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

A characterization of the class Eχ,loc(Ω) 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. 1 Introduction 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) ϕj χ(ϕj )(ddc ϕj )n < +∞}, ϕ, sup j≥1 Ω where E0 (Ω) is the cone of bounded plurisubharmonic functions ϕ defined on Ω with finite total Monge-Amp`ere mass and lim ϕ(z) = 0 for all ξ ∈ ∂Ω. In [HH], z→ξ 2010 Mathematics Subject Classification: 32U05, 32U15, 32U40, 32W20. Key words and phrases: plurisubharmonic functions, weighted energy classes, complex Monge-Amp`ere operator 1 2 V. V. Hung 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 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 generalization of Theorem 1.1 in [Bl2]. Namely, we have following result. Theorem 3.2. Let Ω ⊂ Cn−1 , Ω ⊂ C are bounded hyperconvex domains and u : Ω → [−∞, +∞) is a function in P SH − (Ω ) and χ ∈ K. Then u ∈ Eχ(t),loc (Ω × Ω ) if and only if u ∈ Eχ(t)|t|,loc (Ω ). 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. 2 Preliminairies 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 = ddc z 2 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} Ω. Set E0 = E0 (Ω) = {ϕ ∈ P SH − (Ω) ∩ L∞ (Ω) : lim ϕ(z) = 0, ∀ξ ∈ ∂Ω, (ddc ϕ)n < ∞} z→ξ Ω F = F(Ω) = ϕ ∈ P SH − (Ω) : ∃ E0 (Ω) ϕj (ddc ϕj )n < ∞ , ϕ, sup j Ω A characterization of the class Eχ,loc (Ω) 3 and E = E(Ω) = ϕ ∈ P SH − (Ω) :∀z0 ∈ Ω, ∃ a neighbourhood D E0 (Ω) ϕj z0 , (ddc ϕj )n < ∞ . ϕ on D, sup j Ω 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 − t2 χ (t) tχ (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 tχ (t) ≤ Cχ(t), C = constant > 0 we set s(t) = χ(t) . (−t)C Then s (t) ≥ 0, ∀t < 0, hence s(t) is an increasing function. This implies that s(2t) ≤ s(t) and we have χ(2t) ≤ 2C χ(t). Some notes in the class Ep,loc 3 We set z = (z1 , . . . , zn−1 , zn ) = (z , zn ) ∈ Cn . First, we have the main result Theorem 3.1. Let Ω ⊂ Cn−1 , Ω ⊂ C are bounded hyperconvex domains and u : Ω → [−∞, +∞) is a function in P SH − (Ω ) and χ ∈ K. Then u ∈ Eχ(t),loc (Ω × Ω ) if and only if u ∈ Eχ(t)|t|,loc (Ω ). Proof. We set uj = max(u, −j). Then we have {uj } ⊂ P SH − ∩ L∞ (Ω ), uj u on Ω . We have (ddc uj )n = 0 on Ω × Ω and (ddc uj )n−p ∧ (ddc z 2 )p = (ddc uj )n−p ∧ (ddc z for all 1 ≤ p ≤ n. Next, we take ∀K Ω , ∀K 2 p−1 ) ⊗ dV2 (zn ), Ω . From (1) we have χ(uj )|uj |p (ddc uj )n−p ∧ (ddc z 2 )p K ×K χ(uj )|uj |p (ddc uj )n−p ∧ (ddc z = 2 p−1 ) ⊗ dV2 (zn ) K ×K χ(uj )|uj |p (ddc uj )n−p ∧ (ddc z = K 2 p−1 ) K dV2 (zn ), (1) 4 V. V. Hung for all 1 ≤ p ≤ n. Hence χ(uj )|uj |p (ddc uj )n−p ∧ (ddc z 2 )p < +∞ sup j≥1 K ×K ⇔ sup χ(uj )|uj |p (ddc uj )n−p ∧ (ddc z 2 p−1 ) < +∞ j≥1 K ⇔ sup χ(uj )|uj ||uj |p−1 (ddc uj )(n−1)−(p−1) ∧ (ddc z 2 p−1 ) < +∞ j≥1 K for all 1 ≤ p ≤ n. Finally, we infer that u ∈ Eχ(t),loc (Ω × Ω ) if and only if u ∈ Eχ(t)|t|,loc (Ω ) as desired. Corollary 3.2. Let Ω ⊂ Cn−1 , Ω ⊂ C are bounded hyperconvex domains and u ∈ P SH − (Ω ). Then u ∈ Ep,loc (Ω × Ω ) if and only if u ∈ Ep+1,loc (Ω ). Proof. Using Theorem 3.1 for χ(t) = (−t)p , ∀t < 0 and we get the desired conclusion. Remark 3.3. As Example 2.3 in [Ce1], for each 0 < α < 1 we consider functions uα (z ) = −(− log z )α + (log 2)α , on Ω = B(0, 21 ) = {z ∈ Cn−1 : z if p < n α < 12 }. By [Ce1], we have uα ∈ Ep (Ω ) if only − n. By Proposition 2.3 in [CKZ], we have uα ∈ E(Ω × ∆), ∀0 < α < n1 . Since (ddc uα )n = 0 on Ω = Ω × ∆, we have χ(uα )(ddc uα )n = 0. Ω However, we will show that uα ∈ Ep,loc (Ω) for p = that uα ∈ Ep,loc (Ω) for p = n α n α − n − 1. Indeed, we assume − n − 1. By Corollary 4.2, we get uα ∈ Ep+1,loc (Ω ). Moreover, since lim uα (z ) = 0, ∀ξ ∈ ∂Ω we obtain uα ∈ Ep+1 (Ω ). This is a z →ξ contradiction. Acknowledgement The author would like to thank Professor Pham Hoang Hiep for valuable comments during the preparation of this work. 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