Loading Web-Font TeX/Main/Regular
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

2. Obstruction to Compactness


    1. Add remark

      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].


        • Add remark

          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?


            • Add remark

              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?


                • Add remark

                  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.