Pome3DD22F1D6AC3Since (15.29.2) expresses $\rho _ h(p)$ as a supremum of positive linear functionals evaluated at $|p|$, the root functional is homogeneous, monotone and subadditive. Subadditivity shows that the sum of two densities with positive threshold again has positive threshold. In the statements below, $f$ and $g$ are nonzero nonnegative $L^1$ densities, and $p$ and $q$ are real or complex functions in $L^ n$.
Proposition 15.32.4.D697A55DE07C The functional $\rho _ h$ satisfies $\rho _ h(tp)=|t|\rho _ h(p)$ for scalars $t\ne 0$, $\rho _ h(p)\leq \rho _ h(q)$ whenever $|p|\leq |q|$, and
In particular, if $b_ h(f),b_ h(g)>0$, then
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 $b_ h(f+g)$. $\square$
Proposition 15.32.5.4C06D1B94BD1 The real or complex vector space
is complete for the norm $\| p\| _{\mathcal D_ h}=\| p\| _{L^ n}+\rho _ h(p)$. It is an order ideal in $L^ n$: if $q\in \mathcal D_ h$ and $|p|\leq |q|$, then $p\in \mathcal D_ h$. It contains $L^\infty $ and is therefore dense in $L^ n$ for the $L^ n$ norm.
Moreover, $\mathcal D_ h=L^ n$ if and only if $v_-(h)>0$, and in that case the norm $\| \cdot \| _{\mathcal D_ h}$ is equivalent to the $L^ n$ norm. If $h$ is Kähler, then
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. $\square$
Thus, when the lower mass vanishes, $\mathcal D_ h$ is a proper dense subspace of $L^ n$, complete for the stronger norm $\| \cdot \| _{\mathcal D_ h}$; its nonnegative elements are the $n$th roots of the densities admitting a positive scaled subsolution. A finite sum of such densities again admits a positive scaled subsolution, but a convergent infinite sum need not: the smooth summands in theorem 15.27.3 give an example. By proposition 15.29.5, the space $\mathcal D_ h$ does not depend on the Hermitian form $h$, and the norms $\| \cdot \| _{\mathcal D_ h}$ for different Hermitian forms are equivalent. The closure of the bounded functions for this norm is studied in 15.32.7.
The restriction of $\rho _ h$ to $\mathcal D_ h$ is itself a norm: if $p\ne 0$, then $\rho _ h(p)\geq L_{h,\gamma }(|p|)>0$ for any Gauduchon form $\gamma $. Whether this weaker norm is complete depends on the upper mass.
Proposition 15.32.6.C926F63790E3 The norm $\rho _ h$ is complete on $\mathcal D_ h$ if and only if $v_+(h)<+\infty $. When the upper mass is finite,
so $\rho _ h$ and $\| \cdot \| _{\mathcal D_ h}$ are equivalent norms.
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. $\square$
The two mass properties play separate roles: positive lower mass makes every $L^ n$ function admissible, whereas finite upper mass makes the root norm control the $L^ n$ norm. The norm $\| \cdot \| _{\mathcal D_ h}$ is complete without either assumption.
All four combinations occur. On a compact Kähler manifold both mass properties hold. The Hopf threefold has positive lower mass and infinite upper mass (corollary 23.1.40). The elliptic threefold of theorem 12.19.1 has zero lower mass, and finite upper mass for its balanced metric (proposition 12.19.18). The product manifolds of corollary 12.19.16 have zero lower mass and infinite upper mass. Since neither $\mathcal D_ h$ nor the completeness of $\rho _ h$ depends on the choice of $h$ (proposition 15.29.5, proposition 13.14.15), we obtain the following table.
Manifold | Domain $\mathcal D_ h$ | Is $\rho _ h$ complete? |
Kähler manifold | $L^ n$ | Yes |
Hopf threefold | $L^ n$ | No |
Elliptic threefold | $\mathcal D_ h\subsetneq L^ n$ | Yes |
Product example | $\mathcal D_ h\subsetneq L^ n$ | No |
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