Potential theory on non-Archimedean analytic curves (over arbitrary complete valuation fields) was developed in detail by A.Thuillier in [Thu05]. We only indicate how to recover Theorem A’ when $\dim X=1$ following his approach.

Let $X$ be a smooth projective curve over $K$. Thuillier defined spaces $D^{0}(X)$ and $D^{1}(X)$ of distributions and currents on $X$ as follows. An element of $D^{0}(X)$ is an arbitrary function $X^{\mathrm{qm}}\to\mathbf{R}$ [Thu05, Proposition 3.3.3]. The $dd^{c}$-operator extends to $dd^{c}:D^{0}(X)\to D^{1}(X)$, and its image is exactly the set of currents $\rho\in D^{1}(X)$ such that $\int_{X}\rho=0$ [Thu05, Théorème 3.3.13]. By linearity, this fact easily reduces to the existence, for any two $x,y\in X^{\mathrm{div}}$, of a ’Green function’, i.e. a model function $g_{x,y}$ such that $dd^{c}g_{x,y}=\delta_{x}-\delta_{y}$. The existence of $g_{x,y}$ is in turn a consequence of the intersection form being negative definite on $\Div_{0}(\mathcal{X})_{\mathbf{R}}/\mathbf{R}\mathcal{X}_{0}$, for a model $\mathcal{X}$ such that $x$ and $y$ correspond to components of $\mathcal{X}_{0}$.

Now let $\omega$ be a $(1,1)$-form with $\int\omega>0$, and let $\mu$ be an arbitrary positive Radon measure on $X$ such that $\int\mu=\int\omega$. The previous result shows the existence of a distribution $\varphi_{\mu}$ such that

(9.1)

$$\omega+dd^{c}\varphi_{\mu}=\mu.$$

By [Thu05, Lemme 3.4.1] the positivity of the current $\omega+dd^{c}\varphi_{\mu}$ shows that $\varphi_{\mu}$ uniquely extends to a $\omega$-psh function, and we conclude that any positive Radon measure $\mu$ with $\int\mu=\int\omega$ satisfies (9.1) for some $\varphi\in\PSH(X,\omega)$, unique up to an additive constant.

Finally, assume that $\mu$ is supported on a dual complex $\Delta_{\mathcal{X}}$. In order to see that $\varphi_{\mu}\in C^{0}(X)$, we may assume that $\mathcal{X}$ is also a determination of $\omega$. In this one-dimensional setting, it is easy to check that composing with the retraction $p_{\mathcal{X}}:X\to\Delta_{\mathcal{X}}$ preserves $\omega$-psh functions, i.e. $\varphi\circ p_{\mathcal{X}}$ is $\omega$-psh for every $\omega$-psh function $\varphi$. Since $\mu$ is supported on $\Delta_{\mathcal{X}}$ we have $\left(p_{\mathcal{X}}\right)_{*}\mu=\mu$, hence $\theta+dd^{c}(\varphi_{\mu}\circ p_{\mathcal{X}})=\mu$. It follows that $\varphi_{\mu}\circ p_{\mathcal{X}}=\varphi_{\mu}$ by uniqueness up to an additive constant, since the two functions coincide on $\Delta_{\mathcal{X}}$. Now $\varphi_{\mu}|_{\Delta_{\mathcal{X}}}$ is continuous, hence the continuity of $\varphi_{\mu}$.

Let us now make the connection with the approach we followed in higher dimensions. In dimension $1$, the energy is equal to $E(\varphi)=2\int\varphi\omega+\int\varphi dd^{c}\varphi$ so that a $\omega$-psh function $\varphi$ has finite energy iff $\varphi$ is integrable with respect to the trace measure of $dd^{c}\varphi$.

Now fix a positive Radon measure $\mu$ such that the solution $\varphi_{\mu}$ to (9.1) has finite energy. Then $\varphi_{\mu}$ is the unique $\omega$-psh function realizing the infimum of the functional $E(\varphi)-\int\varphi\mu$, by [Thu05, Proposition 3.5.9].

Observe that the assumption on $\mu$ is automatically satisfied when $\mu$ is supported in some dual complex $\Delta_{\mathcal{X}}$ whence Thuillier’s result gives a stronger version than our result in dimension $1$.

We refer to [Thu05] for more on potential theory on non-Archimedean curves including the notion of harmonic functions, capacity, and the study of polar sets. See also [BR10] for the case of the projective line.