Hardy spaces and Szeg\H{o} projection on quotient domains

The Hardy spaces are defined on the quotient domain of a bounded complete Reinhardt domain by a finite subgroup of $U(n)$. The Szeg\H{o} projection on the quotient domain can be studied by lifting to the covering space. This setting builds on the solution of a boundary value problem for holomorphic functions. In particular, when the covering space is either the polydisc or the unit ball in $\mathbb{C}^n$, the boundary value problem can be solved. Applying this theory in $\mathbb{C}^2$, we further obtain sharp results on the $L^p$ regularity of the Szeg\H{o} projection on the symmetrized bidisc, generalized Thullen domains, and the minimal ball.