The Hardy boundary space, Szegő projection and Szegő kernel on a smoothly bounded domain

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $m\ge 1$, let $\mathrm{\Omega }\subset {\mathbb{C}}^m$ be a nonempty bounded connected open set with $C^1$ boundary (Bounded C1 domains and their outward normals), let $dS$ be its boundary surface measure (Surface integration on compact C1 hypersurfaces), and fix a normalization $\sigma =cdS$ with $c>0$. Give $L^2(\mathrm{\partial }\mathrm{\Omega },\sigma ;\mathbb{C})$ the first-variable-linear pairing from The complex $L^2$ pairing on equivalence classes. For $f\in \mathcal{O}(\mathrm{\Omega })\cap C(\bar{\mathrm{\Omega }})$, its boundary trace ${\mathrm{tr}}_{\sigma }f:=[f{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}]$ is an $L^2$ class because $\mathrm{\partial }\mathrm{\Omega }$ is compact and $\sigma$ is finite. Set

$\mathcal{T}(\mathrm{\Omega },\sigma ):=\{{\mathrm{tr}}_{\sigma }f:f\in \mathcal{O}(\mathrm{\Omega })\cap C(\bar{\mathrm{\Omega }})\},H^2(\mathrm{\partial }\mathrm{\Omega },\sigma ):={\overline{\mathcal{T}(\mathrm{\Omega },\sigma )}}^{L^2(\sigma )}\mathrm{.}$

The space $\mathcal{T}(\mathrm{\Omega },\sigma )$ is linear, and $H^2(\mathrm{\partial }\mathrm{\Omega },\sigma )$ is a closed linear subspace of the Hilbert space $L^2(\mathrm{\partial }\mathrm{\Omega },\sigma ;\mathbb{C})$ by [F1]. The Szegő projection is the orthogonal projection

$P_{\sigma }:L^2(\mathrm{\partial }\mathrm{\Omega },\sigma ;\mathbb{C})⟶H^2(\mathrm{\partial }\mathrm{\Omega },\sigma ),$

which exists by [F2].

Call $(\mathrm{\Omega },\sigma )$ Szegő-regular if, for every $w\in \mathrm{\Omega }$, the rule ${\mathrm{tr}}_{\sigma }f\mapsto f(w)$ is well-defined and bounded on $\mathcal{T}(\mathrm{\Omega },\sigma )$ in the $L^2$ norm, and its unique continuous extension $E_w:H^2(\mathrm{\partial }\mathrm{\Omega },\sigma )\to \mathbb{C}$ has the property that $z\mapsto E_z(h)$ is holomorphic on $\mathrm{\Omega }$ for every $h\in H^2(\mathrm{\partial }\mathrm{\Omega },\sigma )$. Boundedness and density make each $E_w$ unique. By [F3], there is a unique $S_w\in H^2(\mathrm{\partial }\mathrm{\Omega },\sigma )$ such that

$E_w(h)=⟨h,S_w{⟩}_{L^2(\sigma )}(h\in H^2(\mathrm{\partial }\mathrm{\Omega },\sigma ))\mathrm{.}$

For a Szegő-regular pair define its Szegő kernel by $S_{\sigma }(z,w):=E_z(S_w)$. Then $E_w(h)=⟨h,S_w⟩$ is the reproducing identity, $S_{\sigma }(z,w)=⟨S_w,S_z⟩$, and conjugate symmetry of the inner product gives $S_{\sigma }(w,z)=\overline{S_{\sigma }(z,w)}$. The regularity condition makes the kernel holomorphic in $z$; conjugate symmetry makes it antiholomorphic in $w$. The ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption is used through the Hilbert-space, projection and Riesz suppliers in [F1]–[F3]; no stronger choice principle is asserted. No Szegő construction is claimed for boundaries that are not $C^1$ hypersurfaces, including the polydisc when $m\ge 2$.