Lelong–Demailly Numbers and Wedge Product of S-Plurisubahrmonic Currents

Abstract

In this paper, we show the existence of the Lelong–Demailly numbers of positive S-plurisubharmonic currents. Moreover, a valid definition of $\mathrm{{dd}}^{c}g \wedge T$ is obtained for plurisubharmonic currents T and unbounded plurisubharmonic functions g. The importance of this definition comes from the sharpness of our condition on the Hausdorff dimension of the locus points of g.

1 Introduction

Let $\Omega$ be an open subset of $\mathbb {C}^{n}$ and T be a positive current of bi-dimension (p, p). Recall that T is said to be closed if $\mathrm{{d}}T=0$, and is said to be S-plurisubharmonic (resp. S-plurisuperharmonic) if there exists a positive current S on $\Omega$ such that $\mathrm{{dd}}^{c}T\ge -S$ (resp. $\mathrm{{dd}}^{c}T\le S$). Consider a non-negative plurisubharmonic function $\varphi$ of class $\mathcal {C}^{2}$ on $\Omega$ such that $\log {\varphi }$ is plurisubharmonic and set the following notations for every $0<r_{1}<r_{2}$

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

Throughout this paper, we assume that $\varphi$ is semi-exhaustive, which means that there exists $R_{\varphi }>0$ so that $B_{\varphi }(R_{\varphi })$ is relatively compact in $\Omega$. The Lelong–Demailly number of T with respect to the weight $\varphi$ is $\nu (T,\varphi ):=\displaystyle {\lim \nolimits _{r\rightarrow 0^{+}}\nu (T,\varphi ,r)}$, where $\nu (T,\varphi ,r)= \displaystyle {\frac{1}{r^{p}}\int _{B_{\varphi (r)}} T \wedge \beta ^{p}_{\varphi }}, r \in (0,R_{\varphi })$. By $Psh^{-}(\Omega )$ we denote the set of all negative plurisubharmonic functions on $\Omega$. For a function $g \in Psh^{-}(\Omega )$, put $L_{g}$ to be the set of all locus points of g which consists of the points $z \in \Omega$ where g is unbounded in every neighborhood of z.

The first part of this paper deals with Lelong–Demailly numbers. Actually, for S-plurisubharmonic currents T we give sufficient conditions on S that yield to the existence of $\nu (T,\varphi )$.

Theorem (Theorem 2.2) Let T be a positive S-plurisubharmonic current of bi-dimension (p, p) on $\Omega$. If the function $t \mapsto \displaystyle {\frac{\nu (S,\varphi ,t)}{t}}$ is integrable in a neighborhood of 0, then the Lelong–Demailly number $\nu (T,\varphi )$ exists.

The result generalizes essential results due to Pierre Lelong and Henri Skoda [11].

The second part treats the wedge product of currents. More precisely, we study existence of the current $\mathrm{{dd}}^{c}g \wedge T$ in both cases when T is either plurisubharmonic or plurisuperharmonic.

Let T be a positive current of bi-dimension (p, p) on $\Omega$ and $g \in Psh^{-}(\Omega ) \cap \mathcal {C}^{1}(\Omega {\setminus } L_g)$. Assume that $(g_{j})_{j}$ is a decreasing sequence of smooth plurisubharmonic functions on $\Omega$ converging to g in $\mathcal {C}^{1}(\Omega {\setminus } L_g)$. Then $\mathrm{{dd}}^{c}g \wedge T$ is a well defined current as a limit of $\mathrm{{dd}}^{c}g_{j}\wedge T$ on $\Omega$ in each of the following cases.

• (1) $\mathrm{{dd}}^{c}T \ge 0$ and $\mathcal {H}_{2p-2}(L_g \cap \mathrm {Supp}T)$ is locally finite. (Theorem 3.8.)

• (2) $\mathrm{{dd}}^{c}T \le 0$ and $\mathcal {H}_{2p-2}(L_g \cap \mathrm {Supp}T)=0$. (Theorem 3.5.)

The definition of $\mathrm{{dd}}^{c}g \wedge T$ was considered before in the particular case when T is pluriharmonic. In fact, the case when $\Omega$ is a compact Kähler manifold and g is continuous on $\Omega$ is due to Dinh and Sibony [6]. One year later, Alessandrini and Bassanelli [2] defined $\mathrm{{dd}}^{c}g \wedge T$ in the case where $L_g$ is a proper analytic set with $dim(L_g)< p$. In [1] the author generalized the latter work to the case of closed obstacle $L_g$ when $\mathcal {H}_{2p-2}(L_g \cap \mathrm {Supp}T)$ is locally finite. We sholud point out to the inspiring works of Demailly [5] and Fornæss-Sibony [7] where they established fabulous techniques to tackle the case of closed currents.

In order to grasp the above definitions, new versions of Chern–Levine–Nirenberg inequalities induced to the considered objects T and g. Namely, we prove what follows.

Lemma (Lemma 3.3) Let T be a positive plurisubharmonic current of bi-dimension (p, p) on $\Omega$. Let K and L be compact sets of $\Omega$ with $L\subset \overset{\circ }{K}$. Then there exist positive constant $C_{K,L}$, and a neighborhood V of $K\cap L_g$ such that for all $g \in Psh^{-}(\Omega ) \cap \mathcal {C}^{1}(\Omega {\setminus } L_g)$ so that $\mathcal {H}_{2p-1}(L_g \cap {\text {Supp} \ T})=0$ we have

$$\begin{aligned} \Vert \mathrm{{dd}}^{c}g\wedge T\Vert _{L{\setminus } L_g}\le C_{K,L}\Vert g\Vert _{\mathcal {L}^{\infty }(K{\setminus } V)}\Vert T\Vert _{K{\setminus } V}. \end{aligned}$$

We also deduce a similar estimation when T and $\mathrm{{dd}}^{c}T$ alternate in the sign as soon as $\mathcal {H}_{2p-2}(L_g \cap \mathrm{Supp} T)=0$.

2 Lelong–Demailly Numbers

We start here with a very famous version of Lelong–Jensen formula.

Lemma 2.1

Let T be a positive or negative plurisubharmonic current of bi-dimension (p, p) on $\Omega$. Then for all $0<r_{1}<r_{2}<R_{\varphi }$ we have

$$\begin{aligned} \begin{aligned}&\nu (T,\varphi ,r_{2})-\nu (T,\varphi ,r_{1})=\int _{B_{\varphi }(r_{2},r_{1})} T\wedge \alpha _{\varphi }^{p}\\&\qquad + \int _{r_{1}}^{r_{2}}\left( \frac{1}{t^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{B_{\varphi }(t)} \mathrm{{dd}}^{c}T \wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t\\&\qquad + \left( \frac{1}{r_{1}^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{0}^{r_{1}} \int _{B_{\varphi }(t)} \mathrm{{dd}}^{c}T \wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t \end{aligned} \end{aligned}$$

(2.1)

Theorem 2.2

Let T be a positive S-plurisubharmonic current of bi-dimension (p, p) on $\Omega$. If the function

$$\begin{aligned} t \mapsto \displaystyle {\frac{\nu (S,\varphi ,t)}{t}} \end{aligned}$$

(2.2)

is integrable in a neighborhood of 0, then the Lelong–Demailly number $\nu (T,\varphi )$ exists.

Proof

We follow a similar technique as in [8]. For all $0<r<R_{\varphi }$, let us define

$$\begin{aligned} \Gamma (r)=\nu (T,\varphi ,r)+\int _{0}^{r} \left( 1-\frac{t^{p}}{r^{p}}\right) \frac{1}{t^{p}} \int _{B_{\varphi }(t)}S \wedge \beta _{\varphi }^{p-1} \, \mathrm{{d}}t. \end{aligned}$$

(2.3)

By the properties of S, the function $\Gamma$ is well-defined and non-negative. Now, for $0<r_{1}<r_{2}<R_{\varphi }$, we have

$$\begin{aligned} \begin{aligned} \Gamma (r_{2})-\Gamma (r_{1})&=\nu (T,\varphi ,r_{2})-\nu (T,\varphi ,r_{1}) + \int _{0}^{r_{2}} \left( 1-\frac{t^{p}}{r_{2}^{p}}\right) \frac{1}{t^{p}} \int _{B_{\varphi }(t)}S \wedge \beta _{\varphi }^{p-1} \ \mathrm{{d}}t \\&\quad \ -\int _{0}^{r_{1}} \left( 1-\frac{t^{p}}{r_{1}^{p}}\right) \frac{1}{t^{p}} \int _{B_{\varphi }(t)}S \wedge \beta _{\varphi }^{p-1} \ \mathrm{{d}}t \\&= \nu (T,\varphi ,r_{2})-\nu (T,\varphi ,r_{1}) + \int _{r_{1}}^{r_{2}} \frac{1}{t^{p}} \int _{B_{\varphi }(t)} S\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t \\&\quad \ - \int _{0}^{r_{2}} \frac{1}{r_{2}^{p}} \int _{B_{\varphi }(t)} S\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t + \int _{0}^{r_{1}} \frac{1}{r_{1}^{p}} \int _{B_{\varphi }(t)} S\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t. \end{aligned} \end{aligned}$$

(2.4)

Using Lemma 2.1, one has

$$\begin{aligned} \Gamma (r_{2})-\Gamma (r_{1})= & {} \int _{r_{1}}^{r_{2}} \frac{1}{t^{p}} \int _{B_{\varphi }(t)} \mathrm{{dd}}^{c}T\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t - \int _{0}^{r_{2}} \frac{1}{r_{2}^{p}} \int _{B_{\varphi }(t)} \mathrm{{dd}}^{c}T\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t \nonumber \\&+ \int _{0}^{r_{1}} \frac{1}{r_{1}^{p}} \int _{B_{\varphi }(t)} \mathrm{{dd}}^{c}T\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t + \int _{r_{1}}^{r_{2}} \frac{1}{t^{p}} \int _{B_{\varphi }(t)} S\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t \nonumber \\&- \int _{0}^{r_{2}} \frac{1}{r_{2}^{p}} \int _{B_{\varphi }(t)} S\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t + \int _{0}^{r_{1}} \frac{1}{r_{1}^{p}} \int _{B_{\varphi }(t)} S\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t\nonumber \\&+ \int _{B_{\varphi }(r_{2},r_{1})} T\wedge \alpha _{\varphi }^{p}\nonumber \\= & {} \int _{B_{\varphi }(r_{2},r_{1})} T\wedge \alpha _{\varphi }^{p} + \int _{r_{1}}^{r_{2}} \left( \frac{1}{t^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{B_{\varphi }(t)} (\mathrm{{dd}}^{c}T+S)\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t \nonumber \\&+ \int _{0}^{r_{1}} \left( \frac{1}{r_{1}^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{B_{\varphi }(t)} (\mathrm{{dd}}^{c}T+S)\wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t \ge 0. \end{aligned}$$

(2.5)

This shows that the function $\Gamma$ is increasing, and therefore the limit $\lambda =\displaystyle {\lim \nolimits _{r \rightarrow 0^{+}}} \Gamma (r)$ exists. Now, the integrability of $\displaystyle {\frac{\nu (S,\varphi ,t)}{t}}$ together with uniform boundedness of $\displaystyle {\left( 1-\frac{t^{p}}{r^{p}}\right) }$ imply that

$$\begin{aligned} \lim _{r \rightarrow 0^{+}} \int _{0}^{r} \left( 1-\frac{t^{p}}{r^{p}}\right) \frac{1}{t^{p}} \int _{B_{\varphi }(t)}S \wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t =0. \end{aligned}$$

(2.6)

Hence, $\nu (T,\varphi )=\displaystyle {\lim \nolimits _{r \rightarrow 0^{+}}}\nu (T,\varphi ,r)=\displaystyle {\lim \nolimits _{r \rightarrow 0^{+}}} \Gamma (r)=\lambda$. $\square$

We should point out to the work of Ghiloufi [8] where he made a nice contribution in the case when $\mathrm{{dd}}^{c}T \le 0$.

Application For positive closed current S, if the Lelong–Demailly number $\nu (U,\varphi )$ of the potential U of S is obtained, then $\nu (T,\varphi )$ exists. Indeed, the fulfillment of the condition (2.2) can be checked by applying Lemma 2.1 to U and letting $r_{1} \rightarrow 0^+$. More precisely, we have

$$\begin{aligned} \begin{aligned}&0 \le \int _{0}^{r_{2}}\left( \frac{1}{t^{p}}-\frac{1}{r_{2}^{p}}\right) \int _{B_{\varphi }(t)} S \wedge \beta _{\varphi }^{p-1} \mathrm{{d}}t \le \nu (U,\varphi ,r_{2})-\nu (U,\varphi ) \\&\quad -\int _{B_{\varphi }(r_{2})} U \wedge \alpha _{\varphi }^{p} \\&\quad < \ \infty \end{aligned} \end{aligned}$$

(2.7)

Another useful application of Theorem 2.2 occurs when we consider a positive function $h\in \mathcal {C}^{2}(\Omega )$ alongside a positive closed current T. In such a situation, $\nu (hT,\varphi )$ exists regardless the fact that $\mathrm{{dd}}^{c}(hT)$ may not be positive nor negative. Actually, one can, locally, find a positive constant M such that

$$\begin{aligned} \mathrm{{dd}}^{c}(hT) \ge -M T\wedge \beta . \end{aligned}$$

Now we achieve the required Lelong–Demailly number by taking $S=M T\wedge \beta$.

3 Wedge Product of Positive Currents

In this section we continue the work in [1]. From now on, in this section, we assume that $g\in Psh^{-}(\Omega )\cap \mathcal {C}^{1}(\Omega {\setminus } L_g)$. Here we start by recalling a classical result due to Fornæss and Sibony [7].

Lemma 3.1

Let T be a positive closed current on $\Omega$ and $u \in Psh^{-}(\Omega )$. If $\mathcal {H}_{2p}(L_u \cap \mathrm {Supp}T)=0$, then the currents uT together with $\mathrm{{dd}}^{c}u\wedge T$ are well defined.

The result was first considered by Demailly in [5] where he studied the case when $\mathcal {H}_{2p-1}(L_u \cap \text {Supp}T)=0$.

We proceed by studying the case of compact obstacles. More precisely, for positive plurisubharmonic currents T one can obtain the current $T \wedge \mathrm{{dd}}^{c}g$.

Theorem 3.2

Let T be a positive plurisubharmonic current of bi-dimension (p, p) on $\Omega$. If $L_g$ is compact, then

• (1) $g \mathrm{{dd}}^{c}T$ is a well defined current on $\Omega$, and the trivial extension $\widetilde{\mathrm{{dd}}^{c}g \wedge T}$ exists.

• (2) $\mathrm{{dd}}^{c}g \wedge T$ is a well defined current as soon as $L_g$, in addition, is complete pluripolar and $p\ge 2$.

• (3) $\mathrm{{dd}}^{c}g \wedge T$ is a well defined current when $L_g$ is considered to be a single point.

Proof

Assume that $(g_{j})_j$ is a decreasing sequence of smooth plurisubharmonic functions on $\Omega$ converging to g in $\mathcal {C}^{1}(\Omega {\setminus } L_g)$. Let W and $W^{'}$ be neighborhoods of $L_g$ such that $W \Subset W^{'}$, and take a positive function $f \in \mathcal {C}^{\infty }_{0}(W^{'})$ so that $f=1$ on a neighborhood of W. Then we have

$$\begin{aligned} \begin{aligned} \int _{W^{'}} \mathrm{{dd}}^{c}(f g_j)\wedge T\wedge \beta ^{p-1}= \int _{W^{'}} f g_j\wedge \mathrm{{dd}}^{c}T \wedge \beta ^{p-1}\le 0. \end{aligned} \end{aligned}$$

(3.1)

This implies that

$$\begin{aligned} 0\le & {} \int _{W^{'}}f \mathrm{{dd}}^{c}g_j \wedge T \wedge \beta ^{p-1} - \int _{W^{'}} f g_j \ \mathrm{{dd}}^{c}T \wedge \beta ^{p-1} \nonumber \\\le & {} \left| \int _{W^{'}}\mathrm{{d}}g_j \wedge \mathrm{{d}}^{c}f \wedge T \wedge \beta ^{p-1}\right| + \left| \int _{W^{'}}\mathrm{{d}}f \wedge \mathrm{{d}}^{c}g_j \wedge T \wedge \beta ^{p-1} \right| \nonumber \\&\quad \ + \left| \int _{W^{'}} g_j \mathrm{{dd}}^{c}f \wedge T \wedge \beta ^{p-1}\right| . \end{aligned}$$

(3.2)

Thanks to the properties of f, each term of the first line integrals of (3.2) is uniformly bounded. Therefore, one can infer the existence of both extensions $\widetilde{g\mathrm{{dd}}^{c}T}$ and $\widetilde{\mathrm{{dd}}^{c}g \wedge T}$. Notice that, the current $g \mathrm{{dd}}^{c}T$ is well defined by the monotone convergence, and by Banach-Alaoglu, the sequence $(\mathrm{{dd}}^{c}g_{j}\wedge T)$ has a subsequence $(\mathrm{{dd}}^{c}g_{j_{s}}\wedge T)$ converges weakly$^{*}$ to a current denoted by $\mathrm{{dd}}^{c}g \wedge T$. To show (2), we first note that $\mathrm{{dd}}^{c}g \wedge \mathrm{{dd}}^{c}T$ is well defined as well, thanks to the continuity of the operators $\mathrm{{d}}$ and $\mathrm{{d}}^{c}$. Hence by [4] the residual current $R=\widetilde{\mathrm{{dd}}^{c}g \wedge \mathrm{{dd}}^{c}T}- \mathrm{{dd}}^{c}(\widetilde{\mathrm{{dd}}^{c}g \wedge T})$ is positive and supported in $L_g$. Now, if we set $F:= \mathrm{{dd}}^{c}g \wedge T- \widetilde{\mathrm{{dd}}^{c}g \wedge T}$ we find clearly that F is a positive current where

$$\begin{aligned} \mathrm{{dd}}^{c}F = \mathrm{{dd}}^{c}g \wedge \mathrm{{dd}}^{c}T- \mathrm{{dd}}^{c}(\widetilde{\mathrm{{dd}}^{c}g \wedge T}) \ge \mathrm{{dd}}^{c}g \wedge \mathrm{{dd}}^{c}T- \widetilde{\mathrm{{dd}}^{c}g \wedge \mathrm{{dd}}^{c}T} \ge 0. \end{aligned}$$

As F is a compactly supported current, one can infer that $F \equiv 0$ using [9]. The third statement comes immediately from the fact that the distribution $\mu := (\mathrm{{dd}}^{c}g \wedge T - \widetilde{ \mathrm{{dd}}^{c}g \wedge T})\wedge \beta ^{p-1}$ is positive and supported in $L_g$. Indeed, $L_g$ can be assumed to be the origin, and hence there exists a positive constant c such that $\mu =c\delta _{0}$ where $\delta _{0}$ is the Dirac measure. Clearly, the constant c is independent from the choice of $j_{s}$ since

$$\begin{aligned} \begin{aligned} c=&\,\mu (f)=\lim _{s \rightarrow \infty } \int _{W^{'}} f \mathrm{{dd}}^{c}g_{j_{s}} \wedge T \wedge \beta ^{p-1} - \int _{W^{'}} f \widetilde{ \mathrm{{dd}}^{c}g \wedge T} \wedge \beta ^{p-1}\\ =&\, \int _{W^{'}}( g \mathrm{{dd}}^{c}f +2 \mathrm{{d}}g \wedge \mathrm{{d}}^{c} f ) \wedge T \wedge \beta ^{p-1} \\&+\, \int _{W^{'}} f g \mathrm{{dd}}^{c}T \wedge \beta ^{p-1} - \int _{W^{'}} f \widetilde{ \mathrm{{dd}}^{c}g \wedge T} \wedge \beta ^{p-1}. \end{aligned} \end{aligned}$$

In other words, $\mathrm{{dd}}^{c}g \wedge T$ is well defined. $\square$

Lemma 3.3

Let T be a positive plurisubharmonic current of bi-dimension (p, p) on $\Omega$. Let K and L be compact sets of $\Omega$ with $L\subset \overset{\circ }{K}$. Then there exist positive constant $C_{K,L}$, and a neighborhood V of $K\cap L_g$ such that for all $g \in Psh^{-}(\Omega ) \cap \mathcal {C}^{1}(\Omega {\setminus } L_g)$ so that $\mathcal {H}_{2p-1}(L_g \cap {\text {Supp} \ T})=0$ we have

$$\begin{aligned} \Vert \mathrm{{dd}}^{c}g\wedge T\Vert _{L{\setminus } L_g}\le C_{K,L}\Vert g\Vert _{\mathcal {L}^{\infty }(K{\setminus } V)}\Vert T\Vert _{K{\setminus } V} \end{aligned}$$

(3.3)

Proof

Assume that g is a smooth negative function and $0 \in {\text {Supp} \ T}\cap L_g$. Since $\mathcal {H}_{2p-1}(L_g \cap \text {Supp} \ T)=0$, by Bishop [3] and Shiffman [10], there exist a system of coordinates $(z^{\prime },z^{\prime \prime })\in \mathbb {C}^{s}\times \mathbb {C}^{n-s}$, $s=p-1$ and a polydisk $\bigtriangleup ^{n}=\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }$ such that $\overline{\bigtriangleup ^{\prime }}\times \partial \bigtriangleup ^{\prime \prime }\cap ({\text {Supp} \ T}\cap L_g)=\emptyset$. Now, take $0<t<1$ so that $\bigtriangleup ^{\prime }\times \{ z^{\prime \prime }, t<\vert z^{\prime \prime }\vert <1\} \cap ({\text {Supp} \ T}\cap L_g)=\emptyset$. As $\overline{ \bigtriangleup ^{n}}\cap L_g$ is a compact set, one can find a neighborhood $\omega$ of $\overline{ \bigtriangleup ^{n}}\cap L_g$ such that $\omega \cap (\bigtriangleup ^{\prime }\times \{ z^{\prime \prime }, t<\vert z^{\prime \prime }\vert <1\})=\emptyset$. Let $a\in (t,1)$ and choose $\rho (z^{\prime })\in \mathcal {C}_{0}^{\infty }(a\bigtriangleup ^{\prime })$ such that $0\le \rho \le 1$ and $\rho =1$ on $\frac{1}{2} a\bigtriangleup ^{\prime }$. Take $\chi \in \mathcal {C}_{0}^{\infty }(\omega )$ such that $0\le \chi \le 1$ and $\chi =1$ on a neighborhood $\omega _{0}$ of $\overline{ \bigtriangleup ^{n}}\cap L_g$. Obviously, the function $\chi (z) \rho (z^{\prime })$ is positive smooth and compactly supported in $a\bigtriangleup ^{n}$. For convenience, we set $\beta ^{\prime }=\mathrm{{dd}}^{c}\vert z^{\prime }\vert ^{2}$, $\beta ^{\prime \prime }=\mathrm{{dd}}^{c}\vert z^{\prime \prime }\vert ^{2}$ and $\alpha (z^{\prime })=\rho (z^{\prime })\beta ^{\prime s}$. By using Stokes’ formula

$$\begin{aligned} \begin{aligned}&\int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \mathrm{{dd}}^{c}(\chi g)\wedge T\wedge \alpha (z^{\prime }) = \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \chi g \mathrm{{dd}}^{c}T\wedge \alpha (z^{\prime }). \end{aligned} \end{aligned}$$

(3.4)

Then (3.4) implies that

$$\begin{aligned} \begin{aligned}&0 \le \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \chi \mathrm{{dd}}^{c}g\wedge T\wedge \alpha (z^{\prime })- \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \chi g \mathrm{{dd}}^{c}T\wedge \alpha (z^{\prime })\\&\quad \le \left| \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} g \mathrm{{dd}}^{c}\chi \wedge T\wedge \alpha (z^{\prime })\right| \\&\quad \ \ + \left| \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} dg\wedge d^{c}\chi \wedge T\wedge \alpha (z^{\prime })\right| \\&\quad \ \ + \left| \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} d\chi \wedge d^{c}g\wedge T\wedge \alpha (z^{\prime })\right| . \end{aligned} \end{aligned}$$

(3.5)

Using the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \begin{aligned}&\left| \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \mathrm{{d}}g\wedge \mathrm{{d}}^{c}\chi \wedge T\wedge \alpha (z^{\prime })\right| \\&\quad \le \left( \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \mathrm{{d}}g\wedge \mathrm{{d}}^{c} g \wedge T\wedge \alpha (z^{\prime })\right) ^\frac{1}{2} \left( \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \mathrm{{d}}\chi \wedge \mathrm{{d}}^{c}\chi \wedge T\wedge \alpha (z^{\prime })\right) ^\frac{1}{2}. \end{aligned} \end{aligned}$$

As the forms $\mathrm{{d}}\chi$, $\mathrm{{d}}^{c}\chi$ and $\mathrm{{dd}}^{c}\chi$ vanish on some neighborhood $V^{\prime }$ of $\overline{\bigtriangleup ^{n}}\cap L_g$, the limitation of the right hand side integrals of (3.5) is easily achieved. Therefore, by [1], there exists a constant $C\ge 0$ such that

$$\begin{aligned} \int _{ \omega _{0} \cap \frac{1}{2}a \bigtriangleup ^{n}}\mathrm{{dd}}^{c}g \wedge T \wedge \beta ^{p-1} \le C \Vert g \Vert _{\mathcal {L}^{\infty }(\bigtriangleup ^{n} {\setminus } V)} \Vert T\Vert _{\bigtriangleup ^{n} {\setminus } V}. \end{aligned}$$

(3.6)

Now, consider $g\in Psh^{-}(\Omega )\cap \mathcal {C}^{1}(\Omega {\setminus } L_g)$. Set $\Gamma _{m}=\lbrace z\in \overline{ \bigtriangleup ^{n}}, d(z,L_g)<\frac{1}{m}\rbrace$ and put

$$\begin{aligned} a_{m}=\inf _{z\in \overline{ \bigtriangleup ^{n}} {\setminus } \Gamma _{m}} g(z) \end{aligned}$$

(3.7)

Since g is continuous on $\Omega {\setminus } L_g$, then $a_{m}> -\infty$ for all m. For $\varepsilon _{m}>0$ small enough, set

$$\begin{aligned} g_{m}=\hbox { }\ \max _{\varepsilon _{m}} \left( g,a_{m}-\frac{1}{2^{m}}\right) \end{aligned}$$

(3.8)

Observe that, $g_{m}$ is smooth plurisubharmonic function and $g_{m}=g$ on $\overline{ \bigtriangleup ^{n}} {\setminus } \Gamma _{m}$. Then by the previous part, for m sufficiently large we have

$$\begin{aligned} \begin{aligned} \int _{a \overline{ \bigtriangleup ^{n}} {\setminus } \Gamma _{m}}\mathrm{{dd}}^{c}g\wedge T \wedge \beta ^{p-1}&\le \int _{\overline{ \bigtriangleup ^{n}}}\mathrm{{dd}}^{c}g_{m}\wedge T \wedge \beta ^{p-1} \\&\le C^{\prime } \Vert g_{m}\Vert _{\mathcal {L}^{\infty }(\bigtriangleup ^{n}{\setminus } V)} \Vert T\Vert _{\bigtriangleup ^{n}{\setminus } V}. \end{aligned} \end{aligned}$$

(3.9)

Clearly, the result follows by taking the limit over m as what we have shown is true for almost all choice of unitary coordinates $(z^{\prime },z^{\prime \prime })$. $\square$

The above version of Chern–Levine–Nirenberg inequality implies the existence of the trivial extension $\widetilde{\mathrm{{dd}}^{c}g \wedge T}$ across $L_g$. Moreover, under the hypotheses of Theorem 3.3, the current $g \mathrm{{dd}}^{c}T$ is well defined thanks to (3.5). Analogously, one can induce another version to plurisuperharmonic currents.

Corollary 3.4

Let T be a positive plurisuperharmonic current of bi-dimension (p, p) on $\Omega$. Let K and L be compact sets of $\Omega$ with $L\subset \overset{\circ }{K}$. Then there exist positive constant $C_{K,L}$, and a neighborhood V of $K\cap L_g$ such that for all $g \in Psh^{-}(\Omega ) \cap \mathcal {C}^{1}(\Omega {\setminus } L_g)$ so that $\mathcal {H}_{2p-2}(L_g \cap {\text {Supp} \ T})=0$ we have

$$\begin{aligned} \Vert \mathrm{{dd}}^{c}g\wedge T\Vert _{L{\setminus } L_g}\le C_{K,L}\Vert g\Vert _{\mathcal {L}^{\infty }(K{\setminus } V)} \Vert T\Vert _{K{\setminus } V}. \end{aligned}$$

(3.10)

Notice that the term $\Vert \mathrm{{dd}}^{c}T \Vert$ is neglected in (3.10) as this term is dominated by $\Vert T \Vert$.

Proof

As $\mathcal {H}_{2p-2}(L_g \cap {\text {Supp} \ T})=0$, Lemma 3.1 shows that the current $g \mathrm{{dd}}^{c}T$ is well defined. Now the statement follows by reenacting a similar technique as in the proof of Lemma 3.3$\square$

Theorem 3.5

Let T be a positive plurisuperharmonic current of bi-dimension (p, p) on $\Omega$ and $g \in Psh^{-}(\Omega ) \cap \mathcal {C}^{1}(\Omega {\setminus } L_g)$. If $\mathcal {H}_{2p-2}(L_g \cap \mathrm {Supp}T)=0$ and $(g_{j})_j$ converges to g in $\mathcal {C}^{1}(\Omega {\setminus } L_g)$, then $\mathrm{{dd}}^{c}g \wedge T$ is a well defined current as a limit of $\mathrm{{dd}}^{c}g_{j}\wedge T$ on $\Omega$.

Proof

In virtue of Corollary 3.4 alongside Banach-Alaoglu, there exists a subsequence $\mathrm{{dd}}^{c}g_{j_{s}} \wedge T$ converges to a current denoted by $\mathrm{{dd}}^{c}g \wedge T$. Consider now the current $R:= \mathrm{{dd}}^{c}g \wedge T- \widetilde{\mathrm{{dd}}^{c}g \wedge T}$. It is clear that R is a $\mathbb {C}$-flat current and supported in $L_g$. Therefore, $R \equiv 0$ by the support theorem. This means that $\mathrm{{dd}}^{c}g \wedge T$ does not depend on the choice of $j_{s}$ proving our statement. $\square$

Remark 3.6

Theorem 3.5 is fulfilled for the case when T is defined on $\Omega {\setminus } L_g$. Indeed, by [4], the extension $\widetilde{T}$ exists and is positive plurisuperharmonic.

The condition on $L_g$ in Theorem 3.5 is sharp and the thickness can not be reduced for anymore as illustrated in the following example.

Example 3.7

Let $T=\sum _{k=2}^{\infty } \frac{-1}{k \log ^{2}k} \log \vert z_{1}-\frac{1}{k}\vert ^{2}$ and $g(z)=\log \vert z_{1}\vert ^{2}$, then T is a positive $\mathrm{{dd}}^{c}$-negative current on $\frac{1}{2}\bigtriangleup ^{n}$ of bi-dimension (n, n), and $g\in Psh(\bigtriangleup ^{n})\cap \mathcal {C}^{\infty }(\bigtriangleup ^{n}{\setminus } \{ z_{1}=0\})$. In spite that $\mathcal {H}_{2n-2}(\{ z_{1}=0\})$ is locally finite, the mass of $\mathrm{{dd}}^{c}g\wedge T$ explodes near $\{ z_{1}=0 \}$.

The cases studied previously can be employed to generalize the wedge products in [1, 2] to the following assertion.

Theorem 3.8

Let T be a positive plurisubharmonic current of bi-dimension (p, p) on $\Omega$ and $g \in Psh^{-}(\Omega ) \cap \mathcal {C}^{1}(\Omega {\setminus } L_g)$. If $\mathcal {H}_{2p-2}(L_g \cap \mathrm {Supp}T)$ is locally finite and $(g_{j})_j$ is a decreasing sequence of smooth plurisubharmonic functions converging to g in $\mathcal {C}^{1}(\Omega {\setminus } L_g)$, then $\mathrm{{dd}}^{c}g \wedge T$ is a well defined current as a limit of $\mathrm{{dd}}^{c}g_{j}\wedge T$ on $\Omega$.

Proof

We keep the notations of the proof of Lemma 3.3. For each $z^{\prime }$ we set $A_{z^{\prime }}=({\text {Supp} \ T}\cap L_g)\cap (\lbrace z^{\prime }\rbrace \times \bigtriangleup ^{\prime \prime })$. Since $\mathcal {H}_{2p-2}({\text {Supp} \ T}\cap L_g)$ is locally finite, then by Shiffman [10] the set $A_{z^{\prime }}$ is a discrete subset for a.e. $z^{\prime }$. Without loss of generality, we may assume that $A_{z^{\prime }}$ is reduced to a single point $(z^{\prime },0)$. On the other hand, T is $\mathbb {C}$-flat on $\Omega$. Thus, the slice $\langle T,\pi ,z^{\prime }\rangle$ exists for a.e. $z^{\prime }$, and is a positive plurisubharmonic current of bidimension (1, 1) on $\Omega$, supported in $\lbrace z^{\prime }\rbrace \times \bigtriangleup ^{n-p+1}$. Hence, by Theorem 3.2, the sequence $\langle \mathrm{{dd}}^{c}g_{j}\wedge T, \pi , z^{\prime }\rangle$ is weakly$^{*}$ convergent since $A_{z^{\prime }}$ is a single point. So by applying the slice formula we have

$$\begin{aligned} \begin{aligned} \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \mathrm{{dd}}^{c}g\wedge T\wedge \pi ^{*}\beta ^{\prime p-1}&= \lim _{s \rightarrow \infty }\int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \mathrm{{dd}}^{c}g_{j_{s}}\wedge T\wedge \pi ^{*}\beta ^{\prime p-1}\\&=\lim _{s \rightarrow \infty } \int _{z^{\prime }} \langle \mathrm{{dd}}^{c}g_{j_{s}}\wedge T, \pi ,z^{\prime }\rangle \beta ^{\prime p-1}\\ {}&=\lim _{j \rightarrow \infty } \int _{z^{\prime }} \langle \mathrm{{dd}}^{c}g_{j}\wedge T, \pi ,z^{\prime }\rangle \beta ^{\prime p-1}\\ {}&= \lim _{j \rightarrow \infty } \int _{\bigtriangleup ^{\prime }\times \bigtriangleup ^{\prime \prime }} \mathrm{{dd}}^{c}g_{j}\wedge T\wedge \pi ^{*}\beta ^{\prime p-1}. \end{aligned} \end{aligned}$$

This completes the proof. $\square$

As a consequence of the precedent result, one can prolong Alessandirini-Bassanelli study [2] to plurisubharmonic currents.

Corollary 3.9

Let T be a positive plurisubharmonic current of bidimension (p, p) on an open subset $\Omega$ of $\mathbb {C}^{n}$ and A be an analytic subset of $\Omega$, $dimA<p$. Let F be a positive closed current of bidimension $(n-1,n-1)$ on $\Omega$ and smooth on $\Omega {\setminus } A$. Then there exists a unique current on $\Omega$ denoted by $F \wedge T$ with the following property.

If g is a solution of $\mathrm{{dd}}^{c}g=F$ on an open subset $U\subset \Omega$, and $(g_{j})_j$ is a decreasing sequence of smooth plurisubharmonic functions on U converging pointwise to g on $U{\setminus } A$ such that $g_{j}$ converges to g in $\mathcal {C}^{1}(\Omega {\setminus } A)$, then $\mathbf{F} \wedge T = \lim \nolimits _{j \rightarrow \infty } \mathrm{{dd}}^{c}g_{j}\wedge T$ on U.

Another useful application appears when we take $g=\log {|f|^{2}}$, where f is a holomorphic function on $\Omega$. In such a situation, the current $\mathrm{{dd}}^{c}g \wedge T$ is well defined as soon as the zero divisors $Z_{f}$ of f are at most of dimension $p-1$. Notice that slice of T is related to the current $\mathrm{{dd}}^{c}g \wedge T$.

Remark 3.10

The previous proof can be implemented to relax the condition of Lemma 3.1 for a special case. In fact, the statement of that lemma remains true when $\mathcal {H}_{2p}$ is locally finite as soon as T has a locally bounded potential U. The proof comes as an application of Theorem 3.8 on the current U.

We finally, by induction, deduce some consequences for longer wedge product.

Theorem 3.11

Let T be a positive plurisubharmonic (resp. plurisuperharmonic) current of bi-dimension (p, p) on an open subset $\Omega$ of $\mathbb {C}^{n}$ and let $A_{1},\ldots , A_{q}$ be closed subsets of $\Omega$ such that $\mathcal {H}_{2(p-m)+1}(\mathrm{{Supp} \ T}\cap A_{j_{1}}\cap \cdots \cap A_{j_{m}})=0$ (resp.$\mathcal {H}_{2(p-m)-1}({\text {Supp}T}\cap A_{j_{1}}\cap \cdots \cap A_{j_{m}})=0$ ) for all choices of $j_{1}<\cdots <j_{m}$ in $\lbrace 1,\ldots ,q\rbrace$. Let K and L be compact sets of $\Omega$ with $L\subset \overset{\circ }{K}$. Then there exist a positive constant $I_{K,L}$ and a neighborhoods $V_{j}$ of $K\cap A_{j}$ such that for all $g_{j}\in Psh^{-}(\Omega )\cap \mathcal {C}^{1}(\Omega {\setminus } A_{j}), \ L_{g_{j}} \subset A_{j}$ we have

$$\begin{aligned} \left\| \bigwedge _{j=1}^{q} \mathrm{{dd}}^{c}g_{j} \wedge T\right\| _{L {\setminus } {\bigcup _{j=1}^{q}A_{j}}} \le I_{K,L} \prod _{j=1}^{q} \Vert g_{j}\Vert _{\mathcal {L}^{\infty }(K{\setminus } V_{j})}\Vert T\Vert _{K}. \end{aligned}$$

(3.11)

Theorem 3.12

Under the same hypotheses of Theorem 3.11, if T is positive plurisubharmonic and each $A_{j}$ is analytic so that $\mathrm{{Supp} \ T}\cap A_{j_{1}}\cap \cdots \cap A_{j_{m}}$ is at most of dimension $p-m$ for all choices of $j_{1}<\cdots <j_{m}$ in $\lbrace 1,\ldots ,q\rbrace$, then $\bigwedge _{j=1}^{q} \mathrm{{dd}}^{c}g_{j} \wedge T$ is well defined.

These results generalize the case of closed current which was proved by Demailly [5].