Let $\overline{X}$ be a compact Kähler manifold of complex dimention $n$ with normal crossing divisors $D=\sum_{i=1}^{m}=D_{i}$. For $0<\alpha< 2$, we can construct a conic Kähler metric by setting $$
\omega_{\alpha}=\sqrt{-1}(\frac{2}{2-\alpha})\sum_{i=1}^{m}\partial\bar{\partial}|\sigma_{i}|^{2-\alpha}+C_{\alpha}\omega
$$
where $\sigma_{i}$'s are the canonical sections of linebundles $[D_{i}]$'s and $C_{\alpha}$ is a constant large enough so that $\omega_{\alpha}$ is a Kähler metric on $X=\overline{X}-D$. For any point $p\in D$ we choose a neighborhood $U_{p}$ centered at $p$.

Assume that $(z_{1},\cdots,z_{n})$ is a coordinate chart such that $U_{p}\cup D=\{z_{1}z_{2}\cdots z_{j}=0\}$. Then we see that in $U_{p}-D$, $\omega_{\alpha}$ is quasi isometric to 
$$
\sum_{i=1}^{j}|\sigma_{i}|^{-\alpha}dz_{i}\wedge d\bar{z_{i}}+\sum_{i=j+1}^{n}dz_{i}\wedge d\bar{z_{i}}
$$
For a bounded smooth function $f$ in $X$ with $\Delta_{\alpha}f\leq B$(B is a constant), do we have an estimation of Moser's type? i.e. $$
\lVert f\rVert_{L^{\infty}(X)}\leq C(\lVert B\rVert_{L^{p}(X)}+\lVert f\rVert_{L^{1}(X)})(p>n)
$$ where $\Delta_{\alpha}$ is the negative Laplace operator with resepect to $\omega_{\alpha}$ and the measure is given by the volume form of the conic Kähler metric $\omega_{\alpha}$.

If we restrict a Kähler metric $\omega$ on $\overline{X}$ to $X$ rather than the conic metric constructed above, this is true. Indeed, it is possible to show that $\Delta_{\omega}f\leq B$ holds weakly on the compact manifold $\overline{X}$ in the sense of distribution, i.e. for arbitrary smooth function $\eta$ on $\overline{X}$ we have
$$
\int_{\overline{X}}f\Delta_{\omega}\eta dV_{\omega} \leq \int_{\overline{X}}B\eta dV_{\omega}.
$$
And then we may use heat kernel $H_{t}$ of $\overline{X}$ to smooth out $f$ and $B$, by the weak inequality we should obtain $$
\Delta_{\omega}f_{t}\leq B_{t}
$$
in the stronge sense, where $f_{t}=H_{t}\cdot f$ and $B_{t}=H_{t}\cdot B$.Then we use Moser's iteration arguements and let $t$ tends to $0$ to get the desired estimate.

For singular metrics, I know that if $X$ is a quasi-projective curve and $\overline{X}$ is its completion, and if $B$ is a constant on $X$, this is still true. Near a singular point the conic metric looks like $|z|^{-\alpha}dz\wedge d\bar{z}$. Let $\Delta_{0}$ be the Laplace of Eucildean metric, then $\Delta_{\alpha}\leq B$ implies $\Delta_{0}f\leq B|z|^{-\alpha}$. Since $B$ is a constant and $0<\alpha<2$, we have $B|z|^{-\alpha}\in L^{p}(X)(p>1=dimX)$. Then we can apply the above arguements. 

But I don't have a clear idea for general case. Indeed, in section 4 of this [paper][1], the author mentioned that for $f\in C^{\infty}(\overline{X})$, we have a weighted Sobolev inequality $$
\lVert f\rVert_{L^{r}(X,\omega_{\alpha})}\leq C_{\alpha}(\lVert \nabla_{\alpha}f\rVert_{L^{2}(X,\omega_{\alpha})}+\lVert f\rVert_{L^{2}(X,\omega_{\alpha})})
$$
if $2\leq r \leq \frac{2n-\alpha}{n-1}$.
Then he concludes directly that the Moser's iterative arguments work on this setting, hence we can get above estimate.


  [1]: https://intlpress.com/site/pub/files/_fulltext/journals/cag/2000/0008/0003/CAG-2000-0008-0003-a001.pdf