Paper detail

Hardy spaces and the Szegő projection of the non-smooth worm domain $D'_β$

We define Hardy spaces $H^p(D'_β)$ on the non-smooth worm domain $D'_β=\{(z_1,z_2)\in\mathbb{C}^2:|Im z_1-\log |z_2|^2|<\fracπ{2}, |\log |z_2|^2|<β-\fracπ{2}\}$ and we prove a series of related results such as the existence of boundary values on the distinguished boundary $\partial D'_β$ of the domain and a Fatou-type theorem (i.e. pointwise convergence to the boundary values). Thus, we study the Szegő projection operator $\widetilde{S}$ and the associated Szegő kernel $K_{D'_β}$. More precisely, if $H^p(\partial D'_β)$ denotes the space of functions which are boundary values for functions in $H^p(D'_β)$, we prove that the operator $\widetilde{S}$ extends to a bounded linear operator $$ \widetilde{S}: L^p(\partial D'_β)\to H^p(\partial D'_β) $$ for every $p\in(1,+\infty)$ and $$ \widetilde{S}: W^{k,p}(\partial D'_β)\to W^{k,p}(\partial D'_β) $$ for every $k>0$. Here $W^{k,p}$ denotes the Sobolev space of order $k$ and underlying $L^p$ norm. As a consequence of the $L^p$ boundedness of $\widetilde{S}$, we prove that $H^p(D'_β)\cap\mathcal{C}(\overline{D'_β})$ is a dense subspace of $H^p(D'_β)$.