PerronBremermann envelopes and pluricomplex Green functions with poles lying in a complex hypersurface

Let Ω be a bounded domain in C n and let P SH(Ω) be the cone of plurisubharmonic functions on Ω. Recall that a function u : Ω → R∪{−∞} is called plurisubharmonic if u is upper semicontinuous and its restriction on every complex line is subharmonic (we regard the function which is identically −∞ as plurisubharmonic). Let A be a complex hypersurface of Ω. Suppose that A is the zero set of a holomorphic function f defined on an open neighbourhood of Ω such that f generates the ideal sheaf of A i.e., df ̸= 0 on a dense subset of A. Following F. L´arusson and R. Sigurdsson in LS1 we define the pluricomplex Green function of Ω as

with poles lying in a complex hypersurface

This work is dedicated to the memory of Professor Nguyen Thanh Van

Nguyen Quang Dieu and Dau Hoang Hung Let Ω be a bounded domain inCn and let P SH(Ω) be the cone of plurisubharmonic functions on Ω Recall that a function u : Ω → R∪{−∞} is called plurisubharmonic

if u is upper semicontinuous and its restriction on every complex line is

subhar-monic (we regard the function which is identically−∞ as plurisubharmonic) Let

A be a complex hypersurface of Ω Suppose that A is the zero set of a holomor-phic function f defined on an open neighbourhood of Ω such that f generates the ideal sheaf of A i.e., df ̸= 0 on a dense subset of A Following F L´arusson and

R Sigurdsson in [LS1] we define the pluricomplex Green function of Ω as

G A,Ω (z) = sup {u(z) : u ∈ P SH(Ω), u < 0, u − log |f| = O(1)}.

Actually in [LS1], the authors also consider Green functions with poles along complex analytic sets where Lelong numbers of plurisubharmonic functions and multiplicity of the analytic sets are taken into account This includes the multipole pluricomplex Green functions introduced and investigated by Lelong in [Le] and Demailly in [De] as a special case A basic fact regarding multipole Green func-tion is that they provide fundamental solufunc-tions to the (complex) Monge-Amp`ere

operator (dd c)n Moreover, if the domain is hyperconvex i.e, there exists a

nega-tive plurisubharmonic exhaustion function, then the multipole Green function is continuous away from the poles and has (continuous) zero boundary values Nevertheless, for Green functions initiated by L´arusson and Sigurdsson, not much has been understood even in the special case we consider above More precisely, it is quite routine to check that (see Proposition 2.4 and Proposition 3.2 in [LS1]) the function ˜G A,Ω := G A,Ω −log |f| is maximal plurisubharmonic and locally bounded Furthermore, if there exists a strong plurisubharmonic barrier at

ξ ∈ (∂Ω) \ A i.e., there exists u ∈ P SH(Ω), u < 0 such that lim

z →ξ u(z) = 0 whereas

lim sup

z →t u(z) < 0 if t ∈ (∂Ω) \ {ξ}, then lim

z →ξ

˜

G A,Ω (z) = − log |f(ξ)| Thus, at least

in the case where Ω is strictly pseudoconvex (eg Ω is a ball), we may consider

˜

G A,Ω as a solution of the equation

u ∈ PSH(Ω) ∈ L ∞

loc (Ω), (dd c u) n = 0, lim

z →ξ u(z) = − log |f(ξ)|, ∀ξ ∈ (∂Ω) \ A Here (dd c)n is the complex Monge-Amp`ere operator which is defined as positive Borel measures over the class of locally bounded plurisubharmonic functions (see [Kl]) However, the continuity of ˜G A,Ω on Ω is not known yet even in the case

when Ω is a ball and A is a smooth hypersurface Since there is no version of the

comparison principle for plurisubharmonic functions which may tend to +∞ near

the boundary, it is also not clear that the function ˜G A,Ω is the unique solution to the above equation

The main aim of this paper is to address these two problem simultaneously First we discuss continuity of ˜G A,Ω and hence G A,Ω and then apply it to obtain

a sort of uniqueness of the Monge-Ampere equation with unbounded boundary

1

values The first result on continuity of G A,Ω is due to the first named author

in [Di1], where the special case A is a finite union of complex hyperplanes is

treated This partial result might be considered as an attempt to verify an earlier statement about ”almost” continuity stated in Theorem 3.9 in [LS1] (see also [LS2] for an erratum) For the reader convenience, we record below Theorem 2.2

of [Di1]

Theorem 1.1 Let Ω b U be pseudoconvex domains in C n and A be a finite union of complex hyperplanes passing through 0 ∈ Ω Assume further that the following conditions are satisfied:

(1.1.a) Ω is holomorphically convex in U and Int (Ω) = Ω.

(1.1.b) There exists α < 1 and a (relatively) open subset V of A ∩ ∂Ω on ∂Ω such that tV b Ω for α < t < 1.

(1.1.c) Every ξ ∈ (∂Ω) \ V is a good boundary point.

Then ˜ G A,Ω is continuous on Ω.

Recall that ξ ∈ ∂Ω is called a good boundary point (see [Di1] p.184) if there exists

a strong plurisubharmonic barrier u at ξ such that u ∗ is the pointwise limit on Ω

of a uniformly bounded sequence of plurisubharmonic functions u j that defined on

neighborhoods of Ω We should say that the assumption on uniform boundedness

of u j (unfortunately missing from [Di1]) is needed to guarantee the passage to the limit in the remark following Lemma 2.6 in [Di1]

It is not realized in [Di1] that the method of the proof actually gives a descrip-tion of ˜G A,Ω in terms of a limit of Perron-Bremermann envelopes associating to a sequence of a continuous real-valued functions that increases to − log |f| on ∂Ω.

For the reader convenience, we recall briefly the Perron-Bremermann envelope, which play an important role in solving the Dirichlet problem for the complex Monge-Amp`ere operator Let φ : ∂Ω → R, the Perron-Bremermann envelope of

φ (relative to Ω) is defined by

(1) P B φ,Ω (z) := sup {u(z) : u ∈ P SH(Ω), u ∗ ≤ φ on ∂Ω}.

Here u ∗ , the upper-regularization of a function u : Ω → R, is defined as

u ∗ (ξ) = lim sup

z →ξ u(z), ∀ξ ∈ Ω.

It is well known that if Ω is regular (in the real sense) i.e., continuous function on

∂Ω can be extended continuously to a harmonic function on Ω, and if φ ∈ C(∂Ω), the set of real-valued continuous functions on ∂Ω, then (P B φ,Ω)∗ ≤ φ on ∂Ω, and so P B φ,Ω ∈ P SH(Ω) satisfying (dd c P B φ,Ω)n = 0 on Ω Furthermore if Ω is

B −regular in the sense of Sibony (cf [Si2]) then P B φ,Ωextends continuously to Ω

and P B φ,Ω = φ on ∂Ω At this point, we note that, by Edwards’ duality theorem (cf [Ed], [Si2], [Wi] and [DW]), P B φ,Ω may be expressed as lower envelopes

of integrals with respect to Jensen measures Thus, Jensen measures (see the next section for details) come into play and provide us with sufficient conditions

on continuity of P B φ,Ω We should say that, earlier results about continuity of Perron-Bremermann envelopes have been obtained by Walsh in [Wa] without reference to Jensen measures

We recall the following terminology essentially devised by the first named author in [DW] which involves in the statement of Theorem 1.4

Definition 1.2 Let Ω be a domain inCn By an isotopy family of biholomorphic mappings on Ω we mean a continuous map Φ : [0, ε) × Ω → C n (ε > 0), such that the following statements hold.

(1.2.a) For each t ∈ [0, ε), Φ t(·) = Φ(t, ·) is a homeomorphism between Ω and

Φt (Ω); moreover, Φ t maps Ω biholomorphically onto Φ t (Ω).

(1.2.b) For all z ∈ Ω, t 7→ Φ −1

t (z) is real-analytic on [0, ε).

(1.2.c) Φ −1 t converges uniformly to Φ −10 = Id on Ω as t → 0.

An isotopy family of biholomorphic mappings on Ω leads to the following

interest-ing subset of ∂Ω which enables us to decide when two classes of Jensen measures

introduced above coincide

Definition 1.3 If Φ t is an isotopy family of biholomorphic mappings on Ω, we define the boundary cluster set of Φ t (relative to Ω) as the limit points of sequences

of elements in Ω ∩ Φ t (∂Ω) as t → 0.

Now we are able to formulate the first main result of the paper

Theorem 1.4 Let Ω b U be pseudoconvex domains in C n such that (Ω, U )

is a Runge pair Let A be a complex hypersurface defined by the zero set of a holomorphic function f on U such that A ∩ Ω ̸= ∅ and that f generates the ideal sheaf of A Assume that there exists an isotopy family of biholomorphic mappings

Φt defined on Ω satisfying the following conditions:

(1.4.a) Φ t(Ω)b U and (Φ t (Ω), U ) is a Runge pair for every t ∈ [0, ε).

(1.4.b) A is invariant under Φ t i.e., Φ t (A ∩ Ω) = A ∩ Φ t (Ω) for every t.

(1.4.c) The boundary cluster set of Φ t relative to ∂Ω when t → 0, denoted by S

is disjoint from A ∩ ∂Ω In other words, there exists an open neighbourhood V of

A ∩ ∂Ω such that Φ t(Ω) ⊃ V for every t > 0 close enough to 0.

(1.4.d) For every point ξ ∈ S, the set of Jensen measures with barycenter at ξ relative to the cone P SH(Ω) ∩ C(Ω) reduces to the Dirac mass {δ ξ }.

Then the following assertions hold:

(i) For every sequence {f j } j≥1 ⊂ C(∂Ω) such that f j converges locally uniformly to

− log |f| on (∂Ω) \ A, f j < − log |f| on ∂Ω and inf ∂Ω f j > −M for some constant

M > 0, the sequence P B f j ,Ω converges pointwise to ˜ G A,Ω on Ω.

(ii) ˜ G A,Ω is continuous on Ω.

We collect below several remarks regarding Theorem 1.1 and Theorem 1.4

Remarks (i) First, if we assume that Ω is holomorphically convex in U and

Int(Ω) = Ω as in Theorem 1.1 then (Ω, U ) is a Runge pair (see Lemma 2.7

of [Di1]) Second, the hypothesis (1.1.b) in Theorem 1.1 is equivalent to (1.4.c)

if we consider Φt (z) := (1 + t)z Finally, a standard reasoning using Lebesgue

dominated convergence theorem (see the remark after Lemma 2.1) shows that at

a good boundary point ξ ∈ ∂Ω, the only Jensen measures with barycenter at ξ relative to the cone P SH(Ω) ∩ C(Ω) is exactly the Dirac mass δ ξ Summing up,

even in the special case where A is finite union of complex hyperplanes, Theorem

1.2 is still stronger than Theorem 1.1 since it requires slightly weaker assumptions (ii) The assumption (1.4.d) is satisfied if there exists a strong plurisubharmonic

barrier u at ξ which is continuous on Ω i.e., u ∈ P SH(Ω) ∩ C(Ω), u(ξ) = 0, u < 0

on Ω\ {ξ} We do not know if the existence of a strong plurisubharmonic barrier

at ξ implies that J c

ξ ={δ ξ }.

(iii) Theorem 1.4 should be compared to Theorem 3.2 in [Di2] where it shows that

if a uniformly bounded sequence {f j } j ≥1 ⊂ C(∂Ω) converges locally uniformly to a continuous function f away from a compact pluripolar subset of ∂Ω then P B f j ,Ω

is pointwise convergence outside a pluripolar subset of Ω

(iv) The assumption (1.4.a) makes sense in view of a classical example of Wermer about a domain which is biholomorphically equivalent to the bidisk but is not

Runge inC2 On the other hand, (1.4.a) is realized under the following additional conditions: Ω is holomorphically convex in U, Φ t is a biholomorphic mapping from

a fixed neighbourhood ˜Ω b U of Ω onto Φ t( ˜Ω) b U for every t To see this, we

choose pseudoconvex domains Ωb Ω′ b Ω′′ b ˜Ω such that Ω′ is holomorphically

convex in U and

Φt(Ω) b Ω′ b Φt(Ω′′ ), ∀0 ≤ t < ε.

Here we may decrease ε if necessary This implies that every component of the

map Φ−1 t : Φt(Ω′′ → Ω ′′ can be approximated uniformly on Φ

t(Ω) by holomor-phic functions can be approximated uniformly on compact sets by holomorholomor-phic

functions on U It is then easy to see that (Φ t (Ω), U ) is a Runge pair for every t Under stronger convexity conditions of Ω and the boundary cluster set S, we also

able to describe, in our second main result, holomorphical hull of certain compact sets with disks as fibers

Theorem 1.5 Let Ω, U, A, f, Φ t be as in Theorem 1.4 Assume further that the following conditions are met:

(1.5.a) Ω is holomorphically convex in U, Int (Ω) = Ω.

(1.5.b) Every ξ ∈ S(⊂ ∂Ω) is a good boundary point.

(1.5.c) Φ t is a biholomorphic map from a fixed neighbourhood ˜Ωb U of Ω onto

Φt( ˜Ω)b U for every t.

Set

K := {(z, w) : z ∈ ∂Ω, |w| ≤ |f(z)|}.

Then ˜ G A,Ω is continuous on Ω and

ˆ

K U ×C ∩ (Ω × C) = {(z, w) : z ∈ Ω, log |w| + ˜ G A,Ω (z) ≤ 0}.

We use the above theorem to give a short of continuity for our Green functions when the pole sets and the domains both vary

Corollary 1.6 Let Ω, U, A, f, Φ t be as in Theorem 1.5 Let {h j } j ≥1 be a sequence

of holomorphic functions on U such that dh j ̸= 0 on a dense subset of the hy-persurface A j := {z ∈ U : h j (z) = 0 } and h j converges uniformly to f on Ω Let {Ω j } j ≥1 b U be a sequence pseudoconvex domains such that every compact subset of Ω is included in Ω j for j large enough and Ω j , U, A j satisfies the as-sumptions of Theorem 1.5 for every j ≥ 1 Then the sequence ˜ G A j ,Ω j converges locally uniformly to ˜ G A,Ω on Ω.

Observe that Ωj is not assumed relatively compact in Ω Next, we present another

application of Theorem 1.2 to existence and uniqueness of solution of certain Monge-Amp`ere equations with unbounded boundary values.

Theorem 1.7 Let Ω, U, A, f, Φ t be as in Theorem 1.4 Suppose further that S =

∅ i.e., Φ t(Ω) b Ω for t > 0 near 0 and that every ξ ∈ (∂Ω) \ A admits a

strong continuous plurisubharmonic barrier Then u = ˜ G A,Ω is the unique function satisfying the following conditions:

(1.7.a) u ∈ P SH(Ω) ∩ C(Ω), infΩu > −∞;

(1.7.b) (dd c u) n = 0, u < − log |f|on Ω;

(1.7.c) lim

z →ξ u(z) = − log |f(ξ)| for every ξ ∈ (∂Ω) \ A.

Remarks 1 The preceding theorem can be applied to the situation where Ω

is a bounded strictly convex domain in Cn that contains the origin and A is a finite union of d complex hyperplane passing through the origin In that case, the defining function f can be taken as a homogeneous polynomial of degree d.

2 Consider the case where Ω is the open unit ball in C2 and A is the (possibly

singular) complex curve {(z, w) ∈ C2 : z m = w n } where m, n are non-negative

integers Then by setting

Φt (z, w) = ((1 − t) n z, (1 − t) m w), 0 ≤ t < 1.

We obtain an isotopy family of biholomorphic maps on Ω More elementary

com-putations show that Φt (A) = A and that Φ t(Ω) b Ω for t ∈ (0, 1) Thus we may

apply again Theorem 1.7 to deduce the existence and uniqueness solution to the mentioned above Monge-Amp`ere equation

Finally, beside the special examples given in the above remarks, it is useful to

have substantial classes of domains Ω and complex hypersurfaces A verifying the

technical assumptions given in Theorem 1.4 For this purpose, we first present

a situation in which it is easy to locate the boundary cluster set of an isotopy family Φt

Proposition 1.8 Let Ω be a bounded domain in Cn and ρ be a real-valued C1

smooth function defined on an open neighbourhood of Ω such that Ω := {ρ < 0} Let Φ t be a family isotopy of biholomorphic maps on Ω Then for every boundary cluster point ξ of this family we have

ℜ[

n

∑

i=1

∂ρ

∂z i (ξ)

∂Φ i

∂t (0, ξ)

]

= 0.

Observe that the boundary of Ω is not supposed to be smooth

On the other hand, it is not easy in general to construct an isotopy family (which is different from the identity) that fixes a complex hypersurface Nev-ertheless, by invoking a extension theorem for holomorphic functions defined

on complex submanifolds of pseudoconvex domains in Cn, we are able to prove

the following result In what follows, we will denote by π the projection C n →

Cn −1 , (z

1, · · · , z n)7→ z ′ := (z

1, · · · , z n −1 ).

Proposition 1.9 Let U ′ b U be pseudoconvex domains in C n and f1, · · · , f k

be holomorphic functions on π(U ) Assume that f i (z ′) ̸= f j (z ′ ) for every i ̸= j and z ∈ π(U ′ ) Then there exist ε > 0 and a holomorphic function φ on V

ε :=

U ′ × {t ∈ C : |t| < ε} such that the map

(2) Φt (z ′ , z n ) := ((1 + t)z ′ , z n + tφ(z ′ , z n , t))

defines an isotopy family of biholomorphic mappings on every domain Ω relatively compact in U ′ Moreover, we have

(3) Φt (z ′ , f j (z ′ )) = ((1 + t)z ′ , f j ((1 + t)z ′)) ∀1 ≤ j ≤ k, ∀(z ′ , t) ∈ V ε

By combination of Proposition 1.8 and Proposition 1.9 we immediately get the

following result which partially deals with the case where A is a finite union of disjoint holomorphic graphs.

Proposition 1.10 Let U ′ , U, f1, · · · , f k be as in Proposition 1.9 Let A := A1∪

· · · ∪ A k , where

A j :={(z ′ , f

j (z ′ )) : z ′ ∈ π(U ′)}, 1 ≤ j ≤ k.

Suppose that Ω is a pseudoconvex domain with compact closure in U ′ and satisfies the following conditions:

(1.10.a) (Ω, U ′ ) and (U ′ , U ) are Runge pairs.

(1.10.b) S j ∩ A = ∅ for every 1 ≤ j ≤ k,where

S j :=

{

(ξ ′ , ξ n)∈ ∂Ω : ℜ[

n −1

∑

i=1

ξ i ∂ρ

∂z i

(ξ) + g j (ξ ′)∂ρ

∂z n

(ξ)

]

= 0 }

and

g j (z ′) := lim

λ →1

f j (λz ′)− f j (z ′)

λ − 1 , ∀z ′ ∈ U ′ . (1.10.c) Every point ξ ∈ S admits a strong plurisubharmonic barrier.

Then ˜ G A,Ω is continuous on Ω.

Remark We consider below a situation where Proposition 1.10 is applicable.

Let k ≥ 1 and

A := {(z, w) ∈ C2 : w = z j , 1 ≤ j ≤ k}.

Let Ω be an open ball in C2 with center at (a, b) ∈ C2 such that the disk π(U )

lies outside the closed unit disk in C Suppose that

ℜ[z(¯z − ¯a) + jz j( ¯w − ¯b)] ̸= 0, ∀(z, w) ∈ ∂Ω, ∀1 ≤ j ≤ k.

By considering U ′ b U be larger balls that contain Ω and ρ(z, w) := |z − a|2+

|w −b|2−r2 where r is the radius of Ω we may apply Proposition 1.10 to conclude

that ˜G A,Ω is continuous on Ω In fact, with a little more effort we can check that

all the conditions in Theorem 1.5 are also satisfied

Acknowledgements This article has been written during a stay of the

first-named authors at VIASM in the winter of 2014 He wishes wish to thank VIASM for financial support and the warm hospitality This work is dedicated to the memory of Professor Nguyen Thanh Van who has been a guiding light for our research The project is supported by the grant 101.02-2013.11 from the NAFOS-TED program

II Necessary background facts

The aim of this section is to introduce notions and quickly review facts that we

will make use of later on Throughout this paper, by U we mean a pseudoconvex

domain in Cn For a compact subset K of U , we let ˆ K U be the holomorphic hull

of K in U i.e.,

ˆ

K U :={z ∈ U : |f(z)| ≤ ∥f∥ K for all f holomorphic on U }.

The solution to the Levi problem (see Theorem 4.3.4 in [H¨o]) implies that ˆK U

coincides with the ˆK P SH(U ) , the plurisubharmonic hull of K which is defined by

ˆ

K P SH(U ) ={z ∈ U : u(z) ≤ sup

x ∈K u(x) for every u ∈ P SH(U)}.

Recall that if Ω is a pseudoconvex domain contained in U then we say that (Ω, U ) is a Runge pair if holomorphic functions on Ω can be locally uniformly approximated by holomorphic functions on U It is easy to see that if (Ω, U ) is

a Runge pair and K is a compact subset of Ω then ˆ KΩ = ˆK U The following result of Bishop ([Bi]) about representing measure is also quite helpful: For every

a ∈ ˆ K U , there exists a probability measure µ supported on K such that

log|f(a)| ≤

∫

K

log|f|dµ for all f holomorphic on U.

By a classical approximation of Bremermann (see Theorem 9 in [Si1]), the above inequality in fact holds not only for functions of the form log|f| but also for con-tinuous plurisubharmonic function on U Since every concon-tinuous plurisubharmonic

on a small neighbourhood of ˆK U may be continued to a continuous plurisubharmonic

function on U , the above inequality holds true for functions which are only

con-tinuous and plurisubharmonic near ˆK U This information will be important for

us in the proof of Theorem 1.3

The cluster set of a sequence {K j } j ≥1 ⊂ C n is denoted by cl{K j } j ≥1 More

precisely, we define

cl{K j } j ≥1 :={z ∈ C n:∃z j k ∈ K j k , z j k → z as k → ∞}.

The following property about lower semicontinuity of holomorphic hulls is

straight-forward: If {K j } j ≥1 is a sequence of compact subsets of a pseudoconvex domain

U ⊂ C n then the following inclusion holds

(4) cl{[ (K j)U } j ≥1 ⊂ ˆ K U

This fact will be used in the proof of Theorem 1.2 and Theorem 1.3

The following types of measures are quite useful in investigating Perron-Bremermann

envelopes constructed in the preceding section For a point z ∈ Ω and a compact subset K of Ω, following [Wi] (see also [DW]) we define two classes of Jensen measures,

J z(Ω) =

{

µ ∈ B(Ω) : u(z) ≤

∫

Ω

u ∗ dµ ∀u ∈ P SH(Ω), sup u < ∞},

J z c(Ω) =

{

µ ∈ B(Ω) : u(z) ≤

∫

Ω

u ∗ dµ ∀u ∈ P SH(Ω) ∩ C(Ω)},

where B(Ω) is the set of Borel probability measures with compact support in Ω.

By an abuse of notation, we will drop Ω in the notation of Jensen measures if there is no risk of confusion A general duality of Edwards in [Ed] implies the following connections between plurisubharmonic envelopes and Jensen measures See [Wi] and [DW] for details

Lemma 2.0 Let Ω be a bounded domain in Cn and φ be a lower semicontinuous function on Ω Then we have

sup{u(z) : u ∈ P SH(Ω), u ∗ ≤ φ on ∂Ω} = inf{ ∫ φdµ : µ ∈ J z

}

∀z ∈ Ω,

sup{u(z) : u ∈ P SH(Ω) ∩ C(Ω), u ≤ φ on ∂Ω} = inf{ ∫ φdµ : µ ∈ J c

z

}

, ∀z ∈ Ω

The above duality relations are exploited in [DW] in studying approximation problems of plurisubharmonic functions Using the same lines of argument as

in Theorem 3.1 in [DW], we have the following result on continuity of Perron-Bremermann envelopes

Lemma 2.1 Let Ω be a bounded domain (not necessarily regular) such that

J z = J z c for every z ∈ Ω Let φ be a real-valued continuous function defined on

∂Ω Then the following assertions hold.

(i) P B φ,Ω (z) = v(z) := sup {u(z) : u ∈ P SH(Ω) ∩ C(Ω), u ≤ φ on ∂Ω} for every

z ∈ Ω In particular P B φ,Ω is lower semicontinuous on Ω.

(ii) For every ξ ∈ ∂Ω such that J c

ξ ={δ ξ } we have

lim inf

z →ξ P B φ,Ω ≥ f(ξ).

(iii) There exists a sequence {v j } j ≥1 ⊂ P SH(Ω) ∩ C(Ω) such that v j ↑ P B φ on

Ω Furthermore, v j (ξ) ↑ φ(ξ) at every point ξ ∈ ∂Ω such that J c

ξ ={δ ξ }.

Proof (i) Extend φ to a lower semicontinuous function on Ω by setting φ = + ∞

on Ω By Lemma 2.0 and (1) we obtain

{ ∫

Ω

φdµ : µ ∈ J c

z

}

, ∀z ∈ Ω

(6) P B φ,Ω (z) = inf

{ ∫

Ω

φdµ : µ ∈ J z

}

, ∀z ∈ Ω

Combining (5), (6) and using the assumption that J z = J z c for every z ∈ Ω we conclude that v = P B φ,Ω on Ω In particular, P B φ,Ω is lower semicontinuous on

Ω.

(ii) If ξ ∈ ∂Ω satisfies J c

ξ ={δ ξ } then by (3) we obtain v(ξ) = φ(ξ) Combining

this with (i) we infer that

lim inf

z →ξ,z∈Ω P B φ,Ω = lim infz →ξ,z∈Ω v(z) ≥ v(ξ) = φ(ξ).

(iii) By Choquet’s topological lemma (cf Lemma 2.3.4 in [Kl]), there exists a sequence {v j } j ≥1 ⊂ P SH(Ω) ∩ C(Ω) such that v j ↑ v on Ω By (i) and (ii), we infer that v j ↑ P B φ on Ω and v j (ξ) ↑ φ(ξ) for every ξ ∈ ∂Ω such that J c

ξ ={δ ξ }.

The proof is complete

Remarks (a) It is clear that if the boundary point ξ ∈ ∂Ω admits a strong plurisubharmonic barrier which is continuous on Ω then J c

ξ ={δ ξ } Conversely, if

we assume that there exists a strong plurisubharmonic barrier u at ξ such that u ∗

is the pointwise limit on Ω of a uniformly bounded sequence of plurisubharmonic functions {u j } j ≥1 defined on neighbourhoods of Ω then J ξ c = {δ ξ } Indeed, take

µ ∈ J c

ξ (Ω) Fix j ≥ 1, by convolving u j with smoothing kernels, we obtain a sequence of uniformly bounded smooth plurisubharmonic functions defined on

neighbourhoods of Ω that decreases to u j on Ω It follows, using Lebesgue’s

dom-inated convergence theorem, that u j (ξ) ≤ ∫Ωu j dµ Applying again Lebesgue’s dominated convergence theorem and letting j → ∞ we infer that u(ξ) ≤∫Ωu ∗ dµ Since u is a strong plurisubharmonic barrier at ξ, this forces µ = δ ξ We are done.

(b) Under the additional condition that Ω is regular, by Lemma 2.1 (a), the

envelope P B φ,Ω is continuous and plurisubharmonic on Ω.

(c) The lemma remains true under the weaker assumptions that J z (∂Ω) = J z c (∂Ω) for all z ∈ Ω, where J z (∂Ω) and J z c (∂Ω) denote the subsets of J z and J z ccontaining

elements with compact support in ∂Ω Indeed, in the application of Edward’s duality theorem, only Jensen measures supported on ∂Ω are relevant since φ =

+∞ on Ω.

We also need the following fact which is essentially contained in [Di1] This crucial result relates holomorphic hull of certain compact sets and Perron-Bremermann envelopes

Lemma 2.2 Let Ω be a relatively compact domain of a pseudoconvex domain

U ⊂ C n Let φ : ∂Ω → R ∪ {+∞} be a lower semicontinuous function Set

K := {(z, w) ∈ C n+1 : z ∈ ∂Ω, log |w| + φ(z) ≤ 0}.

Then the following assertions hold.

(i) {(z, w) : z ∈ Ω, log |w| + P B φ,Ω (z) ≤ 0} ⊂ ˆ K U ×C

(ii) Suppose, in addition that Ω is strictly pseudoconvex i.e., Ω = {ρ < 0} where

ρ is a continuous strictly plurisubharmonic function on a neighbourhood of Ω and

Ω is holomorphically convex in U Then for every sequence of C2 smooth functions {φ j } j ≥1 defined on neighbourhoods of Ω such that φ j ↑ φ on ∂Ω we have

{(z, w) : z ∈ Ω, log |w| + ˆφ(z) ≤ 0} = ˆ K U ×C ∩ (Ω × C),

where ˆ φ := lim j →∞ P B φ j ,Ω

Proof (i) This is exactly Lemma 2.5 in [Di1] In fact, we only need lower

semi-continuity of φ to guarantee compactness of K.

(ii) We use the same argument as the one given in Step 1 of Lemma 2.6 in [Di1] For the reader convenience, we sketch some details Set

K j :={(z, w) ∈ C n+1

: z ∈ ∂Ω, log |w| + φ j (z) ≤ 0}.

Using strict pseudoconvexity of Ω, we infer that P B φ j ,Ω ∈ P SH(Ω) ∩ C(Ω) and

P B φ j ,Ω = φ j on ∂Ω Moreover, a gluing process using the exhaustion function for Ω, we may extend P B φ j ,Ω to a continuous plurisubharmonic function near Ω Combing this with Theorem 4.3.4 in [H¨o] and recalling that Ω is holomorphically

convex in U, we obtain

ˆ

K U×C ∩ (Ω × C) ⊂ {(z, w) : z ∈ Ω, log |w| + P B φ j ,Ω (z) ≤ 0}.

Thus, in view of (i) we have

{(z, w) : z ∈ Ω, log |w| + P B φ j ,Ω (z) ≤ 0} = ˆ K U ×C ∩ (Ω × C).

Finally, we note that K j ↓ K Therefore [ (K j)U ×C ↓ ˆ K U ×C The desired conclusion

now follows

We have end up this section with another auxiliary fact that will be used in the proof of the Theorem 1.4 and Theorem 1.5

Lemma 2.5 Let Ω ′ ⊂ Ω ⊂ C n be bounded domains and φ be a real-valued continuous function defined on ∂Ω ′ ∪ ∂Ω Assume that J c

z = J z for every z ∈ Ω and J z c ={δ z } for every z ∈ ∂Ω ∩ ∂Ω ′ Suppose that for some point z

0 ∈ Ω ′ and

ε > 0 we have

P B φ,Ω (z0) + ε < P B φ,Ω ′ (z0).

Then there exists ξ ∈ Ω ∩ ∂Ω ′ such that

P B φ,Ω (ξ) + ε < f (ξ).

Proof By Lemma 2.1, we can find a sequence {v j } j ≥1 ⊂ P SH(Ω) ∩ C(Ω) such that v j ↑ P B φ on Ω, v j ≤ φ on ∂Ω and that v j (ξ) ↑ φ(ξ) at every point

ξ ∈ ∂Ω ∩ ∂Ω ′ Fix j ≥ 1, we claim that there exists ξ j ∈ Ω ∩ ∂Ω ′ such that

v j (ξ j ) + ε < φ(ξ j ).

Assume otherwise, then for every t ∈ Ω ∩ ∂Ω ′ we have

v j (t) + ε ≥ φ(t).

Notice that, from (1) and the assumption, there exists u ∈ P SH(Ω) with u ∗ ≤ φ

on ∂Ω and

(7) v j (z0) + ε ≤ P B φ,Ω (z0) + ε < u(z0).

Consider the function

u j (z) =

{

v j (z) for z ∈ Ω \ Ω ′

max{v j (z), u(z) − ε} for z ∈ Ω ′ .

Then u j ∈ P SH(Ω) and it is also easy to check that

lim sup

z →t u j (z) ≤ φ(t) ∀t ∈ ∂Ω.

Therefore u j ≤ P B φ,Ω entirely on Ω In particular, this is true at z0 By letting

j → ∞ we get a contradiction to (7) The claim follows Next, by passing to subsequence, we may assume that ξ j → ξ ∈ ∂Ω ′ By continuity of v

j , φ and the

fact that the sequence {v j } j ≥1 is increasing on Ω we obtain

lim

j →∞ v j (ξ) ≤ lim sup

j →∞ v j (ξ j)≤ φ(ξ) − ε.

If ξ ∈ ∂Ω∩∂Ω ′ then by the assumption J c

ξ ={δ ξ } Hence, by the choice of {v j } j ≥1

we have v j (ξ) ↑ f(ξ) This is absurd Thus ξ ∈ Ω ∩ ∂Ω ′ Since v

j (ξ) ↑ P B φ,Ω (ξ),

we are done

III Proofs of the results

We claim no originality for the first result of this section This lemma will be needed in the proof of our key Lemma 3.1

Lemma 3.0 Let K be a compact subset of a pseudoconvex domain U ⊂ C n Let f be a holomorphic function on U and L be a compact subset of the complex hypersurface A := {z ∈ U : f(z) = 0} Then

\

(K ∪ L) U ⊂ ˆ K U ∪ A.