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## 2. Obstruction to Compactness

1. #### Problem 2.1.

[Şahutoğlu] Let $\Omega$ be a smooth bounded pseudoconvex domain, let $\mathbf{B}_{\Omega}$ denote the Bergman projection operator and $M_{\psi}$ denote the multiplication operator by $\psi$. Suppose that $[\mathbf{B}_{\Omega},M_{\psi}]$ is compact on $L^2(\Omega)$ for all $\psi\in C(\overline{\Omega})$. Is $N_1$ compact on $L^2_{(0,1)}(\Omega)$?
$\bullet$ Yes, if $\Omega$ is convex [MR1659575].

$\bullet$ No, if $\Omega$ is not pseudoconvex [MR3095048].
• #### Problem 2.2.

[Şahutoğlu] Let $\Omega_2\subset \Omega_1$ be two smooth bounded pseudoconvex domains that share a boundary point $p$. If the restriction map from $A^2(\Omega_1)$ into $A^2(\Omega_2)$ is not compact, is the D’Angelo type of $p$ the same with respect to $\Omega_1$ and $\Omega_2$?

$\bullet$ When the inside domain $\Omega_2$ is a ball, is $p$ strictly pseudoconvex with respect to $\Omega_1$?
• #### Problem 2.3.

[Straube] Describe the relation between compactness of the $\overline{\partial}$-Neumann operator and existence of a Stein neighborhood basis. In particular, does the geometric sufficient condition for compactness in [MR2097419] imply existence of a Stein neighborhood basis?
• #### Problem 2.4.

[Şahutoğlu] Is there a smooth bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^n~ (n\geq 3$) that contains a non-trivial analytic disc in $b\Omega$ and yet the $\overline{\partial}$-Neumann operator $N_1$ is compact? When $n=2$, this is not possible [MR2603659].

Cite this as: AimPL: Cauchy-Riemann equations in several variables, available at http://aimpl.org/crscv.