The uniqueness easily follows from the comparison principle as we explain in proposition 4.3 below. We are going to prove the existence by a fixed point method.

Fix $\psi\in{\mathcal{E}}^{1}(X,\omega)$ such that $\int_{X}\psi d\mu=0$, and let us consider the equation

$$MA(\psi)\hskip 56.9055pt(\omega+dd^{c}\varphi)^{n}=e^{t\psi-c_{\psi}}\mu,$$

where the constant $c_{\psi}:=\log[\int_{X}e^{t\psi}d\mu]$ is chosen so that

$$1=\int_{X}(\omega+dd^{c}\varphi)^{n}=e^{-c_{\psi}}\int_{X}e^{t\psi}d\mu.$$

Observe that $\mu_{\psi}:=e^{t\psi-c_{\psi}}\mu$ satisfies condition ${\mathcal{H}}(\alpha,A_{\psi},\omega)$, where $A_{\psi}=\exp(t\sup_{X}\psi-c_{\psi})$. It follows therefore from Theorem 2.1 that there exists a unique continuous function $\varphi\in PSH(X,\omega)$ solution to $MA(\psi)$ and normalized by $\int_{X}\varphi d\mu=0$. We use here this linear normalization rather than the non-linear $\sup$-normalization: they are comparable thanks to proposition 2.7 in [GZ 1], which shows that

$$-M_{\mu}\leq\int_{X}ud\mu-\sup_{X}u\leq 0,$$

for all functions $u\in PSH(X,\omega)$ and for some uniform constant $M_{\mu}>0$. Since $\int_{X}\psi d\mu=0$, we infer

(6)

$$0\leq{\mathcal{E}}_{\omega}(\varphi):=\int_{X}|\varphi|\omega_{\varphi}^{n}=e^{-c_{\psi}}\int_{X}|\varphi|e^{t\psi}d\mu\leq 2M_{\mu}e^{tM_{\mu}},$$

by observing that $c_{\psi}\geq 0$ since $t\geq 0$, and

$$\int_{X}|\varphi|d\mu\leq\int_{X}|\varphi-\sup_{X}\varphi|d\mu+\sup_{X}\varphi\leq 2M_{\mu},$$

since $\int_{X}\varphi d\mu=0$.

The important fact here is that the energy ${\mathcal{E}}_{\omega}(\varphi)$ of $\varphi$ is bounded from above by a constant $M_{0}:=2M_{\mu}e^{tM_{\mu}}$ which is independent of $\psi$. We have thus defined an operator

$$T:\psi\in{\mathcal{C}}_{M}\mapsto\varphi\in{\mathcal{C}}_{M_{0}}$$

which associates to $\psi\in{\mathcal{C}}_{M}$ the unique solution $\varphi\in{\mathcal{C}}_{M_{0}}$ to $MA(\psi)$, where

$${\mathcal{C}}_{M}:=\left\{\psi\in{\mathcal{E}}^{1}(X,\omega)\,/\,\int_{X}\psi d\mu=0\text{ and }{\mathcal{E}}_{\omega}(\psi)\leq M\right\}.$$

It follows from proposition 3.2.3 in [GZ 2] that ${\mathcal{E}}^{1}(X,\omega)$ is convex. So is the subset of functions $\psi\in{\mathcal{E}}^{1}(X,\omega)$ such that $\int_{X}\psi d\mu=0$. The set ${\mathcal{C}}_{M}$ is not convex, but it is relatively compact in $L^{1}(X)$ and its closed convex hull $\hat{{\mathcal{C}}}_{M}$ is contained in ${\mathcal{C}}_{\kappa_{n}M}$ for some uniform constant $\kappa_{n}$ which only depends on the dimension of $X$: this follows from easy computations (see lemma 7.2 and the proof of proposition 3.2 in [GZ 2]). Therefore $T$ maps the compact convex set $\hat{{\mathcal{C}}}_{M}$ into itself if $M$ is large enough.

We claim that $T$ is continuous. Let $(\psi_{j})\in{\mathcal{C}}_{M}^{\mathbb{N}}$ be a sequence of functions which converges in $L^{1}(X)$ towards $\psi\in{\mathcal{C}}_{M}$. We need to show that $\varphi_{j}:=T(\psi_{j})$ converges in $L^{1}(X)$ towards $T(\psi)$. Since the set $\{u\in PSH(X,\omega)\,/$ $\,\int_{X}ud\mu=0\}$ is relatively compact in $L^{1}(X)$ (see proposition 2.7 in [GZ 1]), we can assume – relabelling if necessary – that $(\varphi_{j})$ converges in $L^{1}(X)$ towards a function $\varphi\in PSH(X,\omega)$. We show in lemma 4.2 below that $(\varphi_{j})$ converges in $L^{1}(\mu)$ towards $\varphi$. In particular $\int_{X}\varphi d\mu=0$ and, passing to a subsequence if necessary, we can assume that $e^{t\psi_{j}(x)}\rightarrow e^{t\psi(x)}$ for $\mu$ almost every point $x$. Set

$$\hat{\varphi}_{j}:=\left(\sup_{l\geq j}\varphi_{l}\right)^{*}\;\;\text{ and }\;\;\check{\psi}_{j}:=\inf_{l\geq j}\psi_{l}.$$

Observe that $(\hat{\varphi}_{j})$ decreases towards $\varphi$, while $(e^{t\check{\psi}_{j}})$ increases towards $e^{t\psi}$ at $\mu$ almost every point. The energy of $\hat{\varphi}_{j}$ is controlled by that of $\varphi_{j}$ since $\hat{\varphi}_{j}\geq\varphi_{j}$ (see lemma 7.2 in [GZ 2]), and ${\mathcal{E}}_{\omega}(\varphi_{j})\leq M_{0}$ by (5), therefore $\varphi\in{\mathcal{E}}^{1}(X,\omega)$ and $(\omega+dd^{c}\hat{\varphi}_{j})^{n}\rightarrow(\omega+dd^{c}\varphi)^{n}$. It follows from an inequality of J.-P.Demailly [Dem 1] that

$$(\omega+dd^{c}\hat{\varphi}_{j})^{n}\geq e^{t\check{\psi}_{j}-\hat{c}_{j}}\mu,$$

where $\hat{c}_{j}:=\sup_{l\geq j}c_{\psi_{l}}$. Observe that $\hat{c}_{j}\rightarrow c_{\psi}$, thus

$$(\omega+dd^{c}\varphi)^{n}\geq e^{t\psi-c_{\psi}}\mu.$$

Since these are two probability measures, there is actually equality hence $\varphi=T(\psi)$: this shows that $T$ is continuous.

We can now invoke Schauder fixed point theorem, which yields a fixed point $\varphi=T(\varphi),\varphi\in{\mathcal{C}}_{M}$. The function $\varphi$ is automatically continuous (by Theorem 2.1, since $e^{t\varphi-c_{\varphi}}\mu$ satisfies ${\mathcal{H}}(\alpha,A^{\prime},\omega))$, hence $\Phi:=\varphi-t^{-1}c_{\varphi}$ is the solution we were looking for. ∎