A THEOREM ON THE APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS IN LELONG CLASSES

The paper is dedicated to approximation of plurisubharmonic functions in the Lelong class L(C n) by the functions of the form 1 dj log |pj |, where pj are polynomials in C n 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

A THEOREM ON THE APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS IN LELONG

CLASSES

KIEU PHUONG CHI

Abstract The paper is dedicated to approximation of

plurisub-harmonic functions in the Lelong class L(C n ) by the functions of

the form 1

d j log |pj|, where p j are polynomials in C n with degree deg pj ≤ d j 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) L(Cn) = {u plurisubharmonic on Cn : u(z) ≤ 1

2log(1+|z|

2

)+Cu}, where Cu is a constant depending only on u We also define a more restricted class

L+

(Cn) = {u plurisubharmonic on Cn :

u(z) − 1

2log(1 + |z|

2)

≤ Cu} L(Cn), L+(Cn) 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

|λ|→∞,λ∈Csup u(λz) − log+|λz|

The Robin function plays an important role in approximation prob-lems of holomorphic functions by polynomials (see [1],[17] and the ref-erence 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,L 2 -estimates, pluripolar.

1

Theorem 1.1 (H¨ormander’s theorem) Let PA be the set of all func-tions of the form 1

Nlog |f (z)|, where N is a positive integer and f 6= 0

is an entire function Then the closure of PA in L1

loc(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)

1≤j≤s ν

1

nj log |pj|, 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 approximated in the L1loc(Cn) topology by quotients log |p|

d , where

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

2 A variation of H¨ormander’s theorem

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, for all m ≥ 1 and ϕem = 1

dmlog |ϕm|, the following hold (a) ϕem → u in L1

loc(Cn), (b) ϕem → u pointwise on K,

(c) lim sup

ϕm ≤ u on Cn, (d) lim sup

m→∞

ρ

e

ϕ m ≤ ρu on Cn, (e) For every r > 0 and every z ∈ K \ {0}

ρu(z) ≤ lim inf

|λ|>r

 e

ϕm(λz) − log+|λz|

!

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

(4)

Z

Cn

|f (z)|2

(1 + |z|2)adλn(z) := C < ∞

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

Z

|w−z|<r

|f (w)|2dλn(w) (Cn is independent of z)

≤Cn

r2n

Z

|w−z|<r

|f (w)|2

(1 + |w|2)adλn(w)maxn 1 + |w|2)a: |w − z| < ro

≤ CCnr−2nmaxn 1 + |w|)2a : |w − z| < ro

≤ C0r−2n(r + 2|z|)2a,

where C0 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

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

(5) ρu(z) = lim

j→∞(u(λz,jz) − log+|λz,jz|)

Choose a sequence K0 := {zj}j≥1 with zj 6= zj0 for j 6= j0, which is dense in Cn and contains K ∪ {λz,jz : 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

um(z) = (u ∗ ρ1/m)(z) + 1

2mlog(1 + |z|

2),

where ρ(·) = 1

nρ(·/) for all  > 0 and ρ is a non negative ra-dial test function with support in the unit ball of Cn satisfying Z

Cn

ρ(z)dλn(z) = 1 It is well known that umis C∞smooth and strictly plurisubharmonic on Cn for all m ≥ 1 We also have

um(z) ≤ 1

2

Z

|w|<1/m

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

2) 2m + Cu

≤ 1

2log



1 + 2|z|2+ 2

m2

 +log(1 + |z|

2) 2m + Cu

≤ 1

2 +

1 m

 log(1 + |z|2) + Cu0, ∀z ∈ Cn, where Cu0 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, choose δ > 0 such that δ < minj=1,2, ,νrj,m and δ < 1

2|zj − zj0| when

1 ≤ j < j0 ≤ ν Set αν,m = min1≤j≤νεj,m and

fk,m,ν(z) =

ν

X

j=1

χ((z − zj)/δ)ekAj,m (z) Clearly

∂fk,m,ν(z) =

ν

X

j=1

∂χ((z − zj)/δ)ekAj,m (z) := gk,m,ν(z)

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

(7)

Z

Cn

|gk,m,ν|2e−2kumdλn ≤ C1δ−2e−αν,m kδ 2 /2

and

(8)

Z

Cn

|fk,m,ν|2e−2kumdλn ≤ C2(2kαν,m)−n

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

(9)

Z

Cn

|uk,m,ν(z)|2e−2kum (z)

(1 + |z|2)2 dλn(z) ≤ C1δ−2e−αν,m kδ 2 /2

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 and the oscillation of um on the ball is smaller than αν,m δ 2

8 , from (9) we get for all 1 ≤ j ≤ ν

|uk,m,ν(zj)|2 ≤ C3

t2n

Z

B(z j ,t)

|uk,m,ν|2dλn

≤ C4

t2nδ2e−αν,m kδ 2 /2

e2k(um (z j )+αν,mδ8 2)

,

where C3, C4 are positive constants independent of k and t Since

fk,m,ν(zj) = eku m (z j ) for all 1 ≤ j ≤ ν, we infer from the above inequalities that for all k large enough

(10) |uk,m,ν(zj)| < e

ku m (z j )

2 , hence |pk,m,ν(zj)| >

ekum (z j )

2 , ∀1 ≤ j ≤ ν, 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)

Z

Cn

|pk,m,ν(z)|2e−2kum (z)

(1 + |z|2)2 dλn(z) ≤ C5, where C5 > 0 is independent of k Since um ≤ (1/2 + 1/m) log(1 +

|z|2) + Cu0 on Cn, we deduce from (11) that

(12)

Z

Cn

|pk,m,ν(z)|2

(1 + |z|2)2+k+2k/mdλn(z) ≤ C6

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 greater than k + 2 + 2k

m − n

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

t2n(1 + |z|2+ t2)2e2k supB(z,t) u m

This implies that

lim sup

k→∞

log |pk,m,ν(z)|

B(z,t)

um

By letting t tend to 0 we obtain

k→∞

log |pk,mν(z)|

k ≤ um(z), ∀z ∈ Cn

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 2

k(m, ν) ≤ log |pk(m,ν),m,ν(zj)|

k(m, ν) . For simplicity of notation, we put qm,ν = pk(m,ν),m,ν From the last inequality and (13) we obtain

lim

ν→∞

log |qm,ν(zj)|

k(m, ν) = um(zj), ∀j ≥ 1.

Since um is continuous and plurisubharmonic on Cn and K0 is dense

in Cn, by Lemma 15.1.7 in [7] we have log |qm,ν|

k(m, ν) converges to um in

L1

loc(Cn) when ν tends to ∞ So we can choose ν(m) ≥ m so large that

Z

B m

log |qm,ν(m)| k(m, ν(m)) − um

dλn < 1

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 L1

loc(Cn),

we infer that log |ϕm|

lm

converges to u in L1

loc(Cn) Because u is, in particular, subharmonic on Cn, by Theorem 3.2.13 in [8] we get

k→∞

log |ϕm(z)|

k ≤ u(z), ∀z ∈ Cn

6

Putting (10) and (14) together we get

m→∞

log |ϕm(zj)|

lm = u(zj), ∀j ≥ 1.

Since deg ϕm ≤ dm := b(lm, m) (where b(k, m) is the largest integer not greater than k + 2 + 2k

m − n) and limm→∞dm/lm = 1, from (14) 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

|λ|→∞

(ϕ(λ) − log |λ|) = inf

r≥1(max

|λ|=rϕ(λ) − log r)

To see this, for λ 6= 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

λ→0

ψ(λ) = lim sup

|λ|→∞

(ϕ(λ) − log |λ|)

On the other hand, by the maximum principle we obtain

ψ(0) = inf

0<r<1max{ψ(z) : |z| = r} = inf

r>1(max

|λ|=rϕ(λ) − log r) The claim now follows Fix z ∈ C \ {0} and r > 1, applying the claim just proven to the function λ 7→ϕem(λz), where ϕem = 1

dm log |ϕm|, we obtain

ρϕem(z) + log |z| ≤ max

|λ|=r(ϕem(λz) − log 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

m→∞

ρ

e

ϕ m ≤ max

|λ|=r(u(λz) − log r)

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

j→∞(u(λz∗ ,jz∗) − log+|λz∗ ,jz∗|)

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)| > δd m, where dm = deg pm

(ii) ||pm||U ∩E ≤ (1/m)d m, ||pm||U < 1, λn{z ∈ U : |pm(z)| < δd m} < ε

Recall that a subset E of Cnis 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 6≡ −∞ 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 6≡ −∞ 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

of D which is of Fσ and Gδ type Assume that ED∗, the pluripolar hull

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 that 1

dm log |pm| < − log m on U ∩ E and that

1

dm log |pm(z0)| > c − 1,

Z

L

1

dm log |pm| − u

dλ < ε/2,

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)| < δdm} < ε/2

8

It is now clear that pm, dm satisfy (i), (ii).

It remains to show that (b) ⇒ (a) We write E = ∪m≥1Km 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≥1Bm = 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

dm

log |pm(z0)| > log δ, sup

B m

1

dm

log |pm| < 0, sup

K m

1

dm

log |pm| < − log m

It follows that

u(z) = X

m≥1

1

2mdm log |pm(z)|

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)| > δdm; (ii)

||pm||U ∩E ≤ (1/m)d m, ||pm||U < 1, λn{z ∈ U : |pm(z)| < δdm} < 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 e

ϕm= 1

dm log |ϕm| → u in L1

loc(Cn) and lim supm→∞ϕ˜m = u on Cn

9

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[3] J Duval and N Sibony, Polynomial convexity, rational convexity and cur-rents, Duke Math Journal, 79 (1995), 487-513.

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[5] N Q Dieu and F Wilkstr¨ om, Jensen measures and approximation of plurisubharmonic functions, Michigan Math J 53 (2005), no 3, 529-544 [6] A Edigarian and J Wiegerinck, The pluripolar hull of the graph of a holomorphic function with polar singularities, Indiana Math Journal 52 (6) (2003), 1663-1680.

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[12] Le Mau Hai, Nguyen Quang Dieu and Tang Van Long, Remarks on pluripolar hulls,Annales Polon Math 84 (2004), No3, 225-236.

[13] E Poletsky, Approximation of plurisubharmonic functions by multipole Green functions, Trans Amer Math Soc 355 (2003) 1579-1591.

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Kieu Phuong Chi,

Department of Mathematics, Vinh University,182 Le Duan, Vinh City, Vietnam

E-mail address: kpchidhv@yahoo.com

10

[1] T Bloom, Some applications of the Robin functions to multivariable approx-imation theory, Journal of Approxapprox-imation Theory 92 (1998), 1-21.... important role in approximation prob-lems of holomorphic functions by polynomials (see [1],[17] and the ref-erence therein) One of important results concerning approximation of plurisubharmonic functions. .. 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