Pome

Proposition 11.31.423E45E6251B0

Proposition 11.31.4.23E45E6251B0 A positive measure $\nu $ of mass $M_{\theta ,\gamma }$ belongs to the image of $\mathcal T_\gamma $ if and only if the distribution

\[ w=G_\gamma \bigl (\nu -\theta \wedge \gamma ^{n-1}\bigr ) \]

satisfies $\theta +\mathrm{dd}^{\mathrm{c}}w\geq 0$. In that case the $\theta $-plurisubharmonic representative of $w$ is the unique $u\in \mathcal K^0_{\theta ,\mathrm{d}V}$ with $\mathcal T_\gamma (u)=\nu $.

Proof. If $\nu =\mathcal T_\gamma (u)$ with $u\in \mathcal K^0_{\theta ,\mathrm{d}V}$, then $u$ is a mean-zero solution of (11.31.3) with $\sigma =\nu -\theta \wedge \gamma ^{n-1}$, so uniqueness gives $w=u$ as distributions. Conversely, if $\theta +\mathrm{dd}^{\mathrm{c}}w\geq 0$, then proposition 5.22.2 gives a $\theta $-plurisubharmonic representative $u$ of $w$. It has mean zero by construction, and (11.31.3) gives $\mathcal T_\gamma (u)=\nu $. $\square$

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