Proposition 15.35.5AC5E6031A12C

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

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

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

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

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 $v$,

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 $O(\lambda )$. $\square$