Regularity of projection operators attached to worm domains

This paper considers the non-smooth unbounded worm domains Dβ={(z1,z2)∈ℂ2:Re(z1e−ilogz2z⎯⎯⎯2)>0,|logz2z⎯⎯2|<β−π2}, where β>π2. These model domains were important when the first author [Acta Math. 168 (1992), no. 1-2, 1–10; MR1149863] used them to show that on the Diederich-Fornæss worm domains [K. Diederich and J. E. Fornæss, Math. Ann. 225 (1977), no. 3, 275–292; MR0430315] the Bergman projection does not map the Sobolev space Wk into itself when k≥π/(total amount of winding). In the paper under review, the authors construct an oblique projection operator on Dβ which preserves the level of the Sobolev spaces. More precisely, let L2j(Dβ)={f∈L2(Dβ):f∘ρθ=eijθf}, where ρθ=(z1,eiθz2) is a rotation on Dβ. Define Bj(Dβ):=L2j(Dβ)∩{holomorphic functions on Dβ}, Wsj(Dβ):=L2j(Dβ)∩Ws(Dβ), and Ws(Dβ) the closure of :=C∞0(Dβ) in Ws(Dβ). The main theorem of the paper shows that for all j∈ℤ there exists a bounded linear projection Tj:=L2(Dβ)→Bj(Dβ) which satisfies Tj:Ws(Dβ)→Wsj(Dβ)for every s≥0.