About top-degree $L^2$- and $L^{2,\mathrm{loc}}$-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 $\mathscr{C}^3$-smooth boundary and embedded in a domain of the complex number space, we prove that the $L^2$- and $L^{2,\mathrm{loc}}$-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 $L^{2,\mathrm{loc}}$-Dolbeault $(n,q)$-cohomology groups of relatively compact domains $\Omega=\{\rho<0\}$ in $X$ which are defined by a $\mathscr{C}^3$-smooth function $\rho$ which is strictly plurisubharmonic on a neighbourhood of $\partial\Omega$ except of finitely many points.