L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

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L’INSTITUT FOURIER

Quang Dieu NGUYEN & Dau Hoang HUNG

Jensen measures and unboundedB−regular domains inCⁿ Tome 58, n^o4 (2008), p. 1383-1406.

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JENSEN MEASURES AND UNBOUNDED B−REGULAR DOMAINS IN C

by Quang Dieu NGUYEN & Dau Hoang HUNG

Abstract. — Following Sibony, we say that a bounded domain Ω inCⁿ is B-regular if every continuous real valued function on the boundary ofΩcan be extended continuously to a plurisubharmonic function onΩ. The aim of this paper is to study an analogue of this concept in the category of unbounded domains in Cⁿ. The use of Jensen measures relative to classes of plurisubharmonic functions plays a key role in our work

Résumé. — En suivant Sibony, nous dirons qu’un domaine borneΩdeCⁿest B−régulier si toute fonction continue à valeurs réelles sur la frontière deΩpeut être prolongée continûment à une fonction plurisousharmonique surΩ. Le but de ce papier est d’étudier une notion analogue dans la catégorie des domaines non bornés dansCⁿ. L’usage des mesures de Jensen relatives à des classes de fonctions plurisousharmoniques jouent un rôle clé dans notre travail.

1. Introduction

Let Ω ⊂ Cⁿ be an open set. An upper semicontinuous function u : Ω → [−∞,∞) is said to be plurisubharmonic if the restriction of u on the intersection ofΩwith every complex line is subharmonic (the function identically−∞is considered to be plurisubharmonic). ByPSH(Ω) we de- note the set of plurisubharmonic functions onΩand byPSH^c(Ω)we mean the set of continuous functions onΩwhich are plurisubharmonic onΩ. We say thatu∈PSH(Ω)ismaximalif for every relatively compact subdomain Ω⁰ of Ωand every v ∈PSH(Ω)satisfyingv 6uon ∂Ω we havev 6uon Ω⁰. Denote byMPSH(Ω)the class of maximal plurisubharmonic functions

Keywords: Plurisubharmonic function, Dirichlet-Bremermann problem, B-regular domain.

Math. classification:32T27.

onΩ. It is well known that in pluripotential theory maximal plurisubhar- monic functions play the role as harmonic functions do in classical potential theory.

In this paper, we concern with the following problem: Givenh∈ C(∂Ω), the space of real valued continuous function ∂Ω, we study conditions on Ωandh to ensure the existence of u∈MPSH(Ω) which is continuous on Ωsuch that u= h on ∂Ω. This problem, which is usually referred to as the Dirichlet-Bremermann problem, in the case Ω is bounded, has been initiated in a classical work of Bremermann (cf. [3]). Bremermann proved among other things that if Ω is bounded and h ∈ C(∂Ω) then there is at most one continuous function ϕ on Ω with boundary data h which is maximal plurisubharmonic onΩ. This solution, if exists, is given by (1.1) ϕh,Ω(z) = sup{u(z) :u∈PSH(Ω), u^∗6hon∂Ω}, whereu^∗ is theupper regularizationofuwhich is defined onΩas

u^∗(z) = lim sup

x→z

u(x)∀z∈Ω.

Notice that ifΩisregular in the real sense i.e.,for everyh∈ C(∂Ω)there isH ∈C(Ω)harmonic onΩsuch thatH =hon∂Ω,thenϕh,Ω∈PSH(Ω).

On the other hand, the continuity ofϕ_h,Ω is more delicate, Bremermann showed that if Ω is strictly pseudoconvex (e.g., Ω is an open ball) then ϕ_h,Ω is continuous at every boundary point of Ω and coincides with h there. Later on, Walsh proved in [16] that ϕh,Ω is indeed continuous on Ω. He also gives an example where ϕ_h,Ω is not continuous. More recently, in [14] Sibony characterizes bounded domains Ωin Cⁿ such that ϕh,Ω is continuous on Ωand satisfies ϕh,Ω =hon∂Ω for everyh∈ C(∂Ω). This class of domains is calledB−regulardomains. It should be remarked that the class of B− regular domains properly contains the class of strictly pseudoconvex domains.

The Dirichlet-Bremermann problem forunboundeddomainsΩwas stud- ied only recently in [13] and [15]. In [13], the authors pointed out thatϕ_h,Ω may be identically−∞ even ifh ∈ C(∂Ω)and Ω is a strictly convex do- main with smooth boundary (cf. [13] and [15]). On the other hand, it is shown in Proposition 19 in [15] that ifh∈ C(∂Ω)is bounded and ifΩis an unbounded strictly convex domain withC² smooth boundary, then there existsu∈ C(Ω), maximal plurisubharmonic onΩsuch thatu=hon∂Ω.

In Theorem 4.4 and Proposition 5.1 we generalize this nice result to the caseΩis onlylocallyB−regulari.e.,for every z∈∂Ωthere is a bounded neighbourhoodU ofzsuch thatU∩ΩisB−regular.

The principal tool in this paper is the use of Jensen measures relative to classes of plurisubharmonic functions. Roughly speaking, we express upper envelopes of type (1) in terms of lower envelopes of integral relative to different classes of Jensen measures. Then equality of these classes of measures will give continuity of the corresponding upper envelopes. This approach has been used in [17], [11] and [9]. During the course of our development, we also obtain in Theorem 4.1 a global approximation result for bounded plurisubharmonic functions on unbounded domains.

Acknowledgment. This work is essentially done during the first named author’s stay at the Abdus Salam International Center Theoretical Physic, Trieste, Italy in the Summer of 2007 and finally completed during his visit to the Department of Mathematics, Seoul National University in the Fall of 2007 as a post doc fellow under the Korea Research Foundation Grant 2005-070-C00007. He would like to thank Professor Le Dung Trang (ICTP) and Professor Chong-Kyu Han (SNU) for the kind invitations and the staffs of these institutions for their warm hospitality. We also like to thank the referee for fruitful comments, especially for drawing our attention to a gap in the proof of (an earlier version of) Proposition 2.7. Last but not least, we are indebted to Professor Giuseppe Tomassini for his interest in our work and stimulating talks on the article [15].

2. Necessary Background

We first fix notation and terminology that will be used in this paper. The set of (real valued) continuous functions on a subsetX of Cⁿ is denoted byC(X). Denote byC0(X)the set of functionsu∈ C(X)that vanishes at infinity i.e., for everyε > 0 there is a compactK ⊂X such that |u| < ε onX\K.We writeUSC_∗(X)for the set of bounded upper semicontinuous functionsuonX that is non positive at infinityi.e.,for everyε >0, there is a compactK ⊂X such thatu < ε onX\K. We also denote by Cc(X) the subset ofC(X)consisting functions of compact support.

Given a subset AofUSC_∗(X)andz∈X we denote byJ_z(A)the class of of positive regular Borel measuresµonX satisfyingµ(X)61 and

u(z)6 Z

X

udµ∀u∈ A.

It is customary to call Jz(A) the class of Jensen measures relative to A with barycentre atz.

Now we recall the following result of Sibony (cf.Theorem 2.1 in [14]).

Theorem 2.1. — LetΩbe a bounded domain inCⁿ. Then the following conditions are equivalent.

(a) For everyh∈ C(∂Ω)there exists u∈ MPSH(Ω)∩ C(Ω) satisfying u=hon∂Ω.

(b) For every z ∈ ∂Ω, there exists u ∈ PSH^c(Ω) satisfying u(z) = 0 whileu <0elsewhere.

(c) There exists ϕ ∈ C²(Ω)∩PSH(Ω) and λ > 0 such that {ϕ < c}

is relatively compact in Ω for every c < 0 and ϕ(z)−λ|z|² is plurisubharmonic onΩ.

(d) For every h ∈ C(∂Ω) there exists u ∈ PSH(Ω)∩ C(Ω) satisfying u=hon∂Ω.

(e) For everyz∈∂Ω, there exists a negative functionu∈PSH(Ω)such thatu^∗<0 on∂Ω\{z}whilelim_x→zu(x) = 0.

Following Sibony, a bounded domain Ωin Cⁿ is calledB−regular if Ω satisfies one of above equivalent conditions. For more details on bounded B−regular domains, the reader is invited to the original article [14] (see also [2] and [10] for more recent developments).

We now extend some notions introduced at the beginning of the article to the context ofunboundeddomains.

Definition 2.2. — An unbounded domainΩ⊂Cⁿis calledB−regular if for every bounded functionh∈ C(∂Ω)there is a bounded functionu∈ MPSH(Ω)∩ C(Ω)such thatu=hon∂Ω.

By considering the functions h_ξ(z) := max(−|z−ξ|,−1) for ξ ∈ ∂Ω we deduce that if an unbounded domainΩisB−regular and has an iso- lated boundary point then it is not locally B−regular. Recall that Ω is said to be locallyB−regular if for everyξ ∈ ∂Ω, there exists a bounded neighbourhoodU ofξsuch that U∩ΩisB−regular. Notice also that the restriction on boundedness ofhin Definition 2.2 seems natural, in view of the mentioned above example due to Shcherbina and Tomassini. However, it is unclear that a result analogous to Theorem 2.1 still holds in the case whereΩis unbounded.

Definition 2.3. — An unbounded domain Ω is called regular in the real sense if for everyz∈∂Ω, there is an open neighbourhoodz∈U such thatU∩Ωis regular in the classical sense.

Observe that Ωis regular in the real sense if and only if every z ∈∂Ω is aregular boundary point of Ωi.e.,there exists a subharmonic function uonΩsatisfyingu^∗<0 on∂Ω\{z} while limx→zu(x) = 0. To check the

regularity ofz ∈ ∂Ω we frequently appeal to Theorem 2.11 in [7] which asserts that if n > 2 and if there exists a real cone lying in Cⁿ\Ω with vertex atz,thenz is a regular boundary point ofΩ. It is also easy to see that every (unbounded) locally B−regular domain is regular in the real sense but the reverse implication does not hold in general.

Throughout this paper, unless otherwise specified, byΩwe always mean anunbounded domain inCⁿ.

The simple fact below explains the role of regular domains in the real sense in the study of Dirichlet-Bremermann problem.

Lemma 2.4. — Assume thatΩis regular in the real sense. Leth∈ C(∂Ω) be a bounded function. Define˜h=hon∂Ωand˜h=M := sup_∂ΩhonΩ.

Set

ϕ(z) := sup{u(z) :u∈PSH(Ω), u^∗6h˜ onΩ}.

Thenϕ∈PSH(Ω)andϕ^∗6˜honΩ.

Proof. — Clearly ϕ^∗ 6 M on Ω, so ϕ = ϕ^∗ ∈ PSH(Ω). It remains to showϕ^∗ 6hon∂Ω. Fix z∈∂Ω. Choose an open ballB ⊂Cⁿ such that z∈B. LetΩ⁰ := Ω∩B. ThenΩ⁰ is bounded regular in the real sense. Let H be the solution to the (classical) Dirichlet problem onΩ⁰ with boundary datah. It follows that˜ ϕ6H onΩ. Sincezis a regular boundary point of Ω⁰ andh is continuous nearz, we inferϕ^∗(z)6H^∗(z) =h(z).The proof

is complete.

The following classical fact due to Choquet(cf.[8], Lemma 2.3.4) is very useful while working with upper envelopes .

Choquet’s lemma. — Let{u_α}_α∈Abe a family of functions which are locally bounded from above on a subsetXofCⁿ. Then there is a countable subfamily{αj}j>1⊂Asuch that

(sup{uα:α∈A})^∗= (sup{uα_j :j>1})^∗.

Furthermore, if uα is lower semicontinuous for every α∈ A, then we can choose{αj}j>1such that

sup{uα:α∈A}= sup{uα_j :j>1}.

We also need some elements from pluripotential theory. For more com- plete treatments on this subject we refer the reader to the excellent ac- counts [1] and [8]. First recall that maximality forlocally bounded pluri- subharmonic function is alocal property, and if {u_j}_j>1 is a sequence in MPSH(Ω)that converges monotonically almost everywhere tou∈PSH(Ω)

thenu∈MPSH(Ω). These properties are applications of the highly nontriv- ial theory of the complex Monge-Ampere operator developed by Bedford and Taylor. Next, a subsetF in Cⁿ is calledpluripolar if for everyz∈Ω there exists an open neighbourhoodU of z and u∈PSH(U), u6≡ −∞ on any connected component ofU such that u=−∞onF∩U.A basic the- orem of Josefson states that for every pluripolar setF in Cⁿ we can find u ∈ PSH(Cⁿ), u 6≡ −∞ such that u = −∞ on F. In particular, if F is bounded thenu can be made to be negative on every fixed ball contain- ing F. It also follows from Josefson’s theorem that a countable union of pluripolar set is pluripolar.

We continue this preparatory section by showing that the class of un- bounded locally B−regular domains is much richer than that of strictly convex domains. Before formulating it, recall that a compact K ⊂Cⁿ is called B−regular if every function in C(K) can be approximated by con- tinuous plurisubharmonic functions on neighbourhood s ofK. This notion is also introduced by Sibony in [14]. We also say that a closed set K in Cⁿ islocally B−regular if for everyz ∈K there exists an open bounded neighbourhoodU ofzsuch thatU∩KisB−regular. For a detailed treat- ment ofB−regular compact sets, the reader may consult Section 1 in [14].

In this work, we only use the following properties.

(i) A compactK⊂Cⁿ isB−regular if and only ifKis locally B−re- gular.

(ii) If the compactKis a countable union ofB−regular compacts then KisB−regular.

Proposition 2.5. — Ωis locallyB−regular if the following conditions are verified.

(a) For everyz∈∂Ω, there exists an open ball U aroundz such that Ω∩U is hyperconvex i.e., there exists a negative plurisubharmonic exhaustion function onΩ∩U.

(b) There exists a locallyB−regular closed subsetK of∂Ωsuch that Ωis strictly pseudoconvex near every point of ∂Ω\K.

Here by strict pseudoconvexity ofΩnearz0∈∂Ωwe mean the following:

there is an open neighbourhoodU ofz₀and aC²smooth plurisubharmonic functionuon U such that U∩Ω ={z :ρ(z)<0}, dρ(z0)6= 0and ρ(z)− λ|z|²∈PSH(U)for some constantλ >0.In this case, it is well known that there isv∈PSH^c(U ∩Ω)such that v(z0) = 0whereasv <0 elsewhere.

Proof. — Fix a pointz₀∈Ωand an open ballU containingz₀such that Ω⁰ := Ω∩U is hyperconvex. Set K⁰ =K∩∂Ω⁰. Then the compactK⁰ is

B−regular. It follows from (b) that∂Ω⁰ is a countable union ofB−regular compact sets. Therefore, the compact∂Ω⁰ isB−regular as well. SinceΩ⁰ is hyperconvex, we may apply Lemma 2.8 in [10] to get thatΩ⁰ isB−regular.

The proof is complete.

The following type of unbounded domains is quite important in our study.

Definition 2.6. — We say thatΩis of bounded type if there exists a real valued function ψ ∈ PSH(Ω) such that ψ < 0 and lim_|z|→∞ψ(z) =

−∞.

In particular, Ωcan not contain any (non constant) biholomorphic im- age of the complex plane. It is immediate to note that the bounded type property is invariant under biholomorphic maps. Now we construct a class of unbounded domains of bounded type.

Proposition 2.7. — Assume that there arencomplex hyperplaneH1,

· · ·, H_n lying in Cⁿ\Ω such that Tn

i=1H_i is a singleton and that dist (Sn

i=1Hi, ∂Ω)>0.ThenΩis of bounded type.

Proof. — After a linear change of coordinates, we may arrange thatH_i= {zi= 0} fori= 1,· · ·, n. It is easy to check that for large constantA,the function

ψ(z1,· · · , zn) =−log|z1· · ·zn| −A

is negative plurisubharmonic onΩ. Moreover,lim_|z|→∞ψ(z) =−∞.Thus

Ωis of bounded type.

Finally, we introduce a class of (unbounded) domains which enable us to control the growth to infinity of upper envelopes like (1).

Definition 2.8. — If n> 2 then we say that Ωcontains no complex hyperplane at infinity if there is a compactKofΩsuch thatΩ\Kcontains no complex hyperplane.

Consider the domainΩ :=B×C, whereB is the open unit ball inC². It is easy to see thatΩ containes no complex hypersurface whereas Ω is not of bounded type.

Proposition 2.9. — Assume thatΩ contains no complex hyperplane at infinity and h ∈ C(∂Ω) is bounded. Then for every bounded function u∈PSH(Ω)satisfyingu^∗6hon∂Ωwe have

lim sup

|z|→∞

u(z)6lim sup

|z|→∞

h(z).

Proof. — Choose a compact K ⊂ Ω such that Ω\K contains no com- plex hyperplane. Denoteα:= lim sup_|z|→∞h(z). Fixε > 0, then there is compact L ⊂ ∂Ω such that h < α+ε on ∂Ω\L. We are going to show that u(z)< α+ε for allz outside the convex hull ofK∪L. Fix such a pointz. Then there is a complex hyperplaneH passing throughzwhich is disjoint fromK∪L. Pickz⁰∈H\Ω. Letl be the complex line connecting zandz⁰. SetU =l∩Ω. ThenU∩L=∅. Thush < α+εon∂U. We may viewU as a nonempty open subset (possibly unbounded) of C. Since uis bounded,u^∗ 6h < α+ε on∂U and U is notdense inC, we may apply the maximum principle to reachu(z)< α+ε. The proof is complete.

3. Duality theorem and unbounded B− regular domains We start with the following variation on a basic duality theorem of Ed- wards (cf. [5] and [17]).

Theorem 3.1. — Let X be a closed subset of Cⁿ and A be a convex cone ofUSC_∗(X). Letg:X→(−∞,∞]be a lower semicontinuous function which is increasing limit of a sequence inC0(X). Then for everyz∈X we have

sup{u(z) :u6g, u∈ A}= infnZ

X

gdµ:µ∈Jz(A)o .

We require the following elementary fact.

Lemma 3.2. — Let u ∈ USC_∗(X) then there is a uniformly bounded sequence{uj} ⊂ Cc(X)such that:

(a) u_j →upointwise onX.

(b) uj >u−1/jfor everyj >1.

Proof. — Define forj>1 the function

vj(z) = sup{u(x)−j|x−z|:x∈X}.

Clearly the sequence{vj}j>1 is uniformly bounded. It is also well known thatv_j ∈ C(X)and thatv_j↓uonX asj → ∞. Next, we claim thatv_j is non positive at infinity for everyj. Fixj >1 then for everyε >0, we can find a compactK⊂X such thatu < εonX\K.Set

Kj={z:z∈X,dist(z, K)6M/j},

whereM = sup_Xu. ObviouslyKj is a compact subset ofX. Furthermore, from the definition ofv_j we can check easily thatv_j(z)6εforz∈X\K_j. This proves the claims.

Now we choose a sequence of compacts sets P_j ↑ X and a sequence of positive numbersλj ↑ ∞such that Qj :=X∩ {z:|z|>λj} ∩Pj =∅.By Tietze’s extension theorem, we can find˜v_j ∈ C_c(X) such thatv˜_j =v_j on Pj,v˜j = 0onQj and|˜vj|6||vj||X. Set

uj= max(˜vj, u−1/j)∀j>1.

It is easy to check thatuj is the desired sequence.

Remark. — It follows easily from Lemma 3.2 that Jz(A) is a closed convex subset of the space of positive regular Borel measures onX with total mass61.

Proof of Theorem 3.1. — Fixz∈Ω. Givenu∈ A, u6gandµ∈Jz(A) we have

u(z)6 Z

X

udµ6 Z

X

gdµ.

To prove the reverse direction, first we consider the case g ∈C0(X). For ϕ∈USC_∗(X)we set

Sϕ:= sup{u(z) :u6ϕ, u∈ A}.

Clearly Sϕ is of real value for all ϕ ∈ USC_∗(X). Moreover, since A is a convex cone we have

(a) S(λϕ) =λS(ϕ)forλ >0 andϕ∈USC_∗(X).

(b) S(ϕ₁) +S(ϕ₂)6S(ϕ₁+ϕ₂)forϕ₁, ϕ₂∈USC_∗(X).

In view of Hahn-Banach’s theorem we can find a real linear func- tionalS˜ onC₀(X)such that

(c) Sg˜ =Sg.

(d) S(ϕ)6Sϕ˜ 6−S(−ϕ)for allϕ∈ C0(X).

Since |Sϕ| 6 ||ϕ|| we infer that S˜ is a continuous linear functional on C0(X)and ||S||˜ 61.By Riesz’s representation theorem (cf. Theorem 6.19 in [12]), there is a Borel measureµonX satisfying

Sϕ˜ = Z

X

ϕdµ, ∀ϕ∈ C0(X).

Sinceϕ >0 impliesS(ϕ)˜ > 0 we infer that µ is a positive regular Borel measure on X satisfying µ(X) 61. It remains to check that µ ∈ J_z(A).

For this, fixu∈ A.By Lemma 3.2, there is a uniformly bounded sequence u_j ⊂ C_c(X)satisfyingu_j→uasj → ∞andu_j >u−1/j.By Lebesgue’s convergence theorem we have

Z

X

udµ= lim

j→∞

Z

X

ujdµ> lim

j→∞Suj(z)>u(z).

Now for generalg. Take a sequence{gj}j>1⊂ C0(X)such thatg_j↑gon X. By the previous paragraph, we can find a sequenceµj ∈Jz(A)satisfying (3.1) sup{u(z) :u6gj, u∈ A}=

Z

Ω

gjdµj.

Letµbe a weak^∗−limit ofµ_j, then by the Remark following Lemma 3.2, µ∈Jz(A).Observe that for any fixedk>1

lim sup

j→∞

Z

Ω

g_jdµ_j > lim

j→∞

Z

Ω

g_kdµ_j= Z

Ω

g_kdµ.

Lettingk → ∞ and using Lebesgue’s monotone convergence theorem we get

(3.2) lim sup

j→∞

Z

Ω

gjdµj >

Z

Ω

˜hdµ.

Putting (3.1) and (3.2) together we arrive at the desired conclusion. This

completes the proof.

The following result is an application of the above duality theorem to the case whereX is the closure of an unbounded domain inCⁿ.

Proposition 3.3. — Leth∈ C(∂Ω)be a bounded, non negative func- tion and A ⊂USC_∗(Ω) be a convex cone. Let ˜hbe the function equal to hon∂Ω and to M on Ω, where M is some positive constant larger than sup_∂Ωh. Then forz∈Ω

sup{u(z) :u6˜h, u∈ A}= infnZ

Ω

˜hdµ:µ∈J_z(A)o .

Proof. — According to Theorem 3.1, it suffices to construct a sequence {hj} ∈ Cc(Ω) such that hj ↑ ˜h. To do this, we first choose sequences of compact setsKj ↑Ω, Lj ↑∂Ωand a sequence of positive numbersλj ↑ ∞ such thatPj∩(Lj∪Kj) =∅, where Pj = Ω∩ {z:|z|>λj}.Define

g_j=

h on∂Ω\Pj

0 onP_j, and

˜ gj =

h onLj

0 on(∂Ω∪Pj)\Lj.

Thengjis lower semicontinuous on∂Ω∪Pjandg˜j is upper semicontinuous on∂Ω∪P_j.Notice that˜g_j6g_jon∂Ω∪P_j.Thus by the Hahn interpolation theorem (cf.[4], Proposition 7.2.1), we can find ˜hj ∈ C(∂Ω∪Pj)satisfying

˜

gj6˜hj6gj.In particular˜hj 6hon∂Ω,˜hj =honLj and˜hj= 0 onPj.

By Tietze’s extension theorem we may extend˜h_j to ˆh_j ∈ Cc(Ω)such that 06ˆhj6M onΩand that ˆhj =M onKj. Set

hj = max(ˆh1,· · ·,ˆhj).

It is easy to check thathj is the desired sequence. We are done.

From now on, we only interested in the case where X = Ω and A is one of the following convex subcones ofUSC_∗(Ω) :A₁,the set of functions having compact support inPSH^c(Ω),A2,the set of functions inUSC∗(Ω) which are plurisubharmonic onΩ.For any bounded functionϕ onΩ and 1 6 i 6 2 we also define upper envelopes relative to the convex cone introduced above.

S_iϕ(z) = sup{u(z) :u6ϕ, u∈ A_i}, z∈Ω.

The result below is a geometric interpretation of the situation when the class of Jensen measures at every boundary point ofΩis trivial.

Proposition 3.4. — The following assertion are equivalent.

(a) For everyz ∈ ∂Ω, there is an open neighbourhoodU of z and a function u ∈ PSH^c(U ∩Ω) satisfying u(z) = 0 whereas u < 0 elsewhere.

(b) Ωis regular in the real sense andJ_z(A₁) ={δ_z} for everyz∈∂Ω, whereδz is the Dirac mass atz.

(c) Ωis locally B−regular.

Proof. — (a) ⇒(b). Clearly Ωis regular in the real sense. Fix z ∈∂Ω andµ ∈Jz(A1). LetV be an open neighbourhood of z lying compactly inU. Setα= sup_Ω∩∂V u. Since α <0, the functionv equals to0onΩ\V and tomax(u−α,0)onΩ∩V belongs toA1. It follows that

−α=v(z)6 Z

Ω

vdµ= Z

Ω∩V

vdµ.

Sinceµis a positive regular Borel measure onΩwith total mass 61 and sinceα <0 we deduce that the support ofµis contained inV. Since V is arbitraryµmust be supported atz. Thusµ=δz. The proof is complete.

(b) ⇒ (c). In view of Theorem 2.1, it is enough to show that for every z0 ∈ ∂Ω there is a negative function u ∈ P SH(Ω) such that u^∗ < 0 on

∂Ω\{z0}whilelim_x→z0u(x) = 0. For this, fix such a pointz₀. Define h(z) =

max(−|z−z0|,−1) z∈∂Ω

0 z∈Ω.

By Proposition 3.3 we have S₁h = hon ∂Ω and S₁h 6 0 on Ω. On the other hand, by Lemma 2.4 we obtain(S1h)^∗6hon∂Ω. SinceS1his lower

semicontinuous onΩ,by the choice of h,we can check that u:= (S₁h)^∗ is the desired function.

(c) ⇒(a) follows immediately from Theorem 2.1.

Remark. — It is important to note thatJ_z(A₂) ={δ_z}, for everyz ∈

∂Ω. Indeed, forz0∈∂Ω, consider the functionh(z) = max(−|z−z0|−1,−2) forz ∈∂Ω and h(z) = −2 onΩ. Since h∈ A2, h(z0) = −1 and h < −1 elsewhere, we see immediately thatJ_z0(A2) ={δz0}.

In view of Theorem 3.1, if Jz(A1) ={δz} for every z ∈ ∂Ω then Ω is locallyB−regular. Our next result is a partial generalization of Proposi- tion 19 in [15].

Proposition 3.5. — Assume that Ω is locally B−regular. Let h ∈ C(∂Ω) be bounded. Then there exists ϕ ∈ MPSH(Ω) having the follow- ing properties.

(a) inf∂Ωh6ϕ6sup_∂Ωh,lim_x→zϕ(x) =h(z)for allz∈∂Ω.

(b) There is a pluripolar subset F of Ω such that ϕ is continuous on Ω\F.

Moreover, if h∈ C0(∂Ω)then the pluripolar setF can be constructed to be independent ofh.

Proof. — We may assume thath>0on∂Ω.Define the function˜has˜h= hon∂Ωandh˜= sup_∂ΩhonΩ. Then by Theorem 3.1 and Proposition 3.4 we haveS1˜h=hon∂Ω.Since∂Ωis regular in the real sense, Lemma 2.4 impliesϕ:= (S₁h)˜ ^∗ ∈PSH(Ω) and0 6ϕ6˜honΩ. Thusϕ satisfies(a) of the theorem. For (b), fix an open ball B ⊂ Ω it suffices to show the maximality ofϕonB. By Choquet’s lemma, there is a sequence{vj}j>1⊂ PSH^c(Ω) such thatvj↑S1˜honΩ.Letv˜j be the solution of the Dirichlet- Bremermann problem on B with boundary data v_j. Then the function v^∗_j = ˜vj on B and v^∗_j = vj on Ω\B belongs to PSH^c(Ω) and satisfies vj 6 v^∗_j 6 h˜ on Ω. Thus v˜j ↑ S1˜h on Ω. It follows that v˜j ↑ ϕ almost everywhere onB. Thusϕis maximal onB. Finally, by the solution to the second problem of Lelong (cf.[1]) the setF :={z∈Ω :S1˜h(z)< ϕ(z)} is pluripolar, thus the functionϕ,being lower semicontinuous onΩ\F, must be continuous onΩ\F.

Next, observe thatC0(∂Ω)is a separable Banach space with the sup norm.

Choose a countable dense sequence{hj}j>1 in C0(∂Ω). For eachj >1 we seth˜_j =h_j on∂Ωandh_j:= sup_∂Ωh_j onΩ. By the above argument, there exist a pluripolar subsetFj of Ωandϕj ∈MPSH(Ω)such that

(a) lim_x→zϕ_j(x) =h_j(z)for allz∈∂Ω.

(b) ϕj is continuous onΩ\Fj.

Thus F := ∪Fj is pluripolar. Now given h ∈ C0(∂Ω), we can choose a subsequence hj_k that converges tohuniformly on ∂Ω. It is easy to check that˜hj_k converges to˜huniformly onΩ.Thereforeϕj_kconverges uniformly toϕonΩ. Thus ϕ∈MPSH(Ω),lim_x→zϕ(x) =h(z)for allz ∈∂Ω andϕ is continuous onΩ\F.

It would be interesting to know if every unbounded locally B−regular domain is B−regular. Unfortunately, we can only prove this statement under an additional assumption on the domain (cf.Theorem 4.4). However, if we discard the requirement on maximality of the solution in Definition 2.2 then a satisfactory answer can be obtained. More precisely, we have the following generalization of Theorem 2.1 in [15].

Proposition 3.6. — Assume that Ω is locally B−regular. Then for everyh∈ C(∂Ω), h>0and every closed setK ofΩsuch thatK∩∂Ω =∅, there existsu∈P SH^c(Ω) satisfyingu>0, u= 0onKandu=hon∂Ω.

Proof. — We use the same idea as in [15]. SinceΩis locallyB−regular, we can choose locally finite open coverings{Ui}i,{U_i⁰}i>1 of∂Ωconsisting of open balls such thatΩ_i:= Ω∩U_iisB−regular,U_i⁰⊂⊂U_iandK∩U_i=∅ for alli>1. Let{χi}i>1be a partition of unity subordinating to{U_i⁰}i>1. By Tietze’s extension theorem, for every i > 1 there exists hi ∈ C(∂Ωi) such that06hion∂Ωi, hi= 0onΩ∩∂Ωiandhi=hχionU_i⁰∩∂Ω. Letui

be the solution to the Dirichlet-Bremermann problem onΩiwith boundary valuesh_i. Clearlyu_i>0onΩ_iandu_i= 0onΩ∩∂Ui. Thus we may extend ui to u˜i ∈ PSH^c(Ω) by setting u˜i = 0 out of Ωi. Set u(z) =P

i>1u˜i(z) forz∈Ω. By the construction ofu˜i,locally this sum is taken over afinite number of indicesi. It is not hard to see thatuis the desired function.

4. Equality of Jensen measures

We start with a simple relation between equality of Jensen measures and the possibility of global approximation of bounded plurisubharmonic func- tions by continuous ones.

Theorem 4.1. — The following assertions are equivalent.

(a) Jz(A1) =Jz(A2)for every z∈Ω.

(b) S1ϕ=S2ϕonΩfor every ϕ∈ C0(Ω).

(c) For everyu∈ A2there is a uniformly bounded sequence {vj}j>1∈ A1 such thatvj →uonΩandlim sup_j→∞6u^∗ on∂Ω.

Proof. — We follow the lines of the proof of Theorem 3.2 in [9] (see also Theorem 2.1 in [11]).

(a) ⇒(b). This is a direct application of the duality Theorem 3.1.

(b) ⇒(c). By Lemma 3.2 we can find a sequenceuj ∈ C0(Ω) such that uj →uanduj>u−1/j.Clearly for allj>1

u−1/j6S₂u_j 6u_j onΩ.

Notice that (S2uj)^∗ 6 uj, so S2uj ∈ A2 for all j and S2uj → u on Ω.

Moreover,S1uj=S2uj onΩ. In particularS2uj is continuous onΩfor all j. By Choquet’s lemma we, can find a sequence{vj,k}k>1⊂ A1 increasing to S2uj on Ω. Let Kj ↑ Ωbe an increasing sequence of compact sets. In view of Dini’s lemma and continuity ofS₂u_j, we can find a sequencek_j↑ ∞ such that|vj,k_j −S2uj|<1/j onKj. It is easy to check that vj,k_j is the desired sequence.

(c) ⇒(a). Givenz ∈Ωand µ∈Jz(A1)we must showµ∈Jz(A2). Fix u∈ A2, then there is a sequence{vj}j>1⊂ A1satisfyingvj→uonΩand lim sup_j→∞v_j 6u^∗ on∂Ω. Notice that for allj>1

vj(z)6 Z

Ω

vjdµ.

Letting j → ∞ and using Fatou’s lemma we get u(z) 6 R

Ωudµ. Thus

µ∈Jz(A2).

Remarks. — (a) By Dini’s lemma and the construction of the sequence {vj}j>1 we see that if uis continuous on a compact subset K of Ω then, in addition, the sequences {vj}j>1 in (c) can be chosen to converge tou uniformly onK.

(b) Building on a previous example of Fornaess and Wiegerinck (cf. [6]

p. 260) we will construct a unbounded domainΩsuch that any of equiv- alent assertions in Theorem 4.1 does not hold. Recall that Fornaess and Wiegerinck consider a smoothly bounded domainD inC²defined by

D={(z, w) :|w−e^iϕ(|z|)|²< r(|z|)},

where r and ϕ are C^∞ smooth functions on R. Moreover, Fornaess and Wiegerinck show that ifrandϕare well chosen thenD has the following properties.

(i) The projection of D onto the first coordinate is the annulus {z : 1<|z|<17}.

(ii) The compact A:= {(z,0) : |z| = 2} ∪ {(z,0) :|z|= 9} ∪ {(z,0) :

|z|= 16} is contained inD.

(iii) The annulusB:={(z,0) : 26|z|615} lies inΩ(but not inΩ).

(iv) There is u∈PSH^c(D) such thatu(z, w) = 0if |z|64 or|z|>14 and that there is no continuous plurisubharmonic functiongon a neighbourhood ofD verifying|g−u| < 1 on A, since otherwise there would be a violation to the maximum principle onB.

Now denote by f the holomorphic mapping f(z, w) = (_z−a¹ , w), where ais any number in(16,17). Then Ω :=f(D⁰)is an unbounded domain in C²with smooth boundary, whereD⁰=D\{{a} ×C}. Defineu˜:=u◦f⁻¹, then u˜ is continuous and plurisubharmonic on Ω. Notice that, in view of (iv), the function u^∗ has compact support in Ω. Assume that (a) of Theorem 4.1 holds onΩ, then by Remark (a) there is ag ∈ A₁ satisfying

|g−u|˜ <1onf(A). Setg˜=g◦f⁻¹. Then|˜g−u|<1onA. Observe that

˜

gis plurisubharmonic onD⁰,continuous at every point of∂D\{{a} ×C}, and equals to0 near the complex linez=a. Thus we may extend˜g to an element of PSH^c(D) by setting ˜g = 0 onD∩({a} ×C). By Theorem 1 in [6], the function˜g can be approximated uniformly on D by continuous plurisubharmonic functions on neighbourhood s of D. Thus we can find such a functiong^∗ satisfying|g^∗−u|<1 onA. This is a contradiction to the property(iv).

(c) Similar statements to the equivalence between (a)and (b) in Theo- rem 4.1 have been claimed in Section 2 in [9] and at the end of the proof of Theorem 4.3 in [11]. The proofs given in these references contain gaps due to incorrect applications of Hahn-Banach’s separation theorem. How- ever, we can give honest (and simpler) proofs by repeating the proof of Theorem 4.1.

It remains to decide when one of the equivalent conditions in Theorem 4.1 holds. For this, we recall the following terminology from Section 3 of [11]

(see also [9]).

Definition 4.2. — By an isotopy family of biholomorphic mappings defined onΩ, we mean a continuous mapΦ : Ω×[0,1]→Cⁿ having the following properties.

(a) Φt := Φ(t,·)maps Ω biholomorphically onto its image; moreover, Φ_t is a homeomorphism betweenΩandΦ_t(Ω).

(b) Φ⁻¹_t (z)is real analytic inton a neighbourhood of 0for allz∈Ω.

(c) Φ⁻¹_t converges uniformly to Φ⁻¹₀ = Id on compact subsets of Ω whent→0.

Definition 4.3. — LetΦtbe an isotopy family of biholomorphic map- pings on Ω. Then by the boundary cluster set of Φ_t we mean the set of limit points of sequence of elements inΩ∩Φt(∂Ω)whent→0.

Theorem 4.4. — Assume thatΩis of bounded type and that there is an isotopy familyΦtof biholomorphic mappings on Ωsuch that for every zlying in the boundary cluster setX ofΦ_twe haveJ_z(A₁) ={δ_z}. Then the following assertions hold.

(a) J_z(A₁) =J_z(A₂)for allz∈Ω.

(b) Assume in addition that Ω is regular in the real sense. Then for every bounded functionh∈ C(∂Ω)there exists a bounded function Ψ∈MPSH(Ω)∩ C(Ω)satisfying the following properties.

(i) Ψ^∗6hon∂Ω.

(ii) lim_x→zΨ(x) =h(z)for allz∈∂Ωthat satisfiesJ_z(A1) ={δz}.

Moreover, if Jz(A1) ={δz} for everyz ∈∂Ωthen there exists a unique bounded functionΨ∈MPSH(Ω)∩ C(Ω) such thatu=hon∂Ω.

Proof. — (a) We will use some ideas in the proof of Theorem 4.4 in [9]

(see also Theorem 3.5 in [11]). Fix ϕ ∈ C0(Ω) and z₀ ∈ Ω, we will show thatS1ϕ(z0) =S2ϕ(z0). For this, notice thatinfΩϕ6S2ϕ6(S2ϕ)^∗6ϕ.

This implies thatS₂ϕ= (S₂ϕ)^∗∈ A₂.By Choquet’s lemma we can choose a sequence {vm}m>1 ⊂ A1 such that vm ↑ S1ϕ on Ω. Observe that, by Theorem 3.1 we also have

(4.1) S1ϕ≡S2ϕ≡ϕonX.

In particularvm↑ϕonX. Now for t∈(0,1)we define ut= (S2ϕ)◦Φ⁻¹_t . It is clear thatu_t is non negative, plurisubharmonic onΦ_t(Ω). Fork >0 we set

Ωk :={z:z∈Ω,|z|6k,dist(z, ∂Ω)>1/k}.

Letρ∈ C^∞₀ (Cⁿ)be a nonnegative, radial function with support in the unit ball and satisfiesR

ρ dV = 1, where dVCⁿ is the volume form in Cⁿ. Set ρ_δ :=δ⁻ⁿρ(z/δ)forδ >0.

Fixε >0, we claim that there ist0∈(0,1), k0>1andm>1 such that for allt∈(0, t₀)andk>k₀there is

0< δk < ak:= min06t6t₀dist(Φt(∂Ω),Φt(∂Ωk)) satisfying

ut∗ρδ_k+ψ∗ρδ_k−ε6vmonΩ∩Φt(∂Ωk),

where the convolution of a locally integrable functionuwith ρδ is defined by

(u∗ρ_δ)(ξ) :=

Z

|t|<δ

u(ξ−t)ρ_δ(t)dV(t).

Assume otherwise; then we obtain sequences t_j ↓ 0, k_j ↑ ∞, mj ↑ ∞, {ξj}j>1⊂Ω∩Φt(∂Ωk), and0< δk_j < ak_j such that

(ut_j∗ρδ_kj)(ξj) + (ψ∗ρδ_kj)(ξj)−ε > vm_j(ξj)∀j.

After passing to a subsequence we may assume that either |ξ_j| → ∞ or ξj → ξ^∗ ∈ X. The first possibility is excluded since ψ tends to −∞ at infinity andv_mj are uniformly bounded from below. The second possibility can not occur either, since

(4.2) lim inf

j→∞ vm_j(ξj)> lim

j→∞vm_j(ξ^∗) =ϕ(ξ^∗).

On the other hand, from (6) and the condition (c) in Definition 4.2 we obtain

(4.3) lim sup

j→∞

ut_j ∗ρδ_kj

(ξj)−ε6ϕ(ξ^∗)−ε.

Combining (4.2) and (4.3) we get a contradiction. Thus the claim follows.

By shrinkingt0 and increasing k0 we may obtain thatz0 ∈Φt(Ωk)for all k>k0 andt∈(0, t0). Consider the function

˜ v_t,k,m=

max{vm, ut∗ρδ_k+ψ∗ρδ_k−ε} onΩ∩Φt(Ωk)

vm onΩ\Φt(Ωk).

By the claim proven above we can check that˜vt,k,m∈ A1and thatv˜t,k,m6 ϕonΩ. It follows that˜vt,k,m6S1ϕonΩ, in particular

S₁ϕ(z₀)>˜v_t,k,m(z₀)>(u_t∗ρ_δk)(z₀)−ε>u_t(z₀)−ε.

Taking the limsup of the rightmost term whent → 0, and observing the curvet7→Φ⁻¹_t (z0),being real analytic near0, is not plurithin at0, we infer that S₁ϕ(z₀)> S₂ϕ(z₀)−ε. Since ε > 0 is arbitrary we have S₁ϕ(z₀) = S2ϕ(z0). Now by Theorem 4.1 we conclude that Jz(A1) = Jz(A2)for all z∈Ω.

(b) After adding a constant to h, we may assume that h > 0 on ∂Ω.

Let˜hbe the function equal to hon∂Ωand to M = sup_∂ΩhonΩ. Since Jz(A1) =Jz(A2)on Ω, by Proposition 3.3 we have S1˜h=S2˜honΩ and S1h(z) =˜ h(z)for all z∈∂ΩsatisfyingJz(A1) ={δz}. We will show that Ψ :=S₂˜his the desired function. Since Ωis regular in the real sense, by Lemma 2.4 we have0 6Ψ^∗ 6˜h onΩ. The proof of Proposition 3.5 also implies thatΨ^∗ ∈ MPSH(Ω). Now we claim that Ψ^∗ = Ψ on Ω. For this, fixε >0and define onΩthe function

uε:= max(Ψ^∗+εψ,0).

Observe thatu^∗_ε∈ A₂ and satisfiesu_ε6h˜ onΩ. It follows thatu_ε6S₂˜h onΩ. Letting ε→ 0, one obtains Ψ^∗ 6S2˜h onΩ. The claim follows. In

particular Ψ ∈ MPSH(Ω)∩ C(Ω). Finally, fix a point z ∈ ∂Ω such that Jz(A1) ={δz}then

lim inf

x→z Ψ(x)>lim inf

x→z S1˜h(x)>h(z).

Thereforelim_x→zΨ(x) =h(z).

Finally, assume that there are functions Ψ,Ψ⁰ ∈MPSH(Ω)∩ C(Ω) such that Ψ = Ψ⁰ =h on∂Ω. Let{Ωk}k>1 be a sequence of subdomains of Ω such thatΩ_k ↑Ω.Fixε >0. By the assumptions on boundary values ofΨ andΨ⁰ we see that there isk0 so large thatΨ +εψ 6Ψ⁰+ε on∂Ωk for allk>k₀. Using maximality ofΨ⁰ one obtainsΨ +εψ6Ψ⁰ onΩ_k for all k>k0. Lettingk→ ∞andε→0we have Ψ6Ψ⁰ onΩ. By changing the role ofΨandΨ⁰ we getΨ = Ψ⁰ onΩ. The proof is thereby concluded.

Remarks. — (a) IfJ_z(A1) ={δz}for everyz∈∂Ωthen by considering the familyΦt=Idfor all t, we deduce from Theorem 4.4 that Jz(A1) = J_z(A₂) for all z ∈ Ω. The reverse implication does not hold in general.

Indeed, consider the following pseudoconvex domain of bounded typeΩ :=

{(z, w) : 1<|z|<|w|}.Since through every boundary point of∂Ωwe can find a non constant complex analytic disk lying in∂Ωand passes through this point, by Proposition 3.4 we see thatJz(A1)6={δz}foreveryz∈∂Ω.

On the other hand, by considering the familyΦ_t(z, w) := (z,(1 +t)w)and applying Theorem 4.4 we infer thatJz(A1) =Jz(A2)for allz∈Ω.It is an open problem if such an example can be found in the class of unbounded pseudoconvex domains withsmoothboundaries.

(b) It is proved in Theorem 26 of [15] that an unbounded strictly pseu- doconvex domainΩisB−regular if Ωsatisfies the following condition:

(L⁰)there exists a holomorphic polynomialpsuch that:

|p(z)|²>(1 +|z|²)^degp ∀z∈Ω.

We claim that if the domain Ω satisfies the condition (L⁰) then it is of bounded type. Indeed, pick a polynomialpsuch that

|p(z)|²>(1 +|z|²)^degp ∀z∈Ω.

Then we havelog|p|>0 onΩ andlog|p(z)| → ∞as |z| → ∞. It follows that the function

ψ(z) := 1

4log(1 +|z|²)− 1

degplog|p(z)|

is negative, plurisubharmonic onΩand tends to−∞when|z|goes to ∞.

This proves the claim. We own this elegant proof to the referee.

In the next result, we deal with domains which are not necessarily of bounded type but stronger conditions on boundary data are imposed.

Proposition 4.5. — Assume thatΩsatisfies the following conditions.

(i) Ωis regular in the real sense.

(ii) There is an isotopy familyΦtof biholomorphic mappings onΩsuch that for everyz lying in the boundary cluster setX ofΦ_twe have Jz(A1) ={δz}.

(iii) Ωcontains no complex hyperplane at infinity.

Then for everyh∈ C0(∂Ω), h>0,there isu∈MPSH(Ω)∩C(Ω)satisfying (i)and(ii)in Theorem 4.4 (b).

Proof. — Set M = sup_∂Ωh. Let h˜ = h on ∂Ω and ˜h = M on Ω. By theproofof Proposition 3.5,S2˜h∈MPSH(Ω)and satisfies(S2˜h)^∗6hon

∂Ω. We will show that u:=S2˜his the desired function. For this, we first observe that by Proposition 2.9

|z|→∞lim u(z) = 0.

Next, we apply Choquet’s lemma to get a sequence {v_m}_m>1 ⊂ A₁ such that06vm6˜handvm↑S1˜honΩ. Define fort∈[0,1]the function

ut:=u◦Φ⁻¹_t .

Given ε >0, using (10) and the same reasoning as in the proof of The- orem 4.4 (a), we can find t₀ ∈ (0,1), k₀ >1 and m >1 such that for all t∈(0, t0)andk>k0 there is

0< δk < ak := min

06t6t0

dist(Φt(∂Ω),Φt(∂Ωk)) satisfying

ut∗ρδ_k−ε6vm onΩ∩Φt(∂Ωk),

whereΩ_k is defined as in Theorem 4.4. Now we proceed exactly as in the proof of Theorem 4.4.(a) to reach u = S1h˜ on Ω. Thus u ∈ C(Ω) and satisfies lim_x→zu(x) = h(z) for every z ∈ ∂Ω satisfying Jz(A1) = {δz}.

The proof is complete.

5. Examples of unbounded B−regular domains

We start with the following generalization of Proposition 19 in [15].

Proposition 5.1. — Assume that Ω is an unbounded convex domain in Cⁿ with C¹ smooth boundary. For every z ∈ ∂Ω, denote by Kz the intersection between T_z∂Ω the real tangent space at z and ∂Ω. Assume that for every z ∈ ∂Ω there is a real hyperplane Lz ⊂ Tz∂Ω satisfying Lz∩Kz ={z}. ThenΩisB−regular. In particular, every strictly convex domain withC² smooth boundary isB−regular.

Proof. — First we must show that Ωis locally B−regular. Fix z ∈∂Ω and an open ball U around z. Let µ be a Jensen measure relative to P SH^c(U ∩Ω) with barycentre at z. Since Ω is convex, we see that the support ofµis contained inKz∩∂Ω∩U. From the existence ofLz, we can find a pluriharmonic functionuonCⁿ such thatu(z) = 0whereasu <0 onKz\{0}. It follows that µ ={δz}. Thus Ω is locally B−regular. Now we claim thatΩis of bounded type. Fix a pointz₀∈∂Ω. Letρbe a local defining function forΩon a neighbourhood U ofz0. Define the map

Φ(z) :=

∂ρ

∂z₁(z),· · ·, ∂ρ ρz_n(z)

, ∀z∈∂Ω∩U.

The following claim is crucial. There existsa₁,· · · , a_n ∈∂Ω∩U such that n vectors Φ(a1),· · · ,Φ(an) are linearly independent. Assume otherwise, thenΦ(∂Ω∩U)is contained in a complex hyperplane. Therefore, there is a vectorλ:= (λ1,· · · , λn)satisfyingPn

k=1λk∂ρ

∂z1(z) = 0for allz∈∂Ω∩U.

This means thatλ ∈ TzΩfor all z ∈ ∂Ω. If λ 6∈ ∂Ω, then there is some pointλ⁰∈∂Ωwhich is closest toλ. Obviouslyλ6∈T_λ⁰∂Ω.A contradiction.

Thusλ∈∂Ω. It follows thatTλ∂Ω contains an open piece of∂Ωwhich is absurd. The claim follows. Then we can push those pointsa_i slightlyinto the half spaces separated byTa_i∂Ωand disjoint from Ω to get points a⁰_i. LetHi be the complex hyperplane passing througha⁰_i and parallel to the complex tangent space at ai. Clearly Hi∩Ω = ∅ and dist (Hi, ∂Ω) >0.

By the choices of a⁰_i we also have Tn

i=1Hi is a singleton. Now the claim follows from Proposition 2.7. Finally, the proof is completed by applying

Theorem 4.4.

The next result is another easy application of Theorem 4.4.

Proposition 5.2. — Assume thatΩis of bounded type and regular in the real sense and that there is an open subsetAof∂Dhaving the following properties.

(a) tA∩Ω =∅ for everyt >1.

(b) Jz(A1) ={δz}for every z∈∂Ω\A.

Then for every bounded function h∈ C(∂Ω), there isΨ∈MPSH(Ω)∩ C(Ω) such that Ψ^∗ 6 h on ∂Ω and lim_x→zΨ(x) = h(z) for all z ∈ ∂Ω satisfyingJ_z(A₁) ={δ_z}.

Proof. — Denote byΦt(z) = (1 +t)zfort>0andz∈Cⁿ. ClearlyΦtis an isotopy of biholomorphic mappings onΩ. We can check, in view of the assumptions(a)that the boundary cluster ofΦ_tis contained∂D\A. Thus using (b), we conclude the proof by invoking Theorem 4.4.

Remarks. — (a) As an a concrete application of Proposition 5.2, con- sider the following unbounded domain inC²

Ω ={(z, w) :<z60,−1<<w <1} ∪ {(z, w) :|z|²+|w|²<1,<z >0}.

It is easy to check thatΩis convex and satisfiest∂Ω∩Ω =∅ fort >1. In particular,Ωis regular in the real sense. Since the hyperplanesz= 2and w= 2 have positive distances to Ω, by Proposition 2.7, Ω is of bounded type. Notice also thatA:=∂Ω∩ {<z >0}is strictly pseudoconvex. Thus for every bounded functionh∈ C(∂Ω), there existsu∈MPSH(Ω)∩ C(Ω) satisfyingu=honA.By pushing Aslightly inside the unit ball, we may construct similar examples whereΩis not convex.

(b) It is not hard to check that the domain Ω given in the above re- mark also satisfies the assumptions of Proposition 4.5. Thus, for everyh∈ C0(∂Ω), h >0, there exists a non negative functionu∈MPSH(Ω)∩ C(Ω) such thatlim_x→zu(x) =h(z)for allz∈A andlim_|z|→∞u(z) = 0.

The final result of the section is an invariant property for a class of unbounded B−regular domains. Before formulating it, we introduce the following terminology. A surjective holomorphic mapf between domains ΩandΩ⁰in Cⁿ is said to have the property (P) if the following conditions hold.

(a) f extends continuously to ∂Ω andf : Ω→Ω⁰ is an open surjective continuous map.

(b) There is (possibly empty) a complex subvariety E of Ω⁰ such that f : Ω\f⁻¹(E)→Ω⁰\E is a holomorphic coveringi.e.,for everyw∈Ω⁰\E, there is a neighbourhood U of wsuch thatf⁻¹(U)is a union of at most countably disjoint open setsVi andhis a biholomorphism fromVi ontoU. It is immediate to check that iff : Ω→Ω⁰ is a holomorphic proper map which extends holomorphically to a neighbourhood of Ω then f has the property (P).

Proposition 5.3. — LetΩ,Ω⁰ be unbounded domains in Cⁿ and f : Ω→Ω⁰ be a holomorphic map having the property (P). Assume thatΩis B−regular and thatΩdoes not contain any complex hyperplane at infinity.

ThenΩ⁰ is locallyB−regular. Moreover, if we assume in addition that Ω is of bounded type thenΩ⁰ isB−regular.

Proof. — (a) Fixz0∈∂Ω. Define

h(z) = max(−|f(z)−f(z0)|,−1)∀z∈∂Ω.

SinceΩ is B−regular, we can find u∈ PSH^c(Ω) such thatu=h on∂Ω.

SinceΩcontains no complex hyperplane at infinity, by Proposition 2.9 we haveu <0 onΩand

lim sup

|z|→∞

u(z)6−1.

Define

u(w) = sup{u(ξ) :˜ ξ∈f⁻¹(w)} ∀w∈Ω⁰\E.

We claim that ˜u^∗ <0 on ∂Ω⁰\{f(z0)} whilelim_x→f(z0)u(x) = 0.˜ Indeed, givenw^∗∈∂Ω\{f(z0)}, thenw^∗=f(ξ^∗)whereξ^∗∈∂Ω\{z^∗}.Consider an arbitrary sequence w_j ∈Ω⁰\E, wj → w^∗ such that u(w˜ _j)→ α. It suffices to check thatα <0.To this end, we choose forj >1a pointξj ∈Ωsuch that

f(ξj) =wj andα6u(ξj) + 1/j.

After passing to a subsequence we may assume that either |ξj| → ∞ or ξj → ξ⁰ ∈ Ω. In either case, it is easy to check by using properties of u that α < 0. Now we deal with the point f(z₀). Since f is open, we can find a small open neighbourhood U of z0 inΩsuch that f(U)is an open neighbourhood off(z₀)in Ω⁰. By the choice ofu,u˜ and h, it is not hard to show

lim

x→f(z0),x∈f(U)\Eu(x) = 0.˜ Putting all this together, the claim follows.

Next, since f : Ω\f⁻¹(E) → Ω⁰\E is a holomorphic covering, we see that locally onΩ⁰\E, the function u˜ is supremum of a family of negative plurisubharmonic functions. Thusu˜^∗is plurisubharmonic on Ω⁰\E. Notice that f⁻¹(E) is pluripolar in Ω, so u˜^∗ is in fact plurisubharmonic on Ω.

This statement is implicit in the proof of the theorem on extending locally bounded plurisubharmonic functions through pluripolar sets (cf.Chapter 2 in [8]). By the claim proven above, we have also

lim sup

x→f(z0),x∈Ω⁰

˜

u^∗(x)<0 on∂Ω⁰\{f(z0}.

Sincez₀ is arbitrary, we may apply Theorem 2.1 to find that Ω is locally B−regular.