∂Sq( )= and Sq( )∈Ck+1/2
0,q−1(D) (see o example [RA] and he
e e ences gi en he e). I ∂D is o class C∞and Dis s ongly pseudo-
con ex, ano he well-known solu ion ope a o is gi en by he Neumann
ope a o N ela ed o .I ∂ = 0, hen ∂∗N( ) is he unique solu ion
o minimal L2(D)-no m o he equa ion ∂u = .
Theo em 2. Le Dbe a smoo hly bounded s ongly pseudocon ex do-
main. Assume ha H1(D,R)=0.Le ∈C0
0,1(D)and le gbe a eal
Cλ unc ion on ∂D (λ>0). Then he e exis s u∈C1(D)such ha
∂u = on D,Re u=gon ∂D i and only i ∂ =0and o e e y
H∈Ha m1
0(D)
∂D
g ∂nHdσ=ReD
∧∗∂H.
P oo : Le α<λbe a posi i e numbe wi h α<1
2. The unc ion g−
Re S1( )∈Cα(∂D) is he ace o a plu iha monic unc ion i and only
i i sa isfies he ace condi ion (). This ollows om Theo ems 1 and
3in[P]. Fo H∈Ha m1
0(D), we ans o m he in eg al on ∂D in ol ing
in an in eg al on D:
∂D
S1( )∂nHdσ=∂D
S1( )∗∂H =D
∂S1( )∧∗∂H =D
∧∗∂H.
He e we ha e used he ac ha ∗∂H is a closed (n−1,n)- o m, since
∗∂(∗∂H)=−∂∗∂H =−H=2∆H=0.
Some Applica ions o The T ace Condi ion 455
The las in eg al is he Hodge p oduc ( ,∂H)onD. Since ∂nHis
eal on ∂D, he ace condi ion becomes
0=∂D
(g−Re S1( )) ∂nHdσ
=∂D
g ∂nHdσ−Re ∂D
S1( )∂nHdσ
=∂D
g ∂nHdσ−Re D
∧∗∂H.
No e ha Phα(D)=ReAα(D), since he fi s de i a i es o U∈Phα(D)
sa is y a Ha dy-Li lewood es ima e (see [L,§2] o example) and hen
he same holds o he ha monic conjuga e.
4.2. I Dis he uni ball, he condi ion gi en in he heo em has he
ollowing mo e explici o m.
Co olla y 2. The e exis s u∈C1(B)such ha ∂u = on B,Re u=g
on Si and only i ∂ =0and o e e y s, > 0and Ps, ∈H(s, )
Re S
gPs, dσ =ReB
n

k=1
k
∂
∂zkPs,
+Ps,
sd
Im S
gPs, dσ =ImB
n

k=1
k
∂
∂zkPs,
−Ps,
sd .
P oo : Fo H=N0(Ps, ) he le in eg al in Theo em 2 becomes
Sg2s
s+ Re Ps, dσ, while he igh in eg and ∧∗∂N0(Ps, ) is equal o

k
kdzk∧2
s+ i
2n

k
(−1)k−1∂
∂zk
(sPs, + Ps, )dz[k]∧dz.
Since d =i
2ndz ∧dz, we ge he fi s condi ion. Replacing Ps, wi h
iPs, we ge he second one.
Re e ences
[D1] A. Dzhu ae , On Riemann-Hilbe bounda y p oblem in se e al
complex a iables, Complex Va iables Theo y Appl. 29(4) (1996),
287–303.
[D2] A. Dzhu ae , On linea bounda y alue p oblems in he uni ball
o Cn,J. Ma h. Sci. Uni . Tokyo 3(2) (1996), 271–295.
456 A. Pe o i
[F1] G. Fiche a, Bounda y alues o analy ic unc ions o se e al com-
plex a iables, in: “Complex analysis and applica ions ’81 (Va na,
1981)”, Bulga . Acad. Sci., Sofia, 1984, pp. 167–177.
[F2] G. Fiche a, Bounda y p oblems o plu iha monic unc ions,
(I alian), in: “P oceedings o he Con e ence held in hono
o he 80 h Anni e sa y o he Bi h o Rena o Calapso
(Messina/Tao mina, 1981)”, Veschi, Rome, 1981, pp. 127–152.
[F3] G. Fiche a, Bounda y alue p oblems o plu iha monic unc-
ions, (I alian), in: “Ma hema ics oday (Luxembou g, 1981)”,
Gau hie Villa s, Pa is, 1982, pp. 139–151.
[K] A. M. Ky mano ,“The Bochne -Ma inelli in eg al and i s ap-
plica ions”, Bi kh¨ause Ve lag, Basel, 1995.
[KA] A. M. Ky mano and L. A. Aizenbe g, The holomo phy
o con inuous unc ions ha a e ep esen able by he Bochne -
Ma inelli in eg al, (Russian), Iz . Akad. Nauk A mjan. SSR Se .
Ma . 13(2) (1978), 158–169, 173.
[L] E. Ligocka, The H¨olde duali y o ha monic unc ions, S udia
Ma h. 84(3) (1986), 269–277.
[NR] A. Nagel and W. Rudin, Moebius-in a ian unc ion spaces on
balls and sphe es, Duke Ma h. J. 43(4) (1976), 841–865.
[P] A. Pe o i, Di ichle p oblem o plu iha monic unc ions o
se e al complex a iables, Comm. Pa ial Diffe en ial Equa ions
24(3–4) (1999), 707–717.
[RA] R. M. Range,“Holomo phic unc ions and in eg al ep esen a-
ions in se e al complex a iables”, G adua e Tex s in Ma hema ics
108, Sp inge -Ve lag, New Yo k-Be lin, 1986.
[R] W. Rudin,“Func ion heo y in he uni ball o Cn”, G undleh en
de Ma hema ischen Wissenscha en [Fundamen al P inciples o
Ma hema ical Science] 241, Sp inge -Ve lag, New Yo k-Be lin,
1980.
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