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

Perron-Bremermann envelopes and pluricomplex Green functions 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 in Cn 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 GA,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 function is that they provide fundamental solutions to the (complex) Monge-Amp`ere operator (ddc )n . Moreover, if the domain is hyperconvex i.e, there exists a negative 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 ˜ A,Ω := GA,Ω −log |f | is maximal plurisubharmonic and 3.2 in [LS1]) the function G 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 u(z) = 0 whereas z→ξ ˜ A,Ω (z) = − log |f (ξ)|. Thus, at least lim sup u(z) < 0 if t ∈ (∂Ω) \ {ξ}, then lim G z→t z→ξ in the case where Ω is strictly pseudoconvex (eg. Ω is a ball), we may consider ˜ A,Ω as a solution of the equation G c n u ∈ PSH(Ω) ∈ L∞ loc (Ω), (dd u) = 0, lim u(z) = − log |f (ξ)|, ∀ξ ∈ (∂Ω) \ A. z→ξ Here (ddc )n is the complex Monge-Amp`ere operator which is defined as positive Borel measures over the class of locally bounded plurisubharmonic functions (see ˜ A,Ω on Ω is not known yet even in the case [Kl]). However, the continuity of G 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 ˜ A,Ω is the unique solution to the boundary, it is also not clear that the function G the above equation. The main aim of this paper is to address these two problem simultaneously. ˜ A,Ω and hence GA,Ω and then apply it to obtain First we discuss continuity of G a sort of uniqueness of the Monge-Ampere equation with unbounded boundary 1 2 values. The first result on continuity of GA,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 U be pseudoconvex domains in Cn 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 Ω for α < t < 1. (1.1.c) Every ξ ∈ (∂Ω) \ V is a good boundary point. ˜ A,Ω is continuous on Ω. Then G 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 uj that defined on neighborhoods of Ω. We should say that the assumption on uniform boundedness of uj (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˜ A,Ω in terms of a limit of Perron-Bremermann envelopes associating to a tion of G 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 Bφ,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 u(z), ∀ξ ∈ Ω. z→ξ 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 Bφ,Ω )∗ ≤ φ on ∂Ω, and so P Bφ,Ω ∈ P SH(Ω) satisfying (ddc P Bφ,Ω )n = 0 on Ω. Furthermore if Ω is B−regular in the sense of Sibony (cf. [Si2]) then P Bφ,Ω extends continuously to Ω and P Bφ,Ω = φ on ∂Ω. At this point, we note that, by Edwards’ duality theorem (cf. [Ed], [Si2], [Wi] and [DW]), P Bφ,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 Bφ,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. 3 Definition 1.2. Let Ω be a domain in Cn . By an isotopy family of biholomorphic mappings on Ω we mean a continuous map Φ : [0, ε) × Ω → Cn (ε > 0), such that the following statements hold. (1.2.a) For each t ∈ [0, ε), Φt (·) = Φ(t, ·) is a homeomorphism between Ω and Φt (Ω); moreover, Φt maps Ω biholomorphically onto Φt (Ω). (1.2.b) For all z ∈ Ω, t → Φ−1 t (z) is real-analytic on [0, ε). (1.2.c) Φ−1 converges uniformly to Φ−1 t 0 = Id on Ω as t → 0. An isotopy family of biholomorphic mappings on Ω leads to the following interesting subset of ∂Ω which enables us to decide when two classes of Jensen measures introduced above coincide. Definition 1.3. If Φt is an isotopy family of biholomorphic mappings on Ω, we define the boundary cluster set of Φt (relative to Ω) as the limit points of sequences of elements in Ω ∩ Φt (∂Ω) as t → 0. Now we are able to formulate the first main result of the paper. Theorem 1.4. Let Ω U be pseudoconvex domains in Cn 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) Φt (Ω) U and (Φt (Ω), U ) is a Runge pair for every t ∈ [0, ε). (1.4.b) A is invariant under Φt i.e., Φt (A ∩ Ω) = A ∩ Φt (Ω) for every t. (1.4.c) The boundary cluster set of Φt 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 Φt (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 {δξ }. Then the following assertions hold: (i) For every sequence {fj }j≥1 ⊂ C(∂Ω) such that fj converges locally uniformly to − log |f | on (∂Ω) \ A, fj < − log |f | on ∂Ω and inf ∂Ω fj > −M for some constant ˜ A,Ω on Ω. M > 0, the sequence P Bfj ,Ω converges pointwise to G ˜ A,Ω is continuous on Ω. (ii) G 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 δξ . 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 Jξc = {δξ }. 4 (iii) Theorem 1.4 should be compared to Theorem 3.2 in [Di2] where it shows that if a uniformly bounded sequence {fj }j≥1 ⊂ C(∂Ω) converges locally uniformly to a continuous function f away from a compact pluripolar subset of ∂Ω then P Bfj ,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 in C2 . On the other hand, (1.4.a) is realized under the following additional conditions: Ω is holomorphically convex in U, Φt is a biholomorphic mapping from ˜ ˜ a fixed neighbourhood Ω U of Ω onto Φt (Ω) U for every t. To see this, we ′ ′′ ˜ choose pseudoconvex domains Ω Ω Ω Ω such that Ω′ is holomorphically convex in U and Φt (Ω) Ω′ Φt (Ω′′ ), ∀0 ≤ t < ε. Here we may decrease ε if necessary. This implies that every component of the map Φ−1 : Φt (Ω′′ ) → Ω′′ can be approximated uniformly on Φt (Ω) by holomort phic functions can be approximated uniformly on compact sets by holomorphic functions on U. It is then easy to see that (Φt (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, Φt 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) Φt is a biholomorphic map from a fixed neighbourhood Ω U of Ω onto ˜ Φt (Ω) U for every t. Set K := {(z, w) : z ∈ ∂Ω, |w| ≤ |f (z)|}. ˜ A,Ω is continuous on Ω and Then G ˆ U ×C ∩ (Ω × C) = {(z, w) : z ∈ Ω, log |w| + G ˜ A,Ω (z) ≤ 0}. K 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, Φt be as in Theorem 1.5. Let {hj }j≥1 be a sequence of holomorphic functions on U such that dhj ̸= 0 on a dense subset of the hypersurface Aj := {z ∈ U : hj (z) = 0} and hj converges uniformly to f on Ω. Let {Ωj }j≥1 U be a sequence pseudoconvex domains such that every compact subset of Ω is included in Ωj for j large enough and Ωj , U, Aj satisfies the as˜ A ,Ω converges sumptions of Theorem 1.5 for every j ≥ 1. Then the sequence G j j ˜ A,Ω on Ω. locally uniformly to G 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, Φt be as in Theorem 1.4. Suppose further that S = ∅ i.e., Φt (Ω) Ω for t > 0 near 0 and that every ξ ∈ (∂Ω) \ A admits a 5 ˜ A,Ω is the unique function strong continuous plurisubharmonic barrier. Then u = G satisfying the following conditions: (1.7.a) u ∈ P SH(Ω) ∩ C(Ω), inf Ω u > −∞; (1.7.b) (ddc u)n = 0, u < − log |f |on Ω; (1.7.c) lim u(z) = − log |f (ξ)| for every ξ ∈ (∂Ω) \ A. z→ξ 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 = wn } 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 computations show that Φt (A) = A and that Φt (Ω) 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 C 1 smooth function defined on an open neighbourhood of Ω such that Ω := {ρ < 0}. Let Φt be a family isotopy of biholomorphic maps on Ω. Then for every boundary cluster point ξ of this family we have n [∑ ] ∂ρ ∂Φi ℜ (ξ) (0, ξ) = 0. ∂zi ∂t i=1 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. Nevertheless, 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 Cn → Cn−1 , (z1 , · · · , zn ) → z ′ := (z1 , · · · , zn−1 ). Proposition 1.9. Let U ′ U be pseudoconvex domains in Cn and f1 , · · · , fk be holomorphic functions on π(U ). Assume that fi (z ′ ) ̸= fj (z ′ ) for every i ̸= j and z ∈ π(U ′ ). Then there exist ε > 0 and a holomorphic function φ on Vε := U ′ × {t ∈ C : |t| < ε} such that the map (2) Φt (z ′ , zn ) := ((1 + t)z ′ , zn + tφ(z ′ , zn , t)) defines an isotopy family of biholomorphic mappings on every domain Ω relatively compact in U ′ . Moreover, we have (3) Φt (z ′ , fj (z ′ )) = ((1 + t)z ′ , fj ((1 + t)z ′ )) ∀1 ≤ j ≤ k, ∀(z ′ , t) ∈ Vε . 6 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 , · · · , fk be as in Proposition 1.9. Let A := A1 ∪ · · · ∪ Ak , where Aj := {(z ′ , fj (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) Sj ∩ A = ∅ for every 1 ≤ j ≤ k,where n−1 ] } { [∑ ∂ρ ∂ρ ′ (ξ) + gj (ξ ′ ) (ξ) = 0 Sj := (ξ , ξn ) ∈ ∂Ω : ℜ ξi ∂zi ∂zn i=1 and fj (λz ′ ) − fj (z ′ ) , ∀z ′ ∈ U ′ . λ→1 λ−1 (1.10.c) Every point ξ ∈ S admits a strong plurisubharmonic barrier. ˜ A,Ω is continuous on Ω. Then G gj (z ′ ) := lim 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 ′ 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 ˜ A,Ω is continuous on Ω. In fact, with a little more effort we can check that that G all the conditions in Theorem 1.5 are also satisfied. Acknowledgements. This article has been written during a stay of the firstnamed 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 NAFOSTED 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 ˆ U be the holomorphic hull domain in Cn . For a compact subset K of U , we let K of K in U i.e., ˆ U := {z ∈ U : |f (z)| ≤ ∥f ∥K for all f holomorphic on U }. K 7 ˆU The solution to the Levi problem (see Theorem 4.3.4 in [H¨o]) implies that K ˆ coincides with the KP SH(U ) , the plurisubharmonic hull of K which is defined by ˆ P SH(U ) = {z ∈ U : u(z) ≤ sup u(x) for every u ∈ P SH(U )}. K x∈K 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 ˆΩ = K ˆ U . The following a Runge pair and K is a compact subset of Ω then K result of Bishop ([Bi]) about representing measure is also quite helpful: For every ˆ U , there exists a probability measure µ supported on K such that a∈K ∫ log |f (a)| ≤ log |f |dµ for all f holomorphic on U. K 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 continuous plurisubharmonic function on U . Since every continuous plurisubharmonic ˆ U may be continued to a continuous plurisubharmonic on a small neighbourhood of K function on U , the above inequality holds true for functions which are only conˆ U . This information will be important for tinuous and plurisubharmonic near K us in the proof of Theorem 1.3. The cluster set of a sequence {Kj }j≥1 ⊂ Cn is denoted by cl{Kj }j≥1 . More precisely, we define cl{Kj }j≥1 := {z ∈ Cn : ∃zjk ∈ Kjk , zjk → z as k → ∞}. The following property about lower semicontinuity of holomorphic hulls is straightforward: If {Kj }j≥1 is a sequence of compact subsets of a pseudoconvex domain U ⊂ Cn then the following inclusion holds ˆU . cl{(Kj )U }j≥1 ⊂ K (4) 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, ∫ { } Jz (Ω) = µ ∈ B(Ω) : u(z) ≤ u∗ dµ ∀u ∈ P SH(Ω), sup u < ∞ , Ω Jzc (Ω) ∫ { } ∗ = µ ∈ B(Ω) : u(z) ≤ u dµ ∀u ∈ P SH(Ω) ∩ C(Ω) , 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. 8 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 ∈ Jz ∀z ∈ Ω, {∫ } sup{u(z) : u ∈ P SH(Ω) ∩ C(Ω), u ≤ φ on ∂Ω} = inf φdµ : µ ∈ Jzc , ∀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 PerronBremermann envelopes. Lemma 2.1. Let Ω be a bounded domain (not necessarily regular) such that Jz = Jzc for every z ∈ Ω. Let φ be a real-valued continuous function defined on ∂Ω. Then the following assertions hold. (i) P Bφ,Ω (z) = v(z) := sup{u(z) : u ∈ P SH(Ω) ∩ C(Ω), u ≤ φ on ∂Ω} for every z ∈ Ω. In particular P Bφ,Ω is lower semicontinuous on Ω. (ii) For every ξ ∈ ∂Ω such that Jξc = {δξ } we have lim inf P Bφ,Ω ≥ f (ξ). z→ξ (iii) There exists a sequence {vj }j≥1 ⊂ P SH(Ω) ∩ C(Ω) such that vj ↑ P Bφ on Ω. Furthermore, vj (ξ) ↑ φ(ξ) at every point ξ ∈ ∂Ω such that Jξc = {δξ }. Proof. (i) Extend φ to a lower semicontinuous function on Ω by setting φ = +∞ on Ω. By Lemma 2.0 and (1) we obtain {∫ } (5) v(z) = inf φdµ : µ ∈ Jzc , ∀z ∈ Ω Ω {∫ (6) } φdµ : µ ∈ Jz , ∀z ∈ Ω P Bφ,Ω (z) = inf Ω Combining (5), (6) and using the assumption that Jz = Jzc for every z ∈ Ω we conclude that v = P Bφ,Ω on Ω. In particular, P Bφ,Ω is lower semicontinuous on Ω. (ii) If ξ ∈ ∂Ω satisfies Jξc = {δξ } then by (3) we obtain v(ξ) = φ(ξ). Combining this with (i) we infer that lim inf P Bφ,Ω = lim inf v(z) ≥ v(ξ) = φ(ξ). z→ξ,z∈Ω z→ξ,z∈Ω (iii) By Choquet’s topological lemma (cf. Lemma 2.3.4 in [Kl]), there exists a sequence {vj }j≥1 ⊂ P SH(Ω) ∩ C(Ω) such that vj ↑ v on Ω. By (i) and (ii), we infer that vj ↑ P Bφ on Ω and vj (ξ) ↑ φ(ξ) for every ξ ∈ ∂Ω such that Jξc = {δξ }. 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 Jξc = {δξ }. 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 {uj }j≥1 defined on neighbourhoods of Ω then Jξc = {δξ }. Indeed, take µ ∈ Jξc (Ω). Fix j ≥ 1, by convolving uj with smoothing kernels, we obtain a sequence of uniformly bounded smooth plurisubharmonic functions defined on 9 neighbourhoods of Ω that decreases to uj on ∫Ω. It follows, using Lebesgue’s dominated convergence theorem, that uj (ξ) ≤ Ω uj dµ. Applying again Lebesgue’s ∫ dominated convergence theorem and letting j → ∞ we infer that u(ξ) ≤ Ω u∗ dµ. Since u is a strong plurisubharmonic barrier at ξ, this forces µ = δξ . We are done. (b) Under the additional condition that Ω is regular, by Lemma 2.1 (a), the envelope P Bφ,Ω is continuous and plurisubharmonic on Ω. (c) The lemma remains true under the weaker assumptions that Jz (∂Ω) = Jzc (∂Ω) for all z ∈ Ω, where Jz (∂Ω) and Jzc (∂Ω) denote the subsets of Jz and Jzc containing 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 ⊂ Cn . Let φ : ∂Ω → R ∪ {+∞} be a lower semicontinuous function. Set K := {(z, w) ∈ Cn+1 : z ∈ ∂Ω, log |w| + φ(z) ≤ 0}. Then the following assertions hold. ˆ U ×C . (i) {(z, w) : z ∈ Ω, log |w| + P Bφ,Ω (z) ≤ 0} ⊂ K (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 C 2 smooth functions {φj }j≥1 defined on neighbourhoods of Ω such that φj ↑ φ on ∂Ω we have ˆ U ×C ∩ (Ω × C), {(z, w) : z ∈ Ω, log |w| + φ(z) ˆ ≤ 0} = K where φˆ := limj→∞ P Bφj ,Ω . Proof. (i) This is exactly Lemma 2.5 in [Di1]. In fact, we only need lower semicontinuity 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 Kj := {(z, w) ∈ Cn+1 : z ∈ ∂Ω, log |w| + φj (z) ≤ 0}. Using strict pseudoconvexity of Ω, we infer that P Bφj ,Ω ∈ P SH(Ω) ∩ C(Ω) and P Bφj ,Ω = φj on ∂Ω. Moreover, a gluing process using the exhaustion function for Ω, we may extend P Bφj ,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 U ×C ∩ (Ω × C) ⊂ {(z, w) : z ∈ Ω, log |w| + P Bφ ,Ω (z) ≤ 0}. K j Thus, in view of (i) we have ˆ U ×C ∩ (Ω × C). {(z, w) : z ∈ Ω, log |w| + P Bφj ,Ω (z) ≤ 0} = K ˆ U ×C . The desired conclusion Finally, we note that Kj ↓ K. Therefore (Kj )U ×C ↓ K now follows. 10 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 ⊂ Cn be bounded domains and φ be a real-valued continuous function defined on ∂Ω′ ∪ ∂Ω. Assume that Jzc = Jz for every z ∈ Ω and Jzc = {δz } for every z ∈ ∂Ω ∩ ∂Ω′ . Suppose that for some point z0 ∈ Ω′ and ε > 0 we have P Bφ,Ω (z0 ) + ε < P Bφ,Ω′ (z0 ). Then there exists ξ ∈ Ω ∩ ∂Ω′ such that P Bφ,Ω (ξ) + ε < f (ξ). Proof. By Lemma 2.1, we can find a sequence {vj }j≥1 ⊂ P SH(Ω) ∩ C(Ω) such that vj ↑ P Bφ on Ω, vj ≤ φ on ∂Ω and that vj (ξ) ↑ φ(ξ) at every point ξ ∈ ∂Ω ∩ ∂Ω′ . Fix j ≥ 1, we claim that there exists ξj ∈ Ω ∩ ∂Ω′ such that vj (ξj ) + ε < φ(ξj ). Assume otherwise, then for every t ∈ Ω ∩ ∂Ω′ we have vj (t) + ε ≥ φ(t). Notice that, from (1) and the assumption, there exists u ∈ P SH(Ω) with u∗ ≤ φ on ∂Ω and vj (z0 ) + ε ≤ P Bφ,Ω (z0 ) + ε < u(z0 ). (7) Consider the function { vj (z) for z ∈ Ω \ Ω′ uj (z) = max{vj (z), u(z) − ε} for z ∈ Ω′ . Then uj ∈ P SH(Ω) and it is also easy to check that lim sup uj (z) ≤ φ(t) ∀t ∈ ∂Ω. z→t Therefore uj ≤ P Bφ,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 ξj → ξ ∈ ∂Ω′ . By continuity of vj , φ and the fact that the sequence {vj }j≥1 is increasing on Ω we obtain lim vj (ξ) ≤ lim sup vj (ξj ) ≤ φ(ξ) − ε. j→∞ j→∞ If ξ ∈ ∂Ω∩∂Ω′ then by the assumption Jξc = {δξ }. Hence, by the choice of {vj }j≥1 we have vj (ξ) ↑ f (ξ). This is absurd. Thus ξ ∈ Ω ∩ ∂Ω′ . Since vj (ξ) ↑ P Bφ,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 ⊂ Cn . 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 ˆ U ∪ A. (K ∪ L)U ⊂ K 11 Proof. Fix a point ξ ∈ (K ∪ L)U \ A. Let µ be a representing measure for ξ. Then we have ∫ log |f (ξ)| ≤ log |f |dµ. K∪L ˆ U . We are done. It follows that µ must be supported on K. Thus ξ ∈ K For the proof of Theorem 1.5, we need the following lemma which allows us to compare the Green functions GA,Ω with Perron-Bremermann envelopes of continuous functions defined on slightly smaller domains. Lemma 3.1. Let Ω function on U . Set U ⊂ Cn be pseudoconvex domains and f be a holomorphic Kf,Ω := {(z, w) ∈ Cn+1 : z ∈ ∂Ω, |w| ≤ |f (z)|}. Then the following assertions hold. (i) (Kf,Ω )U ×C ∩ (Ω × C) = {(z, w) ∈ Cn+1 : z ∈ Ω, log |w| + F (z) ≤ 0}, where F : Ω → R ∪ {+∞} is a lower semi-continuous function. (ii) For every sequence {fj }j≥1 ⊂ C(∂Ω) such that and fj < − log |f | on ∂Ω, inf ∂Ω fj > −M for some constant M > 0 and fj converges locally uniformly to − log |f | on (∂Ω) \ A, we have (8) F (z) ≤ lim sup P Bfj ,Ω (z), ∀z ∈ Ω \ A j→∞ (9) ˜ A,Ω (z), ∀z ∈ Ω. lim sup P Bfj ,Ω (z) ≤ G j→∞ In particular F is locally bounded on Ω. ˜ such that Ω Ω ˜ U and that holomor(iii) For every pseudoconvex domain Ω ˜ by holomorphic functions ˜ can be approximated uniformly on Ω phic functions on Ω on U we have ˜ ˜ < F (z), ∀z ∈ Ω. G A,Ω Proof. (i) Since Kf,Ω is invariant under the action (z, w) → (z, eiθ w), θ ∈ R, we deduce that the same also holds for (Kf,Ω )U ×C Now, it is enough to define F : Ω → R ∪ {+∞} such that e−F (z) := sup{|w| : (z, w) ∈ (Kf,Ω )U ×C , z ∈ Ω}. The lower semi-continuity of F follows easily from compactness of (Kf,Ω )U ×C . (ii) Set Kj := {(z, w) : z ∈ ∂Ω, log |w| + fj (z) ≤ 0}. By the assumption, it is clear that Kf,Ω ⊂ Kj for every j. Furthermore, the set K ′ := cl{Kj }j≥1 of limits points {Kj } is included in K ∪ L, where L := {(z, w) : f (z) = 0, |w| ≤ eM }. By Lemma 3.0 we deduce that K ′ U ×C ⊂ (Kf,Ω )U ×C ∪ L ∀j ≥ 1. 12 On the other hand, by Lemma 2.2 (i) we have {(z, w) : z ∈ Ω, log |w| + P Bfj ,Ω (z) ≤ 0} ⊂ (Kj )U ×C . It follows, using the lower semi-continuity of hulls (2), that {(z, w) : z ∈ Ω, log |w| + lim sup P Bfj ,Ω (z) ≤ 0} ⊂ (K ′ )U ×C . j→∞ Combining the above relations we arrive at {(z, w) : z ∈ Ω \ A, log |w| + lim sup P Bfj ,Ω (z) ≤ 0} ⊂ (Kf,Ω )U ×C . j→∞ Thus, by applying the expression of (Kf,Ω )U ×C in terms of the function F obtained in (i), we obtain F (z) ≤ lim sup P Bfj ,Ω (z), ∀z ∈ Ω \ A. j→∞ This proves (7). For (8), it suffices to show that if g is a real-valued continuous function on ∂Ω with g < − log |f | then ˜ A,Ω (z), ∀z ∈ Ω. P Bg,Ω (z) < G To this end, we fix u ∈ P SH(Ω) such that u∗ ≤ g on ∂Ω. Then we have u + log |f | < 0 on ∂Ω. By the maximum principles we infer v := u + log |f | < 0 on Ω. By the definition of GA,Ω we must have v ≤ GA,Ω on Ω. The desired conclusion now follows from the definition of Perron-Bremermann envelopes (1). (iii) We note that the function ˜ ˜ (z) u(z, w) := log |w| + G A,Ω ˜ × C, an open is negative and plurisubharmonic on the pseudoconvex domain Ω neighbourhood of the compact set Kf,Ω . Thus, from Theorem 4.3.4 in [H¨o] we obtain the inclusion ˜ u(z, w) < 0}. (Kf,Ω ) ˜ ⊂ {(z, w) : z ∈ Ω, Ω×C ˜ U ) is a Runge pair, the same property must be true for the pair (Ω ˜× Since (Ω, ˜ × C admits the Hartogs C, U × C). Indeed, ∑ a holomorphic function h on Ω j expansion h(z, w) = j≥0 w hj (z), where hj is holomorphic on Ω. Since hj can ˜ by holomorphic function on be approximated uniformly on compact sets of Ω ˜ × C by holomorphic U , we infer that h is approximated on compact sets of Ω functions in U × C. In particular, (Kf,Ω )Ω×C = (Kf,Ω )U ×C . ˜ Summing up we obtain (Kf,Ω )U ×C ∩ (Ω × C) ⊂ {(z, w) : z ∈ Ω, u(z, w) < 0}. Applying again (i) we arrive at the desired estimate. We need the following elementary fact which may be of independent interest. Lemma 3.2. Let K be a compact subset of an open set U ⊂ Rn and {fj }j≥1 be a sequence of real-valued continuous functions on K. Assume that u : U → R ∪ +∞ is a continuous function function on U satisfying the following conditions: (a) inf U u > −∞; 13 (b) fj < u on K, inf K fj ≥ −M for some constant M > 0 for every j ≥ 1; (c) {fj }j≥1 converges locally uniformly to u on K \ {f = +∞}; Then there exist a sequence {kj }j≥1 of positive integers and a sequence {Fj }j≥1 of real-valued continuous functions on U such that (i) Fj = fkj on K. (ii) Fj < u on U and inf U Fj ≥ −M ′ for some constant M ′ > 0 for every j ≥ 1. (iii) {Fj }j≥1 converges locally uniformly to u on U \ {u = +∞}. It is crucial in the above lemma that u is allowed to be +∞ somewhere on U . Proof. First we set Kj := {x ∈ U : u(x) ≤ j}. Let αj := min (u(x) − fj (x)), βj := max (u(x) − fj (x)). x∈Kj ∩K x∈Kj ∩K Then we can choose a sequence {kj }j≥1 of positive integers such that 0 < αkj ≤ βkj < 1/j ∀j ≥ 1. Next we define the following real-valued functions on U   on K  fk j φj = u − αkj on Kj \ K  j on U \ (Kj ∪ K) and   fkj ψj = u − βkj  −M ′ on K on Kj \ K on U \ (Kj ∪ K), where −M ′ := min{−M, − inf U f − 1}. It is easy to check that φj (resp. ψj ) is real-valued lower (resp. upper) semicontinuous function on U and satisfies −M ′ ≤ ψj ≤ φj , ∀j ≥ 1. By Hahn’s interpolation theorem (cf. Proposition 7.21 in [BP]), we can find a real-valued continuous function Fj on U such that −M ′ < ψj ≤ Fj ≤ φj , ∀j ≥ 1. Obviously Fj = fkj on K, Fj < u on U . It is also easy to check that Fj converges locally uniformly to u on U \ {u = +∞}. Proof of Theorem 1.4. (i) By Lemma 3.1 we have ˜ A,Ω on Ω. lim sup P Bf ,Ω ≤ G j→∞ j It remains to show ˜ A,Ω on Ω. lim inf P Bfj ,Ω ≥ G j→∞ Suppose otherwise, then we can find z0 ∈ Ω, ε > 0 and a sequence {kj }j≥1 ↑ ∞ such that ˜ A,Ω (z0 ), ∀j ≥ 1. P Bfkj ,Ω (z0 ) + ε < G For simplicity of notation, we may assume that kj = j for every j. By the invariant property for Green functions we have GA,Ω ◦ Φ−1 t = GA,Φt (Ω) , ∀t ∈ [0, 1]. 14 Since the curve t → Φ−1 t (z0 ) is not pluri-thin at t = 0 (as we assumed that t → Φ−1 is real-analytic), we obtain t GA,Ω (z0 ) = lim sup GA,Ω (Φ−1 t (z0 )) = lim sup GA,Φt (Ω) (z0 ). t→0 t→0 This implies that ˜ A,Ω (z0 ) = lim sup G ˜ A,Φt (Ω) (z0 ). G t→0 Therefore, we can choose a sequence tj ↓ 0 such that ˜ 0) + ε < G ˜ A,Φt (Ω) (z0 ), ∀j ≥ 1. G(z j By (1.4.c) we can find V ⊂ ∂Ω an open neighbourhood of A ∩ ∂Ω such that V ∩ S = ∅. Choose a sequence {Ωj }j≥1 of strictly pseudoconvex domains with smooth boundaries such that Ωj ↑ Ω and that Ωj contains a connected open neighbourhood Wj of Φ−1 tj ({z0 } ∪ V ). This choice is possible in view of (1.2.c). ˜ Next, we let Ωj be the connected component of Ω∩Φtj (Ωj ) that includes {z0 }∪V . Notice that the domain Ω∗j := Φtj (Ω) is pseudoconvex and by the assumption (a), ˜ j is relatively compact in Ω∗ . (Ω∗j , U ) is a Runge pair for every j ≥ 1. Moreover, Ω j Now, we apply Lemma 3.2 to find that, after passing to a subsequence, we may extend fj to a continuous real-valued function (still denoted by fj ) defined on U such that such that fj < − log |f | on U, {fj }j≥1 converges locally uniformly to − log |f | on U \A and inf j≥1 Fj ≥ −M on U for some constant M > 0 independent of j. Thus, by Lemma 3.1 (ii) and (iii) we have ˜ A,Ω∗ (z0 ) < lim sup P B ˜ (z0 ), ∀j ≥ 1. G j k→∞ fk ,Ωj So for each j ≥ 1 we can find kj ≥ 1 such that ˜ 0 ) + ε < P B ˜ (z0 ). G(z fk ,Ωj j It follows that P Bfkj ,Ω (z0 ) + ε < P Bfk j ˜ j (z0 ), ,Ω ∀j ≥ 1. By the assumption (d) and Theorem 3.5 in [DW] we have Jz (Ω) = Jzc (Ω) for every z ∈ Ω. So we may apply Lemma 2.5 to find, for each j ≥ 1, some point ˜ j such that ξj ∈ ∂ Ω P Bfkj ,Ω (ξj ) + ε < fkj (ξj ). By passing to a subsequence we may assume that {ξj }j≥1 ⊂ Ω \ A and ξj → ξ ∈ S, the boundary cluster set of {Φt }. Fix l ≥ 1, since the sequence P Bfkj ,Ω is increasing, by Lemma 2.1 (ii) and the assumption that ξ admits a continuous strong plurisubharmonic barrier we obtain fkl (ξ) ≤ lim inf P Bfkl ,Ω (ξj ) ≤ lim sup P Bfkj ,Ω (ξj ). j→∞ j→∞ By letting l → ∞ and combining with the above inequality we deduce − log |f (ξ)| ≤ lim sup P Bfkj ,Ω (ξj ) ≤ lim sup fkj (ξj ) − ε = − log |f (ξ)| − ε. j→∞ j→∞ Here, the last equality follows from uniform convergence of {fkj }j≥1 to − log |f | on the compact set {ξkj }j≥1 ∪ {ξ}. Thus we arrived at a contradiction. The proof is complete. 15 (ii) Let fj := − log(|f | + 1/j). It is clear that e−fj converges uniformly to |f | on ∂Ω and fj is increasing to − log |f | on ∂Ω. By the assumption (d) and Theorem 3.5 in [DW] we have Jz (Ω) = Jzc (Ω) for every z ∈ Ω. Thus, using Lemma 2.1 (i) we conclude that P Bfj ,Ω is lower semicontinuous functions for every j. On the other ˜ A,Ω . Thus G ˜ A,Ω is lower semicontinuous on Ω. The desired hand, by (a) P Bfj ,Ω ↑ G conclusion now follows since this function is already upper semicontinuous. Proof of Theorem 1.5. The first conclusion of the theorem is just a special case of Theorem 1.4. For the second one, we proceed as follows. By Lemma 3.2 we have ˆ U ×C ∩ (Ω × C) = {(z, w) : z ∈ Ω, log |w| + F (z) ≤ 0}, K ˜ A,Ω where F : Ω → R∪{∞} is a lower semicontinuous function that satisfies F ≤ G on Ω \ A. Suppose that there exists z0 ∈ Ω and ε > 0 such that (10) ˜ A,Ω (z0 ). F (z0 ) + ε < G Let {Ωk }k≥1 be a sequence of strictly pseudoconvex domains with smooth boundary such that Ωk ↓ Ω and Ωk is holomorphically convex in U . In particular, (Ωk , U ) is a Runge pair for every k ≥ 1. For a proof of this standard fact, see Lemma 2.4 in [Di]. By Lemma 3.2 we have (11) ˜ A,Ω (z0 ) < F (z0 ), ∀k ≥ 1. G k By the same argument as in the proof of Theorem 1.4, using the fact that the real analytic curve t → Φ−1 t (z0 ) is not pluri-thin at t = 0 and the invariant property under biholomorphic transformations of GA,Ω , we get (12) ˜ A,Ω (z0 ) = lim sup G ˜ A,Φt (Ω) (z0 ). G t→0 Combining (10) and (11), we obtain a sequence tk ↓ 0 having the following properties: ˜ k := Φt (Ω) contains a connected neighbourhood of V ∪ {z0 } where V is an (i) Ω k open neighbourhood of A ∩ ∂Ω which is disjoint from S. ˜ A,Ω (z0 ) + ε < G ˜ ˜ (z0 ), ∀k, j ≥ 1. (ii) G k A,Ωk ˜ k that includes V ∪ (iii) If we denote by Ω∗k the connected component of Ωk ∩ Ω {z0 } then Ωk and isotopy family Φt satisfies the condition (1.4.c) after possibly shrinking the parameter ε. We note that Ω∗k is pseudoconvex and (Ω∗k , U ) is a Runge pair for every k ≥ 1. In view of the remark (iv) that follows Theorem 1.4 and the assumption (15.c), we see that (Φt (Ω∗k ), U ) is also a Runge pair for all parameters t close enough to 0. Furthermore, we have ˜ ˜ (z0 ) < G ˜ A,Ω∗ (z0 ). G A,Ωk k Now we let {fj }j≥1 be a sequence of continuous functions that increases to ˜ A,Ω∗ on Ω∗ . It follows, − log |f | on U . By Theorem 1.4 we have P Bfj ,Ω∗k ↑ G k k using (ii) and (8) in Lemma 3.1, that for j large enough we have P Bfj ,Ωk (z0 ) + ε < P Bfj ,Ω∗k (z0 ). 16 Since Ωk is strictly pseudoconvex, we may apply Lemma 2.5 to get a point ξk,j ∈ Ωk ∩ ∂Ω∗k such that P Bfj ,Ωk (ξk,j ) + ε < fj (ξk,j ). (13) After switching to subsequences we may assume that ξk,j → ξk as j → ∞. It is easy to see that ξk ̸∈ A. We claim that ξk ̸∈ ∂Ωk ∩ ∂Ω∗k . Assume otherwise, then we note that, since Ωk is strictly pseudoconvex, P Bfj ,Ωk is continuous on Ωk with boundary values fj . It follows, using (12) that − log |f (ξk )| = lim supP Bfm ,Ωk (ξk ) ≤ lim supP Bfj ,Ωk (ξk,j ) ≤ − log |f (ξk )| − ε. m→∞ j→∞ ∂Ω∗k as claimed. Furthermore, by passing to a This is absurd. Thus ξk,j ∈ Ωk ∩ subsequence we may assume that ξk → ξ ∈ S. Next, for k ≥ 1 we set Kk := {(z, w) : z ∈ ∂Ωk , |w| ≤ |f (z)|}. Moreover, by Lemma 3.1 (i) we have (Kk )U ×C ∩ (Ωk × C) = {(z, w) : z ∈ Ωk , log |w| + Fk (z) ≤ 0}, where Fk : Ωk → R is a lower semicontinuous function. By letting j → ∞ in (12) and invoking (8) in Lemma 3.1 we obtain Fk (ξk ) ≤ lim supP Bfm ,Ωk (ξk ) ≤ lim supP Bfj ,Ωk (ξk,j ) ≤ − log |f (ξk )| − ε. m→∞ j→∞ Observe that Kk converges to K in the Hausdorff metric. Therefore, by the lower semicontinuity of holomorphic hulls (4) we obtain F (ξ) ≤ lim supFk (ξk ). k→∞ Putting all this together, we have (14) F (ξ) ≤ − log |f (ξ)| − ε. On the other hand, fix a point (ξ, η) ∈ KU ×C . Since ξ is a good boundary point, we can find a strong plurisubharmonic barrier φ at ξ and a uniformly bounded sequence of plurisubharmonic functions on neighbourhoods of Ω that converges pointwise to φ on Ω. Since Ω is holomorphically convex in U , for each j ≥ 1 we may regard φj as a plurisubharmonic function on some neighbourhood of KU ×C . Let ∫ µ be a representing measure of ξ with compact support in K. Then u(ξ, η) ≤ udµ for every plurisubharmonic function u defined on a neighbourhood of K KU ×C . In particular ∫ φj (ξ, η) ≤ φj dµ. K By passing∫ j → ∞ and using Lebesgue’s dominated convergence theorem we get φ(ξ, η) ≤ K φdµ. Since φ peaks at ξ, we deduce that µ must be supported on {ξ} × C. Finally, by considering u(z, w) = w we end up with |η| ≤ |f (ξ)|. Hence F (ξ) = − log |f (ξ)|. We arrived at a contradiction to (13). Thus (9) is false and ˜ A,Ω ≤ F entirely on Ω. It follows that F = G ˜ A,Ω on Ω \ A. Finally, for z ∈ A, so G by lower semi-continuity of F we have ˜ A,Ω (ξ) = G ˜ A,Ω (z). F (z) ≤ lim inf F (ξ) ≤ lim sup G ξ→z,ξ̸∈A ξ→z,ξ̸∈A ˜ A,Ω (z) for every z ∈ Ω. The proof is complete. This implies that F (z) = G 17 ˜ A ,Ω does not converge Proof of Corollary 1.6. Suppose that the sequence G j j ˜ A,Ω on some compact L of Ω. It follows, after passing to a subseuniformly to G quence, that for some ε > 0 ˜ A ,Ω − G ˜ A,Ω ∥L > ε ∀j ≥ 1. ∥G j j Now we let ˜ A ,Ω on Ω. u := lim sup G j j j→∞ ˜ A ,Ω < − log |hj | on Ω and since hj converges uniformly to f on Ω, the Since G j ˜ A,Ω . function u∗ ∈ P SH(Ω) and u∗ + log |f | < 0 on Ω. It follows that u∗ ≤ G Therefore, by Hartogs’ lemma, the following estimates hold on L for all j large enough ˜ A ,Ω ≤ G ˜ A,Ω + ε. G j j ˜ A,Ω (by Thus, after passing again to a subsequence and using continuity of G ∗ Theorem 1.2), we can find a sequence {zj } ⊂ Ω, zj → z ∈ L such that ˜ A ,Ω (zj ) + ε < G ˜ A,Ω (z ∗ ) ∀j ≥ 1. G j j For j ≥ 1 we let Kj := {(z, w) : z ∈ ∂Ωj , |w| ≤ |hj (z)|}, K := {(z, w) : z ∈ ∂Ω, |w| ≤ |f (z)|}. By Theorem 1.5 and the assumptions made on Ω and Ωj , for every j ≥ 1 we have ˜ A ,Ω (z) ≤ 0} (Kj )U ×C ∩ (Ωj × C) = {(z, w) : z ∈ Ωj , log |w| + G j j and ˆ U ×C ∩ (Ω × C) = {(z, w) : z ∈ Ω, log |w| + G ˜ A,Ω (z) ≤ 0}. K Since hj converges uniformly to f on Ω we infer that K = cl({Kj }j≥1 ). This ˆ U ×C includes the implies, in view of the lower semi-continuity of hulls, that K cluster set of the sequence (Kj )U ×C . But in view of () we deduce that any limit ˜ ˆ U ×C . This contradiction point of the sequence {(zj , e−GAj ,Ωj (zj ) )} lies outside K finishes the proof of our corollary. Proof of Theorem 1.7. By previously known results of Larusson and Sig˜ A,Ω satisfies urdsson mentioned in the introduction we see that the function G the conditions (1.7.b) and (1.7.c). Furthermore, the condition (1.7.a) also holds by Theorem 1.4. Conversely, let u ∈ P SH(Ω) ∩ C(Ω) be a function satisfying ˜ A,Ω . Fix a sequence {tj }j≥1 ↓ 0 such these conditions. We must show that u = G −1 that Φtj (Ω) is relatively compact in Ω. Since A is invariant under Φtj we have A ⊂ {z ∈ Ω : f ◦ Φtj (z) = 0}. Since f is a principal defining function for A we can write f ◦ Φtj = f gj , where gj is holomorphic on Ω. We claim that the sequence {gj }j≥1 converges uniformly to 1 on compact sets of Ω. First, we show that the sequence is locally uniformly bounded on Ω. The assertion is clear near every point of Ω \ A. Fix z0 ∈ Reg A, the regular locus of A. After composing with a local biholomorphic 18 mapping, we may assume that f (z) = z1 on a small ball V Ω around z0 . This implies that Φ1tj (z) gj (z) = ∀z ∈ V \ A, z1 where Φ1tj is the first component of Φtj . So there exists ξ ∈ V (depending on z) such that ∂Φ1tj gj (z) = (ξ). ∂z1 Thus, by Weierstrass’s theorem, the sequence {gj }j≥1 is uniformly bounded on V . Since Sing A := A \ Reg A is of codimension at least 2, a routine argument using the maximum principle implies that {gj }j≥1 is uniformly bounded near every point of Sing A. Thus the sequence {gj }j≥1 is locally uniformly bounded on Ω. On the other hand, by property (c) of isotopy family, {gj }j≥1 converges pointwise to 1 on Ω \ A. Therefore, by Vitali’s convergence theorem, we conclude that {gj }j≥1 converges to 1 uniformly on compact sets of Ω. This is our claim. Next, for each j ≥ 1, we define uj (z) := (u ◦ Φtj )(z) − aj , ∀z ∈ Ω and fj := uj |∂Ω where eaj := ∥gj ∥Ω . Then by the conditions (1.7.a) and (1.7.b) we have uj ∈ P SH(Ω) ∩ C(Ω) and satisfies (ddc uj )n = 0 on Ω. So, by the comparison principle we infer that uj = P Bfj ,Ω . Furthermore, by the assumption (1.7. b) and the choice of aj we have fj < − log |f ◦ Φtj (z)| − aj ≤ − log |f | on ∂Ω. By the claim proved above we also have fj converges locally uniformly to − log |f | on (∂Ω) \ A and inf ∂Ω fj > −M for some M > 0 independent of j. Therefore, an application of Theorem 1.4 yields ˜ A,Ω (z) = lim P Bf ,Ω (z) = lim u ◦ Φt (z) = u(z) ∀z ∈ Ω. G j j j→∞ j→∞ This is the desired conclusion. The proof is complete. Proof of Proposition 1.8.Let ξ ∈ ∂Ω be a point lying in the boundary cluster set of Φt . Then there exists a sequence {tj }j≥1 ↓ 0 and a sequence {ξj }j≥1 ⊂ ∂Ω such that Φtj (ξj ) → ξ and Φtj (ξj ) ∈ Ω. For j ≥ 1, we define fj (t) := ρ(Φ(t, ξj )). Since Ω is defines by ρ we have fj (0) = fj (tj ) = 0. This implies that fj′ (ηj ) = 0 for some ηj ∈ (0, tj ). An application of the chain rule gives n [∑ ] ∂Φi ∂ρ ℜ (Φ(ηj , ξj )) (ηj , ξj ) = 0. ∂zi ∂t i=1 By passing j → ∞ and observing that ηj → 0, ξj → ξ, we obtain n ] [∑ ∂Φi ∂ρ (ξ) (0, ξ) = 0. ℜ ∂zi ∂t i=1 This is the desired conclusion and we are done. 19 Proof of Proposition 1.9. For ε > 0 we set ∆ε := {λ ∈ C : |t| < ε}. Choose ε > 0 so small such that U ′ + ∆ε U ′ U. We let Hj := Aj × ∆ε = {(z ′ , fj (z ′ ), t) : z ∈ π(U ′ ), |t| < ε} 1 ≤ j ≤ k. Then H1 , · · · , Hk are pairwise disjoint, complex submanifolds of the pseudoconvex domain Vε := U ′ × ∆ε . Observe that for 1 ≤ j ≤ k, (z ′ , t) ∈ Vε we may write fj (z ′ + tz ′ ) − fj (z ′ ) = t˜ gj (z ′ , t) where ′ g˜j (z , t) := n ∫ ∑ i=1 0 1 ∂fj ′ (z + tz ′ λ)dλ ∂zi is holomorphic on Vε for every 1 ≤ j ≤ n. Notice that gj (z ′ , 0) = gj (z ′ ) on U ′ . Next, we let H := H1 ∪ · · · Hk . Then H is a complex submanifolds of Vε . Let φ : H → C be defined by φ(z ′ , fj (z ′ ), t) := g˜j (z ′ , t), ∀(z ′ , t) ∈ Hj . Since Hj are pairwise disjoint, the function φ is holomorphic on H. Thus, by Theorem 7.4.8 in [H¨o], we may extend φ to a holomorphic function, still denoted by φ, on Vε . It is then easy to check that the map Φt defined in (2) satisfies (3). Finally, since Ω U ′ we may shrink ε such that |t| ∂φ ′ (z , zn , t) < 1 ∀(z ′ , zn ) ∈ Ω, |t| < ε. ∂zn It follows that Φt , (0 ≤ t < ε) is now an isotopy family of biholomorphic maps on Ω. The proof is then complete. Proof of Proposition 1.10. By Proposition 1.9 we see that Φt is an isotopy family of biholomorphic maps on Ω that satisfies the conditions (1.4.b). In view of Proposition 1.8 and the assumption (1.10.b), the condition (1.4.c) holds also. Next, by the assumption (1.10.c) implies (1.4.d) by the remark following Theorem 1.4. References [BP] A. Brown and C. Pearcy, An Introduction to Real Analysis, Graduate Texts in Mathematics, vol. 154, Springer-Verlag, 1995. [De] J. -P. Demailly, Mesures de de Monge-Amp`ere et mesures pluriharmoniques, Math. Z. 194 (1987), no. 4, 519-564. [Di1] N. Q. Dieu, Continuity of pluricomplex Green functions with poles along a complex hypersurface, Ark. Mat. 43 (2005), 181-200. [Di2] N. Q. Dieu, Perron-Bremermann envelopes on bounded domains in Cn ,Internat. J. Math., 18 (10) (2007), 1169-1185. [DW] N. Q. Dieu and F. Wikstr¨om, Jensen measures and approximation of plurisubharmonic functions, Michigan Math. J. 53 (2005), no. 3, 529-544. [Kl] M. Klimek, Pluripotential Theory, Oxford 1991. [Le] P. Lelong, Fonction de Green pluricomplexe et lemmes de Schwarz dans les espaces de Banach, J. Math. Pures Appl. 68 (1989), 319-347. 20 [LS1] F. L´arusson and R. Sigurdsson, Plurisubharmonic extremal functions, Lelong numbers and coherent ideal sheaves, Indiana Univ. Math. J. 48 (1999), no. 4, 1513-1534. [LS2] F. L´arusson and R. Sigurdsson, Erratum: ”Plurisubharmonic extremal functions, Lelong numbers and coherent ideal sheaves”, Indiana Univ. Math. J. 50 (2001), no. 4, 1705. [Si1] N. Sibony, Prolongement des fonctions holomorphes born´ees et metrique Carath´eodory, Invent. Math. 29 (1975), 205-230. [Si2] N. Sibony, Une classe de domaines pseudoconvexes, Duke Math. J. 55 (1987), 299-319. [Wa] J. Walsh, Continuity of envelopes of plurisubharmonic functions, J. Math. Mech. 18 (1968/1969), 143-148. [Wi] F. Wikstr¨om, Jensen measures and boundary values of plurisubharmonic functions, Ark. Mat. 39 (2001), 181-200. Nguyen Quang Dieu Hanoi National University of Education, 136 Xuan Thuy street, Cau Giay, Hanoi, Vietnam Email: dieu vn@yahoo.com Dau Hoang Hung Phan Boi Chi high school, Vinh, Viet Nam Email dhhungk9@yahoo.com [...]... Green pluricomplexe et lemmes de Schwarz dans les espaces de Banach, J Math Pures Appl 68 (1989), 319-347 20 [LS1] F L´arusson and R Sigurdsson, Plurisubharmonic extremal functions, Lelong numbers and coherent ideal sheaves, Indiana Univ Math J 48 (1999), no 4, 1513-1534 [LS2] F L´arusson and R Sigurdsson, Erratum: ”Plurisubharmonic extremal functions, Lelong numbers and coherent ideal sheaves”, Indiana... Dieu, Continuity of pluricomplex Green functions with poles along a complex hypersurface, Ark Mat 43 (2005), 181-200 [Di2] N Q Dieu, Perron-Bremermann envelopes on bounded domains in Cn ,Internat J Math., 18 (10) (2007), 1169-1185 [DW] N Q Dieu and F Wikstr¨om, Jensen measures and approximation of plurisubharmonic functions, Michigan Math J 53 (2005), no 3, 529-544 [Kl] M Klimek, Pluripotential Theory,... is a Runge pair for every k ≥ 1 For a proof of this standard fact, see Lemma 2.4 in [Di] By Lemma 3.2 we have (11) ˜ A, Ω (z0 ) < F (z0 ), ∀k ≥ 1 G k By the same argument as in the proof of Theorem 1.4, using the fact that the real analytic curve t → Φ−1 t (z0 ) is not pluri-thin at t = 0 and the invariant property under biholomorphic transformations of GA,Ω , we get (12) ˜ A, Ω (z0 ) = lim sup G ˜ A, Φt... compact sets of Ω ˜ × C by holomorphic U , we infer that h is approximated on compact sets of Ω functions in U × C In particular, (Kf,Ω )Ω×C = (Kf,Ω )U ×C ˜ Summing up we obtain (Kf,Ω )U ×C ∩ (Ω × C) ⊂ {(z, w) : z ∈ Ω, u(z, w) < 0} Applying again (i) we arrive at the desired estimate We need the following elementary fact which may be of independent interest Lemma 3.2 Let K be a compact subset of an... j j j→∞ ˜ A ,Ω < − log |hj | on Ω and since hj converges uniformly to f on Ω, the Since G j ˜ A, Ω function u∗ ∈ P SH(Ω) and u∗ + log |f | < 0 on Ω It follows that u∗ ≤ G Therefore, by Hartogs’ lemma, the following estimates hold on L for all j large enough ˜ A ,Ω ≤ G ˜ A, Ω + ε G j j ˜ A, Ω (by Thus, after passing again to a subsequence and using continuity of G ∗ Theorem 1.2), we can find a sequence... we may assume that {ξj }j≥1 ⊂ Ω \ A and ξj → ξ ∈ S, the boundary cluster set of {Φt } Fix l ≥ 1, since the sequence P Bfkj ,Ω is increasing, by Lemma 2.1 (ii) and the assumption that ξ admits a continuous strong plurisubharmonic barrier we obtain fkl (ξ) ≤ lim inf P Bfkl ,Ω (ξj ) ≤ lim sup P Bfkj ,Ω (ξj ) j→∞ j→∞ By letting l → ∞ and combining with the above inequality we deduce − log |f (ξ)| ≤ lim... (1.4.c) after possibly shrinking the parameter ε We note that Ω∗k is pseudoconvex and (Ω∗k , U ) is a Runge pair for every k ≥ 1 In view of the remark (iv) that follows Theorem 1.4 and the assumption (15.c), we see that (Φt (Ω∗k ), U ) is also a Runge pair for all parameters t close enough to 0 Furthermore, we have ˜ ˜ (z0 ) < G ˜ A, Ω∗ (z0 ) G A, Ωk k Now we let {fj }j≥1 be a sequence of continuous functions. .. Since f is a principal defining function for A we can write f ◦ Φtj = f gj , where gj is holomorphic on Ω We claim that the sequence {gj }j≥1 converges uniformly to 1 on compact sets of Ω First, we show that the sequence is locally uniformly bounded on Ω The assertion is clear near every point of Ω \ A Fix z0 ∈ Reg A, the regular locus of A After composing with a local biholomorphic 18 mapping, we may... of Ω that converges pointwise to φ on Ω Since Ω is holomorphically convex in U , for each j ≥ 1 we may regard φj as a plurisubharmonic function on some neighbourhood of KU ×C Let ∫ µ be a representing measure of ξ with compact support in K Then u(ξ, η) ≤ udµ for every plurisubharmonic function u defined on a neighbourhood of K KU ×C In particular ∫ φj (ξ, η) ≤ φj dµ K By passing∫ j → ∞ and using Lebesgue’s... view of Proposition 1.8 and the assumption (1.10.b), the condition (1.4.c) holds also Next, by the assumption (1.10.c) implies (1.4.d) by the remark following Theorem 1.4 References [BP] A Brown and C Pearcy, An Introduction to Real Analysis, Graduate Texts in Mathematics, vol 154, Springer-Verlag, 1995 [De] J -P Demailly, Mesures de de Monge-Amp`ere et mesures pluriharmoniques, Math Z 194 (1987), no ... pseudoconvex domains in Cn and A be a finite union of complex hyperplanes passing through ∈ Ω Assume further that the following conditions are satisfied: (1.1 .a) Ω is holomorphically convex in U and Int... 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... Continuity of pluricomplex Green functions with poles along a complex hypersurface, Ark Mat 43 (2005), 181-200 [Di2] N Q Dieu, Perron-Bremermann envelopes on bounded domains in Cn ,Internat J Math.,