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

We define Hardy spaces H-p(D'(beta)), p is an element of (1, infinity), on the non-smooth worm domain D'(beta) = {(z(1),z(2)) is an element of C-2 : vertical bar Imz(1) - log vertical bar z(2 vertical bar)2 vertical bar < pi/2, vertical bar log vertical bar z(2 vertical bar)2 vertical bar < beta - pi/2} and we prove a series of related results such as the existence of boundary values on the distinguished boundary partial derivative D'(beta) of the domain and a Fatou-type theorem (i.e., pointwise convergence to the boundary values). Thus, we study the Szego projection operator (S) over tilde and the associated Szego kernel K-D'beta. More precisely, if H-p(partial derivative D'beta) denotes the space of functions which are boundary values for functions in (H(D)-D-p'beta), we prove that the operator (S) over tilde extends to a bounded linear operator(S) over tilde : L-p (partial derivative D'(beta)) -> H-p (partial derivative D'(beta))for every p is an element of (1, + infinity) and(S) over tilde : W-k,W-p (partial derivative D'(beta)) -> W-k,W-p (partial derivative D'(beta))for every k > 0. Here W-k,W-p denotes the Sobolev space of order k and underlying L-p norm, p is an element of (1, infinity). As a consequence of the L-p boundedness of (S) over tilde, we prove that H-p (D'(beta)) boolean AND C <((D'(beta)))over bar>is a dense subspace of H-p (D'(beta)). (C) 2015 Elsevier Inc. All rights reserved.