Corollary 13.30.5.8918C681057E For $f\in C^\infty (Y,\mathbb R)$, the map
\[ \mathcal G_ f(\sigma ,p) =(\sigma +\mathrm{dd}^{\mathrm{c}}f,P_{\sigma +\mathrm{dd}^{\mathrm{c}}f}[p-f]) \]
is an isometry of $\mathscr M(Y)$, with inverse $\mathcal G_{-f}$ and $\mathcal G_ g\mathcal G_ f=\mathcal G_{f+g}$. It preserves analytic models and commutes with $R$.
For $\lambda >0$, the bijection $\mathcal H_\lambda (\sigma ,p)=(\lambda \sigma ,\lambda p)$ commutes with $R$ and satisfies
\[ \min \{ 1,\lambda \} D(x,y) \leq D(\mathcal H_\lambda x,\mathcal H_\lambda y) \leq \max \{ 1,\lambda \} D(x,y). \]
Proof.
The gauge transport in proposition 13.29.1 preserves the polar measure and, after subtraction of $f$, the ordinary type. In the coordinates $(\sigma ,\mu )$ of the proof of theorem 13.30.3, $\mathcal G_ f$ is therefore the map $(\sigma ,\mu )\mapsto (\sigma +\mathrm{dd}^{\mathrm{c}}f,\mu )$. It does not change the area, and $b(\sigma +\mathrm{dd}^{\mathrm{c}}f,\tau +\mathrm{dd}^{\mathrm{c}}f)=b(\sigma ,\tau )$. This proves that $\mathcal G_ f$ is an isometry and the laws for inverses and compositions. Taking the atomic part commutes with this map, and subtraction of a smooth function and bounded changes preserve analytic singularities.
Scaling the competitors of the ordinary envelope shows that $\lambda p$ is an ordinary model for $\lambda \sigma $. In the coordinates $(\sigma ,\mu )$, $\mathcal H_\lambda $ is the map $(\sigma ,\mu )\mapsto (\lambda \sigma ,\lambda \mu )$, with inverse $\mathcal H_{\lambda ^{-1}}$, and taking the atomic part commutes with scaling. For $r\geq 0$,
\[ \min \{ 1,\lambda \} \min \{ 1,r\} \leq \min \{ 1,\lambda r\} \leq \max \{ 1,\lambda \} \min \{ 1,r\} . \]
Apply this to every $C^ k$ norm in (13.30.1); the total-variation term is multiplied by $\lambda $. The two bounds follow.
$\square$
Pome
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