Hardy spaces and the Szegő projection of the non-smooth worm domain $D_{\beta }^{\prime }$

Abstract

We define Hardy spaces $H^p(D_{\beta }^{\prime })$ on the non-smooth worm domain $D_{\beta }^{\prime }=\{(z_1,z_2)\in {\mathbb{C}}^2:|Im{z}_1-\log |z_2{|}^2|<\frac{\pi }{2},|\log |z_2{|}^2|<\beta -\frac{\pi }{2}\}$ and we prove a series of related results such as the existence of boundary values on the distinguished boundary $\partial D_{\beta }^{\prime }$ of the domain and a Fatou-type theorem (i.e. pointwise convergence to the boundary values). Thus, we study the Szegő projection operator $\tilde{S}$ and the associated Szegő kernel $K_{D_{\beta }^{\prime }}$. More precisely, if $H^p(\partial D_{\beta }^{\prime })$ denotes the space of functions which are boundary values for functions in $H^p(D_{\beta }^{\prime })$, we prove that the operator $\tilde{S}$ extends to a bounded linear operator $$\tilde{S}:L^p(\partial D_{\beta }^{\prime })\to H^p(\partial D_{\beta }^{\prime })$$ for every $p\in (1,+\infty )$ and $$\tilde{S}:W^{k,p}(\partial D_{\beta }^{\prime })\to W^{k,p}(\partial D_{\beta }^{\prime })$$ 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 $H^p(D_{\beta }^{\prime })\cap \mathcal{C}(\overline{D_{\beta }^{\prime }})$ is a dense subspace of $H^p(D_{\beta }^{\prime })$.