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Theorem 14.19.21BF849D15DF41

Theorem 14.19.21.BF849D15DF41 Suppose $u_ j\in \mathrm{PSH}(X,\omega )\cap C(X)$, $\sup _ Xu_ j=0$, and

\[ \omega _{u_ j}^ n=F_ j\, \mathrm{d}V,\qquad F_ j\geq 0,\qquad \sup _ j\| F_ j\| _ p<\infty \]

for the fixed $p>1$. If $(u_ j)$ is Cauchy in $L^1(X,\mathrm{d}V)$, then it is Cauchy uniformly. In particular, every such family has a subsequence converging uniformly to a continuous $\omega $-plurisubharmonic function.

Proof. By theorem 14.19.19, there is a common $H\geq 1$ with $-H\leq u_ j\leq 0$. Suppose that the sequence is not uniformly Cauchy. After exchanging indices if necessary, we find $a>0$ and arbitrarily large $j,k$ with

\[ S:=\inf _ X(u_ k-u_ j)\leq -a. \]

Choose $0<\varepsilon <1$ such that $\varepsilon H<a/4$, and choose a fixed $D>0$ with $2D<\delta _\varepsilon $ and $2D<a/4$. Apply the preceding sublevel lemmas with $u=u_ k$ and $v=u_ j$. Since $v\leq 0$, $m=\inf (u-(1-\varepsilon )v)\leq S$. Thus

\[ U_{2D}\subset \{ u-v<S+2D+\varepsilon H\} \subset \{ |u-v|>a/2\} . \]

The capacity lower bound (14.19.18), with a constant $A$ independent of $j,k$ given by corollary 5.29.5, yields $\operatorname {Cap}(U_ D)\geq c>0$. On the other hand, (14.19.15) and the inequalities of Hölder and Chebyshev give

\[ \begin{split} D^ n\operatorname {Cap}(U_ D) & \leq C_ H\int _{U_{2D}}F_ k\, \mathrm{d}V\\ & \leq C_ H\| F_ k\| _ p \left(\frac{2\| u_ k-u_ j\| _1}{a}\right)^{1/p'} \longrightarrow 0. \end{split} \]

This contradiction proves that the sequence is uniformly Cauchy. For the last assertion, the normalized compactness theorem, theorem 5.25.2, gives an $L^1$-Cauchy subsequence of any such family. Its uniform limit is continuous, and distributional convergence preserves $\omega $-plurisubharmonicity. $\square$

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