Irregularity of the Bergman Projection on Smooth Unbounded Worm Domains

In this work, we consider smooth unbounded worm domains Zλ in C2 and show that the Bergman projection, densely defined on the Sobolev spaces Hs,p(Zλ), p ∈ (1, ∞), s ≥ 0, does not extend to a bounded operator Pλ : Hs,p(Zλ) → Hs,p(Zλ) when s > 0 or p ≠ 2. The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.