1. Estimates for \overline{\partial}
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Problem 1.1.
[Varolin] Let \Omega be a pseudoconvex domain in \mathbb{C}^n that contains 0 and let \phi\in L^1_{loc}(\Omega) be a weight function on \Omega. Let H be a hyperplane through 0 and f be a holomorphic function on H\cap\Omega such that \int_{H\cap\Omega}|f|^2e^{-\phi}dV<\infty.Suppose that for some constant C, i\partial\overline{\partial}\phi\geq -Ci\partial\overline{\partial}\log B_{\Omega}(z,z)where B_{\Omega}(z,z) denotes the Bergman kernel of \Omega on the diagonal. How big can C be so that there exists a holomorphic function F on \Omega such that F=f \text{ on }H\cap\Omega \text{ and }\int_{\Omega}|F|^2e^{-\phi}dV\leq \int_{H\cap\Omega}|f|^2e^{-\phi}dV?Is it possible to find a universal constant C?
\bulletWhen C=0, this is the Ohsawa-Takegoshi extension theorem (see [MR2743817]).
\bullet If \Omega is strictly pseudoconvex then C\geq\frac{1}{n+1} works (see [MR2743817]); however, it is not known whether this is sharp.
\bullet It is known that such a constant C exists when the domain is convex and finite type in \mathbb{C}^n or just finite type in \mathbb{C}^2. -
Problem 1.2.
[Dragomir] Let \mathcal{W} be the smooth Diederich–Fornaess worm domain in \mathbb{C}^2 (see [MR0430315]) and \rho be a defining function for \mathcal{W}. Let g_{\rho} be the Kähler metric on \mathcal{W} whose Kähler 2-form is i\partial\overline{\partial}\log(-\rho). Study the \mathcal{C}^{\infty}-regularity up the boundary for the Dirichlet problem \begin{align*} \Delta_{\rho}u&=0 \text{ in } \mathcal{W},\\ u&=f \text{ on } b\mathcal{W}, \end{align*}where f\in \mathcal{C}^{\infty}(b\mathcal{W}) and \Delta_{\rho} is the Laplace-Beltrami operator corresponding to g_{\rho}. Start by analyzing the pseudohermitian invariants of the leaves of the foliation given by the level sets of \rho and in particular understand the geometry of the weakly pseudoconvex locus of this foliation. -
Problem 1.3.
[Catlin] Let \Omega be a smooth bounded domain in \mathbb{C}^n such that the automorphism group of \Omega is non-compact. What can be said about \Omega? See [MR1706680]. -
Problem 1.4.
[Straube] Let \mathcal{W} be the smooth Diederich–Fornaess worm domain in \mathbb{C}^2 and \mathbf{B}_{\mathcal{W}} denote the Bergman projection operator on \mathcal{W}. Prove or disprove that \mathbf{B}_{\mathcal{W}}(\log \overline{z_1}) is in \mathcal{C}^{\infty}(\overline{\mathcal{W}}). Note that if it fails to be smooth up to the boundary then an alternative proof is obtained for the fact that the Condition R fails on \mathcal{W}. See [MR3130312] and [MR1370592]. -
Problem 1.5.
[Harrington] Let \Omega be a smooth bounded pseudoconvex domain such that for any 0<\eta<1, there exists a defining function \rho_{\eta} such that -(-\rho_{\eta})^{\eta} is plurisubharmonic on \Omega. Note that such an \eta always exists [MR0430315]; however, it could be small. Determine minimal assumptions on \Omega to show that \overline{\partial}-Neumann operator N (or the Bergman projection operator \mathbf{B}) is globally regular. Determine minimal assumptions on \Omega to show existence of a Stein neighborhood basis for the closure. Are there any domains such that \mathbf{B} is globally regular but no such a family of defining functions exists?
Cite this as: AimPL: Cauchy-Riemann equations in several variables, available at http://aimpl.org/crscv.