Theorem 7.2.5 (Chern–Levine–Nirenberg).2A84C325AB58 Let $K,L\subset \Omega $ be compact sets with $K\subset \operatorname{Int}L$, and let $0\le q\le r$ be an integer. There is a constant $C_{K,L,q}$, depending only on $K$, $L$, $n$ and $q$, such that
\[ \| \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge T\| _ K \le C_{K,L,q}\| T\| _ L\prod _{j=1}^ q\| u_ j\| _{L^\infty (L)} \]
for every closed positive current $T$ of bidimension $(r,r)$ on $\Omega $ and all locally bounded plurisubharmonic functions $u_1,\ldots ,u_ q$ on $\Omega $. Here $\| u_ j\| _{L^\infty (L)}=\sup _ L|u_ j|$ is the pointwise supremum.
Proof.
For $q=0$ the estimate holds with constant $1$, since $K\subset L$. Let $q=1$ and write $u=u_1$. By lemma 2.5.1 there is a smooth function $0\le \chi \le 1$ with compact support in $\operatorname{Int}L$ which equals $1$ on a neighborhood of $K$; it depends only on $K$ and $L$. Since $\mathrm{dd}^{\mathrm{c}}u\wedge T\wedge \beta ^{r-1}$ is a positive measure, $\beta $ is closed, and $(\mathrm{dd}^{\mathrm{c}}S)(\chi )=S(\mathrm{dd}^{\mathrm{c}}\chi )$ for currents $S$ of bidimension $(1,1)$, we have
\[ \begin{aligned} \int _ K\mathrm{dd}^{\mathrm{c}}u\wedge T\wedge \beta ^{r-1} & \le \int \chi \, \mathrm{dd}^{\mathrm{c}}(uT)\wedge \beta ^{r-1} =\bigl (\mathrm{dd}^{\mathrm{c}}(uT\wedge \beta ^{r-1})\bigr )(\chi )\\ & =\int uT\wedge \mathrm{dd}^{\mathrm{c}}\chi \wedge \beta ^{r-1} \le C\| u\| _{L^\infty (L)}\| T\| _ L. \end{aligned} \]
The form $\mathrm{dd}^{\mathrm{c}}\chi \wedge \beta ^{r-1}$ is smooth with support in $L$, though in general not positive; its coefficients are bounded by a constant depending only on $\chi $ and $n$. The last inequality therefore holds because the total variations of the coefficient measures of $T$ on $L$ are bounded by a constant multiple of $\| T\| _ L$, the multiple depending only on $n$ and $r$ (theorem 5.3.7). Dividing by $(r-1)!$ and taking the largest of the constants obtained for $r=1,\ldots ,n$, we obtain the case $q=1$ with a constant depending only on $K$, $L$ and $n$.
For $q>1$, let $\delta >0$ be the distance from $K$ to $\mathbb C^ n\setminus \operatorname{Int}L$ (if $K=\varnothing $ there is nothing to prove), and put $L_ j=\{ z:\operatorname {dist}(z,K)\le j\delta /(q+1)\} $ for $0\le j<q$ and $L_ q=L$. These are compact sets, depending only on $K$, $L$ and $q$, with $K=L_0$ and $L_{j-1}\subset \operatorname{Int}L_ j$ for $1\le j\le q$. By proposition 7.2.2, the currents $S_ j=\mathrm{dd}^{\mathrm{c}}u_{j+1}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge T$, $0\le j\le q$, are positive and closed, and $S_{j-1}=\mathrm{dd}^{\mathrm{c}}u_ j\wedge S_ j$. The case $q=1$, applied to $S_ j$ and the pair $L_{j-1}\subset \operatorname{Int}L_ j$, gives $\| S_{j-1}\| _{L_{j-1}}\le C_ j\| u_ j\| _{L^\infty (L)}\| S_ j\| _{L_ j}$ for $j=1,\ldots ,q$, since $\| u_ j\| _{L^\infty (L_ j)}\le \| u_ j\| _{L^\infty (L)}$. Combining these $q$ inequalities, we obtain the estimate with $C_{K,L,q}=C_1\cdots C_ q$.
$\square$
Pome
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