About top-degree $L^2$- and $L^{2,\mathrm{l}\mathrm{o}\mathrm{c}}$-Dolbeault cohomologies of complex spaces with pseudoconvex boundary

Let $X$ be a complex space of pure-dimension $n$. For a pseudoconvex relatively compact domain in $X$ with ${\mathcal{C}}^3$-smooth boundary and embedded in a domain of the complex number space, we prove that the $L^2$- and $L^{2,\mathrm{l}\mathrm{o}\mathrm{c}}$-Dolbeault $(n,q)$-cohomology groups are vanishing for $q>0$. Thereby, we include the case that the forms have values in a Nakano semi-positive holomorphic vector bundle. Using this local vanishing theorem, we also prove the equivalence of the $L^2$- and $L^{2,\mathrm{l}\mathrm{o}\mathrm{c}}$-Dolbeault $(n,q)$-cohomology groups of relatively compact domains $\mathrm{\Omega }=\{\rho <0\}$ in $X$ which are defined by a ${\mathcal{C}}^3$-smooth function $\rho$ which is strictly plurisubharmonic on a neighbourhood of $\mathrm{\partial }\mathrm{\Omega }$ except of finitely many points.