$\Delta f_n \longrightarrow 0 \Rightarrow f_n \longrightarrow 0$ Let $X$ be a compact Riemann surface, $\omega$ be a positive (1,1)-form and $f_n: X \longrightarrow \mathbb{R}$ be a sequence of smooth functions. We normalize $f_n$ such that $\displaystyle \int_X f_n \; \omega=0$.
Then does $\mathrm{dd}^cf_n \longrightarrow 0$ implies $f_n \longrightarrow 0$.
The idea is that harmonic functions on compact Riemann surfaces are constant. But I don't know much analysis. I would appreciate references on this subject.
 A: On a compact Kähler manifold, there is an estimate for the Green's function of the Laplacian, in term of Hoelder norms
\begin{equation}
\left|\left|G(\varphi)\right|\right|_{k+2,\alpha}\leq C\,\left|\left|\varphi\right|\right|_{k,\alpha}
\end{equation}
for some $C$ depending only on $k$ and the dimension of the manifold. You can obtain this inequality as a consequence of the usual Schauder estimates and Ascoli-Arzelà, see for example Proposition $2.3$ chapter $4$ in [Kodaira-Morrow][1].
If we plug in $\Delta f_n$ in the inequality we get
\begin{equation}
\left|\left|G(\Delta f_n)\right|\right|_{k+2,\alpha}\leq C\,\left|\left|\Delta f_n\right|\right|_{k,\alpha}
\end{equation}
but since we are normalizing $f_n$ to be orthogonal to Harmonic functions (i.e. the constants) we have $f_n=G(\Delta f_n)$, so that choosing $k=0$ we have
\begin{equation}
\left|\left|f_n\right|\right|_{2,\alpha}\leq C\,\left|\left|\Delta f_n\right|\right|_{0,\alpha}.
\end{equation}
This reasoninig shows that if $\mathrm{d}\mathrm{d}^cf$ converges to zero in $\mathcal{C}^{0,\alpha}$-norm, then also $f_n$ will go to zero. I am not sure if this can be improved, requiring less regularity of the convergence $\mathrm{d}\mathrm{d}^cf\to 0$. Anyway, I hope this is helpful.
