Precise subelliptic estimates for a class of special domains

For the $\bar{\mathrm{\partial }}$-Neumann problem on a regular coordinate domain $\mathrm{\Omega }\subset {\text{C}}^{n+1}$, we prove $ϵ$-subelliptic estimates for an index $ϵ$ which is in some cases better than $ϵ=\frac{1}{2m}$ ($m$ being the {\it multiplicity}) as it was previously proved by Catlin and Cho in \cite{CC08}. This also supplies a much simplified proof of the existing literature. Our approach is founded on the method by Catlin in \cite{C87} which consists in constructing a family of weights $\{{\varphi }^{\delta }\}$ whose Levi form is bigger than ${\delta }^{-2ϵ}$ on the $\delta$-strip around $\mathrm{\partial }\mathrm{\Omega }$.