We prove that compactness of the canonical solution operator to $\overline{\partial }$ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[𝒫,\overline{M}]$ defined on the whole $L_{(0,1)}^2\left(\Omega \right),$ where $\overline{M}$ is the multiplication by $\overline{z}$ and $𝒫$ is the orthogonal projection of $L_{(0,1)}^2\left(\Omega \right)$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\overline{\partial }$-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications...

On a bounded $q$-pseudoconvex domain $\Omega$ in ${ℂ}^n$ with a Lipschitz boundary, we prove that the $\overline{\partial }$-Neumann operator $N$ satisfies a subelliptic $(1/2)$-estimate on $\Omega$ and $N$ can be extended as a bounded operator from Sobolev $(-1/2)$-spaces to Sobolev $(1/2)$-spaces.

Pseudoconvex domains are exhausted in such a way that we keep a part of the boundary fixed in all the domains of the exhaustion. This is used to solve a problem concerning whether the generators for the ideal of either the holomorphic functions continuous up to the boundary or the bounded holomorphic functions, vanishing at a point in ${ℂ}^n$ where the fibre is nontrivial, has to exceed $n$. This is shown not to be the case.