7.10 Bounded measurable densitiesD4D4A913D86D

Proof. Let $w\in \mathrm{PSH}(D)$ with $-1\le w\le 0$. Then

so the middle set is relatively compact in $D$. Hence every $x\in \partial D$ has a neighborhood on whose intersection with $D$ we have $u\ge v-s+tw$, and in particular $\varliminf _{D\ni z\to x}\bigl (u(z)-(v(z)-s+tw(z))\bigr )\ge 0$. Expanding $(\mathrm{dd}^{\mathrm{c}}v+t\mathrm{dd}^{\mathrm{c}}w)^ n$ by additivity and symmetry of the products (proposition 7.2.2, theorem 7.3.7), and using that all the mixed products of the locally bounded plurisubharmonic functions $v$ and $w$ are positive, we get $\bigl (\mathrm{dd}^{\mathrm{c}}(v-s+tw)\bigr )^ n\ge t^ n(\mathrm{dd}^{\mathrm{c}}w)^ n$. The comparison principle theorem 7.3.11, applied to $u$ and $v-s+tw$, therefore gives

As $w$ ranges over these functions, $w+1$ ranges over the competitors for $\operatorname {Cap}_ D$ (definition 7.4.2). Taking the supremum over $w$ proves the lemma. $\square$

Proof. Throughout the proof, $H_{g,\chi }$ denotes the solution of theorem 7.9.2 for continuous data $(g,\chi )$.

Step 1. Approximation and barriers. Extend $f$ by zero outside $B$, let $M=\| f\| _{L^\infty (B)}$, and convolve $f$ with mollifiers. The resulting smooth functions $f_ j$ on $\mathbb C^ n$ satisfy $0\le f_ j\le M$, and $f_ j\to f$ in $L^1(B)$ by lemma 2.5.1(2), applied on $\mathbb C^ n$ with the compact set $\overline B$; thus, with $\varphi _ j=\varphi $, they satisfy the hypotheses of the second part of the theorem. Let $u_ j=H_{f_ j,\varphi }$. The constant density $M$ and the zero density give two barriers. Indeed, since $M\eta ^ n\ge f_ j\eta ^ n\ge 0$, the function $H_{M,\varphi }$ belongs to the family whose supremum is $u_ j$ in theorem 7.9.2, and $u_ j$ belongs to the family whose supremum is $H_{0,\varphi }$. Hence

Both barriers are continuous on $\overline B$ and equal $\varphi $ on $\partial B$.

By theorem 5.10.4, after passing to a subsequence (not relabeled), $u_ j$ converges in $L^1_{\mathrm{loc}}(B)$ to a plurisubharmonic function $U$; convergence to $-\infty $ is excluded by the lower barrier. For every compact set $K\subset B$ we have $(H_{M,\varphi }-U)_+\le |u_ j-U|$ and $(U-H_{0,\varphi })_+\le |u_ j-U|$, so the integrals of these two functions over $K$ are bounded by $\| u_ j-U\| _{L^1(K)}\to 0$ and hence vanish. Thus $H_{M,\varphi }\le U\le H_{0,\varphi }$ almost everywhere, and hence everywhere by the uniqueness assertion of theorem 5.9.1, applied to $\max \{ U,H_{M,\varphi }\} $ and $U$, and to $\max \{ U,H_{0,\varphi }\} $ and $H_{0,\varphi }$. Consequently $U$ is bounded and $\lim _{B\ni z\to x}U(z)=\varphi (x)$ for every $x\in \partial B$. Since $u_ j$ and $U$ lie between the two bounded barriers, $|u_ j-U|$ is bounded independently of $j$, so the convergence also holds in $L^1(B)$: given $\varepsilon >0$, choose a compact set $K\subset B$ with $|B\setminus K|<\varepsilon $ and use the convergence in $L^1(K)$.

Step 2. Convergence in capacity. Fix a compact set $K\subset B$ and $\delta >0$. We first show that the outer capacity of $K\cap \{ u_ j>U+\delta \} $ tends to zero. Let $\varepsilon >0$. By theorem 7.5.2, there is an open set $G\subset B$ with $\operatorname {Cap}_ B(G)<\varepsilon $ such that the restriction of $U$ to $B\setminus G$ is continuous. We claim that $u_ j<U+\delta $ on the compact set $K\setminus G$ for all sufficiently large $j$. Let $x\in K\setminus G$. By upper semicontinuity, $U<U(x)+\delta /4$ on a neighborhood of a closed ball $\overline{B'}$ centered at $x$, so Hartogs' lemma (theorem 5.10.4), applied with the constant majorant $U(x)+\delta /4$, gives $u_ j<U(x)+\delta /4$ on $\overline{B'}$ for all large $j$. Shrinking $B'$, we may also assume that $U>U(x)-\delta /4$ on $(K\setminus G)\cap B'$, since the restriction of $U$ to $B\setminus G$ is continuous. Hence $u_ j-U<\delta /2$ on $(K\setminus G)\cap B'$ for all large $j$, and a finite cover of $K\setminus G$ by such balls proves the claim. Consequently $K\cap \{ u_ j>U+\delta \} \subset G$ for all large $j$, so that $\varlimsup _ j\operatorname {Cap}_ B^*(K\cap \{ u_ j>U+\delta \} )\le \operatorname {Cap}_ B(G)<\varepsilon $. Since $\varepsilon >0$ was arbitrary, the limit is zero.

Next we consider the set where $u_ j$ lies below $U$. Since $H_{M,\varphi }\le u_ j$ and $U\le H_{0,\varphi }$, the set $\{ u_ j<U-\delta \} $ is contained in $\{ H_{M,\varphi }-H_{0,\varphi }\le -\delta \} $, which is a compact subset of $B$ because the continuous function $H_{M,\varphi }-H_{0,\varphi }$ on $\overline B$ vanishes on $\partial B$. Lemma 7.10.1, applied with $s=t=\delta $ to $u_ j$ and $U$, gives

because $(\mathrm{dd}^{\mathrm{c}}u_ j)^ n=f_ j\eta ^ n\le M\eta ^ n$, $U-u_ j>\delta $ on the domain of integration, $\eta ^ n=n!\pi ^{-n}\, \mathrm dV_{2n}$, and $u_ j\to U$ in $L^1(B)$. To pass from capacity to the outer capacity of definition 7.5.5, we use quasi-continuity. The function $(U-u_ j)_+$ is bounded and quasi-continuous: $U$ and $u_ j$ are real and quasi-continuous by theorem 7.5.2, so their difference, and its composition with the continuous map $t\mapsto t_+$, are quasi-continuous by the remark following definition 7.5.1. Let $L\subset B$ be a compact neighborhood of $K$. By lemma 7.5.4,

As $K$ and $\delta $ are arbitrary, $u_ j\to U$ locally in capacity in the sense of definition 7.5.5.

Step 3. The equation, the envelope and uniqueness. The functions $u_ j$ are uniformly bounded and converge to $U$ locally in capacity, so theorem 7.5.8 gives $(\mathrm{dd}^{\mathrm{c}}u_ j)^ n\to (\mathrm{dd}^{\mathrm{c}}U)^ n$ weakly. Since $f_ j\to f$ in $L^1(B)$, also $f_ j\eta ^ n\to f\eta ^ n$ weakly, and therefore $(\mathrm{dd}^{\mathrm{c}}U)^ n=f\eta ^ n$.

Let $v$ be a bounded plurisubharmonic function on $B$ with $(\mathrm{dd}^{\mathrm{c}}v)^ n\ge f\eta ^ n$ and boundary upper limits at most $\varphi $, and put $v_\varepsilon =v+\varepsilon (|z-a|^2-R^2)$ for $\varepsilon >0$. Expanding $(\mathrm{dd}^{\mathrm{c}}v+\varepsilon \eta )^ n$ and using positivity of the mixed products, we get $(\mathrm{dd}^{\mathrm{c}}v_\varepsilon )^ n\ge (f+\varepsilon ^ n)\eta ^ n$. Since $\varliminf _{z\to x}(U(z)-v_\varepsilon (z))\ge 0$ for every $x\in \partial B$, the comparison principle theorem 7.3.11 gives

These integrals are finite, so $\{ U<v_\varepsilon \} $ is Lebesgue null. By the uniqueness assertion of theorem 5.9.1, applied to $\max \{ U,v_\varepsilon \} $ and $U$, we obtain $U\ge v_\varepsilon $ everywhere, and letting $\varepsilon \downarrow 0$ gives $U\ge v$. Since $U$ itself is such a function $v$, it is the pointwise supremum in the statement; in particular, it does not depend on the subsequence chosen in Step 1. If $U'$ is another bounded solution with boundary limits $\varphi $, then $U'\le U$, and the same argument with $U'$ in place of $U$ gives $U\le U'$. This proves existence and uniqueness, and we write $U_{f,\varphi }=U$.

Step 4. Stability. Let $f_ j$, $\varphi _ j$ and $M$ be as in the second part of the theorem, and suppose first that $\varphi _ j=\varphi $ for all $j$. By the envelope characterization just proved, $H_{M,\varphi }\le U_{f_ j,\varphi }\le H_{0,\varphi }$. Steps 1–3 apply without change to the sequence $U_{f_ j,\varphi }$, since they used only these barrier inequalities, the equations $(\mathrm{dd}^{\mathrm{c}}U_{f_ j,\varphi })^ n=f_ j\eta ^ n$ with $0\le f_ j\le M$, and the convergence $f_ j\to f$ in $L^1(B)$. Hence every subsequence of $U_{f_ j,\varphi }$ has a further subsequence converging in $L^1(B)$ and locally in capacity to a bounded solution with data $(f,\varphi )$, that is, to $U_{f,\varphi }$. Therefore the whole sequence converges to $U_{f,\varphi }$ in $L^1(B)$ and locally in capacity.

For varying boundary data we use that, for a fixed density $f$, the function $U_{f,\chi }-\| \varphi -\chi \| _{L^\infty (\partial B)}$ belongs to the family whose supremum is $U_{f,\varphi }$; by symmetry, $\| U_{f,\varphi }-U_{f,\chi }\| _{L^\infty (B)}\le \| \varphi -\chi \| _{L^\infty (\partial B)}$. Applied with $f_ j$ in place of $f$, this gives $\| U_{f_ j,\varphi _ j}-U_{f_ j,\varphi }\| _{L^\infty (B)}\le \| \varphi _ j-\varphi \| _{L^\infty (\partial B)}\to 0$. Since $B$ is bounded, it follows that $U_{f_ j,\varphi _ j}-U_{f_ j,\varphi }\to 0$ in $L^1(B)$. Moreover, for every $\delta >0$ and all large $j$, the set $\{ U_{f_ j,\varphi _ j}>U_{f,\varphi }+\delta \} $ is contained in $\{ U_{f_ j,\varphi }>U_{f,\varphi }+\delta /2\} $, and similarly for the sets where the functions lie below $U_{f,\varphi }-\delta $. The general case therefore follows from the case of fixed boundary data.

Finally, we prove (7.9.3) for bounded measurable densities. Let $f,g\in L^\infty (B)$ be nonnegative and $\varphi ,\chi \in C(\partial B)$. Extend $f^{1/n}$ and $g^{1/n}$ by zero outside $B$, and convolve them with the same mollifiers. The resulting smooth functions $F_ j,G_ j\ge 0$ satisfy $\| F_ j-G_ j\| _{L^\infty (B)}\le \| f^{1/n}-g^{1/n}\| _{L^\infty (B)}$. Moreover $F_ j\to f^{1/n}$ and $G_ j\to g^{1/n}$ in $L^1(B)$ with a common bound, so $F_ j^ n\to f$ and $G_ j^ n\to g$ in $L^1(B)$ with a common bound. By theorem 7.9.2,

By uniqueness, $H_{F_ j^ n,\varphi }=U_{F_ j^ n,\varphi }$ and $H_{G_ j^ n,\chi }=U_{G_ j^ n,\chi }$, and by the stability just proved these functions converge in $L^1(B)$ to $U_{f,\varphi }$ and $U_{g,\chi }$. Since $|H_{F_ j^ n,\varphi }-H_{G_ j^ n,\chi }|\le c$ on $B$, the triangle inequality gives

and integrating over $B$ and letting $j\to \infty $, we obtain $|U_{f,\varphi }-U_{g,\chi }|\le c$ almost everywhere. The uniqueness assertion of theorem 5.9.1, applied to $\max \{ U_{f,\varphi },U_{g,\chi }+c\} $ and $U_{g,\chi }+c$, and to $\max \{ U_{g,\chi },U_{f,\varphi }+c\} $ and $U_{f,\varphi }+c$, shows that this inequality holds everywhere. $\square$