Weighted lp estimates for the ∂-equation on convex domains of finite type

Publ. Ma . 48 (2004), 139–157
WEIGHTED LpESTIMATES FOR THE ∂-EQUATION
ON CONVEX DOMAINS OF FINITE TYPE
Heungju Ahn
Abs ac
We p o e non-iso opic Lp(1 ≤p≤ ∞) es ima es wi h weigh s
o solu ions o he Cauchy-Riemann equa ion on bounded con ex
domains o ini e ype in Cnusing he in eg al ke nel me hod.
We also gi e an example which gua an ees he op imum o he
es ima es.
1. In oduc ion and s a emen o esul s
Le Ω ⊂Cnbe a smoo hly bounded con ex domain o ini e ype m
wi h a de ining unc ion ρ. In his pape we ea a ce ain weigh ed
Lpes ima es o he ∂-equa ion on Ω. The no a ion δ(ζ) will s and
o he dis ance om ζ o he bounda y o Ω, bΩ, which is up o con-
s an s |ρ(ζ)|. Wi h solu ions using he in eg al ke nel in oduced by
Cumenge [Cum01a] we can p o e he ollowing heo em.
Theo em 1.1. Fo he domain Ωas abo e he equa ion ∂u = has a
solu ion uin Ωsuch ha o 1≤p < ∞
(1) ZΩ
δ(ζ)α−1|u(ζ)|pdV (ζ)≤Cp,αZΩ
δ(ζ)α−1+p|| (ζ)||pdV (ζ), α > 0
and
(2) sup
ζ∈Ω
δ(ζ)α−1|u(ζ)| ≤ Cαsup
ζ∈Ω
δ(ζ)α|| (ζ)||, α > 1,
i is a smoo h (n, 1)- o m wi h ∂ = 0 and he igh hand sides o (1)
and (2) a e ini e. He e he non-iso opic no m o o ms, || · || =|| · ||Ω
is de ined by || (ζ)|| = sup ∈Cn {0}| (ζ)( )|/k(ζ, ), whe e k(ζ, )−1is
a weigh ed bounda y dis ance o ζ∈Ωin he di ec ion .
2000 Ma hema ics Subjec Classi ica ion. P ima y: 32W05, 32A26.
Key wo ds. Cauchy-Riemann equa ion, Lpes ima es, con ex domain, ini e ype.
The au ho was pa ially suppo ed by he Pos -doc o al Fellowship P og am o Ko ea
Science and Enginee ing Founda ion and he Pos -doc o al Fellowship o Uni e si y
o Padua in I aly.
140 H. Ahn
Rema k. (i) The non-iso opic no m || · || was i s in oduced by B una,
Cha pen ie and Dupain [BCD98]. The de ini ion o he quan i y k(ζ, )
is a he complica e e en hough k(ζ, )−1is a na u al weigh ed bound-
a y dis ance, so we pos pone he p ecise de ini ion in he nex sec ion.
(ii) Theo em 1.1 implies ha o gi en a smoo h (0,1)- o m wi h
∂ = 0 one can ind a solu ion o he ∂-equa ion on Ω and ha so-
lu ion sa is ies he inequali ies (1) and (2).
I Ω is s ongly pseudocon ex, (1) and (2) we e p o ed by Ahn-
Cho [AC02] and Dau o -Henkin [DH79]. When he domain is con ex
o ini e ype, Cumenge [Cum01a] p o ed (1) in case p= 1. The e-
o e i seems na u al o ex end Cumenge’s esul o o he Lp-no ms,
1≤p≤ ∞. The e a e a numbe o pape s ela ed o he ∂-equa ion on
con ex domains o ini e ype m. We men ion a ew o hem, which
a e closely ela ed o ou wo k; Diede ich-Fische -Fo næss [DFF99],
Cumenge [Cum01a] independen ly p o ed 1/m-H¨olde es ima es and
Cumenge [Cum01a], Fische [Fis01] ob ained he bes possible Lpes-
ima es wi h espec o he iso opic no m. Diede ich-Mazzilli [DM01]
( esp. Cumenge [Cum01b]) ob ained he cha ac e iza ion o he ze o
se s o unc ions in he Ne anlinna ( esp. Ne anlinna-Dj bachian) classes
using non-iso opic L1(bΩ) ( esp. weigh ed noniso opic L1(Ω)) es ima es
o he solu ion o he ∂-equa ion on Ω.
He e we b ie ly ske ch he me hods used o p o e Theo em 1.1. Fi s ,
o ob ain he solu ion o ∂on Ω we use an in eg al ep esen a ion in o-
duced by Be nd sson-Ande sson [BA82]. In he in eg al ep esen a ion
he key pa is a ke nel wi h a weigh con aining he Be gman ke nel o Ω.
Second, o he es ima es o he ke nel and in eg als ele an o he ke -
nel we use non-iso opic polydiscs and ε-ex emal coo dina es o McNeal
on Ω [McN94]. Las , in o de o pass in o Lpes ima es om L1es i-
ma es we employ a a ia ion o H¨olde ’s inequali y used in [AC02].
2. P elimina ies
2.1. In eg al ke nels and solu ion ope a o s. Le B(z, ζ) be he
Be gman ke nel o he domain Ω. Then B(z, ζ) is holomo phic in zand
an iholomo phic in ζ. Mo eo e B(z, ζ)∈C∞(Ω×Ω {(ζ,ζ), ζ ∈bΩ}) and
he bounda y beha io o B(z, ζ) is now well unde s ood by [McN94].
We de ine
Q=Q(z, ζ) = 1
B(ζ,ζ)
n
X
j=1 Z1
0
∂B
∂zj
(z , ζ)d dzj, z =ζ+ (z−ζ).
Weigh ed LpEs ima es o he ∂-Equa ion 141
Fo su icien ly la ge in ege N o be de e mined and ixed la e we de ine
he weighed ke nel on (z, ζ)∈Ω×Ω {(ζ,ζ), ζ ∈bΩ},
K(z, ζ)=
n−1
X
k=0
cn,k,NB(z, ζ)
B(ζ,ζ)N−k∂z|z−ζ|2∧(∂ζQ)k∧(d∂z|ζ−z|)n−k−1
|ζ−z|2n−2k
=
n−1
X
k=0
cn,k,N K(k)(z, ζ),
whe e cn,k,N =−(−1)n(n−1)/2N
nand K(k)(z, ζ) is he k- h e m in he
summa ion. Fo ∈C1
(n,1)(Ω) wi h ∂ = 0, i we de ine
(3) u(z) = CZΩ
K(z, ζ)∧ (ζ), z ∈Ω
hen i is known ha ∂u = [Cum01a].
2.2. McNeal’s esul on he geome y o con ex domains o
ini e ype. We adap o he no a ion o [Cum01a] and [McN94].
2.2.1. Weigh ed bounda y dis ance. De ine he adius o he la ges
complex disc cen e ed a zin he di ec ion ha i s in he domain
{z:ρ(z)< ρ(ζ) + ε},
τ(z, , ε) = sup{ > 0 : |ρ(z+λ )−ρ(z)| ≤ ε, |λ| ≤ , λ ∈C}.
In oduce a weigh ed bounda y dis ance k(z, , ε) = δ(z)/τ(z, , ε) and
w i e k(z, ) when ε=δ(z)/2.
2.2.2. Non-iso opic polydisc. I { 1,..., n}is an ε-ex emal basis
o McNeal a z(see [McN94] and [BCD98] o he p ecise de ini ion),
hen he non-iso opic polydisc a zwi h adius εis de ined by
P(z, ε) = 


w=z+
n
X
j=1
wj j,|wj| ≤ cτ(z, j, ε)


,
whe e cis chosen so ha w∈P(z, ε) implies |ρ(w)−ρ(z)|< ε. The
p ope ies o τwhich we e p o ed by McNeal [McN94] a e he ollowing:
(4) τ(z, 1, ε)≈ε.τ(z, n, ε)≤ · · · ≤ τ(z, 2, ε).ε1
m
and o 0 < ε1≤ε2
(5) ε1
ε2τ(z, , ε2).τ(z, , ε1).ε1
ε21
m
τ(z, , ε2).
142 H. Ahn
P oposi ion 2.1 ([McN94]).
(i) Fo all C > 0, ol P(z, Cε)≈ ol P(z, ε)uni o mly in z,εwi h
cons an s depending on C.
(ii) ol P(z, ε)≈ ol P(ζ, ε)i P(z, ε)∩P(ζ, ε)6=∅.
(iii) τ(ζ, , ε)≈τ(z, , ε) o ζ∈P(z, ε).
(i ) I { 1,..., n}is an ε-ex emal basis a z, hen we ha e ol P(z, ε)≈
Qn
i=1 τ(z, i, ε)2.
2.2.3. Ten and quasi dis ance. Fo z∈Ω close o he bounda y
and η > 0, T(z, η) = P(π(z), η)∩Ω is called he en a zo adius η,
whe e π(z) is he p ojec ion o z o he bounda y. The quasi dis ance o
McNeal is de ined as ollows:
M(z, ζ)≈ M(ζ, z) = in {η:ζ∈P(z, η)}
o |ζ−z|  1 and zclose o bΩ. Fo ∈Cnwi h | |= 1, ϕ∈C∞(Ω)
le D ϕdeno e he di ec ional de i a i es o ϕin he di ec ion . F om
now on se η=η(z, ζ) = |ρ(z)|+|ρ(ζ)|+M(z, ζ). Then he ollowing
p oposi ion is p o ed by McNeal [McN94].
P oposi ion 2.2. Fo e e y p∈bΩ he e exis s a neighbo hood Uo p
such ha o ζ, z ∈U∩Ω,µ, ν ∈N, , 0∈Cnwi h | |=| 0|= 1,
(i) |Dµ
Dν
0B(z, ζ)| ≤ C(µ, ν)τ(ζ, , η)−µτ(ζ, 0, η)−ν( ol Tζ,z)−1. He e
ol Tζ,z is he olume o he smalles en con aining bo h z,ζ.
(ii) Fo ζ∈U∩Ω,B(ζ,ζ)≥( ol P(ζ, δ))−1,δ=δ(ζ) = |ρ(ζ)|/2.
2.2.4. Co e ings. The e exis s a cons an β > 1 such ha
M(z, ζ)< ε ⇒ζ∈P(z, βε), z ∈U∩Ω,0< ε 1,
whe e Uis some neighbo hood de ined in P oposi ion 2.2. Fo he in e-
g al es ima es we de ine he co e ing o W∩Ω
C0(z) = P(z, β d(z)) ∩W∩Ω,
C`(z) = {ζ∈Ω∩W: 2`−1d(z)≤ M(z, ζ)<2`d(z)}, ` ≥1,
whe e W= 1/2U.
3. Ke nel es ima es
3.1. Es ima e o he e m K(k)(z, ζ), 0 ≤k≤n−1. Now we wan
o w i e all o ms wi h espec o ex emal coo dina es o McNeal a ζ
o z. Le {e(`)
j=e(`)
j(ζ),1≤j≤n}be a β2`δ-ex emal basis a ζand
(e`
j(z))jaβ2`d-ex emal basis a z, espec i ely. I j= (`)
j(ζ) is he
Weigh ed LpEs ima es o he ∂-Equa ion 143
j- h componen o hei coo dina es, we deno e L(`)
j=∂/∂ jand L(`)∗
j
which is he dual o L(`)
j. To simpli y no a ions, in any ambiguous case,
we w i e L(z)
j,L(ζ)
j,L∗(z)
j,L∗(ζ)
j o L(`)(z)
j,L(`)(ζ)
j,L(`)∗(z)
j,L(`)∗(ζ)
j,
1≤j≤n, whe e he supe sc ip s z,ζmean he de i a ions ac on
he a iables z,ζ, espec i ely. To sa e us om con usion, we deno e
dis (ζ, Ω) and dis (z, bΩ) by δ=δ(ζ) and d=d(z), espec i ely.
Fi s we es ima e K(k)(z, ζ)∧ (ζ), 1 ≤k≤n−1. Le R=
R1
0∂zB(z , ζ)d . Then compu ing he k- h ex e io p oduc we ob ain
∂ζQk=ck
∂ζB(ζ,ζ)
B(ζ,ζ)k+1 ∧R∧(∂ζR)k−1+∂ζRk
B(ζ,ζ)k.
Using his and exp essing all o ms in e ms o L(ζ)
jand L(z)
j’s, we ha e
K(k)(z, ζ)∧ (ζ) = B(z, ζ)N−k
B(ζ,ζ)N
1
|ζ−z|2n−2kH1+H2,
whe e
H1=1
B(ζ,ζ)X
I,J
L(z)
i0|ζ−z|2L(ζ)
j1B(ζ,ζ)Z1
0L(z)
i1B(z , ζ)d
×
k
Y
ν=2 Z1
0L(ζ)
jνL(z)
iνB(z , ζ)d  (ζ)(e(`)
jn)
×AIJS(z, ζ)L∗(ζ)
S∧L∗(ζ)
J∧L∗(z)
I
H2=X
I,J
L(z)
i0|ζ−z|2
k
Y
ν=1 Z1
0L(ζ)
jνL(z)
iνB(z , ζ)d  (ζ)(e(`)
jn)
×AIJS(z, ζ)L∗(ζ)
S∧L∗(ζ)
J∧L∗(z)
I.
He e I=I(k) = Ik∪Ik0,Ik={i0, i1,...,ik}and Ik0={ik+1,...,in−1};
J=J(k) = Jk∪Jk0,Jk={j1, j2,...,jk}and Jk0={jk+1,...,jn},
iν, jν∈S={1,2,...,n},L∗(z)
I=L∗(z)
i0∧ · · · ∧ L∗(z)
in−1, e c. and AIJS
is uni o mly bounded on Ω ×Ω. No e ha i k= 1, hen he p oduc
e m Qk
ν=2(...) o H1does no appea .
Nex we no e he ollowing inequali y
(6) | (ζ)(e(`)
j(ζ))| ≤ || (ζ)|| δ(ζ)
τ(ζ, e(`)
j(ζ), δ).|| (ζ)||.

144 H. Ahn
Then i is easy o see ha
K(0)(z, ζ)∧ (ζ)≤ || (ζ)||K(0)(z, ζ)
≤ || (ζ)|| |B(z, ζ)|
B(ζ,ζ)N1
|ζ−z|2n−1.
Fi s we es ima e |B(z, ζ)|/B(ζ,ζ), R1
0L(z)
i1B(z , ζ)d and
R1
0L(ζ)
jnuL(z)
iνB(z , ζ)d on he neighbo hood Uo p∈bΩ, whe e Uis he
neighbo hood de ined in P oposi ion 2.2. By P oposi ion 2.2 (ii), (4)
and (5) he ollowing es ima es can be p o ed:
|B(z, ζ)|
B(ζ,ζ)≤δ(ζ)
η(z, ζ)
Z1
0
L(z)
i1B(z , ζ)d 
. ol P(ζ, δ)τ(ζ, e(`)
i1, δ)−1
Z1
0
L(ζ)
jνL(z)
iνB(z , ζ)d 
. ol P(ζ, δ)τ(ζ, e(`)
jν, δ)τ(ζ, e(`)
iν, δ)−1.
(Fo de ails see [Cum01a].) Combining abo e es ima es and he in-
equali y (6) we ha e o 1 ≤k≤n−1
K(k)(z, ζ)∧ (ζ)
.X
Ik,Jk
δ(ζ)N−k
η(z, ζ)N−k
δ(ζ)|| (ζ)||
|ζ−z|2n−2k−1
1
Qk
ν=1
τ(ζ, e(`)
jν, δ)τ(ζ, e(`)
iν, δ)
1
τ(ζ, e(`)
jn, δ)
and
K(0)(z, ζ)∧ (ζ).δ(ζ)N−1
η(z, ζ)N
δ(ζ)|| (ζ)||
|ζ−z|2n−1.
Fo he simpli ica ion o no a ion we w i e o z, ζ ∈U
K(0)
+(z, ζ) = δ(ζ)N−1
η(z, ζ)N
1
|ζ−z|2n−1
K(k)
+(z, ζ) = δ(ζ)N−k
η(z, ζ)N−k
1
|ζ−z|2n−2k−1
×1
Qk
ν=1 τ(ζ, e(`)
jν, δ)τ(ζ, e(`)
iν, δ)
1
τ(ζ, e(`)
jn, δ).
Weigh ed LpEs ima es o he ∂-Equa ion 145
3.2. Es ima es o K(k)
+(z, ζ) on co e ings {C`(z)}and {C`(ζ)}.
We only conside he es ima es o K(k)
+(z, ζ) o k= 1, n −1. In he
in eg al es ima es o he cases can be educed o cases k= 1, n −1. I
k= 0, we can di ec ly es ima e in eg als on he neighbo hood W. Fi s
assume ha ζ∈Uis ixed and we will es ima e K(k)
+(z, ζ) on C`(ζ). By
he de ini ion o C`(ζ), M(z, ζ) and (5), i is easy o see ha
η(z, ζ)≈2`δ(ζ), z ∈ C`(ζ)
τ(ζ, e(`)
j(ζ), δ)&2−`τ(ζ, e(`)
j(ζ), β2`δ).
No e ha τ(`)
jn(ζ)≤τ(`)
2(ζ). The e o e we ha e o z∈ C`(ζ)
K(n−1)
+(z, ζ).τ(`)
2(ζ)
2(N−3n+2)`|ζ−z|Qn
j=1 τ(`)
j(ζ)2
(7)
K(1)
+(z, ζ).X
I10,J10
τ(`)
i0Qn−1
ν=2 τ(`)
iν(ζ)τ(`)
jν(ζ)
2(N−4)`Qn
j=1 τ(`)
j(ζ)2|ζ−z|2n−3,(8)
whe e τ(`)
j(ζ) = τ(ζ, e(`)
j(ζ), β2`δ(ζ)). Nex assume ha z∈Uis ixed
and we will es ima e K(k)(z, ζ) on C`(z). By P oposi ion 2.1 (ii) and (iii),
we ha e
τ(ζ, e(`)
j, δ)&δ
2`dτ(ζ, e(`)
j, β2`δ)≈δ
2`dτ(z, e(`)
j(z), β2`d).
Since η(z, ζ)≈2`d(z), ζ∈ C`(z) we ha e
(9) K(n−1)
+(z, ζ).δ
2`dN−3n+2 1
|ζ−z|
τ(`)
2(z)
Qn
j=1 τ(`)
j(z)2
and
K(1)
+(z, ζ).
n
X
i,j,p=1
i6=p
δ
2`dN−41
|ζ−z|2n−3
1
τ(`)
j(z)τ(`)
i(z)τ(`)
p(z)
.X
I10,J10
δ
2`dN−41
|ζ−z|2n−3
τ(`)
i0(z)Qn−1
ν=2 τ(`)
iν(z)τ(`)
jν(z)
Qn
j=1 τ(`)
j(z)2.
(10)
146 H. Ahn
4. In eg al es ima es
In his sec ion we e i y p elimina y in eg al es ima es ha a e an
essen ial s ep o p o e ou Theo em 1.1.
Lemma 4.1. Le α > 0and ε > 0wi h α−1−ε > −1. Then o
k= 0,1,...,n−1we ha e
ZΩ
d(z)α−1−εK(k)
+(z, ζ)dV (z)≤Cα,εδ(ζ)α−ε−1
(11)
ZΩ
δ(ζ)−εK(k)
+(z, ζ)dV (ζ)≤Cεd(z)−ε.(12)
P oo : We p o e (11) and (12) o k= 0,1, n −1. The o he cases can
be educed o he cases k= 0,1, n −1. Since he only singula i y is o
he o m |ζ−z|−j, we may assume ha z, ζ ∈W.Wcan be co e ed
by ∪`C`(z) and ∪`C`(ζ) so basically we ha e o deal wi h he domain o
he o m C`(z) o C`(ζ).
(i) By he es ima e (7) we ha e o any in ege `≥0
(13) ZC`(ζ)
d(z)α−1−εK(n−1)
+(z, ζ)dV (z)
.τ(`)
2(ζ)
2(N−3n+2)`Qn
j=1 τ(`)
j(ζ)2ZP(ζ,β2`δ)
d(z)α−1−ε
|ζ−z|dV (z).
To ob ain a desi ed es ima e, we use he sys em o coo dina e associa ed
o he basis (e(`)
1(ζ),...,e(`)
n(ζ)). We se
wk=hζ−z, e(`)
ki,1≤k≤n, 1=−ρ(z), 2= Im w1
(14)
and o 2 ≤k≤n,
2k−1= Re wk,
2k= Im wk.
Weigh ed LpEs ima es o he ∂-Equa ion 147
Since τ(`)
1(ζ)≈2`δand d(z).2`d o z∈P(ζ, β2`d), by he coo dina es
change (14) we ha e
ZP(ζ,β2`δ)
d(z)α−1−ε
|ζ−z|dV (z)
.Z| j|<2`δ, j=1,2
|wj|<τ(`)
j, j≥2
α−1−ε
1d 1d 2dV (w2,...,wn)
|w2|
.1
α−ε(2`δ)α−ε+1 Z|wj|<τ(`)
j, j≥2
dV (w2,...,wn)
|w2|
.1
α−ε(2`δ)α−ε−1
n
Y
j=1
τ(`)
j(ζ)2
τ(`)
2(ζ).
(15)
I we choose an in ege Nso ha N−3n+3−α > 0, hen (13) and (15)
gi e
ZW∩Ω
d(z)α−1−εK(n−1)
+(z, ζ)dV (z)
.X
`
2−(N−3n+3−α+ε)`δ(ζ)α−ε−1.δ(ζ)α−ε−1.
To p o e (11) o k= 1 we ha e o conside he in eg al
ZP(ζ,β2`δ)
d(z)α−1−ε
|ζ−z|2n−3dV (z).
To change coo dina es we again use he coo dina es (14). He e we may
assume ha i0< i1,ν= min(i0, j1), µ= max(i0, j1) and i1=|wi1|.
We i s conside 1, 2ν, 2i2−1, 2i2 a iables and hen we in eg a e wi h
he emaining (2n−4) a iables, 0. Since τ(`)
1≈2`δand 2`δ.τ(`)
µ, we
154 H. Ahn
whe e we se s= /(1 − |z1|2)1/m.To calcula e he uppe bound o (22)
we need he ollowing lemma:
Lemma 6.2 ([Rud80]).Fo z∈Bn={z∈Cn:|z|<1},c eal,
η > −1, de ine
Jc,η(z) = ZBn
(1 − |ζ|2)η
|1−ζ·z|n+1+η+cdV (ζ).
When c < 0, hen Jc,η is bounded in Bn. When c > 0, hen Jc,η(z)≈
(1 − |z|2)−c. Finally, J0,η ≈−log(1 − |z|2).
F om (22), (23) and by Lemma 6.2 i ollows ha i γ > α, hen
k kp
p,γ,Em.Z|z1|<1
dA(z1)
|1−z1|dp−γ+1−p/m−2/m
= lim
→1−Z|z1|<1
dA(z1)
|1−z1 |dp−γ+1−p/m−2/m .1
since dp −γ+ 1 −p/m −2/m < 2. Le (z1, z2) = z2/(1 −z1)d. Then
i is clea ha ∂ = on Em. On he o he hand, we ha e
(24) k kp
p,α =Z|z1|<1
dA(z1)
|1−z1|dp Z|z2|<bz1
|z2|p(1−|z1|2−|z2|m)α−1dA(z2).
By pola coo dina e change, we ha e
J(z1) = Z|z2|<bz1
|z2|p(1 − |z1|2− |z2|m)α−1dA(z2)
= 2π(1 − |z1|2)α−1Zbz1
0
p+1 1− m
1− |z1|2α−1
d
= 2π(1 − |z1|2)α−1+ (p+2)
mZ1
0
(1 −sm)α−1sp+1 ds
&(1 − |z1|2)α−1+ (p+2)
m.
(25)

Weigh ed LpEs ima es o he ∂-Equa ion 155
F om (24) and (25) we ha e
k kp
p,α &Z|z1|<1
(1 − |z1|2)α−1+ (p+2)
m
|1−z1|dp dA(z1)
= lim
→1−Z|z1|<1
(1 − |z1|2)α−1+ (p+2)
m
|1−z1 |dp dA(z1)
≈lim
→1−log 1
1− 2=∞,
(26)
by Lemma 6.2. Thus, /∈Lp
α(B2).
Nex we conside he inne p oduc hh, iα o e e y h∈L2
α(B2)∩
O(B2). By Fubini’s heo em, we ha e
hh, iα=ZEm
h(ζ) (ζ)|ρ(ζ1, ζ2)|α−1dV (ζ)
=Z|ζ1|<1
dA(ζ1)
(1 −ζ1)dZ|ζ2|<bζ1
ζ2h(ζ1, ζ2)(1 − |ζ1|2− |ζ2|m)α−1dA(ζ2).
Pu ing ζ2= eiθ, we see
Z|ζ2|<bζ1
ζ2h(ζ1, ζ2)(1 − |ζ1|2− |ζ2|m)α−1dA(ζ2)
=Zbζ1
0Z2π
0
2eiθh(ζ1, eiθ)(1 − |ζ1|2− m)α−1dθ d
=Zbζ1
0
2(1 − |ζ1|2− m)α−1Z2π
0
eiθh(ζ1, eiθ)dθd
=Zbζ1
0
2(1 − |ζ1|2− m)α−1·0·d = 0,
since h(ζ1,·) is holomo phic. Thus is o hogonal o L2
α(B2)∩ O(Em),
i.e., is he canonical solu ion o ∂u = . To comple e ou heo em we
need ano he well-known heo em on he boundedness o he weigh ed
Be gman p ojec ions on Em.
P oposi ion 6.3 ([Cho], [LS92]).Le Bα:L2
α(Em)→L2
α(Em)∩O(Em)
be he o hogonal p ojec ion, α > 0. Then Bα:Lp
α(Em)→Lp
α(Em)∩
O(Em)is a bounded ope a o o e e y 1<p<∞.
Assume ha ∂u = ∈Lp
γ(Em,|| · ||) and u∈Lp
α(Em), γ > α. Then
by he P oposi ion 6.3, =u−Bα(u) and i would be in Lp
α(Em).
156 H. Ahn
By (26) his is impossible. Hence he e is no solu ion uin Lp
α(Em) o
he equa ion ∂u = .
I < p, i.e. d −α− /2<2, hen i also ollows by a simila
calcula ion o he abo e ha ∈L
α(Em,|| · ||) and no solu ion u o
∂u = belongs o Lp
α(Em).
Re e ences
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[BA82] B. Be nd sson and M. Ande sson, Henkin-Rami ez o -
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32(3) (1982), 91–110.
[BCD98] J. B una, P. Cha pen ie and Y. Dupain, Ze o a i-
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[Cho] H. R. Cho, Holomo phic Sobole spaces in con ex domains
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he Ne anlinna-Dj bachian classes, Paci ic J. Ma h. 199(1)
(2001), 79–92.
[DH79] ˇ
S. A. Dau o and G. M. Henkin, Ze os o holomo phic
unc ions o ini e o de and weigh ed es ima es o he solu-
ions o he ∂-equa ion, (Russian), Ma . Sb. (N.S.) 107(149),
no. 2 (1978), 163–174, 317.
[DFF99] K. Diede ich, B. Fische and J. E. Fo næss, H¨olde
es ima es on con ex domains o ini e ype, Ma h. Z. 232(1)
(1999), 43–61.
[DM01] K. Diede ich and E. Mazzilli, Ze o a ie ies o he
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Ma h. J. 163 (2001), 215–227.
[Fis01] B. Fische ,Lpes ima es on con ex domains o ini e ype,
Ma h. Z. 236(2) (2001), 401–418.
[LS92] S. H. Liu and M. S oll, P ojec ions on spaces o holomo -
phic unc ions on ce ain domains in C2,Complex Va iables
Theo y Appl. 17(3–4) (1992), 223–233.
[McN94] J. D. McNeal, Es ima es on he Be gman ke nels o con ex
domains, Ad . Ma h. 109(1) (1994), 108–139.
Weigh ed LpEs ima es o he ∂-Equa ion 157
[Rud80] W. Rudin,“Func ion heo y in he uni ball o Cn”,
G undleh en de Ma hema ischen Wissenscha en 241,
Sp inge -Ve lag, New Yo k-Be lin, 1980.
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