Đang tải... (xem toàn văn)
Let Ω be a hyperconvex domain in C n . By PSH(Ω) (resp. PSH−(Ω)) we denote the cone of plurisubharmonic functions (resp. negative plurisubharmonic functions) on Ω. In 17, the authors introduced and investigated the notion of local class as follows. A class J (Ω) ⊂ PSH−(Ω) is said to be a local class if ϕ ∈ J (Ω) then ϕ ∈ J (D) for all hyperconvex domains D b Ω and if ϕ ∈ PSH−(Ω),ϕ|Ωi ∈ J (Ωi),∀i ∈ I with Ω = S i∈I Ωi then ϕ ∈ J (Ω). As is well known, Błocki (see 9) proved the class E (Ω) introduced and investigated by Cegrell in 11, is a local class. Moreover, in 11 Cegrell has proved this class is the biggest on which the complex MongeAmpere operator ` (ddc .) n is well defined as a Radon measure and it is continuous under decreasing sequences. On the other hand, another weighted energy class Eχ (Ω) which extends the classes Ep(Ω) and F(Ω) in 10 and 11 introduced and investigated recently by Benelkourchi, Guedj and Zeriahi in 5 is
Noname manuscript No. (will be inserted by the editor) Local property of a class of m-subharmonic functions Vu Viet Hung Received: date / Accepted: date 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. Keywords m-subharmonic functions, weighted energy classes of m-subharmonic functions, complex m-Hessian, local property. Mathematics Subject Classification (2010) 32U05, 32U15, 32U40, 32W20. 1 Introduction Let Ω be a hyperconvex domain in Cn . By PSH(Ω ) (resp. PSH − (Ω )) we denote the cone of plurisubharmonic functions (resp. negative plurisubharmonic functions) on Ω . In [17], the authors introduced and investigated the notion of local class as follows. A class J (Ω ) ⊂ PSH − (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 ∈ PSH − (Ω ), ϕ|Ωi ∈ J (Ωi ), ∀i ∈ I with Ω = Ωi i∈I then ϕ ∈ J (Ω ). As is well known, Błocki (see [9]) proved the class E (Ω ) introduced and investigated by Cegrell in [11], is a local class. Moreover, in [11] Cegrell has proved this class is the biggest on which the complex Monge-Amp`ere operator (dd c .)n is well defined as a Radon measure and it is continuous under decreasing sequences. On the other hand, another weighted energy class Eχ (Ω ) which extends the classes E p (Ω ) and F (Ω ) in [10] and [11] introduced and investigated recently by Benelkourchi, Guedj and Zeriahi in [5] is Vu Viet Hung Department of Mathematics, Physics and Informatics, Tay Bac University, Son La, Viet Nam E-mail: viethungtbu@gmail.com 2 V. V. Hung as follows. Let χ : R− −→ R+ be a decreasing function. Then, as in [5], we define Eχ (Ω ) = {ϕ ∈ PSH − (Ω ) : ∃ E0 (Ω ) ϕj χ(ϕ j )(dd c ϕ j )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 . Note that from Corollary 4.4 in z→ξ [4], it follows that if ϕ ∈ Eχ (Ω ) then lim ϕ(z) = 0 for all ξ ∈ ∂ Ω . Hence if ϕ ∈ Eχ (Ω ) then z→ξ 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 from the paper of Benelkourchi, Guedj and Zeriahi in [5] and on Cegrell classes of m-subharmonic functions introduced and studied recently in [13] we introduce weighted energy classes of m-subharmonic functions Fm,χ (Ω ) and Em,χ (Ω ). Under slight hypotheses for weights χ we achieve that the class Fm,χ (Ω ) is a convex cone (see Proposition 2 below). We also show that the complex Hessian operator Hm (u) = (dd c u)m ∧ β n−m is well defined on the class Em,χ (Ω ) where β = dd c 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 2 in Section 4 below). In this article, we prove the following main result. Theorem 4.6. Let Ω be a hyperconvex domain in Cn and m be an integer with 1 ≤ m ≤ n. Assume that u ∈ SHm− (Ω ) and χ ∈ K such that χ (t) ≥ 0, ∀t < 0. Then the following statements are equivalent. a) u ∈ Em,χ (Ω ). b) For all K Ω , there exists a sequence {u j } ⊂ Em0 (Ω ) ∩ C (Ω ), u j χ(u j )|u j | p (dd c u j )m−p ∧ β n−m+p < ∞, sup j u on K such that K for every p = 0, . . . , m. c) For every W Ω such that W is a hyperconvex domain, we have u|W ∈ Em,χ (W ). d) For every z ∈ Ω there exists a hyperconvex domain Vz Ω such that z ∈ Vz and u|Vz ∈ Em,χ (Vz ). Finally, using the main results above, we prove an interesting corollary. Namely, we have Corollary 4.7. Assume that Ω is a bounded hyperconvex domain and χ ∈ K satisfies all hypotheses of Theorem 2. Then Em,χ (Ω ) ⊂ Em−1,χ (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 m-subharmonic functions which were introduced and investigated intensively in recent years by many authors, see [6], [14], [24]. We also recall the Cegrell classes of m-subharmonic functions Fm (Ω ) and Em (Ω ) Local property of a class of m-subharmonic functions 3 introduced and studied in [13]. In Section 3 we introduce two new weighted energy classes of m-subharmonic functions Fm,χ (Ω ) and Em,χ (Ω ). Section 4 is devoted to the proof of the local property of the class Em,χ (Ω ) under some extra assumptions on weights χ. To show this property of the class Em,χ (Ω ) we need a result about subextension for the class Fm,χ (Ω ) (see Lemma 5 below) which is of independent interest. Finally, by relying on the local property of the class Em,χ (Ω ), we prove a corollary for this class. 2 Preliminairies Some elements of pluripotential theory that will be used throughout the paper can be found in [1], [19], [20], [23], while elements of the theory of m-subharmonic functions and the complex Hessian operator can be found in [6], [14], [24]. Now we recall the definition of some Cegrell classes of plurisubharmonic functions (see [10] and [11]), as well as, the class of m-subharmonic functions introduced by Błocki in [6] and the classes Em0 (Ω ) and Fm (Ω ) introduced and investigated by Lu Hoang Chinh in [13] recently. Let Ω be an open subset in Cn . By β = dd c z dVn = 1 n n! β 2 we denote the canonical K¨ahler form of Cn with the volume element where d = ∂ + ∂ and d c = ∂ −∂ 4i , hence, dd c = 2i ∂ ∂ . 2.1. As in [10] and [11] 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 ρ such that for all c < 0 the set Ωc = {z ∈ Ω : ρ(z) < c} Ω . Set E0 = E0 (Ω ) = {ϕ ∈ PSH − (Ω ) ∩ L∞ (Ω ) : lim ϕ(z) = 0, ∀ξ ∈ ∂ Ω , (dd c ϕ)n < ∞} z→ξ Ω and F = F (Ω ) = ϕ ∈ PSH − (Ω ) : ∃ E0 ϕj (dd c ϕ j )n < ∞ . ϕ, sup j Ω 2.2. We recall the class of m-subharmonic functions introduced and investigated in [6] 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. 4 V. V. Hung Definition 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 dd c u ∧ η1 ∧ . . . ∧ ηm−1 ∧ β n−m ≥ 0, holds in the sense of currents. By SHm (Ω ) (resp. SHm− (Ω )) we denote the cone of m−subharmonic functions (resp. negative m−subharmonic functions) on Ω . Before formulating basic properties of m-subharmonic, we recall the following (see [6]). For λ = (λ1 , . . . , λn ) ∈ Rn and 1 ≤ m ≤ n, define Sm (λ ) = ∑ λ j1 · · · λ jm . 1≤ j1 0 then αu + β v ∈ SH m (Ω ). d) If u, v ∈ SHm (Ω ) then so is max{u, v}. e) If {u j }∞j=1 is a family of m-subharmonic functions, u = sup u j < +∞ and u is upper j semicontinuous then u is a m-subharmonic function. f) If {u j }∞j=1 is a decreasing sequence of m-subharmonic functions then so is u = lim u j . j→+∞ Local property of a class of m-subharmonic functions 5 g) Let ρ ≥ 0 be a smooth radial function in Cn vanishing outside the unit ball and satisfying we define Cn ρdVn = 1, where dVn denotes the Lebesgue measure of Cn . For u ∈ SH m (Ω ) uε (z) := (u ∗ ρε )(z) = u(z − ξ )ρε (ξ )dVn (ξ ), ∀z ∈ Ωε , B(0,ε) where ρε (z) := 1 ε 2n ρ(z/ε) and Ωε = {z ∈ Ω : d(z, ∂ Ω ) > ε}. Then uε ∈ SHm (Ωε )∩C ∞ (Ωε ) and uε ↓ u as ε ↓ 0. h) Let u1 , . . . , u p ∈ SHm (Ω ) and χ : R p → 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 , . . . , u p ) ∈ SHm (Ω ). Example 1 Let u(z1 , z2 , z3 ) = 5|z1 |2 + 4|z2 |2 − |z3 |2 . By using b) of Proposition 1 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 [6] and [14], we define the complex Hessian operator of locally bounded msubharmonic functions as follows. ∞ (Ω ). Then the complex Hessian operDefinition 2 Assume that u1 , . . . , u p ∈ SHm (Ω ) ∩ Lloc ator Hm (u1 , . . . , u p ) is defined inductively by dd c u p ∧ · · · ∧ dd c u1 ∧ β n−m = dd c (u p dd c u p−1 ∧ · · · ∧ dd c u1 ∧ β n−m ). From the definition of m-subharmonic functions and using arguments as in the proof of Theorem 2.1 in [1] we note that Hm (u1 , . . . , u p ) 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, dd c u1 ∧ · · · ∧ dd c um ∧ β n−m is a ∞ (Ω ) the nonnegative Borel measure. In particular, when u = u1 = · · · = um ∈ SHm (Ω ) ∩ Lloc Borel measure Hm (u) = (dd c u)m ∧ β n−m , is well defined and is called the complex Hessian of u. 2.3. Similarly to in pluripotential theory now we recall a class of m-subharmonic functions introduced and investigated in [6] recently. Definition 3 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 . 6 V. V. Hung By MSHm (Ω ) we denote the set of m-maximal functions on Ω . Theorem 3.6 in [6] claims that a locally bounded m-subharmonic function u on a bounded domain Ω ⊂ Cn belongs to MSHm (Ω ) if and only if it solves the homogeneous Hessian equation Hm (u) = (dd c u)m ∧ β n−m = 0. 2.4. Next, we recall the classes Em0 (Ω ) and Fm (Ω ) introduced and investigated in [13]. 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. Set Em0 = Em0 (Ω ) = {u ∈ SHm− (Ω ) ∩ L∞ (Ω ) : lim u(z) = 0, Hm (u) < ∞}, z→∂ Ω Ω Fm = Fm (Ω ) = u ∈ SHm− (Ω ) : ∃ Em0 uj Hm (u j ) < ∞ , u, sup j Ω and Em = Em (Ω ) = u ∈ SHm− (Ω ) : ∀z0 ∈ Ω , ∃ a neighborhood ω Em0 uj z0 , and Hm (u j ) < ∞ , u on ω, sup j Ω where Hm (u) = (dd c u)m ∧ β n−m denotes the Hessian measure of u ∈ SHm− (Ω ) ∩L∞ (Ω ). From Theorem 3.14 in [13] it follows that if u ∈ Em (Ω ), the complex Hessian Hm (u) = (dd c u)m ∧β n−m is well defined and is a Radon measure on Ω . On the other hand, by Remark 3.6 in [13] we may give the following description of the class Em (Ω ): Em = Em (Ω ) = u ∈ SHm− (Ω ) : ∀ U Ω , ∃ v ∈ Fm (Ω ), v = u on U . 2.5. We recall the notion of m-capacity introduced in [13]. Definition 4 Let E ⊂ Ω be a Borel subset. The m-capacity of E with respect to Ω is defined by (dd c u)m ∧ β n−m : u ∈ SHm (Ω ), −1 ≤ u ≤ 0 . Cm (E) = Cm (E, Ω ) = sup E Local property of a class of m-subharmonic functions 7 Proposition 2.10 in [13] gives some elementary properties of the m-capacity similar as the capacity presented in [1]. Namely, we have: ∞ a) Cm ( j=1 b) If E j ∞ E j ) ≤ ∑ Cm (E j ). j=1 E then Cm (E j ) Cm (E). We need the following lemma which is used in the proof for the convexity of the class Em,χ (Ω ). Lemma 1 Assume that ϕ ∈ Em0 (Ω ). Then (dd c ϕ)m ∧ β n−m {ϕ < −t} ≤ t mCm {ϕ < −t} and t mCm {ϕ < −2t} ≤ (dd c ϕ) ∧ β n−m {ϕ < −t} . 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 By the comparision principle (Theorem 1.4 in [14]), we get (dd c v)m ∧ β n−m ≤ (dd c v)m ∧ β n−m { ϕt 1. Then for every u ∈ Fm,χ (Ω ), there exists a sequence {u j } ⊂ Em0 (Ω ) ∩ C (Ω ) such that u j u and χ(u j )(dd c u j )m ∧ β n−m < ∞. sup j Ω ∞ Proof Let Ω j Ω j+1 Ω such that Ω = j=1 Ω j and let {v j } ⊂ Em0 (Ω ) such that v j u and χ(v j )(dd c u j )m ∧ β n−m < ∞. sup j Ω Theorem 3.1 in [13] implies that there exists a sequence {w j } ⊂ Em0 (Ω ) ∩ C (Ω ) such that wj u. Set u j = sup{ϕ ∈ SHm− (Ω ) : ϕ ≤ It is easy to see that u j j−1 w j on Ω j }. j u on Ω . By Theorem 1.2.7 in [7] and Proposition 3.2 in [6] we get u j ∈ C (Ω ). Moreover, since w j ≤ u j so u j ∈ Em0 (Ω ) ∩ C (Ω ). Now, since v j u as 12 V. V. Hung k−1 k wk j → ∞ and u ≤ wk so there exists j0 such that v j0 ≤ on Ωk . Therefore, v j0 ≤ uk on Ω . Lemma 2 implies that χ(uk )(dd c uk )m ∧ β n−m ≤ 2m max(a, 2) Ω χ(v j0 )(dd c v j0 )m ∧ β n−m Ω m χ(v j )(dd c v j )m ∧ β n−m . ≤ 2 max(a, 2) sup j Ω Thus, χ(uk )dd c uk )m ∧ β n−m ≤ 2m max(a, 2) sup sup χ(v j )(dd c v j )m ∧ β n−m < ∞, j k Ω Ω The following proposition shows that the Hessian operator is well defined on the class Em,χ (Ω ). Proposition 4 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) = (dd c u)m ∧ β n−m is well defined as a positive Radon measure on Ω . Proof Without loss of generality we can assume that χ(t) > 0 for every t < 0. Let u ∈ Em,χ (Ω ) and z0 ∈ Ω . Take a neighbourhood ω such that sup u1 < 0, u j Ω of z0 and a sequence {u j } ⊂ Em0 (Ω ) u on ω and ω χ(u j )Hm (u j ) < ∞. sup j Ω For each j ≥ 1, set u j = sup{u ∈ SHm− (Ω ) : u|ω ≤ u j |ω }. Then u j ≤ u j on Ω and u j = u j on ω and, by using arguments as in [8], we arrive that u j ∈ MSHm (Ω \ ω). This yields that u j ∈ Em0 (Ω ) and Hm (u j ) = 0 on Ω \ ω. Moreover, it is easy to see that u j u on Ω . On the other hand, as in the proof of Lemma 2, we have χ(u j )Hm (u j ) < ∞. sup j Ω Moreover, we may assume that inf χ(u1 ) = c1 > 0. Then ω Hm (u j ) = c1 sup c1 sup j Hm (u j ) j Ω ω ≤ sup χ(u1 )Hm (u j ) ≤ sup j χ(u j )Hm (u j ) < ∞. j ω Ω Local property of a class of m-subharmonic functions 13 Hence, Hm (u j ) < ∞, 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 [13] implies that Hm (u) is a positive Radon measure on Ω . The proof is complete. Now we prove our main result about the local property of the class Em,χ (Ω ). 4 The local property of the class Em,χ (Ω ) First we give the following definition which is similar as in [17] for plurisubharmonic functions. Definition 6 A class J (Ω ) ⊂ SHm− (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 ∈ SHm− (Ω ), ϕ|Ω j ∈ J (Ω j ), ∀ j ∈ I with Ω j , then ϕ ∈ J (Ω ). Ω= j∈I In [17] the authors introduced the class Eχ,loc (Ω ) and established the local property for this class. This 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 5 Set K = {χ : R− −→ R+ , χ is deareasing and − t 2 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 χ p ∈ K for all p > 0. d) If χ ∈ K then (−t)χ(t) ∈ K . More generally |t k |χ(t) ∈ K for all k = 0, 1, 2, ... Proof The proof is standard hence we omit it. Remark 2 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). 14 V. V. Hung The following result is necessary for the proof of the local property of the class Em,χ (Ω ). ∞ c Lemma 3 Let u, v ∈ SH − m (Ω ) ∩ L (Ω ) with u ≤ v on Ω , χ ∈ K and T = dd ϕ1 ∧ · · · ∧ dd c ϕm−1 ∧ β n−m with ϕ j ∈ SHm− (Ω ) ∩ L∞ (Ω ), j = 1 . . . , m − 1. Then for every p ≥ 0 we have χ(u)dd c v ∧ T Ω where Ω Ω χ(u)(dd c u + |u|β ) ∧ T, c Ω Ω and c is a constant only depending on Ω , Ω , Ω and χ. Proof Choose Φ ∈ C0∞ (Ω ), 0 ≤ Φ ≤ 1 and Φ|Ω = 1, supp Φ Ω . Then, by inte- Ω gration by parts χ(u)dd c v ∧ T = Ω Φ χ(u)dd c v ∧ T ≤ Ω Ω vdd c (Φ χ(u)) ∧ T. Φ χ(u)dd c v ∧ T = Ω On the other hand dd c (Φ χ(u)) = d(d c (Φ χ(u))) = χ(u)dd c Φ + Φ(χ (u)dd c u + χ (u)du ∧ d c u) + χ (u)(dΦ ∧ d c u + du ∧ d c Φ). Since ∀t, d(u + tΦ) ∧ d c (u + tΦ) ∧ T ≥ 0 hence we have ±u(du ∧ d c Φ + dΦ ∧ d c u) ∧ T ≤ (du ∧ d c u + u2 dΦ ∧ d c Φ) ∧ T and 1 χ (u)(dΦ ∧ d c u + du ∧ Φ) ∧ T ≥ −χ (u)(udΦ ∧ d c Φ + du ∧ d c u) ∧ T. u Now, we can choose A > 0 sufficiently large such that dd c Φ ≥ −Add c z 2 , dΦ ∧ d c Φ ≤ Add c z 2 . Thus, we have the following estimates dd c (Φ χ(u)) ∧ T ≥ −Aχ(u)dd c z 2 ∧ T + Φ χ (u)dd c u ∧ T + Φ χ (u)du ∧ d c u ∧ T 1 − χ (u)(udΦ ∧ d c Φ + du ∧ d c u) ∧ T. u In the case χ (u) ≤ 0, we have the following vdd c (Φ χ(u)) ∧ T ≤ −Auχ(u)dd c z 2 ∧ T + uχ (u)dd c u ∧ T + u min{χ (u), 0}du ∧ d c u ∧ T − u2 χ (u)dΦ ∧ d c Φ ∧ T − χ (u)du ∧ d c u ∧ T. (1) Local property of a class of m-subharmonic functions 15 In the case χ (u) ≥ 0, from (1) and we note that Φvχ (u)du ∧ d c u ∧ T ≤ 0 and it is easy to obtain above estimates. Now, we have the following estimates χ(u)dd c v ∧ T ≤ A Ω −uχ(u)dd c z 2 uχ (u)dd c u ∧ T ∧T + Ω Ω −u2 χ (u)dΦ ∧ d c Φ ∧ T c u min{χ (u), 0}du ∧ d u ∧ T + + Ω Ω c −χ (u)du ∧ d u ∧ T. + Ω On other hand, by hypothesis about the class K we have uχ (u) ≤ c1 χ(u) and (−u2 )χ (u) ≤ c1 (−u)χ(u), uχ (u) ≤ c2 (−χ (u)). Therefore χ(u)dd c v ∧ T ≤ A Ω 2 −uχ(u)dd c z ∧ T + c1 Ω χ(u)dd c u ∧ T Ω χ (u)du ∧ d c u ∧ T + Ac1 − (c2 + 1) Ω 2 ∧T Ω |u|χ(u)dd c z = A(c1 + 1) 2 ∧ T + c1 Ω − (c2 + 1) χ(u)dΦ ∧ dd c z χ(u)dd c u ∧ T Ω c χ (u)du ∧ d u ∧ T. Ω t Set χ1 (t) = − χ(x)dx then 0 t |t| χ1 (t) = −χ(t); χ1 (t) = −χ (t); χ(t)|t| ≥ χ1 (t) ≥ χ( ) . 2 2 Now we choose ψ ∈ C0∞ , ψ|Ω = 1, supp ψ χ (u)du ∧ d c u ∧ T = − − Ω Ω , then we have dχ(u) ∧ du ∧ d c u ∧ T ≤ Ω Ω c c χ(u)dψ ∧ d u ∧ T + = Ω dψd c χ1 (u) ∧ T + Ω ≤B Ω ψ χ(u)dd c u ∧ T = χ(u)|u|dd z 2 ∧T + Ω with B > 0 sufficiently large. Ω c ψ χ(u)dd u ∧ T ψ χ(u)dd c u ∧ T Ω χ1 (u)dd c ψ ∧ T + Ω Ω c χ(u)dψ ∧ d c u ∧ T + ψ χ(u)dd u ∧ T = Ω =− ψdχ(u) ∧ d c u ∧ T Ω ψ χ(u)dd c u ∧ T 16 V. V. Hung Finally, we have χ(u)dd c v ∧ T ≤ A(c1 + 1) Ω |u|χ(u)dd c z 2 Ω + (c2 + 1)B χ(u)dd c u ∧ T ∧ T + c1 Ω c χ(u)|u|dd z 2 ∧ T + (c2 + 1) Ω Ω c ≤c χ(u)dd c u ∧ T c χ(u)dd u ∧ T + Ω χ(u)|u|dd z 2 ∧T . Ω The next lemma is a crucial tool for the proof of the local property of the class Em,χ (Ω ). Lemma 4 Let Ω be a hyperconvex domain in Cn and 1 ≤ m ≤ n. Assume that u ∈ Em0 (Ω ) and χ ∈ K such that χ (t) ≥ 0, ∀t < 0. Then for Ω Ω there exists a constant C = C(Ω ) such that the following holds: χ(u)(dd c u)m ∧ β n−m < +∞. χ(u)|u| p (dd c u)m−p ∧ β n−m+p ≤ C (1) Ω Ω Futhermore, if u ∈ Fm,χ (Ω ) then χ(u)|u| p (dd c u)m−p ∧ β n−m+p < +∞ Ω for all p = 1, . . . , m. t Proof Set χ0 (t) = χ(t) and for each k ≥ 1, let χk (t) = − χk−1 (x)dx. From the hypothesis 0 χ ∈ K then χ(2t) ≤ aχ(t) and it is easy to check that χk ∈ K and χ(t)(−t)k χk (t) χ(t)(−t)k . Now, choose R > 0 large enough such that z such that z dd c z 2 2 − R2 2 ≥ Aϕ on Ω . Set h = max( z ≤ R2 on Ω . Let ϕ ∈ Em0 (Ω ) and A > 0 2 − R2 ; Aϕ) then h ∈ Em0 (Ω ) and dd c h = = β on Ω . First, we claim that (1) holds for u ∈ Em0 (Ω ). Indeed, we have χ(u)|u| p (dd c u)m−p ∧ (dd c h) p ∧ β n−m χ(u)|u| p (dd c u)m−p ∧ (dd c h) p ∧ β n−m Ω Ω χ p (u)(dd c u)m−p ∧ (dd c h) p ∧ β n−m . ≈ Ω Local property of a class of m-subharmonic functions 17 By integration by parts we have χ p (u)(dd c u)m−p ∧ (dd c h) p ∧ β n−m Ω h(dd c u)m−p dd c χ p (u) ∧ (dd c h) p−1 ∧ β n−m = Ω h(dd c u)m−p [χ p (u)du ∧ d c u + χ p (u)dd c u] ∧ (dd c h) p−1 ∧ β n−m = Ω hχ p (u)(dd c u)m−p+1 ∧ (dd c h) p−1 ∧ β n−m ≤ Ω ≤ h χ p−1 (dd c u)m−p+1 ∧ (dd c h) p−1 ∧ β n−m L∞ (Ω ) Ω ≤ ········· ≤ h p L∞ (Ω ) χ(u)(dd c u)m ∧ β n−m < +∞. Ω Hence, if we set C = C(Ω ) = p! h p L∞ (Ω ) then χ(u)(dd c u)m ∧ β n−m ≥ +∞ > C Ω χ(u)|u| p (dd c u)m−p ∧ (dd c h) p ∧ β n−m Ω χ(u)|u| p (dd c u)m−p ∧ (dd c h) p ∧ β n−m ≥ Ω χ(u)|u| p (dd c u)m−p ∧ (dd c z 2 ) p ∧ β n−m . = Ω Finally, we prove (1) holds for u ∈ Fm,χ (Ω ). Indeed, we take u j ∈ Em0 (Ω ), u j such that χ(u j )(dd c u j )m ∧ β n−m < +∞. sup j≥1 Ω u on Ω 18 V. V. Hung By dominated convergence theorem and (dd c u j )m−p ∧ (dd c z 2 )n−m+p is weakly convergent to (dd c u)m−p ∧ (dd c z 2 )n−m+p in the sense of currents χ(u)|u| p (dd c u)m−p ∧ (dd c z 2 )n−m+p Ω χ(u j )|u j | p (dd c u j )m−p ∧ (dd c z 2 )n−m+p ≤ lim inf j Ω χ(u j )|u j | p (dd c u j )m−p ∧ (dd c h) p ∧ (dd c z 2 )n−m ≤ lim inf j Ω χ(u j )(dd c u j )m ∧ (dd c z 2 )n−m < +∞. ≤ C sup j Ω We also need a following result on subextension for the class Fm,χ (Ω ). Lemma 5 Assume that Ω Ω and u ∈ Fm,χ (Ω ). Then there exists a u ∈ Fm,χ (Ω ) such that u ≤ u on Ω . Proof We split the proof into three steps. Step 1. We prove that if v ∈ C (Ω ), v ≤ 0, suppv Ω then v := sup{w ∈ SHm− (Ω ) : w ≤ v on Ω } ∈ Em0 (Ω ) ∩ C (Ω ) and (dd c v)m ∧ β n−m = 0 on {v < v}. Indeed, let ϕ ∈ Em0 (Ω ) ∩ C (Ω ) such that ϕ ≤ inf v on suppv. Since ϕ ≤ v so v ∈ Em0 (Ω ). Moreover, by Proposition Ω 3.2 in [6] we have v ∈ C (Ω ). Let w ∈ SHm ({v < v}) such that w ≤ v outside a compact subset K of {v < v}. Set w1 = max(w, v) v on {v < v} on Ω \({v < v}). Since v and v is continuous so ε = − sup(v − v) > 0. Choose δ ∈ (0, 1) such that −δ infΩ v < K ε. We have (1 − δ )v ≤ v + ε ≤ v on K. Hence, (1 − δ )v + δ w1 ≤ v on Ω , and, we get (1 − δ )v + δ w1 = v. Thus, w ≤ v on {v < v}. Hence, v is m-maximal in {v < v}. By [6] we get (dd c v)m ∧ β n−m = 0 on {v < v}. Step 2. Next, we prove that if u ∈ Em0 (Ω ) ∩ C (Ω ) then there exists u ∈ Em0 (Ω ), (dd c u)m ∧ β n−m = 0 on (Ω \Ω ) ∪ ({u < u} ∩ Ω ) and (dd c u)m ∧ β n−m ≤ (dd c u)m ∧ β n−m on {u = u} ∩ Ω . Indeed, set v= u 0 It is easy to see that v ∈ C (Ω ) and suppv ⊂ Ω on Ω on Ω \Ω . Ω . Hence, we have u = v ∈ Em0 (Ω ) ∩ C (Ω ) and (dd c u)m ∧ β n−m = 0 on {v < v} ∩ Ω = (Ω \Ω ) ∪ ({u < u} ∩ Ω ). Let K be a compact Local property of a class of m-subharmonic functions 19 set in {u = u} ∩ Ω . Then for ε > 0 we have K (dd c u)m ∧ β n−m = K {u + ε > u} ∩ Ω so we have 1{u+ε>u} (dd c u)m ∧ β n−m K 1{u+ε>u} (dd c max(u + ε, u))m ∧ β n−m = K (dd c max(u + ε, u))m ∧ β n−m , ≤ K where the equality in the second line follows by using the same arguments as in [2]( also see the proof of Theorem 3.23 in [13]). However, max(u + ε, u) [24] it follows that (dd c max(u + ε, u))m ∧ β n−m u on Ω as ε → 0 so by is weakly convergent to (dd c 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 ϕ j . Hence, we infer that 1K (dd c max(u + ε, u))m ∧ β n−m lim sup ε→0 Ω ϕ j (dd c max(u + ε, u))m ∧ β n−m = lim sup lim ε→0 j Ω ϕ j (dd c max(u + ε, u))m ∧ β n−m ≤ lim sup ε→0 Ω ϕ j (dd c u)m ∧ β n−m ≤ (dd c u)m ∧ β n−m . K Ω as j → ∞. This yields that (dd c u)m ∧ β n−m ≤ (dd c u)m ∧ β n−m Step 3. Now, let u j ∈ Em0 (Ω ) ∩ C (Ω ) such that u j on {u = u} ∩ Ω . u and χ(u j )(dd c u j )m ∧ β n−m < ∞. sup j Ω By Step 2, we have χ(u j )(dd c u j )m ∧ β n−m = χ(u j )(dd c u j )m ∧ β n−m {u j =u j }∩Ω Ω χ(u j )(dd c u j )m ∧ β n−m ≤ {u j =u j }∩Ω χ(u j )(dd c u j )m ∧ β n−m . ≤ Ω Hence, χ(u j )(dd c u j )m ∧ β n−m ≤ sup sup j χ(u j )(dd c u j )m ∧ β n−m < ∞. j Ω Ω 20 V. V. Hung Thus, u := lim u j ∈ Fm,χ (Ω ) and u ≤ u on Ω . j→∞ The following result deals with the local property of the class Em,χ (Ω ). Namely, we have the following. Theorem 2 Let Ω be a hyperconvex domain in Cn and m be an integer with 1 ≤ m ≤ n. Assume that u ∈ SHm− (Ω ) and χ ∈ K such that χ (t) ≥ 0, ∀t < 0. Then the following statements are equivalent. a) u ∈ Em,χ (Ω ). Ω , there exists a sequence {u j } ⊂ Em0 (Ω ) ∩ C (Ω ), u j b) For all K χ(u j )|u j | p (dd c u j )m−p ∧ β n−m+p < ∞, sup j u on K such that K for every p = 0, . . . , m. c) For every W Ω such that W is a hyperconvex domain, we have u|W ∈ Em,χ (W ). d) For every z ∈ Ω there exists a hyperconvex domain Vz Ω such that z ∈ Vz and u|Vz ∈ Em,χ (Vz ). Proof Let χk as in Lemma 4. a)=⇒b) Let K Ω be given. Since u ∈ Em,χ (Ω ) then there exists v ∈ Fm,χ (Ω ) with v = u on K. By the definition of the class Fm,χ (Ω ) there exists a sequence {u j } ⊂ Em0 (Ω ) ∩ C (Ω ), u j v on Ω with χ(u j )(dd c u j )m ∧ β n−m < ∞. sup (3.1) j Ω Then u j u on K. We have to prove χ(u j )|u j | p (dd c u j )m−p ∧ β n−m+p < ∞, sup j K for p = 0, 1, . . . , m. It is obvious the conclusion holds for p = 0. Assume that 1 ≤ p ≤ m. Then, by Lemma 4 we get that χ(u j )|u j | p (dd c u j )m−p ∧ β n−m+p ≤ C sup sup j χ(u j )(dd c u j )m ∧ β n−m < ∞, j K Ω and the desired conclusion follows. b)=⇒c) Let W Em0 (Ω ) uj Ω be a hyperconvex domain. Take U W u on W such that sup j W χ(u j )|u j | p (dd c u j )m−p ∧ β n−m+p < ∞, Ω and a sequence Local property of a class of m-subharmonic functions 21 for p = 0, 1, . . . , m. Set u j = sup{ϕ ∈ SHm− (W ) : ϕ ≤ u j on U} ∈ Em0 (W ). Next, choose U ... Ω1 W . Since u j ≤ u j on W and (dd c u j )m ∧ β n−m = 0 on W \U so by applying Ωm Lemma 3 many times we arrive that χ(u j )(dd c u j )m ∧ β n−m W χ(u j )(dd c u j )m ∧ β n−m = U χ(u j )(dd c u j + |u j |β ) ∧ (dd c u j )m−1 ∧ β n−m Ω1 χ(u j )dd c u j ∧ (dd c u j )m−2 ∧ dd c u j ∧ β n−m + Ω1 χ1 (u j )|u j |dd c u j ∧ (dd c u j )m−2 ∧ β n−m+1 Ω1 χ(u j )(dd c u j + |u j |β ) ∧ (dd c u j )m−2 ∧ dd c u j ∧ β n−m Ω2 χ1 (u j )|u j |(dd c u j + |u j |β ) ∧ (dd c u j )m−2 ∧ β n−m+1 + Ω2 χ(u j ) |u j |2 β 2 + |u j |β ∧ dd c u j + (dd c u j )2 ∧(dd c u j )m−2 ∧ β n−m Ω2 .................................................................. χ(u j )[|u j |m β m + |u j |m−1 dd c u j ∧ β m−1 + · · · + (dd c u j )m ] ∧ β n−m . Ωm Hence, χ(u j )(dd c u j )m ∧ β n−m sup j W sup χ(u j ) [|u j |m β m + |u j |m−1 dd c u j ∧ β m−1 + · · · + (dd c u j )m ] ∧ β n−m j Ωm sup j χ(u j )[|u j |m β m + |u j |m−1 dd c u j ∧ β m−1 + · · · + (dd c u j )m ] ∧ β n−m W < ∞. Thus, uU,W := lim u j ∈ Fm,χ (W ). Since U W is arbitrary and uU,W = u on U so u ∈ Em (W ). c)=⇒d) It is obvious. d)=⇒a) Assume that Ω Ω . Choose z j ∈ Ω , j = 1, 2, . . . , s such that Ω where Vz j are hyperconvex domains. Let Wz j Vz j such that Ω s j=1 Wz j . s j=1 Vz j , Since u|Vz j ∈ Em,χ (Vz j ) so there exists v j ∈ Fm,χ (Vz j ) such that v j = u on Wz j . By Lemma 5 there exists v j ∈ Fm,χ (Ω ) such that v j ≤ v j on Vz j . Then by Proposition 2 we have v := v1 + · · · + 22 V. V. Hung vs ∈ Fm,χ (Ω ) and, hence, max(v, u) ∈ Fm,χ (Ω ). However, max(v, u) = u on Ω then u ∈ Em,χ (Ω ). The proof is complete. From the above theorem we get the following property of the class Em,χ (Ω ). Corollary 1 Assume that Ω is a bounded hyperconvex domain and χ ∈ K satisfies all hypotheses of Theorem 2. Then Em,χ (Ω ) ⊂ Em−1,χ (Ω ). Proof Assume that u ∈ Em,χ (Ω ). Let K Ω . Take a domain Ω with Ω 2 there exists a sequence {u j } ⊂ Em0 (Ω ) ∩ C (Ω ) such that u j sup Ω . By Theorem u on Ω and χ(u j )[|u j |m β m + |u j |m−1 dd c u j ∧ β m−1 + · · · + (dd c u j )m ] ∧ β n−m < ∞. j Ω Let h ∈ 0 (Ω ) Em−1 be chosen. For each j > 0 take m j > 0 such that u j ≥ m j h on Ω . Set 0 (Ω ) and v = u on Ω . Note that v v j = max(u j , m j h) ∈ Em−1 j j j u on Ω and (dd c v j ) p ∧ β q = (dd c u j ) p ∧ β q on Ω for 1 ≤ p ≤ m − 1 and 1 ≤ q ≤ n − m + 1. We may assume that u|Ω ≤ −1. By Hartogs’ lemma (see Theorem 3.2.13 in [18]) we conclude that v j |Ω ≤ −1 for j ≥ j0 with some j0 . Without loss of generality, we may assume that v j |Ω ≤ −1 for j ≥ 1. Hence, |v j |m ≥ |v j |m−1 on Ω for all j ≥ 1. Now we have χ(u j )[|u j |m β m + |u j |m−1 dd c u j ∧ β m−1 + · · · + |u j |(dd c u j )m−1 ∧ β + (dd c u j )m ] ∧ β n−m Ω χ(u j )[|u j |m β m + |u j |m−1 dd c u j ∧ β m−1 + · · · + |u j |(dd c u j )m−1 ∧ β ] ∧ β n−m ≥ Ω χ(v j )[|v j |m β m + |v j |m−1 dd c v j ∧ β m−1 + · · · + |v j |(dd c v j )m−1 ∧ β ] ∧ β n−m = Ω χ(v j )[|v j |m β m−1 + |v j |m−1 dd c v j ∧ β m−2 + · · · + |v j |(dd c v j )m−1 ] ∧ β n−m+1 = Ω χ(v j )[|v j |m−1 β m−1 + |v j |m−2 dd c v j ∧ β m−2 + · · · + (dd c v j )m−1 ] ∧ β n−m+1 . ≥ Ω Note that v j sup u on Ω and χ(v j )[|v j |m−1 β m−1 + |v j |m−2 dd c v j ∧ β m−2 + · · · + (dd c v j )m−1 ] ∧ β n−m+1 < ∞. j Ω Moreover, by Theorem 2 we get u ∈ Em−1,χ (Ω ). Acknowledgements The paper was done while the author was visiting to Vietnam Institute for Advanced Study in Mathematics (VIASM) from May to June 2013. The author would like to thank the VIASM for hospitality and support. The author would like to thank Prof. Le Mau Hai for useful discussions which led to the improvement of the exposition of the paper. The author is also indebted to the referees for their useful comments that led to improvements in the exposition of the paper. Local property of a class of m-subharmonic functions 23 References 1. Bedford, E., Taylor, B. A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982). ˘ 2. Bedford, E., Taylor, B. A.: Fine topology, Silov boundary, and (dd c )n . J. Funct. Anal. 72, 225–251 (1987). 3. Bedford, E., Taylor, B. A.: Plurisubharmonic functions with logarithmic singularities. Ann. Inst. Fourier 38, 133–171 (1988). 4. Benelkourchi, S.: Weighted Pluricomplex Energy. Potential Analysis 31, 1–20 (2009). 5. Benelkourchi, S., Guedj, V., Zeriahi, A.: Plurisubharmonic functions with weak singularities. Complex Analysis and Digital Geometry Proceedings from the Kiselmanfest, Uppsala Universitet. ISSN 05027454, 57–73 (2007). 6. 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Soc. 178, 64p (2005). 24. Sadullaev, A. S., Abullaev, B. I.: Potential theory in the class of m-subharmonic functions. Trudy Mathematicheskogo Instituta imeni V. A. Steklova 279, 166–192 (2012). 25. Yau, S. T.: On the Ricci curvature of a compact K¨ahler manifold and the complex Monge-Amp`ere equation. Com. Pure Appl. Math. 31, 339-411 (1978). [...]... j ω Ω Local property of a class of m-subharmonic functions 13 Hence, Hm (u j ) < ∞, 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 [13] implies that Hm (u) is a positive Radon measure on Ω The proof is complete Now we prove our main result about the local property of the class Em,χ (Ω ) 4 The local property of the class Em,χ... would like to thank Prof Le Mau Hai for useful discussions which led to the improvement of the exposition of the paper The author is also indebted to the referees for their useful comments that led to improvements in the exposition of the paper Local property of a class of m-subharmonic functions 23 References 1 Bedford, E., Taylor, B A. : A new capacity for plurisubharmonic functions Acta Math 149, 1–40... range of the complex Monge-Amp`ere operator II Indiana Univ Math J 44, 765– 782 (1995) 21 Kołodziej, S.: The Monge-Amp`ere equation Acta Math 180, 69-117 (1998) 22 Kołodziej, S.: The Monge-Amp`ere equation on compact K¨ahler manifolds Indiana Univ Math J 52, 667-686 (2003) 23 Kołodziej, S.: The complex Monge-Amp`ere equation and pluripotential theory Mem Amer Math Soc 178, 64p (2005) 24 Sadullaev, A. .. similar as in [17] for plurisubharmonic functions Definition 6 A class J (Ω ) ⊂ SHm− (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 ∈ SHm− (Ω ), ϕ|Ω j ∈ J (Ω j ), ∀ j ∈ I with Ω j , then ϕ ∈ J (Ω ) Ω= j∈I In [17] the authors introduced the class Eχ,loc (Ω ) and established the local property for this class This section is devoted to study the local. .. Some Weighted Energy Classes of Plurisubharmonic Functions Potential Analysis 34, 43–56 (2011) 17 Hai, M L, Hiep, P H, Quy, H N: Local property of the class Eχ,loc J Math Anal Appl 402, 440–445 (2013) 18 H¨ormander, L.: Notions of convexity Prog Math., Birkh¨auser (1994) 19 Klimek, M.: Pluripotential Theory, The Clarendon Press Oxford University Press, New York, Oxford Science Publications (1991) 20 Kołodziej,... (1) Local property of a class of m-subharmonic functions 15 In the case χ (u) ≥ 0, from (1) and we note that Φvχ (u)du ∧ d c u ∧ T ≤ 0 and it is easy to obtain above estimates Now, we have the following estimates χ(u)dd c v ∧ T ≤ A Ω −uχ(u)dd c z 2 uχ (u)dd c u ∧ T ∧T + Ω Ω −u2 χ (u)dΦ ∧ d c Φ ∧ T c u min{χ (u), 0}du ∧ d u ∧ T + + Ω Ω c −χ (u)du ∧ d u ∧ T + Ω On other hand, by hypothesis about the class. .. Subextension of plurisubharmonic functions with weak singularities Math Z 250, 7–22 (2005) 13 Chinh, L H: On Cegrell’s classes of m-subharmonic functions arXiv 1301.6502v1 (2013) 14 Dinew, S., Kołodziej, S.: A priori estimates for the complex Hessian equations Analysis & PDE 7, 227–244 (2014) 15 G˚arding, L.: An inequality for hyperbolic polynomials J Math and Mec 8, 957–965 (1959) 16 Hai, M L, Hiep,... pluripotential theory Mem Amer Math Soc 178, 64p (2005) 24 Sadullaev, A S., Abullaev, B I.: Potential theory in the class of m-subharmonic functions Trudy Mathematicheskogo Instituta imeni V A Steklova 279, 166–192 (2012) 25 Yau, S T.: On the Ricci curvature of a compact K¨ahler manifold and the complex Monge-Amp`ere equation Com Pure Appl Math 31, 339-411 (1978) ... ) and u ≤ u on Ω j→∞ The following result deals with the local property of the class Em,χ (Ω ) Namely, we have the following Theorem 2 Let Ω be a hyperconvex domain in Cn and m be an integer with 1 ≤ m ≤ n Assume that u ∈ SHm− (Ω ) and χ ∈ K such that χ (t) ≥ 0, ∀t < 0 Then the following statements are equivalent a) u ∈ Em,χ (Ω ) Ω , there exists a sequence {u j } ⊂ Em0 (Ω ) ∩ C (Ω ), u j b) For all... 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 5 Set K = {χ : R− −→ R+ , χ is deareasing and − t 2 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 ... investigated intensively in recent years by many authors, see [6], [14], [24] We also recall the Cegrell classes of m-subharmonic functions Fm (Ω ) and Em (Ω ) Local property of a class of m-subharmonic... comments that led to improvements in the exposition of the paper Local property of a class of m-subharmonic functions 23 References Bedford, E., Taylor, B A. : A new capacity for plurisubharmonic functions. .. Energy Classes of Plurisubharmonic Functions Potential Analysis 34, 43–56 (2011) 17 Hai, M L, Hiep, P H, Quy, H N: Local property of the class Eχ,loc J Math Anal Appl 402, 440–445 (2013) 18 H¨ormander,