Vietnam Journal of Mathematics

, Volume 44, Issue 3, pp 603–621 | Cite as

Local Property of a Class of m-Subharmonic Functions

Article

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 \(\mathbb {C}^{n}\). By PSH(Ω) (resp. PSH(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 (ddc.)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
$$\mathcal{E}_{\chi}({\Omega}) = \left\{\varphi\in PSH^{-}({\Omega}):\exists \mathcal{E}_{0}({\Omega})\ni\varphi_{j}\searrow\varphi,~ \sup_{j\geq 1}{\int}_{\Omega}\chi(\varphi_{j})(dd^{c}\varphi_{j})^{n} < +\infty\right\}, $$
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 ξΩ. 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 ξΩ. 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 Hm(u) = (ddcu)mβnm is well defined on the class \(\mathcal {E}_{m,\chi }({\Omega })\) where β = ddcz2 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 ≤ mn. 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.
  1. a)

    \(u\in \mathcal E_{m,\chi }({\Omega })\)

     
  2. 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.
     
  3. c)

    For every\(W\Subset {\Omega }\)such that W is a hyperconvex domain, we have\(u|_{W}\in \mathcal E_{m,\chi }(W)\).

     
  4. d)

    For every z ∈ Ω, there exists a hyperconvex domain\(V_{z}\Subset {\Omega }\)such that z ∈ Vzand\(u|_{V_{z}}\in \mathcal E_{m,\chi }(V_{z})\).

     

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 = ddcz2, 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
$$\mathcal{E}_{0}=\mathcal{E}_{0}({\Omega}) = \left\{\varphi\in {PSH}^{-}({\Omega})\cap {L}^{\infty}({\Omega}): \underset{z\to\xi} \lim\varphi(z) = 0~ \forall \xi\in\partial{\Omega}, ~{\int}_{\Omega}(dd^{c}\varphi)^{n} <\infty\right\} $$
and
$$\mathcal{F}=\mathcal{F}({\Omega}) = \left\{\varphi\in PSH^{-}({\Omega}):\exists \mathcal{E}_{0}\ni\varphi_{j}\searrow\varphi,~\sup_{j}{\int}_{\Omega}(dd^{c}\varphi_{j})^{n}<\infty\right\}. $$

2.2 m-Subharmonic Functions

We recall the class of m-subharmonic functions introduced and investigated in [5] recently. For 1 ≤ mn, we define
$$\widehat{\Gamma}_{m}=\left\{\eta\in \mathbb{C}_{(1,1)}: \eta\wedge \beta^{n-1}\geq 0,\ldots, \eta^{m}\wedge \beta^{n-m}\geq 0\right\}, $$
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 η1, … , ηm−1 in \(\widehat {\Gamma }_{m}\) the inequality
$$dd^{c} u \wedge\eta_{1}\wedge\cdots\wedge\eta_{m-1}\wedge\beta^{n-m}\geq 0 $$
holds in the sense of currents.

By SHm(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 ≤ mn, define
$$S_{m}(\lambda) = \sum\limits_{1\leq j_{1}<\cdots<j_{m}\leq n} \lambda_{j_{1}}\cdots\lambda_{j_{m}}. $$
Set
$${\Gamma}_{m}=\{S_{1}\geq 0\}\cap\{S_{2}\geq 0\}\cap\cdots\cap\{S_{m}\geq 0\}. $$
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
$$\widetilde{S}_{m}(A) = S_{m}(\lambda(A)). $$
As in [14], we define
$$\widetilde{\Gamma}_{m}=\left\{A\in\mathcal{H}: \lambda(A)\in{\Gamma}_{m}\right\}=\left\{\widetilde{S}_{1}\geq 0\right\}\cap\cdots\cap\left\{\widetilde{S}_{m}\geq 0\right\}. $$

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
  1. a)

    \(PSH({\Omega })\,=\, SH_{n}({\Omega })\subset SH_{n-1}({\Omega })\subset {\cdots } \subset SH_{1}({\Omega })\!= \!SH({\Omega })\). Hence, uSHm(Ω), 1 ≤ mn, then uSHr(Ω) for every 1 ≤ rm.

     
  2. b)

    If u is C2smooth then it is m-subharmonic if and only if the form ddcu is pointwise in\(\widehat {\Gamma }_{m}\).

     
  3. c)

    If u, vSHm(Ω) and α, β > 0 then αu + βv ∈ SHm(Ω).

     
  4. d)

    If u, vSHm(Ω) then so is\(\max \{u,v\}\).

     
  5. 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.

     
  6. f)

    If\(\{u_{j}\}_{j=1}^{\infty }\)is a decreasing sequence of m-subharmonic functions then so is\(u\,=\,\lim _{j\to +\infty }u_{j}\).

     
  7. 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 dVndenotes the Lebesgue measure of\(\mathbb {C}^{n}\). For uSHm(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 Ωε = {z ∈ Ω :d(z, ∂Ω) > ε}. Then\(u_{\varepsilon } \in SH_{m} ({\Omega }_{\varepsilon }) \cap \mathcal C^{\infty } ({\Omega }_{\varepsilon })\)and uεu as ε ↓ 0.
     
  8. h)

    Let u1, … , upSHm(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(u1, … , up) ∈ 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 \(u\in SH_{2}(\mathbb {C}^{3})\). However, u is not a plurisubharmonic function in \(\mathbb C^{3}\) because the restriction of u on the line (0, 0, z3) 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 Hm(u1, … , up) is defined inductively by
$$dd^{c}u_{p}\wedge\cdots\wedge dd^{c}u_{1}\wedge\beta^{n-m}= dd^{c}\left( u_{p} dd^{c}u_{p-1}\wedge\cdots\wedge dd^{c}u_{1}\wedge\beta^{n-m}\right). $$
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, … , up) is a closed positive current of bidegree (nm + p, nm + p), and this operator is continuous under decreasing sequences of locally bounded m-subharmonic functions. Hence, for p = m, ddcu1 ∧ ⋯ ∧ ddcumβnm 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
$$H_{m}(u) = (dd^{c} u)^{m}\wedge\beta^{n-m} $$
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 uSHm(Ω) is called m-maximal if every vSHm(Ω), vu outside a compact subset of Ω implies that vu on Ω.

By MSHm(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 MSHm(Ω) if and only if it solves the homogeneous Hessian equation Hm(u) = (ddcu)mβnm = 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
$$\begin{array}{@{}rcl@{}} \mathcal{E}^{0}_{m}&=&\mathcal{E}^{0}_{m}({\Omega}) = \left\{u\in SH^{-}_{m}({\Omega})\cap{L}^{\infty}({\Omega}): \lim_{z\to\partial{\Omega}} u(z) = 0,~ {\int}_{\Omega}H_{m}(u) <\infty\right\},\\ \mathcal{F}_{m}&=&\mathcal{F}_{m}({\Omega}) = \left\{u\in SH^{-}_{m}({\Omega}): \exists \mathcal{E}^{0}_{m}\ni u_{j}\searrow u,~ \sup_{j}{\int}_{\Omega}H_{m}(u_{j})<\infty\right\}, \end{array} $$
and
$$\begin{array}{@{}rcl@{}} \mathcal{E}_{m}=\mathcal{E}_{m}({\Omega})&=&\left\{u\in SH^{-}_{m}({\Omega}):\forall z_{0}\in{\Omega}, \exists \text{ a neighborhood } \omega\ni z_{0},~ \text{ and }{\phantom{\mathcal{E}^{0}_{m}\ni u_{j}\searrow u~ \text{ on }~ \omega, \sup_{j}{\int}_{\Omega}H_{m}(u_{j}) <\infty}}\right.\\ &&\qquad\qquad\qquad~\left. \mathcal{E}^{0}_{m}\ni u_{j}\searrow u~ \text{ on }~ \omega, \sup_{j}{\int}_{\Omega}H_{m}(u_{j}) <\infty\right\}, \end{array} $$
where Hm(u) = (ddcu)mβnm 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 Hm(u) = (ddcu)mβnm 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 \(\mathcal {E}_{m}({\Omega })\):
$$\mathcal{E}_{m}=\mathcal{E}_{m}({\Omega}) = \left\{u\in SH^{-}_{m}({\Omega}):\forall~ U\Subset{\Omega}, \exists v\in\mathcal{F}_{m}({\Omega}),~ v=u~ \text{ on }~ U\right\}. $$

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
$$C_{m}(E) = C_{m}(E,{\Omega}) = \sup\left\{{\int}_{E}(dd^{c} u)^{m}\wedge\beta^{n-m}: u\in SH_{m}({\Omega}), -1\leq u\leq 0\right\}. $$
Proposition 2.10 in [12] gives some elementary properties of the m-capacity similar as the capacity presented in [1]. Namely, we have
  1. a)

    \(C_{m}\left (\bigcup _{j=1}^{\infty } E_{j}\right )\leq {\sum }_{j=1}^{\infty } C_{m}(E_{j})\).

     
  2. b)

    If EjE then Cm(Ej)↗Cm(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
$$(dd^{c} \varphi)^{m}\wedge\beta^{n-m}(\{\varphi<-t\})\leq t^{m}C_{m}(\{\varphi<-t\}) $$
and
$$t^{m}C_{m}(\{\varphi<-2t\})\leq (dd^{c}\varphi)\wedge\beta^{n-m}(\{\varphi<-t\}). $$

Proof

Let vSHm(Ω), −1 < v < 0. For all t > 0, we have the following inclusion:
$$\{\varphi<-2t\}\subset\left\{\frac{\varphi}{t}< v-1\right\}\subset\{\varphi<-t\}. $$
By the comparison principle (Theorem 1.4 in [13]), we get
$$\begin{array}{@{}rcl@{}} {\int}_{\{\varphi<-2t\}}(dd^{c} v)^{m}\wedge\beta^{n-m}&\leq& {\int}_{\{\frac{\varphi}{t}< v-1\}}(dd^{c} v)^{m}\wedge\beta^{n-m}\\ &\leq& {\int}_{\{\frac{\varphi}{t}< v-1\}}\frac{1}{t^{m}}(dd^{c} \varphi)^{m}\wedge\beta^{n-m}\\ &\leq& \frac{1}{t^{m}}{\int}_{\{\varphi<-t\}}(dd^{c} \varphi)^{m}\wedge\beta^{n-m}. \end{array} $$
Hence, taking the supremum over all v, we obtain
$$t^{m}C_{m}(\{\varphi<-2t\})\leq (dd^{c} \varphi)^{m}\wedge\beta^{n-m}(\{\varphi<-t\}). $$
By similar arguments as in the proof of Proposition 3.4 in [11], it follows that
$$(dd^{c} \varphi)^{m}\wedge\beta^{n-m}(\{\varphi<-t\}) = {\int}_{\{\varphi<-t\}}(dd^{c} \varphi)^{m}\wedge\beta^{n-m}\leq t^{m}C_{m}(\{\varphi<-t\}). $$
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 ≤ mn. We define
$$\begin{array}{@{}rcl@{}} \mathcal{F}_{m,\chi}({\Omega})&=&\left\{u\in SH^{-}_{m}({\Omega}): \exists \{u_{j}\}\subset\mathcal{E}^{0}_{m}({\Omega}),~ u_{j}\searrow u~ \text{ on }~ {\Omega}{\phantom{\sup_{j}{\int}_{\Omega}}}\right.\\ &&\left.\quad \sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}< +\infty\right\} \end{array} $$
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

  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].

     
  2. (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].

     
  3. (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) ≤ (t) with some a > 1. Assume that 1 ≤ mn and\(u,v\in \mathcal {E}^{0}_{m}({\Omega })\). Then the following hold:
  1. (a)
    If uv, then
    $${\int}_{\Omega}\chi(v)(dd^{c} v)^{m}\wedge\beta^{n-m}\leq 2^{m}\max(a,2){\int}_{\Omega}\chi(u)(dd^{c} u)^{m}\wedge\beta^{n-m}. $$
     
  2. (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
$$\chi_{j}(t):=\chi(t)+\frac{(1-e^{t})}{j},\quad t<0. $$
Then χj 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
$$\begin{array}{@{}rcl@{}} {\int}_{\Omega}\chi_{j}(v)(dd^{c} v)^{m}\wedge\beta^{n-m} &=&- {\int}_{0}^{+\infty}\chi_{j}^{\prime}(-t)(dd^{c} v)^{m}\wedge\beta^{n-m}(\{v<-t\})dt\\ &\leq& - {\int}_{0}^{+\infty}t^{m}\chi_{j}^{\prime}(-t)C_{m}(\{v<-t\})dt\\ &\leq& - {\int}_{0}^{+\infty}t^{m}\chi_{j}^{\prime}(-t)C_{m}(\{u<-t\})dt\\ &\leq& -2^{m}{\int}_{0}^{+\infty}\chi_{j}^{\prime}(-t)(dd^{c} u)^{m}\wedge\beta^{n-m}(\{u<-t/2\})dt\\ &=&{\int}_{\Omega}\chi_{j}(2u)(dd^{c} (2u))^{m}\wedge\beta^{n-m}\\ &\leq& 2^{m}\max(a,2){\int}_{\Omega}\chi_{j}(u)(dd^{c} u)^{m}\wedge\beta^{n-m}\\ &\leq& 2^{m}\max(a,2)\left( {\int}_{\Omega}\left( \chi(u)+\frac{1}{j}\right)(dd^{c} u)^{m}\wedge\beta^{n-m}\right). \end{array} $$
Letting \(j\to \infty \), we get
$${\int}_{\Omega}\chi(v)(dd^{c} v)^{m}\wedge\beta^{n-m}\leq 2^{m}\max(a,2){\int}_{\Omega}\chi(u)(dd^{c} u)^{m}\wedge\beta^{n-m}. $$
In the general case, we set \({\Phi }_{j}(t) = \min (\chi (t); -jt)\). Then Φj are decreasing functions such that Φj(0) = 0 and Φjχ on \((-\infty , 0)\). By the first case, we have
$${\int}_{\Omega}{\Phi}_{j}(v)(dd^{c} v)^{m}\wedge\beta^{n-m}\leq 2^{m}\max(a,2){\int}_{\Omega}{\Phi}_{j}(u)(dd^{c} u)^{m}\wedge\beta^{n-m}. $$
Letting \(j\to \infty \), we obtain
$${\int}_{\Omega}\chi(v)(dd^{c} v)^{m}\wedge\beta^{n-m}\leq 2^{m}\max(a,2){\int}_{\Omega}\chi(u)(dd^{c} u)^{m}\wedge\beta^{n-m}. $$
(b) 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
$$\begin{array}{@{}rcl@{}} &&{\int}_{\Omega}\chi(\lambda u + (1-\lambda) v )(dd^{c} (\lambda u + (1-\lambda) v ))^{m}\wedge\beta^{n-m}\\ &&\quad\leq {\int}_{\Omega}\chi_{j}(\lambda u + (1-\lambda) v)(dd^{c} (\lambda u + (1-\lambda) v))^{m}\wedge\beta^{n-m}\\ &&\quad\leq -{\int}_{0}^{+\infty} t^{m}\chi_{j}^{\prime}(-t)C_{m}(\{u<-t\})dt -{\int}_{0}^{+\infty} t^{m}\chi_{j}^{\prime}(-t)C_{m}(\{v<-t\})dt\\ &&\quad\leq 2^{m}\max(a,2)\left( {\int}_{\Omega}\left( \chi(u)+\frac{1}{j}\right)(dd^{c} u)^{m}\wedge\beta^{n-m}+ {\int}_{\Omega}\left( \chi(v)+\frac{1}{j}\right)(dd^{c} v)^{m}\wedge\beta^{n-m}\right). \end{array} $$
Letting \(j\to \infty \), we get
$$\begin{array}{@{}rcl@{}} &&{\int}_{\Omega}\chi(\lambda u+(1-\lambda )v)(dd^{c} (\lambda u+(1-\lambda)v))^{m}\wedge\beta^{n-m}\\ &&\qquad \leq 2^{m}\max(a,2)\left( {\int}_{\Omega}\chi(u)(dd^{c} u)^{m}\wedge\beta^{n-m}+ {\int}_{\Omega}\chi(v)(dd^{c} v)^{m}\wedge\beta^{n-m}\right). \end{array} $$

Proposition 2

Let\(\chi :\mathbb {R}^{- }\longrightarrow \mathbb {R}^{+}\)be a decreasing function such that χ(2t) ≤ (t) with some a > 1. Then the following hold:
  1. (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.\({ \mathcal {E}_{m, \chi }({\Omega })}\)).

     
  2. (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

  1. (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 uv, \(v\in SH^{-}_{m}({\Omega })\). 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 \(v_{j}=\max (u_{j}, v)\in \mathcal {E}^{0}_{m}({\Omega })\), vjv on Ω and ujvj. 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, \(v\in \mathcal {F}_{m, \chi }({\Omega })\).
     
  2. (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 \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\), uju on Ω with
    $$\sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}<\infty. $$
    It is clear that \(\{\alpha u_{j}\}\!\subset \! \mathcal {E}^{0}_{m}({\Omega })\), αujαu on Ω. Moreover, since χ(αuj) ≤ χ(2kuj) ≤ akχ(uj) 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 {uj}, \(\{v_{j}\}\!\subset \mathcal {E}^{0}_{m}({\Omega })\), uju on Ω, vju 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) ≤ (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 uju and
$$\sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}< \infty. $$

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 vju and
$$\sup_{j}{\int}_{\Omega}\chi(v_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}< \infty. $$
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 wju. Set
$$u_{j}=\sup\left\{\varphi\in SH_{m}^{-}({\Omega}): \varphi\leq \frac{j-1}{j} w_{j}\text{ on } {\Omega}_{j}\right\}. $$
It is easy to see that uju 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 wjuj so \(u_{j}\in \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C({\Omega })\). Now, since vju as \(j\to \infty \) and uwk so there exists j0 such that \(v_{j_{0}}\leq \frac {k-1}{k} w_{k}\) on Ωk. Therefore, \(v_{j_{0}}\leq u_{k}\) on Ω. Lemma 2 implies that
$$\begin{array}{@{}rcl@{}} {\int}_{\Omega}\chi(u_{k})(dd^{c} u_{k})^{m}\wedge\beta^{n-m}&\leq& 2^{m}\max(a,2){\int}_{\Omega}\chi(v_{j_{0}})(dd^{c} v_{j_{0}})^{m}\wedge\beta^{n-m} \\ &\leq& 2^{m}\max(a,2)\sup_{j}{\int}_{\Omega}\chi(v_{j})(dd^{c} v_{j})^{m}\wedge\beta^{n-m}. \end{array} $$
Thus,
$$\sup_{k}{\int}_{\Omega}\chi(u_{k})dd^{c} u_{k})^{m}\wedge\beta^{n-m}\leq 2^{m}\max(a,2)\sup_{j}{\int}_{\Omega}\chi(v_{j})(dd^{c} v_{j})^{m}\wedge\beta^{n-m}<\infty. $$

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) ≤ (t) for all t < 0 with some a > 1. Then\(\mathcal {E}_{m,\chi }({\Omega })\subset \mathcal {E}_{m}({\Omega })\), and hence, the Hessian Hm(u) = (ddcu)mβn−mis 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 z0 ∈ Ω. Take a neighborhood \(\omega \Subset {\Omega }\) of z0 and a sequence \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\) such that \(\sup _{\overline {\omega }}{u}_{1} <0\), uju on ω and
$$\sup_{j}{\int}_{\Omega}\chi(u_{j})H_{m}(u_{j})< \infty. $$
For each j ≥ 1, set
$$\widetilde{u}_{j}=\sup\{u\in SH^{-}_{m}({\Omega}): u|_{\omega}\leq u_{j}|_{\omega}\}. $$
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
$$\sup_{j}\int\limits_{\Omega}\chi(\widetilde{u}_{j})H_{m}(\widetilde{u}_{j}) <\infty. $$
Moreover, we may assume that \(\inf _{\overline {\omega }}\chi (\widetilde {u}_{1}) = c_{1}>0\). Then
$$\begin{array}{@{}rcl@{}} c_{1}\sup_{j}{\int}_{\Omega}H_{m}(\widetilde{u}_{j}) &=& c_{1}\sup_{j}{\int}_{\overline{\omega}}H_{m}(\widetilde{u}_{j})\\ &\leq&\sup_{j}{\int}_{\overline{\omega}}\chi(\widetilde{u}_{1})H_{m}(\widetilde{u}_{j})\leq\sup_{j}{\int}_{\Omega}\chi(\widetilde{u}_{j}) H_{m}(\widetilde{u}_{j})<\infty. \end{array} $$
Hence,
$$\sup_{j}{\int}_{\Omega}H_{m}(\widetilde{u}_{j})<\infty $$
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 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 \(\mathcal {E}_{m,\chi }({\Omega })\).

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 ACB.

Proposition 5

Set
$$\mathcal{K}=\{\chi: \mathbb{R}^{-}\longrightarrow\mathbb{R}^{+}, \chi \text{ is decreasing and } -t^{2}\chi^{\prime\prime}(t)\lesssim t\chi^{\prime}(t)\lesssim \chi(t)~ \forall t<0\}. $$
Then the class\(\mathcal {K}\)has the following properties.
  1. (a)

    If\(\chi _{1},\chi _{2}\in \mathcal {K}\)and a1,a2 ≥ 0 then\(a_{1}\chi _{1} + a_{2}\chi _{2}\in \mathcal {K}\).

     
  2. (b)

    If\(\chi _{1},\chi _{2}\in \mathcal {K}\)then\(\chi _{1}\cdot \chi _{2}\in \mathcal {K}\).

     
  3. (c)

    If\(\chi \in \mathcal {K}\)then\(\chi ^{p}\in \mathcal {K}\)for all p > 0.

     
  4. (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) ≤ aχ(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) ≤ 2Cχ(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 uv on Ω, \(\chi \!\in \!\mathcal {K}\)and T = ddcφ1 ∧ ⋯ ∧ ddcφm−1βn−mwith\(\varphi _{j}\in SH^{-}_{m}({\Omega })\cap L^{\infty }({\Omega })\), j = 1 … , m − 1. Then for every p ≥ 0, we have
$${\int}_{{\Omega}^{\prime}}\chi(u)dd^{c}v\wedge T\leq c{\int}_{{\Omega}^{\prime\prime}}\chi(u)(dd^{c}u+|u|\beta)\wedge T, $$
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
$${\int}_{{\Omega}^{\prime}}\chi(u)dd^{c}v\wedge T = {\int}_{{\Omega}^{\prime}}{\Phi}\chi(u)dd^{c}v\wedge T \leq {\int}_{\Omega}{\Phi}\chi(u)dd^{c}v\wedge T = {\int}_{\Omega}vdd^{c}({\Phi}\chi(u))\wedge T. $$
On the other hand,
$$\begin{array}{@{}rcl@{}} dd^{c}({\Phi}\chi(u)) &=& d(d^{c}({\Phi}\chi(u)))\\ &=& \chi(u)dd^{c}{\Phi}+{\Phi}(\chi^{\prime}(u)dd^{c}u+\chi^{\prime\prime}(u)du\wedge d^{c}u)+ \chi^{\prime}(u)(d{\Phi}\wedge d^{c}u+du\wedge d^{c}{\Phi}). \end{array} $$
Since ∀t, d(u + tΦ) ∧ dc(u + tΦ) ∧ T ≥ 0, we have
$$\pm u(du\wedge d^{c}{\Phi}+d{\Phi}\wedge d^{c}u)\wedge T\leq (du\wedge d^{c}u+u^{2}d{\Phi}\wedge d^{c}{\Phi})\wedge T $$
and
$$\chi^{\prime}(u)(d{\Phi}\wedge d^{c}u+du\wedge {\Phi})\wedge T\geq -\chi^{\prime}(u)\left( ud{\Phi}\wedge d^{c}{\Phi}+\frac{1}{u}du\wedge d^{c}u\right)\wedge T. $$
Now, we can choose A > 0 sufficiently large such that ddcΦ ≥ −Addcz2, dΦ ∧ dcΦ ≤ Addcz2. Thus, we have the following estimates
$$\begin{array}{@{}rcl@{}} dd^{c}({\Phi}\chi(u))\wedge T&\geq& -A\chi(u)dd^{c}\|z\|^{2}\wedge T+{\Phi}\chi^{\prime}(u)dd^{c}u\wedge T+{\Phi}\chi^{\prime\prime}(u)du\wedge d^{c}u\!\wedge T\\ &&-\chi^{\prime}(u)(ud{\Phi}\wedge d^{c}{\Phi}\,+\,\frac{1}{u}du\!\wedge d^{c}u)\!\wedge T. \end{array} $$
(1)
In the case \(\chi ^{\prime \prime }(u)\leq 0\), we have the following
$$\begin{array}{@{}rcl@{}} vdd^{c}({\Phi}\chi(u))\wedge T &\leq& -Au\chi(u)dd^{c}\|z\|^{2}\wedge T+u\chi^{\prime}(u)dd^{c}u\wedge T\\ &&+u\min\{\chi^{\prime\prime}(u), 0\}du\wedge d^{c}u\wedge T - u^{2}\chi^{\prime}(u)d{\Phi}\wedge d^{c}{\Phi}\wedge T \\ &&- \chi^{\prime}(u)du\wedge d^{c}u\wedge T. \end{array} $$
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
$$\begin{array}{@{}rcl@{}} {\int}_{{\Omega}^{\prime}}\chi(u)dd^{c}v\wedge T &\leq& A{\int}_{{\Omega}^{\prime\prime\prime}}-u\chi(u)dd^{c}\|z\|^{2}\wedge T\,+\,{\int}_{{\Omega}^{\prime\prime\prime}}u\chi^{\prime}(u)dd^{c}u\wedge T\\ &&+\!{\int}_{{\Omega}^{\prime\prime\prime}}u\min\{\chi^{\prime\prime}(u), 0\}du\wedge d^{c}u\wedge T\,+\, {\int}_{{\Omega}^{\prime\prime\prime}}\!- u^{2}\chi^{\prime}(u)d{\Phi}\!\wedge d^{c}{\Phi}\!\wedge T \\ &&+{\int}_{{\Omega}^{\prime\prime\prime}}- \chi^{\prime}(u)du\wedge d^{c}u\wedge T. \end{array} $$
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,
$$\begin{array}{@{}rcl@{}} {\int}_{{\Omega}^{\prime}}\chi(u)dd^{c}v\wedge T &\leq& A{\int}_{{\Omega}^{\prime\prime\prime}}-u\chi(u)dd^{c}\|z\|^{2}\wedge T+c_{1}{\int}_{{\Omega}^{\prime\prime\prime}}\chi(u)dd^{c}u\wedge T\\ &&\!-(c_{2}\,+\,1){\int}_{{\Omega}^{\prime\prime\prime}}\chi^{\prime}(u)du\!\wedge d^{c}u\!\wedge T\,+\, Ac_{1}{\int}_{{\Omega}^{\prime\prime\prime}}\!\chi(u)d{\Phi}\!\wedge dd^{c}\|z\|^{2}\!\wedge T\\ &=& A(c_{1}+1){\int}_{{\Omega}^{\prime\prime\prime}}|u|\chi(u)dd^{c}\|z\|^{2}\wedge T+c_{1}{\int}_{{\Omega}^{\prime\prime\prime}}\chi(u)dd^{c}u\wedge T\\ &&-(c_{2}+1){\int}_{{\Omega}^{\prime\prime\prime}}\chi^{\prime}(u)du\wedge d^{c}u\wedge T. \end{array} $$
Set \( \chi _{1}(t) = -{{\int }^{t}_{0}}\chi (x)dx \) then
$$\chi_{1}^{\prime}(t)\,=\,-\chi(t);\quad \chi_{1}^{\prime\prime}(t)\,=\,-\chi^{\prime}(t);\quad \chi(t)|t|\!\geq\! \chi_{1}(t)\!\geq\! \chi\left( \frac{t}{2}\right)\frac{|t|}{2}. $$
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
$$\begin{array}{@{}rcl@{}} -{\int}_{{\Omega}^{\prime\prime\prime}}\chi^{\prime}(u)du\wedge d^{c}u\wedge T&=& -{\int}_{{\Omega}^{\prime\prime\prime}}d\chi(u)\wedge du\wedge d^{c}u\wedge T\leq {\int}_{\Omega}\psi d\chi(u)\wedge d^{c}u\wedge T\\ &=& {\int}_{\Omega}\chi(u)d\psi\wedge d^{c}u\wedge T+{\int}_{\Omega}\psi \chi(u)dd^{c}u\wedge T\\ &=& {\int}_{\Omega}\chi(u)d\psi\wedge d^{c}u\wedge T+{\int}_{{\Omega}^{\prime\prime}}\psi \chi(u)dd^{c}u\wedge T\\ &=& -{\int}_{\Omega}d\psi d^{c}\chi_{1}(u)\wedge T+{\int}_{{\Omega}^{\prime\prime}}\psi \chi(u)dd^{c}u\wedge T\\ &=& {\int}_{\Omega}\chi_{1}(u)dd^{c}\psi\wedge T+{\int}_{{\Omega}^{\prime\prime}}\psi \chi(u)dd^{c}u\wedge T\\ &\leq& B{\int}_{{\Omega}^{\prime\prime}}\chi(u)|u|dd^{c}\|z\|^{2}\wedge T+{\int}_{{\Omega}^{\prime\prime}}\psi \chi(u)dd^{c}u\wedge T \end{array} $$
with B > 0 sufficiently large.
Finally, we have
$$\begin{array}{@{}rcl@{}} {\int}_{{\Omega}^{\prime}}\chi(u)dd^{c}v\wedge T &\leq& A(c_{1}+1){\int}_{{\Omega}^{\prime\prime\prime}}|u|\chi(u)dd^{c}\|z\|^{2}\wedge T+c_{1}{\int}_{{\Omega}^{\prime\prime\prime}}\chi(u)dd^{c}u\wedge T\\ &&+(c_{2}\,+\,1)B{\int}_{{\Omega}^{\prime\prime\prime}}\chi(u)|u|dd^{c}\|z\|^{2}\wedge T\,+\, (c_{2}\,+\,1){\int}_{{\Omega}^{\prime\prime\prime}}\chi(u)dd^{c}u\!\wedge T\\ &\leq& c\left[{\int}_{{\Omega}^{\prime\prime\prime}}\chi(u)dd^{c}u\wedge T+{\int}_{{\Omega}^{\prime\prime\prime}}\chi(u)|u|dd^{c}\|z\|^{2}\wedge T\right]. \end{array} $$

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 ≤ mn. 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:
$$ {\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}(dd^{c}u)^{m-p}\wedge \beta^{n-m+p} \leq C{\int}_{\Omega}\chi(u)(dd^{c}u)^{m}\wedge \beta^{n-m}< +\infty. $$
(2)
Furthermore, if\(u\in \mathcal {F}_{m, \chi }({\Omega })\)then
$${\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}(dd^{c}u)^{m-p}\wedge \beta^{n-m+p}<+\infty $$
for all p = 1, … , m.

Proof

Set χ0(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) ≤ aχ(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 ∥z2R2 on Ω. Let \(\varphi \in \mathcal {E}^{0}_{m}({\Omega })\) and A > 0 such that ∥z2R2Aφ on \({\Omega }^{\prime }\). Set \(h=\max (\|z\|^{2}-R^{2}; A\varphi )\) then \(h\in \mathcal {E}^{0}_{m}({\Omega })\) and ddch = ddcz2 = β on \({\Omega }^{\prime }\). First, we claim that (2) holds for \(u\in \mathcal {E}^{0}_{m}({\Omega })\). Indeed, we have
$$\begin{array}{@{}rcl@{}} {\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}(dd^{c}u)^{m-p} \wedge (dd^{c}h)^{p}\wedge \beta^{n-m}&\lesssim& {\int}_{\Omega} \chi(u)|u|^{p}(dd^{c}u)^{m-p}\!\wedge (dd^{c}h)^{p} \!\wedge \beta^{n\!-m}\\ &\thickapprox& {\int}_{\Omega} \chi_{p}(u)(dd^{c}u)^{m-p}\wedge (dd^{c}h)^{p} \wedge \beta^{n-m}. \end{array} $$
Integrating by parts, we have
$$\begin{array}{@{}rcl@{}} {\int}_{\Omega}\chi_{p}(u)(dd^{c}u)^{m-p}\wedge (dd^{c}h)^{p} \wedge \beta^{n-m}&=& {\int}_{\Omega}h(dd^{c}u)^{m-p} dd^{c}\chi_{p}(u)\wedge (dd^{c}h)^{p-1}\wedge \beta^{n-m}\\ &=& {\int}_{\Omega}h(dd^{c}u)^{m-p} \left[\chi^{\prime\prime}_{p}(u)du\wedge d^{c}u +\chi^{\prime}_{p}(u)dd^{c}u\right]\\ &&\wedge (dd^{c}h)^{p-1}\wedge \beta^{n-m}\\ &\leq& {\int}_{\Omega}h\chi^{\prime}_{p}(u)(dd^{c}u)^{m-p+1}\wedge (dd^{c}h)^{p-1}\wedge \beta^{n-m}\\ &\leq& \|h\|_{L^{\infty}({\Omega})}{\int}_{\Omega}\chi_{p-1}(dd^{c}u)^{m-p+1}\wedge (dd^{c}h)^{p-1}\!\wedge \beta^{n-m}\\ &\leq& \cdots\\ &\leq& \|h\|^{p}_{L^{\infty}({\Omega})}{\int}_{\Omega}\chi(u)(dd^{c}u)^{m}\wedge \beta^{n-m}<+\infty. \end{array} $$
Hence, if we set \(C=C({\Omega }^{\prime }) = p!\|h\|^{p}_{L^{\infty }({\Omega })}\) then
$$\begin{array}{@{}rcl@{}} +\infty &>& C{\int}_{\Omega}\chi(u)(dd^{c}u)^{m}\wedge \beta^{n-m}\geq {\int}_{\Omega}\chi(u)|u|^{p}(dd^{c}u)^{m-p}\wedge (dd^{c}h)^{p} \wedge \beta^{n-m}\\ &\geq& {\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}(dd^{c}u)^{m-p} \wedge (dd^{c}h)^{p}\wedge \beta^{n-m}\\ &=& {\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}(dd^{c}u)^{m-p} \wedge (dd^{c}\|z\|^{2})^{p}\wedge \beta^{n-m}. \end{array} $$
Finally, we prove (2) holds for \(u\in \mathcal {F}_{m, \chi }({\Omega })\). Indeed, we take \(u_{j}\in \mathcal {E}^{0}_{m}({\Omega })\), uju on Ω such that
$$\sup_{j\geq 1}{\int}_{\Omega} \chi(u_{j})(dd^{c}u_{j})^{m}\wedge\beta^{n-m}<+\infty. $$
By dominated convergence theorem and (ddcuj)mp ∧ (ddcz2)nm + p is weakly convergent to (ddcu)mp ∧ (ddcz2)nm + p in the sense of currents
$$\begin{array}{@{}rcl@{}} &&{\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}\left( dd^{c}u\right)^{m-p} \wedge \left( dd^{c}\|z\|^{2}\right)^{n-m+p}\\ &&\qquad\leq \liminf_{j}{\int}_{{\Omega}^{\prime}}\chi(u_{j})|u_{j}|^{p}\left( dd^{c}u_{j}\right)^{m-p} \wedge \left( dd^{c}\|z\|^{2}\right)^{n-m+p}\\ &&\qquad\leq \liminf_{j}{\int}_{\Omega}\chi(u_{j})|u_{j}|^{p}\left( dd^{c}u_{j}\right)^{m-p} \wedge \left( dd^{c}h\right)^{p} \wedge \left( dd^{c}\|z\|^{2}\right)^{n-m}\\ &&\qquad\leq C \sup_{j}{\int}_{\Omega}\chi(u_{j})\left( dd^{c}u_{j}\right)^{m} \wedge \left( dd^{c}\|z\|^{2}\right)^{n-m}<+\infty. \end{array} $$

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 (ddcu)mβnm 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
    $$\begin{array}{@{}rcl@{}} &&\limsup_{\varepsilon\to 0}{\int}_{\Omega} 1_{K} (dd^{c} \max(\widetilde{u}+\varepsilon,u))^{m}\wedge\beta^{n-m}\\ &&\qquad= \limsup_{\varepsilon\to 0}\left[\lim_{j}{\int}_{\Omega}\varphi_{j} (dd^{c} \max(\widetilde{u}+\varepsilon,u))^{m}\wedge\beta^{n-m}\right]\\ &&\qquad\leq \limsup_{\varepsilon\to 0}\left( {\int}_{\Omega}\varphi_{j} (dd^{c} \max(\widetilde{u}+\varepsilon,u))^{m}\wedge\beta^{n-m}\right)\\ &&\qquad\leq{\int}_{\Omega}\varphi_{j} (dd^{c} u)^{m}\wedge\beta^{n-m}\searrow {\int}_{K}(dd^{c} u)^{m}\wedge\beta^{n-m}. \end{array} $$
    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 uju 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 ≤ mn. 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.
  1. a)

    \(u\in \mathcal E_{m,\chi }({\Omega })\).

     
  2. b)
    For all\(K\Subset {\Omega }\), there exists a sequence\(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal C({\Omega })\), uju 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.
     
  3. c)

    For every\(W\Subset {\Omega }\)such that W is a hyperconvex domain, we have\(u|_{W}\in \mathcal E_{m,\chi }(W)\).

     
  4. d)

    For every z ∈ Ω, there exists a hyperconvex domain\(V_{z}\Subset {\Omega }\)such that zVzand\(u|_{V_{z}}\in \mathcal E_{m,\chi }(V_{z})\).

     

Proof

Let χk 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 \(\{u_{j}\}\subset \mathcal {E}^{0}_{m}({\Omega })\cap \mathcal {C}({\Omega })\), ujv on Ω with
$$ \sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}<\infty. $$
(3)
Then uju on K. We have to prove
$$\sup_{j}{\int}_{K}\chi(u_{j})|u_{j}|^{p}(dd^{c} u_{j})^{m-p}\wedge\beta^{n-m+p}<\infty $$
for p=0,1, … , m. It is obvious that the conclusion holds for p=0. Assume that 1 ≤ pm. Then, by Lemma 4, we get that
$$\sup_{j}{\int}_{K}\chi(u_{j})|u_{j}|^{p}(dd^{c} u_{j})^{m-p}\wedge\beta^{n-m+p}\leq C\sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge \beta^{n-m}<\infty $$
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
$$\sup_{j}{\int}_{W}\chi(u_{j})|u_{j}|^{p}(dd^{c} u_{j})^{m-p}\wedge\beta^{n-m+p}<\infty $$
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
$$\begin{array}{@{}rcl@{}} &&{\int}_{W}\chi(\widetilde{u}_{j})\left( dd^{c}\widetilde{u}_{j}\right)^{m}\wedge \beta^{n-m}= {\int}_{\overline{U}}\chi(\widetilde{u}_{j})\left( dd^{c}\widetilde{u}_{j}\right)^{m}\wedge \beta^{n-m}\\ &&\quad\lesssim {\int}_{{\Omega}_{1}}\chi(u_{j})\left( dd^{c}u_{j}+|u_{j}|\beta\right)\wedge\left( dd^{c}\widetilde{u}_{j}\right)^{m-1}\wedge\beta^{n-m}\\ &&\quad\lesssim {\int}_{{\Omega}_{1}}\chi(u_{j})dd^{c}\widetilde{u}_{j}\wedge \left( dd^{c}\widetilde{u}_{j}\right)^{m-2}\wedge dd^{c} u_{j}\wedge\beta^{n-m}\\ &&\qquad+{\int}_{{\Omega}_{1}}\chi_{1}(u_{j})|u_{j}|dd^{c}\widetilde{u}_{j}\wedge\left( dd^{c}\widetilde{u}_{j}\right)^{m-2}\wedge\beta^{n-m+1}\\ &&\quad\lesssim {\int}_{{\Omega}_{2}}\chi(u_{j})\left( dd^{c}u_{j}+|u_{j}|\beta\right)\wedge\left( dd^{c}\widetilde{u}_{j}\right)^{m-2}\wedge dd^{c}{u}_{j}\wedge\beta^{n-m}\\ &&\qquad+ {\int}_{{\Omega}_{2}}\chi_{1}(u_{j})|u_{j}|\left( dd^{c}u_{j}+|u_{j}|\beta\right)\wedge\left( dd^{c}\widetilde{u}_{j}\right)^{m-2}\wedge\beta^{n-m+1}\\ &&\quad\lesssim{\int}_{{\Omega}_{2}}\chi(u_{j})\left[|u_{j}|^{2} \beta^{2} +|u_{j}|\beta\wedge dd^{c} u_{j} + \left( dd^{c} u_{j}\right)^{2}\right]\wedge \left( dd^{c}\widetilde{u}_{j}\right)^{m-2}\wedge\beta^{n-m}\\ &&\quad\lesssim \cdots\\ &&\quad \lesssim {\int}_{{\Omega}_{m}}\chi(u_{j})\left[|u_{j}|^{m}\beta^{m}+|u_{j}|^{m-1}dd^{c}u_{j}\wedge\beta^{m-1}+\cdots+\left( dd^{c}u_{j}\right)^{m}\right]\wedge\beta^{n-m}. \end{array} $$
Hence,
$$\begin{array}{@{}rcl@{}} &&\sup_{j}{\int}_{W}\chi(u_{j})\left( dd^{c}\widetilde{u}_{j}\right)^{m}\wedge \beta^{n-m}\\ &&\qquad \lesssim \sup_{j}\chi(u_{j}){\int}_{{\Omega}_{m}}\left[|u_{j}|^{m}\beta^{m}+|u_{j}|^{m-1}dd^{c}u_{j}\wedge\beta^{m-1}+\cdots+\left( dd^{c}u_{j}\right)^{m}\right]\wedge\beta^{n-m}\\ &&\qquad \lesssim \!\sup_{j}{\int}_{W}\chi(u_{j})\left[|u_{j}|^{m}\beta^{m}\,+\,|u_{j}|^{m-1}dd^{c}u_{j}\!\wedge\beta^{m-1}\,+\,\cdots\,+\,\left( dd^{c}u_{j}\right)^{m}\right]\wedge\beta^{n\!-m}\!<\!\infty. \end{array} $$
Thus, \(u_{U,W}\!\!:=\!\lim \widetilde {u}_{j}\!\!\in \!\mathcal {F}_{m, \chi }(W)\). Since \(U\!\!\Subset \! \!W\) is arbitrary and uU, 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 zj ∈ 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 vj = 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 uju on \({\Omega }^{\prime }\) and
$$\sup_{j}{\int}_{{\Omega}^{\prime}}\chi(u_{j})\left[|u_{j}|^{m}\beta^{m}+|u_{j}|^{m-1}dd^{c}u_{j}\wedge\beta^{m-1}+\cdots+\left( dd^{c}u_{j}\right)^{m}\right]\wedge \beta^{n-m}<\infty. $$
Let \(h\!\in \!\mathcal {E}^{0}_{m-1}({\Omega })\) be chosen. For each j > 0, take mj > 0 such that ujmjh on \({\Omega }^{\prime }\). Set \(v_{j} \,=\, \max (u_{j}, m_{j} h)\!\in \!\mathcal {E}^{0}_{m-1}({\Omega })\) and vj = uj on \({\Omega }^{\prime }\). Note that vju on \({\Omega }^{\prime }\) and (ddcvj)pβq = (ddcuj)pβq on \({\Omega }^{\prime }\) for 1 ≤ pm−1 and 1 ≤ qnm+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 jj0 with some j0. Without loss of generality, we may assume that \(v_{j}|_{{\Omega }^{\prime }}\leq -1\) for j ≥ 1. Hence, |vj|m ≥ |vj|m−1 on \({\Omega }^{\prime }\) for all j ≥ 1. Now, we have
$$\begin{array}{@{}rcl@{}} && {\int}_{{\Omega}^{\prime}}\chi(u_{j})\left[|u_{j}|^{m}\beta^{m}\,+\,|u_{j}|^{m-1}dd^{c}u_{j}\wedge\beta^{m-1}\,+\,\cdots\,+\,|u_{j}|\left( dd^{c}u_{j}\right)^{m-1}\wedge\beta+\left( dd^{c}u_{j}\right)^{m}\right]\wedge\beta^{n-m}\\ &&\quad \geq {\int}_{{\Omega}^{\prime}}\chi(u_{j})\left[|u_{j}|^{m}\beta^{m}+|u_{j}|^{m-1}dd^{c}u_{j}\wedge\beta^{m-1}+\cdots+|u_{j}|\left( dd^{c}u_{j}\right)^{m-1}\wedge\beta\right]\wedge\beta^{n-m}\\ &&\quad= {\int}_{{\Omega}^{\prime}}\chi(v_{j})\left[|v_{j}|^{m}\beta^{m}+|v_{j}|^{m-1}dd^{c}v_{j}\wedge\beta^{m-1}+\cdots+|v_{j}|\left( dd^{c}v_{j}\right)^{m-1}\wedge\beta\right]\wedge\beta^{n-m}\\ &&\quad = {\int}_{{\Omega}^{\prime}}\chi(v_{j})\left[|v_{j}|^{m}\beta^{m-1}+|v_{j}|^{m-1}dd^{c}v_{j}\wedge\beta^{m-2}+\cdots+|v_{j}|\left( dd^{c}v_{j}\right)^{m-1}\right]\wedge\beta^{n-m+1}\\ &&\quad\geq {\int}_{{\Omega}^{\prime}}\chi(v_{j})\left[|v_{j}|^{m-1}\beta^{m-1}+|v_{j}|^{m-2}dd^{c}v_{j}\wedge\beta^{m-2}+\cdots+\left( dd^{c}v_{j}\right)^{m-1}\right]\wedge\beta^{n-m+1}. \end{array} $$
Note that vju on \({\Omega }^{\prime }\) and
$$\sup_{j} {\int}_{{\Omega}^{\prime}}\chi(v_{j})\left[|v_{j}|^{m-1}\beta^{m-1}\,+\,|v_{j}|^{m-2}dd^{c}v_{j}\!\wedge\beta^{m-2}\,+\,\cdots\,+\,\left( dd^{c}v_{j}\right)^{m\,-\,1}\right]\!\wedge\!\beta^{n-m+1}\!<\! \infty. $$
Moreover, by Theorem 1, we get \(u\in \mathcal E_{m-1,\chi }({\Omega })\). □

Notes

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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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Physics and InformaticsTay Bac UniversitySon LaVietnam

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