%% ARTICLE "Potential theory in several complex variables" %%%

\input arma.tex
%\anglais

\let\ov=\overline
\let\wt=\widetilde
\def\proof{{\sl Proof}\pointir}
\def\ssubset{\subset\!\subset}
\def\bu{\bullet}
\def\ci{C^\infty}
\def\bfun{{\bf 1}}
\def\gge{{\scriptscriptstyle\ge}}
\def\cb{{\Bbb C}}
\def\rb{{\Bbb R}}
\def\re{\mathop{\rm Re}\nolimits}
\def\bulle{\hbox{$\scriptstyle\bullet$}}

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\vfill\eject
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{~}\vskip100pt
\centerline{{\bftwelve POTENTIAL THEORY IN}}
\centerline{{\bftwelve SEVERAL COMPLEX VARIABLES}}
\vskip100pt

{\noindent\hangindent3.5cm\hangafter-4{\sl 
Course of Jean-Pierre DEMAILLY\hfill\break
at the ICPAM Summer School\hfill\break
on Complex Analysis,\hfill\break
Nice, France, July 3--7, 1989

}}
\vfill\eject

\pav 1. Monge-Amp\`ere operators|

Let $X$ be a complex manifold of dimension $n$~. We denote as usual
$d=d'+d''$ the exterior derivative and we set
$$d^c={1\over2i\pi}(d'-d'')~,$$
so that $dd^c={i\over\pi}d'd''$~. In this context, we have the following
integration by parts formula.

\tha Formula 1.1|Let $\Omega\ssubset X$ be a smooth open subset of $X$ and
$f,g$ forms of class $C^2$ on $\smash{\ov\Omega}$ of pure bidegrees 
$(p,p)$ and $(q,q)$ with $p+q=n-1$~. Then
$$\int_\Omega f\wedge dd^c g-dd^c f\wedge g=
\int_{\partial\Omega} f\wedge d^c g-d^c f\wedge g~.$$\fintha

\proof By Stokes' theorem the right hand side is the integral over $\Omega$ of
$$d(f\wedge d^c g-d^c f\wedge g)= f\wedge dd^c g-dd^c f\wedge g+
(df\wedge d^c g+d^c f\wedge dg)~.$$
As all forms of total degree $2n$ and bidegree $\ne(n,n)$ are zero, we get
$$df\wedge d^cg={i\over 2\pi}(d'f\wedge d''g-d''f\wedge d'g)=-d^cf\wedge dg~.~
\finpr$$

Let $u$ be a psh function on $X$ and $T$ a closed positive current of
bidimension $(p,p)$, \ie of bidegree $(n-p,n-p)$~.  Our desire is to
define the wedge product $dd^cu\wedge T$ even when neither $u$ nor $T$
are smooth.  A priori, this product makes no sense since $dd^cu$ and
$T$ have measure coefficients in general.  Assume that $u$ is a {\sl
locally bounded} psh function.  Then the current $uT$ is well defined
since $u$ is a locally bounded Borel function and $T$ has measure
coefficients.  According to Bedford-Taylor [B-T2] one defines
    $$dd^cu\wedge T=dd^c(uT)$$
where $dd^c(~~)$ is taken in the sense of distribution (or current) theory.

\tha Proposition 1.2|The wedge product $dd^c u\wedge T$ is again a closed
positive current.\fintha

\proof The result is local. We may assume that $X$ is an open set 
$\Omega\subset\cb^n$~, and after shrinking $\Omega$~, that
$|u|\le M$ on $\Omega$~.
Let $(\rho_\varepsilon)$ be a family of regularizing kernels with
${\rm Supp}\,\rho_\varepsilon\subset B(0,\varepsilon)$ and
$\int\rho_\varepsilon=1$~. The sequence of convolutions 
$u_k=u\star\rho_{1/k}$ is decreasing, bounded by $M$~, and converges
pointwise to $u$ as\break $k\to+\infty$~. By Lebesgue's dominated convergence
theorem $u_k T$ converges weakly to $uT$~, thus $dd^c(u_k T)$ converges
weakly to $dd^c(u T)$~. However, since $u_k$ is smooth,
$dd^c(u_k T)$ coincides with the product $dd^c u_k\wedge T$ in its usual
sense. As $T\ge0$ and as $dd^cu_k$ is a positive $(1,1)$--form, we have 
$dd^c u_k\wedge T\ge 0$~, hence
the weak limit $dd^c u\wedge T$ is $\ge 0$ (and obviously closed).~\finpr

Given locally bounded psh functions $u_1\ld u_q$~, one defines inductively
$$dd^c u_1\wedge dd^cu_2\wedge\ldots\wedge dd^c u_q\wedge T=
dd^c(u_1dd^c u_2\ldots\wedge dd^c u_q\wedge T)$$
and the result is a closed positive current. In particular, when $u$ is
a locally bounded psh function, there is a well defined positive measure
$(dd^cu)^n$~. If $u$ is of class $C^2$, a computation in local
coordinates gives
$$(dd^cu)^n=\det\big({\partial^2u\over\partial z_j\partial\ov z_k}\Big)
\cdot{n!\over\pi^n}\,idz_1\wedge d\ov z_1\wedge\ldots\wedge
idz_n\wedge d\ov z_n~.$$
The expression ``Monge-Amp\`ere operator" refers generally to the non-linear
partial differential operator $u\longmapsto\det(\partial^2u/\partial z_j
\partial\ov z_k)$~.

Now, let $\Theta$ be a current of order $0$~. If $K$ is a compact
subset contained in a coordinate patch of $X$~, we define the {\sl mass}
of $\Theta=\sum\Theta_{I,J}\,dz_I\wedge d\ov z_J$ on $K$ by
$$||\Theta||_K=\int_K~~\sum_{I,J}|\Theta_{I,J}|$$
where $|\Theta_{I,J}|$ is the absolute value of the measure $\Theta_{I,J}$~.
When $\Theta\ge 0$ we have $|\Theta_{I,J}|\le C.\Theta\wedge\beta^p$
with $\beta=dd^c|z|^2$~; up to constants, the mass $||\Theta||_K$ is 
then equivalent to the integral $\int_K\Theta\wedge\beta^p$~. When
$K\ssubset X$ is arbitrary, we take a partition
$K=\bigcup K_j$ where each $\smash{\ov K_j}$ is contained in a coordinate 
patch and write
$$||\Theta||_K=\sum||\Theta||_{K_j}~.$$
Up to constants, the semi-norm $||\Theta||_K$ does not depend on the choice
of the coordinate systems involved.

\tha Chern-Levine-Nirenberg inequalities 1.3 ([C-L-N])|For all compact sets
$K,L$ of $X$ with $L\subset K^\circ$~, there exists a constant $C_{K,L}\ge0$ 
such that
$$||dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T||_L\le C_{K,L}~
||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}\,||T||_K~.$$
\fintha

\proof By induction, it is sufficient to prove the result for $q=1$ and
$u_1=u$~. There is a covering of $L$ by a family of balls
$B'_j\ssubset B_j\subset K$ contained in coordinate patches of $X$~.
Let $\chi\in\dr(B_j)$ be equal to $1$ on $\smash{\ov B'_j}$~. Then
$$||dd^cu\wedge T||_{L\cap\ov B'_j}\le C\int_{\ov B'_j}dd^c u\wedge T
\wedge\beta^{p-1}\le C\int_{B_j}\chi\,dd^c u\wedge T\wedge\beta^{p-1}~.$$
As $T$ and $\beta$ are closed, an integration by parts yields
$$||dd^cu\wedge T||_{L\cap\ov B'_j}\le C\int_{B_j}u\,T\wedge dd^c\chi
\wedge\beta^{p-1}\le C'||u||_{L^\infty(K)}||T||_K$$
where $C'$ is equal to $C$ multiplied by a bound for the coefficients
of $dd^c\chi\wedge\beta^{p-1}$~.~\finpr

\tha Exercise 1.4|Denote by $L^1(K)$ the space of integrable
functions with respect to some smooth positive density on $K$~.
For any $V$ psh on $X$ show

\noindent{\rm(a)}~~$||dd^cV||_L\le C_{K,L}\,||V||_{L^1(K)}$~.

\noindent{\rm(b)}~~${\displaystyle\sup_L}\,V_+\le C_{K,L}\,||V||_{L^1(K)}$~.
\eject\fintha

Now, we prove a rather important continuity theorem due to [B-T2].

\tha Theorem 1.5|Let $u_1\ld u_q$ be locally bounded psh functions and
let $u_1^k\ld u_q^k$ be decreasing sequences of psh functions 
converging pointwise to\break $u_1\ld u_q$~.  Then 

\noindent{\rm(a)}~~$u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k
\wedge T\la u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\sl weakly}.$

\noindent{\rm(b)}~~$dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T\la
dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\sl weakly}.$\fintha

\proof As the sequence $\smash{(u_j^k)}$ is non increasing and as $u_j$ is
locally bounded, the family $\smash{(u_j^k)_{k\in\nb}}$ is locally uniformly 
bounded. The result is local, so we can work on a strongly pseudoconvex open 
set $\Omega\ssubset X$~.  Let $\psi$ be a strongly psh function of class
$\ci$ near $\smash{\ov\Omega}$ with $\psi<0$ on $\Omega$~, $\psi=0$ and
$d\psi\ne 0$ on $\partial\Omega$~.  After addition of a constant we can
assume that $-M\le\smash{u_j^k}\le-1$ near $\smash{\ov\Omega}$~. Let us denote
by $\smash{(u_j^{k,\varepsilon})}$~, $\varepsilon\in]0,\varepsilon_0]$~, an
increasing family of regularizations converging to $\smash{u_j^k}$ as 
$\varepsilon\to 0$ and such that $-M\le\smash{u^{k,\varepsilon}_j}\le-1$
on $\smash{\ov\Omega}$~. Set $A=M/\delta$ with $\delta>0$ small and replace
$\smash{u_j^k}$ by $\smash{v_j^k=\max\{A\psi,u_j^k\}}$~, 
$\smash{u_j^{k,\varepsilon}}$ by $\smash{v_j^{k,\varepsilon}=
\max_\varepsilon\{A\psi,u_j^{k,\varepsilon}\}}$
where $\max_\varepsilon=\max~\star~\rho_\varepsilon$ is a regularized 
max function.

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\centerline{{\bf Fig.~1} Construction of $v_j^k$}
\vskip4mm


\noindent Then $\smash{v_j^k}$ coincides with $\smash{u_j^k}$ on 
$\Omega_\delta=\{\psi<-\delta\}$ since $A\psi<-A\delta=-M$ on
$\Omega_\delta$~, and $\smash{v_j^k}$ is equal to 
$A\psi$ on the corona $\Omega\setminus\Omega_{\delta/M}$~.  Without loss 
of generality, we can therefore assume that all $\smash{u_j^k}$ 
(and similarly all $\smash{u_j^{k,\varepsilon}})$ coincide with $A\psi$ 
on a fixed neighborhood of $\partial\Omega$~.

Now, we argue by induction on $q$ and observe that (b) is an immediate
consequence of (a). When $q=1$~, (a) follows directly from the bounded
convergence theorem. We need a lemma.

\tha Lemma 1.6|Let $f_k$ be a non-increasing sequence of upper
semi-continuous functions converging to $f$ on some separable locally
compact space $X$ and $\mu_k$ a sequence of positive measures
converging weakly to $\mu$ on $X$~. Then every weak limit $\nu$ of $f_k\mu_k$
satisfies $\nu\le f\mu$~.\fintha

Indeed if $(g_p)$ is a decreasing sequence of continuous functions
converging to $f_{k_0}$ for some $k_0$~, then $f_k\mu_k\le f_{k_0}\mu_k
\le g_p\mu_k$ for $k\ge k_0$~, thus $\nu\le g_p\mu$ as $k\to+\infty$~.
The monotone convergence theorem then gives $\nu\le f_{k_0}\mu$ as
$p\to+\infty$ and $\nu\le f\mu$ as $k_0\to+\infty$~.~\finpr

\remarque End of proof of theorem 1.5|Assume that (a) has been proved
for $q-1$~.  Then 
$$S^k=dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k\wedge T\la 
S=dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~.$$ 
By 1.3 the sequence $(u_1^k S^k)$ has locally bounded mass, hence is 
relatively compact for the weak topology.  In order to prove (a), we only
have to show that every weak limit $\Theta$ of $u_1^k S^k$ is equal to 
$u_1 S$~.  Let $(m,m)$ be the bidimension of $S$ and let $\gamma$ be an
arbitrary smooth and strongly positive form of bidegree $(m,m)$~.  Then
the positive measures $S^k\wedge\gamma$ converge weakly to
$S\wedge\gamma$ and lemma 1.6 shows that $\Theta\wedge\gamma\le
u_1S\wedge\gamma$~, hence $\Theta\le u_1S$~.  To get the equality, we set
$\beta=dd^c\psi>0$ and show that
$\int_\Omega u_1 S\wedge\beta^m\le\int_\Omega\Theta\wedge\beta^m$~, \ie 
$$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\le
\liminf\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k\wedge
T\wedge\beta^m~.$$ 
As $u_1\le u_1^k\le u_1^{k,\varepsilon_1}$ for every $\varepsilon_1>0$ we get
$$\eqalign{\int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge
T\wedge\beta^m\cr
&\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2
\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr 
&=\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge u_2dd^cu_3
\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr}$$ 
after an integration by parts (there is no boundary term because
$\smash{u_1^{k,\varepsilon_1}}$ and $u_2$ vanish on $\partial\Omega$).  
Repeating this argument with $u_2\ld u_q$~, we obtain
$$\eqalign{
\int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr 
&\le\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge\ldots\wedge dd^c
u_{q-1}^{k,\varepsilon_{q-1}}\wedge u_q T\wedge\beta^m\cr
&\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2^{k,\varepsilon_2}
\wedge\ldots\wedge dd^c u_q^{k,\varepsilon_q}\wedge T\wedge\beta^m~.\cr}$$ 
Now let $\varepsilon_q\to 0\ld\varepsilon_1\to 0$ in this order.  We
have weak convergence at each step and $\smash{u_1^{k,\varepsilon_1}}=0$
on the boundary~; therefore the last integral converges and we
get the desired inequality 
$$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge
T\wedge\beta^m\le\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge
dd^cu_q^k\wedge T\wedge\beta^m~.~\finpr$$

\tha Corollary 1.7|The product $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$
is symmetric with respect to $u_1\ld u_q$~.\fintha

Observe that the definition was unsymmetric. The
result is true when $u_1\ld u_q$ are smooth and follows in general from
theorem 1.5 applied to $u_1^k=u_1\star\rho_{1/k}$~.~\finpr

\tha Theorem 1.8|Let $K,L$ be compact subsets of $X$ such that
$L\subset K^\circ$~. For any psh functions $V,u_1\ld u_q$ on $X$ 
such that $u_1\ld u_q$ are locally bounded, there is an inequality
$$||Vdd^cu_1\wedge\ldots\wedge dd^cu_q||_L\le 
C_{K,L}\,||V||_{L^1(K)}||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}~.$$
\fintha

\proof First, we may assume that $L$ is contained in a strictly
pseudoconvex open set $\Omega=\{\psi<0\}\subset K$ (otherwise cover $L$ by
small balls contained in $K$).  A suitable normalization gives
$-2\le u_j\le -1$ on $K$~;  then we can modify $u_j$ on 
$\Omega\setminus L$ so that $u_j=A\psi$ on
$\Omega\setminus\Omega_\delta$ with a fixed constant $A$ and
$\delta>0$ such that $L\subset\Omega_\delta$~.  Let $\chi\ge 0$
be a smooth function equal to $-\psi$ on $\Omega_\delta$ with compact
support in $\Omega$~.  If we take $||V||_{L^1(K)}=1$~, we see that $V_+$
is uniformly bounded on $\Omega_\delta$ by 1.4 (b); after subtraction
of a fixed constant we get $V\le 0$ on $\Omega_\delta$~.  As
$u_j=A\psi$ on $\Omega\setminus\Omega_\delta$~, we find for $q\le n-1$~:
$$\eqalign{
\int_{\Omega_\delta}&-V\,dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q}\cr
&=\int_\Omega V\,dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q-1}\wedge dd^c\chi-A^q\int_{\Omega\setminus
\Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi\cr
&=\int_\Omega\chi\,dd^cV\wedge dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q-1}-A^q\int_{\Omega\setminus
\Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi~.\cr}$$
The first integral of the last line is uniformly bounded thanks to
1.3 and 1.4 (a), and the second one is bounded by $||V||_{L^1(\Omega)}\le$
constant. Inequality 1.8 follows if $q\le n-1$~. If $q=n$~,
we can work instead on $X\times\cb$ and consider $V,u_1\ld u_q$ as
functions on $X\times\cb$ independent of the extra factor $\cb$~.~\finpr
\vskip5pt

Now, we would like to define $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$
also in some cases when $u_1\ld u_q$ are not bounded below everywhere. 
Consider first the case $q=1$ and let $u$ be a plurisubharmonic function
on $X$~.  The polar set of $u$ is by definition $u^{-1}(-\infty)$~. 

\tha Assumptions 1.9|We make two additional assumptions~:

\noindent{\rm(a)}~~$T$ has non zero bidimension $(p,p)$ (\ie degree of $T<2n$)~.

\noindent{\rm(b)}~~$X$ is covered by a family of strongly pseudoconvex open
sets $\Omega=\{\psi<0\}$~, $\Omega\ssubset X$~, with the following property~:
there is an open set $\omega_T$ containing ${\rm Supp}\,T\cap\Omega$ and
an open set $\omega_u$ containing $u^{-1}(-\infty)\cap\Omega$ such that
$\ov\omega_T\cap\ov\omega_u$ is compact in $\Omega$ and
$u$ is bounded on $\omega_T\setminus\omega_u$~.\fintha

\remarque Example|For any $T$~, hypothesis 1.9 (b) is clearly satisfied 
if $u$ has a discrete set $P$ of poles~; an interesting example is
$u=\log|F|$ where $F=(F_1\ld F_N)$ are holomorphic functions having
a discrete set of common zeroes.~\finpr

Let us replace $u$ by the everywhere finite function
$$u_{\gge s}(z)=\max\{u(z),s\}~.$$ 
We shall let hereafter $s$ tend to $-\infty$~. Let $\beta=dd^c\psi$ and
let $s_0$ be a lower bound for $u$ on a neighborhood of $\partial\Omega
\cap{\rm Supp}\,T$~. For $s<s_0$~, the integral $\int_\Omega dd^c u_{\gge s}
\wedge T\wedge\beta^{p-1}$ does not depend on $s$~; in fact, Stokes' 
theorem shows that
$$\int_\Omega(dd^c u_{\gge r}-dd^c u_{\gge s})\wedge T\wedge(dd^c\psi)^{p-1}=
\int_\Omega dd^c\big[(u_{\gge r}-u_{\gge s})\,T\wedge(dd^c\psi)^{p-1}\big]=0
$$
because $u_{\gge r}$ and $u_{\gge s}$ both coincide with $u$ near 
$\partial\Omega\cap{\rm Supp}\,T$~, hence the current $[\ldots]$ has 
compact support in $\Omega$~. This shows that the mass of 
$dd^c u_{\gge s}\wedge T$ is uniformly bounded on $\Omega$~. 
Now let $\chi$ be a function with compact support in $\Omega$ equal to
$\psi$ on a neighborhood $\Omega'$ of $\ov\omega_T\cap\ov\omega_u$~.
As $u$ is bounded on $(\Omega\setminus\Omega')\cap{\rm Supp}\,T$~, we have
$$\int_\Omega\kern-1.5pt\chi dd^cu_{\gge s}\wedge T\wedge(dd^c\psi)^{p-1}
=\int_\Omega\kern-1.5pt u_{\gge s}T\wedge(dd^c\psi)^{p-1}\wedge dd^c\chi
\le C+\int_{\Omega'}\kern-1.5pt u_{\gge s}T\wedge(dd^c\psi)^p.$$
The first integral remains bounded as $s\to-\infty$~. Hence the last
integral cannot decrease to $-\infty$ and we see that $uT$ has bounded 
mass on $\Omega'$~. We can therefore define $dd^c u\wedge T=dd^c(uT)$
as before.

\remarque Remark 1.10|The current $uT$ has not necessarily a finite
mass when $T$ has degree $2n$ (\ie $T$ is a measure); example~:
$T=\delta_0$ and $u(z)=\log|z|$ in $\cb^n$~.~\finpr

Assume now that $u_1\ld u_q$ are psh functions on $X$ that are
bounded on $\omega_T\setminus\omega_u$~, where $\omega_u$ is an open
set containing all polar sets $u_j^{-1}(-\infty)$ such that
$\ov\omega_T\cap\ov\omega_u\ssubset\Omega$~. 
One can again use induction to define
$$dd^c u_1\wedge dd^cu_2\wedge\ldots\wedge dd^c u_q\wedge T=
dd^c(u_1dd^c u_2\ldots\wedge dd^c u_q\wedge T)~.\leqno(1.11)$$

\tha Theorem 1.12|If $u_1^k\ld u_q^k$ are non-increasing
sequences converging pointwise to $u_1\ld u_q$~, then
$$\eqalign{
&u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k
\wedge T\la u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\sl weakly},\cr
&dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T\la
dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\sl weakly}.\cr}$$
\fintha

\proof Same proof as in theorem 1.5, with the following minor
modification~: the max procedure $\max\{u_j^k,A\psi\}$ is applied
only on $\omega_T\setminus\Omega_\delta$ and $u_j^k$ is left
unchanged on $\omega_T\cap\Omega_\delta$~, assuming that 
$\Omega_\delta\supset\ov\omega_T\cap\ov\omega_u$~; observe that the
functions $u_j^k$ and $\smash{u_j^{k,\varepsilon}}$ are needed only 
on $\omega_T$~.

\tha Theorem 1.13|Let $P$ be a compact subset of a strongly pseudoconvex
open set $\Omega\subset X$~. If $V$ is a psh function on $X$ and 
$u_1\ld u_q$~, $1\le q\le n-1$~, are psh functions that
are locally bounded on $\ov\Omega\setminus P$~, then $Vdd^cu_1\wedge\ldots
\wedge dd^cu_q$ has finite mass on $\Omega$~.\fintha

\proof Same proof as 1.8, taking $P\subset\Omega_\delta$~.~\finpr

\pav 2. Generalized Lelong numbers|

Assume from now on that $X$ is a Stein manifold, \ie that $X$
has a strictly psh exhaustion function.
Let $\varphi:X\longrightarrow[-\infty,+\infty[$ 
be a continuous psh function. The sets
$$\leqalignno{
S(r)&=\{x\in X\,;\,\varphi(x)=r\}~,&(2.1)\cr
B(r)&=\{x\in X\,;\,\varphi(x)<r\}~,&(2.1')\cr
\ov B(r)&=\{x\in X\,;\,\varphi(x)\le r\}&(2.1'')\cr}$$
will be called {\sl pseudo-spheres} and {\sl pseudo-balls}
associated to $\varphi$~. It may happen in some cases that $\ov B(r)$ 
is distinct from the closure of $B(r)$~. For simplicity, we sometimes denote
$$\alpha=dd^c\varphi~,~~~~\beta={1\over 2}dd^c(e^{2\varphi})~.\leqno(2.2)$$
The most simple example we have in mind is $\varphi(z)=\log|z-a|$ on
an open subset $X\subset\cb^n$~; in this case $B(r)$ is the euclidean
ball of center $a$ and radius $e^r$~, and $\beta$ is the usual
hermitian metric ${i\over 2\pi}d'd''|z|^2$ of $\cb^n$~. When $a=0$~,
$\alpha$ is the pull back on $\cb^n$ of the standard Fubini-Study
metric on $\pb^{n-1}$~.

\tha Definition 2.3|We say that $\varphi$ is semi-exhaustive
if there exists a real number $R$ such that $B(R)\ssubset X$~. 
Similarly, $\varphi$ is said to be
semi-exhaustive on a closed subset $A\subset X$ if there exists
$R$ such that $A\cap B(R)\ssubset X$~.\fintha

We are interested especially in the set of poles $S(-\infty)=
\{\varphi=-\infty\}$ and in the behaviour of $\varphi$ near $S(-\infty)$~.
Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$~.
Assume that $\varphi$ is semi-exhaustive on ${\rm Supp}\,T$ and that
$B(R)\cap{\rm Supp}\,T\ssubset X$~.
Then $P=S(-\infty)\cap{\rm Supp T}$ is compact and the results
of \S 1 show that the measure $T\wedge(dd^c\varphi)^p$ 
is well defined.

\tha Definition 2.4|For $r\in]-\infty,R[$~, we set
$$\eqalign{
\nu(T,\varphi,r)&=\int_{B(r)}T\wedge(dd^c\varphi)^p~,\cr
\nu(T,\varphi)&=\int_{S(-\infty)}T\wedge(dd^c\varphi)^p=\lim_{r\to-\infty}
\nu(T,\varphi,r)~.\cr}$$
The number $\nu(T,\varphi)$ will be called the (generalized) Lelong
number of $T$ with respect to the weight $\varphi$~.\fintha

It is clear that $r\longmapsto\nu(T,\varphi,r)$ is an increasing
function of $r$~. Before giving an example, we need a formula.

\tha Formula 2.5|For any convex increasing function $\chi:\rb\la\rb$ one has
$$\int_{B(r)}T\wedge(dd^c\chi\circ\varphi)^p=
\chi'(r-0)^p\,\nu(T,\varphi,r)$$
where $\chi'(r-0)$ denotes the left derivative of $\chi$ at $r$~.\fintha

\proof Let $\chi_\varepsilon$ be the convex function equal to
$\chi$ on $[r-\varepsilon,+\infty[$ and to
a linear function of slope $\chi'(r-\varepsilon-0)$
on $]-\infty,r-\varepsilon]$~. We get
$dd^c(\chi_\varepsilon\circ\varphi)=\chi'(r-\varepsilon-0)dd^c\varphi$
on $B(r-\varepsilon)$ and Stokes' theorem implies
$$\eqalign{
\int_{B(r)}T\wedge(dd^c\chi\circ\varphi)^p
&=\int_{B(r)}T\wedge(dd^c\chi_\varepsilon\circ\varphi)^p\cr
&\ge\int_{B(r-\varepsilon)}T\wedge(dd^c\chi_\varepsilon\circ\varphi)^p
=\chi'(r-\varepsilon-0)^p\nu(T,\varphi,r-\varepsilon)~.\cr}$$
Similarly, taking $\wt\chi_\varepsilon$ equal to $\chi$ on 
$]-\infty,r-\varepsilon]$ and linear on $[r-\varepsilon,r]$~, we obtain
$$\int_{B(r-\varepsilon)}T\wedge(dd^c\chi\circ\varphi)^p
\le\int_{B(r)}T\wedge(dd^c\wt\chi_\varepsilon\circ\varphi)^p
=\chi'(r-\varepsilon-0)^p\nu(T,\varphi,r)~.$$
The expected formula follows when $\varepsilon$ tends to $0$~.~\finpr

We get in particular
$\int_{B(r)}T\wedge(dd^ce^{2\varphi})^p=(2e^{2r})^p\nu(T,\varphi,r)$~,
whence the formula
$$\nu(T,\varphi,r)=e^{-2pr}\int_{B(r)}T\wedge\beta^p~.\leqno(2.6)$$

Now, assume that $X$ is an open subset of $\cb^n$ and that 
$\varphi(z)=\log|z-a|$ for some $a\in X$~. Formula (2.6) gives
$$\nu(T,\varphi,\log r)=r^{-2p}\int_{|z-a|<r}T\wedge\Big({i\over 2\pi}
d'd''|z|^2\Big)^p~.$$
The positive measure $\sigma_T={1\over p!}T\wedge({i\over 2}d'd''|z|^2)^p
=2^{-p}\sum T_{I,I}\,.\,i^n dz_1\wedge d\ov z_1\ldots d\ov z_n$ 
is called the {\sl trace measure} of $T$~. We get
$$\nu(T,\varphi,\log r)={\sigma_T\big(B(a,r)\big)\over
\pi^p r^{2p}/p!}\leqno(2.7)$$
and $\nu(T,\varphi)$ is the limit of this ratio as $r\to 0$~. This
limit is called the Lelong number of $T$ at point $a$ and denoted
$\nu(T,a)$~. This was precisely the original definition of Lelong 
(\cf [Le3]). Let us mention an important consequence.

\tha Consequence 2.8|The ratio $\sigma_T\big(B(a,r)\big)/r^{2p}$ is an
increasing function of the radius $r$~. In particular, we have
$$\sigma_T\big(B(a,r)\big)\le Cr^{2p}$$
for $r<r_0$ small enough.\fintha

All these results are particularly interesting when $T=[A]$ is the 
current of integration
over an analytic subset $A\subset X$ of pure dimension $p$~. Then
$\sigma_T\big(B(a,r)\big)$ is the euclidean area of $A\cap B(a,r)$~,
and $\nu(T,\varphi,\log r)$ is the ratio of this area to the area
of a ball of radius $r$ in $\cb^p$~.

\tha Exercise 2.9|When $A$ is a smooth submanifold of $X$~, show that 
$$\nu([A],x)=\cases{1&for $x\in A$\cr
                    0&for $x\notin A$~.\cr}$$\fintha

\remarque Remark 2.10|When $X=\cb^n$~, $\varphi(z)=\log|z-a|$ and 
$A=X$ (\ie $T=1$), we obtain in particular $\int_{B(a,r)}
(dd^c\log|z-a|)^n=1$ for all $r$~. This implies 
$$(dd^c\log|z-a|)^n=\delta_a~.$$
This fundamental formula can be viewed as a higher dimensional analogue
of the usual formula $\Delta\log|z-a|=2\pi\delta_a$ in $\cb$~.

\pav 3. The Lelong-Jensen formula|

Assume in this paragraph that $\varphi$ is {\sl semi-exhaustive} on $X$
and that $B(R)\ssubset X$~.  For every $r\in]-\infty,R[$~, the measures
$dd^c(\varphi_{\gge r})^n$ are well defined.  The map $r\longmapsto
(dd^c\varphi_{\gge r})^n$ is continuous on $]-\infty,R[$ with respect to
the weak topology~: right continuity follows immediately from theorem
1.5, while left continuity is obtained similarly from the equality
$(dd^c\varphi_{\gge r})^n=(dd^c\max\{\varphi-r,0\})^n$~.  As
$(dd^c\varphi_{\gge r})^n=(dd^c\varphi)^n$ on $X\setminus\ov B(r)$
and $\varphi_{\gge r}\equiv r$~, $(dd^c\varphi_{\gge r})^n=0$ on $B(r)$~,
the left continuity implies 
$(dd^c\varphi_{\gge r})^n\ge\bfun_{X\setminus B(r)}(dd^c\varphi)^n~.$ 
Here $\bfun_A$ denotes the characteristic function of any subset
$A\subset X$~. According to the definition introduced in [De2], the
collection of {\sl Monge-Amp\`ere measures} associated to $\varphi$ is 
the family of positive measures $\mu_r$ such that
$$\mu_r=(dd^c\varphi_{\gge r})^n-\bfun_{X\setminus B(r)}(dd^c\varphi)^n~,
~~~~r\in]-\infty,R[~.\leqno(3.1)$$ 
The measure $\mu_r$ is supported on $S(r)$ and $r\longmapsto\mu_r$ is 
weakly continuous on the left by the bounded convergence theorem. 
Stokes' formula shows
that $\int_{B(s)}(dd^c\varphi_{\gge r})^n-(dd^c\varphi)^n=0$ for $s>r$~,
hence the total mass $\mu_r\big(S(r)\big)=\mu_r\big(B(s)\big)$ is equal
to the difference between the masses of $(dd^c\varphi)^n$ and 
$\bfun_{X\setminus B(r)}(dd^c\varphi)^n$ over $B(s)$~, \ie
$$\mu_r\big(S(r)\big)=\int_{B(r)}(dd^c\varphi)^n~.\leqno(3.2)$$

\remarque Example 3.3|When $(dd^c\varphi)^n=0$ on
$X\setminus\varphi^{-1}(-\infty)$, formula (3.1) simplifies into
$\mu_r=(dd^c\varphi_{\gge r})^n$~. This is so for $\varphi(z)=\log|z|$~.
In this case, the invariance of $\varphi$
under unitary transformations implies that $\mu_r$ is also invariant.
As the total mass of $\mu_r$ is equal to $1$ by 2.10 and (3.2), we see
that $\mu_r$ is the invariant measure of mass $1$ on the euclidean sphere
of radius $e^r$~. 

\tha Theorem 3.4|Assume that $\varphi$ is smooth near $S(r)$ and that
$d\varphi\ne 0$ on $S(r)$~, \ie $r$ is a non critical value. Then
$S(r)=\partial B(r)$ is a smooth oriented real hypersurface and
$\mu_r$ is given by the $(2n-1)$--volume form
$(dd^c\varphi)^{n-1}\wedge d^c\varphi_{\restriction S(r)}$~.\fintha

\proof Write $\max\{t,r\}=\lim_{k\to+\infty}\chi_k(t)$ where $\chi$ is
a non-increasing sequence of smooth convex functions with
$\chi_k(t)=r$ for $t\le r-1/k$~, $\chi_k(t)=t$ for $t\ge r+1/k$~.
Theorem 1.5 shows that $(dd^c\chi_k\circ\varphi)^n$ converges
weakly to $(dd^c\varphi_{\gge r})^n$~.
Let $h$ be a smooth function $h$ with compact support near $S(r)$~.
Let us apply Stokes' theorem with $S(r)$ considered as the boundary
of $X\setminus B(r)$~:
$$\leqalignno{
\int_X h(dd^c\varphi_{\gge r})^n&=\lim_{k\to+\infty}
\int_X h(dd^c\chi_k\circ\varphi)^n\cr
&=\lim_{k\to+\infty}\int_X -dh\wedge(dd^c\chi_k\circ\varphi)^{n-1}
\wedge d^c(\chi_k\circ\varphi)\cr
&=\lim_{k\to+\infty}\int_X -\chi'_k(t)^n\,dh\wedge(dd^c\varphi)^{n-1}
\wedge d^c\varphi\cr
&=\int_{X\setminus B(r)}-dh\wedge(dd^c\varphi)^{n-1}\wedge d^c\varphi\cr
&=\int_{S(r)}h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi+
\int_{X\setminus B(r)}h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi~.\cr}$$
Near $S(r)$ we thus have an equality of measures
$$(dd^c\varphi_{\gge r})^n=(dd^c\varphi)^{n-1}\wedge d^c
\varphi_{\restriction S(r)}+\bfun_{X\setminus B(r)}(dd^c\varphi)^n~.~\finpr$$

\tha Lelong-Jensen formula 3.5|Let $V$ be any psh function on $X$~. Then
$V$ is $\mu_r$--integrable for every $r\in]-\infty,R[$ and
$$\mu_r(V)-\int_{B(r)}V(dd^c\varphi)^n=
\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt~.$$
\fintha

\proof Theorem 1.8 shows that $V$ is integrable with respect to
$(dd^c\varphi_{\gge r})^n$~, hence $V$ is $\mu_r$--integrable.
By definition $\nu(dd^cV,\varphi,t)=\int_{\varphi(z)<t}
dd^cV\wedge(dd^c\varphi)^{n-1}$ and Fubini's theorem gives
$$\leqalignno{
\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt
&=\int\!\!\!\int_{\varphi(z)<t<r}dd^cV(z)\wedge(dd^c\varphi(z))^{n-1}\,dt\cr
&=\int_{B(r)}(r-\varphi)dd^cV\wedge(dd^c\varphi)^{n-1}~.&(3.6)\cr}$$
We first show that formula 3.5 is true when $\varphi$ and $V$ are
smooth. As both members of the formula are left continuous with
respect to $r$ and as almost all values of $\varphi$ are non critical
by Sard's theorem, we may assume $r$ non critical. Formula 1.1 applied
with $f=(r-\varphi)(dd^c\varphi)^{n-1}$ and $g=V$ shows 
that integral $(3.6)$ is equal to
$$\int_{S(r)}V(dd^c\varphi)^{n-1}\wedge d^c\varphi-\int_{B(r)}
V\,(dd^c\varphi)^n=\mu_r(V)-\int_{B(r)}V\,(dd^c\varphi)^n~.$$
Formula 3.5 is thus proved when $\varphi$ and $V$ are smooth.
If $V$ is smooth and $\varphi$ merely continuous and finite,
one can write $\varphi=\lim\varphi_k$ where $\varphi_k$ is a 
non-increasing sequence of smooth plurisubharmonic functions 
(because $X$ is Stein). Then $dd^cV\wedge(dd^c\varphi_k)^{n-1}$
converges weakly to $dd^cV\wedge(dd^c\varphi)^{n-1}$ and (3.6)
converges, since $\bfun_{B(r)}(r-\varphi)$ is continuous with compact
support on $X$~. The left hand side of formula 3.5 also
converges because the definition of $\mu_r$ implies
$$\mu_{k,r}(V)-\int_{\varphi_k<r}V(dd^c\varphi_k)^n=
\int_X V\big((dd^c\varphi_{k,\gge r})^n-(dd^c\varphi_k)^n\big)$$
and we can apply again weak convergence on a neighborhood of
$\ov B(r)$~. If $\varphi$ takes $-\infty$ values, replace
$\varphi$ by $\varphi_{\gge-k}$ where $k\to+\infty$~. Then $\mu_r(V)$
is unchanged, $\int_{B(r)}V(dd^c\varphi_{\gge -k})^n$ converges
to $\int_{B(r)}V(dd^c\varphi)^n$ and the right hand side of formula 3.5
is replaced by $\int_{-k}^r\nu(dd^cV,\varphi,t)\,dt$~.
Finally, for $V$ arbitrary, write $V=\lim\downarrow V_k$\break
with a sequence of smooth functions $V_k$~.
Then $dd^cV_k\wedge(dd^c\varphi)^{n-1}$ converges weakly to 
$dd^cV\wedge(dd^c\varphi)^{n-1}$ by theorem 1.13, thus integral
(3.6) converges to the expected limit, and the same is true for the
left hand side of 3.5 by the monotone convergence theorem.~\finpr

For $r<r_0<R$~, the Lelong-Jensen formula implies
$$\mu_r(V)-\mu_{r_0}(V)+\int_{B(r_0)\setminus B(r)}V(dd^c\varphi)^n=
\int_{r_0}^r\nu(dd^cV,\varphi,t)\,dt~.\leqno(3.7)$$

\tha Corollary 3.8|Assume that $(dd^c\varphi)^n=0$ on $X\setminus
S(-\infty)$~. Then $r\longmapsto\mu_r(V)$ is a convex increasing
function of $r$ and the lelong number $\nu(dd^cV,\varphi)$ is given by
$$\nu(dd^cV,\varphi)=\lim_{r\to-\infty}{\mu_r(V)\over r}~.$$
\fintha

\proof By (3.7) we have
$$\mu_r(V)=\mu_{r_0}(V)+\int_{r_0}^r\nu(dd^cV,\varphi,t)\,dt~.$$
As $\nu(dd^cV,\varphi,t)$ is increasing and non-negative, it follows
that $r\longmapsto\mu_r(V)$ is convex and increasing. The formula
for $\nu(dd^cV,\varphi)=\lim_{t\to-\infty}\nu(dd^cV,\varphi,t)$ is then
obvious.~\finpr

\remarque Example 3.9|Take $\varphi(z)=\log|z-a|$ on an open subset of
$\cb^n$ containing the point $a$~. The Lelong-Jensen formula becomes
$$\mu_r(V)=V(a)+\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt~.$$
As $\mu_r$ is the mean value measure on the sphere $S(a,e^r)$~, we
make the change of variables $r\mapsto\log r$~, $t\mapsto\log t$ and
obtain the more familiar formula
$$\mu(V,S(a,r))=V(a)+\int_0^r\nu(dd^cV,t)\,{dt\over t}$$
where $\nu(dd^cV,t)=\nu(dd^cV,\varphi,\log t)$ is given by (2.7)~:
$$\nu(dd^cV,t)=
{1\over\pi^{n-1}t^{2n-2}/(n-1)!}\int_{B(a,t)}{1\over2\pi}\Delta V~.$$
In particular, take $V=\log|f|$ where $f$ is a holomorphic function
on $X$~. The Poincar\'e-Lelong formula shows that $dd^c\log|f|$ is
equal to the zero divisor $[Z_f]=\sum m_j[H_j]$~, where $H_j$ are the
irreducible components of $f^{-1}(0)$ and $m_j$ the multiplicity of
$f$ on $H_j$~. The trace ${1\over 2\pi}\Delta f$ is then the euclidean
area measure of $Z_f$ (with corresponding multiplicities $m_j$). In
dimension $n=1$~, we have ${1\over2\pi}\Delta f=\sum m_j\delta_{a_j}$~.
Then we get the usual Jensen formula
$$\mu\big(\log|f|,S(0,r)\big)-\log|f(0)|=\int_0^r\nu(t){dt\over t}=
\sum m_j\log{r\over|a_j|}$$
where $\nu(t)$ is the number of zeroes $a_j$ in the disk $D(0,t)$~,
counted with multi\-plicities $m_j$~.

\remarque Example 3.10|Take $\varphi(z)=\log\max|z_j|^{\lambda_j}$~.
where $\lambda_j>0$~. Then $B(r)$ is the polydisk of
radii $(e^{r/\lambda_1}\ld e^{r/\lambda_n})$~. If some coordinate $z_j$
is non zero, say $z_1$~, we can write $\varphi(z)$ as
$\lambda_1\log|z_1|$ plus some function depending only on
the $(n-1)$ variables $z_j/z_1^{\lambda_1/\lambda_j}$~. Hence
$(dd^c\varphi)^n=0$ on $\cb^n\setminus\{0\}$~. It will be shown
later that 
$$(dd^c\varphi)^n=\lambda_1\ldots\lambda_n\,\delta_0~.\leqno(3.11)$$
We now determine the measures $\mu_r$~.
At any point $z$ where not all terms $|z_j|^{\lambda_j}$
are equal, the smallest one can be omitted without changing $\varphi$
in a neighborhood of~$z$~. Thus $\varphi$ depends only on
$(n-1)$--variables and $(dd^c\varphi_{\ge r})^n=0$~, $\mu_r=0$
near~$z$~. It follows that $\mu_r$ is supported by the
distinguished boundary $|z_j|=e^{r/\lambda_j}$ of the polydisk $B(r)$~.
As $\varphi$ is invariant by all rotations $z_j\longmapsto
e^{i\theta_j}z_j$~, the measure $\mu_r$ is also invariant and we see that
$\mu_r$ is a constant multiple of $d\theta_1\ldots d\theta_n$~.
By formula (3.2) and (3.11) we get
$$\mu_r=\lambda_1\ldots\lambda_n\,(2\pi)^{-n}d\theta_1\ldots d\theta_n~.$$
In particular, the Lelong number $\nu(dd^cV,\varphi)$ is given by
$$\nu(dd^cV,\varphi)=\lambda_1\ldots\lambda_n\,\lim_{r\to-\infty}
\int_{\theta_j\in[0,2\pi]}V(e^{r/\lambda_1+i\theta_1}
\ld e^{r/\lambda_n+i\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}~.$$
These numbers have been introduced and studied by Kiselman [Ki3]~;
we shall denote them $\nu(T,x,\lambda)$~, where $\lambda=(\lambda_1\ld
\lambda_n)$~.

\pav 4. Comparison theorem for Lelong numbers|

We show here that the Lelong numbers $\nu(T,\varphi)$ only depend 
on the asymptotic behaviour of $\varphi$ near the polar set
$S(-\infty)$~. In a precise way~:

\tha Theorem 4.1|Let $\varphi,\psi:X\longrightarrow[-\infty,+\infty[$
be continuous psh functions. We assume that $\varphi,\psi$ are 
semi-exhaustive on ${\rm Supp}\,T$ and that
$$l:=\limsup{{\psi(x)}\over{\varphi(x)}}<+\infty~~~~{\sl as}~~
x\in{\rm Supp}\,T~~{\sl and}~~\varphi(x)\to -\infty~.$$
Then $\ \nu(T,\psi)\le l^p\nu(T,\varphi)\ ,$ and the equality holds if
$l=\lim\psi/\varphi~.$\fintha

\proof Definition 2.4 gives immediately
$$\nu(T,\lambda\varphi)=\lambda^p\nu(T,\varphi)$$
for every scalar $\lambda>0$~. It is thus sufficient to verify the
inequality $\nu(T,\psi)\le\nu(T,\varphi)$ under tha hypothesis 
$\limsup\psi/\varphi<1$~. For all $c>0$~, consider the psh
function 
$$u_c=\max(\psi-c,\varphi)~.$$
Let $R_\varphi$ and $R_\psi$
be such that $B_\varphi(R_\varphi)\cap{\rm Supp}\,T$ and
$B_\psi(R_\psi)\cap{\rm Supp}\,T$ be relatively compact in $X$~.
Let $r<R_\varphi$ be fixed and $a<r$~. For $c>0$ large enough, we have 
$u_c=\varphi$ on $\varphi^{-1}([a,r])$ and Stokes' formula gives
$$\nu(T,\varphi,r)=\nu(T,u_c,r)\ge\nu(T,u_c)~.$$
The hypothesis $\limsup\psi/\varphi<1$ implies on the other hand that
there exists $t_0<0$ such that $u_c=\psi-c$ on $\{u_c<t_0\}\cap
{\rm Supp}\,T$~. We infer 
$$\nu(T,u_c)=\nu(T,\psi-c)=\nu(T,\psi)~,$$
hence $\nu(T,\psi)\le\nu(T,\varphi)$~. The equality case is obtained by
reversing the roles of $\varphi$ and $\psi$ and observing that
$\lim\varphi/\psi=1/l$~.~\finpr

  Assume in particular that $z^k=(z^k_1\ld z^k_n)$~, $k=1,2$~, are
coordinate systems centered at a point $x\in X$ and let
   $$\varphi_k(z)=\log|z^k|=\log\bigl(|z^k_1|^2+\ldots+|z^k_n|^2\bigr)^{1/2}.$$
We have $\lim_{z\to x}{\varphi_2(z)/\varphi_1(z)}=1$~, hence 
$\nu(T,\varphi_1)=\nu(T,\varphi_2)$ by theorem $4.1$~. 

\tha Corollary 4.2|The usual Lelong numbers $\nu(T,x)$ are independent
of the choice of local coordinates.~\finpr\fintha

This result had been originally proved by [Siu] with a much more
delicate proof. Another interesting consequence is~:

\tha Corollary 4.3|On an open subset of $\cb^n$~, the Lelong numbers
and Kiselman numbers are related by
     $$\nu(T,x)=\nu\big(T,x,(1\ld 1)\big)~.$$\fintha

\proof By definition, the number $\nu(T,x)$ is associated to the weight\break
$\varphi(z)=\log|z-x|$ and $\nu\big(T,x,(1\ld 1)\big)$ to the weight
$\psi(z)=\log\max|z_j-x_j|$~. It is clear that
$\lim_{z\to x}\psi(z)/\varphi(z)=1$~, whence the conclusion.~\finpr

Another consequence of theorem 4.1 is that $\nu(T,x,\lambda)$ is an
increasing function of each variable $\lambda_j$~. Moreover, if
$\lambda_1\le\ldots\le\lambda_n$~, we get the inequalities
$$\lambda_1^p\nu(T,x)\le\nu(T,x,\lambda)\le\lambda_n^p\nu(T,x)~.$$
It can be shown ([De1]) that the stronger inequalities
$$\lambda_1\ldots\lambda_p\nu(T,x)\le\nu(T,x,\lambda)\le
\lambda_{n-p+1}\ldots\lambda_n\nu(T,x)$$
hold for every current $T$ of bidimension $(p,p)$~. By formula (3.11),
this is easily checked for any $p$--plane of coordinates
$T=[\cb e_{i_1}\oplus\ldots\oplus\cb e_{i_p}]$~. The general case can
be deduced from this special case.\vskip5pt

Now, we assume that $T=[A]$ is the current of integration over an
analytic set $A\subset X$ of pure dimension $p$ (\cf P. Lelong[Le1]).
The above comparison theorem will enable us to give a simple
proof of P.~Thie's main result~[Th]~: the Lelong number $\nu([A],x)$ 
can be interpreted as the multiplicity of the analytic set $A$ at point
$x$~.

   Let $x\in A$ be a given point and ${\cal I}_{A,x}$ the ideal of
germs of holomorphic functions at $x$ vanishing on $A$~.
Then, one can find local coordinates $z=(z_1,\ldots,z_n)$ on $X$
centered at $x$ such that there exist distinguished Weierstrass
polynomials $P_j\in{\cal I}_{A,x}$ in the variable $z_j$~, $p<j\le n$~,
of the type
$$P_j(z)=z_j^{d_j}+\sum_{k=1}^{d_j}a_{j,k}(z_1,\ldots,z_{j-1})z_j^{d_j-k}\ ,\ 
                a_{j,k}\in{\cal M}^k_{\cb^{j-1},0}\leqno(4.4)$$
where ${\cal M}_{X,x}$ is the maximal ideal of $X$ at $x$~.

Indeed, let us prove this property by induction on codim$\,X=n-p$~.
We fix a coordinate system $(w_1\ld w_n)$ by which we identify
the germ $(X,x)$ to $(\cb^n,0)$~.

   If $n-p\ge 1$~, there exists a non zero element $f\in{\cal I}_{A,x}$~.
Let $d$ be the smallest integer such that $f\in{\cal M}_{\cb^n,0}^d$ and
let $e_n\in\cb^n$ be a non zero vector such that 
$\lim_{t\to 0}{f(te_n)/t^d\ne 0}$~. Complete $e_n$ into a basis 
$(\tilde e_1,\ldots,\tilde e_{n-1},e_n)$ of $\cb^n$ and denote by
$(\tilde z_1,\ldots,\tilde z_{n-1},z_n)$ the corresponding coordinates.
The Weierstrass preparation theorem gives a factorization 
$f=gP$ where $P$ is a distinguished polynomial of type (4.4) in the
variable $z_n$ and where $g$ is an invertible holomorphic function 
at point $x$~. If $n-p=1$~, the polynomial $P_n=P$ satisfies the
requirements.

    If $n-p\ge 2$~, ${\cal O}_{A,x}={\cal O}_{X,x}/{\cal I}_{A,x}$ is a
${\cal O}_{\cb^{n-1},0}=\cb\{\tilde z_1,\ldots,\tilde z_{n-1}\}$-module
of finite type, \ie the projection ${\rm pr}:(X,x)\approx
(\cb^n,0)\longrightarrow(\cb^{n-1},0)$ is a finite morphism 
of $(A,x)$ onto a germ $(Z,0)\subset(\cb^{n-1},0)$ of dimension
$p$~. The induction hypothesis applied to
${\cal I}_{Z,0}={\cal O}_{\cb^{n-1},0}\cap{\cal I}_{A,x}$ implies the
existence of a new basis $(e_1,\ldots,e_{n-1})$ of $\cb^{n-1}$ and
polynomials $P_{p+1},\ldots,P_{n-1}\in{\cal I}_{Z,0}$ of type (4.4)
in the coordinates $(z_1,\ldots,z_{n-1})$ associated
to the basis $(e_1,\ldots,e_{n-1})$~. If we choose $P_n=P$~,
the expected property is proved in codimension $n-p$~.

  For any polynomial $Q(w)=w^d+a_1w^{d-1}+\ldots+a_d\in\cb[w]$~,
the roots $w$ of $Q$ satisfy
         $$|w|\le 2\max_{1\le k\le d}|a_k|^{1/k},\leqno(4.5)$$
otherwise $Q(w)w^{-d}=1+a_1w^{-1}+\ldots+a_dw^{-d}$ would have a
modulus larger than $1-(2^{-1}+\ldots+2^{-d})=2^{-d}$~, a contradiction.
Let us denote $z=(z',z'')$ with $z'=(z_1,\ldots,z_p)$ and
$z''=(z_{p+1},\ldots,z_n)$~. As $a_{j,k}\in{\cal M}^k_{\cb^{j-1},0}$,
we get
   $$|a_{j,k}(z_1,\ldots,z_{j-1})|=
    {\rm O}\big((|z_1|+\ldots+|z_{j-1}|)^k\big)~~~{\rm if}~~j>p~,$$
and we deduce from (4.4), (4.5) that $|z_j|=O(|z_1|+\ldots+|z_{j-1}|)$
on $(A,x)$~. Therefore, the germ $(A,x)$ is contained in a cone
$|z''|\le C|z'|$~.

   We shall use this property in order to compute the Lelong number
of $[A]$ at point $x$~. When $z\in A$ tends to $x$ , the functions
$$\varphi(z)=\log|z|=\log(|z'|^2+|z''|^2)^{1/2}~,~~~~\psi(z)=\log|z'|~.$$
are equivalent. As $\varphi,\psi$ are semi-exhaustive on $A$~, 
theorem 4.1 implies
        $$\nu([A],x)=\nu([A],\varphi)=\nu([A],\psi).$$
Let $B'\subset\cb^p$ the ball of center $0$ and radius $r'$~, 
$B''\subset\cb^{n-p}$ the ball of center $0$ and radius $r''=Cr'$~. 
The inclusion of germ $(A,x)$ in the cone $|z''|\le C|z'|$ shows that
for $r'$ small enough the projection
      $${\rm pr}:A\cap(B'\times B'')\longrightarrow B'$$
is proper. The fibers are finite by (4.4). Hence this projection
is a ramified covering with a finite sheet number $m$~. 

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\vskip5mm
\centerline{{\bf Fig.~2} Ramified covering from $A$ to 
$\Delta'\subset\cb^p$.}
\vskip5mm

\noindent Let us apply formula (2.6) to $\psi$~: for every $t<r'$ we get
$$\eqalign{\nu([A],\psi,\log t)
&=t^{-2p}\int_{\{\psi<\log t\}}[A]\wedge\Big({1\over 2}dd^ce^{2\psi}\Big)^p\cr
&=t^{-2p}\int_{A\cap\{|z'|<t\}}\Big({1\over 2}{\rm pr}^\star dd^c|z'|^2\Big)^p\cr
&=m\,t^{-2p}\int_{\cb^p\cap\{|z'|<t\}}\Big({1\over 2}dd^c|z'|^2\Big)^p=m~,\cr}$$
hence $\nu(T,\psi)=m$~.  Here, we used the fact that pr is actually a
covering with m sheets over the complement of the ramification locus
$S\subset B'$~, which is of zero Lebesgue measure.  We thus obtain~:

\tha Theorem 4.6 ({\rm P. Thie [Th])}|Let $A$ be an analytic set of
dimension $p$ in a complex manifold of dimension $p$~. For every point
$x\in A$~, there exist local coordinates
    $$z=(z',z'')~,~~~z'=(z_1,\ldots,z_p)~,~~~z''=(z_{p+1},\ldots,z_n)$$
centered at $x$ and balls $B'\subset\cb^p$~, $B''\subset\cb^{n-p}$ in
of radii $r',r''$ in these coordinates, such that
$A\cap(B'\times B'')$ is contained in the cone $|z''|\le(r''/r')|z'|$~.
The multiplicity of $A$ at $x$ is defined as the number $m$ of
sheets of the ramified covering map $A\cap(B'\times B'')\longrightarrow B'$~.
Then $\nu([A],x)=m$~.
\fintha

\tha Exercise 4.7|Show that the measures
$$\big(dd^c\log\max|z_j|^{\lambda_j}\big)^n~,~~~~
\Big(dd^c\log\sum|z_j|^{\lambda_j}\Big)^n$$
are equal to $c(\lambda)\delta_0$ with the same coefficient 
$c(\lambda)$ in both cases. When all $\lambda_j$ are even integers, 
compute explicitly $c(\lambda)$ by means of formula (2.6).
Extend the result to arbitrary rational (resp. real) numbers
$\lambda_j>0$~.\fintha

\pav 5. Siu's semi-continuity theorem|

  Let $X,Y$ be complex manifolds of dimension $n,n'$ such that $X$ is Stein.
Let $\varphi:X\times Y\longrightarrow[-\infty,+\infty[$ be a continuous 
psh function. We assume that $\varphi$ is {\sl semi-exhaustive}
with respect to ${\rm Supp}\,T$ , \ie that for every compact subset 
$L\subset Y$ there exists $R=R(L)<0$ such that
 $$\{(x,y)\in{\rm Supp}\,T\times L\,;
            \,\varphi(x,y)\le R\}\ssubset X\times Y.\leqno(5.1)$$
Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$~. 
For every point $y\in Y$~, the function $\varphi_y(x):=\varphi(x,y)$ 
is semi-exhaustive on ${\rm Supp}\,T$~; one can therefore associate to $y$ 
a generalized Lelong number $\nu(T,\varphi_y)$~.

\tha Lemma 5.2|{\rm(a)}~~The measure $T\wedge(dd^c\varphi_{y,\gge t})^p$
depends continuously~on~$y$.

\noindent{\rm(b)}~~For all $r_1<r_2<R(L)$ and $y,y_0\in L$ one has
$$\limsup_{y\to y_0}~\nu(T,\varphi_y,r_1)\le\nu(T,\varphi_{y_0},r_2)~.$$

\noindent{\rm(c)}~~$\,$The map $y\mapsto\nu(T,\varphi_y)$ is upper
semi-continuous.\fintha

\proof (a) We prove by induction on $q$ that
$T\wedge(dd^c\varphi_{y,\gge t})^q$ is weakly continuous in $y$~.
Let $h$ be a smooth form on $X$~. Then
$$\int_X h\wedge T\wedge(dd^c\varphi_{y,\gge t})^q=
\int_X dd^ch\wedge T\wedge\varphi_{y,\gge t}(dd^c\varphi_{y,\gge t})^{q-1}~.$$
Taking the difference for two points $y,y_0$~, we get
$$\eqalign{
\int_X h\wedge T\wedge&\big((dd^c\varphi_{y,\gge t})^q-
(dd^c\varphi_{y_0,\gge t})^q\big)\cr
&=\int_X dd^ch\wedge T\wedge(\varphi_{y,\gge t}-\varphi_{y_0,\gge t})
(dd^c\varphi_{y,\gge t})^{q-1}\big)\cr
&~~~~~~~~+\int_X\varphi_{y_0,\gge t}dd^ch\wedge T\wedge
\big((dd^c\varphi_{y,\gge t})^{q-1}-(dd^c\varphi_{y_0,\gge t})^{q-1}\big)~.
\cr}$$
The last integral tends to $0$ thanks to the induction hypothesis,
and the previous one thanks to the Chern-Levine Nirenberg inequalities
and the uniform convergence of $\varphi_{y,\gge t}$ to
$\varphi_{y_0,\gge t}$~.

\noindent (b) As $\nu(T,\varphi_y,r)=\int_{B(r)}T\wedge
(dd^c\varphi_{y,\gge t})^p$ when $t<r$~, (b) follows from (a).

\noindent(c) Letting $r_1,r_2\to-\infty$~, we get easily
$\limsup_{y\to y_0}~\nu(T,\varphi_y)\le\nu(T,\varphi_{y_0})$~.~\finpr

As a consequence of 5.2 (c), we see that the sublevel sets
  $$E_c=\{y\in Y ;\nu(T,\varphi_y)\ge c\}\ ,\ c>0\leqno(5.3)$$
are closed. Under mild additional hypotheses, we are going to show that 
the sets $E_c$ are in fact analytic subsets of $Y$~. 

\tha Definition 5.3|We say that a function $f(x,y)$ is locally
H\"older continuous with respect to $y$ on $X\times Y$
if every point of $X\times Y$ has a neighborhood $\Omega$ on which
$$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$
for all $(x,y_1)\in\Omega$~, $(x,y_2)\in\Omega$~,
for some constants $M>0$~, $\gamma\in]0,1]$~, and for suitable
coordinates on $Y$~.\fintha

\tha Theorem 5.4|Let $T$ be a closed positive current on $X$ and 
       $$\varphi:X\times Y\longrightarrow[-\infty,+\infty[$$
a continuous psh function. Assume that $\varphi$ is semi-exhaustive 
on ${\rm Supp}\,T$ and that $e^{\varphi(x,y)}$ is locally H\"older
continuous with respect to $y$ on $X\times Y$~. Then the sublevel
sets
          $$E_c=\{y\in Y;\nu(T,\varphi_y)\ge c\}$$
are analytic subsets of $Y$~.\fintha

This theorem, proved in [De3], can be rephrased by saying that
$y\longmapsto\nu(T,\varphi_y)$ is upper semi-continuous with respect
to the analytic Zariski topology. As a special case, we get the
following important result of [Siu]~:

\tha Corollary 5.5|If $T$ is a closed positive current on a manifold
$X$~, the sublevel sets $E_c=\{x\in X\,;\,\nu(T,x)\ge c\}$
of the usual Lelong numbers are analytic.\fintha

\proof The result is local, so we may assume that $X\subset\cb^n$ is an open
subset. Then apply theorem 5.4 with $Y=X$ and
$\varphi(x,y)=\log|x-y|$~.~\finpr

If $X$ is an open subset of $\cb^n$~, the sublevel sets for
Kiselman's numbers $\nu(T,x,\lambda)$ are also analytic in $X$~.
However, this result is not intrinsically significant on a manifold,
because Kiselman's numbers depend on the choice of coordinates.
To give another application, set $u_\lambda(z)=\log\max|z_j|^{\lambda_j}$
and set $\varphi(x,y,g)=u_\lambda\big(g(x-y)\big)$ where $x,y\in\cb^n$
and $g\in{\rm Gl}(\cb^n)$~. Then $\nu(T,\varphi_{y,g})$ 
is the Kiselman number of $T$ at $y$ when the coordinates have been rotated
by $g$~. It is clear that $\varphi$ is psh in $(x,y,g)$ and
semi-exhaustive with respect to $x$~, and 
that $e^\varphi$ is H\"older continuous with exponent 
$\gamma=\min\{1,\lambda_j\}$~. Thus the sublevel sets
$$E_c=\{(y,g)\in X\times{\rm Gl}(\cb^n)\,;\,\nu(T,\varphi_{y,g})\ge c\}$$
are analytic in $X\times{\rm Gl}(\cb^n)$~. Theorem 5.4 can be applied
more generally to weight functions of the type
  $$\varphi=\max_j~\log\Big(\sum_k|F_{j,k}|^{\lambda_{j,k}}\Big)$$
where $F_{j,k}$ are holomorphic functions on $X\times Y$ and
$\gamma_{j,k}$ positive real constants~; in this case $e^\varphi$ is
H\"older continuous of exponent $\gamma=\min\{\lambda_{j,k},1\}$~.

Now let us prove theorem 5.4. As the
result is local on $Y$~, we may assume without loss of
generality that $Y$ is a ball in $\cb^{n'}$~. After addition of
a constant to $\varphi$~, we may also assume that
there exists a compact subset $K\subset X$ such that
$$\{(x,y)\in X\times Y ;\varphi(x,y)\le 0\}\subset K\times Y.$$
By theorem 4.1, the Lelong numbers depend only on the
asymptotic behaviour of $\varphi$ near the (compact) polar set 
$\varphi^{-1}(-\infty)\cap({\rm Supp T}\times Y)$~. We can add 
a smooth strictly plurisubharmonic function on $X\times Y$ to make
$\varphi$ strictly plurisuharmonic. Then Richberg's approximation
theorem for continuous psh functions shows that there exists
a smooth psh function $\wt\varphi$ such that
$\varphi\le\wt\varphi\le\varphi+1$~. We may therefore assume that
$\varphi$ is smooth on $(X\times Y)\setminus\varphi^{-1}(-\infty)$~.
\vskip7pt

$\bullet$ {\bf First step} : {\sl construction of a local psh potential.}

   Our goal is to generalize the usual construction of psh
potentials associated to a closed positive current (\cf P.~Lelong[Le2]
and H.~Skoda[Sk]). We replace here the usual kernel
$|z-\zeta|^{-2p}$ arising from the hermitian metric of $\cb^n$
by a kernel depending on the weight $\varphi$~.
Let $\chi\in C^\infty(\rb,\rb)$ be an increasing function such that
$\chi(t)=t$ for $t\le-1$ and $\chi(t)=0$ for $t\ge 0$~. We consider
the half-plane $H=\{z\in\cb\,;\,{\rm Re}z<-1\}$ and associate to
$T$ the potential function $V$ on $Y\times H$ defined by
$$V(y,z)=-\int^0_{{\rm Re}z}\nu(T,\varphi_y,t)\chi'(t)\,dt~.\leqno(5.6)$$
For every $t>\re z$~, Stokes' formula gives
$$\nu(T,\varphi_y,t)=
       \int_{\varphi(x,y)<t}T(x)\wedge(dd^c_x\wt\varphi(x,y,z))^p$$
with $\wt\varphi(x,y,z):=\max\{\varphi(x,y),{\rm Re}z\}$~. Fubini's
theorem applied to (5.6) gives
$$\eqalign{V(y,z)
&=-\int_{{\scriptstyle x\in X,\varphi(x,y)<t}\atop{\scriptstyle\re z<t<0}}
 T(x)\wedge(dd^c_x\wt\varphi(x,y,z))^p~\chi'(t)dt\cr
&=\int_{x\in X}T(x)\wedge\chi(\wt\varphi(x,y,z))\,(dd^c_x\wt\varphi(x,y,z))^p~.
\cr}$$
For all $(n-1,n-1)$--form $h$ of class $C^{\infty}$ with compact support 
in $Y\times H$~, we get
$$\langle dd^cV,h\rangle=\langle V,dd^ch\rangle=
\int_{X\times Y\times H}T(x)\wedge\chi(\wt\varphi(x,y,z))
(dd^c\wt\varphi(x,y,z))^p\wedge dd^ch(y,z).$$
Observe that the replacement of $dd^c_x$ by the total differentiation
$dd^c=dd^c_{x,y,z}$ does not modify the integrand, because the
terms in $dx$~, $d\ov x$ must have total bidegree $(n,n)$~.
The current $T(x)\wedge\chi(\wt\varphi(x,y,z))h(y,z)$ has compact support in
$X\times Y\times H$. An integration by parts can thus be performed
to obtain
$$\langle dd^cV,h\rangle=(2\pi)^{-p}\int_{X\times Y\times H}T(x)\wedge
  dd^c(\chi\circ\wt\varphi(x,y,z))\wedge(dd^c\wt\varphi(x,y,z))^p.h(y,z).$$
On the corona $\{-1\le\varphi(x,y)\le0\}$ we have $\wt\varphi(x,y,z)
=\varphi(x,y)$~, whereas for $\varphi(x,y)<-1$ we get
$\wt\varphi<1$ and $\chi\circ\wt\varphi=\wt\varphi$~. As $\wt\varphi$
is psh, we see that $dd^cV(y,z)$ is the sum of the positive $(1,1)$--form
   $$(y,z)\longmapsto\int_{\{x\in X;\varphi(x,y)<-1\}}
                      T(x)\wedge(dd^c_{x,y,z}\wt\varphi(x,y,z))^{p+1}$$
and of the $(1,1)$--form independent of $z$
   $$y\longmapsto\int_{\{x\in X;-1\le\varphi(x,y)\le0\}}T\wedge 
                 dd^c_{x,y}(\chi\circ\varphi)\wedge(dd^c_{x,y}\varphi)^p~;$$
as $\varphi$ is smooth outside $\varphi^{-1}(-\infty)$~, this last
form has locally bounded coefficients. We obtain therefore the following
result.

\tha Theorem 5.7|There exists a positive psh function $\rho\in C^\infty(Y)$
such that $\rho(y)+V(y,z)$ is psh on $Y\times H$~.\fintha

To be quite complete, we must observe in addition that $V$ is
continuous on $Y\times H$ because
$T\wedge(dd^c\wt\varphi_{y,z})^p$ is weakly continuous in the
variables $(y,z)$ by lemma 5.1 (a). 

If we let $\re z$ tend to $-\infty$~, we see that the function
    $$U_0(y)=\rho(y)+V(y,-\infty)
            =\rho(y)-\int_{-\infty}^0\nu(T,\varphi_y,t)\chi'(t)dt$$
is locally psh or $\equiv-\infty$ on $Y$~. Moreover, it is clear
that $U_0(y)=-\infty$ at every point $y$ such that
$\nu(T,\varphi_y)>0$~. If $Y$ is connected and $U_0\not\equiv-\infty$~,
we already conclude that the density set $\bigcup_{c>0}E_c$
is pluripolar in $Y$~.
\vskip7pt

\noindent$\bullet$ {\bf Second step} : {\sl application of
                                           Kiselman's minimum principle.}

\tha Kiselman's minimum principle 5.8 {\rm([Ki1])}|Let $M$ be a complex
manifold, $\omega\subset\rb^n$ a convex open subset and
$\Omega$ the ``tube domain" $\Omega=\omega+i\rb^n$~. For
every plurisubharmonic function $v(\zeta,z)$ on $M\times\Omega$ that
does not depend on $\im z$~, the function
$$u(\zeta)=\inf_{z\in\Omega}~v(\zeta,z)$$
is plurisubharmonic or locally $\equiv-\infty$ on $M$~.\fintha

\proof Without loss of generality, we may assume that $M$ is a ball and
that $0\in M\times\Omega$~. The hypothesis implies that $v(\zeta,z)$ is convex
in $x=\re z$~. Let $\tau\ge 0$ be a smooth strictly convex exhaustion
function on $\omega$ with $\tau(0)=0$~. We approximate $v$ by the
sequence of smooth functions defined on $(1-\varepsilon)M\times\Omega$ by
$$v_\varepsilon=v\star\rho_{\varepsilon^2}+
\varepsilon\tau\big((1-\varepsilon)^{-1}x\big)$$
Then $u_\varepsilon(\zeta)=\inf_{z\in\Omega}v_\varepsilon(\zeta,z)$ 
is increasing in $\varepsilon$ and converges to $u$~.
We may therefore assume that $v$ has all properties of $v_\varepsilon$~,
\ie $v$ is smooth, plurisubharmonic in $(\zeta,z)$~, strictly convex in $x$
and $\lim_{|x|\to+\infty}v(\zeta,x)=+\infty$ for every $\zeta\in M$~.
Then $x\longmapsto v(\zeta,x)$ has a unique minimum point $x=g(\zeta)$~,
solution of the equations $\partial v/\partial x_j(x,\zeta)=0$~.
As the matrix $(\partial^2 v/\partial x_j\partial x_k)$ is positive 
definite, the implicit function theorem shows that $g$ is smooth.
Now, if $w\longmapsto\zeta_0+wa$~, $a\in\cb^n$~, $|w|\le 1$ is
a complex disk $\Delta$ contained in $M$~, there exists a holomorphic
function $f$ on the unit disk and smooth up to the boundary, whose
real part solves the Dirichlet problem
$$\re f(e^{i\theta})=g(\zeta_0+e^{i\theta}a)~.$$
Since $v(\zeta_0+wa,f(w))$ is subharmonic in $w$~, we get the
mean value inequality
$$v(\zeta_0,f(0))\le{1\over 2\pi}\int_0^{2\pi}v\big(\zeta_0+e^{i\theta}a,
f(e^{i\theta})\big)d\theta=
{1\over 2\pi}\int_{\partial\Delta}v(\zeta,g(\zeta))d\theta~.$$
The last equality holds because $\re f=g$ on $\partial\Delta$ and
$v(\zeta,z)=v(\zeta,\re z)$ by hypothesis.
As $u(\zeta_0)\le v(\zeta_0,f(0))$ and $u(\zeta)=v(\zeta,g(\zeta))$~,
we see that $u$ satisfies the mean value inequality.~\finpr

   Let $a\ge 0$ be arbitrary. The function
     $$Y\times H\ni(y,z)\longmapsto\rho(y)+V(y,z)-a{\rm Re}z$$
is psh and independent of ${\rm Im}\,z$~. By 5.8, the Legendre transform
     $$U_a(y)=\inf_{r<-1}[\rho(y)+V(y,r)-a r]$$
is locally psh or $\equiv-\infty$ on $Y$~.

\tha Lemma 5.9|Let $y_0\in Y$ be a given point.

\noindent{\rm(a)}~~If $a>\nu(T,\varphi_{y_0})$~, then $U_a$ is bounded
below on a neighborhood of $y_0$~.

\noindent{\rm(b)}~~If $a<\nu(T,\varphi_{y_0})$~, then $U_a(y_0)=-\infty$~.
\fintha

\proof By definition of $V$ (\cf $(5.6)$) we have
$$V(y,r)\le-\nu(T,\varphi_y,r)\int_r^0\chi'(t)dt=r\nu(T,\varphi_y,r)
                 \le r\nu(T,\varphi_y)~.\leqno(5.10)$$
Then clearly $U_a(y_0)=-\infty$ if $a<\nu(T,\varphi_{y_0})$~. On the
other hand, if $\nu(T,\varphi_{y_0})<a$~, there exists $t_0<0$ 
such that $\nu(T,\varphi_{y_0},t_0)<a$~. Fix $r_0<t_0$~. 
The semi-continuity property 5.2 (b) shows that there exists a
neighborhood $\omega$ of $y_0$ such that $\sup_{y\in\omega}~
\nu(T,\varphi_y,r_0)<a$~. For all $y\in\omega$~, we get
$$V(y,r)\ge-C-a\int_r^{r_0}\chi'(t)dt=-C+a(r-r_0)~,$$
and this implies $U_a(y)\ge-C-a r_0$~.~\finpr

\tha Theorem 5.11|If $Y$ is connected and if $E_c\ne Y$~, then $E_c$
is a closed complete pluripolar subset of $Y$~, \ie there exists
a continuous psh function $w:Y\la[-\infty,+\infty[$ such that
$E_c=w^{-1}(-\infty)$~.\fintha

\proof We first observe that the family $(U_a)$ is increasing in $a$~,
that $U_a=-\infty$ on $E_c$ for all $a<c$ and that
$\sup_{a<c}U_a(y)>-\infty$ if $y\in Y\setminus E_c$
(apply lemma 5.9). For any integer $k\ge 1$~, let $w_k\in\ci(Y)$ be a
psh regularization of $U_{c-1/k}$ such that
$w_k\ge U_{c-1/k}$ on $Y$ and $w_k\le -2^k$ on
$E_c\cap Y_k$~. Then lemma 5.10 (a) shows that the family
$(w_k)$ is uniformly bounded below on every compact subset of
$Y\setminus E_c$~. We can also choose $w_k$ uniformly
bounded above on every compact subset of $Y$ because
$U_{c-1/k}\le U_c$~. The function
$$w=\sum_{k=1}^{+\infty}~2^{-k}w_k$$
satifies our requirements.~\finpr
\vskip7pt

\noindent$\bullet$ {\bf Third step} : {\sl estimation of the singularities of
                                          the potentials} $U_a$.

\tha Lemma 5.12|Let $y_0\in Y$ be a given point, $L$ a compact
neighborhood of $y_0$ , $K\subset X$ a compact subset and $r_0$
a real number $<-1$ such that
    $$\{(x,y)\in X\times L;\varphi(x,y)\le r_0\}\subset K\times L~.$$
Assume that $e^\varphi(x,y)$ is locally H\"older continuous in $y$
and that
$$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$
for all $(x,y_1,y_2)\in K\times L\times L$~. Then, for all 
$\varepsilon\in]0,1[$~, there exists a real number
$\eta(\varepsilon)>0$ such that all $y\in Y$ with 
$|y-y_0|<\eta(\varepsilon)$ satisfy
    $$U_a(y)\le\rho(y)+\big((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\big)
        \Big(\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\Big)~.$$
\fintha

\proof First, we try to estimate $\nu(T,\varphi_y,r)$ when $y\in L$ is
near $y_0$~. Set
$$\left\{ 
\eqalign{\psi(x)&=(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2\cr
         \psi(x)&=\max\bigl(\varphi_y(x),(1-\varepsilon)\varphi_{y_0}(x)+
                  \varepsilon r-\varepsilon/2\bigr)\cr
         \psi(x)&=\varphi_y(x)\cr}
\eqalign{&{\quad\rm if\quad}\cr
         &{\quad\rm if\quad}\cr
         &{\quad\rm if\quad}\cr}
\eqalign{&\varphi_{y_0}(x)\le r-1\cr
   r-1\le&\varphi_{y_0}(x)\le r\cr
     r\le&\varphi_{y_0}(x)\le r_0\cr}\right.$$
and verify that this definition is coherent when $|y-y_0|$ is small enough.
By hypothesis
$$|e^{\varphi_y(x)}-e^{\varphi_{y_0}(x)}|\le M|y-y_0|^\gamma.$$
This inequality implies
$$\eqalign{\varphi_y(x)&\le\varphi_{y_0}(x)+\log\bigl(1+M|y-y_0|^\gamma 
                          e^{-\varphi_{y_0}(x)}\bigr)\cr
           \varphi_y(x)&\ge\varphi_{y_0}(x)+\log\bigl(1-M|y-y_0|^\gamma
                          e^{-\varphi_{y_0}(x)}\bigr).\cr}$$
In particular, for $\varphi_{y_0}(x)=r$~, we have
$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2=r-\varepsilon/2$~,
thus
      $$\varphi_y(x)\ge r+\log(1-M|y-y_0|^\gamma e^{-r})~.$$
Similarly, for $\varphi_{y_0}(x)=r-1$~, we have
$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2=
r-1+\varepsilon/2$~, thus
      $$\varphi_y(x)\le r-1+\log(1+M|y-y_0|^\gamma e^{1-r}).$$
The definition of $\psi$ is thus coherent as soon as
$M|y-y_0|^\gamma e^{1-r}\le\varepsilon/2$ , \ie
      $$\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\le r~.$$
In this case $\psi$ coincides with $\varphi_y$ on a neighborhood of 
$\{\psi=r\}$ , and with
         $$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2$$
on a neighborhood of the polar set $\psi^{-1}(-\infty)$~. By Stokes'
formula applied to $\nu(T,\psi,r)$~, we infer
  $$\nu(T,\varphi_y,r)=\nu(T,\psi,r)\ge\nu(T,\psi)=
                 (1-\varepsilon)^p\nu(T,\varphi_{y_0}).$$
From (5.10) we get $V(y,r)\le r\nu(T,\varphi_y,r)$~, hence
$$\leqalignno{
     U_a(y)&\le\rho(y)+V(y,r)-ar\le 
                   \rho(y)+r\big(\nu(T,\varphi_y,r)-a\big)~,\cr
     U_a(y)&\le\rho(y)+r\bigl((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\bigr)~.
                   &(5.13)\cr}$$
Suppose $\gamma\log|y-y_0|+\log(2eM/\varepsilon)\le r_0$ , \ie 
$|y-y_0|\le(\varepsilon/2eM)^{1/\gamma}e^{r_0/\gamma}$~;\break 
one can then choose $r=\gamma\log|y-y_0|+\log(2eM/\varepsilon)$~, and by
$(5.13)$ this yields the inequality asserted in theorem 5.12~.~\finpr

\noindent $\bullet$ {\bf Fourth step}~:
{\sl application of H\"ormander's $L^2$ estimates.}

The end of the proof rests upon the following crucial result, known as
the H\"ormander-Bombieri-Skoda theorem (\cf [H\"o]~, [Bo] and [Sk]).

\tha Theorem 5.14|Let $u$ be a psh function on a pseudoconvex open set 
$\Omega\subset\cb^n$~. For every point $z_0\in\Omega$ such that $e^{-u}$
is integrable in a neighborhood\break of $z_0$~, there exists a holomorphic
function $F$ on $\Omega$ such that $F(z_0)=1$ and
$$\int_\Omega{|F(z)|^2e^{-u(z)}\over(1+|z|^2)^{n+\varepsilon}}d\lambda(z)
<+\infty~.$$\fintha


\tha Corollary 5.15|Let $u$ be a psh function on a complex manifold
$Y.$ The set of points in a neighborhood of which $e^{-u}$ is not
integrable is an analytic subset of $Y.$\fintha

\remarque Proof|The result is local, so we may assume that $Y$ is a ball
in ${\Bbb C}^n.$ Then the set of non integrability points of $e^{-u}$ is
the intersection of all hypersurfaces $F^{-1}(0)$ defined by the
holomorphic functions $F$ such that
$\int_Y|F|^2e^{-u}d\lambda<+\infty.$
Indeed $F$ must vanish at any non integrability point, and on the other
hand theorem 5.14 shows that one can choose $F(z_0)=1$ at any
integrability point $z_0.$ \finpr

The main idea in what follows is due to Kiselman [Ki2]. For all real
numbers $a,b>0,$ we let $Z_{a,b}$ be the set of points in a neighborhood
of which $\exp(-U_a/b)$ is not integrable. Then $Z_{a,b}$ is analytic,
and as the family $(U_a)$ is increasing in $a$, we have
$Z_{a',b'}\supset Z_{a'',b''}$
if $a'\le a'',$ $b'\le b''.$

Let $y_0\in Y$ be a given point. If $y_0\notin E_c,$ then $\nu
(T,\varphi_{y_0})<c$ by definition of $E_c.$ Choose $a$ such that $\nu
(T,\varphi_{y_0})<a<c.$
Lemma 5.9 (a) implies that $U_a$ is bounded below in a neighborhood of
$y_0$, thus $\exp(-U_a/b)$ is integrable and $y_0\notin Z_{a,b}$ for all
$b>0.$

On the other hand, if $y_0\in E_c$ and if $a<c,$ then lemma 5.12 implies
for all $\varepsilon>0$ that
$$U_a(y)\le(1-\varepsilon)(c-a)\gamma\log|y-y_0|+C(\varepsilon)$$
on a neighborhood of $y_0.$ Hence $\exp(-U_a/b)$ is non integrable at
$y_0$ as soon as $b<(c-a)\gamma/2n',$ where $n'=\dim Y.$ We obtain
therefore
$$E_c=\bigcap_{{\scriptstyle a<c}\atop{\scriptstyle b<(c-a)\gamma/2n'}}
Z_{a,b}.$$
This proves that $E_c$ is an analytic subset of $Y$. \finpr

\tha Exercise 5.16|Combine El Mir's extension theorem and Siu's theorem
in order to get the following result~:
let $P$ be a complete pluripolar set in a complex manifold $X$ and $A$
an analytic subset of $X\setminus P.$ If $A$ has finite mass near every
point of $P,$ then $\overline A$ is analytic in $X.$\fintha

This result has been first obtained by E. Bishop when $P$ is an analytic
subset of $X.$ The general case is due to Siu. 

\pav 6. Monge-Amp\`ere capacities and quasi-continuity|

Let $\Omega$ be a bounded open subset of $\cb^n.$ We denote by
$P(\Omega)$ the set of all psh functions that are $\not\equiv
-\infty$ on each connected component of $\Omega.$ The following
fundamental definition has been introduced in [B--T2].

\tha D\'efinition 6.1|For every Borel subset $E\subset\Omega ,$ we set 
$$c(E,\Omega)=\sup\big\{\int_E(dd^cu)^n~;~
u\in P(\Omega),~0\le u\le1\big\}.$$\fintha

The Chern-Levine-Nirenberg inequalities show that
$c(E,\Omega)<+\infty$ as soon as $E\ssubset\Omega.$ If $\Omega
\subset B(z_0,R),$ we can choose $u(z)=R^{-2}|z-z_0|^2$ and we obtain
therefore
$$c(E,\Omega)\ge{2^nn!\over\pi^nR^{2n}}\,\lambda(E)$$
where $\lambda$ is the Lebesgue measure. From the standard properties
of measures (countable additivity, monotone convergence theorem), we
immediately deduce~:

\tha Properties 6.2|Let $\Omega\ssubset\cb^n$ and
$E,E_1,E_2,\ldots$ Borel subsets of $\Omega.$ Denote $c(E)=c(E,\Omega
)$ for simplicity.

\noindent{\rm(a)}~~ If $E_1\subset E_2$ then $c(E_1)\le c(E_2).$

\noindent{\rm(b)}~~ $c(\bigcup_{j\ge1}E_j)\le\sum_{j\ge1}c(E_j).$

\noindent{\rm(c)}~~ If $E_1\subset E_2\subset\ldots$ then $c(\bigcup
E_j)\le\lim_{j\to+\infty}c(E_j).$ \finpr\fintha

A set function $c:E\mapsto c(E)$ defined on all Borel subsets
$E\subset\Omega$ with values in $[0,+\infty]$ is called a capacity,
resp. a subadditive capacity, if $c$ satisfies the axioms (a,c), resp.
(a,b,c) and $c(\emptyset)=0.$ The capacity is said to be inner regular
if all Borel subsets satisfy
$$c(E)=\sup_{K\,{\rm compact}\,\subset E} c(K).\leqno{\rm(d)}$$
Similarly, $c$ is said to be outer regular if all Borel subsets $E$
satisfy
$$c(E)=\inf_{G~{\rm open}~\supset E} c(G)\leqno{\rm(e)}$$

\remarque Example 6.3|If $(\mu_\alpha)$ is a family of positive Radon
measures on $\Omega ,$ then $c(E)=\sup\mu_\alpha(E)$ is a subadditive
capacity. In general, $c$ does not satisfy the additivity property
$$E_1,E_2~{\rm disjoint}~\Rightarrow
c(E_1\cup E_2)=c(E_1)+c(E_2)~;$$
for a specific example, consider the measures
$\mu_1=\delta_0,$ $\mu_2=d\lambda$ on ${\Bbb R}$ and the sets
$E_1=\{0\},$ $E_2=]0,1]~;$ then
$$c(\{0\})=1~,\quad
c(]0,1])=1~,\quad
c([0,1])=1.$$
Moreover, the capacity $c=\sup\mu_\alpha$ is inner regular because all
Radon measures on a separable locally compact space are inner regular.
However, $c$ need not be outer regular~: for instance, take 
$d\mu_\alpha(x)=\alpha^{-1}\rho(x/\alpha)dx$
on ${\Bbb R},$ $\alpha>0,$ where $\rho\ge0$ is a function with support
in $[-1,1]$ and $\int_{\Bbb R}\rho(x)dx=1~;$ then $c(\{0\})=0$ but every
neighborhood of $0$ has capacity $1.$ \finpr

The capacity $c(\bulle,\Omega)$ defined in 6.1 is called the relative
Monge-Amp\`ere capacity on $\Omega.$ It is associated to the family of
measures $\mu_u=(dd^cu)^n,$ $u\in P(\Omega),$ $0\le u\le 1.$
In particular $c(\bulle,\Omega)$ is inner regular. It is also outer
regular, but this fact is non trivial and will be proved only in \S~9.

When $c$ is a capacity and $E\subset\Omega$ an arbitrary subset, we
define the inner capacity $c_\star(E)$ and outer capacity $c^\star(E)$ by 
$$c_\star(E)=\sup_{K\,{\rm compact}\,\subset E} c(K),\leqno(6.4')$$
$$c^\star(E)=\inf_{G\,{\rm open}\,\supset E} c(G),\leqno(6.4'')$$
A set $E\subset\Omega$ is said to be $c$-capacitable if
$c_\star(E)=c^\star(E).$ By definition, $c$ is thus regular if and only if all
Borel subsets are $c$-capacitable.

Now we compare capacities associated to different open sets $\Omega$.

\tha Theorem 6.5|Let $\Omega_1\subset\Omega_2\ssubset\cb^n.$
Then

\noindent{\rm(a)}~~ $c(E,\Omega_1)\ge c(E,\Omega_2)$ for all Borel subsets
$E\subset\Omega_1.$

\noindent{\rm(b)}~~ Let $\omega\ssubset\Omega_1.$ There exists a constant $A>0$
such that $c(E,\Omega_1)\le A\,c(E,\Omega_2)$ for all Borel subsets
$E\subset\omega.$\fintha

\proof Since every psh function $u\in P(\Omega_2)$ with $0\le u\le 1$
induces a psh function in $P(\Omega_1)$ with the same property, (a) is
clear.

(b) Use a finite covering of $\overline\omega$ by open balls contained
in $\Omega_1$ and cut $E$ into pieces. The proof is then reduced to the
case when $\omega\ssubset\Omega_1$ are concentric balls, say
$\Omega_1=B(0,r)$ and $\omega=B(0,r-\varepsilon).$ For every $u\in
P(\Omega_1)$ such that $0\le u\le 1,$ set
$$\tilde u(z)=\cases{
\max\{u(z),\lambda(|z|^2-r^2)+2\}~ &\quad on $\;\;\Omega_1,$\cr
\lambda(|z|^2-r^2)+2~ &\quad on $\;\;\Omega_2\setminus\Omega
_1.$\cr}$$
Choose $\lambda$ so large that $\lambda((r-\varepsilon)^2-r^2)\le
-2.$ Then $\tilde u\in P(\Omega_2)$ and $\tilde u=u$ on $\omega.$
Moreover $0\le\tilde u\le M$ for some constant $M>0$, thus for
$E\subset\omega$ we get
$$\int_E(dd^cu)^n=\int_E(dd^c\tilde u)^n\le M^nc(E,\Omega_2).$$
Therefore $c(E,\Omega_1)\le M^nc(E,\Omega_2).$ \finpr

As a consequence of theorem 6.5, it is in general harmless to shrink the
domain $\Omega$ when one wants to estimate capacities.

\tha Theorem 6.6|Let $K$ be a compact subset of $\Omega$ and $\omega
\ssubset\Omega$ a neighborhood of $K.$ There is a constant $A>0$
such that for every $v\in P(\Omega)$
$$c(K\cap\{v<-m\},\Omega)\le A\|v\|_{L^1(\overline\omega
)}\cdot{1\over m}.$$\fintha

\proof For every $u\in P(\Omega),~0\le u\le 1,$ theorem 1.8 implies
$$\int_{K\cap\{v<-m\}}(dd^cu)^n\le
{1\over m}\int_K|v|(dd^cu)^n\le
{1\over m}C_{K,\overline\omega}\|v\|_{L^1(\overline\omega)}.
\finpr$$

\tha Definition 6.7|A set $P\subset\Omega$ is said to be pluripolar in
$\Omega$ if there exists $v\in P(\Omega)$ such that
$P\subset\{v=-\infty\}.$\fintha

\tha Corollary 6.8|If $P$ is pluripolar in $\Omega ,$ then
$$c^\star(P,\Omega)=0.$$\fintha

\proof Write $P\subset\{v=-\infty\}$ and
$\Omega=\bigcup_{j\ge1}\Omega_j$ with $\Omega_j\ssubset\Omega.$
Theorem 6.4 shows that there is an open set 
$G_j=\Omega_j\cap\{v<-m_j\}$
such that $c(G_j,\Omega)<\varepsilon 2^{-j}.$ Then
$\{v=-\infty\}\subset G=\bigcup G_j$
and $c(G,\Omega)<\varepsilon.$ \finpr

\tha Theorem 6.9|Let $v_k,~v\in P(\Omega)$ be locally bounded psh
functions such that $(v_k)$ decreases to $v.$ Then for every compact
subset $K\subset\Omega$ and every $\delta>0$
$$\lim_{k\to+\infty}c(K\cap\{v_k>v+\delta\},\Omega)=0.$$\fintha

\proof It is sufficient to show that
$$\sup_{u\in P(\Omega),0\le u\le1}\int_K(v_k-v)(dd^cu)^n$$
tends to 0, because this supremum is larger than
$\delta.c(K\cap\{v_k>v+\delta\},\Omega).$
By cutting $K$ into pieces and modifying $v,v_k,u$ with the max
construction, we may assume that $K\subset\Omega=B(0,r)$ are concentric
balls and that all functions $v,v_k,u$ are equal to $\lambda
(|z|^2-r^2)+2)$ on the corona $\Omega\setminus\omega ,$ $\omega
=B(0,r-\varepsilon).$ An integration by parts yields 
$$\int_\Omega(v_k-v)(dd^cu)^n=
-\int_\Omega d(v_k-v)\wedge d^cu\wedge(dd^cu)^{n-1}.$$
The Cauchy-Schwarz inequality implies that this integral is bounded by
$$A\Big(\int_\Omega d(v_k-v)\wedge d^c(v_k-v)\wedge(dd^cu)^{n-1}\Big)^{1/2}$$
where
$$A^2=\int_\Omega du\wedge d^cu\wedge(dd^cu)^{n-1}\le
\int_\Omega dd^c(u^2)\wedge(dd^cu)^{n-1}$$
and this last integral depends only on the constants $\lambda ,r.$
Another integration by parts yields
$$\eqalign{
\int_\Omega d(v_k-v)\wedge d^c(v_k-v)\wedge(dd^cu)^{n-1}&=
-\int_\Omega(v_k-v)dd^c(v_k-v)\wedge(dd^cu)^{n-1}\cr
&\le\int_\Omega(v_k-v)dd^cv\wedge(dd^cu)^{n-1}.\cr}$$
We have thus replaced one factor $dd^cu$ by $dd^cv$ in the integral.
Repeating the argument $(n-1)$ times we get
$$\int_\Omega(v_k-v)(dd^cu)^n\le
C\big(\int_\Omega(v_k-v)(dd^cv)^n\big)^{1/2^n}$$
and the last integral converges to 0 by the bounded convergence theorem.
\finpr

\tha Theorem 6.10 {\rm(Quasi-consinuity of psh functions)}|Let $\Omega
\ssubset\cb^n$ and $v\in P(\Omega).$ Then for each $\varepsilon
>0,$ there is an open subset $G$ of $\Omega$ such that $c(G,\Omega
)<\varepsilon $ and $v$ is continuous on $\Omega\setminus G.$\fintha

\proof Let $\omega\ssubset\Omega$ be arbitrary. We first show that there
exists $G\subset\omega$ such that $c(G,\Omega)<\varepsilon $ and $v$
continuous on $\omega\setminus G.$ For $m>0$ large enough, the set
$G_0=\omega\cap\{v<-m\}$ has capacity $<\varepsilon/2$ by theorem 6.6.
On $\omega\setminus G_0$ we have $v\ge -m,$ thus $\tilde
v=\max\{v,-m\}$ coincides with $v$ there and $\tilde v$ is locally
bounded on $\Omega.$ Let $(v_k)$ be a sequence of smooth psh functions
which decrease to $\tilde v$ in a neighborhood of $\overline\omega.$
For each $\ell\ge1,$ theorem 6.9 shows that there is an index $k(\ell)$
and an open set
$$G_{k(\ell)}=\omega\cap\{v_{k(\ell)}>\tilde v+1/\ell\}$$
such that $c(G_{k(\ell)},\Omega)<\varepsilon 2^{-\ell-1}.$
Then $G=G_0\cup\bigcup G_{k(\ell)}$ has capacity $c(G,\Omega
)<\varepsilon $ by 6.2 (b) and $(v_{k(\ell)})$ converges uniformly to
$\tilde v=v$ on $\omega\setminus G.$ Hence $v$ is continuous on $\omega
\setminus G.$ Now, take an increasing sequence $\omega_1\subset\omega
_2\subset\ldots$ with $\bigcup\omega_j=\Omega$ and $G_j\subset\omega_j$
such that $c(G_j,\Omega)<\varepsilon 2^{-j}$ and $v$ continuous on
$\omega_j\setminus G_j.$ The set $G=\bigcup G_j$ satisfies all
requirements. \finpr

As an example of application, we prove an interesting inequality for the
Monge-Amp\`ere operator.

\tha Proposition 6.11|Let $u,v$ be locally bounded psh functions on
$\Omega.$ Then we have an inequality of measures
$$(dd^c\max\{u,v\})^n\ge
{\bf 1}_{\{u\ge v\}}(dd^cu)^n+{\bf 1}_{\{u<v\}}(dd^cv)^n.$$\fintha


\proof It is enough to check that
$$\int_K(dd^c\max\{u,v\})^n\ge\int_K(dd^cu)^n$$
for every compact set $K\subset\{u\ge v\}~;$ the other term is then
obtained by reversing the roles of $u$ and $v.$ By shrinking $\Omega ,$
adding and multiplying with constants, we may assume that $0\le u,v\le
1$ and that $u,v$ have regularizations $u_\varepsilon ,v_\varepsilon $
with $0\le u_\varepsilon ,$ $v_\varepsilon\le 1$ on $\Omega.$
Let $G\subset\Omega$ be an open set of small capacity such that $u,v$
are continuous on $\Omega\setminus G.$ By Dini's lemma,
$u_\varepsilon ,v_\varepsilon $ converge uniformly to $u,v$ on $\Omega
\setminus G.$ Hence for any $\delta>0,$ we can find an arbitrarily
small neighborhood $L$ of $K$ such that $u_\varepsilon>v_\varepsilon
-\delta$ on $L\setminus G$ for $\varepsilon $ small enough. As
$(dd^cu_\varepsilon)^n$ converges weakly to $(dd^cu)^n$ on $\Omega ,$
we get
$$\eqalign{
\int_K(dd^cu)^n 
&\le\liminf_{\varepsilon\to 0}\int_L(dd^cu_\varepsilon)^n\cr
&\le\liminf_{\varepsilon\to 0}\big(\int_G(dd^cu_\varepsilon)^n
+\int_{L\setminus G}(dd^cu_\varepsilon)^n\big)\cr
&\le c(G,\Omega)+\liminf_{\varepsilon\to 0}\int_{L\setminus
G}(dd^c\max\{u_\varepsilon+\delta ,v_\varepsilon\})^n.\cr}$$
Observe that $\max\{u_\varepsilon+\delta ,v_\varepsilon\}$ coincides
with $u_\varepsilon+\delta$ on a neighborhood of $L\setminus G.$ By
weak convergence again, we get
$$\int_K(dd^cu)^n\le
c(G,\Omega)+\int_{L\setminus G}(dd^c\max\{u+\delta ,v\})^n.$$
By taking $L$ very close to $K$ and $c(G,\Omega)$ arbitrarily small,
this implies
$$\int_K(dd^cu)^n\le\int_K(dd^c\max\{u+\delta ,v\})^n$$
and the desired conclusion follows by letting $\delta$ tend to 0.~\finpr

\pav 7. Extremal functions and negligible sets|

Let $(u_\alpha)$ be a family of upper semi-continuous functions on
$\Omega$ which is locally bounded from above. Then the upper envelope
$$u=\sup_\alpha u_\alpha(z)$$
need not be upper semi-continuous, so we consider its ``upper
semi-continuous regularization"
$$u^\star(z)=\lim_{\varepsilon\to0}\sup_{B(z,\varepsilon)}u\ge u(z).$$
It is easy to check that $u^\star$ is upper semi-continuous and that $u^\star$
is the smallest upper semi-continuous function $\ge u.$

Let $B(z_j,\varepsilon_j)$ be a denumerable basis of the topology of
$\Omega.$
For each $j,$ let $(z_{jk})$ be a sequence in $B(z_j,\varepsilon_j)$
such that
$$\sup_k u(z_{jk})=\sup_{B(z_j,\varepsilon_j)}u,$$
and for each $(j,k),$ let $\alpha(j,k,\ell)$ be a sequence of indices
$\alpha$ such that $u(z_{jk})=\sup_\ell u_{\alpha(j,k,\ell)}(z_{jk}).$
Set
$$v=\sup_{j,k,\ell}u_{\alpha(j,k,\ell)}.$$
Then $v\le u$ and $v^\star\le u^\star.$ On the other hand
$$\sup_{B(z_j,\varepsilon_j)}v 
\ge\sup_k v(z_{jk})\ge\sup_{k,\ell}u_{\alpha(j,k,\ell)}(z_{jk})
=\sup_k u(z_{jk})=\sup_{B(z_j,\varepsilon_j)}u.$$
As every ball $B(z,\varepsilon)$ is a union of balls $B(z_j,\varepsilon_j),$
we easily conclude that $v^\star\ge u^\star,$ hence $v^\star=u^\star.$ 
Therefore~:

\tha Choquet's lemma 7.1|Every family $(u_\alpha)$ has a denumerable
subfamily  $(u_{\alpha(j)})$ whose upper envelope $v$ satisfies $v\le
u\le u^\star=v^\star.$\fintha

\tha Proposition 7.2|If all $u_\alpha$ are psh, then $u^\star$ is psh and
equal almost everywhere to $u.$\fintha

\proof By Choquet's lemma one may assume that $(u_\alpha)$ is
denumerable. Then $u=\sup u_\alpha$ is a Borel function. For every
$(z_0,a)\in\Omega\times\cb^n,$ $u_\alpha$ satisfies the mean value
inequality on circles, hence
$$u(z_0)=\sup u_\alpha(z_0)\le 
\sup\int^{2\pi}_0u_\alpha(z_0+ae^{i\theta}){d\theta\over 2\pi}
\le\int^{2\pi}_0u(z_0+ae^{i\theta}){d\theta\over 2\pi}.$$
It follows easily that each convolution $u\star\rho_\varepsilon $ also
satisfies the mean value inequality, thus $u\star\rho_\varepsilon $ is
smooth and psh.
Therefore $(u\star\rho_\varepsilon)\star\rho_\eta$ is increasing in $\eta.$
Letting $\varepsilon $ tends to 0, we see that $u\star\rho_\eta$ in
increasing in $\eta.$ Since $u\star\rho_\varepsilon $ is smooth and
$u\star\rho_\varepsilon\ge u$ by the mean value inequality, we also have
$u\star\rho_\varepsilon\ge u^\star.$ By the upper semi-continuity we get
$\lim_{\varepsilon\to 0}u\star\rho_\varepsilon=u^\star,$ in particular
$u^\star $ is psh and coincides almost everywhere with the $L^1_{\rm loc}$
limit $u.$ \finpr

A set of the form
$$N=\{z\in\Omega ~;~u(z)<u^\star(z)\}\leqno(7.3)$$
is called negligible. Every pluripolar set $P=\{v=-\infty\}$ is
negligible~: let $w\in P(\Omega)\cap\ci(\Omega)$ such that $w\ge v$
and $u_\alpha=(1-\alpha)v+\alpha w,$ $\alpha\in ]0,1[.$ Then
$u_\alpha$ is increasing in $\alpha$ and $u=\sup_\alpha u_\alpha$
satisfies
$$\matrix{
&u=-\infty\quad\hfill &{\rm on}\;\;\{v=-\infty\},\hfill\cr
&u=w\quad\hfill &{\rm on}\;\;\{v>-\infty\}.\hfill\cr}$$
Hence $u^\star=w$ and $\{u<u^\star\}=\{v=-\infty\}.$

To study further properties of the capacity, we consider the extremal
function associated to a subset $E$ of $\Omega ~:$
$$u_E(z)=\sup\{v(z);v\in P(\Omega),~v\le-1~{\rm on}~E,~v\le0~{\rm
on}~\Omega\}.\leqno(7.4)$$
Proposition 7.2 implies $u^\star_E\in P(\Omega)$ and $-1\le u^\star_E\le 0.$ In
the sequel, we need the fundamental result of Bedford-Taylor [B--T1] on
the solution of the Dirichlet problem for complex Monge-Amp\`ere
equations.

\tha Theorem 7.5|Let $\Omega\ssubset\cb^n$ be a smooth strongly
pseudoconvex domain and let $f\in C^0(\partial\Omega)$ be a continuous
function on the boundary. Then
$$u(z)=\sup\{v(z)~;~v\in P(\Omega)\cap C^0(\overline\Omega),~v\le
f~{\rm on}~\partial\Omega\}$$
is continuous on $\overline\Omega$ and psh on $\Omega ,$ and solves
the Dirichlet problem
$$(dd^cu)^n=0~{\rm on}~\Omega ~,\quad u=f~{\rm on}~\partial\Omega.$$\fintha

\remarque Sketch of proof| 
Let $g\in C^2(\overline\Omega)$ be an approximate extension of $f$ such that
$|g-f|<\varepsilon $ on $\partial\Omega$ and let $\psi<0$ be a smooth
strongly psh exhaustion of $\Omega.$ Then $g-\varepsilon+A\psi $ is
psh for $A>0$ large enough and $g-\varepsilon+A\psi=g-\varepsilon\le
f$ on $\partial\Omega ,$ hence $g-\varepsilon+A\psi\le u$ on
$\overline\Omega.$  Similarly, for all $v\in P(\Omega)\cap
C^0(\overline\Omega)$ with $v\le f$ on $\partial\Omega ,$ the function
$v-g-\varepsilon+A\psi $ equals $v-g-\varepsilon\le 0$ on $\partial\Omega
$ and is psh for $A$ large, thus $v-g-\varepsilon+A\psi\le 0$ on
$\overline\Omega$ by the maximum principle. Therefore we get $u\le
g+\varepsilon -A\psi ~;$ as $\varepsilon $ tends to 0, we see that $u=f$
on $\partial\Omega$ and that $u$ is continuous at every point of
$\partial\Omega.$ Since $g+\varepsilon+A\psi=g+\varepsilon>f$ on 
$\partial\Omega,$ there exists $\delta>0$ such that $u^\star<g+\varepsilon
+A\psi $ on $\Omega\setminus\Omega_\delta ,$ where 
$\Omega_\delta=\{\psi<-\delta\}.$
For  $\eta>0$ small enough, the regularizations of $u^\star$ satisfy
$u^\star\star\rho_\eta<g+\varepsilon+A\psi $ on a neighborhood of
$\partial\Omega_\delta.$ Then we let
$$v_\varepsilon=\cases{
\max\{u^\star\star\rho_\eta -2\varepsilon ,g-\varepsilon+A\psi\}&\quad on
$\;\;\Omega_\delta$\cr
g-\varepsilon+A\psi  &\quad on $\;\;\overline\Omega\setminus\Omega
_\delta.$\cr}$$
It is clear that $v_\varepsilon $ is psh and continuous on 
$\overline\Omega$ and that $v_\varepsilon=g-\varepsilon\le f$ on
$\partial\Omega,$ hence $v_\varepsilon\le u$ on $\overline\Omega.$ We
obtain therefore $u^\star\star\rho_\eta\le u+2\varepsilon $ on $\Omega_\delta
.$ As $u\le u^\star\le u^\star\star\rho_\eta ,$ we see that $u^\star\star\rho_\eta$
converges uniformly to $u$ on every compact subset of $\Omega.$ Hence
$u$ is psh and continuous on $\overline\Omega.$

We shall complete the proof under the following additional assumptions~:
$f\in C^2(\partial\Omega)$ and $u\in C^2(\Omega)~;$ the general case
is difficult and rather technical (\cf [B--T1]).
The plurisubharmonicity of $u$ implies
$\det(\partial^2u/\partial z_j\partial\overline z_k)\ge0.$
If we had a strict inequality at one point $z_0\in\Omega ,$ say $z_0=0$
for simplicity, the Taylor expansion of $u$ at $z_0$ would give
$$u(z)=\re P(z)+\sum c_{jk}z_j\overline z_k+o(|z|^2)$$
where $P$ is a holomorphic polynomial of degree 2 and $(c_{jk})$ a
positive definite hermitian matrix. Hence we would have $u>\re
P+\varepsilon $ on a small sphere $S(0,r)$ with $\overline
B(0,r)\subset\Omega.$ The function
$$v=\cases{
\max\{u,\re P+\varepsilon\} &\quad on $\;\; B(0,r)$\cr
u &\quad on $\;\;\overline\Omega\setminus B(0,r)$\cr}$$
is then continuous on $\overline\Omega$ and psh, and satisfies
$$v=u\le f~~{\rm on}~\partial\Omega.$$
By the definition of $u,$ we thus have $u\ge v$ on $\overline\Omega.$
This is a contradiction, because
$$v(0)>\re P(0)=u(0). \finpr$$

\tha Corollary 7.6|Fix a ball $\overline B(z_0,r)\subset\Omega$ and let
$f\in P(\Omega)$ be locally bounded. There exists a function $\tilde
f\in P(\Omega)$ such that $\tilde f\ge f$ on $\Omega ,$ $\tilde f=f$ on
$\Omega\setminus B(0,r)$ and $(dd^c\tilde f)^n=0$ on $B(z_0,r).$
Moreover, for $f_1\le f_2$ we have $\tilde f_1\le\tilde f_2.$\fintha

\proof Assume first that $f\in C^0(\Omega).$ By theorem 7.5 applied on
$B(z_0,r)$, there exists $u$ psh and continuous on $\overline B(z_0,r)$
with $u=f$ on $S(z_0,r)$ and $(dd^cu)^n=0$ on $B(z_0,r).$ Set
$$\tilde f=\cases{
u &\quad on $\;\; B(z_0,r)$\cr
f &\quad on $\;\;\Omega\setminus B(z_0,r)$.\cr}$$
By definition of $u,$ we have $\tilde f=u\ge f$ on $B(z_0,r).$ Moreover,
$\tilde f$ is the decreasing limit of the psh functions 
$$g_k=\cases{
\max\{u,f+{1\over k}\} &\quad on $\;\; B(z_0,r)$\cr
f+{1\over k} &\quad near $\;\;\Omega\setminus B(z_0,r)$\cr}$$
hence $\tilde f$ is psh. Also clearly, for $f_1\le f_2$ we have $u_1\le
u_2,$ hence $\tilde f_1\le\tilde f_2.$ For an arbitrary locally bounded
function $f\in P(\Omega),$ write $f$ as a decreasing limit of smooth
psh functions $f_k=f\star\rho_{1/k}$ and set $\tilde
f=\lim_{k\to+\infty}\downarrow\tilde f_k.$ Then $\tilde f$ has all
required properties. \finpr

Now we prove the following three fundamental results by a simultaneous
induction on $n.$

\tha Proposition 7.7|Let $u,u_j\in P(\Omega)$ be locally bounded
functions such that $u_j$ increases to $u$ almost everywhere. Then the
measure $(dd^cu_j)^n$ converges weakly to $(dd^cu)^n$ on $\Omega
.$\fintha

\tha Proposition 7.8|Let $\Omega$ be a strongly pseudoconvex smooth
open subset of $\cb^n.$ If $K\subset\Omega$ is compact, then 

\noindent{\rm(a)}~~ $(dd^cu^\star_K)^n=0$ on $\Omega\setminus K.$

\noindent{\rm(b)}~~ $c(K,\Omega)=\int_K(dd^cu^\star_K)^n=\int_\Omega 
(dd^cu^\star_K)^n.$\fintha

\tha Proposition 7.9|If a Borel set $N\subset\Omega$ is negligible,
then
$$c(N,\Omega)=0.$$\fintha

The inductive proof is made in three steps.

\noindent {\bf Step 1}~: 7.7 in $\cb^n\Rightarrow$ 7.8 in $\cb^n.$\hfill\break
{\bf Step 2}~: 7.8 in $\cb^n\Rightarrow$ 7.9 in $\cb^n.$\hfill\break
{\bf Step 3}~: 7.7 and 7.9 in $\cb^n\Rightarrow$ 7.7 in $\cb^{n+1}.$

\noindent In the case $n=1,$ proposition 7.7 is a well-known fact of
distribution theory~: $u_j$ converges to $u$ in $L^1_{\rm loc}(\Omega
),$ thus $dd^cu_j$ converges weakly to $dd^cu.$ By the inductive
argument, propositions 7.7, 7.8, 7.9 hold in all dimensions.

\remarque Proof of step 1|By Choquet's lemma, there is a sequence of
functions $v_j\in P(\Omega)$ such that $v_j\le0$ on $\Omega ,$ $v_j\le
-1$ on $K$ and $v^\star=u^\star_K.$ If we replace $v_j$ by
$\max\{-1,v_1,\ldots,v_j\},$ we see that we may assume $v_j\ge-1$ for
all $j$ and $v_j$ increasing. Then fix an arbitrary ball
$B(z_0,r)\subset\Omega\setminus K$ and consider the increasing sequence
$\tilde v_j$ given by corollary 7.6. We still have $\tilde v_j\le 0$ on
$\Omega$ and $\tilde v_j\le -1$ on $K,$ thus $v_j\le\tilde v_j\le u_K$
and $\tilde v=\lim\tilde v_j$ satisfies $v^\star=\tilde v^\star=u^\star_K,$ in
particular $\lim\tilde v_j=\lim v_j=u^\star_K$ almost everywhere. Since
$(dd^c\tilde v_j)^n=0$ on $B(z_0,r),$ we conclude by 7.7 that
$(dd^cu^\star_K)^n=0$ on $B(z_0,r)$ and 7.8 (a) is proved.

To prove 7.8 (b), observe first that $-1\le u^\star_K\le 0$ on $\Omega ,$
hence $c(K,\Omega)\ge\int_K(dd^cu^\star_K)^n$ by definition of the
capacity. If $\psi<0$ is a smooth strictly psh exhaustion function of
$\Omega ,$ we have $A\psi\le -1$ on $K$ for $A$ large enough.
We can clearly assume $v_j\ge A\psi $ on $\Omega ~;$ otherwise replace
$v_j$ by $\max\{v_j,A\psi\}.$ Now, let $w\in P(\Omega)$ be such that
$0\le w\le 1$ and set
$$w'=(1-\varepsilon)w-1+\varepsilon/2~,~~~~
w_j=\max\{w',v_j\}.$$
Since $-1+\varepsilon/2\le w'\le -\varepsilon/2$
on $\Omega ,$ we have
$w_j=v_j$ as soon as $A\psi>-\varepsilon/2,$ whereas
$w_j=w'\ge -1+\varepsilon/2>v_j$ on a
neighborhood of $K.$ Hence for $\delta>0$ small enough Stokes' theorem
implies
$$\int_{\Omega_\delta}(dd^cv_j)^n=\int_{\Omega_\delta}(dd^cw_j)^n
\ge\int_K(dd^cw_j)^n=(1-\varepsilon)^n\int_K(dd^cw)^n.$$
By 7.7 $(dd^cv_j)^n$ converges weakly to $(dd^cu^\star_K)^n$ and we get 
$$\limsup_{j\to+\infty}\int_{\Omega_\delta}(dd^cv_j)^n\le
\int_{\overline\Omega_\delta}
(dd^cu^\star_K)^n=\int_K(dd^cu^\star_K)^n.$$
Therefore $\int_K(dd^cw)^n\le\int_K(dd^cu^\star_K)^n$ and $c(K,\Omega
)\le\int_K(dd^cu^\star_K)^n.$ \finpr

\remarque Proof of step 2|Let $N=\{v<v^\star\}$ with $v=\sup v_\alpha.$ By
Choquet's lemma, we may assume that $v_\alpha$ is an increasing
sequence of psh functions. The theorem of quasi-continuity shows that
there exists an open set $G\subset\Omega$ such that all functions
$v_\alpha$ and $v^\star$ are continuous on $\Omega\setminus G$ and
$c(G,\Omega)<\varepsilon.$ Write
$$N\subset G\cup(N\cap(\Omega\setminus G))=G\cup\bigcup_{\delta
,\lambda ,\mu\in{\Bbb Q}}K_{\delta\lambda_\mu}$$
where $\delta>0,$ $~\lambda<\mu $ and
$$K_{\delta\lambda\mu}=\big\{z\in\overline\Omega_\delta\setminus
G~;~ v(z)\le\lambda<\mu\le v^\star(z)\big\}.$$
As $v^\star$ is continuous and $v$ lower semi-continuous on $\Omega
\setminus G,$ we see that $K_{\delta\lambda\mu}$ is compact. We only
have to prove that $c(K_{\delta\lambda\mu},\Omega)=0.$ Set
$K=K_{\delta\lambda\mu}$ for simplicity and take an open set $\omega
\ssubset\Omega.$ By subtracting a large constant, we may assume
$v^\star\le 0$ on $\overline\omega.$

Multiplying by another constant, we may set $\lambda=-1.$ Then all
$v_\alpha$ satisfy  $v_\alpha\le 0$ on $\omega$ and $v_\alpha\le v\le
-1$ on $K.$
We infer that the extremal function $u_K$ on $\omega$ satisfies $u_K\ge
v,$ $u^\star_K\ge v^\star$, in particular $u^\star_K\ge\mu>-1$ on $K.$ By
proposition 6.11 we obtain
$$c(K,\omega)=\int_K(dd^cu^\star_K)^n\le
\int_K(dd^c\max\{u^\star_K,\mu\})^n\le
|\mu |^nc(K,\omega)$$
because $-1\le|\mu |^{-1}\max\{u^\star_K,\mu\}\le0.$
As $|\mu |<1,$ we conclude that $c(K,\omega)=0,$ hence $c(K,\Omega
)=0.$ \finpr

\remarque Proof of step 3|We have to show that if $\Omega\subset\cb^{n+1},$ 
$$\lim_{j\to+\infty}\int_\Omega\chi(dd^cu_j)^{n+1}=\int_\Omega\chi
(dd^cu)^{n+1}$$
for all test functions $\chi\in C^\infty_0(\Omega).$ That is,
$$\lim_{j\to+\infty}\int_\Omega u_j(dd^cu_j)^{n+1}\wedge\gamma
=\int_\Omega u(dd^cu)^n\wedge\gamma\leqno(7.10)$$
with $\gamma=dd^c\chi.$ As all $(1,1)$-forms $\gamma$ can be written
as linear combinations of forms of the type $i\alpha\wedge\overline
\alpha ,$ $\alpha\in\Lambda^{1,0}(\cb^n)^\star,$ it is sufficient,
after a change of coordinates, to consider forms of the type 
$\gamma={i\over 2}\chi(z)dz_{n+1}\wedge d\overline z_{n+1},$
$\chi\in C^\infty_0(\Omega).$
In this case, for any locally bounded psh function $v$ on $\Omega ,$ the
Fubini theorem yields
$$\int_\Omega v(dd^cv)^n\wedge\gamma=
\int_\cb d\lambda(z_{n+1})\int_{\Omega(z_{n+1})}
\chi(\bulle,z_{n+1})(dd^cv(\bulle,z_{n+1}))^n$$
where $\Omega(z_{n+1})=\{z\in\cb^n;(z,z_{n+1})\in\Omega\}$
and $f(\bulle,z_{n+1})$ denotes the function $z\mapsto f(z,z_{n+1})$ on
$\Omega(z_{n+1}).$
Indeed, the result is clearly true if $v$ is smooth. The general case
follows by taking smooth psh functions $v_j$ decreasing to $v.$ The
convergence of both terms in the equality is guaranteed by theorem 1.5
(a), combined with 1.3 and the bounded convergence theorem for the right
hand side.

In order to prove (7.10), we thus have to show
$$\lim_{j\to+\infty}\int_\omega\chi u_j(dd^cu_j)^n=\int_\omega\chi 
u(dd^cu)^n\leqno(7.11)$$
for $\omega\subset\cb^n,$ $\chi\in C^\infty_0(\omega)$ and 
$u_j\in P(\omega)\cap L^\infty_{\rm loc}(\omega)$ increasing
to $u\in P(\omega)$ almost everywhere. To prove (7.11), we can clearly
assume $0\le\chi\le 1$ and $0\le u_j\le u\le 1$ on $\Omega.$ By our
inductive hypothesis 7.7, $(dd^cu_j)^n$ converges weakly to $(dd^cu)^n.$
As $u_j\le u\le u_\varepsilon=u\star\rho_\varepsilon ,$ we get
$$\eqalign{
\limsup_{j\to+\infty}\int_\omega\chi u_j(dd^cu_j)^n 
&\le\lim_{\varepsilon\to 0}\lim_{j\to+\infty}\int_\omega\chi 
u_\varepsilon(dd^cu_j)^n\cr
&=\lim_{\varepsilon\to0}\int_\omega\chi u_\varepsilon
(dd^cu)^n=\int_\omega\chi  u(dd^cu)^n.\cr}$$
To prove the other inequality, let $\varepsilon>0$ and choose an open
set $G\subset\omega$ such that $c(G,\omega)<\varepsilon $ and $u,u_j$
are all continuous on $\omega\setminus G.$ Let $v=\sup u_j.$ Then
$v^\star=u$ because $v^\star$ and $u$ are psh and coincide almost everywhere.
Let $\tilde u_j$ be a continuous extension of $u_j|_{\omega\setminus
G}$ to $\omega$ such that $0\le\tilde u_j\le 1.$ For $j\ge k$ we have
$u_j\ge u_k,$ hence
$$\eqalign{
\int_\omega\chi u_j(dd^cu_j)^n 
&\ge\int_{\omega\setminus G}\chi\tilde u_k(dd^cu_j)^n\cr
&\ge\int_\omega\chi\tilde u_k(dd^cu_j)^n-\int_G(dd^cu_j)^n.\cr}$$
The last integral on the right is
$\le c(G,\omega)<\varepsilon.$ Taking the limit as $j$ tends to
$+\infty,$ we obtain
$$\eqalign{
\liminf_{j\to+\infty}\int_\omega\chi  u_j(dd^cu_j)^n
&\ge\int_\omega\chi\tilde u_k(dd^cu)^n-\varepsilon\cr
&\ge\int_\omega\chi u_k(dd^cu)^n-2\varepsilon.\cr}$$
The second term $\varepsilon $ comes from
$\int_G(dd^cu)^n\le c(G,\omega)<\varepsilon.$
Now let $k\to+\infty$ and $\varepsilon\to 0$ to get
$$\liminf_{j\to+\infty}\int_\omega\chi  u_j(dd^cu_j)^n\ge
\int_\omega\chi  v(dd^cu)^n.$$
Moreover, the Borel set $N=\{v<u=v^\star\}$ is negligible and the inductive
hypothesis 7.9 implies $c(N,\omega)=0.$ Therefore 
$$\int_\omega\chi(u-v)(dd^cu)^n\le\int_N(dd^cu)^n=0$$
and the proof is complete. \finpr

\tha Theorem 7.12|For each $j=1\ld q,$ let $u^k_j$ be an increasing
sequence of locally bounded psh functions such that $u^k_j$ converges
almost everywhere to $u_j\in P(\Omega).$ Then 

\noindent{\rm(a)}~~ $dd^cu^k_1\wedge\ldots\wedge dd^cu^k_q\to
dd^cu_1\wedge\ldots\wedge dd^cu_q$
weakly.

\noindent{\rm(b)}~~ $u^k_1dd^cu^k_2\wedge\ldots\wedge dd^cu^k_q\to
u_1dd^cu_2\wedge\ldots\wedge dd^cu_q$ weakly.\fintha

\remarque Proof|
(a) Without loss of generality, we may assume $q=n,$ otherwise we
complete with additional stationary sequences
$u^k_{q+1}=u_{q+1}\ld u^k_n=u_n$
where
$u_{q+1}\ld u_n$ are chosen arbitrarily in
$P(\Omega)\cap C^\infty(\Omega)$. Now apply proposition 7.7 to 
$u_k=\lambda_1u^k_1+\cdots+\lambda_nu^k_n,$
$\lambda_j>0,$ and consider the coefficient of 
$\lambda_1\ldots\lambda_n$ in $(dd^cu_k)^n.$

(b) Same proof as for (7.11). \finpr

\tha Exercise 7.13|Use formula 7.8 (b) in order to compute the capacity
of $K=\overline B(0,r)$ in a ball $\Omega=B(0,R)~;$ show that the
extremal function $u_K$ is 
$$u_K(z)=\Big(\log{R\over r}\Big)^{-1}\max
\Big\{\log{|z|\over R},\log{r\over R}\Big\}.$$\fintha

\pav 8. Characterization of pluripolar sets|

First we quote a few elementary properties of the extremal functions
$u^\star_E$.

\tha Properties 8.1|

\noindent{\rm(a)}~~ 
if $E_1\subset E_2\subset\Omega ,$ then $u^\star_{E_1}\ge u^\star_{E_2}.$

\noindent{\rm(b)}~~ 
if $E\subset\Omega_1\subset\Omega_2,$ then $u^\star_{E,\Omega_1}\ge
u^\star_{E,\Omega_2}.$

\noindent{\rm(c)}~~
 if $E\subset\Omega ,$ then $u^\star_E=u_E=-1$ on $E^0$ and
$(dd^cu^\star_E)^n=0$ on $\Omega\setminus\overline E~;$
hence $(dd^cu^\star_E)^n$ is supported by $\partial E.$

\noindent{\rm(d)}~~
$u^\star_E=0$ if and only if there exists $v\in P(\Omega),$ $v\le 0$
such that $E\subset\{v=-\infty\}.$

\noindent{\rm(e)}~~
if $E\ssubset\Omega$ and if $\Omega$ is strongly pseudoconvex
with exhaustion $\psi<0,$ then $u^\star_E\ge A\psi $ for some $A>0.$\fintha

\proof 
(a), (b) are obvious from definition (7.4)~; (e) is true as soon as
$A\psi\le-1$ on $E~;$ the equality 
$(dd^cu^\star_E)^n=0$ on $\Omega\setminus\overline E$ in (c) is proved
exactly in the same way as 7.8 (a) in step 1.

(d) If $E\subset\{v=-\infty\},$ $v\in P(\Omega),$ $v\le 0,$ then for
every $\varepsilon>0$ we have $\varepsilon v\le u_E,$ hence $u_E=0$ on
$\Omega\setminus\{v=-\infty\}$ and $u^\star_E=0.$

Conversely, Choquet's lemma shows that there is an increasing sequence
$v_j\in P(\Omega),$ $-1\le v_j\le u_E,$ converging almost everywhere to
$u^\star_E.$ If $u^\star_E=0,$ we can extract a subsequence in such a way that
$\int_\Omega |v_j|d\lambda<2^{-j}.$
As $v_j\le 0$ and $v_j\le-1$ on $E,$ the function $v=\sum v_j$ is psh
$\le0$ and $v=-\infty$ on $E.$ \finpr

If $G\ssubset\Omega$ is an open subset, $K_1\subset K_2\subset\ldots$
compact subsets of $G$ with $K_j\subset K^0_{j+1}$ and $\bigcup K_j=G,$
then $u^\star_{K_j}=-1$ on $K^0_j\supset K_{j-1},$  $\lim\downarrow
u^\star_{K_j}=-1$ on $G.$
Therefore $u^\star_G\le\lim u^\star_{K_j}\le u_G\le u^\star_G$ and theorems 1.5,
7.8, 8.1 (c) show that
$$c(G,\Omega)=\int_{\overline G}(dd^cu^\star_G)^n=\int_\Omega
(dd^cu^\star_G)^n.\leqno(8.2)$$

\tha Proposition 8.3|Let $\Omega\ssubset\cb^n$ and $K_1\supset
K_2\supset\ldots,$ $K=\bigcap K_j$ compact subsets of $\Omega.$ Then

\noindent{\rm(a)}~~ $\big(\buildu{j\to+\infty}\under\lim 
\uparrow u^\star_{K_j}\big)^\star=u^\star_K.$

\noindent{\rm(b)}~~ $\lim c(K_j,\Omega)=c(K,\Omega).$

\noindent{\rm(c)}~~ $c^\star(K,\Omega)=c(K,\Omega).$\fintha

\proof We have $\lim\uparrow u^\star_{K_j}\le u^\star_K$ by 8.1 (a). On the
other hand, let $v\in P(\Omega)$ be such that $v\le 0$ on $\Omega$ and
$v\le -1$ on $K.$ For every $\varepsilon>0$ the open set
$\{v<-1+\varepsilon\}$ is a neighborhood of $K,$ thus
$K\subset\{v<-1+\varepsilon\}$ for $j$ large. We obtain therefore
$v-\varepsilon\le u^\star_{K_j}$ and $u_K=\sup\{v\}\le\lim u^\star_{K_j},$
whence equality (a). Property (b) follows now from theorems 7.7 and 7.8
(b), and (c) is a consequence of (b) when $K_j$ are neighborhoods of
$K.$ \finpr

\tha Lemma 8.4|Let $\Omega\ssubset\cb^n$ and $u,v\in P(\Omega)$
locally bounded psh functions such that $u\le v\le 0$ and 
$\buildu z\to\partial\Omega\under\lim u(z)=0.$
Then
$$\int_\Omega(dd^cv)^n\le\int_\Omega(dd^cu)^n.$$
Moreover $\int_\Omega(dd^cu)^n=0$ if and only if $u=0.$\fintha

\proof As $\max\{u+\varepsilon ,v\}=u+\varepsilon $ near $\partial\Omega
,$ we get 
$$\int_\Omega(dd^cu)^n=\int_\Omega(dd^c\max\{u+\varepsilon ,v\})^n.$$
Let $\varepsilon $ tend to 0, and observe that the integrand on the
right hand side converges weakly to $(dd^cv)^n$ by theorem 1.5.
The asserted inequality follows.

Now, assume that $u(z_0)<0$ at some point. Then
$$v(z)=\max\{u(z),\varepsilon^2|z|^2-\varepsilon\}$$
coincides with $u$ near $\partial\Omega$ and with $\varepsilon
^2|z|^2-\varepsilon $ on a neighborhood $\omega$ of $z_0.$ We get
therefore
$$\int_\Omega(dd^cu)^n=\int_\Omega(dd^cv)^n\ge
\int_\omega(dd^cv)^n>0. \finpr$$

\tha Proposition 8.5|Let $\Omega\ssubset\cb^n$ be strongly
pseudoconvex. If $E\ssubset\Omega$ is an arbitrary subset, then
$$c^\star(E,\Omega)=\int_\Omega(dd^cu^\star_E)^n.$$\fintha

\proof Let $\psi<0$ be a strictly psh exhaustion function on $\Omega.$
For every open set $G\supset E,$
we have $u^\star_E\ge u^\star_G\ge A\psi $ by
8.1, and lemma 8.4 implies
$$\int_\Omega(dd^cu^\star_E)^n\le\int_\Omega(dd^cu^\star_G)^n=c(G,\Omega),$$
thus $\int_\Omega(dd^cu^\star_E)^n\le c^\star(E,\Omega).$

Conversely, Choquet's lemma shows that there exists an increasing
sequence $v_j\in P(\Omega)$ with $-1\le v_j\le0,$ $v_j\ge A\psi $ on
$\Omega$ and $\lim v_j=u_E$ almost everywhere. If
$$G_j=\{z\in\Omega ;(1+1/j)v_j(z)<-1\}$$
then $G_j\supset E,$ $G_j$ is decreasing and $(1+1/j)v_j\le u_{G_j}.$
Thus $\lim\uparrow u^\star_{G_j}=u^\star_E$ almost everywhere and theorem 7.7
gives 
$$\lim_{j\to+\infty}\int_\Omega(dd^cu^\star_{G_j})^n=\int_\Omega
(dd^cu^\star_E)^n.\finpr$$

\tha Corollary 8.6|Let $\Omega\ssubset\cb^n$ be strongly
pseudoconvex. If $E\ssubset\Omega ,$ then $c^\star(E,\Omega)=0$ if and only
if $u^\star_E=0.$ \finpr\fintha

Now, we prove an important result due to Josefson [Jo]. A set $P$ in
$\cb^n$ is said to be {\sl locally pluripolar} if for each $z\in P$
there is an open neighborhood $\Omega$ of $z$ and $v\in P(\Omega)$
such that $P\cap\Omega\subset\{v=-\infty\}.$

\tha Theorem 8.7 {\rm(Josefson)}|If $P\subset\cb^n$ is locally
pluripolar, there exists $v\in P(\cb^n)$ with
$P\subset\{v=-\infty\},$ \ie $P$ is globally pluripolar in $\cb^n.$\fintha

\proof By the definition of locally pluripolar, one can find sets
$P_j,\Omega_j$ with $P_j\ssubset\Omega_j\ssubset\cb^n,$
$\bigcup_{j\ge1}P_j=P,$ each $\Omega_j$ strongly pseudoconvex. By 8.1 (d)
and 8.5, we have $c^\star(P_j,\Omega_j)=0.$

Let $B_k$ be the ball of center 0 and radius $k$ in $\cb^n$ and
$j(k)$ a sequence of integers such that each integer is repeated
infinitely many times and $\Omega_{j(k)}\subset B_k.$
By the comparison theorem 6.5 we have
$c^\star(P_{j(k)},B_{k+2})=0,$
hence the extremal function
$u^\star_{P_{j(k)}}$ in $B_{k+2}$ is zero and we can find $v_k\in
P(B_{k+2})$ with $-1\le v_k\le 0,$ $v_k=-1$ on $P_{j(k)}$ and
$\int_{B_k}|v_k|d\lambda<2^{-k}.$
Now set
$$\tilde v_k(z)=\cases{
v_k(z) &\quad on $\;\;B_k$\cr
\max\{v_k(z),|z|^2-(k+1)^2\} &\quad on $\;\; B_{k+2}\setminus B_k$\cr
|z|^2-(k+1)^2 &\quad on $\;\;\cb^n\setminus B_{k+2}.$\cr}$$
As $\tilde v_k\le0$ on $B_k$ and $\int_{B_k}|\tilde v_k|d\lambda
<2^{-k},$
the series $v=\sum\tilde v_k$ defines a global psh function on $\cb^n.$
Moreover $\tilde v_k=-1$ on $P_{j(k)}$ and each $P_j$ is repeated
infinitely many times, therefore
$$v=-\infty\quad{\rm on}\;\;\bigcup P_j=P.\finpr$$

\tha Corollary 8.8|Let $\Omega\subset\cb^n$ and $P\subset\Omega.$
Then $P$ is pluripolar in $\Omega$ if and only if $c^\star(P,\Omega
)=0.$\fintha

\proof That $P$ is pluripolar implies $c^\star(P,\Omega)=0$ was proved in
corollary 6.8. Conversely, if $c^\star(P,\Omega)=0$ then $c^\star(P\cap\omega
',\omega)=0$ for all concentric balls $\omega '\ssubset\omega
\ssubset\Omega$ and corollary 8.6 combined with 8.1 (d) shows that
$P\cap\omega'$ is pluripolar in $\omega.$ Josefson's theorem implies
that $P$ is globally pluripolar in $\cb^n.$ \finpr

\tha Corollary 8.9|Negligible sets are pluripolar.\fintha

\proof By Choquet's lemma every negligible set is contained in a Borel
negligible set $N=\{v<v^\star\}$ with $v=\sup v_j.$ However, in step 2 of
\S~7, we showed that $N\subset G\cup\bigcup K_{\delta\lambda\mu}$
with $G$ open, $c(G,\Omega)<\varepsilon $
and $c(K_{\delta\lambda\mu},\Omega)=0.$ By 8.3 (c), we have
$c^\star(N,\Omega)<\varepsilon $ for all $\varepsilon>0.$ Therefore
$c^\star(N,\Omega)=0$ and $N$ is pluripolar. \finpr

\pav 9. Capacitability and outer regularity|

First we introduce some definitions and prove a general capacitability
result due to G. Choquet. All topological spaces occurring here are
assumed to be Hausdorff.

\tha Definition 9.1|Let $X$ be a topological space.

\noindent $\bullet$ A $F_\sigma$ subset of $X$ is a countable union of
closed subsets of $X~;$

\noindent $\bullet$ A $F_{\sigma\delta}$ subset of $X$ is a 
countable intersection of $F_\sigma$ subsets of $X.$ 

\noindent $\bullet$ A space $X$ is said to be a $K_\sigma$ 
$(\resp K_{\sigma\delta})$ space if it is homeomorphic to some 
$F_\sigma$ $(\resp F_{\sigma\delta})$ subset of a compact space
$W.$\fintha

\tha Properties 9.2|

\noindent{\rm(a)}~~ 
Every closed subset $F$ of a $K_{\sigma\delta}$ space $X$ is a
$K_{\sigma\delta}$ space.

\noindent{\rm(b)}~~ 
Every countable disjoint sum $\coprod X_j$ of $K_{\sigma\delta}$ spaces is
a $K_{\sigma\delta}$ space.

\noindent{\rm(c)}~~ 
Every countable product $\prod X_j$ of $K_{\sigma\delta}$ spaces is a
$K_{\sigma\delta}$ space.\fintha

\proof (a) Write $X=\bigcap_{\ell\ge1}G_\ell$ and
$G_\ell=\bigcup_{m\ge1}K_{\ell m}$ where $K_{\ell m}$ are closed subsets of
a compact space $W$. If $\ov F$ is the closure of $F$ in $W$, we have
$$F=X\cap\overline F=\bigcap_{\ell\ge1}G_\ell\cap\overline F,\qquad
G_\ell\cap\overline F=\bigcup_{m\ge1}K_{\ell m}\cap\overline F.$$

(b) Let $X_j$, $j\ge 1$, be $K_{\sigma\delta}$ spaces and write for each $j$
$$X_j=\bigcap_{\ell\ge1}G^j_\ell,\qquad
G^j_\ell=\bigcup_{m\ge1}K^j_{\ell m}$$
where $K^j_{\ell m}$ is a closed subspace of a compact space $W_j.$
Then $\coprod W_j$ can be embedded in the compact space 
$W=\prod(W_j\amalg\{\star\})$
via the obvious map that sends $w\in W_j$ to $(\star\ld\star,w,\star,\ldots)$ where
$w$ is in the $j$-th position.
Now $X=\coprod X_j$ can be written
$$X=\bigcap_{\ell\ge1}G_\ell,\qquad
G_\ell=\bigcup_{m\ge1}\coprod_{j\ge1}K^j_{\ell m}.$$
As $K^j_{\ell m}$ is sent onto a closed set by the embedding $\coprod W_j
\la W$, we conclude that $X$ is a $K_{\sigma\delta}$ space.

(c) With the notations of (b), write $X=\prod X_j$ as
$$\eqalign{
X=\bigcap_{\ell\ge1}G_\ell,\qquad
G_\ell=G^1_\ell\times G^2_{\ell-1}\times\ldots\times G^\ell_1&{}\times
W_{\ell+1}\times\ldots\times W_j\times\ldots,\cr
G_\ell=\bigcup_{m_1\ld m_\ell\ge1}K^1_{\ell\,m_1}\times K^2_{\ell-1\,m_2}
\times\ldots\times K^\ell_{1\,m_\ell}&{}\times
W_{\ell+1}\times\ldots\times  W_j\times\ldots\cr}$$
where each term in the union is closed in
$W=\prod W_j.$ \finpr

\tha Definition 9.3|A space $E$ is said to be $K$-analytic if $E$ is a
continuous image of a $K_{\sigma\delta}$ space $X.$\fintha

\tha Theorem 9.4|Let $\Omega$ be a topological space and
$E_1\subset E_2\subset\ldots$ $K$-analytic subsets of $\Omega.$ Then
$\bigcup E_j$ and $\bigcap E_j$ are $K$-analytic.\fintha

\proof Let $f_j:X_j\to E_j$ be a continuous map from a 
$K_{\sigma\delta}$ space onto $E_j.$ Set $X=\coprod X_j$ and
$f=\coprod f_j:X\to\Omega.$
Then $X$ is a $K_{\sigma\delta}$ space, $f$ is continuous and
$f(X)=\bigcup E_j$. Now set
$$X=\{x=(x_1,x_2,\ldots)\in\prod X_j~;~
f_1(x_1)=f_2(x_2)=\cdots\},$$
$$f:X\to\Omega ~,\qquad f(x)=f_1(x_1)=f_2(x_2)=\cdots~.$$
Then $X$ is closed in $\prod X_j,$ so $X$ is a $K_{\sigma\delta}$
space by 9.2 (a,c) and $f(X)=\bigcap E_j.$ \finpr

\tha Corollary 9.5|Let $\Omega$ be a separable locally compact space.
Then all Borel subsets of $\Omega$ are $K$-analytic.\fintha

\proof Any open or closed open set in $\Omega$ is a countable union of
compact subsets, hence $K$-analytic. On the other hand, theorem 9.4 shows that
$${\cal A}=\{E\subset\Omega ~;~E~{\rm and}~\Omega\setminus E~{\rm
are}~K-{\rm analytic}\}$$
is a $\sigma$-algebra. Since ${\cal A}$ contains all open sets in $E,$
${\cal A}$ must also contain all Borel subsets. \finpr

Before going further, we need a simple lemma.

\tha Lemma 9.6|Let $E$ be a relatively compact $K$-analytic subset of a
topological space $\Omega.$ There exists a compact space $T,$ a
continuous map $g:T\to\Omega$ and a $F_{\sigma\delta}$ subset
$Y\subset T$ such that $g(Y)=E.$\fintha

\proof There is a compact space $W,$ a $F_{\sigma\delta}$ subset
$X\subset W$ and a continuous map $f:X\to E$ onto $E.$ Let
$$Y=\{(x,f(x))\,;\,x\in X\}\subset X\times E$$
be the graph of $f$ and $T=\overline Y$ the closure of $Y$ in the
compact space $\overline X\times\overline E.$ As $f$ is continuous, $Y$
is closed in $X\times\overline E,$ thus $Y=T\cap(X\times\overline E).$
Now, $X$ is a $F_{\sigma\delta}$ subset of $\overline X$, so $X\times\ov E$
is a $F_{\sigma\delta}$ subset of $\ov X\times\ov E$ and
$Y$ is a $F_{\sigma\delta}$ subset of $T$.
Finally $E$ is the image of $Y$ by the second projection
$g:T\to\overline E.$ \finpr

\tha Definition 9.7|Let $\Omega$ be a topological space. A generalized
capacity is a set function $\overline c$ defined on all subsets
$E\subset\Omega$ satisfying the following axioms~:

\noindent{\rm(a)}~~ If $E_1\subset E_2\subset\Omega ,$ then 
$\overline c(E_1)\le\overline c(E_2).$

\noindent{\rm(b)}~~ If $E_1\subset E_2\subset\ldots\subset\Omega ,$ then
$\overline c(\bigcup E_j)=\lim_{j\to+\infty}\overline c(E_j).$

\noindent{\rm(c)}
If $K_1\supset K_2\supset\ldots$ are compact subsets of $\Omega ,$
then
$\overline c(\bigcap K_j)=\lim_{j\to+\infty}\overline c(K_j).$
\fintha

\tha Choquet's capacitability theorem 9.8|Let $\Omega$ be a $K_\sigma$
space and let $\overline c$ be a generalized capacity on $\Omega$. Then
every $K$-analytic subset $E\subset\Omega$ satisfies
$$\overline c(E)=\sup_{K\,{\rm compact}\,\subset E}
\overline c(K).$$\fintha

\proof As $\Omega$ is an increasing union of compact sets $L_j,$ we
have $\overline c(E)=\lim_{j\to+\infty}\overline c(E\cap L_j)$ by axiom
(b)~; we may therefore assume that $E$ is relatively compact in $\Omega
.$ Then lemma 9.6 shows that there is a $F_{\sigma\delta}$ subset $Y$
in a compact space $T$ and a continuous map $g:T\to\Omega$ such that
$g(Y)=E.$ It is immediate to check that $\overline c\circ g$ is a
generalized capacity on $T.$ Hence we are reduced to proving the theorem
when $\Omega$ is a compact space and $E$ is a $F_{\sigma\delta}$ subset
of $\Omega.$ Then write
$$E=\bigcap_{\ell\ge1}G_\ell,\qquad
G_\ell=\bigcup_{m\ge1}K_{\ell m}$$
where $K_{\ell m}$ is a closed subset of $\Omega.$ Without loss of
generality, we can arrange that $K_{\ell m}$ is increasing in $m.$ Fix
$\lambda<\overline c(E).$ Then
$$E=G_1\cap\bigcap_{\ell\ge2}G_\ell=
\bigcup_{m\ge1}(K_{1m}\cap\bigcap_{\ell\ge2}G_\ell)$$
and axiom (b) implies that exists a subset
$E_1=K_{1m_1}\cap\bigcap_{\ell\ge2}G_\ell$
of $E$ such that $\overline c(E_1)>\lambda.$
By induction, there is a decreasing sequence
$E\supset E_1\supset\ldots\supset E_s$ with
$$E_s=K_{1m_1}\cap\ldots\cap K_{sm_s}\cap\bigcap_{\ell\ge s+1}G_\ell$$
and $\overline c(E_s)>\lambda.$
Set $K=\bigcap K_{sm_s}=\bigcap E_s\subset E.$
Axiom (c) implies
$$\overline c(K)=\lim_{s\to+\infty}\overline c
\big(K_{1m_s}\cap\ldots\cap K_{sm_s}\big)\ge
\lim_{s\to+\infty}\overline c(E_s)\ge\lambda$$
and the theorem is proved. \finpr

Now, we apply these general results to the outer Monge-Amp\`ere capacity
$c^\star$ introduced in \S~6.

\tha Theorem 9.9|Let $\Omega\ssubset\cb^n$ be strongly
pseudoconvex.
Then the outer capacity $c^\star(\bulle,\Omega)$ is a generalized capacity
in the sense of 9.7.\fintha

\proof Axiom 9.7 (a) is clear, and 9.7 (c) is a consequence of 8.3. To
prove 9.7 (b), we only have to show that $c^\star(\bigcup E_j,\Omega
)\le\lim_{j\to+\infty}c^\star(E_j,\Omega).$
It is no loss of generality to assume that $E_j\ssubset\Omega.$ Let
$N_j$ be the negligible set $N_j=\{u_{E_j}<u^\star_{E_j}\}$ and $G_0$ an
open subset of $\Omega$ with $G_0\supset\bigcup N_j$ and $c(G_0,\Omega
)<\varepsilon.$ Consider the open sets $V_j=\{u^\star_{E_j}<-1+\eta\}$ and
$G_j=G_0\cup V_j\supset E_j.$ Then $(1-\eta)^{-1}u^\star_{E_j}\le
u^\star_{V_j}\le0$ and lemma 8.4 implies
$$\eqalign{
c(G_j,\Omega) &\le\varepsilon+c(V_j,\Omega)
=\varepsilon+\int_\Omega(dd^cu^\star_{V_j})^n\cr
&\le\varepsilon+(1-\eta)^{-n}\int_\Omega(dd^cu^\star_{E_j})^n=
\varepsilon+(1-\eta)^{-n}c^\star(E_j,\Omega),\cr}$$
thanks to proposition 8.5. Further $E_j\subset G_j$ and $G_1\subset
G_2\subset\ldots $ since $u^\star_{E_j}$ is decreasing. Thus $G=\bigcup
G_j\supset E=\bigcup E_j$ and
$$c(G,\Omega)=\lim_{j\to+\infty}c(G_j,\Omega)\le
\varepsilon+(1-\eta)^{-n}\lim_{j\to+\infty}c^\star(E_j,\Omega).$$
Letting $\varepsilon ,\eta\to 0$ we get the desired inequality
$$c^\star(E,\Omega)\le\lim_{j\to+\infty}c^\star(E_j,\Omega).\finpr$$

Choquet's capacitability theorem combined with 8.3 (c) implies that
every $K$-analytic subset $E\subset\Omega$ satisfies 
$$c^\star(E,\Omega)=\sup_{K\,{\rm compact}\,\subset E}c(K,\Omega)=
c_\star(E,\Omega).\leqno(9.10)$$
When $E$ is a Borel set, we thus have $c^\star(E,\Omega)=c(E,\Omega).$
These results can be restated as~:

\tha Theorem 9.11|Every $K$-analytic (in particular every Borel) subset
of $\Omega$ is $c$-capacitable. Consequently $c$ is outer regular.
\finpr\fintha


\pav 10. Siciak's extremal function and Alexander's capacity|

We work here on the whole space $\cb^n$ rather than on a bounded
open subset $\Omega.$ In this case, the relevant class of psh functions
to consider is the set $P_{\rm log}(\cb^n)$ of psh functions $v$
with logarithmic growth at infinity, \ie such that 
$$v(z)\le\log_+|z|+C\leqno(10.1)$$
for some real constant $C.$ Let $E$ be a bounded subset of $\cb^n.$
We consider the global extremal function introduced by Siciak [Sic]~:
$$U_E(z)=\sup\{v(z)~;~v\in P_{\log}(\cb^n),\; v\le0~{\rm
on}~E\}.\leqno(10.2)$$

\tha Theorem 10.3|If $U^\star_E$ is not identically $+\infty,$ then
$U^\star_E\in P_{\log}(\cb^n)$ and $U^\star_E$ satisfies an inequality
$$\log_+(|z|/R)\le U^\star_E(z)\le\log_+|z|+C$$
for suitable constants $C,R>0.$ Moreover $U^\star_E=0$ on $E^0,$
$$(dd^cU^\star_E)=0~{\rm on}~\cb^n\setminus\overline E,\qquad
\int_{\overline E}(dd^cU^\star_E)^n=1.$$
\fintha

\proof Assume that $U^\star_E$ is not identically $+\infty.$ Then
$U^\star_E(z_0)<+\infty$ for some point $z_0\in\cb^n,$ say $z_0=0$ for
simplicity. By the upper semi-continuity of $U^\star_E,$ this implies that
all functions $v\in P_{\log}(\cb^n)$ such that $v\le0$ on $E$ are
uniformly bounded above by some constant $M_0$ on a small ball
$B(0,r_0).$ As $\sup_{B(0,r)}v=\chi(\log r)$
where $\chi $ is a convex increasing function, (10.1) implies 
$\chi '\le1,$ thus
$$\chi(\log r)\le\chi(\log r_0)+\log r/r_0\quad{\rm for}~r\ge r_0.$$
Therefore all functions $v$ under consideration satisfy 
$v(z)\le\log_+(|z|/r_0)+M_0.$ 
In particular these functions are uniformly bounded above everywhere and
$U^\star_E=(\sup\{v\})^\star$ is psh of logarithmic growth on $\cb^n.$ On
the other hand, $E$ is contained in a ball $\overline B(0,R)$ so 
$\log_+|z|/R\le 0$ on $E$ and we get $U_E(z)\ge\log_+|z|/R.$ The equality
$(dd^cU^\star_E)^n=0$ on $\cb^n\setminus\overline E$ is verified exactly
in the same way as 7.8 (a). The integral over $\overline E$ is obtained
by the following lemma. \finpr

\tha Lemma 10.4|Let $v\in P_{\log}(\cb^n)$ be such that
$$\log_+|z|-C_1\le v(z)\le\log_+|z|+C_2$$
for some constants. Then $\int_{\cb^n}(dd^cv)^n=1.$ \fintha

\proof It is sufficient to check that
$$\int_{\cb^n}(dd^cv_1)^n\le\int_{\cb^n}(dd^cv_2)^n$$
when $v_1,v_2$ are two such functions. Indeed, we have
$$\int_{\cb^n}(dd^c\log_+|z|)^n=
\int_{\cb^n}(dd^c\log|z|)^n=1$$
by Stokes' theorem and remark 2.10, and we only have to choose $v_1(z)$
or $v_2(z)=\log_+|z|$ and the other function equal to $v.$ To prove the
inequality, fix $r,\varepsilon>0$ and choose $C>0$ large enough so that
$(1-\varepsilon)v_1>v_2-C$ on $B(0,r).$
As the function
$u=\max\{(1-\varepsilon)v_1,v_2-C\}$ is equal to $v_2-C$ for $|z|=R$
large, we get 
$$(1-\varepsilon)^n\int_{B(0,r)}(dd^cv_1)^n=
\int_{B(0,r)}(dd^cu)^n\le
\int_{B(0,R)}(dd^cu)^n=\int_{B(0,R)}(dd^cv_2)^n$$
and the expected inequality follows as
$\varepsilon\to0$ and $r\to+\infty.$ \finpr

\tha Theorem 10.5|Let $E,E_1,E_2,\ldots\subset B(0,R)\subset\cb^n.$

\noindent{\rm(a)}~~ If $E_1\subset E_2,$ then $U^\star_{E_1}\ge U^\star_{E_2}.$

\noindent{\rm(b)}~~ If $E_1\subset E_2\subset\ldots$ and $E=\bigcup E_j,$ then
$U^\star_E=\lim\downarrow U^\star_{E_j}.$

\noindent{\rm(c)}~~ If $K_1\supset K_2\supset\ldots$ and $K=\bigcap K_j,$ then
$U^\star_K=(\lim\uparrow U^\star_{K_j})^\star.$

\noindent{\rm(d)}~~ For every set $E,$ there exists a decreasing
sequence of open sets $G_j\supset E$ such that $U^\star_E=(\lim\uparrow
U^\star_{G_j})^\star.$\fintha

\proof (a) is obvious and the proof of (c) is similar to that of 8.3 (a).

(d) By Choquet's lemma, there is an increasing sequence $v_j\in
P_{\log}(\cb^n)$ with $U^\star_E=(\lim v_j)^\star$ and
$v_j(z)\ge\log_+|z|/R.$
Set $G_j=\{v_j<1/j\}$ and observe that $U^\star_{G_j}\ge v_j-1/j.$

(b) Set $v=\lim\downarrow U^\star_{E_j}.$ Then $v\in P_{\log}(\cb^n)$
and $v=0$ on $E,$ except on the negligible set
$N=\bigcup\{U_{E_j}<U^\star_{E_j}\}.$
By Josefson's theorem, there exists $w\in P(\cb^n)$ such that 
$N\subset\{w=-\infty\}.$ We set
$$v_j(z)=\cases{
\big(1-{1\over j}\big)v(z)+\max\{\varepsilon
_jw(z),{1\over j}\log|z|-j\}~& on $\;\;B(0,e^{j^3})$\cr
\big(1-{1\over j}\big)v(z)+{1\over j}\log|z|-j~& on $\;\;\cb^n\setminus
B(0,e^{j^3})$\cr}$$
where $\varepsilon_j$ is chosen such that $\varepsilon
_jw<j^2-j$ on $S(0,e^{j^3}).$ Then $v_j\in P_{\log}(\cb^n)$ and
$v_j\le 0$ everywhere on $E$ for $j$ large. Therefore
$$U^\star_E\ge U_E\ge v_j\ge\big(1-{1\over j}\big)v+
\varepsilon_jw~~{\rm on}~B(0,e^{j^3})$$
and letting $j\to+\infty$ we obtain $U^\star_E\ge v.$ The other inequality
is clear. \finpr

Now, we show that the extremal function of a compact set can be computed
in terms of polynomials. We denote by ${\cal P}_d$ the space of
polynomials of degree $\le d$ in $\cb[z_1\ld z_n].$

\tha Theorem 10.6|Let $K$ be a compact subset of $\cb^n.$ Then
$$U_K(z)=\sup\Big\{
{1\over d}\log|P(z)|\,;\,d\ge 1,\,
P\in{\cal P}_d,\,\|P\|_{L^\infty(K)}\le1\Big\}.$$
\fintha

\proof For any of the polynomials $P$ involved in the above formula, we
clearly have ${1\over d}\log|P|\in P_{\log}(\cb^n)$ and this function is
$\le 0$ on $K.$ Hence
$${1\over d}\log|P|\le U_K.$$
Conversely, fix a point $z_0\in\cb^n$ and a real number $a<U_K(z_0).$ Then
there exists $v\in P_{\log}(\cb^n)$ such that $v\le 0$ on $K$ and
$v(z_0)>a.$ Replacing $v$ by $v\star\rho_\delta -\varepsilon $ with $\delta
\ll\varepsilon\ll 1,$ we may assume that $v\in
P_{\log}(\cb^n)\cap\ci(\cb^n),$ $v<0$ on $K$ and $v(z_0)>a.$ Choose a ball
$B(z_0,r)$ on which $v>a,$ a smooth function $\chi $ with compact support
in $B(z_0,r)$ such that $\chi=1$ on $B(z_0,r/2)$ and apply H\"ormander's
$L^2$ estimates to the closed $(0,1)$-form $d''\chi$ 
and to the weight
$$\varphi(z)=2dv(z)+2n\log|z-z_0|+\varepsilon\log(1+|z|^2).$$
We find a solution $f$ of $d''f=d''\chi$ such that
$$\eqalign{
\int_{\cb^n}|f|^2e^{-2dv}&|z-z_0|^{-2n}(1+|z|^2)^{-\varepsilon}d\lambda
\cr
&\le\int_{B(z_0,r)}|d''\chi|^2e^{-2dv}|z-z_0|^{-2n}
(1+|z|^2)^{2-\varepsilon}d\lambda\le C_1e^{-2da}.\cr}$$
We thus have $f(z_0)=0$ and $F=\chi-f$ is a holomorphic function on
$\cb^n$ such that $F(z_0)=1.$ In addition we get
$$\int_{\cb^n}|F|^2e^{-2dv}(1+|z|^2)^{-2n-2\varepsilon}d\lambda\le
C_2e^{-2da}$$
where $C_1,C_2>0$ are constants independent of $d.$ As
$v(z)\le\log_+|z|+C_3,$ it follows that $F\in{\cal P}_d.$
Moreover, since $v>0$ on a neighborhood of $K,$ the mean value
inequality applied to the subharmonic function $|F|^2$ gives
$$\sup_K|F|^2\le C_4e^{-2da}.$$
The polynomial $P=C^{-1/2}_4e^{da}F\in{\cal P}_d$ is such that
$\|P\|_K=1$ and we have\break $\log|P(z_0)|\ge da-C_5,$ whence 
$$\sup\Big\{{1\over d}\log|P(z_0)|\,;\,d\ge 1,\,
P\in{\cal P}_d,\,\|P\|_{L^\infty(K)}\le1\Big\}\ge a.$$
As $a$ was an arbitrary number $< U_K(z_0),$ the proof is complete. \finpr

Now, we introduce a few concepts related to extremal polynomials. Let
$B$ be the unit ball of $\cb^n$ and $K$ a compact subset of $B.$
The Chebishev constants $M_d(K)$ are defined by 
$$M_d(K)=\inf\big\{\|P\|_{L^\infty(K)}\; ;\;
P\in{\cal P}_d,\,\|P\|_{L^\infty(B)}=1\big\}.\leqno(10.7)$$
It is clear that $M_d(K)\le 1$ and that $M_d(K)$ satisfies
$$M_{d+d'}(K)\le M_d(K)M_{d'}(K).$$
The Alexander capacity is defined by
$$T(K)=\inf_{d\ge 1}M_d(K)^{1/d}.\leqno(10.8)$$
It is easy to see that we have in fact
$T(K)=\lim_{d\to+\infty}M_d(K)^{1/d}~:$
for any integer $\delta\ge1$, write $\delta=qd+r$ with $0\le r<d$ and
observe that 
$$M_\delta(K)^{1/\delta}\le M_{qd}(K)^{1/(qd+r)}\le
M_d(K)^{q/(qd+r)}~;$$
letting $\delta\to+\infty$ with $d$ fixed, we get
$$T(K)\le\liminf_{\delta\to+\infty}M_\delta(K)^{1/\delta}\le
\limsup_{\delta\to+\infty}M_\delta(K)^{1/\delta}\le
M_d(K)^{1/d},$$
whence the equality. Now, for an arbitrary subset $E\subset B,$ we set
$$T_\star(E)=\sup_{K\subset E}T(K)\, ,\qquad
T^\star(E)=\inf_{G\,{\rm open}\,\supset E}T_\star(G).\leqno(10.9)$$

\tha Siciak's theorem 10.10|For every set $E\subset B,$
$$T^\star(E)=\exp(-\sup_BU^\star_E).$$
\fintha

\proof The main step is to show that the equality holds for compact
subsets $K\subset B,$ \ie that
$$T(K)=\exp(-\sup_BU^\star_K).\leqno(10.11)$$
Indeed, it is clear that $\sup_BU^\star_K=\sup_BU_K$ and theorem 10.6 gives
$$\eqalign{
\sup_BU_K &=\sup\big\{
{1\over d}\log\|P\|_{L^\infty(B)}\; ;\;
d\ge1,\, P\in{\cal P}_d,\,\|P\|_{L^\infty(K)}=1\big\}\cr
&=\sup\big\{
-{1\over d}\log\|P\|_{L^\infty(K)}\; ;\;
d\ge1,\, P\in{\cal P}_d,\,\|P\|_{L^\infty(B)}=1\big\}\cr}$$
after an obvious rescaling argument $P\mapsto\alpha P.$ Taking the
exponential, we get
$$\eqalign{
\exp(-\sup_BU^\star_K) &=
\inf_{d\ge1}\inf\big\{\|P\|^{1/d}_{L^\infty(K)}\; ;\;
P\in{\cal P}_d,\,\|P\|_{L^\infty(B)}=1\big\}\cr
&=\inf_{d\ge1}M_d(K)^{1/d}=T(K).\cr}$$

Next, let $G$ be an open subset of $B$ and $K_j$ an increasing sequence
of compact sets such that $G=\bigcup K_j$ and
$T_\star(G)=\lim T(K_j).$ Then 10.5 (b) implies $U^\star_G=\lim\downarrow
U^\star_{K_j},$ hence
$$\lim_{j\to+\infty}\sup_B U^\star_{K_j}=
\sup_B U^\star_G=\sup_{\overline B}U^\star_G$$
by Dini's lemma. Taking the limit in (10.11), we get 
$$T_\star(G)=\exp(-\sup_BU^\star_G).$$
Finally, 10.5 (d) shows that there exists a decreasing sequence of open
sets $G_j\supset E$ such that $U^\star_E=(\lim\uparrow U^\star_{G_j})^\star.$
We may take $G_j$ so small that $T^\star(E)=\lim T_\star(G_j).$
Theorem 10.9 follows. \finpr

\tha Corollary 10.12|The set function $T^\star$ is a generalized capacity in
the sense of definition 9.7 and we have
$T^\star(E)=T_\star(E)$ for every $K$-analytic set $E\subset B.$\fintha

\proof Axioms 9.7 (a,b,c) are immediate consequences of properties 10.5
(a,b,c) respectively. In addition, formulas 10.10 and 10.11 show that
$T^\star(K)=T(K)$ for every compact set $K\subset B.$ The last statement is
then a consequence of Choquet's capacitability theorem. \finpr

To conclude this section, we show that $1/|\log T^\star|$ is not very far
from being subadditive. We need a lemma.

\tha Lemma 10.13|For every $P\in{\cal P}_d,$ one has
$$\log\|P\|_{L^\infty(B)}-c_nd\le
\int_{\partial B}\log|P(z)|d\sigma(z)\le\log\|P\|_{L^\infty(B)}$$
where $d\sigma$ is the unit invariant measure on the sphere and $c_n$ a
constant such that $c_n\sim\log(2n)$ as $n\to+\infty.$\fintha

\proof Without loss of generality, we may assume that
$\|P\|_{L^\infty(B)}=1.$ Since
${1\over d}\log|P|\in P_{\log}(\cb^n),$
the logarithmic convexity property already used implies that
$$\sup_{B(0,r)}{1\over d}\log|P|\ge\log r\quad{\rm for}~r<1.$$
The Harnack inequality for the Poisson kernel implies now
$$\sup_{B(0,r)}\log|P|\le
{1-r^2\over(1+r)^{2n}}\int_{\partial B}\log|P|d\sigma ,$$
$$\int_{\partial B}\log|P|d\sigma\ge
{1+r^{2n}\over 1-r^2}\log r.d.$$
The lemma follows with
$$c_n=\inf_{r\in]0,1[}
{(1+r)^{2n}\over 1-r^2}\log{1\over r}~;$$
the infimum is attained approximately for 
$r=1/(2n\log 2n).$ \finpr

\tha Corollary 10.14|For
$P_j\in{\cal P}_{d_j},\; 1\le j\le N,$
$$\|P_1\ldots P_N\|_{L^\infty(B)}\ge
e^{-c_n(d_1+\cdots+d_N)}
\|P_1\|_{L^\infty(B)}\ldots\|P_N\|_{L^\infty(B)}.$$\fintha

\proof Apply lemma 10.13 to each $P_j$ and observe that
$$\int_{\partial B}\log|P_1\ldots P_N|d\sigma=
\sum_{1\le j\le N}\int_{\partial B}\log|P_j|d\sigma.\finpr$$

\tha Theorem 10.15|For any set $E=\bigcup_{j\ge1}E_j,$ one has
$${1\over c_n-\log T^\star(E)}\le\sum_{j\ge1}
{1\over |\log T^\star(E_j)|}.$$
\fintha

\proof It is sufficient to check the inequality for a finite union
$K=\bigcup K_j$ of compacts sets $K_j\subset B,$ $1\le j\le N.$ Select
$P_j\in{\cal P}_{d_j}$ such that
$$\|P_j\|_{L^\infty(B)}=1,\qquad
\|P_j\|_{L^\infty(K_j)}=M_{d_j}(K_j),$$
and set $P=P_1\ldots P_N,\; d=d_1+\cdots+d_N.$
Then corollary 10.14 shows that $\|P_j\|_{L^\infty(B)}\ge e^{-c_nd},$
thus
$$M_d(K)\le e^{c_nd}\|P\|_{L^\infty(K)}.$$
If $z\in K$ is in $K_j,$ then
$|P(z)|\le|P_j(z)|\le\|P_j\|_{L^\infty(K_j)}$
because all other factors are $\le 1.$ Thus
$$M_d(K)\le e^{c_nd}\max\{\|P_j\|_{L^\infty(K_j)}\},$$
$$T(K)\le M_d(K)^{1/d}\le e^{c_n}\max\{M_{d_j}(K)^{1/d_j\cdot
d_j/d}\}.$$
Take $d_j=[k\alpha_j]$ with arbitrary $\alpha_j>0$ and let
$k\to+\infty.$
It follows that
$$T(K)\le e^{c_n}\max\{T(K_j)^{\alpha_j/\alpha}\}$$
where $\alpha=\sum\alpha_j.$ The inequality asserted in theorem
10.15 is obtained for the special choice
$\alpha_j=1/|\log T(K_j)|>0$ which makes all terms in
$\max\{\ldots\}$ equal.

\pav 11. Comparison of capacities and El Mir's theorem|

We first prove a comparison theorem for the capacities $c(\bulle,\Omega
)$ and $T,$ due to Alexander and Taylor [A--T].

\tha Theorem 11.1|Let $K$ be a compact subset of the unit ball
$B\subset\cb^n.$ Then

\noindent{\rm(a)}~~ $T(K)\le\exp(-c(K,B)^{-1/n}).$

\noindent For each $r<1,$ there is a constant $A(r)$ such that

\noindent{\rm(b)}~~ $T(K)\ge\exp(-A(r)\,c(K,B)^{-1})$
when $K\subset B(0,r).$\fintha

\remarque Remark 11.2|Both the set functions $c^\star(\bulle,\Omega)$ and
$T^\star$ are generalized capacities. Hence, the estimates of the theorem
also hold for all $K$-analytic sets, in particular all Borel sets.

\remarque Remark 11.3|The inequalities are sharp, at least as far as the
exponents on $c(K,B)$ are concerned. For if $K=\overline B(0,\varepsilon
),$ then it is easy to check that $T(K)=\varepsilon $ and exercise 7.13
gives $c(K,B)=(\log\,1/\varepsilon)^{-n}.$ Hence, equality holds in
(11.2). On the other hand, if $K$ is a small polydisc
$$K=\{(z_1\ld z_n)\in\cb^n\; ;\;
|z_1|\le\delta ,\,
|z_j|\le 1/n,\,
1<j\le n\}$$
and $\delta\le 1/n,$ then $T(K)\le\delta ,$ while $c(K,B)\ge
C(\log\,1/\delta)^{-1}.$
To check the last inequality, put
$$u(z)=\Big(\log{1\over\delta}\Big)^{-1}\log_+{|z_1|\over\delta}+
(\log n)^{-1}\sum^n_{j=2}\log_+(n|z_j|).$$
Then $u=0$ and $u\le n$ on $B,$ hence
$$c(K,B)\ge\int_B\Big({1\over n}dd^cu\Big)^n=
{n!\over n^n}(\log n)^{-(n-1)}(\log\,1/\delta)^{-1}$$
because all measures $dd^c\log_+(|z_j|/r)$ have total mass 1 in $\cb.$

\remarque Proof of 11.1 (a)|Set $M=\sup_BU^\star_K~;$ then $T(K)=e^{-M}$ by
Siciak's theorem. Since $u=U^\star_K/M\in P(B)$ and $0\le u\le1$ on $B,$ we
get
$$c(K,B)\ge M^{-n}\int_K(dd^cU^\star_K)^n=M^{-n}$$
by theorem 10.3. This inequality is equivalent to 11.1 (a). \finpr

\remarque Proof of 11.1 (b)|Let $u^\star_K$ be the extremal function for $K$
relative to the ball $B'=B(0,e)\supset\!\supset B.$ For any
$v\in P_{\log}(\cb^n)$ such that $v\le 0$ on $K,$ we have
$v\le U^\star_K\le M+1$ on $B',$ hence the function
$$w={v-M-1\over M+1}$$
satisfies $w\le 0$ on $B'$ and $w\le -1$ on $K.$ We infer
$w\le u^\star_K~;$ by taking the supremum over all choices of $v,$ we get
$$u^\star_K\ge{U^\star_K-M-1\over M+1}.$$
Now, there is a point $z_0\in\overline B$ such that $U^\star_K(z_0)=M,$ thus
$$u^\star_K(z_0)\ge -{1\over M+1}.$$
As $u^\star_K\le0$ on $B',$ the mean value and Harnack inequalities show that
$$u^\star_K(z_0)\le C_1\int_{B'}u^\star_Kd\lambda\Longrightarrow
\|u^\star_K\|_{L^1(B')}\le-{1\over C_1}u^\star_K(z_0)\le
{C_2\over M}.$$
The Chern-Levine-Nirenberg inequalities 1.3 and 1.4 (a) imply now
$$c(K,B')=\int_B(dd^cu^\star_K)^n\le
C_3\|u^\star_K\|_{L^1(B')}\,\|u^\star_K\|^{n-1}_{L^\infty(B')}
\le{C_4\over M}~.$$
As $K\subset B(0,r)\ssubset B,$ theorem 6.5 (b) gives
$$c(K,B)\le C_5(r)c(K,B')\le A(r)M^{-1}$$
and inequality 1.11 (b) follows. \finpr

We now prove El Mir's theorem [E--M]. This result is an effective
version of Josefson's theorem~: given a psh function in the ball,
a subextension can be found with prescribed singularities of poles
and slow growth at infinity.

\tha El Mir's theorem 11.4|Let $v\in P(B)$ with $v\le -1,$ 
$\varepsilon\in~]0,1/n[$ and $r<1.$ Then there exists $u\in
P_{\log}(\cb^n)$ such that $u\le -|v|^{{1\over n}-\varepsilon}$ on
$B(0,r).$\fintha

\proof For $t\ge1,$ set $G_t=\{z\in B(0,r);v(z)<-t\}$ and let $U^\star_t\in
P_{\log}(\cb^n)$ be the Siciak extremal function of $G_t.$ Since $G_t$ is
open, we have $U^\star_t=0$ on $G_t.$ We set $M(t)=\sup_BU^\star_t$ and
$$u(z)=\varepsilon^{-1}\int^{+\infty}_1t^{-1-\varepsilon
}(U^\star_t(z)-M(t))dt.$$
Theorem 6.6 shows that $c(G_t,B)\le C_1/t,$ therefore
$$M(t)=-\log T^\star(G_t)\ge c(G_t,B)^{-1/n}\ge C_2t^{1/n}$$
by inequality 1.11 (a). As $U^\star_t-M(t)\le0$ on $B,$ we get
$U^\star_t(z)-M(t)\le\log_+|z|$ by logarithmic convexity, thus
$$u(z)\le\log_+|z|.$$
For $z\in B(0,r)$ we have $U^\star_t(z)=0$ as soon as $G_t\ni z,$ \ie
$t<-v(z).$ Hence
$$u(z)\le-\varepsilon^{-1}\int^{|v(z)|}_1t^{-1-\varepsilon}M(t)dt\le
-C_3\int^{|v(z)|}_1t^{-1-\varepsilon+{1\over n}}dt=
-C_4|v(z)|^{{1\over n}-\varepsilon}.$$
Starting if necessary with a smaller value of $\varepsilon $ and
subtracting a constant to $u,$ we can actually get
$$u\le-|v|^{{1\over n}-\varepsilon}~~{\rm on}~B(0,r).$$
It remains to check that $u$ is not identically $-\infty.$ By
logarithmic convexity again, we have
$$\sup_{\overline B(0,1/2)}U^\star_t\ge M(t)-\log 2$$
and there exists $z_0\in S(0;1/2)$ such that 
$U^\star_t(z_0)-M(t)\ge-\log 2.$
The Harnack inequality shows that
$${1-1/4\over(1+1/2)^{2n}}\int_{\partial B}
\big(U^\star_t(z)-M(t)\big)d\sigma(z)\ge
U^\star_t(z_0)-M(t)\ge -\log 2$$
and integration with respect to $t$ yields
$$\int_{\partial B}u(z)d\sigma(z)\ge -4/3(3/2)^{2n}\log 2>-\infty.\finpr$$


\titre References|

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\vskip15pt
{\noindent\hangindent4cm\hangafter-5{\sl
Jean-Pierre Demailly\hfill\break
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
Laboratoire associ\'e au C.N.R.S.
n\kern-2pt{\rm\char'27}188,\hfill\break
F-38402 Saint-Martin d'H\`eres

}}

\end
