Polynomial traces, monomial basis and bounded evaluation for the ball Hardy space

The segment from $0$ to each $z\in {\mathbb{B}}^m$ stays in ${\mathbb{B}}^m$, so [F2] puts ${\mathbb{B}}^m$ in the domain class of The Hardy boundary space, Szegő projection and Szegő kernel on a smoothly bounded domain. By [F2], its normalized polar surface measure is an allowed positive multiple of chart surface measure.

Let $g$ be holomorphic on an open neighbourhood $U$ of $\overline{{\mathbb{B}}^m}$. Consider the family of metric balls $B(a,\epsilon )$ with $a\in \overline{{\mathbb{B}}^m}$, $\epsilon >0$, and $\bar{B}(a,2\epsilon )\subset U$. It covers $\overline{{\mathbb{B}}^m}$: openness supplies such a radius at each point, and the family is defined by this property, so no uncountable choice of radii is made. The ambient-cover implication of A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it gives a finite subcover $B(a_i,{\epsilon }_i)$, and let $\delta ={\min }_i{\epsilon }_i>0$. If $\mathrm{\mid }y\mathrm{\mid }\le 1+\delta$, put $x=y\mathrm{/}(1+\delta )$, so $\mathrm{\mid }x\mathrm{\mid }\le 1$ and $\mathrm{\mid }y-x\mathrm{\mid }\le \delta$. Choose an index $i$ whose covering ball contains $x$; then $\mathrm{\mid }y-a_i\mathrm{\mid }<\delta +{\epsilon }_i\le 2{\epsilon }_i$, hence $y\in U$. Therefore $\bar{B}(0,1+\delta )\subset U$.

Fix $R$ with $1<R<1+\delta$. Continuity of $g$ and the modulus inequality in [F7] make $\mathrm{\mid }g\mathrm{\mid }$ continuous on the compact closed ball of radius $R$; let $M_R={\max }_{\mathrm{\mid }u\mathrm{\mid }\le R}\mathrm{\mid }g(u)\mathrm{\mid }<\mathrm{\infty }$. For each $\mathrm{\mid }z\mathrm{\mid }\le 1$, the function ${\varphi }_z(t)=g(tz)$ is holomorphic on $\mathrm{\mid }t\mathrm{\mid }<1+\delta$: if $z\mathrm{\ne }0$, complex differentiability of $g$ gives ${\varphi }_z(t+h)-{\varphi }_z(t)=Dg(tz)(hz)+o(\mathrm{\mid }h\mathrm{\mid }\mathrm{\mid }z\mathrm{\mid })$, and for $z=0$ it is constant.

The local power series of $g$ at $0$ is ${\sum }_{\alpha }{c}_{\alpha }{u}^{\alpha }$. For each fixed $\mathrm{\mid }z\mathrm{\mid }\le 1$ and sufficiently small $\mathrm{\mid }t\mathrm{\mid }$, absolute convergence lets us group $g(tz)={\sum }_{k\ge 0}{P}_k(z)t^k$, where $P_k(z)={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=k}{c}_{\alpha }{z}^{\alpha }$. Uniqueness of power-series coefficients identifies $P_k(z)$ as the $k$-th Taylor coefficient of ${\varphi }_z$. The one-variable Cauchy estimate [F6], applied on $\mathrm{\mid }t\mathrm{\mid }=R$, gives $\mathrm{\mid }{P}_k(z)\mathrm{\mid }\le M_R\mathrm{/}{R}^k$, uniformly for $\mathrm{\mid }z\mathrm{\mid }\le 1$. Thus ${\sup }_{\mathrm{\mid }z\mathrm{\mid }\le 1}\mathrm{\mid }{\sum }_{k>N}{P}_k(z)\mathrm{\mid }\le M_R{\sum }_{k>N}{R}^{-k}\to 0$; the Taylor polynomials ${\sum }_{k=0}^N{P}_k$ converge uniformly to $g$ on the closed ball.

Let $f\in \mathcal{O}({\mathbb{B}}^m)\cap C(\overline{{\mathbb{B}}^m})$. For $0<\rho <1$, $f_{\rho }(z)=f(\rho z)$ is holomorphic on the neighbourhood ${\rho }^{-1}{\mathbb{B}}^m$ of the closed unit ball. Uniform continuity of $f$ on that compact ball gives $f_{\rho }\to f$ uniformly there as $\rho \uparrow 1$. For each positive integer $n$, choose ${\rho }_n$ close enough to $1$ that $\mathrm{\parallel }{f}_{{\rho }_n}-f{\mathrm{\parallel }}_{C(\overline{{\mathbb{B}}^m})}<1\mathrm{/}(2n)$, then use step 2.1 to choose a polynomial $p_n$ with $\mathrm{\parallel }{p}_n-f_{{\rho }_n}{\mathrm{\parallel }}_{C(\overline{{\mathbb{B}}^m})}<1\mathrm{/}(2n)$. The countable selection is allowed by [A1], and $\mathrm{\parallel }{p}_n-f{\mathrm{\parallel }}_C<1\mathrm{/}n$. Since $\sigma (S)=1$, uniform convergence implies ${\mathrm{tr}}_{\sigma }{p}_n\to {\mathrm{tr}}_{\sigma }f$ in $L^2(S,\sigma )$. Hence polynomial traces are dense in $\mathcal{T}$, and by the definition of $H^2$ in [F3] their closure is all of $H^2(S,\sigma )$.

By [F4], distinct monomial traces are orthogonal and $\mathrm{\parallel }{\zeta }^{\alpha }{\mathrm{\parallel }}_2^2=w_{\alpha }>0$. Thus $e_{\alpha }={\zeta }^{\alpha }\mathrm{/}\sqrt{w_{\alpha }}$ is an orthonormal family. Its finite linear span is exactly the polynomial traces, dense by step 3.1; the definition in [F11] therefore makes it a complete orthonormal system. In particular the zero multi-index has norm squared $w_0=1$, and for $m=1$ every monomial has norm squared $1$.

If $F=\mathrm{\varnothing }$, then $p=0$ and the bound is immediate. Otherwise write $p(z)={\sum }_{\alpha \in F}{c}_{\alpha }{z}^{\alpha }$ for a finite set $F$. Orthogonality in [F4] gives $\mathrm{\parallel }{\mathrm{tr}}_{\sigma }p{\mathrm{\parallel }}_2^2={\sum }_{\alpha \in F}\mathrm{\mid }{c}_{\alpha }{\mathrm{\mid }}^2{w}_{\alpha }$. Cauchy–Schwarz and [F8] give, for $\mathrm{\mid }z\mathrm{\mid }\le r$, $\mathrm{\mid }p(z){\mathrm{\mid }}^2\le \mathrm{\parallel }{\mathrm{tr}}_{\sigma }p{\mathrm{\parallel }}_2^2{\sum }_{\alpha \in F}\frac{\mathrm{\mid }{z}^{\alpha }{\mathrm{\mid }}^2}{w_{\alpha }}\le \mathrm{\parallel }{\mathrm{tr}}_{\sigma }p{\mathrm{\parallel }}_2^2{\sum }_{k=0}^{\mathrm{\infty }}\frac{(m-1+k)!}{(m-1)!k!}{r}^{2k}$. Indeed, the degree-$k$ part of the full sum is $\frac{(m-1+k)!}{(m-1)!k!}({\sum }_{j<m}\mathrm{\mid }{z}_j{\mathrm{\mid }}^2{)}^k$, by the multinomial identity. When $m=1$, this coefficient is $1$ for every $k$ and the series is ${\sum }_{k\ge 0}{r}^{2k}$. The final series converges by [F9], proving the claimed bound with the displayed $C_r$.

If $f\in \mathcal{O}({\mathbb{B}}^m)\cap C(\overline{{\mathbb{B}}^m})$, choose the uniform polynomial approximants from step 3.1. For each $\mathrm{\mid }z\mathrm{\mid }\le r$, step 5.1 bounds $\mathrm{\mid }{p}_n(z)\mathrm{\mid }$ by $C_r\mathrm{\parallel }{\mathrm{tr}}_{\sigma }{p}_n{\mathrm{\parallel }}_2$. Uniform convergence gives $p_n(z)\to f(z)$, and $L^2$ convergence gives $\mathrm{\parallel }{\mathrm{tr}}_{\sigma }{p}_n{\mathrm{\parallel }}_2\to \mathrm{\parallel }{\mathrm{tr}}_{\sigma }f{\mathrm{\parallel }}_2$. Taking the limit pointwise and then the supremum over $\mathrm{\mid }z\mathrm{\mid }\le r$ proves ${\sup }_{\mathrm{\mid }z\mathrm{\mid }\le r}\mathrm{\mid }f(z)\mathrm{\mid }\le C_r\mathrm{\parallel }{\mathrm{tr}}_{\sigma }f{\mathrm{\parallel }}_2$. If two trace generators determine the same $L^2$ class, apply this bound to their difference for every $r<1$; their holomorphic functions agree throughout ${\mathbb{B}}^m$. Thus trace evaluation is well-defined, and for any $w\in {\mathbb{B}}^m$, choose $\mathrm{\mid }w\mathrm{\mid }<r<1$ to get $\mathrm{\mid }f(w)\mathrm{\mid }\le C_r\mathrm{\parallel }{\mathrm{tr}}_{\sigma }f{\mathrm{\parallel }}_2$.

Given $h\in H^2(S,\sigma )$, [F3] and [A1] give a sequence $f_n\in \mathcal{O}({\mathbb{B}}^m)\cap C(\overline{{\mathbb{B}}^m})$ whose traces converge to $h$ in $L^2$. If a compact $K\subset {\mathbb{B}}^m$ is nonempty, the norm attains a maximum $s<1$ on $K$ by [F7]; choose $s<r<1$, so $K\subset r{\mathbb{B}}^m$. For empty $K$ the convergence assertion is automatic. Step 6.1 applied to $f_n-f_j$ then shows that $(f_n)$ is uniformly Cauchy on $K$. Its limit $\tilde{h}$ is holomorphic by [F10], independent of the approximating sequence by the same estimate, and agrees with every trace generator by applying it to the constant sequence at that generator. Hence it is the unique extension represented by the convergent sequence. Passing the bound in step 6.1 to the limit gives $\mathrm{\mid }\tilde{h}(w)\mathrm{\mid }\le C_r\mathrm{\parallel }h{\mathrm{\parallel }}_2$ whenever $\mathrm{\mid }w\mathrm{\mid }<r<1$. At each $w$, this limit extends the well-defined bounded linear trace evaluation from step 6.1; the extension is linear because the trace subspace is dense and linearity passes to limits.

The Hilbert-space and pairing assumptions for $H^2(S,\sigma )$ are [F3]. The Riesz theorem [F11] therefore supplies a unique $S_w\in H^2(S,\sigma )$ with $E_w(h)=⟨h,S_w⟩$ for every $h\in H^2(S,\sigma )$. Step 7.1 makes $w\mapsto E_w(h)$ holomorphic, so the pair $({\mathbb{B}}^m,\sigma )$ is Szegő-regular by The Hardy boundary space, Szegő projection and Szegő kernel on a smoothly bounded domain.