Proof.
First observe that by proposition 18.28.45, for any $k\geq 1$, $N^ k_{w,\omega ^ n}$ and $N^ k_{w',\omega ^ n}$ are both norms, hence the expressions inside the limit in (18.28.60) make sense.
We first assume that $w,w'\in C^\infty (X)$. To start, we make the following observation:
\[ \begin{aligned} \mathcal{L}_{k,\varphi }(N^ k_{w,\omega ^ n}) - \mathcal{L}_{k,\varphi }(N^ k_{w',\omega ^ n})& = \int _0^1 \frac{\mathrm{d}}{\mathrm{d}t} \mathcal{L}_{k,\varphi }(N^ k_{w' + t(w-w'),\omega ^ n})\, \mathrm{d}t\\ & = \int _0^1 \int _ X (w-w') \, \beta ^ k_{w' + t(w-w'),\varphi ,\omega ^ n}\, \mathrm{d}t. \end{aligned} \]
The second equality follows from differentiating the Gram matrix of the Hilbert norm $N^ k_{w'+t(w-w'),\omega ^ n}$:
\[ \frac{\mathrm d}{\mathrm dt}\mathcal L_{k,\varphi } \left(N^ k_{w'+t(w-w'),\omega ^ n}\right)=\int _ X (w-w')\, \beta ^ k_{w'+t(w-w'),\varphi ,\omega ^ n}. \]
By proposition 18.28.51, we have
\[ \lim _{k\to \infty }\int _ X (w-w') \, \beta ^ k_{w' + t(w-w'),\varphi ,\omega ^ n}=\int _ X (w-w')\, \theta _{P_{\theta ,X}[\varphi ](w' + t(w-w'))}^ n. \]
By theorem 18.1.46, we have
\[ \bigl |\int _ X (w-w')\beta ^ k_{w' + t(w-w'),\varphi ,\omega ^ n}\bigr | \leq C \sup _ X |w - w'|. \]
Hence, by the dominated convergence theorem we obtain that
\[ \begin{aligned} \lim _{k\to \infty } \left(\mathcal{L}_{k,\varphi }(N^ k_{w,\omega ^ n}) - \mathcal{L}_{k,\varphi }(N^ k_{w',\omega ^ n})\right) & = \int _0^1 \int _ X (w-w') \, \theta _{P_{\theta ,X}[\varphi ](w' + t(w-w'))}^ n \, \mathrm{d}t\\ & = \mathcal{E}_{\theta ,X}^{\varphi }(w)-\mathcal{E}_{\theta ,X}^{\varphi }(w'), \end{aligned} \]
where in the last line we have used proposition 18.28.27.
This proves (18.28.60) for smooth weights.
For general $w,w'\in C^0(X)$, choose smooth functions $w_ j,w'_ j$ converging uniformly to $w,w'$, respectively. If $\| w_ j-w\| _{C^0}\leq \delta _ j$, then
\[ \mathrm{e}^{-k\delta _ j/2}N^ k_{w,\omega ^ n}\leq N^ k_{w_ j,\omega ^ n}\leq \mathrm{e}^{k\delta _ j/2}N^ k_{w,\omega ^ n}, \]
and hence
\[ \left|\mathcal{L}_{k,\varphi }(N^ k_{w_ j,\omega ^ n})-\mathcal{L}_{k,\varphi }(N^ k_{w,\omega ^ n})\right| \leq \frac{n!}{k^ n}h^0\left(X,L^ k\otimes T\otimes \mathcal{I}(k\varphi )\right)\delta _ j. \]
The same estimate holds for $w'_ j$ and $w'$. Moreover, proposition 18.28.27 shows that the energy side is Lipschitz with respect to the uniform norm of the weight. Letting first $k\to \infty $ in the smooth case, and then $j\to \infty $, proves (18.28.60) for continuous weights. Equation (18.28.61) again follows by taking $w'=0$.
$\square$
Pome
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