h1 is isometric to finite regular complex boundary measures

Converse direction, norm bound. Let $\mu$ be a finite regular complex Borel measure and put $v:=P[\mu ]$. First establish harmonicity. Identifying $\eta \in \mathbb{T}$ with its unit-circle point, the geometric series gives $P(z,\eta )=1+{\sum }_{k\ge 1}(z^k{\bar{\eta }}^k+{\bar{z}}^k{\eta }^k)\mathrm{.}$ Put $a_k:=\int {\bar{\eta }}^{k}d\mu (\eta )$ and $b_k:=\int {\eta }^{k}d\mu (\eta )$. These integrals exist by [L3]. The functions $v_N(z):=\mu (\mathbb{T})+{\sum }_{k=1}^N(a_k{z}^k+b_k{\bar{z}}^k)$ are complex harmonic by [L11] and linearity of the Laplacian. For $\mathrm{\mid }z\mathrm{\mid }\le \rho <1$, the kernel remainder and [L3] give $\mathrm{\mid }v(z)-v_N(z)\mathrm{\mid }\le \frac{2\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T}){\rho }^{N+1}}{1-\rho }\mathrm{.}$ Thus $v_N\to v$ locally uniformly; [L11], applied to real and imaginary parts under the given Countable Choice, proves that $v$ is complex harmonic. For $0\le r<1$ and $\zeta \in \mathbb{T}$, [L1] and [L3] give $\mathrm{\mid }{v}_r(\zeta )\mathrm{\mid }\le {\int }_{\mathbb{T}}P(r\zeta ,\eta )d\mathrm{\mid }\mu \mathrm{\mid }(\eta )$; integrating over $\zeta$ and applying Tonelli's theorem [L4] to the nonnegative integrand, together with translation invariance of $m$ and the unit mass of the kernel from [L2], yields ${\int }_{\mathbb{T}}\mathrm{\mid }{v}_r\mathrm{\mid }dm\le {\int }_{\mathbb{T}}({\int }_{\mathbb{T}}P(r\zeta ,\eta )dm(\zeta ))d\mathrm{\mid }\mu \mathrm{\mid }(\eta )=\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})<+\mathrm{\infty }\mathrm{.}$ Hence ${\sup }_{0\le r<1}\mathrm{\parallel }{v}_r{\mathrm{\parallel }}_1\le \mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})$ and $v\in h^1(\mathbb{D})$ with $\mathrm{\parallel }v{\mathrm{\parallel }}_{h^1}\le \mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})$ by [L5].

Converse direction, weak-star convergence. Let $g\in C(\mathbb{T},\mathbb{C})$. The function $(\zeta ,\eta )\mapsto g(\zeta )P(r\zeta ,\eta )$ is bounded by $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}{\sup }_{\zeta }P(r\zeta ,\eta )$ and is integrable for the product of the probability measure $m$ and the finite measure $\mathrm{\mid }\mu \mathrm{\mid }$; Fubini's theorem [L4] therefore gives ${\int }_{\mathbb{T}}g{v}_{r}dm={\int }_{\mathbb{T}}({\int }_{\mathbb{T}}g(\zeta )P(r\zeta ,\eta )dm(\zeta ))d\mu (\eta )\mathrm{.}$ The inner integral equals $(P_r\ast g)(\eta )$ by translation invariance of $m$ and the symmetry $P_r(-\theta )=P_r(\theta )$, and $P_r\ast g\to g$ uniformly by [L2]; hence $\mathrm{\mid }\int g{v}_{r}dm-\int gd\mu \mathrm{\mid }=\mathrm{\mid }\int (P_r\ast g-g)d\mu \mathrm{\mid }\le \mathrm{\parallel }{P}_r\ast g-g{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})\to 0$. Thus against $C(\mathbb{T})$ as $r\uparrow 1$.

Extraction of a boundary measure. Let $u\in h^1(\mathbb{D})$ with $M=\mathrm{\parallel }u{\mathrm{\parallel }}_{h^1}<\mathrm{\infty }$ and put $r_j:=1-1\mathrm{/}(j+1)\uparrow 1$. Each $u_{r_j}m$ is a finite complex measure with $\mathrm{\mid }{u}_{r_j}m\mathrm{\mid }(\mathbb{T})=\mathrm{\parallel }{u}_{r_j}{\mathrm{\parallel }}_1\le M$ by [L5], so $(u_{r_j}m)$ is a bounded sequence in the dual of the separable space $C(\mathbb{T},\mathbb{C})$ by [L6]. The ultrafilter lemma gives a subsequence, relabelled $(u_{r_j}m)$, and a finite regular complex Borel measure $\mu$ with by [L7] and [L8]. For every $g\in C(\mathbb{T})$ with $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$, $\mathrm{\mid }\int gd\mu \mathrm{\mid }={\lim }_j\mathrm{\mid }\int g{u}_{r_j}dm\mathrm{\mid }\le {\mathrm{lim\hspace{0.17em}inf}}_j\mathrm{\parallel }{u}_{r_j}{\mathrm{\parallel }}_1\le M,$ and taking the supremum over such $g$ gives $\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})\le M$ by the norm formula of [L8].

The Dirac measure is not a density. Fix ${\zeta }_0\in \mathbb{T}$. By [L10], ${\delta }_{{\zeta }_0}$ is a finite regular complex Borel measure with ${\delta }_{{\zeta }_0}(\{{\zeta }_0\})=1$, while $(fm)(\{{\zeta }_0\})={\int }_{\{{\zeta }_0\}}fdm=0$ for every $f\in L^1(\mathbb{T},m)$ because $m(\{{\zeta }_0\})=0$. Hence there is no $f\in L^1(\mathbb{T},m)$ with $fm={\delta }_{{\zeta }_0}$: the boundary measure ${\delta }_{{\zeta }_0}$ admits no $L^1$ density.

Converse direction, reverse norm inequality. For a finite regular complex Borel measure $\mu$ and $v=P[\mu ]$, the norm formula of [L8] gives $\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})=\sup \{\mathrm{\mid }\int gd\mu \mathrm{\mid }:\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\}$; for each such $g$, step 1.2 yields $\mathrm{\mid }\int gd\mu \mathrm{\mid }={\lim }_{r\uparrow 1}\mathrm{\mid }\int g{v}_{r}dm\mathrm{\mid }\le {\sup }_{r<1}\mathrm{\parallel }{v}_r{\mathrm{\parallel }}_1=\mathrm{\parallel }v{\mathrm{\parallel }}_{h^1}$. Taking the supremum over $g$ and combining with step 1.1 gives $\mathrm{\parallel }P[\mu ]{\mathrm{\parallel }}_{h^1}=\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})$.

Identification of the Poisson integral. Let $u$ and $\mu$ be as in step 1.3 and fix $z\in \mathbb{D}$. For all large $j$ one has $r_j>\mathrm{\mid }z\mathrm{\mid }$, and [L9] applied to the harmonic function $u$ on the disc of radius $r_j$ gives, in the torus normalization, $u(z)={\int }_{\mathbb{T}}\frac{r_j^2-\mathrm{\mid }z{\mathrm{\mid }}^2}{\mathrm{\mid }{r}_j\eta -z{\mathrm{\mid }}^2}{u}_{r_j}(\eta )dm(\eta )={\int }_{\mathbb{T}}P(z,\eta )u_{r_j}(\eta )dm(\eta )+{\int }_{\mathbb{T}}(P^{(j)}(z,\eta )-P(z,\eta ))u_{r_j}(\eta )dm(\eta ),$ where $P^{(j)}(z,\eta ):=(r_j^2-\mathrm{\mid }z{\mathrm{\mid }}^2)\mathrm{/}\mathrm{\mid }{r}_j\eta -z{\mathrm{\mid }}^2\to P(z,\eta )$ uniformly in $\eta$ as $j\to \mathrm{\infty }$ because $\mathrm{\mid }z\mathrm{\mid }<1$ stays a positive distance from the boundary circle; the second term is bounded by $\mathrm{\parallel }{P}^{(j)}(z,\cdot )-P(z,\cdot ){\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{u}_{r_j}{\mathrm{\parallel }}_1\to 0$. The first term tends to ${\int }_{\mathbb{T}}P(z,\eta )d\mu (\eta )=P[\mu ](z)$ by step 1.3, since $\eta \mapsto P(z,\eta )$ is continuous. Hence $u(z)=P[\mu ](z)$ for every $z\in \mathbb{D}$, that is, $u=P[\mu ]$.

Uniqueness of the boundary measure. If $P[\mu ]=P[\nu ]$, then for every $g\in C(\mathbb{T})$ step 1.2 applied to $\mu$ and to $\nu$ gives $\int gd\mu ={\lim }_r\int g(P[\mu ]{)}_{r}dm={\lim }_r\int g(P[\nu ]{)}_{r}dm=\int gd\nu$. The uniqueness clause of the representation theorem [L8] then gives $\mu =\nu$. In particular the measure produced by the weak-star subsequence in step 1.3 is the unique representing measure of $u$, independently of the subsequence.

Norm equality on $h^1$. Let $u\in h^1(\mathbb{D})$ with representing measure $\mu$ as in steps 1.3 and 2.2. Step 1.3 gives $\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})\le \mathrm{\parallel }u{\mathrm{\parallel }}_{h^1}$, and step 2.2 gives $u=P[\mu ]$, so step 2.1 yields $\mathrm{\parallel }u{\mathrm{\parallel }}_{h^1}=\mathrm{\parallel }P[\mu ]{\mathrm{\parallel }}_{h^1}=\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})$. Therefore the representation is an isometry, and every $h^1$ function has the same norm as its boundary measure.

Full-net weak-star convergence. Since $u=P[\mu ]$ by step 2.2, step 1.2 applied to the measure $\mu$ shows that against $C(\mathbb{T})$ along the whole net $r\uparrow 1$, not merely along the subsequence selected in step 1.3.

Assembly. (i) If $u\in h^1(\mathbb{D})$, steps 1.3 and 2.2 produce a finite regular complex Borel measure $\mu$ with $u=P[\mu ]$, step 2.3 shows it is unique, step 3.1 gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{h^1}=\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})$, and step 3.2 gives . Conversely, if $\mu$ is a finite regular complex Borel measure, step 1.1 puts $P[\mu ]$ in $h^1(\mathbb{D})$ and step 2.1 gives $\mathrm{\parallel }P[\mu ]{\mathrm{\parallel }}_{h^1}=\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})$, while step 1.2 gives the weak-star convergence of the radial measures; this proves both directions of the asserted isometric correspondence. (ii) For the final clause, the measure ${\delta }_{{\zeta }_0}$ of step 1.4 is finite and regular, so $P[{\delta }_{{\zeta }_0}]$ is an $h^1$ function whose boundary measure is not of the form $fm$; hence a general $h^1$ function need not have an $L^1$ density. (iii) The Axiom of Choice is used exactly as recorded: it gives the ultrafilter lemma used in the separable-predual sequential compactness theorem and Dependent Choice for the Riesz representation theorem, both cited in [L7] and [L8] and carried in the dependency list of this item.