Đang tải... (xem toàn văn)
In the paper we introduce a new class of msubharmonic functions with finite weighted complex mHessian. We prove that this class has local property. As an application, we give a lower estimate for the log canonical threshold of plurisubharmonic functions in the class Em(Ω).
Local property of a class of m-subharmonic functions and applications Le Mau Hai*, Nguyen Xuan Hong* and Vu Viet Hung** *Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam **Department of Mathematics, Physics and Informatics, Tay Bac University, Son La, Viet Nam E-mail: mauhai@fpt.vn, xuanhongdhsp@yahoo.com and viethungtbu@gmail.com Abstract In the paper we introduce a new class of m-subharmonic functions with finite weighted complex m-Hessian. We prove that this class has local property. As an application, we give a lower estimate for the log canonical threshold of plurisubharmonic functions in the class Em (Ω). 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. We recall the notion of a local class introduced and investigated in [HHQ]. A class J (Ω) ⊂ P SH − (Ω) is said to be a local class if ϕ ∈ J (Ω) then ϕ ∈ J (D) for all hyperconvex domains D Ω and if ϕ ∈ P SH − (Ω), ϕ|Ωi ∈ J (Ωi ), ∀i ∈ I with Ωi then ϕ ∈ J (Ω). As well known that, the class E(Ω) introduced and Ω= i∈I investigated in [Ce2] is a local class. Moreover, in [Ce2] Cegrell 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. 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. Then, as in [BGZ], we define Eχ (Ω) = {ϕ ∈ P SH − (Ω) : ∃ E0 (Ω) ϕj χ(ϕj )(ddc ϕj )n < +∞}, ϕ, sup j≥1 Ω 2010 Mathematics Subject Classification: 32U05, 32U15, 32U40, 32W20. Key words and phrases: m-subharmonic functions, weighted energy classes of msubharmonic functions, complex m-Hessian, local property, the log canonical threshold. 1 2 L. M. Hai, N. X. Hong and V. V. Hung 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. Note z→ξ that from Corollary 4.4 in [Be], it follows that if ϕ ∈ Eχ (Ω) then lim ϕ(z) = 0 z→ξ for all ξ ∈ ∂Ω. Hence if ϕ ∈ Eχ (Ω) then ϕ ∈ / Eχ (D) with D a relatively compact hyperconvex domain in Ω. Thus the class Eχ (Ω) is not a ”local” one. In this paper by relying on ideas of the paper of Benelkourchi, Guedj and Zeriahi in [BGZ] and on Cegrell’s classes of m-subharmonic functions introduced and studied recently in [Ch] we introduce weighted energy classes of m-subharmonic functions Fm,χ (Ω) and Em,χ (Ω). However, under slight hypotheses for weights χ we achieve that the class Fm,χ (Ω) is a convex cone (see Proposition 3.4 below). We also show that the complex Hessian operator Hm (u) = (ddc u)m ∧ β n−m is well defined on the class Em,χ (Ω) where β = ddc z 2 denotes the canonical K¨ahler form of Cn . Futhermore, we prove that the class Em,χ (Ω) is a local class (see Theorem 4.6 in Section 4 below). Next, based on a recent result of J.-P. Demailly and Pham Hoang Hiep on a lower bound of the log canonical threshold for the class E(Ω) in [DH] we establish a lower estimate for the log canonical threshold of plurisubharmonic functions in the class Em (Ω). The paper is organized as follows. Beside the introduction the paper has three sections. In Section 2 we recall the definitions and results concerning to msubharmonic functions which were introduced and investigated intensively in recent years by many authors, such as, in [Bl1], [DiKo], [SA]. We also recall the Cegrell’s classes of m-subharmonic functions Fm (Ω) and Em (Ω) introduced and studied in [Ch]. In Section 3 we introduce the new two weighted energy classes of m-subharmonic functions Fm,χ (Ω) and Em,χ (Ω). Section 4 is devoted to prove the local property of the class Em,χ (Ω) under some extra assumptions on weights χ. To arrive this property of the class Em,χ (Ω) we need a result about subextension for the class Fm,χ (Ω) (see Lemma 4.5 below) which is of interest. Finally, by relying on the local property of the class Em,χ (Ω), in section 5 we give a lower estimate for the log canonical threshold of plurisubharmonic functions in the class Em (Ω). 2 Preliminairies Some elements of pluripotential theory that will be used throughout the paper can be found in [BT1], [Kl], [Ko1], [Ko2] while elements of the theory of msubharmonic functions and the complex Hessian operator can be found in [Bl1], [DiKo], [SA]. Now we recall the definition of some Cegrell classes of plurisubharmonic functions (see [Ce1] and [Ce2]), as well as, the class of m-subharmonic 0 (Ω) and F (Ω) introfunctions introduced by Blocki in [Bl1] and the classes Em m duced and investigated by Lu Hoang Chinh in [Ch] recently. Let Ω be an open subset in Cn . By β = ddc z 2 we denote the canonical K¨ahler form of Cn with 1 n i c β where d = ∂ + ∂ and dc = ∂−∂ the form of volume dVn = n! 4i , hence, dd = 2 ∂∂. 2.1. As in [Ce1] and [Ce2] we define the classes E0 (Ω) and F(Ω) as follows. Let Ω be a bounded hyperconvex domain. That means that Ω is a connected, bounded open subset and there exists a negative plurisubharmonic function such that Local property of a class of m-subharmonic functions and applications for all c < 0 the set Ωc = {z ∈ Ω : (z) < c} 3 Ω. Set E0 = E0 (Ω) = {ϕ ∈ P SH − (Ω) ∩ L∞ (Ω) : lim ϕ(z) = 0, ∀ξ ∈ ∂Ω, (ddc ϕ)n < ∞} z→ξ Ω and F = F(Ω) = ϕ ∈ P SH − (Ω) : ∃ E0 ϕj (ddc ϕj )n < ∞ , ϕ, sup j Ω 2.2. We recall the class of m-subharmonic functions introduced and investigated in [Bl1] recently. For 1 ≤ m ≤ n, we define Γm = {η ∈ C(1,1) : η ∧ β n−1 ≥ 0, . . . , η m ∧ β n−m ≥ 0}, where C(1,1) denotes the space of (1, 1)-forms with constant coefficients. Definition 2.1. Let u be a subharmonic function on an open subset Ω ⊂ Cn . u is said to be a m-subharmonic function on Ω if for every η1 , . . . , ηm−1 in Γm the inequality ddc u ∧ η1 ∧ . . . ∧ ηm−1 ∧ β n−m ≥ 0, holds in the sense of currents. − (Ω) By SHm (Ω) we denote the set of m-subharmonic functions on Ω while SHm denotes the set of negative m-subharmonic functions on Ω. Before to formulate basic properties of m-subharmonic functions, as in [Bl1], we recall the following. For λ = (λ1 , . . . , λn ) ∈ Rn and 1 ≤ m ≤ n, define λj1 · · · λjm . Sm (λ) = 1≤j1 ε}. Then uε ∈ SHm (Ωε )∩ ∞ C (Ωε ) and uε ↓ u as ε ↓ 0. h) Let u1 , . . . , up ∈ SHm (Ω) and χ : Rp → R be a convex function which is non decreasing in each variable. If χ is extended by continuity to a function [−∞, +∞)p → [−∞, ∞), then χ(u1 , . . . , up ) ∈ SHm (Ω). Example 2.3. Let u(z1 , z2 , z3 ) = 5|z1 |2 +4|z2 |2 −|z3 |2 . By using b) of Proposition 2.2 it is easy to see that u ∈ SH2 (C3 ). However, u is not a plurisubharmonic function in C3 because the restriction of u on the line (0, 0, z3 ) is not subharmonic. Now as in [Bl1] and [DiKo] we define the complex Hessian operator of locally bounded m-subharmonic functions as follows. Definition 2.4. Assume that u1 , . . . , up ∈ SHm (Ω) ∩ L∞ loc (Ω). Then the complex Hessian operator Hm (u1 , . . . , up ) is defined inductively by ddc up ∧ · · · ∧ ddc u1 ∧ β n−m = ddc (up ddc up−1 ∧ · · · ∧ ddc u1 ∧ β n−m ). From the definition of m-subharmonic functions and using arguments as in the proof of Theorem 2.1 in [BT1] we note that Hm (u1 , . . . , up ) is a closed positive current of bidegree (n − m + p, n − m + p) and this operator in continuous under decreasing sequences of locally bounded m-subharmonic functions. Hence, for p = m, ddc u1 ∧ · · · ∧ ddc um ∧ β n−m is a nonnegative Borel measure. In particular, when u = u1 = · · · = um ∈ SHm (Ω) ∩ L∞ loc (Ω) the Borel measure Hm (u) = (ddc u)m ∧ β n−m , is well defined and is called the complex Hessian of u. 2.3. Similar as in pluripotential theory now we recall a class of m-subharmonic functions introduced and investigated in [Bl1] recently. Definition 2.5. A m-subharmonic function u ∈ SHm (Ω) is called m-maximal if every v ∈ SHm (Ω), v ≤ u outside a compact subset of Ω implies that v ≤ u on Ω. Local property of a class of m-subharmonic functions and applications 5 By M SHm (Ω) we denote the set of m-maximal functions on Ω. One of essential results in [Bl1] is Theorem 3.6 saying that a m-subharmonic function on a bounded domain Ω ⊂ Cn belongs to M SHm (Ω) if and only if it solves the homogeneous Hessian equation Hm (u) = (ddc u)m ∧ β n−m = 0. 0 (Ω) and F (Ω) introduced and investigated in 2.4. Next, we recall the classes Em m [Ch]. First we give the following. Let Ω be a bounded domain in Cn . Ω is said to be m-hyperconvex if there exists a continuous m-subharmonic function u : Ω −→ R− such that Ωc = {u < c} Ω for every c < 0. As above, every plurisubharmonic function is m-subharmonic with m ≥ 1 then every hyperconvex domain in Cn is m-hyperconvex. Let Ω ⊂ Cn be a m-hyperconvex domain. Put 0 0 − Em = Em (Ω) = {u ∈ SHm (Ω) ∩ L∞ (Ω) : lim u(z) = 0, Hm (u) < ∞}, z→∂Ω Ω − 0 Fm = Fm (Ω) = u ∈ SHm (Ω) : ∃ Em uj Hm (uj ) < ∞ , u, sup j Ω and − Em = Em (Ω) = u ∈ SHm (Ω) : ∀z0 ∈ Ω, ∃ a neighborhood ω 0 Em uj z0 , and Hm (uj ) < ∞ , u on ω, sup j Ω − (Ω) where Hm (u) = (ddc u)m ∧ β n−m denotes the Hessian measure of u ∈ SHm ∞ ∩L (Ω). From Theorem 3.14 in [Ch] it follows that if u ∈ Fm (Ω), the complex Hessian Hm (u) = (ddc u)m ∧ β n−m is well defined and is a Radon measure on Ω 2.5. We recall the notion of m-capacity introduced in [Ch]. Definition 2.6. Let E ⊂ Ω be a Borel subset. The m-capacity of E with respect to Ω is defined by (ddc u)m ∧ β n−m : u ∈ SHm (Ω), −1 ≤ u ≤ 0 . Cm (E) = Cm (E, Ω) = sup E Proposition 2.10 in [Ch] gives some elementary properties of the m-capacity similar as the capacity presented in [BT1]. Namely, we have: ∞ a) Cm ( j=1 ∞ Ej ) ≤ Cm (Ej ). j=1 b) If Ej E then Cm (Ej ) Cm (E). We need the following lemma which is used in the proof for the convexity of the class Em,χ (Ω) late. 0 (Ω). Then Lemma 2.7. Assume that ϕ ∈ Em (ddc ϕ)m ∧ β n−m {ϕ < −t} ≤ tm Cm {ϕ < −t} and tm Cm {ϕ < −2t} ≤ (ddc ϕ) ∧ β n−m {ϕ < −t} . 6 L. M. Hai, N. X. Hong and V. V. Hung Proof. Let v ∈ SHm (Ω), −1 < v < 0. For all t > 0 we have the following inclusion: {ϕ < −2t} ⊂ { ϕ < v − 1} ⊂ {ϕ < −t}. t Thus (ddc v)m ∧ β n−m ≤ (ddc v)m ∧ β n−m { ϕt 0 then αu + γv ∈ Fm,χ (Ω) (resp. Em,χ (Ω)). Proof. a) It suffices to prove that the conclusion holds for the class Fm,χ (Ω). − (Ω). From the Definition 3.1, Assume that u ∈ Fm,χ (Ω) and u ≤ v, v ∈ SHm 0 (Ω), u there exists a sequence {uj } ⊂ Em u on Ω with j χ(uj )(ddc uj )m ∧ β n−m < +∞. sup j Ω 0 (Ω), v Put vj = max(uj , v) ∈ Em j v on Ω and uj ≤ vj . By Lemma 3.3 we have χ(vj )(ddc vj )m ∧ β n−m sup j Ω ≤ 2m max(a, 2) sup χ(uj )(ddc uj )m ∧ β n−m < +∞. j Ω Local property of a class of m-subharmonic functions and applications 9 Hence, we get v ∈ Fm,χ (Ω). b) First, we prove that if u ∈ Fm,χ (Ω) and α > 0 then αu ∈ Fm,χ (Ω). Indeed, 0 (Ω), u let k ∈ N∗ with 2k > α and let {uj } ⊂ Em u on Ω with j χ(uj )(ddc uj )m ∧ β n−m < +∞. sup j Ω 0 (Ω), αu It is clear that {αuj } ⊂ Em j k k χ(2 uj ) ≤ a χ(uj ) so αu on Ω. Moreover, since χ(αuj ) ≤ χ(αuj )(ddc αuj )m ∧ β n−m sup j Ω χ(uj )(ddc uj )m ∧ β n−m < +∞. ≤ ak αm sup j Ω Hence, αu ∈ Fm,χ (Ω). Now, assume that α, γ ≥ 0, α+γ > 0. By the above proof, we can assume that α+ 0 (Ω), u u on Ω, vj u on Ω, sup χ(uj )(ddc uj )m ∧ γ = 1. Let {uj }, {vj } ⊂ Em j j β n−m < +∞ and sup j Ω χ(vj )(ddc uj )m ∧ β n−m < +∞. By Lemma 3.3 we have Ω χ(αuj + γvj )(ddc (αuj + γvj ))m ∧ β n−m sup j Ω ≤ 2m max(a, 2) sup χ(uj )(ddc uj )m ∧ β n−m + sup j χ(vj )(ddc uj )m ∧ β n−m j Ω Ω < +∞. Hence, the desired conclusion follows. Proposition 3.5. Let χ : R− −→ R+ be a decreasing function such that χ(2t) ≤ aχ(t) for all t < 0 with some a > 1. Then for every u ∈ Fm,χ (Ω), there exists a 0 (Ω) ∩ C(Ω) such that u sequence {uj } ⊂ Em u and j χ(uj )(ddc uj )m ∧ β n−m < +∞. sup j Ω ∞ Proof. Let Ωj Ωj+1 Ω such that Ω = 0 (Ω) such that Ωj and let {vj } ⊂ Em j=1 vj u and χ(vj )(ddc uj )m ∧ β n−m < +∞. sup j Ω 0 (Ω) ∩ C(Ω) Theorem 3.1 in [Ch] implies that there exists a sequence {wj } ⊂ Em such that wj u. Put − uj = sup{ϕ ∈ SHm (Ω) : ϕ ≤ j−1 wj on Ωj }. j 10 L. M. Hai, N. X. Hong and V. V. Hung It is easy to see that uj u on Ω. By Theorem 1.2.7 in [Bl3] and Proposition 3.2 0 (Ω) ∩ C(Ω). Now, in [Bl1] we get uj ∈ C(Ω). Moreover, since wj ≤ uj so uj ∈ Em since vj u as j → ∞ and u ≤ wk so there exists j0 such that vj0 ≤ k−1 k wk on Ωk . Therefore, vj0 ≤ uk on Ω. Lemma 3.3 implies that χ(vj0 )(ddc vj0 )m ∧ β n−m χ(uk )(ddc uk )m ∧ β n−m ≤ 2m max(a, 2) Ω Ω χ(vj )(ddc vj )m ∧ β n−m . ≤ 2m max(a, 2) sup j Ω Thus, χ(vj )(ddc vj )m ∧ β n−m < +∞, χ(uk )ddc uk )m ∧ β n−m ≤ 2m max(a, 2) sup sup j k Ω Ω as we wanted. The following proposition shows that the Hessian operator is well defined on the class Em,χ (Ω). Proposition 3.6. Let χ : R− −→ R+ be a decreasing function such that χ ≡ 0 and χ(2t) ≤ aχ(t) for all t < 0 with some a > 1. Then Em,χ (Ω) ⊂ Em (Ω) and, hence, the Hessian Hm (u) = (ddc u)m ∧ β n−m is well defined as a positive Radon measure on Ω. Proof. Without loss of generallity we can assume that χ(t) > 0 for every t < 0. Let u ∈ Em,χ (Ω) and z0 ∈ Ω. Take a neighbourhood ω Ω of z0 and a sequence 0 (Ω) such that sup u < 0, u {uj } ⊂ Em u on ω and 1 j ω sup χ(uj )Hm (uj ) < +∞. j Ω For each j ≥ 1, put − uj = sup{u ∈ SHm (Ω) : u|ω ≤ uj |ω }. Then uj ≤ uj on Ω and uj = uj on ω and, by using arguments as in [Bl4], we 0 (Ω) and H (u ) = 0 on arrive that uj ∈ M SHm (Ω \ ω). This yields that uj ∈ Em m j Ω \ ω. Moreover, it is easy to see that uj u on Ω. On the other hand, as in the proof of Lemma 3.3, we have sup χ(uj )Hm (uj ) < +∞. j Ω Moreover, we may assume that inf χ(u1 ) = c1 > 0. Then ω c1 sup Hm (uj ) = c1 sup j Hm (uj ) j ω Ω ≤ sup χ(u1 )Hm (uj ) ≤ sup j χ(uj )Hm (uj ) < +∞. j ω Ω Local property of a class of m-subharmonic functions and applications 11 Hence, Hm (uj ) < +∞, sup j Ω and it follows that u ∈ Fm (Ω). It is easy to see that u = u on ω and this yields that u ∈ Em (Ω). Theorem 3.14 in [Ch] implies that Hm (u) is a positive Radon measure on Ω. The proof is complete. 4 The local property of the class Em,χ (Ω) Now we give the following definition which is similar as in [HHQ] for plurisubharmonic functions. − (Ω) is said to be a local class if ϕ ∈ J (Ω) Definition 4.1. A class J (Ω) ⊂ SHm − (Ω), ϕ| then ϕ ∈ J (D) for all hyperconvex domain D Ω and if ϕ ∈ SHm Ωj ∈ J (Ωj ), ∀j ∈ I with Ω = Ωj , then ϕ ∈ J (Ω). j∈I In [HHQ] the authors introduced the class Eχ,loc (Ω) and established the local property for this class. The section is devoted to study the local property of the class Em,χ (Ω). In the sequel of the paper we will use the following notation. We will write ”A B” if there exists a constant C such that A ≤ CB. Proposition 4.2. Put K = {χ : R− −→ R+ : χ is deareasing and − t2 χ (t) tχ (t) χ(t), ∀t < 0}. Then the class K has the following properties. a) If χ1 , χ2 ∈ K and a1 , a2 ≥ 0 then a1 χ1 + a2 χ2 ∈ K. b) If χ1 , χ2 ∈ K then χ1 .χ2 ∈ K. c) If χ ∈ K then χp ∈ K for all p > 0. d) If χ ∈ K then (−t)χ(t) ∈ K and, more general, |tk |χ(t) ∈ K for all k = 0, 1, 2, ... Proof. The proof is easy from direct calculations and we omit. The following result is necessary for the proof of the local property of the class Em,χ (Ω). c Lemma 4.3. Let u, v ∈ SH − m (Ω) ∩ C(Ω) with u ≤ v on Ω and T = dd ϕ1 ∧ · · · ∧ − (Ω) ∩ C(Ω), j = 1 . . . , m − 1. Suppose that χ ∈ K. ddc ϕm−1 ∧ β n−m with ϕj ∈ SHm Then χ(u)ddc v ∧ T c χ(u)(ddc u + |u|β) ∧ T, Ω where Ω Ω Ω Ω and c is a constant only depending on Ω , Ω , Ω and χ. Proof. Repeating the proof of Lemma 3.2 in [HHQ]. The next lemma is a crucial tool for the proof of the local property of the class Em,χ and for lower bounds of the log canonical threshold in the class Em (Ω) in section 5. 12 L. M. Hai, N. X. Hong and V. V. Hung − (Ω) ∩ Lemma 4.4. Let Ω be a hyperconvex domain in Cn . Assume that u ∈ SHm C(Ω) and χ ∈ K with χ (t) ≥ 0 for every t < 0. Then for every open ball B(x, r) Ω and 1 ≤ m ≤ n the following holds: |χ (u)|(ddc u)m ∧ β n−m . χ(u)(ddc u)m−1 ∧ β n−m+1 ≤ r2 B(x,r) B(x,r) Moreover, if u ∈ P SH − (Ω) ∩ C(Ω) then for every nonnegative closed current T of bidegree (n − m, n − m) we have |χ (u)|(ddc u)m ∧ T. χ(u)(ddc u)m−1 ∧ ddc |z|2 ∧ T ≤ r2 B(x,r) B(x,r) Proof. It suffices to prove lemma holds for the case u ∈ P SH − (Ω) ∩ C(Ω). For ε > 0, put hε (z) := |z − x|2 − r2 − max(|z − x|2 − r2 , −ε). Using integration by parts we infer that χ(u)(ddc u)m−1 ∧ ddc hε ∧ T B(x,r) hε (ddc u)m−1 ∧ ddc χ(u) ∧ T = B(x,r) hε (ddc u)m−1 ∧ [χ (u)du ∧ dc u + χ (u)ddc u] ∧ T = B(x,r) hε χ (u)(ddc u)m ∧ T ≤ B(x,r) ≤ r2 |χ (u)|(ddc u)m ∧ T. B(x,r) Let ε 0 we get |χ (u)|(ddc u)m ∧ T, χ(u)(ddc u)m−1 ∧ ddc |z|2 ∧ T ≤ r2 B(x,r) B(x,r) and we are done. We also need a following result on subextension for the class Fm,χ (Ω). Lemma 4.5. Assume that Ω Fm,χ (Ω) such that u ≤ u on Ω. Ω and u ∈ Fm,χ (Ω). Then there exists a u ∈ Proof. We split the proof into three steps. − (Ω) : Step 1. We prove that if v ∈ C(Ω), v ≤ 0, suppv Ω then v := sup{w ∈ SHm 0 c m n−m w ≤ v on Ω} ∈ Em (Ω) ∩ C(Ω) and (dd v) ∧ β = 0 on {v < v} ∩ Ω. Indeed, 0 0 (Ω). let ϕ ∈ Em (Ω) ∩ C(Ω) such that ϕ ≤ inf v on suppv. Since ϕ ≤ v so v ∈ Em Ω Moreover, by Proposition 3.2 in [Bl1] we have v ∈ C(Ω). Let w ∈ SHm ({v < v} ∩ Ω) such that w ≤ v outside a compact subset K of {v < v} ∩ Ω. Put w1 = max(w, v) v on {v < v} ∩ Ω on Ω\({v < v} ∩ Ω). Local property of a class of m-subharmonic functions and applications 13 Since v is upper semicontinuous so ε = − sup(v − v) > 0. Choose δ ∈ (0, 1) such K that −δ inf Ω v < ε. We have (1 − δ)v ≤ v + ε ≤ v on K. Hence, (1 − δ)v + δw1 ≤ v on Ω, and, we get (1 − δ)v + δw1 = v. Thus, w ≤ v on {v < v} ∩ Ω. Hence, v is m-maximal in {v < v} ∩ Ω. By [Bl1] we get (ddc v)m ∧ β n−m = 0 on {v < v} ∩ Ω. 0 (Ω) ∩ C(Ω) then there exists u ∈ E 0 (Ω), Step 2. Next, we prove that if u ∈ Em m c m n−m (dd u) ∧ β = 0 on (Ω\Ω) ∪ ({u < u} ∩ Ω) and (ddc u)m ∧ β n−m ≤ (ddc u)m ∧ β n−m on {u = u} ∩ Ω. Indeed, put u 0 v= on Ω on Ω\Ω. It is easy to see that v ∈ C(Ω) and suppv ⊂ Ω Ω. Hence, we have u = v ∈ 0 (Ω)∩C(Ω) and (ddc u)m ∧β n−m = 0 on {v < v}∩ Ω = (Ω\Ω)∪({u < u}∩Ω). Let Em K be a compact set in {u = u} ∩ Ω. Then for ε > 0 we have K {u + ε > u} ∩ Ω so we have (ddc u)m ∧ β n−m = K 1{u+ε>u} (ddc u)m ∧ β n−m K 1{u+ε>u} (ddc max(u + ε, u))m ∧ β n−m = K (ddc max(u + ε, u))m ∧ β n−m , ≤ K where the equality in the second line follows by using the same arguments as in [BT2]( also see the proof of Theorem 3.23 in [Ch]). However, max(u + ε, u) u on Ω as ε → 0 so by [SA] it follows that (ddc max(u + ε, u))m ∧ β n−m is weakly convergent to (ddc u)m ∧ β n−m as ε → 0. On the other hand, 1K is upper semicontinuous on Ω so we can approximate 1K with a decreasing sequence of continuous functions ϕj . Hence, we infer that 1K (ddc max(u + ε, u))m ∧ β n−m lim sup ε→0 Ω ϕj (ddc max(u + ε, u))m ∧ β n−m = lim sup lim j ε→0 Ω ϕj (ddc max(u + ε, u))m ∧ β n−m ≤ lim sup ε→0 Ω c ϕj (dd u)m ∧ β n−m ≤ Ω (ddc u)m ∧ β n−m . K as j → +∞. This yields that (ddc u)m ∧ β n−m ≤ (ddc u)m ∧ β n−m on {u = u} ∩ Ω. 0 (Ω) ∩ C(Ω) such that u Step 3. Now, let uj ∈ Em u and j χ(uj )(ddc uj )m ∧ β n−m < +∞. sup j Ω 14 L. M. Hai, N. X. Hong and V. V. Hung By Step 2, we have χ(uj )(ddc uj )m ∧ β n−m = χ(uj )(ddc uj )m ∧ β n−m {uj =uj }∩Ω Ω χ(uj )(ddc uj )m ∧ β n−m ≤ {uj =uj }∩Ω χ(uj )(ddc uj )m ∧ β n−m . ≤ Ω Hence, j χ(uj )(ddc uj )m ∧ β n−m < +∞. χ(uj )(ddc uj )m ∧ β n−m ≤ sup sup j Ω Ω Thus, u := lim uj ∈ Fm,χ (Ω) and u ≤ u on Ω. The proof is complete. j→∞ The following result deal with the local property of the class Em,χ (Ω). Namely, we have the following: Theorem 4.6. Let Ω be a hyperconvex domain in Cn and m be an integer with − (Ω) and χ ∈ K such that χ(2t) ≤ aχ(t) with 1 ≤ m ≤ n. Assume that u ∈ SHm some a > 1. Then the following statements are equivalent. a) u ∈ Em,χ (Ω). 0 (Ω) ∩ C(Ω) such that u b) For every K Ω Ω and every {uj } ⊂ Em u j on Ω and lim uj ∈ Fm,χ (Ω) we have j χ(uj )|uj |p (ddc uj )m−p ∧ β n−m+p < +∞, sup j K for every p = 0, 1, . . . , m. − (Ω) ∩ C(Ω) such that u c) For every sequence {uj } ⊂ SHm j every K Ω we have u on Ω and for χ(uj )|uj |p (ddc uj )m−p ∧ β n−m+p < +∞, sup j K for every p = 0, 1, . . . , m. d) For every U Ω such that U is a hyperconvex domain, we have u|U ∈ Em,χ (U ). e) For every z ∈ Ω there exists a hyperconvex domain Vz Ω such that z ∈ Vz and u|Vz ∈ Em,χ (Vz ). t Proof. Set χ0 (t) = χ(t) and for each k ≥ 1 , let χk (t) = − χk−1 (x)dx. From the 0 hypothesis χ(2t) ≤ aχ(t) it is easy to check that χk ∈ K and χk (t) ≈ χ(t)(−t)k . Local property of a class of m-subharmonic functions and applications 15 N a)⇒b) Let r > 0 and x1 , . . . , xN ∈ K such that K B(xj , r) Ω . By j=1 Lemma 4.4 we have N p c m−p χ(uj )|uj | (dd uj ) ∧β n−m+p K χp (uj )(ddc uj )m−p ∧ β n−m+p B(xj ,r) k=1 N r2 |χp (uj )|(ddc uj )m−p+1 ∧ β n−m+p−1 k=1 N B(xj ,r) χp−1 (uj )(ddc uj )m−p+1 ∧ β n−m+p−1 = r2 =r k=1 N 2p B(xj ,r) χ0 (uj )(ddc uj )m ∧ β n−m B(xj ,r) k=1 χ(uj )(ddc uj )m ∧ β n−m . ≤ N r2p Ω Hence, we get χ(uj )|uj |p (ddc uj )m−p ∧ β n−m+p sup j K ≤ N r2p sup χ(uj )(ddc uj )m ∧ β n−m < +∞, j Ω for every p = 0, 1, . . . , m. b)⇒c) Assume that K U V Ω such that U, V are hyperconvex. Choose 0 (Ω) ∩ C(Ω) such that v a sequence {vk } ⊂ Em u on V , v = lim vk ∈ Fm,χ (Ω) k and χ(vk )|vk |m−p (ddc vk )p ∧ β n−p < +∞, sup k V for every p = 0, 1, . . . , m. For each j, choose kj +∞ such that 2vkj ≤ uj on U . 0 (Ω) ∩ C(Ω), w u on Put wj = max(vkj , uj ). It is easy to see that {wj } ⊂ Em j U , w = lim wj ∈ Fm,χ (Ω). Hence, by Lemma 3.3 and by the above proof we get χ(uj )|uj |p (ddc uj )m−p ∧ β n−m+p sup j K χ(uj )|wj |p (ddc wj )m−p ∧ β n−m+p = sup j K ≤ N r2p sup j χ(wj )(ddc wj )m ∧ β n−m < +∞, Ω for every p = 0, 1, . . . , m. c)⇒d) Assume that U Ω such that U is hyperconvex. Let Ω 0 (Ω) ∩ C(Ω) such that u Choose a sequence {uj } ⊂ Em u on K and j χ(uj )|uj |p (ddc uj )m−p ∧ β n−m+p < +∞, sup j K K U. 16 L. M. Hai, N. X. Hong and V. V. Hung − (U ) : ϕ ≤ u on Ω } ∈ E 0 (U ). for every p = 0, 1, . . . , m. Put uj = sup{ϕ ∈ SHm j m Let Ω Ω1 . . . Ωm K. Since uj ≤ uj on U and (ddc uj )m ∧ β n−m = 0 on U \Ω so by applying Lemma 4.4 many times we arrive that χ(uj )(ddc uj )m ∧ β n−m U χ(uj )(ddc uj )m ∧ β n−m ≤ Ω χ(uj )(ddc uj + |uj |β) ∧ (ddc uj )m−1 ∧ β n−m Ω1 [χ(uj )ddc uj + χ1 (uj )β] ∧ (ddc uj )m−1 ∧ β n−m Ω1 χ(uj )(ddc uj + |uj |β) ∧ (ddc uj )m−2 ∧ ddc uj ∧ β n−m Ω2 χ1 (uj )(ddc uj + |uj |β) ∧ (ddc uj )m−2 ∧ β n−m+1 + Ω2 ............ χ(uj )[|uj |m β m + |uj |m−1 ddc uj ∧ β m−1 + · · · + (ddc uj )m ] ∧ β n−m . Ωm Hence, χ(uj )(ddc uj )m ∧ β n−m sup j U χ(uj )[|uj |m β m + |uj |m−1 ddc uj ∧ β m−1 + · · · + (ddc uj )m ] ∧ β n−m sup j Ωm χ(uj )[|uj |m β m + |uj |m−1 ddc uj ∧ β m−1 + · · · + (ddc uj )m ] ∧ β n−m sup j K < +∞. Thus, uΩ ,U := lim uj ∈ Fm,χ (U ). Since Ω U is arbitrary and uΩ ,U = u on Ω so u ∈ Em,χ (U ). d)⇒e) It is clear. e)⇒a) Assume that Ω Ω. Choose zj ∈ Ω, j = 1, 2, . . . , s such that Ω s s V . Let W V such that Ω j j j=1 Wzj . Since u|Vzj ∈ Em,χ (Vzj ) so j=1 zj there exists vj ∈ Fm,χ (Vzj ) such that vj = u on Wj . By Lemma 4.5 there exists vj ∈ Fm,χ (Ω) such that vj ≤ vj . Proposition 3.4 implies that v := v1 + · · · + vs ∈ Fm,χ (Ω) and, hence, max(v, u) ∈ Fm,χ (Ω). Moreover, since max(v, u) = u on Ω so u ∈ Em,χ (Ω). The proof is complete. From the above theorem we find out an interesting property of the class Em,χ (Ω). Corollary 4.7. Assume that Ω is a hyperconvex domain and χ ∈ K satisfies all hypotheses of Theorem 4.6. Then Em,χ (Ω) ⊂ Em−1,χ (Ω). Proof. Assume that u ∈ Em,χ (Ω). Let K Ω. Take a domain Ω with K Ω 0 (Ω)∩C(Ω) such that u Ω. By Theorem 4.6 there exists a sequence {uj } ⊂ Em u j on Ω and χ(uj )[|uj |m β m + |uj |m−1 ddc uj ∧ β m−1 + · · · + (ddc uj )m ] ∧ β n−m < +∞. sup j Ω Local property of a class of m-subharmonic functions and applications 17 0 Let h ∈ Em−1 (Ω) be chosen. For each j > 0 take mj > 0 such that uj ≥ mj h on 0 Ω . Put vj = max(uj , mj h) ∈ Em−1 (Ω) and vj = uj on Ω . Note that vj u on c p q c p q Ω and (dd vj ) ∧ β = (dd uj ) ∧ β on Ω for 1 ≤ p ≤ m and 1 ≤ q ≤ m − 1. Futhermore, we may assume that vj |Ω ≤ −1. Hence, |vj |m ≥ |vj |m−1 on Ω for all j ≥ 1. Now we have χ(uj )[|uj |m β m + |uj |m−1 ddc uj ∧ β m−1 + · · · + (ddc uj )m−1 ] ∧ β n−m Ω χ(vj )[|vj |m β m + |vj |m−1 ddc vj ∧ β m−1 + · · · + (ddc vj )m−1 ] ∧ β n−m = Ω χ(vj )[|vj |m−1 β m−1 + |vj |m−2 ddc vj ∧ β m−2 + · · · + (ddc vj )m−1 ] ∧ β n−m+1 . ≥ Ω Note that vj χ(vj )[|vj |m−1 β m−1 +|vj |m−2 ddc vj ∧β m−2 +· · ·+(ddc vj )m−1 ]∧β n−m+1 < +∞. sup j u on K and K Therefore, again by Theorem 4.6 we get u ∈ Em−1,χ (Ω). The proof is complete. 5 Applications for lower bounds of the log canonical threshold in the class Em (Ω) ∩ P SH(Ω) In this section by relying on the local property of the class Em (Ω) in the previous section and a recent result about a lower bound for the log canonical threshold of plurisubharmonic functions in the class E(Ω) presented in [DH] we give a lower bound for the log canonical threshold of plurisubharmonic functions in the class Em (Ω) ∩ P SH(Ω). First we need the following lemma. Lemma 5.1. Assume that u ∈ Em (Ω). Then for every 1 ≤ p ≤ m − 1 and for every K Ω we have u(ddc u)p ∧ β n−p > −∞. K Proof. By Corollary 4.7 we have Ep+1 (Ω) ⊂ Ep (Ω) for 1 ≤ p ≤ m − 1. Hence, it suffices to prove the lemma holds for p = m − 1. Moreover, we can assume that K = B(0, r0 ) Ω. Let r0 < r1 < r2 such that B(0, r2 ) Ω. By Theorem 4.6 we have u ∈ Em (B(0, r2 )). Choose v ∈ Fm (B(0, r2 )) such that u = v in B(0, r1 ). We have u(ddc u)m−1 ∧ β n−m+1 ≥ B(0,r0 ) v(ddc v)m−1 ∧ β n−m+1 B(0,r2 ) (|z|2 − r22 )(ddc v)m ∧ β n−m = B(0,r2 ) ≥ −r22 (ddc v)m ∧ β n−m > −∞. B(0,r2 ) The proof is complete. 18 L. M. Hai, N. X. Hong and V. V. Hung The following proposition is used for further results. Proposition 5.2. Assume that u ∈ Em (Ω) ∩ P SH − (Ω). Then (ddc u)p is a closed nonnegative current in Ω for p = 1, 2, . . . , m. Proof. For p = 1 the statement is clear. Assume that for 2 ≤ p ≤ m, (ddc u)p−1 is well defined as a closed nonnegative current. Since u(ddc u)p−1 has locally bounded mass and coefficients of (ddc u)p−1 are complex measures so u is locally integrable for these measures. Hence, as in [BT1] we can define (ddc u)p := ddc (u(ddc u)p−1 ). Thus (ddc u)p defines a closed current. Moreover, since u ∈ P SH(Ω) so (ddc u)p is nonnegative and the desired conclusion follows. Now we recall the following definitions introduced and investigated in [DeKo] and [DH]. Definition 5.3. Let u ∈ P SH(Ω) and 0 ∈ Ω. As in [DeKo] the log canonical threshold at 0 ∈ Ω of u is defined c(u) = sup{c > 0 : e−2cu is L1 on a neighbourhood of 0}. Moreover, for u ∈ P SH(Ω) ∩ Em (Ω) we define the intersection numbers (ddc u)j ∧ (ddc log|z|)n−j , j = 1, . . . , m. ej (u) = {0} Note that by Proposition 5.2 and Proposition 2.4 in [De] we have ej (u) < +∞. The following lemma is a crucial tool in this section. Lemma 5.4. Assume that 1 ≤ m < n and u ∈ Em (B(0, r))∩P SH(B(0, r)). Then there exists a subspace H of (n − 1)-dimension such that u|H ∈ Em (B(0, r) ∩ H) ∩ P SH(B(0, r) ∩ H) and ej (u) = ej (u|H ), j = 1, 2, . . . , m. Proof. By G(n − 1, n) we denote the Grassmann space of (n − 1)-dimensional subspaces in Cn . For each H ∈ G(n − 1, n) by [H] we denote the current induced by a hyperplane H. By the hypothesis and Crofton’s formula and the proof of Theorem 5.14 in [De] and Fubini’s theorem we infer that dV (H) H∈G(n−1,n) (ddc u)m ∧ β n−m−1 ∧ [H] B(0,r) (ddc u)m ∧ β n−m−1 ∧ = [H]dV (H) H∈G(n−1,n) B(0,r) (ddc u)m ∧ β n−m−1 ∧ ddc log |z| < +∞. = B(0,r) Hence, we have (ddc u)m ∧ β n−m−1 ∧ [H] < +∞, B(0,r) Local property of a class of m-subharmonic functions and applications 19 for almost H ∈ G(n − 1, n). On the other hand, Theorem 5.14 in [De] implies that ej (u|H ) = ej (u), j = 1, . . . , m for almost all (n − 1)-dimension subspaces H. Therefore, there is at least a (n − 1)-dimension subspace H such that ej (u|H ) = ej (u) for j = 1, . . . , m and (ddc u)m ∧ β n−m−1 ∧ [H] < +∞. B(0,r) Without loss of generality we can assume that H = {zn = 0} and u ∈ Fm (B(0, r))∩ P SH(B(0, r)). Let uj ∈ E0 (B(0, r)), uj u on B(0, r). Put vj (z ) = uj (z , 0), z = (z1 , z2 , . . . , zn−1 ). For 0 < t < r and by applying Lemma 4.4 to each p = 0, 1, 2, . . . , m − 1, we have (−uj )m−p (ddc uj )p ∧ β n−p−1 ∧ log |zn | B(0,t) (−uj )m−p−1 (ddc uj )p+1 ∧ β n−p−2 ∧ log |zn | ≤ r2 (m − p) B(0,t) ≤ ............ ≤ r2(m−p) (m − p)! (ddc uj )m ∧ β n−m−1 ∧ log |zn |. B(0,t) Therefore, we get (−vj )m−p (ddc vj )p ∧ (ddc |z |2 )n−p−1 lim sup j B(0,t)∩{zn =0} ≤ lim sup (−uj )m−p (ddc uj )p ∧ β n−p−1 ∧ log |zn | j B(0,t) ≤ r2(m−p) (m − p)! lim sup (ddc uj )m ∧ β n−m−1 ∧ log |zn | j B(0,t) ≤ r2(m−p) (m − p)! (ddc u)m ∧ β n−m−1 ∧ log |zn | < +∞. B(0,t) Thus it follows that (−vj )m−p (ddc vj )p ∧ (ddc |z |2 )n−p−1 < +∞. sup j B(0,t)∩{zn =0} Hence, by Theorem 4.6 we get u|{zn =0} ∈ Em (B(0, r) ∩ {zn = 0}). The proof is complete. Now we need the following which is a special case of the fundamental L2 extension theorem due to Ohsawa-Takeghoshi in [OT]. Proposition 5.5. Let ϕ ∈ P SH − (∆n ), ∆n = ∆n (0, 1) = {z ∈ Cn : |z| = max |zj | < 1} be a negative plurisubharmonic function with 1≤j≤n e−ϕ(z1 ,...,zm ,0,...,0) dV (z1 , . . . , zm , 0, . . . , 0) < +∞, 1 ≤ m ≤ n. {|z1 |[...]... for the complex Hessian equations, arXiv: 1112.3063v1 [G˚ a] L G˚ arding, An inequality for hyperbolic polynomials, J Math and Mec, Vol 8(1959), No.6, 957–965 [HH] Le Mau Hai, Pham Hoang Hiep, Some Weighted Energy Classes of Plurisubharmonic Functions, Potential Anal, 34(1)(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),... Ω Ω Ω and c is a constant only depending on Ω , Ω , Ω and χ Proof Repeating the proof of Lemma 3.2 in [HHQ] The next lemma is a crucial tool for the proof of the local property of the class Em,χ and for lower bounds of the log canonical threshold in the class Em (Ω) in section 5 12 L M Hai, N X Hong and V V Hung − (Ω) ∩ Lemma 4.4 Let Ω be a hyperconvex domain in Cn Assume that u ∈ SHm C(Ω) and χ.. .Local property of a class of m- subharmonic functions and applications 11 Hence, Hm (uj ) < +∞, sup j Ω and it follows that u ∈ Fm (Ω) It is easy to see that u = u on ω and this yields that u ∈ Em (Ω) Theorem 3.14 in [Ch] implies that Hm (u) is a positive Radon measure on Ω The proof is complete 4 The local property of the class Em,χ (Ω) Now we give the following definition which is similar as in... Local property of a class of m- subharmonic functions and applications 21 Acknowledgement The authors would like to thank Prof Pham Hoang Hiep for useful discusses which led to the improvement of the exposition of the paper The paper was done while the authors were visit to Vietnam Institute for Advanced Study in Mathematics (VIASM) The authors would like to thank the VIASM for hospitality and support... L M Hai, N X Hong and V V Hung [DeKo] J.-P Demailly and J Kollar, Semi-continuity of complex singularity exponents and K¨ ahler- Einstein metric on Fano orbifolds, Ann Sci Ecole Norm Sup., (4)34(2001), 525–556 [DH] J.-P Demailly and Pham Hoang Hiep, A sharp lower bound for the log canonical threshold, arXiv: 1201.4086v1, to appear in Acta Mathematica [DiKo] S Dinew and S Kolodziej, A priori estimates... Theorem 4.6 there exists a sequence {uj } ⊂ Em u j on Ω and χ(uj )[|uj |m β m + |uj |m 1 ddc uj ∧ β m 1 + · · · + (ddc uj )m ] ∧ β n m < +∞ sup j Ω Local property of a class of m- subharmonic functions and applications 17 0 Let h ∈ Em−1 (Ω) be chosen For each j > 0 take mj > 0 such that uj ≥ mj h on 0 Ω Put vj = max(uj , mj h) ∈ Em−1 (Ω) and vj = uj on Ω Note that vj u on c p q c p q Ω and (dd vj ) ∧ β =... the complex Monge-Amp`ere operator, Ann Inst Fourier (Grenoble), 54(2004), 159–179 [CKZ] U Cegrell, S Kolodziej and A Zeriahi, Subextension of plurisubharmonic functions with weak singularities, Math Z, 250(1)(2005), 7–22 [De] J.-P Demailly, Monge-Ampe`re operator, Lelong numbers and intersection theory, Complex Analysis and Geometry, Univ Series in Math, edited by V Ancona and A Silva, Plenum Press,... such that U is a hyperconvex domain, we have u|U ∈ Em,χ (U ) e) For every z ∈ Ω there exists a hyperconvex domain Vz Ω such that z ∈ Vz and u|Vz ∈ Em,χ (Vz ) t Proof Set χ0 (t) = χ(t) and for each k ≥ 1 , let χk (t) = − χk−1 (x)dx From the 0 hypothesis χ(2t) ≤ a (t) it is easy to check that χk ∈ K and χk (t) ≈ χ(t)(−t)k Local property of a class of m- subharmonic functions and applications 15 N a) ⇒b)... u) ∈ Fm,χ (Ω) Moreover, since max(v, u) = u on Ω so u ∈ Em,χ (Ω) The proof is complete From the above theorem we find out an interesting property of the class Em,χ (Ω) Corollary 4.7 Assume that Ω is a hyperconvex domain and χ ∈ K satisfies all hypotheses of Theorem 4.6 Then Em,χ (Ω) ⊂ Em−1,χ (Ω) Proof Assume that u ∈ Em,χ (Ω) Let K Ω Take a domain Ω with K Ω 0 (Ω)∩C(Ω) such that u Ω By Theorem 4.6... u )m ∧ β n m 1 ∧ ddc log |z| < +∞ = B(0,r) Hence, we have (ddc u )m ∧ β n m 1 ∧ [H] < +∞, B(0,r) Local property of a class of m- subharmonic functions and applications 19 for almost H ∈ G(n − 1, n) On the other hand, Theorem 5.14 in [De] implies that ej (u|H ) = ej (u), j = 1, , m for almost all (n − 1)-dimension subspaces H Therefore, there is at least a (n − 1)-dimension subspace H such that ej (u|H ... a compact subset of Ω implies that v ≤ u on Ω Local property of a class of m- subharmonic functions and applications By M SHm (Ω) we denote the set of m- maximal functions on Ω One of essential... uj ≤ vj By Lemma 3.3 we have χ(vj )(ddc vj )m ∧ β n m sup j Ω ≤ 2m max (a, 2) sup χ(uj )(ddc uj )m ∧ β n m < +∞ j Ω Local property of a class of m- subharmonic functions and applications Hence,... case m = n and χ(t) ≡ for all t < the classes Fn,χ (Ω) and En,χ (Ω) coincide with the classes F(Ω) and E(Ω) introduced and investigated in [Ce2] Local property of a class of m- subharmonic functions