Hardy spaces and the Szegő projection of the non-smooth worm domain Dβ′

On Hardy spaces on worm domains

“…The results regarding D 0 are proved in [41] together with a detailed analysis of the space H 2 .D 0 /, whereas the results on Dǎ re new and greater details will appear in a forthcoming paper [42].…”

“…In [41] it can be seen that .S D 0 /`;`D 1; : : : ; 4, is given by a sum similar to the one in(31), but with more addends. Although, each of these addends is a minor modification of the ones already appearing in (31).…”

“…Now, we write an explicit formula for .S D 0 F / 1 assuming that the starting function F is of the form F D .F 1 ; 0; 0; 0/, that is, F is constantly zero on the components E 2 ; E 3 and E 4 of @D 0 . We refer the reader to [41] for the details on how to obtain this formula and for its most general version.…”

“…Moreover, we only write explicitly the component .S DˇF / C of S DˇF . The expression of S DˇF for the most general F can be deduced from (35) and the formula for S D 0 contained in [41].…”

Regularity of the Szegö projection on model worm domains

“…We mention that on the domains D ′ β and D β it is possible to introduce and study also Hardy spaces defined in terms of the induced surface measure on the distinguished boundary, that is, in the case of the domain D ′ β , the four vertices in Figure 1. This has been done by the first author [19] in the case of D ′ β , and in [20] in the case of D β . More precisely, denote by ∂D ′ β and ∂D β the distinguished boundaries of D ′ β and D β , resp., and by P ′ and P the corresponding projections, resp.…”