Ball monomial norms, Bergman and Szegő kernels of the ball

By [F10] and [F11], the multi-index factorials, lengths and powers in the formulas have their stated meanings. The ball moment in [F1] gives $\mathrm{\parallel }{z}^{\alpha }{\mathrm{\parallel }}_{A^2({\mathbb{B}}^m)}^2=c_{\alpha }$, and [F1] gives completeness of the normalized monomials.

By [F2], $\mathrm{\parallel }{\zeta }^{\alpha }{\mathrm{\parallel }}_{L^2(S,{\sigma }_1)}^2=w_{\alpha }$ and the normalized boundary monomials form a complete orthonormal system of the Hardy space; their evaluations are bounded.

By [F12], $z,w\in {\mathbb{B}}^m$ have Euclidean norms below $1$; Cauchy–Schwarz [F9] gives $\mathrm{\mid }⟨z,w⟩\mathrm{\mid }\le \mathrm{\parallel }z\mathrm{\parallel }\mathrm{\parallel }w\mathrm{\parallel }<1$, so both denominators are nonzero. The model-domain theorem [F3], with Lebesgue measure and normalized polar boundary measure, gives the displayed closed-form kernels.

Fix $w\in {\mathbb{B}}^m$ and let $K_w=K_{{\mathbb{B}}^m}(\cdot ,w)$. By [F4], its Fourier sums over finite $F\subset {\mathbb{N}}^m$ converge in $A^2({\mathbb{B}}^m)$ to $K_w$. For each $\beta$, [F5] gives $⟨e_{\beta },K_w⟩=e_{\beta }(w)$, so conjugate symmetry [F8] makes its Fourier coefficient $⟨K_w,e_{\beta }⟩=\overline{e_{\beta }(w)}$. If $F$ contains $\alpha$, orthonormality gives $⟨e_{\alpha },{\sum }_{\beta \in F}\overline{e_{\beta }(w)}{e}_{\beta }⟩={\sum }_{\beta \in F}{e}_{\beta }(w)⟨e_{\alpha },e_{\beta }⟩=e_{\alpha }(w)\mathrm{.}$ Passing to the norm limit using [F9] proves ${\int }_{{\mathbb{B}}^m}{e}_{\alpha }(z)\stackrel{‾}{K_{{\mathbb{B}}^m}(z,w)}d\lambda (z)=e_{\alpha }(w)$.

Fix $w\in {\mathbb{B}}^m$ and let $S_w$ be the Riesz representer of the bounded Hardy evaluation at $w$. By [F4], its finite-subset Fourier sums in the complete system $e_{\beta }^{\mathrm{\partial }}$ converge to $S_w$. By [F6], $⟨e_{\beta }^{\mathrm{\partial }},S_w⟩=e_{\beta }^{\mathrm{\partial }}(w)$, so conjugate symmetry [F8] gives Fourier coefficient $⟨S_w,e_{\beta }^{\mathrm{\partial }}⟩=\overline{e_{\beta }^{\mathrm{\partial }}(w)}$. For a finite $F$ containing $\alpha$, orthonormality gives $⟨e_{\alpha }^{\mathrm{\partial }},{\sum }_{\beta \in F}\overline{e_{\beta }^{\mathrm{\partial }}(w)}{e}_{\beta }^{\mathrm{\partial }}⟩=e_{\alpha }^{\mathrm{\partial }}(w)\mathrm{.}$ Passing to the norm limit using [F9] proves $⟨e_{\alpha }^{\mathrm{\partial }},S_w⟩=e_{\alpha }^{\mathrm{\partial }}(w)$, the Hardy reproducing identity for that basis vector. This uses the boundary Hilbert-space representer $S_w$; the interior kernel is not evaluated at a boundary point.

The finite linear spans of the complete systems in [F1] and [F2] are dense in their respective Hilbert spaces. Bounded evaluation from [F5] and [F2], and continuity of inner products from [F9], extend the identities of steps 1.4 and 1.5 from monomials to the full Bergman and Hardy spaces. At $\alpha =0$, $c_0={\pi }^m\mathrm{/}m!$ and $w_0=1$; setting $z=0$ in the kernels gives $m!\mathrm{/}{\pi }^m$ and $1$. When $m=1$, ${\mathbb{B}}^1=\mathbb{D}$ and the formulas specialize to the disc kernels.