A THEOREM ON THE APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS IN LELONG CLASSES

A THEOREM ON THE APPROXIMATION OF
PLURISUBHARMONIC FUNCTIONS IN LELONG
CLASSES
KIEU PHUONG CHI
Abstract. The paper is dedicated to approximation of plurisubharmonic functions in the Lelong class L(Cn ) by the functions of
the form d1j log |pj |, where pj are polynomials in Cn with degree
deg pj ≤ dj . We must say that this result is inspired by Theorem
15.1.6 in [7]. Some applications to pluripolar sets are also given.

1. Introduction
The set of plurisubharmonic functions with logarithmic growth
is defined by
1
(1) L(Cn ) = {u plurisubharmonic on Cn : u(z) ≤ log(1+|z|2 )+Cu },
2
where Cu is a constant depending only on u. We also define a more
restricted class
1
L+ (Cn ) = {u plurisubharmonic on Cn : u(z) − log(1 + |z|2 ) ≤ Cu }.
2
n
+
n
L(C ), L (C ) are sometimes referred to as the Lelong classes in
Cn . These classes have been studied by many authors, including
Leja, Lelong, Sadullaev, Siciak, Zaharjuta, Bedford and Taylor,...
in connection with problems concerning polynomials in n complex
variables (see [2, 9, 15] and the references given therein). Of particular
interest for the Lelong classes is the Robin function defined by
(2)

ρu (z) =

lim

|λ|→∞,λ∈C

sup u(λz) − log+ |λz| .

The Robin function plays an important role in approximation problems of holomorphic functions by polynomials (see [1],[17] and the reference therein). One of important results concerning approximation of
plurisubharmonic functions is the following theorem due to H¨ormander
in [7] (Theorem 15.1.6).
2010 Mathematics Subject Classification. 32U05, 32E99, 32U15.
Key words and phrases. plurisubharmonic, H¨ormander’s theorem, Lelong
classes,L2 -estimates, pluripolar.
1


Theorem 1.1. (H¨ormander’s theorem) Let PA be the set of all func1
tions of the form log |f (z)|, where N is a positive integer and f = 0
N
is an entire function. Then the closure of PA in L1loc (Cn ) consists of
all plurisubharmonic functions.
Some refinements of the above result were given for the functions in
L(Cn ). In fact, if u ∈ L(Cn ) then we are able to obtain approximation
results in precise form. One of them is due to Siciak [16, 17].
Theorem 1.2. (Siciak’s theorem) Let u ∈ L(Cn ). One can find for
every positive integer ν a function uν ∈ L(Cn ) with the following
properties:
a)
(3)

uν := sup
1≤j≤sν

1
log |pj |,
nj

where pj are holomorphic polynomials in Cn , nj are positive integers
satisfying deg pj ≤ nj .
b) uν+1 ≤ uν and limν→∞ uν = u.
The main goal of the present work is to explore a variation of
H¨ormander’s result on approximation in the Lelong classes. More
precisely, in Theorem 2.1 we show that every u ∈ L(Cn ) can be
log |p|
approximated in the L1loc (Cn ) topology by quotients
, where
d
p is a polynomial and d is some integer larger than deg p. The
proof relies heavily on the solution to the ∂−problem and H¨ormander
L2 −estimates. As a consequence of the theorem, we give in Corollary
2.3 a characterization of closed complete pluripolar sets in Cn . In the
same vein, a complete description of closed complete pluripolar subsets
of a pseudoconvex domain in Cn is also given in Proposition 2.5.
Acknowledgment. The author would like to thank Professor
Nguyen Quang Dieu for his valuable suggestions. This paper was
revised during a stay of the author at Vietnam Institute for Advance
Study in Mathematics. He wishes to express his gratitude to the
institute for the support.
¨ rmander’s theorem

2. A variation of Ho
The main result of this paper is the following theorem.
Theorem 2.1. Let u ∈ L(Cn ) and K a subset of Cn which is at most
countable. Then there exist a sequence {ϕm }m≥1 of polynomials in Cn ,
2


a sequence {dm }m≥1 of positive integers with deg ϕm ≤ dm such that,
1
for all m ≥ 1 and ϕm =
log |ϕm |, the following hold
dm
(a) ϕm → u in L1loc (Cn ),
(b) ϕm → u pointwise on K,
(c) lim sup ϕm ≤ u on Cn ,
m→∞

(d) lim sup ρϕm ≤ ρu on Cn ,
m→∞

(e) For every r > 0 and every z ∈ K \ {0}
ρu (z) ≤ lim inf
m→∞

sup ϕm (λz) − log+ |λz|

.

|λ|>r

We need the following elementary fact, which may not be original.
Lemma 2.2. Let f be a holomorphic function on Cn . Assume that
there exists a > 0 such that
|f (z)|2
dλn (z) := C < ∞.
(1 + |z|2 )a

(4)
Cn

where λn is the Lebesgue measure in Cn . Then f is a polynomial of
degree not greater than a − n.
Proof. By subharmonicity of |f |2 on Cn , we have for any z ∈ Cn with
|z| ≥ 1 and r > 0,
|f (z)|2 ≤
≤

Cn
r2n
Cn
r2n

|f (w)|2 dλn (w) (Cn is independent of z)
|w−z|
|w−z|
|f (w)|2
dλn (w) max
(1 + |w|2 )a

≤ CCn r−2n max

1 + |w|2 )a : |w − z| < r

1 + |w|)2a : |w − z| < r

≤ C r−2n (r + 2|z|)2a ,
where C is independent of z. Choosing here r = |z|, we arrive at
|f (z)| ≤ M |z|a−n
for all z ∈ Cn and |z| ≥ 1, where M is independent of z. By Cauchy’s
inequalities, we can deduce that f is a polynomial of degree not greater
than a − n.
Proof of Theorem 2.1. For each z ∈ K\{0}, in view of (2) there
exists a sequence {λz,j }j≥1 ⊂ C such that |λz,j | ↑ ∞ and
3


ρu (z) = lim (u(λz,j z) − log+ |λz,j z|).

(5)

j→∞

Choose a sequence K := {zj }j≥1 with zj = zj for j = j , which is
dense in Cn and contains K ∪ {λz,j z : z ∈ K, j ≥ 1}. The rest of the
proof follows closely the lines of Theorem 15.1.6 in [7]. For each m ≥ 1,
we set
1
log(1 + |z|2 ),

um (z) = (u ∗ ρ1/m )(z) +
2m
1
where ρ (·) = n ρ(·/ ) for all > 0 and ρ is a non negative radial test function with support in the unit ball of Cn satisfying
ρ(z)dλn (z) = 1. It is well known that um is C ∞ smooth and strictly
Cn

plurisubharmonic on Cn for all m ≥ 1. We also have
1
um (z) ≤
2

log(1 + |z|2 )
log(1 + |z − w| )ρ1/m (w)dλn (w) +
+ Cu
2m
|w|<1/m
2

2
1
log(1 + |z|2 )
log 1 + 2|z|2 + 2 +
+ Cu
2
m
2m
1
1
≤

+
log(1 + |z|2 ) + Cu , ∀z ∈ Cn ,
2 m
≤

where Cu is a constant depending only on u. Fix m ≥ 1, then from the
strict plurisubharmonicity of um we obtain
(6)

um (z) ≥ Re Aj,m (z) + εj,m |z − zj |2 , |z − zj | < rj,m ,

where εj,m > 0, rj,m > 0, and Aj,m is a quadratic holomorphic
polynomial satisfying Aj,m (zj ) = um (zj ). Let χ be a test function in Cn
with compact support in |z| < 1 and equal to 1 if |z| < 1/2. Fix ν ≥ 1,
1
choose δ > 0 such that δ < minj=1,2,...,ν rj,m and δ < |zj − zj | when
2
1 ≤ j < j ≤ ν. Set αν,m = min1≤j≤ν εj,m and
ν

χ((z − zj )/δ)ekAj,m (z) .

fk,m,ν (z) =
j=1

Clearly
ν

∂χ((z − zj )/δ)ekAj,m (z) := gk,m,ν (z).

∂fk,m,ν (z) =
j=1

We are going to solve the equation ∂uk,m,ν = gk,m,ν with respect to the
plurisubharmonic weight 2kum . Observe that, by the choice of χ and
4


(6), we can find C1 > 0, C2 > 0 independent of k such that
|gk,m,ν |2 e−2kum dλn ≤ C1 δ −2 e−αν,m kδ

(7)

2 /2

Cn

and
|fk,m,ν |2 e−2kum dλn ≤ C2 (2kαν,m )−n .

(8)
Cn

By (7) and Theorem 15.1.2 in [7], we can find a solution uk,m,ν such
that
(9)
Cn

|uk,m,ν (z)|2 e−2kum (z)
2

dλn (z) ≤ C1 δ −2 e−αν,m kδ /2 .
2
2
(1 + |z| )

Since uk,m,ν is holomorphic on the ball |z − zj | < δ/2, if j ≤ ν, by
applying the submean value inequality to the subharmonic function
|uk,m,ν |2 and the ball B(zj , t), where t is chosen such that 0 < t < δ/2
δ2
, from (9) we
and the oscillation of um on the ball is smaller than αν,m
8
get for all 1 ≤ j ≤ ν
C3
|uk,m,ν |2 dλn
2n
t
B(zj ,t)
αν,m δ 2
C4
2
≤ 2n 2 e−αν,m kδ /2 e2k(um (zj )+ 8 ) ,
t δ

|uk,m,ν (zj )|2 ≤

where C3 , C4 are positive constants independent of k and t. Since
fk,m,ν (zj ) = ekum (zj ) for all 1 ≤ j ≤ ν, we infer from the above
inequalities that for all k large enough

(10) |uk,m,ν (zj )| <

ekum (zj )
ekum (zj )
, hence |pk,m,ν (zj )| >
, ∀1 ≤ j ≤ ν,
2
2

where pk,m,ν := fk,m,ν − uk,m,ν . Combining the elementary inequality
|pk,m,ν |2 ≤ 2(|fk,mν |2 + |uk,m,ν |2 ) with (8) and (9) we infer for all k large
enough
(11)
Cn

|pk,m,ν (z)|2 e−2kum (z)
dλn (z) ≤ C5 ,
(1 + |z|2 )2

where C5 > 0 is independent of k. Since um ≤ (1/2 + 1/m) log(1 +
|z|2 ) + Cu on Cn , we deduce from (11) that
(12)
Cn

|pk,m,ν (z)|2
dλn (z) ≤ C6 .
(1 + |z|2 )2+k+2k/m
5

Here C6 > 0 is independent of k. By Lemma 2.2, pk,m,ν is a polynomial
of degree not exceeding b(k, m), where b(k, m) is the largest integer not
2k
greater than k + 2 +
− n.
m
Next, we let z be an arbitrary point in Cn . Using (11) and the
subharmonicity of the function |pk,m,ν |2 , for every t > 0 we obtain
|pk,m,ν (z)|2 ≤

C6
(1 + |z|2 + t2 )2 e2k supB(z,t) um
t2n

This implies that
lim sup
k→∞

log |pk,m,ν (z)|
≤ sup um .
k
B(z,t)

By letting t tend to 0 we obtain
(13)

lim sup
k→∞

log |pk,mν (z)|

≤ um (z), ∀z ∈ Cn .
k

In view of (10) we can choose k = k(m, ν) ≥ m + ν so large that for
1 ≤ j ≤ ν the following inequalities hold
um (zj ) −

log |pk(m,ν),m,ν (zj )|
log 2
≤
.
k(m, ν)
k(m, ν)

For simplicity of notation, we put qm,ν = pk(m,ν),m,ν . From the last
inequality and (13) we obtain
log |qm,ν (zj )|
= um (zj ), ∀j ≥ 1.
ν→∞
k(m, ν)
lim

Since um is continuous and plurisubharmonic on Cn and K is dense
log |qm,ν |
converges to um in
in Cn , by Lemma 15.1.7 in [7] we have
k(m, ν)
L1loc (Cn ) when ν tends to ∞. So we can choose ν(m) ≥ m so large that

Bm

log |qm,ν(m) |
1
− um dλn < ,
k(m, ν(m))
m

where Bm is the ball of radius m centered at 0. Set ϕm = qm,ν(m) and
lm = k(m, ν(m)). Since um ↓ u, in particular um → u in L1loc (Cn ),
log |ϕm |
we infer that
converges to u in L1loc (Cn ). Because u is, in
lm
particular, subharmonic on Cn , by Theorem 3.2.13 in [8] we get
(14)

lim sup
k→∞

log |ϕm (z)|
≤ u(z), ∀z ∈ Cn .
k
6


Putting (10) and (14) together we get
log |ϕm (zj )|
(15)
lim
= u(zj ), ∀j ≥ 1.

m→∞
lm
Since deg ϕm ≤ dm := b(lm , m) (where b(k, m) is the largest integer
2k
not greater than k + 2 +
− n) and limm→∞ dm /lm = 1, from (14)
m
and (15) we conclude that the sequences {ϕm }m≥1 and {dm }m≥1 satisfy
(a), (b) and (c) of the theorem.
For (d), we are going to repeat a reasoning due to Bloom and Zeriahi
(see Theorem 2.1 in [1]) for the readers convenience. First, we claim
that if ϕ ∈ L(C) then
lim sup(ϕ(λ) − log |λ|) = inf (max ϕ(λ) − log r)
r≥1 |λ|=r

|λ|→∞

To see this, for λ = 0 we set ψ(λ) = ϕ(1/λ) + log |λ|. It is easy to see
that ψ is a subharmonic function on C\{0}, which is also bounded from
above near 0. Thus ψ extends through 0 to a subharmonic function on
C. We still denote this extension by ψ. On one hand we have
ψ(0) = lim sup ψ(λ) = lim sup(ϕ(λ) − log |λ|).
|λ|→∞

λ→0

On the other hand, by the maximum principle we obtain
ψ(0) = inf max{ψ(z) : |z| = r} = inf (max ϕ(λ) − log r).
0

r>1 |λ|=r

The claim now follows. Fix z ∈ C \ {0} and r > 1, applying the claim
1
just proven to the function λ → ϕm (λz), where ϕm =
log |ϕm |, we
dm
obtain
ρϕm (z) + log |z| ≤ max(ϕm (λz) − log r).
|λ|=r

Taking lim sup of both sides when m → ∞ and using Hartogs’ lemma
([8, 9]) and (c) of the theorem on the righthand side we obtain
log |z| + lim sup ρϕm ≤ max(u(λz) − log r).
|λ|=r

m→∞

Letting r → ∞, we get (d) of the theorem.
Finally, fix z ∗ ∈ K \ {0}. Then by (5) we can choose a sequence
{λz∗ ,j }j≥1 ↑ ∞ such that
ρu (z ∗ ) = lim (u(λz∗ ,j z ∗ ) − log+ |λz∗ ,j z ∗ |).
j→∞

This implies (e). The proof of the theorem is complete.
As a consequence of the theorem, we have a characterization of
complete pluripolar sets in Cn .
7

Corollary 2.3. Let E be a closed set in Cn . Then the following
assertions are equivalent
(a) E is complete pluripolar in Cn .
(b) For every closed ball U in Cn , every point z0 ∈ U \ E, and every
> 0, there exists a constant δ > 0 such that for all m ≥ 1 we can find
a polynomial pm satisfying
(i) |pm (z0 )| > δ dm , where dm = deg pm .
(ii) ||pm ||U ∩E ≤ (1/m)dm , ||pm ||U < 1, λn {z ∈ U : |pm (z)| < δ dm } <
ε.
Recall that a subset E of Cn is called pluripolar if for every a ∈ E and
every neighbourhood U of a we can find a plurisubharmonic function
u on U such that u ≡ −∞ on every connected component of U and
u|E∩U ≡ −∞. A subset E of an open set D ⊂ Cn is said complete
pluripolar in D if there exists a plurisubharmonic function u in D such
that u ≡ −∞ and E = {z ∈ D : u(z) = −∞}.
We need the following lemma due to Zeriahi (Proposition 2.1 in [18],
see also Proposition 3.1 in [12]).
Lemma 2.4. Let D be a pseudoconvex domain in Cn and E be a subset
∗
, the pluripolar hull
of D which is of Fσ and Gδ type. Assume that ED
of E relative to D, coincides with E. Then E is complete pluripolar in
D.
Recall that if E is a pluripolar subset of D then
∗
ED
:= {z ∈ D : u(z) = −∞ if u is plurisubharmonic on D, u ≡ −∞ on E}.

Proof of Corollary 2.3. (a) ⇒ (b). According to a theorem of
Siciak in [15] (see also Theorem 7.1 in [2]), we can find u ∈ L(Cn ) such

that {u = −∞} = E. By subtracting a large constant, we may assume
that u < 0 on a neighbourhood of U . Since λn (E) = 0, we may choose
c < u(z0 ) such that λn {z ∈ U : u(z) < c} < ε/2. Given m ≥ 1, by
Hartogs’ lemma and Theorem 2.1, we can find a polynomial pm such
1
that
log |pm | < − log m on U ∩ E and that
dm
1
log |pm (z0 )| > c − 1,
dm

L

1
log |pm | − u dλ < ε/2,
dm

where dm is some integer deg pm ≤ dm and L is the compact {z ∈
U : u(z) ≥ c}. After adding, if necessary, a homogeneous polynomial
of degree dm to pm , we may assume that deg pm = dm . By setting
δ = ec−1 we infer that
λn {z ∈ L : |pm (z)| < δ dm } < ε/2.
8


It is now clear that pm , dm satisfy (i), (ii).
It remains to show that (b) ⇒ (a). We write E = ∪m≥1 Km where
Km is an increasing sequence of compact sets. Let z0 ∈ Cn \ E. Choose
an increasing sequence of closed balls {Bm }m≥1 such that

z0 ∈ B1 , ∪m≥1 Bm = Cn , Km ⊂ Bm ∀m ≥ 1.
By (b), there exists δ > 0 such that for each m ≥ 1 there are a
polynomial pm on Cn and an integer dm such that
1
1
1
log |pm (z0 )| > log δ, sup
log |pm | < 0, sup
log |pm | < − log m.
dm
Bm dm
K m dm
It follows that

1

u(z) =
m≥1

2m d

log |pm (z)|
m

defines a plurisubharmonic function on Cn and that satisfies u ≡ −∞
on E whereas u(z0 ) > −1. By Lemma 2.4 and since E is closed in
Cn , we conclude that E is complete pluripolar in Cn . The proof is
complete.
Using Theorem 2.1 and the lines of the proof of Corollary 2.4, we
easily prove the following characterization of closed complete pluripolar

sets in an arbitrary pseudoconvex domain.
Proposition 2.5. Let E be a closed subset of a pseudoconvex domain
D in Cn . The following assertions are equivalent.
(a) E is complete pluripolar in D.
(b) For every relatively compact subdomain U of D, every z0 ∈ U \E
and every ε > 0, there exists a constant δ > 0 such that we can find
a sequence {pm }m≥1 of holomorphic functions on D and a sequence
{dm }m≥1 of positive integers satisfying
(i)
|pm (z0 )| > δ dm ;
(ii)
||pm ||U ∩E ≤ (1/m)dm , ||pm ||U < 1, λn {z ∈ U : |pm (z)| < δ dm } < ε.
In the proof of Theorem 2.1, if we let K be a countable dense set of
Cn then we obtain the following result whose proof we omitted.
Corollary 2.6. Let u ∈ L(Cn ) be continuous. Then there exists a
sequence {ϕm }m≥1 of polynomials with degree dm ≥ 1 in Cn such that
1
ϕm =
log |ϕm | → u in L1loc (Cn ) and lim supm→∞ ϕ˜m = u on Cn .
dm
9


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Kieu Phuong Chi,
Department of Mathematics, Vinh University,182 Le Duan, Vinh City,
Vietnam
E-mail address:

10