A UNIFIED APPROACH TO THE THEORY OF SEPARATELY
HOLOMORPHIC MAPPINGS
VIÊT-ANH NGUYÊN
Abstract. We extend the theory of separately holomorphic mappings between
complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay
Theorem on holomorphic discs and our recent joint-work with Pflug on boundary
cross theorems in dimension 1. It also relies on our new technique of conformal
mappings and a generalization of Siciak’s relative extremal function. Our approach
illustrates the unified character: “From local informations to global extensions”.
Moreover, it avoids systematically the use of the classical method of doubly or-
thogonal bases of Bergman type.
1. Introduction
In this article all complex manifolds are supposed to be of finite dimension and
countable at infinity, and all complex analytic spaces are supposed to be reduced,
irreducible, of finite dimension and countable at infinity. For a subset S of a topo-
logical space M, S denotes the closure of S in M, and the set ∂S := S ∩ M \ S
denotes, as usual, the boundary of S in M.
The main purpose of this work is to investigate the following
PROBLEM. Let X, Y be two complex manifolds, let D (resp. G) be an open subset
of X (resp. Y ), let A (resp. B) be a subset of D (resp. G) and let Z be a complex
analytic space. Define the cross
A× (G ∪B)
(D ∪ A)× B
We want to determine the “envelope of holomorphy” of the cross W, that is, an
“optimal” open subset of X×Y, denoted by
W, which is characterized by the following
properties:
Let f : W −→ Z be a mapping that satisfies, in essence, the following two
conditions:
• f(a, ·) is holomorphic on G for all a ∈ A, f(·, b) is holomorphic on D for all
b ∈ B;
• f(a, ·) is continuous on G ∪ B for all a ∈ A, f(·, b) is continuous on D ∪ A
for all b ∈ B.
2000 Mathematics Subject Classification. Primary 32D15, 32D10.
Key words and phrases. Hartogs’ theorem, holomorphic extension, Poletsky theory of discs,
Rosay Theorem on holomorphic discs.
http://arxiv.org/abs/0704.0897v2
2 VIÊT-ANH NGUYÊN
Then there is a holomorphic mapping f̂ defined on
W such that for every (ζ, η) ∈ W,
f̂(z, w) tends to f(ζ, η) as (z, w) ∈
W tends, in some sense, to (ζ, η).
Now we recall briefly the main developments around this problem. All the results
obtained so far may be divided into two directions. The first direction investigates
the results in the “interior” context: A ⊂ D and B ⊂ G, while the second one
explores the “boundary” context: A ⊂ ∂D and B ⊂ ∂G.
The first fundamental result in the field of separate holomorphy is the well-known
Hartogs extension theorem for separately holomorphic functions (see [14]). In the
language of the PROBLEM the following case X = Cn, Y = Cm, A = D, B =
G, Z = C has been solved and the result is
W = D×G. In particular, this theorem
may be considered as the first main result in the first direction. In his famous
article [8] Bernstein obtained some positive results for the PROBLEM in certain
cases where A ⊂ D, B ⊂ G, X = Y = C and Z = C.
More than 60 years later, a next important impetus was made by Siciak (see
[43, 44]) in 1969–1970, where he established some significant generalizations of the
Hartogs extension theorem. In fact, Siciak’s formulation of these generalizations
gives rise to the above PROBLEM: to determine the envelope of holomorphy for sep-
arately holomorphic functions defined on some cross sets W. The theorems obtained
under this formulation are often called cross theorems. Using the so-called relative
extremal function, Siciak completed the PROBLEM for the case where A ⊂ D,
B ⊂ G, X = Y = C and Z = C.
The next deep steps were initiated by Zahariuta in 1976 (see [45]) when he started
to use the method of common bases of Hilbert spaces. This original approach per-
mitted him to obtain new cross theorems for some cases where A ⊂ D, B ⊂ G and
D = X, G = Y are Stein manifolds. As a consequence, he was able to generalize
the result of Siciak in higher dimensions.
Later, Nguyên Thanh Vân and Zeriahi (see [25, 26, 27]) developed the method
of doubly orthogonal bases of Bergman type in order to generalize the result of Za-
hariuta. This is a significantly simpler and more constructive version of Zahariuta’s
original method. Nguyên Thanh Vân and Zeriahi have recently achieved an elegant
improvement of their method (see [24], [47]).
Using Siciak’s method, Shiffman (see [41]) was the first to generalize some Siciak’s
results to separately holomorphic mappings with values in a complex analytic space
Z. Shiffman’s result (see [42]) shows that the natural “target spaces” for obtaining
satisfactory generalizations of cross theorems are the ones which possess the Hartogs
extension property (see Subsection 2.4 below for more explanations).
In 2001 Alehyane and Zeriahi solved the PROBLEM for the case where A ⊂ D,
B ⊂ G and X, Y are Stein manifolds, and Z is a complex analytic space which
possesses the Hartogs extension property (see Theorem 2.2.4 in [5]).
In a recent work (see [28]) we complete, in some sense, the PROBLEM for the
case where A ⊂ D, B ⊂ G and X, Y are arbitrary complex manifolds. The main
ingredients in our approach are Poletsky theory of discs developed in [37, 38], Rosay’s
A UNIFIED APPROACH 3
Theorem on holomorphic discs (see [40]), the above mentioned result of Alehyane–
Zeriahi and the technique of level sets of the plurisubharmonic measure which was
previously introduced in our joint-work with Pflug (see [33]).
To conclude the first direction of research we mention the survey articles by
Nguyên Thanh Vân [23] and Peter Pflug [32] which give nice accounts on this sub-
ject.
The first result in the second direction (i.e. “boundary context”) was established
in the work of Malgrange–Zerner [46] in the 1960s. Further results in this direction
were obtained by Komatsu [21] and Drużkowski [9], but only for some special cases.
Recently, Gonchar [12, 13] has proved a more general result where the following
case has been solved: X = Y = C, D and G are Jordan domains, A (resp. B) is
an open boundary subset of ∂D (resp. ∂G), and Z = C. It should be noted that
Airapetyan and Henkin published a general version of the edge-of-the-wedge theorem
for CR manifolds (see [1] for a brief version and [2] for a complete proof). Gonchar’s
result could be deduced from the latter works. In our joint-articles with Pflug (see
[33, 34, 35]), Gonchar’s result has been generalized considerably. More precisely, the
work in [35] treats the case where the “source spaces” X, Y are arbitrary complex
manifolds, A (resp. B) is an open boundary subset of ∂D (resp. ∂G), and Z =
C. The work in [34] solves the case where the “source spaces” X, Y are Riemann
surfaces, A (resp. B) is a measurable (boundary) subset of ∂D (resp. ∂G), and
Z = C.
The main purpose of this article is to give a new version of the Hartogs extension
theorem which unifies all results up to now. Namely, we are able to give a reason-
able solution to the PROBLEM when the “target space” Z possesses the Hartogs
extension property. Our method is based on a systematic application of Poletsky
theory of discs, Rosay Theorem on holomorphic discs and our joint-work with Pflug
on boundary cross theorems in dimension 1 (see [34]). It also relies on our new
technique of conformal mappings and a generalization of Siciak’s relative extremal
function. The approach illustrates the unified character in the theory of extension
of holomorphic mappings:
One can deduce the global extension from local informations.
Moreover, the novelty of this new approach is that one does not use the classical
method of doubly orthogonal bases of Bergman type.
We close the introduction with a brief outline of the paper to follow.
In Section 2 we formulate the main results.
The tools which are needed for the proof of the main results are developed in
Section 3, 4, 5 and 7.
The proof of the main results is divided into three parts, which correspond to
Section 6, 8 and 9. Section 10 concludes the article with various applications of our
results.
Acknowledgment. The paper was written while the author was visiting the
Abdus Salam International Centre for Theoretical Physics in Trieste. He wishes to
express his gratitude to this organization.
4 VIÊT-ANH NGUYÊN
2. Preliminaries and statement of the main result
First we develop some new notions such as system of approach regions for an open
set in a complex manifold, and the corresponding plurisubharmonic measure. These
will provide the framework for an exact formulation of the PROBLEM and for our
solution.
2.1. Approach regions, local pluripolarity and plurisubharmonic measure.
Definition 2.1. Let X be a complex manifold and let D ⊂ X be an open subset.
A system of approach regions for D is a collection A =
Aα(ζ)
ζ∈D, α∈Iζ
of open
subsets of D with the following properties:
(i) For all ζ ∈ D, the system
Aα(ζ)
forms a basis of open neighborhoods
of ζ (i.e., for any open neighborhood U of a point ζ ∈ D, there is α ∈ Iζ
such that ζ ∈ Aα(ζ) ⊂ U).
(ii) For all ζ ∈ ∂D and α ∈ Iζ , ζ ∈ Aα(ζ).
Aα(ζ) is often called an approach region at ζ.
A is said to be canonical if it satisfies (i) and the following property (which is
stronger than (ii)):
(ii’) For every point ζ ∈ ∂D, there is a basis of open neighborhoods (Uα)α∈Iζ of ζ
in X such that Aα(ζ) = Uα ∩D, α ∈ Iζ.
It is possible that Iζ = ∅ for some ζ ∈ ∂D.
Various systems of approach regions which one often encounters in Complex Anal-
ysis will be described in the next subsection. Systems of approach regions for
D are used to deal with the limit at points in D of mappings defined on some
open subsets of D. Consequently, we deduce from Definition 2.1 that the subfamily(
Aα(ζ)
ζ∈D, α∈Iζ
is, in a certain sense, independent of the choice of a system of ap-
proach regions A. In addition, any two canonical systems of approach regions are,
in some sense, equivalent. These observations lead us to use, throughout the paper,
the following convention:
We fix, for every open set D ⊂ X, a canonical system of approach regions.
When we want to define a system of approach regions A for an open set D ⊂ X, we
only need to specify the subfamily
Aα(ζ)
ζ∈∂D, α∈Iζ
In what follows we fix an open subset D ⊂ X and a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
for D.
For every function u : D −→ [−∞,∞), let
(A− lim sup u)(z) :=
lim sup
w∈Aα(z), w→z
u(w), z ∈ D, Iz 6= ∅,
lim sup
w∈D, w→z
u(w), z ∈ ∂D, Iz = ∅.
By Definition 2.1 (i), (A−lim sup u)|D coincides with the usual upper semicontinuous
regularization of u.
For a set A ⊂ D put
hA,D := sup {u : u ∈ PSH(D), u ≤ 1 on D, A− lim sup u ≤ 0 on A} ,
A UNIFIED APPROACH 5
where PSH(D) denotes the cone of all functions plurisubharmonic on D.
A is said to be pluripolar inD if there is u ∈ PSH(D) such that u is not identically
−∞ on every connected component of D and A ⊂ {z ∈ D : u(z) = −∞} . A is said
to be locally pluripolar in D if for any z ∈ A, there is an open neighborhood V ⊂ D
of z such that A ∩ V is pluripolar in V. A is said to be nonpluripolar (resp. non
locally pluripolar) if it is not pluripolar (resp. not locally pluripolar). According to
a classical result of Josefson and Bedford (see [16], [6]), if D is a Riemann domain
over a Stein manifold, then A ⊂ D is locally pluripolar if and only if it is pluripolar.
Definition 2.2. The relative extremal function of A relative to D is the function
ω(·, A,D) defined by
ω(z, A,D) = ωA(z, A,D) := (A− lim sup hA,D)(z), z ∈ D.
Note that when A ⊂ D, Definition 2.2 coincides with the classical definition of
Siciak’s relative extremal function.
Next, we say that a set A ⊂ D is locally pluriregular at a point a ∈ A if ω(a, A ∩
U,D ∩ U) = 0 for all open neighborhoods U of a. Moreover, A is said to be locally
pluriregular if it is locally pluriregular at all points a ∈ A. It should be noted from
Definition 2.1 that if a ∈ A ∩D then the property of local pluriregularity of A at a
does not depend on any particular choices of a system of approach regions A, while
the situation is different when a ∈ A ∩ ∂D : the property does depend on A.
We denote by A∗ the following set
(A ∩ ∂D)
a ∈ A ∩D : A is locally pluriregular at a
If A ⊂ D is non locally pluripolar, then a classical result of Bedford and Taylor (see
[6, 7]) says that A∗ is locally pluriregular and A\A∗ is locally pluripolar. Moreover,
A∗ is locally of type Gδ, that is, for every a ∈ A
∗ there is an open neighborhood
U ⊂ D of a such that A∗ ∩ U is a countable intersection of open sets.
Now we are in the position to formulate the following version of the plurisubhar-
monic measure.
Definition 2.3. For a set A ⊂ D, let Ã = Ã(A) :=
P∈E(A)
P, where
E(A) = E(A,A) :=
P ⊂ D : P is locally pluriregular, P ⊂ A∗
The plurisubharmonic measure of A relative to D is the function ω̃(·, A,D) defined
ω̃(z, A,D) := ω(z, Ã, D), z ∈ D.
It is worthy to remark that ω̃(·, A,D) ∈ PSH(D) and 0 ≤ ω̃(z, A,D) ≤ 1, z ∈ D.
Moreover,
(2.1)
A− lim sup ω̃(·, A,D)
(z) = 0, z ∈ Ã.
An example in [3] shows that in general, ω(·, A,D) 6= ω̃(·, A,D) on D. Section 10
below is devoted to the study of ω̃(·, A,D) in some important cases.
1Observe that this function depends on the system of approach regions.
6 VIÊT-ANH NGUYÊN
Now we compare the plurisubharmonic measure ω̃(·, A,D) with Siciak’s relative
extremal function ω(·, A,D). We only consider two important special cases: A ⊂ D
and A ⊂ ∂D. For the moment, we only focus on the case where A ⊂ D. The latter
one will be discussed in Section 10 below.
If A is an open subset of an arbitrary complex manifold D, then it is easy to see
ω̃(z, A,D) = ω(z, A,D), z ∈ D.
If A is a (not necessarily open) subset of an arbitrary complex manifold D, then we
will prove in Proposition 7.1 below that
ω̃(z, A,D) = ω(z, A∗, D), z ∈ D.
On the other hand, if, morever, D is a bounded open subset of Cn then we have (see,
for example, Lemma 3.5.3 in [18]) ω(z, A,D) = ω(z, A∗, D), z ∈ D. Consequently,
under the last assumption,
ω̃(z, A,D) = ω(z, A,D), z ∈ D.
Our discussion shows that at least in the case where A ⊂ D, the notion of the
plurisubharmonic measure is a good candidate for generalizing Siciak’s relative ex-
tremal function to the manifold context in the theory of separate holomorphy.
For a good background of the pluripotential theory, see the books [18] or [20].
2.2. Examples of systems of approach regions. There are many systems of
approach regions which are very useful in Complex Analysis. In this subsection we
present some of them.
1. Canonical system of approach regions. It has been given by Definition 2.1
(i)–(ii’).
2. System of angular (or Stolz) approach regions for the open unit disc.
Let E be the open unit disc of C. Put
Aα(ζ) :=
t ∈ E :
∣∣∣∣arg
ζ − t
)∣∣∣∣ < α
, ζ ∈ ∂E, 0 < α <
where arg : C −→ (−π, π] is as usual the argument function. A =
(Aα(ζ))ζ∈∂E, 0<α<π
is referred to as the system of angular (or Stolz) approach regions
for E. In this context A− lim is also called angular limit.
3. System of angular approach regions for certain “good” open subsets
of Riemann surfaces. Now we generalize the previous construction (for the open
unit disc) to a global situation. More precisely, we will use as the local model the
system of angular approach regions for E. Let X be a complex manifold of dimension
1, in other words, X is a Riemann surface, and D ⊂ X an open set. Then D is said
to be good at a point ζ ∈ ∂D2 if there is a Jordan domain U ⊂ X such that ζ ∈ U
and U ∩ ∂D is the interior of a Jordan curve.
Suppose that D is good at ζ. This point is said to be of type 1 if there is a
neighborhood V of ζ such that V0 = V ∩D is a Jordan domain. Otherwise, ζ is said to
be of type 2. We see easily that if ζ is of type 2, then there are an open neighborhood
2 In the work [34] we use the more appealing word Jordan-curve-like for this notion.
A UNIFIED APPROACH 7
V of ζ and two disjoint Jordan domains V1, V2 such that V ∩D = V1∪V2. Moreover,
D is said to be good on a subset A of ∂D if D is good at all points of A.
Here is a simple example which may clarify the above definitions. Let G be the
open square in C with vertices 1 + i, −1 + i, −1− i, and 1− i. Define the domain
D := G \
Then D is good on ∂G ∪
. All points of ∂G are of type 1 and all points of(
are of type 2.
Suppose now that D is good on a nonempty subset A of ∂D.We define the system
of angular approach regions supported on A: A =
Aα(ζ)
ζ∈D, α∈Iζ
as follows:
• If ζ ∈ D \A, then
Aα(ζ)
coincide with the canonical approach regions.
• If ζ ∈ A, then by using a conformal mapping Φ from V0 (resp. V1 and V2)
onto E when ζ is of type 1 (resp. 2), we can “transfer” the angular approach
regions at the point Φ(ζ) ∈ ∂E : (Aα(Φ(ζ)))0<α<π
to those at the point
ζ ∈ ∂D (see [34] for more detailed explanations).
Making use of conformal mappings in a local way, we can transfer, in the same way,
many notions which exist on E (resp. ∂E) to those on D (resp. ∂D).
4. System of conical approach regions.
Let D ⊂ Cn be a domain and A ⊂ ∂D. Suppose in addition that for every point
ζ ∈ A there exists the (real) tangent space Tζ to ∂D at ζ. We define the system of
conical approach regions supported on A: A =
Aα(ζ)
ζ∈D, α∈Iζ
as follows:
• If ζ ∈ D \A, then
Aα(ζ)
coincide with the canonical approach regions.
• If ζ ∈ A, then
Aα(ζ) := {z ∈ D : |z − ζ | < α · dist(z, Tζ)} ,
where Iζ := (1,∞) and dist(z, Tζ) denotes the Euclidean distance from the
point z to Tζ .
We can also generalize the previous construction to a global situation:
X is an arbitrary complex manifold, D ⊂ X is an open set and A ⊂ ∂D is a
subset with the property that at every point ζ ∈ A there exists the (real) tangent
space Tζ to ∂D.
We can also formulate the notion of points of type 1 or 2 in this general context
in the same way as we have already done in Paragraph 3 above.
2.3. Cross and separate holomorphicity and A-limit. Let X, Y be two com-
plex manifolds, let D ⊂ X, G ⊂ Y be two nonempty open sets, let A ⊂ D and
B ⊂ G. Moreover, D (resp. G) is equipped with a system of approach regions
A(D) =
Aα(ζ)
ζ∈D, α∈Iζ
(resp. A(G) =
Aα(η)
η∈G, α∈Iη
). We define a 2-fold
cross W, its interior W o and its regular part W̃ (with respect to A(D) and A(G))
8 VIÊT-ANH NGUYÊN
W = X(A,B;D,G) :=
(D ∪A)×B
A× (B ∪G)
W o = Xo(A,B;D,G) := (A×G) ∪ (D × B),
W̃ = X̃(A,B;D,G) :=
(D ∪ Ã)× B̃
Ã× (G ∪ B̃)
Moreover, put
ω(z, w) := ω(z, A,D) + ω(w,B,G), (z, w) ∈ D ×G,
ω̃(z, w) := ω̃(z, A,D) + ω̃(w,B,G), (z, w) ∈ D ×G.
For a 2-fold cross W := X(A,B;D,G) let
Ŵ := X̂(A,B;D,G) = {(z, w) ∈ D ×G : ω(z, w) < 1} ,
W := X̂(Ã, B̃;D,G) = {(z, w) ∈ D ×G : ω̃(z, w) < 1} .
Let Z be a complex analytic space. We say that a mapping f : W o −→ Z is
separately holomorphic and write f ∈ Os(W
o, Z), if, for any a ∈ A (resp. b ∈ B)
the restricted mapping f(a, ·) (resp. f(·, b)) is holomorphic on G (resp. on D).
We say that a mapping f : W −→ Z is separately continuous and write f ∈
if, for any a ∈ A (resp. b ∈ B) the restricted mapping f(a, ·) (resp.
f(·, b)) is continuous on G ∪ B (resp. on D ∪A).
In virtue of (2.1), for every (ζ, η) ∈ W̃ and every α ∈ Iζ , β ∈ Iη, there are open
neighborhoods U of ζ and V of η such that
U ∩ Aα(ζ)
V ∩Aβ(η)
Then a mapping f :
W −→ Z is said to admit A-limit λ at (ζ, η) ∈ W̃ , and one
writes
(A− lim f)(ζ, η) = λ, 3
if, for all α ∈ Iζ, β ∈ Iη,
cfW∋(z,w)→(ζ,η), z∈Aα(ζ), w∈Aβ(η)
f(z, w) = λ.
Throughout the paper, for a topological space M, C(M, Z) denotes the set of all
continuous mappings f : M −→ Z. If, moreover, Z = C, then C(M,C) is equipped
with the “sup-norm” |f |M := supM |f | ∈ [0,∞]. A mapping f : M −→ Z is said to
be bounded if there exist an open neighborhood U of f(M) in Z and a holomorphic
embedding φ of U into a polydisc of Ck such that φ(U) is an analytic set in this
polydisc. f is said to be locally bounded along N ⊂ M if for every point z ∈ N ,
there is an open neighborhood U of z (in M) such that f |U : U −→ Z is bounded.
f is said to be locally bounded if it is so for N = M. It is clear that if Z = C then
the above notions of boundedness coincide with the usual ones.
3Note that here A = A(D)×A(G).
A UNIFIED APPROACH 9
2.4. Hartogs extension property. The following example (see Shiffman [42])
shows that an additional hypothesis on the “target space” Z is necessary in or-
der that the PROBLEM makes sense. Consider the mapping f : C2 −→ P1 given
f(z, w) :=
[(z + w)2 : (z − w)2], (z, w) 6= (0, 0),
[1 : 1], (z, w) = (0, 0).
Then f ∈ Os
o(C,C;C,C),P1
, but f is not continuous at (0, 0).
We recall here the following notion (see, for example, Shiffman [41]). Let p ≥ 2
be an integer. For 0 < r < 1, the Hartogs figure in dimension p, denoted by Hp(r),
is given by
Hp(r) :=
, zp) ∈ E
p : ‖z
‖ < r or |zp| > 1− r
where E is the open unit disc of C and z
= (z1, . . . , zp−1), ‖z
‖ := max
1≤j≤p−1
|zj|.
Definition 2.4. A complex analytic space Z is said to possess the Hartogs extension
property in dimension p if every mapping f ∈ O(Hp(r), Z) extends to a mapping
f̂ ∈ O(Ep, Z). Moreover, Z is said to possess the Hartogs extension property if it
does in any dimension p ≥ 2.
It is a classical result of Ivashkovich (see [17]) that if Z possesses the Hartogs
extension property in dimension 2, then it does in all dimensions p ≥ 2. Some typical
examples of complex analytic spaces possessing the Hartogs extension property are
the complex Lie groups (see [4]), the taut spaces (see [48]), the Hermitian manifold
with negative holomorphic sectional curvature (see [41]), the holomorphically convex
Kähler manifold without rational curves (see [17]).
Here we mention an important characterization due to Shiffman (see [41]).
Theorem 2.5. A complex analytic space Z possesses the Hartogs extension property
if and only if for every domain D of any Stein manifold M, every mapping f ∈
O(D,Z) extends to a mapping f̂ ∈ O(D̂, Z), where D̂ is the envelope of holomorphy
of D.
In the light of Definition 2.4 and Shiffman’s Theorem, the natural “target spaces”
Z for obtaining satisfactory answers to the PROBLEM are the complex analytic
spaces which possess the Hartogs extension property.
2.5. Statement of the main results. We are now ready to state the main results.
Theorem A. Let X, Y be two complex manifolds, let D ⊂ X, G ⊂ Y be two open
sets, let A (resp. B) be a subset of D (resp. G). D (resp. G) is equipped with a
system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let Z be a
complex analytic space possessing the Hartogs extension property. Then, for every
mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
10 VIÊT-ANH NGUYÊN
• f is locally bounded along X
A ∩ ∂D,B ∩ ∂G;D,G
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique mapping f̂ ∈ O(
W,Z) which admits A-limit f(ζ, η) at every
point (ζ, η) ∈ W ∩ W̃ .
If, moreover, Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−eω(z,w)
A×B |f |
eω(z,w)
W , (z, w) ∈
Theorem A has an important corollary. Before stating this, we need to introduce
a terminology. A complex manifoldM is said to be a Liouville manifold if PSH(M)
does not contain any non-constant bounded above functions. We see clearly that
the class of Liouville manifolds contains the class of connected compact manifolds.
Corollary B. We keep the hypothesis and the notation in Theorem A. Suppose in
addition that G is a Liouville manifold and that Ã, B̃ 6= ∅. Then, for every mapping
f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along X
A ∩ ∂D,B ∩ ∂G;D,G
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there is a unique mapping f̂ ∈ O(D × G,Z) which admits A-limit f(ζ, η) at every
point (ζ, η) ∈ W ∩ W̃ .
Corollary B follows immediately from Theorem A since ω̃(·, B,G) ≡ 0.
We will see in Section 10 below that Theorem A and Corollary B generalizes all
the results discussed in Section 1 above. Moreover, they also give many new results.
Although our main results have been stated only for the case of a 2-fold cross, they
can be formulated for the general case of an N -fold cross with N ≥ 2 (see also
[28, 33]).
3. Holomorphic discs and a Two-Constant Theorem
We recall here some elements of Poletsky theory of discs, some background of the
pluripotential theory and auxiliary results needed for the proof of Theorem A.
3.1. Poletsky theory of discs and Rosay Theorem on holomorphic discs.
Let E denote as usual the open unit disc in C. For a complex manifold M, let
O(E,M) denote the set of all holomorphic mappings φ : E −→ M which extend
holomorphically to a neighborhood of E. Such a mapping φ is called a holomorphic
disc on M. Moreover, for a subset A of M, let
1A,M(z) :=
1, z ∈ A,
0, z ∈ M \ A.
4 It follows from Subsection 2.3 that
A ∩ ∂D,B ∩ ∂G;D,G
(A ∩ ∂D)× (G ∪B)
(D ∪ A)× (B ∩ ∂G)
A UNIFIED APPROACH 11
In the work [40] Rosay proved the following remarkable result.
Theorem 3.1. Let u be an upper semicontinuous function on a complex manifold
M. Then the Poisson functional of u defined by
P[u](z) := inf
u(φ(eiθ))dθ : φ ∈ O(E,M), φ(0) = z
is plurisubharmonic on M.
Rosay Theorem may be viewed as an important development in Poletsky theory
of discs. Observe that special cases of Theorem 3.1 have been considered by Poletsky
(see [37, 38]), Lárusson–Sigurdsson (see [22]) and Edigarian (see [10]).
The following Rosay type result gives the connections between the Poisson func-
tional of the characteristic function 1M\A,M and holomorphic discs.
Lemma 3.2. Let M be a complex manifold and let A be a nonempty open subset of
M. Then for any ǫ > 0 and any z0 ∈ M, there are an open neighborhood U of z0,
an open subset T of C, and a family of holomorphic discs (φz)z∈U ⊂ O(E,M) with
the following properties:
(i) Φ ∈ O(U ×E,M), where Φ(z, t) := φz(t), (z, t) ∈ U × E;
(ii) φz(0) = z, z ∈ U ;
(iii) φz(t) ∈ A, t ∈ T ∩ E, z ∈ U ;
(iv) 1
1∂E\T,∂E(e
iθ)dθ < P[1M\A,M](z0) + ǫ.
Proof. See Lemma 3.2 in [28]. �
The next result describes the situation in dimension 1. It will be very useful later
Lemma 3.3. Let T be an open subset of E. Then
ω(0, T ∩ E,E) ≤
1∂E\T,T (e
iθ)dθ.
Proof. See, for example, Lemma 3.3 in [28]. �
The last result, which is an important consequence of Rosay’s Theorem, gives the
connection between the Poisson functional and the plurisubharmonic measure.
Proposition 3.4. Let M be a complex manifold and A a nonempty open subset of
M. Then ω(z, A,M) = P[1M\A,M](z), z ∈ M.
Proof. See, for example, the proof of Proposition 3.4 in [28]. �
12 VIÊT-ANH NGUYÊN
3.2. Level sets of the relative extremal functions and a Two-Constant
Theorem. Let X be a complex manifold and D ⊂ X an open set. Suppose that
D is equipped with a system of approach regions A =
Aα(ζ)
ζ∈D, α∈Iζ
. For every
open subset G of D, there is a natural system of approach regions for G which is
called the induced system of approach regions A
ζ∈G, α∈I
of A onto G.
It is given by
α(ζ) := Aα(ζ) ∩G, ζ ∈ G, α ∈ I
where I
α ∈ Iζ : ζ ∈ Aα(ζ) ∩G
Proposition 3.5. Under the above hypothesis and notation, let A ⊂ D be a locally
pluriregular set (relative to A). For 0 < δ < 1, define the δ-level set of D relative
to A as follows
Dδ,A := {z ∈ D : ω(z, A,D) < 1− δ} .
We equip Dδ,A with the induced system of approach regions A
of A onto Dδ,A (see
Subsection 2.1 above). Then A ⊂ Dδ,A and
(3.1) ω(z, A,Dδ,A) =
ω(z, A,D)
, z ∈ Dδ,A.
Moreover, A is locally pluriregular relative to A
Proof. Since A is locally pluriregular, we see that
(3.2)
A− lim supω(·, A,D)
(z) = 0, z ∈ A.
Therefore, for every z ∈ A and α ∈ Iz, there is an open neighborhood U of z such
that ∅ 6= Aα(z) ∩ U ⊂ Dδ,A. Hence, A ⊂ Dδ,A.
Next, we turn to the proof of identity (3.1). Observe that 0 ≤
ω(·,A,D)
≤ 1 on
Dδ,A by definition. This, combined with (3.2), implies that
(3.3)
ω(z, A,D)
≤ ω(z, A,Dδ,A), z ∈ Dδ,A.
To prove the converse inequality of (3.3), let u ∈ PSH(Dδ,A) be such that u ≤ 1 on
Dδ,A and A
− lim sup u ≤ 0 on A. Consider the following function
(3.4) û(z) :=
max {(1− δ)u(z), ω(z, A,D)} , z ∈ Dδ,A,
ω(z, A,D), z ∈ D \Dδ,A.
It can be checked that û ∈ PSH(D) and 0 ≤ û ≤ 1. Moreover, in virtue of the
assumption on u and (3.2) and (3.4), we have that
(A−lim sup û)(a) ≤ max
(1− δ)(A
− lim sup u)(a),
A− lim supω(·, A,D)
for all a ∈ A. Consequently, û ≤ ω(·, A,D). In particular, one gets from (3.4) that
u(z) ≤
ω(z, A,D)
, z ∈ Dδ,A.
Since u is arbitrary, we deduce from the latter estimate that the converse inequality
of (3.3) also holds. This, combined with (3.3), completes the proof of (3.1).
A UNIFIED APPROACH 13
To prove the last conclusion of the proposition, fix a point a ∈ A and an open
neighborhood U of a. Then we have
A− lim supω(·, A∩U,Dδ,A∩U)
(a) ≤
A− lim supω(·, A∩U, (D∩U)δ,A∩U )
A− lim supω(·, A ∩ U,D ∩ U)
(a) = 0,
where the first equality follows from identity (3.1) and the second one from the
hypothesis that A is locally pluriregular. �
The following Two-Constant Theorem for plurisubharmonic functions will play
an important role in the proof of the estimate in Theorem A.
Theorem 3.6. Let X be a complex manifold and D ⊂ X an open subset. Suppose
that D is equipped with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
. Let A ⊂ D
be a locally pluriregular set. Let m,M ∈ R and u ∈ PSH(D) such that u(z) ≤ M
for z ∈ D, and (A− lim sup u)(z) ≤ m for z ∈ A. Then
u(z) ≤ m(1− ω(z, A,D)) +M · ω(z, A,D), z ∈ D.
Proof. It follows immediately from Definition 2.2. �
Theorem 3.7. We keep the hypotheses and notation of Theorem 3.6. Let f be a
bounded function in O(D,C) such that (A − lim f)(ζ) = 0, ζ ∈ A. Then f(z) = 0
for all z ∈ D such that ω(z, A,D) 6= 1.
Proof. Fix a finite positive constant M such that |f |D < M. Consequently, the
desired conclusion follows from applying Theorem 3.6 to the function u := log |f |.
3.3. Construction of discs. In this subsection we present the construction of discs
à la Poletsky (see [38]). This is one of the main ingredients in the proof of Theorem
Let mes denote the Lebesgue measure on the unit circle ∂E. For a bounded
mapping φ ∈ O(E,Cn) and ζ ∈ ∂E, f(ζ) denotes the angular limit value of f at
ζ if it exists. A classical theorem of Fatou says that mes ({ζ ∈ ∂E : ∃f(ζ)}) = 2π.
For z ∈ Cn and r > 0, let B(z, r) denote the open ball centered at z with radius r.
Theorem 3.8. Let D be a bounded open set in Cn, A ⊂ D, z0 ∈ D and ǫ > 0.
Let A be a system of approach regions for D. Suppose in addition that A is locally
pluriregular (relative to A). Then there exist a bounded mapping φ ∈ O(E,Cn) and
a measurable subset Γ0 ⊂ ∂E with the following properties:
1) Γ0 is pluriregular (with respect to the system of angular approach regions),
φ(0) = z0, φ(E) ⊂ D, Γ0 ⊂
ζ ∈ ∂E : φ(ζ) ∈ A
, and
·mes(Γ0) < ω(z0, A,D) + ǫ.
14 VIÊT-ANH NGUYÊN
2) Let f ∈ C(D ∪ A,C) ∩ O(D,C) be such that f(D) is bounded. Then there
exist a bounded function g ∈ O(E,C) such that g = f ◦ φ in a neighborhood
of 0 ∈ E and5 g(ζ) = (f ◦ φ)(ζ) for all ζ ∈ Γ0. Moreover, g|Γ0 ∈ C(Γ0,C).
This theorem motivates the following
Definition 3.9. We keep the hypothesis and notation of Theorem 3.8. Then ev-
ery pair (φ,Γ0) satisfying the conclusions 1)–2) of this theorem is said to be an
ǫ-candidate for the triplet (z0, A,D).
Theorem 3.8 says that there always exist ǫ-candidates for all triplets (z, A,D).
Proof. First we will construct φ. To do this we will construct by induction a sequence
k=1 ⊂ O(E,D) which approximates φ as k ր ∞. This will allow to define the
desired mapping as φ := lim
φk. The construction of such a sequence is divided into
three steps.
For 0 < δ, r < 1 let
Da,r := D ∩ B(a, r), a ∈ A.
Aa,r,δ := {z ∈ Da,r : ω(z, A ∩ B(a, r), Da,r) < δ} , a ∈ A,
Ar,δ :=
Aa,r,δ,
(3.5)
where in the second “:=” Da,r is equipped with the induced system of approach
regions of A onto Da,r (see Subsection 3.2 above).
Suppose without loss of generality that D ⊂ B(0, 1).
Step 1: Construction of φ1.
Let δ0 :=
and r0 := 1. Fix 0 < δ1 <
and 0 < r1 <
. Applying Proposition
3.4, we obtain φ1 ∈ O(E,D) such that φ1(0) = z0 and
∂E ∩ φ−11 (Ar1,δ1)
≤ ω(z0, Ar1,δ1, D) + δ0.
On the other hand, using (3.5) and Definition 2.2 and the hypothesis that A is
locally pluriregular, we obtain
ω(z0, Ar1,δ1 , D) ≤ ω(z0, A,D).
Consequently, we may choose a subset Γ1 of Γ0 := ∂E ∩ φ
1 (Ar1,δ1) which consists
of finite disjoint closed arcs (Γ1j)j∈J1 so that
(3.6) 1−
·mes(Γ1) < ω(z0, Ar1,δ1, D) + 2δ0 ≤ ω(z0, A,D) + 2δ0,
t,τ∈Γ1j
|t− τ | < 2δ1, sup
t,τ∈Γ1j
|φ1(t)− φ1(τ)| < 2r1, j ∈ J1.
Step 2: Construction of φk+1 from φk for all k ≥ 1.
By the inductive construction we have 0 < δk <
and 0 < rk <
φk ∈ O(E,D) such that φk(0) = z0 and there exists a closed subset Γk of ∂E ∩
5 Note here that by Part 1), (f ◦ φ)(ζ) exists for all ζ ∈ Γ0.
A UNIFIED APPROACH 15
(Ark,δk) ∩ Γk−1 which consists of finite closed arcs (Γk,j)j∈Jk such that Γk is
relatively compact in the interior of Γk−1, and
(3.7) 1−
·mes(Γk) < 1−
·mes(Γk−1) + 2δk−1,
t,τ∈Γk,j
|t− τ | < 2δk, sup
t,τ∈Γk,j
|φk(t)− φk(τ)| < 2rk, j ∈ Jk,
|φk − φk−1|Γk < 2rk−1.
Here we make the convention that the last inequality is empty when k = 1.
In particular, we have that φk(Γk) ⊂ Ark,δk . Therefore, by (3.5), for every ζ ∈
φk(Γk) there is a ∈ A such that ζ ∈ Aa,rk,δk , that is,
ω(ζ, A∩ B(a, rk), Da,rk) < δk.
Using the hypothesis that A is locally pluriregular and (3.5) we see that
ω(z, Ar,δ ∩Da,rk , Da,rk) ≤ ω(z, A ∩ B(a, rk), Da,rk), 0 < δ, r < 1.
Consequently, for every ζ ∈ φk(Γk) there is a ∈ A such that
ω(ζ, Ar,δ ∩Da,rk , Da,rk) < δk, 0 < δ, r < 1.
Using the last estimate and arguing as in [38, p. 120–121] (see also the proof of
Theorem 1.10.7 in [19] for a nice presentation), we can choose 0 < δk+1 <
0 < rk+1 <
and φk+1 ∈ O(E,D) such that φk+1(0) = z0, and there exists a
closed subset Γk+1 of ∂E ∩ φ
k+1(Ark+1,δk+1) ∩ Γk which consists of finite closed arcs
(Γk+1,j)j∈Jk+1 such that Γk+1 is relatively compact in the interior of Γk, and
(3.8) 1−
·mes(Γk+1) < 1−
·mes(Γk) + 2δk,
t,τ∈Γk+1,j
|t− τ | < 2δk+1, sup
t,τ∈Γk+1,j
|φk+1(t)− φk+1(τ)| < 2rk+1, j ∈ Jk+1,
|φk+1 − φk|Γk+1 < 2rk.
Step 3: Construction of φ from the sequence (φk)
In summary, we have constructed a decreasing sequence (Γk)
k=1 of closed subsets
of ∂E. Consider the new closed set
By (3.7)–(3.8),
·mes(Γ) =
mes(Γ1)− 2
mes(Γ1)− 3δ1.
16 VIÊT-ANH NGUYÊN
This, combined with (3.6), implies the following property
·mes(Γ) < 1−
·mes(Γ1)+3δ1 ≤ ω(z0, A,D)+2δ0+3δ1 < ω(z0, A,D)+ ǫ.
On the other hand, we recall from the above construction the following properties:
(ii) φk(Γ) ⊂ φk(Γk) ⊂ Ark,δk .
(iii) δ0 =
, r0 = 1, 0 < δk+1 <
, 0 < rk+1 <
and |φk+1−φk|Γ ≤ |φk+1−φk|Γk+1 <
(iv) sup
t,τ∈Γkj
|t− τ | < 2δk and sup
t,τ∈Γk,j
|φk(t)− φk(τ)| < 2rk, j ∈ Jk.
(v) For every ζ ∈ Γ there exists a sequence (jk)k≥1 such that jk ∈ Jk, and ζ is an
interior point of Γk,jk , and Γk+1,jk+1 ⋐ Γk,jk , and ζ =
Γk,jk .
Therefore, we are able to apply the Khinchin–Ostrowski Theorem (see [11, The-
orem 4, p. 397]) to the sequence (φk)
k=1. Consequently, this sequence converges
uniformly on compact subsets of E to a mapping φ ∈ O(E,D). Moreover, φ admits
(angular) boundary values at all points of Γ and φ(Γ) ⊂
Ark,δk ⊂ A.
Observe that since φk(0) = φ(0) = z0 ∈ D and f ∈ C(D ∪ A,C) ∩ O(D,C), the
sequence (f ◦ φk)
k=1 converges to f ◦ φ uniformly on a neighborhood of 0 ∈ E. On
the other hand, f(D) is bounded by the hypothesis. Thus by Montel Theorem, the
family (f ◦ φk)
k=1 ⊂ O(E,C) is normal. Consequently, the sequence (f ◦ φk)
converges uniformly on compact subsets of E. Let g be the limit mapping. Then
g ∈ O(E,C) and g = f ◦ φ in a neighborhood of 0 ∈ E. Moreover, it follows
from (i)–(iii) above and the hypothesis f ∈ C(D ∪ A,C) that g(ζ) = (f ◦ φ)(ζ)
for all ζ ∈ Γ. We deduce from (iii)–(v) above that g|Γ ∈ C(Γ,C) Finally, applying
Lemma 4.1 below we may choose a locally pluriregular subset Γ0 ⊂ Γ (relative to
the system of angular approach regions) such that mes(Γ0) = mes(Γ). Hence, the
proof is finished. �
It is worthy to remark that φ(E) ⊂ D; but in general, φ(E) 6⊂ D !
The last result of this section sharpens Theorem 3.8.
Theorem 3.10. Let D be a bounded open set in Cn, A ⊂ D, and ǫ > 0. Let A be a
system of approach regions for D. Suppose in addition that A is locally pluriregular
(relative to A). Then there exists a Borel mapping Φ : D × E −→ Cn with the
following property: for every z ∈ D, there is a measurable subset Γz of ∂E such that
(Φ(z, ·),Γz) is an ǫ-candidate for the triplet (z, A,D).
Roughly speaking, this result says that one can construct ǫ-candidates for (z, A,D)
so that they depend in a Borel-measurable way on z ∈ D.
Proof. Observe that in Proposition 3.4 we can construct ǫ-candidates for (z, A,M)
so that they depend in a Borel-measurable way on z ∈ M. Here an ǫ-candidate
for (z, A,M) is a holomorphic disc φ ∈ O(E,M) such that φ(0) = z and
1∂E\φ−1(A),∂E(e
iθ)dθ < P[1M\A,M](z) + ǫ.
A UNIFIED APPROACH 17
Using this we can adapt the proof of Theorem 3.8 in order to obtain the desired
result. �
4. A mixed cross theorem
Let E be as usual the open unit disc in C. Let B be a measurable subset of ∂E
and ω(·, B, E) the relative extremal function of B relative to E (with respect to the
canonical system of approach regions). Then it is well-known (see [39]) that
(4.1) ω(z, B, E) =
1− |z|2
|eiθ − z|2
· 1∂E\B,∂E(e
iθ)dθ.
The following elementary lemma will be very useful.
Lemma 4.1. We keep the above hypotheses and notation.
1) Let u be a subharmonic function defined on E with u ≤ 1 and let α ∈ (0, π
be such that
lim sup
z→ζ, z∈Aα(ζ)
u(z) ≤ 0 for a.e. ζ ∈ B,
where A = (Aα(ζ)) is the system of angular approach regions defined in
Subsection 2.2. Then u ≤ ω(·, B, E) on E.
2) ω(·, B, E) is also the relative extremal function of B relative to E (with
respect to the system of angular approach regions).
3) For all subsets N ⊂ ∂E with mes(N ) = 0, ω(·, B, E) = ω(·, B ∪N , E).
4) Let B
be the set of all density points of B. Then
z→ζ, z∈Aα(ζ)
ω(z, B, E) = 0, ζ ∈ B
, 0 < α <
In particular, B
is locally pluriregular (with respect to the system of angular
approach regions).
5) ω(·, B, E) = ω̃c(·, B, E) = ω̃a(·, B, E) on E, where ω̃c(·, B, E) (resp.
ω̃a(·, B, E)) is given by Definition 2.3 relative to the system of canonical
approach regions (resp. angular approach regions).
Proof. It follows immediately from the explicit formula (4.1). �
The main ingredient in the proof of Theorem A is the following mixed cross
theorem.
Theorem 4.2. Let D be a complex manifold and E as usual the open unit disc in
C. D (resp. E) is equipped with the canonical system of approach regions (resp.
the system of angular approach regions). Let A be an open subset of D and B a
measurable subset of ∂E such that B is locally pluriregular (relative to the system of
angular approach regions). For 0 ≤ δ < 1 put G := {w ∈ E : ω(w,B,E) < 1− δ} .
18 VIÊT-ANH NGUYÊN
Let W := X(A,B;D,G), W o := Xo(A,B;D,G), and6
Ŵ = X̂(A,B;D,G) :=
(z, w) ∈ D ×G : ω(z, A,D) +
ω(w,B,E)
Let f : W −→ C be such that
(i) f ∈ Os(W
o,C);
(ii) f is locally bounded on W, f |A×B is a Borel function;
(iii) for all z ∈ A,
w→η, w∈Aα(η)
f(z, w) = f(z, η), η ∈ B, 0 < α <
Then there is a unique function f̂ ∈ O(Ŵ ,C) such that f̂ = f on A×G. Moreover,
|f |W = |f̂ |cW .
The proof of this theorem will occupy the present and the next sections. Our
approach here avoid completely the classical method of doubly orthogonal bases
of Bergman type. For the proof we need the following “measurable” version of
Gonchar’s Theorem.
Theorem 4.3. Let D = G := E be equipped with the system of angular approach
regions. Let A (resp. B) be a Borel measurable subset of ∂D (resp. ∂G) such
that A and B are locally pluriregular and that mes(A), mes(B) > 0. Put W :=
X(A,B;D,G) and define W o, Ŵ , ω(z, w) as in Subsection 2.3. Let f : W −→ C
be such that:
(i) f is locally bounded on W and f ∈ Os(W
o,C);
(ii) f |A×B is a Borel function;
(iii) for all a ∈ A (resp. b ∈ B), f(a, ·)|G (resp. f(·, b)|D) admits A-limit
7 f(a, b)
at all b ∈ B (resp. at all a ∈ A).
Then there exists a unique function f̂ ∈ O(Ŵ ,C) which admits A-limit f(ζ, η) at
all points (ζ, η) ∈ W o. If, moreover, |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,w)
A×B |f |
ω(z,w)
W , (z, w) ∈ Ŵ .
Proof. It follows from Steps 1–3 of Section 6 in [34]. �
The above theorem is also true in the context of an N -fold cross W (N ≥ 2). We
give here a version of a special 3-fold cross which is needed for the proof of Theorem
Theorem 4.4. Let D = G := E be equipped with the system of angular approach
regions. Let A (resp. B) be a Borel measurable subset of ∂D (resp. ∂G) such that
6 In fact, Theorem 4.10 in [34] says that ω(·, B,G) =
ω(·,B,E)
on G, where ω(·, B,G) is the
relative extremal function with respect to the system of angular approach regions induced onto G.
7 that is, the angular limit
A UNIFIED APPROACH 19
A and B are locally pluriregular and that mes(A), mes(B) > 0. Define W, W o, Ŵ
as follows:
W = X(A, ∂E,B;D,E,G) := A× ∂E × (G ∪B)
A× E × B
(D ∪ A)× ∂E × B,
W o = Xo(A, ∂E,B;D,E,G) := A× ∂E ×G
A× E × B
D × ∂E × B,
Ŵ = X̂(A, ∂E,B;D,E,G) := {(z, t, w) ∈ D × E ×G : ω(z, A,D) + ω(w,B,G) < 1} .
Let f : W −→ C be such that:
(i) f is locally bounded on W and f ∈ Os(W
o,C)8;
(ii) f |A×∂E×B is a Borel function;
(iii) for all (a, λ) ∈ A × ∂E (resp. (a, b) ∈ A × B) (resp. (λ, b) ∈ ∂E × B),
f(a, λ, ·)|G (resp. f(a, ·, b)|E) (resp. f(·, λ, b)|D) admits the angular limit
f(a, λ, b) at all b ∈ B (resp. at all λ ∈ ∂E) (resp. at all a ∈ A).
Then there exists a unique function f̂ ∈ O(Ŵ ,C) such that
cW∋(z,t,w)→(ζ,τ,η),w∈Aα(η)
f̂(z, t, w) =
f(ζ, λ, η)
(ζ, τ, η) ∈ D × E × B, 0 < α <
If, moreover, |f |W <∞, then
|f̂(z, t, w)| ≤ |f |
1−ω(z,A,D)−ω(w,B,G)
A×∂E×B |f |
ω(z,A,D)+ω(w,B,G)
W , (z, t, w) ∈ Ŵ .
Proof. We refer the reader to Subsections 5.2 and 5.3 in [34].
Let ω̂(·, A,D) (resp. ω̂(·, B,G)) be the conjugate harmonic function of ω(·, A,D)
(resp. ω(·, B,G) ) such that ω̂(z0, A,D) = 0 (resp. ω̂(w0, B,G) = 0) for a certain
fixed point z0 ∈ D (resp. w0 ∈ G). Thus we define the holomorphic functions
g1(z) := ω(z, A,D) + iω̂(z, A,D), g2(w) := ω(w,B,G) + iω̂(w,B,G), and
g(z, w) := g1(z) + g2(w), (z, w) ∈ D ×G.
Each function e−g1 (resp. e−g2) is bounded on D (resp. on G). Therefore, in
virtue of [11, p. 439], we may define e−g1(a) (resp. e−g2(b)) for a.e. a ∈ A (resp. for
a.e. b ∈ B) to be the angular limit of e−g1 at a (resp. e−g2 at b).
In virtue of (i), for each positive integer N, we define, as in [12, 13] (see also [34]),
the Gonchar–Carleman operator as follows
(4.2) KN(z, t, w) = KN [f ](z, t, w) :=
(2πi)2
−N(g(a,b)−g(z,w)) f(a, t, b)dadb
(a− z)(b− w)
for (z, t, w) ∈ D × ∂E × G. Reasoning as in [13] and using (i)–(iii) above, we see
that the following limit
(4.3) K(z, t, w) = K[f ](z, t, w) := lim
KN(z, t, w)
8 This notation means that for all (a, λ) ∈ A×∂E (resp. (a, b) ∈ A×B) (resp. (λ, b) ∈ ∂E×B),
the function f(a, λ, ·)|G (resp. f(a, ·, b)|E) (resp. f(·, λ, b)|D) is holomorphic.
20 VIÊT-ANH NGUYÊN
exists for all points in the set
(z, t, w) : t ∈ ∂E, (z, w) ∈ X̂(A,B;D,G)
, and its
limit is uniform on compact subsets of the latter set.
Observe that for n = 0, 1, 2, . . . , and N = 1, 2, . . . ,
KN(z, t, w)dt =
(2πi)2
f(a, t, b)dt
e−N(g(a,b)−g(z,w))dadb
(a− z)(b− w)
where the first equality follows from (4.2), the second one from the equality∫
tnf(a, t, b)dt = 0 which itself is an immediate consequence of (i). Therefore,
we deduce from (4.3) that
tnK(z, t, w)dt = 0, (z, w) ∈ X̂(A,B;D,G), n = 0, 1, 2, . . . .
On the other hand,
(z, t, w) : t ∈ E, (z, w) ∈ X̂(A,B;D,G)
Hence, we are able to define the desired extension function
f̂(z, t, w) :=
K(z, λ, w)
dλ, (z, t, w) ∈ Ŵ .
Recall from Steps 1–3 of Section 6 in [34] that
cW∋(z,w)→(ζ,η),w∈Aα(η)
K(z, t, w) = f(ζ, t, η), (ζ, t, η) ∈ D × ∂E × B, 0 < α <
Inserting this into the above formula of f̂ , the desired conclusion of the theorem
follows. �
We break the proof of Theorem 4.2 into two cases.
CASE 1: δ = 0 (that is G = E).
We follow essentially the arguments presented in Section 4 of [28]. For the sake
of clarity and completeness we give here the most basic arguments.
We begin the proof with the following lemma.
Lemma 4.5. We keep the hypothesis of Theorem 4.2. For j ∈ {1, 2}, let φj ∈
O(E,D) be a holomorphic disc, and let tj ∈ E such that φ1(t1) = φ2(t2) and
1D\A,D(φj(e
iθ))dθ < 1. Then:
1) For j ∈ {1, 2}, the function (t, w) 7→ f(φ(t), w) defined on X(φ−1j (A) ∩
∂E,B;E,G) satisfies the hypothesis of Theorem 4.3, where φ−1j (A) := {t ∈
E : φj(t) ∈ A}.
A UNIFIED APPROACH 21
2) For j ∈ {1, 2}, in virtue of Part 1), let f̂j be the unique function in
X̂(φ−1j (A) ∩ ∂E,B;E,G),C
given by Theorem 4.3. Then
f̂1(t1, w) = f̂2(t2, w),
for all w ∈ G such that (tj, w) ∈ X̂
φ−1j (A) ∩ ∂E,B;E,G
, j ∈ {1, 2}.
Proof of Lemma 4.5. Part 1) follows immediately from the hypothesis. There-
fore, it remains to prove Part 2). To do this fix w0 ∈ G such that (tj, w0) ∈
φ−1j (A) ∩ E,B;E,G
for j ∈ {1, 2}.We need to show that f̂1(t1, w0) = f̂2(t2, w0).
Observe that both functions w ∈ G 7→ f̂1(t1, w) and w ∈ G 7→ f̂2(t2, w) belong to
O(G,C), where G is the connected component which contains w0 of the following
open set
w ∈ G : ω(w,B,G) < 1− max
j∈{1,2}
ω(tj, φ
j (A) ∩ ∂E,E)
Since φ1(t1) = φ2(t2), it follows from Theorem 4.3 and the hypothesis of Part 2)
(A− lim f̂1)(t1, η) = f(φ1(t1), η) = f(φ2(t2), η) = (A− lim f̂2)(t2, η), η ∈ B.
Therefore, by Theorem 3.7, f̂1(t1, w) = f̂2(t2, w), w ∈ G. Hence, f̂1(t1, w0) =
f̂2(t2, w0), which completes the proof of the lemma. �
Now we return to the proof of the theorem in CASE 1 which is divided into two
steps.
Step 1: Construction of the extension function f̂ on Ŵ and its uniqueness.
Proof of Step 1. We define f̂ as follows: Let W be the set of all pairs (z, w) ∈ D×G
with the property that there are a holomorphic disc φ ∈ O(E,D) and t ∈ E such
that φ(t) = z and (t, w) ∈ X̂ (φ−1(A) ∩ ∂E,B;E,G) . By Part 1) of Lemma 4.5 and
Theorem 4.3, let f̂φ be the unique function in O
X̂(φ−1(A) ∩ ∂E,B;E,G),C
(4.4) (A− lim f̂φ)(t, w) = f(φ(t), w), (t, w) ∈ X
−1(A) ∩ ∂E,B;E,G
Then the desired extension function f̂ is given by
(4.5) f̂(z, w) := f̂φ(t, w).
In virtue of Part 2) of Lemma 4.5, f̂ is well-defined on W. We next prove that
(4.6) W = Ŵ .
Taking (4.6) for granted, then f̂ is well-defined on Ŵ .
Now we return to (4.6). To prove the inclusion W ⊂ Ŵ , let (z, w) ∈ W. By the
above definition of W, one may find a holomorphic disc φ ∈ O(E,D), a point t ∈ E
22 VIÊT-ANH NGUYÊN
such that φ(t) = z and (t, w) ∈ X̂ (φ−1(A) ∩ ∂E,B;E,G) . Since ω(φ(t), A,D) ≤
ω(t, φ−1(A) ∩ ∂E,E), it follows that
ω(z, A,D) + ω(w,B,G) ≤ ω(t, φ−1(A) ∩ ∂E,E) + ω(w,B,G) < 1,
Hence (z, w) ∈ Ŵ . This proves the above mentioned inclusion.
To finish the proof of (4.6), it suffices to show that Ŵ ⊂ W. To do this, let
(z, w) ∈ Ŵ and fix any ǫ > 0 such that
(4.7) ǫ < 1− ω(z, A,D)− ω(w,B,G).
Applying Theorem 3.1 and Proposition 3.4, there is a holomorphic disc φ ∈ O(E,D)
such that φ(0) = z and
(4.8)
1D\A,D(φ(e
iθ))dθ < ω(z, A,D) + ǫ.
Observe that
ω(0, φ−1(A) ∩ ∂E,E) + ω(w,B,G) =
1D\A,D(φ(e
iθ))dθ + ω(w,B,G)
< ω(z, A,D) + ω(w,B,G) + ǫ < 1,
where the equality follows from (4.1), the first inequality holds by (4.8), and the
last one by (4.7). Hence, (0, w) ∈ X̂ (φ−1(A) ∩ ∂E,B;E,G) , which implies that
(z, w) ∈ W. This completes the proof of (4.6). Hence, the construction of f̂ on Ŵ
has been completed.
Next we show that f̂ = f on A×G. To this end let (z0, w0) be an arbitrary point
of A × G. Choose the holomorphic disc φ ∈ O(E,D) given by φ(t) := z0, t ∈ E.
Then by formula (4.5),
f̂(z0, w0) = f̂φ(0, w0) = f(φ(0), w0) = f(z0, w0).
If g ∈ O(Ŵ ,C) satisfies g = f on A × G, then we deduce from (4.4)–(4.5) that
g = f̂ . This proves the uniqueness of f̂ . �
Finally, we conclude the proof of CASE 1 by the following
Step 2: Proof of the fact that f̂ ∈ O(Ŵ ,C).
Proof of Step 2. Fix an arbitrary point (z0, w0) ∈ Ŵ and let ǫ > 0 be so small such
(4.9) 2ǫ < 1− ω(z0, A,D)− ω(w0, B,G).
Since ω(·, B,G) ∈ PSH(G), one may find an open neighborhood V of w0 such that
(4.10) ω(w,B,D) < ω(w0, B,G) + ǫ, w ∈ V.
A UNIFIED APPROACH 23
Let n be the dimension of D at the point z0. Applying Lemma 3.2 and Proposition
3.4, we obtain an open set T in C, an open neighborhood U of z0, and a family of
holomorphic discs (φz)z∈U ⊂ O(E,D) with the following properties:
the mapping (z, t) ∈ U × E 7→ φz(t) is holomorphic;(4.11)
φz(0) = z, z ∈ U ;(4.12)
φz(t) ∈ A, t ∈ T ∩ E, z ∈ U ;(4.13)
1∂E\T,∂E(e
iθ)dθ < ω(z0, A,D) + ǫ.(4.14)
By shrinking U (if necessary), we may assume without loss of generality that in a
chart, z0 = 0 ∈ C
n and
(4.15) U =
z = (z1, . . . , zn) = (z
, zn) ∈ Cn : z
∈ S, |zn| < 2
where S ⊂ Cn−1 is an open set.
Consider the 3-fold cross (compared with the notation in Theorem 4.4)
X (T ∩ ∂E, U,B;E,U,G) := (T ∩ ∂E)× U × (G ∪ B)
(T ∩ ∂E)× U × B
(E ∪ (T ∩ ∂E))× U ×B,
and the function g : X (T ∩ ∂E, U,B;E,U,G) −→ C given by
(4.16) g(t, z, w) := f(φz(t), w), (t, z, w) ∈ X (T ∩ ∂E, U,B;E,U,G) .
We make the following observations:
Let t ∈ T ∩ ∂E. Then, in virtue of (4.13) we have φz(t) ∈ A for z ∈ U. Con-
sequently, in virtue of (4.11), (4.16) and the hypothesis f ∈ Os(W
o,C), we con-
clude that g(t, z, ·)|G ∈ O(G,C)
resp. g(t, ·, w)|U ∈ O(U,C)
for any z ∈ U (resp.
w ∈ B). Analogously, for any z ∈ U, w ∈ B, we can show that g(·, z, w)|E ∈ O(E,C).
In summary, we have shown that g is separately holomorphic. In addition,
it follows from hypothesis (ii) and (4.11)–(4.13) that g is locally bounded and
g|(T∩∂E)×U×B is a Borel function.
For z
∈ S write Ez′ :=
z = (z
, zn) ∈ C
n : |zn| < 1
. Then by (4.15),⋃
Ez′ ⊂ U. Consequently, for all z
∈ S, using hypothesis (iii) we are
able to apply Theorem 4.4 to g in order to obtain a unique function ĝ ∈
X̂ (T ∩ ∂E, ∂Ez′ , B;E,Ez′ , G) ,C
9 such that
(t,z,w)→(τ,ζ,η), w∈Aα(η)
ĝ(t, z, w) =
g(τ, ζ
, λ, η)
λ− ζn
(τ, ζ, η) ∈ E ×Ez′ × B, z
∈ S, 0 < α <
9 In fact, we identify Ez′ with E in an obvious way.
24 VIÊT-ANH NGUYÊN
Using (4.11) and (4.15)–(4.16) and the Cauchy’s formula, we see that the right hand
side is equal to g(τ, ζ, η). Hence, we have shown that
(4.17)
(t,z,w)→(τ,ζ,η), w∈Aα(η)
ĝ(t, z, w) = g(τ, ζ, η), (τ, ζ, η) ∈ E×Ez′×B, z
∈ S, 0 < α <
Observe that
X̂ (T ∩ ∂E, ∂Ez′ , B;E,Ez′ , G) = {(t, z, w) ∈ E × Ez′ ×G : ω(t, T ∩ ∂E,E) + ω(w,B,G) < 1} .
On the other hand, for any w ∈ V,
ω(0, T ∩ ∂E,E) + ω(w,B,G) ≤
1∂E\T,∂E(e
iθ)dθ + ω(w0, B,G) + ǫ
< ω(z0, A,D) + ω(w0, B,G) + 2ǫ < 1,
(4.18)
where the first inequality follows from (4.1) and (4.10), the second one from (4.14),
and the last one from (4.9). Consequently,
(4.19) (0, z, w) ∈ X̂ (T ∩ ∂E, ∂Ez′ , B;E,Ez′ , G) , (z, w) ∈ Ez′ × V, z
It follows from (4.5), (4.12), (4.13) and (4.18) that, for z
∈ S and z ∈ Ez′ , f̂φz is
well-defined and holomorphic on X̂(T ∩ ∂E,B;E,G), and
(4.20) f̂(z, w) = f̂φz(0, w), w ∈ V.
On the other hand, it follows from (4.4), (4.16) and (4.17) that
(t,w)→(τ,η), w∈Aα(η)
f̂φz(t, w) = lim
(t,w)→(τ,η), w∈Aα(η)
ĝ(t, z, w),
(τ, η) ∈ E ×B, z ∈ Ez′ , z
∈ S, 0 < α <
Since, for fixed z ∈ Ez′ , the restricted functions (t, w) 7→ ĝ(t, z, w) and f̂φz are
holomorphic on X̂(T ∩ ∂E,B;E,G), we deduce from the latter equality and the
uniqueness of Theorem 4.3 that
ĝ(t, z, w) = f̂φz(t, w), (t, w) ∈ X̂ (T ∩ ∂E,B;E,G) , z ∈ Ez′ , z
In particular, using (4.5), (4.19) and (4.20),
ĝ(0, z, w) = f̂φz(0, w) = f̂(z, w), (z, w) ∈ Ez′ × V, z
Since we know from (4.19) that ĝ is holomorphic in the variables zn and w on a
neighborhood of (0, z0, w0), it follows that f̂ is holomorphic in the variables z
n and
w on a neighborhood of (z0, w0). Exchanging the role of z
n and any other variable
zj , j = 1, . . . , n − 1, we see that f̂ is separetely holomorphic on a neighborhood
of (z0, w0). In addition, f̂ is locally bounded. Consequently, we conclude, by the
classical Hartogs extension Theorem, that f̂ is holomorphic on a neighborhood of
(z0, w0). Since (z0, w0) ∈ Ŵ is arbitrary, it follows that f̂ ∈ O(Ŵ ,C). �
Combining Steps 1–2, CASE 1 follows. �
A UNIFIED APPROACH 25
5. Completion of the proof of Theorem 4.2
In this section we introduce the new technique of conformal mappings. This
technique will allow us to pass from CASE 1 to the general case. We recall a notion
from Definition 4.8 in [34] which will be relevant for our further study.
Definition 5.1. Let A be the system of angular approach regions for E, let Ω be an
open subset of the unit disc E and ζ a point in ∂E. Then the point ζ is said to be
an end-point of Ω if, for every 0 < α < π
, there is an open neighborhood U = Uα of
ζ such that U ∩Aα(ζ) ⊂ Ω. The set of all end-points of Ω is denoted by End(Ω).
The main idea of the technique of conformal mappings is described below.
Proposition 5.2. Let B be a measurable subset of ∂E with mes(B) > 0. For 0 ≤
δ < 1 put G := {w ∈ E : ω(w,B,E) < 1− δ} . Let Ω be an arbitrary connected
component of G. Then
1) End(Ω) is a measurable subset of ∂E and mes(End(Ω)) > 0. Moreover, Ω is
a simply connected domain.
In virtue of Part 1) and the Riemann mapping theorem, let Φ be a confor-
mal mapping of Ω onto E.
2) For every ζ ∈ End(Ω), there is η ∈ ∂E such that
z→ζ, z∈Ω∩Aα(ζ)
Φ(z) = η, 0 < α <
η is called the limit of Φ at the end-point ζ and it is denoted by Φ(ζ).
Moreover, Φ|End(Ω) is one-to-one.
3) Let f be a bounded holomorphic function on Ω, ζ ∈ End(Ω) and λ ∈ C such
that lim
z→ζ, z∈Ω∩Aα(ζ)
f(z) = λ for some 0 < α < π
. Then f ◦ Φ−1 ∈ O(E,C)
admits the angular limit λ at Φ(ζ).
4) Let ∆ be a subset of End(Ω) such that mes(∆) = mes(End(Ω)). Put Φ(∆) :=
{Φ(ζ), ζ ∈ ∆}, where Φ(ζ) is given by Part 2). Then Φ(∆) is a measurable
subset of of ∂E with mes
> 0. and
ω(Φ(z),Φ(∆), E) =
ω(z, B, E)
, z ∈ Ω.
Proof. The first assertions of Part 1) follows from Theorem 4.9 in [34]. To show
that Ω is simply connected, take an arbitrary Jordan domain D such that ∂D ⊂ Ω.
We need to prove that D ⊂ Ω. Observe that D ⊂ E and ω(z, B, E) < 1 − δ for all
z ∈ ∂D ⊂ Ω ⊂ G. By the Maximum Principle, we deduce that ω(z, B, E) < 1 − δ
for all z ∈ D. Hence, D ⊂ G, which, in turn, implies that D ⊂ Ω. This completes
Part 1).
Part 2) follows from the “end-point” version of Theorem 4.4.13 in [39] (that is,
we replace the hypothesis “accessible point” therein by end-point).
Applying the classical Lindelöf’s Theorem to f ◦ Φ−1 ∈ O(E,C), Part 3) follows.
26 VIÊT-ANH NGUYÊN
It remains to prove Part 4). A straightforward argument shows that Φ(∆) is a
measurable subset of ∂E. Next, we show that
(5.1) ω(Φ(z),Φ(∆), E) ≤
ω(z, B, E)
, z ∈ Ω.
To do this pick any u ∈ PSH(E) such that u ≤ 1 and
lim sup
u(w) ≤ 0, η ∈ Φ(∆).
Consequently, Part 2) gives that
(5.2) lim sup
z→ζ, z∈Ω∩Aα(ζ)
u ◦ Φ(z) = 0, ζ ∈ ∆, 0 < α <
Next, consider the following function
(5.3) ũ(z) :=
max{(1− δ) · (u ◦ Φ)(z), ω(z, B, E)}, z ∈ Ω,
ω(z, B, E), z ∈ E \ Ω.
Then it can be checked that ũ is subharmonic and ũ ≤ 1 in E. In addition, for
every density point ζ of B such that ζ 6∈ End(Ω), we know from Theorem 4.9 in [34]
that there is a connected component Ωζ of G other than Ω such that ζ ∈ End(Ωζ).
Consequently, Part 4) of Lemma 4.1 gives, for such a point ζ, that
lim sup
z→ζ, z∈Aα(ζ)
ũ(z) = lim sup
z→ζ, z∈Aα(ζ)
ω(z, B, E) = 0, 0 < α <
This, combined with (5.2), implies that
lim sup
z→ζ, z∈Aα(ζ)
ũ(z) = 0, 0 < α <
, for a.e. ζ ∈ B.
Consequently, applying Part 1) of Lemma 4.1 yields that ũ ≤ ω(·, B, E) on E.
Hence, by (5.3), (u ◦Φ)(z) ≤
ω(z,B,E)
, z ∈ Ω, which completes the proof of (5.1). In
particular, we obtain that mes (Φ(∆)) > 0.
To prove the opposite inequality of (5.1), let u be an arbitrary function in PSH(E)
such that u ≤ 1 and
lim sup
u(z) ≤ 0, ζ ∈ B.
Applying Part 3) to the function f(z) := z, we obtain that
lim sup
w→η, w∈Aα(η)
(u ◦ Φ−1) (w)
≤ 0, η ∈ Φ(∆), 0 < α <
On the other hand, since u ≤ ω(·, B, E) on E, one gets that
(u◦Φ−1)(w)
≤ 1, w ∈ E.
Therefore, applying Part 1) of Lemma 4.1 yields that
(u ◦ Φ−1) (w)
≤ ω(w,Φ(∆), E), w ∈ E,
which, in turn, implies the converse inequality of (5.1). Hence, the proof of Part 4)
is complete. �
A UNIFIED APPROACH 27
Now we are in the position to complete the proof of Theorem 4.2:
CASE 2: 0 < δ < 1.
Let (Gk)k∈K be the family of all connected components of G, where K is an (at
most) countable index set. By Proposition 5.2, we may fix a conformal mapping Φk
from Gk onto E for every k ∈ K. Put
Bk :=
End(Gk) ∩ B
, Wk := X(A,B
k;D,E),
W ok := X
o(A,B
k;D,E), Ŵk := X̂(A,B
k;D,E), k ∈ K.
(5.4)
where [T ]
(or simply T
) for T ⊂ ∂E is, following the notation of Lemma 4.1, the
set of all density points of T.
Recall from the hypotheses of Theorem 4.2 that for every fixed z ∈ A, the holo-
morphic function f(z, ·)|G is bounded and that for every η ∈ B,
w→η, w∈Ω∩Aα(η)
f(z, w) = f(z, η), 0 < α <
Consequently, Part 3) of Proposition 5.2, applied to f(z, ·)|Gk with k ∈ K, implies
that for every fixed z ∈ D, f(z,Φ−1k (·)) ∈ O(E,C) admits the angular limit f(z, η)
at Φk(η) for all η ∈ B ∩ End(Gk). By Part 1) of that proposition, we know that
B ∩ End(Gk)
> 0. This discussion and the hypothesis allow us to apply the
result of CASE 1 to the function gk : Wk −→ C defined by
(5.5) gk(z, w) :=
f(z,Φ−1
(w)), (z, w) ∈ D ×Gk,
f(z,Φ−1k (w)) (z, w) ∈ D × B
where in the second line we have used the definition of Φk|End(Gk) and its one-to-one
property proved by Part 2) of Proposition 5.2.
Consequently, we obtain an extension function ĝk ∈ O(Ŵk,C) such that
(5.6) ĝk(z, w) = gk(z, w), (z, w) ∈ A× E.
Ŵk :=
(z,Φ−1
(w)), (z, w) ∈ Ŵk
, k ∈ K.
Observe that the open sets (Ŵk)k∈K are pairwise disjoint. Moreover, by (5.4),
Ŵk = {(z, w) ∈ D ×E : w ∈ Gk and
ω(z, A,D) + ω
Φk(w),Φk(End(Gk)), E
< 1 for some k ∈ K
(z, w) ∈ D × E : w ∈ Gk and ω(z, A,D) +
ω(w,B,E)
< 1 for some k ∈ K
= Ŵ ,
28 VIÊT-ANH NGUYÊN
where the second equality follows from Part 4) of Proposition 5.2. Therefore, we
can define the desired extension function f̂ ∈ O(Ŵ ,C) by the formula
f̂(z, w) := ĝk(z,Φk(w)), (z, w) ∈ Ŵk, k ∈ K.
This, combined with (5.4)–(5.6), implies that f̂ = f on A×G. The uniqueness of f̂
follows from that of ĝk, k ∈ K. Hence, the proof of the theorem is complete. �
6. A local version of Theorem A
The main purpose of the section is to prove the following result.
Theorem 6.1. Let D ⊂ Cn, G ⊂ Cm be bounded open sets. D (resp. G) is equipped
with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aα(η)
η∈G, α∈Iη
). Let
A (resp. B) be a subset of D (resp. G) such that A and B are locally pluriregular.
W := X(A,B;D,G), W := X(A,B;D,G),
:= Xo(A,B;D,G), Ŵ := X̂(A,B;D,G).
Then, for every bounded function f : W −→ C such that f ∈ Cs(W,C)∩Os(W
and that f |A×B is continuous at all points of (A ∩ ∂D) × (B ∩ ∂G), there exists a
unique bounded function f̂ ∈ O(Ŵ ,C) which admits A-limit f(ζ, η) at all points
(ζ, η) ∈ W. Moreover,
(6.1) |f̂(z, w)| ≤ |f |
1−ω(z,w)
A×B |f |
ω(z,w)
W , (z, w) ∈ Ŵ .
The core of our unified approach will be presented in the proof below. Our idea is
to use Theorem 3.8 in order to reduce Theorem 6.1 to the case of bidisk, that is, the
case of Theorem 4.3. This reduction is based on Theorem 4.2 and on the technique
of level sets.
Proof. It is divided into four steps.
Step 1: Construction of the desired function f̂ ∈ O(Ŵ ,C) and proof of the estimate
|f̂ |cW ≤ |f |W .
Proof of Step 1. We define f̂ at an arbitrary point (z, w) ∈ Ŵ as follows: Let ǫ > 0
be such that
(6.2) ω(z, A,D) + ω(w,B,G) + 2ǫ < 1.
By Theorem 3.8 and Definition 3.9, there is an ǫ-candidate (φ,Γ) (resp. (ψ,∆))
for (z, A,D) (resp. (w,B,G)). Moreover, using the hypotheses, we see that the
function fφ,ψ, defined by
(6.3) fφ,ψ(t, τ) := f(φ(t), ψ(τ)), (t, τ) ∈ X (Γ,∆;E,E) ,
satisfies the hypotheses of Theorem 4.3. By this theorem, let f̂φ,ψ be the unique
function in X̂ (Γ,∆;E,E) such that
(6.4) (A− lim f̂φ,ψ)(t, τ) = fφ,ψ(t, τ), (t, τ) ∈ X
o (Γ,∆;E,E) .
A UNIFIED APPROACH 29
In virtue of (6.2) and Theorem 3.8 and Lemma 3.3, (0, 0) ∈ X̂ (Γ,∆;E,E) . Then
we can define the value of the desired extension function f̂ at (z, w) as follows
(6.5) f̂(z, w) := f̂φ,ψ(0, 0).
The remaining part of this step is devoted to showing that f̂ is well-defined and
holomorphic on Ŵ .
To this end we fix an arbitrary point w0 ∈ G, a number ǫ0 : 0 < ǫ0 < 1 −
ω(w0, B,G), and an arbitrary ǫ0-candidate (ψ0,∆0) for (w0, B,G).
(6.6) Ŵ0 := {(z, τ) ∈ D × E : ω(z, A,D) + ω(τ,∆0, E) < 1} .
Inspired by formula (6.5) we define a function f̂0 : Ŵ0 −→ C as follows
(6.7) f̂0(z, τ) := f̂φ,ψ0(0, τ).
Here we have used an ǫ-candidate (φ,Γ) for (z, A,D), where ǫ is arbitrarily chosen
so that 0 < ǫ < 1− ω(z, A,D)− ω(τ,∆0, E).
Using (6.3)–(6.4) and (6.7) and arguing as in Part 2) of Lemma 4.5, one can show
that f̂0 is well-defined on Ŵ0.
For all 0 < δ < 1 let
(6.8) Aδ := {z ∈ D : ω(z, A,D) < δ} and Eδ := {w ∈ E : ω(w,∆0, E) < 1− δ} .
Then by the construction in (6.7), we remark that f̂0(z, ·) is holomorphic on Eδ for
every fixed z ∈ Aδ. We are able to define a new function f̃δ on X (Aδ, B;D,Eδ) as
follows
(6.9) f̃δ(z, τ) :=
f̂0(z, τ) (z, τ) ∈ Aδ × Eδ,
f(z, ψ0(τ)) (z, τ) ∈ D ×∆0.
Using the hypotheses on f and the previous remark, we see that f̃δ ∈
o (Aδ, B;D,Eδ) ,C
Observe that Aδ is an open set in D. Consequently, f̃δ satisfies the hypothe-
ses of Theorem 4.2. Applying this theorem yields a unique function f̂δ ∈
X̂ (Aδ, B;D,Eδ) ,C
such that
f̂δ(z, w) = f̃δ(z, w), (z, w) ∈ Aδ × Eδ.
This, combined with (6.9), implies that f̂0 is holomorphic on Aδ ×Gδ. On the other
hand, it follows from (6.6) and (6.8) that
Ŵ0 = X̂ (A,∆0;D,E) =
0<δ<1
Aδ ×Gδ.
Hence, f̂0 ∈ O(Ŵ0,C).
In summary, we have shown that f̂0, given by (6.7), is well-defined and holomor-
phic on Ŵ0.
30 VIÊT-ANH NGUYÊN
Now we are able to prove that f̂ , given by (6.5), is well-defined. To this end we fix
an arbitrary point (z0, w0) ∈ Ŵ , an ǫ0 : 0 < ǫ0 < 1− ω(z0, D,G), and two arbitrary
ǫ0-candidates (ψ1,∆1) and (ψ2,∆2) for (w0, B,G). Let
Ŵj := {(z, τ) ∈ D ×E : ω(z, A,D) + ω(τ,∆j, E) < 1} , j ∈ {1, 2}.
Using formula (6.7) define, for j ∈ {1, 2}, a function f̂j : Ŵj −→ C as follows
(6.10) f̂j(z, τ) := f̂φ,ψj(0, τ).
Here we have used any ǫ-candidate (φ,Γ) for (z, A,D) with a suitable ǫ > 0. Let
τj ∈ E be such that ψj(τj) = w0, j ∈ {1, 2}. Then, in virtue of (6.5) and (6.10) and
the result of the previous paragraph on the well-definedness of f̂0, the well-defined
property of f̂ is reduced to showing that
(6.11) f̂1(φ(t), τ1) = f̂2(φ(t), τ2)
for all t ∈ E and all ǫ-candidates (φ,Γ) for (φ(t), A,D), such that
ω(t,Γ, A) < ǫ := 1− max
j∈{1,2}
{ω(τ1,∆1, E), ω(τ2,∆2, E)} .
Observe that (6.11) follows from an argument based on Part 2) of Lemma 4.5. Hence,
f̂ is well-defined on Ŵ .
As in (6.8), for all 0 < δ < 1 let
Aδ := {z ∈ D : ω(z, A,D) < δ} , Bδ := {w ∈ G : ω(w,B,G) < δ} ,
Dδ := {z ∈ D : ω(z, A,D) < 1− δ} , Gδ := {w ∈ G : ω(w,B,G) < 1− δ} .
(6.12)
Now we combine (6.8) and (6.12) and the result that f̂0, given by (6.7), is well-defined
and holomorphic on Ŵ0, and the result that f̂ is well-defined on Ŵ . Consequently,
we obtain that
f̂(·, w) ∈ O(Dδ,C), w ∈ Bδ, 0 < δ < 1.
Since the formula (6.5) for f̂ is symmetric in two variables (z, w), one also gets that
f̂(z, ·) ∈ O(Gδ,C), z ∈ Aδ, 0 < δ < 1.
Since by (6.12),
0<δ<1
Aδ ×Gδ =
0<δ<1
Dδ × Bδ,
it follows from the previous conclusions that, for all points (z, w) ∈ Ŵ , there is an
open neighborhood U of z (resp. V of w) such that f ∈ Os(X
o(U, V ;U, V ),C). By
the classical Hartogs extension theorem, f ∈ O(U × V,C). Hence, f̂ ∈ O(Ŵ ,C).
On the other hand, it follows from (6.5) and the estimate in Theorem 4.3 that
(6.13) |f̂ |cW ≤ |f |W .
This completes Step 1. �
Step 2: f |A×B ∈ C(A× B,C).
A UNIFIED APPROACH 31
Proof of Step 2. Using the hypotheses we only need to check the continuity of f |A×B
at every point (a0, w0) ∈ A × (G ∩ B) and at every point (z0, b0) ∈ (D ∩ A) × B.
We will verify the first assertion. To do this let (ak)
k=1 ⊂ A and (wk)
k=1 ⊂ (G∩B)
such that lim
ak = a0 and lim
wk = w0. We need to show that
(6.14) lim
f(ak, wk) = f(a0, w0).
Since f |W is locally bounded, we may choose an open connected neighborhood V
of w0 such that sup
|f(ak, ·)|V <∞. Consequently, by Montel’s Theorem, there is a
sequence (kp)
p=1 such that (f(akp, ·)) converges uniformly on compact subsets of V
to a function g ∈ O(V ). Equality (6.14) is reduced to showing that g = f(a0, ·) on
V. Since f ∈ Cs(W,C), we deduce that f(a0, ·) = g on B ∩ V. On the other hand,
B ∩V is non locally pluripolar because B is locally pluriregular and w0 ∈ B. Hence,
we conclude by the uniqueness principle that g = f(a0, ·) on V. �
Step 3: f̂ admits A-limit f(ζ, η) at all points (ζ, η) ∈ W.
Proof of Step 3. To this end we only need to prove that
(6.15)
A− lim sup |f̂ − f(ζ0, η0)|
(ζ0, η0) < ǫ0
for an arbitrary fixed point (ζ0, η0) ∈ W and an arbitrary fixed 0 < ǫ0 < 1. Suppose
without loss of generality that
(6.16) |f |W ≤
First consider (ζ0, η0) ∈ A × B. Since f ∈ C(A × B,C), one may find an open
neighborhood U of ζ0 in C
n (resp. V of η0 in C
m) so that
(6.17) |f − f(ζ0, η0)|(A∩U)×(B∩V ) <
Consider the open sets
(6.18)
z ∈ D : ω(z, A ∩ U,D) <
and G
w ∈ G : ω(w,B ∩ V,G) <
In virtue of (6.16)–(6.18), an application of Theorem 3.6 gives that
|f(ζ, w)− f(ζ, η0)| ≤ (
)1−ω(w,B∩V,G) ≤
, ζ ∈ A ∩ U, w ∈ G
Hence,
(6.19) |f − f(ζ0, η0)|X(A∩U,B∩V ;D′ ,G′ ) ≤
Consider the function g : X(A ∩ U,B ∩ V ;D
) −→ C, given by
(6.20) g(z, w) := f(z, w)− f(ζ0, η0).
32 VIÊT-ANH NGUYÊN
Applying the result of Step 1, we can construct a function ĝ ∈ O(X̂(A ∩ U,B ∩
),C) from g in exactly the same way as we obtain f̂ ∈ O(Ŵ ,C) from f.
Moreover, combining (6.5) and (6.20), we see that
(6.21) ĝ = f̂ − f(ζ0, η0) on X̂(A ∩ U,B ∩ V ;D
On the other hand, it follows from formula (6.20), estimate (6.19), and estimate
(6.13) that
|ĝ|bX(A∩U,B∩V ;D′ ,G′) ≤
This, combined with (6.21) and (6.18), implies that
A− lim sup |f̂(z, w)− f(ζ0, η0)|
(ζ0, η0) ≤
Hence, (6.15) follows. In summary, we have shown that A− lim f̂ = f on A× B.
Now it remains to consider (ζ0, η0) ∈ A ×G. Using the last limit and arguing as
in Step 2, one can show that A− lim f̂(ζ0, η0) = f(ζ0, η0). �
Step 4: Proof of the uniqueness of f̂ and (6.1).
Proof of Step 4. To prove the uniqueness of f̂ suppose that ĝ ∈ O(Ŵ ,C) is a
bounded function which admits A-limit f(ζ, η) at all points (ζ, η) ∈ W. Fix an
arbitrary point (z0, w0) ∈ Ŵ , it suffices to show that f̂(z0, w0) = ĝ(z0, w0). Observe
that both functions f̂(z0, ·) and ĝ(z0, ·) are bounded and holomorphic on the δ-level
set of G relative to B :
Gδ,B := {w ∈ G : ω(w,B,G) < 1− ω(z0, A,D)} ,
where δ := ω(z0, A,D). On the other hand, they admit A-limit f(z0, η) at all
points η ∈ B. Consequently, applying Proposition 3.5 and Theorem 3.7 yields that
f̂(z0, ·) = ĝ(z0, ·) on Gδ,B. Hence, f̂(z0, w0) = ĝ(z0, w0).
To prove (6.1) fix an arbitrary point (z0, w0) ∈ Ŵ . For every η ∈ B, applying
Theorem 3.6 to log |f(·, η)| defined on D, we obtain that
(6.22) |f(z0, η)| ≤ |f |
1−ω(z0,A,D)
A×B |f |
ω(z0,A,D)
Applying Theorem 3.6 again to log |f̂(z0, ·)| defined on Gδ,B of the preceeding para-
graph, one gets that
|f̂(z0, w0)| ≤ |f(z0, ·)|
1−ω(w0,B,G)
B |f̂ |
ω(w0,B,G)
Inserting (6.13) and (6.22) into the right hand side of the latter estimate, (6.1)
follows. Hence Step 4 is finished. �
This completes the proof. �
In the sequel we will need the following refined version of Theorem 6.1.
Theorem 6.2. Let D ⊂ Cn, G ⊂ Cm be bounded open sets. D (resp. G) is equipped
with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aα(η)
η∈G, α∈Iη
). Let
A UNIFIED APPROACH 33
A, A0 (resp. B, B0) be subsets of D (resp. G) such that A0 and B0 are locally
pluriregular and that A0 ⊂ A
∗ and B0 ⊂ B
∗. Put
W := X(A,B;D,G) and W0 := X(A0, B0;D,G).
Then, for every bounded function f : W −→ C which satisfies the following condi-
tions:
• f ∈ Cs(W,C) ∩ Os(W
o,C);
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique bounded function f̂ ∈ O(Ŵ0,C) which admits A-limit f(ζ, η)
at all points (ζ, η) ∈ W0. Moreover,
(6.23) |f̂(z, w)| ≤ |f |
1−ω(z,A0,D)−ω(w,B0,G)
A0×B0
ω(z,A0,D)+ω(w,B0,G)
W , (z, w) ∈ Ŵ0.
Proof. Using the hypotheses and applying Part 1) of Theorem 7.2 below we can ex-
tend f to a locally bounded function (still denoted by) f defined on X(A∗, B∗, D,G)
such that f ∈ Os
o(A∗, B∗, D,G),C
and that f |X(A∗∩D,B∗∩G;D,G) is continuous.
Therefore, the newly defined function f satisfies
(6.24) f(a, b) := lim
f(ak, b),
where (a, b) is an arbitrary point of A∗ × (G∪B∗) and (ak)
k=1 ⊂ A
∗ is an arbitrary
sequence with lim
ak = a. Since f |W is bounded, it follows that the newly defined
function f is also bounded. In virtue of the definition of A∗ and B∗ we have
(6.25) ∂D ∩ A = ∂D ∩A∗ and ∂G ∩ B = ∂G ∩ B∗.
Using the second • in the hypotheses and formula (6.24) we see that f |A∗×B∗ is
continuous at all points all (∂D ∩ A) × (∂G ∩ B). Consequently, arguing as in the
proof of Step 2 of Theorem 6.1 and using (6.25), we can show that f ∈ C
. In summary, the newly defined function f which is defined and bounded on
X(A∗, B∗, D,G) satisfies
(6.26) f ∈ Os
o(A∗, B∗, D,G),C
and f ∈ C
A∗ × B∗,C
Observe that f is only separately continuous on X(A,B;D,G), but it is not nec-
essarily so on the cross X
A∗, B∗, D,G
. However, we will show that one can adapt
the argument of Theorem 6.1 in order to prove Theorem 6.2.
We define f̂ at an arbitrary point (z0, w0) ∈ Ŵ0 as follows: Let ǫ > 0 be such that
ω(z0, A0, D) + ω(w0, B0, G) + 2ǫ < 1.
By Theorem 3.8 and Definition 3.9, there is an ǫ-candidate (φ,Γ) (resp. (ψ,∆)) for
(z0, A0, D) (resp. (w0, B,G)). To conclude the proof we only need to prove that the
function fφ,ψ, defined by
fφ,ψ(t, τ) := f(φ(t), ψ(τ)), (t, τ) ∈ X (Γ,∆;E,E) ,
satisfies the hypotheses of Theorem 4.3. Indeed, having proved this assertion, the
proof will follow along the same lines as those given in Theorem 6.1. This assertion
is again reduced to showing that for each fixed t ∈ Γ, the function fφ,ψ(t, ·) admits
the angular limit f(φ(t), ψ(τ)) for every point τ ∈ ∆. We will prove the last claim.
34 VIÊT-ANH NGUYÊN
Using the first • and Theorem 3.8, we see that for every a ∈ A, the function
f(a, ψ(·)) ∈ O(E,C) admits the angular limit f(a, ψ(τ)) for every point τ ∈ ∆.
Next, using the hypothesis A0 ⊂ A
∗ we may choose a sequence (ak)
k=1 ⊂ A ∩ A
such that lim
ak = φ(t) ∈ A0. Observe from (6.26) that for every k the uniformly
bounded function f(ak, ψ(·)) ∈ O(E,C) admits the angular limit f(ak, ψ(τ)) and
that lim
f(ak, ψ(τ)) = f(φ(t), ψ(τ)) for every point τ ∈ ∆. Consequently, by the
Khinchin–Ostrowski Theorem (see [11, Theorem 4, p. 397]), the above claim follows.
7. Preparatory results
The first result of this section shows that the two definitions of plurisubharmonic
measure ω̃(·, A,D), given respectively in Definition 2.3 and in Subsection 2.1 of [28],
coincide in the case when A ⊂ D.
Proposition 7.1. Let X be a complex manifold and D ⊂ X an open set. D is
equipped with the canonical system A of approach regions. Let A be a subset of D.
Then ω̃(z, A,D) = ω(z, A∗, D).
Proof. Let P ∈ E(A). Then by Definition 2.3, P ⊂ A∗ and P is locally pluriregular.
Hence, P ⊂ (A∗)∗ = A∗. Since P ∈ E(A) is arbitrary, it follows from Definition 2.3
that Ã is locally pluriregular and Ã ⊂ A∗. In particular, (Ã)∗ ⊂ A∗ and
(7.1) ω̃(z, A,D) = ω(z, Ã, D) ≥ ω(z, A∗, D).
In the sequel we will show that
(7.2) A∗ ⊂ (Ã)∗.
Taking (7.2) for granted, we have that A∗ = (Ã)∗. Consequently,
ω̃(z, A,D) = ω(z, Ã, D) ≤ ω(z, A∗, D).
This, coupled with (7.1), completes the proof.
To prove (7.2) fix an arbitrary point a ∈ A∗ and an arbitrary but sufficiently small
neighborhood U ⊂ X of a such that U is biholomorphic to a bounded open set in
n, where n is the dimension of X at a. Since A∗ is a Borel subset of D, Theorem
8.5 in [7] provides a subset P ⊂ A∗ ∩ U of type Fσ
10 such that
(7.3) ω(z, P, U) = ω(z, A∗ ∩ U, U), z ∈ U.
Write P =
Pn, where Pn is closed. Observe that Pn ∩ P
n is locally pluriregular,
Pn \ (Pn ∩ P
n) is locally pluripolar and Pn ∩ P
n ⊂ Pn ⊂ A
∗ ∩ P. Consequently,⋃
(Pn ∩ P
n) ⊂ Ã ∩ P and P \
(Pn ∩ P
n) is locally pluripolar. This implies that
ω(z, Ã ∩ U, U) ≤ ω
(Pn ∩ P
n), U
= ω(z, P, U),
10 This means that P is a countable (or finite) union of relatively closed subsets of U.
A UNIFIED APPROACH 35
where the equality holds by applying Lemma 3.5.3 in [18] and by using the fact that
U is biholomorphic to a bounded open set in Cn. This, combined with (7.3) and the
assumption a ∈ A∗, implies that ω(a, Ã ∩ U, U) = 0. Thus (7.2) follows. �
The main purpose of this and the next sections is to generalize Theorem 6.1 to the
case where the “target space” Z is an arbitrary complex analytic space possessing
the Hartogs extension property.
Theorem 7.2. Let D ⊂ Cn, G ⊂ Cm be two bounded open sets. D (resp. G) is
equipped with the canonical system of approach regions. Let Z be a complex analytic
space possessing the Hartogs extension property. Let A (resp. B) be a subset of D
(resp. G). Put W := X(A,B;D,G) and Ŵ := X̂(A,B;D,G). Let f ∈ Os(W
o, Z).
1) Then f extends to a mapping (still denoted by) f defined on Xo(A∪A∗, B ∪
B∗;D,G) such that f is separately holomorphic on Xo(A∪A∗, B∪B∗;D,G)
and that f |Xo(A∗,B∗;D,G) is continuous.
2) Suppose in addition that A and B are locally pluriregular. Then f extends
to a unique mapping f̂ ∈ O(Ŵ , Z) such that f̂ = f on W.
Proof. This result has already been proved in Théorème 2.2.4 in [5] starting from
Proposition 3.2.1 therein. In the latter proposition Alehyane and Zeriahi make use
of the method of doubly orthogonal bases of Bergman type. We can avoid this
method by simply replacing every application of this proposition by Theorem 6.1.
Keeping this change in mind and using Proposition 7.1, the remaining part of the
proof follows along the same lines as that of Théorème 2.2.4 in [5]. �
Theorem 7.3. Let D, G be complex manifolds, and let A ⊂ D, B ⊂ G be open
subsets. Let Z be a complex analytic space possessing the Hartogs extension property.
Put W := X(A,B;D,G) and Ŵ := X̂(A,B;D,G). Then for any mapping f ∈
Os(W,Z), there is a unique mapping f̂ ∈ O(Ŵ , Z) such that f̂ = f on W.
Proof. It has already been proved in Theorem 5.1 of [28]. The only places where the
method of doubly orthogonal bases of Bergman type is involved is the applications
of Théorème 2.2.4 in [5]. As we already pointed out in Theorem 7.2, one can avoid
this method by using Theorem 6.1 instead. �
We are ready to formulate a slight generalization of Theorems 6.2 and 7.2.
Theorem 7.4. Let D ⊂ Cn, G ⊂ Cm be bounded open sets. D (resp. G) is equipped
with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let A
and A0 (resp. B and B0) be two subsets of D (resp. G) such that A0 and B0 are
locally pluriregular and that A0 ⊂ A
∗ and B0 ⊂ B
∗. Let Z be a complex analytic
space possessing the Hartogs extension property. Put
W := X(A,B;D,G) and W0 := X(A0, B0;D,G).
Then, for every bounded mapping f : W −→ Z which satisfies the following condi-
tions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
36 VIÊT-ANH NGUYÊN
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique bounded mapping f̂ ∈ O(Ŵ0,C) which admits A-limit f(ζ, η)
at all points (ζ, η) ∈ W0.
Proof. Since f is bounded, one may find an open neighborhood U of f(W ) in Z and
a holomorphic embedding φ of U into the polydisc Ek of Ck such that φ(U) is an
analytic set in Ek. Now we are able to apply Theorem 6.2 to the mapping φ ◦ f :
W −→ Ck. Consequently, one obtains a unique bounded mapping F ∈ O(Ŵ ,Ck)
which admits A-limit (φ ◦ f)(ζ, η) at all points (ζ, η) ∈ W. Using estimate (6.23)
one can show that F ∈ O(Ŵ , Ek). Now using Theorem 3.7 it is not difficult to see
that F (Ŵ ) ⊂ φ(U). Consequently, one can define the desired extension mapping f̂
as follows:
f̂(z, w) := (φ−1 ◦ F )(z, w), (z, w) ∈ Ŵ .
The following Uniqueness Theorem for holomorphic mappings generalizes Theo-
rem 3.7.
Theorem 7.5. Let X be a complex manifold, D ⊂ X an open subset and Z a
complex analytic space. Suppose that D is equipped with a system of approach regions(
Aα(ζ)
ζ∈D, α∈Iζ
. Let A ⊂ D be a locally pluriregular set. Let f1, f2 : D∪A −→ Z
be locally bounded mappings such that f1|D, f2|D ∈ O(D,Z) and A− lim f1 = A −
lim f2 on A. Then f1(z) = f2(z) for all z ∈ D such that ω(z, A,D) 6= 1.
We leave the proof to the interested reader. Finally, we conclude this section with
the following Gluing Lemma.
Lemma 7.6. Let D and G be open subsets of some complex manifolds and Z a com-
plex analytic space. Suppose that D (resp. G) is equipped with a system of approach
regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let (Dk)
(resp. (Gk)
a family of open subsets of D (resp. G) equipped with the induced system of approach
regions. Let (Pk)
(resp. (Qk)
) be a family of locally pluriregular subsets of
D (resp. G). Suppose, in addition, that
(i) Pk ⊂ Pk0, Dk0 ⊂ Dk, and Pk is locally pluriregular relative to Dk0 . Similarly,
Qk ⊂ Qk0, Gk0 ⊂ Gk, and Qk is locally pluriregular relative to Gk0 .
(ii) There are a family of locally bounded mappings (fk)
such that fk :
o (Pk,Qk;Dk,Gk) −→ Z verifies fk = fk0 on X
o (Pk,Qk;Dk0,Gk0) ,
and a family of holomorphic mappings (f̂k)
such that f̂k ∈
X̂ (Pk,Qk;Dk,Gk) , Z
, and
(A− lim f̂k)(z, w) = fk(z, w), (z, w) ∈ X
o (Pk,Qk;Dk0,Gk0) .
(iii) There are open subsets U of D and V of G such that ω̃(z,Pk,Dk0) +
ω̃(w,Qk,Gk0) < 1 for all (z, w) ∈ U × V and k ≥ k0.
Then f̂k(z, w) = f̂k0(z, w) for all (z, w) ∈ U × V and k ≥ k0.
A UNIFIED APPROACH 37
Proof. By (iii), we have that
(7.4) U × V ⊂ H := X̂ (Pk,Qk;Dk0,Gk0) .
On the other hand, using (i) we see that
(7.5) H ⊂ X̂ (Pk,Qk;Dk,Gk) ∩ X̂ (Pk0 ,Qk0;Dk0,Gk0) .
Fix arbitrary (z0, w0) ∈ H and k ≥ k0. Observe that both mappings f̂k(·, w0) and
f̂k0(·, w0) are defined on {z ∈ Dk0 : ω(z,Pk,Dk0) < 1− ω(w0,Qk,Gk0)} . Using (ii)
and Proposition 3.5, we may apply Theorem 7.5 to these mappings and conclude
that f̂k(z0, w0) = f̂k0(z0, w0). �
8. Local and semi-local versions of Theorem A
The aim of this section is to generalize Theorem 6.2 to some cases where the
“target space” Z is a complex analytic space possessing the Hartogs extension prop-
erty. Our philosophy is the following: we first apply Theorem 6.2 locally in order to
obtain various local extension mappings, then we glue them together. The gluing
process needs the following
Definition 8.1. Let M be a complex manifold and Z a complex space. Let (Uj)j∈J
be a family of open subsets of M, and (fj)j∈J a family of mappings such that fj ∈
O(Uj , Z). We say that the family (fj)j∈J is collective if, for any j, k ∈ J, fj = fk
on Uj ∩ Uk. The unique holomorphic mapping f :
Uj −→ Z, defined by f := fj
on Uj, j ∈ J, is called the collected mapping of (fj)j∈J .
We arrive at the following local version of Theorem A.
Theorem 8.2. Let D ⊂ Cp, G ⊂ Cq be bounded open sets and Z a complex analytic
space possessing the Hartogs extension property. D (resp. G) is equipped with a
system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let A, A0
(resp. B, B0) be subsets of D (resp. G) such that A0 and B0 are locally pluriregular
and that A0 ⊂ A
∗ and B0 ⊂ B
∗. Put
W := X(A,B;D,G) and W0 := X(A0, B0;D,G).
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along X
A ∩ ∂D,B ∩ ∂G;D,G
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique mapping f̂ ∈ O(Ŵ0, Z) which admits A-limit f(ζ, η) at all
points (ζ, η) ∈ W0.
Theorem 8.2 generalizes Theorem 6.2 to the case where the “target space” Z is an
arbitrary complex analytic space possessing the Hartogs extension property. Since
the proof is somewhat technical, the reader may skip it at the first reading.
38 VIÊT-ANH NGUYÊN
Proof. Recall that for a ∈ Ck and r > 0, B(a, r) denotes the open ball centered at a
with radius r. For 0 < δ < 1 and 0 < r put
Da,δ,r := {z ∈ D ∩ B(a, r) : ω(A0 ∩ B(a, r), D ∩ B(a, r)) < δ} , a ∈ A0,
Gb,δ,r := {w ∈ G ∩ B(b, r) : ω(B0 ∩ B(b, r), G ∩ B(b, r)) < δ} , b ∈ B0.
(8.1)
Applying Part 1) of Theorem 7.2 and using the hypotheses on f, we see that f
extends to a mapping defined on X(A ∪A∗, B ∪B∗;D,G) such that f is separately
holomorphic on Xo(A∪A∗, B ∪B∗;D,G) and that f |X(A∗,B∗;D,G) is locally bounded.
Therefore, using the compactness of A0 and B0, one may find a real number r0 > 0
such that
(8.2) fa,b := f |X(A0∩B(a,r),B0∩B(b,r);D∩B(a,r),G∩B(b,r))
is bounded for all 0 < r ≤ r0 and a ∈ A0, b ∈ B0. Applying Theorem 7.4 to fa,b ,
one obtains a mapping
(8.3) f̂a,b ∈ O
A0 ∩ B(a, r), B0 ∩ B(b, r);D ∩ B(a, r), G ∩ B(b, r)
which admits A-limit f on X
A0 ∩ B(a, r), B0 ∩ B(b, r);D ∩ B(a, r), G ∩ B(b, r)
Fix 0 < δ0 <
. Then it follows from (8.1) that for 0 < r ≤ r0, a ∈ A0, b ∈ B0.
Da,δ0,r ×Gb,δ0,r ⊂ X̂
A0 ∩ B(a, r), B0 ∩ B(b, r);D ∩ B(a, r), G ∩ B(b, r)
This, combined with (8.3), implies that
(8.4) f̂a,b ∈ O (Da,δ0,r ×Gb,δ0,r, Z) , 0 < r ≤ r0, a ∈ A0, b ∈ B0.
Next we fix a finite covering (A0 ∩ B(am, r))
m=1 of A0 and (B0 ∩ B(bn, r))
n=1 of B0,
where (am)
m=1 ⊂ A0 and (bn)
n=1 ⊂ B0.
We divide the proof into two steps.
Step 1: Fix an open set G
⋐ G. Then there exists r1: 0 < r1 < r0 with the following
property: for every a ∈ A0 there exist an open subset Aa of D and a mapping
f̂ = f̂a ∈ O
Gbn,δ0,r0
such that
f̂(z, w) = f̂a,bn(z, w), (z, w) ∈ (Aa ∩Da,δ0,r0)×Gbn,δ0,r0, n = 1, . . . , N ;
and that Aa is of the form {z ∈ D∩B(a, r1) : ω(z, A0∩B(a, r1), D∩B(a, r1)) < δa}
for some 0 < δa < δ0.
Proof of Step 1. Fix an arbitrary point a0 ∈ A0. First we claim that there are a
sufficiently small number r1 : 0 < r1 < r0 and a finite number of open subsets
n=1 of G with the following properties:
(a) V1 = Gb1,δ0,r0 and (Gbn,δ0,r0)
⊂ (Vn)
n=1 (see the notation in (8.1));
(b) f |(A0∩B(a,r1))×Vn is bounded, n = 1, . . . , N0;
(c) G
A UNIFIED APPROACH 39
(d) Vn ∩ Vn+1 6= ∅, n = 1, . . . , N0 − 1.
Indeed, we first start with the test r1 := r0 and N0 := N and (Vn)
n=1 :=
(Gbn,δ0)
. In virtue of (8.2) we see that our choice satisfies (a)–(b). If (c)–(d)
are satisfied then we are done. Otherwise, we will make the following procedure.
Fix a point w0 ∈ G
. For n = 1, . . . , N, let γn : [0, 1] → G be a continuous
one-to-one map such that
γn(0) = w0 and γn(1) ∈ Gbn,δ0,r0.
Since f is locally bounded, there exist sufficiently small numbers r1, s : 0 < r1 ≤
r0 and 0 < s such that f |(A0∩B(a,r1))×B(w,s) is bounded for all a ∈ A0 and w ∈
γn([0, 1]). Therefore, we may add to the starting collection (Vn)
n=1 some balls
of the form B(w, s), where w ∈ G
γn([0, 1]), and the new collection (Vn)
still satisfies (a)–(b). Now it remains to show that by adding a finite number of
suitable balls B(w, s), (c)–(d) are also satisfied. But this assertion follows from an
almost obvious geometric argument. In fact, we may renumber the collection (Vn)
if necessary. Hence, the above claim has been shown.
Using (c)–(d) above we may fix open sets Un ⋐ Vn for n = 1, . . . , N0, such that
(8.5) G
Un and Un ∩ Un−1 6= ∅, 1 < n ≤ N0.
In what follows we will find the desired set Aa0 and the desired holomorphic mapping
f̂ after N0 steps. Namely, after the n-th step (1 ≤ n ≤ N0), we construct an
open subset An of D in the form Da0,δn,r1 for a suitable δn > 0, and a mapping
f̂n ∈ O
. Finally, we obtain Aa0 := AN0 and f̂ := f̂N0. Now we
carry out this construction.
In the first step, using (8.1), (8.3), (8.4) and (a), we define
δ1 := δ0, A1 := Da0,δ1,r1 and f̂1(z, w) := f̂a0,b1(z, w), (z, w) ∈ A1 × U1.
Suppose that we have constructed an open subset An−1 of D and a mapping f̂n−1 ∈
An−1 ×
( n−1⋃
for some n : 2 ≤ n ≤ N0. We wish to construct an open
subset An of D and a mapping f̂n ∈ O
. There are two cases to
consider.
Case Vn = Gbm,δ0 for some 1 ≤ m ≤ N.
In this case let δn := δn−1 and An := An−1 = Da0,δn−1,r1, and
f̂n :=
f̂n−1, on An ×
( n−1⋃
f̂a0,bm, on An × Un.
40 VIÊT-ANH NGUYÊN
Case Vn 6∈
Gbm,δ0
By (8.5) fix a nonempty open set K ⋐ Un ∩ Un−1. Then by the induction, f̂n−1 ∈
O (An−1 ×K,Z) . Recall from (b) that f : (A0 ∩ B(a0, r1))× Vn −→ Z is bounded.
Since f is locally bounded, by decreasing r1 > 0 (if necessary) we may assume that
g := f |
X(A0∩B(a0,r1),K;D∩B(a0,r1),Vn)
is bounded. Applying Theorem 7.4 to g, we obtain
ĝ ∈ O
X̂(A0 ∩ B(a0, r1), K;D ∩ B(a0, r1), Vn), Z
which extends g. Since Un ⋐ Vn, we may choose δn such that 0 < δn < 1 −
ω(w,K, Vn). Using this and (8.1), it follows that
Da0,δn,r1 × Un ⊂ X̂(A0 ∩ B(a0, r1), K;D ∩ B(a0, r1), Vn).
Therefore, let An := Da0,δn,r1 and define
f̂n :=
f̂n−1, on An ×
( n−1⋃
ĝ, on An × Un.
This completes our construction in the n-step. Finally, we put Aa0 := AN0 and
f̂a0 := f̂N0 . Using this and (8.3) and (8.5) and (a), the desired conclusion of Step 1
follows. �
Step 2: Completion of the proof.
Proof of Step 2. Fix a sequence of relatively compact open subsets (D
k=1 of D
(resp. (G
k=1 of G) such that D
k ր D and G
k ր G as k ր ∞. Put
(8.6) Dk := D
Dam,δ0,r0, Gk := G
Gbn,δ0,r0, k ≥ 1.
Using the result of Step 1, we may find, for every k, a number 0 < rk < r0 with the
following properties:
• for every a ∈ A0, there is 0 < δa,k < δ0 such that by considering the open set
Aa,k := {z ∈ D ∩ B(a, rk) : ω (z, A0 ∩ B(a, rk), D ∩ B(a, rk)) < δa,k}
one can find a mapping f̂a,k ∈ O (Aa,k ×Gk, Z) satisfying
(8.7) f̂a,k = f̂a,bn on (Aa,k ∩Da,δ0,rk)×Gbn,δ0,rk , n = 1, . . . , N ;
• for every b ∈ B, there is 0 < δb,k < δ0 such that by considering the open set
Bb,k := {w ∈ G ∩ B(b, rk) : ω (z, B0 ∩ B(b, rk), G ∩ B(b, rk)) < δb,k}
one can find a mapping f̂b,k ∈ O (Dk ×Bb,k, Z) satisfying
(8.8) f̂b,k = f̂am,b on Dam,δ0,rk × (Bb,k ∩Gb,δ0,rk), m = 1, . . . ,M.
A UNIFIED APPROACH 41
Next using the compactness of A0 and B0, one may find, for every k, two fi-
nite coverings (A0 ∩ B(a
m, rk))
of A0 and (B0 ∩ B(bn′ , rk))
of B0, where
(am′ )
⊂ A0 and (bn′ )
⊂ B0. Put
(8.9) Ak :=
′ ,k and Bk :=
′ ,k, k ≥ 1.
In virtue of (8.6)–(8.9) and (8.2)–(8.4), the family (f̂a
′ ,k)
is col-
lective for every k ≥ 1. Let
(8.10) f̂k ∈ O
X(Ak, Bk;Dk, Gk), Z
denote the collected mapping of this family.
Next, we show that
(8.11)
ω(z, A0, Dk) = ω(z, A0, D) and lim
ω(w,B0, Gk) = ω(z, B0, G), z ∈ D, w ∈ G.
It is sufficient to prove the first identity in (8.11) since the proof of the second one
is similar. Observe that there is u ∈ PSH(D) such that ω(·, A0, Dk) ց u as k ր ∞
and u ≥ ω(·, A0, D) on D. Therefore, the proof of (8.11) will be complete if one can
show that u ≤ ω(·, A0, D) on D.
To this end observe that for every a ∈ A0 there is 1 ≤ m ≤ M such that
a ∈ B(am, r0). Consequently, using (8.6),
(A− lim sup u)(a) ≤
A− lim supω(·, A0 ∩ B(am, r0), Dam,δ0,r0)
(a) = 0,
where the equality follows from an application of Proposition 3.5. This, combined
with the obvious inequality u ≤ 1, implies that u ≤ ω(·, A0, D). Hence, (8.11)
follows.
We are now in the position to define the desired extension mapping f̂ . Indeed,
one glues
given in (8.10) together to obtain f̂ in the following way
f̂ := lim
f̂k on Ŵ0.
One needs to check that the last limit exists and possesses all the required properties.
In virtue of (8.7)–(8.11), and the Gluing Lemma 7.6, the proof will be complete if
we can show the following
Claim. For every (z0, w0) ∈ Ŵ0, there are an open neighborhood U × V of (z0, w0)
and δ0 > 0 such that the hypotheses of Lemma 7.6 is fulfilled with
D := D, G := G, Pk := Ak, Qk := Bk, Dk := Dk, Gk := Gk, k ≥ 1.
To this end let
δ0 :=
1− ω(z0, A0, D)− ω(w0, B0, G)
and let U × V be an open neighborhood of (z0, w0) such that
ω(z, A0, D) + ω(w,B0, G) < ω(z0, A0, D) + ω(w0, B0, G) + δ0.
42 VIÊT-ANH NGUYÊN
Then using these inequalities and (8.11), we see that there is a sufficiently big q0 ∈ N
such that for q0 ≤ q ≤ p and (z, w) ∈ U × V,
ω(z, Ap, Dq) + ω(w,Bp, Dq) ≤ ω(z, A0, Dq) + ω(w,B0, Gq)
≤ ω(z, A0, D) + ω(w,B0, G) + δ0 < 1.
This proves the above claim. Hence, the proof of the theorem is finished. �
Now we are able to formulate the following semi-local result.
Theorem 8.3. Let D be an open subset of a complex manifold and G ⊂ Cm a
bounded open set and Z a complex analytic space possessing the Hartogs extension
property. D (resp. G) is equipped with the canonical system of approach regions
(resp. the system of approach regions
Aβ(η)
η∈G, α∈Iη
). Let A be an open subset of
D and let B, B0 be subsets of G such that B0 is locally pluriregular and B0 ⊂ B
W := X(A,B;D,G) and W0 := X(A,B0;D,G).
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along D × (B ∩ ∂G),
there exists a unique mapping f̂ ∈ O(Ŵ0, Z) which admits A-limit f(ζ, η) at all
points (ζ, η) ∈ W0.
Proof. First, applying Part 1) of Theorem 7.2 and using the hypotheses on f, we
see that f extends to a mapping (still denoted by) f defined on X(A,B ∪B∗;D,G)
such that f is separately holomorphic on Xo(A,B ∪B∗;D,G) and that f |X(A,B∗;D,G)
is locally bounded.
We define f̂ at a point (z0, w0) ∈ Ŵ0 as follows: Let ǫ > 0 be such that
(8.12) ω(z0, A,D) + ω(w0, B0, G) + ǫ < 1.
By Theorem 3.1 and Proposition 3.4, there is a holomorphic disc φ ∈ O(E,D) such
that φ(0) = z0 and
(8.13) 1−
·mes(φ−1(A) ∩ ∂E) < ω(z0, A,D) + ǫ.
Moreover, using the hypotheses, we see that the mapping fφ, defined by
(8.14) fφ(t, w) := f(φ(t), w), (t, w) ∈ X
φ−1(A) ∩ ∂E,B;E,G
satisfies the hypotheses of Theorem 8.2. By this theorem, let f̂φ be the unique
mapping in X̂ (φ−1(A) ∩ ∂E,B0;E,G) such that
(8.15) (A− lim f̂φ)(t, w) = fφ(t, w), (t, w) ∈ X
φ−1(A) ∩ ∂E,B0;E,G
In virtue of (8.12)–(8.13), (0, w0) ∈ X̂ (φ
−1(A) ∩ ∂E,B0;E,G) . Then the value at
(z0, w0) of the desired extension mapping f̂ is given by
f̂(z0, w0) := f̂φ(0, w0).
A UNIFIED APPROACH 43
Using this and (8.14)–(8.15), and arguing as in Part 2) of Lemma 4.5, one can show
that f̂ is well-defined on Ŵ0.
To show that f̂ is holomorphic, one argues as in Step 1 of the proof of Theorem
6.1. To show that f̂ admits A-limit f(ζ, η) at all points (ζ, η) ∈ W0 and that it is
uniquely defined, one proceeds as in Step 2–4 of the proof of Theorem 6.1 making
the obviously necessary changes and adaptations. Hence, the proof is finished. �
9. The proof of Theorem A
First we need a variant of Definition 2.3. For a set A ⊂ D, Let Ẽ(A) be the set of
all elements P ∈ E(A) with the property that there is an open neighborhood U ⊂ X
of P such that U is biholomorphic to a domain in some Cn. Then it can be checked
(9.1) Ã :=
P∈eE(A)
This identity will allow us to pass from “local informations” to “global extensions”.
For the proof we need to develop some preparatory results.
In virtue of (9.1), for any P ∈ Ẽ(A) (resp. Q ∈ Ẽ(B)) fix an open neighborhood
UP of P (resp. VQ of Q) such that UP (resp. VQ) is biholomorphic to a domain in
dP (resp. in CdQ), where dP (resp. dQ) is the dimension of D (resp. G) at points
of P (resp. Q). For any 0 < δ ≤ 1
define
UP,δ := {z ∈ UP : ω(z, P, UP ) < δ} , P ∈ Ẽ(A),
VQ,δ := {w ∈ VQ : ω(w,Q, VQ) < δ} , Q ∈ Ẽ(B),
Aδ :=
P∈eE(A)
UP,δ, Bδ :=
Q∈eE(B)
VQ,δ,
Dδ := {z ∈ D : ω̃(z, A,D) < 1− δ} , Gδ := {w ∈ G : ω̃(w,B,G) < 1− δ} .
(9.2)
Lemma 9.1. We keep the above notation. Then:
(1) For every ζ ∈ Ã and α ∈ Iζ, there is an open neighborhood U of ζ such that
U ∩ Aα(ζ) ⊂ Aδ.
(2) Aδ is an open subset of D and Aδ ⊂ D1−δ ⊂ Dδ.
(3) ω̃(z, A,D)− δ ≤ ω(z, Aδ, D) ≤ ω̃(z, A,D), z ∈ D.
Proof of Lemma 9.1. To prove Part (1) fix, in view of (9.1)–(9.2), P ∈ Ẽ(A),
ζ ∈ P and α ∈ Iζ . Using the definition of local pluriregularity, we see that
lim sup
z→ζ, z∈Aα(ζ)
ω(z, P, UP ) = 0. Hence, Part (1) follows.
The assertion that Aδ is open follows immediately from (9.2). Since 0 < δ ≤
the second inclusion in Part (2) is clear. To prove the first inclusion let z be an
arbitrary point of Aδ. Then there is P ∈ Ẽ(A) such that z ∈ UP,δ. Using (9.2) and
Definition 2.3 we obtain
(9.3) ω̃(z, A,D) = ω(z, Ã, D) ≤ ω(z, P, UP ) < δ.
44 VIÊT-ANH NGUYÊN
Hence, z ∈ D1−δ, which in turn implies that Aδ ⊂ D1−δ.
It follows from Part (1) that
ω(z, Aδ, D) ≤ ω(z, Ã, D) = ω̃(z, A,D), z ∈ D,
which proves the second estimate in Part (3). To complete the proof let P ∈ Ẽ(A)
and 0 < δ ≤ 1
. We deduce from (9.3) that ω̃(z, A,D) − δ ≤ 0 for z ∈ UP,δ. Hence,
by (9.2),
ω̃(z, A,D)− δ ≤ 0, z ∈ Aδ.
On the other hand, ω̃(z, A,D) − δ < 1, z ∈ D. Recall from Part (2) that Aδ is an
open subset of Dδ. Consequently, the first estimate of Part (3) follows. �
Now we are able to to prove Theorem A in the following special case.
Proposition 9.2. Let D be an open subset of a complex manifold and G a bounded
open subset of Cm and Z a complex analytic space possessing the Hartogs extension
property. D (resp. G) is equipped with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let A be a subset of D, let B, B0 be subsets of G such
that B0 is locally pluriregular and B0 ⊂ B
∗. Put
W := X(A,B;D,G), W0 := X(A,B0;D,G), W̃
(D ∪ Ã)× B0
Ã× (G ∪ B0)
Ŵ o := {(z, w) ∈ D ×G : ω̃(z, A,D) + ω(w,B0, G) < 1} .
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along X
A ∩ ∂D,B ∩ ∂G;D,G
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique mapping f̂ ∈ O(Ŵ o, Z) which admits A-limit f(ζ, η) at all
points (ζ, η) ∈ W̃ o.
Proof of Proposition 9.2. First, applying Part 1) of Theorem 7.2 and using the
hypotheses on f, we see that f extends to a mapping (still denoted by f) defined on
X(A ∪ A∗, B ∪ B∗;D,G) such that f is separately holomorphic on Xo(A ∪ A∗, B ∪
B∗;D,G) and that f |X(A∗,B∗;D,G) is locally bounded.
For each P ∈ Ẽ(A), UP (resp. G) is biholomorphic to an open set in C
dP (resp.
in Cm). Consequently, the mapping fP := f |X(P ,B;UP ,G) satisfies the hypotheses of
Theorem 8.2. Hence, we obtain a unique mapping f̂P ∈ O
X̂ (P,B0;UP , G) , Z
(9.4) (A− lim f̂P )(z, w) = fP (z, w) = f(z, w), (z, w) ∈ X (P,B0;UP , G) .
Let 0 < δ ≤ 1
and G
δ := {w ∈ G : ω(w,B0, G) < 1 − δ}. We will show that the
family
f̂P |UP,δ×G
P∈eE(A)
is collective in the sense of Definition 8.1, where UP,δ is
given in (9.2).
A UNIFIED APPROACH 45
To prove this assertion let P1, P2 be arbitrary elements of Ẽ(A). By (9.4), we have
(9.5)
(A− lim f̂P1)(z, w) = f(z, w) = (A− lim f̂P2)(z, w), (z, w) ∈ (UP1 ∩ UP2)× B0.
The assertion is reduced to showing that
(9.6) f̂P1(z, w) = f̂P2(z, w), (z, w) ∈ X̂ (P1, B0;UP1, G) ∩ X̂ (P2, B0;UP2, G) .
To this end fix (z0, w0) ∈ X̂ (P1, B0;UP1, G) ∩ X̂ (P2, B0;UP2 , G) . Observe that both
mappings w 7→ f̂P1(z0, w) and w 7→ f̂P2(z0, w) belong to O(G, Z), where G is the
connected component which contains w0 of the following open set{
w ∈ G : ω(w,B0, G) < 1− max
j∈{1,2}
ω(z0, Pj, Uj)
Applying Theorem 7.5 to these mappings using (9.5), Proposition 3.5 and (9.6), the
above assertion follows.
In virtue of (9.2) let
(9.7)
fδ ∈ O(Aδ ×G
δ, Z)
denote the collected mapping of the family
f̂P |UP,δ×G
P∈eE(A)
. In virtue of (9.4)
and (9.7), we are able to define a new mapping f̃δ on X
Aδ, B;D,G
as follows
f̃δ :=
fδ, on Aδ ×G
f, on D × B.
Using this and (9.4)–(9.7), we see that
(9.8) A− lim f̃δ = f on X(A ∩ Ã, B0;D,G
Since Aδ is an open subset of X and G
δ is a bounded open set in C
m, we are able to
apply Theorem 8.3 to f̃δ in order to obtain a mapping f̂δ ∈ O
Aδ, B0;D,G
such that
(9.9) A− lim f̂δ = f̃δ on X(Aδ, B0;D,G
We are now in a position to define the desired extension mapping f̂ . Indeed, one
glues
0<δ≤ 1
together to obtain f̂ in the following way
f̂ := lim
on Ŵ o.
One needs to check that the last limit exists and possesses all the required properties.
In virtue of (9.8)–(9.9) and Lemma 7.6, the proof will be complete if one can show
that for every (z0, w0) ∈ Ŵ
o, there are an open neighborhood U × V of (z0, w0) and
δ0 > 0 such that hypothesis (iii) of Lemma 7.6 is fulfilled with
D := D, G := G, Pk := A 1
, Qk := B0, Dk := D, Gk := G
, k > 2.
To this end let
δ0 :=
1− ω̃(z0, A,D)− ω(w0, B0, G)
46 VIÊT-ANH NGUYÊN
and let U × V be an open neighborhood of (z0, w0) such that
ω̃(z, A,D) + ω(w,B0, G) < ω̃(z0, A,D) + ω(w0, B0, G) + δ0.
Then for k > 1
and for (z, w) ∈ U ×V, using the last inequality, and applying Part
(3) of Lemma 9.1 and Proposition 3.5, we see that
ω̃(z, A 1
, D) + ω(w,B0, G
) ≤ ω̃(z, A,D) +
ω(w,B0, G)
1− δ0
ω̃(z, A,D) + ω(w,B0, G)
1− δ0
This proves the above assertion. Hence, the proof of the proposition is finished. �
We now arrive at
Proof of Theorem A. First, applying Part 1) of Theorem 7.2 and using the
hypotheses on f, we see that f extends to a mapping (still denoted by) f defined on
X(A ∪ A∗, B ∪ B∗;D,G) such that f is separately holomorphic on Xo(A ∪ A∗, B ∪
B∗;D,G) and that f |X(A∗,B∗;D,G) is locally bounded.
For each P ∈ Ẽ(A), UP is biholomorphic to an open set in C
dP . Consequently, the
mapping fP := f |X(P,B;UP ,G) satisfies the hypotheses of Proposition 9.2. Hence, we
obtain a unique mapping f̂P ∈ O
o (P,B;UP , G) , Z
11 such that
(9.10) (A− lim f̂P )(z, w) = f(z, w), (z, w) ∈ X
P, B̃ ∩B;UP , G
Let 0 < δ ≤ 1
. Using (9.10) and arguing as in the proof of Proposition 9.2, we
may collect the family
f̂P |UP,δ×Gδ
P∈eE(A)
in order to obtain the collected mapping
f̃Aδ ∈ O(Aδ ×Gδ, Z).
Similarly, for each Q ∈ Ẽ(B), one obtains a unique mapping f̂Q ∈
o (A,Q;D, VQ) , Z
12 such that
(9.11) (A− lim f̂Q)(z, w) = f(z, w), (z, w) ∈ X
A ∩ Ã, Q;D, VQ
Moreover, one can collect the family
f̂Q|Dδ×VQ,δ
Q∈eE(B)
in order to obtain the col-
lected mapping f̃Bδ ∈ O(Dδ × Bδ, Z).
Next, we prove that
(9.12) f̃Aδ = f̃
δ on Aδ × Bδ.
Indeed, in virtue of (9.10)–(9.11) it suffices to show that for any P ∈ Ẽ(A) and
Q ∈ Ẽ(B) and any 0 < δ ≤ 1
(9.13) f̂P (z, w) = f̂Q(z, w), (z, w) ∈ UP,δ × VQ,δ.
Observe that in virtue of (9.10)–(9.11) one has that
(A− lim f̂P )(z, w) = (A− lim f̂Q)(z, w) = f(z, w), (z, w) ∈ X (P,Q;UP , VQ) .
11 Here X̂o (P,B;UP , G) := {(z, w) ∈ UP ×G : ω(z, P, UP ) + ω̃(w,B,G) < 1} .
12 Here X̂o (A,Q;D,VQ) := {(z, w) ∈ D × VQ : ω̃(z, A,D) + ω(w,Q, VQ) < 1} .
A UNIFIED APPROACH 47
Recall that UP (resp. VQ) is biholomorphic to a domain in C
dP (resp. CdQ). Con-
sequently, applying the uniqueness of Theorem 8.2 yields that
f̂P (z, w) = f̂Q(z, w), (z, w) ∈ X̂ (P,Q;UP , VQ) .
Hence, the proof of (9.13) and then the proof of (9.12) are finished.
In virtue of (9.12), we are able to define a new mapping f̃δ :
o (Aδ, Bδ;Dδ, Gδ) −→ Z as follows
(9.14) f̃δ :=
f̃Aδ , on Aδ ×Gδ,
f̃Bδ , on Dδ ×Bδ.
Using formula (9.14) it can be readily checked that f̃δ ∈ Os
o (Aδ, Bδ;Dδ, Gδ) , Z
Since we know from Part (2) of Lemma 9.1 that Aδ (resp. Bδ) is an open subset
of Dδ (resp. Gδ), we are able to apply Theorem 7.3 to f̃δ for every 0 < δ ≤
Consequently, one obtains a unique mapping f̂δ ∈ O
X̂ (Aδ, Bδ;Dδ, Gδ) , Z
(9.15) f̂δ = f̃δ on X
o (Aδ, Bδ;Dδ, Gδ) .
It follows from (9.10)–(9.11) and (9.14)–(9.15) that
(9.16) A− lim f̂δ = f on X
A ∩ Ã, B ∩ B̃;Dδ, Gδ
In addition, for any 0 < δ ≤ δ0 ≤
, and any (z, w) ∈ Aδ × Bδ, there is P ∈ Ẽ(A)
such that z ∈ UP,δ0. Therefore, it follows from the construction of f̃
δ , (9.14) and
(9.15) that
f̂δ(z, w) = f̂P (z, w) = f̂δ0(z, w).
This proves that f̂δ = f̂δ0 on Aδ × Bδ for 0 < δ ≤ δ0 ≤
. Hence,
(9.17) f̂δ = f̂δ0 on X(Aδ, Bδ;Dδ0, Gδ0), 0 < δ ≤ δ0 ≤
We are now in a position to define the desired extension mapping f̂ .
f̂ := lim
To prove that f̂ satisfies the desired conclusion of the theorem one proceeds as in
the end of the proof of Proposition 9.2. In virtue of (9.16)–(9.17) and Lemma 7.6,
the proof will be complete if we can verify that for every (z0, w0) ∈ Ŵ , there are an
open neighborhood U×V of (z0, w0) and δ0 > 0 such that hypothesis (iii) of Lemma
7.6 is fulfilled with
D := D, G := G, Pk := A 1
, Qk := B 1
, Dk := D 1
, Gk := G 1
, k > 2.
Since the verification follows along almost the same lines as that of Proposition 9.2,
it is, therefore, left to the interested reader.
Hence, the proof of Theorem A is finished. �
48 VIÊT-ANH NGUYÊN
10. Applications
In this section we give various applications of Theorem A using different systems
of approach regions defined in Subsection 2.2.
10.1. Canonical system of approach regions. For every open subset U ⊂ R2n−1
and every continuous function h : U −→ R, the graph
z = (z
, zn) = (z
, xn + iyn) ∈ C
n : (z
, xn) ∈ U and yn = h(z
, xn)
is called a topological hypersurface in Cn.
Let X be a complex manifold of dimension n. A subset A ⊂ X is said to be a
topological hypersurface if, for every point a ∈ A, there is a local chart (U, φ : U →
n) around a such that φ(A ∩ U) is a topological hypersurface in Cn
Now let D ⊂ X be an open subset and let A ⊂ ∂D be an open subset (with
respect to the topology induced on ∂D). Suppose in addition that A is a topological
hypersurface. A point a ∈ A is said to be of type 1 (with respect to D) if, for every
neighborhood U of a there is an open neighborhood V of a such that V ⊂ U and
V ∩D is a domain. Otherwise, a is said to be of type 2. We see easily that if a is of
type 2, then for every neighborhood U of a, there are an open neighborhood V of a
and two domains V1, V2 such that V ⊂ U, V ∩D = V1 ∪ V2 and all points in A ∩ V
are of type 1 with respect to V1 and V2.
In virtue of Proposition 3.7 in [35] we have the following
Proposition 10.1. Let X be a complex manifold and D an open subset of X. D
is equipped with the canonical system of approach regions. Suppose that A ⊂ ∂D is
an open boundary subset which is also a topological hypersurface. Then A is locally
pluriregular and A ⊂ Ã.
This, combined with Theorem A, implies the following result.
Theorem 10.2. Let X, Y be two complex manifolds, and D ⊂ X, G ⊂ Y two
nonempty open sets. D (resp. G) is equipped with the canonical system of approach
regions. Let A (resp. B) be a nonempty open subset of ∂D (resp. ∂G) which is also
a topological hypersurface. Let Z be a complex analytic space possessing the Hartogs
extension property. Define
W := X(A,B;D,G),
Ŵ := {(z, w) ∈ D ×G : ω(z, A,D) + ω(w,B,G) < 1} .
Let f : W −→ Z be such that:
(i) f ∈ Cs(W,Z) ∩ Os(W
o, Z);
(ii) f is locally bounded on W ;
(iii) f |A×B is continuous.
Then there exists a unique mapping f̂ ∈ O(Ŵ ) such that
cW∋(z,w)→(ζ,η)
f̂(z, w) = f(ζ, η), (ζ, η) ∈ W.
A UNIFIED APPROACH 49
If, moreover, Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,w)
A×B |f |
ω(z,w)
W , (z, w) ∈ Ŵ .
The special case where Z = C has been proved in [35].
10.2. System of angular approach regions. We will use the terminology and
the notation in Paragraph 3 of Subsection 2.2. More precisely, if D is an open set
of a Riemann surface such that D is good on a nonempty part of ∂D, we equip D
with the system of angular approach regions supported on this part. Moreover, the
notions such as set of positive length, set of zero length, locally pluriregular point
which exist on ∂E can be transferred to ∂D using conformal mappings in a local
way (see [34] for more details).
Theorem 10.3. Let X, Y be Riemann surfaces and D ⊂ X, G ⊂ Y open subsets
and A (resp. B) a subset of ∂D (resp. ∂G) such that D (resp. G) is good on A
(resp. B) and that both A and B are of positive length. Let Z be a complex analytic
space possessing the Hartogs extension property. Define
W := X(A,B;D,G), W
:= X(A
;D,G),
Ŵ := {(z, w) ∈ D ×G : ω(z, A,D) + ω(w,B,G) < 1} ,
(z, w) ∈ D ×G : ω(z, A
, D) + ω(w,B
, G) < 1
where A
(resp. B
) is the set of points at which A (resp. B) is locally pluriregular
with respect to the system of angular approach regions supported on A (resp. B),
and ω(·, A,D), ω(·, A
, D) (resp. ω(·, B,G), ω(·, B
, G)) are calculated using the
canonical system of approach regions.
Then for every mapping f : W −→ Z which satisfies the following conditions:
(i) f ∈ Cs(W,Z) ∩ Os(W
o, Z);
(ii) f is locally bounded;
(iii) f |A×B is continuous,
there exists a unique mapping f̂ ∈ O(Ŵ
, Z) which admits the angular limit f at all
points of W ∩W
If A and B are Borel sets or if X = Y = C then Ŵ = Ŵ
If Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,A
,D)−ω(w,B
A×B |f |
ω(z,A
,D)+ω(w,B
W , (z, w) ∈ Ŵ
Theorem 10.3 generalizes, in some sense, the result of [34].
In the above theorem we have used the equality
W = Ŵ
when either A and B
are Borel sets or X = Y = C. This follows from the identity ω(·, A,D) = ω̃(·, A,D)
when either A is a Borel set or D ⊂ C (see Theorem 4.6 in [34]). On the other hand,
we can sharpen Theorem 10.3 further, namely, hypothesis (i) can be replaced by a
weaker hypothesis (i’) as follows:
50 VIÊT-ANH NGUYÊN
(i’) for any a ∈ A the mapping f(a, ·)|G is holomorphic and has angular limit
f(a, b) at all points b ∈ B, and for any b ∈ B the mapping f(·, b)|D is
holomorphic and has angular limit f(a, b) at all points a ∈ A.
To see this it suffices to observe that the hypotheses of Theorem 3.8 and Theorem
6.1 can be weakened considerably when the bounded open set D therein is just
one-dimensional.
10.3. System of conical approach regions. The remaining part of this section
is devoted to two important applications of Theorem A: a boundary cross theorem
and a mixed cross theorem. In order to formulate them, we need to introduce some
terminology and notation.
Let X be an arbitrary complex manifold and D ⊂ X an open subset. We say
that a set A ⊂ ∂D is locally contained in a generating manifold if there exist an
(at most countable) index set J 6= ∅, a family of open subsets (Uj)j∈J of X and
a family of generating manifolds13 (Mj)j∈J such that A ∩ Uj ⊂ Mj, j ∈ J, and
that A ⊂
j∈J Uj . The dimensions of Mj may vary according to j ∈ J. Given a
set A ⊂ ∂D which is locally contained in a generating manifold, we say that A is
of positive size if under the above notation
j∈J mesMj (A∩ Uj) > 0, where mesMj
denotes the Lebesgue measure on Mj. A point a ∈ A is said to be a density point
of A if it is a density point of A ∩ Uj on Mj for some j ∈ J. Denote by A
the set
of density points of A.
Suppose now that A ⊂ ∂D is of positive size. We equip D with the system
of conical approach regions supported on A. Using the work of B. Jöricke (see,
for example, Theorem 3, pages 44–45 in [15]), one can show that14 A is locally
pluriregular at all density points of A. Observe that mesMj
(A \ A
) ∩ Uj
= 0 for
j ∈ J. Therefore, it is not difficult to show that A
is locally pluriregular. Choose
an increasing sequence (An)
n=1 of subsets of A such that An ∩ Uj is closed and
mesMj
An) ∩ Uj
= 0 for j ∈ J. Observe that A
n is locally pluriregular,
n ∩ Uj ⊂ A for j ∈ J and that Â :=
n is locally pluriregular and that Â is
locally pluriregular at all points of A
. Consequently, it follows from Definition 2.3
ω̃(z, A,D) ≤ ω(z, A
, D), z ∈ D.
This estimate, combined with Theorem A, implies the following result which is a
generalization in higher dimensions of Theorem 10.3.
Theorem 10.4. Let X, Y be two complex manifolds, let D ⊂ X, G ⊂ Y be two
open sets, and let A (resp. B) be a subset of ∂D (resp. ∂G). D (resp. G) is equipped
with a system of conical approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
13 A differentiable submanifold M of a complex manifold X is said to be a generating manifold
if for all ζ ∈ M, every complex vector subspace of TζX containing TζM coincides with TζX.
14 A complete proof will be available in [29].
A UNIFIED APPROACH 51
supported on A (resp. on B). Suppose in addition that A and B are of positive size.
Let Z be a complex analytic space possessing the Hartogs extension property. Define
:= X(A
;D,G),
(z, w) ∈ D ×G : ω(z, A
, D) + ω(w,B
, G) < 1
where A
(resp. B
) is the set of density points of A (resp. B).
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded;
• f |A×Bis continuous,
there exists a unique mapping f̂ ∈ O(Ŵ
, Z) which admits A-limit f(ζ, η) at every
point (ζ, η) ∈ W ∩W
If, moreover, Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,A
,D)−ω(w,B
A×B |f |
ω(z,A
,D)+ω(w,B
W , (z, w) ∈ Ŵ
The second application is a very general mixed cross theorem.
Theorem 10.5. Let X, Y be two complex manifolds, let D ⊂ X, G ⊂ Y be open
sets, let A be a subset of ∂D, and let B be a subset of G. D is equipped with
the system of conical approach regions
Aα(ζ)
ζ∈D, α∈Iζ
supported on A and G is
equipped with the canonical system of approach regions
Aβ(η)
η∈G, β∈Iη
. Suppose
in addition that A is of positive size. Let Z be a complex analytic space possessing
the Hartogs extension property. Define
:= X(A
, B∗;D,G),
(z, w) ∈ D ×G : ω(z, A
, D) + ω(w,B∗, G) < 1
where A
is the set of density points of A and B∗ denotes, as usual (see Subsection
2.1 above), the set of points in B ∩G at which B is locally pluriregular.
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along A×G,
there exists a unique mapping f̂ ∈ O(Ŵ
, Z) which admits A-limit f(ζ, η) at every
point (ζ, η) ∈ W ∩W
If, moreover, Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,A
,D)−ω(w,B∗,G)
A×B |f |
ω(z,A
,D)+ω(w,B∗,G)
W , (z, w) ∈ Ŵ
Concluding remarks. In ongoing joint-works with Pflug [30, 31] we develop new
cross theorems with singularities. On the other hand, in [36] the problem of opti-
mality of the envelope of holomorphy
W in Theorem A has been investigated.
52 VIÊT-ANH NGUYÊN
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sional subsets of Cn, Ann. Polon. Math., 22, (1970), 145–171.
[45] V. P. Zahariuta, Separately analytic functions, generalizations of the Hartogs theorem and
envelopes of holomorphy, Math. USSR-Sb., 30, (1976), 51–67.
[46] M. Zerner, Quelques résultats sur le prolongement analytique des fonctions de variables com-
plexes, Séminaire de Physique Mathématique.
[47] A. Zeriahi, Comportement asymptotique des systèmes doublement orthogonaux de Bergman:
Une approche élémentaire, Vietnam J. Math., 30, No.2, (2002), 177–188.
[48] H. Wu, Normal families of holomorphic mappings, Acta Math., 119, (1967), 193–233.
http://arxiv.org/abs/0705.4649
54 VIÊT-ANH NGUYÊN
Viêt-Anh Nguyên, Mathematics Section, The Abdus Salam international centre
for theoretical physics, Strada costiera, 11, 34014 Trieste, Italy
E-mail address : vnguyen0@ictp.trieste.it
	1. Introduction
	2. Preliminaries and statement of the main result
	2.1. Approach regions, local pluripolarity and plurisubharmonic measure
	2.2. Examples of systems of approach regions
	2.3. Cross and separate holomorphicity and A-limit.
	2.4. Hartogs extension property.
	2.5. Statement of the main results
	3. Holomorphic discs and a Two-Constant Theorem
	3.1. Poletsky theory of discs and Rosay Theorem on holomorphic discs
	3.2. Level sets of the relative extremal functions and a Two-Constant Theorem
	3.3. Construction of discs
	4. A mixed cross theorem
	5. Completion of the proof of Theorem 4.2
	6. A local version of Theorem A
	7. Preparatory results
	8. Local and semi-local versions of Theorem A
	9. The proof of Theorem A
	10. Applications
	10.1. Canonical system of approach regions
	10.2. System of angular approach regions
	10.3. System of conical approach regions
	References

  We extend the theory of separately holomorphic mappings between complex
analytic spaces. Our method is based on Poletsky theory of discs, Rosay Theorem
on holomorphic discs and our recent joint-work with Pflug on cross theorems in
dimension 1. It also relies on our new technique of conformal mappings and a
generalization of Siciak's relative extremal function.
  Our approach illustrates the unified character: ``From local informations to
global extensions". Moreover, it avoids systematically the use of the classical
method of doubly orthogonal bases of Bergman type.

Introduction
In this article all complex manifolds are supposed to be of finite dimension and
countable at infinity, and all complex analytic spaces are supposed to be reduced,
irreducible, of finite dimension and countable at infinity. For a subset S of a topo-
logical space M, S denotes the closure of S in M, and the set ∂S := S ∩ M \ S
denotes, as usual, the boundary of S in M.
The main purpose of this work is to investigate the following
PROBLEM. Let X, Y be two complex manifolds, let D (resp. G) be an open subset
of X (resp. Y ), let A (resp. B) be a subset of D (resp. G) and let Z be a complex
analytic space. Define the cross
A× (G ∪B)
(D ∪ A)× B
We want to determine the “envelope of holomorphy” of the cross W, that is, an
“optimal” open subset of X×Y, denoted by
W, which is characterized by the following
properties:
Let f : W −→ Z be a mapping that satisfies, in essence, the following two
conditions:
• f(a, ·) is holomorphic on G for all a ∈ A, f(·, b) is holomorphic on D for all
b ∈ B;
• f(a, ·) is continuous on G ∪ B for all a ∈ A, f(·, b) is continuous on D ∪ A
for all b ∈ B.
2000 Mathematics Subject Classification. Primary 32D15, 32D10.
Key words and phrases. Hartogs’ theorem, holomorphic extension, Poletsky theory of discs,
Rosay Theorem on holomorphic discs.
http://arxiv.org/abs/0704.0897v2
2 VIÊT-ANH NGUYÊN
Then there is a holomorphic mapping f̂ defined on
W such that for every (ζ, η) ∈ W,
f̂(z, w) tends to f(ζ, η) as (z, w) ∈
W tends, in some sense, to (ζ, η).
Now we recall briefly the main developments around this problem. All the results
obtained so far may be divided into two directions. The first direction investigates
the results in the “interior” context: A ⊂ D and B ⊂ G, while the second one
explores the “boundary” context: A ⊂ ∂D and B ⊂ ∂G.
The first fundamental result in the field of separate holomorphy is the well-known
Hartogs extension theorem for separately holomorphic functions (see [14]). In the
language of the PROBLEM the following case X = Cn, Y = Cm, A = D, B =
G, Z = C has been solved and the result is
W = D×G. In particular, this theorem
may be considered as the first main result in the first direction. In his famous
article [8] Bernstein obtained some positive results for the PROBLEM in certain
cases where A ⊂ D, B ⊂ G, X = Y = C and Z = C.
More than 60 years later, a next important impetus was made by Siciak (see
[43, 44]) in 1969–1970, where he established some significant generalizations of the
Hartogs extension theorem. In fact, Siciak’s formulation of these generalizations
gives rise to the above PROBLEM: to determine the envelope of holomorphy for sep-
arately holomorphic functions defined on some cross sets W. The theorems obtained
under this formulation are often called cross theorems. Using the so-called relative
extremal function, Siciak completed the PROBLEM for the case where A ⊂ D,
B ⊂ G, X = Y = C and Z = C.
The next deep steps were initiated by Zahariuta in 1976 (see [45]) when he started
to use the method of common bases of Hilbert spaces. This original approach per-
mitted him to obtain new cross theorems for some cases where A ⊂ D, B ⊂ G and
D = X, G = Y are Stein manifolds. As a consequence, he was able to generalize
the result of Siciak in higher dimensions.
Later, Nguyên Thanh Vân and Zeriahi (see [25, 26, 27]) developed the method
of doubly orthogonal bases of Bergman type in order to generalize the result of Za-
hariuta. This is a significantly simpler and more constructive version of Zahariuta’s
original method. Nguyên Thanh Vân and Zeriahi have recently achieved an elegant
improvement of their method (see [24], [47]).
Using Siciak’s method, Shiffman (see [41]) was the first to generalize some Siciak’s
results to separately holomorphic mappings with values in a complex analytic space
Z. Shiffman’s result (see [42]) shows that the natural “target spaces” for obtaining
satisfactory generalizations of cross theorems are the ones which possess the Hartogs
extension property (see Subsection 2.4 below for more explanations).
In 2001 Alehyane and Zeriahi solved the PROBLEM for the case where A ⊂ D,
B ⊂ G and X, Y are Stein manifolds, and Z is a complex analytic space which
possesses the Hartogs extension property (see Theorem 2.2.4 in [5]).
In a recent work (see [28]) we complete, in some sense, the PROBLEM for the
case where A ⊂ D, B ⊂ G and X, Y are arbitrary complex manifolds. The main
ingredients in our approach are Poletsky theory of discs developed in [37, 38], Rosay’s
A UNIFIED APPROACH 3
Theorem on holomorphic discs (see [40]), the above mentioned result of Alehyane–
Zeriahi and the technique of level sets of the plurisubharmonic measure which was
previously introduced in our joint-work with Pflug (see [33]).
To conclude the first direction of research we mention the survey articles by
Nguyên Thanh Vân [23] and Peter Pflug [32] which give nice accounts on this sub-
ject.
The first result in the second direction (i.e. “boundary context”) was established
in the work of Malgrange–Zerner [46] in the 1960s. Further results in this direction
were obtained by Komatsu [21] and Drużkowski [9], but only for some special cases.
Recently, Gonchar [12, 13] has proved a more general result where the following
case has been solved: X = Y = C, D and G are Jordan domains, A (resp. B) is
an open boundary subset of ∂D (resp. ∂G), and Z = C. It should be noted that
Airapetyan and Henkin published a general version of the edge-of-the-wedge theorem
for CR manifolds (see [1] for a brief version and [2] for a complete proof). Gonchar’s
result could be deduced from the latter works. In our joint-articles with Pflug (see
[33, 34, 35]), Gonchar’s result has been generalized considerably. More precisely, the
work in [35] treats the case where the “source spaces” X, Y are arbitrary complex
manifolds, A (resp. B) is an open boundary subset of ∂D (resp. ∂G), and Z =
C. The work in [34] solves the case where the “source spaces” X, Y are Riemann
surfaces, A (resp. B) is a measurable (boundary) subset of ∂D (resp. ∂G), and
Z = C.
The main purpose of this article is to give a new version of the Hartogs extension
theorem which unifies all results up to now. Namely, we are able to give a reason-
able solution to the PROBLEM when the “target space” Z possesses the Hartogs
extension property. Our method is based on a systematic application of Poletsky
theory of discs, Rosay Theorem on holomorphic discs and our joint-work with Pflug
on boundary cross theorems in dimension 1 (see [34]). It also relies on our new
technique of conformal mappings and a generalization of Siciak’s relative extremal
function. The approach illustrates the unified character in the theory of extension
of holomorphic mappings:
One can deduce the global extension from local informations.
Moreover, the novelty of this new approach is that one does not use the classical
method of doubly orthogonal bases of Bergman type.
We close the introduction with a brief outline of the paper to follow.
In Section 2 we formulate the main results.
The tools which are needed for the proof of the main results are developed in
Section 3, 4, 5 and 7.
The proof of the main results is divided into three parts, which correspond to
Section 6, 8 and 9. Section 10 concludes the article with various applications of our
results.
Acknowledgment. The paper was written while the author was visiting the
Abdus Salam International Centre for Theoretical Physics in Trieste. He wishes to
express his gratitude to this organization.
4 VIÊT-ANH NGUYÊN
2. Preliminaries and statement of the main result
First we develop some new notions such as system of approach regions for an open
set in a complex manifold, and the corresponding plurisubharmonic measure. These
will provide the framework for an exact formulation of the PROBLEM and for our
solution.
2.1. Approach regions, local pluripolarity and plurisubharmonic measure.
Definition 2.1. Let X be a complex manifold and let D ⊂ X be an open subset.
A system of approach regions for D is a collection A =
Aα(ζ)
ζ∈D, α∈Iζ
of open
subsets of D with the following properties:
(i) For all ζ ∈ D, the system
Aα(ζ)
forms a basis of open neighborhoods
of ζ (i.e., for any open neighborhood U of a point ζ ∈ D, there is α ∈ Iζ
such that ζ ∈ Aα(ζ) ⊂ U).
(ii) For all ζ ∈ ∂D and α ∈ Iζ , ζ ∈ Aα(ζ).
Aα(ζ) is often called an approach region at ζ.
A is said to be canonical if it satisfies (i) and the following property (which is
stronger than (ii)):
(ii’) For every point ζ ∈ ∂D, there is a basis of open neighborhoods (Uα)α∈Iζ of ζ
in X such that Aα(ζ) = Uα ∩D, α ∈ Iζ.
It is possible that Iζ = ∅ for some ζ ∈ ∂D.
Various systems of approach regions which one often encounters in Complex Anal-
ysis will be described in the next subsection. Systems of approach regions for
D are used to deal with the limit at points in D of mappings defined on some
open subsets of D. Consequently, we deduce from Definition 2.1 that the subfamily(
Aα(ζ)
ζ∈D, α∈Iζ
is, in a certain sense, independent of the choice of a system of ap-
proach regions A. In addition, any two canonical systems of approach regions are,
in some sense, equivalent. These observations lead us to use, throughout the paper,
the following convention:
We fix, for every open set D ⊂ X, a canonical system of approach regions.
When we want to define a system of approach regions A for an open set D ⊂ X, we
only need to specify the subfamily
Aα(ζ)
ζ∈∂D, α∈Iζ
In what follows we fix an open subset D ⊂ X and a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
for D.
For every function u : D −→ [−∞,∞), let
(A− lim sup u)(z) :=
lim sup
w∈Aα(z), w→z
u(w), z ∈ D, Iz 6= ∅,
lim sup
w∈D, w→z
u(w), z ∈ ∂D, Iz = ∅.
By Definition 2.1 (i), (A−lim sup u)|D coincides with the usual upper semicontinuous
regularization of u.
For a set A ⊂ D put
hA,D := sup {u : u ∈ PSH(D), u ≤ 1 on D, A− lim sup u ≤ 0 on A} ,
A UNIFIED APPROACH 5
where PSH(D) denotes the cone of all functions plurisubharmonic on D.
A is said to be pluripolar inD if there is u ∈ PSH(D) such that u is not identically
−∞ on every connected component of D and A ⊂ {z ∈ D : u(z) = −∞} . A is said
to be locally pluripolar in D if for any z ∈ A, there is an open neighborhood V ⊂ D
of z such that A ∩ V is pluripolar in V. A is said to be nonpluripolar (resp. non
locally pluripolar) if it is not pluripolar (resp. not locally pluripolar). According to
a classical result of Josefson and Bedford (see [16], [6]), if D is a Riemann domain
over a Stein manifold, then A ⊂ D is locally pluripolar if and only if it is pluripolar.
Definition 2.2. The relative extremal function of A relative to D is the function
ω(·, A,D) defined by
ω(z, A,D) = ωA(z, A,D) := (A− lim sup hA,D)(z), z ∈ D.
Note that when A ⊂ D, Definition 2.2 coincides with the classical definition of
Siciak’s relative extremal function.
Next, we say that a set A ⊂ D is locally pluriregular at a point a ∈ A if ω(a, A ∩
U,D ∩ U) = 0 for all open neighborhoods U of a. Moreover, A is said to be locally
pluriregular if it is locally pluriregular at all points a ∈ A. It should be noted from
Definition 2.1 that if a ∈ A ∩D then the property of local pluriregularity of A at a
does not depend on any particular choices of a system of approach regions A, while
the situation is different when a ∈ A ∩ ∂D : the property does depend on A.
We denote by A∗ the following set
(A ∩ ∂D)
a ∈ A ∩D : A is locally pluriregular at a
If A ⊂ D is non locally pluripolar, then a classical result of Bedford and Taylor (see
[6, 7]) says that A∗ is locally pluriregular and A\A∗ is locally pluripolar. Moreover,
A∗ is locally of type Gδ, that is, for every a ∈ A
∗ there is an open neighborhood
U ⊂ D of a such that A∗ ∩ U is a countable intersection of open sets.
Now we are in the position to formulate the following version of the plurisubhar-
monic measure.
Definition 2.3. For a set A ⊂ D, let Ã = Ã(A) :=
P∈E(A)
P, where
E(A) = E(A,A) :=
P ⊂ D : P is locally pluriregular, P ⊂ A∗
The plurisubharmonic measure of A relative to D is the function ω̃(·, A,D) defined
ω̃(z, A,D) := ω(z, Ã, D), z ∈ D.
It is worthy to remark that ω̃(·, A,D) ∈ PSH(D) and 0 ≤ ω̃(z, A,D) ≤ 1, z ∈ D.
Moreover,
(2.1)
A− lim sup ω̃(·, A,D)
(z) = 0, z ∈ Ã.
An example in [3] shows that in general, ω(·, A,D) 6= ω̃(·, A,D) on D. Section 10
below is devoted to the study of ω̃(·, A,D) in some important cases.
1Observe that this function depends on the system of approach regions.
6 VIÊT-ANH NGUYÊN
Now we compare the plurisubharmonic measure ω̃(·, A,D) with Siciak’s relative
extremal function ω(·, A,D). We only consider two important special cases: A ⊂ D
and A ⊂ ∂D. For the moment, we only focus on the case where A ⊂ D. The latter
one will be discussed in Section 10 below.
If A is an open subset of an arbitrary complex manifold D, then it is easy to see
ω̃(z, A,D) = ω(z, A,D), z ∈ D.
If A is a (not necessarily open) subset of an arbitrary complex manifold D, then we
will prove in Proposition 7.1 below that
ω̃(z, A,D) = ω(z, A∗, D), z ∈ D.
On the other hand, if, morever, D is a bounded open subset of Cn then we have (see,
for example, Lemma 3.5.3 in [18]) ω(z, A,D) = ω(z, A∗, D), z ∈ D. Consequently,
under the last assumption,
ω̃(z, A,D) = ω(z, A,D), z ∈ D.
Our discussion shows that at least in the case where A ⊂ D, the notion of the
plurisubharmonic measure is a good candidate for generalizing Siciak’s relative ex-
tremal function to the manifold context in the theory of separate holomorphy.
For a good background of the pluripotential theory, see the books [18] or [20].
2.2. Examples of systems of approach regions. There are many systems of
approach regions which are very useful in Complex Analysis. In this subsection we
present some of them.
1. Canonical system of approach regions. It has been given by Definition 2.1
(i)–(ii’).
2. System of angular (or Stolz) approach regions for the open unit disc.
Let E be the open unit disc of C. Put
Aα(ζ) :=
t ∈ E :
∣∣∣∣arg
ζ − t
)∣∣∣∣ < α
, ζ ∈ ∂E, 0 < α <
where arg : C −→ (−π, π] is as usual the argument function. A =
(Aα(ζ))ζ∈∂E, 0<α<π
is referred to as the system of angular (or Stolz) approach regions
for E. In this context A− lim is also called angular limit.
3. System of angular approach regions for certain “good” open subsets
of Riemann surfaces. Now we generalize the previous construction (for the open
unit disc) to a global situation. More precisely, we will use as the local model the
system of angular approach regions for E. Let X be a complex manifold of dimension
1, in other words, X is a Riemann surface, and D ⊂ X an open set. Then D is said
to be good at a point ζ ∈ ∂D2 if there is a Jordan domain U ⊂ X such that ζ ∈ U
and U ∩ ∂D is the interior of a Jordan curve.
Suppose that D is good at ζ. This point is said to be of type 1 if there is a
neighborhood V of ζ such that V0 = V ∩D is a Jordan domain. Otherwise, ζ is said to
be of type 2. We see easily that if ζ is of type 2, then there are an open neighborhood
2 In the work [34] we use the more appealing word Jordan-curve-like for this notion.
A UNIFIED APPROACH 7
V of ζ and two disjoint Jordan domains V1, V2 such that V ∩D = V1∪V2. Moreover,
D is said to be good on a subset A of ∂D if D is good at all points of A.
Here is a simple example which may clarify the above definitions. Let G be the
open square in C with vertices 1 + i, −1 + i, −1− i, and 1− i. Define the domain
D := G \
Then D is good on ∂G ∪
. All points of ∂G are of type 1 and all points of(
are of type 2.
Suppose now that D is good on a nonempty subset A of ∂D.We define the system
of angular approach regions supported on A: A =
Aα(ζ)
ζ∈D, α∈Iζ
as follows:
• If ζ ∈ D \A, then
Aα(ζ)
coincide with the canonical approach regions.
• If ζ ∈ A, then by using a conformal mapping Φ from V0 (resp. V1 and V2)
onto E when ζ is of type 1 (resp. 2), we can “transfer” the angular approach
regions at the point Φ(ζ) ∈ ∂E : (Aα(Φ(ζ)))0<α<π
to those at the point
ζ ∈ ∂D (see [34] for more detailed explanations).
Making use of conformal mappings in a local way, we can transfer, in the same way,
many notions which exist on E (resp. ∂E) to those on D (resp. ∂D).
4. System of conical approach regions.
Let D ⊂ Cn be a domain and A ⊂ ∂D. Suppose in addition that for every point
ζ ∈ A there exists the (real) tangent space Tζ to ∂D at ζ. We define the system of
conical approach regions supported on A: A =
Aα(ζ)
ζ∈D, α∈Iζ
as follows:
• If ζ ∈ D \A, then
Aα(ζ)
coincide with the canonical approach regions.
• If ζ ∈ A, then
Aα(ζ) := {z ∈ D : |z − ζ | < α · dist(z, Tζ)} ,
where Iζ := (1,∞) and dist(z, Tζ) denotes the Euclidean distance from the
point z to Tζ .
We can also generalize the previous construction to a global situation:
X is an arbitrary complex manifold, D ⊂ X is an open set and A ⊂ ∂D is a
subset with the property that at every point ζ ∈ A there exists the (real) tangent
space Tζ to ∂D.
We can also formulate the notion of points of type 1 or 2 in this general context
in the same way as we have already done in Paragraph 3 above.
2.3. Cross and separate holomorphicity and A-limit. Let X, Y be two com-
plex manifolds, let D ⊂ X, G ⊂ Y be two nonempty open sets, let A ⊂ D and
B ⊂ G. Moreover, D (resp. G) is equipped with a system of approach regions
A(D) =
Aα(ζ)
ζ∈D, α∈Iζ
(resp. A(G) =
Aα(η)
η∈G, α∈Iη
). We define a 2-fold
cross W, its interior W o and its regular part W̃ (with respect to A(D) and A(G))
8 VIÊT-ANH NGUYÊN
W = X(A,B;D,G) :=
(D ∪A)×B
A× (B ∪G)
W o = Xo(A,B;D,G) := (A×G) ∪ (D × B),
W̃ = X̃(A,B;D,G) :=
(D ∪ Ã)× B̃
Ã× (G ∪ B̃)
Moreover, put
ω(z, w) := ω(z, A,D) + ω(w,B,G), (z, w) ∈ D ×G,
ω̃(z, w) := ω̃(z, A,D) + ω̃(w,B,G), (z, w) ∈ D ×G.
For a 2-fold cross W := X(A,B;D,G) let
Ŵ := X̂(A,B;D,G) = {(z, w) ∈ D ×G : ω(z, w) < 1} ,
W := X̂(Ã, B̃;D,G) = {(z, w) ∈ D ×G : ω̃(z, w) < 1} .
Let Z be a complex analytic space. We say that a mapping f : W o −→ Z is
separately holomorphic and write f ∈ Os(W
o, Z), if, for any a ∈ A (resp. b ∈ B)
the restricted mapping f(a, ·) (resp. f(·, b)) is holomorphic on G (resp. on D).
We say that a mapping f : W −→ Z is separately continuous and write f ∈
if, for any a ∈ A (resp. b ∈ B) the restricted mapping f(a, ·) (resp.
f(·, b)) is continuous on G ∪ B (resp. on D ∪A).
In virtue of (2.1), for every (ζ, η) ∈ W̃ and every α ∈ Iζ , β ∈ Iη, there are open
neighborhoods U of ζ and V of η such that
U ∩ Aα(ζ)
V ∩Aβ(η)
Then a mapping f :
W −→ Z is said to admit A-limit λ at (ζ, η) ∈ W̃ , and one
writes
(A− lim f)(ζ, η) = λ, 3
if, for all α ∈ Iζ, β ∈ Iη,
cfW∋(z,w)→(ζ,η), z∈Aα(ζ), w∈Aβ(η)
f(z, w) = λ.
Throughout the paper, for a topological space M, C(M, Z) denotes the set of all
continuous mappings f : M −→ Z. If, moreover, Z = C, then C(M,C) is equipped
with the “sup-norm” |f |M := supM |f | ∈ [0,∞]. A mapping f : M −→ Z is said to
be bounded if there exist an open neighborhood U of f(M) in Z and a holomorphic
embedding φ of U into a polydisc of Ck such that φ(U) is an analytic set in this
polydisc. f is said to be locally bounded along N ⊂ M if for every point z ∈ N ,
there is an open neighborhood U of z (in M) such that f |U : U −→ Z is bounded.
f is said to be locally bounded if it is so for N = M. It is clear that if Z = C then
the above notions of boundedness coincide with the usual ones.
3Note that here A = A(D)×A(G).
A UNIFIED APPROACH 9
2.4. Hartogs extension property. The following example (see Shiffman [42])
shows that an additional hypothesis on the “target space” Z is necessary in or-
der that the PROBLEM makes sense. Consider the mapping f : C2 −→ P1 given
f(z, w) :=
[(z + w)2 : (z − w)2], (z, w) 6= (0, 0),
[1 : 1], (z, w) = (0, 0).
Then f ∈ Os
o(C,C;C,C),P1
, but f is not continuous at (0, 0).
We recall here the following notion (see, for example, Shiffman [41]). Let p ≥ 2
be an integer. For 0 < r < 1, the Hartogs figure in dimension p, denoted by Hp(r),
is given by
Hp(r) :=
, zp) ∈ E
p : ‖z
‖ < r or |zp| > 1− r
where E is the open unit disc of C and z
= (z1, . . . , zp−1), ‖z
‖ := max
1≤j≤p−1
|zj|.
Definition 2.4. A complex analytic space Z is said to possess the Hartogs extension
property in dimension p if every mapping f ∈ O(Hp(r), Z) extends to a mapping
f̂ ∈ O(Ep, Z). Moreover, Z is said to possess the Hartogs extension property if it
does in any dimension p ≥ 2.
It is a classical result of Ivashkovich (see [17]) that if Z possesses the Hartogs
extension property in dimension 2, then it does in all dimensions p ≥ 2. Some typical
examples of complex analytic spaces possessing the Hartogs extension property are
the complex Lie groups (see [4]), the taut spaces (see [48]), the Hermitian manifold
with negative holomorphic sectional curvature (see [41]), the holomorphically convex
Kähler manifold without rational curves (see [17]).
Here we mention an important characterization due to Shiffman (see [41]).
Theorem 2.5. A complex analytic space Z possesses the Hartogs extension property
if and only if for every domain D of any Stein manifold M, every mapping f ∈
O(D,Z) extends to a mapping f̂ ∈ O(D̂, Z), where D̂ is the envelope of holomorphy
of D.
In the light of Definition 2.4 and Shiffman’s Theorem, the natural “target spaces”
Z for obtaining satisfactory answers to the PROBLEM are the complex analytic
spaces which possess the Hartogs extension property.
2.5. Statement of the main results. We are now ready to state the main results.
Theorem A. Let X, Y be two complex manifolds, let D ⊂ X, G ⊂ Y be two open
sets, let A (resp. B) be a subset of D (resp. G). D (resp. G) is equipped with a
system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let Z be a
complex analytic space possessing the Hartogs extension property. Then, for every
mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
10 VIÊT-ANH NGUYÊN
• f is locally bounded along X
A ∩ ∂D,B ∩ ∂G;D,G
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique mapping f̂ ∈ O(
W,Z) which admits A-limit f(ζ, η) at every
point (ζ, η) ∈ W ∩ W̃ .
If, moreover, Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−eω(z,w)
A×B |f |
eω(z,w)
W , (z, w) ∈
Theorem A has an important corollary. Before stating this, we need to introduce
a terminology. A complex manifoldM is said to be a Liouville manifold if PSH(M)
does not contain any non-constant bounded above functions. We see clearly that
the class of Liouville manifolds contains the class of connected compact manifolds.
Corollary B. We keep the hypothesis and the notation in Theorem A. Suppose in
addition that G is a Liouville manifold and that Ã, B̃ 6= ∅. Then, for every mapping
f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along X
A ∩ ∂D,B ∩ ∂G;D,G
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there is a unique mapping f̂ ∈ O(D × G,Z) which admits A-limit f(ζ, η) at every
point (ζ, η) ∈ W ∩ W̃ .
Corollary B follows immediately from Theorem A since ω̃(·, B,G) ≡ 0.
We will see in Section 10 below that Theorem A and Corollary B generalizes all
the results discussed in Section 1 above. Moreover, they also give many new results.
Although our main results have been stated only for the case of a 2-fold cross, they
can be formulated for the general case of an N -fold cross with N ≥ 2 (see also
[28, 33]).
3. Holomorphic discs and a Two-Constant Theorem
We recall here some elements of Poletsky theory of discs, some background of the
pluripotential theory and auxiliary results needed for the proof of Theorem A.
3.1. Poletsky theory of discs and Rosay Theorem on holomorphic discs.
Let E denote as usual the open unit disc in C. For a complex manifold M, let
O(E,M) denote the set of all holomorphic mappings φ : E −→ M which extend
holomorphically to a neighborhood of E. Such a mapping φ is called a holomorphic
disc on M. Moreover, for a subset A of M, let
1A,M(z) :=
1, z ∈ A,
0, z ∈ M \ A.
4 It follows from Subsection 2.3 that
A ∩ ∂D,B ∩ ∂G;D,G
(A ∩ ∂D)× (G ∪B)
(D ∪ A)× (B ∩ ∂G)
A UNIFIED APPROACH 11
In the work [40] Rosay proved the following remarkable result.
Theorem 3.1. Let u be an upper semicontinuous function on a complex manifold
M. Then the Poisson functional of u defined by
P[u](z) := inf
u(φ(eiθ))dθ : φ ∈ O(E,M), φ(0) = z
is plurisubharmonic on M.
Rosay Theorem may be viewed as an important development in Poletsky theory
of discs. Observe that special cases of Theorem 3.1 have been considered by Poletsky
(see [37, 38]), Lárusson–Sigurdsson (see [22]) and Edigarian (see [10]).
The following Rosay type result gives the connections between the Poisson func-
tional of the characteristic function 1M\A,M and holomorphic discs.
Lemma 3.2. Let M be a complex manifold and let A be a nonempty open subset of
M. Then for any ǫ > 0 and any z0 ∈ M, there are an open neighborhood U of z0,
an open subset T of C, and a family of holomorphic discs (φz)z∈U ⊂ O(E,M) with
the following properties:
(i) Φ ∈ O(U ×E,M), where Φ(z, t) := φz(t), (z, t) ∈ U × E;
(ii) φz(0) = z, z ∈ U ;
(iii) φz(t) ∈ A, t ∈ T ∩ E, z ∈ U ;
(iv) 1
1∂E\T,∂E(e
iθ)dθ < P[1M\A,M](z0) + ǫ.
Proof. See Lemma 3.2 in [28]. �
The next result describes the situation in dimension 1. It will be very useful later
Lemma 3.3. Let T be an open subset of E. Then
ω(0, T ∩ E,E) ≤
1∂E\T,T (e
iθ)dθ.
Proof. See, for example, Lemma 3.3 in [28]. �
The last result, which is an important consequence of Rosay’s Theorem, gives the
connection between the Poisson functional and the plurisubharmonic measure.
Proposition 3.4. Let M be a complex manifold and A a nonempty open subset of
M. Then ω(z, A,M) = P[1M\A,M](z), z ∈ M.
Proof. See, for example, the proof of Proposition 3.4 in [28]. �
12 VIÊT-ANH NGUYÊN
3.2. Level sets of the relative extremal functions and a Two-Constant
Theorem. Let X be a complex manifold and D ⊂ X an open set. Suppose that
D is equipped with a system of approach regions A =
Aα(ζ)
ζ∈D, α∈Iζ
. For every
open subset G of D, there is a natural system of approach regions for G which is
called the induced system of approach regions A
ζ∈G, α∈I
of A onto G.
It is given by
α(ζ) := Aα(ζ) ∩G, ζ ∈ G, α ∈ I
where I
α ∈ Iζ : ζ ∈ Aα(ζ) ∩G
Proposition 3.5. Under the above hypothesis and notation, let A ⊂ D be a locally
pluriregular set (relative to A). For 0 < δ < 1, define the δ-level set of D relative
to A as follows
Dδ,A := {z ∈ D : ω(z, A,D) < 1− δ} .
We equip Dδ,A with the induced system of approach regions A
of A onto Dδ,A (see
Subsection 2.1 above). Then A ⊂ Dδ,A and
(3.1) ω(z, A,Dδ,A) =
ω(z, A,D)
, z ∈ Dδ,A.
Moreover, A is locally pluriregular relative to A
Proof. Since A is locally pluriregular, we see that
(3.2)
A− lim supω(·, A,D)
(z) = 0, z ∈ A.
Therefore, for every z ∈ A and α ∈ Iz, there is an open neighborhood U of z such
that ∅ 6= Aα(z) ∩ U ⊂ Dδ,A. Hence, A ⊂ Dδ,A.
Next, we turn to the proof of identity (3.1). Observe that 0 ≤
ω(·,A,D)
≤ 1 on
Dδ,A by definition. This, combined with (3.2), implies that
(3.3)
ω(z, A,D)
≤ ω(z, A,Dδ,A), z ∈ Dδ,A.
To prove the converse inequality of (3.3), let u ∈ PSH(Dδ,A) be such that u ≤ 1 on
Dδ,A and A
− lim sup u ≤ 0 on A. Consider the following function
(3.4) û(z) :=
max {(1− δ)u(z), ω(z, A,D)} , z ∈ Dδ,A,
ω(z, A,D), z ∈ D \Dδ,A.
It can be checked that û ∈ PSH(D) and 0 ≤ û ≤ 1. Moreover, in virtue of the
assumption on u and (3.2) and (3.4), we have that
(A−lim sup û)(a) ≤ max
(1− δ)(A
− lim sup u)(a),
A− lim supω(·, A,D)
for all a ∈ A. Consequently, û ≤ ω(·, A,D). In particular, one gets from (3.4) that
u(z) ≤
ω(z, A,D)
, z ∈ Dδ,A.
Since u is arbitrary, we deduce from the latter estimate that the converse inequality
of (3.3) also holds. This, combined with (3.3), completes the proof of (3.1).
A UNIFIED APPROACH 13
To prove the last conclusion of the proposition, fix a point a ∈ A and an open
neighborhood U of a. Then we have
A− lim supω(·, A∩U,Dδ,A∩U)
(a) ≤
A− lim supω(·, A∩U, (D∩U)δ,A∩U )
A− lim supω(·, A ∩ U,D ∩ U)
(a) = 0,
where the first equality follows from identity (3.1) and the second one from the
hypothesis that A is locally pluriregular. �
The following Two-Constant Theorem for plurisubharmonic functions will play
an important role in the proof of the estimate in Theorem A.
Theorem 3.6. Let X be a complex manifold and D ⊂ X an open subset. Suppose
that D is equipped with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
. Let A ⊂ D
be a locally pluriregular set. Let m,M ∈ R and u ∈ PSH(D) such that u(z) ≤ M
for z ∈ D, and (A− lim sup u)(z) ≤ m for z ∈ A. Then
u(z) ≤ m(1− ω(z, A,D)) +M · ω(z, A,D), z ∈ D.
Proof. It follows immediately from Definition 2.2. �
Theorem 3.7. We keep the hypotheses and notation of Theorem 3.6. Let f be a
bounded function in O(D,C) such that (A − lim f)(ζ) = 0, ζ ∈ A. Then f(z) = 0
for all z ∈ D such that ω(z, A,D) 6= 1.
Proof. Fix a finite positive constant M such that |f |D < M. Consequently, the
desired conclusion follows from applying Theorem 3.6 to the function u := log |f |.
3.3. Construction of discs. In this subsection we present the construction of discs
à la Poletsky (see [38]). This is one of the main ingredients in the proof of Theorem
Let mes denote the Lebesgue measure on the unit circle ∂E. For a bounded
mapping φ ∈ O(E,Cn) and ζ ∈ ∂E, f(ζ) denotes the angular limit value of f at
ζ if it exists. A classical theorem of Fatou says that mes ({ζ ∈ ∂E : ∃f(ζ)}) = 2π.
For z ∈ Cn and r > 0, let B(z, r) denote the open ball centered at z with radius r.
Theorem 3.8. Let D be a bounded open set in Cn, A ⊂ D, z0 ∈ D and ǫ > 0.
Let A be a system of approach regions for D. Suppose in addition that A is locally
pluriregular (relative to A). Then there exist a bounded mapping φ ∈ O(E,Cn) and
a measurable subset Γ0 ⊂ ∂E with the following properties:
1) Γ0 is pluriregular (with respect to the system of angular approach regions),
φ(0) = z0, φ(E) ⊂ D, Γ0 ⊂
ζ ∈ ∂E : φ(ζ) ∈ A
, and
·mes(Γ0) < ω(z0, A,D) + ǫ.
14 VIÊT-ANH NGUYÊN
2) Let f ∈ C(D ∪ A,C) ∩ O(D,C) be such that f(D) is bounded. Then there
exist a bounded function g ∈ O(E,C) such that g = f ◦ φ in a neighborhood
of 0 ∈ E and5 g(ζ) = (f ◦ φ)(ζ) for all ζ ∈ Γ0. Moreover, g|Γ0 ∈ C(Γ0,C).
This theorem motivates the following
Definition 3.9. We keep the hypothesis and notation of Theorem 3.8. Then ev-
ery pair (φ,Γ0) satisfying the conclusions 1)–2) of this theorem is said to be an
ǫ-candidate for the triplet (z0, A,D).
Theorem 3.8 says that there always exist ǫ-candidates for all triplets (z, A,D).
Proof. First we will construct φ. To do this we will construct by induction a sequence
k=1 ⊂ O(E,D) which approximates φ as k ր ∞. This will allow to define the
desired mapping as φ := lim
φk. The construction of such a sequence is divided into
three steps.
For 0 < δ, r < 1 let
Da,r := D ∩ B(a, r), a ∈ A.
Aa,r,δ := {z ∈ Da,r : ω(z, A ∩ B(a, r), Da,r) < δ} , a ∈ A,
Ar,δ :=
Aa,r,δ,
(3.5)
where in the second “:=” Da,r is equipped with the induced system of approach
regions of A onto Da,r (see Subsection 3.2 above).
Suppose without loss of generality that D ⊂ B(0, 1).
Step 1: Construction of φ1.
Let δ0 :=
and r0 := 1. Fix 0 < δ1 <
and 0 < r1 <
. Applying Proposition
3.4, we obtain φ1 ∈ O(E,D) such that φ1(0) = z0 and
∂E ∩ φ−11 (Ar1,δ1)
≤ ω(z0, Ar1,δ1, D) + δ0.
On the other hand, using (3.5) and Definition 2.2 and the hypothesis that A is
locally pluriregular, we obtain
ω(z0, Ar1,δ1 , D) ≤ ω(z0, A,D).
Consequently, we may choose a subset Γ1 of Γ0 := ∂E ∩ φ
1 (Ar1,δ1) which consists
of finite disjoint closed arcs (Γ1j)j∈J1 so that
(3.6) 1−
·mes(Γ1) < ω(z0, Ar1,δ1, D) + 2δ0 ≤ ω(z0, A,D) + 2δ0,
t,τ∈Γ1j
|t− τ | < 2δ1, sup
t,τ∈Γ1j
|φ1(t)− φ1(τ)| < 2r1, j ∈ J1.
Step 2: Construction of φk+1 from φk for all k ≥ 1.
By the inductive construction we have 0 < δk <
and 0 < rk <
φk ∈ O(E,D) such that φk(0) = z0 and there exists a closed subset Γk of ∂E ∩
5 Note here that by Part 1), (f ◦ φ)(ζ) exists for all ζ ∈ Γ0.
A UNIFIED APPROACH 15
(Ark,δk) ∩ Γk−1 which consists of finite closed arcs (Γk,j)j∈Jk such that Γk is
relatively compact in the interior of Γk−1, and
(3.7) 1−
·mes(Γk) < 1−
·mes(Γk−1) + 2δk−1,
t,τ∈Γk,j
|t− τ | < 2δk, sup
t,τ∈Γk,j
|φk(t)− φk(τ)| < 2rk, j ∈ Jk,
|φk − φk−1|Γk < 2rk−1.
Here we make the convention that the last inequality is empty when k = 1.
In particular, we have that φk(Γk) ⊂ Ark,δk . Therefore, by (3.5), for every ζ ∈
φk(Γk) there is a ∈ A such that ζ ∈ Aa,rk,δk , that is,
ω(ζ, A∩ B(a, rk), Da,rk) < δk.
Using the hypothesis that A is locally pluriregular and (3.5) we see that
ω(z, Ar,δ ∩Da,rk , Da,rk) ≤ ω(z, A ∩ B(a, rk), Da,rk), 0 < δ, r < 1.
Consequently, for every ζ ∈ φk(Γk) there is a ∈ A such that
ω(ζ, Ar,δ ∩Da,rk , Da,rk) < δk, 0 < δ, r < 1.
Using the last estimate and arguing as in [38, p. 120–121] (see also the proof of
Theorem 1.10.7 in [19] for a nice presentation), we can choose 0 < δk+1 <
0 < rk+1 <
and φk+1 ∈ O(E,D) such that φk+1(0) = z0, and there exists a
closed subset Γk+1 of ∂E ∩ φ
k+1(Ark+1,δk+1) ∩ Γk which consists of finite closed arcs
(Γk+1,j)j∈Jk+1 such that Γk+1 is relatively compact in the interior of Γk, and
(3.8) 1−
·mes(Γk+1) < 1−
·mes(Γk) + 2δk,
t,τ∈Γk+1,j
|t− τ | < 2δk+1, sup
t,τ∈Γk+1,j
|φk+1(t)− φk+1(τ)| < 2rk+1, j ∈ Jk+1,
|φk+1 − φk|Γk+1 < 2rk.
Step 3: Construction of φ from the sequence (φk)
In summary, we have constructed a decreasing sequence (Γk)
k=1 of closed subsets
of ∂E. Consider the new closed set
By (3.7)–(3.8),
·mes(Γ) =
mes(Γ1)− 2
mes(Γ1)− 3δ1.
16 VIÊT-ANH NGUYÊN
This, combined with (3.6), implies the following property
·mes(Γ) < 1−
·mes(Γ1)+3δ1 ≤ ω(z0, A,D)+2δ0+3δ1 < ω(z0, A,D)+ ǫ.
On the other hand, we recall from the above construction the following properties:
(ii) φk(Γ) ⊂ φk(Γk) ⊂ Ark,δk .
(iii) δ0 =
, r0 = 1, 0 < δk+1 <
, 0 < rk+1 <
and |φk+1−φk|Γ ≤ |φk+1−φk|Γk+1 <
(iv) sup
t,τ∈Γkj
|t− τ | < 2δk and sup
t,τ∈Γk,j
|φk(t)− φk(τ)| < 2rk, j ∈ Jk.
(v) For every ζ ∈ Γ there exists a sequence (jk)k≥1 such that jk ∈ Jk, and ζ is an
interior point of Γk,jk , and Γk+1,jk+1 ⋐ Γk,jk , and ζ =
Γk,jk .
Therefore, we are able to apply the Khinchin–Ostrowski Theorem (see [11, The-
orem 4, p. 397]) to the sequence (φk)
k=1. Consequently, this sequence converges
uniformly on compact subsets of E to a mapping φ ∈ O(E,D). Moreover, φ admits
(angular) boundary values at all points of Γ and φ(Γ) ⊂
Ark,δk ⊂ A.
Observe that since φk(0) = φ(0) = z0 ∈ D and f ∈ C(D ∪ A,C) ∩ O(D,C), the
sequence (f ◦ φk)
k=1 converges to f ◦ φ uniformly on a neighborhood of 0 ∈ E. On
the other hand, f(D) is bounded by the hypothesis. Thus by Montel Theorem, the
family (f ◦ φk)
k=1 ⊂ O(E,C) is normal. Consequently, the sequence (f ◦ φk)
converges uniformly on compact subsets of E. Let g be the limit mapping. Then
g ∈ O(E,C) and g = f ◦ φ in a neighborhood of 0 ∈ E. Moreover, it follows
from (i)–(iii) above and the hypothesis f ∈ C(D ∪ A,C) that g(ζ) = (f ◦ φ)(ζ)
for all ζ ∈ Γ. We deduce from (iii)–(v) above that g|Γ ∈ C(Γ,C) Finally, applying
Lemma 4.1 below we may choose a locally pluriregular subset Γ0 ⊂ Γ (relative to
the system of angular approach regions) such that mes(Γ0) = mes(Γ). Hence, the
proof is finished. �
It is worthy to remark that φ(E) ⊂ D; but in general, φ(E) 6⊂ D !
The last result of this section sharpens Theorem 3.8.
Theorem 3.10. Let D be a bounded open set in Cn, A ⊂ D, and ǫ > 0. Let A be a
system of approach regions for D. Suppose in addition that A is locally pluriregular
(relative to A). Then there exists a Borel mapping Φ : D × E −→ Cn with the
following property: for every z ∈ D, there is a measurable subset Γz of ∂E such that
(Φ(z, ·),Γz) is an ǫ-candidate for the triplet (z, A,D).
Roughly speaking, this result says that one can construct ǫ-candidates for (z, A,D)
so that they depend in a Borel-measurable way on z ∈ D.
Proof. Observe that in Proposition 3.4 we can construct ǫ-candidates for (z, A,M)
so that they depend in a Borel-measurable way on z ∈ M. Here an ǫ-candidate
for (z, A,M) is a holomorphic disc φ ∈ O(E,M) such that φ(0) = z and
1∂E\φ−1(A),∂E(e
iθ)dθ < P[1M\A,M](z) + ǫ.
A UNIFIED APPROACH 17
Using this we can adapt the proof of Theorem 3.8 in order to obtain the desired
result. �
4. A mixed cross theorem
Let E be as usual the open unit disc in C. Let B be a measurable subset of ∂E
and ω(·, B, E) the relative extremal function of B relative to E (with respect to the
canonical system of approach regions). Then it is well-known (see [39]) that
(4.1) ω(z, B, E) =
1− |z|2
|eiθ − z|2
· 1∂E\B,∂E(e
iθ)dθ.
The following elementary lemma will be very useful.
Lemma 4.1. We keep the above hypotheses and notation.
1) Let u be a subharmonic function defined on E with u ≤ 1 and let α ∈ (0, π
be such that
lim sup
z→ζ, z∈Aα(ζ)
u(z) ≤ 0 for a.e. ζ ∈ B,
where A = (Aα(ζ)) is the system of angular approach regions defined in
Subsection 2.2. Then u ≤ ω(·, B, E) on E.
2) ω(·, B, E) is also the relative extremal function of B relative to E (with
respect to the system of angular approach regions).
3) For all subsets N ⊂ ∂E with mes(N ) = 0, ω(·, B, E) = ω(·, B ∪N , E).
4) Let B
be the set of all density points of B. Then
z→ζ, z∈Aα(ζ)
ω(z, B, E) = 0, ζ ∈ B
, 0 < α <
In particular, B
is locally pluriregular (with respect to the system of angular
approach regions).
5) ω(·, B, E) = ω̃c(·, B, E) = ω̃a(·, B, E) on E, where ω̃c(·, B, E) (resp.
ω̃a(·, B, E)) is given by Definition 2.3 relative to the system of canonical
approach regions (resp. angular approach regions).
Proof. It follows immediately from the explicit formula (4.1). �
The main ingredient in the proof of Theorem A is the following mixed cross
theorem.
Theorem 4.2. Let D be a complex manifold and E as usual the open unit disc in
C. D (resp. E) is equipped with the canonical system of approach regions (resp.
the system of angular approach regions). Let A be an open subset of D and B a
measurable subset of ∂E such that B is locally pluriregular (relative to the system of
angular approach regions). For 0 ≤ δ < 1 put G := {w ∈ E : ω(w,B,E) < 1− δ} .
18 VIÊT-ANH NGUYÊN
Let W := X(A,B;D,G), W o := Xo(A,B;D,G), and6
Ŵ = X̂(A,B;D,G) :=
(z, w) ∈ D ×G : ω(z, A,D) +
ω(w,B,E)
Let f : W −→ C be such that
(i) f ∈ Os(W
o,C);
(ii) f is locally bounded on W, f |A×B is a Borel function;
(iii) for all z ∈ A,
w→η, w∈Aα(η)
f(z, w) = f(z, η), η ∈ B, 0 < α <
Then there is a unique function f̂ ∈ O(Ŵ ,C) such that f̂ = f on A×G. Moreover,
|f |W = |f̂ |cW .
The proof of this theorem will occupy the present and the next sections. Our
approach here avoid completely the classical method of doubly orthogonal bases
of Bergman type. For the proof we need the following “measurable” version of
Gonchar’s Theorem.
Theorem 4.3. Let D = G := E be equipped with the system of angular approach
regions. Let A (resp. B) be a Borel measurable subset of ∂D (resp. ∂G) such
that A and B are locally pluriregular and that mes(A), mes(B) > 0. Put W :=
X(A,B;D,G) and define W o, Ŵ , ω(z, w) as in Subsection 2.3. Let f : W −→ C
be such that:
(i) f is locally bounded on W and f ∈ Os(W
o,C);
(ii) f |A×B is a Borel function;
(iii) for all a ∈ A (resp. b ∈ B), f(a, ·)|G (resp. f(·, b)|D) admits A-limit
7 f(a, b)
at all b ∈ B (resp. at all a ∈ A).
Then there exists a unique function f̂ ∈ O(Ŵ ,C) which admits A-limit f(ζ, η) at
all points (ζ, η) ∈ W o. If, moreover, |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,w)
A×B |f |
ω(z,w)
W , (z, w) ∈ Ŵ .
Proof. It follows from Steps 1–3 of Section 6 in [34]. �
The above theorem is also true in the context of an N -fold cross W (N ≥ 2). We
give here a version of a special 3-fold cross which is needed for the proof of Theorem
Theorem 4.4. Let D = G := E be equipped with the system of angular approach
regions. Let A (resp. B) be a Borel measurable subset of ∂D (resp. ∂G) such that
6 In fact, Theorem 4.10 in [34] says that ω(·, B,G) =
ω(·,B,E)
on G, where ω(·, B,G) is the
relative extremal function with respect to the system of angular approach regions induced onto G.
7 that is, the angular limit
A UNIFIED APPROACH 19
A and B are locally pluriregular and that mes(A), mes(B) > 0. Define W, W o, Ŵ
as follows:
W = X(A, ∂E,B;D,E,G) := A× ∂E × (G ∪B)
A× E × B
(D ∪ A)× ∂E × B,
W o = Xo(A, ∂E,B;D,E,G) := A× ∂E ×G
A× E × B
D × ∂E × B,
Ŵ = X̂(A, ∂E,B;D,E,G) := {(z, t, w) ∈ D × E ×G : ω(z, A,D) + ω(w,B,G) < 1} .
Let f : W −→ C be such that:
(i) f is locally bounded on W and f ∈ Os(W
o,C)8;
(ii) f |A×∂E×B is a Borel function;
(iii) for all (a, λ) ∈ A × ∂E (resp. (a, b) ∈ A × B) (resp. (λ, b) ∈ ∂E × B),
f(a, λ, ·)|G (resp. f(a, ·, b)|E) (resp. f(·, λ, b)|D) admits the angular limit
f(a, λ, b) at all b ∈ B (resp. at all λ ∈ ∂E) (resp. at all a ∈ A).
Then there exists a unique function f̂ ∈ O(Ŵ ,C) such that
cW∋(z,t,w)→(ζ,τ,η),w∈Aα(η)
f̂(z, t, w) =
f(ζ, λ, η)
(ζ, τ, η) ∈ D × E × B, 0 < α <
If, moreover, |f |W <∞, then
|f̂(z, t, w)| ≤ |f |
1−ω(z,A,D)−ω(w,B,G)
A×∂E×B |f |
ω(z,A,D)+ω(w,B,G)
W , (z, t, w) ∈ Ŵ .
Proof. We refer the reader to Subsections 5.2 and 5.3 in [34].
Let ω̂(·, A,D) (resp. ω̂(·, B,G)) be the conjugate harmonic function of ω(·, A,D)
(resp. ω(·, B,G) ) such that ω̂(z0, A,D) = 0 (resp. ω̂(w0, B,G) = 0) for a certain
fixed point z0 ∈ D (resp. w0 ∈ G). Thus we define the holomorphic functions
g1(z) := ω(z, A,D) + iω̂(z, A,D), g2(w) := ω(w,B,G) + iω̂(w,B,G), and
g(z, w) := g1(z) + g2(w), (z, w) ∈ D ×G.
Each function e−g1 (resp. e−g2) is bounded on D (resp. on G). Therefore, in
virtue of [11, p. 439], we may define e−g1(a) (resp. e−g2(b)) for a.e. a ∈ A (resp. for
a.e. b ∈ B) to be the angular limit of e−g1 at a (resp. e−g2 at b).
In virtue of (i), for each positive integer N, we define, as in [12, 13] (see also [34]),
the Gonchar–Carleman operator as follows
(4.2) KN(z, t, w) = KN [f ](z, t, w) :=
(2πi)2
−N(g(a,b)−g(z,w)) f(a, t, b)dadb
(a− z)(b− w)
for (z, t, w) ∈ D × ∂E × G. Reasoning as in [13] and using (i)–(iii) above, we see
that the following limit
(4.3) K(z, t, w) = K[f ](z, t, w) := lim
KN(z, t, w)
8 This notation means that for all (a, λ) ∈ A×∂E (resp. (a, b) ∈ A×B) (resp. (λ, b) ∈ ∂E×B),
the function f(a, λ, ·)|G (resp. f(a, ·, b)|E) (resp. f(·, λ, b)|D) is holomorphic.
20 VIÊT-ANH NGUYÊN
exists for all points in the set
(z, t, w) : t ∈ ∂E, (z, w) ∈ X̂(A,B;D,G)
, and its
limit is uniform on compact subsets of the latter set.
Observe that for n = 0, 1, 2, . . . , and N = 1, 2, . . . ,
KN(z, t, w)dt =
(2πi)2
f(a, t, b)dt
e−N(g(a,b)−g(z,w))dadb
(a− z)(b− w)
where the first equality follows from (4.2), the second one from the equality∫
tnf(a, t, b)dt = 0 which itself is an immediate consequence of (i). Therefore,
we deduce from (4.3) that
tnK(z, t, w)dt = 0, (z, w) ∈ X̂(A,B;D,G), n = 0, 1, 2, . . . .
On the other hand,
(z, t, w) : t ∈ E, (z, w) ∈ X̂(A,B;D,G)
Hence, we are able to define the desired extension function
f̂(z, t, w) :=
K(z, λ, w)
dλ, (z, t, w) ∈ Ŵ .
Recall from Steps 1–3 of Section 6 in [34] that
cW∋(z,w)→(ζ,η),w∈Aα(η)
K(z, t, w) = f(ζ, t, η), (ζ, t, η) ∈ D × ∂E × B, 0 < α <
Inserting this into the above formula of f̂ , the desired conclusion of the theorem
follows. �
We break the proof of Theorem 4.2 into two cases.
CASE 1: δ = 0 (that is G = E).
We follow essentially the arguments presented in Section 4 of [28]. For the sake
of clarity and completeness we give here the most basic arguments.
We begin the proof with the following lemma.
Lemma 4.5. We keep the hypothesis of Theorem 4.2. For j ∈ {1, 2}, let φj ∈
O(E,D) be a holomorphic disc, and let tj ∈ E such that φ1(t1) = φ2(t2) and
1D\A,D(φj(e
iθ))dθ < 1. Then:
1) For j ∈ {1, 2}, the function (t, w) 7→ f(φ(t), w) defined on X(φ−1j (A) ∩
∂E,B;E,G) satisfies the hypothesis of Theorem 4.3, where φ−1j (A) := {t ∈
E : φj(t) ∈ A}.
A UNIFIED APPROACH 21
2) For j ∈ {1, 2}, in virtue of Part 1), let f̂j be the unique function in
X̂(φ−1j (A) ∩ ∂E,B;E,G),C
given by Theorem 4.3. Then
f̂1(t1, w) = f̂2(t2, w),
for all w ∈ G such that (tj, w) ∈ X̂
φ−1j (A) ∩ ∂E,B;E,G
, j ∈ {1, 2}.
Proof of Lemma 4.5. Part 1) follows immediately from the hypothesis. There-
fore, it remains to prove Part 2). To do this fix w0 ∈ G such that (tj, w0) ∈
φ−1j (A) ∩ E,B;E,G
for j ∈ {1, 2}.We need to show that f̂1(t1, w0) = f̂2(t2, w0).
Observe that both functions w ∈ G 7→ f̂1(t1, w) and w ∈ G 7→ f̂2(t2, w) belong to
O(G,C), where G is the connected component which contains w0 of the following
open set
w ∈ G : ω(w,B,G) < 1− max
j∈{1,2}
ω(tj, φ
j (A) ∩ ∂E,E)
Since φ1(t1) = φ2(t2), it follows from Theorem 4.3 and the hypothesis of Part 2)
(A− lim f̂1)(t1, η) = f(φ1(t1), η) = f(φ2(t2), η) = (A− lim f̂2)(t2, η), η ∈ B.
Therefore, by Theorem 3.7, f̂1(t1, w) = f̂2(t2, w), w ∈ G. Hence, f̂1(t1, w0) =
f̂2(t2, w0), which completes the proof of the lemma. �
Now we return to the proof of the theorem in CASE 1 which is divided into two
steps.
Step 1: Construction of the extension function f̂ on Ŵ and its uniqueness.
Proof of Step 1. We define f̂ as follows: Let W be the set of all pairs (z, w) ∈ D×G
with the property that there are a holomorphic disc φ ∈ O(E,D) and t ∈ E such
that φ(t) = z and (t, w) ∈ X̂ (φ−1(A) ∩ ∂E,B;E,G) . By Part 1) of Lemma 4.5 and
Theorem 4.3, let f̂φ be the unique function in O
X̂(φ−1(A) ∩ ∂E,B;E,G),C
(4.4) (A− lim f̂φ)(t, w) = f(φ(t), w), (t, w) ∈ X
−1(A) ∩ ∂E,B;E,G
Then the desired extension function f̂ is given by
(4.5) f̂(z, w) := f̂φ(t, w).
In virtue of Part 2) of Lemma 4.5, f̂ is well-defined on W. We next prove that
(4.6) W = Ŵ .
Taking (4.6) for granted, then f̂ is well-defined on Ŵ .
Now we return to (4.6). To prove the inclusion W ⊂ Ŵ , let (z, w) ∈ W. By the
above definition of W, one may find a holomorphic disc φ ∈ O(E,D), a point t ∈ E
22 VIÊT-ANH NGUYÊN
such that φ(t) = z and (t, w) ∈ X̂ (φ−1(A) ∩ ∂E,B;E,G) . Since ω(φ(t), A,D) ≤
ω(t, φ−1(A) ∩ ∂E,E), it follows that
ω(z, A,D) + ω(w,B,G) ≤ ω(t, φ−1(A) ∩ ∂E,E) + ω(w,B,G) < 1,
Hence (z, w) ∈ Ŵ . This proves the above mentioned inclusion.
To finish the proof of (4.6), it suffices to show that Ŵ ⊂ W. To do this, let
(z, w) ∈ Ŵ and fix any ǫ > 0 such that
(4.7) ǫ < 1− ω(z, A,D)− ω(w,B,G).
Applying Theorem 3.1 and Proposition 3.4, there is a holomorphic disc φ ∈ O(E,D)
such that φ(0) = z and
(4.8)
1D\A,D(φ(e
iθ))dθ < ω(z, A,D) + ǫ.
Observe that
ω(0, φ−1(A) ∩ ∂E,E) + ω(w,B,G) =
1D\A,D(φ(e
iθ))dθ + ω(w,B,G)
< ω(z, A,D) + ω(w,B,G) + ǫ < 1,
where the equality follows from (4.1), the first inequality holds by (4.8), and the
last one by (4.7). Hence, (0, w) ∈ X̂ (φ−1(A) ∩ ∂E,B;E,G) , which implies that
(z, w) ∈ W. This completes the proof of (4.6). Hence, the construction of f̂ on Ŵ
has been completed.
Next we show that f̂ = f on A×G. To this end let (z0, w0) be an arbitrary point
of A × G. Choose the holomorphic disc φ ∈ O(E,D) given by φ(t) := z0, t ∈ E.
Then by formula (4.5),
f̂(z0, w0) = f̂φ(0, w0) = f(φ(0), w0) = f(z0, w0).
If g ∈ O(Ŵ ,C) satisfies g = f on A × G, then we deduce from (4.4)–(4.5) that
g = f̂ . This proves the uniqueness of f̂ . �
Finally, we conclude the proof of CASE 1 by the following
Step 2: Proof of the fact that f̂ ∈ O(Ŵ ,C).
Proof of Step 2. Fix an arbitrary point (z0, w0) ∈ Ŵ and let ǫ > 0 be so small such
(4.9) 2ǫ < 1− ω(z0, A,D)− ω(w0, B,G).
Since ω(·, B,G) ∈ PSH(G), one may find an open neighborhood V of w0 such that
(4.10) ω(w,B,D) < ω(w0, B,G) + ǫ, w ∈ V.
A UNIFIED APPROACH 23
Let n be the dimension of D at the point z0. Applying Lemma 3.2 and Proposition
3.4, we obtain an open set T in C, an open neighborhood U of z0, and a family of
holomorphic discs (φz)z∈U ⊂ O(E,D) with the following properties:
the mapping (z, t) ∈ U × E 7→ φz(t) is holomorphic;(4.11)
φz(0) = z, z ∈ U ;(4.12)
φz(t) ∈ A, t ∈ T ∩ E, z ∈ U ;(4.13)
1∂E\T,∂E(e
iθ)dθ < ω(z0, A,D) + ǫ.(4.14)
By shrinking U (if necessary), we may assume without loss of generality that in a
chart, z0 = 0 ∈ C
n and
(4.15) U =
z = (z1, . . . , zn) = (z
, zn) ∈ Cn : z
∈ S, |zn| < 2
where S ⊂ Cn−1 is an open set.
Consider the 3-fold cross (compared with the notation in Theorem 4.4)
X (T ∩ ∂E, U,B;E,U,G) := (T ∩ ∂E)× U × (G ∪ B)
(T ∩ ∂E)× U × B
(E ∪ (T ∩ ∂E))× U ×B,
and the function g : X (T ∩ ∂E, U,B;E,U,G) −→ C given by
(4.16) g(t, z, w) := f(φz(t), w), (t, z, w) ∈ X (T ∩ ∂E, U,B;E,U,G) .
We make the following observations:
Let t ∈ T ∩ ∂E. Then, in virtue of (4.13) we have φz(t) ∈ A for z ∈ U. Con-
sequently, in virtue of (4.11), (4.16) and the hypothesis f ∈ Os(W
o,C), we con-
clude that g(t, z, ·)|G ∈ O(G,C)
resp. g(t, ·, w)|U ∈ O(U,C)
for any z ∈ U (resp.
w ∈ B). Analogously, for any z ∈ U, w ∈ B, we can show that g(·, z, w)|E ∈ O(E,C).
In summary, we have shown that g is separately holomorphic. In addition,
it follows from hypothesis (ii) and (4.11)–(4.13) that g is locally bounded and
g|(T∩∂E)×U×B is a Borel function.
For z
∈ S write Ez′ :=
z = (z
, zn) ∈ C
n : |zn| < 1
. Then by (4.15),⋃
Ez′ ⊂ U. Consequently, for all z
∈ S, using hypothesis (iii) we are
able to apply Theorem 4.4 to g in order to obtain a unique function ĝ ∈
X̂ (T ∩ ∂E, ∂Ez′ , B;E,Ez′ , G) ,C
9 such that
(t,z,w)→(τ,ζ,η), w∈Aα(η)
ĝ(t, z, w) =
g(τ, ζ
, λ, η)
λ− ζn
(τ, ζ, η) ∈ E ×Ez′ × B, z
∈ S, 0 < α <
9 In fact, we identify Ez′ with E in an obvious way.
24 VIÊT-ANH NGUYÊN
Using (4.11) and (4.15)–(4.16) and the Cauchy’s formula, we see that the right hand
side is equal to g(τ, ζ, η). Hence, we have shown that
(4.17)
(t,z,w)→(τ,ζ,η), w∈Aα(η)
ĝ(t, z, w) = g(τ, ζ, η), (τ, ζ, η) ∈ E×Ez′×B, z
∈ S, 0 < α <
Observe that
X̂ (T ∩ ∂E, ∂Ez′ , B;E,Ez′ , G) = {(t, z, w) ∈ E × Ez′ ×G : ω(t, T ∩ ∂E,E) + ω(w,B,G) < 1} .
On the other hand, for any w ∈ V,
ω(0, T ∩ ∂E,E) + ω(w,B,G) ≤
1∂E\T,∂E(e
iθ)dθ + ω(w0, B,G) + ǫ
< ω(z0, A,D) + ω(w0, B,G) + 2ǫ < 1,
(4.18)
where the first inequality follows from (4.1) and (4.10), the second one from (4.14),
and the last one from (4.9). Consequently,
(4.19) (0, z, w) ∈ X̂ (T ∩ ∂E, ∂Ez′ , B;E,Ez′ , G) , (z, w) ∈ Ez′ × V, z
It follows from (4.5), (4.12), (4.13) and (4.18) that, for z
∈ S and z ∈ Ez′ , f̂φz is
well-defined and holomorphic on X̂(T ∩ ∂E,B;E,G), and
(4.20) f̂(z, w) = f̂φz(0, w), w ∈ V.
On the other hand, it follows from (4.4), (4.16) and (4.17) that
(t,w)→(τ,η), w∈Aα(η)
f̂φz(t, w) = lim
(t,w)→(τ,η), w∈Aα(η)
ĝ(t, z, w),
(τ, η) ∈ E ×B, z ∈ Ez′ , z
∈ S, 0 < α <
Since, for fixed z ∈ Ez′ , the restricted functions (t, w) 7→ ĝ(t, z, w) and f̂φz are
holomorphic on X̂(T ∩ ∂E,B;E,G), we deduce from the latter equality and the
uniqueness of Theorem 4.3 that
ĝ(t, z, w) = f̂φz(t, w), (t, w) ∈ X̂ (T ∩ ∂E,B;E,G) , z ∈ Ez′ , z
In particular, using (4.5), (4.19) and (4.20),
ĝ(0, z, w) = f̂φz(0, w) = f̂(z, w), (z, w) ∈ Ez′ × V, z
Since we know from (4.19) that ĝ is holomorphic in the variables zn and w on a
neighborhood of (0, z0, w0), it follows that f̂ is holomorphic in the variables z
n and
w on a neighborhood of (z0, w0). Exchanging the role of z
n and any other variable
zj , j = 1, . . . , n − 1, we see that f̂ is separetely holomorphic on a neighborhood
of (z0, w0). In addition, f̂ is locally bounded. Consequently, we conclude, by the
classical Hartogs extension Theorem, that f̂ is holomorphic on a neighborhood of
(z0, w0). Since (z0, w0) ∈ Ŵ is arbitrary, it follows that f̂ ∈ O(Ŵ ,C). �
Combining Steps 1–2, CASE 1 follows. �
A UNIFIED APPROACH 25
5. Completion of the proof of Theorem 4.2
In this section we introduce the new technique of conformal mappings. This
technique will allow us to pass from CASE 1 to the general case. We recall a notion
from Definition 4.8 in [34] which will be relevant for our further study.
Definition 5.1. Let A be the system of angular approach regions for E, let Ω be an
open subset of the unit disc E and ζ a point in ∂E. Then the point ζ is said to be
an end-point of Ω if, for every 0 < α < π
, there is an open neighborhood U = Uα of
ζ such that U ∩Aα(ζ) ⊂ Ω. The set of all end-points of Ω is denoted by End(Ω).
The main idea of the technique of conformal mappings is described below.
Proposition 5.2. Let B be a measurable subset of ∂E with mes(B) > 0. For 0 ≤
δ < 1 put G := {w ∈ E : ω(w,B,E) < 1− δ} . Let Ω be an arbitrary connected
component of G. Then
1) End(Ω) is a measurable subset of ∂E and mes(End(Ω)) > 0. Moreover, Ω is
a simply connected domain.
In virtue of Part 1) and the Riemann mapping theorem, let Φ be a confor-
mal mapping of Ω onto E.
2) For every ζ ∈ End(Ω), there is η ∈ ∂E such that
z→ζ, z∈Ω∩Aα(ζ)
Φ(z) = η, 0 < α <
η is called the limit of Φ at the end-point ζ and it is denoted by Φ(ζ).
Moreover, Φ|End(Ω) is one-to-one.
3) Let f be a bounded holomorphic function on Ω, ζ ∈ End(Ω) and λ ∈ C such
that lim
z→ζ, z∈Ω∩Aα(ζ)
f(z) = λ for some 0 < α < π
. Then f ◦ Φ−1 ∈ O(E,C)
admits the angular limit λ at Φ(ζ).
4) Let ∆ be a subset of End(Ω) such that mes(∆) = mes(End(Ω)). Put Φ(∆) :=
{Φ(ζ), ζ ∈ ∆}, where Φ(ζ) is given by Part 2). Then Φ(∆) is a measurable
subset of of ∂E with mes
> 0. and
ω(Φ(z),Φ(∆), E) =
ω(z, B, E)
, z ∈ Ω.
Proof. The first assertions of Part 1) follows from Theorem 4.9 in [34]. To show
that Ω is simply connected, take an arbitrary Jordan domain D such that ∂D ⊂ Ω.
We need to prove that D ⊂ Ω. Observe that D ⊂ E and ω(z, B, E) < 1 − δ for all
z ∈ ∂D ⊂ Ω ⊂ G. By the Maximum Principle, we deduce that ω(z, B, E) < 1 − δ
for all z ∈ D. Hence, D ⊂ G, which, in turn, implies that D ⊂ Ω. This completes
Part 1).
Part 2) follows from the “end-point” version of Theorem 4.4.13 in [39] (that is,
we replace the hypothesis “accessible point” therein by end-point).
Applying the classical Lindelöf’s Theorem to f ◦ Φ−1 ∈ O(E,C), Part 3) follows.
26 VIÊT-ANH NGUYÊN
It remains to prove Part 4). A straightforward argument shows that Φ(∆) is a
measurable subset of ∂E. Next, we show that
(5.1) ω(Φ(z),Φ(∆), E) ≤
ω(z, B, E)
, z ∈ Ω.
To do this pick any u ∈ PSH(E) such that u ≤ 1 and
lim sup
u(w) ≤ 0, η ∈ Φ(∆).
Consequently, Part 2) gives that
(5.2) lim sup
z→ζ, z∈Ω∩Aα(ζ)
u ◦ Φ(z) = 0, ζ ∈ ∆, 0 < α <
Next, consider the following function
(5.3) ũ(z) :=
max{(1− δ) · (u ◦ Φ)(z), ω(z, B, E)}, z ∈ Ω,
ω(z, B, E), z ∈ E \ Ω.
Then it can be checked that ũ is subharmonic and ũ ≤ 1 in E. In addition, for
every density point ζ of B such that ζ 6∈ End(Ω), we know from Theorem 4.9 in [34]
that there is a connected component Ωζ of G other than Ω such that ζ ∈ End(Ωζ).
Consequently, Part 4) of Lemma 4.1 gives, for such a point ζ, that
lim sup
z→ζ, z∈Aα(ζ)
ũ(z) = lim sup
z→ζ, z∈Aα(ζ)
ω(z, B, E) = 0, 0 < α <
This, combined with (5.2), implies that
lim sup
z→ζ, z∈Aα(ζ)
ũ(z) = 0, 0 < α <
, for a.e. ζ ∈ B.
Consequently, applying Part 1) of Lemma 4.1 yields that ũ ≤ ω(·, B, E) on E.
Hence, by (5.3), (u ◦Φ)(z) ≤
ω(z,B,E)
, z ∈ Ω, which completes the proof of (5.1). In
particular, we obtain that mes (Φ(∆)) > 0.
To prove the opposite inequality of (5.1), let u be an arbitrary function in PSH(E)
such that u ≤ 1 and
lim sup
u(z) ≤ 0, ζ ∈ B.
Applying Part 3) to the function f(z) := z, we obtain that
lim sup
w→η, w∈Aα(η)
(u ◦ Φ−1) (w)
≤ 0, η ∈ Φ(∆), 0 < α <
On the other hand, since u ≤ ω(·, B, E) on E, one gets that
(u◦Φ−1)(w)
≤ 1, w ∈ E.
Therefore, applying Part 1) of Lemma 4.1 yields that
(u ◦ Φ−1) (w)
≤ ω(w,Φ(∆), E), w ∈ E,
which, in turn, implies the converse inequality of (5.1). Hence, the proof of Part 4)
is complete. �
A UNIFIED APPROACH 27
Now we are in the position to complete the proof of Theorem 4.2:
CASE 2: 0 < δ < 1.
Let (Gk)k∈K be the family of all connected components of G, where K is an (at
most) countable index set. By Proposition 5.2, we may fix a conformal mapping Φk
from Gk onto E for every k ∈ K. Put
Bk :=
End(Gk) ∩ B
, Wk := X(A,B
k;D,E),
W ok := X
o(A,B
k;D,E), Ŵk := X̂(A,B
k;D,E), k ∈ K.
(5.4)
where [T ]
(or simply T
) for T ⊂ ∂E is, following the notation of Lemma 4.1, the
set of all density points of T.
Recall from the hypotheses of Theorem 4.2 that for every fixed z ∈ A, the holo-
morphic function f(z, ·)|G is bounded and that for every η ∈ B,
w→η, w∈Ω∩Aα(η)
f(z, w) = f(z, η), 0 < α <
Consequently, Part 3) of Proposition 5.2, applied to f(z, ·)|Gk with k ∈ K, implies
that for every fixed z ∈ D, f(z,Φ−1k (·)) ∈ O(E,C) admits the angular limit f(z, η)
at Φk(η) for all η ∈ B ∩ End(Gk). By Part 1) of that proposition, we know that
B ∩ End(Gk)
> 0. This discussion and the hypothesis allow us to apply the
result of CASE 1 to the function gk : Wk −→ C defined by
(5.5) gk(z, w) :=
f(z,Φ−1
(w)), (z, w) ∈ D ×Gk,
f(z,Φ−1k (w)) (z, w) ∈ D × B
where in the second line we have used the definition of Φk|End(Gk) and its one-to-one
property proved by Part 2) of Proposition 5.2.
Consequently, we obtain an extension function ĝk ∈ O(Ŵk,C) such that
(5.6) ĝk(z, w) = gk(z, w), (z, w) ∈ A× E.
Ŵk :=
(z,Φ−1
(w)), (z, w) ∈ Ŵk
, k ∈ K.
Observe that the open sets (Ŵk)k∈K are pairwise disjoint. Moreover, by (5.4),
Ŵk = {(z, w) ∈ D ×E : w ∈ Gk and
ω(z, A,D) + ω
Φk(w),Φk(End(Gk)), E
< 1 for some k ∈ K
(z, w) ∈ D × E : w ∈ Gk and ω(z, A,D) +
ω(w,B,E)
< 1 for some k ∈ K
= Ŵ ,
28 VIÊT-ANH NGUYÊN
where the second equality follows from Part 4) of Proposition 5.2. Therefore, we
can define the desired extension function f̂ ∈ O(Ŵ ,C) by the formula
f̂(z, w) := ĝk(z,Φk(w)), (z, w) ∈ Ŵk, k ∈ K.
This, combined with (5.4)–(5.6), implies that f̂ = f on A×G. The uniqueness of f̂
follows from that of ĝk, k ∈ K. Hence, the proof of the theorem is complete. �
6. A local version of Theorem A
The main purpose of the section is to prove the following result.
Theorem 6.1. Let D ⊂ Cn, G ⊂ Cm be bounded open sets. D (resp. G) is equipped
with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aα(η)
η∈G, α∈Iη
). Let
A (resp. B) be a subset of D (resp. G) such that A and B are locally pluriregular.
W := X(A,B;D,G), W := X(A,B;D,G),
:= Xo(A,B;D,G), Ŵ := X̂(A,B;D,G).
Then, for every bounded function f : W −→ C such that f ∈ Cs(W,C)∩Os(W
and that f |A×B is continuous at all points of (A ∩ ∂D) × (B ∩ ∂G), there exists a
unique bounded function f̂ ∈ O(Ŵ ,C) which admits A-limit f(ζ, η) at all points
(ζ, η) ∈ W. Moreover,
(6.1) |f̂(z, w)| ≤ |f |
1−ω(z,w)
A×B |f |
ω(z,w)
W , (z, w) ∈ Ŵ .
The core of our unified approach will be presented in the proof below. Our idea is
to use Theorem 3.8 in order to reduce Theorem 6.1 to the case of bidisk, that is, the
case of Theorem 4.3. This reduction is based on Theorem 4.2 and on the technique
of level sets.
Proof. It is divided into four steps.
Step 1: Construction of the desired function f̂ ∈ O(Ŵ ,C) and proof of the estimate
|f̂ |cW ≤ |f |W .
Proof of Step 1. We define f̂ at an arbitrary point (z, w) ∈ Ŵ as follows: Let ǫ > 0
be such that
(6.2) ω(z, A,D) + ω(w,B,G) + 2ǫ < 1.
By Theorem 3.8 and Definition 3.9, there is an ǫ-candidate (φ,Γ) (resp. (ψ,∆))
for (z, A,D) (resp. (w,B,G)). Moreover, using the hypotheses, we see that the
function fφ,ψ, defined by
(6.3) fφ,ψ(t, τ) := f(φ(t), ψ(τ)), (t, τ) ∈ X (Γ,∆;E,E) ,
satisfies the hypotheses of Theorem 4.3. By this theorem, let f̂φ,ψ be the unique
function in X̂ (Γ,∆;E,E) such that
(6.4) (A− lim f̂φ,ψ)(t, τ) = fφ,ψ(t, τ), (t, τ) ∈ X
o (Γ,∆;E,E) .
A UNIFIED APPROACH 29
In virtue of (6.2) and Theorem 3.8 and Lemma 3.3, (0, 0) ∈ X̂ (Γ,∆;E,E) . Then
we can define the value of the desired extension function f̂ at (z, w) as follows
(6.5) f̂(z, w) := f̂φ,ψ(0, 0).
The remaining part of this step is devoted to showing that f̂ is well-defined and
holomorphic on Ŵ .
To this end we fix an arbitrary point w0 ∈ G, a number ǫ0 : 0 < ǫ0 < 1 −
ω(w0, B,G), and an arbitrary ǫ0-candidate (ψ0,∆0) for (w0, B,G).
(6.6) Ŵ0 := {(z, τ) ∈ D × E : ω(z, A,D) + ω(τ,∆0, E) < 1} .
Inspired by formula (6.5) we define a function f̂0 : Ŵ0 −→ C as follows
(6.7) f̂0(z, τ) := f̂φ,ψ0(0, τ).
Here we have used an ǫ-candidate (φ,Γ) for (z, A,D), where ǫ is arbitrarily chosen
so that 0 < ǫ < 1− ω(z, A,D)− ω(τ,∆0, E).
Using (6.3)–(6.4) and (6.7) and arguing as in Part 2) of Lemma 4.5, one can show
that f̂0 is well-defined on Ŵ0.
For all 0 < δ < 1 let
(6.8) Aδ := {z ∈ D : ω(z, A,D) < δ} and Eδ := {w ∈ E : ω(w,∆0, E) < 1− δ} .
Then by the construction in (6.7), we remark that f̂0(z, ·) is holomorphic on Eδ for
every fixed z ∈ Aδ. We are able to define a new function f̃δ on X (Aδ, B;D,Eδ) as
follows
(6.9) f̃δ(z, τ) :=
f̂0(z, τ) (z, τ) ∈ Aδ × Eδ,
f(z, ψ0(τ)) (z, τ) ∈ D ×∆0.
Using the hypotheses on f and the previous remark, we see that f̃δ ∈
o (Aδ, B;D,Eδ) ,C
Observe that Aδ is an open set in D. Consequently, f̃δ satisfies the hypothe-
ses of Theorem 4.2. Applying this theorem yields a unique function f̂δ ∈
X̂ (Aδ, B;D,Eδ) ,C
such that
f̂δ(z, w) = f̃δ(z, w), (z, w) ∈ Aδ × Eδ.
This, combined with (6.9), implies that f̂0 is holomorphic on Aδ ×Gδ. On the other
hand, it follows from (6.6) and (6.8) that
Ŵ0 = X̂ (A,∆0;D,E) =
0<δ<1
Aδ ×Gδ.
Hence, f̂0 ∈ O(Ŵ0,C).
In summary, we have shown that f̂0, given by (6.7), is well-defined and holomor-
phic on Ŵ0.
30 VIÊT-ANH NGUYÊN
Now we are able to prove that f̂ , given by (6.5), is well-defined. To this end we fix
an arbitrary point (z0, w0) ∈ Ŵ , an ǫ0 : 0 < ǫ0 < 1− ω(z0, D,G), and two arbitrary
ǫ0-candidates (ψ1,∆1) and (ψ2,∆2) for (w0, B,G). Let
Ŵj := {(z, τ) ∈ D ×E : ω(z, A,D) + ω(τ,∆j, E) < 1} , j ∈ {1, 2}.
Using formula (6.7) define, for j ∈ {1, 2}, a function f̂j : Ŵj −→ C as follows
(6.10) f̂j(z, τ) := f̂φ,ψj(0, τ).
Here we have used any ǫ-candidate (φ,Γ) for (z, A,D) with a suitable ǫ > 0. Let
τj ∈ E be such that ψj(τj) = w0, j ∈ {1, 2}. Then, in virtue of (6.5) and (6.10) and
the result of the previous paragraph on the well-definedness of f̂0, the well-defined
property of f̂ is reduced to showing that
(6.11) f̂1(φ(t), τ1) = f̂2(φ(t), τ2)
for all t ∈ E and all ǫ-candidates (φ,Γ) for (φ(t), A,D), such that
ω(t,Γ, A) < ǫ := 1− max
j∈{1,2}
{ω(τ1,∆1, E), ω(τ2,∆2, E)} .
Observe that (6.11) follows from an argument based on Part 2) of Lemma 4.5. Hence,
f̂ is well-defined on Ŵ .
As in (6.8), for all 0 < δ < 1 let
Aδ := {z ∈ D : ω(z, A,D) < δ} , Bδ := {w ∈ G : ω(w,B,G) < δ} ,
Dδ := {z ∈ D : ω(z, A,D) < 1− δ} , Gδ := {w ∈ G : ω(w,B,G) < 1− δ} .
(6.12)
Now we combine (6.8) and (6.12) and the result that f̂0, given by (6.7), is well-defined
and holomorphic on Ŵ0, and the result that f̂ is well-defined on Ŵ . Consequently,
we obtain that
f̂(·, w) ∈ O(Dδ,C), w ∈ Bδ, 0 < δ < 1.
Since the formula (6.5) for f̂ is symmetric in two variables (z, w), one also gets that
f̂(z, ·) ∈ O(Gδ,C), z ∈ Aδ, 0 < δ < 1.
Since by (6.12),
0<δ<1
Aδ ×Gδ =
0<δ<1
Dδ × Bδ,
it follows from the previous conclusions that, for all points (z, w) ∈ Ŵ , there is an
open neighborhood U of z (resp. V of w) such that f ∈ Os(X
o(U, V ;U, V ),C). By
the classical Hartogs extension theorem, f ∈ O(U × V,C). Hence, f̂ ∈ O(Ŵ ,C).
On the other hand, it follows from (6.5) and the estimate in Theorem 4.3 that
(6.13) |f̂ |cW ≤ |f |W .
This completes Step 1. �
Step 2: f |A×B ∈ C(A× B,C).
A UNIFIED APPROACH 31
Proof of Step 2. Using the hypotheses we only need to check the continuity of f |A×B
at every point (a0, w0) ∈ A × (G ∩ B) and at every point (z0, b0) ∈ (D ∩ A) × B.
We will verify the first assertion. To do this let (ak)
k=1 ⊂ A and (wk)
k=1 ⊂ (G∩B)
such that lim
ak = a0 and lim
wk = w0. We need to show that
(6.14) lim
f(ak, wk) = f(a0, w0).
Since f |W is locally bounded, we may choose an open connected neighborhood V
of w0 such that sup
|f(ak, ·)|V <∞. Consequently, by Montel’s Theorem, there is a
sequence (kp)
p=1 such that (f(akp, ·)) converges uniformly on compact subsets of V
to a function g ∈ O(V ). Equality (6.14) is reduced to showing that g = f(a0, ·) on
V. Since f ∈ Cs(W,C), we deduce that f(a0, ·) = g on B ∩ V. On the other hand,
B ∩V is non locally pluripolar because B is locally pluriregular and w0 ∈ B. Hence,
we conclude by the uniqueness principle that g = f(a0, ·) on V. �
Step 3: f̂ admits A-limit f(ζ, η) at all points (ζ, η) ∈ W.
Proof of Step 3. To this end we only need to prove that
(6.15)
A− lim sup |f̂ − f(ζ0, η0)|
(ζ0, η0) < ǫ0
for an arbitrary fixed point (ζ0, η0) ∈ W and an arbitrary fixed 0 < ǫ0 < 1. Suppose
without loss of generality that
(6.16) |f |W ≤
First consider (ζ0, η0) ∈ A × B. Since f ∈ C(A × B,C), one may find an open
neighborhood U of ζ0 in C
n (resp. V of η0 in C
m) so that
(6.17) |f − f(ζ0, η0)|(A∩U)×(B∩V ) <
Consider the open sets
(6.18)
z ∈ D : ω(z, A ∩ U,D) <
and G
w ∈ G : ω(w,B ∩ V,G) <
In virtue of (6.16)–(6.18), an application of Theorem 3.6 gives that
|f(ζ, w)− f(ζ, η0)| ≤ (
)1−ω(w,B∩V,G) ≤
, ζ ∈ A ∩ U, w ∈ G
Hence,
(6.19) |f − f(ζ0, η0)|X(A∩U,B∩V ;D′ ,G′ ) ≤
Consider the function g : X(A ∩ U,B ∩ V ;D
) −→ C, given by
(6.20) g(z, w) := f(z, w)− f(ζ0, η0).
32 VIÊT-ANH NGUYÊN
Applying the result of Step 1, we can construct a function ĝ ∈ O(X̂(A ∩ U,B ∩
),C) from g in exactly the same way as we obtain f̂ ∈ O(Ŵ ,C) from f.
Moreover, combining (6.5) and (6.20), we see that
(6.21) ĝ = f̂ − f(ζ0, η0) on X̂(A ∩ U,B ∩ V ;D
On the other hand, it follows from formula (6.20), estimate (6.19), and estimate
(6.13) that
|ĝ|bX(A∩U,B∩V ;D′ ,G′) ≤
This, combined with (6.21) and (6.18), implies that
A− lim sup |f̂(z, w)− f(ζ0, η0)|
(ζ0, η0) ≤
Hence, (6.15) follows. In summary, we have shown that A− lim f̂ = f on A× B.
Now it remains to consider (ζ0, η0) ∈ A ×G. Using the last limit and arguing as
in Step 2, one can show that A− lim f̂(ζ0, η0) = f(ζ0, η0). �
Step 4: Proof of the uniqueness of f̂ and (6.1).
Proof of Step 4. To prove the uniqueness of f̂ suppose that ĝ ∈ O(Ŵ ,C) is a
bounded function which admits A-limit f(ζ, η) at all points (ζ, η) ∈ W. Fix an
arbitrary point (z0, w0) ∈ Ŵ , it suffices to show that f̂(z0, w0) = ĝ(z0, w0). Observe
that both functions f̂(z0, ·) and ĝ(z0, ·) are bounded and holomorphic on the δ-level
set of G relative to B :
Gδ,B := {w ∈ G : ω(w,B,G) < 1− ω(z0, A,D)} ,
where δ := ω(z0, A,D). On the other hand, they admit A-limit f(z0, η) at all
points η ∈ B. Consequently, applying Proposition 3.5 and Theorem 3.7 yields that
f̂(z0, ·) = ĝ(z0, ·) on Gδ,B. Hence, f̂(z0, w0) = ĝ(z0, w0).
To prove (6.1) fix an arbitrary point (z0, w0) ∈ Ŵ . For every η ∈ B, applying
Theorem 3.6 to log |f(·, η)| defined on D, we obtain that
(6.22) |f(z0, η)| ≤ |f |
1−ω(z0,A,D)
A×B |f |
ω(z0,A,D)
Applying Theorem 3.6 again to log |f̂(z0, ·)| defined on Gδ,B of the preceeding para-
graph, one gets that
|f̂(z0, w0)| ≤ |f(z0, ·)|
1−ω(w0,B,G)
B |f̂ |
ω(w0,B,G)
Inserting (6.13) and (6.22) into the right hand side of the latter estimate, (6.1)
follows. Hence Step 4 is finished. �
This completes the proof. �
In the sequel we will need the following refined version of Theorem 6.1.
Theorem 6.2. Let D ⊂ Cn, G ⊂ Cm be bounded open sets. D (resp. G) is equipped
with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aα(η)
η∈G, α∈Iη
). Let
A UNIFIED APPROACH 33
A, A0 (resp. B, B0) be subsets of D (resp. G) such that A0 and B0 are locally
pluriregular and that A0 ⊂ A
∗ and B0 ⊂ B
∗. Put
W := X(A,B;D,G) and W0 := X(A0, B0;D,G).
Then, for every bounded function f : W −→ C which satisfies the following condi-
tions:
• f ∈ Cs(W,C) ∩ Os(W
o,C);
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique bounded function f̂ ∈ O(Ŵ0,C) which admits A-limit f(ζ, η)
at all points (ζ, η) ∈ W0. Moreover,
(6.23) |f̂(z, w)| ≤ |f |
1−ω(z,A0,D)−ω(w,B0,G)
A0×B0
ω(z,A0,D)+ω(w,B0,G)
W , (z, w) ∈ Ŵ0.
Proof. Using the hypotheses and applying Part 1) of Theorem 7.2 below we can ex-
tend f to a locally bounded function (still denoted by) f defined on X(A∗, B∗, D,G)
such that f ∈ Os
o(A∗, B∗, D,G),C
and that f |X(A∗∩D,B∗∩G;D,G) is continuous.
Therefore, the newly defined function f satisfies
(6.24) f(a, b) := lim
f(ak, b),
where (a, b) is an arbitrary point of A∗ × (G∪B∗) and (ak)
k=1 ⊂ A
∗ is an arbitrary
sequence with lim
ak = a. Since f |W is bounded, it follows that the newly defined
function f is also bounded. In virtue of the definition of A∗ and B∗ we have
(6.25) ∂D ∩ A = ∂D ∩A∗ and ∂G ∩ B = ∂G ∩ B∗.
Using the second • in the hypotheses and formula (6.24) we see that f |A∗×B∗ is
continuous at all points all (∂D ∩ A) × (∂G ∩ B). Consequently, arguing as in the
proof of Step 2 of Theorem 6.1 and using (6.25), we can show that f ∈ C
. In summary, the newly defined function f which is defined and bounded on
X(A∗, B∗, D,G) satisfies
(6.26) f ∈ Os
o(A∗, B∗, D,G),C
and f ∈ C
A∗ × B∗,C
Observe that f is only separately continuous on X(A,B;D,G), but it is not nec-
essarily so on the cross X
A∗, B∗, D,G
. However, we will show that one can adapt
the argument of Theorem 6.1 in order to prove Theorem 6.2.
We define f̂ at an arbitrary point (z0, w0) ∈ Ŵ0 as follows: Let ǫ > 0 be such that
ω(z0, A0, D) + ω(w0, B0, G) + 2ǫ < 1.
By Theorem 3.8 and Definition 3.9, there is an ǫ-candidate (φ,Γ) (resp. (ψ,∆)) for
(z0, A0, D) (resp. (w0, B,G)). To conclude the proof we only need to prove that the
function fφ,ψ, defined by
fφ,ψ(t, τ) := f(φ(t), ψ(τ)), (t, τ) ∈ X (Γ,∆;E,E) ,
satisfies the hypotheses of Theorem 4.3. Indeed, having proved this assertion, the
proof will follow along the same lines as those given in Theorem 6.1. This assertion
is again reduced to showing that for each fixed t ∈ Γ, the function fφ,ψ(t, ·) admits
the angular limit f(φ(t), ψ(τ)) for every point τ ∈ ∆. We will prove the last claim.
34 VIÊT-ANH NGUYÊN
Using the first • and Theorem 3.8, we see that for every a ∈ A, the function
f(a, ψ(·)) ∈ O(E,C) admits the angular limit f(a, ψ(τ)) for every point τ ∈ ∆.
Next, using the hypothesis A0 ⊂ A
∗ we may choose a sequence (ak)
k=1 ⊂ A ∩ A
such that lim
ak = φ(t) ∈ A0. Observe from (6.26) that for every k the uniformly
bounded function f(ak, ψ(·)) ∈ O(E,C) admits the angular limit f(ak, ψ(τ)) and
that lim
f(ak, ψ(τ)) = f(φ(t), ψ(τ)) for every point τ ∈ ∆. Consequently, by the
Khinchin–Ostrowski Theorem (see [11, Theorem 4, p. 397]), the above claim follows.
7. Preparatory results
The first result of this section shows that the two definitions of plurisubharmonic
measure ω̃(·, A,D), given respectively in Definition 2.3 and in Subsection 2.1 of [28],
coincide in the case when A ⊂ D.
Proposition 7.1. Let X be a complex manifold and D ⊂ X an open set. D is
equipped with the canonical system A of approach regions. Let A be a subset of D.
Then ω̃(z, A,D) = ω(z, A∗, D).
Proof. Let P ∈ E(A). Then by Definition 2.3, P ⊂ A∗ and P is locally pluriregular.
Hence, P ⊂ (A∗)∗ = A∗. Since P ∈ E(A) is arbitrary, it follows from Definition 2.3
that Ã is locally pluriregular and Ã ⊂ A∗. In particular, (Ã)∗ ⊂ A∗ and
(7.1) ω̃(z, A,D) = ω(z, Ã, D) ≥ ω(z, A∗, D).
In the sequel we will show that
(7.2) A∗ ⊂ (Ã)∗.
Taking (7.2) for granted, we have that A∗ = (Ã)∗. Consequently,
ω̃(z, A,D) = ω(z, Ã, D) ≤ ω(z, A∗, D).
This, coupled with (7.1), completes the proof.
To prove (7.2) fix an arbitrary point a ∈ A∗ and an arbitrary but sufficiently small
neighborhood U ⊂ X of a such that U is biholomorphic to a bounded open set in
n, where n is the dimension of X at a. Since A∗ is a Borel subset of D, Theorem
8.5 in [7] provides a subset P ⊂ A∗ ∩ U of type Fσ
10 such that
(7.3) ω(z, P, U) = ω(z, A∗ ∩ U, U), z ∈ U.
Write P =
Pn, where Pn is closed. Observe that Pn ∩ P
n is locally pluriregular,
Pn \ (Pn ∩ P
n) is locally pluripolar and Pn ∩ P
n ⊂ Pn ⊂ A
∗ ∩ P. Consequently,⋃
(Pn ∩ P
n) ⊂ Ã ∩ P and P \
(Pn ∩ P
n) is locally pluripolar. This implies that
ω(z, Ã ∩ U, U) ≤ ω
(Pn ∩ P
n), U
= ω(z, P, U),
10 This means that P is a countable (or finite) union of relatively closed subsets of U.
A UNIFIED APPROACH 35
where the equality holds by applying Lemma 3.5.3 in [18] and by using the fact that
U is biholomorphic to a bounded open set in Cn. This, combined with (7.3) and the
assumption a ∈ A∗, implies that ω(a, Ã ∩ U, U) = 0. Thus (7.2) follows. �
The main purpose of this and the next sections is to generalize Theorem 6.1 to the
case where the “target space” Z is an arbitrary complex analytic space possessing
the Hartogs extension property.
Theorem 7.2. Let D ⊂ Cn, G ⊂ Cm be two bounded open sets. D (resp. G) is
equipped with the canonical system of approach regions. Let Z be a complex analytic
space possessing the Hartogs extension property. Let A (resp. B) be a subset of D
(resp. G). Put W := X(A,B;D,G) and Ŵ := X̂(A,B;D,G). Let f ∈ Os(W
o, Z).
1) Then f extends to a mapping (still denoted by) f defined on Xo(A∪A∗, B ∪
B∗;D,G) such that f is separately holomorphic on Xo(A∪A∗, B∪B∗;D,G)
and that f |Xo(A∗,B∗;D,G) is continuous.
2) Suppose in addition that A and B are locally pluriregular. Then f extends
to a unique mapping f̂ ∈ O(Ŵ , Z) such that f̂ = f on W.
Proof. This result has already been proved in Théorème 2.2.4 in [5] starting from
Proposition 3.2.1 therein. In the latter proposition Alehyane and Zeriahi make use
of the method of doubly orthogonal bases of Bergman type. We can avoid this
method by simply replacing every application of this proposition by Theorem 6.1.
Keeping this change in mind and using Proposition 7.1, the remaining part of the
proof follows along the same lines as that of Théorème 2.2.4 in [5]. �
Theorem 7.3. Let D, G be complex manifolds, and let A ⊂ D, B ⊂ G be open
subsets. Let Z be a complex analytic space possessing the Hartogs extension property.
Put W := X(A,B;D,G) and Ŵ := X̂(A,B;D,G). Then for any mapping f ∈
Os(W,Z), there is a unique mapping f̂ ∈ O(Ŵ , Z) such that f̂ = f on W.
Proof. It has already been proved in Theorem 5.1 of [28]. The only places where the
method of doubly orthogonal bases of Bergman type is involved is the applications
of Théorème 2.2.4 in [5]. As we already pointed out in Theorem 7.2, one can avoid
this method by using Theorem 6.1 instead. �
We are ready to formulate a slight generalization of Theorems 6.2 and 7.2.
Theorem 7.4. Let D ⊂ Cn, G ⊂ Cm be bounded open sets. D (resp. G) is equipped
with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let A
and A0 (resp. B and B0) be two subsets of D (resp. G) such that A0 and B0 are
locally pluriregular and that A0 ⊂ A
∗ and B0 ⊂ B
∗. Let Z be a complex analytic
space possessing the Hartogs extension property. Put
W := X(A,B;D,G) and W0 := X(A0, B0;D,G).
Then, for every bounded mapping f : W −→ Z which satisfies the following condi-
tions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
36 VIÊT-ANH NGUYÊN
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique bounded mapping f̂ ∈ O(Ŵ0,C) which admits A-limit f(ζ, η)
at all points (ζ, η) ∈ W0.
Proof. Since f is bounded, one may find an open neighborhood U of f(W ) in Z and
a holomorphic embedding φ of U into the polydisc Ek of Ck such that φ(U) is an
analytic set in Ek. Now we are able to apply Theorem 6.2 to the mapping φ ◦ f :
W −→ Ck. Consequently, one obtains a unique bounded mapping F ∈ O(Ŵ ,Ck)
which admits A-limit (φ ◦ f)(ζ, η) at all points (ζ, η) ∈ W. Using estimate (6.23)
one can show that F ∈ O(Ŵ , Ek). Now using Theorem 3.7 it is not difficult to see
that F (Ŵ ) ⊂ φ(U). Consequently, one can define the desired extension mapping f̂
as follows:
f̂(z, w) := (φ−1 ◦ F )(z, w), (z, w) ∈ Ŵ .
The following Uniqueness Theorem for holomorphic mappings generalizes Theo-
rem 3.7.
Theorem 7.5. Let X be a complex manifold, D ⊂ X an open subset and Z a
complex analytic space. Suppose that D is equipped with a system of approach regions(
Aα(ζ)
ζ∈D, α∈Iζ
. Let A ⊂ D be a locally pluriregular set. Let f1, f2 : D∪A −→ Z
be locally bounded mappings such that f1|D, f2|D ∈ O(D,Z) and A− lim f1 = A −
lim f2 on A. Then f1(z) = f2(z) for all z ∈ D such that ω(z, A,D) 6= 1.
We leave the proof to the interested reader. Finally, we conclude this section with
the following Gluing Lemma.
Lemma 7.6. Let D and G be open subsets of some complex manifolds and Z a com-
plex analytic space. Suppose that D (resp. G) is equipped with a system of approach
regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let (Dk)
(resp. (Gk)
a family of open subsets of D (resp. G) equipped with the induced system of approach
regions. Let (Pk)
(resp. (Qk)
) be a family of locally pluriregular subsets of
D (resp. G). Suppose, in addition, that
(i) Pk ⊂ Pk0, Dk0 ⊂ Dk, and Pk is locally pluriregular relative to Dk0 . Similarly,
Qk ⊂ Qk0, Gk0 ⊂ Gk, and Qk is locally pluriregular relative to Gk0 .
(ii) There are a family of locally bounded mappings (fk)
such that fk :
o (Pk,Qk;Dk,Gk) −→ Z verifies fk = fk0 on X
o (Pk,Qk;Dk0,Gk0) ,
and a family of holomorphic mappings (f̂k)
such that f̂k ∈
X̂ (Pk,Qk;Dk,Gk) , Z
, and
(A− lim f̂k)(z, w) = fk(z, w), (z, w) ∈ X
o (Pk,Qk;Dk0,Gk0) .
(iii) There are open subsets U of D and V of G such that ω̃(z,Pk,Dk0) +
ω̃(w,Qk,Gk0) < 1 for all (z, w) ∈ U × V and k ≥ k0.
Then f̂k(z, w) = f̂k0(z, w) for all (z, w) ∈ U × V and k ≥ k0.
A UNIFIED APPROACH 37
Proof. By (iii), we have that
(7.4) U × V ⊂ H := X̂ (Pk,Qk;Dk0,Gk0) .
On the other hand, using (i) we see that
(7.5) H ⊂ X̂ (Pk,Qk;Dk,Gk) ∩ X̂ (Pk0 ,Qk0;Dk0,Gk0) .
Fix arbitrary (z0, w0) ∈ H and k ≥ k0. Observe that both mappings f̂k(·, w0) and
f̂k0(·, w0) are defined on {z ∈ Dk0 : ω(z,Pk,Dk0) < 1− ω(w0,Qk,Gk0)} . Using (ii)
and Proposition 3.5, we may apply Theorem 7.5 to these mappings and conclude
that f̂k(z0, w0) = f̂k0(z0, w0). �
8. Local and semi-local versions of Theorem A
The aim of this section is to generalize Theorem 6.2 to some cases where the
“target space” Z is a complex analytic space possessing the Hartogs extension prop-
erty. Our philosophy is the following: we first apply Theorem 6.2 locally in order to
obtain various local extension mappings, then we glue them together. The gluing
process needs the following
Definition 8.1. Let M be a complex manifold and Z a complex space. Let (Uj)j∈J
be a family of open subsets of M, and (fj)j∈J a family of mappings such that fj ∈
O(Uj , Z). We say that the family (fj)j∈J is collective if, for any j, k ∈ J, fj = fk
on Uj ∩ Uk. The unique holomorphic mapping f :
Uj −→ Z, defined by f := fj
on Uj, j ∈ J, is called the collected mapping of (fj)j∈J .
We arrive at the following local version of Theorem A.
Theorem 8.2. Let D ⊂ Cp, G ⊂ Cq be bounded open sets and Z a complex analytic
space possessing the Hartogs extension property. D (resp. G) is equipped with a
system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let A, A0
(resp. B, B0) be subsets of D (resp. G) such that A0 and B0 are locally pluriregular
and that A0 ⊂ A
∗ and B0 ⊂ B
∗. Put
W := X(A,B;D,G) and W0 := X(A0, B0;D,G).
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along X
A ∩ ∂D,B ∩ ∂G;D,G
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique mapping f̂ ∈ O(Ŵ0, Z) which admits A-limit f(ζ, η) at all
points (ζ, η) ∈ W0.
Theorem 8.2 generalizes Theorem 6.2 to the case where the “target space” Z is an
arbitrary complex analytic space possessing the Hartogs extension property. Since
the proof is somewhat technical, the reader may skip it at the first reading.
38 VIÊT-ANH NGUYÊN
Proof. Recall that for a ∈ Ck and r > 0, B(a, r) denotes the open ball centered at a
with radius r. For 0 < δ < 1 and 0 < r put
Da,δ,r := {z ∈ D ∩ B(a, r) : ω(A0 ∩ B(a, r), D ∩ B(a, r)) < δ} , a ∈ A0,
Gb,δ,r := {w ∈ G ∩ B(b, r) : ω(B0 ∩ B(b, r), G ∩ B(b, r)) < δ} , b ∈ B0.
(8.1)
Applying Part 1) of Theorem 7.2 and using the hypotheses on f, we see that f
extends to a mapping defined on X(A ∪A∗, B ∪B∗;D,G) such that f is separately
holomorphic on Xo(A∪A∗, B ∪B∗;D,G) and that f |X(A∗,B∗;D,G) is locally bounded.
Therefore, using the compactness of A0 and B0, one may find a real number r0 > 0
such that
(8.2) fa,b := f |X(A0∩B(a,r),B0∩B(b,r);D∩B(a,r),G∩B(b,r))
is bounded for all 0 < r ≤ r0 and a ∈ A0, b ∈ B0. Applying Theorem 7.4 to fa,b ,
one obtains a mapping
(8.3) f̂a,b ∈ O
A0 ∩ B(a, r), B0 ∩ B(b, r);D ∩ B(a, r), G ∩ B(b, r)
which admits A-limit f on X
A0 ∩ B(a, r), B0 ∩ B(b, r);D ∩ B(a, r), G ∩ B(b, r)
Fix 0 < δ0 <
. Then it follows from (8.1) that for 0 < r ≤ r0, a ∈ A0, b ∈ B0.
Da,δ0,r ×Gb,δ0,r ⊂ X̂
A0 ∩ B(a, r), B0 ∩ B(b, r);D ∩ B(a, r), G ∩ B(b, r)
This, combined with (8.3), implies that
(8.4) f̂a,b ∈ O (Da,δ0,r ×Gb,δ0,r, Z) , 0 < r ≤ r0, a ∈ A0, b ∈ B0.
Next we fix a finite covering (A0 ∩ B(am, r))
m=1 of A0 and (B0 ∩ B(bn, r))
n=1 of B0,
where (am)
m=1 ⊂ A0 and (bn)
n=1 ⊂ B0.
We divide the proof into two steps.
Step 1: Fix an open set G
⋐ G. Then there exists r1: 0 < r1 < r0 with the following
property: for every a ∈ A0 there exist an open subset Aa of D and a mapping
f̂ = f̂a ∈ O
Gbn,δ0,r0
such that
f̂(z, w) = f̂a,bn(z, w), (z, w) ∈ (Aa ∩Da,δ0,r0)×Gbn,δ0,r0, n = 1, . . . , N ;
and that Aa is of the form {z ∈ D∩B(a, r1) : ω(z, A0∩B(a, r1), D∩B(a, r1)) < δa}
for some 0 < δa < δ0.
Proof of Step 1. Fix an arbitrary point a0 ∈ A0. First we claim that there are a
sufficiently small number r1 : 0 < r1 < r0 and a finite number of open subsets
n=1 of G with the following properties:
(a) V1 = Gb1,δ0,r0 and (Gbn,δ0,r0)
⊂ (Vn)
n=1 (see the notation in (8.1));
(b) f |(A0∩B(a,r1))×Vn is bounded, n = 1, . . . , N0;
(c) G
A UNIFIED APPROACH 39
(d) Vn ∩ Vn+1 6= ∅, n = 1, . . . , N0 − 1.
Indeed, we first start with the test r1 := r0 and N0 := N and (Vn)
n=1 :=
(Gbn,δ0)
. In virtue of (8.2) we see that our choice satisfies (a)–(b). If (c)–(d)
are satisfied then we are done. Otherwise, we will make the following procedure.
Fix a point w0 ∈ G
. For n = 1, . . . , N, let γn : [0, 1] → G be a continuous
one-to-one map such that
γn(0) = w0 and γn(1) ∈ Gbn,δ0,r0.
Since f is locally bounded, there exist sufficiently small numbers r1, s : 0 < r1 ≤
r0 and 0 < s such that f |(A0∩B(a,r1))×B(w,s) is bounded for all a ∈ A0 and w ∈
γn([0, 1]). Therefore, we may add to the starting collection (Vn)
n=1 some balls
of the form B(w, s), where w ∈ G
γn([0, 1]), and the new collection (Vn)
still satisfies (a)–(b). Now it remains to show that by adding a finite number of
suitable balls B(w, s), (c)–(d) are also satisfied. But this assertion follows from an
almost obvious geometric argument. In fact, we may renumber the collection (Vn)
if necessary. Hence, the above claim has been shown.
Using (c)–(d) above we may fix open sets Un ⋐ Vn for n = 1, . . . , N0, such that
(8.5) G
Un and Un ∩ Un−1 6= ∅, 1 < n ≤ N0.
In what follows we will find the desired set Aa0 and the desired holomorphic mapping
f̂ after N0 steps. Namely, after the n-th step (1 ≤ n ≤ N0), we construct an
open subset An of D in the form Da0,δn,r1 for a suitable δn > 0, and a mapping
f̂n ∈ O
. Finally, we obtain Aa0 := AN0 and f̂ := f̂N0. Now we
carry out this construction.
In the first step, using (8.1), (8.3), (8.4) and (a), we define
δ1 := δ0, A1 := Da0,δ1,r1 and f̂1(z, w) := f̂a0,b1(z, w), (z, w) ∈ A1 × U1.
Suppose that we have constructed an open subset An−1 of D and a mapping f̂n−1 ∈
An−1 ×
( n−1⋃
for some n : 2 ≤ n ≤ N0. We wish to construct an open
subset An of D and a mapping f̂n ∈ O
. There are two cases to
consider.
Case Vn = Gbm,δ0 for some 1 ≤ m ≤ N.
In this case let δn := δn−1 and An := An−1 = Da0,δn−1,r1, and
f̂n :=
f̂n−1, on An ×
( n−1⋃
f̂a0,bm, on An × Un.
40 VIÊT-ANH NGUYÊN
Case Vn 6∈
Gbm,δ0
By (8.5) fix a nonempty open set K ⋐ Un ∩ Un−1. Then by the induction, f̂n−1 ∈
O (An−1 ×K,Z) . Recall from (b) that f : (A0 ∩ B(a0, r1))× Vn −→ Z is bounded.
Since f is locally bounded, by decreasing r1 > 0 (if necessary) we may assume that
g := f |
X(A0∩B(a0,r1),K;D∩B(a0,r1),Vn)
is bounded. Applying Theorem 7.4 to g, we obtain
ĝ ∈ O
X̂(A0 ∩ B(a0, r1), K;D ∩ B(a0, r1), Vn), Z
which extends g. Since Un ⋐ Vn, we may choose δn such that 0 < δn < 1 −
ω(w,K, Vn). Using this and (8.1), it follows that
Da0,δn,r1 × Un ⊂ X̂(A0 ∩ B(a0, r1), K;D ∩ B(a0, r1), Vn).
Therefore, let An := Da0,δn,r1 and define
f̂n :=
f̂n−1, on An ×
( n−1⋃
ĝ, on An × Un.
This completes our construction in the n-step. Finally, we put Aa0 := AN0 and
f̂a0 := f̂N0 . Using this and (8.3) and (8.5) and (a), the desired conclusion of Step 1
follows. �
Step 2: Completion of the proof.
Proof of Step 2. Fix a sequence of relatively compact open subsets (D
k=1 of D
(resp. (G
k=1 of G) such that D
k ր D and G
k ր G as k ր ∞. Put
(8.6) Dk := D
Dam,δ0,r0, Gk := G
Gbn,δ0,r0, k ≥ 1.
Using the result of Step 1, we may find, for every k, a number 0 < rk < r0 with the
following properties:
• for every a ∈ A0, there is 0 < δa,k < δ0 such that by considering the open set
Aa,k := {z ∈ D ∩ B(a, rk) : ω (z, A0 ∩ B(a, rk), D ∩ B(a, rk)) < δa,k}
one can find a mapping f̂a,k ∈ O (Aa,k ×Gk, Z) satisfying
(8.7) f̂a,k = f̂a,bn on (Aa,k ∩Da,δ0,rk)×Gbn,δ0,rk , n = 1, . . . , N ;
• for every b ∈ B, there is 0 < δb,k < δ0 such that by considering the open set
Bb,k := {w ∈ G ∩ B(b, rk) : ω (z, B0 ∩ B(b, rk), G ∩ B(b, rk)) < δb,k}
one can find a mapping f̂b,k ∈ O (Dk ×Bb,k, Z) satisfying
(8.8) f̂b,k = f̂am,b on Dam,δ0,rk × (Bb,k ∩Gb,δ0,rk), m = 1, . . . ,M.
A UNIFIED APPROACH 41
Next using the compactness of A0 and B0, one may find, for every k, two fi-
nite coverings (A0 ∩ B(a
m, rk))
of A0 and (B0 ∩ B(bn′ , rk))
of B0, where
(am′ )
⊂ A0 and (bn′ )
⊂ B0. Put
(8.9) Ak :=
′ ,k and Bk :=
′ ,k, k ≥ 1.
In virtue of (8.6)–(8.9) and (8.2)–(8.4), the family (f̂a
′ ,k)
is col-
lective for every k ≥ 1. Let
(8.10) f̂k ∈ O
X(Ak, Bk;Dk, Gk), Z
denote the collected mapping of this family.
Next, we show that
(8.11)
ω(z, A0, Dk) = ω(z, A0, D) and lim
ω(w,B0, Gk) = ω(z, B0, G), z ∈ D, w ∈ G.
It is sufficient to prove the first identity in (8.11) since the proof of the second one
is similar. Observe that there is u ∈ PSH(D) such that ω(·, A0, Dk) ց u as k ր ∞
and u ≥ ω(·, A0, D) on D. Therefore, the proof of (8.11) will be complete if one can
show that u ≤ ω(·, A0, D) on D.
To this end observe that for every a ∈ A0 there is 1 ≤ m ≤ M such that
a ∈ B(am, r0). Consequently, using (8.6),
(A− lim sup u)(a) ≤
A− lim supω(·, A0 ∩ B(am, r0), Dam,δ0,r0)
(a) = 0,
where the equality follows from an application of Proposition 3.5. This, combined
with the obvious inequality u ≤ 1, implies that u ≤ ω(·, A0, D). Hence, (8.11)
follows.
We are now in the position to define the desired extension mapping f̂ . Indeed,
one glues
given in (8.10) together to obtain f̂ in the following way
f̂ := lim
f̂k on Ŵ0.
One needs to check that the last limit exists and possesses all the required properties.
In virtue of (8.7)–(8.11), and the Gluing Lemma 7.6, the proof will be complete if
we can show the following
Claim. For every (z0, w0) ∈ Ŵ0, there are an open neighborhood U × V of (z0, w0)
and δ0 > 0 such that the hypotheses of Lemma 7.6 is fulfilled with
D := D, G := G, Pk := Ak, Qk := Bk, Dk := Dk, Gk := Gk, k ≥ 1.
To this end let
δ0 :=
1− ω(z0, A0, D)− ω(w0, B0, G)
and let U × V be an open neighborhood of (z0, w0) such that
ω(z, A0, D) + ω(w,B0, G) < ω(z0, A0, D) + ω(w0, B0, G) + δ0.
42 VIÊT-ANH NGUYÊN
Then using these inequalities and (8.11), we see that there is a sufficiently big q0 ∈ N
such that for q0 ≤ q ≤ p and (z, w) ∈ U × V,
ω(z, Ap, Dq) + ω(w,Bp, Dq) ≤ ω(z, A0, Dq) + ω(w,B0, Gq)
≤ ω(z, A0, D) + ω(w,B0, G) + δ0 < 1.
This proves the above claim. Hence, the proof of the theorem is finished. �
Now we are able to formulate the following semi-local result.
Theorem 8.3. Let D be an open subset of a complex manifold and G ⊂ Cm a
bounded open set and Z a complex analytic space possessing the Hartogs extension
property. D (resp. G) is equipped with the canonical system of approach regions
(resp. the system of approach regions
Aβ(η)
η∈G, α∈Iη
). Let A be an open subset of
D and let B, B0 be subsets of G such that B0 is locally pluriregular and B0 ⊂ B
W := X(A,B;D,G) and W0 := X(A,B0;D,G).
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along D × (B ∩ ∂G),
there exists a unique mapping f̂ ∈ O(Ŵ0, Z) which admits A-limit f(ζ, η) at all
points (ζ, η) ∈ W0.
Proof. First, applying Part 1) of Theorem 7.2 and using the hypotheses on f, we
see that f extends to a mapping (still denoted by) f defined on X(A,B ∪B∗;D,G)
such that f is separately holomorphic on Xo(A,B ∪B∗;D,G) and that f |X(A,B∗;D,G)
is locally bounded.
We define f̂ at a point (z0, w0) ∈ Ŵ0 as follows: Let ǫ > 0 be such that
(8.12) ω(z0, A,D) + ω(w0, B0, G) + ǫ < 1.
By Theorem 3.1 and Proposition 3.4, there is a holomorphic disc φ ∈ O(E,D) such
that φ(0) = z0 and
(8.13) 1−
·mes(φ−1(A) ∩ ∂E) < ω(z0, A,D) + ǫ.
Moreover, using the hypotheses, we see that the mapping fφ, defined by
(8.14) fφ(t, w) := f(φ(t), w), (t, w) ∈ X
φ−1(A) ∩ ∂E,B;E,G
satisfies the hypotheses of Theorem 8.2. By this theorem, let f̂φ be the unique
mapping in X̂ (φ−1(A) ∩ ∂E,B0;E,G) such that
(8.15) (A− lim f̂φ)(t, w) = fφ(t, w), (t, w) ∈ X
φ−1(A) ∩ ∂E,B0;E,G
In virtue of (8.12)–(8.13), (0, w0) ∈ X̂ (φ
−1(A) ∩ ∂E,B0;E,G) . Then the value at
(z0, w0) of the desired extension mapping f̂ is given by
f̂(z0, w0) := f̂φ(0, w0).
A UNIFIED APPROACH 43
Using this and (8.14)–(8.15), and arguing as in Part 2) of Lemma 4.5, one can show
that f̂ is well-defined on Ŵ0.
To show that f̂ is holomorphic, one argues as in Step 1 of the proof of Theorem
6.1. To show that f̂ admits A-limit f(ζ, η) at all points (ζ, η) ∈ W0 and that it is
uniquely defined, one proceeds as in Step 2–4 of the proof of Theorem 6.1 making
the obviously necessary changes and adaptations. Hence, the proof is finished. �
9. The proof of Theorem A
First we need a variant of Definition 2.3. For a set A ⊂ D, Let Ẽ(A) be the set of
all elements P ∈ E(A) with the property that there is an open neighborhood U ⊂ X
of P such that U is biholomorphic to a domain in some Cn. Then it can be checked
(9.1) Ã :=
P∈eE(A)
This identity will allow us to pass from “local informations” to “global extensions”.
For the proof we need to develop some preparatory results.
In virtue of (9.1), for any P ∈ Ẽ(A) (resp. Q ∈ Ẽ(B)) fix an open neighborhood
UP of P (resp. VQ of Q) such that UP (resp. VQ) is biholomorphic to a domain in
dP (resp. in CdQ), where dP (resp. dQ) is the dimension of D (resp. G) at points
of P (resp. Q). For any 0 < δ ≤ 1
define
UP,δ := {z ∈ UP : ω(z, P, UP ) < δ} , P ∈ Ẽ(A),
VQ,δ := {w ∈ VQ : ω(w,Q, VQ) < δ} , Q ∈ Ẽ(B),
Aδ :=
P∈eE(A)
UP,δ, Bδ :=
Q∈eE(B)
VQ,δ,
Dδ := {z ∈ D : ω̃(z, A,D) < 1− δ} , Gδ := {w ∈ G : ω̃(w,B,G) < 1− δ} .
(9.2)
Lemma 9.1. We keep the above notation. Then:
(1) For every ζ ∈ Ã and α ∈ Iζ, there is an open neighborhood U of ζ such that
U ∩ Aα(ζ) ⊂ Aδ.
(2) Aδ is an open subset of D and Aδ ⊂ D1−δ ⊂ Dδ.
(3) ω̃(z, A,D)− δ ≤ ω(z, Aδ, D) ≤ ω̃(z, A,D), z ∈ D.
Proof of Lemma 9.1. To prove Part (1) fix, in view of (9.1)–(9.2), P ∈ Ẽ(A),
ζ ∈ P and α ∈ Iζ . Using the definition of local pluriregularity, we see that
lim sup
z→ζ, z∈Aα(ζ)
ω(z, P, UP ) = 0. Hence, Part (1) follows.
The assertion that Aδ is open follows immediately from (9.2). Since 0 < δ ≤
the second inclusion in Part (2) is clear. To prove the first inclusion let z be an
arbitrary point of Aδ. Then there is P ∈ Ẽ(A) such that z ∈ UP,δ. Using (9.2) and
Definition 2.3 we obtain
(9.3) ω̃(z, A,D) = ω(z, Ã, D) ≤ ω(z, P, UP ) < δ.
44 VIÊT-ANH NGUYÊN
Hence, z ∈ D1−δ, which in turn implies that Aδ ⊂ D1−δ.
It follows from Part (1) that
ω(z, Aδ, D) ≤ ω(z, Ã, D) = ω̃(z, A,D), z ∈ D,
which proves the second estimate in Part (3). To complete the proof let P ∈ Ẽ(A)
and 0 < δ ≤ 1
. We deduce from (9.3) that ω̃(z, A,D) − δ ≤ 0 for z ∈ UP,δ. Hence,
by (9.2),
ω̃(z, A,D)− δ ≤ 0, z ∈ Aδ.
On the other hand, ω̃(z, A,D) − δ < 1, z ∈ D. Recall from Part (2) that Aδ is an
open subset of Dδ. Consequently, the first estimate of Part (3) follows. �
Now we are able to to prove Theorem A in the following special case.
Proposition 9.2. Let D be an open subset of a complex manifold and G a bounded
open subset of Cm and Z a complex analytic space possessing the Hartogs extension
property. D (resp. G) is equipped with a system of approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
). Let A be a subset of D, let B, B0 be subsets of G such
that B0 is locally pluriregular and B0 ⊂ B
∗. Put
W := X(A,B;D,G), W0 := X(A,B0;D,G), W̃
(D ∪ Ã)× B0
Ã× (G ∪ B0)
Ŵ o := {(z, w) ∈ D ×G : ω̃(z, A,D) + ω(w,B0, G) < 1} .
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along X
A ∩ ∂D,B ∩ ∂G;D,G
• f |A×B is continuous at all points of (A ∩ ∂D)× (B ∩ ∂G),
there exists a unique mapping f̂ ∈ O(Ŵ o, Z) which admits A-limit f(ζ, η) at all
points (ζ, η) ∈ W̃ o.
Proof of Proposition 9.2. First, applying Part 1) of Theorem 7.2 and using the
hypotheses on f, we see that f extends to a mapping (still denoted by f) defined on
X(A ∪ A∗, B ∪ B∗;D,G) such that f is separately holomorphic on Xo(A ∪ A∗, B ∪
B∗;D,G) and that f |X(A∗,B∗;D,G) is locally bounded.
For each P ∈ Ẽ(A), UP (resp. G) is biholomorphic to an open set in C
dP (resp.
in Cm). Consequently, the mapping fP := f |X(P ,B;UP ,G) satisfies the hypotheses of
Theorem 8.2. Hence, we obtain a unique mapping f̂P ∈ O
X̂ (P,B0;UP , G) , Z
(9.4) (A− lim f̂P )(z, w) = fP (z, w) = f(z, w), (z, w) ∈ X (P,B0;UP , G) .
Let 0 < δ ≤ 1
and G
δ := {w ∈ G : ω(w,B0, G) < 1 − δ}. We will show that the
family
f̂P |UP,δ×G
P∈eE(A)
is collective in the sense of Definition 8.1, where UP,δ is
given in (9.2).
A UNIFIED APPROACH 45
To prove this assertion let P1, P2 be arbitrary elements of Ẽ(A). By (9.4), we have
(9.5)
(A− lim f̂P1)(z, w) = f(z, w) = (A− lim f̂P2)(z, w), (z, w) ∈ (UP1 ∩ UP2)× B0.
The assertion is reduced to showing that
(9.6) f̂P1(z, w) = f̂P2(z, w), (z, w) ∈ X̂ (P1, B0;UP1, G) ∩ X̂ (P2, B0;UP2, G) .
To this end fix (z0, w0) ∈ X̂ (P1, B0;UP1, G) ∩ X̂ (P2, B0;UP2 , G) . Observe that both
mappings w 7→ f̂P1(z0, w) and w 7→ f̂P2(z0, w) belong to O(G, Z), where G is the
connected component which contains w0 of the following open set{
w ∈ G : ω(w,B0, G) < 1− max
j∈{1,2}
ω(z0, Pj, Uj)
Applying Theorem 7.5 to these mappings using (9.5), Proposition 3.5 and (9.6), the
above assertion follows.
In virtue of (9.2) let
(9.7)
fδ ∈ O(Aδ ×G
δ, Z)
denote the collected mapping of the family
f̂P |UP,δ×G
P∈eE(A)
. In virtue of (9.4)
and (9.7), we are able to define a new mapping f̃δ on X
Aδ, B;D,G
as follows
f̃δ :=
fδ, on Aδ ×G
f, on D × B.
Using this and (9.4)–(9.7), we see that
(9.8) A− lim f̃δ = f on X(A ∩ Ã, B0;D,G
Since Aδ is an open subset of X and G
δ is a bounded open set in C
m, we are able to
apply Theorem 8.3 to f̃δ in order to obtain a mapping f̂δ ∈ O
Aδ, B0;D,G
such that
(9.9) A− lim f̂δ = f̃δ on X(Aδ, B0;D,G
We are now in a position to define the desired extension mapping f̂ . Indeed, one
glues
0<δ≤ 1
together to obtain f̂ in the following way
f̂ := lim
on Ŵ o.
One needs to check that the last limit exists and possesses all the required properties.
In virtue of (9.8)–(9.9) and Lemma 7.6, the proof will be complete if one can show
that for every (z0, w0) ∈ Ŵ
o, there are an open neighborhood U × V of (z0, w0) and
δ0 > 0 such that hypothesis (iii) of Lemma 7.6 is fulfilled with
D := D, G := G, Pk := A 1
, Qk := B0, Dk := D, Gk := G
, k > 2.
To this end let
δ0 :=
1− ω̃(z0, A,D)− ω(w0, B0, G)
46 VIÊT-ANH NGUYÊN
and let U × V be an open neighborhood of (z0, w0) such that
ω̃(z, A,D) + ω(w,B0, G) < ω̃(z0, A,D) + ω(w0, B0, G) + δ0.
Then for k > 1
and for (z, w) ∈ U ×V, using the last inequality, and applying Part
(3) of Lemma 9.1 and Proposition 3.5, we see that
ω̃(z, A 1
, D) + ω(w,B0, G
) ≤ ω̃(z, A,D) +
ω(w,B0, G)
1− δ0
ω̃(z, A,D) + ω(w,B0, G)
1− δ0
This proves the above assertion. Hence, the proof of the proposition is finished. �
We now arrive at
Proof of Theorem A. First, applying Part 1) of Theorem 7.2 and using the
hypotheses on f, we see that f extends to a mapping (still denoted by) f defined on
X(A ∪ A∗, B ∪ B∗;D,G) such that f is separately holomorphic on Xo(A ∪ A∗, B ∪
B∗;D,G) and that f |X(A∗,B∗;D,G) is locally bounded.
For each P ∈ Ẽ(A), UP is biholomorphic to an open set in C
dP . Consequently, the
mapping fP := f |X(P,B;UP ,G) satisfies the hypotheses of Proposition 9.2. Hence, we
obtain a unique mapping f̂P ∈ O
o (P,B;UP , G) , Z
11 such that
(9.10) (A− lim f̂P )(z, w) = f(z, w), (z, w) ∈ X
P, B̃ ∩B;UP , G
Let 0 < δ ≤ 1
. Using (9.10) and arguing as in the proof of Proposition 9.2, we
may collect the family
f̂P |UP,δ×Gδ
P∈eE(A)
in order to obtain the collected mapping
f̃Aδ ∈ O(Aδ ×Gδ, Z).
Similarly, for each Q ∈ Ẽ(B), one obtains a unique mapping f̂Q ∈
o (A,Q;D, VQ) , Z
12 such that
(9.11) (A− lim f̂Q)(z, w) = f(z, w), (z, w) ∈ X
A ∩ Ã, Q;D, VQ
Moreover, one can collect the family
f̂Q|Dδ×VQ,δ
Q∈eE(B)
in order to obtain the col-
lected mapping f̃Bδ ∈ O(Dδ × Bδ, Z).
Next, we prove that
(9.12) f̃Aδ = f̃
δ on Aδ × Bδ.
Indeed, in virtue of (9.10)–(9.11) it suffices to show that for any P ∈ Ẽ(A) and
Q ∈ Ẽ(B) and any 0 < δ ≤ 1
(9.13) f̂P (z, w) = f̂Q(z, w), (z, w) ∈ UP,δ × VQ,δ.
Observe that in virtue of (9.10)–(9.11) one has that
(A− lim f̂P )(z, w) = (A− lim f̂Q)(z, w) = f(z, w), (z, w) ∈ X (P,Q;UP , VQ) .
11 Here X̂o (P,B;UP , G) := {(z, w) ∈ UP ×G : ω(z, P, UP ) + ω̃(w,B,G) < 1} .
12 Here X̂o (A,Q;D,VQ) := {(z, w) ∈ D × VQ : ω̃(z, A,D) + ω(w,Q, VQ) < 1} .
A UNIFIED APPROACH 47
Recall that UP (resp. VQ) is biholomorphic to a domain in C
dP (resp. CdQ). Con-
sequently, applying the uniqueness of Theorem 8.2 yields that
f̂P (z, w) = f̂Q(z, w), (z, w) ∈ X̂ (P,Q;UP , VQ) .
Hence, the proof of (9.13) and then the proof of (9.12) are finished.
In virtue of (9.12), we are able to define a new mapping f̃δ :
o (Aδ, Bδ;Dδ, Gδ) −→ Z as follows
(9.14) f̃δ :=
f̃Aδ , on Aδ ×Gδ,
f̃Bδ , on Dδ ×Bδ.
Using formula (9.14) it can be readily checked that f̃δ ∈ Os
o (Aδ, Bδ;Dδ, Gδ) , Z
Since we know from Part (2) of Lemma 9.1 that Aδ (resp. Bδ) is an open subset
of Dδ (resp. Gδ), we are able to apply Theorem 7.3 to f̃δ for every 0 < δ ≤
Consequently, one obtains a unique mapping f̂δ ∈ O
X̂ (Aδ, Bδ;Dδ, Gδ) , Z
(9.15) f̂δ = f̃δ on X
o (Aδ, Bδ;Dδ, Gδ) .
It follows from (9.10)–(9.11) and (9.14)–(9.15) that
(9.16) A− lim f̂δ = f on X
A ∩ Ã, B ∩ B̃;Dδ, Gδ
In addition, for any 0 < δ ≤ δ0 ≤
, and any (z, w) ∈ Aδ × Bδ, there is P ∈ Ẽ(A)
such that z ∈ UP,δ0. Therefore, it follows from the construction of f̃
δ , (9.14) and
(9.15) that
f̂δ(z, w) = f̂P (z, w) = f̂δ0(z, w).
This proves that f̂δ = f̂δ0 on Aδ × Bδ for 0 < δ ≤ δ0 ≤
. Hence,
(9.17) f̂δ = f̂δ0 on X(Aδ, Bδ;Dδ0, Gδ0), 0 < δ ≤ δ0 ≤
We are now in a position to define the desired extension mapping f̂ .
f̂ := lim
To prove that f̂ satisfies the desired conclusion of the theorem one proceeds as in
the end of the proof of Proposition 9.2. In virtue of (9.16)–(9.17) and Lemma 7.6,
the proof will be complete if we can verify that for every (z0, w0) ∈ Ŵ , there are an
open neighborhood U×V of (z0, w0) and δ0 > 0 such that hypothesis (iii) of Lemma
7.6 is fulfilled with
D := D, G := G, Pk := A 1
, Qk := B 1
, Dk := D 1
, Gk := G 1
, k > 2.
Since the verification follows along almost the same lines as that of Proposition 9.2,
it is, therefore, left to the interested reader.
Hence, the proof of Theorem A is finished. �
48 VIÊT-ANH NGUYÊN
10. Applications
In this section we give various applications of Theorem A using different systems
of approach regions defined in Subsection 2.2.
10.1. Canonical system of approach regions. For every open subset U ⊂ R2n−1
and every continuous function h : U −→ R, the graph
z = (z
, zn) = (z
, xn + iyn) ∈ C
n : (z
, xn) ∈ U and yn = h(z
, xn)
is called a topological hypersurface in Cn.
Let X be a complex manifold of dimension n. A subset A ⊂ X is said to be a
topological hypersurface if, for every point a ∈ A, there is a local chart (U, φ : U →
n) around a such that φ(A ∩ U) is a topological hypersurface in Cn
Now let D ⊂ X be an open subset and let A ⊂ ∂D be an open subset (with
respect to the topology induced on ∂D). Suppose in addition that A is a topological
hypersurface. A point a ∈ A is said to be of type 1 (with respect to D) if, for every
neighborhood U of a there is an open neighborhood V of a such that V ⊂ U and
V ∩D is a domain. Otherwise, a is said to be of type 2. We see easily that if a is of
type 2, then for every neighborhood U of a, there are an open neighborhood V of a
and two domains V1, V2 such that V ⊂ U, V ∩D = V1 ∪ V2 and all points in A ∩ V
are of type 1 with respect to V1 and V2.
In virtue of Proposition 3.7 in [35] we have the following
Proposition 10.1. Let X be a complex manifold and D an open subset of X. D
is equipped with the canonical system of approach regions. Suppose that A ⊂ ∂D is
an open boundary subset which is also a topological hypersurface. Then A is locally
pluriregular and A ⊂ Ã.
This, combined with Theorem A, implies the following result.
Theorem 10.2. Let X, Y be two complex manifolds, and D ⊂ X, G ⊂ Y two
nonempty open sets. D (resp. G) is equipped with the canonical system of approach
regions. Let A (resp. B) be a nonempty open subset of ∂D (resp. ∂G) which is also
a topological hypersurface. Let Z be a complex analytic space possessing the Hartogs
extension property. Define
W := X(A,B;D,G),
Ŵ := {(z, w) ∈ D ×G : ω(z, A,D) + ω(w,B,G) < 1} .
Let f : W −→ Z be such that:
(i) f ∈ Cs(W,Z) ∩ Os(W
o, Z);
(ii) f is locally bounded on W ;
(iii) f |A×B is continuous.
Then there exists a unique mapping f̂ ∈ O(Ŵ ) such that
cW∋(z,w)→(ζ,η)
f̂(z, w) = f(ζ, η), (ζ, η) ∈ W.
A UNIFIED APPROACH 49
If, moreover, Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,w)
A×B |f |
ω(z,w)
W , (z, w) ∈ Ŵ .
The special case where Z = C has been proved in [35].
10.2. System of angular approach regions. We will use the terminology and
the notation in Paragraph 3 of Subsection 2.2. More precisely, if D is an open set
of a Riemann surface such that D is good on a nonempty part of ∂D, we equip D
with the system of angular approach regions supported on this part. Moreover, the
notions such as set of positive length, set of zero length, locally pluriregular point
which exist on ∂E can be transferred to ∂D using conformal mappings in a local
way (see [34] for more details).
Theorem 10.3. Let X, Y be Riemann surfaces and D ⊂ X, G ⊂ Y open subsets
and A (resp. B) a subset of ∂D (resp. ∂G) such that D (resp. G) is good on A
(resp. B) and that both A and B are of positive length. Let Z be a complex analytic
space possessing the Hartogs extension property. Define
W := X(A,B;D,G), W
:= X(A
;D,G),
Ŵ := {(z, w) ∈ D ×G : ω(z, A,D) + ω(w,B,G) < 1} ,
(z, w) ∈ D ×G : ω(z, A
, D) + ω(w,B
, G) < 1
where A
(resp. B
) is the set of points at which A (resp. B) is locally pluriregular
with respect to the system of angular approach regions supported on A (resp. B),
and ω(·, A,D), ω(·, A
, D) (resp. ω(·, B,G), ω(·, B
, G)) are calculated using the
canonical system of approach regions.
Then for every mapping f : W −→ Z which satisfies the following conditions:
(i) f ∈ Cs(W,Z) ∩ Os(W
o, Z);
(ii) f is locally bounded;
(iii) f |A×B is continuous,
there exists a unique mapping f̂ ∈ O(Ŵ
, Z) which admits the angular limit f at all
points of W ∩W
If A and B are Borel sets or if X = Y = C then Ŵ = Ŵ
If Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,A
,D)−ω(w,B
A×B |f |
ω(z,A
,D)+ω(w,B
W , (z, w) ∈ Ŵ
Theorem 10.3 generalizes, in some sense, the result of [34].
In the above theorem we have used the equality
W = Ŵ
when either A and B
are Borel sets or X = Y = C. This follows from the identity ω(·, A,D) = ω̃(·, A,D)
when either A is a Borel set or D ⊂ C (see Theorem 4.6 in [34]). On the other hand,
we can sharpen Theorem 10.3 further, namely, hypothesis (i) can be replaced by a
weaker hypothesis (i’) as follows:
50 VIÊT-ANH NGUYÊN
(i’) for any a ∈ A the mapping f(a, ·)|G is holomorphic and has angular limit
f(a, b) at all points b ∈ B, and for any b ∈ B the mapping f(·, b)|D is
holomorphic and has angular limit f(a, b) at all points a ∈ A.
To see this it suffices to observe that the hypotheses of Theorem 3.8 and Theorem
6.1 can be weakened considerably when the bounded open set D therein is just
one-dimensional.
10.3. System of conical approach regions. The remaining part of this section
is devoted to two important applications of Theorem A: a boundary cross theorem
and a mixed cross theorem. In order to formulate them, we need to introduce some
terminology and notation.
Let X be an arbitrary complex manifold and D ⊂ X an open subset. We say
that a set A ⊂ ∂D is locally contained in a generating manifold if there exist an
(at most countable) index set J 6= ∅, a family of open subsets (Uj)j∈J of X and
a family of generating manifolds13 (Mj)j∈J such that A ∩ Uj ⊂ Mj, j ∈ J, and
that A ⊂
j∈J Uj . The dimensions of Mj may vary according to j ∈ J. Given a
set A ⊂ ∂D which is locally contained in a generating manifold, we say that A is
of positive size if under the above notation
j∈J mesMj (A∩ Uj) > 0, where mesMj
denotes the Lebesgue measure on Mj. A point a ∈ A is said to be a density point
of A if it is a density point of A ∩ Uj on Mj for some j ∈ J. Denote by A
the set
of density points of A.
Suppose now that A ⊂ ∂D is of positive size. We equip D with the system
of conical approach regions supported on A. Using the work of B. Jöricke (see,
for example, Theorem 3, pages 44–45 in [15]), one can show that14 A is locally
pluriregular at all density points of A. Observe that mesMj
(A \ A
) ∩ Uj
= 0 for
j ∈ J. Therefore, it is not difficult to show that A
is locally pluriregular. Choose
an increasing sequence (An)
n=1 of subsets of A such that An ∩ Uj is closed and
mesMj
An) ∩ Uj
= 0 for j ∈ J. Observe that A
n is locally pluriregular,
n ∩ Uj ⊂ A for j ∈ J and that Â :=
n is locally pluriregular and that Â is
locally pluriregular at all points of A
. Consequently, it follows from Definition 2.3
ω̃(z, A,D) ≤ ω(z, A
, D), z ∈ D.
This estimate, combined with Theorem A, implies the following result which is a
generalization in higher dimensions of Theorem 10.3.
Theorem 10.4. Let X, Y be two complex manifolds, let D ⊂ X, G ⊂ Y be two
open sets, and let A (resp. B) be a subset of ∂D (resp. ∂G). D (resp. G) is equipped
with a system of conical approach regions
Aα(ζ)
ζ∈D, α∈Iζ
(resp.
Aβ(η)
η∈G, β∈Iη
13 A differentiable submanifold M of a complex manifold X is said to be a generating manifold
if for all ζ ∈ M, every complex vector subspace of TζX containing TζM coincides with TζX.
14 A complete proof will be available in [29].
A UNIFIED APPROACH 51
supported on A (resp. on B). Suppose in addition that A and B are of positive size.
Let Z be a complex analytic space possessing the Hartogs extension property. Define
:= X(A
;D,G),
(z, w) ∈ D ×G : ω(z, A
, D) + ω(w,B
, G) < 1
where A
(resp. B
) is the set of density points of A (resp. B).
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded;
• f |A×Bis continuous,
there exists a unique mapping f̂ ∈ O(Ŵ
, Z) which admits A-limit f(ζ, η) at every
point (ζ, η) ∈ W ∩W
If, moreover, Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,A
,D)−ω(w,B
A×B |f |
ω(z,A
,D)+ω(w,B
W , (z, w) ∈ Ŵ
The second application is a very general mixed cross theorem.
Theorem 10.5. Let X, Y be two complex manifolds, let D ⊂ X, G ⊂ Y be open
sets, let A be a subset of ∂D, and let B be a subset of G. D is equipped with
the system of conical approach regions
Aα(ζ)
ζ∈D, α∈Iζ
supported on A and G is
equipped with the canonical system of approach regions
Aβ(η)
η∈G, β∈Iη
. Suppose
in addition that A is of positive size. Let Z be a complex analytic space possessing
the Hartogs extension property. Define
:= X(A
, B∗;D,G),
(z, w) ∈ D ×G : ω(z, A
, D) + ω(w,B∗, G) < 1
where A
is the set of density points of A and B∗ denotes, as usual (see Subsection
2.1 above), the set of points in B ∩G at which B is locally pluriregular.
Then, for every mapping f : W −→ Z which satisfies the following conditions:
• f ∈ Cs(W,Z) ∩ Os(W
o, Z);
• f is locally bounded along A×G,
there exists a unique mapping f̂ ∈ O(Ŵ
, Z) which admits A-limit f(ζ, η) at every
point (ζ, η) ∈ W ∩W
If, moreover, Z = C and |f |W <∞, then
|f̂(z, w)| ≤ |f |
1−ω(z,A
,D)−ω(w,B∗,G)
A×B |f |
ω(z,A
,D)+ω(w,B∗,G)
W , (z, w) ∈ Ŵ
Concluding remarks. In ongoing joint-works with Pflug [30, 31] we develop new
cross theorems with singularities. On the other hand, in [36] the problem of opti-
mality of the envelope of holomorphy
W in Theorem A has been investigated.
52 VIÊT-ANH NGUYÊN
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http://arxiv.org/abs/0705.4649
54 VIÊT-ANH NGUYÊN
Viêt-Anh Nguyên, Mathematics Section, The Abdus Salam international centre
for theoretical physics, Strada costiera, 11, 34014 Trieste, Italy
E-mail address : vnguyen0@ictp.trieste.it
	1. Introduction
	2. Preliminaries and statement of the main result
	2.1. Approach regions, local pluripolarity and plurisubharmonic measure
	2.2. Examples of systems of approach regions
	2.3. Cross and separate holomorphicity and A-limit.
	2.4. Hartogs extension property.
	2.5. Statement of the main results
	3. Holomorphic discs and a Two-Constant Theorem
	3.1. Poletsky theory of discs and Rosay Theorem on holomorphic discs
	3.2. Level sets of the relative extremal functions and a Two-Constant Theorem
	3.3. Construction of discs
	4. A mixed cross theorem
	5. Completion of the proof of Theorem 4.2
	6. A local version of Theorem A
	7. Preparatory results
	8. Local and semi-local versions of Theorem A
	9. The proof of Theorem A
	10. Applications
	10.1. Canonical system of approach regions
	10.2. System of angular approach regions
	10.3. System of conical approach regions
	References
