Bounded functions are dense in $\mathcal D_ h$ for the $L^ n$ norm, but not obviously for the norm $\| \cdot \| _{\mathcal D_ h}$. Let $\mathcal D_ h^0$ denote the closure of $L^\infty $ in $\mathcal D_ h$ for this norm. We show that a function belongs to $\mathcal D_ h^0$ exactly when its tails are small uniformly over all Gauduchon tests. The basic tool is a subcritical estimate. For every $r>n$,
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15.32.8
\begin{equation} \rho _ h(p)\leq C_ r\| p\| _{L^ r}\qquad (p\in L^ r), \end{equation}
where the constant $C_ r$ depends on $h$, the reference volume and $r$, but not on $p$. Indeed, for nonzero $p\in L^ r$ the density $|p|^ n$ belongs to $L^{r/n}$, so $b_ h(|p|^ n)=c_ h(|p|^ n)$ by proposition 15.28.7, and the lower-density estimate of theorem 14.19.19, with exponent $r/n$, gives $c_ h(|p|^ n)\| |p|^ n\| _{L^{r/n}}\geq c_{r/n}>0$.
Proposition 15.32.9.2210D4123A64 For $p\in \mathcal D_ h$, the following conditions are equivalent:
$p\in \mathcal D_ h^0$;
$\rho _ h(|p|\mathbf1_{\{ |p|>K\} })\to 0$ as $K\to \infty $;
whenever $|q_ j|\leq |p|$ and $q_ j\to 0$ in measure, one has $\| q_ j\| _{\mathcal D_ h}\to 0$.
By (15.29.2), the second condition can be written as
\[ \sup _{\gamma \text{ Gauduchon}} L_{h,\gamma }(|p|\mathbf1_{\{ |p|>K\} })\longrightarrow 0 \qquad (K\to \infty ). \]
The space $\mathcal D_ h^0$ is a closed order ideal in $\mathcal D_ h$. It contains $L^ r$ for every $r>n$, and smooth functions are dense in it. If $v_-(h)>0$, then $\mathcal D_ h^0=\mathcal D_ h=L^ n$.
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
\[ (|p|-K)_+\leq |p|\mathbf1_{\{ |p|>K\} },\qquad |p|\mathbf1_{\{ |p|>2K\} }\leq 2(|p|-K)_+. \]
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 $\mathcal D_ h^0=\mathcal D_ h=L^ n$.
$\square$
By proposition 15.29.5, the space $\mathcal D_ h^0$ does not depend on the choice of the Hermitian form $h$. Convergence in $\mathcal D_ h$ controls the threshold: if $p_ j\to p\ne 0$ in $\mathcal D_ h$, then
\[ |\rho _ h(p_ j)-\rho _ h(p)|\leq \rho _ h(p_ j-p)\longrightarrow 0, \qquad b_ h(|p_ j|^ n)\longrightarrow b_ h(|p|^ n)>0. \]
For $p\in \mathcal D_ h^0$, the uniform tail condition also gives a uniform modulus for exponential damping and error bounds for the scalars of the exponential equations; see proposition 15.35.5. It does not by itself give capacity convergence of the solutions. We do not know whether $\mathcal D_ h^0=\mathcal D_ h$ when $v_-(h)=0$.
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