Sharp Estimates for the Szegő Projection on the Distinguished Boundary of Model Worm Domains

In this paper we study the regularity of the Szegő projection on Lebesgue and Sobolev spaces on the distinguished boundary of the unbounded model worm domain Dβ. We denote by db(Dβ) the distinguished boundary of Dβ and define the corresponding Hardy space H2(Dβ). This can be identified with a closed subspace of L2(db(Dβ) , dσ) , that we denote by H2(db(Dβ)) , where dσ is the naturally induced measure on db(Dβ). The orthogonal Hilbert space projection P: L2(db(Dβ) , dσ) → H2(db(Dβ)) is called the Szegő projection on the distinguished boundary. We prove that P, initially defined on the dense subspace L2∩ Lp(db(Dβ) , dσ) extends to a bounded operator P: Lp(db(Dβ) , dσ) → Lp(db(Dβ) , dσ) if and only if 21+p<21-νβ where νβ=π2β- Furthermore, we also prove that P defines a bounded operator P: Ws,2(db(Dβ) , dσ) → Ws,2(db(Dβ) , dσ) if and only if 0≤s2 where Ws.2(db(Dβ) , dσ) denotes the Sobolev space of order s and underlying L2-norm. Finally, we prove a necessary condition for the boundedness of P on Ws,p(db(Dβ) , dσ) , p∈ (1 , ∞) , the Sobolev space of order s and underlying Lp-norm.