Hardy spaces and the Szeg\H{o} projection of the non-smooth worm domain $D_{\beta }^{\mathrm{\prime }}$

We define Hardy spaces $H^p(D_{\beta }^{\mathrm{\prime }})$ on the non-smooth worm domain $D’\beta={(z_1,z_2)\in$\mathbb{C}$^2:|Im z_1-\log |z_2|^2|<\frac{\pi}{2}, |\log |z_2|^2|<\beta-\frac{\pi}{2}}$ and we prove a series of related results such as the existence of boundary values on the distinguished boundary $\partial D’\beta$ of the domain and a Fatou-type theorem (i.e. pointwise convergence to the boundary values). Thus, we study the Szeg\H{o} projection operator $\tilde{S}$ and the associated Szeg\H{o} kernel $K_{D_{\beta }^{\mathrm{\prime }}}$. More precisely, if $H^p(\mathrm{\partial }{D}_{\beta }^{\mathrm{\prime }})$ denotes the space of functions which are boundary values for functions in $H^p(D_{\beta }^{\mathrm{\prime }})$, we prove that the operator $\tilde{S}$ extends to a bounded linear operator $$\tilde{S}:L^p(\mathrm{\partial }{D}_{\beta }^{\mathrm{\prime }})\to H^p(\mathrm{\partial }{D}_{\beta }^{\mathrm{\prime }})$$ for every $p\in (1,+\mathrm{\infty })$ and $$\tilde{S}:W^{k,p}(\mathrm{\partial }{D}_{\beta }^{\mathrm{\prime }})\to W^{k,p}(\mathrm{\partial }{D}_{\beta }^{\mathrm{\prime }})$$ 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 $\tilde{S}$, we prove that $H^p(D_{\beta }^{\mathrm{\prime }})\cap \mathcal{C}(\overline{D_{\beta }^{\mathrm{\prime }}})$ is a dense subspace of $H^p(D_{\beta }^{\mathrm{\prime }})$.