Hardy spaces and the Szegő projection of the non-smooth worm domain [formula omitted

Abstract

We define Hardy spaces Hp(Dβ′), p∈(1,∞), on the non-smooth worm domain Dβ′={(z1,z2)∈C2:|Imz1−log⁡|z2|2|<π2,|log⁡|z2|2|<β−π2} and we prove a series of related results such as the existence of boundary values on the distinguished boundary ∂Dβ′ of the domain and a Fatou-type theorem (i.e., pointwise convergence to the boundary values). Thus, we study the Szegő projection operator S˜ and the associated Szegő kernel KDβ′. More precisely, if Hp(∂Dβ′) denotes the space of functions which are boundary values for functions in Hp(Dβ′), we prove that the operator S˜ extends to a bounded linear operatorS˜:Lp(∂Dβ′)→Hp(∂Dβ′) for every p∈(1,+∞) andS˜:Wk,p(∂Dβ′)→Wk,p(∂Dβ′) for every k>0. Here Wk,p denotes the Sobolev space of order k and underlying Lp norm, p∈(1,∞). As a consequence of the Lp boundedness of S˜, we prove that Hp(Dβ′)∩C(Dβ′‾) is a dense subspace of Hp(Dβ′).