A characterization of the class EχlocΩ

Let Ω be a hyperconvex domain in C n . By P SH(Ω) (resp. P SH−(Ω)) we denote the set of plurisubharmonic functions (resp. negative plurisubharmonic functions) on Ω. In Ce2, Cegrell introduced a general class E(Ω) of psh functions on which the complex MongeAmp`ere operator (ddc .) n is well defined. Moreover, in Ce2, he has proved the class E(Ω) is the biggest class on which the complex MongeAmp`ere operator (ddc .) n is well defined as a Radon measure and it is continuous under decreasing sequences. This class was explicitly characterized in papers Bl1, Bl2. On the other hand, a

Vu Viet Hung Department of Mathematics, Physics and Informatics,

Tay Bac University, Son La, Viet Nam E-mail: viethungtbu@gmail.com

Abstract The main aim of the present paper is to study the class Eχ,loc(Ω) and some of its consequences.

Let Ω be a hyperconvex domain in Cn By P SH(Ω) (resp P SH−(Ω)) we denote the set of plurisubharmonic functions (resp negative plurisubharmonic functions) on Ω In [Ce2], Cegrell introduced a general class E (Ω) of psh functions

on which the complex Monge-Amp`ere operator (ddc.)n is well defined Moreover,

in [Ce2], he has proved the class E (Ω) is the biggest class on which the complex Monge-Amp`ere operator (ddc.)n is well defined as a Radon measure and it is continuous under decreasing sequences This class was explicitly characterized in papers [Bl1], [Bl2]

On the other hand, an another weighted energy class Eχ(Ω) which extends the classes Ep(Ω) and F (Ω) in [Ce1] and [Ce2] introduced and investigated recently

by Benelkourchi, Guedj and Zeriahi in [BGZ] is as follows Let χ : R− −→ R+ be

a decreasing function Following [BGZ], we introduce the weighted energy class

of plurisubharmonic functions

Eχ(Ω) = {ϕ ∈ P SH−(Ω) : ∃ E0(Ω) 3 ϕj & ϕ, sup

j≥1

Z

Ω

χ(ϕj)(ddcϕj)n< +∞},

where E0(Ω) is the cone of bounded plurisubharmonic functions ϕ defined on Ω with finite total Monge-Amp`ere mass and lim

z→ξϕ(z) = 0 for all ξ ∈ ∂Ω In [HH],

2010 Mathematics Subject Classification: 32U05, 32U15, 32U40, 32W20.

Key words and phrases: plurisubharmonic functions, weighted energy classes, complex

1

Hai and Hiep proved that Eχ(Ω) ⊂ E (Ω) We note that, when χ(t) = (−t)p, Eχ(Ω)

is the class Ep(Ω) studied by Cegrell in [Ce1]

Next, from [HHQ], we introduce the following class Eχ,loc(Ω) First, let H(Ω) ⊂

P SH−(Ω), we say that ϕ ∈ P SH−(Ω) belongs to Hloc(Ω) if for every hyperconvex domain D b Ω there exists ψ ∈ H(Ω) such that ψ = ϕ on D In the case H(Ω) = Eχ(Ω) we obtain the class Eχ,loc(Ω) It is well known that in [HHQ], the authors proved that if χ(2t) ≤ aχ(t) for some a > 1 and χ ∈ K then Eχ,loc is a local class

In this paper, we give a characterization of the class Eχ,loc(Ω), which is a gener-alization of Theorem 1.1 in [Bl2] Namely, we have following result

Theorem 3.2 Let Ω0⊂ Cn−1, Ω00⊂ C are bounded hyperconvex domains and u :

Ω0 → [−∞, +∞) is a function in P SH−(Ω0) and χ ∈ K Then u ∈ Eχ(t),loc(Ω0×

Ω00) if and only if u ∈ Eχ(t)|t|,loc(Ω0)

Next, we shall then use this result to give some consequences in section 4 The paper is organized as follows Beside the introduction the paper has three sections We also recall the Cegrell classes of plurisubharmonic functions F (Ω) and E (Ω) in section 2 In section 3, we prove a characterization of the class

Eχ,loc(Ω) under some extra assumptions on weights χ Finally, by relying on this characterization, we give some consequences

Some elements of pluripotential theory that will be used throughout the paper can be found in [ACCH], [ACH], [Be], [BT1], [BT2], [Ce1], [Ce2], [Kl], [Ko1], [Ko2] Now we recall the definition of some Cegrell classes of plurisubharmonic functions (see [Ce1] and [Ce2]) Let Ω be an open subset in Cn By β = ddckzk2

we denote the canonical K¨ahler form of Cn where d = ∂ + ∂ and dc= i(∂ − ∂), hence, ddc= 2i∂∂

As in [Ce1] and [Ce2] we define the classes E0(Ω), F (Ω) and E (Ω) as follows Let

Ω be a bounded hyperconvex domain in Cn This means that Ω is a connected, bounded open subset and there exists a negative plurisubharmonic function % such that for all c < 0 the set Ωc= {z ∈ Ω : %(z) < c} b Ω Set

E0= E0(Ω) = {ϕ ∈ P SH−(Ω) ∩ L∞(Ω) : lim

z→ξϕ(z) = 0, ∀ξ ∈ ∂Ω,

Z

Ω

(ddcϕ)n< ∞}

F = F (Ω) =ϕ ∈ P SH−(Ω) : ∃ E0(Ω) 3 ϕj & ϕ, sup

j

Z

Ω

(ddcϕj)n< ∞ ,

E = E(Ω) =ϕ ∈ P SH−

(Ω) :∀z0 ∈ Ω, ∃ a neighbourhood D 3 z0,

E0(Ω) 3 ϕj & ϕ on D, sup

j

Z

Ω

(ddcϕj)n< ∞

To simplify the notation we will write ”A B” if there exists a constant C > 0 such that A ≤ CB The following class of functions were introduced in [HHQ]

K = {χ : R−−→ R+ is a decreasing function such that

− t2χ00(t) tχ0(t) χ(t), ∀t < 0} Remark 2.1 (i) If χ(t) = (−t)p then χ ∈ K

(ii) If χ ∈ K then χ(2t) ≤ aχ(t), ∀t < 0 with some a > 1 Indeed, by hypothesis

tχ0(t) ≤ Cχ(t), C = constant > 0 we set s(t) = (−t)χ(t)C Then s0(t) ≥ 0, ∀t < 0, hence s(t) is an increasing function This implies that s(2t) ≤ s(t) and we have χ(2t) ≤ 2Cχ(t)

3 Some notes in the class Ep,loc

We set z = (z1, , zn−1, zn) = (z0, zn) ∈ Cn First, we have the main result Theorem 3.1 Let Ω0 ⊂ Cn−1, Ω00⊂ C are bounded hyperconvex domains and u :

Ω0 → [−∞, +∞) is a function in P SH−(Ω0) and χ ∈ K Then u ∈ Eχ(t),loc(Ω0×

Ω00) if and only if u ∈ Eχ(t)|t|,loc(Ω0)

Proof We set uj = max(u, −j) Then we have {uj} ⊂ P SH−∩ L∞(Ω0), uj & u

on Ω0 We have

(ddcuj)n= 0 on Ω0× Ω00

and

(ddcuj)n−p∧ (ddckzk2)p = (ddcuj)n−p∧ (ddckz0k2)p−1⊗ dV2(zn), (1)

for all 1 ≤ p ≤ n Next, we take ∀K0b Ω0, ∀K00b Ω00 From (1) we have

Z

K 0 ×K 00

χ(uj)|uj|p(ddcuj)n−p∧ (ddckzk2)p

=

Z

K 0 ×K 00

χ(uj)|uj|p(ddcuj)n−p∧ (ddckz0k2)p−1⊗ dV2(zn)

=

Z

K 0

χ(uj)|uj|p(ddcuj)n−p∧ (ddckz0k2)p−1

Z

K 00

dV2(zn),

for all 1 ≤ p ≤ n Hence

sup

j≥1

Z

K 0 ×K 00

χ(uj)|uj|p(ddcuj)n−p∧ (ddckzk2)p < +∞

⇔ sup

j≥1

Z

K 0

χ(uj)|uj|p(ddcuj)n−p∧ (ddckz0k2)p−1< +∞

⇔ sup

j≥1

Z

K 0

χ(uj)|uj||uj|p−1(ddcuj)(n−1)−(p−1)∧ (ddckz0k2)p−1< +∞

for all 1 ≤ p ≤ n

Finally, we infer that u ∈ Eχ(t),loc(Ω0 × Ω00) if and only if u ∈ Eχ(t)|t|,loc(Ω0) as desired

Corollary 3.2 Let Ω0 ⊂ Cn−1, Ω00 ⊂ C are bounded hyperconvex domains and

u ∈ P SH−(Ω0) Then u ∈ Ep,loc(Ω0× Ω00) if and only if u ∈ Ep+1,loc(Ω0)

Proof Using Theorem 3.1 for χ(t) = (−t)p, ∀t < 0 and we get the desired con-clusion

Remark 3.3 As Example 2.3 in [Ce1], for each 0 < α < 1 we consider functions

uα(z0) = −(− log kz0k)α+ (log 2)α,

on Ω0 = B(0,12) = {z0 ∈ Cn−1: kz0k < 12} By [Ce1], we have uα ∈ Ep(Ω0) if only

if p < αn− n By Proposition 2.3 in [CKZ], we have uα∈ E(Ω0× ∆), ∀0 < α < 1

n Since (ddcuα)n= 0 on Ω = Ω0× ∆, we have R

Ω

χ(uα)(ddcuα)n= 0

However, we will show that uα 6∈ Ep,loc(Ω) for p = nα− n − 1 Indeed, we assume that uα ∈ Ep,loc(Ω) for p = nα− n − 1 By Corollary 4.2, we get uα ∈ Ep+1,loc(Ω0) Moreover, since lim

z 0 →ξ 0uα(z0) = 0, ∀ξ0 ∈ ∂Ω0 we obtain uα ∈ Ep+1(Ω0) This is a contradiction

Acknowledgement

The author would like to thank Professor Pham Hoang Hiep for valuable com-ments during the preparation of this work The paper was done while the author were visiting Vietnam Institute for Advanced Study in Mathematics (VIASM) The author would like to thank the VIASM for hospitality and support

References

[ACCH] P Ahag, U Cegrell, R Czyz and Pham Hoang Hiep, Monge-Amp`ere measures on pluripolar sets, J Math Pures Appl., 92 (2009), 613-627

[ACH] P Ahag, R Czyz and Pham Hoang Hiep, Concerning the energy class Ep for 0 < p < 1, Ann Polon Math., 91 (2007), 119-130

[BT1] E Bedford and B A Taylor, A new capacity for plurisubharmonic func-tions, Acta Math., 149 (1982), 1–40

[BT2] E Bedford and B A Taylor, Fine topology, ˘Silov boundary, and (ddc)n,

J Funct Anal., 72 (1987), 225–251

[Be] S Benelkourchi, Weighted Pluricomplex Energy, Potential Analysis, 31 (2009), 1–20

[BGZ] S Benelkourchi, V Guedj and A Zeriahi, Plurisubharmonic functions with weak singularities, Complex Analysis and Digital Geometry Proceed-ings from the Kiselmanfest, Uppsala Universitet, Uppsala, 2007 ISSN

0502-7454, 57–73

[Bl1] Z B locki, On the definition of the Monge-Amp`ere operator in C2, Math Ann., 328 (2004), 415–423

[Bl2] Z B locki, The domain of definition of the complex Monge-Amp`ere opera-tor, Amer J Math., 128 (2006), 519–530

[Ce1] U Cegrell, Pluricomplex energy, Acta Mathematica, 180 (1998), 187–217 [Ce2] U Cegrell, The general definition of the complex Monge-Amp`ere operator, Ann Inst Fourier, 54 (2004), 159–179

[CKZ] U Cegrell, S Kolodziej and A Zeriahi, Subextension of plurisubharmonic functions with weak singularities, Math Z., 250 (2005), 7–22

[HH] Le Mau Hai and Pham Hoang Hiep, Some Weighted Energy Classes of Plurisubharmonic Functions, Potential Analysis, 34 (2011), 43–56 [HHQ] Le Mau Hai, Pham Hoang Hiep and Hoang Nhat Quy, Local property of the class Eχ,loc, J Math Anal Appl., 402 (2013), 440–445

[Kl] M Klimek, Pluripotential Theory, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications

[Ko1] S Ko lodziej, The range of the complex Monge-Amp`ere operator II, Indiana Univ Math J., 44 (1995), 765–782

[Ko2] S Ko lodziej, The complex Monge-Amp`ere equation and pluripotential the-ory, Mem Amer Math Soc., 178 (2005), 64p