15.32 Stability of the subsolution threshold06284CD8A9D4

Proof. For a Gauduchon form $\gamma $ on $X$, write $A_\gamma (f)=\int _ X(f\, \mathrm{d}V/\gamma ^ n)^{1/n}\gamma ^ n$. Since $|s^{1/n}-t^{1/n}|\leq |s-t|^{1/n}$ for $s,t\geq 0$, Hölder's inequality shows that $A_\gamma $ is continuous on $L^1$; it is positive at every nonzero nonnegative density. By theorem 15.29.1, $b_ h^{1/n}$ is the infimum over $\gamma $ of the continuous positive functions $f\mapsto \int _ Xh\wedge \gamma ^{n-1}/A_\gamma (f)$. It is therefore upper semicontinuous, and testing a single $\gamma $ bounds it above on an $L^1$ neighborhood of $f$.

For the second assertion, recall that $b_ h(q)=c_ h(q)$ for bounded $q$ (proposition 15.28.7). By the first assertion, $\limsup _ jc_ h(q_ j)\leq b_ h(f)$ for every sequence of nonzero nonnegative bounded densities $q_ j\to f$ in $L^1$. Hence the limit in (15.32.2) is at most $b_ h(f)$, and the local upper bound makes the suprema finite for small $\delta $. Conversely, the truncations $f_ k$ converge to $f$ in $L^1$ and satisfy $c_ h(f_ k)\geq b_ h(f)$, so every supremum in (15.32.2) is at least $b_ h(f)$. $\square$

Proof. By (15.29.2), $\rho _ h(p)=\sup _\gamma L_{h,\gamma }(|p|)$ is a supremum of positive linear functionals evaluated at $|p|$. This gives homogeneity and monotonicity, and, together with $|p+q|\leq |p|+|q|$, the triangle inequality. Finally, $(f+g)^{1/n}\leq f^{1/n}+g^{1/n}$ gives

which is equivalent to the asserted lower bound for $b_ h(f+g)$. $\square$

Proof. By proposition 15.32.4, $\mathcal D_ h$ is a vector space and an order ideal, and $\| \cdot \| _{\mathcal D_ h}$ is a norm on it. Bounded functions belong to $\mathcal D_ h$, because $b_ h(|p|^ n)=c_ h(|p|^ n)>0$ for bounded nonzero $p$.

We next show that $\rho _ h$ is lower semicontinuous on $L^ n$. If $p_ j\to p$ in $L^ n$, then $|p_ j|^ n\to |p|^ n$ in $L^1$, and the upper semicontinuity of $b_ h$ in theorem 15.32.1 gives

This includes the case $b_ h(|p|^ n)=0$, in which the right side must be infinite; for $p=0$ the inequality is trivial. Now let $(p_ j)$ be a Cauchy sequence in $\mathcal D_ h$. It converges in $L^ n$ to some $p$, and lower semicontinuity gives $\rho _ h(p)\leq \liminf _ j\rho _ h(p_ j)<+\infty $, so $p\in \mathcal D_ h$. Applied to $p_ j-p_ k$ as $k\to \infty $, lower semicontinuity also gives

which tends to zero as $j\to \infty $. Together with $L^ n$ convergence, this proves completeness.

If $v_-(h)>0$, proposition 15.28.7 gives $\rho _ h(p)\leq v_-(h)^{-1/n}\| p\| _{L^ n}$, so $\mathcal D_ h=L^ n$ with an equivalent norm. If $v_-(h)=0$, the density $f$ of theorem 15.27.3 has $b_ h(f)=0$ by theorem 15.28.4; thus $f^{1/n}$ belongs to $L^ n$ but not to $\mathcal D_ h$. The Kähler formula follows from the same proposition. $\square$

Proof. Suppose $v_+(h)<+\infty $. A continuous solution for a bounded density $q$ has mass $c_ h(q)\int _ Xq\, \mathrm{d}V\leq v_+(h)$. Applying this to the truncations of $|p|^ n$ and letting the truncation level increase gives $b_ h(|p|^ n)\| p\| _{L^ n}^ n\leq v_+(h)$, which is the asserted inequality. Hence $\rho _ h$ and $\| \cdot \| _{\mathcal D_ h}$ are equivalent, and completeness follows from proposition 15.32.5.

Suppose instead that $v_+(h)=+\infty $. By theorem 12.15.4, theorem 12.2.3, the normalization scalars of smooth strictly positive probability densities are unbounded above. For each $j$, choose such a density $f_ j$ with

Then $\| p_ j\| _{L^ n}=j$ and $\rho _ h(p_ j)=jc_ h(f_ j)^{-1/n}\leq 2^{-j}$, so the partial sums $P_ N=\sum _{j\leq N}p_ j$ form a Cauchy sequence for $\rho _ h$.

Fix a Gauduchon form $\gamma $. Its coefficient $a_\gamma $ is bounded below by a positive constant, so (15.29.2) gives $\rho _ h(p)\geq c\| p\| _{L^1}$ with a constant $c>0$. Hence $\sum _ jp_ j$ converges in $L^1$ to a nonnegative function $P$. If $P_ N$ converged for $\rho _ h$ to an element of $\mathcal D_ h$, that element would also be the $L^1$ limit, hence equal to $P$ almost everywhere. But $P\geq p_ j$ gives $\| P\| _{L^ n}\geq j$ for every $j$, so $P\notin L^ n$. Therefore $\rho _ h$ is not complete. $\square$

Proof. For $K>0$, set $p_ K=p\min (1,K/|p|)$ where $p\ne 0$, and $p_ K=0$ where $p=0$. Then $|p-p_ K|=(|p|-K)_+$ and

Since the $L^ n$ norms of the tails tend to zero, these inequalities and the monotonicity of $\rho _ h$ show that the second condition is equivalent to $p_ K\to p$ in $\mathcal D_ h$; in particular, it implies the first. Conversely, if $a$ is bounded and $K\geq \| a\| _{L^\infty }$, then $(|p|-K)_+\leq |p-a|$, so $\| p-p_ K\| _{\mathcal D_ h}\leq \| p-a\| _{\mathcal D_ h}$. Approximating $p$ by bounded functions $a$ shows that the first condition implies $p_ K\to p$, hence the second.

Assume the second condition, and let $q_ j$ be as in the third. Fix $K$. On $\{ |p|\leq K\} $, the functions $q_ j$ are bounded by $K$ and tend to zero in measure, hence in $L^{2n}$; by (15.32.8) with $r=2n$, the root functional of $q_ j\mathbf1_{\{ |p|\leq K\} }$ tends to zero. On the complementary set, $\rho _ h(q_ j\mathbf1_{\{ |p|>K\} })\leq \rho _ h(|p|\mathbf1_{\{ |p|>K\} })$. Letting first $j$ and then $K$ tend to infinity gives $\rho _ h(q_ j)\to 0$, and dominated convergence gives $\| q_ j\| _{L^ n}\to 0$. Conversely, the third condition applied to $q_ j=|p|\mathbf1_{\{ |p|>j\} }$ gives the second.

Since the second condition passes from $p$ to every function dominated by $|p|$, the space $\mathcal D_ h^0$ is an order ideal; it is closed by definition. For $p\in L^ r$ with $r>n$, the truncations $p_ K$ converge to $p$ in $L^ r$, hence in $\mathcal D_ h$ by (15.32.8). Bounded functions can be approximated in $L^{2n}$ by smooth functions, which then converge in $\mathcal D_ h$ by the same estimate; hence smooth functions are dense in $\mathcal D_ h^0$. Under positive lower mass, proposition 15.32.5 shows that $\mathcal D_ h=L^ n$ with an equivalent norm, and bounded functions are dense in $L^ n$; hence $\mathcal D_ h^0=\mathcal D_ h=L^ n$. $\square$

Proof. Since $|J\xi |^ N=R|\xi |^ N$, integration against $\mathrm dV$ gives the first norm identity. For nonzero $\xi $, theorem 15.30.6 applied to $g=|\xi |^ N$ gives the root-functional identity; for $\xi =0$ it holds by definition. Applying the two identities to differences yields

Bounded approximations of $\xi $ in $L^ N$ have bounded images because $R$ is bounded, so $J\xi \in \mathcal D_ H^0$. Completeness of $L^ N(Y,\kappa ^ k)$ and the displayed identity make the image closed.

On $\{ |J\xi |>K\} $ one has $g>K^ N/A$, and hence

Monotonicity of the root functional and its identity for factor roots give the estimate, with the value zero when the tail vanishes. Integrability of $g$ makes the bound tend to zero; the $L^ N$ tails also vanish. When $R=1$, the amplitude cutoff itself is pulled back from $Y$, so the same norm identity gives equality. $\square$