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 $X=X-D$. 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 $\Delta_{\alpha}f\leq B(B\in L^{p}(X)),p>n$, do we have estimation of Moser's type? i.e. $$ \lVert f\rVert_{L^{\infty}(X)}\leq C(\lVert B\rVert_{L^{p}(X)}+\lVert f\rVert_{L^{1}(X)}) $$ 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 desired estimates.

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 if 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. I don't even know if it is true if we only require $B$ to be a constant as above.