Pome5F7DB1E779E6Corollary 9.11.72.5F7DB1E779E6 For the same $X$ and $h$, every nonzero nonnegative $f\in L^ p(Y,\Omega )$, $p>1$, admits a bounded $u\in \mathrm{PSH}(X,h)$ with $\sup _ Xu=0$ and
Proof. The torus $Y$ is Kähler, so it has bounded mass by lemma 19.9.12. Since $k>0$, zero is an ordinary model for $k$: $P_ k[0]=0$. Its lower volume is $M>0$ by lemma 9.11.68, because the potentials of the same ordinary type as zero are exactly the bounded ones. Hence theorem 10.31.3, applied on $Y$ with background $k$ and reference zero, gives a bounded $k$-plurisubharmonic solution for the measure $f\Omega $, and its scalar is $M/\int f\Omega $. Adding $v_0$ and using theorem 9.11.5, theorem 9.11.26 gives the normalized equation on $X$. The datum is non-pluripolar as in theorem 9.11.71. Neither smallness nor strict positivity of $f$ is needed for this existence statement. $\square$
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