Proposition 15.35.5.AC5E6031A12C Let $0\leq p\in \mathcal D_ h^0$ be nonzero, and put $f=p^ n$, $R=\rho _ h(p)$ and $b=R^{-n}=b_ h(f)$. There is a constant $C_ h$, depending only on $h$ and $\mathrm{d}V$, such that the function
\[ \eta _ p(\lambda )=\inf _{K>0} \left(\| p\mathbf1_{\{ p>K\} }\| _{\mathcal D_ h}+C_ hK\lambda \right) \]
tends to zero as $\lambda \downarrow 0$ and satisfies
747F5A0D1221
15.35.6
\begin{equation} \big \| p(1-\mathrm{e}^{\lambda v/n})\big \| _{\mathcal D_ h} \leq \eta _ p(\lambda ) \qquad \bigl (v\in \mathrm{PSH}(X,h),\ \sup _ Xv=0\bigr ). \end{equation}
Let $u_\lambda $, $v_\lambda $ and $c_\lambda $ be the greatest exponential solutions and their normalizations from corollary 15.35.3. If $\eta _ p(\lambda )<R$, then
\[ b\leq c_\lambda \leq \bigl (R-\eta _ p(\lambda )\bigr )^{-n} \]
and
\begin{align*} \big \| \mathrm{MA}^{\mathrm{ext}}_ h(v_\lambda ) -bf\, \mathrm{d}V\big \| _{\mathrm{TV}} & \leq (c_\lambda -b)\| p\| _{L^ n}^{n} +nb\| p\| _{L^ n}^{n-1}\eta _ p(\lambda ). \end{align*}
If $p\in L^ r$ for some finite $r>n$, the same bounds hold with $\eta _ p(\lambda )$ replaced by $C_{h,r}\| p\| _{L^ r}\lambda $, where $C_{h,r}$ depends only on $h$, $\mathrm{d}V$ and $r$. In particular, for each such $p$ the scalar and total-variation errors are $O(\lambda )$.
Proof.
By lemma 5.26.2, normalized $h$-plurisubharmonic functions are uniformly bounded in every $L^ s$ with $s<+\infty $. By (15.32.8) with $r=2n$, their norms $\| v\| _{\mathcal D_ h}$ are therefore uniformly bounded; let $C_ h$ be $1/n$ times this bound. Let $v$ be normalized and $K>0$. On $\{ p\leq K\} $ we have
\[ 0\leq p(1-\mathrm{e}^{\lambda v/n}) \leq K\lambda (-v)/n, \]
since $1-\mathrm{e}^{-x}\leq x$ for $x\geq 0$, while on $\{ p>K\} $ we use the bound $p(1-\mathrm{e}^{\lambda v/n})\leq p$. The monotonicity of $\| \cdot \| _{\mathcal D_ h}$ and the triangle inequality give
\[ \big \| p(1-\mathrm{e}^{\lambda v/n})\big \| _{\mathcal D_ h} \leq \| p\mathbf1_{\{ p>K\} }\| _{\mathcal D_ h}+C_ hK\lambda , \]
and taking the infimum over $K$ proves (15.35.6). By proposition 15.32.9, and since the $L^ n$ tails of $p$ tend to zero, $\| p\mathbf1_{\{ p>K\} }\| _{\mathcal D_ h}\to 0$ as $K\to \infty $. Choosing $K$ first and then letting $\lambda \downarrow 0$ shows that $\eta _ p(\lambda )\to 0$.
Now set $q_\lambda =p\, \mathrm{e}^{\lambda v_\lambda /n}$, so that $q_\lambda ^ n=g_\lambda =\mathrm{e}^{\lambda v_\lambda }f$, with $g_\lambda $ as in the proof of corollary 15.35.3. Since $0\leq q_\lambda \leq p$ and $\rho _ h(p-q_\lambda )\leq \eta _ p(\lambda )$ by (15.35.6), the triangle inequality and monotonicity give
\[ R-\eta _ p(\lambda )\leq \rho _ h(q_\lambda )\leq R. \]
The same proof gives $b\leq c_\lambda \leq b_ h(g_\lambda )=\rho _ h(q_\lambda )^{-n}$, which yields the scalar bound. For the measures, the factorization of $p^ n-q_\lambda ^ n$ gives $0\leq p^ n-q_\lambda ^ n\leq np^{n-1}(p-q_\lambda )$, and Hölder's inequality implies
\[ \| f-g_\lambda \| _{L^1} \leq n\| p\| _{L^ n}^{n-1}\| p-q_\lambda \| _{L^ n} \leq n\| p\| _{L^ n}^{n-1}\eta _ p(\lambda ). \]
Since $\mathrm{MA}^{\mathrm{ext}}_ h(v_\lambda )=c_\lambda g_\lambda \, \mathrm{d}V$, the identity $c_\lambda g_\lambda -bf=(c_\lambda -b)g_\lambda -b(f-g_\lambda )$, together with $\| g_\lambda \| _{L^1}\leq \| f\| _{L^1}=\| p\| _{L^ n}^ n$, proves the total-variation bound.
Finally, suppose $p\in L^ r$ with $n<r<+\infty $, and choose $n<s<r$. By (15.32.8) with exponent $s$, and since $\| \cdot \| _{L^ n}\leq \| \cdot \| _{L^ s}$, we have $\| \cdot \| _{\mathcal D_ h}\leq (1+C_ s)\| \cdot \| _{L^ s}$. Hölder's inequality and lemma 5.26.2 then give, for every normalized $v$,
\[ \| p(-v)\| _{\mathcal D_ h} \leq (1+C_ s)\| p(-v)\| _{L^ s} \leq (1+C_ s)\| p\| _{L^ r}\| v\| _{L^{rs/(r-s)}} \leq nC_{h,r}\| p\| _{L^ r}. \]
Since $0\leq 1-\mathrm{e}^{\lambda v/n}\leq \lambda (-v)/n$, the left side of (15.35.6) is therefore at most $C_{h,r}\| p\| _{L^ r}\lambda $, and the two estimates above can be repeated with this bound in place of $\eta _ p(\lambda )$. For $\lambda $ so small that $C_{h,r}\| p\| _{L^ r}\lambda \leq R/2$, the function $t\mapsto (R-t)^{-n}$ has bounded derivative on $[0,R/2]$, so the scalar error $c_\lambda -b$, and hence the total-variation error, is $O(\lambda )$.
$\square$
For a general $p\in \mathcal D_ h^0$ the modulus $\eta _ p(\lambda )$ tends to zero, but we do not know whether it is $O(\lambda )$. Combined with the lower bound $v_\lambda \geq v-\lambda ^{-1}\log (c_\lambda /b)$ for normalized threshold subsolutions $v$, the proposition gives
\[ 0\leq \lambda ^{-1}\log (c_\lambda /b) \leq -\frac n\lambda \log \bigl (1-\eta _ p(\lambda )/R\bigr ), \]
which stays bounded as $\lambda \downarrow 0$ when $\eta _ p(\lambda )=O(\lambda )$, in particular for $p\in L^ r$ with $r>n$; root tails alone do not give such a bound. Even when a uniform lower barrier for $v_\lambda $ is available, passing to the limit in capacity requires the singularity-type or lower-mass hypotheses of a continuity theorem. The proposition estimates only scalars and measures; when the lower mass vanishes, it does not give an exact equation at the threshold.
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