On Hardy spaces on worm domains

In this review article we present the problem of studying Hardy spaces and the related Szego projection on worm domains. We review the importance of the Diederich-Fornaess worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szego projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth worm domains