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.
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1 Introduction
Let Ω be a hyperconvex domain in \(\mathbb {C}^{n}\). By P S H(Ω) (resp. P S H −(Ω)), we denote the cone of plurisubharmonic functions (resp. negative plurisubharmonic functions) on Ω. In [15], the authors introduced and investigated the notion of local class as follows. A class \(\mathcal {J}({\Omega })\subset PSH^{-}({\Omega })\) is said to be a local class if \(\varphi \!\in \!\mathcal {J}({\Omega })\) then \(\varphi \!\in \!\mathcal {J}(D)\) for all hyperconvex domains D ⋐ Ω and if \(\varphi \!\in \! PSH^{-}({\Omega }), \varphi |_{{\Omega }_{i}}\!\in \!\mathcal {J}({\Omega }_{i})~\forall i\!\in \! I\) with \({\Omega }\,=\,\bigcup _{i\in I}{\Omega }_{i}\) then \(\varphi \!\in \!\mathcal {J}({\Omega })\). As is well known, Błocki (see [8]) proved the class \(\mathcal {E}({\Omega })\) introduced and investigated by Cegrell in [10], is a local class. Moreover, in [10], Cegrell has proved this class is the biggest on which the complex Monge–Ampère operator (d d c.)n is well defined as a Radon measure, and it is continuous under decreasing sequences. On the other hand, another weighted energy class \(\mathcal {E}_{\chi }({\Omega })\) which extends the classes \(\mathcal {E}_{p}({\Omega })\) and \(\mathcal {F}({\Omega })\) in [9] and [10] introduced and investigated recently by Benelkourchi et al. [4] is as follows. Let χ : ℝ− → ℝ+ be a decreasing function. Then, as in [5], we define
where \(\mathcal {E}_{0}({\Omega })\) is the cone of bounded plurisubharmonic functions φ defined on Ω with finite total Monge–Ampère mass and \(\lim _{z\to \xi }\varphi (z) = 0\) for all ξ ∈ ∂Ω. Note that from Corollary 4.4 in [3], it follows that if \(\varphi \in \mathcal {E}_{\chi }({\Omega })\) then \(\lim _{z\to \xi }\varphi (z) =0\) for all ξ ∈ ∂Ω. Hence, if \(\varphi \in \mathcal {E}_{\chi }({\Omega })\), then \(\varphi \notin \mathcal {E}_{\chi }(D)\) with D a relatively compact hyperconvex domain in Ω. Thus, the class \(\mathcal {E}_{\chi }({\Omega })\) is not a “local” one. In this paper, by relying on ideas from the paper of Benelkourchi et al. [4] and on Cegrell classes of m-subharmonic functions introduced and studied recently in [12], we introduce weighted energy classes of m-subharmonic functions \(\mathcal {F}_{m,\chi }({\Omega })\) and \(\mathcal {E}_{m,\chi }({\Omega })\). Under slight hypotheses for weights χ, we achieve that the class \(\mathcal {F}_{m,\chi }({\Omega })\) is a convex cone (see Proposition 2 below). We also show that the complex Hessian operator H m (u) = (d d c u)m ∧ β n−m is well defined on the class \(\mathcal {E}_{m,\chi }({\Omega })\) where β = d d c∥z∥2 denotes the canonical Kähler form of \(\mathbb {C}^{n}\). Furthermore, we prove that the class \(\mathcal {E}_{m,\chi }({\Omega })\) is a local class (see Theorem 1 in Section 4 below). In this article, we prove the following main result.
Theorem 1
Let Ω be a hyperconvex domain in \(\mathbb C^{n}\) and m be an integer with 1 ≤ m ≤ n. Assume that \(u\in SH^{-}_{m}({\Omega })\) and \(\chi \in \mathcal K\) such that \(\chi ^{\prime \prime }(t)\!\geq \! 0~ \forall t<0\) . Then the following statements are equivalent.
-
a)
\(u\in \mathcal E_{m,\chi }({\Omega })\)
-
b)
.For all \(K\Subset {\Omega }\) , there exists a sequence \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C({\Omega }), u_{j}\searrow u\) on K such that
$$\sup_{j}{\int}_{K}\chi(u_{j})|u_{j}|^{p}(dd^{c}u_{j})^{m-p}\wedge\beta^{n-m+p}<\infty $$for every p = 0, … , m.
-
c)
For every \(W\Subset {\Omega }\) such that W is a hyperconvex domain, we have \(u|_{W}\in \mathcal E_{m,\chi }(W)\).
-
d)
For every z ∈ Ω, there exists a hyperconvex domain \(V_{z}\Subset {\Omega }\) such that z ∈ V z and \(u|_{V_{z}}\in \mathcal E_{m,\chi }(V_{z})\).
Finally, using the main results above, we prove an interesting corollary. Namely, we have
Corollary 1
Assume that Ω is a bounded hyperconvex domain, and \(\chi \in \mathcal K\) satisfies all hypotheses of Theorem 1. Then \(\mathcal E_{m,\chi }({\Omega })\subset \mathcal E_{m-1,\chi }({\Omega })\).
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 [5, 13, 21]. We also recall the Cegrell classes of m-subharmonic functions \(\mathcal {F}_{m}({\Omega })\) and \(\mathcal {E}_{m}({\Omega })\) introduced and studied in [12]. In Section 3, we introduce two new weighted energy classes of m-subharmonic functions \(\mathcal {F}_{m,\chi }({\Omega })\) and \(\mathcal {E}_{m,\chi }({\Omega })\). Section 4 is devoted to the proof of the local property of the class \(\mathcal {E}_{m,\chi }({\Omega })\) under some extra assumptions on weights χ. To show this property of the class \(\mathcal {E}_{m,\chi }({\Omega })\), we need a result about subextension for the class \(\mathcal {F}_{m,\chi }({\Omega })\) (see Lemma 5 below) which is of independent interest. Finally, by relying on the local property of the class \(\mathcal {E}_{m,\chi }({\Omega })\), we prove a corollary for this class.
2 Preliminaries
Some elements of pluripotential theory that will be used throughout the paper can be found in [1, 17, 18, 20], while elements of the theory of m-subharmonic functions and the complex Hessian operator can be found in [5, 13, 21]. Now, we recall the definition of some Cegrell classes of plurisubharmonic functions (see [9] and [10]), as well as the class of m-subharmonic functions introduced by Błocki in [5] and the classes \(\mathcal {E}^{0}_{m}({\Omega })\) and \(\mathcal {F}_{m}({\Omega })\) introduced and investigated by Chinh in [12] recently. Let Ω be an open subset in \(\mathbb {C}^{n}\). By β = d d c∥z∥2, we denote the canonical Kähler form of \(\mathbb {C}^{n}\) with the volume element \(dV_{n}= \frac {1}{n!}\beta ^{n}\) where \(d= \partial +\overline {\partial }\) and \(d^{c} =\frac {\partial - \overline {\partial }}{4i}\), hence \(dd^{c} = \frac {i}{2}\partial \overline {\partial }\).
2.1 The Cegrell Classes
As in [9, 10], we define the classes \(\mathcal {E}_{0}({\Omega })\) and \(\mathcal {F}({\Omega })\) 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 \({\Omega }_{c}=\{z\in {\Omega }: \varrho (z)< c\}\Subset {\Omega }\). Set
and
2.2 m-Subharmonic Functions
We recall the class of m-subharmonic functions introduced and investigated in [5] recently. For 1 ≤ m ≤ n, we define
where \(\mathbb {C}_{(1,1)}\) denotes the space of (1,1)-forms with constant coefficients.
Definition 1
Let u be a subharmonic function on an open subset \({\Omega }\subset \mathbb {C}^{n}\). u is said to be a m-subharmonic function on Ω if for every η 1, … , η m−1 in \(\widehat {\Gamma }_{m}\) the inequality
holds in the sense of currents.
By S H m (Ω) (resp. \(SH_{m}^{-}({\Omega })\)), we denote the cone of m-subharmonic functions (resp. negative m-subharmonic functions) on Ω. Before formulating the basic properties of m-subharmonic functions, we recall the following (see [5]).
For \(\lambda =(\lambda _{1},\ldots ,\lambda _{n})\in \mathbb {R}^{n}\) and 1 ≤ m ≤ n, define
Set
By \(\mathcal {H}\), we denote the vector space of complex hermitian n × n matrices over \(\mathbb {R}\). For A ∈ \(\mathcal {H}\), let \(\lambda (A) = (\lambda _{1},\ldots ,\lambda _{n})\in \mathbb {R}^{n}\) be the eigenvalues of A. Set
As in [14], we define
Now, we list the basic properties of m-subharmonic functions whose proofs repeat analogous reasonings for plurisubharmonic functions, hence we omit them.
Proposition 1
Let Ω be an open set in \(\mathbb {C}^{n}\). Then we have
-
a)
\(PSH({\Omega })\,=\, SH_{n}({\Omega })\subset SH_{n-1}({\Omega })\subset {\cdots } \subset SH_{1}({\Omega })\!= \!SH({\Omega })\) . Hence, u ∈ SH m (Ω), 1 ≤ m ≤ n, then u ∈ SH r (Ω) for every 1 ≤ r ≤ m.
-
b)
If u is C 2 smooth then it is m-subharmonic if and only if the form dd c u is pointwise in \(\widehat {\Gamma }_{m}\).
-
c)
If u, v ∈ SH m (Ω) and α, β > 0 then αu + βv ∈ SH m (Ω).
-
d)
If u, v ∈ SH m (Ω) then so is \(\max \{u,v\}\).
-
e)
If \(\{u_{j}\}_{j=1}^{\infty }\) is a family of m-subharmonic functions, \(u\,=\,\sup _{j}u_{j}\!<\!+\infty \) and u is upper semicontinuous then u is a m-subharmonic function.
-
f)
If \(\{u_{j}\}_{j=1}^{\infty }\) is a decreasing sequence of m-subharmonic functions then so is \(u\,=\,\lim _{j\to +\infty }u_{j}\).
-
g)
Let ρ ≥ 0 be a smooth radial function in \(\mathbb {C}^{n}\) vanishing outside the unit ball and satisfying \({\int }_{\mathbb {C}^{n}}\rho dV_{n}=1\), where dV n denotes the Lebesgue measure of \(\mathbb {C}^{n}\) . For u ∈ SH m (Ω), we define
$$u_{\varepsilon} (z):= (u*\rho_{\varepsilon}) (z) = {\int}_{\mathbb{B}(0,\varepsilon)} u(z-\xi)\rho_{\varepsilon} (\xi)dV_{n}(\xi)\quad \forall z \in {\Omega}_{\varepsilon}, $$where \(\rho _{\varepsilon }(z):=\frac 1{\varepsilon ^{2n}} \rho (z/\varepsilon )\) and Ω ε = {z ∈ Ω :d(z, ∂Ω) > ε}. Then \(u_{\varepsilon } \in SH_{m} ({\Omega }_{\varepsilon }) \cap \mathcal C^{\infty } ({\Omega }_{\varepsilon })\) and u ε ↓ u as ε ↓ 0.
-
h)
Let u 1, … , u p ∈ SH m (Ω) and \(\chi : \mathbb R^{p} \to \mathbb R\) be a convex function which is non decreasing in each variable. If χ is extended by continuity to a function \([-\infty , +\infty )^{p} \to [-\infty , \infty )\), then χ(u 1, … , u p ) ∈ SH m (Ω).
Example 1
Let u(z 1, z 2, z 3) = 5|z 1|2 + 4|z 2|2 − |z 3|2. By using (b) of Proposition 1, it is easy to see that \(u\in SH_{2}(\mathbb {C}^{3})\). However, u is not a plurisubharmonic function in \(\mathbb C^{3}\) because the restriction of u on the line (0, 0, z 3) is not subharmonic.
Now, as in [5, 13], we define the complex Hessian operator of locally bounded m-subharmonic functions as follows.
Definition 2
Assume that \(u_{1},\ldots , u_{p}\in SH_{m}({\Omega })\cap L^{\infty }_{loc}({\Omega })\). Then the complex Hessian operator H m (u 1, … , u p ) is defined inductively by
From the definition of m-subharmonic functions and using arguments as in the proof of Theorem 2.1 in [1], we note that H m (u 1, … , u p ) is a closed positive current of bidegree (n − m + p, n − m + p), and this operator is continuous under decreasing sequences of locally bounded m-subharmonic functions. Hence, for p = m, d d c u 1 ∧ ⋯ ∧ d d c u m ∧ β n−m is a nonnegative Borel measure. In particular, when \(u\,=\,u_{1}\,=\,\cdots \,=\,u_{m}\!\in \! SH_{m}({\Omega })\cap L^{\infty }_{loc}({\Omega })\), the Borel measure
is well defined and is called the complex Hessian of u.
2.3 m-Maximal Functions
Similarly in pluripotential theory now we recall a class of m-maximal functions introduced and investigated in [12] recently.
Definition 3
A m-subharmonic function u ∈ S H m (Ω) is called m-maximal if every v ∈ S H m (Ω), v ≤ u outside a compact subset of Ω implies that v ≤ u on Ω.
By M S H m (Ω) we denote the set of m-maximal functions on Ω. Theorem 3.6 in [5] claims that a locally bounded m-subharmonic function u on a bounded domain \({\Omega }\subset \mathbb {C}^{n}\) belongs to M S H m (Ω) if and only if it solves the homogeneous Hessian equation H m (u) = (d d c u)m ∧ β n − m = 0.
2.4 The \(\mathcal {E}^0_{m}({\Omega })\) and \(\mathcal {F}_{m}({\Omega })\) Classes
Next, we recall the classes \(\mathcal {E}^{0}_{m}({\Omega })\) and \(\mathcal {F}_{m}({\Omega })\) introduced and investigated in [13]. First, we give the following.
Let Ω be a bounded domain in \(\mathbb {C}^{n}\). Ω is said to be m-hyperconvex if there exists a continuous m-subharmonic function \(u:{\Omega }\longrightarrow \mathbb {R}^{-}\) such that \({\Omega }_{c}\,=\,\{u\!<\! c\}\!\Subset \!{\Omega }\) for every c < 0. As above, every plurisubharmonic function is m-subharmonic with m ≥ 1 then every hyperconvex domain in \(\mathbb {C}^{n}\) is m-hyperconvex. Let \({\Omega }\subset \mathbb {C}^{n}\) be a m-hyperconvex domain. Set
and
where H m (u) = (d d c u)m ∧ β n − m denotes the Hessian measure of \(u\in SH_{m}^{-}({\Omega })\cap L^{\infty }({\Omega })\). From Theorem 3.14 in [5], it follows that if \(u\in \mathcal {E}_{m}({\Omega })\), the complex Hessian H m (u) = (d d 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 [5], we may give the following description of the class \(\mathcal {E}_{m}({\Omega })\):
2.5 m-Capacity
We recall the notion of m-capacity introduced in [5].
Definition 4
Let \(E \subset {\Omega }\) be a Borel subset. The m-capacity of E with respect to Ω is defined by
Proposition 2.10 in [12] gives some elementary properties of the m-capacity similar as the capacity presented in [1]. Namely, we have
-
a)
\(C_{m}\left (\bigcup _{j=1}^{\infty } E_{j}\right )\leq {\sum }_{j=1}^{\infty } C_{m}(E_{j})\).
-
b)
If E j ↗E then C m (E j )↗C m (E).
We need the following lemma which is used in the proof for the convexity of the class \(\mathcal {E}_{m,\chi }({\Omega })\).
Lemma 1
Assume that \(\varphi \in \mathcal {E}^{0}_{m}({\Omega })\). Then
and
Proof
Let v ∈ S H m (Ω), −1 < v < 0. For all t > 0, we have the following inclusion:
By the comparison principle (Theorem 1.4 in [13]), we get
Hence, taking the supremum over all v, we obtain
By similar arguments as in the proof of Proposition 3.4 in [11], it follows that
The proof is complete. □
3 The Classes \(\mathcal {F}_{m, \chi }({\Omega })\), \(\mathcal {E}_{m, \chi }({\Omega })\)
In what follows, we assume that Ω is a bounded hyperconvex domain in \(\mathbb {C}^{n}\). Now, we introduce two weighted pluricomplex energy classes of m-subharmonic functions defined as follows.
Definition 5
Let \(\chi :\mathbb {R}^{-}\lg \mathbb {R}^{+}\) be a decreasing function and 1 ≤ m ≤ n. We define
and \(\mathcal {E}_{m,\chi }({\Omega }) = \{u\in SH^{-}_{m}({\Omega }): \forall K\Subset {\Omega }, \exists v\in \mathcal {F}_{m,\chi }({\Omega }), v=u ~ \text { on }~ K\}\).
Remark 1
-
(a)
From the above definitions of the two classes \(\mathcal {F}_{m,\chi }({\Omega })\) and \(\mathcal {E}_{m,\chi }({\Omega })\), we note that in the case χ(t) ≡ 1 for all t < 0 we get the pluricomplex energy classes \(\mathcal {F}_{m}({\Omega })\) and \(\mathcal {E}_{m}({\Omega })\) introduced and investigated in [12].
-
(b)
In the case m = n, the class \(\mathcal {F}_{n, \chi }({\Omega })\) coincides with the class of plurisubharmonic functions with weak singularities \(\mathcal {E}_{-\chi }({\Omega })\) erase early introduced and investigated in [4].
-
(c)
In the case m = n and χ(t) ≡ 1 for all t < 0, the classes \(\mathcal {F}_{n,\chi }({\Omega })\) and \(\mathcal {E}_{n,\chi }({\Omega })\) coincide with the classes \(\mathcal {F}({\Omega })\) and \(\mathcal {E}({\Omega })\) in [10].
We need the following lemma.
Lemma 2
Let \(\chi :\mathbb {R}^{-}\to \mathbb {R}^{+}\) be a decreasing function such that χ(2t) ≤ aχ(t) with some a > 1. Assume that 1 ≤ m ≤ n and \(u,v\in \mathcal {E}^{0}_{m}({\Omega })\). Then the following hold:
-
(a)
If u ≤ v, then
$${\int}_{\Omega}\chi(v)(dd^{c} v)^{m}\wedge\beta^{n-m}\leq 2^{m}\max(a,2){\int}_{\Omega}\chi(u)(dd^{c} u)^{m}\wedge\beta^{n-m}. $$ -
(b)
For every 0 ≤ λ ≤ 1, we have
$$\begin{array}{@{}rcl@{}} &&{\int}_{\Omega}\chi(\lambda u+(1-\lambda )v)(dd^{c} (\lambda u+(1-\lambda)v))^{m}\wedge\beta^{n-m}\\ &&\qquad\leq 2^{m}\max(a,2)\left( {\int}_{\Omega}\chi(u)(dd^{c} u)^{m}\wedge\beta^{n-m}+ {\int}_{\Omega}\chi(v)(dd^{c} v)^{m}\wedge\beta^{n-m}\right). \end{array} $$
Proof
(a) First, we assume that χ(0) = 0. Set
Then χ j is a strictly decreasing function, \(\chi <\chi _{j}\!<\!\chi +\frac {1}{j}\) and \(\chi _{j}(2t)\!\leq \! \max (a,2)\cdot \chi _{j}(t)\) for every t < 0. Moreover, since \(\{v<-t\}\subset \{u<-t\}\) for every t > 0 so by Lemma 1, we have
Letting \(j\to \infty \), we get
In the general case, we set \({\Phi }_{j}(t) = \min (\chi (t); -jt)\). Then Φ j are decreasing functions such that Φ j (0) = 0 and Φ j ↗χ on \((-\infty , 0)\). By the first case, we have
Letting \(j\to \infty \), we obtain
(b) As in the proof of (a), we can assume that χ(0) = 0. Since \(\{\lambda u + (1-\lambda ) v <-t\}\subset \{ u<-t\}\cup \{v<-t\}\), so we have
Letting \(j\to \infty \), we get
□
Proposition 2
Let \(\chi :\mathbb {R}^{- }\longrightarrow \mathbb {R}^{+}\) be a decreasing function such that χ(2t) ≤ aχ(t) with some a > 1. Then the following hold:
-
(a)
If \(u\in \mathcal {F}_{m, \chi }({\Omega })\) (resp. \({\mathcal {E}_{m, \chi }({\Omega })}\)) and \(v\in SH^{-}_{m}({\Omega })\) with u ≤ v then \(v\in \mathcal {F}_{m, \chi }({\Omega })\) (resp. \({ \mathcal {E}_{m, \chi }({\Omega })}\)).
-
(b)
If \(u,v\in \mathcal {F}_{m, \chi }({\Omega })\) (resp. \({ \mathcal {E}_{m, \chi }({\Omega }) }\)) and α, γ ≥ 0 then \(\alpha u+\gamma v\in \mathcal {F}_{m, \chi }({\Omega })\) (resp. \({ \mathcal {E}_{m, \chi }({\Omega })}\)).
Proof
-
(a)
It suffices to prove that the conclusion holds for the class \(\mathcal {F}_{m, \chi }({\Omega })\). Assume that \(u\in \mathcal {F}_{m, \chi }({\Omega })\) and u ≤ v, \(v\in SH^{-}_{m}({\Omega })\). From Definition 5, there exists a sequence \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega }), u_{j}\searrow u\) on Ω with
$$\sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}<\infty. $$Set \(v_{j}=\max (u_{j}, v)\in \mathcal {E}^{0}_{m}({\Omega })\), v j ↘ v on Ω and u j ≤ v j . By Lemma 2, we have
$$\sup_{j}{\int}_{\Omega}\chi(v_{j})(dd^{c} v_{j})^{m}\wedge\beta^{n-m}\!\leq\! 2^{m}\max(a,2)\sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\!\wedge\beta^{n-m}\!<\!+\infty. $$Hence, \(v\in \mathcal {F}_{m, \chi }({\Omega })\).
-
(b)
First, we prove that if \(u\in \mathcal {F}_{m, \chi }({\Omega })\) then \(\alpha u\in \mathcal {F}_{m, \chi }({\Omega })\). Indeed, let \(k\in \mathbb N^{*}\) with 2k > α and let \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\), u j ↘ u on Ω with
$$\sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}<\infty. $$It is clear that \(\{\alpha u_{j}\}\!\subset \! \mathcal {E}^{0}_{m}({\Omega })\), α u j ↘ α u on Ω. Moreover, since χ(α u j ) ≤ χ(2k u j ) ≤ a k χ(u j ) so
$$\sup_{j}{\int}_{\Omega}\chi(\alpha u_{j})(dd^{c} \alpha u_{j})^{m}\wedge\beta^{n-m} \leq a^{k} \alpha^{m} \sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}<\infty. $$Hence, \(\alpha u\in \mathcal {F}_{m, \chi }({\Omega })\). By the above proof, we can assume that α + γ=1. Let {u j }, \(\{v_{j}\}\!\subset \mathcal {E}^{0}_{m}({\Omega })\), u j ↘ u on Ω, v j ↘ u on Ω, \(\sup _{j}{\int }_{\Omega }\chi (u_{j})(dd^{c} u_{j})^{m}\wedge \beta ^{n-m}\!<\!\infty \) and \( \sup _{j}{\int }_{\Omega }\chi (v_{j})\allowbreak (dd^{c} u_{j})^{m}\wedge \beta ^{n-m}\!<\!\infty \). By Lemma 2, we have
$$\begin{array}{@{}rcl@{}} &&\sup_{j} {\int}_{\Omega}\chi(\alpha u_{j} + \gamma v_{j})(dd^{c} (\alpha u_{j} +\gamma v_{j}))^{m}\wedge\beta^{n-m}\\ &&\qquad\leq 2^{m}\max(a,2)\left( \sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}+\sup_{j}{\int}_{\Omega}\chi(v_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}\right)\\ &&\qquad<\infty. \end{array} $$Hence, the desired conclusion follows.
□
Proposition 3
Let \(\chi :\mathbb {R}^{-}\!\longrightarrow \! \mathbb {R}^{+}\) be a decreasing function such that χ(2t) ≤ aχ(t) for all t < 0 with some a > 1. Then for every \(u\!\in \! \mathcal {F}_{m,\chi }({\Omega })\) , there exists a sequence \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C ({\Omega })\) such that u j ↘ u and
Proof
Let \({\Omega }_{j}\Subset {\Omega }_{j+1}\Subset {\Omega }\) be such that \({\Omega }=\bigcup _{j=1}^{\infty }{\Omega }_{j}\) and let \(\{v_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\) be such that v j ↘ u and
Theorem 3.1 in [12] implies that there exists a sequence \(\{w_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C ({\Omega })\) such that w j ↘ u. Set
It is easy to see that u j ↘ u on Ω. By Theorem 1.2.7 in [6] and Proposition 3.2 in [5], we get \(u_{j}\in \mathcal C({\Omega })\). Moreover, since w j ≤ u j so \(u_{j}\in \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C({\Omega })\). Now, since v j ↘ u as \(j\to \infty \) and u ≤ w k so there exists j 0 such that \(v_{j_{0}}\leq \frac {k-1}{k} w_{k}\) on Ω k . Therefore, \(v_{j_{0}}\leq u_{k}\) on Ω. Lemma 2 implies that
Thus,
□
The following proposition shows that the Hessian operator is well defined on the class \(\mathcal {E}_{m,\chi }({\Omega })\).
Proposition 4
Let \(\chi :\mathbb {R}^{-}\longrightarrow \mathbb {R}^{+}\) be a decreasing function such that χ ≢ 0 and χ(2t) ≤ aχ(t) for all t < 0 with some a > 1. Then \(\mathcal {E}_{m,\chi }({\Omega })\subset \mathcal {E}_{m}({\Omega })\), and hence, the Hessian H m (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 \(u\in \mathcal {E}_{m,\chi }({\Omega })\) and z 0 ∈ Ω. Take a neighborhood \(\omega \Subset {\Omega }\) of z 0 and a sequence \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\) such that \(\sup _{\overline {\omega }}{u}_{1} <0\), u j ↘ u on ω and
For each j ≥ 1, set
Then \(u_{j}\leq \widetilde {u}_{j}\) on Ω and \(u_{j} = \widetilde {u}_{j}\) on ω and, by using arguments as in [7], we arrive at \(\widetilde {u}_{j}\in MSH_{m}({\Omega }\setminus \overline \omega )\). This yields that \(\widetilde {u}_{j}\in \mathcal {E}^{0}_{m}({\Omega })\) and \(H_{m}(\widetilde {u}_{j}) =0\) on \({\Omega }\setminus \overline \omega \). Moreover, it is easy to see that \(\widetilde {u}_{j}\searrow \widetilde {u}\) on Ω. On the other hand, as in the proof of Lemma 2, we have
Moreover, we may assume that \(\inf _{\overline {\omega }}\chi (\widetilde {u}_{1}) = c_{1}>0\). Then
Hence,
and it follows that \(\widetilde {u}\!\in \!\mathcal {F}_{m}({\Omega })\). It is easy to see that \(\widetilde {u}\,=\, u\) on ω, and this yields that \(u\in \mathcal {E}_{m}({\Omega })\). Theorem 3.14 in [12] implies that H m (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 \(\mathcal {E}_{m,\chi }({\Omega })\).
4 The Local Property of the Class \(\mathcal {E}_{m, \chi }({\Omega })\)
First, we give the following definition which is similar as in [15] for plurisubharmonic functions.
Definition 6
A class \(\mathcal {J}({\Omega })\subset SH^{-}_{m}({\Omega })\) is said to be a local class if \(\varphi \in \mathcal {J}({\Omega })\) then \(\varphi \in \mathcal {J}(D)\) for all hyperconvex domains \(D\Subset {\Omega }\) and if \(\varphi \in SH^{-}_{m}({\Omega }), \varphi |_{{\Omega }_{j}}\in \mathcal {J}({\Omega }_{j})~ \forall j\in I\) with \({\Omega }=\bigcup _{j\in I}{\Omega }_{j}\), then \(\varphi \in \mathcal {J}({\Omega })\).
In [15], the authors introduced the class \(\mathcal {E}_{\chi ,loc}({\Omega })\) and established the local property for this class. This section is devoted to study the local property of the class \(\mathcal {E}_{m, \chi }({\Omega })\).
In the sequel of the paper, we will use the following notation. We will write “\(A\lesssim B\)” if there exists a constant C such that A ≤ C B.
Proposition 5
Set
Then the class \(\mathcal {K}\) has the following properties.
-
(a)
If \(\chi _{1},\chi _{2}\in \mathcal {K}\) and a 1 ,a 2 ≥ 0 then \(a_{1}\chi _{1} + a_{2}\chi _{2}\in \mathcal {K}\).
-
(b)
If \(\chi _{1},\chi _{2}\in \mathcal {K}\) then \(\chi _{1}\cdot \chi _{2}\in \mathcal {K}\).
-
(c)
If \(\chi \in \mathcal {K}\) then \(\chi ^{p}\in \mathcal {K}\) for all p > 0.
-
(d)
If \(\chi \in \mathcal {K}\) , then \((-t)\chi (t) \in \mathcal {K}\) . More generally \(|t^{k}|\chi (t) \in \mathcal {K}\) for all \(k=0,1,2,\dots \).
Proof
The proof is standard hence we omit it. □
Remark 2
If \(\chi \in \mathcal {K}\), then χ(2t) ≤ a χ(t) ∀t < 0 with some a > 1. Indeed, by hypothesis \(t\chi ^{\prime }(t)\leq C\chi (t), C= \text {constant} >0\). We set \(s(t) = \frac {\chi (t)}{(-t)^{C}}\). Then \(s^{\prime }(t)\geq 0~ \forall t<0\), hence s(t) is an increasing function. This implies that s(2t) ≤ s(t), and we have χ(2t) ≤ 2C χ(t).
The following result is necessary for the proof of the local property of the class \(\mathcal {E}_{m,\chi }({\Omega })\).
Lemma 3
Let \(u, v\!\in \! {SH}^{-}_{m}({\Omega })\cap L^{\infty }({\Omega })\) with u ≤ v on Ω, \(\chi \!\in \!\mathcal {K}\) and T = dd c φ 1 ∧ ⋯ ∧ dd c φ m−1 ∧ β n−m with \(\varphi _{j}\in SH^{-}_{m}({\Omega })\cap L^{\infty }({\Omega })\), j = 1 … , m − 1. Then for every p ≥ 0, we have
where \({\Omega }^{\prime }\Subset {\Omega }^{\prime \prime }\Subset {\Omega }\) and c is a constant only depending on \({\Omega }^{\prime },{\Omega }^{\prime \prime },{\Omega }\) and χ.
Proof
Choose \({\Phi }\in \mathcal {C}^{\infty }_{0}({\Omega }), 0\leq {\Phi }\leq 1\) and \({\Phi }|_{{\Omega }^{\prime }}=1, \mathrm {supp\,}{\Phi }\Subset {\Omega }^{\prime \prime \prime }\Subset {\Omega }^{\prime \prime }\). Then, by integration by parts
On the other hand,
Since ∀t, d(u + tΦ) ∧ d c(u + tΦ) ∧ T ≥ 0, we have
and
Now, we can choose A > 0 sufficiently large such that d d cΦ ≥ −A d d c∥z∥2, dΦ ∧ d cΦ ≤ A d d c∥z∥2. Thus, we have the following estimates
In the case \(\chi ^{\prime \prime }(u)\leq 0\), we have the following
In the case \(\chi ^{\prime \prime }(u)\!\geq \!0\), from (1), we note that \({\Phi } v\chi ^{\prime \prime }(u)du\wedge d^{c}u\wedge T\!\leq \! 0\), and it is easy to obtain the above estimates. Now, we have the following estimates
On the other hand, by hypothesis about the class \(\mathcal {K}\), we have \(u\chi ^{\prime }(u)\!\leq \! c_{1}\chi (u)\) and \((-u^{2})\chi ^{\prime }(u)\!\leq \! c_{1}(-u)\chi (u)\), \(u\chi ^{\prime \prime }(u)\!\leq \! c_{2}(-\chi ^{\prime }(u))\). Therefore,
Set \( \chi _{1}(t) = -{{\int }^{t}_{0}}\chi (x)dx \) then
Now, we choose \(\psi \in \mathcal {C}^{\infty }_{0}, \psi |_{{\Omega }^{\prime \prime \prime }}=1\), \(\mathrm {supp\,}\psi \Subset {\Omega }^{\prime \prime }\), then we have
with B > 0 sufficiently large.
Finally, we have
□
The next lemma is a crucial tool for the proof of the local property of the class \(\mathcal {E}_{m,\chi }({\Omega })\).
Lemma 4
Let Ω be a hyperconvex domain in \(\mathbb C^{n}\) and 1 ≤ m ≤ n. Assume that \(u\!\in \! \mathcal {E}^{0}_{m}({\Omega })\) and \(\chi \in \mathcal {K}\) such that \(\chi ^{\prime \prime }(t)\geq 0~ \forall t<0\) . Then for \({\Omega }^{\prime }\Subset {\Omega }\) , there exists a constant \(C\,=\,C({\Omega }^{\prime })\) such that the following holds:
Furthermore, if \(u\in \mathcal {F}_{m, \chi }({\Omega })\) then
for all p = 1, … , m.
Proof
Set χ 0(t) = χ(t) and for each k ≥ 1, let \(\chi _{k}(t)\!=-{{\int }_{0}^{t}} \chi _{k-1} (x) dx\). From the hypothesis \(\chi \!\in \!\mathcal {K}\), then χ(2t) ≤ a χ(t) and it is easy to check that \(\chi _{k}\in \mathcal K\) and \(\chi (t) (-t)^{k} \!\lesssim \! \chi _{k} (t) \!\lesssim \chi (t) (-t)^{k}\).
Now, choose R > 0 large enough such that ∥z∥2 ≤ R 2 on Ω. Let \(\varphi \in \mathcal {E}^{0}_{m}({\Omega })\) and A > 0 such that ∥z∥2−R 2 ≥ A φ on \({\Omega }^{\prime }\). Set \(h=\max (\|z\|^{2}-R^{2}; A\varphi )\) then \(h\in \mathcal {E}^{0}_{m}({\Omega })\) and d d c h = d d c∥z∥2 = β on \({\Omega }^{\prime }\). First, we claim that (2) holds for \(u\in \mathcal {E}^{0}_{m}({\Omega })\). Indeed, we have
Integrating by parts, we have
Hence, if we set \(C=C({\Omega }^{\prime }) = p!\|h\|^{p}_{L^{\infty }({\Omega })}\) then
Finally, we prove (2) holds for \(u\in \mathcal {F}_{m, \chi }({\Omega })\). Indeed, we take \(u_{j}\in \mathcal {E}^{0}_{m}({\Omega })\), u j ↘ u on Ω such that
By dominated convergence theorem and (d d c u j )m−p ∧ (d d c∥z∥2)n−m + p is weakly convergent to (d d c u)m−p ∧ (d d c∥z∥2)n−m + p in the sense of currents
□
We also need the following result on subextension for the class \(\mathcal F_{m,\chi }({\Omega })\).
Lemma 5
Assume that \({\Omega }\Subset \widetilde {\Omega }\) and \(u\in \mathcal F_{m,\chi }({\Omega })\) . Then there exists \(\widetilde {u}\in \mathcal F_{m,\chi }(\widetilde {\Omega })\) such that \(\widetilde {u}\leq u\) on Ω.
Proof
We split the proof into three steps.
-
Step 1. We prove that if \(v\!\in \! \mathcal C(\widetilde {\Omega })\), v ≤ 0, \(\operatorname {supp}\, v\!\Subset \!\widetilde {\Omega }\) then \(\widetilde {v}\!:=\!\sup \{w\!\in \! SH^{-}_{m}(\widetilde {\Omega })\!: w\!\leq v \text { on }\widetilde {\Omega } \}\in \mathcal {E}^{0}_{m}(\widetilde {\Omega })\cap \mathcal C(\widetilde {\Omega })\) and \((dd^{c} \widetilde {v})^{m}\wedge \beta ^{n-m}=0\) on \(\{\widetilde {v}<v\}\). Indeed, let \(\varphi \in \mathcal {E}^{0}_{m}(\widetilde {\Omega })\cap \mathcal C(\widetilde {\Omega })\) be such that \(\varphi \leq \inf _{\widetilde {\Omega }}v\) on supp v. Since \(\varphi \leq \widetilde {v}\) so \(\widetilde {v}\in \mathcal {E}^{0}_{m}(\widetilde {\Omega })\). Moreover, by Proposition 3.2 in [5], we have \(\widetilde {v}\in \mathcal C(\widetilde {\Omega })\). Let \(w\in SH_{m}(\{\widetilde {v}<v\})\) be such that \(w\leq \widetilde {v}\) outside a compact subset K of \(\{\widetilde {v}<v\}\). Set
$$w_{1}=\left\{\begin{array}{llllllll} \max(w,\widetilde{v})& \text{ on }~\{\widetilde{v}<v\},\\ \widetilde{v}& \text{ on }~\widetilde{\Omega}\backslash(\{\widetilde{v}<v\}). \end{array}\right. $$Since \(\widetilde {v}\) and v are continuous so \(\varepsilon \,=\,-\sup _{K}(\widetilde {v}\,-\,v)\!>\!0\). Choose δ ∈ (0,1) such that \(-\delta \inf _{\widetilde {\Omega }} \widetilde {v}<\varepsilon \). We have \((1\!-\delta ) \widetilde {v}\!\leq \! \widetilde {v}\,+\,\varepsilon \!\leq \!v\) on K. Hence, \((1\,-\,\delta ) \widetilde {v}+\delta w_{1}\!\leq \! v\) on \(\widetilde {\Omega }\) and we get \((1\,-\,\delta ) \widetilde {v}\,+\,\delta w_{1}\!=\widetilde {\!v}\). Thus, \(w\!\leq \! \widetilde {v}\) on \(\{\widetilde {v}\!<\!v\}\). Hence, \(\widetilde {v}\) is m-maximal in \(\{\widetilde {v}\!<\!v\}\). By [5], we get \((dd^{c} \widetilde {v})^{m}\wedge \beta ^{n-m}\,=\,0\) on \(\{\widetilde {v}\!<\!v\}\).
-
Step 2. Next, we prove that if \(u\in \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C({\Omega })\) then there exists \(\widetilde {u}\in \mathcal {E}^{0}_{m}(\widetilde {\Omega })\), \((dd^{c} \widetilde {u})^{m}\wedge \beta ^{n-m}=0\) on \((\widetilde {\Omega }\backslash {\Omega })\cup (\{\widetilde {u}<u\}\cap {\Omega })\) and \((dd^{c} \widetilde {u})^{m}\wedge \beta ^{n-m}\leq (dd^{c} {u})^{m}\wedge \beta ^{n-m}\) on \(\{\widetilde {u}=u\}\cap {\Omega }\). Indeed, set
$$v=\left\{\begin{array}{llllllll} u& \text{ on }~{\Omega},\\ 0& \text{ on }~\widetilde{\Omega}\backslash{\Omega}. \end{array}\right. $$It is easy to see that \(v\in \mathcal C(\widetilde {\Omega })\) and \(\text {supp}v\subset {\Omega }\Subset \widetilde {\Omega }\). Hence, we have \(\widetilde {u}=\widetilde {v}\in \mathcal {E}^{0}_{m}(\widetilde {\Omega })\cap \mathcal C(\widetilde {\Omega })\) and \((dd^{c} \widetilde {u})^{m}\wedge \beta ^{n-m}=0\) on \(\{\widetilde {v}<v\}\cap \widetilde {\Omega } =(\widetilde {\Omega }\backslash {\Omega })\cup (\{\widetilde {u}<u\}\cap {\Omega })\). Let K be a compact set in \(\{\widetilde {u}=u\}\cap {\Omega }\). Then for ε > 0, we have \(K \Subset \{\widetilde {u}+\varepsilon >u\}\cap {\Omega }\) so we have
$$\begin{array}{@{}rcl@{}} {\int}_{K}(dd^{c} \widetilde{u})^{m}\wedge\beta^{n-m}&=&{\int}_{K} 1_{\{\widetilde{u}+\varepsilon>u\}}(dd^{c}\widetilde{u})^{m}\wedge\beta^{n-m}\\ &=&{\int}_{K}1_{\{\widetilde{u}+\varepsilon>u\}}(dd^{c} \max(\widetilde{u}+\varepsilon,u))^{m}\wedge\beta^{n-m}\\ &\leq&{\int}_{K}(dd^{c} \max(\widetilde{u}+\varepsilon,u))^{m}\wedge\beta^{n-m}, \end{array} $$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 [12]). However, \(\max (\widetilde {u}+\varepsilon ,u)\searrow u\) on Ω as ε→0 so by [21] it follows that \((dd^{c} \max (\widetilde {u}+\varepsilon ,u))^{m}\wedge \beta ^{n-m}\) is weakly convergent to (d d c u)m ∧ β n−m as ε→0. On the other hand, 1 K is upper semicontinuous on Ω so we can approximate 1 K with a decreasing sequence of continuous functions φ j . Hence, we infer that
$$\begin{array}{@{}rcl@{}} &&\limsup_{\varepsilon\to 0}{\int}_{\Omega} 1_{K} (dd^{c} \max(\widetilde{u}+\varepsilon,u))^{m}\wedge\beta^{n-m}\\ &&\qquad= \limsup_{\varepsilon\to 0}\left[\lim_{j}{\int}_{\Omega}\varphi_{j} (dd^{c} \max(\widetilde{u}+\varepsilon,u))^{m}\wedge\beta^{n-m}\right]\\ &&\qquad\leq \limsup_{\varepsilon\to 0}\left( {\int}_{\Omega}\varphi_{j} (dd^{c} \max(\widetilde{u}+\varepsilon,u))^{m}\wedge\beta^{n-m}\right)\\ &&\qquad\leq{\int}_{\Omega}\varphi_{j} (dd^{c} u)^{m}\wedge\beta^{n-m}\searrow {\int}_{K}(dd^{c} u)^{m}\wedge\beta^{n-m}. \end{array} $$as \(j\to \infty \). This yields that \((dd^{c} \widetilde {u})^{m}\wedge \beta ^{n-m}\leq (dd^{c} {u})^{m}\wedge \beta ^{n-m}\) on \(\{\widetilde {u}=u\}\cap {\Omega }\).
-
Step 3. Now, let \(u_{j}\in \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C({\Omega })\) be such that u j ↘ u and
$$\sup_{j}{\int}_{\Omega} \chi(u_{j})\left( dd^{c}u_{j}\right)^{m}\wedge \beta^{n-m}<\infty. $$By Step 2, we have
$$\begin{array}{@{}rcl@{}} {\int}_{\widetilde{\Omega}} \chi(\widetilde{u}_{j})\left( dd^{c} \widetilde{u}_{j}\right)^{m}\wedge\beta^{n-m}&=&{\int}_{\{\widetilde{u}_{j}=u_{j}\}\cap{\Omega}} \chi(\widetilde{u}_{j}) (dd^{c} \widetilde{u}_{j})^{m}\wedge\beta^{n-m}\\ &\leq&{\int}_{\{\widetilde{u}_{j}=u_{j}\}\cap{\Omega}} \chi({u}_{j}) (dd^{c} {u}_{j})^{m}\wedge\beta^{n-m}\\ &\leq&{\int}_{\Omega} \chi({u}_{j}) (dd^{c} {u}_{j})^{m}\wedge\beta^{n-m}. \end{array} $$Hence,
$$\sup_{j}\int\limits_{\widetilde{\Omega}} \chi(\widetilde{u}_{j})(dd^{c} \widetilde{u}_{j})^{m}\wedge\beta^{n-m} \leq \sup_{j}\int\limits_{\Omega} \chi({u}_{j}) (dd^{c} {u}_{j})^{m}\wedge\beta^{n-m}<\infty. $$Thus, \(\widetilde {u}:=\lim _{j\to \infty }\widetilde {u}_{j}\in \mathcal F_{m,\chi }(\widetilde {\Omega })\) and \(\widetilde {u}\leq u\) on Ω.
□
The following result deals with the local property of the class \(\mathcal {E}_{m, \chi }({\Omega })\). Namely, we have the following.
Theorem 1
Let Ω be a hyperconvex domain in \(\mathbb C^{n}\) and m be an integer with 1 ≤ m ≤ n. Assume that \(u\in SH^{-}_{m}({\Omega })\) and \(\chi \in \mathcal K\) such that \(\chi ^{\prime \prime }(t)\geq 0~ \forall t<0\) . Then the following statements are equivalent.
-
a)
\(u\in \mathcal E_{m,\chi }({\Omega })\).
-
b)
For all \(K\Subset {\Omega }\) , there exists a sequence \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C({\Omega })\) , u j ↘ u on K such that
$$\sup_{j}{\int}_{K}\chi(u_{j})|u_{j}|^{p}(dd^{c}u_{j})^{m-p}\wedge\beta^{n-m+p}<\infty $$for every p = 0, … , m.
-
c)
For every \(W\Subset {\Omega }\) such that W is a hyperconvex domain, we have \(u|_{W}\in \mathcal E_{m,\chi }(W)\).
-
d)
For every z ∈ Ω, there exists a hyperconvex domain \(V_{z}\Subset {\Omega }\) such that z ∈ V z and \(u|_{V_{z}}\in \mathcal E_{m,\chi }(V_{z})\).
Proof
Let χ k be as in Lemma 4.
“ a) ⇒ b)” Let \(K\Subset {\Omega }\) be given. Since \(u\in \mathcal {E}_{m, \chi }({\Omega })\), then there exists \(v\in \mathcal {F}_{m, \chi }({\Omega })\) with v = u on K. By the definition of the class \(\mathcal {F}_{m, \chi }({\Omega })\), there exists a sequence \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal {C}({\Omega })\), u j ↘ v on Ω with
Then u j ↘ u on K. We have to prove
for p=0,1, … , m. It is obvious that the conclusion holds for p=0. Assume that 1 ≤ p ≤ m. Then, by Lemma 4, we get that
and the desired conclusion follows.
“ b) ⇒ c)” Let \(W\Subset {\Omega }\) be a hyperconvex domain. Take \(U\Subset W \Subset {\Omega }\) and a sequence \(\mathcal {E}^{0}_{m}({\Omega })\ni u_{j}\searrow u\) on W such that
for p=0,1, … , m. Set \(\widetilde {u}_{j}=\sup \{\varphi \in SH^{-}_{m}(W): \varphi \leq u_{j}\text { on } U \}\in \mathcal {E}^{0}_{m}(W)\). Next, choose \( U\Subset {\Omega }_{1}\Subset \ldots \Subset {\Omega }_{m}\Subset W\). Since \(u_{j}\!\leq \! \widetilde {u}_{j}\) on W and \((dd^{c} \widetilde {u}_{j})^{m}\wedge \beta ^{n-m}\,=\,0\) on \(W\backslash \overline {U}\) so by applying Lemma 3 many times, we arrive at
Hence,
Thus, \(u_{U,W}\!\!:=\!\lim \widetilde {u}_{j}\!\!\in \!\mathcal {F}_{m, \chi }(W)\). Since \(U\!\!\Subset \! \!W\) is arbitrary and u U, W = u on U so \(u\!\in \!\mathcal E_{m}(W)\).
“ c) ⇒ d)” It is obvious.
“ d) ⇒ a)” Assume that \({\Omega }^{\prime }\!\!\Subset \!\!{\Omega }\). Choose z j ∈ Ω, j = 1,2, … , s such that \({\Omega }^{\prime }\!\Subset \! \bigcup _{j=1}^{s} V_{z_{j}}\), where \(V_{z_{j}}\) are hyperconvex domains. Let \(W_{z_{j}}\!\Subset \! V_{z_{j}}\) be such that \({\Omega }^{\prime }\!\Subset \! \bigcup _{j=1}^{s} W_{z_{j}}\). Since \(u|_{V_{z_{j}}}\!\in \!\mathcal {E}_{m, \chi }(V_{z_{j}})\) so there exists \(v_{j}\!\in \!\mathcal F_{m, \chi }(V_{z_{j}})\) such that v j = u on \(W_{z_{j}}\). By Lemma 5, there exists \(\widetilde {v}_{j}\in \mathcal F_{m, \chi }({\Omega })\) such that \(\widetilde {v}_{j}\leq v_{j}\) on \(V_{z_{j}}\). Then by Proposition 2, we have \(\widetilde {v}:=\widetilde {v}_{1}+\cdots +\widetilde {v}_{s}\in \mathcal F_{m, \chi }({\Omega })\) and, hence, \(\max (\widetilde {v},u) \in \mathcal F_{m, \chi }({\Omega })\). However, \(\max (\widetilde {v},u) = u\) on \({\Omega }^{\prime }\), then \(u\in \mathcal E_{m, \chi }({\Omega })\). The proof is complete. □
From the above theorem, we get the following property of the class \(\mathcal {E}_{m, \chi }({\Omega })\).
Corollary 1
Assume that Ω is a bounded hyperconvex domain, and \(\chi \!\in \!\mathcal K\) satisfies all hypotheses of Theorem 1. Then \(\mathcal E_{m,\chi }({\Omega })\!\subset \! \mathcal E_{m-1,\chi }({\Omega })\).
Proof
Assume that \(u\in \mathcal E_{m,\chi }({\Omega })\). Let \(K\!\Subset \!{\Omega }\). Take a domain \({\Omega }^{\prime }\) with \({\Omega }^{\prime }\!\Subset \!{\Omega }\). By Theorem 1, there exists a sequence \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C({\Omega })\) such that u j ↘ u on \({\Omega }^{\prime }\) and
Let \(h\!\in \!\mathcal {E}^{0}_{m-1}({\Omega })\) be chosen. For each j > 0, take m j > 0 such that u j ≥ m j h on \({\Omega }^{\prime }\). Set \(v_{j} \,=\, \max (u_{j}, m_{j} h)\!\in \!\mathcal {E}^{0}_{m-1}({\Omega })\) and v j = u j on \({\Omega }^{\prime }\). Note that v j ↘ u on \({\Omega }^{\prime }\) and (d d c v j )p ∧ β q = (d d c u j )p ∧ β q on \({\Omega }^{\prime }\) for 1 ≤ p ≤ m−1 and 1 ≤ q ≤ n−m+1. We may assume that \(u|_{{\Omega }^{\prime }}\leq -1\). By Hartogs’ lemma (see Theorem 3.2.13 in [18]), we conclude that \(v_{j}|_{{\Omega }^{\prime }}\leq -1\) for j ≥ j 0 with some j 0. Without loss of generality, we may assume that \(v_{j}|_{{\Omega }^{\prime }}\leq -1\) for j ≥ 1. Hence, |v j |m ≥ |v j |m−1 on \({\Omega }^{\prime }\) for all j ≥ 1. Now, we have
Note that v j ↘ u on \({\Omega }^{\prime }\) and
Moreover, by Theorem 1, we get \(u\in \mathcal E_{m-1,\chi }({\Omega })\). □
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Acknowledgments
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.
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Hung, V.V. Local Property of a Class of m-Subharmonic Functions. Vietnam J. Math. 44, 603–621 (2016). https://doi.org/10.1007/s10013-015-0176-5
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DOI: https://doi.org/10.1007/s10013-015-0176-5
Keywords
- m-Subharmonic functions
- Weighted energy classes of m-subharmonic functions
- Complex m-Hessian
- Local property