PomeLemma 8.6.4.EBF9BA687F39 Every bounded upper semicontinuous real function $u$ on $X$ is the pointwise decreasing limit of smooth functions $h_ j>u$.
8.6.3 Decreasing approximation of bounded potentialsABD764A7564F
Upper semicontinuity supplies smooth obstacles decreasing to a bounded potential. The same penalty equation gives admissible functions between the potential and each obstacle, up to an error tending to zero. Choosing successive obstacles below the preceding smooth approximation makes the admissible approximations decrease as well.
Proof. Fix a Riemannian distance $d$ on $X$ and put
\[ c_ j(x)=\sup _{y\in X}\{ u(y)-j d(x,y)\} . \]
Each $c_ j$ is Lipschitz, $c_ j\geq u$, and $c_{j+1}\leq c_ j$. The supremum is attained. If $y_ j$ attains it at a fixed $x$, then $j d(x,y_ j)\leq \sup _ X u-u(x)$, so $y_ j\to x$. Upper semicontinuity gives $\limsup _ j c_ j(x)\leq u(x)$, and hence $c_ j\downarrow u$ pointwise.
Set $\delta _ j=4^{-j}$. Smooth approximation of continuous functions on a compact manifold gives $s_ j\in C^\infty (X)$ with $\| s_ j-c_ j\| _\infty \leq \delta _ j/4$. Take $h_ j=s_ j+\delta _ j$. Then
\[ u+\tfrac 34\delta _ j\leq h_ j\leq c_ j+\tfrac 54\delta _ j, \qquad h_{j+1}\leq c_ j+\tfrac 5{16}\delta _ j < c_ j+\tfrac 34\delta _ j\leq h_ j. \]
The bounds also imply $h_ j(x)\to u(x)$ at every point. $\square$
Lemma 8.6.5.DFE78235C5E9 Let $u$ be bounded admissible and $h\geq u$ smooth. Choose
\[ b>\max \{ 1,\| H_ m(h)/\omega ^ n\| _\infty \} . \]
For each positive integer $N$, let $w_ N$ be the smooth strictly admissible solution of
\[ H_ m(w_ N)=b e^{N(w_ N-h)}\omega ^ n. \]
With
\[ M_ N=\| h\| _\infty +\frac{\log b}{N},\qquad E_ N=\frac{m\log (N+1)+\log b}{N}, \]
one has $\| w_ N\| _\infty \leq M_ N$ and
\[ u-E_ N-2M_ N/N\leq w_ N\leq h. \]
Proof. Existence follows from 8.6.1. The maximum comparison with $h$ in the proof of 8.6.2 gives $w_ N\leq h$; it involves only the smooth obstacle and solution. The zero-order bound from 8.6.1 gives $\| w_ N\| _\infty \leq M_ N$.
The upper semicontinuous function $u-(1+N^{-1})w_ N$ attains its maximum at some $x_ N$. The smooth function $(1+N^{-1})w_ N$ plus this maximum is an upper test for $u$ there. Put
\[ T_ N=(\omega +\mathrm{dd}^{\mathrm{c}}w_ N)^{m-1}\wedge \omega ^{n-m}, \qquad d_ N=H_ m(w_ N)/\omega ^ n>0. \]
Condition (8.2.2) and polarized positivity yield, at $x_ N$,
\[ (N+1)d_ N\geq \omega \wedge T_ N/\omega ^ n \geq d_ N^{(m-1)/m}. \]
Consequently $d_ N\geq (N+1)^{-m}$ and $w_ N-h\geq -E_ N$ there. Using $u\leq h$, the maximum is at most $E_ N+M_ N/N$. On all of $X$ this gives
\[ u\leq (1+N^{-1})w_ N+E_ N+M_ N/N \leq w_ N+E_ N+2M_ N/N. \]
$\square$
Theorem 8.6.6.68BE4F589DCE Every bounded admissible potential $u$ on $X$ is the pointwise decreasing limit of smooth strictly admissible potentials. The sequence may be chosen uniformly bounded.
Proof. Choose smooth $h_ k\downarrow u$ by 8.6.4 and put $\varepsilon _ j=8^{-j}$. We construct smooth strictly admissible $u_ j$ with $u_ j\geq u+\varepsilon _ j$. For $j=1$, take $k_1=1$. At a subsequent step choose $k_ j\geq \max \{ j,k_{j-1}+1\} $ so large that
\[ h_{k_ j}<u_{j-1}-\tfrac 12\varepsilon _{j-1}\quad \hbox{on }X. \]
Such an index exists: the open sets $\{ h_ k<u_{j-1}-\varepsilon _{j-1}/2\} $ increase with $k$ and cover $X$, since $h_ k\downarrow u\leq u_{j-1}-\varepsilon _{j-1}$. Compactness gives a single index for all of $X$.
For this fixed obstacle $h_{k_ j}$ choose $b_ j$ as in 8.6.5, and then choose $N_ j$ so large that its error $E_{N_ j}+2M_{N_ j}/N_ j$ is at most $\varepsilon _ j$. The resulting smooth solution $w_ j$ satisfies
\[ u-\varepsilon _ j\leq w_ j\leq h_{k_ j}. \]
Put $u_ j=w_ j+2\varepsilon _ j$. Adding a constant preserves strict admissibility, and $u_ j\geq u+\varepsilon _ j$. For $j\geq 2$,
\[ u_ j\leq h_{k_ j}+2\varepsilon _ j <u_{j-1}-\tfrac 12\varepsilon _{j-1}+2\varepsilon _ j =u_{j-1}-\tfrac 14\varepsilon _{j-1}. \]
Finally, $u\leq u_ j\leq h_{k_ j}+2\varepsilon _ j$ and $k_ j\geq j$ give pointwise convergence to $u$. The bounds $\inf _ X u\leq u_ j\leq \sup _ X u_1$ give a common uniform bound. $\square$
For continuous $u$, Dini's theorem makes these decreasing approximations converge uniformly on $X$. This is the convergence used for the continuous mixed products in the next section.
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