RAPID CONVERGENCE OF RATIONAL FUNCTIONS AND PLURIPOLAR HULLS

ABSTRACT. Let f be a holomorphic function on a domain D ⊂ C n . Using the method of rapidly convergence, we study pluripolar hull in C n+1 of Γf(D) := {(z, f(z) : z ∈ D}. The first main result of this work (Theorem 3.1) states that if D is an analytic polyhedron then there exists a holomorphic function f such that Γf(D) is complete pluripolar in C n+1 . Proposition 4.2 gives a description of (Γf(D))∗ Cn+1 in case f can be rapidly approximated in capacity by a sequence of rational functions. In the course of our research, we obtain in Theorem 4.1 an improvement of an earlier result of Bloom with a simpler proof

RAPID CONVERGENCE OF RATIONAL FUNCTIONS AND PLURIPOLAR

HULLS NGUYEN QUANG DIEU AND PHUNG VAN MANH

A BSTRACT Let f be a holomorphic function on a domain D ⊂ C n Using the method of rapidly

convergence, we study pluripolar hull in Cn+1of Γf (D) := {(z, f (z) : z ∈ D} The first main

result of this work (Theorem 3.1) states that if D is an analytic polyhedron then there exists a

holomorphic function f such thatΓf (D) is complete pluripolar inCn+1 Proposition 4.2 gives a

description of ( Γf (D)) ∗

Cn+1 in case f can be rapidly approximated in capacity by a sequence of

rational functions In the course of our research, we obtain in Theorem 4.1 an improvement of

an earlier result of Bloom with a simpler proof.

I Introduction

One of traditional problems in complex analysis is the question of holomorphic propagation

e.g., find the maximal holomorphic object containing a given one For example, let f be a holomorphic function defined on a domain D inCnwe search for its holomorphic continuation

on a larger domain A natural counterpart of holomorphic propagation in pluripotential theory

is the theory of pluripolar hull We now recall briefly some elements of pluripotential theory

leading to the concept of pluripolar hull An upper semicontinuous function u, u ̸≡ −∞, defined

on a domain D ⊂ C n is said to be plurisubharmonic if the restriction of u on every complex line is subharmonic function as a function of one variable A subset E of D is said to be pluripolar if locally E is included in the −∞ locus of plurisubharmonic functions According

to a well-known result of Josefson, every pluripolar set E is contained in the singular locus of some global plurisubharmonic function u onCn Given a pluripolar subset E of D, following Poletsky and Levenberg in [LP] we define the pluripolar hull of E relative to D as follows

E ∗

D:={z ∈ D : ∀u ∈ PSH(D),u|E ≡ −∞ ⇒ u(z) = −∞}.

Here PSH(D) denotes the cone of plurisubharmonic functions on D It is clear that E ∗

D is

pluripolar It is also obvious that if E is complete pluripolar in D i.e., E coincides with the

−∞ locus of an element u ∈ PSH(D) then E ∗

D = E Conversely, Zeriahi proved in [Ze] that if

E ∗

D = E, E is Fσ and Gδ and if D is pseudoconvex then E must be complete pluripolar in D.

For a simple explicit example of pluripolar hull we can take E = {0} × {|z| = 1} ⊂ C2 and

D =C2 Then it is easy to check, using the fact that the circle|z| = 1 is non polar in C, that

the pluripolar hull E ∗

D={0} × C Nevertheless, given a pluripolar set E ⊂ D, it is quite hard

in general to determine E ∗

D A typical object in the study of pluripolar hull is the graphΓf (D)

of a holomorphic function f over a domain D ⊂ C n In the case where D =∆, the unit disk

in C, Levenberg, Martin and Poletsky in [LMP], continuing preceding work of Sadullaev in

[Sad], showed that if f (z) =∑k ≥0 a n(k) z n(k) is a gap series with radius of convergence equal

to 1 and with gaps satisfying limk→∞ n(k)/n(k + 1) = 0, then Γf (D) is complete pluripolar in

C2 A notable property of gap series emerging from the work [Sad] and [LMP] is the fact that

the partial sums f m (z) =∑0≤k≤m a n(k) z n(k) converges rapidly uniformly to f on compact sets

of∆ i.e., limm →∞ ∥ f − fm∥ 1/n(m)

K = 0 for every compact subset K in ∆ Later, using the rapidly

2000 Mathematics Subject Classification Primary 41A05, 41A63, 46A32.

Key words and phrases Pluripolar hull, Rapid convergence, complete pluripolar.

1

convergence method, Poletsky and Wiegerinck constructed in Theorem 3.6 of [PW], a Cantor

compact set K ⊂ C and a holomorphic function f on D := C \ K such that (Γ f (D)) ∗

C 2 is two

sheeted over D The function f is essentially constructed as a square root of a holomorphic function g which can be rapidly approximated by a sequence g m of rational functions whose zeros and poles are related to the endpoints of the intervals removed from the construction of

the Cantor set K.

Motivated by the previous work of Poletsky and Wiegerinck we consider the problem of describing (Γf (D)) ∗

Cn+1 with D is a domain inCn and f is holomorphic on D such that f is the

limit of a sequence{rm}m ≥1 of rational functions which converged rapidly to f on D, see the

next section for the definition of these types of convergence Here, we always assume that the

numerator and denominator of r m are polynomials of degree at most m Our first main result is

Theorem 3.1 which is a generalization of the mentioned above theorem of Levenberg, Martin

and Poletsky More precisely, for any connected analytic polyhedron D ⊂ C n we construct a

holormorphic function g on D such thatΓg (D) is complete pluripolar inCn+1 For the proof of

this result, we first deal with the special case where D =∆nis the unit polydisk in Cn Using

gap series, we construct in this special case a holomorphic function h on∆n continuous up to the boundary such thatΓh(∆n) is complete pluripolar inCn+1 Next we let g be the composition

of h with the analytic mapping that defines D as an analytic polyhedron Then by a theorem of

Coltoiu on the equivalence between closed locally complete pluripolar sets and closed globally complete pluripolar sets we conclude thatΓg (D) is complete pluripolar inCn+1 Finally, we invoke a version of Theorem 4.6 in [EW] to conclude that the pluripolar hull ofΓg (D) disjoints

from the cylinder∂D × C So by Zeriahi’s theorem we get complete pluripolarity of Γg (D) in

Cn+1

The next section deals with the situation when f is the limit of a sequence r m of rational

functions which converged rapidly to f on D We show in Proposition 4.2 that there exists

an ”exceptional” pluripolar set E ⊂ C n which can be somehow determined by the poles of r m

such that (Γf (D)) ∗

Cn+1 has exactly one sheet over D \ E Moreover, if the sequence {rm}m ≥1

converges as a point z0∈ E which may lie outside D then (Γ f (D)) ∗

Cn+1 contains at most one

point over z0 As an immediate step, we prove in Theorem 4.1 a result which is of independent interest, saying roughly that pointwise rapidly convergence on a non-pluripolar set implies

rapidly convergence in capacity on the whole domain D We must stress that this result has

been essentially proved by Bloom (see Theorem 2.1 in [Bl]) Our point is to offer a simpler

proof using compactness of PSH(D) in the topology L1loc (D) together with the

Chern-Levine-Nirenberg inequality instead of the more delicate comparison result of Alexander and Taylor between relative capacity and global capacity as done in [Bl] A natural question arises is

whether the exceptional set E appears in Proposition 4.2 can really occur It seems to be a hard problem and we are able to solve this in a very particular case when D is a planar domain and the set E is presumably at most countable We close this article by providing an explicit example of a planar domain D, a holomorphic function f on D and a sequence of rational

functions{rm}m ≥1 such that{rm}m ≥1 converges to f rapidly uniformly on compact sets of D

and{rm}m ≥1 goes rapidly pointwise on∂D except for a countable set to a bounded function.

The interesting question of determining precisely (Γf (D)) ∗

C 2 is unfortunately left open by our method

Acknowledgements This work has been done during a visit of the first named author at the Vietnam Institute for Advanced Mathematics in the winter of 2012 He wishes to thank this institution for financial support and the warm hospitality that he receives We also would like to thank Professor Pascal Thomas for a useful suggestion in the construction of Proposition 4.7

II Preliminaries

The following sort of compactness in PSH(D) will be useful in dealing with convergence of

holomorphic functions

Lemma 2.1 Let {um}m≥1 be a sequence of plurisubharmonic functions defined on a domain

D inCn If the sequence is uniformly bounded from above on compact subsets of D then either {um}m≥1 goes to −∞ uniformly on compacta or there exists a subsequence {um j } j≥1 converges

to a plurisubharmonic function u almost everywhere on D In the latter case lim sup

j →∞ um j = u outside a pluripolar subset of D In particular, in this case, the set {z ∈ D : limm→∞ u m (z) =

−∞} is pluripolar.

Proof It follows from Theorem 3.2.12 in [H¨o] that either {um}m ≥1 goes to−∞ uniformly on

compacta or there exists a subsequence{um j } j ≥1 converges to a plurisubharmonic function u

in L1loc (D) If the convergence in L loc1 occurs, then we can find a subsequence of{um j } j ≥1 that

converges almost everywhere on D to u The first assertion follows For the second one, we note that u = (lim sup

j→∞ u m j)

∗ everywhere on D Using a result of Bedford-Taylor which asserts

that the set (lim sup

j→∞ u m j)

∗ ̸= limsup j→∞ u m j is pluripolar, we conclude the lemma. 

Lemma 2.1’ Let {um}m≥1 be a sequence of pluriharmonic functions defined on a domain D

inCn If the sequence is uniformly bounded from above on compact subsets of D then either {um}m≥1 goes to −∞ uniformly on compacta or there exists a subsequence {um j } j≥1 converges locally uniformly to a pluriharmonic function u.

Proof Assume that {um}m≥1 does not go to−∞ uniformly on compacta, then by Lemma 2.1,

there exists a subsequence{um j } j≥1 tends to a subharmonic function u in L1loc Using Corollary

3.1.4 in [H¨o] we see that the convergence must be uniform on compact subsets in D. 

For a Borel subset E in a domain D ⊂ C n , following Bedford and Taylor (see [Kl] p.120) we

let cap(E, D) be the relative capacity of a Borel subset E in D which is defined as

cap(E, D) = sup {∫

E

(dd c u) n : u ∈ PSH(D),−1 < u < 0}.

It is well known that relative capacity enjoys some important properties such as sub-additivity

and monotone under increasing sequences Moreover, pluripolar subsets of D are exactly

sub-sets with vanishing relative capacity We recall the following types of convergence for of mea-surable functions which are essentially taken from [Bl]

Definition 2.2 Let { fm}m≥1 , f be Borel measurable functions defined on a domain D ⊂ C n

We say that the sequence { fm}m≥1

(i) converges rapidly pointwise to f on a Borel subset X ⊂ D if | f − fm| 1/m goes pointwise to 0

on X ;

(ii) converges in capacity to f on X if for everyε > 0 we have

lim

m →∞ cap(X m,ε, D) = 0,

where Xm,ε:={x ∈ X : | fm (x) − f (x)| >ε};

(iii) converges rapidly in capacity to f on X if | f − fm| 1/m converges in capacity to 0 on X ; (iv) converges rapidly in capacity to f on D if the property (iii) holds true for every compact subset X of D.

As in classical measure theory, we have the following relation between convergence in capacity and pointwise convergence

Proposition 2.3 Let { fm}m≥1 , f be Borel measurable functions defined on a domain D ⊂ C n

If { fm}m≥1 converges in capacity (resp rapidly in capacity) to f on a Borel subset X of D, then there exists a subsequence { fm j } j ≥1 and a pluripolar subset E ⊂ X such that { fm j } j ≥1 converges pointwise (resp rapidly pointwise) to f on X \ E.

Proof. Suppose that{ fm}m ≥1 is a sequence that converges in capacity to f on X For every

ε> 0, the hypothesis gives

lim

m →∞ cap(X m,ε, D) = 0,

where X m,ε :={x ∈ X : | fm (x) − f (x)| >ε} Hence, we can find a strictly increasing sequence {mk} such that

cap(X m,1

2k

, D) < 1

2k , ∀m ≥ mk

Let us set E j := ∪∞

k= j X m

k ,1

2k

and X := ∩∞

j=1 E j By the subadditive property of the relative capacity we have

cap(E, D) ≤ cap(Ej , D) ≤ ∑∞

k= j

cap(K m

k ,1

2k

, D) <

∞

∑

k= j

1

2k = 1

2j−1 , ∀ j ≥ 1.

It follows that cap(E, D) = 0 and hence E is a pluripolar set Now, for z ∈ X \ E, it is easy to

check using the definition of E j that limk →∞ f m k (z) = f (z) We are done Finally, if { fm}m≥1

converges rapidly in capacity to f on X then by setting f ′

m:=| fm − f | 1/m and f ′:= 0 we reduce

We should remark that there exists pointwise convergence sequence that contains no sub-sequence that converges in capacity Indeed, let {Am}m ≥1 be a sequence of pairwise disjoint subset of the unit disk∆ ⊂ C such that inf m ≥1 cap (A m , ∆) > 0 Then the sequence {χA m }m ≥1

provides the desired example

Next, we recall some major tools that will be used in the study of pluripolar hulls According

to Levenberg and Poletsky in [LP], the negative pluripolar hull E −

D of a pluripolar set E ⊂ D is

defined as follows

E −

D =∩{z ∈ D : u(z) = −∞,u ∈ PSH(D),u < 0,u|E =−∞}.

If D is bounded and hyperconvex i.e., D admits a negative continuous exhaustion function, then

by Theorem 2.4 in [LP] for every sequence{D j} of relatively compact domains in D such that

D j ↑ D we have

E −

D =∪ j≥1 (E ∩ D j)−

D j

Now for every subset E of D we define the pluriharmonic measure of E relative to D as follows

ω(z, E, D) = −sup{u(z) : u ∈ PSH(D),u ≤ 0 on D

and lim sup

D∋ ξ→w u(ξ)≤ −1 for w ∈ E}, z ∈ D.

The following connection between the pluriharmonic measure and negative pluripolar hull is again due to Levenberg and Poletsky (see [LP]),

E −

D ={z ∈ D :ω(z, E, D) > 0 }.

Using these tools together with some technical facts from classical potential theory, Edigarian and Wiegerinck in [EW] completely solve the problem of describing the pluripolar hull of

{(z, f (z)) : z ∈ D \ E} in D × C where E is a closed polar subset of a domain D in C and

f is holomorphic on D \ E However, up to now the case where E is non-polar seems to be

untractable by these methods At the end of this article, we give an example of a holomorphic

function f defined on a domain D ⊂ C where D = C \ A,A is a countable set with A \ A is

non-polar such that (Γf (D)) ∗

C 2 has at most one sheet over every point ofC except for a polar

subset of A Even in this concrete case, we are unable to compute exactly (Γf (D)) ∗

C 2

III Graphs of holomorphic functions over analytic polyhedron

The main result of this section is the following generalization of the mentioned above theorem due to Levenberg, Martin and Poletsky in [LMP] It is an open and interesting problem to see

if our theorem is still true for every pseudoconvex domain D in C n

Theorem 3.1 Let Ω be a domain in C n and D be a connected analytic polyhedron relatively compact in Ω that is defined by holomorphic functions on Ω Then there exists a holomorphic

function g on D such thatΓg (D) is complete pluripolar inCn+1

We first prove the following special case of Theorem 3.1 This is also a high dimension ana-logue of an earlier result due to Levenberg-Martin-Poletsky (see Proposition 2.15 in [LMP])

Lemma 3.2 Let ∆ be the unit disk in C Then for every n ≥ 1, there exists a holomorphic

function h on the polydisks ∆n , continuous up to the boundary such that Γh(∆n ) is complete

pluripolar inCn+1

Proof We rely heavily on the techniques given in Section 2 of [LMP] in which the case n = 1

has been proved For the clarity of the exposition we consider first the case where n = 2, the necessary modifications for general n will be given at the end Let z = (z1, z2) be coordinates

in C2 According to [LMP] p.526 we can find two sequences {am} and { jm} satisfying the

following conditions:

1) limm→∞ j m+1 j m = 0;

2) c |am | ≥ ∑k≥m+1 |ak|,∀m ≥ 1, where c > 0 is independent of m;

3)∑m ≥1 j1mlog|am| = −∞;

4) limm→∞ |am| 1/ jm= 1

From (2), (4) we infer that the series f (ξ) :=∑m ≥1 amξj m has radius of convergence equal to 1 and defines a holomorphic function on∆, continuous on ∆ By the argument in [LMP] p 519

we also have

lim

k→∞ |pk(ξ)| = ∞, ∀|ξ| > 1,

where p k(ξ) :=∑1≤m≤k a mξj m We set

h(z1, z2) := 2 f (z1) + f (z2), (z1, z2)∈ ∆2.

It is clear that h is continuous on∆2and holomorphic on∆2 We will construct a plurisubharmonic

function u on C3 such that u = −∞ exactly on Γh(∆2) To this end, we first note that p k

converges uniformly on ∆ to f So by Bernstein-Walsh inequality, the sequence 1

j klog|pk| is

uniformly upper bounded on compact sets ofC It follows that the functions

u k (z, w) := 1

j k+1log|w − 2pk (z1)− pk (z2)|

are plurisubharmonic on C3 and uniformly bounded from above on compacta On the other hand, from (1) we deduce that∑k≥1 j1k <∞, so the series

u(z, w) := ∑

k ≥1

max{uk (z, w), −1}

defines a global plurisubharmonic function on C3 Now we show that u(z, h(z)) = −∞ if z =

(z1, z2)∈ ∆2 Fix such a point z For k ≥ 1 andξ ∈ ∆, using (2) we have

| f (ξ)− pk(ξ)| = ∑

m>k

amξj m ≤ ∑

m>k

|am| ≤ (1 + c)|ak+1|.

From this we deduce easily the following estimate

|h(z1, z2)− 2pk (z1)− pk (z2)| ≤ 3(c + 1)|ak+1|.

Therefore

u k (z, h(z)) ≤ 1

j k+1(log|ak+1| + log(3(c + 1))), ∀k ≥ 1,

so that u(z, h(z)) = −∞ by (3) Let z ∈ ∆2and w ̸= h(z) we will show that u(z,w) > −∞ Since

|w − 2pk (z1)− pk (z2)| ≥ |w − h(z1, z2)| − 2| f (z1)− pk (z1)| − | f (z2)− pk (z2)|,

we can chooseδ > 0 and k0≥ 1 such that

|w − 2pk (z1)− pk (z2)| >δ, ∀k ≥ k0.

This implies

u(z, w) ≥ −(k0− 1) + (logδ)(

∑

k≥k0

1

j k

)

> −∞.

Next we prove that u(z, w) > −∞ if z = (z1, z2)̸∈ ∆2and w ∈ C For this, we claim that

lim

k→∞ |2pk (z1) + p k (z2)| = ∞, ∀(z1, z2)̸∈ ∆2.

This is clear if either|z1| ≤ 1 or |z2| ≤ 1, since limk →∞ |pk(ξ)| = ∞ if |ξ| > 1 Now we consider

the remaining cases

Case 1 |z2| > |z1| > 1 Chooseε> 0 such that

|z1| > 1 +ε

(1−ε)2.

Choose m0such that

(1−ε)j m < |am| < (1 +ε)j m ∀m > m0.

So for k > m0we obtain the following estimates

|pk (z1)| ≤ A + k((1 +ε)|z1|) j k , |pk (z2)| ≤ A + k((1 +ε)|z2|) j k ,

where A > 0 is independent of k It follows that

|2pk (z1) + p k (z2)| ≥ |pk (z2)| − 2|pk (z1)| ≥ |ak||z2| j k − |pk −1 (z2)| − 2|ak||z1| j k − 2|pk −1 (z1)|

≥ |ak|

2 (|z2| j k − 4|z1| j k) +|ak|

2 |z2| j k − 3(k − 1)((1 +ε)|z2|) j k −1 − 3A

≥ |ak|

2 (|z2| j k − 4|z1| j k) +1

2((1−ε)|z2|) j k − 3(k − 1)((1 +ε)|z2|) j k −1 − 3A

≥ |ak|

2 (|z2| j k − 4|z1| j k) +1

2((1 +ε)|z2|) j k /2 − 3(k − 1)((1 +ε)|z2|) j k −1 − 3A.

Since|z2| > |z1| and jk / j k −1 → ∞ the last expression goes to ∞ as k tends to ∞.

Case 2 |z1| ≥ |z2| > 1 By a similar and somewhat simpler argument than the above case we

also get that limk→∞ |2pk (z1) + p k (z2)| = ∞.

The claim follows So we can find k0≥ 1 such that

|w − 2pk (z1)− pk (z2)| ≥ |2pk (z1) + p k (z2)| − |w| > 1 ∀k > k0.

This implies that u(z, w) > −∞.

All in all, we have shown that u = −∞ exactly on Γh(∆2) This concludes the proof for the

case n = 2 For general n, we modify the function h as follows Let λ1, ··· ,λn be positive numbers such that if 1≤ i ≤ n − 1 then

|λi| > ∑

i+1 ≤ j≤n |λj|.

Now we set

h(z1, ··· ,zn) = ∑

1≤i≤nλi f (z i ).

Then h is continuous on∆nand holomorphic on∆n Next we let

u(z, w) = ∑

k≥1

max{ 1

j k+1log|w − ∑

1≤i≤nλi p k (z i)|,−1}.

In a similar fashion as in the case n = 2 we can show that u is plurisubharmonic onCn+1 , u =

−∞ on Γh(∆n ) Using the condition (1) we can check that u(z, w) > −∞ if z ̸∈ ∆ n and w ∈ C,

We also need the following fact

Lemma 3.3 Let E be an Fσ-pluripolar subset ofCn and let F be a holomorphic mapping from

an open neighborhood U of E ∗

Cn to C such that F(E) ⊂ ∆ and F(E ∗

Cn)⊂ ∆ Then F(E ∗

Cn)⊂ ∆ Proof This lemma can be deduced from from the proof of Theorem 4.6 in [EW] For the

reader convenience, we give some details Assume that there exists some point z0∈ EC∗ n such

that F(z0)∈∂∆ Fix 0 < r < 1,R > 1 For δ > 0 we let∆δ :={z ∈ C : |z| <δ} and Uδ :=

{z ∈ U : F(z) ∈ ∆1+ δ} The key observation is that ifε > 0 then Uε is an open neighborhood

of E ∗

Cn In particular∂Uε∩EC∗ n = /0 SetBR:={z ∈ C n:|z| < R} By applying the localization

principle of Edigarian and Wiegerinck (Theorem 4.1 in [EW]) we have for everyε > 0

0≤ω(z0, E ∩ F −1(∆r)∩ BR ,BR) =ω(z0, E ∩ F −1(∆r)∩ BR ,BR ∩Uε)≤ω(F(z0),∆r ,∆1+ ε).

By lettingε→ 0 we get

ω(z0, E ∩ F −1(∆r)∩ BR ,BR ) = 0.

So

z0∈ (E ∩ F / −1(∆r)∩ BR ,BR)−

BR.

Since F(E) ⊂ ∆, by letting r → 1 we infer that z0 ̸∈ (E ∩ BR ,BR)−

BR Finally, since R > 1

is arbitrarily large, by Theorem 2.4 in [LP] we get z0 ̸∈ E ∗

Cn , a contradiction The proof is

Proof of Theorem 3.1 We have D = {z ∈ Ω : | f1(z) | < 1,··· ,| fk (z) | < 1}, where f1, ··· , fk

are holomorphic onΩ According to Lemma 3.2, we can find h a continuous function on ∆,

holomorphic on∆ such that Γh(∆k) is complete pluripolar inCk+1 Let F := ( f1, ··· , fk) and

g := h ◦F We will show that X := Γg (D) is complete pluripolar inCn+1 For this, we first claim

that X =Γg (D) is complete pluripolar inCn+1 Indeed, let u be a plurisubharmonic function

on Ck+1 such that u = −∞ precisely on Γh(∆k

) Consider φ(z, w) := u(F(z), w) Then φ is plurisubharmonic onΩ × C andφ=−∞ exactly on X Since X is closed in C n+1 By a result

of Coltoiu about equivalence between locally complete pluripolarity and globally complete

pluripolarity we see that X is complete pluripolar in Cn+1 This proves the claim Next, we

show X ∗

Cn+1 = X By the above reasoning X ∗

Cn+1 ⊂ X For every 1 ≤ j ≤ k, consider the map

Fj:Ω × C → C defined by

Fj (z1, ··· ,zn , w) = f j (z1, ··· ,zn ).

Then F j is holomorphic on an open neighborhood of X ∗

Cn+1 and

F j (X ∗

Cn+1)⊂ Fj (X ) = ∆.

Observe that F j (X ) ⊂ ∆, so we may apply Lemma 3.3 to conclude that

Fj (X ∗

Cn+1)⊂ ∆.

Since it is true for every 1≤ j ≤ k we infer XC∗ n+1 = X Since X = X \Γg(∂D) and sinceΓg(∂D)

is closed we deduce that X is a Gδ set Finally, since X is also Fσ, we apply Zeriahi’s theorem

to get that X is complete pluripolar inCn+1 The proof is complete  Remark If D is a simply connected, proper subdomain inC with real analytic boundary then

by Riemann’s mapping theorem we can find a holomorphic bijection map f from D onto∆

Since D has real analytic boundary, the map f extends to a larger neighborhood Ω of D Thus

D is an analytic polyhedron inΩ, and so Theorem 3.1 applies in this case

IV Graphs of holomorphic functions which can be rapidly approximated by rational functions

It is proved in Theorem 2 of [Go] that if a sequence {rm}m≥1 of rational functions converges

rapidly in measure on an open set X to a holomorphic function f defined on a domain D(X ⊂ D)

then {rm}m≥1 must converge rapidly in measure to f on D Much later, by using delicate

techniques of pluripotential theory, Bloom was able to prove an analogous result in which

rapidly convergence in measure is replaced by rapidly convergence in capacity and the set X is

only required to be compact and non-pluripolar (see Theorem 2.1 in [Bl]) Along this line, we will prove the following refinement of the above results

Theorem 4.1 Let f be a holomorphic function on a domain D ⊂ C n and X a Borel non-pluripolar subset of D Suppose that there exists a sequence {rm}m ≥1 of rational functions with poles off X that converges rapidly pointwise to f on X Then the following assertions hold: (a) There exists a pluripolar subset (possibly empty) E ofCn and a sequence {qm} of polyno-mials such that E = {z ∈ C n: limm→∞ m1log|qm (z) | = −∞} and for every point z ∈ D \ E we have

lim inf

m →∞ | f (z) − rm (z) | 1/m = 0.

(b) The sequence {rm}m≥1 converges rapidly in capacity to f on D.

(c) If the poles of {rm}m≥1 are all disjoint from D then {rm}m≥1 goes to f rapidly uniformly on compacta i.e., for every compact K ⊂ D

lim

m →∞ ∥ f (z) − rm (z) ∥ 1/m

The above theorem demonstrates a sharp contrast between uniform convergence and rapidly uniform convergence Indeed, there exists a sequence of polynomials that converges on com-pact subsets of the unit disk ∆ ⊂ C but is not bounded on any open neighborhood of ∆ (e.g.,

rm (z) = z m)

Proof (a) First, we claim that there exists a non-pluripolar subset X ′ of X such that {rm}m ≥1

is uniformly bounded on X ′ For this, observe that since the sequence {rm}m≥1 converges

pointwise to f on X , we have sup m≥1 |rm (z) | < ∞ for every z ∈ X Now we set

X N ={z ∈ X : |rm (z) | ≤ N,∀m ≥ 1},N ≥ 1.

Since∪N≥1 XN = X and X is non-pluripolar, we infer that there must exist N0such that X N0 is

non-pluripolar The claim follows by choosing X ′ := X

N Now we write r m = p m /q m, where

pm , q m are polynomials of degree ≤ m and ∥qm∥X ′ = 1 This implies ∥pm∥X ′ ≤ N0 for every

m Since X ′is non-pluripolar, by Bernstein-Walsh inequality we deduce that the two sequences

1

mlog|pm|,1

mlog|qm| are uniformly bounded from above on compact sets of C n The next step

is to prove the following claim: For every compact subset K of D we have

1

mlog∥qm f − pm∥K → −∞.

To see this, for m ≥ 1 we set

u m (z) := 1

mlog

f (z) − p m (z)

qm (z)

, ∀z ∈ D \ q −1 m (0).

Then for every z ∈ D we have

v m (z) := u m (z) + 1

mlog|qm (z) | = 1

mlog|qm (z) f (z) − pm (z) |.

Since m1log|pm|, 1

mlog|qm| are uniformly bounded from above on compact sets of D, the

func-tions v m are plurisubharmonic on D and uniformly bounded from above on compacta in D.

By Lemma 2.1, either there exists a subsequence v m converges almost everywhere to v ∈ PSH(D), v ̸≡ −∞ or vm goes to−∞ uniformly on compact subsets of D We will show that

the first case does not occur For the sake of seeking a contradiction we can assume that the

se-quence v m itself converges almost everywhere to v ∈ PSH(D),v ̸≡ −∞ Applying again Lemma

2.1 we have lim sup

m →∞ vm = v outside a pluripolar set Since lim m →∞ um (z) = −∞ for every z ∈ X

and since

sup

m ≥1

1

mlog|qm (z) | < ∞ ∀z ∈ C n

we infer

lim

m →∞ v m (z) = −∞,∀z ∈ X.

This implies v = −∞ on a non-pluripolar subset of D, a contradiction It follows that vm con-verges uniformly to −∞ on compact subsets of D This proves our assertion Next, we note

that the sequence m1log|qm| is uniformly bounded from above on compact sets of C n So using

Lemma 2.1 and the fact that supX ′ m1log|qm| = 0 for every m, we deduce that

E := {z ∈ C n: lim

m →∞

1

mlog|qm (z) | = −∞}

is pluripolar For z ̸∈ E, we can choose a subsequence {mk} such that

inf

k ≥1

1

m klog|qm k (z0)| > −∞.

On the other hand, by above arguments, we have

lim

k→∞ v m k (z) = −∞, ∀z ∈ D \ E.

It follows that limk→∞ u m k (z) = −∞ We are done.

(b) Assume that{rm}m ≥1 does not converge in capacity to f on D Then there exists a compact

subset K of D, a subsequence {m j} and positive constantsε,δ > 0 such that

cap(K j , D) >δ ∀ j ≥ 1,

where K j:={z ∈ K : | f (z) − rm j (z) | 1/m j >ε} It follows from the proof of (a) that for every

M > 0 there exists j M ≥ 1 such that

v m j (z) = 1

m jlog|qm j (z) f (z) − pm j (z) | < −M ∀ j ≥ jM , z ∈ K.

So for j ≥ jMwe have the following inclusion

K j ⊂ L j:={z ∈ K : h j (z) := 1

m jlog|qm j (z) | < −M − logε}.

This implies

cap(L j , D) ≥ cap(Kj , D) >δ, ∀ j ≥ jM

Moreover, by passing to a subsequence we may assume that h j converges in L1loc (D) to a plurisubharmonic function as j → ∞ Letω be a small neighborhood of K which is relatively compact in D Then we have

sup

j≥1 ∥h j∥ L1 ( ω)< ∞.

Take u j ∈ PSH(D),−1 < uj < 0 such that

∫

L j

(dd c u j ) >δ.

By a version of the Chern-Levine-Nirenberg’s inequality (see the proof of Proposition 2.2.3 in

[Bl]) we obtain for every j ≥ jM the following estimate

δ <

∫

L j

(dd c u j)n ≤ 1

M + logε

∫

K |h j|(dd c u j)n ≤ C K,ω

M + logε∥h j∥L1 ( ω).

Here C K,ω is a positive constant depends only on K,ω By letting M → ∞ we get a contradiction.

(c) If the polynomials q m are nowhere vanishing on D then the functions h m:= m1log|qm| are

pluriharmonic on D By the proof of (a), this sequence is also bounded from above on compacta

in D On the other hand, since sup X ′ h m = 0 for every m, by Lemma 2.1’, we deduce that the

sequence {hm} is uniformly bounded on compact sets of D It then follows from (a) that

∥ f − rm∥ 1/m

K goes to 0 for every compact K in D This proves our theorem.  Remarks 1) Under the assumption of Theorem 4.1, we do not know if f can be rapidly point-wise approximated everywhere on D by a sequence of rational functions (possibly different

from{rm}m≥1)

2) It is of interest to see if we can replace ”liminf” in Theorem 4.1(a) by ”lim”

3) Combining Proposition 2.3 and Theorem 4.1(b) we obtain Theorem 2.1 in [Bl] which says

that rapid convergence in capacity on some non-pluripolar Borel set X ⊂ D implies rapid

con-vergence in capacity on D On the other hand, in view of the remark following Proposition 2.3,

it is not clear whether our result follows from Bloom’s theorem

The following result contains some information about pluripolar hulls of Γf (D) where f is a

holomorphic function satisfying the assumption of Theorem 4.1

Proposition 4.2 Let f , D, X , E be as in the theorem and (z0, w0) be a point in (Γf (D)) ∗

Cn+1 Then the following assertions hold:

(a) If z0∈ D \ E then w0= f (z0);

(b) If z0∈ C n \ (D ∪ E) then

lim inf

m →∞ |w0− rm (z0)| 1/m = 0.

In particular, if lim

m →∞ r m (z0) =λ, then w0=λ Proof (a) Define the plurisubharmonic functions

φm (z, w) = 1

mlog|qm (z)w − pm (z) |

onCn+1 Since m1log|pm|,1

mlog|qm| are uniformly bounded from above on compact sets of C n,

by easy estimates we deduce thatφm are uniformly bounded from above on compact subsets

D is an analytic polyhedron inΩ, and so Theorem 3.1 applies in this case

IV Graphs of holomorphic functions which can be rapidly approximated by rational functions. .. by rapidly convergence in capacity and the set X is

only required to be compact and non -pluripolar (see Theorem 2.1 in [Bl]) Along this line, we will prove the following refinement of