Monge–Ampère measures associated with semi-exhaustive functions

Abstract

In this paper, we study the current $T \wedge \text {{dd}}^{c}\psi$ for positive currents T and semi-exhaustive, not necessarily plurisubharmonic, functions $\psi$. The study leads to new definitions of capacity and Lelong–Demailly numbers with respect to the weight $\psi$.

1 Introduction

Let $\Omega$ be a domain in ${\mathbb {C}}^{n}$ and T be a current of bi-dimension (p, p) on $\Omega$. Recall that T is said to be closed if $\text {{d}}T=0$, and is said to be plurisubharmonic (resp. plurisuperharmonic) if $\text {{dd}}^{c}T\ge 0$ (resp. $\text {{dd}}^{c}T\le 0$). Consider a non-negative function $\psi$ of class $\mathcal {C}^{2}$ on $\Omega$ and set the following notations for every reals $r_{1}<r_{2}$

$$\begin{aligned} \begin{aligned}&B_{\psi }(r_{1}):=\{ z \in \Omega ; \ \psi (z)<r_{1} \},\\&S_{\psi }(r_{1}):=\{ z \in \Omega ; \ \psi (z)=r_{1} \},\\&B_{\psi }(r_{2},r_{1}):= B_{\psi }(r_{2}) \setminus B_{\psi }(r_{1}),\\&\beta _{\psi }:= \text {{dd}}^{c}\psi , \ \alpha _{\psi }= \text {{dd}}^{c}\log \psi . \end{aligned} \end{aligned}$$

Throughout this paper, we assume that $\text {{d}}\psi (z) \ne 0$ on $\{ z \in \Omega , \psi (z) \ne 0 \}$ and that $\psi$ is semi-exhaustive, which means that there exists $R_{\psi }>0$ so that $B_{\psi }(R_{\psi })$ is relatively compact in $\Omega$. The paper consists of two parts. The first one concerns with obtaining Lelong–Jensen formula and Lelong–Demailly numbers related to $\psi$. More precisely, we show the following result.

Theorem. (Theorem 3.7) If T and $\text {{dd}}^{c}T$ are of order zero and $0<r_1<r_2<R_{\psi }$, then

$$\begin{aligned} \frac{1}{r^p_2} \int _{B_{\psi }(r_2)} T \wedge \beta _{\psi }^p-\frac{1}{r^p_1} \int _{B_{\psi }(r_1)} T\wedge \beta _{\psi }^p&= \int _{r_{1}}^{r_{2}}\left( \frac{1}{t^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1} \text {{d}}t\nonumber \\ {}&\ \ + \left( \frac{1}{r_{1}^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{0}^{r_{1}} \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1} \text {{d}}t \nonumber \\ {}&\ \ + \int _{B_{\psi }(r_{2},r_{1})} T\wedge \alpha _{\psi }^{p}. \end{aligned}$$

(1.1)

Moreover, Theorem 3.8 shows that the previous formula remains true when T is positive (or negative) plurisubharmonic and $\psi$ is plurisubharmonic of class $\mathcal {C}^{1}$. These results generalize some classical conclusions of [2, 5, 8]. As a consequence of these formulas, one can obtain the Lelong–Demailly number $\nu (T,\psi )$ with respect to the weight $\psi$ for positive plurisubharmonic current T and plurisubharmonic function $\psi$ of class $\mathcal {C}^{1}$.

The second part is devoted to study the Monge–Ampère measure $T\wedge \text {{dd}}^{c}\psi$. Namely, the contribution of this section is stated as follows.

Theorem. (Theorem 4.1) Let T be a positive current. If $\psi$ is of class $\mathcal {C}^{1}$ and $\text {{d}}^{c}\psi \wedge T$ is well defined on $S_{\psi }(r)$ for all $0<r<R_{\psi }$. Then we have

$$\begin{aligned} \int _{S_{\psi }(r)} T \wedge \text {{d}}^{c}\psi \wedge \beta ^{p-1} \ge 0, \ \beta =\text {{dd}}^{c}|z|^{2}. \end{aligned}$$

(1.2)

If, in addition, T is plurisuperharmonic, then $\displaystyle {\int _{B_{\psi }(r)}T \wedge \text {{dd}}^{c}\psi \wedge \beta ^{p-1} \ge 0}$.

The above inequalities make possible to introduce different capacities, each originating from a different source.

2 Preliminaries and notations

Let $\mathcal {D}_{p,q}(\Omega ,k)$ be the space of $\mathcal {C}^k$ compactly supported differential forms of bi-degree (p, q) on $\Omega$. A form $\varphi \in \mathcal {D}_{p,p}(\Omega ,k)$ is said to be strongly positive form if $\varphi$ can be written as

$$\begin{aligned} \varphi (z)=\sum _{j=1}^N \gamma _{j}(z) \ i\alpha _{1,j}\wedge \overline{\alpha }_{1,j}\wedge ...\wedge i\alpha _{p,j}\wedge \overline{\alpha }_{p,j}, \end{aligned}$$

where $\gamma _{j}\ge 0$ and $\alpha _{s,j}\in \mathcal {D}_{1,0}(\Omega ,k)$. Then, $\mathcal {D}_{p,p}(\Omega ,k)$ admits a basis consisting of strongly positive forms. The dual space $\mathcal {D}_{p,q}^{\prime }(\Omega ,k)$ is the space of currents of bi-dimension (p, q) or bi-degree $(n-p,n-q)$ and of order k. If $T\in \mathcal {D}_{p,p}^{\prime }(\Omega ,k)$, then it can be written as

$$\begin{aligned} T=i^{(n-p)^2} \sum _{\vert I\vert =\vert J\vert = n-p} T_{I,J} dz_{I}\wedge d{\overline{z}}_{J}, \end{aligned}$$

where the coefficients $T_{I,J}$ are distributions on $\Omega$. If these coefficients are measures, then T is called of order zero. Remember that when T and $\text {{dd}}^{c}T$ are of order zero, then T is called ${\mathbb {C}}$-normal. The current $T\in \mathcal {D}_{p,p}^{\prime }(\Omega ,k)$ is said to be positive if $\langle {T,\varphi }\rangle \ge 0$ for all forms $\varphi \in \mathcal {D}_{p,p}(\Omega ,k)$ that are strongly positive. For such currents T, the mass is denoted by $\Vert T\Vert$ and defined by $\sum \vert T_{I,J}\vert$, where $\vert T_{I,J}\vert$ are the total variations of the measures $T_{I,J}$. Let $\beta =\text {{dd}}^{c}\vert z\vert ^{2}$ be the Kähler form on ${\mathbb {C}}^n$ ( where $\text {{d}}=\partial +\overline{\partial }$ and $\text {{d}}^{c}=i(-\partial +\overline{\partial })$, thus $\text {{dd}}^{c}=2i\partial \overline{\partial }$), then for each open subset $\Omega _{1}\subset \Omega$, there exists a constant $C>0$ depends only on n and p such that

$$\begin{aligned} T\wedge \frac{\beta ^p}{2^{p}p!}(\Omega _{1})\le \Vert T\Vert _{\Omega _{1}}\le C \ T\wedge \beta ^p(\Omega _{1}). \end{aligned}$$

3 Lelong–Jensen Formula

We start this section with some basic facts that will be used frequently in this paper.

Lemma 3.1

Let E be a domain in ${\mathbb {R}}^{n}$ and $f:E \rightarrow {\mathbb {R}}$ be a function of class $\mathcal {C}^{1}$ so that $\text {{d}}f(x) \ne 0$ for all $x \in E$. If $\varphi$ is a locally bounded $(n-1)$-form and compactly supported, then

$$\begin{aligned} \int _{E} \text {{d}}f \wedge \varphi =\int _{-\infty }^{\infty } \text {{d}}t \int _{f=t} \varphi . \end{aligned}$$

Lemma 3.2

Let $\Omega$ be a domain in ${\mathbb {C}}^{n}$ and $\varphi : \Omega \rightarrow [0,\infty )$ be a function of class $\mathcal {C}^{2}$. Let $t>0$ be a regular value of $\varphi$ and set $S(r)=\{ z \in \Omega , \ \varphi (z)=r \}$. Then,

$$\begin{aligned} j_{t}^{*} \text {{dd}}^{c}(Log \varphi )=\frac{1}{t}j_{t}^{*} \text {{dd}}^{c}\varphi , \end{aligned}$$

where $j_{t}^{*}:S(t) \rightarrow \Omega$ is the canonical injection.

Lemma 3.3

Let $\varphi$ be a function of class $\mathcal {C}^{1}$. If T and $\gamma$ are two $\mathcal {C}^{1}$-form of bi-degree $(n-p,n-p)$ and $(p-1,p-1)$, respectively, then

$$\begin{aligned} \text {{d}}\varphi \wedge \text {{d}}^{c}T \wedge \gamma =- \text {{d}}^{c}\varphi \wedge \text {{d}}T \wedge \gamma . \end{aligned}$$

Lemma 3.4

Let u be a $\mathcal {C}^{1}$-function on $\Omega$. If T is a ${\mathbb {C}}$-normal current of bi-dimension (p, p), then the current $T \wedge \text {{dd}}^{c}u$ is well defined.

Proof

Take a test form $\varphi$ in $\Omega$ and let $(u_{j})_{j \in {\mathbb {N}}}$ be a sequence of smooth functions converges in $\mathcal {C}^{1}(\Omega )$ to u. Then,

$$\begin{aligned} \int _{\Omega } \text {{dd}}^{c}(u_{j} \varphi ) \wedge T \wedge \beta ^{p-1}= \int _{\Omega } u_{j} \varphi \text {{dd}}^{c}T \wedge \beta ^{p-1}. \end{aligned}$$

(3.1)

Hence, by a simple computation, one can deduce that

$$\begin{aligned} \begin{aligned} \int _{\Omega } \varphi \text {{dd}}^{c}u_{j} \wedge T \wedge \beta ^{p-1}&= \int _{\Omega } u_{j} \varphi \text {{dd}}^{c}T \wedge \beta ^{p-1} - 2 \int _{\Omega } \text {{d}}u_{j} \wedge \text {{d}}^{c} \varphi \wedge T \wedge \beta ^{p-1}\\&\ \ -\int _{\Omega } u_{j} \text {{dd}}^{c}\varphi \wedge T \wedge \beta ^{p-1}. \end{aligned} \end{aligned}$$

(3.2)

This shows that $\displaystyle {\lim _{j \rightarrow \infty } \text {{dd}}^{c}u_{j} \wedge T}$ exists as the right-hand side terms of the previous equality are convergent. $\square$

Lemma 3.5

Let $u_{1}, ..., u_{q}, \ 1 \le q \le p$ be plurisubharmonic functions of class $\mathcal {C}^{1}$ on $\Omega$. If T is positive (or negative) plurisubharmonic, then the current $T \wedge \text {{dd}}^{c}u_1 \wedge \dots \wedge \text {{dd}}^{c}u_{q}$ is well defined.

Proof

By the precedent lemma, $T \wedge \text {{dd}}^{c}u_{j}$ is well defined for all $j \in \{ 1, \dots , q \}$. Now, the result is induced by induction and the fact that each $T \wedge \text {{dd}}^{c}u_{j}$ is positive (or negative) plurisubharmonic. $\square$

Theorem 3.6

(See [6]) Let T be an $(n-p,n-p)$-form of class $\mathcal {C}^{2}$ on $\Omega$. Then for all $0<r_{1}<r_{2}<R_{\psi }$, we have

$$\begin{aligned} \begin{aligned} \int _{r_{1}}^{r_{2}} \frac{\text {{d}}t}{t^{p}} \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1}&=\frac{1}{r_{2}^{p}} \int _{S_{\psi }(r_{2})} T \wedge \text {{d}}^{c}\psi \wedge \beta _{\psi }^{p-1}\\&\ - \frac{1}{r_{1}^{p}} \int _{S_{\psi }(r_{1})}T \wedge \text {{d}}^{c}\psi \wedge \beta _{\psi }^{p-1} -\int _{B_{\psi }(r_2,r_1)} T \wedge \alpha _{\psi }^{p}. \end{aligned} \end{aligned}$$

(3.3)

Proof

By Stokes’ theorem, we have

$$\begin{aligned} \begin{aligned} \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1} =\int _{B_{\psi }(t)} \text {{d}}(\text {{d}}^{c}T \wedge \beta _{\psi }^{p-1})&=\int _{S_{\psi }(t)} \text {{d}}^{c}T \wedge \beta _{\psi }^{p-1}\\ {}&= t^{p-1} \int _{S_{\psi }(t)} \text {{d}}^{c}T \wedge \alpha _{\psi }^{p-1}. \end{aligned} \end{aligned}$$

(3.4)

Therefore,

$$\begin{aligned} \int _{r_{1}}^{r_{2}}\frac{\text {{d}}t}{t} \int _{S_{\psi }(t)} \text {{d}}^{c}T \wedge \alpha _{\psi }^{p-1}= & {} \int _{r_{1}}^{r_{2}}\frac{\text {{d}}t}{t^{p}} \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1} \nonumber \\= & {} \int _{B_{\psi }(r_{2},r_{1})} \text {{d}}Log \psi \wedge \text {{d}}^{c}T \wedge \alpha _{\psi }^{p-1}\nonumber \\= & {} \int _{B_{\psi }(r_{2},r_{1})} \text {{d}}T \wedge \text {{d}}^{c}Log \psi \wedge \alpha _{\psi }^{p-1}\nonumber \\= & {} \int _{S_{\psi }(r_{2})} T \wedge \text {{d}}^{c}Log \psi \wedge \alpha _{\psi }^{p-1} - \int _{S_{\psi }(r_{1})} T \wedge \text {{d}}^{c}Log \psi \wedge \alpha _{\psi }^{p-1} \nonumber \\{} & {} - \int _{B_{\psi }(r_{2},r_{1})} T \wedge \alpha _{\psi }^{p-1}. \end{aligned}$$

(3.5)

Now, (3.3) follows by applying Lemma 3.2. $\square$

Theorem 3.7

If T is ${\mathbb {C}}$-normal and $0<r_1<r_2<R_{\psi }$, then

$$\begin{aligned} \frac{1}{r^p_2} \int _{B_{\psi }(r_2)} T \wedge \beta _{\psi }^p-\frac{1}{r^p_1} \int _{B_{\psi }(r_1)} T\wedge \beta _{\psi }^p= & {} \int _{r_{1}}^{r_{2}}\left( \frac{1}{t^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1} \text {{d}}t\nonumber \\ {}{} & {} + \left( \frac{1}{r_{1}^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{0}^{r_{1}} \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1} \text {{d}}t \nonumber \\ {}{} & {} + \int _{B_{\psi }(r_{2},r_{1})} T\wedge \alpha _{\psi }^{p}. \end{aligned}$$

(3.6)

Notice that the previous formula is obtained without constraint on $\text {{d}}T$ as required in [8] and [6].

Proof

We first assume that T of class $\mathcal {C}^{2}$. Then by the previous lemma, one has

$$\begin{aligned} \int _{r_{1}}^{r_{2}} \frac{\text {{d}}t}{t^{p}} \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1}= & {} \frac{1}{r_{2}^{p}} \int _{S_{\psi }(r_{2})} T \wedge \text {{d}}^{c}\psi \wedge \beta _{\psi }^{p-1}\nonumber \\{} & {} - \frac{1}{r_{1}^{p}} \int _{S_{\psi }(r_{1})}T \wedge \text {{d}}^{c}\psi \wedge \beta _{\psi }^{p-1} -\int _{B_{\psi }(r_2,r_1)} T \wedge \alpha _{\psi }^{p}. \end{aligned}$$

(3.7)

But

$$\begin{aligned} \frac{1}{r_{2}^{p}} \int _{S_{\psi }(r_{2})} T \wedge \text {{d}}^{c}\psi \wedge \beta _{\psi }^{p-1}= & {} \frac{1}{r_{2}^{p}} \int _{B_{\psi }(r_{2})} T \wedge \beta _{\psi }^{p}+\frac{1}{r_{2}^{p}} \int _{B_{\psi }(r_{2})} \text {{d}}T \wedge \text {{d}}^{c}\psi \wedge \beta _{\psi }^{p-1} \nonumber \\= & {} \frac{1}{r_{2}^{p}} \int _{B_{\psi }(r_{2})} T \wedge \beta _{\psi }^{p}+ \frac{1}{r_{2}^{p}} \int _{0}^{r_{2}}\text {{d}}t \int _{B_{\psi }(r_{2})} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1}. \end{aligned}$$

(3.8)

Similarly, we have

$$\begin{aligned} \frac{1}{r_{1}^{p}} \int _{S_{\psi }(r_{1})} T \wedge \text {{d}}^{c}\psi \wedge \beta _{\psi }^{p-1}= \frac{1}{r_{1}^{p}} \int _{B_{\psi }(r_{1})} T \wedge \beta _{\psi }^{p}+ \frac{1}{r_{1}^{p}} \int _{0}^{r_{1}}\text {{d}}t \int _{B_{\psi }(r_{2})} \text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1}. \end{aligned}$$

(3.9)

Thus, the result is verified for $\mathcal {C}^{2}$ currents T by combining the latter equalities. Now, for ${\mathbb {C}}$-normal currents T, set $E_{T}= \{ r \in {\mathbb {R}}, \ ||T||_{S(r)}*|| \text {{dd}}^{c}T||_{S(r)} \ne 0 \}$. By the assumptions of T and $\text {{dd}}^{c}T$, it is clear that $||T||_{K}$ and $|| \text {{dd}}^{c}T||_{K}$ are bounded for all compact subset K of $\Omega$. Hence, the set $E_{T}$ is countable. Consider a regularization $\rho _{\varepsilon }$. Then for all $t \in {\mathbb {R}}\setminus E_{T}$, we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _{B_{\psi }(t)} T*\rho _{\varepsilon } \wedge \beta _{\psi }^{p}=\lim _{\varepsilon \rightarrow 0} \int _{{\mathbb {C}}^{n}} \mathbbm {1}_{B_{\psi }(t)} T*\rho _{\varepsilon } \wedge \beta _{\psi }^{p}= \int _{B_{\psi }(t)} T \wedge \beta _{\psi }^{p}, \end{aligned}$$

(3.10)

where $\mathbbm {1}_{B_{\psi }(t)}$ is the characteristic function of $B_{\psi }(t)$. If $r_{1}, \ r_{2}$ are elements of $E_T$ one can take $(r_{1}^{(j)})_{j \in {\mathbb {N}}}$ increasing to $r_{1}$ and $(r_{2}^{(j)})_{j \in {\mathbb {N}}}$ increasing to $r_{2}$ so that $r_{k}^{(j)} \in {\mathbb {R}}\setminus E_T$. The result is achieved by taking the limits.

$\square$

Theorem 3.8

If T is positive (or negative) plurisubharmonic current and $\psi$ is plurisubharmonic and of class $\mathcal {C}^{1}$, then Lelong–Jensen formula (3.6) remains valid.

This result generalizes the formulas in [2] to the case of $\mathcal {C}^{1}$ functions.

Proof

By regularizing $\psi$, one can assume that $\psi$ is smooth. Now the result follows by applying, first, Theorem 3.7 and, second, Lemma 3.5. $\square$

Remark 3.9

According to Theorem 3.7 and Theorem 3.8, if $T\wedge \alpha _{\psi }^{p}$ and $\text {{dd}}^{c}T \wedge \beta _{\psi }^{p-1}$ are positive measures, then the function $r \mapsto \displaystyle {\frac{1}{r^{p}}\int _{B_{\psi }(r)} T \wedge \beta _{\psi }^{p}}$ is positive and increasing on $(0,R_{\psi })$. Therefore, $\displaystyle {\lim _{r \rightarrow 0^{+}}\frac{1}{r^{p}}\int _{B_{\psi }(r)} T \wedge \beta _{\psi }^{p}}$ exists, and is denoted by $\nu (T,\psi )$ the Demailly–Lelong number of T with respect to the weight $\psi$. This show that $\nu (T,\psi )$ exists in the particular case when T is positive plurisubharmonic and $\psi$ is plurisubharmonic and of class $\mathcal {C}^{1}$.

4 Capacity related to semi-exhaustive functions

In this section, we study the current $\text {{dd}}^{c}\psi \wedge T$. From now on, we relax the classification of $\psi$ to $\mathcal {C}^{1}$.

Theorem 4.1

If T is positive and $\text {{d}}^{c}\psi \wedge T$ is well defined on $S_{\psi }(r)$ for all $0<r<R_{\psi }$, then we have

$$\begin{aligned} \int _{S_{\psi }(r)} \text {{d}}^{c}\psi \wedge T \wedge \beta ^{p-1} \ge 0. \end{aligned}$$

(4.1)

If, in addition, T is plurisuperharmonic, then $\displaystyle {\int _{B_{\psi }(r)}T \wedge \text {{dd}}^{c}\psi \wedge \beta ^{p-1} \ge 0}$.

Proof

Notice first that $\text {{d}}\psi \wedge \text {{d}}^{c}\psi \wedge T$ is a positive current. Hence, the function ${f(r)\!=\!\int _{B_{\psi }(r)} \text {{d}}\psi \!\wedge \! \text {{d}}^{c}\psi \!\wedge \! T \!\wedge \! \beta ^{p-1}}$ is non decreasing. So, $f^{'}(r) \ge 0$. But

$$\begin{aligned} f^{'}(r)= & {} \left[ \int _{B_{\psi }(r)} \text {{d}}\psi \wedge \text {{d}}^{c}\psi \wedge T \wedge \beta ^{p-1} \right] ^{'} \nonumber \\= & {} \left[ \int _{0}^{r} \text {{d}}t \int _{S_{\psi }(t)} \text {{d}}^{c}\psi \wedge T \wedge \beta ^{p-1} \right] ^{'}\nonumber \\= & {} \int _{S_{\psi }(r)} \text {{d}}^{c}\psi \wedge T \wedge \beta ^{p-1}. \end{aligned}$$

(4.2)

Now, assume that $\text {{dd}}^{c}T \le 0$. By Stokes’ formula, we have

$$\begin{aligned} \int _{S_{\psi }(r)} \text {{d}}^{c}\psi \wedge T \wedge \beta ^{p-1}= & {} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}- \int _{B_{\psi }(r)} \text {{d}}^{c}\psi \wedge \text {{d}}T \wedge \beta ^{p-1}\nonumber \\= & {} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}+ \int _{B_{\psi }(r)} \text {{d}}\psi \wedge \text {{d}}^{c}T \wedge \beta ^{p-1}\nonumber \\= & {} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}+ \int _{0}^{r} \text {{d}}t \int _{S_{\psi }(t)} \text {{d}}^{c}T \wedge \beta ^{p-1}\nonumber \\= & {} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}+ \int _{0}^{r} \text {{d}}t \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta ^{p-1}. \end{aligned}$$

(4.3)

This shows that

$$\begin{aligned} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}= \int _{S_{\psi }(r)} \text {{d}}^{c}\psi \wedge T \wedge \beta ^{p-1}- \int _{0}^{r} \text {{d}}t \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta ^{p-1} \ge 0. \end{aligned}$$

(4.4)

$\square$

Remark 4.2

If T is ${\mathbb {C}}$-normal on $\Omega$, then the current $\text {{d}}^{c}\psi \wedge T$ is well defined on $S_{\psi }(r)$. Indeed, the wedge product $\text {{dd}}^{c}\psi \wedge T$ is achieved by Lemma 3.4. Hence, we set

$$\begin{aligned} \int _{S_{\psi }(r)} \text {{d}}^{c}\psi \wedge T \wedge \beta ^{p-1}= \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1} + \int _{0}^{r} \text {{d}}t \int _{B_{\psi }(t)} \text {{dd}}^{c}T \wedge \beta ^{p-1}. \end{aligned}$$

(4.5)

As shown above, semi-exhaustive functions have things in common with plurisubharmonic functions. Despite this, we must be cautious once we deal with these semi-exhaustive functions as some important properties of Psh are not applicable to this type of functions. For example, if $\psi$ is plurisubharmonic, then it is so obvious that $\displaystyle {r \mapsto \int _{B_{\psi }(r)}\text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}}$ is increasing in r. This fact is not valid when the plurisubharmonicity is omitted. The following example shows this.

Example 4.3

In ${\mathbb {C}}$, set $\Omega = B(0,1)$ and put $T=1$. Now, take $\psi (z)= \sin {(\frac{\pi }{2}|z|^{2})}$. Clearly, $\psi$ is an semi-exhaustive function on $\Omega$ where $R_{\psi }=1$. By a simple computation, we have

$$\begin{aligned} \text {{dd}}^{c}\psi =[-(\frac{\pi }{2})^{2}|z|^{2} \sin {(\frac{\pi }{2}|z|^{2})}+\frac{\pi }{2}\cos {(\frac{\pi }{2}|z|^{2})}] \ \beta . \end{aligned}$$

(4.6)

Notice that $\text {{dd}}^{c}\psi$ tends to $\frac{\pi }{2} \beta$ when $|z| \rightarrow 0$, while $\text {{dd}}^{c}\psi$ tends to $-(\frac{\pi }{2})^{2} \beta$ as $|z| \rightarrow 1^{-}$.

Let us recall a very fundamental fact about currents. When g is a locally bounded plurisubharmonic function on $\Omega$ and T is positive and closed, the current gT is well defined. The exterior derivatives lead to the current $\text {{dd}}^{c}g \wedge T$ as it is defined by $\text {{dd}}^{c}(gT)$.

Proposition 4.4

Let T be a positive closed current of bi-dimension (p, p) on $\Omega$ and g be a locally bounded plurisubharmonic function. Then,

$$\begin{aligned} \int _{B_{\psi }(r)} \text {{d}}\psi \wedge \text {{d}}^{c}g \wedge T \wedge \beta ^{p-1} \ge 0. \end{aligned}$$

(4.7)

If g is positive, then

$$\begin{aligned} \int _{B_{\psi }(r)} \text {{dd}}^{c}(\psi g) \wedge T \wedge \beta ^{p-1} \ge 0 . \end{aligned}$$

(4.8)

Proof

First, we show that the quantity $\displaystyle {\int _{B_{\psi }(r)} \text {{d}}\psi \wedge \text {{d}}^{c}g \wedge T \wedge \beta ^{p-1}}$ is non-negative and increasing in r. By Lemma 3.1, we have

$$\begin{aligned} \begin{aligned} \int _{B_{\psi }(r)} \text {{d}}\psi \wedge \text {{d}}^{c}g \wedge T \wedge \beta ^{p-1}&= \int _{0}^{r} \text {{d}}t \int _{S_{\psi }(t)} \text {{d}}^{c}g \wedge T \wedge \beta ^{p-1}\\ {}&= \int _{0}^{r} \text {{d}}t \int _{B_{\psi }(t)} \text {{dd}}^{c}g \wedge T \wedge \beta ^{p-1} \ge 0. \end{aligned} \end{aligned}$$

(4.9)

Now, assume that g is positive. Then the current gT is ${\mathbb {C}}$-normal. Hence by Lemma 3.1 and Theorem 4.1, we have

$$\begin{aligned} \int _{B_{\psi }(r)} \text {{dd}}^{c}(\psi g) \wedge T \wedge \beta ^{p-1}= & {} \int _{S_{\psi }(r)} \text {{d}}^{c}(\psi g) \wedge T \wedge \beta ^{p-1}\nonumber \\ {}= & {} \int _{S_{\psi }(r)} g \text {{d}}^{c}\psi \wedge T \wedge \beta ^{p-1}+ r \int _{B_{\psi }(r)} \text {{dd}}^{c}g \wedge T \wedge \beta ^{p-1} \ge 0. \end{aligned}$$

(4.10)

$\square$

In virtue of [7], if $g \in Psh(\Omega ) \cap L^{\infty }_{loc}(\Omega \setminus K)$ for some compact subset K of $\Omega$, then $\text {{dd}}^{c}g \wedge T$ is well defined. Therefore, by following a similar technique as in [1], the current gT in this case can be deduced. Indeed, take neighborhoods V and W so that $K\Subset V \Subset W \subset \Omega$, and $\chi \in \mathcal {C}^{\infty }_{0}(W)$ such that $\chi =1$ on V. Now, construct a decreasing sequence of smooth plurisubharmonic functions $(g_{j})$ converges point-wise to g on $\Omega$. Then,

$$\begin{aligned} \int _{W} \text {{dd}}^{c}(\chi |z|^{2}) g_{j}T \wedge \beta ^{p-1}=\int _{W} \chi |z|^{2} \text {{dd}}^{c}g_{j} \wedge T \wedge \beta ^{p-1}. \end{aligned}$$

This implies that,

$$\begin{aligned} \int _{W} -\chi g_{j}T\wedge \beta ^{p}= & {} -\int _{W} \chi |z|^{2} \text {{dd}}^{c}g_{j} \wedge T \wedge \beta ^{p-1} \nonumber \\{} & {} +2 \int _{W} g_{j}\text {{d}}\chi \wedge \text {{d}}^{c} |z|^{2} \wedge T \wedge \beta ^{p-1} \nonumber \\{} & {} + \int _{W} g_{j}|z|^{2}\text {{dd}}^{c}\chi \wedge T \wedge \beta ^{p-1}. \end{aligned}$$

Thus,

$$\begin{aligned} \sup _{j} \int _{V} |g_{j}| T \wedge \beta ^{p} < \infty . \end{aligned}$$

The previous discussion yields to the fulfillment of Proposition 4.4 in the case of unbounded functions g.

Definition 4.5

A real-valued function f on $\Omega$ is called a T-Monge-Ampère of degree $q, \ 0 \le q \le p$ on $\Omega$ (for short $f \in {\mathcal {M}}{\mathcal {A}}^{q} (T,\Omega ))$ if the current $(\text {{dd}}^{c}f)^{q} \wedge T$ is well defined on $\Omega$. If in addition $\int _{\Omega } (\text {{dd}}^{c}f)^{q} \wedge T \wedge \beta ^{p-q}\ge 0$, then f is said to be of class $\mathcal {P}^{q}(T,\Omega )$.

Clearly, the set $\mathcal {C}^{2} \cap Psh (\Omega ) \subseteq \mathcal {P}^{p}(T,\Omega )$. Moreover, the early studies of currents lead to many cases where the previous inclusion is proper. For instant, if T is positive and closed, then we already know that $Psh(\Omega ) \cap L^{\infty }_{loc}(\Omega ) \subset \mathcal {P}^{p}(T,\Omega )$. Also, the above study shows that the $\mathcal {C}^{1}$ semi-exhaustive function $\psi \in \mathcal {P}^{1}(T,B_{\psi }(r))$.

Definition 4.6

Let S be a positive current of bi-dimension (p, p) on $\Omega$. We define the capacity $\mathcal {C}^{q}_{S}(O,\Omega )$ for all Borel set $O \Subset \Omega$ by

$$\begin{aligned} \mathcal {C}^{q}_{S}(O,\Omega )= \sup \left\{ \int _{O} (\text {{dd}}^{c}f)^{q} \wedge S \wedge \beta ^{p-q}, \ f \in \mathcal {P}^{q}(S, \Omega ), \ 0 \le f \le 1 \right\} . \end{aligned}$$

Observe that for positive and closed currents S, the capacity $\mathcal {C}_{S}$, which is introduced in [4], is dominated by $\mathcal {C}^{p}_{S}$. This is an obvious inclusion from the fact that $Psh(\Omega ) \cap L^{\infty }_{loc}(\Omega ) \subset \mathcal {P}^{p}(T,\Omega )$. We give an example where $\mathcal {C}_{S}<\mathcal {C}^{q}_{S}$.

Example 4.7

In ${\mathbb {C}}^{1}$, set $\Omega =B(0,1)$ and $S=1$. From [3] it is very well known that $\displaystyle {\mathcal {C}_{S}(B(0,\frac{1}{2}))=\frac{1}{\log {2}}}$. Now construct a positive smooth semi-exhaustive function $\psi$ on B(0, 1) so that $\psi (z)=\frac{2}{3} |z|^{2}$ on $B(0,\frac{1}{2})$, and $\psi (z)=\sin {(\frac{\pi }{2}|z|^{2})}$ on an appropriate neighborhood of $\{ |z|=1 \}$. Clearly,

$$\begin{aligned} \int _{B(0,\frac{1}{2})} \text {{dd}}^{c}\psi =\frac{2}{3}\pi . \end{aligned}$$

(4.11)

This show that $\displaystyle {\mathcal {C}_{S}(B(0,\frac{1}{2}))<\mathcal {C}^{1}_{S}(B(0,\frac{1}{2}))}$.

Another definition of capacity is given as follows.

Definition 4.8

Let S be a positive closed current of bi-dimension (p, p) on $\Omega$. We define the capacity $\mathcal {C}_{S}(\text {{d}}\psi ,r,r^{\prime })$ for all $0<r<r^{\prime }<R_{\psi }$ by

$$\begin{aligned} \sup \left\{ \int _{B_{\psi }(r)} \text {{d}}\psi \wedge \text {{d}}^{c}g \wedge (\text {{dd}}^{c}g)^{p-1} \wedge S , \ g \in Psh(B_{\psi }(r^{\prime })), \ 0 \le g \le 1 \right\} . \end{aligned}$$

The above definitions together with Proposition 4.4 yield to the next properties.

Proposition 4.9

Let S be a positive closed current of bi-dimension (p, p) on $\Omega$. Then for all $0<r<r^{\prime }<r^{\prime \prime }<R_{\psi }$, we have

• (1) $\mathcal {C}_{S}(\text {{d}}\psi ,r,r^{\prime \prime })\le \mathcal {C}_{S}(\text {{d}}\psi ,r^{\prime },r^{\prime \prime })$.

• (2) $\mathcal {C}_{S}(\text {{d}}\psi ,r,r^{\prime })\ge \mathcal {C}_{S}(\text {{d}}\psi ,r,r^{\prime \prime })$.

• (3) $\frac{1}{r}\mathcal {C}_{S}(\text {{d}}\psi ,r,R_{\psi })\le \mathcal {C}_{S}(B_{\psi }(r),B_{\psi }(R_{\psi })) \le \mathcal {C}^{p}_{S}(B_{\psi }(r),B_{\psi }(R_{\psi }))$.

We end this paper with a version of Chern–Levine–Nirenberg inequality in the case of semi-exhaustive functions.

Proposition 4.10

Let K be a compact subset of $\Omega$ so that $B_{\psi }(r) \Subset K \Subset \Omega$. If T is positive and plurisuperharmonic, then there exists a constant $C_{K}(r)>0$ such that

$$\begin{aligned} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1} \le C_{K}(r) \Vert \psi \Vert _{\mathcal {L}^{\infty }(K)} \Vert T\Vert _{K}. \end{aligned}$$

(4.12)

Proof

By similar arguments as above, one can assume that $\psi$ is of class $\mathcal {C}^{2}$. Now, set $O= \{ z \in B_{\psi }(r), \text {{dd}}^{c}\psi (z)>0\}$. Clearly, O is an open subset of $B_{\psi }(r)$, and

$$\begin{aligned} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1} \le \int _{O} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}. \end{aligned}$$

(4.13)

Thus, for $\varepsilon >0$, there exists an open subset $O_{\varepsilon }$ of O so that

$$\begin{aligned} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1} \le \int _{O_{\varepsilon }} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}+\varepsilon . \end{aligned}$$

(4.14)

But Chern–Lieven–Nirenberg shows that

$$\begin{aligned} \int _{B_{\psi }(r)} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}\le & {} \int _{O_{\varepsilon }} \text {{dd}}^{c}\psi \wedge T \wedge \beta ^{p-1}+\varepsilon \nonumber \\\le & {} C_{K}(r) \Vert \psi \Vert _{\mathcal {L}^{\infty }(K)} \Vert T\Vert _{K} . \end{aligned}$$

(4.15)

$\square$