Elfadil Mohammed Shomaim
Abderouf Elbdree Hassan
University of Blue Nile, Faculty of Education, Department of mathematics, Ad-Damzin, Sudan
abstract
We define Hardy spaces H^(1+ε) (D_(2π+ε)^’ ),0<ε<∞, on the non-smooth worm domain
D_(2π+ε)^’={(z_r,z_(r+1) )∈C^2:|Imz_r-log〖|z_(r+1) |^2 〗 |<π/2,|log〖|z_(r+1) |^2 〗 |<3/2 π+ε}
Following the work of [36], and we prove a series of related results such as the existence of boundary values on the distinguished boundary ∂D_(2π+ε)^’ of the domain and a Fatou-type theorem. Thus, we study the Sequences of Szegő projection operator S ̃ and the associated Szegő kernel K_(D_(2π+ε)^’ ). if H^(1+ε) (〖∂D〗(2π+ε)^’ ) denotes the space of functions which are boundary values for functions in H^(1+ε) (D(2π+ε)^’ ), we prove that the operator S ̃ extends to a bounded linear operator S ̃:L^(1+ε) (〖∂D〗(2π+ε)^’ )→H^(1+ε) (〖∂D〗(2π+ε)^’ ) for every 0<ε<∞ and
S ̃:W^(1+ε,1+ε) (〖∂D〗(2π+ε)^’ )→W^(1+ε,1+ε) (〖∂D〗(2π+ε)^’ ) for every ε≥0. Here W^(1+ε,1+ε) denotes the Sobolev space of order 1+ε and underlying L^(1+ε) norm, 0<ε<∞. As a consequence of the L^(1+ε) boundedness of S ̃, we prove that H^(1+ε) (D_(2π+ε)^’ )∩C(¯(D_(2π+ε)^’ )) is a dense subspace of H^(1+ε) (D_(2π+ε)^’ ).