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 DβDβ . We denote by db(Dβ)db(Dβ) the distinguished boundary of DβDβ and define the corresponding Hardy space ℋ2(Dβ)H2(Dβ) . This can be identified with a closed subspace of L2(db(Dβ),dσ)L2(db(Dβ),dσ) , that we denote by ℋ2(db(Dβ))H2(db(Dβ)) , where dσdσ is the naturally induced measure on db(Dβ)db(Dβ) . The orthogonal Hilbert space projection :L2(db(Dβ),dσ)→ℋ2(db(Dβ))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)L2∩Lp(db(Dβ),dσ) extends to a bounded operator :Lp(db(Dβ),dσ)→Lp(db(Dβ),dσ)P:Lp(db(Dβ),dσ)→Lp(db(Dβ),dσ) if and only if 21+νβ<p<21−νβ21+νβ<p<21−νβ where νβ=π2β−π,β>πνβ=π2β−π,β>π . Furthermore, we also prove that P defines a bounded operator :Ws,2(db(Dβ),dσ)→Ws,2(db(Dβ),dσ)P:Ws,2(db(Dβ),dσ)→Ws,2(db(Dβ),dσ) if and only if 0≤s<νβ20≤s<νβ2 where Ws.2(db(Dβ),dσ)Ws.2(db(Dβ),dσ) denotes the Sobolev space of order s and underlying L2L2 -norm. Finally, we prove a necessary condition for the boundedness of P on Ws,p(db(Dβ),dσ)Ws,p(db(Dβ),dσ) , p∈(1,∞)p∈(1,∞) , the Sobolev space of order s and underlying LpLp -norm.