The Bergman kernel and projection on non-smooth worm domains

Abstract

This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.

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• Bergman subspaces and subkernels: Degenerate $L^p$ mapping and zeroes
• Irregularity of the Bergman projection on worm domains in C^n
• Weighted Bergman Projection on the Hartogs Triangle
• Weak-type estimates for the Bergman projection on the polydisc and the Hartogs triangle
• Sharp estimates for the Szegő projection on the distinguished boundary of model worm domains
• Bergman kernel and projection on the unbounded worm domain
• Essential norm estimates for the $\bar{\partial }$-Neumann operator on convex domains and worm domains
• Completeness on the worm domain and the Müntz-Szász problem for the Bergman space
• Nonabelian ramified coverings and $L^p$-boundedness of Bergman projections in ${\mathbb{C}}^2$