Proposition 15.29.5.429BB0FF98A9 Let $f$ be a nonzero nonnegative function in $L^1(\mathrm{d}V)$ and let $p\in L^ n(\mathrm{d}V)$ be real or complex. If $h_1,h_2$ are Hermitian forms on $X$ and $m h_1\leq h_2\leq M h_1$ for positive constants $m,M$, then
\[ m^ n b_{h_1}(f)\leq b_{h_2}(f)\leq M^ n b_{h_1}(f), \qquad M^{-1}\rho _{h_1}(p)\leq \rho _{h_2}(p)\leq m^{-1}\rho _{h_1}(p). \]
If $h_2-h_1=\mathrm{dd}^{\mathrm{c}}a$ for a smooth real function $a$, then $b_{h_1}(f)=b_{h_2}(f)$ and $\rho _{h_1}(p)=\rho _{h_2}(p)$.
Proof.
For every Gauduchon form $\gamma $, the numerators in (15.29.3) satisfy
\[ m\int _ Xh_1\wedge \gamma ^{n-1}\leq \int _ Xh_2\wedge \gamma ^{n-1} \leq M\int _ Xh_1\wedge \gamma ^{n-1}, \]
while the denominators do not depend on the background. Taking the infimum over $\gamma $ and then $n$th powers proves the bounds for the thresholds; taking reciprocal $n$th roots gives the bounds for $\rho $, with the stated conventions at zero and infinity. If $h_2-h_1=\mathrm{dd}^{\mathrm{c}}a$, the numerators agree by Stokes' theorem, since $\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$.
$\square$
Pome
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