Maximal subextensions of plurisubharmonic functions

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

U. CEGRELL, S. KOŁODZIEJ, A. ZERIAHI

Maximal subextensions of plurisubharmonic functions

Tome XX, n^oS2 (2011), p. 101-122.

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Maximal subextensions of plurisubharmonic functions

U. Cegrell⁽¹⁾, S. Kolodziej⁽²⁾ and A. Zeriahi⁽³⁾ Dedicated to Professor Nguyen Thanh Van

on the occasion of his retirement

ABSTRACT.— In our earlier [CKZ], we proved that any plurisubharmonic function on a bounded hyperconvex domain inCⁿ with zero boundary values in a quite general sense, admits a plurisubharmonic subextension to a larger hyperconvex domain. Here we study important properties of its maximal subextension and give informations on its Monge-Amp`ere mea- sure. More generally, given a quasi-plurisubharmonic functionϕon a given quasi-hyperconvex domainD⊂X of a compact K¨ahler manifold (X, ω), with well defined Monge-Amp`ere measure such that

D(ω+dd^cϕ)ⁿ

Xωⁿ, we prove thatϕadmits a global quasi-plurisubharmonic subex- tension ˜ϕ to the whole manifold X. If moreover (ω+dd^cϕ)ⁿ puts no mass on pluripolar sets ofD, the maximal subextension is shown to have a well defined global Monge-Amp`ere measure on X. Moreover we give a good control on the weigthed energy of the subextension in terms of the weigthed energy of the original function. Finally we provide an exem- ple inP<sup>2</sup> which shows that in general the maximal subextension do not have a well defined Monge-Amp`ere measure onP²if the original function concentrates some mass in an analytic disc.

R^´ESUM´E. — Dans notre travail pr´ec´edent paper [CKZ], nous avions d´emontr´e que toute fonction plurisousharmonique sur un ouvert hyper- convexe born´e deCⁿ ayant des valeurs au bord nulles en un sens assez g´en´eral poss`ede une sous-extension plurisousharmonic dans un dmaine hyperconvex plus grand. D’une fa¸con plus g´en´erale, ´etant donn´ee une fonction quasi-plurisousharmoniqueϕsur un domaine quasi-hyperconvex

(1) Department of Mathematics, University of Umea, S-90187 Umea, Sweden.

(2) Jagiellonian University, Institute of Mathematics, @Lojasiewicza 6, 30-348 Krak´ow, Poland.

(3) Institut de Math´ematiques de Toulouse, UPS, 118 Route de Narbonne, 31062 Toulouse cedex, France.

D ⊂ X d’une vari´et´e k¨ahlerienne compacte (X, ω), de valeurs au bord nulle en un sens g´en´eralis´e et ayant une mesure de Monge-Amp`ere bien d´efinie surDet v´erifiant

D(ω+dd^cϕ)ⁿ

Xωⁿ, nous d´emontrons que ϕ admet une sousextension ˜ϕ `a la vari´et´e X toute enti`ere. Si de plus (ω+dd^cϕ)ⁿne charge pas les ensembles pluripolaires deD, la sousexten- sion maximale poss`ede une mesure de Monge-Amp`ere globale bien d´efinie surXdont nous ´etudions la mesure de Monge-Amp`ere. De plus nous don- nons un contrˆole pr´ecis en terme d’energie de Monge-Amp`ere pond´er´ee de la sousextension maximale en fonction de l’energie pond´er´ee de la donn´ee ϕ. Enfin nous donnons un exemple dans P² qui montre qu’en g´en´eral la sousextension maximale n’a pas une mesure de Monge-Amp`ere globale bien d´efinie si la mesure de Monge-Amp`ere de la fonction donn´e concentre de la masse sur un disque analytique.

1. Introduction

This is the sequel to our earlier paper [CKZ]. There we proved that given a plurisubharmonic function ϕ from the class F(Ω) (see the next section for definitions) in a hyperconvex domain Ω Cⁿ one can find its maximal subextension ϕ which is plurisubharmonic in Cⁿ and which has logarithmic growth at infinity. If, in addition, the Monge-Amp`ere measure of ϕ vanishes on pluripolar sets then the Monge-Amp`ere of ϕ is a well defined positive measure on Cⁿ in the sense that it is the weak limit of the sequence ofpositive measures (dd^cϕ^j)ⁿ for any sequence of continuous plurisubharmonic functionsϕ^j ↓ϕhaving the same rate ofgrowth at infinity as ϕ. In Subsection 4.3 ofthis article we complete this picture studying in more detail the Monge-Amp`ere measures ofmaximal subextensions ˜ϕ. If the sublevel sets ofthose subextensions are bounded then such a measure can be split into µ1, dominated by (dd<sup>c</sup>ϕ)<sup>n</sup> and essentially supported on the contact set whereϕ= ˜ϕ, and µ2living on the set∂{ϕ <˜ 0}.In general the maximal global subextension of a function from the class F(Ω) may not have well defined Monge-Amp`ere measure. It is the case for generic multipole Green function as we show in the last section.

Now, a subextension of a plurisubharmonic function from a domainDin Cⁿ to a function defined in the whole space and of logarithmic growth can be viewed upon as a subextension ofanω-plurisubharmonic function (with ω a multiple of the Fubini-Study form) from a subset ofCPⁿ to the whole manifold. Here the domainDis special since there exists a potential forωin D.If, for instance,D⊂CPⁿ contains an algebraic set ofpositive dimension then there are no strictly plurisubharmonic functions inD.Thus on a com- pact K¨ahler manifoldX we face a more general problem of subextension of anω-plurisubharmonic function inD⊂X to anω-plurisubharmonic func-

tion inX.In Section 3 we introduce classes ofω-plurisubharmonic functions onD⊂X modelled on the classes defined by Cegrell and prove the subex- tension results which are generalizations ofthe ones on global subextensions inCⁿ. We refer to [CKZ] for a historical account on subextension problems.

Acknowledgements. — The authors whould like to thank the referee for his careful reading and his/her suggestions which permits to improve the exposition.

D´edicace. — C’est avec un grand plaisir que nous apportons cette con- tribution au volume sp´ecial en l’honneur du Professeur Nguyen Thanh Van

`

a l’occasion de sa retraite. Ses travaux de recherche notamment en Th´eorie du Pluripotentiel et ses applications `a la th´eorie de l’approximation font partie de ceux nombreux qui ont contribu´e `a la naissance de cette ”belle th´eorie” dans les ann´ees 1980.

2. Monge-Amp`ere measure of maximal subextensions We assume the notational convention d^c = _2πⁱ ( ¯∂−∂). Let us recall some definitions from ([Ce1], [Ce2]). Let D Cⁿ be a hyperconvex do- main. We denote by E⁰(D) the set ofnegative and bounded plurisubhar- monic functions ϕ on D which tend to zero at the boundary and satisfy

D(dd^cϕ)ⁿ<+∞.

Let us denote byF(D) the set ofallϕ∈P SH(D) such that there exists a sequence (ϕj) ofplurisubharmonic functions in E⁰(D) such thatϕj ϕ and sup_j

D(dd^cϕj)ⁿ<+∞.

Before we consider the subextensions from a hyperconvex domain toCⁿ, we first need a result on subextensions to just a larger hyperconvex set. Let D ΩCⁿ be two bounded hyperconvex domains (open and connected) and and letu∈ F(D) be a given function. Thenu admits a subextension

˜

u∈ F(Ω) i.e. ˜uuonD (see [CZ]). Therefore we can define the maximal subextension ofuto Ω by

() u˜= sup{v∈P SH(Ω);v <0, v|^Du}.

It follows from [Ce2] that ˜u ∈ F(Ω). The following theorem provides a description ofthe Monge-Amp`ere measure ofthe maximal subextension.

Theorem 2.1. — LetD⊂⊂Ω.For everyu∈ F(D),u˜∈ F(Ω),(dd^cu)˜ ⁿ χD(dd^cu)ⁿ and

{˜u<u}

(dd^cu)˜ ⁿ= 0.

For the proofofthe last equality we need the following elementary lemma.

Lemma 2.2. — Suppose (µj) is a sequence of positive measures on D with uniformly bounded mass and that to every >0 there is a δ >0 such that to every E ⊂ D with cap(E) < δ we have µj(E) < for all j. If limµj =µand iff, g∈P SH(D)then

{f <g}

dµlim inf

j

{f <g}

dµj.

To prove the lemma, one can use Bedford-Taylor capacity and the qua- sicontinuity ofg(see [BT2]).

Proof (Of the theorem). — The first statement ofthe theorem was proved in [CH].

Observe that the function ˜u defined by () is plurisubharmonic if uis just any continuous function onD. Using the balayage procedure, it is easy to show that in that case we have

{˜u<u}

(dd^cu)˜ ⁿ= 0.

Assume now that u ∈ F ∩L^∞(D) and take a sequence ofcontinuous functionsuj onDdecreasing tou.Then ˜uj decreases to ˜uand the sequence (˜uj) is uniformly bounded on Ω since ˜uu˜j 0 on Ω. Therefore the Monge- Amp`ere measures (dd<sup>c</sup>u˜j)<sup>n</sup>are uniformly dominated by the Monge-Amp`ere capacity.

So ifwe putµj= (dd^cu˜j)ⁿ we can apply the lemma to conclude that for every s0:

{˜us<u}

(dd^cu)˜ ⁿlim inf

j

{˜us<u}

(dd^cu˜j)ⁿlim inf

j

{˜uj<uj}

(dd^cu˜j)ⁿ= 0,

since by the remark at the beginning ofthis proof

{˜uj<uj}

(dd^cu˜j)ⁿ = 0.To complete the proofin this case, we letstend to +∞.

Ifu∈ F(D) only, consideruj = max{u,−j}.Then, fort >0 fixed (1 + max{u/t,−1}) (dd^cuj)ⁿ→(1 + max{u/t,−1}) (dd^cu)ⁿ, j→+∞. Observe that the function (1 + max{u/t,−1}) vanishes on {u −t} and is bounded from above by 1. Moreover for any j > t we have{u >−t} ⊂ {u >−j} and the sequence ofmeasures1_{u>−j}(dd^cuj)ⁿ increases to the measure1_{u>−∞}(dd^cu)ⁿ (see [BGZ]). Therefore we obtain forj > t

(1 + max{u/t,−1}) (dd^cuj)ⁿ 1_{u>−j}(dd^cuj)ⁿ 1_{u>−∞}(dd^cu)ⁿ.

It follows that, for every fixedt, the sequence ofmeasures µj:= (1 + max{u/t,−1}) (dd^cuj)ⁿ

and therefore (1 + max{u/t,−1}) (dd^cu˜j)ⁿ satisfy the requirements of the lemma, so we get for every fixedsandt:

{u˜s<u}

(1 + max{u/t,−1}) (dd^cu)˜ ⁿlim inf

j

{˜us<u}

(1 + max{u/t,−1}) (dd^cu˜j)ⁿ

lim inf

j

{u˜s<u}

(dd^cu˜j)ⁿlim inf

j

{u˜j<uj}

(dd^cu˜j)ⁿ= 0.

We now let t tend to +∞. Then since 1 + max{u/t,−1} 1_{u>−∞} as t+∞, it follows from the previous inequalities that

{u˜s<u}

(dd^cu)˜ ⁿ = 0.

To complete the proof, we letstend to +∞.

Remark 2.3. — Independently the above theorem was proved in [P], Lemma 4:5.

Remark 2.4. — It follows that

1_{u=˜ −∞}(dd^cu)˜ ⁿ =1_{u=−∞}(dd^cu)ⁿ.

Indeed, the inequality ”” follows from Theorem 2.1 and the other one from Demailly’s inequality [D] (see also [ACCP], Lemma 4.1).

3. Potentials on K¨ahler domains

Here we want to establish some elementary facts in pluripotential theory on compact K¨ahler manifolds with boundary i.e. on domains in a compact K¨ahler manifold.

3.1. The comparison principle

The aim ofthis section is to give a semi global version ofthe comparison principle which contains the local one from pluripotential theory on bounded hyperconvex domains in Cⁿ as well as the global one from the theory on compact K¨ahler manifolds (see [GZ2]).

LetX be a K¨ahler manifold of dimensionnandωthe K¨ahler form onX. We want to consider boundedω−plurisubharmonic functions on K¨ahler do- mains inX with boundary. For any domainD⊂X, denote byP SH(D, ω) the set ofω−plurisubharmonic functions onD.

By definition if ϕ is ω−plurisubharmonic on D then locally in D the function u := ϕ+p is a local plurisubharmonic function, where p is a local plurisubharmonic potential ofthe formω i.e.dd^cp=ω. Therefore by Bedford and Taylor [BT] the curvature currentωϕ:=dd^cϕ+ω associated toϕis a globally defined closed positive current onDwhich can be witten locally asωϕ=dd^cu. Hence by Bedford and Taylor [BT], the wedge power ω_ϕ^p is a well defined closed positive current ofbidegree (p, p) onD. More generally, if ϕ1,· · ·, ϕq are bounded ω−plurisubharmonic functions on D, we can define inductively the wedge intersection product

T(ϕ1,· · ·, ϕq) :=ωϕ1∧ · · · ∧ωϕq (3.1) as a closed positive current ofbidimension (n−q, n−q) on D. Moreover these currents put no mass on pluripolar sets.

Actually all local results from pluripotential theory concerning bounded plurisubharmonic functions on domains in Cⁿ are valid in the situation considered here. We will refer to these results as results from the ”local theory”.

Here we use ideas from the global case (see [GZ2]). Our starting point is the following “local version” of the comparison principle which follows from quasi-continuity ofplurisubharmonic functions (see [BT2],[BT3]).

Proposition 3.1. — LetT be a closed positive current of bidimension (p, p)(1pn) of type (3.1) andϕ, ψ∈P SH(D, ω)∩L^∞(D). Then

1_{ϕ<ψ}(ω+dd^csup{ϕ, ψ})^p∧T =1_{ϕ<ψ}(ω+dd^cψ)^p∧T, (3.2) in the weak sense of Borel measures on D. In particular

1_{ϕψ}(ω+dd^csup{ϕ, ψ})^p∧T 1_{ϕψ}(ω+dd^cψ)^p∧T, (3.3) in the weak sense of Borel measures on D.

To perform a useful integration by parts formula, we need to consider special domains.

Definition 3.2. — We will say that a domainD⊂Xis quasi-hypercon- vex ifDadmits a continuous negativeω−plurisubharmonic exhaustion func- tionρ:D−→[−1,0[.

Observe that any domain D ⊂ X with smooth boundary given by D := {r < 0}, where r is smooth in a neighbourhood of D, is quasi- hyperconvex since forε >0 small enough, the functionρ:=ε risω−pluri- subharmonic on a neighbourhood ofDand is a bounded exhaustion forD.

Observe that such a domain can be pseudoconcave.

Here we will consider only quasi-hyperconvex domainsD satisfying

D

ωⁿ <

X

ωⁿ. (3.4)

Definition 3.3. — Given a quasi-hyperconvex domainD, we define the class of test functionsP⁰(D, ω)to be the class of functionsϕ∈P SH⁻(D, ω)∩ L^∞(D)such thatlim_z→∂Dϕ= 0 and

D(ω+dd^cϕ)ⁿ<+∞.

Observe that for any negative smooth functionhwith compact support in D, the functionεhis inP⁰(D, ω) f orε >0 small enough. Moreover, ifρ is an ω−plurisubharmonic defining function for D then for any 0t1, tρ∈ P⁰(D, ω).

Lemma 3.4. — Let T be a closed positive current of bidimension (p, p) (1pn) and ϕ, ψ∈P SH(D, ω)∩L^∞(D)be such that (ϕ−ψ)0 on

∂D. Assume that

D(dd^cϕ)^p∧T <+∞. Then we have

{ϕ<ψ}

ω^p_ψ∧T

{ϕ<ψ}

ω^p_ϕ∧T,

and

{ϕψ}

ω^p_ψ∧T

{ϕψ}

ω_ϕ^p∧T.

and if ϕψ onD then

D

ω^p_ψ∧T

D

ω_ϕ^p∧T.

In particular ifϕ∈P SH⁻(D, ω)∩L^∞(D)andϕ→0at the boundary, then

D

ω^p∧T

D

ω_ϕ^p∧T.

Proof. —Recall that the condition (ϕ−ψ) 0 means that for any ε > 0, {ϕ < ψ−ε} D. So replacing ψ by ψ−ε and letting ε 0, we can assume that {ϕ < ψ} D. Then the function ϑ := sup{ϕ, ψ} ∈ P SH(D, ω)∩L^∞(D) coincides withϕnear the boundary ofD. This implies

that

D

(ω+dd^cϑ)^p∧T =

D

(ω+dd^cϕ)^p∧T. (3.5) Indeed, using local regularization ofplurisubharmonic functions, we see that (ω+dd^cϑ)^p∧T−(ω+dd^cϕ)^p∧T =dS,in the sense ofcurrents inD, where

S:=d^c(ϑ−ϕ)

(ω+dd^c ϑ)^p−1+· · ·+ (ω+dd^cϕ)^p−1

∧T is a well defined current with measure coefficients and with compact support inD. Therefore, by definition ofthe differential ofa current, we get

DχdS= 0 for any test function χ which is identically 1 in a neignbourhood ofthe support ofS.

This implies the identity (3.5).

Now by Proposition 3.1, we get

{ϕ<ψ}

ω_ψ^p∧T =

{ϕ<ψ}

ω_ϑ^p∧T.

Then using the identity (3.5) and again Proposition 3.1, we deduce

{ϕ<ψ}

ω_ψ^p ∧T =

D

ω_ϑ^p∧T−

{ϕψ}

ω^p_ϑ∧T

D

ω_ϕ^p∧T −

{ϕ>ψ}

ω_ϑ^p∧T

=

D

ω_ϕ^p∧T −

{ϕ>ψ}

ω_ϕ^p∧T,

which implies

{ϕ<ψ}

ω^p_ψ∧T

{ϕψ}

ω^p_ϕ∧T.

Applying this result toϕ+εandψand lettingε→0, we obtain the required inequality.

To obtain the second inequality, we can assumeϕ, ψ < 0 on D. Now apply the above inequality to ϕ and tψ with 0 < t < 1 and observe that (dd^c(tψ) +ω)ⁿtⁿωⁿ_ψ. Then lettingt→1, we obtain the required inequal-

ity.

Ifm = 0 we setT0 = 1 and for m 1 we set Tm :=ωu1 ∧ · · · ∧ωum, where u1,· · ·um ∈ P⁰(D, ω). Thus Tm is a closed positive current on D.

Then we have the following important result.

Corollary 3.5. — 1) The class P⁰(D, ω) is convex and satisfies the lattice condition:

ϕ∈ P⁰(D, ω), u∈P SH⁻(D, ω) =⇒sup{ϕ, u} ∈P0(D, ω).

2) Let1p, qbe integers such thatp+qnand denote bym:=n−p−q.

Then for any ϕ, ψ∈P0(D, ω),

D

ω^p_ϕ∧ω_ψ^q ∧Tm

D

ω^p+q_ϕ ∧Tm+

D

ω_ψ^p+q∧Tm. (3.6)

3) If ϕ1,· · ·, ϕn ∈ P⁰(D, ω), then

D

ωϕ1∧ · · · ∧ωϕn2ⁿ⁻¹

n

j=1

D

ω_ϕⁿ_j.

Proof. —Let ϕ ∈ P⁰(D, ω) and u ∈ P SH⁻(D, ω) and denote by σ(ϕ, u) := sup{ϕ, u}. Since ϕ σ(ϕ, u) 0, it is clear from the lemma

above that

D

ωⁿ

D

(ω+dd^cσ(ϕ, u))ⁿ

D

ωⁿ_ϕ, which implies thatσ(ϕ, u)∈P0(D, ω).

We first prove the inequality (3.6) whenm= 0,Tm= 1 andp+q=n.

Indeed, by Lemma 3.4 we get

{ϕ+<ψ}

ω_ϕ^p∧ω^q_ψ

{ϕ<ψ}

ω_ϕ^p+q

D

ω_ϕ^p+q. Applying this result withψ= 0 we deduce that

D

ω^p_ϕ∧ω^q

D

ω_ϕ^p+q. (3.7)

In the same way we obtain

{ψ<ϕ}

ω_ϕ^p∧ω^q_ψ

D

ω^p+q_ψ .

Therefore

D

ω^p_ϕ∧ω_ψ^q

D

ω_ϕ^p+q+

D

ω^p+q_ψ , ifwe choose >0 such that

{ψ+=ϕ}ω_ϕ^p∧ω^q_ψ= 0 and letdecrease to 0.

Then the convexity of P⁰(D, ω) follows since for ϕ, ψ ∈ P⁰(D, ω) and 0< t <1, we have

(ω+dd^c(tϕ+ (1−t)ψ))ⁿ =

n

p=0

n

p

t^p(1−t)^n−pω^p_ϕ∧ω_ψ^n−p, which implies by the previous inequality that

D

(ω+dd^c(tϕ+ (1−t)ψ))ⁿ

D

ω_ϕⁿ+

D

ω_ψⁿ,

which is finite and thustϕ+ (1−t)ψ∈ P⁰(D, ω). From this, it follows that for anyϕ1,· · ·, ϕn ∈ P⁰(D), we haveu:= (ϕ1+· · ·+ϕn)/n∈ P⁰(D) and from the last inequality we deduce that

D

(ω+dd^cϕ1)∧ · · · ∧(ω+dd^cϕn)nⁿ

D

(ω+dd^cu)ⁿ<+∞. Therefore we can apply Lemma 3.4 forTm and use the same argument as before to get the inequality (3.6) in the general case.

3.2. Integration by parts formula

To prove the integration by parts formula (IBP) which will be crucial for our considerations, we need a semi-global version of the classical (local) convergence theorem of Bedfod and Taylor for our classP⁰(D, ω).

Proposition 3.6. — Let(ϕ⁰_j),· · ·(ϕⁿ_j)be sequences of locally uniformly bounded ω−plurisubharmonic functions in the class P⁰(D, ω) converging monotonically toϕ⁰,· · ·, ϕⁿ ∈ P⁰(D, ω)respectively. Then the positive cur- rents Sj := (dd^cϕ¹_j+ω)∧ · · · ∧(dd^cϕⁿ_j +ω)and S := (dd^cϕ¹+ω)∧ · · · ∧ (dd^cϕⁿ+ω)have uniformly bounded total masses inD and

j→lim+∞

D

(−ϕ⁰_j)(dd^cϕ¹_j+ω)∧ · · · ∧(dd^cϕⁿ_j +ω) =

D

(−ϕ⁰)(dd^cϕ¹+ω)∧ · · · ∧(dd^cϕⁿ+ω).

Proof. —Observe first that the local theory ofBedford and Taylor im- plies that (−ϕ⁰_j)Sj→(−ϕ⁰)S weakly onD(see [BT2]). It follows from our hypothesis that given ε > 0, there exists an open set D D such that

−εϕ⁰_j 0 and−εϕ⁰0 on D\D. Then

D

(−ϕ⁰_j)Sj−

D

(−ϕ⁰)S=

D

(−ϕ⁰_j)Sj−

D

(−ϕ⁰)S0 + O(ε), (3.8) uniformly in j ∈ N. Here we have used the fact that the currents Sj have uniformly bounded mass on D by Lemma 3.4. Now observe that we can always choose the domain D so that the positive measure µ0 := (−ϕ⁰)S puts no mass on its boundary∂D. Then since the positive measuresµj:=

(−ϕ⁰_j)Sj converge weakly toµ0 inD, it follows that µ0(D)lim inf

j µj(D)lim sup

j µj(D)µ0(D) =µ0(D),

which proves that the first integral on the right hand side converges to 0 and the proposition is proved.

Now we can prove the following integration by parts formula which will be useful in the sequel.

Lemma 3.7. — LetT := (ω+dd^cu1)∧ · · · ∧(ω+dd^cun−1), whereu1,· · · un−1∈ P⁰(D, ω). Let u, v∈ P⁰(D, ω). Then

D

udd^cv∧T =

D

vdd^cu∧T, (3.9)

and

D

uωv∧T−

D

vωu∧T =

D

(u−v)ω∧T. (3.10) Proof. —Denote byH(u, v) :=udd^cv∧T−vdd^cu∧T. Then by Propo- sition 3.1 the current H(u, v) has finite total mass in D. It follows from Stokes formula that if ¯u,v¯ are boundedω−plurisubharmonic functions on D such that ¯u=uand ¯v=v near the boundary∂Dthen

D

H(¯u,v) =¯

D

H(u, v).

Indeed observe that since u, v,u,¯ v¯ are boundedω−quasiplurisubharmonic functions onD, it follows from the local theory that the currentsS:=ud^cv∧ T−vd^cu∧T and ¯S:=:= ¯ud^cv¯∧T−vd¯ ^cu¯∧T are well defined currents with measure coefficients onD such thatdS=udd^cv∧T−vdd^cu∧T =H(u, v) anddS¯= ¯udd^c¯v∧T−vdd¯ ^cu¯∧T =H(¯u,v) in the weak sense ofcurrents on¯ D.

Now sinceS−S¯is ofcompact support inD, it follows that

Dd(S−S) = 0¯

and then

D

H(u, v) =

D

H(¯u,¯v).

Now forε >0 small enough, setuε:= sup{u, v−ε}andvε:= sup{v, u−ε} and observe thatuε=uandvε=vnear∂D. Thus by the previous remark, we have for ε >0 small enough

D

H(uε, vε) =

D

H(u, v). (3.11)

We want to pass to the limit. Here we must use the fact thatu=v= 0 on

∂D, which implies thatuε =vε= 0 on ∂D. Now for ε > 0 small enough, we have

H(uε, vε) =uεdd^cvε∧T−vεdd^cuε∧T.

Sinceuεg:= max{u, v}andvεg, it follows from Proposition 3.1 that

εlim→0

D

uεdd^cvε∧T =

D

gdd^cg∧T = lim

ε→0

D

vεdd^cuε∧T,

which implies the required integration by parts formula.

4. Subextension of quasi-plurisubharmonic functions 4.1. WeightedMonge-Amp`ere energy classes

In the contrast to the local case, the domain ofdefinition ofthe com- plex Monge-Amp`ere operator is not well understood in the global case.

Interesting classes have been investigated in [GZ2] and [CGZ]. We are go- ing to introduce similar classes in the semi-global case where the complex Monge-Amp`ere operator is well defined and continuous under deacreasing sequences. The first class is modeled on the class defined by Cegrell in ([Ce2]) as follows.

Definition 4.1. — We say thatϕ∈ F(D, ω)if there exists a decreasing sequence(ϕj)from the classP⁰(D, ω)which converges toϕonD such that

sup

j

D

ω_ϕⁿ_j <+∞.

Observe thatF(D, ω) is a convex set andP⁰(D, ω)⊂ F(D, ω). The class F(D, ω) is the counterpart ofthe class defined by Cegrell in [Ce2]. LetDbe a hyperconvex domain where the formωhas a plurisubharmonic potentialq onD with boundary values 0 and letF(D) the class defined in [Ce2]. Then ifϕ∈ F(D, ω) iffu:=ϕ+q∈ F(D).

We do not know at the moment ifthe Monge-Amp`ere operator is well defined on the classF(D, ω) but we can define the Monge-Amp`ere mass of a functionϕ∈ F(D, ω) thanks to the following lemma.

Lemma 4.2. — Letϕ∈ F(D, ω)be a fixed function. Then the constant MD(ϕ) := lim

j

D

(ω+dd^cϕj)ⁿ= sup

j

D

(ω+dd^cϕj)ⁿ

is independant of the decreasing sequence (ϕj) from P⁰(D, ω) converging toϕ.

Moreover ifψ∈P SH(D, ω)andϕψ0 thenψ∈ F(D, ω).

Proof. —Take a defining sequence (ϕj)j forϕ. By Lemma 3.4 we know that the sequence {

D(ω+dd^cϕj)ⁿ}^j is increasing and by definition it is bounded so the limitMD(ϕ) exists. We only need to show that it does not depend on the sequence. Let (ψj) another decreasing sequence offunctions in the classP⁰(D, ω) converging toϕinD. Fixε >0 andj. Since by Bedford- Taylor continuity theorem ([BT2]), (ω+dd^csup{ψj, ϕk})ⁿ→(ω+dd^cψj)ⁿ weakly onD ask→ ∞, it follows that there existskj such that

D

(ω+dd^csup{ψj, ϕkj})ⁿ >

D

(ω+dd^cψj)ⁿ−ε.

By Lemma 3.4, we have

D

(ω+dd^csup{ψj, ϕkj})ⁿ

D

(ω+dd^cϕkj)ⁿ MD(ϕ).

Therefore it follows that

D(ω+dd^cψj)ⁿ−εMD(ϕ), which implies that sup_j

D(ω+dd^cψj)ⁿMD(ϕ) and proves the first part ofthe lemma.

Now setψj:=sup{ψ, ϕj}. Then by Lemma 3.4,ψj ∈ P⁰(D) and

D(ω+ dd^cψj)ⁿ

D(ω+dd^cϕj)ⁿMD(ϕ). Since (ψj) decreases toψ, it follows that ψ ∈ F(D, ω) and from the first part of the proof we deduce that MD(ψ)MD(ϕ).

Let us introduce the following classes of finite weighted Monge-Amp`ere energy (see [Ce1], [GZ2], [BGZ]). A weight function is by definition an in- creasing functionχ:R−→Rsuch thatχ(t) =tist0 andχ(−∞) =−∞. To any weight function we associate the classE^χ(D, ω) of ofω−plurisubhar- monic functions ϕ∈P SH(D, ω) for which there exists a sequence (ϕj)∈ P⁰(D, ω),ϕj ϕsuch that

sup

j

D|χ(ϕj)|ωⁿ_ϕj <+∞.

In our case the weight functionχwill be convex. From the (IBP) formula, we can derive the following fundamental inequality which will be useful (see [GZ2]).

Proposition 4.3 Let χ : R−→ R be a convex weight function. Then for any ϕ, ψ∈ P⁰(D, ω)withϕψ, we have

D|χ(ψ)|ω_ψⁿ2ⁿ

D|χ(ϕ)|ω_ϕⁿ. (4.1)

We can prove that the complex Monge-Amp`ere operator is well defined and continuous on decreasing sequences in the classE^χ(D, ω),whereχ is a convex increasing functionsR−→R(see [GZ2], [CGZ]).

Proposition 4.4. — The complex Monge-Amp`ere operator is well defined on the class E<sup>χ</sup>(D, ω). Moreover if (ϕj) is a decreasing sequence from the class E<sup>χ</sup>(D, ω)which converges toϕ∈ E<sup>χ</sup>(D, ω), then the Monge- Amp`ere measures (ωⁿ_ϕj) converge to ω_ϕⁿ weakly on D. Moreover for any h∈P SH(D, ω)∩L^∞(D)

limj

D

hωⁿ_ϕj =

D

hω_ϕⁿ.

Using the integration by parts formula, the fundamental inequality and following the same arguments as [GZ2], it is possible to prove the following result.

Proposition 4.5. — Let ϕ ∈ PSH(D, ω). Assume there exists a de- creasing sequence (ϕ)j∈N in P⁰(D, ω) which converges to ϕ∈ P SH(D, ω) and satisfiessup_j

D|χ(ϕj)|ωⁿ_ϕj <+∞. Thenϕ∈ E^χ(D, ω) and

j→lim+∞

D|χ(ϕj)|ωⁿ_ϕj =

D|χ(ϕ)|ωⁿ_ϕ.

4.2. A general subextension theorem

We now prove the following general subextension result which generalizes our previous result with a new proof(see [CKZ]).

Theorem 4.6. — LetD⊂X be a quasi-hyperconvex domain satisfying the condition(3.4). Letϕ∈ F(D, ω)such thatMD(ϕ)

Xωⁿ. Then there exists a function ϕ∈P SH(X, ω)such that ϕϕonD.

Proof. —Let (ϕj) be a decreasing sequence from the class P⁰(D, ω) which converges toϕonD. By Lemma 4.2 we have

D

(ω+dd^cϕj)ⁿ MD(ϕ).

First assume that MD(ϕ) <

Xωⁿ. Then by [GZ2] there exists uj ∈ E¹(X, ω) with sup_Xuj=−1 such that

(ω+dd^cuj)ⁿ =1D(ω+dd^cϕj)ⁿ+εjωⁿ

onX, whereεj >0 is chosen so that the total mass ofboth sides are equal.

Fixj ∈N. Since{ϕj < uj}:={x∈D;ϕj < uj}D,andϕj is bounded, it follows that for s > 1 large enough, {ϕj < u^s_j} = {ϕj < uj} D, where u^s_j := sup{uj,−s}. Then by the comparison principle (Lemma 3.4), it follows that

{ϕj<u^s_j}

(ω+dd^cu^s_j)ⁿ

{ϕj<u^s_j}

(ω+dd^cϕj)ⁿ.

Recall that 1_{uj>−s}(ω +dd^cu^s_j)ⁿ = 1_{uj>−s}(ω +dd^cuj)ⁿ (see [GZ2]).

Therefore

{ϕj<uj}

(ω+dd^cuj)ⁿ

{ϕj<uj}

(ω+dd^cϕj)ⁿ,

which implies that V olω({ϕj < uj}) = 0 and then uj ϕj onD. Due to the normalization ofuj, the functionu:= (lim sup_j→+∞uj)^∗∈P SH(X, ω) and satisfiesuϕonD and maxXu=−1 (see [GZ1]).

Now assumeϕ∈ F(D, ω) withMD(ϕ) =

Xωⁿ and consider a decreas- ing sequence (ϕj) in P⁰(D, ω) converging to ϕ with uniformly bounded Monge-Amp`ere masses. Then it follows that for any 0< t <1 the function tϕj ∈ P⁰(D, ω) and

D(ω+dd^ctϕj)ⁿ =

D(tωϕj + (1−t)ω)ⁿ. By Lemma 3.4 we have

Dω^p_ϕj ∧ω^n−p

Dω_ϕⁿ_j. Therefore since

Dωⁿ <

Xωⁿ, it follows thatMD(tϕj) =

D(ω+dd^ctϕj)ⁿ <

Xωⁿ. By the first part we can find a subextensionψ^t_j∈P SH(X, ω) oftϕj toX satisying maxXψ^t_j=−1.

Therefore the function ψj := (lim sup_{t 1}ψjt)^∗ is an ω−plurisubharmonic subextension of ϕj to X with maxXψj =−1 . Now observe that, as be- fore,ψ:= (lim sup_j→+∞ψj)^∗∈P SH(X, ω) and satisfies maxXψ=−1 and ψϕonD.

It follows from the above theorem that given ϕ ∈ F(D, ω) such that MD(ϕ)

Xωⁿ, the following function

ϕ=ϕD:= sup{ψ∈P SH(X, ω);ψϕonD}

is a well defined ω−plurisubharmonic function on X and will be called the maximal subextension ofϕfromD toX.

The example below shows that in general the maximal subextension does not belong to the global domain ofdefinition ofthe complex Monge- Amp`ere operator on X since it may have positive Lelong number along a hypersurface.

However ifthe given function has a finite weighted Monge-Amp`ere en- ergy in the sense of[GZ2], we will prove that the maximal subextension satisfies the same property.

Theorem 4.7. — LetD⊂X be an quasi-hyperconvex domain satisfying the condition(3.4)and letϕ∈ E^χ(D, ω)be such that

Dω_ϕⁿ

Xωⁿ, where χ:R−→R is a convex weight function. Then the maximal subextensionϕ˜ of ϕfromD toX exists and has the following properties:

(i)ϕ∈ E^χ(X, ω) and

X|χ◦ϕ|(ω+dd^cϕ)ⁿ

D|χ◦ϕ|(ω+dd^cϕ)ⁿ, (ii)1D(ω+dd^cϕ)ⁿ1D(ω+dd^cϕ)ⁿ holds in the sense of measures onX, (iii) the measure(ω+dd^cϕ)ⁿ is carried by the Borel set{ϕ=ϕ} ∪∂D.

We will need the following lemma which can be proved using the argu- ment from the first part ofthe proofofTheorem 2.1.

Lemma 4.8. — LetDbe as above andϕ∈ P⁰(D, ω)be such that

Dω_ϕⁿ

Xωⁿ, thenϕ∈P SH(X, ω)∩L^∞(X)and1D(ω+dd^cϕ)ⁿ1D(ω+dd^cϕ)ⁿ in the sense of measures onX. Moreover the measure(ω+dd^cϕ)ⁿis carried by the Borel set{x∈D; ˜¯ ϕ(x) =ϕ(x)}.

Proof of the theorem. — Let (ϕj) a sequence (ϕj) ∈ P⁰(D, ω) which decreases toϕonD. Define ˜ϕj to be the maximal subextension ofϕj from D to X. Then by the previous lemma ˜ϕj ∈ P SH(X, ω)∩L^∞(X) and (ω+dd^cϕj)ⁿ is supported on the contact set {x ∈ D¯ : ˜ϕj(x) = ϕj(x)}. Hence (−χ◦ϕ˜j)(ω+dd^cϕ˜j)ⁿ 1D(−χ◦ϕj)(ω+dd^cϕj)ⁿ in the sense of measures onX. Therefore there is a uniform constant C >0 such that for anyj∈N,

X

(−χ◦ϕ˜j)(ω+dd^cϕ˜j)ⁿ

D

(−χ◦ϕj)(ω+dd^cϕj)ⁿC.

Since ( ˜ϕj) ϕ on X it follows from [GZ2] that ϕ ∈ E^χ(X, ω). Moreover by the convergence theorem ([GZ2], [CGZ]) it follows that 1D|χ◦ϕ˜|(ω+ dd^cϕ)˜ ⁿ 1D|χ◦ϕ|(ω+dd^cϕ)ⁿ in the sense ofmeasures onX.

The third part ofthe theorem is proved along the same lines as the last part ofthe proofofTheorem 2.1 using Lemma 4.8 and Proposition 4.2.

Remark 4.9. — In contrast to the local case it may happen that a part ofthe Monge-Amp`ere measure of˜ϕlives on the boundary ofD.

As we already said before, the example in the last section shows that the maximal subextension ofa given functionϕ∈ F(D, ω) may have not a well defined Monge-Amp`ere measure. However the following property may be useful.

Proposition 4.10. — Let ϕ ∈ F(D, ω) be a given function. Then if (ϕj)is a decreasing sequence of functions in the classP⁰(D, ω)converging to ϕthen the sequence(ϕj)decreases to ϕonX. Moreover any Borel measure µon X which is a limit point of the sequence of measures(ω+dd^cϕj)ⁿ on X satisfies the inequality1Dµ1D(ω+dd^cϕ)ⁿ in the sense of measures on X.

Proof. —Observe that for eachj ∈N,ϕis a global subextension ofϕj

to X and thenϕ ϕj on X. Therefore it is clear that the sequence (ϕj) decreases to an ω−plurisubharmonic function ψ on X which satisties the inequalityϕψonX. This shows thatψ∈ PSH(X, ω). On the other hand sinceψϕj ϕj onDwe infer thatψϕonD, which proves thatψis a subextension ofϕtoX and thenψϕonD. We conclude thatψ=ϕon X. We know from the last lemma that1D(ω+dd^cϕj)ⁿ1D(ω+dd^cϕj)ⁿ in the sense ofmeasures on X, which implies the last statement ofthe proposition.

4.3. Subextension in Cⁿ

Now we pass to subextensions from a hyperconvex domain D Cⁿ to Cⁿ,considered as an open subset ofthe complex projective spacePⁿ. Recall that the Lelong class is defined by

L(Cⁿ) :={u∈P SH(Cⁿ); sup{u(z)−log⁺|z|<+∞}.

Letω=ωF S be the normalized Fubini-Study metric onPⁿdefiend in affine coordinates by

ω:=dd^clog|ζ|,

where ζ:= [ζ0:· · ·:ζn] are the homogeneous coordinates onPⁿ. As usual we will considerCⁿ =Pⁿ\{ζ0= 0}with the affine coordinates defined as by zj :=ζj/ζ0(1jn). With these notations we have ω|Cⁿ =dd^c6, where 6(z) := (1/2) log(1 +|z|²). Therefore given any u ∈ L(Cⁿ), the function defined by

ϕ(ζ) :=u(z)−(1/2) log(1 +|z|²), ζ0= 0

is ω−plurisubharmonic on Pⁿ\ {ζ0 = 0} and locally upper bounded in a neighbourhood ofthe hyperplane at infinity H_∞ := {ζ0 = 0} so that it extends to anω−plurisubharmonic function onPⁿwhich we also denote by ϕ. It follows that the correspondanceu−→ϕis a bijection betweenL(Cⁿ) andP SH(Pⁿ, ω) such thatω+dd^cϕ=dd^cuonCⁿ.

From the last theorem we can deduce a generalization ofour earlier result (see [CKZ], Theorem 5.3).

Theorem 4.11. — Let D Cⁿ be a hyperconvex domain and let u ∈ F(D)be such that(dd^cu)ⁿdoes not put any mass on pluripolar sets inDand

D(dd^cu)ⁿ 1. Then its maximal subextension u˜ from D to Cⁿ belongs to L(Cⁿ)and has a well defined global Monge-Amp`ere measure(dd^cu)˜ ⁿ which is carried by the set{u˜=u} ∪∂D and satsifies the inequality1D(dd^cu)˜ ⁿ 1D(dd^cu)ⁿ.

Proof. —Assume first that D=BR is an euclidean ball with center at the origin and radiusR > 0. Then the function q:= (1/2) log(1 +|z|²)− (1/2)log(1 +R²) is a potential ofthe normalized Fubini-Study form ω on Cⁿ which vanishes on ∂D. In this case ϕ := u−q ∈ F(D, ω). From our hypothesis (ω+dd^cϕ)ⁿ({ϕ=−∞}) = (dd^cu)ⁿ({u=−∞}) = 0. It follows from standard fact in measure theory that there exists a convex inceasing functionχ:]− ∞,0]−→]− ∞,0] such that

D(−χ◦ϕ)(ω+dd^cϕ)ⁿ <+∞ (see [GZ 2]. It easily follows that ϕ∈ E^χ(D, ω) and then we can apply the last result to find a subextension ˜ϕ∈ E(Pⁿ, ω) ofϕtoPⁿ. Then ˜u:= ˜ϕ+q is the maximal subextension ofuto Cⁿ.

Now in the general case consider an euclidean ballB such thatD ⊂B and use Theorem 2.1 to produce a subextension v ∈ F(B) of u. Then by the previous casevhas a subextension ˜v such thatψ:= ˜v−qis a function in E(Pn, ω) which is a subextension ofϕ:=u−qfromD to Pⁿ. Therefore the maximal subextension ˜ϕ of ϕ exists and since ψ ϕ˜ it follows that

˜

ϕ∈ E(Pⁿ, ω). Thus ˜u:=ϕ+q∈ L(Cⁿ) is the maximal subextension ofuto Cⁿ. The other properties follow in the same way as in the proofofTheorem

4.8.

Now we consider an arbitrary functionu∈ F(D) and a positiveγsatis- fying

γⁿ

D

(dd^cu)ⁿ.

Then from Theorem 4.6 the set ofentire subextensions oflogarithmic growth {v∈P SH(Cⁿ);v|^Du, v(z)av+γlog⁺|z|}

is not empty. Thus, using notation

L^γ(Cⁿ) ={v∈P SH(Cⁿ);v(z)av+γlog⁺|z|}

one can choose the maximal subextension ofuoflogarithmic growth related toγ

˜

uγ = sup{v∈ L^γ(Cⁿ);v|^Du}.

As we shall see the Monge-Amp`ere measure ofthis subextension may not exist. Ifit exists however, one can deduce some information on the support ofsuch measure.

Define

Nu={z∈Cⁿ; ˜uγ <0}.

Proposition 4.12. — Assume that u∈ F(D) and let γⁿ =

D

(dd^cu)ⁿ. Then for any sequenceuj∈ E⁰(D)∩C( ¯D),decreasing touifµis an accumu- lation point of (dd^cu˜j,γ)ⁿ thenµ=f(dd^cu)ⁿ+ν where0f 1 is a func- tion vanishing outsideD and whereν is a positive measure,suppν ⊂∂Nu. Proof. —Assume first thatu ∈ E⁰(D)∩C( ¯D). Then ˜uγ is continuous and the zero sublevel set of˜uγ, Nu is hyperconvex.

By definition,D⊂Nuand by Theorem 5.1 in [CKZ]Dis not relatively compact inNu. There are two possibilities:

1)D=Nu.

2)D=Nu⊂⊂Cⁿ.

If1) occurs then ˜uγ extends uto a function inL^γ∩L^∞_locand 1Nu(dd^cu˜γ)ⁿ=1D(dd^cu˜γ)ⁿ=1D(dd^cu)ⁿ. In particular, ifγⁿ =

D

(dd^cu)ⁿ then (dd^cu˜γ)ⁿ=1D(dd^cu)ⁿ onCⁿ.

Generically we have 2). Then on Nu, ˜uγ is equal to ˜u, the maximal local subextension of ufrom D to Nu. Consider Dj ⊂⊂ Dj+1 ⊂⊂ D an exhaustion sequence of D.Denote by ˜uj the corresponding local maximal subextension toNuofthe solutionuj∈ E⁰(D) to (dd^cuj)ⁿ=1D_j−1(dd^cu)ⁿ. Then ˜uu˜j and (dd^cu˜j)ⁿ 1D_j−1(dd^cu)ⁿ on Nu by Theorem 2.1 and so (dd^cu)˜ ⁿ1D(dd^cu)ⁿ onNu.

Therefore, (dd^cu˜γ)ⁿ = f(dd^cu)ⁿ +ν where 0 f 1 is a function vanishing outsideDand whereν is a positive measure, suppν⊂∂Nu∩∂D.

Now consider the general case. Choose a decreasing sequence (uj) in E⁰(D)∩C( ¯D),decreasing tou.Then ˜uj,γ decreases to ˜uγ and (dd^cu˜j,γ)ⁿ = fj(dd^cuj)ⁿ+νj where 0 fj 1 is a function vanishing outside D and where νj is a positive measure, suppνj⊂∂Nuj.Also

(dd^cu˜j,γ)ⁿ =γⁿ.So ifµis any weak limit of(dd^cu˜j,γ)ⁿ,thenµ=f(dd^cu)ⁿ+ν where 0f 1 is a function vanishing outsideD and whereν is a positive measure carried by∂Nu.

Corollary 4.13. — If, for u ∈ F(D), the set Nu is bounded then the Monge-Amp`ere measure of uγ is well defined and equal to the limit of (dd^cu˜j,γ)ⁿ.

IfNuis not a bounded hyperconvex set,uγneed not to be in the domain ofdefinition ofthe Monge-Amp`ere operator. This is shown in the following example.

Example 4.14. — The maximal entire subextension of a function from the classF(B) may not have well defined global Monge-Amp`ere measure on C².

Consider the Green functiong in the ballB(0,2)⊂C² with two poles at (−1,0) and (1,0) ofweight ^√1₂ each. Then

B^(0,2)(dd^cg)²= 1.

So there exists the maximal entire subextension ˜g = ˜gt in the Lelong class L^t(C²),1t <√

2.Note that √¹

2log||^z2²||is a subextension. By the defini- tion of the Green function we have for someR∈(0,1), A >0 the following inequalities

|g(z)−^√1₂log||(z1+ 1, z2)|| |< A in B((−1,0), R)

|g(z)−^√1₂log||(z1−1, z2)|| |< A in B((1,0), R).

Let 0< r < ₁₆^R be fixed and letz2=wbe fixed with 0<|w|< r.

Consider the restriction ˜g: ˜gw(z) = ˜g(z, w). If |z−1|r or|z+ 1|r then||(z, w)||<2 so ˜g(z, w)0 on{|z−1|r} and{|z+ 1|r}.

If−∞ ≡g˜w ∈ L^t(C) one concludes that the total mass of _2π¹ ∆˜gw does not exceedt.By symmerty one can assume that

B^(1,R)∆˜gwt/2. (4.2)

(Otherwise considerB(−1, R) in place ofB(1, R).) If|z−1||w|we then have

˜

gw(z) = ˜g(z, w) 1

√2log||(z−1, w)||+A 1

√2log|w|+A+ 1. (4.3)

Letzbe any point on{|w|<|z−1|r}. Denote byB¹the diskB(z,2r) and by B² the diskB(1, r). ThenB²⊂B¹ ⊂B(1, R). IfJ(K) denotes the average value of˜gwover a setK⊂Cthen

˜

gw(z)J(B1) 1

4J(B2). (4.4)

Since J(B²) is dominated by the average of˜gw over the boundary of B² one obtains from Riesz representation formula, using that ˜gw0 and (4.2), (4.3) :

J(B²) max_B(1,|w|)g˜w−

{|w|<|x−1|<r}log|x−1|∆˜gw

^√1₂log|w|+A+ 1−2^tlog|w| (√¹

2−2^t) log|w|+A+ 1.

Therefore

˜

g(z, w)

√2−t

8 log|w|+A+ 1

for ||(z−1, w)||< r. Since the Monge-Amp`ere operator cannot be defined forv(z, w) = log|w|it follows that the same goes for the function ˜g.

Remark 4.15. — The above example relies on a geometrical effect which is also responsible for nonexistence of solutions to the Monge-Amp`ere equa- tions inCPⁿ where we have on the right hand side a generic combination ofDirac measures (cf. [Co]).

Remark 4.16. — It follows from [S] that ˜g(z, w)−^√2−t8 log|w|is plurisub- harmonic on C².For an elementary proof, see [Ce4].

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