Hardy spaces and the Szego projection of the non-smooth worm domain D′β

We define Hardy spaces Hp(Dβ'), p∈(1, ∞), on the non-smooth worm domain Dβ'=[(z1,z2)∈C2:|Imz1-log|z2|2|<π/2,|log|z2|2|<β-π/2] 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 Szego projection operator S and the associated Szego 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').