Corollary 10.3.2384717733620

Proof. (If $\mathrm{PSH}(X,\theta )$ is empty, there is nothing to prove.) Choose finitely many coordinate polydiscs $D_ k$ such that the polydiscs $\frac14D_ k$ with the same centers and a quarter of the radii cover $X$, and smooth functions $h_ k$ near $\overline{D_ k}$ with $\mathrm{dd}^{\mathrm{c}}h_ k=\theta $. For $u$ as in the statement, $v_ k=u+h_ k-\sup _{D_ k}h_ k$ is a nonpositive plurisubharmonic function on $D_ k$. By proposition 10.3.1 and the boundedness of the $h_ k$ and of the coordinate volume densities, the norms $\| v_ k\| _{L^1(D_ k)}$ are bounded independently of $u$. After rescaling $D_ k$ to the unit polydisc, (5.10.9) bounds $\| v_ k\| _{L^ p(\frac14D_ k)}$ independently of $u$. Since $u=v_ k-h_ k+\sup _{D_ k}h_ k$ on $D_ k$, the bounded functions $h_ k$ and the bounded volume densities give the assertion. $\square$