Pome

Theorem 12.9.2E7F890105D42

Theorem 12.9.2.E7F890105D42 With the extended value $+\infty $ allowed, one has

\[ \sup _{T\in \mathscr G_\omega } \| k_ T\| _{L^{n/(n-1)}(\mathrm{d}V)}=v_-(\omega )^{-1/n}, \]

where the right side is interpreted as $+\infty $ when $v_-(\omega )=0$. The supremum does not change if $T$ is restricted to the smooth strictly positive members of $\mathscr G_\omega $. Thus a uniform estimate at the endpoint exponent holds if and only if $v_-(\omega )>0$.

Proof. Suppose first that $v_-(\omega )>0$. For the smooth equation in the preceding proof, the definition of $v_-$ gives

\[ c_ h\int _ Xh^ n\, \mathrm{d}V=\int _ X\Omega _ h^ n\geq v_-(\omega ). \]

The same pointwise inequality and Stokes' theorem therefore give

\[ \int _ Xh\, \mathrm{d}\Gamma _0(T) \leq v_-(\omega )^{-1/n}\| h\| _{L^ n(\mathrm{d}V)}. \]

By duality, $\| k_ T\| _{L^{n/(n-1)}}\leq v_-(\omega )^{-1/n}$ for every $T\in \mathscr G_\omega $.

For the reverse bound, let $g$ be any smooth positive probability density relative to $\mathrm{d}V$, and solve

\[ \Omega _ g^ n=c(g)g\, \mathrm{d}V. \]

Let $T_ g$ be the trace-normalized $(n-1)$st power of the Gauduchon metric conformal to $\Omega _ g$. Equality in the pointwise matrix inequality gives

\[ \int _ Xg^{1/n}\, \mathrm{d}\Gamma _0(T_ g)=c(g)^{-1/n}. \]

Since $\| g^{1/n}\| _{L^ n}=1$, Hölder's inequality gives $\| k_{T_ g}\| _{L^{n/(n-1)}}\geq c(g)^{-1/n}$. By theorem 12.15.4, theorem 12.2.3,

\[ \inf _{g>0\ \mathrm{smooth},\ \int _ Xg\, \mathrm{d}V=1}c(g)=v_-(\omega ). \]

Taking the supremum over these smooth currents $T_ g$ proves the matching lower bound; if $v_-(\omega )=0$, the same argument makes this supremum infinite. Since the currents $T_ g$ are smooth, this also proves that allowing nonsmooth currents does not change the supremum. $\square$

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