Irregularity of the Bergman Projection on Smooth Unbounded Worm Domains

Abstract

AbstractIn this work, we consider smooth unbounded worm domains $${\mathcal {Z}}_\lambda $$ Z λ in $${\mathbb {C}}^2$$ C 2 and show that the Bergman projection, densely defined on the Sobolev spaces $$H^{s,p}({\mathcal {Z}}_\lambda ),$$ H s , p ( Z λ ) , $$p\in (1,\infty ),$$ p ∈ ( 1 , ∞ ) , $$s\ge 0,$$ s ≥ 0 , does not extend to a bounded operator $$P_\lambda :H^{s,p}({\mathcal {Z}}_\lambda )\rightarrow H^{s,p}({\mathcal {Z}}_\lambda )$$ P λ : H s , p ( Z λ ) → H s , p ( Z λ ) when $$s>0$$ s > 0 or $$p\ne 2.$$ 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.