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We compute the Bergman kernel functions of the unbounded domains ${\Omega }_p=(z^{\text{'}},z)\in ℂ²:z>p\left(z^{\text{'}}\right)$, where $p\left(z^{\text{'}}\right)={\left|z^{\text{'}}\right|}^{\alpha }/\alpha$. It is also shown that these kernel functions have no zeros in ${\Omega }_p$. We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.

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...