Local Property of a Class of m-Subharmonic Functions

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.

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

$$\mathcal{E}_{\chi}({\Omega}) = \left\{\varphi\in PSH^{-}({\Omega}):\exists \mathcal{E}_{0}({\Omega})\ni\varphi_{j}\searrow\varphi,~ \sup_{j\geq 1}{\int}_{\Omega}\chi(\varphi_{j})(dd^{c}\varphi_{j})^{n} < +\infty\right\}, $$

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 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

$$\mathcal{E}_{0}=\mathcal{E}_{0}({\Omega}) = \left\{\varphi\in {PSH}^{-}({\Omega})\cap {L}^{\infty}({\Omega}): \underset{z\to\xi} \lim\varphi(z) = 0~ \forall \xi\in\partial{\Omega}, ~{\int}_{\Omega}(dd^{c}\varphi)^{n} <\infty\right\} $$

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 ≤ m ≤ n, 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 η ₁, … , η _m−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 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

$$S_{m}(\lambda) = \sum\limits_{1\leq j_{1}<\cdots<j_{m}\leq n} \lambda_{j_{1}}\cdots\lambda_{j_{m}}. $$

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

• 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

$$dd^{c}u_{p}\wedge\cdots\wedge dd^{c}u_{1}\wedge\beta^{n-m}= dd^{c}\left( u_{p} dd^{c}u_{p-1}\wedge\cdots\wedge dd^{c}u_{1}\wedge\beta^{n-m}\right). $$

From the definition of m-subharmonic functions and using arguments as in the proof of Theorem 2.1 in [1], we note that 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

$$H_{m}(u) = (dd^{c} u)^{m}\wedge\beta^{n-m} $$

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 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

$$\begin{array}{@{}rcl@{}} \mathcal{E}^{0}_{m}&=&\mathcal{E}^{0}_{m}({\Omega}) = \left\{u\in SH^{-}_{m}({\Omega})\cap{L}^{\infty}({\Omega}): \lim_{z\to\partial{\Omega}} u(z) = 0,~ {\int}_{\Omega}H_{m}(u) <\infty\right\},\\ \mathcal{F}_{m}&=&\mathcal{F}_{m}({\Omega}) = \left\{u\in SH^{-}_{m}({\Omega}): \exists \mathcal{E}^{0}_{m}\ni u_{j}\searrow u,~ \sup_{j}{\int}_{\Omega}H_{m}(u_{j})<\infty\right\}, \end{array} $$

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 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 $\mathcal {E}_{m}({\Omega })$:

$$\mathcal{E}_{m}=\mathcal{E}_{m}({\Omega}) = \left\{u\in SH^{-}_{m}({\Omega}):\forall~ U\Subset{\Omega}, \exists v\in\mathcal{F}_{m}({\Omega}),~ v=u~ \text{ on }~ U\right\}. $$

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

• 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

$$(dd^{c} \varphi)^{m}\wedge\beta^{n-m}(\{\varphi<-t\})\leq t^{m}C_{m}(\{\varphi<-t\}) $$

and

$$t^{m}C_{m}(\{\varphi<-2t\})\leq (dd^{c}\varphi)\wedge\beta^{n-m}(\{\varphi<-t\}). $$

Proof

Let v ∈ S H _m (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 ≤ m ≤ n. 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

• (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

$$\chi_{j}(t):=\chi(t)+\frac{(1-e^{t})}{j},\quad t<0. $$

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

$$\begin{array}{@{}rcl@{}} {\int}_{\Omega}\chi_{j}(v)(dd^{c} v)^{m}\wedge\beta^{n-m} &=&- {\int}_{0}^{+\infty}\chi_{j}^{\prime}(-t)(dd^{c} v)^{m}\wedge\beta^{n-m}(\{v<-t\})dt\\ &\leq& - {\int}_{0}^{+\infty}t^{m}\chi_{j}^{\prime}(-t)C_{m}(\{v<-t\})dt\\ &\leq& - {\int}_{0}^{+\infty}t^{m}\chi_{j}^{\prime}(-t)C_{m}(\{u<-t\})dt\\ &\leq& -2^{m}{\int}_{0}^{+\infty}\chi_{j}^{\prime}(-t)(dd^{c} u)^{m}\wedge\beta^{n-m}(\{u<-t/2\})dt\\ &=&{\int}_{\Omega}\chi_{j}(2u)(dd^{c} (2u))^{m}\wedge\beta^{n-m}\\ &\leq& 2^{m}\max(a,2){\int}_{\Omega}\chi_{j}(u)(dd^{c} u)^{m}\wedge\beta^{n-m}\\ &\leq& 2^{m}\max(a,2)\left( {\int}_{\Omega}\left( \chi(u)+\frac{1}{j}\right)(dd^{c} u)^{m}\wedge\beta^{n-m}\right). \end{array} $$

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 Φ _j are decreasing functions such that Φ _j (0) = 0 and Φ _j ↗χ 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) ≤ 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 2^k > 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(2^k 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

$$\sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}< \infty. $$

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 v _j ↘ u 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 w _j ↘ u. 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 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

$$\begin{array}{@{}rcl@{}} {\int}_{\Omega}\chi(u_{k})(dd^{c} u_{k})^{m}\wedge\beta^{n-m}&\leq& 2^{m}\max(a,2){\int}_{\Omega}\chi(v_{j_{0}})(dd^{c} v_{j_{0}})^{m}\wedge\beta^{n-m} \\ &\leq& 2^{m}\max(a,2)\sup_{j}{\int}_{\Omega}\chi(v_{j})(dd^{c} v_{j})^{m}\wedge\beta^{n-m}. \end{array} $$

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) ≤ 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

$$\sup_{j}{\int}_{\Omega}\chi(u_{j})H_{m}(u_{j})< \infty. $$

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 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

$$\mathcal{K}=\{\chi: \mathbb{R}^{-}\longrightarrow\mathbb{R}^{+}, \chi \text{ is decreasing and } -t^{2}\chi^{\prime\prime}(t)\lesssim t\chi^{\prime}(t)\lesssim \chi(t)~ \forall t<0\}. $$

Then the class $\mathcal {K}$ has the following properties.

• (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) ≤ 2^C 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

$${\int}_{{\Omega}^{\prime}}\chi(u)dd^{c}v\wedge T\leq c{\int}_{{\Omega}^{\prime\prime}}\chi(u)(dd^{c}u+|u|\beta)\wedge T, $$

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) ∧ d ^c(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 d d ^cΦ ≥ −A d d ^c∥z∥², dΦ ∧ d ^cΦ ≤ A d d ^c∥z∥². 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 ≤ 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:

$$ {\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}(dd^{c}u)^{m-p}\wedge \beta^{n-m+p} \leq C{\int}_{\Omega}\chi(u)(dd^{c}u)^{m}\wedge \beta^{n-m}< +\infty. $$

(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 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

$$\begin{array}{@{}rcl@{}} {\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}(dd^{c}u)^{m-p} \wedge (dd^{c}h)^{p}\wedge \beta^{n-m}&\lesssim& {\int}_{\Omega} \chi(u)|u|^{p}(dd^{c}u)^{m-p}\!\wedge (dd^{c}h)^{p} \!\wedge \beta^{n\!-m}\\ &\thickapprox& {\int}_{\Omega} \chi_{p}(u)(dd^{c}u)^{m-p}\wedge (dd^{c}h)^{p} \wedge \beta^{n-m}. \end{array} $$

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 $u_{j}\in \mathcal {E}^{0}_{m}({\Omega })$, u _j ↘ u 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 (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

$$\begin{array}{@{}rcl@{}} &&{\int}_{{\Omega}^{\prime}}\chi(u)|u|^{p}\left( dd^{c}u\right)^{m-p} \wedge \left( dd^{c}\|z\|^{2}\right)^{n-m+p}\\ &&\qquad\leq \liminf_{j}{\int}_{{\Omega}^{\prime}}\chi(u_{j})|u_{j}|^{p}\left( dd^{c}u_{j}\right)^{m-p} \wedge \left( dd^{c}\|z\|^{2}\right)^{n-m+p}\\ &&\qquad\leq \liminf_{j}{\int}_{\Omega}\chi(u_{j})|u_{j}|^{p}\left( dd^{c}u_{j}\right)^{m-p} \wedge \left( dd^{c}h\right)^{p} \wedge \left( dd^{c}\|z\|^{2}\right)^{n-m}\\ &&\qquad\leq C \sup_{j}{\int}_{\Omega}\chi(u_{j})\left( dd^{c}u_{j}\right)^{m} \wedge \left( dd^{c}\|z\|^{2}\right)^{n-m}<+\infty. \end{array} $$

□

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 (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

$$ \sup_{j}{\int}_{\Omega}\chi(u_{j})(dd^{c} u_{j})^{m}\wedge\beta^{n-m}<\infty. $$

(3)

Then u _j ↘ u 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 ≤ p ≤ m. 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 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

$$\sup_{j}{\int}_{{\Omega}^{\prime}}\chi(u_{j})\left[|u_{j}|^{m}\beta^{m}+|u_{j}|^{m-1}dd^{c}u_{j}\wedge\beta^{m-1}+\cdots+\left( dd^{c}u_{j}\right)^{m}\right]\wedge \beta^{n-m}<\infty. $$

Let $h\!\in \!\mathcal {E}^{0}_{m-1}({\Omega })$ be chosen. For each j > 0, take 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

$$\begin{array}{@{}rcl@{}} && {\int}_{{\Omega}^{\prime}}\chi(u_{j})\left[|u_{j}|^{m}\beta^{m}\,+\,|u_{j}|^{m-1}dd^{c}u_{j}\wedge\beta^{m-1}\,+\,\cdots\,+\,|u_{j}|\left( dd^{c}u_{j}\right)^{m-1}\wedge\beta+\left( dd^{c}u_{j}\right)^{m}\right]\wedge\beta^{n-m}\\ &&\quad \geq {\int}_{{\Omega}^{\prime}}\chi(u_{j})\left[|u_{j}|^{m}\beta^{m}+|u_{j}|^{m-1}dd^{c}u_{j}\wedge\beta^{m-1}+\cdots+|u_{j}|\left( dd^{c}u_{j}\right)^{m-1}\wedge\beta\right]\wedge\beta^{n-m}\\ &&\quad= {\int}_{{\Omega}^{\prime}}\chi(v_{j})\left[|v_{j}|^{m}\beta^{m}+|v_{j}|^{m-1}dd^{c}v_{j}\wedge\beta^{m-1}+\cdots+|v_{j}|\left( dd^{c}v_{j}\right)^{m-1}\wedge\beta\right]\wedge\beta^{n-m}\\ &&\quad = {\int}_{{\Omega}^{\prime}}\chi(v_{j})\left[|v_{j}|^{m}\beta^{m-1}+|v_{j}|^{m-1}dd^{c}v_{j}\wedge\beta^{m-2}+\cdots+|v_{j}|\left( dd^{c}v_{j}\right)^{m-1}\right]\wedge\beta^{n-m+1}\\ &&\quad\geq {\int}_{{\Omega}^{\prime}}\chi(v_{j})\left[|v_{j}|^{m-1}\beta^{m-1}+|v_{j}|^{m-2}dd^{c}v_{j}\wedge\beta^{m-2}+\cdots+\left( dd^{c}v_{j}\right)^{m-1}\right]\wedge\beta^{n-m+1}. \end{array} $$

Note that v _j ↘ u 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 $u\in \mathcal E_{m-1,\chi }({\Omega })$. □