17.4 Continuity and compactness3B8286222A1B

Proof. By lemma 16.22.2, $u_ i\leq \phi +\sup _ Xu_ i$, and the suprema $\sup _ Xu_ i$ are bounded above by hypothesis; hence $u\preceq \phi $. Since $u$ dominates every $u_ i$, monotonicity of the energy and theorem 17.3.14 show that $u\in \mathcal E^1(X,\theta ,\phi )$.

Bounded differences. Consider first an increasing sequence $(u_ j)$ and a constant $C\geq 1$ such that $\phi -C\leq u_ j\leq u\leq \phi $ for all $j$. The pole sets of $u_ j$ and $u$ coincide with that of $\phi $. Let $f_ j$ be the function equal to $u-u_ j$ outside this pole set and to $0$ on it; it is a Borel function with values in $[0,C]$. By the tangent bound (17.3.10),

and we show that the right side tends to zero by applying lemma 17.2.2 with $\nu _ j=\mu _{u_ j}$, $\nu =\mu _ u$, the weights $f_ j$ and the limit weight $0$. For each $j$ the potential $C^{-1}u_ j+(1-C^{-1})\phi $ lies between $\phi -1$ and $\phi $ and its measure is at least $C^{-n}\mu _{u_ j}$; hence $\mu _{u_ j}\leq C^ n\operatorname {Cap}_{\theta ,\phi }$ on Borel sets, and likewise $\mu _ u\leq C^ n\operatorname {Cap}_{\theta ,\phi }$. By theorem 16.27.17, the measures $\mu _{u_ j}$ and $\mu _ u$ are therefore uniformly absolutely continuous with respect to $\operatorname {Cap}_\omega $. They converge weakly by corollary 13.8.9, since $u_ j$ increases to $u$ quasi-everywhere (corollary 5.47.11).

It remains to show that $f_ j\to 0$ in capacity. Fix finitely many pairs of concentric coordinate balls $B_\alpha \Subset B'_\alpha $ as in proposition 16.27.18, chosen so that $\theta $ has a smooth potential $h_\alpha $ on a neighborhood of $\overline{B'_\alpha }$ (this is possible by lemma 6.14.2, applied on slightly larger concentric balls), and let $c$ be the constant of that proposition. Let $\delta ,\eta >0$. By proposition 16.27.18(3) there is an open set $G$ with $\operatorname {Cap}_\omega (G)<\eta $ which contains the pole set of $\phi $ and such that $\phi $ is finite and continuous on the compact set $X\setminus G$. Let $L$ be a common bound for $|\phi |$ on $X\setminus G$ and for the functions $|h_\alpha |$ on $B'_\alpha $. The functions

are bounded plurisubharmonic functions on $B'_\alpha $, and $w_{\alpha ,j}$ increases to $w_\alpha $ almost everywhere, because $u_ j$ increases to $u$ quasi-everywhere and pluripolar sets have Lebesgue measure zero. By theorem 7.6.1, $w_{\alpha ,j}\to w_\alpha $ locally in capacity on $B'_\alpha $. On $B_\alpha \setminus G$ we have $u_ j+h_\alpha \geq \phi -C-L\geq -2L-C$, so the truncations are inactive there and $w_\alpha -w_{\alpha ,j}=u-u_ j$. Since the balls $B_\alpha $ cover $X$,

By the countable subadditivity of $\operatorname {Cap}_\omega $ (proposition 16.25.6) and the bound $\operatorname {Cap}_\omega (E)\leq c\operatorname {Cap}_{B'_\alpha }(E)$ for Borel sets $E\subset B'_\alpha $ in proposition 16.27.18,

where we also used that the capacity of a Borel set is at most its outer capacity. By definition 7.5.5, the sum tends to zero as $j\to \infty $. Hence $\varlimsup _ j\operatorname {Cap}_\omega (\{ f_ j>\delta \} )\leq \eta $, and since $\eta $ is arbitrary, $f_ j\to 0$ in capacity. Lemma 17.2.2 now gives $\int _ Xf_ j\, \mu _{u_ j}\to 0$, and hence $E_\theta ^\phi (u_ j)\to E_\theta ^\phi (u)$.

Sequences. Now let $(u_ j)$ be an arbitrary increasing sequence; subtracting a common constant, we may assume $u_ j\leq u\leq \phi $. Put $u_{j,k}=\max \{ u_ j,\phi -k\} $ and $u_ k=\max \{ u,\phi -k\} $. For fixed $k$, the sequence $u_{j,k}$ increases to $\max \{ \sup _ ju_ j,\phi -k\} $, whose upper semicontinuous regularization is $\max \{ (\sup _ ju_ j)^*,\phi -k\} =u_ k$, because $\phi -k$ is upper semicontinuous; moreover $\phi -k\leq u_{j,k}\leq u_ k\leq \phi $. The preceding case, with $\max \{ k,1\} $ in place of $C$, gives $E_\theta ^\phi (u_{j,k})\to E_\theta ^\phi (u_ k)$ as $j\to \infty $. We bound the truncation error uniformly in $j$. The tangent bound of proposition 17.3.19 gives

For $t\geq k$ put $q_ t=(u_ j+\phi -t)/2$. On $\{ u_ j<\phi -t\} $ we have $u_1\leq u_ j<q_ t$, and $q_ t\leq \phi -t/2$ on $X$. Since $\mu _{q_ t}\geq 2^{-n}\mu _{u_ j}$ by multilinearity, and $q_ t,u_1\in \mathcal E(X,\theta ,\phi )$, the comparison principle of proposition 16.25.4 gives

Consequently

as $k\to \infty $, uniformly in $j$; the limit is zero by dominated convergence, since $\phi -u_1$ is $\mu _{u_1}$-integrable. Since $E_\theta ^\phi (u_ k)\geq E_\theta ^\phi (u)$, letting first $j\to \infty $ and then $k\to \infty $ gives $\lim _ jE_\theta ^\phi (u_ j)\geq E_\theta ^\phi (u)$, and monotonicity gives the reverse inequality.

Nets. Let $(u_ i)_{i\in I}$ be a net as in the statement. Cover $X$ by finitely many coordinate charts $D_\beta $ on which $\theta $ has smooth potentials $h_\beta $. By theorem 5.10.2, applied to the functions $u_ i+h_\beta $ on $D_\beta $, there are countable sets $J_\beta \subset I$ such that $(\sup _{i\in J_\beta }u_ i)^*=u$ on $D_\beta $. The countable set $J=\bigcup _\beta J_\beta $ then satisfies $(\sup _{i\in J}u_ i)^*=u$ on $X$, since on each $D_\beta $ this envelope lies between the envelopes over $J_\beta $ and over $I$. Since the index set is directed, we can choose indices $i_1\leq i_2\leq \cdots $ such that $i_ k$ dominates the first $k$ elements of an enumeration of $J$. Since the net is increasing, $\sup _ ku_{i_ k}\geq \sup _{i\in J}u_ i$, so the increasing sequence $(u_{i_ k})$ has regularized upper envelope $u$, and the sequence case gives $E_\theta ^\phi (u_{i_ k})\to E_\theta ^\phi (u)$. Each of these energies is at most $\sup _ iE_\theta ^\phi (u_ i)$, which in turn is at most $E_\theta ^\phi (u)$ by monotonicity. Hence $\sup _ iE_\theta ^\phi (u_ i)=E_\theta ^\phi (u)$, and since the net of energies is increasing, its limit is this supremum. (The selected indices need not be cofinal in $I$, which is why we compare suprema.) $\square$

Proof. By proposition 10.3.1 the suprema $\sup _ Xq_ k$ converge to $\sup _ Xq$, so they are bounded. Hence the tail envelopes $a_ j$ are $\theta $-psh, by theorem 5.10.2 applied in charts after adding smooth local potentials of $\theta $, and they decrease as $j$ increases. A subsequence of $(q_ k)$ converges to $q$ almost everywhere, so $a_ j\geq q$ almost everywhere, hence everywhere, since both functions are $\theta $-psh (section 16.11).

Conversely, let $x\in X$ and $\varepsilon >0$, let $h$ be a smooth local potential of $\theta $ on a coordinate ball $\Omega $ around $x$, and let $M>q(x)+h(x)$ be a real number. By upper semicontinuity there is a closed coordinate ball $K\subset \Omega $ around $x$ on which $q+h<M$ and $|h-h(x)|<\varepsilon $. The plurisubharmonic functions $q_ k+h$ converge to $q+h$ in $L^1_{\mathrm{loc}}(\Omega )$, so Hartogs' lemma in the form of theorem 5.10.4, applied with the constant function $M$ on $K$, gives $q_ k+h<M$ on $K$ for all large $k$. For such $k$ we have $q_ k<M-h(x)+\varepsilon $ on $K$. Since $K$ is a neighborhood of $x$, it follows that $a_ j(x)\leq M-h(x)+\varepsilon $ for all large $j$. Letting $\varepsilon \downarrow 0$ and $M\downarrow q(x)+h(x)$, or $M\to -\infty $ if $q(x)=-\infty $, we obtain $\lim _ ja_ j(x)\leq q(x)$. Thus $a_ j\downarrow q$.

Finally, if $q_ k\leq \psi $ for all $k$ with $\psi $ upper semicontinuous, then $\sup _{k\geq j}q_ k\leq \psi $, so $a_ j\leq \psi $ for every $j$, and therefore $q=\lim _ ja_ j\leq \psi $. $\square$

Proof. Let $u_ j\preceq \phi $ converge to $u$ in $L^1(X)$. By lemma 17.4.3, the $u_ j$ are uniformly bounded above, so lemma 16.22.2 gives a constant $C$ with $u_ j\leq \phi +C$ for all $j$. Since $\phi +C$ is upper semicontinuous, the same lemma gives $u\leq \phi +C$, which proves closedness. The lemma also shows that the tail envelopes $v_ j=(\sup _{k\geq j}u_ k)^*$ are $\theta $-psh potentials which decrease pointwise to $u$; moreover $v_ j\leq \phi +C$, because $\phi +C$ is upper semicontinuous and dominates every $u_ k$. Since $u_ j\leq v_ j$, monotonicity and proposition 17.3.13 give

Since $L^1(X)$ is a metric space, sequential upper semicontinuity is upper semicontinuity. $\square$

Proof. For $u$ in the first set, lemma 16.22.2 gives $u\leq \phi +\sup _ Xu$, so monotonicity and the shift formula give $-B\leq E_\theta ^\phi (u)\leq m\sup _ Xu$. Also $\sup _ Xu\leq \sup _ X\phi =0$. Hence the set lies in $\{ u\in \mathrm{PSH}(X,\theta ):-B/m\leq \sup _ Xu\leq 0\} $, which is compact in $L^1(X)$ by proposition 10.3.1. The set is closed in $L^1(X)$: the inequality $u\leq \phi $ passes to $L^1$ limits by lemma 17.4.3, since $\phi $ is upper semicontinuous, and the energy bound passes to limits by upper semicontinuity (proposition 17.4.4). It is therefore compact. Its members have finite energy, so they lie in $\mathcal E^1(X,\theta ,\phi )$ by theorem 17.3.14.

The set $K_\phi $ is contained in $\{ u\in \mathrm{PSH}(X,\theta ):\sup _ Xu=0\} $, which is compact in $L^1(X)$ by proposition 10.3.1, and it is closed in $L^1(X)$, because the supremum is continuous for $L^1$ convergence of potentials (by the same proposition) and the inequality $u\leq \phi $ passes to $L^1$ limits by lemma 17.4.3. Hence $K_\phi $ is compact. $\square$

Proof. By lemma 16.22.2, every member of the family is bounded above by $\phi +C$ for a common constant $C$, and so is the regularized upper envelope $U$. Fix a member $u_0$ of the family; then $u_0\leq U\preceq \phi $. For $\mathcal F=\mathcal E(X,\theta ,\phi )$, mass monotonicity gives $m=\int _ X\langle \theta _{u_0}^ n\rangle \leq \int _ X\langle \theta _ U^ n\rangle \leq m$. For $\mathcal F=\mathcal E^1(X,\theta ,\phi )$, the last assertion of corollary 17.3.17, applied to $u_0\leq U$, gives $U\in \mathcal F$. For $\mathcal F=\mathcal E^\infty (X,\theta ,\phi )$, the member $u_0$ supplies a lower bound $U\geq u_0\geq \phi -C'$. $\square$

Proof. Since $u\leq \max \{ u,v\} \preceq \phi $, mass monotonicity gives $\int _ X\langle \theta _{\max \{ u,v\} }^ n\rangle =m$. Thus $\int _ X\langle \theta _ u^ n\rangle +\int _ X\langle \theta _ v^ n\rangle =2m>\int _ X\langle \theta _{\max \{ u,v\} }^ n\rangle $, and proposition 16.23.1 shows that $w\neq -\infty $. Theorem 16.23.5 gives $2m\leq m+\int _ X\langle \theta _ w^ n\rangle $, while $\int _ X\langle \theta _ w^ n\rangle \leq m$ by mass monotonicity, since $w\leq u$. Hence $w\in \mathcal E(X,\theta ,\phi )$.

Now let $u,v\in \mathcal E^1(X,\theta ,\phi )$. The rooftop commutes with adding a common constant to $u$ and $v$ (proposition 16.11.4(3)), so we may assume $u,v\leq \phi $; then $w\leq \phi $. The contact inequality of corollary 16.13.5 gives

Integrate the nonnegative weight $\min \{ \phi -w,N\} $ against both sides. On the first contact set the weight equals $\min \{ \phi -u,N\} $, and on the second it equals $\min \{ \phi -v,N\} $; pole sets carry no mass. Letting $N\to \infty $ gives the displayed bound. Both terms on the right are finite, so $w\in \mathcal E^1(X,\theta ,\phi )$. $\square$

Proof. By proposition 17.4.7, the rooftop $w=P_\theta (u,v)$ belongs to $\mathcal E(X,\theta ,\phi )$. Since $w\preceq v$ and $\int _ X\langle \theta _ w^ n\rangle =\int _ X\langle \theta _ v^ n\rangle >0$, proposition 16.22.14 gives $P_\theta [w](v)=v$. Every competitor for $P_\theta [w](v)$ is more singular than $w$, hence than $u$, so it is a competitor for $P_\theta [u](v)$; and $P_\theta [u](v)\leq v$ quasi-everywhere by proposition 16.25.2(1), hence everywhere. Therefore $v=P_\theta [w](v)\leq P_\theta [u](v)\leq v$. $\square$

Proof. By lemma 17.3.18 every integral $\int _ X(v-u)\langle \theta _ v^ a\wedge \theta _ u^{n-a}\rangle $ is finite. Expanding $\mu _{u_ t}$ by multilinearity shows that the right side is a polynomial in $t$ with these integrals as coefficients. The potentials $u_ t$ lie in $\mathcal E^1(X,\theta ,\phi )$ by corollary 17.3.17. Apply the cocycle formula of proposition 17.3.19 to $u_{t+s}$ and $u_ t$, with $s\neq 0$ and $t,t+s\in [0,1]$, and use $u_{t+s}-u_ t=s(v-u)$:

Each mixed integral on the right is a polynomial in $s$ with the finite coefficients above, and at $s=0$ each of the $n+1$ terms equals $\int _ X(v-u)\mu _{u_ t}$. Letting $s\to 0$ proves the formula for the derivative, including the one-sided derivatives at the endpoints, and its continuity. $\square$

Proof. The first display is the tangent bound (17.3.10), which holds on $\mathcal E^1(X,\theta ,\phi )$ by proposition 17.3.19. By the same proposition, the cocycle formula (17.3.6) holds for $u$ and $v$. If $u\leq v$, all its terms are nonnegative, and retaining the term $j=0$, whose measure is $\mu _ u$, gives the second display. $\square$