Behavior of the Bergman projection on the Diederich-Fornæss worm (Q1202518)

Define \(\Omega=\{(z_ 1,z_ 2)/| z_ 1+e^{i\log z_ 2\overline z_ 2}|^ 2<1-\varphi(\log z_ 2\overline z_ 2)\) where \(\varphi\) is a smooth nonnegative even function choosen such that \(\Omega\) is a smooth bounded, connected pseudoconvex domain, and moreover \(\varphi^{- 1}(0)=[-\beta+\pi/2,\beta-\pi/2]\). The following result is proven: Theorem. The Bergmann projection operator attached to \(\Omega\) does not map \(W^ k(\Omega)\) into \(W^ k(\Omega)\) when \(k\geq/(2\beta-\pi)\). This settles negatively a long standing problem about the behaviour of the Bergman projection for weakly pseudoconvex domains.