Proposition 18.31.15E124FAEA311

Proof. Let $\nu $ be a smooth positive probability measure, and fix $\epsilon >0$. Cover $X$ by finitely many coordinate charts on which $L$ has holomorphic frames, and choose relatively compact open subsets of these charts that still cover $X$. In a frame $e$ write $|e|_ h^2=\mathrm{e}^{-\chi }$ and $s=f e^{\otimes m}$, so that $|s|_{h^ m}^2\mathrm{e}^{-mv}=|f|^2\mathrm{e}^{-m(\chi +v)}$. By compactness and uniform continuity of the finitely many functions $\chi +v$, there is a radius $\delta >0$ with the following property: every point $x$ lies in one of the smaller sets, the coordinate ball $B(x,\delta )$ lies in the corresponding chart, and the oscillation of $\chi +v$ on $B(x,\delta )$ is at most $2\epsilon $. On these balls the density of $\nu $ with respect to coordinate Lebesgue measure is bounded below by a positive constant independent of $x$.

The submean inequality for the holomorphic function $f$ on $B(x,\delta )$ therefore gives

with $C_\delta $ independent of $x$ and $m$. Taking the supremum over $x$ and then the square root gives the Bernstein–Markov inequality with $C_\epsilon =C_\delta ^{1/2}$. $\square$