Lemma 14.19.26B2B0EF9B5807

Proof. Absorbing the fixed smooth positive ratio $\mathrm{d}V/\omega ^ n$ into $f_ j$ and $f$ preserves their $L^ p$ convergence, so we may assume $\mathrm{d}V=\omega ^ n$. Choose $\rho >0$ smooth with $\mathrm{dd}^{\mathrm{c}}(\rho \omega ^{n-1})=0$, using theorem 11.4.2; for $n=1$, take $\rho =1$. Put

Set $A=\int _ X f^{1/n}\rho \omega ^ n$. The inequality for $n$th roots used in the preceding existence proof gives $A_ j\to A>0$. Since $\mathrm{dd}^{\mathrm{c}}(\rho \omega ^{n-1})=0$, the trace $\int _ X\omega _{u_ j}\wedge \rho \omega ^{n-1}$ equals $\alpha $; the arithmetic–geometric mean inequality therefore implies

Applying Jensen's inequality to the probability measure $A_ j^{-1}f_ j^{1/n}\rho \omega ^ n$ and taking logarithms, we obtain

By lemma 5.26.2, the normalized potentials $\psi _ j$ have uniform $L^ q$ bounds for every finite $q$. With $q=np/(np-1)$, Hölder's inequality gives a uniform bound for the last integral, since $\| f_ j^{1/n}\| _{np}=\| f_ j\| _ p^{1/n}$. This proves the upper bound for $M_ j$.

Finally, the density $F_ j=\mathrm{e}^{\lambda u_ j}f_ j$ satisfies $\| F_ j\| _ p\leq \mathrm{e}^{\lambda M_ j}\| f_ j\| _ p$. The lower-density bound of theorem 14.19.19, applied to $\psi _ j$, gives $c_ p\leq \mathrm{e}^{\lambda M_ j}\| f_ j\| _ p$, which bounds $M_ j$ below. $\square$