Theorem 5.10.10F2A3CA649325

Proof. If $X$ has dimension zero, every compact subset of $X$ is finite, plurisubharmonic functions are real-valued, and convergence in $L^ p_{\mathrm{loc}}$ is pointwise convergence; so both assertions hold. Let $X$ have dimension $n\geq 1$. Every compact subset of $X$ is covered by finitely many coordinate polydiscs, so it suffices to work near a point $x\in X$ in a chart in which $x=0$ and the closed polydisc $\overline{(RD)^ n}$ lies in the chart, for some $R>0$.

Let $v$ be plurisubharmonic. Being upper semicontinuous, $v$ is bounded above on $\overline{(RD)^ n}$ by some constant $A$. The function $\zeta \mapsto v(R\zeta )-A$ is plurisubharmonic and nonpositive on $D^ n$, and its $L^1$ norm on $D^ n$ is finite because $v$ is locally integrable. By (5.10.9), $v$ belongs to $L^ p$ on the polydisc $(\tfrac R4D)^ n$. This proves the first assertion.

Now let $(u_ j)$ be a sequence of plurisubharmonic functions converging in $L^1_{\mathrm{loc}}(X)$. For $p=1$ there is nothing to prove, so let $p>1$. By theorem 5.10.4 applied in charts, together with the uniqueness in theorem 5.9.1, the limit is represented by a plurisubharmonic function $u$. The norms $\| u_ j\| _{L^1((RD)^ n)}$ are bounded because the sequence converges. If $y\in (\tfrac R2D)^ n$, the ball $B(y,R/2)$ lies in $(RD)^ n$, so the ball submean inequality gives $u_ j(y)\leq |B(y,R/2)|^{-1}\| u_ j\| _{L^1((RD)^ n)}$. Hence there is a constant $M$, independent of $j$, with $u_ j\leq M$ and $u\leq M$ on $(\tfrac R2D)^ n$. Applying (5.10.9) to $\zeta \mapsto u_ j(\tfrac R2\zeta )-M$ and to $\zeta \mapsto u(\tfrac R2\zeta )-M$, whose $L^1$ norms on $D^ n$ are bounded independently of $j$, we obtain a bound for the $L^{2p}$ norms of $u_ j$ and $u$ on $(\tfrac R8D)^ n$ that is uniform in $j$. By a finite covering, for every compact set $K\subset X$ the norms $\| u_ j\| _{L^{2p}(K)}$ and $\| u\| _{L^{2p}(K)}$ are bounded independently of $j$.

We interpolate between $L^1$ and $L^{2p}$. For a function $f$ on $K$, write $|f|^ p=|f|^{p/(2p-1)}\cdot |f|^{p(2p-2)/(2p-1)}$ and apply Hölder's inequality with the conjugate exponents $(2p-1)/p$ and $(2p-1)/(p-1)$; since $\frac{p(2p-2)}{2p-1}\cdot \frac{2p-1}{p-1}=2p$, this gives

that is, $\| f\| _{L^ p(K)}\leq \| f\| _{L^1(K)}^{1/(2p-1)}\| f\| _{L^{2p}(K)}^{(2p-2)/(2p-1)}$. With $f=u_ j-u$, whose $L^{2p}(K)$ norms are bounded independently of $j$, we obtain